[{"prompt": "Given Q{(r)} = e^{r}, then obtain Q^{r}{(r)} + \\frac{d}{d r} \\frac{Q^{r}{(r)}}{Q{(r)}} + \\frac{Q^{r}{(r)}}{Q{(r)}} = Q^{r}{(r)} + \\frac{d}{d r} \\frac{(e^{r})^{r}}{Q{(r)}} + \\frac{Q^{r}{(r)}}{Q{(r)}}", "derivation": "Q{(r)} = e^{r} and Q^{r}{(r)} = (e^{r})^{r} and \\frac{Q^{r}{(r)}}{Q{(r)}} = \\frac{(e^{r})^{r}}{Q{(r)}} and \\frac{d}{d r} \\frac{Q^{r}{(r)}}{Q{(r)}} = \\frac{d}{d r} \\frac{(e^{r})^{r}}{Q{(r)}} and Q^{r}{(r)} + \\frac{d}{d r} \\frac{Q^{r}{(r)}}{Q{(r)}} + \\frac{Q^{r}{(r)}}{Q{(r)}} = Q^{r}{(r)} + \\frac{d}{d r} \\frac{(e^{r})^{r}}{Q{(r)}} + \\frac{Q^{r}{(r)}}{Q{(r)}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["divide", 2, "Function('Q')(Symbol('r', commutative=True))"], "Equality(Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["add", 4, "Add(Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], "Equality(Add(Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Derivative(Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)))), Add(Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Derivative(Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Pow(Function('Q')(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('Q')(Symbol('r', commutative=True)), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(c,F_{N})} = \\frac{\\partial}{\\partial c} (F_{N} - c), then derive \\operatorname{A_{x}}^{c}{(c,F_{N})} = (-1)^{c}, then obtain \\int (\\frac{\\partial}{\\partial c} (F_{N} - c))^{c} dc = \\int (-1)^{c} dc", "derivation": "\\operatorname{A_{x}}{(c,F_{N})} = \\frac{\\partial}{\\partial c} (F_{N} - c) and \\operatorname{A_{x}}^{c}{(c,F_{N})} = (\\frac{\\partial}{\\partial c} (F_{N} - c))^{c} and \\operatorname{A_{x}}^{c}{(c,F_{N})} = (-1)^{c} and \\int \\operatorname{A_{x}}^{c}{(c,F_{N})} dc = \\int (-1)^{c} dc and \\int (\\frac{\\partial}{\\partial c} (F_{N} - c))^{c} dc = \\int (-1)^{c} dc", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('c', commutative=True), Symbol('F_N', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('c', commutative=True), Symbol('F_N', commutative=True)), Symbol('c', commutative=True)), Pow(Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('A_x')(Symbol('c', commutative=True), Symbol('F_N', commutative=True)), Symbol('c', commutative=True)), Pow(Integer(-1), Symbol('c', commutative=True)))"], [["integrate", 3, "Symbol('c', commutative=True)"], "Equality(Integral(Pow(Function('A_x')(Symbol('c', commutative=True), Symbol('F_N', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Integer(-1), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Pow(Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Integer(-1), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given T{(v_{2},q)} = q v_{2}, then obtain (\\frac{\\partial}{\\partial q} \\int T{(v_{2},q)} dv_{2})^{v_{2}} = (\\frac{\\partial}{\\partial q} \\int q v_{2} dv_{2})^{v_{2}}", "derivation": "T{(v_{2},q)} = q v_{2} and \\int T{(v_{2},q)} dv_{2} = \\int q v_{2} dv_{2} and \\frac{\\partial}{\\partial q} \\int T{(v_{2},q)} dv_{2} = \\frac{\\partial}{\\partial q} \\int q v_{2} dv_{2} and (\\frac{\\partial}{\\partial q} \\int T{(v_{2},q)} dv_{2})^{v_{2}} = (\\frac{\\partial}{\\partial q} \\int q v_{2} dv_{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('v_2', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('T')(Symbol('v_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Symbol('q', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(Function('T')(Symbol('v_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('q', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('v_2', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('T')(Symbol('v_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('v_2', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('q', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then obtain (- \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)})^{- \\hat{H}_l} \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)} = (- \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)})^{- \\hat{H}_l} \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "derivation": "\\theta_{1}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}^{\\hat{H}_l} and - \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)} = - \\log{(\\hat{H}_l)}^{\\hat{H}_l} and (- \\log{(\\hat{H}_l)}^{\\hat{H}_l})^{- \\hat{H}_l} \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)} = (- \\log{(\\hat{H}_l)}^{\\hat{H}_l})^{- \\hat{H}_l} \\log{(\\hat{H}_l)}^{\\hat{H}_l} and (- \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)})^{- \\hat{H}_l} \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)} = (- \\theta_{1}^{\\hat{H}_l}{(\\hat{H}_l)})^{- \\hat{H}_l} \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 2, "Pow(Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given i{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}, then obtain \\frac{2 i^{2}{(\\Psi^{\\dagger})}}{\\sin{(\\Psi^{\\dagger})}} = \\frac{(i{(\\Psi^{\\dagger})} + \\sin{(\\Psi^{\\dagger})}) i{(\\Psi^{\\dagger})}}{\\sin{(\\Psi^{\\dagger})}}", "derivation": "i{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})} and 2 i{(\\Psi^{\\dagger})} = i{(\\Psi^{\\dagger})} + \\sin{(\\Psi^{\\dagger})} and 2 i^{2}{(\\Psi^{\\dagger})} = (i{(\\Psi^{\\dagger})} + \\sin{(\\Psi^{\\dagger})}) i{(\\Psi^{\\dagger})} and \\frac{2 i^{2}{(\\Psi^{\\dagger})}}{\\sin{(\\Psi^{\\dagger})}} = \\frac{(i{(\\Psi^{\\dagger})} + \\sin{(\\Psi^{\\dagger})}) i{(\\Psi^{\\dagger})}}{\\sin{(\\Psi^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["add", 1, "Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["times", 2, "Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Add(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["divide", 3, "sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Add(Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('i')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then derive \\frac{d}{d \\mathbf{B}} \\phi_{1}{(\\mathbf{B})} - 1 = \\cos{(\\mathbf{B})} - 1, then obtain (\\frac{d}{d \\mathbf{B}} \\phi_{1}{(\\mathbf{B})} - 1) \\operatorname{n_{2}}{(\\lambda)} = (\\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} - 1) \\operatorname{n_{2}}{(\\lambda)}", "derivation": "\\phi_{1}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and - \\mathbf{B} + \\phi_{1}{(\\mathbf{B})} = - \\mathbf{B} + \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} (- \\mathbf{B} + \\phi_{1}{(\\mathbf{B})}) = \\frac{d}{d \\mathbf{B}} (- \\mathbf{B} + \\sin{(\\mathbf{B})}) and \\frac{d}{d \\mathbf{B}} \\phi_{1}{(\\mathbf{B})} - 1 = \\cos{(\\mathbf{B})} - 1 and \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} - 1 = \\cos{(\\mathbf{B})} - 1 and \\frac{d}{d \\mathbf{B}} \\phi_{1}{(\\mathbf{B})} - 1 = \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} - 1 and (\\frac{d}{d \\mathbf{B}} \\phi_{1}{(\\mathbf{B})} - 1) \\operatorname{n_{2}}{(\\lambda)} = (\\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} - 1) \\operatorname{n_{2}}{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)))"], [["times", 6, "Function('n_2')(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Function('n_2')(Symbol('\\\\lambda', commutative=True))), Mul(Add(Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Function('n_2')(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\hat{p}_0)} = \\log{(e^{\\hat{p}_0})}, then derive \\int \\operatorname{M_{E}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{\\hat{p}_0^{2}}{2} + \\theta_1, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} (\\frac{\\hat{p}_0^{2}}{2} + \\theta_1) = \\frac{d}{d \\hat{p}_0} \\int \\log{(e^{\\hat{p}_0})} d\\hat{p}_0", "derivation": "\\operatorname{M_{E}}{(\\hat{p}_0)} = \\log{(e^{\\hat{p}_0})} and \\int \\operatorname{M_{E}}{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\log{(e^{\\hat{p}_0})} d\\hat{p}_0 and \\frac{d}{d \\hat{p}_0} \\int \\operatorname{M_{E}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{d}{d \\hat{p}_0} \\int \\log{(e^{\\hat{p}_0})} d\\hat{p}_0 and \\int \\operatorname{M_{E}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{\\hat{p}_0^{2}}{2} + \\theta_1 and \\frac{\\partial}{\\partial \\hat{p}_0} (\\frac{\\hat{p}_0^{2}}{2} + \\theta_1) = \\frac{d}{d \\hat{p}_0} \\int \\log{(e^{\\hat{p}_0})} d\\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\hat{p}_0', commutative=True)), log(exp(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(log(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Integral(Function('M_E')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integral(log(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integral(log(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\psi{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain - \\log{(q{(\\mathbf{J}_M)})} + \\int \\log{(\\psi{(\\mathbf{J}_M)})} d\\mathbf{J}_M = - \\log{(q{(\\mathbf{J}_M)})} + \\int \\log{(\\cos{(\\mathbf{J}_M)})} d\\mathbf{J}_M", "derivation": "q{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\psi{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\log{(\\psi{(\\mathbf{J}_M)})} = \\log{(\\cos{(\\mathbf{J}_M)})} and \\int \\log{(\\psi{(\\mathbf{J}_M)})} d\\mathbf{J}_M = \\int \\log{(\\cos{(\\mathbf{J}_M)})} d\\mathbf{J}_M and - \\log{(\\cos{(\\mathbf{J}_M)})} + \\int \\log{(\\psi{(\\mathbf{J}_M)})} d\\mathbf{J}_M = - \\log{(\\cos{(\\mathbf{J}_M)})} + \\int \\log{(\\cos{(\\mathbf{J}_M)})} d\\mathbf{J}_M and - \\log{(q{(\\mathbf{J}_M)})} + \\int \\log{(\\psi{(\\mathbf{J}_M)})} d\\mathbf{J}_M = - \\log{(q{(\\mathbf{J}_M)})} + \\int \\log{(\\cos{(\\mathbf{J}_M)})} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["log", 2], "Equality(log(Function('\\\\psi')(Symbol('\\\\mathbf{J}_M', commutative=True))), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(log(Function('\\\\psi')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 4, "log(cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(log(Function('\\\\psi')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), log(Function('q')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(log(Function('\\\\psi')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Mul(Integer(-1), log(Function('q')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\mathbb{I}{(\\hat{\\mathbf{x}})} = \\frac{1}{\\mathbf{P}{(\\hat{\\mathbf{x}})}}, then obtain - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} + \\frac{1}{\\sin{(\\hat{\\mathbf{x}})}} = - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} + \\frac{1}{\\mathbf{P}{(\\hat{\\mathbf{x}})}}", "derivation": "\\mathbf{P}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\mathbb{I}{(\\hat{\\mathbf{x}})} = \\frac{1}{\\mathbf{P}{(\\hat{\\mathbf{x}})}} and \\mathbb{I}{(\\hat{\\mathbf{x}})} = \\frac{1}{\\sin{(\\hat{\\mathbf{x}})}} and \\mathbb{I}{(\\hat{\\mathbf{x}})} - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} = - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} + \\frac{1}{\\mathbf{P}{(\\hat{\\mathbf{x}})}} and - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} + \\frac{1}{\\sin{(\\hat{\\mathbf{x}})}} = - \\sin^{\\hat{\\mathbf{x}}}{(\\hat{\\mathbf{x}})} + \\frac{1}{\\mathbf{P}{(\\hat{\\mathbf{x}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)))"], [["minus", 2, "Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\chi{(k)} = \\log{(k)} and \\operatorname{g_{\\varepsilon}}{(k)} = (- k + \\chi{(k)}) (- k + \\cos{(\\log{(k)})}), then obtain (- k + \\chi{(k)}) (- k + \\cos{(\\log{(k)})}) = (- k + \\log{(k)}) (- k + \\cos{(\\log{(k)})})", "derivation": "\\chi{(k)} = \\log{(k)} and - k + \\chi{(k)} = - k + \\log{(k)} and \\cos{(\\chi{(k)})} = \\cos{(\\log{(k)})} and \\operatorname{g_{\\varepsilon}}{(k)} = (- k + \\chi{(k)}) (- k + \\cos{(\\log{(k)})}) and \\operatorname{g_{\\varepsilon}}{(k)} = (- k + \\chi{(k)}) (- k + \\cos{(\\chi{(k)})}) and \\operatorname{g_{\\varepsilon}}{(k)} = (- k + \\log{(k)}) (- k + \\cos{(\\chi{(k)})}) and (- k + \\chi{(k)}) (- k + \\cos{(\\chi{(k)})}) = (- k + \\log{(k)}) (- k + \\cos{(\\chi{(k)})}) and (- k + \\chi{(k)}) (- k + \\cos{(\\log{(k)})}) = (- k + \\log{(k)}) (- k + \\cos{(\\log{(k)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["minus", 1, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\chi')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\chi')(Symbol('k', commutative=True))), cos(log(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('k', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\chi')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(log(Symbol('k', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('k', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\chi')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Function('\\\\chi')(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('g_{\\\\varepsilon}')(Symbol('k', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Function('\\\\chi')(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\chi')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Function('\\\\chi')(Symbol('k', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Function('\\\\chi')(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\chi')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(log(Symbol('k', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(log(Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain \\frac{d}{d \\mathbf{B}} (\\hat{x}_0^{\\mathbf{B}}{(\\mathbf{B})} - (e^{\\mathbf{B}})^{\\mathbf{B}}) = \\frac{d}{d \\mathbf{B}} 0", "derivation": "\\hat{x}_0{(\\mathbf{B})} = e^{\\mathbf{B}} and \\hat{x}_0^{\\mathbf{B}}{(\\mathbf{B})} = (e^{\\mathbf{B}})^{\\mathbf{B}} and \\hat{x}_0^{\\mathbf{B}}{(\\mathbf{B})} - (e^{\\mathbf{B}})^{\\mathbf{B}} = 0 and \\frac{d}{d \\mathbf{B}} (\\hat{x}_0^{\\mathbf{B}}{(\\mathbf{B})} - (e^{\\mathbf{B}})^{\\mathbf{B}}) = \\frac{d}{d \\mathbf{B}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 2, "Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\rho_b,A_{2})} = \\frac{\\partial}{\\partial \\rho_b} (A_{2} + \\rho_b), then derive \\operatorname{z^{*}}{(\\rho_b,A_{2})} = 1, then derive A_{2} + \\operatorname{z^{*}}{(\\rho_b,A_{2})} = A_{2} + 1, then obtain \\log{(A_{2} + \\operatorname{z^{*}}{(\\rho_b,A_{2})})} = \\log{(A_{2} + 1)}", "derivation": "\\operatorname{z^{*}}{(\\rho_b,A_{2})} = \\frac{\\partial}{\\partial \\rho_b} (A_{2} + \\rho_b) and \\operatorname{z^{*}}{(\\rho_b,A_{2})} = 1 and A_{2} \\operatorname{z^{*}}{(\\rho_b,A_{2})} = A_{2} and A_{2} \\frac{\\partial}{\\partial \\rho_b} (A_{2} + \\rho_b) = A_{2} and \\operatorname{z^{*}}{(\\rho_b,A_{2} \\frac{\\partial}{\\partial \\rho_b} (A_{2} + \\rho_b))} = 1 and A_{2} + \\operatorname{z^{*}}{(\\rho_b,A_{2} \\frac{\\partial}{\\partial \\rho_b} (A_{2} + \\rho_b))} = A_{2} + 1 and A_{2} + \\operatorname{z^{*}}{(\\rho_b,A_{2})} = A_{2} + 1 and \\log{(A_{2} + \\operatorname{z^{*}}{(\\rho_b,A_{2})})} = \\log{(A_{2} + 1)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('A_2', commutative=True)), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('A_2', commutative=True)), Integer(1))"], [["times", 2, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A_2', commutative=True), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Symbol('A_2', commutative=True))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Mul(Symbol('A_2', commutative=True), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))), Integer(1))"], [["add", 5, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Function('z^*')(Symbol('\\\\rho_b', commutative=True), Mul(Symbol('A_2', commutative=True), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))), Add(Symbol('A_2', commutative=True), Integer(1)))"], [["evaluate_derivatives", 6], "Equality(Add(Symbol('A_2', commutative=True), Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('A_2', commutative=True))), Add(Symbol('A_2', commutative=True), Integer(1)))"], [["log", 7], "Equality(log(Add(Symbol('A_2', commutative=True), Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('A_2', commutative=True)))), log(Add(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(G)} = \\sin{(G)} and \\nabla{(G)} = \\mathbf{F}^{2}{(G)}, then obtain \\frac{\\mathbf{F}^{2}{(G)}}{G} = \\frac{\\nabla{(G)}}{G}", "derivation": "\\mathbf{F}{(G)} = \\sin{(G)} and \\mathbf{F}^{2}{(G)} = \\mathbf{F}{(G)} \\sin{(G)} and \\frac{\\mathbf{F}^{2}{(G)}}{G} = \\frac{\\mathbf{F}{(G)} \\sin{(G)}}{G} and \\nabla{(G)} = \\mathbf{F}^{2}{(G)} and \\nabla{(G)} = \\mathbf{F}{(G)} \\sin{(G)} and \\frac{\\mathbf{F}^{2}{(G)}}{G} = \\frac{\\nabla{(G)}}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{F}')(Symbol('G', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True))))"], [["divide", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), Integer(2))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('G', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\nabla')(Symbol('G', commutative=True)), Mul(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('G', commutative=True)), Integer(2))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\phi{(E,\\Omega)} = E - \\Omega, then obtain \\frac{\\partial}{\\partial \\Omega} - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} \\phi{(E,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} (E - \\Omega)", "derivation": "\\phi{(E,\\Omega)} = E - \\Omega and \\frac{\\partial}{\\partial E} \\phi{(E,\\Omega)} = \\frac{\\partial}{\\partial E} (E - \\Omega) and - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} \\phi{(E,\\Omega)} = - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} (E - \\Omega) and \\frac{\\partial}{\\partial \\Omega} - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} \\phi{(E,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} - \\phi{(E,\\Omega)} \\frac{\\partial}{\\partial E} (E - \\Omega)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})} = (\\frac{\\mathbf{J}_P}{Q})^{t_{2}}, then derive 0 = - \\frac{\\partial}{\\partial Q} \\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})} - \\frac{t_{2} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}}}{Q}, then obtain 0 = - \\frac{\\partial}{\\partial Q} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}} - \\frac{t_{2} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}}}{Q}", "derivation": "\\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})} = (\\frac{\\mathbf{J}_P}{Q})^{t_{2}} and 0 = (\\frac{\\mathbf{J}_P}{Q})^{t_{2}} - \\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})} and \\frac{d}{d Q} 0 = \\frac{\\partial}{\\partial Q} ((\\frac{\\mathbf{J}_P}{Q})^{t_{2}} - \\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})}) and 0 = - \\frac{\\partial}{\\partial Q} \\mathbf{F}{(Q,\\mathbf{J}_P,t_{2})} - \\frac{t_{2} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}}}{Q} and 0 = - \\frac{\\partial}{\\partial Q} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}} - \\frac{t_{2} (\\frac{\\mathbf{J}_P}{Q})^{t_{2}}}{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('t_2', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{F}')(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('t_2', commutative=True)))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(A_{y},F_{H})} = A_{y} - F_{H} and \\chi{(A_{y},F_{H})} = A_{y} - 2 F_{H}, then derive \\frac{\\partial}{\\partial A_{y}} \\chi{(A_{y},F_{H})} = 1, then obtain \\frac{\\partial}{\\partial A_{y}} (A_{y} - 2 F_{H}) = 1", "derivation": "\\hat{p}_0{(A_{y},F_{H})} = A_{y} - F_{H} and - F_{H} + \\hat{p}_0{(A_{y},F_{H})} = A_{y} - 2 F_{H} and \\chi{(A_{y},F_{H})} = A_{y} - 2 F_{H} and \\frac{\\partial}{\\partial A_{y}} \\chi{(A_{y},F_{H})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} - 2 F_{H}) and \\frac{\\partial}{\\partial A_{y}} \\chi{(A_{y},F_{H})} = 1 and - F_{H} + \\hat{p}_0{(A_{y},F_{H})} = \\chi{(A_{y},F_{H})} and \\frac{\\partial}{\\partial A_{y}} (- F_{H} + \\hat{p}_0{(A_{y},F_{H})}) = 1 and \\frac{\\partial}{\\partial A_{y}} (A_{y} - 2 F_{H}) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))))"], [["differentiate", 3, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\chi')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True))), Function('\\\\chi')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\Omega)} = \\Omega, then derive \\int \\mathbf{J}_f{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\mu, then obtain \\int \\Omega d\\Omega = \\frac{\\Omega^{2}}{2} + \\mu", "derivation": "\\mathbf{J}_f{(\\Omega)} = \\Omega and \\int \\mathbf{J}_f{(\\Omega)} d\\Omega = \\int \\Omega d\\Omega and \\int \\mathbf{J}_f{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\mu and \\int \\Omega d\\Omega = \\frac{\\Omega^{2}}{2} + \\mu", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given L{(P_{e})} = \\log{(P_{e})}, then obtain L{(P_{e})} \\int 1 dP_{e} = \\log{(P_{e})} \\int 1 dP_{e}", "derivation": "L{(P_{e})} = \\log{(P_{e})} and 1 = \\frac{\\log{(P_{e})}}{L{(P_{e})}} and 1 = (\\frac{\\log{(P_{e})}}{L{(P_{e})}})^{P_{e}} and \\int 1 dP_{e} = \\int (\\frac{\\log{(P_{e})}}{L{(P_{e})}})^{P_{e}} dP_{e} and L{(P_{e})} \\int (\\frac{\\log{(P_{e})}}{L{(P_{e})}})^{P_{e}} dP_{e} = \\log{(P_{e})} \\int (\\frac{\\log{(P_{e})}}{L{(P_{e})}})^{P_{e}} dP_{e} and L{(P_{e})} \\int 1 dP_{e} = \\log{(P_{e})} \\int 1 dP_{e}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["divide", 1, "Function('L')(Symbol('P_e', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('P_e', commutative=True))), Integral(Pow(Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["times", 1, "Integral(Pow(Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))"], "Equality(Mul(Function('L')(Symbol('P_e', commutative=True)), Integral(Pow(Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Mul(log(Symbol('P_e', commutative=True)), Integral(Pow(Mul(Pow(Function('L')(Symbol('P_e', commutative=True)), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('L')(Symbol('P_e', commutative=True)), Integral(Integer(1), Tuple(Symbol('P_e', commutative=True)))), Mul(log(Symbol('P_e', commutative=True)), Integral(Integer(1), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given y{(\\hbar,\\eta^{\\prime})} = \\eta^{\\prime} \\hbar, then obtain \\frac{y{(\\hbar,\\eta^{\\prime})}}{\\eta^{\\prime} \\hbar + y{(\\hbar,\\eta^{\\prime})}} - \\frac{1}{\\eta^{\\prime}} = \\frac{1}{2} - \\frac{1}{\\eta^{\\prime}}", "derivation": "y{(\\hbar,\\eta^{\\prime})} = \\eta^{\\prime} \\hbar and \\eta^{\\prime} \\hbar + y{(\\hbar,\\eta^{\\prime})} = 2 \\eta^{\\prime} \\hbar and \\frac{y{(\\hbar,\\eta^{\\prime})}}{2 \\eta^{\\prime} \\hbar} = \\frac{1}{2} and \\frac{y{(\\hbar,\\eta^{\\prime})}}{\\eta^{\\prime} \\hbar + y{(\\hbar,\\eta^{\\prime})}} = \\frac{1}{2} and \\frac{y{(\\hbar,\\eta^{\\prime})}}{\\eta^{\\prime} \\hbar + y{(\\hbar,\\eta^{\\prime})}} - \\frac{1}{\\eta^{\\prime}} = \\frac{1}{2} - \\frac{1}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Rational(1, 2))"], [["minus", 4, "Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Function('y')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)))), Add(Rational(1, 2), Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(c_{0},A_{x})} = \\frac{e^{A_{x}}}{c_{0}} and \\mathbf{J}{(c_{0},A_{x})} = \\frac{e^{A_{x}}}{c_{0}}, then obtain A_{x} + \\operatorname{n_{2}}^{A_{x}}{(c_{0},A_{x})} = A_{x} + \\mathbf{J}^{A_{x}}{(c_{0},A_{x})}", "derivation": "\\operatorname{n_{2}}{(c_{0},A_{x})} = \\frac{e^{A_{x}}}{c_{0}} and \\operatorname{n_{2}}^{A_{x}}{(c_{0},A_{x})} = (\\frac{e^{A_{x}}}{c_{0}})^{A_{x}} and A_{x} + \\operatorname{n_{2}}^{A_{x}}{(c_{0},A_{x})} = A_{x} + (\\frac{e^{A_{x}}}{c_{0}})^{A_{x}} and \\mathbf{J}{(c_{0},A_{x})} = \\frac{e^{A_{x}}}{c_{0}} and A_{x} + \\operatorname{n_{2}}^{A_{x}}{(c_{0},A_{x})} = A_{x} + \\mathbf{J}^{A_{x}}{(c_{0},A_{x})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Symbol('A_x', commutative=True))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["add", 2, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Pow(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('A_x', commutative=True), Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given t{(\\delta,\\mathbf{B})} = \\frac{\\log{(\\mathbf{B})}}{\\delta}, then obtain - \\mathbf{B} + (\\iint t{(\\delta,\\mathbf{B})} d\\delta d\\mathbf{B})^{\\delta} = - \\mathbf{B} + (\\iint \\frac{\\log{(\\mathbf{B})}}{\\delta} d\\delta d\\mathbf{B})^{\\delta}", "derivation": "t{(\\delta,\\mathbf{B})} = \\frac{\\log{(\\mathbf{B})}}{\\delta} and \\int t{(\\delta,\\mathbf{B})} d\\delta = \\int \\frac{\\log{(\\mathbf{B})}}{\\delta} d\\delta and \\iint t{(\\delta,\\mathbf{B})} d\\delta d\\mathbf{B} = \\iint \\frac{\\log{(\\mathbf{B})}}{\\delta} d\\delta d\\mathbf{B} and (\\iint t{(\\delta,\\mathbf{B})} d\\delta d\\mathbf{B})^{\\delta} = (\\iint \\frac{\\log{(\\mathbf{B})}}{\\delta} d\\delta d\\mathbf{B})^{\\delta} and - \\mathbf{B} + (\\iint t{(\\delta,\\mathbf{B})} d\\delta d\\mathbf{B})^{\\delta} = - \\mathbf{B} + (\\iint \\frac{\\log{(\\mathbf{B})}}{\\delta} d\\delta d\\mathbf{B})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["minus", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given C{(F_{g},t_{2})} = \\frac{\\log{(F_{g})}}{t_{2}}, then derive \\frac{\\partial}{\\partial t_{2}} C{(F_{g},t_{2})} = - \\frac{\\log{(F_{g})}}{t_{2}^{2}}, then obtain - \\frac{C{(F_{g},t_{2})}}{t_{2}} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\log{(F_{g})}}{t_{2}}", "derivation": "C{(F_{g},t_{2})} = \\frac{\\log{(F_{g})}}{t_{2}} and \\frac{\\partial}{\\partial t_{2}} C{(F_{g},t_{2})} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\log{(F_{g})}}{t_{2}} and \\frac{\\partial}{\\partial t_{2}} C{(F_{g},t_{2})} = - \\frac{\\log{(F_{g})}}{t_{2}^{2}} and \\frac{\\partial}{\\partial t_{2}} C{(F_{g},t_{2})} = - \\frac{C{(F_{g},t_{2})}}{t_{2}} and - \\frac{C{(F_{g},t_{2})}}{t_{2}} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\log{(F_{g})}}{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-2)), log(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('C')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}}, then derive - \\mathbf{H} + \\rho_{f}{(\\mathbf{H})} = - \\mathbf{H} + e^{\\mathbf{H}}, then obtain (- \\mathbf{H} + e^{\\mathbf{H}}) (- \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}}) = (- \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}})^{2}", "derivation": "\\rho_{f}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}} and - \\mathbf{H} + \\rho_{f}{(\\mathbf{H})} = - \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}} and - \\mathbf{H} + \\rho_{f}{(\\mathbf{H})} = - \\mathbf{H} + e^{\\mathbf{H}} and - \\mathbf{H} + e^{\\mathbf{H}} = - \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}} and (- \\mathbf{H} + e^{\\mathbf{H}}) (- \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}}) = (- \\mathbf{H} + \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{A}{(q,\\hat{x}_0)} = - \\hat{x}_0 + q, then obtain \\frac{\\int \\frac{\\int \\mathbf{A}{(q,\\hat{x}_0)} dq}{- \\hat{x}_0 + q} dq}{- \\hat{x}_0 + q} = \\frac{\\int \\frac{\\int (- \\hat{x}_0 + q) dq}{- \\hat{x}_0 + q} dq}{- \\hat{x}_0 + q}", "derivation": "\\mathbf{A}{(q,\\hat{x}_0)} = - \\hat{x}_0 + q and \\int \\mathbf{A}{(q,\\hat{x}_0)} dq = \\int (- \\hat{x}_0 + q) dq and \\frac{\\int \\mathbf{A}{(q,\\hat{x}_0)} dq}{- \\hat{x}_0 + q} = \\frac{\\int (- \\hat{x}_0 + q) dq}{- \\hat{x}_0 + q} and \\int \\frac{\\int \\mathbf{A}{(q,\\hat{x}_0)} dq}{- \\hat{x}_0 + q} dq = \\int \\frac{\\int (- \\hat{x}_0 + q) dq}{- \\hat{x}_0 + q} dq and \\frac{\\int \\frac{\\int \\mathbf{A}{(q,\\hat{x}_0)} dq}{- \\hat{x}_0 + q} dq}{- \\hat{x}_0 + q} = \\frac{\\int \\frac{\\int (- \\hat{x}_0 + q) dq}{- \\hat{x}_0 + q} dq}{- \\hat{x}_0 + q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(r_{0},A_{x})} = \\frac{r_{0}}{A_{x}}, then obtain 2 \\log{(\\frac{r_{0}}{A_{x}})} + \\log{(\\varepsilon_{0}{(r_{0},A_{x})})} = 3 \\log{(\\frac{r_{0}}{A_{x}})}", "derivation": "\\varepsilon_{0}{(r_{0},A_{x})} = \\frac{r_{0}}{A_{x}} and \\log{(\\varepsilon_{0}{(r_{0},A_{x})})} = \\log{(\\frac{r_{0}}{A_{x}})} and \\log{(\\frac{r_{0}}{A_{x}})} + \\log{(\\varepsilon_{0}{(r_{0},A_{x})})} = 2 \\log{(\\frac{r_{0}}{A_{x}})} and 2 \\log{(\\frac{r_{0}}{A_{x}})} + \\log{(\\varepsilon_{0}{(r_{0},A_{x})})} = 3 \\log{(\\frac{r_{0}}{A_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True))), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))"], [["add", 2, "log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], "Equality(Add(log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), log(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)))), Mul(Integer(2), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))))"], [["add", 3, "log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(2), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))), log(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)))), Mul(Integer(3), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(m_{s})} = \\sin{(m_{s})}, then derive \\int \\operatorname{A_{2}}{(m_{s})} dm_{s} = \\hbar - \\cos{(m_{s})}, then obtain \\operatorname{A_{2}}{(m_{s})} \\int \\sin{(m_{s})} dm_{s} = (\\hbar - \\cos{(m_{s})}) \\operatorname{A_{2}}{(m_{s})}", "derivation": "\\operatorname{A_{2}}{(m_{s})} = \\sin{(m_{s})} and \\int \\operatorname{A_{2}}{(m_{s})} dm_{s} = \\int \\sin{(m_{s})} dm_{s} and \\int \\operatorname{A_{2}}{(m_{s})} dm_{s} = \\hbar - \\cos{(m_{s})} and \\int \\sin{(m_{s})} dm_{s} = \\hbar - \\cos{(m_{s})} and \\operatorname{A_{2}}{(m_{s})} \\int \\sin{(m_{s})} dm_{s} = (\\hbar - \\cos{(m_{s})}) \\operatorname{A_{2}}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_2')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["times", 4, "Function('A_2')(Symbol('m_s', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('m_s', commutative=True)), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))), Function('A_2')(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\log{(A_{y})} and \\Omega{(A_{y},\\Psi_{\\lambda})} = (- \\Psi_{\\lambda} + \\log{(A_{y})})^{2}, then obtain \\frac{\\partial}{\\partial A_{y}} (- \\Psi_{\\lambda} + \\log{(A_{y})}) \\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial A_{y}} \\Omega{(A_{y},\\Psi_{\\lambda})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\log{(A_{y})} and (- \\Psi_{\\lambda} + \\log{(A_{y})}) \\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = (- \\Psi_{\\lambda} + \\log{(A_{y})})^{2} and \\frac{\\partial}{\\partial A_{y}} (- \\Psi_{\\lambda} + \\log{(A_{y})}) \\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial A_{y}} (- \\Psi_{\\lambda} + \\log{(A_{y})})^{2} and \\Omega{(A_{y},\\Psi_{\\lambda})} = (- \\Psi_{\\lambda} + \\log{(A_{y})})^{2} and \\frac{\\partial}{\\partial A_{y}} (- \\Psi_{\\lambda} + \\log{(A_{y})}) \\operatorname{g_{\\varepsilon}}{(A_{y},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial A_{y}} \\Omega{(A_{y},\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Integer(2)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('A_y', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Function('\\\\Omega')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(H,\\tilde{g})} = H - \\tilde{g}, then obtain \\frac{\\partial}{\\partial H} (\\frac{e^{\\mathbf{f}{(H,\\tilde{g})}}}{\\tilde{g}})^{\\tilde{g}} = \\frac{\\partial}{\\partial H} (\\frac{e^{H - \\tilde{g}}}{\\tilde{g}})^{\\tilde{g}}", "derivation": "\\mathbf{f}{(H,\\tilde{g})} = H - \\tilde{g} and e^{\\mathbf{f}{(H,\\tilde{g})}} = e^{H - \\tilde{g}} and \\frac{e^{\\mathbf{f}{(H,\\tilde{g})}}}{\\tilde{g}} = \\frac{e^{H - \\tilde{g}}}{\\tilde{g}} and (\\frac{e^{\\mathbf{f}{(H,\\tilde{g})}}}{\\tilde{g}})^{\\tilde{g}} = (\\frac{e^{H - \\tilde{g}}}{\\tilde{g}})^{\\tilde{g}} and \\frac{\\partial}{\\partial H} (\\frac{e^{\\mathbf{f}{(H,\\tilde{g})}}}{\\tilde{g}})^{\\tilde{g}} = \\frac{\\partial}{\\partial H} (\\frac{e^{H - \\tilde{g}}}{\\tilde{g}})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{f}')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))))"], [["divide", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Function('\\\\mathbf{f}')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))))"], [["power", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Function('\\\\mathbf{f}')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 4, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Function('\\\\mathbf{f}')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\nabla,F_{N})} = F_{N} + \\nabla, then obtain (F_{N} + P_{g} + \\nabla \\log{(F_{N})}) \\int \\frac{\\tilde{g}^*{(\\nabla,F_{N})}}{F_{N}} dF_{N} = (F_{N} + P_{g} + \\nabla \\log{(F_{N})})^{2}", "derivation": "\\tilde{g}^*{(\\nabla,F_{N})} = F_{N} + \\nabla and \\frac{\\tilde{g}^*{(\\nabla,F_{N})}}{F_{N}} = \\frac{F_{N} + \\nabla}{F_{N}} and \\int \\frac{\\tilde{g}^*{(\\nabla,F_{N})}}{F_{N}} dF_{N} = \\int \\frac{F_{N} + \\nabla}{F_{N}} dF_{N} and (\\int \\frac{F_{N} + \\nabla}{F_{N}} dF_{N}) \\int \\frac{\\tilde{g}^*{(\\nabla,F_{N})}}{F_{N}} dF_{N} = (\\int \\frac{F_{N} + \\nabla}{F_{N}} dF_{N})^{2} and (F_{N} + P_{g} + \\nabla \\log{(F_{N})}) \\int \\frac{\\tilde{g}^*{(\\nabla,F_{N})}}{F_{N}} dF_{N} = (F_{N} + P_{g} + \\nabla \\log{(F_{N})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["divide", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["times", 3, "Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Mul(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Pow(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(2)))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('F_N', commutative=True), Symbol('P_g', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), log(Symbol('F_N', commutative=True)))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Pow(Add(Symbol('F_N', commutative=True), Symbol('P_g', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), log(Symbol('F_N', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given T{(v_{x},L)} = L + v_{x}, then obtain \\int - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL dL = \\int (e^{L + v_{x}} - e^{T{(v_{x},L)}} - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL) dL", "derivation": "T{(v_{x},L)} = L + v_{x} and e^{T{(v_{x},L)}} = e^{L + v_{x}} and e^{T{(v_{x},L)}} = e^{L} e^{v_{x}} and e^{L} e^{v_{x}} = e^{L + v_{x}} and 0 = e^{L} e^{v_{x}} - e^{T{(v_{x},L)}} and 0 = e^{L + v_{x}} - e^{T{(v_{x},L)}} and - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL = e^{L + v_{x}} - e^{T{(v_{x},L)}} - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL and \\int - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL dL = \\int (e^{L + v_{x}} - e^{T{(v_{x},L)}} - \\int (e^{L} e^{v_{x}} - e^{T{(v_{x},L)}}) dL) dL", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True)))"], [["exp", 1], "Equality(exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))), exp(Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True))))"], [["expand", 2], "Equality(exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))), Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), exp(Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True))))"], [["minus", 3, "exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True)))"], "Equality(Integer(0), Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Add(exp(Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))))"], [["minus", 6, "Integral(Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True)))), Add(exp(Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True))))))"], [["integrate", 7, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integral(Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True))), Integral(Add(exp(Add(Symbol('L', commutative=True), Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(exp(Symbol('L', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Function('T')(Symbol('v_x', commutative=True), Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)}, then obtain \\int \\frac{\\operatorname{t_{1}}^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} d\\mathbf{J}_f = \\int \\frac{\\cos^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} d\\mathbf{J}_f", "derivation": "\\operatorname{t_{1}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\operatorname{t_{1}}^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = \\cos^{\\mathbf{J}_f}{(\\mathbf{J}_f)} and \\frac{\\operatorname{t_{1}}^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} = \\frac{\\cos^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} and \\int \\frac{\\operatorname{t_{1}}^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} d\\mathbf{J}_f = \\int \\frac{\\cos^{\\mathbf{J}_f}{(\\mathbf{J}_f)}}{\\cos{(\\mathbf{J}_f)}} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 2, "cos(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Pow(Function('t_1')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Mul(Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(Pow(Function('t_1')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Mul(Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\psi^*)} = \\sin{(\\cos{(\\psi^*)})}, then obtain - u + \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})} = - u + \\frac{d}{d \\psi^*} \\frac{\\sin^{2}{(\\cos{(\\psi^*)})}}{\\operatorname{m_{s}}{(\\psi^*)}}", "derivation": "\\operatorname{m_{s}}{(\\psi^*)} = \\sin{(\\cos{(\\psi^*)})} and 1 = \\frac{\\sin{(\\cos{(\\psi^*)})}}{\\operatorname{m_{s}}{(\\psi^*)}} and \\sin{(\\cos{(\\psi^*)})} = \\frac{\\sin^{2}{(\\cos{(\\psi^*)})}}{\\operatorname{m_{s}}{(\\psi^*)}} and \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})} = \\frac{d}{d \\psi^*} \\frac{\\sin^{2}{(\\cos{(\\psi^*)})}}{\\operatorname{m_{s}}{(\\psi^*)}} and - u + \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})} = - u + \\frac{d}{d \\psi^*} \\frac{\\sin^{2}{(\\cos{(\\psi^*)})}}{\\operatorname{m_{s}}{(\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), sin(cos(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Function('m_s')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\psi^*', commutative=True)))))"], [["times", 2, "sin(cos(Symbol('\\\\psi^*', commutative=True)))"], "Equality(sin(cos(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('\\\\psi^*', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(sin(cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('\\\\psi^*', commutative=True))), Integer(2))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(sin(cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Pow(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('\\\\psi^*', commutative=True))), Integer(2))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{A}{(f^{\\prime})} = \\cos{(f^{\\prime})}, then obtain \\frac{d}{d f^{\\prime}} (\\mathbf{A}{(f^{\\prime})} - \\cos{(f^{\\prime})})^{f^{\\prime}} = \\frac{d}{d f^{\\prime}} 0^{f^{\\prime}}", "derivation": "\\mathbf{A}{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\mathbf{A}{(f^{\\prime})} - \\cos{(f^{\\prime})} = 0 and (\\mathbf{A}{(f^{\\prime})} - \\cos{(f^{\\prime})})^{f^{\\prime}} = 0^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} (\\mathbf{A}{(f^{\\prime})} - \\cos{(f^{\\prime})})^{f^{\\prime}} = \\frac{d}{d f^{\\prime}} 0^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "cos(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{A}')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\mathbf{A}')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(x)} = e^{x}, then obtain 1 = \\frac{e^{x}}{\\frac{d}{d x} n{(x)}}", "derivation": "n{(x)} = e^{x} and \\frac{d}{d x} n{(x)} = \\frac{d}{d x} e^{x} and 1 = \\frac{\\frac{d}{d x} e^{x}}{\\frac{d}{d x} n{(x)}} and 1 = \\frac{e^{x}}{\\frac{d}{d x} n{(x)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('n')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('n')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(exp(Symbol('x', commutative=True)), Pow(Derivative(Function('n')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given E{(\\eta,g^{\\prime}_{\\varepsilon})} = \\eta g^{\\prime}_{\\varepsilon} and \\operatorname{F_{g}}{(\\psi)} = \\sin{(\\psi)}, then obtain \\operatorname{F_{g}}^{\\psi}{(\\psi)} + \\cos{(\\cos{(\\eta g^{\\prime}_{\\varepsilon})})} = \\sin^{\\psi}{(\\psi)} + \\cos{(\\cos{(\\eta g^{\\prime}_{\\varepsilon})})}", "derivation": "E{(\\eta,g^{\\prime}_{\\varepsilon})} = \\eta g^{\\prime}_{\\varepsilon} and \\cos{(E{(\\eta,g^{\\prime}_{\\varepsilon})})} = \\cos{(\\eta g^{\\prime}_{\\varepsilon})} and \\operatorname{F_{g}}{(\\psi)} = \\sin{(\\psi)} and \\operatorname{F_{g}}^{\\psi}{(\\psi)} = \\sin^{\\psi}{(\\psi)} and \\operatorname{F_{g}}^{\\psi}{(\\psi)} + \\cos{(\\cos{(E{(\\eta,g^{\\prime}_{\\varepsilon})})})} = \\sin^{\\psi}{(\\psi)} + \\cos{(\\cos{(E{(\\eta,g^{\\prime}_{\\varepsilon})})})} and \\operatorname{F_{g}}^{\\psi}{(\\psi)} + \\cos{(\\cos{(\\eta g^{\\prime}_{\\varepsilon})})} = \\sin^{\\psi}{(\\psi)} + \\cos{(\\cos{(\\eta g^{\\prime}_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), cos(Mul(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["add", 4, "cos(cos(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], "Equality(Add(Pow(Function('F_g')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), cos(cos(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Add(Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), cos(cos(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Function('F_g')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), cos(cos(Mul(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Add(Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), cos(cos(Mul(Symbol('\\\\eta', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given A{(\\mathbf{S},L_{\\varepsilon})} = - L_{\\varepsilon} + \\mathbf{S} and \\operatorname{v_{t}}{(\\mathbf{S})} = - \\mathbf{S}, then obtain L_{\\varepsilon} - 2 \\mathbf{S} + A{(\\mathbf{S},L_{\\varepsilon})} = - \\mathbf{S}", "derivation": "A{(\\mathbf{S},L_{\\varepsilon})} = - L_{\\varepsilon} + \\mathbf{S} and L_{\\varepsilon} - \\mathbf{S} + A{(\\mathbf{S},L_{\\varepsilon})} = 0 and \\operatorname{v_{t}}{(\\mathbf{S})} = - \\mathbf{S} and L_{\\varepsilon} - \\mathbf{S} + A{(\\mathbf{S},L_{\\varepsilon})} + \\operatorname{v_{t}}{(\\mathbf{S})} = \\operatorname{v_{t}}{(\\mathbf{S})} and L_{\\varepsilon} - 2 \\mathbf{S} + A{(\\mathbf{S},L_{\\varepsilon})} = - \\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('A')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 2, "Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('A')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True))), Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Function('A')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given v{(\\phi)} = \\log{(\\phi)}, then obtain \\log{(2 v{(\\phi)} - \\log{(\\phi)})} + \\log{(v{(\\phi)})} = \\log{(4 v{(\\phi)} - 3 \\log{(\\phi)})} + \\log{(v{(\\phi)})}", "derivation": "v{(\\phi)} = \\log{(\\phi)} and \\log{(v{(\\phi)})} = \\log{(\\log{(\\phi)})} and 2 v{(\\phi)} - \\log{(\\phi)} = v{(\\phi)} and \\log{(2 v{(\\phi)} - \\log{(\\phi)})} = \\log{(\\log{(\\phi)})} and \\log{(2 v{(\\phi)} - \\log{(\\phi)})} + \\log{(v{(\\phi)})} = \\log{(v{(\\phi)})} + \\log{(\\log{(\\phi)})} and \\log{(4 v{(\\phi)} - 3 \\log{(\\phi)})} = \\log{(\\log{(\\phi)})} and \\log{(2 v{(\\phi)} - \\log{(\\phi)})} + \\log{(v{(\\phi)})} = \\log{(4 v{(\\phi)} - 3 \\log{(\\phi)})} + \\log{(v{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["log", 1], "Equality(log(Function('v')(Symbol('\\\\phi', commutative=True))), log(log(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Function('v')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True)))), Function('v')(Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(log(Add(Mul(Integer(2), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True))))), log(log(Symbol('\\\\phi', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), log(Function('v')(Symbol('\\\\phi', commutative=True))))"], "Equality(Add(log(Add(Mul(Integer(2), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True))))), log(Function('v')(Symbol('\\\\phi', commutative=True)))), Add(log(Function('v')(Symbol('\\\\phi', commutative=True))), log(log(Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(log(Add(Mul(Integer(4), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('\\\\phi', commutative=True))))), log(log(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(log(Add(Mul(Integer(2), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True))))), log(Function('v')(Symbol('\\\\phi', commutative=True)))), Add(log(Add(Mul(Integer(4), Function('v')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('\\\\phi', commutative=True))))), log(Function('v')(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\hat{X},b)} = \\hat{X} b and \\mathbf{S}{(\\hat{X},b)} = \\mathbf{v}^{b}{(\\hat{X},b)}, then obtain (\\mathbf{v}^{b}{(\\hat{X},b)})^{b} - \\mathbf{S}^{b}{(\\hat{X},b)} = 0", "derivation": "\\mathbf{v}{(\\hat{X},b)} = \\hat{X} b and \\mathbf{v}^{b}{(\\hat{X},b)} = (\\hat{X} b)^{b} and (\\mathbf{v}^{b}{(\\hat{X},b)})^{b} = ((\\hat{X} b)^{b})^{b} and \\mathbf{S}{(\\hat{X},b)} = \\mathbf{v}^{b}{(\\hat{X},b)} and \\mathbf{S}{(\\hat{X},b)} = (\\hat{X} b)^{b} and - ((\\hat{X} b)^{b})^{b} + (\\mathbf{v}^{b}{(\\hat{X},b)})^{b} = 0 and (\\mathbf{v}^{b}{(\\hat{X},b)})^{b} - \\mathbf{S}^{b}{(\\hat{X},b)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Pow(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["minus", 3, "Pow(Pow(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{P}{(v_{2},f)} = e^{f v_{2}}, then obtain \\frac{\\int \\mathbf{P}{(v_{2},f)} dv_{2}}{\\mathbf{P}{(v_{2},f)} \\int e^{f v_{2}} dv_{2}} = \\frac{1}{\\mathbf{P}{(v_{2},f)}}", "derivation": "\\mathbf{P}{(v_{2},f)} = e^{f v_{2}} and \\int \\mathbf{P}{(v_{2},f)} dv_{2} = \\int e^{f v_{2}} dv_{2} and \\frac{\\int \\mathbf{P}{(v_{2},f)} dv_{2}}{\\int e^{f v_{2}} dv_{2}} = 1 and \\frac{\\int \\mathbf{P}{(v_{2},f)} dv_{2}}{\\mathbf{P}{(v_{2},f)} \\int e^{f v_{2}} dv_{2}} = \\frac{1}{\\mathbf{P}{(v_{2},f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), exp(Mul(Symbol('f', commutative=True), Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(exp(Mul(Symbol('f', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["divide", 2, "Integral(exp(Mul(Symbol('f', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Pow(Integral(exp(Mul(Symbol('f', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 3, "Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Pow(Integral(exp(Mul(Symbol('f', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))), Pow(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('f', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(C,l)} = C + l and \\operatorname{E_{n}}{(C,l)} = \\frac{\\operatorname{A_{1}}{(C,l)}}{C}, then obtain \\frac{1}{C + l} + \\frac{\\operatorname{A_{1}}{(C,l)}}{C} = \\frac{1}{C + l} + \\frac{C + l}{C}", "derivation": "\\operatorname{A_{1}}{(C,l)} = C + l and \\frac{\\operatorname{A_{1}}{(C,l)}}{C} = \\frac{C + l}{C} and \\operatorname{E_{n}}{(C,l)} = \\frac{\\operatorname{A_{1}}{(C,l)}}{C} and \\operatorname{E_{n}}{(C,l)} = \\frac{C + l}{C} and \\operatorname{E_{n}}{(C,l)} + \\frac{1}{C + l} = \\frac{1}{C + l} + \\frac{C + l}{C} and \\frac{1}{C + l} + \\frac{\\operatorname{A_{1}}{(C,l)}}{C} = \\frac{1}{C + l} + \\frac{C + l}{C}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Add(Symbol('C', commutative=True), Symbol('l', commutative=True)))"], [["divide", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('A_1')(Symbol('C', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('A_1')(Symbol('C', commutative=True), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_n')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('l', commutative=True))))"], [["add", 4, "Pow(Add(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integer(-1))"], "Equality(Add(Function('E_n')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integer(-1))), Add(Pow(Add(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Add(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('A_1')(Symbol('C', commutative=True), Symbol('l', commutative=True)))), Add(Pow(Add(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f)} = \\sin{(f)}, then derive \\frac{d}{d f} \\hat{\\mathbf{r}}{(f)} = \\cos{(f)}, then obtain \\frac{\\partial}{\\partial f} (\\phi + \\hat{\\mathbf{r}}{(f)}) = \\frac{\\partial}{\\partial f} (\\hbar + \\sin{(f)})", "derivation": "\\hat{\\mathbf{r}}{(f)} = \\sin{(f)} and \\frac{d}{d f} \\hat{\\mathbf{r}}{(f)} = \\frac{d}{d f} \\sin{(f)} and \\frac{d}{d f} \\hat{\\mathbf{r}}{(f)} = \\cos{(f)} and \\int \\frac{d}{d f} \\hat{\\mathbf{r}}{(f)} df = \\int \\cos{(f)} df and \\frac{d}{d f} \\int \\frac{d}{d f} \\hat{\\mathbf{r}}{(f)} df = \\frac{d}{d f} \\int \\cos{(f)} df and \\frac{\\partial}{\\partial f} (\\phi + \\hat{\\mathbf{r}}{(f)}) = \\frac{\\partial}{\\partial f} (\\hbar + \\sin{(f)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), cos(Symbol('f', commutative=True)))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\phi', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(f_{E})} = \\sin{(f_{E})}, then derive \\int \\operatorname{n_{2}}{(f_{E})} df_{E} = k - \\cos{(f_{E})}, then derive \\mathbf{A} - \\cos{(f_{E})} = k - \\cos{(f_{E})}, then obtain (\\mathbf{A} - \\cos{(f_{E})}) (k - \\cos{(f_{E})}) = (k - \\cos{(f_{E})}) \\int \\operatorname{n_{2}}{(f_{E})} df_{E}", "derivation": "\\operatorname{n_{2}}{(f_{E})} = \\sin{(f_{E})} and \\int \\operatorname{n_{2}}{(f_{E})} df_{E} = \\int \\sin{(f_{E})} df_{E} and \\int \\operatorname{n_{2}}{(f_{E})} df_{E} = k - \\cos{(f_{E})} and \\int \\sin{(f_{E})} df_{E} = k - \\cos{(f_{E})} and \\mathbf{A} - \\cos{(f_{E})} = k - \\cos{(f_{E})} and \\mathbf{A} - \\cos{(f_{E})} = \\int \\operatorname{n_{2}}{(f_{E})} df_{E} and (\\mathbf{A} - \\cos{(f_{E})}) (k - \\cos{(f_{E})}) = (k - \\cos{(f_{E})}) \\int \\operatorname{n_{2}}{(f_{E})} df_{E}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Integral(Function('n_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["times", 6, "Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True))))), Mul(Add(Symbol('k', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Integral(Function('n_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given n{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)}, then obtain \\frac{\\iint n{(\\psi)} d\\psi d\\psi}{\\int n{(\\psi)} d\\psi} = \\frac{\\iint \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi d\\psi}{\\int n{(\\psi)} d\\psi}", "derivation": "n{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)} and \\int n{(\\psi)} d\\psi = \\int \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi and \\iint n{(\\psi)} d\\psi d\\psi = \\iint \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi d\\psi and \\frac{\\iint n{(\\psi)} d\\psi d\\psi}{\\int \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi} = \\frac{\\iint \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi d\\psi}{\\int \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi} and \\frac{\\iint n{(\\psi)} d\\psi d\\psi}{\\int n{(\\psi)} d\\psi} = \\frac{\\iint \\frac{d}{d \\psi} \\sin{(\\psi)} d\\psi d\\psi}{\\int n{(\\psi)} d\\psi}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\psi', commutative=True)), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 3, "Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Pow(Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)), Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)), Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)), Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)), Integral(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain \\frac{\\sigma_p + 2 \\log{(\\sigma_p)}}{2 (- \\mathbf{A}{(\\sigma_p)} + 2 \\log{(\\sigma_p)})} = \\frac{\\sigma_p - 2 \\mathbf{A}{(\\sigma_p)} + 4 \\log{(\\sigma_p)}}{2 (- \\mathbf{A}{(\\sigma_p)} + 2 \\log{(\\sigma_p)})}", "derivation": "\\mathbf{A}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\mathbf{A}{(\\sigma_p)} + \\log{(\\sigma_p)} = 2 \\log{(\\sigma_p)} and \\log{(\\sigma_p)} = - \\mathbf{A}{(\\sigma_p)} + 2 \\log{(\\sigma_p)} and \\sigma_p + \\mathbf{A}{(\\sigma_p)} + \\log{(\\sigma_p)} = \\sigma_p + 2 \\log{(\\sigma_p)} and \\frac{\\sigma_p + \\mathbf{A}{(\\sigma_p)} + \\log{(\\sigma_p)}}{2 \\log{(\\sigma_p)}} = \\frac{\\sigma_p + 2 \\log{(\\sigma_p)}}{2 \\log{(\\sigma_p)}} and \\frac{\\sigma_p + 2 \\log{(\\sigma_p)}}{2 (- \\mathbf{A}{(\\sigma_p)} + 2 \\log{(\\sigma_p)})} = \\frac{\\sigma_p - 2 \\mathbf{A}{(\\sigma_p)} + 4 \\log{(\\sigma_p)}}{2 (- \\mathbf{A}{(\\sigma_p)} + 2 \\log{(\\sigma_p)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(log(Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["add", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Rational(1, 2), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Integer(-1))), Mul(Rational(1, 2), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Integer(-1)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(4), log(Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{M}{(\\dot{z},\\hat{H})} = \\frac{\\dot{z}}{\\hat{H}}, then obtain - \\frac{e^{\\mathbf{M}{(\\dot{z},\\hat{H})}}}{\\mathbf{M}{(\\dot{z},\\hat{H})}} = - \\frac{e^{\\frac{\\dot{z}}{\\hat{H}}}}{\\mathbf{M}{(\\dot{z},\\hat{H})}}", "derivation": "\\mathbf{M}{(\\dot{z},\\hat{H})} = \\frac{\\dot{z}}{\\hat{H}} and e^{\\mathbf{M}{(\\dot{z},\\hat{H})}} = e^{\\frac{\\dot{z}}{\\hat{H}}} and - e^{\\mathbf{M}{(\\dot{z},\\hat{H})}} = - e^{\\frac{\\dot{z}}{\\hat{H}}} and - \\frac{e^{\\mathbf{M}{(\\dot{z},\\hat{H})}}}{\\mathbf{M}{(\\dot{z},\\hat{H})}} = - \\frac{e^{\\frac{\\dot{z}}{\\hat{H}}}}{\\mathbf{M}{(\\dot{z},\\hat{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(-1), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))))"], [["divide", 3, "Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\dot{z}{(V_{\\mathbf{E}},T)} = - T + e^{V_{\\mathbf{E}}}, then obtain 1 = ((\\int (T + \\dot{z}{(V_{\\mathbf{E}},T)}) dV_{\\mathbf{E}})^{- V_{\\mathbf{E}}}) (\\int e^{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}}", "derivation": "\\dot{z}{(V_{\\mathbf{E}},T)} = - T + e^{V_{\\mathbf{E}}} and T + \\dot{z}{(V_{\\mathbf{E}},T)} = e^{V_{\\mathbf{E}}} and \\int (T + \\dot{z}{(V_{\\mathbf{E}},T)}) dV_{\\mathbf{E}} = \\int e^{V_{\\mathbf{E}}} dV_{\\mathbf{E}} and (\\int (T + \\dot{z}{(V_{\\mathbf{E}},T)}) dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} = (\\int e^{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} and 1 = ((\\int (T + \\dot{z}{(V_{\\mathbf{E}},T)}) dV_{\\mathbf{E}})^{- V_{\\mathbf{E}}}) (\\int e^{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["power", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('T', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Integral(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["divide", 4, "Pow(Integral(Add(Symbol('T', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integral(Add(Symbol('T', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Integral(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then derive \\int \\dot{z}{(\\mathbf{S})} d\\mathbf{S} = \\hat{\\mathbf{r}} + \\sin{(\\mathbf{S})}, then obtain \\int (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{S})}) d\\hat{\\mathbf{r}} = \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\hat{\\mathbf{r}}", "derivation": "\\dot{z}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\int \\dot{z}{(\\mathbf{S})} d\\mathbf{S} = \\int \\cos{(\\mathbf{S})} d\\mathbf{S} and \\int \\dot{z}{(\\mathbf{S})} d\\mathbf{S} = \\hat{\\mathbf{r}} + \\sin{(\\mathbf{S})} and \\hat{\\mathbf{r}} + \\sin{(\\mathbf{S})} = \\int \\cos{(\\mathbf{S})} d\\mathbf{S} and \\int (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{S})}) d\\hat{\\mathbf{r}} = \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{z}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\eta,F_{g})} = e^{F_{g} + \\eta}, then derive \\frac{\\partial}{\\partial F_{g}} \\nabla{(\\eta,F_{g})} = e^{F_{g} + \\eta}, then obtain (\\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\nabla{(\\eta,F_{g})})^{\\eta} = (e^{F_{g} + \\eta})^{\\eta}", "derivation": "\\nabla{(\\eta,F_{g})} = e^{F_{g} + \\eta} and \\frac{\\partial}{\\partial F_{g}} \\nabla{(\\eta,F_{g})} = \\frac{\\partial}{\\partial F_{g}} e^{F_{g} + \\eta} and \\frac{\\partial}{\\partial F_{g}} \\nabla{(\\eta,F_{g})} = e^{F_{g} + \\eta} and \\frac{\\partial}{\\partial F_{g}} e^{F_{g} + \\eta} = e^{F_{g} + \\eta} and \\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\nabla{(\\eta,F_{g})} = \\frac{\\partial}{\\partial F_{g}} \\nabla{(\\eta,F_{g})} and \\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\nabla{(\\eta,F_{g})} = e^{F_{g} + \\eta} and (\\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\nabla{(\\eta,F_{g})})^{\\eta} = (e^{F_{g} + \\eta})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))), Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["power", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\nabla')(Symbol('\\\\eta', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))), Symbol('\\\\eta', commutative=True)), Pow(exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\varphi^*)} = \\int \\cos{(\\varphi^*)} d\\varphi^* and \\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} = \\Psi_{nl}, then derive \\operatorname{F_{N}}{(\\varphi^*)} = \\Psi_{nl} + \\sin{(\\varphi^*)}, then obtain (C_{d} + \\sin{(\\varphi^*)})^{\\varphi^*} = (\\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} + \\sin{(\\varphi^*)})^{\\varphi^*}", "derivation": "\\operatorname{F_{N}}{(\\varphi^*)} = \\int \\cos{(\\varphi^*)} d\\varphi^* and \\operatorname{F_{N}}{(\\varphi^*)} = \\Psi_{nl} + \\sin{(\\varphi^*)} and \\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} = \\Psi_{nl} and \\operatorname{F_{N}}{(\\varphi^*)} = \\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} + \\sin{(\\varphi^*)} and \\operatorname{F_{N}}^{\\varphi^*}{(\\varphi^*)} = (\\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} + \\sin{(\\varphi^*)})^{\\varphi^*} and (\\int \\cos{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (\\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} + \\sin{(\\varphi^*)})^{\\varphi^*} and (C_{d} + \\sin{(\\varphi^*)})^{\\varphi^*} = (\\operatorname{J_{\\varepsilon}}{(\\Psi_{nl})} + \\sin{(\\varphi^*)})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\varphi^*', commutative=True)), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('F_N')(Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_N')(Symbol('\\\\varphi^*', commutative=True)), Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('C_d', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given W{(p)} = \\cos{(p)}, then obtain - p \\cos{(p)} + W{(p)} = - p \\cos{(p)} + \\cos{(p)}", "derivation": "W{(p)} = \\cos{(p)} and p W{(p)} = p \\cos{(p)} and - p W{(p)} + W{(p)} = - p W{(p)} + \\cos{(p)} and - p \\cos{(p)} + W{(p)} = - p \\cos{(p)} + \\cos{(p)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('W')(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), cos(Symbol('p', commutative=True))))"], [["minus", 1, "Mul(Symbol('p', commutative=True), Function('W')(Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), Function('W')(Symbol('p', commutative=True))), Function('W')(Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True), Function('W')(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), cos(Symbol('p', commutative=True))), Function('W')(Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True), cos(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\Psi^{\\dagger}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain (\\frac{1}{2})^{\\mathbf{A}} = (\\frac{\\sin{(\\mathbf{A})}}{2 \\Psi^{\\dagger}{(\\mathbf{A})}})^{\\mathbf{A}}", "derivation": "\\rho{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\rho^{2}{(\\mathbf{A})} = \\rho{(\\mathbf{A})} \\sin{(\\mathbf{A})} and \\frac{1}{2} = \\frac{\\sin{(\\mathbf{A})}}{2 \\rho{(\\mathbf{A})}} and \\Psi^{\\dagger}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\Psi^{\\dagger}{(\\mathbf{A})} = \\rho{(\\mathbf{A})} and \\frac{1}{2} = \\frac{\\sin{(\\mathbf{A})}}{2 \\Psi^{\\dagger}{(\\mathbf{A})}} and (\\frac{1}{2})^{\\mathbf{A}} = (\\frac{\\sin{(\\mathbf{A})}}{2 \\Psi^{\\dagger}{(\\mathbf{A})}})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 6, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Rational(1, 2), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\ddot{x},\\mathbf{S})} = \\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}), then derive \\hat{\\mathbf{r}}^{\\ddot{x}}{(\\ddot{x},\\mathbf{S})} = 1, then obtain \\sin{(((\\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}))^{\\ddot{x}})^{\\mathbf{S}})} = \\sin{(1)}", "derivation": "\\hat{\\mathbf{r}}{(\\ddot{x},\\mathbf{S})} = \\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}) and \\hat{\\mathbf{r}}^{\\ddot{x}}{(\\ddot{x},\\mathbf{S})} = (\\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}))^{\\ddot{x}} and \\hat{\\mathbf{r}}^{\\ddot{x}}{(\\ddot{x},\\mathbf{S})} = 1 and (\\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}))^{\\ddot{x}} = 1 and ((\\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}))^{\\ddot{x}})^{\\mathbf{S}} = 1 and \\sin{(((\\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\mathbf{S}))^{\\ddot{x}})^{\\mathbf{S}})} = \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Symbol('\\\\ddot{x}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Symbol('\\\\ddot{x}', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1))"], [["sin", 5], "Equality(sin(Pow(Pow(Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), sin(Integer(1)))"]]}, {"prompt": "Given \\dot{y}{(U)} = \\log{(U)}, then obtain \\frac{d}{d U} 1 - 1 = \\frac{d}{d U} \\frac{\\int \\log{(U)} dU}{\\int \\dot{y}{(U)} dU} - 1", "derivation": "\\dot{y}{(U)} = \\log{(U)} and \\int \\dot{y}{(U)} dU = \\int \\log{(U)} dU and e^{- \\mathbf{D}} \\int \\dot{y}{(U)} dU = e^{- \\mathbf{D}} \\int \\log{(U)} dU and 1 = \\frac{\\int \\log{(U)} dU}{\\int \\dot{y}{(U)} dU} and \\frac{d}{d U} 1 = \\frac{d}{d U} \\frac{\\int \\log{(U)} dU}{\\int \\dot{y}{(U)} dU} and \\frac{d}{d U} 1 - 1 = \\frac{d}{d U} \\frac{\\int \\log{(U)} dU}{\\int \\dot{y}{(U)} dU} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["divide", 2, "exp(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["divide", 3, "Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["differentiate", 4, "Symbol('U', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Pow(Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 5, 1], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Pow(Integral(Function('\\\\dot{y}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(v_{y},s)} = s^{v_{y}}, then obtain - \\frac{\\partial}{\\partial v_{y}} (- v_{y} + \\operatorname{v_{z}}{(v_{y},s)}) = - \\frac{\\partial}{\\partial v_{y}} (s^{v_{y}} - v_{y})", "derivation": "\\operatorname{v_{z}}{(v_{y},s)} = s^{v_{y}} and - v_{y} + \\operatorname{v_{z}}{(v_{y},s)} = s^{v_{y}} - v_{y} and \\frac{\\partial}{\\partial v_{y}} (- v_{y} + \\operatorname{v_{z}}{(v_{y},s)}) = \\frac{\\partial}{\\partial v_{y}} (s^{v_{y}} - v_{y}) and - \\frac{\\partial}{\\partial v_{y}} (- v_{y} + \\operatorname{v_{z}}{(v_{y},s)}) = - \\frac{\\partial}{\\partial v_{y}} (s^{v_{y}} - v_{y})", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('v_y', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_z')(Symbol('v_y', commutative=True), Symbol('s', commutative=True))), Add(Pow(Symbol('s', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_z')(Symbol('v_y', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('s', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_z')(Symbol('v_y', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Pow(Symbol('s', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(M)} = \\log{(M)}, then obtain \\frac{\\int \\mathbf{M}{(M)} \\cos^{M}{(\\mathbf{M}{(M)})} dM}{\\hat{p}_0 \\sin{(v_{2})}} = \\frac{\\int \\mathbf{M}{(M)} \\cos^{M}{(\\log{(M)})} dM}{\\hat{p}_0 \\sin{(v_{2})}}", "derivation": "\\mathbf{M}{(M)} = \\log{(M)} and \\cos{(\\mathbf{M}{(M)})} = \\cos{(\\log{(M)})} and \\cos^{M}{(\\mathbf{M}{(M)})} = \\cos^{M}{(\\log{(M)})} and \\mathbf{M}{(M)} \\cos^{M}{(\\mathbf{M}{(M)})} = \\mathbf{M}{(M)} \\cos^{M}{(\\log{(M)})} and \\int \\mathbf{M}{(M)} \\cos^{M}{(\\mathbf{M}{(M)})} dM = \\int \\mathbf{M}{(M)} \\cos^{M}{(\\log{(M)})} dM and \\frac{\\int \\mathbf{M}{(M)} \\cos^{M}{(\\mathbf{M}{(M)})} dM}{\\hat{p}_0 \\sin{(v_{2})}} = \\frac{\\int \\mathbf{M}{(M)} \\cos^{M}{(\\log{(M)})} dM}{\\hat{p}_0 \\sin{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{M}')(Symbol('M', commutative=True))), cos(log(Symbol('M', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(cos(Function('\\\\mathbf{M}')(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(cos(log(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["times", 3, "Function('\\\\mathbf{M}')(Symbol('M', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(Function('\\\\mathbf{M}')(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(log(Symbol('M', commutative=True))), Symbol('M', commutative=True))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(Function('\\\\mathbf{M}')(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(log(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["divide", 5, "Mul(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(Function('\\\\mathbf{M}')(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('M', commutative=True)), Pow(cos(log(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then obtain (\\frac{d}{d \\varepsilon_0} \\frac{\\operatorname{v_{t}}{(\\varepsilon_0)}}{\\varepsilon_0})^{\\varepsilon_0} = (\\frac{d}{d \\varepsilon_0} \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0})^{\\varepsilon_0}", "derivation": "\\operatorname{v_{t}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\frac{\\operatorname{v_{t}}{(\\varepsilon_0)}}{\\varepsilon_0} = \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} \\frac{\\operatorname{v_{t}}{(\\varepsilon_0)}}{\\varepsilon_0} = \\frac{d}{d \\varepsilon_0} \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0} and (\\frac{d}{d \\varepsilon_0} \\frac{\\operatorname{v_{t}}{(\\varepsilon_0)}}{\\varepsilon_0})^{\\varepsilon_0} = (\\frac{d}{d \\varepsilon_0} \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_t')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_t')(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_t')(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(c,n)} = \\frac{c}{n}, then obtain (\\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)})^{- c} ((\\frac{c}{n})^{c} + \\Psi_{nl}{(c,n)})^{c} = (\\frac{c}{n} + (\\frac{c}{n})^{c})^{c} (\\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)})^{- c}", "derivation": "\\Psi_{nl}{(c,n)} = \\frac{c}{n} and \\Psi_{nl}^{c}{(c,n)} = (\\frac{c}{n})^{c} and \\Psi_{nl}{(c,n)} + \\Psi_{nl}^{c}{(c,n)} = \\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)} and (\\Psi_{nl}{(c,n)} + \\Psi_{nl}^{c}{(c,n)})^{c} = (\\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)})^{c} and ((\\frac{c}{n})^{c} + \\Psi_{nl}{(c,n)})^{c} = (\\frac{c}{n} + (\\frac{c}{n})^{c})^{c} and (\\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)})^{- c} ((\\frac{c}{n})^{c} + \\Psi_{nl}{(c,n)})^{c} = (\\frac{c}{n} + (\\frac{c}{n})^{c})^{c} (\\frac{c}{n} + \\Psi_{nl}^{c}{(c,n)})^{- c}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True)), Pow(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('c', commutative=True)))"], [["add", 1, "Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Pow(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('c', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True))), Symbol('c', commutative=True)), Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["divide", 5, "Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Add(Pow(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('c', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True))), Symbol('c', commutative=True))), Mul(Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(C,Q)} = Q^{C} and \\varepsilon_{0}{(C,Q)} = (\\psi^{*}^{C}{(C,Q)})^{Q}, then obtain \\varepsilon_{0}{(C,Q)} + \\frac{1}{\\varepsilon_{0}{(C,Q)}} = ((Q^{C})^{C})^{Q} + \\frac{1}{\\varepsilon_{0}{(C,Q)}}", "derivation": "\\psi^{*}{(C,Q)} = Q^{C} and \\psi^{*}^{C}{(C,Q)} = (Q^{C})^{C} and (\\psi^{*}^{C}{(C,Q)})^{Q} = ((Q^{C})^{C})^{Q} and \\varepsilon_{0}{(C,Q)} = (\\psi^{*}^{C}{(C,Q)})^{Q} and \\varepsilon_{0}{(C,Q)} = ((Q^{C})^{C})^{Q} and \\varepsilon_{0}{(C,Q)} + \\frac{1}{\\varepsilon_{0}{(C,Q)}} = ((Q^{C})^{C})^{Q} + \\frac{1}{\\varepsilon_{0}{(C,Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('C', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Symbol('C', commutative=True)), Pow(Pow(Symbol('Q', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Symbol('C', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Pow(Symbol('Q', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Pow(Pow(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Symbol('C', commutative=True)), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Pow(Pow(Pow(Symbol('Q', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('Q', commutative=True)))"], [["add", 5, "Pow(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Add(Pow(Pow(Pow(Symbol('Q', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('Q', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(n)} = \\sin{(\\log{(n)})} and i{(n)} = \\sin{(\\log{(n)})}, then obtain \\frac{d}{d n} (i{(n)} + \\operatorname{v_{t}}{(n)}) = \\frac{d}{d n} (\\operatorname{v_{t}}{(n)} + \\sin{(\\log{(n)})})", "derivation": "\\operatorname{v_{t}}{(n)} = \\sin{(\\log{(n)})} and 2 \\operatorname{v_{t}}{(n)} = \\operatorname{v_{t}}{(n)} + \\sin{(\\log{(n)})} and i{(n)} = \\sin{(\\log{(n)})} and 2 \\operatorname{v_{t}}{(n)} = i{(n)} + \\operatorname{v_{t}}{(n)} and i{(n)} + \\operatorname{v_{t}}{(n)} = \\operatorname{v_{t}}{(n)} + \\sin{(\\log{(n)})} and \\frac{d}{d n} (i{(n)} + \\operatorname{v_{t}}{(n)}) = \\frac{d}{d n} (\\operatorname{v_{t}}{(n)} + \\sin{(\\log{(n)})})", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('n', commutative=True)), sin(log(Symbol('n', commutative=True))))"], [["add", 1, "Function('v_t')(Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('v_t')(Symbol('n', commutative=True))), Add(Function('v_t')(Symbol('n', commutative=True)), sin(log(Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('i')(Symbol('n', commutative=True)), sin(log(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('v_t')(Symbol('n', commutative=True))), Add(Function('i')(Symbol('n', commutative=True)), Function('v_t')(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('i')(Symbol('n', commutative=True)), Function('v_t')(Symbol('n', commutative=True))), Add(Function('v_t')(Symbol('n', commutative=True)), sin(log(Symbol('n', commutative=True)))))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('i')(Symbol('n', commutative=True)), Function('v_t')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Function('v_t')(Symbol('n', commutative=True)), sin(log(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(p)} = e^{p}, then derive \\int \\operatorname{z^{*}}{(p)} dp = v + e^{p}, then obtain \\int (\\int \\operatorname{z^{*}}{(p)} dp)^{p} dv = \\int (v + \\operatorname{z^{*}}{(p)})^{p} dv", "derivation": "\\operatorname{z^{*}}{(p)} = e^{p} and \\int \\operatorname{z^{*}}{(p)} dp = \\int e^{p} dp and \\int \\operatorname{z^{*}}{(p)} dp = v + e^{p} and (\\int \\operatorname{z^{*}}{(p)} dp)^{p} = (v + e^{p})^{p} and \\int (\\int \\operatorname{z^{*}}{(p)} dp)^{p} dv = \\int (v + e^{p})^{p} dv and \\int (\\int \\operatorname{z^{*}}{(p)} dp)^{p} dv = \\int (v + \\operatorname{z^{*}}{(p)})^{p} dv", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('v', commutative=True), exp(Symbol('p', commutative=True))))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Integral(Function('z^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Symbol('v', commutative=True), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Integral(Function('z^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Symbol('v', commutative=True), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Pow(Integral(Function('z^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Symbol('v', commutative=True), Function('z^*')(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given Q{(l)} = \\log{(l)}, then obtain \\frac{\\frac{d}{d l} (Q{(l)} - \\log{(Q{(l)})})}{\\hat{H}_l} = \\frac{\\frac{d}{d l} (\\log{(l)} - \\log{(Q{(l)})})}{\\hat{H}_l}", "derivation": "Q{(l)} = \\log{(l)} and \\log{(Q{(l)})} = \\log{(\\log{(l)})} and Q{(l)} - \\log{(Q{(l)})} = \\log{(l)} - \\log{(Q{(l)})} and Q{(l)} - \\log{(\\log{(l)})} = \\log{(l)} - \\log{(\\log{(l)})} and \\frac{d}{d l} (Q{(l)} - \\log{(\\log{(l)})}) = \\frac{d}{d l} (\\log{(l)} - \\log{(\\log{(l)})}) and \\frac{d}{d l} (Q{(l)} - \\log{(Q{(l)})}) = \\frac{d}{d l} (\\log{(l)} - \\log{(Q{(l)})}) and \\frac{\\frac{d}{d l} (Q{(l)} - \\log{(Q{(l)})})}{\\hat{H}_l} = \\frac{\\frac{d}{d l} (\\log{(l)} - \\log{(Q{(l)})})}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["log", 1], "Equality(log(Function('Q')(Symbol('l', commutative=True))), log(log(Symbol('l', commutative=True))))"], [["minus", 1, "log(Function('Q')(Symbol('l', commutative=True)))"], "Equality(Add(Function('Q')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))), Add(log(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('Q')(Symbol('l', commutative=True)), Mul(Integer(-1), log(log(Symbol('l', commutative=True))))), Add(log(Symbol('l', commutative=True)), Mul(Integer(-1), log(log(Symbol('l', commutative=True))))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Function('Q')(Symbol('l', commutative=True)), Mul(Integer(-1), log(log(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(log(Symbol('l', commutative=True)), Mul(Integer(-1), log(log(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Add(Function('Q')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(log(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 6, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Add(Function('Q')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Add(log(Symbol('l', commutative=True)), Mul(Integer(-1), log(Function('Q')(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})}, then derive \\frac{d}{d L_{\\varepsilon}} W{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then obtain \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}", "derivation": "W{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} W{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} W{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given g{(H)} = \\sin{(H)}, then obtain \\frac{H (- \\frac{d}{d H} \\frac{g{(H)}}{H} + \\frac{g{(H)}}{H})}{\\sin{(H)}} = \\frac{H (- \\frac{d}{d H} \\frac{\\sin{(H)}}{H} + \\frac{g{(H)}}{H})}{\\sin{(H)}}", "derivation": "g{(H)} = \\sin{(H)} and \\frac{g{(H)}}{H} = \\frac{\\sin{(H)}}{H} and \\frac{d}{d H} \\frac{g{(H)}}{H} = \\frac{d}{d H} \\frac{\\sin{(H)}}{H} and - \\frac{d}{d H} \\frac{g{(H)}}{H} = - \\frac{d}{d H} \\frac{\\sin{(H)}}{H} and - \\frac{d}{d H} \\frac{g{(H)}}{H} + \\frac{g{(H)}}{H} = - \\frac{d}{d H} \\frac{\\sin{(H)}}{H} + \\frac{g{(H)}}{H} and \\frac{H (- \\frac{d}{d H} \\frac{g{(H)}}{H} + \\frac{g{(H)}}{H})}{\\sin{(H)}} = \\frac{H (- \\frac{d}{d H} \\frac{\\sin{(H)}}{H} + \\frac{g{(H)}}{H})}{\\sin{(H)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["add", 4, "Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True)))))"], [["divide", 5, "Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True)))"], "Equality(Mul(Symbol('H', commutative=True), Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True)))), Pow(sin(Symbol('H', commutative=True)), Integer(-1))), Mul(Symbol('H', commutative=True), Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('g')(Symbol('H', commutative=True)))), Pow(sin(Symbol('H', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(E,\\theta_2)} = \\theta_2^{E}, then obtain \\frac{\\mathbf{r}{(E,\\theta_2)} - \\frac{\\mathbf{r}{(E,\\theta_2)}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}} = \\frac{\\theta_2^{E} - \\frac{\\mathbf{r}{(E,\\theta_2)}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}}", "derivation": "\\mathbf{r}{(E,\\theta_2)} = \\theta_2^{E} and \\frac{\\mathbf{r}{(E,\\theta_2)}}{\\theta_2} = \\frac{\\theta_2^{E}}{\\theta_2} and \\mathbf{r}{(E,\\theta_2)} - \\frac{\\theta_2^{E}}{\\theta_2} = \\theta_2^{E} - \\frac{\\theta_2^{E}}{\\theta_2} and \\frac{\\mathbf{r}{(E,\\theta_2)} - \\frac{\\theta_2^{E}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}} = \\frac{\\theta_2^{E} - \\frac{\\theta_2^{E}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}} and \\frac{\\mathbf{r}{(E,\\theta_2)} - \\frac{\\mathbf{r}{(E,\\theta_2)}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}} = \\frac{\\theta_2^{E} - \\frac{\\mathbf{r}{(E,\\theta_2)}}{\\theta_2}}{\\mathbf{r}{(E,\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))), Add(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))))"], [["divide", 3, "Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))), Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Mul(Add(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)))), Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Mul(Add(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} = \\log{(\\sin{(E_{x})})}, then derive \\frac{d}{d E_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} - \\frac{\\cos{(E_{x})}}{\\sin{(E_{x})}} = 0, then obtain (\\frac{d}{d E_{x}} \\log{(\\sin{(E_{x})})} - \\frac{\\cos{(E_{x})}}{\\sin{(E_{x})}}) \\log{(\\sin{(E_{x})})} = 0", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} = \\log{(\\sin{(E_{x})})} and \\frac{d}{d E_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} = \\frac{d}{d E_{x}} \\log{(\\sin{(E_{x})})} and \\frac{d}{d E_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} - \\frac{d}{d E_{x}} \\log{(\\sin{(E_{x})})} = 0 and \\frac{d}{d E_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(E_{x})} - \\frac{\\cos{(E_{x})}}{\\sin{(E_{x})}} = 0 and \\frac{d}{d E_{x}} \\log{(\\sin{(E_{x})})} - \\frac{\\cos{(E_{x})}}{\\sin{(E_{x})}} = 0 and (\\frac{d}{d E_{x}} \\log{(\\sin{(E_{x})})} - \\frac{\\cos{(E_{x})}}{\\sin{(E_{x})}}) \\log{(\\sin{(E_{x})})} = 0", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), log(sin(Symbol('E_x', commutative=True))))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(log(sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('E_x', commutative=True)), Integer(-1)), cos(Symbol('E_x', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(log(sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('E_x', commutative=True)), Integer(-1)), cos(Symbol('E_x', commutative=True)))), Integer(0))"], [["times", 5, "log(sin(Symbol('E_x', commutative=True)))"], "Equality(Mul(Add(Derivative(log(sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('E_x', commutative=True)), Integer(-1)), cos(Symbol('E_x', commutative=True)))), log(sin(Symbol('E_x', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\phi{(n)} = \\sin{(n)}, then obtain \\frac{\\frac{d}{d n} 0}{- \\phi{(n)} + \\sin{(n)}} = \\frac{\\frac{d}{d n} (- \\phi{(n)} + \\sin{(n)})}{- \\phi{(n)} + \\sin{(n)}}", "derivation": "\\phi{(n)} = \\sin{(n)} and 0 = - \\phi{(n)} + \\sin{(n)} and \\frac{d}{d n} 0 = \\frac{d}{d n} (- \\phi{(n)} + \\sin{(n)}) and \\frac{\\frac{d}{d n} 0}{- \\phi{(n)} + \\sin{(n)}} = \\frac{\\frac{d}{d n} (- \\phi{(n)} + \\sin{(n)})}{- \\phi{(n)} + \\sin{(n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["minus", 1, "Function('\\\\phi')(Symbol('n', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(z^{*})} = z^{*}, then derive \\int H{(z^{*})} dz^{*} = F_{x} + \\frac{(z^{*})^{2}}{2}, then derive F_{x} + \\frac{(z^{*})^{2}}{2} = v + \\frac{(z^{*})^{2}}{2}, then obtain (\\frac{\\partial}{\\partial v} (F_{x} + \\frac{(z^{*})^{2}}{2}))^{2} = (\\frac{\\partial}{\\partial v} (v + \\frac{(z^{*})^{2}}{2}))^{2}", "derivation": "H{(z^{*})} = z^{*} and \\int H{(z^{*})} dz^{*} = \\int z^{*} dz^{*} and \\int H{(z^{*})} dz^{*} = F_{x} + \\frac{(z^{*})^{2}}{2} and F_{x} + \\frac{(z^{*})^{2}}{2} = \\int z^{*} dz^{*} and F_{x} + \\frac{(z^{*})^{2}}{2} = v + \\frac{(z^{*})^{2}}{2} and \\frac{\\partial}{\\partial v} (F_{x} + \\frac{(z^{*})^{2}}{2}) = \\frac{\\partial}{\\partial v} (v + \\frac{(z^{*})^{2}}{2}) and (\\frac{\\partial}{\\partial v} (F_{x} + \\frac{(z^{*})^{2}}{2}))^{2} = (\\frac{\\partial}{\\partial v} (v + \\frac{(z^{*})^{2}}{2}))^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('H')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('H')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Symbol('z^*', commutative=True), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('H')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Integral(Symbol('z^*', commutative=True), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Add(Symbol('v', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"], [["differentiate", 5, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["power", 6, 2], "Equality(Pow(Derivative(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Symbol('v', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\dot{x}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}}, then obtain \\frac{d}{d \\Psi_{\\lambda}} \\int \\dot{x}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\frac{d}{d \\Psi_{\\lambda}} \\int 1 d\\Psi_{\\lambda}", "derivation": "\\dot{x}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\dot{x}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} = 1 and \\int \\dot{x}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\int 1 d\\Psi_{\\lambda} and \\frac{d}{d \\Psi_{\\lambda}} \\int \\dot{x}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\frac{d}{d \\Psi_{\\lambda}} \\int 1 d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(A_{2},\\mathbf{g})} = \\mathbf{g}^{A_{2}} and \\tilde{g}^*{(A_{2},\\mathbf{g})} = \\operatorname{v_{x}}^{\\mathbf{g}}{(A_{2},\\mathbf{g})}, then obtain \\tilde{g}^*^{\\mathbf{g}}{(A_{2},\\mathbf{g})} = ((\\mathbf{g}^{A_{2}})^{\\mathbf{g}})^{\\mathbf{g}}", "derivation": "\\operatorname{v_{x}}{(A_{2},\\mathbf{g})} = \\mathbf{g}^{A_{2}} and \\operatorname{v_{x}}^{\\mathbf{g}}{(A_{2},\\mathbf{g})} = (\\mathbf{g}^{A_{2}})^{\\mathbf{g}} and \\tilde{g}^*{(A_{2},\\mathbf{g})} = \\operatorname{v_{x}}^{\\mathbf{g}}{(A_{2},\\mathbf{g})} and \\tilde{g}^*^{\\mathbf{g}}{(A_{2},\\mathbf{g})} = (\\operatorname{v_{x}}^{\\mathbf{g}}{(A_{2},\\mathbf{g})})^{\\mathbf{g}} and \\tilde{g}^*^{\\mathbf{g}}{(A_{2},\\mathbf{g})} = ((\\mathbf{g}^{A_{2}})^{\\mathbf{g}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('A_2', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('v_x')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Function('v_x')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(y,A_{x})} = e^{A_{x}^{y}} and L{(y,A_{x})} = - \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})}, then derive \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} = \\frac{A_{x}^{y} y e^{A_{x}^{y}}}{A_{x}}, then obtain L{(y,A_{x})} = - \\frac{A_{x}^{y} y \\theta_{1}{(y,A_{x})}}{A_{x}}", "derivation": "\\theta_{1}{(y,A_{x})} = e^{A_{x}^{y}} and \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} = \\frac{\\partial}{\\partial A_{x}} e^{A_{x}^{y}} and \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} = \\frac{A_{x}^{y} y e^{A_{x}^{y}}}{A_{x}} and \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} = \\frac{A_{x}^{y} y \\theta_{1}{(y,A_{x})}}{A_{x}} and - \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} = - \\frac{A_{x}^{y} y \\theta_{1}{(y,A_{x})}}{A_{x}} and L{(y,A_{x})} = - \\frac{\\partial}{\\partial A_{x}} \\theta_{1}{(y,A_{x})} and L{(y,A_{x})} = - \\frac{A_{x}^{y} y \\theta_{1}{(y,A_{x})}}{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), exp(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), exp(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('L')(Symbol('y', commutative=True), Symbol('A_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\theta_1')(Symbol('y', commutative=True), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given L{(\\varepsilon,\\dot{y})} = \\dot{y} \\varepsilon, then obtain ((\\frac{\\int L{(\\varepsilon,\\dot{y})} d\\varepsilon}{\\dot{y}})^{\\dot{y}})^{\\dot{y}} = ((\\frac{\\int \\dot{y} \\varepsilon d\\varepsilon}{\\dot{y}})^{\\dot{y}})^{\\dot{y}}", "derivation": "L{(\\varepsilon,\\dot{y})} = \\dot{y} \\varepsilon and \\int L{(\\varepsilon,\\dot{y})} d\\varepsilon = \\int \\dot{y} \\varepsilon d\\varepsilon and \\frac{\\int L{(\\varepsilon,\\dot{y})} d\\varepsilon}{\\dot{y}} = \\frac{\\int \\dot{y} \\varepsilon d\\varepsilon}{\\dot{y}} and (\\frac{\\int L{(\\varepsilon,\\dot{y})} d\\varepsilon}{\\dot{y}})^{\\dot{y}} = (\\frac{\\int \\dot{y} \\varepsilon d\\varepsilon}{\\dot{y}})^{\\dot{y}} and ((\\frac{\\int L{(\\varepsilon,\\dot{y})} d\\varepsilon}{\\dot{y}})^{\\dot{y}})^{\\dot{y}} = ((\\frac{\\int \\dot{y} \\varepsilon d\\varepsilon}{\\dot{y}})^{\\dot{y}})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Function('L')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Function('L')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Function('L')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given v{(c_{0},M_{E})} = \\frac{M_{E}}{c_{0}}, then derive \\frac{c_{0} v^{c_{0}}{(c_{0},M_{E})} \\frac{\\partial}{\\partial M_{E}} v{(c_{0},M_{E})}}{v{(c_{0},M_{E})}} = \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}}}{M_{E}}, then obtain \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}} \\frac{\\partial}{\\partial M_{E}} v{(c_{0},M_{E})}}{v{(c_{0},M_{E})}} = \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}}}{M_{E}}", "derivation": "v{(c_{0},M_{E})} = \\frac{M_{E}}{c_{0}} and v^{c_{0}}{(c_{0},M_{E})} = (\\frac{M_{E}}{c_{0}})^{c_{0}} and \\frac{\\partial}{\\partial M_{E}} v^{c_{0}}{(c_{0},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (\\frac{M_{E}}{c_{0}})^{c_{0}} and \\frac{c_{0} v^{c_{0}}{(c_{0},M_{E})} \\frac{\\partial}{\\partial M_{E}} v{(c_{0},M_{E})}}{v{(c_{0},M_{E})}} = \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}}}{M_{E}} and \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}} \\frac{\\partial}{\\partial M_{E}} v{(c_{0},M_{E})}}{v{(c_{0},M_{E})}} = \\frac{c_{0} (\\frac{M_{E}}{c_{0}})^{c_{0}}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Symbol('c_0', commutative=True)), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('c_0', commutative=True)))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Pow(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('c_0', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('c_0', commutative=True), Pow(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Symbol('c_0', commutative=True)), Derivative(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('c_0', commutative=True), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('c_0', commutative=True), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('c_0', commutative=True)), Pow(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('c_0', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('c_0', commutative=True), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(i)} = \\frac{d}{d i} \\log{(i)}, then derive \\mu_{0}{(i)} = \\frac{1}{i}, then derive \\frac{d}{d i} \\int \\mu_{0}{(i)} di = \\frac{\\partial}{\\partial i} (\\mathbf{r} + \\log{(i)}), then obtain \\frac{d}{d i} \\int \\frac{1}{i} di = \\frac{1}{i}", "derivation": "\\mu_{0}{(i)} = \\frac{d}{d i} \\log{(i)} and \\mu_{0}{(i)} = \\frac{1}{i} and \\int \\mu_{0}{(i)} di = \\int \\frac{1}{i} di and \\frac{d}{d i} \\int \\mu_{0}{(i)} di = \\frac{d}{d i} \\int \\frac{1}{i} di and \\frac{d}{d i} \\int \\mu_{0}{(i)} di = \\frac{\\partial}{\\partial i} (\\mathbf{r} + \\log{(i)}) and \\frac{d}{d i} \\int \\frac{1}{i} di = \\frac{\\partial}{\\partial i} (\\mathbf{r} + \\log{(i)}) and \\frac{d}{d i} \\int \\frac{1}{i} di = \\frac{1}{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('i', commutative=True)), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu_0')(Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mu_0')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('\\\\mu_0')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Pow(Symbol('i', commutative=True), Integer(-1)))"]]}, {"prompt": "Given t{(\\eta)} = \\log{(\\eta)} and H{(\\eta)} = \\frac{d}{d \\eta} \\log{(\\eta)}, then derive \\frac{d}{d \\eta} t{(\\eta)} = \\frac{1}{\\eta}, then obtain \\psi + H{(\\eta)} + \\log{(\\eta)} = \\psi + \\log{(\\eta)} + \\frac{1}{\\eta}", "derivation": "t{(\\eta)} = \\log{(\\eta)} and \\frac{d}{d \\eta} t{(\\eta)} = \\frac{d}{d \\eta} \\log{(\\eta)} and H{(\\eta)} = \\frac{d}{d \\eta} \\log{(\\eta)} and \\frac{d}{d \\eta} t{(\\eta)} = \\frac{1}{\\eta} and H{(\\eta)} = \\frac{d}{d \\eta} t{(\\eta)} and H{(\\eta)} = \\frac{1}{\\eta} and H{(\\eta)} + \\int \\frac{1}{\\eta} d\\eta = \\int \\frac{1}{\\eta} d\\eta + \\frac{1}{\\eta} and \\psi + H{(\\eta)} + \\log{(\\eta)} = \\psi + \\log{(\\eta)} + \\frac{1}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\eta', commutative=True)), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('H')(Symbol('\\\\eta', commutative=True)), Derivative(Function('t')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('H')(Symbol('\\\\eta', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))"], [["add", 6, "Integral(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Function('H')(Symbol('\\\\eta', commutative=True)), Integral(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Integral(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\psi', commutative=True), Function('H')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\psi', commutative=True), log(Symbol('\\\\eta', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(A_{x})} = \\frac{d}{d A_{x}} \\sin{(A_{x})}, then derive \\tilde{g}^*{(A_{x})} = \\cos{(A_{x})}, then obtain \\tilde{g}^*^{2}{(A_{x})} = \\tilde{g}^*{(A_{x})} \\frac{d}{d A_{x}} \\sin{(A_{x})}", "derivation": "\\tilde{g}^*{(A_{x})} = \\frac{d}{d A_{x}} \\sin{(A_{x})} and \\tilde{g}^*{(A_{x})} = \\cos{(A_{x})} and \\tilde{g}^*{(A_{x})} \\cos{(A_{x})} = \\cos{(A_{x})} \\frac{d}{d A_{x}} \\sin{(A_{x})} and \\tilde{g}^*^{2}{(A_{x})} = \\tilde{g}^*{(A_{x})} \\frac{d}{d A_{x}} \\sin{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True)), Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["times", 1, "cos(Symbol('A_x', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))), Mul(cos(Symbol('A_x', commutative=True)), Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True)), Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and u{(Z)} = \\cos{(Z)}, then obtain \\frac{u^{2}{(Z)} + y^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})}}{y{(f_{\\mathbf{p}})}} = \\frac{u{(Z)} \\cos{(Z)} + y^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})}}{y{(f_{\\mathbf{p}})}}", "derivation": "y{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and u{(Z)} = \\cos{(Z)} and u^{2}{(Z)} = u{(Z)} \\cos{(Z)} and u^{2}{(Z)} + \\log{(f_{\\mathbf{p}})}^{f_{\\mathbf{p}}} = u{(Z)} \\cos{(Z)} + \\log{(f_{\\mathbf{p}})}^{f_{\\mathbf{p}}} and \\frac{u^{2}{(Z)} + \\log{(f_{\\mathbf{p}})}^{f_{\\mathbf{p}}}}{\\log{(f_{\\mathbf{p}})}} = \\frac{u{(Z)} \\cos{(Z)} + \\log{(f_{\\mathbf{p}})}^{f_{\\mathbf{p}}}}{\\log{(f_{\\mathbf{p}})}} and \\frac{u^{2}{(Z)} + y^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})}}{y{(f_{\\mathbf{p}})}} = \\frac{u{(Z)} \\cos{(Z)} + y^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})}}{y{(f_{\\mathbf{p}})}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["get_premise", "Equality(Function('u')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["times", 2, "Function('u')(Symbol('Z', commutative=True))"], "Equality(Pow(Function('u')(Symbol('Z', commutative=True)), Integer(2)), Mul(Function('u')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))))"], [["add", 3, "Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Pow(Function('u')(Symbol('Z', commutative=True)), Integer(2)), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Function('u')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["divide", 4, "log(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Add(Pow(Function('u')(Symbol('Z', commutative=True)), Integer(2)), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Add(Mul(Function('u')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Pow(Function('u')(Symbol('Z', commutative=True)), Integer(2)), Pow(Function('y')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Pow(Function('y')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Add(Mul(Function('u')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))), Pow(Function('y')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Pow(Function('y')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given C{(C_{d})} = e^{C_{d}} and A{(F_{c})} = e^{F_{c}}, then obtain A{(F_{c})} \\int 0 dC_{d} = e^{F_{c}} \\int 0 dC_{d}", "derivation": "C{(C_{d})} = e^{C_{d}} and \\frac{d}{d C_{d}} C{(C_{d})} = \\frac{d}{d C_{d}} e^{C_{d}} and \\frac{d}{d C_{d}} C{(C_{d})} - \\frac{d}{d C_{d}} e^{C_{d}} = 0 and \\int (\\frac{d}{d C_{d}} C{(C_{d})} - \\frac{d}{d C_{d}} e^{C_{d}}) dC_{d} = \\int 0 dC_{d} and A{(F_{c})} = e^{F_{c}} and A{(F_{c})} \\int (\\frac{d}{d C_{d}} C{(C_{d})} - \\frac{d}{d C_{d}} e^{C_{d}}) dC_{d} = e^{F_{c}} \\int (\\frac{d}{d C_{d}} C{(C_{d})} - \\frac{d}{d C_{d}} e^{C_{d}}) dC_{d} and A{(F_{c})} \\int 0 dC_{d} = e^{F_{c}} \\int 0 dC_{d}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Integer(0))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Tuple(Symbol('C_d', commutative=True))), Integral(Integer(0), Tuple(Symbol('C_d', commutative=True))))"], ["get_premise", "Equality(Function('A')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["times", 5, "Integral(Add(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Tuple(Symbol('C_d', commutative=True)))"], "Equality(Mul(Function('A')(Symbol('F_c', commutative=True)), Integral(Add(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Tuple(Symbol('C_d', commutative=True)))), Mul(exp(Symbol('F_c', commutative=True)), Integral(Add(Derivative(Function('C')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Tuple(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Function('A')(Symbol('F_c', commutative=True)), Integral(Integer(0), Tuple(Symbol('C_d', commutative=True)))), Mul(exp(Symbol('F_c', commutative=True)), Integral(Integer(0), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given A{(A_{x})} = A_{x}, then obtain 1 = \\frac{1}{\\frac{d}{d A_{x}} A{(A_{x})}}", "derivation": "A{(A_{x})} = A_{x} and \\frac{d}{d A_{x}} A{(A_{x})} = \\frac{d}{d A_{x}} A_{x} and 1 = \\frac{\\frac{d}{d A_{x}} A_{x}}{\\frac{d}{d A_{x}} A{(A_{x})}} and 1 = \\frac{1}{\\frac{d}{d A_{x}} A{(A_{x})}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Symbol('A_x', commutative=True), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('A')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Symbol('A_x', commutative=True), Tuple(Symbol('A_x', commutative=True), Integer(1))), Pow(Derivative(Function('A')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Pow(Derivative(Function('A')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given b{(\\mathbf{S})} = \\int e^{\\mathbf{S}} d\\mathbf{S}, then derive b{(\\mathbf{S})} = \\sigma_x + e^{\\mathbf{S}}, then derive (z^{*} + e^{\\mathbf{S}})^{2} = (\\sigma_x + e^{\\mathbf{S}}) (z^{*} + e^{\\mathbf{S}}), then obtain \\frac{(z^{*} + e^{\\mathbf{S}})^{2}}{b{(\\mathbf{S})}} = \\frac{(\\sigma_x + e^{\\mathbf{S}}) (z^{*} + e^{\\mathbf{S}})}{b{(\\mathbf{S})}}", "derivation": "b{(\\mathbf{S})} = \\int e^{\\mathbf{S}} d\\mathbf{S} and b{(\\mathbf{S})} = \\sigma_x + e^{\\mathbf{S}} and b{(\\mathbf{S})} \\int e^{\\mathbf{S}} d\\mathbf{S} = (\\sigma_x + e^{\\mathbf{S}}) \\int e^{\\mathbf{S}} d\\mathbf{S} and (\\int e^{\\mathbf{S}} d\\mathbf{S})^{2} = (\\sigma_x + e^{\\mathbf{S}}) \\int e^{\\mathbf{S}} d\\mathbf{S} and (z^{*} + e^{\\mathbf{S}})^{2} = (\\sigma_x + e^{\\mathbf{S}}) (z^{*} + e^{\\mathbf{S}}) and \\frac{(z^{*} + e^{\\mathbf{S}})^{2}}{b{(\\mathbf{S})}} = \\frac{(\\sigma_x + e^{\\mathbf{S}}) (z^{*} + e^{\\mathbf{S}})}{b{(\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 2, "Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('z^*', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('z^*', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["divide", 5, "Function('b')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('z^*', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integer(2)), Pow(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('z^*', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Pow(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given A{(z^{*},h,\\varepsilon)} = \\varepsilon^{h} z^{*}, then obtain (\\varepsilon^{h} h A{(z^{*},h,\\varepsilon)})^{z^{*}} A{(z^{*},h,\\varepsilon)} = (\\varepsilon^{2 h} h z^{*})^{z^{*}} A{(z^{*},h,\\varepsilon)}", "derivation": "A{(z^{*},h,\\varepsilon)} = \\varepsilon^{h} z^{*} and \\frac{A{(z^{*},h,\\varepsilon)}}{z^{*}} = \\varepsilon^{h} and \\varepsilon^{h} h A{(z^{*},h,\\varepsilon)} = \\varepsilon^{2 h} h z^{*} and (\\varepsilon^{h} h A{(z^{*},h,\\varepsilon)})^{z^{*}} = (\\varepsilon^{2 h} h z^{*})^{z^{*}} and (\\varepsilon^{h} h A{(z^{*},h,\\varepsilon)})^{z^{*}} A{(z^{*},h,\\varepsilon)} = (\\varepsilon^{2 h} h z^{*})^{z^{*}} A{(z^{*},h,\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Symbol('z^*', commutative=True)))"], [["divide", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))), Symbol('h', commutative=True), Symbol('z^*', commutative=True)))"], [["power", 3, "Symbol('z^*', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('z^*', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))), Symbol('h', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["times", 4, "Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('z^*', commutative=True)), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))), Symbol('h', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Function('A')(Symbol('z^*', commutative=True), Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\dot{z})} = \\log{(\\dot{z})} and \\dot{\\mathbf{r}}{(\\dot{z})} = \\int \\log{(\\dot{z})} d\\dot{z}, then obtain 2 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{\\mathbf{r}}{(\\dot{z})} + \\int \\sigma_{p}{(\\dot{z})} d\\dot{z}", "derivation": "\\sigma_{p}{(\\dot{z})} = \\log{(\\dot{z})} and \\int \\sigma_{p}{(\\dot{z})} d\\dot{z} = \\int \\log{(\\dot{z})} d\\dot{z} and \\dot{\\mathbf{r}}{(\\dot{z})} = \\int \\log{(\\dot{z})} d\\dot{z} and 2 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{\\mathbf{r}}{(\\dot{z})} + \\int \\log{(\\dot{z})} d\\dot{z} and 2 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{\\mathbf{r}}{(\\dot{z})} + \\int \\sigma_{p}{(\\dot{z})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)), Integral(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)), Integral(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)), Integral(Function('\\\\sigma_p')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain (\\int (\\operatorname{A_{1}}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}) dV_{\\mathbf{B}})^{V_{\\mathbf{B}}} = (\\int 0 dV_{\\mathbf{B}})^{V_{\\mathbf{B}}}", "derivation": "\\operatorname{A_{1}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\operatorname{A_{1}}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})} = 0 and \\int (\\operatorname{A_{1}}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}) dV_{\\mathbf{B}} = \\int 0 dV_{\\mathbf{B}} and (\\int (\\operatorname{A_{1}}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}) dV_{\\mathbf{B}})^{V_{\\mathbf{B}}} = (\\int 0 dV_{\\mathbf{B}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Add(Function('A_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Integral(Add(Function('A_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(E)} = e^{\\cos{(E)}} and \\operatorname{A_{2}}{(E)} = - \\cos{(E)}, then obtain (e^{\\operatorname{A_{2}}{(E)}} e^{\\cos{(E)}})^{E} = 1", "derivation": "\\mathbf{J}_M{(E)} = e^{\\cos{(E)}} and \\mathbf{J}_M{(E)} e^{- \\cos{(E)}} = 1 and \\operatorname{A_{2}}{(E)} = - \\cos{(E)} and (\\mathbf{J}_M{(E)} e^{- \\cos{(E)}})^{E} = 1 and (\\mathbf{J}_M{(E)} e^{\\operatorname{A_{2}}{(E)}})^{E} = 1 and (e^{\\operatorname{A_{2}}{(E)}} e^{\\cos{(E)}})^{E} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True)), exp(cos(Symbol('E', commutative=True))))"], [["divide", 1, "exp(cos(Symbol('E', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('E', commutative=True))))), Integer(1))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('E', commutative=True)), Mul(Integer(-1), cos(Symbol('E', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('E', commutative=True))))), Symbol('E', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True)), exp(Function('A_2')(Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(exp(Function('A_2')(Symbol('E', commutative=True))), exp(cos(Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(l)} = e^{l}, then derive \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = e^{l}, then obtain \\frac{(\\frac{d}{d l} \\operatorname{C_{2}}{(l)})^{l}}{l} = \\frac{(e^{l})^{l}}{l}", "derivation": "\\operatorname{C_{2}}{(l)} = e^{l} and \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = \\frac{d}{d l} e^{l} and \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = e^{l} and (\\frac{d}{d l} \\operatorname{C_{2}}{(l)})^{l} = (e^{l})^{l} and \\frac{(\\frac{d}{d l} \\operatorname{C_{2}}{(l)})^{l}}{l} = \\frac{(e^{l})^{l}}{l}", "srepr_derivation": [["get_premise", "Equality(Function('C_2')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), exp(Symbol('l', commutative=True)))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Symbol('l', commutative=True)), Pow(exp(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 4, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Symbol('l', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(exp(Symbol('l', commutative=True)), Symbol('l', commutative=True))))"]]}, {"prompt": "Given n{(M_{E})} = e^{M_{E}}, then obtain ((n{(M_{E})} \\int (n{(M_{E})} - e^{M_{E}}) dM_{E})^{M_{E}})^{M_{E}} = ((n{(M_{E})} \\int 0 dM_{E})^{M_{E}})^{M_{E}}", "derivation": "n{(M_{E})} = e^{M_{E}} and n{(M_{E})} - e^{M_{E}} = 0 and \\int (n{(M_{E})} - e^{M_{E}}) dM_{E} = \\int 0 dM_{E} and e^{M_{E}} \\int (n{(M_{E})} - e^{M_{E}}) dM_{E} = e^{M_{E}} \\int 0 dM_{E} and n{(M_{E})} \\int (n{(M_{E})} - e^{M_{E}}) dM_{E} = n{(M_{E})} \\int 0 dM_{E} and (n{(M_{E})} \\int (n{(M_{E})} - e^{M_{E}}) dM_{E})^{M_{E}} = (n{(M_{E})} \\int 0 dM_{E})^{M_{E}} and ((n{(M_{E})} \\int (n{(M_{E})} - e^{M_{E}}) dM_{E})^{M_{E}})^{M_{E}} = ((n{(M_{E})} \\int 0 dM_{E})^{M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["minus", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True))))"], [["times", 3, "exp(Symbol('M_E', commutative=True))"], "Equality(Mul(exp(Symbol('M_E', commutative=True)), Integral(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Mul(exp(Symbol('M_E', commutative=True)), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True)))))"], [["power", 5, "Symbol('M_E', commutative=True)"], "Equality(Pow(Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Pow(Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)))"], [["power", 6, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Add(Function('n')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Mul(Function('n')(Symbol('M_E', commutative=True)), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\nabla,M)} = M^{\\nabla}, then derive \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)} = M^{\\nabla} \\log{(M)}, then obtain 1 = \\frac{M^{\\nabla} \\log{(M)}}{\\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)}}", "derivation": "\\hat{p}_0{(\\nabla,M)} = M^{\\nabla} and \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)} = \\frac{\\partial}{\\partial \\nabla} M^{\\nabla} and 1 = \\frac{\\frac{\\partial}{\\partial \\nabla} M^{\\nabla}}{\\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)}} and \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)} = M^{\\nabla} \\log{(M)} and \\frac{\\partial}{\\partial \\nabla} M^{\\nabla} = M^{\\nabla} \\log{(M)} and 1 = \\frac{M^{\\nabla} \\log{(M)}}{\\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,M)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(1), Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('M', commutative=True)), Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(C_{d})} = \\log{(C_{d})} and \\operatorname{E_{\\lambda}}{(C_{d})} = \\Omega{(C_{d})} \\log{(C_{d})}, then obtain \\frac{d}{d C_{d}} \\operatorname{E_{\\lambda}}^{2}{(C_{d})} = \\frac{d}{d C_{d}} \\operatorname{E_{\\lambda}}{(C_{d})} \\log{(C_{d})}^{2}", "derivation": "\\Omega{(C_{d})} = \\log{(C_{d})} and \\Omega{(C_{d})} \\log{(C_{d})} = \\log{(C_{d})}^{2} and \\Omega^{2}{(C_{d})} \\log{(C_{d})}^{2} = \\Omega{(C_{d})} \\log{(C_{d})}^{3} and \\operatorname{E_{\\lambda}}{(C_{d})} = \\Omega{(C_{d})} \\log{(C_{d})} and \\operatorname{E_{\\lambda}}^{2}{(C_{d})} = \\operatorname{E_{\\lambda}}{(C_{d})} \\log{(C_{d})}^{2} and \\frac{d}{d C_{d}} \\operatorname{E_{\\lambda}}^{2}{(C_{d})} = \\frac{d}{d C_{d}} \\operatorname{E_{\\lambda}}{(C_{d})} \\log{(C_{d})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["times", 1, "log(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True))), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))"], [["times", 2, "Mul(Function('\\\\Omega')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\Omega')(Symbol('C_d', commutative=True)), Integer(2)), Pow(log(Symbol('C_d', commutative=True)), Integer(2))), Mul(Function('\\\\Omega')(Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(3))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('C_d', commutative=True)), Mul(Function('\\\\Omega')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('C_d', commutative=True)), Integer(2)), Mul(Function('E_{\\\\lambda}')(Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Pow(Function('E_{\\\\lambda}')(Symbol('C_d', commutative=True)), Integer(2)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Function('E_{\\\\lambda}')(Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(2))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(M,\\hbar)} = \\sin{(M + \\hbar)} and \\operatorname{F_{g}}{(M,\\hbar)} = \\sin{(M + \\hbar)}, then obtain \\frac{\\phi_{1}^{M}{(M,\\hbar)}}{\\sin{(M + \\hbar)}} = \\frac{\\operatorname{F_{g}}^{M}{(M,\\hbar)}}{\\sin{(M + \\hbar)}}", "derivation": "\\phi_{1}{(M,\\hbar)} = \\sin{(M + \\hbar)} and \\phi_{1}^{M}{(M,\\hbar)} = \\sin^{M}{(M + \\hbar)} and \\frac{\\phi_{1}^{M}{(M,\\hbar)}}{\\sin{(M + \\hbar)}} = \\frac{\\sin^{M}{(M + \\hbar)}}{\\sin{(M + \\hbar)}} and \\operatorname{F_{g}}{(M,\\hbar)} = \\sin{(M + \\hbar)} and \\phi_{1}^{M}{(M,\\hbar)} = \\operatorname{F_{g}}^{M}{(M,\\hbar)} and \\sin^{M}{(M + \\hbar)} = \\operatorname{F_{g}}^{M}{(M,\\hbar)} and \\frac{\\phi_{1}^{M}{(M,\\hbar)}}{\\sin{(M + \\hbar)}} = \\frac{\\operatorname{F_{g}}^{M}{(M,\\hbar)}}{\\sin{(M + \\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)), Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('M', commutative=True)))"], [["divide", 2, "sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\phi_1')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)), Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1))), Mul(Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\phi_1')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)), Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('M', commutative=True)), Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Mul(Pow(Function('\\\\phi_1')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)), Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1))), Mul(Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('M', commutative=True)), Pow(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} = \\mathbf{J}_M + \\phi_1, then obtain \\frac{\\int 2 \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} d\\phi_1}{2 \\mathbf{J}_M + 2 \\phi_1} = \\frac{\\int (\\mathbf{J}_M + \\phi_1 + \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)}) d\\phi_1}{2 \\mathbf{J}_M + 2 \\phi_1}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} = \\mathbf{J}_M + \\phi_1 and \\mathbf{J}_M + \\phi_1 + \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} = 2 \\mathbf{J}_M + 2 \\phi_1 and 2 \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} = 2 \\mathbf{J}_M + 2 \\phi_1 and 2 \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} = \\mathbf{J}_M + \\phi_1 + \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} and \\int 2 \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} d\\phi_1 = \\int (\\mathbf{J}_M + \\phi_1 + \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)}) d\\phi_1 and \\frac{\\int 2 \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)} d\\phi_1}{2 \\mathbf{J}_M + 2 \\phi_1} = \\frac{\\int (\\mathbf{J}_M + \\phi_1 + \\operatorname{f^{*}}{(\\mathbf{J}_M,\\phi_1)}) d\\phi_1}{2 \\mathbf{J}_M + 2 \\phi_1}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["divide", 5, "Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Integral(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\varphi^*,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}), then obtain 0 = - \\rho{(\\varphi^*,m_{s})} \\sin{(\\rho{(\\varphi^*,m_{s})} - \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}))}", "derivation": "\\rho{(\\varphi^*,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}) and - m_{s} + \\rho{(\\varphi^*,m_{s})} = - m_{s} + \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}) and 0 = - \\rho{(\\varphi^*,m_{s})} + \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}) and 0 = - \\sin{(\\rho{(\\varphi^*,m_{s})} - \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}))} and 0 = - \\rho{(\\varphi^*,m_{s})} \\sin{(\\rho{(\\varphi^*,m_{s})} - \\frac{\\partial}{\\partial m_{s}} (\\varphi^* - m_{s}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True)), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True))), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))))))"], [["times", 4, "Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True)), sin(Add(Function('\\\\rho')(Symbol('\\\\varphi^*', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given g{(\\nabla)} = \\log{(\\nabla)}, then obtain ((- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} e^{- f_{\\mathbf{p}}} - e^{- f_{\\mathbf{p}}}) (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} = 0", "derivation": "g{(\\nabla)} = \\log{(\\nabla)} and 0 = - g{(\\nabla)} + \\log{(\\nabla)} and 0^{\\nabla} = (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} and 0^{\\nabla} e^{- f_{\\mathbf{p}}} = (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} e^{- f_{\\mathbf{p}}} and 0^{\\nabla} e^{- f_{\\mathbf{p}}} - (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} e^{- f_{\\mathbf{p}}} = 0 and (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} e^{- f_{\\mathbf{p}}} - e^{- f_{\\mathbf{p}}} = 0 and ((- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} e^{- f_{\\mathbf{p}}} - e^{- f_{\\mathbf{p}}}) (- g{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} = 0", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Function('g')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["divide", 3, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["minus", 4, "Mul(Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], "Equality(Add(Mul(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Integer(0))"], [["times", 6, "Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Add(Mul(Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\Psi{(a^{\\dagger})} = \\sin{(\\log{(a^{\\dagger})})}, then obtain \\int (a^{\\dagger} + \\Psi{(a^{\\dagger})} + \\frac{2}{\\Psi{(a^{\\dagger})}}) da^{\\dagger} = \\int (a^{\\dagger} + \\sin{(\\log{(a^{\\dagger})})} + \\frac{2}{\\Psi{(a^{\\dagger})}}) da^{\\dagger}", "derivation": "\\Psi{(a^{\\dagger})} = \\sin{(\\log{(a^{\\dagger})})} and \\Psi{(a^{\\dagger})} + \\frac{1}{\\Psi{(a^{\\dagger})}} = \\sin{(\\log{(a^{\\dagger})})} + \\frac{1}{\\Psi{(a^{\\dagger})}} and a^{\\dagger} + \\Psi{(a^{\\dagger})} + \\frac{1}{\\Psi{(a^{\\dagger})}} = a^{\\dagger} + \\sin{(\\log{(a^{\\dagger})})} + \\frac{1}{\\Psi{(a^{\\dagger})}} and a^{\\dagger} + \\Psi{(a^{\\dagger})} + \\frac{2}{\\Psi{(a^{\\dagger})}} = a^{\\dagger} + \\sin{(\\log{(a^{\\dagger})})} + \\frac{2}{\\Psi{(a^{\\dagger})}} and \\int (a^{\\dagger} + \\Psi{(a^{\\dagger})} + \\frac{2}{\\Psi{(a^{\\dagger})}}) da^{\\dagger} = \\int (a^{\\dagger} + \\sin{(\\log{(a^{\\dagger})})} + \\frac{2}{\\Psi{(a^{\\dagger})}}) da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), sin(log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Add(sin(log(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Add(Symbol('a^{\\\\dagger}', commutative=True), sin(log(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["add", 3, "Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))), Add(Symbol('a^{\\\\dagger}', commutative=True), sin(log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))))"], [["integrate", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), Pow(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(u,n_{2})} = - n_{2} + u and \\theta{(u,n_{2})} = - n_{2} + u, then obtain (\\frac{\\operatorname{v_{t}}{(u,n_{2})}}{- n_{2} + u})^{u} = 1", "derivation": "\\operatorname{v_{t}}{(u,n_{2})} = - n_{2} + u and \\theta{(u,n_{2})} = - n_{2} + u and \\operatorname{v_{t}}{(u,n_{2})} = \\theta{(u,n_{2})} and \\frac{\\operatorname{v_{t}}{(u,n_{2})}}{- n_{2} + u} = \\frac{\\theta{(u,n_{2})}}{- n_{2} + u} and \\frac{\\operatorname{v_{t}}{(u,n_{2})}}{\\theta{(u,n_{2})}} = 1 and \\frac{\\operatorname{v_{t}}{(u,n_{2})}}{- n_{2} + u} = 1 and (\\frac{\\operatorname{v_{t}}{(u,n_{2})}}{- n_{2} + u})^{u} = 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('\\\\theta')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('\\\\theta')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)), Integer(-1)), Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Integer(1))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('v_t')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Symbol('u', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{g}{(\\delta)} = e^{\\delta}, then derive \\frac{d}{d \\delta} \\mathbf{g}{(\\delta)} + 1 = e^{\\delta} + 1, then obtain \\frac{d^{2}}{d \\delta^{2}} (\\mathbf{g}{(\\delta)} + 1) = \\frac{d^{2}}{d \\delta^{2}} (e^{\\delta} + 1)", "derivation": "\\mathbf{g}{(\\delta)} = e^{\\delta} and \\delta + \\mathbf{g}{(\\delta)} = \\delta + e^{\\delta} and \\frac{d}{d \\delta} (\\delta + \\mathbf{g}{(\\delta)}) = \\frac{d}{d \\delta} (\\delta + e^{\\delta}) and \\frac{d}{d \\delta} \\mathbf{g}{(\\delta)} + 1 = e^{\\delta} + 1 and \\frac{d}{d \\delta} e^{\\delta} + 1 = e^{\\delta} + 1 and \\frac{d}{d \\delta} (\\frac{d}{d \\delta} \\mathbf{g}{(\\delta)} + 1) = \\frac{d}{d \\delta} (e^{\\delta} + 1) and \\frac{d^{2}}{d \\delta^{2}} (\\frac{d}{d \\delta} \\mathbf{g}{(\\delta)} + 1) = \\frac{d^{2}}{d \\delta^{2}} (e^{\\delta} + 1) and \\frac{d}{d \\delta} \\mathbf{g}{(\\delta)} + 1 = \\mathbf{g}{(\\delta)} + 1 and \\frac{d^{2}}{d \\delta^{2}} (\\mathbf{g}{(\\delta)} + 1) = \\frac{d^{2}}{d \\delta^{2}} (e^{\\delta} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), exp(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\delta', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\delta', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\delta', commutative=True)), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Derivative(Add(exp(Symbol('\\\\delta', commutative=True)), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Derivative(Add(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True)), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Derivative(Add(exp(Symbol('\\\\delta', commutative=True)), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\sigma_{p}{(\\nabla)} = \\cos{(\\nabla)}, then obtain - \\cos{(\\nabla)} + \\int e^{\\sigma_{p}^{\\nabla}{(\\nabla)}} d\\nabla = - \\cos{(\\nabla)} + \\int e^{\\cos^{\\nabla}{(\\nabla)}} d\\nabla", "derivation": "\\sigma_{p}{(\\nabla)} = \\cos{(\\nabla)} and \\sigma_{p}^{\\nabla}{(\\nabla)} = \\cos^{\\nabla}{(\\nabla)} and e^{\\sigma_{p}^{\\nabla}{(\\nabla)}} = e^{\\cos^{\\nabla}{(\\nabla)}} and \\int e^{\\sigma_{p}^{\\nabla}{(\\nabla)}} d\\nabla = \\int e^{\\cos^{\\nabla}{(\\nabla)}} d\\nabla and - \\cos{(\\nabla)} + \\int e^{\\sigma_{p}^{\\nabla}{(\\nabla)}} d\\nabla = - \\cos{(\\nabla)} + \\int e^{\\cos^{\\nabla}{(\\nabla)}} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["power", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), exp(Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(exp(Pow(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["minus", 4, "cos(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True))), Integral(exp(Pow(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True))), Integral(exp(Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(M_{E},I)} = \\frac{\\partial}{\\partial I} (I - M_{E}), then obtain \\frac{(\\psi^{*}{(M_{E},I)} + 1)^{M_{E}} (\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{M_{E}}}{M_{E}^{2}} = \\frac{(\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{2 M_{E}}}{M_{E}^{2}}", "derivation": "\\psi^{*}{(M_{E},I)} = \\frac{\\partial}{\\partial I} (I - M_{E}) and \\psi^{*}{(M_{E},I)} + 1 = \\frac{\\partial}{\\partial I} (I - M_{E}) + 1 and (\\psi^{*}{(M_{E},I)} + 1)^{M_{E}} = (\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{M_{E}} and \\frac{(\\psi^{*}{(M_{E},I)} + 1)^{M_{E}}}{M_{E}} = \\frac{(\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{M_{E}}}{M_{E}} and \\frac{(\\psi^{*}{(M_{E},I)} + 1)^{M_{E}} (\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{M_{E}}}{M_{E}^{2}} = \\frac{(\\frac{\\partial}{\\partial I} (I - M_{E}) + 1)^{2 M_{E}}}{M_{E}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\psi^*')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Integer(1)), Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Function('\\\\psi^*')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Integer(1)), Symbol('M_E', commutative=True)), Pow(Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Symbol('M_E', commutative=True)))"], [["divide", 3, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Add(Function('\\\\psi^*')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Integer(1)), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Symbol('M_E', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Symbol('M_E', commutative=True)))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Add(Function('\\\\psi^*')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Integer(1)), Symbol('M_E', commutative=True)), Pow(Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Add(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Mul(Integer(2), Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given b{(\\hat{p}_0)} = \\sin{(e^{\\hat{p}_0})}, then obtain b^{\\hat{p}_0}{(\\hat{p}_0)} \\int \\sin{(e^{\\hat{p}_0})} d\\hat{p}_0 = \\sin^{\\hat{p}_0}{(e^{\\hat{p}_0})} \\int \\sin{(e^{\\hat{p}_0})} d\\hat{p}_0", "derivation": "b{(\\hat{p}_0)} = \\sin{(e^{\\hat{p}_0})} and \\int b{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\sin{(e^{\\hat{p}_0})} d\\hat{p}_0 and b^{\\hat{p}_0}{(\\hat{p}_0)} = \\sin^{\\hat{p}_0}{(e^{\\hat{p}_0})} and b^{\\hat{p}_0}{(\\hat{p}_0)} \\int b{(\\hat{p}_0)} d\\hat{p}_0 = \\sin^{\\hat{p}_0}{(e^{\\hat{p}_0})} \\int b{(\\hat{p}_0)} d\\hat{p}_0 and b^{\\hat{p}_0}{(\\hat{p}_0)} \\int \\sin{(e^{\\hat{p}_0})} d\\hat{p}_0 = \\sin^{\\hat{p}_0}{(e^{\\hat{p}_0})} \\int \\sin{(e^{\\hat{p}_0})} d\\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), sin(exp(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 3, "Integral(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Pow(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Integral(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Pow(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Integral(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('b')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Integral(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Pow(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Integral(sin(exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} = \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f}, then obtain \\frac{\\partial}{\\partial E_{x}} (- \\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} - 1) = \\frac{\\partial}{\\partial E_{x}} (- \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f} - 1)", "derivation": "\\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} = \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f} and - \\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} = - \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f} and - \\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} - 1 = - \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f} - 1 and \\frac{\\partial}{\\partial E_{x}} (- \\Psi_{\\lambda}{(E_{x},\\mathbf{J}_f,V_{\\mathbf{E}})} - 1) = \\frac{\\partial}{\\partial E_{x}} (- \\frac{E_{x} V_{\\mathbf{E}}}{\\mathbf{J}_f} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Integer(-1)))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(-1)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(i,C_{d})} = - C_{d} + i, then derive \\frac{\\partial}{\\partial i} A{(i,C_{d})} = 1, then obtain \\frac{\\partial}{\\partial C_{d}} \\frac{(1 - C_{d}) \\frac{\\partial}{\\partial i} (- C_{d} + i)}{2} = \\frac{d}{d C_{d}} (\\frac{1}{2} - \\frac{C_{d}}{2})", "derivation": "A{(i,C_{d})} = - C_{d} + i and \\frac{\\partial}{\\partial i} A{(i,C_{d})} = \\frac{\\partial}{\\partial i} (- C_{d} + i) and \\frac{\\partial}{\\partial i} A{(i,C_{d})} = 1 and \\frac{\\partial}{\\partial i} (- C_{d} + i) = 1 and \\frac{\\partial}{\\partial i} (- C_{d} + i) + 1 = 2 and (1 - C_{d}) \\frac{\\partial}{\\partial i} (- C_{d} + i) = 1 - C_{d} and \\frac{(1 - C_{d}) \\frac{\\partial}{\\partial i} (- C_{d} + i)}{\\frac{\\partial}{\\partial i} (- C_{d} + i) + 1} = \\frac{1 - C_{d}}{\\frac{\\partial}{\\partial i} (- C_{d} + i) + 1} and \\frac{(1 - C_{d}) \\frac{\\partial}{\\partial i} (- C_{d} + i)}{2} = \\frac{1}{2} - \\frac{C_{d}}{2} and \\frac{\\partial}{\\partial C_{d}} \\frac{(1 - C_{d}) \\frac{\\partial}{\\partial i} (- C_{d} + i)}{2} = \\frac{d}{d C_{d}} (\\frac{1}{2} - \\frac{C_{d}}{2})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('i', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('i', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('i', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["times", 4, "Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True)))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))))"], [["divide", 6, "Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))), Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))), Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Rational(1, 2), Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), Symbol('C_d', commutative=True))))"], [["differentiate", 8, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(\\dot{z},\\rho_f,J)} = - \\dot{z} + \\frac{\\rho_f}{J}, then derive (\\frac{\\partial}{\\partial \\rho_f} U{(\\dot{z},\\rho_f,J)} - \\frac{1}{J})^{2} = 0, then obtain (\\frac{\\partial}{\\partial \\rho_f} (- \\dot{z} + \\frac{\\rho_f}{J}) - \\frac{1}{J})^{2} = 0", "derivation": "U{(\\dot{z},\\rho_f,J)} = - \\dot{z} + \\frac{\\rho_f}{J} and \\frac{\\partial}{\\partial \\rho_f} U{(\\dot{z},\\rho_f,J)} = \\frac{\\partial}{\\partial \\rho_f} (- \\dot{z} + \\frac{\\rho_f}{J}) and - \\frac{\\partial}{\\partial \\rho_f} (- \\dot{z} + \\frac{\\rho_f}{J}) + \\frac{\\partial}{\\partial \\rho_f} U{(\\dot{z},\\rho_f,J)} = 0 and (- \\frac{\\partial}{\\partial \\rho_f} (- \\dot{z} + \\frac{\\rho_f}{J}) + \\frac{\\partial}{\\partial \\rho_f} U{(\\dot{z},\\rho_f,J)})^{2} = 0 and (\\frac{\\partial}{\\partial \\rho_f} U{(\\dot{z},\\rho_f,J)} - \\frac{1}{J})^{2} = 0 and (\\frac{\\partial}{\\partial \\rho_f} (- \\dot{z} + \\frac{\\rho_f}{J}) - \\frac{1}{J})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Derivative(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Integer(0))"], [["times", 3, "Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Derivative(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], "Equality(Pow(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Derivative(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Integer(2)), Integer(0))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('U')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)))), Integer(2)), Integer(0))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)))), Integer(2)), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(E)} = \\frac{d}{d E} e^{E}, then derive \\operatorname{t_{1}}{(E)} = e^{E}, then obtain 1 = \\frac{\\int \\frac{d}{d E} \\operatorname{t_{1}}{(E)} dE}{\\int \\operatorname{t_{1}}{(E)} dE}", "derivation": "\\operatorname{t_{1}}{(E)} = \\frac{d}{d E} e^{E} and \\operatorname{t_{1}}{(E)} = e^{E} and \\operatorname{t_{1}}{(E)} = \\frac{d}{d E} \\operatorname{t_{1}}{(E)} and \\int \\operatorname{t_{1}}{(E)} dE = \\int \\frac{d}{d E} \\operatorname{t_{1}}{(E)} dE and 1 = \\frac{\\int \\frac{d}{d E} \\operatorname{t_{1}}{(E)} dE}{\\int \\operatorname{t_{1}}{(E)} dE}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t_1')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('t_1')(Symbol('E', commutative=True)), Derivative(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Derivative(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))))"], [["divide", 4, "Integral(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(-1)), Integral(Derivative(Function('t_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(A_{1},\\chi)} = e^{\\chi^{A_{1}}}, then obtain - \\sin{(\\frac{\\chi^{- A_{1}} \\hat{H}_l{(A_{1},\\chi)}}{A_{1}})} = - \\sin{(\\frac{\\chi^{- A_{1}} e^{\\chi^{A_{1}}}}{A_{1}})}", "derivation": "\\hat{H}_l{(A_{1},\\chi)} = e^{\\chi^{A_{1}}} and \\chi^{- A_{1}} \\hat{H}_l{(A_{1},\\chi)} = \\chi^{- A_{1}} e^{\\chi^{A_{1}}} and - \\frac{\\chi^{- A_{1}} \\hat{H}_l{(A_{1},\\chi)}}{A_{1}} = - \\frac{\\chi^{- A_{1}} e^{\\chi^{A_{1}}}}{A_{1}} and - \\sin{(\\frac{\\chi^{- A_{1}} \\hat{H}_l{(A_{1},\\chi)}}{A_{1}})} = - \\sin{(\\frac{\\chi^{- A_{1}} e^{\\chi^{A_{1}}}}{A_{1}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Pow(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), exp(Pow(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Symbol('A_1', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), exp(Pow(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))))), Mul(Integer(-1), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), exp(Pow(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\theta)} = \\cos{(\\theta)} and B{(\\Psi^{\\dagger},\\mathbf{p})} = \\Psi^{\\dagger} \\mathbf{p}, then obtain - \\Psi^{\\dagger} \\mathbf{p} + 2 \\hat{H}_{\\lambda}{(\\theta)} + \\cos{(\\theta)} = - \\Psi^{\\dagger} \\mathbf{p} + \\hat{H}_{\\lambda}{(\\theta)} + 2 \\cos{(\\theta)}", "derivation": "\\hat{H}_{\\lambda}{(\\theta)} = \\cos{(\\theta)} and B{(\\Psi^{\\dagger},\\mathbf{p})} = \\Psi^{\\dagger} \\mathbf{p} and \\hat{H}_{\\lambda}{(\\theta)} + \\cos{(\\theta)} = 2 \\cos{(\\theta)} and - B{(\\Psi^{\\dagger},\\mathbf{p})} + 2 \\hat{H}_{\\lambda}{(\\theta)} + \\cos{(\\theta)} = - B{(\\Psi^{\\dagger},\\mathbf{p})} + \\hat{H}_{\\lambda}{(\\theta)} + 2 \\cos{(\\theta)} and - \\Psi^{\\dagger} \\mathbf{p} + 2 \\hat{H}_{\\lambda}{(\\theta)} + \\cos{(\\theta)} = - \\Psi^{\\dagger} \\mathbf{p} + \\hat{H}_{\\lambda}{(\\theta)} + 2 \\cos{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], ["get_premise", "Equality(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))"], [["minus", 3, "Add(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True))), cos(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True))), cos(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\mu{(i,J)} = i^{J}, then obtain (\\frac{\\partial^{2}}{\\partial i^{2}} \\mu^{i}{(i,J)})^{i} = (\\frac{\\partial^{2}}{\\partial i^{2}} (i^{J})^{i})^{i}", "derivation": "\\mu{(i,J)} = i^{J} and \\mu^{i}{(i,J)} = (i^{J})^{i} and \\frac{\\partial}{\\partial i} \\mu^{i}{(i,J)} = \\frac{\\partial}{\\partial i} (i^{J})^{i} and \\frac{\\partial^{2}}{\\partial i^{2}} \\mu^{i}{(i,J)} = \\frac{\\partial^{2}}{\\partial i^{2}} (i^{J})^{i} and (\\frac{\\partial^{2}}{\\partial i^{2}} \\mu^{i}{(i,J)})^{i} = (\\frac{\\partial^{2}}{\\partial i^{2}} (i^{J})^{i})^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Pow(Pow(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mu')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True)), Pow(Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('J', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given S{(f,\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}} f)}, then obtain (- f + S{(f,\\hat{\\mathbf{r}})}) (\\int S{(f,\\hat{\\mathbf{r}})} df - \\int \\log{(\\hat{\\mathbf{r}} f)} df) = 0", "derivation": "S{(f,\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}} f)} and \\int S{(f,\\hat{\\mathbf{r}})} df = \\int \\log{(\\hat{\\mathbf{r}} f)} df and - \\hat{\\mathbf{r}} + \\int S{(f,\\hat{\\mathbf{r}})} df = - \\hat{\\mathbf{r}} + \\int \\log{(\\hat{\\mathbf{r}} f)} df and \\int S{(f,\\hat{\\mathbf{r}})} df - \\int \\log{(\\hat{\\mathbf{r}} f)} df = 0 and (- f + \\log{(\\hat{\\mathbf{r}} f)}) (\\int S{(f,\\hat{\\mathbf{r}})} df - \\int \\log{(\\hat{\\mathbf{r}} f)} df) = 0 and (- f + S{(f,\\hat{\\mathbf{r}})}) (\\int S{(f,\\hat{\\mathbf{r}})} df - \\int \\log{(\\hat{\\mathbf{r}} f)} df) = 0", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["minus", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], "Equality(Add(Integral(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))), Integer(0))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('f', commutative=True)), log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)))), Add(Integral(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Integral(Function('S')(Symbol('f', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Integral(log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\omega,\\hbar)} = - \\hbar + \\cos{(\\omega)}, then obtain - \\hbar + \\cos{(\\operatorname{L_{\\varepsilon}}^{\\omega}{(\\omega,\\hbar)})} = - \\hbar + \\cos{((- \\hbar + \\cos{(\\omega)})^{\\omega})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\omega,\\hbar)} = - \\hbar + \\cos{(\\omega)} and \\operatorname{L_{\\varepsilon}}^{\\omega}{(\\omega,\\hbar)} = (- \\hbar + \\cos{(\\omega)})^{\\omega} and \\cos{(\\operatorname{L_{\\varepsilon}}^{\\omega}{(\\omega,\\hbar)})} = \\cos{((- \\hbar + \\cos{(\\omega)})^{\\omega})} and - \\hbar + \\cos{(\\operatorname{L_{\\varepsilon}}^{\\omega}{(\\omega,\\hbar)})} = - \\hbar + \\cos{((- \\hbar + \\cos{(\\omega)})^{\\omega})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\omega', commutative=True))), cos(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain - \\dot{z} + \\frac{\\dot{z} - \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\rho{(\\dot{z})} = - \\dot{z} + \\frac{\\dot{z} - \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\sin{(\\dot{z})}", "derivation": "\\rho{(\\dot{z})} = \\sin{(\\dot{z})} and - \\dot{z} + \\rho{(\\dot{z})} = - \\dot{z} + \\sin{(\\dot{z})} and \\frac{- \\dot{z} + \\rho{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{- \\dot{z} + \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} and - \\dot{z} - \\frac{- \\dot{z} + \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\rho{(\\dot{z})} = - \\dot{z} - \\frac{- \\dot{z} + \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\sin{(\\dot{z})} and - \\dot{z} + \\frac{\\dot{z} - \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\rho{(\\dot{z})} = - \\dot{z} + \\frac{\\dot{z} - \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} + \\sin{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), sin(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given Z{(g_{\\varepsilon},I)} = g_{\\varepsilon} + \\sin{(I)}, then obtain I Z^{2}{(g_{\\varepsilon},I)} \\int 2 I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} dI = I (g_{\\varepsilon} + \\sin{(I)})^{2} \\int 2 I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} dI", "derivation": "Z{(g_{\\varepsilon},I)} = g_{\\varepsilon} + \\sin{(I)} and I Z{(g_{\\varepsilon},I)} = I (g_{\\varepsilon} + \\sin{(I)}) and I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} = I (g_{\\varepsilon} + \\sin{(I)})^{2} and I Z^{2}{(g_{\\varepsilon},I)} = I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} and I Z^{2}{(g_{\\varepsilon},I)} = I (g_{\\varepsilon} + \\sin{(I)})^{2} and I Z^{2}{(g_{\\varepsilon},I)} \\int 2 I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} dI = I (g_{\\varepsilon} + \\sin{(I)})^{2} \\int 2 I (g_{\\varepsilon} + \\sin{(I)}) Z{(g_{\\varepsilon},I)} dI", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True)))))"], [["times", 1, "Mul(Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))))"], "Equality(Mul(Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('I', commutative=True), Pow(Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True)), Integer(2))), Mul(Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('I', commutative=True), Pow(Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True)), Integer(2))), Mul(Symbol('I', commutative=True), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Integer(2))))"], [["times", 5, "Integral(Mul(Integer(2), Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(Symbol('I', commutative=True), Pow(Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True)), Integer(2)), Integral(Mul(Integer(2), Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Integer(2)), Integral(Mul(Integer(2), Symbol('I', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('I', commutative=True))), Function('Z')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then obtain \\theta_1 + \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} = \\theta_1 - \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} + 2 \\sin{(\\hat{\\mathbf{x}})}", "derivation": "\\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\theta_1 + \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} = \\theta_1 + \\sin{(\\hat{\\mathbf{x}})} and \\theta_1 = \\theta_1 - \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} + \\sin{(\\hat{\\mathbf{x}})} and \\theta_1 + \\sin{(\\hat{\\mathbf{x}})} = \\theta_1 - \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} + 2 \\sin{(\\hat{\\mathbf{x}})} and \\theta_1 + \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} = \\theta_1 - \\operatorname{A_{1}}{(\\hat{\\mathbf{x}})} + 2 \\sin{(\\hat{\\mathbf{x}})}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["minus", 2, "Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Symbol('\\\\theta_1', commutative=True), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given Q{(I,\\hat{\\mathbf{x}},z)} = I^{\\hat{\\mathbf{x}}} - z, then obtain - \\frac{I \\int Q{(I,\\hat{\\mathbf{x}},z)} dI}{z} = - \\frac{I \\int (I^{\\hat{\\mathbf{x}}} - z) dI}{z}", "derivation": "Q{(I,\\hat{\\mathbf{x}},z)} = I^{\\hat{\\mathbf{x}}} - z and \\int Q{(I,\\hat{\\mathbf{x}},z)} dI = \\int (I^{\\hat{\\mathbf{x}}} - z) dI and - \\frac{\\int Q{(I,\\hat{\\mathbf{x}},z)} dI}{z} = - \\frac{\\int (I^{\\hat{\\mathbf{x}}} - z) dI}{z} and - \\frac{I \\int Q{(I,\\hat{\\mathbf{x}},z)} dI}{z} = - \\frac{I \\int (I^{\\hat{\\mathbf{x}}} - z) dI}{z}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('z', commutative=True)), Add(Pow(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Add(Pow(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('z', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Integral(Function('Q')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["times", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Integral(Function('Q')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(i,A_{1},n_{2})} = A_{1}^{i} + n_{2}, then derive \\frac{\\partial}{\\partial i} (A_{1} + \\tilde{g}^*) = \\frac{\\partial}{\\partial i} \\int \\frac{A_{1}^{i} + n_{2}}{\\dot{y}{(i,A_{1},n_{2})}} dA_{1}, then obtain 0 = \\frac{d}{d i} \\int 1 dA_{1}", "derivation": "\\dot{y}{(i,A_{1},n_{2})} = A_{1}^{i} + n_{2} and 1 = \\frac{A_{1}^{i} + n_{2}}{\\dot{y}{(i,A_{1},n_{2})}} and \\int 1 dA_{1} = \\int \\frac{A_{1}^{i} + n_{2}}{\\dot{y}{(i,A_{1},n_{2})}} dA_{1} and \\frac{d}{d i} \\int 1 dA_{1} = \\frac{\\partial}{\\partial i} \\int \\frac{A_{1}^{i} + n_{2}}{\\dot{y}{(i,A_{1},n_{2})}} dA_{1} and \\frac{\\partial}{\\partial i} (A_{1} + \\tilde{g}^*) = \\frac{\\partial}{\\partial i} \\int \\frac{A_{1}^{i} + n_{2}}{\\dot{y}{(i,A_{1},n_{2})}} dA_{1} and \\frac{\\partial}{\\partial i} (A_{1} + \\tilde{g}^*) = \\frac{d}{d i} \\int 1 dA_{1} and 0 = \\frac{d}{d i} \\int 1 dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True)), Add(Pow(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Symbol('n_2', commutative=True)))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Integer(1), Mul(Add(Pow(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Add(Pow(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Mul(Add(Pow(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Mul(Add(Pow(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Derivative(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(i)} = \\log{(i)}, then obtain - 2 \\log{(i)} + \\frac{d}{d i} \\ddot{x}{(i)} = - 2 \\log{(i)} + \\frac{1}{i}", "derivation": "\\ddot{x}{(i)} = \\log{(i)} and \\frac{d}{d i} \\ddot{x}{(i)} = \\frac{d}{d i} \\log{(i)} and - \\log{(i)} + \\frac{d}{d i} \\ddot{x}{(i)} = - \\log{(i)} + \\frac{d}{d i} \\log{(i)} and - 2 \\log{(i)} + \\frac{d}{d i} \\ddot{x}{(i)} = - 2 \\log{(i)} + \\frac{d}{d i} \\log{(i)} and - 2 \\log{(i)} + \\frac{d}{d i} \\ddot{x}{(i)} = - 2 \\log{(i)} + \\frac{1}{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["minus", 2, "log(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('i', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('i', commutative=True))), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["minus", 3, "log(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), log(Symbol('i', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), log(Symbol('i', commutative=True))), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Integer(2), log(Symbol('i', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), log(Symbol('i', commutative=True))), Pow(Symbol('i', commutative=True), Integer(-1))))"]]}, {"prompt": "Given k{(y)} = \\int \\cos{(y)} dy, then derive k{(y)} = \\psi^* + \\sin{(y)}, then obtain \\frac{\\frac{d}{d y} (k{(y)} - \\cos{(y)})}{\\psi^*} = \\frac{\\frac{\\partial}{\\partial y} (\\psi^* + \\sin{(y)} - \\cos{(y)})}{\\psi^*}", "derivation": "k{(y)} = \\int \\cos{(y)} dy and k{(y)} = \\psi^* + \\sin{(y)} and k{(y)} - \\cos{(y)} = - \\cos{(y)} + \\int \\cos{(y)} dy and \\psi^* + \\sin{(y)} = \\int \\cos{(y)} dy and \\frac{d}{d y} (k{(y)} - \\cos{(y)}) = \\frac{d}{d y} (- \\cos{(y)} + \\int \\cos{(y)} dy) and \\frac{\\frac{d}{d y} (k{(y)} - \\cos{(y)})}{\\psi^*} = \\frac{\\frac{d}{d y} (- \\cos{(y)} + \\int \\cos{(y)} dy)}{\\psi^*} and \\frac{\\frac{d}{d y} (k{(y)} - \\cos{(y)})}{\\psi^*} = \\frac{\\frac{\\partial}{\\partial y} (\\psi^* + \\sin{(y)} - \\cos{(y)})}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('k')(Symbol('y', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('y', commutative=True))))"], [["minus", 1, "cos(Symbol('y', commutative=True))"], "Equality(Add(Function('k')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Function('k')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Derivative(Add(Function('k')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Derivative(Add(Function('k')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then derive \\frac{\\int g{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} = \\frac{a - \\cos{(\\mathbf{v})}}{\\mathbf{v}}, then obtain \\frac{\\int \\sin{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} = \\frac{a - \\cos{(\\mathbf{v})}}{\\mathbf{v}}", "derivation": "g{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\int g{(\\mathbf{v})} d\\mathbf{v} = \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\frac{\\int g{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} = \\frac{\\int \\sin{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} and \\frac{\\int g{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} = \\frac{a - \\cos{(\\mathbf{v})}}{\\mathbf{v}} and \\frac{\\int \\sin{(\\mathbf{v})} d\\mathbf{v}}{\\mathbf{v}} = \\frac{a - \\cos{(\\mathbf{v})}}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Integral(Function('g')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Integral(Function('g')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(r_{0})} = \\cos{(r_{0})}, then obtain 2 \\sigma_{p}{(r_{0})} \\cos{(r_{0})} - \\cos^{2}{(r_{0})} = \\sigma_{p}{(r_{0})} \\cos{(r_{0})}", "derivation": "\\sigma_{p}{(r_{0})} = \\cos{(r_{0})} and \\sigma_{p}{(r_{0})} \\cos{(r_{0})} = \\cos^{2}{(r_{0})} and \\sigma_{p}{(r_{0})} \\cos{(r_{0})} - \\cos^{2}{(r_{0})} = 0 and 2 \\sigma_{p}{(r_{0})} \\cos{(r_{0})} - \\cos^{2}{(r_{0})} = \\sigma_{p}{(r_{0})} \\cos{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["times", 1, "cos(Symbol('r_0', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Integer(2)))"], [["minus", 2, "Pow(cos(Symbol('r_0', commutative=True)), Integer(2))"], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('r_0', commutative=True)), Integer(2)))), Integer(0))"], [["add", 3, "Mul(Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('r_0', commutative=True)), Integer(2)))), Mul(Function('\\\\sigma_p')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(i,t,\\psi^*)} = \\frac{i}{\\psi^* t}, then obtain (2 - t)^{\\psi^*} = (- t + \\frac{2 i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}})^{\\psi^*}", "derivation": "\\psi^{*}{(i,t,\\psi^*)} = \\frac{i}{\\psi^* t} and 1 = \\frac{i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}} and 2 = 1 + \\frac{i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}} and 2 - t = - t + 1 + \\frac{i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}} and (2 - t)^{\\psi^*} = (- t + 1 + \\frac{i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}})^{\\psi^*} and (- t + 1 + \\frac{i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}})^{\\psi^*} = (- t + \\frac{2 i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}})^{\\psi^*} and (2 - t)^{\\psi^*} = (- t + \\frac{2 i}{\\psi^* t \\psi^{*}{(i,t,\\psi^*)}})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["divide", 1, "Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))))"], [["minus", 3, "Symbol('t', commutative=True)"], "Equality(Add(Integer(2), Mul(Integer(-1), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))))"], [["power", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Integer(2), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Integer(2), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('i', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('i', commutative=True), Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}}^{a})} and z{(a,V_{\\mathbf{B}})} = \\frac{a^{2} \\log{(V_{\\mathbf{B}})}}{2}, then derive \\int \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} da = S + \\frac{a^{2} \\log{(V_{\\mathbf{B}})}}{2}, then obtain \\int \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} da = S + z{(a,V_{\\mathbf{B}})}", "derivation": "\\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}}^{a})} and \\int \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} da = \\int \\log{(V_{\\mathbf{B}}^{a})} da and \\int \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} da = S + \\frac{a^{2} \\log{(V_{\\mathbf{B}})}}{2} and z{(a,V_{\\mathbf{B}})} = \\frac{a^{2} \\log{(V_{\\mathbf{B}})}}{2} and \\int \\operatorname{z^{*}}{(a,V_{\\mathbf{B}})} da = S + z{(a,V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('S', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('z^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('S', commutative=True), Function('z')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given n{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\pi{(f,\\mathbf{F})} = \\mathbf{F} + \\log{(f)}, then obtain (\\int (\\pi{(f,\\mathbf{F})} + n{(\\mathbf{H})}) df)^{\\mathbf{F}} = (\\int (\\mathbf{F} + n{(\\mathbf{H})} + \\log{(f)}) df)^{\\mathbf{F}}", "derivation": "n{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\pi{(f,\\mathbf{F})} = \\mathbf{F} + \\log{(f)} and \\pi{(f,\\mathbf{F})} + \\sin{(\\mathbf{H})} = \\mathbf{F} + \\log{(f)} + \\sin{(\\mathbf{H})} and \\int (\\pi{(f,\\mathbf{F})} + \\sin{(\\mathbf{H})}) df = \\int (\\mathbf{F} + \\log{(f)} + \\sin{(\\mathbf{H})}) df and \\int (\\pi{(f,\\mathbf{F})} + n{(\\mathbf{H})}) df = \\int (\\mathbf{F} + n{(\\mathbf{H})} + \\log{(f)}) df and (\\int (\\pi{(f,\\mathbf{F})} + n{(\\mathbf{H})}) df)^{\\mathbf{F}} = (\\int (\\mathbf{F} + n{(\\mathbf{H})} + \\log{(f)}) df)^{\\mathbf{F}}", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\pi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('f', commutative=True))))"], [["add", 2, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('f', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Function('\\\\pi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('f', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Function('\\\\pi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('n')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\pi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('n')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{v})} = e^{e^{\\mathbf{v}}}, then derive \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})} = e^{\\mathbf{v}} e^{e^{\\mathbf{v}}}, then obtain 2 \\mathbf{P}{(\\mathbf{v})} e^{\\mathbf{v}} = \\mathbf{P}{(\\mathbf{v})} e^{\\mathbf{v}} + \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})}", "derivation": "\\mathbf{P}{(\\mathbf{v})} = e^{e^{\\mathbf{v}}} and \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} e^{e^{\\mathbf{v}}} and \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})} = e^{\\mathbf{v}} e^{e^{\\mathbf{v}}} and e^{\\mathbf{v}} e^{e^{\\mathbf{v}}} = \\frac{d}{d \\mathbf{v}} e^{e^{\\mathbf{v}}} and \\mathbf{P}{(\\mathbf{v})} e^{\\mathbf{v}} = \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})} and 2 \\mathbf{P}{(\\mathbf{v})} e^{\\mathbf{v}} = \\mathbf{P}{(\\mathbf{v})} e^{\\mathbf{v}} + \\frac{d}{d \\mathbf{v}} \\mathbf{P}{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\mathbf{v}', commutative=True)), exp(exp(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(exp(Symbol('\\\\mathbf{v}', commutative=True)), exp(exp(Symbol('\\\\mathbf{v}', commutative=True)))), Derivative(exp(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True))), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["add", 5, "Mul(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True))), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(A,g^{\\prime}_{\\varepsilon})} = (g^{\\prime}_{\\varepsilon})^{A}, then obtain - ((g^{\\prime}_{\\varepsilon})^{A})^{A} + (e^{Q^{A}{(A,g^{\\prime}_{\\varepsilon})}})^{A} = - ((g^{\\prime}_{\\varepsilon})^{A})^{A} + (e^{((g^{\\prime}_{\\varepsilon})^{A})^{A}})^{A}", "derivation": "Q{(A,g^{\\prime}_{\\varepsilon})} = (g^{\\prime}_{\\varepsilon})^{A} and Q^{A}{(A,g^{\\prime}_{\\varepsilon})} = ((g^{\\prime}_{\\varepsilon})^{A})^{A} and e^{Q^{A}{(A,g^{\\prime}_{\\varepsilon})}} = e^{((g^{\\prime}_{\\varepsilon})^{A})^{A}} and (e^{Q^{A}{(A,g^{\\prime}_{\\varepsilon})}})^{A} = (e^{((g^{\\prime}_{\\varepsilon})^{A})^{A}})^{A} and - ((g^{\\prime}_{\\varepsilon})^{A})^{A} + (e^{Q^{A}{(A,g^{\\prime}_{\\varepsilon})}})^{A} = - ((g^{\\prime}_{\\varepsilon})^{A})^{A} + (e^{((g^{\\prime}_{\\varepsilon})^{A})^{A}})^{A}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('A', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('A', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('Q')(Symbol('A', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True))), exp(Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(exp(Pow(Function('Q')(Symbol('A', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(exp(Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["minus", 4, "Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Pow(exp(Pow(Function('Q')(Symbol('A', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True))), Symbol('A', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Pow(exp(Pow(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Symbol('A', commutative=True))))"]]}, {"prompt": "Given Z{(\\mathbf{P},\\sigma_p)} = \\cos{(\\mathbf{P} - \\sigma_p)}, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} Z{(\\mathbf{P},\\sigma_p)} + \\sigma_p) = \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\cos{(\\mathbf{P} - \\sigma_p)} + \\sigma_p)", "derivation": "Z{(\\mathbf{P},\\sigma_p)} = \\cos{(\\mathbf{P} - \\sigma_p)} and \\mathbf{P} Z{(\\mathbf{P},\\sigma_p)} = \\mathbf{P} \\cos{(\\mathbf{P} - \\sigma_p)} and \\mathbf{P} Z{(\\mathbf{P},\\sigma_p)} + \\sigma_p = \\mathbf{P} \\cos{(\\mathbf{P} - \\sigma_p)} + \\sigma_p and \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} Z{(\\mathbf{P},\\sigma_p)} + \\sigma_p) = \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\cos{(\\mathbf{P} - \\sigma_p)} + \\sigma_p)", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))))"], [["times", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))))"], [["add", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\varepsilon)} = e^{\\varepsilon}, then obtain \\hat{p}_0{(\\varepsilon)} \\frac{d}{d \\varepsilon} \\hat{p}_0{(\\varepsilon)} = e^{\\varepsilon} \\frac{d}{d \\varepsilon} \\hat{p}_0{(\\varepsilon)}", "derivation": "\\hat{p}_0{(\\varepsilon)} = e^{\\varepsilon} and \\frac{d}{d \\varepsilon} \\hat{p}_0{(\\varepsilon)} = \\frac{d}{d \\varepsilon} e^{\\varepsilon} and \\hat{p}_0{(\\varepsilon)} \\frac{d}{d \\varepsilon} e^{\\varepsilon} = e^{\\varepsilon} \\frac{d}{d \\varepsilon} e^{\\varepsilon} and \\hat{p}_0{(\\varepsilon)} \\frac{d}{d \\varepsilon} \\hat{p}_0{(\\varepsilon)} = e^{\\varepsilon} \\frac{d}{d \\varepsilon} \\hat{p}_0{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["times", 1, "Derivative(exp(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), Derivative(exp(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\varepsilon', commutative=True)), Derivative(exp(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(\\eta,P_{e})} = P_{e} + \\eta, then obtain \\eta + (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int (- \\eta + g{(\\eta,P_{e})}) d\\eta - \\int P_{e} d\\eta = \\eta + (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int P_{e} d\\eta - \\int P_{e} d\\eta", "derivation": "g{(\\eta,P_{e})} = P_{e} + \\eta and - \\eta + g{(\\eta,P_{e})} = P_{e} and \\int (- \\eta + g{(\\eta,P_{e})}) d\\eta = \\int P_{e} d\\eta and (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int (- \\eta + g{(\\eta,P_{e})}) d\\eta = (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int P_{e} d\\eta and \\eta + (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int (- \\eta + g{(\\eta,P_{e})}) d\\eta - \\int P_{e} d\\eta = \\eta + (P_{e} - \\eta + g{(\\eta,P_{e})}) \\int P_{e} d\\eta - \\int P_{e} d\\eta", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))"], [["times", 3, "Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True)))"], "Equality(Mul(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))"], "Equality(Add(Symbol('\\\\eta', commutative=True), Mul(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))), Add(Symbol('\\\\eta', commutative=True), Mul(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\eta', commutative=True), Symbol('P_e', commutative=True))), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Integral(Symbol('P_e', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mu)} = e^{\\sin{(\\mu)}}, then obtain \\Psi_{\\lambda}{(\\mu)} + e^{\\Psi_{\\lambda}{(\\mu)}} - e^{\\sin{(\\mu)}} = e^{\\Psi_{\\lambda}{(\\mu)}}", "derivation": "\\Psi_{\\lambda}{(\\mu)} = e^{\\sin{(\\mu)}} and e^{\\Psi_{\\lambda}{(\\mu)}} = e^{e^{\\sin{(\\mu)}}} and \\Psi_{\\lambda}{(\\mu)} - e^{\\sin{(\\mu)}} = 0 and \\Psi_{\\lambda}{(\\mu)} + e^{e^{\\sin{(\\mu)}}} - e^{\\sin{(\\mu)}} = e^{e^{\\sin{(\\mu)}}} and \\Psi_{\\lambda}{(\\mu)} + e^{\\Psi_{\\lambda}{(\\mu)}} - e^{\\sin{(\\mu)}} = e^{\\Psi_{\\lambda}{(\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True)), exp(sin(Symbol('\\\\mu', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True))), exp(exp(sin(Symbol('\\\\mu', commutative=True)))))"], [["minus", 1, "exp(sin(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('\\\\mu', commutative=True))))), Integer(0))"], [["add", 3, "exp(exp(sin(Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True)), exp(exp(sin(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), exp(sin(Symbol('\\\\mu', commutative=True))))), exp(exp(sin(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True)), exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('\\\\mu', commutative=True))))), exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\theta,T)} = \\log{(\\theta)}^{T}, then obtain \\cos{(\\int \\frac{\\operatorname{F_{H}}{(\\theta,T)}}{T} d\\theta)} = \\cos{(\\int \\frac{\\log{(\\theta)}^{T}}{T} d\\theta)}", "derivation": "\\operatorname{F_{H}}{(\\theta,T)} = \\log{(\\theta)}^{T} and \\frac{\\operatorname{F_{H}}{(\\theta,T)}}{T} = \\frac{\\log{(\\theta)}^{T}}{T} and \\int \\frac{\\operatorname{F_{H}}{(\\theta,T)}}{T} d\\theta = \\int \\frac{\\log{(\\theta)}^{T}}{T} d\\theta and \\cos{(\\int \\frac{\\operatorname{F_{H}}{(\\theta,T)}}{T} d\\theta)} = \\cos{(\\int \\frac{\\log{(\\theta)}^{T}}{T} d\\theta)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), cos(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain (\\int 0 d\\rho_f)^{\\rho_f} = (\\int (- 2 \\rho_{b}{(\\rho_f)} + 2 \\sin{(\\rho_f)}) d\\rho_f)^{\\rho_f}", "derivation": "\\rho_{b}{(\\rho_f)} = \\sin{(\\rho_f)} and 0 = - \\rho_{b}{(\\rho_f)} + \\sin{(\\rho_f)} and - \\rho_{b}{(\\rho_f)} = - 2 \\rho_{b}{(\\rho_f)} + \\sin{(\\rho_f)} and 0 = - 2 \\rho_{b}{(\\rho_f)} + 2 \\sin{(\\rho_f)} and \\int 0 d\\rho_f = \\int (- 2 \\rho_{b}{(\\rho_f)} + 2 \\sin{(\\rho_f)}) d\\rho_f and (\\int 0 d\\rho_f)^{\\rho_f} = (\\int (- 2 \\rho_{b}{(\\rho_f)} + 2 \\sin{(\\rho_f)}) d\\rho_f)^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), sin(Symbol('\\\\rho_f', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), sin(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 5, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given C{(\\mathbf{r},s)} = \\log{(s^{\\mathbf{r}})} and \\hat{X}{(\\mathbf{r},s)} = \\log{(s^{\\mathbf{r}})}, then obtain \\cos{(\\hat{X}{(\\mathbf{r},s)})} = \\cos{(C{(\\mathbf{r},s)})}", "derivation": "C{(\\mathbf{r},s)} = \\log{(s^{\\mathbf{r}})} and \\hat{X}{(\\mathbf{r},s)} = \\log{(s^{\\mathbf{r}})} and \\cos{(\\hat{X}{(\\mathbf{r},s)})} = \\cos{(\\log{(s^{\\mathbf{r}})})} and \\cos{(\\hat{X}{(\\mathbf{r},s)})} = \\cos{(C{(\\mathbf{r},s)})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('s', commutative=True)), log(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('s', commutative=True)), log(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["cos", 2], "Equality(cos(Function('\\\\hat{X}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('s', commutative=True))), cos(log(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(cos(Function('\\\\hat{X}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('s', commutative=True))), cos(Function('C')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\delta)} = e^{\\delta} and \\mathbf{r}{(v_{1})} = \\log{(\\sin{(v_{1})})}, then obtain \\frac{\\partial}{\\partial v_{1}} (\\delta + \\mathbf{r}{(v_{1})} + \\sigma_{x}{(\\delta)}) = \\frac{\\partial}{\\partial v_{1}} (\\delta + \\sigma_{x}{(\\delta)} + \\log{(\\sin{(v_{1})})})", "derivation": "\\sigma_{x}{(\\delta)} = e^{\\delta} and \\mathbf{r}{(v_{1})} = \\log{(\\sin{(v_{1})})} and \\delta + \\mathbf{r}{(v_{1})} = \\delta + \\log{(\\sin{(v_{1})})} and \\delta + \\mathbf{r}{(v_{1})} + e^{\\delta} = \\delta + e^{\\delta} + \\log{(\\sin{(v_{1})})} and \\delta + \\mathbf{r}{(v_{1})} + \\sigma_{x}{(\\delta)} = \\delta + \\sigma_{x}{(\\delta)} + \\log{(\\sin{(v_{1})})} and \\frac{\\partial}{\\partial v_{1}} (\\delta + \\mathbf{r}{(v_{1})} + \\sigma_{x}{(\\delta)}) = \\frac{\\partial}{\\partial v_{1}} (\\delta + \\sigma_{x}{(\\delta)} + \\log{(\\sin{(v_{1})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True)), log(sin(Symbol('v_1', commutative=True))))"], [["add", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True))), Add(Symbol('\\\\delta', commutative=True), log(sin(Symbol('v_1', commutative=True)))))"], [["add", 3, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), exp(Symbol('\\\\delta', commutative=True)), log(sin(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True)), log(sin(Symbol('v_1', commutative=True)))))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True)), log(sin(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\theta)} = \\log{(\\theta)}, then derive \\int \\operatorname{F_{x}}{(\\theta)} d\\theta = \\theta \\log{(\\theta)} - \\theta + v, then derive \\theta \\log{(\\theta)} - \\theta + v = \\mathbf{J}_f + \\theta \\log{(\\theta)} - \\theta, then obtain \\mathbf{J}_f + \\theta \\log{(\\theta)} - \\theta = \\int \\log{(\\theta)} d\\theta", "derivation": "\\operatorname{F_{x}}{(\\theta)} = \\log{(\\theta)} and \\int \\operatorname{F_{x}}{(\\theta)} d\\theta = \\int \\log{(\\theta)} d\\theta and \\int \\operatorname{F_{x}}{(\\theta)} d\\theta = \\theta \\log{(\\theta)} - \\theta + v and \\theta \\log{(\\theta)} - \\theta + v = \\int \\log{(\\theta)} d\\theta and \\theta \\log{(\\theta)} - \\theta + v = \\mathbf{J}_f + \\theta \\log{(\\theta)} - \\theta and \\mathbf{J}_f + \\theta \\log{(\\theta)} - \\theta = \\int \\log{(\\theta)} d\\theta", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v', commutative=True)), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\psi^*)} = - \\psi^* and \\operatorname{f_{\\mathbf{p}}}{(\\psi^*)} = (- \\psi^*)^{\\psi^*}, then obtain \\sin^{\\psi^*}{(\\operatorname{f_{\\mathbf{p}}}{(\\psi^*)})} = \\sin^{\\psi^*}{(\\operatorname{f^{\\prime}}^{\\psi^*}{(\\psi^*)})}", "derivation": "\\operatorname{f^{\\prime}}{(\\psi^*)} = - \\psi^* and \\operatorname{f^{\\prime}}^{\\psi^*}{(\\psi^*)} = (- \\psi^*)^{\\psi^*} and \\operatorname{f_{\\mathbf{p}}}{(\\psi^*)} = (- \\psi^*)^{\\psi^*} and \\operatorname{f_{\\mathbf{p}}}{(\\psi^*)} = \\operatorname{f^{\\prime}}^{\\psi^*}{(\\psi^*)} and \\sin{(\\operatorname{f_{\\mathbf{p}}}{(\\psi^*)})} = \\sin{(\\operatorname{f^{\\prime}}^{\\psi^*}{(\\psi^*)})} and \\sin^{\\psi^*}{(\\operatorname{f_{\\mathbf{p}}}{(\\psi^*)})} = \\sin^{\\psi^*}{(\\operatorname{f^{\\prime}}^{\\psi^*}{(\\psi^*)})}", "srepr_derivation": [["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["sin", 4], "Equality(sin(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True))), sin(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["power", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(sin(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\dot{z},E_{\\lambda})} = \\frac{E_{\\lambda}}{\\dot{z}}, then obtain - \\frac{E_{\\lambda}}{\\dot{z} \\rho_f} - \\frac{d}{d \\rho_f} e^{\\rho_f} + \\frac{\\hat{p}{(\\dot{z},E_{\\lambda})}}{\\rho_f} = - \\frac{d}{d \\rho_f} e^{\\rho_f}", "derivation": "\\hat{p}{(\\dot{z},E_{\\lambda})} = \\frac{E_{\\lambda}}{\\dot{z}} and \\frac{\\hat{p}{(\\dot{z},E_{\\lambda})}}{\\rho_f} = \\frac{E_{\\lambda}}{\\dot{z} \\rho_f} and - \\frac{d}{d \\rho_f} e^{\\rho_f} + \\frac{\\hat{p}{(\\dot{z},E_{\\lambda})}}{\\rho_f} = \\frac{E_{\\lambda}}{\\dot{z} \\rho_f} - \\frac{d}{d \\rho_f} e^{\\rho_f} and - \\frac{E_{\\lambda}}{\\dot{z} \\rho_f} - \\frac{d}{d \\rho_f} e^{\\rho_f} + \\frac{\\hat{p}{(\\dot{z},E_{\\lambda})}}{\\rho_f} = - \\frac{d}{d \\rho_f} e^{\\rho_f}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["minus", 2, "Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))))"], [["minus", 3, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} = \\log{(- \\mathbf{v} + \\mu_0)}, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} + \\frac{1}{- \\mathbf{v} + \\mu_0}}{2 \\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)}}", "derivation": "\\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} = \\log{(- \\mathbf{v} + \\mu_0)} and 2 \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} = \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} + \\log{(- \\mathbf{v} + \\mu_0)} and \\frac{\\partial}{\\partial \\mu_0} 2 \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (\\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} + \\log{(- \\mathbf{v} + \\mu_0)}) and 1 = \\frac{\\frac{\\partial}{\\partial \\mu_0} (\\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} + \\log{(- \\mathbf{v} + \\mu_0)})}{\\frac{\\partial}{\\partial \\mu_0} 2 \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)}} and 1 = \\frac{\\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)} + \\frac{1}{- \\mathbf{v} + \\mu_0}}{2 \\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mathbf{v},\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Integer(2), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Mul(Integer(2), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Derivative(Mul(Integer(2), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Mul(Rational(1, 2), Add(Derivative(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Pow(Derivative(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\phi{(\\Psi)} = \\sin{(\\Psi)}, then derive \\int \\phi{(\\Psi)} d\\Psi = x^\\prime - \\cos{(\\Psi)}, then obtain \\frac{\\int \\phi{(\\Psi)} d\\Psi}{x^\\prime - \\cos{(\\Psi)}} - \\frac{1}{x^\\prime - \\cos{(\\Psi)}} = 1 - \\frac{1}{x^\\prime - \\cos{(\\Psi)}}", "derivation": "\\phi{(\\Psi)} = \\sin{(\\Psi)} and \\int \\phi{(\\Psi)} d\\Psi = \\int \\sin{(\\Psi)} d\\Psi and \\int \\phi{(\\Psi)} d\\Psi = x^\\prime - \\cos{(\\Psi)} and x^\\prime - \\cos{(\\Psi)} = \\int \\sin{(\\Psi)} d\\Psi and \\frac{x^\\prime - \\cos{(\\Psi)}}{\\int \\sin{(\\Psi)} d\\Psi} = 1 and \\frac{x^\\prime - \\cos{(\\Psi)}}{\\int \\sin{(\\Psi)} d\\Psi} - \\frac{1}{\\int \\sin{(\\Psi)} d\\Psi} = 1 - \\frac{1}{\\int \\sin{(\\Psi)} d\\Psi} and \\frac{\\int \\phi{(\\Psi)} d\\Psi}{\\int \\sin{(\\Psi)} d\\Psi} - \\frac{1}{\\int \\sin{(\\Psi)} d\\Psi} = 1 - \\frac{1}{\\int \\sin{(\\Psi)} d\\Psi} and \\frac{\\int \\phi{(\\Psi)} d\\Psi}{x^\\prime - \\cos{(\\Psi)}} - \\frac{1}{x^\\prime - \\cos{(\\Psi)}} = 1 - \\frac{1}{x^\\prime - \\cos{(\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 4, "Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 5, "Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Mul(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(-1), Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Integer(-1)))))"]]}, {"prompt": "Given \\rho_{f}{(i)} = \\sin{(i)} and \\varepsilon{(i)} = - \\sin{(i)}, then obtain \\frac{\\varepsilon{(i)} - \\sin{(i)}}{\\sin{(i)}} = -2", "derivation": "\\rho_{f}{(i)} = \\sin{(i)} and \\varepsilon{(i)} = - \\sin{(i)} and \\varepsilon{(i)} = - \\rho_{f}{(i)} and - \\rho_{f}{(i)} = - \\sin{(i)} and - \\rho_{f}{(i)} + \\varepsilon{(i)} = - 2 \\rho_{f}{(i)} and \\varepsilon{(i)} - \\sin{(i)} = - 2 \\sin{(i)} and \\frac{\\varepsilon{(i)} - \\sin{(i)}}{\\rho_{f}{(i)}} = - \\frac{2 \\sin{(i)}}{\\rho_{f}{(i)}} and \\frac{\\varepsilon{(i)} - \\sin{(i)}}{\\sin{(i)}} = -2", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\varepsilon')(Symbol('i', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('i', commutative=True))), Mul(Integer(-1), sin(Symbol('i', commutative=True))))"], [["minus", 3, "Function('\\\\rho_f')(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('i', commutative=True))), Function('\\\\varepsilon')(Symbol('i', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\rho_f')(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Function('\\\\varepsilon')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Mul(Integer(-1), Integer(2), sin(Symbol('i', commutative=True))))"], [["divide", 6, "Function('\\\\rho_f')(Symbol('i', commutative=True))"], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Pow(Function('\\\\rho_f')(Symbol('i', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(Function('\\\\rho_f')(Symbol('i', commutative=True)), Integer(-1)), sin(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Pow(sin(Symbol('i', commutative=True)), Integer(-1))), Integer(-2))"]]}, {"prompt": "Given m{(\\varepsilon,n)} = \\frac{\\varepsilon}{n}, then obtain \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{\\partial}{\\partial \\varepsilon} m{(\\varepsilon,n)} d\\varepsilon = \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{n} d\\varepsilon", "derivation": "m{(\\varepsilon,n)} = \\frac{\\varepsilon}{n} and \\frac{\\partial}{\\partial \\varepsilon} m{(\\varepsilon,n)} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{n} and \\int \\frac{\\partial}{\\partial \\varepsilon} m{(\\varepsilon,n)} d\\varepsilon = \\int \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{n} d\\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{\\partial}{\\partial \\varepsilon} m{(\\varepsilon,n)} d\\varepsilon = \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{n} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(r,\\psi)} = \\log{(\\psi r)}, then derive \\frac{\\partial}{\\partial \\psi} \\hat{\\mathbf{r}}{(r,\\psi)} = \\frac{1}{\\psi}, then obtain - 2 \\psi - 2 \\log{(\\psi r)} + \\frac{\\partial}{\\partial \\psi} \\log{(\\psi r)} + \\frac{1}{\\psi} = - 2 \\psi - 2 \\log{(\\psi r)} + \\frac{2}{\\psi}", "derivation": "\\hat{\\mathbf{r}}{(r,\\psi)} = \\log{(\\psi r)} and \\frac{\\partial}{\\partial \\psi} \\hat{\\mathbf{r}}{(r,\\psi)} = \\frac{\\partial}{\\partial \\psi} \\log{(\\psi r)} and \\frac{\\partial}{\\partial \\psi} \\hat{\\mathbf{r}}{(r,\\psi)} = \\frac{1}{\\psi} and \\frac{\\partial}{\\partial \\psi} \\log{(\\psi r)} = \\frac{1}{\\psi} and - \\psi - \\log{(\\psi r)} + \\frac{\\partial}{\\partial \\psi} \\log{(\\psi r)} = - \\psi - \\log{(\\psi r)} + \\frac{1}{\\psi} and - 2 \\psi - 2 \\log{(\\psi r)} + \\frac{\\partial}{\\partial \\psi} \\log{(\\psi r)} + \\frac{1}{\\psi} = - 2 \\psi - 2 \\log{(\\psi r)} + \\frac{2}{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r', commutative=True), Symbol('\\\\psi', commutative=True)), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))"], [["minus", 4, "Add(Symbol('\\\\psi', commutative=True), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True)))), Derivative(log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True)))), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True)))), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Integer(2), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True)))), Derivative(log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Integer(2), log(Mul(Symbol('\\\\psi', commutative=True), Symbol('r', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{H},B)} = \\frac{\\mathbf{H}}{B}, then derive \\frac{\\partial}{\\partial B} \\operatorname{E_{x}}{(\\mathbf{H},B)} = - \\frac{\\mathbf{H}}{B^{2}}, then obtain \\frac{\\partial}{\\partial B} \\frac{\\mathbf{H}}{B} = \\frac{\\partial}{\\partial B} \\operatorname{E_{x}}{(\\mathbf{H},B)}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{H},B)} = \\frac{\\mathbf{H}}{B} and \\operatorname{E_{x}}{(\\mathbf{H},B)} - 1 = -1 + \\frac{\\mathbf{H}}{B} and \\frac{\\partial}{\\partial B} (\\operatorname{E_{x}}{(\\mathbf{H},B)} - 1) = \\frac{\\partial}{\\partial B} (-1 + \\frac{\\mathbf{H}}{B}) and \\frac{\\partial}{\\partial B} \\operatorname{E_{x}}{(\\mathbf{H},B)} = - \\frac{\\mathbf{H}}{B^{2}} and \\frac{\\partial}{\\partial B} \\operatorname{E_{x}}{(\\mathbf{H},B)} = - \\frac{\\operatorname{E_{x}}{(\\mathbf{H},B)}}{B} and \\frac{\\partial}{\\partial B} \\frac{\\mathbf{H}}{B} = - \\frac{\\mathbf{H}}{B^{2}} and \\frac{\\partial}{\\partial B} \\frac{\\mathbf{H}}{B} = \\frac{\\partial}{\\partial B} \\operatorname{E_{x}}{(\\mathbf{H},B)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)} and \\mathbf{H}{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)}, then obtain \\mathbf{H}{(\\psi)} + h{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)} + h{(\\psi)}", "derivation": "h{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)} and h{(\\psi)} + 1 = \\frac{d}{d \\psi} \\sin{(\\psi)} + 1 and (h{(\\psi)} + 1) h{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)} and \\mathbf{H}{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)} and (h{(\\psi)} + 1) h{(\\psi)} + h{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)} + h{(\\psi)} and \\mathbf{H}{(\\psi)} = (h{(\\psi)} + 1) h{(\\psi)} and \\mathbf{H}{(\\psi)} + h{(\\psi)} = (\\frac{d}{d \\psi} \\sin{(\\psi)} + 1) h{(\\psi)} + h{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\psi', commutative=True)), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('h')(Symbol('\\\\psi', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1)))"], [["times", 2, "Function('h')(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Add(Function('h')(Symbol('\\\\psi', commutative=True)), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))), Mul(Add(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True)), Mul(Add(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))))"], [["add", 3, "Function('h')(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Add(Function('h')(Symbol('\\\\psi', commutative=True)), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))), Function('h')(Symbol('\\\\psi', commutative=True))), Add(Mul(Add(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))), Function('h')(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True)), Mul(Add(Function('h')(Symbol('\\\\psi', commutative=True)), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True)), Function('h')(Symbol('\\\\psi', commutative=True))), Add(Mul(Add(Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1)), Function('h')(Symbol('\\\\psi', commutative=True))), Function('h')(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(c)} = \\cos{(\\log{(c)})} and \\operatorname{v_{y}}{(c)} = \\cos{(\\log{(c)})}, then obtain \\frac{\\operatorname{v_{y}}{(c)} \\cos{(\\log{(c)})}}{\\log{(c)}^{2}} = \\frac{\\operatorname{C_{1}}{(c)} \\operatorname{v_{y}}{(c)}}{\\log{(c)}^{2}}", "derivation": "\\operatorname{C_{1}}{(c)} = \\cos{(\\log{(c)})} and \\operatorname{v_{y}}{(c)} = \\cos{(\\log{(c)})} and \\operatorname{C_{1}}{(c)} = \\operatorname{v_{y}}{(c)} and - \\frac{\\operatorname{C_{1}}{(c)}}{\\log{(c)}} = - \\frac{\\operatorname{v_{y}}{(c)}}{\\log{(c)}} and - \\frac{\\cos{(\\log{(c)})}}{\\log{(c)}} = - \\frac{\\operatorname{v_{y}}{(c)}}{\\log{(c)}} and - \\frac{\\cos{(\\log{(c)})}}{\\log{(c)}} = - \\frac{\\operatorname{C_{1}}{(c)}}{\\log{(c)}} and \\frac{\\operatorname{v_{y}}{(c)} \\cos{(\\log{(c)})}}{\\log{(c)}^{2}} = \\frac{\\operatorname{C_{1}}{(c)} \\operatorname{v_{y}}{(c)}}{\\log{(c)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_1')(Symbol('c', commutative=True)), Function('v_y')(Symbol('c', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), log(Symbol('c', commutative=True)))"], "Equality(Mul(Integer(-1), Function('C_1')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('v_y')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Pow(log(Symbol('c', commutative=True)), Integer(-1)), cos(log(Symbol('c', commutative=True)))), Mul(Integer(-1), Function('v_y')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Pow(log(Symbol('c', commutative=True)), Integer(-1)), cos(log(Symbol('c', commutative=True)))), Mul(Integer(-1), Function('C_1')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1))))"], [["times", 6, "Mul(Integer(-1), Function('v_y')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('v_y')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-2)), cos(log(Symbol('c', commutative=True)))), Mul(Function('C_1')(Symbol('c', commutative=True)), Function('v_y')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\psi{(J,k)} = J k, then obtain \\log{(- \\psi{(J,k)} \\psi^{- J}{(J,k)} + 2 \\psi^{J}{(J,k)} - 1)} = \\log{((J k)^{J} - \\psi{(J,k)} \\psi^{- J}{(J,k)} + \\psi^{J}{(J,k)} - 1)}", "derivation": "\\psi{(J,k)} = J k and \\psi^{J}{(J,k)} = (J k)^{J} and 2 \\psi^{J}{(J,k)} = (J k)^{J} + \\psi^{J}{(J,k)} and 2 \\psi^{J}{(J,k)} - 1 = (J k)^{J} + \\psi^{J}{(J,k)} - 1 and - \\psi{(J,k)} \\psi^{- J}{(J,k)} + 2 \\psi^{J}{(J,k)} - 1 = (J k)^{J} - \\psi{(J,k)} \\psi^{- J}{(J,k)} + \\psi^{J}{(J,k)} - 1 and \\log{(- \\psi{(J,k)} \\psi^{- J}{(J,k)} + 2 \\psi^{J}{(J,k)} - 1)} = \\log{((J k)^{J} - \\psi{(J,k)} \\psi^{- J}{(J,k)} + \\psi^{J}{(J,k)} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Pow(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)))"], [["add", 2, "Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))), Add(Pow(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), Add(Pow(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Integer(-1)))"], [["minus", 4, "Mul(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(2), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), Add(Pow(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Mul(Integer(-1), Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)))), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Integer(-1)))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(2), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True))), Integer(-1))), log(Add(Pow(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Mul(Integer(-1), Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)))), Pow(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Symbol('J', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(c,\\omega)} = \\omega^{c} and T{(c,\\omega)} = (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)}, then obtain ((-1 + (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)}) (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)})^{c} = 0^{c}", "derivation": "\\operatorname{v_{y}}{(c,\\omega)} = \\omega^{c} and T{(c,\\omega)} = (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)} and T{(c,\\omega)} - 1 = -1 + (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)} and T{(c,\\omega)} - 1 = 0 and (T{(c,\\omega)} - 1) T{(c,\\omega)} = 0 and (-1 + (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)}) (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)} = 0 and ((-1 + (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)}) (\\omega^{c})^{- c} \\operatorname{v_{y}}^{c}{(c,\\omega)})^{c} = 0^{c}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Integer(0))"], [["times", 4, "Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Add(Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Function('T')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Integer(-1), Mul(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True)))), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True))), Integer(0))"], [["power", 6, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Add(Integer(-1), Mul(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True)))), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('v_y')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Integer(0), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{X},c_{0})} = \\log{(\\hat{X} c_{0})} and \\bar{\\h}{(c_{0})} = 0^{c_{0}}, then obtain 1 = \\bar{\\h}{(c_{0})}", "derivation": "\\operatorname{z^{*}}{(\\hat{X},c_{0})} = \\log{(\\hat{X} c_{0})} and \\operatorname{z^{*}}{(\\hat{X},c_{0})} - \\log{(\\hat{X} c_{0})} = 0 and (\\operatorname{z^{*}}{(\\hat{X},c_{0})} - \\log{(\\hat{X} c_{0})})^{c_{0}} = 0^{c_{0}} and \\bar{\\h}{(c_{0})} = 0^{c_{0}} and 1 = \\bar{\\h}{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True)), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True))))"], [["minus", 1, "log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True)))"], "Equality(Add(Function('z^*')(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('c_0', commutative=True)"], "Equality(Pow(Add(Function('z^*')(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('c_0', commutative=True))))), Symbol('c_0', commutative=True)), Pow(Integer(0), Symbol('c_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('c_0', commutative=True)), Pow(Integer(0), Symbol('c_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Function('\\\\hbar')(Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\Psi{(a^{\\dagger},\\hbar)} = \\hbar^{a^{\\dagger}}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)} = \\hbar^{a^{\\dagger}} \\log{(\\hbar)}, then obtain (\\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)})^{\\hbar} = (\\Psi{(a^{\\dagger},\\hbar)} \\log{(\\hbar)})^{\\hbar}", "derivation": "\\Psi{(a^{\\dagger},\\hbar)} = \\hbar^{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)} = \\frac{\\partial}{\\partial a^{\\dagger}} \\hbar^{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)} = \\hbar^{a^{\\dagger}} \\log{(\\hbar)} and \\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)} = \\Psi{(a^{\\dagger},\\hbar)} \\log{(\\hbar)} and (\\frac{\\partial}{\\partial a^{\\dagger}} \\Psi{(a^{\\dagger},\\hbar)})^{\\hbar} = (\\Psi{(a^{\\dagger},\\hbar)} \\log{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Function('\\\\Psi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given L{(r_{0})} = e^{r_{0}}, then obtain 0 = \\frac{(- L{(r_{0})} + e^{r_{0}}) e^{- r_{0}}}{2}", "derivation": "L{(r_{0})} = e^{r_{0}} and L{(r_{0})} + e^{r_{0}} = 2 e^{r_{0}} and 0 = - L{(r_{0})} + e^{r_{0}} and 0 = \\frac{- L{(r_{0})} + e^{r_{0}}}{L{(r_{0})} + e^{r_{0}}} and 0 = \\frac{(- L{(r_{0})} + e^{r_{0}}) e^{- r_{0}}}{2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], [["add", 1, "exp(Symbol('r_0', commutative=True))"], "Equality(Add(Function('L')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True))))"], [["minus", 1, "Function('L')(Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), exp(Symbol('r_0', commutative=True))))"], [["divide", 3, "Add(Function('L')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), exp(Symbol('r_0', commutative=True))), Pow(Add(Function('L')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Rational(1, 2), Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), exp(Symbol('r_0', commutative=True))), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\psi^*)} = e^{e^{\\psi^*}}, then obtain (\\psi^*)^{2} \\hat{\\mathbf{r}}^{4}{(\\psi^*)} = (\\psi^*)^{2} \\hat{\\mathbf{r}}^{3}{(\\psi^*)} e^{e^{\\psi^*}}", "derivation": "\\hat{\\mathbf{r}}{(\\psi^*)} = e^{e^{\\psi^*}} and \\psi^* \\hat{\\mathbf{r}}{(\\psi^*)} = \\psi^* e^{e^{\\psi^*}} and (\\psi^*)^{2} \\hat{\\mathbf{r}}^{2}{(\\psi^*)} = (\\psi^*)^{2} \\hat{\\mathbf{r}}{(\\psi^*)} e^{e^{\\psi^*}} and (\\psi^*)^{2} \\hat{\\mathbf{r}}^{4}{(\\psi^*)} = (\\psi^*)^{2} \\hat{\\mathbf{r}}^{3}{(\\psi^*)} e^{e^{\\psi^*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), exp(exp(Symbol('\\\\psi^*', commutative=True))))"], [["times", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), exp(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["times", 2, "Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), exp(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["times", 3, "Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Integer(4))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Integer(3)), exp(exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given q{(a)} = e^{a} and G{(a)} = - e^{a} + \\frac{d}{d a} q{(a)}, then derive - e^{a} + \\frac{d}{d a} q{(a)} = 0, then obtain e^{G{(a)}} = 1", "derivation": "q{(a)} = e^{a} and 2 q{(a)} = q{(a)} + e^{a} and q{(a)} - e^{a} = 0 and \\frac{d}{d a} (q{(a)} - e^{a}) = \\frac{d}{d a} 0 and - e^{a} + \\frac{d}{d a} q{(a)} = 0 and G{(a)} = - e^{a} + \\frac{d}{d a} q{(a)} and G{(a)} = 0 and e^{G{(a)}} = 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["add", 1, "Function('q')(Symbol('a', commutative=True))"], "Equality(Mul(Integer(2), Function('q')(Symbol('a', commutative=True))), Add(Function('q')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))))"], [["minus", 2, "Add(Function('q')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], "Equality(Add(Function('q')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Function('q')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), Derivative(Function('q')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(0))"], ["renaming_premise", "Equality(Function('G')(Symbol('a', commutative=True)), Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), Derivative(Function('q')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('G')(Symbol('a', commutative=True)), Integer(0))"], [["exp", 7], "Equality(exp(Function('G')(Symbol('a', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(I,\\chi)} = \\frac{\\partial}{\\partial I} (I + \\chi) and M{(I,\\chi)} = \\frac{\\partial}{\\partial I} (\\operatorname{t_{1}}{(I,\\chi)} - 1), then derive 0 = - M{(I,\\chi)}, then obtain 1 = \\cos{(\\frac{\\partial}{\\partial I} (\\frac{\\partial}{\\partial I} (I + \\chi) - 1))}", "derivation": "\\operatorname{t_{1}}{(I,\\chi)} = \\frac{\\partial}{\\partial I} (I + \\chi) and M{(I,\\chi)} = \\frac{\\partial}{\\partial I} (\\operatorname{t_{1}}{(I,\\chi)} - 1) and M{(I,\\chi)} = \\frac{\\partial}{\\partial I} (\\frac{\\partial}{\\partial I} (I + \\chi) - 1) and 0 = - M{(I,\\chi)} + \\frac{\\partial}{\\partial I} (\\frac{\\partial}{\\partial I} (I + \\chi) - 1) and 0 = - M{(I,\\chi)} and 0 = - \\frac{\\partial}{\\partial I} (\\frac{\\partial}{\\partial I} (I + \\chi) - 1) and 1 = \\cos{(\\frac{\\partial}{\\partial I} (\\frac{\\partial}{\\partial I} (I + \\chi) - 1))}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Add(Function('t_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('M')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Add(Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["minus", 3, "Function('M')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))), Derivative(Add(Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Integer(-1), Function('M')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Mul(Integer(-1), Derivative(Add(Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["cos", 6], "Equality(Integer(1), cos(Derivative(Add(Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(g)} = g, then obtain \\frac{2^{Z{(g)}} Z^{Z{(g)}}{(g)}}{(2 Z{(g)})^{g} + Z^{Z{(g)}}{(g)}} = \\frac{(g + Z{(g)})^{Z{(g)}}}{(2 Z{(g)})^{g} + Z^{Z{(g)}}{(g)}}", "derivation": "Z{(g)} = g and 2 Z{(g)} = g + Z{(g)} and (2 Z{(g)})^{g} = (g + Z{(g)})^{g} and 2^{g} Z^{g}{(g)} = (g + Z{(g)})^{g} and 2^{g} g^{g} = (2 g)^{g} and 2^{Z{(g)}} Z^{Z{(g)}}{(g)} = (2 Z{(g)})^{Z{(g)}} and 2^{Z{(g)}} Z^{Z{(g)}}{(g)} = (g + Z{(g)})^{Z{(g)}} and \\frac{2^{Z{(g)}} Z^{Z{(g)}}{(g)}}{(2 Z{(g)})^{g} + Z^{Z{(g)}}{(g)}} = \\frac{(g + Z{(g)})^{Z{(g)}}}{(2 Z{(g)})^{g} + Z^{Z{(g)}}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('g', commutative=True)), Symbol('g', commutative=True))"], [["add", 1, "Function('Z')(Symbol('g', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), Function('Z')(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Symbol('g', commutative=True), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["expand", 3], "Equality(Mul(Pow(Integer(2), Symbol('g', commutative=True)), Pow(Function('Z')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Pow(Add(Symbol('g', commutative=True), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Integer(2), Symbol('g', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('g', commutative=True))), Pow(Mul(Integer(2), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Integer(2), Function('Z')(Symbol('g', commutative=True))), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True)))), Pow(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Function('Z')(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Integer(2), Function('Z')(Symbol('g', commutative=True))), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True)))), Pow(Add(Symbol('g', commutative=True), Function('Z')(Symbol('g', commutative=True))), Function('Z')(Symbol('g', commutative=True))))"], [["divide", 7, "Add(Pow(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True))))"], "Equality(Mul(Pow(Integer(2), Function('Z')(Symbol('g', commutative=True))), Pow(Add(Pow(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True)))), Integer(-1)), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True)))), Mul(Pow(Add(Symbol('g', commutative=True), Function('Z')(Symbol('g', commutative=True))), Function('Z')(Symbol('g', commutative=True))), Pow(Add(Pow(Mul(Integer(2), Function('Z')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Function('Z')(Symbol('g', commutative=True)), Function('Z')(Symbol('g', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\hat{X}{(C,\\nabla)} = e^{C + \\nabla}, then obtain (\\nabla + e^{C + \\nabla})^{- \\nabla} \\hat{X}{(C,\\nabla)} = (\\nabla + e^{C + \\nabla})^{- \\nabla} e^{C + \\nabla}", "derivation": "\\hat{X}{(C,\\nabla)} = e^{C + \\nabla} and \\nabla + \\hat{X}{(C,\\nabla)} = \\nabla + e^{C + \\nabla} and (\\nabla + \\hat{X}{(C,\\nabla)})^{- \\nabla} \\hat{X}{(C,\\nabla)} = (\\nabla + \\hat{X}{(C,\\nabla)})^{- \\nabla} e^{C + \\nabla} and (\\nabla + e^{C + \\nabla})^{- \\nabla} \\hat{X}{(C,\\nabla)} = (\\nabla + e^{C + \\nabla})^{- \\nabla} e^{C + \\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["add", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["divide", 1, "Pow(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\nabla', commutative=True), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Add(Symbol('\\\\nabla', commutative=True), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), exp(Add(Symbol('C', commutative=True), Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given c{(J,f^{*})} = J + f^{*}, then derive \\frac{\\partial}{\\partial f^{*}} \\int c{(J,f^{*})} dJ = \\frac{\\partial}{\\partial f^{*}} (\\frac{J^{2}}{2} + J f^{*} + \\psi), then obtain \\frac{\\partial}{\\partial f^{*}} \\int (J + f^{*}) dJ = \\frac{\\partial}{\\partial f^{*}} (\\frac{J^{2}}{2} + J f^{*} + \\psi)", "derivation": "c{(J,f^{*})} = J + f^{*} and \\int c{(J,f^{*})} dJ = \\int (J + f^{*}) dJ and \\frac{\\partial}{\\partial f^{*}} \\int c{(J,f^{*})} dJ = \\frac{\\partial}{\\partial f^{*}} \\int (J + f^{*}) dJ and \\frac{\\partial}{\\partial f^{*}} \\int c{(J,f^{*})} dJ = \\frac{\\partial}{\\partial f^{*}} (\\frac{J^{2}}{2} + J f^{*} + \\psi) and \\frac{\\partial}{\\partial f^{*}} \\int (J + f^{*}) dJ = \\frac{\\partial}{\\partial f^{*}} (\\frac{J^{2}}{2} + J f^{*} + \\psi)", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('J', commutative=True), Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('c')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('c')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integral(Add(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(H,\\varepsilon)} = \\frac{e^{\\varepsilon}}{H} and U{(\\varepsilon)} = \\varepsilon, then obtain (\\int - i^{2}{(H,\\varepsilon)} dU{(\\varepsilon)})^{H} = (\\int - \\frac{i{(H,\\varepsilon)} e^{\\varepsilon}}{H} dU{(\\varepsilon)})^{H}", "derivation": "i{(H,\\varepsilon)} = \\frac{e^{\\varepsilon}}{H} and U{(\\varepsilon)} = \\varepsilon and - i^{2}{(H,\\varepsilon)} = - \\frac{i{(H,\\varepsilon)} e^{\\varepsilon}}{H} and \\int - i^{2}{(H,\\varepsilon)} d\\varepsilon = \\int - \\frac{i{(H,\\varepsilon)} e^{\\varepsilon}}{H} d\\varepsilon and (\\int - i^{2}{(H,\\varepsilon)} d\\varepsilon)^{H} = (\\int - \\frac{i{(H,\\varepsilon)} e^{\\varepsilon}}{H} d\\varepsilon)^{H} and (\\int - i^{2}{(H,\\varepsilon)} dU{(\\varepsilon)})^{H} = (\\int - \\frac{i{(H,\\varepsilon)} e^{\\varepsilon}}{H} dU{(\\varepsilon)})^{H}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["times", 1, "Mul(Integer(-1), Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 4, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Tuple(Function('U')(Symbol('\\\\varepsilon', commutative=True)))), Symbol('H', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('i')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Function('U')(Symbol('\\\\varepsilon', commutative=True)))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(C_{1})} = e^{e^{C_{1}}} and \\operatorname{r_{0}}{(C_{1})} = 2 e^{e^{C_{1}}}, then obtain 2 \\mathbf{M}{(C_{1})} + e^{C_{1}} = \\operatorname{r_{0}}{(C_{1})} + e^{C_{1}}", "derivation": "\\mathbf{M}{(C_{1})} = e^{e^{C_{1}}} and \\mathbf{M}{(C_{1})} + e^{C_{1}} = e^{C_{1}} + e^{e^{C_{1}}} and 2 \\mathbf{M}{(C_{1})} + e^{C_{1}} = \\mathbf{M}{(C_{1})} + e^{C_{1}} + e^{e^{C_{1}}} and 2 \\mathbf{M}{(C_{1})} + e^{C_{1}} = e^{C_{1}} + 2 e^{e^{C_{1}}} and \\operatorname{r_{0}}{(C_{1})} = 2 e^{e^{C_{1}}} and 2 \\mathbf{M}{(C_{1})} + e^{C_{1}} = \\operatorname{r_{0}}{(C_{1})} + e^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True))))"], [["add", 1, "exp(Symbol('C_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))), Add(exp(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))))"], [["add", 2, "Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True))), exp(Symbol('C_1', commutative=True))), Add(Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True))), exp(Symbol('C_1', commutative=True))), Add(exp(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True))))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C_1', commutative=True))), exp(Symbol('C_1', commutative=True))), Add(Function('r_0')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(Z,v_{x})} = Z \\sin{(v_{x})}, then derive \\frac{\\partial}{\\partial v_{x}} \\mathbf{H}{(Z,v_{x})} = Z \\cos{(v_{x})}, then obtain Z \\cos{(v_{x})} + \\int Z \\cos{(v_{x})} dv_{x} = Z \\cos{(v_{x})} + \\int \\frac{\\partial}{\\partial v_{x}} Z \\sin{(v_{x})} dv_{x}", "derivation": "\\mathbf{H}{(Z,v_{x})} = Z \\sin{(v_{x})} and \\frac{\\partial}{\\partial v_{x}} \\mathbf{H}{(Z,v_{x})} = \\frac{\\partial}{\\partial v_{x}} Z \\sin{(v_{x})} and \\frac{\\partial}{\\partial v_{x}} \\mathbf{H}{(Z,v_{x})} = Z \\cos{(v_{x})} and Z \\cos{(v_{x})} = \\frac{\\partial}{\\partial v_{x}} Z \\sin{(v_{x})} and \\int Z \\cos{(v_{x})} dv_{x} = \\int \\frac{\\partial}{\\partial v_{x}} Z \\sin{(v_{x})} dv_{x} and Z \\cos{(v_{x})} + \\int Z \\cos{(v_{x})} dv_{x} = Z \\cos{(v_{x})} + \\int \\frac{\\partial}{\\partial v_{x}} Z \\sin{(v_{x})} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('Z', commutative=True), sin(Symbol('v_x', commutative=True))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))), Derivative(Mul(Symbol('Z', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Derivative(Mul(Symbol('Z', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))))"], [["add", 5, "Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True)))"], "Equality(Add(Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Symbol('Z', commutative=True), cos(Symbol('v_x', commutative=True))), Integral(Derivative(Mul(Symbol('Z', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\omega{(F_{c})} = \\int e^{F_{c}} dF_{c}, then derive \\omega{(F_{c})} = \\sigma_p + e^{F_{c}}, then derive \\omega{(F_{c})} \\int \\omega{(F_{c})} dF_{c} = (v + e^{F_{c}}) \\int \\omega{(F_{c})} dF_{c}, then obtain (\\int e^{F_{c}} dF_{c}) \\iint e^{F_{c}} dF_{c} dF_{c} = (v + e^{F_{c}}) \\iint e^{F_{c}} dF_{c} dF_{c}", "derivation": "\\omega{(F_{c})} = \\int e^{F_{c}} dF_{c} and \\omega{(F_{c})} = \\sigma_p + e^{F_{c}} and \\omega{(F_{c})} \\int \\omega{(F_{c})} dF_{c} = (\\sigma_p + e^{F_{c}}) \\int \\omega{(F_{c})} dF_{c} and \\int e^{F_{c}} dF_{c} = \\sigma_p + e^{F_{c}} and \\omega{(F_{c})} \\int \\omega{(F_{c})} dF_{c} = (\\int \\omega{(F_{c})} dF_{c}) \\int e^{F_{c}} dF_{c} and \\omega{(F_{c})} \\int \\omega{(F_{c})} dF_{c} = (v + e^{F_{c}}) \\int \\omega{(F_{c})} dF_{c} and (\\int e^{F_{c}} dF_{c}) \\iint e^{F_{c}} dF_{c} dF_{c} = (v + e^{F_{c}}) \\iint e^{F_{c}} dF_{c} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('F_c', commutative=True)), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\omega')(Symbol('F_c', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('F_c', commutative=True))))"], [["times", 2, "Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('F_c', commutative=True)), Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('F_c', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('\\\\omega')(Symbol('F_c', commutative=True)), Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Function('\\\\omega')(Symbol('F_c', commutative=True)), Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Add(Symbol('v', commutative=True), exp(Symbol('F_c', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Add(Symbol('v', commutative=True), exp(Symbol('F_c', commutative=True))), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(n_{2})} = \\cos{(n_{2})}, then obtain \\frac{\\Psi^{\\dagger}^{3}{(n_{2})} \\cos{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}} = \\frac{\\Psi^{\\dagger}^{2}{(n_{2})} \\cos^{2}{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}}", "derivation": "\\Psi^{\\dagger}{(n_{2})} = \\cos{(n_{2})} and \\Psi^{\\dagger}{(n_{2})} \\cos{(n_{2})} = \\cos^{2}{(n_{2})} and \\Psi^{\\dagger}^{2}{(n_{2})} \\cos^{2}{(n_{2})} = \\cos^{4}{(n_{2})} and \\frac{\\Psi^{\\dagger}^{2}{(n_{2})} \\cos^{2}{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}} = \\frac{\\cos^{4}{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}} and \\frac{\\Psi^{\\dagger}^{3}{(n_{2})} \\cos{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}} = \\frac{\\Psi^{\\dagger}^{2}{(n_{2})} \\cos^{2}{(n_{2})}}{\\mathbf{S}{(\\sigma_p)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["times", 1, "cos(Symbol('n_2', commutative=True))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))), Pow(cos(Symbol('n_2', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), Integer(2)), Pow(cos(Symbol('n_2', commutative=True)), Integer(2))), Pow(cos(Symbol('n_2', commutative=True)), Integer(4)))"], [["divide", 3, "Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Pow(cos(Symbol('n_2', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Pow(cos(Symbol('n_2', commutative=True)), Integer(4))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), Integer(3)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), cos(Symbol('n_2', commutative=True))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_2', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Pow(cos(Symbol('n_2', commutative=True)), Integer(2))))"]]}, {"prompt": "Given H{(q,\\hat{p}_0)} = \\cos{(\\hat{p}_0 - q)}, then obtain - q + (q + H{(q,\\hat{p}_0)})^{\\hat{p}_0} + 1 = - q + (q + \\cos{(\\hat{p}_0 - q)})^{\\hat{p}_0} + 1", "derivation": "H{(q,\\hat{p}_0)} = \\cos{(\\hat{p}_0 - q)} and q + H{(q,\\hat{p}_0)} = q + \\cos{(\\hat{p}_0 - q)} and (q + H{(q,\\hat{p}_0)})^{\\hat{p}_0} = (q + \\cos{(\\hat{p}_0 - q)})^{\\hat{p}_0} and - q + (q + H{(q,\\hat{p}_0)})^{\\hat{p}_0} = - q + (q + \\cos{(\\hat{p}_0 - q)})^{\\hat{p}_0} and - q + (q + H{(q,\\hat{p}_0)})^{\\hat{p}_0} + 1 = - q + (q + \\cos{(\\hat{p}_0 - q)})^{\\hat{p}_0} + 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), cos(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Add(Symbol('q', commutative=True), Function('H')(Symbol('q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('q', commutative=True), cos(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Add(Symbol('q', commutative=True), Function('H')(Symbol('q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Symbol('q', commutative=True), cos(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 3, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Add(Symbol('q', commutative=True), Function('H')(Symbol('q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Add(Symbol('q', commutative=True), cos(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Add(Symbol('q', commutative=True), Function('H')(Symbol('q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Add(Symbol('q', commutative=True), cos(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\phi_{2}{(g_{\\varepsilon})} = \\cos{(\\log{(g_{\\varepsilon})})}, then obtain - g_{\\varepsilon} + \\frac{d}{d g_{\\varepsilon}} \\phi_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = - g_{\\varepsilon} + \\frac{d}{d g_{\\varepsilon}} \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})}", "derivation": "\\phi_{2}{(g_{\\varepsilon})} = \\cos{(\\log{(g_{\\varepsilon})})} and \\phi_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})} and \\frac{d}{d g_{\\varepsilon}} \\phi_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})} and - g_{\\varepsilon} + \\frac{d}{d g_{\\varepsilon}} \\phi_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = - g_{\\varepsilon} + \\frac{d}{d g_{\\varepsilon}} \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\phi_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Pow(Function('\\\\phi_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\pi)} = \\sin{(e^{\\pi})}, then derive (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi} \\int \\operatorname{v_{z}}{(\\pi)} d\\pi = (C_{d} + \\operatorname{Si}{(e^{\\pi})}) (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi}, then obtain (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi} \\int \\sin{(e^{\\pi})} d\\pi = (C_{d} + \\operatorname{Si}{(e^{\\pi})}) (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi}", "derivation": "\\operatorname{v_{z}}{(\\pi)} = \\sin{(e^{\\pi})} and \\int \\operatorname{v_{z}}{(\\pi)} d\\pi = \\int \\sin{(e^{\\pi})} d\\pi and (\\int \\operatorname{v_{z}}{(\\pi)} d\\pi) (\\int \\sin{(e^{\\pi})} d\\pi)^{\\pi} = (\\int \\sin{(e^{\\pi})} d\\pi) (\\int \\sin{(e^{\\pi})} d\\pi)^{\\pi} and (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi} \\int \\operatorname{v_{z}}{(\\pi)} d\\pi = (C_{d} + \\operatorname{Si}{(e^{\\pi})}) (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi} and (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi} \\int \\sin{(e^{\\pi})} d\\pi = (C_{d} + \\operatorname{Si}{(e^{\\pi})}) (C_{d} + \\operatorname{Si}{(e^{\\pi})})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\pi', commutative=True)), sin(exp(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["times", 2, "Pow(Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integral(Function('v_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Pow(Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Pow(Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Integral(Function('v_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Pow(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Integral(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Pow(Add(Symbol('C_d', commutative=True), Si(exp(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{1},y)} = - v_{1} + y, then obtain \\frac{\\partial}{\\partial y} (v_{1} + (v_{1} + \\operatorname{x^{{\\}'}}{(v_{1},y)})^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)}) = \\frac{\\partial}{\\partial y} (v_{1} + y^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)})", "derivation": "\\operatorname{x^{{\\}'}}{(v_{1},y)} = - v_{1} + y and v_{1} + \\operatorname{x^{{\\}'}}{(v_{1},y)} = y and (v_{1} + \\operatorname{x^{{\\}'}}{(v_{1},y)})^{v_{1}} = y^{v_{1}} and v_{1} + (v_{1} + \\operatorname{x^{{\\}'}}{(v_{1},y)})^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)} = v_{1} + y^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)} and \\frac{\\partial}{\\partial y} (v_{1} + (v_{1} + \\operatorname{x^{{\\}'}}{(v_{1},y)})^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)}) = \\frac{\\partial}{\\partial y} (v_{1} + y^{v_{1}} + \\operatorname{x^{{\\}'}}{(v_{1},y)})", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Symbol('v_1', commutative=True), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Symbol('v_1', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('v_1', commutative=True)))"], [["add", 3, "Add(Symbol('v_1', commutative=True), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Symbol('v_1', commutative=True), Pow(Add(Symbol('v_1', commutative=True), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Symbol('v_1', commutative=True)), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Add(Symbol('v_1', commutative=True), Pow(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Symbol('v_1', commutative=True), Pow(Add(Symbol('v_1', commutative=True), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Symbol('v_1', commutative=True)), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('v_1', commutative=True), Pow(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Function('x^\\\\prime')(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\nabla)} = \\cos{(\\nabla)}, then obtain (\\int \\operatorname{E_{x}}{(\\nabla)} d\\nabla)^{\\nabla} = (W + \\sin{(\\nabla)})^{\\nabla}", "derivation": "\\operatorname{E_{x}}{(\\nabla)} = \\cos{(\\nabla)} and \\int \\operatorname{E_{x}}{(\\nabla)} d\\nabla = \\int \\cos{(\\nabla)} d\\nabla and (\\int \\operatorname{E_{x}}{(\\nabla)} d\\nabla)^{\\nabla} = (\\int \\cos{(\\nabla)} d\\nabla)^{\\nabla} and (\\int \\operatorname{E_{x}}{(\\nabla)} d\\nabla)^{\\nabla} = (W + \\sin{(\\nabla)})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Integral(Function('E_x')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('E_x')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('W', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\eta)} = e^{\\eta}, then obtain 0 = - (\\frac{d^{2}}{d \\eta^{2}} \\mathbf{g}{(\\eta)})^{2} + (\\frac{d^{2}}{d \\eta^{2}} e^{\\eta})^{2}", "derivation": "\\mathbf{g}{(\\eta)} = e^{\\eta} and \\frac{d}{d \\eta} \\mathbf{g}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\frac{d^{2}}{d \\eta^{2}} \\mathbf{g}{(\\eta)} = \\frac{d^{2}}{d \\eta^{2}} e^{\\eta} and (\\frac{d^{2}}{d \\eta^{2}} \\mathbf{g}{(\\eta)})^{2} = (\\frac{d^{2}}{d \\eta^{2}} e^{\\eta})^{2} and 0 = - (\\frac{d^{2}}{d \\eta^{2}} \\mathbf{g}{(\\eta)})^{2} + (\\frac{d^{2}}{d \\eta^{2}} e^{\\eta})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(2)))"], [["minus", 4, "Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(2))), Pow(Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(2))))"]]}, {"prompt": "Given \\hat{H}_l{(t,\\psi^*)} = \\cos{(\\psi^* + t)}, then derive \\frac{\\partial}{\\partial \\psi^*} \\hat{H}_l{(t,\\psi^*)} = - \\sin{(\\psi^* + t)}, then obtain \\frac{\\partial}{\\partial \\psi^*} \\hat{H}_l{(t,\\psi^*)} = \\frac{\\partial}{\\partial \\psi^*} \\cos{(\\psi^* + t)}", "derivation": "\\hat{H}_l{(t,\\psi^*)} = \\cos{(\\psi^* + t)} and t + \\hat{H}_l{(t,\\psi^*)} = t + \\cos{(\\psi^* + t)} and \\frac{\\partial}{\\partial \\psi^*} (t + \\hat{H}_l{(t,\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (t + \\cos{(\\psi^* + t)}) and \\frac{\\partial}{\\partial \\psi^*} \\hat{H}_l{(t,\\psi^*)} = - \\sin{(\\psi^* + t)} and \\frac{\\partial}{\\partial \\psi^*} \\cos{(\\psi^* + t)} = - \\sin{(\\psi^* + t)} and \\frac{\\partial}{\\partial \\psi^*} \\hat{H}_l{(t,\\psi^*)} = \\frac{\\partial}{\\partial \\psi^*} \\cos{(\\psi^* + t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), cos(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True))))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\hat{H}_l')(Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True))), Add(Symbol('t', commutative=True), cos(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Function('\\\\hat{H}_l')(Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('t', commutative=True), cos(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\eta)} = e^{\\eta}, then derive \\int \\operatorname{n_{1}}{(\\eta)} d\\eta = \\tilde{g} + e^{\\eta}, then obtain 1 = \\frac{\\int \\operatorname{n_{1}}{(\\eta)} d\\eta}{\\int e^{\\eta} d\\eta}", "derivation": "\\operatorname{n_{1}}{(\\eta)} = e^{\\eta} and \\int \\operatorname{n_{1}}{(\\eta)} d\\eta = \\int e^{\\eta} d\\eta and \\int \\operatorname{n_{1}}{(\\eta)} d\\eta = \\tilde{g} + e^{\\eta} and \\frac{\\int \\operatorname{n_{1}}{(\\eta)} d\\eta}{\\int e^{\\eta} d\\eta} = \\frac{\\tilde{g} + e^{\\eta}}{\\int e^{\\eta} d\\eta} and 1 = \\frac{\\tilde{g} + e^{\\eta}}{\\int \\operatorname{n_{1}}{(\\eta)} d\\eta} and 1 = \\frac{\\tilde{g} + e^{\\eta}}{\\int e^{\\eta} d\\eta} and 1 = \\frac{\\int \\operatorname{n_{1}}{(\\eta)} d\\eta}{\\int e^{\\eta} d\\eta}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))"], [["divide", 3, "Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integral(Function('n_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Pow(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Pow(Integral(Function('n_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Pow(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), Mul(Integral(Function('n_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} = A (\\mathbf{P} + \\varepsilon), then derive \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = A \\mathbf{P} \\varepsilon + \\frac{A \\varepsilon^{2}}{2} + J_{\\varepsilon}, then derive - A \\mathbf{P} \\varepsilon - \\frac{A \\varepsilon^{2}}{2} - S + \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = 0, then obtain J_{\\varepsilon} - S = 0", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} = A (\\mathbf{P} + \\varepsilon) and \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = \\int A (\\mathbf{P} + \\varepsilon) d\\varepsilon and - \\int A (\\mathbf{P} + \\varepsilon) d\\varepsilon + \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = 0 and \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = A \\mathbf{P} \\varepsilon + \\frac{A \\varepsilon^{2}}{2} + J_{\\varepsilon} and - A \\mathbf{P} \\varepsilon - \\frac{A \\varepsilon^{2}}{2} - S + \\int \\operatorname{L_{\\varepsilon}}{(\\mathbf{P},\\varepsilon,A)} d\\varepsilon = 0 and J_{\\varepsilon} - S = 0", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('A', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('A', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('S', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\theta{(F_{g})} = \\cos{(\\log{(F_{g})})} and S{(\\mathbf{P},M)} = M \\mathbf{P}, then obtain S{(\\mathbf{P},M)} - \\frac{1}{\\theta{(F_{g})}} = M \\mathbf{P} - \\frac{1}{\\theta{(F_{g})}}", "derivation": "\\theta{(F_{g})} = \\cos{(\\log{(F_{g})})} and S{(\\mathbf{P},M)} = M \\mathbf{P} and S{(\\mathbf{P},M)} - \\frac{1}{\\cos{(\\log{(F_{g})})}} = M \\mathbf{P} - \\frac{1}{\\cos{(\\log{(F_{g})})}} and S{(\\mathbf{P},M)} - \\frac{1}{\\theta{(F_{g})}} = M \\mathbf{P} - \\frac{1}{\\theta{(F_{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('F_g', commutative=True)), cos(log(Symbol('F_g', commutative=True))))"], ["get_premise", "Equality(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 2, "Pow(cos(log(Symbol('F_g', commutative=True))), Integer(-1))"], "Equality(Add(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(cos(log(Symbol('F_g', commutative=True))), Integer(-1)))), Add(Mul(Symbol('M', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(cos(log(Symbol('F_g', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('F_g', commutative=True)), Integer(-1)))), Add(Mul(Symbol('M', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('F_g', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\hat{x}{(\\phi)} = \\sin{(\\log{(\\phi)})} and \\delta{(\\phi)} = \\phi (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})}), then obtain \\frac{d}{d \\phi} \\delta{(\\phi)} = \\frac{d}{d \\phi} (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})})", "derivation": "\\hat{x}{(\\phi)} = \\sin{(\\log{(\\phi)})} and 0 = - \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})} and \\frac{d}{d \\phi} 0 = \\frac{d}{d \\phi} (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})}) and 0 = \\phi (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})}) and \\delta{(\\phi)} = \\phi (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})}) and \\frac{d}{d \\phi} 0 = \\frac{d}{d \\phi} \\phi (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})}) and \\frac{d}{d \\phi} 0 = \\frac{d}{d \\phi} \\delta{(\\phi)} and \\frac{d}{d \\phi} \\delta{(\\phi)} = \\frac{d}{d \\phi} (- \\hat{x}{(\\phi)} + \\sin{(\\log{(\\phi)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True)), sin(log(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True))))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 7], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\phi', commutative=True))), sin(log(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(n_{2})} = \\int e^{n_{2}} dn_{2}, then obtain e^{n_{2}} + \\cos{(q{(n_{2})})} = e^{n_{2}} + \\cos{(\\chi + e^{n_{2}})}", "derivation": "q{(n_{2})} = \\int e^{n_{2}} dn_{2} and \\cos{(q{(n_{2})})} = \\cos{(\\int e^{n_{2}} dn_{2})} and e^{n_{2}} + \\cos{(q{(n_{2})})} = e^{n_{2}} + \\cos{(\\int e^{n_{2}} dn_{2})} and e^{n_{2}} + \\cos{(q{(n_{2})})} = e^{n_{2}} + \\cos{(\\chi + e^{n_{2}})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('n_2', commutative=True)), Integral(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["cos", 1], "Equality(cos(Function('q')(Symbol('n_2', commutative=True))), cos(Integral(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["add", 2, "exp(Symbol('n_2', commutative=True))"], "Equality(Add(exp(Symbol('n_2', commutative=True)), cos(Function('q')(Symbol('n_2', commutative=True)))), Add(exp(Symbol('n_2', commutative=True)), cos(Integral(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))))"], [["evaluate_integrals", 3], "Equality(Add(exp(Symbol('n_2', commutative=True)), cos(Function('q')(Symbol('n_2', commutative=True)))), Add(exp(Symbol('n_2', commutative=True)), cos(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{S}{(a,F_{N})} = \\frac{F_{N}}{a}, then obtain \\frac{\\partial}{\\partial a} (F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\mathbf{S}{(a,F_{N})}) = \\frac{\\partial}{\\partial a} (F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{a})", "derivation": "\\mathbf{S}{(a,F_{N})} = \\frac{F_{N}}{a} and - \\mathbf{S}{(a,F_{N})} = - \\frac{F_{N}}{a} and \\frac{\\partial}{\\partial F_{N}} - \\mathbf{S}{(a,F_{N})} = \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{a} and F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\mathbf{S}{(a,F_{N})} = F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{a} and \\frac{\\partial}{\\partial a} (F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\mathbf{S}{(a,F_{N})}) = \\frac{\\partial}{\\partial a} (F_{N} + \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{a})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 3, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Derivative(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Symbol('F_N', commutative=True), Derivative(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Derivative(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Derivative(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(S)} = \\log{(S)}, then derive \\int \\operatorname{x^{{\\}'}}{(S)} dS = B + S \\log{(S)} - S, then obtain \\int \\operatorname{x^{{\\}'}}{(S)} dS = \\frac{\\partial}{\\partial S} \\int (B + S \\log{(S)} - S) dS", "derivation": "\\operatorname{x^{{\\}'}}{(S)} = \\log{(S)} and \\int \\operatorname{x^{{\\}'}}{(S)} dS = \\int \\log{(S)} dS and \\int \\operatorname{x^{{\\}'}}{(S)} dS = B + S \\log{(S)} - S and \\iint \\operatorname{x^{{\\}'}}{(S)} dS dS = \\int (B + S \\log{(S)} - S) dS and \\frac{d}{d S} \\iint \\operatorname{x^{{\\}'}}{(S)} dS dS = \\frac{\\partial}{\\partial S} \\int (B + S \\log{(S)} - S) dS and \\int \\operatorname{x^{{\\}'}}{(S)} dS = \\frac{\\partial}{\\partial S} \\int (B + S \\log{(S)} - S) dS", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x^\\\\prime')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('B', commutative=True), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["differentiate", 4, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Function('x^\\\\prime')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('B', commutative=True), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integral(Function('x^\\\\prime')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Derivative(Integral(Add(Symbol('B', commutative=True), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(u)} = u and \\hat{X}{(\\nabla)} = e^{\\sin{(\\nabla)}}, then obtain ((\\hat{X}{(\\nabla)} - \\int u du)^{u})^{\\nabla} = ((e^{\\sin{(\\nabla)}} - \\int u du)^{u})^{\\nabla}", "derivation": "q{(u)} = u and \\int q{(u)} du = \\int u du and \\int q{(u)} dq{(u)} = \\int u dq{(u)} and \\hat{X}{(\\nabla)} = e^{\\sin{(\\nabla)}} and \\hat{X}{(\\nabla)} - \\int q{(u)} dq{(u)} = e^{\\sin{(\\nabla)}} - \\int q{(u)} dq{(u)} and \\hat{X}{(\\nabla)} - \\int u dq{(u)} = e^{\\sin{(\\nabla)}} - \\int u dq{(u)} and (\\hat{X}{(\\nabla)} - \\int u dq{(u)})^{u} = (e^{\\sin{(\\nabla)}} - \\int u dq{(u)})^{u} and ((\\hat{X}{(\\nabla)} - \\int u dq{(u)})^{u})^{\\nabla} = ((e^{\\sin{(\\nabla)}} - \\int u dq{(u)})^{u})^{\\nabla} and ((\\hat{X}{(\\nabla)} - \\int u du)^{u})^{\\nabla} = ((e^{\\sin{(\\nabla)}} - \\int u du)^{u})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('q')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('q')(Symbol('u', commutative=True)), Tuple(Function('q')(Symbol('u', commutative=True)))), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), exp(sin(Symbol('\\\\nabla', commutative=True))))"], [["minus", 4, "Integral(Function('q')(Symbol('u', commutative=True)), Tuple(Function('q')(Symbol('u', commutative=True))))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Function('q')(Symbol('u', commutative=True)), Tuple(Function('q')(Symbol('u', commutative=True)))))), Add(exp(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integral(Function('q')(Symbol('u', commutative=True)), Tuple(Function('q')(Symbol('u', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))), Add(exp(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))), Symbol('u', commutative=True)), Pow(Add(exp(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))), Symbol('u', commutative=True)))"], [["power", 7, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))), Symbol('u', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Pow(Add(exp(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Function('q')(Symbol('u', commutative=True)))))), Symbol('u', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Pow(Pow(Add(Function('\\\\hat{X}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True))))), Symbol('u', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Pow(Add(exp(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True))))), Symbol('u', commutative=True)), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(f,\\mathbf{B})} = \\frac{\\mathbf{B}}{f}, then obtain \\frac{\\frac{\\partial}{\\partial f} (\\operatorname{r_{0}}^{\\mathbf{B}}{(f,\\mathbf{B})})^{f}}{\\mathbf{B}} = \\frac{\\frac{\\partial}{\\partial f} ((\\frac{\\mathbf{B}}{f})^{\\mathbf{B}})^{f}}{\\mathbf{B}}", "derivation": "\\operatorname{r_{0}}{(f,\\mathbf{B})} = \\frac{\\mathbf{B}}{f} and \\operatorname{r_{0}}^{\\mathbf{B}}{(f,\\mathbf{B})} = (\\frac{\\mathbf{B}}{f})^{\\mathbf{B}} and (\\operatorname{r_{0}}^{\\mathbf{B}}{(f,\\mathbf{B})})^{f} = ((\\frac{\\mathbf{B}}{f})^{\\mathbf{B}})^{f} and \\frac{\\partial}{\\partial f} (\\operatorname{r_{0}}^{\\mathbf{B}}{(f,\\mathbf{B})})^{f} = \\frac{\\partial}{\\partial f} ((\\frac{\\mathbf{B}}{f})^{\\mathbf{B}})^{f} and \\frac{\\frac{\\partial}{\\partial f} (\\operatorname{r_{0}}^{\\mathbf{B}}{(f,\\mathbf{B})})^{f}}{\\mathbf{B}} = \\frac{\\frac{\\partial}{\\partial f} ((\\frac{\\mathbf{B}}{f})^{\\mathbf{B}})^{f}}{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Function('r_0')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('r_0')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["divide", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Derivative(Pow(Pow(Function('r_0')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Derivative(Pow(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(z)} = \\int \\sin{(z)} dz, then derive \\mathbf{B}{(z)} = T - \\cos{(z)}, then obtain \\frac{d}{d z} \\mathbf{B}{(z)} = \\sin{(z)}", "derivation": "\\mathbf{B}{(z)} = \\int \\sin{(z)} dz and \\mathbf{B}{(z)} = T - \\cos{(z)} and \\frac{d}{d z} \\mathbf{B}{(z)} = \\frac{\\partial}{\\partial z} (T - \\cos{(z)}) and \\frac{d}{d z} \\mathbf{B}{(z)} = \\sin{(z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), sin(Symbol('z', commutative=True)))"]]}, {"prompt": "Given s{(\\hbar)} = \\sin{(\\sin{(\\hbar)})}, then obtain 2 s{(\\hbar)} + \\sin{(\\hbar)} = - 2 s{(\\hbar)} + \\sin{(\\hbar)} + 4 \\sin{(\\sin{(\\hbar)})}", "derivation": "s{(\\hbar)} = \\sin{(\\sin{(\\hbar)})} and s{(\\hbar)} + \\sin{(\\hbar)} = \\sin{(\\hbar)} + \\sin{(\\sin{(\\hbar)})} and s{(\\hbar)} + \\sin{(\\hbar)} + \\sin{(\\sin{(\\hbar)})} = \\sin{(\\hbar)} + 2 \\sin{(\\sin{(\\hbar)})} and 2 s{(\\hbar)} + \\sin{(\\hbar)} = \\sin{(\\hbar)} + 2 \\sin{(\\sin{(\\hbar)})} and \\sin{(\\sin{(\\hbar)})} = - s{(\\hbar)} + 2 \\sin{(\\sin{(\\hbar)})} and 2 s{(\\hbar)} + \\sin{(\\hbar)} = - 2 s{(\\hbar)} + \\sin{(\\hbar)} + 4 \\sin{(\\sin{(\\hbar)})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hbar', commutative=True)), sin(sin(Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('s')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Add(sin(Symbol('\\\\hbar', commutative=True)), sin(sin(Symbol('\\\\hbar', commutative=True)))))"], [["add", 2, "sin(sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('s')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)), sin(sin(Symbol('\\\\hbar', commutative=True)))), Add(sin(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), sin(sin(Symbol('\\\\hbar', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('s')(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Add(sin(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), sin(sin(Symbol('\\\\hbar', commutative=True))))))"], [["minus", 3, "Add(Function('s')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(sin(sin(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Function('s')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), sin(sin(Symbol('\\\\hbar', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(2), Function('s')(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('s')(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)), Mul(Integer(4), sin(sin(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(\\phi_2)} = \\phi_2, then obtain \\phi_2 (\\phi_2 + \\mathbf{r}{(\\phi_2)}) - \\phi_2 = 2 \\phi_2^{2} - \\phi_2", "derivation": "\\mathbf{r}{(\\phi_2)} = \\phi_2 and \\phi_2 + \\mathbf{r}{(\\phi_2)} = 2 \\phi_2 and \\phi_2 (\\phi_2 + \\mathbf{r}{(\\phi_2)}) = 2 \\phi_2^{2} and \\phi_2 (\\phi_2 + \\mathbf{r}{(\\phi_2)}) - \\phi_2 = 2 \\phi_2^{2} - \\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["add", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)))"], [["times", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))))"], [["minus", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)}, then obtain \\frac{d}{d \\mathbf{J}_P} \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\eta,U)} = U - \\eta, then obtain - U = - U - \\sin{(\\frac{(U - \\eta) \\frac{\\partial}{\\partial \\eta} \\operatorname{F_{g}}{(\\eta,U)}}{\\operatorname{F_{g}}^{2}{(\\eta,U)}} + \\frac{1}{\\operatorname{F_{g}}{(\\eta,U)}})}", "derivation": "\\operatorname{F_{g}}{(\\eta,U)} = U - \\eta and 1 = \\frac{U - \\eta}{\\operatorname{F_{g}}{(\\eta,U)}} and \\frac{d}{d \\eta} 1 = \\frac{\\partial}{\\partial \\eta} \\frac{U - \\eta}{\\operatorname{F_{g}}{(\\eta,U)}} and \\sin{(\\frac{d}{d \\eta} 1)} = \\sin{(\\frac{\\partial}{\\partial \\eta} \\frac{U - \\eta}{\\operatorname{F_{g}}{(\\eta,U)}})} and - U + \\sin{(\\frac{d}{d \\eta} 1)} = - U + \\sin{(\\frac{\\partial}{\\partial \\eta} \\frac{U - \\eta}{\\operatorname{F_{g}}{(\\eta,U)}})} and - U = - U - \\sin{(\\frac{(U - \\eta) \\frac{\\partial}{\\partial \\eta} \\operatorname{F_{g}}{(\\eta,U)}}{\\operatorname{F_{g}}^{2}{(\\eta,U)}} + \\frac{1}{\\operatorname{F_{g}}{(\\eta,U)}})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Integer(1), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), sin(Derivative(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["minus", 4, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), sin(Derivative(Integer(1), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), sin(Derivative(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Add(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-2)), Derivative(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Pow(Function('F_g')(Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Integer(-1)))))))"]]}, {"prompt": "Given B{(l)} = e^{\\sin{(l)}}, then obtain \\int\\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + B{(l)}} dl dl = \\int\\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + e^{\\sin{(l)}}} dl dl", "derivation": "B{(l)} = e^{\\sin{(l)}} and l + B{(l)} = l + e^{\\sin{(l)}} and e^{l + B{(l)}} = e^{l + e^{\\sin{(l)}}} and l + B{(l)} - e^{\\sin{(l)}} = l and \\int e^{l + B{(l)}} dl = \\int e^{l + e^{\\sin{(l)}}} dl and \\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + B{(l)}} dl = \\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + e^{\\sin{(l)}}} dl and \\int\\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + B{(l)}} dl dl = \\int\\int\\limits^{l + B{(l)} - e^{\\sin{(l)}}} e^{l + e^{\\sin{(l)}}} dl dl", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True))))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), exp(sin(Symbol('l', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)))), exp(Add(Symbol('l', commutative=True), exp(sin(Symbol('l', commutative=True))))))"], [["minus", 2, "exp(sin(Symbol('l', commutative=True)))"], "Equality(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('l', commutative=True))))), Symbol('l', commutative=True))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(exp(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Integral(exp(Add(Symbol('l', commutative=True), exp(sin(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(exp(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('l', commutative=True))))))), Integral(exp(Add(Symbol('l', commutative=True), exp(sin(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('l', commutative=True))))))))"], [["integrate", 6, "Symbol('l', commutative=True)"], "Equality(Integral(exp(Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('l', commutative=True)))))), Tuple(Symbol('l', commutative=True))), Integral(exp(Add(Symbol('l', commutative=True), exp(sin(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('B')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('l', commutative=True)))))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given l{(J,U)} = J U, then obtain \\frac{\\partial}{\\partial U} \\int U l{(J,U)} dU = \\frac{\\partial}{\\partial U} \\int J U^{2} dU", "derivation": "l{(J,U)} = J U and U l{(J,U)} = J U^{2} and \\int U l{(J,U)} dU = \\int J U^{2} dU and \\frac{\\partial}{\\partial U} \\int U l{(J,U)} dU = \\frac{\\partial}{\\partial U} \\int J U^{2} dU", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('J', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('U', commutative=True)))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('l')(Symbol('J', commutative=True), Symbol('U', commutative=True))), Mul(Symbol('J', commutative=True), Pow(Symbol('U', commutative=True), Integer(2))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(Symbol('U', commutative=True), Function('l')(Symbol('J', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Pow(Symbol('U', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('U', commutative=True), Function('l')(Symbol('J', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J', commutative=True), Pow(Symbol('U', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\hbar)} = e^{\\hbar} and \\varepsilon{(\\hbar)} = e^{2 \\hbar} + e^{\\hbar} \\frac{d}{d \\hbar} e^{\\hbar}, then derive \\frac{d}{d \\hbar} b{(\\hbar)} = e^{\\hbar}, then obtain - \\varepsilon{(\\hbar)} \\frac{d}{d \\hbar} e^{\\hbar} = - ((\\frac{d}{d \\hbar} e^{\\hbar})^{2} + \\frac{d}{d \\hbar} e^{\\hbar} \\frac{d^{2}}{d \\hbar^{2}} e^{\\hbar}) \\frac{d}{d \\hbar} e^{\\hbar}", "derivation": "b{(\\hbar)} = e^{\\hbar} and \\frac{d}{d \\hbar} b{(\\hbar)} = \\frac{d}{d \\hbar} e^{\\hbar} and \\frac{d}{d \\hbar} b{(\\hbar)} = e^{\\hbar} and \\frac{d}{d \\hbar} e^{\\hbar} = e^{\\hbar} and \\varepsilon{(\\hbar)} = e^{2 \\hbar} + e^{\\hbar} \\frac{d}{d \\hbar} e^{\\hbar} and \\varepsilon{(\\hbar)} = (\\frac{d}{d \\hbar} e^{\\hbar})^{2} + \\frac{d}{d \\hbar} e^{\\hbar} \\frac{d^{2}}{d \\hbar^{2}} e^{\\hbar} and - \\varepsilon{(\\hbar)} \\frac{d}{d \\hbar} e^{\\hbar} = - ((\\frac{d}{d \\hbar} e^{\\hbar})^{2} + \\frac{d}{d \\hbar} e^{\\hbar} \\frac{d^{2}}{d \\hbar^{2}} e^{\\hbar}) \\frac{d}{d \\hbar} e^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), exp(Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), exp(Symbol('\\\\hbar', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\hbar', commutative=True)), Add(exp(Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Mul(exp(Symbol('\\\\hbar', commutative=True)), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\varepsilon')(Symbol('\\\\hbar', commutative=True)), Add(Pow(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))))"], [["times", 6, "Mul(Integer(-1), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\hbar', commutative=True)), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Pow(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{P},A_{2})} = \\mathbf{P}^{A_{2}}, then obtain \\sin{(\\frac{\\mathbf{P}^{A_{2}} \\sin{(\\mathbf{P}^{A_{2}})}}{\\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})}})} = \\sin{(\\mathbf{P}^{A_{2}})}", "derivation": "\\Psi_{nl}{(\\mathbf{P},A_{2})} = \\mathbf{P}^{A_{2}} and \\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})} = \\sin{(\\mathbf{P}^{A_{2}})} and 1 = \\frac{\\sin{(\\mathbf{P}^{A_{2}})}}{\\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})}} and \\mathbf{P}^{A_{2}} = \\frac{\\mathbf{P}^{A_{2}} \\sin{(\\mathbf{P}^{A_{2}})}}{\\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})}} and \\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})} = \\sin{(\\frac{\\mathbf{P}^{A_{2}} \\sin{(\\mathbf{P}^{A_{2}})}}{\\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})}})} and \\sin{(\\frac{\\mathbf{P}^{A_{2}} \\sin{(\\mathbf{P}^{A_{2}})}}{\\sin{(\\Psi_{nl}{(\\mathbf{P},A_{2})})}})} = \\sin{(\\mathbf{P}^{A_{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))))"], [["divide", 2, "sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)))"], "Equality(Integer(1), Mul(sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Integer(-1))))"], [["times", 3, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), sin(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(sin(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))), Integer(-1)))), sin(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain e^{\\mathbf{S}} \\frac{d}{d \\mathbf{S}} \\mathbf{s}{(\\mathbf{S})} = e^{2 \\mathbf{S}}", "derivation": "\\mathbf{s}{(\\mathbf{S})} = e^{\\mathbf{S}} and \\frac{d}{d \\mathbf{S}} \\mathbf{s}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} e^{\\mathbf{S}} and e^{\\mathbf{S}} \\frac{d}{d \\mathbf{S}} \\mathbf{s}{(\\mathbf{S})} = e^{\\mathbf{S}} \\frac{d}{d \\mathbf{S}} e^{\\mathbf{S}} and e^{\\mathbf{S}} \\frac{d}{d \\mathbf{S}} \\mathbf{s}{(\\mathbf{S})} = e^{2 \\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["times", 2, "exp(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(x)} = e^{x} and \\operatorname{E_{n}}{(x)} = \\frac{d}{d x} (x + e^{x}), then obtain \\frac{d}{d x} (x + \\tilde{g}^*{(x)}) = \\operatorname{E_{n}}{(x)}", "derivation": "\\tilde{g}^*{(x)} = e^{x} and x + \\tilde{g}^*{(x)} = x + e^{x} and \\frac{d}{d x} (x + \\tilde{g}^*{(x)}) = \\frac{d}{d x} (x + e^{x}) and \\operatorname{E_{n}}{(x)} = \\frac{d}{d x} (x + e^{x}) and \\frac{d}{d x} (x + \\tilde{g}^*{(x)}) = \\operatorname{E_{n}}{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True))), Add(Symbol('x', commutative=True), exp(Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Symbol('x', commutative=True), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('x', commutative=True), exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('x', commutative=True)), Derivative(Add(Symbol('x', commutative=True), exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('x', commutative=True), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Function('E_n')(Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(V)} = \\cos{(V)}, then obtain \\frac{d}{d V} \\int (\\mathbf{H}{(V)} + \\cos{(V)}) dV = \\frac{d}{d V} \\int 2 \\cos{(V)} dV", "derivation": "\\mathbf{H}{(V)} = \\cos{(V)} and \\mathbf{H}{(V)} + \\cos{(V)} = 2 \\cos{(V)} and \\int (\\mathbf{H}{(V)} + \\cos{(V)}) dV = \\int 2 \\cos{(V)} dV and \\frac{d}{d V} \\int (\\mathbf{H}{(V)} + \\cos{(V)}) dV = \\frac{d}{d V} \\int 2 \\cos{(V)} dV", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["add", 1, "cos(Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Mul(Integer(2), cos(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(A)} = e^{A}, then obtain \\hat{X}^{2 A}{(A)} (e^{A})^{- 2 A} = 1", "derivation": "\\hat{X}{(A)} = e^{A} and \\hat{X}^{A}{(A)} = (e^{A})^{A} and A \\hat{X}^{A}{(A)} = A (e^{A})^{A} and A \\hat{X}^{A}{(A)} (e^{A})^{A} = A (e^{A})^{2 A} and A \\hat{X}^{2 A}{(A)} = A \\hat{X}^{A}{(A)} (e^{A})^{A} and A \\hat{X}^{2 A}{(A)} = A (e^{A})^{2 A} and \\hat{X}^{2 A}{(A)} (e^{A})^{- 2 A} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["times", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True))))"], [["times", 3, "Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True))"], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Pow(exp(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True)))), Mul(Symbol('A', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True)))), Mul(Symbol('A', commutative=True), Pow(exp(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True)))))"], [["divide", 6, "Mul(Symbol('A', commutative=True), Pow(exp(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Mul(Integer(2), Symbol('A', commutative=True))), Pow(exp(Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(r,u)} = r u and \\operatorname{F_{g}}{(r,u)} = \\int (\\operatorname{r_{0}}^{u}{(r,u)})^{r} dr, then obtain \\int \\operatorname{F_{g}}{(r,u)} du = \\iint ((r u)^{u})^{r} dr du", "derivation": "\\operatorname{r_{0}}{(r,u)} = r u and \\operatorname{r_{0}}^{u}{(r,u)} = (r u)^{u} and (\\operatorname{r_{0}}^{u}{(r,u)})^{r} = ((r u)^{u})^{r} and \\int (\\operatorname{r_{0}}^{u}{(r,u)})^{r} dr = \\int ((r u)^{u})^{r} dr and \\operatorname{F_{g}}{(r,u)} = \\int (\\operatorname{r_{0}}^{u}{(r,u)})^{r} dr and \\operatorname{F_{g}}{(r,u)} = \\int ((r u)^{u})^{r} dr and \\int \\operatorname{F_{g}}{(r,u)} du = \\iint ((r u)^{u})^{r} dr du", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('r_0')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Pow(Function('r_0')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Pow(Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Integral(Pow(Pow(Function('r_0')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('F_g')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Integral(Pow(Pow(Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["integrate", 6, "Symbol('u', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(Pow(Mul(Symbol('r', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{x})} = e^{v_{x}} and \\Psi_{nl}{(v_{x})} = e^{v_{x}}, then obtain \\int \\mathbf{J}_P{(v_{x})} dv_{x} = \\int \\Psi_{nl}{(v_{x})} dv_{x}", "derivation": "\\mathbf{J}_P{(v_{x})} = e^{v_{x}} and \\Psi_{nl}{(v_{x})} = e^{v_{x}} and \\mathbf{J}_P{(v_{x})} = \\Psi_{nl}{(v_{x})} and \\int \\mathbf{J}_P{(v_{x})} dv_{x} = \\int \\Psi_{nl}{(v_{x})} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('v_x', commutative=True)))"], [["integrate", 3, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\hat{p}_0)} = \\log{(\\hat{p}_0)} and \\operatorname{t_{1}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\operatorname{f^{\\prime}}{(\\hat{p}_0)}, then obtain \\operatorname{t_{1}}{(\\hat{p}_0)} e^{- v_{z}} = e^{- v_{z}} \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\log{(\\hat{p}_0)}", "derivation": "\\operatorname{f^{\\prime}}{(\\hat{p}_0)} = \\log{(\\hat{p}_0)} and \\hat{p}_0 \\operatorname{f^{\\prime}}{(\\hat{p}_0)} = \\hat{p}_0 \\log{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\operatorname{f^{\\prime}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\log{(\\hat{p}_0)} and \\operatorname{t_{1}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\operatorname{f^{\\prime}}{(\\hat{p}_0)} and \\operatorname{t_{1}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\log{(\\hat{p}_0)} and \\operatorname{t_{1}}{(\\hat{p}_0)} e^{- v_{z}} = e^{- v_{z}} \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\log{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hat{p}_0', commutative=True)), log(Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["divide", 5, "exp(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Mul(Integer(-1), Symbol('v_z', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given k{(m,\\mathbf{p},\\chi)} = (\\mathbf{p} - m)^{\\chi}, then obtain k{(m,\\mathbf{p},\\chi)} + \\frac{(\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)}}{\\mathbf{p} - m} = (\\mathbf{p} - m)^{\\chi} + \\frac{(\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)}}{\\mathbf{p} - m}", "derivation": "k{(m,\\mathbf{p},\\chi)} = (\\mathbf{p} - m)^{\\chi} and (\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)} = (\\mathbf{p} - m)^{2 \\chi} and \\frac{(\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)}}{\\mathbf{p} - m} = \\frac{(\\mathbf{p} - m)^{2 \\chi}}{\\mathbf{p} - m} and k{(m,\\mathbf{p},\\chi)} + \\frac{(\\mathbf{p} - m)^{2 \\chi}}{\\mathbf{p} - m} = (\\mathbf{p} - m)^{\\chi} + \\frac{(\\mathbf{p} - m)^{2 \\chi}}{\\mathbf{p} - m} and k{(m,\\mathbf{p},\\chi)} + \\frac{(\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)}}{\\mathbf{p} - m} = (\\mathbf{p} - m)^{\\chi} + \\frac{(\\mathbf{p} - m)^{\\chi} k{(m,\\mathbf{p},\\chi)}}{\\mathbf{p} - m}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True)))))"], [["add", 1, "Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True))))"], "Equality(Add(Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True))))), Add(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('k')(Symbol('m', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(E_{n},u)} = E_{n} - u, then obtain ((E_{n} - u)^{E_{n}})^{u} \\operatorname{c_{0}}^{E_{n}}{(E_{n},u)} = (E_{n} - u)^{E_{n}} ((E_{n} - u)^{E_{n}})^{u}", "derivation": "\\operatorname{c_{0}}{(E_{n},u)} = E_{n} - u and \\operatorname{c_{0}}^{E_{n}}{(E_{n},u)} = (E_{n} - u)^{E_{n}} and (\\operatorname{c_{0}}^{E_{n}}{(E_{n},u)})^{u} = ((E_{n} - u)^{E_{n}})^{u} and (\\operatorname{c_{0}}^{E_{n}}{(E_{n},u)})^{u} \\operatorname{c_{0}}^{E_{n}}{(E_{n},u)} = (E_{n} - u)^{E_{n}} (\\operatorname{c_{0}}^{E_{n}}{(E_{n},u)})^{u} and ((E_{n} - u)^{E_{n}})^{u} \\operatorname{c_{0}}^{E_{n}}{(E_{n},u)} = (E_{n} - u)^{E_{n}} ((E_{n} - u)^{E_{n}})^{u}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True)), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)), Symbol('u', commutative=True)))"], [["times", 2, "Pow(Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True)), Symbol('u', commutative=True)), Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True))), Mul(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)), Pow(Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True)), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)), Symbol('u', commutative=True)), Pow(Function('c_0')(Symbol('E_n', commutative=True), Symbol('u', commutative=True)), Symbol('E_n', commutative=True))), Mul(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)), Pow(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('E_n', commutative=True)), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\mathbf{E},y)} = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} - y), then derive \\rho{(\\mathbf{E},y)} - 1 = 0, then obtain \\frac{y^{2}}{4 (\\rho{(\\mathbf{E},y)} - 1)^{2}} = \\frac{y^{2}}{(\\rho{(\\mathbf{E},y)} - 1)^{2}}", "derivation": "\\rho{(\\mathbf{E},y)} = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} - y) and \\rho{(\\mathbf{E},y)} - \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} - y) = 0 and \\rho{(\\mathbf{E},y)} - 1 = 0 and - \\frac{\\rho{(\\mathbf{E},y)} - 1}{y} = 0 and - \\frac{2 (\\rho{(\\mathbf{E},y)} - 1)}{y} = - \\frac{\\rho{(\\mathbf{E},y)} - 1}{y} and \\frac{y^{2}}{4 (\\rho{(\\mathbf{E},y)} - 1)^{2}} = \\frac{y^{2}}{(\\rho{(\\mathbf{E},y)} - 1)^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Integer(0))"], [["divide", 3, "Mul(Integer(-1), Symbol('y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1))), Integer(0))"], [["add", 4, "Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Integer(2), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], [["power", 5, "Integer(-2)"], "Equality(Mul(Rational(1, 4), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Integer(-2))), Mul(Pow(Symbol('y', commutative=True), Integer(2)), Pow(Add(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\pi)} = \\sin{(\\pi)}, then derive \\int \\operatorname{t_{1}}{(\\pi)} d\\pi = \\hbar - \\cos{(\\pi)}, then obtain - \\hbar + (- \\hbar + \\cos{(\\pi)} + \\int \\operatorname{t_{1}}{(\\pi)} d\\pi)^{\\pi} = 0^{\\pi} - \\hbar", "derivation": "\\operatorname{t_{1}}{(\\pi)} = \\sin{(\\pi)} and \\int \\operatorname{t_{1}}{(\\pi)} d\\pi = \\int \\sin{(\\pi)} d\\pi and \\int \\operatorname{t_{1}}{(\\pi)} d\\pi = \\hbar - \\cos{(\\pi)} and \\int \\sin{(\\pi)} d\\pi = \\hbar - \\cos{(\\pi)} and - \\hbar + \\cos{(\\pi)} + \\int \\sin{(\\pi)} d\\pi = 0 and - \\hbar + \\cos{(\\pi)} + \\int \\operatorname{t_{1}}{(\\pi)} d\\pi = 0 and (- \\hbar + \\cos{(\\pi)} + \\int \\operatorname{t_{1}}{(\\pi)} d\\pi)^{\\pi} = 0^{\\pi} and - \\hbar + (- \\hbar + \\cos{(\\pi)} + \\int \\operatorname{t_{1}}{(\\pi)} d\\pi)^{\\pi} = 0^{\\pi} - \\hbar", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"], [["minus", 4, "Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Integral(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Integer(0))"], [["power", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Integral(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Pow(Integer(0), Symbol('\\\\pi', commutative=True)))"], [["add", 7, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Integral(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given p{(v_{2},\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f - v_{2})}, then obtain - (p{(v_{2},\\mathbf{J}_f)} - \\log{(\\mathbf{J}_f - v_{2})}) \\log{(\\mathbf{J}_f - v_{2})}^{2} = 0", "derivation": "p{(v_{2},\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f - v_{2})} and p{(v_{2},\\mathbf{J}_f)} + \\log{(\\mathbf{J}_f - v_{2})} = 2 \\log{(\\mathbf{J}_f - v_{2})} and p{(v_{2},\\mathbf{J}_f)} - \\log{(\\mathbf{J}_f - v_{2})} = 0 and - (p{(v_{2},\\mathbf{J}_f)} - \\log{(\\mathbf{J}_f - v_{2})}) \\log{(\\mathbf{J}_f - v_{2})} = 0 and - (p{(v_{2},\\mathbf{J}_f)} - \\log{(\\mathbf{J}_f - v_{2})}) \\log{(\\mathbf{J}_f - v_{2})}^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["add", 1, "log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], "Equality(Add(Function('p')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Mul(Integer(2), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"], [["minus", 2, "Mul(Integer(2), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], "Equality(Add(Function('p')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))), Integer(0))"], [["times", 3, "Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], "Equality(Mul(Integer(-1), Add(Function('p')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Integer(0))"], [["times", 4, "log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('p')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))), Pow(log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Integer(2))), Integer(0))"]]}, {"prompt": "Given E{(F_{g})} = \\log{(e^{F_{g}})}, then derive \\log{(\\frac{d}{d F_{g}} E{(F_{g})})} + 1 = 1, then obtain \\log{(\\frac{d}{d F_{g}} \\log{(e^{F_{g}})})} + 1 = 1", "derivation": "E{(F_{g})} = \\log{(e^{F_{g}})} and \\frac{d}{d F_{g}} E{(F_{g})} = \\frac{d}{d F_{g}} \\log{(e^{F_{g}})} and \\log{(\\frac{d}{d F_{g}} E{(F_{g})})} = \\log{(\\frac{d}{d F_{g}} \\log{(e^{F_{g}})})} and \\log{(\\frac{d}{d F_{g}} E{(F_{g})})} + 1 = \\log{(\\frac{d}{d F_{g}} \\log{(e^{F_{g}})})} + 1 and \\log{(\\frac{d}{d F_{g}} E{(F_{g})})} + 1 = 1 and \\log{(\\frac{d}{d F_{g}} \\log{(e^{F_{g}})})} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('F_g', commutative=True)), log(exp(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(log(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('E')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), log(Derivative(log(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(log(Derivative(Function('E')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1)), Add(log(Derivative(log(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(log(Derivative(Function('E')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(log(Derivative(log(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1)), Integer(1))"]]}, {"prompt": "Given r{(\\phi)} = e^{\\phi}, then obtain \\frac{d}{d \\phi} (r^{\\phi}{(\\phi)} + (e^{\\phi})^{\\phi}) = \\frac{d}{d \\phi} 2 (e^{\\phi})^{\\phi}", "derivation": "r{(\\phi)} = e^{\\phi} and r^{\\phi}{(\\phi)} = (e^{\\phi})^{\\phi} and r^{\\phi}{(\\phi)} + (e^{\\phi})^{\\phi} = 2 (e^{\\phi})^{\\phi} and \\frac{d}{d \\phi} (r^{\\phi}{(\\phi)} + (e^{\\phi})^{\\phi}) = \\frac{d}{d \\phi} 2 (e^{\\phi})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('r')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["add", 2, "Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Pow(Function('r')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Integer(2), Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Pow(Function('r')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(exp(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\frac{\\mathbf{g}}{\\mathbf{v}}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{v}\\partial \\mathbf{g}} \\mathbf{v} \\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{v}d \\mathbf{g}} \\mathbf{g}", "derivation": "\\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\frac{\\mathbf{g}}{\\mathbf{v}} and \\mathbf{v} \\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\mathbf{g} and \\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{v} \\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\mathbf{g} and \\frac{\\partial^{2}}{\\partial \\mathbf{v}\\partial \\mathbf{g}} \\mathbf{v} \\hat{p}_0{(\\mathbf{v},\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{v}d \\mathbf{g}} \\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(P_{e},M)} = P_{e}^{M} and \\operatorname{A_{x}}{(P_{e},M)} = P_{e}^{M} + \\frac{\\partial}{\\partial P_{e}} \\hat{X}{(P_{e},M)}, then obtain \\hat{X}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} = \\operatorname{A_{x}}{(P_{e},M)}", "derivation": "\\hat{X}{(P_{e},M)} = P_{e}^{M} and \\frac{\\partial}{\\partial P_{e}} \\hat{X}{(P_{e},M)} = \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} and \\hat{X}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} \\hat{X}{(P_{e},M)} = P_{e}^{M} + \\frac{\\partial}{\\partial P_{e}} \\hat{X}{(P_{e},M)} and \\operatorname{A_{x}}{(P_{e},M)} = P_{e}^{M} + \\frac{\\partial}{\\partial P_{e}} \\hat{X}{(P_{e},M)} and \\operatorname{A_{x}}{(P_{e},M)} = P_{e}^{M} + \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} and \\hat{X}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} = P_{e}^{M} + \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} and \\hat{X}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} P_{e}^{M} = \\operatorname{A_{x}}{(P_{e},M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Add(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('A_x')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Add(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Function('A_x')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\pi{(E_{\\lambda})} = \\cos{(e^{E_{\\lambda}})}, then obtain \\tilde{g} + \\int 1 dE_{\\lambda} = \\tilde{g} + \\int \\frac{\\cos{(e^{E_{\\lambda}})}}{\\pi{(E_{\\lambda})}} dE_{\\lambda}", "derivation": "\\pi{(E_{\\lambda})} = \\cos{(e^{E_{\\lambda}})} and 1 = \\frac{\\cos{(e^{E_{\\lambda}})}}{\\pi{(E_{\\lambda})}} and \\int 1 dE_{\\lambda} = \\int \\frac{\\cos{(e^{E_{\\lambda}})}}{\\pi{(E_{\\lambda})}} dE_{\\lambda} and \\tilde{g} + \\int 1 dE_{\\lambda} = \\tilde{g} + \\int \\frac{\\cos{(e^{E_{\\lambda}})}}{\\pi{(E_{\\lambda})}} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)), cos(exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 1, "Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), cos(exp(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["integrate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Mul(Pow(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), cos(exp(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Integral(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Add(Symbol('\\\\tilde{g}', commutative=True), Integral(Mul(Pow(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), cos(exp(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given E{(\\hat{p},\\chi)} = \\chi + \\hat{p}, then obtain \\frac{4 E{(\\hat{p},\\chi)}}{2 \\chi + 2 \\hat{p} + 2 E{(\\hat{p},\\chi)}} = 1", "derivation": "E{(\\hat{p},\\chi)} = \\chi + \\hat{p} and 2 E{(\\hat{p},\\chi)} = \\chi + \\hat{p} + E{(\\hat{p},\\chi)} and \\chi + \\hat{p} + 3 E{(\\hat{p},\\chi)} = 2 \\chi + 2 \\hat{p} + 2 E{(\\hat{p},\\chi)} and 4 E{(\\hat{p},\\chi)} = 2 \\chi + 2 \\hat{p} + 2 E{(\\hat{p},\\chi)} and 4 E{(\\hat{p},\\chi)} = \\chi + \\hat{p} + 3 E{(\\hat{p},\\chi)} and \\frac{\\chi + \\hat{p} + 3 E{(\\hat{p},\\chi)}}{2 \\chi + 2 \\hat{p} + 2 E{(\\hat{p},\\chi)}} = 1 and \\frac{4 E{(\\hat{p},\\chi)}}{2 \\chi + 2 \\hat{p} + 2 E{(\\hat{p},\\chi)}} = 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["add", 1, "Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(3), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(4), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(4), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(3), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(3), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))), Pow(Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)))), Integer(-1)), Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\phi_{2}{(s)} = e^{s}, then obtain \\frac{((\\int \\phi_{2}{(s)} ds)^{s}) (\\int e^{s} ds)^{s}}{U^{2}{(\\mathbb{I},c,\\eta^{\\prime})}} = \\frac{(\\int e^{s} ds)^{2 s}}{U^{2}{(\\mathbb{I},c,\\eta^{\\prime})}}", "derivation": "\\phi_{2}{(s)} = e^{s} and \\int \\phi_{2}{(s)} ds = \\int e^{s} ds and (\\int \\phi_{2}{(s)} ds)^{s} = (\\int e^{s} ds)^{s} and \\frac{(\\int \\phi_{2}{(s)} ds)^{s}}{U{(\\mathbb{I},c,\\eta^{\\prime})}} = \\frac{(\\int e^{s} ds)^{s}}{U{(\\mathbb{I},c,\\eta^{\\prime})}} and \\frac{((\\int \\phi_{2}{(s)} ds)^{s}) (\\int e^{s} ds)^{s}}{U^{2}{(\\mathbb{I},c,\\eta^{\\prime})}} = \\frac{(\\int e^{s} ds)^{2 s}}{U^{2}{(\\mathbb{I},c,\\eta^{\\prime})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Function('\\\\phi_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["divide", 3, "Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Integral(Function('\\\\phi_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))), Mul(Pow(Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["times", 4, "Mul(Pow(Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(Integral(Function('\\\\phi_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))), Mul(Pow(Function('U')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Mul(Integer(2), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given S{(\\pi)} = \\log{(\\pi)}, then obtain 1 = S^{- \\pi}{(\\pi)} \\log{(\\pi)}^{\\pi}", "derivation": "S{(\\pi)} = \\log{(\\pi)} and S^{\\pi}{(\\pi)} = \\log{(\\pi)}^{\\pi} and \\frac{S^{\\pi}{(\\pi)}}{S{(\\pi)} \\log{(\\pi)}} = \\frac{\\log{(\\pi)}^{\\pi}}{S{(\\pi)} \\log{(\\pi)}} and 1 = S^{- \\pi}{(\\pi)} \\log{(\\pi)}^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["divide", 2, "Mul(Function('S')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["divide", 3, "Mul(Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Symbol('\\\\pi', commutative=True)), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Function('S')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(r,E_{n})} = \\log{(- E_{n} + r)} and \\hat{\\mathbf{x}}{(r,E_{n})} = \\frac{\\partial}{\\partial E_{n}} \\operatorname{P_{g}}{(r,E_{n})}, then obtain \\hat{\\mathbf{x}}^{r}{(r,E_{n})} = (\\frac{\\partial}{\\partial E_{n}} \\log{(- E_{n} + r)})^{r}", "derivation": "\\operatorname{P_{g}}{(r,E_{n})} = \\log{(- E_{n} + r)} and \\frac{\\partial}{\\partial E_{n}} \\operatorname{P_{g}}{(r,E_{n})} = \\frac{\\partial}{\\partial E_{n}} \\log{(- E_{n} + r)} and (\\frac{\\partial}{\\partial E_{n}} \\operatorname{P_{g}}{(r,E_{n})})^{r} = (\\frac{\\partial}{\\partial E_{n}} \\log{(- E_{n} + r)})^{r} and \\hat{\\mathbf{x}}{(r,E_{n})} = \\frac{\\partial}{\\partial E_{n}} \\operatorname{P_{g}}{(r,E_{n})} and \\hat{\\mathbf{x}}^{r}{(r,E_{n})} = (\\frac{\\partial}{\\partial E_{n}} \\log{(- E_{n} + r)})^{r}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), log(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('P_g')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Derivative(Function('P_g')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Symbol('r', commutative=True)), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('r', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(G,l)} = G - l, then derive \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)} = -1, then obtain e^{G - l - 1} + e^{G - l + \\frac{\\partial}{\\partial l} (G - l)} = 2 e^{G - l + \\frac{\\partial}{\\partial l} (G - l)}", "derivation": "\\Psi_{nl}{(G,l)} = G - l and \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)} = \\frac{\\partial}{\\partial l} (G - l) and G - l + \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)} = G - l + \\frac{\\partial}{\\partial l} (G - l) and e^{G - l + \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)}} = e^{G - l + \\frac{\\partial}{\\partial l} (G - l)} and e^{G - l + \\frac{\\partial}{\\partial l} (G - l)} + e^{G - l + \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)}} = 2 e^{G - l + \\frac{\\partial}{\\partial l} (G - l)} and \\frac{\\partial}{\\partial l} \\Psi_{nl}{(G,l)} = -1 and e^{G - l - 1} + e^{G - l + \\frac{\\partial}{\\partial l} (G - l)} = 2 e^{G - l + \\frac{\\partial}{\\partial l} (G - l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 2, "Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["exp", 3], "Equality(exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["add", 4, "exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], "Equality(Add(exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))), exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))), Mul(Integer(2), exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Integer(-1))), exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))), Mul(Integer(2), exp(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\mathbf{r}{(C)} = \\log{(C)}, then derive (\\int \\mathbf{r}{(C)} dC)^{C} = (C \\log{(C)} - C + V_{\\mathbf{E}})^{C}, then obtain (C \\log{(C)} - C + \\psi^*)^{C} = (C \\log{(C)} - C + V_{\\mathbf{E}})^{C}", "derivation": "\\mathbf{r}{(C)} = \\log{(C)} and \\int \\mathbf{r}{(C)} dC = \\int \\log{(C)} dC and (\\int \\mathbf{r}{(C)} dC)^{C} = (\\int \\log{(C)} dC)^{C} and (\\int \\mathbf{r}{(C)} dC)^{C} = (C \\log{(C)} - C + V_{\\mathbf{E}})^{C} and (\\int \\log{(C)} dC)^{C} = (C \\log{(C)} - C + V_{\\mathbf{E}})^{C} and (C \\log{(C)} - C + \\psi^*)^{C} = (C \\log{(C)} - C + V_{\\mathbf{E}})^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('C', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Symbol('C', commutative=True)), Pow(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(M)} = \\cos{(M)}, then obtain \\frac{\\int \\operatorname{x^{{\\}'}}{(M)} \\cos{(M)} dM}{\\operatorname{x^{{\\}'}}{(M)} \\cos{(M)}} = \\frac{\\int \\cos^{2}{(M)} dM}{\\operatorname{x^{{\\}'}}{(M)} \\cos{(M)}}", "derivation": "\\operatorname{x^{{\\}'}}{(M)} = \\cos{(M)} and \\operatorname{x^{{\\}'}}{(M)} \\cos{(M)} = \\cos^{2}{(M)} and \\int \\operatorname{x^{{\\}'}}{(M)} \\cos{(M)} dM = \\int \\cos^{2}{(M)} dM and \\frac{\\int \\operatorname{x^{{\\}'}}{(M)} \\cos{(M)} dM}{\\cos^{2}{(M)}} = \\frac{\\int \\cos^{2}{(M)} dM}{\\cos^{2}{(M)}} and \\frac{\\int \\operatorname{x^{{\\}'}}{(M)} \\cos{(M)} dM}{\\operatorname{x^{{\\}'}}{(M)} \\cos{(M)}} = \\frac{\\int \\cos^{2}{(M)} dM}{\\operatorname{x^{{\\}'}}{(M)} \\cos{(M)}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["times", 1, "cos(Symbol('M', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Pow(cos(Symbol('M', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Function('x^\\\\prime')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Pow(cos(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True))))"], [["divide", 3, "Pow(cos(Symbol('M', commutative=True)), Integer(2))"], "Equality(Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-2)), Integral(Mul(Function('x^\\\\prime')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-2)), Integral(Pow(cos(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('M', commutative=True)), Integer(-1)), Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Integral(Mul(Function('x^\\\\prime')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Function('x^\\\\prime')(Symbol('M', commutative=True)), Integer(-1)), Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Integral(Pow(cos(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\phi{(h)} = \\log{(\\cos{(h)})} and \\phi_{1}{(h)} = \\int (- h + \\phi{(h)}) dh, then obtain \\frac{\\int (- h + \\phi{(h)}) dh}{- h + \\log{(\\cos{(h)})}} = \\frac{\\phi_{1}{(h)}}{- h + \\log{(\\cos{(h)})}}", "derivation": "\\phi{(h)} = \\log{(\\cos{(h)})} and \\phi_{1}{(h)} = \\int (- h + \\phi{(h)}) dh and \\phi_{1}{(h)} = \\int (- h + \\log{(\\cos{(h)})}) dh and \\frac{\\phi_{1}{(h)}}{- h + \\log{(\\cos{(h)})}} = \\frac{\\int (- h + \\log{(\\cos{(h)})}) dh}{- h + \\log{(\\cos{(h)})}} and \\frac{\\int (- h + \\phi{(h)}) dh}{- h + \\log{(\\cos{(h)})}} = \\frac{\\int (- h + \\log{(\\cos{(h)})}) dh}{- h + \\log{(\\cos{(h)})}} and \\frac{\\int (- h + \\phi{(h)}) dh}{- h + \\log{(\\cos{(h)})}} = \\frac{\\phi_{1}{(h)}}{- h + \\log{(\\cos{(h)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('h', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\phi')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\phi_1')(Symbol('h', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Function('\\\\phi_1')(Symbol('h', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\phi')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\phi')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(cos(Symbol('h', commutative=True)))), Integer(-1)), Function('\\\\phi_1')(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\hat{x})} = \\log{(e^{\\hat{x}})}, then obtain \\int \\mathbf{f}{(\\hat{x})} d\\hat{x} + \\frac{\\log{(e^{\\hat{x}})}}{\\hat{x}} = \\int \\log{(e^{\\hat{x}})} d\\hat{x} + \\frac{\\log{(e^{\\hat{x}})}}{\\hat{x}}", "derivation": "\\mathbf{f}{(\\hat{x})} = \\log{(e^{\\hat{x}})} and \\int \\mathbf{f}{(\\hat{x})} d\\hat{x} = \\int \\log{(e^{\\hat{x}})} d\\hat{x} and \\frac{\\mathbf{f}{(\\hat{x})}}{\\hat{x}} = \\frac{\\log{(e^{\\hat{x}})}}{\\hat{x}} and \\int \\mathbf{f}{(\\hat{x})} d\\hat{x} + \\frac{\\mathbf{f}{(\\hat{x})}}{\\hat{x}} = \\int \\log{(e^{\\hat{x}})} d\\hat{x} + \\frac{\\mathbf{f}{(\\hat{x})}}{\\hat{x}} and \\int \\mathbf{f}{(\\hat{x})} d\\hat{x} + \\frac{\\log{(e^{\\hat{x}})}}{\\hat{x}} = \\int \\log{(e^{\\hat{x}})} d\\hat{x} + \\frac{\\log{(e^{\\hat{x}})}}{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)), log(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\hat{x}', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)))), Add(Integral(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\hat{x}', commutative=True))))), Add(Integral(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\hat{x}', commutative=True))))))"]]}, {"prompt": "Given n{(\\eta,E,s)} = E s - \\eta, then obtain - (4 E s - 4 \\eta - 2 n{(\\eta,E,s)}) n{(\\eta,E,s)} = - (2 E s - 2 \\eta) n{(\\eta,E,s)}", "derivation": "n{(\\eta,E,s)} = E s - \\eta and - \\eta + n{(\\eta,E,s)} = E s - 2 \\eta and E s - \\eta + n{(\\eta,E,s)} = 2 E s - 2 \\eta and 2 n{(\\eta,E,s)} = 2 E s - 2 \\eta and E s - \\eta = 2 E s - 2 \\eta - n{(\\eta,E,s)} and n{(\\eta,E,s)} = 2 E s - 2 \\eta - n{(\\eta,E,s)} and 4 E s - 4 \\eta - 2 n{(\\eta,E,s)} = 2 E s - 2 \\eta and - (4 E s - 4 \\eta - 2 n{(\\eta,E,s)}) n{(\\eta,E,s)} = - (2 E s - 2 \\eta) n{(\\eta,E,s)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)), Add(Mul(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Mul(Symbol('E', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))))"], [["minus", 3, "Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(4), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Integer(2), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))))"], [["times", 7, "Mul(Integer(-1), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(4), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Integer(2), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True)))), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))), Function('n')(Symbol('\\\\eta', commutative=True), Symbol('E', commutative=True), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\varphi{(Z)} = \\cos{(Z)} and \\phi_{1}{(Z)} = \\frac{d}{d Z} \\varphi{(Z)}, then derive \\frac{d}{d Z} \\varphi{(Z)} = - \\sin{(Z)}, then obtain \\phi_{1}{(Z)} = - \\sin{(Z)}", "derivation": "\\varphi{(Z)} = \\cos{(Z)} and \\frac{d}{d Z} \\varphi{(Z)} = \\frac{d}{d Z} \\cos{(Z)} and \\phi_{1}{(Z)} = \\frac{d}{d Z} \\varphi{(Z)} and \\frac{d}{d Z} \\varphi{(Z)} = - \\sin{(Z)} and \\phi_{1}{(Z)} = - \\sin{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Mul(Integer(-1), sin(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(y,C_{d})} = - C_{d} + y, then obtain (- C_{d} + y - \\operatorname{v_{1}}{(y,C_{d})})^{y} \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{1}}{(y,C_{d})} = 0^{y} \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{1}}{(y,C_{d})}", "derivation": "\\operatorname{v_{1}}{(y,C_{d})} = - C_{d} + y and - \\operatorname{v_{1}}{(y,C_{d})} = C_{d} - y and - C_{d} + y - \\operatorname{v_{1}}{(y,C_{d})} = 0 and (- C_{d} + y - \\operatorname{v_{1}}{(y,C_{d})})^{y} = 0^{y} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{1}}{(y,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + y) and (- C_{d} + y - \\operatorname{v_{1}}{(y,C_{d})})^{y} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + y) = 0^{y} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + y) and (- C_{d} + y - \\operatorname{v_{1}}{(y,C_{d})})^{y} \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{1}}{(y,C_{d})} = 0^{y} \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{1}}{(y,C_{d})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)))), Symbol('y', commutative=True)), Pow(Integer(0), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)))), Symbol('y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Integer(0), Symbol('y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('y', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)))), Symbol('y', commutative=True)), Derivative(Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Integer(0), Symbol('y', commutative=True)), Derivative(Function('v_1')(Symbol('y', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(\\dot{y})} = \\log{(\\cos{(\\dot{y})})}, then obtain \\int (\\frac{d}{d \\dot{y}} Z{(\\dot{y})} \\log{(\\cos{(\\dot{y})})})^{\\dot{y}} d\\dot{y} = \\int (\\frac{d}{d \\dot{y}} \\log{(\\cos{(\\dot{y})})}^{2})^{\\dot{y}} d\\dot{y}", "derivation": "Z{(\\dot{y})} = \\log{(\\cos{(\\dot{y})})} and Z{(\\dot{y})} \\log{(\\cos{(\\dot{y})})} = \\log{(\\cos{(\\dot{y})})}^{2} and \\frac{d}{d \\dot{y}} Z{(\\dot{y})} \\log{(\\cos{(\\dot{y})})} = \\frac{d}{d \\dot{y}} \\log{(\\cos{(\\dot{y})})}^{2} and (\\frac{d}{d \\dot{y}} Z{(\\dot{y})} \\log{(\\cos{(\\dot{y})})})^{\\dot{y}} = (\\frac{d}{d \\dot{y}} \\log{(\\cos{(\\dot{y})})}^{2})^{\\dot{y}} and \\int (\\frac{d}{d \\dot{y}} Z{(\\dot{y})} \\log{(\\cos{(\\dot{y})})})^{\\dot{y}} d\\dot{y} = \\int (\\frac{d}{d \\dot{y}} \\log{(\\cos{(\\dot{y})})}^{2})^{\\dot{y}} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), log(cos(Symbol('\\\\dot{y}', commutative=True))))"], [["times", 1, "log(cos(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), log(cos(Symbol('\\\\dot{y}', commutative=True)))), Pow(log(cos(Symbol('\\\\dot{y}', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Mul(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), log(cos(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(log(cos(Symbol('\\\\dot{y}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), log(cos(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)), Pow(Derivative(Pow(log(cos(Symbol('\\\\dot{y}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Pow(Derivative(Mul(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), log(cos(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Pow(Derivative(Pow(log(cos(Symbol('\\\\dot{y}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given V{(\\mathbf{v},A)} = \\int (A + \\mathbf{v}) d\\mathbf{v}, then derive \\int V^{A}{(\\mathbf{v},A)} dA = n + \\operatorname{NonElementaryIntegral}{(2^{- A} (\\mathbf{v} (2 A + \\mathbf{v}))^{A},( A))}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} \\int V^{A}{(\\mathbf{v},A)} dA = \\frac{\\partial}{\\partial \\mathbf{v}} (n + \\operatorname{NonElementaryIntegral}{(2^{- A} (\\mathbf{v} (2 A + \\mathbf{v}))^{A},( A))})", "derivation": "V{(\\mathbf{v},A)} = \\int (A + \\mathbf{v}) d\\mathbf{v} and V^{A}{(\\mathbf{v},A)} = (\\int (A + \\mathbf{v}) d\\mathbf{v})^{A} and \\int V^{A}{(\\mathbf{v},A)} dA = \\int (\\int (A + \\mathbf{v}) d\\mathbf{v})^{A} dA and \\int V^{A}{(\\mathbf{v},A)} dA = n + \\operatorname{NonElementaryIntegral}{(2^{- A} (\\mathbf{v} (2 A + \\mathbf{v}))^{A},( A))} and \\frac{\\partial}{\\partial \\mathbf{v}} \\int V^{A}{(\\mathbf{v},A)} dA = \\frac{\\partial}{\\partial \\mathbf{v}} (n + \\operatorname{NonElementaryIntegral}{(2^{- A} (\\mathbf{v} (2 A + \\mathbf{v}))^{A},( A))})", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True)), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Pow(Function('V')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Pow(Function('V')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('n', commutative=True), NonElementaryIntegral(Mul(Pow(Integer(2), Mul(Integer(-1), Symbol('A', commutative=True))), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('V')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), NonElementaryIntegral(Mul(Pow(Integer(2), Mul(Integer(-1), Symbol('A', commutative=True))), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(\\mathbf{D},\\theta_1)} = \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\theta_1, then derive \\frac{\\partial}{\\partial \\theta_1} z{(\\mathbf{D},\\theta_1)} = 1, then obtain \\mathbf{D} \\theta_1 + \\frac{\\partial}{\\partial \\theta_1} z{(\\mathbf{D},\\theta_1)} = \\mathbf{D} \\theta_1 + 1", "derivation": "z{(\\mathbf{D},\\theta_1)} = \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\theta_1 and z{(\\mathbf{D},\\theta_1)} + 1 = \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\theta_1 + 1 and \\frac{\\partial}{\\partial \\theta_1} (z{(\\mathbf{D},\\theta_1)} + 1) = \\frac{\\partial}{\\partial \\theta_1} (\\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\theta_1 + 1) and \\frac{\\partial}{\\partial \\theta_1} z{(\\mathbf{D},\\theta_1)} = 1 and \\mathbf{D} \\theta_1 + \\frac{\\partial}{\\partial \\theta_1} z{(\\mathbf{D},\\theta_1)} = \\mathbf{D} \\theta_1 + 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(1)), Add(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(1)))"]]}, {"prompt": "Given M{(\\delta)} = e^{\\delta}, then obtain M{(\\delta)} + \\frac{d}{d \\delta} e^{\\delta} = e^{\\delta} + \\frac{d}{d \\delta} e^{\\delta}", "derivation": "M{(\\delta)} = e^{\\delta} and \\frac{d}{d \\delta} M{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta} and M{(\\delta)} + \\frac{d}{d \\delta} M{(\\delta)} = e^{\\delta} + \\frac{d}{d \\delta} M{(\\delta)} and M{(\\delta)} + \\frac{d}{d \\delta} e^{\\delta} = e^{\\delta} + \\frac{d}{d \\delta} e^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Add(Function('M')(Symbol('\\\\delta', commutative=True)), Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\delta', commutative=True)), Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('M')(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(c_{0})} = e^{c_{0}}, then derive \\int \\operatorname{C_{2}}{(c_{0})} dc_{0} = \\mathbf{P} + e^{c_{0}}, then obtain e^{c_{0}} = \\frac{d}{d c_{0}} \\int e^{c_{0}} dc_{0}", "derivation": "\\operatorname{C_{2}}{(c_{0})} = e^{c_{0}} and \\int \\operatorname{C_{2}}{(c_{0})} dc_{0} = \\int e^{c_{0}} dc_{0} and \\frac{d}{d c_{0}} \\int \\operatorname{C_{2}}{(c_{0})} dc_{0} = \\frac{d}{d c_{0}} \\int e^{c_{0}} dc_{0} and \\int \\operatorname{C_{2}}{(c_{0})} dc_{0} = \\mathbf{P} + e^{c_{0}} and \\frac{\\partial}{\\partial c_{0}} (\\mathbf{P} + e^{c_{0}}) = \\frac{d}{d c_{0}} \\int e^{c_{0}} dc_{0} and e^{c_{0}} = \\frac{d}{d c_{0}} \\int e^{c_{0}} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Integral(Function('C_2')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(exp(Symbol('c_0', commutative=True)), Derivative(Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(H,\\mathbf{D})} = \\frac{H}{\\mathbf{D}}, then derive \\frac{\\partial}{\\partial H} \\rho{(H,\\mathbf{D})} = \\frac{1}{\\mathbf{D}}, then obtain (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{\\partial}{\\partial H} \\frac{H}{\\mathbf{D}})^{H} = (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{1}{\\mathbf{D}})^{H}", "derivation": "\\rho{(H,\\mathbf{D})} = \\frac{H}{\\mathbf{D}} and \\frac{\\partial}{\\partial H} \\rho{(H,\\mathbf{D})} = \\frac{\\partial}{\\partial H} \\frac{H}{\\mathbf{D}} and \\frac{\\partial}{\\partial H} \\rho{(H,\\mathbf{D})} = \\frac{1}{\\mathbf{D}} and (\\frac{\\partial}{\\partial H} \\rho{(H,\\mathbf{D})})^{H} = (\\frac{1}{\\mathbf{D}})^{H} and (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{\\partial}{\\partial H} \\rho{(H,\\mathbf{D})})^{H} = (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{1}{\\mathbf{D}})^{H} and (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{\\partial}{\\partial H} \\frac{H}{\\mathbf{D}})^{H} = (\\frac{H}{\\mathbf{D}} - \\mathbf{D}) (\\frac{1}{\\mathbf{D}})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('H', commutative=True)))"], [["times", 4, "Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Derivative(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{2},\\Psi)} = \\Psi - v_{2}, then obtain (\\Psi - v_{2})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (v_{2} + \\operatorname{n_{1}}^{v_{2}}{(v_{2},\\Psi)}) = (\\Psi - v_{2})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (v_{2} + (\\Psi - v_{2})^{v_{2}})", "derivation": "\\operatorname{n_{1}}{(v_{2},\\Psi)} = \\Psi - v_{2} and \\operatorname{n_{1}}^{v_{2}}{(v_{2},\\Psi)} = (\\Psi - v_{2})^{v_{2}} and v_{2} + \\operatorname{n_{1}}^{v_{2}}{(v_{2},\\Psi)} = v_{2} + (\\Psi - v_{2})^{v_{2}} and \\frac{\\partial}{\\partial v_{2}} (v_{2} + \\operatorname{n_{1}}^{v_{2}}{(v_{2},\\Psi)}) = \\frac{\\partial}{\\partial v_{2}} (v_{2} + (\\Psi - v_{2})^{v_{2}}) and (\\Psi - v_{2})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (v_{2} + \\operatorname{n_{1}}^{v_{2}}{(v_{2},\\Psi)}) = (\\Psi - v_{2})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (v_{2} + (\\Psi - v_{2})^{v_{2}})", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('v_2', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"], [["add", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Symbol('v_2', commutative=True), Pow(Function('n_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('v_2', commutative=True))), Add(Symbol('v_2', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Add(Symbol('v_2', commutative=True), Pow(Function('n_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('v_2', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 4, "Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Derivative(Add(Symbol('v_2', commutative=True), Pow(Function('n_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Derivative(Add(Symbol('v_2', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(n_{1},A_{z})} = - A_{z} + \\cos{(n_{1})}, then obtain 0^{A_{z}} - \\cos{(n_{1})} = 1 - \\cos{(n_{1})}", "derivation": "\\operatorname{A_{y}}{(n_{1},A_{z})} = - A_{z} + \\cos{(n_{1})} and 0 = - A_{z} - \\operatorname{A_{y}}{(n_{1},A_{z})} + \\cos{(n_{1})} and 0^{A_{z}} = (- A_{z} - \\operatorname{A_{y}}{(n_{1},A_{z})} + \\cos{(n_{1})})^{A_{z}} and 0^{A_{z}} - \\cos{(n_{1})} = (- A_{z} - \\operatorname{A_{y}}{(n_{1},A_{z})} + \\cos{(n_{1})})^{A_{z}} - \\cos{(n_{1})} and (- A_{z} - \\operatorname{A_{y}}{(n_{1},A_{z})} + \\cos{(n_{1})})^{A_{z}} - \\cos{(n_{1})} = 1 - \\cos{(n_{1})} and 0^{A_{z}} - \\cos{(n_{1})} = 1 - \\cos{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), cos(Symbol('n_1', commutative=True))))"], [["minus", 1, "Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True))), cos(Symbol('n_1', commutative=True))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True))), cos(Symbol('n_1', commutative=True))), Symbol('A_z', commutative=True)))"], [["minus", 3, "cos(Symbol('n_1', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True))), cos(Symbol('n_1', commutative=True))), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('n_1', commutative=True), Symbol('A_z', commutative=True))), cos(Symbol('n_1', commutative=True))), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Add(Integer(1), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Integer(0), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Add(Integer(1), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} = F_{x} S, then obtain F_{x} S - \\int F_{x} S dF_{x} + \\int \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} dF_{x} = \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} = F_{x} S and \\int \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} dF_{x} = \\int F_{x} S dF_{x} and \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} - \\int F_{x} S dF_{x} = F_{x} S - \\int F_{x} S dF_{x} and \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} - \\int F_{x} S dF_{x} + \\int \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} dF_{x} = \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} and F_{x} S - \\int F_{x} S dF_{x} + \\int \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})} dF_{x} = \\operatorname{f_{\\mathbf{p}}}{(S,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["minus", 1, "Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True))))), Add(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True))))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('F_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Function('f_{\\\\mathbf{p}}')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} = \\cos{(E \\mathbf{J}_P)} and \\Psi{(c,l)} = \\frac{l}{c}, then obtain \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} + 2 \\Psi{(c,l)} + \\cos{(E \\mathbf{J}_P)} = 2 \\Psi{(c,l)} + 2 \\cos{(E \\mathbf{J}_P)}", "derivation": "\\operatorname{E_{x}}{(E,\\mathbf{J}_P)} = \\cos{(E \\mathbf{J}_P)} and \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} + \\cos{(E \\mathbf{J}_P)} = 2 \\cos{(E \\mathbf{J}_P)} and \\Psi{(c,l)} = \\frac{l}{c} and \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} + \\cos{(E \\mathbf{J}_P)} + \\frac{l}{c} = 2 \\cos{(E \\mathbf{J}_P)} + \\frac{l}{c} and \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} + \\cos{(E \\mathbf{J}_P)} + \\frac{2 l}{c} = 2 \\cos{(E \\mathbf{J}_P)} + \\frac{2 l}{c} and \\operatorname{E_{x}}{(E,\\mathbf{J}_P)} + 2 \\Psi{(c,l)} + \\cos{(E \\mathbf{J}_P)} = 2 \\Psi{(c,l)} + 2 \\cos{(E \\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 1, "cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(2), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\Psi')(Symbol('c', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Add(Mul(Integer(2), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))))"], [["add", 4, "Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Add(Mul(Integer(2), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('c', commutative=True), Symbol('l', commutative=True))), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Mul(Integer(2), Function('\\\\Psi')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Mul(Integer(2), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))))"]]}, {"prompt": "Given s{(y,\\mu_0)} = \\int (\\mu_0 + y) dy, then derive s{(y,\\mu_0)} = \\mu_0 y + r + \\frac{y^{2}}{2}, then obtain y + s{(y,\\mu_0)} = \\mu_0 y + r + \\frac{y^{2}}{2} + y", "derivation": "s{(y,\\mu_0)} = \\int (\\mu_0 + y) dy and s{(y,\\mu_0)} = \\mu_0 y + r + \\frac{y^{2}}{2} and \\mu_0 y + r + \\frac{y^{2}}{2} = \\int (\\mu_0 + y) dy and y + s{(y,\\mu_0)} = y + \\int (\\mu_0 + y) dy and y + s{(y,\\mu_0)} = \\mu_0 y + r + \\frac{y^{2}}{2} + y", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('s')(Symbol('y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Symbol('r', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Symbol('r', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('s')(Symbol('y', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Symbol('y', commutative=True), Integral(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('y', commutative=True), Function('s')(Symbol('y', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), Symbol('r', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given q{(A_{2})} = \\sin{(A_{2})} and \\varepsilon{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain \\frac{\\int \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{\\sin{(A_{2})} - 1} = \\frac{\\int \\cos^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{\\sin{(A_{2})} - 1}", "derivation": "q{(A_{2})} = \\sin{(A_{2})} and \\varepsilon{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} = \\cos^{\\mathbf{A}}{(\\mathbf{A})} and \\int \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A} = \\int \\cos^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A} and \\frac{\\int \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{q{(A_{2})} - 1} = \\frac{\\int \\cos^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{q{(A_{2})} - 1} and \\frac{\\int \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{\\sin{(A_{2})} - 1} = \\frac{\\int \\cos^{\\mathbf{A}}{(\\mathbf{A})} d\\mathbf{A}}{\\sin{(A_{2})} - 1}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 4, "Add(Function('q')(Symbol('A_2', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Function('q')(Symbol('A_2', commutative=True)), Integer(-1)), Integer(-1)), Integral(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Pow(Add(Function('q')(Symbol('A_2', commutative=True)), Integer(-1)), Integer(-1)), Integral(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Add(sin(Symbol('A_2', commutative=True)), Integer(-1)), Integer(-1)), Integral(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Pow(Add(sin(Symbol('A_2', commutative=True)), Integer(-1)), Integer(-1)), Integral(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given n{(\\psi)} = e^{\\psi}, then derive \\frac{d}{d \\psi} n{(\\psi)} = e^{\\psi}, then obtain \\frac{d^{2}}{d \\psi^{2}} n{(\\psi)} = e^{\\psi}", "derivation": "n{(\\psi)} = e^{\\psi} and \\frac{d}{d \\psi} n{(\\psi)} = \\frac{d}{d \\psi} e^{\\psi} and \\frac{d}{d \\psi} n{(\\psi)} = e^{\\psi} and e^{\\psi} = \\frac{d}{d \\psi} e^{\\psi} and \\frac{d}{d \\psi} n{(\\psi)} = \\frac{d^{2}}{d \\psi^{2}} n{(\\psi)} and \\frac{d^{2}}{d \\psi^{2}} n{(\\psi)} = \\frac{d}{d \\psi} e^{\\psi} and \\frac{d^{2}}{d \\psi^{2}} n{(\\psi)} = e^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), exp(Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\psi', commutative=True)), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(2))), exp(Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\theta{(F_{x})} = \\log{(F_{x})} and \\operatorname{a^{\\dagger}}{(F_{x})} = F_{x} + \\frac{d}{d F_{x}} \\theta{(F_{x})}, then obtain \\operatorname{a^{\\dagger}}{(F_{x})} = F_{x} + \\frac{d}{d F_{x}} \\log{(F_{x})}", "derivation": "\\theta{(F_{x})} = \\log{(F_{x})} and \\frac{d}{d F_{x}} \\theta{(F_{x})} = \\frac{d}{d F_{x}} \\log{(F_{x})} and F_{x} + \\frac{d}{d F_{x}} \\theta{(F_{x})} = F_{x} + \\frac{d}{d F_{x}} \\log{(F_{x})} and \\operatorname{a^{\\dagger}}{(F_{x})} = F_{x} + \\frac{d}{d F_{x}} \\theta{(F_{x})} and \\operatorname{a^{\\dagger}}{(F_{x})} = F_{x} + \\frac{d}{d F_{x}} \\log{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["add", 2, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Derivative(Function('\\\\theta')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Add(Symbol('F_x', commutative=True), Derivative(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Derivative(Function('\\\\theta')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Derivative(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(J,g^{\\prime}_{\\varepsilon})} = - J + g^{\\prime}_{\\varepsilon} and \\mu{(J,g^{\\prime}_{\\varepsilon})} = \\frac{- 2 J + g^{\\prime}_{\\varepsilon}}{- J + \\hat{\\mathbf{r}}{(J,g^{\\prime}_{\\varepsilon})}}, then obtain \\int \\mu{(J,g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int 1 dg^{\\prime}_{\\varepsilon}", "derivation": "\\hat{\\mathbf{r}}{(J,g^{\\prime}_{\\varepsilon})} = - J + g^{\\prime}_{\\varepsilon} and - J + \\hat{\\mathbf{r}}{(J,g^{\\prime}_{\\varepsilon})} = - 2 J + g^{\\prime}_{\\varepsilon} and \\mu{(J,g^{\\prime}_{\\varepsilon})} = \\frac{- 2 J + g^{\\prime}_{\\varepsilon}}{- J + \\hat{\\mathbf{r}}{(J,g^{\\prime}_{\\varepsilon})}} and \\mu{(J,g^{\\prime}_{\\varepsilon})} = 1 and \\int \\mu{(J,g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int 1 dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\Psi)} = e^{\\Psi} and \\operatorname{C_{d}}{(\\Psi)} = \\frac{\\Psi + e^{\\Psi}}{\\Psi + \\rho{(\\Psi)}}, then obtain 0 = - \\frac{\\frac{d}{d \\Psi} \\operatorname{C_{d}}{(\\Psi)}}{\\operatorname{C_{d}}^{2}{(\\Psi)}}", "derivation": "\\rho{(\\Psi)} = e^{\\Psi} and \\operatorname{C_{d}}{(\\Psi)} = \\frac{\\Psi + e^{\\Psi}}{\\Psi + \\rho{(\\Psi)}} and \\operatorname{C_{d}}{(\\Psi)} = 1 and 1 = \\frac{1}{\\operatorname{C_{d}}{(\\Psi)}} and \\frac{d}{d \\Psi} 1 = \\frac{d}{d \\Psi} \\frac{1}{\\operatorname{C_{d}}{(\\Psi)}} and 0 = - \\frac{\\frac{d}{d \\Psi} \\operatorname{C_{d}}{(\\Psi)}}{\\operatorname{C_{d}}^{2}{(\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\rho')(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Integer(1))"], [["divide", 3, "Function('C_d')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(1), Pow(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Pow(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Mul(Integer(-1), Pow(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Integer(-2)), Derivative(Function('C_d')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})} = \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}}, then obtain \\int (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})})^{2} dm_{s} = \\int (m_{s} + \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}}) (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})}) dm_{s}", "derivation": "v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})} = \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}} and m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})} = m_{s} + \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}} and (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})})^{2} = (m_{s} + \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}}) (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})}) and \\int (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})})^{2} dm_{s} = \\int (m_{s} + \\frac{m_{s}^{\\hat{\\mathbf{r}}}}{\\tilde{g}}) (m_{s} + v{(\\tilde{g},m_{s},\\hat{\\mathbf{r}})}) dm_{s}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('m_s', commutative=True), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["times", 2, "Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Pow(Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)), Mul(Add(Symbol('m_s', commutative=True), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["integrate", 3, "Symbol('m_s', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Add(Symbol('m_s', commutative=True), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('m_s', commutative=True), Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varphi)} = \\int \\cos{(\\varphi)} d\\varphi, then derive \\operatorname{A_{x}}{(\\varphi)} = C + \\sin{(\\varphi)}, then derive C + \\sin{(\\varphi)} = \\hat{X} + \\sin{(\\varphi)}, then obtain \\operatorname{A_{x}}{(\\varphi)} = \\hat{X} + \\sin{(\\varphi)}", "derivation": "\\operatorname{A_{x}}{(\\varphi)} = \\int \\cos{(\\varphi)} d\\varphi and \\operatorname{A_{x}}{(\\varphi)} = C + \\sin{(\\varphi)} and C + \\sin{(\\varphi)} = \\int \\cos{(\\varphi)} d\\varphi and C + \\sin{(\\varphi)} = \\hat{X} + \\sin{(\\varphi)} and \\operatorname{A_{x}}{(\\varphi)} = \\hat{X} + \\sin{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varphi', commutative=True)), Integral(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('A_x')(Symbol('\\\\varphi', commutative=True)), Add(Symbol('C', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('C', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Integral(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('C', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Function('A_x')(Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(S,z)} = - S + z, then obtain (- S + z) ((\\frac{\\mathbf{J}{(S,z)}}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S}) = (- S + z) ((\\frac{- S + z}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S})", "derivation": "\\mathbf{J}{(S,z)} = - S + z and \\frac{\\mathbf{J}{(S,z)}}{S} = \\frac{- S + z}{S} and (\\frac{\\mathbf{J}{(S,z)}}{S})^{S} = (\\frac{- S + z}{S})^{S} and (\\frac{\\mathbf{J}{(S,z)}}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S} = (\\frac{- S + z}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S} and (- S + z) ((\\frac{\\mathbf{J}{(S,z)}}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S}) = (- S + z) ((\\frac{- S + z}{S})^{S} - \\frac{\\mathbf{J}{(S,z)}}{S})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True)))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))), Symbol('S', commutative=True)), Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True))), Symbol('S', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True)))"], "Equality(Add(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True)))), Add(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True))), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True)), Add(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True)), Add(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('z', commutative=True))), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('S', commutative=True), Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\ddot{x}{(S)} = \\frac{d}{d S} \\cos{(S)}, then derive \\ddot{x}{(S)} = - \\sin{(S)}, then derive \\int - \\sin{(S)} dS = t_{1} + \\cos{(S)}, then derive t_{1} + \\cos{(S)} = \\sigma_x + \\cos{(S)}, then obtain \\int \\ddot{x}{(S)} dS = \\sigma_x + \\cos{(S)}", "derivation": "\\ddot{x}{(S)} = \\frac{d}{d S} \\cos{(S)} and \\int \\ddot{x}{(S)} dS = \\int \\frac{d}{d S} \\cos{(S)} dS and \\ddot{x}{(S)} = - \\sin{(S)} and \\int - \\sin{(S)} dS = \\int \\frac{d}{d S} \\cos{(S)} dS and \\int - \\sin{(S)} dS = t_{1} + \\cos{(S)} and t_{1} + \\cos{(S)} = \\int \\frac{d}{d S} \\cos{(S)} dS and t_{1} + \\cos{(S)} = \\sigma_x + \\cos{(S)} and \\int \\frac{d}{d S} \\cos{(S)} dS = \\sigma_x + \\cos{(S)} and \\int \\ddot{x}{(S)} dS = \\sigma_x + \\cos{(S)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('S', commutative=True)), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\ddot{x}')(Symbol('S', commutative=True)), Mul(Integer(-1), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Mul(Integer(-1), sin(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Integer(-1), sin(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Add(Symbol('t_1', commutative=True), cos(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('t_1', commutative=True), cos(Symbol('S', commutative=True))), Integral(Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('t_1', commutative=True), cos(Symbol('S', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integral(Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(t,A_{y})} = A_{y}^{t}, then derive \\frac{\\partial}{\\partial t} \\operatorname{x^{{\\}'}}{(t,A_{y})} = A_{y}^{t} \\log{(A_{y})}, then obtain A_{y}^{t} \\log{(A_{y})} = \\frac{\\partial}{\\partial t} A_{y}^{t}", "derivation": "\\operatorname{x^{{\\}'}}{(t,A_{y})} = A_{y}^{t} and \\frac{\\partial}{\\partial t} \\operatorname{x^{{\\}'}}{(t,A_{y})} = \\frac{\\partial}{\\partial t} A_{y}^{t} and \\frac{\\partial}{\\partial t} \\operatorname{x^{{\\}'}}{(t,A_{y})} = A_{y}^{t} \\log{(A_{y})} and A_{y}^{t} \\log{(A_{y})} = \\frac{\\partial}{\\partial t} A_{y}^{t}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('t', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('A_y', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('t', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('t', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Mul(Pow(Symbol('A_y', commutative=True), Symbol('t', commutative=True)), log(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Symbol('t', commutative=True)), log(Symbol('A_y', commutative=True))), Derivative(Pow(Symbol('A_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{x},\\theta_1)} = e^{- \\hat{x} + \\theta_1}, then obtain \\hat{x}^{2} \\operatorname{A_{y}}{(\\hat{x},\\theta_1)} e^{- 2 \\hat{x} + 2 \\theta_1} e^{- \\hat{x} + \\theta_1} = \\hat{x}^{2} e^{- 4 \\hat{x} + 4 \\theta_1}", "derivation": "\\operatorname{A_{y}}{(\\hat{x},\\theta_1)} = e^{- \\hat{x} + \\theta_1} and \\hat{x} \\operatorname{A_{y}}{(\\hat{x},\\theta_1)} = \\hat{x} e^{- \\hat{x} + \\theta_1} and \\hat{x}^{2} \\operatorname{A_{y}}{(\\hat{x},\\theta_1)} e^{- \\hat{x} + \\theta_1} = \\hat{x}^{2} e^{- 2 \\hat{x} + 2 \\theta_1} and \\hat{x}^{2} \\operatorname{A_{y}}{(\\hat{x},\\theta_1)} e^{- 2 \\hat{x} + 2 \\theta_1} e^{- \\hat{x} + \\theta_1} = \\hat{x}^{2} e^{- 4 \\hat{x} + 4 \\theta_1}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('A_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\theta_1', commutative=True)))))"], [["times", 2, "Mul(Symbol('\\\\hat{x}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), Function('A_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), exp(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))))"], [["times", 3, "exp(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), Function('A_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), exp(Add(Mul(Integer(-1), Integer(4), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(4), Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(G,k)} = G - k, then derive - G \\frac{\\partial}{\\partial k} \\varepsilon{(G,k)} = G, then obtain \\int - G \\frac{\\partial}{\\partial k} \\varepsilon{(G,k)} dG = \\int G dG", "derivation": "\\varepsilon{(G,k)} = G - k and G \\varepsilon{(G,k)} = G (G - k) and - G \\varepsilon{(G,k)} = - G (G - k) and \\frac{\\partial}{\\partial k} - G \\varepsilon{(G,k)} = \\frac{\\partial}{\\partial k} - G (G - k) and - G \\frac{\\partial}{\\partial k} \\varepsilon{(G,k)} = G and \\int - G \\frac{\\partial}{\\partial k} \\varepsilon{(G,k)} dG = \\int G dG", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True))), Mul(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('G', commutative=True), Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Symbol('G', commutative=True), Derivative(Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Symbol('G', commutative=True))"], [["integrate", 5, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('G', commutative=True), Derivative(Function('\\\\varepsilon')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True))), Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(u,\\hat{p})} = \\hat{p} - u and \\operatorname{F_{N}}{(u,\\hat{p})} = - u (\\hat{p} - u), then obtain \\cos{(\\operatorname{F_{N}}{(u,\\hat{p})})} = \\cos{(u (\\hat{p} - u))}", "derivation": "\\operatorname{C_{d}}{(u,\\hat{p})} = \\hat{p} - u and \\operatorname{F_{N}}{(u,\\hat{p})} = - u (\\hat{p} - u) and \\operatorname{F_{N}}{(u,\\hat{p})} = - u \\operatorname{C_{d}}{(u,\\hat{p})} and \\cos{(\\operatorname{F_{N}}{(u,\\hat{p})})} = \\cos{(u \\operatorname{C_{d}}{(u,\\hat{p})})} and \\cos{(\\operatorname{F_{N}}{(u,\\hat{p})})} = \\cos{(u (\\hat{p} - u))}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True), Function('C_d')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["cos", 3], "Equality(cos(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True))), cos(Mul(Symbol('u', commutative=True), Function('C_d')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\hat{p}', commutative=True))), cos(Mul(Symbol('u', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(f,W)} = W + f, then derive \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = 0, then obtain \\frac{\\partial}{\\partial f} \\frac{\\partial}{\\partial f} \\mathbf{f}{(f,W)} \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = \\frac{d}{d f} 0", "derivation": "\\mathbf{f}{(f,W)} = W + f and \\frac{\\partial}{\\partial f} \\mathbf{f}{(f,W)} = \\frac{\\partial}{\\partial f} (W + f) and \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = \\frac{\\partial^{2}}{\\partial f^{2}} (W + f) and \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = 0 and \\frac{\\partial^{2}}{\\partial f^{2}} (W + f) = 0 and \\frac{\\partial}{\\partial f} (W + f) \\frac{\\partial^{2}}{\\partial f^{2}} (W + f) = 0 and \\frac{\\partial}{\\partial f} (W + f) \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = 0 and \\frac{\\partial}{\\partial f} \\mathbf{f}{(f,W)} \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = 0 and \\frac{\\partial}{\\partial f} \\frac{\\partial}{\\partial f} \\mathbf{f}{(f,W)} \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{f}{(f,W)} = \\frac{d}{d f} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Integer(0))"], [["times", 5, "Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Derivative(Add(Symbol('W', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Integer(0))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Mul(Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Integer(0))"], [["differentiate", 8, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(n)} = e^{n}, then derive e^{n} \\int \\operatorname{F_{g}}{(n)} dn = (\\theta_1 + e^{n}) e^{n}, then obtain - (\\theta_1 + e^{n}) e^{n} + e^{n} \\int e^{n} dn = 0", "derivation": "\\operatorname{F_{g}}{(n)} = e^{n} and \\int \\operatorname{F_{g}}{(n)} dn = \\int e^{n} dn and e^{n} \\int \\operatorname{F_{g}}{(n)} dn = e^{n} \\int e^{n} dn and e^{n} \\int \\operatorname{F_{g}}{(n)} dn = (\\theta_1 + e^{n}) e^{n} and \\operatorname{F_{g}}{(n)} \\int \\operatorname{F_{g}}{(n)} dn = (\\theta_1 + \\operatorname{F_{g}}{(n)}) \\operatorname{F_{g}}{(n)} and - (\\theta_1 + e^{n}) e^{n} + \\operatorname{F_{g}}{(n)} \\int \\operatorname{F_{g}}{(n)} dn = (\\theta_1 + \\operatorname{F_{g}}{(n)}) \\operatorname{F_{g}}{(n)} - (\\theta_1 + e^{n}) e^{n} and - (\\theta_1 + e^{n}) e^{n} + e^{n} \\int e^{n} dn = 0", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["times", 2, "exp(Symbol('n', commutative=True))"], "Equality(Mul(exp(Symbol('n', commutative=True)), Integral(Function('F_g')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(exp(Symbol('n', commutative=True)), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(exp(Symbol('n', commutative=True)), Integral(Function('F_g')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('F_g')(Symbol('n', commutative=True)), Integral(Function('F_g')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('F_g')(Symbol('n', commutative=True))), Function('F_g')(Symbol('n', commutative=True))))"], [["minus", 5, "Mul(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True))), Mul(Function('F_g')(Symbol('n', commutative=True)), Integral(Function('F_g')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Add(Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('F_g')(Symbol('n', commutative=True))), Function('F_g')(Symbol('n', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True))), Mul(exp(Symbol('n', commutative=True)), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(A_{1},\\phi)} = \\int A_{1} \\phi d\\phi, then derive \\int (A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)}) dA_{1} = A_{1}^{2} (\\frac{\\phi^{2}}{4} + \\frac{\\phi}{2}) + \\mathbf{p}, then obtain (\\int A_{1} \\phi d\\phi) \\int (A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)}) dA_{1} = (A_{1}^{2} (\\frac{\\phi^{2}}{4} + \\frac{\\phi}{2}) + \\mathbf{p}) \\int A_{1} \\phi d\\phi", "derivation": "\\operatorname{F_{N}}{(A_{1},\\phi)} = \\int A_{1} \\phi d\\phi and A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)} = A_{1} \\phi + \\int A_{1} \\phi d\\phi and \\int (A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)}) dA_{1} = \\int (A_{1} \\phi + \\int A_{1} \\phi d\\phi) dA_{1} and \\int (A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)}) dA_{1} = A_{1}^{2} (\\frac{\\phi^{2}}{4} + \\frac{\\phi}{2}) + \\mathbf{p} and (\\int A_{1} \\phi d\\phi) \\int (A_{1} \\phi + \\operatorname{F_{N}}{(A_{1},\\phi)}) dA_{1} = (A_{1}^{2} (\\frac{\\phi^{2}}{4} + \\frac{\\phi}{2}) + \\mathbf{p}) \\int A_{1} \\phi d\\phi", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Function('F_N')(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Function('F_N')(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Function('F_N')(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Add(Mul(Rational(1, 4), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 4, "Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Function('F_N')(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), Mul(Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Add(Mul(Rational(1, 4), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given c{(\\rho_b,V)} = \\int \\frac{V}{\\rho_b} dV, then obtain \\cos^{\\rho_b}{(c{(\\rho_b,V)} - 1)} = \\cos^{\\rho_b}{(\\int \\frac{V}{\\rho_b} dV - 1)}", "derivation": "c{(\\rho_b,V)} = \\int \\frac{V}{\\rho_b} dV and c{(\\rho_b,V)} - 1 = \\int \\frac{V}{\\rho_b} dV - 1 and \\cos{(c{(\\rho_b,V)} - 1)} = \\cos{(\\int \\frac{V}{\\rho_b} dV - 1)} and \\cos^{\\rho_b}{(c{(\\rho_b,V)} - 1)} = \\cos^{\\rho_b}{(\\int \\frac{V}{\\rho_b} dV - 1)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('V', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Integer(-1)), Add(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integer(-1)))"], [["cos", 2], "Equality(cos(Add(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Integer(-1))), cos(Add(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(cos(Add(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Pow(cos(Add(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integer(-1))), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(p)} = e^{p}, then derive \\cos{(\\operatorname{C_{1}}{(p)})} \\frac{d}{d p} \\operatorname{C_{1}}{(p)} = e^{p} \\cos{(e^{p})}, then obtain \\frac{d}{d p} e^{p} \\cos{(e^{p})} = \\frac{d}{d p} \\operatorname{C_{1}}{(p)} \\cos{(\\operatorname{C_{1}}{(p)})}", "derivation": "\\operatorname{C_{1}}{(p)} = e^{p} and \\sin{(\\operatorname{C_{1}}{(p)})} = \\sin{(e^{p})} and \\frac{d}{d p} \\sin{(\\operatorname{C_{1}}{(p)})} = \\frac{d}{d p} \\sin{(e^{p})} and \\cos{(\\operatorname{C_{1}}{(p)})} \\frac{d}{d p} \\operatorname{C_{1}}{(p)} = e^{p} \\cos{(e^{p})} and \\cos{(e^{p})} \\frac{d}{d p} e^{p} = e^{p} \\cos{(e^{p})} and \\cos{(\\operatorname{C_{1}}{(p)})} \\frac{d}{d p} \\operatorname{C_{1}}{(p)} = \\operatorname{C_{1}}{(p)} \\cos{(\\operatorname{C_{1}}{(p)})} and e^{p} \\cos{(e^{p})} = \\operatorname{C_{1}}{(p)} \\cos{(\\operatorname{C_{1}}{(p)})} and \\frac{d}{d p} e^{p} \\cos{(e^{p})} = \\frac{d}{d p} \\operatorname{C_{1}}{(p)} \\cos{(\\operatorname{C_{1}}{(p)})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["sin", 1], "Equality(sin(Function('C_1')(Symbol('p', commutative=True))), sin(exp(Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(sin(Function('C_1')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('C_1')(Symbol('p', commutative=True))), Derivative(Function('C_1')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(exp(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(cos(exp(Symbol('p', commutative=True))), Derivative(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(exp(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(cos(Function('C_1')(Symbol('p', commutative=True))), Derivative(Function('C_1')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Function('C_1')(Symbol('p', commutative=True)), cos(Function('C_1')(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(exp(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True)))), Mul(Function('C_1')(Symbol('p', commutative=True)), cos(Function('C_1')(Symbol('p', commutative=True)))))"], [["differentiate", 7, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(exp(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Function('C_1')(Symbol('p', commutative=True)), cos(Function('C_1')(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\varphi^*)} = \\log{(\\varphi^*)}, then obtain 0 = \\frac{- (- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)})^{\\varphi^*} + (- \\varphi^* + \\log{(\\varphi^*)})^{\\varphi^*}}{- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\varphi^*)} = \\log{(\\varphi^*)} and - \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)} = - \\varphi^* + \\log{(\\varphi^*)} and (- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)})^{\\varphi^*} = (- \\varphi^* + \\log{(\\varphi^*)})^{\\varphi^*} and 0 = - (- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)})^{\\varphi^*} + (- \\varphi^* + \\log{(\\varphi^*)})^{\\varphi^*} and 0 = \\frac{- (- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)})^{\\varphi^*} + (- \\varphi^* + \\log{(\\varphi^*)})^{\\varphi^*}}{- \\varphi^* + \\operatorname{x^{{\\}'}}{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{H}_l)} = e^{\\hat{H}_l}, then obtain 1 = \\log{(\\frac{d}{d \\hat{H}_l} \\mathbf{J}_M{(\\hat{H}_l)})}^{- \\hat{H}_l} \\log{(\\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l})}^{\\hat{H}_l}", "derivation": "\\mathbf{J}_M{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} \\mathbf{J}_M{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\log{(\\frac{d}{d \\hat{H}_l} \\mathbf{J}_M{(\\hat{H}_l)})} = \\log{(\\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l})} and \\log{(\\frac{d}{d \\hat{H}_l} \\mathbf{J}_M{(\\hat{H}_l)})}^{\\hat{H}_l} = \\log{(\\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l})}^{\\hat{H}_l} and 1 = \\log{(\\frac{d}{d \\hat{H}_l} \\mathbf{J}_M{(\\hat{H}_l)})}^{- \\hat{H}_l} \\log{(\\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l})}^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), log(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(log(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["divide", 4, "Pow(log(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(1), Mul(Pow(log(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(log(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given L{(\\mathbf{J}_f,\\rho)} = \\rho \\sin{(\\mathbf{J}_f)} and \\Omega{(\\mathbf{J}_f,\\rho)} = \\rho \\sin{(\\mathbf{J}_f)} + L{(\\mathbf{J}_f,\\rho)}, then obtain \\rho \\sin{(\\mathbf{J}_f)} - \\sin{(\\mathbf{J}_f)} = - \\rho \\sin{(\\mathbf{J}_f)} + \\Omega{(\\mathbf{J}_f,\\rho)} - \\sin{(\\mathbf{J}_f)}", "derivation": "L{(\\mathbf{J}_f,\\rho)} = \\rho \\sin{(\\mathbf{J}_f)} and \\Omega{(\\mathbf{J}_f,\\rho)} = \\rho \\sin{(\\mathbf{J}_f)} + L{(\\mathbf{J}_f,\\rho)} and - \\rho \\sin{(\\mathbf{J}_f)} + \\Omega{(\\mathbf{J}_f,\\rho)} = L{(\\mathbf{J}_f,\\rho)} and - L{(\\mathbf{J}_f,\\rho)} + \\Omega{(\\mathbf{J}_f,\\rho)} = L{(\\mathbf{J}_f,\\rho)} and \\rho \\sin{(\\mathbf{J}_f)} = - \\rho \\sin{(\\mathbf{J}_f)} + \\Omega{(\\mathbf{J}_f,\\rho)} and \\rho \\sin{(\\mathbf{J}_f)} - \\sin{(\\mathbf{J}_f)} = - \\rho \\sin{(\\mathbf{J}_f)} + \\Omega{(\\mathbf{J}_f,\\rho)} - \\sin{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True))), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True))), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["minus", 5, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\theta_1)} = \\theta_1, then obtain - 2 \\theta_1 + e^{\\theta_1 + \\mathbf{P}{(\\theta_1)} + \\cos{(2 \\theta_1)}} - \\cos{(2 \\theta_1)} = - 2 \\theta_1 + e^{2 \\theta_1 + \\cos{(2 \\theta_1)}} - \\cos{(2 \\theta_1)}", "derivation": "\\mathbf{P}{(\\theta_1)} = \\theta_1 and \\theta_1 + \\mathbf{P}{(\\theta_1)} = 2 \\theta_1 and \\theta_1 + \\mathbf{P}{(\\theta_1)} + \\cos{(2 \\theta_1)} = 2 \\theta_1 + \\cos{(2 \\theta_1)} and e^{\\theta_1 + \\mathbf{P}{(\\theta_1)} + \\cos{(2 \\theta_1)}} = e^{2 \\theta_1 + \\cos{(2 \\theta_1)}} and - 2 \\theta_1 + e^{\\theta_1 + \\mathbf{P}{(\\theta_1)} + \\cos{(2 \\theta_1)}} - \\cos{(2 \\theta_1)} = - 2 \\theta_1 + e^{2 \\theta_1 + \\cos{(2 \\theta_1)}} - \\cos{(2 \\theta_1)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))"], [["add", 2, "cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))), exp(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))))"], [["minus", 4, "Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), exp(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))), Mul(Integer(-1), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), exp(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))), Mul(Integer(-1), cos(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{J}_f)} = e^{\\cos{(\\mathbf{J}_f)}}, then obtain e^{\\cos{(\\mathbf{J}_f)}} + \\log{(2 \\operatorname{F_{c}}{(\\mathbf{J}_f)})} + \\cos{(\\mathbf{J}_f)} = e^{\\cos{(\\mathbf{J}_f)}} + \\log{(\\operatorname{F_{c}}{(\\mathbf{J}_f)} + e^{\\cos{(\\mathbf{J}_f)}})} + \\cos{(\\mathbf{J}_f)}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{J}_f)} = e^{\\cos{(\\mathbf{J}_f)}} and 2 \\operatorname{F_{c}}{(\\mathbf{J}_f)} = \\operatorname{F_{c}}{(\\mathbf{J}_f)} + e^{\\cos{(\\mathbf{J}_f)}} and \\log{(2 \\operatorname{F_{c}}{(\\mathbf{J}_f)})} = \\log{(\\operatorname{F_{c}}{(\\mathbf{J}_f)} + e^{\\cos{(\\mathbf{J}_f)}})} and \\log{(2 \\operatorname{F_{c}}{(\\mathbf{J}_f)})} + \\cos{(\\mathbf{J}_f)} = \\log{(\\operatorname{F_{c}}{(\\mathbf{J}_f)} + e^{\\cos{(\\mathbf{J}_f)}})} + \\cos{(\\mathbf{J}_f)} and e^{\\cos{(\\mathbf{J}_f)}} + \\log{(2 \\operatorname{F_{c}}{(\\mathbf{J}_f)})} + \\cos{(\\mathbf{J}_f)} = e^{\\cos{(\\mathbf{J}_f)}} + \\log{(\\operatorname{F_{c}}{(\\mathbf{J}_f)} + e^{\\cos{(\\mathbf{J}_f)}})} + \\cos{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)))), log(Add(Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["add", 3, "cos(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(log(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(log(Add(Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 4, "exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))), log(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))), log(Add(Function('F_c')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}_f', commutative=True))))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given l{(A,g)} = \\frac{g}{A}, then derive \\frac{A \\frac{\\partial}{\\partial A} l{(A,g)}}{g^{2}} + \\frac{l{(A,g)}}{g^{2}} = 0, then obtain (\\frac{\\partial}{\\partial g} (\\frac{A \\frac{\\partial}{\\partial A} l{(A,g)}}{g^{2}} + \\frac{l{(A,g)}}{g^{2}}))^{g} = (\\frac{d}{d g} 0)^{g}", "derivation": "l{(A,g)} = \\frac{g}{A} and \\frac{A l{(A,g)}}{g^{2}} = \\frac{1}{g} and \\frac{\\partial}{\\partial A} \\frac{A l{(A,g)}}{g^{2}} = \\frac{d}{d A} \\frac{1}{g} and \\frac{A \\frac{\\partial}{\\partial A} l{(A,g)}}{g^{2}} + \\frac{l{(A,g)}}{g^{2}} = 0 and \\frac{\\partial}{\\partial g} (\\frac{A \\frac{\\partial}{\\partial A} l{(A,g)}}{g^{2}} + \\frac{l{(A,g)}}{g^{2}}) = \\frac{d}{d g} 0 and (\\frac{\\partial}{\\partial g} (\\frac{A \\frac{\\partial}{\\partial A} l{(A,g)}}{g^{2}} + \\frac{l{(A,g)}}{g^{2}}))^{g} = (\\frac{d}{d g} 0)^{g}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('g', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(2)))"], "Equality(Mul(Symbol('A', commutative=True), Pow(Symbol('g', commutative=True), Integer(-2)), Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Symbol('A', commutative=True), Pow(Symbol('g', commutative=True), Integer(-2)), Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Symbol('g', commutative=True), Integer(-1)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('g', commutative=True), Integer(-2)), Derivative(Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('g', commutative=True), Integer(-2)), Derivative(Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('g', commutative=True), Integer(-2)), Derivative(Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Function('l')(Symbol('A', commutative=True), Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\mathbf{H},c)} = \\mathbf{H} c, then obtain - \\frac{2 \\mathbf{H} c + 2 \\mathbf{H} + \\operatorname{y^{\\prime}}{(\\mathbf{H},c)}}{2 \\mathbf{H}} = - \\frac{3 \\mathbf{H} c + 2 \\mathbf{H}}{2 \\mathbf{H}}", "derivation": "\\operatorname{y^{\\prime}}{(\\mathbf{H},c)} = \\mathbf{H} c and 2 \\mathbf{H} + \\operatorname{y^{\\prime}}{(\\mathbf{H},c)} = \\mathbf{H} c + 2 \\mathbf{H} and 4 \\mathbf{H} + \\operatorname{y^{\\prime}}{(\\mathbf{H},c)} = \\mathbf{H} c + 4 \\mathbf{H} and 2 \\mathbf{H} c + 2 \\mathbf{H} + \\operatorname{y^{\\prime}}{(\\mathbf{H},c)} = 3 \\mathbf{H} c + 2 \\mathbf{H} and - \\frac{2 \\mathbf{H} c + 2 \\mathbf{H} + \\operatorname{y^{\\prime}}{(\\mathbf{H},c)}}{2 \\mathbf{H}} = - \\frac{3 \\mathbf{H} c + 2 \\mathbf{H}}{2 \\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(4), Symbol('\\\\mathbf{H}', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(4), Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 3, "Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} = e^{\\mathbf{D}^{\\mathbf{J}}}, then obtain \\mathbf{D} \\int \\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} e^{- \\mathbf{D}^{\\mathbf{J}}} d\\mathbf{J} = \\mathbf{D} \\int 1 d\\mathbf{J}", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} = e^{\\mathbf{D}^{\\mathbf{J}}} and \\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} e^{- \\mathbf{D}^{\\mathbf{J}}} = 1 and \\int \\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} e^{- \\mathbf{D}^{\\mathbf{J}}} d\\mathbf{J} = \\int 1 d\\mathbf{J} and \\mathbf{D} \\int \\hat{H}_{\\lambda}{(\\mathbf{J},\\mathbf{D})} e^{- \\mathbf{D}^{\\mathbf{J}}} d\\mathbf{J} = \\mathbf{D} \\int 1 d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 1, "exp(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Integral(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\varphi)} = \\log{(\\varphi)}, then obtain \\operatorname{t_{1}}^{4}{(\\varphi)} = \\operatorname{t_{1}}^{3}{(\\varphi)} \\log{(\\varphi)}", "derivation": "\\operatorname{t_{1}}{(\\varphi)} = \\log{(\\varphi)} and \\operatorname{t_{1}}^{2}{(\\varphi)} = \\operatorname{t_{1}}{(\\varphi)} \\log{(\\varphi)} and \\operatorname{t_{1}}^{3}{(\\varphi)} \\log{(\\varphi)} = \\operatorname{t_{1}}^{2}{(\\varphi)} \\log{(\\varphi)}^{2} and \\operatorname{t_{1}}^{3}{(\\varphi)} \\log{(\\varphi)} = \\operatorname{t_{1}}{(\\varphi)} \\log{(\\varphi)}^{3} and \\operatorname{t_{1}}^{4}{(\\varphi)} = \\operatorname{t_{1}}^{2}{(\\varphi)} \\log{(\\varphi)}^{2} and \\operatorname{t_{1}}^{4}{(\\varphi)} = \\operatorname{t_{1}}^{3}{(\\varphi)} \\log{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Function('t_1')(Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Function('t_1')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))))"], [["times", 2, "Mul(Function('t_1')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(3)), log(Symbol('\\\\varphi', commutative=True))), Mul(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(3)), log(Symbol('\\\\varphi', commutative=True))), Mul(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(4)), Mul(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(4)), Mul(Pow(Function('t_1')(Symbol('\\\\varphi', commutative=True)), Integer(3)), log(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given E{(y^{\\prime})} = \\cos{(y^{\\prime})}, then derive \\int E{(y^{\\prime})} dy^{\\prime} = \\phi_1 + \\sin{(y^{\\prime})}, then obtain q + \\sin{(y^{\\prime})} = \\phi_1 + \\sin{(y^{\\prime})}", "derivation": "E{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\int E{(y^{\\prime})} dy^{\\prime} = \\int \\cos{(y^{\\prime})} dy^{\\prime} and \\int E{(y^{\\prime})} dy^{\\prime} = \\phi_1 + \\sin{(y^{\\prime})} and \\int \\cos{(y^{\\prime})} dy^{\\prime} = \\phi_1 + \\sin{(y^{\\prime})} and q + \\sin{(y^{\\prime})} = \\phi_1 + \\sin{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('q', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given M{(P_{e},g)} = \\cos{(P_{e} - g)}, then derive \\log{(\\int M{(P_{e},g)} dP_{e})} = \\log{(\\mathbf{H} + \\sin{(P_{e} - g)})}, then obtain \\int M{(P_{e},g)} dP_{e} = \\mathbf{H} + \\sin{(P_{e} - g)}", "derivation": "M{(P_{e},g)} = \\cos{(P_{e} - g)} and \\int M{(P_{e},g)} dP_{e} = \\int \\cos{(P_{e} - g)} dP_{e} and \\log{(\\int M{(P_{e},g)} dP_{e})} = \\log{(\\int \\cos{(P_{e} - g)} dP_{e})} and \\log{(\\int M{(P_{e},g)} dP_{e})} = \\log{(\\mathbf{H} + \\sin{(P_{e} - g)})} and \\int M{(P_{e},g)} dP_{e} = \\mathbf{H} + \\sin{(P_{e} - g)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), cos(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('M')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(cos(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))), Tuple(Symbol('P_e', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('M')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), log(Integral(cos(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))), Tuple(Symbol('P_e', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('M')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), log(Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))))"], [["exp", 4], "Equality(Integral(Function('M')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(\\hat{p})} = \\sin{(\\hat{p})}, then derive \\frac{d}{d \\hat{p}} \\mathbf{P}{(\\hat{p})} = \\cos{(\\hat{p})}, then obtain \\frac{d}{d \\hat{p}} \\int (\\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} - 1) d\\hat{p} = \\frac{d}{d \\hat{p}} \\int (\\cos{(\\hat{p})} - 1) d\\hat{p}", "derivation": "\\mathbf{P}{(\\hat{p})} = \\sin{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\mathbf{P}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\mathbf{P}{(\\hat{p})} = \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} = \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} - 1 = \\cos{(\\hat{p})} - 1 and \\int (\\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} - 1) d\\hat{p} = \\int (\\cos{(\\hat{p})} - 1) d\\hat{p} and \\frac{d}{d \\hat{p}} \\int (\\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} - 1) d\\hat{p} = \\frac{d}{d \\hat{p}} \\int (\\cos{(\\hat{p})} - 1) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Integral(Add(Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Integral(Add(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{g})} = \\log{(\\mathbf{g})}, then derive 1 + \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{F_{c} + \\mathbf{g} \\log{(\\mathbf{g})} - \\mathbf{g}} = 2, then obtain t (1 + \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{F_{c} + \\mathbf{g} \\log{(\\mathbf{g})} - \\mathbf{g}}) = 2 t", "derivation": "\\operatorname{A_{y}}{(\\mathbf{g})} = \\log{(\\mathbf{g})} and \\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g} = \\int \\log{(\\mathbf{g})} d\\mathbf{g} and \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{\\int \\log{(\\mathbf{g})} d\\mathbf{g}} = 1 and \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{\\int \\log{(\\mathbf{g})} d\\mathbf{g}} + 1 = 2 and 1 + \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{F_{c} + \\mathbf{g} \\log{(\\mathbf{g})} - \\mathbf{g}} = 2 and t (1 + \\frac{\\int \\operatorname{A_{y}}{(\\mathbf{g})} d\\mathbf{g}}{F_{c} + \\mathbf{g} \\log{(\\mathbf{g})} - \\mathbf{g}}) = 2 t", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(log(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 2, "Integral(log(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Integral(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Integral(log(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integral(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Integral(log(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Integer(1)), Integer(2))"], [["evaluate_integrals", 4], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('F_c', commutative=True), Mul(Symbol('\\\\mathbf{g}', commutative=True), log(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)), Integral(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))), Integer(2))"], [["times", 5, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Add(Integer(1), Mul(Pow(Add(Symbol('F_c', commutative=True), Mul(Symbol('\\\\mathbf{g}', commutative=True), log(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)), Integral(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))), Mul(Integer(2), Symbol('t', commutative=True)))"]]}, {"prompt": "Given H{(r)} = e^{e^{r}}, then obtain H{(r)} e^{- e^{r}} - 1 = 0", "derivation": "H{(r)} = e^{e^{r}} and r H{(r)} = r e^{e^{r}} and H{(r)} e^{- e^{r}} = 1 and H{(r)} e^{- e^{r}} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('r', commutative=True)), exp(exp(Symbol('r', commutative=True))))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('H')(Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), exp(exp(Symbol('r', commutative=True)))))"], [["divide", 2, "Mul(Symbol('r', commutative=True), exp(exp(Symbol('r', commutative=True))))"], "Equality(Mul(Function('H')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('r', commutative=True))))), Integer(1))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Function('H')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('r', commutative=True))))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\nabla)} = \\sin{(\\nabla)}, then obtain \\frac{d}{d \\nabla} \\cos{(\\operatorname{A_{1}}{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})} = \\frac{d}{d \\nabla} \\cos{(\\sin{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})}", "derivation": "\\operatorname{A_{1}}{(\\nabla)} = \\sin{(\\nabla)} and \\operatorname{A_{1}}{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)} = \\sin{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)} and \\cos{(\\operatorname{A_{1}}{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})} = \\cos{(\\sin{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})} and \\frac{d}{d \\nabla} \\cos{(\\operatorname{A_{1}}{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})} = \\frac{d}{d \\nabla} \\cos{(\\sin{(\\nabla)} - \\sin^{\\nabla}{(\\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))), Add(sin(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Function('A_1')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))), cos(Add(sin(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(cos(Add(Function('A_1')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(cos(Add(sin(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\psi)} = e^{\\psi}, then derive \\int \\operatorname{f_{E}}{(\\psi)} d\\psi = C_{2} + e^{\\psi}, then obtain - C_{2} - (\\frac{- \\Psi_{\\lambda} + s}{m})^{m} + \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\int \\operatorname{f_{E}}{(\\psi)} d\\psi) = - C_{2} - (\\frac{- \\Psi_{\\lambda} + s}{m})^{m} + \\frac{d}{d C_{2}} e^{\\psi}", "derivation": "\\operatorname{f_{E}}{(\\psi)} = e^{\\psi} and \\int \\operatorname{f_{E}}{(\\psi)} d\\psi = \\int e^{\\psi} d\\psi and \\int \\operatorname{f_{E}}{(\\psi)} d\\psi = C_{2} + e^{\\psi} and - C_{2} + \\int \\operatorname{f_{E}}{(\\psi)} d\\psi = e^{\\psi} and \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\int \\operatorname{f_{E}}{(\\psi)} d\\psi) = \\frac{d}{d C_{2}} e^{\\psi} and - C_{2} - (\\frac{- \\Psi_{\\lambda} + s}{m})^{m} + \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\int \\operatorname{f_{E}}{(\\psi)} d\\psi) = - C_{2} - (\\frac{- \\Psi_{\\lambda} + s}{m})^{m} + \\frac{d}{d C_{2}} e^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\psi', commutative=True))))"], [["minus", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Integral(Function('f_E')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), exp(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 4, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Integral(Function('f_E')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["minus", 5, "Add(Symbol('C_2', commutative=True), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('s', commutative=True))), Symbol('m', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('s', commutative=True))), Symbol('m', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Integral(Function('f_E')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('s', commutative=True))), Symbol('m', commutative=True))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}}, then obtain \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} 1 = \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} \\frac{e^{e^{\\eta^{\\prime}}}}{\\omega{(\\eta^{\\prime})}}", "derivation": "\\omega{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}} and 1 = \\frac{e^{e^{\\eta^{\\prime}}}}{\\omega{(\\eta^{\\prime})}} and \\frac{d}{d \\eta^{\\prime}} 1 = \\frac{d}{d \\eta^{\\prime}} \\frac{e^{e^{\\eta^{\\prime}}}}{\\omega{(\\eta^{\\prime})}} and \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} 1 = \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} \\frac{e^{e^{\\eta^{\\prime}}}}{\\omega{(\\eta^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('\\\\omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))), Derivative(Mul(Pow(Function('\\\\omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\varepsilon{(\\phi,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\phi and \\psi{(\\phi,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\phi, then obtain \\frac{\\cos{(\\varepsilon{(\\phi,\\Psi_{\\lambda})})}}{\\varepsilon{(\\phi,\\Psi_{\\lambda})}} = \\frac{\\cos{(\\Psi_{\\lambda} + \\phi)}}{\\varepsilon{(\\phi,\\Psi_{\\lambda})}}", "derivation": "\\varepsilon{(\\phi,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\phi and \\psi{(\\phi,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\phi and \\cos{(\\psi{(\\phi,\\Psi_{\\lambda})})} = \\cos{(\\Psi_{\\lambda} + \\phi)} and \\varepsilon{(\\phi,\\Psi_{\\lambda})} = \\psi{(\\phi,\\Psi_{\\lambda})} and \\frac{\\cos{(\\psi{(\\phi,\\Psi_{\\lambda})})}}{\\psi{(\\phi,\\Psi_{\\lambda})}} = \\frac{\\cos{(\\Psi_{\\lambda} + \\phi)}}{\\psi{(\\phi,\\Psi_{\\lambda})}} and \\frac{\\cos{(\\varepsilon{(\\phi,\\Psi_{\\lambda})})}}{\\varepsilon{(\\phi,\\Psi_{\\lambda})}} = \\frac{\\cos{(\\Psi_{\\lambda} + \\phi)}}{\\varepsilon{(\\phi,\\Psi_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["cos", 2], "Equality(cos(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 3, "Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Pow(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(E_{x},G,\\pi)} = \\frac{E_{x} G}{\\pi}, then obtain G \\frac{\\partial}{\\partial G} \\hat{x}{(E_{x},G,\\pi)} + \\hat{x}{(E_{x},G,\\pi)} = \\frac{2 E_{x} G}{\\pi}", "derivation": "\\hat{x}{(E_{x},G,\\pi)} = \\frac{E_{x} G}{\\pi} and G \\hat{x}{(E_{x},G,\\pi)} = \\frac{E_{x} G^{2}}{\\pi} and \\frac{\\partial}{\\partial G} G \\hat{x}{(E_{x},G,\\pi)} = \\frac{\\partial}{\\partial G} \\frac{E_{x} G^{2}}{\\pi} and G \\frac{\\partial}{\\partial G} \\hat{x}{(E_{x},G,\\pi)} + \\hat{x}{(E_{x},G,\\pi)} = \\frac{2 E_{x} G}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('E_x', commutative=True), Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_x', commutative=True), Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('G', commutative=True), Derivative(Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('G', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('G', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(r_{0},\\phi_1)} = e^{\\phi_1 r_{0}}, then obtain \\frac{\\partial}{\\partial r_{0}} (- \\phi_1 r_{0} + \\operatorname{v_{x}}{(r_{0},\\phi_1)} + e^{\\phi_1 r_{0}}) = \\frac{\\partial}{\\partial r_{0}} (- \\phi_1 r_{0} + 2 e^{\\phi_1 r_{0}})", "derivation": "\\operatorname{v_{x}}{(r_{0},\\phi_1)} = e^{\\phi_1 r_{0}} and \\operatorname{v_{x}}{(r_{0},\\phi_1)} + e^{\\phi_1 r_{0}} = 2 e^{\\phi_1 r_{0}} and - \\phi_1 r_{0} + \\operatorname{v_{x}}{(r_{0},\\phi_1)} + e^{\\phi_1 r_{0}} = - \\phi_1 r_{0} + 2 e^{\\phi_1 r_{0}} and \\frac{\\partial}{\\partial r_{0}} (- \\phi_1 r_{0} + \\operatorname{v_{x}}{(r_{0},\\phi_1)} + e^{\\phi_1 r_{0}}) = \\frac{\\partial}{\\partial r_{0}} (- \\phi_1 r_{0} + 2 e^{\\phi_1 r_{0}})", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('r_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 1, "exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Add(Function('v_x')(Symbol('r_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)))), Mul(Integer(2), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)))))"], [["minus", 2, "Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)), Function('v_x')(Symbol('r_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True))))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)), Function('v_x')(Symbol('r_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(v_{1},\\Psi^{\\dagger})} = \\sin^{v_{1}}{(\\Psi^{\\dagger})}, then obtain \\int 0^{v_{1}} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\mu", "derivation": "\\chi{(v_{1},\\Psi^{\\dagger})} = \\sin^{v_{1}}{(\\Psi^{\\dagger})} and 0 = - \\chi{(v_{1},\\Psi^{\\dagger})} + \\sin^{v_{1}}{(\\Psi^{\\dagger})} and 0^{v_{1}} = (- \\chi{(v_{1},\\Psi^{\\dagger})} + \\sin^{v_{1}}{(\\Psi^{\\dagger})})^{v_{1}} and \\int 0^{v_{1}} d\\Psi^{\\dagger} = \\int (- \\chi{(v_{1},\\Psi^{\\dagger})} + \\sin^{v_{1}}{(\\Psi^{\\dagger})})^{v_{1}} d\\Psi^{\\dagger} and \\int (- \\chi{(v_{1},\\Psi^{\\dagger})} + \\sin^{v_{1}}{(\\Psi^{\\dagger})})^{v_{1}} d\\Psi^{\\dagger} = \\int 1 d\\Psi^{\\dagger} and \\int 0^{v_{1}} d\\Psi^{\\dagger} = \\int 1 d\\Psi^{\\dagger} and \\int 0^{v_{1}} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_1', commutative=True)))"], [["minus", 1, "Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_1', commutative=True))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('v_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(a^{\\dagger})} = e^{\\cos{(a^{\\dagger})}}, then obtain ((a^{\\dagger} + \\mu_{0}{(a^{\\dagger})})^{a^{\\dagger}})^{a^{\\dagger}} = ((a^{\\dagger} + e^{\\cos{(a^{\\dagger})}})^{a^{\\dagger}})^{a^{\\dagger}}", "derivation": "\\mu_{0}{(a^{\\dagger})} = e^{\\cos{(a^{\\dagger})}} and a^{\\dagger} + \\mu_{0}{(a^{\\dagger})} = a^{\\dagger} + e^{\\cos{(a^{\\dagger})}} and (a^{\\dagger} + \\mu_{0}{(a^{\\dagger})})^{a^{\\dagger}} = (a^{\\dagger} + e^{\\cos{(a^{\\dagger})}})^{a^{\\dagger}} and ((a^{\\dagger} + \\mu_{0}{(a^{\\dagger})})^{a^{\\dagger}})^{a^{\\dagger}} = ((a^{\\dagger} + e^{\\cos{(a^{\\dagger})}})^{a^{\\dagger}})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('a^{\\\\dagger}', commutative=True)), exp(cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mu_0')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), exp(cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mu_0')(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), exp(cos(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mu_0')(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), exp(cos(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given v{(M)} = \\cos{(M)} and \\operatorname{P_{g}}{(M)} = \\int \\cos{(M)} dM, then obtain \\int v{(M)} dM + \\int \\cos{(M)} dM = \\operatorname{P_{g}}{(M)} + \\int v{(M)} dM", "derivation": "v{(M)} = \\cos{(M)} and \\int v{(M)} dM = \\int \\cos{(M)} dM and \\operatorname{P_{g}}{(M)} = \\int \\cos{(M)} dM and 2 \\int v{(M)} dM = \\int v{(M)} dM + \\int \\cos{(M)} dM and 2 \\int v{(M)} dM = \\operatorname{P_{g}}{(M)} + \\int v{(M)} dM and \\int v{(M)} dM + \\int \\cos{(M)} dM = \\operatorname{P_{g}}{(M)} + \\int v{(M)} dM", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('M', commutative=True)), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["add", 2, "Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Add(Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Add(Function('P_g')(Symbol('M', commutative=True)), Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Add(Function('P_g')(Symbol('M', commutative=True)), Integral(Function('v')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given I{(r)} = \\cos{(r)}, then derive \\frac{I{(r)} \\sin{(r)}}{\\cos^{2}{(r)}} + \\frac{\\frac{d}{d r} I{(r)}}{\\cos{(r)}} = 0, then obtain (\\frac{\\sin{(r)}}{I{(r)}} + \\frac{\\frac{d}{d r} I{(r)}}{I{(r)}})^{r} = 0^{r}", "derivation": "I{(r)} = \\cos{(r)} and \\frac{I{(r)}}{\\cos{(r)}} = 1 and \\frac{d}{d r} \\frac{I{(r)}}{\\cos{(r)}} = \\frac{d}{d r} 1 and \\frac{I{(r)} \\sin{(r)}}{\\cos^{2}{(r)}} + \\frac{\\frac{d}{d r} I{(r)}}{\\cos{(r)}} = 0 and \\frac{\\sin{(r)}}{I{(r)}} + \\frac{\\frac{d}{d r} I{(r)}}{I{(r)}} = 0 and (\\frac{\\sin{(r)}}{I{(r)}} + \\frac{\\frac{d}{d r} I{(r)}}{I{(r)}})^{r} = 0^{r}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["divide", 1, "cos(Symbol('r', commutative=True))"], "Equality(Mul(Function('I')(Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Function('I')(Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('I')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Integer(-2))), Mul(Pow(cos(Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('I')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Pow(Function('I')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))), Mul(Pow(Function('I')(Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('I')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Integer(0))"], [["power", 5, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Function('I')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))), Mul(Pow(Function('I')(Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('I')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Symbol('r', commutative=True)), Pow(Integer(0), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\eta{(F_{x})} = e^{\\cos{(F_{x})}}, then obtain - \\frac{\\frac{d}{d F_{x}} \\eta{(F_{x})} e^{\\cos{(F_{x})}}}{\\cos{(F_{x})}} = - \\frac{\\frac{d}{d F_{x}} e^{2 \\cos{(F_{x})}}}{\\cos{(F_{x})}}", "derivation": "\\eta{(F_{x})} = e^{\\cos{(F_{x})}} and \\eta{(F_{x})} e^{\\cos{(F_{x})}} = e^{2 \\cos{(F_{x})}} and \\frac{d}{d F_{x}} \\eta{(F_{x})} e^{\\cos{(F_{x})}} = \\frac{d}{d F_{x}} e^{2 \\cos{(F_{x})}} and - \\frac{\\frac{d}{d F_{x}} \\eta{(F_{x})} e^{\\cos{(F_{x})}}}{\\cos{(F_{x})}} = - \\frac{\\frac{d}{d F_{x}} e^{2 \\cos{(F_{x})}}}{\\cos{(F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True))))"], [["times", 1, "exp(cos(Symbol('F_x', commutative=True)))"], "Equality(Mul(Function('\\\\eta')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('F_x', commutative=True)))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\eta')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(cos(Symbol('F_x', commutative=True)), Integer(-1)), Derivative(Mul(Function('\\\\eta')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(cos(Symbol('F_x', commutative=True)), Integer(-1)), Derivative(exp(Mul(Integer(2), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given t{(\\eta,g_{\\varepsilon},s)} = \\eta + g_{\\varepsilon} - s, then obtain \\frac{s - (\\eta + g_{\\varepsilon} - s)^{s} + t^{s}{(\\eta,g_{\\varepsilon},s)}}{\\eta} = \\frac{s}{\\eta}", "derivation": "t{(\\eta,g_{\\varepsilon},s)} = \\eta + g_{\\varepsilon} - s and t^{s}{(\\eta,g_{\\varepsilon},s)} = (\\eta + g_{\\varepsilon} - s)^{s} and s + t^{s}{(\\eta,g_{\\varepsilon},s)} = s + (\\eta + g_{\\varepsilon} - s)^{s} and s - (\\eta + g_{\\varepsilon} - s)^{s} + t^{s}{(\\eta,g_{\\varepsilon},s)} = s and \\frac{s - (\\eta + g_{\\varepsilon} - s)^{s} + t^{s}{(\\eta,g_{\\varepsilon},s)}}{\\eta} = \\frac{s}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Pow(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["minus", 3, "Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], "Equality(Add(Symbol('s', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Pow(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], [["divide", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Pow(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given I{(\\rho_b,L)} = e^{L \\rho_b} and \\phi{(\\rho_b,L)} = e^{L \\rho_b}, then derive \\frac{\\partial}{\\partial \\rho_b} \\phi{(\\rho_b,L)} = L e^{L \\rho_b}, then obtain \\frac{\\partial}{\\partial \\rho_b} e^{L \\rho_b} = L e^{L \\rho_b}", "derivation": "I{(\\rho_b,L)} = e^{L \\rho_b} and \\phi{(\\rho_b,L)} = e^{L \\rho_b} and \\phi{(\\rho_b,L)} = I{(\\rho_b,L)} and \\frac{\\partial}{\\partial \\rho_b} \\phi{(\\rho_b,L)} = \\frac{\\partial}{\\partial \\rho_b} I{(\\rho_b,L)} and \\frac{\\partial}{\\partial \\rho_b} \\phi{(\\rho_b,L)} = \\frac{\\partial}{\\partial \\rho_b} e^{L \\rho_b} and \\frac{\\partial}{\\partial \\rho_b} \\phi{(\\rho_b,L)} = L e^{L \\rho_b} and \\frac{\\partial}{\\partial \\rho_b} e^{L \\rho_b} = L e^{L \\rho_b}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), Function('I')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Function('I')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Symbol('L', commutative=True), exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Symbol('L', commutative=True), exp(Mul(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(V)} = \\log{(\\sin{(V)})}, then obtain \\log{(\\sin{(V)} - \\int \\log{(\\sin{(V)})} dV + 1)} = \\log{(\\sin{(V)} + \\cos{(\\int \\operatorname{F_{c}}{(V)} dV - \\int \\log{(\\sin{(V)})} dV)} - \\int \\log{(\\sin{(V)})} dV)}", "derivation": "\\operatorname{F_{c}}{(V)} = \\log{(\\sin{(V)})} and \\int \\operatorname{F_{c}}{(V)} dV = \\int \\log{(\\sin{(V)})} dV and - \\sin{(V)} + \\int \\operatorname{F_{c}}{(V)} dV = - \\sin{(V)} + \\int \\log{(\\sin{(V)})} dV and 0 = - \\int \\operatorname{F_{c}}{(V)} dV + \\int \\log{(\\sin{(V)})} dV and 1 = \\cos{(\\int \\operatorname{F_{c}}{(V)} dV - \\int \\log{(\\sin{(V)})} dV)} and \\sin{(V)} - \\int \\log{(\\sin{(V)})} dV + 1 = \\sin{(V)} + \\cos{(\\int \\operatorname{F_{c}}{(V)} dV - \\int \\log{(\\sin{(V)})} dV)} - \\int \\log{(\\sin{(V)})} dV and \\log{(\\sin{(V)} - \\int \\log{(\\sin{(V)})} dV + 1)} = \\log{(\\sin{(V)} + \\cos{(\\int \\operatorname{F_{c}}{(V)} dV - \\int \\log{(\\sin{(V)})} dV)} - \\int \\log{(\\sin{(V)})} dV)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('V', commutative=True)), log(sin(Symbol('V', commutative=True))))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["minus", 2, "sin(Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('V', commutative=True))), Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('V', commutative=True))), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), sin(Symbol('V', commutative=True))), Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))"], [["cos", 4], "Equality(Integer(1), cos(Add(Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))))"], [["minus", 5, "Add(Mul(Integer(-1), sin(Symbol('V', commutative=True))), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], "Equality(Add(sin(Symbol('V', commutative=True)), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))), Integer(1)), Add(sin(Symbol('V', commutative=True)), cos(Add(Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))))"], [["log", 6], "Equality(log(Add(sin(Symbol('V', commutative=True)), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))), Integer(1))), log(Add(sin(Symbol('V', commutative=True)), cos(Add(Integral(Function('F_c')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))), Mul(Integer(-1), Integral(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))))"]]}, {"prompt": "Given \\varphi{(\\mu_0)} = e^{\\mu_0} and \\mu{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\operatorname{x^{{\\}'}}{(\\mu_0)} = \\frac{\\frac{d}{d \\mu_0} \\varphi{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}}, then obtain (e^{\\operatorname{x^{{\\}'}}{(\\mu_0)}})^{\\mu_0} = (e^{\\frac{\\mu{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}}})^{\\mu_0}", "derivation": "\\varphi{(\\mu_0)} = e^{\\mu_0} and \\frac{d}{d \\mu_0} \\varphi{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\mu{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\frac{d}{d \\mu_0} \\varphi{(\\mu_0)} = \\mu{(\\mu_0)} and \\operatorname{x^{{\\}'}}{(\\mu_0)} = \\frac{\\frac{d}{d \\mu_0} \\varphi{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}} and e^{\\operatorname{x^{{\\}'}}{(\\mu_0)}} = e^{\\frac{\\frac{d}{d \\mu_0} \\varphi{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}}} and (e^{\\operatorname{x^{{\\}'}}{(\\mu_0)}})^{\\mu_0} = (e^{\\frac{\\frac{d}{d \\mu_0} \\varphi{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}}})^{\\mu_0} and (e^{\\operatorname{x^{{\\}'}}{(\\mu_0)}})^{\\mu_0} = (e^{\\frac{\\mu{(\\mu_0)}}{\\mu_0 + e^{\\mu_0}}})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True)), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["exp", 5], "Equality(exp(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True))), exp(Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["power", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(exp(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(exp(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(x)} = \\log{(x)}, then obtain \\frac{\\int - \\frac{\\operatorname{C_{1}}{(x)}}{x} dx}{- x + \\log{(x)}} = \\frac{\\int - \\frac{\\log{(x)}}{x} dx}{- x + \\log{(x)}}", "derivation": "\\operatorname{C_{1}}{(x)} = \\log{(x)} and - \\frac{\\operatorname{C_{1}}{(x)}}{x} = - \\frac{\\log{(x)}}{x} and \\int - \\frac{\\operatorname{C_{1}}{(x)}}{x} dx = \\int - \\frac{\\log{(x)}}{x} dx and \\frac{\\int - \\frac{\\operatorname{C_{1}}{(x)}}{x} dx}{- x + \\log{(x)}} = \\frac{\\int - \\frac{\\log{(x)}}{x} dx}{- x + \\log{(x)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Function('C_1')(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Function('C_1')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Function('C_1')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(b)} = b, then obtain \\int (- \\hat{\\mathbf{r}}{(b)} + \\hat{\\mathbf{r}}^{b}{(b)}) db = \\int (b^{b} - \\hat{\\mathbf{r}}{(b)}) db", "derivation": "\\hat{\\mathbf{r}}{(b)} = b and \\hat{\\mathbf{r}}^{b}{(b)} = b^{b} and - \\hat{\\mathbf{r}}{(b)} + \\hat{\\mathbf{r}}^{b}{(b)} = b^{b} - \\hat{\\mathbf{r}}{(b)} and \\int (- \\hat{\\mathbf{r}}{(b)} + \\hat{\\mathbf{r}}^{b}{(b)}) db = \\int (b^{b} - \\hat{\\mathbf{r}}{(b)}) db", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Symbol('b', commutative=True), Symbol('b', commutative=True)))"], [["minus", 2, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Add(Pow(Symbol('b', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)))))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Add(Pow(Symbol('b', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} = \\frac{F_{g} \\lambda}{Q}, then obtain \\sin{(\\lambda \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)})} = \\sin{(\\frac{F_{g} \\lambda^{2}}{Q} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)})}", "derivation": "\\Psi^{\\dagger}{(Q,F_{g},\\lambda)} = \\frac{F_{g} \\lambda}{Q} and \\lambda \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} = \\frac{F_{g} \\lambda^{2}}{Q} and \\lambda \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} = \\frac{F_{g} \\lambda^{2}}{Q} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} and \\sin{(\\lambda \\Psi^{\\dagger}{(Q,F_{g},\\lambda)} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)})} = \\sin{(\\frac{F_{g} \\lambda^{2}}{Q} + 2 \\Psi^{\\dagger}{(Q,F_{g},\\lambda)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))))"], [["add", 2, "Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True)))), Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["sin", 3], "Equality(sin(Add(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True))))), sin(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(v_{x},\\rho_f)} = \\rho_f v_{x} and \\dot{y}{(v_{x},\\rho_f)} = \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)}, then obtain \\dot{y}{(v_{x},\\rho_f)} + \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)} = 2 \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)}", "derivation": "\\hat{H}_{\\lambda}{(v_{x},\\rho_f)} = \\rho_f v_{x} and \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)} = (\\rho_f v_{x})^{v_{x}} and (\\rho_f v_{x})^{v_{x}} + \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)} = 2 (\\rho_f v_{x})^{v_{x}} and \\dot{y}{(v_{x},\\rho_f)} = \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)} and (\\rho_f v_{x})^{v_{x}} + \\dot{y}{(v_{x},\\rho_f)} = 2 (\\rho_f v_{x})^{v_{x}} and \\dot{y}{(v_{x},\\rho_f)} + \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)} = 2 \\hat{H}_{\\lambda}^{v_{x}}{(v_{x},\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('v_x', commutative=True)), Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["add", 2, "Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('v_x', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('v_x', commutative=True))), Mul(Integer(2), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(Q,s)} = \\frac{s}{Q}, then obtain \\int (s + 2 \\operatorname{F_{g}}{(Q,s)}) dQ = \\int (s + \\operatorname{F_{g}}{(Q,s)} + \\frac{s}{Q}) dQ", "derivation": "\\operatorname{F_{g}}{(Q,s)} = \\frac{s}{Q} and 2 \\operatorname{F_{g}}{(Q,s)} = \\operatorname{F_{g}}{(Q,s)} + \\frac{s}{Q} and s + 2 \\operatorname{F_{g}}{(Q,s)} = s + \\operatorname{F_{g}}{(Q,s)} + \\frac{s}{Q} and \\int (s + 2 \\operatorname{F_{g}}{(Q,s)}) dQ = \\int (s + \\operatorname{F_{g}}{(Q,s)} + \\frac{s}{Q}) dQ", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["add", 1, "Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Integer(2), Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True))), Add(Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Mul(Integer(2), Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)))), Add(Symbol('s', commutative=True), Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('s', commutative=True), Mul(Integer(2), Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('s', commutative=True), Function('F_g')(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\rho{(t_{2})} = e^{t_{2}}, then derive \\int \\rho{(t_{2})} dt_{2} = F_{N} + e^{t_{2}}, then obtain \\frac{\\int \\rho{(t_{2})} dt_{2}}{\\mathbf{A}} = \\frac{F_{N} + \\rho{(t_{2})}}{\\mathbf{A}}", "derivation": "\\rho{(t_{2})} = e^{t_{2}} and \\int \\rho{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2} and \\int \\rho{(t_{2})} dt_{2} = F_{N} + e^{t_{2}} and \\int \\rho{(t_{2})} dt_{2} = F_{N} + \\rho{(t_{2})} and \\frac{\\int \\rho{(t_{2})} dt_{2}}{\\mathbf{A}} = \\frac{F_{N} + \\rho{(t_{2})}}{\\mathbf{A}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\rho')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('F_N', commutative=True), Function('\\\\rho')(Symbol('t_2', commutative=True))))"], [["divide", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Integral(Function('\\\\rho')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Function('\\\\rho')(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given t{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\sigma_{x}{(\\mathbb{I})} = - \\frac{\\cos{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}}, then obtain \\sigma_{x}{(\\mathbb{I})} + t{(\\mathbb{I})} - \\cos{(\\mathbb{I})} = t{(\\mathbb{I})} - \\cos{(\\mathbb{I})} - \\frac{t{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}}", "derivation": "t{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\frac{t{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}} = \\frac{\\cos{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}} and \\sigma_{x}{(\\mathbb{I})} = - \\frac{\\cos{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}} and \\sigma_{x}{(\\mathbb{I})} = - \\frac{t{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}} and \\sigma_{x}{(\\mathbb{I})} + t{(\\mathbb{I})} - \\cos{(\\mathbb{I})} = t{(\\mathbb{I})} - \\cos{(\\mathbb{I})} - \\frac{t{(\\mathbb{I})}}{t{(\\mathbb{I})} - \\cos{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], "Equality(Mul(Pow(Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(-1)), Function('t')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(-1)), Function('t')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 4, "Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\mathbb{I}', commutative=True)), Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Add(Function('t')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(-1)), Function('t')(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(u,\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\tilde{g}^* u, then derive \\operatorname{v_{y}}{(u,\\tilde{g}^*)} = u, then obtain \\frac{\\partial}{\\partial u} \\sin{(\\int \\operatorname{v_{y}}{(u,\\tilde{g}^*)} du)} = \\frac{d}{d u} \\sin{(\\int u du)}", "derivation": "\\operatorname{v_{y}}{(u,\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\tilde{g}^* u and \\int \\operatorname{v_{y}}{(u,\\tilde{g}^*)} du = \\int \\frac{\\partial}{\\partial \\tilde{g}^*} \\tilde{g}^* u du and \\operatorname{v_{y}}{(u,\\tilde{g}^*)} = u and \\sin{(\\int \\operatorname{v_{y}}{(u,\\tilde{g}^*)} du)} = \\sin{(\\int \\frac{\\partial}{\\partial \\tilde{g}^*} \\tilde{g}^* u du)} and \\int u du = \\int \\frac{\\partial}{\\partial \\tilde{g}^*} \\tilde{g}^* u du and \\sin{(\\int \\operatorname{v_{y}}{(u,\\tilde{g}^*)} du)} = \\sin{(\\int u du)} and \\frac{\\partial}{\\partial u} \\sin{(\\int \\operatorname{v_{y}}{(u,\\tilde{g}^*)} du)} = \\frac{d}{d u} \\sin{(\\int u du)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True))"], [["sin", 2], "Equality(sin(Integral(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('u', commutative=True)))), sin(Integral(Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(sin(Integral(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('u', commutative=True)))), sin(Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True)))))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(sin(Integral(Function('v_y')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(sin(Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{2})} = e^{v_{2}}, then derive \\int \\mathbf{f}{(v_{2})} dv_{2} = \\tilde{g}^* + e^{v_{2}}, then obtain \\tilde{g}^* + \\mathbf{f}{(v_{2})} = \\tilde{g}^* + e^{v_{2}}", "derivation": "\\mathbf{f}{(v_{2})} = e^{v_{2}} and \\int \\mathbf{f}{(v_{2})} dv_{2} = \\int e^{v_{2}} dv_{2} and \\int \\mathbf{f}{(v_{2})} dv_{2} = \\tilde{g}^* + e^{v_{2}} and \\int \\mathbf{f}{(v_{2})} dv_{2} = \\tilde{g}^* + \\mathbf{f}{(v_{2})} and \\tilde{g}^* + \\mathbf{f}{(v_{2})} = \\tilde{g}^* + e^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{f}')(Symbol('v_2', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\rho_b,l)} = \\frac{l}{\\rho_b} and \\operatorname{z^{*}}{(\\rho_b,l)} = \\frac{l}{\\rho_b}, then derive \\frac{\\frac{\\partial}{\\partial l} \\phi{(\\rho_b,l)}}{\\rho_b} = \\frac{1}{\\rho_b^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial l} \\frac{l}{\\rho_b}}{\\rho_b} = \\frac{1}{\\rho_b^{2}}", "derivation": "\\phi{(\\rho_b,l)} = \\frac{l}{\\rho_b} and \\frac{\\partial}{\\partial l} \\phi{(\\rho_b,l)} = \\frac{\\partial}{\\partial l} \\frac{l}{\\rho_b} and \\frac{\\frac{\\partial}{\\partial l} \\phi{(\\rho_b,l)}}{\\rho_b} = \\frac{\\frac{\\partial}{\\partial l} \\frac{l}{\\rho_b}}{\\rho_b} and \\frac{\\frac{\\partial}{\\partial l} \\phi{(\\rho_b,l)}}{\\rho_b} = \\frac{1}{\\rho_b^{2}} and \\operatorname{z^{*}}{(\\rho_b,l)} = \\frac{l}{\\rho_b} and \\operatorname{z^{*}}{(\\rho_b,l)} = \\phi{(\\rho_b,l)} and \\frac{\\frac{\\partial}{\\partial l} \\operatorname{z^{*}}{(\\rho_b,l)}}{\\rho_b} = \\frac{1}{\\rho_b^{2}} and \\frac{\\frac{\\partial}{\\partial l} \\frac{l}{\\rho_b}}{\\rho_b} = \\frac{1}{\\rho_b^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 2, "Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2)))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('z^*')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(n)} = e^{n}, then obtain (\\frac{d}{d n} \\hat{\\mathbf{r}}{(n)} + 1)^{n} = (e^{n} + 1)^{n}", "derivation": "\\hat{\\mathbf{r}}{(n)} = e^{n} and n + \\hat{\\mathbf{r}}{(n)} = n + e^{n} and \\frac{d}{d n} (n + \\hat{\\mathbf{r}}{(n)}) = \\frac{d}{d n} (n + e^{n}) and (\\frac{d}{d n} (n + \\hat{\\mathbf{r}}{(n)}))^{n} = (\\frac{d}{d n} (n + e^{n}))^{n} and (\\frac{d}{d n} \\hat{\\mathbf{r}}{(n)} + 1)^{n} = (e^{n} + 1)^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('n', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('n', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)), Symbol('n', commutative=True)), Pow(Add(exp(Symbol('n', commutative=True)), Integer(1)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(C_{1},F_{c})} = \\log{(F_{c}^{C_{1}})}, then derive \\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(C_{1},F_{c})} = \\log{(F_{c})}, then obtain (\\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(C_{1},F_{c})})^{F_{c}} = \\log{(F_{c})}^{F_{c}}", "derivation": "\\mathbb{I}{(C_{1},F_{c})} = \\log{(F_{c}^{C_{1}})} and \\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(C_{1},F_{c})} = \\frac{\\partial}{\\partial C_{1}} \\log{(F_{c}^{C_{1}})} and \\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(C_{1},F_{c})} = \\log{(F_{c})} and \\frac{\\partial}{\\partial C_{1}} \\log{(F_{c}^{C_{1}})} = \\log{(F_{c})} and (\\frac{\\partial}{\\partial C_{1}} \\log{(F_{c}^{C_{1}})})^{F_{c}} = \\log{(F_{c})}^{F_{c}} and (\\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(C_{1},F_{c})})^{F_{c}} = \\log{(F_{c})}^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), log(Pow(Symbol('F_c', commutative=True), Symbol('C_1', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('F_c', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), log(Symbol('F_c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Pow(Symbol('F_c', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), log(Symbol('F_c', commutative=True)))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Derivative(log(Pow(Symbol('F_c', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\phi)} = \\log{(\\sin{(\\phi)})}, then obtain \\psi^* \\int \\frac{d}{d \\phi} \\operatorname{A_{z}}{(\\phi)} d\\phi = \\psi^* \\int \\frac{d}{d \\phi} \\log{(\\sin{(\\phi)})} d\\phi", "derivation": "\\operatorname{A_{z}}{(\\phi)} = \\log{(\\sin{(\\phi)})} and \\frac{d}{d \\phi} \\operatorname{A_{z}}{(\\phi)} = \\frac{d}{d \\phi} \\log{(\\sin{(\\phi)})} and \\int \\frac{d}{d \\phi} \\operatorname{A_{z}}{(\\phi)} d\\phi = \\int \\frac{d}{d \\phi} \\log{(\\sin{(\\phi)})} d\\phi and \\psi^* \\int \\frac{d}{d \\phi} \\operatorname{A_{z}}{(\\phi)} d\\phi = \\psi^* \\int \\frac{d}{d \\phi} \\log{(\\sin{(\\phi)})} d\\phi", "srepr_derivation": [["get_premise", "Equality(Function('A_z')(Symbol('\\\\phi', commutative=True)), log(sin(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Derivative(Function('A_z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Derivative(log(sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["times", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Integral(Derivative(Function('A_z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Symbol('\\\\psi^*', commutative=True), Integral(Derivative(log(sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\theta_1)} = \\int \\log{(\\theta_1)} d\\theta_1, then obtain \\frac{\\partial}{\\partial \\theta_1} (\\delta \\operatorname{C_{2}}{(\\theta_1)} + \\delta \\int \\log{(\\theta_1)} d\\theta_1) = \\frac{\\partial}{\\partial \\theta_1} 2 \\delta \\int \\log{(\\theta_1)} d\\theta_1", "derivation": "\\operatorname{C_{2}}{(\\theta_1)} = \\int \\log{(\\theta_1)} d\\theta_1 and - \\operatorname{C_{2}}{(\\theta_1)} = - \\int \\log{(\\theta_1)} d\\theta_1 and \\delta \\operatorname{C_{2}}{(\\theta_1)} = \\delta \\int \\log{(\\theta_1)} d\\theta_1 and \\delta \\operatorname{C_{2}}{(\\theta_1)} + \\delta \\int \\log{(\\theta_1)} d\\theta_1 = 2 \\delta \\int \\log{(\\theta_1)} d\\theta_1 and \\frac{\\partial}{\\partial \\theta_1} (\\delta \\operatorname{C_{2}}{(\\theta_1)} + \\delta \\int \\log{(\\theta_1)} d\\theta_1) = \\frac{\\partial}{\\partial \\theta_1} 2 \\delta \\int \\log{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('C_2')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('C_2')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 3, "Mul(Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('C_2')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), Mul(Integer(2), Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\delta', commutative=True), Function('C_2')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(Z,z,C_{d})} = C_{d} z + Z, then obtain \\frac{- C_{d} z + f{(Z,z,C_{d})}}{\\frac{\\partial}{\\partial C_{d}} (C_{d} z + Z)} = \\frac{Z}{\\frac{\\partial}{\\partial C_{d}} (C_{d} z + Z)}", "derivation": "f{(Z,z,C_{d})} = C_{d} z + Z and - C_{d} z + f{(Z,z,C_{d})} = Z and \\frac{\\partial}{\\partial C_{d}} f{(Z,z,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (C_{d} z + Z) and \\frac{- C_{d} z + f{(Z,z,C_{d})}}{\\frac{\\partial}{\\partial C_{d}} f{(Z,z,C_{d})}} = \\frac{Z}{\\frac{\\partial}{\\partial C_{d}} f{(Z,z,C_{d})}} and \\frac{- C_{d} z + f{(Z,z,C_{d})}}{\\frac{\\partial}{\\partial C_{d}} (C_{d} z + Z)} = \\frac{Z}{\\frac{\\partial}{\\partial C_{d}} (C_{d} z + Z)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('Z', commutative=True)))"], [["minus", 1, "Mul(Symbol('C_d', commutative=True), Symbol('z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True))), Symbol('Z', commutative=True))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True))), Pow(Derivative(Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('Z', commutative=True), Pow(Derivative(Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Function('f')(Symbol('Z', commutative=True), Symbol('z', commutative=True), Symbol('C_d', commutative=True))), Pow(Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('Z', commutative=True), Pow(Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(A)} = e^{A} and \\mathbf{f}{(A)} = 2 \\tilde{g}^*{(A)}, then obtain \\log{(2 (\\tilde{g}^*{(A)} + e^{A}) e^{A})} = \\log{(2 (\\tilde{g}^*{(A)} + e^{A}) \\tilde{g}^*{(A)})}", "derivation": "\\tilde{g}^*{(A)} = e^{A} and \\mathbf{f}{(A)} = 2 \\tilde{g}^*{(A)} and (\\tilde{g}^*{(A)} + e^{A}) \\mathbf{f}{(A)} = 2 (\\tilde{g}^*{(A)} + e^{A}) \\tilde{g}^*{(A)} and \\mathbf{f}{(A)} = 2 e^{A} and \\log{((\\tilde{g}^*{(A)} + e^{A}) \\mathbf{f}{(A)})} = \\log{(2 (\\tilde{g}^*{(A)} + e^{A}) \\tilde{g}^*{(A)})} and \\log{(2 (\\tilde{g}^*{(A)} + e^{A}) e^{A})} = \\log{(2 (\\tilde{g}^*{(A)} + e^{A}) \\tilde{g}^*{(A)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('A', commutative=True))))"], [["times", 2, "Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], "Equality(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('\\\\mathbf{f}')(Symbol('A', commutative=True))), Mul(Integer(2), Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Mul(Integer(2), exp(Symbol('A', commutative=True))))"], [["log", 3], "Equality(log(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('\\\\mathbf{f}')(Symbol('A', commutative=True)))), log(Mul(Integer(2), Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(log(Mul(Integer(2), Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), exp(Symbol('A', commutative=True)))), log(Mul(Integer(2), Add(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{v})} = \\log{(\\mathbf{v})}, then derive - \\mathbf{v} \\log{(\\mathbf{v})} + \\mathbf{v} - n + \\int \\mathbf{f}{(\\mathbf{v})} d\\mathbf{v} = 0, then obtain - \\mathbf{v} \\log{(\\mathbf{v})} + \\mathbf{v} - n + \\int \\log{(\\mathbf{v})} d\\mathbf{v} = 0", "derivation": "\\mathbf{f}{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\int \\mathbf{f}{(\\mathbf{v})} d\\mathbf{v} = \\int \\log{(\\mathbf{v})} d\\mathbf{v} and \\int \\mathbf{f}{(\\mathbf{v})} d\\mathbf{v} - \\int \\log{(\\mathbf{v})} d\\mathbf{v} = 0 and - \\mathbf{v} \\log{(\\mathbf{v})} + \\mathbf{v} - n + \\int \\mathbf{f}{(\\mathbf{v})} d\\mathbf{v} = 0 and - \\mathbf{v} \\log{(\\mathbf{v})} + \\mathbf{v} - n + \\int \\log{(\\mathbf{v})} d\\mathbf{v} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 2, "Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given C{(E,\\hbar)} = \\sin{(E^{\\hbar})}, then obtain - \\frac{(C{(E,\\hbar)} - \\frac{C{(E,\\hbar)}}{\\hbar}) \\sin{(E^{\\hbar})}}{\\hbar^{2}} = - \\frac{(\\sin{(E^{\\hbar})} - \\frac{C{(E,\\hbar)}}{\\hbar}) \\sin{(E^{\\hbar})}}{\\hbar^{2}}", "derivation": "C{(E,\\hbar)} = \\sin{(E^{\\hbar})} and \\frac{C{(E,\\hbar)}}{\\hbar} = \\frac{\\sin{(E^{\\hbar})}}{\\hbar} and C{(E,\\hbar)} - \\frac{\\sin{(E^{\\hbar})}}{\\hbar} = \\sin{(E^{\\hbar})} - \\frac{\\sin{(E^{\\hbar})}}{\\hbar} and \\frac{C{(E,\\hbar)} - \\frac{\\sin{(E^{\\hbar})}}{\\hbar}}{\\hbar} = \\frac{\\sin{(E^{\\hbar})} - \\frac{\\sin{(E^{\\hbar})}}{\\hbar}}{\\hbar} and \\frac{C{(E,\\hbar)} - \\frac{C{(E,\\hbar)}}{\\hbar}}{\\hbar} = \\frac{\\sin{(E^{\\hbar})} - \\frac{C{(E,\\hbar)}}{\\hbar}}{\\hbar} and - \\frac{(C{(E,\\hbar)} - \\frac{C{(E,\\hbar)}}{\\hbar}) \\sin{(E^{\\hbar})}}{\\hbar^{2}} = - \\frac{(\\sin{(E^{\\hbar})} - \\frac{C{(E,\\hbar)}}{\\hbar}) \\sin{(E^{\\hbar})}}{\\hbar^{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))"], "Equality(Add(Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))), Add(sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))))"], [["times", 3, "Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))))"], [["times", 5, "Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Add(Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Add(sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('C')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given B{(\\mathbf{D},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}), then derive B{(\\mathbf{D},\\Omega)} = 1, then derive \\mathbf{D} + \\sigma_x = \\mathbf{D} + r_{0}, then obtain \\mathbf{D} + \\sigma_x + (\\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}))^{\\mathbf{D}} = \\mathbf{D} + r_{0} + (\\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}))^{\\mathbf{D}}", "derivation": "B{(\\mathbf{D},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}) and B{(\\mathbf{D},\\Omega)} = 1 and \\int B{(\\mathbf{D},\\Omega)} d\\mathbf{D} = \\int 1 d\\mathbf{D} and \\int \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}) d\\mathbf{D} = \\int 1 d\\mathbf{D} and \\mathbf{D} + \\sigma_x = \\mathbf{D} + r_{0} and \\mathbf{D} + \\sigma_x + (\\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}))^{\\mathbf{D}} = \\mathbf{D} + r_{0} + (\\frac{\\partial}{\\partial \\Omega} (\\Omega + \\mathbf{D}))^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('B')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 5, "Pow(Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Pow(Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('r_0', commutative=True), Pow(Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(v_{x})} = \\sin{(v_{x})}, then obtain \\frac{- \\operatorname{v_{z}}{(v_{x})} + \\sin{(\\operatorname{v_{z}}{(v_{x})})} + \\sin{(\\sin{(v_{x})})}}{\\sin{(v_{x})}} = \\frac{- \\operatorname{v_{z}}{(v_{x})} + 2 \\sin{(\\sin{(v_{x})})}}{\\sin{(v_{x})}}", "derivation": "\\operatorname{v_{z}}{(v_{x})} = \\sin{(v_{x})} and \\sin{(\\operatorname{v_{z}}{(v_{x})})} = \\sin{(\\sin{(v_{x})})} and - \\operatorname{v_{z}}{(v_{x})} + \\sin{(\\operatorname{v_{z}}{(v_{x})})} + \\sin{(\\sin{(v_{x})})} = - \\operatorname{v_{z}}{(v_{x})} + 2 \\sin{(\\sin{(v_{x})})} and \\frac{- \\operatorname{v_{z}}{(v_{x})} + \\sin{(\\operatorname{v_{z}}{(v_{x})})} + \\sin{(\\sin{(v_{x})})}}{\\sin{(v_{x})}} = \\frac{- \\operatorname{v_{z}}{(v_{x})} + 2 \\sin{(\\sin{(v_{x})})}}{\\sin{(v_{x})}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["sin", 1], "Equality(sin(Function('v_z')(Symbol('v_x', commutative=True))), sin(sin(Symbol('v_x', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('v_z')(Symbol('v_x', commutative=True))), sin(sin(Symbol('v_x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('v_z')(Symbol('v_x', commutative=True))), sin(Function('v_z')(Symbol('v_x', commutative=True))), sin(sin(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Function('v_z')(Symbol('v_x', commutative=True))), Mul(Integer(2), sin(sin(Symbol('v_x', commutative=True))))))"], [["divide", 3, "sin(Symbol('v_x', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('v_z')(Symbol('v_x', commutative=True))), sin(Function('v_z')(Symbol('v_x', commutative=True))), sin(sin(Symbol('v_x', commutative=True)))), Pow(sin(Symbol('v_x', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Function('v_z')(Symbol('v_x', commutative=True))), Mul(Integer(2), sin(sin(Symbol('v_x', commutative=True))))), Pow(sin(Symbol('v_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho{(C,u)} = C u and \\operatorname{V_{\\mathbf{B}}}{(C,u)} = C u, then obtain C u + \\operatorname{V_{\\mathbf{B}}}{(C,u)} = 2 \\operatorname{V_{\\mathbf{B}}}{(C,u)}", "derivation": "\\rho{(C,u)} = C u and C u + \\rho{(C,u)} = 2 C u and \\operatorname{V_{\\mathbf{B}}}{(C,u)} = C u and \\operatorname{V_{\\mathbf{B}}}{(C,u)} + \\rho{(C,u)} = 2 \\operatorname{V_{\\mathbf{B}}}{(C,u)} and C u + \\operatorname{V_{\\mathbf{B}}}{(C,u)} = 2 \\operatorname{V_{\\mathbf{B}}}{(C,u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Symbol('C', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Function('\\\\rho')(Symbol('C', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('C', commutative=True), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Function('\\\\rho')(Symbol('C', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given h{(y)} = e^{y}, then obtain \\frac{\\partial}{\\partial W} \\cos{(\\int \\frac{\\frac{d}{d y} h^{y}{(y)}}{W} dy)} = \\frac{\\partial}{\\partial W} \\cos{(\\int \\frac{\\frac{d}{d y} (e^{y})^{y}}{W} dy)}", "derivation": "h{(y)} = e^{y} and h^{y}{(y)} = (e^{y})^{y} and \\frac{d}{d y} h^{y}{(y)} = \\frac{d}{d y} (e^{y})^{y} and \\frac{\\frac{d}{d y} h^{y}{(y)}}{W} = \\frac{\\frac{d}{d y} (e^{y})^{y}}{W} and \\int \\frac{\\frac{d}{d y} h^{y}{(y)}}{W} dy = \\int \\frac{\\frac{d}{d y} (e^{y})^{y}}{W} dy and \\cos{(\\int \\frac{\\frac{d}{d y} h^{y}{(y)}}{W} dy)} = \\cos{(\\int \\frac{\\frac{d}{d y} (e^{y})^{y}}{W} dy)} and \\frac{\\partial}{\\partial W} \\cos{(\\int \\frac{\\frac{d}{d y} h^{y}{(y)}}{W} dy)} = \\frac{\\partial}{\\partial W} \\cos{(\\int \\frac{\\frac{d}{d y} (e^{y})^{y}}{W} dy)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True))))"], [["cos", 5], "Equality(cos(Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True)))), cos(Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True)))))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(cos(Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(cos(Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(A_{1})} = \\frac{d}{d A_{1}} e^{A_{1}}, then derive I{(A_{1})} = e^{A_{1}}, then obtain I^{2 A_{1}}{(A_{1})} = (\\frac{d^{2}}{d A_{1}^{2}} e^{A_{1}})^{2 A_{1}}", "derivation": "I{(A_{1})} = \\frac{d}{d A_{1}} e^{A_{1}} and I{(A_{1})} = e^{A_{1}} and e^{A_{1}} = \\frac{d}{d A_{1}} e^{A_{1}} and I{(A_{1})} = \\frac{d^{2}}{d A_{1}^{2}} e^{A_{1}} and I^{A_{1}}{(A_{1})} = (\\frac{d^{2}}{d A_{1}^{2}} e^{A_{1}})^{A_{1}} and I^{2 A_{1}}{(A_{1})} = (\\frac{d^{2}}{d A_{1}^{2}} e^{A_{1}})^{2 A_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('A_1', commutative=True)), Derivative(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('I')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('A_1', commutative=True)), Derivative(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('I')(Symbol('A_1', commutative=True)), Derivative(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('I')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))), Symbol('A_1', commutative=True)))"], [["power", 5, 2], "Equality(Pow(Function('I')(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True))), Pow(Derivative(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(2), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(A_{z})} = \\frac{d}{d A_{z}} \\log{(A_{z})}, then derive \\operatorname{v_{1}}{(A_{z})} = \\frac{1}{A_{z}}, then obtain \\operatorname{v_{1}}{(A_{z})} \\log{(A_{z})} = \\frac{\\log{(A_{z})}}{A_{z}}", "derivation": "\\operatorname{v_{1}}{(A_{z})} = \\frac{d}{d A_{z}} \\log{(A_{z})} and \\operatorname{v_{1}}{(A_{z})} \\log{(A_{z})} = \\log{(A_{z})} \\frac{d}{d A_{z}} \\log{(A_{z})} and \\operatorname{v_{1}}{(A_{z})} = \\frac{1}{A_{z}} and \\frac{d}{d A_{z}} \\log{(A_{z})} = \\frac{1}{A_{z}} and \\operatorname{v_{1}}{(A_{z})} \\log{(A_{z})} = \\frac{\\log{(A_{z})}}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('A_z', commutative=True)), Derivative(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["times", 1, "log(Symbol('A_z', commutative=True))"], "Equality(Mul(Function('v_1')(Symbol('A_z', commutative=True)), log(Symbol('A_z', commutative=True))), Mul(log(Symbol('A_z', commutative=True)), Derivative(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('v_1')(Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Pow(Symbol('A_z', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Function('v_1')(Symbol('A_z', commutative=True)), log(Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), log(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(t)} = e^{t}, then obtain \\int \\frac{(\\operatorname{n_{1}}{(t)} + e^{t}) e^{- t}}{2} dt = \\int 1 dt", "derivation": "\\operatorname{n_{1}}{(t)} = e^{t} and \\operatorname{n_{1}}{(t)} + e^{t} = 2 e^{t} and \\frac{(\\operatorname{n_{1}}{(t)} + e^{t}) e^{- t}}{2} = 1 and \\int \\frac{(\\operatorname{n_{1}}{(t)} + e^{t}) e^{- t}}{2} dt = \\int 1 dt", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["add", 1, "exp(Symbol('t', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))), Mul(Integer(2), exp(Symbol('t', commutative=True))))"], [["divide", 2, "Mul(Integer(2), exp(Symbol('t', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('n_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))), exp(Mul(Integer(-1), Symbol('t', commutative=True)))), Integer(1))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Add(Function('n_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))), exp(Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Integer(1), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\pi{(v_{2},r_{0})} = \\frac{v_{2}}{r_{0}}, then obtain - \\sin{(\\pi{(v_{2},r_{0})})} \\frac{\\partial}{\\partial r_{0}} \\pi{(v_{2},r_{0})} = \\frac{v_{2} \\sin{(\\frac{v_{2}}{r_{0}})}}{r_{0}^{2}}", "derivation": "\\pi{(v_{2},r_{0})} = \\frac{v_{2}}{r_{0}} and \\cos{(\\pi{(v_{2},r_{0})})} = \\cos{(\\frac{v_{2}}{r_{0}})} and \\frac{\\partial}{\\partial r_{0}} \\cos{(\\pi{(v_{2},r_{0})})} = \\frac{\\partial}{\\partial r_{0}} \\cos{(\\frac{v_{2}}{r_{0}})} and - \\sin{(\\pi{(v_{2},r_{0})})} \\frac{\\partial}{\\partial r_{0}} \\pi{(v_{2},r_{0})} = \\frac{v_{2} \\sin{(\\frac{v_{2}}{r_{0}})}}{r_{0}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_2', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\pi')(Symbol('v_2', commutative=True), Symbol('r_0', commutative=True))), cos(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(cos(Function('\\\\pi')(Symbol('v_2', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), sin(Function('\\\\pi')(Symbol('v_2', commutative=True), Symbol('r_0', commutative=True))), Derivative(Function('\\\\pi')(Symbol('v_2', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-2)), Symbol('v_2', commutative=True), sin(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(W)} = \\sin{(W)} and p{(W)} = \\sin{(W)}, then obtain \\frac{d}{d W} \\sin^{W}{(W)} = \\frac{d}{d W} \\mathbf{J}_P^{W}{(W)}", "derivation": "\\mathbf{J}_P{(W)} = \\sin{(W)} and p{(W)} = \\sin{(W)} and p^{W}{(W)} = \\sin^{W}{(W)} and \\frac{d}{d W} p^{W}{(W)} = \\frac{d}{d W} \\sin^{W}{(W)} and \\frac{d}{d W} p^{W}{(W)} = \\frac{d}{d W} \\mathbf{J}_P^{W}{(W)} and \\frac{d}{d W} \\sin^{W}{(W)} = \\frac{d}{d W} \\mathbf{J}_P^{W}{(W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Function('p')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Function('p')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Pow(Function('p')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(u,x,C_{2})} = C_{2} x - u and V{(u,x,C_{2})} = C_{2} x - u, then obtain (- u)^{u} + ((C_{2} x - u)^{C_{2}})^{C_{2}} = (- u)^{u} + (V^{C_{2}}{(u,x,C_{2})})^{C_{2}}", "derivation": "\\Psi{(u,x,C_{2})} = C_{2} x - u and \\Psi^{C_{2}}{(u,x,C_{2})} = (C_{2} x - u)^{C_{2}} and (\\Psi^{C_{2}}{(u,x,C_{2})})^{C_{2}} = ((C_{2} x - u)^{C_{2}})^{C_{2}} and V{(u,x,C_{2})} = C_{2} x - u and (\\Psi^{C_{2}}{(u,x,C_{2})})^{C_{2}} = (V^{C_{2}}{(u,x,C_{2})})^{C_{2}} and (- u)^{u} + (\\Psi^{C_{2}}{(u,x,C_{2})})^{C_{2}} = (- u)^{u} + (V^{C_{2}}{(u,x,C_{2})})^{C_{2}} and (- u)^{u} + ((C_{2} x - u)^{C_{2}})^{C_{2}} = (- u)^{u} + (V^{C_{2}}{(u,x,C_{2})})^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Symbol('C_2', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Add(Mul(Symbol('C_2', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('C_2', commutative=True)))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Pow(Add(Mul(Symbol('C_2', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Symbol('C_2', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('\\\\Psi')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Pow(Function('V')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["add", 5, "Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Add(Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Function('\\\\Psi')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Function('V')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Add(Mul(Symbol('C_2', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Function('V')(Symbol('u', commutative=True), Symbol('x', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(f,\\mathbf{B})} = \\log{(\\frac{f}{\\mathbf{B}})}, then derive \\int \\mathbf{r}{(f,\\mathbf{B})} df = \\dot{y} + f \\log{(\\frac{f}{\\mathbf{B}})} - f, then derive f \\log{(\\frac{f}{\\mathbf{B}})} - f + n_{1} = \\dot{y} + f \\log{(\\frac{f}{\\mathbf{B}})} - f, then obtain f \\mathbf{r}{(f,\\mathbf{B})} - f + n_{1} = \\dot{y} + f \\mathbf{r}{(f,\\mathbf{B})} - f", "derivation": "\\mathbf{r}{(f,\\mathbf{B})} = \\log{(\\frac{f}{\\mathbf{B}})} and \\int \\mathbf{r}{(f,\\mathbf{B})} df = \\int \\log{(\\frac{f}{\\mathbf{B}})} df and \\int \\mathbf{r}{(f,\\mathbf{B})} df = \\dot{y} + f \\log{(\\frac{f}{\\mathbf{B}})} - f and \\int \\log{(\\frac{f}{\\mathbf{B}})} df = \\dot{y} + f \\log{(\\frac{f}{\\mathbf{B}})} - f and f \\log{(\\frac{f}{\\mathbf{B}})} - f + n_{1} = \\dot{y} + f \\log{(\\frac{f}{\\mathbf{B}})} - f and f \\mathbf{r}{(f,\\mathbf{B})} - f + n_{1} = \\dot{y} + f \\mathbf{r}{(f,\\mathbf{B})} - f", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Symbol('f', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Symbol('f', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('f', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('n_1', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Symbol('f', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Symbol('f', commutative=True), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('n_1', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Symbol('f', commutative=True), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(z^{*})} = \\cos{(\\cos{(z^{*})})}, then derive - \\sin{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{V_{\\mathbf{B}}}{(z^{*})} = \\sin{(z^{*})} \\sin{(\\cos{(z^{*})})} - \\sin{(z^{*})}, then obtain - \\sin{(z^{*})} + \\frac{d}{d z^{*}} \\cos{(\\cos{(z^{*})})} = \\sin{(z^{*})} \\sin{(\\cos{(z^{*})})} - \\sin{(z^{*})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(z^{*})} = \\cos{(\\cos{(z^{*})})} and \\operatorname{V_{\\mathbf{B}}}{(z^{*})} + \\cos{(z^{*})} = \\cos{(z^{*})} + \\cos{(\\cos{(z^{*})})} and \\frac{d}{d z^{*}} (\\operatorname{V_{\\mathbf{B}}}{(z^{*})} + \\cos{(z^{*})}) = \\frac{d}{d z^{*}} (\\cos{(z^{*})} + \\cos{(\\cos{(z^{*})})}) and - \\sin{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{V_{\\mathbf{B}}}{(z^{*})} = \\sin{(z^{*})} \\sin{(\\cos{(z^{*})})} - \\sin{(z^{*})} and - \\sin{(z^{*})} + \\frac{d}{d z^{*}} \\cos{(\\cos{(z^{*})})} = \\sin{(z^{*})} \\sin{(\\cos{(z^{*})})} - \\sin{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('z^*', commutative=True)), cos(cos(Symbol('z^*', commutative=True))))"], [["add", 1, "cos(Symbol('z^*', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True))), Add(cos(Symbol('z^*', commutative=True)), cos(cos(Symbol('z^*', commutative=True)))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Add(Function('V_{\\\\mathbf{B}}')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('z^*', commutative=True)), cos(cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('z^*', commutative=True))), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(Mul(sin(Symbol('z^*', commutative=True)), sin(cos(Symbol('z^*', commutative=True)))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('z^*', commutative=True))), Derivative(cos(cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(Mul(sin(Symbol('z^*', commutative=True)), sin(cos(Symbol('z^*', commutative=True)))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(H,E_{\\lambda})} = \\log{(E_{\\lambda} H)}, then obtain ((- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\mathbf{p}{(H,E_{\\lambda})})^{H})^{H} = ((- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} H)})^{H})^{H}", "derivation": "\\mathbf{p}{(H,E_{\\lambda})} = \\log{(E_{\\lambda} H)} and \\frac{\\partial}{\\partial E_{\\lambda}} \\mathbf{p}{(H,E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} H)} and - H + \\frac{\\partial}{\\partial E_{\\lambda}} \\mathbf{p}{(H,E_{\\lambda})} = - H + \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} H)} and (- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\mathbf{p}{(H,E_{\\lambda})})^{H} = (- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} H)})^{H} and ((- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\mathbf{p}{(H,E_{\\lambda})})^{H})^{H} = ((- H + \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} H)})^{H})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('H', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), log(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('H', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('H', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(log(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('H', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(log(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('H', commutative=True)))"], [["power", 4, "Symbol('H', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('H', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(log(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(v_{t},V)} = V - v_{t}, then obtain \\operatorname{L_{\\varepsilon}}{(v_{t},V)} + \\int \\frac{\\partial}{\\partial V} (- \\operatorname{L_{\\varepsilon}}{(v_{t},V)} - 1) dV + 1 = \\operatorname{L_{\\varepsilon}}{(v_{t},V)} + \\int \\frac{\\partial}{\\partial V} (- V + v_{t} - 1) dV + 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(v_{t},V)} = V - v_{t} and - \\operatorname{L_{\\varepsilon}}{(v_{t},V)} = - V + v_{t} and - \\operatorname{L_{\\varepsilon}}{(v_{t},V)} - 1 = - V + v_{t} - 1 and \\frac{\\partial}{\\partial V} (- \\operatorname{L_{\\varepsilon}}{(v_{t},V)} - 1) = \\frac{\\partial}{\\partial V} (- V + v_{t} - 1) and \\int \\frac{\\partial}{\\partial V} (- \\operatorname{L_{\\varepsilon}}{(v_{t},V)} - 1) dV = \\int \\frac{\\partial}{\\partial V} (- V + v_{t} - 1) dV and \\operatorname{L_{\\varepsilon}}{(v_{t},V)} + \\int \\frac{\\partial}{\\partial V} (- \\operatorname{L_{\\varepsilon}}{(v_{t},V)} - 1) dV + 1 = \\operatorname{L_{\\varepsilon}}{(v_{t},V)} + \\int \\frac{\\partial}{\\partial V} (- V + v_{t} - 1) dV + 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_t', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_t', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_t', commutative=True), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_t', commutative=True), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["minus", 5, "Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Integer(-1))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True)), Integral(Derivative(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integer(1)), Add(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True), Symbol('V', commutative=True)), Integral(Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_t', commutative=True), Integer(-1)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integer(1)))"]]}, {"prompt": "Given J{(v,\\rho_f)} = - v + \\sin{(\\rho_f)}, then derive \\int J{(v,\\rho_f)} dv = b - \\frac{v^{2}}{2} + v \\sin{(\\rho_f)}, then obtain \\frac{\\int J{(v,\\rho_f)} dv}{v} = \\frac{b - \\frac{v^{2}}{2} + v \\sin{(\\rho_f)}}{v}", "derivation": "J{(v,\\rho_f)} = - v + \\sin{(\\rho_f)} and \\int J{(v,\\rho_f)} dv = \\int (- v + \\sin{(\\rho_f)}) dv and \\int J{(v,\\rho_f)} dv = b - \\frac{v^{2}}{2} + v \\sin{(\\rho_f)} and b - \\frac{v^{2}}{2} + v \\sin{(\\rho_f)} = \\int (- v + \\sin{(\\rho_f)}) dv and \\frac{\\int J{(v,\\rho_f)} dv}{v} = \\frac{\\int (- v + \\sin{(\\rho_f)}) dv}{v} and \\frac{\\int J{(v,\\rho_f)} dv}{v} = \\frac{b - \\frac{v^{2}}{2} + v \\sin{(\\rho_f)}}{v}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('J')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Symbol('v', commutative=True), sin(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('b', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Symbol('v', commutative=True), sin(Symbol('\\\\rho_f', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["divide", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('J')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('J')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('b', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Symbol('v', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(p)} = \\sin{(p)} and \\hat{H}{(p)} = \\Psi_{\\lambda}{(p)} - \\sin{(p)}, then obtain 1 = \\hat{H}^{p}{(p)}", "derivation": "\\Psi_{\\lambda}{(p)} = \\sin{(p)} and \\Psi_{\\lambda}{(p)} - \\sin{(p)} = 0 and \\hat{H}{(p)} = \\Psi_{\\lambda}{(p)} - \\sin{(p)} and \\hat{H}^{p}{(p)} = (\\Psi_{\\lambda}{(p)} - \\sin{(p)})^{p} and \\hat{H}{(p)} = 0 and \\sin{(\\hat{H}{(p)})} = 0 and \\sin^{p}{(\\hat{H}{(p)})} = 0^{p} and \\hat{H}^{p}{(p)} = 0^{p} and 1 = \\hat{H}^{p}{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["minus", 1, "sin(Symbol('p', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Integer(0))"], [["sin", 5], "Equality(sin(Function('\\\\hat{H}')(Symbol('p', commutative=True))), Integer(0))"], [["power", 6, "Symbol('p', commutative=True)"], "Equality(Pow(sin(Function('\\\\hat{H}')(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Integer(0), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Integer(0), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Integer(1), Pow(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = \\mu + \\phi^{\\mathbf{s}}, then derive \\frac{\\partial}{\\partial \\mu} \\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = 1, then obtain - \\mathbf{s} + \\frac{\\partial}{\\partial \\mu} (\\mu + \\phi^{\\mathbf{s}}) = 1 - \\mathbf{s}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = \\mu + \\phi^{\\mathbf{s}} and \\frac{\\partial}{\\partial \\mu} \\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = \\frac{\\partial}{\\partial \\mu} (\\mu + \\phi^{\\mathbf{s}}) and \\frac{\\partial}{\\partial \\mu} \\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = 1 and - \\mathbf{s} + \\frac{\\partial}{\\partial \\mu} \\operatorname{c_{0}}{(\\mathbf{s},\\phi,\\mu)} = 1 - \\mathbf{s} and - \\mathbf{s} + \\frac{\\partial}{\\partial \\mu} (\\mu + \\phi^{\\mathbf{s}}) = 1 - \\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain (\\frac{d}{d f^{\\prime}} \\sigma_{p}{(f^{\\prime})} + 1) e^{f^{\\prime}} = (\\frac{d}{d f^{\\prime}} e^{f^{\\prime}} + 1) e^{f^{\\prime}}", "derivation": "\\sigma_{p}{(f^{\\prime})} = e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\sigma_{p}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\sigma_{p}{(f^{\\prime})} + 1 = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} + 1 and (\\frac{d}{d f^{\\prime}} \\sigma_{p}{(f^{\\prime})} + 1) e^{f^{\\prime}} = (\\frac{d}{d f^{\\prime}} e^{f^{\\prime}} + 1) e^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\sigma_p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1)))"], [["times", 3, "exp(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Derivative(Function('\\\\sigma_p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1)), exp(Symbol('f^{\\\\prime}', commutative=True))), Mul(Add(Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1)), exp(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(G,C_{2})} = G^{C_{2}}, then obtain 1 = \\frac{- (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} G^{C_{2}}}{- (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} \\operatorname{z^{*}}{(G,C_{2})}}", "derivation": "\\operatorname{z^{*}}{(G,C_{2})} = G^{C_{2}} and \\operatorname{z^{*}}^{C_{2}}{(G,C_{2})} = (G^{C_{2}})^{C_{2}} and \\frac{\\partial}{\\partial G} \\operatorname{z^{*}}{(G,C_{2})} = \\frac{\\partial}{\\partial G} G^{C_{2}} and - \\operatorname{z^{*}}^{C_{2}}{(G,C_{2})} + \\frac{\\partial}{\\partial G} \\operatorname{z^{*}}{(G,C_{2})} = - \\operatorname{z^{*}}^{C_{2}}{(G,C_{2})} + \\frac{\\partial}{\\partial G} G^{C_{2}} and - (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} \\operatorname{z^{*}}{(G,C_{2})} = - (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} G^{C_{2}} and 1 = \\frac{- (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} G^{C_{2}}}{- (G^{C_{2}})^{C_{2}} + \\frac{\\partial}{\\partial G} \\operatorname{z^{*}}{(G,C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["minus", 3, "Pow(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["divide", 5, "Add(Mul(Integer(-1), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Pow(Add(Mul(Integer(-1), Pow(Pow(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Derivative(Function('z^*')(Symbol('G', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} = \\chi + e^{\\mathbf{S}}, then obtain \\log{(\\frac{d}{d \\chi} 0)} = \\log{(\\frac{\\partial}{\\partial \\chi} (\\chi - \\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} + e^{\\mathbf{S}}))}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} = \\chi + e^{\\mathbf{S}} and 0 = \\chi - \\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} + e^{\\mathbf{S}} and \\frac{d}{d \\chi} 0 = \\frac{\\partial}{\\partial \\chi} (\\chi - \\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} + e^{\\mathbf{S}}) and \\log{(\\frac{d}{d \\chi} 0)} = \\log{(\\frac{\\partial}{\\partial \\chi} (\\chi - \\operatorname{a^{\\dagger}}{(\\mathbf{S},\\chi)} + e^{\\mathbf{S}}))}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), log(Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}} h)}, then obtain \\frac{\\cos{(\\hat{\\mathbf{r}} + \\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})})}}{\\hat{\\mathbf{r}}} = \\frac{\\cos{(\\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}} h)})}}{\\hat{\\mathbf{r}}}", "derivation": "\\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}} h)} and \\hat{\\mathbf{r}} + \\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}} h)} and \\cos{(\\hat{\\mathbf{r}} + \\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})})} = \\cos{(\\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}} h)})} and \\frac{\\cos{(\\hat{\\mathbf{r}} + \\operatorname{c_{0}}{(h,\\hat{\\mathbf{r}})})}}{\\hat{\\mathbf{r}}} = \\frac{\\cos{(\\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}} h)})}}{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('h', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True))))))"], [["divide", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)))))))"]]}, {"prompt": "Given k{(\\phi,\\mathbf{v})} = \\mathbf{v} - \\phi, then obtain - k{(\\phi,\\mathbf{v})} = - \\frac{k{(\\phi,\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} - \\phi + k{(\\phi,\\mathbf{v})})}{\\frac{\\partial}{\\partial \\mathbf{v}} 2 k{(\\phi,\\mathbf{v})}}", "derivation": "k{(\\phi,\\mathbf{v})} = \\mathbf{v} - \\phi and 2 k{(\\phi,\\mathbf{v})} = \\mathbf{v} - \\phi + k{(\\phi,\\mathbf{v})} and \\frac{\\partial}{\\partial \\mathbf{v}} 2 k{(\\phi,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} - \\phi + k{(\\phi,\\mathbf{v})}) and 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} - \\phi + k{(\\phi,\\mathbf{v})})}{\\frac{\\partial}{\\partial \\mathbf{v}} 2 k{(\\phi,\\mathbf{v})}} and - k{(\\phi,\\mathbf{v})} = - \\frac{k{(\\phi,\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} - \\phi + k{(\\phi,\\mathbf{v})})}{\\frac{\\partial}{\\partial \\mathbf{v}} 2 k{(\\phi,\\mathbf{v})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Mul(Integer(2), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Derivative(Mul(Integer(2), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Integer(-1))))"], [["times", 4, "Mul(Integer(-1), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Derivative(Mul(Integer(2), Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given n{(Z)} = e^{e^{Z}}, then obtain \\frac{d}{d Z} n{(Z)} + \\frac{d}{d Z} e^{e^{Z}} + 1 = 2 \\frac{d}{d Z} e^{e^{Z}} + 1", "derivation": "n{(Z)} = e^{e^{Z}} and \\frac{d}{d Z} n{(Z)} = \\frac{d}{d Z} e^{e^{Z}} and \\frac{d}{d Z} n{(Z)} + 1 = \\frac{d}{d Z} e^{e^{Z}} + 1 and \\frac{d}{d Z} n{(Z)} + \\frac{d}{d Z} e^{e^{Z}} + 1 = 2 \\frac{d}{d Z} e^{e^{Z}} + 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('Z', commutative=True)), exp(exp(Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)))"], [["add", 3, "Derivative(exp(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(2), Derivative(exp(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given y{(F_{x})} = F_{x}, then obtain \\frac{d}{d F_{x}} \\int \\frac{2 y{(F_{x})}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x} = \\frac{d}{d F_{x}} \\int \\frac{2 F_{x}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x}", "derivation": "y{(F_{x})} = F_{x} and 2 y{(F_{x})} = 2 F_{x} and \\frac{2 y{(F_{x})}}{y{(F_{x})} + \\int F_{x} dF_{x}} = \\frac{2 F_{x}}{y{(F_{x})} + \\int F_{x} dF_{x}} and \\int \\frac{2 y{(F_{x})}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x} = \\int \\frac{2 F_{x}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x} and \\frac{d}{d F_{x}} \\int \\frac{2 y{(F_{x})}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x} = \\frac{d}{d F_{x}} \\int \\frac{2 F_{x}}{y{(F_{x})} + \\int F_{x} dF_{x}} dF_{x}", "srepr_derivation": [["renaming_premise", "Equality(Function('y')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], [["divide", 1, "Rational(1, 2)"], "Equality(Mul(Integer(2), Function('y')(Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('F_x', commutative=True)))"], [["divide", 2, "Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1)), Function('y')(Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('F_x', commutative=True), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1)), Function('y')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Integer(2), Symbol('F_x', commutative=True), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1))), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 4, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(2), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1)), Function('y')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), Symbol('F_x', commutative=True), Pow(Add(Function('y')(Symbol('F_x', commutative=True)), Integral(Symbol('F_x', commutative=True), Tuple(Symbol('F_x', commutative=True)))), Integer(-1))), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M}, then obtain \\frac{\\chi{(\\mathbf{J}_M)} - 1}{\\frac{d}{d \\mathbf{J}_M} \\chi{(\\mathbf{J}_M)}} = \\frac{e^{\\mathbf{J}_M} - 1}{\\frac{d}{d \\mathbf{J}_M} \\chi{(\\mathbf{J}_M)}}", "derivation": "\\chi{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} \\chi{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M} and \\chi{(\\mathbf{J}_M)} - 1 = e^{\\mathbf{J}_M} - 1 and \\frac{\\chi{(\\mathbf{J}_M)} - 1}{\\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M}} = \\frac{e^{\\mathbf{J}_M} - 1}{\\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M}} and \\frac{\\chi{(\\mathbf{J}_M)} - 1}{\\frac{d}{d \\mathbf{J}_M} \\chi{(\\mathbf{J}_M)}} = \\frac{e^{\\mathbf{J}_M} - 1}{\\frac{d}{d \\mathbf{J}_M} \\chi{(\\mathbf{J}_M)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))"], [["divide", 3, "Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(i,W)} = i^{W}, then obtain (i^{W})^{- W} (\\operatorname{g^{\\prime}_{\\varepsilon}}^{W}{(i,W)})^{W} = (i^{W})^{- W} ((i^{W})^{W})^{W}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(i,W)} = i^{W} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{W}{(i,W)} = (i^{W})^{W} and (\\operatorname{g^{\\prime}_{\\varepsilon}}^{W}{(i,W)})^{W} = ((i^{W})^{W})^{W} and (i^{W})^{- W} (\\operatorname{g^{\\prime}_{\\varepsilon}}^{W}{(i,W)})^{W} = (i^{W})^{- W} ((i^{W})^{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["divide", 3, "Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(Pow(Pow(Symbol('i', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True))))"]]}, {"prompt": "Given Z{(\\rho_f)} = \\sin{(\\rho_f)}, then derive \\int \\rho_f \\int Z{(\\rho_f)} d\\rho_f d\\rho_f = \\nabla - \\rho_f \\sin{(\\rho_f)} - \\cos{(\\rho_f)}, then obtain \\int \\rho_f \\int \\sin{(\\rho_f)} d\\rho_f d\\rho_f = \\nabla - \\rho_f Z{(\\rho_f)} - \\cos{(\\rho_f)}", "derivation": "Z{(\\rho_f)} = \\sin{(\\rho_f)} and \\int Z{(\\rho_f)} d\\rho_f = \\int \\sin{(\\rho_f)} d\\rho_f and \\rho_f \\int Z{(\\rho_f)} d\\rho_f = \\rho_f \\int \\sin{(\\rho_f)} d\\rho_f and \\int \\rho_f \\int Z{(\\rho_f)} d\\rho_f d\\rho_f = \\int \\rho_f \\int \\sin{(\\rho_f)} d\\rho_f d\\rho_f and \\int \\rho_f \\int Z{(\\rho_f)} d\\rho_f d\\rho_f = \\nabla - \\rho_f \\sin{(\\rho_f)} - \\cos{(\\rho_f)} and \\int \\rho_f \\int \\sin{(\\rho_f)} d\\rho_f d\\rho_f = \\nabla - \\rho_f \\sin{(\\rho_f)} - \\cos{(\\rho_f)} and \\int \\rho_f \\int \\sin{(\\rho_f)} d\\rho_f d\\rho_f = \\nabla - \\rho_f Z{(\\rho_f)} - \\cos{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["times", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Symbol('\\\\rho_f', commutative=True), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\rho_f', commutative=True), Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Symbol('\\\\rho_f', commutative=True), Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Symbol('\\\\rho_f', commutative=True), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integral(Mul(Symbol('\\\\rho_f', commutative=True), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Function('Z')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\operatorname{A_{y}}{(\\hat{\\mathbf{r}})} = 2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}}, then obtain ((2 \\hat{\\mathbf{r}} + \\chi{(\\hat{\\mathbf{r}})})^{2})^{\\hat{\\mathbf{r}}} = ((2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}})^{2})^{\\hat{\\mathbf{r}}}", "derivation": "\\chi{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\operatorname{A_{y}}{(\\hat{\\mathbf{r}})} = 2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}} and \\operatorname{A_{y}}^{2}{(\\hat{\\mathbf{r}})} = (2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}})^{2} and \\operatorname{A_{y}}{(\\hat{\\mathbf{r}})} = 2 \\hat{\\mathbf{r}} + \\chi{(\\hat{\\mathbf{r}})} and (2 \\hat{\\mathbf{r}} + \\chi{(\\hat{\\mathbf{r}})})^{2} = (2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}})^{2} and ((2 \\hat{\\mathbf{r}} + \\chi{(\\hat{\\mathbf{r}})})^{2})^{\\hat{\\mathbf{r}}} = ((2 \\hat{\\mathbf{r}} + e^{\\hat{\\mathbf{r}}})^{2})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('A_y')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_y')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)))"], [["power", 5, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Pow(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(2)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given c{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain c^{2}{(\\sigma_p)} + c{(\\sigma_p)} = c^{2}{(\\sigma_p)} + \\log{(\\sigma_p)}", "derivation": "c{(\\sigma_p)} = \\log{(\\sigma_p)} and c^{2}{(\\sigma_p)} = c{(\\sigma_p)} \\log{(\\sigma_p)} and c{(\\sigma_p)} \\log{(\\sigma_p)} + c{(\\sigma_p)} = c{(\\sigma_p)} \\log{(\\sigma_p)} + \\log{(\\sigma_p)} and c^{2}{(\\sigma_p)} + c{(\\sigma_p)} = c^{2}{(\\sigma_p)} + \\log{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Function('c')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Pow(Function('c')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 1, "Mul(Function('c')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Function('c')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), Function('c')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Function('c')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Pow(Function('c')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Function('c')(Symbol('\\\\sigma_p', commutative=True))), Add(Pow(Function('c')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), log(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given E{(W)} = e^{W} and \\hat{\\mathbf{x}}{(W)} = e^{W}, then obtain \\frac{d^{2}}{d W^{2}} (2 E{(W)})^{W} = \\frac{d^{2}}{d W^{2}} (E{(W)} + e^{W})^{W}", "derivation": "E{(W)} = e^{W} and \\hat{\\mathbf{x}}{(W)} = e^{W} and 2 E{(W)} = E{(W)} + e^{W} and E{(W)} + \\hat{\\mathbf{x}}{(W)} = E{(W)} + e^{W} and (2 E{(W)})^{W} = (E{(W)} + e^{W})^{W} and (2 E{(W)})^{W} = (E{(W)} + \\hat{\\mathbf{x}}{(W)})^{W} and \\frac{d}{d W} (2 E{(W)})^{W} = \\frac{d}{d W} (E{(W)} + \\hat{\\mathbf{x}}{(W)})^{W} and \\frac{d^{2}}{d W^{2}} (2 E{(W)})^{W} = \\frac{d^{2}}{d W^{2}} (E{(W)} + \\hat{\\mathbf{x}}{(W)})^{W} and \\frac{d^{2}}{d W^{2}} (2 E{(W)})^{W} = \\frac{d^{2}}{d W^{2}} (E{(W)} + e^{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["add", 1, "Function('E')(Symbol('W', commutative=True))"], "Equality(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Add(Function('E')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))))"], [["add", 2, "Function('E')(Symbol('W', commutative=True))"], "Equality(Add(Function('E')(Symbol('W', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True))), Add(Function('E')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Function('E')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Function('E')(Symbol('W', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Add(Function('E')(Symbol('W', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Derivative(Pow(Add(Function('E')(Symbol('W', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Derivative(Pow(Mul(Integer(2), Function('E')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Derivative(Pow(Add(Function('E')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} = (e^{p})^{\\sigma_p}, then obtain (\\frac{\\partial}{\\partial \\sigma_p} \\int \\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} d\\sigma_p)^{\\sigma_p} = (\\frac{\\partial}{\\partial \\sigma_p} \\int (e^{p})^{\\sigma_p} d\\sigma_p)^{\\sigma_p}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} = (e^{p})^{\\sigma_p} and \\int \\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} d\\sigma_p = \\int (e^{p})^{\\sigma_p} d\\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} \\int \\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} d\\sigma_p = \\frac{\\partial}{\\partial \\sigma_p} \\int (e^{p})^{\\sigma_p} d\\sigma_p and (\\frac{\\partial}{\\partial \\sigma_p} \\int \\operatorname{f_{\\mathbf{p}}}{(p,\\sigma_p)} d\\sigma_p)^{\\sigma_p} = (\\frac{\\partial}{\\partial \\sigma_p} \\int (e^{p})^{\\sigma_p} d\\sigma_p)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Pow(exp(Symbol('p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Pow(exp(Symbol('p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integral(Pow(exp(Symbol('p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\nabla,\\hbar)} = \\frac{e^{\\hbar}}{\\nabla} and \\operatorname{r_{0}}{(\\nabla,\\hbar)} = \\cos{(2 \\operatorname{M_{E}}{(\\nabla,\\hbar)})}, then obtain \\operatorname{r_{0}}{(\\nabla,\\hbar)} = \\cos{(\\operatorname{M_{E}}{(\\nabla,\\hbar)} + \\frac{e^{\\hbar}}{\\nabla})}", "derivation": "\\operatorname{M_{E}}{(\\nabla,\\hbar)} = \\frac{e^{\\hbar}}{\\nabla} and 2 \\operatorname{M_{E}}{(\\nabla,\\hbar)} = \\operatorname{M_{E}}{(\\nabla,\\hbar)} + \\frac{e^{\\hbar}}{\\nabla} and \\cos{(2 \\operatorname{M_{E}}{(\\nabla,\\hbar)})} = \\cos{(\\operatorname{M_{E}}{(\\nabla,\\hbar)} + \\frac{e^{\\hbar}}{\\nabla})} and \\operatorname{r_{0}}{(\\nabla,\\hbar)} = \\cos{(2 \\operatorname{M_{E}}{(\\nabla,\\hbar)})} and \\operatorname{r_{0}}{(\\nabla,\\hbar)} = \\cos{(\\operatorname{M_{E}}{(\\nabla,\\hbar)} + \\frac{e^{\\hbar}}{\\nabla})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\hbar', commutative=True)))))"], [["cos", 2], "Equality(cos(Mul(Integer(2), Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)))), cos(Add(Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\hbar', commutative=True))))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Integer(2), Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('r_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Add(Function('M_E')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given q{(f^{*})} = e^{f^{*}}, then derive \\frac{d}{d f^{*}} q{(f^{*})} + 1 = e^{f^{*}} + 1, then obtain e^{f^{*}} + 1 = q{(f^{*})} + 1", "derivation": "q{(f^{*})} = e^{f^{*}} and f^{*} + q{(f^{*})} = f^{*} + e^{f^{*}} and \\frac{d}{d f^{*}} (f^{*} + q{(f^{*})}) = \\frac{d}{d f^{*}} (f^{*} + e^{f^{*}}) and \\frac{d}{d f^{*}} q{(f^{*})} + 1 = e^{f^{*}} + 1 and \\frac{d}{d f^{*}} q{(f^{*})} + 1 = q{(f^{*})} + 1 and e^{f^{*}} + 1 = q{(f^{*})} + 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["add", 1, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Function('q')(Symbol('f^*', commutative=True))), Add(Symbol('f^*', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Add(Symbol('f^*', commutative=True), Function('q')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('f^*', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)), Add(Function('q')(Symbol('f^*', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(exp(Symbol('f^*', commutative=True)), Integer(1)), Add(Function('q')(Symbol('f^*', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\eta{(E)} = \\sin{(E)}, then obtain - E (E + \\sin{(E)}) + 1 = - E (E + \\sin{(E)}) - \\eta{(E)} + \\sin{(E)} + 1", "derivation": "\\eta{(E)} = \\sin{(E)} and E + \\eta{(E)} = E + \\sin{(E)} and E (E + \\eta{(E)}) = E (E + \\sin{(E)}) and 0 = - \\eta{(E)} + \\sin{(E)} and - E (E + \\eta{(E)}) = - E (E + \\eta{(E)}) - \\eta{(E)} + \\sin{(E)} and - E (E + \\eta{(E)}) + 1 = - E (E + \\eta{(E)}) - \\eta{(E)} + \\sin{(E)} + 1 and - E (E + \\sin{(E)}) + 1 = - E (E + \\sin{(E)}) - \\eta{(E)} + \\sin{(E)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True))), Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))))"], [["times", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))))"], [["minus", 2, "Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('E', commutative=True))), sin(Symbol('E', commutative=True))))"], [["minus", 4, "Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))), Mul(Integer(-1), Function('\\\\eta')(Symbol('E', commutative=True))), sin(Symbol('E', commutative=True))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))), Integer(1)), Add(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\eta')(Symbol('E', commutative=True)))), Mul(Integer(-1), Function('\\\\eta')(Symbol('E', commutative=True))), sin(Symbol('E', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Integer(1)), Add(Mul(Integer(-1), Symbol('E', commutative=True), Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Mul(Integer(-1), Function('\\\\eta')(Symbol('E', commutative=True))), sin(Symbol('E', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\operatorname{c_{0}}{(\\mathbf{v})} = \\psi^{*}{(\\mathbf{v})} \\log{(\\mathbf{v})}, then obtain \\int \\operatorname{c_{0}}{(\\mathbf{v})} d\\mathbf{v} = \\int \\log{(\\mathbf{v})}^{2} d\\mathbf{v}", "derivation": "\\psi^{*}{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\operatorname{c_{0}}{(\\mathbf{v})} = \\psi^{*}{(\\mathbf{v})} \\log{(\\mathbf{v})} and \\operatorname{c_{0}}{(\\mathbf{v})} = \\log{(\\mathbf{v})}^{2} and \\int \\operatorname{c_{0}}{(\\mathbf{v})} d\\mathbf{v} = \\int \\log{(\\mathbf{v})}^{2} d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Function('\\\\psi^*')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('c_0')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given z{(E)} = \\log{(e^{E})} and G{(E)} = e^{E}, then obtain G{(E)} z{(E)} - \\log{(G{(E)})} = G{(E)} \\log{(G{(E)})} - \\log{(G{(E)})}", "derivation": "z{(E)} = \\log{(e^{E})} and z{(E)} e^{E} = e^{E} \\log{(e^{E})} and G{(E)} = e^{E} and G{(E)} z{(E)} = G{(E)} \\log{(G{(E)})} and G{(E)} z{(E)} - \\log{(G{(E)})} = G{(E)} \\log{(G{(E)})} - \\log{(G{(E)})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E', commutative=True)), log(exp(Symbol('E', commutative=True))))"], [["times", 1, "exp(Symbol('E', commutative=True))"], "Equality(Mul(Function('z')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Mul(exp(Symbol('E', commutative=True)), log(exp(Symbol('E', commutative=True)))))"], ["renaming_premise", "Equality(Function('G')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('G')(Symbol('E', commutative=True)), Function('z')(Symbol('E', commutative=True))), Mul(Function('G')(Symbol('E', commutative=True)), log(Function('G')(Symbol('E', commutative=True)))))"], [["minus", 4, "log(Function('G')(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Function('G')(Symbol('E', commutative=True)), Function('z')(Symbol('E', commutative=True))), Mul(Integer(-1), log(Function('G')(Symbol('E', commutative=True))))), Add(Mul(Function('G')(Symbol('E', commutative=True)), log(Function('G')(Symbol('E', commutative=True)))), Mul(Integer(-1), log(Function('G')(Symbol('E', commutative=True))))))"]]}, {"prompt": "Given c{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and \\hat{x}_0{(\\mathbf{s})} = \\mathbf{s}, then obtain - (\\mathbf{s} + \\hat{x}_0{(\\mathbf{s})} - \\sin{(\\mathbf{s})}) \\sin{(\\mathbf{s})} = - (2 \\mathbf{s} - \\sin{(\\mathbf{s})}) \\sin{(\\mathbf{s})}", "derivation": "c{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and \\hat{x}_0{(\\mathbf{s})} = \\mathbf{s} and \\mathbf{s} + \\hat{x}_0{(\\mathbf{s})} - c{(\\mathbf{s})} = 2 \\mathbf{s} - c{(\\mathbf{s})} and - (\\mathbf{s} + \\hat{x}_0{(\\mathbf{s})} - c{(\\mathbf{s})}) c{(\\mathbf{s})} = - (2 \\mathbf{s} - c{(\\mathbf{s})}) c{(\\mathbf{s})} and - (\\mathbf{s} + \\hat{x}_0{(\\mathbf{s})} - \\sin{(\\mathbf{s})}) \\sin{(\\mathbf{s})} = - (2 \\mathbf{s} - \\sin{(\\mathbf{s})}) \\sin{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))), Function('c')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{s}', commutative=True)))), Function('c')(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True)))), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True)))), sin(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(l,\\hat{H})} = \\hat{H} + l, then derive \\frac{\\int \\operatorname{v_{1}}{(l,\\hat{H})} dl}{\\hat{H} + l} = \\frac{\\hat{H} l + \\hat{x} + \\frac{l^{2}}{2}}{\\hat{H} + l}, then obtain \\hat{H} l + \\frac{\\int (\\hat{H} + l) dl}{\\hat{H} + l} = \\hat{H} l + \\frac{\\hat{H} l + \\hat{x} + \\frac{l^{2}}{2}}{\\hat{H} + l}", "derivation": "\\operatorname{v_{1}}{(l,\\hat{H})} = \\hat{H} + l and \\int \\operatorname{v_{1}}{(l,\\hat{H})} dl = \\int (\\hat{H} + l) dl and \\frac{\\int \\operatorname{v_{1}}{(l,\\hat{H})} dl}{\\hat{H} + l} = \\frac{\\int (\\hat{H} + l) dl}{\\hat{H} + l} and \\frac{\\int \\operatorname{v_{1}}{(l,\\hat{H})} dl}{\\hat{H} + l} = \\frac{\\hat{H} l + \\hat{x} + \\frac{l^{2}}{2}}{\\hat{H} + l} and \\frac{\\int (\\hat{H} + l) dl}{\\hat{H} + l} = \\frac{\\hat{H} l + \\hat{x} + \\frac{l^{2}}{2}}{\\hat{H} + l} and \\hat{H} l + \\frac{\\int (\\hat{H} + l) dl}{\\hat{H} + l} = \\hat{H} l + \\frac{\\hat{H} l + \\hat{x} + \\frac{l^{2}}{2}}{\\hat{H} + l}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('l', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('l', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Integral(Function('v_1')(Symbol('l', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Integral(Function('v_1')(Symbol('l', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))))))"], [["add", 5, "Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))))))"]]}, {"prompt": "Given m{(v_{y},p)} = \\int (p - v_{y}) dv_{y}, then derive m{(v_{y},p)} = \\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2}, then obtain \\frac{\\partial}{\\partial \\rho_f} (\\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2}) = \\frac{\\partial}{\\partial \\rho_f} (m_{s} + p v_{y} - \\frac{v_{y}^{2}}{2})", "derivation": "m{(v_{y},p)} = \\int (p - v_{y}) dv_{y} and m{(v_{y},p)} = \\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2} and \\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2} = \\int (p - v_{y}) dv_{y} and \\frac{\\partial}{\\partial \\rho_f} (\\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2}) = \\frac{\\partial}{\\partial \\rho_f} \\int (p - v_{y}) dv_{y} and \\frac{\\partial}{\\partial \\rho_f} (\\rho_f + p v_{y} - \\frac{v_{y}^{2}}{2}) = \\frac{\\partial}{\\partial \\rho_f} (m_{s} + p v_{y} - \\frac{v_{y}^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Integral(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('m')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Integral(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Symbol('m_s', commutative=True), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\eta,\\theta_2)} = - \\eta + \\theta_2, then obtain \\mathbf{A}^{2 \\eta}{(\\eta,\\theta_2)} = (- \\eta + \\theta_2)^{2 \\eta}", "derivation": "\\mathbf{A}{(\\eta,\\theta_2)} = - \\eta + \\theta_2 and \\mathbf{A}^{\\eta}{(\\eta,\\theta_2)} = (- \\eta + \\theta_2)^{\\eta} and (- \\eta + \\theta_2)^{\\eta} \\mathbf{A}^{\\eta}{(\\eta,\\theta_2)} = (- \\eta + \\theta_2)^{2 \\eta} and \\mathbf{A}^{2 \\eta}{(\\eta,\\theta_2)} = (- \\eta + \\theta_2)^{\\eta} \\mathbf{A}^{\\eta}{(\\eta,\\theta_2)} and \\mathbf{A}^{2 \\eta}{(\\eta,\\theta_2)} = (- \\eta + \\theta_2)^{2 \\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["times", 2, "Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\chi{(\\ddot{x})} = \\phi{(\\ddot{x})} + \\sin{(\\ddot{x})}, then obtain \\int 2 \\phi{(\\ddot{x})} d\\ddot{x} = \\int (\\phi{(\\ddot{x})} + \\sin{(\\ddot{x})}) d\\ddot{x}", "derivation": "\\phi{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\chi{(\\ddot{x})} = \\phi{(\\ddot{x})} + \\sin{(\\ddot{x})} and \\chi{(\\ddot{x})} = 2 \\phi{(\\ddot{x})} and 2 \\phi{(\\ddot{x})} = \\phi{(\\ddot{x})} + \\sin{(\\ddot{x})} and \\int 2 \\phi{(\\ddot{x})} d\\ddot{x} = \\int (\\phi{(\\ddot{x})} + \\sin{(\\ddot{x})}) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\ddot{x}', commutative=True)), Add(Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True))), Add(Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Function('\\\\phi')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(r_{0})} = \\sin{(r_{0})}, then obtain \\operatorname{M_{E}}^{3}{(r_{0})} \\sin{(r_{0})} = \\operatorname{M_{E}}^{2}{(r_{0})} \\sin^{2}{(r_{0})}", "derivation": "\\operatorname{M_{E}}{(r_{0})} = \\sin{(r_{0})} and \\operatorname{M_{E}}{(r_{0})} \\sin{(r_{0})} = \\sin^{2}{(r_{0})} and \\operatorname{M_{E}}^{2}{(r_{0})} \\sin^{2}{(r_{0})} = \\sin^{4}{(r_{0})} and \\operatorname{M_{E}}^{3}{(r_{0})} \\sin{(r_{0})} = \\operatorname{M_{E}}^{2}{(r_{0})} \\sin^{2}{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], [["times", 1, "sin(Symbol('r_0', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))), Pow(sin(Symbol('r_0', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('M_E')(Symbol('r_0', commutative=True)), Integer(2)), Pow(sin(Symbol('r_0', commutative=True)), Integer(2))), Pow(sin(Symbol('r_0', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('M_E')(Symbol('r_0', commutative=True)), Integer(3)), sin(Symbol('r_0', commutative=True))), Mul(Pow(Function('M_E')(Symbol('r_0', commutative=True)), Integer(2)), Pow(sin(Symbol('r_0', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(E,y)} = e^{E y}, then obtain - \\log{(E_{\\lambda})} + \\int \\operatorname{f_{E}}^{y}{(E,y)} dy = - \\log{(E_{\\lambda})} + \\int (e^{E y})^{y} dy", "derivation": "\\operatorname{f_{E}}{(E,y)} = e^{E y} and \\operatorname{f_{E}}^{y}{(E,y)} = (e^{E y})^{y} and \\int \\operatorname{f_{E}}^{y}{(E,y)} dy = \\int (e^{E y})^{y} dy and - \\log{(E_{\\lambda})} + \\int \\operatorname{f_{E}}^{y}{(E,y)} dy = - \\log{(E_{\\lambda})} + \\int (e^{E y})^{y} dy", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('E', commutative=True), Symbol('y', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Mul(Symbol('E', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Pow(exp(Mul(Symbol('E', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["minus", 3, "log(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Pow(exp(Mul(Symbol('E', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(h)} = \\cos{(h)} and Z{(h)} = \\int \\cos{(h)} dh, then obtain s{(r,\\rho_b,L_{\\varepsilon})} + \\int \\cos{(h)} dh = Z{(h)} + s{(r,\\rho_b,L_{\\varepsilon})}", "derivation": "\\theta_{1}{(h)} = \\cos{(h)} and \\int \\theta_{1}{(h)} dh = \\int \\cos{(h)} dh and Z{(h)} = \\int \\cos{(h)} dh and \\int \\theta_{1}{(h)} dh = Z{(h)} and s{(r,\\rho_b,L_{\\varepsilon})} + \\int \\theta_{1}{(h)} dh = Z{(h)} + s{(r,\\rho_b,L_{\\varepsilon})} and s{(r,\\rho_b,L_{\\varepsilon})} + \\int \\theta_{1}{(h)} dh = s{(r,\\rho_b,L_{\\varepsilon})} + \\int \\cos{(h)} dh and s{(r,\\rho_b,L_{\\varepsilon})} + \\int \\cos{(h)} dh = Z{(h)} + s{(r,\\rho_b,L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('h', commutative=True)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\theta_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Function('Z')(Symbol('h', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Function('Z')(Symbol('h', commutative=True)), Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Function('Z')(Symbol('h', commutative=True)), Function('s')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given L{(F_{x})} = \\cos{(F_{x})} and \\varphi{(F_{x})} = \\int \\cos{(F_{x})} dF_{x}, then obtain \\frac{\\varphi^{2}{(F_{x})}}{F_{x}} = \\frac{\\varphi{(F_{x})} \\int \\cos{(F_{x})} dF_{x}}{F_{x}}", "derivation": "L{(F_{x})} = \\cos{(F_{x})} and \\int L{(F_{x})} dF_{x} = \\int \\cos{(F_{x})} dF_{x} and \\varphi{(F_{x})} = \\int \\cos{(F_{x})} dF_{x} and \\varphi{(F_{x})} \\int L{(F_{x})} dF_{x} = \\varphi{(F_{x})} \\int \\cos{(F_{x})} dF_{x} and \\varphi{(F_{x})} = \\int L{(F_{x})} dF_{x} and \\frac{\\varphi{(F_{x})} \\int L{(F_{x})} dF_{x}}{F_{x}} = \\frac{\\varphi{(F_{x})} \\int \\cos{(F_{x})} dF_{x}}{F_{x}} and \\frac{\\varphi^{2}{(F_{x})}}{F_{x}} = \\frac{\\varphi{(F_{x})} \\int \\cos{(F_{x})} dF_{x}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('L')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["times", 2, "Function('\\\\varphi')(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(Function('L')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(Function('L')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["divide", 4, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(Function('L')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integer(2))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True)), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given C{(\\eta^{\\prime},J_{\\varepsilon})} = J_{\\varepsilon} + \\eta^{\\prime}, then obtain \\frac{\\int 1 d\\eta^{\\prime}}{\\frac{d}{d \\eta^{\\prime}} 0} = \\frac{\\int \\cos{(J_{\\varepsilon} + \\eta^{\\prime} - C{(\\eta^{\\prime},J_{\\varepsilon})})} d\\eta^{\\prime}}{\\frac{d}{d \\eta^{\\prime}} 0}", "derivation": "C{(\\eta^{\\prime},J_{\\varepsilon})} = J_{\\varepsilon} + \\eta^{\\prime} and 0 = J_{\\varepsilon} + \\eta^{\\prime} - C{(\\eta^{\\prime},J_{\\varepsilon})} and 1 = \\cos{(J_{\\varepsilon} + \\eta^{\\prime} - C{(\\eta^{\\prime},J_{\\varepsilon})})} and \\int 1 d\\eta^{\\prime} = \\int \\cos{(J_{\\varepsilon} + \\eta^{\\prime} - C{(\\eta^{\\prime},J_{\\varepsilon})})} d\\eta^{\\prime} and \\frac{\\int 1 d\\eta^{\\prime}}{\\frac{d}{d \\eta^{\\prime}} 0} = \\frac{\\int \\cos{(J_{\\varepsilon} + \\eta^{\\prime} - C{(\\eta^{\\prime},J_{\\varepsilon})})} d\\eta^{\\prime}}{\\frac{d}{d \\eta^{\\prime}} 0}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["cos", 2], "Equality(Integer(1), cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 4, "Derivative(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('C')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\pi,E_{\\lambda})} = E_{\\lambda} \\pi, then obtain E_{\\lambda} \\pi - \\Psi^{\\dagger}{(\\pi,E_{\\lambda})} = 2 E_{\\lambda} \\pi - 2 \\Psi^{\\dagger}{(\\pi,E_{\\lambda})}", "derivation": "\\Psi^{\\dagger}{(\\pi,E_{\\lambda})} = E_{\\lambda} \\pi and \\pi + \\Psi^{\\dagger}{(\\pi,E_{\\lambda})} = E_{\\lambda} \\pi + \\pi and 0 = E_{\\lambda} \\pi - \\Psi^{\\dagger}{(\\pi,E_{\\lambda})} and E_{\\lambda} \\pi - \\Psi^{\\dagger}{(\\pi,E_{\\lambda})} = 2 E_{\\lambda} \\pi - 2 \\Psi^{\\dagger}{(\\pi,E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "Add(Symbol('\\\\pi', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["add", 3, "Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\pi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(C_{1})} = e^{C_{1}}, then obtain \\int ((\\frac{d}{d C_{1}} \\operatorname{A_{x}}{(C_{1})})^{C_{1}})^{C_{1}} dC_{1} = \\int ((\\frac{d}{d C_{1}} e^{C_{1}})^{C_{1}})^{C_{1}} dC_{1}", "derivation": "\\operatorname{A_{x}}{(C_{1})} = e^{C_{1}} and \\frac{d}{d C_{1}} \\operatorname{A_{x}}{(C_{1})} = \\frac{d}{d C_{1}} e^{C_{1}} and (\\frac{d}{d C_{1}} \\operatorname{A_{x}}{(C_{1})})^{C_{1}} = (\\frac{d}{d C_{1}} e^{C_{1}})^{C_{1}} and ((\\frac{d}{d C_{1}} \\operatorname{A_{x}}{(C_{1})})^{C_{1}})^{C_{1}} = ((\\frac{d}{d C_{1}} e^{C_{1}})^{C_{1}})^{C_{1}} and \\int ((\\frac{d}{d C_{1}} \\operatorname{A_{x}}{(C_{1})})^{C_{1}})^{C_{1}} dC_{1} = \\int ((\\frac{d}{d C_{1}} e^{C_{1}})^{C_{1}})^{C_{1}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Derivative(Function('A_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('A_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["integrate", 4, "Symbol('C_1', commutative=True)"], "Equality(Integral(Pow(Pow(Derivative(Function('A_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(Pow(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\dot{x},\\hat{p})} = \\hat{p} \\log{(\\dot{x})}, then obtain ((\\hat{p} \\log{(\\dot{x})})^{\\hat{p}})^{- \\dot{x}} (\\varepsilon_{0}^{\\hat{p}}{(\\dot{x},\\hat{p})})^{\\dot{x}} = 1", "derivation": "\\varepsilon_{0}{(\\dot{x},\\hat{p})} = \\hat{p} \\log{(\\dot{x})} and \\varepsilon_{0}^{\\hat{p}}{(\\dot{x},\\hat{p})} = (\\hat{p} \\log{(\\dot{x})})^{\\hat{p}} and (\\varepsilon_{0}^{\\hat{p}}{(\\dot{x},\\hat{p})})^{\\dot{x}} = ((\\hat{p} \\log{(\\dot{x})})^{\\hat{p}})^{\\dot{x}} and ((\\hat{p} \\log{(\\dot{x})})^{\\hat{p}})^{- \\dot{x}} (\\varepsilon_{0}^{\\hat{p}}{(\\dot{x},\\hat{p})})^{\\dot{x}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 3, "Pow(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Pow(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\operatorname{M_{E}}{(\\rho_f)} = \\sin{(\\log{(\\rho_f)})}, then obtain \\operatorname{M_{E}}{(\\rho_f)} + \\iint \\theta_{2}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\operatorname{M_{E}}{(\\rho_f)} + \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S}", "derivation": "\\theta_{2}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\int \\theta_{2}{(\\mathbf{S})} d\\mathbf{S} = \\int \\cos{(\\mathbf{S})} d\\mathbf{S} and \\operatorname{M_{E}}{(\\rho_f)} = \\sin{(\\log{(\\rho_f)})} and \\iint \\theta_{2}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} and \\sin{(\\log{(\\rho_f)})} + \\iint \\theta_{2}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\sin{(\\log{(\\rho_f)})} + \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} and \\operatorname{M_{E}}{(\\rho_f)} + \\iint \\theta_{2}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\operatorname{M_{E}}{(\\rho_f)} + \\iint \\cos{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], ["get_premise", "Equality(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), sin(log(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 4, "sin(log(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(sin(log(Symbol('\\\\rho_f', commutative=True))), Integral(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(sin(log(Symbol('\\\\rho_f', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\chi)} = \\log{(\\chi)}, then obtain \\frac{\\mathbf{p}{(\\chi)} \\mathbf{p}^{\\chi}{(\\chi)}}{\\chi} = \\frac{\\mathbf{p}{(\\chi)} \\log{(\\chi)}^{\\chi}}{\\chi}", "derivation": "\\mathbf{p}{(\\chi)} = \\log{(\\chi)} and \\mathbf{p}^{\\chi}{(\\chi)} = \\log{(\\chi)}^{\\chi} and \\frac{\\mathbf{p}{(\\chi)}}{\\chi} = \\frac{\\log{(\\chi)}}{\\chi} and \\frac{\\mathbf{p}^{\\chi}{(\\chi)} \\log{(\\chi)}}{\\chi} = \\frac{\\log{(\\chi)} \\log{(\\chi)}^{\\chi}}{\\chi} and \\frac{\\mathbf{p}{(\\chi)} \\mathbf{p}^{\\chi}{(\\chi)}}{\\chi} = \\frac{\\mathbf{p}{(\\chi)} \\log{(\\chi)}^{\\chi}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), log(Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), log(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), log(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(v_{2},F_{N})} = - F_{N} + v_{2}, then obtain \\int\\limits^{\\frac{v_{2} (- F_{N} + v_{2})}{\\mathbf{p}{(v_{2},F_{N})}}} \\mathbf{p}^{F_{N}}{(v_{2},F_{N})} dv_{2} = \\int\\limits^{\\frac{v_{2} (- F_{N} + v_{2})}{\\mathbf{p}{(v_{2},F_{N})}}} (- F_{N} + v_{2})^{F_{N}} dv_{2}", "derivation": "\\mathbf{p}{(v_{2},F_{N})} = - F_{N} + v_{2} and 1 = \\frac{- F_{N} + v_{2}}{\\mathbf{p}{(v_{2},F_{N})}} and \\mathbf{p}^{F_{N}}{(v_{2},F_{N})} = (- F_{N} + v_{2})^{F_{N}} and \\int \\mathbf{p}^{F_{N}}{(v_{2},F_{N})} dv_{2} = \\int (- F_{N} + v_{2})^{F_{N}} dv_{2} and v_{2} = \\frac{v_{2} (- F_{N} + v_{2})}{\\mathbf{p}{(v_{2},F_{N})}} and \\int\\limits^{\\frac{v_{2} (- F_{N} + v_{2})}{\\mathbf{p}{(v_{2},F_{N})}}} \\mathbf{p}^{F_{N}}{(v_{2},F_{N})} dv_{2} = \\int\\limits^{\\frac{v_{2} (- F_{N} + v_{2})}{\\mathbf{p}{(v_{2},F_{N})}}} (- F_{N} + v_{2})^{F_{N}} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Symbol('F_N', commutative=True)))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["times", 2, "Symbol('v_2', commutative=True)"], "Equality(Symbol('v_2', commutative=True), Mul(Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_2', commutative=True), Mul(Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_2', commutative=True), Mul(Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('v_2', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\chi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\chi and \\mathbf{p}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}}, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- \\chi + \\mathbf{s}{(\\chi,V_{\\mathbf{E}})}) + 1 = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}} + 1", "derivation": "\\mathbf{s}{(\\chi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\chi and - \\chi + \\mathbf{s}{(\\chi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- \\chi + \\mathbf{s}{(\\chi,V_{\\mathbf{E}})}) = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}} and \\mathbf{p}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}} and \\mathbf{p}{(V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- \\chi + \\mathbf{s}{(\\chi,V_{\\mathbf{E}})}) and \\mathbf{p}{(V_{\\mathbf{E}})} + 1 = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}} + 1 and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- \\chi + \\mathbf{s}{(\\chi,V_{\\mathbf{E}})}) + 1 = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Symbol('V_{\\\\mathbf{E}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Symbol('V_{\\\\mathbf{E}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)), Add(Derivative(Symbol('V_{\\\\mathbf{E}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('V_{\\\\mathbf{E}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(P_{g},A)} = \\frac{A}{P_{g}} and \\operatorname{n_{2}}{(\\eta,v_{z})} = e^{\\frac{v_{z}}{\\eta}}, then obtain (0^{A} + \\frac{P_{g} \\operatorname{n_{2}}{(\\eta,v_{z})}}{A})^{\\eta} = (0^{A} + \\frac{P_{g} e^{\\frac{v_{z}}{\\eta}}}{A})^{\\eta}", "derivation": "\\operatorname{m_{s}}{(P_{g},A)} = \\frac{A}{P_{g}} and \\operatorname{n_{2}}{(\\eta,v_{z})} = e^{\\frac{v_{z}}{\\eta}} and \\frac{\\operatorname{n_{2}}{(\\eta,v_{z})}}{\\operatorname{m_{s}}{(P_{g},A)}} = \\frac{e^{\\frac{v_{z}}{\\eta}}}{\\operatorname{m_{s}}{(P_{g},A)}} and \\frac{P_{g} \\operatorname{n_{2}}{(\\eta,v_{z})}}{A} = \\frac{P_{g} e^{\\frac{v_{z}}{\\eta}}}{A} and 0^{A} + \\frac{P_{g} \\operatorname{n_{2}}{(\\eta,v_{z})}}{A} = 0^{A} + \\frac{P_{g} e^{\\frac{v_{z}}{\\eta}}}{A} and (0^{A} + \\frac{P_{g} \\operatorname{n_{2}}{(\\eta,v_{z})}}{A})^{\\eta} = (0^{A} + \\frac{P_{g} e^{\\frac{v_{z}}{\\eta}}}{A})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('n_2')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["divide", 2, "Function('m_s')(Symbol('P_g', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Pow(Function('m_s')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Integer(-1)), Function('n_2')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Function('m_s')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Integer(-1)), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Function('n_2')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))))"], [["add", 4, "Pow(Integer(0), Symbol('A', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Function('n_2')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)))), Add(Pow(Integer(0), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))))"], [["power", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Add(Pow(Integer(0), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Function('n_2')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)))), Symbol('\\\\eta', commutative=True)), Pow(Add(Pow(Integer(0), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(\\pi)} = \\cos{(\\pi)}, then obtain (\\pi + \\cos^{2}{(\\pi)}) \\theta_{2}^{2}{(\\pi)} \\cos^{2}{(\\pi)} = (\\pi + \\cos^{2}{(\\pi)}) \\cos^{4}{(\\pi)}", "derivation": "\\theta_{2}{(\\pi)} = \\cos{(\\pi)} and \\theta_{2}{(\\pi)} \\cos{(\\pi)} = \\cos^{2}{(\\pi)} and \\pi + \\theta_{2}{(\\pi)} \\cos{(\\pi)} = \\pi + \\cos^{2}{(\\pi)} and \\theta_{2}^{2}{(\\pi)} \\cos^{2}{(\\pi)} = \\cos^{4}{(\\pi)} and (\\pi + \\theta_{2}{(\\pi)} \\cos{(\\pi)}) \\theta_{2}^{2}{(\\pi)} \\cos^{2}{(\\pi)} = (\\pi + \\theta_{2}{(\\pi)} \\cos{(\\pi)}) \\cos^{4}{(\\pi)} and (\\pi + \\cos^{2}{(\\pi)}) \\theta_{2}^{2}{(\\pi)} \\cos^{2}{(\\pi)} = (\\pi + \\cos^{2}{(\\pi)}) \\cos^{4}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2)))"], [["add", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(4)))"], [["times", 4, "Add(Symbol('\\\\pi', commutative=True), Mul(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Mul(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\pi', commutative=True), Mul(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(4))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))), Pow(Function('\\\\theta_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(4))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(p)} = \\cos{(p)}, then obtain 1 = \\frac{\\int \\cos{(p)} dp}{\\int \\operatorname{F_{g}}{(p)} dp}", "derivation": "\\operatorname{F_{g}}{(p)} = \\cos{(p)} and \\int \\operatorname{F_{g}}{(p)} dp = \\int \\cos{(p)} dp and \\iint \\operatorname{F_{g}}{(p)} dp dp = \\iint \\cos{(p)} dp dp and (\\int \\operatorname{F_{g}}{(p)} dp) \\iint \\operatorname{F_{g}}{(p)} dp dp = (\\int \\cos{(p)} dp) \\iint \\operatorname{F_{g}}{(p)} dp dp and (\\int \\operatorname{F_{g}}{(p)} dp) \\iint \\cos{(p)} dp dp = (\\int \\cos{(p)} dp) \\iint \\cos{(p)} dp dp and \\frac{\\iint \\cos{(p)} dp dp}{\\iint \\operatorname{F_{g}}{(p)} dp dp} = \\frac{(\\int \\cos{(p)} dp) \\iint \\cos{(p)} dp dp}{(\\int \\operatorname{F_{g}}{(p)} dp) \\iint \\operatorname{F_{g}}{(p)} dp dp} and 1 = \\frac{\\int \\cos{(p)} dp}{\\int \\operatorname{F_{g}}{(p)} dp}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["times", 2, "Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Mul(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["divide", 5, "Mul(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Mul(Pow(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Pow(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(1), Mul(Pow(Integral(Function('F_g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given r{(H)} = e^{H}, then derive \\int (- H + r{(H)}) dH = - \\frac{H^{2}}{2} + m + e^{H}, then obtain - \\frac{H^{2}}{2} = - \\frac{H^{2} (- \\frac{H^{2}}{2} + m + r{(H)})}{2 (- \\frac{H^{2}}{2} + \\rho + e^{H})}", "derivation": "r{(H)} = e^{H} and - H + r{(H)} = - H + e^{H} and \\int (- H + r{(H)}) dH = \\int (- H + e^{H}) dH and \\int (- H + r{(H)}) dH = - \\frac{H^{2}}{2} + m + e^{H} and \\frac{\\int (- H + r{(H)}) dH}{\\int (- H + e^{H}) dH} = \\frac{- \\frac{H^{2}}{2} + m + e^{H}}{\\int (- H + e^{H}) dH} and 1 = \\frac{- \\frac{H^{2}}{2} + m + e^{H}}{\\int (- H + e^{H}) dH} and 1 = \\frac{- \\frac{H^{2}}{2} + m + r{(H)}}{\\int (- H + e^{H}) dH} and - \\frac{H^{2}}{2} = - \\frac{H^{2} (- \\frac{H^{2}}{2} + m + r{(H)})}{2 \\int (- H + e^{H}) dH} and - \\frac{H^{2}}{2} = - \\frac{H^{2} (- \\frac{H^{2}}{2} + m + r{(H)})}{2 (- \\frac{H^{2}}{2} + \\rho + e^{H})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('r')(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('r')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('r')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), exp(Symbol('H', commutative=True))))"], [["divide", 4, "Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('r')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), exp(Symbol('H', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), exp(Symbol('H', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), Function('r')(Symbol('H', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["times", 7, "Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), Function('r')(Symbol('H', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 8], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True), exp(Symbol('H', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('m', commutative=True), Function('r')(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\phi_2,\\varepsilon_0)} = \\sin{(\\phi_2^{\\varepsilon_0})} and T{(\\phi_2,\\varepsilon_0)} = \\phi_2^{\\varepsilon_0}, then obtain e^{\\sin{(T{(\\phi_2,\\varepsilon_0)})}} = e^{\\sin{(\\phi_2^{\\varepsilon_0})}}", "derivation": "\\mathbf{r}{(\\phi_2,\\varepsilon_0)} = \\sin{(\\phi_2^{\\varepsilon_0})} and T{(\\phi_2,\\varepsilon_0)} = \\phi_2^{\\varepsilon_0} and \\mathbf{r}{(\\phi_2,\\varepsilon_0)} = \\sin{(T{(\\phi_2,\\varepsilon_0)})} and \\sin{(T{(\\phi_2,\\varepsilon_0)})} = \\sin{(\\phi_2^{\\varepsilon_0})} and e^{\\sin{(T{(\\phi_2,\\varepsilon_0)})}} = e^{\\sin{(\\phi_2^{\\varepsilon_0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), sin(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), sin(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(sin(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), sin(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["exp", 4], "Equality(exp(sin(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), exp(sin(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\phi{(m_{s})} = \\sin{(\\log{(m_{s})})}, then derive \\int \\phi{(m_{s})} dm_{s} = \\frac{m_{s} \\sin{(\\log{(m_{s})})}}{2} - \\frac{m_{s} \\cos{(\\log{(m_{s})})}}{2} + t, then obtain \\frac{\\int \\phi{(m_{s})} dm_{s}}{\\frac{m_{s} \\phi{(m_{s})}}{2} - \\frac{m_{s} \\cos{(\\log{(m_{s})})}}{2} + t} = 1", "derivation": "\\phi{(m_{s})} = \\sin{(\\log{(m_{s})})} and \\int \\phi{(m_{s})} dm_{s} = \\int \\sin{(\\log{(m_{s})})} dm_{s} and \\int \\phi{(m_{s})} dm_{s} = \\frac{m_{s} \\sin{(\\log{(m_{s})})}}{2} - \\frac{m_{s} \\cos{(\\log{(m_{s})})}}{2} + t and \\frac{\\int \\phi{(m_{s})} dm_{s}}{\\frac{m_{s} \\sin{(\\log{(m_{s})})}}{2} - \\frac{m_{s} \\cos{(\\log{(m_{s})})}}{2} + t} = 1 and \\frac{\\int \\phi{(m_{s})} dm_{s}}{\\frac{m_{s} \\phi{(m_{s})}}{2} - \\frac{m_{s} \\cos{(\\log{(m_{s})})}}{2} + t} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('m_s', commutative=True)), sin(log(Symbol('m_s', commutative=True))))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(sin(log(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Mul(Rational(1, 2), Symbol('m_s', commutative=True), sin(log(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('m_s', commutative=True), cos(log(Symbol('m_s', commutative=True)))), Symbol('t', commutative=True)))"], [["divide", 3, "Add(Mul(Rational(1, 2), Symbol('m_s', commutative=True), sin(log(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('m_s', commutative=True), cos(log(Symbol('m_s', commutative=True)))), Symbol('t', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Symbol('m_s', commutative=True), sin(log(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('m_s', commutative=True), cos(log(Symbol('m_s', commutative=True)))), Symbol('t', commutative=True)), Integer(-1)), Integral(Function('\\\\phi')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Symbol('m_s', commutative=True), Function('\\\\phi')(Symbol('m_s', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('m_s', commutative=True), cos(log(Symbol('m_s', commutative=True)))), Symbol('t', commutative=True)), Integer(-1)), Integral(Function('\\\\phi')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{p},C_{d})} = \\log{(C_{d}^{\\mathbf{p}})}, then obtain (\\int \\operatorname{n_{2}}{(\\mathbf{p},C_{d})} dC_{d})^{C_{d}} = (- C_{d} \\mathbf{p} + C_{d} \\log{(C_{d}^{\\mathbf{p}})} + S)^{C_{d}}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{p},C_{d})} = \\log{(C_{d}^{\\mathbf{p}})} and \\int \\operatorname{n_{2}}{(\\mathbf{p},C_{d})} dC_{d} = \\int \\log{(C_{d}^{\\mathbf{p}})} dC_{d} and (\\int \\operatorname{n_{2}}{(\\mathbf{p},C_{d})} dC_{d})^{C_{d}} = (\\int \\log{(C_{d}^{\\mathbf{p}})} dC_{d})^{C_{d}} and (\\int \\operatorname{n_{2}}{(\\mathbf{p},C_{d})} dC_{d})^{C_{d}} = (- C_{d} \\mathbf{p} + C_{d} \\log{(C_{d}^{\\mathbf{p}})} + S)^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('C_d', commutative=True)), log(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(log(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Integral(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Pow(Integral(log(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('C_d', commutative=True), log(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('S', commutative=True)), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given f{(\\theta_1)} = e^{\\sin{(\\theta_1)}} and \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} = - \\theta_1, then obtain f{(\\theta_1)} + \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} + \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)})} = \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} + e^{\\sin{(\\theta_1)}} + \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)})}", "derivation": "f{(\\theta_1)} = e^{\\sin{(\\theta_1)}} and f{(\\theta_1)} - \\sin{(\\theta_1)} = e^{\\sin{(\\theta_1)}} - \\sin{(\\theta_1)} and - \\theta_1 + f{(\\theta_1)} - \\sin{(\\theta_1)} = - \\theta_1 + e^{\\sin{(\\theta_1)}} - \\sin{(\\theta_1)} and \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} = - \\theta_1 and f{(\\theta_1)} + \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} + \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)})} = \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} + e^{\\sin{(\\theta_1)}} + \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\theta_1', commutative=True)), exp(sin(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('f')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Add(exp(sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('f')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('f')(Symbol('\\\\theta_1', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), exp(sin(Symbol('\\\\theta_1', commutative=True))), sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\Psi^{\\dagger},n)} = \\log{(\\Psi^{\\dagger} + n)}, then obtain \\frac{\\int \\frac{\\partial}{\\partial n} \\hat{H}_l{(\\Psi^{\\dagger},n)} dn}{\\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)}} = \\frac{\\int \\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)} dn}{\\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)}}", "derivation": "\\hat{H}_l{(\\Psi^{\\dagger},n)} = \\log{(\\Psi^{\\dagger} + n)} and \\frac{\\partial}{\\partial n} \\hat{H}_l{(\\Psi^{\\dagger},n)} = \\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)} and \\int \\frac{\\partial}{\\partial n} \\hat{H}_l{(\\Psi^{\\dagger},n)} dn = \\int \\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)} dn and \\frac{\\int \\frac{\\partial}{\\partial n} \\hat{H}_l{(\\Psi^{\\dagger},n)} dn}{\\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)}} = \\frac{\\int \\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)} dn}{\\frac{\\partial}{\\partial n} \\log{(\\Psi^{\\dagger} + n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))))"], [["divide", 3, "Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Mul(Pow(Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(log(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(i,t)} = \\sin{(i + t)}, then obtain t ((i + t) \\Psi_{nl}{(i,t)})^{i} (i + t) \\Psi_{nl}{(i,t)} \\sin{(i + t)} = t ((i + t) \\sin{(i + t)})^{i} (i + t) \\Psi_{nl}{(i,t)} \\sin{(i + t)}", "derivation": "\\Psi_{nl}{(i,t)} = \\sin{(i + t)} and (i + t) \\Psi_{nl}{(i,t)} = (i + t) \\sin{(i + t)} and ((i + t) \\Psi_{nl}{(i,t)})^{i} = ((i + t) \\sin{(i + t)})^{i} and ((i + t) \\Psi_{nl}{(i,t)})^{i} \\sin{(i + t)} = ((i + t) \\sin{(i + t)})^{i} \\sin{(i + t)} and t ((i + t) \\Psi_{nl}{(i,t)})^{i} \\sin{(i + t)} = t ((i + t) \\sin{(i + t)})^{i} \\sin{(i + t)} and t ((i + t) \\Psi_{nl}{(i,t)})^{i} (i + t) \\Psi_{nl}{(i,t)} \\sin{(i + t)} = t ((i + t) \\sin{(i + t)})^{i} (i + t) \\Psi_{nl}{(i,t)} \\sin{(i + t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True))))"], [["times", 1, "Add(Symbol('i', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True))), Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True))), Symbol('i', commutative=True)), Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Symbol('i', commutative=True)))"], [["times", 3, "sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))"], "Equality(Mul(Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True))), Symbol('i', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Mul(Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Symbol('i', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))))"], [["times", 4, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True))), Symbol('i', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Symbol('i', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))))"], [["times", 5, "Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True)))"], "Equality(Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True))), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), Symbol('t', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mu{(Q)} = \\log{(e^{Q})}, then obtain (\\frac{Q \\frac{d}{d Q} \\mu{(Q)}}{\\mu{(Q)}} + \\log{(\\mu{(Q)})}) \\mu^{Q}{(Q)} = (\\frac{Q}{\\log{(e^{Q})}} + \\log{(\\log{(e^{Q})})}) \\log{(e^{Q})}^{Q}", "derivation": "\\mu{(Q)} = \\log{(e^{Q})} and \\mu^{Q}{(Q)} = \\log{(e^{Q})}^{Q} and \\frac{d}{d Q} \\mu^{Q}{(Q)} = \\frac{d}{d Q} \\log{(e^{Q})}^{Q} and (\\frac{Q \\frac{d}{d Q} \\mu{(Q)}}{\\mu{(Q)}} + \\log{(\\mu{(Q)})}) \\mu^{Q}{(Q)} = (\\frac{Q}{\\log{(e^{Q})}} + \\log{(\\log{(e^{Q})})}) \\log{(e^{Q})}^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('Q', commutative=True)), log(exp(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(log(exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(log(exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('Q', commutative=True), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), log(Function('\\\\mu')(Symbol('Q', commutative=True)))), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Add(Mul(Symbol('Q', commutative=True), Pow(log(exp(Symbol('Q', commutative=True))), Integer(-1))), log(log(exp(Symbol('Q', commutative=True))))), Pow(log(exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(v_{t},W)} = W \\log{(v_{t})}, then obtain \\cos{(\\int v_{t} \\mathbb{I}{(v_{t},W)} dW)} = \\cos{(\\int W v_{t} \\log{(v_{t})} dW)}", "derivation": "\\mathbb{I}{(v_{t},W)} = W \\log{(v_{t})} and v_{t} \\mathbb{I}{(v_{t},W)} = W v_{t} \\log{(v_{t})} and \\int v_{t} \\mathbb{I}{(v_{t},W)} dW = \\int W v_{t} \\log{(v_{t})} dW and \\cos{(\\int v_{t} \\mathbb{I}{(v_{t},W)} dW)} = \\cos{(\\int W v_{t} \\log{(v_{t})} dW)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v_t', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), log(Symbol('v_t', commutative=True))))"], [["times", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_t', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Symbol('v_t', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_t', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('W', commutative=True), Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Mul(Symbol('v_t', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_t', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), cos(Integral(Mul(Symbol('W', commutative=True), Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given x{(\\Omega)} = \\int \\sin{(\\Omega)} d\\Omega, then derive x{(\\Omega)} = V_{\\mathbf{E}} - \\cos{(\\Omega)}, then obtain \\int (V_{\\mathbf{E}} - \\cos{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega = \\int (x{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega", "derivation": "x{(\\Omega)} = \\int \\sin{(\\Omega)} d\\Omega and x{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega = 2 \\int \\sin{(\\Omega)} d\\Omega and x{(\\Omega)} = V_{\\mathbf{E}} - \\cos{(\\Omega)} and \\int (x{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega = \\int 2 \\int \\sin{(\\Omega)} d\\Omega d\\Omega and \\int (V_{\\mathbf{E}} - \\cos{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega = \\int 2 \\int \\sin{(\\Omega)} d\\Omega d\\Omega and \\int (V_{\\mathbf{E}} - \\cos{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega = \\int (x{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('x')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('x')(Symbol('\\\\Omega', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Function('x')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(2), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(2), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Function('x')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given x{(v)} = e^{v} and \\operatorname{t_{1}}{(v)} = v e^{v}, then obtain - v x{(v)} + v e^{v} - x{(v)} + \\int v x{(v)} dv = - v x{(v)} + v e^{v} - x{(v)} + \\int v e^{v} dv", "derivation": "x{(v)} = e^{v} and v x{(v)} = v e^{v} and \\operatorname{t_{1}}{(v)} = v e^{v} and \\int \\operatorname{t_{1}}{(v)} dv = \\int v e^{v} dv and \\operatorname{t_{1}}{(v)} = v x{(v)} and \\int v x{(v)} dv = \\int v e^{v} dv and - x{(v)} + \\int v x{(v)} dv = - x{(v)} + \\int v e^{v} dv and - v x{(v)} + v e^{v} - x{(v)} + \\int v x{(v)} dv = - v x{(v)} + v e^{v} - x{(v)} + \\int v e^{v} dv", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["times", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('t_1')(Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Mul(Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["minus", 6, "Function('x')(Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('x')(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('x')(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))))"], [["add", 7, "Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Mul(Integer(-1), Function('x')(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('x')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Mul(Integer(-1), Function('x')(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(L_{\\varepsilon})} = \\sin{(\\log{(L_{\\varepsilon})})}, then obtain \\frac{d}{d L_{\\varepsilon}} \\hat{p}{(L_{\\varepsilon})} \\sin^{3}{(\\log{(L_{\\varepsilon})})} = \\frac{d}{d L_{\\varepsilon}} \\sin^{4}{(\\log{(L_{\\varepsilon})})}", "derivation": "\\hat{p}{(L_{\\varepsilon})} = \\sin{(\\log{(L_{\\varepsilon})})} and \\hat{p}{(L_{\\varepsilon})} \\sin{(\\log{(L_{\\varepsilon})})} = \\sin^{2}{(\\log{(L_{\\varepsilon})})} and \\hat{p}{(L_{\\varepsilon})} \\sin^{3}{(\\log{(L_{\\varepsilon})})} = \\sin^{4}{(\\log{(L_{\\varepsilon})})} and \\frac{d}{d L_{\\varepsilon}} \\hat{p}{(L_{\\varepsilon})} \\sin^{3}{(\\log{(L_{\\varepsilon})})} = \\frac{d}{d L_{\\varepsilon}} \\sin^{4}{(\\log{(L_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "sin(log(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(log(Symbol('L_{\\\\varepsilon}', commutative=True)))), Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(2)))"], [["times", 2, "Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(3))), Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(4)))"], [["differentiate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(3))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(sin(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(4)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain (\\int (\\mu_0 + \\int (l{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0) d\\mu_0)^{\\mu_0} = (\\int (\\mu_0 + \\int 2 \\cos{(\\mu_0)} d\\mu_0) d\\mu_0)^{\\mu_0}", "derivation": "l{(\\mu_0)} = \\cos{(\\mu_0)} and l{(\\mu_0)} + \\cos{(\\mu_0)} = 2 \\cos{(\\mu_0)} and \\int (l{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0 = \\int 2 \\cos{(\\mu_0)} d\\mu_0 and \\mu_0 + \\int (l{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0 = \\mu_0 + \\int 2 \\cos{(\\mu_0)} d\\mu_0 and \\int (\\mu_0 + \\int (l{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0) d\\mu_0 = \\int (\\mu_0 + \\int 2 \\cos{(\\mu_0)} d\\mu_0) d\\mu_0 and (\\int (\\mu_0 + \\int (l{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0) d\\mu_0)^{\\mu_0} = (\\int (\\mu_0 + \\int 2 \\cos{(\\mu_0)} d\\mu_0) d\\mu_0)^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Integral(Add(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu_0', commutative=True), Integral(Add(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["power", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\mu_0', commutative=True), Integral(Add(Function('l')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given I{(i)} = e^{i} and f{(i)} = \\frac{1}{I{(i)}}, then derive - \\frac{e^{i}}{I{(i)}} + \\frac{e^{i} \\frac{d}{d i} I{(i)}}{I^{2}{(i)}} = 0, then obtain f{(i)} \\frac{d}{d i} \\frac{1}{f{(i)}} - 1 = 0", "derivation": "I{(i)} = e^{i} and 1 = \\frac{e^{i}}{I{(i)}} and 1 - \\frac{e^{i}}{I{(i)}} = 0 and \\frac{d}{d i} (1 - \\frac{e^{i}}{I{(i)}}) = \\frac{d}{d i} 0 and - \\frac{e^{i}}{I{(i)}} + \\frac{e^{i} \\frac{d}{d i} I{(i)}}{I^{2}{(i)}} = 0 and -1 + e^{- i} \\frac{d}{d i} e^{i} = 0 and f{(i)} = \\frac{1}{I{(i)}} and f{(i)} = e^{- i} and f{(i)} \\frac{d}{d i} \\frac{1}{f{(i)}} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["divide", 1, "Function('I')(Symbol('i', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)), exp(Symbol('i', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)), exp(Symbol('i', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)), exp(Symbol('i', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)), exp(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)), exp(Symbol('i', commutative=True))), Mul(Pow(Function('I')(Symbol('i', commutative=True)), Integer(-2)), exp(Symbol('i', commutative=True)), Derivative(Function('I')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(-1), Mul(exp(Mul(Integer(-1), Symbol('i', commutative=True))), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Integer(0))"], ["renaming_premise", "Equality(Function('f')(Symbol('i', commutative=True)), Pow(Function('I')(Symbol('i', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Function('f')(Symbol('i', commutative=True)), exp(Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Add(Mul(Function('f')(Symbol('i', commutative=True)), Derivative(Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1)), Tuple(Symbol('i', commutative=True), Integer(1)))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{F}{(s)} = \\frac{d}{d s} \\sin{(s)}, then obtain - (\\mathbf{F}{(s)} + \\frac{d}{d s} \\sin{(s)})^{2} + 4 \\mathbf{F}^{2}{(s)} - \\sin{(s)} \\frac{d}{d s} \\sin{(s)} = - \\sin{(s)} \\frac{d}{d s} \\sin{(s)}", "derivation": "\\mathbf{F}{(s)} = \\frac{d}{d s} \\sin{(s)} and 2 \\mathbf{F}{(s)} = \\mathbf{F}{(s)} + \\frac{d}{d s} \\sin{(s)} and 4 \\mathbf{F}^{2}{(s)} = (\\mathbf{F}{(s)} + \\frac{d}{d s} \\sin{(s)})^{2} and 4 \\mathbf{F}^{2}{(s)} - \\sin{(s)} \\frac{d}{d s} \\sin{(s)} = (\\mathbf{F}{(s)} + \\frac{d}{d s} \\sin{(s)})^{2} - \\sin{(s)} \\frac{d}{d s} \\sin{(s)} and - (\\mathbf{F}{(s)} + \\frac{d}{d s} \\sin{(s)})^{2} + 4 \\mathbf{F}^{2}{(s)} - \\sin{(s)} \\frac{d}{d s} \\sin{(s)} = - \\sin{(s)} \\frac{d}{d s} \\sin{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["add", 1, "Function('\\\\mathbf{F}')(Symbol('s', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('s', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Integer(2))), Pow(Add(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(2)))"], [["add", 3, "Mul(Integer(-1), sin(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Pow(Add(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(-1), sin(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))))"], [["minus", 4, "Pow(Add(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(2))), Mul(Integer(4), Pow(Function('\\\\mathbf{F}')(Symbol('s', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Mul(Integer(-1), sin(Symbol('s', commutative=True)), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(y,A)} = \\sin{(\\frac{A}{y})}, then obtain \\int (\\int \\operatorname{J_{\\varepsilon}}{(y,A)} dA)^{A} dy = \\int (\\int \\sin{(\\frac{A}{y})} dA)^{A} dy", "derivation": "\\operatorname{J_{\\varepsilon}}{(y,A)} = \\sin{(\\frac{A}{y})} and \\int \\operatorname{J_{\\varepsilon}}{(y,A)} dA = \\int \\sin{(\\frac{A}{y})} dA and (\\int \\operatorname{J_{\\varepsilon}}{(y,A)} dA)^{A} = (\\int \\sin{(\\frac{A}{y})} dA)^{A} and \\int (\\int \\operatorname{J_{\\varepsilon}}{(y,A)} dA)^{A} dy = \\int (\\int \\sin{(\\frac{A}{y})} dA)^{A} dy", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('A', commutative=True)), sin(Mul(Symbol('A', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(sin(Mul(Symbol('A', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(sin(Mul(Symbol('A', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Pow(Integral(sin(Mul(Symbol('A', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given l{(r,\\omega)} = e^{\\omega r} and m{(r,\\omega)} = - \\omega r, then derive - \\omega r + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} l{(r,\\omega)} = - \\omega r + (\\omega r + 1) e^{\\omega r}, then obtain m{(r,\\omega)} + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} l{(r,\\omega)} = (\\omega r + 1) l{(r,\\omega)} + m{(r,\\omega)}", "derivation": "l{(r,\\omega)} = e^{\\omega r} and \\frac{\\partial}{\\partial r} l{(r,\\omega)} = \\frac{\\partial}{\\partial r} e^{\\omega r} and \\frac{\\partial^{2}}{\\partial \\omega\\partial r} l{(r,\\omega)} = \\frac{\\partial^{2}}{\\partial \\omega\\partial r} e^{\\omega r} and - \\omega r + \\frac{\\partial^{2}}{\\partial \\omega\\partial r} l{(r,\\omega)} = - \\omega r + \\frac{\\partial^{2}}{\\partial \\omega\\partial r} e^{\\omega r} and m{(r,\\omega)} = - \\omega r and - \\omega r + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} l{(r,\\omega)} = - \\omega r + (\\omega r + 1) e^{\\omega r} and m{(r,\\omega)} + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} l{(r,\\omega)} = (\\omega r + 1) e^{\\omega r} + m{(r,\\omega)} and m{(r,\\omega)} + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} e^{\\omega r} = (\\omega r + 1) e^{\\omega r} + m{(r,\\omega)} and m{(r,\\omega)} + \\frac{\\partial^{2}}{\\partial r\\partial \\omega} l{(r,\\omega)} = (\\omega r + 1) l{(r,\\omega)} + m{(r,\\omega)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Derivative(exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Mul(Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Integer(1)), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Integer(1)), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)))), Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Integer(1)), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)))), Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Add(Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('r', commutative=True)), Integer(1)), Function('l')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True))), Function('m')(Symbol('r', commutative=True), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} = \\hat{\\mathbf{x}} n_{2} and Z{(\\mu)} = \\sin{(\\mu)}, then obtain (\\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} + \\sin{(\\mu)})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} n_{2} + \\sin{(\\mu)})^{\\hat{\\mathbf{x}}}", "derivation": "\\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} = \\hat{\\mathbf{x}} n_{2} and Z{(\\mu)} = \\sin{(\\mu)} and \\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} + \\sin{(\\mu)} = \\hat{\\mathbf{x}} n_{2} + \\sin{(\\mu)} and Z{(\\mu)} + \\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} = \\hat{\\mathbf{x}} n_{2} + Z{(\\mu)} and (Z{(\\mu)} + \\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} n_{2} + Z{(\\mu)})^{\\hat{\\mathbf{x}}} and (\\mathbf{f}{(\\hat{\\mathbf{x}},n_{2})} + \\sin{(\\mu)})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} n_{2} + \\sin{(\\mu)})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)))"], ["get_premise", "Equality(Function('Z')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), sin(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('Z')(Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Function('Z')(Symbol('\\\\mu', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Add(Function('Z')(Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Function('Z')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Function('\\\\mathbf{f}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{F})} = e^{\\mathbf{F}}, then obtain \\frac{\\mathbf{F} e^{\\mathbf{F}} \\frac{d}{d \\mathbf{F}} \\operatorname{A_{2}}^{\\mathbf{F}}{(\\mathbf{F})}}{\\operatorname{A_{2}}^{2}{(\\mathbf{F})}} = \\frac{\\mathbf{F} e^{\\mathbf{F}} \\frac{d}{d \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}}}{\\operatorname{A_{2}}^{2}{(\\mathbf{F})}}", "derivation": "\\operatorname{A_{2}}{(\\mathbf{F})} = e^{\\mathbf{F}} and \\operatorname{A_{2}}^{\\mathbf{F}}{(\\mathbf{F})} = (e^{\\mathbf{F}})^{\\mathbf{F}} and \\frac{d}{d \\mathbf{F}} \\operatorname{A_{2}}^{\\mathbf{F}}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}} and \\frac{\\mathbf{F} e^{\\mathbf{F}} \\frac{d}{d \\mathbf{F}} \\operatorname{A_{2}}^{\\mathbf{F}}{(\\mathbf{F})}}{\\operatorname{A_{2}}^{2}{(\\mathbf{F})}} = \\frac{\\mathbf{F} e^{\\mathbf{F}} \\frac{d}{d \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}}}{\\operatorname{A_{2}}^{2}{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["times", 3, "Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('A_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(\\mathbf{A})} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\mathbf{J}_P{(\\mathbf{A})} = \\int z{(\\mathbf{A})} d\\mathbf{A}, then obtain 2 \\mathbf{J}_P^{2}{(\\mathbf{A})} = \\mathbf{J}_P^{2}{(\\mathbf{A})} + \\mathbf{J}_P{(\\mathbf{A})} \\iint \\log{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A}", "derivation": "z{(\\mathbf{A})} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\int z{(\\mathbf{A})} d\\mathbf{A} = \\iint \\log{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} and (\\int z{(\\mathbf{A})} d\\mathbf{A})^{2} = (\\int z{(\\mathbf{A})} d\\mathbf{A}) \\iint \\log{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} and \\mathbf{J}_P{(\\mathbf{A})} = \\int z{(\\mathbf{A})} d\\mathbf{A} and 2 (\\int z{(\\mathbf{A})} d\\mathbf{A})^{2} = (\\int z{(\\mathbf{A})} d\\mathbf{A})^{2} + (\\int z{(\\mathbf{A})} d\\mathbf{A}) \\iint \\log{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} and 2 \\mathbf{J}_P^{2}{(\\mathbf{A})} = \\mathbf{J}_P^{2}{(\\mathbf{A})} + \\mathbf{J}_P{(\\mathbf{A})} \\iint \\log{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 2, "Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Pow(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)), Mul(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 3, "Pow(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2))"], "Equality(Mul(Integer(2), Pow(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2))), Add(Pow(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)), Mul(Integral(Function('z')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Add(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))))"]]}, {"prompt": "Given H{(C,v_{y})} = \\frac{\\partial}{\\partial v_{y}} (C + v_{y}), then derive \\int e^{H{(C,v_{y})}} dv_{y} = C_{d} + e v_{y}, then obtain v_{y} \\int e^{H{(C,v_{y})}} dv_{y} = v_{y} (C_{d} + e v_{y})", "derivation": "H{(C,v_{y})} = \\frac{\\partial}{\\partial v_{y}} (C + v_{y}) and e^{H{(C,v_{y})}} = e^{\\frac{\\partial}{\\partial v_{y}} (C + v_{y})} and \\int e^{H{(C,v_{y})}} dv_{y} = \\int e^{\\frac{\\partial}{\\partial v_{y}} (C + v_{y})} dv_{y} and v_{y} \\int e^{H{(C,v_{y})}} dv_{y} = v_{y} \\int e^{\\frac{\\partial}{\\partial v_{y}} (C + v_{y})} dv_{y} and \\int e^{H{(C,v_{y})}} dv_{y} = C_{d} + e v_{y} and \\int e^{\\frac{\\partial}{\\partial v_{y}} (C + v_{y})} dv_{y} = C_{d} + e v_{y} and v_{y} \\int e^{H{(C,v_{y})}} dv_{y} = v_{y} (C_{d} + e v_{y})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Derivative(Add(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["exp", 1], "Equality(exp(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True))), exp(Derivative(Add(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(exp(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Derivative(Add(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True))))"], [["times", 3, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Integral(exp(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True)))), Mul(Symbol('v_y', commutative=True), Integral(exp(Derivative(Add(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(exp(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(E, Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(exp(Derivative(Add(Symbol('C', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(E, Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Symbol('v_y', commutative=True), Integral(exp(Function('H')(Symbol('C', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True)))), Mul(Symbol('v_y', commutative=True), Add(Symbol('C_d', commutative=True), Mul(E, Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\delta{(l)} = \\sin{(l)} and \\operatorname{P_{g}}{(l)} = (\\delta{(l)} + \\sin{(l)})^{2}, then obtain \\frac{\\operatorname{P_{g}}{(l)} - \\delta{(l)}}{(\\delta{(l)} + \\sin{(l)})^{2}} = \\frac{(\\delta{(l)} + \\sin{(l)})^{2} - \\delta{(l)}}{(\\delta{(l)} + \\sin{(l)})^{2}}", "derivation": "\\delta{(l)} = \\sin{(l)} and 2 \\delta{(l)} = \\delta{(l)} + \\sin{(l)} and 4 \\delta^{2}{(l)} = (\\delta{(l)} + \\sin{(l)})^{2} and \\operatorname{P_{g}}{(l)} = (\\delta{(l)} + \\sin{(l)})^{2} and \\operatorname{P_{g}}{(l)} - \\delta{(l)} = (\\delta{(l)} + \\sin{(l)})^{2} - \\delta{(l)} and \\frac{\\operatorname{P_{g}}{(l)} - \\delta{(l)}}{4 \\delta^{2}{(l)}} = \\frac{(\\delta{(l)} + \\sin{(l)})^{2} - \\delta{(l)}}{4 \\delta^{2}{(l)}} and \\frac{\\operatorname{P_{g}}{(l)} - \\delta{(l)}}{(\\delta{(l)} + \\sin{(l)})^{2}} = \\frac{(\\delta{(l)} + \\sin{(l)})^{2} - \\delta{(l)}}{(\\delta{(l)} + \\sin{(l)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["add", 1, "Function('\\\\delta')(Symbol('l', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('l', commutative=True))), Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Integer(2))), Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('l', commutative=True)), Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(2)))"], [["minus", 4, "Function('\\\\delta')(Symbol('l', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))), Add(Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))))"], [["divide", 5, "Mul(Integer(4), Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 4), Add(Function('P_g')(Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))), Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Integer(-2))), Mul(Rational(1, 4), Add(Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))), Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Add(Function('P_g')(Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))), Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(-2))), Mul(Add(Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\delta')(Symbol('l', commutative=True)))), Pow(Add(Function('\\\\delta')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(n_{1},v)} = v \\log{(n_{1})}, then derive \\frac{\\frac{\\partial}{\\partial n_{1}} \\operatorname{F_{N}}{(n_{1},v)}}{\\log{(n_{1})}} = \\frac{v}{n_{1} \\log{(n_{1})}}, then obtain \\frac{\\frac{\\partial}{\\partial n_{1}} v \\log{(n_{1})}}{\\log{(n_{1})}^{2}} = \\frac{v}{n_{1} \\log{(n_{1})}^{2}}", "derivation": "\\operatorname{F_{N}}{(n_{1},v)} = v \\log{(n_{1})} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{F_{N}}{(n_{1},v)} = \\frac{\\partial}{\\partial n_{1}} v \\log{(n_{1})} and \\frac{\\frac{\\partial}{\\partial n_{1}} \\operatorname{F_{N}}{(n_{1},v)}}{\\log{(n_{1})}} = \\frac{\\frac{\\partial}{\\partial n_{1}} v \\log{(n_{1})}}{\\log{(n_{1})}} and \\frac{\\frac{\\partial}{\\partial n_{1}} \\operatorname{F_{N}}{(n_{1},v)}}{\\log{(n_{1})}} = \\frac{v}{n_{1} \\log{(n_{1})}} and \\frac{\\frac{\\partial}{\\partial n_{1}} v \\log{(n_{1})}}{\\log{(n_{1})}} = \\frac{v}{n_{1} \\log{(n_{1})}} and \\frac{\\frac{\\partial}{\\partial n_{1}} v \\log{(n_{1})}}{\\log{(n_{1})}^{2}} = \\frac{v}{n_{1} \\log{(n_{1})}^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('n_1', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), log(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('n_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('v', commutative=True), log(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 2, "Pow(log(Symbol('n_1', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(log(Symbol('n_1', commutative=True)), Integer(-1)), Derivative(Function('F_N')(Symbol('n_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('n_1', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('v', commutative=True), log(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(log(Symbol('n_1', commutative=True)), Integer(-1)), Derivative(Function('F_N')(Symbol('n_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(log(Symbol('n_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(log(Symbol('n_1', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('v', commutative=True), log(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(log(Symbol('n_1', commutative=True)), Integer(-1))))"], [["times", 5, "Pow(log(Symbol('n_1', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(log(Symbol('n_1', commutative=True)), Integer(-2)), Derivative(Mul(Symbol('v', commutative=True), log(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(log(Symbol('n_1', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\nabla{(y)} = \\int \\sin{(y)} dy, then derive y \\nabla{(y)} = y (C_{1} - \\cos{(y)}), then derive y (t_{1} - \\cos{(y)}) = y (C_{1} - \\cos{(y)}), then derive y (\\sigma_p - \\cos{(y)}) = y (t_{1} - \\cos{(y)}), then obtain y (\\sigma_p - \\cos{(y)}) = y (g - \\cos{(y)})", "derivation": "\\nabla{(y)} = \\int \\sin{(y)} dy and y \\nabla{(y)} = y \\int \\sin{(y)} dy and y \\nabla{(y)} = y (C_{1} - \\cos{(y)}) and y \\int \\sin{(y)} dy = y (C_{1} - \\cos{(y)}) and y (t_{1} - \\cos{(y)}) = y (C_{1} - \\cos{(y)}) and y \\int \\sin{(y)} dy = y (t_{1} - \\cos{(y)}) and y (\\sigma_p - \\cos{(y)}) = y (t_{1} - \\cos{(y)}) and y (\\sigma_p - \\cos{(y)}) = y \\int \\sin{(y)} dy and y (\\sigma_p - \\cos{(y)}) = y (g - \\cos{(y)})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('y', commutative=True)), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\nabla')(Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\nabla')(Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('y', commutative=True), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(Symbol('y', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('y', commutative=True), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Symbol('y', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('y', commutative=True), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(Symbol('y', commutative=True), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Mul(Symbol('y', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Symbol('y', commutative=True), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Symbol('y', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Symbol('y', commutative=True), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 8], "Equality(Mul(Symbol('y', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Symbol('y', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(y)} = y, then derive 0 = 1 - \\frac{d}{d y} \\operatorname{m_{s}}{(y)}, then obtain 0 = 1 - \\frac{d}{d y} y", "derivation": "\\operatorname{m_{s}}{(y)} = y and 0 = y - \\operatorname{m_{s}}{(y)} and \\frac{d}{d y} 0 = \\frac{d}{d y} (y - \\operatorname{m_{s}}{(y)}) and 0 = 1 - \\frac{d}{d y} \\operatorname{m_{s}}{(y)} and 0 = 1 - \\frac{d}{d y} y", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('y', commutative=True)), Symbol('y', commutative=True))"], [["minus", 1, "Function('m_s')(Symbol('y', commutative=True))"], "Equality(Integer(0), Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('y', commutative=True)))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('m_s')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given G{(E_{x},\\omega)} = \\log{(\\omega^{E_{x}})} and \\operatorname{P_{g}}{(V)} = \\sin{(V)}, then obtain \\int (G^{\\omega}{(E_{x},\\omega)} + \\operatorname{P_{g}}{(V)}) dV = \\int (G^{\\omega}{(E_{x},\\omega)} + \\sin{(V)}) dV", "derivation": "G{(E_{x},\\omega)} = \\log{(\\omega^{E_{x}})} and G^{\\omega}{(E_{x},\\omega)} = \\log{(\\omega^{E_{x}})}^{\\omega} and \\operatorname{P_{g}}{(V)} = \\sin{(V)} and G^{\\omega}{(E_{x},\\omega)} + \\operatorname{P_{g}}{(V)} = G^{\\omega}{(E_{x},\\omega)} + \\sin{(V)} and \\operatorname{P_{g}}{(V)} + \\log{(\\omega^{E_{x}})}^{\\omega} = \\log{(\\omega^{E_{x}})}^{\\omega} + \\sin{(V)} and \\int (\\operatorname{P_{g}}{(V)} + \\log{(\\omega^{E_{x}})}^{\\omega}) dV = \\int (\\log{(\\omega^{E_{x}})}^{\\omega} + \\sin{(V)}) dV and \\int (G^{\\omega}{(E_{x},\\omega)} + \\operatorname{P_{g}}{(V)}) dV = \\int (G^{\\omega}{(E_{x},\\omega)} + \\sin{(V)}) dV", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))), Symbol('\\\\omega', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["add", 3, "Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Function('P_g')(Symbol('V', commutative=True))), Add(Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), sin(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('P_g')(Symbol('V', commutative=True)), Pow(log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))), Symbol('\\\\omega', commutative=True))), Add(Pow(log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))), Symbol('\\\\omega', commutative=True)), sin(Symbol('V', commutative=True))))"], [["integrate", 5, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('P_g')(Symbol('V', commutative=True)), Pow(log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Add(Pow(log(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_x', commutative=True))), Symbol('\\\\omega', commutative=True)), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integral(Add(Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Function('P_g')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Add(Pow(Function('G')(Symbol('E_x', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(y)} = e^{y}, then obtain \\cos{(\\mathbf{J}_P{(y)})} + \\int \\log{(\\mathbf{J}_P^{y}{(y)})} dy = \\cos{(\\mathbf{J}_P{(y)})} + \\int \\log{((e^{y})^{y})} dy", "derivation": "\\mathbf{J}_P{(y)} = e^{y} and \\mathbf{J}_P^{y}{(y)} = (e^{y})^{y} and \\log{(\\mathbf{J}_P^{y}{(y)})} = \\log{((e^{y})^{y})} and \\int \\log{(\\mathbf{J}_P^{y}{(y)})} dy = \\int \\log{((e^{y})^{y})} dy and \\cos{(\\mathbf{J}_P{(y)})} + \\int \\log{(\\mathbf{J}_P^{y}{(y)})} dy = \\cos{(\\mathbf{J}_P{(y)})} + \\int \\log{((e^{y})^{y})} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), log(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(log(Pow(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(log(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["add", 4, "cos(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)))"], "Equality(Add(cos(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True))), Integral(log(Pow(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))), Add(cos(Function('\\\\mathbf{J}_P')(Symbol('y', commutative=True))), Integral(log(Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\dot{\\mathbf{r}},P_{e})} = \\sin{(P_{e} \\dot{\\mathbf{r}})}, then derive \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},P_{e})} = P_{e} \\cos{(P_{e} \\dot{\\mathbf{r}})}, then obtain (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\sin{(P_{e} \\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}} = (P_{e} \\cos{(P_{e} \\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}}", "derivation": "\\mathbf{B}{(\\dot{\\mathbf{r}},P_{e})} = \\sin{(P_{e} \\dot{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},P_{e})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\sin{(P_{e} \\dot{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},P_{e})} = P_{e} \\cos{(P_{e} \\dot{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\sin{(P_{e} \\dot{\\mathbf{r}})} = P_{e} \\cos{(P_{e} \\dot{\\mathbf{r}})} and (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\sin{(P_{e} \\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}} = (P_{e} \\cos{(P_{e} \\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('P_e', commutative=True)), sin(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('P_e', commutative=True), cos(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('P_e', commutative=True), cos(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["power", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Derivative(sin(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Mul(Symbol('P_e', commutative=True), cos(Mul(Symbol('P_e', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(E_{n})} = e^{E_{n}} and \\operatorname{f_{E}}{(E_{n})} = (e^{E_{n}})^{E_{n}}, then obtain \\operatorname{P_{e}}^{E_{n}}{(E_{n})} = \\operatorname{f_{E}}{(E_{n})}", "derivation": "\\operatorname{P_{e}}{(E_{n})} = e^{E_{n}} and \\operatorname{P_{e}}^{E_{n}}{(E_{n})} = (e^{E_{n}})^{E_{n}} and \\operatorname{f_{E}}{(E_{n})} = (e^{E_{n}})^{E_{n}} and \\operatorname{P_{e}}^{E_{n}}{(E_{n})} = \\operatorname{f_{E}}{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(exp(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('E_n', commutative=True)), Pow(exp(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('P_e')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Function('f_E')(Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then obtain \\frac{d^{3}}{d (a^{\\dagger})^{3}} \\frac{\\operatorname{x^{{\\}'}}{(a^{\\dagger})}}{\\cos{(a^{\\dagger})}} = \\frac{d^{3}}{d (a^{\\dagger})^{3}} 1", "derivation": "\\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\frac{\\operatorname{x^{{\\}'}}{(a^{\\dagger})}}{\\cos{(a^{\\dagger})}} = 1 and \\frac{d}{d a^{\\dagger}} \\frac{\\operatorname{x^{{\\}'}}{(a^{\\dagger})}}{\\cos{(a^{\\dagger})}} = \\frac{d}{d a^{\\dagger}} 1 and \\frac{d^{2}}{d (a^{\\dagger})^{2}} \\frac{\\operatorname{x^{{\\}'}}{(a^{\\dagger})}}{\\cos{(a^{\\dagger})}} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} 1 and \\frac{d^{3}}{d (a^{\\dagger})^{3}} \\frac{\\operatorname{x^{{\\}'}}{(a^{\\dagger})}}{\\cos{(a^{\\dagger})}} = \\frac{d^{3}}{d (a^{\\dagger})^{3}} 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "cos(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(3))), Derivative(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(3))))"]]}, {"prompt": "Given L{(h,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + \\log{(h)}, then obtain (\\int L{(h,f_{\\mathbf{v}})} L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}})^{h} = (\\int (- f_{\\mathbf{v}} + \\log{(h)}) L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}})^{h}", "derivation": "L{(h,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + \\log{(h)} and L{(h,f_{\\mathbf{v}})} L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} = (- f_{\\mathbf{v}} + \\log{(h)}) L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} and \\int L{(h,f_{\\mathbf{v}})} L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int (- f_{\\mathbf{v}} + \\log{(h)}) L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}} and (\\int L{(h,f_{\\mathbf{v}})} L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}})^{h} = (\\int (- f_{\\mathbf{v}} + \\log{(h)}) L^{- f_{\\mathbf{v}}}{(h,f_{\\mathbf{v}})} df_{\\mathbf{v}})^{h}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('h', commutative=True))))"], [["divide", 1, "Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('h', commutative=True))), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Mul(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('h', commutative=True))), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Mul(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('h', commutative=True))), Pow(Function('L')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(m_{s},A)} = \\cos{(A + m_{s})}, then derive \\int \\operatorname{v_{x}}{(m_{s},A)} dm_{s} = c_{0} + \\sin{(A + m_{s})}, then obtain c_{0} + \\int \\operatorname{v_{x}}{(m_{s},A)} dm_{s} = 2 c_{0} + \\sin{(A + m_{s})}", "derivation": "\\operatorname{v_{x}}{(m_{s},A)} = \\cos{(A + m_{s})} and \\int \\operatorname{v_{x}}{(m_{s},A)} dm_{s} = \\int \\cos{(A + m_{s})} dm_{s} and \\int \\operatorname{v_{x}}{(m_{s},A)} dm_{s} = c_{0} + \\sin{(A + m_{s})} and c_{0} + \\int \\operatorname{v_{x}}{(m_{s},A)} dm_{s} = 2 c_{0} + \\sin{(A + m_{s})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('m_s', commutative=True), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('m_s', commutative=True))))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('m_s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(cos(Add(Symbol('A', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_x')(Symbol('m_s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('c_0', commutative=True), sin(Add(Symbol('A', commutative=True), Symbol('m_s', commutative=True)))))"], [["add", 3, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Integral(Function('v_x')(Symbol('m_s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Add(Mul(Integer(2), Symbol('c_0', commutative=True)), sin(Add(Symbol('A', commutative=True), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given V{(\\phi)} = e^{\\phi}, then derive \\int \\phi V{(\\phi)} d\\phi = \\hat{H}_l + (\\phi - 1) e^{\\phi}, then obtain (\\hat{H}_l + (\\phi - 1) e^{\\phi} - \\int \\phi e^{\\phi} d\\phi)^{\\phi} = 0^{\\phi}", "derivation": "V{(\\phi)} = e^{\\phi} and \\phi V{(\\phi)} = \\phi e^{\\phi} and \\int \\phi V{(\\phi)} d\\phi = \\int \\phi e^{\\phi} d\\phi and \\int \\phi V{(\\phi)} d\\phi - \\int \\phi e^{\\phi} d\\phi = 0 and (\\int \\phi V{(\\phi)} d\\phi - \\int \\phi e^{\\phi} d\\phi)^{\\phi} = 0^{\\phi} and \\int \\phi V{(\\phi)} d\\phi = \\hat{H}_l + (\\phi - 1) e^{\\phi} and (\\hat{H}_l + (\\phi - 1) e^{\\phi} - \\int \\phi e^{\\phi} d\\phi)^{\\phi} = 0^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["minus", 3, "Integral(Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))), Symbol('\\\\phi', commutative=True)), Pow(Integer(0), Symbol('\\\\phi', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Add(Symbol('\\\\phi', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Add(Symbol('\\\\phi', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))), Symbol('\\\\phi', commutative=True)), Pow(Integer(0), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(E_{x},\\theta_1)} = \\log{(\\frac{E_{x}}{\\theta_1})}, then obtain Z + \\operatorname{F_{c}}{(E_{x},\\theta_1)} = \\mathbf{s} - \\log{(\\theta_1)}", "derivation": "\\operatorname{F_{c}}{(E_{x},\\theta_1)} = \\log{(\\frac{E_{x}}{\\theta_1})} and \\frac{\\partial}{\\partial \\theta_1} \\operatorname{F_{c}}{(E_{x},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\log{(\\frac{E_{x}}{\\theta_1})} and \\int \\frac{\\partial}{\\partial \\theta_1} \\operatorname{F_{c}}{(E_{x},\\theta_1)} d\\theta_1 = \\int \\frac{\\partial}{\\partial \\theta_1} \\log{(\\frac{E_{x}}{\\theta_1})} d\\theta_1 and Z + \\operatorname{F_{c}}{(E_{x},\\theta_1)} = \\mathbf{s} - \\log{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Function('F_c')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('Z', commutative=True), Function('F_c')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(v_{y})} = \\log{(v_{y})} and \\Psi_{nl}{(v_{y})} = \\log{(\\log{(v_{y})})}^{v_{y}}, then obtain \\Psi_{nl}{(v_{y})} = \\log{(\\rho_{b}{(v_{y})})}^{v_{y}}", "derivation": "\\rho_{b}{(v_{y})} = \\log{(v_{y})} and \\log{(\\rho_{b}{(v_{y})})} = \\log{(\\log{(v_{y})})} and \\log{(\\rho_{b}{(v_{y})})}^{v_{y}} = \\log{(\\log{(v_{y})})}^{v_{y}} and \\Psi_{nl}{(v_{y})} = \\log{(\\log{(v_{y})})}^{v_{y}} and \\Psi_{nl}{(v_{y})} = \\log{(\\rho_{b}{(v_{y})})}^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\rho_b')(Symbol('v_y', commutative=True))), log(log(Symbol('v_y', commutative=True))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(log(Function('\\\\rho_b')(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(log(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True)), Pow(log(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True)), Pow(log(Function('\\\\rho_b')(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{D})} = \\mathbf{D}, then derive \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})} = 1, then obtain - \\frac{\\sin{(\\frac{d}{d \\mathbf{D}} \\mathbf{D})}}{\\mathbf{D} \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}} = - \\frac{\\sin{(1)}}{\\mathbf{D} \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}}", "derivation": "\\Psi_{nl}{(\\mathbf{D})} = \\mathbf{D} and \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\mathbf{D} and \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})} = 1 and \\frac{d}{d \\mathbf{D}} \\mathbf{D} = 1 and \\sin{(\\frac{d}{d \\mathbf{D}} \\mathbf{D})} = \\sin{(1)} and \\frac{\\sin{(\\frac{d}{d \\mathbf{D}} \\mathbf{D})}}{\\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}} = \\frac{\\sin{(1)}}{\\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}} and - \\frac{\\sin{(\\frac{d}{d \\mathbf{D}} \\mathbf{D})}}{\\mathbf{D} \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}} = - \\frac{\\sin{(1)}}{\\mathbf{D} \\frac{d}{d \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))"], [["sin", 4], "Equality(sin(Derivative(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), sin(Integer(1)))"], [["divide", 5, "Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))"], "Equality(Mul(sin(Derivative(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Integer(1)), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 6, "Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), sin(Derivative(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), sin(Integer(1)), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(h,A_{z})} = \\sin^{A_{z}}{(h)}, then obtain \\iint (\\operatorname{c_{0}}^{2}{(h,A_{z})})^{A_{z}} dA_{z} dA_{z} = \\iint (\\operatorname{c_{0}}{(h,A_{z})} \\sin^{A_{z}}{(h)})^{A_{z}} dA_{z} dA_{z}", "derivation": "\\operatorname{c_{0}}{(h,A_{z})} = \\sin^{A_{z}}{(h)} and \\operatorname{c_{0}}^{2}{(h,A_{z})} = \\operatorname{c_{0}}{(h,A_{z})} \\sin^{A_{z}}{(h)} and (\\operatorname{c_{0}}^{2}{(h,A_{z})})^{A_{z}} = (\\operatorname{c_{0}}{(h,A_{z})} \\sin^{A_{z}}{(h)})^{A_{z}} and \\int (\\operatorname{c_{0}}^{2}{(h,A_{z})})^{A_{z}} dA_{z} = \\int (\\operatorname{c_{0}}{(h,A_{z})} \\sin^{A_{z}}{(h)})^{A_{z}} dA_{z} and \\iint (\\operatorname{c_{0}}^{2}{(h,A_{z})})^{A_{z}} dA_{z} dA_{z} = \\iint (\\operatorname{c_{0}}{(h,A_{z})} \\sin^{A_{z}}{(h)})^{A_{z}} dA_{z} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('A_z', commutative=True)))"], [["times", 1, "Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Pow(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('A_z', commutative=True))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Symbol('A_z', commutative=True)), Pow(Mul(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Pow(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(Mul(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["integrate", 4, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Pow(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(Mul(Function('c_0')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\chi,\\sigma_x)} = - \\chi + \\sigma_x and s{(\\chi,\\sigma_x)} = - \\chi + \\sigma_x, then obtain - \\chi + (- \\frac{\\int s{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x} = - \\chi + (- \\frac{\\int \\operatorname{v_{2}}{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x}", "derivation": "\\operatorname{v_{2}}{(\\chi,\\sigma_x)} = - \\chi + \\sigma_x and s{(\\chi,\\sigma_x)} = - \\chi + \\sigma_x and s{(\\chi,\\sigma_x)} = \\operatorname{v_{2}}{(\\chi,\\sigma_x)} and \\int s{(\\chi,\\sigma_x)} d\\chi = \\int \\operatorname{v_{2}}{(\\chi,\\sigma_x)} d\\chi and - \\frac{\\int s{(\\chi,\\sigma_x)} d\\chi}{2 \\chi} = - \\frac{\\int \\operatorname{v_{2}}{(\\chi,\\sigma_x)} d\\chi}{2 \\chi} and (- \\frac{\\int s{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x} = (- \\frac{\\int \\operatorname{v_{2}}{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x} and - \\chi + (- \\frac{\\int s{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x} = - \\chi + (- \\frac{\\int \\operatorname{v_{2}}{(\\chi,\\sigma_x)} d\\chi}{2 \\chi})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Integer(2), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["power", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Integral(Function('v_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given S{(y^{\\prime})} = \\sin{(y^{\\prime})}, then derive \\int \\frac{S{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} = \\eta^{\\prime} - \\log{(\\cos{(y^{\\prime})})}, then obtain \\iint \\frac{\\sin{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} dy^{\\prime} = \\int (\\eta^{\\prime} - \\log{(\\cos{(y^{\\prime})})}) dy^{\\prime}", "derivation": "S{(y^{\\prime})} = \\sin{(y^{\\prime})} and \\frac{S{(y^{\\prime})}}{\\cos{(y^{\\prime})}} = \\frac{\\sin{(y^{\\prime})}}{\\cos{(y^{\\prime})}} and \\int \\frac{S{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} = \\int \\frac{\\sin{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} and \\int \\frac{S{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} = \\eta^{\\prime} - \\log{(\\cos{(y^{\\prime})})} and \\int \\frac{\\sin{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} = \\eta^{\\prime} - \\log{(\\cos{(y^{\\prime})})} and \\iint \\frac{\\sin{(y^{\\prime})}}{\\cos{(y^{\\prime})}} dy^{\\prime} dy^{\\prime} = \\int (\\eta^{\\prime} - \\log{(\\cos{(y^{\\prime})})}) dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 1, "cos(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Mul(sin(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(sin(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), log(cos(Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(sin(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), log(cos(Symbol('y^{\\\\prime}', commutative=True))))))"], [["integrate", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(sin(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), log(cos(Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain \\frac{d}{d \\sigma_p} - \\int \\frac{\\operatorname{A_{2}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = \\frac{d}{d \\sigma_p} - \\int \\frac{\\cos{(\\sigma_p)}}{\\sigma_p} d\\sigma_p", "derivation": "\\operatorname{A_{2}}{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\frac{\\operatorname{A_{2}}{(\\sigma_p)}}{\\sigma_p} = \\frac{\\cos{(\\sigma_p)}}{\\sigma_p} and \\int \\frac{\\operatorname{A_{2}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = \\int \\frac{\\cos{(\\sigma_p)}}{\\sigma_p} d\\sigma_p and - \\int \\frac{\\operatorname{A_{2}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = - \\int \\frac{\\cos{(\\sigma_p)}}{\\sigma_p} d\\sigma_p and \\frac{d}{d \\sigma_p} - \\int \\frac{\\operatorname{A_{2}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = \\frac{d}{d \\sigma_p} - \\int \\frac{\\cos{(\\sigma_p)}}{\\sigma_p} d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\nabla,\\varphi^*)} = \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*, then derive \\sin{(\\Psi_{\\lambda}{(\\nabla,\\varphi^*)})} = \\sin{(\\varphi^*)}, then obtain \\sin{(\\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*)} \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^* = \\sin{(\\varphi^*)} \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*", "derivation": "\\Psi_{\\lambda}{(\\nabla,\\varphi^*)} = \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^* and \\sin{(\\Psi_{\\lambda}{(\\nabla,\\varphi^*)})} = \\sin{(\\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*)} and \\sin{(\\Psi_{\\lambda}{(\\nabla,\\varphi^*)})} = \\sin{(\\varphi^*)} and \\Psi_{\\lambda}{(\\nabla,\\varphi^*)} \\sin{(\\Psi_{\\lambda}{(\\nabla,\\varphi^*)})} = \\Psi_{\\lambda}{(\\nabla,\\varphi^*)} \\sin{(\\varphi^*)} and \\sin{(\\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*)} \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^* = \\sin{(\\varphi^*)} \\frac{\\partial}{\\partial \\nabla} \\nabla \\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["sin", 1], "Equality(sin(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True))), sin(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(sin(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 3, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), sin(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(sin(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(sin(Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given B{(q)} = \\sin{(q)}, then derive \\frac{d}{d q} B{(q)} = \\cos{(q)}, then obtain \\cos^{- q}{(q)} \\frac{d}{d q} B{(q)} = \\cos{(q)} \\cos^{- q}{(q)}", "derivation": "B{(q)} = \\sin{(q)} and \\frac{d}{d q} B{(q)} = \\frac{d}{d q} \\sin{(q)} and \\frac{d}{d q} B{(q)} = \\cos{(q)} and \\cos{(q)} = \\frac{d}{d q} \\sin{(q)} and \\cos^{- q}{(q)} \\frac{d}{d q} B{(q)} = \\cos^{- q}{(q)} \\frac{d}{d q} \\sin{(q)} and \\cos^{- q}{(q)} \\frac{d}{d q} B{(q)} = \\cos{(q)} \\cos^{- q}{(q)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), cos(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('q', commutative=True)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 2, "Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Derivative(Function('B')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(cos(Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Derivative(Function('B')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(cos(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(A_{z})} = e^{A_{z}} and f{(A_{z})} = \\frac{d}{d A_{z}} \\mathbf{p}{(A_{z})}, then obtain - f{(A_{z})} = - \\frac{d}{d A_{z}} \\mathbf{p}{(A_{z})}", "derivation": "\\mathbf{p}{(A_{z})} = e^{A_{z}} and \\frac{d}{d A_{z}} \\mathbf{p}{(A_{z})} = \\frac{d}{d A_{z}} e^{A_{z}} and f{(A_{z})} = \\frac{d}{d A_{z}} \\mathbf{p}{(A_{z})} and f{(A_{z})} = \\frac{d}{d A_{z}} e^{A_{z}} and - f{(A_{z})} = - \\frac{d}{d A_{z}} e^{A_{z}} and - f{(A_{z})} = - \\frac{d}{d A_{z}} \\mathbf{p}{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f')(Symbol('A_z', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('f')(Symbol('A_z', commutative=True)), Derivative(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f')(Symbol('A_z', commutative=True))), Mul(Integer(-1), Derivative(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Function('f')(Symbol('A_z', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{p}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi{(r,T)} = T + r, then obtain r + \\Psi{(r,T)} = - T + 2 \\Psi{(r,T)}", "derivation": "\\Psi{(r,T)} = T + r and T + r + \\Psi{(r,T)} = 2 T + 2 r and 2 \\Psi{(r,T)} = 2 T + 2 r and T + r + \\Psi{(r,T)} = 2 \\Psi{(r,T)} and r + \\Psi{(r,T)} = - T + 2 \\Psi{(r,T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('r', commutative=True)))"], [["add", 1, "Add(Symbol('T', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Symbol('r', commutative=True), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('T', commutative=True), Symbol('r', commutative=True), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True))))"], [["minus", 4, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('r', commutative=True), Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(q,r_{0})} = \\cos{(q - r_{0})}, then obtain 0 = - \\operatorname{F_{H}}{(q,r_{0})} + \\cos{(q - r_{0})}", "derivation": "\\operatorname{F_{H}}{(q,r_{0})} = \\cos{(q - r_{0})} and r_{0} + \\operatorname{F_{H}}{(q,r_{0})} = r_{0} + \\cos{(q - r_{0})} and 2 r_{0} + \\operatorname{F_{H}}{(q,r_{0})} = 2 r_{0} + \\cos{(q - r_{0})} and 0 = - \\operatorname{F_{H}}{(q,r_{0})} + \\cos{(q - r_{0})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('q', commutative=True), Symbol('r_0', commutative=True)), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('r_0', commutative=True), Function('F_H')(Symbol('q', commutative=True), Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"], [["minus", 2, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Function('F_H')(Symbol('q', commutative=True), Symbol('r_0', commutative=True))), Add(Mul(Integer(2), Symbol('r_0', commutative=True)), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"], [["minus", 3, "Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Function('F_H')(Symbol('q', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_H')(Symbol('q', commutative=True), Symbol('r_0', commutative=True))), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\hat{x},B)} = B \\hat{x}, then obtain \\frac{- B + \\eta^{\\prime}{(\\hat{x},B)}}{B \\hat{x} \\eta^{\\prime}{(\\hat{x},B)}} = \\frac{B \\hat{x} - B}{B \\hat{x} \\eta^{\\prime}{(\\hat{x},B)}}", "derivation": "\\eta^{\\prime}{(\\hat{x},B)} = B \\hat{x} and B \\hat{x} \\eta^{\\prime}{(\\hat{x},B)} = B^{2} \\hat{x}^{2} and - B + \\eta^{\\prime}{(\\hat{x},B)} = B \\hat{x} - B and \\frac{- B + \\eta^{\\prime}{(\\hat{x},B)}}{B^{2} \\hat{x}^{2}} = \\frac{B \\hat{x} - B}{B^{2} \\hat{x}^{2}} and \\frac{- B + \\eta^{\\prime}{(\\hat{x},B)}}{B \\hat{x} \\eta^{\\prime}{(\\hat{x},B)}} = \\frac{B \\hat{x} - B}{B \\hat{x} \\eta^{\\prime}{(\\hat{x},B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(2)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('B', commutative=True), Integer(2)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), Add(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given a{(n)} = \\log{(e^{n})} and \\psi{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)}, then obtain \\psi{(\\mathbf{J}_P)} + \\log{(e^{n})} = \\log{(e^{n})} + \\cos{(\\mathbf{J}_P)}", "derivation": "a{(n)} = \\log{(e^{n})} and \\psi{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\psi{(\\mathbf{J}_P)} + a{(n)} = a{(n)} + \\cos{(\\mathbf{J}_P)} and \\psi{(\\mathbf{J}_P)} + \\log{(e^{n})} = \\log{(e^{n})} + \\cos{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('n', commutative=True)), log(exp(Symbol('n', commutative=True))))"], ["get_premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 2, "Function('a')(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True)), Function('a')(Symbol('n', commutative=True))), Add(Function('a')(Symbol('n', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(exp(Symbol('n', commutative=True)))), Add(log(exp(Symbol('n', commutative=True))), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(P_{g},Q)} = P_{g} Q, then obtain 1 = \\frac{\\int P_{g} Q dQ}{\\int \\phi_{1}{(P_{g},Q)} dQ}", "derivation": "\\phi_{1}{(P_{g},Q)} = P_{g} Q and \\int \\phi_{1}{(P_{g},Q)} dQ = \\int P_{g} Q dQ and \\frac{\\int \\phi_{1}{(P_{g},Q)} dQ}{P_{g}} = \\frac{\\int P_{g} Q dQ}{P_{g}} and 1 = \\frac{\\int P_{g} Q dQ}{\\int \\phi_{1}{(P_{g},Q)} dQ}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_1')(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["divide", 2, "Symbol('P_g', commutative=True)"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Integral(Mul(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], "Equality(Integer(1), Mul(Integral(Mul(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Pow(Integral(Function('\\\\phi_1')(Symbol('P_g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given H{(\\rho)} = \\cos{(e^{\\rho})}, then derive - e^{\\rho} + \\frac{d}{d \\rho} H{(\\rho)} = - e^{\\rho} \\sin{(e^{\\rho})} - e^{\\rho}, then obtain \\rho - e^{\\rho} + \\frac{d}{d \\rho} \\cos{(e^{\\rho})} = \\rho - e^{\\rho} \\sin{(e^{\\rho})} - e^{\\rho}", "derivation": "H{(\\rho)} = \\cos{(e^{\\rho})} and H{(\\rho)} - e^{\\rho} = - e^{\\rho} + \\cos{(e^{\\rho})} and \\frac{d}{d \\rho} (H{(\\rho)} - e^{\\rho}) = \\frac{d}{d \\rho} (- e^{\\rho} + \\cos{(e^{\\rho})}) and - e^{\\rho} + \\frac{d}{d \\rho} H{(\\rho)} = - e^{\\rho} \\sin{(e^{\\rho})} - e^{\\rho} and - e^{\\rho} + \\frac{d}{d \\rho} \\cos{(e^{\\rho})} = - e^{\\rho} \\sin{(e^{\\rho})} - e^{\\rho} and \\rho - e^{\\rho} + \\frac{d}{d \\rho} \\cos{(e^{\\rho})} = \\rho - e^{\\rho} \\sin{(e^{\\rho})} - e^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\rho', commutative=True)), cos(exp(Symbol('\\\\rho', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('H')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))), cos(exp(Symbol('\\\\rho', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Add(Function('H')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))), cos(exp(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))), Derivative(Function('H')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)), sin(exp(Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))), Derivative(cos(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)), sin(exp(Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)))))"], [["add", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))), Derivative(cos(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)), sin(exp(Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given W{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and \\operatorname{v_{y}}{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})}, then obtain \\int (W{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1) dE_{x} = \\int (\\operatorname{v_{y}}{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1) dE_{x}", "derivation": "W{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and \\operatorname{v_{y}}{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and W{(E_{x})} = \\operatorname{v_{y}}{(E_{x})} and W{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1 = \\operatorname{v_{y}}{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1 and \\int (W{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1) dE_{x} = \\int (\\operatorname{v_{y}}{(E_{x})} - \\frac{d}{d E_{x}} \\log{(E_{x})} + 1) dE_{x}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('E_x', commutative=True)), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('E_x', commutative=True)), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('W')(Symbol('E_x', commutative=True)), Function('v_y')(Symbol('E_x', commutative=True)))"], [["minus", 3, "Add(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Function('W')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1)), Add(Function('v_y')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1)))"], [["integrate", 4, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Function('W')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Function('v_y')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} = F_{c} + \\frac{y^{\\prime}}{m}, then obtain \\int (\\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} - \\frac{1}{m}) (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m}) dy^{\\prime} = \\int (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m})^{2} dy^{\\prime}", "derivation": "\\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} = F_{c} + \\frac{y^{\\prime}}{m} and \\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} - \\frac{1}{m} = F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m} and (\\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} - \\frac{1}{m}) (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m}) = (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m})^{2} and \\int (\\operatorname{P_{e}}{(m,F_{c},y^{\\prime})} - \\frac{1}{m}) (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m}) dy^{\\prime} = \\int (F_{c} + \\frac{y^{\\prime}}{m} - \\frac{1}{m})^{2} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('m', commutative=True), Symbol('F_c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "Pow(Symbol('m', commutative=True), Integer(-1))"], "Equality(Add(Function('P_e')(Symbol('m', commutative=True), Symbol('F_c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["times", 2, "Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Function('P_e')(Symbol('m', commutative=True), Symbol('F_c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))), Pow(Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Integer(2)))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Add(Function('P_e')(Symbol('m', commutative=True), Symbol('F_c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Pow(Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Integer(2)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(b,s)} = b^{s} and \\eta{(\\lambda,\\mathbf{r})} = e^{\\lambda \\mathbf{r}}, then obtain b^{s} + e^{\\lambda \\mathbf{r}} = b^{s} - \\eta{(\\lambda,\\mathbf{r})} + 2 e^{\\lambda \\mathbf{r}}", "derivation": "\\mathbf{F}{(b,s)} = b^{s} and \\eta{(\\lambda,\\mathbf{r})} = e^{\\lambda \\mathbf{r}} and \\eta{(\\lambda,\\mathbf{r})} + \\mathbf{F}{(b,s)} = \\mathbf{F}{(b,s)} + e^{\\lambda \\mathbf{r}} and b^{s} + \\eta{(\\lambda,\\mathbf{r})} = b^{s} + e^{\\lambda \\mathbf{r}} and b^{s} = b^{s} - \\eta{(\\lambda,\\mathbf{r})} + e^{\\lambda \\mathbf{r}} and b^{s} + e^{\\lambda \\mathbf{r}} = b^{s} - \\eta{(\\lambda,\\mathbf{r})} + 2 e^{\\lambda \\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["minus", 4, "Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), Add(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Add(Pow(Symbol('b', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(2), exp(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(x^\\prime)} = e^{e^{x^\\prime}}, then obtain (\\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + \\operatorname{a^{\\dagger}}{(x^\\prime)})^{x^\\prime} = (\\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + e^{e^{x^\\prime}})^{x^\\prime}", "derivation": "\\operatorname{a^{\\dagger}}{(x^\\prime)} = e^{e^{x^\\prime}} and \\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} = e^{2 e^{x^\\prime}} and \\operatorname{a^{\\dagger}}{(x^\\prime)} + e^{2 e^{x^\\prime}} = e^{2 e^{x^\\prime}} + e^{e^{x^\\prime}} and \\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + \\operatorname{a^{\\dagger}}{(x^\\prime)} = \\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + e^{e^{x^\\prime}} and (\\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + \\operatorname{a^{\\dagger}}{(x^\\prime)})^{x^\\prime} = (\\operatorname{a^{\\dagger}}{(x^\\prime)} e^{e^{x^\\prime}} + e^{e^{x^\\prime}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True))))"], [["times", 1, "exp(exp(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('x^\\\\prime', commutative=True)))))"], [["add", 1, "exp(Mul(Integer(2), exp(Symbol('x^\\\\prime', commutative=True))))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(Mul(Integer(2), exp(Symbol('x^\\\\prime', commutative=True))))), Add(exp(Mul(Integer(2), exp(Symbol('x^\\\\prime', commutative=True)))), exp(exp(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True))), Add(Mul(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), exp(exp(Symbol('x^\\\\prime', commutative=True)))))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Mul(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), exp(exp(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(M)} = \\log{(M)}, then obtain (\\iint \\rho_{f}{(M)} dM dM)^{M} = (\\iint \\log{(M)} dM dM)^{M}", "derivation": "\\rho_{f}{(M)} = \\log{(M)} and \\int \\rho_{f}{(M)} dM = \\int \\log{(M)} dM and \\iint \\rho_{f}{(M)} dM dM = \\iint \\log{(M)} dM dM and (\\iint \\rho_{f}{(M)} dM dM)^{M} = (\\iint \\log{(M)} dM dM)^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given u{(\\mathbf{J}_M)} = \\mathbf{J}_M, then obtain \\mathbf{J}_M^{2} (u^{2}{(\\mathbf{J}_M)})^{2 \\mathbf{J}_M} u^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M^{2} (\\mathbf{J}_M u{(\\mathbf{J}_M)})^{\\mathbf{J}_M} (u^{2}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} u^{2}{(\\mathbf{J}_M)}", "derivation": "u{(\\mathbf{J}_M)} = \\mathbf{J}_M and u^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M u{(\\mathbf{J}_M)} and (u^{2}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} = (\\mathbf{J}_M u{(\\mathbf{J}_M)})^{\\mathbf{J}_M} and \\mathbf{J}_M (u^{2}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} u{(\\mathbf{J}_M)} = \\mathbf{J}_M (\\mathbf{J}_M u{(\\mathbf{J}_M)})^{\\mathbf{J}_M} u{(\\mathbf{J}_M)} and \\mathbf{J}_M^{2} (u^{2}{(\\mathbf{J}_M)})^{2 \\mathbf{J}_M} u^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M^{2} (\\mathbf{J}_M u{(\\mathbf{J}_M)})^{\\mathbf{J}_M} (u^{2}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} u^{2}{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], [["times", 1, "Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 3, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 4, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Pow(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Function('u')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}{(h,A)} = A^{h}, then obtain \\frac{\\partial}{\\partial A} \\int A^{h} (h \\tilde{g}{(h,A)})^{h} dA = \\frac{\\partial}{\\partial A} \\int A^{h} (A^{h} h)^{h} dA", "derivation": "\\tilde{g}{(h,A)} = A^{h} and h \\tilde{g}{(h,A)} = A^{h} h and (h \\tilde{g}{(h,A)})^{h} = (A^{h} h)^{h} and A^{h} (h \\tilde{g}{(h,A)})^{h} = A^{h} (A^{h} h)^{h} and \\int A^{h} (h \\tilde{g}{(h,A)})^{h} dA = \\int A^{h} (A^{h} h)^{h} dA and \\frac{\\partial}{\\partial A} \\int A^{h} (h \\tilde{g}{(h,A)})^{h} dA = \\frac{\\partial}{\\partial A} \\int A^{h} (A^{h} h)^{h} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Symbol('h', commutative=True), Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["times", 3, "Pow(Symbol('A', commutative=True), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Symbol('h', commutative=True), Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Symbol('h', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["integrate", 4, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Symbol('h', commutative=True), Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["differentiate", 5, "Symbol('A', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Symbol('h', commutative=True), Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(f_{E})} = \\log{(f_{E})}, then obtain \\frac{f_{E} + 2 k{(f_{E})}}{f_{E} + k{(f_{E})} + \\log{(f_{E})}} = 1", "derivation": "k{(f_{E})} = \\log{(f_{E})} and f_{E} + k{(f_{E})} = f_{E} + \\log{(f_{E})} and f_{E} + 2 k{(f_{E})} = f_{E} + k{(f_{E})} + \\log{(f_{E})} and f_{E} + 2 k{(f_{E})} = f_{E} + 2 \\log{(f_{E})} and \\frac{f_{E} + 2 k{(f_{E})}}{f_{E} + 2 \\log{(f_{E})}} = \\frac{f_{E} + k{(f_{E})} + \\log{(f_{E})}}{f_{E} + 2 \\log{(f_{E})}} and f_{E} + k{(f_{E})} + \\log{(f_{E})} = f_{E} + 2 \\log{(f_{E})} and \\frac{f_{E} + 2 k{(f_{E})}}{f_{E} + k{(f_{E})} + \\log{(f_{E})}} = 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["add", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True))), Add(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True))))"], [["add", 1, "Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True)))"], "Equality(Add(Symbol('f_E', commutative=True), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True)))), Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('f_E', commutative=True), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True)))), Add(Symbol('f_E', commutative=True), Mul(Integer(2), log(Symbol('f_E', commutative=True)))))"], [["divide", 3, "Add(Symbol('f_E', commutative=True), Mul(Integer(2), log(Symbol('f_E', commutative=True))))"], "Equality(Mul(Add(Symbol('f_E', commutative=True), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True)))), Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(2), log(Symbol('f_E', commutative=True)))), Integer(-1))), Mul(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(2), log(Symbol('f_E', commutative=True)))), Integer(-1)), Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Integer(2), log(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Add(Symbol('f_E', commutative=True), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True)))), Pow(Add(Symbol('f_E', commutative=True), Function('k')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given Q{(\\delta,\\phi_1)} = \\log{(\\phi_1)}^{\\delta} and \\mathbf{s}{(\\hat{x},M)} = - M + \\hat{x}, then obtain - \\frac{M Q{(\\delta,\\phi_1)} \\mathbf{s}{(\\hat{x},M)}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1} = - \\frac{M (- M + \\hat{x}) Q{(\\delta,\\phi_1)}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1}", "derivation": "Q{(\\delta,\\phi_1)} = \\log{(\\phi_1)}^{\\delta} and \\mathbf{s}{(\\hat{x},M)} = - M + \\hat{x} and - M \\mathbf{s}{(\\hat{x},M)} = - M (- M + \\hat{x}) and - \\frac{M \\mathbf{s}{(\\hat{x},M)} \\log{(\\phi_1)}^{\\delta}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1} = - \\frac{M (- M + \\hat{x}) \\log{(\\phi_1)}^{\\delta}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1} and - \\frac{M Q{(\\delta,\\phi_1)} \\mathbf{s}{(\\hat{x},M)}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1} = - \\frac{M (- M + \\hat{x}) Q{(\\delta,\\phi_1)}}{\\int Q{(\\delta,\\phi_1)} d\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('M', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('M', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True), Symbol('M', commutative=True))), Mul(Integer(-1), Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\hat{x}', commutative=True))))"], [["times", 3, "Mul(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(-1), Symbol('M', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True), Symbol('M', commutative=True)), Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('M', commutative=True), Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True), Symbol('M', commutative=True)), Pow(Integral(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Integral(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{1},\\sigma_p)} = \\frac{v_{1}}{\\sigma_p}, then derive \\frac{\\partial}{\\partial v_{1}} \\operatorname{F_{c}}{(v_{1},\\sigma_p)} = \\frac{1}{\\sigma_p}, then obtain \\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\sigma_p} - \\frac{v_{1}}{\\sigma_p} = - \\frac{v_{1}}{\\sigma_p} + \\frac{1}{\\sigma_p}", "derivation": "\\operatorname{F_{c}}{(v_{1},\\sigma_p)} = \\frac{v_{1}}{\\sigma_p} and \\frac{\\partial}{\\partial v_{1}} \\operatorname{F_{c}}{(v_{1},\\sigma_p)} = \\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\sigma_p} and \\frac{\\partial}{\\partial v_{1}} \\operatorname{F_{c}}{(v_{1},\\sigma_p)} = \\frac{1}{\\sigma_p} and \\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\sigma_p} = \\frac{1}{\\sigma_p} and \\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\sigma_p} - \\frac{v_{1}}{\\sigma_p} = - \\frac{v_{1}}{\\sigma_p} + \\frac{1}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('v_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('v_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["minus", 4, "Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))"], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(I,B)} = \\log{(B^{I})}, then obtain 8 B^{I} \\hat{\\mathbf{r}}^{3}{(I,B)} = 4 B^{I} (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})}) \\hat{\\mathbf{r}}^{2}{(I,B)}", "derivation": "\\hat{\\mathbf{r}}{(I,B)} = \\log{(B^{I})} and 2 \\hat{\\mathbf{r}}{(I,B)} = \\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})} and 2 B^{I} \\hat{\\mathbf{r}}{(I,B)} = B^{I} (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})}) and 2 (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})}) \\hat{\\mathbf{r}}{(I,B)} = (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})})^{2} and 4 B^{I} (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})}) \\hat{\\mathbf{r}}^{2}{(I,B)} = 2 B^{I} (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})})^{2} \\hat{\\mathbf{r}}{(I,B)} and 8 B^{I} \\hat{\\mathbf{r}}^{3}{(I,B)} = 4 B^{I} (\\hat{\\mathbf{r}}{(I,B)} + \\log{(B^{I})}) \\hat{\\mathbf{r}}^{2}{(I,B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True))))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))))"], [["times", 2, "Pow(Symbol('B', commutative=True), Symbol('I', commutative=True))"], "Equality(Mul(Integer(2), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True))))))"], [["times", 2, "Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True))))"], "Equality(Mul(Integer(2), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True))), Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))), Integer(2)))"], [["times", 4, "Mul(Integer(2), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(2), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))), Integer(2)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(8), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(3))), Mul(Integer(4), Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('I', commutative=True)))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\phi_2,\\pi)} = e^{\\phi_2 + \\pi}, then obtain (- \\pi + \\operatorname{c_{0}}^{\\phi_2}{(\\phi_2,\\pi)} - (e^{\\phi_2 + \\pi})^{\\phi_2})^{\\phi_2} = (- \\pi)^{\\phi_2}", "derivation": "\\operatorname{c_{0}}{(\\phi_2,\\pi)} = e^{\\phi_2 + \\pi} and \\operatorname{c_{0}}^{\\phi_2}{(\\phi_2,\\pi)} = (e^{\\phi_2 + \\pi})^{\\phi_2} and - \\pi + \\operatorname{c_{0}}^{\\phi_2}{(\\phi_2,\\pi)} = - \\pi + (e^{\\phi_2 + \\pi})^{\\phi_2} and - \\pi + \\operatorname{c_{0}}^{\\phi_2}{(\\phi_2,\\pi)} - (e^{\\phi_2 + \\pi})^{\\phi_2} = - \\pi and (- \\pi + \\operatorname{c_{0}}^{\\phi_2}{(\\phi_2,\\pi)} - (e^{\\phi_2 + \\pi})^{\\phi_2})^{\\phi_2} = (- \\pi)^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True)), exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(Function('c_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"], [["minus", 3, "Pow(exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(Function('c_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(Function('c_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(c_{0},\\pi)} = \\frac{\\partial}{\\partial \\pi} \\pi c_{0}, then derive \\operatorname{f_{\\mathbf{p}}}{(c_{0},\\pi)} = c_{0}, then obtain \\operatorname{f_{\\mathbf{p}}}^{\\pi}{(\\frac{\\partial}{\\partial \\pi} \\pi c_{0},\\pi)} = (\\frac{\\partial}{\\partial \\pi} \\pi c_{0})^{\\pi}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(c_{0},\\pi)} = \\frac{\\partial}{\\partial \\pi} \\pi c_{0} and \\operatorname{f_{\\mathbf{p}}}{(c_{0},\\pi)} = c_{0} and \\frac{\\partial}{\\partial \\pi} \\pi c_{0} = c_{0} and \\operatorname{f_{\\mathbf{p}}}^{\\pi}{(c_{0},\\pi)} = c_{0}^{\\pi} and \\operatorname{f_{\\mathbf{p}}}^{\\pi}{(\\frac{\\partial}{\\partial \\pi} \\pi c_{0},\\pi)} = (\\frac{\\partial}{\\partial \\pi} \\pi c_{0})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('c_0', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('c_0', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('c_0', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('c_0', commutative=True))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('c_0', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(H)} = \\int \\sin{(H)} dH, then obtain ((\\ddot{x}{(H)} + 1) \\frac{d}{d H} \\ddot{x}{(H)})^{H} = ((\\int \\sin{(H)} dH + 1) \\frac{d}{d H} \\ddot{x}{(H)})^{H}", "derivation": "\\ddot{x}{(H)} = \\int \\sin{(H)} dH and \\frac{d}{d H} \\ddot{x}{(H)} = \\frac{d}{d H} \\int \\sin{(H)} dH and \\ddot{x}{(H)} + 1 = \\int \\sin{(H)} dH + 1 and (\\ddot{x}{(H)} + 1) \\frac{d}{d H} \\int \\sin{(H)} dH = (\\int \\sin{(H)} dH + 1) \\frac{d}{d H} \\int \\sin{(H)} dH and (\\ddot{x}{(H)} + 1) \\frac{d}{d H} \\ddot{x}{(H)} = (\\int \\sin{(H)} dH + 1) \\frac{d}{d H} \\ddot{x}{(H)} and ((\\ddot{x}{(H)} + 1) \\frac{d}{d H} \\ddot{x}{(H)})^{H} = ((\\int \\sin{(H)} dH + 1) \\frac{d}{d H} \\ddot{x}{(H)})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Integer(1)), Add(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(1)))"], [["times", 3, "Derivative(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Integer(1)), Derivative(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(1)), Derivative(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Integer(1)), Derivative(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(1)), Derivative(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Integer(1)), Derivative(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Pow(Mul(Add(Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(1)), Derivative(Function('\\\\ddot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(z)} = \\sin{(z)}, then derive \\frac{d}{d z} \\phi_{1}{(z)} + 1 = \\cos{(z)} + 1, then obtain \\frac{d}{d z} (\\frac{d}{d z} \\phi_{1}{(z)} + 1) = \\frac{d}{d z} (\\cos{(z)} + 1)", "derivation": "\\phi_{1}{(z)} = \\sin{(z)} and z + \\phi_{1}{(z)} = z + \\sin{(z)} and \\frac{d}{d z} (z + \\phi_{1}{(z)}) = \\frac{d}{d z} (z + \\sin{(z)}) and \\frac{d}{d z} \\phi_{1}{(z)} + 1 = \\cos{(z)} + 1 and \\frac{d}{d z} (\\frac{d}{d z} \\phi_{1}{(z)} + 1) = \\frac{d}{d z} (\\cos{(z)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["add", 1, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('z', commutative=True))), Add(Symbol('z', commutative=True), sin(Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('z', commutative=True), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\phi_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('z', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\phi_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('z', commutative=True)), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(\\rho_b)} = \\frac{d}{d \\rho_b} \\sin{(\\rho_b)}, then obtain \\rho_b \\frac{d}{d \\rho_b} \\frac{E{(\\rho_b)}}{\\sin{(\\rho_b)}} = \\rho_b \\frac{d}{d \\rho_b} \\frac{\\frac{d}{d \\rho_b} \\sin{(\\rho_b)}}{\\sin{(\\rho_b)}}", "derivation": "E{(\\rho_b)} = \\frac{d}{d \\rho_b} \\sin{(\\rho_b)} and \\frac{E{(\\rho_b)}}{\\sin{(\\rho_b)}} = \\frac{\\frac{d}{d \\rho_b} \\sin{(\\rho_b)}}{\\sin{(\\rho_b)}} and \\frac{d}{d \\rho_b} \\frac{E{(\\rho_b)}}{\\sin{(\\rho_b)}} = \\frac{d}{d \\rho_b} \\frac{\\frac{d}{d \\rho_b} \\sin{(\\rho_b)}}{\\sin{(\\rho_b)}} and \\rho_b \\frac{d}{d \\rho_b} \\frac{E{(\\rho_b)}}{\\sin{(\\rho_b)}} = \\rho_b \\frac{d}{d \\rho_b} \\frac{\\frac{d}{d \\rho_b} \\sin{(\\rho_b)}}{\\sin{(\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\rho_b', commutative=True)), Derivative(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["divide", 1, "sin(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('E')(Symbol('\\\\rho_b', commutative=True)), Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Function('E')(Symbol('\\\\rho_b', commutative=True)), Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Derivative(Mul(Function('E')(Symbol('\\\\rho_b', commutative=True)), Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho_b', commutative=True), Derivative(Mul(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(g_{\\varepsilon})} = \\sin{(\\sin{(g_{\\varepsilon})})}, then obtain z{(g_{\\varepsilon})} + \\sin{(\\sin{(g_{\\varepsilon})})} - \\sin^{g_{\\varepsilon}}{(\\sin{(g_{\\varepsilon})})} = 2 \\sin{(\\sin{(g_{\\varepsilon})})} - \\sin^{g_{\\varepsilon}}{(\\sin{(g_{\\varepsilon})})}", "derivation": "z{(g_{\\varepsilon})} = \\sin{(\\sin{(g_{\\varepsilon})})} and z^{g_{\\varepsilon}}{(g_{\\varepsilon})} = \\sin^{g_{\\varepsilon}}{(\\sin{(g_{\\varepsilon})})} and z{(g_{\\varepsilon})} + \\sin{(\\sin{(g_{\\varepsilon})})} = 2 \\sin{(\\sin{(g_{\\varepsilon})})} and z{(g_{\\varepsilon})} - z^{g_{\\varepsilon}}{(g_{\\varepsilon})} + \\sin{(\\sin{(g_{\\varepsilon})})} = - z^{g_{\\varepsilon}}{(g_{\\varepsilon})} + 2 \\sin{(\\sin{(g_{\\varepsilon})})} and z{(g_{\\varepsilon})} + \\sin{(\\sin{(g_{\\varepsilon})})} - \\sin^{g_{\\varepsilon}}{(\\sin{(g_{\\varepsilon})})} = 2 \\sin{(\\sin{(g_{\\varepsilon})})} - \\sin^{g_{\\varepsilon}}{(\\sin{(g_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["minus", 3, "Pow(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('z')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Pow(sin(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given i{(\\mathbf{H},\\Psi_{nl})} = \\frac{\\mathbf{H}}{\\Psi_{nl}}, then derive \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})} = \\frac{1}{\\Psi_{nl}}, then obtain \\int (- i{(\\mathbf{H},\\Psi_{nl})} + \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})}) d\\mathbf{H} = \\int (- i{(\\mathbf{H},\\Psi_{nl})} + \\frac{1}{\\Psi_{nl}}) d\\mathbf{H}", "derivation": "i{(\\mathbf{H},\\Psi_{nl})} = \\frac{\\mathbf{H}}{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\frac{\\mathbf{H}}{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})} = \\frac{1}{\\Psi_{nl}} and - i{(\\mathbf{H},\\Psi_{nl})} + \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})} = - i{(\\mathbf{H},\\Psi_{nl})} + \\frac{1}{\\Psi_{nl}} and \\int (- i{(\\mathbf{H},\\Psi_{nl})} + \\frac{\\partial}{\\partial \\mathbf{H}} i{(\\mathbf{H},\\Psi_{nl})}) d\\mathbf{H} = \\int (- i{(\\mathbf{H},\\Psi_{nl})} + \\frac{1}{\\Psi_{nl}}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))"], [["minus", 3, "Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Derivative(Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Derivative(Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given H{(u)} = e^{u}, then obtain 2 \\frac{d}{d u} H{(u)} = e^{u} + \\frac{d}{d u} H{(u)}", "derivation": "H{(u)} = e^{u} and \\frac{d}{d u} H{(u)} = \\frac{d}{d u} e^{u} and 2 \\frac{d}{d u} H{(u)} = \\frac{d}{d u} H{(u)} + \\frac{d}{d u} e^{u} and 2 \\frac{d}{d u} H{(u)} = e^{u} + \\frac{d}{d u} H{(u)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(exp(Symbol('u', commutative=True)), Derivative(Function('H')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\nabla)} = \\log{(\\nabla)}, then obtain \\int (\\operatorname{F_{c}}^{\\nabla}{(\\nabla)} + \\log{(\\nabla)}) d\\nabla = \\int (\\log{(\\nabla)} + \\log{(\\nabla)}^{\\nabla}) d\\nabla", "derivation": "\\operatorname{F_{c}}{(\\nabla)} = \\log{(\\nabla)} and \\operatorname{F_{c}}^{\\nabla}{(\\nabla)} = \\log{(\\nabla)}^{\\nabla} and \\operatorname{F_{c}}^{\\nabla}{(\\nabla)} + \\log{(\\nabla)} = \\log{(\\nabla)} + \\log{(\\nabla)}^{\\nabla} and \\int (\\operatorname{F_{c}}^{\\nabla}{(\\nabla)} + \\log{(\\nabla)}) d\\nabla = \\int (\\log{(\\nabla)} + \\log{(\\nabla)}^{\\nabla}) d\\nabla", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["power", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["add", 2, "log(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Pow(Function('F_c')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Add(log(Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Add(Pow(Function('F_c')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(log(Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given t{(S)} = e^{e^{S}}, then derive 0 = L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (t{(S)} + e^{S}) dS, then obtain \\int 0^{L} dL = \\int (L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (t{(S)} + e^{S}) dS)^{L} dL", "derivation": "t{(S)} = e^{e^{S}} and t{(S)} + e^{S} = e^{S} + e^{e^{S}} and \\int (t{(S)} + e^{S}) dS = \\int (e^{S} + e^{e^{S}}) dS and 0 = - \\int (t{(S)} + e^{S}) dS + \\int (e^{S} + e^{e^{S}}) dS and 0 = L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (t{(S)} + e^{S}) dS and 0 = L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (e^{S} + e^{e^{S}}) dS and 0^{L} = (L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (e^{S} + e^{e^{S}}) dS)^{L} and \\int 0^{L} dL = \\int (L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (e^{S} + e^{e^{S}}) dS)^{L} dL and \\int 0^{L} dL = \\int (L + e^{S} + \\operatorname{Ei}{(e^{S})} - \\int (t{(S)} + e^{S}) dS)^{L} dL", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True))))"], [["add", 1, "exp(Symbol('S', commutative=True))"], "Equality(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["minus", 3, "Integral(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))), Integral(Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Integer(0), Add(Symbol('L', commutative=True), exp(Symbol('S', commutative=True)), Ei(exp(Symbol('S', commutative=True))), Mul(Integer(-1), Integral(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Symbol('L', commutative=True), exp(Symbol('S', commutative=True)), Ei(exp(Symbol('S', commutative=True))), Mul(Integer(-1), Integral(Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))))"], [["power", 6, "Symbol('L', commutative=True)"], "Equality(Pow(Integer(0), Symbol('L', commutative=True)), Pow(Add(Symbol('L', commutative=True), exp(Symbol('S', commutative=True)), Ei(exp(Symbol('S', commutative=True))), Mul(Integer(-1), Integral(Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))), Symbol('L', commutative=True)))"], [["integrate", 7, "Symbol('L', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(Add(Symbol('L', commutative=True), exp(Symbol('S', commutative=True)), Ei(exp(Symbol('S', commutative=True))), Mul(Integer(-1), Integral(Add(exp(Symbol('S', commutative=True)), exp(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Integral(Pow(Integer(0), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(Add(Symbol('L', commutative=True), exp(Symbol('S', commutative=True)), Ei(exp(Symbol('S', commutative=True))), Mul(Integer(-1), Integral(Add(Function('t')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{S},f)} = f e^{\\mathbf{S}}, then obtain (f \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{P}{(\\mathbf{S},f)} + 2 \\frac{\\partial}{\\partial f} \\mathbf{P}{(\\mathbf{S},f)}) e^{\\mathbf{S}} = 2 e^{2 \\mathbf{S}}", "derivation": "\\mathbf{P}{(\\mathbf{S},f)} = f e^{\\mathbf{S}} and f \\mathbf{P}{(\\mathbf{S},f)} e^{\\mathbf{S}} = f^{2} e^{2 \\mathbf{S}} and \\frac{\\partial}{\\partial f} f \\mathbf{P}{(\\mathbf{S},f)} e^{\\mathbf{S}} = \\frac{\\partial}{\\partial f} f^{2} e^{2 \\mathbf{S}} and \\frac{\\partial^{2}}{\\partial f^{2}} f \\mathbf{P}{(\\mathbf{S},f)} e^{\\mathbf{S}} = \\frac{\\partial^{2}}{\\partial f^{2}} f^{2} e^{2 \\mathbf{S}} and (f \\frac{\\partial^{2}}{\\partial f^{2}} \\mathbf{P}{(\\mathbf{S},f)} + 2 \\frac{\\partial}{\\partial f} \\mathbf{P}{(\\mathbf{S},f)}) e^{\\mathbf{S}} = 2 e^{2 \\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Mul(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Symbol('f', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Symbol('f', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Symbol('f', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Mul(Symbol('f', commutative=True), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), exp(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given Z{(\\lambda)} = \\sin{(\\lambda)} and \\operatorname{A_{z}}{(\\lambda)} = - Z{(\\lambda)} + \\sin^{2}{(\\lambda)} - \\sin{(\\lambda)}, then obtain \\operatorname{A_{z}}{(\\lambda)} = Z{(\\lambda)} \\sin{(\\lambda)} - Z{(\\lambda)} - \\sin{(\\lambda)}", "derivation": "Z{(\\lambda)} = \\sin{(\\lambda)} and Z{(\\lambda)} + \\sin{(\\lambda)} = 2 \\sin{(\\lambda)} and Z{(\\lambda)} \\sin{(\\lambda)} = \\sin^{2}{(\\lambda)} and Z{(\\lambda)} \\sin{(\\lambda)} - 2 \\sin{(\\lambda)} = \\sin^{2}{(\\lambda)} - 2 \\sin{(\\lambda)} and Z{(\\lambda)} \\sin{(\\lambda)} - Z{(\\lambda)} - \\sin{(\\lambda)} = - Z{(\\lambda)} + \\sin^{2}{(\\lambda)} - \\sin{(\\lambda)} and \\operatorname{A_{z}}{(\\lambda)} = - Z{(\\lambda)} + \\sin^{2}{(\\lambda)} - \\sin{(\\lambda)} and \\operatorname{A_{z}}{(\\lambda)} = Z{(\\lambda)} \\sin{(\\lambda)} - Z{(\\lambda)} - \\sin{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2)))"], [["minus", 3, "Mul(Integer(2), sin(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\lambda', commutative=True)))), Add(Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Function('Z')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Integer(-1), Function('Z')(Symbol('\\\\lambda', commutative=True))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Function('Z')(Symbol('\\\\lambda', commutative=True))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('A_z')(Symbol('\\\\lambda', commutative=True)), Add(Mul(Function('Z')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Function('Z')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{s})} = \\mathbf{s}, then obtain \\frac{d}{d \\mathbf{s}} (\\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s} + 1) = \\frac{d}{d \\mathbf{s}} ((\\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}})^{\\mathbf{s}} + \\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s})", "derivation": "\\operatorname{v_{1}}{(\\mathbf{s})} = \\mathbf{s} and 1 = \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} and 1 = (\\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}})^{\\mathbf{s}} and \\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s} + 1 = (\\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}})^{\\mathbf{s}} + \\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s} and \\frac{d}{d \\mathbf{s}} (\\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s} + 1) = \\frac{d}{d \\mathbf{s}} ((\\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}})^{\\mathbf{s}} + \\int \\frac{\\mathbf{s}}{\\operatorname{v_{1}}{(\\mathbf{s})}} d\\mathbf{s})", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["divide", 1, "Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 3, "Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Add(Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('v_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(r_{0},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1 + r_{0}), then derive b{(r_{0},\\theta_1)} = 1, then obtain \\theta_1 + r_{0} + \\frac{\\partial^{2}}{\\partial r_{0}\\partial \\theta_1} (\\theta_1 + r_{0}) = \\theta_1 + r_{0} + \\frac{d}{d r_{0}} 1", "derivation": "b{(r_{0},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1 + r_{0}) and b{(r_{0},\\theta_1)} = 1 and \\frac{\\partial}{\\partial \\theta_1} (\\theta_1 + r_{0}) = 1 and \\frac{\\partial^{2}}{\\partial r_{0}\\partial \\theta_1} (\\theta_1 + r_{0}) = \\frac{d}{d r_{0}} 1 and \\theta_1 + r_{0} + \\frac{\\partial^{2}}{\\partial r_{0}\\partial \\theta_1} (\\theta_1 + r_{0}) = \\theta_1 + r_{0} + \\frac{d}{d r_{0}} 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('r_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('b')(Symbol('r_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["add", 4, "Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Symbol('\\\\theta_1', commutative=True), Symbol('r_0', commutative=True), Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(Z,A_{y})} = A_{y} - Z, then derive (\\frac{\\partial}{\\partial A_{y}} \\operatorname{a^{\\dagger}}{(Z,A_{y})})^{A_{y}} = 1, then obtain (\\frac{\\partial}{\\partial A_{y}} (A_{y} - Z))^{A_{y}} = 1", "derivation": "\\operatorname{a^{\\dagger}}{(Z,A_{y})} = A_{y} - Z and \\frac{\\partial}{\\partial A_{y}} \\operatorname{a^{\\dagger}}{(Z,A_{y})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} - Z) and (\\frac{\\partial}{\\partial A_{y}} \\operatorname{a^{\\dagger}}{(Z,A_{y})})^{A_{y}} = (\\frac{\\partial}{\\partial A_{y}} (A_{y} - Z))^{A_{y}} and (\\frac{\\partial}{\\partial A_{y}} \\operatorname{a^{\\dagger}}{(Z,A_{y})})^{A_{y}} = 1 and (\\frac{\\partial}{\\partial A_{y}} (A_{y} - Z))^{A_{y}} = 1", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Symbol('A_y', commutative=True)), Pow(Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Symbol('A_y', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Symbol('A_y', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Symbol('A_y', commutative=True)), Integer(1))"]]}, {"prompt": "Given H{(v_{y},\\mathbf{D})} = \\mathbf{D} \\log{(v_{y})} and \\Psi_{\\lambda}{(v_{y},\\mathbf{D})} = \\mathbf{D} \\log{(v_{y})}, then obtain (\\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\log{(v_{y})})^{\\mathbf{D}} = (\\frac{\\partial}{\\partial \\mathbf{D}} H{(v_{y},\\mathbf{D})})^{\\mathbf{D}}", "derivation": "H{(v_{y},\\mathbf{D})} = \\mathbf{D} \\log{(v_{y})} and \\Psi_{\\lambda}{(v_{y},\\mathbf{D})} = \\mathbf{D} \\log{(v_{y})} and \\Psi_{\\lambda}{(v_{y},\\mathbf{D})} = H{(v_{y},\\mathbf{D})} and \\frac{\\partial}{\\partial \\mathbf{D}} \\Psi_{\\lambda}{(v_{y},\\mathbf{D})} = \\frac{\\partial}{\\partial \\mathbf{D}} H{(v_{y},\\mathbf{D})} and \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\log{(v_{y})} = \\frac{\\partial}{\\partial \\mathbf{D}} H{(v_{y},\\mathbf{D})} and (\\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} \\log{(v_{y})})^{\\mathbf{D}} = (\\frac{\\partial}{\\partial \\mathbf{D}} H{(v_{y},\\mathbf{D})})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Symbol('v_y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(u)} = \\sin{(u)}, then obtain \\frac{q}{\\mathbf{v}^{3}{(u)}} = \\frac{q}{\\mathbf{v}^{2}{(u)} \\sin{(u)}}", "derivation": "\\mathbf{v}{(u)} = \\sin{(u)} and \\mathbf{v}^{2}{(u)} = \\mathbf{v}{(u)} \\sin{(u)} and \\mathbf{v}^{4}{(u)} = \\mathbf{v}^{2}{(u)} \\sin^{2}{(u)} and q \\mathbf{v}{(u)} = q \\sin{(u)} and \\frac{q}{\\mathbf{v}^{3}{(u)}} = \\frac{q \\sin{(u)}}{\\mathbf{v}^{4}{(u)}} and \\frac{q}{\\mathbf{v}^{3}{(u)}} = \\frac{q}{\\mathbf{v}^{2}{(u)} \\sin{(u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{v}')(Symbol('u', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('\\\\mathbf{v}')(Symbol('u', commutative=True))), Mul(Symbol('q', commutative=True), sin(Symbol('u', commutative=True))))"], [["divide", 4, "Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(4))"], "Equality(Mul(Symbol('q', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(-3))), Mul(Symbol('q', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(-4)), sin(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('q', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(-3))), Mul(Symbol('q', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('u', commutative=True)), Integer(-2)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)}, then derive \\hat{p}{(\\varepsilon)} = - \\sin{(\\varepsilon)}, then obtain \\hat{p}{(\\varepsilon)} - \\cos{(\\varepsilon)} = - \\cos{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)}", "derivation": "\\hat{p}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and \\hat{p}{(\\varepsilon)} = - \\sin{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} = - \\sin{(\\varepsilon)} and - \\cos{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} = - \\sin{(\\varepsilon)} - \\cos{(\\varepsilon)} and \\hat{p}{(\\varepsilon)} - \\cos{(\\varepsilon)} = - \\sin{(\\varepsilon)} - \\cos{(\\varepsilon)} and \\hat{p}{(\\varepsilon)} - \\cos{(\\varepsilon)} = - \\cos{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{p}')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 3, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(r_{0},\\omega)} = \\omega - r_{0}, then obtain (- r_{0} + \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\omega)})^{r_{0}} = (- r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\omega - r_{0}))^{r_{0}}", "derivation": "\\hat{x}_0{(r_{0},\\omega)} = \\omega - r_{0} and \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\omega)} = \\frac{\\partial}{\\partial r_{0}} (\\omega - r_{0}) and - r_{0} + \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\omega)} = - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\omega - r_{0}) and (- r_{0} + \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\omega)})^{r_{0}} = (- r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\omega - r_{0}))^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["add", 2, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Symbol('r_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(n_{1},y)} = - n_{1} + y, then obtain y (1 - \\frac{\\operatorname{t_{1}}{(n_{1},y)}}{n_{1} y}) = y (1 - \\frac{- n_{1} + y}{n_{1} y})", "derivation": "\\operatorname{t_{1}}{(n_{1},y)} = - n_{1} + y and \\frac{\\operatorname{t_{1}}{(n_{1},y)}}{y} = \\frac{- n_{1} + y}{y} and - \\frac{\\operatorname{t_{1}}{(n_{1},y)}}{n_{1} y} = - \\frac{- n_{1} + y}{n_{1} y} and 1 - \\frac{\\operatorname{t_{1}}{(n_{1},y)}}{n_{1} y} = 1 - \\frac{- n_{1} + y}{n_{1} y} and y (1 - \\frac{\\operatorname{t_{1}}{(n_{1},y)}}{n_{1} y}) = y (1 - \\frac{- n_{1} + y}{n_{1} y})", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('y', commutative=True)))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('t_1')(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('y', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('n_1', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('t_1')(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('y', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('t_1')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('y', commutative=True)))))"], [["times", 4, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('t_1')(Symbol('n_1', commutative=True), Symbol('y', commutative=True))))), Mul(Symbol('y', commutative=True), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('y', commutative=True))))))"]]}, {"prompt": "Given M{(F_{N})} = \\int e^{F_{N}} dF_{N}, then derive \\sin^{F_{N}}{(\\sin{(M{(F_{N})})})} = \\sin^{F_{N}}{(\\sin{(\\hat{H} + e^{F_{N}})})}, then obtain \\sin^{F_{N}}{(\\sin{(\\int e^{F_{N}} dF_{N})})} = \\sin^{F_{N}}{(\\sin{(\\hat{H} + e^{F_{N}})})}", "derivation": "M{(F_{N})} = \\int e^{F_{N}} dF_{N} and \\sin{(M{(F_{N})})} = \\sin{(\\int e^{F_{N}} dF_{N})} and \\sin{(\\sin{(M{(F_{N})})})} = \\sin{(\\sin{(\\int e^{F_{N}} dF_{N})})} and \\sin^{F_{N}}{(\\sin{(M{(F_{N})})})} = \\sin^{F_{N}}{(\\sin{(\\int e^{F_{N}} dF_{N})})} and \\sin^{F_{N}}{(\\sin{(M{(F_{N})})})} = \\sin^{F_{N}}{(\\sin{(\\hat{H} + e^{F_{N}})})} and \\sin^{F_{N}}{(\\sin{(\\int e^{F_{N}} dF_{N})})} = \\sin^{F_{N}}{(\\sin{(\\hat{H} + e^{F_{N}})})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_N', commutative=True)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["sin", 1], "Equality(sin(Function('M')(Symbol('F_N', commutative=True))), sin(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"], [["sin", 2], "Equality(sin(sin(Function('M')(Symbol('F_N', commutative=True)))), sin(sin(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(sin(sin(Function('M')(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)), Pow(sin(sin(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(sin(sin(Function('M')(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)), Pow(sin(sin(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(sin(sin(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)), Pow(sin(sin(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\eta{(G)} = \\log{(\\log{(G)})}, then derive \\frac{d}{d G} \\eta{(G)} = \\frac{1}{G \\log{(G)}}, then obtain \\int \\cos{(\\frac{1}{G \\log{(G)}})} dG = \\int \\cos{(\\frac{d}{d G} \\log{(\\log{(G)})})} dG", "derivation": "\\eta{(G)} = \\log{(\\log{(G)})} and \\frac{d}{d G} \\eta{(G)} = \\frac{d}{d G} \\log{(\\log{(G)})} and \\frac{d}{d G} \\eta{(G)} = \\frac{1}{G \\log{(G)}} and \\cos{(\\frac{d}{d G} \\eta{(G)})} = \\cos{(\\frac{d}{d G} \\log{(\\log{(G)})})} and \\cos{(\\frac{1}{G \\log{(G)}})} = \\cos{(\\frac{d}{d G} \\log{(\\log{(G)})})} and \\int \\cos{(\\frac{1}{G \\log{(G)}})} dG = \\int \\cos{(\\frac{d}{d G} \\log{(\\log{(G)})})} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('G', commutative=True)), log(log(Symbol('G', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(log(log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\eta')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), cos(Derivative(log(log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1)))), cos(Derivative(log(log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('G', commutative=True)"], "Equality(Integral(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1)))), Tuple(Symbol('G', commutative=True))), Integral(cos(Derivative(log(log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(M)} = e^{M}, then obtain \\int \\log{(\\dot{y}{(M)} + \\int \\dot{y}{(M)} e^{M} dM)} dM = \\int \\log{(e^{M} + \\int \\dot{y}{(M)} e^{M} dM)} dM", "derivation": "\\dot{y}{(M)} = e^{M} and \\dot{y}^{2}{(M)} = \\dot{y}{(M)} e^{M} and \\int \\dot{y}^{2}{(M)} dM = \\int \\dot{y}{(M)} e^{M} dM and \\dot{y}{(M)} + \\int \\dot{y}^{2}{(M)} dM = e^{M} + \\int \\dot{y}^{2}{(M)} dM and \\dot{y}{(M)} + \\int \\dot{y}{(M)} e^{M} dM = e^{M} + \\int \\dot{y}{(M)} e^{M} dM and \\log{(\\dot{y}{(M)} + \\int \\dot{y}{(M)} e^{M} dM)} = \\log{(e^{M} + \\int \\dot{y}{(M)} e^{M} dM)} and \\int \\log{(\\dot{y}{(M)} + \\int \\dot{y}{(M)} e^{M} dM)} dM = \\int \\log{(e^{M} + \\int \\dot{y}{(M)} e^{M} dM)} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('M', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integer(2)), Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True))), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["add", 1, "Integral(Pow(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True)))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integral(Pow(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True)))), Add(exp(Symbol('M', commutative=True)), Integral(Pow(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integer(2)), Tuple(Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Add(exp(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"], [["log", 5], "Equality(log(Add(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))), log(Add(exp(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))))"], [["integrate", 6, "Symbol('M', commutative=True)"], "Equality(Integral(log(Add(Function('\\\\dot{y}')(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True))), Integral(log(Add(exp(Symbol('M', commutative=True)), Integral(Mul(Function('\\\\dot{y}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\phi{(x)} = \\sin{(x)} and \\rho_{b}{(x)} = \\phi{(x)} + \\sin{(x)} - 1, then obtain (\\theta \\rho_{b}{(x)})^{\\theta} = (\\theta (2 \\phi{(x)} - 1))^{\\theta}", "derivation": "\\phi{(x)} = \\sin{(x)} and 2 \\phi{(x)} = \\phi{(x)} + \\sin{(x)} and 2 \\phi{(x)} - 1 = \\phi{(x)} + \\sin{(x)} - 1 and \\rho_{b}{(x)} = \\phi{(x)} + \\sin{(x)} - 1 and \\theta \\rho_{b}{(x)} = \\theta (\\phi{(x)} + \\sin{(x)} - 1) and (\\theta \\rho_{b}{(x)})^{\\theta} = (\\theta (\\phi{(x)} + \\sin{(x)} - 1))^{\\theta} and (\\theta \\rho_{b}{(x)})^{\\theta} = (\\theta (2 \\phi{(x)} - 1))^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('x', commutative=True))), Add(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Function('\\\\phi')(Symbol('x', commutative=True))), Integer(-1)), Add(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('x', commutative=True)), Add(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)), Integer(-1)))"], [["times", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\rho_b')(Symbol('x', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Add(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\rho_b')(Symbol('x', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Add(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)), Integer(-1))), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\rho_b')(Symbol('x', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Add(Mul(Integer(2), Function('\\\\phi')(Symbol('x', commutative=True))), Integer(-1))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}} and \\operatorname{f_{E}}{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain \\cos{((\\theta_{2}{(\\Psi^{\\dagger})} - \\sin{(L)}) e^{a^{\\dagger}})} = \\cos{((e^{\\Psi^{\\dagger}} - \\sin{(L)}) e^{a^{\\dagger}})}", "derivation": "\\theta_{2}{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}} and \\theta_{2}{(\\Psi^{\\dagger})} - \\sin{(L)} = e^{\\Psi^{\\dagger}} - \\sin{(L)} and \\operatorname{f_{E}}{(a^{\\dagger})} = e^{a^{\\dagger}} and (\\theta_{2}{(\\Psi^{\\dagger})} - \\sin{(L)}) \\operatorname{f_{E}}{(a^{\\dagger})} = (e^{\\Psi^{\\dagger}} - \\sin{(L)}) \\operatorname{f_{E}}{(a^{\\dagger})} and \\cos{((\\theta_{2}{(\\Psi^{\\dagger})} - \\sin{(L)}) \\operatorname{f_{E}}{(a^{\\dagger})})} = \\cos{((e^{\\Psi^{\\dagger}} - \\sin{(L)}) \\operatorname{f_{E}}{(a^{\\dagger})})} and \\cos{((\\theta_{2}{(\\Psi^{\\dagger})} - \\sin{(L)}) e^{a^{\\dagger}})} = \\cos{((e^{\\Psi^{\\dagger}} - \\sin{(L)}) e^{a^{\\dagger}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 1, "sin(Symbol('L', commutative=True))"], "Equality(Add(Function('\\\\theta_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), Add(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))))"], ["get_premise", "Equality(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 2, "Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Function('\\\\theta_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Add(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True))))"], [["cos", 4], "Equality(cos(Mul(Add(Function('\\\\theta_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True)))), cos(Mul(Add(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(cos(Mul(Add(Function('\\\\theta_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), exp(Symbol('a^{\\\\dagger}', commutative=True)))), cos(Mul(Add(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True)))), exp(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\dot{z},f)} = \\dot{z} - f, then obtain 2 \\int \\frac{\\phi_{1}{(\\dot{z},f)}}{f} d\\dot{z} = \\int \\frac{\\dot{z} - f}{f} d\\dot{z} + \\int \\frac{\\phi_{1}{(\\dot{z},f)}}{f} d\\dot{z}", "derivation": "\\phi_{1}{(\\dot{z},f)} = \\dot{z} - f and \\frac{\\phi_{1}{(\\dot{z},f)}}{f} = \\frac{\\dot{z} - f}{f} and \\int \\frac{\\phi_{1}{(\\dot{z},f)}}{f} d\\dot{z} = \\int \\frac{\\dot{z} - f}{f} d\\dot{z} and 2 \\int \\frac{\\phi_{1}{(\\dot{z},f)}}{f} d\\dot{z} = \\int \\frac{\\dot{z} - f}{f} d\\dot{z} + \\int \\frac{\\phi_{1}{(\\dot{z},f)}}{f} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 3, "Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Add(Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\dot{z}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given z{(\\mathbf{g},s)} = s^{\\mathbf{g}}, then derive \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)} = \\frac{\\mathbf{g} s^{\\mathbf{g}}}{s}, then obtain \\int 1 ds = \\int \\frac{s s^{- \\mathbf{g}} \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)}}{\\mathbf{g}} ds", "derivation": "z{(\\mathbf{g},s)} = s^{\\mathbf{g}} and \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)} = \\frac{\\partial}{\\partial s} s^{\\mathbf{g}} and \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)} = \\frac{\\mathbf{g} s^{\\mathbf{g}}}{s} and \\frac{\\mathbf{g} s^{\\mathbf{g}}}{s} = \\frac{\\partial}{\\partial s} s^{\\mathbf{g}} and 1 = \\frac{s s^{- \\mathbf{g}} \\frac{\\partial}{\\partial s} s^{\\mathbf{g}}}{\\mathbf{g}} and 1 = \\frac{s s^{- \\mathbf{g}} \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)}}{\\mathbf{g}} and \\int 1 ds = \\int \\frac{s s^{- \\mathbf{g}} \\frac{\\partial}{\\partial s} z{(\\mathbf{g},s)}}{\\mathbf{g}} ds", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Function('z')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["integrate", 6, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Function('z')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(U,A_{y},\\hbar)} = (A_{y} - U)^{\\hbar} and \\operatorname{c_{0}}{(U,A_{y},\\hbar)} = \\int (A_{y} - U)^{\\hbar} dU, then obtain e^{\\operatorname{c_{0}}{(U,A_{y},\\hbar)}} = e^{\\int \\eta^{\\prime}{(U,A_{y},\\hbar)} dU}", "derivation": "\\eta^{\\prime}{(U,A_{y},\\hbar)} = (A_{y} - U)^{\\hbar} and \\int \\eta^{\\prime}{(U,A_{y},\\hbar)} dU = \\int (A_{y} - U)^{\\hbar} dU and \\operatorname{c_{0}}{(U,A_{y},\\hbar)} = \\int (A_{y} - U)^{\\hbar} dU and e^{\\operatorname{c_{0}}{(U,A_{y},\\hbar)}} = e^{\\int (A_{y} - U)^{\\hbar} dU} and e^{\\operatorname{c_{0}}{(U,A_{y},\\hbar)}} = e^{\\int \\eta^{\\prime}{(U,A_{y},\\hbar)} dU}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["exp", 3], "Equality(exp(Function('c_0')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True))), exp(Integral(Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(exp(Function('c_0')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True))), exp(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(E_{n},x)} = E_{n} x, then obtain m_{s} + \\phi_{1}{(E_{n},x)} = E_{n} x + W", "derivation": "\\phi_{1}{(E_{n},x)} = E_{n} x and \\frac{\\partial}{\\partial x} \\phi_{1}{(E_{n},x)} = \\frac{\\partial}{\\partial x} E_{n} x and \\int \\frac{\\partial}{\\partial x} \\phi_{1}{(E_{n},x)} dx = \\int \\frac{\\partial}{\\partial x} E_{n} x dx and m_{s} + \\phi_{1}{(E_{n},x)} = E_{n} x + W", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Derivative(Mul(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('m_s', commutative=True), Function('\\\\phi_1')(Symbol('E_n', commutative=True), Symbol('x', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Symbol('x', commutative=True)), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\lambda,A_{x})} = A_{x} + \\lambda, then obtain \\frac{\\frac{\\partial}{\\partial A_{x}} \\operatorname{t_{2}}{(\\lambda,A_{x})}}{\\operatorname{t_{2}}{(\\lambda,A_{x})}} = \\frac{1}{\\operatorname{t_{2}}{(\\lambda,A_{x})}}", "derivation": "\\operatorname{t_{2}}{(\\lambda,A_{x})} = A_{x} + \\lambda and \\frac{\\partial}{\\partial A_{x}} \\operatorname{t_{2}}{(\\lambda,A_{x})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\lambda) and \\frac{\\frac{\\partial}{\\partial A_{x}} \\operatorname{t_{2}}{(\\lambda,A_{x})}}{\\operatorname{t_{2}}{(\\lambda,A_{x})}} = \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} + \\lambda)}{\\operatorname{t_{2}}{(\\lambda,A_{x})}} and \\frac{\\frac{\\partial}{\\partial A_{x}} \\operatorname{t_{2}}{(\\lambda,A_{x})}}{\\operatorname{t_{2}}{(\\lambda,A_{x})}} = \\frac{1}{\\operatorname{t_{2}}{(\\lambda,A_{x})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 2, "Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Derivative(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Pow(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Derivative(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Pow(Function('t_2')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given L{(\\hat{X})} = \\cos{(e^{\\hat{X}})}, then obtain 1 = \\frac{\\cos{(\\hat{X} + \\cos{(e^{\\hat{X}})})}}{\\cos{(\\hat{X} + L{(\\hat{X})})}}", "derivation": "L{(\\hat{X})} = \\cos{(e^{\\hat{X}})} and \\hat{X} + L{(\\hat{X})} = \\hat{X} + \\cos{(e^{\\hat{X}})} and \\cos{(\\hat{X} + L{(\\hat{X})})} = \\cos{(\\hat{X} + \\cos{(e^{\\hat{X}})})} and 1 = \\frac{\\cos{(\\hat{X} + \\cos{(e^{\\hat{X}})})}}{\\cos{(\\hat{X} + L{(\\hat{X})})}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{X}', commutative=True)), cos(exp(Symbol('\\\\hat{X}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Function('L')(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), cos(exp(Symbol('\\\\hat{X}', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\hat{X}', commutative=True), Function('L')(Symbol('\\\\hat{X}', commutative=True)))), cos(Add(Symbol('\\\\hat{X}', commutative=True), cos(exp(Symbol('\\\\hat{X}', commutative=True))))))"], [["divide", 3, "cos(Add(Symbol('\\\\hat{X}', commutative=True), Function('L')(Symbol('\\\\hat{X}', commutative=True))))"], "Equality(Integer(1), Mul(Pow(cos(Add(Symbol('\\\\hat{X}', commutative=True), Function('L')(Symbol('\\\\hat{X}', commutative=True)))), Integer(-1)), cos(Add(Symbol('\\\\hat{X}', commutative=True), cos(exp(Symbol('\\\\hat{X}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\rho)} = e^{\\rho}, then obtain \\operatorname{y^{\\prime}}^{3}{(\\rho)} = \\operatorname{y^{\\prime}}{(\\rho)} e^{2 \\rho}", "derivation": "\\operatorname{y^{\\prime}}{(\\rho)} = e^{\\rho} and \\operatorname{y^{\\prime}}^{2}{(\\rho)} = \\operatorname{y^{\\prime}}{(\\rho)} e^{\\rho} and \\operatorname{y^{\\prime}}^{3}{(\\rho)} = \\operatorname{y^{\\prime}}^{2}{(\\rho)} e^{\\rho} and \\operatorname{y^{\\prime}}^{3}{(\\rho)} = \\operatorname{y^{\\prime}}{(\\rho)} e^{2 \\rho}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), Integer(2)), Mul(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"], [["times", 2, "Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), Integer(3)), Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), Integer(2)), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), Integer(3)), Mul(Function('y^{\\\\prime}')(Symbol('\\\\rho', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)}, then derive \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = M_{E} + \\sin{(\\hat{x}_0)}, then derive \\frac{d}{d M_{E}} \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = 1, then obtain (\\frac{d}{d M_{E}} \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0) \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0", "derivation": "\\mathbf{g}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)} and \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = \\int \\cos{(\\hat{x}_0)} d\\hat{x}_0 and \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = M_{E} + \\sin{(\\hat{x}_0)} and \\frac{d}{d M_{E}} \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{\\partial}{\\partial M_{E}} (M_{E} + \\sin{(\\hat{x}_0)}) and \\frac{d}{d M_{E}} \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = 1 and (\\frac{d}{d M_{E}} \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0) \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0 = \\int \\mathbf{g}{(\\hat{x}_0)} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Symbol('M_E', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1))"], [["times", 5, "Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Integral(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\phi{(C,\\hat{p}_0)} = C + \\hat{p}_0, then obtain \\frac{2 (C + \\hat{p}_0)}{C} = \\frac{2 C + 2 \\hat{p}_0}{C}", "derivation": "\\phi{(C,\\hat{p}_0)} = C + \\hat{p}_0 and 2 \\phi{(C,\\hat{p}_0)} = C + \\hat{p}_0 + \\phi{(C,\\hat{p}_0)} and \\frac{2 \\phi{(C,\\hat{p}_0)}}{C} = \\frac{C + \\hat{p}_0 + \\phi{(C,\\hat{p}_0)}}{C} and \\frac{2 (C + \\hat{p}_0)}{C} = \\frac{2 C + 2 \\hat{p}_0}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\phi')(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('C', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(a,g)} = \\log{(a + g)} and \\ddot{x}{(\\dot{z})} = \\cos{(e^{\\dot{z}})}, then obtain \\frac{\\partial}{\\partial a} 2 (\\mathbf{r}{(a,g)} - \\log{(a + g)}) \\ddot{x}{(\\dot{z})} = \\frac{d}{d a} 0", "derivation": "\\mathbf{r}{(a,g)} = \\log{(a + g)} and g + \\mathbf{r}{(a,g)} = g + \\log{(a + g)} and \\mathbf{r}{(a,g)} - \\log{(a + g)} = 0 and \\ddot{x}{(\\dot{z})} = \\cos{(e^{\\dot{z}})} and (\\ddot{x}{(\\dot{z})} + \\cos{(e^{\\dot{z}})}) (\\mathbf{r}{(a,g)} - \\log{(a + g)}) = 0 and 2 (\\mathbf{r}{(a,g)} - \\log{(a + g)}) \\ddot{x}{(\\dot{z})} = 0 and \\frac{\\partial}{\\partial a} 2 (\\mathbf{r}{(a,g)} - \\log{(a + g)}) \\ddot{x}{(\\dot{z})} = \\frac{d}{d a} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True))))"], [["add", 1, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True)))))"], [["minus", 2, "Add(Symbol('g', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True))))), Integer(0))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True)), cos(exp(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 3, "Add(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True)), cos(exp(Symbol('\\\\dot{z}', commutative=True))))"], "Equality(Mul(Add(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True)), cos(exp(Symbol('\\\\dot{z}', commutative=True)))), Add(Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Add(Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True))))), Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True))), Integer(0))"], [["differentiate", 6, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Add(Function('\\\\mathbf{r}')(Symbol('a', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), log(Add(Symbol('a', commutative=True), Symbol('g', commutative=True))))), Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(c,y)} = c^{y}, then derive \\frac{\\partial}{\\partial y} \\operatorname{V_{\\mathbf{E}}}{(c,y)} = c^{y} \\log{(c)}, then obtain \\frac{c^{y} (1 - \\frac{\\partial}{\\partial y} c^{y}) \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} = \\frac{c^{y} (\\frac{c^{y} \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} - \\frac{\\partial}{\\partial y} c^{y}) \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(c,y)} = c^{y} and \\frac{\\partial}{\\partial y} \\operatorname{V_{\\mathbf{E}}}{(c,y)} = \\frac{\\partial}{\\partial y} c^{y} and \\frac{\\partial}{\\partial y} \\operatorname{V_{\\mathbf{E}}}{(c,y)} = c^{y} \\log{(c)} and \\frac{\\partial}{\\partial y} c^{y} = c^{y} \\log{(c)} and 1 = \\frac{c^{y} \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} and 1 - \\frac{\\partial}{\\partial y} c^{y} = \\frac{c^{y} \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} - \\frac{\\partial}{\\partial y} c^{y} and \\frac{c^{y} (1 - \\frac{\\partial}{\\partial y} c^{y}) \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} = \\frac{c^{y} (\\frac{c^{y} \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}} - \\frac{\\partial}{\\partial y} c^{y}) \\log{(c)}}{\\frac{\\partial}{\\partial y} c^{y}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('c', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True))))"], [["divide", 4, "Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 5, "Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))))"], [["times", 6, "Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Add(Integer(1), Mul(Integer(-1), Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Add(Mul(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), log(Symbol('c', commutative=True)), Pow(Derivative(Pow(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mu_0)} = \\log{(\\mu_0)}, then derive \\mu_0 + \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\mu_0 + \\frac{1}{\\mu_0}, then obtain (\\mu_0 + \\frac{1}{\\mu_0}) (\\mu_0 + \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)}) = (\\mu_0 + \\frac{1}{\\mu_0})^{2}", "derivation": "\\operatorname{v_{z}}{(\\mu_0)} = \\log{(\\mu_0)} and \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and \\mu_0 + \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\mu_0 + \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and \\mu_0 + \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\mu_0 + \\frac{1}{\\mu_0} and (\\mu_0 + \\frac{1}{\\mu_0}) (\\mu_0 + \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)}) = (\\mu_0 + \\frac{1}{\\mu_0})^{2}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Symbol('\\\\mu_0', commutative=True), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["times", 4, "Add(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))"], "Equality(Mul(Add(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Pow(Add(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(f_{E})} = \\cos{(f_{E})}, then derive \\int \\operatorname{C_{2}}{(f_{E})} df_{E} = s + \\sin{(f_{E})}, then derive \\cos{(f_{E})} = \\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E}, then obtain (\\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E})^{f_{E}} = \\cos^{f_{E}}{(f_{E})}", "derivation": "\\operatorname{C_{2}}{(f_{E})} = \\cos{(f_{E})} and \\operatorname{C_{2}}^{f_{E}}{(f_{E})} = \\cos^{f_{E}}{(f_{E})} and \\int \\operatorname{C_{2}}{(f_{E})} df_{E} = \\int \\cos{(f_{E})} df_{E} and \\int \\operatorname{C_{2}}{(f_{E})} df_{E} = s + \\sin{(f_{E})} and s + \\sin{(f_{E})} = \\int \\cos{(f_{E})} df_{E} and \\frac{\\partial}{\\partial f_{E}} (s + \\sin{(f_{E})}) = \\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E} and \\cos{(f_{E})} = \\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E} and \\operatorname{C_{2}}{(f_{E})} = \\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E} and (\\frac{d}{d f_{E}} \\int \\cos{(f_{E})} df_{E})^{f_{E}} = \\cos^{f_{E}}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('C_2')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('s', commutative=True), sin(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('s', commutative=True), sin(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["differentiate", 5, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Symbol('s', commutative=True), sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(cos(Symbol('f_E', commutative=True)), Derivative(Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Function('C_2')(Symbol('f_E', commutative=True)), Derivative(Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 8], "Equality(Pow(Derivative(Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('f_E', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(M_{E},\\Psi_{nl})} = M_{E} + \\Psi_{nl}, then derive \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(M_{E},\\Psi_{nl})} = 1, then obtain \\frac{\\partial}{\\partial \\Psi_{nl}} (M_{E} + \\Psi_{nl}) = 1", "derivation": "\\operatorname{x^{{\\}'}}{(M_{E},\\Psi_{nl})} = M_{E} + \\Psi_{nl} and \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(M_{E},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} (M_{E} + \\Psi_{nl}) and \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(M_{E},\\Psi_{nl})} = 1 and \\frac{\\partial}{\\partial \\Psi_{nl}} (M_{E} + \\Psi_{nl}) = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given E{(M)} = e^{\\sin{(M)}}, then obtain (e^{\\sin{(M)}} + \\iint (M + E{(M)}) dM dM) \\iint (M + e^{\\sin{(M)}}) dM dM = (e^{\\sin{(M)}} + \\iint (M + e^{\\sin{(M)}}) dM dM) \\iint (M + e^{\\sin{(M)}}) dM dM", "derivation": "E{(M)} = e^{\\sin{(M)}} and M + E{(M)} = M + e^{\\sin{(M)}} and \\int (M + E{(M)}) dM = \\int (M + e^{\\sin{(M)}}) dM and \\iint (M + E{(M)}) dM dM = \\iint (M + e^{\\sin{(M)}}) dM dM and e^{\\sin{(M)}} + \\iint (M + E{(M)}) dM dM = e^{\\sin{(M)}} + \\iint (M + e^{\\sin{(M)}}) dM dM and (e^{\\sin{(M)}} + \\iint (M + E{(M)}) dM dM) \\iint (M + e^{\\sin{(M)}}) dM dM = (e^{\\sin{(M)}} + \\iint (M + e^{\\sin{(M)}}) dM dM) \\iint (M + e^{\\sin{(M)}}) dM dM", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True))))"], [["add", 1, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True))), Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["add", 4, "exp(sin(Symbol('M', commutative=True)))"], "Equality(Add(exp(sin(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Add(exp(sin(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["times", 5, "Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Add(exp(sin(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Add(exp(sin(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Integral(Add(Symbol('M', commutative=True), exp(sin(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(f)} = \\sin{(f)} and \\mathbf{H}{(f)} = 2 \\sin{(f)}, then obtain \\frac{d}{d f} \\frac{\\mathbf{H}{(f)} + 3 \\sin{(f)}}{3 \\sin{(f)}} = \\frac{d}{d f} \\frac{5}{3}", "derivation": "\\mathbf{g}{(f)} = \\sin{(f)} and \\mathbf{g}{(f)} + \\sin{(f)} = 2 \\sin{(f)} and \\mathbf{g}{(f)} + 2 \\sin{(f)} = 3 \\sin{(f)} and \\mathbf{H}{(f)} = 2 \\sin{(f)} and 2 \\mathbf{g}{(f)} + \\sin{(f)} = 3 \\sin{(f)} and \\mathbf{H}{(f)} + 3 \\sin{(f)} = 5 \\sin{(f)} and \\frac{\\mathbf{H}{(f)} + 3 \\sin{(f)}}{2 \\mathbf{g}{(f)} + \\sin{(f)}} = \\frac{5 \\sin{(f)}}{2 \\mathbf{g}{(f)} + \\sin{(f)}} and \\frac{d}{d f} \\frac{\\mathbf{H}{(f)} + 3 \\sin{(f)}}{2 \\mathbf{g}{(f)} + \\sin{(f)}} = \\frac{d}{d f} \\frac{5 \\sin{(f)}}{2 \\mathbf{g}{(f)} + \\sin{(f)}} and \\frac{d}{d f} \\frac{\\mathbf{H}{(f)} + 3 \\sin{(f)}}{3 \\sin{(f)}} = \\frac{d}{d f} \\frac{5}{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["add", 1, "sin(Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Mul(Integer(2), sin(Symbol('f', commutative=True))))"], [["add", 2, "sin(Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('f', commutative=True)), Mul(Integer(2), sin(Symbol('f', commutative=True)))), Mul(Integer(3), sin(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f', commutative=True)), Mul(Integer(2), sin(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Mul(Integer(3), sin(Symbol('f', commutative=True))))"], [["add", 4, "Mul(Integer(3), sin(Symbol('f', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('f', commutative=True)), Mul(Integer(3), sin(Symbol('f', commutative=True)))), Mul(Integer(5), sin(Symbol('f', commutative=True))))"], [["divide", 6, "Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{H}')(Symbol('f', commutative=True)), Mul(Integer(3), sin(Symbol('f', commutative=True)))), Pow(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Integer(-1))), Mul(Integer(5), Pow(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Integer(-1)), sin(Symbol('f', commutative=True))))"], [["differentiate", 7, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\mathbf{H}')(Symbol('f', commutative=True)), Mul(Integer(3), sin(Symbol('f', commutative=True)))), Pow(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Integer(5), Pow(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Integer(-1)), sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Derivative(Mul(Rational(1, 3), Add(Function('\\\\mathbf{H}')(Symbol('f', commutative=True)), Mul(Integer(3), sin(Symbol('f', commutative=True)))), Pow(sin(Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Rational(5, 3), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(I)} = \\cos{(I)}, then derive \\int \\operatorname{v_{z}}{(I)} dI = \\hat{H}_{\\lambda} + \\sin{(I)}, then obtain \\int \\operatorname{v_{z}}{(I)} dI + \\int \\cos{(I)} dI = 2 \\int \\cos{(I)} dI", "derivation": "\\operatorname{v_{z}}{(I)} = \\cos{(I)} and \\int \\operatorname{v_{z}}{(I)} dI = \\int \\cos{(I)} dI and \\int \\operatorname{v_{z}}{(I)} dI = \\hat{H}_{\\lambda} + \\sin{(I)} and \\hat{H}_{\\lambda} + \\sin{(I)} = \\int \\cos{(I)} dI and \\hat{H}_{\\lambda} + \\sin{(I)} + \\int \\cos{(I)} dI = 2 \\int \\cos{(I)} dI and \\int \\operatorname{v_{z}}{(I)} dI + \\int \\cos{(I)} dI = 2 \\int \\cos{(I)} dI", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_z')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 4, "Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('I', commutative=True)), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(Function('v_z')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\log{(\\hbar \\mathbf{J})}, then obtain 2 \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} + \\frac{1}{\\hbar}", "derivation": "\\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\log{(\\hbar \\mathbf{J})} and 2 \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} + \\log{(\\hbar \\mathbf{J})} and \\frac{\\partial}{\\partial \\hbar} 2 \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (\\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} + \\log{(\\hbar \\mathbf{J})}) and 2 \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{1}}{(\\mathbf{J},\\hbar)} + \\frac{1}{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Derivative(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{H},f^{*})} = \\frac{f^{*}}{\\mathbf{H}} and b{(\\mathbf{H},f^{*})} = \\operatorname{M_{E}}^{f^{*}}{(\\mathbf{H},f^{*})}, then obtain \\frac{b{(\\mathbf{H},f^{*})}}{\\mathbf{H}} = \\frac{(\\frac{f^{*}}{\\mathbf{H}})^{f^{*}}}{\\mathbf{H}}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{H},f^{*})} = \\frac{f^{*}}{\\mathbf{H}} and \\operatorname{M_{E}}^{f^{*}}{(\\mathbf{H},f^{*})} = (\\frac{f^{*}}{\\mathbf{H}})^{f^{*}} and b{(\\mathbf{H},f^{*})} = \\operatorname{M_{E}}^{f^{*}}{(\\mathbf{H},f^{*})} and \\frac{\\operatorname{M_{E}}^{f^{*}}{(\\mathbf{H},f^{*})}}{\\mathbf{H}} = \\frac{(\\frac{f^{*}}{\\mathbf{H}})^{f^{*}}}{\\mathbf{H}} and \\frac{b{(\\mathbf{H},f^{*})}}{\\mathbf{H}} = \\frac{(\\frac{f^{*}}{\\mathbf{H}})^{f^{*}}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True)), Pow(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["times", 2, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\phi_2,h,W)} = h (W + \\phi_2), then obtain \\int (1 - W) d\\phi_2 = \\mathbf{g} + \\int \\frac{W h - W \\hat{H}_l{(\\phi_2,h,W)} + \\phi_2 h}{\\hat{H}_l{(\\phi_2,h,W)}} d\\phi_2", "derivation": "\\hat{H}_l{(\\phi_2,h,W)} = h (W + \\phi_2) and 1 = \\frac{h (W + \\phi_2)}{\\hat{H}_l{(\\phi_2,h,W)}} and 1 - W = - W + \\frac{h (W + \\phi_2)}{\\hat{H}_l{(\\phi_2,h,W)}} and \\int (1 - W) d\\phi_2 = \\int (- W + \\frac{h (W + \\phi_2)}{\\hat{H}_l{(\\phi_2,h,W)}}) d\\phi_2 and \\int (1 - W) d\\phi_2 = \\mathbf{g} + \\int \\frac{W h - W \\hat{H}_l{(\\phi_2,h,W)} + \\phi_2 h}{\\hat{H}_l{(\\phi_2,h,W)}} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('h', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(1), Mul(Symbol('h', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True)), Integer(-1))))"], [["minus", 2, "Symbol('W', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Symbol('h', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True)), Integer(-1)))))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Symbol('h', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Integral(Mul(Add(Mul(Symbol('W', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True), Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\dot{z},\\theta_2)} = \\sin{(\\frac{\\dot{z}}{\\theta_2})}, then derive \\frac{\\partial}{\\partial \\theta_2} \\operatorname{P_{g}}{(\\dot{z},\\theta_2)} = - \\frac{\\dot{z} \\cos{(\\frac{\\dot{z}}{\\theta_2})}}{\\theta_2^{2}}, then obtain - b{(\\dot{z},\\theta_2)} + \\frac{\\partial}{\\partial \\theta_2} \\sin{(\\frac{\\dot{z}}{\\theta_2})} = - \\frac{\\dot{z} \\cos{(\\frac{\\dot{z}}{\\theta_2})}}{\\theta_2^{2}} - b{(\\dot{z},\\theta_2)}", "derivation": "\\operatorname{P_{g}}{(\\dot{z},\\theta_2)} = \\sin{(\\frac{\\dot{z}}{\\theta_2})} and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{P_{g}}{(\\dot{z},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\sin{(\\frac{\\dot{z}}{\\theta_2})} and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{P_{g}}{(\\dot{z},\\theta_2)} = - \\frac{\\dot{z} \\cos{(\\frac{\\dot{z}}{\\theta_2})}}{\\theta_2^{2}} and \\frac{\\partial}{\\partial \\theta_2} \\sin{(\\frac{\\dot{z}}{\\theta_2})} = - \\frac{\\dot{z} \\cos{(\\frac{\\dot{z}}{\\theta_2})}}{\\theta_2^{2}} and - b{(\\dot{z},\\theta_2)} + \\frac{\\partial}{\\partial \\theta_2} \\sin{(\\frac{\\dot{z}}{\\theta_2})} = - \\frac{\\dot{z} \\cos{(\\frac{\\dot{z}}{\\theta_2})}}{\\theta_2^{2}} - b{(\\dot{z},\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-2)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-2)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))))"], [["minus", 4, "Function('b')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Derivative(sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-2)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))), Mul(Integer(-1), Function('b')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(Z)} = Z, then obtain \\frac{d}{d Z} \\tilde{\\infty} \\sin{(\\rho_{b}{(Z)})} = \\frac{d}{d Z} \\tilde{\\infty} \\sin{(Z)}", "derivation": "\\rho_{b}{(Z)} = Z and \\sin{(\\rho_{b}{(Z)})} = \\sin{(Z)} and \\tilde{\\infty} \\sin{(\\rho_{b}{(Z)})} = \\tilde{\\infty} \\sin{(Z)} and \\frac{d}{d Z} \\tilde{\\infty} \\sin{(\\rho_{b}{(Z)})} = \\frac{d}{d Z} \\tilde{\\infty} \\sin{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], [["sin", 1], "Equality(sin(Function('\\\\rho_b')(Symbol('Z', commutative=True))), sin(Symbol('Z', commutative=True)))"], [["divide", 2, 0], "Equality(Mul(zoo, sin(Function('\\\\rho_b')(Symbol('Z', commutative=True)))), Mul(zoo, sin(Symbol('Z', commutative=True))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(zoo, sin(Function('\\\\rho_b')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(zoo, sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\omega)} = \\cos{(\\omega)} and \\operatorname{n_{1}}{(\\omega)} = \\int (- \\operatorname{A_{z}}{(\\omega)})^{\\omega} d\\omega, then obtain - \\operatorname{A_{z}}{(\\omega)} + \\operatorname{n_{1}}{(\\omega)} = - \\operatorname{A_{z}}{(\\omega)} + \\int (- \\cos{(\\omega)})^{\\omega} d\\omega", "derivation": "\\operatorname{A_{z}}{(\\omega)} = \\cos{(\\omega)} and - \\operatorname{A_{z}}{(\\omega)} = - \\cos{(\\omega)} and (- \\operatorname{A_{z}}{(\\omega)})^{\\omega} = (- \\cos{(\\omega)})^{\\omega} and \\int (- \\operatorname{A_{z}}{(\\omega)})^{\\omega} d\\omega = \\int (- \\cos{(\\omega)})^{\\omega} d\\omega and \\operatorname{n_{1}}{(\\omega)} = \\int (- \\operatorname{A_{z}}{(\\omega)})^{\\omega} d\\omega and - \\operatorname{A_{z}}{(\\omega)} + \\operatorname{n_{1}}{(\\omega)} = - \\operatorname{A_{z}}{(\\omega)} + \\int (- \\operatorname{A_{z}}{(\\omega)})^{\\omega} d\\omega and - \\operatorname{A_{z}}{(\\omega)} + \\operatorname{n_{1}}{(\\omega)} = - \\operatorname{A_{z}}{(\\omega)} + \\int (- \\cos{(\\omega)})^{\\omega} d\\omega", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\omega', commutative=True)), Integral(Pow(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["add", 5, "Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Function('n_1')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Integral(Pow(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Function('n_1')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True))), Integral(Pow(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{H}_l,W)} = \\frac{W}{\\hat{H}_l} and z{(\\hat{H}_l,W)} = - \\frac{2 W}{\\hat{H}_l} + \\hat{H}_l + \\mathbf{H}{(\\hat{H}_l,W)}, then obtain \\frac{\\partial}{\\partial \\hat{H}_l} z{(\\hat{H}_l,W)} = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l - \\mathbf{H}{(\\hat{H}_l,W)})", "derivation": "\\mathbf{H}{(\\hat{H}_l,W)} = \\frac{W}{\\hat{H}_l} and \\hat{H}_l + \\mathbf{H}{(\\hat{H}_l,W)} = \\frac{W}{\\hat{H}_l} + \\hat{H}_l and - \\frac{2 W}{\\hat{H}_l} + \\hat{H}_l + \\mathbf{H}{(\\hat{H}_l,W)} = - \\frac{W}{\\hat{H}_l} + \\hat{H}_l and z{(\\hat{H}_l,W)} = - \\frac{2 W}{\\hat{H}_l} + \\hat{H}_l + \\mathbf{H}{(\\hat{H}_l,W)} and z{(\\hat{H}_l,W)} = - \\frac{W}{\\hat{H}_l} + \\hat{H}_l and \\frac{\\partial}{\\partial \\hat{H}_l} z{(\\hat{H}_l,W)} = \\frac{\\partial}{\\partial \\hat{H}_l} (- \\frac{W}{\\hat{H}_l} + \\hat{H}_l) and \\frac{\\partial}{\\partial \\hat{H}_l} z{(\\hat{H}_l,W)} = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l - \\mathbf{H}{(\\hat{H}_l,W)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 2, "Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(\\eta^{\\prime})} = \\sin{(\\sin{(\\eta^{\\prime})})} and \\operatorname{v_{1}}{(\\theta,\\eta^{\\prime})} = - \\theta \\sin{(\\sin{(\\eta^{\\prime})})}, then obtain \\operatorname{v_{1}}{(\\theta,\\eta^{\\prime})} = - \\theta \\mathbf{v}{(\\eta^{\\prime})}", "derivation": "\\mathbf{v}{(\\eta^{\\prime})} = \\sin{(\\sin{(\\eta^{\\prime})})} and 0 = - \\mathbf{v}{(\\eta^{\\prime})} + \\sin{(\\sin{(\\eta^{\\prime})})} and - \\sin{(\\sin{(\\eta^{\\prime})})} = - \\mathbf{v}{(\\eta^{\\prime})} and - \\theta \\sin{(\\sin{(\\eta^{\\prime})})} = - \\theta \\mathbf{v}{(\\eta^{\\prime})} and \\operatorname{v_{1}}{(\\theta,\\eta^{\\prime})} = - \\theta \\sin{(\\sin{(\\eta^{\\prime})})} and \\operatorname{v_{1}}{(\\theta,\\eta^{\\prime})} = - \\theta \\mathbf{v}{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["minus", 2, "sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Integer(-1), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\theta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('v_1')(Symbol('\\\\theta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given h{(i)} = \\sin{(\\cos{(i)})}, then obtain \\frac{d}{d i} \\int \\frac{(i + h{(i)}) h{(i)}}{\\sin{(\\cos{(i)})}} di = \\frac{d}{d i} \\int \\frac{(i + \\sin{(\\cos{(i)})}) h{(i)}}{\\sin{(\\cos{(i)})}} di", "derivation": "h{(i)} = \\sin{(\\cos{(i)})} and i + h{(i)} = i + \\sin{(\\cos{(i)})} and \\frac{(i + h{(i)}) h{(i)}}{\\sin{(\\cos{(i)})}} = \\frac{(i + \\sin{(\\cos{(i)})}) h{(i)}}{\\sin{(\\cos{(i)})}} and \\int \\frac{(i + h{(i)}) h{(i)}}{\\sin{(\\cos{(i)})}} di = \\int \\frac{(i + \\sin{(\\cos{(i)})}) h{(i)}}{\\sin{(\\cos{(i)})}} di and \\frac{d}{d i} \\int \\frac{(i + h{(i)}) h{(i)}}{\\sin{(\\cos{(i)})}} di = \\frac{d}{d i} \\int \\frac{(i + \\sin{(\\cos{(i)})}) h{(i)}}{\\sin{(\\cos{(i)})}} di", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('i', commutative=True)), sin(cos(Symbol('i', commutative=True))))"], [["add", 1, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Function('h')(Symbol('i', commutative=True))), Add(Symbol('i', commutative=True), sin(cos(Symbol('i', commutative=True)))))"], [["divide", 2, "Mul(Pow(Function('h')(Symbol('i', commutative=True)), Integer(-1)), sin(cos(Symbol('i', commutative=True))))"], "Equality(Mul(Add(Symbol('i', commutative=True), Function('h')(Symbol('i', commutative=True))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))), Mul(Add(Symbol('i', commutative=True), sin(cos(Symbol('i', commutative=True)))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('i', commutative=True), Function('h')(Symbol('i', commutative=True))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))), Tuple(Symbol('i', commutative=True))), Integral(Mul(Add(Symbol('i', commutative=True), sin(cos(Symbol('i', commutative=True)))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Mul(Add(Symbol('i', commutative=True), Function('h')(Symbol('i', commutative=True))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Mul(Add(Symbol('i', commutative=True), sin(cos(Symbol('i', commutative=True)))), Function('h')(Symbol('i', commutative=True)), Pow(sin(cos(Symbol('i', commutative=True))), Integer(-1))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\Omega)} = \\sin{(\\Omega)}, then derive - (- \\frac{I{(\\Omega)} \\cos{(\\Omega)}}{\\sin^{2}{(\\Omega)}} + \\frac{\\frac{d}{d \\Omega} I{(\\Omega)}}{\\sin{(\\Omega)}}) \\sin{(\\frac{I{(\\Omega)}}{\\sin{(\\Omega)}})} = 0, then obtain \\frac{d}{d \\Omega} - (- \\frac{\\cos{(\\Omega)}}{\\sin{(\\Omega)}} + \\frac{\\frac{d}{d \\Omega} \\sin{(\\Omega)}}{\\sin{(\\Omega)}}) \\sin{(1)} = \\frac{d}{d \\Omega} 0", "derivation": "I{(\\Omega)} = \\sin{(\\Omega)} and \\frac{I{(\\Omega)}}{\\sin{(\\Omega)}} = 1 and \\cos{(\\frac{I{(\\Omega)}}{\\sin{(\\Omega)}})} = \\cos{(1)} and \\frac{d}{d \\Omega} \\cos{(\\frac{I{(\\Omega)}}{\\sin{(\\Omega)}})} = \\frac{d}{d \\Omega} \\cos{(1)} and - (- \\frac{I{(\\Omega)} \\cos{(\\Omega)}}{\\sin^{2}{(\\Omega)}} + \\frac{\\frac{d}{d \\Omega} I{(\\Omega)}}{\\sin{(\\Omega)}}) \\sin{(\\frac{I{(\\Omega)}}{\\sin{(\\Omega)}})} = 0 and - (- \\frac{\\cos{(\\Omega)}}{\\sin{(\\Omega)}} + \\frac{\\frac{d}{d \\Omega} \\sin{(\\Omega)}}{\\sin{(\\Omega)}}) \\sin{(1)} = 0 and \\frac{d}{d \\Omega} - (- \\frac{\\cos{(\\Omega)}}{\\sin{(\\Omega)}} + \\frac{\\frac{d}{d \\Omega} \\sin{(\\Omega)}}{\\sin{(\\Omega)}}) \\sin{(1)} = \\frac{d}{d \\Omega} 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Integer(1))"], [["cos", 2], "Equality(cos(Mul(Function('I')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)))), cos(Integer(1)))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(cos(Mul(Function('I')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('I')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-2)), cos(Symbol('\\\\Omega', commutative=True))), Mul(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(Function('I')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), sin(Mul(Function('I')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Mul(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), sin(Integer(1))), Integer(0))"], [["differentiate", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Mul(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), sin(Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(\\hat{p},n)} = \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}}, then derive \\frac{n^{\\hat{p}} + Z{(\\hat{p},n)}}{n^{\\hat{p}} \\log{(n)} + n^{\\hat{p}}} = 1, then obtain \\frac{n^{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}}}{n^{\\hat{p}} \\log{(n)} + n^{\\hat{p}}} = 1", "derivation": "Z{(\\hat{p},n)} = \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}} and n^{\\hat{p}} + Z{(\\hat{p},n)} = n^{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}} and \\frac{n^{\\hat{p}} + Z{(\\hat{p},n)}}{\\hat{p}} = \\frac{n^{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}}}{\\hat{p}} and \\frac{n^{\\hat{p}} + Z{(\\hat{p},n)}}{n^{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}}} = 1 and \\frac{n^{\\hat{p}} + Z{(\\hat{p},n)}}{n^{\\hat{p}} \\log{(n)} + n^{\\hat{p}}} = 1 and \\frac{n^{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} n^{\\hat{p}}}{n^{\\hat{p}} \\log{(n)} + n^{\\hat{p}}} = 1", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["add", 1, "Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Function('Z')(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True))), Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"], [["divide", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Function('Z')(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"], "Equality(Mul(Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Function('Z')(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True))), Pow(Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Function('Z')(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True))), Pow(Add(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Pow(Add(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(u)} = \\log{(u)} and Q{(u)} = \\int \\log{(u)} du, then obtain u + Q{(u)} + \\iint \\log{(u)} du du = u + \\int \\log{(u)} du + \\iint \\log{(u)} du du", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(u)} = \\log{(u)} and \\int \\operatorname{V_{\\mathbf{B}}}{(u)} du = \\int \\log{(u)} du and \\iint \\operatorname{V_{\\mathbf{B}}}{(u)} du du = \\iint \\log{(u)} du du and Q{(u)} = \\int \\log{(u)} du and Q{(u)} + \\iint \\operatorname{V_{\\mathbf{B}}}{(u)} du du = \\int \\log{(u)} du + \\iint \\operatorname{V_{\\mathbf{B}}}{(u)} du du and Q{(u)} + \\iint \\log{(u)} du du = \\int \\log{(u)} du + \\iint \\log{(u)} du du and u + Q{(u)} + \\iint \\log{(u)} du du = u + \\int \\log{(u)} du + \\iint \\log{(u)} du du", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('u', commutative=True)), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["add", 4, "Integral(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Add(Function('Q')(Symbol('u', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('Q')(Symbol('u', commutative=True)), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["add", 6, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('Q')(Symbol('u', commutative=True)), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Symbol('u', commutative=True), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\delta,r_{0})} = - \\delta + r_{0}, then obtain 0 = - \\frac{\\partial}{\\partial \\delta} \\varepsilon{(\\delta,r_{0})} - 1", "derivation": "\\varepsilon{(\\delta,r_{0})} = - \\delta + r_{0} and \\frac{\\partial}{\\partial \\delta} \\varepsilon{(\\delta,r_{0})} = \\frac{\\partial}{\\partial \\delta} (- \\delta + r_{0}) and 0 = \\frac{\\partial}{\\partial \\delta} (- \\delta + r_{0}) - \\frac{\\partial}{\\partial \\delta} \\varepsilon{(\\delta,r_{0})} and 0 = - \\frac{\\partial}{\\partial \\delta} \\varepsilon{(\\delta,r_{0})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given \\Omega{(Q,\\psi^*)} = Q \\psi^* and \\operatorname{f^{\\prime}}{(Q,\\psi^*)} = \\int Q \\psi^* d\\psi^*, then obtain (\\int \\Omega{(Q,\\psi^*)} d\\psi^*)^{\\psi^*} = \\operatorname{f^{\\prime}}^{\\psi^*}{(Q,\\psi^*)}", "derivation": "\\Omega{(Q,\\psi^*)} = Q \\psi^* and \\int \\Omega{(Q,\\psi^*)} d\\psi^* = \\int Q \\psi^* d\\psi^* and (\\int \\Omega{(Q,\\psi^*)} d\\psi^*)^{\\psi^*} = (\\int Q \\psi^* d\\psi^*)^{\\psi^*} and \\operatorname{f^{\\prime}}{(Q,\\psi^*)} = \\int Q \\psi^* d\\psi^* and (\\int \\Omega{(Q,\\psi^*)} d\\psi^*)^{\\psi^*} = \\operatorname{f^{\\prime}}^{\\psi^*}{(Q,\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Omega')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('\\\\Omega')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(v_{x})} = \\cos{(v_{x})}, then derive \\frac{d}{d v_{x}} \\mathbf{J}{(v_{x})} = - \\sin{(v_{x})}, then obtain - (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} = \\sin{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})}", "derivation": "\\mathbf{J}{(v_{x})} = \\cos{(v_{x})} and \\frac{d}{d v_{x}} \\mathbf{J}{(v_{x})} = \\frac{d}{d v_{x}} \\cos{(v_{x})} and \\frac{d}{d v_{x}} \\mathbf{J}{(v_{x})} = - \\sin{(v_{x})} and - \\sin{(v_{x})} = \\frac{d}{d v_{x}} \\cos{(v_{x})} and - \\sin^{2}{(v_{x})} = \\sin{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})} and - (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} = \\sin{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('v_x', commutative=True))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["times", 4, "sin(Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(-1), Pow(sin(Symbol('v_x', commutative=True)), Integer(2))), Mul(sin(Symbol('v_x', commutative=True)), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(2))), Mul(sin(Symbol('v_x', commutative=True)), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(J,n_{2})} = - J + \\sin{(n_{2})}, then obtain \\frac{\\partial}{\\partial n_{2}} (J + \\hat{p}_0^{n_{2}}{(J,n_{2})}) = \\frac{\\partial}{\\partial n_{2}} (J + (- J + \\sin{(n_{2})})^{n_{2}})", "derivation": "\\hat{p}_0{(J,n_{2})} = - J + \\sin{(n_{2})} and \\hat{p}_0^{n_{2}}{(J,n_{2})} = (- J + \\sin{(n_{2})})^{n_{2}} and J + \\hat{p}_0^{n_{2}}{(J,n_{2})} = J + (- J + \\sin{(n_{2})})^{n_{2}} and \\frac{\\partial}{\\partial n_{2}} (J + \\hat{p}_0^{n_{2}}{(J,n_{2})}) = \\frac{\\partial}{\\partial n_{2}} (J + (- J + \\sin{(n_{2})})^{n_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('J', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), sin(Symbol('n_2', commutative=True))))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('J', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), sin(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('J', commutative=True))"], "Equality(Add(Symbol('J', commutative=True), Pow(Function('\\\\hat{p}_0')(Symbol('J', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Add(Symbol('J', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), sin(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True))))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Add(Symbol('J', commutative=True), Pow(Function('\\\\hat{p}_0')(Symbol('J', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), sin(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(I,\\mathbf{E})} = \\log{(I + \\mathbf{E})}, then derive \\frac{\\partial}{\\partial I} \\mathbf{r}{(I,\\mathbf{E})} = \\frac{1}{I + \\mathbf{E}}, then obtain e^{\\frac{\\varphi^{*}{(\\mathbf{p})}}{I + \\mathbf{E}}} = e^{\\varphi^{*}{(\\mathbf{p})} \\frac{\\partial}{\\partial I} \\log{(I + \\mathbf{E})}}", "derivation": "\\mathbf{r}{(I,\\mathbf{E})} = \\log{(I + \\mathbf{E})} and \\frac{\\partial}{\\partial I} \\mathbf{r}{(I,\\mathbf{E})} = \\frac{\\partial}{\\partial I} \\log{(I + \\mathbf{E})} and \\frac{\\partial}{\\partial I} \\mathbf{r}{(I,\\mathbf{E})} = \\frac{1}{I + \\mathbf{E}} and \\frac{1}{I + \\mathbf{E}} = \\frac{\\partial}{\\partial I} \\log{(I + \\mathbf{E})} and \\frac{\\varphi^{*}{(\\mathbf{p})}}{I + \\mathbf{E}} = \\varphi^{*}{(\\mathbf{p})} \\frac{\\partial}{\\partial I} \\log{(I + \\mathbf{E})} and e^{\\frac{\\varphi^{*}{(\\mathbf{p})}}{I + \\mathbf{E}}} = e^{\\varphi^{*}{(\\mathbf{p})} \\frac{\\partial}{\\partial I} \\log{(I + \\mathbf{E})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), log(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(log(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Derivative(log(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 4, "Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(log(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["exp", 5], "Equality(exp(Mul(Pow(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True)))), exp(Mul(Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(log(Add(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))))"]]}, {"prompt": "Given W{(\\mathbf{p})} = e^{e^{\\mathbf{p}}}, then derive (\\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})})^{2} = e^{\\mathbf{p}} e^{e^{\\mathbf{p}}} \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})}, then obtain (\\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})})^{2} = W{(\\mathbf{p})} e^{\\mathbf{p}} \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})}", "derivation": "W{(\\mathbf{p})} = e^{e^{\\mathbf{p}}} and \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} e^{e^{\\mathbf{p}}} and (\\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})})^{2} = \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} e^{e^{\\mathbf{p}}} and (\\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})})^{2} = e^{\\mathbf{p}} e^{e^{\\mathbf{p}}} \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})} and (\\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})})^{2} = W{(\\mathbf{p})} e^{\\mathbf{p}} \\frac{d}{d \\mathbf{p}} W{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), exp(exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Integer(2)), Mul(exp(Symbol('\\\\mathbf{p}', commutative=True)), exp(exp(Symbol('\\\\mathbf{p}', commutative=True))), Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Integer(2)), Mul(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{1},n_{1})} = n_{1} v_{1} and \\hat{x}{(v_{1})} = v_{1}, then obtain \\frac{\\int - \\mathbf{E}{(v_{1},n_{1})} d\\hat{x}{(v_{1})}}{\\cos{(\\varepsilon)}} = \\frac{\\int - n_{1} v_{1} d\\hat{x}{(v_{1})}}{\\cos{(\\varepsilon)}}", "derivation": "\\mathbf{E}{(v_{1},n_{1})} = n_{1} v_{1} and - \\mathbf{E}{(v_{1},n_{1})} = - n_{1} v_{1} and \\int - \\mathbf{E}{(v_{1},n_{1})} dv_{1} = \\int - n_{1} v_{1} dv_{1} and \\hat{x}{(v_{1})} = v_{1} and \\frac{\\int - \\mathbf{E}{(v_{1},n_{1})} dv_{1}}{\\cos{(\\varepsilon)}} = \\frac{\\int - n_{1} v_{1} dv_{1}}{\\cos{(\\varepsilon)}} and \\frac{\\int - \\mathbf{E}{(v_{1},n_{1})} d\\hat{x}{(v_{1})}}{\\cos{(\\varepsilon)}} = \\frac{\\int - n_{1} v_{1} d\\hat{x}{(v_{1})}}{\\cos{(\\varepsilon)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True), Symbol('n_1', commutative=True)), Mul(Symbol('n_1', commutative=True), Symbol('v_1', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True), Symbol('n_1', commutative=True))), Mul(Integer(-1), Symbol('n_1', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 2, "Symbol('v_1', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Integer(-1), Symbol('n_1', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))"], [["divide", 3, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))), Mul(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Symbol('n_1', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True), Symbol('n_1', commutative=True))), Tuple(Function('\\\\hat{x}')(Symbol('v_1', commutative=True))))), Mul(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Symbol('n_1', commutative=True), Symbol('v_1', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\chi,T)} = T \\chi, then obtain \\dot{\\mathbf{r}}^{T}{(\\chi,T)} + e^{\\dot{\\mathbf{r}}{(\\chi,T)}} = \\dot{\\mathbf{r}}^{T}{(\\chi,T)} + e^{T \\chi}", "derivation": "\\dot{\\mathbf{r}}{(\\chi,T)} = T \\chi and e^{\\dot{\\mathbf{r}}{(\\chi,T)}} = e^{T \\chi} and \\dot{\\mathbf{r}}^{T}{(\\chi,T)} = (T \\chi)^{T} and (T \\chi)^{T} + e^{\\dot{\\mathbf{r}}{(\\chi,T)}} = (T \\chi)^{T} + e^{T \\chi} and \\dot{\\mathbf{r}}^{T}{(\\chi,T)} + e^{\\dot{\\mathbf{r}}{(\\chi,T)}} = \\dot{\\mathbf{r}}^{T}{(\\chi,T)} + e^{T \\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True))), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)))"], [["add", 2, "Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), exp(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)))), Add(Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)))), Add(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\mathbf{P},A_{x})} = A_{x} + \\mathbf{P}, then derive \\frac{\\partial}{\\partial \\mathbf{P}} \\pi{(\\mathbf{P},A_{x})} = 1, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} (A_{x} + \\mathbf{P}) = 1", "derivation": "\\pi{(\\mathbf{P},A_{x})} = A_{x} + \\mathbf{P} and \\frac{\\partial}{\\partial \\mathbf{P}} \\pi{(\\mathbf{P},A_{x})} = \\frac{\\partial}{\\partial \\mathbf{P}} (A_{x} + \\mathbf{P}) and \\frac{\\partial}{\\partial \\mathbf{P}} \\pi{(\\mathbf{P},A_{x})} = 1 and \\frac{\\partial}{\\partial \\mathbf{P}} (A_{x} + \\mathbf{P}) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\rho_{b}{(\\varphi^*)} = \\cos{(\\varphi^*)}, then obtain \\cos{(\\rho_{b}^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^*)} = \\cos{(\\cos^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^*)}", "derivation": "\\rho_{b}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\rho_{b}^{\\varphi^*}{(\\varphi^*)} = \\cos^{\\varphi^*}{(\\varphi^*)} and \\rho_{b}^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^* = \\cos^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^* and \\cos{(\\rho_{b}^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^*)} = \\cos{(\\cos^{\\varphi^*}{(\\varphi^*)} + \\int \\rho_{b}^{\\varphi^*}{(\\varphi^*)} d\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 2, "Integral(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), cos(Add(Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))))"]]}, {"prompt": "Given y{(v_{y})} = \\log{(v_{y})}, then obtain (y{(v_{y})} + \\log{(v_{y})})^{2} - 2 \\log{(v_{y})} = 2 (y{(v_{y})} + \\log{(v_{y})}) \\log{(v_{y})} - 2 \\log{(v_{y})}", "derivation": "y{(v_{y})} = \\log{(v_{y})} and y{(v_{y})} + \\log{(v_{y})} = 2 \\log{(v_{y})} and (y{(v_{y})} + \\log{(v_{y})})^{2} = 2 (y{(v_{y})} + \\log{(v_{y})}) \\log{(v_{y})} and (y{(v_{y})} + \\log{(v_{y})})^{2} - 2 \\log{(v_{y})} = 2 (y{(v_{y})} + \\log{(v_{y})}) \\log{(v_{y})} - 2 \\log{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["add", 1, "log(Symbol('v_y', commutative=True))"], "Equality(Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Mul(Integer(2), log(Symbol('v_y', commutative=True))))"], [["times", 2, "Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], "Equality(Pow(Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Integer(2)), Mul(Integer(2), Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), log(Symbol('v_y', commutative=True))))"], [["minus", 3, "Mul(Integer(2), log(Symbol('v_y', commutative=True)))"], "Equality(Add(Pow(Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Integer(2)), Mul(Integer(-1), Integer(2), log(Symbol('v_y', commutative=True)))), Add(Mul(Integer(2), Add(Function('y')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), log(Symbol('v_y', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given f{(\\pi)} = \\cos{(\\pi)}, then derive 1 = e^{- \\pi (- \\sin{(\\pi)} - \\frac{d}{d \\pi} f{(\\pi)})}, then obtain 1 = e^{- \\pi (- \\sin{(\\pi)} - \\frac{d}{d \\pi} \\cos{(\\pi)})}", "derivation": "f{(\\pi)} = \\cos{(\\pi)} and 0 = - f{(\\pi)} + \\cos{(\\pi)} and \\frac{d}{d \\pi} 0 = \\frac{d}{d \\pi} (- f{(\\pi)} + \\cos{(\\pi)}) and - \\pi \\frac{d}{d \\pi} 0 = - \\pi \\frac{d}{d \\pi} (- f{(\\pi)} + \\cos{(\\pi)}) and e^{- \\pi \\frac{d}{d \\pi} 0} = e^{- \\pi \\frac{d}{d \\pi} (- f{(\\pi)} + \\cos{(\\pi)})} and 1 = e^{- \\pi (- \\sin{(\\pi)} - \\frac{d}{d \\pi} f{(\\pi)})} and 1 = e^{- \\pi (- \\sin{(\\pi)} - \\frac{d}{d \\pi} \\cos{(\\pi)})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Function('f')(Symbol('\\\\pi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Derivative(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["exp", 4], "Equality(exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Derivative(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Integer(1), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(1), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given \\nabla{(\\mathbf{H},A_{x})} = A_{x} \\mathbf{H} and \\hat{H}{(\\mathbf{H},A_{x})} = \\frac{1}{\\nabla{(\\mathbf{H},A_{x})}}, then obtain A_{x} \\mathbf{H} + \\hat{H}{(\\mathbf{H},A_{x})} + \\nabla{(\\mathbf{H},A_{x})} = A_{x} \\mathbf{H} + \\nabla{(\\mathbf{H},A_{x})} + \\frac{1}{A_{x} \\mathbf{H}}", "derivation": "\\nabla{(\\mathbf{H},A_{x})} = A_{x} \\mathbf{H} and A_{x} \\mathbf{H} + \\nabla{(\\mathbf{H},A_{x})} = 2 A_{x} \\mathbf{H} and \\hat{H}{(\\mathbf{H},A_{x})} = \\frac{1}{\\nabla{(\\mathbf{H},A_{x})}} and 2 A_{x} \\mathbf{H} + \\hat{H}{(\\mathbf{H},A_{x})} = 2 A_{x} \\mathbf{H} + \\frac{1}{\\nabla{(\\mathbf{H},A_{x})}} and 2 A_{x} \\mathbf{H} + \\hat{H}{(\\mathbf{H},A_{x})} = 2 A_{x} \\mathbf{H} + \\frac{1}{A_{x} \\mathbf{H}} and A_{x} \\mathbf{H} + \\hat{H}{(\\mathbf{H},A_{x})} + \\nabla{(\\mathbf{H},A_{x})} = A_{x} \\mathbf{H} + \\nabla{(\\mathbf{H},A_{x})} + \\frac{1}{A_{x} \\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)))"], [["add", 3, "Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True))), Add(Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True))), Add(Mul(Integer(2), Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True))), Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{p}{(x,y)} = x y, then obtain \\frac{x y + x + y + \\mathbf{p}{(x,y)}}{y} = \\frac{2 x y + x + y}{y}", "derivation": "\\mathbf{p}{(x,y)} = x y and y + \\mathbf{p}{(x,y)} = x y + y and y + 2 \\mathbf{p}{(x,y)} = x y + y + \\mathbf{p}{(x,y)} and x + y + 2 \\mathbf{p}{(x,y)} = x y + x + y + \\mathbf{p}{(x,y)} and x + y + 2 \\mathbf{p}{(x,y)} = 2 x y + x + y and x y + x + y + \\mathbf{p}{(x,y)} = 2 x y + x + y and \\frac{x y + x + y + \\mathbf{p}{(x,y)}}{y} = \\frac{2 x y + x + y}{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["add", 1, "Add(Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True))))"], [["add", 3, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('x', commutative=True), Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(2), Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(2), Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True)))"], [["divide", 6, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('x', commutative=True), Symbol('y', commutative=True)), Symbol('x', commutative=True), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and \\mathbf{E}{(g_{\\varepsilon})} = \\int \\hat{X}^{g_{\\varepsilon}}{(g_{\\varepsilon})} dg_{\\varepsilon}, then obtain \\mathbf{E}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = (\\int (e^{g_{\\varepsilon}})^{g_{\\varepsilon}} dg_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "\\hat{X}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and \\hat{X}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = (e^{g_{\\varepsilon}})^{g_{\\varepsilon}} and \\int \\hat{X}^{g_{\\varepsilon}}{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int (e^{g_{\\varepsilon}})^{g_{\\varepsilon}} dg_{\\varepsilon} and \\mathbf{E}{(g_{\\varepsilon})} = \\int \\hat{X}^{g_{\\varepsilon}}{(g_{\\varepsilon})} dg_{\\varepsilon} and \\mathbf{E}{(g_{\\varepsilon})} = \\int (e^{g_{\\varepsilon}})^{g_{\\varepsilon}} dg_{\\varepsilon} and \\mathbf{E}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = (\\int (e^{g_{\\varepsilon}})^{g_{\\varepsilon}} dg_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Pow(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Pow(Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Pow(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 5, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Integral(Pow(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given x{(\\hat{H}_l,\\varphi)} = \\hat{H}_l + \\varphi, then obtain ((x^{2}{(\\hat{H}_l,\\varphi)})^{\\varphi})^{\\hat{H}_l} = (((\\hat{H}_l + \\varphi) x{(\\hat{H}_l,\\varphi)})^{\\varphi})^{\\hat{H}_l}", "derivation": "x{(\\hat{H}_l,\\varphi)} = \\hat{H}_l + \\varphi and x^{2}{(\\hat{H}_l,\\varphi)} = (\\hat{H}_l + \\varphi) x{(\\hat{H}_l,\\varphi)} and (x^{2}{(\\hat{H}_l,\\varphi)})^{\\varphi} = ((\\hat{H}_l + \\varphi) x{(\\hat{H}_l,\\varphi)})^{\\varphi} and ((x^{2}{(\\hat{H}_l,\\varphi)})^{\\varphi})^{\\hat{H}_l} = (((\\hat{H}_l + \\varphi) x{(\\hat{H}_l,\\varphi)})^{\\varphi})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Pow(Pow(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(Q,f_{\\mathbf{p}})} = Q f_{\\mathbf{p}}, then obtain \\frac{\\partial}{\\partial Q} \\theta_{1}{(Q,f_{\\mathbf{p}})} + \\frac{1}{- f_{\\mathbf{p}} + \\theta_{1}{(Q,f_{\\mathbf{p}})}} = \\frac{\\partial}{\\partial Q} Q f_{\\mathbf{p}} + \\frac{1}{- f_{\\mathbf{p}} + \\theta_{1}{(Q,f_{\\mathbf{p}})}}", "derivation": "\\theta_{1}{(Q,f_{\\mathbf{p}})} = Q f_{\\mathbf{p}} and - f_{\\mathbf{p}} + \\theta_{1}{(Q,f_{\\mathbf{p}})} = Q f_{\\mathbf{p}} - f_{\\mathbf{p}} and \\frac{\\partial}{\\partial Q} \\theta_{1}{(Q,f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial Q} Q f_{\\mathbf{p}} and \\frac{\\partial}{\\partial Q} \\theta_{1}{(Q,f_{\\mathbf{p}})} + \\frac{1}{Q f_{\\mathbf{p}} - f_{\\mathbf{p}}} = \\frac{\\partial}{\\partial Q} Q f_{\\mathbf{p}} + \\frac{1}{Q f_{\\mathbf{p}} - f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial Q} \\theta_{1}{(Q,f_{\\mathbf{p}})} + \\frac{1}{- f_{\\mathbf{p}} + \\theta_{1}{(Q,f_{\\mathbf{p}})}} = \\frac{\\partial}{\\partial Q} Q f_{\\mathbf{p}} + \\frac{1}{- f_{\\mathbf{p}} + \\theta_{1}{(Q,f_{\\mathbf{p}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["minus", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 3, "Pow(Add(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Pow(Add(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))), Add(Derivative(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Pow(Add(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Pow(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))), Add(Derivative(Mul(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Pow(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\theta_1')(Symbol('Q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\dot{z},B,L)} = (\\frac{\\dot{z}}{L})^{B}, then obtain 1 - (\\frac{\\dot{z}}{L})^{B} = - (\\frac{\\dot{z}}{L})^{B} + (\\frac{(\\frac{\\dot{z}}{L})^{B}}{\\operatorname{x^{{\\}'}}{(\\dot{z},B,L)}})^{\\dot{z}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\dot{z},B,L)} = (\\frac{\\dot{z}}{L})^{B} and 1 = \\frac{(\\frac{\\dot{z}}{L})^{B}}{\\operatorname{x^{{\\}'}}{(\\dot{z},B,L)}} and 1 = (\\frac{(\\frac{\\dot{z}}{L})^{B}}{\\operatorname{x^{{\\}'}}{(\\dot{z},B,L)}})^{\\dot{z}} and 1 - (\\frac{\\dot{z}}{L})^{B} = - (\\frac{\\dot{z}}{L})^{B} + (\\frac{(\\frac{\\dot{z}}{L})^{B}}{\\operatorname{x^{{\\}'}}{(\\dot{z},B,L)}})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True), Symbol('L', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True)))"], [["divide", 1, "Function('x^\\\\prime')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 3, "Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True))), Pow(Mul(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)), Symbol('B', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given q{(\\ddot{x})} = \\int e^{\\ddot{x}} d\\ddot{x}, then derive - \\rho + q{(\\ddot{x})} - e^{\\ddot{x}} = 0, then obtain - \\frac{- \\rho - e^{\\ddot{x}} + \\int e^{\\ddot{x}} d\\ddot{x}}{\\rho} = 0", "derivation": "q{(\\ddot{x})} = \\int e^{\\ddot{x}} d\\ddot{x} and q{(\\ddot{x})} - \\int e^{\\ddot{x}} d\\ddot{x} = 0 and - \\rho + q{(\\ddot{x})} - e^{\\ddot{x}} = 0 and - \\rho - e^{\\ddot{x}} + \\int e^{\\ddot{x}} d\\ddot{x} = 0 and - \\frac{- \\rho - e^{\\ddot{x}} + \\int e^{\\ddot{x}} d\\ddot{x}}{\\rho} = 0", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\ddot{x}', commutative=True)), Integral(exp(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 1, "Integral(exp(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Function('q')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('q')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\ddot{x}', commutative=True))), Integral(exp(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\ddot{x}', commutative=True))), Integral(exp(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})}, then obtain \\frac{d}{d \\Psi_{nl}} (\\operatorname{t_{2}}{(\\Psi_{nl})} - 1) - \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} + 1 = \\frac{d}{d \\Psi_{nl}} (\\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} - 1) - \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} + 1", "derivation": "\\operatorname{t_{2}}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} and \\operatorname{t_{2}}{(\\Psi_{nl})} - 1 = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} - 1 and \\frac{d}{d \\Psi_{nl}} (\\operatorname{t_{2}}{(\\Psi_{nl})} - 1) = \\frac{d}{d \\Psi_{nl}} (\\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} - 1) and \\frac{d}{d \\Psi_{nl}} (\\operatorname{t_{2}}{(\\Psi_{nl})} - 1) - \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} + 1 = \\frac{d}{d \\Psi_{nl}} (\\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} - 1) - \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} + 1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('t_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Add(Function('t_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 3, "Add(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Add(Function('t_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Integer(1)), Add(Derivative(Add(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\phi{(\\dot{y})} = \\log{(\\dot{y})}, then obtain \\frac{d}{d \\dot{y}} \\frac{1}{\\log{(\\dot{y})}} = \\frac{d}{d \\dot{y}} \\frac{\\phi^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}}}{\\log{(\\dot{y})}}", "derivation": "\\phi{(\\dot{y})} = \\log{(\\dot{y})} and \\phi^{\\dot{y}}{(\\dot{y})} = \\log{(\\dot{y})}^{\\dot{y}} and \\frac{1}{\\log{(\\dot{y})}} = \\frac{\\phi^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}}}{\\log{(\\dot{y})}} and \\frac{d}{d \\dot{y}} \\frac{1}{\\log{(\\dot{y})}} = \\frac{d}{d \\dot{y}} \\frac{\\phi^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}}}{\\log{(\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 2, "Mul(Pow(Function('\\\\phi')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\phi')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\phi')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(V_{\\mathbf{B}},\\mathbf{r})} = \\log{(\\mathbf{r})}^{V_{\\mathbf{B}}}, then obtain 2 \\int \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} dV_{\\mathbf{B}} = \\int (\\log{(\\mathbf{r})}^{V_{\\mathbf{B}}})^{\\mathbf{r}} dV_{\\mathbf{B}} + \\int \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} dV_{\\mathbf{B}}", "derivation": "\\operatorname{E_{\\lambda}}{(V_{\\mathbf{B}},\\mathbf{r})} = \\log{(\\mathbf{r})}^{V_{\\mathbf{B}}} and \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} = (\\log{(\\mathbf{r})}^{V_{\\mathbf{B}}})^{\\mathbf{r}} and \\int \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} dV_{\\mathbf{B}} = \\int (\\log{(\\mathbf{r})}^{V_{\\mathbf{B}}})^{\\mathbf{r}} dV_{\\mathbf{B}} and 2 \\int \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} dV_{\\mathbf{B}} = \\int (\\log{(\\mathbf{r})}^{V_{\\mathbf{B}}})^{\\mathbf{r}} dV_{\\mathbf{B}} + \\int \\operatorname{E_{\\lambda}}^{\\mathbf{r}}{(V_{\\mathbf{B}},\\mathbf{r})} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Pow(Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 3, "Integral(Pow(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Pow(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Integral(Pow(Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Pow(Function('E_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\lambda)} = \\frac{d}{d \\lambda} \\log{(\\lambda)}, then derive \\operatorname{f^{\\prime}}{(\\lambda)} = \\frac{1}{\\lambda}, then obtain \\int \\operatorname{f^{\\prime}}{(\\frac{1}{\\frac{d}{d \\lambda} \\log{(\\lambda)}})} d\\lambda = \\int \\frac{d}{d \\lambda} \\log{(\\lambda)} d\\lambda", "derivation": "\\operatorname{f^{\\prime}}{(\\lambda)} = \\frac{d}{d \\lambda} \\log{(\\lambda)} and \\operatorname{f^{\\prime}}{(\\lambda)} = \\frac{1}{\\lambda} and \\frac{d}{d \\lambda} \\log{(\\lambda)} = \\frac{1}{\\lambda} and \\operatorname{f^{\\prime}}{(\\frac{1}{\\frac{d}{d \\lambda} \\log{(\\lambda)}})} = \\frac{d}{d \\lambda} \\log{(\\lambda)} and \\int \\operatorname{f^{\\prime}}{(\\frac{1}{\\frac{d}{d \\lambda} \\log{(\\lambda)}})} d\\lambda = \\int \\frac{d}{d \\lambda} \\log{(\\lambda)} d\\lambda", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('f^{\\\\prime}')(Pow(Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1))), Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Pow(Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Derivative(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\pi)} = e^{\\pi}, then derive \\frac{d}{d \\pi} \\operatorname{C_{1}}{(\\pi)} = e^{\\pi}, then obtain (\\frac{d}{d \\pi} e^{\\pi})^{2} = e^{\\pi} \\frac{d}{d \\pi} e^{\\pi}", "derivation": "\\operatorname{C_{1}}{(\\pi)} = e^{\\pi} and \\frac{d}{d \\pi} \\operatorname{C_{1}}{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} and \\frac{d}{d \\pi} \\operatorname{C_{1}}{(\\pi)} = e^{\\pi} and (\\frac{d}{d \\pi} \\operatorname{C_{1}}{(\\pi)})^{2} = e^{\\pi} \\frac{d}{d \\pi} \\operatorname{C_{1}}{(\\pi)} and (\\frac{d}{d \\pi} e^{\\pi})^{2} = e^{\\pi} \\frac{d}{d \\pi} e^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), exp(Symbol('\\\\pi', commutative=True)))"], [["times", 3, "Derivative(Function('C_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('C_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(2)), Mul(exp(Symbol('\\\\pi', commutative=True)), Derivative(Function('C_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(2)), Mul(exp(Symbol('\\\\pi', commutative=True)), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} \\Psi_{nl})}, then obtain \\frac{f_{\\mathbf{p}} - \\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})}}{\\cos{(\\sin{(\\Psi_{\\lambda} \\Psi_{nl})})}} = \\frac{f_{\\mathbf{p}} - \\sin{(\\Psi_{\\lambda} \\Psi_{nl})}}{\\cos{(\\sin{(\\Psi_{\\lambda} \\Psi_{nl})})}}", "derivation": "\\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} \\Psi_{nl})} and - f_{\\mathbf{p}} + \\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})} = - f_{\\mathbf{p}} + \\sin{(\\Psi_{\\lambda} \\Psi_{nl})} and f_{\\mathbf{p}} - \\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})} = f_{\\mathbf{p}} - \\sin{(\\Psi_{\\lambda} \\Psi_{nl})} and \\frac{f_{\\mathbf{p}} - \\operatorname{C_{2}}{(\\Psi_{nl},\\Psi_{\\lambda})}}{\\cos{(\\sin{(\\Psi_{\\lambda} \\Psi_{nl})})}} = \\frac{f_{\\mathbf{p}} - \\sin{(\\Psi_{\\lambda} \\Psi_{nl})}}{\\cos{(\\sin{(\\Psi_{\\lambda} \\Psi_{nl})})}}", "srepr_derivation": [["get_premise", "Equality(Function('C_2')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('C_2')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["divide", 3, "cos(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], "Equality(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Pow(cos(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(-1))), Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))), Pow(cos(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(P_{e},\\mathbf{B})} = \\frac{\\partial}{\\partial P_{e}} P_{e}^{\\mathbf{B}}, then derive \\theta_{2}{(P_{e},\\mathbf{B})} = \\frac{P_{e}^{\\mathbf{B}} \\mathbf{B}}{P_{e}}, then obtain \\theta_{2}^{2}{(P_{e},\\mathbf{B})} = \\frac{P_{e}^{\\mathbf{B}} \\mathbf{B} \\theta_{2}{(P_{e},\\mathbf{B})}}{P_{e}}", "derivation": "\\theta_{2}{(P_{e},\\mathbf{B})} = \\frac{\\partial}{\\partial P_{e}} P_{e}^{\\mathbf{B}} and \\theta_{2}{(P_{e},\\mathbf{B})} = \\frac{P_{e}^{\\mathbf{B}} \\mathbf{B}}{P_{e}} and \\theta_{2}{(P_{e},\\mathbf{B})} \\frac{\\partial}{\\partial P_{e}} P_{e}^{\\mathbf{B}} = \\frac{P_{e}^{\\mathbf{B}} \\mathbf{B} \\frac{\\partial}{\\partial P_{e}} P_{e}^{\\mathbf{B}}}{P_{e}} and \\theta_{2}^{2}{(P_{e},\\mathbf{B})} = \\frac{P_{e}^{\\mathbf{B}} \\mathbf{B} \\theta_{2}{(P_{e},\\mathbf{B})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta_2')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["times", 2, "Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('\\\\theta_2')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\theta_2')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given Z{(\\delta,\\hat{x}_0)} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\delta + \\hat{x}_0), then derive Z{(\\delta,\\hat{x}_0)} = 1, then obtain 2 Z{(\\delta,\\hat{x}_0)} = 2", "derivation": "Z{(\\delta,\\hat{x}_0)} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\delta + \\hat{x}_0) and Z{(\\delta,\\hat{x}_0)} = 1 and 2 Z{(\\delta,\\hat{x}_0)} \\frac{\\partial}{\\partial \\hat{x}_0} (\\delta + \\hat{x}_0) = 2 \\frac{\\partial}{\\partial \\hat{x}_0} (\\delta + \\hat{x}_0) and 2 Z{(\\delta,\\hat{x}_0)} = 2", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Z')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(1))"], [["times", 2, "Mul(Integer(2), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Integer(2))"]]}, {"prompt": "Given \\mathbf{M}{(C,\\mathbf{s},\\mathbb{I})} = \\frac{C \\mathbf{s}}{\\mathbb{I}} and y{(C,\\mathbf{s},\\mathbb{I})} = (\\int \\mathbf{M}^{\\mathbb{I}}{(C,\\mathbf{s},\\mathbb{I})} dC)^{\\mathbb{I}}, then obtain y{(C,\\mathbf{s},\\mathbb{I})} = (\\int (\\frac{C \\mathbf{s}}{\\mathbb{I}})^{\\mathbb{I}} dC)^{\\mathbb{I}}", "derivation": "\\mathbf{M}{(C,\\mathbf{s},\\mathbb{I})} = \\frac{C \\mathbf{s}}{\\mathbb{I}} and \\mathbf{M}^{\\mathbb{I}}{(C,\\mathbf{s},\\mathbb{I})} = (\\frac{C \\mathbf{s}}{\\mathbb{I}})^{\\mathbb{I}} and \\int \\mathbf{M}^{\\mathbb{I}}{(C,\\mathbf{s},\\mathbb{I})} dC = \\int (\\frac{C \\mathbf{s}}{\\mathbb{I}})^{\\mathbb{I}} dC and (\\int \\mathbf{M}^{\\mathbb{I}}{(C,\\mathbf{s},\\mathbb{I})} dC)^{\\mathbb{I}} = (\\int (\\frac{C \\mathbf{s}}{\\mathbb{I}})^{\\mathbb{I}} dC)^{\\mathbb{I}} and y{(C,\\mathbf{s},\\mathbb{I})} = (\\int \\mathbf{M}^{\\mathbb{I}}{(C,\\mathbf{s},\\mathbb{I})} dC)^{\\mathbb{I}} and y{(C,\\mathbf{s},\\mathbb{I})} = (\\int (\\frac{C \\mathbf{s}}{\\mathbb{I}})^{\\mathbb{I}} dC)^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given C{(\\mathbf{f},C_{d})} = C_{d} + \\mathbf{f}, then derive - C_{d} \\mathbf{f} - \\frac{\\mathbf{f}^{2}}{2} - c + \\int C{(\\mathbf{f},C_{d})} d\\mathbf{f} = 0, then obtain - C_{d} \\mathbf{f} - \\frac{\\mathbf{f}^{2}}{2} - c + \\int (C_{d} + \\mathbf{f}) d\\mathbf{f} = 0", "derivation": "C{(\\mathbf{f},C_{d})} = C_{d} + \\mathbf{f} and \\int C{(\\mathbf{f},C_{d})} d\\mathbf{f} = \\int (C_{d} + \\mathbf{f}) d\\mathbf{f} and - \\int (C_{d} + \\mathbf{f}) d\\mathbf{f} + \\int C{(\\mathbf{f},C_{d})} d\\mathbf{f} = 0 and - C_{d} \\mathbf{f} - \\frac{\\mathbf{f}^{2}}{2} - c + \\int C{(\\mathbf{f},C_{d})} d\\mathbf{f} = 0 and - C_{d} \\mathbf{f} - \\frac{\\mathbf{f}^{2}}{2} - c + \\int (C_{d} + \\mathbf{f}) d\\mathbf{f} = 0", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integral(Function('C')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Function('C')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{J}{(F_{c})} = \\sin{(F_{c})}, then derive \\int \\mathbf{J}{(F_{c})} dF_{c} = \\hat{\\mathbf{r}} - \\cos{(F_{c})}, then obtain (\\int \\sin{(F_{c})} dF_{c})^{\\hat{\\mathbf{r}}} = (\\hat{\\mathbf{r}} - \\cos{(F_{c})})^{\\hat{\\mathbf{r}}}", "derivation": "\\mathbf{J}{(F_{c})} = \\sin{(F_{c})} and \\int \\mathbf{J}{(F_{c})} dF_{c} = \\int \\sin{(F_{c})} dF_{c} and \\int \\mathbf{J}{(F_{c})} dF_{c} = \\hat{\\mathbf{r}} - \\cos{(F_{c})} and (\\int \\mathbf{J}{(F_{c})} dF_{c})^{\\hat{\\mathbf{r}}} = (\\hat{\\mathbf{r}} - \\cos{(F_{c})})^{\\hat{\\mathbf{r}}} and (\\int \\sin{(F_{c})} dF_{c})^{\\hat{\\mathbf{r}}} = (\\hat{\\mathbf{r}} - \\cos{(F_{c})})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{J}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\omega{(v,W)} = W - v, then derive S + \\omega{(v,W)} = J - v, then obtain J + S + W - v = 2 J - v", "derivation": "\\omega{(v,W)} = W - v and \\frac{\\partial}{\\partial v} \\omega{(v,W)} = \\frac{\\partial}{\\partial v} (W - v) and \\int \\frac{\\partial}{\\partial v} \\omega{(v,W)} dv = \\int \\frac{\\partial}{\\partial v} (W - v) dv and S + \\omega{(v,W)} = J - v and S + W - v = J - v and J + S + W - v = 2 J - v", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Integral(Derivative(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('S', commutative=True), Function('\\\\omega')(Symbol('v', commutative=True), Symbol('W', commutative=True))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('S', commutative=True), Symbol('W', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["add", 5, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Symbol('S', commutative=True), Symbol('W', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Add(Mul(Integer(2), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(a)} = \\sin{(a)}, then derive - \\cos{(a)} - \\frac{d}{d a} \\mu_{0}{(a)} + 2 = 2 - 2 \\cos{(a)}, then obtain \\int (- \\cos{(a)} - \\frac{d}{d a} \\sin{(a)} + 2) da = \\int (- \\cos{(a)} - \\frac{d}{d a} \\mu_{0}{(a)} + 2) da", "derivation": "\\mu_{0}{(a)} = \\sin{(a)} and - a + \\mu_{0}{(a)} = - a + \\sin{(a)} and a - \\mu_{0}{(a)} = a - \\sin{(a)} and \\frac{d}{d a} (a - \\mu_{0}{(a)}) = \\frac{d}{d a} (a - \\sin{(a)}) and \\frac{d}{d a} (a - \\mu_{0}{(a)}) + \\frac{d}{d a} (a - \\sin{(a)}) = 2 \\frac{d}{d a} (a - \\sin{(a)}) and - \\cos{(a)} - \\frac{d}{d a} \\mu_{0}{(a)} + 2 = 2 - 2 \\cos{(a)} and \\int (- \\cos{(a)} - \\frac{d}{d a} \\mu_{0}{(a)} + 2) da = \\int (2 - 2 \\cos{(a)}) da and \\int (- \\cos{(a)} - \\frac{d}{d a} \\sin{(a)} + 2) da = \\int (2 - 2 \\cos{(a)}) da and \\int (- \\cos{(a)} - \\frac{d}{d a} \\sin{(a)} + 2) da = \\int (- \\cos{(a)} - \\frac{d}{d a} \\mu_{0}{(a)} + 2) da", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\mu_0')(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Add(Integer(2), Mul(Integer(-1), Integer(2), cos(Symbol('a', commutative=True)))))"], [["integrate", 6, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Tuple(Symbol('a', commutative=True))), Integral(Add(Integer(2), Mul(Integer(-1), Integer(2), cos(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integral(Add(Mul(Integer(-1), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Tuple(Symbol('a', commutative=True))), Integral(Add(Integer(2), Mul(Integer(-1), Integer(2), cos(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Integral(Add(Mul(Integer(-1), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Tuple(Symbol('a', commutative=True))), Integral(Add(Mul(Integer(-1), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(Q,x)} = \\frac{x}{Q} and \\operatorname{t_{2}}{(Q,x)} = - x + \\frac{x}{Q}, then obtain \\frac{\\partial}{\\partial Q} - x (- x + \\dot{z}{(Q,x)}) = \\frac{\\partial}{\\partial Q} - x (- x + \\frac{x}{Q})", "derivation": "\\dot{z}{(Q,x)} = \\frac{x}{Q} and - x + \\dot{z}{(Q,x)} = - x + \\frac{x}{Q} and \\operatorname{t_{2}}{(Q,x)} = - x + \\frac{x}{Q} and - x \\operatorname{t_{2}}{(Q,x)} = - x (- x + \\frac{x}{Q}) and \\frac{\\partial}{\\partial Q} - x \\operatorname{t_{2}}{(Q,x)} = \\frac{\\partial}{\\partial Q} - x (- x + \\frac{x}{Q}) and \\operatorname{t_{2}}{(Q,x)} = - x + \\dot{z}{(Q,x)} and \\frac{\\partial}{\\partial Q} - x (- x + \\dot{z}{(Q,x)}) = \\frac{\\partial}{\\partial Q} - x (- x + \\frac{x}{Q})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('Q', commutative=True), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\dot{z}')(Symbol('Q', commutative=True), Symbol('x', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('Q', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Function('t_2')(Symbol('Q', commutative=True), Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True)))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('x', commutative=True), Function('t_2')(Symbol('Q', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('t_2')(Symbol('Q', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\dot{z}')(Symbol('Q', commutative=True), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\dot{z}')(Symbol('Q', commutative=True), Symbol('x', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('x', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(h,C_{2})} = - C_{2} + h, then obtain C_{2} + \\frac{1}{4 \\operatorname{n_{2}}^{4}{(h,C_{2})}} = C_{2} + \\frac{1}{(- 2 C_{2} + 2 h)^{2} \\operatorname{n_{2}}^{2}{(h,C_{2})}}", "derivation": "\\operatorname{n_{2}}{(h,C_{2})} = - C_{2} + h and - C_{2} + h + \\operatorname{n_{2}}{(h,C_{2})} = - 2 C_{2} + 2 h and (- C_{2} + h + \\operatorname{n_{2}}{(h,C_{2})}) \\operatorname{n_{2}}{(h,C_{2})} = (- 2 C_{2} + 2 h) \\operatorname{n_{2}}{(h,C_{2})} and \\frac{1}{(- C_{2} + h + \\operatorname{n_{2}}{(h,C_{2})})^{2} \\operatorname{n_{2}}^{2}{(h,C_{2})}} = \\frac{1}{(- 2 C_{2} + 2 h)^{2} \\operatorname{n_{2}}^{2}{(h,C_{2})}} and \\frac{1}{4 \\operatorname{n_{2}}^{4}{(h,C_{2})}} = \\frac{1}{(- 2 C_{2} + 2 h)^{2} \\operatorname{n_{2}}^{2}{(h,C_{2})}} and C_{2} + \\frac{1}{4 \\operatorname{n_{2}}^{4}{(h,C_{2})}} = C_{2} + \\frac{1}{(- 2 C_{2} + 2 h)^{2} \\operatorname{n_{2}}^{2}{(h,C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('h', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('h', commutative=True), Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True))))"], [["times", 2, "Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('h', commutative=True), Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))), Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True))), Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))))"], [["power", 3, "Integer(-2)"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('h', commutative=True), Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True))), Integer(-2)), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-2))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True))), Integer(-2)), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Rational(1, 4), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-4))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True))), Integer(-2)), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-2))))"], [["add", 5, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Mul(Rational(1, 4), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-4)))), Add(Symbol('C_2', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True))), Integer(-2)), Pow(Function('n_2')(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given W{(I,p)} = e^{I p} and J{(I,p)} = e^{I p}, then obtain e^{W^{I}{(I,p)}} = e^{J^{I}{(I,p)}}", "derivation": "W{(I,p)} = e^{I p} and W^{I}{(I,p)} = (e^{I p})^{I} and J{(I,p)} = e^{I p} and W^{I}{(I,p)} = J^{I}{(I,p)} and e^{W^{I}{(I,p)}} = e^{J^{I}{(I,p)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('I', commutative=True), Symbol('p', commutative=True)), exp(Mul(Symbol('I', commutative=True), Symbol('p', commutative=True))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('W')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Symbol('I', commutative=True)), Pow(exp(Mul(Symbol('I', commutative=True), Symbol('p', commutative=True))), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('I', commutative=True), Symbol('p', commutative=True)), exp(Mul(Symbol('I', commutative=True), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('W')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Symbol('I', commutative=True)), Pow(Function('J')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Symbol('I', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Function('W')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Symbol('I', commutative=True))), exp(Pow(Function('J')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(r_{0},l)} = - l + r_{0}, then obtain \\int (3 l - 4 r_{0} + 4 \\varphi^{*}{(r_{0},l)}) dr_{0} = \\int (- r_{0} + \\varphi^{*}{(r_{0},l)}) dr_{0}", "derivation": "\\varphi^{*}{(r_{0},l)} = - l + r_{0} and l - r_{0} + \\varphi^{*}{(r_{0},l)} = 0 and - r_{0} + \\varphi^{*}{(r_{0},l)} = - l and l - 2 r_{0} + 2 \\varphi^{*}{(r_{0},l)} = - r_{0} + \\varphi^{*}{(r_{0},l)} and l - 2 r_{0} + \\varphi^{*}{(r_{0},l)} = - r_{0} and 3 l - 4 r_{0} + 4 \\varphi^{*}{(r_{0},l)} = l - 2 r_{0} + 2 \\varphi^{*}{(r_{0},l)} and 3 l - 4 r_{0} + 4 \\varphi^{*}{(r_{0},l)} = - r_{0} + \\varphi^{*}{(r_{0},l)} and \\int (3 l - 4 r_{0} + 4 \\varphi^{*}{(r_{0},l)}) dr_{0} = \\int (- r_{0} + \\varphi^{*}{(r_{0},l)}) dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))), Integer(0))"], [["minus", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)))"], [["add", 3, "Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(3), Symbol('l', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('r_0', commutative=True)), Mul(Integer(4), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Mul(Integer(3), Symbol('l', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('r_0', commutative=True)), Mul(Integer(4), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))))"], [["integrate", 7, "Symbol('r_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(3), Symbol('l', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('r_0', commutative=True)), Mul(Integer(4), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\varphi^*')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(F_{H})} = \\log{(F_{H})}, then derive \\frac{d}{d F_{H}} \\varepsilon{(F_{H})} = \\frac{1}{F_{H}}, then obtain \\frac{1}{F_{H}} = \\frac{d}{d F_{H}} \\log{(F_{H})}", "derivation": "\\varepsilon{(F_{H})} = \\log{(F_{H})} and \\frac{d}{d F_{H}} \\varepsilon{(F_{H})} = \\frac{d}{d F_{H}} \\log{(F_{H})} and \\frac{d}{d F_{H}} \\varepsilon{(F_{H})} = \\frac{1}{F_{H}} and \\frac{1}{F_{H}} = \\frac{d}{d F_{H}} \\log{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('F_H', commutative=True), Integer(-1)), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{M})} = \\log{(\\mathbf{M})}, then obtain 3 \\lambda{(\\mathbf{M})} = \\lambda{(\\mathbf{M})} + 2 \\log{(\\mathbf{M})}", "derivation": "\\lambda{(\\mathbf{M})} = \\log{(\\mathbf{M})} and 2 \\lambda{(\\mathbf{M})} = \\lambda{(\\mathbf{M})} + \\log{(\\mathbf{M})} and 3 \\lambda{(\\mathbf{M})} = 2 \\lambda{(\\mathbf{M})} + \\log{(\\mathbf{M})} and 3 \\lambda{(\\mathbf{M})} = \\lambda{(\\mathbf{M})} + 2 \\log{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 1, "Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 2, "Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Function('\\\\lambda')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} = m^{\\ddot{x}} and Q{(m,\\ddot{x})} = m^{\\ddot{x}} + 1, then obtain - \\ddot{x} Q{(m,\\ddot{x})} \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} = - \\ddot{x} (\\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} + 1) \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} = m^{\\ddot{x}} and \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} + 1 = m^{\\ddot{x}} + 1 and Q{(m,\\ddot{x})} = m^{\\ddot{x}} + 1 and Q{(m,\\ddot{x})} = \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} + 1 and - \\ddot{x} Q{(m,\\ddot{x})} \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} = - \\ddot{x} (\\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})} + 1) \\operatorname{f_{\\mathbf{p}}}{(m,\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(1)), Add(Pow(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Pow(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(1)))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Add(Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(1)), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\nabla)} = \\log{(\\nabla)} and \\psi^{*}{(\\nabla)} = 2 \\operatorname{m_{s}}^{2}{(\\nabla)} and \\chi{(\\nabla)} = \\frac{d}{d \\nabla} (\\operatorname{m_{s}}^{2}{(\\nabla)} + \\operatorname{m_{s}}{(\\nabla)} \\log{(\\nabla)}), then obtain \\chi{(\\nabla)} = \\frac{d}{d \\nabla} \\psi^{*}{(\\nabla)}", "derivation": "\\operatorname{m_{s}}{(\\nabla)} = \\log{(\\nabla)} and \\operatorname{m_{s}}^{2}{(\\nabla)} = \\operatorname{m_{s}}{(\\nabla)} \\log{(\\nabla)} and 2 \\operatorname{m_{s}}^{2}{(\\nabla)} = \\operatorname{m_{s}}^{2}{(\\nabla)} + \\operatorname{m_{s}}{(\\nabla)} \\log{(\\nabla)} and \\psi^{*}{(\\nabla)} = 2 \\operatorname{m_{s}}^{2}{(\\nabla)} and \\chi{(\\nabla)} = \\frac{d}{d \\nabla} (\\operatorname{m_{s}}^{2}{(\\nabla)} + \\operatorname{m_{s}}{(\\nabla)} \\log{(\\nabla)}) and \\psi^{*}{(\\nabla)} = \\operatorname{m_{s}}^{2}{(\\nabla)} + \\operatorname{m_{s}}{(\\nabla)} \\log{(\\nabla)} and \\chi{(\\nabla)} = \\frac{d}{d \\nabla} \\psi^{*}{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "Function('m_s')(Symbol('\\\\nabla', commutative=True))"], "Equality(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2))), Add(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\nabla', commutative=True)), Derivative(Add(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\psi^*')(Symbol('\\\\nabla', commutative=True)), Add(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\chi')(Symbol('\\\\nabla', commutative=True)), Derivative(Function('\\\\psi^*')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} (b + x^\\prime) and \\operatorname{f_{E}}{(x^\\prime,b)} = b + x^\\prime, then obtain \\operatorname{v_{t}}^{x^\\prime}{(x^\\prime,b)} = (\\frac{\\partial}{\\partial x^\\prime} \\operatorname{f_{E}}{(x^\\prime,b)})^{x^\\prime}", "derivation": "\\operatorname{v_{t}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} (b + x^\\prime) and \\operatorname{f_{E}}{(x^\\prime,b)} = b + x^\\prime and \\operatorname{v_{t}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} \\operatorname{f_{E}}{(x^\\prime,b)} and \\operatorname{v_{t}}^{x^\\prime}{(x^\\prime,b)} = (\\frac{\\partial}{\\partial x^\\prime} \\operatorname{f_{E}}{(x^\\prime,b)})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Derivative(Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_t')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Derivative(Function('f_E')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Function('f_E')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\rho)} = \\sin{(\\log{(\\rho)})} and \\mathbf{P}{(\\rho)} = \\log{(\\rho)}, then obtain \\operatorname{C_{1}}{(\\rho)} + e^{\\sin{(\\log{(\\rho)})}} = e^{\\sin{(\\log{(\\rho)})}} + \\sin{(\\log{(\\rho)})}", "derivation": "\\operatorname{C_{1}}{(\\rho)} = \\sin{(\\log{(\\rho)})} and \\mathbf{P}{(\\rho)} = \\log{(\\rho)} and e^{\\operatorname{C_{1}}{(\\rho)}} = e^{\\sin{(\\log{(\\rho)})}} and \\operatorname{C_{1}}{(\\rho)} = \\sin{(\\mathbf{P}{(\\rho)})} and \\operatorname{C_{1}}{(\\rho)} + e^{\\operatorname{C_{1}}{(\\rho)}} = e^{\\operatorname{C_{1}}{(\\rho)}} + \\sin{(\\mathbf{P}{(\\rho)})} and \\operatorname{C_{1}}{(\\rho)} + e^{\\operatorname{C_{1}}{(\\rho)}} = e^{\\operatorname{C_{1}}{(\\rho)}} + \\sin{(\\log{(\\rho)})} and \\operatorname{C_{1}}{(\\rho)} + e^{\\sin{(\\log{(\\rho)})}} = e^{\\sin{(\\log{(\\rho)})}} + \\sin{(\\log{(\\rho)})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\rho', commutative=True)), sin(log(Symbol('\\\\rho', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["exp", 1], "Equality(exp(Function('C_1')(Symbol('\\\\rho', commutative=True))), exp(sin(log(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_1')(Symbol('\\\\rho', commutative=True)), sin(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True))))"], [["add", 4, "exp(Function('C_1')(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Function('C_1')(Symbol('\\\\rho', commutative=True)), exp(Function('C_1')(Symbol('\\\\rho', commutative=True)))), Add(exp(Function('C_1')(Symbol('\\\\rho', commutative=True))), sin(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('C_1')(Symbol('\\\\rho', commutative=True)), exp(Function('C_1')(Symbol('\\\\rho', commutative=True)))), Add(exp(Function('C_1')(Symbol('\\\\rho', commutative=True))), sin(log(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Function('C_1')(Symbol('\\\\rho', commutative=True)), exp(sin(log(Symbol('\\\\rho', commutative=True))))), Add(exp(sin(log(Symbol('\\\\rho', commutative=True)))), sin(log(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\theta)} = e^{\\theta}, then obtain (2 \\varphi{(\\theta)} - e^{\\theta}) e^{- \\theta} = \\varphi{(\\theta)} e^{- \\theta}", "derivation": "\\varphi{(\\theta)} = e^{\\theta} and \\varphi{(\\theta)} + e^{\\theta} = 2 e^{\\theta} and 2 \\varphi{(\\theta)} + 2 e^{\\theta} = \\varphi{(\\theta)} + 3 e^{\\theta} and 2 \\varphi{(\\theta)} - e^{\\theta} = \\varphi{(\\theta)} and (2 \\varphi{(\\theta)} - e^{\\theta}) e^{- \\theta} = \\varphi{(\\theta)} e^{- \\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\theta', commutative=True))))"], [["add", 2, "Add(Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\theta', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\theta', commutative=True)))), Add(Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\theta', commutative=True)))))"], [["minus", 3, "Mul(Integer(3), exp(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True)))), Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)))"], [["divide", 4, "exp(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Function('\\\\varphi')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(C)} = \\cos{(C)}, then derive \\int \\varphi{(C)} dC = \\lambda + \\sin{(C)}, then obtain \\cos{(C)} + \\int \\cos{(C)} dC = \\lambda + \\sin{(C)} + \\cos{(C)}", "derivation": "\\varphi{(C)} = \\cos{(C)} and \\int \\varphi{(C)} dC = \\int \\cos{(C)} dC and \\int \\varphi{(C)} dC = \\lambda + \\sin{(C)} and \\int \\cos{(C)} dC = \\lambda + \\sin{(C)} and \\cos{(C)} + \\int \\cos{(C)} dC = \\lambda + \\sin{(C)} + \\cos{(C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('C', commutative=True))))"], [["add", 4, "cos(Symbol('C', commutative=True))"], "Equality(Add(cos(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{1},W)} = \\int (W + v_{1}) dv_{1}, then derive W + v_{1} + \\operatorname{F_{H}}{(v_{1},W)} = W v_{1} + W + t + \\frac{v_{1}^{2}}{2} + v_{1}, then obtain \\frac{\\partial}{\\partial v_{1}} (2 W + 2 v_{1} + 2 \\int (W + v_{1}) dv_{1}) = \\frac{\\partial}{\\partial v_{1}} (W v_{1} + 2 W + t + \\frac{v_{1}^{2}}{2} + 2 v_{1} + \\int (W + v_{1}) dv_{1})", "derivation": "\\operatorname{F_{H}}{(v_{1},W)} = \\int (W + v_{1}) dv_{1} and W + v_{1} + \\operatorname{F_{H}}{(v_{1},W)} = W + v_{1} + \\int (W + v_{1}) dv_{1} and W + v_{1} + \\operatorname{F_{H}}{(v_{1},W)} = W v_{1} + W + t + \\frac{v_{1}^{2}}{2} + v_{1} and W + v_{1} + \\int (W + v_{1}) dv_{1} = W v_{1} + W + t + \\frac{v_{1}^{2}}{2} + v_{1} and 2 W + 2 v_{1} + 2 \\int (W + v_{1}) dv_{1} = W v_{1} + 2 W + t + \\frac{v_{1}^{2}}{2} + 2 v_{1} + \\int (W + v_{1}) dv_{1} and \\frac{\\partial}{\\partial v_{1}} (2 W + 2 v_{1} + 2 \\int (W + v_{1}) dv_{1}) = \\frac{\\partial}{\\partial v_{1}} (W v_{1} + 2 W + t + \\frac{v_{1}^{2}}{2} + 2 v_{1} + \\int (W + v_{1}) dv_{1})", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["add", 1, "Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True), Function('F_H')(Symbol('v_1', commutative=True), Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True), Function('F_H')(Symbol('v_1', commutative=True), Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Symbol('W', commutative=True), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Mul(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Symbol('W', commutative=True), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Symbol('v_1', commutative=True)))"], [["add", 4, "Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Add(Mul(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True)), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Mul(Integer(2), Symbol('v_1', commutative=True)), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True)), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Mul(Integer(2), Symbol('v_1', commutative=True)), Integral(Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then obtain (\\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\hat{x}_0{(\\mathbf{g})})^{\\mathbf{g}} = (\\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})})^{\\mathbf{g}}", "derivation": "\\hat{x}_0{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} \\hat{x}_0{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\hat{x}_0{(\\mathbf{g})} = \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and (\\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\hat{x}_0{(\\mathbf{g})})^{\\mathbf{g}} = (\\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(S,p)} = \\frac{\\partial}{\\partial S} (S + p) and x{(S,p)} = \\operatorname{f^{\\prime}}{(S,p)} - 1, then obtain (\\frac{\\partial}{\\partial S} x^{S}{(S,p)})^{S} = (\\frac{\\partial}{\\partial S} (\\operatorname{f^{\\prime}}{(S,p)} - 1)^{S})^{S}", "derivation": "\\operatorname{f^{\\prime}}{(S,p)} = \\frac{\\partial}{\\partial S} (S + p) and \\operatorname{f^{\\prime}}{(S,p)} - 1 = \\frac{\\partial}{\\partial S} (S + p) - 1 and x{(S,p)} = \\operatorname{f^{\\prime}}{(S,p)} - 1 and (\\operatorname{f^{\\prime}}{(S,p)} - 1)^{S} = (\\frac{\\partial}{\\partial S} (S + p) - 1)^{S} and \\frac{\\partial}{\\partial S} (\\operatorname{f^{\\prime}}{(S,p)} - 1)^{S} = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S + p) - 1)^{S} and \\frac{\\partial}{\\partial S} x^{S}{(S,p)} = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S + p) - 1)^{S} and \\frac{\\partial}{\\partial S} x^{S}{(S,p)} = \\frac{\\partial}{\\partial S} (\\operatorname{f^{\\prime}}{(S,p)} - 1)^{S} and (\\frac{\\partial}{\\partial S} x^{S}{(S,p)})^{S} = (\\frac{\\partial}{\\partial S} (\\operatorname{f^{\\prime}}{(S,p)} - 1)^{S})^{S}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Add(Derivative(Add(Symbol('S', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('x')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Symbol('S', commutative=True)), Pow(Add(Derivative(Add(Symbol('S', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)))"], [["differentiate", 4, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Add(Derivative(Add(Symbol('S', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Pow(Function('x')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Add(Derivative(Add(Symbol('S', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Pow(Function('x')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["power", 7, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('x')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Pow(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain 2 e = e^{e^{- \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})}}} + e", "derivation": "\\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and 0 = - \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} and \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and 0 = - \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} and 1 = e^{- \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})}} and e = e^{e^{- \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})}}} and 2 e = e^{e^{- \\operatorname{C_{d}}{(g^{\\prime}_{\\varepsilon})} + \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})}}} + e", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["exp", 4], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["exp", 5], "Equality(E, exp(exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"], [["add", 6, "E"], "Equality(Mul(Integer(2), E), Add(exp(exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), E))"]]}, {"prompt": "Given \\mathbf{J}_f{(s)} = e^{e^{s}}, then derive \\frac{d^{2}}{d s^{2}} \\mathbf{J}_f{(s)} = (e^{s} + 1) e^{s} e^{e^{s}}, then obtain (e^{s} + 1)^{4} + (\\frac{d^{2}}{d s^{2}} e^{e^{s}})^{2} = (e^{s} + 1)^{4} + (e^{s} + 1)^{2} e^{2 s} e^{2 e^{s}}", "derivation": "\\mathbf{J}_f{(s)} = e^{e^{s}} and \\frac{d}{d s} \\mathbf{J}_f{(s)} = \\frac{d}{d s} e^{e^{s}} and \\frac{d^{2}}{d s^{2}} \\mathbf{J}_f{(s)} = \\frac{d^{2}}{d s^{2}} e^{e^{s}} and \\frac{d^{2}}{d s^{2}} \\mathbf{J}_f{(s)} = (e^{s} + 1) e^{s} e^{e^{s}} and (\\frac{d^{2}}{d s^{2}} \\mathbf{J}_f{(s)})^{2} = (e^{s} + 1)^{2} e^{2 s} e^{2 e^{s}} and (e^{s} + 1)^{4} + (\\frac{d^{2}}{d s^{2}} \\mathbf{J}_f{(s)})^{2} = (e^{s} + 1)^{4} + (e^{s} + 1)^{2} e^{2 s} e^{2 e^{s}} and (e^{s} + 1)^{4} + (\\frac{d^{2}}{d s^{2}} e^{e^{s}})^{2} = (e^{s} + 1)^{4} + (e^{s} + 1)^{2} e^{2 s} e^{2 e^{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Derivative(exp(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Mul(Add(exp(Symbol('s', commutative=True)), Integer(1)), exp(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2)), Mul(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(2)), exp(Mul(Integer(2), Symbol('s', commutative=True))), exp(Mul(Integer(2), exp(Symbol('s', commutative=True))))))"], [["add", 5, "Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(4))"], "Equality(Add(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(4)), Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2))), Add(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(4)), Mul(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(2)), exp(Mul(Integer(2), Symbol('s', commutative=True))), exp(Mul(Integer(2), exp(Symbol('s', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(4)), Pow(Derivative(exp(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2))), Add(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(4)), Mul(Pow(Add(exp(Symbol('s', commutative=True)), Integer(1)), Integer(2)), exp(Mul(Integer(2), Symbol('s', commutative=True))), exp(Mul(Integer(2), exp(Symbol('s', commutative=True)))))))"]]}, {"prompt": "Given \\chi{(x,\\theta_1)} = - \\theta_1 + x and \\dot{z}{(x,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x), then obtain - \\frac{\\dot{z}{(x,\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} \\chi{(x,\\theta_1)}}{\\theta_1} = - \\frac{2 \\frac{\\partial}{\\partial \\theta_1} \\chi{(x,\\theta_1)}}{\\theta_1}", "derivation": "\\chi{(x,\\theta_1)} = - \\theta_1 + x and \\frac{\\partial}{\\partial \\theta_1} \\chi{(x,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x) and \\dot{z}{(x,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x) and \\dot{z}{(x,\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x) = 2 \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x) and - \\frac{\\dot{z}{(x,\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x)}{\\theta_1} = - \\frac{2 \\frac{\\partial}{\\partial \\theta_1} (- \\theta_1 + x)}{\\theta_1} and - \\frac{\\dot{z}{(x,\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} \\chi{(x,\\theta_1)}}{\\theta_1} = - \\frac{2 \\frac{\\partial}{\\partial \\theta_1} \\chi{(x,\\theta_1)}}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Add(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Add(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(F_{x},E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}}, then derive \\operatorname{A_{z}}{(F_{x},E_{\\lambda})} = \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}}, then obtain E_{\\lambda} \\frac{\\partial^{2}}{\\partial F_{x}\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}} = E_{\\lambda} \\frac{\\partial}{\\partial F_{x}} \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}}", "derivation": "\\operatorname{A_{z}}{(F_{x},E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}} and \\operatorname{A_{z}}{(F_{x},E_{\\lambda})} = \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}} and \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}} = \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}} and \\frac{\\partial^{2}}{\\partial F_{x}\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}} = \\frac{\\partial}{\\partial F_{x}} \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}} and E_{\\lambda} \\frac{\\partial^{2}}{\\partial F_{x}\\partial E_{\\lambda}} E_{\\lambda}^{F_{x}} = E_{\\lambda} \\frac{\\partial}{\\partial F_{x}} \\frac{E_{\\lambda}^{F_{x}} F_{x}}{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('F_x', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_z')(Symbol('F_x', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(x^\\prime,f^{*})} = \\sin{(f^{*} x^\\prime)}, then obtain \\int r^{- f^{*}}{(x^\\prime,f^{*})} \\int r^{f^{*}}{(x^\\prime,f^{*})} df^{*} df^{*} = \\int r^{- f^{*}}{(x^\\prime,f^{*})} \\int \\sin^{f^{*}}{(f^{*} x^\\prime)} df^{*} df^{*}", "derivation": "r{(x^\\prime,f^{*})} = \\sin{(f^{*} x^\\prime)} and r^{f^{*}}{(x^\\prime,f^{*})} = \\sin^{f^{*}}{(f^{*} x^\\prime)} and \\int r^{f^{*}}{(x^\\prime,f^{*})} df^{*} = \\int \\sin^{f^{*}}{(f^{*} x^\\prime)} df^{*} and r^{- f^{*}}{(x^\\prime,f^{*})} \\int r^{f^{*}}{(x^\\prime,f^{*})} df^{*} = r^{- f^{*}}{(x^\\prime,f^{*})} \\int \\sin^{f^{*}}{(f^{*} x^\\prime)} df^{*} and \\int r^{- f^{*}}{(x^\\prime,f^{*})} \\int r^{f^{*}}{(x^\\prime,f^{*})} df^{*} df^{*} = \\int r^{- f^{*}}{(x^\\prime,f^{*})} \\int \\sin^{f^{*}}{(f^{*} x^\\prime)} df^{*} df^{*}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), sin(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(sin(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f^*', commutative=True)))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Pow(sin(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["divide", 3, "Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))), Integral(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))), Integral(Pow(sin(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["integrate", 4, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))), Integral(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Pow(Function('r')(Symbol('x^\\\\prime', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))), Integral(Pow(sin(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mu_0)} = e^{\\mu_0}, then derive \\frac{\\frac{d}{d \\mu_0} \\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} - \\frac{\\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0^{2}} = \\frac{e^{\\mu_0}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}}, then obtain \\frac{\\frac{d}{d \\mu_0} \\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}} = \\frac{e^{\\mu_0}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}}", "derivation": "\\operatorname{v_{x}}{(\\mu_0)} = e^{\\mu_0} and \\frac{\\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} = \\frac{e^{\\mu_0}}{\\mu_0} and \\frac{d}{d \\mu_0} \\frac{\\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} = \\frac{d}{d \\mu_0} \\frac{e^{\\mu_0}}{\\mu_0} and \\frac{\\frac{d}{d \\mu_0} \\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} - \\frac{\\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0^{2}} = \\frac{e^{\\mu_0}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}} and \\frac{\\frac{d}{d \\mu_0} \\operatorname{v_{x}}{(\\mu_0)}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}} = \\frac{e^{\\mu_0}}{\\mu_0} - \\frac{e^{\\mu_0}}{\\mu_0^{2}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(Function('v_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), Function('v_x')(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), exp(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(Function('v_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), exp(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given g{(T)} = \\log{(T)}, then obtain ((T + g{(T)}) g^{T}{(T)})^{T} = ((T + \\log{(T)}) g^{T}{(T)})^{T}", "derivation": "g{(T)} = \\log{(T)} and T + g{(T)} = T + \\log{(T)} and g^{T}{(T)} = \\log{(T)}^{T} and (T + g{(T)}) \\log{(T)}^{T} = (T + \\log{(T)}) \\log{(T)}^{T} and (T + g{(T)}) g^{T}{(T)} = (T + \\log{(T)}) g^{T}{(T)} and ((T + g{(T)}) g^{T}{(T)})^{T} = ((T + \\log{(T)}) g^{T}{(T)})^{T}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('g')(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), log(Symbol('T', commutative=True))))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('g')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["times", 2, "Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Mul(Add(Symbol('T', commutative=True), Function('g')(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Add(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('T', commutative=True), Function('g')(Symbol('T', commutative=True))), Pow(Function('g')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Add(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Pow(Function('g')(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["power", 5, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('T', commutative=True), Function('g')(Symbol('T', commutative=True))), Pow(Function('g')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Symbol('T', commutative=True)), Pow(Mul(Add(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Pow(Function('g')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mu{(I,F_{c},q)} = \\frac{F_{c} I}{q}, then derive \\frac{\\partial}{\\partial F_{c}} \\mu{(I,F_{c},q)} = \\frac{I}{q}, then obtain 0 = F_{c} \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} I}{q} - \\mu{(I,F_{c},q)}", "derivation": "\\mu{(I,F_{c},q)} = \\frac{F_{c} I}{q} and \\frac{\\partial}{\\partial F_{c}} \\mu{(I,F_{c},q)} = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} I}{q} and 0 = \\frac{F_{c} I}{q} - \\mu{(I,F_{c},q)} and \\frac{\\partial}{\\partial F_{c}} \\mu{(I,F_{c},q)} = \\frac{I}{q} and 0 = F_{c} \\frac{\\partial}{\\partial F_{c}} \\mu{(I,F_{c},q)} - \\mu{(I,F_{c},q)} and 0 = F_{c} \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} I}{q} - \\mu{(I,F_{c},q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('I', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_c', commutative=True), Symbol('I', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('F_c', commutative=True), Symbol('I', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Symbol('I', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Mul(Symbol('F_c', commutative=True), Derivative(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Mul(Symbol('F_c', commutative=True), Derivative(Mul(Symbol('F_c', commutative=True), Symbol('I', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('F_c', commutative=True), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\psi,\\theta_2)} = \\psi - \\theta_2, then obtain \\int \\frac{\\partial}{\\partial \\psi} (\\psi (\\theta_2 + \\pi{(\\psi,\\theta_2)}) - (- \\psi + \\theta_2) (\\psi - \\theta_2)) d\\theta_2 = \\int \\frac{\\partial}{\\partial \\psi} (\\psi^{2} - (- \\psi + \\theta_2) (\\psi - \\theta_2)) d\\theta_2", "derivation": "\\pi{(\\psi,\\theta_2)} = \\psi - \\theta_2 and \\theta_2 + \\pi{(\\psi,\\theta_2)} = \\psi and \\psi (\\theta_2 + \\pi{(\\psi,\\theta_2)}) = \\psi^{2} and \\psi (\\theta_2 + \\pi{(\\psi,\\theta_2)}) - (- \\psi + \\theta_2) (\\psi - \\theta_2) = \\psi^{2} - (- \\psi + \\theta_2) (\\psi - \\theta_2) and \\frac{\\partial}{\\partial \\psi} (\\psi (\\theta_2 + \\pi{(\\psi,\\theta_2)}) - (- \\psi + \\theta_2) (\\psi - \\theta_2)) = \\frac{\\partial}{\\partial \\psi} (\\psi^{2} - (- \\psi + \\theta_2) (\\psi - \\theta_2)) and \\int \\frac{\\partial}{\\partial \\psi} (\\psi (\\theta_2 + \\pi{(\\psi,\\theta_2)}) - (- \\psi + \\theta_2) (\\psi - \\theta_2)) d\\theta_2 = \\int \\frac{\\partial}{\\partial \\psi} (\\psi^{2} - (- \\psi + \\theta_2) (\\psi - \\theta_2)) d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\psi', commutative=True))"], [["times", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Pow(Symbol('\\\\psi', commutative=True), Integer(2)))"], [["add", 3, "Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Add(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Add(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(z)} = \\sin{(z)}, then derive \\frac{d}{d z} \\operatorname{F_{N}}{(z)} = \\cos{(z)}, then obtain \\frac{\\frac{d}{d z} \\operatorname{F_{N}}{(z)} + 1}{\\cos{(z)}} = \\frac{\\cos{(z)} + 1}{\\cos{(z)}}", "derivation": "\\operatorname{F_{N}}{(z)} = \\sin{(z)} and \\frac{d}{d z} \\operatorname{F_{N}}{(z)} = \\frac{d}{d z} \\sin{(z)} and \\frac{d}{d z} \\operatorname{F_{N}}{(z)} = \\cos{(z)} and \\frac{d}{d z} \\operatorname{F_{N}}{(z)} + 1 = \\cos{(z)} + 1 and \\frac{\\frac{d}{d z} \\operatorname{F_{N}}{(z)} + 1}{\\frac{d}{d z} \\sin{(z)}} = \\frac{\\cos{(z)} + 1}{\\frac{d}{d z} \\sin{(z)}} and \\frac{\\frac{d}{d z} \\operatorname{F_{N}}{(z)} + 1}{\\cos{(z)}} = \\frac{\\cos{(z)} + 1}{\\cos{(z)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), cos(Symbol('z', commutative=True)))"], [["add", 3, 1], "Equality(Add(Derivative(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('z', commutative=True)), Integer(1)))"], [["divide", 4, "Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)), Pow(Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))), Mul(Add(cos(Symbol('z', commutative=True)), Integer(1)), Pow(Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Derivative(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('z', commutative=True)), Integer(1)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain 3 \\hat{p}{(E_{\\lambda})} = \\hat{p}{(E_{\\lambda})} + 2 e^{E_{\\lambda}}", "derivation": "\\hat{p}{(E_{\\lambda})} = e^{E_{\\lambda}} and 2 \\hat{p}{(E_{\\lambda})} = \\hat{p}{(E_{\\lambda})} + e^{E_{\\lambda}} and 3 \\hat{p}{(E_{\\lambda})} = 2 \\hat{p}{(E_{\\lambda})} + e^{E_{\\lambda}} and 3 \\hat{p}{(E_{\\lambda})} = \\hat{p}{(E_{\\lambda})} + 2 e^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), exp(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(v_{1},v_{x})} = e^{v_{x}^{v_{1}}}, then obtain - \\operatorname{A_{y}}{(v_{1},v_{x})} + \\int \\operatorname{A_{y}}^{v_{x}}{(v_{1},v_{x})} dv_{1} = - \\operatorname{A_{y}}{(v_{1},v_{x})} + \\int (e^{v_{x}^{v_{1}}})^{v_{x}} dv_{1}", "derivation": "\\operatorname{A_{y}}{(v_{1},v_{x})} = e^{v_{x}^{v_{1}}} and \\operatorname{A_{y}}^{v_{x}}{(v_{1},v_{x})} = (e^{v_{x}^{v_{1}}})^{v_{x}} and \\int \\operatorname{A_{y}}^{v_{x}}{(v_{1},v_{x})} dv_{1} = \\int (e^{v_{x}^{v_{1}}})^{v_{x}} dv_{1} and - \\operatorname{A_{y}}{(v_{1},v_{x})} + \\int \\operatorname{A_{y}}^{v_{x}}{(v_{1},v_{x})} dv_{1} = - \\operatorname{A_{y}}{(v_{1},v_{x})} + \\int (e^{v_{x}^{v_{1}}})^{v_{x}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True)), exp(Pow(Symbol('v_x', commutative=True), Symbol('v_1', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(exp(Pow(Symbol('v_x', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_x', commutative=True)))"], [["integrate", 2, "Symbol('v_1', commutative=True)"], "Equality(Integral(Pow(Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Pow(exp(Pow(Symbol('v_x', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["minus", 3, "Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True))), Integral(Pow(Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), Function('A_y')(Symbol('v_1', commutative=True), Symbol('v_x', commutative=True))), Integral(Pow(exp(Pow(Symbol('v_x', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\tilde{g}^*,h)} = - h + e^{\\tilde{g}^*} and n{(\\tilde{g}^*)} = \\tilde{g}^*, then derive \\frac{\\partial}{\\partial h} \\operatorname{E_{x}}{(\\tilde{g}^*,h)} = -1, then obtain \\frac{\\partial}{\\partial h} (- h + e^{n{(\\tilde{g}^*)}}) = -1", "derivation": "\\operatorname{E_{x}}{(\\tilde{g}^*,h)} = - h + e^{\\tilde{g}^*} and n{(\\tilde{g}^*)} = \\tilde{g}^* and - \\tilde{g}^* + \\operatorname{E_{x}}{(\\tilde{g}^*,h)} = - \\tilde{g}^* - h + e^{\\tilde{g}^*} and \\frac{\\partial}{\\partial h} (- \\tilde{g}^* + \\operatorname{E_{x}}{(\\tilde{g}^*,h)}) = \\frac{\\partial}{\\partial h} (- \\tilde{g}^* - h + e^{\\tilde{g}^*}) and \\frac{\\partial}{\\partial h} \\operatorname{E_{x}}{(\\tilde{g}^*,h)} = -1 and \\frac{\\partial}{\\partial h} (- h + e^{\\tilde{g}^*}) = -1 and \\frac{\\partial}{\\partial h} (- h + e^{n{(\\tilde{g}^*)}}) = -1", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], [["minus", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('E_x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('E_x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('E_x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Function('n')(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given y{(n_{1})} = \\sin{(n_{1})}, then obtain - \\frac{(y{(n_{1})} - \\sin{(n_{1})} + 1)^{8}}{\\sin{(n_{1})}} = - \\frac{1}{\\sin{(n_{1})}}", "derivation": "y{(n_{1})} = \\sin{(n_{1})} and y{(n_{1})} - \\sin{(n_{1})} = 0 and y{(n_{1})} - \\sin{(n_{1})} + 1 = 1 and (y{(n_{1})} - \\sin{(n_{1})} + 1)^{2} = y{(n_{1})} - \\sin{(n_{1})} + 1 and (y{(n_{1})} - \\sin{(n_{1})} + 1)^{2} = 1 and (y{(n_{1})} - \\sin{(n_{1})} + 1)^{4} = 1 and - \\frac{(y{(n_{1})} - \\sin{(n_{1})} + 1)^{4}}{\\sin{(n_{1})}} = - \\frac{1}{\\sin{(n_{1})}} and - \\frac{(y{(n_{1})} - \\sin{(n_{1})} + 1)^{8}}{\\sin{(n_{1})}} = - \\frac{1}{\\sin{(n_{1})}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('n_1', commutative=True)), sin(Symbol('n_1', commutative=True)))"], [["minus", 1, "sin(Symbol('n_1', commutative=True))"], "Equality(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(1))"], [["times", 3, "Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1))"], "Equality(Pow(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(2)), Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(2)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(4)), Integer(1))"], [["divide", 6, "Mul(Integer(-1), sin(Symbol('n_1', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(4)), Pow(sin(Symbol('n_1', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('n_1', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(-1), Pow(Add(Function('y')(Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Symbol('n_1', commutative=True))), Integer(1)), Integer(8)), Pow(sin(Symbol('n_1', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('n_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\varphi^*)} = e^{\\varphi^*} and W{(\\varphi^*)} = - (\\varphi^* e^{\\varphi^*})^{\\varphi^*} + e^{\\varphi^*}, then obtain W^{\\varphi^*}{(\\varphi^*)} = (- (\\varphi^* \\hat{H}_{\\lambda}{(\\varphi^*)})^{\\varphi^*} + e^{\\varphi^*})^{\\varphi^*}", "derivation": "\\hat{H}_{\\lambda}{(\\varphi^*)} = e^{\\varphi^*} and \\varphi^* \\hat{H}_{\\lambda}{(\\varphi^*)} = \\varphi^* e^{\\varphi^*} and (\\varphi^* \\hat{H}_{\\lambda}{(\\varphi^*)})^{\\varphi^*} = (\\varphi^* e^{\\varphi^*})^{\\varphi^*} and W{(\\varphi^*)} = - (\\varphi^* e^{\\varphi^*})^{\\varphi^*} + e^{\\varphi^*} and W{(\\varphi^*)} = - (\\varphi^* \\hat{H}_{\\lambda}{(\\varphi^*)})^{\\varphi^*} + e^{\\varphi^*} and W^{\\varphi^*}{(\\varphi^*)} = (- (\\varphi^* \\hat{H}_{\\lambda}{(\\varphi^*)})^{\\varphi^*} + e^{\\varphi^*})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('W')(Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), exp(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('W')(Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), exp(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), exp(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given n{(\\phi)} = e^{\\phi}, then derive \\int n{(\\phi)} d\\phi = G + e^{\\phi}, then derive \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi = \\frac{d}{d \\phi} n{(\\phi)}, then obtain \\int \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi d\\phi = F_{c} + n{(\\phi)}", "derivation": "n{(\\phi)} = e^{\\phi} and \\int n{(\\phi)} d\\phi = \\int e^{\\phi} d\\phi and \\int n{(\\phi)} d\\phi = G + e^{\\phi} and \\int n{(\\phi)} d\\phi = G + n{(\\phi)} and \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi = \\frac{\\partial}{\\partial \\phi} (G + n{(\\phi)}) and \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi = \\frac{d}{d \\phi} n{(\\phi)} and \\int \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi d\\phi = \\int \\frac{d}{d \\phi} n{(\\phi)} d\\phi and \\int \\frac{d}{d \\phi} \\int n{(\\phi)} d\\phi d\\phi = F_{c} + n{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('G', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('G', commutative=True), Function('n')(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Symbol('G', commutative=True), Function('n')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Derivative(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Integral(Derivative(Integral(Function('n')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('F_c', commutative=True), Function('n')(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given E{(\\rho_f)} = e^{\\rho_f} and \\operatorname{r_{0}}{(\\rho_f)} = - E{(\\rho_f)} + e^{\\rho_f}, then obtain e^{\\rho_f} + \\int 0 d\\rho_f = e^{\\rho_f} + \\int \\operatorname{r_{0}}{(\\rho_f)} d\\rho_f", "derivation": "E{(\\rho_f)} = e^{\\rho_f} and 0 = - E{(\\rho_f)} + e^{\\rho_f} and \\operatorname{r_{0}}{(\\rho_f)} = - E{(\\rho_f)} + e^{\\rho_f} and 0 = \\operatorname{r_{0}}{(\\rho_f)} and \\int 0 d\\rho_f = \\int \\operatorname{r_{0}}{(\\rho_f)} d\\rho_f and e^{\\rho_f} + \\int 0 d\\rho_f = e^{\\rho_f} + \\int \\operatorname{r_{0}}{(\\rho_f)} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Function('E')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E')(Symbol('\\\\rho_f', commutative=True))), exp(Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Function('E')(Symbol('\\\\rho_f', commutative=True))), exp(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Function('r_0')(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Function('r_0')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["add", 5, "exp(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(exp(Symbol('\\\\rho_f', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\rho_f', commutative=True)))), Add(exp(Symbol('\\\\rho_f', commutative=True)), Integral(Function('r_0')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(x^\\prime,\\theta_2)} = \\theta_2 + x^\\prime, then obtain \\sigma_{p}^{\\theta_2}{(x^\\prime,\\theta_2)} - \\frac{\\partial}{\\partial \\theta_2} \\sigma_{p}{(x^\\prime,\\theta_2)} = (\\theta_2 + x^\\prime)^{\\theta_2} - \\frac{\\partial}{\\partial \\theta_2} \\sigma_{p}{(x^\\prime,\\theta_2)}", "derivation": "\\sigma_{p}{(x^\\prime,\\theta_2)} = \\theta_2 + x^\\prime and \\sigma_{p}^{\\theta_2}{(x^\\prime,\\theta_2)} = (\\theta_2 + x^\\prime)^{\\theta_2} and \\frac{\\partial}{\\partial \\theta_2} \\sigma_{p}{(x^\\prime,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + x^\\prime) and \\sigma_{p}^{\\theta_2}{(x^\\prime,\\theta_2)} - \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + x^\\prime) = (\\theta_2 + x^\\prime)^{\\theta_2} - \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + x^\\prime) and \\sigma_{p}^{\\theta_2}{(x^\\prime,\\theta_2)} - \\frac{\\partial}{\\partial \\theta_2} \\sigma_{p}{(x^\\prime,\\theta_2)} = (\\theta_2 + x^\\prime)^{\\theta_2} - \\frac{\\partial}{\\partial \\theta_2} \\sigma_{p}{(x^\\prime,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\phi_{2}{(\\psi,\\sigma_x)} = \\psi - \\sigma_x and \\mathbf{r}{(\\psi,\\sigma_x)} = \\frac{\\phi_{2}{(\\psi,\\sigma_x)} + 1}{\\psi - \\sigma_x}, then obtain \\mathbf{r}{(\\psi,\\sigma_x)} = \\frac{\\psi - \\sigma_x + 1}{\\psi - \\sigma_x}", "derivation": "\\phi_{2}{(\\psi,\\sigma_x)} = \\psi - \\sigma_x and \\phi_{2}{(\\psi,\\sigma_x)} + 1 = \\psi - \\sigma_x + 1 and \\frac{\\phi_{2}{(\\psi,\\sigma_x)} + 1}{\\psi - \\sigma_x} = \\frac{\\psi - \\sigma_x + 1}{\\psi - \\sigma_x} and \\mathbf{r}{(\\psi,\\sigma_x)} = \\frac{\\phi_{2}{(\\psi,\\sigma_x)} + 1}{\\psi - \\sigma_x} and \\mathbf{r}{(\\psi,\\sigma_x)} = \\frac{\\psi - \\sigma_x + 1}{\\psi - \\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Integer(1)))"], [["divide", 2, "Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1))), Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\nabla)} = e^{\\nabla}, then derive \\int \\lambda{(\\nabla)} d\\nabla = C + e^{\\nabla}, then obtain C + \\lambda{(\\nabla)} + \\frac{1}{\\hat{H}_l} = S + e^{\\nabla} + \\frac{1}{\\hat{H}_l}", "derivation": "\\lambda{(\\nabla)} = e^{\\nabla} and \\int \\lambda{(\\nabla)} d\\nabla = \\int e^{\\nabla} d\\nabla and \\int \\lambda{(\\nabla)} d\\nabla = C + e^{\\nabla} and \\int \\lambda{(\\nabla)} d\\nabla = C + \\lambda{(\\nabla)} and C + \\lambda{(\\nabla)} = \\int e^{\\nabla} d\\nabla and C + e^{\\nabla} = \\int e^{\\nabla} d\\nabla and C + e^{\\nabla} + \\frac{1}{\\hat{H}_l} = \\int e^{\\nabla} d\\nabla + \\frac{1}{\\hat{H}_l} and C + \\lambda{(\\nabla)} + \\frac{1}{\\hat{H}_l} = \\int e^{\\nabla} d\\nabla + \\frac{1}{\\hat{H}_l} and C + \\lambda{(\\nabla)} + \\frac{1}{\\hat{H}_l} = S + e^{\\nabla} + \\frac{1}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('C', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('C', commutative=True), Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('C', commutative=True), Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('C', commutative=True), exp(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 6, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))"], "Equality(Add(Symbol('C', commutative=True), exp(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Add(Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Symbol('C', commutative=True), Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Add(Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"], [["evaluate_integrals", 8], "Equality(Add(Symbol('C', commutative=True), Function('\\\\lambda')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Add(Symbol('S', commutative=True), exp(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(b,W)} = - W + b and \\operatorname{n_{2}}{(b,W)} = \\mathbf{J}_P^{b}{(b,W)}, then obtain \\frac{\\partial^{2}}{\\partial W\\partial b} \\operatorname{n_{2}}{(b,W)} = \\frac{\\partial^{2}}{\\partial W\\partial b} (- W + b)^{b}", "derivation": "\\mathbf{J}_P{(b,W)} = - W + b and \\mathbf{J}_P^{b}{(b,W)} = (- W + b)^{b} and \\frac{\\partial}{\\partial b} \\mathbf{J}_P^{b}{(b,W)} = \\frac{\\partial}{\\partial b} (- W + b)^{b} and \\operatorname{n_{2}}{(b,W)} = \\mathbf{J}_P^{b}{(b,W)} and \\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)} = \\frac{\\partial}{\\partial b} (- W + b)^{b} and \\frac{\\partial^{2}}{\\partial W\\partial b} \\operatorname{n_{2}}{(b,W)} = \\frac{\\partial^{2}}{\\partial W\\partial b} (- W + b)^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(\\mathbf{s},E_{\\lambda})} = \\sin{(E_{\\lambda}^{\\mathbf{s}})} and r{(\\mathbf{s},E_{\\lambda})} = \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s}, then obtain r{(\\mathbf{s},E_{\\lambda})} - \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s} = \\int u{(\\mathbf{s},E_{\\lambda})} d\\mathbf{s} - \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s}", "derivation": "u{(\\mathbf{s},E_{\\lambda})} = \\sin{(E_{\\lambda}^{\\mathbf{s}})} and \\int u{(\\mathbf{s},E_{\\lambda})} d\\mathbf{s} = \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s} and r{(\\mathbf{s},E_{\\lambda})} = \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s} and r{(\\mathbf{s},E_{\\lambda})} = \\int u{(\\mathbf{s},E_{\\lambda})} d\\mathbf{s} and r{(\\mathbf{s},E_{\\lambda})} - \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s} = \\int u{(\\mathbf{s},E_{\\lambda})} d\\mathbf{s} - \\int \\sin{(E_{\\lambda}^{\\mathbf{s}})} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Function('u')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 4, "Integral(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(Integral(Function('u')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Integral(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given L{(\\mathbf{S},s)} = \\mathbf{S} s and \\sigma_{p}{(\\mathbf{S},s)} = \\mathbf{S}^{2} s, then obtain \\mathbf{S} L{(\\mathbf{S},s)} = \\sigma_{p}{(\\mathbf{S},s)}", "derivation": "L{(\\mathbf{S},s)} = \\mathbf{S} s and \\mathbf{S} L{(\\mathbf{S},s)} = \\mathbf{S}^{2} s and \\sigma_{p}{(\\mathbf{S},s)} = \\mathbf{S}^{2} s and \\mathbf{S} L{(\\mathbf{S},s)} = \\sigma_{p}{(\\mathbf{S},s)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('L')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('L')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True))), Function('\\\\sigma_p')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(m_{s})} = \\log{(\\cos{(m_{s})})}, then obtain \\frac{e^{4 \\mathbf{g}{(m_{s})}}}{\\cos^{2}{(m_{s})}} = e^{2 \\mathbf{g}{(m_{s})}}", "derivation": "\\mathbf{g}{(m_{s})} = \\log{(\\cos{(m_{s})})} and 2 \\mathbf{g}{(m_{s})} = \\mathbf{g}{(m_{s})} + \\log{(\\cos{(m_{s})})} and e^{2 \\mathbf{g}{(m_{s})}} = e^{\\mathbf{g}{(m_{s})}} \\cos{(m_{s})} and \\frac{e^{2 \\mathbf{g}{(m_{s})}}}{\\cos{(m_{s})}} = e^{\\mathbf{g}{(m_{s})}} and \\frac{e^{4 \\mathbf{g}{(m_{s})}}}{\\cos^{2}{(m_{s})}} = e^{2 \\mathbf{g}{(m_{s})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)), log(cos(Symbol('m_s', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True))), Add(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)), log(cos(Symbol('m_s', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)))), Mul(exp(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True))), cos(Symbol('m_s', commutative=True))))"], [["divide", 3, "cos(Symbol('m_s', commutative=True))"], "Equality(Mul(exp(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)))), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1))), exp(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(exp(Mul(Integer(4), Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)))), Pow(cos(Symbol('m_s', commutative=True)), Integer(-2))), exp(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})}, then obtain ((\\frac{d}{d \\varepsilon} \\int \\operatorname{v_{z}}{(\\varepsilon)} d\\varepsilon)^{\\varepsilon})^{\\varepsilon} = ((\\frac{d}{d \\varepsilon} \\int \\log{(\\cos{(\\varepsilon)})} d\\varepsilon)^{\\varepsilon})^{\\varepsilon}", "derivation": "\\operatorname{v_{z}}{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})} and \\int \\operatorname{v_{z}}{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\cos{(\\varepsilon)})} d\\varepsilon and \\frac{d}{d \\varepsilon} \\int \\operatorname{v_{z}}{(\\varepsilon)} d\\varepsilon = \\frac{d}{d \\varepsilon} \\int \\log{(\\cos{(\\varepsilon)})} d\\varepsilon and (\\frac{d}{d \\varepsilon} \\int \\operatorname{v_{z}}{(\\varepsilon)} d\\varepsilon)^{\\varepsilon} = (\\frac{d}{d \\varepsilon} \\int \\log{(\\cos{(\\varepsilon)})} d\\varepsilon)^{\\varepsilon} and ((\\frac{d}{d \\varepsilon} \\int \\operatorname{v_{z}}{(\\varepsilon)} d\\varepsilon)^{\\varepsilon})^{\\varepsilon} = ((\\frac{d}{d \\varepsilon} \\int \\log{(\\cos{(\\varepsilon)})} d\\varepsilon)^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\varepsilon', commutative=True)), log(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integral(Function('v_z')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('v_z')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(Integral(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Derivative(Integral(Function('v_z')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Derivative(Integral(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(m,v_{2})} = m^{v_{2}} and \\theta_{1}{(m,v_{2})} = \\int \\operatorname{A_{1}}{(m,v_{2})} dm, then obtain \\theta_{1}{(m,v_{2})} = \\int m^{v_{2}} dm", "derivation": "\\operatorname{A_{1}}{(m,v_{2})} = m^{v_{2}} and \\int \\operatorname{A_{1}}{(m,v_{2})} dm = \\int m^{v_{2}} dm and \\theta_{1}{(m,v_{2})} = \\int \\operatorname{A_{1}}{(m,v_{2})} dm and \\theta_{1}{(m,v_{2})} = \\int m^{v_{2}} dm", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integral(Function('A_1')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\theta_1')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integral(Pow(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(u,\\mathbf{J})} = \\log{(\\mathbf{J} - u)} and \\mathbf{v}{(u,\\mathbf{J})} = \\log{(\\mathbf{J} - u)}, then obtain 1 = e^{- (\\mathbf{v}{(u,\\mathbf{J})} - \\operatorname{t_{2}}{(u,\\mathbf{J})}) \\operatorname{t_{2}}{(u,\\mathbf{J})}}", "derivation": "\\operatorname{t_{2}}{(u,\\mathbf{J})} = \\log{(\\mathbf{J} - u)} and 0 = - \\operatorname{t_{2}}{(u,\\mathbf{J})} + \\log{(\\mathbf{J} - u)} and \\mathbf{v}{(u,\\mathbf{J})} = \\log{(\\mathbf{J} - u)} and 0 = \\mathbf{v}{(u,\\mathbf{J})} - \\operatorname{t_{2}}{(u,\\mathbf{J})} and 0 = - (\\mathbf{v}{(u,\\mathbf{J})} - \\operatorname{t_{2}}{(u,\\mathbf{J})}) \\log{(\\mathbf{J} - u)} and 0 = - (\\mathbf{v}{(u,\\mathbf{J})} - \\operatorname{t_{2}}{(u,\\mathbf{J})}) \\operatorname{t_{2}}{(u,\\mathbf{J})} and 1 = e^{- (\\mathbf{v}{(u,\\mathbf{J})} - \\operatorname{t_{2}}{(u,\\mathbf{J})}) \\operatorname{t_{2}}{(u,\\mathbf{J})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))))"], [["minus", 1, "Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('\\\\mathbf{v}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))))"], "Equality(Integer(0), Mul(Integer(-1), Add(Function('\\\\mathbf{v}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Mul(Integer(-1), Add(Function('\\\\mathbf{v}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["exp", 6], "Equality(Integer(1), exp(Mul(Integer(-1), Add(Function('\\\\mathbf{v}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Function('t_2')(Symbol('u', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(r,n_{1},\\mathbf{s})} = \\mathbf{s} + n_{1} + r and I{(\\mathbf{s})} = \\int 1 d\\mathbf{s}, then obtain 1 - I{(\\mathbf{s})} = \\frac{\\mathbf{s} + n_{1} + r}{\\dot{x}{(r,n_{1},\\mathbf{s})}} - I{(\\mathbf{s})}", "derivation": "\\dot{x}{(r,n_{1},\\mathbf{s})} = \\mathbf{s} + n_{1} + r and 1 = \\frac{\\mathbf{s} + n_{1} + r}{\\dot{x}{(r,n_{1},\\mathbf{s})}} and 1 - \\int 1 d\\mathbf{s} = \\frac{\\mathbf{s} + n_{1} + r}{\\dot{x}{(r,n_{1},\\mathbf{s})}} - \\int 1 d\\mathbf{s} and I{(\\mathbf{s})} = \\int 1 d\\mathbf{s} and 1 - I{(\\mathbf{s})} = \\frac{\\mathbf{s} + n_{1} + r}{\\dot{x}{(r,n_{1},\\mathbf{s})}} - I{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('r', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True), Symbol('r', commutative=True)))"], [["divide", 1, "Function('\\\\dot{x}')(Symbol('r', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('r', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], [["minus", 2, "Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('r', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbf{s}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Integer(1), Mul(Integer(-1), Function('I')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('r', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('I')(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given g{(\\phi)} = \\sin{(\\phi)}, then obtain \\int \\phi g{(\\phi)} d\\phi = \\Omega - \\phi \\cos{(\\phi)} + \\sin{(\\phi)}", "derivation": "g{(\\phi)} = \\sin{(\\phi)} and \\phi g{(\\phi)} = \\phi \\sin{(\\phi)} and \\int \\phi g{(\\phi)} d\\phi = \\int \\phi \\sin{(\\phi)} d\\phi and \\int \\phi g{(\\phi)} d\\phi = \\Omega - \\phi \\cos{(\\phi)} + \\sin{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('g')(Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), sin(Symbol('\\\\phi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('g')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('g')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), sin(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\sin{(x^\\prime)} and \\operatorname{A_{z}}{(x^\\prime)} = \\sin{(x^\\prime)}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} dx^\\prime = I - \\cos{(x^\\prime)}, then obtain (I - \\cos{(x^\\prime)})^{x^\\prime} = (\\int \\sin{(x^\\prime)} dx^\\prime)^{x^\\prime}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\sin{(x^\\prime)} and \\operatorname{A_{z}}{(x^\\prime)} = \\sin{(x^\\prime)} and \\int \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} dx^\\prime = \\int \\sin{(x^\\prime)} dx^\\prime and \\int \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} dx^\\prime = I - \\cos{(x^\\prime)} and \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\operatorname{A_{z}}{(x^\\prime)} and \\int \\operatorname{A_{z}}{(x^\\prime)} dx^\\prime = \\int \\sin{(x^\\prime)} dx^\\prime and (\\int \\operatorname{A_{z}}{(x^\\prime)} dx^\\prime)^{x^\\prime} = (\\int \\sin{(x^\\prime)} dx^\\prime)^{x^\\prime} and (\\int \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} dx^\\prime)^{x^\\prime} = (\\int \\sin{(x^\\prime)} dx^\\prime)^{x^\\prime} and (I - \\cos{(x^\\prime)})^{x^\\prime} = (\\int \\sin{(x^\\prime)} dx^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Function('A_z')(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Function('A_z')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["power", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integral(Function('A_z')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Pow(Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(C)} = \\log{(C)} and \\operatorname{P_{e}}{(C)} = \\log{(C)}, then obtain \\frac{l \\int \\operatorname{f_{E}}{(C)} dC}{\\frac{d}{d l} \\frac{1}{\\sin{(n_{2})}}} = \\frac{l \\int \\operatorname{P_{e}}{(C)} dC}{\\frac{d}{d l} \\frac{1}{\\sin{(n_{2})}}}", "derivation": "\\operatorname{f_{E}}{(C)} = \\log{(C)} and \\int \\operatorname{f_{E}}{(C)} dC = \\int \\log{(C)} dC and \\operatorname{P_{e}}{(C)} = \\log{(C)} and \\operatorname{P_{e}}{(C)} = \\operatorname{f_{E}}{(C)} and \\int \\operatorname{P_{e}}{(C)} dC = \\int \\log{(C)} dC and \\int \\operatorname{f_{E}}{(C)} dC = \\int \\operatorname{P_{e}}{(C)} dC and l \\int \\operatorname{f_{E}}{(C)} dC = l \\int \\operatorname{P_{e}}{(C)} dC and \\frac{l \\int \\operatorname{f_{E}}{(C)} dC}{\\frac{d}{d l} \\frac{1}{\\sin{(n_{2})}}} = \\frac{l \\int \\operatorname{P_{e}}{(C)} dC}{\\frac{d}{d l} \\frac{1}{\\sin{(n_{2})}}}", "srepr_derivation": [["get_premise", "Equality(Function('f_E')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('P_e')(Symbol('C', commutative=True)), Function('f_E')(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Integral(Function('f_E')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["times", 6, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Integral(Function('f_E')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('l', commutative=True), Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["divide", 7, "Derivative(Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('l', commutative=True), Pow(Derivative(Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Function('f_E')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('l', commutative=True), Pow(Derivative(Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\omega)} = \\omega, then derive \\int \\operatorname{g_{\\varepsilon}}{(\\omega)} d\\omega = \\hat{\\mathbf{r}} + \\frac{\\omega^{2}}{2}, then obtain W + \\frac{\\omega^{2}}{2} = \\hat{\\mathbf{r}} + \\frac{\\omega^{2}}{2}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\omega)} = \\omega and \\int \\operatorname{g_{\\varepsilon}}{(\\omega)} d\\omega = \\int \\omega d\\omega and \\int \\operatorname{g_{\\varepsilon}}{(\\omega)} d\\omega = \\hat{\\mathbf{r}} + \\frac{\\omega^{2}}{2} and \\int \\omega d\\omega = \\hat{\\mathbf{r}} + \\frac{\\omega^{2}}{2} and W + \\frac{\\omega^{2}}{2} = \\hat{\\mathbf{r}} + \\frac{\\omega^{2}}{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('W', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then obtain \\int (\\mathbb{I}{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} + 1) d\\hat{\\mathbf{x}} = \\int 1 d\\hat{\\mathbf{x}}", "derivation": "\\mathbb{I}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and \\mathbb{I}{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} = 0 and \\mathbb{I}{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} + 1 = 1 and \\int (\\mathbb{I}{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} + 1) d\\hat{\\mathbf{x}} = \\int 1 d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(1)), Integer(1))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(L)} = e^{\\cos{(L)}}, then obtain - \\frac{4 \\operatorname{v_{1}}^{2}{(L)} e^{- 2 \\cos{(L)}}}{\\cos{(L)}} = - \\frac{(\\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}})^{2} e^{- 2 \\cos{(L)}}}{\\cos{(L)}}", "derivation": "\\operatorname{v_{1}}{(L)} = e^{\\cos{(L)}} and 2 \\operatorname{v_{1}}{(L)} = \\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}} and 2 \\operatorname{v_{1}}{(L)} e^{- \\cos{(L)}} = (\\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}}) e^{- \\cos{(L)}} and 4 \\operatorname{v_{1}}^{2}{(L)} e^{- 2 \\cos{(L)}} = (\\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}})^{2} e^{- 2 \\cos{(L)}} and - \\frac{4 \\operatorname{v_{1}}^{2}{(L)} e^{- 2 \\cos{(L)}}}{\\cos{(L)}} = - \\frac{(\\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}})^{2} e^{- 2 \\cos{(L)}}}{\\cos{(L)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True))))"], [["add", 1, "Function('v_1')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('v_1')(Symbol('L', commutative=True))), Add(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True)))))"], [["divide", 2, "exp(cos(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(2), Function('v_1')(Symbol('L', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('L', commutative=True))))), Mul(Add(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True)))), exp(Mul(Integer(-1), cos(Symbol('L', commutative=True))))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('v_1')(Symbol('L', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('L', commutative=True))))), Mul(Pow(Add(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True)))), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('L', commutative=True))))))"], [["divide", 4, "Mul(Integer(-1), cos(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(4), Pow(Function('v_1')(Symbol('L', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('L', commutative=True)))), Pow(cos(Symbol('L', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True)))), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('L', commutative=True)))), Pow(cos(Symbol('L', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} = - F_{g} + \\hat{H} + v, then derive \\frac{\\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H}}{F_{g}} = \\frac{\\frac{\\hat{H}^{2}}{2} + \\hat{H} (- F_{g} + v) + p}{F_{g}}, then obtain (\\frac{\\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H}}{F_{g}})^{F_{g}} = (\\frac{\\frac{\\hat{H}^{2}}{2} + \\hat{H} (- F_{g} + v) + p}{F_{g}})^{F_{g}}", "derivation": "\\operatorname{v_{x}}{(F_{g},v,\\hat{H})} = - F_{g} + \\hat{H} + v and \\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H} = \\int (- F_{g} + \\hat{H} + v) d\\hat{H} and \\frac{\\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H}}{F_{g}} = \\frac{\\int (- F_{g} + \\hat{H} + v) d\\hat{H}}{F_{g}} and \\frac{\\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H}}{F_{g}} = \\frac{\\frac{\\hat{H}^{2}}{2} + \\hat{H} (- F_{g} + v) + p}{F_{g}} and (\\frac{\\int \\operatorname{v_{x}}{(F_{g},v,\\hat{H})} d\\hat{H}}{F_{g}})^{F_{g}} = (\\frac{\\frac{\\hat{H}^{2}}{2} + \\hat{H} (- F_{g} + v) + p}{F_{g}})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('F_g', commutative=True), Symbol('v', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('F_g', commutative=True), Symbol('v', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Integral(Function('v_x')(Symbol('F_g', commutative=True), Symbol('v', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Integral(Function('v_x')(Symbol('F_g', commutative=True), Symbol('v', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('v', commutative=True))), Symbol('p', commutative=True))))"], [["power", 4, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Integral(Function('v_x')(Symbol('F_g', commutative=True), Symbol('v', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Symbol('F_g', commutative=True)), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('v', commutative=True))), Symbol('p', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})} = C_{d} \\hat{x}, then obtain 0 = \\frac{C_{d}^{2}}{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})}} - C_{d}", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{x},C_{d})} = C_{d} \\hat{x} and \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})} = \\frac{\\partial}{\\partial \\hat{x}} C_{d} \\hat{x} and C_{d} \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})} = C_{d} \\frac{\\partial}{\\partial \\hat{x}} C_{d} \\hat{x} and C_{d} = \\frac{C_{d} \\frac{\\partial}{\\partial \\hat{x}} C_{d} \\hat{x}}{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})}} and 0 = \\frac{C_{d} \\frac{\\partial}{\\partial \\hat{x}} C_{d} \\hat{x}}{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})}} - C_{d} and 0 = \\frac{C_{d}^{2}}{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{y^{\\prime}}{(\\hat{x},C_{d})}} - C_{d}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('C_d', commutative=True)"], "Equality(Mul(Symbol('C_d', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(Symbol('C_d', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["divide", 3, "Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))"], "Equality(Symbol('C_d', commutative=True), Mul(Symbol('C_d', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 4, "Symbol('C_d', commutative=True)"], "Equality(Integer(0), Add(Mul(Symbol('C_d', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Symbol('C_d', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Add(Mul(Pow(Symbol('C_d', commutative=True), Integer(2)), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)}, then obtain \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{g}{(\\mathbf{J}_f)} + \\sin{(\\mathbf{J}_f)}} = \\frac{d}{d \\mathbf{J}_f} e^{2 \\sin{(\\mathbf{J}_f)}}", "derivation": "\\mathbf{g}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)} and \\mathbf{g}{(\\mathbf{J}_f)} + \\sin{(\\mathbf{J}_f)} = 2 \\sin{(\\mathbf{J}_f)} and e^{\\mathbf{g}{(\\mathbf{J}_f)} + \\sin{(\\mathbf{J}_f)}} = e^{2 \\sin{(\\mathbf{J}_f)}} and \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{g}{(\\mathbf{J}_f)} + \\sin{(\\mathbf{J}_f)}} = \\frac{d}{d \\mathbf{J}_f} e^{2 \\sin{(\\mathbf{J}_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["exp", 2], "Equality(exp(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(exp(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(\\Psi_{nl},b)} = \\cos{(\\Psi_{nl}^{b})} and \\mu_{0}{(\\Psi_{nl},b)} = \\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}}, then obtain \\mu_{0}^{b}{(\\Psi_{nl},b)} = (\\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}})^{b}", "derivation": "\\ddot{x}{(\\Psi_{nl},b)} = \\cos{(\\Psi_{nl}^{b})} and \\mu_{0}{(\\Psi_{nl},b)} = \\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}} and \\mu_{0}{(\\Psi_{nl},b)} = \\frac{1}{\\cos^{2}{(\\Psi_{nl}^{b})}} and \\frac{1}{\\cos^{2}{(\\Psi_{nl}^{b})}} = \\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}} and (\\frac{1}{\\cos^{2}{(\\Psi_{nl}^{b})}})^{b} = (\\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}})^{b} and (\\frac{1}{\\cos^{2}{(\\Psi_{nl}^{b})}})^{b} = \\mu_{0}^{b}{(\\Psi_{nl},b)} and \\mu_{0}^{b}{(\\Psi_{nl},b)} = (\\frac{1}{\\ddot{x}^{2}{(\\Psi_{nl},b)}})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), cos(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Integer(-2)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mu_0')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Pow(cos(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True))), Integer(-2)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(cos(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True))), Integer(-2)), Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Integer(-2)))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(cos(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True))), Integer(-2)), Symbol('b', commutative=True)), Pow(Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Integer(-2)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Pow(cos(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True))), Integer(-2)), Symbol('b', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('b', commutative=True)), Integer(-2)), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\Omega,s)} = \\frac{s}{\\Omega}, then obtain \\int \\frac{\\eta{(\\Omega,s)}}{- s + \\frac{s}{\\Omega}} ds - \\frac{1}{- s + \\frac{s}{\\Omega}} = \\int \\frac{s}{\\Omega (- s + \\frac{s}{\\Omega})} ds - \\frac{1}{- s + \\frac{s}{\\Omega}}", "derivation": "\\eta{(\\Omega,s)} = \\frac{s}{\\Omega} and \\frac{\\eta{(\\Omega,s)}}{- s + \\frac{s}{\\Omega}} = \\frac{s}{\\Omega (- s + \\frac{s}{\\Omega})} and \\int \\frac{\\eta{(\\Omega,s)}}{- s + \\frac{s}{\\Omega}} ds = \\int \\frac{s}{\\Omega (- s + \\frac{s}{\\Omega})} ds and \\int \\frac{\\eta{(\\Omega,s)}}{- s + \\frac{s}{\\Omega}} ds - \\frac{1}{- s + \\frac{s}{\\Omega}} = \\int \\frac{s}{\\Omega (- s + \\frac{s}{\\Omega})} ds - \\frac{1}{- s + \\frac{s}{\\Omega}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True))))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1)))), Add(Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(y^{\\prime})} = \\log{(y^{\\prime})} and \\operatorname{v_{t}}{(y^{\\prime})} = y^{\\prime} and \\Psi_{\\lambda}{(y^{\\prime})} = \\frac{1}{\\dot{\\mathbf{r}}{(y^{\\prime})}}, then obtain y^{\\prime} \\Psi_{\\lambda}{(y^{\\prime})} = \\frac{y^{\\prime}}{\\dot{\\mathbf{r}}{(y^{\\prime})}}", "derivation": "\\dot{\\mathbf{r}}{(y^{\\prime})} = \\log{(y^{\\prime})} and \\operatorname{v_{t}}{(y^{\\prime})} = y^{\\prime} and \\Psi_{\\lambda}{(y^{\\prime})} = \\frac{1}{\\dot{\\mathbf{r}}{(y^{\\prime})}} and \\frac{\\Psi_{\\lambda}{(y^{\\prime})} \\operatorname{v_{t}}{(y^{\\prime})} \\log{(y^{\\prime})}}{\\dot{\\mathbf{r}}{(y^{\\prime})}} = \\frac{\\operatorname{v_{t}}{(y^{\\prime})} \\log{(y^{\\prime})}}{\\dot{\\mathbf{r}}^{2}{(y^{\\prime})}} and \\Psi_{\\lambda}{(y^{\\prime})} \\operatorname{v_{t}}{(y^{\\prime})} = \\frac{\\operatorname{v_{t}}{(y^{\\prime})}}{\\dot{\\mathbf{r}}{(y^{\\prime})}} and y^{\\prime} \\Psi_{\\lambda}{(y^{\\prime})} = \\frac{y^{\\prime}}{\\dot{\\mathbf{r}}{(y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)))"], [["times", 3, "Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('v_t')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('v_t')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-2)), Function('v_t')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Function('v_t')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('v_t')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C_{1},\\sigma_x)} = \\log{(\\sigma_x^{C_{1}})}, then obtain - \\log{(\\sigma_x^{C_{1}})} + \\frac{\\sigma_x \\operatorname{M_{E}}{(C_{1},\\sigma_x)} - \\sigma_x \\log{(\\sigma_x^{C_{1}})}}{\\sigma_x} = - \\log{(\\sigma_x^{C_{1}})}", "derivation": "\\operatorname{M_{E}}{(C_{1},\\sigma_x)} = \\log{(\\sigma_x^{C_{1}})} and \\sigma_x \\operatorname{M_{E}}{(C_{1},\\sigma_x)} = \\sigma_x \\log{(\\sigma_x^{C_{1}})} and \\sigma_x \\operatorname{M_{E}}{(C_{1},\\sigma_x)} - \\sigma_x \\log{(\\sigma_x^{C_{1}})} = 0 and \\frac{\\sigma_x \\operatorname{M_{E}}{(C_{1},\\sigma_x)} - \\sigma_x \\log{(\\sigma_x^{C_{1}})}}{\\sigma_x} = 0 and - \\log{(\\sigma_x^{C_{1}})} + \\frac{\\sigma_x \\operatorname{M_{E}}{(C_{1},\\sigma_x)} - \\sigma_x \\log{(\\sigma_x^{C_{1}})}}{\\sigma_x} = - \\log{(\\sigma_x^{C_{1}})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True))))"], [["times", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True)))))"], [["minus", 2, "Mul(Symbol('\\\\sigma_x', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True))))), Integer(0))"], [["divide", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True)))))), Integer(0))"], [["minus", 4, "log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True)))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True))))))), Mul(Integer(-1), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given Z{(\\chi)} = \\log{(\\chi)} and \\operatorname{r_{0}}{(\\chi)} = \\chi, then obtain \\frac{\\log{(\\chi)} (\\int \\operatorname{r_{0}}{(\\chi)} d\\chi)^{\\chi}}{\\int \\chi d\\chi} = \\frac{\\log{(\\chi)} (\\int \\chi d\\chi)^{\\chi}}{\\int \\chi d\\chi}", "derivation": "Z{(\\chi)} = \\log{(\\chi)} and \\operatorname{r_{0}}{(\\chi)} = \\chi and \\int \\operatorname{r_{0}}{(\\chi)} d\\chi = \\int \\chi d\\chi and (\\int \\operatorname{r_{0}}{(\\chi)} d\\chi)^{\\chi} = (\\int \\chi d\\chi)^{\\chi} and Z{(\\chi)} (\\int \\operatorname{r_{0}}{(\\chi)} d\\chi)^{\\chi} = Z{(\\chi)} (\\int \\chi d\\chi)^{\\chi} and \\frac{Z{(\\chi)} (\\int \\operatorname{r_{0}}{(\\chi)} d\\chi)^{\\chi}}{\\int \\chi d\\chi} = \\frac{Z{(\\chi)} (\\int \\chi d\\chi)^{\\chi}}{\\int \\chi d\\chi} and \\frac{\\log{(\\chi)} (\\int \\operatorname{r_{0}}{(\\chi)} d\\chi)^{\\chi}}{\\int \\chi d\\chi} = \\frac{\\log{(\\chi)} (\\int \\chi d\\chi)^{\\chi}}{\\int \\chi d\\chi}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Function('r_0')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["times", 4, "Function('Z')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Function('r_0')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Mul(Function('Z')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"], [["divide", 5, "Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Function('Z')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Pow(Integral(Function('r_0')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Mul(Function('Z')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(log(Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Pow(Integral(Function('r_0')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Mul(log(Symbol('\\\\chi', commutative=True)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Pow(Integral(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(q)} = \\cos{(q)}, then derive V - \\int q \\operatorname{v_{x}}{(q)} dq - \\int - \\operatorname{v_{x}}{(q)} dq = \\int (- q \\operatorname{v_{x}}{(q)} + \\cos{(q)}) dq, then obtain V - \\int q \\operatorname{v_{x}}{(q)} dq - \\int - \\operatorname{v_{x}}{(q)} dq = \\int (- q \\operatorname{v_{x}}{(q)} + \\operatorname{v_{x}}{(q)}) dq", "derivation": "\\operatorname{v_{x}}{(q)} = \\cos{(q)} and q \\operatorname{v_{x}}{(q)} = q \\cos{(q)} and - q \\cos{(q)} + \\operatorname{v_{x}}{(q)} = - q \\cos{(q)} + \\cos{(q)} and - q \\operatorname{v_{x}}{(q)} + \\operatorname{v_{x}}{(q)} = - q \\operatorname{v_{x}}{(q)} + \\cos{(q)} and \\int (- q \\operatorname{v_{x}}{(q)} + \\operatorname{v_{x}}{(q)}) dq = \\int (- q \\operatorname{v_{x}}{(q)} + \\cos{(q)}) dq and V - \\int q \\operatorname{v_{x}}{(q)} dq - \\int - \\operatorname{v_{x}}{(q)} dq = \\int (- q \\operatorname{v_{x}}{(q)} + \\cos{(q)}) dq and V - \\int q \\operatorname{v_{x}}{(q)} dq - \\int - \\operatorname{v_{x}}{(q)} dq = \\int (- q \\operatorname{v_{x}}{(q)} + \\operatorname{v_{x}}{(q)}) dq", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True))))"], [["minus", 1, "Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True), cos(Symbol('q', commutative=True))), Function('v_x')(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True), cos(Symbol('q', commutative=True))), cos(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Function('v_x')(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), cos(Symbol('q', commutative=True))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('V', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('V', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True), Function('v_x')(Symbol('q', commutative=True))), Function('v_x')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given V{(\\mathbf{J}_P,\\omega)} = (e^{\\mathbf{J}_P})^{\\omega}, then obtain - \\int \\frac{\\int V{(\\mathbf{J}_P,\\omega)} d\\omega}{\\int (e^{\\mathbf{J}_P})^{\\omega} d\\omega} d\\omega = - \\int 1 d\\omega", "derivation": "V{(\\mathbf{J}_P,\\omega)} = (e^{\\mathbf{J}_P})^{\\omega} and \\int V{(\\mathbf{J}_P,\\omega)} d\\omega = \\int (e^{\\mathbf{J}_P})^{\\omega} d\\omega and \\frac{\\int V{(\\mathbf{J}_P,\\omega)} d\\omega}{\\int (e^{\\mathbf{J}_P})^{\\omega} d\\omega} = 1 and \\int \\frac{\\int V{(\\mathbf{J}_P,\\omega)} d\\omega}{\\int (e^{\\mathbf{J}_P})^{\\omega} d\\omega} d\\omega = \\int 1 d\\omega and - \\int \\frac{\\int V{(\\mathbf{J}_P,\\omega)} d\\omega}{\\int (e^{\\mathbf{J}_P})^{\\omega} d\\omega} d\\omega = - \\int 1 d\\omega", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('V')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 2, "Integral(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Integral(Function('V')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Pow(Integral(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Integral(Function('V')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Pow(Integral(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\omega', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Integral(Function('V')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Pow(Integral(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J,f^{\\prime})} = J f^{\\prime}, then obtain 0^{J} = (J - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{t_{1}}{(J,f^{\\prime})})^{J}", "derivation": "\\operatorname{t_{1}}{(J,f^{\\prime})} = J f^{\\prime} and 0 = J f^{\\prime} - \\operatorname{t_{1}}{(J,f^{\\prime})} and \\frac{d}{d f^{\\prime}} 0 = \\frac{\\partial}{\\partial f^{\\prime}} (J f^{\\prime} - \\operatorname{t_{1}}{(J,f^{\\prime})}) and (\\frac{d}{d f^{\\prime}} 0)^{J} = (\\frac{\\partial}{\\partial f^{\\prime}} (J f^{\\prime} - \\operatorname{t_{1}}{(J,f^{\\prime})}))^{J} and 0^{J} = (J - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{t_{1}}{(J,f^{\\prime})})^{J}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(Add(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Integer(0), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Derivative(Function('t_1')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\lambda{(A,z)} = \\sin{(A z)}, then obtain (\\lambda{(A,z)} + \\int \\sin{(A z)} dA)^{2} = (\\lambda{(A,z)} + \\int \\sin{(A z)} dA) (\\sin{(A z)} + \\int \\sin{(A z)} dA)", "derivation": "\\lambda{(A,z)} = \\sin{(A z)} and \\int \\lambda{(A,z)} dA = \\int \\sin{(A z)} dA and \\lambda{(A,z)} + \\int \\lambda{(A,z)} dA = \\sin{(A z)} + \\int \\lambda{(A,z)} dA and \\lambda{(A,z)} + \\int \\sin{(A z)} dA = \\sin{(A z)} + \\int \\sin{(A z)} dA and (\\lambda{(A,z)} + \\int \\sin{(A z)} dA)^{2} = (\\lambda{(A,z)} + \\int \\sin{(A z)} dA) (\\sin{(A z)} + \\int \\sin{(A z)} dA)", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["add", 1, "Integral(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Integral(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Integral(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True)))), Add(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["times", 4, "Add(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True))))"], "Equality(Pow(Add(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True)))), Integer(2)), Mul(Add(Function('\\\\lambda')(Symbol('A', commutative=True), Symbol('z', commutative=True)), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True)))), Add(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('A', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(h,A_{y})} = A_{y} h and \\Omega{(A_{y})} = A_{y}, then obtain \\log{(h + 2 \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})}^{h} = \\log{(A_{y} + h + \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})}^{h}", "derivation": "\\operatorname{a^{\\dagger}}{(h,A_{y})} = A_{y} h and \\Omega{(A_{y})} = A_{y} and - A_{y} h + h + \\Omega{(A_{y})} = - A_{y} h + A_{y} + h and - A_{y} h + h + 2 \\Omega{(A_{y})} = - A_{y} h + A_{y} + h + \\Omega{(A_{y})} and \\log{(- A_{y} h + h + 2 \\Omega{(A_{y})})} = \\log{(- A_{y} h + A_{y} + h + \\Omega{(A_{y})})} and \\log{(h + 2 \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})} = \\log{(A_{y} + h + \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})} and \\log{(h + 2 \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})}^{h} = \\log{(A_{y} + h + \\Omega{(A_{y})} - \\operatorname{a^{\\dagger}}{(h,A_{y})})}^{h}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["minus", 2, "Add(Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('A_y', commutative=True), Symbol('h', commutative=True)))"], [["add", 3, "Function('\\\\Omega')(Symbol('A_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\Omega')(Symbol('A_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('A_y', commutative=True), Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('A_y', commutative=True))))"], [["log", 4], "Equality(log(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\Omega')(Symbol('A_y', commutative=True))))), log(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Symbol('A_y', commutative=True), Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('A_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(log(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\Omega')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('A_y', commutative=True))))), log(Add(Symbol('A_y', commutative=True), Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('A_y', commutative=True))))))"], [["power", 6, "Symbol('h', commutative=True)"], "Equality(Pow(log(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\Omega')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('A_y', commutative=True))))), Symbol('h', commutative=True)), Pow(log(Add(Symbol('A_y', commutative=True), Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('A_y', commutative=True))))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(L)} = \\sin{(L)}, then obtain (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + (\\int (- \\varphi^{*}{(L)} + \\sin{(L)}) dL)^{L} = (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + 1", "derivation": "\\varphi^{*}{(L)} = \\sin{(L)} and 0 = - \\varphi^{*}{(L)} + \\sin{(L)} and \\int 0 dL = \\int (- \\varphi^{*}{(L)} + \\sin{(L)}) dL and (\\int 0 dL)^{L} = (\\int (- \\varphi^{*}{(L)} + \\sin{(L)}) dL)^{L} and (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + (\\int 0 dL)^{L} = (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + (\\int (- \\varphi^{*}{(L)} + \\sin{(L)}) dL)^{L} and (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + (\\int (- \\varphi^{*}{(L)} + \\sin{(L)}) dL)^{L} = (- \\varphi^{*}{(L)} + \\sin{(L)}) \\sin{(L)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["minus", 1, "Function('\\\\varphi^*')(Symbol('L', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["add", 4, "Mul(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Pow(Integral(Integer(0), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), sin(Symbol('L', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and \\mathbf{f}{(g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})}, then obtain - g^{\\prime}_{\\varepsilon} + \\mathbf{f}{(g^{\\prime}_{\\varepsilon})} = - 2 g^{\\prime}_{\\varepsilon} + \\sin{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and - g^{\\prime}_{\\varepsilon} + \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\sin{(g^{\\prime}_{\\varepsilon})} and \\mathbf{f}{(g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} and \\mathbf{f}{(g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\sin{(g^{\\prime}_{\\varepsilon})} and - g^{\\prime}_{\\varepsilon} + \\mathbf{f}{(g^{\\prime}_{\\varepsilon})} = - 2 g^{\\prime}_{\\varepsilon} + \\sin{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{f}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v)} = \\int \\cos{(v)} dv, then derive - v + \\operatorname{x^{{\\}'}}{(v)} = \\psi - v + \\sin{(v)}, then derive \\psi - v + \\sin{(v)} = \\pi - v + \\sin{(v)}, then derive (\\tilde{g} - v + \\sin{(v)})^{\\psi} = (\\psi - v + \\sin{(v)})^{\\psi}, then obtain (\\tilde{g} - v + \\sin{(v)})^{\\psi} = (\\pi - v + \\sin{(v)})^{\\psi}", "derivation": "\\operatorname{x^{{\\}'}}{(v)} = \\int \\cos{(v)} dv and - v + \\operatorname{x^{{\\}'}}{(v)} = - v + \\int \\cos{(v)} dv and - v + \\operatorname{x^{{\\}'}}{(v)} = \\psi - v + \\sin{(v)} and \\psi - v + \\sin{(v)} = - v + \\int \\cos{(v)} dv and (- v + \\operatorname{x^{{\\}'}}{(v)})^{\\psi} = (\\psi - v + \\sin{(v)})^{\\psi} and \\psi - v + \\sin{(v)} = \\pi - v + \\sin{(v)} and (- v + \\int \\cos{(v)} dv)^{\\psi} = (\\psi - v + \\sin{(v)})^{\\psi} and (\\tilde{g} - v + \\sin{(v)})^{\\psi} = (\\psi - v + \\sin{(v)})^{\\psi} and (\\tilde{g} - v + \\sin{(v)})^{\\psi} = (\\pi - v + \\sin{(v)})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('x^\\\\prime')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('x^\\\\prime')(Symbol('v', commutative=True))), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('x^\\\\prime')(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Symbol('\\\\psi', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given c{(t_{2})} = e^{t_{2}}, then obtain \\frac{d}{d t_{2}} (c{(t_{2})} - \\frac{d}{d t_{2}} e^{t_{2}}) = \\frac{d}{d t_{2}} (e^{t_{2}} - \\frac{d}{d t_{2}} e^{t_{2}})", "derivation": "c{(t_{2})} = e^{t_{2}} and \\frac{d}{d t_{2}} c{(t_{2})} = \\frac{d}{d t_{2}} e^{t_{2}} and c{(t_{2})} - \\frac{d}{d t_{2}} c{(t_{2})} = e^{t_{2}} - \\frac{d}{d t_{2}} c{(t_{2})} and \\frac{d}{d t_{2}} (c{(t_{2})} - \\frac{d}{d t_{2}} c{(t_{2})}) = \\frac{d}{d t_{2}} (e^{t_{2}} - \\frac{d}{d t_{2}} c{(t_{2})}) and \\frac{d}{d t_{2}} (c{(t_{2})} - \\frac{d}{d t_{2}} e^{t_{2}}) = \\frac{d}{d t_{2}} (e^{t_{2}} - \\frac{d}{d t_{2}} e^{t_{2}})", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))"], "Equality(Add(Function('c')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), Add(exp(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Function('c')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(Function('c')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Function('c')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(\\phi_2,\\sigma_x)} = \\phi_2 - \\sigma_x, then obtain - \\phi_2 + \\sigma_x + \\frac{\\partial}{\\partial \\phi_2} (\\sigma_x + \\psi^{*}{(\\phi_2,\\sigma_x)}) = - \\phi_2 + \\sigma_x + \\frac{d}{d \\phi_2} \\phi_2", "derivation": "\\psi^{*}{(\\phi_2,\\sigma_x)} = \\phi_2 - \\sigma_x and \\sigma_x + \\psi^{*}{(\\phi_2,\\sigma_x)} = \\phi_2 and \\frac{\\partial}{\\partial \\phi_2} (\\sigma_x + \\psi^{*}{(\\phi_2,\\sigma_x)}) = \\frac{d}{d \\phi_2} \\phi_2 and - \\phi_2 + \\sigma_x + \\frac{\\partial}{\\partial \\phi_2} (\\sigma_x + \\psi^{*}{(\\phi_2,\\sigma_x)}) = - \\phi_2 + \\sigma_x + \\frac{d}{d \\phi_2} \\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{2})} = \\cos{(C_{2})}, then derive \\frac{d}{d C_{2}} \\Psi^{\\dagger}{(C_{2})} = - \\sin{(C_{2})}, then obtain \\cos{(\\int \\cos{(C_{2})} dC_{2})} + \\frac{d}{d C_{2}} \\cos{(C_{2})} = - \\sin{(C_{2})} + \\cos{(\\int \\cos{(C_{2})} dC_{2})}", "derivation": "\\Psi^{\\dagger}{(C_{2})} = \\cos{(C_{2})} and \\int \\Psi^{\\dagger}{(C_{2})} dC_{2} = \\int \\cos{(C_{2})} dC_{2} and \\frac{d}{d C_{2}} \\Psi^{\\dagger}{(C_{2})} = \\frac{d}{d C_{2}} \\cos{(C_{2})} and \\frac{d}{d C_{2}} \\Psi^{\\dagger}{(C_{2})} = - \\sin{(C_{2})} and \\frac{d}{d C_{2}} \\cos{(C_{2})} = - \\sin{(C_{2})} and \\cos{(\\int \\Psi^{\\dagger}{(C_{2})} dC_{2})} + \\frac{d}{d C_{2}} \\cos{(C_{2})} = - \\sin{(C_{2})} + \\cos{(\\int \\Psi^{\\dagger}{(C_{2})} dC_{2})} and \\cos{(\\int \\cos{(C_{2})} dC_{2})} + \\frac{d}{d C_{2}} \\cos{(C_{2})} = - \\sin{(C_{2})} + \\cos{(\\int \\cos{(C_{2})} dC_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"], [["add", 5, "cos(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], "Equality(Add(cos(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), cos(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(cos(Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), cos(Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{2},c)} = c v_{2}, then obtain (c \\operatorname{y^{\\prime}}^{2}{(v_{2},c)})^{v_{2}} = (c^{3} v_{2}^{2})^{v_{2}}", "derivation": "\\operatorname{y^{\\prime}}{(v_{2},c)} = c v_{2} and c \\operatorname{y^{\\prime}}{(v_{2},c)} = c^{2} v_{2} and c^{2} v_{2} \\operatorname{y^{\\prime}}{(v_{2},c)} = c^{3} v_{2}^{2} and c \\operatorname{y^{\\prime}}^{2}{(v_{2},c)} = c^{2} v_{2} \\operatorname{y^{\\prime}}{(v_{2},c)} and c \\operatorname{y^{\\prime}}^{2}{(v_{2},c)} = c^{3} v_{2}^{2} and (c \\operatorname{y^{\\prime}}^{2}{(v_{2},c)})^{v_{2}} = (c^{3} v_{2}^{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True))), Mul(Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_2', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_2', commutative=True))"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True))), Mul(Pow(Symbol('c', commutative=True), Integer(3)), Pow(Symbol('v_2', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('c', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True)), Integer(2))), Mul(Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('c', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True)), Integer(2))), Mul(Pow(Symbol('c', commutative=True), Integer(3)), Pow(Symbol('v_2', commutative=True), Integer(2))))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(Mul(Symbol('c', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('c', commutative=True)), Integer(2))), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('c', commutative=True), Integer(3)), Pow(Symbol('v_2', commutative=True), Integer(2))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(n_{1},\\omega)} = \\cos^{\\omega}{(n_{1})} and \\operatorname{L_{\\varepsilon}}{(n_{1})} = 0^{n_{1}}, then obtain 0^{n_{1}} \\cos^{\\omega}{(n_{1})} = (- \\Psi_{\\lambda}{(n_{1},\\omega)} + \\cos^{\\omega}{(n_{1})})^{n_{1}} \\cos^{\\omega}{(n_{1})}", "derivation": "\\Psi_{\\lambda}{(n_{1},\\omega)} = \\cos^{\\omega}{(n_{1})} and 0 = - \\Psi_{\\lambda}{(n_{1},\\omega)} + \\cos^{\\omega}{(n_{1})} and 0^{n_{1}} = (- \\Psi_{\\lambda}{(n_{1},\\omega)} + \\cos^{\\omega}{(n_{1})})^{n_{1}} and \\operatorname{L_{\\varepsilon}}{(n_{1})} = 0^{n_{1}} and \\operatorname{L_{\\varepsilon}}{(n_{1})} \\cos^{\\omega}{(n_{1})} = 0^{n_{1}} \\cos^{\\omega}{(n_{1})} and \\operatorname{L_{\\varepsilon}}{(n_{1})} \\cos^{\\omega}{(n_{1})} = (- \\Psi_{\\lambda}{(n_{1},\\omega)} + \\cos^{\\omega}{(n_{1})})^{n_{1}} \\cos^{\\omega}{(n_{1})} and 0^{n_{1}} \\cos^{\\omega}{(n_{1})} = (- \\Psi_{\\lambda}{(n_{1},\\omega)} + \\cos^{\\omega}{(n_{1})})^{n_{1}} \\cos^{\\omega}{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True)), Pow(Integer(0), Symbol('n_1', commutative=True)))"], [["times", 4, "Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Integer(0), Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Integer(0), Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then obtain (3 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + \\sin{(\\mathbf{J}_M)})^{3} = (\\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + 3 \\sin{(\\mathbf{J}_M)})^{3}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and 2 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} = \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + \\sin{(\\mathbf{J}_M)} and 3 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + \\sin{(\\mathbf{J}_M)} = 2 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + 2 \\sin{(\\mathbf{J}_M)} and 3 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + \\sin{(\\mathbf{J}_M)} = \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + 3 \\sin{(\\mathbf{J}_M)} and (3 \\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + \\sin{(\\mathbf{J}_M)})^{3} = (\\dot{\\mathbf{r}}{(\\mathbf{J}_M)} + 3 \\sin{(\\mathbf{J}_M)})^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 2, "Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(3), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["power", 4, 3], "Equality(Pow(Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True))), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(3)), Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(3), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(3)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(z)} = \\operatorname{x^{{\\}'}}^{2}{(z)}, then obtain \\iiint (m + \\operatorname{v_{y}}{(z)}) dm dz dm = \\iiint (m + \\operatorname{x^{{\\}'}}^{2}{(z)}) dm dz dm", "derivation": "\\operatorname{v_{y}}{(z)} = \\operatorname{x^{{\\}'}}^{2}{(z)} and m + \\operatorname{v_{y}}{(z)} = m + \\operatorname{x^{{\\}'}}^{2}{(z)} and \\int (m + \\operatorname{v_{y}}{(z)}) dm = \\int (m + \\operatorname{x^{{\\}'}}^{2}{(z)}) dm and \\iint (m + \\operatorname{v_{y}}{(z)}) dm dz = \\iint (m + \\operatorname{x^{{\\}'}}^{2}{(z)}) dm dz and \\iiint (m + \\operatorname{v_{y}}{(z)}) dm dz dm = \\iiint (m + \\operatorname{x^{{\\}'}}^{2}{(z)}) dm dz dm", "srepr_derivation": [["renaming_premise", "Equality(Function('v_y')(Symbol('z', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('z', commutative=True)), Integer(2)))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('v_y')(Symbol('z', commutative=True))), Add(Symbol('m', commutative=True), Pow(Function('x^\\\\prime')(Symbol('z', commutative=True)), Integer(2))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Symbol('m', commutative=True), Function('v_y')(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('m', commutative=True), Pow(Function('x^\\\\prime')(Symbol('z', commutative=True)), Integer(2))), Tuple(Symbol('m', commutative=True))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Symbol('m', commutative=True), Function('v_y')(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Symbol('m', commutative=True), Pow(Function('x^\\\\prime')(Symbol('z', commutative=True)), Integer(2))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Symbol('m', commutative=True), Function('v_y')(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('m', commutative=True), Pow(Function('x^\\\\prime')(Symbol('z', commutative=True)), Integer(2))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} = \\mathbf{B} \\mathbf{P}, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{B} + (\\int \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} d\\mathbf{P})^{\\mathbf{B}}) = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{B} + (\\int \\mathbf{B} \\mathbf{P} d\\mathbf{P})^{\\mathbf{B}})", "derivation": "\\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} = \\mathbf{B} \\mathbf{P} and \\int \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} d\\mathbf{P} = \\int \\mathbf{B} \\mathbf{P} d\\mathbf{P} and (\\int \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} d\\mathbf{P})^{\\mathbf{B}} = (\\int \\mathbf{B} \\mathbf{P} d\\mathbf{P})^{\\mathbf{B}} and - \\mathbf{B} + (\\int \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} d\\mathbf{P})^{\\mathbf{B}} = - \\mathbf{B} + (\\int \\mathbf{B} \\mathbf{P} d\\mathbf{P})^{\\mathbf{B}} and \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{B} + (\\int \\operatorname{P_{g}}{(\\mathbf{P},\\mathbf{B})} d\\mathbf{P})^{\\mathbf{B}}) = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{B} + (\\int \\mathbf{B} \\mathbf{P} d\\mathbf{P})^{\\mathbf{B}})", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integral(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(q,\\Omega)} = \\frac{\\Omega}{q}, then obtain \\frac{\\int\\limits^{q h{(q,\\Omega)}} \\frac{\\partial}{\\partial \\Omega} q h{(q,\\Omega)} d\\Omega}{\\log{(g^{\\prime}_{\\varepsilon})}} = \\frac{\\int\\limits^{q h{(q,\\Omega)}} \\frac{d}{d \\Omega} \\Omega d\\Omega}{\\log{(g^{\\prime}_{\\varepsilon})}}", "derivation": "h{(q,\\Omega)} = \\frac{\\Omega}{q} and q h{(q,\\Omega)} = \\Omega and \\frac{\\partial}{\\partial \\Omega} q h{(q,\\Omega)} = \\frac{d}{d \\Omega} \\Omega and \\int \\frac{\\partial}{\\partial \\Omega} q h{(q,\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} \\Omega d\\Omega and \\int\\limits^{q h{(q,\\Omega)}} \\frac{\\partial}{\\partial \\Omega} q h{(q,\\Omega)} d\\Omega = \\int\\limits^{q h{(q,\\Omega)}} \\frac{d}{d \\Omega} \\Omega d\\Omega and \\frac{\\int\\limits^{q h{(q,\\Omega)}} \\frac{\\partial}{\\partial \\Omega} q h{(q,\\Omega)} d\\Omega}{\\log{(g^{\\prime}_{\\varepsilon})}} = \\frac{\\int\\limits^{q h{(q,\\Omega)}} \\frac{d}{d \\Omega} \\Omega d\\Omega}{\\log{(g^{\\prime}_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Derivative(Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True), Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))))), Integral(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True), Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["divide", 5, "log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Integral(Derivative(Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True), Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Mul(Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Integral(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True), Mul(Symbol('q', commutative=True), Function('h')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)))))))"]]}, {"prompt": "Given q{(f_{\\mathbf{p}})} = \\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}}, then obtain \\frac{(J + f_{\\mathbf{p}} \\log{(f_{\\mathbf{p}})} - f_{\\mathbf{p}}) q{(f_{\\mathbf{p}})}}{\\mathbf{H}} = \\frac{(J + f_{\\mathbf{p}} \\log{(f_{\\mathbf{p}})} - f_{\\mathbf{p}})^{2}}{\\mathbf{H}}", "derivation": "q{(f_{\\mathbf{p}})} = \\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}} and q{(f_{\\mathbf{p}})} \\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = (\\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}})^{2} and \\frac{q{(f_{\\mathbf{p}})} \\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}}}{\\mathbf{H}} = \\frac{(\\int \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}})^{2}}{\\mathbf{H}} and \\frac{(J + f_{\\mathbf{p}} \\log{(f_{\\mathbf{p}})} - f_{\\mathbf{p}}) q{(f_{\\mathbf{p}})}}{\\mathbf{H}} = \\frac{(J + f_{\\mathbf{p}} \\log{(f_{\\mathbf{p}})} - f_{\\mathbf{p}})^{2}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["times", 1, "Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Pow(Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2)))"], [["divide", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('q')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Integral(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Function('q')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Add(Symbol('J', commutative=True), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{B})} = \\cos{(\\mathbf{B})}, then derive \\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})} = - \\sin{(\\mathbf{B})}, then obtain - \\frac{\\sin{(\\mathbf{B})}}{\\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})}} = 1", "derivation": "\\mathbf{r}{(\\mathbf{B})} = \\cos{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})} and \\frac{\\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})}}{\\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})}} = 1 and \\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})} = - \\sin{(\\mathbf{B})} and - \\frac{\\sin{(\\mathbf{B})}}{\\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})}} = 1 and - \\frac{\\sin{(\\mathbf{B})}}{\\frac{d}{d \\mathbf{B}} \\mathbf{r}{(\\mathbf{B})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\phi_{1}{(t_{1})} = \\sin{(t_{1})} and x{(t_{1})} = - \\sin{(t_{1})}, then obtain - \\phi_{1}{(t_{1})} + \\sin{(t_{1})} = 0", "derivation": "\\phi_{1}{(t_{1})} = \\sin{(t_{1})} and x{(t_{1})} = - \\sin{(t_{1})} and x{(t_{1})} = - \\phi_{1}{(t_{1})} and \\phi_{1}{(t_{1})} + x{(t_{1})} = \\phi_{1}{(t_{1})} - \\sin{(t_{1})} and \\phi_{1}{(t_{1})} + x{(t_{1})} = 0 and x{(t_{1})} + \\sin{(t_{1})} = 0 and - \\phi_{1}{(t_{1})} + \\sin{(t_{1})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('t_1', commutative=True)), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('x')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('t_1', commutative=True))))"], [["add", 2, "Function('\\\\phi_1')(Symbol('t_1', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('t_1', commutative=True)), Function('x')(Symbol('t_1', commutative=True))), Add(Function('\\\\phi_1')(Symbol('t_1', commutative=True)), Mul(Integer(-1), sin(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\phi_1')(Symbol('t_1', commutative=True)), Function('x')(Symbol('t_1', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('x')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('t_1', commutative=True))), sin(Symbol('t_1', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(g,y)} = y^{g}, then derive \\frac{\\partial}{\\partial y} \\operatorname{g_{\\varepsilon}}{(g,y)} = \\frac{g y^{g}}{y}, then obtain \\frac{\\partial}{\\partial y} \\operatorname{g_{\\varepsilon}}{(g,y)} = \\frac{g \\operatorname{g_{\\varepsilon}}{(g,y)}}{y}", "derivation": "\\operatorname{g_{\\varepsilon}}{(g,y)} = y^{g} and \\frac{\\partial}{\\partial y} \\operatorname{g_{\\varepsilon}}{(g,y)} = \\frac{\\partial}{\\partial y} y^{g} and \\frac{\\partial}{\\partial y} \\operatorname{g_{\\varepsilon}}{(g,y)} = \\frac{g y^{g}}{y} and \\frac{\\partial}{\\partial y} \\operatorname{g_{\\varepsilon}}{(g,y)} = \\frac{g \\operatorname{g_{\\varepsilon}}{(g,y)}}{y}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Symbol('y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})}, then obtain 0 = (- \\operatorname{C_{2}}{(E_{\\lambda})} + \\sin{(E_{\\lambda})}) \\frac{d}{d E_{\\lambda}} (E_{\\lambda} + \\sin{(E_{\\lambda})})", "derivation": "\\operatorname{C_{2}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and E_{\\lambda} + \\operatorname{C_{2}}{(E_{\\lambda})} = E_{\\lambda} + \\sin{(E_{\\lambda})} and 0 = - \\operatorname{C_{2}}{(E_{\\lambda})} + \\sin{(E_{\\lambda})} and 0 = (- \\operatorname{C_{2}}{(E_{\\lambda})} + \\sin{(E_{\\lambda})}) \\frac{d}{d E_{\\lambda}} (E_{\\lambda} + \\sin{(E_{\\lambda})})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('E_{\\\\lambda}', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True)))"], [["add", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Function('C_2')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 2, "Add(Symbol('E_{\\\\lambda}', commutative=True), Function('C_2')(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_2')(Symbol('E_{\\\\lambda}', commutative=True))), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 3, "Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('C_2')(Symbol('E_{\\\\lambda}', commutative=True))), sin(Symbol('E_{\\\\lambda}', commutative=True))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(I)} = e^{I}, then derive \\frac{d}{d I} \\mathbf{s}{(I)} = e^{I}, then obtain v_{1} \\frac{d^{2}}{d I^{2}} \\mathbf{s}{(I)} = v_{1} e^{I}", "derivation": "\\mathbf{s}{(I)} = e^{I} and \\frac{d}{d I} \\mathbf{s}{(I)} = \\frac{d}{d I} e^{I} and \\frac{d}{d I} \\mathbf{s}{(I)} = e^{I} and \\frac{d}{d I} \\mathbf{s}{(I)} = \\mathbf{s}{(I)} and \\frac{d^{2}}{d I^{2}} \\mathbf{s}{(I)} = e^{I} and v_{1} \\frac{d^{2}}{d I^{2}} \\mathbf{s}{(I)} = v_{1} e^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), exp(Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Function('\\\\mathbf{s}')(Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), exp(Symbol('I', commutative=True)))"], [["times", 5, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2)))), Mul(Symbol('v_1', commutative=True), exp(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\psi,A_{x})} = A_{x} + \\psi and \\mathbf{r}{(\\psi,A_{x})} = \\frac{1}{\\operatorname{M_{E}}{(\\psi,A_{x})}}, then obtain \\frac{1}{A_{x} + \\psi} = \\mathbf{r}{(\\psi,A_{x})}", "derivation": "\\operatorname{M_{E}}{(\\psi,A_{x})} = A_{x} + \\psi and \\frac{1}{A_{x} + \\psi} = \\frac{1}{\\operatorname{M_{E}}{(\\psi,A_{x})}} and \\mathbf{r}{(\\psi,A_{x})} = \\frac{1}{\\operatorname{M_{E}}{(\\psi,A_{x})}} and \\frac{1}{A_{x} + \\psi} = \\mathbf{r}{(\\psi,A_{x})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\psi', commutative=True)), Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)))"], "Equality(Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)), Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)), Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\psi', commutative=True), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\mathbf{S}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then obtain \\frac{\\operatorname{f_{\\mathbf{p}}}{(f_{\\mathbf{v}})} e^{- f_{\\mathbf{v}}}}{2} = \\frac{1}{2}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\mathbf{S}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(f_{\\mathbf{v}})}}{\\mathbf{S}{(f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}}} = \\frac{e^{f_{\\mathbf{v}}}}{\\mathbf{S}{(f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}}} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(f_{\\mathbf{v}})} e^{- f_{\\mathbf{v}}}}{2} = \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 1, "Add(Function('\\\\mathbf{S}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{S}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Pow(Add(Function('\\\\mathbf{S}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Rational(1, 2), Function('f_{\\\\mathbf{p}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Rational(1, 2))"]]}, {"prompt": "Given \\omega{(\\nabla)} = \\log{(\\nabla)} and J{(\\nabla)} = 2 \\nabla, then obtain \\omega^{4 \\nabla}{(\\nabla)} = \\omega^{2 J{(\\nabla)}}{(\\nabla)}", "derivation": "\\omega{(\\nabla)} = \\log{(\\nabla)} and \\omega^{\\nabla}{(\\nabla)} = \\log{(\\nabla)}^{\\nabla} and \\omega^{2 \\nabla}{(\\nabla)} = \\omega^{\\nabla}{(\\nabla)} \\log{(\\nabla)}^{\\nabla} and J{(\\nabla)} = 2 \\nabla and \\omega^{J{(\\nabla)}}{(\\nabla)} = \\omega^{\\nabla}{(\\nabla)} \\log{(\\nabla)}^{\\nabla} and \\omega^{2 \\nabla}{(\\nabla)} = \\omega^{J{(\\nabla)}}{(\\nabla)} and \\omega^{4 \\nabla}{(\\nabla)} = \\omega^{2 J{(\\nabla)}}{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["power", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["times", 2, "Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Function('J')(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(log(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Function('J')(Symbol('\\\\nabla', commutative=True))))"], [["power", 6, 2], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(4), Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\omega')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Function('J')(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given G{(\\hat{X})} = e^{\\hat{X}}, then obtain \\frac{d}{d \\hat{X}} \\cos{(\\int \\log{(G^{\\hat{X}}{(\\hat{X})})} d\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\int \\log{((e^{\\hat{X}})^{\\hat{X}})} d\\hat{X})}", "derivation": "G{(\\hat{X})} = e^{\\hat{X}} and G^{\\hat{X}}{(\\hat{X})} = (e^{\\hat{X}})^{\\hat{X}} and \\log{(G^{\\hat{X}}{(\\hat{X})})} = \\log{((e^{\\hat{X}})^{\\hat{X}})} and \\int \\log{(G^{\\hat{X}}{(\\hat{X})})} d\\hat{X} = \\int \\log{((e^{\\hat{X}})^{\\hat{X}})} d\\hat{X} and \\cos{(\\int \\log{(G^{\\hat{X}}{(\\hat{X})})} d\\hat{X})} = \\cos{(\\int \\log{((e^{\\hat{X}})^{\\hat{X}})} d\\hat{X})} and \\frac{d}{d \\hat{X}} \\cos{(\\int \\log{(G^{\\hat{X}}{(\\hat{X})})} d\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\int \\log{((e^{\\hat{X}})^{\\hat{X}})} d\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\hat{X}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('G')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), log(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(log(Pow(Function('G')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(log(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["cos", 4], "Equality(cos(Integral(log(Pow(Function('G')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), cos(Integral(log(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(cos(Integral(log(Pow(Function('G')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(cos(Integral(log(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(s)} = \\sin{(\\sin{(s)})}, then obtain (\\operatorname{f^{*}}{(s)} \\int \\operatorname{f^{*}}{(s)} ds)^{s} + (\\sin{(\\sin{(s)})} \\int \\operatorname{f^{*}}{(s)} ds)^{s} = 2 (\\sin{(\\sin{(s)})} \\int \\operatorname{f^{*}}{(s)} ds)^{s}", "derivation": "\\operatorname{f^{*}}{(s)} = \\sin{(\\sin{(s)})} and \\int \\operatorname{f^{*}}{(s)} ds = \\int \\sin{(\\sin{(s)})} ds and \\operatorname{f^{*}}{(s)} \\int \\sin{(\\sin{(s)})} ds = \\sin{(\\sin{(s)})} \\int \\sin{(\\sin{(s)})} ds and (\\operatorname{f^{*}}{(s)} \\int \\sin{(\\sin{(s)})} ds)^{s} = (\\sin{(\\sin{(s)})} \\int \\sin{(\\sin{(s)})} ds)^{s} and (\\operatorname{f^{*}}{(s)} \\int \\sin{(\\sin{(s)})} ds)^{s} + (\\sin{(\\sin{(s)})} \\int \\sin{(\\sin{(s)})} ds)^{s} = 2 (\\sin{(\\sin{(s)})} \\int \\sin{(\\sin{(s)})} ds)^{s} and (\\operatorname{f^{*}}{(s)} \\int \\operatorname{f^{*}}{(s)} ds)^{s} + (\\sin{(\\sin{(s)})} \\int \\operatorname{f^{*}}{(s)} ds)^{s} = 2 (\\sin{(\\sin{(s)})} \\int \\operatorname{f^{*}}{(s)} ds)^{s}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('s', commutative=True)), sin(sin(Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["times", 1, "Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))"], "Equality(Mul(Function('f^*')(Symbol('s', commutative=True)), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Mul(sin(sin(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Function('f^*')(Symbol('s', commutative=True)), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True)))"], [["add", 4, "Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True))"], "Equality(Add(Pow(Mul(Function('f^*')(Symbol('s', commutative=True)), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True))), Mul(Integer(2), Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Mul(Function('f^*')(Symbol('s', commutative=True)), Integral(Function('f^*')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(Function('f^*')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True))), Mul(Integer(2), Pow(Mul(sin(sin(Symbol('s', commutative=True))), Integral(Function('f^*')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\varepsilon,\\mathbf{r})} = \\mathbf{r}^{\\varepsilon}, then obtain ((- \\mathbf{r}^{\\varepsilon} + \\psi^{*}{(\\varepsilon,\\mathbf{r})})^{\\mathbf{r}})^{\\varepsilon} = (0^{\\mathbf{r}})^{\\varepsilon}", "derivation": "\\psi^{*}{(\\varepsilon,\\mathbf{r})} = \\mathbf{r}^{\\varepsilon} and - \\mathbf{r}^{\\varepsilon} + \\psi^{*}{(\\varepsilon,\\mathbf{r})} = 0 and (- \\mathbf{r}^{\\varepsilon} + \\psi^{*}{(\\varepsilon,\\mathbf{r})})^{\\mathbf{r}} = 0^{\\mathbf{r}} and ((- \\mathbf{r}^{\\varepsilon} + \\psi^{*}{(\\varepsilon,\\mathbf{r})})^{\\mathbf{r}})^{\\varepsilon} = (0^{\\mathbf{r}})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{r}', commutative=True)))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\psi)} = e^{\\psi} and \\varphi{(S,\\lambda)} = \\cos{(S - \\lambda)}, then obtain \\frac{\\partial}{\\partial S} \\varphi{(S,\\lambda)} \\frac{d}{d \\psi} e^{\\psi} = \\frac{d}{d \\psi} e^{\\psi} \\frac{\\partial}{\\partial S} \\cos{(S - \\lambda)}", "derivation": "\\operatorname{z^{*}}{(\\psi)} = e^{\\psi} and \\frac{d}{d \\psi} \\operatorname{z^{*}}{(\\psi)} = \\frac{d}{d \\psi} e^{\\psi} and \\varphi{(S,\\lambda)} = \\cos{(S - \\lambda)} and \\frac{\\partial}{\\partial S} \\varphi{(S,\\lambda)} = \\frac{\\partial}{\\partial S} \\cos{(S - \\lambda)} and \\frac{\\partial}{\\partial S} \\varphi{(S,\\lambda)} \\frac{d}{d \\psi} \\operatorname{z^{*}}{(\\psi)} = \\frac{d}{d \\psi} \\operatorname{z^{*}}{(\\psi)} \\frac{\\partial}{\\partial S} \\cos{(S - \\lambda)} and \\frac{\\partial}{\\partial S} \\varphi{(S,\\lambda)} \\frac{d}{d \\psi} e^{\\psi} = \\frac{d}{d \\psi} e^{\\psi} \\frac{\\partial}{\\partial S} \\cos{(S - \\lambda)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\varphi')(Symbol('S', commutative=True), Symbol('\\\\lambda', commutative=True)), cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('S', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\varphi')(Symbol('S', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Derivative(Function('\\\\varphi')(Symbol('S', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(\\tilde{g})} = \\cos{(\\log{(\\tilde{g})})}, then obtain \\int (- \\eta{(\\tilde{g})} + 2 \\cos{(\\log{(\\tilde{g})})}) d\\tilde{g} = \\int (- 2 \\eta{(\\tilde{g})} + 3 \\cos{(\\log{(\\tilde{g})})}) d\\tilde{g}", "derivation": "\\eta{(\\tilde{g})} = \\cos{(\\log{(\\tilde{g})})} and 0 = - \\eta{(\\tilde{g})} + \\cos{(\\log{(\\tilde{g})})} and \\cos{(\\log{(\\tilde{g})})} = - \\eta{(\\tilde{g})} + 2 \\cos{(\\log{(\\tilde{g})})} and - \\eta{(\\tilde{g})} + 2 \\cos{(\\log{(\\tilde{g})})} = - 2 \\eta{(\\tilde{g})} + 3 \\cos{(\\log{(\\tilde{g})})} and \\int (- \\eta{(\\tilde{g})} + 2 \\cos{(\\log{(\\tilde{g})})}) d\\tilde{g} = \\int (- 2 \\eta{(\\tilde{g})} + 3 \\cos{(\\log{(\\tilde{g})})}) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True)), cos(log(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 1, "Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), cos(log(Symbol('\\\\tilde{g}', commutative=True)))))"], [["add", 2, "cos(log(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(cos(log(Symbol('\\\\tilde{g}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\tilde{g}', commutative=True))))))"], [["add", 3, "Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), cos(log(Symbol('\\\\tilde{g}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\tilde{g}', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(3), cos(log(Symbol('\\\\tilde{g}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\tilde{g}', commutative=True))))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Function('\\\\eta')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(3), cos(log(Symbol('\\\\tilde{g}', commutative=True))))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\mu{(h)} = \\int \\log{(h)} dh, then derive \\frac{d}{d h} \\mu{(h)} = \\frac{\\partial}{\\partial h} (J_{\\varepsilon} + h \\log{(h)} - h), then obtain (\\frac{\\partial}{\\partial h} (J_{\\varepsilon} + h \\log{(h)} - h))^{h} = (\\frac{d}{d h} \\int \\log{(h)} dh)^{h}", "derivation": "\\mu{(h)} = \\int \\log{(h)} dh and \\frac{d}{d h} \\mu{(h)} = \\frac{d}{d h} \\int \\log{(h)} dh and \\frac{d}{d h} \\mu{(h)} = \\frac{\\partial}{\\partial h} (J_{\\varepsilon} + h \\log{(h)} - h) and \\frac{\\partial}{\\partial h} (J_{\\varepsilon} + h \\log{(h)} - h) = \\frac{d}{d h} \\int \\log{(h)} dh and (\\frac{\\partial}{\\partial h} (J_{\\varepsilon} + h \\log{(h)} - h))^{h} = (\\frac{d}{d h} \\int \\log{(h)} dh)^{h}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('h', commutative=True)), Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)} = e^{f_{E}^{\\theta_1}} and \\hat{H}{(f_{E},\\theta_1)} = \\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1}, then obtain (\\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)}}{\\theta_1})^{\\theta_1} = (\\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1})^{\\theta_1}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)} = e^{f_{E}^{\\theta_1}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)}}{\\theta_1} = \\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1} and \\hat{H}{(f_{E},\\theta_1)} = \\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1} and \\hat{H}^{\\theta_1}{(f_{E},\\theta_1)} = (\\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1})^{\\theta_1} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)}}{\\theta_1} = \\hat{H}{(f_{E},\\theta_1)} and (\\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(f_{E},\\theta_1)}}{\\theta_1})^{\\theta_1} = (\\frac{e^{f_{E}^{\\theta_1}}}{\\theta_1})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Pow(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Pow(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Pow(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Pow(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True))), Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Pow(Symbol('f_E', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given S{(F_{N})} = \\int e^{F_{N}} dF_{N} and \\operatorname{C_{2}}{(F_{N})} = F_{N}, then derive S{(F_{N})} = \\psi^* + e^{F_{N}}, then obtain e^{\\operatorname{C_{2}}{(F_{N})}} \\frac{\\partial}{\\partial F_{N}} \\int e^{F_{N}} d\\operatorname{C_{2}}{(F_{N})} = e^{\\operatorname{C_{2}}{(F_{N})}} \\frac{\\partial}{\\partial F_{N}} (\\psi^* + e^{\\operatorname{C_{2}}{(F_{N})}})", "derivation": "S{(F_{N})} = \\int e^{F_{N}} dF_{N} and S{(F_{N})} = \\psi^* + e^{F_{N}} and \\int e^{F_{N}} dF_{N} = \\psi^* + e^{F_{N}} and \\operatorname{C_{2}}{(F_{N})} = F_{N} and \\int e^{F_{N}} d\\operatorname{C_{2}}{(F_{N})} = \\psi^* + e^{\\operatorname{C_{2}}{(F_{N})}} and \\frac{\\partial}{\\partial F_{N}} \\int e^{F_{N}} d\\operatorname{C_{2}}{(F_{N})} = \\frac{\\partial}{\\partial F_{N}} (\\psi^* + e^{\\operatorname{C_{2}}{(F_{N})}}) and e^{\\operatorname{C_{2}}{(F_{N})}} \\frac{\\partial}{\\partial F_{N}} \\int e^{F_{N}} d\\operatorname{C_{2}}{(F_{N})} = e^{\\operatorname{C_{2}}{(F_{N})}} \\frac{\\partial}{\\partial F_{N}} (\\psi^* + e^{\\operatorname{C_{2}}{(F_{N})}})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('F_N', commutative=True)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('S')(Symbol('F_N', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), exp(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), exp(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Function('C_2')(Symbol('F_N', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), exp(Function('C_2')(Symbol('F_N', commutative=True)))))"], [["differentiate", 5, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Function('C_2')(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Function('C_2')(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["times", 6, "exp(Function('C_2')(Symbol('F_N', commutative=True)))"], "Equality(Mul(exp(Function('C_2')(Symbol('F_N', commutative=True))), Derivative(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Function('C_2')(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(exp(Function('C_2')(Symbol('F_N', commutative=True))), Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Function('C_2')(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(\\theta)} = e^{\\theta} and \\operatorname{J_{\\varepsilon}}{(\\theta)} = \\theta^{2} e^{\\theta}, then obtain \\theta^{2} Q{(\\theta)} = \\operatorname{J_{\\varepsilon}}{(\\theta)}", "derivation": "Q{(\\theta)} = e^{\\theta} and \\theta Q{(\\theta)} = \\theta e^{\\theta} and \\theta^{2} Q{(\\theta)} = \\theta^{2} e^{\\theta} and \\operatorname{J_{\\varepsilon}}{(\\theta)} = \\theta^{2} e^{\\theta} and \\theta^{2} Q{(\\theta)} = \\operatorname{J_{\\varepsilon}}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('Q')(Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\theta', commutative=True))))"], [["times", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), exp(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), exp(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\theta', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then derive (\\int \\operatorname{A_{z}}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{P})})^{\\mathbf{P}}, then obtain (\\int \\cos{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{P})})^{\\mathbf{P}}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\int \\operatorname{A_{z}}{(\\mathbf{P})} d\\mathbf{P} = \\int \\cos{(\\mathbf{P})} d\\mathbf{P} and (\\int \\operatorname{A_{z}}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\int \\cos{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} and (\\int \\operatorname{A_{z}}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{P})})^{\\mathbf{P}} and (\\int \\cos{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\hat{\\mathbf{r}} + \\sin{(\\mathbf{P})})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Integral(Function('A_z')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('A_z')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(W)} = \\cos{(W)}, then obtain \\frac{d}{d W} W \\operatorname{n_{2}}^{- W}{(W)} \\int \\operatorname{n_{2}}^{W}{(W)} dW = \\frac{d}{d W} W \\operatorname{n_{2}}^{- W}{(W)} \\int \\cos^{W}{(W)} dW", "derivation": "\\operatorname{n_{2}}{(W)} = \\cos{(W)} and \\operatorname{n_{2}}^{W}{(W)} = \\cos^{W}{(W)} and \\int \\operatorname{n_{2}}^{W}{(W)} dW = \\int \\cos^{W}{(W)} dW and W \\cos^{- W}{(W)} \\int \\operatorname{n_{2}}^{W}{(W)} dW = W \\cos^{- W}{(W)} \\int \\cos^{W}{(W)} dW and W \\operatorname{n_{2}}^{- W}{(W)} \\int \\operatorname{n_{2}}^{W}{(W)} dW = W \\operatorname{n_{2}}^{- W}{(W)} \\int \\cos^{W}{(W)} dW and \\frac{d}{d W} W \\operatorname{n_{2}}^{- W}{(W)} \\int \\operatorname{n_{2}}^{W}{(W)} dW = \\frac{d}{d W} W \\operatorname{n_{2}}^{- W}{(W)} \\int \\cos^{W}{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('n_2')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], "Equality(Mul(Symbol('W', commutative=True), Pow(cos(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(Function('n_2')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('W', commutative=True), Pow(cos(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('W', commutative=True), Pow(Function('n_2')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(Function('n_2')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('W', commutative=True), Pow(Function('n_2')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Symbol('W', commutative=True), Pow(Function('n_2')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(Function('n_2')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Symbol('W', commutative=True), Pow(Function('n_2')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\pi,y)} = \\pi + y, then obtain \\int (\\frac{y}{- \\pi + t{(\\pi,y)}} + \\int 1 dy) dy = \\int (\\frac{y}{- \\pi + t{(\\pi,y)}} + \\int \\frac{y}{- \\pi + t{(\\pi,y)}} dy) dy", "derivation": "t{(\\pi,y)} = \\pi + y and - \\pi + t{(\\pi,y)} = y and 1 = \\frac{y}{- \\pi + t{(\\pi,y)}} and \\int 1 dy = \\int \\frac{y}{- \\pi + t{(\\pi,y)}} dy and \\frac{y}{- \\pi + t{(\\pi,y)}} + \\int 1 dy = \\frac{y}{- \\pi + t{(\\pi,y)}} + \\int \\frac{y}{- \\pi + t{(\\pi,y)}} dy and \\int (\\frac{y}{- \\pi + t{(\\pi,y)}} + \\int 1 dy) dy = \\int (\\frac{y}{- \\pi + t{(\\pi,y)}} + \\int \\frac{y}{- \\pi + t{(\\pi,y)}} dy) dy", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)))"], "Equality(Integer(1), Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('y', commutative=True))), Integral(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Tuple(Symbol('y', commutative=True))))"], [["add", 4, "Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Integral(Integer(1), Tuple(Symbol('y', commutative=True)))), Add(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Integral(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Tuple(Symbol('y', commutative=True)))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Integral(Integer(1), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Integral(Mul(Symbol('y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('t')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\eta)} = e^{\\eta}, then obtain \\frac{d}{d \\eta} \\int \\pi{(\\eta)} d\\eta = \\frac{\\partial}{\\partial \\eta} (\\tilde{g} + e^{\\eta})", "derivation": "\\pi{(\\eta)} = e^{\\eta} and \\int \\pi{(\\eta)} d\\eta = \\int e^{\\eta} d\\eta and \\frac{d}{d \\eta} \\int \\pi{(\\eta)} d\\eta = \\frac{d}{d \\eta} \\int e^{\\eta} d\\eta and \\frac{d}{d \\eta} \\int \\pi{(\\eta)} d\\eta = \\frac{\\partial}{\\partial \\eta} (\\tilde{g} + e^{\\eta})", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(Q)} = \\cos{(Q)} and \\mathbf{E}{(Q)} = \\cos{(Q)} + \\int \\sigma_{x}{(Q)} dQ and \\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} = \\pi + \\sin{(Q)}, then derive \\int \\sigma_{x}{(Q)} dQ = \\pi + \\sin{(Q)}, then obtain \\int \\mathbf{E}{(Q)} d\\pi = \\int (\\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} + \\sigma_{x}{(Q)}) d\\pi", "derivation": "\\sigma_{x}{(Q)} = \\cos{(Q)} and \\int \\sigma_{x}{(Q)} dQ = \\int \\cos{(Q)} dQ and \\int \\sigma_{x}{(Q)} dQ = \\pi + \\sin{(Q)} and \\mathbf{E}{(Q)} = \\cos{(Q)} + \\int \\sigma_{x}{(Q)} dQ and \\mathbf{E}{(Q)} = \\sigma_{x}{(Q)} + \\int \\sigma_{x}{(Q)} dQ and \\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} = \\pi + \\sin{(Q)} and \\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} = \\int \\sigma_{x}{(Q)} dQ and \\mathbf{E}{(Q)} = \\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} + \\sigma_{x}{(Q)} and \\int \\mathbf{E}{(Q)} d\\pi = \\int (\\operatorname{V_{\\mathbf{E}}}{(Q,\\pi)} + \\sigma_{x}{(Q)}) d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('Q', commutative=True)), Add(cos(Symbol('Q', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{E}')(Symbol('Q', commutative=True)), Add(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Function('\\\\mathbf{E}')(Symbol('Q', commutative=True)), Add(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\sigma_x')(Symbol('Q', commutative=True))))"], [["integrate", 8, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\sigma_x')(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\varphi)} = \\frac{1}{\\varphi}, then obtain \\int \\frac{d}{d \\varphi} \\operatorname{v_{y}}^{\\varphi}{(\\varphi)} d\\varphi = \\int \\frac{d}{d \\varphi} (\\frac{1}{\\varphi})^{\\varphi} d\\varphi", "derivation": "\\operatorname{v_{y}}{(\\varphi)} = \\frac{1}{\\varphi} and \\operatorname{v_{y}}^{\\varphi}{(\\varphi)} = (\\frac{1}{\\varphi})^{\\varphi} and \\frac{d}{d \\varphi} \\operatorname{v_{y}}^{\\varphi}{(\\varphi)} = \\frac{d}{d \\varphi} (\\frac{1}{\\varphi})^{\\varphi} and \\int \\frac{d}{d \\varphi} \\operatorname{v_{y}}^{\\varphi}{(\\varphi)} d\\varphi = \\int \\frac{d}{d \\varphi} (\\frac{1}{\\varphi})^{\\varphi} d\\varphi", "srepr_derivation": [["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Pow(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hat{X})} = \\log{(\\hat{X})}, then derive \\int \\operatorname{F_{c}}{(\\hat{X})} d\\hat{X} = \\hat{X} \\log{(\\hat{X})} - \\hat{X} + v_{y}, then obtain \\frac{\\partial}{\\partial v_{y}} (\\hat{X} \\log{(\\hat{X})} - \\hat{X} + v_{y}) = \\frac{d}{d v_{y}} \\int \\log{(\\hat{X})} d\\hat{X}", "derivation": "\\operatorname{F_{c}}{(\\hat{X})} = \\log{(\\hat{X})} and \\int \\operatorname{F_{c}}{(\\hat{X})} d\\hat{X} = \\int \\log{(\\hat{X})} d\\hat{X} and \\int \\operatorname{F_{c}}{(\\hat{X})} d\\hat{X} = \\hat{X} \\log{(\\hat{X})} - \\hat{X} + v_{y} and \\hat{X} \\log{(\\hat{X})} - \\hat{X} + v_{y} = \\int \\log{(\\hat{X})} d\\hat{X} and \\frac{\\partial}{\\partial v_{y}} (\\hat{X} \\log{(\\hat{X})} - \\hat{X} + v_{y}) = \\frac{d}{d v_{y}} \\int \\log{(\\hat{X})} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_c')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Symbol('\\\\hat{X}', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\hat{X}', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('v_y', commutative=True)), Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 4, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\hat{X}', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\psi)} = \\sin{(\\cos{(\\psi)})}, then obtain (\\phi^{\\psi}{(\\psi)})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\phi{(\\psi)})} = (\\phi^{\\psi}{(\\psi)})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\sin{(\\cos{(\\psi)})})}", "derivation": "\\phi{(\\psi)} = \\sin{(\\cos{(\\psi)})} and \\cos{(\\phi{(\\psi)})} = \\cos{(\\sin{(\\cos{(\\psi)})})} and \\frac{d}{d \\psi} \\cos{(\\phi{(\\psi)})} = \\frac{d}{d \\psi} \\cos{(\\sin{(\\cos{(\\psi)})})} and \\phi^{\\psi}{(\\psi)} = \\sin^{\\psi}{(\\cos{(\\psi)})} and (\\sin^{\\psi}{(\\cos{(\\psi)})})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\phi{(\\psi)})} = (\\sin^{\\psi}{(\\cos{(\\psi)})})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\sin{(\\cos{(\\psi)})})} and (\\phi^{\\psi}{(\\psi)})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\phi{(\\psi)})} = (\\phi^{\\psi}{(\\psi)})^{\\psi} \\frac{d}{d \\psi} \\cos{(\\sin{(\\cos{(\\psi)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\psi', commutative=True)), sin(cos(Symbol('\\\\psi', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\phi')(Symbol('\\\\psi', commutative=True))), cos(sin(cos(Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(cos(Function('\\\\phi')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(sin(cos(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(sin(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["times", 3, "Pow(Pow(sin(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Pow(sin(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(cos(Function('\\\\phi')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Pow(Pow(sin(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(cos(sin(cos(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Pow(Function('\\\\phi')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(cos(Function('\\\\phi')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Pow(Pow(Function('\\\\phi')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(cos(sin(cos(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = s^{\\mathbf{M}}, then derive - \\frac{\\mathbf{M} s^{\\mathbf{M}}}{s} + \\frac{\\partial}{\\partial s} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = 0, then obtain - \\frac{\\mathbf{M} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)}}{s} + \\frac{\\partial}{\\partial s} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = 0", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = s^{\\mathbf{M}} and - s^{\\mathbf{M}} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = 0 and \\frac{\\partial}{\\partial s} (- s^{\\mathbf{M}} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)}) = \\frac{d}{d s} 0 and - \\frac{\\mathbf{M} s^{\\mathbf{M}}}{s} + \\frac{\\partial}{\\partial s} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = 0 and - \\frac{\\mathbf{M} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)}}{s} + \\frac{\\partial}{\\partial s} \\operatorname{L_{\\varepsilon}}{(\\mathbf{M},s)} = 0", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 1, "Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True))), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain - \\operatorname{x^{{\\}'}}{(\\dot{x})} + \\int \\dot{x} \\int \\operatorname{x^{{\\}'}}{(\\dot{x})} d\\dot{x} d\\dot{x} = - \\operatorname{x^{{\\}'}}{(\\dot{x})} + \\int \\dot{x} \\int \\sin{(\\dot{x})} d\\dot{x} d\\dot{x}", "derivation": "\\operatorname{x^{{\\}'}}{(\\dot{x})} = \\sin{(\\dot{x})} and \\int \\operatorname{x^{{\\}'}}{(\\dot{x})} d\\dot{x} = \\int \\sin{(\\dot{x})} d\\dot{x} and \\dot{x} \\int \\operatorname{x^{{\\}'}}{(\\dot{x})} d\\dot{x} = \\dot{x} \\int \\sin{(\\dot{x})} d\\dot{x} and \\int \\dot{x} \\int \\operatorname{x^{{\\}'}}{(\\dot{x})} d\\dot{x} d\\dot{x} = \\int \\dot{x} \\int \\sin{(\\dot{x})} d\\dot{x} d\\dot{x} and - \\operatorname{x^{{\\}'}}{(\\dot{x})} + \\int \\dot{x} \\int \\operatorname{x^{{\\}'}}{(\\dot{x})} d\\dot{x} d\\dot{x} = - \\operatorname{x^{{\\}'}}{(\\dot{x})} + \\int \\dot{x} \\int \\sin{(\\dot{x})} d\\dot{x} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Symbol('\\\\dot{x}', commutative=True), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 4, "Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\eta{(v_{2})} = \\frac{d}{d v_{2}} \\sin{(v_{2})}, then derive v_{2} \\eta{(v_{2})} = v_{2} \\cos{(v_{2})}, then obtain 0^{v_{2}} = (v_{2} \\eta{(v_{2})} - v_{2} \\cos{(v_{2})})^{v_{2}}", "derivation": "\\eta{(v_{2})} = \\frac{d}{d v_{2}} \\sin{(v_{2})} and v_{2} \\eta{(v_{2})} = v_{2} \\frac{d}{d v_{2}} \\sin{(v_{2})} and v_{2} \\eta{(v_{2})} = v_{2} \\cos{(v_{2})} and 0 = - v_{2} \\eta{(v_{2})} + v_{2} \\frac{d}{d v_{2}} \\sin{(v_{2})} and 0 = - v_{2} \\cos{(v_{2})} + v_{2} \\frac{d}{d v_{2}} \\sin{(v_{2})} and 0^{v_{2}} = (- v_{2} \\cos{(v_{2})} + v_{2} \\frac{d}{d v_{2}} \\sin{(v_{2})})^{v_{2}} and 0^{v_{2}} = (v_{2} \\eta{(v_{2})} - v_{2} \\cos{(v_{2})})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('v_2', commutative=True)), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["times", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Function('\\\\eta')(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('v_2', commutative=True), Function('\\\\eta')(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))))"], [["minus", 2, "Mul(Symbol('v_2', commutative=True), Function('\\\\eta')(Symbol('v_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Function('\\\\eta')(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Integer(0), Symbol('v_2', commutative=True)), Pow(Add(Mul(Symbol('v_2', commutative=True), Function('\\\\eta')(Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\varphi)} = \\varphi, then obtain \\delta + \\frac{\\operatorname{v_{2}}^{2}{(\\varphi)}}{2} = \\int \\varphi d\\operatorname{v_{2}}{(\\varphi)}", "derivation": "\\operatorname{v_{2}}{(\\varphi)} = \\varphi and \\int \\operatorname{v_{2}}{(\\varphi)} d\\varphi = \\int \\varphi d\\varphi and \\int \\operatorname{v_{2}}{(\\varphi)} d\\operatorname{v_{2}}{(\\varphi)} = \\int \\varphi d\\operatorname{v_{2}}{(\\varphi)} and \\delta + \\frac{\\operatorname{v_{2}}^{2}{(\\varphi)}}{2} = \\int \\varphi d\\operatorname{v_{2}}{(\\varphi)}", "srepr_derivation": [["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('v_2')(Symbol('\\\\varphi', commutative=True)), Tuple(Function('v_2')(Symbol('\\\\varphi', commutative=True)))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Function('v_2')(Symbol('\\\\varphi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Pow(Function('v_2')(Symbol('\\\\varphi', commutative=True)), Integer(2)))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Function('v_2')(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\hat{p}_0)} = e^{\\hat{p}_0}, then derive \\frac{d}{d \\hat{p}_0} \\int \\operatorname{A_{x}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{\\partial}{\\partial \\hat{p}_0} (\\mathbf{s} + e^{\\hat{p}_0}), then obtain \\frac{d}{d \\hat{p}_0} \\int e^{\\hat{p}_0} d\\hat{p}_0 = \\frac{\\partial}{\\partial \\hat{p}_0} (\\mathbf{s} + e^{\\hat{p}_0})", "derivation": "\\operatorname{A_{x}}{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\int \\operatorname{A_{x}}{(\\hat{p}_0)} d\\hat{p}_0 = \\int e^{\\hat{p}_0} d\\hat{p}_0 and \\frac{d}{d \\hat{p}_0} \\int \\operatorname{A_{x}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{d}{d \\hat{p}_0} \\int e^{\\hat{p}_0} d\\hat{p}_0 and \\frac{d}{d \\hat{p}_0} \\int \\operatorname{A_{x}}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{\\partial}{\\partial \\hat{p}_0} (\\mathbf{s} + e^{\\hat{p}_0}) and \\frac{d}{d \\hat{p}_0} \\int e^{\\hat{p}_0} d\\hat{p}_0 = \\frac{\\partial}{\\partial \\hat{p}_0} (\\mathbf{s} + e^{\\hat{p}_0})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Integral(Function('A_x')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('A_x')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(f_{E})} = \\cos{(f_{E})} and \\lambda{(f_{E})} = \\cos{(f_{E})}, then obtain \\frac{d^{2}}{d f_{E}^{2}} \\operatorname{J_{\\varepsilon}}{(f_{E})} = \\frac{d^{2}}{d f_{E}^{2}} \\lambda{(f_{E})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(f_{E})} = \\cos{(f_{E})} and \\frac{d}{d f_{E}} \\operatorname{J_{\\varepsilon}}{(f_{E})} = \\frac{d}{d f_{E}} \\cos{(f_{E})} and \\frac{d^{2}}{d f_{E}^{2}} \\operatorname{J_{\\varepsilon}}{(f_{E})} = \\frac{d^{2}}{d f_{E}^{2}} \\cos{(f_{E})} and \\lambda{(f_{E})} = \\cos{(f_{E})} and \\frac{d^{2}}{d f_{E}^{2}} \\operatorname{J_{\\varepsilon}}{(f_{E})} = \\frac{d^{2}}{d f_{E}^{2}} \\lambda{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(2))), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(2))), Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{s}{(k,b)} = k^{b} and \\nabla{(k)} = \\frac{1}{k}, then obtain \\frac{\\nabla{(k)}}{k} = \\frac{1}{k^{2}}", "derivation": "\\mathbf{s}{(k,b)} = k^{b} and \\frac{\\mathbf{s}{(k,b)}}{k} = \\frac{k^{b}}{k} and \\nabla{(k)} = \\frac{1}{k} and \\frac{k^{- b} \\mathbf{s}{(k,b)} \\nabla{(k)}}{k} = \\frac{k^{- b} \\mathbf{s}{(k,b)}}{k^{2}} and \\frac{\\nabla{(k)}}{k} = \\frac{1}{k^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('k', commutative=True), Symbol('b', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('b', commutative=True)))"], [["divide", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('k', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('k', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1)))"], [["times", 3, "Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Function('\\\\mathbf{s}')(Symbol('k', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Function('\\\\mathbf{s}')(Symbol('k', commutative=True), Symbol('b', commutative=True)), Function('\\\\nabla')(Symbol('k', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Function('\\\\mathbf{s}')(Symbol('k', commutative=True), Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True))), Pow(Symbol('k', commutative=True), Integer(-2)))"]]}, {"prompt": "Given \\rho_{b}{(\\sigma_p,L)} = \\cos^{\\sigma_p}{(L)}, then obtain 2 \\int \\cos{(\\rho_{b}{(\\sigma_p,L)})} dL = \\int \\cos{(\\rho_{b}{(\\sigma_p,L)})} dL + \\int \\cos{(\\cos^{\\sigma_p}{(L)})} dL", "derivation": "\\rho_{b}{(\\sigma_p,L)} = \\cos^{\\sigma_p}{(L)} and \\cos{(\\rho_{b}{(\\sigma_p,L)})} = \\cos{(\\cos^{\\sigma_p}{(L)})} and \\int \\cos{(\\rho_{b}{(\\sigma_p,L)})} dL = \\int \\cos{(\\cos^{\\sigma_p}{(L)})} dL and 2 \\int \\cos{(\\rho_{b}{(\\sigma_p,L)})} dL = \\int \\cos{(\\rho_{b}{(\\sigma_p,L)})} dL + \\int \\cos{(\\cos^{\\sigma_p}{(L)})} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True))), cos(Pow(cos(Symbol('L', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(cos(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(cos(Pow(cos(Symbol('L', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["add", 3, "Integral(cos(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(2), Integral(cos(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))), Add(Integral(cos(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(cos(Pow(cos(Symbol('L', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(F_{g},\\mathbf{J})} = \\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}) and \\phi_{1}{(A,V)} = A + V, then derive \\operatorname{A_{z}}^{\\mathbf{J}}{(F_{g},\\mathbf{J})} = 1, then obtain \\mathbf{J} \\phi_{1}{(A,V)} (\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}))^{\\mathbf{J}} = \\mathbf{J} \\phi_{1}{(A,V)}", "derivation": "\\operatorname{A_{z}}{(F_{g},\\mathbf{J})} = \\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}) and \\operatorname{A_{z}}^{\\mathbf{J}}{(F_{g},\\mathbf{J})} = (\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}))^{\\mathbf{J}} and \\operatorname{A_{z}}^{\\mathbf{J}}{(F_{g},\\mathbf{J})} = 1 and \\mathbf{J} \\operatorname{A_{z}}^{\\mathbf{J}}{(F_{g},\\mathbf{J})} = \\mathbf{J} and \\phi_{1}{(A,V)} = A + V and \\mathbf{J} (\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}))^{\\mathbf{J}} = \\mathbf{J} and \\mathbf{J} (A + V) (\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}))^{\\mathbf{J}} = \\mathbf{J} (A + V) and \\mathbf{J} \\phi_{1}{(A,V)} (\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{J}))^{\\mathbf{J}} = \\mathbf{J} \\phi_{1}{(A,V)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('A_z')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1))"], [["times", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('A_z')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], ["get_premise", "Equality(Function('\\\\phi_1')(Symbol('A', commutative=True), Symbol('V', commutative=True)), Add(Symbol('A', commutative=True), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], [["times", 6, "Add(Symbol('A', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('A', commutative=True), Symbol('V', commutative=True)), Pow(Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('A', commutative=True), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True), Symbol('V', commutative=True)), Pow(Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})} = - \\dot{z} + \\mathbf{M}, then obtain (1 - \\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})}) (- \\dot{z} + \\mathbf{M} - 1) = (- \\dot{z} + \\mathbf{M} - 1) (\\dot{z} - \\mathbf{M} + 1)", "derivation": "\\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})} = - \\dot{z} + \\mathbf{M} and \\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})} - 1 = - \\dot{z} + \\mathbf{M} - 1 and 1 - \\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})} = \\dot{z} - \\mathbf{M} + 1 and (1 - \\dot{\\mathbf{r}}{(\\dot{z},\\mathbf{M})}) (- \\dot{z} + \\mathbf{M} - 1) = (- \\dot{z} + \\mathbf{M} - 1) (\\dot{z} - \\mathbf{M} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Integer(1)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Integer(-1))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Integer(1))))"]]}, {"prompt": "Given c{(\\pi)} = \\int \\log{(\\pi)} d\\pi, then obtain - \\pi c{(\\pi)} - 1 = - \\pi (\\pi \\log{(\\pi)} - \\pi + \\theta) - 1", "derivation": "c{(\\pi)} = \\int \\log{(\\pi)} d\\pi and - \\pi c{(\\pi)} = - \\pi \\int \\log{(\\pi)} d\\pi and - \\pi c{(\\pi)} - 1 = - \\pi \\int \\log{(\\pi)} d\\pi - 1 and - \\pi c{(\\pi)} - 1 = - \\pi (\\pi \\log{(\\pi)} - \\pi + \\theta) - 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('c')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('c')(Symbol('\\\\pi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Integer(-1)))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('c')(Symbol('\\\\pi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given l{(A_{1})} = \\log{(A_{1})} and \\mathbf{A}{(z^{*},\\psi,P_{g})} = \\frac{- P_{g} + \\psi}{z^{*}}, then obtain \\mathbf{A}{(z^{*},\\psi,P_{g})} + \\frac{d}{d A_{1}} l{(A_{1})} = \\frac{d}{d A_{1}} l{(A_{1})} + \\frac{- P_{g} + \\psi}{z^{*}}", "derivation": "l{(A_{1})} = \\log{(A_{1})} and \\frac{d}{d A_{1}} l{(A_{1})} = \\frac{d}{d A_{1}} \\log{(A_{1})} and \\mathbf{A}{(z^{*},\\psi,P_{g})} = \\frac{- P_{g} + \\psi}{z^{*}} and \\mathbf{A}{(z^{*},\\psi,P_{g})} + \\frac{d}{d A_{1}} \\log{(A_{1})} = \\frac{d}{d A_{1}} \\log{(A_{1})} + \\frac{- P_{g} + \\psi}{z^{*}} and \\mathbf{A}{(z^{*},\\psi,P_{g})} + \\frac{d}{d A_{1}} l{(A_{1})} = \\frac{d}{d A_{1}} l{(A_{1})} + \\frac{- P_{g} + \\psi}{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('z^*', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["add", 3, "Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('z^*', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('P_g', commutative=True)), Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('z^*', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('P_g', commutative=True)), Derivative(Function('l')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Derivative(Function('l')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\delta,\\phi_1)} = e^{\\delta + \\phi_1}, then obtain (\\hat{H}^{\\delta}{(\\delta,\\phi_1)} - e^{e^{\\delta + \\phi_1}})^{\\phi_1} = ((e^{\\delta + \\phi_1})^{\\delta} - e^{e^{\\delta + \\phi_1}})^{\\phi_1}", "derivation": "\\hat{H}{(\\delta,\\phi_1)} = e^{\\delta + \\phi_1} and e^{\\hat{H}{(\\delta,\\phi_1)}} = e^{e^{\\delta + \\phi_1}} and \\hat{H}^{\\delta}{(\\delta,\\phi_1)} = (e^{\\delta + \\phi_1})^{\\delta} and \\hat{H}^{\\delta}{(\\delta,\\phi_1)} - e^{\\hat{H}{(\\delta,\\phi_1)}} = (e^{\\delta + \\phi_1})^{\\delta} - e^{\\hat{H}{(\\delta,\\phi_1)}} and \\hat{H}^{\\delta}{(\\delta,\\phi_1)} - e^{e^{\\delta + \\phi_1}} = (e^{\\delta + \\phi_1})^{\\delta} - e^{e^{\\delta + \\phi_1}} and (\\hat{H}^{\\delta}{(\\delta,\\phi_1)} - e^{e^{\\delta + \\phi_1}})^{\\phi_1} = ((e^{\\delta + \\phi_1})^{\\delta} - e^{e^{\\delta + \\phi_1}})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))), exp(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["minus", 3, "exp(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Pow(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))))), Add(Pow(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))))), Add(Pow(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))))))"], [["power", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Pow(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(exp(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_1', commutative=True)))))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(M)} = \\cos{(M)}, then obtain e^{\\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\operatorname{n_{2}}^{M}{(M)} + 1)} = e^{\\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\cos^{M}{(M)} + 1)}", "derivation": "\\operatorname{n_{2}}{(M)} = \\cos{(M)} and \\operatorname{n_{2}}^{M}{(M)} = \\cos^{M}{(M)} and \\operatorname{n_{2}}^{M}{(M)} + 1 = \\cos^{M}{(M)} + 1 and M + \\operatorname{n_{2}}^{M}{(M)} + 1 = M + \\cos^{M}{(M)} + 1 and \\frac{d}{d M} (M + \\operatorname{n_{2}}^{M}{(M)} + 1) = \\frac{d}{d M} (M + \\cos^{M}{(M)} + 1) and \\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\operatorname{n_{2}}^{M}{(M)} + 1) = \\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\cos^{M}{(M)} + 1) and e^{\\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\operatorname{n_{2}}^{M}{(M)} + 1)} = e^{\\dot{x} + v{(\\dot{x})} + \\frac{d}{d M} (M + \\cos^{M}{(M)} + 1)}", "srepr_derivation": [["get_premise", "Equality(Function('n_2')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["add", 2, 1], "Equality(Add(Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Add(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)))"], [["add", 3, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Add(Symbol('M', commutative=True), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Symbol('M', commutative=True), Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["add", 5, "Add(Symbol('\\\\dot{x}', commutative=True), Function('v')(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Function('v')(Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Symbol('M', commutative=True), Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{x}', commutative=True), Function('v')(Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Symbol('M', commutative=True), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["exp", 6], "Equality(exp(Add(Symbol('\\\\dot{x}', commutative=True), Function('v')(Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Symbol('M', commutative=True), Pow(Function('n_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))))), exp(Add(Symbol('\\\\dot{x}', commutative=True), Function('v')(Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Symbol('M', commutative=True), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{v},n_{2})} = \\mathbf{v} + n_{2}, then obtain (n_{2} - (\\mathbf{v} + n_{2})^{2} + (\\mathbf{v} + n_{2}) \\operatorname{f^{*}}{(\\mathbf{v},n_{2})})^{\\mathbf{v}} = n_{2}^{\\mathbf{v}}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{v},n_{2})} = \\mathbf{v} + n_{2} and (\\mathbf{v} + n_{2}) \\operatorname{f^{*}}{(\\mathbf{v},n_{2})} = (\\mathbf{v} + n_{2})^{2} and n_{2} + (\\mathbf{v} + n_{2}) \\operatorname{f^{*}}{(\\mathbf{v},n_{2})} = n_{2} + (\\mathbf{v} + n_{2})^{2} and n_{2} - (\\mathbf{v} + n_{2})^{2} + (\\mathbf{v} + n_{2}) \\operatorname{f^{*}}{(\\mathbf{v},n_{2})} = n_{2} and (n_{2} - (\\mathbf{v} + n_{2})^{2} + (\\mathbf{v} + n_{2}) \\operatorname{f^{*}}{(\\mathbf{v},n_{2})})^{\\mathbf{v}} = n_{2}^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Function('f^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True))), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Integer(2)))"], [["add", 2, "Symbol('n_2', commutative=True)"], "Equality(Add(Symbol('n_2', commutative=True), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Function('f^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)))), Add(Symbol('n_2', commutative=True), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Integer(2))))"], [["minus", 3, "Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Integer(2))"], "Equality(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Function('f^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True))"], [["power", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)), Function('f^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('n_2', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given Z{(z^{*})} = \\log{(\\sin{(z^{*})})}, then obtain ((Z^{2}{(z^{*})})^{z^{*}} - Z{(z^{*})})^{z^{*}} = ((Z{(z^{*})} \\log{(\\sin{(z^{*})})})^{z^{*}} - Z{(z^{*})})^{z^{*}}", "derivation": "Z{(z^{*})} = \\log{(\\sin{(z^{*})})} and Z^{2}{(z^{*})} = Z{(z^{*})} \\log{(\\sin{(z^{*})})} and (Z^{2}{(z^{*})})^{z^{*}} = (Z{(z^{*})} \\log{(\\sin{(z^{*})})})^{z^{*}} and (Z^{2}{(z^{*})})^{z^{*}} - Z{(z^{*})} = (Z{(z^{*})} \\log{(\\sin{(z^{*})})})^{z^{*}} - Z{(z^{*})} and ((Z^{2}{(z^{*})})^{z^{*}} - Z{(z^{*})})^{z^{*}} = ((Z{(z^{*})} \\log{(\\sin{(z^{*})})})^{z^{*}} - Z{(z^{*})})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True))))"], [["times", 1, "Function('Z')(Symbol('z^*', commutative=True))"], "Equality(Pow(Function('Z')(Symbol('z^*', commutative=True)), Integer(2)), Mul(Function('Z')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True)))))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Pow(Function('Z')(Symbol('z^*', commutative=True)), Integer(2)), Symbol('z^*', commutative=True)), Pow(Mul(Function('Z')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)))"], [["minus", 3, "Function('Z')(Symbol('z^*', commutative=True))"], "Equality(Add(Pow(Pow(Function('Z')(Symbol('z^*', commutative=True)), Integer(2)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('z^*', commutative=True)))), Add(Pow(Mul(Function('Z')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('z^*', commutative=True)))))"], [["power", 4, "Symbol('z^*', commutative=True)"], "Equality(Pow(Add(Pow(Pow(Function('Z')(Symbol('z^*', commutative=True)), Integer(2)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Pow(Add(Pow(Mul(Function('Z')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(F_{g},\\mathbf{J}_f)} = \\mathbf{J}_f + e^{F_{g}}, then obtain ((\\mathbf{J}_f + e^{F_{g}})^{F_{g}})^{\\mathbf{J}_f} (\\hat{\\mathbf{r}}^{F_{g}}{(F_{g},\\mathbf{J}_f)})^{\\mathbf{J}_f} = ((\\mathbf{J}_f + e^{F_{g}})^{F_{g}})^{2 \\mathbf{J}_f}", "derivation": "\\hat{\\mathbf{r}}{(F_{g},\\mathbf{J}_f)} = \\mathbf{J}_f + e^{F_{g}} and \\hat{\\mathbf{r}}^{F_{g}}{(F_{g},\\mathbf{J}_f)} = (\\mathbf{J}_f + e^{F_{g}})^{F_{g}} and (\\hat{\\mathbf{r}}^{F_{g}}{(F_{g},\\mathbf{J}_f)})^{\\mathbf{J}_f} = ((\\mathbf{J}_f + e^{F_{g}})^{F_{g}})^{\\mathbf{J}_f} and ((\\mathbf{J}_f + e^{F_{g}})^{F_{g}})^{\\mathbf{J}_f} (\\hat{\\mathbf{r}}^{F_{g}}{(F_{g},\\mathbf{J}_f)})^{\\mathbf{J}_f} = ((\\mathbf{J}_f + e^{F_{g}})^{F_{g}})^{2 \\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('F_g', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 3, "Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\phi)} = \\log{(\\phi)} and \\dot{z}{(\\phi)} = \\log{(\\phi)}, then obtain \\frac{\\frac{d}{d \\phi} \\mathbf{B}{(\\phi)}}{\\phi} - \\frac{1}{\\phi^{2}} = \\frac{\\frac{d}{d \\phi} \\dot{z}{(\\phi)}}{\\phi} - \\frac{1}{\\phi^{2}}", "derivation": "\\mathbf{B}{(\\phi)} = \\log{(\\phi)} and \\frac{d}{d \\phi} \\mathbf{B}{(\\phi)} = \\frac{d}{d \\phi} \\log{(\\phi)} and \\dot{z}{(\\phi)} = \\log{(\\phi)} and \\frac{\\frac{d}{d \\phi} \\mathbf{B}{(\\phi)}}{\\phi} = \\frac{\\frac{d}{d \\phi} \\log{(\\phi)}}{\\phi} and \\frac{\\frac{d}{d \\phi} \\mathbf{B}{(\\phi)}}{\\phi} = \\frac{\\frac{d}{d \\phi} \\dot{z}{(\\phi)}}{\\phi} and - (\\frac{d}{d \\phi} \\log{(\\phi)})^{2} + \\frac{\\frac{d}{d \\phi} \\mathbf{B}{(\\phi)}}{\\phi} = - (\\frac{d}{d \\phi} \\log{(\\phi)})^{2} + \\frac{\\frac{d}{d \\phi} \\dot{z}{(\\phi)}}{\\phi} and \\frac{\\frac{d}{d \\phi} \\mathbf{B}{(\\phi)}}{\\phi} - \\frac{1}{\\phi^{2}} = \\frac{\\frac{d}{d \\phi} \\dot{z}{(\\phi)}}{\\phi} - \\frac{1}{\\phi^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["divide", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["minus", 5, "Pow(Derivative(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Derivative(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(Derivative(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given x{(\\varphi^*)} = \\log{(\\log{(\\varphi^*)})}, then derive \\frac{d}{d \\varphi^*} x{(\\varphi^*)} = \\frac{1}{\\varphi^* \\log{(\\varphi^*)}}, then obtain (x{(\\varphi^*)} - \\frac{1}{\\varphi^* \\log{(\\varphi^*)}})^{\\varphi^*} = (\\log{(\\log{(\\varphi^*)})} - \\frac{1}{\\varphi^* \\log{(\\varphi^*)}})^{\\varphi^*}", "derivation": "x{(\\varphi^*)} = \\log{(\\log{(\\varphi^*)})} and \\frac{d}{d \\varphi^*} x{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\log{(\\log{(\\varphi^*)})} and \\frac{d}{d \\varphi^*} x{(\\varphi^*)} = \\frac{1}{\\varphi^* \\log{(\\varphi^*)}} and x{(\\varphi^*)} - \\frac{d}{d \\varphi^*} x{(\\varphi^*)} = \\log{(\\log{(\\varphi^*)})} - \\frac{d}{d \\varphi^*} x{(\\varphi^*)} and (x{(\\varphi^*)} - \\frac{d}{d \\varphi^*} x{(\\varphi^*)})^{\\varphi^*} = (\\log{(\\log{(\\varphi^*)})} - \\frac{d}{d \\varphi^*} x{(\\varphi^*)})^{\\varphi^*} and (x{(\\varphi^*)} - \\frac{1}{\\varphi^* \\log{(\\varphi^*)}})^{\\varphi^*} = (\\log{(\\log{(\\varphi^*)})} - \\frac{1}{\\varphi^* \\log{(\\varphi^*)}})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\varphi^*', commutative=True)), log(log(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["minus", 1, "Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Add(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Add(log(log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Add(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(log(log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Derivative(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('x')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(log(log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(i,\\Omega)} = i^{\\Omega}, then obtain \\int (- \\Omega + 2 \\Psi^{\\dagger}{(i,\\Omega)}) di = \\int (- \\Omega + i^{\\Omega} + \\Psi^{\\dagger}{(i,\\Omega)}) di", "derivation": "\\Psi^{\\dagger}{(i,\\Omega)} = i^{\\Omega} and - \\Omega + \\Psi^{\\dagger}{(i,\\Omega)} = - \\Omega + i^{\\Omega} and - \\Omega + i^{\\Omega} + \\Psi^{\\dagger}{(i,\\Omega)} = - \\Omega + 2 i^{\\Omega} and - \\Omega + 2 \\Psi^{\\dagger}{(i,\\Omega)} = - \\Omega + 2 i^{\\Omega} and - \\Omega + 2 \\Psi^{\\dagger}{(i,\\Omega)} = - \\Omega + i^{\\Omega} + \\Psi^{\\dagger}{(i,\\Omega)} and \\int (- \\Omega + 2 \\Psi^{\\dagger}{(i,\\Omega)}) di = \\int (- \\Omega + i^{\\Omega} + \\Psi^{\\dagger}{(i,\\Omega)}) di", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["minus", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(n_{2})} = \\int \\sin{(n_{2})} dn_{2}, then obtain \\operatorname{n_{1}}^{n_{2}}{(n_{2})} \\sin{(\\int \\sin{(n_{2})} dn_{2})} = (\\sin{(\\int \\sin{(n_{2})} dn_{2})}) (\\int \\sin{(n_{2})} dn_{2})^{n_{2}}", "derivation": "\\operatorname{n_{1}}{(n_{2})} = \\int \\sin{(n_{2})} dn_{2} and \\operatorname{n_{1}}^{n_{2}}{(n_{2})} = (\\int \\sin{(n_{2})} dn_{2})^{n_{2}} and \\sin{(\\operatorname{n_{1}}{(n_{2})})} = \\sin{(\\int \\sin{(n_{2})} dn_{2})} and \\operatorname{n_{1}}^{n_{2}}{(n_{2})} \\sin{(\\operatorname{n_{1}}{(n_{2})})} = \\sin{(\\operatorname{n_{1}}{(n_{2})})} (\\int \\sin{(n_{2})} dn_{2})^{n_{2}} and \\operatorname{n_{1}}^{n_{2}}{(n_{2})} \\sin{(\\int \\sin{(n_{2})} dn_{2})} = (\\sin{(\\int \\sin{(n_{2})} dn_{2})}) (\\int \\sin{(n_{2})} dn_{2})^{n_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('n_1')(Symbol('n_2', commutative=True)), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"], [["sin", 1], "Equality(sin(Function('n_1')(Symbol('n_2', commutative=True))), sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["times", 2, "sin(Function('n_1')(Symbol('n_2', commutative=True)))"], "Equality(Mul(Pow(Function('n_1')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), sin(Function('n_1')(Symbol('n_2', commutative=True)))), Mul(sin(Function('n_1')(Symbol('n_2', commutative=True))), Pow(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('n_1')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Mul(sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Pow(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\nabla{(f^{\\prime},\\rho)} = \\log{(\\frac{\\rho}{f^{\\prime}})} and S{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})}, then obtain \\log{(\\log{(\\mathbb{I})})} + \\frac{f^{\\prime} e^{\\nabla{(f^{\\prime},\\rho)}}}{\\rho} = \\log{(\\log{(\\mathbb{I})})} + 1", "derivation": "\\nabla{(f^{\\prime},\\rho)} = \\log{(\\frac{\\rho}{f^{\\prime}})} and \\nabla{(f^{\\prime},\\rho)} + 1 = \\log{(\\frac{\\rho}{f^{\\prime}})} + 1 and \\nabla{(f^{\\prime},\\rho)} - \\log{(\\frac{\\rho}{f^{\\prime}})} = 0 and S{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})} and \\frac{f^{\\prime} e^{\\nabla{(f^{\\prime},\\rho)}}}{\\rho} = 1 and S{(\\mathbb{I})} + \\frac{f^{\\prime} e^{\\nabla{(f^{\\prime},\\rho)}}}{\\rho} = S{(\\mathbb{I})} + 1 and \\log{(\\log{(\\mathbb{I})})} + \\frac{f^{\\prime} e^{\\nabla{(f^{\\prime},\\rho)}}}{\\rho} = \\log{(\\log{(\\mathbb{I})})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), log(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(1)), Add(log(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Integer(1)))"], [["minus", 2, "Add(log(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Integer(1))"], "Equality(Add(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))), Integer(0))"], ["get_premise", "Equality(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), log(log(Symbol('\\\\mathbb{I}', commutative=True))))"], [["exp", 3], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), exp(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)))), Integer(1))"], [["add", 5, "Function('S')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), exp(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True))))), Add(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(log(log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), exp(Function('\\\\nabla')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True))))), Add(log(log(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{v}{(y^{\\prime},\\phi)} = \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}}, then derive \\mathbf{v}{(y^{\\prime},\\phi)} = \\phi^{y^{\\prime}} \\log{(\\phi)}, then obtain \\cos{(\\phi^{y^{\\prime}} \\log{(\\phi)} - y^{\\prime})} = \\cos{(y^{\\prime} - \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}})}", "derivation": "\\mathbf{v}{(y^{\\prime},\\phi)} = \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}} and - y^{\\prime} + \\mathbf{v}{(y^{\\prime},\\phi)} = - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}} and \\mathbf{v}{(y^{\\prime},\\phi)} = \\phi^{y^{\\prime}} \\log{(\\phi)} and \\phi^{y^{\\prime}} \\log{(\\phi)} - y^{\\prime} = - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}} and \\cos{(\\phi^{y^{\\prime}} \\log{(\\phi)} - y^{\\prime})} = \\cos{(y^{\\prime} - \\frac{\\partial}{\\partial y^{\\prime}} \\phi^{y^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\mathbf{v}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{v}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["cos", 4], "Equality(cos(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given n{(v,s)} = s v, then obtain - s v - \\operatorname{F_{x}}{(v,s)} + n{(v,s)} - \\frac{d}{d v} 0 = - \\operatorname{F_{x}}{(v,s)} - \\frac{d}{d v} 0", "derivation": "n{(v,s)} = s v and - s v + n{(v,s)} = 0 and - s v + n{(v,s)} - \\frac{d}{d v} 0 = - \\frac{d}{d v} 0 and - s v - \\operatorname{F_{x}}{(v,s)} + n{(v,s)} - \\frac{d}{d v} 0 = - \\operatorname{F_{x}}{(v,s)} - \\frac{d}{d v} 0", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('v', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v', commutative=True)))"], [["minus", 1, "Mul(Symbol('s', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True), Symbol('v', commutative=True)), Function('n')(Symbol('v', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["minus", 2, "Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True), Symbol('v', commutative=True)), Function('n')(Symbol('v', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["minus", 3, "Function('F_x')(Symbol('v', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('F_x')(Symbol('v', commutative=True), Symbol('s', commutative=True))), Function('n')(Symbol('v', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('F_x')(Symbol('v', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))))"]]}, {"prompt": "Given r{(\\dot{x},\\mathbf{f},\\pi)} = \\frac{\\dot{x} + \\pi}{\\mathbf{f}}, then obtain \\int r{(\\dot{x},\\mathbf{f},\\pi)} \\int r{(\\dot{x},\\mathbf{f},\\pi)} d\\dot{x} d\\mathbf{f} = \\int r{(\\dot{x},\\mathbf{f},\\pi)} \\int \\frac{\\dot{x} + \\pi}{\\mathbf{f}} d\\dot{x} d\\mathbf{f}", "derivation": "r{(\\dot{x},\\mathbf{f},\\pi)} = \\frac{\\dot{x} + \\pi}{\\mathbf{f}} and \\int r{(\\dot{x},\\mathbf{f},\\pi)} d\\dot{x} = \\int \\frac{\\dot{x} + \\pi}{\\mathbf{f}} d\\dot{x} and r{(\\dot{x},\\mathbf{f},\\pi)} \\int r{(\\dot{x},\\mathbf{f},\\pi)} d\\dot{x} = r{(\\dot{x},\\mathbf{f},\\pi)} \\int \\frac{\\dot{x} + \\pi}{\\mathbf{f}} d\\dot{x} and \\int r{(\\dot{x},\\mathbf{f},\\pi)} \\int r{(\\dot{x},\\mathbf{f},\\pi)} d\\dot{x} d\\mathbf{f} = \\int r{(\\dot{x},\\mathbf{f},\\pi)} \\int \\frac{\\dot{x} + \\pi}{\\mathbf{f}} d\\dot{x} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Function('r')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(v_{x},\\sigma_x)} = \\cos{(\\frac{\\sigma_x}{v_{x}})} and \\operatorname{f_{\\mathbf{v}}}{(v_{x},\\sigma_x)} = \\frac{\\sigma_x}{v_{x}}, then obtain \\frac{\\partial}{\\partial v_{x}} \\mathbf{v}{(v_{x},\\sigma_x)} = \\frac{\\partial}{\\partial v_{x}} \\cos{(\\operatorname{f_{\\mathbf{v}}}{(v_{x},\\sigma_x)})}", "derivation": "\\mathbf{v}{(v_{x},\\sigma_x)} = \\cos{(\\frac{\\sigma_x}{v_{x}})} and \\frac{\\partial}{\\partial v_{x}} \\mathbf{v}{(v_{x},\\sigma_x)} = \\frac{\\partial}{\\partial v_{x}} \\cos{(\\frac{\\sigma_x}{v_{x}})} and \\operatorname{f_{\\mathbf{v}}}{(v_{x},\\sigma_x)} = \\frac{\\sigma_x}{v_{x}} and \\frac{\\partial}{\\partial v_{x}} \\mathbf{v}{(v_{x},\\sigma_x)} = \\frac{\\partial}{\\partial v_{x}} \\cos{(\\operatorname{f_{\\mathbf{v}}}{(v_{x},\\sigma_x)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Function('f_{\\\\mathbf{v}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(l)} = \\cos{(l)}, then derive \\frac{d}{d l} \\Omega{(l)} - 1 = - \\sin{(l)} - 1, then obtain \\frac{d}{d l} (\\frac{d}{d l} \\cos{(l)} - 1) = \\frac{d}{d l} (- \\sin{(l)} - 1)", "derivation": "\\Omega{(l)} = \\cos{(l)} and - l + \\Omega{(l)} = - l + \\cos{(l)} and \\frac{d}{d l} (- l + \\Omega{(l)}) = \\frac{d}{d l} (- l + \\cos{(l)}) and \\frac{d}{d l} \\Omega{(l)} = \\frac{d}{d l} \\cos{(l)} and \\frac{d}{d l} \\Omega{(l)} - 1 = - \\sin{(l)} - 1 and \\frac{d}{d l} \\cos{(l)} - 1 = - \\sin{(l)} - 1 and \\frac{d}{d l} (\\frac{d}{d l} \\cos{(l)} - 1) = \\frac{d}{d l} (- \\sin{(l)} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\Omega')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\Omega')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Integer(-1)))"], [["differentiate", 6, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\psi,n_{1})} = \\psi + n_{1} and \\operatorname{n_{1}}{(\\psi)} = \\psi, then obtain (\\psi + n_{1})^{- n_{1}} \\operatorname{n_{1}}{(\\psi)} = \\psi (\\psi + n_{1})^{- n_{1}}", "derivation": "\\operatorname{A_{2}}{(\\psi,n_{1})} = \\psi + n_{1} and \\operatorname{n_{1}}{(\\psi)} = \\psi and \\operatorname{A_{2}}^{- n_{1}}{(\\psi,n_{1})} \\operatorname{n_{1}}{(\\psi)} = \\psi \\operatorname{A_{2}}^{- n_{1}}{(\\psi,n_{1})} and (\\psi + n_{1})^{- n_{1}} \\operatorname{n_{1}}{(\\psi)} = \\psi (\\psi + n_{1})^{- n_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["divide", 2, "Pow(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True))), Function('n_1')(Symbol('\\\\psi', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True))), Function('n_1')(Symbol('\\\\psi', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} = \\frac{\\lambda}{\\mathbf{J}_f}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\frac{\\lambda}{\\mathbf{J}_f} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} - 1) = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\frac{2 \\lambda}{\\mathbf{J}_f} - 1)", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} = \\frac{\\lambda}{\\mathbf{J}_f} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} - 1 = \\frac{\\lambda}{\\mathbf{J}_f} - 1 and \\frac{\\lambda}{\\mathbf{J}_f} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} - 1 = \\frac{2 \\lambda}{\\mathbf{J}_f} - 1 and \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\frac{\\lambda}{\\mathbf{J}_f} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J}_f,\\lambda)} - 1) = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\frac{2 \\lambda}{\\mathbf{J}_f} - 1)", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Integer(-1)))"], [["add", 2, "Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(\\varepsilon)} = \\log{(\\varepsilon)}, then derive \\int \\pi{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + \\varphi^*, then obtain \\frac{\\partial^{2}}{\\partial \\varphi^*\\partial \\varepsilon} (\\varepsilon \\log{(\\varepsilon)} - \\varepsilon + \\varphi^*) = \\frac{d^{2}}{d \\varphi^*d \\varepsilon} \\int \\log{(\\varepsilon)} d\\varepsilon", "derivation": "\\pi{(\\varepsilon)} = \\log{(\\varepsilon)} and \\int \\pi{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\varepsilon)} d\\varepsilon and \\int \\pi{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + \\varphi^* and \\frac{d}{d \\varepsilon} \\int \\pi{(\\varepsilon)} d\\varepsilon = \\frac{d}{d \\varepsilon} \\int \\log{(\\varepsilon)} d\\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} (\\varepsilon \\log{(\\varepsilon)} - \\varepsilon + \\varphi^*) = \\frac{d}{d \\varepsilon} \\int \\log{(\\varepsilon)} d\\varepsilon and \\frac{\\partial^{2}}{\\partial \\varphi^*\\partial \\varepsilon} (\\varepsilon \\log{(\\varepsilon)} - \\varepsilon + \\varphi^*) = \\frac{d^{2}}{d \\varphi^*d \\varepsilon} \\int \\log{(\\varepsilon)} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(V,n)} = n + \\sin{(V)}, then obtain \\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}) - (\\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}))^{V} = - (\\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}))^{V} + \\frac{\\partial}{\\partial n} (V + n + \\sin{(V)})", "derivation": "\\mathbb{I}{(V,n)} = n + \\sin{(V)} and V + \\mathbb{I}{(V,n)} = V + n + \\sin{(V)} and \\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}) = \\frac{\\partial}{\\partial n} (V + n + \\sin{(V)}) and (\\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}))^{V} = (\\frac{\\partial}{\\partial n} (V + n + \\sin{(V)}))^{V} and \\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}) - (\\frac{\\partial}{\\partial n} (V + n + \\sin{(V)}))^{V} = \\frac{\\partial}{\\partial n} (V + n + \\sin{(V)}) - (\\frac{\\partial}{\\partial n} (V + n + \\sin{(V)}))^{V} and \\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}) - (\\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}))^{V} = - (\\frac{\\partial}{\\partial n} (V + \\mathbb{I}{(V,n)}))^{V} + \\frac{\\partial}{\\partial n} (V + n + \\sin{(V)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), sin(Symbol('V', commutative=True))))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["minus", 3, "Pow(Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True))"], "Equality(Add(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True)))), Add(Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Pow(Derivative(Add(Symbol('V', commutative=True), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('V', commutative=True))), Derivative(Add(Symbol('V', commutative=True), Symbol('n', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v_{y})} = \\sin{(v_{y})}, then derive \\frac{d}{d v_{y}} \\operatorname{v_{t}}{(v_{y})} = \\cos{(v_{y})}, then obtain \\cos^{v_{y}}{(v_{y})} = (\\frac{d}{d v_{y}} \\sin{(v_{y})})^{v_{y}}", "derivation": "\\operatorname{v_{t}}{(v_{y})} = \\sin{(v_{y})} and \\frac{d}{d v_{y}} \\operatorname{v_{t}}{(v_{y})} = \\frac{d}{d v_{y}} \\sin{(v_{y})} and (\\frac{d}{d v_{y}} \\operatorname{v_{t}}{(v_{y})})^{v_{y}} = (\\frac{d}{d v_{y}} \\sin{(v_{y})})^{v_{y}} and \\frac{d}{d v_{y}} \\operatorname{v_{t}}{(v_{y})} = \\cos{(v_{y})} and \\cos^{v_{y}}{(v_{y})} = (\\frac{d}{d v_{y}} \\sin{(v_{y})})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(Derivative(Function('v_t')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Pow(Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), cos(Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(cos(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given n{(q,v)} = q + v, then derive - m - \\frac{q^{2}}{2} - q v + \\int n{(q,v)} dq = 0, then obtain \\frac{\\partial}{\\partial m} \\cos{(m + \\frac{q^{2}}{2} + q v - \\int (q + v) dq)} = \\frac{d}{d m} 1", "derivation": "n{(q,v)} = q + v and \\int n{(q,v)} dq = \\int (q + v) dq and - \\int (q + v) dq + \\int n{(q,v)} dq = 0 and - m - \\frac{q^{2}}{2} - q v + \\int n{(q,v)} dq = 0 and - m - \\frac{q^{2}}{2} - q v + \\int (q + v) dq = 0 and \\cos{(m + \\frac{q^{2}}{2} + q v - \\int (q + v) dq)} = 1 and \\frac{\\partial}{\\partial m} \\cos{(m + \\frac{q^{2}}{2} + q v - \\int (q + v) dq)} = \\frac{d}{d m} 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('q', commutative=True), Symbol('v', commutative=True)), Add(Symbol('q', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('n')(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Function('n')(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('q', commutative=True), Symbol('v', commutative=True)), Integral(Function('n')(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('q', commutative=True), Symbol('v', commutative=True)), Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integer(0))"], [["cos", 5], "Equality(cos(Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Mul(Symbol('q', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))))), Integer(1))"], [["differentiate", 6, "Symbol('m', commutative=True)"], "Equality(Derivative(cos(Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Mul(Symbol('q', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('q', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('q', commutative=True)))))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(G,s)} = G - s, then derive - \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)} = -1, then derive (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)})^{G} = (-1)^{G}, then obtain (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)} \\frac{\\partial^{\\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)}}}{\\partial G^{\\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)}}} \\mathbb{I}{(G,s)})^{G} = (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)})^{G}", "derivation": "\\mathbb{I}{(G,s)} = G - s and - \\mathbb{I}{(G,s)} = - G + s and \\frac{\\partial}{\\partial G} - \\mathbb{I}{(G,s)} = \\frac{\\partial}{\\partial G} (- G + s) and (\\frac{\\partial}{\\partial G} - \\mathbb{I}{(G,s)})^{G} = (\\frac{\\partial}{\\partial G} (- G + s))^{G} and - \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)} = -1 and (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)})^{G} = (-1)^{G} and (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)} \\frac{\\partial^{\\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)}}}{\\partial G^{\\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)}}} \\mathbb{I}{(G,s)})^{G} = (- \\frac{\\partial}{\\partial G} \\mathbb{I}{(G,s)})^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('s', commutative=True)))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Integer(-1))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True)), Pow(Integer(-1), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))), Symbol('G', commutative=True)), Pow(Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('G', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(J,\\dot{y})} = \\frac{J}{\\dot{y}}, then obtain \\int (- \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{- \\dot{y} + \\operatorname{v_{t}}{(J,\\dot{y})}}{\\dot{y}}) dJ = \\int (- \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{\\frac{J}{\\dot{y}} - \\dot{y}}{\\dot{y}}) dJ", "derivation": "\\operatorname{v_{t}}{(J,\\dot{y})} = \\frac{J}{\\dot{y}} and - \\dot{y} + \\operatorname{v_{t}}{(J,\\dot{y})} = \\frac{J}{\\dot{y}} - \\dot{y} and \\frac{- \\dot{y} + \\operatorname{v_{t}}{(J,\\dot{y})}}{\\dot{y}} = \\frac{\\frac{J}{\\dot{y}} - \\dot{y}}{\\dot{y}} and - \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{- \\dot{y} + \\operatorname{v_{t}}{(J,\\dot{y})}}{\\dot{y}} = - \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{\\frac{J}{\\dot{y}} - \\dot{y}}{\\dot{y}} and \\int (- \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{- \\dot{y} + \\operatorname{v_{t}}{(J,\\dot{y})}}{\\dot{y}}) dJ = \\int (- \\frac{J}{\\dot{y}} + \\dot{y} + \\frac{\\frac{J}{\\dot{y}} - \\dot{y}}{\\dot{y}}) dJ", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], [["divide", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))))"], [["minus", 3, "Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Symbol('\\\\dot{y}', commutative=True), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Symbol('\\\\dot{y}', commutative=True), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Symbol('\\\\dot{y}', commutative=True), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Symbol('\\\\dot{y}', commutative=True), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\varphi^*,U)} = \\frac{\\varphi^*}{U} and \\hat{X}{(\\varphi^*,U)} = - \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^*, then obtain (- \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^*)^{\\varphi^*} = (- \\int \\frac{\\varphi^*}{U} d\\varphi^*)^{\\varphi^*}", "derivation": "\\mathbf{P}{(\\varphi^*,U)} = \\frac{\\varphi^*}{U} and \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^* = \\int \\frac{\\varphi^*}{U} d\\varphi^* and \\hat{X}{(\\varphi^*,U)} = - \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^* and \\hat{X}^{\\varphi^*}{(\\varphi^*,U)} = (- \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^*)^{\\varphi^*} and \\hat{X}^{\\varphi^*}{(\\varphi^*,U)} = (- \\int \\frac{\\varphi^*}{U} d\\varphi^*)^{\\varphi^*} and (- \\int \\mathbf{P}{(\\varphi^*,U)} d\\varphi^*)^{\\varphi^*} = (- \\int \\frac{\\varphi^*}{U} d\\varphi^*)^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["power", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Integer(-1), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Integer(-1), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Mul(Integer(-1), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Integer(-1), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(p)} = e^{p}, then obtain \\frac{d}{d p} \\int (2 \\Psi_{nl}{(p)} + (e^{p})^{p}) dp = \\frac{d}{d p} \\int (\\Psi_{nl}{(p)} + e^{p} + (e^{p})^{p}) dp", "derivation": "\\Psi_{nl}{(p)} = e^{p} and \\Psi_{nl}^{p}{(p)} = (e^{p})^{p} and 2 \\Psi_{nl}{(p)} = \\Psi_{nl}{(p)} + e^{p} and 2 \\Psi_{nl}{(p)} + \\Psi_{nl}^{p}{(p)} = \\Psi_{nl}{(p)} + \\Psi_{nl}^{p}{(p)} + e^{p} and 2 \\Psi_{nl}{(p)} + (e^{p})^{p} = \\Psi_{nl}{(p)} + e^{p} + (e^{p})^{p} and \\int (2 \\Psi_{nl}{(p)} + (e^{p})^{p}) dp = \\int (\\Psi_{nl}{(p)} + e^{p} + (e^{p})^{p}) dp and \\frac{d}{d p} \\int (2 \\Psi_{nl}{(p)} + (e^{p})^{p}) dp = \\frac{d}{d p} \\int (\\Psi_{nl}{(p)} + e^{p} + (e^{p})^{p}) dp", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True))))"], [["add", 3, "Pow(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Pow(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), exp(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["integrate", 5, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 6, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(f)} = \\sin{(e^{f})} and \\omega{(f)} = (\\delta^{f}{(f)})^{f} \\delta^{f}{(f)}, then obtain - U + \\rho_f + \\omega^{f}{(f)} = - U + \\rho_f + ((\\sin^{f}{(e^{f})})^{f} \\sin^{f}{(e^{f})})^{f}", "derivation": "\\delta{(f)} = \\sin{(e^{f})} and \\delta^{f}{(f)} = \\sin^{f}{(e^{f})} and \\omega{(f)} = (\\delta^{f}{(f)})^{f} \\delta^{f}{(f)} and \\omega^{f}{(f)} = ((\\delta^{f}{(f)})^{f} \\delta^{f}{(f)})^{f} and \\omega^{f}{(f)} = ((\\sin^{f}{(e^{f})})^{f} \\sin^{f}{(e^{f})})^{f} and - U + \\rho_f + \\omega^{f}{(f)} = - U + \\rho_f + ((\\sin^{f}{(e^{f})})^{f} \\sin^{f}{(e^{f})})^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('f', commutative=True)), sin(exp(Symbol('f', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('f', commutative=True)), Mul(Pow(Pow(Function('\\\\delta')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\delta')(Symbol('f', commutative=True)), Symbol('f', commutative=True))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Mul(Pow(Pow(Function('\\\\delta')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\delta')(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Mul(Pow(Pow(sin(exp(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["minus", 5, "Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_f', commutative=True), Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_f', commutative=True), Pow(Mul(Pow(Pow(sin(exp(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Symbol('f', commutative=True))), Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(m_{s},z)} = \\frac{m_{s}}{z} and \\mathbf{D}{(m_{s},z)} = \\frac{\\partial}{\\partial z} \\operatorname{F_{N}}{(m_{s},z)}, then obtain (\\mathbf{D}^{z}{(m_{s},z)})^{z} = ((\\frac{\\partial}{\\partial z} \\frac{m_{s}}{z})^{z})^{z}", "derivation": "\\operatorname{F_{N}}{(m_{s},z)} = \\frac{m_{s}}{z} and \\mathbf{D}{(m_{s},z)} = \\frac{\\partial}{\\partial z} \\operatorname{F_{N}}{(m_{s},z)} and \\mathbf{D}{(m_{s},z)} = \\frac{\\partial}{\\partial z} \\frac{m_{s}}{z} and \\mathbf{D}^{z}{(m_{s},z)} = (\\frac{\\partial}{\\partial z} \\frac{m_{s}}{z})^{z} and (\\mathbf{D}^{z}{(m_{s},z)})^{z} = ((\\frac{\\partial}{\\partial z} \\frac{m_{s}}{z})^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('m_s', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Derivative(Function('F_N')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{D}')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Derivative(Mul(Symbol('m_s', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Derivative(Mul(Symbol('m_s', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{D}')(Symbol('m_s', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Derivative(Mul(Symbol('m_s', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(p,\\mathbf{r})} = \\mathbf{r} + p, then obtain \\ddot{x}{(p,\\mathbf{r})} \\frac{\\partial}{\\partial \\mathbf{r}} \\ddot{x}{(p,\\mathbf{r})} = \\ddot{x}{(p,\\mathbf{r})}", "derivation": "\\ddot{x}{(p,\\mathbf{r})} = \\mathbf{r} + p and \\frac{\\partial}{\\partial \\mathbf{r}} \\ddot{x}{(p,\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} (\\mathbf{r} + p) and \\ddot{x}{(p,\\mathbf{r})} \\frac{\\partial}{\\partial \\mathbf{r}} \\ddot{x}{(p,\\mathbf{r})} = \\ddot{x}{(p,\\mathbf{r})} \\frac{\\partial}{\\partial \\mathbf{r}} (\\mathbf{r} + p) and \\ddot{x}{(p,\\mathbf{r})} \\frac{\\partial}{\\partial \\mathbf{r}} \\ddot{x}{(p,\\mathbf{r})} = \\ddot{x}{(p,\\mathbf{r})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Function('\\\\ddot{x}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(q,\\rho)} = \\frac{\\rho}{q}, then derive \\frac{\\partial}{\\partial q} \\operatorname{E_{\\lambda}}{(q,\\rho)} = - \\frac{\\rho}{q^{2}}, then obtain (- \\frac{\\operatorname{E_{\\lambda}}{(q,\\rho)}}{q})^{\\rho} = (\\frac{\\partial}{\\partial q} \\frac{\\rho}{q})^{\\rho}", "derivation": "\\operatorname{E_{\\lambda}}{(q,\\rho)} = \\frac{\\rho}{q} and \\frac{\\operatorname{E_{\\lambda}}{(q,\\rho)}}{q} = \\frac{\\rho}{q^{2}} and \\frac{\\partial}{\\partial q} \\operatorname{E_{\\lambda}}{(q,\\rho)} = \\frac{\\partial}{\\partial q} \\frac{\\rho}{q} and (\\frac{\\partial}{\\partial q} \\operatorname{E_{\\lambda}}{(q,\\rho)})^{\\rho} = (\\frac{\\partial}{\\partial q} \\frac{\\rho}{q})^{\\rho} and \\frac{\\partial}{\\partial q} \\operatorname{E_{\\lambda}}{(q,\\rho)} = - \\frac{\\rho}{q^{2}} and (- \\frac{\\rho}{q^{2}})^{\\rho} = (\\frac{\\partial}{\\partial q} \\frac{\\rho}{q})^{\\rho} and (- \\frac{\\operatorname{E_{\\lambda}}{(q,\\rho)}}{q})^{\\rho} = (\\frac{\\partial}{\\partial q} \\frac{\\rho}{q})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-2))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Derivative(Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-2))), Symbol('\\\\rho', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(V)} = e^{e^{V}}, then obtain \\int \\frac{d}{d V} \\operatorname{f_{\\mathbf{v}}}^{V}{(V)} dV = \\int \\frac{d}{d V} (e^{e^{V}})^{V} dV", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(V)} = e^{e^{V}} and \\operatorname{f_{\\mathbf{v}}}^{V}{(V)} = (e^{e^{V}})^{V} and \\frac{d}{d V} \\operatorname{f_{\\mathbf{v}}}^{V}{(V)} = \\frac{d}{d V} (e^{e^{V}})^{V} and \\int \\frac{d}{d V} \\operatorname{f_{\\mathbf{v}}}^{V}{(V)} dV = \\int \\frac{d}{d V} (e^{e^{V}})^{V} dV", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True)), exp(exp(Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(exp(exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(exp(exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Pow(exp(exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then obtain \\frac{\\frac{d}{d \\eta^{\\prime}} \\lambda^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\frac{d}{d \\eta^{\\prime}} \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}}", "derivation": "\\lambda{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and \\lambda^{\\eta^{\\prime}}{(\\eta^{\\prime})} = \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\lambda^{\\eta^{\\prime}}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})} and \\frac{\\frac{d}{d \\eta^{\\prime}} \\lambda^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\frac{d}{d \\eta^{\\prime}} \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Derivative(Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(x^\\prime,y^{\\prime})} = x^\\prime y^{\\prime}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{g}{(x^\\prime,y^{\\prime})} = x^\\prime, then obtain 2 x^\\prime - y^{\\prime} = x^\\prime - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} x^\\prime y^{\\prime}", "derivation": "\\mathbf{g}{(x^\\prime,y^{\\prime})} = x^\\prime y^{\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{g}{(x^\\prime,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} x^\\prime y^{\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{g}{(x^\\prime,y^{\\prime})} = x^\\prime and x^\\prime + \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{g}{(x^\\prime,y^{\\prime})} = x^\\prime + \\frac{\\partial}{\\partial y^{\\prime}} x^\\prime y^{\\prime} and 2 x^\\prime = x^\\prime + \\frac{\\partial}{\\partial y^{\\prime}} x^\\prime y^{\\prime} and 2 x^\\prime - y^{\\prime} = x^\\prime - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} x^\\prime y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))"], [["add", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Derivative(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Add(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["minus", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('x^\\\\prime', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(A_{z},\\mathbf{M})} = \\mathbf{M}^{A_{z}} and y{(A_{z},\\mathbf{M})} = 2 \\mathbf{g}{(A_{z},\\mathbf{M})}, then obtain \\int \\cos{(y{(A_{z},\\mathbf{M})})} dA_{z} = \\int \\cos{(2 \\mathbf{M}^{A_{z}})} dA_{z}", "derivation": "\\mathbf{g}{(A_{z},\\mathbf{M})} = \\mathbf{M}^{A_{z}} and y{(A_{z},\\mathbf{M})} = 2 \\mathbf{g}{(A_{z},\\mathbf{M})} and \\cos{(y{(A_{z},\\mathbf{M})})} = \\cos{(2 \\mathbf{g}{(A_{z},\\mathbf{M})})} and y{(A_{z},\\mathbf{M})} = 2 \\mathbf{M}^{A_{z}} and 2 \\mathbf{g}{(A_{z},\\mathbf{M})} = 2 \\mathbf{M}^{A_{z}} and \\cos{(y{(A_{z},\\mathbf{M})})} = \\cos{(2 \\mathbf{M}^{A_{z}})} and \\int \\cos{(y{(A_{z},\\mathbf{M})})} dA_{z} = \\int \\cos{(2 \\mathbf{M}^{A_{z}})} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["cos", 2], "Equality(cos(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), cos(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(cos(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), cos(Mul(Integer(2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_z', commutative=True)))))"], [["integrate", 6, "Symbol('A_z', commutative=True)"], "Equality(Integral(cos(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Integral(cos(Mul(Integer(2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})} = t_{2}^{n_{1}}, then obtain n_{1} (t_{2}^{n_{1}} + 2 \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})}) = n_{1} (2 t_{2}^{n_{1}} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})})", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})} = t_{2}^{n_{1}} and t_{2}^{n_{1}} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})} = 2 t_{2}^{n_{1}} and t_{2}^{n_{1}} + 2 \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})} = 2 t_{2}^{n_{1}} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})} and n_{1} (t_{2}^{n_{1}} + 2 \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})}) = n_{1} (2 t_{2}^{n_{1}} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},t_{2})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(2), Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True))))"], [["add", 2, "Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True)))), Add(Mul(Integer(2), Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True))))"], [["times", 3, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Add(Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True))))), Mul(Symbol('n_1', commutative=True), Add(Mul(Integer(2), Pow(Symbol('t_2', commutative=True), Symbol('n_1', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(U)} = \\cos{(e^{U})}, then obtain U (\\frac{\\int \\operatorname{A_{y}}{(U)} dU}{\\int \\cos{(e^{U})} dU} - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU}) = U (1 - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU})", "derivation": "\\operatorname{A_{y}}{(U)} = \\cos{(e^{U})} and \\int \\operatorname{A_{y}}{(U)} dU = \\int \\cos{(e^{U})} dU and \\frac{\\int \\operatorname{A_{y}}{(U)} dU}{\\int \\cos{(e^{U})} dU} = 1 and \\frac{\\int \\operatorname{A_{y}}{(U)} dU}{\\int \\cos{(e^{U})} dU} - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU} = 1 - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU} and U (\\frac{\\int \\operatorname{A_{y}}{(U)} dU}{\\int \\cos{(e^{U})} dU} - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU}) = U (1 - \\frac{\\int \\cos{(e^{U})} dU}{\\int \\operatorname{A_{y}}{(U)} dU})", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('U', commutative=True)), cos(exp(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["divide", 2, "Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Pow(Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Mul(Pow(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], "Equality(Add(Mul(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Pow(Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Pow(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))))"], [["times", 4, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Add(Mul(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Pow(Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))), Mul(Symbol('U', commutative=True), Add(Integer(1), Mul(Integer(-1), Pow(Integral(Function('A_y')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))))"]]}, {"prompt": "Given C{(v_{2},\\phi_2)} = \\sin{(\\phi_2 - v_{2})}, then obtain - \\frac{(\\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)})^{2}}{v_{2}} = - \\frac{\\cos{(\\phi_2 - v_{2})} \\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)}}{v_{2}}", "derivation": "C{(v_{2},\\phi_2)} = \\sin{(\\phi_2 - v_{2})} and \\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)} = \\frac{\\partial}{\\partial \\phi_2} \\sin{(\\phi_2 - v_{2})} and - \\frac{(\\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)})^{2}}{v_{2}} = - \\frac{\\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)} \\frac{\\partial}{\\partial \\phi_2} \\sin{(\\phi_2 - v_{2})}}{v_{2}} and - \\frac{(\\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)})^{2}}{v_{2}} = - \\frac{\\cos{(\\phi_2 - v_{2})} \\frac{\\partial}{\\partial \\phi_2} C{(v_{2},\\phi_2)}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["times", 2, "Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Derivative(Function('C')(Symbol('v_2', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(g_{\\varepsilon},A)} = A g_{\\varepsilon}, then obtain - (- A g_{\\varepsilon} \\operatorname{F_{x}}{(g_{\\varepsilon},A)} + \\operatorname{F_{x}}^{2}{(g_{\\varepsilon},A)}) \\psi{(g_{\\varepsilon},A)} = 0", "derivation": "\\operatorname{F_{x}}{(g_{\\varepsilon},A)} = A g_{\\varepsilon} and \\operatorname{F_{x}}^{2}{(g_{\\varepsilon},A)} = A g_{\\varepsilon} \\operatorname{F_{x}}{(g_{\\varepsilon},A)} and - A g_{\\varepsilon} \\operatorname{F_{x}}{(g_{\\varepsilon},A)} + \\operatorname{F_{x}}^{2}{(g_{\\varepsilon},A)} = 0 and (- A g_{\\varepsilon} \\operatorname{F_{x}}{(g_{\\varepsilon},A)} + \\operatorname{F_{x}}^{2}{(g_{\\varepsilon},A)}) \\psi{(g_{\\varepsilon},A)} = 0 and - (- A g_{\\varepsilon} \\operatorname{F_{x}}{(g_{\\varepsilon},A)} + \\operatorname{F_{x}}^{2}{(g_{\\varepsilon},A)}) \\psi{(g_{\\varepsilon},A)} = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Integer(2)), Mul(Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))))"], [["minus", 2, "Mul(Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))), Pow(Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Integer(2))), Integer(0))"], [["times", 3, "Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))), Pow(Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Integer(2))), Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))), Integer(0))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))), Pow(Function('F_x')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True)), Integer(2))), Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})}, then derive \\operatorname{A_{y}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then obtain (\\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})})^{2} = \\cos{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})} and \\operatorname{A_{y}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and \\operatorname{A_{y}}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})} = \\cos{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})} and \\operatorname{A_{y}}^{2}{(\\mathbf{E})} = \\operatorname{A_{y}}{(\\mathbf{E})} \\cos{(\\mathbf{E})} and (\\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})})^{2} = \\cos{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\sin{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 2, "Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), Mul(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2)), Mul(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(J)} = \\log{(J)} and \\operatorname{c_{0}}{(J)} = \\frac{\\hat{H}_l{(J)} + \\log{(J)}}{J}, then obtain - \\hat{H}_l{(J)} + \\frac{2 \\hat{H}_l{(J)}}{J} = - \\hat{H}_l{(J)} + \\operatorname{c_{0}}{(J)}", "derivation": "\\hat{H}_l{(J)} = \\log{(J)} and 2 \\hat{H}_l{(J)} = \\hat{H}_l{(J)} + \\log{(J)} and \\frac{2 \\hat{H}_l{(J)}}{J} = \\frac{\\hat{H}_l{(J)} + \\log{(J)}}{J} and - \\hat{H}_l{(J)} + \\frac{2 \\hat{H}_l{(J)}}{J} = - \\hat{H}_l{(J)} + \\frac{\\hat{H}_l{(J)} + \\log{(J)}}{J} and \\operatorname{c_{0}}{(J)} = \\frac{\\hat{H}_l{(J)} + \\log{(J)}}{J} and - \\hat{H}_l{(J)} + \\frac{2 \\hat{H}_l{(J)}}{J} = - \\hat{H}_l{(J)} + \\operatorname{c_{0}}{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}_l')(Symbol('J', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Add(Function('\\\\hat{H}_l')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True))))"], [["divide", 2, "Symbol('J', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_l')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))))"], [["minus", 3, "Function('\\\\hat{H}_l')(Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Mul(Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_l')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True))))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_l')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Mul(Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('J', commutative=True))), Function('c_0')(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(v_{2},t)} = t - v_{2}, then derive - v_{2} + (\\frac{\\partial}{\\partial t} \\operatorname{A_{x}}{(v_{2},t)})^{v_{2}} = 1 - v_{2}, then obtain - v_{2} + (\\frac{\\partial}{\\partial t} (t - v_{2}))^{v_{2}} = 1 - v_{2}", "derivation": "\\operatorname{A_{x}}{(v_{2},t)} = t - v_{2} and \\frac{\\partial}{\\partial t} \\operatorname{A_{x}}{(v_{2},t)} = \\frac{\\partial}{\\partial t} (t - v_{2}) and (\\frac{\\partial}{\\partial t} \\operatorname{A_{x}}{(v_{2},t)})^{v_{2}} = (\\frac{\\partial}{\\partial t} (t - v_{2}))^{v_{2}} and - v_{2} + (\\frac{\\partial}{\\partial t} \\operatorname{A_{x}}{(v_{2},t)})^{v_{2}} = - v_{2} + (\\frac{\\partial}{\\partial t} (t - v_{2}))^{v_{2}} and - v_{2} + (\\frac{\\partial}{\\partial t} \\operatorname{A_{x}}{(v_{2},t)})^{v_{2}} = 1 - v_{2} and - v_{2} + (\\frac{\\partial}{\\partial t} (t - v_{2}))^{v_{2}} = 1 - v_{2}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('v_2', commutative=True), Symbol('t', commutative=True)), Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('v_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_2', commutative=True)"], "Equality(Pow(Derivative(Function('A_x')(Symbol('v_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True)), Pow(Derivative(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Derivative(Function('A_x')(Symbol('v_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Derivative(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Derivative(Function('A_x')(Symbol('v_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Derivative(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('v_2', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(A_{2},n)} = A_{2} + n, then obtain (A_{2} + \\mathbf{H}{(A_{2},n)})^{2} \\mathbf{H}^{2}{(A_{2},n)} = (2 A_{2} + n)^{2} \\mathbf{H}^{2}{(A_{2},n)}", "derivation": "\\mathbf{H}{(A_{2},n)} = A_{2} + n and A_{2} + \\mathbf{H}{(A_{2},n)} = 2 A_{2} + n and (A_{2} + n) (A_{2} + \\mathbf{H}{(A_{2},n)}) = (A_{2} + n) (2 A_{2} + n) and (A_{2} + \\mathbf{H}{(A_{2},n)}) \\mathbf{H}{(A_{2},n)} = (2 A_{2} + n) \\mathbf{H}{(A_{2},n)} and (A_{2} + \\mathbf{H}{(A_{2},n)})^{2} \\mathbf{H}^{2}{(A_{2},n)} = (2 A_{2} + n)^{2} \\mathbf{H}^{2}{(A_{2},n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('n', commutative=True)))"], [["times", 2, "Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)), Add(Symbol('A_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))), Mul(Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Symbol('A_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True))), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('n', commutative=True)), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True))), Integer(2)), Pow(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('n', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('n', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\ddot{x}{(\\theta,F_{H})} = F_{H} \\theta, then obtain ((\\frac{\\partial}{\\partial F_{H}} \\log{(\\theta \\ddot{x}{(\\theta,F_{H})})})^{F_{H}})^{\\theta} = ((\\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} \\theta^{2})})^{F_{H}})^{\\theta}", "derivation": "\\ddot{x}{(\\theta,F_{H})} = F_{H} \\theta and \\theta \\ddot{x}{(\\theta,F_{H})} = F_{H} \\theta^{2} and \\log{(\\theta \\ddot{x}{(\\theta,F_{H})})} = \\log{(F_{H} \\theta^{2})} and \\frac{\\partial}{\\partial F_{H}} \\log{(\\theta \\ddot{x}{(\\theta,F_{H})})} = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} \\theta^{2})} and (\\frac{\\partial}{\\partial F_{H}} \\log{(\\theta \\ddot{x}{(\\theta,F_{H})})})^{F_{H}} = (\\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} \\theta^{2})})^{F_{H}} and ((\\frac{\\partial}{\\partial F_{H}} \\log{(\\theta \\ddot{x}{(\\theta,F_{H})})})^{F_{H}})^{\\theta} = ((\\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} \\theta^{2})})^{F_{H}})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2))))"], [["log", 2], "Equality(log(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), log(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(log(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["power", 4, "Symbol('F_H', commutative=True)"], "Equality(Pow(Derivative(log(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Pow(Derivative(log(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)))"], [["power", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Pow(Derivative(log(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Pow(Derivative(log(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(a,F_{x},I)} = \\frac{- F_{x} + I}{a} and \\psi^{*}{(a,F_{x},I)} = \\varepsilon_{0}^{I}{(a,F_{x},I)}, then obtain ((\\frac{- F_{x} + I}{a})^{I})^{I} = (\\varepsilon_{0}^{I}{(a,F_{x},I)})^{I}", "derivation": "\\varepsilon_{0}{(a,F_{x},I)} = \\frac{- F_{x} + I}{a} and \\varepsilon_{0}^{I}{(a,F_{x},I)} = (\\frac{- F_{x} + I}{a})^{I} and \\psi^{*}{(a,F_{x},I)} = \\varepsilon_{0}^{I}{(a,F_{x},I)} and \\psi^{*}^{I}{(a,F_{x},I)} = (\\varepsilon_{0}^{I}{(a,F_{x},I)})^{I} and \\psi^{*}{(a,F_{x},I)} = (\\frac{- F_{x} + I}{a})^{I} and ((\\frac{- F_{x} + I}{a})^{I})^{I} = (\\varepsilon_{0}^{I}{(a,F_{x},I)})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('I', commutative=True))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Pow(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('I', commutative=True))), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Pow(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('F_x', commutative=True), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(C_{2})} = \\log{(C_{2})} and C{(C_{2})} = C_{2}, then obtain \\log{(C_{2})} - \\cos{(C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})})} = \\log{(C_{2})} - \\cos{(2 C_{2}^{C_{2}})}", "derivation": "\\operatorname{f^{\\prime}}{(C_{2})} = \\log{(C_{2})} and C{(C_{2})} = C_{2} and C^{C_{2}}{(C_{2})} = C_{2}^{C_{2}} and C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})} = 2 C_{2}^{C_{2}} and \\cos{(C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})})} = \\cos{(2 C_{2}^{C_{2}})} and - \\operatorname{f^{\\prime}}{(C_{2})} + \\cos{(C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})})} = - \\operatorname{f^{\\prime}}{(C_{2})} + \\cos{(2 C_{2}^{C_{2}})} and - \\log{(C_{2})} + \\cos{(C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})})} = - \\log{(C_{2})} + \\cos{(2 C_{2}^{C_{2}})} and \\log{(C_{2})} - \\cos{(C_{2}^{C_{2}} + C^{C_{2}}{(C_{2})})} = \\log{(C_{2})} - \\cos{(2 C_{2}^{C_{2}})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)))"], [["add", 3, "Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Mul(Integer(2), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True))))"], [["cos", 4], "Equality(cos(Add(Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), cos(Mul(Integer(2), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)))))"], [["minus", 5, "Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))), cos(Add(Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))), cos(Mul(Integer(2), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), log(Symbol('C_2', commutative=True))), cos(Add(Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))), Add(Mul(Integer(-1), log(Symbol('C_2', commutative=True))), cos(Mul(Integer(2), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True))))))"], [["divide", 7, "Integer(-1)"], "Equality(Add(log(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Add(Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))), Add(log(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Mul(Integer(2), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(b,\\sigma_p)} = \\sigma_p^{b}, then derive \\frac{\\partial^{2}}{\\partial b^{2}} \\operatorname{r_{0}}{(b,\\sigma_p)} = \\sigma_p^{b} \\log{(\\sigma_p)}^{2}, then obtain \\frac{\\partial^{2}}{\\partial b^{2}} \\sigma_p^{b} = \\sigma_p^{b} \\log{(\\sigma_p)}^{2}", "derivation": "\\operatorname{r_{0}}{(b,\\sigma_p)} = \\sigma_p^{b} and \\frac{\\partial}{\\partial b} \\operatorname{r_{0}}{(b,\\sigma_p)} = \\frac{\\partial}{\\partial b} \\sigma_p^{b} and \\frac{\\partial^{2}}{\\partial b^{2}} \\operatorname{r_{0}}{(b,\\sigma_p)} = \\frac{\\partial^{2}}{\\partial b^{2}} \\sigma_p^{b} and \\frac{\\partial^{2}}{\\partial b^{2}} \\operatorname{r_{0}}{(b,\\sigma_p)} = \\sigma_p^{b} \\log{(\\sigma_p)}^{2} and \\frac{\\partial^{2}}{\\partial b^{2}} \\sigma_p^{b} = \\sigma_p^{b} \\log{(\\sigma_p)}^{2}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(2))))"]]}, {"prompt": "Given i{(\\chi)} = \\sin{(\\chi)} and \\mathbf{S}{(\\chi)} = - \\chi - 2 i{(\\chi)} + 2 \\sin{(\\chi)}, then obtain - \\mathbf{S}{(\\chi)} - 2 i{(\\chi)} = - \\mathbf{S}{(\\chi)} - 2 \\sin{(\\chi)}", "derivation": "i{(\\chi)} = \\sin{(\\chi)} and - \\chi + i{(\\chi)} = - \\chi + \\sin{(\\chi)} and - \\chi + 2 i{(\\chi)} = - \\chi + i{(\\chi)} + \\sin{(\\chi)} and - \\chi + 2 i{(\\chi)} = - \\chi + 2 \\sin{(\\chi)} and - \\chi = - \\chi - 2 i{(\\chi)} + 2 \\sin{(\\chi)} and \\mathbf{S}{(\\chi)} = - \\chi - 2 i{(\\chi)} + 2 \\sin{(\\chi)} and - \\chi = \\mathbf{S}{(\\chi)} and \\mathbf{S}{(\\chi)} + 2 i{(\\chi)} = \\mathbf{S}{(\\chi)} + 2 \\sin{(\\chi)} and - \\mathbf{S}{(\\chi)} - 2 i{(\\chi)} = - \\mathbf{S}{(\\chi)} - 2 \\sin{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('i')(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Function('i')(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('i')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('i')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('i')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"], [["minus", 4, "Mul(Integer(2), Function('i')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('i')(Symbol('\\\\chi', commutative=True)))), Add(Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"], [["times", 8, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(z^{*},\\mathbf{s})} = \\int (z^{*})^{\\mathbf{s}} d\\mathbf{s}, then obtain - \\mathbf{s} + z^{*} + \\frac{\\operatorname{F_{x}}{(z^{*},\\mathbf{s})}}{\\int (z^{*})^{\\mathbf{s}} d\\mathbf{s}} = - \\mathbf{s} + z^{*} + 1", "derivation": "\\operatorname{F_{x}}{(z^{*},\\mathbf{s})} = \\int (z^{*})^{\\mathbf{s}} d\\mathbf{s} and \\frac{\\operatorname{F_{x}}{(z^{*},\\mathbf{s})}}{\\int (z^{*})^{\\mathbf{s}} d\\mathbf{s}} = 1 and z^{*} + \\frac{\\operatorname{F_{x}}{(z^{*},\\mathbf{s})}}{\\int (z^{*})^{\\mathbf{s}} d\\mathbf{s}} = z^{*} + 1 and - \\mathbf{s} + z^{*} + \\frac{\\operatorname{F_{x}}{(z^{*},\\mathbf{s})}}{\\int (z^{*})^{\\mathbf{s}} d\\mathbf{s}} = - \\mathbf{s} + z^{*} + 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Pow(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 1, "Integral(Pow(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Function('F_x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Pow(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Mul(Function('F_x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Pow(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)))), Add(Symbol('z^*', commutative=True), Integer(1)))"], [["minus", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('z^*', commutative=True), Mul(Function('F_x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Pow(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('z^*', commutative=True), Integer(1)))"]]}, {"prompt": "Given L{(i,a)} = \\sin{(\\frac{i}{a})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(i,a)} = (\\frac{L{(i,a)}}{i} - \\frac{1}{a})^{i} L{(i,a)}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(i,a)} = (\\frac{\\sin{(\\frac{i}{a})}}{i} - \\frac{1}{a})^{i} L{(i,a)}", "derivation": "L{(i,a)} = \\sin{(\\frac{i}{a})} and \\frac{L{(i,a)}}{i} = \\frac{\\sin{(\\frac{i}{a})}}{i} and \\frac{L{(i,a)}}{i} - \\frac{1}{a} = \\frac{\\sin{(\\frac{i}{a})}}{i} - \\frac{1}{a} and (\\frac{L{(i,a)}}{i} - \\frac{1}{a})^{i} = (\\frac{\\sin{(\\frac{i}{a})}}{i} - \\frac{1}{a})^{i} and (\\frac{L{(i,a)}}{i} - \\frac{1}{a})^{i} L{(i,a)} = (\\frac{\\sin{(\\frac{i}{a})}}{i} - \\frac{1}{a})^{i} L{(i,a)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(i,a)} = (\\frac{L{(i,a)}}{i} - \\frac{1}{a})^{i} L{(i,a)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(i,a)} = (\\frac{\\sin{(\\frac{i}{a})}}{i} - \\frac{1}{a})^{i} L{(i,a)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True))))"], [["divide", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True)))))"], [["minus", 2, "Pow(Symbol('a', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)))"], [["times", 4, "Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Symbol('i', commutative=True)), Function('L')(Symbol('i', commutative=True), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(H,\\mathbf{H})} = \\log{(H \\mathbf{H})}, then obtain (H \\mathbf{H} + \\operatorname{r_{0}}{(H,\\mathbf{H})}) \\operatorname{r_{0}}{(H,\\mathbf{H})} = (H \\mathbf{H} + \\operatorname{r_{0}}{(H,\\mathbf{H})}) \\log{(H \\mathbf{H})}", "derivation": "\\operatorname{r_{0}}{(H,\\mathbf{H})} = \\log{(H \\mathbf{H})} and H \\mathbf{H} + \\operatorname{r_{0}}{(H,\\mathbf{H})} = H \\mathbf{H} + \\log{(H \\mathbf{H})} and (H \\mathbf{H} + \\log{(H \\mathbf{H})}) \\operatorname{r_{0}}{(H,\\mathbf{H})} = (H \\mathbf{H} + \\log{(H \\mathbf{H})}) \\log{(H \\mathbf{H})} and (H \\mathbf{H} + \\operatorname{r_{0}}{(H,\\mathbf{H})}) \\operatorname{r_{0}}{(H,\\mathbf{H})} = (H \\mathbf{H} + \\operatorname{r_{0}}{(H,\\mathbf{H})}) \\log{(H \\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["times", 1, "Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('r_0')(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), log(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given a{(\\mathbf{P},G,p)} = G \\mathbf{P} p and m{(\\mathbf{P},G,p)} = - G \\mathbf{P} p, then obtain G + a^{G}{(\\mathbf{P},G,p)} + m{(\\mathbf{P},G,p)} = G + (G \\mathbf{P} p)^{G} + m{(\\mathbf{P},G,p)}", "derivation": "a{(\\mathbf{P},G,p)} = G \\mathbf{P} p and a^{G}{(\\mathbf{P},G,p)} = (G \\mathbf{P} p)^{G} and - G \\mathbf{P} p + G + a^{G}{(\\mathbf{P},G,p)} = - G \\mathbf{P} p + G + (G \\mathbf{P} p)^{G} and m{(\\mathbf{P},G,p)} = - G \\mathbf{P} p and G + a^{G}{(\\mathbf{P},G,p)} + m{(\\mathbf{P},G,p)} = G + (G \\mathbf{P} p)^{G} + m{(\\mathbf{P},G,p)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)))"], [["minus", 2, "Add(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True), Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('G', commutative=True), Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Function('m')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True))), Add(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Function('m')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('G', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(n_{2})} = e^{n_{2}} and \\operatorname{V_{\\mathbf{E}}}{(n_{2})} = n_{2} e^{n_{2}} and \\psi^{*}{(n_{2})} = n_{2} e^{n_{2}}, then obtain \\frac{d}{d n_{2}} n_{2} \\operatorname{z^{*}}{(n_{2})} = \\frac{d}{d n_{2}} \\psi^{*}{(n_{2})}", "derivation": "\\operatorname{z^{*}}{(n_{2})} = e^{n_{2}} and n_{2} \\operatorname{z^{*}}{(n_{2})} = n_{2} e^{n_{2}} and \\operatorname{V_{\\mathbf{E}}}{(n_{2})} = n_{2} e^{n_{2}} and n_{2} \\operatorname{z^{*}}{(n_{2})} = \\operatorname{V_{\\mathbf{E}}}{(n_{2})} and \\psi^{*}{(n_{2})} = n_{2} e^{n_{2}} and \\operatorname{V_{\\mathbf{E}}}{(n_{2})} = \\psi^{*}{(n_{2})} and n_{2} \\operatorname{z^{*}}{(n_{2})} = \\psi^{*}{(n_{2})} and \\frac{d}{d n_{2}} n_{2} \\operatorname{z^{*}}{(n_{2})} = \\frac{d}{d n_{2}} \\psi^{*}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["times", 1, "Symbol('n_2', commutative=True)"], "Equality(Mul(Symbol('n_2', commutative=True), Function('z^*')(Symbol('n_2', commutative=True))), Mul(Symbol('n_2', commutative=True), exp(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), exp(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('n_2', commutative=True), Function('z^*')(Symbol('n_2', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), exp(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)), Function('\\\\psi^*')(Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Symbol('n_2', commutative=True), Function('z^*')(Symbol('n_2', commutative=True))), Function('\\\\psi^*')(Symbol('n_2', commutative=True)))"], [["differentiate", 7, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('n_2', commutative=True), Function('z^*')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Function('\\\\psi^*')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(r,u)} = \\frac{r}{u}, then derive \\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)} = - \\frac{r}{u^{2}}, then obtain r + \\frac{r}{(\\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)})^{2}} = r + \\frac{u^{4}}{r}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(r,u)} = \\frac{r}{u} and \\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)} = \\frac{\\partial}{\\partial u} \\frac{r}{u} and \\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)} = - \\frac{r}{u^{2}} and \\frac{1}{(\\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)})^{2}} = \\frac{u^{4}}{r^{2}} and \\frac{r}{(\\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)})^{2}} = \\frac{u^{4}}{r} and r + \\frac{r}{(\\frac{\\partial}{\\partial u} \\operatorname{f_{\\mathbf{v}}}{(r,u)})^{2}} = r + \\frac{u^{4}}{r}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('r', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Symbol('r', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('r', commutative=True), Pow(Symbol('u', commutative=True), Integer(-2))))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-2)), Mul(Pow(Symbol('r', commutative=True), Integer(-2)), Pow(Symbol('u', commutative=True), Integer(4))))"], [["times", 4, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-2))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(4))))"], [["add", 5, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Mul(Symbol('r', commutative=True), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-2)))), Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(4)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbb{I},v_{1})} = v_{1}^{\\mathbb{I}}, then obtain ((\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{2}) \\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I} = (\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{3}", "derivation": "\\mathbf{J}_P{(\\mathbb{I},v_{1})} = v_{1}^{\\mathbb{I}} and \\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I} = \\int v_{1}^{\\mathbb{I}} d\\mathbb{I} and (\\int v_{1}^{\\mathbb{I}} d\\mathbb{I}) \\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I} = (\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{2} and (\\int v_{1}^{\\mathbb{I}} d\\mathbb{I}) (\\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I})^{2} = ((\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{2}) \\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I} and ((\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{2}) \\int \\mathbf{J}_P{(\\mathbb{I},v_{1})} d\\mathbb{I} = (\\int v_{1}^{\\mathbb{I}} d\\mathbb{I})^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 2, "Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Pow(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], "Equality(Mul(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Pow(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2))), Mul(Pow(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Pow(Integral(Pow(Symbol('v_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(3)))"]]}, {"prompt": "Given \\mathbf{A}{(M)} = \\sin{(M)} and \\operatorname{A_{z}}{(M)} = \\frac{1}{\\mathbf{A}{(M)}}, then obtain \\int \\operatorname{A_{z}}{(M)} dM = E_{n} + \\frac{\\log{(\\cos{(M)} - 1)}}{2} - \\frac{\\log{(\\cos{(M)} + 1)}}{2}", "derivation": "\\mathbf{A}{(M)} = \\sin{(M)} and \\operatorname{A_{z}}{(M)} = \\frac{1}{\\mathbf{A}{(M)}} and \\operatorname{A_{z}}{(M)} = \\frac{1}{\\sin{(M)}} and \\int \\operatorname{A_{z}}{(M)} dM = \\int \\frac{1}{\\sin{(M)}} dM and \\int \\operatorname{A_{z}}{(M)} dM = E_{n} + \\frac{\\log{(\\cos{(M)} - 1)}}{2} - \\frac{\\log{(\\cos{(M)} + 1)}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('M', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('M', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_z')(Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(sin(Symbol('M', commutative=True)), Integer(-1)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), log(Add(cos(Symbol('M', commutative=True)), Integer(-1)))), Mul(Integer(-1), Rational(1, 2), log(Add(cos(Symbol('M', commutative=True)), Integer(1))))))"]]}, {"prompt": "Given \\rho_{b}{(v_{y},\\hat{p},t)} = \\hat{p} + v_{y}^{t}, then obtain \\frac{(\\frac{\\partial}{\\partial v_{y}} t \\rho_{b}{(v_{y},\\hat{p},t)})^{t}}{v_{y}} = \\frac{(\\frac{\\partial}{\\partial v_{y}} t (\\hat{p} + v_{y}^{t}))^{t}}{v_{y}}", "derivation": "\\rho_{b}{(v_{y},\\hat{p},t)} = \\hat{p} + v_{y}^{t} and t \\rho_{b}{(v_{y},\\hat{p},t)} = t (\\hat{p} + v_{y}^{t}) and \\frac{\\partial}{\\partial v_{y}} t \\rho_{b}{(v_{y},\\hat{p},t)} = \\frac{\\partial}{\\partial v_{y}} t (\\hat{p} + v_{y}^{t}) and (\\frac{\\partial}{\\partial v_{y}} t \\rho_{b}{(v_{y},\\hat{p},t)})^{t} = (\\frac{\\partial}{\\partial v_{y}} t (\\hat{p} + v_{y}^{t}))^{t} and \\frac{(\\frac{\\partial}{\\partial v_{y}} t \\rho_{b}{(v_{y},\\hat{p},t)})^{t}}{v_{y}} = \\frac{(\\frac{\\partial}{\\partial v_{y}} t (\\hat{p} + v_{y}^{t}))^{t}}{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('t', commutative=True))))"], [["times", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\rho_b')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('t', commutative=True)))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Symbol('t', commutative=True), Function('\\\\rho_b')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('t', commutative=True), Function('\\\\rho_b')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('t', commutative=True)))"], [["divide", 4, "Symbol('v_y', commutative=True)"], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('t', commutative=True), Function('\\\\rho_b')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('t', commutative=True))), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\varphi)} = \\log{(\\varphi)}, then obtain \\int \\frac{\\log{(\\varphi)}^{- \\varphi} \\sin{(\\mathbf{v}{(\\varphi)})}}{\\sin{(\\log{(\\varphi)})}} d\\varphi = \\int \\log{(\\varphi)}^{- \\varphi} d\\varphi", "derivation": "\\mathbf{v}{(\\varphi)} = \\log{(\\varphi)} and \\sin{(\\mathbf{v}{(\\varphi)})} = \\sin{(\\log{(\\varphi)})} and \\frac{\\sin{(\\mathbf{v}{(\\varphi)})}}{\\sin{(\\log{(\\varphi)})}} = 1 and \\frac{\\log{(\\varphi)}^{- \\varphi} \\sin{(\\mathbf{v}{(\\varphi)})}}{\\sin{(\\log{(\\varphi)})}} = \\log{(\\varphi)}^{- \\varphi} and \\int \\frac{\\log{(\\varphi)}^{- \\varphi} \\sin{(\\mathbf{v}{(\\varphi)})}}{\\sin{(\\log{(\\varphi)})}} d\\varphi = \\int \\log{(\\varphi)}^{- \\varphi} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))), sin(log(Symbol('\\\\varphi', commutative=True))))"], [["divide", 2, "sin(log(Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(sin(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))), Pow(sin(log(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 3, "Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), sin(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))), Pow(sin(log(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Mul(Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), sin(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))), Pow(sin(log(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\hat{p},k)} = \\hat{p} k and \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} = \\hat{p} k, then obtain \\iint \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} dk d\\hat{p} = \\iint \\hat{p} k dk d\\hat{p}", "derivation": "\\mathbf{B}{(\\hat{p},k)} = \\hat{p} k and \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} = \\hat{p} k and \\int \\mathbf{B}{(\\hat{p},k)} dk = \\int \\hat{p} k dk and \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} = \\mathbf{B}{(\\hat{p},k)} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} dk = \\int \\hat{p} k dk and \\iint \\operatorname{f_{\\mathbf{p}}}{(\\hat{p},k)} dk d\\hat{p} = \\iint \\hat{p} k dk d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["integrate", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(A_{x})} = \\log{(A_{x})}, then derive \\sin{(\\frac{d}{d A_{x}} \\operatorname{V_{\\mathbf{E}}}{(A_{x})})} = \\sin{(\\frac{1}{A_{x}})}, then obtain \\sin{(\\frac{1}{A_{x}})} = \\sin{(\\frac{d}{d A_{x}} \\log{(A_{x})})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(A_{x})} = \\log{(A_{x})} and \\frac{d}{d A_{x}} \\operatorname{V_{\\mathbf{E}}}{(A_{x})} = \\frac{d}{d A_{x}} \\log{(A_{x})} and \\sin{(\\frac{d}{d A_{x}} \\operatorname{V_{\\mathbf{E}}}{(A_{x})})} = \\sin{(\\frac{d}{d A_{x}} \\log{(A_{x})})} and \\sin{(\\frac{d}{d A_{x}} \\operatorname{V_{\\mathbf{E}}}{(A_{x})})} = \\sin{(\\frac{1}{A_{x}})} and \\sin{(\\frac{1}{A_{x}})} = \\sin{(\\frac{d}{d A_{x}} \\log{(A_{x})})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), sin(Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), sin(Pow(Symbol('A_x', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(sin(Pow(Symbol('A_x', commutative=True), Integer(-1))), sin(Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}, then obtain \\cos^{2}{(\\eta^{\\prime})} = (- \\mathbf{B}{(\\eta^{\\prime})} + 2 \\cos{(\\eta^{\\prime})}) \\cos{(\\eta^{\\prime})}", "derivation": "\\mathbf{B}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and 0 = - \\mathbf{B}{(\\eta^{\\prime})} + \\cos{(\\eta^{\\prime})} and \\cos{(\\eta^{\\prime})} = - \\mathbf{B}{(\\eta^{\\prime})} + 2 \\cos{(\\eta^{\\prime})} and \\cos^{2}{(\\eta^{\\prime})} = (- \\mathbf{B}{(\\eta^{\\prime})} + 2 \\cos{(\\eta^{\\prime})}) \\cos{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["times", 3, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Pow(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given n{(\\varepsilon,U)} = U + e^{\\varepsilon}, then obtain (n^{U}{(\\varepsilon,U)} e^{- \\varepsilon})^{U} - n^{U}{(\\varepsilon,U)} = ((U + e^{\\varepsilon})^{U} e^{- \\varepsilon})^{U} - n^{U}{(\\varepsilon,U)}", "derivation": "n{(\\varepsilon,U)} = U + e^{\\varepsilon} and n^{U}{(\\varepsilon,U)} = (U + e^{\\varepsilon})^{U} and n^{U}{(\\varepsilon,U)} e^{- \\varepsilon} = (U + e^{\\varepsilon})^{U} e^{- \\varepsilon} and (n^{U}{(\\varepsilon,U)} e^{- \\varepsilon})^{U} = ((U + e^{\\varepsilon})^{U} e^{- \\varepsilon})^{U} and (n^{U}{(\\varepsilon,U)} e^{- \\varepsilon})^{U} - n^{U}{(\\varepsilon,U)} = ((U + e^{\\varepsilon})^{U} e^{- \\varepsilon})^{U} - n^{U}{(\\varepsilon,U)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Add(Symbol('U', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)))"], [["divide", 2, "exp(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Mul(Pow(Add(Symbol('U', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Symbol('U', commutative=True)), Pow(Mul(Pow(Add(Symbol('U', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Symbol('U', commutative=True)))"], [["minus", 4, "Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)))), Add(Pow(Mul(Pow(Add(Symbol('U', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('n')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given Q{(\\mathbf{A},p)} = \\mathbf{A}^{p}, then obtain \\int (\\mathbf{A} Q{(\\mathbf{A},p)} - Q{(\\mathbf{A},p)}) dp = \\int (\\mathbf{A} \\mathbf{A}^{p} - Q{(\\mathbf{A},p)}) dp", "derivation": "Q{(\\mathbf{A},p)} = \\mathbf{A}^{p} and \\mathbf{A} Q{(\\mathbf{A},p)} = \\mathbf{A} \\mathbf{A}^{p} and \\mathbf{A} Q{(\\mathbf{A},p)} - Q{(\\mathbf{A},p)} = \\mathbf{A} \\mathbf{A}^{p} - Q{(\\mathbf{A},p)} and \\int (\\mathbf{A} Q{(\\mathbf{A},p)} - Q{(\\mathbf{A},p)}) dp = \\int (\\mathbf{A} \\mathbf{A}^{p} - Q{(\\mathbf{A},p)}) dp", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))))"], [["minus", 2, "Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Function('Q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given t{(\\theta_2)} = \\sin{(\\theta_2)}, then derive \\frac{d}{d \\theta_2} \\int t{(\\theta_2)} d\\theta_2 = \\frac{\\partial}{\\partial \\theta_2} (i - \\cos{(\\theta_2)}), then obtain \\frac{\\partial}{\\partial \\theta_2} (i - \\cos{(\\theta_2)}) = \\frac{\\partial}{\\partial \\theta_2} (\\ddot{x} - \\cos{(\\theta_2)})", "derivation": "t{(\\theta_2)} = \\sin{(\\theta_2)} and \\int t{(\\theta_2)} d\\theta_2 = \\int \\sin{(\\theta_2)} d\\theta_2 and \\frac{d}{d \\theta_2} \\int t{(\\theta_2)} d\\theta_2 = \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 and \\frac{d}{d \\theta_2} \\int t{(\\theta_2)} d\\theta_2 = \\frac{\\partial}{\\partial \\theta_2} (i - \\cos{(\\theta_2)}) and \\frac{\\partial}{\\partial \\theta_2} (i - \\cos{(\\theta_2)}) = \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 and \\frac{\\partial}{\\partial \\theta_2} (i - \\cos{(\\theta_2)}) = \\frac{\\partial}{\\partial \\theta_2} (\\ddot{x} - \\cos{(\\theta_2)})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{r},C_{d})} = C_{d} + \\log{(\\mathbf{r})}, then obtain \\int \\frac{\\partial}{\\partial C_{d}} (\\Omega{(\\mathbf{r},C_{d})} - \\log{(\\mathbf{r})}) dC_{d} = \\int \\frac{d}{d C_{d}} C_{d} dC_{d}", "derivation": "\\Omega{(\\mathbf{r},C_{d})} = C_{d} + \\log{(\\mathbf{r})} and \\Omega{(\\mathbf{r},C_{d})} - \\log{(\\mathbf{r})} = C_{d} and \\frac{\\partial}{\\partial C_{d}} (\\Omega{(\\mathbf{r},C_{d})} - \\log{(\\mathbf{r})}) = \\frac{d}{d C_{d}} C_{d} and \\int \\frac{\\partial}{\\partial C_{d}} (\\Omega{(\\mathbf{r},C_{d})} - \\log{(\\mathbf{r})}) dC_{d} = \\int \\frac{d}{d C_{d}} C_{d} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('C_d', commutative=True))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))), Integral(Derivative(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\dot{\\mathbf{r}},H)} = H + \\dot{\\mathbf{r}}, then obtain - (H + \\dot{\\mathbf{r}})^{H} + \\int \\omega{(\\dot{\\mathbf{r}},H)} dH = - (H + \\dot{\\mathbf{r}})^{H} + \\int (H + \\dot{\\mathbf{r}}) dH", "derivation": "\\omega{(\\dot{\\mathbf{r}},H)} = H + \\dot{\\mathbf{r}} and \\omega^{H}{(\\dot{\\mathbf{r}},H)} = (H + \\dot{\\mathbf{r}})^{H} and \\int \\omega{(\\dot{\\mathbf{r}},H)} dH = \\int (H + \\dot{\\mathbf{r}}) dH and - \\omega^{H}{(\\dot{\\mathbf{r}},H)} + \\int \\omega{(\\dot{\\mathbf{r}},H)} dH = - \\omega^{H}{(\\dot{\\mathbf{r}},H)} + \\int (H + \\dot{\\mathbf{r}}) dH and - (H + \\dot{\\mathbf{r}})^{H} + \\int \\omega{(\\dot{\\mathbf{r}},H)} dH = - (H + \\dot{\\mathbf{r}})^{H} + \\int (H + \\dot{\\mathbf{r}}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))), Integral(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('H', commutative=True))), Integral(Function('\\\\omega')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(y^{\\prime})} = \\cos{(y^{\\prime})}, then obtain \\mathbf{M}^{4}{(y^{\\prime})} = \\mathbf{M}^{3}{(y^{\\prime})} \\cos{(y^{\\prime})}", "derivation": "\\mathbf{M}{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\mathbf{M}^{2}{(y^{\\prime})} = \\mathbf{M}{(y^{\\prime})} \\cos{(y^{\\prime})} and \\mathbf{M}^{3}{(y^{\\prime})} \\cos{(y^{\\prime})} = \\mathbf{M}^{2}{(y^{\\prime})} \\cos^{2}{(y^{\\prime})} and \\mathbf{M}^{3}{(y^{\\prime})} \\cos{(y^{\\prime})} = \\mathbf{M}{(y^{\\prime})} \\cos^{3}{(y^{\\prime})} and \\mathbf{M}^{4}{(y^{\\prime})} = \\mathbf{M}^{2}{(y^{\\prime})} \\cos^{2}{(y^{\\prime})} and \\mathbf{M}^{4}{(y^{\\prime})} = \\mathbf{M}^{3}{(y^{\\prime})} \\cos{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 2, "Mul(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(3)), cos(Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(3)), cos(Symbol('y^{\\\\prime}', commutative=True))), Mul(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{M}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(3)), cos(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given E{(f^{\\prime})} = \\log{(f^{\\prime})} and \\operatorname{f^{*}}{(f^{\\prime})} = \\log{(f^{\\prime})}, then obtain \\operatorname{f^{*}}{(f^{\\prime})} - \\log{(f^{\\prime})} = 0", "derivation": "E{(f^{\\prime})} = \\log{(f^{\\prime})} and \\operatorname{f^{*}}{(f^{\\prime})} = \\log{(f^{\\prime})} and E{(f^{\\prime})} = \\operatorname{f^{*}}{(f^{\\prime})} and E{(f^{\\prime})} - \\log{(f^{\\prime})} = \\operatorname{f^{*}}{(f^{\\prime})} - \\log{(f^{\\prime})} and E{(f^{\\prime})} - \\log{(f^{\\prime})} = 0 and \\operatorname{f^{*}}{(f^{\\prime})} - \\log{(f^{\\prime})} = 0", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E')(Symbol('f^{\\\\prime}', commutative=True)), Function('f^*')(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 3, "log(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('E')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Add(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('E')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(A_{z})} = \\cos{(A_{z})} and \\mathbf{J}_f{(A_{z})} = \\cos^{2}{(A_{z})}, then obtain \\operatorname{f_{\\mathbf{p}}}{(A_{z})} \\cos{(A_{z})} = \\operatorname{f_{\\mathbf{p}}}^{2}{(A_{z})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(A_{z})} = \\cos{(A_{z})} and \\operatorname{f_{\\mathbf{p}}}{(A_{z})} \\cos{(A_{z})} = \\cos^{2}{(A_{z})} and \\mathbf{J}_f{(A_{z})} = \\cos^{2}{(A_{z})} and \\mathbf{J}_f{(A_{z})} = \\operatorname{f_{\\mathbf{p}}}^{2}{(A_{z})} and \\mathbf{J}_f{(A_{z})} = \\operatorname{f_{\\mathbf{p}}}{(A_{z})} \\cos{(A_{z})} and \\operatorname{f_{\\mathbf{p}}}{(A_{z})} \\cos{(A_{z})} = \\operatorname{f_{\\mathbf{p}}}^{2}{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["times", 1, "cos(Symbol('A_z', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True))), Pow(cos(Symbol('A_z', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_z', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_z', commutative=True)), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True))), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('A_z', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{F}{(c_{0},f)} = c_{0} f, then obtain - 2 c_{0} + (- \\mathbf{F}{(c_{0},f)})^{c_{0}} + \\int - c_{0} f df = - 2 c_{0} + (- c_{0} f)^{c_{0}} + \\int - c_{0} f df", "derivation": "\\mathbf{F}{(c_{0},f)} = c_{0} f and - \\mathbf{F}{(c_{0},f)} = - c_{0} f and (- \\mathbf{F}{(c_{0},f)})^{c_{0}} = (- c_{0} f)^{c_{0}} and - c_{0} + (- \\mathbf{F}{(c_{0},f)})^{c_{0}} + \\int - c_{0} f df = - c_{0} + (- c_{0} f)^{c_{0}} + \\int - c_{0} f df and - 2 c_{0} + (- \\mathbf{F}{(c_{0},f)})^{c_{0}} + \\int - c_{0} f df = - 2 c_{0} + (- c_{0} f)^{c_{0}} + \\int - c_{0} f df", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('f', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)))"], [["power", 2, "Symbol('c_0', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True), Symbol('f', commutative=True))), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Symbol('c_0', commutative=True)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True), Symbol('f', commutative=True))), Symbol('c_0', commutative=True)), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Symbol('c_0', commutative=True)), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True), Symbol('f', commutative=True))), Symbol('c_0', commutative=True)), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Symbol('c_0', commutative=True)), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\theta)} = \\sin{(\\theta)}, then derive \\sigma_{x}{(\\theta)} \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = \\sigma_{x}{(\\theta)} \\cos{(\\theta)}, then obtain \\int \\sin{(\\theta)} \\frac{d}{d \\theta} \\sin{(\\theta)} d\\theta = \\int \\sin{(\\theta)} \\cos{(\\theta)} d\\theta", "derivation": "\\sigma_{x}{(\\theta)} = \\sin{(\\theta)} and \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\theta)} and \\sigma_{x}{(\\theta)} \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = \\sigma_{x}{(\\theta)} \\frac{d}{d \\theta} \\sin{(\\theta)} and \\sigma_{x}{(\\theta)} \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = \\sigma_{x}{(\\theta)} \\cos{(\\theta)} and \\int \\sigma_{x}{(\\theta)} \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} d\\theta = \\int \\sigma_{x}{(\\theta)} \\cos{(\\theta)} d\\theta and \\int \\sin{(\\theta)} \\frac{d}{d \\theta} \\sin{(\\theta)} d\\theta = \\int \\sin{(\\theta)} \\cos{(\\theta)} d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(sin(Symbol('\\\\theta', commutative=True)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(sin(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\theta{(t_{1})} = t_{1}, then derive g^{\\prime}_{\\varepsilon} + \\frac{\\theta^{2}{(t_{1})}}{2} = \\int t_{1} d\\theta{(t_{1})}, then derive g^{\\prime}_{\\varepsilon} + \\frac{t_{1}^{2}}{2} = \\rho + \\frac{t_{1}^{2}}{2}, then obtain \\int t_{1} dt_{1} = \\rho + \\frac{t_{1}^{2}}{2}", "derivation": "\\theta{(t_{1})} = t_{1} and \\int \\theta{(t_{1})} dt_{1} = \\int t_{1} dt_{1} and \\int \\theta{(t_{1})} d\\theta{(t_{1})} = \\int t_{1} d\\theta{(t_{1})} and g^{\\prime}_{\\varepsilon} + \\frac{\\theta^{2}{(t_{1})}}{2} = \\int t_{1} d\\theta{(t_{1})} and g^{\\prime}_{\\varepsilon} + \\frac{t_{1}^{2}}{2} = \\int t_{1} dt_{1} and g^{\\prime}_{\\varepsilon} + \\frac{t_{1}^{2}}{2} = \\rho + \\frac{t_{1}^{2}}{2} and \\int t_{1} dt_{1} = \\rho + \\frac{t_{1}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\theta')(Symbol('t_1', commutative=True)), Tuple(Function('\\\\theta')(Symbol('t_1', commutative=True)))), Integral(Symbol('t_1', commutative=True), Tuple(Function('\\\\theta')(Symbol('t_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\theta')(Symbol('t_1', commutative=True)), Integer(2)))), Integral(Symbol('t_1', commutative=True), Tuple(Function('\\\\theta')(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))), Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))), Add(Symbol('\\\\rho', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given M{(\\mathbf{s})} = \\sin{(\\sin{(\\mathbf{s})})}, then derive \\frac{d}{d \\mathbf{s}} M{(\\mathbf{s})} = \\cos{(\\mathbf{s})} \\cos{(\\sin{(\\mathbf{s})})}, then obtain - (e^{F_{c}})^{y^{\\prime}} + \\frac{d}{d \\mathbf{s}} \\sin{(\\sin{(\\mathbf{s})})} = - (e^{F_{c}})^{y^{\\prime}} + \\cos{(\\mathbf{s})} \\cos{(\\sin{(\\mathbf{s})})}", "derivation": "M{(\\mathbf{s})} = \\sin{(\\sin{(\\mathbf{s})})} and \\frac{d}{d \\mathbf{s}} M{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\sin{(\\sin{(\\mathbf{s})})} and \\frac{d}{d \\mathbf{s}} M{(\\mathbf{s})} = \\cos{(\\mathbf{s})} \\cos{(\\sin{(\\mathbf{s})})} and \\frac{d}{d \\mathbf{s}} \\sin{(\\sin{(\\mathbf{s})})} = \\cos{(\\mathbf{s})} \\cos{(\\sin{(\\mathbf{s})})} and - (e^{F_{c}})^{y^{\\prime}} + \\frac{d}{d \\mathbf{s}} \\sin{(\\sin{(\\mathbf{s})})} = - (e^{F_{c}})^{y^{\\prime}} + \\cos{(\\mathbf{s})} \\cos{(\\sin{(\\mathbf{s})})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{s}', commutative=True)), sin(sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{s}', commutative=True)), cos(sin(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{s}', commutative=True)), cos(sin(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["minus", 4, "Pow(exp(Symbol('F_c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(exp(Symbol('F_c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Derivative(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(exp(Symbol('F_c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(cos(Symbol('\\\\mathbf{s}', commutative=True)), cos(sin(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(i,C_{1})} = \\frac{\\partial}{\\partial C_{1}} C_{1} i and i{(\\mu_0)} = \\log{(\\sin{(\\mu_0)})}, then derive \\hat{p}{(i,C_{1})} - i{(\\mu_0)} = i - i{(\\mu_0)}, then obtain (- \\log{(\\sin{(\\mu_0)})} + \\frac{\\partial}{\\partial C_{1}} C_{1} i)^{\\mu_0} = (i - \\log{(\\sin{(\\mu_0)})})^{\\mu_0}", "derivation": "\\hat{p}{(i,C_{1})} = \\frac{\\partial}{\\partial C_{1}} C_{1} i and i{(\\mu_0)} = \\log{(\\sin{(\\mu_0)})} and \\hat{p}{(i,C_{1})} - i{(\\mu_0)} = - i{(\\mu_0)} + \\frac{\\partial}{\\partial C_{1}} C_{1} i and \\hat{p}{(i,C_{1})} - i{(\\mu_0)} = i - i{(\\mu_0)} and - i{(\\mu_0)} + \\frac{\\partial}{\\partial C_{1}} C_{1} i = i - i{(\\mu_0)} and - \\log{(\\sin{(\\mu_0)})} + \\frac{\\partial}{\\partial C_{1}} C_{1} i = i - \\log{(\\sin{(\\mu_0)})} and (- \\log{(\\sin{(\\mu_0)})} + \\frac{\\partial}{\\partial C_{1}} C_{1} i)^{\\mu_0} = (i - \\log{(\\sin{(\\mu_0)})})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('i', commutative=True), Symbol('C_1', commutative=True)), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('i')(Symbol('\\\\mu_0', commutative=True)), log(sin(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 1, "Function('i')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('i', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('\\\\hat{p}')(Symbol('i', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True)))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('i')(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True)))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True))))))"], [["power", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True)))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('i', commutative=True), Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(n)} = n, then derive \\frac{d}{d n} \\theta_{1}{(n)} = 1, then obtain \\log{(\\frac{\\frac{d}{d n} \\theta_{1}{(n)}}{- n - \\theta_{1}{(n)} + 1})} = \\log{(\\frac{1}{- n - \\theta_{1}{(n)} + 1})}", "derivation": "\\theta_{1}{(n)} = n and \\frac{d}{d n} \\theta_{1}{(n)} = \\frac{d}{d n} n and 2 \\theta_{1}{(n)} = n + \\theta_{1}{(n)} and \\frac{d}{d n} \\theta_{1}{(n)} = 1 and \\frac{d}{d n} n = 1 and \\frac{\\frac{d}{d n} n}{1 - 2 \\theta_{1}{(n)}} = \\frac{1}{1 - 2 \\theta_{1}{(n)}} and \\log{(\\frac{\\frac{d}{d n} n}{1 - 2 \\theta_{1}{(n)}})} = \\log{(\\frac{1}{1 - 2 \\theta_{1}{(n)}})} and \\log{(\\frac{\\frac{d}{d n} \\theta_{1}{(n)}}{1 - 2 \\theta_{1}{(n)}})} = \\log{(\\frac{1}{1 - 2 \\theta_{1}{(n)}})} and \\log{(\\frac{\\frac{d}{d n} \\theta_{1}{(n)}}{- n - \\theta_{1}{(n)} + 1})} = \\log{(\\frac{1}{- n - \\theta_{1}{(n)} + 1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('n', commutative=True)), Symbol('n', commutative=True))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 1, "Function('\\\\theta_1')(Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), Function('\\\\theta_1')(Symbol('n', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1))"], [["divide", 5, "Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True))))"], "Equality(Mul(Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1)), Derivative(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1)))"], [["log", 6], "Equality(log(Mul(Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1)), Derivative(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))))), log(Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(log(Mul(Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), log(Pow(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\theta_1')(Symbol('n', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 8, 3], "Equality(log(Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('n', commutative=True))), Integer(1)), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), log(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('n', commutative=True))), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(L)} = \\cos{(L)}, then derive \\frac{d}{d L} \\mathbf{p}{(L)} = - \\sin{(L)}, then obtain (\\int \\frac{d}{d L} \\cos{(L)} dL)^{L} = (\\int - \\sin{(L)} dL)^{L}", "derivation": "\\mathbf{p}{(L)} = \\cos{(L)} and \\frac{d}{d L} \\mathbf{p}{(L)} = \\frac{d}{d L} \\cos{(L)} and \\frac{d}{d L} \\mathbf{p}{(L)} = - \\sin{(L)} and \\int \\frac{d}{d L} \\mathbf{p}{(L)} dL = \\int - \\sin{(L)} dL and \\int \\frac{d}{d L} \\cos{(L)} dL = \\int - \\sin{(L)} dL and (\\int \\frac{d}{d L} \\cos{(L)} dL)^{L} = (\\int - \\sin{(L)} dL)^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(cos(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('L', commutative=True))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Derivative(cos(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)}, then derive \\Psi_{\\lambda}{(\\Omega)} - 1 = -1 + \\frac{1}{\\Omega}, then obtain \\frac{(- \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)})^{\\Omega}}{\\frac{d}{d \\Omega} \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)}} = \\frac{(- \\cos{(1 - \\frac{1}{\\Omega})})^{\\Omega}}{\\frac{d}{d \\Omega} \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)}}", "derivation": "\\Psi_{\\lambda}{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and \\Psi_{\\lambda}{(\\Omega)} - 1 = \\frac{d}{d \\Omega} \\log{(\\Omega)} - 1 and \\Psi_{\\lambda}{(\\Omega)} - 1 = -1 + \\frac{1}{\\Omega} and \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)} = \\cos{(1 - \\frac{1}{\\Omega})} and - \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)} = - \\cos{(1 - \\frac{1}{\\Omega})} and (- \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)})^{\\Omega} = (- \\cos{(1 - \\frac{1}{\\Omega})})^{\\Omega} and \\frac{(- \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)})^{\\Omega}}{\\frac{d}{d \\Omega} \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)}} = \\frac{(- \\cos{(1 - \\frac{1}{\\Omega})})^{\\Omega}}{\\frac{d}{d \\Omega} \\cos{(\\Psi_{\\lambda}{(\\Omega)} - 1)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))"], [["cos", 3], "Equality(cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), cos(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)))), Mul(Integer(-1), cos(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Mul(Integer(-1), cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)))), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Integer(-1), cos(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))))), Symbol('\\\\Omega', commutative=True)))"], [["divide", 6, "Derivative(cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Mul(Integer(-1), cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Mul(Integer(-1), cos(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(cos(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(A)} = \\log{(A)}, then obtain A (- \\log{(A)} + \\frac{d}{d A} \\log{(A)}) \\frac{d}{d A} \\tilde{g}^*{(A)} = A (- \\log{(A)} + \\frac{d}{d A} \\log{(A)}) \\frac{d}{d A} \\log{(A)}", "derivation": "\\tilde{g}^*{(A)} = \\log{(A)} and \\frac{d}{d A} \\tilde{g}^*{(A)} = \\frac{d}{d A} \\log{(A)} and A \\frac{d}{d A} \\tilde{g}^*{(A)} = A \\frac{d}{d A} \\log{(A)} and A (- \\log{(A)} + \\frac{d}{d A} \\log{(A)}) \\frac{d}{d A} \\tilde{g}^*{(A)} = A (- \\log{(A)} + \\frac{d}{d A} \\log{(A)}) \\frac{d}{d A} \\log{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Derivative(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Symbol('A', commutative=True), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["times", 3, "Add(Mul(Integer(-1), log(Symbol('A', commutative=True))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('A', commutative=True), Add(Mul(Integer(-1), log(Symbol('A', commutative=True))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Derivative(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Symbol('A', commutative=True), Add(Mul(Integer(-1), log(Symbol('A', commutative=True))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(t,h)} = - h + \\sin{(t)}, then obtain \\chi^{- h}{(t,h)} \\frac{\\partial}{\\partial t} - \\frac{\\chi^{2}{(t,h)}}{- h + \\sin{(t)}} = \\chi^{- h}{(t,h)} \\frac{\\partial}{\\partial t} (h - \\sin{(t)})", "derivation": "\\chi{(t,h)} = - h + \\sin{(t)} and \\frac{\\chi{(t,h)}}{t} = \\frac{- h + \\sin{(t)}}{t} and - \\frac{\\chi^{2}{(t,h)}}{t} = - \\frac{(- h + \\sin{(t)}) \\chi{(t,h)}}{t} and - \\frac{(- h + \\sin{(t)}) \\chi{(t,h)}}{t} = - \\frac{(- h + \\sin{(t)})^{2}}{t} and - \\frac{\\chi^{2}{(t,h)}}{t} = - \\frac{(- h + \\sin{(t)})^{2}}{t} and - \\frac{\\chi^{2}{(t,h)}}{- h + \\sin{(t)}} = h - \\sin{(t)} and \\frac{\\partial}{\\partial t} - \\frac{\\chi^{2}{(t,h)}}{- h + \\sin{(t)}} = \\frac{\\partial}{\\partial t} (h - \\sin{(t)}) and \\chi^{- h}{(t,h)} \\frac{\\partial}{\\partial t} - \\frac{\\chi^{2}{(t,h)}}{- h + \\sin{(t)}} = \\chi^{- h}{(t,h)} \\frac{\\partial}{\\partial t} (h - \\sin{(t)})", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))))"], [["divide", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Integer(2))))"], [["divide", 5, "Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Integer(2))), Add(Symbol('h', commutative=True), Mul(Integer(-1), sin(Symbol('t', commutative=True)))))"], [["differentiate", 6, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('h', commutative=True), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["divide", 7, "Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Derivative(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('t', commutative=True))), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Derivative(Add(Symbol('h', commutative=True), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}{(v_{x})} = \\log{(v_{x})}, then obtain 2 (\\frac{d}{d v_{x}} \\tilde{g}{(v_{x})})^{v_{x}} = (\\frac{d}{d v_{x}} \\tilde{g}{(v_{x})})^{v_{x}} + (\\frac{d}{d v_{x}} \\log{(v_{x})})^{v_{x}}", "derivation": "\\tilde{g}{(v_{x})} = \\log{(v_{x})} and \\frac{d}{d v_{x}} \\tilde{g}{(v_{x})} = \\frac{d}{d v_{x}} \\log{(v_{x})} and (\\frac{d}{d v_{x}} \\tilde{g}{(v_{x})})^{v_{x}} = (\\frac{d}{d v_{x}} \\log{(v_{x})})^{v_{x}} and 2 (\\frac{d}{d v_{x}} \\tilde{g}{(v_{x})})^{v_{x}} = (\\frac{d}{d v_{x}} \\tilde{g}{(v_{x})})^{v_{x}} + (\\frac{d}{d v_{x}} \\log{(v_{x})})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True)), Pow(Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True)))"], [["add", 3, "Pow(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True))), Add(Pow(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True)), Pow(Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{r})} = \\log{(\\sin{(\\mathbf{r})})}, then derive \\frac{d}{d \\mathbf{r}} \\hat{x}{(\\mathbf{r})} = \\frac{\\cos{(\\mathbf{r})}}{\\sin{(\\mathbf{r})}}, then obtain \\log{(\\frac{\\cos{(\\mathbf{r})}}{\\sin{(\\mathbf{r})}})} = \\log{(\\frac{d}{d \\mathbf{r}} \\log{(\\sin{(\\mathbf{r})})})}", "derivation": "\\hat{x}{(\\mathbf{r})} = \\log{(\\sin{(\\mathbf{r})})} and \\frac{d}{d \\mathbf{r}} \\hat{x}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\log{(\\sin{(\\mathbf{r})})} and \\log{(\\frac{d}{d \\mathbf{r}} \\hat{x}{(\\mathbf{r})})} = \\log{(\\frac{d}{d \\mathbf{r}} \\log{(\\sin{(\\mathbf{r})})})} and \\frac{d}{d \\mathbf{r}} \\hat{x}{(\\mathbf{r})} = \\frac{\\cos{(\\mathbf{r})}}{\\sin{(\\mathbf{r})}} and \\log{(\\frac{\\cos{(\\mathbf{r})}}{\\sin{(\\mathbf{r})}})} = \\log{(\\frac{d}{d \\mathbf{r}} \\log{(\\sin{(\\mathbf{r})})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), log(Derivative(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(log(Mul(Pow(sin(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{r}', commutative=True)))), log(Derivative(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(l)} = \\sin{(l)}, then obtain (\\operatorname{C_{1}}^{l}{(l)})^{l} - \\operatorname{C_{1}}^{l}{(l)} = (\\sin^{l}{(l)})^{l} - \\operatorname{C_{1}}^{l}{(l)}", "derivation": "\\operatorname{C_{1}}{(l)} = \\sin{(l)} and \\operatorname{C_{1}}^{l}{(l)} = \\sin^{l}{(l)} and (\\operatorname{C_{1}}^{l}{(l)})^{l} = (\\sin^{l}{(l)})^{l} and (\\operatorname{C_{1}}^{l}{(l)})^{l} - \\operatorname{C_{1}}^{l}{(l)} = (\\sin^{l}{(l)})^{l} - \\operatorname{C_{1}}^{l}{(l)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["minus", 3, "Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True))"], "Equality(Add(Pow(Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Add(Pow(Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Function('C_1')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\sigma_p,s)} = \\sigma_p - s, then obtain \\mathbb{I} - \\sigma_p - \\int 0 ds + 2 \\int s^{2} ds + 2 \\int - \\sigma_p s ds + 2 \\int s \\operatorname{v_{2}}{(\\sigma_p,s)} ds = - \\sigma_p", "derivation": "\\operatorname{v_{2}}{(\\sigma_p,s)} = \\sigma_p - s and - \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)} = 0 and s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) = 0 and 2 s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) = s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) and 2 s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) = 0 and \\int 2 s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) ds = \\int 0 ds and - \\sigma_p - \\int 0 ds + \\int 2 s (- \\sigma_p + s + \\operatorname{v_{2}}{(\\sigma_p,s)}) ds = - \\sigma_p and \\mathbb{I} - \\sigma_p - \\int 0 ds + 2 \\int s^{2} ds + 2 \\int - \\sigma_p s ds + 2 \\int s \\operatorname{v_{2}}{(\\sigma_p,s)} ds = - \\sigma_p", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["times", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))), Integer(0))"], [["add", 3, "Mul(Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True))))"], "Equality(Mul(Integer(2), Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))), Mul(Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))), Integer(0))"], [["integrate", 5, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Integer(2), Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Integral(Integer(0), Tuple(Symbol('s', commutative=True))))"], [["minus", 6, "Add(Symbol('\\\\sigma_p', commutative=True), Integral(Integer(0), Tuple(Symbol('s', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('s', commutative=True)))), Integral(Mul(Integer(2), Symbol('s', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Add(Integral(Pow(Symbol('s', commutative=True), Integer(2)), Tuple(Symbol('s', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Mul(Symbol('s', commutative=True), Function('v_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('s', commutative=True))))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given W{(c_{0},s)} = c_{0}^{s}, then obtain \\int (s W{(c_{0},s)})^{s} ds = \\int (c_{0}^{s} s)^{s} ds", "derivation": "W{(c_{0},s)} = c_{0}^{s} and - c_{0}^{s} + W{(c_{0},s)} = 0 and - c_{0}^{s} + s + W{(c_{0},s)} = s and (- c_{0}^{s} + s + W{(c_{0},s)}) W{(c_{0},s)} = c_{0}^{s} (- c_{0}^{s} + s + W{(c_{0},s)}) and s W{(c_{0},s)} = c_{0}^{s} s and (s W{(c_{0},s)})^{s} = (c_{0}^{s} s)^{s} and \\int (s W{(c_{0},s)})^{s} ds = \\int (c_{0}^{s} s)^{s} ds", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], [["times", 1, "Add(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["power", 5, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Mul(Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 6, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('s', commutative=True), Function('W')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Mul(Pow(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given J{(\\psi^*)} = e^{\\psi^*}, then derive \\int J{(\\psi^*)} d\\psi^* = \\sigma_x + e^{\\psi^*}, then obtain V_{\\mathbf{E}} + e^{\\psi^*} = \\sigma_x + e^{\\psi^*}", "derivation": "J{(\\psi^*)} = e^{\\psi^*} and \\int J{(\\psi^*)} d\\psi^* = \\int e^{\\psi^*} d\\psi^* and \\int J{(\\psi^*)} d\\psi^* = \\sigma_x + e^{\\psi^*} and \\int e^{\\psi^*} d\\psi^* = \\sigma_x + e^{\\psi^*} and V_{\\mathbf{E}} + e^{\\psi^*} = \\sigma_x + e^{\\psi^*}", "srepr_derivation": [["get_premise", "Equality(Function('J')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(r,v_{2})} = r + v_{2}, then obtain 2 r \\mathbf{A}{(r,v_{2})} + \\mathbf{A}{(r,v_{2})} = r (r + v_{2} + \\mathbf{A}{(r,v_{2})}) + \\mathbf{A}{(r,v_{2})}", "derivation": "\\mathbf{A}{(r,v_{2})} = r + v_{2} and 2 \\mathbf{A}{(r,v_{2})} = r + v_{2} + \\mathbf{A}{(r,v_{2})} and 2 r \\mathbf{A}{(r,v_{2})} = r (r + v_{2} + \\mathbf{A}{(r,v_{2})}) and 2 r \\mathbf{A}{(r,v_{2})} + \\mathbf{A}{(r,v_{2})} = r (r + v_{2} + \\mathbf{A}{(r,v_{2})}) + \\mathbf{A}{(r,v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('r', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))), Add(Symbol('r', commutative=True), Symbol('v_2', commutative=True), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))))"], [["times", 2, "Symbol('r', commutative=True)"], "Equality(Mul(Integer(2), Symbol('r', commutative=True), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))), Mul(Symbol('r', commutative=True), Add(Symbol('r', commutative=True), Symbol('v_2', commutative=True), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True)))))"], [["add", 3, "Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('r', commutative=True), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Symbol('r', commutative=True), Add(Symbol('r', commutative=True), Symbol('v_2', commutative=True), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True)))), Function('\\\\mathbf{A}')(Symbol('r', commutative=True), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given n{(g)} = e^{g}, then derive \\frac{d^{2}}{d g^{2}} n{(g)} = e^{g}, then obtain B{(g)} + \\int \\frac{d^{2}}{d g^{2}} n{(g)} dg = B{(g)} + \\int \\frac{d^{4}}{d g^{4}} e^{g} dg", "derivation": "n{(g)} = e^{g} and \\frac{d}{d g} n{(g)} = \\frac{d}{d g} e^{g} and \\frac{d^{2}}{d g^{2}} n{(g)} = \\frac{d^{2}}{d g^{2}} e^{g} and \\frac{d^{2}}{d g^{2}} n{(g)} = e^{g} and \\frac{d^{2}}{d g^{2}} e^{g} = e^{g} and \\frac{d^{2}}{d g^{2}} n{(g)} = \\frac{d^{4}}{d g^{4}} e^{g} and \\int \\frac{d^{2}}{d g^{2}} n{(g)} dg = \\int \\frac{d^{4}}{d g^{4}} e^{g} dg and B{(g)} + \\int \\frac{d^{2}}{d g^{2}} n{(g)} dg = B{(g)} + \\int \\frac{d^{4}}{d g^{4}} e^{g} dg", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), exp(Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), exp(Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(4))))"], [["integrate", 6, "Symbol('g', commutative=True)"], "Equality(Integral(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), Tuple(Symbol('g', commutative=True))), Integral(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(4))), Tuple(Symbol('g', commutative=True))))"], [["add", 7, "Function('B')(Symbol('g', commutative=True))"], "Equality(Add(Function('B')(Symbol('g', commutative=True)), Integral(Derivative(Function('n')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), Tuple(Symbol('g', commutative=True)))), Add(Function('B')(Symbol('g', commutative=True)), Integral(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(4))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain \\frac{2 \\nabla{(V_{\\mathbf{E}})} \\rho_{f}{(V_{\\mathbf{E}})}}{V_{\\mathbf{E}}} = \\frac{2 \\rho_{f}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})}}{V_{\\mathbf{E}}}", "derivation": "\\nabla{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\nabla{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} = 2 \\sin{(V_{\\mathbf{E}})} and 2 \\nabla{(V_{\\mathbf{E}})} = \\nabla{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} and 2 \\nabla{(V_{\\mathbf{E}})} = 2 \\sin{(V_{\\mathbf{E}})} and 2 \\nabla{(V_{\\mathbf{E}})} \\rho_{f}{(V_{\\mathbf{E}})} = 2 \\rho_{f}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})} and \\frac{2 \\nabla{(V_{\\mathbf{E}})} \\rho_{f}{(V_{\\mathbf{E}})}}{V_{\\mathbf{E}}} = \\frac{2 \\rho_{f}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})}}{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, "sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["add", 1, "Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 4, "Function('\\\\rho_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('\\\\rho_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["divide", 5, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('\\\\rho_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(2), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\operatorname{C_{2}}{(\\mathbf{H})} = \\int (\\mathbf{H} + \\cos{(\\mathbf{H})}) d\\mathbf{H}, then obtain \\int (\\mathbf{H} + p{(\\mathbf{H})}) d\\mathbf{H} = \\operatorname{C_{2}}{(\\mathbf{H})}", "derivation": "p{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\mathbf{H} + p{(\\mathbf{H})} = \\mathbf{H} + \\cos{(\\mathbf{H})} and \\int (\\mathbf{H} + p{(\\mathbf{H})}) d\\mathbf{H} = \\int (\\mathbf{H} + \\cos{(\\mathbf{H})}) d\\mathbf{H} and \\operatorname{C_{2}}{(\\mathbf{H})} = \\int (\\mathbf{H} + \\cos{(\\mathbf{H})}) d\\mathbf{H} and \\int (\\mathbf{H} + p{(\\mathbf{H})}) d\\mathbf{H} = \\operatorname{C_{2}}{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('p')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('p')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('p')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given G{(E_{n})} = \\sin{(E_{n})}, then derive \\log{(\\frac{d}{d E_{n}} G{(E_{n})})} = \\log{(\\cos{(E_{n})})}, then obtain \\log{(\\cos{(E_{n})})} - \\cos{(E_{n})} = \\log{(\\frac{d}{d E_{n}} \\sin{(E_{n})})} - \\cos{(E_{n})}", "derivation": "G{(E_{n})} = \\sin{(E_{n})} and \\frac{d}{d E_{n}} G{(E_{n})} = \\frac{d}{d E_{n}} \\sin{(E_{n})} and \\log{(\\frac{d}{d E_{n}} G{(E_{n})})} = \\log{(\\frac{d}{d E_{n}} \\sin{(E_{n})})} and \\log{(\\frac{d}{d E_{n}} G{(E_{n})})} = \\log{(\\cos{(E_{n})})} and \\log{(\\cos{(E_{n})})} = \\log{(\\frac{d}{d E_{n}} \\sin{(E_{n})})} and \\log{(\\cos{(E_{n})})} - \\cos{(E_{n})} = \\log{(\\frac{d}{d E_{n}} \\sin{(E_{n})})} - \\cos{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('G')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), log(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('G')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), log(cos(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(log(cos(Symbol('E_n', commutative=True))), log(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["minus", 5, "cos(Symbol('E_n', commutative=True))"], "Equality(Add(log(cos(Symbol('E_n', commutative=True))), Mul(Integer(-1), cos(Symbol('E_n', commutative=True)))), Add(log(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(v_{y},\\Omega)} = - \\Omega + v_{y} and \\operatorname{t_{2}}{(v_{y},\\Omega)} = \\Omega v_{y}, then derive \\int \\hat{H}{(v_{y},\\Omega)} d\\Omega = - \\frac{\\Omega^{2}}{2} + \\Omega v_{y} + g, then obtain \\log{(\\int (- \\Omega + v_{y}) d\\Omega)} - 1 = \\log{(- \\frac{\\Omega^{2}}{2} + g + \\operatorname{t_{2}}{(v_{y},\\Omega)})} - 1", "derivation": "\\hat{H}{(v_{y},\\Omega)} = - \\Omega + v_{y} and \\int \\hat{H}{(v_{y},\\Omega)} d\\Omega = \\int (- \\Omega + v_{y}) d\\Omega and \\int \\hat{H}{(v_{y},\\Omega)} d\\Omega = - \\frac{\\Omega^{2}}{2} + \\Omega v_{y} + g and \\int (- \\Omega + v_{y}) d\\Omega = - \\frac{\\Omega^{2}}{2} + \\Omega v_{y} + g and \\operatorname{t_{2}}{(v_{y},\\Omega)} = \\Omega v_{y} and \\int (- \\Omega + v_{y}) d\\Omega = - \\frac{\\Omega^{2}}{2} + g + \\operatorname{t_{2}}{(v_{y},\\Omega)} and \\log{(\\int (- \\Omega + v_{y}) d\\Omega)} = \\log{(- \\frac{\\Omega^{2}}{2} + g + \\operatorname{t_{2}}{(v_{y},\\Omega)})} and \\log{(\\int (- \\Omega + v_{y}) d\\Omega)} - 1 = \\log{(- \\frac{\\Omega^{2}}{2} + g + \\operatorname{t_{2}}{(v_{y},\\Omega)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_y', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_y', commutative=True)), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('g', commutative=True), Function('t_2')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["log", 6], "Equality(log(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), log(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('g', commutative=True), Function('t_2')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["add", 7, "Integer(-1)"], "Equality(Add(log(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Add(log(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('g', commutative=True), Function('t_2')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})}, then derive \\int \\tilde{g}^*{(\\mathbb{I})} d\\mathbb{I} = \\frac{\\mathbb{I}^{2}}{2} + z, then obtain - \\frac{\\mathbb{I}^{2}}{2} + e^{\\frac{\\mathbb{I}^{2}}{2} + z} = - \\frac{\\mathbb{I}^{2}}{2} + e^{\\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I}}", "derivation": "\\tilde{g}^*{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})} and \\int \\tilde{g}^*{(\\mathbb{I})} d\\mathbb{I} = \\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I} and \\int \\tilde{g}^*{(\\mathbb{I})} d\\mathbb{I} = \\frac{\\mathbb{I}^{2}}{2} + z and e^{\\int \\tilde{g}^*{(\\mathbb{I})} d\\mathbb{I}} = e^{\\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I}} and e^{\\frac{\\mathbb{I}^{2}}{2} + z} = e^{\\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I}} and - \\frac{\\mathbb{I}^{2}}{2} + e^{\\frac{\\mathbb{I}^{2}}{2} + z} = - \\frac{\\mathbb{I}^{2}}{2} + e^{\\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbb{I}', commutative=True)), log(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), exp(Integral(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Symbol('z', commutative=True))), exp(Integral(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["minus", 5, "Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), exp(Integral(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{x}{(\\eta)} = e^{e^{\\eta}}, then obtain (\\frac{d}{d \\eta} \\sigma_{x}^{\\eta}{(\\eta)} + \\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta})^{2} = 4 (\\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta})^{2}", "derivation": "\\sigma_{x}{(\\eta)} = e^{e^{\\eta}} and \\sigma_{x}^{\\eta}{(\\eta)} = (e^{e^{\\eta}})^{\\eta} and \\frac{d}{d \\eta} \\sigma_{x}^{\\eta}{(\\eta)} = \\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta} and \\frac{d}{d \\eta} \\sigma_{x}^{\\eta}{(\\eta)} + \\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta} = 2 \\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta} and (\\frac{d}{d \\eta} \\sigma_{x}^{\\eta}{(\\eta)} + \\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta})^{2} = 4 (\\frac{d}{d \\eta} (e^{e^{\\eta}})^{\\eta})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True))))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["power", 4, 2], "Equality(Pow(Add(Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(4), Pow(Derivative(Pow(exp(exp(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\rho{(\\varepsilon,l)} = \\varepsilon + l and \\operatorname{P_{e}}{(l)} = 2 l, then obtain \\varepsilon + 2 l = \\varepsilon + \\operatorname{P_{e}}{(l)}", "derivation": "\\rho{(\\varepsilon,l)} = \\varepsilon + l and l + \\rho{(\\varepsilon,l)} = \\varepsilon + 2 l and \\operatorname{P_{e}}{(l)} = 2 l and l + \\rho{(\\varepsilon,l)} = \\varepsilon + \\operatorname{P_{e}}{(l)} and \\varepsilon + 2 l = \\varepsilon + \\operatorname{P_{e}}{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('l', commutative=True)))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True), Symbol('l', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('l', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('l', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True), Symbol('l', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Function('P_e')(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('l', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Function('P_e')(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(S)} = \\cos{(S)} and \\chi{(S)} = \\frac{d}{d S} \\mathbf{A}{(S)}, then obtain \\frac{d^{2}}{d S^{2}} \\mathbf{A}{(S)} = \\frac{d}{d S} \\chi{(S)}", "derivation": "\\mathbf{A}{(S)} = \\cos{(S)} and \\chi{(S)} = \\frac{d}{d S} \\mathbf{A}{(S)} and \\chi{(S)} = \\frac{d}{d S} \\cos{(S)} and \\frac{d}{d S} \\chi{(S)} = \\frac{d^{2}}{d S^{2}} \\cos{(S)} and \\frac{d^{2}}{d S^{2}} \\mathbf{A}{(S)} = \\frac{d^{2}}{d S^{2}} \\cos{(S)} and \\frac{d^{2}}{d S^{2}} \\mathbf{A}{(S)} = \\frac{d}{d S} \\chi{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('S', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('S', commutative=True)), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(Function('\\\\chi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(u)} = \\frac{d}{d u} \\sin{(u)}, then derive \\int (A{(u)} - \\sin{(u)}) du = f_{E} + \\sin{(u)} + \\cos{(u)}, then derive \\mathbf{v} + f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)} = 2 f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)}, then obtain \\mathbf{v} + \\sin{(u)} + \\cos{(u)} = f_{E} + \\sin{(u)} + \\cos{(u)}", "derivation": "A{(u)} = \\frac{d}{d u} \\sin{(u)} and A{(u)} - \\sin{(u)} = - \\sin{(u)} + \\frac{d}{d u} \\sin{(u)} and \\int (A{(u)} - \\sin{(u)}) du = \\int (- \\sin{(u)} + \\frac{d}{d u} \\sin{(u)}) du and \\int (A{(u)} - \\sin{(u)}) du = f_{E} + \\sin{(u)} + \\cos{(u)} and f_{E} + \\sin{(u)} + \\cos{(u)} + \\int (A{(u)} - \\sin{(u)}) du = 2 f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)} and f_{E} + \\sin{(u)} + \\cos{(u)} + \\int (- \\sin{(u)} + \\frac{d}{d u} \\sin{(u)}) du = 2 f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)} and \\mathbf{v} + f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)} = 2 f_{E} + 2 \\sin{(u)} + 2 \\cos{(u)} and \\mathbf{v} + \\sin{(u)} + \\cos{(u)} = f_{E} + \\sin{(u)} + \\cos{(u)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('u', commutative=True)), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["minus", 1, "sin(Symbol('u', commutative=True))"], "Equality(Add(Function('A')(Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('u', commutative=True))), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Function('A')(Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('u', commutative=True))), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('A')(Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True))))"], [["add", 4, "Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], "Equality(Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)), Integral(Add(Function('A')(Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), sin(Symbol('u', commutative=True))), Mul(Integer(2), cos(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), sin(Symbol('u', commutative=True))), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), sin(Symbol('u', commutative=True))), Mul(Integer(2), cos(Symbol('u', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('f_E', commutative=True), Mul(Integer(2), sin(Symbol('u', commutative=True))), Mul(Integer(2), cos(Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), sin(Symbol('u', commutative=True))), Mul(Integer(2), cos(Symbol('u', commutative=True)))))"], [["minus", 7, "Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True))), Add(Symbol('f_E', commutative=True), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then derive \\frac{d}{d L_{\\varepsilon}} \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (m + \\sin{(L_{\\varepsilon})}), then obtain \\frac{d}{d L_{\\varepsilon}} \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\cos{(L_{\\varepsilon})}", "derivation": "\\mathbf{p}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and \\frac{d}{d L_{\\varepsilon}} \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\frac{d}{d L_{\\varepsilon}} \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and \\frac{d}{d L_{\\varepsilon}} \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (m + \\sin{(L_{\\varepsilon})}) and \\frac{d}{d L_{\\varepsilon}} \\int \\mathbf{p}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\cos{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(A)} = \\cos{(A)}, then derive \\cos{(A)} + \\frac{d}{d A} \\mathbf{S}{(A)} = - \\sin{(A)} + \\cos{(A)}, then obtain \\sin{(A)} + \\cos{(A)} + \\frac{d}{d A} (\\sin{(A)} + \\cos{(A)} + \\frac{d}{d A} \\cos{(A)}) + \\frac{d}{d A} \\cos{(A)} = \\cos{(A)} + \\frac{d}{d A} \\cos{(A)}", "derivation": "\\mathbf{S}{(A)} = \\cos{(A)} and \\frac{d}{d A} \\mathbf{S}{(A)} = \\frac{d}{d A} \\cos{(A)} and \\cos{(A)} + \\frac{d}{d A} \\mathbf{S}{(A)} = \\cos{(A)} + \\frac{d}{d A} \\cos{(A)} and \\cos{(A)} + \\frac{d}{d A} \\mathbf{S}{(A)} = - \\sin{(A)} + \\cos{(A)} and \\cos{(A)} + \\frac{d}{d A} \\cos{(A)} = - \\sin{(A)} + \\cos{(A)} and \\sin{(A)} + \\cos{(A)} + \\frac{d}{d A} \\cos{(A)} = \\cos{(A)} and \\sin{(A)} + \\cos{(A)} + \\frac{d}{d A} (\\sin{(A)} + \\cos{(A)} + \\frac{d}{d A} \\cos{(A)}) + \\frac{d}{d A} \\cos{(A)} = \\cos{(A)} + \\frac{d}{d A} \\cos{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["add", 2, "cos(Symbol('A', commutative=True))"], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(cos(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('A', commutative=True))), cos(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('A', commutative=True))), cos(Symbol('A', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), sin(Symbol('A', commutative=True)))"], "Equality(Add(sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), cos(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)), Derivative(Add(sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(cos(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{f}{(A)} = \\cos{(A)}, then derive \\frac{d}{d A} \\mathbf{f}{(A)} = - \\sin{(A)}, then obtain \\frac{d}{d A} ((- \\sin{(A)})^{A} + \\frac{d}{d A} \\cos{(A)}) = \\frac{d}{d A} ((- \\sin{(A)})^{A} - \\sin{(A)})", "derivation": "\\mathbf{f}{(A)} = \\cos{(A)} and \\frac{d}{d A} \\mathbf{f}{(A)} = \\frac{d}{d A} \\cos{(A)} and \\frac{d}{d A} \\mathbf{f}{(A)} = - \\sin{(A)} and \\frac{d}{d A} \\cos{(A)} = - \\sin{(A)} and (\\frac{d}{d A} \\mathbf{f}{(A)})^{A} + \\frac{d}{d A} \\cos{(A)} = - \\sin{(A)} + (\\frac{d}{d A} \\mathbf{f}{(A)})^{A} and (- \\sin{(A)})^{A} + \\frac{d}{d A} \\cos{(A)} = (- \\sin{(A)})^{A} - \\sin{(A)} and \\frac{d}{d A} ((- \\sin{(A)})^{A} + \\frac{d}{d A} \\cos{(A)}) = \\frac{d}{d A} ((- \\sin{(A)})^{A} - \\sin{(A)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A', commutative=True))))"], [["add", 4, "Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True))"], "Equality(Add(Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(-1), sin(Symbol('A', commutative=True)))))"], [["differentiate", 6, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(-1), sin(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(\\rho_b,t_{1})} = \\rho_b^{t_{1}}, then obtain \\frac{\\partial}{\\partial t_{1}} (- t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(l{(\\rho_b,t_{1})})}) = \\frac{\\partial}{\\partial t_{1}} (- t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(\\rho_b^{t_{1}})})", "derivation": "l{(\\rho_b,t_{1})} = \\rho_b^{t_{1}} and \\log{(l{(\\rho_b,t_{1})})} = \\log{(\\rho_b^{t_{1}})} and \\frac{\\partial}{\\partial t_{1}} \\log{(l{(\\rho_b,t_{1})})} = \\frac{\\partial}{\\partial t_{1}} \\log{(\\rho_b^{t_{1}})} and - t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(l{(\\rho_b,t_{1})})} = - t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(\\rho_b^{t_{1}})} and \\frac{\\partial}{\\partial t_{1}} (- t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(l{(\\rho_b,t_{1})})}) = \\frac{\\partial}{\\partial t_{1}} (- t_{1} + \\frac{\\partial}{\\partial t_{1}} \\log{(\\rho_b^{t_{1}})})", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True)), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True)))"], [["log", 1], "Equality(log(Function('l')(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(log(Function('l')(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('t_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(log(Function('l')(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(log(Function('l')(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(q,\\hat{H})} = \\int (\\hat{H} - q) d\\hat{H}, then obtain \\eta{(q,\\hat{H})} + \\frac{d}{d \\hat{H}} 1 = \\eta{(q,\\hat{H})} + \\frac{\\partial}{\\partial \\hat{H}} \\eta^{- q}{(q,\\hat{H})} (\\int (\\hat{H} - q) d\\hat{H})^{q}", "derivation": "\\eta{(q,\\hat{H})} = \\int (\\hat{H} - q) d\\hat{H} and \\eta^{q}{(q,\\hat{H})} = (\\int (\\hat{H} - q) d\\hat{H})^{q} and 1 = \\eta^{- q}{(q,\\hat{H})} (\\int (\\hat{H} - q) d\\hat{H})^{q} and \\frac{d}{d \\hat{H}} 1 = \\frac{\\partial}{\\partial \\hat{H}} \\eta^{- q}{(q,\\hat{H})} (\\int (\\hat{H} - q) d\\hat{H})^{q} and \\eta{(q,\\hat{H})} + \\frac{d}{d \\hat{H}} 1 = \\eta{(q,\\hat{H})} + \\frac{\\partial}{\\partial \\hat{H}} \\eta^{- q}{(q,\\hat{H})} (\\int (\\hat{H} - q) d\\hat{H})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('q', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('q', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('q', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["add", 4, "Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Pow(Function('\\\\eta')(Symbol('q', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(U,n_{2})} = n_{2} \\log{(U)}, then derive 0 = \\log{(U)} - \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_f{(U,n_{2})}, then obtain 0 = \\log{(U)} - \\frac{\\partial}{\\partial n_{2}} n_{2} \\log{(U)}", "derivation": "\\mathbf{J}_f{(U,n_{2})} = n_{2} \\log{(U)} and \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_f{(U,n_{2})} = \\frac{\\partial}{\\partial n_{2}} n_{2} \\log{(U)} and 0 = \\frac{\\partial}{\\partial n_{2}} n_{2} \\log{(U)} - \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_f{(U,n_{2})} and 0 = \\log{(U)} - \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_f{(U,n_{2})} and 0 = \\log{(U)} - \\frac{\\partial}{\\partial n_{2}} n_{2} \\log{(U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('U', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), log(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('U', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('n_2', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mathbf{J}_f')(Symbol('U', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('n_2', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_f')(Symbol('U', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(log(Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_f')(Symbol('U', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(log(Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('n_2', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} = \\mathbf{P} - i, then derive (\\frac{\\partial}{\\partial i} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} + 1)^{i} = 0^{i}, then obtain \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} + 1)^{i} = \\frac{d}{d i} 0^{i}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} = \\mathbf{P} - i and i + \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} = \\mathbf{P} and \\frac{\\partial}{\\partial i} (i + \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)}) = \\frac{d}{d i} \\mathbf{P} and (\\frac{\\partial}{\\partial i} (i + \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)}))^{i} = (\\frac{d}{d i} \\mathbf{P})^{i} and (\\frac{\\partial}{\\partial i} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} + 1)^{i} = 0^{i} and \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},i)} + 1)^{i} = \\frac{d}{d i} 0^{i}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["add", 1, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Symbol('i', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('i', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Derivative(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1)), Symbol('i', commutative=True)), Pow(Integer(0), Symbol('i', commutative=True)))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Add(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(\\omega)} = \\frac{d}{d \\omega} \\sin{(\\omega)}, then derive \\mathbf{v}{(\\omega)} = \\cos{(\\omega)}, then derive - \\cos{(\\omega)} = \\frac{d^{2}}{d \\omega^{2}} \\mathbf{v}{(\\omega)}, then obtain - \\mathbf{v}{(\\omega)} = \\frac{d^{2}}{d \\omega^{2}} \\mathbf{v}{(\\omega)}", "derivation": "\\mathbf{v}{(\\omega)} = \\frac{d}{d \\omega} \\sin{(\\omega)} and \\mathbf{v}{(\\omega)} + 1 = \\frac{d}{d \\omega} \\sin{(\\omega)} + 1 and \\mathbf{v}{(\\omega)} = \\cos{(\\omega)} and \\cos{(\\omega)} + 1 = \\frac{d}{d \\omega} \\sin{(\\omega)} + 1 and \\cos{(\\omega)} + 1 = \\mathbf{v}{(\\omega)} + 1 and \\frac{d}{d \\omega} (\\cos{(\\omega)} + 1) = \\frac{d}{d \\omega} (\\mathbf{v}{(\\omega)} + 1) and \\frac{d^{2}}{d \\omega^{2}} (\\cos{(\\omega)} + 1) = \\frac{d^{2}}{d \\omega^{2}} (\\mathbf{v}{(\\omega)} + 1) and - \\cos{(\\omega)} = \\frac{d^{2}}{d \\omega^{2}} \\mathbf{v}{(\\omega)} and - \\mathbf{v}{(\\omega)} = \\frac{d^{2}}{d \\omega^{2}} \\mathbf{v}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(cos(Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(cos(Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Integer(1)))"], [["differentiate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(cos(Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(cos(Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Derivative(Add(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["evaluate_derivatives", 7], "Equality(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(g,L,t_{2})} = - L + g - t_{2}, then derive \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})} = 1, then obtain \\int \\sin{(- L + g + \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})})} dg = \\int \\sin{(- L + g + 1)} dg", "derivation": "\\operatorname{v_{x}}{(g,L,t_{2})} = - L + g - t_{2} and \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})} = \\frac{\\partial}{\\partial g} (- L + g - t_{2}) and \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})} = 1 and - L + \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})} = 1 - L and - L + g + \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})} = - L + g + 1 and \\sin{(- L + g + \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})})} = \\sin{(- L + g + 1)} and \\int \\sin{(- L + g + \\frac{\\partial}{\\partial g} \\operatorname{v_{x}}{(g,L,t_{2})})} dg = \\int \\sin{(- L + g + 1)} dg", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Mul(Integer(-1), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('L', commutative=True))))"], [["add", 4, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Integer(1)))"], [["sin", 5], "Equality(sin(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), sin(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('g', commutative=True)"], "Equality(Integral(sin(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Derivative(Function('v_x')(Symbol('g', commutative=True), Symbol('L', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), Tuple(Symbol('g', commutative=True))), Integral(sin(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given J{(l,\\lambda)} = \\log{(\\lambda l)}, then obtain - (\\frac{\\partial}{\\partial l} \\lambda J^{\\lambda}{(l,\\lambda)})^{2} = - \\frac{\\partial}{\\partial l} \\lambda J^{\\lambda}{(l,\\lambda)} \\frac{\\partial}{\\partial l} \\lambda \\log{(\\lambda l)}^{\\lambda}", "derivation": "J{(l,\\lambda)} = \\log{(\\lambda l)} and J^{\\lambda}{(l,\\lambda)} = \\log{(\\lambda l)}^{\\lambda} and \\lambda J^{\\lambda}{(l,\\lambda)} = \\lambda \\log{(\\lambda l)}^{\\lambda} and \\frac{\\partial}{\\partial l} \\lambda J^{\\lambda}{(l,\\lambda)} = \\frac{\\partial}{\\partial l} \\lambda \\log{(\\lambda l)}^{\\lambda} and - (\\frac{\\partial}{\\partial l} \\lambda J^{\\lambda}{(l,\\lambda)})^{2} = - \\frac{\\partial}{\\partial l} \\lambda J^{\\lambda}{(l,\\lambda)} \\frac{\\partial}{\\partial l} \\lambda \\log{(\\lambda l)}^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), log(Mul(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Mul(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["times", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Pow(log(Mul(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(log(Mul(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 4, "Mul(Integer(-1), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Pow(Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('J')(Symbol('l', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(log(Mul(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(f)} = \\log{(f)}, then derive (\\frac{\\int G{(f)} df}{f})^{f} = (\\frac{M + f \\log{(f)} - f}{f})^{f}, then obtain (\\frac{M + f \\log{(f)} - f}{f})^{f} = (\\frac{\\int \\log{(f)} df}{f})^{f}", "derivation": "G{(f)} = \\log{(f)} and \\int G{(f)} df = \\int \\log{(f)} df and \\frac{\\int G{(f)} df}{f} = \\frac{\\int \\log{(f)} df}{f} and (\\frac{\\int G{(f)} df}{f})^{f} = (\\frac{\\int \\log{(f)} df}{f})^{f} and (\\frac{\\int G{(f)} df}{f})^{f} = (\\frac{M + f \\log{(f)} - f}{f})^{f} and (\\frac{M + f \\log{(f)} - f}{f})^{f} = (\\frac{\\int \\log{(f)} df}{f})^{f}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('G')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Function('G')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Function('G')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Symbol('f', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Function('G')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True)))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(A)} = \\frac{d}{d A} \\cos{(A)}, then derive \\phi_{1}{(A)} + \\sin{(A)} = 0, then obtain \\sin{(A)} + \\frac{d}{d A} \\cos{(A)} = 0", "derivation": "\\phi_{1}{(A)} = \\frac{d}{d A} \\cos{(A)} and \\phi_{1}{(A)} - \\frac{d}{d A} \\cos{(A)} = 0 and \\phi_{1}{(A)} + \\sin{(A)} = 0 and \\sin{(A)} + \\frac{d}{d A} \\cos{(A)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(sin(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\varphi{(n)} = e^{n}, then derive e^{n} + \\frac{d}{d n} \\varphi{(n)} = 2 e^{n}, then obtain \\frac{e^{n} + \\frac{d}{d n} \\varphi{(n)}}{\\frac{d}{d n} e^{n}} = \\frac{e^{n} + \\frac{d}{d n} e^{n}}{\\frac{d}{d n} e^{n}}", "derivation": "\\varphi{(n)} = e^{n} and \\frac{d}{d n} \\varphi{(n)} = \\frac{d}{d n} e^{n} and \\frac{d}{d n} \\varphi{(n)} + \\frac{d}{d n} e^{n} = 2 \\frac{d}{d n} e^{n} and e^{n} + \\frac{d}{d n} \\varphi{(n)} = 2 e^{n} and e^{n} + \\frac{d}{d n} e^{n} = 2 e^{n} and e^{n} + \\frac{d}{d n} \\varphi{(n)} = e^{n} + \\frac{d}{d n} e^{n} and \\frac{e^{n} + \\frac{d}{d n} \\varphi{(n)}}{\\frac{d}{d n} e^{n}} = \\frac{e^{n} + \\frac{d}{d n} e^{n}}{\\frac{d}{d n} e^{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 2, "Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\varphi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('n', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(exp(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(exp(Symbol('n', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(exp(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["divide", 6, "Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Add(exp(Symbol('n', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(v_{x},y)} = - y + e^{v_{x}}, then obtain \\frac{\\cos{(\\sin{(\\varphi{(v_{x},y)})})}}{- y + \\varphi{(v_{x},y)}} = \\frac{\\cos{(\\sin{(y - e^{v_{x}})})}}{- y + \\varphi{(v_{x},y)}}", "derivation": "\\varphi{(v_{x},y)} = - y + e^{v_{x}} and \\sin{(\\varphi{(v_{x},y)})} = - \\sin{(y - e^{v_{x}})} and \\cos{(\\sin{(\\varphi{(v_{x},y)})})} = \\cos{(\\sin{(y - e^{v_{x}})})} and - y + \\varphi{(v_{x},y)} = - 2 y + e^{v_{x}} and \\frac{\\cos{(\\sin{(\\varphi{(v_{x},y)})})}}{- 2 y + e^{v_{x}}} = \\frac{\\cos{(\\sin{(y - e^{v_{x}})})}}{- 2 y + e^{v_{x}}} and \\frac{\\cos{(\\sin{(\\varphi{(v_{x},y)})})}}{- y + \\varphi{(v_{x},y)}} = \\frac{\\cos{(\\sin{(y - e^{v_{x}})})}}{- y + \\varphi{(v_{x},y)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))))))"], [["cos", 2], "Equality(cos(sin(Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))), cos(sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))))))"], [["add", 1, "Mul(Integer(-1), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('y', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('y', commutative=True)), exp(Symbol('v_x', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('y', commutative=True)), exp(Symbol('v_x', commutative=True))), Integer(-1)), cos(sin(Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('y', commutative=True)), exp(Symbol('v_x', commutative=True))), Integer(-1)), cos(sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), exp(Symbol('v_x', commutative=True))))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Integer(-1)), cos(sin(Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\varphi')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Integer(-1)), cos(sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), exp(Symbol('v_x', commutative=True))))))))"]]}, {"prompt": "Given \\delta{(z,\\dot{y})} = \\dot{y} - z, then derive \\int \\delta{(z,\\dot{y})} d\\dot{y} = \\frac{\\dot{y}^{2}}{2} - \\dot{y} z + \\hbar, then obtain \\frac{\\int (\\dot{y} - z) d\\dot{y}}{\\dot{y}^{2}} = \\frac{\\int \\delta{(z,\\dot{y})} d\\dot{y}}{\\dot{y}^{2}}", "derivation": "\\delta{(z,\\dot{y})} = \\dot{y} - z and \\int \\delta{(z,\\dot{y})} d\\dot{y} = \\int (\\dot{y} - z) d\\dot{y} and \\int \\delta{(z,\\dot{y})} d\\dot{y} = \\frac{\\dot{y}^{2}}{2} - \\dot{y} z + \\hbar and \\int (\\dot{y} - z) d\\dot{y} = \\frac{\\dot{y}^{2}}{2} - \\dot{y} z + \\hbar and \\frac{\\int (\\dot{y} - z) d\\dot{y}}{\\dot{y}^{2}} = \\frac{\\frac{\\dot{y}^{2}}{2} - \\dot{y} z + \\hbar}{\\dot{y}^{2}} and \\frac{\\int (\\dot{y} - z) d\\dot{y}}{\\dot{y}^{2}} = \\frac{\\int \\delta{(z,\\dot{y})} d\\dot{y}}{\\dot{y}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["divide", 4, "Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Integral(Function('\\\\delta')(Symbol('z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(S)} = \\sin{(S)}, then obtain \\frac{d}{d S} 0 + \\frac{1}{\\varepsilon_0} = \\frac{d}{d S} (- \\theta_{1}^{S}{(S)} + \\sin^{S}{(S)}) + \\frac{1}{\\varepsilon_0}", "derivation": "\\theta_{1}{(S)} = \\sin{(S)} and \\theta_{1}^{S}{(S)} = \\sin^{S}{(S)} and 0 = - \\theta_{1}^{S}{(S)} + \\sin^{S}{(S)} and \\frac{d}{d S} 0 = \\frac{d}{d S} (- \\theta_{1}^{S}{(S)} + \\sin^{S}{(S)}) and \\frac{d}{d S} 0 + \\frac{1}{\\varepsilon_0} = \\frac{d}{d S} (- \\theta_{1}^{S}{(S)} + \\sin^{S}{(S)}) + \\frac{1}{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 4, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Add(Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given W{(\\mathbf{D},\\hbar)} = \\hbar \\cos{(\\mathbf{D})}, then derive \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = - \\sin{(\\mathbf{D})}, then derive \\frac{\\partial}{\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = \\cos{(\\mathbf{D})}, then obtain - \\sin{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})}", "derivation": "W{(\\mathbf{D},\\hbar)} = \\hbar \\cos{(\\mathbf{D})} and \\frac{\\partial}{\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\hbar \\cos{(\\mathbf{D})} and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial \\hbar} \\hbar \\cos{(\\mathbf{D})} and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = - \\sin{(\\mathbf{D})} and \\frac{\\partial}{\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = \\cos{(\\mathbf{D})} and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial \\hbar} W{(\\mathbf{D},\\hbar)} = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} and - \\sin{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\varphi^*,\\dot{z},t_{2})} = t_{2} (\\dot{z} + \\varphi^*) and \\rho_{f}{(\\varphi^*,\\dot{z},t_{2})} = \\frac{\\operatorname{f_{E}}{(\\varphi^*,\\dot{z},t_{2})}}{\\dot{z} + \\varphi^*}, then obtain t_{2}^{\\dot{z}} + \\rho_{f}^{\\dot{z}}{(\\varphi^*,\\dot{z},t_{2})} = 2 t_{2}^{\\dot{z}}", "derivation": "\\operatorname{f_{E}}{(\\varphi^*,\\dot{z},t_{2})} = t_{2} (\\dot{z} + \\varphi^*) and \\frac{\\operatorname{f_{E}}{(\\varphi^*,\\dot{z},t_{2})}}{\\dot{z} + \\varphi^*} = t_{2} and \\rho_{f}{(\\varphi^*,\\dot{z},t_{2})} = \\frac{\\operatorname{f_{E}}{(\\varphi^*,\\dot{z},t_{2})}}{\\dot{z} + \\varphi^*} and \\rho_{f}{(\\varphi^*,\\dot{z},t_{2})} = t_{2} and \\rho_{f}^{\\dot{z}}{(\\varphi^*,\\dot{z},t_{2})} = t_{2}^{\\dot{z}} and t_{2}^{\\dot{z}} + \\rho_{f}^{\\dot{z}}{(\\varphi^*,\\dot{z},t_{2})} = 2 t_{2}^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('t_2', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["power", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["add", 5, "Pow(Symbol('t_2', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Pow(Symbol('t_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), Pow(Symbol('t_2', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(\\eta,\\mathbf{E},\\pi)} = - \\mathbf{E} + \\pi^{\\eta}, then obtain \\int \\frac{2 \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} d\\pi = \\int \\frac{- \\mathbf{E} + \\pi^{\\eta} + \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} d\\pi", "derivation": "\\dot{y}{(\\eta,\\mathbf{E},\\pi)} = - \\mathbf{E} + \\pi^{\\eta} and - \\mathbf{E} + \\pi^{\\eta} + \\dot{y}{(\\eta,\\mathbf{E},\\pi)} = - 2 \\mathbf{E} + 2 \\pi^{\\eta} and 2 \\dot{y}{(\\eta,\\mathbf{E},\\pi)} = - 2 \\mathbf{E} + 2 \\pi^{\\eta} and 2 \\dot{y}{(\\eta,\\mathbf{E},\\pi)} = - \\mathbf{E} + \\pi^{\\eta} + \\dot{y}{(\\eta,\\mathbf{E},\\pi)} and \\frac{2 \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} = \\frac{- \\mathbf{E} + \\pi^{\\eta} + \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} and \\int \\frac{2 \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} d\\pi = \\int \\frac{- \\mathbf{E} + \\pi^{\\eta} + \\dot{y}{(\\eta,\\mathbf{E},\\pi)}}{\\pi} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["divide", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})}, then obtain \\frac{\\int \\mathbf{E}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} d\\lambda}{\\mathbf{E}{(\\lambda)}} = \\frac{\\int \\cos^{2}{(\\cos{(\\lambda)})} d\\lambda}{\\mathbf{E}{(\\lambda)}}", "derivation": "\\mathbf{E}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})} and \\mathbf{E}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} = \\cos^{2}{(\\cos{(\\lambda)})} and \\int \\mathbf{E}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} d\\lambda = \\int \\cos^{2}{(\\cos{(\\lambda)})} d\\lambda and \\frac{\\int \\mathbf{E}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} d\\lambda}{\\cos{(\\cos{(\\lambda)})}} = \\frac{\\int \\cos^{2}{(\\cos{(\\lambda)})} d\\lambda}{\\cos{(\\cos{(\\lambda)})}} and \\frac{\\int \\mathbf{E}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} d\\lambda}{\\mathbf{E}{(\\lambda)}} = \\frac{\\int \\cos^{2}{(\\cos{(\\lambda)})} d\\lambda}{\\mathbf{E}{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "cos(cos(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 3, "cos(cos(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\hat{X})} = \\cos{(\\hat{X})}, then derive - \\cos{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = - \\sin{(\\hat{X})} - \\cos{(\\hat{X})}, then obtain - \\hat{\\mathbf{x}}{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = - \\hat{\\mathbf{x}}{(\\hat{X})} - \\sin{(\\hat{X})}", "derivation": "\\hat{\\mathbf{x}}{(\\hat{X})} = \\cos{(\\hat{X})} and \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\hat{X})} and - \\cos{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = - \\cos{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\cos{(\\hat{X})} and - \\cos{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = - \\sin{(\\hat{X})} - \\cos{(\\hat{X})} and - \\hat{\\mathbf{x}}{(\\hat{X})} + \\frac{d}{d \\hat{X}} \\hat{\\mathbf{x}}{(\\hat{X})} = - \\hat{\\mathbf{x}}{(\\hat{X})} - \\sin{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["minus", 2, "cos(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True))), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True))), Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True))), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True))), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(x)} = \\cos{(x)} and \\dot{z}{(x)} = \\frac{1}{\\cos{(x)}}, then obtain \\dot{y}^{4}{(x)} \\dot{z}^{2}{(x)} - \\dot{z}^{3}{(x)} = \\dot{y}^{2}{(x)} - \\dot{z}^{3}{(x)}", "derivation": "\\dot{y}{(x)} = \\cos{(x)} and \\frac{\\dot{y}^{2}{(x)}}{\\cos{(x)}} = \\dot{y}{(x)} and \\dot{z}{(x)} = \\frac{1}{\\cos{(x)}} and \\frac{\\dot{y}^{4}{(x)}}{\\cos^{2}{(x)}} = \\dot{y}^{2}{(x)} and \\frac{\\dot{y}^{4}{(x)}}{\\cos^{2}{(x)}} - \\frac{1}{\\cos^{3}{(x)}} = \\dot{y}^{2}{(x)} - \\frac{1}{\\cos^{3}{(x)}} and \\dot{y}^{4}{(x)} \\dot{z}^{2}{(x)} - \\dot{z}^{3}{(x)} = \\dot{y}^{2}{(x)} - \\dot{z}^{3}{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["divide", 1, "Mul(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(-1)), cos(Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(2)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Function('\\\\dot{y}')(Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(4)), Pow(cos(Symbol('x', commutative=True)), Integer(-2))), Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(2)))"], [["minus", 4, "Pow(cos(Symbol('x', commutative=True)), Integer(-3))"], "Equality(Add(Mul(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(4)), Pow(cos(Symbol('x', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(cos(Symbol('x', commutative=True)), Integer(-3)))), Add(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(cos(Symbol('x', commutative=True)), Integer(-3)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(4)), Pow(Function('\\\\dot{z}')(Symbol('x', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('x', commutative=True)), Integer(3)))), Add(Pow(Function('\\\\dot{y}')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('x', commutative=True)), Integer(3)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(J)} = \\sin{(J)}, then obtain \\frac{d^{3}}{d J^{3}} \\operatorname{C_{d}}{(J)} = \\frac{d^{3}}{d J^{3}} \\sin{(J)}", "derivation": "\\operatorname{C_{d}}{(J)} = \\sin{(J)} and \\frac{d}{d J} \\operatorname{C_{d}}{(J)} = \\frac{d}{d J} \\sin{(J)} and \\frac{d^{2}}{d J^{2}} \\operatorname{C_{d}}{(J)} = \\frac{d^{2}}{d J^{2}} \\sin{(J)} and \\frac{d^{3}}{d J^{3}} \\operatorname{C_{d}}{(J)} = \\frac{d^{3}}{d J^{3}} \\sin{(J)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(3))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\Psi)} = \\cos{(e^{\\Psi})}, then obtain \\Psi + \\frac{\\operatorname{C_{2}}{(\\Psi)}}{\\int \\cos{(e^{\\Psi})} d\\Psi} = \\Psi + \\frac{\\cos{(e^{\\Psi})}}{\\int \\cos{(e^{\\Psi})} d\\Psi}", "derivation": "\\operatorname{C_{2}}{(\\Psi)} = \\cos{(e^{\\Psi})} and \\int \\operatorname{C_{2}}{(\\Psi)} d\\Psi = \\int \\cos{(e^{\\Psi})} d\\Psi and \\frac{\\operatorname{C_{2}}{(\\Psi)}}{\\int \\operatorname{C_{2}}{(\\Psi)} d\\Psi} = \\frac{\\cos{(e^{\\Psi})}}{\\int \\operatorname{C_{2}}{(\\Psi)} d\\Psi} and \\frac{\\operatorname{C_{2}}{(\\Psi)}}{\\int \\cos{(e^{\\Psi})} d\\Psi} = \\frac{\\cos{(e^{\\Psi})}}{\\int \\cos{(e^{\\Psi})} d\\Psi} and \\Psi + \\frac{\\operatorname{C_{2}}{(\\Psi)}}{\\int \\cos{(e^{\\Psi})} d\\Psi} = \\Psi + \\frac{\\cos{(e^{\\Psi})}}{\\int \\cos{(e^{\\Psi})} d\\Psi}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\Psi', commutative=True)), cos(exp(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(cos(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "Integral(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Pow(Integral(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(cos(exp(Symbol('\\\\Psi', commutative=True))), Pow(Integral(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Pow(Integral(cos(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(cos(exp(Symbol('\\\\Psi', commutative=True))), Pow(Integral(cos(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))))"], [["add", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Function('C_2')(Symbol('\\\\Psi', commutative=True)), Pow(Integral(cos(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))), Add(Symbol('\\\\Psi', commutative=True), Mul(cos(exp(Symbol('\\\\Psi', commutative=True))), Pow(Integral(cos(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\sigma_x)} = \\log{(e^{\\sigma_x})}, then obtain \\frac{d^{2}}{d \\sigma_x^{2}} \\operatorname{E_{x}}{(\\sigma_x)} = 0", "derivation": "\\operatorname{E_{x}}{(\\sigma_x)} = \\log{(e^{\\sigma_x})} and \\frac{d}{d \\sigma_x} \\operatorname{E_{x}}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(e^{\\sigma_x})} and \\frac{d^{2}}{d \\sigma_x^{2}} \\operatorname{E_{x}}{(\\sigma_x)} = \\frac{d^{2}}{d \\sigma_x^{2}} \\log{(e^{\\sigma_x})} and \\frac{d^{2}}{d \\sigma_x^{2}} \\operatorname{E_{x}}{(\\sigma_x)} = 0", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), log(exp(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Derivative(log(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(U)} = U, then obtain \\mathbf{P}{(U)} \\int (\\int U dU) \\int \\operatorname{C_{2}}{(U)} dU d\\operatorname{C_{2}}{(U)} = \\mathbf{P}{(U)} \\int (\\int U dU)^{2} d\\operatorname{C_{2}}{(U)}", "derivation": "\\operatorname{C_{2}}{(U)} = U and \\int \\operatorname{C_{2}}{(U)} dU = \\int U dU and (\\int U dU) \\int \\operatorname{C_{2}}{(U)} dU = (\\int U dU)^{2} and \\int (\\int U dU) \\int \\operatorname{C_{2}}{(U)} dU dU = \\int (\\int U dU)^{2} dU and \\int (\\int U dU) \\int \\operatorname{C_{2}}{(U)} dU d\\operatorname{C_{2}}{(U)} = \\int (\\int U dU)^{2} d\\operatorname{C_{2}}{(U)} and \\mathbf{P}{(U)} \\int (\\int U dU) \\int \\operatorname{C_{2}}{(U)} dU d\\operatorname{C_{2}}{(U)} = \\mathbf{P}{(U)} \\int (\\int U dU)^{2} d\\operatorname{C_{2}}{(U)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('U', commutative=True)), Symbol('U', commutative=True))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))))"], [["times", 2, "Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Function('C_2')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Pow(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integer(2)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Function('C_2')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Integral(Pow(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integer(2)), Tuple(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Function('C_2')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Function('C_2')(Symbol('U', commutative=True)))), Integral(Pow(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integer(2)), Tuple(Function('C_2')(Symbol('U', commutative=True)))))"], [["times", 5, "Function('\\\\mathbf{P}')(Symbol('U', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), Integral(Mul(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Function('C_2')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Function('C_2')(Symbol('U', commutative=True))))), Mul(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), Integral(Pow(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Integer(2)), Tuple(Function('C_2')(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\mathbb{I},\\delta)} = \\delta \\mathbb{I}, then obtain \\frac{\\partial}{\\partial \\delta} (- \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}} + \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta^{2} \\mathbb{I}}) = \\frac{\\partial}{\\partial \\delta} (\\frac{1}{\\delta} - \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}})", "derivation": "\\varepsilon_{0}{(\\mathbb{I},\\delta)} = \\delta \\mathbb{I} and \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}} = 1 and \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta^{2} \\mathbb{I}} = \\frac{1}{\\delta} and - \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}} + \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta^{2} \\mathbb{I}} = \\frac{1}{\\delta} - \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}} and \\frac{\\partial}{\\partial \\delta} (- \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}} + \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta^{2} \\mathbb{I}}) = \\frac{\\partial}{\\partial \\delta} (\\frac{1}{\\delta} - \\frac{\\varepsilon_{0}{(\\mathbb{I},\\delta)}}{\\delta \\mathbb{I}})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(1))"], [["divide", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True))), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["minus", 3, "Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)))), Add(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(A_{y},\\Omega,g)} = - A_{y} + \\Omega + g, then obtain (\\int T{(A_{y},\\Omega,g)} d\\Omega)^{A_{y}} = (\\frac{\\Omega^{2}}{2} + \\Omega (- A_{y} + g) + \\mu_0)^{A_{y}}", "derivation": "T{(A_{y},\\Omega,g)} = - A_{y} + \\Omega + g and \\int T{(A_{y},\\Omega,g)} d\\Omega = \\int (- A_{y} + \\Omega + g) d\\Omega and (\\int T{(A_{y},\\Omega,g)} d\\Omega)^{A_{y}} = (\\int (- A_{y} + \\Omega + g) d\\Omega)^{A_{y}} and (\\int T{(A_{y},\\Omega,g)} d\\Omega)^{A_{y}} = (\\frac{\\Omega^{2}}{2} + \\Omega (- A_{y} + g) + \\mu_0)^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('A_y', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('T')(Symbol('A_y', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Integral(Function('T')(Symbol('A_y', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('A_y', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('A_y', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('T')(Symbol('A_y', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('g', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(V)} = \\cos{(\\log{(V)})} and \\Psi_{nl}{(V)} = \\log{(V)}, then obtain \\log{(V)} + \\cos{(\\log{(V)})} = \\log{(V)} + \\cos{(\\Psi_{nl}{(V)})}", "derivation": "\\varphi^{*}{(V)} = \\cos{(\\log{(V)})} and \\Psi_{nl}{(V)} = \\log{(V)} and \\varphi^{*}{(V)} = \\cos{(\\Psi_{nl}{(V)})} and \\varphi^{*}{(V)} + \\log{(V)} = \\log{(V)} + \\cos{(\\Psi_{nl}{(V)})} and \\log{(V)} + \\cos{(\\log{(V)})} = \\log{(V)} + \\cos{(\\Psi_{nl}{(V)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True)), cos(log(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True)), cos(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True))))"], [["add", 3, "log(Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Add(log(Symbol('V', commutative=True)), cos(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('V', commutative=True)), cos(log(Symbol('V', commutative=True)))), Add(log(Symbol('V', commutative=True)), cos(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given J{(\\phi_1,p)} = p^{\\phi_1} and \\theta_{2}{(\\phi_1,p,s)} = \\frac{p^{\\phi_1}}{s}, then obtain (p - J^{\\phi_1}{(\\phi_1,p)} + \\frac{J{(\\phi_1,p)}}{s}) e^{- s} = (p - J^{\\phi_1}{(\\phi_1,p)} + \\theta_{2}{(\\phi_1,p,s)}) e^{- s}", "derivation": "J{(\\phi_1,p)} = p^{\\phi_1} and \\frac{J{(\\phi_1,p)}}{s} = \\frac{p^{\\phi_1}}{s} and p - J^{\\phi_1}{(\\phi_1,p)} + \\frac{J{(\\phi_1,p)}}{s} = p + \\frac{p^{\\phi_1}}{s} - J^{\\phi_1}{(\\phi_1,p)} and \\theta_{2}{(\\phi_1,p,s)} = \\frac{p^{\\phi_1}}{s} and (p - J^{\\phi_1}{(\\phi_1,p)} + \\frac{J{(\\phi_1,p)}}{s}) e^{- s} = (p + \\frac{p^{\\phi_1}}{s} - J^{\\phi_1}{(\\phi_1,p)}) e^{- s} and (p - J^{\\phi_1}{(\\phi_1,p)} + \\frac{J{(\\phi_1,p)}}{s}) e^{- s} = (p - J^{\\phi_1}{(\\phi_1,p)} + \\theta_{2}{(\\phi_1,p,s)}) e^{- s}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Symbol('p', commutative=True), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)))), Add(Symbol('p', commutative=True), Mul(Pow(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["divide", 3, "exp(Symbol('s', commutative=True))"], "Equality(Mul(Add(Symbol('p', commutative=True), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)))), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Mul(Add(Symbol('p', commutative=True), Mul(Pow(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True)))), exp(Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Symbol('p', commutative=True), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)))), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Mul(Add(Symbol('p', commutative=True), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True))), exp(Mul(Integer(-1), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then derive \\int \\operatorname{F_{N}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + v_{x}, then obtain \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{B} \\operatorname{F_{N}}{(\\mathbf{B})} - \\mathbf{B} + v_{x}) = \\frac{d}{d \\mathbf{B}} \\int \\log{(\\mathbf{B})} d\\mathbf{B}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\int \\operatorname{F_{N}}{(\\mathbf{B})} d\\mathbf{B} = \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\frac{d}{d \\mathbf{B}} \\int \\operatorname{F_{N}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{d}{d \\mathbf{B}} \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\int \\operatorname{F_{N}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + v_{x} and \\int \\operatorname{F_{N}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\operatorname{F_{N}}{(\\mathbf{B})} - \\mathbf{B} + v_{x} and \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{B} \\operatorname{F_{N}}{(\\mathbf{B})} - \\mathbf{B} + v_{x}) = \\frac{d}{d \\mathbf{B}} \\int \\log{(\\mathbf{B})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})} = y^{\\prime} + \\cos{(J)}, then derive \\log{(\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})})} = 0, then obtain \\frac{(y^{\\prime} + \\cos{(J)}) \\log{(\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\cos{(J)}))}}{y^{\\prime} + \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})} + \\cos{(J)}} = 0", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})} = y^{\\prime} + \\cos{(J)} and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\cos{(J)}) and \\log{(\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})})} = \\log{(\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\cos{(J)}))} and \\log{(\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})})} = 0 and \\log{(\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\cos{(J)}))} = 0 and \\frac{(y^{\\prime} + \\cos{(J)}) \\log{(\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\cos{(J)}))}}{y^{\\prime} + \\operatorname{V_{\\mathbf{B}}}{(J,y^{\\prime})} + \\cos{(J)}} = 0", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), log(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Integer(0))"], [["divide", 5, "Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Integer(-1)), Add(Symbol('y^{\\\\prime}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('J', commutative=True))))"], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('J', commutative=True))), Integer(-1)), log(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('J', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given Q{(m,W)} = \\frac{\\partial}{\\partial m} W m and \\pi{(m,W)} = \\frac{Q^{W}{(m,W)}}{W}, then derive Q{(m,W)} = W, then obtain \\frac{\\pi{(m,\\frac{\\partial}{\\partial m} W m)}}{W} = \\frac{(\\frac{\\partial}{\\partial m} W m)^{\\frac{\\partial}{\\partial m} W m}}{W \\frac{\\partial}{\\partial m} W m}", "derivation": "Q{(m,W)} = \\frac{\\partial}{\\partial m} W m and Q{(m,W)} = W and \\pi{(m,W)} = \\frac{Q^{W}{(m,W)}}{W} and \\pi{(m,W)} = \\frac{W^{W}}{W} and \\frac{\\partial}{\\partial m} W m = W and \\pi{(m,\\frac{\\partial}{\\partial m} W m)} = \\frac{(\\frac{\\partial}{\\partial m} W m)^{\\frac{\\partial}{\\partial m} W m}}{\\frac{\\partial}{\\partial m} W m} and \\frac{\\pi{(m,\\frac{\\partial}{\\partial m} W m)}}{W} = \\frac{(\\frac{\\partial}{\\partial m} W m)^{\\frac{\\partial}{\\partial m} W m}}{W \\frac{\\partial}{\\partial m} W m}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\pi')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('W', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\pi')(Symbol('m', commutative=True), Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))))"], [["times", 6, "Pow(Symbol('W', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('m', commutative=True), Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Symbol('W', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\nabla{(y)} = \\cos{(\\log{(y)})}, then derive \\int \\nabla{(y)} dy = \\hat{H}_l + \\frac{y \\sin{(\\log{(y)})}}{2} + \\frac{y \\cos{(\\log{(y)})}}{2}, then obtain y \\int \\cos{(\\log{(y)})} dy = y (\\hat{H}_l + \\frac{y \\sin{(\\log{(y)})}}{2} + \\frac{y \\cos{(\\log{(y)})}}{2})", "derivation": "\\nabla{(y)} = \\cos{(\\log{(y)})} and \\int \\nabla{(y)} dy = \\int \\cos{(\\log{(y)})} dy and \\int \\nabla{(y)} dy = \\hat{H}_l + \\frac{y \\sin{(\\log{(y)})}}{2} + \\frac{y \\cos{(\\log{(y)})}}{2} and y \\int \\nabla{(y)} dy = y (\\hat{H}_l + \\frac{y \\sin{(\\log{(y)})}}{2} + \\frac{y \\cos{(\\log{(y)})}}{2}) and y \\int \\cos{(\\log{(y)})} dy = y (\\hat{H}_l + \\frac{y \\sin{(\\log{(y)})}}{2} + \\frac{y \\cos{(\\log{(y)})}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Symbol('y', commutative=True), sin(log(Symbol('y', commutative=True)))), Mul(Rational(1, 2), Symbol('y', commutative=True), cos(log(Symbol('y', commutative=True))))))"], [["times", 3, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Integral(Function('\\\\nabla')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(Symbol('y', commutative=True), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Symbol('y', commutative=True), sin(log(Symbol('y', commutative=True)))), Mul(Rational(1, 2), Symbol('y', commutative=True), cos(log(Symbol('y', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('y', commutative=True), Integral(cos(log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))), Mul(Symbol('y', commutative=True), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Symbol('y', commutative=True), sin(log(Symbol('y', commutative=True)))), Mul(Rational(1, 2), Symbol('y', commutative=True), cos(log(Symbol('y', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{J}_f,r_{0})} = r_{0}^{\\mathbf{J}_f} and u{(\\mathbf{J}_f)} = \\mathbf{J}_f, then obtain 1 = \\cos^{u{(\\mathbf{J}_f)}}{(\\int (r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})}) du{(\\mathbf{J}_f)})}", "derivation": "\\mathbf{H}{(\\mathbf{J}_f,r_{0})} = r_{0}^{\\mathbf{J}_f} and 0 = r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})} and u{(\\mathbf{J}_f)} = \\mathbf{J}_f and \\int 0 d\\mathbf{J}_f = \\int (r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})}) d\\mathbf{J}_f and 1 = \\cos{(\\int (r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})}) d\\mathbf{J}_f)} and 1 = \\cos^{\\mathbf{J}_f}{(\\int (r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})}) d\\mathbf{J}_f)} and 1 = \\cos^{u{(\\mathbf{J}_f)}}{(\\int (r_{0}^{\\mathbf{J}_f} - \\mathbf{H}{(\\mathbf{J}_f,r_{0})}) du{(\\mathbf{J}_f)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["cos", 4], "Equality(Integer(1), cos(Integral(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integer(1), Pow(cos(Integral(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), Pow(cos(Integral(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Function('u')(Symbol('\\\\mathbf{J}_f', commutative=True))))), Function('u')(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\mu)} = \\mu and \\hat{H}_l{(\\mu)} = \\mu^{\\mu}, then obtain \\frac{d}{d \\mu} 2 \\mu^{\\mu} = \\frac{d}{d \\mu} (\\mu^{\\mu} + \\hat{H}_l{(\\mu)})", "derivation": "\\varepsilon{(\\mu)} = \\mu and \\varepsilon^{\\mu}{(\\mu)} = \\mu^{\\mu} and 2 \\varepsilon^{\\mu}{(\\mu)} = \\mu^{\\mu} + \\varepsilon^{\\mu}{(\\mu)} and \\frac{d}{d \\mu} 2 \\varepsilon^{\\mu}{(\\mu)} = \\frac{d}{d \\mu} (\\mu^{\\mu} + \\varepsilon^{\\mu}{(\\mu)}) and \\hat{H}_l{(\\mu)} = \\mu^{\\mu} and \\frac{d}{d \\mu} 2 \\varepsilon^{\\mu}{(\\mu)} = \\frac{d}{d \\mu} (\\hat{H}_l{(\\mu)} + \\varepsilon^{\\mu}{(\\mu)}) and \\frac{d}{d \\mu} 2 \\mu^{\\mu} = \\frac{d}{d \\mu} (\\mu^{\\mu} + \\hat{H}_l{(\\mu)})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 2, "Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Add(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Mul(Integer(2), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})} = - \\mathbf{H} + n_{1} + z^{*}, then obtain ((- \\mathbf{H} + n_{1} + z^{*}) \\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})})^{z^{*}} + (- \\mathbf{H} + n_{1} + z^{*})^{2} = (- \\mathbf{H} + n_{1} + z^{*})^{2} + ((- \\mathbf{H} + n_{1} + z^{*})^{2})^{z^{*}}", "derivation": "\\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})} = - \\mathbf{H} + n_{1} + z^{*} and (- \\mathbf{H} + n_{1} + z^{*}) \\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})} = (- \\mathbf{H} + n_{1} + z^{*})^{2} and ((- \\mathbf{H} + n_{1} + z^{*}) \\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})})^{z^{*}} = ((- \\mathbf{H} + n_{1} + z^{*})^{2})^{z^{*}} and ((- \\mathbf{H} + n_{1} + z^{*}) \\mathbf{J}_M{(z^{*},n_{1},\\mathbf{H})})^{z^{*}} + (- \\mathbf{H} + n_{1} + z^{*})^{2} = (- \\mathbf{H} + n_{1} + z^{*})^{2} + ((- \\mathbf{H} + n_{1} + z^{*})^{2})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('z^*', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2)), Symbol('z^*', commutative=True)))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2))"], "Equality(Add(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('z^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Integer(2)), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(r)} = \\sin{(\\sin{(r)})}, then obtain \\frac{W (\\mathbf{A}{(r)} + \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}} = \\frac{W (2 \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}}", "derivation": "\\mathbf{A}{(r)} = \\sin{(\\sin{(r)})} and \\mathbf{A}{(r)} + \\sin{(\\sin{(r)})} = 2 \\sin{(\\sin{(r)})} and (\\mathbf{A}{(r)} + \\sin{(\\sin{(r)})})^{r} = (2 \\sin{(\\sin{(r)})})^{r} and \\frac{(\\mathbf{A}{(r)} + \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}} = \\frac{(2 \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}} and \\frac{W (\\mathbf{A}{(r)} + \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}} = \\frac{W (2 \\sin{(\\sin{(r)})})^{r}}{2 \\mathbf{A}{(r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True))))"], [["add", 1, "sin(sin(Symbol('r', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True)))), Mul(Integer(2), sin(sin(Symbol('r', commutative=True)))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Mul(Integer(2), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["divide", 3, "Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('r', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Add(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Mul(Integer(2), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Integer(-1))))"], [["times", 4, "Symbol('W', commutative=True)"], "Equality(Mul(Rational(1, 2), Symbol('W', commutative=True), Pow(Add(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Symbol('W', commutative=True), Pow(Mul(Integer(2), sin(sin(Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\Psi,V)} = \\frac{\\partial}{\\partial \\Psi} (V - \\Psi), then obtain \\int \\frac{\\Psi \\int \\mathbf{J}_P{(\\Psi,V)} d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} d\\Psi = \\int \\frac{\\Psi \\int \\frac{\\partial}{\\partial \\Psi} (V - \\Psi) d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} d\\Psi", "derivation": "\\mathbf{J}_P{(\\Psi,V)} = \\frac{\\partial}{\\partial \\Psi} (V - \\Psi) and \\int \\mathbf{J}_P{(\\Psi,V)} d\\Psi = \\int \\frac{\\partial}{\\partial \\Psi} (V - \\Psi) d\\Psi and \\frac{\\Psi \\int \\mathbf{J}_P{(\\Psi,V)} d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} = \\frac{\\Psi \\int \\frac{\\partial}{\\partial \\Psi} (V - \\Psi) d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} and \\int \\frac{\\Psi \\int \\mathbf{J}_P{(\\Psi,V)} d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} d\\Psi = \\int \\frac{\\Psi \\int \\frac{\\partial}{\\partial \\Psi} (V - \\Psi) d\\Psi}{\\frac{\\partial}{\\partial \\Psi} (V - \\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\Psi', commutative=True), Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Symbol('\\\\Psi', commutative=True), Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(v_{2})} = \\cos{(v_{2})}, then obtain (- v_{2} + \\int \\operatorname{A_{1}}{(v_{2})} dv_{2})^{v_{2}} = (- v_{2} + \\int \\cos{(v_{2})} dv_{2})^{v_{2}}", "derivation": "\\operatorname{A_{1}}{(v_{2})} = \\cos{(v_{2})} and \\int \\operatorname{A_{1}}{(v_{2})} dv_{2} = \\int \\cos{(v_{2})} dv_{2} and - v_{2} + \\int \\operatorname{A_{1}}{(v_{2})} dv_{2} = - v_{2} + \\int \\cos{(v_{2})} dv_{2} and (- v_{2} + \\int \\operatorname{A_{1}}{(v_{2})} dv_{2})^{v_{2}} = (- v_{2} + \\int \\cos{(v_{2})} dv_{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Function('A_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))))"], [["power", 3, "Symbol('v_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Function('A_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(t_{1},v_{t})} = t_{1} v_{t}, then derive \\frac{\\partial}{\\partial t_{1}} \\varepsilon{(t_{1},v_{t})} = v_{t}, then obtain \\frac{\\partial}{\\partial t_{1}} t_{1} v_{t} = v_{t}", "derivation": "\\varepsilon{(t_{1},v_{t})} = t_{1} v_{t} and \\frac{\\partial}{\\partial t_{1}} \\varepsilon{(t_{1},v_{t})} = \\frac{\\partial}{\\partial t_{1}} t_{1} v_{t} and \\frac{\\partial}{\\partial t_{1}} \\varepsilon{(t_{1},v_{t})} = v_{t} and \\frac{\\partial}{\\partial t_{1}} t_{1} v_{t} = v_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Symbol('v_t', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Symbol('v_t', commutative=True))"]]}, {"prompt": "Given \\mathbf{g}{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and A{(\\tilde{g}^*)} = \\tilde{g}^* \\mathbf{g}{(\\tilde{g}^*)}, then obtain - \\tilde{g}^* e^{\\tilde{g}^*} - \\tilde{g}^* + A{(\\tilde{g}^*)} - \\mathbf{g}{(\\tilde{g}^*)} = - \\tilde{g}^* - \\mathbf{g}{(\\tilde{g}^*)}", "derivation": "\\mathbf{g}{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and \\tilde{g}^* \\mathbf{g}{(\\tilde{g}^*)} = \\tilde{g}^* e^{\\tilde{g}^*} and \\tilde{g}^* \\mathbf{g}{(\\tilde{g}^*)} - \\tilde{g}^* e^{\\tilde{g}^*} = 0 and A{(\\tilde{g}^*)} = \\tilde{g}^* \\mathbf{g}{(\\tilde{g}^*)} and - \\tilde{g}^* e^{\\tilde{g}^*} + A{(\\tilde{g}^*)} = 0 and - \\tilde{g}^* e^{\\tilde{g}^*} - \\tilde{g}^* + A{(\\tilde{g}^*)} - \\mathbf{g}{(\\tilde{g}^*)} = - \\tilde{g}^* - \\mathbf{g}{(\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Function('A')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(0))"], [["minus", 5, "Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('A')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(x^\\prime)} = \\sin{(x^\\prime)}, then derive (\\tilde{g}^* - \\cos{(x^\\prime)}) \\int \\rho_{f}{(x^\\prime)} dx^\\prime = (\\tilde{g}^* - \\cos{(x^\\prime)})^{2}, then obtain \\log{((\\tilde{g}^* - \\cos{(x^\\prime)}) \\int \\sin{(x^\\prime)} dx^\\prime)} = \\log{((\\tilde{g}^* - \\cos{(x^\\prime)})^{2})}", "derivation": "\\rho_{f}{(x^\\prime)} = \\sin{(x^\\prime)} and \\int \\rho_{f}{(x^\\prime)} dx^\\prime = \\int \\sin{(x^\\prime)} dx^\\prime and (\\int \\rho_{f}{(x^\\prime)} dx^\\prime) \\int \\sin{(x^\\prime)} dx^\\prime = (\\int \\sin{(x^\\prime)} dx^\\prime)^{2} and (\\tilde{g}^* - \\cos{(x^\\prime)}) \\int \\rho_{f}{(x^\\prime)} dx^\\prime = (\\tilde{g}^* - \\cos{(x^\\prime)})^{2} and \\log{((\\tilde{g}^* - \\cos{(x^\\prime)}) \\int \\rho_{f}{(x^\\prime)} dx^\\prime)} = \\log{((\\tilde{g}^* - \\cos{(x^\\prime)})^{2})} and \\log{((\\tilde{g}^* - \\cos{(x^\\prime)}) \\int \\sin{(x^\\prime)} dx^\\prime)} = \\log{((\\tilde{g}^* - \\cos{(x^\\prime)})^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Pow(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integral(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integer(2)))"], [["log", 4], "Equality(log(Mul(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integral(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), log(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), log(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given Z{(\\psi,r_{0})} = \\psi + r_{0}, then obtain 2 r_{0} \\frac{\\partial}{\\partial \\psi} Z{(\\psi,r_{0})} = r_{0} (\\frac{\\partial}{\\partial \\psi} Z{(\\psi,r_{0})} + 1)", "derivation": "Z{(\\psi,r_{0})} = \\psi + r_{0} and 2 Z{(\\psi,r_{0})} = \\psi + r_{0} + Z{(\\psi,r_{0})} and 2 r_{0} Z{(\\psi,r_{0})} = r_{0} (\\psi + r_{0} + Z{(\\psi,r_{0})}) and \\frac{\\partial}{\\partial \\psi} 2 r_{0} Z{(\\psi,r_{0})} = \\frac{\\partial}{\\partial \\psi} r_{0} (\\psi + r_{0} + Z{(\\psi,r_{0})}) and 2 r_{0} \\frac{\\partial}{\\partial \\psi} Z{(\\psi,r_{0})} = r_{0} (\\frac{\\partial}{\\partial \\psi} Z{(\\psi,r_{0})} + 1)", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 1, "Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True))), Add(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True))))"], [["times", 2, "Symbol('r_0', commutative=True)"], "Equality(Mul(Integer(2), Symbol('r_0', commutative=True), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Symbol('r_0', commutative=True), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True), Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Symbol('r_0', commutative=True), Derivative(Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Symbol('r_0', commutative=True), Add(Derivative(Function('Z')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\delta)} = \\cos{(\\delta)}, then obtain (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 = (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 - \\frac{\\mathbf{S}{(\\delta)}}{\\delta} + \\frac{\\cos{(\\delta)}}{\\delta}", "derivation": "\\mathbf{S}{(\\delta)} = \\cos{(\\delta)} and \\frac{\\mathbf{S}{(\\delta)}}{\\delta} = \\frac{\\cos{(\\delta)}}{\\delta} and -1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta} = -1 + \\frac{\\cos{(\\delta)}}{\\delta} and (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta} = (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 + \\frac{\\cos{(\\delta)}}{\\delta} and (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 = (-1 + \\frac{\\mathbf{S}{(\\delta)}}{\\delta})^{\\delta} - 1 - \\frac{\\mathbf{S}{(\\delta)}}{\\delta} + \\frac{\\cos{(\\delta)}}{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["divide", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), cos(Symbol('\\\\delta', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), cos(Symbol('\\\\delta', commutative=True)))))"], [["add", 3, "Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Add(Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), cos(Symbol('\\\\delta', commutative=True)))))"], [["minus", 4, "Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Integer(-1)), Add(Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), cos(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given c{(F_{g})} = e^{F_{g}}, then obtain (c{(F_{g})} e^{F_{g}} + e^{F_{g}}) e^{- F_{g}} - e^{2 F_{g}} - e^{F_{g}} = (e^{2 F_{g}} + e^{F_{g}}) e^{- F_{g}} - e^{2 F_{g}} - e^{F_{g}}", "derivation": "c{(F_{g})} = e^{F_{g}} and c{(F_{g})} e^{F_{g}} = e^{2 F_{g}} and c{(F_{g})} e^{F_{g}} + e^{F_{g}} = e^{2 F_{g}} + e^{F_{g}} and (c{(F_{g})} e^{F_{g}} + e^{F_{g}}) e^{- F_{g}} = (e^{2 F_{g}} + e^{F_{g}}) e^{- F_{g}} and (c{(F_{g})} e^{F_{g}} + e^{F_{g}}) e^{- F_{g}} - e^{2 F_{g}} - e^{F_{g}} = (e^{2 F_{g}} + e^{F_{g}}) e^{- F_{g}} - e^{2 F_{g}} - e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["times", 1, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('c')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(2), Symbol('F_g', commutative=True))))"], [["add", 2, "exp(Symbol('F_g', commutative=True))"], "Equality(Add(Mul(Function('c')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))), Add(exp(Mul(Integer(2), Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))))"], [["divide", 3, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Add(Mul(Function('c')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Add(exp(Mul(Integer(2), Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))))"], [["minus", 4, "Add(exp(Mul(Integer(2), Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Function('c')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Add(Mul(Add(exp(Mul(Integer(2), Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given k{(\\hat{p}_0)} = e^{\\hat{p}_0}, then obtain \\hat{p}_0 + (- k{(\\hat{p}_0)})^{\\hat{p}_0} = \\hat{p}_0 + (- e^{\\hat{p}_0})^{\\hat{p}_0}", "derivation": "k{(\\hat{p}_0)} = e^{\\hat{p}_0} and - k{(\\hat{p}_0)} = - e^{\\hat{p}_0} and (- k{(\\hat{p}_0)})^{\\hat{p}_0} = (- e^{\\hat{p}_0})^{\\hat{p}_0} and \\hat{p}_0 + (- k{(\\hat{p}_0)})^{\\hat{p}_0} = \\hat{p}_0 + (- e^{\\hat{p}_0})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('k')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('k')(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Pow(Mul(Integer(-1), Function('k')(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Pow(Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\varepsilon)} = \\varepsilon and U{(\\varepsilon)} = \\varepsilon, then derive \\int U{(\\varepsilon)} d\\varepsilon = \\frac{\\varepsilon^{2}}{2} + \\varphi^*, then obtain - y^{\\prime} + \\int \\varepsilon d\\varepsilon = \\frac{\\varepsilon^{2}}{2} + \\varphi^* - y^{\\prime}", "derivation": "\\mathbf{E}{(\\varepsilon)} = \\varepsilon and U{(\\varepsilon)} = \\varepsilon and \\int U{(\\varepsilon)} d\\varepsilon = \\int \\varepsilon d\\varepsilon and \\int U{(\\varepsilon)} d\\varepsilon = \\frac{\\varepsilon^{2}}{2} + \\varphi^* and \\int \\varepsilon d\\varepsilon = \\frac{\\varepsilon^{2}}{2} + \\varphi^* and \\int \\varepsilon d\\mathbf{E}{(\\varepsilon)} = \\varphi^* + \\frac{\\mathbf{E}^{2}{(\\varepsilon)}}{2} and - y^{\\prime} + \\int \\varepsilon d\\mathbf{E}{(\\varepsilon)} = \\varphi^* - y^{\\prime} + \\frac{\\mathbf{E}^{2}{(\\varepsilon)}}{2} and - y^{\\prime} + \\int \\varepsilon d\\varepsilon = \\frac{\\varepsilon^{2}}{2} + \\varphi^* - y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('U')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon', commutative=True)))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)))))"], [["minus", 6, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon', commutative=True))))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\Omega)} = \\sin{(e^{\\Omega})}, then derive P_{e} + 2 \\int \\Omega d\\Omega + 2 \\int \\operatorname{v_{1}}{(\\Omega)} d\\Omega = \\int (2 \\Omega + \\operatorname{v_{1}}{(\\Omega)} + \\sin{(e^{\\Omega})}) d\\Omega, then obtain \\frac{P_{e} + 2 \\int \\Omega d\\Omega + 2 \\int \\sin{(e^{\\Omega})} d\\Omega}{\\frac{d}{d \\Omega} \\operatorname{v_{1}}{(\\Omega)}} = \\frac{\\int (2 \\Omega + 2 \\sin{(e^{\\Omega})}) d\\Omega}{\\frac{d}{d \\Omega} \\operatorname{v_{1}}{(\\Omega)}}", "derivation": "\\operatorname{v_{1}}{(\\Omega)} = \\sin{(e^{\\Omega})} and \\Omega + \\operatorname{v_{1}}{(\\Omega)} = \\Omega + \\sin{(e^{\\Omega})} and 2 \\Omega + 2 \\operatorname{v_{1}}{(\\Omega)} = 2 \\Omega + \\operatorname{v_{1}}{(\\Omega)} + \\sin{(e^{\\Omega})} and \\int (2 \\Omega + 2 \\operatorname{v_{1}}{(\\Omega)}) d\\Omega = \\int (2 \\Omega + \\operatorname{v_{1}}{(\\Omega)} + \\sin{(e^{\\Omega})}) d\\Omega and P_{e} + 2 \\int \\Omega d\\Omega + 2 \\int \\operatorname{v_{1}}{(\\Omega)} d\\Omega = \\int (2 \\Omega + \\operatorname{v_{1}}{(\\Omega)} + \\sin{(e^{\\Omega})}) d\\Omega and P_{e} + 2 \\int \\Omega d\\Omega + 2 \\int \\sin{(e^{\\Omega})} d\\Omega = \\int (2 \\Omega + 2 \\sin{(e^{\\Omega})}) d\\Omega and \\frac{P_{e} + 2 \\int \\Omega d\\Omega + 2 \\int \\sin{(e^{\\Omega})} d\\Omega}{\\frac{d}{d \\Omega} \\operatorname{v_{1}}{(\\Omega)}} = \\frac{\\int (2 \\Omega + 2 \\sin{(e^{\\Omega})}) d\\Omega}{\\frac{d}{d \\Omega} \\operatorname{v_{1}}{(\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\Omega', commutative=True)), sin(exp(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('v_1')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), sin(exp(Symbol('\\\\Omega', commutative=True)))))"], [["add", 2, "Add(Symbol('\\\\Omega', commutative=True), Function('v_1')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Function('v_1')(Symbol('\\\\Omega', commutative=True)), sin(exp(Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Function('v_1')(Symbol('\\\\Omega', commutative=True)), sin(exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('P_e', commutative=True), Mul(Integer(2), Add(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))), Integral(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Function('v_1')(Symbol('\\\\Omega', commutative=True)), sin(exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('P_e', commutative=True), Mul(Integer(2), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Integer(2), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Integral(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), sin(exp(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 6, "Derivative(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('P_e', commutative=True), Mul(Integer(2), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Integer(2), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Pow(Derivative(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Derivative(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), sin(exp(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(L,\\mathbf{M})} = \\mathbf{M}^{L}, then obtain (\\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})})^{\\mathbf{M}})})^{L} = (\\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{2 L})^{\\mathbf{M}})})^{L}", "derivation": "\\sigma_{x}{(L,\\mathbf{M})} = \\mathbf{M}^{L} and \\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})} = \\mathbf{M}^{2 L} and (\\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})})^{\\mathbf{M}} = (\\mathbf{M}^{2 L})^{\\mathbf{M}} and \\sin{((\\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})})^{\\mathbf{M}})} = \\sin{((\\mathbf{M}^{2 L})^{\\mathbf{M}})} and \\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})})^{\\mathbf{M}})} = \\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{2 L})^{\\mathbf{M}})} and (\\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{L} \\sigma_{x}{(L,\\mathbf{M})})^{\\mathbf{M}})})^{L} = (\\frac{\\partial}{\\partial L} \\sin{((\\mathbf{M}^{2 L})^{\\mathbf{M}})})^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), sin(Pow(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(sin(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Pow(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(Derivative(sin(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Function('\\\\sigma_x')(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)), Pow(Derivative(sin(Pow(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(v_{t})} = \\log{(v_{t})}, then obtain \\frac{\\hat{x}^{3}{(v_{t})}}{v_{t}} = \\frac{\\hat{x}^{2}{(v_{t})} \\log{(v_{t})}}{v_{t}}", "derivation": "\\hat{x}{(v_{t})} = \\log{(v_{t})} and \\frac{\\hat{x}{(v_{t})}}{v_{t}} = \\frac{\\log{(v_{t})}}{v_{t}} and \\frac{\\hat{x}^{2}{(v_{t})}}{v_{t}} = \\frac{\\hat{x}{(v_{t})} \\log{(v_{t})}}{v_{t}} and \\frac{\\hat{x}^{3}{(v_{t})}}{v_{t}} = \\frac{\\hat{x}^{2}{(v_{t})} \\log{(v_{t})}}{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["divide", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True))))"], [["times", 2, "Function('\\\\hat{x}')(Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), Integer(2))), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True))))"], [["times", 3, "Function('\\\\hat{x}')(Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), Integer(3))), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), Integer(2)), log(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given u{(g_{\\varepsilon})} = g_{\\varepsilon}, then derive \\int u{(g_{\\varepsilon})} dg_{\\varepsilon} = M_{E} + \\frac{g_{\\varepsilon}^{2}}{2}, then obtain (M_{E} + \\frac{u^{2}{(g_{\\varepsilon})}}{2}) \\int u{(g_{\\varepsilon})} du{(g_{\\varepsilon})} = (M_{E} + \\frac{u^{2}{(g_{\\varepsilon})}}{2})^{2}", "derivation": "u{(g_{\\varepsilon})} = g_{\\varepsilon} and \\int u{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int g_{\\varepsilon} dg_{\\varepsilon} and \\int u{(g_{\\varepsilon})} dg_{\\varepsilon} = M_{E} + \\frac{g_{\\varepsilon}^{2}}{2} and \\int u{(g_{\\varepsilon})} du{(g_{\\varepsilon})} = M_{E} + \\frac{u^{2}{(g_{\\varepsilon})}}{2} and (M_{E} + \\frac{u^{2}{(g_{\\varepsilon})}}{2}) \\int u{(g_{\\varepsilon})} du{(g_{\\varepsilon})} = (M_{E} + \\frac{u^{2}{(g_{\\varepsilon})}}{2})^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)))))"], [["times", 4, "Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2))))"], "Equality(Mul(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)))), Integral(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True))))), Pow(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Function('u')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(B)} = \\cos{(B)}, then derive (\\frac{d}{d B} \\operatorname{v_{z}}{(B)})^{B} = (- \\sin{(B)})^{B}, then obtain - \\cos{(B)} + (\\frac{d}{d B} \\cos{(B)})^{B} = (- \\sin{(B)})^{B} - \\cos{(B)}", "derivation": "\\operatorname{v_{z}}{(B)} = \\cos{(B)} and \\frac{d}{d B} \\operatorname{v_{z}}{(B)} = \\frac{d}{d B} \\cos{(B)} and (\\frac{d}{d B} \\operatorname{v_{z}}{(B)})^{B} = (\\frac{d}{d B} \\cos{(B)})^{B} and (\\frac{d}{d B} \\operatorname{v_{z}}{(B)})^{B} = (- \\sin{(B)})^{B} and (\\frac{d}{d B} \\cos{(B)})^{B} = (- \\sin{(B)})^{B} and - \\cos{(B)} + (\\frac{d}{d B} \\cos{(B)})^{B} = (- \\sin{(B)})^{B} - \\cos{(B)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Derivative(Function('v_z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('v_z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["add", 5, "Mul(Integer(-1), cos(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('B', commutative=True))), Pow(Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True))), Add(Pow(Mul(Integer(-1), sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then derive \\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain \\log{(\\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}})} = \\log{(\\cos{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}})}", "derivation": "\\theta_{1}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}} = \\cos{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}} and \\log{(\\frac{d}{d \\mathbb{I}} \\theta_{1}{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}})} = \\log{(\\cos{(\\mathbb{I})} + \\frac{1}{\\Psi^{\\dagger}{(E_{x},v_{z})}})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 3, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_x', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_x', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Add(cos(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_x', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["log", 4], "Equality(log(Add(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_x', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)))), log(Add(cos(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_x', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\psi{(\\hat{p},\\phi,A)} = - A - \\hat{p} + \\phi, then derive \\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} = \\hat{p} - 1, then obtain - \\hat{p} (\\hat{p} + 2 \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} + 1) = - \\hat{p} (\\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)})", "derivation": "\\psi{(\\hat{p},\\phi,A)} = - A - \\hat{p} + \\phi and \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} = \\frac{\\partial}{\\partial \\hat{p}} (- A - \\hat{p} + \\phi) and \\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} = \\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} (- A - \\hat{p} + \\phi) and \\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} = \\hat{p} - 1 and - \\hat{p} (\\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)}) = - \\hat{p} (\\hat{p} - 1) and - \\hat{p} (\\hat{p} + 2 \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)} + 1) = - \\hat{p} (\\hat{p} + \\frac{\\partial}{\\partial \\hat{p}} \\psi{(\\hat{p},\\phi,A)})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(2), Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given J{(\\tilde{g}^*,G)} = \\frac{\\partial}{\\partial G} G \\tilde{g}^*, then derive J{(\\tilde{g}^*,G)} = \\tilde{g}^*, then obtain \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* = \\frac{\\partial}{\\partial \\tilde{g}^*} J{(\\tilde{g}^*,G)}", "derivation": "J{(\\tilde{g}^*,G)} = \\frac{\\partial}{\\partial G} G \\tilde{g}^* and J{(\\tilde{g}^*,G)} = \\tilde{g}^* and \\frac{\\partial}{\\partial \\tilde{g}^*} J{(\\tilde{g}^*,G)} = \\frac{\\partial^{2}}{\\partial \\tilde{g}^*\\partial G} G \\tilde{g}^* and \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* = \\frac{\\partial^{2}}{\\partial \\tilde{g}^*\\partial G} G \\tilde{g}^* and \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* = \\frac{\\partial}{\\partial \\tilde{g}^*} J{(\\tilde{g}^*,G)}", "srepr_derivation": [["get_premise", "Equality(Function('J')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('G', commutative=True)), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('J')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('G', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('\\\\tilde{g}^*', commutative=True), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Symbol('\\\\tilde{g}^*', commutative=True), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Function('J')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\mathbf{s},\\mathbf{f})} = \\mathbf{f} + \\mathbf{s}, then obtain (\\iint (- \\mathbf{f} - \\mathbf{s} + h{(\\mathbf{s},\\mathbf{f})}) d\\mathbf{s} d\\mathbf{f})^{\\mathbf{f}} = (\\iint 0 d\\mathbf{s} d\\mathbf{f})^{\\mathbf{f}}", "derivation": "h{(\\mathbf{s},\\mathbf{f})} = \\mathbf{f} + \\mathbf{s} and - \\mathbf{f} - \\mathbf{s} + h{(\\mathbf{s},\\mathbf{f})} = 0 and \\int (- \\mathbf{f} - \\mathbf{s} + h{(\\mathbf{s},\\mathbf{f})}) d\\mathbf{s} = \\int 0 d\\mathbf{s} and \\iint (- \\mathbf{f} - \\mathbf{s} + h{(\\mathbf{s},\\mathbf{f})}) d\\mathbf{s} d\\mathbf{f} = \\iint 0 d\\mathbf{s} d\\mathbf{f} and (\\iint (- \\mathbf{f} - \\mathbf{s} + h{(\\mathbf{s},\\mathbf{f})}) d\\mathbf{s} d\\mathbf{f})^{\\mathbf{f}} = (\\iint 0 d\\mathbf{s} d\\mathbf{f})^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(\\eta,c,\\lambda)} = - \\eta + \\lambda - c, then obtain (- c \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda)^{c} + \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda = (- c \\int (- \\eta + \\lambda - c) d\\lambda)^{c} + \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda", "derivation": "\\bar{\\h}{(\\eta,c,\\lambda)} = - \\eta + \\lambda - c and \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda = \\int (- \\eta + \\lambda - c) d\\lambda and - c \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda = - c \\int (- \\eta + \\lambda - c) d\\lambda and (- c \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda)^{c} = (- c \\int (- \\eta + \\lambda - c) d\\lambda)^{c} and (- c \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda)^{c} + \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda = (- c \\int (- \\eta + \\lambda - c) d\\lambda)^{c} + \\int \\bar{\\h}{(\\eta,c,\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('c', commutative=True), Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('c', commutative=True), Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Symbol('c', commutative=True)))"], [["add", 4, "Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(-1), Symbol('c', commutative=True), Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Symbol('c', commutative=True)), Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Pow(Mul(Integer(-1), Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Symbol('c', commutative=True)), Integral(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True), Symbol('c', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given L{(x^\\prime,A_{1})} = \\frac{x^\\prime}{A_{1}}, then derive x^\\prime \\frac{\\partial}{\\partial A_{1}} L{(x^\\prime,A_{1})} = - \\frac{(x^\\prime)^{2}}{A_{1}^{2}}, then obtain x^\\prime \\frac{\\partial}{\\partial A_{1}} \\frac{x^\\prime}{A_{1}} = - \\frac{(x^\\prime)^{2}}{A_{1}^{2}}", "derivation": "L{(x^\\prime,A_{1})} = \\frac{x^\\prime}{A_{1}} and x^\\prime L{(x^\\prime,A_{1})} = \\frac{(x^\\prime)^{2}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} x^\\prime L{(x^\\prime,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\frac{(x^\\prime)^{2}}{A_{1}} and x^\\prime \\frac{\\partial}{\\partial A_{1}} L{(x^\\prime,A_{1})} = - \\frac{(x^\\prime)^{2}}{A_{1}^{2}} and x^\\prime \\frac{\\partial}{\\partial A_{1}} \\frac{x^\\prime}{A_{1}} = - \\frac{(x^\\prime)^{2}}{A_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('x^\\\\prime', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('L')(Symbol('x^\\\\prime', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Mul(Symbol('x^\\\\prime', commutative=True), Function('L')(Symbol('x^\\\\prime', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Derivative(Function('L')(Symbol('x^\\\\prime', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\dot{z}{(g_{\\varepsilon})} = \\sin{(\\cos{(g_{\\varepsilon})})} and \\pi{(g_{\\varepsilon})} = \\dot{z}{(g_{\\varepsilon})} + \\sin{(\\cos{(g_{\\varepsilon})})}, then obtain \\frac{d}{d g_{\\varepsilon}} (\\dot{z}{(g_{\\varepsilon})} + \\sin{(\\cos{(g_{\\varepsilon})})}) = \\frac{d}{d g_{\\varepsilon}} 2 \\sin{(\\cos{(g_{\\varepsilon})})}", "derivation": "\\dot{z}{(g_{\\varepsilon})} = \\sin{(\\cos{(g_{\\varepsilon})})} and \\pi{(g_{\\varepsilon})} = \\dot{z}{(g_{\\varepsilon})} + \\sin{(\\cos{(g_{\\varepsilon})})} and \\pi{(g_{\\varepsilon})} = 2 \\sin{(\\cos{(g_{\\varepsilon})})} and \\dot{z}{(g_{\\varepsilon})} + \\sin{(\\cos{(g_{\\varepsilon})})} = 2 \\sin{(\\cos{(g_{\\varepsilon})})} and \\frac{d}{d g_{\\varepsilon}} (\\dot{z}{(g_{\\varepsilon})} + \\sin{(\\cos{(g_{\\varepsilon})})}) = \\frac{d}{d g_{\\varepsilon}} 2 \\sin{(\\cos{(g_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Function('\\\\dot{z}')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\pi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\dot{z}')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["differentiate", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{z}')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(F_{H},\\omega)} = F_{H}^{\\omega}, then derive \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} = \\frac{F_{H}^{\\omega} \\omega}{F_{H}}, then derive \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} + \\frac{F_{H}^{\\omega} \\omega}{F_{H}} = \\frac{2 F_{H}^{\\omega} \\omega}{F_{H}}, then obtain \\frac{F_{H}^{\\omega} \\omega}{F_{H}} + \\frac{\\omega h{(F_{H},\\omega)}}{F_{H}} = \\frac{2 F_{H}^{\\omega} \\omega}{F_{H}}", "derivation": "h{(F_{H},\\omega)} = F_{H}^{\\omega} and \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} = \\frac{\\partial}{\\partial F_{H}} F_{H}^{\\omega} and \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} = \\frac{F_{H}^{\\omega} \\omega}{F_{H}} and \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} = \\frac{\\omega h{(F_{H},\\omega)}}{F_{H}} and \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} + \\frac{F_{H}^{\\omega} \\omega}{F_{H}} = \\frac{\\partial}{\\partial F_{H}} F_{H}^{\\omega} + \\frac{F_{H}^{\\omega} \\omega}{F_{H}} and \\frac{\\partial}{\\partial F_{H}} h{(F_{H},\\omega)} + \\frac{F_{H}^{\\omega} \\omega}{F_{H}} = \\frac{2 F_{H}^{\\omega} \\omega}{F_{H}} and \\frac{F_{H}^{\\omega} \\omega}{F_{H}} + \\frac{\\omega h{(F_{H},\\omega)}}{F_{H}} = \\frac{2 F_{H}^{\\omega} \\omega}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Derivative(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Derivative(Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Function('h')(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)))), Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given f{(Z)} = e^{Z}, then derive \\frac{d}{d Z} \\int f{(Z)} dZ = \\frac{\\partial}{\\partial Z} (\\mathbf{J}_f + e^{Z}), then derive \\frac{\\partial}{\\partial Z} (\\tilde{g}^* + e^{Z}) = \\frac{\\partial}{\\partial Z} (\\mathbf{J}_f + e^{Z}), then obtain \\frac{\\partial^{2}}{\\partial Z^{2}} (\\tilde{g}^* + e^{Z}) = \\frac{d^{2}}{d Z^{2}} \\int f{(Z)} dZ", "derivation": "f{(Z)} = e^{Z} and \\int f{(Z)} dZ = \\int e^{Z} dZ and \\frac{d}{d Z} \\int f{(Z)} dZ = \\frac{d}{d Z} \\int e^{Z} dZ and \\frac{d}{d Z} \\int f{(Z)} dZ = \\frac{\\partial}{\\partial Z} (\\mathbf{J}_f + e^{Z}) and \\frac{d}{d Z} \\int e^{Z} dZ = \\frac{\\partial}{\\partial Z} (\\mathbf{J}_f + e^{Z}) and \\frac{\\partial}{\\partial Z} (\\tilde{g}^* + e^{Z}) = \\frac{\\partial}{\\partial Z} (\\mathbf{J}_f + e^{Z}) and \\frac{\\partial}{\\partial Z} (\\tilde{g}^* + e^{Z}) = \\frac{d}{d Z} \\int f{(Z)} dZ and \\frac{\\partial^{2}}{\\partial Z^{2}} (\\tilde{g}^* + e^{Z}) = \\frac{d^{2}}{d Z^{2}} \\int f{(Z)} dZ", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('f')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integral(Function('f')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('f')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(Function('f')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2))), Derivative(Integral(Function('f')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{p}{(g_{\\varepsilon},\\dot{\\mathbf{r}})} = e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}}, then obtain -1 = - \\sin{(e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}} - e^{- \\dot{\\mathbf{r}}} e^{g_{\\varepsilon}})} - 1", "derivation": "\\mathbf{p}{(g_{\\varepsilon},\\dot{\\mathbf{r}})} = e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}} and 0 = - \\mathbf{p}{(g_{\\varepsilon},\\dot{\\mathbf{r}})} + e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}} and 0 = - \\sin{(\\mathbf{p}{(g_{\\varepsilon},\\dot{\\mathbf{r}})} - e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}})} and 0 = - \\sin{(\\mathbf{p}{(g_{\\varepsilon},\\dot{\\mathbf{r}})} - e^{- \\dot{\\mathbf{r}}} e^{g_{\\varepsilon}})} and 0 = - \\sin{(e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}} - e^{- \\dot{\\mathbf{r}}} e^{g_{\\varepsilon}})} and -1 = - \\sin{(e^{- \\dot{\\mathbf{r}} + g_{\\varepsilon}} - e^{- \\dot{\\mathbf{r}}} e^{g_{\\varepsilon}})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{p}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["sin", 2], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('\\\\mathbf{p}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))))))"], [["expand", 3], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('\\\\mathbf{p}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Mul(Integer(-1), sin(Add(exp(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))))"], [["minus", 5, 1], "Equality(Integer(-1), Add(Mul(Integer(-1), sin(Add(exp(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))), Integer(-1)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} = g^{\\mathbf{J}_f}, then obtain (g^{2 \\mathbf{J}_f} + g^{\\mathbf{J}_f}) (g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + \\Psi^{\\dagger}{(\\mathbf{J}_f,g)}) = (g^{2 \\mathbf{J}_f} + g^{\\mathbf{J}_f}) (g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + g^{\\mathbf{J}_f})", "derivation": "\\Psi^{\\dagger}{(\\mathbf{J}_f,g)} = g^{\\mathbf{J}_f} and g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} = g^{2 \\mathbf{J}_f} and g^{2 \\mathbf{J}_f} + \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} = g^{2 \\mathbf{J}_f} + g^{\\mathbf{J}_f} and g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} = g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + g^{\\mathbf{J}_f} and (g^{2 \\mathbf{J}_f} + g^{\\mathbf{J}_f}) (g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + \\Psi^{\\dagger}{(\\mathbf{J}_f,g)}) = (g^{2 \\mathbf{J}_f} + g^{\\mathbf{J}_f}) (g^{\\mathbf{J}_f} \\Psi^{\\dagger}{(\\mathbf{J}_f,g)} + g^{\\mathbf{J}_f})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Add(Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["times", 4, "Add(Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Mul(Add(Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True)))), Mul(Add(Pow(Symbol('g', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(m_{s})} = \\sin{(m_{s})}, then obtain (\\sin^{2}{(m_{s})} + \\sin{(m_{s})})^{m_{s}} = (\\tilde{g}{(m_{s})} \\sin{(m_{s})} + \\sin{(m_{s})})^{m_{s}}", "derivation": "\\tilde{g}{(m_{s})} = \\sin{(m_{s})} and \\tilde{g}{(m_{s})} \\sin{(m_{s})} = \\sin^{2}{(m_{s})} and \\tilde{g}^{2}{(m_{s})} = \\tilde{g}{(m_{s})} \\sin{(m_{s})} and \\tilde{g}^{2}{(m_{s})} = \\sin^{2}{(m_{s})} and \\tilde{g}^{2}{(m_{s})} + \\sin{(m_{s})} = \\tilde{g}{(m_{s})} \\sin{(m_{s})} + \\sin{(m_{s})} and (\\tilde{g}^{2}{(m_{s})} + \\sin{(m_{s})})^{m_{s}} = (\\tilde{g}{(m_{s})} \\sin{(m_{s})} + \\sin{(m_{s})})^{m_{s}} and (\\sin^{2}{(m_{s})} + \\sin{(m_{s})})^{m_{s}} = (\\tilde{g}{(m_{s})} \\sin{(m_{s})} + \\sin{(m_{s})})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["times", 1, "sin(Symbol('m_s', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))), Pow(sin(Symbol('m_s', commutative=True)), Integer(2)))"], [["times", 1, "Function('\\\\tilde{g}')(Symbol('m_s', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2)))"], [["add", 3, "sin(Symbol('m_s', commutative=True))"], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), Integer(2)), sin(Symbol('m_s', commutative=True))), Add(Mul(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))), sin(Symbol('m_s', commutative=True))))"], [["power", 5, "Symbol('m_s', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), Integer(2)), sin(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Pow(Add(Mul(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))), sin(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Add(Pow(sin(Symbol('m_s', commutative=True)), Integer(2)), sin(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Pow(Add(Mul(Function('\\\\tilde{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))), sin(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given x{(A_{z})} = \\cos{(A_{z})}, then obtain -1 + \\frac{\\cos{(A_{z})}}{x^{2}{(A_{z})}} = -1 + \\frac{\\cos^{3}{(A_{z})}}{x^{4}{(A_{z})}}", "derivation": "x{(A_{z})} = \\cos{(A_{z})} and \\frac{1}{\\cos{(A_{z})}} = \\frac{1}{x{(A_{z})}} and \\frac{1}{x{(A_{z})}} = \\frac{\\cos{(A_{z})}}{x^{2}{(A_{z})}} and -1 + \\frac{1}{\\cos{(A_{z})}} = -1 + \\frac{1}{x{(A_{z})}} and -1 + \\frac{1}{\\cos{(A_{z})}} = -1 + \\frac{\\cos{(A_{z})}}{x^{2}{(A_{z})}} and -1 + \\frac{1}{x{(A_{z})}} = -1 + \\frac{\\cos{(A_{z})}}{x^{2}{(A_{z})}} and -1 + \\frac{\\cos{(A_{z})}}{x^{2}{(A_{z})}} = -1 + \\frac{\\cos^{3}{(A_{z})}}{x^{4}{(A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["divide", 1, "Mul(Function('x')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], "Equality(Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)), Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-1)))"], [["divide", 2, "Mul(Function('x')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))"], "Equality(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-2)), cos(Symbol('A_z', commutative=True))))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Add(Integer(-1), Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integer(-1), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Add(Integer(-1), Mul(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-2)), cos(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Integer(-1), Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-1))), Add(Integer(-1), Mul(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-2)), cos(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Integer(-1), Mul(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-2)), cos(Symbol('A_z', commutative=True)))), Add(Integer(-1), Mul(Pow(Function('x')(Symbol('A_z', commutative=True)), Integer(-4)), Pow(cos(Symbol('A_z', commutative=True)), Integer(3)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(E)} = \\log{(E)}, then obtain \\frac{d}{d E} \\operatorname{P_{e}}^{3}{(E)} = \\frac{d}{d E} \\operatorname{P_{e}}{(E)} \\log{(E)}^{2}", "derivation": "\\operatorname{P_{e}}{(E)} = \\log{(E)} and \\operatorname{P_{e}}^{2}{(E)} = \\operatorname{P_{e}}{(E)} \\log{(E)} and \\operatorname{P_{e}}^{2}{(E)} \\log{(E)} = \\operatorname{P_{e}}{(E)} \\log{(E)}^{2} and \\operatorname{P_{e}}^{3}{(E)} = \\operatorname{P_{e}}^{2}{(E)} \\log{(E)} and \\operatorname{P_{e}}^{3}{(E)} = \\operatorname{P_{e}}{(E)} \\log{(E)}^{2} and \\frac{d}{d E} \\operatorname{P_{e}}^{3}{(E)} = \\frac{d}{d E} \\operatorname{P_{e}}{(E)} \\log{(E)}^{2}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('E', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2)), Mul(Function('P_e')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))))"], [["times", 2, "log(Symbol('E', commutative=True))"], "Equality(Mul(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2)), log(Symbol('E', commutative=True))), Mul(Function('P_e')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(3)), Mul(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2)), log(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(3)), Mul(Function('P_e')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('E', commutative=True)"], "Equality(Derivative(Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(3)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Function('P_e')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(2))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\chi,J_{\\varepsilon})} = J_{\\varepsilon} \\chi and E{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} J_{\\varepsilon} \\chi and \\Omega{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} \\psi{(\\chi,J_{\\varepsilon})}, then obtain \\Omega{(\\chi,J_{\\varepsilon})} = E{(\\chi,J_{\\varepsilon})}", "derivation": "\\psi{(\\chi,J_{\\varepsilon})} = J_{\\varepsilon} \\chi and \\frac{\\partial}{\\partial \\chi} \\psi{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} J_{\\varepsilon} \\chi and E{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} J_{\\varepsilon} \\chi and \\Omega{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} \\psi{(\\chi,J_{\\varepsilon})} and \\frac{\\partial}{\\partial \\chi} \\psi{(\\chi,J_{\\varepsilon})} = E{(\\chi,J_{\\varepsilon})} and \\Omega{(\\chi,J_{\\varepsilon})} = E{(\\chi,J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given r{(F_{H})} = \\frac{d}{d F_{H}} \\sin{(F_{H})}, then derive - 2 B - M_{E} + r{(F_{H})} = - 2 B - M_{E} + \\cos{(F_{H})}, then obtain \\frac{1}{(- 2 B - M_{E} + \\cos{(F_{H})})^{2}} = \\frac{1}{(- 2 B - M_{E} + r{(F_{H})})^{2}}", "derivation": "r{(F_{H})} = \\frac{d}{d F_{H}} \\sin{(F_{H})} and - 2 B - M_{E} + r{(F_{H})} = - 2 B - M_{E} + \\frac{d}{d F_{H}} \\sin{(F_{H})} and \\frac{1}{(- 2 B - M_{E} + r{(F_{H})})^{2}} = \\frac{1}{(- 2 B - M_{E} + \\frac{d}{d F_{H}} \\sin{(F_{H})})^{2}} and - 2 B - M_{E} + r{(F_{H})} = - 2 B - M_{E} + \\cos{(F_{H})} and - 2 B - M_{E} + \\cos{(F_{H})} = - 2 B - M_{E} + \\frac{d}{d F_{H}} \\sin{(F_{H})} and \\frac{1}{(- 2 B - M_{E} + \\cos{(F_{H})})^{2}} = \\frac{1}{(- 2 B - M_{E} + \\frac{d}{d F_{H}} \\sin{(F_{H})})^{2}} and \\frac{1}{(- 2 B - M_{E} + \\cos{(F_{H})})^{2}} = \\frac{1}{(- 2 B - M_{E} + r{(F_{H})})^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('F_H', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["minus", 1, "Add(Mul(Integer(2), Symbol('B', commutative=True)), Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('r')(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('r')(Symbol('F_H', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Integer(-2)))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('r')(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), cos(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), cos(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["power", 5, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), cos(Symbol('F_H', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Integer(-2)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), cos(Symbol('F_H', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('r')(Symbol('F_H', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given Z{(C_{1},\\chi)} = \\sin{(\\chi^{C_{1}})} and \\hat{p}{(C_{1},\\chi)} = (C_{1} + Z{(C_{1},\\chi)}) (2 C_{1} + \\sin{(\\chi^{C_{1}})}), then obtain \\hat{p}{(C_{1},\\chi)} = (C_{1} + \\sin{(\\chi^{C_{1}})}) (2 C_{1} + \\sin{(\\chi^{C_{1}})})", "derivation": "Z{(C_{1},\\chi)} = \\sin{(\\chi^{C_{1}})} and C_{1} + Z{(C_{1},\\chi)} = C_{1} + \\sin{(\\chi^{C_{1}})} and (C_{1} + Z{(C_{1},\\chi)}) (2 C_{1} + \\sin{(\\chi^{C_{1}})}) = (C_{1} + \\sin{(\\chi^{C_{1}})}) (2 C_{1} + \\sin{(\\chi^{C_{1}})}) and \\hat{p}{(C_{1},\\chi)} = (C_{1} + Z{(C_{1},\\chi)}) (2 C_{1} + \\sin{(\\chi^{C_{1}})}) and \\hat{p}{(C_{1},\\chi)} = (C_{1} + \\sin{(\\chi^{C_{1}})}) (2 C_{1} + \\sin{(\\chi^{C_{1}})})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))"], [["add", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Symbol('C_1', commutative=True), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(2), Symbol('C_1', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))), Mul(Add(Symbol('C_1', commutative=True), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True)))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Add(Symbol('C_1', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\hat{p}')(Symbol('C_1', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Add(Symbol('C_1', commutative=True), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True)))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), sin(Pow(Symbol('\\\\chi', commutative=True), Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given z{(\\chi,\\mathbf{f},\\mathbf{S})} = \\chi \\mathbf{f} - \\mathbf{S} and \\Psi{(\\mathbf{S},\\mathbf{f},\\chi)} = \\frac{\\chi \\mathbf{f} - \\mathbf{S}}{\\chi}, then obtain \\frac{z{(\\chi,\\mathbf{f},\\mathbf{S})}}{\\chi} = \\Psi{(\\mathbf{S},\\mathbf{f},\\chi)}", "derivation": "z{(\\chi,\\mathbf{f},\\mathbf{S})} = \\chi \\mathbf{f} - \\mathbf{S} and \\frac{z{(\\chi,\\mathbf{f},\\mathbf{S})}}{\\chi \\mathbf{f}} = \\frac{\\chi \\mathbf{f} - \\mathbf{S}}{\\chi \\mathbf{f}} and \\frac{z{(\\chi,\\mathbf{f},\\mathbf{S})}}{\\chi} = \\frac{\\chi \\mathbf{f} - \\mathbf{S}}{\\chi} and \\Psi{(\\mathbf{S},\\mathbf{f},\\chi)} = \\frac{\\chi \\mathbf{f} - \\mathbf{S}}{\\chi} and \\frac{z{(\\chi,\\mathbf{f},\\mathbf{S})}}{\\chi} = \\Psi{(\\mathbf{S},\\mathbf{f},\\chi)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["divide", 2, "Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\phi)} = e^{\\cos{(\\phi)}}, then derive \\frac{d}{d \\phi} \\mathbf{J}_M{(\\phi)} - 1 = - e^{\\cos{(\\phi)}} \\sin{(\\phi)} - 1, then obtain (\\frac{d}{d \\phi} \\mathbf{J}_M{(\\phi)} - 1)^{\\phi} = (- e^{\\cos{(\\phi)}} \\sin{(\\phi)} - 1)^{\\phi}", "derivation": "\\mathbf{J}_M{(\\phi)} = e^{\\cos{(\\phi)}} and - \\phi + \\mathbf{J}_M{(\\phi)} = - \\phi + e^{\\cos{(\\phi)}} and \\frac{d}{d \\phi} (- \\phi + \\mathbf{J}_M{(\\phi)}) = \\frac{d}{d \\phi} (- \\phi + e^{\\cos{(\\phi)}}) and \\frac{d}{d \\phi} \\mathbf{J}_M{(\\phi)} - 1 = - e^{\\cos{(\\phi)}} \\sin{(\\phi)} - 1 and (\\frac{d}{d \\phi} \\mathbf{J}_M{(\\phi)} - 1)^{\\phi} = (- e^{\\cos{(\\phi)}} \\sin{(\\phi)} - 1)^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), exp(cos(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(cos(Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(cos(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), exp(cos(Symbol('\\\\phi', commutative=True))), sin(Symbol('\\\\phi', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), exp(cos(Symbol('\\\\phi', commutative=True))), sin(Symbol('\\\\phi', commutative=True))), Integer(-1)), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given u{(\\dot{z},\\psi^*)} = \\frac{\\partial}{\\partial \\dot{z}} (\\psi^*)^{\\dot{z}}, then derive u^{\\dot{z}}{(\\dot{z},\\psi^*)} = ((\\psi^*)^{\\dot{z}} \\log{(\\psi^*)})^{\\dot{z}}, then obtain (\\frac{u^{\\dot{z}}{(\\dot{z},\\psi^*)}}{\\log{(\\psi^*)}})^{\\psi^*} = (\\frac{((\\psi^*)^{\\dot{z}} \\log{(\\psi^*)})^{\\dot{z}}}{\\log{(\\psi^*)}})^{\\psi^*}", "derivation": "u{(\\dot{z},\\psi^*)} = \\frac{\\partial}{\\partial \\dot{z}} (\\psi^*)^{\\dot{z}} and u^{\\dot{z}}{(\\dot{z},\\psi^*)} = (\\frac{\\partial}{\\partial \\dot{z}} (\\psi^*)^{\\dot{z}})^{\\dot{z}} and u^{\\dot{z}}{(\\dot{z},\\psi^*)} = ((\\psi^*)^{\\dot{z}} \\log{(\\psi^*)})^{\\dot{z}} and \\frac{u^{\\dot{z}}{(\\dot{z},\\psi^*)}}{\\log{(\\psi^*)}} = \\frac{((\\psi^*)^{\\dot{z}} \\log{(\\psi^*)})^{\\dot{z}}}{\\log{(\\psi^*)}} and (\\frac{u^{\\dot{z}}{(\\dot{z},\\psi^*)}}{\\log{(\\psi^*)}})^{\\psi^*} = (\\frac{((\\psi^*)^{\\dot{z}} \\log{(\\psi^*)})^{\\dot{z}}}{\\log{(\\psi^*)}})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('u')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('\\\\dot{z}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('u')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 3, "log(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Function('u')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Mul(Pow(Function('u')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Pow(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(S,\\Psi)} = S \\Psi and \\operatorname{a^{\\dagger}}{(S,\\Psi)} = S \\Psi, then derive \\frac{\\partial}{\\partial S} \\operatorname{a^{\\dagger}}{(S,\\Psi)} = \\Psi, then obtain \\frac{\\partial}{\\partial S} \\operatorname{J_{\\varepsilon}}{(S,\\Psi)} = \\Psi", "derivation": "\\operatorname{J_{\\varepsilon}}{(S,\\Psi)} = S \\Psi and \\operatorname{a^{\\dagger}}{(S,\\Psi)} = S \\Psi and \\frac{\\partial}{\\partial S} \\operatorname{a^{\\dagger}}{(S,\\Psi)} = \\frac{\\partial}{\\partial S} S \\Psi and \\frac{\\partial}{\\partial S} \\operatorname{a^{\\dagger}}{(S,\\Psi)} = \\Psi and \\frac{\\partial}{\\partial S} S \\Psi = \\Psi and \\frac{\\partial}{\\partial S} \\operatorname{J_{\\varepsilon}}{(S,\\Psi)} = \\Psi", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\psi^*)} = \\cos{(\\psi^*)}, then obtain \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} + \\int \\cos^{\\psi^*}{(\\psi^*)} d\\psi^* = \\cos^{\\psi^*}{(\\psi^*)} + \\int \\cos^{\\psi^*}{(\\psi^*)} d\\psi^*", "derivation": "\\operatorname{c_{0}}{(\\psi^*)} = \\cos{(\\psi^*)} and \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} = \\cos^{\\psi^*}{(\\psi^*)} and \\int \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} d\\psi^* = \\int \\cos^{\\psi^*}{(\\psi^*)} d\\psi^* and \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} + \\int \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} d\\psi^* = \\cos^{\\psi^*}{(\\psi^*)} + \\int \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} d\\psi^* and \\operatorname{c_{0}}^{\\psi^*}{(\\psi^*)} + \\int \\cos^{\\psi^*}{(\\psi^*)} d\\psi^* = \\cos^{\\psi^*}{(\\psi^*)} + \\int \\cos^{\\psi^*}{(\\psi^*)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Integral(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integral(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integral(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Function('c_0')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integral(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integral(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\varepsilon,A_{z})} = A_{z} \\varepsilon, then derive \\frac{\\partial}{\\partial A_{z}} \\eta{(\\varepsilon,A_{z})} = \\varepsilon, then obtain \\int (A_{z} \\varepsilon + \\frac{\\partial}{\\partial A_{z}} A_{z} \\varepsilon) dA_{z} = \\int (A_{z} \\varepsilon + \\varepsilon) dA_{z}", "derivation": "\\eta{(\\varepsilon,A_{z})} = A_{z} \\varepsilon and \\frac{\\partial}{\\partial A_{z}} \\eta{(\\varepsilon,A_{z})} = \\frac{\\partial}{\\partial A_{z}} A_{z} \\varepsilon and \\frac{\\partial}{\\partial A_{z}} \\eta{(\\varepsilon,A_{z})} = \\varepsilon and A_{z} \\varepsilon + \\frac{\\partial}{\\partial A_{z}} \\eta{(\\varepsilon,A_{z})} = A_{z} \\varepsilon + \\varepsilon and \\eta{(\\varepsilon,A_{z})} + \\frac{\\partial}{\\partial A_{z}} \\eta{(\\varepsilon,A_{z})} = \\varepsilon + \\eta{(\\varepsilon,A_{z})} and A_{z} \\varepsilon + \\frac{\\partial}{\\partial A_{z}} A_{z} \\varepsilon = A_{z} \\varepsilon + \\varepsilon and \\int (A_{z} \\varepsilon + \\frac{\\partial}{\\partial A_{z}} A_{z} \\varepsilon) dA_{z} = \\int (A_{z} \\varepsilon + \\varepsilon) dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))"], [["add", 3, "Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Derivative(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 6, "Symbol('A_z', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\mathbf{B}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then obtain \\frac{\\mathbf{B}{(\\mathbf{v})}}{\\mathbf{p}{(\\mathbf{P})}} = \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{p}{(\\mathbf{P})}}", "derivation": "\\mathbf{p}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\mathbf{B}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\frac{\\mathbf{B}{(\\mathbf{v})}}{\\log{(\\mathbf{P})}} = \\frac{\\sin{(\\mathbf{v})}}{\\log{(\\mathbf{P})}} and \\frac{\\mathbf{B}{(\\mathbf{v})}}{\\mathbf{p}{(\\mathbf{P})}} = \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{p}{(\\mathbf{P})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 2, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given H{(P_{e},A_{2})} = \\cos{(A_{2} + P_{e})}, then obtain \\int (- A_{2} + H{(P_{e},A_{2})}) dA_{2} = - \\frac{A_{2}^{2}}{2} + \\mathbf{J}_P + \\sin{(A_{2} + P_{e})}", "derivation": "H{(P_{e},A_{2})} = \\cos{(A_{2} + P_{e})} and - A_{2} + H{(P_{e},A_{2})} = - A_{2} + \\cos{(A_{2} + P_{e})} and \\int (- A_{2} + H{(P_{e},A_{2})}) dA_{2} = \\int (- A_{2} + \\cos{(A_{2} + P_{e})}) dA_{2} and \\int (- A_{2} + H{(P_{e},A_{2})}) dA_{2} = - \\frac{A_{2}^{2}}{2} + \\mathbf{J}_P + \\sin{(A_{2} + P_{e})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('P_e', commutative=True), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True))))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('H')(Symbol('P_e', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)))))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('H')(Symbol('P_e', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('H')(Symbol('P_e', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('\\\\mathbf{J}_P', commutative=True), sin(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(F_{g},G)} = \\cos^{G}{(F_{g})} and \\theta_{2}{(F_{g},G)} = \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})}, then obtain \\theta_{2}{(F_{g},G)} \\frac{\\partial}{\\partial F_{g}} \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} = \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} \\frac{\\partial}{\\partial F_{g}} \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})}", "derivation": "\\mathbf{S}{(F_{g},G)} = \\cos^{G}{(F_{g})} and \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} = 1 and \\frac{\\partial}{\\partial F_{g}} \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} = \\frac{d}{d F_{g}} 1 and \\theta_{2}{(F_{g},G)} = \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} and \\theta_{2}{(F_{g},G)} \\frac{d}{d F_{g}} 1 = \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} \\frac{d}{d F_{g}} 1 and \\theta_{2}{(F_{g},G)} \\frac{\\partial}{\\partial F_{g}} \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} = \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})} \\frac{\\partial}{\\partial F_{g}} \\mathbf{S}{(F_{g},G)} \\cos^{- G}{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Symbol('G', commutative=True)))"], [["divide", 1, "Pow(cos(Symbol('F_g', commutative=True)), Symbol('G', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))))"], [["times", 4, "Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))), Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Derivative(Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))), Derivative(Mul(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('G', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(I)} = \\sin{(I)}, then obtain \\int 0 dI = v - y{(I)} + \\sin{(I)}", "derivation": "y{(I)} = \\sin{(I)} and I + y{(I)} = I + \\sin{(I)} and \\frac{d}{d I} (I + y{(I)}) = \\frac{d}{d I} (I + \\sin{(I)}) and 0 = - \\frac{d}{d I} (I + y{(I)}) + \\frac{d}{d I} (I + \\sin{(I)}) and \\int 0 dI = \\int (- \\frac{d}{d I} (I + y{(I)}) + \\frac{d}{d I} (I + \\sin{(I)})) dI and \\int 0 dI = v - y{(I)} + \\sin{(I)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('y')(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Symbol('I', commutative=True), Function('y')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Add(Symbol('I', commutative=True), Function('y')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Function('y')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Derivative(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('I', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Function('y')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Derivative(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('y')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(t,k)} = \\sin{(\\frac{k}{t})}, then derive \\frac{\\partial}{\\partial t} \\operatorname{V_{\\mathbf{B}}}{(t,k)} = - \\frac{k \\cos{(\\frac{k}{t})}}{t^{2}}, then obtain - \\operatorname{V_{\\mathbf{B}}}{(t,k)} + \\frac{\\partial}{\\partial t} \\operatorname{V_{\\mathbf{B}}}{(t,k)} = - \\frac{k \\cos{(\\frac{k}{t})}}{t^{2}} - \\operatorname{V_{\\mathbf{B}}}{(t,k)}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(t,k)} = \\sin{(\\frac{k}{t})} and \\frac{\\partial}{\\partial t} \\operatorname{V_{\\mathbf{B}}}{(t,k)} = \\frac{\\partial}{\\partial t} \\sin{(\\frac{k}{t})} and \\frac{\\partial}{\\partial t} \\operatorname{V_{\\mathbf{B}}}{(t,k)} = - \\frac{k \\cos{(\\frac{k}{t})}}{t^{2}} and - \\operatorname{V_{\\mathbf{B}}}{(t,k)} + \\frac{\\partial}{\\partial t} \\operatorname{V_{\\mathbf{B}}}{(t,k)} = - \\frac{k \\cos{(\\frac{k}{t})}}{t^{2}} - \\operatorname{V_{\\mathbf{B}}}{(t,k)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True)), sin(Mul(Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-2)), cos(Mul(Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"], [["minus", 3, "Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True))), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-2)), cos(Mul(Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('t', commutative=True), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(t_{1})} = \\cos{(t_{1})}, then obtain 0 = - \\frac{\\sin{(t_{1})}}{\\dot{y}{(t_{1})}} - \\frac{\\cos{(t_{1})} \\frac{d}{d t_{1}} \\dot{y}{(t_{1})}}{\\dot{y}^{2}{(t_{1})}}", "derivation": "\\dot{y}{(t_{1})} = \\cos{(t_{1})} and 1 = \\frac{\\cos{(t_{1})}}{\\dot{y}{(t_{1})}} and \\frac{d}{d t_{1}} 1 = \\frac{d}{d t_{1}} \\frac{\\cos{(t_{1})}}{\\dot{y}{(t_{1})}} and 0 = - \\frac{\\sin{(t_{1})}}{\\dot{y}{(t_{1})}} - \\frac{\\cos{(t_{1})} \\frac{d}{d t_{1}} \\dot{y}{(t_{1})}}{\\dot{y}^{2}{(t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('t_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), Integer(-1)), cos(Symbol('t_1', commutative=True))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), Integer(-1)), cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), Integer(-1)), sin(Symbol('t_1', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), Integer(-2)), cos(Symbol('t_1', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given J{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and \\Psi{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})}, then obtain 1 = \\frac{\\Psi^{2}{(A_{2})}}{(\\frac{d}{d A_{2}} \\log{(A_{2})})^{2}}", "derivation": "J{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and 1 = \\frac{\\frac{d}{d A_{2}} \\log{(A_{2})}}{J{(A_{2})}} and \\frac{\\frac{d}{d A_{2}} \\log{(A_{2})}}{J{(A_{2})}} = \\frac{(\\frac{d}{d A_{2}} \\log{(A_{2})})^{2}}{J^{2}{(A_{2})}} and \\Psi{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and 1 = \\frac{(\\frac{d}{d A_{2}} \\log{(A_{2})})^{2}}{J^{2}{(A_{2})}} and 1 = \\frac{\\Psi^{2}{(A_{2})}}{J^{2}{(A_{2})}} and 1 = \\frac{\\Psi^{2}{(A_{2})}}{(\\frac{d}{d A_{2}} \\log{(A_{2})})^{2}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('A_2', commutative=True)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["divide", 1, "Function('J')(Symbol('A_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["times", 2, "Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-2)), Pow(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('A_2', commutative=True)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-2)), Pow(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(1), Mul(Pow(Function('J')(Symbol('A_2', commutative=True)), Integer(-2)), Pow(Function('\\\\Psi')(Symbol('A_2', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi')(Symbol('A_2', commutative=True)), Integer(2)), Pow(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-2))))"]]}, {"prompt": "Given \\bar{\\h}{(\\Omega)} = e^{e^{\\Omega}}, then obtain \\cos{(4 \\bar{\\h}{(\\Omega)} e^{e^{\\Omega}})} = \\cos{(4 \\bar{\\h}^{2}{(\\Omega)})}", "derivation": "\\bar{\\h}{(\\Omega)} = e^{e^{\\Omega}} and 2 \\bar{\\h}{(\\Omega)} = \\bar{\\h}{(\\Omega)} + e^{e^{\\Omega}} and \\bar{\\h}^{2}{(\\Omega)} = \\bar{\\h}{(\\Omega)} e^{e^{\\Omega}} and 4 \\bar{\\h}^{2}{(\\Omega)} = (\\bar{\\h}{(\\Omega)} + e^{e^{\\Omega}})^{2} and 4 \\bar{\\h}{(\\Omega)} e^{e^{\\Omega}} = (\\bar{\\h}{(\\Omega)} + e^{e^{\\Omega}})^{2} and \\cos{(4 \\bar{\\h}{(\\Omega)} e^{e^{\\Omega}})} = \\cos{((\\bar{\\h}{(\\Omega)} + e^{e^{\\Omega}})^{2})} and \\cos{(4 \\bar{\\h}{(\\Omega)} e^{e^{\\Omega}})} = \\cos{(4 \\bar{\\h}^{2}{(\\Omega)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True))), Add(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))))"], [["times", 1, "Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Pow(Add(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(4), Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))), Pow(Add(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))), Integer(2)))"], [["cos", 5], "Equality(cos(Mul(Integer(4), Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True))))), cos(Pow(Add(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(cos(Mul(Integer(4), Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), exp(exp(Symbol('\\\\Omega', commutative=True))))), cos(Mul(Integer(4), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(E,P_{e})} = E + P_{e}, then derive \\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})} = 1, then derive \\int (\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})})^{E} dE = C_{2} + E, then obtain \\frac{\\partial}{\\partial P_{e}} (C_{2} + E) = \\frac{d}{d P_{e}} \\int 1 dE", "derivation": "\\hat{H}_{\\lambda}{(E,P_{e})} = E + P_{e} and \\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})} = \\frac{\\partial}{\\partial E} (E + P_{e}) and \\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})} = 1 and (\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})})^{E} = 1 and \\int (\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})})^{E} dE = \\int 1 dE and \\int (\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})})^{E} dE = C_{2} + E and \\frac{\\partial}{\\partial P_{e}} \\int (\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,P_{e})})^{E} dE = \\frac{d}{d P_{e}} \\int 1 dE and \\frac{\\partial}{\\partial P_{e}} (C_{2} + E) = \\frac{d}{d P_{e}} \\int 1 dE", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('E', commutative=True), Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Integer(1), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Pow(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Symbol('C_2', commutative=True), Symbol('E', commutative=True)))"], [["differentiate", 5, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integral(Pow(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Derivative(Add(Symbol('C_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}, then derive \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})} = 1, then obtain (\\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} = 1", "derivation": "\\psi{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} and \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} and \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})} = 1 and (\\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{X})} = \\cos{(\\hat{X})}, then obtain \\hat{X} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\hat{X}}{(\\hat{X})} + \\cos{(\\hat{X})} = \\hat{X} \\cos^{\\hat{X}}{(\\hat{X})} + \\cos{(\\hat{X})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{X})} = \\cos{(\\hat{X})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\hat{X}}{(\\hat{X})} = \\cos^{\\hat{X}}{(\\hat{X})} and \\hat{X} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\hat{X}}{(\\hat{X})} = \\hat{X} \\cos^{\\hat{X}}{(\\hat{X})} and \\hat{X} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\hat{X}}{(\\hat{X})} + \\cos{(\\hat{X})} = \\hat{X} \\cos^{\\hat{X}}{(\\hat{X})} + \\cos{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["add", 3, "cos(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), cos(Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), cos(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given a{(h)} = \\cos{(h)}, then derive \\frac{d}{d h} a{(h)} = - \\sin{(h)}, then obtain (a{(h)} \\frac{d}{d h} a{(h)})^{- h} \\int - a{(h)} \\sin{(h)} dh = (a{(h)} \\frac{d}{d h} a{(h)})^{- h} \\int - \\sin{(h)} \\cos{(h)} dh", "derivation": "a{(h)} = \\cos{(h)} and \\frac{d}{d h} a{(h)} = \\frac{d}{d h} \\cos{(h)} and \\frac{d}{d h} a{(h)} = - \\sin{(h)} and - a{(h)} \\sin{(h)} = - \\sin{(h)} \\cos{(h)} and \\int - a{(h)} \\sin{(h)} dh = \\int - \\sin{(h)} \\cos{(h)} dh and (- a{(h)} \\sin{(h)})^{- h} \\int - a{(h)} \\sin{(h)} dh = (- a{(h)} \\sin{(h)})^{- h} \\int - \\sin{(h)} \\cos{(h)} dh and (a{(h)} \\frac{d}{d h} a{(h)})^{- h} \\int - a{(h)} \\sin{(h)} dh = (a{(h)} \\frac{d}{d h} a{(h)})^{- h} \\int - \\sin{(h)} \\cos{(h)} dh", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('h', commutative=True))))"], [["times", 1, "Mul(Integer(-1), sin(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), sin(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["divide", 5, "Pow(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Integral(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Mul(Function('a')(Symbol('h', commutative=True)), Derivative(Function('a')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('h', commutative=True))), Integral(Mul(Integer(-1), Function('a')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Mul(Function('a')(Symbol('h', commutative=True)), Derivative(Function('a')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('h', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(C_{1},c)} = C_{1}^{c}, then derive 0 = C_{1}^{c} \\log{(C_{1})} - \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)}, then obtain 0 = \\mathbf{g}{(C_{1},c)} \\log{(C_{1})} - \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)}", "derivation": "\\mathbf{g}{(C_{1},c)} = C_{1}^{c} and \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)} = \\frac{\\partial}{\\partial c} C_{1}^{c} and 0 = \\frac{\\partial}{\\partial c} C_{1}^{c} - \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)} and 0 = C_{1}^{c} \\log{(C_{1})} - \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)} and 0 = \\mathbf{g}{(C_{1},c)} \\log{(C_{1})} - \\frac{\\partial}{\\partial c} \\mathbf{g}{(C_{1},c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), log(Symbol('C_1', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), log(Symbol('C_1', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\phi_2}{(\\phi_2)} = \\cos^{\\phi_2}{(\\phi_2)}", "derivation": "\\operatorname{y^{\\prime}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\operatorname{y^{\\prime}}^{\\phi_2}{(\\phi_2)} = \\cos^{\\phi_2}{(\\phi_2)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\operatorname{y^{\\prime}}{(\\phi_2)} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2)} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\phi_2}{(\\phi_2)} = \\cos^{\\phi_2}{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given T{(B)} = \\log{(\\cos{(B)})}, then derive \\mathbf{J}_P + T{(B)} = I + \\log{(\\cos{(B)})}, then obtain (I + \\log{(\\cos{(B)})})^{B} = (I + T{(B)})^{B}", "derivation": "T{(B)} = \\log{(\\cos{(B)})} and \\frac{d}{d B} T{(B)} = \\frac{d}{d B} \\log{(\\cos{(B)})} and \\int \\frac{d}{d B} T{(B)} dB = \\int \\frac{d}{d B} \\log{(\\cos{(B)})} dB and \\mathbf{J}_P + T{(B)} = I + \\log{(\\cos{(B)})} and \\mathbf{J}_P + \\log{(\\cos{(B)})} = I + \\log{(\\cos{(B)})} and (\\mathbf{J}_P + \\log{(\\cos{(B)})})^{B} = (I + \\log{(\\cos{(B)})})^{B} and (\\mathbf{J}_P + \\log{(\\cos{(B)})})^{B} = (\\mathbf{J}_P + T{(B)})^{B} and (I + \\log{(\\cos{(B)})})^{B} = (\\mathbf{J}_P + T{(B)})^{B} and (\\mathbf{J}_P + T{(B)})^{B} = (I + T{(B)})^{B} and (I + \\log{(\\cos{(B)})})^{B} = (I + T{(B)})^{B}", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('B', commutative=True)), log(cos(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Function('T')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(log(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('T')(Symbol('B', commutative=True))), Add(Symbol('I', commutative=True), log(cos(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(cos(Symbol('B', commutative=True)))), Add(Symbol('I', commutative=True), log(cos(Symbol('B', commutative=True)))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(cos(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Pow(Add(Symbol('I', commutative=True), log(cos(Symbol('B', commutative=True)))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(cos(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('T')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Add(Symbol('I', commutative=True), log(cos(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('T')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('T')(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Symbol('I', commutative=True), Function('T')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 9], "Equality(Pow(Add(Symbol('I', commutative=True), log(cos(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Pow(Add(Symbol('I', commutative=True), Function('T')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given u{(F_{N})} = \\log{(\\cos{(F_{N})})} and \\hat{\\mathbf{x}}{(F_{N})} = \\log{(\\cos{(F_{N})})}, then obtain 1 = \\frac{\\log{(\\cos{(F_{N})})}}{\\hat{\\mathbf{x}}{(F_{N})}}", "derivation": "u{(F_{N})} = \\log{(\\cos{(F_{N})})} and 1 = \\frac{\\log{(\\cos{(F_{N})})}}{u{(F_{N})}} and \\hat{\\mathbf{x}}{(F_{N})} = \\log{(\\cos{(F_{N})})} and u{(F_{N})} = \\hat{\\mathbf{x}}{(F_{N})} and 1 = \\frac{\\log{(\\cos{(F_{N})})}}{\\hat{\\mathbf{x}}{(F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('F_N', commutative=True)), log(cos(Symbol('F_N', commutative=True))))"], [["divide", 1, "Function('u')(Symbol('F_N', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('u')(Symbol('F_N', commutative=True)), Integer(-1)), log(cos(Symbol('F_N', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_N', commutative=True)), log(cos(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('u')(Symbol('F_N', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_N', commutative=True)), Integer(-1)), log(cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given J{(\\delta,a)} = \\delta + a, then obtain \\int \\frac{\\partial}{\\partial \\delta} J{(\\delta,a)} da = \\mathbf{M} + a", "derivation": "J{(\\delta,a)} = \\delta + a and \\frac{\\partial}{\\partial \\delta} J{(\\delta,a)} = \\frac{\\partial}{\\partial \\delta} (\\delta + a) and \\int \\frac{\\partial}{\\partial \\delta} J{(\\delta,a)} da = \\int \\frac{\\partial}{\\partial \\delta} (\\delta + a) da and \\int \\frac{\\partial}{\\partial \\delta} J{(\\delta,a)} da = \\mathbf{M} + a", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Derivative(Function('J')(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Integral(Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('J')(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(b)} = \\log{(\\sin{(b)})}, then obtain b + \\operatorname{F_{c}}{(b)} - 2 \\log{(\\sin{(b)})} = b - \\log{(\\sin{(b)})}", "derivation": "\\operatorname{F_{c}}{(b)} = \\log{(\\sin{(b)})} and b + \\operatorname{F_{c}}{(b)} = b + \\log{(\\sin{(b)})} and b + 2 \\operatorname{F_{c}}{(b)} = b + \\operatorname{F_{c}}{(b)} + \\log{(\\sin{(b)})} and b + 2 \\operatorname{F_{c}}{(b)} - 2 \\log{(\\sin{(b)})} = b + \\operatorname{F_{c}}{(b)} - \\log{(\\sin{(b)})} and b + \\operatorname{F_{c}}{(b)} - 2 \\log{(\\sin{(b)})} = b - \\log{(\\sin{(b)})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('b', commutative=True)), log(sin(Symbol('b', commutative=True))))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('F_c')(Symbol('b', commutative=True))), Add(Symbol('b', commutative=True), log(sin(Symbol('b', commutative=True)))))"], [["add", 2, "Function('F_c')(Symbol('b', commutative=True))"], "Equality(Add(Symbol('b', commutative=True), Mul(Integer(2), Function('F_c')(Symbol('b', commutative=True)))), Add(Symbol('b', commutative=True), Function('F_c')(Symbol('b', commutative=True)), log(sin(Symbol('b', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), log(sin(Symbol('b', commutative=True))))"], "Equality(Add(Symbol('b', commutative=True), Mul(Integer(2), Function('F_c')(Symbol('b', commutative=True))), Mul(Integer(-1), Integer(2), log(sin(Symbol('b', commutative=True))))), Add(Symbol('b', commutative=True), Function('F_c')(Symbol('b', commutative=True)), Mul(Integer(-1), log(sin(Symbol('b', commutative=True))))))"], [["add", 4, "Mul(Integer(-1), Function('F_c')(Symbol('b', commutative=True)))"], "Equality(Add(Symbol('b', commutative=True), Function('F_c')(Symbol('b', commutative=True)), Mul(Integer(-1), Integer(2), log(sin(Symbol('b', commutative=True))))), Add(Symbol('b', commutative=True), Mul(Integer(-1), log(sin(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(c_{0})} = \\log{(c_{0})}, then obtain c_{0} \\tilde{g}^*{(c_{0})} = c_{0} \\log{(c_{0})}", "derivation": "\\tilde{g}^*{(c_{0})} = \\log{(c_{0})} and \\frac{d}{d c_{0}} \\tilde{g}^*{(c_{0})} = \\frac{d}{d c_{0}} \\log{(c_{0})} and \\frac{\\tilde{g}^*{(c_{0})}}{\\frac{d}{d c_{0}} \\tilde{g}^*{(c_{0})}} = \\frac{\\log{(c_{0})}}{\\frac{d}{d c_{0}} \\tilde{g}^*{(c_{0})}} and \\frac{\\tilde{g}^*{(c_{0})}}{\\frac{d}{d c_{0}} \\log{(c_{0})}} = \\frac{\\log{(c_{0})}}{\\frac{d}{d c_{0}} \\log{(c_{0})}} and c_{0} \\tilde{g}^*{(c_{0})} = c_{0} \\log{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('c_0', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True)), Pow(Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('c_0', commutative=True)), Pow(Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('c_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('c_0', commutative=True))), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\eta{(Z,\\pi)} = \\cos^{\\pi}{(Z)}, then obtain \\frac{\\partial^{2}}{\\partial \\pi^{2}} \\int (\\eta{(Z,\\pi)} - \\sin{(\\dot{z})}) d\\pi = \\frac{\\partial^{2}}{\\partial \\pi^{2}} \\int (- \\sin{(\\dot{z})} + \\cos^{\\pi}{(Z)}) d\\pi", "derivation": "\\eta{(Z,\\pi)} = \\cos^{\\pi}{(Z)} and \\eta{(Z,\\pi)} - \\sin{(\\dot{z})} = - \\sin{(\\dot{z})} + \\cos^{\\pi}{(Z)} and \\int (\\eta{(Z,\\pi)} - \\sin{(\\dot{z})}) d\\pi = \\int (- \\sin{(\\dot{z})} + \\cos^{\\pi}{(Z)}) d\\pi and \\frac{\\partial}{\\partial \\pi} \\int (\\eta{(Z,\\pi)} - \\sin{(\\dot{z})}) d\\pi = \\frac{\\partial}{\\partial \\pi} \\int (- \\sin{(\\dot{z})} + \\cos^{\\pi}{(Z)}) d\\pi and \\frac{\\partial^{2}}{\\partial \\pi^{2}} \\int (\\eta{(Z,\\pi)} - \\sin{(\\dot{z})}) d\\pi = \\frac{\\partial^{2}}{\\partial \\pi^{2}} \\int (- \\sin{(\\dot{z})} + \\cos^{\\pi}{(Z)}) d\\pi", "srepr_derivation": [["get_premise", "Equality(Function('\\\\eta')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('Z', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(cos(Symbol('Z', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(cos(Symbol('Z', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\eta')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(cos(Symbol('Z', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\eta')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Pow(cos(Symbol('Z', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))))"]]}, {"prompt": "Given L{(\\Omega,C_{2})} = C_{2} + \\Omega and Z{(\\Omega,C_{2})} = \\cos{(L^{\\Omega}{(\\Omega,C_{2})})}, then obtain \\frac{\\partial}{\\partial \\Omega} Z{(\\Omega,C_{2})} = \\frac{\\partial}{\\partial \\Omega} \\cos{((C_{2} + \\Omega)^{\\Omega})}", "derivation": "L{(\\Omega,C_{2})} = C_{2} + \\Omega and L^{\\Omega}{(\\Omega,C_{2})} = (C_{2} + \\Omega)^{\\Omega} and \\cos{(L^{\\Omega}{(\\Omega,C_{2})})} = \\cos{((C_{2} + \\Omega)^{\\Omega})} and Z{(\\Omega,C_{2})} = \\cos{(L^{\\Omega}{(\\Omega,C_{2})})} and Z{(\\Omega,C_{2})} = \\cos{((C_{2} + \\Omega)^{\\Omega})} and \\frac{\\partial}{\\partial \\Omega} Z{(\\Omega,C_{2})} = \\frac{\\partial}{\\partial \\Omega} \\cos{((C_{2} + \\Omega)^{\\Omega})}", "srepr_derivation": [["get_premise", "Equality(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), Symbol('\\\\Omega', commutative=True))), cos(Pow(Add(Symbol('C_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), cos(Pow(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('Z')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), cos(Pow(Add(Symbol('C_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\Omega', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(Pow(Add(Symbol('C_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(C_{1},\\rho,\\varphi)} = - \\rho + \\varphi^{C_{1}} and m{(C_{1},\\rho,\\varphi)} = - \\rho + \\varphi^{C_{1}}, then obtain \\frac{\\partial}{\\partial \\varphi} - \\frac{m{(C_{1},\\rho,\\varphi)}}{\\rho} = \\frac{\\partial}{\\partial \\varphi} - \\frac{\\mathbf{g}{(C_{1},\\rho,\\varphi)}}{\\rho}", "derivation": "\\mathbf{g}{(C_{1},\\rho,\\varphi)} = - \\rho + \\varphi^{C_{1}} and - \\mathbf{g}{(C_{1},\\rho,\\varphi)} = \\rho - \\varphi^{C_{1}} and m{(C_{1},\\rho,\\varphi)} = - \\rho + \\varphi^{C_{1}} and \\mathbf{g}{(C_{1},\\rho,\\varphi)} = m{(C_{1},\\rho,\\varphi)} and - m{(C_{1},\\rho,\\varphi)} = \\rho - \\varphi^{C_{1}} and - m{(C_{1},\\rho,\\varphi)} = - \\mathbf{g}{(C_{1},\\rho,\\varphi)} and - \\frac{m{(C_{1},\\rho,\\varphi)}}{\\rho} = - \\frac{\\mathbf{g}{(C_{1},\\rho,\\varphi)}}{\\rho} and \\frac{\\partial}{\\partial \\varphi} - \\frac{m{(C_{1},\\rho,\\varphi)}}{\\rho} = \\frac{\\partial}{\\partial \\varphi} - \\frac{\\mathbf{g}{(C_{1},\\rho,\\varphi)}}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["divide", 6, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('m')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(C_{2})} = \\log{(\\cos{(C_{2})})} and \\varphi^{*}{(C_{2})} = \\frac{\\mathbf{F}{(C_{2})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}}, then obtain \\varphi^{*}{(C_{2})} = \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}}", "derivation": "\\mathbf{F}{(C_{2})} = \\log{(\\cos{(C_{2})})} and \\frac{\\mathbf{F}{(C_{2})}}{C_{2}} = \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}} and \\frac{\\mathbf{F}{(C_{2})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}} = \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}} and \\varphi^{*}{(C_{2})} = \\frac{\\mathbf{F}{(C_{2})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}} and \\varphi^{*}{(C_{2})} = \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}} + \\frac{\\log{(\\cos{(C_{2})})}}{C_{2}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True)), log(cos(Symbol('C_2', commutative=True))))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), log(cos(Symbol('C_2', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), log(cos(Symbol('C_2', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), log(cos(Symbol('C_2', commutative=True))))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), log(cos(Symbol('C_2', commutative=True)))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), log(cos(Symbol('C_2', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('C_2', commutative=True)), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), log(cos(Symbol('C_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varphi^*')(Symbol('C_2', commutative=True)), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), log(cos(Symbol('C_2', commutative=True)))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), log(cos(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{D}{(C,h)} = \\cos{(\\frac{C}{h})} and \\operatorname{n_{1}}{(m,\\omega)} = \\cos{(\\omega m)} and \\mathbf{J}_M{(C,m,h,\\omega)} = \\operatorname{n_{1}}{(m,\\omega)} + \\cos^{C}{(\\frac{C}{h})}, then obtain \\mathbf{J}_M{(C,m,h,\\omega)} + \\cos{(\\omega m)} - \\frac{1}{C} = \\mathbf{D}^{C}{(C,h)} + 2 \\cos{(\\omega m)} - \\frac{1}{C}", "derivation": "\\mathbf{D}{(C,h)} = \\cos{(\\frac{C}{h})} and \\mathbf{D}^{C}{(C,h)} = \\cos^{C}{(\\frac{C}{h})} and \\operatorname{n_{1}}{(m,\\omega)} = \\cos{(\\omega m)} and \\operatorname{n_{1}}{(m,\\omega)} + \\cos^{C}{(\\frac{C}{h})} = \\cos^{C}{(\\frac{C}{h})} + \\cos{(\\omega m)} and \\mathbf{J}_M{(C,m,h,\\omega)} = \\operatorname{n_{1}}{(m,\\omega)} + \\cos^{C}{(\\frac{C}{h})} and \\mathbf{J}_M{(C,m,h,\\omega)} = \\cos^{C}{(\\frac{C}{h})} + \\cos{(\\omega m)} and \\mathbf{J}_M{(C,m,h,\\omega)} = \\mathbf{D}^{C}{(C,h)} + \\cos{(\\omega m)} and \\mathbf{J}_M{(C,m,h,\\omega)} + \\cos{(\\omega m)} - \\frac{1}{C} = \\mathbf{D}^{C}{(C,h)} + 2 \\cos{(\\omega m)} - \\frac{1}{C}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('C', commutative=True), Symbol('h', commutative=True)), cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('C', commutative=True), Symbol('h', commutative=True)), Symbol('C', commutative=True)), Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True)))"], ["get_premise", "Equality(Function('n_1')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True))))"], [["add", 3, "Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True))), Add(Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('C', commutative=True), Symbol('m', commutative=True), Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Function('n_1')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mathbf{J}_M')(Symbol('C', commutative=True), Symbol('m', commutative=True), Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Pow(cos(Mul(Symbol('C', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Symbol('C', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Function('\\\\mathbf{J}_M')(Symbol('C', commutative=True), Symbol('m', commutative=True), Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Pow(Function('\\\\mathbf{D}')(Symbol('C', commutative=True), Symbol('h', commutative=True)), Symbol('C', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True)))))"], [["add", 7, "Add(cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1))))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('C', commutative=True), Symbol('m', commutative=True), Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Add(Pow(Function('\\\\mathbf{D}')(Symbol('C', commutative=True), Symbol('h', commutative=True)), Symbol('C', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('\\\\omega', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\rho{(\\varphi)} = \\log{(\\log{(\\varphi)})}, then obtain \\int \\frac{d}{d \\varphi} (\\rho{(\\varphi)} \\log{(\\log{(\\varphi)})} + \\log{(\\log{(\\varphi)})}^{2}) d\\varphi = \\int \\frac{d}{d \\varphi} 2 \\log{(\\log{(\\varphi)})}^{2} d\\varphi", "derivation": "\\rho{(\\varphi)} = \\log{(\\log{(\\varphi)})} and \\rho{(\\varphi)} \\log{(\\log{(\\varphi)})} = \\log{(\\log{(\\varphi)})}^{2} and \\rho{(\\varphi)} \\log{(\\log{(\\varphi)})} + \\log{(\\log{(\\varphi)})}^{2} = 2 \\log{(\\log{(\\varphi)})}^{2} and \\frac{d}{d \\varphi} (\\rho{(\\varphi)} \\log{(\\log{(\\varphi)})} + \\log{(\\log{(\\varphi)})}^{2}) = \\frac{d}{d \\varphi} 2 \\log{(\\log{(\\varphi)})}^{2} and \\int \\frac{d}{d \\varphi} (\\rho{(\\varphi)} \\log{(\\log{(\\varphi)})} + \\log{(\\log{(\\varphi)})}^{2}) d\\varphi = \\int \\frac{d}{d \\varphi} 2 \\log{(\\log{(\\varphi)})}^{2} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\varphi', commutative=True)), log(log(Symbol('\\\\varphi', commutative=True))))"], [["times", 1, "log(log(Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\varphi', commutative=True)), log(log(Symbol('\\\\varphi', commutative=True)))), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2)))"], [["add", 2, "Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))"], "Equality(Add(Mul(Function('\\\\rho')(Symbol('\\\\varphi', commutative=True)), log(log(Symbol('\\\\varphi', commutative=True)))), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))), Mul(Integer(2), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Add(Mul(Function('\\\\rho')(Symbol('\\\\varphi', commutative=True)), log(log(Symbol('\\\\varphi', commutative=True)))), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Function('\\\\rho')(Symbol('\\\\varphi', commutative=True)), log(log(Symbol('\\\\varphi', commutative=True)))), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Mul(Integer(2), Pow(log(log(Symbol('\\\\varphi', commutative=True))), Integer(2))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(s)} = \\cos{(s)}, then obtain \\int 0 ds = \\int \\frac{\\mathbf{D}{(s)} - \\cos{(s)}}{\\cos{(s)}} ds", "derivation": "\\mathbf{D}{(s)} = \\cos{(s)} and 0 = - \\mathbf{D}{(s)} + \\cos{(s)} and \\cos{(s)} = - \\mathbf{D}{(s)} + 2 \\cos{(s)} and \\mathbf{D}{(s)} = - \\mathbf{D}{(s)} + 2 \\cos{(s)} and 0 = \\mathbf{D}{(s)} - \\cos{(s)} and 0 = \\frac{\\mathbf{D}{(s)} - \\cos{(s)}}{\\cos{(s)}} and \\int 0 ds = \\int \\frac{\\mathbf{D}{(s)} - \\cos{(s)}}{\\cos{(s)}} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{D}')(Symbol('s', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('s', commutative=True))), cos(Symbol('s', commutative=True))))"], [["add", 2, "cos(Symbol('s', commutative=True))"], "Equality(cos(Symbol('s', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('s', commutative=True))), Mul(Integer(2), cos(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\mathbf{D}')(Symbol('s', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('s', commutative=True))), Mul(Integer(2), cos(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Add(Function('\\\\mathbf{D}')(Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True)))))"], [["times", 5, "Pow(cos(Symbol('s', commutative=True)), Integer(-1))"], "Equality(Integer(0), Mul(Add(Function('\\\\mathbf{D}')(Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Pow(cos(Symbol('s', commutative=True)), Integer(-1))))"], [["integrate", 6, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Integral(Mul(Add(Function('\\\\mathbf{D}')(Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Pow(cos(Symbol('s', commutative=True)), Integer(-1))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given z{(x^\\prime,F_{c})} = e^{- F_{c} + x^\\prime}, then obtain \\int \\cos{(\\frac{\\int z{(x^\\prime,F_{c})} dx^\\prime}{\\int e^{- F_{c}} e^{x^\\prime} dx^\\prime})} dx^\\prime = \\int \\cos{(1)} dx^\\prime", "derivation": "z{(x^\\prime,F_{c})} = e^{- F_{c} + x^\\prime} and \\int z{(x^\\prime,F_{c})} dx^\\prime = \\int e^{- F_{c} + x^\\prime} dx^\\prime and \\frac{\\int z{(x^\\prime,F_{c})} dx^\\prime}{\\int e^{- F_{c} + x^\\prime} dx^\\prime} = 1 and \\cos{(\\frac{\\int z{(x^\\prime,F_{c})} dx^\\prime}{\\int e^{- F_{c} + x^\\prime} dx^\\prime})} = \\cos{(1)} and \\int \\cos{(\\frac{\\int z{(x^\\prime,F_{c})} dx^\\prime}{\\int e^{- F_{c} + x^\\prime} dx^\\prime})} dx^\\prime = \\int \\cos{(1)} dx^\\prime and \\int \\cos{(\\frac{\\int z{(x^\\prime,F_{c})} dx^\\prime}{\\int e^{- F_{c}} e^{x^\\prime} dx^\\prime})} dx^\\prime = \\int \\cos{(1)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 2, "Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Integral(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1))"], [["cos", 3], "Equality(cos(Mul(Integral(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)))), cos(Integer(1)))"], [["integrate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(cos(Mul(Integral(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["expand", 5], "Equality(Integral(cos(Mul(Pow(Integral(Mul(exp(Mul(Integer(-1), Symbol('F_c', commutative=True))), exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Integral(Function('z')(Symbol('x^\\\\prime', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given b{(t,y)} = \\cos{(t + y)}, then derive \\int \\frac{\\partial}{\\partial t} b{(t,y)} dy = v_{z} + \\cos{(t + y)}, then obtain \\int \\frac{\\partial}{\\partial t} \\cos{(t + y)} dy = v_{z} + \\cos{(t + y)}", "derivation": "b{(t,y)} = \\cos{(t + y)} and \\frac{\\partial}{\\partial t} b{(t,y)} = \\frac{\\partial}{\\partial t} \\cos{(t + y)} and \\int \\frac{\\partial}{\\partial t} b{(t,y)} dy = \\int \\frac{\\partial}{\\partial t} \\cos{(t + y)} dy and \\int \\frac{\\partial}{\\partial t} b{(t,y)} dy = v_{z} + \\cos{(t + y)} and \\int \\frac{\\partial}{\\partial t} b{(t,y)} dy = v_{z} + b{(t,y)} and \\int \\frac{\\partial}{\\partial t} \\cos{(t + y)} dy = v_{z} + \\cos{(t + y)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True)), cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Add(Symbol('v_z', commutative=True), cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Derivative(Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Add(Symbol('v_z', commutative=True), Function('b')(Symbol('t', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Add(Symbol('v_z', commutative=True), cos(Add(Symbol('t', commutative=True), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\mathbf{E},\\phi_1)} = \\frac{\\mathbf{E}}{\\phi_1}, then obtain \\frac{\\phi_1 (\\frac{\\mathbf{E}}{\\phi_1} - \\phi_1)}{2 \\mathbf{E}} = \\frac{\\phi_1 (\\frac{2 \\mathbf{E}}{\\phi_1} - \\phi_1 - \\Psi{(\\mathbf{E},\\phi_1)})}{2 \\mathbf{E}}", "derivation": "\\Psi{(\\mathbf{E},\\phi_1)} = \\frac{\\mathbf{E}}{\\phi_1} and - \\phi_1 + \\Psi{(\\mathbf{E},\\phi_1)} = \\frac{\\mathbf{E}}{\\phi_1} - \\phi_1 and - \\phi_1 = \\frac{\\mathbf{E}}{\\phi_1} - \\phi_1 - \\Psi{(\\mathbf{E},\\phi_1)} and \\frac{\\mathbf{E}}{\\phi_1} - \\phi_1 = \\frac{2 \\mathbf{E}}{\\phi_1} - \\phi_1 - \\Psi{(\\mathbf{E},\\phi_1)} and \\frac{\\phi_1 (\\frac{\\mathbf{E}}{\\phi_1} - \\phi_1)}{2 \\mathbf{E}} = \\frac{\\phi_1 (\\frac{2 \\mathbf{E}}{\\phi_1} - \\phi_1 - \\Psi{(\\mathbf{E},\\phi_1)})}{2 \\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 2, "Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_1', commutative=True))))))"]]}, {"prompt": "Given y{(V)} = \\sin{(V)}, then derive A_{z} + V = \\int \\frac{\\sin{(V)}}{y{(V)}} dV, then derive A_{z} + V = V + W, then obtain A_{z} + V + c - \\operatorname{J_{\\varepsilon}}{(V)} = c - \\operatorname{J_{\\varepsilon}}{(V)} + \\int 1 dV", "derivation": "y{(V)} = \\sin{(V)} and 1 = \\frac{\\sin{(V)}}{y{(V)}} and \\int 1 dV = \\int \\frac{\\sin{(V)}}{y{(V)}} dV and A_{z} + V = \\int \\frac{\\sin{(V)}}{y{(V)}} dV and A_{z} + V = \\int 1 dV and A_{z} + V = V + W and A_{z} + V + c = V + W + c and \\int 1 dV = V + W and A_{z} + V + c - \\operatorname{J_{\\varepsilon}}{(V)} = V + W + c - \\operatorname{J_{\\varepsilon}}{(V)} and A_{z} + V + c - \\operatorname{J_{\\varepsilon}}{(V)} = c - \\operatorname{J_{\\varepsilon}}{(V)} + \\int 1 dV", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["divide", 1, "Function('y')(Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('y')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Function('y')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True)), Integral(Mul(Pow(Function('y')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True)), Integral(Integer(1), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Symbol('W', commutative=True)))"], [["add", 6, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True), Symbol('c', commutative=True)), Add(Symbol('V', commutative=True), Symbol('W', commutative=True), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Add(Symbol('V', commutative=True), Symbol('W', commutative=True)))"], [["minus", 7, "Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True))"], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True)))), Add(Symbol('V', commutative=True), Symbol('W', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 9, 8], "Equality(Add(Symbol('A_z', commutative=True), Symbol('V', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True)))), Add(Symbol('c', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True))), Integral(Integer(1), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\dot{x})} = - \\dot{x} and \\mathbf{f}{(\\dot{x},S)} = - S + \\phi{(\\dot{x})}, then obtain \\mathbf{f}{(\\dot{x},S)} = - S - \\dot{x}", "derivation": "\\phi{(\\dot{x})} = - \\dot{x} and - S + \\phi{(\\dot{x})} = - S - \\dot{x} and \\mathbf{f}{(\\dot{x},S)} = - S + \\phi{(\\dot{x})} and \\mathbf{f}{(\\dot{x},S)} = - S - \\dot{x}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\phi')(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\phi')(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\dot{z})} = e^{\\dot{z}}, then obtain 0 = - \\frac{\\varphi^{*}^{2}{(\\dot{z})}}{\\dot{z}} + \\frac{\\varphi^{*}{(\\dot{z})} e^{\\dot{z}}}{\\dot{z}}", "derivation": "\\varphi^{*}{(\\dot{z})} = e^{\\dot{z}} and \\frac{\\varphi^{*}{(\\dot{z})}}{\\dot{z}} = \\frac{e^{\\dot{z}}}{\\dot{z}} and \\frac{\\varphi^{*}^{2}{(\\dot{z})}}{\\dot{z}} = \\frac{\\varphi^{*}{(\\dot{z})} e^{\\dot{z}}}{\\dot{z}} and 0 = - \\frac{\\varphi^{*}^{2}{(\\dot{z})}}{\\dot{z}} + \\frac{\\varphi^{*}{(\\dot{z})} e^{\\dot{z}}}{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 2, "Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\tilde{g},t)} = \\sin{(\\tilde{g} t)}, then obtain - 3 \\mathbf{J}_f{(\\tilde{g},t)} + 3 \\sin{(\\tilde{g} t)} = 0", "derivation": "\\mathbf{J}_f{(\\tilde{g},t)} = \\sin{(\\tilde{g} t)} and \\mathbf{J}_f{(\\tilde{g},t)} - \\sin{(\\tilde{g} t)} = 0 and \\sin{(\\tilde{g} t)} = - \\mathbf{J}_f{(\\tilde{g},t)} + 2 \\sin{(\\tilde{g} t)} and \\mathbf{J}_f{(\\tilde{g},t)} = - \\mathbf{J}_f{(\\tilde{g},t)} + 2 \\sin{(\\tilde{g} t)} and \\mathbf{J}_f{(\\tilde{g},t)} = - 3 \\mathbf{J}_f{(\\tilde{g},t)} + 4 \\sin{(\\tilde{g} t)} and - 3 \\mathbf{J}_f{(\\tilde{g},t)} + 3 \\sin{(\\tilde{g} t)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))"], [["minus", 1, "sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))), Integer(0))"], [["minus", 1, "Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)))))"], "Equality(sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(4), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(3), sin(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('t', commutative=True))))), Integer(0))"]]}, {"prompt": "Given W{(\\hat{p}_0)} = \\log{(\\cos{(\\hat{p}_0)})} and \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0)} = \\log{(\\cos{(\\hat{p}_0)})}, then obtain \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0)} - \\log{(\\cos{(\\hat{p}_0)})} = 0", "derivation": "W{(\\hat{p}_0)} = \\log{(\\cos{(\\hat{p}_0)})} and \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0)} = \\log{(\\cos{(\\hat{p}_0)})} and W{(\\hat{p}_0)} - \\log{(\\cos{(\\hat{p}_0)})} = 0 and W{(\\hat{p}_0)} = \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0)} and \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0)} - \\log{(\\cos{(\\hat{p}_0)})} = 0", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), log(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True)), log(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 1, "log(cos(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\hat{p}_0', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\hat{p}_0', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\Psi{(\\nabla,\\mathbf{M})} = e^{- \\mathbf{M} + \\nabla}, then derive \\int \\Psi{(\\nabla,\\mathbf{M})} d\\mathbf{M} = \\pi - e^{- \\mathbf{M} + \\nabla}, then obtain \\int \\Psi{(\\nabla,\\mathbf{M})} d\\mathbf{M} = \\pi - \\Psi{(\\nabla,\\mathbf{M})}", "derivation": "\\Psi{(\\nabla,\\mathbf{M})} = e^{- \\mathbf{M} + \\nabla} and \\int \\Psi{(\\nabla,\\mathbf{M})} d\\mathbf{M} = \\int e^{- \\mathbf{M} + \\nabla} d\\mathbf{M} and \\int \\Psi{(\\nabla,\\mathbf{M})} d\\mathbf{M} = \\pi - e^{- \\mathbf{M} + \\nabla} and \\int e^{- \\mathbf{M} + \\nabla} d\\mathbf{M} = \\pi - e^{- \\mathbf{M} + \\nabla} and \\int e^{- \\mathbf{M} + \\nabla} d\\mathbf{M} = \\pi - \\Psi{(\\nabla,\\mathbf{M})} and \\int \\Psi{(\\nabla,\\mathbf{M})} d\\mathbf{M} = \\pi - \\Psi{(\\nabla,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given c{(Z,I)} = I^{Z}, then obtain 2 (\\int I^{Z} dI) \\int c{(Z,I)} dZ + \\int c{(Z,I)} dZ = (\\int I^{Z} dI) \\int I^{Z} dZ + (\\int I^{Z} dI) \\int c{(Z,I)} dZ + \\int c{(Z,I)} dZ", "derivation": "c{(Z,I)} = I^{Z} and \\int c{(Z,I)} dZ = \\int I^{Z} dZ and (\\int I^{Z} dI) \\int c{(Z,I)} dZ = (\\int I^{Z} dI) \\int I^{Z} dZ and 2 (\\int I^{Z} dI) \\int c{(Z,I)} dZ = (\\int I^{Z} dI) \\int I^{Z} dZ + (\\int I^{Z} dI) \\int c{(Z,I)} dZ and 2 (\\int I^{Z} dI) \\int c{(Z,I)} dZ + \\int c{(Z,I)} dZ = (\\int I^{Z} dI) \\int I^{Z} dZ + (\\int I^{Z} dI) \\int c{(Z,I)} dZ + \\int c{(Z,I)} dZ", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["times", 2, "Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["add", 3, "Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], "Equality(Mul(Integer(2), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True))))))"], [["add", 4, "Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(2), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Integral(Pow(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Integral(Function('c')(Symbol('Z', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given h{(U)} = \\cos{(U)}, then derive r_{0} + \\cos{(h{(U)})} - \\cos{(\\cos{(U)})} = \\int 0 dU, then obtain r_{0} + \\cos{(h{(U)})} - \\cos{(\\cos{(U)})} - \\int 0 dU = 0", "derivation": "h{(U)} = \\cos{(U)} and \\cos{(h{(U)})} = \\cos{(\\cos{(U)})} and \\frac{d}{d U} \\cos{(h{(U)})} = \\frac{d}{d U} \\cos{(\\cos{(U)})} and \\frac{d}{d U} \\cos{(h{(U)})} - \\frac{d}{d U} \\cos{(\\cos{(U)})} = 0 and \\int (\\frac{d}{d U} \\cos{(h{(U)})} - \\frac{d}{d U} \\cos{(\\cos{(U)})}) dU = \\int 0 dU and r_{0} + \\cos{(h{(U)})} - \\cos{(\\cos{(U)})} = \\int 0 dU and r_{0} + \\cos{(h{(U)})} - \\cos{(\\cos{(U)})} - \\int 0 dU = 0", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["cos", 1], "Equality(cos(Function('h')(Symbol('U', commutative=True))), cos(cos(Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(cos(Function('h')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(cos(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Add(Derivative(cos(Function('h')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))), Integer(0))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Derivative(cos(Function('h')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('r_0', commutative=True), cos(Function('h')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(cos(Symbol('U', commutative=True))))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["minus", 6, "Integral(Integer(0), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Symbol('r_0', commutative=True), cos(Function('h')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(cos(Symbol('U', commutative=True)))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\hat{X}{(v_{x},\\hat{H}_l)} = \\log{(\\frac{v_{x}}{\\hat{H}_l})} and \\mu_{0}{(v_{x},\\hat{H}_l)} = \\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)}, then obtain \\frac{\\log{(\\frac{v_{x}}{\\hat{H}_l})}^{v_{x}}}{\\hat{H}_l} = \\frac{\\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)}}{\\hat{H}_l}", "derivation": "\\hat{X}{(v_{x},\\hat{H}_l)} = \\log{(\\frac{v_{x}}{\\hat{H}_l})} and \\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)} = \\log{(\\frac{v_{x}}{\\hat{H}_l})}^{v_{x}} and \\mu_{0}{(v_{x},\\hat{H}_l)} = \\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)} and \\frac{\\mu_{0}{(v_{x},\\hat{H}_l)}}{\\hat{H}_l} = \\frac{\\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)}}{\\hat{H}_l} and \\mu_{0}{(v_{x},\\hat{H}_l)} = \\log{(\\frac{v_{x}}{\\hat{H}_l})}^{v_{x}} and \\frac{\\log{(\\frac{v_{x}}{\\hat{H}_l})}^{v_{x}}}{\\hat{H}_l} = \\frac{\\hat{X}^{v_{x}}{(v_{x},\\hat{H}_l)}}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), log(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('v_x', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_x', commutative=True)))"], [["divide", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('\\\\hat{X}')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mu_0')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(log(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('\\\\hat{X}')(Symbol('v_x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain \\log{(\\int \\frac{1}{E_{\\lambda}} dE_{\\lambda})} = \\log{(\\int \\frac{\\log{(E_{\\lambda})}}{E_{\\lambda} \\mathbf{F}{(E_{\\lambda})}} dE_{\\lambda})}", "derivation": "\\mathbf{F}{(E_{\\lambda})} = \\log{(E_{\\lambda})} and \\frac{\\mathbf{F}{(E_{\\lambda})}}{E_{\\lambda}} = \\frac{\\log{(E_{\\lambda})}}{E_{\\lambda}} and \\frac{1}{E_{\\lambda}} = \\frac{\\log{(E_{\\lambda})}}{E_{\\lambda} \\mathbf{F}{(E_{\\lambda})}} and \\int \\frac{1}{E_{\\lambda}} dE_{\\lambda} = \\int \\frac{\\log{(E_{\\lambda})}}{E_{\\lambda} \\mathbf{F}{(E_{\\lambda})}} dE_{\\lambda} and \\log{(\\int \\frac{1}{E_{\\lambda}} dE_{\\lambda})} = \\log{(\\int \\frac{\\log{(E_{\\lambda})}}{E_{\\lambda} \\mathbf{F}{(E_{\\lambda})}} dE_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["log", 4], "Equality(log(Integral(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), log(Integral(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\Omega)} = \\sin{(\\Omega)}, then obtain -1 + \\frac{\\int \\phi_{1}{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega} = -1 + \\frac{\\int \\sin{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega}", "derivation": "\\phi_{1}{(\\Omega)} = \\sin{(\\Omega)} and \\int \\phi_{1}{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\frac{\\int \\phi_{1}{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega} = \\frac{\\int \\sin{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega} and -1 + \\frac{\\int \\phi_{1}{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega} = -1 + \\frac{\\int \\sin{(\\Omega)} d\\Omega}{\\frac{d}{d \\Omega} \\int \\phi_{1}{(\\Omega)} d\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 2, "Derivative(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Derivative(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Derivative(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Integer(-1), Mul(Pow(Derivative(Integral(Function('\\\\phi_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given n{(V,\\eta^{\\prime})} = \\eta^{\\prime} + e^{V}, then obtain n{(V,\\eta^{\\prime})} + \\int (V n{(V,\\eta^{\\prime})} + e^{V}) d\\eta^{\\prime} = n{(V,\\eta^{\\prime})} + \\int (V (\\eta^{\\prime} + e^{V}) + e^{V}) d\\eta^{\\prime}", "derivation": "n{(V,\\eta^{\\prime})} = \\eta^{\\prime} + e^{V} and V n{(V,\\eta^{\\prime})} = V (\\eta^{\\prime} + e^{V}) and V n{(V,\\eta^{\\prime})} + e^{V} = V (\\eta^{\\prime} + e^{V}) + e^{V} and \\int (V n{(V,\\eta^{\\prime})} + e^{V}) d\\eta^{\\prime} = \\int (V (\\eta^{\\prime} + e^{V}) + e^{V}) d\\eta^{\\prime} and \\eta^{\\prime} + e^{V} + \\int (V n{(V,\\eta^{\\prime})} + e^{V}) d\\eta^{\\prime} = \\eta^{\\prime} + e^{V} + \\int (V (\\eta^{\\prime} + e^{V}) + e^{V}) d\\eta^{\\prime} and n{(V,\\eta^{\\prime})} + \\int (V n{(V,\\eta^{\\prime})} + e^{V}) d\\eta^{\\prime} = n{(V,\\eta^{\\prime})} + \\int (V (\\eta^{\\prime} + e^{V}) + e^{V}) d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True))))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Symbol('V', commutative=True), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))))"], [["add", 2, "exp(Symbol('V', commutative=True))"], "Equality(Add(Mul(Symbol('V', commutative=True), Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('V', commutative=True))), Add(Mul(Symbol('V', commutative=True), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))), exp(Symbol('V', commutative=True))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('V', commutative=True), Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Add(Mul(Symbol('V', commutative=True), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 4, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)), Integral(Add(Mul(Symbol('V', commutative=True), Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)), Integral(Add(Mul(Symbol('V', commutative=True), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Add(Mul(Symbol('V', commutative=True), Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Function('n')(Symbol('V', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Add(Mul(Symbol('V', commutative=True), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('V', commutative=True)))), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given S{(a^{\\dagger},\\pi)} = \\pi^{a^{\\dagger}} and \\mu{(a^{\\dagger},\\pi)} = \\pi^{a^{\\dagger}}, then obtain (\\pi^{a^{\\dagger}})^{a^{\\dagger}} = \\mu^{a^{\\dagger}}{(a^{\\dagger},\\pi)}", "derivation": "S{(a^{\\dagger},\\pi)} = \\pi^{a^{\\dagger}} and \\mu{(a^{\\dagger},\\pi)} = \\pi^{a^{\\dagger}} and S{(a^{\\dagger},\\pi)} = \\mu{(a^{\\dagger},\\pi)} and S^{a^{\\dagger}}{(a^{\\dagger},\\pi)} = \\mu^{a^{\\dagger}}{(a^{\\dagger},\\pi)} and S^{a^{\\dagger}}{(a^{\\dagger},\\pi)} = (\\pi^{a^{\\dagger}})^{a^{\\dagger}} and (\\pi^{a^{\\dagger}})^{a^{\\dagger}} = \\mu^{a^{\\dagger}}{(a^{\\dagger},\\pi)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('S')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\mu')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('S')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\mu')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('S')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Pow(Symbol('\\\\pi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Symbol('\\\\pi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\mu')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{X},G)} = G \\hat{X} and Q{(\\hat{X},G)} = G \\hat{X}, then obtain \\iint (- G \\hat{X} + \\operatorname{F_{N}}{(\\hat{X},G)}) d\\hat{X} dG = \\iint 0 d\\hat{X} dG", "derivation": "\\operatorname{F_{N}}{(\\hat{X},G)} = G \\hat{X} and Q{(\\hat{X},G)} = G \\hat{X} and \\operatorname{F_{N}}{(\\hat{X},G)} = Q{(\\hat{X},G)} and - G + Q{(\\hat{X},G)} = G \\hat{X} - G and - G \\hat{X} + Q{(\\hat{X},G)} = 0 and - G \\hat{X} + \\operatorname{F_{N}}{(\\hat{X},G)} = 0 and \\int (- G \\hat{X} + \\operatorname{F_{N}}{(\\hat{X},G)}) d\\hat{X} = \\int 0 d\\hat{X} and \\iint (- G \\hat{X} + \\operatorname{F_{N}}{(\\hat{X},G)}) d\\hat{X} dG = \\iint 0 d\\hat{X} dG", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_N')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Function('Q')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)))"], [["minus", 2, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('Q')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))))"], [["minus", 4, "Add(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('Q')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('F_N')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True))), Integer(0))"], [["integrate", 6, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('F_N')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 7, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('F_N')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(f_{\\mathbf{p}},A_{x})} = A_{x} + f_{\\mathbf{p}} and \\Psi_{nl}{(f_{\\mathbf{p}},A_{x})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\bar{\\h}{(f_{\\mathbf{p}},A_{x})}, then obtain \\Psi_{nl}{(f_{\\mathbf{p}},A_{x})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (A_{x} + f_{\\mathbf{p}})", "derivation": "\\bar{\\h}{(f_{\\mathbf{p}},A_{x})} = A_{x} + f_{\\mathbf{p}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\bar{\\h}{(f_{\\mathbf{p}},A_{x})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (A_{x} + f_{\\mathbf{p}}) and \\Psi_{nl}{(f_{\\mathbf{p}},A_{x})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\bar{\\h}{(f_{\\mathbf{p}},A_{x})} and \\Psi_{nl}{(f_{\\mathbf{p}},A_{x})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (A_{x} + f_{\\mathbf{p}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A_x', commutative=True)), Derivative(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A_x', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(f^{\\prime})} = \\log{(\\sin{(f^{\\prime})})} and \\varphi^{*}{(A_{2})} = \\sin{(e^{A_{2}})}, then obtain \\log{(U^{2}{(f^{\\prime})})} - \\sin^{A_{2}}{(e^{A_{2}})} = \\log{(U{(f^{\\prime})} \\log{(\\sin{(f^{\\prime})})})} - \\sin^{A_{2}}{(e^{A_{2}})}", "derivation": "U{(f^{\\prime})} = \\log{(\\sin{(f^{\\prime})})} and U^{2}{(f^{\\prime})} = U{(f^{\\prime})} \\log{(\\sin{(f^{\\prime})})} and \\log{(U^{2}{(f^{\\prime})})} = \\log{(U{(f^{\\prime})} \\log{(\\sin{(f^{\\prime})})})} and \\varphi^{*}{(A_{2})} = \\sin{(e^{A_{2}})} and \\varphi^{*}^{A_{2}}{(A_{2})} = \\sin^{A_{2}}{(e^{A_{2}})} and - \\varphi^{*}^{A_{2}}{(A_{2})} + \\log{(U^{2}{(f^{\\prime})})} = - \\varphi^{*}^{A_{2}}{(A_{2})} + \\log{(U{(f^{\\prime})} \\log{(\\sin{(f^{\\prime})})})} and \\log{(U^{2}{(f^{\\prime})})} - \\sin^{A_{2}}{(e^{A_{2}})} = \\log{(U{(f^{\\prime})} \\log{(\\sin{(f^{\\prime})})})} - \\sin^{A_{2}}{(e^{A_{2}})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), log(sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "Function('U')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), Integer(2)), Mul(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), log(sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["log", 2], "Equality(log(Pow(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), Integer(2))), log(Mul(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), log(sin(Symbol('f^{\\\\prime}', commutative=True))))))"], ["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('A_2', commutative=True)), sin(exp(Symbol('A_2', commutative=True))))"], [["power", 4, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(sin(exp(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))"], [["minus", 3, "Pow(Function('\\\\varphi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\varphi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), log(Pow(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('\\\\varphi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), log(Mul(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), log(sin(Symbol('f^{\\\\prime}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(log(Pow(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(sin(exp(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))), Add(log(Mul(Function('U')(Symbol('f^{\\\\prime}', commutative=True)), log(sin(Symbol('f^{\\\\prime}', commutative=True))))), Mul(Integer(-1), Pow(sin(exp(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})} = - \\delta + \\mathbf{f}, then obtain 0^{\\delta} + 1 = (- \\delta + \\mathbf{f} - \\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})})^{\\delta} + 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})} = - \\delta + \\mathbf{f} and 0 = - \\delta + \\mathbf{f} - \\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})} and 0^{\\delta} = (- \\delta + \\mathbf{f} - \\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})})^{\\delta} and 0^{\\delta} + 1 = (- \\delta + \\mathbf{f} - \\operatorname{L_{\\varepsilon}}{(\\delta,\\mathbf{f})})^{\\delta} + 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\delta', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\delta', commutative=True)), Integer(1)))"]]}, {"prompt": "Given T{(E_{n},\\mathbf{B})} = E_{n} + \\mathbf{B}, then obtain \\log{(\\int (- \\mathbf{B} + \\int T{(E_{n},\\mathbf{B})} d\\mathbf{B}) dE_{n})} = \\log{(\\int (- \\mathbf{B} + \\int (E_{n} + \\mathbf{B}) d\\mathbf{B}) dE_{n})}", "derivation": "T{(E_{n},\\mathbf{B})} = E_{n} + \\mathbf{B} and \\int T{(E_{n},\\mathbf{B})} d\\mathbf{B} = \\int (E_{n} + \\mathbf{B}) d\\mathbf{B} and - \\mathbf{B} + \\int T{(E_{n},\\mathbf{B})} d\\mathbf{B} = - \\mathbf{B} + \\int (E_{n} + \\mathbf{B}) d\\mathbf{B} and \\int (- \\mathbf{B} + \\int T{(E_{n},\\mathbf{B})} d\\mathbf{B}) dE_{n} = \\int (- \\mathbf{B} + \\int (E_{n} + \\mathbf{B}) d\\mathbf{B}) dE_{n} and \\log{(\\int (- \\mathbf{B} + \\int T{(E_{n},\\mathbf{B})} d\\mathbf{B}) dE_{n})} = \\log{(\\int (- \\mathbf{B} + \\int (E_{n} + \\mathbf{B}) d\\mathbf{B}) dE_{n})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('T')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Function('T')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Function('T')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('E_n', commutative=True))))"], [["log", 4], "Equality(log(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Function('T')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('E_n', commutative=True)))), log(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(b,M)} = M b, then derive \\frac{\\partial}{\\partial b} \\dot{x}{(b,M)} = M, then obtain \\frac{(M + 1) \\dot{x}{(b,M)}}{b} = \\frac{(\\frac{\\partial}{\\partial b} M b + 1) \\dot{x}{(b,M)}}{b}", "derivation": "\\dot{x}{(b,M)} = M b and \\frac{\\partial}{\\partial b} \\dot{x}{(b,M)} = \\frac{\\partial}{\\partial b} M b and \\frac{\\partial}{\\partial b} \\dot{x}{(b,M)} = M and \\frac{\\partial}{\\partial b} \\dot{x}{(b,M)} + 1 = \\frac{\\partial}{\\partial b} M b + 1 and M + 1 = \\frac{\\partial}{\\partial b} M b + 1 and \\frac{(M + 1) \\dot{x}{(b,M)}}{b} = \\frac{(\\frac{\\partial}{\\partial b} M b + 1) \\dot{x}{(b,M)}}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('M', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('M', commutative=True))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Symbol('M', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('M', commutative=True), Integer(1)), Add(Derivative(Mul(Symbol('M', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"], [["times", 5, "Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Integer(1)), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Derivative(Mul(Symbol('M', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(x)} = e^{\\sin{(x)}}, then obtain \\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{y^{\\prime}}{(x)}}{- x + \\operatorname{y^{\\prime}}{(x)}} = \\frac{d^{2}}{d x^{2}} \\frac{e^{\\sin{(x)}}}{- x + \\operatorname{y^{\\prime}}{(x)}}", "derivation": "\\operatorname{y^{\\prime}}{(x)} = e^{\\sin{(x)}} and - x + \\operatorname{y^{\\prime}}{(x)} = - x + e^{\\sin{(x)}} and \\frac{\\operatorname{y^{\\prime}}{(x)}}{- x + e^{\\sin{(x)}}} = \\frac{e^{\\sin{(x)}}}{- x + e^{\\sin{(x)}}} and \\frac{\\operatorname{y^{\\prime}}{(x)}}{- x + \\operatorname{y^{\\prime}}{(x)}} = \\frac{e^{\\sin{(x)}}}{- x + \\operatorname{y^{\\prime}}{(x)}} and \\frac{d}{d x} \\frac{\\operatorname{y^{\\prime}}{(x)}}{- x + \\operatorname{y^{\\prime}}{(x)}} = \\frac{d}{d x} \\frac{e^{\\sin{(x)}}}{- x + \\operatorname{y^{\\prime}}{(x)}} and \\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{y^{\\prime}}{(x)}}{- x + \\operatorname{y^{\\prime}}{(x)}} = \\frac{d^{2}}{d x^{2}} \\frac{e^{\\sin{(x)}}}{- x + \\operatorname{y^{\\prime}}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True))))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True)))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True)))), Integer(-1)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True)))), Integer(-1)), exp(sin(Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), exp(sin(Symbol('x', commutative=True)))))"], [["differentiate", 4, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), exp(sin(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True))), Integer(-1)), exp(sin(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(t)} = \\cos{(t)}, then derive \\int \\operatorname{P_{g}}{(t)} dt = \\eta^{\\prime} + \\sin{(t)}, then derive c + \\sin{(t)} = \\eta^{\\prime} + \\sin{(t)}, then obtain - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda} + \\int \\cos{(t)} dt = c + \\sin{(t)} - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda}", "derivation": "\\operatorname{P_{g}}{(t)} = \\cos{(t)} and \\int \\operatorname{P_{g}}{(t)} dt = \\int \\cos{(t)} dt and \\int \\operatorname{P_{g}}{(t)} dt = \\eta^{\\prime} + \\sin{(t)} and \\int \\cos{(t)} dt = \\eta^{\\prime} + \\sin{(t)} and c + \\sin{(t)} = \\eta^{\\prime} + \\sin{(t)} and \\int \\operatorname{P_{g}}{(t)} dt = c + \\sin{(t)} and \\int \\operatorname{P_{g}}{(t)} dt - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda} = c + \\sin{(t)} - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda} and - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda} + \\int \\cos{(t)} dt = c + \\sin{(t)} - \\int \\operatorname{v_{x}}{(E_{\\lambda})} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('c', commutative=True), sin(Symbol('t', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('P_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('c', commutative=True), sin(Symbol('t', commutative=True))))"], [["minus", 6, "Integral(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Integral(Function('P_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Mul(Integer(-1), Integral(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))), Add(Symbol('c', commutative=True), sin(Symbol('t', commutative=True)), Mul(Integer(-1), Integral(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Integer(-1), Integral(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('c', commutative=True), sin(Symbol('t', commutative=True)), Mul(Integer(-1), Integral(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(F_{x})} = \\log{(e^{F_{x}})}, then derive \\int \\operatorname{F_{H}}{(F_{x})} dF_{x} = B + \\frac{F_{x}^{2}}{2}, then obtain \\iint \\log{(e^{F_{x}})} dF_{x} dB = \\iint \\operatorname{F_{H}}{(F_{x})} dF_{x} dB", "derivation": "\\operatorname{F_{H}}{(F_{x})} = \\log{(e^{F_{x}})} and \\int \\operatorname{F_{H}}{(F_{x})} dF_{x} = \\int \\log{(e^{F_{x}})} dF_{x} and \\int \\operatorname{F_{H}}{(F_{x})} dF_{x} = B + \\frac{F_{x}^{2}}{2} and \\iint \\operatorname{F_{H}}{(F_{x})} dF_{x} dB = \\int (B + \\frac{F_{x}^{2}}{2}) dB and \\iint \\log{(e^{F_{x}})} dF_{x} dB = \\int (B + \\frac{F_{x}^{2}}{2}) dB and \\iint \\log{(e^{F_{x}})} dF_{x} dB = \\iint \\operatorname{F_{H}}{(F_{x})} dF_{x} dB", "srepr_derivation": [["get_premise", "Equality(Function('F_H')(Symbol('F_x', commutative=True)), log(exp(Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(log(exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2)))), Tuple(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2)))), Tuple(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(log(exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Function('F_H')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(x^\\prime,y)} = e^{y^{x^\\prime}}, then obtain \\rho_{b}{(x^\\prime,y)} + e^{y^{x^\\prime} + \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}}} = 2 e^{y^{x^\\prime} + \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}}}", "derivation": "\\rho_{b}{(x^\\prime,y)} = e^{y^{x^\\prime}} and \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}} = 0 and \\rho_{b}{(x^\\prime,y)} + e^{y^{x^\\prime}} = 2 e^{y^{x^\\prime}} and y^{x^\\prime} + \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}} = y^{x^\\prime} and \\rho_{b}{(x^\\prime,y)} + e^{y^{x^\\prime} + \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}}} = 2 e^{y^{x^\\prime} + \\rho_{b}{(x^\\prime,y)} - e^{y^{x^\\prime}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 1, "exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Integer(0))"], [["add", 1, "exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(2), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["add", 2, "Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), exp(Add(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))))))), Mul(Integer(2), exp(Add(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('y', commutative=True), Symbol('x^\\\\prime', commutative=True))))))))"]]}, {"prompt": "Given \\hat{p}_0{(U)} = \\sin{(U)}, then obtain \\frac{d^{2}}{d U^{2}} (U + 2 \\hat{p}_0{(U)}) = \\frac{d^{2}}{d U^{2}} (U + 2 \\sin{(U)})", "derivation": "\\hat{p}_0{(U)} = \\sin{(U)} and U + \\hat{p}_0{(U)} = U + \\sin{(U)} and U + 2 \\hat{p}_0{(U)} = U + \\hat{p}_0{(U)} + \\sin{(U)} and \\frac{d}{d U} (U + 2 \\hat{p}_0{(U)}) = \\frac{d}{d U} (U + \\hat{p}_0{(U)} + \\sin{(U)}) and \\frac{d}{d U} (U + 2 \\hat{p}_0{(U)}) = \\frac{d}{d U} (U + 2 \\sin{(U)}) and \\frac{d}{d U} (U + \\hat{p}_0{(U)} + \\sin{(U)}) = \\frac{d}{d U} (U + 2 \\sin{(U)}) and \\frac{d^{2}}{d U^{2}} (U + \\hat{p}_0{(U)} + \\sin{(U)}) = \\frac{d^{2}}{d U^{2}} (U + 2 \\sin{(U)}) and \\frac{d^{2}}{d U^{2}} (U + 2 \\hat{p}_0{(U)}) = \\frac{d^{2}}{d U^{2}} (U + 2 \\sin{(U)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('\\\\hat{p}_0')(Symbol('U', commutative=True))), Add(Symbol('U', commutative=True), sin(Symbol('U', commutative=True))))"], [["add", 2, "Function('\\\\hat{p}_0')(Symbol('U', commutative=True))"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))), Add(Symbol('U', commutative=True), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('U', commutative=True), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Symbol('U', commutative=True), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(2))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(2))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(m,t)} = - t + \\sin{(m)} and \\eta^{\\prime}{(t)} = - t, then obtain \\Psi_{\\lambda}^{m}{(m,t)} = (\\eta^{\\prime}{(t)} + \\sin{(m)})^{m}", "derivation": "\\Psi_{\\lambda}{(m,t)} = - t + \\sin{(m)} and \\Psi_{\\lambda}^{m}{(m,t)} = (- t + \\sin{(m)})^{m} and \\eta^{\\prime}{(t)} = - t and \\Psi_{\\lambda}^{m}{(m,t)} = (\\eta^{\\prime}{(t)} + \\sin{(m)})^{m}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), sin(Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\theta)} = \\cos{(\\theta)}, then derive \\int \\operatorname{F_{N}}{(\\theta)} d\\theta = \\rho_b + \\sin{(\\theta)}, then derive \\frac{d}{d \\theta} \\int \\cos{(\\theta)} d\\theta = \\cos{(\\theta)}, then obtain \\frac{e^{- 3 \\mathbf{D} v_{z}} \\frac{d}{d \\theta} \\int \\cos{(\\theta)} d\\theta}{\\operatorname{A_{z}}{(v_{z},\\mathbf{D})}} = \\frac{e^{- 3 \\mathbf{D} v_{z}} \\cos{(\\theta)}}{\\operatorname{A_{z}}{(v_{z},\\mathbf{D})}}", "derivation": "\\operatorname{F_{N}}{(\\theta)} = \\cos{(\\theta)} and \\int \\operatorname{F_{N}}{(\\theta)} d\\theta = \\int \\cos{(\\theta)} d\\theta and \\int \\operatorname{F_{N}}{(\\theta)} d\\theta = \\rho_b + \\sin{(\\theta)} and \\int \\cos{(\\theta)} d\\theta = \\rho_b + \\sin{(\\theta)} and \\frac{d}{d \\theta} \\int \\cos{(\\theta)} d\\theta = \\frac{\\partial}{\\partial \\theta} (\\rho_b + \\sin{(\\theta)}) and \\frac{d}{d \\theta} \\int \\cos{(\\theta)} d\\theta = \\cos{(\\theta)} and \\frac{e^{- 3 \\mathbf{D} v_{z}} \\frac{d}{d \\theta} \\int \\cos{(\\theta)} d\\theta}{\\operatorname{A_{z}}{(v_{z},\\mathbf{D})}} = \\frac{e^{- 3 \\mathbf{D} v_{z}} \\cos{(\\theta)}}{\\operatorname{A_{z}}{(v_{z},\\mathbf{D})}}", "srepr_derivation": [["get_premise", "Equality(Function('F_N')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_N')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), cos(Symbol('\\\\theta', commutative=True)))"], [["divide", 6, "Mul(Function('A_z')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(3), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_z', commutative=True))))"], "Equality(Mul(Pow(Function('A_z')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Integer(3), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_z', commutative=True))), Derivative(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Pow(Function('A_z')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Integer(3), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_z', commutative=True))), cos(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\eta)} = \\log{(\\eta)}, then derive \\int \\operatorname{t_{2}}{(\\eta)} d\\eta = C_{d} + \\eta \\log{(\\eta)} - \\eta, then obtain \\frac{d}{d \\eta} \\log{(\\int \\operatorname{t_{2}}{(\\eta)} d\\eta)} = \\frac{\\partial}{\\partial \\eta} \\log{(C_{d} + \\eta \\operatorname{t_{2}}{(\\eta)} - \\eta)}", "derivation": "\\operatorname{t_{2}}{(\\eta)} = \\log{(\\eta)} and \\int \\operatorname{t_{2}}{(\\eta)} d\\eta = \\int \\log{(\\eta)} d\\eta and \\int \\operatorname{t_{2}}{(\\eta)} d\\eta = C_{d} + \\eta \\log{(\\eta)} - \\eta and \\log{(\\int \\operatorname{t_{2}}{(\\eta)} d\\eta)} = \\log{(C_{d} + \\eta \\log{(\\eta)} - \\eta)} and \\frac{d}{d \\eta} \\log{(\\int \\operatorname{t_{2}}{(\\eta)} d\\eta)} = \\frac{\\partial}{\\partial \\eta} \\log{(C_{d} + \\eta \\log{(\\eta)} - \\eta)} and \\frac{d}{d \\eta} \\log{(\\int \\operatorname{t_{2}}{(\\eta)} d\\eta)} = \\frac{\\partial}{\\partial \\eta} \\log{(C_{d} + \\eta \\operatorname{t_{2}}{(\\eta)} - \\eta)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], [["log", 3], "Equality(log(Integral(Function('t_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), log(Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(log(Integral(Function('t_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(log(Integral(Function('t_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Function('t_2')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\psi^*)} = \\sin{(\\psi^*)} and p{(\\psi^*)} = \\psi^*, then obtain \\int \\frac{d}{d \\psi^*} \\mathbf{r}{(\\psi^*)} \\sin{(\\psi^*)} dp{(\\psi^*)} = \\int \\frac{d}{d \\psi^*} \\sin^{2}{(\\psi^*)} dp{(\\psi^*)}", "derivation": "\\mathbf{r}{(\\psi^*)} = \\sin{(\\psi^*)} and p{(\\psi^*)} = \\psi^* and \\mathbf{r}{(\\psi^*)} \\sin{(\\psi^*)} = \\sin^{2}{(\\psi^*)} and \\frac{d}{d \\psi^*} \\mathbf{r}{(\\psi^*)} \\sin{(\\psi^*)} = \\frac{d}{d \\psi^*} \\sin^{2}{(\\psi^*)} and \\int \\frac{d}{d \\psi^*} \\mathbf{r}{(\\psi^*)} \\sin{(\\psi^*)} d\\psi^* = \\int \\frac{d}{d \\psi^*} \\sin^{2}{(\\psi^*)} d\\psi^* and \\int \\frac{d}{d \\psi^*} \\mathbf{r}{(\\psi^*)} \\sin{(\\psi^*)} dp{(\\psi^*)} = \\int \\frac{d}{d \\psi^*} \\sin^{2}{(\\psi^*)} dp{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["divide", 1, "Pow(sin(Symbol('\\\\psi^*', commutative=True)), Integer(-1))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Derivative(Pow(sin(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Derivative(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Function('p')(Symbol('\\\\psi^*', commutative=True)))), Integral(Derivative(Pow(sin(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Function('p')(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} = \\frac{\\hat{x}_0}{\\mathbb{I} y}, then obtain ((\\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} + \\frac{1}{y})^{\\hat{x}_0})^{y} = ((\\frac{\\hat{x}_0}{\\mathbb{I} y} + \\frac{1}{y})^{\\hat{x}_0})^{y}", "derivation": "\\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} = \\frac{\\hat{x}_0}{\\mathbb{I} y} and \\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} + \\frac{1}{y} = \\frac{\\hat{x}_0}{\\mathbb{I} y} + \\frac{1}{y} and (\\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} + \\frac{1}{y})^{\\hat{x}_0} = (\\frac{\\hat{x}_0}{\\mathbb{I} y} + \\frac{1}{y})^{\\hat{x}_0} and ((\\hat{x}{(\\mathbb{I},y,\\hat{x}_0)} + \\frac{1}{y})^{\\hat{x}_0})^{y} = ((\\frac{\\hat{x}_0}{\\mathbb{I} y} + \\frac{1}{y})^{\\hat{x}_0})^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('y', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\dot{x})} = e^{\\sin{(\\dot{x})}} and k{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain e^{u} + e^{\\sin{(\\dot{x})}} = e^{u} + e^{k{(\\dot{x})}}", "derivation": "\\rho{(\\dot{x})} = e^{\\sin{(\\dot{x})}} and k{(\\dot{x})} = \\sin{(\\dot{x})} and \\rho{(\\dot{x})} = e^{k{(\\dot{x})}} and e^{\\sin{(\\dot{x})}} = e^{k{(\\dot{x})}} and e^{u} + e^{\\sin{(\\dot{x})}} = e^{u} + e^{k{(\\dot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\rho')(Symbol('\\\\dot{x}', commutative=True)), exp(Function('k')(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(sin(Symbol('\\\\dot{x}', commutative=True))), exp(Function('k')(Symbol('\\\\dot{x}', commutative=True))))"], [["add", 4, "exp(Symbol('u', commutative=True))"], "Equality(Add(exp(Symbol('u', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))), Add(exp(Symbol('u', commutative=True)), exp(Function('k')(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(f)} = \\sin{(f)} and q{(f)} = \\sin{(f)}, then obtain - f + (f + \\sin{(f)}) \\int \\operatorname{E_{\\lambda}}^{f}{(f)} df - \\operatorname{E_{\\lambda}}{(f)} = - f + (f + \\sin{(f)}) \\int q^{f}{(f)} df - \\operatorname{E_{\\lambda}}{(f)}", "derivation": "\\operatorname{E_{\\lambda}}{(f)} = \\sin{(f)} and \\operatorname{E_{\\lambda}}^{f}{(f)} = \\sin^{f}{(f)} and q{(f)} = \\sin{(f)} and \\operatorname{E_{\\lambda}}{(f)} = q{(f)} and q^{f}{(f)} = \\sin^{f}{(f)} and \\operatorname{E_{\\lambda}}^{f}{(f)} = q^{f}{(f)} and \\int \\operatorname{E_{\\lambda}}^{f}{(f)} df = \\int q^{f}{(f)} df and (f + \\sin{(f)}) \\int \\operatorname{E_{\\lambda}}^{f}{(f)} df = (f + \\sin{(f)}) \\int q^{f}{(f)} df and - f + (f + \\sin{(f)}) \\int \\operatorname{E_{\\lambda}}^{f}{(f)} df - \\operatorname{E_{\\lambda}}{(f)} = - f + (f + \\sin{(f)}) \\int q^{f}{(f)} df - \\operatorname{E_{\\lambda}}{(f)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('q')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Function('q')(Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('q')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('q')(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["integrate", 6, "Symbol('f', commutative=True)"], "Equality(Integral(Pow(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(Function('q')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["times", 7, "Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True)))"], "Equality(Mul(Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True))), Integral(Pow(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True))), Integral(Pow(Function('q')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["minus", 8, "Add(Symbol('f', commutative=True), Function('E_{\\\\lambda}')(Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True))), Integral(Pow(Function('E_{\\\\lambda}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True))), Integral(Pow(Function('q')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given f{(\\hat{\\mathbf{r}},J_{\\varepsilon})} = J_{\\varepsilon} \\cos{(\\hat{\\mathbf{r}})}, then obtain 1 = \\frac{J_{\\varepsilon} \\cos{(\\hat{\\mathbf{r}})}}{f{(\\hat{\\mathbf{r}},J_{\\varepsilon})}}", "derivation": "f{(\\hat{\\mathbf{r}},J_{\\varepsilon})} = J_{\\varepsilon} \\cos{(\\hat{\\mathbf{r}})} and \\frac{f{(\\hat{\\mathbf{r}},J_{\\varepsilon})}}{\\cos{(\\hat{\\mathbf{r}})}} = J_{\\varepsilon} and \\frac{f{(\\hat{\\mathbf{r}},J_{\\varepsilon})}}{\\cos^{2}{(\\hat{\\mathbf{r}})}} = \\frac{J_{\\varepsilon}}{\\cos{(\\hat{\\mathbf{r}})}} and 1 = \\frac{J_{\\varepsilon} \\cos{(\\hat{\\mathbf{r}})}}{f{(\\hat{\\mathbf{r}},J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 1, "cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Function('f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["times", 2, "Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))"], "Equality(Mul(Function('f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-2))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Function('f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-2)))"], "Equality(Integer(1), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(P_{e},\\mu)} = - P_{e} + \\log{(\\mu)}, then obtain P_{e} + \\operatorname{t_{1}}{(P_{e},\\mu)} = \\log{(\\mu)}", "derivation": "\\operatorname{t_{1}}{(P_{e},\\mu)} = - P_{e} + \\log{(\\mu)} and - P_{e} + \\operatorname{t_{1}}{(P_{e},\\mu)} = - 2 P_{e} + \\log{(\\mu)} and - 2 P_{e} + \\operatorname{t_{1}}{(P_{e},\\mu)} = - 3 P_{e} + \\log{(\\mu)} and P_{e} + \\operatorname{t_{1}}{(P_{e},\\mu)} = \\log{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Function('t_1')(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('P_e', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Symbol('P_e', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('P_e', commutative=True)), Function('t_1')(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Integer(3), Symbol('P_e', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Integer(3), Symbol('P_e', commutative=True))"], "Equality(Add(Symbol('P_e', commutative=True), Function('t_1')(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), log(Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{g},C_{d})} = C_{d} \\mathbf{g}, then obtain \\frac{\\partial}{\\partial C_{d}} \\dot{z}{(\\mathbf{g},C_{d})} - 1 = \\mathbf{g} - 1", "derivation": "\\dot{z}{(\\mathbf{g},C_{d})} = C_{d} \\mathbf{g} and - C_{d} + \\dot{z}{(\\mathbf{g},C_{d})} = C_{d} \\mathbf{g} - C_{d} and \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\dot{z}{(\\mathbf{g},C_{d})}) = \\frac{\\partial}{\\partial C_{d}} (C_{d} \\mathbf{g} - C_{d}) and \\frac{\\partial}{\\partial C_{d}} \\dot{z}{(\\mathbf{g},C_{d})} - 1 = \\mathbf{g} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True))))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given a{(P_{e})} = \\cos{(P_{e})}, then obtain - P_{e} + (\\frac{a{(P_{e})}}{P_{e}})^{P_{e}} = - P_{e} + (\\frac{\\cos{(P_{e})}}{P_{e}})^{P_{e}}", "derivation": "a{(P_{e})} = \\cos{(P_{e})} and \\frac{a{(P_{e})}}{P_{e}} = \\frac{\\cos{(P_{e})}}{P_{e}} and (\\frac{a{(P_{e})}}{P_{e}})^{P_{e}} = (\\frac{\\cos{(P_{e})}}{P_{e}})^{P_{e}} and - P_{e} + (\\frac{a{(P_{e})}}{P_{e}})^{P_{e}} = - P_{e} + (\\frac{\\cos{(P_{e})}}{P_{e}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["divide", 1, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('a')(Symbol('P_e', commutative=True))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('a')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["minus", 3, "Symbol('P_e', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('a')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(C)} = \\cos{(C)}, then derive \\frac{C \\int \\operatorname{A_{x}}{(C)} dC}{\\operatorname{A_{x}}{(C)}} = \\frac{C (\\hat{x}_0 + \\sin{(C)})}{\\operatorname{A_{x}}{(C)}}, then obtain \\frac{C \\int \\cos{(C)} dC}{\\operatorname{A_{x}}{(C)}} = \\frac{C (\\hat{x}_0 + \\sin{(C)})}{\\operatorname{A_{x}}{(C)}}", "derivation": "\\operatorname{A_{x}}{(C)} = \\cos{(C)} and \\int \\operatorname{A_{x}}{(C)} dC = \\int \\cos{(C)} dC and \\frac{C \\int \\operatorname{A_{x}}{(C)} dC}{\\operatorname{A_{x}}{(C)}} = \\frac{C \\int \\cos{(C)} dC}{\\operatorname{A_{x}}{(C)}} and \\frac{C \\int \\operatorname{A_{x}}{(C)} dC}{\\operatorname{A_{x}}{(C)}} = \\frac{C (\\hat{x}_0 + \\sin{(C)})}{\\operatorname{A_{x}}{(C)}} and \\frac{C \\int \\cos{(C)} dC}{\\operatorname{A_{x}}{(C)}} = \\frac{C (\\hat{x}_0 + \\sin{(C)})}{\\operatorname{A_{x}}{(C)}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('A_x')(Symbol('C', commutative=True)))"], "Equality(Mul(Symbol('C', commutative=True), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1)), Integral(Function('A_x')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('C', commutative=True), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1)), Integral(Function('A_x')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Add(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('C', commutative=True))), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('C', commutative=True), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Add(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('C', commutative=True))), Pow(Function('A_x')(Symbol('C', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(C_{2})} = \\int \\cos{(C_{2})} dC_{2}, then derive \\delta{(C_{2})} = r_{0} + \\sin{(C_{2})}, then obtain C_{2} + (r_{0} + \\sin{(C_{2})})^{2} = C_{2} + (A_{1} + \\sin{(C_{2})}) (r_{0} + \\sin{(C_{2})})", "derivation": "\\delta{(C_{2})} = \\int \\cos{(C_{2})} dC_{2} and \\delta{(C_{2})} = r_{0} + \\sin{(C_{2})} and r_{0} + \\sin{(C_{2})} = \\int \\cos{(C_{2})} dC_{2} and (r_{0} + \\sin{(C_{2})})^{2} = (r_{0} + \\sin{(C_{2})}) \\int \\cos{(C_{2})} dC_{2} and C_{2} + (r_{0} + \\sin{(C_{2})})^{2} = C_{2} + (r_{0} + \\sin{(C_{2})}) \\int \\cos{(C_{2})} dC_{2} and C_{2} + (r_{0} + \\sin{(C_{2})})^{2} = C_{2} + (A_{1} + \\sin{(C_{2})}) (r_{0} + \\sin{(C_{2})})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('C_2', commutative=True)), Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\delta')(Symbol('C_2', commutative=True)), Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["times", 3, "Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True)))"], "Equality(Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integer(2)), Mul(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"], [["add", 4, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integer(2))), Add(Symbol('C_2', commutative=True), Mul(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integral(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('C_2', commutative=True), Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))), Integer(2))), Add(Symbol('C_2', commutative=True), Mul(Add(Symbol('A_1', commutative=True), sin(Symbol('C_2', commutative=True))), Add(Symbol('r_0', commutative=True), sin(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(p)} = \\int \\sin{(p)} dp, then derive \\hat{H}{(p)} = \\mathbf{p} - \\cos{(p)}, then obtain n_{1} - \\cos{(p)} + \\frac{\\partial}{\\partial p} (\\mathbf{p} - n_{1} + \\cos{(p)}) = n_{1} - \\cos{(p)} + \\frac{d}{d p} \\cos{(p)}", "derivation": "\\hat{H}{(p)} = \\int \\sin{(p)} dp and \\hat{H}{(p)} = \\mathbf{p} - \\cos{(p)} and \\mathbf{p} - \\cos{(p)} = \\int \\sin{(p)} dp and \\mathbf{p} - \\hat{H}{(p)} = - \\hat{H}{(p)} + \\cos{(p)} + \\int \\sin{(p)} dp and \\frac{\\partial}{\\partial p} (\\mathbf{p} - \\hat{H}{(p)}) = \\frac{d}{d p} (- \\hat{H}{(p)} + \\cos{(p)} + \\int \\sin{(p)} dp) and \\frac{\\partial}{\\partial p} (\\mathbf{p} - \\int \\sin{(p)} dp) = \\frac{d}{d p} \\cos{(p)} and \\frac{\\partial}{\\partial p} (\\mathbf{p} - \\int \\sin{(p)} dp) + \\int \\sin{(p)} dp = \\frac{d}{d p} \\cos{(p)} + \\int \\sin{(p)} dp and n_{1} - \\cos{(p)} + \\frac{\\partial}{\\partial p} (\\mathbf{p} - n_{1} + \\cos{(p)}) = n_{1} - \\cos{(p)} + \\frac{d}{d p} \\cos{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), cos(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["minus", 3, "Add(Function('\\\\hat{H}')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True))))"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True)), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["differentiate", 4, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True)), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 6, "Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('n_1', commutative=True), Mul(Integer(-1), cos(Symbol('p', commutative=True))), Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)), cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Symbol('n_1', commutative=True), Mul(Integer(-1), cos(Symbol('p', commutative=True))), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(\\hbar)} = \\cos{(\\hbar)}, then derive \\int \\theta{(\\hbar)} d\\hbar = x^\\prime + \\sin{(\\hbar)}, then obtain (x^\\prime \\int \\theta{(\\hbar)} d\\hbar - x^\\prime)^{x^\\prime} = (x^\\prime (x^\\prime + \\sin{(\\hbar)}) - x^\\prime)^{x^\\prime}", "derivation": "\\theta{(\\hbar)} = \\cos{(\\hbar)} and \\int \\theta{(\\hbar)} d\\hbar = \\int \\cos{(\\hbar)} d\\hbar and \\int \\theta{(\\hbar)} d\\hbar = x^\\prime + \\sin{(\\hbar)} and x^\\prime \\int \\theta{(\\hbar)} d\\hbar = x^\\prime (x^\\prime + \\sin{(\\hbar)}) and x^\\prime \\int \\theta{(\\hbar)} d\\hbar - x^\\prime = x^\\prime (x^\\prime + \\sin{(\\hbar)}) - x^\\prime and (x^\\prime \\int \\theta{(\\hbar)} d\\hbar - x^\\prime)^{x^\\prime} = (x^\\prime (x^\\prime + \\sin{(\\hbar)}) - x^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["times", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Integral(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\hbar', commutative=True)))))"], [["minus", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Mul(Symbol('x^\\\\prime', commutative=True), Integral(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], [["power", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('x^\\\\prime', commutative=True), Integral(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then obtain (2 \\operatorname{r_{0}}{(\\Psi_{nl})} + e^{\\Psi_{nl}})^{\\Psi_{nl}} = (\\operatorname{r_{0}}{(\\Psi_{nl})} + 2 e^{\\Psi_{nl}})^{\\Psi_{nl}}", "derivation": "\\operatorname{r_{0}}{(\\Psi_{nl})} = e^{\\Psi_{nl}} and 2 \\operatorname{r_{0}}{(\\Psi_{nl})} = \\operatorname{r_{0}}{(\\Psi_{nl})} + e^{\\Psi_{nl}} and 2 \\operatorname{r_{0}}{(\\Psi_{nl})} + e^{\\Psi_{nl}} = \\operatorname{r_{0}}{(\\Psi_{nl})} + 2 e^{\\Psi_{nl}} and (2 \\operatorname{r_{0}}{(\\Psi_{nl})} + e^{\\Psi_{nl}})^{\\Psi_{nl}} = (\\operatorname{r_{0}}{(\\Psi_{nl})} + 2 e^{\\Psi_{nl}})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["add", 1, "Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Integer(2), Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True))), exp(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["power", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True))), exp(Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Function('r_0')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{J}_f)} = e^{\\sin{(\\mathbf{J}_f)}} and \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = - \\hat{x}{(\\mathbf{J}_f)} + e^{\\sin{(\\mathbf{J}_f)}}, then derive \\frac{d}{d \\mathbf{J}_f} \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = 0, then obtain \\sin{(\\mathbf{J}_f)} \\frac{d}{d \\mathbf{J}_f} \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = 0", "derivation": "\\hat{x}{(\\mathbf{J}_f)} = e^{\\sin{(\\mathbf{J}_f)}} and \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = - \\hat{x}{(\\mathbf{J}_f)} + e^{\\sin{(\\mathbf{J}_f)}} and \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = 0 and \\frac{d}{d \\mathbf{J}_f} \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} 0 and \\frac{d}{d \\mathbf{J}_f} \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = 0 and \\sin{(\\mathbf{J}_f)} \\frac{d}{d \\mathbf{J}_f} \\operatorname{E_{\\lambda}}{(\\mathbf{J}_f)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True))), exp(sin(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(0))"], [["times", 5, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\hbar)} = \\log{(e^{\\hbar})}, then derive \\frac{d}{d \\hbar} \\operatorname{c_{0}}{(\\hbar)} - 1 = 0, then obtain \\int (\\frac{d}{d \\hbar} \\operatorname{c_{0}}{(\\hbar)} - 1)^{\\hbar} d\\hbar = \\int 0^{\\hbar} d\\hbar", "derivation": "\\operatorname{c_{0}}{(\\hbar)} = \\log{(e^{\\hbar})} and - \\hbar + \\operatorname{c_{0}}{(\\hbar)} = - \\hbar + \\log{(e^{\\hbar})} and \\frac{d}{d \\hbar} (- \\hbar + \\operatorname{c_{0}}{(\\hbar)}) = \\frac{d}{d \\hbar} (- \\hbar + \\log{(e^{\\hbar})}) and \\frac{d}{d \\hbar} \\operatorname{c_{0}}{(\\hbar)} - 1 = 0 and (\\frac{d}{d \\hbar} \\operatorname{c_{0}}{(\\hbar)} - 1)^{\\hbar} = 0^{\\hbar} and \\int (\\frac{d}{d \\hbar} \\operatorname{c_{0}}{(\\hbar)} - 1)^{\\hbar} d\\hbar = \\int 0^{\\hbar} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\hbar', commutative=True)), log(exp(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('c_0')(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(exp(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('c_0')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('c_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Derivative(Function('c_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Add(Derivative(Function('c_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(G,p)} = \\frac{G}{p} and \\operatorname{y^{\\prime}}{(G,p)} = \\int \\phi_{1}{(G,p)} dG, then obtain \\int (- p - \\int \\phi_{1}{(G,p)} dG) dp = \\int (- p - \\operatorname{y^{\\prime}}{(G,p)}) dp", "derivation": "\\phi_{1}{(G,p)} = \\frac{G}{p} and \\int \\phi_{1}{(G,p)} dG = \\int \\frac{G}{p} dG and - \\int \\phi_{1}{(G,p)} dG = - \\int \\frac{G}{p} dG and - p - \\int \\phi_{1}{(G,p)} dG = - p - \\int \\frac{G}{p} dG and \\operatorname{y^{\\prime}}{(G,p)} = \\int \\phi_{1}{(G,p)} dG and \\operatorname{y^{\\prime}}{(G,p)} = \\int \\frac{G}{p} dG and - p - \\int \\phi_{1}{(G,p)} dG = - p - \\operatorname{y^{\\prime}}{(G,p)} and \\int (- p - \\int \\phi_{1}{(G,p)} dG) dp = \\int (- p - \\operatorname{y^{\\prime}}{(G,p)}) dp", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Symbol('G', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('G', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True)))))"], [["minus", 3, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('G', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True))))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('y^{\\\\prime}')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Symbol('G', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('G', commutative=True), Symbol('p', commutative=True)))))"], [["integrate", 7, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\phi_1')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('G', commutative=True))))), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('G', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)} = e^{\\cos{(\\omega)}} and b{(\\omega)} = \\cos{(\\omega)}, then obtain \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{b{(\\omega)} + \\frac{e^{b{(\\omega)}}}{\\omega}} = \\frac{e^{b{(\\omega)}}}{b{(\\omega)} + \\frac{e^{b{(\\omega)}}}{\\omega}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)} = e^{\\cos{(\\omega)}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{\\omega} = \\frac{e^{\\cos{(\\omega)}}}{\\omega} and b{(\\omega)} = \\cos{(\\omega)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{\\omega} = \\frac{e^{b{(\\omega)}}}{\\omega} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)} = e^{b{(\\omega)}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{b{(\\omega)} + \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{\\omega}} = \\frac{e^{b{(\\omega)}}}{b{(\\omega)} + \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{\\omega}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\omega)}}{b{(\\omega)} + \\frac{e^{b{(\\omega)}}}{\\omega}} = \\frac{e^{b{(\\omega)}}}{b{(\\omega)} + \\frac{e^{b{(\\omega)}}}{\\omega}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(cos(Symbol('\\\\omega', commutative=True))))"], [["divide", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(cos(Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Function('b')(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(Function('b')(Symbol('\\\\omega', commutative=True))))"], [["divide", 5, "Add(Function('b')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Pow(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)))), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)))), Integer(-1)), exp(Function('b')(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Function('b')(Symbol('\\\\omega', commutative=True))))), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Function('b')(Symbol('\\\\omega', commutative=True))))), Integer(-1)), exp(Function('b')(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given h{(J,\\Psi)} = e^{\\Psi^{J}}, then derive e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} = \\Psi^{J} e^{2 \\Psi^{J}} \\log{(\\Psi)}, then obtain \\Psi^{J} e^{2 \\Psi^{J}} \\log{(\\Psi)} + 1 = e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} + 1", "derivation": "h{(J,\\Psi)} = e^{\\Psi^{J}} and \\frac{\\partial}{\\partial J} h{(J,\\Psi)} = \\frac{\\partial}{\\partial J} e^{\\Psi^{J}} and e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} = e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} e^{\\Psi^{J}} and e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} = \\Psi^{J} e^{2 \\Psi^{J}} \\log{(\\Psi)} and e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} + 1 = e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} e^{\\Psi^{J}} + 1 and \\Psi^{J} e^{2 \\Psi^{J}} \\log{(\\Psi)} + 1 = e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} e^{\\Psi^{J}} + 1 and \\Psi^{J} e^{2 \\Psi^{J}} \\log{(\\Psi)} + 1 = e^{\\Psi^{J}} \\frac{\\partial}{\\partial J} h{(J,\\Psi)} + 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["divide", 2, "exp(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))))"], "Equality(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)), exp(Mul(Integer(2), Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(1)), Add(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)), exp(Mul(Integer(2), Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Integer(1)), Add(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)), exp(Mul(Integer(2), Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Integer(1)), Add(Mul(exp(Pow(Symbol('\\\\Psi', commutative=True), Symbol('J', commutative=True))), Derivative(Function('h')(Symbol('J', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g})} = \\sin{(\\tilde{g})}, then obtain \\cos{(\\operatorname{g^{\\prime}_{\\varepsilon}}^{\\tilde{g}}{(\\tilde{g})} - \\sin{(\\tilde{g})})} = \\cos{(\\sin{(\\tilde{g})} - \\sin^{\\tilde{g}}{(\\tilde{g})})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\tilde{g}}{(\\tilde{g})} = \\sin^{\\tilde{g}}{(\\tilde{g})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\tilde{g}}{(\\tilde{g})} - \\sin{(\\tilde{g})} = - \\sin{(\\tilde{g})} + \\sin^{\\tilde{g}}{(\\tilde{g})} and \\cos{(\\operatorname{g^{\\prime}_{\\varepsilon}}^{\\tilde{g}}{(\\tilde{g})} - \\sin{(\\tilde{g})})} = \\cos{(\\sin{(\\tilde{g})} - \\sin^{\\tilde{g}}{(\\tilde{g})})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 2, "sin(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True))), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))))"], [["cos", 3], "Equality(cos(Add(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True))))), cos(Add(sin(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))))))"]]}, {"prompt": "Given \\dot{z}{(f_{E})} = \\cos{(f_{E})} and \\operatorname{V_{\\mathbf{B}}}{(f_{E})} = \\frac{\\cos{(\\frac{d}{d f_{E}} \\dot{z}{(f_{E})})}}{\\cos{(\\frac{d}{d f_{E}} \\cos{(f_{E})})}}, then obtain \\operatorname{V_{\\mathbf{B}}}{(f_{E})} = 1", "derivation": "\\dot{z}{(f_{E})} = \\cos{(f_{E})} and \\frac{d}{d f_{E}} \\dot{z}{(f_{E})} = \\frac{d}{d f_{E}} \\cos{(f_{E})} and \\cos{(\\frac{d}{d f_{E}} \\dot{z}{(f_{E})})} = \\cos{(\\frac{d}{d f_{E}} \\cos{(f_{E})})} and \\frac{\\cos{(\\frac{d}{d f_{E}} \\dot{z}{(f_{E})})}}{\\cos{(\\frac{d}{d f_{E}} \\cos{(f_{E})})}} = 1 and \\operatorname{V_{\\mathbf{B}}}{(f_{E})} = \\frac{\\cos{(\\frac{d}{d f_{E}} \\dot{z}{(f_{E})})}}{\\cos{(\\frac{d}{d f_{E}} \\cos{(f_{E})})}} and \\operatorname{V_{\\mathbf{B}}}{(f_{E})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\dot{z}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), cos(Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["divide", 3, "cos(Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], "Equality(Mul(cos(Derivative(Function('\\\\dot{z}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(cos(Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f_E', commutative=True)), Mul(cos(Derivative(Function('\\\\dot{z}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(cos(Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f_E', commutative=True)), Integer(1))"]]}, {"prompt": "Given A{(x)} = \\cos{(e^{x})} and \\mathbf{v}{(x)} = A{(x)} e^{x}, then obtain \\mathbf{v}^{x}{(x)} = (e^{x} \\cos{(e^{x})})^{x}", "derivation": "A{(x)} = \\cos{(e^{x})} and A{(x)} e^{x} = e^{x} \\cos{(e^{x})} and \\mathbf{v}{(x)} = A{(x)} e^{x} and \\mathbf{v}{(x)} = e^{x} \\cos{(e^{x})} and \\mathbf{v}^{x}{(x)} = (e^{x} \\cos{(e^{x})})^{x}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('x', commutative=True)), cos(exp(Symbol('x', commutative=True))))"], [["times", 1, "exp(Symbol('x', commutative=True))"], "Equality(Mul(Function('A')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True))), Mul(exp(Symbol('x', commutative=True)), cos(exp(Symbol('x', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('x', commutative=True)), Mul(Function('A')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('x', commutative=True)), Mul(exp(Symbol('x', commutative=True)), cos(exp(Symbol('x', commutative=True)))))"], [["power", 4, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Mul(exp(Symbol('x', commutative=True)), cos(exp(Symbol('x', commutative=True)))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\ddot{x})} = \\sin{(\\ddot{x})}, then obtain 2 = 1 + \\frac{\\sin{(\\ddot{x})}}{\\eta{(\\ddot{x})}}", "derivation": "\\eta{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\eta^{2}{(\\ddot{x})} = \\eta{(\\ddot{x})} \\sin{(\\ddot{x})} and 1 = \\frac{\\sin{(\\ddot{x})}}{\\eta{(\\ddot{x})}} and 2 = 1 + \\frac{\\sin{(\\ddot{x})}}{\\eta{(\\ddot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 1, "Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Pow(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), Integer(2)), Mul(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 3, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('\\\\eta')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given x{(q,\\theta_1)} = \\theta_1^{q}, then obtain \\frac{x{(q,\\theta_1)}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} + (\\int \\theta_1^{q} d\\theta_1)^{2} = \\frac{\\theta_1^{q}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} + (\\int \\theta_1^{q} d\\theta_1)^{2}", "derivation": "x{(q,\\theta_1)} = \\theta_1^{q} and \\int x{(q,\\theta_1)} d\\theta_1 = \\int \\theta_1^{q} d\\theta_1 and \\frac{x{(q,\\theta_1)}}{(\\int \\theta_1^{q} d\\theta_1)^{2}} = \\frac{\\theta_1^{q}}{(\\int \\theta_1^{q} d\\theta_1)^{2}} and \\frac{x{(q,\\theta_1)}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} = \\frac{\\theta_1^{q}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} and \\frac{x{(q,\\theta_1)}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} + (\\int \\theta_1^{q} d\\theta_1)^{2} = \\frac{\\theta_1^{q}}{(\\int x{(q,\\theta_1)} d\\theta_1)^{2}} + (\\int \\theta_1^{q} d\\theta_1)^{2}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))"], "Equality(Mul(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Integral(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Pow(Integral(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))))"], [["add", 4, "Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))"], "Equality(Add(Mul(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Integral(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))), Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))), Add(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Pow(Integral(Function('x')(Symbol('q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-2))), Pow(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given L{(\\Psi_{nl},f)} = f^{\\Psi_{nl}}, then obtain 2 f + \\frac{((f^{\\Psi_{nl}})^{f} + L^{f}{(\\Psi_{nl},f)})^{2} (f^{\\Psi_{nl}})^{- f}}{2} = 2 f + 2 (f^{\\Psi_{nl}})^{f}", "derivation": "L{(\\Psi_{nl},f)} = f^{\\Psi_{nl}} and L^{f}{(\\Psi_{nl},f)} = (f^{\\Psi_{nl}})^{f} and (f^{\\Psi_{nl}})^{f} + L^{f}{(\\Psi_{nl},f)} = 2 (f^{\\Psi_{nl}})^{f} and ((f^{\\Psi_{nl}})^{f} + L^{f}{(\\Psi_{nl},f)})^{2} = 4 (f^{\\Psi_{nl}})^{2 f} and \\frac{((f^{\\Psi_{nl}})^{f} + L^{f}{(\\Psi_{nl},f)})^{2} (f^{\\Psi_{nl}})^{- f}}{2} = 2 (f^{\\Psi_{nl}})^{f} and 2 f + \\frac{((f^{\\Psi_{nl}})^{f} + L^{f}{(\\Psi_{nl},f)})^{2} (f^{\\Psi_{nl}})^{- f}}{2} = 2 f + 2 (f^{\\Psi_{nl}})^{f}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Add(Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Integer(2)), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(2), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True))))"], [["add", 5, "Mul(Integer(2), Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Rational(1, 2), Pow(Add(Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Integer(2)), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True))))), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(2), Pow(Pow(Symbol('f', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(\\theta_2,C_{2})} = \\frac{C_{2}}{\\theta_2} and E{(\\theta_2,C_{2})} = \\dot{x}{(\\theta_2,C_{2})} + \\frac{1}{\\theta_2}, then obtain E{(\\theta_2,C_{2})} = \\frac{C_{2}}{\\theta_2} + \\frac{1}{\\theta_2}", "derivation": "\\dot{x}{(\\theta_2,C_{2})} = \\frac{C_{2}}{\\theta_2} and \\dot{x}{(\\theta_2,C_{2})} + \\frac{1}{\\theta_2} = \\frac{C_{2}}{\\theta_2} + \\frac{1}{\\theta_2} and E{(\\theta_2,C_{2})} = \\dot{x}{(\\theta_2,C_{2})} + \\frac{1}{\\theta_2} and E{(\\theta_2,C_{2})} = \\frac{C_{2}}{\\theta_2} + \\frac{1}{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('C_2', commutative=True)), Add(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"]]}, {"prompt": "Given h{(Z,p)} = \\log{(Z + p)}, then derive \\frac{\\partial}{\\partial Z} h{(Z,p)} - 2 = -2 + \\frac{1}{Z + p}, then obtain ((\\frac{\\partial}{\\partial Z} h{(Z,p)} - 2)^{Z})^{p} = ((\\frac{\\partial}{\\partial Z} \\log{(Z + p)} - 2)^{Z})^{p}", "derivation": "h{(Z,p)} = \\log{(Z + p)} and - Z + h{(Z,p)} = - Z + \\log{(Z + p)} and - 2 Z + h{(Z,p)} = - 2 Z + \\log{(Z + p)} and \\frac{\\partial}{\\partial Z} (- 2 Z + h{(Z,p)}) = \\frac{\\partial}{\\partial Z} (- 2 Z + \\log{(Z + p)}) and \\frac{\\partial}{\\partial Z} h{(Z,p)} - 2 = -2 + \\frac{1}{Z + p} and \\frac{\\partial}{\\partial Z} \\log{(Z + p)} - 2 = -2 + \\frac{1}{Z + p} and (\\frac{\\partial}{\\partial Z} h{(Z,p)} - 2)^{Z} = (-2 + \\frac{1}{Z + p})^{Z} and (\\frac{\\partial}{\\partial Z} h{(Z,p)} - 2)^{Z} = (\\frac{\\partial}{\\partial Z} \\log{(Z + p)} - 2)^{Z} and ((\\frac{\\partial}{\\partial Z} h{(Z,p)} - 2)^{Z})^{p} = ((\\frac{\\partial}{\\partial Z} \\log{(Z + p)} - 2)^{Z})^{p}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True))))"], [["minus", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Add(Integer(-2), Pow(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Add(Integer(-2), Pow(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Symbol('Z', commutative=True)), Pow(Add(Integer(-2), Pow(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Symbol('Z', commutative=True)), Pow(Add(Derivative(log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Symbol('Z', commutative=True)))"], [["power", 8, "Symbol('p', commutative=True)"], "Equality(Pow(Pow(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Symbol('Z', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(Add(Derivative(log(Add(Symbol('Z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-2)), Symbol('Z', commutative=True)), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\Psi_{nl} v_{t}, then derive \\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\Psi_{nl}, then obtain \\int \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} d\\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\int \\Psi_{nl} v_{t} d\\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})}", "derivation": "\\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\Psi_{nl} v_{t} and \\int \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} d\\Psi_{nl} = \\int \\Psi_{nl} v_{t} d\\Psi_{nl} and \\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\frac{\\partial}{\\partial v_{t}} \\Psi_{nl} v_{t} and \\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\Psi_{nl} and \\int \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} d\\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})} = \\int \\Psi_{nl} v_{t} d\\frac{\\partial}{\\partial v_{t}} \\hat{\\mathbf{x}}{(\\Psi_{nl},v_{t})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\rho_b)} = \\sin{(e^{\\rho_b})}, then obtain \\frac{4 \\Psi_{nl}^{2}{(\\rho_b)}}{\\sin^{2}{(e^{\\rho_b})}} = \\frac{(\\Psi_{nl}{(\\rho_b)} + \\sin{(e^{\\rho_b})})^{2}}{\\sin^{2}{(e^{\\rho_b})}}", "derivation": "\\Psi_{nl}{(\\rho_b)} = \\sin{(e^{\\rho_b})} and 2 \\Psi_{nl}{(\\rho_b)} = \\Psi_{nl}{(\\rho_b)} + \\sin{(e^{\\rho_b})} and \\frac{2 \\Psi_{nl}{(\\rho_b)}}{\\sin{(e^{\\rho_b})}} = \\frac{\\Psi_{nl}{(\\rho_b)} + \\sin{(e^{\\rho_b})}}{\\sin{(e^{\\rho_b})}} and \\frac{4 \\Psi_{nl}^{2}{(\\rho_b)}}{\\sin^{2}{(e^{\\rho_b})}} = \\frac{(\\Psi_{nl}{(\\rho_b)} + \\sin{(e^{\\rho_b})})^{2}}{\\sin^{2}{(e^{\\rho_b})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), sin(exp(Symbol('\\\\rho_b', commutative=True))))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), sin(exp(Symbol('\\\\rho_b', commutative=True)))))"], [["divide", 2, "sin(exp(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), Pow(sin(exp(Symbol('\\\\rho_b', commutative=True))), Integer(-1))), Mul(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), sin(exp(Symbol('\\\\rho_b', commutative=True)))), Pow(sin(exp(Symbol('\\\\rho_b', commutative=True))), Integer(-1))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Pow(sin(exp(Symbol('\\\\rho_b', commutative=True))), Integer(-2))), Mul(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\rho_b', commutative=True)), sin(exp(Symbol('\\\\rho_b', commutative=True)))), Integer(2)), Pow(sin(exp(Symbol('\\\\rho_b', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{E}{(g)} = e^{\\cos{(g)}}, then derive \\frac{\\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{(\\sin^{2}{(g)} - \\cos{(g)}) e^{\\cos{(g)}}}{\\cos{(g)}}, then obtain \\frac{\\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{(\\sin^{2}{(g)} - \\cos{(g)}) \\mathbf{E}{(g)}}{\\cos{(g)}}", "derivation": "\\mathbf{E}{(g)} = e^{\\cos{(g)}} and \\frac{\\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{e^{\\cos{(g)}}}{\\cos{(g)}} and \\frac{d}{d g} \\mathbf{E}{(g)} = \\frac{d}{d g} e^{\\cos{(g)}} and \\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)} = \\frac{d^{2}}{d g^{2}} e^{\\cos{(g)}} and \\frac{\\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{\\frac{d^{2}}{d g^{2}} e^{\\cos{(g)}}}{\\cos{(g)}} and \\frac{\\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{(\\sin^{2}{(g)} - \\cos{(g)}) e^{\\cos{(g)}}}{\\cos{(g)}} and \\frac{\\frac{d^{2}}{d g^{2}} \\mathbf{E}{(g)}}{\\cos{(g)}} = \\frac{(\\sin^{2}{(g)} - \\cos{(g)}) \\mathbf{E}{(g)}}{\\cos{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), exp(cos(Symbol('g', commutative=True))))"], [["divide", 1, "cos(Symbol('g', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), Mul(exp(cos(Symbol('g', commutative=True))), Pow(cos(Symbol('g', commutative=True)), Integer(-1))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))), Derivative(exp(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(2))))"], [["times", 4, "Pow(cos(Symbol('g', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2)))), Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Derivative(exp(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2)))), Mul(Add(Pow(sin(Symbol('g', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('g', commutative=True)))), exp(cos(Symbol('g', commutative=True))), Pow(cos(Symbol('g', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2)))), Mul(Add(Pow(sin(Symbol('g', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('g', commutative=True)))), Function('\\\\mathbf{E}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(v_{t})} = \\cos{(v_{t})}, then obtain 1 = - \\sin{(v_{t})} - \\frac{d}{d v_{t}} \\hat{H}_{\\lambda}{(v_{t})} + 1", "derivation": "\\hat{H}_{\\lambda}{(v_{t})} = \\cos{(v_{t})} and \\frac{d}{d v_{t}} \\hat{H}_{\\lambda}{(v_{t})} = \\frac{d}{d v_{t}} \\cos{(v_{t})} and \\frac{d}{d v_{t}} \\hat{H}_{\\lambda}{(v_{t})} + 1 = \\frac{d}{d v_{t}} \\cos{(v_{t})} + 1 and 1 = - \\frac{d}{d v_{t}} \\hat{H}_{\\lambda}{(v_{t})} + \\frac{d}{d v_{t}} \\cos{(v_{t})} + 1 and 1 = - \\sin{(v_{t})} - \\frac{d}{d v_{t}} \\hat{H}_{\\lambda}{(v_{t})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)))"], [["minus", 3, "Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))"], "Equality(Integer(1), Add(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\varphi^{*}{(A_{2},f^{*})} = \\log{(A_{2} + f^{*})}, then derive \\frac{\\partial}{\\partial f^{*}} \\varphi^{*}{(A_{2},f^{*})} = \\frac{1}{A_{2} + f^{*}}, then obtain \\frac{\\partial}{\\partial f^{*}} \\log{(A_{2} + f^{*})} = \\frac{1}{A_{2} + f^{*}}", "derivation": "\\varphi^{*}{(A_{2},f^{*})} = \\log{(A_{2} + f^{*})} and \\frac{\\partial}{\\partial f^{*}} \\varphi^{*}{(A_{2},f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\log{(A_{2} + f^{*})} and \\frac{\\partial}{\\partial f^{*}} \\varphi^{*}{(A_{2},f^{*})} = \\frac{1}{A_{2} + f^{*}} and \\frac{\\partial}{\\partial f^{*}} \\log{(A_{2} + f^{*})} = \\frac{1}{A_{2} + f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True)), log(Add(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(log(Add(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Add(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Add(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Add(Symbol('A_2', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\phi{(\\tilde{g})} = \\sin{(\\cos{(\\tilde{g})})} and \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} = \\hat{p}_0^{\\sigma_x}, then obtain - E_{n} + \\phi{(\\tilde{g})} + \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} = - E_{n} + \\hat{p}_0^{\\sigma_x} + \\phi{(\\tilde{g})}", "derivation": "\\phi{(\\tilde{g})} = \\sin{(\\cos{(\\tilde{g})})} and \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} = \\hat{p}_0^{\\sigma_x} and \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} + \\sin{(\\cos{(\\tilde{g})})} = \\hat{p}_0^{\\sigma_x} + \\sin{(\\cos{(\\tilde{g})})} and \\phi{(\\tilde{g})} + \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} = \\hat{p}_0^{\\sigma_x} + \\phi{(\\tilde{g})} and - E_{n} + \\phi{(\\tilde{g})} + \\varepsilon_{0}{(\\hat{p}_0,\\sigma_x)} = - E_{n} + \\hat{p}_0^{\\sigma_x} + \\phi{(\\tilde{g})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi')(Symbol('\\\\tilde{g}', commutative=True)), sin(cos(Symbol('\\\\tilde{g}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["add", 2, "sin(cos(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)), sin(cos(Symbol('\\\\tilde{g}', commutative=True)))), Add(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)), sin(cos(Symbol('\\\\tilde{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\phi')(Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\phi')(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 4, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\phi')(Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\phi')(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\rho_b)} = e^{\\rho_b}, then derive e^{- \\rho_b} \\frac{d}{d \\rho_b} \\operatorname{f_{E}}{(\\rho_b)} = 1, then obtain \\int e^{- \\rho_b} \\frac{d}{d \\rho_b} e^{\\rho_b} d\\rho_b = \\int 1 d\\rho_b", "derivation": "\\operatorname{f_{E}}{(\\rho_b)} = e^{\\rho_b} and \\frac{d}{d \\rho_b} \\operatorname{f_{E}}{(\\rho_b)} = \\frac{d}{d \\rho_b} e^{\\rho_b} and \\frac{\\frac{d}{d \\rho_b} \\operatorname{f_{E}}{(\\rho_b)}}{\\frac{d}{d \\rho_b} e^{\\rho_b}} = 1 and e^{- \\rho_b} \\frac{d}{d \\rho_b} \\operatorname{f_{E}}{(\\rho_b)} = 1 and \\int e^{- \\rho_b} \\frac{d}{d \\rho_b} \\operatorname{f_{E}}{(\\rho_b)} d\\rho_b = \\int 1 d\\rho_b and \\int e^{- \\rho_b} \\frac{d}{d \\rho_b} e^{\\rho_b} d\\rho_b = \\int 1 d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('f_E')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Derivative(Function('f_E')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Integer(1))"], [["integrate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(exp(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Derivative(Function('f_E')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(exp(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given A{(v_{1})} = \\log{(v_{1})}, then derive \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} \\log{(v_{1})} - v_{1}, then obtain - v_{1} + \\int E_{x} dE_{x} + \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} A{(v_{1})} - 2 v_{1} + \\int E_{x} dE_{x}", "derivation": "A{(v_{1})} = \\log{(v_{1})} and \\int A{(v_{1})} dv_{1} = \\int \\log{(v_{1})} dv_{1} and \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} \\log{(v_{1})} - v_{1} and - v_{1} + \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} \\log{(v_{1})} - 2 v_{1} and - v_{1} + \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} A{(v_{1})} - 2 v_{1} and - v_{1} + \\int E_{x} dE_{x} + \\int A{(v_{1})} dv_{1} = E_{x} + v_{1} A{(v_{1})} - 2 v_{1} + \\int E_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('A')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["minus", 3, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Function('A')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('E_x', commutative=True), Mul(Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Function('A')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('E_x', commutative=True), Mul(Symbol('v_1', commutative=True), Function('A')(Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))))"], [["add", 5, "Integral(Symbol('E_x', commutative=True), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Symbol('E_x', commutative=True), Tuple(Symbol('E_x', commutative=True))), Integral(Function('A')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('E_x', commutative=True), Mul(Symbol('v_1', commutative=True), Function('A')(Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Integral(Symbol('E_x', commutative=True), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(c)} = \\int \\sin{(c)} dc and L{(\\mathbb{I},a^{\\dagger})} = \\mathbb{I} - a^{\\dagger}, then derive \\mathbf{D}{(c)} = F_{g} - \\cos{(c)}, then obtain F_{g} - \\mathbb{I} + L{(\\mathbb{I},a^{\\dagger})} - \\cos{(c)} = F_{g} - a^{\\dagger} - \\cos{(c)}", "derivation": "\\mathbf{D}{(c)} = \\int \\sin{(c)} dc and \\mathbf{D}{(c)} = F_{g} - \\cos{(c)} and \\int \\sin{(c)} dc = F_{g} - \\cos{(c)} and L{(\\mathbb{I},a^{\\dagger})} = \\mathbb{I} - a^{\\dagger} and - \\mathbb{I} + L{(\\mathbb{I},a^{\\dagger})} = - a^{\\dagger} and - \\mathbb{I} + L{(\\mathbb{I},a^{\\dagger})} + \\int \\sin{(c)} dc = - a^{\\dagger} + \\int \\sin{(c)} dc and F_{g} - \\mathbb{I} + L{(\\mathbb{I},a^{\\dagger})} - \\cos{(c)} = F_{g} - a^{\\dagger} - \\cos{(c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{D}')(Symbol('c', commutative=True)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 5, "Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\varepsilon_0,f)} = f + \\sin{(\\varepsilon_0)}, then obtain (\\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)})^{f} \\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)} = ((f + \\sin{(\\varepsilon_0)})^{\\varepsilon_0})^{f} \\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)}", "derivation": "\\mathbf{s}{(\\varepsilon_0,f)} = f + \\sin{(\\varepsilon_0)} and \\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)} = (f + \\sin{(\\varepsilon_0)})^{\\varepsilon_0} and (\\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)})^{f} = ((f + \\sin{(\\varepsilon_0)})^{\\varepsilon_0})^{f} and (\\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)})^{f} \\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)} = ((f + \\sin{(\\varepsilon_0)})^{\\varepsilon_0})^{f} \\mathbf{s}^{\\varepsilon_0}{(\\varepsilon_0,f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Add(Symbol('f', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('f', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Add(Symbol('f', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('f', commutative=True)))"], [["times", 3, "Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Pow(Add(Symbol('f', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given f{(\\chi)} = e^{\\chi} and y{(\\chi)} = e^{- \\chi}, then derive \\int (\\chi + y{(\\chi)} e^{\\chi}) d\\chi = L + \\frac{\\chi^{2}}{2} + \\chi, then obtain L + \\frac{\\chi^{2}}{2} + \\chi = \\int (\\chi + 1) d\\chi", "derivation": "f{(\\chi)} = e^{\\chi} and f{(\\chi)} e^{- \\chi} = 1 and y{(\\chi)} = e^{- \\chi} and f{(\\chi)} y{(\\chi)} = 1 and y{(\\chi)} e^{\\chi} = 1 and \\chi + y{(\\chi)} e^{\\chi} = \\chi + 1 and \\int (\\chi + y{(\\chi)} e^{\\chi}) d\\chi = \\int (\\chi + 1) d\\chi and \\int (\\chi + y{(\\chi)} e^{\\chi}) d\\chi = L + \\frac{\\chi^{2}}{2} + \\chi and L + \\frac{\\chi^{2}}{2} + \\chi = \\int (\\chi + 1) d\\chi", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('f')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('f')(Symbol('\\\\chi', commutative=True)), Function('y')(Symbol('\\\\chi', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('y')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))), Integer(1))"], [["minus", 5, "Mul(Integer(-1), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Function('y')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Integer(1)))"], [["integrate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\chi', commutative=True), Mul(Function('y')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Integral(Add(Symbol('\\\\chi', commutative=True), Mul(Function('y')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('L', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2))), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Add(Symbol('L', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2))), Symbol('\\\\chi', commutative=True)), Integral(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)} = \\log{(\\mathbf{p} x^\\prime)}, then obtain 3 \\mathbf{p} + (\\mathbf{p} + \\frac{\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)}}{\\log{(\\mathbf{p} x^\\prime)}})^{\\mathbf{p}} + 1 = 3 \\mathbf{p} + (\\mathbf{p} + 1)^{\\mathbf{p}} + 1", "derivation": "\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)} = \\log{(\\mathbf{p} x^\\prime)} and \\frac{\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)}}{\\log{(\\mathbf{p} x^\\prime)}} = 1 and \\mathbf{p} + \\frac{\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)}}{\\log{(\\mathbf{p} x^\\prime)}} = \\mathbf{p} + 1 and (\\mathbf{p} + \\frac{\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)}}{\\log{(\\mathbf{p} x^\\prime)}})^{\\mathbf{p}} = (\\mathbf{p} + 1)^{\\mathbf{p}} and 3 \\mathbf{p} + (\\mathbf{p} + \\frac{\\Psi_{\\lambda}{(\\mathbf{p},x^\\prime)}}{\\log{(\\mathbf{p} x^\\prime)}})^{\\mathbf{p}} + 1 = 3 \\mathbf{p} + (\\mathbf{p} + 1)^{\\mathbf{p}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["divide", 1, "log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integer(-1)))), Add(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], [["power", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integer(-1)))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 4, "Add(Mul(Integer(3), Symbol('\\\\mathbf{p}', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integer(-1)))), Symbol('\\\\mathbf{p}', commutative=True)), Integer(1)), Add(Mul(Integer(3), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)), Symbol('\\\\mathbf{p}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\eta^{\\prime}{(v_{1})} = \\sin{(\\log{(v_{1})})}, then obtain \\frac{d}{d v_{1}} \\eta^{\\prime}^{3}{(v_{1})} \\sin{(\\log{(v_{1})})} = \\frac{d}{d v_{1}} \\eta^{\\prime}^{2}{(v_{1})} \\sin^{2}{(\\log{(v_{1})})}", "derivation": "\\eta^{\\prime}{(v_{1})} = \\sin{(\\log{(v_{1})})} and \\eta^{\\prime}^{2}{(v_{1})} = \\eta^{\\prime}{(v_{1})} \\sin{(\\log{(v_{1})})} and \\eta^{\\prime}^{3}{(v_{1})} \\sin{(\\log{(v_{1})})} = \\eta^{\\prime}^{2}{(v_{1})} \\sin^{2}{(\\log{(v_{1})})} and \\frac{d}{d v_{1}} \\eta^{\\prime}^{3}{(v_{1})} \\sin{(\\log{(v_{1})})} = \\frac{d}{d v_{1}} \\eta^{\\prime}^{2}{(v_{1})} \\sin^{2}{(\\log{(v_{1})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), sin(log(Symbol('v_1', commutative=True))))"], [["times", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True))"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), Integer(2)), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), sin(log(Symbol('v_1', commutative=True)))))"], [["times", 2, "Mul(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), sin(log(Symbol('v_1', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), Integer(3)), sin(log(Symbol('v_1', commutative=True)))), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), Integer(2)), Pow(sin(log(Symbol('v_1', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), Integer(3)), sin(log(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True)), Integer(2)), Pow(sin(log(Symbol('v_1', commutative=True))), Integer(2))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(Z,L_{\\varepsilon})} = L_{\\varepsilon} Z and \\operatorname{P_{e}}{(Z,L_{\\varepsilon})} = \\int x{(Z,L_{\\varepsilon})} dZ, then obtain 1 = \\frac{\\operatorname{P_{e}}{(Z,L_{\\varepsilon})}}{\\int x{(Z,L_{\\varepsilon})} dZ}", "derivation": "x{(Z,L_{\\varepsilon})} = L_{\\varepsilon} Z and \\int x{(Z,L_{\\varepsilon})} dZ = \\int L_{\\varepsilon} Z dZ and 1 = \\frac{\\int L_{\\varepsilon} Z dZ}{\\int x{(Z,L_{\\varepsilon})} dZ} and \\operatorname{P_{e}}{(Z,L_{\\varepsilon})} = \\int x{(Z,L_{\\varepsilon})} dZ and \\operatorname{P_{e}}{(Z,L_{\\varepsilon})} = \\int L_{\\varepsilon} Z dZ and 1 = \\frac{\\operatorname{P_{e}}{(Z,L_{\\varepsilon})}}{\\int x{(Z,L_{\\varepsilon})} dZ}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["divide", 2, "Integral(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Pow(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('P_e')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(1), Mul(Function('P_e')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(t_{2})} = e^{t_{2}}, then derive Q + t_{2} = \\int \\frac{e^{t_{2}}}{\\operatorname{M_{E}}{(t_{2})}} dt_{2}, then obtain - t_{2} + \\int 1 dt_{2} = Q", "derivation": "\\operatorname{M_{E}}{(t_{2})} = e^{t_{2}} and 1 = \\frac{e^{t_{2}}}{\\operatorname{M_{E}}{(t_{2})}} and \\int 1 dt_{2} = \\int \\frac{e^{t_{2}}}{\\operatorname{M_{E}}{(t_{2})}} dt_{2} and - t_{2} + \\int 1 dt_{2} = - t_{2} + \\int \\frac{e^{t_{2}}}{\\operatorname{M_{E}}{(t_{2})}} dt_{2} and Q + t_{2} = \\int \\frac{e^{t_{2}}}{\\operatorname{M_{E}}{(t_{2})}} dt_{2} and - t_{2} + \\int 1 dt_{2} = Q", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["divide", 1, "Function('M_E')(Symbol('t_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), exp(Symbol('t_2', commutative=True))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["minus", 3, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Integral(Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('Q', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True)))), Symbol('Q', commutative=True))"]]}, {"prompt": "Given \\bar{\\h}{(J,\\mathbf{J})} = \\log{(J + \\mathbf{J})} and \\mathbb{I}{(Z,F_{H})} = \\log{(- F_{H} + Z)}, then obtain - (\\bar{\\h}^{J}{(J,\\mathbf{J})})^{J} = - (\\bar{\\h}^{J}{(J,\\mathbf{J})})^{J} - \\mathbb{I}{(Z,F_{H})} + \\log{(- F_{H} + Z)}", "derivation": "\\bar{\\h}{(J,\\mathbf{J})} = \\log{(J + \\mathbf{J})} and \\bar{\\h}^{J}{(J,\\mathbf{J})} = \\log{(J + \\mathbf{J})}^{J} and \\mathbb{I}{(Z,F_{H})} = \\log{(- F_{H} + Z)} and 0 = - \\mathbb{I}{(Z,F_{H})} + \\log{(- F_{H} + Z)} and - (\\log{(J + \\mathbf{J})}^{J})^{J} = - (\\log{(J + \\mathbf{J})}^{J})^{J} - \\mathbb{I}{(Z,F_{H})} + \\log{(- F_{H} + Z)} and - (\\bar{\\h}^{J}{(J,\\mathbf{J})})^{J} = - (\\bar{\\h}^{J}{(J,\\mathbf{J})})^{J} - \\mathbb{I}{(Z,F_{H})} + \\log{(- F_{H} + Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), log(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('J', commutative=True)), Pow(log(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('J', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), log(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('Z', commutative=True))))"], [["minus", 3, "Function('\\\\mathbb{I}')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True))), log(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('Z', commutative=True)))))"], [["minus", 4, "Pow(Pow(log(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Pow(log(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(log(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True))), log(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True))), log(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\omega{(u)} = \\log{(\\cos{(u)})} and \\mu{(J)} = \\log{(\\log{(J)})}, then obtain \\cos^{u}{(\\frac{\\mu{(J)} \\omega{(u)}}{\\cos{(u)}})} = \\cos^{u}{(\\frac{\\omega{(u)} \\log{(\\log{(J)})}}{\\cos{(u)}})}", "derivation": "\\omega{(u)} = \\log{(\\cos{(u)})} and \\mu{(J)} = \\log{(\\log{(J)})} and \\mu{(J)} \\log{(\\cos{(u)})} = \\log{(\\log{(J)})} \\log{(\\cos{(u)})} and \\frac{\\mu{(J)} \\log{(\\cos{(u)})}}{\\cos{(u)}} = \\frac{\\log{(\\log{(J)})} \\log{(\\cos{(u)})}}{\\cos{(u)}} and \\cos{(\\frac{\\mu{(J)} \\log{(\\cos{(u)})}}{\\cos{(u)}})} = \\cos{(\\frac{\\log{(\\log{(J)})} \\log{(\\cos{(u)})}}{\\cos{(u)}})} and \\cos{(\\frac{\\mu{(J)} \\omega{(u)}}{\\cos{(u)}})} = \\cos{(\\frac{\\omega{(u)} \\log{(\\log{(J)})}}{\\cos{(u)}})} and \\cos^{u}{(\\frac{\\mu{(J)} \\omega{(u)}}{\\cos{(u)}})} = \\cos^{u}{(\\frac{\\omega{(u)} \\log{(\\log{(J)})}}{\\cos{(u)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('u', commutative=True)), log(cos(Symbol('u', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True))))"], [["times", 2, "log(cos(Symbol('u', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('J', commutative=True)), log(cos(Symbol('u', commutative=True)))), Mul(log(log(Symbol('J', commutative=True))), log(cos(Symbol('u', commutative=True)))))"], [["divide", 3, "cos(Symbol('u', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('J', commutative=True)), log(cos(Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Mul(log(log(Symbol('J', commutative=True))), log(cos(Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1))))"], [["cos", 4], "Equality(cos(Mul(Function('\\\\mu')(Symbol('J', commutative=True)), log(cos(Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))), cos(Mul(log(log(Symbol('J', commutative=True))), log(cos(Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(cos(Mul(Function('\\\\mu')(Symbol('J', commutative=True)), Function('\\\\omega')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))), cos(Mul(Function('\\\\omega')(Symbol('u', commutative=True)), log(log(Symbol('J', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(cos(Mul(Function('\\\\mu')(Symbol('J', commutative=True)), Function('\\\\omega')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))), Symbol('u', commutative=True)), Pow(cos(Mul(Function('\\\\omega')(Symbol('u', commutative=True)), log(log(Symbol('J', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\Omega{(B)} = \\sin{(e^{B})}, then obtain - \\Omega{(B)} + e^{B} + 2 = - \\Omega{(B)} + e^{B} + 1 + \\frac{- e^{B} + \\sin{(e^{B})}}{\\Omega{(B)} - e^{B}}", "derivation": "\\Omega{(B)} = \\sin{(e^{B})} and \\Omega{(B)} - e^{B} = - e^{B} + \\sin{(e^{B})} and 1 = \\frac{- e^{B} + \\sin{(e^{B})}}{\\Omega{(B)} - e^{B}} and 2 = 1 + \\frac{- e^{B} + \\sin{(e^{B})}}{\\Omega{(B)} - e^{B}} and - \\Omega{(B)} + e^{B} + 2 = - \\Omega{(B)} + e^{B} + 1 + \\frac{- e^{B} + \\sin{(e^{B})}}{\\Omega{(B)} - e^{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('B', commutative=True)), sin(exp(Symbol('B', commutative=True))))"], [["minus", 1, "exp(Symbol('B', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), sin(exp(Symbol('B', commutative=True)))))"], [["divide", 2, "Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), sin(exp(Symbol('B', commutative=True))))))"], [["add", 3, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), sin(exp(Symbol('B', commutative=True)))))))"], [["minus", 4, "Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True)), Integer(2)), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True)), Integer(1), Mul(Pow(Add(Function('\\\\Omega')(Symbol('B', commutative=True)), Mul(Integer(-1), exp(Symbol('B', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), sin(exp(Symbol('B', commutative=True)))))))"]]}, {"prompt": "Given \\nabla{(C_{2})} = \\cos{(\\log{(C_{2})})}, then obtain \\int \\frac{\\nabla^{6}{(C_{2})}}{\\cos^{5}{(\\log{(C_{2})})}} dC_{2} = \\int \\frac{\\nabla^{3}{(C_{2})}}{\\cos^{2}{(\\log{(C_{2})})}} dC_{2}", "derivation": "\\nabla{(C_{2})} = \\cos{(\\log{(C_{2})})} and \\frac{\\nabla{(C_{2})}}{\\cos{(\\log{(C_{2})})}} = 1 and \\frac{\\nabla^{2}{(C_{2})}}{\\cos{(\\log{(C_{2})})}} = \\nabla{(C_{2})} and \\frac{\\nabla^{3}{(C_{2})}}{\\cos{(\\log{(C_{2})})}} = \\nabla^{2}{(C_{2})} and \\int \\frac{\\nabla^{2}{(C_{2})}}{\\cos{(\\log{(C_{2})})}} dC_{2} = \\int \\nabla{(C_{2})} dC_{2} and \\frac{\\nabla^{3}{(C_{2})}}{\\cos^{2}{(\\log{(C_{2})})}} = \\nabla{(C_{2})} and \\int \\frac{\\nabla^{6}{(C_{2})}}{\\cos^{5}{(\\log{(C_{2})})}} dC_{2} = \\int \\frac{\\nabla^{3}{(C_{2})}}{\\cos^{2}{(\\log{(C_{2})})}} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('C_2', commutative=True)), cos(log(Symbol('C_2', commutative=True))))"], [["divide", 1, "cos(log(Symbol('C_2', commutative=True)))"], "Equality(Mul(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Function('\\\\nabla')(Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(2)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-1))), Function('\\\\nabla')(Symbol('C_2', commutative=True)))"], [["times", 2, "Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(3)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-1))), Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(2)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-1))), Tuple(Symbol('C_2', commutative=True))), Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(3)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-2))), Function('\\\\nabla')(Symbol('C_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(6)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-5))), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Integer(3)), Pow(cos(log(Symbol('C_2', commutative=True))), Integer(-2))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given u{(\\mathbf{D})} = \\mathbf{D}, then derive \\int u{(\\mathbf{D})} d\\mathbf{D} = \\frac{\\mathbf{D}^{2}}{2} + \\phi_1, then obtain (\\int u{(\\mathbf{D})} d\\mathbf{D})^{2} = (\\int \\mathbf{D} d\\mathbf{D})^{2}", "derivation": "u{(\\mathbf{D})} = \\mathbf{D} and \\int u{(\\mathbf{D})} d\\mathbf{D} = \\int \\mathbf{D} d\\mathbf{D} and \\int u{(\\mathbf{D})} d\\mathbf{D} = \\frac{\\mathbf{D}^{2}}{2} + \\phi_1 and \\int \\mathbf{D} d\\mathbf{D} = \\frac{\\mathbf{D}^{2}}{2} + \\phi_1 and (\\int u{(\\mathbf{D})} d\\mathbf{D})^{2} = (\\frac{\\mathbf{D}^{2}}{2} + \\phi_1)^{2} and (\\int u{(\\mathbf{D})} d\\mathbf{D})^{2} = (\\int \\mathbf{D} d\\mathbf{D})^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('u')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Symbol('\\\\phi_1', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Integral(Function('u')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Symbol('\\\\phi_1', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(Function('u')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(2)), Pow(Integral(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{A}{(U)} = \\log{(U)} and \\theta_{1}{(a,U)} = a + \\log{(U)}, then obtain U + \\mathbf{J}_f = \\int \\frac{\\partial}{\\partial a} \\theta_{1}{(a,U)} dU", "derivation": "\\mathbf{A}{(U)} = \\log{(U)} and a + \\mathbf{A}{(U)} = a + \\log{(U)} and \\theta_{1}{(a,U)} = a + \\log{(U)} and \\frac{\\partial}{\\partial a} \\theta_{1}{(a,U)} = \\frac{\\partial}{\\partial a} (a + \\log{(U)}) and \\theta_{1}{(a,U)} = a + \\mathbf{A}{(U)} and \\frac{\\partial}{\\partial a} (a + \\mathbf{A}{(U)}) = \\frac{\\partial}{\\partial a} (a + \\log{(U)}) and \\frac{\\partial}{\\partial a} (a + \\mathbf{A}{(U)}) = \\frac{\\partial}{\\partial a} \\theta_{1}{(a,U)} and \\int \\frac{\\partial}{\\partial a} (a + \\mathbf{A}{(U)}) dU = \\int \\frac{\\partial}{\\partial a} \\theta_{1}{(a,U)} dU and U + \\mathbf{J}_f = \\int \\frac{\\partial}{\\partial a} \\theta_{1}{(a,U)} dU", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('\\\\mathbf{A}')(Symbol('U', commutative=True))), Add(Symbol('a', commutative=True), log(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Add(Symbol('a', commutative=True), log(Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Add(Symbol('a', commutative=True), Function('\\\\mathbf{A}')(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Add(Symbol('a', commutative=True), Function('\\\\mathbf{A}')(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Add(Symbol('a', commutative=True), Function('\\\\mathbf{A}')(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["integrate", 7, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('a', commutative=True), Function('\\\\mathbf{A}')(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 8], "Equality(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Derivative(Function('\\\\theta_1')(Symbol('a', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\psi)} = \\log{(\\psi)}, then obtain \\frac{1}{\\psi^{2}} = \\frac{\\log{(\\psi)}}{\\psi^{2} \\operatorname{E_{n}}{(\\psi)}}", "derivation": "\\operatorname{E_{n}}{(\\psi)} = \\log{(\\psi)} and \\frac{\\operatorname{E_{n}}{(\\psi)}}{\\psi} = \\frac{\\log{(\\psi)}}{\\psi} and \\frac{\\operatorname{E_{n}}{(\\psi)}}{\\psi^{2}} = \\frac{\\log{(\\psi)}}{\\psi^{2}} and \\frac{1}{\\psi^{2}} = \\frac{\\log{(\\psi)}}{\\psi^{2} \\operatorname{E_{n}}{(\\psi)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('E_n')(Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Pow(Symbol('\\\\psi', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-2)), Function('E_n')(Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-2)), log(Symbol('\\\\psi', commutative=True))))"], [["divide", 3, "Function('E_n')(Symbol('\\\\psi', commutative=True))"], "Equality(Pow(Symbol('\\\\psi', commutative=True), Integer(-2)), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-2)), Pow(Function('E_n')(Symbol('\\\\psi', commutative=True)), Integer(-1)), log(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(B)} = \\cos{(B)}, then obtain \\int \\frac{d}{d B} \\frac{\\mathbf{F}{(B)}}{B} dB = \\int \\frac{d}{d B} \\frac{\\cos{(B)}}{B} dB", "derivation": "\\mathbf{F}{(B)} = \\cos{(B)} and \\frac{\\mathbf{F}{(B)}}{B} = \\frac{\\cos{(B)}}{B} and \\frac{d}{d B} \\frac{\\mathbf{F}{(B)}}{B} = \\frac{d}{d B} \\frac{\\cos{(B)}}{B} and \\int \\frac{d}{d B} \\frac{\\mathbf{F}{(B)}}{B} dB = \\int \\frac{d}{d B} \\frac{\\cos{(B)}}{B} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), cos(Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(x^\\prime,h)} = \\log{(h x^\\prime)}, then obtain \\frac{h x^\\prime (h x^\\prime + \\operatorname{a^{\\dagger}}{(x^\\prime,h)})}{\\operatorname{a^{\\dagger}}{(x^\\prime,h)}} = \\frac{h x^\\prime (h x^\\prime + \\log{(h x^\\prime)})}{\\operatorname{a^{\\dagger}}{(x^\\prime,h)}}", "derivation": "\\operatorname{a^{\\dagger}}{(x^\\prime,h)} = \\log{(h x^\\prime)} and h x^\\prime + \\operatorname{a^{\\dagger}}{(x^\\prime,h)} = h x^\\prime + \\log{(h x^\\prime)} and h x^\\prime (h x^\\prime + \\operatorname{a^{\\dagger}}{(x^\\prime,h)}) = h x^\\prime (h x^\\prime + \\log{(h x^\\prime)}) and \\frac{h x^\\prime (h x^\\prime + \\operatorname{a^{\\dagger}}{(x^\\prime,h)})}{\\operatorname{a^{\\dagger}}{(x^\\prime,h)}} = \\frac{h x^\\prime (h x^\\prime + \\log{(h x^\\prime)})}{\\operatorname{a^{\\dagger}}{(x^\\prime,h)}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)), log(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["times", 2, "Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)))), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))))))"], [["divide", 3, "Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))"], "Equality(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)), Integer(-1))), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Pow(Function('a^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given V{(\\phi_2,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\mathbf{v} + \\phi_2) and S{(\\phi_2,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\mathbf{v} + \\phi_2), then obtain \\int \\log{(S^{\\phi_2}{(\\phi_2,\\mathbf{v})})} d\\mathbf{v} = \\int \\log{(V^{\\phi_2}{(\\phi_2,\\mathbf{v})})} d\\mathbf{v}", "derivation": "V{(\\phi_2,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\mathbf{v} + \\phi_2) and S{(\\phi_2,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\mathbf{v} + \\phi_2) and S^{\\phi_2}{(\\phi_2,\\mathbf{v})} = (\\frac{\\partial}{\\partial \\mathbf{v}} (- \\mathbf{v} + \\phi_2))^{\\phi_2} and S^{\\phi_2}{(\\phi_2,\\mathbf{v})} = V^{\\phi_2}{(\\phi_2,\\mathbf{v})} and \\log{(S^{\\phi_2}{(\\phi_2,\\mathbf{v})})} = \\log{(V^{\\phi_2}{(\\phi_2,\\mathbf{v})})} and \\int \\log{(S^{\\phi_2}{(\\phi_2,\\mathbf{v})})} d\\mathbf{v} = \\int \\log{(V^{\\phi_2}{(\\phi_2,\\mathbf{v})})} d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Function('V')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["log", 4], "Equality(log(Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True))), log(Pow(Function('V')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(log(Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(log(Pow(Function('V')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + f_{\\mathbf{p}}, then obtain \\hat{p} \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p}", "derivation": "\\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + f_{\\mathbf{p}} and \\hat{p} \\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} (\\hat{p} + f_{\\mathbf{p}}) and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\hat{p} \\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\hat{p} (\\hat{p} + f_{\\mathbf{p}}) and \\hat{p} \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\Psi_{nl}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Symbol('\\\\hat{p}', commutative=True))"]]}, {"prompt": "Given \\mathbf{S}{(\\hat{X},A)} = A + \\hat{X}, then derive \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{S}{(\\hat{X},A)} = 1, then obtain \\frac{\\partial}{\\partial \\hat{X}} (A + \\hat{X}) = 1", "derivation": "\\mathbf{S}{(\\hat{X},A)} = A + \\hat{X} and \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{S}{(\\hat{X},A)} = \\frac{\\partial}{\\partial \\hat{X}} (A + \\hat{X}) and \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{S}{(\\hat{X},A)} = 1 and \\frac{\\partial}{\\partial \\hat{X}} (A + \\hat{X}) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given i{(r_{0})} = \\cos{(e^{r_{0}})}, then obtain r_{0} i^{2}{(r_{0})} - i^{2}{(r_{0})} = r_{0} \\cos^{2}{(e^{r_{0}})} - i^{2}{(r_{0})}", "derivation": "i{(r_{0})} = \\cos{(e^{r_{0}})} and r_{0} i{(r_{0})} = r_{0} \\cos{(e^{r_{0}})} and r_{0} i{(r_{0})} \\cos{(e^{r_{0}})} = r_{0} \\cos^{2}{(e^{r_{0}})} and r_{0} i^{2}{(r_{0})} = r_{0} i{(r_{0})} \\cos{(e^{r_{0}})} and r_{0} i^{2}{(r_{0})} = r_{0} \\cos^{2}{(e^{r_{0}})} and r_{0} i^{2}{(r_{0})} - i^{2}{(r_{0})} = r_{0} \\cos^{2}{(e^{r_{0}})} - i^{2}{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('r_0', commutative=True)), cos(exp(Symbol('r_0', commutative=True))))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('i')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(exp(Symbol('r_0', commutative=True)))))"], [["times", 2, "cos(exp(Symbol('r_0', commutative=True)))"], "Equality(Mul(Symbol('r_0', commutative=True), Function('i')(Symbol('r_0', commutative=True)), cos(exp(Symbol('r_0', commutative=True)))), Mul(Symbol('r_0', commutative=True), Pow(cos(exp(Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('r_0', commutative=True), Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2))), Mul(Symbol('r_0', commutative=True), Function('i')(Symbol('r_0', commutative=True)), cos(exp(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('r_0', commutative=True), Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2))), Mul(Symbol('r_0', commutative=True), Pow(cos(exp(Symbol('r_0', commutative=True))), Integer(2))))"], [["minus", 5, "Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2)))), Add(Mul(Symbol('r_0', commutative=True), Pow(cos(exp(Symbol('r_0', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(Function('i')(Symbol('r_0', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given S{(\\hbar)} = \\sin{(\\hbar)}, then derive - \\cos{(\\hbar)} + \\frac{d}{d \\hbar} S{(\\hbar)} - 1 = -1, then obtain (- \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\sin{(\\hbar)} - 1)^{\\hbar} - \\frac{d^{2}}{d \\hbar^{2}} S{(\\hbar)} = (-1)^{\\hbar} - \\frac{d^{2}}{d \\hbar^{2}} S{(\\hbar)}", "derivation": "S{(\\hbar)} = \\sin{(\\hbar)} and S{(\\hbar)} - \\sin{(\\hbar)} = 0 and \\frac{d}{d \\hbar} (S{(\\hbar)} - \\sin{(\\hbar)}) = \\frac{d}{d \\hbar} 0 and \\frac{d}{d \\hbar} (S{(\\hbar)} - \\sin{(\\hbar)}) - 1 = \\frac{d}{d \\hbar} 0 - 1 and - \\cos{(\\hbar)} + \\frac{d}{d \\hbar} S{(\\hbar)} - 1 = -1 and - \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\sin{(\\hbar)} - 1 = -1 and (- \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\sin{(\\hbar)} - 1)^{\\hbar} = (-1)^{\\hbar} and (- \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\sin{(\\hbar)} - 1)^{\\hbar} - \\frac{d^{2}}{d \\hbar^{2}} S{(\\hbar)} = (-1)^{\\hbar} - \\frac{d^{2}}{d \\hbar^{2}} S{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('S')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hbar', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('S')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Add(Function('S')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True))), Derivative(Function('S')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True))), Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"], [["power", 6, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True))), Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Pow(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], [["add", 7, "Mul(Integer(-1), Derivative(Function('S')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))"], "Equality(Add(Pow(Add(Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True))), Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Derivative(Function('S')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))), Add(Pow(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Derivative(Function('S')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))))"]]}, {"prompt": "Given W{(\\mathbf{f},M)} = M + \\mathbf{f}, then obtain M + 2 \\mathbf{f} + W{(\\mathbf{f},M)} - 1 = 2 M + 3 \\mathbf{f} - 1", "derivation": "W{(\\mathbf{f},M)} = M + \\mathbf{f} and \\mathbf{f} + W{(\\mathbf{f},M)} = M + 2 \\mathbf{f} and M + 2 \\mathbf{f} + W{(\\mathbf{f},M)} = 2 M + 3 \\mathbf{f} and M + 2 \\mathbf{f} + W{(\\mathbf{f},M)} - 1 = 2 M + 3 \\mathbf{f} - 1", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('M', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('W')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('M', commutative=True))), Add(Symbol('M', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 2, "Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Symbol('M', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Function('W')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(2), Symbol('M', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Symbol('M', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Function('W')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('M', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(k,M)} = M - k, then obtain k^{2} (M - k)^{2} \\hat{H}_{\\lambda}^{3}{(k,M)} = k^{2} (M - k)^{4} \\hat{H}_{\\lambda}{(k,M)}", "derivation": "\\hat{H}_{\\lambda}{(k,M)} = M - k and (M - k) \\hat{H}_{\\lambda}{(k,M)} = (M - k)^{2} and - k (M - k) \\hat{H}_{\\lambda}{(k,M)} = - k (M - k)^{2} and k^{2} (M - k)^{2} \\hat{H}_{\\lambda}^{2}{(k,M)} = k^{2} (M - k)^{4} and k^{2} (M - k)^{2} \\hat{H}_{\\lambda}^{3}{(k,M)} = k^{2} (M - k)^{4} \\hat{H}_{\\lambda}{(k,M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True)), Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["times", 1, "Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))"], "Equality(Mul(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True))), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('k', commutative=True), Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(2))))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(2)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(2)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True)), Integer(2))), Mul(Pow(Symbol('k', commutative=True), Integer(2)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(4))))"], [["times", 4, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(2)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(2)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True)), Integer(3))), Mul(Pow(Symbol('k', commutative=True), Integer(2)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(4)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\delta{(\\lambda)} = \\sin{(\\lambda)}, then obtain \\frac{\\delta^{2}{(\\lambda)} \\sin{(\\lambda)} - 2 \\sin^{3}{(\\lambda)}}{\\delta^{2}{(\\lambda)}} = - \\frac{\\sin^{3}{(\\lambda)}}{\\delta^{2}{(\\lambda)}}", "derivation": "\\delta{(\\lambda)} = \\sin{(\\lambda)} and \\delta{(\\lambda)} \\sin{(\\lambda)} = \\sin^{2}{(\\lambda)} and \\delta{(\\lambda)} \\sin^{2}{(\\lambda)} = \\sin^{3}{(\\lambda)} and \\delta^{2}{(\\lambda)} \\sin{(\\lambda)} = \\sin^{3}{(\\lambda)} and \\delta^{2}{(\\lambda)} \\sin{(\\lambda)} - \\delta{(\\lambda)} \\sin^{2}{(\\lambda)} - \\sin^{3}{(\\lambda)} = - \\delta{(\\lambda)} \\sin^{2}{(\\lambda)} and \\delta{(\\lambda)} \\sin^{2}{(\\lambda)} - 2 \\sin^{3}{(\\lambda)} = - \\sin^{3}{(\\lambda)} and \\delta^{2}{(\\lambda)} \\sin{(\\lambda)} - 2 \\sin^{3}{(\\lambda)} = - \\sin^{3}{(\\lambda)} and \\frac{\\delta^{2}{(\\lambda)} \\sin{(\\lambda)} - 2 \\sin^{3}{(\\lambda)}}{\\delta^{2}{(\\lambda)}} = - \\frac{\\sin^{3}{(\\lambda)}}{\\delta^{2}{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2)))"], [["times", 1, "Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))"], [["minus", 4, "Add(Mul(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))"], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(2))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3))))"], [["divide", 7, "Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(2))"], "Equality(Mul(Add(Mul(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3)))), Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(f_{E})} = \\log{(f_{E})} and \\hat{H}{(f_{E})} = f_{E}, then obtain f_{E} \\frac{d}{d f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f_{E})} + \\int f_{E} d\\hat{H}{(f_{E})} = f_{E} \\frac{d}{d f_{E}} \\log{(f_{E})} + \\int f_{E} d\\hat{H}{(f_{E})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(f_{E})} = \\log{(f_{E})} and \\hat{H}{(f_{E})} = f_{E} and \\frac{d}{d f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f_{E})} = \\frac{d}{d f_{E}} \\log{(f_{E})} and \\hat{H}{(f_{E})} \\frac{d}{d f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f_{E})} = \\hat{H}{(f_{E})} \\frac{d}{d f_{E}} \\log{(f_{E})} and f_{E} \\frac{d}{d f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f_{E})} = f_{E} \\frac{d}{d f_{E}} \\log{(f_{E})} and f_{E} \\frac{d}{d f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f_{E})} + \\int f_{E} d\\hat{H}{(f_{E})} = f_{E} \\frac{d}{d f_{E}} \\log{(f_{E})} + \\int f_{E} d\\hat{H}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["times", 3, "Function('\\\\hat{H}')(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('f_E', commutative=True)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Function('\\\\hat{H}')(Symbol('f_E', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('f_E', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Symbol('f_E', commutative=True), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["add", 5, "Integral(Symbol('f_E', commutative=True), Tuple(Function('\\\\hat{H}')(Symbol('f_E', commutative=True))))"], "Equality(Add(Mul(Symbol('f_E', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integral(Symbol('f_E', commutative=True), Tuple(Function('\\\\hat{H}')(Symbol('f_E', commutative=True))))), Add(Mul(Symbol('f_E', commutative=True), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integral(Symbol('f_E', commutative=True), Tuple(Function('\\\\hat{H}')(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(F_{H})} = \\cos{(F_{H})}, then derive \\frac{d}{d F_{H}} \\mathbf{P}{(F_{H})} = - \\sin{(F_{H})}, then obtain \\int - \\sin{(F_{H})} dF_{H} = \\mathbf{M} + \\cos{(F_{H})}", "derivation": "\\mathbf{P}{(F_{H})} = \\cos{(F_{H})} and \\frac{d}{d F_{H}} \\mathbf{P}{(F_{H})} = \\frac{d}{d F_{H}} \\cos{(F_{H})} and \\frac{d}{d F_{H}} \\mathbf{P}{(F_{H})} = - \\sin{(F_{H})} and - \\sin{(F_{H})} = \\frac{d}{d F_{H}} \\cos{(F_{H})} and \\int - \\sin{(F_{H})} dF_{H} = \\int \\frac{d}{d F_{H}} \\cos{(F_{H})} dF_{H} and \\int - \\sin{(F_{H})} dF_{H} = \\mathbf{M} + \\cos{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Derivative(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Derivative(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\theta{(r_{0})} = \\cos{(\\sin{(r_{0})})}, then obtain \\int ((2 \\theta{(r_{0})})^{r_{0}} - (\\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})})^{r_{0}} - \\frac{1}{2}) dr_{0} = \\int (- \\frac{1}{2}) dr_{0}", "derivation": "\\theta{(r_{0})} = \\cos{(\\sin{(r_{0})})} and 2 \\theta{(r_{0})} = \\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})} and (2 \\theta{(r_{0})})^{r_{0}} = (\\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})})^{r_{0}} and (2 \\theta{(r_{0})})^{r_{0}} - (\\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})})^{r_{0}} = 0 and (2 \\theta{(r_{0})})^{r_{0}} - (\\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})})^{r_{0}} - \\frac{1}{2} = - \\frac{1}{2} and \\int ((2 \\theta{(r_{0})})^{r_{0}} - (\\theta{(r_{0})} + \\cos{(\\sin{(r_{0})})})^{r_{0}} - \\frac{1}{2}) dr_{0} = \\int (- \\frac{1}{2}) dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True))))"], [["add", 1, "Function('\\\\theta')(Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta')(Symbol('r_0', commutative=True))), Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\theta')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Pow(Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)))"], [["minus", 3, "Pow(Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True))"], "Equality(Add(Pow(Mul(Integer(2), Function('\\\\theta')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)))), Integer(0))"], [["minus", 4, "Rational(1, 2)"], "Equality(Add(Pow(Mul(Integer(2), Function('\\\\theta')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True))), Rational(-1, 2)), Rational(-1, 2))"], [["integrate", 5, "Symbol('r_0', commutative=True)"], "Equality(Integral(Add(Pow(Mul(Integer(2), Function('\\\\theta')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Add(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True))), Rational(-1, 2)), Tuple(Symbol('r_0', commutative=True))), Integral(Rational(-1, 2), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given W{(\\mathbf{J},\\hat{H}_{\\lambda})} = \\mathbf{J}^{\\hat{H}_{\\lambda}}, then obtain (- \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{J}^{\\hat{H}_{\\lambda}} + \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})}) \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})} = 0", "derivation": "W{(\\mathbf{J},\\hat{H}_{\\lambda})} = \\mathbf{J}^{\\hat{H}_{\\lambda}} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{J}^{\\hat{H}_{\\lambda}} and - \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{J}^{\\hat{H}_{\\lambda}} + \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})} = 0 and (- \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{J}^{\\hat{H}_{\\lambda}} + \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})}) \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} W{(\\mathbf{J},\\hat{H}_{\\lambda})} = 0", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Derivative(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Integer(0))"], [["times", 3, "Derivative(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Derivative(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Derivative(Function('W')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(F_{g})} = e^{F_{g}}, then obtain \\int (\\operatorname{A_{1}}^{F_{g}}{(F_{g})})^{F_{g}} dF_{g} = \\int ((e^{F_{g}})^{F_{g}})^{F_{g}} dF_{g}", "derivation": "\\operatorname{A_{1}}{(F_{g})} = e^{F_{g}} and \\operatorname{A_{1}}^{F_{g}}{(F_{g})} = (e^{F_{g}})^{F_{g}} and (\\operatorname{A_{1}}^{F_{g}}{(F_{g})})^{F_{g}} = ((e^{F_{g}})^{F_{g}})^{F_{g}} and \\int (\\operatorname{A_{1}}^{F_{g}}{(F_{g})})^{F_{g}} dF_{g} = \\int ((e^{F_{g}})^{F_{g}})^{F_{g}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Pow(Function('A_1')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Pow(Pow(Function('A_1')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})} = - E_{\\lambda} + f_{\\mathbf{v}}^{u}, then obtain (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) \\int (f_{\\mathbf{v}} + \\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})}) du = (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) \\int (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) du", "derivation": "\\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})} = - E_{\\lambda} + f_{\\mathbf{v}}^{u} and f_{\\mathbf{v}} + \\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})} = - E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u} and \\int (f_{\\mathbf{v}} + \\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})}) du = \\int (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) du and (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) \\int (f_{\\mathbf{v}} + \\dot{y}{(u,f_{\\mathbf{v}},E_{\\lambda})}) du = (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) \\int (- E_{\\lambda} + f_{\\mathbf{v}} + f_{\\mathbf{v}}^{u}) du", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))))"], [["add", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))), Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbb{I},\\mathbf{S})} = \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} and W{(I)} = \\cos{(I)}, then obtain \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)} + \\frac{d}{d I} W{(I)} = - \\sin{(I)} + \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)}", "derivation": "\\Psi^{\\dagger}{(\\mathbb{I},\\mathbf{S})} = \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} and W{(I)} = \\cos{(I)} and \\frac{d}{d I} W{(I)} = \\frac{d}{d I} \\cos{(I)} and \\cos{(\\mathbf{S} + \\Psi^{\\dagger}{(\\mathbb{I},\\mathbf{S})} - 1)} + \\frac{d}{d I} W{(I)} = \\cos{(\\mathbf{S} + \\Psi^{\\dagger}{(\\mathbb{I},\\mathbf{S})} - 1)} + \\frac{d}{d I} \\cos{(I)} and \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)} + \\frac{d}{d I} W{(I)} = \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)} + \\frac{d}{d I} \\cos{(I)} and \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)} + \\frac{d}{d I} W{(I)} = - \\sin{(I)} + \\cos{(\\mathbf{S} + \\log{(\\frac{\\mathbb{I}}{\\mathbf{S}})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))))"], ["get_premise", "Equality(Function('W')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 3, "cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)))"], "Equality(Add(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Derivative(Function('W')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integer(-1))), Derivative(Function('W')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integer(-1))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integer(-1))), Derivative(Function('W')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('I', commutative=True))), cos(Add(Symbol('\\\\mathbf{S}', commutative=True), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integer(-1)))))"]]}, {"prompt": "Given b{(P_{e})} = \\cos{(P_{e})}, then obtain - b{(P_{e})} + \\frac{b^{P_{e}}{(P_{e})}}{P_{e} + b{(P_{e})}} = - b{(P_{e})} + \\frac{\\cos^{P_{e}}{(P_{e})}}{P_{e} + b{(P_{e})}}", "derivation": "b{(P_{e})} = \\cos{(P_{e})} and P_{e} + b{(P_{e})} = P_{e} + \\cos{(P_{e})} and b^{P_{e}}{(P_{e})} = \\cos^{P_{e}}{(P_{e})} and \\frac{b^{P_{e}}{(P_{e})}}{P_{e} + \\cos{(P_{e})}} = \\frac{\\cos^{P_{e}}{(P_{e})}}{P_{e} + \\cos{(P_{e})}} and - b{(P_{e})} + \\frac{b^{P_{e}}{(P_{e})}}{P_{e} + \\cos{(P_{e})}} = - b{(P_{e})} + \\frac{\\cos^{P_{e}}{(P_{e})}}{P_{e} + \\cos{(P_{e})}} and - b{(P_{e})} + \\frac{b^{P_{e}}{(P_{e})}}{P_{e} + b{(P_{e})}} = - b{(P_{e})} + \\frac{\\cos^{P_{e}}{(P_{e})}}{P_{e} + b{(P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('b')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(cos(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))"], [["divide", 3, "Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Integer(-1)), Pow(Function('b')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Pow(Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))))"], [["minus", 4, "Function('b')(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('P_e', commutative=True))), Mul(Pow(Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Integer(-1)), Pow(Function('b')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('P_e', commutative=True))), Mul(Pow(Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('P_e', commutative=True))), Mul(Pow(Add(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Integer(-1)), Pow(Function('b')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('P_e', commutative=True))), Mul(Pow(Add(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(k,\\sigma_p,\\omega)} = - \\omega + \\sigma_p + k, then obtain e^{\\omega - k + \\frac{\\operatorname{C_{1}}^{\\sigma_p}{(k,\\sigma_p,\\omega)}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}}} = e^{\\omega - k + \\frac{(- \\omega + \\sigma_p + k)^{\\sigma_p}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}}}", "derivation": "\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)} = - \\omega + \\sigma_p + k and \\operatorname{C_{1}}^{\\sigma_p}{(k,\\sigma_p,\\omega)} = (- \\omega + \\sigma_p + k)^{\\sigma_p} and \\frac{\\operatorname{C_{1}}^{\\sigma_p}{(k,\\sigma_p,\\omega)}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}} = \\frac{(- \\omega + \\sigma_p + k)^{\\sigma_p}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}} and \\omega - k + \\frac{\\operatorname{C_{1}}^{\\sigma_p}{(k,\\sigma_p,\\omega)}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}} = \\omega - k + \\frac{(- \\omega + \\sigma_p + k)^{\\sigma_p}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}} and e^{\\omega - k + \\frac{\\operatorname{C_{1}}^{\\sigma_p}{(k,\\sigma_p,\\omega)}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}}} = e^{\\omega - k + \\frac{(- \\omega + \\sigma_p + k)^{\\sigma_p}}{\\operatorname{C_{1}}{(k,\\sigma_p,\\omega)}}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 2, "Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True))"], "Equality(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))))"], [["exp", 4], "Equality(exp(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))), exp(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('C_1')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(m_{s},g)} = \\frac{\\partial}{\\partial g} g m_{s}, then derive 0 = m_{s} - \\operatorname{P_{g}}{(m_{s},g)}, then obtain \\operatorname{P_{g}}^{g}{(m_{s},g)} = \\operatorname{P_{g}}^{g}{(m_{s},g)} \\cos{(m_{s} - \\operatorname{P_{g}}{(m_{s},g)})}", "derivation": "\\operatorname{P_{g}}{(m_{s},g)} = \\frac{\\partial}{\\partial g} g m_{s} and 0 = - \\operatorname{P_{g}}{(m_{s},g)} + \\frac{\\partial}{\\partial g} g m_{s} and 0 = m_{s} - \\operatorname{P_{g}}{(m_{s},g)} and 1 = \\cos{(m_{s} - \\operatorname{P_{g}}{(m_{s},g)})} and \\operatorname{P_{g}}^{g}{(m_{s},g)} = \\operatorname{P_{g}}^{g}{(m_{s},g)} \\cos{(m_{s} - \\operatorname{P_{g}}{(m_{s},g)})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 1, "Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True))), Derivative(Mul(Symbol('g', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)))))"], [["cos", 3], "Equality(Integer(1), cos(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True))))))"], [["times", 4, "Pow(Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))"], "Equality(Pow(Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Mul(Pow(Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), cos(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Function('P_g')(Symbol('m_s', commutative=True), Symbol('g', commutative=True)))))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\varepsilon,U)} = U + \\varepsilon, then obtain \\hat{H} + \\int \\frac{U + \\varepsilon + \\Psi_{nl}{(\\varepsilon,U)}}{U + \\varepsilon} d\\varepsilon = \\int 2 d\\varepsilon", "derivation": "\\Psi_{nl}{(\\varepsilon,U)} = U + \\varepsilon and \\frac{\\Psi_{nl}{(\\varepsilon,U)}}{U + \\varepsilon} = 1 and 1 + \\frac{\\Psi_{nl}{(\\varepsilon,U)}}{U + \\varepsilon} = 2 and \\int (1 + \\frac{\\Psi_{nl}{(\\varepsilon,U)}}{U + \\varepsilon}) d\\varepsilon = \\int 2 d\\varepsilon and \\hat{H} + \\int \\frac{U + \\varepsilon + \\Psi_{nl}{(\\varepsilon,U)}}{U + \\varepsilon} d\\varepsilon = \\int 2 d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True))), Integer(1))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)))), Integer(2))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Integer(2), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Integral(Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(Integer(2), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} = \\sin{(\\mathbf{F} p)}, then obtain \\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} + \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{F}}{(p,\\mathbf{F})} + \\sin{(\\mathbf{F} p)} = \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{F}}{(p,\\mathbf{F})} + 2 \\sin{(\\mathbf{F} p)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} = \\sin{(\\mathbf{F} p)} and \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{F}}{(p,\\mathbf{F})} = \\sin^{\\mathbf{F}}{(\\mathbf{F} p)} and \\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} + \\sin^{\\mathbf{F}}{(\\mathbf{F} p)} = \\sin{(\\mathbf{F} p)} + \\sin^{\\mathbf{F}}{(\\mathbf{F} p)} and \\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} + \\sin{(\\mathbf{F} p)} + \\sin^{\\mathbf{F}}{(\\mathbf{F} p)} = 2 \\sin{(\\mathbf{F} p)} + \\sin^{\\mathbf{F}}{(\\mathbf{F} p)} and \\operatorname{f_{\\mathbf{v}}}{(p,\\mathbf{F})} + \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{F}}{(p,\\mathbf{F})} + \\sin{(\\mathbf{F} p)} = \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{F}}{(p,\\mathbf{F})} + 2 \\sin{(\\mathbf{F} p)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 1, "Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))), Add(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 3, "sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(2), sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)))), Pow(sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)))), Add(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), sin(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = \\cos{(\\hat{x} \\mathbf{B})}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})}, then obtain \\hat{x}^{2} \\sin^{2}{(\\hat{x} \\mathbf{B})} = - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})} \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(\\hat{x} \\mathbf{B})}", "derivation": "\\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = \\cos{(\\hat{x} \\mathbf{B})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(\\hat{x} \\mathbf{B})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})} and - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})} \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{C_{2}}{(\\hat{x},\\mathbf{B})} = - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})} \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(\\hat{x} \\mathbf{B})} and \\hat{x}^{2} \\sin^{2}{(\\hat{x} \\mathbf{B})} = - \\hat{x} \\sin{(\\hat{x} \\mathbf{B})} \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(\\hat{x} \\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), cos(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Derivative(Function('C_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Derivative(cos(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), Pow(sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Derivative(cos(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(J,\\rho)} = \\rho^{J} and \\mathbf{f}{(\\hat{x})} = e^{\\hat{x}}, then obtain \\frac{\\mathbf{f}{(\\hat{x})}}{\\rho + (\\rho^{J})^{J}} = \\frac{e^{\\hat{x}}}{\\rho + (\\rho^{J})^{J}}", "derivation": "Q{(J,\\rho)} = \\rho^{J} and Q^{J}{(J,\\rho)} = (\\rho^{J})^{J} and \\mathbf{f}{(\\hat{x})} = e^{\\hat{x}} and \\rho + Q^{J}{(J,\\rho)} = \\rho + (\\rho^{J})^{J} and \\frac{\\mathbf{f}{(\\hat{x})}}{\\rho + Q^{J}{(J,\\rho)}} = \\frac{e^{\\hat{x}}}{\\rho + Q^{J}{(J,\\rho)}} and \\frac{\\mathbf{f}{(\\hat{x})}}{\\rho + (\\rho^{J})^{J}} = \\frac{e^{\\hat{x}}}{\\rho + (\\rho^{J})^{J}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["add", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Pow(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('J', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\rho', commutative=True), Pow(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Pow(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Pow(Function('Q')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given B{(\\omega)} = \\sin{(\\omega)}, then obtain \\frac{\\omega^{2} B{(\\omega)}}{\\sin{(\\omega)}} = \\omega^{2}", "derivation": "B{(\\omega)} = \\sin{(\\omega)} and \\omega B{(\\omega)} = \\omega \\sin{(\\omega)} and \\omega^{2} B{(\\omega)} \\sin{(\\omega)} = \\omega^{2} \\sin^{2}{(\\omega)} and \\frac{\\omega^{2} B{(\\omega)}}{\\sin{(\\omega)}} = \\omega^{2}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('B')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], [["times", 2, "Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Function('B')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2))))"], [["divide", 3, "Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Function('B')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(E)} = e^{\\cos{(E)}}, then obtain \\frac{\\int \\operatorname{E_{n}}{(E)} dE}{\\operatorname{E_{n}}{(E)} \\cos{(E)}} = \\frac{\\int e^{\\cos{(E)}} dE}{\\operatorname{E_{n}}{(E)} \\cos{(E)}}", "derivation": "\\operatorname{E_{n}}{(E)} = e^{\\cos{(E)}} and \\int \\operatorname{E_{n}}{(E)} dE = \\int e^{\\cos{(E)}} dE and e^{- \\cos{(E)}} \\int \\operatorname{E_{n}}{(E)} dE = e^{- \\cos{(E)}} \\int e^{\\cos{(E)}} dE and \\frac{\\int \\operatorname{E_{n}}{(E)} dE}{\\operatorname{E_{n}}{(E)}} = \\frac{\\int e^{\\cos{(E)}} dE}{\\operatorname{E_{n}}{(E)}} and \\frac{\\int \\operatorname{E_{n}}{(E)} dE}{\\operatorname{E_{n}}{(E)} \\cos{(E)}} = \\frac{\\int e^{\\cos{(E)}} dE}{\\operatorname{E_{n}}{(E)} \\cos{(E)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('E', commutative=True)), exp(cos(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(exp(cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["divide", 2, "exp(cos(Symbol('E', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), cos(Symbol('E', commutative=True)))), Integral(Function('E_n')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(exp(Mul(Integer(-1), cos(Symbol('E', commutative=True)))), Integral(exp(cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('E_n')(Symbol('E', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Pow(Function('E_n')(Symbol('E', commutative=True)), Integer(-1)), Integral(exp(cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["divide", 4, "cos(Symbol('E', commutative=True))"], "Equality(Mul(Pow(Function('E_n')(Symbol('E', commutative=True)), Integer(-1)), Pow(cos(Symbol('E', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Pow(Function('E_n')(Symbol('E', commutative=True)), Integer(-1)), Pow(cos(Symbol('E', commutative=True)), Integer(-1)), Integral(exp(cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(g)} = \\cos{(g)}, then obtain 4 \\phi_{1}^{2}{(g)} + \\phi_{1}{(g)} - \\cos^{2}{(g)} + \\cos{(g)} = (\\phi_{1}{(g)} + \\cos{(g)})^{2} + \\phi_{1}{(g)} - \\cos^{2}{(g)} + \\cos{(g)}", "derivation": "\\phi_{1}{(g)} = \\cos{(g)} and \\phi_{1}{(g)} \\cos{(g)} = \\cos^{2}{(g)} and 2 \\phi_{1}{(g)} = \\phi_{1}{(g)} + \\cos{(g)} and 4 \\phi_{1}^{2}{(g)} = (\\phi_{1}{(g)} + \\cos{(g)})^{2} and 4 \\phi_{1}^{2}{(g)} - \\phi_{1}{(g)} \\cos{(g)} = (\\phi_{1}{(g)} + \\cos{(g)})^{2} - \\phi_{1}{(g)} \\cos{(g)} and 4 \\phi_{1}^{2}{(g)} - \\phi_{1}{(g)} \\cos{(g)} + \\phi_{1}{(g)} + \\cos{(g)} = (\\phi_{1}{(g)} + \\cos{(g)})^{2} - \\phi_{1}{(g)} \\cos{(g)} + \\phi_{1}{(g)} + \\cos{(g)} and 4 \\phi_{1}^{2}{(g)} + \\phi_{1}{(g)} - \\cos^{2}{(g)} + \\cos{(g)} = (\\phi_{1}{(g)} + \\cos{(g)})^{2} + \\phi_{1}{(g)} - \\cos^{2}{(g)} + \\cos{(g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["times", 1, "cos(Symbol('g', commutative=True))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Pow(cos(Symbol('g', commutative=True)), Integer(2)))"], [["add", 1, "Function('\\\\phi_1')(Symbol('g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('g', commutative=True))), Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\phi_1')(Symbol('g', commutative=True)), Integer(2))), Pow(Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Integer(2)))"], [["minus", 4, "Mul(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\phi_1')(Symbol('g', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))), Add(Pow(Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))))"], [["add", 5, "Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\phi_1')(Symbol('g', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(Pow(Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\phi_1')(Symbol('g', commutative=True)), Integer(2))), Function('\\\\phi_1')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('g', commutative=True)), Integer(2))), cos(Symbol('g', commutative=True))), Add(Pow(Add(Function('\\\\phi_1')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Integer(2)), Function('\\\\phi_1')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('g', commutative=True)), Integer(2))), cos(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\sigma_p,L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{\\sigma_p} and \\sigma_{x}{(\\sigma_p,L_{\\varepsilon})} = \\mathbf{p}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})}, then obtain \\sigma_{x}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})} = (\\mathbf{p}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})})^{\\sigma_p}", "derivation": "\\mathbf{p}{(\\sigma_p,L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{\\sigma_p} and \\sigma_{x}{(\\sigma_p,L_{\\varepsilon})} = \\mathbf{p}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})} and \\sigma_{x}{(\\sigma_p,L_{\\varepsilon})} = (\\frac{L_{\\varepsilon}}{\\sigma_p})^{\\sigma_p} and \\sigma_{x}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})} = ((\\frac{L_{\\varepsilon}}{\\sigma_p})^{\\sigma_p})^{\\sigma_p} and \\sigma_{x}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})} = (\\mathbf{p}^{\\sigma_p}{(\\sigma_p,L_{\\varepsilon})})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\sigma_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('\\\\sigma_p', commutative=True)))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\eta{(V_{\\mathbf{B}})} = V_{\\mathbf{B}}, then obtain \\int \\eta^{2}{(V_{\\mathbf{B}})} d\\eta{(V_{\\mathbf{B}})} = \\int V_{\\mathbf{B}} \\eta{(V_{\\mathbf{B}})} d\\eta{(V_{\\mathbf{B}})}", "derivation": "\\eta{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} and \\eta^{2}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\eta{(V_{\\mathbf{B}})} and \\int \\eta^{2}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\int V_{\\mathbf{B}} \\eta{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and \\int \\eta^{2}{(V_{\\mathbf{B}})} d\\eta{(V_{\\mathbf{B}})} = \\int V_{\\mathbf{B}} \\eta{(V_{\\mathbf{B}})} d\\eta{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["times", 1, "Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Pow(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Tuple(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Function('\\\\eta')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(Q)} = \\cos{(\\log{(Q)})}, then obtain - \\sin{((\\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}})^{Q})} - \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})} = - 2 \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})}", "derivation": "\\hat{\\mathbf{x}}{(Q)} = \\cos{(\\log{(Q)})} and \\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}} = \\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}} and (\\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}})^{Q} = (\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q} and \\sin{((\\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}})^{Q})} = \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})} and - \\sin{((\\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}})^{Q})} = - \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})} and - \\sin{((\\frac{\\hat{\\mathbf{x}}{(Q)}}{\\log{(Q)}})^{Q})} - \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})} = - 2 \\sin{((\\frac{\\cos{(\\log{(Q)})}}{\\log{(Q)}})^{Q})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), cos(log(Symbol('Q', commutative=True))))"], [["divide", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Symbol('Q', commutative=True)), Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Symbol('Q', commutative=True))), sin(Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(Pow(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Symbol('Q', commutative=True)))), Mul(Integer(-1), sin(Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))))"], [["minus", 5, "sin(Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Symbol('Q', commutative=True)))), Mul(Integer(-1), sin(Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True))))), Mul(Integer(-1), Integer(2), sin(Pow(Mul(Pow(log(Symbol('Q', commutative=True)), Integer(-1)), cos(log(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given v{(f)} = \\log{(\\cos{(f)})}, then obtain \\frac{d}{d f} \\sin{(\\log{(\\frac{v{(f)}}{\\log{(\\cos{(f)})}})})} = \\frac{d}{d f} 0", "derivation": "v{(f)} = \\log{(\\cos{(f)})} and \\frac{v{(f)}}{\\log{(\\cos{(f)})}} = 1 and \\log{(\\frac{v{(f)}}{\\log{(\\cos{(f)})}})} = 0 and \\sin{(\\log{(\\frac{v{(f)}}{\\log{(\\cos{(f)})}})})} = 0 and \\frac{d}{d f} \\sin{(\\log{(\\frac{v{(f)}}{\\log{(\\cos{(f)})}})})} = \\frac{d}{d f} 0", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('f', commutative=True)), log(cos(Symbol('f', commutative=True))))"], [["divide", 1, "log(cos(Symbol('f', commutative=True)))"], "Equality(Mul(Function('v')(Symbol('f', commutative=True)), Pow(log(cos(Symbol('f', commutative=True))), Integer(-1))), Integer(1))"], [["log", 2], "Equality(log(Mul(Function('v')(Symbol('f', commutative=True)), Pow(log(cos(Symbol('f', commutative=True))), Integer(-1)))), Integer(0))"], [["sin", 3], "Equality(sin(log(Mul(Function('v')(Symbol('f', commutative=True)), Pow(log(cos(Symbol('f', commutative=True))), Integer(-1))))), Integer(0))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(sin(log(Mul(Function('v')(Symbol('f', commutative=True)), Pow(log(cos(Symbol('f', commutative=True))), Integer(-1))))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(n_{1},\\varphi)} = \\varphi n_{1}, then obtain n_{1} e^{\\varphi n_{1}} + n_{1} e^{E{(n_{1},\\varphi)}} = 2 n_{1} e^{\\varphi n_{1}}", "derivation": "E{(n_{1},\\varphi)} = \\varphi n_{1} and e^{E{(n_{1},\\varphi)}} = e^{\\varphi n_{1}} and n_{1} e^{E{(n_{1},\\varphi)}} = n_{1} e^{\\varphi n_{1}} and n_{1} e^{\\varphi n_{1}} + n_{1} e^{E{(n_{1},\\varphi)}} = 2 n_{1} e^{\\varphi n_{1}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('n_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True)))"], [["exp", 1], "Equality(exp(Function('E')(Symbol('n_1', commutative=True), Symbol('\\\\varphi', commutative=True))), exp(Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True))))"], [["times", 2, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), exp(Function('E')(Symbol('n_1', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True)))))"], [["add", 3, "Mul(Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True))))"], "Equality(Add(Mul(Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True)))), Mul(Symbol('n_1', commutative=True), exp(Function('E')(Symbol('n_1', commutative=True), Symbol('\\\\varphi', commutative=True))))), Mul(Integer(2), Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\varphi', commutative=True), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)} = A_{y} + M, then obtain \\frac{M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}}{\\int (A_{y} + 2 M) dA_{y}} = \\frac{A_{y} + 2 M}{\\int (A_{y} + 2 M) dA_{y}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)} = A_{y} + M and M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)} = A_{y} + 2 M and \\int (M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}) dA_{y} = \\int (A_{y} + 2 M) dA_{y} and \\frac{M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}}{\\int (M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}) dA_{y}} = \\frac{A_{y} + 2 M}{\\int (M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}) dA_{y}} and \\frac{M + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y},M)}}{\\int (A_{y} + 2 M) dA_{y}} = \\frac{A_{y} + 2 M}{\\int (A_{y} + 2 M) dA_{y}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))"], [["add", 1, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integral(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], [["divide", 2, "Integral(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True)))"], "Equality(Mul(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Pow(Integral(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))), Mul(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), Pow(Integral(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('M', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))), Pow(Integral(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))), Mul(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), Pow(Integral(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then obtain \\mathbf{S}^{2} \\dot{\\mathbf{r}}^{4}{(\\mathbf{S})} = \\mathbf{S}^{2} \\sin^{4}{(\\mathbf{S})}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and \\mathbf{S} \\dot{\\mathbf{r}}{(\\mathbf{S})} = \\mathbf{S} \\sin{(\\mathbf{S})} and \\mathbf{S} \\dot{\\mathbf{r}}{(\\mathbf{S})} \\sin{(\\mathbf{S})} = \\mathbf{S} \\sin^{2}{(\\mathbf{S})} and \\mathbf{S} \\dot{\\mathbf{r}}^{2}{(\\mathbf{S})} = \\mathbf{S} \\dot{\\mathbf{r}}{(\\mathbf{S})} \\sin{(\\mathbf{S})} and \\mathbf{S} \\dot{\\mathbf{r}}^{2}{(\\mathbf{S})} = \\mathbf{S} \\sin^{2}{(\\mathbf{S})} and \\mathbf{S}^{2} \\dot{\\mathbf{r}}^{4}{(\\mathbf{S})} = \\mathbf{S}^{2} \\sin^{4}{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))))"], [["power", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4))))"]]}, {"prompt": "Given \\mathbf{E}{(\\dot{\\mathbf{r}},s)} = \\dot{\\mathbf{r}}^{s}, then obtain - \\frac{\\partial}{\\partial s} \\sin{(\\dot{\\mathbf{r}}^{s})} + \\frac{\\partial}{\\partial s} \\sin{(\\mathbf{E}{(\\dot{\\mathbf{r}},s)})} = 0", "derivation": "\\mathbf{E}{(\\dot{\\mathbf{r}},s)} = \\dot{\\mathbf{r}}^{s} and \\sin{(\\mathbf{E}{(\\dot{\\mathbf{r}},s)})} = \\sin{(\\dot{\\mathbf{r}}^{s})} and \\frac{\\partial}{\\partial s} \\sin{(\\mathbf{E}{(\\dot{\\mathbf{r}},s)})} = \\frac{\\partial}{\\partial s} \\sin{(\\dot{\\mathbf{r}}^{s})} and - \\frac{\\partial}{\\partial s} \\sin{(\\dot{\\mathbf{r}}^{s})} + \\frac{\\partial}{\\partial s} \\sin{(\\mathbf{E}{(\\dot{\\mathbf{r}},s)})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{E}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(sin(Function('\\\\mathbf{E}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))), Derivative(sin(Function('\\\\mathbf{E}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\chi{(\\mathbf{E},l)} = \\mathbf{E} - l, then obtain (\\mathbf{E} - l)^{\\mathbf{E}} \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\chi{(\\mathbf{E},l)}}{\\mathbf{E} - l} = (\\frac{(\\mathbf{E} - l)^{2}}{\\chi{(\\mathbf{E},l)}})^{\\mathbf{E}} \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\chi{(\\mathbf{E},l)}}{\\mathbf{E} - l}", "derivation": "\\chi{(\\mathbf{E},l)} = \\mathbf{E} - l and \\mathbf{E} - l = \\frac{(\\mathbf{E} - l)^{2}}{\\chi{(\\mathbf{E},l)}} and (\\mathbf{E} - l)^{\\mathbf{E}} = (\\frac{(\\mathbf{E} - l)^{2}}{\\chi{(\\mathbf{E},l)}})^{\\mathbf{E}} and (\\mathbf{E} - l)^{\\mathbf{E}} \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\chi{(\\mathbf{E},l)}}{\\mathbf{E} - l} = (\\frac{(\\mathbf{E} - l)^{2}}{\\chi{(\\mathbf{E},l)}})^{\\mathbf{E}} \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\chi{(\\mathbf{E},l)}}{\\mathbf{E} - l}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["divide", 1, "Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(2)), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(2)), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 3, "Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Pow(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(2)), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(n_{1},B)} = \\sin{(B + n_{1})} and \\theta_{1}{(n_{1},B)} = \\sin{(B + n_{1})}, then obtain - \\frac{H{(n_{1},B)}}{- 2 H{(n_{1},B)} + \\theta_{1}{(n_{1},B)}} = \\frac{- 2 H{(n_{1},B)} + \\sin{(B + n_{1})}}{- 2 H{(n_{1},B)} + \\theta_{1}{(n_{1},B)}}", "derivation": "H{(n_{1},B)} = \\sin{(B + n_{1})} and \\theta_{1}{(n_{1},B)} = \\sin{(B + n_{1})} and H{(n_{1},B)} = \\theta_{1}{(n_{1},B)} and 0 = - H{(n_{1},B)} + \\theta_{1}{(n_{1},B)} and - H{(n_{1},B)} = - 2 H{(n_{1},B)} + \\theta_{1}{(n_{1},B)} and - H{(n_{1},B)} = - 2 H{(n_{1},B)} + \\sin{(B + n_{1})} and - \\frac{H{(n_{1},B)}}{- 2 H{(n_{1},B)} + \\theta_{1}{(n_{1},B)}} = \\frac{- 2 H{(n_{1},B)} + \\sin{(B + n_{1})}}{- 2 H{(n_{1},B)} + \\theta_{1}{(n_{1},B)}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)))"], [["minus", 3, "Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(-1), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), sin(Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True)))))"], [["divide", 6, "Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Integer(-1)), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('H')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))), sin(Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))))))"]]}, {"prompt": "Given \\mu{(\\lambda)} = \\cos{(\\lambda)}, then derive \\frac{d}{d \\lambda} \\mu{(\\lambda)} = - \\sin{(\\lambda)}, then obtain 0 = - \\sin{(\\lambda)} - \\frac{d}{d \\lambda} \\cos{(\\lambda)}", "derivation": "\\mu{(\\lambda)} = \\cos{(\\lambda)} and \\frac{d}{d \\lambda} \\mu{(\\lambda)} = \\frac{d}{d \\lambda} \\cos{(\\lambda)} and \\frac{d}{d \\lambda} \\mu{(\\lambda)} = - \\sin{(\\lambda)} and \\frac{d}{d \\lambda} \\cos{(\\lambda)} = - \\sin{(\\lambda)} and \\frac{d}{d \\lambda} \\cos{(\\lambda)} - 1 = - \\sin{(\\lambda)} - 1 and \\mu{(\\lambda)} + \\frac{d}{d \\lambda} \\cos{(\\lambda)} - 1 = \\mu{(\\lambda)} - \\sin{(\\lambda)} - 1 and 0 = - \\sin{(\\lambda)} - \\frac{d}{d \\lambda} \\cos{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)))"], [["add", 5, "Function('\\\\mu')(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1)), Add(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)))"], [["minus", 6, "Add(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} = \\log{(\\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}})} and \\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})} = \\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}}, then obtain \\log{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})})} + 1 = \\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} + 1", "derivation": "\\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} = \\log{(\\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}})} and \\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} + 1 = \\log{(\\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}})} + 1 and \\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})} = \\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}} and \\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} = \\log{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})})} and \\log{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})})} + 1 = \\log{(\\frac{f_{\\mathbf{p}}}{\\eta^{\\prime}})} + 1 and \\log{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},\\eta^{\\prime})})} + 1 = \\bar{\\h}{(f_{\\mathbf{p}},\\eta^{\\prime})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)), Add(log(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(log(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(1)), Add(log(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(log(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(1)), Add(Function('\\\\hbar')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\Psi{(\\mathbb{I},B)} = B + \\mathbb{I}, then obtain 3 B = 4 B + \\mathbb{I} - \\Psi{(\\mathbb{I},B)}", "derivation": "\\Psi{(\\mathbb{I},B)} = B + \\mathbb{I} and 2 B + \\Psi{(\\mathbb{I},B)} = 3 B + \\mathbb{I} and 3 B + \\Psi{(\\mathbb{I},B)} = 4 B + \\mathbb{I} and 3 B = 4 B + \\mathbb{I} - \\Psi{(\\mathbb{I},B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Mul(Integer(2), Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('B', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(3), Symbol('B', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 2, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(3), Symbol('B', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(4), Symbol('B', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 3, "Function('\\\\Psi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Integer(3), Symbol('B', commutative=True)), Add(Mul(Integer(4), Symbol('B', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given V{(A_{1})} = \\log{(e^{A_{1}})} and \\rho_{b}{(\\hat{x},r_{0},U)} = (\\hat{x}^{r_{0}})^{U}, then obtain - (\\hat{x}^{r_{0}})^{U} + \\rho_{b}{(\\hat{x},r_{0},U)} + \\log{(e^{A_{1}})} - 1 = \\log{(e^{A_{1}})} - 1", "derivation": "V{(A_{1})} = \\log{(e^{A_{1}})} and \\rho_{b}{(\\hat{x},r_{0},U)} = (\\hat{x}^{r_{0}})^{U} and \\rho_{b}{(\\hat{x},r_{0},U)} - 1 = (\\hat{x}^{r_{0}})^{U} - 1 and V{(A_{1})} + \\rho_{b}{(\\hat{x},r_{0},U)} - 1 = (\\hat{x}^{r_{0}})^{U} + V{(A_{1})} - 1 and - (\\hat{x}^{r_{0}})^{U} + V{(A_{1})} + \\rho_{b}{(\\hat{x},r_{0},U)} - 1 = V{(A_{1})} - 1 and - (\\hat{x}^{r_{0}})^{U} + \\rho_{b}{(\\hat{x},r_{0},U)} + \\log{(e^{A_{1}})} - 1 = \\log{(e^{A_{1}})} - 1", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('A_1', commutative=True)), log(exp(Symbol('A_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True), Symbol('U', commutative=True)), Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True)))"], [["minus", 2, 1], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Add(Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True)), Integer(-1)))"], [["add", 3, "Function('V')(Symbol('A_1', commutative=True))"], "Equality(Add(Function('V')(Symbol('A_1', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Add(Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True)), Function('V')(Symbol('A_1', commutative=True)), Integer(-1)))"], [["minus", 4, "Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True))), Function('V')(Symbol('A_1', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Add(Function('V')(Symbol('A_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('U', commutative=True))), Function('\\\\rho_b')(Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True), Symbol('U', commutative=True)), log(exp(Symbol('A_1', commutative=True))), Integer(-1)), Add(log(exp(Symbol('A_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(z)} = \\log{(z)}, then obtain \\frac{\\log{(z)}^{z} \\frac{d}{d z} (\\operatorname{t_{2}}{(z)} - \\log{(z)})}{\\operatorname{t_{2}}{(z)} - \\log{(z)}} = \\frac{\\log{(z)}^{z} \\frac{d}{d z} 0}{\\operatorname{t_{2}}{(z)} - \\log{(z)}}", "derivation": "\\operatorname{t_{2}}{(z)} = \\log{(z)} and \\operatorname{t_{2}}{(z)} - \\log{(z)} = 0 and \\frac{d}{d z} (\\operatorname{t_{2}}{(z)} - \\log{(z)}) = \\frac{d}{d z} 0 and \\operatorname{t_{2}}^{z}{(z)} = \\log{(z)}^{z} and \\frac{\\operatorname{t_{2}}^{z}{(z)} \\frac{d}{d z} (\\operatorname{t_{2}}{(z)} - \\log{(z)})}{\\operatorname{t_{2}}{(z)} - \\log{(z)}} = \\frac{\\operatorname{t_{2}}^{z}{(z)} \\frac{d}{d z} 0}{\\operatorname{t_{2}}{(z)} - \\log{(z)}} and \\frac{\\log{(z)}^{z} \\frac{d}{d z} (\\operatorname{t_{2}}{(z)} - \\log{(z)})}{\\operatorname{t_{2}}{(z)} - \\log{(z)}} = \\frac{\\log{(z)}^{z} \\frac{d}{d z} 0}{\\operatorname{t_{2}}{(z)} - \\log{(z)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["minus", 1, "log(Symbol('z', commutative=True))"], "Equality(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["divide", 3, "Mul(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Pow(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))))"], "Equality(Mul(Pow(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(-1)), Pow(Function('t_2')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Derivative(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(-1)), Pow(Function('t_2')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Derivative(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Add(Function('t_2')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{J}_P,W)} = \\frac{\\mathbf{J}_P}{W}, then obtain 0 = - \\frac{\\operatorname{F_{c}}{(\\mathbf{J}_P,W)} - \\frac{\\mathbf{J}_P}{W}}{\\sin{(\\operatorname{F_{c}}{(\\mathbf{J}_P,W)})}}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{J}_P,W)} = \\frac{\\mathbf{J}_P}{W} and \\operatorname{F_{c}}{(\\mathbf{J}_P,W)} - \\frac{\\mathbf{J}_P}{W} = 0 and \\frac{\\operatorname{F_{c}}{(\\mathbf{J}_P,W)} - \\frac{\\mathbf{J}_P}{W}}{\\sin{(\\operatorname{F_{c}}{(\\mathbf{J}_P,W)})}} = 0 and 0 = - \\frac{\\operatorname{F_{c}}{(\\mathbf{J}_P,W)} - \\frac{\\mathbf{J}_P}{W}}{\\sin{(\\operatorname{F_{c}}{(\\mathbf{J}_P,W)})}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(0))"], [["divide", 2, "sin(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Add(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True))), Integer(-1))), Integer(0))"], [["minus", 3, "Mul(Add(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Function('F_c')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('W', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(r,F_{H})} = F_{H} + r, then derive \\int \\phi_{1}{(r,F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} r + m_{s}, then obtain \\cos{(\\frac{F_{H}^{2}}{2} + F_{H} r + m_{s})} = \\cos{(\\int (F_{H} + r) dF_{H})}", "derivation": "\\phi_{1}{(r,F_{H})} = F_{H} + r and \\int \\phi_{1}{(r,F_{H})} dF_{H} = \\int (F_{H} + r) dF_{H} and \\int \\phi_{1}{(r,F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} r + m_{s} and \\frac{F_{H}^{2}}{2} + F_{H} r + m_{s} = \\int (F_{H} + r) dF_{H} and \\cos{(\\frac{F_{H}^{2}}{2} + F_{H} r + m_{s})} = \\cos{(\\int (F_{H} + r) dF_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Symbol('m_s', commutative=True)), Integral(Add(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["cos", 4], "Equality(cos(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Symbol('m_s', commutative=True))), cos(Integral(Add(Symbol('F_H', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(Q)} = \\log{(Q)}, then obtain \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi{(Q)} \\varphi^{Q}{(Q)}) + 1 = \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi^{Q}{(Q)} \\log{(Q)}) + 1", "derivation": "\\varphi{(Q)} = \\log{(Q)} and \\varphi^{Q}{(Q)} = \\log{(Q)}^{Q} and \\varphi{(Q)} \\log{(Q)}^{Q} = \\log{(Q)} \\log{(Q)}^{Q} and \\varphi{(Q)} \\varphi^{Q}{(Q)} = \\varphi^{Q}{(Q)} \\log{(Q)} and \\frac{d}{d Q} \\varphi{(Q)} \\varphi^{Q}{(Q)} = \\frac{d}{d Q} \\varphi^{Q}{(Q)} \\log{(Q)} and - \\hat{H} + \\frac{d}{d Q} \\varphi{(Q)} \\varphi^{Q}{(Q)} = - \\hat{H} + \\frac{d}{d Q} \\varphi^{Q}{(Q)} \\log{(Q)} and \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi{(Q)} \\varphi^{Q}{(Q)}) = \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi^{Q}{(Q)} \\log{(Q)}) and \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi{(Q)} \\varphi^{Q}{(Q)}) + 1 = \\eta (- \\hat{H} + \\frac{d}{d Q} \\varphi^{Q}{(Q)} \\log{(Q)}) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["times", 1, "Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(log(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["minus", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["times", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["minus", 7, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(1)), Add(Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Pow(Function('\\\\varphi')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(1)))"]]}, {"prompt": "Given z{(\\hbar)} = \\log{(\\hbar)}, then derive \\frac{d}{d \\hbar} z{(\\hbar)} = \\frac{1}{\\hbar}, then obtain \\frac{d^{2}}{d \\hbar^{2}} z{(\\hbar)} = - \\frac{1}{\\hbar^{2}}", "derivation": "z{(\\hbar)} = \\log{(\\hbar)} and \\frac{d}{d \\hbar} z{(\\hbar)} = \\frac{d}{d \\hbar} \\log{(\\hbar)} and \\frac{d}{d \\hbar} z{(\\hbar)} = \\frac{1}{\\hbar} and \\frac{d}{d \\hbar} z{(\\hbar)} - 1 = -1 + \\frac{1}{\\hbar} and \\frac{d}{d \\hbar} (\\frac{d}{d \\hbar} z{(\\hbar)} - 1) = \\frac{d}{d \\hbar} (-1 + \\frac{1}{\\hbar}) and \\frac{d^{2}}{d \\hbar^{2}} z{(\\hbar)} = - \\frac{1}{\\hbar^{2}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('z')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('z')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('z')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{J}_P{(b)} = \\sin{(b)}, then obtain - \\frac{\\frac{d}{d b} 0}{\\mathbf{J}_P{(b)}} = - \\frac{\\frac{d}{d b} (\\mathbf{J}_P{(b)} - \\sin{(b)})}{\\mathbf{J}_P{(b)}}", "derivation": "\\mathbf{J}_P{(b)} = \\sin{(b)} and 0 = - \\mathbf{J}_P{(b)} + \\sin{(b)} and 0 = \\mathbf{J}_P{(b)} - \\sin{(b)} and \\frac{d}{d b} 0 = \\frac{d}{d b} (\\mathbf{J}_P{(b)} - \\sin{(b)}) and - \\frac{\\frac{d}{d b} 0}{\\mathbf{J}_P{(b)}} = - \\frac{\\frac{d}{d b} (\\mathbf{J}_P{(b)} - \\sin{(b)})}{\\mathbf{J}_P{(b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True))), sin(Symbol('b', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), Mul(Integer(-1), sin(Symbol('b', commutative=True)))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), Mul(Integer(-1), sin(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), Integer(-1)), Derivative(Add(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True)), Mul(Integer(-1), sin(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(C,H)} = e^{C H}, then obtain \\frac{\\partial}{\\partial H} (\\mathbf{B}{(C,H)} e^{C H})^{H} = \\frac{\\partial}{\\partial H} (e^{2 C H})^{H}", "derivation": "\\mathbf{B}{(C,H)} = e^{C H} and \\mathbf{B}{(C,H)} e^{C H} = e^{2 C H} and (\\mathbf{B}{(C,H)} e^{C H})^{H} = (e^{2 C H})^{H} and \\frac{\\partial}{\\partial H} (\\mathbf{B}{(C,H)} e^{C H})^{H} = \\frac{\\partial}{\\partial H} (e^{2 C H})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('H', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('H', commutative=True))))"], [["times", 1, "exp(Mul(Symbol('C', commutative=True), Symbol('H', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('H', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('H', commutative=True)))), exp(Mul(Integer(2), Symbol('C', commutative=True), Symbol('H', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('H', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Pow(exp(Mul(Integer(2), Symbol('C', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('H', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(exp(Mul(Integer(2), Symbol('C', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(\\rho_b)} = \\cos{(\\rho_b)}, then obtain \\iiint \\frac{d}{d \\rho_b} \\sigma_{p}{(\\rho_b)} d\\rho_b d\\rho_b d\\rho_b = \\iiint \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} d\\rho_b d\\rho_b d\\rho_b", "derivation": "\\sigma_{p}{(\\rho_b)} = \\cos{(\\rho_b)} and \\frac{d}{d \\rho_b} \\sigma_{p}{(\\rho_b)} = \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} and \\int \\frac{d}{d \\rho_b} \\sigma_{p}{(\\rho_b)} d\\rho_b = \\int \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} d\\rho_b and \\iint \\frac{d}{d \\rho_b} \\sigma_{p}{(\\rho_b)} d\\rho_b d\\rho_b = \\iint \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} d\\rho_b d\\rho_b and \\iiint \\frac{d}{d \\rho_b} \\sigma_{p}{(\\rho_b)} d\\rho_b d\\rho_b d\\rho_b = \\iiint \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} d\\rho_b d\\rho_b d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Derivative(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Derivative(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Derivative(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\mathbf{P},\\hat{p}_0)} = \\mathbf{P}^{\\hat{p}_0}, then obtain - \\hat{p}_0 \\mu{(\\mathbf{P},\\hat{p}_0)} + \\mathbf{P}^{\\hat{p}_0} \\mu{(\\mathbf{P},\\hat{p}_0)} = - \\hat{p}_0 \\mu{(\\mathbf{P},\\hat{p}_0)} + \\mathbf{P}^{2 \\hat{p}_0}", "derivation": "\\mu{(\\mathbf{P},\\hat{p}_0)} = \\mathbf{P}^{\\hat{p}_0} and \\hat{p}_0 \\mu{(\\mathbf{P},\\hat{p}_0)} = \\hat{p}_0 \\mathbf{P}^{\\hat{p}_0} and \\mathbf{P}^{\\hat{p}_0} \\mu{(\\mathbf{P},\\hat{p}_0)} = \\mathbf{P}^{2 \\hat{p}_0} and - \\hat{p}_0 \\mathbf{P}^{\\hat{p}_0} + \\mathbf{P}^{\\hat{p}_0} \\mu{(\\mathbf{P},\\hat{p}_0)} = - \\hat{p}_0 \\mathbf{P}^{\\hat{p}_0} + \\mathbf{P}^{2 \\hat{p}_0} and - \\hat{p}_0 \\mu{(\\mathbf{P},\\hat{p}_0)} + \\mathbf{P}^{\\hat{p}_0} \\mu{(\\mathbf{P},\\hat{p}_0)} = - \\hat{p}_0 \\mu{(\\mathbf{P},\\hat{p}_0)} + \\mathbf{P}^{2 \\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(f_{E},\\sigma_x)} = - \\sigma_x + f_{E} and \\mathbf{E}{(A,V)} = A V, then obtain - 2 \\sigma_x + 2 f_{E} + \\mathbf{E}{(A,V)} = A V - 2 \\sigma_x + 2 f_{E}", "derivation": "\\operatorname{a^{\\dagger}}{(f_{E},\\sigma_x)} = - \\sigma_x + f_{E} and \\mathbf{E}{(A,V)} = A V and \\mathbf{E}{(A,V)} + 2 \\operatorname{a^{\\dagger}}{(f_{E},\\sigma_x)} = A V + 2 \\operatorname{a^{\\dagger}}{(f_{E},\\sigma_x)} and - 2 \\sigma_x + 2 f_{E} + \\mathbf{E}{(A,V)} = A V - 2 \\sigma_x + 2 f_{E}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('A', commutative=True), Symbol('V', commutative=True)), Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('f_E', commutative=True)), Function('\\\\mathbf{E}')(Symbol('A', commutative=True), Symbol('V', commutative=True))), Add(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(i,f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}}, then derive f_{\\mathbf{v}} \\operatorname{v_{t}}{(i,f_{\\mathbf{v}})} = 1, then obtain \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}} = \\frac{1}{f_{\\mathbf{v}}}", "derivation": "\\operatorname{v_{t}}{(i,f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}} and f_{\\mathbf{v}} \\operatorname{v_{t}}{(i,f_{\\mathbf{v}})} = f_{\\mathbf{v}} \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}} and f_{\\mathbf{v}} \\operatorname{v_{t}}{(i,f_{\\mathbf{v}})} = 1 and f_{\\mathbf{v}} \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}} = 1 and \\frac{\\partial}{\\partial i} \\frac{i}{f_{\\mathbf{v}}} = \\frac{1}{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["divide", 1, "Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('v_t')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('v_t')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Integer(1))"], [["times", 4, "Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))"], "Equality(Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then derive (\\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (a + \\sin{(\\mathbf{H})})^{\\mathbf{H}}, then obtain a (\\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = a (a + \\sin{(\\mathbf{H})})^{\\mathbf{H}}", "derivation": "\\sigma_{p}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H} = \\int \\cos{(\\mathbf{H})} d\\mathbf{H} and (\\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\int \\cos{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} and (\\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (a + \\sin{(\\mathbf{H})})^{\\mathbf{H}} and a (\\int \\sigma_{p}{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = a (a + \\sin{(\\mathbf{H})})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('a', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 4, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('a', commutative=True), Pow(Add(Symbol('a', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{S},C_{d})} = \\sin{(C_{d} - \\mathbf{S})}, then derive \\cos{(C_{d} - \\mathbf{S})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} = \\cos^{2}{(C_{d} - \\mathbf{S})}, then obtain \\mathbf{S} + \\cos{(C_{d} - \\mathbf{S})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} = \\mathbf{S} + \\cos^{2}{(C_{d} - \\mathbf{S})}", "derivation": "\\mathbf{A}{(\\mathbf{S},C_{d})} = \\sin{(C_{d} - \\mathbf{S})} and \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\sin{(C_{d} - \\mathbf{S})} and \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} \\frac{\\partial}{\\partial C_{d}} \\sin{(C_{d} - \\mathbf{S})} = (\\frac{\\partial}{\\partial C_{d}} \\sin{(C_{d} - \\mathbf{S})})^{2} and \\cos{(C_{d} - \\mathbf{S})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} = \\cos^{2}{(C_{d} - \\mathbf{S})} and \\mathbf{S} + \\cos{(C_{d} - \\mathbf{S})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(\\mathbf{S},C_{d})} = \\mathbf{S} + \\cos^{2}{(C_{d} - \\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["times", 2, "Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Pow(Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Pow(cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Integer(2)))"], [["add", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\phi_{1}{(v_{y},z)} = z + \\log{(v_{y})} and T{(v_{y},z)} = - \\frac{\\partial}{\\partial z} (z + \\log{(v_{y})}), then derive T{(v_{y},z)} = -1, then obtain - \\frac{1}{v_{y}} = - \\frac{\\frac{\\partial}{\\partial z} \\phi_{1}{(v_{y},z)}}{v_{y}}", "derivation": "\\phi_{1}{(v_{y},z)} = z + \\log{(v_{y})} and T{(v_{y},z)} = - \\frac{\\partial}{\\partial z} (z + \\log{(v_{y})}) and T{(v_{y},z)} = -1 and T{(v_{y},z)} = - \\frac{\\partial}{\\partial z} \\phi_{1}{(v_{y},z)} and \\frac{T{(v_{y},z)}}{v_{y}} = - \\frac{\\frac{\\partial}{\\partial z} \\phi_{1}{(v_{y},z)}}{v_{y}} and - \\frac{1}{v_{y}} = - \\frac{\\frac{\\partial}{\\partial z} \\phi_{1}{(v_{y},z)}}{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Add(Symbol('z', commutative=True), log(Symbol('v_y', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('z', commutative=True), log(Symbol('v_y', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Function('T')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('T')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["divide", 4, "Symbol('v_y', commutative=True)"], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Function('T')(Symbol('v_y', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Derivative(Function('\\\\phi_1')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Derivative(Function('\\\\phi_1')(Symbol('v_y', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(b)} = \\cos{(b)} and \\operatorname{P_{g}}{(E_{x},M)} = E_{x} \\sin{(M)}, then obtain - \\mathbf{p}{(b)} + \\cos{(\\operatorname{P_{g}}{(E_{x},M)} \\mathbf{p}{(b)})} = - \\mathbf{p}{(b)} + \\cos{(E_{x} \\mathbf{p}{(b)} \\sin{(M)})}", "derivation": "\\mathbf{p}{(b)} = \\cos{(b)} and \\operatorname{P_{g}}{(E_{x},M)} = E_{x} \\sin{(M)} and \\operatorname{P_{g}}{(E_{x},M)} \\cos{(b)} = E_{x} \\sin{(M)} \\cos{(b)} and \\cos{(\\operatorname{P_{g}}{(E_{x},M)} \\cos{(b)})} = \\cos{(E_{x} \\sin{(M)} \\cos{(b)})} and \\cos{(\\operatorname{P_{g}}{(E_{x},M)} \\mathbf{p}{(b)})} = \\cos{(E_{x} \\mathbf{p}{(b)} \\sin{(M)})} and - \\mathbf{p}{(b)} + \\cos{(\\operatorname{P_{g}}{(E_{x},M)} \\mathbf{p}{(b)})} = - \\mathbf{p}{(b)} + \\cos{(E_{x} \\mathbf{p}{(b)} \\sin{(M)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('E_x', commutative=True), sin(Symbol('M', commutative=True))))"], [["times", 2, "cos(Symbol('b', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), cos(Symbol('b', commutative=True))), Mul(Symbol('E_x', commutative=True), sin(Symbol('M', commutative=True)), cos(Symbol('b', commutative=True))))"], [["cos", 3], "Equality(cos(Mul(Function('P_g')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), cos(Symbol('b', commutative=True)))), cos(Mul(Symbol('E_x', commutative=True), sin(Symbol('M', commutative=True)), cos(Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(cos(Mul(Function('P_g')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('b', commutative=True)))), cos(Mul(Symbol('E_x', commutative=True), Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), sin(Symbol('M', commutative=True)))))"], [["minus", 5, "Function('\\\\mathbf{p}')(Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), cos(Mul(Function('P_g')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), cos(Mul(Symbol('E_x', commutative=True), Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), sin(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\sigma_p)} = \\sin{(\\sigma_p)}, then obtain (\\frac{d}{d \\sigma_p} 1)^{\\sigma_p} = (\\frac{d}{d \\sigma_p} \\frac{\\sin{(\\sigma_p)}}{\\operatorname{P_{e}}{(\\sigma_p)}})^{\\sigma_p}", "derivation": "\\operatorname{P_{e}}{(\\sigma_p)} = \\sin{(\\sigma_p)} and 1 = \\frac{\\sin{(\\sigma_p)}}{\\operatorname{P_{e}}{(\\sigma_p)}} and \\frac{d}{d \\sigma_p} 1 = \\frac{d}{d \\sigma_p} \\frac{\\sin{(\\sigma_p)}}{\\operatorname{P_{e}}{(\\sigma_p)}} and (\\frac{d}{d \\sigma_p} 1)^{\\sigma_p} = (\\frac{d}{d \\sigma_p} \\frac{\\sin{(\\sigma_p)}}{\\operatorname{P_{e}}{(\\sigma_p)}})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "Function('P_e')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_e')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('P_e')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Mul(Pow(Function('P_e')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given B{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}}, then obtain 0 = - \\frac{\\dot{\\mathbf{r}} (\\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} - 1)}{B{(\\dot{\\mathbf{r}})}}", "derivation": "B{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} and 1 = \\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} and - \\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} + 1 = 0 and - \\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} + 2 = 1 and 0 = \\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} - 1 and 0 = - \\frac{\\dot{\\mathbf{r}} (\\frac{\\dot{\\mathbf{r}}}{B{(\\dot{\\mathbf{r}})}} - 1)}{B{(\\dot{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["divide", 1, "Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(1)), Integer(0))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(2)), Integer(1))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(2))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(-1)))"], [["times", 5, "Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(-1)), Pow(Function('B')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given r{(k)} = \\cos{(k)}, then obtain \\int \\frac{r{(k)}}{k \\cos^{2}{(k)}} dk = \\int \\frac{1}{k r{(k)}} dk", "derivation": "r{(k)} = \\cos{(k)} and k r{(k)} = k \\cos{(k)} and \\frac{r{(k)}}{k \\cos^{2}{(k)}} = \\frac{1}{k \\cos{(k)}} and \\frac{1}{k \\cos{(k)}} = \\frac{1}{k r{(k)}} and \\frac{r{(k)}}{k \\cos^{2}{(k)}} = \\frac{1}{k r{(k)}} and \\int \\frac{r{(k)}}{k \\cos^{2}{(k)}} dk = \\int \\frac{1}{k r{(k)}} dk", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('r')(Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), cos(Symbol('k', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('k', commutative=True), Integer(2)), Pow(cos(Symbol('k', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('r')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-2))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('k', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('r')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-2))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('k', commutative=True)), Integer(-1))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('r')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-2))), Tuple(Symbol('k', commutative=True))), Integral(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\eta^{\\prime})} = \\sin{(\\sin{(\\eta^{\\prime})})}, then obtain \\sin{(\\frac{\\sin{(\\operatorname{t_{1}}{(\\eta^{\\prime})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}})} = \\sin{(\\frac{\\sin{(\\sin{(\\sin{(\\eta^{\\prime})})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}})}", "derivation": "\\operatorname{t_{1}}{(\\eta^{\\prime})} = \\sin{(\\sin{(\\eta^{\\prime})})} and \\sin{(\\operatorname{t_{1}}{(\\eta^{\\prime})})} = \\sin{(\\sin{(\\sin{(\\eta^{\\prime})})})} and \\frac{\\sin{(\\operatorname{t_{1}}{(\\eta^{\\prime})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}} = \\frac{\\sin{(\\sin{(\\sin{(\\eta^{\\prime})})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}} and \\sin{(\\frac{\\sin{(\\operatorname{t_{1}}{(\\eta^{\\prime})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}})} = \\sin{(\\frac{\\sin{(\\sin{(\\sin{(\\eta^{\\prime})})})}}{\\operatorname{t_{1}}{(\\eta^{\\prime})}})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["sin", 1], "Equality(sin(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), sin(sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["divide", 2, "Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Pow(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))))"], [["sin", 3], "Equality(sin(Mul(Pow(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))), sin(Mul(Pow(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(sin(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))))"]]}, {"prompt": "Given t{(m,H)} = \\cos{(\\frac{H}{m})} and \\sigma_{x}{(m,H)} = \\cos{(\\frac{H}{m})}, then obtain (H \\operatorname{Si}{(\\frac{H}{m})} + f^{*} + m \\cos{(\\frac{H}{m})}) \\sigma_{x}{(m,H)} = (H \\operatorname{Si}{(\\frac{H}{m})} + f^{*} + m \\cos{(\\frac{H}{m})}) t{(m,H)}", "derivation": "t{(m,H)} = \\cos{(\\frac{H}{m})} and \\sigma_{x}{(m,H)} = \\cos{(\\frac{H}{m})} and \\sigma_{x}{(m,H)} = t{(m,H)} and \\sigma_{x}{(m,H)} \\int \\cos{(\\frac{H}{m})} dm = t{(m,H)} \\int \\cos{(\\frac{H}{m})} dm and (H \\operatorname{Si}{(\\frac{H}{m})} + f^{*} + m \\cos{(\\frac{H}{m})}) \\sigma_{x}{(m,H)} = (H \\operatorname{Si}{(\\frac{H}{m})} + f^{*} + m \\cos{(\\frac{H}{m})}) t{(m,H)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('m', commutative=True), Symbol('H', commutative=True)), cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('m', commutative=True), Symbol('H', commutative=True)), cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\sigma_x')(Symbol('m', commutative=True), Symbol('H', commutative=True)), Function('t')(Symbol('m', commutative=True), Symbol('H', commutative=True)))"], [["times", 3, "Integral(cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('m', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('m', commutative=True), Symbol('H', commutative=True)), Integral(cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('m', commutative=True)))), Mul(Function('t')(Symbol('m', commutative=True), Symbol('H', commutative=True)), Integral(cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('m', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Si(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Symbol('f^*', commutative=True), Mul(Symbol('m', commutative=True), cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))), Function('\\\\sigma_x')(Symbol('m', commutative=True), Symbol('H', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), Si(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Symbol('f^*', commutative=True), Mul(Symbol('m', commutative=True), cos(Mul(Symbol('H', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))), Function('t')(Symbol('m', commutative=True), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\Psi_{nl})} = \\log{(\\Psi_{nl})}, then obtain \\iint \\operatorname{n_{2}}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl} d\\Psi_{nl} = \\iint \\log{(\\Psi_{nl})}^{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl}", "derivation": "\\operatorname{n_{2}}{(\\Psi_{nl})} = \\log{(\\Psi_{nl})} and \\operatorname{n_{2}}^{\\Psi_{nl}}{(\\Psi_{nl})} = \\log{(\\Psi_{nl})}^{\\Psi_{nl}} and \\int \\operatorname{n_{2}}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl} = \\int \\log{(\\Psi_{nl})}^{\\Psi_{nl}} d\\Psi_{nl} and \\iint \\operatorname{n_{2}}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl} d\\Psi_{nl} = \\iint \\log{(\\Psi_{nl})}^{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\Psi_{nl}', commutative=True)), log(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(log(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Pow(Function('n_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(log(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Pow(Function('n_2')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(log(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\mathbf{D}{(\\mathbf{J})} = \\log{(\\mathbf{J})}, then obtain \\frac{d}{d \\mathbf{J}} (- \\frac{d}{d \\mathbf{J}} \\mathbf{D}{(\\mathbf{J})} + \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})}) = \\frac{d}{d \\mathbf{J}} 0", "derivation": "\\mathbf{M}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})} - 1 = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} - 1 and \\mathbf{D}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})} - 1 = \\frac{d}{d \\mathbf{J}} \\mathbf{D}{(\\mathbf{J})} - 1 and - \\frac{d}{d \\mathbf{J}} \\mathbf{D}{(\\mathbf{J})} + \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})} = 0 and \\frac{d}{d \\mathbf{J}} (- \\frac{d}{d \\mathbf{J}} \\mathbf{D}{(\\mathbf{J})} + \\frac{d}{d \\mathbf{J}} \\mathbf{M}{(\\mathbf{J})}) = \\frac{d}{d \\mathbf{J}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 5, "Add(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 6, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(C,b)} = C b, then obtain \\frac{\\operatorname{v_{1}}{(C,b)}}{- 2 C b + 2 \\operatorname{v_{1}}{(C,b)}} = \\frac{C b}{- 2 C b + 2 \\operatorname{v_{1}}{(C,b)}}", "derivation": "\\operatorname{v_{1}}{(C,b)} = C b and - C b + \\operatorname{v_{1}}{(C,b)} = 0 and - 2 C b + \\operatorname{v_{1}}{(C,b)} = - C b and \\frac{\\operatorname{v_{1}}{(C,b)}}{- C b + \\operatorname{v_{1}}{(C,b)}} = \\frac{C b}{- C b + \\operatorname{v_{1}}{(C,b)}} and \\frac{\\operatorname{v_{1}}{(C,b)}}{- 2 C b + 2 \\operatorname{v_{1}}{(C,b)}} = \\frac{C b}{- 2 C b + 2 \\operatorname{v_{1}}{(C,b)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('b', commutative=True)))"], [["minus", 1, "Mul(Symbol('C', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Integer(0))"], [["add", 2, "Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C', commutative=True), Symbol('b', commutative=True)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('C', commutative=True), Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('b', commutative=True)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)))), Integer(-1)), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('C', commutative=True), Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given m{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + \\mathbf{F} and H{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + \\mathbf{F}, then obtain - \\frac{\\sin{(\\mathbf{F} - H{(\\mathbf{F},V_{\\mathbf{B}})})}}{\\cos{(\\hat{x})}} = \\frac{\\sin{(V_{\\mathbf{B}})}}{\\cos{(\\hat{x})}}", "derivation": "m{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + \\mathbf{F} and - \\mathbf{F} + m{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} and H{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + \\mathbf{F} and m{(\\mathbf{F},V_{\\mathbf{B}})} = H{(\\mathbf{F},V_{\\mathbf{B}})} and - \\mathbf{F} + H{(\\mathbf{F},V_{\\mathbf{B}})} = V_{\\mathbf{B}} and - \\sin{(\\mathbf{F} - H{(\\mathbf{F},V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}})} and - \\frac{\\sin{(\\mathbf{F} - H{(\\mathbf{F},V_{\\mathbf{B}})})}}{\\cos{(\\hat{x})}} = \\frac{\\sin{(V_{\\mathbf{B}})}}{\\cos{(\\hat{x})}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('H')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('H')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["sin", 5], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('H')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))))), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 6, "cos(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('H')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Mul(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(v_{z},z)} = v_{z}^{z}, then obtain \\frac{\\operatorname{C_{2}}{(v_{z},z)}}{\\int \\operatorname{C_{2}}{(v_{z},z)} dz} = \\frac{v_{z}^{z}}{\\int \\operatorname{C_{2}}{(v_{z},z)} dz}", "derivation": "\\operatorname{C_{2}}{(v_{z},z)} = v_{z}^{z} and \\int \\operatorname{C_{2}}{(v_{z},z)} dz = \\int v_{z}^{z} dz and \\frac{\\operatorname{C_{2}}{(v_{z},z)}}{\\int v_{z}^{z} dz} = \\frac{v_{z}^{z}}{\\int v_{z}^{z} dz} and \\frac{\\operatorname{C_{2}}{(v_{z},z)}}{\\int \\operatorname{C_{2}}{(v_{z},z)} dz} = \\frac{v_{z}^{z}}{\\int \\operatorname{C_{2}}{(v_{z},z)} dz}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["divide", 1, "Integral(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Mul(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Pow(Integral(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))), Mul(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Pow(Integral(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Pow(Integral(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))), Mul(Pow(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Pow(Integral(Function('C_2')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} = \\int (\\hat{\\mathbf{r}} + \\phi_2) d\\hat{\\mathbf{r}}, then derive \\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} = \\frac{\\hat{\\mathbf{r}}^{2}}{2} + \\hat{\\mathbf{r}} \\phi_2 + \\varphi^*, then obtain (\\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} - \\frac{1}{2})^{\\phi_2} = (\\frac{\\hat{\\mathbf{r}}^{2}}{2} + \\hat{\\mathbf{r}} \\phi_2 + \\varphi^* - \\frac{1}{2})^{\\phi_2}", "derivation": "\\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} = \\int (\\hat{\\mathbf{r}} + \\phi_2) d\\hat{\\mathbf{r}} and \\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} = \\frac{\\hat{\\mathbf{r}}^{2}}{2} + \\hat{\\mathbf{r}} \\phi_2 + \\varphi^* and \\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} - \\frac{1}{2} = \\frac{\\hat{\\mathbf{r}}^{2}}{2} + \\hat{\\mathbf{r}} \\phi_2 + \\varphi^* - \\frac{1}{2} and (\\dot{y}{(\\phi_2,\\hat{\\mathbf{r}})} - \\frac{1}{2})^{\\phi_2} = (\\frac{\\hat{\\mathbf{r}}^{2}}{2} + \\hat{\\mathbf{r}} \\phi_2 + \\varphi^* - \\frac{1}{2})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{y}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 2, "Rational(1, 2)"], "Equality(Add(Function('\\\\dot{y}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Rational(-1, 2)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Rational(-1, 2)))"], [["power", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{y}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Rational(-1, 2)), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Rational(-1, 2)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(P_{g},y)} = \\int \\frac{P_{g}}{y} dy, then obtain - \\dot{x}{(P_{g},y)} + \\cos{(P_{g} + \\dot{x}{(P_{g},y)})} = - \\dot{x}{(P_{g},y)} + \\cos{(P_{g} + \\int \\frac{P_{g}}{y} dy)}", "derivation": "\\dot{x}{(P_{g},y)} = \\int \\frac{P_{g}}{y} dy and P_{g} + \\dot{x}{(P_{g},y)} = P_{g} + \\int \\frac{P_{g}}{y} dy and \\cos{(P_{g} + \\dot{x}{(P_{g},y)})} = \\cos{(P_{g} + \\int \\frac{P_{g}}{y} dy)} and - \\dot{x}{(P_{g},y)} + \\cos{(P_{g} + \\dot{x}{(P_{g},y)})} = - \\dot{x}{(P_{g},y)} + \\cos{(P_{g} + \\int \\frac{P_{g}}{y} dy)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True))))"], [["add", 1, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True))), Add(Symbol('P_g', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Symbol('P_g', commutative=True), Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)))), cos(Add(Symbol('P_g', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True))))))"], [["minus", 3, "Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True))), cos(Add(Symbol('P_g', commutative=True), Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True))), cos(Add(Symbol('P_g', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True)))))))"]]}, {"prompt": "Given U{(\\mathbf{J},B)} = \\log{(\\mathbf{J})}^{B}, then derive \\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)} = \\frac{B \\log{(\\mathbf{J})}^{B}}{\\mathbf{J} \\log{(\\mathbf{J})}}, then obtain \\frac{\\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)}}{\\mathbf{J}} = \\frac{B U{(\\mathbf{J},B)}}{\\mathbf{J}^{2} \\log{(\\mathbf{J})}}", "derivation": "U{(\\mathbf{J},B)} = \\log{(\\mathbf{J})}^{B} and \\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\log{(\\mathbf{J})}^{B} and \\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)} = \\frac{B \\log{(\\mathbf{J})}^{B}}{\\mathbf{J} \\log{(\\mathbf{J})}} and \\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)} = \\frac{B U{(\\mathbf{J},B)}}{\\mathbf{J} \\log{(\\mathbf{J})}} and \\frac{\\frac{\\partial}{\\partial \\mathbf{J}} U{(\\mathbf{J},B)}}{\\mathbf{J}} = \\frac{B U{(\\mathbf{J},B)}}{\\mathbf{J}^{2} \\log{(\\mathbf{J})}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))))"], [["times", 4, "Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)), Function('U')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given g{(Q)} = \\sin{(Q)}, then obtain \\iint 0 dQ dQ = \\iint (- g{(Q)} + \\sin{(Q)}) dQ dQ", "derivation": "g{(Q)} = \\sin{(Q)} and 0 = - g{(Q)} + \\sin{(Q)} and \\int 0 dQ = \\int (- g{(Q)} + \\sin{(Q)}) dQ and \\iint 0 dQ dQ = \\iint (- g{(Q)} + \\sin{(Q)}) dQ dQ", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["minus", 1, "Function('g')(Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(-1), Function('g')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(-1), Function('g')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(f)} = \\log{(f)}, then obtain - 2 \\mu_{0}{(f)} + \\int (\\mu_{0}{(f)} + \\log{(f)})^{2} df = - 2 \\mu_{0}{(f)} + \\int 4 \\log{(f)}^{2} df", "derivation": "\\mu_{0}{(f)} = \\log{(f)} and \\mu_{0}{(f)} + \\log{(f)} = 2 \\log{(f)} and (\\mu_{0}{(f)} + \\log{(f)})^{2} = 4 \\log{(f)}^{2} and \\int (\\mu_{0}{(f)} + \\log{(f)})^{2} df = \\int 4 \\log{(f)}^{2} df and - 2 \\log{(f)} + \\int (\\mu_{0}{(f)} + \\log{(f)})^{2} df = - 2 \\log{(f)} + \\int 4 \\log{(f)}^{2} df and - \\mu_{0}{(f)} - \\log{(f)} + \\int (\\mu_{0}{(f)} + \\log{(f)})^{2} df = - \\mu_{0}{(f)} - \\log{(f)} + \\int 4 \\log{(f)}^{2} df and - 2 \\mu_{0}{(f)} + \\int (\\mu_{0}{(f)} + \\log{(f)})^{2} df = - 2 \\mu_{0}{(f)} + \\int 4 \\log{(f)}^{2} df", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["add", 1, "log(Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Mul(Integer(2), log(Symbol('f', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)), Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True))))"], [["minus", 4, "Mul(Integer(2), log(Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), log(Symbol('f', commutative=True))), Integral(Pow(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Integer(2), log(Symbol('f', commutative=True))), Integral(Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True))), Mul(Integer(-1), log(Symbol('f', commutative=True))), Integral(Pow(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True))), Mul(Integer(-1), log(Symbol('f', commutative=True))), Integral(Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\mu_0')(Symbol('f', commutative=True))), Integral(Pow(Add(Function('\\\\mu_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mu_0')(Symbol('f', commutative=True))), Integral(Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(A_{y})} = \\log{(\\sin{(A_{y})})}, then obtain A_{y} + \\log{(\\sigma_{x}{(A_{y})} - \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})})} + \\sin{(A_{y})} = A_{y} + \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})}", "derivation": "\\sigma_{x}{(A_{y})} = \\log{(\\sin{(A_{y})})} and A_{y} + \\sigma_{x}{(A_{y})} = A_{y} + \\log{(\\sin{(A_{y})})} and A_{y} + \\sigma_{x}{(A_{y})} + \\sin{(A_{y})} = A_{y} + \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})} and \\sigma_{x}{(A_{y})} - \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})} = \\sin{(A_{y})} and A_{y} + \\sigma_{x}{(A_{y})} = A_{y} + \\log{(\\sigma_{x}{(A_{y})} - \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})})} and A_{y} + \\log{(\\sigma_{x}{(A_{y})} - \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})})} + \\sin{(A_{y})} = A_{y} + \\log{(\\sin{(A_{y})})} + \\sin{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('A_y', commutative=True)), log(sin(Symbol('A_y', commutative=True))))"], [["add", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), log(sin(Symbol('A_y', commutative=True)))))"], [["add", 2, "sin(Symbol('A_y', commutative=True))"], "Equality(Add(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), log(sin(Symbol('A_y', commutative=True))), sin(Symbol('A_y', commutative=True))))"], [["minus", 3, "Add(Symbol('A_y', commutative=True), log(sin(Symbol('A_y', commutative=True))))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('A_y', commutative=True)), Mul(Integer(-1), log(sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))), sin(Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), log(Add(Function('\\\\sigma_x')(Symbol('A_y', commutative=True)), Mul(Integer(-1), log(sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('A_y', commutative=True), log(Add(Function('\\\\sigma_x')(Symbol('A_y', commutative=True)), Mul(Integer(-1), log(sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), log(sin(Symbol('A_y', commutative=True))), sin(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given k{(F_{g})} = \\sin{(F_{g})} and \\operatorname{A_{2}}{(F_{g})} = (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} \\sin{(F_{g})}, then obtain \\operatorname{A_{2}}{(F_{g})} = (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} (\\frac{(\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{- F_{g}} k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} \\sin{(F_{g})}", "derivation": "k{(F_{g})} = \\sin{(F_{g})} and \\frac{k{(F_{g})}}{\\sin{(F_{g})}} = 1 and (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} = 1 and (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} \\sin{(F_{g})} = \\sin{(F_{g})} and \\operatorname{A_{2}}{(F_{g})} = (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} \\sin{(F_{g})} and \\operatorname{A_{2}}{(F_{g})} = (\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} (\\frac{(\\frac{k{(F_{g})}}{\\sin{(F_{g})}})^{- F_{g}} k{(F_{g})}}{\\sin{(F_{g})}})^{F_{g}} \\sin{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["divide", 1, "sin(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Symbol('F_g', commutative=True)), Integer(1))"], [["times", 3, "sin(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))), sin(Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('F_g', commutative=True)), Mul(Pow(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('A_2')(Symbol('F_g', commutative=True)), Mul(Pow(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Symbol('F_g', commutative=True)), Pow(Mul(Pow(Mul(Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('F_g', commutative=True))), Function('k')(Symbol('F_g', commutative=True)), Pow(sin(Symbol('F_g', commutative=True)), Integer(-1))), Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given z{(\\omega,\\eta^{\\prime})} = \\log{(\\eta^{\\prime} + \\omega)}, then obtain (\\frac{z^{\\eta^{\\prime}}{(\\omega,\\eta^{\\prime})}}{\\log{(\\eta^{\\prime} + \\omega)}})^{\\eta^{\\prime}} = (\\frac{\\log{(\\eta^{\\prime} + \\omega)}^{\\eta^{\\prime}}}{\\log{(\\eta^{\\prime} + \\omega)}})^{\\eta^{\\prime}}", "derivation": "z{(\\omega,\\eta^{\\prime})} = \\log{(\\eta^{\\prime} + \\omega)} and z^{\\eta^{\\prime}}{(\\omega,\\eta^{\\prime})} = \\log{(\\eta^{\\prime} + \\omega)}^{\\eta^{\\prime}} and \\frac{z^{\\eta^{\\prime}}{(\\omega,\\eta^{\\prime})}}{\\log{(\\eta^{\\prime} + \\omega)}} = \\frac{\\log{(\\eta^{\\prime} + \\omega)}^{\\eta^{\\prime}}}{\\log{(\\eta^{\\prime} + \\omega)}} and (\\frac{z^{\\eta^{\\prime}}{(\\omega,\\eta^{\\prime})}}{\\log{(\\eta^{\\prime} + \\omega)}})^{\\eta^{\\prime}} = (\\frac{\\log{(\\eta^{\\prime} + \\omega)}^{\\eta^{\\prime}}}{\\log{(\\eta^{\\prime} + \\omega)}})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 2, "log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1))), Mul(Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\operatorname{v_{2}}{(V_{\\mathbf{B}})} = \\frac{\\hat{H}_{\\lambda}{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}}, then obtain \\operatorname{v_{2}}{(V_{\\mathbf{B}})} = 1", "derivation": "\\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\frac{\\hat{H}_{\\lambda}{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} = 1 and \\operatorname{v_{2}}{(V_{\\mathbf{B}})} = \\frac{\\hat{H}_{\\lambda}{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} and \\operatorname{v_{2}}{(V_{\\mathbf{B}})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 1, "cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('v_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\rho{(\\eta,\\mathbf{D})} = \\eta \\mathbf{D}, then obtain \\mathbf{D} (- \\mathbf{D} + \\rho{(\\eta,\\mathbf{D})})^{\\eta} = \\mathbf{D} (\\eta \\mathbf{D} - \\mathbf{D})^{\\eta}", "derivation": "\\rho{(\\eta,\\mathbf{D})} = \\eta \\mathbf{D} and - \\mathbf{D} + \\rho{(\\eta,\\mathbf{D})} = \\eta \\mathbf{D} - \\mathbf{D} and (- \\mathbf{D} + \\rho{(\\eta,\\mathbf{D})})^{\\eta} = (\\eta \\mathbf{D} - \\mathbf{D})^{\\eta} and \\mathbf{D} (- \\mathbf{D} + \\rho{(\\eta,\\mathbf{D})})^{\\eta} = \\mathbf{D} (\\eta \\mathbf{D} - \\mathbf{D})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["times", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\eta', commutative=True))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\varphi)} = \\sin{(\\varphi)}, then obtain y^{\\prime} + \\bar{\\h}{(\\varphi)} - \\int (y^{\\prime} + \\sin{(\\varphi)}) d\\varphi + 1 = y^{\\prime} + \\sin{(\\varphi)} - \\int (y^{\\prime} + \\sin{(\\varphi)}) d\\varphi + 1", "derivation": "\\bar{\\h}{(\\varphi)} = \\sin{(\\varphi)} and y^{\\prime} + \\bar{\\h}{(\\varphi)} = y^{\\prime} + \\sin{(\\varphi)} and \\int (y^{\\prime} + \\bar{\\h}{(\\varphi)}) d\\varphi = \\int (y^{\\prime} + \\sin{(\\varphi)}) d\\varphi and y^{\\prime} + \\bar{\\h}{(\\varphi)} + 1 = y^{\\prime} + \\sin{(\\varphi)} + 1 and y^{\\prime} + \\bar{\\h}{(\\varphi)} - \\int (y^{\\prime} + \\bar{\\h}{(\\varphi)}) d\\varphi + 1 = y^{\\prime} + \\sin{(\\varphi)} - \\int (y^{\\prime} + \\bar{\\h}{(\\varphi)}) d\\varphi + 1 and y^{\\prime} + \\bar{\\h}{(\\varphi)} - \\int (y^{\\prime} + \\sin{(\\varphi)}) d\\varphi + 1 = y^{\\prime} + \\sin{(\\varphi)} - \\int (y^{\\prime} + \\sin{(\\varphi)}) d\\varphi + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True)), Integer(1)))"], [["minus", 4, "Integral(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Integer(1)), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hbar')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Integer(1)), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\nabla{(\\Psi_{nl},n)} = \\Psi_{nl} n, then obtain \\frac{\\partial}{\\partial n} (- (2 \\Psi_{nl} n)^{\\Psi_{nl}} + (\\Psi_{nl} n + \\nabla{(\\Psi_{nl},n)})^{\\Psi_{nl}}) = \\frac{d}{d n} 0", "derivation": "\\nabla{(\\Psi_{nl},n)} = \\Psi_{nl} n and \\Psi_{nl} n + \\nabla{(\\Psi_{nl},n)} = 2 \\Psi_{nl} n and (\\Psi_{nl} n + \\nabla{(\\Psi_{nl},n)})^{\\Psi_{nl}} = (2 \\Psi_{nl} n)^{\\Psi_{nl}} and - (2 \\Psi_{nl} n)^{\\Psi_{nl}} + (\\Psi_{nl} n + \\nabla{(\\Psi_{nl},n)})^{\\Psi_{nl}} = 0 and \\frac{\\partial}{\\partial n} (- (2 \\Psi_{nl} n)^{\\Psi_{nl}} + (\\Psi_{nl} n + \\nabla{(\\Psi_{nl},n)})^{\\Psi_{nl}}) = \\frac{d}{d n} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Function('\\\\nabla')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)))"], [["power", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Function('\\\\nabla')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 3, "Pow(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Function('\\\\nabla')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True)), Function('\\\\nabla')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(\\Omega)} = \\cos{(\\Omega)}, then derive C_{1} + \\Omega = \\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega, then obtain \\sin{(1)} = \\sin{(\\frac{\\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega}{C_{1} + \\Omega})}", "derivation": "\\pi{(\\Omega)} = \\cos{(\\Omega)} and 1 = \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} and \\int 1 d\\Omega = \\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega and C_{1} + \\Omega = \\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega and C_{1} + \\Omega = \\int 1 d\\Omega and 1 = \\frac{\\int 1 d\\Omega}{C_{1} + \\Omega} and 1 = \\frac{\\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega}{C_{1} + \\Omega} and \\sin{(1)} = \\sin{(\\frac{\\int \\frac{\\cos{(\\Omega)}}{\\pi{(\\Omega)}} d\\Omega}{C_{1} + \\Omega})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "Function('\\\\pi')(Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 5, "Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(1), Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["sin", 7], "Equality(sin(Integer(1)), sin(Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given C{(\\mathbf{s},\\hat{H}_{\\lambda})} = \\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})}, then obtain \\int (C{(\\mathbf{s},\\hat{H}_{\\lambda})} + 1)^{\\mathbf{s}} d\\mathbf{s} + 1 = \\int (\\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} + 1)^{\\mathbf{s}} d\\mathbf{s} + 1", "derivation": "C{(\\mathbf{s},\\hat{H}_{\\lambda})} = \\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} and C{(\\mathbf{s},\\hat{H}_{\\lambda})} + 1 = \\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} + 1 and (C{(\\mathbf{s},\\hat{H}_{\\lambda})} + 1)^{\\mathbf{s}} = (\\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} + 1)^{\\mathbf{s}} and \\int (C{(\\mathbf{s},\\hat{H}_{\\lambda})} + 1)^{\\mathbf{s}} d\\mathbf{s} = \\int (\\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} + 1)^{\\mathbf{s}} d\\mathbf{s} and \\int (C{(\\mathbf{s},\\hat{H}_{\\lambda})} + 1)^{\\mathbf{s}} d\\mathbf{s} + 1 = \\int (\\cos{(\\frac{\\mathbf{s}}{\\hat{H}_{\\lambda}})} + 1)^{\\mathbf{s}} d\\mathbf{s} + 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('C')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Add(cos(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Function('C')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(cos(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Pow(Add(Function('C')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Pow(Add(cos(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Pow(Add(Function('C')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Add(Integral(Pow(Add(cos(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{v}{(f_{\\mathbf{p}},s)} = f_{\\mathbf{p}} s, then obtain \\frac{\\partial}{\\partial s} (s + \\mathbf{v}^{2}{(f_{\\mathbf{p}},s)}) = \\frac{\\partial}{\\partial s} (f_{\\mathbf{p}} s \\mathbf{v}{(f_{\\mathbf{p}},s)} + s)", "derivation": "\\mathbf{v}{(f_{\\mathbf{p}},s)} = f_{\\mathbf{p}} s and \\mathbf{v}^{2}{(f_{\\mathbf{p}},s)} = f_{\\mathbf{p}} s \\mathbf{v}{(f_{\\mathbf{p}},s)} and s + \\mathbf{v}^{2}{(f_{\\mathbf{p}},s)} = f_{\\mathbf{p}} s \\mathbf{v}{(f_{\\mathbf{p}},s)} + s and \\frac{\\partial}{\\partial s} (s + \\mathbf{v}^{2}{(f_{\\mathbf{p}},s)}) = \\frac{\\partial}{\\partial s} (f_{\\mathbf{p}} s \\mathbf{v}{(f_{\\mathbf{p}},s)} + s)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True)), Integer(2)), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True), Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True))))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Add(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True), Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Symbol('s', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True), Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{A})} = \\sin{(e^{\\mathbf{A}})}, then obtain - \\mathbf{A} + (\\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} - \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})})^{2} + 1 = 1 - \\mathbf{A}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{A})} = \\sin{(e^{\\mathbf{A}})} and \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} = \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})} and \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} - \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})} = 0 and (\\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} - \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})})^{2} = 0 and (\\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} - \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})})^{2} + 1 = 1 and - \\mathbf{A} + (\\operatorname{V_{\\mathbf{B}}}^{\\mathbf{A}}{(\\mathbf{A})} - \\sin^{\\mathbf{A}}{(e^{\\mathbf{A}})})^{2} + 1 = 1 - \\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(exp(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 2, "Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"], [["times", 3, "Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"], "Equality(Pow(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(2)), Integer(0))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Pow(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(2)), Integer(1)), Integer(1))"], [["minus", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(2)), Integer(1)), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\theta)} = \\cos{(\\theta)} and \\operatorname{y^{\\prime}}{(\\theta)} = \\cos{(\\theta)}, then obtain - e^{\\cos{(\\theta)}} + \\frac{d}{d \\theta} e^{\\cos{(\\theta)}} = - e^{\\cos{(\\theta)}} + \\frac{d}{d \\theta} e^{\\hat{H}{(\\theta)}}", "derivation": "\\hat{H}{(\\theta)} = \\cos{(\\theta)} and \\operatorname{y^{\\prime}}{(\\theta)} = \\cos{(\\theta)} and \\operatorname{y^{\\prime}}{(\\theta)} = \\hat{H}{(\\theta)} and e^{\\operatorname{y^{\\prime}}{(\\theta)}} = e^{\\hat{H}{(\\theta)}} and \\frac{d}{d \\theta} e^{\\operatorname{y^{\\prime}}{(\\theta)}} = \\frac{d}{d \\theta} e^{\\hat{H}{(\\theta)}} and \\frac{d}{d \\theta} e^{\\cos{(\\theta)}} = \\frac{d}{d \\theta} e^{\\hat{H}{(\\theta)}} and - e^{\\cos{(\\theta)}} + \\frac{d}{d \\theta} e^{\\cos{(\\theta)}} = - e^{\\cos{(\\theta)}} + \\frac{d}{d \\theta} e^{\\hat{H}{(\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\theta', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True)))"], [["exp", 3], "Equality(exp(Function('y^{\\\\prime}')(Symbol('\\\\theta', commutative=True))), exp(Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(exp(Function('y^{\\\\prime}')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(exp(cos(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 6, "exp(cos(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(cos(Symbol('\\\\theta', commutative=True)))), Derivative(exp(cos(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(cos(Symbol('\\\\theta', commutative=True)))), Derivative(exp(Function('\\\\hat{H}')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain (\\frac{d}{d a^{\\dagger}} x{(a^{\\dagger})} + \\frac{1}{a^{\\dagger}})^{2 a^{\\dagger}} = (\\frac{2}{a^{\\dagger}})^{2 a^{\\dagger}}", "derivation": "x{(a^{\\dagger})} = \\log{(a^{\\dagger})} and x{(a^{\\dagger})} + \\log{(a^{\\dagger})} = 2 \\log{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} (x{(a^{\\dagger})} + \\log{(a^{\\dagger})}) = \\frac{d}{d a^{\\dagger}} 2 \\log{(a^{\\dagger})} and (\\frac{d}{d a^{\\dagger}} (x{(a^{\\dagger})} + \\log{(a^{\\dagger})}))^{a^{\\dagger}} = (\\frac{d}{d a^{\\dagger}} 2 \\log{(a^{\\dagger})})^{a^{\\dagger}} and (\\frac{d}{d a^{\\dagger}} (x{(a^{\\dagger})} + \\log{(a^{\\dagger})}))^{2 a^{\\dagger}} = (\\frac{d}{d a^{\\dagger}} 2 \\log{(a^{\\dagger})})^{2 a^{\\dagger}} and (\\frac{d}{d a^{\\dagger}} x{(a^{\\dagger})} + \\frac{1}{a^{\\dagger}})^{2 a^{\\dagger}} = (\\frac{2}{a^{\\dagger}})^{2 a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 1, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Derivative(Add(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(Mul(Integer(2), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Derivative(Add(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Pow(Derivative(Mul(Integer(2), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Derivative(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Pow(Mul(Integer(2), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(v_{t})} = \\log{(\\sin{(v_{t})})}, then obtain \\frac{d^{2}}{d v_{t}^{2}} \\mu_{0}{(v_{t})} = -1 - \\frac{\\cos^{2}{(v_{t})}}{\\sin^{2}{(v_{t})}}", "derivation": "\\mu_{0}{(v_{t})} = \\log{(\\sin{(v_{t})})} and \\frac{d}{d v_{t}} \\mu_{0}{(v_{t})} = \\frac{d}{d v_{t}} \\log{(\\sin{(v_{t})})} and \\frac{d^{2}}{d v_{t}^{2}} \\mu_{0}{(v_{t})} = \\frac{d^{2}}{d v_{t}^{2}} \\log{(\\sin{(v_{t})})} and \\frac{d^{2}}{d v_{t}^{2}} \\mu_{0}{(v_{t})} = -1 - \\frac{\\cos^{2}{(v_{t})}}{\\sin^{2}{(v_{t})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_t', commutative=True)), log(sin(Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(log(sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(2))), Derivative(log(sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mu_0')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(2))), Mul(Integer(-1), Add(Integer(1), Mul(Pow(sin(Symbol('v_t', commutative=True)), Integer(-2)), Pow(cos(Symbol('v_t', commutative=True)), Integer(2))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\pi)} = \\int \\log{(\\pi)} d\\pi, then derive \\operatorname{J_{\\varepsilon}}{(\\pi)} = S + \\pi \\log{(\\pi)} - \\pi, then derive S - \\pi = \\hat{x}_0 - \\pi, then derive \\hat{x}_0 - \\pi + \\operatorname{J_{\\varepsilon}}{(\\pi)} = \\hat{x}_0 + \\phi_1 + \\pi \\log{(\\pi)} - 2 \\pi, then obtain \\hat{x}_0 + \\phi_1 + \\pi \\log{(\\pi)} - 2 \\pi = \\hat{x}_0 - \\pi + \\int \\log{(\\pi)} d\\pi", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\pi)} = \\int \\log{(\\pi)} d\\pi and \\operatorname{J_{\\varepsilon}}{(\\pi)} = S + \\pi \\log{(\\pi)} - \\pi and S + \\pi \\log{(\\pi)} - \\pi = \\int \\log{(\\pi)} d\\pi and S - \\pi = - \\pi \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi and S - \\pi + \\operatorname{J_{\\varepsilon}}{(\\pi)} = S - \\pi + \\int \\log{(\\pi)} d\\pi and S - \\pi = \\hat{x}_0 - \\pi and \\hat{x}_0 - \\pi + \\operatorname{J_{\\varepsilon}}{(\\pi)} = \\hat{x}_0 - \\pi + \\int \\log{(\\pi)} d\\pi and \\hat{x}_0 - \\pi + \\operatorname{J_{\\varepsilon}}{(\\pi)} = \\hat{x}_0 + \\phi_1 + \\pi \\log{(\\pi)} - 2 \\pi and \\hat{x}_0 + \\phi_1 + \\pi \\log{(\\pi)} - 2 \\pi = \\hat{x}_0 - \\pi + \\int \\log{(\\pi)} d\\pi", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Add(Symbol('S', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('S', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["add", 1, "Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain \\frac{d}{d \\mathbf{J}_M} 0^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} 1", "derivation": "\\hat{H}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and 0 = - \\hat{H}{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)} and 0^{\\mathbf{J}_M} = (- \\hat{H}{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)})^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} 0^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} (- \\hat{H}{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)})^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} (- \\hat{H}{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)})^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} 1 and \\frac{d}{d \\mathbf{J}_M} 0^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}, then derive r{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}} = 0, then obtain (\\frac{d}{d \\mathbf{F}} (\\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}}))^{\\mathbf{F}} = (\\frac{d}{d \\mathbf{F}} 0)^{\\mathbf{F}}", "derivation": "r{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and r{(\\mathbf{F})} - \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} = 0 and r{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}} = 0 and \\frac{d}{d \\mathbf{F}} (r{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}}) = \\frac{d}{d \\mathbf{F}} 0 and \\frac{d}{d \\mathbf{F}} (\\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}}) = \\frac{d}{d \\mathbf{F}} 0 and (\\frac{d}{d \\mathbf{F}} (\\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} - \\frac{1}{\\mathbf{F}}))^{\\mathbf{F}} = (\\frac{d}{d \\mathbf{F}} 0)^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))"], "Equality(Add(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Derivative(Add(Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(v_{x})} = \\sin{(\\log{(v_{x})})} and A{(v_{x})} = \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}}, then obtain - \\operatorname{J_{\\varepsilon}}{(v_{x})} + \\int A{(v_{x})} dv_{x} = - \\operatorname{J_{\\varepsilon}}{(v_{x})} + \\int \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}} dv_{x}", "derivation": "\\operatorname{J_{\\varepsilon}}{(v_{x})} = \\sin{(\\log{(v_{x})})} and A{(v_{x})} = \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}} and \\int A{(v_{x})} dv_{x} = \\int \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}} dv_{x} and - \\sin{(\\log{(v_{x})})} + \\int A{(v_{x})} dv_{x} = - \\sin{(\\log{(v_{x})})} + \\int \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}} dv_{x} and - \\operatorname{J_{\\varepsilon}}{(v_{x})} + \\int A{(v_{x})} dv_{x} = - \\operatorname{J_{\\varepsilon}}{(v_{x})} + \\int \\frac{\\sin{(\\log{(v_{x})})}}{\\log{(v_{x})}} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), sin(log(Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('v_x', commutative=True)), Mul(Pow(log(Symbol('v_x', commutative=True)), Integer(-1)), sin(log(Symbol('v_x', commutative=True)))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('A')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Pow(log(Symbol('v_x', commutative=True)), Integer(-1)), sin(log(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["minus", 3, "sin(log(Symbol('v_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(log(Symbol('v_x', commutative=True)))), Integral(Function('A')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), sin(log(Symbol('v_x', commutative=True)))), Integral(Mul(Pow(log(Symbol('v_x', commutative=True)), Integer(-1)), sin(log(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Integral(Function('A')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Integral(Mul(Pow(log(Symbol('v_x', commutative=True)), Integer(-1)), sin(log(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(I,\\mathbf{E})} = I - \\mathbf{E} and q{(v_{t},\\rho)} = v_{t}^{\\rho}, then obtain - \\mathbf{E} q{(v_{t},\\rho)} + \\mathbf{E} + \\varepsilon_{0}{(I,\\mathbf{E})} = - \\mathbf{E} v_{t}^{\\rho} + \\mathbf{E} + \\varepsilon_{0}{(I,\\mathbf{E})}", "derivation": "\\varepsilon_{0}{(I,\\mathbf{E})} = I - \\mathbf{E} and \\mathbf{E} + \\varepsilon_{0}{(I,\\mathbf{E})} = I and q{(v_{t},\\rho)} = v_{t}^{\\rho} and - \\mathbf{E} q{(v_{t},\\rho)} = - \\mathbf{E} v_{t}^{\\rho} and I - \\mathbf{E} q{(v_{t},\\rho)} = I - \\mathbf{E} v_{t}^{\\rho} and - \\mathbf{E} q{(v_{t},\\rho)} + \\mathbf{E} + \\varepsilon_{0}{(I,\\mathbf{E})} = - \\mathbf{E} v_{t}^{\\rho} + \\mathbf{E} + \\varepsilon_{0}{(I,\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\varepsilon_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('I', commutative=True))"], ["get_premise", "Equality(Function('q')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Function('q')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["add", 4, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Function('q')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Function('q')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\varepsilon_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\varepsilon_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given h{(U)} = \\log{(U)}, then obtain \\frac{\\frac{d}{d U} h{(U)} - \\frac{1}{U}}{\\varphi} = 0", "derivation": "h{(U)} = \\log{(U)} and h{(U)} - \\log{(U)} = 0 and \\frac{d}{d U} (h{(U)} - \\log{(U)}) = \\frac{d}{d U} 0 and \\frac{\\frac{d}{d U} (h{(U)} - \\log{(U)})}{\\varphi} = \\frac{\\frac{d}{d U} 0}{\\varphi} and \\frac{\\frac{d}{d U} h{(U)} - \\frac{1}{U}}{\\varphi} = 0", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["minus", 1, "log(Symbol('U', commutative=True))"], "Equality(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Derivative(Function('h')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1))))), Integer(0))"]]}, {"prompt": "Given W{(\\nabla,\\dot{y})} = \\dot{y} \\nabla, then derive \\frac{\\frac{\\partial}{\\partial \\dot{y}} W{(\\nabla,\\dot{y})}}{\\dot{y} \\nabla} - \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla} = 0, then obtain \\frac{\\frac{\\partial}{\\partial \\dot{y}} W{(\\nabla,\\dot{y})}}{W{(\\nabla,\\dot{y})}} - \\frac{1}{\\dot{y}} + \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla} = \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla}", "derivation": "W{(\\nabla,\\dot{y})} = \\dot{y} \\nabla and \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y} \\nabla} = 1 and \\frac{\\partial}{\\partial \\dot{y}} \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y} \\nabla} = \\frac{d}{d \\dot{y}} 1 and \\frac{\\frac{\\partial}{\\partial \\dot{y}} W{(\\nabla,\\dot{y})}}{\\dot{y} \\nabla} - \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla} = 0 and \\frac{\\frac{\\partial}{\\partial \\dot{y}} W{(\\nabla,\\dot{y})}}{W{(\\nabla,\\dot{y})}} - \\frac{1}{\\dot{y}} = 0 and \\frac{\\frac{\\partial}{\\partial \\dot{y}} W{(\\nabla,\\dot{y})}}{W{(\\nabla,\\dot{y})}} - \\frac{1}{\\dot{y}} + \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla} = \\frac{W{(\\nabla,\\dot{y})}}{\\dot{y}^{2} \\nabla}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Pow(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))), Integer(0))"], [["minus", 5, "Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Pow(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\varphi^*,A_{y})} = \\frac{\\partial}{\\partial \\varphi^*} A_{y} \\varphi^*, then obtain A_{y} \\varphi^* + \\frac{\\frac{\\partial}{\\partial A_{y}} \\phi_{2}{(\\varphi^*,A_{y})}}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}} = A_{y} \\varphi^* + \\frac{\\frac{\\partial^{2}}{\\partial A_{y}\\partial \\varphi^*} A_{y} \\varphi^*}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}}", "derivation": "\\phi_{2}{(\\varphi^*,A_{y})} = \\frac{\\partial}{\\partial \\varphi^*} A_{y} \\varphi^* and \\frac{\\partial}{\\partial A_{y}} \\phi_{2}{(\\varphi^*,A_{y})} = \\frac{\\partial^{2}}{\\partial A_{y}\\partial \\varphi^*} A_{y} \\varphi^* and \\frac{\\frac{\\partial}{\\partial A_{y}} \\phi_{2}{(\\varphi^*,A_{y})}}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}} = \\frac{\\frac{\\partial^{2}}{\\partial A_{y}\\partial \\varphi^*} A_{y} \\varphi^*}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}} and A_{y} \\varphi^* + \\frac{\\frac{\\partial}{\\partial A_{y}} \\phi_{2}{(\\varphi^*,A_{y})}}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}} = A_{y} \\varphi^* + \\frac{\\frac{\\partial^{2}}{\\partial A_{y}\\partial \\varphi^*} A_{y} \\varphi^*}{\\mathbf{v}{(\\dot{\\mathbf{r}},\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_y', commutative=True)), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\mathbf{v}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["add", 3, "Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))), Add(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('A_y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(Z)} = \\frac{d}{d Z} \\log{(Z)}, then derive \\operatorname{M_{E}}{(Z)} + \\log{(Z)} = \\log{(Z)} + \\frac{1}{Z}, then obtain \\log{(\\log{(Z)} + \\frac{d}{d Z} \\log{(Z)})} = \\log{(\\log{(Z)} + \\frac{1}{Z})}", "derivation": "\\operatorname{M_{E}}{(Z)} = \\frac{d}{d Z} \\log{(Z)} and \\operatorname{M_{E}}{(Z)} + \\log{(Z)} = \\log{(Z)} + \\frac{d}{d Z} \\log{(Z)} and \\log{(\\operatorname{M_{E}}{(Z)} + \\log{(Z)})} = \\log{(\\log{(Z)} + \\frac{d}{d Z} \\log{(Z)})} and \\operatorname{M_{E}}{(Z)} + \\log{(Z)} = \\log{(Z)} + \\frac{1}{Z} and \\log{(Z)} + \\frac{d}{d Z} \\log{(Z)} = \\log{(Z)} + \\frac{1}{Z} and \\log{(\\operatorname{M_{E}}{(Z)} + \\log{(Z)})} = \\log{(\\log{(Z)} + \\frac{1}{Z})} and \\log{(\\log{(Z)} + \\frac{d}{d Z} \\log{(Z)})} = \\log{(\\log{(Z)} + \\frac{1}{Z})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 1, "log(Symbol('Z', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Add(log(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["log", 2], "Equality(log(Add(Function('M_E')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))), log(Add(log(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('M_E')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Add(log(Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(log(Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Add(Function('M_E')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))), log(Add(log(Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(log(Add(log(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))), log(Add(log(Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given k{(\\varphi)} = \\log{(e^{\\varphi})} and \\hat{p}_0{(\\varphi)} = \\int k^{2}{(\\varphi)} d\\varphi, then obtain \\frac{\\hat{p}_0{(\\varphi)}}{k^{2}{(\\varphi)}} = \\frac{\\int k{(\\varphi)} \\log{(e^{\\varphi})} d\\varphi}{k^{2}{(\\varphi)}}", "derivation": "k{(\\varphi)} = \\log{(e^{\\varphi})} and k^{2}{(\\varphi)} = k{(\\varphi)} \\log{(e^{\\varphi})} and \\int k^{2}{(\\varphi)} d\\varphi = \\int k{(\\varphi)} \\log{(e^{\\varphi})} d\\varphi and \\hat{p}_0{(\\varphi)} = \\int k^{2}{(\\varphi)} d\\varphi and \\frac{\\int k^{2}{(\\varphi)} d\\varphi}{k^{2}{(\\varphi)}} = \\frac{\\int k{(\\varphi)} \\log{(e^{\\varphi})} d\\varphi}{k^{2}{(\\varphi)}} and \\frac{\\hat{p}_0{(\\varphi)}}{k^{2}{(\\varphi)}} = \\frac{\\int k{(\\varphi)} \\log{(e^{\\varphi})} d\\varphi}{k^{2}{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True))))"], [["times", 1, "Function('k')(Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Function('k')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\varphi', commutative=True)), Integral(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["divide", 3, "Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(-2)), Integral(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(-2)), Integral(Mul(Function('k')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\varphi', commutative=True)), Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(-2))), Mul(Pow(Function('k')(Symbol('\\\\varphi', commutative=True)), Integer(-2)), Integral(Mul(Function('k')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(A_{x})} = \\log{(A_{x})}, then derive (v_{1} + \\mathbf{P}{(A_{x})})^{A_{x}} = (r + \\log{(A_{x})})^{A_{x}}, then obtain (v_{1} + \\log{(A_{x})})^{A_{x}} = (r + \\log{(A_{x})})^{A_{x}}", "derivation": "\\mathbf{P}{(A_{x})} = \\log{(A_{x})} and \\frac{d}{d A_{x}} \\mathbf{P}{(A_{x})} = \\frac{d}{d A_{x}} \\log{(A_{x})} and \\int \\frac{d}{d A_{x}} \\mathbf{P}{(A_{x})} dA_{x} = \\int \\frac{d}{d A_{x}} \\log{(A_{x})} dA_{x} and (\\int \\frac{d}{d A_{x}} \\mathbf{P}{(A_{x})} dA_{x})^{A_{x}} = (\\int \\frac{d}{d A_{x}} \\log{(A_{x})} dA_{x})^{A_{x}} and (v_{1} + \\mathbf{P}{(A_{x})})^{A_{x}} = (r + \\log{(A_{x})})^{A_{x}} and (v_{1} + \\log{(A_{x})})^{A_{x}} = (r + \\log{(A_{x})})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Integral(Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integral(Derivative(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('v_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Add(Symbol('r', commutative=True), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Symbol('v_1', commutative=True), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Add(Symbol('r', commutative=True), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given E{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}, then derive E{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain \\frac{d}{d a^{\\dagger}} E{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} E{(a^{\\dagger})}", "derivation": "E{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and E{(a^{\\dagger})} = e^{a^{\\dagger}} and E{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} E{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} e^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} E{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} E{(a^{\\dagger})}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\varphi{(z)} = e^{z}, then derive p + \\int \\varphi{(z)} dz = \\hat{p}_0 + p + e^{z}, then obtain \\frac{\\partial}{\\partial p} (p + \\int \\varphi{(z)} dz) = 1", "derivation": "\\varphi{(z)} = e^{z} and \\int \\varphi{(z)} dz = \\int e^{z} dz and p + \\int \\varphi{(z)} dz = p + \\int e^{z} dz and p + \\int \\varphi{(z)} dz = \\hat{p}_0 + p + e^{z} and p + \\int \\varphi{(z)} dz = \\hat{p}_0 + p + \\varphi{(z)} and \\frac{\\partial}{\\partial p} (p + \\int \\varphi{(z)} dz) = \\frac{\\partial}{\\partial p} (\\hat{p}_0 + p + \\varphi{(z)}) and \\frac{\\partial}{\\partial p} (p + \\int \\varphi{(z)} dz) = 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('p', commutative=True))"], "Equality(Add(Symbol('p', commutative=True), Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('p', commutative=True), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('p', commutative=True), Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('p', commutative=True), exp(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('p', commutative=True), Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('p', commutative=True), Function('\\\\varphi')(Symbol('z', commutative=True))))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Symbol('p', commutative=True), Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('p', commutative=True), Function('\\\\varphi')(Symbol('z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Add(Symbol('p', commutative=True), Integral(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given n{(p)} = \\cos{(p)} and \\operatorname{C_{d}}{(p)} = \\frac{d}{d p} n{(p)}, then obtain \\operatorname{C_{d}}{(p)} = \\frac{d}{d p} \\cos{(p)}", "derivation": "n{(p)} = \\cos{(p)} and \\frac{d}{d p} n{(p)} = \\frac{d}{d p} \\cos{(p)} and \\operatorname{C_{d}}{(p)} = \\frac{d}{d p} n{(p)} and \\operatorname{C_{d}}{(p)} = \\frac{d}{d p} \\cos{(p)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('p', commutative=True)), Derivative(Function('n')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('C_d')(Symbol('p', commutative=True)), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(\\varphi^*,\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\varphi^* and \\mathbf{M}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}, then obtain \\frac{\\mathbf{M}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - \\varphi^* + (- \\hat{\\mathbf{r}} + \\varphi^*)^{\\hat{\\mathbf{r}}}} = \\frac{\\hat{\\mathbf{r}}}{\\hat{\\mathbf{r}} - \\varphi^* + (- \\hat{\\mathbf{r}} + \\varphi^*)^{\\hat{\\mathbf{r}}}}", "derivation": "M{(\\varphi^*,\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\varphi^* and \\mathbf{M}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} and \\frac{\\mathbf{M}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - \\varphi^* + M^{\\hat{\\mathbf{r}}}{(\\varphi^*,\\hat{\\mathbf{r}})}} = \\frac{\\hat{\\mathbf{r}}}{\\hat{\\mathbf{r}} - \\varphi^* + M^{\\hat{\\mathbf{r}}}{(\\varphi^*,\\hat{\\mathbf{r}})}} and \\frac{\\mathbf{M}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - \\varphi^* + (- \\hat{\\mathbf{r}} + \\varphi^*)^{\\hat{\\mathbf{r}}}} = \\frac{\\hat{\\mathbf{r}}}{\\hat{\\mathbf{r}} - \\varphi^* + (- \\hat{\\mathbf{r}} + \\varphi^*)^{\\hat{\\mathbf{r}}}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], [["divide", 2, "Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('M')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('M')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('M')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given L{(\\hat{p})} = \\log{(\\hat{p})}, then obtain (\\hat{p} L{(\\hat{p})} + 2 \\log{(\\hat{p})})^{2} = (\\hat{p} \\log{(\\hat{p})} + 2 \\log{(\\hat{p})})^{2}", "derivation": "L{(\\hat{p})} = \\log{(\\hat{p})} and L{(\\hat{p})} + \\log{(\\hat{p})} = 2 \\log{(\\hat{p})} and \\hat{p} L{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} and \\hat{p} L{(\\hat{p})} + L{(\\hat{p})} + \\log{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} + L{(\\hat{p})} + \\log{(\\hat{p})} and \\hat{p} L{(\\hat{p})} + 2 \\log{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} + 2 \\log{(\\hat{p})} and (\\hat{p} L{(\\hat{p})} + 2 \\log{(\\hat{p})})^{2} = (\\hat{p} \\log{(\\hat{p})} + 2 \\log{(\\hat{p})})^{2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Function('L')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{p}', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 3, "Add(Function('L')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True))), Function('L')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Function('L')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{p}', commutative=True)))))"], [["power", 5, 2], "Equality(Pow(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{p}', commutative=True)))), Integer(2)), Pow(Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{p}', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given a{(P_{e})} = \\cos{(P_{e})}, then obtain P_{e} + \\int \\sin{((a{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e} = P_{e} + \\int \\sin{((\\cos{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e}", "derivation": "a{(P_{e})} = \\cos{(P_{e})} and a{(P_{e})} - 1 = \\cos{(P_{e})} - 1 and (a{(P_{e})} - 1) \\cos{(P_{e})} = (\\cos{(P_{e})} - 1) \\cos{(P_{e})} and \\sin{((a{(P_{e})} - 1) \\cos{(P_{e})})} = \\sin{((\\cos{(P_{e})} - 1) \\cos{(P_{e})})} and \\int \\sin{((a{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e} = \\int \\sin{((\\cos{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e} and P_{e} + \\int \\sin{((a{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e} = P_{e} + \\int \\sin{((\\cos{(P_{e})} - 1) \\cos{(P_{e})})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('a')(Symbol('P_e', commutative=True)), Integer(-1)), Add(cos(Symbol('P_e', commutative=True)), Integer(-1)))"], [["times", 2, "cos(Symbol('P_e', commutative=True))"], "Equality(Mul(Add(Function('a')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))), Mul(Add(cos(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))))"], [["sin", 3], "Equality(sin(Mul(Add(Function('a')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), sin(Mul(Add(cos(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(sin(Mul(Add(Function('a')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True))), Integral(sin(Mul(Add(cos(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True))))"], [["add", 5, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Integral(sin(Mul(Add(Function('a')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True)))), Add(Symbol('P_e', commutative=True), Integral(sin(Mul(Add(cos(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(v_{x},\\lambda)} = \\sin^{\\lambda}{(v_{x})} and \\mathbf{A}{(v_{x},\\lambda)} = 2 \\sin^{\\lambda}{(v_{x})}, then obtain \\frac{\\partial}{\\partial \\lambda} \\mathbf{A}{(v_{x},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} (\\mathbf{J}{(v_{x},\\lambda)} + \\sin^{\\lambda}{(v_{x})})", "derivation": "\\mathbf{J}{(v_{x},\\lambda)} = \\sin^{\\lambda}{(v_{x})} and \\mathbf{J}{(v_{x},\\lambda)} + \\sin^{\\lambda}{(v_{x})} = 2 \\sin^{\\lambda}{(v_{x})} and \\mathbf{A}{(v_{x},\\lambda)} = 2 \\sin^{\\lambda}{(v_{x})} and \\mathbf{A}{(v_{x},\\lambda)} = \\mathbf{J}{(v_{x},\\lambda)} + \\sin^{\\lambda}{(v_{x})} and \\frac{\\partial}{\\partial \\lambda} \\mathbf{A}{(v_{x},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} (\\mathbf{J}{(v_{x},\\lambda)} + \\sin^{\\lambda}{(v_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('v_x', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{E},A_{2})} = A_{2} \\mathbf{E} and G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} = \\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}}, then obtain - A_{2} \\mathbf{E} + A_{2} + \\int G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} d\\mathbf{S} = - A_{2} \\mathbf{E} + A_{2} + \\int (\\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}}) d\\mathbf{S}", "derivation": "\\mu_{0}{(\\mathbf{E},A_{2})} = A_{2} \\mathbf{E} and G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} = \\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}} and \\int G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} d\\mathbf{S} = \\int (\\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}}) d\\mathbf{S} and A_{2} - \\mu_{0}{(\\mathbf{E},A_{2})} + \\int G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} d\\mathbf{S} = A_{2} - \\mu_{0}{(\\mathbf{E},A_{2})} + \\int (\\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}}) d\\mathbf{S} and - A_{2} \\mathbf{E} + A_{2} + \\int G{(\\mathbf{H},\\mathbf{S},\\mathbf{M})} d\\mathbf{S} = - A_{2} \\mathbf{E} + A_{2} + \\int (\\mathbf{S} + \\frac{\\mathbf{M}}{\\mathbf{H}}) d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], ["get_premise", "Equality(Function('G')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_2', commutative=True)))"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_2', commutative=True))), Integral(Function('G')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_2', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('A_2', commutative=True), Integral(Function('G')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('A_2', commutative=True), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\ddot{x})} = \\log{(\\ddot{x})} and b{(\\ddot{x})} = \\frac{\\log{(\\ddot{x})}}{\\mathbf{J}_M{(\\ddot{x})}}, then derive - \\sin{(b{(\\ddot{x})})} \\frac{d}{d \\ddot{x}} b{(\\ddot{x})} = 0, then obtain \\mathbf{J}_M{(\\ddot{x})} - \\sin{(b{(\\ddot{x})})} \\frac{d}{d \\ddot{x}} b{(\\ddot{x})} = \\mathbf{J}_M{(\\ddot{x})}", "derivation": "\\mathbf{J}_M{(\\ddot{x})} = \\log{(\\ddot{x})} and b{(\\ddot{x})} = \\frac{\\log{(\\ddot{x})}}{\\mathbf{J}_M{(\\ddot{x})}} and b{(\\ddot{x})} = 1 and \\cos{(b{(\\ddot{x})})} = \\cos{(1)} and \\frac{d}{d \\ddot{x}} \\cos{(b{(\\ddot{x})})} = \\frac{d}{d \\ddot{x}} \\cos{(1)} and - \\sin{(b{(\\ddot{x})})} \\frac{d}{d \\ddot{x}} b{(\\ddot{x})} = 0 and \\mathbf{J}_M{(\\ddot{x})} - \\sin{(b{(\\ddot{x})})} \\frac{d}{d \\ddot{x}} b{(\\ddot{x})} = \\mathbf{J}_M{(\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), log(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('b')(Symbol('\\\\ddot{x}', commutative=True)), Integer(1))"], [["cos", 3], "Equality(cos(Function('b')(Symbol('\\\\ddot{x}', commutative=True))), cos(Integer(1)))"], [["differentiate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(cos(Function('b')(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(cos(Integer(1)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), sin(Function('b')(Symbol('\\\\ddot{x}', commutative=True))), Derivative(Function('b')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Integer(0))"], [["add", 6, "Function('\\\\mathbf{J}_M')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Function('b')(Symbol('\\\\ddot{x}', commutative=True))), Derivative(Function('b')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))), Function('\\\\mathbf{J}_M')(Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(r_{0})} = \\sin{(r_{0})} and \\hat{x}{(r_{0})} = \\sin{(r_{0})}, then obtain -2 = - \\frac{\\operatorname{f^{*}}{(r_{0})} + \\sin{(r_{0})}}{\\sin{(r_{0})}}", "derivation": "\\operatorname{f^{*}}{(r_{0})} = \\sin{(r_{0})} and \\hat{x}{(r_{0})} = \\sin{(r_{0})} and \\hat{x}{(r_{0})} = \\operatorname{f^{*}}{(r_{0})} and \\hat{x}{(r_{0})} + \\sin{(r_{0})} = \\operatorname{f^{*}}{(r_{0})} + \\sin{(r_{0})} and 2 \\sin{(r_{0})} = \\operatorname{f^{*}}{(r_{0})} + \\sin{(r_{0})} and -2 = - \\frac{\\operatorname{f^{*}}{(r_{0})} + \\sin{(r_{0})}}{\\sin{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), Function('f^*')(Symbol('r_0', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))), Add(Function('f^*')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), sin(Symbol('r_0', commutative=True))), Add(Function('f^*')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))"], "Equality(Integer(-2), Mul(Integer(-1), Add(Function('f^*')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))), Pow(sin(Symbol('r_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\theta_{1}{(J,\\mathbf{S})} = J^{\\mathbf{S}} and A{(J,\\mathbf{S})} = \\frac{\\partial}{\\partial J} \\int J^{\\mathbf{S}} dJ, then obtain A{(J,\\mathbf{S})} = \\frac{\\partial}{\\partial J} \\int \\theta_{1}{(J,\\mathbf{S})} dJ", "derivation": "\\theta_{1}{(J,\\mathbf{S})} = J^{\\mathbf{S}} and \\int \\theta_{1}{(J,\\mathbf{S})} dJ = \\int J^{\\mathbf{S}} dJ and A{(J,\\mathbf{S})} = \\frac{\\partial}{\\partial J} \\int J^{\\mathbf{S}} dJ and A{(J,\\mathbf{S})} = \\frac{\\partial}{\\partial J} \\int \\theta_{1}{(J,\\mathbf{S})} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('A')(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Integral(Function('\\\\theta_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(P_{g},y)} = \\int \\frac{y}{P_{g}} dy, then obtain \\frac{P_{g} \\log{(\\operatorname{V_{\\mathbf{E}}}^{y}{(P_{g},y)})}}{\\dot{z} + \\eta^{\\prime}} = \\frac{P_{g} \\log{((\\int \\frac{y}{P_{g}} dy)^{y})}}{\\dot{z} + \\eta^{\\prime}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(P_{g},y)} = \\int \\frac{y}{P_{g}} dy and \\operatorname{V_{\\mathbf{E}}}^{y}{(P_{g},y)} = (\\int \\frac{y}{P_{g}} dy)^{y} and \\log{(\\operatorname{V_{\\mathbf{E}}}^{y}{(P_{g},y)})} = \\log{((\\int \\frac{y}{P_{g}} dy)^{y})} and P_{g} \\log{(\\operatorname{V_{\\mathbf{E}}}^{y}{(P_{g},y)})} = P_{g} \\log{((\\int \\frac{y}{P_{g}} dy)^{y})} and \\frac{P_{g} \\log{(\\operatorname{V_{\\mathbf{E}}}^{y}{(P_{g},y)})}}{\\dot{z} + \\eta^{\\prime}} = \\frac{P_{g} \\log{((\\int \\frac{y}{P_{g}} dy)^{y})}}{\\dot{z} + \\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), log(Pow(Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True))))"], [["divide", 3, "Pow(Symbol('P_g', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('P_g', commutative=True), log(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))), Mul(Symbol('P_g', commutative=True), log(Pow(Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"], [["divide", 4, "Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), log(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('P_g', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))), Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), log(Pow(Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(I,s)} = I + s, then derive 0 = 1 - \\frac{\\partial}{\\partial I} \\operatorname{c_{0}}{(I,s)}, then obtain F_{g}^{s} = A_{x}^{s}", "derivation": "\\operatorname{c_{0}}{(I,s)} = I + s and 0 = I + s - \\operatorname{c_{0}}{(I,s)} and \\frac{d}{d I} 0 = \\frac{\\partial}{\\partial I} (I + s - \\operatorname{c_{0}}{(I,s)}) and 0 = 1 - \\frac{\\partial}{\\partial I} \\operatorname{c_{0}}{(I,s)} and 0^{s} = (1 - \\frac{\\partial}{\\partial I} \\operatorname{c_{0}}{(I,s)})^{s} and 0^{s} = (1 - \\frac{\\partial}{\\partial I} (I + s))^{s} and \\int 0^{s} ds = \\int (1 - \\frac{\\partial}{\\partial I} (I + s))^{s} ds and (\\int 0^{s} ds)^{s} = (\\int (1 - \\frac{\\partial}{\\partial I} (I + s))^{s} ds)^{s} and F_{g}^{s} = A_{x}^{s}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Add(Symbol('I', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True))"], "Equality(Integer(0), Add(Symbol('I', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True)))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Symbol('I', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('s', commutative=True)"], "Equality(Pow(Integer(0), Symbol('s', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integer(0), Symbol('s', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Symbol('s', commutative=True)))"], [["integrate", 6, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 7, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Pow(Integer(0), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["evaluate_integrals", 8], "Equality(Pow(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\mathbf{S})} = \\log{(\\mathbf{S})} and \\operatorname{v_{x}}{(\\mathbf{S})} = 0^{\\mathbf{S}}, then obtain \\tilde{\\infty}^{\\mathbf{S}} (\\varepsilon_{0}{(\\mathbf{S})} - \\log{(\\mathbf{S})})^{\\mathbf{S}} = 0^{\\mathbf{S}} \\tilde{\\infty}^{\\mathbf{S}}", "derivation": "\\varepsilon_{0}{(\\mathbf{S})} = \\log{(\\mathbf{S})} and \\varepsilon_{0}{(\\mathbf{S})} - \\log{(\\mathbf{S})} = 0 and (\\varepsilon_{0}{(\\mathbf{S})} - \\log{(\\mathbf{S})})^{\\mathbf{S}} = 0^{\\mathbf{S}} and \\operatorname{v_{x}}{(\\mathbf{S})} = 0^{\\mathbf{S}} and \\frac{(\\varepsilon_{0}{(\\mathbf{S})} - \\log{(\\mathbf{S})})^{\\mathbf{S}}}{\\operatorname{v_{x}}{(\\mathbf{S})}} = \\frac{0^{\\mathbf{S}}}{\\operatorname{v_{x}}{(\\mathbf{S})}} and \\tilde{\\infty}^{\\mathbf{S}} (\\varepsilon_{0}{(\\mathbf{S})} - \\log{(\\mathbf{S})})^{\\mathbf{S}} = 0^{\\mathbf{S}} \\tilde{\\infty}^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)))"], [["divide", 3, "Function('v_x')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('v_x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('v_x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(zoo, Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Pow(zoo, Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given J{(T)} = \\cos{(T)}, then obtain - T + J{(T)} \\cos{(T)} - \\cos{(T)} = - T + \\cos^{2}{(T)} - \\cos{(T)}", "derivation": "J{(T)} = \\cos{(T)} and J{(T)} \\cos{(T)} = \\cos^{2}{(T)} and T + J{(T)} = T + \\cos{(T)} and - T + J{(T)} \\cos{(T)} - J{(T)} = - T - J{(T)} + \\cos^{2}{(T)} and - T + J{(T)} \\cos{(T)} - \\cos{(T)} = - T + \\cos^{2}{(T)} - \\cos{(T)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["times", 1, "cos(Symbol('T', commutative=True))"], "Equality(Mul(Function('J')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Pow(cos(Symbol('T', commutative=True)), Integer(2)))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('J')(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True))))"], [["minus", 2, "Add(Symbol('T', commutative=True), Function('J')(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Function('J')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), Pow(cos(Symbol('T', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Function('J')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(f_{\\mathbf{p}},y^{\\prime},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} f_{\\mathbf{p}} y^{\\prime}, then derive \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{E}{(f_{\\mathbf{p}},y^{\\prime},\\hat{H}_{\\lambda})} = f_{\\mathbf{p}} y^{\\prime}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} f_{\\mathbf{p}} y^{\\prime} = f_{\\mathbf{p}} y^{\\prime}", "derivation": "\\mathbf{E}{(f_{\\mathbf{p}},y^{\\prime},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} f_{\\mathbf{p}} y^{\\prime} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{E}{(f_{\\mathbf{p}},y^{\\prime},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} f_{\\mathbf{p}} y^{\\prime} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{E}{(f_{\\mathbf{p}},y^{\\prime},\\hat{H}_{\\lambda})} = f_{\\mathbf{p}} y^{\\prime} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} f_{\\mathbf{p}} y^{\\prime} = f_{\\mathbf{p}} y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(h,c)} = c + h, then derive \\frac{\\partial}{\\partial h} \\mathbf{H}{(h,c)} = 1, then derive \\frac{\\partial^{2}}{\\partial h\\partial c} \\mathbf{H}{(h,c)} = 0, then obtain (\\frac{\\partial}{\\partial h} (c + h))^{h} \\frac{\\partial^{2}}{\\partial h\\partial c} (c + h) = 0", "derivation": "\\mathbf{H}{(h,c)} = c + h and \\frac{\\partial}{\\partial h} \\mathbf{H}{(h,c)} = \\frac{\\partial}{\\partial h} (c + h) and \\frac{\\partial}{\\partial h} \\mathbf{H}{(h,c)} = 1 and \\frac{\\partial^{2}}{\\partial c\\partial h} \\mathbf{H}{(h,c)} = \\frac{d}{d c} 1 and \\frac{\\partial^{2}}{\\partial h\\partial c} \\mathbf{H}{(h,c)} = 0 and \\frac{\\partial^{2}}{\\partial h\\partial c} (c + h) = 0 and (\\frac{\\partial}{\\partial h} (c + h))^{h} \\frac{\\partial^{2}}{\\partial h\\partial c} (c + h) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Symbol('c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(0))"], [["times", 6, "Pow(Derivative(Add(Symbol('c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Derivative(Add(Symbol('c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Derivative(Add(Symbol('c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\varepsilon{(\\phi,l)} = e^{\\phi + l}, then obtain (\\int \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl) \\iint 1 dl dl = (\\int \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl) \\iint e^{- \\phi - l} e^{\\phi + l} dl dl", "derivation": "\\varepsilon{(\\phi,l)} = e^{\\phi + l} and 1 = \\frac{e^{\\phi + l}}{\\varepsilon{(\\phi,l)}} and 1 = e^{- \\phi - l} e^{\\phi + l} and 1 = \\varepsilon{(\\phi,l)} e^{- \\phi - l} and \\int 1 dl = \\int \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl and \\iint 1 dl dl = \\iint \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl dl and \\iint 1 dl dl = \\iint e^{- \\phi - l} e^{\\phi + l} dl dl and (\\int \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl) \\iint 1 dl dl = (\\int \\varepsilon{(\\phi,l)} e^{- \\phi - l} dl) \\iint e^{- \\phi - l} e^{\\phi + l} dl dl", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integer(1), Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), exp(Add(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('l', commutative=True))), Integral(Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))))"], [["integrate", 5, "Symbol('l', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Integer(1), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), exp(Add(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 7, "Integral(Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Integral(Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))), Integral(Integer(1), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integral(Mul(Function('\\\\varepsilon')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), exp(Add(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\Psi^{\\dagger})} = \\cos{(\\log{(\\Psi^{\\dagger})})}, then derive \\int (\\mathbf{S}{(\\Psi^{\\dagger})} - 1) d\\Psi^{\\dagger} = V_{\\mathbf{B}} + \\frac{\\Psi^{\\dagger} \\sin{(\\log{(\\Psi^{\\dagger})})}}{2} + \\frac{\\Psi^{\\dagger} \\cos{(\\log{(\\Psi^{\\dagger})})}}{2} - \\Psi^{\\dagger}, then obtain \\int (\\mathbf{S}{(\\Psi^{\\dagger})} - 1) d\\Psi^{\\dagger} = V_{\\mathbf{B}} + \\frac{\\Psi^{\\dagger} \\mathbf{S}{(\\Psi^{\\dagger})}}{2} + \\frac{\\Psi^{\\dagger} \\sin{(\\log{(\\Psi^{\\dagger})})}}{2} - \\Psi^{\\dagger}", "derivation": "\\mathbf{S}{(\\Psi^{\\dagger})} = \\cos{(\\log{(\\Psi^{\\dagger})})} and \\mathbf{S}{(\\Psi^{\\dagger})} - 1 = \\cos{(\\log{(\\Psi^{\\dagger})})} - 1 and \\int (\\mathbf{S}{(\\Psi^{\\dagger})} - 1) d\\Psi^{\\dagger} = \\int (\\cos{(\\log{(\\Psi^{\\dagger})})} - 1) d\\Psi^{\\dagger} and \\int (\\mathbf{S}{(\\Psi^{\\dagger})} - 1) d\\Psi^{\\dagger} = V_{\\mathbf{B}} + \\frac{\\Psi^{\\dagger} \\sin{(\\log{(\\Psi^{\\dagger})})}}{2} + \\frac{\\Psi^{\\dagger} \\cos{(\\log{(\\Psi^{\\dagger})})}}{2} - \\Psi^{\\dagger} and \\int (\\mathbf{S}{(\\Psi^{\\dagger})} - 1) d\\Psi^{\\dagger} = V_{\\mathbf{B}} + \\frac{\\Psi^{\\dagger} \\mathbf{S}{(\\Psi^{\\dagger})}}{2} + \\frac{\\Psi^{\\dagger} \\sin{(\\log{(\\Psi^{\\dagger})})}}{2} - \\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Add(cos(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Add(cos(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), sin(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Rational(1, 2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), cos(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Rational(1, 2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), sin(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})}, then obtain \\iint \\operatorname{f_{\\mathbf{p}}}^{2}{(r_{0})} dr_{0} dr_{0} = \\iint \\operatorname{f_{\\mathbf{p}}}{(r_{0})} \\frac{d}{d r_{0}} \\cos{(r_{0})} dr_{0} dr_{0}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})} and \\operatorname{f_{\\mathbf{p}}}^{2}{(r_{0})} = \\operatorname{f_{\\mathbf{p}}}{(r_{0})} \\frac{d}{d r_{0}} \\cos{(r_{0})} and \\int \\operatorname{f_{\\mathbf{p}}}^{2}{(r_{0})} dr_{0} = \\int \\operatorname{f_{\\mathbf{p}}}{(r_{0})} \\frac{d}{d r_{0}} \\cos{(r_{0})} dr_{0} and \\iint \\operatorname{f_{\\mathbf{p}}}^{2}{(r_{0})} dr_{0} dr_{0} = \\iint \\operatorname{f_{\\mathbf{p}}}{(r_{0})} \\frac{d}{d r_{0}} \\cos{(r_{0})} dr_{0} dr_{0}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["times", 1, "Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Integer(2)), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Tuple(Symbol('r_0', commutative=True))))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Integer(2)), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\theta_1,M)} = \\frac{\\sin{(\\theta_1)}}{M} and \\operatorname{V_{\\mathbf{E}}}{(M)} = \\frac{1}{M}, then obtain - \\mathbf{S}{(\\theta_1,M)} - 1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(M)}}{\\mathbf{S}{(\\theta_1,M)} + 1} = - \\mathbf{S}{(\\theta_1,M)} - 1 + \\frac{1}{M (\\mathbf{S}{(\\theta_1,M)} + 1)}", "derivation": "\\mathbf{S}{(\\theta_1,M)} = \\frac{\\sin{(\\theta_1)}}{M} and \\operatorname{V_{\\mathbf{E}}}{(M)} = \\frac{1}{M} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(M)}}{1 + \\frac{\\sin{(\\theta_1)}}{M}} = \\frac{1}{M (1 + \\frac{\\sin{(\\theta_1)}}{M})} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(M)}}{\\mathbf{S}{(\\theta_1,M)} + 1} = \\frac{1}{M (\\mathbf{S}{(\\theta_1,M)} + 1)} and -1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(M)}}{\\mathbf{S}{(\\theta_1,M)} + 1} - \\frac{\\sin{(\\theta_1)}}{M} = -1 - \\frac{\\sin{(\\theta_1)}}{M} + \\frac{1}{M (\\mathbf{S}{(\\theta_1,M)} + 1)} and - \\mathbf{S}{(\\theta_1,M)} - 1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(M)}}{\\mathbf{S}{(\\theta_1,M)} + 1} = - \\mathbf{S}{(\\theta_1,M)} - 1 + \\frac{1}{M (\\mathbf{S}{(\\theta_1,M)} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["divide", 2, "Add(Integer(1), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Integer(1), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1))))"], [["minus", 4, "Add(Integer(1), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Integer(-1), Mul(Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True))), Integer(-1), Mul(Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True))), Integer(-1), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\theta_1', commutative=True), Symbol('M', commutative=True)), Integer(1)), Integer(-1)))))"]]}, {"prompt": "Given \\omega{(G,\\mu)} = \\cos{(\\frac{\\mu}{G})}, then derive \\int \\omega{(G,\\mu)} dG = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a, then obtain G \\omega{(G,\\mu)} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a - 1 = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a - 1", "derivation": "\\omega{(G,\\mu)} = \\cos{(\\frac{\\mu}{G})} and \\int \\omega{(G,\\mu)} dG = \\int \\cos{(\\frac{\\mu}{G})} dG and \\int \\omega{(G,\\mu)} dG = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a and \\int \\cos{(\\frac{\\mu}{G})} dG = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a and \\int \\cos{(\\frac{\\mu}{G})} dG - 1 = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a - 1 and \\int \\cos{(\\frac{\\mu}{G})} dG = G \\omega{(G,\\mu)} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a and G \\omega{(G,\\mu)} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a - 1 = G \\cos{(\\frac{\\mu}{G})} + \\mu \\operatorname{Si}{(\\frac{\\mu}{G})} + a - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('G', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('G', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('G', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integral(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('G', commutative=True))), Integer(-1)), Add(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Function('\\\\omega')(Symbol('G', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Symbol('G', commutative=True), Function('\\\\omega')(Symbol('G', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True), Integer(-1)), Add(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Si(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))), Symbol('a', commutative=True), Integer(-1)))"]]}, {"prompt": "Given i{(Q)} = \\cos{(Q)}, then obtain (8 i{(Q)} - 7 \\cos{(Q)})^{2} = i^{2}{(Q)}", "derivation": "i{(Q)} = \\cos{(Q)} and 2 i{(Q)} - \\cos{(Q)} = i{(Q)} and (2 i{(Q)} - \\cos{(Q)})^{2} = i^{2}{(Q)} and (4 i{(Q)} - 3 \\cos{(Q)})^{2} = (2 i{(Q)} - \\cos{(Q)})^{2} and (4 i{(Q)} - 3 \\cos{(Q)})^{2} = i^{2}{(Q)} and (8 i{(Q)} - 7 \\cos{(Q)})^{2} = (4 i{(Q)} - 3 \\cos{(Q)})^{2} and (8 i{(Q)} - 7 \\cos{(Q)})^{2} = i^{2}{(Q)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["add", 1, "Add(Function('i')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Function('i')(Symbol('Q', commutative=True)))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Integer(2), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Integer(2)), Pow(Function('i')(Symbol('Q', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Add(Mul(Integer(4), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(3), cos(Symbol('Q', commutative=True)))), Integer(2)), Pow(Add(Mul(Integer(2), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Mul(Integer(4), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(3), cos(Symbol('Q', commutative=True)))), Integer(2)), Pow(Function('i')(Symbol('Q', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Mul(Integer(8), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(7), cos(Symbol('Q', commutative=True)))), Integer(2)), Pow(Add(Mul(Integer(4), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(3), cos(Symbol('Q', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Mul(Integer(8), Function('i')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(7), cos(Symbol('Q', commutative=True)))), Integer(2)), Pow(Function('i')(Symbol('Q', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\psi{(f,\\sigma_x)} = \\sigma_x - f, then obtain \\frac{(\\psi^{f}{(f,\\sigma_x)})^{f}}{f (\\sigma_x - f)} = \\frac{((\\sigma_x - f)^{f})^{f}}{f (\\sigma_x - f)}", "derivation": "\\psi{(f,\\sigma_x)} = \\sigma_x - f and \\psi^{f}{(f,\\sigma_x)} = (\\sigma_x - f)^{f} and (\\psi^{f}{(f,\\sigma_x)})^{f} = ((\\sigma_x - f)^{f})^{f} and \\frac{(\\psi^{f}{(f,\\sigma_x)})^{f}}{f} = \\frac{((\\sigma_x - f)^{f})^{f}}{f} and \\frac{(\\psi^{f}{(f,\\sigma_x)})^{f}}{f (\\sigma_x - f)} = \\frac{((\\sigma_x - f)^{f})^{f}}{f (\\sigma_x - f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["divide", 3, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Symbol('f', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Integer(-1)), Pow(Pow(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Integer(-1)), Pow(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Symbol('f', commutative=True))))"]]}, {"prompt": "Given H{(\\mathbb{I},E,v_{1})} = - E + \\mathbb{I} - v_{1}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})} = 1, then obtain \\frac{\\partial}{\\partial v_{1}} \\log{(\\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})})} = \\frac{d}{d v_{1}} 0", "derivation": "H{(\\mathbb{I},E,v_{1})} = - E + \\mathbb{I} - v_{1} and \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})} = \\frac{\\partial}{\\partial \\mathbb{I}} (- E + \\mathbb{I} - v_{1}) and \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})} = 1 and \\log{(\\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})})} = 0 and \\frac{\\partial}{\\partial v_{1}} \\log{(\\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},E,v_{1})})} = \\frac{d}{d v_{1}} 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1))"], [["log", 3], "Equality(log(Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 4, "Symbol('v_1', commutative=True)"], "Equality(Derivative(log(Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} = \\frac{\\mathbf{F} + q}{q}, then obtain \\int (q^{2} \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} - \\frac{1}{q}) d\\mathbf{F} = \\int (q (\\mathbf{F} + q) - \\frac{1}{q}) d\\mathbf{F}", "derivation": "\\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} = \\frac{\\mathbf{F} + q}{q} and q \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} = \\mathbf{F} + q and q^{2} \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} = q (\\mathbf{F} + q) and q^{2} \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} - \\frac{1}{q} = q (\\mathbf{F} + q) - \\frac{1}{q} and \\int (q^{2} \\operatorname{g_{\\varepsilon}}{(q,\\mathbf{F})} - \\frac{1}{q}) d\\mathbf{F} = \\int (q (\\mathbf{F} + q) - \\frac{1}{q}) d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True)))"], [["times", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(2)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('q', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))))"], [["minus", 3, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('q', commutative=True), Integer(2)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))), Add(Mul(Symbol('q', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))))"], [["integrate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('q', commutative=True), Integer(2)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Mul(Symbol('q', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(C_{1},h)} = h + \\cos{(C_{1})} and \\operatorname{g_{\\varepsilon}}{(C_{1},h)} = \\operatorname{C_{d}}^{C_{1}}{(C_{1},h)}, then obtain \\operatorname{g_{\\varepsilon}}{(C_{1},h)} = (h + \\cos{(C_{1})})^{C_{1}}", "derivation": "\\operatorname{C_{d}}{(C_{1},h)} = h + \\cos{(C_{1})} and \\operatorname{C_{d}}^{C_{1}}{(C_{1},h)} = (h + \\cos{(C_{1})})^{C_{1}} and \\operatorname{g_{\\varepsilon}}{(C_{1},h)} = \\operatorname{C_{d}}^{C_{1}}{(C_{1},h)} and \\operatorname{g_{\\varepsilon}}{(C_{1},h)} = (h + \\cos{(C_{1})})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), cos(Symbol('C_1', commutative=True))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Symbol('C_1', commutative=True)), Pow(Add(Symbol('h', commutative=True), cos(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Pow(Function('C_d')(Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Pow(Add(Symbol('h', commutative=True), cos(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\mu)} = \\cos{(\\mu)}, then obtain (\\ddot{x}{(\\mu)} \\cos{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)} = (\\cos^{2}{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)}", "derivation": "\\ddot{x}{(\\mu)} = \\cos{(\\mu)} and \\ddot{x}{(\\mu)} \\cos{(\\mu)} = \\cos^{2}{(\\mu)} and (\\ddot{x}{(\\mu)} \\cos{(\\mu)})^{\\mu} = (\\cos^{2}{(\\mu)})^{\\mu} and (\\ddot{x}{(\\mu)} \\cos{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)} = (\\cos^{2}{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)), Symbol('\\\\mu', commutative=True)))"], [["add", 3, "Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Add(Pow(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)), Symbol('\\\\mu', commutative=True)), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(t_{2})} = e^{t_{2}}, then obtain \\frac{(- t_{2} + S{(t_{2})} - 1) S{(t_{2})}}{(- t_{2} + e^{t_{2}})^{2}} = \\frac{(- t_{2} + e^{t_{2}} - 1) S{(t_{2})}}{(- t_{2} + e^{t_{2}})^{2}}", "derivation": "S{(t_{2})} = e^{t_{2}} and - t_{2} + S{(t_{2})} = - t_{2} + e^{t_{2}} and - t_{2} + S{(t_{2})} - 1 = - t_{2} + e^{t_{2}} - 1 and \\frac{- t_{2} + S{(t_{2})} - 1}{- t_{2} + e^{t_{2}}} = \\frac{- t_{2} + e^{t_{2}} - 1}{- t_{2} + e^{t_{2}}} and \\frac{(- t_{2} + S{(t_{2})} - 1) S{(t_{2})}}{- t_{2} + e^{t_{2}}} = \\frac{(- t_{2} + e^{t_{2}} - 1) S{(t_{2})}}{- t_{2} + e^{t_{2}}} and \\frac{(- t_{2} + S{(t_{2})} - 1) S{(t_{2})}}{(- t_{2} + e^{t_{2}})^{2}} = \\frac{(- t_{2} + e^{t_{2}} - 1) S{(t_{2})}}{(- t_{2} + e^{t_{2}})^{2}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["minus", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('S')(Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('S')(Symbol('t_2', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(-1)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('S')(Symbol('t_2', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(-1))))"], [["times", 4, "Function('S')(Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('S')(Symbol('t_2', commutative=True)), Integer(-1)), Function('S')(Symbol('t_2', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(-1)), Function('S')(Symbol('t_2', commutative=True))))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('S')(Symbol('t_2', commutative=True)), Integer(-1)), Function('S')(Symbol('t_2', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(-1)), Function('S')(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(v_{y},u)} = e^{u^{v_{y}}}, then obtain \\frac{\\int u^{- v_{y}} \\mathbf{B}{(v_{y},u)} dv_{y}}{\\operatorname{A_{2}}{(v_{y},u)}} = \\frac{\\int u^{- v_{y}} e^{u^{v_{y}}} dv_{y}}{\\operatorname{A_{2}}{(v_{y},u)}}", "derivation": "\\mathbf{B}{(v_{y},u)} = e^{u^{v_{y}}} and u^{- v_{y}} \\mathbf{B}{(v_{y},u)} = u^{- v_{y}} e^{u^{v_{y}}} and \\int u^{- v_{y}} \\mathbf{B}{(v_{y},u)} dv_{y} = \\int u^{- v_{y}} e^{u^{v_{y}}} dv_{y} and \\frac{\\int u^{- v_{y}} \\mathbf{B}{(v_{y},u)} dv_{y}}{\\operatorname{A_{2}}{(v_{y},u)}} = \\frac{\\int u^{- v_{y}} e^{u^{v_{y}}} dv_{y}}{\\operatorname{A_{2}}{(v_{y},u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True), Symbol('u', commutative=True)), exp(Pow(Symbol('u', commutative=True), Symbol('v_y', commutative=True))))"], [["divide", 1, "Pow(Symbol('u', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), exp(Pow(Symbol('u', commutative=True), Symbol('v_y', commutative=True)))))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Integral(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), exp(Pow(Symbol('u', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))))"], [["divide", 3, "Function('A_2')(Symbol('v_y', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('A_2')(Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('v_y', commutative=True)))), Mul(Pow(Function('A_2')(Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), exp(Pow(Symbol('u', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given L{(\\nabla)} = \\cos{(\\nabla)}, then derive \\int L{(\\nabla)} d\\nabla = P_{e} + \\sin{(\\nabla)}, then obtain \\int L{(\\nabla)} d\\nabla + \\frac{\\int L{(\\nabla)} d\\nabla}{L{(\\nabla)}} = \\frac{P_{e} + \\sin{(\\nabla)}}{L{(\\nabla)}} + \\int L{(\\nabla)} d\\nabla", "derivation": "L{(\\nabla)} = \\cos{(\\nabla)} and \\int L{(\\nabla)} d\\nabla = \\int \\cos{(\\nabla)} d\\nabla and \\int L{(\\nabla)} d\\nabla = P_{e} + \\sin{(\\nabla)} and \\frac{\\int L{(\\nabla)} d\\nabla}{L{(\\nabla)}} = \\frac{P_{e} + \\sin{(\\nabla)}}{L{(\\nabla)}} and \\int L{(\\nabla)} d\\nabla + \\frac{\\int L{(\\nabla)} d\\nabla}{L{(\\nabla)}} = \\frac{P_{e} + \\sin{(\\nabla)}}{L{(\\nabla)}} + \\int L{(\\nabla)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('P_e', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["divide", 3, "Function('L')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Function('L')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Add(Symbol('P_e', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Pow(Function('L')(Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["add", 4, "Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Function('L')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))), Add(Mul(Add(Symbol('P_e', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Pow(Function('L')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Integral(Function('L')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(f,\\eta^{\\prime})} = \\cos{(\\frac{\\eta^{\\prime}}{f})} and \\mathbf{g}{(f,\\eta^{\\prime})} = \\cos{(\\frac{\\eta^{\\prime}}{f})}, then obtain 0^{f} = (- \\hat{H}{(f,\\eta^{\\prime})} + \\mathbf{g}{(f,\\eta^{\\prime})})^{f}", "derivation": "\\hat{H}{(f,\\eta^{\\prime})} = \\cos{(\\frac{\\eta^{\\prime}}{f})} and 0 = - \\hat{H}{(f,\\eta^{\\prime})} + \\cos{(\\frac{\\eta^{\\prime}}{f})} and \\mathbf{g}{(f,\\eta^{\\prime})} = \\cos{(\\frac{\\eta^{\\prime}}{f})} and 0 = - \\hat{H}{(f,\\eta^{\\prime})} + \\mathbf{g}{(f,\\eta^{\\prime})} and 0^{f} = (- \\hat{H}{(f,\\eta^{\\prime})} + \\mathbf{g}{(f,\\eta^{\\prime})})^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), cos(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('\\\\mathbf{g}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('\\\\mathbf{g}')(Symbol('f', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given Q{(M_{E})} = \\sin{(M_{E})}, then obtain \\frac{(M_{E} + Q{(M_{E})}) (- \\cos{(M_{E})} - 1)}{(M_{E} + \\sin{(M_{E})})^{2}} + \\frac{\\frac{d}{d M_{E}} Q{(M_{E})} + 1}{M_{E} + \\sin{(M_{E})}} = 0", "derivation": "Q{(M_{E})} = \\sin{(M_{E})} and M_{E} + Q{(M_{E})} = M_{E} + \\sin{(M_{E})} and \\frac{M_{E} + Q{(M_{E})}}{M_{E} + \\sin{(M_{E})}} = 1 and \\frac{M_{E} + Q{(M_{E})}}{M_{E} + \\sin{(M_{E})}} - 1 = 0 and \\frac{d}{d M_{E}} (\\frac{M_{E} + Q{(M_{E})}}{M_{E} + \\sin{(M_{E})}} - 1) = \\frac{d}{d M_{E}} 0 and \\frac{(M_{E} + Q{(M_{E})}) (- \\cos{(M_{E})} - 1)}{(M_{E} + \\sin{(M_{E})})^{2}} + \\frac{\\frac{d}{d M_{E}} Q{(M_{E})} + 1}{M_{E} + \\sin{(M_{E})}} = 0", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('Q')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))))"], [["divide", 2, "Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Symbol('M_E', commutative=True), Function('Q')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, 1], "Equality(Add(Mul(Add(Symbol('M_E', commutative=True), Function('Q')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))), Integer(-1))), Integer(-1)), Integer(0))"], [["differentiate", 4, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Symbol('M_E', commutative=True), Function('Q')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))), Integer(-1))), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Add(Symbol('M_E', commutative=True), Function('Q')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), cos(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('M_E', commutative=True), sin(Symbol('M_E', commutative=True))), Integer(-1)), Add(Derivative(Function('Q')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1)))), Integer(0))"]]}, {"prompt": "Given p{(\\eta)} = \\cos{(\\eta)} and m{(\\eta)} = \\cos{(\\eta)}, then obtain - p{(\\eta)} + \\frac{3 \\cos{(\\eta)}}{p{(\\eta)}} = \\frac{(\\frac{m{(\\eta)}}{p{(\\eta)}} + 1) \\cos{(\\eta)}}{p{(\\eta)}} - p{(\\eta)} + \\frac{\\cos{(\\eta)}}{p{(\\eta)}}", "derivation": "p{(\\eta)} = \\cos{(\\eta)} and 1 = \\frac{\\cos{(\\eta)}}{p{(\\eta)}} and 2 = 1 + \\frac{\\cos{(\\eta)}}{p{(\\eta)}} and m{(\\eta)} = \\cos{(\\eta)} and 2 = \\frac{m{(\\eta)}}{p{(\\eta)}} + 1 and \\frac{2 \\cos{(\\eta)}}{p{(\\eta)}} = \\frac{(\\frac{m{(\\eta)}}{p{(\\eta)}} + 1) \\cos{(\\eta)}}{p{(\\eta)}} and - p{(\\eta)} + \\frac{3 \\cos{(\\eta)}}{p{(\\eta)}} = \\frac{(\\frac{m{(\\eta)}}{p{(\\eta)}} + 1) \\cos{(\\eta)}}{p{(\\eta)}} - p{(\\eta)} + \\frac{\\cos{(\\eta)}}{p{(\\eta)}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "Function('p')(Symbol('\\\\eta', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(2), Add(Mul(Function('m')(Symbol('\\\\eta', commutative=True)), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(1)))"], [["times", 5, "Mul(Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True))), Mul(Add(Mul(Function('m')(Symbol('\\\\eta', commutative=True)), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(1)), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True))))"], [["minus", 6, "Add(Function('p')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('p')(Symbol('\\\\eta', commutative=True))), Mul(Integer(3), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True)))), Add(Mul(Add(Mul(Function('m')(Symbol('\\\\eta', commutative=True)), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(1)), Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Function('p')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Function('p')(Symbol('\\\\eta', commutative=True)), Integer(-1)), cos(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(z^{*},\\eta^{\\prime})} = \\frac{\\eta^{\\prime}}{z^{*}}, then derive \\frac{\\partial}{\\partial z^{*}} \\tilde{g}^*{(z^{*},\\eta^{\\prime})} = - \\frac{\\eta^{\\prime}}{(z^{*})^{2}}, then obtain \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial z^{*}} \\frac{\\eta^{\\prime}}{z^{*}} = \\frac{\\partial}{\\partial \\eta^{\\prime}} - \\frac{\\eta^{\\prime}}{(z^{*})^{2}}", "derivation": "\\tilde{g}^*{(z^{*},\\eta^{\\prime})} = \\frac{\\eta^{\\prime}}{z^{*}} and \\frac{\\partial}{\\partial z^{*}} \\tilde{g}^*{(z^{*},\\eta^{\\prime})} = \\frac{\\partial}{\\partial z^{*}} \\frac{\\eta^{\\prime}}{z^{*}} and \\frac{\\partial}{\\partial z^{*}} \\tilde{g}^*{(z^{*},\\eta^{\\prime})} = - \\frac{\\eta^{\\prime}}{(z^{*})^{2}} and \\frac{\\partial}{\\partial z^{*}} \\frac{\\eta^{\\prime}}{z^{*}} = - \\frac{\\eta^{\\prime}}{(z^{*})^{2}} and \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial z^{*}} \\frac{\\eta^{\\prime}}{z^{*}} = \\frac{\\partial}{\\partial \\eta^{\\prime}} - \\frac{\\eta^{\\prime}}{(z^{*})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-2))))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(\\Psi_{\\lambda},V)} = \\Psi_{\\lambda}^{V} and \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} = \\Psi_{\\lambda}^{V}, then obtain 2 V + \\Psi_{\\lambda}^{V} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} = 2 V + \\Psi_{\\lambda}^{V} \\Omega{(\\Psi_{\\lambda},V)}", "derivation": "\\Omega{(\\Psi_{\\lambda},V)} = \\Psi_{\\lambda}^{V} and \\Psi_{\\lambda}^{V} \\Omega{(\\Psi_{\\lambda},V)} = \\Psi_{\\lambda}^{2 V} and 2 V + \\Psi_{\\lambda}^{V} \\Omega{(\\Psi_{\\lambda},V)} = 2 V + \\Psi_{\\lambda}^{2 V} and \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} = \\Psi_{\\lambda}^{V} and \\Omega{(\\Psi_{\\lambda},V)} = \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} and 2 V + \\Psi_{\\lambda}^{V} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} = 2 V + \\Psi_{\\lambda}^{2 V} and 2 V + \\Psi_{\\lambda}^{V} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda},V)} = 2 V + \\Psi_{\\lambda}^{V} \\Omega{(\\Psi_{\\lambda},V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))))"], [["add", 2, "Mul(Integer(2), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))), Add(Mul(Integer(2), Symbol('V', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\Omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))), Add(Mul(Integer(2), Symbol('V', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))), Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))))"]]}, {"prompt": "Given l{(k)} = \\cos{(k)} and \\mathbf{M}{(k)} = l^{k}{(k)}, then obtain - \\frac{d}{d k} l{(k)} + \\int (2 k + \\mathbf{M}{(k)} + \\cos^{k}{(k)}) dk = - \\frac{d}{d k} l{(k)} + \\int (2 k + 2 \\cos^{k}{(k)}) dk", "derivation": "l{(k)} = \\cos{(k)} and l^{k}{(k)} = \\cos^{k}{(k)} and \\mathbf{M}{(k)} = l^{k}{(k)} and \\mathbf{M}{(k)} = \\cos^{k}{(k)} and k + \\mathbf{M}{(k)} = k + \\cos^{k}{(k)} and 2 k + \\mathbf{M}{(k)} + \\cos^{k}{(k)} = 2 k + 2 \\cos^{k}{(k)} and \\int (2 k + \\mathbf{M}{(k)} + \\cos^{k}{(k)}) dk = \\int (2 k + 2 \\cos^{k}{(k)}) dk and - \\frac{d}{d k} l{(k)} + \\int (2 k + \\mathbf{M}{(k)} + \\cos^{k}{(k)}) dk = - \\frac{d}{d k} l{(k)} + \\int (2 k + 2 \\cos^{k}{(k)}) dk", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('l')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Pow(Function('l')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["add", 4, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\mathbf{M}')(Symbol('k', commutative=True))), Add(Symbol('k', commutative=True), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))))"], [["add", 5, "Add(Symbol('k', commutative=True), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))))"], [["integrate", 6, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["minus", 7, "Derivative(Function('l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given L{(\\pi,A_{x})} = \\cos{(A_{x}^{\\pi})}, then obtain \\frac{\\partial}{\\partial A_{x}} (L^{A_{x}}{(\\pi,A_{x})})^{\\pi} + \\frac{\\partial}{\\partial \\pi} \\cos{(A_{x}^{\\pi})} = \\frac{\\partial}{\\partial A_{x}} (\\cos^{A_{x}}{(A_{x}^{\\pi})})^{\\pi} + \\frac{\\partial}{\\partial \\pi} \\cos{(A_{x}^{\\pi})}", "derivation": "L{(\\pi,A_{x})} = \\cos{(A_{x}^{\\pi})} and L^{A_{x}}{(\\pi,A_{x})} = \\cos^{A_{x}}{(A_{x}^{\\pi})} and (L^{A_{x}}{(\\pi,A_{x})})^{\\pi} = (\\cos^{A_{x}}{(A_{x}^{\\pi})})^{\\pi} and \\frac{\\partial}{\\partial A_{x}} (L^{A_{x}}{(\\pi,A_{x})})^{\\pi} = \\frac{\\partial}{\\partial A_{x}} (\\cos^{A_{x}}{(A_{x}^{\\pi})})^{\\pi} and \\frac{\\partial}{\\partial A_{x}} (L^{A_{x}}{(\\pi,A_{x})})^{\\pi} + \\frac{\\partial}{\\partial \\pi} \\cos{(A_{x}^{\\pi})} = \\frac{\\partial}{\\partial A_{x}} (\\cos^{A_{x}}{(A_{x}^{\\pi})})^{\\pi} + \\frac{\\partial}{\\partial \\pi} \\cos{(A_{x}^{\\pi})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('A_x', commutative=True)), cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('A_x', commutative=True)))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Pow(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Pow(Pow(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["add", 4, "Derivative(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Pow(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Derivative(Pow(Pow(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('A_x', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(v_{2})} = e^{e^{v_{2}}}, then obtain - \\mathbf{s}{(v_{2})} e^{e^{v_{2}}} - \\frac{\\mathbf{s}{(v_{2})} e^{- v_{2}} e^{e^{v_{2}}}}{v_{2}} = - \\mathbf{s}{(v_{2})} e^{e^{v_{2}}} - \\frac{e^{- v_{2}} e^{2 e^{v_{2}}}}{v_{2}}", "derivation": "\\mathbf{s}{(v_{2})} = e^{e^{v_{2}}} and \\mathbf{s}{(v_{2})} e^{e^{v_{2}}} = e^{2 e^{v_{2}}} and \\mathbf{s}{(v_{2})} e^{- v_{2}} e^{e^{v_{2}}} = e^{- v_{2}} e^{2 e^{v_{2}}} and - \\frac{\\mathbf{s}{(v_{2})} e^{- v_{2}} e^{e^{v_{2}}}}{v_{2}} = - \\frac{e^{- v_{2}} e^{2 e^{v_{2}}}}{v_{2}} and - \\mathbf{s}{(v_{2})} e^{e^{v_{2}}} - \\frac{\\mathbf{s}{(v_{2})} e^{- v_{2}} e^{e^{v_{2}}}}{v_{2}} = - \\mathbf{s}{(v_{2})} e^{e^{v_{2}}} - \\frac{e^{- v_{2}} e^{2 e^{v_{2}}}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True))))"], [["times", 1, "exp(exp(Symbol('v_2', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('v_2', commutative=True)))))"], [["divide", 2, "exp(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(Mul(Integer(2), exp(Symbol('v_2', commutative=True))))))"], [["divide", 3, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(Mul(Integer(2), exp(Symbol('v_2', commutative=True))))))"], [["minus", 4, "Mul(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), exp(Mul(Integer(2), exp(Symbol('v_2', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(g)} = \\cos{(g)}, then obtain \\int \\operatorname{A_{y}}^{g}{(g)} dg + \\int \\cos^{g}{(g)} dg = 2 \\int \\cos^{g}{(g)} dg", "derivation": "\\operatorname{A_{y}}{(g)} = \\cos{(g)} and \\operatorname{A_{y}}^{g}{(g)} = \\cos^{g}{(g)} and \\int \\operatorname{A_{y}}^{g}{(g)} dg = \\int \\cos^{g}{(g)} dg and \\int \\operatorname{A_{y}}^{g}{(g)} dg + \\int \\cos^{g}{(g)} dg = 2 \\int \\cos^{g}{(g)} dg", "srepr_derivation": [["renaming_premise", "Equality(Function('A_y')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Pow(Function('A_y')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["add", 3, "Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Integral(Pow(Function('A_y')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(2), Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given q{(\\psi,n)} = n^{\\psi} and \\dot{y}{(\\psi,n)} = n^{\\psi}, then obtain n^{\\psi} q{(\\psi,n)} = n^{\\psi} \\dot{y}{(\\psi,n)}", "derivation": "q{(\\psi,n)} = n^{\\psi} and \\dot{y}{(\\psi,n)} = n^{\\psi} and q{(\\psi,n)} = \\dot{y}{(\\psi,n)} and n^{\\psi} q{(\\psi,n)} = n^{\\psi} \\dot{y}{(\\psi,n)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True)))"], [["times", 3, "Pow(Symbol('n', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\psi', commutative=True)), Function('q')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\psi', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(c,\\mathbf{v})} = \\mathbf{v} + \\log{(c)}, then derive \\frac{\\partial^{2}}{\\partial c^{2}} \\operatorname{C_{d}}{(c,\\mathbf{v})} = - \\frac{1}{c^{2}}, then obtain \\frac{\\partial^{2}}{\\partial c^{2}} \\operatorname{C_{d}}{(c,\\mathbf{v})} = \\frac{\\partial^{2}}{\\partial c^{2}} (\\mathbf{v} + \\log{(c)})", "derivation": "\\operatorname{C_{d}}{(c,\\mathbf{v})} = \\mathbf{v} + \\log{(c)} and c + \\operatorname{C_{d}}{(c,\\mathbf{v})} = \\mathbf{v} + c + \\log{(c)} and \\frac{\\partial}{\\partial c} (c + \\operatorname{C_{d}}{(c,\\mathbf{v})}) = \\frac{\\partial}{\\partial c} (\\mathbf{v} + c + \\log{(c)}) and \\frac{\\partial^{2}}{\\partial c^{2}} (c + \\operatorname{C_{d}}{(c,\\mathbf{v})}) = \\frac{\\partial^{2}}{\\partial c^{2}} (\\mathbf{v} + c + \\log{(c)}) and \\frac{\\partial^{2}}{\\partial c^{2}} \\operatorname{C_{d}}{(c,\\mathbf{v})} = - \\frac{1}{c^{2}} and \\frac{\\partial^{2}}{\\partial c^{2}} (\\mathbf{v} + \\log{(c)}) = - \\frac{1}{c^{2}} and \\frac{\\partial^{2}}{\\partial c^{2}} \\operatorname{C_{d}}{(c,\\mathbf{v})} = \\frac{\\partial^{2}}{\\partial c^{2}} (\\mathbf{v} + \\log{(c)})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('c', commutative=True))))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c', commutative=True), log(Symbol('c', commutative=True))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Symbol('c', commutative=True), Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Symbol('c', commutative=True), Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Function('C_d')(Symbol('c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{M}{(\\omega)} = e^{\\omega}, then derive \\frac{d}{d \\omega} \\mathbf{M}{(\\omega)} = e^{\\omega}, then obtain \\frac{G \\mathbf{M}{(\\omega)}}{\\frac{d}{d \\omega} \\mathbf{M}{(\\omega)}} = \\frac{G \\frac{d}{d \\omega} e^{\\omega}}{\\frac{d}{d \\omega} \\mathbf{M}{(\\omega)}}", "derivation": "\\mathbf{M}{(\\omega)} = e^{\\omega} and G \\mathbf{M}{(\\omega)} = G e^{\\omega} and \\frac{d}{d \\omega} \\mathbf{M}{(\\omega)} = \\frac{d}{d \\omega} e^{\\omega} and \\frac{d}{d \\omega} \\mathbf{M}{(\\omega)} = e^{\\omega} and e^{\\omega} = \\frac{d}{d \\omega} e^{\\omega} and G \\mathbf{M}{(\\omega)} = G \\frac{d}{d \\omega} e^{\\omega} and \\frac{G \\mathbf{M}{(\\omega)}}{\\frac{d}{d \\omega} \\mathbf{M}{(\\omega)}} = \\frac{G \\frac{d}{d \\omega} e^{\\omega}}{\\frac{d}{d \\omega} \\mathbf{M}{(\\omega)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('G', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), exp(Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('G', commutative=True), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["divide", 6, "Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Pow(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('G', commutative=True), Pow(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(t_{2})} = \\cos{(t_{2})}, then derive \\frac{d}{d t_{2}} \\mathbf{F}{(t_{2})} = - \\sin{(t_{2})}, then obtain - \\sin{(\\sin{(t_{2})})} = \\sin{(\\frac{d}{d t_{2}} \\cos{(t_{2})})}", "derivation": "\\mathbf{F}{(t_{2})} = \\cos{(t_{2})} and \\frac{d}{d t_{2}} \\mathbf{F}{(t_{2})} = \\frac{d}{d t_{2}} \\cos{(t_{2})} and \\frac{d}{d t_{2}} \\mathbf{F}{(t_{2})} = - \\sin{(t_{2})} and - \\sin{(t_{2})} = \\frac{d}{d t_{2}} \\cos{(t_{2})} and - \\sin{(\\sin{(t_{2})})} = \\sin{(\\frac{d}{d t_{2}} \\cos{(t_{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('t_2', commutative=True))), Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(sin(Symbol('t_2', commutative=True)))), sin(Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(\\delta)} = e^{\\delta}, then obtain Z{(\\delta)} + 3 e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} 1 = Z{(\\delta)} + 3 e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} \\frac{4 e^{\\delta}}{Z{(\\delta)} + 3 e^{\\delta}}", "derivation": "Z{(\\delta)} = e^{\\delta} and Z{(\\delta)} + e^{\\delta} = 2 e^{\\delta} and Z{(\\delta)} + 3 e^{\\delta} = 4 e^{\\delta} and 1 = \\frac{4 e^{\\delta}}{Z{(\\delta)} + 3 e^{\\delta}} and \\frac{d}{d \\delta} 1 = \\frac{d}{d \\delta} \\frac{4 e^{\\delta}}{Z{(\\delta)} + 3 e^{\\delta}} and \\frac{d^{2}}{d \\delta^{2}} 1 = \\frac{d^{2}}{d \\delta^{2}} \\frac{4 e^{\\delta}}{Z{(\\delta)} + 3 e^{\\delta}} and Z{(\\delta)} + 3 e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} 1 = Z{(\\delta)} + 3 e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} \\frac{4 e^{\\delta}}{Z{(\\delta)} + 3 e^{\\delta}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))))"], [["add", 2, "Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True)))), Mul(Integer(4), exp(Symbol('\\\\delta', commutative=True))))"], [["divide", 3, "Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True))))"], "Equality(Integer(1), Mul(Integer(4), Pow(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Integer(4), Pow(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Derivative(Mul(Integer(4), Pow(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"], [["add", 6, "Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True))))"], "Equality(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True))), Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))), Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True))), Derivative(Mul(Integer(4), Pow(Add(Function('Z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\mathbf{B}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} 2 \\operatorname{n_{2}}{(\\mathbf{D})}, then obtain \\mathbf{B}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} + \\frac{d}{d \\mathbf{D}} \\operatorname{n_{2}}{(\\mathbf{D})}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and 2 \\operatorname{n_{2}}{(\\mathbf{D})} = \\operatorname{n_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} 2 \\operatorname{n_{2}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} (\\operatorname{n_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}) and \\mathbf{B}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} 2 \\operatorname{n_{2}}{(\\mathbf{D})} and \\mathbf{B}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} (\\operatorname{n_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}) and \\mathbf{B}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} + \\frac{d}{d \\mathbf{D}} \\operatorname{n_{2}}{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True)), Add(cos(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbb{I}{(r,v_{1})} = r + v_{1}, then derive \\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})} = 1, then obtain \\frac{\\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})}}{v_{1}} + \\frac{1}{v_{1}} = \\frac{2}{v_{1}}", "derivation": "\\mathbb{I}{(r,v_{1})} = r + v_{1} and \\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})} = \\frac{\\partial}{\\partial r} (r + v_{1}) and \\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})} = 1 and \\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})} + 1 = 2 and \\frac{\\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})} + 1}{v_{1}} = \\frac{2}{v_{1}} and \\frac{\\frac{\\partial}{\\partial r} \\mathbb{I}{(r,v_{1})}}{v_{1}} + \\frac{1}{v_{1}} = \\frac{2}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('r', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["divide", 4, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Derivative(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1))), Mul(Integer(2), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["expand", 5], "Equality(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('v_1', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\chi{(T)} = \\sin{(T)}, then derive \\frac{\\frac{d}{d T} \\chi{(T)}}{\\sin{(T)}} = \\frac{\\cos{(T)}}{\\sin{(T)}}, then obtain g^{\\prime}_{\\varepsilon} + \\log{(\\chi{(T)})} = \\int \\frac{\\cos{(T)}}{\\chi{(T)}} dT", "derivation": "\\chi{(T)} = \\sin{(T)} and \\frac{d}{d T} \\chi{(T)} = \\frac{d}{d T} \\sin{(T)} and \\frac{\\frac{d}{d T} \\chi{(T)}}{\\sin{(T)}} = \\frac{\\frac{d}{d T} \\sin{(T)}}{\\sin{(T)}} and \\frac{\\frac{d}{d T} \\chi{(T)}}{\\sin{(T)}} = \\frac{\\cos{(T)}}{\\sin{(T)}} and \\frac{\\frac{d}{d T} \\chi{(T)}}{\\chi{(T)}} = \\frac{\\cos{(T)}}{\\chi{(T)}} and \\int \\frac{\\frac{d}{d T} \\chi{(T)}}{\\chi{(T)}} dT = \\int \\frac{\\cos{(T)}}{\\chi{(T)}} dT and g^{\\prime}_{\\varepsilon} + \\log{(\\chi{(T)})} = \\int \\frac{\\cos{(T)}}{\\chi{(T)}} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 2, "sin(Symbol('T', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), cos(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\chi')(Symbol('T', commutative=True)), Integer(-1)), cos(Symbol('T', commutative=True))))"], [["integrate", 5, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\chi')(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Function('\\\\chi')(Symbol('T', commutative=True)), Integer(-1)), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Function('\\\\chi')(Symbol('T', commutative=True)))), Integral(Mul(Pow(Function('\\\\chi')(Symbol('T', commutative=True)), Integer(-1)), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given v{(\\dot{z})} = \\cos{(\\dot{z})} and E{(\\dot{z})} = \\dot{z}, then obtain \\dot{z} + v{(\\dot{z})} + \\cos{(\\dot{z})} = \\dot{z} + 2 \\cos{(\\dot{z})}", "derivation": "v{(\\dot{z})} = \\cos{(\\dot{z})} and E{(\\dot{z})} = \\dot{z} and v{(\\dot{z})} + \\cos{(\\dot{z})} = 2 \\cos{(\\dot{z})} and E{(\\dot{z})} + v{(\\dot{z})} + \\cos{(\\dot{z})} = E{(\\dot{z})} + 2 \\cos{(\\dot{z})} and \\dot{z} + v{(\\dot{z})} + \\cos{(\\dot{z})} = \\dot{z} + 2 \\cos{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))"], [["add", 1, "cos(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('v')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 3, "Function('E')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('E')(Symbol('\\\\dot{z}', commutative=True)), Function('v')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Add(Function('E')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('v')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(U,\\rho)} = \\frac{\\rho}{U}, then obtain \\frac{2 \\mathbf{p}{(U,\\rho)}}{-1 + \\frac{\\rho}{U}} = \\frac{2 \\rho}{U (-1 + \\frac{\\rho}{U})}", "derivation": "\\mathbf{p}{(U,\\rho)} = \\frac{\\rho}{U} and \\mathbf{p}{(U,\\rho)} - 1 = -1 + \\frac{\\rho}{U} and \\mathbf{p}{(U,\\rho)} + \\frac{\\rho}{U} = \\frac{2 \\rho}{U} and \\frac{\\mathbf{p}{(U,\\rho)} + \\frac{\\rho}{U}}{\\mathbf{p}{(U,\\rho)} - 1} = \\frac{2 \\rho}{U (\\mathbf{p}{(U,\\rho)} - 1)} and 2 \\mathbf{p}{(U,\\rho)} = \\mathbf{p}{(U,\\rho)} + \\frac{\\rho}{U} and \\frac{2 \\mathbf{p}{(U,\\rho)}}{\\mathbf{p}{(U,\\rho)} - 1} = \\frac{2 \\rho}{U (\\mathbf{p}{(U,\\rho)} - 1)} and \\frac{2 \\mathbf{p}{(U,\\rho)}}{-1 + \\frac{\\rho}{U}} = \\frac{2 \\rho}{U (-1 + \\frac{\\rho}{U})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["divide", 3, "Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integer(-1)), Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integer(-1))))"], [["minus", 3, "Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Pow(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(2), Pow(Add(Integer(-1), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True), Pow(Add(Integer(-1), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given J{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})} and \\operatorname{r_{0}}{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})}, then obtain 0 = - \\operatorname{r_{0}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})}", "derivation": "J{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})} and \\operatorname{r_{0}}{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})} and J{(\\mathbf{B})} - \\operatorname{r_{0}}{(\\mathbf{B})} = - \\operatorname{r_{0}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})} and J{(\\mathbf{B})} = \\operatorname{r_{0}}{(\\mathbf{B})} and 0 = - \\operatorname{r_{0}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 1, "Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Function('J')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Mul(Integer(-1), Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True))), sin(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('J')(Symbol('\\\\mathbf{B}', commutative=True)), Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('r_0')(Symbol('\\\\mathbf{B}', commutative=True))), sin(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\Psi - \\eta^{\\prime} + \\phi_1, then derive \\frac{\\partial}{\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} (\\Psi - \\eta^{\\prime} + \\phi_1)", "derivation": "\\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\Psi - \\eta^{\\prime} + \\phi_1 and \\frac{\\partial}{\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\phi_1} (\\Psi - \\eta^{\\prime} + \\phi_1) and \\frac{\\partial}{\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = 1 and \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} 1 and \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} (\\Psi - \\eta^{\\prime} + \\phi_1) = \\frac{d}{d \\eta^{\\prime}} 1 and \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} \\dot{z}{(\\Psi,\\phi_1,\\eta^{\\prime})} = \\frac{\\partial^{2}}{\\partial \\eta^{\\prime}\\partial \\phi_1} (\\Psi - \\eta^{\\prime} + \\phi_1)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\Psi)} = \\log{(\\Psi)}, then obtain \\Psi \\int \\mathbf{J}_M{(\\Psi)} d\\Psi = \\Psi (U + \\Psi \\log{(\\Psi)} - \\Psi)", "derivation": "\\mathbf{J}_M{(\\Psi)} = \\log{(\\Psi)} and \\int \\mathbf{J}_M{(\\Psi)} d\\Psi = \\int \\log{(\\Psi)} d\\Psi and \\Psi \\int \\mathbf{J}_M{(\\Psi)} d\\Psi = \\Psi \\int \\log{(\\Psi)} d\\Psi and \\Psi \\int \\mathbf{J}_M{(\\Psi)} d\\Psi = \\Psi (U + \\Psi \\log{(\\Psi)} - \\Psi)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\Psi', commutative=True), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('U', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\cos{(v_{1})} and u{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})}, then obtain \\frac{d}{d v_{1}} 0 = \\frac{d}{d v_{1}} (- u{(v_{1})} + \\frac{d}{d v_{1}} \\cos{(v_{1})})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\cos{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and 0 = - \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} + \\frac{d}{d v_{1}} \\cos{(v_{1})} and \\frac{d}{d v_{1}} 0 = \\frac{d}{d v_{1}} (- \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} + \\frac{d}{d v_{1}} \\cos{(v_{1})}) and u{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and u{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} and \\frac{d}{d v_{1}} 0 = \\frac{d}{d v_{1}} (- u{(v_{1})} + \\frac{d}{d v_{1}} \\cos{(v_{1})})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('u')(Symbol('v_1', commutative=True)), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('u')(Symbol('v_1', commutative=True)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('u')(Symbol('v_1', commutative=True))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(Z,M_{E})} = M_{E} Z, then obtain (M_{E} + \\iint \\varepsilon{(Z,M_{E})} dM_{E} dM_{E})^{M_{E}} = (M_{E} + \\iint M_{E} Z dM_{E} dM_{E})^{M_{E}}", "derivation": "\\varepsilon{(Z,M_{E})} = M_{E} Z and \\int \\varepsilon{(Z,M_{E})} dM_{E} = \\int M_{E} Z dM_{E} and \\iint \\varepsilon{(Z,M_{E})} dM_{E} dM_{E} = \\iint M_{E} Z dM_{E} dM_{E} and M_{E} + \\iint \\varepsilon{(Z,M_{E})} dM_{E} dM_{E} = M_{E} + \\iint M_{E} Z dM_{E} dM_{E} and (M_{E} + \\iint \\varepsilon{(Z,M_{E})} dM_{E} dM_{E})^{M_{E}} = (M_{E} + \\iint M_{E} Z dM_{E} dM_{E})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["add", 3, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Integral(Function('\\\\varepsilon')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Symbol('M_E', commutative=True), Integral(Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Symbol('M_E', commutative=True), Integral(Function('\\\\varepsilon')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), Integral(Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(m,\\omega)} = \\cos^{\\omega}{(m)}, then obtain (\\dot{z}{(m,\\omega)} \\cos^{2}{(m)} \\cos^{\\omega}{(m)} - \\cos^{\\omega}{(m)})^{2} = (\\cos^{2}{(m)} \\cos^{2 \\omega}{(m)} - \\cos^{\\omega}{(m)})^{2}", "derivation": "\\dot{z}{(m,\\omega)} = \\cos^{\\omega}{(m)} and \\dot{z}{(m,\\omega)} \\cos{(m)} = \\cos{(m)} \\cos^{\\omega}{(m)} and \\dot{z}{(m,\\omega)} \\cos^{2}{(m)} \\cos^{\\omega}{(m)} = \\cos^{2}{(m)} \\cos^{2 \\omega}{(m)} and \\dot{z}{(m,\\omega)} \\cos^{2}{(m)} \\cos^{\\omega}{(m)} - \\cos^{\\omega}{(m)} = \\cos^{2}{(m)} \\cos^{2 \\omega}{(m)} - \\cos^{\\omega}{(m)} and (\\dot{z}{(m,\\omega)} \\cos^{2}{(m)} \\cos^{\\omega}{(m)} - \\cos^{\\omega}{(m)})^{2} = (\\cos^{2}{(m)} \\cos^{2 \\omega}{(m)} - \\cos^{\\omega}{(m)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["times", 1, "cos(Symbol('m', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Symbol('m', commutative=True))), Mul(cos(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["times", 2, "Mul(cos(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True)))))"], [["minus", 3, "Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Mul(Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(2)), Pow(Add(Mul(Pow(cos(Symbol('m', commutative=True)), Integer(2)), Pow(cos(Symbol('m', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(g,\\mu)} = \\sin{(g^{\\mu})}, then obtain (- g + \\sin{(g^{\\mu})})^{2} \\operatorname{A_{z}}{(g,\\mu)} = (- g + \\sin{(g^{\\mu})})^{2} \\sin{(g^{\\mu})}", "derivation": "\\operatorname{A_{z}}{(g,\\mu)} = \\sin{(g^{\\mu})} and - g + \\operatorname{A_{z}}{(g,\\mu)} = - g + \\sin{(g^{\\mu})} and (- g + \\operatorname{A_{z}}{(g,\\mu)}) \\operatorname{A_{z}}{(g,\\mu)} = (- g + \\operatorname{A_{z}}{(g,\\mu)}) \\sin{(g^{\\mu})} and (- g + \\sin{(g^{\\mu})}) \\operatorname{A_{z}}{(g,\\mu)} = (- g + \\sin{(g^{\\mu})}) \\sin{(g^{\\mu})} and (- g + \\operatorname{A_{z}}{(g,\\mu)}) (- g + \\sin{(g^{\\mu})}) \\operatorname{A_{z}}{(g,\\mu)} = (- g + \\operatorname{A_{z}}{(g,\\mu)}) (- g + \\sin{(g^{\\mu})}) \\sin{(g^{\\mu})} and (- g + \\sin{(g^{\\mu})})^{2} \\operatorname{A_{z}}{(g,\\mu)} = (- g + \\sin{(g^{\\mu})})^{2} \\sin{(g^{\\mu})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(2)), Function('A_z')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(2)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(r)} = \\cos{(r)}, then obtain \\int (r + \\operatorname{E_{n}}{(r)}) dr = A_{x} + \\frac{r^{2}}{2} + \\sin{(r)}", "derivation": "\\operatorname{E_{n}}{(r)} = \\cos{(r)} and r + \\operatorname{E_{n}}{(r)} = r + \\cos{(r)} and \\int (r + \\operatorname{E_{n}}{(r)}) dr = \\int (r + \\cos{(r)}) dr and \\int (r + \\operatorname{E_{n}}{(r)}) dr = A_{x} + \\frac{r^{2}}{2} + \\sin{(r)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["add", 1, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Function('E_n')(Symbol('r', commutative=True))), Add(Symbol('r', commutative=True), cos(Symbol('r', commutative=True))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Symbol('r', commutative=True), Function('E_n')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('r', commutative=True), cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('r', commutative=True), Function('E_n')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), sin(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{2})} = \\cos{(A_{2})}, then obtain A_{2} + \\int (- A_{2} - 2 \\cos{(A_{2})}) dA_{2} = A_{2} + \\int (- A_{2} - \\operatorname{A_{x}}{(A_{2})} - \\cos{(A_{2})}) dA_{2}", "derivation": "\\operatorname{A_{x}}{(A_{2})} = \\cos{(A_{2})} and - A_{2} + \\operatorname{A_{x}}{(A_{2})} - \\cos{(A_{2})} = - A_{2} and - A_{2} - \\cos{(A_{2})} = - A_{2} - \\operatorname{A_{x}}{(A_{2})} and - A_{2} - 2 \\cos{(A_{2})} = - A_{2} - \\operatorname{A_{x}}{(A_{2})} - \\cos{(A_{2})} and \\int (- A_{2} - 2 \\cos{(A_{2})}) dA_{2} = \\int (- A_{2} - \\operatorname{A_{x}}{(A_{2})} - \\cos{(A_{2})}) dA_{2} and A_{2} + \\int (- A_{2} - 2 \\cos{(A_{2})}) dA_{2} = A_{2} + \\int (- A_{2} - \\operatorname{A_{x}}{(A_{2})} - \\cos{(A_{2})}) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["minus", 1, "Add(Symbol('A_2', commutative=True), cos(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('A_x')(Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Mul(Integer(-1), Symbol('A_2', commutative=True)))"], [["minus", 2, "Function('A_x')(Symbol('A_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('A_2', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('A_2', commutative=True))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('A_2', commutative=True))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Symbol('A_2', commutative=True))"], "Equality(Add(Symbol('A_2', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))), Add(Symbol('A_2', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('A_2', commutative=True))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{S},\\chi)} = \\chi^{\\mathbf{S}}, then obtain \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} \\hat{p}{(\\mathbf{S},\\chi)}}{\\mathbf{S}} = \\frac{\\chi^{\\mathbf{S}} \\log{(\\chi)}}{\\mathbf{S}}", "derivation": "\\hat{p}{(\\mathbf{S},\\chi)} = \\chi^{\\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} \\hat{p}{(\\mathbf{S},\\chi)} = \\frac{\\partial}{\\partial \\mathbf{S}} \\chi^{\\mathbf{S}} and \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} \\hat{p}{(\\mathbf{S},\\chi)}}{\\mathbf{S}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} \\chi^{\\mathbf{S}}}{\\mathbf{S}} and \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} \\hat{p}{(\\mathbf{S},\\chi)}}{\\mathbf{S}} = \\frac{\\chi^{\\mathbf{S}} \\log{(\\chi)}}{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then obtain \\int (\\mu{(\\mathbf{E})} \\cos{(\\mathbf{E})} + \\cos^{2}{(\\mathbf{E})}) d\\mathbf{E} = \\int 2 \\cos^{2}{(\\mathbf{E})} d\\mathbf{E}", "derivation": "\\mu{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and \\mu{(\\mathbf{E})} \\cos{(\\mathbf{E})} = \\cos^{2}{(\\mathbf{E})} and \\mu{(\\mathbf{E})} \\cos{(\\mathbf{E})} + \\cos^{2}{(\\mathbf{E})} = 2 \\cos^{2}{(\\mathbf{E})} and \\int (\\mu{(\\mathbf{E})} \\cos{(\\mathbf{E})} + \\cos^{2}{(\\mathbf{E})}) d\\mathbf{E} = \\int 2 \\cos^{2}{(\\mathbf{E})} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)))"], [["add", 2, "Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))"], "Equality(Add(Mul(Function('\\\\mu')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Mul(Function('\\\\mu')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(x,\\Omega)} = \\frac{\\Omega}{x}, then obtain - \\int (- x + \\operatorname{r_{0}}^{x}{(x,\\Omega)}) d\\Omega = - \\int (- x + (\\frac{\\Omega}{x})^{x}) d\\Omega", "derivation": "\\operatorname{r_{0}}{(x,\\Omega)} = \\frac{\\Omega}{x} and \\operatorname{r_{0}}^{x}{(x,\\Omega)} = (\\frac{\\Omega}{x})^{x} and - x + \\operatorname{r_{0}}^{x}{(x,\\Omega)} = - x + (\\frac{\\Omega}{x})^{x} and \\int (- x + \\operatorname{r_{0}}^{x}{(x,\\Omega)}) d\\Omega = \\int (- x + (\\frac{\\Omega}{x})^{x}) d\\Omega and - \\int (- x + \\operatorname{r_{0}}^{x}{(x,\\Omega)}) d\\Omega = - \\int (- x + (\\frac{\\Omega}{x})^{x}) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True)))"], [["minus", 2, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - \\mathbf{J}_f + \\mu - \\phi, then obtain \\int \\tilde{\\infty} (- \\mu + 2 \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)}) d\\mathbf{J}_f = \\int \\tilde{\\infty} (- 2 \\mathbf{J}_f + \\mu - 2 \\phi) d\\mathbf{J}_f", "derivation": "\\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - \\mathbf{J}_f + \\mu - \\phi and \\mu + \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - \\mathbf{J}_f + 2 \\mu - \\phi and - \\mu + \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - \\mathbf{J}_f - \\phi and - \\mathbf{J}_f - \\phi + \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - 2 \\mathbf{J}_f + \\mu - 2 \\phi and - \\mu + 2 \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)} = - 2 \\mathbf{J}_f + \\mu - 2 \\phi and \\tilde{\\infty} (- \\mu + 2 \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)}) = \\tilde{\\infty} (- 2 \\mathbf{J}_f + \\mu - 2 \\phi) and \\int \\tilde{\\infty} (- \\mu + 2 \\Psi^{\\dagger}{(\\mathbf{J}_f,\\phi,\\mu)}) d\\mathbf{J}_f = \\int \\tilde{\\infty} (- 2 \\mathbf{J}_f + \\mu - 2 \\phi) d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True))))"], [["divide", 5, 0], "Equality(Mul(zoo, Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True))))), Mul(zoo, Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(zoo, Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Mul(zoo, Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(E_{n},\\chi)} = E_{n} + \\chi, then obtain \\operatorname{f^{\\prime}}{(E_{n},\\chi)} + \\int \\operatorname{f^{\\prime}}{(E_{n},\\chi)} d\\chi = E_{n} + \\chi + \\int \\operatorname{f^{\\prime}}{(E_{n},\\chi)} d\\chi", "derivation": "\\operatorname{f^{\\prime}}{(E_{n},\\chi)} = E_{n} + \\chi and \\int \\operatorname{f^{\\prime}}{(E_{n},\\chi)} d\\chi = \\int (E_{n} + \\chi) d\\chi and \\operatorname{f^{\\prime}}{(E_{n},\\chi)} + \\int (E_{n} + \\chi) d\\chi = E_{n} + \\chi + \\int (E_{n} + \\chi) d\\chi and \\operatorname{f^{\\prime}}{(E_{n},\\chi)} + \\int \\operatorname{f^{\\prime}}{(E_{n},\\chi)} d\\chi = E_{n} + \\chi + \\int \\operatorname{f^{\\prime}}{(E_{n},\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True), Integral(Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True), Integral(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}}, then obtain 2 g_{\\varepsilon} + \\hat{H}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} = 2 g_{\\varepsilon} + 2 e^{g_{\\varepsilon}}", "derivation": "\\hat{H}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and g_{\\varepsilon} + \\hat{H}{(g_{\\varepsilon})} = g_{\\varepsilon} + e^{g_{\\varepsilon}} and 2 g_{\\varepsilon} + 2 \\hat{H}{(g_{\\varepsilon})} = 2 g_{\\varepsilon} + \\hat{H}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} and 2 g_{\\varepsilon} + \\hat{H}{(g_{\\varepsilon})} = 2 g_{\\varepsilon} + e^{g_{\\varepsilon}} and 2 g_{\\varepsilon} + 2 \\hat{H}{(g_{\\varepsilon})} = 2 g_{\\varepsilon} + 2 e^{g_{\\varepsilon}} and 2 g_{\\varepsilon} + \\hat{H}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} = 2 g_{\\varepsilon} + 2 e^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 2, "Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{H}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain \\frac{d}{d \\hat{H}_{\\lambda}} (\\frac{\\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}{\\log{(\\hat{H}_{\\lambda})}} - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}}) = \\frac{d}{d \\hat{H}_{\\lambda}} (1 - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}})", "derivation": "\\operatorname{m_{s}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\frac{\\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}{\\log{(\\hat{H}_{\\lambda})}} = 1 and \\frac{\\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}{\\log{(\\hat{H}_{\\lambda})}} - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}} = 1 - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}} and \\frac{d}{d \\hat{H}_{\\lambda}} (\\frac{\\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}{\\log{(\\hat{H}_{\\lambda})}} - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}}) = \\frac{d}{d \\hat{H}_{\\lambda}} (1 - \\frac{1}{\\log{(\\hat{H}_{\\lambda})}})", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))))"], [["differentiate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(k,T)} = - T + k, then derive \\frac{\\int \\hat{p}_0{(k,T)} dT}{k} = \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k}, then derive T^{2} + \\frac{- \\frac{T^{2}}{2} + T k + \\mathbf{v}}{k} = T^{2} + \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k}, then obtain T^{2} + \\frac{- \\frac{T^{2}}{2} + T k + \\mathbf{v}}{k} = T^{2} + \\frac{\\int \\hat{p}_0{(k,T)} dT}{k}", "derivation": "\\hat{p}_0{(k,T)} = - T + k and \\int \\hat{p}_0{(k,T)} dT = \\int (- T + k) dT and \\frac{\\int \\hat{p}_0{(k,T)} dT}{k} = \\frac{\\int (- T + k) dT}{k} and \\frac{\\int \\hat{p}_0{(k,T)} dT}{k} = \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k} and T^{2} + \\frac{\\int \\hat{p}_0{(k,T)} dT}{k} = T^{2} + \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k} and T^{2} + \\frac{\\int (- T + k) dT}{k} = T^{2} + \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k} and T^{2} + \\frac{- \\frac{T^{2}}{2} + T k + \\mathbf{v}}{k} = T^{2} + \\frac{A_{z} - \\frac{T^{2}}{2} + T k}{k} and T^{2} + \\frac{- \\frac{T^{2}}{2} + T k + \\mathbf{v}}{k} = T^{2} + \\frac{\\int \\hat{p}_0{(k,T)} dT}{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["divide", 2, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True)))))"], [["add", 4, "Pow(Symbol('T', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))), Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))), Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('k', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))))"]]}, {"prompt": "Given \\mathbb{I}{(G)} = \\cos{(G)}, then obtain 1 + \\frac{1}{\\cos{(G)}} + \\frac{\\int \\mathbb{I}{(G)} dG}{\\mathbb{I}{(G)}} = 1 + \\frac{1}{\\cos{(G)}} + \\frac{\\int \\cos{(G)} dG}{\\mathbb{I}{(G)}}", "derivation": "\\mathbb{I}{(G)} = \\cos{(G)} and \\int \\mathbb{I}{(G)} dG = \\int \\cos{(G)} dG and \\frac{\\int \\mathbb{I}{(G)} dG}{\\cos{(G)}} = \\frac{\\int \\cos{(G)} dG}{\\cos{(G)}} and 1 + \\frac{\\int \\mathbb{I}{(G)} dG}{\\cos{(G)}} = 1 + \\frac{\\int \\cos{(G)} dG}{\\cos{(G)}} and 1 + \\frac{\\int \\mathbb{I}{(G)} dG}{\\mathbb{I}{(G)}} = 1 + \\frac{\\int \\cos{(G)} dG}{\\mathbb{I}{(G)}} and 1 + \\frac{1}{\\cos{(G)}} + \\frac{\\int \\mathbb{I}{(G)} dG}{\\mathbb{I}{(G)}} = 1 + \\frac{1}{\\cos{(G)}} + \\frac{\\int \\cos{(G)} dG}{\\mathbb{I}{(G)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["divide", 2, "cos(Symbol('G', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Integer(1), Mul(Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integer(1), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Integer(1), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Integer(-1)), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))))"], [["add", 5, "Pow(cos(Symbol('G', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Integer(1), Pow(cos(Symbol('G', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Integer(-1)), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))))"]]}, {"prompt": "Given \\pi{(S,\\mathbf{f},t_{2})} = (S \\mathbf{f})^{t_{2}} and \\mathbf{s}{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})}, then obtain S \\mathbf{f} + S + \\mathbf{s}{(L_{\\varepsilon})} - \\pi{(S,\\mathbf{f},t_{2})} = S \\mathbf{f} + S - \\pi{(S,\\mathbf{f},t_{2})} + \\log{(L_{\\varepsilon})}", "derivation": "\\pi{(S,\\mathbf{f},t_{2})} = (S \\mathbf{f})^{t_{2}} and \\mathbf{s}{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})} and S - (S \\mathbf{f})^{t_{2}} + \\mathbf{s}{(L_{\\varepsilon})} = S - (S \\mathbf{f})^{t_{2}} + \\log{(L_{\\varepsilon})} and S \\mathbf{f} + S - (S \\mathbf{f})^{t_{2}} + \\mathbf{s}{(L_{\\varepsilon})} = S \\mathbf{f} + S - (S \\mathbf{f})^{t_{2}} + \\log{(L_{\\varepsilon})} and S \\mathbf{f} + S + \\mathbf{s}{(L_{\\varepsilon})} - \\pi{(S,\\mathbf{f},t_{2})} = S \\mathbf{f} + S - \\pi{(S,\\mathbf{f},t_{2})} + \\log{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True)))"], "Equality(Add(Symbol('S', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), log(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["add", 3, "Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), log(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('S', commutative=True), Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)))), Add(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('S', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True))), log(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\dot{x},t)} = t^{\\dot{x}}, then obtain \\Psi_{\\lambda}{(\\dot{x},t)} = 4 t^{\\dot{x}} - 3 \\Psi_{\\lambda}{(\\dot{x},t)}", "derivation": "\\Psi_{\\lambda}{(\\dot{x},t)} = t^{\\dot{x}} and 0 = t^{\\dot{x}} - \\Psi_{\\lambda}{(\\dot{x},t)} and t^{\\dot{x}} = 2 t^{\\dot{x}} - \\Psi_{\\lambda}{(\\dot{x},t)} and \\Psi_{\\lambda}{(\\dot{x},t)} = 2 t^{\\dot{x}} - \\Psi_{\\lambda}{(\\dot{x},t)} and \\Psi_{\\lambda}{(\\dot{x},t)} = 4 t^{\\dot{x}} - 3 \\Psi_{\\lambda}{(\\dot{x},t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)))))"], [["add", 2, "Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Mul(Integer(2), Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(2), Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(4), Pow(Symbol('t', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Integer(3), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\phi)} = \\cos{(\\phi)} and \\mathbf{P}{(\\tilde{g},F_{H},\\sigma_x)} = F_{H} \\tilde{g} - \\sigma_x, then obtain \\frac{\\mathbf{P}{(\\tilde{g},F_{H},\\sigma_x)}}{- \\phi + \\hat{H}_l{(\\phi)}} = \\frac{F_{H} \\tilde{g} - \\sigma_x}{- \\phi + \\hat{H}_l{(\\phi)}}", "derivation": "\\hat{H}_l{(\\phi)} = \\cos{(\\phi)} and - \\phi + \\hat{H}_l{(\\phi)} = - \\phi + \\cos{(\\phi)} and \\mathbf{P}{(\\tilde{g},F_{H},\\sigma_x)} = F_{H} \\tilde{g} - \\sigma_x and \\frac{\\mathbf{P}{(\\tilde{g},F_{H},\\sigma_x)}}{- \\phi + \\cos{(\\phi)}} = \\frac{F_{H} \\tilde{g} - \\sigma_x}{- \\phi + \\cos{(\\phi)}} and \\frac{\\mathbf{P}{(\\tilde{g},F_{H},\\sigma_x)}}{- \\phi + \\hat{H}_l{(\\phi)}} = \\frac{F_{H} \\tilde{g} - \\sigma_x}{- \\phi + \\hat{H}_l{(\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Symbol('F_H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True))), Integer(-1)), Add(Mul(Symbol('F_H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True))), Integer(-1)), Add(Mul(Symbol('F_H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hat{x},y^{\\prime})} = \\log{(- \\hat{x} + y^{\\prime})} and \\operatorname{A_{z}}{(\\hat{x},y^{\\prime})} = \\log{(- \\hat{x} + y^{\\prime})}, then obtain \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(\\hat{x},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\log{(- \\hat{x} + y^{\\prime})}", "derivation": "\\mathbf{J}_f{(\\hat{x},y^{\\prime})} = \\log{(- \\hat{x} + y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{J}_f{(\\hat{x},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\log{(- \\hat{x} + y^{\\prime})} and \\operatorname{A_{z}}{(\\hat{x},y^{\\prime})} = \\log{(- \\hat{x} + y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{J}_f{(\\hat{x},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(\\hat{x},y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(\\hat{x},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\log{(- \\hat{x} + y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(y^{\\prime})} = \\cos{(y^{\\prime})}, then obtain I^{2}{(y^{\\prime})} = I^{2}{(y^{\\prime})} - 1 + \\frac{\\cos{(y^{\\prime})}}{I{(y^{\\prime})}}", "derivation": "I{(y^{\\prime})} = \\cos{(y^{\\prime})} and 1 = \\frac{\\cos{(y^{\\prime})}}{I{(y^{\\prime})}} and 0 = -1 + \\frac{\\cos{(y^{\\prime})}}{I{(y^{\\prime})}} and I^{2}{(y^{\\prime})} = I^{2}{(y^{\\prime})} - 1 + \\frac{\\cos{(y^{\\prime})}}{I{(y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 1, "Function('I')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 3, "Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2))"], "Equality(Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Add(Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Integer(-1), Mul(Pow(Function('I')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), cos(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given B{(\\chi)} = e^{\\chi}, then obtain \\frac{\\partial}{\\partial \\chi} (\\mathbf{S} - \\int \\frac{B{(\\chi)}}{B{(\\chi)} - e^{\\chi}} d\\chi) = \\frac{\\partial}{\\partial \\chi} (\\mathbf{A} - \\int \\frac{e^{\\chi}}{B{(\\chi)} - e^{\\chi}} d\\chi)", "derivation": "B{(\\chi)} = e^{\\chi} and \\frac{B{(\\chi)}}{- B{(\\chi)} + e^{\\chi}} = \\frac{e^{\\chi}}{- B{(\\chi)} + e^{\\chi}} and \\int \\frac{B{(\\chi)}}{- B{(\\chi)} + e^{\\chi}} d\\chi = \\int \\frac{e^{\\chi}}{- B{(\\chi)} + e^{\\chi}} d\\chi and \\frac{d}{d \\chi} \\int \\frac{B{(\\chi)}}{- B{(\\chi)} + e^{\\chi}} d\\chi = \\frac{d}{d \\chi} \\int \\frac{e^{\\chi}}{- B{(\\chi)} + e^{\\chi}} d\\chi and \\frac{\\partial}{\\partial \\chi} (\\mathbf{S} - \\int \\frac{B{(\\chi)}}{B{(\\chi)} - e^{\\chi}} d\\chi) = \\frac{\\partial}{\\partial \\chi} (\\mathbf{A} - \\int \\frac{e^{\\chi}}{B{(\\chi)} - e^{\\chi}} d\\chi)", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), Function('B')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), Function('B')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), Function('B')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Integral(Mul(Pow(Add(Function('B')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Integer(-1)), Function('B')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integral(Mul(Pow(Add(Function('B')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(\\theta,\\hat{H})} = \\log{(- \\hat{H} + \\theta)}, then obtain (\\theta + \\varphi^{*}{(\\theta,\\hat{H})})^{\\theta} = (\\theta + \\log{(- \\hat{H} + \\theta)})^{\\theta}", "derivation": "\\varphi^{*}{(\\theta,\\hat{H})} = \\log{(- \\hat{H} + \\theta)} and \\theta + \\varphi^{*}{(\\theta,\\hat{H})} = \\theta + \\log{(- \\hat{H} + \\theta)} and e^{\\varphi^{*}{(\\theta,\\hat{H})}} = - \\hat{H} + \\theta and \\theta + \\varphi^{*}{(\\theta,\\hat{H})} = \\theta + \\log{(e^{\\varphi^{*}{(\\theta,\\hat{H})}})} and \\theta + \\log{(e^{\\varphi^{*}{(\\theta,\\hat{H})}})} = \\theta + \\log{(- \\hat{H} + \\theta)} and (\\theta + \\log{(e^{\\varphi^{*}{(\\theta,\\hat{H})}})})^{\\theta} = (\\theta + \\log{(- \\hat{H} + \\theta)})^{\\theta} and (\\theta + \\varphi^{*}{(\\theta,\\hat{H})})^{\\theta} = (\\theta + \\log{(- \\hat{H} + \\theta)})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\theta', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True)))))"], [["exp", 1], "Equality(exp(Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\theta', commutative=True), log(exp(Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('\\\\theta', commutative=True), log(exp(Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Add(Symbol('\\\\theta', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True)))))"], [["power", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta', commutative=True), log(exp(Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Symbol('\\\\theta', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(l)} = \\sin{(l)}, then obtain (\\frac{d}{d l} 1 + 1) \\frac{d}{d l} 1 = (\\frac{d}{d l} 1 + 1) \\frac{d}{d l} (\\frac{\\sin{(l)}}{\\hat{\\mathbf{r}}{(l)}})^{l}", "derivation": "\\hat{\\mathbf{r}}{(l)} = \\sin{(l)} and 1 = \\frac{\\sin{(l)}}{\\hat{\\mathbf{r}}{(l)}} and 1 = (\\frac{\\sin{(l)}}{\\hat{\\mathbf{r}}{(l)}})^{l} and \\frac{d}{d l} 1 = \\frac{d}{d l} (\\frac{\\sin{(l)}}{\\hat{\\mathbf{r}}{(l)}})^{l} and (\\frac{d}{d l} 1 + 1) \\frac{d}{d l} 1 = (\\frac{d}{d l} 1 + 1) \\frac{d}{d l} (\\frac{\\sin{(l)}}{\\hat{\\mathbf{r}}{(l)}})^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["divide", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 4, "Add(Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Add(Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)), Derivative(Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(P_{g})} = \\cos{(P_{g})} and \\theta_{1}{(P_{g})} = 2 a{(P_{g})}, then obtain \\frac{d}{d P_{g}} \\theta_{1}{(P_{g})} = \\frac{d}{d P_{g}} (a{(P_{g})} + \\cos{(P_{g})})", "derivation": "a{(P_{g})} = \\cos{(P_{g})} and 2 a{(P_{g})} = a{(P_{g})} + \\cos{(P_{g})} and \\frac{d}{d P_{g}} 2 a{(P_{g})} = \\frac{d}{d P_{g}} (a{(P_{g})} + \\cos{(P_{g})}) and \\theta_{1}{(P_{g})} = 2 a{(P_{g})} and \\frac{d}{d P_{g}} \\theta_{1}{(P_{g})} = \\frac{d}{d P_{g}} (a{(P_{g})} + \\cos{(P_{g})})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True)))"], [["add", 1, "Function('a')(Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(2), Function('a')(Symbol('P_g', commutative=True))), Add(Function('a')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True))))"], [["differentiate", 2, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('a')(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Function('a')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('P_g', commutative=True)), Mul(Integer(2), Function('a')(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\theta_1')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Function('a')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(t_{2})} = e^{t_{2}}, then derive (\\delta + e^{t_{2}}) \\mathbf{J}_P{(t_{2})} = (\\delta + e^{t_{2}}) e^{t_{2}}, then obtain \\int (\\delta + e^{t_{2}}) \\mathbf{J}_P{(t_{2})} dt_{2} = \\int (\\delta + e^{t_{2}}) e^{t_{2}} dt_{2}", "derivation": "\\mathbf{J}_P{(t_{2})} = e^{t_{2}} and \\mathbf{J}_P{(t_{2})} \\int e^{t_{2}} dt_{2} = e^{t_{2}} \\int e^{t_{2}} dt_{2} and (\\delta + e^{t_{2}}) \\mathbf{J}_P{(t_{2})} = (\\delta + e^{t_{2}}) e^{t_{2}} and \\int (\\delta + e^{t_{2}}) \\mathbf{J}_P{(t_{2})} dt_{2} = \\int (\\delta + e^{t_{2}}) e^{t_{2}} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["times", 1, "Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True)), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(exp(Symbol('t_2', commutative=True)), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('\\\\delta', commutative=True), exp(Symbol('t_2', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True))), Mul(Add(Symbol('\\\\delta', commutative=True), exp(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\delta', commutative=True), exp(Symbol('t_2', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Add(Symbol('\\\\delta', commutative=True), exp(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given J{(\\theta_1,A_{x})} = A_{x} + \\sin{(\\theta_1)}, then obtain e^{\\int (- \\theta_1 - 1) d\\theta_1} = e^{\\int (A_{x} - \\theta_1 - J{(\\theta_1,A_{x})} + \\sin{(\\theta_1)} - 1) d\\theta_1}", "derivation": "J{(\\theta_1,A_{x})} = A_{x} + \\sin{(\\theta_1)} and - \\theta_1 + J{(\\theta_1,A_{x})} = A_{x} - \\theta_1 + \\sin{(\\theta_1)} and - \\theta_1 + J{(\\theta_1,A_{x})} - 1 = A_{x} - \\theta_1 + \\sin{(\\theta_1)} - 1 and - \\theta_1 - 1 = A_{x} - \\theta_1 - J{(\\theta_1,A_{x})} + \\sin{(\\theta_1)} - 1 and \\int (- \\theta_1 - 1) d\\theta_1 = \\int (A_{x} - \\theta_1 - J{(\\theta_1,A_{x})} + \\sin{(\\theta_1)} - 1) d\\theta_1 and e^{\\int (- \\theta_1 - 1) d\\theta_1} = e^{\\int (A_{x} - \\theta_1 - J{(\\theta_1,A_{x})} + \\sin{(\\theta_1)} - 1) d\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)))"], [["minus", 3, "Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["exp", 5], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('\\\\theta_1', commutative=True), Symbol('A_x', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given q{(\\Psi)} = \\cos{(e^{\\Psi})}, then obtain \\int (0^{\\Psi})^{\\Psi} d\\Psi = \\int ((- q{(\\Psi)} + \\cos{(e^{\\Psi})})^{\\Psi})^{\\Psi} d\\Psi", "derivation": "q{(\\Psi)} = \\cos{(e^{\\Psi})} and 0 = - q{(\\Psi)} + \\cos{(e^{\\Psi})} and 0^{\\Psi} = (- q{(\\Psi)} + \\cos{(e^{\\Psi})})^{\\Psi} and (0^{\\Psi})^{\\Psi} = ((- q{(\\Psi)} + \\cos{(e^{\\Psi})})^{\\Psi})^{\\Psi} and \\int (0^{\\Psi})^{\\Psi} d\\Psi = \\int ((- q{(\\Psi)} + \\cos{(e^{\\Psi})})^{\\Psi})^{\\Psi} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\Psi', commutative=True)), cos(exp(Symbol('\\\\Psi', commutative=True))))"], [["minus", 1, "Function('q')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi', commutative=True))), cos(exp(Symbol('\\\\Psi', commutative=True)))))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi', commutative=True))), cos(exp(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi', commutative=True))), cos(exp(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Pow(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Pow(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi', commutative=True))), cos(exp(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\hat{p},C_{2})} = C_{2} - \\hat{p}, then obtain - \\frac{1}{\\hat{p}} = - \\frac{((C_{2} - \\hat{p})^{C_{2}})^{\\hat{p}} (\\hat{H}^{C_{2}}{(\\hat{p},C_{2})})^{- \\hat{p}}}{\\hat{p}}", "derivation": "\\hat{H}{(\\hat{p},C_{2})} = C_{2} - \\hat{p} and \\hat{H}^{C_{2}}{(\\hat{p},C_{2})} = (C_{2} - \\hat{p})^{C_{2}} and (\\hat{H}^{C_{2}}{(\\hat{p},C_{2})})^{\\hat{p}} = ((C_{2} - \\hat{p})^{C_{2}})^{\\hat{p}} and - \\frac{(\\hat{H}^{C_{2}}{(\\hat{p},C_{2})})^{\\hat{p}}}{\\hat{p}} = - \\frac{((C_{2} - \\hat{p})^{C_{2}})^{\\hat{p}}}{\\hat{p}} and - \\frac{1}{\\hat{p}} = - \\frac{((C_{2} - \\hat{p})^{C_{2}})^{\\hat{p}} (\\hat{H}^{C_{2}}{(\\hat{p},C_{2})})^{- \\hat{p}}}{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('C_2', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 4, "Pow(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('C_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\phi_1)} = e^{\\phi_1}, then obtain \\frac{d}{d \\phi_1} ((\\phi{(\\phi_1)} + e^{\\phi_1}) \\phi{(\\phi_1)})^{\\phi_1} = \\frac{d}{d \\phi_1} ((\\phi{(\\phi_1)} + e^{\\phi_1}) e^{\\phi_1})^{\\phi_1}", "derivation": "\\phi{(\\phi_1)} = e^{\\phi_1} and 2 \\phi{(\\phi_1)} = \\phi{(\\phi_1)} + e^{\\phi_1} and 2 \\phi^{2}{(\\phi_1)} = 2 \\phi{(\\phi_1)} e^{\\phi_1} and (2 \\phi^{2}{(\\phi_1)})^{\\phi_1} = (2 \\phi{(\\phi_1)} e^{\\phi_1})^{\\phi_1} and ((\\phi{(\\phi_1)} + e^{\\phi_1}) \\phi{(\\phi_1)})^{\\phi_1} = ((\\phi{(\\phi_1)} + e^{\\phi_1}) e^{\\phi_1})^{\\phi_1} and \\frac{d}{d \\phi_1} ((\\phi{(\\phi_1)} + e^{\\phi_1}) \\phi{(\\phi_1)})^{\\phi_1} = \\frac{d}{d \\phi_1} ((\\phi{(\\phi_1)} + e^{\\phi_1}) e^{\\phi_1})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True))), Add(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Add(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Add(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Mul(Add(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Add(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{H},z)} = \\frac{\\partial}{\\partial \\mathbf{H}} z^{\\mathbf{H}}, then derive \\eta^{\\prime}^{\\mathbf{H}}{(\\mathbf{H},z)} = (z^{\\mathbf{H}} \\log{(z)})^{\\mathbf{H}}, then obtain (\\frac{\\partial}{\\partial \\mathbf{H}} z^{\\mathbf{H}})^{\\mathbf{H}} = (z^{\\mathbf{H}} \\log{(z)})^{\\mathbf{H}}", "derivation": "\\eta^{\\prime}{(\\mathbf{H},z)} = \\frac{\\partial}{\\partial \\mathbf{H}} z^{\\mathbf{H}} and \\eta^{\\prime}^{\\mathbf{H}}{(\\mathbf{H},z)} = (\\frac{\\partial}{\\partial \\mathbf{H}} z^{\\mathbf{H}})^{\\mathbf{H}} and \\eta^{\\prime}^{\\mathbf{H}}{(\\mathbf{H},z)} = (z^{\\mathbf{H}} \\log{(z)})^{\\mathbf{H}} and (\\frac{\\partial}{\\partial \\mathbf{H}} z^{\\mathbf{H}})^{\\mathbf{H}} = (z^{\\mathbf{H}} \\log{(z)})^{\\mathbf{H}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), Derivative(Pow(Symbol('z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(Pow(Symbol('z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Pow(Symbol('z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('z', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(Pow(Symbol('z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Pow(Symbol('z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('z', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(f)} = \\cos{(f)}, then derive F_{x} + \\int (\\operatorname{n_{2}}{(f)} + \\cos{(f)} - 1) (\\operatorname{n_{2}}{(f)} + \\cos{(f)} + 1) df = \\mu_0 + f + 2 \\sin{(f)} \\cos{(f)}, then obtain (F_{x} + \\int (\\operatorname{n_{2}}{(f)} + \\cos{(f)} - 1) (\\operatorname{n_{2}}{(f)} + \\cos{(f)} + 1) df) \\cos{(f)} = (\\mu_0 + f + 2 \\sin{(f)} \\cos{(f)}) \\cos{(f)}", "derivation": "\\operatorname{n_{2}}{(f)} = \\cos{(f)} and \\operatorname{n_{2}}{(f)} + \\cos{(f)} = 2 \\cos{(f)} and (\\operatorname{n_{2}}{(f)} + \\cos{(f)})^{2} = 4 \\cos^{2}{(f)} and (\\operatorname{n_{2}}{(f)} + \\cos{(f)})^{2} - 1 = 4 \\cos^{2}{(f)} - 1 and \\int ((\\operatorname{n_{2}}{(f)} + \\cos{(f)})^{2} - 1) df = \\int (4 \\cos^{2}{(f)} - 1) df and F_{x} + \\int (\\operatorname{n_{2}}{(f)} + \\cos{(f)} - 1) (\\operatorname{n_{2}}{(f)} + \\cos{(f)} + 1) df = \\mu_0 + f + 2 \\sin{(f)} \\cos{(f)} and (F_{x} + \\int (\\operatorname{n_{2}}{(f)} + \\cos{(f)} - 1) (\\operatorname{n_{2}}{(f)} + \\cos{(f)} + 1) df) \\cos{(f)} = (\\mu_0 + f + 2 \\sin{(f)} \\cos{(f)}) \\cos{(f)}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["add", 1, "cos(Symbol('f', commutative=True))"], "Equality(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True))), Mul(Integer(2), cos(Symbol('f', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True))), Integer(2)), Mul(Integer(4), Pow(cos(Symbol('f', commutative=True)), Integer(2))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True))), Integer(2)), Integer(-1)), Add(Mul(Integer(4), Pow(cos(Symbol('f', commutative=True)), Integer(2))), Integer(-1)))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Pow(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True))), Integer(2)), Integer(-1)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(4), Pow(cos(Symbol('f', commutative=True)), Integer(2))), Integer(-1)), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_x', commutative=True), Integral(Mul(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)), Integer(-1)), Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)), Integer(1))), Tuple(Symbol('f', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Symbol('f', commutative=True), Mul(Integer(2), sin(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))))"], [["times", 6, "cos(Symbol('f', commutative=True))"], "Equality(Mul(Add(Symbol('F_x', commutative=True), Integral(Mul(Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)), Integer(-1)), Add(Function('n_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)), Integer(1))), Tuple(Symbol('f', commutative=True)))), cos(Symbol('f', commutative=True))), Mul(Add(Symbol('\\\\mu_0', commutative=True), Symbol('f', commutative=True), Mul(Integer(2), sin(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))), cos(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{t},f^{*})} = \\int (- f^{*} + v_{t}) dv_{t}, then derive \\operatorname{t_{2}}^{f^{*}}{(v_{t},f^{*})} = (- f^{*} v_{t} + t_{1} + \\frac{v_{t}^{2}}{2})^{f^{*}}, then obtain (\\int (- f^{*} + v_{t}) dv_{t})^{f^{*}} = (- f^{*} v_{t} + t_{1} + \\frac{v_{t}^{2}}{2})^{f^{*}}", "derivation": "\\operatorname{t_{2}}{(v_{t},f^{*})} = \\int (- f^{*} + v_{t}) dv_{t} and \\operatorname{t_{2}}^{f^{*}}{(v_{t},f^{*})} = (\\int (- f^{*} + v_{t}) dv_{t})^{f^{*}} and \\operatorname{t_{2}}^{f^{*}}{(v_{t},f^{*})} = (- f^{*} v_{t} + t_{1} + \\frac{v_{t}^{2}}{2})^{f^{*}} and (\\int (- f^{*} + v_{t}) dv_{t})^{f^{*}} = (- f^{*} v_{t} + t_{1} + \\frac{v_{t}^{2}}{2})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_t', commutative=True), Symbol('f^*', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('v_t', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('f^*', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('t_2')(Symbol('v_t', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('v_t', commutative=True)), Symbol('t_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_t', commutative=True), Integer(2)))), Symbol('f^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('v_t', commutative=True)), Symbol('t_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_t', commutative=True), Integer(2)))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(t_{2})} = \\log{(t_{2})}, then obtain \\frac{- \\mathbf{H}{(t_{2})} + \\int \\mathbf{H}^{t_{2}}{(t_{2})} dt_{2}}{t_{2}} = \\frac{- \\mathbf{H}{(t_{2})} + \\int \\log{(t_{2})}^{t_{2}} dt_{2}}{t_{2}}", "derivation": "\\mathbf{H}{(t_{2})} = \\log{(t_{2})} and \\mathbf{H}^{t_{2}}{(t_{2})} = \\log{(t_{2})}^{t_{2}} and \\int \\mathbf{H}^{t_{2}}{(t_{2})} dt_{2} = \\int \\log{(t_{2})}^{t_{2}} dt_{2} and - \\mathbf{H}{(t_{2})} + \\int \\mathbf{H}^{t_{2}}{(t_{2})} dt_{2} = - \\mathbf{H}{(t_{2})} + \\int \\log{(t_{2})}^{t_{2}} dt_{2} and \\frac{- \\mathbf{H}{(t_{2})} + \\int \\mathbf{H}^{t_{2}}{(t_{2})} dt_{2}}{t_{2}} = \\frac{- \\mathbf{H}{(t_{2})} + \\int \\log{(t_{2})}^{t_{2}} dt_{2}}{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(log(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Pow(log(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["minus", 3, "Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True))), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True))), Integral(Pow(log(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["divide", 4, "Symbol('t_2', commutative=True)"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True))), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True))), Integral(Pow(log(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{v})} = \\log{(e^{\\mathbf{v}})} and \\operatorname{n_{1}}{(\\mathbf{v})} = (\\int \\log{(e^{\\mathbf{v}})} d\\mathbf{v})^{\\mathbf{v}}, then obtain (\\int \\operatorname{F_{N}}{(\\mathbf{v})} d\\mathbf{v})^{\\mathbf{v}} = \\operatorname{n_{1}}{(\\mathbf{v})}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{v})} = \\log{(e^{\\mathbf{v}})} and \\int \\operatorname{F_{N}}{(\\mathbf{v})} d\\mathbf{v} = \\int \\log{(e^{\\mathbf{v}})} d\\mathbf{v} and (\\int \\operatorname{F_{N}}{(\\mathbf{v})} d\\mathbf{v})^{\\mathbf{v}} = (\\int \\log{(e^{\\mathbf{v}})} d\\mathbf{v})^{\\mathbf{v}} and \\operatorname{n_{1}}{(\\mathbf{v})} = (\\int \\log{(e^{\\mathbf{v}})} d\\mathbf{v})^{\\mathbf{v}} and (\\int \\operatorname{F_{N}}{(\\mathbf{v})} d\\mathbf{v})^{\\mathbf{v}} = \\operatorname{n_{1}}{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{v}', commutative=True)), log(exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(log(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Integral(Function('F_N')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Integral(log(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(Integral(log(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('F_N')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Function('n_1')(Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(J,\\mathbf{H})} = \\cos{(J + \\mathbf{H})} and \\lambda{(J,\\mathbf{H})} = \\cos{(J + \\mathbf{H})} and \\mathbf{A}{(J,\\mathbf{H})} = \\cos^{2}{(J + \\mathbf{H})}, then obtain \\mathbf{A}{(J,\\mathbf{H})} = \\operatorname{n_{2}}{(J,\\mathbf{H})} \\cos{(J + \\mathbf{H})}", "derivation": "\\operatorname{n_{2}}{(J,\\mathbf{H})} = \\cos{(J + \\mathbf{H})} and \\lambda{(J,\\mathbf{H})} = \\cos{(J + \\mathbf{H})} and \\lambda{(J,\\mathbf{H})} = \\operatorname{n_{2}}{(J,\\mathbf{H})} and \\lambda{(J,\\mathbf{H})} \\cos{(J + \\mathbf{H})} = \\cos^{2}{(J + \\mathbf{H})} and \\mathbf{A}{(J,\\mathbf{H})} = \\cos^{2}{(J + \\mathbf{H})} and \\operatorname{n_{2}}{(J,\\mathbf{H})} \\cos{(J + \\mathbf{H})} = \\cos^{2}{(J + \\mathbf{H})} and \\mathbf{A}{(J,\\mathbf{H})} = \\operatorname{n_{2}}{(J,\\mathbf{H})} \\cos{(J + \\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 2, "cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\mathbf{A}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(a,\\Psi,n_{2})} = \\Psi^{n_{2}} + a, then derive \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{s}{(a,\\Psi,n_{2})} da = \\frac{\\partial}{\\partial n_{2}} (\\Psi^{n_{2}} a + \\frac{a^{2}}{2} + x^\\prime), then obtain 2 \\frac{\\partial}{\\partial n_{2}} (\\Psi^{n_{2}} a + \\frac{a^{2}}{2} + x^\\prime) = 2 \\frac{\\partial}{\\partial n_{2}} \\int (\\Psi^{n_{2}} + a) da", "derivation": "\\mathbf{s}{(a,\\Psi,n_{2})} = \\Psi^{n_{2}} + a and \\int \\mathbf{s}{(a,\\Psi,n_{2})} da = \\int (\\Psi^{n_{2}} + a) da and \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{s}{(a,\\Psi,n_{2})} da = \\frac{\\partial}{\\partial n_{2}} \\int (\\Psi^{n_{2}} + a) da and \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{s}{(a,\\Psi,n_{2})} da = \\frac{\\partial}{\\partial n_{2}} (\\Psi^{n_{2}} a + \\frac{a^{2}}{2} + x^\\prime) and 2 \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{s}{(a,\\Psi,n_{2})} da = 2 \\frac{\\partial}{\\partial n_{2}} \\int (\\Psi^{n_{2}} + a) da and 2 \\frac{\\partial}{\\partial n_{2}} (\\Psi^{n_{2}} a + \\frac{a^{2}}{2} + x^\\prime) = 2 \\frac{\\partial}{\\partial n_{2}} \\int (\\Psi^{n_{2}} + a) da", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["divide", 3, "Rational(1, 2)"], "Equality(Mul(Integer(2), Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Derivative(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given m{(\\mathbf{J},n_{2})} = \\frac{\\mathbf{J}}{n_{2}}, then obtain n_{2} = \\frac{\\mathbf{J}}{m{(\\mathbf{J},n_{2})}}", "derivation": "m{(\\mathbf{J},n_{2})} = \\frac{\\mathbf{J}}{n_{2}} and \\mathbf{J} m{(\\mathbf{J},n_{2})} = \\frac{\\mathbf{J}^{2}}{n_{2}} and \\mathbf{J} = \\frac{\\mathbf{J}^{2}}{n_{2} m{(\\mathbf{J},n_{2})}} and n_{2} = \\frac{\\mathbf{J}}{m{(\\mathbf{J},n_{2})}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('m')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["divide", 2, "Function('m')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Symbol('\\\\mathbf{J}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Function('m')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))"], "Equality(Symbol('n_2', commutative=True), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('m')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given l{(G)} = \\log{(G)}, then obtain e^{\\int l^{G}{(G)} \\log{(G)}^{2} \\log{(G)}^{G} dG} = e^{\\int \\log{(G)}^{2} \\log{(G)}^{2 G} dG}", "derivation": "l{(G)} = \\log{(G)} and l^{G}{(G)} = \\log{(G)}^{G} and l^{G}{(G)} \\log{(G)} = \\log{(G)} \\log{(G)}^{G} and l^{G}{(G)} \\log{(G)}^{2} \\log{(G)}^{G} = \\log{(G)}^{2} \\log{(G)}^{2 G} and \\int l^{G}{(G)} \\log{(G)}^{2} \\log{(G)}^{G} dG = \\int \\log{(G)}^{2} \\log{(G)}^{2 G} dG and e^{\\int l^{G}{(G)} \\log{(G)}^{2} \\log{(G)}^{G} dG} = e^{\\int \\log{(G)}^{2} \\log{(G)}^{2 G} dG}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('l')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["times", 2, "log(Symbol('G', commutative=True))"], "Equality(Mul(Pow(Function('l')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), log(Symbol('G', commutative=True))), Mul(log(Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True))))"], [["times", 3, "Mul(log(Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], "Equality(Mul(Pow(Function('l')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Mul(Integer(2), Symbol('G', commutative=True)))))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(Function('l')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Mul(Integer(2), Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["exp", 5], "Equality(exp(Integral(Mul(Pow(Function('l')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), exp(Integral(Mul(Pow(log(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Mul(Integer(2), Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\mu)} = \\sin{(\\log{(\\mu)})}, then derive \\int \\tilde{g}{(\\mu)} d\\mu = \\frac{\\mu \\sin{(\\log{(\\mu)})}}{2} - \\frac{\\mu \\cos{(\\log{(\\mu)})}}{2}, then obtain \\int \\sin{(\\log{(\\mu)})} d\\mu = \\frac{\\mu \\tilde{g}{(\\mu)}}{2} - \\frac{\\mu \\cos{(\\log{(\\mu)})}}{2}", "derivation": "\\tilde{g}{(\\mu)} = \\sin{(\\log{(\\mu)})} and \\int \\tilde{g}{(\\mu)} d\\mu = \\int \\sin{(\\log{(\\mu)})} d\\mu and \\iint \\tilde{g}{(\\mu)} d\\mu d\\mu = \\iint \\sin{(\\log{(\\mu)})} d\\mu d\\mu and \\frac{d}{d \\mu} \\iint \\tilde{g}{(\\mu)} d\\mu d\\mu = \\frac{d}{d \\mu} \\iint \\sin{(\\log{(\\mu)})} d\\mu d\\mu and \\int \\tilde{g}{(\\mu)} d\\mu = \\frac{\\mu \\sin{(\\log{(\\mu)})}}{2} - \\frac{\\mu \\cos{(\\log{(\\mu)})}}{2} and \\int \\sin{(\\log{(\\mu)})} d\\mu = \\frac{\\mu \\sin{(\\log{(\\mu)})}}{2} - \\frac{\\mu \\cos{(\\log{(\\mu)})}}{2} and \\int \\sin{(\\log{(\\mu)})} d\\mu = \\frac{\\mu \\tilde{g}{(\\mu)}}{2} - \\frac{\\mu \\cos{(\\log{(\\mu)})}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True)), sin(log(Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(sin(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(sin(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integral(sin(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mu', commutative=True), sin(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\mu', commutative=True), cos(log(Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(sin(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mu', commutative=True), sin(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\mu', commutative=True), cos(log(Symbol('\\\\mu', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integral(sin(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mu', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\mu', commutative=True), cos(log(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(v_{z},\\varepsilon)} = \\varepsilon - v_{z}, then obtain \\frac{\\partial}{\\partial \\varepsilon} \\frac{v_{z} + \\operatorname{f^{*}}{(v_{z},\\varepsilon)}}{\\varepsilon - v_{z}} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{\\varepsilon - v_{z}}", "derivation": "\\operatorname{f^{*}}{(v_{z},\\varepsilon)} = \\varepsilon - v_{z} and v_{z} + \\operatorname{f^{*}}{(v_{z},\\varepsilon)} = \\varepsilon and \\frac{v_{z} + \\operatorname{f^{*}}{(v_{z},\\varepsilon)}}{\\varepsilon - v_{z}} = \\frac{\\varepsilon}{\\varepsilon - v_{z}} and \\frac{\\partial}{\\partial \\varepsilon} \\frac{v_{z} + \\operatorname{f^{*}}{(v_{z},\\varepsilon)}}{\\varepsilon - v_{z}} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\varepsilon}{\\varepsilon - v_{z}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('v_z', commutative=True))"], "Equality(Add(Symbol('v_z', commutative=True), Function('f^*')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))"], [["divide", 2, "Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Integer(-1)), Add(Symbol('v_z', commutative=True), Function('f^*')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Integer(-1)), Add(Symbol('v_z', commutative=True), Function('f^*')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(I,P_{e})} = I P_{e}, then obtain \\iint (P_{e} + \\psi{(I,P_{e})}) dP_{e} dP_{e} = \\iint (I P_{e} + P_{e}) dP_{e} dP_{e}", "derivation": "\\psi{(I,P_{e})} = I P_{e} and P_{e} + \\psi{(I,P_{e})} = I P_{e} + P_{e} and \\int (P_{e} + \\psi{(I,P_{e})}) dP_{e} = \\int (I P_{e} + P_{e}) dP_{e} and \\iint (P_{e} + \\psi{(I,P_{e})}) dP_{e} dP_{e} = \\iint (I P_{e} + P_{e}) dP_{e} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('P_e', commutative=True)))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('P_e', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Symbol('P_e', commutative=True), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Symbol('P_e', commutative=True), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\rho)} = \\cos{(\\sin{(\\rho)})} and t{(\\rho)} = \\operatorname{f_{\\mathbf{p}}}{(\\rho)} \\cos{(\\sin{(\\rho)})}, then obtain 2 t{(\\rho)} = t{(\\rho)} + \\cos^{2}{(\\sin{(\\rho)})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\rho)} = \\cos{(\\sin{(\\rho)})} and \\operatorname{f_{\\mathbf{p}}}{(\\rho)} \\cos{(\\sin{(\\rho)})} = \\cos^{2}{(\\sin{(\\rho)})} and 2 \\operatorname{f_{\\mathbf{p}}}{(\\rho)} \\cos{(\\sin{(\\rho)})} = \\operatorname{f_{\\mathbf{p}}}{(\\rho)} \\cos{(\\sin{(\\rho)})} + \\cos^{2}{(\\sin{(\\rho)})} and t{(\\rho)} = \\operatorname{f_{\\mathbf{p}}}{(\\rho)} \\cos{(\\sin{(\\rho)})} and 2 t{(\\rho)} = t{(\\rho)} + \\cos^{2}{(\\sin{(\\rho)})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True))))"], [["times", 1, "cos(sin(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True)))), Pow(cos(sin(Symbol('\\\\rho', commutative=True))), Integer(2)))"], [["add", 2, "Mul(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True))))"], "Equality(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True)))), Add(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True)))), Pow(cos(sin(Symbol('\\\\rho', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Function('t')(Symbol('\\\\rho', commutative=True))), Add(Function('t')(Symbol('\\\\rho', commutative=True)), Pow(cos(sin(Symbol('\\\\rho', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{E}{(q,C_{1},C_{2})} = \\frac{C_{2} q}{C_{1}}, then obtain \\mathbf{E}{(q,C_{1},C_{2})} + \\frac{q \\mathbf{E}{(q,C_{1},C_{2})}}{C_{1} C_{2}} = \\mathbf{E}{(q,C_{1},C_{2})} + \\frac{q^{2}}{C_{1}^{2}}", "derivation": "\\mathbf{E}{(q,C_{1},C_{2})} = \\frac{C_{2} q}{C_{1}} and \\frac{C_{2} q \\mathbf{E}{(q,C_{1},C_{2})}}{C_{1}} = \\frac{C_{2}^{2} q^{2}}{C_{1}^{2}} and \\frac{q \\mathbf{E}{(q,C_{1},C_{2})}}{C_{1} C_{2}} = \\frac{q^{2}}{C_{1}^{2}} and \\mathbf{E}{(q,C_{1},C_{2})} + \\frac{q \\mathbf{E}{(q,C_{1},C_{2})}}{C_{1} C_{2}} = \\mathbf{E}{(q,C_{1},C_{2})} + \\frac{q^{2}}{C_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('C_2', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('C_2', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('C_2', commutative=True), Symbol('q', commutative=True), Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["divide", 2, "Pow(Symbol('C_2', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('q', commutative=True), Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["add", 3, "Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('q', commutative=True), Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True)))), Add(Function('\\\\mathbf{E}')(Symbol('q', commutative=True), Symbol('C_1', commutative=True), Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('q', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\varphi^{*}{(G)} = \\sin{(\\log{(G)})}, then obtain \\int \\frac{\\sin{(\\log{(G)})}}{\\log{(G)}} dG = \\int \\frac{\\sin^{2}{(\\log{(G)})}}{\\varphi^{*}{(G)} \\log{(G)}} dG", "derivation": "\\varphi^{*}{(G)} = \\sin{(\\log{(G)})} and \\frac{\\varphi^{*}{(G)}}{\\log{(G)}} = \\frac{\\sin{(\\log{(G)})}}{\\log{(G)}} and \\frac{\\varphi^{*}{(G)} \\log{(G)}}{\\sin{(\\log{(G)})}} = \\log{(G)} and \\frac{\\sin{(\\log{(G)})}}{\\log{(G)}} = \\frac{\\sin{(\\frac{\\varphi^{*}{(G)} \\log{(G)}}{\\sin{(\\log{(G)})}})} \\sin{(\\log{(G)})}}{\\varphi^{*}{(G)} \\log{(G)}} and \\frac{\\sin{(\\log{(G)})}}{\\log{(G)}} = \\frac{\\sin^{2}{(\\log{(G)})}}{\\varphi^{*}{(G)} \\log{(G)}} and \\int \\frac{\\sin{(\\log{(G)})}}{\\log{(G)}} dG = \\int \\frac{\\sin^{2}{(\\log{(G)})}}{\\varphi^{*}{(G)} \\log{(G)}} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('G', commutative=True)), sin(log(Symbol('G', commutative=True))))"], [["divide", 1, "log(Symbol('G', commutative=True))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(log(Symbol('G', commutative=True)))))"], [["divide", 1, "Mul(Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(log(Symbol('G', commutative=True))))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)), Pow(sin(log(Symbol('G', commutative=True))), Integer(-1))), log(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(log(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(Mul(Function('\\\\varphi^*')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)), Pow(sin(log(Symbol('G', commutative=True))), Integer(-1)))), sin(log(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(log(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('G', commutative=True))), Integer(2))))"], [["integrate", 5, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(log(Symbol('G', commutative=True)), Integer(-1)), sin(log(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Integer(-1)), Pow(log(Symbol('G', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('G', commutative=True))), Integer(2))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given h{(\\mathbf{B})} = \\log{(\\mathbf{B})} and B{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\Psi_{\\lambda}{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then obtain \\Psi_{\\lambda}{(\\mathbf{B})} + \\log{(\\mathbf{B})} = 2 \\log{(\\mathbf{B})}", "derivation": "h{(\\mathbf{B})} = \\log{(\\mathbf{B})} and B{(\\mathbf{B})} = \\log{(\\mathbf{B})} and h{(\\mathbf{B})} = B{(\\mathbf{B})} and \\Psi_{\\lambda}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and B{(\\mathbf{B})} + \\Psi_{\\lambda}{(\\mathbf{B})} + h{(\\mathbf{B})} - \\log{(\\mathbf{B})} = B{(\\mathbf{B})} + h{(\\mathbf{B})} and 2 B{(\\mathbf{B})} + \\Psi_{\\lambda}{(\\mathbf{B})} - \\log{(\\mathbf{B})} = 2 B{(\\mathbf{B})} and \\Psi_{\\lambda}{(\\mathbf{B})} + \\log{(\\mathbf{B})} = 2 \\log{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Function('B')(Symbol('\\\\mathbf{B}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 4, "Add(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Add(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Function('h')(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Function('B')(Symbol('\\\\mathbf{B}', commutative=True))), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(2), Function('B')(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(F_{c},n_{2})} = \\cos{(F_{c} n_{2})}, then derive \\frac{\\partial}{\\partial F_{c}} \\tilde{g}{(F_{c},n_{2})} = - n_{2} \\sin{(F_{c} n_{2})}, then obtain n_{2} \\sin{(F_{c} n_{2})} + 2 \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} = \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})}", "derivation": "\\tilde{g}{(F_{c},n_{2})} = \\cos{(F_{c} n_{2})} and \\frac{\\partial}{\\partial F_{c}} \\tilde{g}{(F_{c},n_{2})} = \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} and \\frac{\\partial}{\\partial F_{c}} \\tilde{g}{(F_{c},n_{2})} = - n_{2} \\sin{(F_{c} n_{2})} and \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} = - n_{2} \\sin{(F_{c} n_{2})} and 2 \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} = - n_{2} \\sin{(F_{c} n_{2})} + \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} and n_{2} \\sin{(F_{c} n_{2})} + 2 \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})} = \\frac{\\partial}{\\partial F_{c}} \\cos{(F_{c} n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)), cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)))))"], [["add", 4, "Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)))), Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["minus", 5, "Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))))"], "Equality(Add(Mul(Symbol('n_2', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True)))), Mul(Integer(2), Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))), Derivative(cos(Mul(Symbol('F_c', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(C_{1},\\phi)} = C_{1} + \\phi, then obtain \\frac{\\int C_{1} dC_{1} + \\int \\phi dC_{1}}{(C_{1} + \\phi) (\\sigma_{p}{(C_{1})} - \\log{(C_{1})})} = \\frac{\\int (C_{1} + \\phi) dC_{1}}{(C_{1} + \\phi) (\\sigma_{p}{(C_{1})} - \\log{(C_{1})})}", "derivation": "\\phi_{2}{(C_{1},\\phi)} = C_{1} + \\phi and \\int \\phi_{2}{(C_{1},\\phi)} dC_{1} = \\int (C_{1} + \\phi) dC_{1} and \\int \\phi_{2}{(C_{1},\\phi)} dC_{1} = \\int C_{1} dC_{1} + \\int \\phi dC_{1} and \\int C_{1} dC_{1} + \\int \\phi dC_{1} = \\int (C_{1} + \\phi) dC_{1} and \\frac{\\int C_{1} dC_{1} + \\int \\phi dC_{1}}{\\sigma_{p}{(C_{1})} - \\log{(C_{1})}} = \\frac{\\int (C_{1} + \\phi) dC_{1}}{\\sigma_{p}{(C_{1})} - \\log{(C_{1})}} and \\frac{\\int C_{1} dC_{1} + \\int \\phi dC_{1}}{(C_{1} + \\phi) (\\sigma_{p}{(C_{1})} - \\log{(C_{1})})} = \\frac{\\int (C_{1} + \\phi) dC_{1}}{(C_{1} + \\phi) (\\sigma_{p}{(C_{1})} - \\log{(C_{1})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('C_1', commutative=True)))), Integral(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["divide", 4, "Add(Function('\\\\sigma_p')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\sigma_p')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True)))), Integer(-1)), Add(Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('C_1', commutative=True))))), Mul(Pow(Add(Function('\\\\sigma_p')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True)))), Integer(-1)), Integral(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True)))))"], [["divide", 5, "Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(Add(Function('\\\\sigma_p')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True)))), Integer(-1)), Add(Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('C_1', commutative=True))))), Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(Add(Function('\\\\sigma_p')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True)))), Integer(-1)), Integral(Add(Symbol('C_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given A{(r,\\dot{y})} = e^{r^{\\dot{y}}} and \\mathbb{I}{(r,\\dot{y})} = A{(r,\\dot{y})} e^{- r^{\\dot{y}}}, then obtain \\dot{y} A{(r,\\dot{y})} + \\mathbb{I}{(r,\\dot{y})} = \\dot{y} A{(r,\\dot{y})} + 1", "derivation": "A{(r,\\dot{y})} = e^{r^{\\dot{y}}} and \\dot{y} A{(r,\\dot{y})} = \\dot{y} e^{r^{\\dot{y}}} and A{(r,\\dot{y})} e^{- r^{\\dot{y}}} = 1 and \\mathbb{I}{(r,\\dot{y})} = A{(r,\\dot{y})} e^{- r^{\\dot{y}}} and \\mathbb{I}{(r,\\dot{y})} = 1 and \\dot{y} A{(r,\\dot{y})} + \\mathbb{I}{(r,\\dot{y})} = \\dot{y} A{(r,\\dot{y})} + 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["times", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), exp(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["divide", 2, "Mul(Symbol('\\\\dot{y}', commutative=True), exp(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Mul(Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1))"], [["add", 5, "Mul(Symbol('\\\\dot{y}', commutative=True), Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('\\\\mathbb{I}')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Function('A')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\varepsilon_{0}{(t,\\theta_1)} = t + \\sin{(\\theta_1)}, then derive \\int \\varepsilon_{0}{(t,\\theta_1)} dt = \\mathbf{J}_f + \\frac{t^{2}}{2} + t \\sin{(\\theta_1)}, then obtain 0 = - \\mathbf{J}_f - \\frac{t^{2}}{2} - t \\sin{(\\theta_1)} + \\int \\varepsilon_{0}{(t,\\theta_1)} dt", "derivation": "\\varepsilon_{0}{(t,\\theta_1)} = t + \\sin{(\\theta_1)} and \\int \\varepsilon_{0}{(t,\\theta_1)} dt = \\int (t + \\sin{(\\theta_1)}) dt and \\int \\varepsilon_{0}{(t,\\theta_1)} dt = \\mathbf{J}_f + \\frac{t^{2}}{2} + t \\sin{(\\theta_1)} and 0 = \\int (t + \\sin{(\\theta_1)}) dt - \\int \\varepsilon_{0}{(t,\\theta_1)} dt and 0 = - \\mathbf{J}_f - \\frac{t^{2}}{2} - t \\sin{(\\theta_1)} + \\int (t + \\sin{(\\theta_1)}) dt and 0 = - \\mathbf{J}_f - \\frac{t^{2}}{2} - t \\sin{(\\theta_1)} + \\int \\varepsilon_{0}{(t,\\theta_1)} dt", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Mul(Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 2, "Integral(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('t', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('t', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Integral(Function('\\\\varepsilon_0')(Symbol('t', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\omega{(a^{\\dagger},\\phi_2)} = \\phi_2 - a^{\\dagger}, then obtain a^{\\dagger} - \\omega{(a^{\\dagger},\\phi_2)} + \\int (- a^{\\dagger} + \\omega{(a^{\\dagger},\\phi_2)}) da^{\\dagger} = a^{\\dagger} - \\omega{(a^{\\dagger},\\phi_2)} + \\int (\\phi_2 - 2 a^{\\dagger}) da^{\\dagger}", "derivation": "\\omega{(a^{\\dagger},\\phi_2)} = \\phi_2 - a^{\\dagger} and - a^{\\dagger} + \\omega{(a^{\\dagger},\\phi_2)} = \\phi_2 - 2 a^{\\dagger} and \\int (- a^{\\dagger} + \\omega{(a^{\\dagger},\\phi_2)}) da^{\\dagger} = \\int (\\phi_2 - 2 a^{\\dagger}) da^{\\dagger} and - \\phi_2 + 2 a^{\\dagger} + \\int (- a^{\\dagger} + \\omega{(a^{\\dagger},\\phi_2)}) da^{\\dagger} = - \\phi_2 + 2 a^{\\dagger} + \\int (\\phi_2 - 2 a^{\\dagger}) da^{\\dagger} and a^{\\dagger} - \\omega{(a^{\\dagger},\\phi_2)} + \\int (- a^{\\dagger} + \\omega{(a^{\\dagger},\\phi_2)}) da^{\\dagger} = a^{\\dagger} - \\omega{(a^{\\dagger},\\phi_2)} + \\int (\\phi_2 - 2 a^{\\dagger}) da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('\\\\omega')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(u,q)} = q u and \\mathbf{S}{(u,q)} = \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)}, then obtain \\frac{\\mathbf{S}{(u,q)} \\int \\mathbf{S}{(u,q)} du}{q} = \\frac{\\mathbf{S}{(u,q)} \\int \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} du}{q}", "derivation": "\\hat{p}{(u,q)} = q u and \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} = \\frac{\\partial}{\\partial q} q u and \\mathbf{S}{(u,q)} = \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} and \\mathbf{S}{(u,q)} = \\frac{\\partial}{\\partial q} q u and \\int \\mathbf{S}{(u,q)} du = \\int \\frac{\\partial}{\\partial q} q u du and \\int \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} du = \\int \\frac{\\partial}{\\partial q} q u du and \\int \\mathbf{S}{(u,q)} du = \\int \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} du and \\frac{\\mathbf{S}{(u,q)} \\int \\mathbf{S}{(u,q)} du}{q} = \\frac{\\mathbf{S}{(u,q)} \\int \\frac{\\partial}{\\partial q} \\hat{p}{(u,q)} du}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('q', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Derivative(Mul(Symbol('q', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Symbol('q', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Symbol('q', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["times", 7, "Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Integral(Derivative(Function('\\\\hat{p}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{D},M)} = \\frac{\\mathbf{D}}{M}, then obtain \\int 0 d\\mathbf{D} = \\int (- \\operatorname{n_{2}}{(\\mathbf{D},M)} + \\frac{\\mathbf{D}}{M}) \\operatorname{n_{2}}{(\\mathbf{D},M)} d\\mathbf{D}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{D},M)} = \\frac{\\mathbf{D}}{M} and 0 = - \\operatorname{n_{2}}{(\\mathbf{D},M)} + \\frac{\\mathbf{D}}{M} and 0 = (- \\operatorname{n_{2}}{(\\mathbf{D},M)} + \\frac{\\mathbf{D}}{M}) \\operatorname{n_{2}}{(\\mathbf{D},M)} and \\int 0 d\\mathbf{D} = \\int (- \\operatorname{n_{2}}{(\\mathbf{D},M)} + \\frac{\\mathbf{D}}{M}) \\operatorname{n_{2}}{(\\mathbf{D},M)} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True))), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True))), Function('n_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(f_{E})} = \\sin{(f_{E})}, then obtain \\int f_{E} (f_{E} + \\Omega{(f_{E})}) df_{E} = \\int f_{E} (f_{E} + \\sin{(f_{E})}) df_{E}", "derivation": "\\Omega{(f_{E})} = \\sin{(f_{E})} and f_{E} + \\Omega{(f_{E})} = f_{E} + \\sin{(f_{E})} and f_{E} (f_{E} + \\Omega{(f_{E})}) = f_{E} (f_{E} + \\sin{(f_{E})}) and \\int f_{E} (f_{E} + \\Omega{(f_{E})}) df_{E} = \\int f_{E} (f_{E} + \\sin{(f_{E})}) df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["add", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Add(Symbol('f_E', commutative=True), sin(Symbol('f_E', commutative=True))))"], [["times", 2, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Add(Symbol('f_E', commutative=True), Function('\\\\Omega')(Symbol('f_E', commutative=True)))), Mul(Symbol('f_E', commutative=True), Add(Symbol('f_E', commutative=True), sin(Symbol('f_E', commutative=True)))))"], [["integrate", 3, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('f_E', commutative=True), Function('\\\\Omega')(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('f_E', commutative=True), sin(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(S,J_{\\varepsilon})} = J_{\\varepsilon} - S, then derive S + \\int \\bar{\\h}{(S,J_{\\varepsilon})} dS = J_{\\varepsilon} S - \\frac{S^{2}}{2} + S + v_{2}, then obtain - S (S + \\int (J_{\\varepsilon} - S) dS) = - S (J_{\\varepsilon} S - \\frac{S^{2}}{2} + S + v_{2})", "derivation": "\\bar{\\h}{(S,J_{\\varepsilon})} = J_{\\varepsilon} - S and \\int \\bar{\\h}{(S,J_{\\varepsilon})} dS = \\int (J_{\\varepsilon} - S) dS and S + \\int \\bar{\\h}{(S,J_{\\varepsilon})} dS = S + \\int (J_{\\varepsilon} - S) dS and S + \\int \\bar{\\h}{(S,J_{\\varepsilon})} dS = J_{\\varepsilon} S - \\frac{S^{2}}{2} + S + v_{2} and S + \\int (J_{\\varepsilon} - S) dS = J_{\\varepsilon} S - \\frac{S^{2}}{2} + S + v_{2} and - S (S + \\int (J_{\\varepsilon} - S) dS) = - S (J_{\\varepsilon} S - \\frac{S^{2}}{2} + S + v_{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Add(Symbol('S', commutative=True), Integral(Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Symbol('S', commutative=True), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('S', commutative=True), Integral(Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('S', commutative=True), Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('S', commutative=True), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('S', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 5, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('S', commutative=True), Add(Symbol('S', commutative=True), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))), Mul(Integer(-1), Symbol('S', commutative=True), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('S', commutative=True), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} = \\frac{M_{E} + \\mathbf{D}}{B}, then derive \\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E} = a + \\frac{M_{E}^{2}}{2 B} + \\frac{M_{E} \\mathbf{D}}{B}, then obtain \\frac{B^{2} (\\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E})^{2}}{M_{E}^{2} \\mathbf{D}^{2}} = \\frac{B^{2} (a + \\frac{M_{E}^{2}}{2 B} + \\frac{M_{E} \\mathbf{D}}{B})^{2}}{M_{E}^{2} \\mathbf{D}^{2}}", "derivation": "\\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} = \\frac{M_{E} + \\mathbf{D}}{B} and \\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E} = \\int \\frac{M_{E} + \\mathbf{D}}{B} dM_{E} and \\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E} = a + \\frac{M_{E}^{2}}{2 B} + \\frac{M_{E} \\mathbf{D}}{B} and \\frac{B \\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E}}{M_{E} \\mathbf{D}} = \\frac{B (a + \\frac{M_{E}^{2}}{2 B} + \\frac{M_{E} \\mathbf{D}}{B})}{M_{E} \\mathbf{D}} and \\frac{B^{2} (\\int \\operatorname{A_{z}}{(M_{E},\\mathbf{D},B)} dM_{E})^{2}}{M_{E}^{2} \\mathbf{D}^{2}} = \\frac{B^{2} (a + \\frac{M_{E}^{2}}{2 B} + \\frac{M_{E} \\mathbf{D}}{B})^{2}}{M_{E}^{2} \\mathbf{D}^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_z')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Symbol('B', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Integral(Function('A_z')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('B', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(2)), Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-2)), Pow(Integral(Function('A_z')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(2)), Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-2)), Pow(Add(Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given a{(a^{\\dagger},x)} = - a^{\\dagger} + x and q{(x,a^{\\dagger},\\varepsilon)} = \\frac{a{(a^{\\dagger},x)}}{\\varepsilon}, then obtain -1 + \\frac{2 a{(a^{\\dagger},x)}}{\\varepsilon} = -1 + \\frac{- 2 a^{\\dagger} + 2 x}{\\varepsilon}", "derivation": "a{(a^{\\dagger},x)} = - a^{\\dagger} + x and \\frac{a{(a^{\\dagger},x)}}{\\varepsilon} = \\frac{- a^{\\dagger} + x}{\\varepsilon} and q{(x,a^{\\dagger},\\varepsilon)} = \\frac{a{(a^{\\dagger},x)}}{\\varepsilon} and q{(x,a^{\\dagger},\\varepsilon)} = \\frac{- a^{\\dagger} + x}{\\varepsilon} and \\frac{- a^{\\dagger} + x}{\\varepsilon} + \\frac{a{(a^{\\dagger},x)}}{\\varepsilon} = \\frac{2 (- a^{\\dagger} + x)}{\\varepsilon} and q{(x,a^{\\dagger},\\varepsilon)} + \\frac{a{(a^{\\dagger},x)}}{\\varepsilon} = \\frac{- 2 a^{\\dagger} + 2 x}{\\varepsilon} and \\frac{2 a{(a^{\\dagger},x)}}{\\varepsilon} = \\frac{- 2 a^{\\dagger} + 2 x}{\\varepsilon} and -1 + \\frac{2 a{(a^{\\dagger},x)}}{\\varepsilon} = -1 + \\frac{- 2 a^{\\dagger} + 2 x}{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('q')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('q')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True)))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)))))"], [["add", 7, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('a')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\lambda,\\mathbf{g})} = \\frac{\\mathbf{g}}{\\lambda}, then obtain - \\int \\lambda \\mathbf{J}_P{(\\lambda,\\mathbf{g})} d\\lambda = - \\int \\mathbf{g} d\\lambda", "derivation": "\\mathbf{J}_P{(\\lambda,\\mathbf{g})} = \\frac{\\mathbf{g}}{\\lambda} and \\lambda \\mathbf{J}_P{(\\lambda,\\mathbf{g})} = \\mathbf{g} and \\int \\lambda \\mathbf{J}_P{(\\lambda,\\mathbf{g})} d\\lambda = \\int \\mathbf{g} d\\lambda and - \\int \\lambda \\mathbf{J}_P{(\\lambda,\\mathbf{g})} d\\lambda = - \\int \\mathbf{g} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(E)} = e^{\\cos{(E)}}, then obtain \\frac{d}{d E} \\phi_{1}{(E)} + 1 = - e^{\\cos{(E)}} \\sin{(E)} + 1", "derivation": "\\phi_{1}{(E)} = e^{\\cos{(E)}} and E + \\phi_{1}{(E)} = E + e^{\\cos{(E)}} and \\frac{d}{d E} (E + \\phi_{1}{(E)}) = \\frac{d}{d E} (E + e^{\\cos{(E)}}) and \\frac{d}{d E} \\phi_{1}{(E)} + 1 = - e^{\\cos{(E)}} \\sin{(E)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('E', commutative=True)), exp(cos(Symbol('E', commutative=True))))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\phi_1')(Symbol('E', commutative=True))), Add(Symbol('E', commutative=True), exp(cos(Symbol('E', commutative=True)))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Symbol('E', commutative=True), Function('\\\\phi_1')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), exp(cos(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\phi_1')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), exp(cos(Symbol('E', commutative=True))), sin(Symbol('E', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mu{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\operatorname{A_{z}}{(V_{\\mathbf{B}})} = \\frac{1}{\\mu{(V_{\\mathbf{B}})}}, then obtain \\operatorname{A_{z}}{(V_{\\mathbf{B}})} - \\frac{1}{\\sin{(V_{\\mathbf{B}})}} = 0", "derivation": "\\mu{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\operatorname{A_{z}}{(V_{\\mathbf{B}})} = \\frac{1}{\\mu{(V_{\\mathbf{B}})}} and \\operatorname{A_{z}}{(V_{\\mathbf{B}})} = \\frac{1}{\\sin{(V_{\\mathbf{B}})}} and \\frac{1}{\\sin{(V_{\\mathbf{B}})}} = \\frac{1}{\\mu{(V_{\\mathbf{B}})}} and \\operatorname{A_{z}}{(V_{\\mathbf{B}})} - \\frac{1}{\\mu{(V_{\\mathbf{B}})}} = \\frac{1}{\\sin{(V_{\\mathbf{B}})}} - \\frac{1}{\\mu{(V_{\\mathbf{B}})}} and \\operatorname{A_{z}}{(V_{\\mathbf{B}})} - \\frac{1}{\\sin{(V_{\\mathbf{B}})}} = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))"], [["minus", 3, "Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))"], "Equality(Add(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))), Add(Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\lambda{(\\mathbf{J}_f,\\varphi)} = \\mathbf{J}_f \\varphi and \\operatorname{m_{s}}{(S,\\mathbf{H})} = (e^{\\mathbf{H}})^{S}, then derive \\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)} = \\mathbf{J}_f, then obtain \\operatorname{m_{s}}{(S,\\mathbf{H})} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)} = (e^{\\mathbf{H}})^{S} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)}", "derivation": "\\lambda{(\\mathbf{J}_f,\\varphi)} = \\mathbf{J}_f \\varphi and \\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)} = \\frac{\\partial}{\\partial \\varphi} \\mathbf{J}_f \\varphi and \\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)} = \\mathbf{J}_f and \\operatorname{m_{s}}{(S,\\mathbf{H})} = (e^{\\mathbf{H}})^{S} and \\operatorname{m_{s}}{(S,\\mathbf{H})} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\mathbf{J}_f = (e^{\\mathbf{H}})^{S} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\mathbf{J}_f and \\operatorname{m_{s}}{(S,\\mathbf{H})} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)} = (e^{\\mathbf{H}})^{S} - \\int \\lambda{(\\mathbf{J}_f,\\varphi)} d\\frac{\\partial}{\\partial \\varphi} \\lambda{(\\mathbf{J}_f,\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True))"], ["get_premise", "Equality(Function('m_s')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)))"], [["minus", 4, "Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Function('m_s')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))), Add(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('m_s')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))))), Add(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(g,\\hat{H}_l)} = \\cos{(\\frac{\\hat{H}_l}{g})}, then obtain g \\cos{(\\frac{\\hat{H}_l}{g} + \\operatorname{F_{H}}{(g,\\hat{H}_l)} - \\cos{(\\frac{\\hat{H}_l}{g})})} = g \\cos{(\\frac{\\hat{H}_l}{g})}", "derivation": "\\operatorname{F_{H}}{(g,\\hat{H}_l)} = \\cos{(\\frac{\\hat{H}_l}{g})} and g \\operatorname{F_{H}}{(g,\\hat{H}_l)} = g \\cos{(\\frac{\\hat{H}_l}{g})} and \\frac{\\hat{H}_l}{g} + \\operatorname{F_{H}}{(g,\\hat{H}_l)} = \\frac{\\hat{H}_l}{g} + \\cos{(\\frac{\\hat{H}_l}{g})} and \\frac{\\hat{H}_l}{g} + \\operatorname{F_{H}}{(g,\\hat{H}_l)} - \\cos{(\\frac{\\hat{H}_l}{g})} = \\frac{\\hat{H}_l}{g} and \\operatorname{F_{H}}{(g,\\hat{H}_l)} = \\cos{(\\frac{\\hat{H}_l}{g} + \\operatorname{F_{H}}{(g,\\hat{H}_l)} - \\cos{(\\frac{\\hat{H}_l}{g})})} and g \\cos{(\\frac{\\hat{H}_l}{g} + \\operatorname{F_{H}}{(g,\\hat{H}_l)} - \\cos{(\\frac{\\hat{H}_l}{g})})} = g \\cos{(\\frac{\\hat{H}_l}{g})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))"], [["add", 1, "Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))"], [["minus", 3, "cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), cos(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Symbol('g', commutative=True), cos(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))))), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(A_{1})} = \\log{(A_{1})} and \\operatorname{n_{1}}{(A_{1})} = \\frac{\\mathbf{J}_P{(A_{1})}}{A_{1}}, then obtain \\operatorname{n_{1}}{(A_{1})} = \\frac{\\log{(A_{1})}}{A_{1}}", "derivation": "\\mathbf{J}_P{(A_{1})} = \\log{(A_{1})} and \\frac{\\mathbf{J}_P{(A_{1})}}{A_{1}} = \\frac{\\log{(A_{1})}}{A_{1}} and \\operatorname{n_{1}}{(A_{1})} = \\frac{\\mathbf{J}_P{(A_{1})}}{A_{1}} and \\operatorname{n_{1}}{(A_{1})} = \\frac{\\log{(A_{1})}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('n_1')(Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(C,F_{x})} = C F_{x}, then derive \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})} = C, then obtain \\frac{\\frac{\\partial^{2}}{\\partial C\\partial F_{x}} \\mathbf{J}{(C,F_{x})}}{\\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})}} = \\frac{\\frac{d}{d C} C}{\\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})}}", "derivation": "\\mathbf{J}{(C,F_{x})} = C F_{x} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})} = \\frac{\\partial}{\\partial F_{x}} C F_{x} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})} = C and \\frac{\\partial^{2}}{\\partial C\\partial F_{x}} \\mathbf{J}{(C,F_{x})} = \\frac{d}{d C} C and \\frac{\\frac{\\partial^{2}}{\\partial C\\partial F_{x}} \\mathbf{J}{(C,F_{x})}}{\\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})}} = \\frac{\\frac{d}{d C} C}{\\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(C,F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('C', commutative=True))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Symbol('C', commutative=True), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Derivative(Symbol('C', commutative=True), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given s{(m,a)} = \\log{(a - m)}, then obtain a - m + 2 \\log{(a - m)} + \\frac{s{(m,a)}}{a - m} = a - m + 2 \\log{(a - m)} + \\frac{\\log{(a - m)}}{a - m}", "derivation": "s{(m,a)} = \\log{(a - m)} and \\frac{s{(m,a)}}{a - m} = \\frac{\\log{(a - m)}}{a - m} and a - m + s{(m,a)} = a - m + \\log{(a - m)} and s{(m,a)} + \\frac{s{(m,a)}}{a - m} = s{(m,a)} + \\frac{\\log{(a - m)}}{a - m} and a - m + s{(m,a)} + \\log{(a - m)} + \\frac{s{(m,a)}}{a - m} = a - m + s{(m,a)} + \\log{(a - m)} + \\frac{\\log{(a - m)}}{a - m} and a - m + 2 \\log{(a - m)} + \\frac{s{(m,a)}}{a - m} = a - m + 2 \\log{(a - m)} + \\frac{\\log{(a - m)}}{a - m}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))"], [["divide", 1, "Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["add", 1, "Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["add", 2, "Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)))), Add(Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))))"], [["add", 4, "Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), Function('s')(Symbol('m', commutative=True), Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Integer(-1)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))))"]]}, {"prompt": "Given U{(A_{z},\\pi)} = - \\pi + \\cos{(A_{z})}, then obtain \\iint U{(A_{z},\\pi)} dA_{z} d\\pi = - \\frac{A_{z} \\pi^{2}}{2} + \\pi \\sin{(A_{z})}", "derivation": "U{(A_{z},\\pi)} = - \\pi + \\cos{(A_{z})} and \\int U{(A_{z},\\pi)} dA_{z} = \\int (- \\pi + \\cos{(A_{z})}) dA_{z} and \\iint U{(A_{z},\\pi)} dA_{z} dA_{z} = \\iint (- \\pi + \\cos{(A_{z})}) dA_{z} dA_{z} and \\iiint U{(A_{z},\\pi)} dA_{z} dA_{z} d\\pi = \\iiint (- \\pi + \\cos{(A_{z})}) dA_{z} dA_{z} d\\pi and \\frac{\\partial}{\\partial A_{z}} \\iiint U{(A_{z},\\pi)} dA_{z} dA_{z} d\\pi = \\frac{\\partial}{\\partial A_{z}} \\iiint (- \\pi + \\cos{(A_{z})}) dA_{z} dA_{z} d\\pi and \\iint U{(A_{z},\\pi)} dA_{z} d\\pi = - \\frac{A_{z} \\pi^{2}}{2} + \\pi \\sin{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), cos(Symbol('A_z', commutative=True))))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), cos(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), cos(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), cos(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 4, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Integral(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), cos(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integral(Function('U')(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Symbol('A_z', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), sin(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\phi_{1}{(\\mathbf{A})} = -1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}}, then obtain \\phi_{1}{(\\mathbf{A})} - 1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}} = -1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\phi_{1}{(\\mathbf{A})} = -1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}} and \\phi_{1}{(\\mathbf{A})} - \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}} = -1 and \\phi_{1}{(\\mathbf{A})} - 1 = -1 and \\phi_{1}{(\\mathbf{A})} - 1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}} = -1 + \\frac{\\cos{(\\mathbf{A})}}{\\operatorname{x^{{\\}'}}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Integer(-1), Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["minus", 2, "Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Integer(-1))"], [["add", 4, "Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1), Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Integer(-1), Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given x{(\\mathbf{D})} = e^{\\mathbf{D}} and \\dot{x}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} (x{(\\mathbf{D})} - e^{\\mathbf{D}}), then obtain \\dot{x}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} 0", "derivation": "x{(\\mathbf{D})} = e^{\\mathbf{D}} and x{(\\mathbf{D})} - e^{\\mathbf{D}} = 0 and \\frac{d}{d \\mathbf{D}} (x{(\\mathbf{D})} - e^{\\mathbf{D}}) = \\frac{d}{d \\mathbf{D}} 0 and \\dot{x}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} (x{(\\mathbf{D})} - e^{\\mathbf{D}}) and \\dot{x}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} 0", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Function('x')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Function('x')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive \\frac{d^{2}}{d \\mathbf{p}^{2}} \\bar{\\h}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain e^{\\mathbf{p}} = \\frac{d^{2}}{d \\mathbf{p}^{2}} e^{\\mathbf{p}}", "derivation": "\\bar{\\h}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} \\bar{\\h}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} and \\frac{d^{2}}{d \\mathbf{p}^{2}} \\bar{\\h}{(\\mathbf{p})} = \\frac{d^{2}}{d \\mathbf{p}^{2}} e^{\\mathbf{p}} and \\frac{d^{2}}{d \\mathbf{p}^{2}} \\bar{\\h}{(\\mathbf{p})} = e^{\\mathbf{p}} and e^{\\mathbf{p}} = \\frac{d^{2}}{d \\mathbf{p}^{2}} e^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(2))), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(2))))"]]}, {"prompt": "Given a{(\\Psi^{\\dagger})} = \\log{(e^{\\Psi^{\\dagger}})}, then derive \\log{(\\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})})} = 0, then obtain \\log{(\\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})})}^{\\Psi^{\\dagger}} = 0^{\\Psi^{\\dagger}}", "derivation": "a{(\\Psi^{\\dagger})} = \\log{(e^{\\Psi^{\\dagger}})} and \\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} \\log{(e^{\\Psi^{\\dagger}})} and \\log{(\\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})})} = \\log{(\\frac{d}{d \\Psi^{\\dagger}} \\log{(e^{\\Psi^{\\dagger}})})} and \\log{(\\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})})} = 0 and \\log{(\\frac{d}{d \\Psi^{\\dagger}} \\log{(e^{\\Psi^{\\dagger}})})} = 0 and \\log{(\\frac{d}{d \\Psi^{\\dagger}} \\log{(e^{\\Psi^{\\dagger}})})}^{\\Psi^{\\dagger}} = 0^{\\Psi^{\\dagger}} and \\log{(\\frac{d}{d \\Psi^{\\dagger}} a{(\\Psi^{\\dagger})})}^{\\Psi^{\\dagger}} = 0^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('a')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), log(Derivative(log(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('a')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(log(Derivative(log(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Integer(0))"], [["power", 5, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(log(Derivative(log(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Integer(0), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(log(Derivative(Function('a')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Integer(0), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\phi_{1}{(\\mathbf{M})} = - 2 \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} + \\int \\cos{(\\mathbf{M})} d\\mathbf{M}, then obtain \\int \\phi_{1}{(\\mathbf{M})} d\\mathbf{M} = \\int - \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} d\\mathbf{M}", "derivation": "\\mathbf{E}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} = \\int \\cos{(\\mathbf{M})} d\\mathbf{M} and 0 = - \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} + \\int \\cos{(\\mathbf{M})} d\\mathbf{M} and - \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} = - 2 \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} + \\int \\cos{(\\mathbf{M})} d\\mathbf{M} and \\phi_{1}{(\\mathbf{M})} = - 2 \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} + \\int \\cos{(\\mathbf{M})} d\\mathbf{M} and \\phi_{1}{(\\mathbf{M})} = - \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} and \\int \\phi_{1}{(\\mathbf{M})} d\\mathbf{M} = \\int - \\int \\mathbf{E}{(\\mathbf{M})} d\\mathbf{M} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Mul(Integer(-1), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Integer(2), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Mul(Integer(-1), Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(F_{g},n)} = \\cos{(F_{g} - n)}, then derive \\int \\operatorname{n_{1}}{(F_{g},n)} dn = B - \\sin{(F_{g} - n)}, then derive \\phi_1 - \\sin{(F_{g} - n)} = B - \\sin{(F_{g} - n)}, then obtain F_{g} + \\phi_1 - \\sin{(F_{g} - n)} = B + F_{g} - \\sin{(F_{g} - n)}", "derivation": "\\operatorname{n_{1}}{(F_{g},n)} = \\cos{(F_{g} - n)} and \\int \\operatorname{n_{1}}{(F_{g},n)} dn = \\int \\cos{(F_{g} - n)} dn and \\int \\operatorname{n_{1}}{(F_{g},n)} dn = B - \\sin{(F_{g} - n)} and \\int \\cos{(F_{g} - n)} dn = B - \\sin{(F_{g} - n)} and \\phi_1 - \\sin{(F_{g} - n)} = B - \\sin{(F_{g} - n)} and F_{g} + \\phi_1 - \\sin{(F_{g} - n)} = B + F_{g} - \\sin{(F_{g} - n)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_1')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('B', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Add(Symbol('B', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), Add(Symbol('B', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"], [["add", 5, "Symbol('F_g', commutative=True)"], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), Add(Symbol('B', commutative=True), Symbol('F_g', commutative=True), Mul(Integer(-1), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"]]}, {"prompt": "Given c{(r_{0},\\eta^{\\prime})} = - \\eta^{\\prime} + r_{0}, then obtain - \\tilde{g}{(\\hat{x}_0)} + \\frac{c^{\\eta^{\\prime}}{(r_{0},\\eta^{\\prime})}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} = \\frac{(- \\eta^{\\prime} + r_{0})^{\\eta^{\\prime}}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} - \\tilde{g}{(\\hat{x}_0)}", "derivation": "c{(r_{0},\\eta^{\\prime})} = - \\eta^{\\prime} + r_{0} and c^{\\eta^{\\prime}}{(r_{0},\\eta^{\\prime})} = (- \\eta^{\\prime} + r_{0})^{\\eta^{\\prime}} and \\frac{c^{\\eta^{\\prime}}{(r_{0},\\eta^{\\prime})}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} = \\frac{(- \\eta^{\\prime} + r_{0})^{\\eta^{\\prime}}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} and - \\tilde{g}{(\\hat{x}_0)} + \\frac{c^{\\eta^{\\prime}}{(r_{0},\\eta^{\\prime})}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} = \\frac{(- \\eta^{\\prime} + r_{0})^{\\eta^{\\prime}}}{\\eta^{\\prime} + c{(r_{0},\\eta^{\\prime})}} - \\tilde{g}{(\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 2, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Pow(Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))))"], [["minus", 3, "Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Pow(Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('c')(Symbol('r_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(E)} = \\cos{(E)}, then obtain \\frac{d}{d E} (\\frac{d^{2}}{d E^{2}} \\mu_{0}{(E)} + \\frac{d^{2}}{d E^{2}} \\cos{(E)}) + \\frac{d}{d E} 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)} = 2 \\frac{d}{d E} 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)}", "derivation": "\\mu_{0}{(E)} = \\cos{(E)} and \\frac{d}{d E} \\mu_{0}{(E)} = \\frac{d}{d E} \\cos{(E)} and \\frac{d^{2}}{d E^{2}} \\mu_{0}{(E)} = \\frac{d^{2}}{d E^{2}} \\cos{(E)} and \\frac{d^{2}}{d E^{2}} \\mu_{0}{(E)} + \\frac{d^{2}}{d E^{2}} \\cos{(E)} = 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)} and \\frac{d}{d E} (\\frac{d^{2}}{d E^{2}} \\mu_{0}{(E)} + \\frac{d^{2}}{d E^{2}} \\cos{(E)}) = \\frac{d}{d E} 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)} and \\frac{d}{d E} (\\frac{d^{2}}{d E^{2}} \\mu_{0}{(E)} + \\frac{d^{2}}{d E^{2}} \\cos{(E)}) + \\frac{d}{d E} 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)} = 2 \\frac{d}{d E} 2 \\frac{d^{2}}{d E^{2}} \\cos{(E)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))))"], [["add", 3, "Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Function('\\\\mu_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\mu_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["add", 5, "Derivative(Mul(Integer(2), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Derivative(Function('\\\\mu_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Integer(2), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}{(V_{\\mathbf{B}},f^{*})} = - V_{\\mathbf{B}} + f^{*} and \\operatorname{A_{z}}{(V_{\\mathbf{B}},f^{*})} = - 2 V_{\\mathbf{B}} + f^{*} - \\hat{x}{(V_{\\mathbf{B}},f^{*})} + \\frac{1}{V_{\\mathbf{B}}}, then obtain \\operatorname{A_{z}}{(V_{\\mathbf{B}},f^{*})} = - V_{\\mathbf{B}} + \\frac{1}{V_{\\mathbf{B}}}", "derivation": "\\hat{x}{(V_{\\mathbf{B}},f^{*})} = - V_{\\mathbf{B}} + f^{*} and 0 = - V_{\\mathbf{B}} + f^{*} - \\hat{x}{(V_{\\mathbf{B}},f^{*})} and \\frac{1}{V_{\\mathbf{B}}} = - V_{\\mathbf{B}} + f^{*} - \\hat{x}{(V_{\\mathbf{B}},f^{*})} + \\frac{1}{V_{\\mathbf{B}}} and - V_{\\mathbf{B}} + \\frac{1}{V_{\\mathbf{B}}} = - 2 V_{\\mathbf{B}} + f^{*} - \\hat{x}{(V_{\\mathbf{B}},f^{*})} + \\frac{1}{V_{\\mathbf{B}}} and \\operatorname{A_{z}}{(V_{\\mathbf{B}},f^{*})} = - 2 V_{\\mathbf{B}} + f^{*} - \\hat{x}{(V_{\\mathbf{B}},f^{*})} + \\frac{1}{V_{\\mathbf{B}}} and \\operatorname{A_{z}}{(V_{\\mathbf{B}},f^{*})} = - V_{\\mathbf{B}} + \\frac{1}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('f^*', commutative=True)))"], [["minus", 1, "Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True)))))"], [["add", 2, "Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))"], "Equality(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"], [["add", 3, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('A_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^*', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(f)} = \\cos{(f)} and \\operatorname{m_{s}}{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain \\frac{\\partial}{\\partial f} (\\mathbf{J}_M{(f)} + \\int \\operatorname{m_{s}}{(\\mu_0)} d\\mu_0) = \\frac{\\partial}{\\partial f} (\\mathbf{J}_M{(f)} + \\int \\cos{(\\mu_0)} d\\mu_0)", "derivation": "\\mathbf{J}_M{(f)} = \\cos{(f)} and \\operatorname{m_{s}}{(\\mu_0)} = \\cos{(\\mu_0)} and \\int \\operatorname{m_{s}}{(\\mu_0)} d\\mu_0 = \\int \\cos{(\\mu_0)} d\\mu_0 and \\cos{(f)} + \\int \\operatorname{m_{s}}{(\\mu_0)} d\\mu_0 = \\cos{(f)} + \\int \\cos{(\\mu_0)} d\\mu_0 and \\mathbf{J}_M{(f)} + \\int \\operatorname{m_{s}}{(\\mu_0)} d\\mu_0 = \\mathbf{J}_M{(f)} + \\int \\cos{(\\mu_0)} d\\mu_0 and \\frac{\\partial}{\\partial f} (\\mathbf{J}_M{(f)} + \\int \\operatorname{m_{s}}{(\\mu_0)} d\\mu_0) = \\frac{\\partial}{\\partial f} (\\mathbf{J}_M{(f)} + \\int \\cos{(\\mu_0)} d\\mu_0)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], ["get_premise", "Equality(Function('m_s')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 3, "cos(Symbol('f', commutative=True))"], "Equality(Add(cos(Symbol('f', commutative=True)), Integral(Function('m_s')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(cos(Symbol('f', commutative=True)), Integral(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), Integral(Function('m_s')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), Integral(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 5, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), Integral(Function('m_s')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), Integral(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(F_{g},t_{2})} = \\cos^{F_{g}}{(t_{2})}, then obtain \\frac{\\partial}{\\partial F_{g}} \\iint L{(F_{g},t_{2})} dF_{g} dF_{g} = \\frac{\\partial}{\\partial F_{g}} \\iint \\cos^{F_{g}}{(t_{2})} dF_{g} dF_{g}", "derivation": "L{(F_{g},t_{2})} = \\cos^{F_{g}}{(t_{2})} and \\int L{(F_{g},t_{2})} dF_{g} = \\int \\cos^{F_{g}}{(t_{2})} dF_{g} and \\iint L{(F_{g},t_{2})} dF_{g} dF_{g} = \\iint \\cos^{F_{g}}{(t_{2})} dF_{g} dF_{g} and \\frac{\\partial}{\\partial F_{g}} \\iint L{(F_{g},t_{2})} dF_{g} dF_{g} = \\frac{\\partial}{\\partial F_{g}} \\iint \\cos^{F_{g}}{(t_{2})} dF_{g} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Pow(cos(Symbol('t_2', commutative=True)), Symbol('F_g', commutative=True)))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('L')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(cos(Symbol('t_2', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('L')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(cos(Symbol('t_2', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Integral(Function('L')(Symbol('F_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(Symbol('t_2', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(b)} = \\log{(b)}, then derive \\frac{d}{d b} M{(b)} = \\frac{1}{b}, then obtain \\frac{d^{2}}{d b^{2}} \\log{(b)} = \\frac{d}{d b} \\frac{1}{b}", "derivation": "M{(b)} = \\log{(b)} and \\frac{d}{d b} M{(b)} = \\frac{d}{d b} \\log{(b)} and \\frac{d}{d b} M{(b)} = \\frac{1}{b} and \\frac{d}{d b} \\log{(b)} = \\frac{1}{b} and \\frac{d^{2}}{d b^{2}} \\log{(b)} = \\frac{d}{d b} \\frac{1}{b}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Pow(Symbol('b', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Pow(Symbol('b', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Pow(Symbol('b', commutative=True), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain (\\frac{d}{d \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sigma_p)}))^{\\sigma_p} = (\\frac{d}{d \\sigma_p} 0)^{\\sigma_p}", "derivation": "\\varepsilon{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\varepsilon{(\\sigma_p)} - \\cos{(\\sigma_p)} = 0 and \\frac{d}{d \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sigma_p)}) = \\frac{d}{d \\sigma_p} 0 and (\\frac{d}{d \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sigma_p)}))^{\\sigma_p} = (\\frac{d}{d \\sigma_p} 0)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(t_{1})} = \\log{(t_{1})}, then derive \\int \\dot{\\mathbf{r}}{(t_{1})} dt_{1} = \\hat{\\mathbf{x}} + t_{1} \\log{(t_{1})} - t_{1}, then obtain \\int \\log{(t_{1})} dt_{1} = \\hat{\\mathbf{x}} + t_{1} \\log{(t_{1})} - t_{1}", "derivation": "\\dot{\\mathbf{r}}{(t_{1})} = \\log{(t_{1})} and \\int \\dot{\\mathbf{r}}{(t_{1})} dt_{1} = \\int \\log{(t_{1})} dt_{1} and \\int \\dot{\\mathbf{r}}{(t_{1})} dt_{1} = \\hat{\\mathbf{x}} + t_{1} \\log{(t_{1})} - t_{1} and \\int \\dot{\\mathbf{r}}{(t_{1})} dt_{1} = \\hat{\\mathbf{x}} + t_{1} \\dot{\\mathbf{r}}{(t_{1})} - t_{1} and \\int \\log{(t_{1})} dt_{1} = \\hat{\\mathbf{x}} + t_{1} \\log{(t_{1})} - t_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Symbol('t_1', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given z{(v_{2},g)} = (e^{v_{2}})^{g}, then obtain (e^{v_{2}} e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}})^{- g} (e^{v_{2}})^{g} = 1", "derivation": "z{(v_{2},g)} = (e^{v_{2}})^{g} and 0 = - z{(v_{2},g)} + (e^{v_{2}})^{g} and 1 = e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}} and e^{v_{2}} = e^{v_{2}} e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}} and (e^{v_{2}})^{g} = (e^{v_{2}} e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}})^{g} and (- z{(v_{2},g)} + (e^{v_{2}})^{g}) (e^{v_{2}})^{g} = (e^{v_{2}} e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}})^{g} (- z{(v_{2},g)} + (e^{v_{2}})^{g}) and (e^{v_{2}} e^{- z{(v_{2},g)} + (e^{v_{2}})^{g}})^{- g} (e^{v_{2}})^{g} = 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True)), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True)))"], [["minus", 1, "Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))"], [["exp", 2], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True)))))"], [["times", 3, "exp(Symbol('v_2', commutative=True))"], "Equality(exp(Symbol('v_2', commutative=True)), Mul(exp(Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(exp(Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))), Symbol('g', commutative=True)))"], [["times", 5, "Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Mul(exp(Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True)))))"], [["divide", 6, "Mul(Pow(Mul(exp(Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))"], "Equality(Mul(Pow(Mul(exp(Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Function('z')(Symbol('v_2', commutative=True), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))))), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(exp(Symbol('v_2', commutative=True)), Symbol('g', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(z)} = e^{\\cos{(z)}} and B{(x,t_{2})} = \\frac{\\log{(x)}}{t_{2}}, then obtain \\cos{(\\frac{B^{x}{(x,t_{2})}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}})} = \\cos{(\\frac{(\\frac{\\log{(x)}}{t_{2}})^{x}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}})}", "derivation": "\\operatorname{F_{c}}{(z)} = e^{\\cos{(z)}} and B{(x,t_{2})} = \\frac{\\log{(x)}}{t_{2}} and B^{x}{(x,t_{2})} = (\\frac{\\log{(x)}}{t_{2}})^{x} and \\frac{B^{x}{(x,t_{2})}}{\\cos{((e^{\\cos{(z)}})^{z})}} = \\frac{(\\frac{\\log{(x)}}{t_{2}})^{x}}{\\cos{((e^{\\cos{(z)}})^{z})}} and \\frac{B^{x}{(x,t_{2})}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}} = \\frac{(\\frac{\\log{(x)}}{t_{2}})^{x}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}} and \\cos{(\\frac{B^{x}{(x,t_{2})}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}})} = \\cos{(\\frac{(\\frac{\\log{(x)}}{t_{2}})^{x}}{\\cos{(\\operatorname{F_{c}}^{z}{(z)})}})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('z', commutative=True)), exp(cos(Symbol('z', commutative=True))))"], ["get_premise", "Equality(Function('B')(Symbol('x', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Function('B')(Symbol('x', commutative=True), Symbol('t_2', commutative=True)), Symbol('x', commutative=True)), Pow(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["divide", 3, "cos(Pow(exp(cos(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], "Equality(Mul(Pow(Function('B')(Symbol('x', commutative=True), Symbol('t_2', commutative=True)), Symbol('x', commutative=True)), Pow(cos(Pow(exp(cos(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(cos(Pow(exp(cos(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('B')(Symbol('x', commutative=True), Symbol('t_2', commutative=True)), Symbol('x', commutative=True)), Pow(cos(Pow(Function('F_c')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(cos(Pow(Function('F_c')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Integer(-1))))"], [["cos", 5], "Equality(cos(Mul(Pow(Function('B')(Symbol('x', commutative=True), Symbol('t_2', commutative=True)), Symbol('x', commutative=True)), Pow(cos(Pow(Function('F_c')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Integer(-1)))), cos(Mul(Pow(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), log(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(cos(Pow(Function('F_c')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(i,b)} = i^{b}, then obtain (- i^{b} + \\operatorname{f^{\\prime}}{(i,b)})^{4} = 0", "derivation": "\\operatorname{f^{\\prime}}{(i,b)} = i^{b} and - i^{b} + \\operatorname{f^{\\prime}}{(i,b)} = 0 and (- i^{b} + \\operatorname{f^{\\prime}}{(i,b)})^{2} = 0 and (- i^{b} + \\operatorname{f^{\\prime}}{(i,b)})^{4} = 0", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('i', commutative=True), Symbol('b', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('b', commutative=True)))"], [["minus", 1, "Pow(Symbol('i', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('b', commutative=True))), Function('f^{\\\\prime}')(Symbol('i', commutative=True), Symbol('b', commutative=True))), Integer(0))"], [["times", 2, "Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('b', commutative=True))), Function('f^{\\\\prime}')(Symbol('i', commutative=True), Symbol('b', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('b', commutative=True))), Function('f^{\\\\prime}')(Symbol('i', commutative=True), Symbol('b', commutative=True))), Integer(2)), Integer(0))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('b', commutative=True))), Function('f^{\\\\prime}')(Symbol('i', commutative=True), Symbol('b', commutative=True))), Integer(4)), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\sigma_p)} = \\sin{(\\cos{(\\sigma_p)})} and \\phi_{2}{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain \\int (\\operatorname{v_{z}}^{\\sigma_p}{(\\sigma_p)} + \\sin{(\\phi_{2}{(\\sigma_p)})}) d\\sigma_p = \\int (\\sin{(\\phi_{2}{(\\sigma_p)})} + \\sin^{\\sigma_p}{(\\phi_{2}{(\\sigma_p)})}) d\\sigma_p", "derivation": "\\operatorname{v_{z}}{(\\sigma_p)} = \\sin{(\\cos{(\\sigma_p)})} and \\operatorname{v_{z}}^{\\sigma_p}{(\\sigma_p)} = \\sin^{\\sigma_p}{(\\cos{(\\sigma_p)})} and \\phi_{2}{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\operatorname{v_{z}}^{\\sigma_p}{(\\sigma_p)} + \\sin{(\\cos{(\\sigma_p)})} = \\sin{(\\cos{(\\sigma_p)})} + \\sin^{\\sigma_p}{(\\cos{(\\sigma_p)})} and \\operatorname{v_{z}}^{\\sigma_p}{(\\sigma_p)} + \\sin{(\\phi_{2}{(\\sigma_p)})} = \\sin{(\\phi_{2}{(\\sigma_p)})} + \\sin^{\\sigma_p}{(\\phi_{2}{(\\sigma_p)})} and \\int (\\operatorname{v_{z}}^{\\sigma_p}{(\\sigma_p)} + \\sin{(\\phi_{2}{(\\sigma_p)})}) d\\sigma_p = \\int (\\sin{(\\phi_{2}{(\\sigma_p)})} + \\sin^{\\sigma_p}{(\\phi_{2}{(\\sigma_p)})}) d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\sigma_p', commutative=True)), sin(cos(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(sin(cos(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 2, "sin(cos(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Pow(Function('v_z')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), sin(cos(Symbol('\\\\sigma_p', commutative=True)))), Add(sin(cos(Symbol('\\\\sigma_p', commutative=True))), Pow(sin(cos(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('v_z')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True)))), Add(sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True))), Pow(sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Add(Pow(Function('v_z')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True))), Pow(sin(Function('\\\\phi_2')(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given U{(\\nabla)} = \\log{(\\nabla)}, then derive - \\log{(\\nabla)} + \\int U{(\\nabla)} d\\nabla = \\nabla \\log{(\\nabla)} - \\nabla + x^\\prime - \\log{(\\nabla)}, then obtain \\frac{d}{d \\nabla} (- \\log{(\\nabla)} + \\int U{(\\nabla)} d\\nabla) = \\frac{\\partial}{\\partial \\nabla} (\\nabla \\log{(\\nabla)} - \\nabla + x^\\prime - \\log{(\\nabla)})", "derivation": "U{(\\nabla)} = \\log{(\\nabla)} and \\int U{(\\nabla)} d\\nabla = \\int \\log{(\\nabla)} d\\nabla and - \\log{(\\nabla)} + \\int U{(\\nabla)} d\\nabla = - \\log{(\\nabla)} + \\int \\log{(\\nabla)} d\\nabla and - \\log{(\\nabla)} + \\int U{(\\nabla)} d\\nabla = \\nabla \\log{(\\nabla)} - \\nabla + x^\\prime - \\log{(\\nabla)} and \\frac{d}{d \\nabla} (- \\log{(\\nabla)} + \\int U{(\\nabla)} d\\nabla) = \\frac{\\partial}{\\partial \\nabla} (\\nabla \\log{(\\nabla)} - \\nabla + x^\\prime - \\log{(\\nabla)})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "log(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True))), Integral(Function('U')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True))), Integral(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True))), Integral(Function('U')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('\\\\nabla', commutative=True), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True))), Integral(Function('U')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\nabla', commutative=True), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), log(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\phi_2,W)} = W \\phi_2 and \\dot{z}{(\\phi_2,W)} = 2 \\operatorname{F_{N}}{(\\phi_2,W)}, then obtain 2 W \\phi_2 + W + \\frac{\\partial}{\\partial \\phi_2} (W + \\dot{z}{(\\phi_2,W)}) = 2 W \\phi_2 + W + \\frac{\\partial}{\\partial \\phi_2} (W + 2 \\operatorname{F_{N}}{(\\phi_2,W)})", "derivation": "\\operatorname{F_{N}}{(\\phi_2,W)} = W \\phi_2 and W + \\operatorname{F_{N}}{(\\phi_2,W)} = W \\phi_2 + W and W \\phi_2 + W + \\operatorname{F_{N}}{(\\phi_2,W)} = 2 W \\phi_2 + W and W + 2 \\operatorname{F_{N}}{(\\phi_2,W)} = 2 W \\phi_2 + W and \\frac{\\partial}{\\partial \\phi_2} (W + 2 \\operatorname{F_{N}}{(\\phi_2,W)}) = \\frac{\\partial}{\\partial \\phi_2} (2 W \\phi_2 + W) and \\dot{z}{(\\phi_2,W)} = 2 \\operatorname{F_{N}}{(\\phi_2,W)} and \\frac{\\partial}{\\partial \\phi_2} (W + \\dot{z}{(\\phi_2,W)}) = \\frac{\\partial}{\\partial \\phi_2} (2 W \\phi_2 + W) and \\frac{\\partial}{\\partial \\phi_2} (W + \\dot{z}{(\\phi_2,W)}) = \\frac{\\partial}{\\partial \\phi_2} (W + 2 \\operatorname{F_{N}}{(\\phi_2,W)}) and 2 W \\phi_2 + W + \\frac{\\partial}{\\partial \\phi_2} (W + \\dot{z}{(\\phi_2,W)}) = 2 W \\phi_2 + W + \\frac{\\partial}{\\partial \\phi_2} (W + 2 \\operatorname{F_{N}}{(\\phi_2,W)})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True)))"], [["add", 1, "Add(Mul(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True))"], "Equality(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Symbol('W', commutative=True), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Derivative(Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 8, "Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True), Derivative(Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('W', commutative=True), Derivative(Add(Symbol('W', commutative=True), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} = \\int (x^\\prime)^{\\varphi^*} dx^\\prime and \\operatorname{n_{2}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} 2 \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)}, then obtain \\operatorname{n_{2}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} 2 \\int (x^\\prime)^{\\varphi^*} dx^\\prime", "derivation": "\\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} = \\int (x^\\prime)^{\\varphi^*} dx^\\prime and 2 \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} = \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} + \\int (x^\\prime)^{\\varphi^*} dx^\\prime and \\frac{\\partial}{\\partial x^\\prime} 2 \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} (\\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} + \\int (x^\\prime)^{\\varphi^*} dx^\\prime) and \\operatorname{n_{2}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} 2 \\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} and \\operatorname{n_{2}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} (\\operatorname{A_{y}}{(x^\\prime,\\varphi^*)} + \\int (x^\\prime)^{\\varphi^*} dx^\\prime) and \\operatorname{n_{2}}{(x^\\prime,\\varphi^*)} = \\frac{\\partial}{\\partial x^\\prime} 2 \\int (x^\\prime)^{\\varphi^*} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Add(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Integer(2), Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Add(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Integer(2), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(F_{x},\\mathbf{M})} = - F_{x} + \\mathbf{M}, then derive \\int \\mathbf{g}{(F_{x},\\mathbf{M})} dF_{x} = - \\frac{F_{x}^{2}}{2} + F_{x} \\mathbf{M} + \\mathbf{D}, then obtain \\int (- F_{x} + \\mathbf{M}) dF_{x} = - \\frac{F_{x}^{2}}{2} + F_{x} \\mathbf{M} + \\mathbf{D}", "derivation": "\\mathbf{g}{(F_{x},\\mathbf{M})} = - F_{x} + \\mathbf{M} and \\int \\mathbf{g}{(F_{x},\\mathbf{M})} dF_{x} = \\int (- F_{x} + \\mathbf{M}) dF_{x} and \\int \\mathbf{g}{(F_{x},\\mathbf{M})} dF_{x} = - \\frac{F_{x}^{2}}{2} + F_{x} \\mathbf{M} + \\mathbf{D} and \\int (- F_{x} + \\mathbf{M}) dF_{x} = - \\frac{F_{x}^{2}}{2} + F_{x} \\mathbf{M} + \\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2))), Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2))), Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given x{(P_{e})} = \\sin{(e^{P_{e}})}, then obtain \\frac{d^{2}}{d P_{e}^{2}} \\int x{(P_{e})} dP_{e} = \\frac{\\partial^{2}}{\\partial P_{e}^{2}} (C_{2} + \\operatorname{Si}{(e^{P_{e}})})", "derivation": "x{(P_{e})} = \\sin{(e^{P_{e}})} and \\int x{(P_{e})} dP_{e} = \\int \\sin{(e^{P_{e}})} dP_{e} and \\frac{d}{d P_{e}} \\int x{(P_{e})} dP_{e} = \\frac{d}{d P_{e}} \\int \\sin{(e^{P_{e}})} dP_{e} and \\frac{d^{2}}{d P_{e}^{2}} \\int x{(P_{e})} dP_{e} = \\frac{d^{2}}{d P_{e}^{2}} \\int \\sin{(e^{P_{e}})} dP_{e} and \\frac{d^{2}}{d P_{e}^{2}} \\int x{(P_{e})} dP_{e} = \\frac{\\partial^{2}}{\\partial P_{e}^{2}} (C_{2} + \\operatorname{Si}{(e^{P_{e}})})", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('P_e', commutative=True)), sin(exp(Symbol('P_e', commutative=True))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(sin(exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integral(Function('x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Integral(sin(exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integral(Function('x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(2))), Derivative(Integral(sin(exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(2))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(2))), Derivative(Add(Symbol('C_2', commutative=True), Si(exp(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(g_{\\varepsilon})} = g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}, then obtain \\sin{(g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}) + g_{\\varepsilon} \\operatorname{v_{y}}{(g_{\\varepsilon})})} = \\sin{(2 g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}))}", "derivation": "\\operatorname{v_{y}}{(g_{\\varepsilon})} = g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} and g_{\\varepsilon} \\operatorname{v_{y}}{(g_{\\varepsilon})} = g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}) and g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}) + g_{\\varepsilon} \\operatorname{v_{y}}{(g_{\\varepsilon})} = 2 g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}) and \\sin{(g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}) + g_{\\varepsilon} \\operatorname{v_{y}}{(g_{\\varepsilon})})} = \\sin{(2 g_{\\varepsilon} (g_{\\varepsilon} + \\sin{(g_{\\varepsilon})}))}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["add", 2, "Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], "Equality(Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["sin", 3], "Equality(sin(Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True))))), sin(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given n{(x)} = \\int \\cos{(x)} dx, then derive n{(x)} = \\mathbf{D} + \\sin{(x)}, then obtain (\\mathbf{D} + \\sin{(x)})^{x} = (\\int \\cos{(x)} dx)^{x}", "derivation": "n{(x)} = \\int \\cos{(x)} dx and n^{x}{(x)} = (\\int \\cos{(x)} dx)^{x} and n{(x)} = \\mathbf{D} + \\sin{(x)} and (\\mathbf{D} + \\sin{(x)})^{x} = (\\int \\cos{(x)} dx)^{x}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('x', commutative=True)), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('n')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('n')(Symbol('x', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given L{(A)} = \\sin{(\\sin{(A)})} and \\operatorname{c_{0}}{(A)} = \\iint L{(A)} dA dA, then obtain \\sin{(\\sin{(A)})} + \\iint \\sin{(\\sin{(A)})} dA dA = \\sin{(\\sin{(A)})} + \\iint L{(A)} dA dA", "derivation": "L{(A)} = \\sin{(\\sin{(A)})} and \\int L{(A)} dA = \\int \\sin{(\\sin{(A)})} dA and \\iint L{(A)} dA dA = \\iint \\sin{(\\sin{(A)})} dA dA and \\operatorname{c_{0}}{(A)} = \\iint L{(A)} dA dA and \\operatorname{c_{0}}{(A)} = \\iint \\sin{(\\sin{(A)})} dA dA and L{(A)} + \\operatorname{c_{0}}{(A)} = L{(A)} + \\iint L{(A)} dA dA and \\operatorname{c_{0}}{(A)} + \\sin{(\\sin{(A)})} = \\sin{(\\sin{(A)})} + \\iint \\sin{(\\sin{(A)})} dA dA and \\operatorname{c_{0}}{(A)} + \\sin{(\\sin{(A)})} = \\sin{(\\sin{(A)})} + \\iint L{(A)} dA dA and \\sin{(\\sin{(A)})} + \\iint \\sin{(\\sin{(A)})} dA dA = \\sin{(\\sin{(A)})} + \\iint L{(A)} dA dA", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('A', commutative=True)), sin(sin(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('A', commutative=True)), Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('c_0')(Symbol('A', commutative=True)), Integral(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["add", 4, "Function('L')(Symbol('A', commutative=True))"], "Equality(Add(Function('L')(Symbol('A', commutative=True)), Function('c_0')(Symbol('A', commutative=True))), Add(Function('L')(Symbol('A', commutative=True)), Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Function('c_0')(Symbol('A', commutative=True)), sin(sin(Symbol('A', commutative=True)))), Add(sin(sin(Symbol('A', commutative=True))), Integral(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Function('c_0')(Symbol('A', commutative=True)), sin(sin(Symbol('A', commutative=True)))), Add(sin(sin(Symbol('A', commutative=True))), Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Add(sin(sin(Symbol('A', commutative=True))), Integral(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(sin(sin(Symbol('A', commutative=True))), Integral(Function('L')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(n)} = \\log{(n)}, then obtain (- 2 \\dot{\\mathbf{r}}{(n)} + 2 \\log{(n)}) (2 \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)}) + \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)} = \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)}", "derivation": "\\dot{\\mathbf{r}}{(n)} = \\log{(n)} and 0 = - \\dot{\\mathbf{r}}{(n)} + \\log{(n)} and \\log{(n)} = - \\dot{\\mathbf{r}}{(n)} + 2 \\log{(n)} and \\dot{\\mathbf{r}}{(n)} - \\log{(n)} = 0 and (- \\dot{\\mathbf{r}}{(n)} + \\log{(n)}) (\\dot{\\mathbf{r}}{(n)} - \\log{(n)}) = 0 and (- \\dot{\\mathbf{r}}{(n)} + \\log{(n)}) (\\dot{\\mathbf{r}}{(n)} - \\log{(n)}) - \\log{(n)} = - \\log{(n)} and (- 2 \\dot{\\mathbf{r}}{(n)} + 2 \\log{(n)}) (2 \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)}) + \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)} = \\dot{\\mathbf{r}}{(n)} - 2 \\log{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["minus", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["add", 2, "log(Symbol('n', commutative=True))"], "Equality(log(Symbol('n', commutative=True)), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Mul(Integer(2), log(Symbol('n', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Mul(Integer(-1), log(Symbol('n', commutative=True)))), Integer(0))"], [["times", 4, "Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Mul(Integer(-1), log(Symbol('n', commutative=True))))), Integer(0))"], [["minus", 5, "log(Symbol('n', commutative=True))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Mul(Integer(-1), log(Symbol('n', commutative=True))))), Mul(Integer(-1), log(Symbol('n', commutative=True)))), Mul(Integer(-1), log(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Mul(Integer(2), log(Symbol('n', commutative=True)))), Add(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('n', commutative=True))))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('n', commutative=True)))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given h{(P_{e},\\pi)} = \\frac{\\partial}{\\partial \\pi} P_{e} \\pi and M{(P_{e},\\pi)} = \\frac{\\partial}{\\partial \\pi} P_{e} \\pi, then derive h{(P_{e},\\pi)} = P_{e}, then obtain M{(\\frac{\\partial}{\\partial \\pi} P_{e} \\pi,\\pi)} = h{(\\frac{\\partial}{\\partial \\pi} P_{e} \\pi,\\pi)}", "derivation": "h{(P_{e},\\pi)} = \\frac{\\partial}{\\partial \\pi} P_{e} \\pi and h{(P_{e},\\pi)} = P_{e} and \\frac{\\partial}{\\partial \\pi} P_{e} \\pi = P_{e} and M{(P_{e},\\pi)} = \\frac{\\partial}{\\partial \\pi} P_{e} \\pi and M{(P_{e},\\pi)} = h{(P_{e},\\pi)} and M{(\\frac{\\partial}{\\partial \\pi} P_{e} \\pi,\\pi)} = h{(\\frac{\\partial}{\\partial \\pi} P_{e} \\pi,\\pi)}", "srepr_derivation": [["get_premise", "Equality(Function('h')(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Mul(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('h')(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('P_e', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('P_e', commutative=True))"], ["renaming_premise", "Equality(Function('M')(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Mul(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('M')(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Function('h')(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('M')(Derivative(Mul(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Function('h')(Derivative(Mul(Symbol('P_e', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(q)} = \\log{(q)}, then derive \\int \\ddot{x}{(q)} dq = q \\log{(q)} - q + v_{1}, then obtain q \\log{(q)} - q + v_{1} - \\int \\log{(q)} dq = 0", "derivation": "\\ddot{x}{(q)} = \\log{(q)} and \\int \\ddot{x}{(q)} dq = \\int \\log{(q)} dq and \\int \\ddot{x}{(q)} dq = q \\log{(q)} - q + v_{1} and q \\log{(q)} - q + v_{1} = \\int \\log{(q)} dq and q \\log{(q)} - q + v_{1} - \\int \\log{(q)} dq = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_1', commutative=True)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["minus", 4, "Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_1', commutative=True), Mul(Integer(-1), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(A,J_{\\varepsilon})} = A + J_{\\varepsilon}, then obtain \\int (\\int \\operatorname{V_{\\mathbf{B}}}^{A}{(A,J_{\\varepsilon})} dA)^{A} dJ_{\\varepsilon} = \\int (\\int (A + J_{\\varepsilon})^{A} dA)^{A} dJ_{\\varepsilon}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(A,J_{\\varepsilon})} = A + J_{\\varepsilon} and \\operatorname{V_{\\mathbf{B}}}^{A}{(A,J_{\\varepsilon})} = (A + J_{\\varepsilon})^{A} and \\int \\operatorname{V_{\\mathbf{B}}}^{A}{(A,J_{\\varepsilon})} dA = \\int (A + J_{\\varepsilon})^{A} dA and (\\int \\operatorname{V_{\\mathbf{B}}}^{A}{(A,J_{\\varepsilon})} dA)^{A} = (\\int (A + J_{\\varepsilon})^{A} dA)^{A} and \\int (\\int \\operatorname{V_{\\mathbf{B}}}^{A}{(A,J_{\\varepsilon})} dA)^{A} dJ_{\\varepsilon} = \\int (\\int (A + J_{\\varepsilon})^{A} dA)^{A} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Pow(Add(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(Add(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(Pow(Add(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["integrate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Integral(Pow(Add(Symbol('A', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(F_{N})} = \\log{(F_{N})}, then derive \\frac{d}{d F_{N}} \\Psi^{\\dagger}{(F_{N})} = \\frac{1}{F_{N}}, then obtain \\frac{d^{2}}{d F_{N}^{2}} \\Psi^{\\dagger}{(F_{N})} \\frac{d}{d F_{N}} \\log{(F_{N})} = \\frac{\\frac{d^{2}}{d F_{N}^{2}} \\Psi^{\\dagger}{(F_{N})}}{F_{N}}", "derivation": "\\Psi^{\\dagger}{(F_{N})} = \\log{(F_{N})} and \\frac{d}{d F_{N}} \\Psi^{\\dagger}{(F_{N})} = \\frac{d}{d F_{N}} \\log{(F_{N})} and \\frac{d}{d F_{N}} \\Psi^{\\dagger}{(F_{N})} = \\frac{1}{F_{N}} and \\frac{d}{d F_{N}} \\log{(F_{N})} = \\frac{1}{F_{N}} and \\frac{d^{2}}{d F_{N}^{2}} \\Psi^{\\dagger}{(F_{N})} \\frac{d}{d F_{N}} \\log{(F_{N})} = \\frac{\\frac{d^{2}}{d F_{N}^{2}} \\Psi^{\\dagger}{(F_{N})}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Symbol('F_N', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Symbol('F_N', commutative=True), Integer(-1)))"], [["times", 4, "Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(2))), Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(v,y,L)} = \\frac{L v}{y} and \\operatorname{v_{t}}{(v,y,L)} = \\frac{L v}{y}, then obtain - v + \\hat{\\mathbf{x}}{(v,y,L)} = - v + \\operatorname{v_{t}}{(v,y,L)}", "derivation": "\\hat{\\mathbf{x}}{(v,y,L)} = \\frac{L v}{y} and - v + \\hat{\\mathbf{x}}{(v,y,L)} = \\frac{L v}{y} - v and \\operatorname{v_{t}}{(v,y,L)} = \\frac{L v}{y} and - v + \\hat{\\mathbf{x}}{(v,y,L)} = - v + \\operatorname{v_{t}}{(v,y,L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('y', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('y', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('v', commutative=True), Symbol('y', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('y', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('v_t')(Symbol('v', commutative=True), Symbol('y', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given J{(\\hat{p},\\chi)} = \\sin{(\\frac{\\hat{p}}{\\chi})}, then obtain ((\\frac{J{(\\hat{p},\\chi)}}{\\sin{(\\frac{\\hat{p}}{\\chi})}})^{\\hat{p}} + J{(\\hat{p},\\chi)})^{\\chi} = (J{(\\hat{p},\\chi)} + 1)^{\\chi}", "derivation": "J{(\\hat{p},\\chi)} = \\sin{(\\frac{\\hat{p}}{\\chi})} and \\frac{J{(\\hat{p},\\chi)}}{\\sin{(\\frac{\\hat{p}}{\\chi})}} = 1 and (\\frac{J{(\\hat{p},\\chi)}}{\\sin{(\\frac{\\hat{p}}{\\chi})}})^{\\hat{p}} = 1 and (\\frac{J{(\\hat{p},\\chi)}}{\\sin{(\\frac{\\hat{p}}{\\chi})}})^{\\hat{p}} + J{(\\hat{p},\\chi)} = J{(\\hat{p},\\chi)} + 1 and ((\\frac{J{(\\hat{p},\\chi)}}{\\sin{(\\frac{\\hat{p}}{\\chi})}})^{\\hat{p}} + J{(\\hat{p},\\chi)})^{\\chi} = (J{(\\hat{p},\\chi)} + 1)^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Mul(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Symbol('\\\\hat{p}', commutative=True)), Integer(1))"], [["add", 3, "Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Pow(Mul(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Symbol('\\\\hat{p}', commutative=True)), Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Pow(Mul(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Symbol('\\\\hat{p}', commutative=True)), Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Add(Function('J')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(1)), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(P_{g},\\pi)} = \\log{(P_{g} \\pi)} and \\operatorname{A_{1}}{(P_{g},\\pi)} = \\log{(P_{g} \\pi)}, then obtain P_{g} \\Psi^{\\dagger}{(P_{g},\\pi)} = P_{g} \\operatorname{A_{1}}{(P_{g},\\pi)}", "derivation": "\\Psi^{\\dagger}{(P_{g},\\pi)} = \\log{(P_{g} \\pi)} and P_{g} \\Psi^{\\dagger}{(P_{g},\\pi)} = P_{g} \\log{(P_{g} \\pi)} and \\operatorname{A_{1}}{(P_{g},\\pi)} = \\log{(P_{g} \\pi)} and P_{g} \\Psi^{\\dagger}{(P_{g},\\pi)} = P_{g} \\operatorname{A_{1}}{(P_{g},\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True)), log(Mul(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('P_g', commutative=True), log(Mul(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True)), log(Mul(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('P_g', commutative=True), Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(r,\\mathbf{S})} = \\mathbf{S} + r and \\operatorname{r_{0}}{(r,\\mathbf{S})} = - \\mathbf{S} - r + \\bar{\\h}{(r,\\mathbf{S})}, then obtain \\operatorname{r_{0}}{(r,\\mathbf{S})} = 0", "derivation": "\\bar{\\h}{(r,\\mathbf{S})} = \\mathbf{S} + r and - \\mathbf{S} - r + \\bar{\\h}{(r,\\mathbf{S})} = 0 and \\operatorname{r_{0}}{(r,\\mathbf{S})} = - \\mathbf{S} - r + \\bar{\\h}{(r,\\mathbf{S})} and \\operatorname{r_{0}}{(r,\\mathbf{S})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('r', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\hbar')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\hbar')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('r_0')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(0))"]]}, {"prompt": "Given \\omega{(\\psi,C_{d})} = C_{d} + \\psi, then obtain 0 = \\frac{- C_{d} - \\psi}{\\psi} + \\frac{\\omega{(\\psi,C_{d})}}{\\psi}", "derivation": "\\omega{(\\psi,C_{d})} = C_{d} + \\psi and - \\frac{\\omega{(\\psi,C_{d})}}{\\psi} = - \\frac{C_{d} + \\psi}{\\psi} and \\frac{\\omega{(\\psi,C_{d})}}{\\psi} = \\frac{C_{d} + \\psi}{\\psi} and 0 = - \\frac{C_{d} + \\psi}{\\psi} + \\frac{\\omega{(\\psi,C_{d})}}{\\psi} and 0 = \\frac{- C_{d} - \\psi}{\\psi} + \\frac{\\omega{(\\psi,C_{d})}}{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given x{(\\mathbf{S})} = e^{\\cos{(\\mathbf{S})}}, then obtain x^{4}{(\\mathbf{S})} = x{(\\mathbf{S})} e^{3 \\cos{(\\mathbf{S})}}", "derivation": "x{(\\mathbf{S})} = e^{\\cos{(\\mathbf{S})}} and x^{2}{(\\mathbf{S})} = x{(\\mathbf{S})} e^{\\cos{(\\mathbf{S})}} and x^{4}{(\\mathbf{S})} = x^{2}{(\\mathbf{S})} e^{2 \\cos{(\\mathbf{S})}} and x^{2}{(\\mathbf{S})} e^{2 \\cos{(\\mathbf{S})}} = x{(\\mathbf{S})} e^{3 \\cos{(\\mathbf{S})}} and x^{4}{(\\mathbf{S})} = x{(\\mathbf{S})} e^{3 \\cos{(\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), exp(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Function('x')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Pow(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Mul(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), exp(cos(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4)), Mul(Pow(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), exp(Mul(Integer(2), cos(Symbol('\\\\mathbf{S}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), exp(Mul(Integer(2), cos(Symbol('\\\\mathbf{S}', commutative=True))))), Mul(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Mul(Integer(3), cos(Symbol('\\\\mathbf{S}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4)), Mul(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Mul(Integer(3), cos(Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given x{(\\Psi^{\\dagger})} = \\log{(\\sin{(\\Psi^{\\dagger})})}, then obtain \\frac{\\frac{x{(\\Psi^{\\dagger})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}}}{x{(\\Psi^{\\dagger})}} = \\frac{\\frac{\\log{(\\sin{(\\Psi^{\\dagger})})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}}}{x{(\\Psi^{\\dagger})}}", "derivation": "x{(\\Psi^{\\dagger})} = \\log{(\\sin{(\\Psi^{\\dagger})})} and \\frac{x{(\\Psi^{\\dagger})}}{\\Psi^{\\dagger}} = \\frac{\\log{(\\sin{(\\Psi^{\\dagger})})}}{\\Psi^{\\dagger}} and \\frac{x{(\\Psi^{\\dagger})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}} = \\frac{\\log{(\\sin{(\\Psi^{\\dagger})})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}} and \\frac{\\frac{x{(\\Psi^{\\dagger})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}}}{x{(\\Psi^{\\dagger})}} = \\frac{\\frac{\\log{(\\sin{(\\Psi^{\\dagger})})}}{\\Psi^{\\dagger}} + \\frac{1}{\\Psi^{\\dagger}}}{x{(\\Psi^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), log(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["add", 2, "Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), log(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["divide", 3, "Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Pow(Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Add(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), log(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Pow(Function('x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given H{(\\delta,t_{2})} = \\frac{\\delta}{t_{2}} and E{(\\delta,t_{2})} = - \\frac{\\delta \\int H{(\\delta,t_{2})} d\\delta}{t_{2}}, then obtain E{(\\delta,t_{2})} = - H{(\\delta,t_{2})} \\int H{(\\delta,t_{2})} d\\delta", "derivation": "H{(\\delta,t_{2})} = \\frac{\\delta}{t_{2}} and \\int H{(\\delta,t_{2})} d\\delta = \\int \\frac{\\delta}{t_{2}} d\\delta and H{(\\delta,t_{2})} \\int H{(\\delta,t_{2})} d\\delta = \\frac{\\delta \\int H{(\\delta,t_{2})} d\\delta}{t_{2}} and H{(\\delta,t_{2})} \\int \\frac{\\delta}{t_{2}} d\\delta = \\frac{\\delta \\int \\frac{\\delta}{t_{2}} d\\delta}{t_{2}} and - H{(\\delta,t_{2})} \\int \\frac{\\delta}{t_{2}} d\\delta = - \\frac{\\delta \\int \\frac{\\delta}{t_{2}} d\\delta}{t_{2}} and E{(\\delta,t_{2})} = - \\frac{\\delta \\int H{(\\delta,t_{2})} d\\delta}{t_{2}} and - H{(\\delta,t_{2})} \\int H{(\\delta,t_{2})} d\\delta = - \\frac{\\delta \\int H{(\\delta,t_{2})} d\\delta}{t_{2}} and E{(\\delta,t_{2})} = - H{(\\delta,t_{2})} \\int H{(\\delta,t_{2})} d\\delta", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 1, "Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('E')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Integral(Function('H')(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(E_{x})} = \\log{(E_{x})}, then derive \\int (E_{x} + \\frac{\\operatorname{J_{\\varepsilon}}{(E_{x})}}{\\log{(E_{x})}}) dE_{x} = \\frac{E_{x}^{2}}{2} + E_{x} + u, then obtain 0 = \\frac{E_{x}^{2}}{2} + E_{x} + u - \\int (E_{x} + 1) dE_{x}", "derivation": "\\operatorname{J_{\\varepsilon}}{(E_{x})} = \\log{(E_{x})} and \\frac{\\operatorname{J_{\\varepsilon}}{(E_{x})}}{\\log{(E_{x})}} = 1 and E_{x} + \\frac{\\operatorname{J_{\\varepsilon}}{(E_{x})}}{\\log{(E_{x})}} = E_{x} + 1 and \\int (E_{x} + \\frac{\\operatorname{J_{\\varepsilon}}{(E_{x})}}{\\log{(E_{x})}}) dE_{x} = \\int (E_{x} + 1) dE_{x} and \\int (E_{x} + \\frac{\\operatorname{J_{\\varepsilon}}{(E_{x})}}{\\log{(E_{x})}}) dE_{x} = \\frac{E_{x}^{2}}{2} + E_{x} + u and \\int (E_{x} + 1) dE_{x} = \\frac{E_{x}^{2}}{2} + E_{x} + u and 0 = \\frac{E_{x}^{2}}{2} + E_{x} + u - \\int (E_{x} + 1) dE_{x}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["divide", 1, "log(Symbol('E_x', commutative=True))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Mul(Function('J_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1)))), Add(Symbol('E_x', commutative=True), Integer(1)))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Symbol('E_x', commutative=True), Mul(Function('J_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1)))), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Integer(1)), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Symbol('E_x', commutative=True), Mul(Function('J_{\\\\varepsilon}')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1)))), Tuple(Symbol('E_x', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_x', commutative=True), Integer(2))), Symbol('E_x', commutative=True), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Symbol('E_x', commutative=True), Integer(1)), Tuple(Symbol('E_x', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_x', commutative=True), Integer(2))), Symbol('E_x', commutative=True), Symbol('u', commutative=True)))"], [["minus", 6, "Integral(Add(Symbol('E_x', commutative=True), Integer(1)), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('E_x', commutative=True), Integer(2))), Symbol('E_x', commutative=True), Symbol('u', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('E_x', commutative=True), Integer(1)), Tuple(Symbol('E_x', commutative=True))))))"]]}, {"prompt": "Given G{(F_{x})} = e^{\\sin{(F_{x})}}, then obtain (\\frac{d^{2}}{d F_{x}^{2}} (G{(F_{x})} - \\sin{(F_{x})}))^{F_{x}} = (\\frac{d^{2}}{d F_{x}^{2}} (e^{\\sin{(F_{x})}} - \\sin{(F_{x})}))^{F_{x}}", "derivation": "G{(F_{x})} = e^{\\sin{(F_{x})}} and G{(F_{x})} - \\sin{(F_{x})} = e^{\\sin{(F_{x})}} - \\sin{(F_{x})} and \\frac{d}{d F_{x}} (G{(F_{x})} - \\sin{(F_{x})}) = \\frac{d}{d F_{x}} (e^{\\sin{(F_{x})}} - \\sin{(F_{x})}) and \\frac{d^{2}}{d F_{x}^{2}} (G{(F_{x})} - \\sin{(F_{x})}) = \\frac{d^{2}}{d F_{x}^{2}} (e^{\\sin{(F_{x})}} - \\sin{(F_{x})}) and (\\frac{d^{2}}{d F_{x}^{2}} (G{(F_{x})} - \\sin{(F_{x})}))^{F_{x}} = (\\frac{d^{2}}{d F_{x}^{2}} (e^{\\sin{(F_{x})}} - \\sin{(F_{x})}))^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('F_x', commutative=True)), exp(sin(Symbol('F_x', commutative=True))))"], [["minus", 1, "sin(Symbol('F_x', commutative=True))"], "Equality(Add(Function('G')(Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Add(exp(sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(exp(sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2))), Derivative(Add(exp(sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2))))"], [["power", 4, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Add(Function('G')(Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2))), Symbol('F_x', commutative=True)), Pow(Derivative(Add(exp(sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(v_{2})} = \\sin{(v_{2})}, then obtain - 2 v_{2} + \\frac{\\mathbf{B}^{2}{(v_{2})}}{\\sin{(v_{2})}} + 2 \\mathbf{B}{(v_{2})} = - 2 v_{2} + 3 \\mathbf{B}{(v_{2})}", "derivation": "\\mathbf{B}{(v_{2})} = \\sin{(v_{2})} and \\frac{\\mathbf{B}^{2}{(v_{2})}}{\\sin{(v_{2})}} = \\mathbf{B}{(v_{2})} and - v_{2} + \\frac{\\mathbf{B}^{2}{(v_{2})}}{\\sin{(v_{2})}} = - v_{2} + \\mathbf{B}{(v_{2})} and - 2 v_{2} + \\frac{\\mathbf{B}^{2}{(v_{2})}}{\\sin{(v_{2})}} + 2 \\mathbf{B}{(v_{2})} = - 2 v_{2} + 3 \\mathbf{B}{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["divide", 1, "Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), Integer(2)), Pow(sin(Symbol('v_2', commutative=True)), Integer(-1))), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), Integer(2)), Pow(sin(Symbol('v_2', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), Integer(2)), Pow(sin(Symbol('v_2', commutative=True)), Integer(-1))), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\varepsilon_0)} = e^{e^{\\varepsilon_0}}, then obtain \\sin{(\\int \\tilde{g}^*{(\\varepsilon_0)} d\\varepsilon_0)} = \\sin{(\\mathbf{P} + \\operatorname{Ei}{(e^{\\varepsilon_0})})}", "derivation": "\\tilde{g}^*{(\\varepsilon_0)} = e^{e^{\\varepsilon_0}} and \\int \\tilde{g}^*{(\\varepsilon_0)} d\\varepsilon_0 = \\int e^{e^{\\varepsilon_0}} d\\varepsilon_0 and \\sin{(\\int \\tilde{g}^*{(\\varepsilon_0)} d\\varepsilon_0)} = \\sin{(\\int e^{e^{\\varepsilon_0}} d\\varepsilon_0)} and \\sin{(\\int \\tilde{g}^*{(\\varepsilon_0)} d\\varepsilon_0)} = \\sin{(\\mathbf{P} + \\operatorname{Ei}{(e^{\\varepsilon_0})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True)), exp(exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(exp(exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), sin(Integral(exp(exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(sin(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), sin(Add(Symbol('\\\\mathbf{P}', commutative=True), Ei(exp(Symbol('\\\\varepsilon_0', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}), then derive \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} - 1 = 0, then obtain - \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} + 2 \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}) - 1 = - \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} + \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl})", "derivation": "\\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}) and \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} - \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}) = 0 and \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} - 1 = 0 and \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}) - 1 = 0 and - \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} + 2 \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl}) - 1 = - \\sigma_{p}{(\\Psi^{\\dagger},\\Psi_{nl})} + \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi^{\\dagger} + \\Psi_{nl})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["minus", 4, "Add(Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(2), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(\\hat{x},\\rho_f)} = e^{\\hat{x}^{\\rho_f}} and L{(\\mathbf{H},U)} = U + \\mathbf{H}, then obtain 2 (\\hat{x} + L{(\\mathbf{H},U)}) s{(\\hat{x},\\rho_f)} = 2 (U + \\hat{x} + \\mathbf{H}) s{(\\hat{x},\\rho_f)}", "derivation": "s{(\\hat{x},\\rho_f)} = e^{\\hat{x}^{\\rho_f}} and L{(\\mathbf{H},U)} = U + \\mathbf{H} and 2 s{(\\hat{x},\\rho_f)} = s{(\\hat{x},\\rho_f)} + e^{\\hat{x}^{\\rho_f}} and \\hat{x} + L{(\\mathbf{H},U)} = U + \\hat{x} + \\mathbf{H} and (\\hat{x} + L{(\\mathbf{H},U)}) (s{(\\hat{x},\\rho_f)} + e^{\\hat{x}^{\\rho_f}}) = (s{(\\hat{x},\\rho_f)} + e^{\\hat{x}^{\\rho_f}}) (U + \\hat{x} + \\mathbf{H}) and 2 (\\hat{x} + L{(\\mathbf{H},U)}) s{(\\hat{x},\\rho_f)} = 2 (U + \\hat{x} + \\mathbf{H}) s{(\\hat{x},\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Integer(2), Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Add(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)))))"], [["add", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Function('L')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True))), Add(Symbol('U', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 4, "Add(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Function('L')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True))), Add(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))))), Mul(Add(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Add(Symbol('U', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Add(Symbol('\\\\hat{x}', commutative=True), Function('L')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True))), Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), Add(Symbol('U', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given B{(\\theta)} = \\log{(\\theta)} and \\lambda{(\\theta)} = - B{(\\theta)} + \\log{(\\theta)} and \\operatorname{v_{1}}{(\\theta)} = - B{(\\theta)} + \\log{(\\theta)}, then obtain \\tilde{\\infty} \\operatorname{v_{1}}^{\\theta}{(\\theta)} = 0^{\\theta} \\tilde{\\infty}", "derivation": "B{(\\theta)} = \\log{(\\theta)} and \\lambda{(\\theta)} = - B{(\\theta)} + \\log{(\\theta)} and \\lambda{(\\theta)} = 0 and \\lambda^{\\theta}{(\\theta)} = 0^{\\theta} and \\tilde{\\infty} \\lambda^{\\theta}{(\\theta)} = 0^{\\theta} \\tilde{\\infty} and \\tilde{\\infty} (- B{(\\theta)} + \\log{(\\theta)})^{\\theta} = 0^{\\theta} \\tilde{\\infty} and \\operatorname{v_{1}}{(\\theta)} = - B{(\\theta)} + \\log{(\\theta)} and \\tilde{\\infty} \\operatorname{v_{1}}^{\\theta}{(\\theta)} = 0^{\\theta} \\tilde{\\infty}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\theta', commutative=True))), log(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Integer(0))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Integer(0), Symbol('\\\\theta', commutative=True)))"], [["divide", 4, 0], "Equality(Mul(zoo, Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), zoo))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(zoo, Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\theta', commutative=True))), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), zoo))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\theta', commutative=True))), log(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(zoo, Pow(Function('v_1')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), zoo))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(f_{E})} = \\log{(f_{E})}, then obtain f_{E} - z^{*} + (- f_{E} + \\operatorname{P_{e}}{(f_{E})}) (- f_{E} + \\log{(f_{E})}) = f_{E} - z^{*} + (- f_{E} + \\log{(f_{E})})^{2}", "derivation": "\\operatorname{P_{e}}{(f_{E})} = \\log{(f_{E})} and - f_{E} + \\operatorname{P_{e}}{(f_{E})} = - f_{E} + \\log{(f_{E})} and (- f_{E} + \\operatorname{P_{e}}{(f_{E})}) (- f_{E} + \\log{(f_{E})}) = (- f_{E} + \\log{(f_{E})})^{2} and f_{E} + (- f_{E} + \\operatorname{P_{e}}{(f_{E})}) (- f_{E} + \\log{(f_{E})}) = f_{E} + (- f_{E} + \\log{(f_{E})})^{2} and f_{E} - z^{*} + (- f_{E} + \\operatorname{P_{e}}{(f_{E})}) (- f_{E} + \\log{(f_{E})}) = f_{E} - z^{*} + (- f_{E} + \\log{(f_{E})})^{2}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["minus", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('P_e')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('P_e')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Integer(2)))"], [["add", 3, "Symbol('f_E', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('P_e')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))), Add(Symbol('f_E', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Integer(2))))"], [["minus", 4, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('P_e')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(a,A_{z})} = A_{z} + a, then obtain 4 (A_{z} + a)^{2} = (2 A_{z} + 2 a)^{2}", "derivation": "\\operatorname{P_{g}}{(a,A_{z})} = A_{z} + a and 2 \\operatorname{P_{g}}{(a,A_{z})} = A_{z} + a + \\operatorname{P_{g}}{(a,A_{z})} and 4 \\operatorname{P_{g}}^{2}{(a,A_{z})} = (A_{z} + a + \\operatorname{P_{g}}{(a,A_{z})})^{2} and 4 (A_{z} + a)^{2} = (2 A_{z} + 2 a)^{2}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('a', commutative=True)))"], [["add", 1, "Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Integer(2), Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('A_z', commutative=True), Symbol('a', commutative=True), Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True)), Integer(2))), Pow(Add(Symbol('A_z', commutative=True), Symbol('a', commutative=True), Function('P_g')(Symbol('a', commutative=True), Symbol('A_z', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('A_z', commutative=True), Symbol('a', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('A_z', commutative=True)), Mul(Integer(2), Symbol('a', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\varepsilon{(\\varphi)} = \\log{(\\varphi)}, then derive \\int \\varepsilon{(\\varphi)} d\\varphi = \\tilde{g} + \\varphi \\log{(\\varphi)} - \\varphi, then obtain \\tilde{g} + \\varphi \\varepsilon{(\\varphi)} - \\varphi + \\frac{d}{d \\varphi} \\log{(\\varphi)} = \\tilde{g} + \\varphi \\log{(\\varphi)} - \\varphi + \\frac{d}{d \\varphi} \\log{(\\varphi)}", "derivation": "\\varepsilon{(\\varphi)} = \\log{(\\varphi)} and \\int \\varepsilon{(\\varphi)} d\\varphi = \\int \\log{(\\varphi)} d\\varphi and \\int \\varepsilon{(\\varphi)} d\\varphi = \\tilde{g} + \\varphi \\log{(\\varphi)} - \\varphi and \\int \\varepsilon{(\\varphi)} d\\varphi = \\tilde{g} + \\varphi \\varepsilon{(\\varphi)} - \\varphi and \\frac{d}{d \\varphi} \\log{(\\varphi)} + \\int \\varepsilon{(\\varphi)} d\\varphi = \\tilde{g} + \\varphi \\log{(\\varphi)} - \\varphi + \\frac{d}{d \\varphi} \\log{(\\varphi)} and \\tilde{g} + \\varphi \\varepsilon{(\\varphi)} - \\varphi + \\frac{d}{d \\varphi} \\log{(\\varphi)} = \\tilde{g} + \\varphi \\log{(\\varphi)} - \\varphi + \\frac{d}{d \\varphi} \\log{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["add", 3, "Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integral(Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\psi)} = \\log{(\\cos{(\\psi)})} and a{(\\psi)} = \\operatorname{f_{\\mathbf{v}}}^{\\psi}{(\\psi)}, then obtain \\cos{(a{(\\psi)})} = \\cos{(\\log{(\\cos{(\\psi)})}^{\\psi})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\psi)} = \\log{(\\cos{(\\psi)})} and \\operatorname{f_{\\mathbf{v}}}^{\\psi}{(\\psi)} = \\log{(\\cos{(\\psi)})}^{\\psi} and a{(\\psi)} = \\operatorname{f_{\\mathbf{v}}}^{\\psi}{(\\psi)} and a{(\\psi)} = \\log{(\\cos{(\\psi)})}^{\\psi} and \\cos{(a{(\\psi)})} = \\cos{(\\log{(\\cos{(\\psi)})}^{\\psi})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True)), log(cos(Symbol('\\\\psi', commutative=True))))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(log(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\psi', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('a')(Symbol('\\\\psi', commutative=True)), Pow(log(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["cos", 4], "Equality(cos(Function('a')(Symbol('\\\\psi', commutative=True))), cos(Pow(log(cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(B)} = \\log{(B)}, then derive \\frac{\\int \\hat{H}_l{(B)} dB}{2 \\hat{H}_l{(B)}} = \\frac{B \\log{(B)} - B + a}{2 \\hat{H}_l{(B)}}, then obtain \\frac{B \\log{(B)} - B + a}{2 \\hat{H}_l{(B)}} = \\frac{A_{1} + B \\log{(B)} - B}{2 \\hat{H}_l{(B)}}", "derivation": "\\hat{H}_l{(B)} = \\log{(B)} and \\int \\hat{H}_l{(B)} dB = \\int \\log{(B)} dB and \\frac{\\int \\hat{H}_l{(B)} dB}{2 \\hat{H}_l{(B)}} = \\frac{\\int \\log{(B)} dB}{2 \\hat{H}_l{(B)}} and \\frac{\\int \\hat{H}_l{(B)} dB}{2 \\hat{H}_l{(B)}} = \\frac{B \\log{(B)} - B + a}{2 \\hat{H}_l{(B)}} and \\frac{B \\log{(B)} - B + a}{2 \\hat{H}_l{(B)}} = \\frac{\\int \\log{(B)} dB}{2 \\hat{H}_l{(B)}} and \\frac{B \\log{(B)} - B + a}{2 \\hat{H}_l{(B)}} = \\frac{A_{1} + B \\log{(B)} - B}{2 \\hat{H}_l{(B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('B', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Rational(1, 2), Add(Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('a', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Rational(1, 2), Add(Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('a', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Rational(1, 2), Add(Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('a', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Symbol('A_1', commutative=True), Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{P}{(H,\\hat{\\mathbf{x}})} = - H + \\hat{\\mathbf{x}} and \\operatorname{V_{\\mathbf{E}}}{(v_{x})} = \\sin{(v_{x})}, then obtain \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{x})}}{\\int \\mathbf{P}{(H,\\hat{\\mathbf{x}})} dH} = \\frac{\\sin{(v_{x})}}{\\int \\mathbf{P}{(H,\\hat{\\mathbf{x}})} dH}", "derivation": "\\mathbf{P}{(H,\\hat{\\mathbf{x}})} = - H + \\hat{\\mathbf{x}} and \\int \\mathbf{P}{(H,\\hat{\\mathbf{x}})} dH = \\int (- H + \\hat{\\mathbf{x}}) dH and \\operatorname{V_{\\mathbf{E}}}{(v_{x})} = \\sin{(v_{x})} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{x})}}{\\int (- H + \\hat{\\mathbf{x}}) dH} = \\frac{\\sin{(v_{x})}}{\\int (- H + \\hat{\\mathbf{x}}) dH} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{x})}}{\\int \\mathbf{P}{(H,\\hat{\\mathbf{x}})} dH} = \\frac{\\sin{(v_{x})}}{\\int \\mathbf{P}{(H,\\hat{\\mathbf{x}})} dH}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["divide", 3, "Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('v_x', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(sin(Symbol('v_x', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('v_x', commutative=True)), Pow(Integral(Function('\\\\mathbf{P}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(sin(Symbol('v_x', commutative=True)), Pow(Integral(Function('\\\\mathbf{P}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(m)} = \\log{(m)}, then obtain ((\\Psi^{\\dagger}{(m)} - \\log{(m)}^{m})^{m})^{m} = ((\\log{(m)} - \\log{(m)}^{m})^{m})^{m}", "derivation": "\\Psi^{\\dagger}{(m)} = \\log{(m)} and \\Psi^{\\dagger}^{m}{(m)} = \\log{(m)}^{m} and \\Psi^{\\dagger}{(m)} - \\Psi^{\\dagger}^{m}{(m)} = - \\Psi^{\\dagger}^{m}{(m)} + \\log{(m)} and (\\Psi^{\\dagger}{(m)} - \\Psi^{\\dagger}^{m}{(m)})^{m} = (- \\Psi^{\\dagger}^{m}{(m)} + \\log{(m)})^{m} and ((\\Psi^{\\dagger}{(m)} - \\Psi^{\\dagger}^{m}{(m)})^{m})^{m} = ((- \\Psi^{\\dagger}^{m}{(m)} + \\log{(m)})^{m})^{m} and ((\\Psi^{\\dagger}{(m)} - \\log{(m)}^{m})^{m})^{m} = ((\\log{(m)} - \\log{(m)}^{m})^{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), log(Symbol('m', commutative=True))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), log(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), log(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Add(log(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(v_{1},x^\\prime)} = e^{(x^\\prime)^{v_{1}}}, then obtain \\frac{\\partial}{\\partial x^\\prime} \\frac{\\hat{x}^{2}{(v_{1},x^\\prime)}}{x^\\prime} = \\frac{\\partial}{\\partial x^\\prime} \\frac{\\hat{x}{(v_{1},x^\\prime)} e^{(x^\\prime)^{v_{1}}}}{x^\\prime}", "derivation": "\\hat{x}{(v_{1},x^\\prime)} = e^{(x^\\prime)^{v_{1}}} and \\frac{\\hat{x}{(v_{1},x^\\prime)}}{x^\\prime} = \\frac{e^{(x^\\prime)^{v_{1}}}}{x^\\prime} and \\frac{\\hat{x}{(v_{1},x^\\prime)} e^{(x^\\prime)^{v_{1}}}}{x^\\prime} = \\frac{e^{2 (x^\\prime)^{v_{1}}}}{x^\\prime} and \\frac{\\hat{x}^{2}{(v_{1},x^\\prime)}}{x^\\prime} = \\frac{\\hat{x}{(v_{1},x^\\prime)} e^{(x^\\prime)^{v_{1}}}}{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\frac{\\hat{x}^{2}{(v_{1},x^\\prime)}}{x^\\prime} = \\frac{\\partial}{\\partial x^\\prime} \\frac{\\hat{x}{(v_{1},x^\\prime)} e^{(x^\\prime)^{v_{1}}}}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)))))"], [["times", 2, "exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Mul(Integer(2), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)))))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(2))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('v_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(H,A_{y})} = H^{A_{y}}, then obtain (\\int \\frac{1}{\\varphi{(H,A_{y})}} dH)^{- 2 H} = (\\int \\frac{H^{A_{y}}}{\\varphi^{2}{(H,A_{y})}} dH)^{- 2 H}", "derivation": "\\varphi{(H,A_{y})} = H^{A_{y}} and \\frac{1}{\\varphi{(H,A_{y})}} = \\frac{H^{A_{y}}}{\\varphi^{2}{(H,A_{y})}} and \\int \\frac{1}{\\varphi{(H,A_{y})}} dH = \\int \\frac{H^{A_{y}}}{\\varphi^{2}{(H,A_{y})}} dH and (\\int \\frac{1}{\\varphi{(H,A_{y})}} dH)^{H} = (\\int \\frac{H^{A_{y}}}{\\varphi^{2}{(H,A_{y})}} dH)^{H} and (\\int \\frac{1}{\\varphi{(H,A_{y})}} dH)^{- 2 H} = (\\int \\frac{H^{A_{y}}}{\\varphi^{2}{(H,A_{y})}} dH)^{- 2 H}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('A_y', commutative=True)))"], [["divide", 1, "Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Mul(Pow(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-2))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Tuple(Symbol('H', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-2))), Tuple(Symbol('H', commutative=True))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Mul(Pow(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-2))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["power", 4, "Integer(-2)"], "Equality(Pow(Integral(Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('H', commutative=True))), Pow(Integral(Mul(Pow(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Pow(Function('\\\\varphi')(Symbol('H', commutative=True), Symbol('A_y', commutative=True)), Integer(-2))), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('H', commutative=True))))"]]}, {"prompt": "Given S{(u,t_{1},v)} = t_{1} v - u and \\mathbf{J}{(t_{1},v)} = t_{1} v, then obtain (t_{1} v (t_{1} v - u - \\mathbf{J}{(t_{1},v)}) + (- u)^{v})^{v} = (- t_{1} u v + (- u)^{v})^{v}", "derivation": "S{(u,t_{1},v)} = t_{1} v - u and - t_{1} v + S{(u,t_{1},v)} = - u and \\mathbf{J}{(t_{1},v)} = t_{1} v and S{(u,t_{1},v)} - \\mathbf{J}{(t_{1},v)} = - u and t_{1} v (S{(u,t_{1},v)} - \\mathbf{J}{(t_{1},v)}) = - t_{1} u v and t_{1} v (S{(u,t_{1},v)} - \\mathbf{J}{(t_{1},v)}) + (- u)^{v} = - t_{1} u v + (- u)^{v} and t_{1} v (t_{1} v - u - \\mathbf{J}{(t_{1},v)}) + (- u)^{v} = - t_{1} u v + (- u)^{v} and (t_{1} v (t_{1} v - u - \\mathbf{J}{(t_{1},v)}) + (- u)^{v})^{v} = (- t_{1} u v + (- u)^{v})^{v}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('u', commutative=True), Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["minus", 1, "Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Function('S')(Symbol('u', commutative=True), Symbol('t_1', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('S')(Symbol('u', commutative=True), Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True)))"], [["times", 4, "Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True), Add(Function('S')(Symbol('u', commutative=True), Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True))))), Mul(Integer(-1), Symbol('t_1', commutative=True), Symbol('u', commutative=True), Symbol('v', commutative=True)))"], [["add", 5, "Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))"], "Equality(Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True), Add(Function('S')(Symbol('u', commutative=True), Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True))))), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Symbol('u', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True), Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True))))), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Symbol('u', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))))"], [["power", 7, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True), Add(Mul(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v', commutative=True))))), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Symbol('u', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Symbol('u', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given b{(E_{n},A_{1})} = A_{1} E_{n}, then obtain \\int (- E_{n} + \\int b{(E_{n},A_{1})} dE_{n}) dA_{1} = \\int (- E_{n} + \\int A_{1} E_{n} dE_{n}) dA_{1}", "derivation": "b{(E_{n},A_{1})} = A_{1} E_{n} and \\int b{(E_{n},A_{1})} dE_{n} = \\int A_{1} E_{n} dE_{n} and - E_{n} + \\int b{(E_{n},A_{1})} dE_{n} = - E_{n} + \\int A_{1} E_{n} dE_{n} and \\int (- E_{n} + \\int b{(E_{n},A_{1})} dE_{n}) dA_{1} = \\int (- E_{n} + \\int A_{1} E_{n} dE_{n}) dA_{1}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E_n', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('E_n', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('b')(Symbol('E_n', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Symbol('A_1', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["minus", 2, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Function('b')(Symbol('E_n', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"], [["integrate", 3, "Symbol('A_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Function('b')(Symbol('E_n', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(n_{2})} = n_{2}, then derive \\log{(\\int \\mu_{0}{(n_{2})} dn_{2})}^{n_{2}} = \\log{(\\frac{n_{2}^{2}}{2} + z^{*})}^{n_{2}}, then obtain \\log{(\\frac{n_{2}^{2}}{2} + t)}^{n_{2}} = \\log{(\\frac{n_{2}^{2}}{2} + z^{*})}^{n_{2}}", "derivation": "\\mu_{0}{(n_{2})} = n_{2} and \\int \\mu_{0}{(n_{2})} dn_{2} = \\int n_{2} dn_{2} and \\log{(\\int \\mu_{0}{(n_{2})} dn_{2})} = \\log{(\\int n_{2} dn_{2})} and \\log{(\\int \\mu_{0}{(n_{2})} dn_{2})}^{n_{2}} = \\log{(\\int n_{2} dn_{2})}^{n_{2}} and \\log{(\\int \\mu_{0}{(n_{2})} dn_{2})}^{n_{2}} = \\log{(\\frac{n_{2}^{2}}{2} + z^{*})}^{n_{2}} and \\log{(\\int n_{2} dn_{2})}^{n_{2}} = \\log{(\\frac{n_{2}^{2}}{2} + z^{*})}^{n_{2}} and \\log{(\\frac{n_{2}^{2}}{2} + t)}^{n_{2}} = \\log{(\\frac{n_{2}^{2}}{2} + z^{*})}^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Symbol('n_2', commutative=True), Tuple(Symbol('n_2', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\mu_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), log(Integral(Symbol('n_2', commutative=True), Tuple(Symbol('n_2', commutative=True)))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(log(Integral(Function('\\\\mu_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Pow(log(Integral(Symbol('n_2', commutative=True), Tuple(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(log(Integral(Function('\\\\mu_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Pow(log(Add(Mul(Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Symbol('z^*', commutative=True))), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(log(Integral(Symbol('n_2', commutative=True), Tuple(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Pow(log(Add(Mul(Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Symbol('z^*', commutative=True))), Symbol('n_2', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(log(Add(Mul(Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Symbol('t', commutative=True))), Symbol('n_2', commutative=True)), Pow(log(Add(Mul(Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Symbol('z^*', commutative=True))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{P},A_{1})} = \\cos{(\\mathbf{P}^{A_{1}})}, then obtain 1 = (\\frac{\\frac{\\partial}{\\partial A_{1}} \\cos{(\\mathbf{P}^{A_{1}})}}{\\frac{\\partial}{\\partial A_{1}} \\operatorname{C_{d}}{(\\mathbf{P},A_{1})}})^{\\mathbf{P}}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{P},A_{1})} = \\cos{(\\mathbf{P}^{A_{1}})} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{C_{d}}{(\\mathbf{P},A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\cos{(\\mathbf{P}^{A_{1}})} and 1 = \\frac{\\frac{\\partial}{\\partial A_{1}} \\cos{(\\mathbf{P}^{A_{1}})}}{\\frac{\\partial}{\\partial A_{1}} \\operatorname{C_{d}}{(\\mathbf{P},A_{1})}} and 1 = (\\frac{\\frac{\\partial}{\\partial A_{1}} \\cos{(\\mathbf{P}^{A_{1}})}}{\\frac{\\partial}{\\partial A_{1}} \\operatorname{C_{d}}{(\\mathbf{P},A_{1})}})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True)), cos(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Derivative(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(k)} = \\cos{(k)}, then obtain \\cos{(k)} (\\int \\frac{\\hat{H}{(k)}}{\\cos{(k)}} dk)^{k} = \\cos{(k)} (\\int 1 dk)^{k}", "derivation": "\\hat{H}{(k)} = \\cos{(k)} and \\frac{\\hat{H}{(k)}}{\\cos{(k)}} = 1 and \\int \\frac{\\hat{H}{(k)}}{\\cos{(k)}} dk = \\int 1 dk and (\\int \\frac{\\hat{H}{(k)}}{\\cos{(k)}} dk)^{k} = (\\int 1 dk)^{k} and \\cos{(k)} (\\int \\frac{\\hat{H}{(k)}}{\\cos{(k)}} dk)^{k} = \\cos{(k)} (\\int 1 dk)^{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["divide", 1, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{H}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('k', commutative=True))), Integral(Integer(1), Tuple(Symbol('k', commutative=True))))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Integral(Mul(Function('\\\\hat{H}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["times", 4, "cos(Symbol('k', commutative=True))"], "Equality(Mul(cos(Symbol('k', commutative=True)), Pow(Integral(Mul(Function('\\\\hat{H}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True))), Mul(cos(Symbol('k', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{M} \\theta, then derive V_{\\mathbf{E}} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{D} + \\mathbf{M} \\theta, then obtain \\mathbf{D} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{D} + \\mathbf{M} \\theta", "derivation": "\\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{M} \\theta and \\frac{\\partial}{\\partial \\mathbf{M}} \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\theta and \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} d\\mathbf{M} = \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\theta d\\mathbf{M} and V_{\\mathbf{E}} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{D} + \\mathbf{M} \\theta and V_{\\mathbf{E}} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{D} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} and \\mathbf{D} + \\operatorname{F_{N}}{(\\theta,\\mathbf{M})} = \\mathbf{D} + \\mathbf{M} \\theta", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Derivative(Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('F_N')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given r{(\\pi,\\mu_0)} = \\mu_0 \\pi, then obtain (\\frac{\\partial}{\\partial \\mu_0} r{(\\pi,\\mu_0)})^{\\mu_0} = \\pi^{\\mu_0}", "derivation": "r{(\\pi,\\mu_0)} = \\mu_0 \\pi and \\frac{\\partial}{\\partial \\mu_0} r{(\\pi,\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\pi and (\\frac{\\partial}{\\partial \\mu_0} r{(\\pi,\\mu_0)})^{\\mu_0} = (\\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\pi)^{\\mu_0} and (\\frac{\\partial}{\\partial \\mu_0} r{(\\pi,\\mu_0)})^{\\mu_0} = \\pi^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Derivative(Function('r')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('r')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(\\varepsilon)} = \\log{(\\log{(\\varepsilon)})} and n{(\\varepsilon)} = \\varepsilon + \\hat{X}{(\\varepsilon)}, then obtain n{(\\varepsilon)} \\log{(\\log{(\\varepsilon)})} = (\\varepsilon + \\hat{X}{(\\varepsilon)}) \\log{(\\log{(\\varepsilon)})}", "derivation": "\\hat{X}{(\\varepsilon)} = \\log{(\\log{(\\varepsilon)})} and \\varepsilon + \\hat{X}{(\\varepsilon)} = \\varepsilon + \\log{(\\log{(\\varepsilon)})} and n{(\\varepsilon)} = \\varepsilon + \\hat{X}{(\\varepsilon)} and n{(\\varepsilon)} = \\varepsilon + \\log{(\\log{(\\varepsilon)})} and n{(\\varepsilon)} \\log{(\\log{(\\varepsilon)})} = (\\varepsilon + \\log{(\\log{(\\varepsilon)})}) \\log{(\\log{(\\varepsilon)})} and n{(\\varepsilon)} \\log{(\\log{(\\varepsilon)})} = (\\varepsilon + \\hat{X}{(\\varepsilon)}) \\log{(\\log{(\\varepsilon)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True)), log(log(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), log(log(Symbol('\\\\varepsilon', commutative=True)))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('n')(Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), log(log(Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 4, "log(log(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Function('n')(Symbol('\\\\varepsilon', commutative=True)), log(log(Symbol('\\\\varepsilon', commutative=True)))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), log(log(Symbol('\\\\varepsilon', commutative=True)))), log(log(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('n')(Symbol('\\\\varepsilon', commutative=True)), log(log(Symbol('\\\\varepsilon', commutative=True)))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True))), log(log(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given c{(P_{e},L_{\\varepsilon})} = L_{\\varepsilon} + P_{e}, then derive \\frac{\\partial}{\\partial L_{\\varepsilon}} c{(P_{e},L_{\\varepsilon})} = 1, then obtain - L_{\\varepsilon} + \\int (\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + P_{e}))^{P_{e}} dP_{e} = - L_{\\varepsilon} + \\int 1 dP_{e}", "derivation": "c{(P_{e},L_{\\varepsilon})} = L_{\\varepsilon} + P_{e} and \\frac{\\partial}{\\partial L_{\\varepsilon}} c{(P_{e},L_{\\varepsilon})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + P_{e}) and \\frac{\\partial}{\\partial L_{\\varepsilon}} c{(P_{e},L_{\\varepsilon})} = 1 and (\\frac{\\partial}{\\partial L_{\\varepsilon}} c{(P_{e},L_{\\varepsilon})})^{P_{e}} = 1 and (\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + P_{e}))^{P_{e}} = 1 and \\int (\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + P_{e}))^{P_{e}} dP_{e} = \\int 1 dP_{e} and - L_{\\varepsilon} + \\int (\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + P_{e}))^{P_{e}} dP_{e} = - L_{\\varepsilon} + \\int 1 dP_{e}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('P_e', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('P_e', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('P_e', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('P_e', commutative=True)"], "Equality(Pow(Derivative(Function('c')(Symbol('P_e', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('P_e', commutative=True)"], "Equality(Integral(Pow(Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_e', commutative=True))))"], [["minus", 6, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Pow(Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Integer(1), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(G,\\Psi)} = G^{\\Psi} and \\mathbf{J}{(\\Psi)} = \\Psi, then obtain \\frac{E + \\mathbf{J}{(\\Psi)}}{- G + G^{\\Psi}} = \\frac{E + \\Psi}{- G + G^{\\Psi}}", "derivation": "\\bar{\\h}{(G,\\Psi)} = G^{\\Psi} and - G + \\bar{\\h}{(G,\\Psi)} = - G + G^{\\Psi} and \\mathbf{J}{(\\Psi)} = \\Psi and E + \\mathbf{J}{(\\Psi)} = E + \\Psi and \\frac{E + \\mathbf{J}{(\\Psi)}}{- G + \\bar{\\h}{(G,\\Psi)}} = \\frac{E + \\Psi}{- G + \\bar{\\h}{(G,\\Psi)}} and \\frac{E + \\mathbf{J}{(\\Psi)}}{- G + G^{\\Psi}} = \\frac{E + \\Psi}{- G + G^{\\Psi}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hbar')(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], [["add", 3, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\Psi', commutative=True))), Add(Symbol('E', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hbar')(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Add(Symbol('E', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\Psi', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hbar')(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(Add(Symbol('E', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hbar')(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Symbol('E', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\Psi', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(Add(Symbol('E', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\psi{(k,V_{\\mathbf{E}})} = (e^{k})^{V_{\\mathbf{E}}}, then obtain V_{\\mathbf{E}} \\psi^{k}{(k,V_{\\mathbf{E}})} - \\psi{(k,V_{\\mathbf{E}})} = V_{\\mathbf{E}} ((e^{k})^{V_{\\mathbf{E}}})^{k} - \\psi{(k,V_{\\mathbf{E}})}", "derivation": "\\psi{(k,V_{\\mathbf{E}})} = (e^{k})^{V_{\\mathbf{E}}} and \\psi^{k}{(k,V_{\\mathbf{E}})} = ((e^{k})^{V_{\\mathbf{E}}})^{k} and V_{\\mathbf{E}} \\psi^{k}{(k,V_{\\mathbf{E}})} = V_{\\mathbf{E}} ((e^{k})^{V_{\\mathbf{E}}})^{k} and V_{\\mathbf{E}} \\psi^{k}{(k,V_{\\mathbf{E}})} - \\psi{(k,V_{\\mathbf{E}})} = V_{\\mathbf{E}} ((e^{k})^{V_{\\mathbf{E}}})^{k} - \\psi{(k,V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(exp(Symbol('k', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(exp(Symbol('k', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True)))"], [["times", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Pow(exp(Symbol('k', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True))))"], [["minus", 3, "Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True))), Mul(Integer(-1), Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Pow(exp(Symbol('k', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('k', commutative=True))), Mul(Integer(-1), Function('\\\\psi')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(v_{2},E,u)} = v_{2} (E + u) and \\operatorname{L_{\\varepsilon}}{(v_{2},E,u)} = v_{2} (E + u), then obtain \\mu_{0}^{E}{(v_{2},E,u)} = (v_{2} (E + u))^{E}", "derivation": "\\mu_{0}{(v_{2},E,u)} = v_{2} (E + u) and \\operatorname{L_{\\varepsilon}}{(v_{2},E,u)} = v_{2} (E + u) and \\operatorname{L_{\\varepsilon}}^{E}{(v_{2},E,u)} = (v_{2} (E + u))^{E} and \\operatorname{L_{\\varepsilon}}^{E}{(v_{2},E,u)} = \\mu_{0}^{E}{(v_{2},E,u)} and \\mu_{0}^{E}{(v_{2},E,u)} = (v_{2} (E + u))^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('v_2', commutative=True), Add(Symbol('E', commutative=True), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('v_2', commutative=True), Add(Symbol('E', commutative=True), Symbol('u', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Symbol('v_2', commutative=True), Add(Symbol('E', commutative=True), Symbol('u', commutative=True))), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Symbol('E', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('E', commutative=True), Symbol('u', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Symbol('v_2', commutative=True), Add(Symbol('E', commutative=True), Symbol('u', commutative=True))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(M)} = \\cos{(\\log{(M)})}, then obtain \\frac{d}{d M} \\mathbf{P}{(M)} + (\\frac{d}{d M} \\cos{(\\log{(M)})})^{M} = \\frac{d}{d M} \\cos{(\\log{(M)})} + (\\frac{d}{d M} \\cos{(\\log{(M)})})^{M}", "derivation": "\\mathbf{P}{(M)} = \\cos{(\\log{(M)})} and \\frac{d}{d M} \\mathbf{P}{(M)} = \\frac{d}{d M} \\cos{(\\log{(M)})} and (\\frac{d}{d M} \\mathbf{P}{(M)})^{M} = (\\frac{d}{d M} \\cos{(\\log{(M)})})^{M} and \\frac{d}{d M} \\mathbf{P}{(M)} + (\\frac{d}{d M} \\mathbf{P}{(M)})^{M} = (\\frac{d}{d M} \\mathbf{P}{(M)})^{M} + \\frac{d}{d M} \\cos{(\\log{(M)})} and \\frac{d}{d M} \\mathbf{P}{(M)} + (\\frac{d}{d M} \\cos{(\\log{(M)})})^{M} = \\frac{d}{d M} \\cos{(\\log{(M)})} + (\\frac{d}{d M} \\cos{(\\log{(M)})})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), cos(log(Symbol('M', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], [["add", 2, "Pow(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))"], "Equality(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))), Add(Pow(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))), Add(Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(cos(log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\Psi)} = e^{\\sin{(\\Psi)}} and \\operatorname{P_{g}}{(\\Psi)} = e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} \\mathbf{B}{(\\Psi)}, then obtain \\operatorname{P_{g}}{(\\Psi)} - \\cos{(t)} = - \\cos{(t)} + e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} e^{\\sin{(\\Psi)}}", "derivation": "\\mathbf{B}{(\\Psi)} = e^{\\sin{(\\Psi)}} and \\frac{d}{d \\Psi} \\mathbf{B}{(\\Psi)} = \\frac{d}{d \\Psi} e^{\\sin{(\\Psi)}} and e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} \\mathbf{B}{(\\Psi)} = e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} e^{\\sin{(\\Psi)}} and \\operatorname{P_{g}}{(\\Psi)} = e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} \\mathbf{B}{(\\Psi)} and - \\cos{(t)} + e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} \\mathbf{B}{(\\Psi)} = - \\cos{(t)} + e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} e^{\\sin{(\\Psi)}} and \\operatorname{P_{g}}{(\\Psi)} - \\cos{(t)} = - \\cos{(t)} + e^{- \\sin{(\\Psi)}} \\frac{d}{d \\Psi} e^{\\sin{(\\Psi)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\Psi', commutative=True)), exp(sin(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["divide", 2, "exp(sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(exp(sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\Psi', commutative=True)), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["minus", 3, "cos(Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(exp(sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('P_g')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Derivative(exp(sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(v,W)} = \\log{(v)}^{W} and \\operatorname{r_{0}}{(v,W)} = \\log{(v)}^{W}, then obtain - \\operatorname{c_{0}}{(v,W)} + 2 \\log{(v)} \\log{(v)}^{W} = (\\operatorname{r_{0}}{(v,W)} + \\log{(v)}^{W}) \\log{(v)} - \\operatorname{c_{0}}{(v,W)}", "derivation": "\\operatorname{v_{1}}{(v,W)} = \\log{(v)}^{W} and 2 \\operatorname{v_{1}}{(v,W)} = \\operatorname{v_{1}}{(v,W)} + \\log{(v)}^{W} and \\operatorname{r_{0}}{(v,W)} = \\log{(v)}^{W} and 2 \\operatorname{v_{1}}{(v,W)} \\log{(v)} = (\\operatorname{v_{1}}{(v,W)} + \\log{(v)}^{W}) \\log{(v)} and 2 \\operatorname{v_{1}}{(v,W)} \\log{(v)} = (\\operatorname{r_{0}}{(v,W)} + \\operatorname{v_{1}}{(v,W)}) \\log{(v)} and - \\operatorname{c_{0}}{(v,W)} + 2 \\operatorname{v_{1}}{(v,W)} \\log{(v)} = (\\operatorname{r_{0}}{(v,W)} + \\operatorname{v_{1}}{(v,W)}) \\log{(v)} - \\operatorname{c_{0}}{(v,W)} and - \\operatorname{c_{0}}{(v,W)} + 2 \\log{(v)} \\log{(v)}^{W} = (\\operatorname{r_{0}}{(v,W)} + \\log{(v)}^{W}) \\log{(v)} - \\operatorname{c_{0}}{(v,W)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True)))"], [["add", 1, "Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True))"], "Equality(Mul(Integer(2), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True))), Add(Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True)))"], [["times", 2, "log(Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), log(Symbol('v', commutative=True))), Mul(Add(Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True))), log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), log(Symbol('v', commutative=True))), Mul(Add(Function('r_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True))), log(Symbol('v', commutative=True))))"], [["minus", 5, "Function('c_0')(Symbol('v', commutative=True), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True), Symbol('W', commutative=True))), Mul(Integer(2), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True)), log(Symbol('v', commutative=True)))), Add(Mul(Add(Function('r_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Function('v_1')(Symbol('v', commutative=True), Symbol('W', commutative=True))), log(Symbol('v', commutative=True))), Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True), Symbol('W', commutative=True))), Mul(Integer(2), log(Symbol('v', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True)))), Add(Mul(Add(Function('r_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('W', commutative=True))), log(Symbol('v', commutative=True))), Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True), Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\sigma_p)} = e^{\\sigma_p}, then obtain \\int - \\frac{\\dot{z}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p - 1 = \\int - \\frac{e^{\\sigma_p}}{\\sigma_p} d\\sigma_p - 1", "derivation": "\\dot{z}{(\\sigma_p)} = e^{\\sigma_p} and \\frac{\\dot{z}{(\\sigma_p)}}{\\sigma_p} = \\frac{e^{\\sigma_p}}{\\sigma_p} and - \\frac{\\dot{z}{(\\sigma_p)}}{\\sigma_p} = - \\frac{e^{\\sigma_p}}{\\sigma_p} and \\int - \\frac{\\dot{z}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = \\int - \\frac{e^{\\sigma_p}}{\\sigma_p} d\\sigma_p and \\int - \\frac{\\dot{z}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p - 1 = \\int - \\frac{e^{\\sigma_p}}{\\sigma_p} d\\sigma_p - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integral(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Add(Integral(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\varphi^*)} = \\cos{(\\varphi^*)}, then derive 2 \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} = - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)}, then obtain 2 \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)}", "derivation": "\\operatorname{M_{E}}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} and 2 \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} and 2 \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} = - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} and \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\operatorname{M_{E}}{(\\varphi^*)} and 2 \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Add(Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\varphi^*', commutative=True))), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\log{(e^{\\hat{x}})} and V{(\\hat{x})} = \\log{(e^{\\hat{x}})}, then obtain - \\frac{d}{d \\hat{x}} V{(\\hat{x})} = - \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\log{(e^{\\hat{x}})} and \\frac{d}{d \\hat{x}} \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})} and V{(\\hat{x})} = \\log{(e^{\\hat{x}})} and V{(\\hat{x})} = \\operatorname{J_{\\varepsilon}}{(\\hat{x})} and \\frac{d}{d \\hat{x}} V{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})} and - \\frac{d}{d \\hat{x}} V{(\\hat{x})} = - \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), log(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\hat{x}', commutative=True)), log(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(\\varepsilon_0,n_{2})} = \\varepsilon_0 + n_{2}, then obtain \\varepsilon_0 + n_{2} + \\hat{H}{(\\varepsilon_0,n_{2})} = 2 \\hat{H}{(\\varepsilon_0,n_{2})}", "derivation": "\\hat{H}{(\\varepsilon_0,n_{2})} = \\varepsilon_0 + n_{2} and \\varepsilon_0 + n_{2} + \\hat{H}{(\\varepsilon_0,n_{2})} = 2 \\varepsilon_0 + 2 n_{2} and 2 \\hat{H}{(\\varepsilon_0,n_{2})} = 2 \\varepsilon_0 + 2 n_{2} and \\varepsilon_0 + n_{2} + \\hat{H}{(\\varepsilon_0,n_{2})} = 2 \\hat{H}{(\\varepsilon_0,n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v_{t})} = v_{t}, then derive \\int v_{t} \\operatorname{M_{E}}{(v_{t})} dv_{t} = \\Psi^{\\dagger} + \\frac{v_{t}^{3}}{3}, then obtain \\int v_{t}^{2} dv_{t} = \\Psi^{\\dagger} + \\frac{v_{t}^{3}}{3}", "derivation": "\\operatorname{M_{E}}{(v_{t})} = v_{t} and v_{t} \\operatorname{M_{E}}{(v_{t})} = v_{t}^{2} and \\int v_{t} \\operatorname{M_{E}}{(v_{t})} dv_{t} = \\int v_{t}^{2} dv_{t} and \\int v_{t} \\operatorname{M_{E}}{(v_{t})} dv_{t} = \\Psi^{\\dagger} + \\frac{v_{t}^{3}}{3} and \\int v_{t}^{2} dv_{t} = \\Psi^{\\dagger} + \\frac{v_{t}^{3}}{3}", "srepr_derivation": [["renaming_premise", "Equality(Function('M_E')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], [["times", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('v_t', commutative=True))), Pow(Symbol('v_t', commutative=True), Integer(2)))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integral(Pow(Symbol('v_t', commutative=True), Integer(2)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Rational(1, 3), Pow(Symbol('v_t', commutative=True), Integer(3)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Pow(Symbol('v_t', commutative=True), Integer(2)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Rational(1, 3), Pow(Symbol('v_t', commutative=True), Integer(3)))))"]]}, {"prompt": "Given \\phi{(q,f_{\\mathbf{p}})} = q^{f_{\\mathbf{p}}}, then obtain \\int \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + \\phi{(q,f_{\\mathbf{p}})}) df_{\\mathbf{p}} = \\int \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + q^{f_{\\mathbf{p}}}) df_{\\mathbf{p}}", "derivation": "\\phi{(q,f_{\\mathbf{p}})} = q^{f_{\\mathbf{p}}} and - f_{\\mathbf{p}} + \\phi{(q,f_{\\mathbf{p}})} = - f_{\\mathbf{p}} + q^{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + \\phi{(q,f_{\\mathbf{p}})}) = \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + q^{f_{\\mathbf{p}}}) and \\int \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + \\phi{(q,f_{\\mathbf{p}})}) df_{\\mathbf{p}} = \\int \\frac{\\partial}{\\partial q} (- f_{\\mathbf{p}} + q^{f_{\\mathbf{p}}}) df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["minus", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\phi')(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\phi')(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\phi')(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given i{(U)} = e^{e^{U}}, then obtain (\\int e^{U} \\iint (i{(U)} - e^{e^{U}}) dU dU dU)^{U} = (\\int e^{U} \\iint 0 dU dU dU)^{U}", "derivation": "i{(U)} = e^{e^{U}} and i{(U)} - e^{e^{U}} = 0 and \\int (i{(U)} - e^{e^{U}}) dU = \\int 0 dU and \\iint (i{(U)} - e^{e^{U}}) dU dU = \\iint 0 dU dU and e^{U} \\iint (i{(U)} - e^{e^{U}}) dU dU = e^{U} \\iint 0 dU dU and \\int e^{U} \\iint (i{(U)} - e^{e^{U}}) dU dU dU = \\int e^{U} \\iint 0 dU dU dU and (\\int e^{U} \\iint (i{(U)} - e^{e^{U}}) dU dU dU)^{U} = (\\int e^{U} \\iint 0 dU dU dU)^{U}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('U', commutative=True)), exp(exp(Symbol('U', commutative=True))))"], [["minus", 1, "exp(exp(Symbol('U', commutative=True)))"], "Equality(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["times", 4, "exp(Symbol('U', commutative=True))"], "Equality(Mul(exp(Symbol('U', commutative=True)), Integral(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(exp(Symbol('U', commutative=True)), Integral(Integer(0), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(exp(Symbol('U', commutative=True)), Integral(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Integral(Mul(exp(Symbol('U', commutative=True)), Integral(Integer(0), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["power", 6, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Mul(exp(Symbol('U', commutative=True)), Integral(Add(Function('i')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Mul(exp(Symbol('U', commutative=True)), Integral(Integer(0), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given n{(J)} = \\cos{(J)}, then derive \\frac{d}{d J} n{(J)} = - \\sin{(J)}, then obtain (\\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J})^{2} = \\frac{d}{d J} (- \\sin{(J)})^{J} \\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J}", "derivation": "n{(J)} = \\cos{(J)} and \\frac{d}{d J} n{(J)} = \\frac{d}{d J} \\cos{(J)} and \\frac{d}{d J} n{(J)} = - \\sin{(J)} and (\\frac{d}{d J} n{(J)})^{J} = (\\frac{d}{d J} \\cos{(J)})^{J} and \\frac{d}{d J} \\cos{(J)} = - \\sin{(J)} and \\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J} = \\frac{d}{d J} (\\frac{d}{d J} \\cos{(J)})^{J} and \\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J} = \\frac{d}{d J} (- \\sin{(J)})^{J} and (\\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J})^{2} = \\frac{d}{d J} (- \\sin{(J)})^{J} \\frac{d}{d J} (\\frac{d}{d J} n{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('J', commutative=True))))"], [["differentiate", 4, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["times", 7, "Derivative(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(\\hat{H}_{\\lambda},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} \\hat{H}_{\\lambda} \\lambda, then derive \\sigma_{p}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda},\\lambda)} = \\hat{H}_{\\lambda}^{\\hat{H}_{\\lambda}}, then obtain - \\hat{H}_{\\lambda} \\lambda + \\sigma_{p}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda},\\lambda)} = - \\hat{H}_{\\lambda} \\lambda + \\hat{H}_{\\lambda}^{\\hat{H}_{\\lambda}}", "derivation": "\\sigma_{p}{(\\hat{H}_{\\lambda},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} \\hat{H}_{\\lambda} \\lambda and \\sigma_{p}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda},\\lambda)} = (\\frac{\\partial}{\\partial \\lambda} \\hat{H}_{\\lambda} \\lambda)^{\\hat{H}_{\\lambda}} and \\sigma_{p}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda},\\lambda)} = \\hat{H}_{\\lambda}^{\\hat{H}_{\\lambda}} and - \\hat{H}_{\\lambda} \\lambda + \\sigma_{p}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda},\\lambda)} = - \\hat{H}_{\\lambda} \\lambda + \\hat{H}_{\\lambda}^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["minus", 3, "Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given U{(T)} = e^{T}, then derive \\int U{(T)} dT = \\Psi + e^{T}, then obtain \\frac{\\Psi + U{(T)}}{\\Psi} = \\frac{\\Psi + e^{T}}{\\Psi}", "derivation": "U{(T)} = e^{T} and \\int U{(T)} dT = \\int e^{T} dT and \\int U{(T)} dT = \\Psi + e^{T} and \\frac{\\int U{(T)} dT}{\\Psi} = \\frac{\\int e^{T} dT}{\\Psi} and \\int U{(T)} dT = \\Psi + U{(T)} and \\int e^{T} dT = \\Psi + e^{T} and \\frac{\\int U{(T)} dT}{\\Psi} = \\frac{\\Psi + e^{T}}{\\Psi} and \\frac{\\Psi + U{(T)}}{\\Psi} = \\frac{\\Psi + e^{T}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('U')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('U')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('T', commutative=True))))"], [["divide", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(Function('U')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('U')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Function('U')(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(Function('U')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), Function('U')(Symbol('T', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\theta,L)} = L + \\theta, then derive \\frac{\\partial^{2}}{\\partial L^{2}} \\phi{(\\theta,L)} = 0, then obtain \\frac{\\partial^{2}}{\\partial L^{2}} (L + \\theta) = 0", "derivation": "\\phi{(\\theta,L)} = L + \\theta and \\frac{\\partial}{\\partial L} \\phi{(\\theta,L)} = \\frac{\\partial}{\\partial L} (L + \\theta) and \\frac{\\partial^{2}}{\\partial L^{2}} \\phi{(\\theta,L)} = \\frac{\\partial^{2}}{\\partial L^{2}} (L + \\theta) and \\frac{\\partial^{2}}{\\partial L^{2}} \\phi{(\\theta,L)} = 0 and \\frac{\\partial^{2}}{\\partial L^{2}} (L + \\theta) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} = \\hat{\\mathbf{r}} + \\mu and \\mathbf{B}{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain (\\hat{\\mathbf{r}} + \\mu) \\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} + \\mathbf{B}{(\\theta_2)} = (\\hat{\\mathbf{r}} + \\mu)^{2} + \\mathbf{B}{(\\theta_2)}", "derivation": "\\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} = \\hat{\\mathbf{r}} + \\mu and (\\hat{\\mathbf{r}} + \\mu) \\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} = (\\hat{\\mathbf{r}} + \\mu)^{2} and \\mathbf{B}{(\\theta_2)} = \\cos{(\\theta_2)} and (\\hat{\\mathbf{r}} + \\mu) \\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} + \\cos{(\\theta_2)} = (\\hat{\\mathbf{r}} + \\mu)^{2} + \\cos{(\\theta_2)} and (\\hat{\\mathbf{r}} + \\mu) \\hat{p}_0{(\\hat{\\mathbf{r}},\\mu)} + \\mathbf{B}{(\\theta_2)} = (\\hat{\\mathbf{r}} + \\mu)^{2} + \\mathbf{B}{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)))"], ["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["add", 2, "cos(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True))), cos(Symbol('\\\\theta_2', commutative=True))), Add(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)), cos(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True))), Add(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)), Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{E},F_{H})} = F_{H} + \\mathbf{E}, then obtain (F_{H} + \\mathbf{E})^{2} \\mathbb{I}{(\\mathbf{E},F_{H})} = (F_{H} + \\mathbf{E})^{3}", "derivation": "\\mathbb{I}{(\\mathbf{E},F_{H})} = F_{H} + \\mathbf{E} and (F_{H} + \\mathbf{E}) \\mathbb{I}{(\\mathbf{E},F_{H})} = (F_{H} + \\mathbf{E})^{2} and (F_{H} + \\mathbf{E}) \\mathbb{I}^{2}{(\\mathbf{E},F_{H})} = (F_{H} + \\mathbf{E})^{2} \\mathbb{I}{(\\mathbf{E},F_{H})} and (F_{H} + \\mathbf{E})^{2} \\mathbb{I}{(\\mathbf{E},F_{H})} = (F_{H} + \\mathbf{E})^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 1, "Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))), Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)))"], [["times", 2, "Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))), Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\mathbf{B}{(l)} = \\cos{(l)}, then derive - \\mathbf{B}{(l)} = - \\mathbf{B}{(l)} + \\frac{\\sin{(l)}}{\\mathbf{B}{(l)}} + \\frac{\\cos{(l)} \\frac{d}{d l} \\mathbf{B}{(l)}}{\\mathbf{B}^{2}{(l)}}, then obtain - \\mathbf{B}{(l)} - \\frac{d}{d l} \\mathbf{B}{(l)} = - \\mathbf{B}{(l)} - \\frac{d}{d l} \\mathbf{B}{(l)} + \\frac{\\sin{(l)}}{\\mathbf{B}{(l)}} + \\frac{\\cos{(l)} \\frac{d}{d l} \\mathbf{B}{(l)}}{\\mathbf{B}^{2}{(l)}}", "derivation": "\\mathbf{B}{(l)} = \\cos{(l)} and -1 = - \\frac{\\cos{(l)}}{\\mathbf{B}{(l)}} and \\frac{d}{d l} (-1) = \\frac{d}{d l} - \\frac{\\cos{(l)}}{\\mathbf{B}{(l)}} and - \\mathbf{B}{(l)} + \\frac{d}{d l} (-1) = - \\mathbf{B}{(l)} + \\frac{d}{d l} - \\frac{\\cos{(l)}}{\\mathbf{B}{(l)}} and - \\mathbf{B}{(l)} = - \\mathbf{B}{(l)} + \\frac{\\sin{(l)}}{\\mathbf{B}{(l)}} + \\frac{\\cos{(l)} \\frac{d}{d l} \\mathbf{B}{(l)}}{\\mathbf{B}^{2}{(l)}} and - \\mathbf{B}{(l)} - \\frac{d}{d l} \\mathbf{B}{(l)} = - \\mathbf{B}{(l)} - \\frac{d}{d l} \\mathbf{B}{(l)} + \\frac{\\sin{(l)}}{\\mathbf{B}{(l)}} + \\frac{\\cos{(l)} \\frac{d}{d l} \\mathbf{B}{(l)}}{\\mathbf{B}^{2}{(l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-1)), cos(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-1)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Derivative(Integer(-1), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Derivative(Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-1)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-2)), cos(Symbol('l', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["minus", 5, "Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-1)), sin(Symbol('l', commutative=True))), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Integer(-2)), cos(Symbol('l', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"]]}, {"prompt": "Given v{(v_{2},U)} = U v_{2}, then derive \\frac{\\partial}{\\partial U} v{(v_{2},U)} = v_{2}, then derive \\int \\frac{\\partial^{2}}{\\partial U^{2}} v{(v_{2},U)} dU = \\int 0 dU, then derive A_{z} + \\frac{\\partial}{\\partial U} v{(v_{2},U)} = \\int 0 dU, then obtain \\frac{\\partial}{\\partial A_{z}} (A_{z} + v_{2}) = \\frac{d}{d A_{z}} \\int 0 dU", "derivation": "v{(v_{2},U)} = U v_{2} and \\frac{\\partial}{\\partial U} v{(v_{2},U)} = \\frac{\\partial}{\\partial U} U v_{2} and \\frac{\\partial}{\\partial U} v{(v_{2},U)} = v_{2} and \\frac{\\partial^{2}}{\\partial U^{2}} v{(v_{2},U)} = \\frac{d}{d U} v_{2} and \\int \\frac{\\partial^{2}}{\\partial U^{2}} v{(v_{2},U)} dU = \\int \\frac{d}{d U} v_{2} dU and \\int \\frac{\\partial^{2}}{\\partial U^{2}} v{(v_{2},U)} dU = \\int 0 dU and A_{z} + \\frac{\\partial}{\\partial U} v{(v_{2},U)} = \\int 0 dU and \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\frac{\\partial}{\\partial U} v{(v_{2},U)}) = \\frac{d}{d A_{z}} \\int 0 dU and \\frac{\\partial}{\\partial A_{z}} (A_{z} + v_{2}) = \\frac{d}{d A_{z}} \\int 0 dU", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('v_2', commutative=True))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2))), Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integral(Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('A_z', commutative=True), Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 7, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('A_z', commutative=True), Derivative(Function('v')(Symbol('v_2', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 3], "Equality(Derivative(Add(Symbol('A_z', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(\\dot{x},A_{2})} = \\sin{(A_{2} + \\dot{x})} and \\bar{\\h}{(\\dot{x},A_{2})} = A_{2} + \\dot{x}, then obtain \\cos{(\\frac{\\sin{(\\bar{\\h}{(\\dot{x},A_{2})})}}{\\sin{(A_{2} + \\dot{x})}})} = \\cos{(1)}", "derivation": "\\theta{(\\dot{x},A_{2})} = \\sin{(A_{2} + \\dot{x})} and \\frac{\\theta{(\\dot{x},A_{2})}}{\\sin{(A_{2} + \\dot{x})}} = 1 and \\bar{\\h}{(\\dot{x},A_{2})} = A_{2} + \\dot{x} and \\theta{(\\dot{x},A_{2})} = \\sin{(\\bar{\\h}{(\\dot{x},A_{2})})} and \\frac{\\sin{(\\bar{\\h}{(\\dot{x},A_{2})})}}{\\sin{(A_{2} + \\dot{x})}} = 1 and \\cos{(\\frac{\\sin{(\\bar{\\h}{(\\dot{x},A_{2})})}}{\\sin{(A_{2} + \\dot{x})}})} = \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True)), sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 1, "sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True)), Pow(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True)), sin(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), sin(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True)))), Integer(1))"], [["cos", 5], "Equality(cos(Mul(Pow(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), sin(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_2', commutative=True))))), cos(Integer(1)))"]]}, {"prompt": "Given \\theta{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then obtain \\theta^{4}{(\\Psi_{nl})} = e^{4 \\Psi_{nl}}", "derivation": "\\theta{(\\Psi_{nl})} = e^{\\Psi_{nl}} and \\theta{(\\Psi_{nl})} e^{\\Psi_{nl}} = e^{2 \\Psi_{nl}} and \\theta^{2}{(\\Psi_{nl})} = \\theta{(\\Psi_{nl})} e^{\\Psi_{nl}} and \\theta^{2}{(\\Psi_{nl})} = e^{2 \\Psi_{nl}} and \\theta^{4}{(\\Psi_{nl})} = e^{4 \\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["times", 1, "Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Mul(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(4)), exp(Mul(Integer(4), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given i{(\\varepsilon)} = \\sin{(\\varepsilon)}, then obtain \\frac{d}{d \\varepsilon} (\\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + 1) = \\frac{d}{d \\varepsilon} (\\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + i^{- \\varepsilon}{(\\varepsilon)} \\sin^{\\varepsilon}{(\\varepsilon)})", "derivation": "i{(\\varepsilon)} = \\sin{(\\varepsilon)} and i^{\\varepsilon}{(\\varepsilon)} = \\sin^{\\varepsilon}{(\\varepsilon)} and 1 = i^{- \\varepsilon}{(\\varepsilon)} \\sin^{\\varepsilon}{(\\varepsilon)} and \\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + 1 = \\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + i^{- \\varepsilon}{(\\varepsilon)} \\sin^{\\varepsilon}{(\\varepsilon)} and \\frac{d}{d \\varepsilon} (\\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + 1) = \\frac{d}{d \\varepsilon} (\\int i^{\\varepsilon}{(\\varepsilon)} d\\varepsilon + i^{- \\varepsilon}{(\\varepsilon)} \\sin^{\\varepsilon}{(\\varepsilon)})", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(sin(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 2, "Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Pow(sin(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"], [["add", 3, "Integral(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Integral(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(1)), Add(Integral(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Pow(sin(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Integral(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(1)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Integral(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Function('i')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Pow(sin(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(C)} = \\sin{(C)}, then obtain m{(C)} + 1 + \\frac{m{(C)}}{C} = \\sin{(C)} + 1 + \\frac{m{(C)}}{C}", "derivation": "m{(C)} = \\sin{(C)} and \\frac{m{(C)}}{C} = \\frac{\\sin{(C)}}{C} and m{(C)} + \\frac{\\sin{(C)}}{C} = \\sin{(C)} + \\frac{\\sin{(C)}}{C} and m{(C)} + 1 + \\frac{\\sin{(C)}}{C} = \\sin{(C)} + 1 + \\frac{\\sin{(C)}}{C} and m{(C)} + 1 + \\frac{m{(C)}}{C} = \\sin{(C)} + 1 + \\frac{m{(C)}}{C}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["divide", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('m')(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True)))"], "Equality(Add(Function('m')(Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True)))), Add(sin(Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('m')(Symbol('C', commutative=True)), Integer(1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True)))), Add(sin(Symbol('C', commutative=True)), Integer(1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('m')(Symbol('C', commutative=True)), Integer(1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('m')(Symbol('C', commutative=True)))), Add(sin(Symbol('C', commutative=True)), Integer(1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('m')(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given B{(Z,\\hat{X})} = \\cos{(Z^{\\hat{X}})}, then obtain \\int (\\frac{B{(Z,\\hat{X})}}{\\cos{(Z^{\\hat{X}})}})^{Z} dZ = \\int 1 dZ", "derivation": "B{(Z,\\hat{X})} = \\cos{(Z^{\\hat{X}})} and \\frac{B{(Z,\\hat{X})}}{\\cos{(Z^{\\hat{X}})}} = 1 and (\\frac{B{(Z,\\hat{X})}}{\\cos{(Z^{\\hat{X}})}})^{Z} = 1 and \\int (\\frac{B{(Z,\\hat{X})}}{\\cos{(Z^{\\hat{X}})}})^{Z} dZ = \\int 1 dZ", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)), cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 1, "cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Function('B')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Function('B')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Pow(Mul(Function('B')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Integer(1), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given c{(v_{y})} = \\log{(v_{y})}, then obtain \\frac{d}{d v_{y}} \\log{(v_{y})} = \\frac{d}{d v_{y}} \\frac{\\log{(v_{y})}^{4}}{c^{3}{(v_{y})}}", "derivation": "c{(v_{y})} = \\log{(v_{y})} and 1 = \\frac{\\log{(v_{y})}}{c{(v_{y})}} and \\log{(v_{y})} = \\frac{\\log{(v_{y})}^{2}}{c{(v_{y})}} and \\frac{d}{d v_{y}} \\log{(v_{y})} = \\frac{d}{d v_{y}} \\frac{\\log{(v_{y})}^{2}}{c{(v_{y})}} and \\frac{d}{d v_{y}} \\frac{\\log{(v_{y})}^{2}}{c{(v_{y})}} = \\frac{d}{d v_{y}} \\frac{\\log{(v_{y})}^{4}}{c^{3}{(v_{y})}} and \\frac{d}{d v_{y}} \\log{(v_{y})} = \\frac{d}{d v_{y}} \\frac{\\log{(v_{y})}^{4}}{c^{3}{(v_{y})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["divide", 1, "Function('c')(Symbol('v_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-1)), log(Symbol('v_y', commutative=True))))"], [["divide", 2, "Pow(log(Symbol('v_y', commutative=True)), Integer(-1))"], "Equality(log(Symbol('v_y', commutative=True)), Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-1)), Pow(log(Symbol('v_y', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-1)), Pow(log(Symbol('v_y', commutative=True)), Integer(2))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-1)), Pow(log(Symbol('v_y', commutative=True)), Integer(2))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-3)), Pow(log(Symbol('v_y', commutative=True)), Integer(4))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('c')(Symbol('v_y', commutative=True)), Integer(-3)), Pow(log(Symbol('v_y', commutative=True)), Integer(4))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})} = \\sin{(\\mathbf{P} g^{\\prime}_{\\varepsilon})}, then obtain - \\frac{\\frac{\\partial}{\\partial \\mathbf{P}} \\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})}}{\\mathbf{P} g^{\\prime}_{\\varepsilon}} = - \\frac{\\cos{(\\mathbf{P} g^{\\prime}_{\\varepsilon})}}{\\mathbf{P}}", "derivation": "\\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})} = \\sin{(\\mathbf{P} g^{\\prime}_{\\varepsilon})} and \\frac{\\partial}{\\partial \\mathbf{P}} \\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} g^{\\prime}_{\\varepsilon})} and - \\frac{\\frac{\\partial}{\\partial \\mathbf{P}} \\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})}}{\\mathbf{P} g^{\\prime}_{\\varepsilon}} = - \\frac{\\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} g^{\\prime}_{\\varepsilon})}}{\\mathbf{P} g^{\\prime}_{\\varepsilon}} and - \\frac{\\frac{\\partial}{\\partial \\mathbf{P}} \\phi_{2}{(\\mathbf{P},g^{\\prime}_{\\varepsilon})}}{\\mathbf{P} g^{\\prime}_{\\varepsilon}} = - \\frac{\\cos{(\\mathbf{P} g^{\\prime}_{\\varepsilon})}}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\dot{x})} = \\cos{(\\dot{x})}, then obtain \\frac{4 \\tilde{g}^*^{2}{(\\dot{x})}}{\\cos^{2}{(\\dot{x})}} = \\frac{(\\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})})^{2}}{\\cos^{2}{(\\dot{x})}}", "derivation": "\\tilde{g}^*{(\\dot{x})} = \\cos{(\\dot{x})} and 2 \\tilde{g}^*{(\\dot{x})} = \\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})} and \\frac{2 \\tilde{g}^*{(\\dot{x})}}{\\cos{(\\dot{x})}} = \\frac{\\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})}}{\\cos{(\\dot{x})}} and \\frac{4 \\tilde{g}^*^{2}{(\\dot{x})}}{\\cos^{2}{(\\dot{x})}} = \\frac{(\\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})})^{2}}{\\cos^{2}{(\\dot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 2, "cos(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True))), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-2))), Mul(Pow(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True))), Integer(2)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given T{(h,v)} = \\frac{h}{v}, then obtain (v (v + T{(h,v)}) - \\cos{(\\frac{h}{v} + v)})^{v} = (v (\\frac{h}{v} + v) - \\cos{(\\frac{h}{v} + v)})^{v}", "derivation": "T{(h,v)} = \\frac{h}{v} and v + T{(h,v)} = \\frac{h}{v} + v and v (v + T{(h,v)}) = v (\\frac{h}{v} + v) and \\cos{(v + T{(h,v)})} = \\cos{(\\frac{h}{v} + v)} and v (v + T{(h,v)}) - \\cos{(\\frac{h}{v} + v)} = v (\\frac{h}{v} + v) - \\cos{(\\frac{h}{v} + v)} and v (v + T{(h,v)}) - \\cos{(v + T{(h,v)})} = v (\\frac{h}{v} + v) - \\cos{(v + T{(h,v)})} and (v (v + T{(h,v)}) - \\cos{(v + T{(h,v)})})^{v} = (v (\\frac{h}{v} + v) - \\cos{(v + T{(h,v)})})^{v} and (v (v + T{(h,v)}) - \\cos{(\\frac{h}{v} + v)})^{v} = (v (\\frac{h}{v} + v) - \\cos{(\\frac{h}{v} + v)})^{v}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True)))"], [["divide", 2, "Pow(Symbol('v', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))"], [["minus", 3, "cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(-1), cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))), Add(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))), Mul(Integer(-1), cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(-1), cos(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))))), Add(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))))))"], [["power", 6, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(-1), cos(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))))), Symbol('v', commutative=True)), Pow(Add(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(Add(Mul(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(-1), cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))), Symbol('v', commutative=True)), Pow(Add(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))), Mul(Integer(-1), cos(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('v', commutative=True))))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(u,n_{2})} = - n_{2} + u, then obtain (\\int \\frac{\\hat{H}_{\\lambda}{(u,n_{2})}}{- n_{2} + u} dn_{2} - 1) \\operatorname{F_{c}}{(\\mathbf{H},\\rho_f)} = (\\int 1 dn_{2} - 1) \\operatorname{F_{c}}{(\\mathbf{H},\\rho_f)}", "derivation": "\\hat{H}_{\\lambda}{(u,n_{2})} = - n_{2} + u and \\frac{\\hat{H}_{\\lambda}{(u,n_{2})}}{- n_{2} + u} = 1 and \\int \\frac{\\hat{H}_{\\lambda}{(u,n_{2})}}{- n_{2} + u} dn_{2} = \\int 1 dn_{2} and \\int \\frac{\\hat{H}_{\\lambda}{(u,n_{2})}}{- n_{2} + u} dn_{2} - 1 = \\int 1 dn_{2} - 1 and (\\int \\frac{\\hat{H}_{\\lambda}{(u,n_{2})}}{- n_{2} + u} dn_{2} - 1) \\operatorname{F_{c}}{(\\mathbf{H},\\rho_f)} = (\\int 1 dn_{2} - 1) \\operatorname{F_{c}}{(\\mathbf{H},\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('n_2', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('n_2', commutative=True))), Integer(-1)))"], [["times", 4, "Function('F_c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Add(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1)), Function('F_c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Add(Integral(Integer(1), Tuple(Symbol('n_2', commutative=True))), Integer(-1)), Function('F_c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given q{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then obtain q{(\\mathbf{E})} + \\log{(\\mathbf{E})}^{\\mathbf{E}} = \\log{(\\mathbf{E})} + \\log{(\\mathbf{E})}^{\\mathbf{E}}", "derivation": "q{(\\mathbf{E})} = \\log{(\\mathbf{E})} and q^{\\mathbf{E}}{(\\mathbf{E})} = \\log{(\\mathbf{E})}^{\\mathbf{E}} and q{(\\mathbf{E})} + q^{\\mathbf{E}}{(\\mathbf{E})} = q^{\\mathbf{E}}{(\\mathbf{E})} + \\log{(\\mathbf{E})} and q{(\\mathbf{E})} + \\log{(\\mathbf{E})}^{\\mathbf{E}} = \\log{(\\mathbf{E})} + \\log{(\\mathbf{E})}^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Add(Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Add(log(Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\rho)} = e^{\\rho}, then derive \\int \\hat{X}{(\\rho)} d\\rho = \\Psi_{\\lambda} + e^{\\rho}, then obtain \\int \\hat{X}{(\\rho)} d\\rho = \\Psi_{\\lambda} + \\hat{X}{(\\rho)}", "derivation": "\\hat{X}{(\\rho)} = e^{\\rho} and \\int \\hat{X}{(\\rho)} d\\rho = \\int e^{\\rho} d\\rho and \\int \\hat{X}{(\\rho)} d\\rho = \\Psi_{\\lambda} + e^{\\rho} and \\int \\hat{X}{(\\rho)} d\\rho = \\Psi_{\\lambda} + \\hat{X}{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(M_{E})} = \\sin{(M_{E})}, then obtain \\iint (\\sigma_{x}{(M_{E})} - \\sin{(M_{E})}) dM_{E} dM_{E} = \\iint 0 dM_{E} dM_{E}", "derivation": "\\sigma_{x}{(M_{E})} = \\sin{(M_{E})} and \\sigma_{x}{(M_{E})} - \\sin{(M_{E})} = 0 and \\int (\\sigma_{x}{(M_{E})} - \\sin{(M_{E})}) dM_{E} = \\int 0 dM_{E} and \\iint (\\sigma_{x}{(M_{E})} - \\sin{(M_{E})}) dM_{E} dM_{E} = \\iint 0 dM_{E} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["minus", 1, "sin(Symbol('M_E', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Symbol('M_E', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\sigma_x')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\sigma_x')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(E_{x},r,\\sigma_x)} = E_{x} - \\sigma_x + r and q{(E_{x},r,\\sigma_x)} = \\log{(E_{x} - \\sigma_x + r)}, then obtain - \\sigma_x q{(E_{x},r,\\sigma_x)} + \\sigma_x = - \\sigma_x \\log{(E_{x} - \\sigma_x + r)} + \\sigma_x", "derivation": "\\operatorname{A_{x}}{(E_{x},r,\\sigma_x)} = E_{x} - \\sigma_x + r and \\log{(\\operatorname{A_{x}}{(E_{x},r,\\sigma_x)})} = \\log{(E_{x} - \\sigma_x + r)} and - \\sigma_x \\log{(\\operatorname{A_{x}}{(E_{x},r,\\sigma_x)})} = - \\sigma_x \\log{(E_{x} - \\sigma_x + r)} and q{(E_{x},r,\\sigma_x)} = \\log{(E_{x} - \\sigma_x + r)} and \\log{(\\operatorname{A_{x}}{(E_{x},r,\\sigma_x)})} = q{(E_{x},r,\\sigma_x)} and - \\sigma_x q{(E_{x},r,\\sigma_x)} = - \\sigma_x \\log{(E_{x} - \\sigma_x + r)} and - \\sigma_x q{(E_{x},r,\\sigma_x)} + \\sigma_x = - \\sigma_x \\log{(E_{x} - \\sigma_x + r)} + \\sigma_x", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True)))"], [["log", 1], "Equality(log(Function('A_x')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True))), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Function('A_x')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True)), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(log(Function('A_x')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Function('q')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('q')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True)))))"], [["add", 6, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('q')(Symbol('E_x', commutative=True), Symbol('r', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('r', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(x,\\lambda)} = \\lambda + x, then obtain - \\lambda \\mathbf{A}{(x,\\lambda)} + \\cos{((\\lambda + x) \\mathbf{A}{(x,\\lambda)})} = - \\lambda \\mathbf{A}{(x,\\lambda)} + \\cos{((\\lambda + x)^{2})}", "derivation": "\\mathbf{A}{(x,\\lambda)} = \\lambda + x and (\\lambda + x) \\mathbf{A}{(x,\\lambda)} = (\\lambda + x)^{2} and \\cos{((\\lambda + x) \\mathbf{A}{(x,\\lambda)})} = \\cos{((\\lambda + x)^{2})} and \\lambda \\mathbf{A}{(x,\\lambda)} = \\lambda (\\lambda + x) and - \\lambda (\\lambda + x) + \\cos{((\\lambda + x) \\mathbf{A}{(x,\\lambda)})} = - \\lambda (\\lambda + x) + \\cos{((\\lambda + x)^{2})} and - \\lambda \\mathbf{A}{(x,\\lambda)} + \\cos{((\\lambda + x) \\mathbf{A}{(x,\\lambda)})} = - \\lambda \\mathbf{A}{(x,\\lambda)} + \\cos{((\\lambda + x)^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))), Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Integer(2)))"], [["cos", 2], "Equality(cos(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True)))), cos(Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Integer(2))))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True))), cos(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True))), cos(Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))), cos(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('\\\\lambda', commutative=True))), cos(Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('x', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given k{(r,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + r, then obtain (2 r + \\int (V_{\\mathbf{E}} + r) dr) \\iint k{(r,V_{\\mathbf{E}})} dr dV_{\\mathbf{E}} = (2 r + \\int (V_{\\mathbf{E}} + r) dr) \\iint (V_{\\mathbf{E}} + r) dr dV_{\\mathbf{E}}", "derivation": "k{(r,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + r and \\int k{(r,V_{\\mathbf{E}})} dr = \\int (V_{\\mathbf{E}} + r) dr and \\iint k{(r,V_{\\mathbf{E}})} dr dV_{\\mathbf{E}} = \\iint (V_{\\mathbf{E}} + r) dr dV_{\\mathbf{E}} and (2 r + \\int (V_{\\mathbf{E}} + r) dr) \\iint k{(r,V_{\\mathbf{E}})} dr dV_{\\mathbf{E}} = (2 r + \\int (V_{\\mathbf{E}} + r) dr) \\iint (V_{\\mathbf{E}} + r) dr dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('k')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 3, "Add(Mul(Integer(2), Symbol('r', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('r', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Integral(Function('k')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Mul(Integer(2), Symbol('r', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\rho{(f_{\\mathbf{p}})} = (\\theta_{1}^{2}{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}}, then obtain \\rho{(f_{\\mathbf{p}})} = (\\cos^{2}{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}}", "derivation": "\\theta_{1}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\theta_{1}^{2}{(f_{\\mathbf{p}})} = \\theta_{1}{(f_{\\mathbf{p}})} \\cos{(f_{\\mathbf{p}})} and (\\theta_{1}^{2}{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} = (\\theta_{1}{(f_{\\mathbf{p}})} \\cos{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} and \\rho{(f_{\\mathbf{p}})} = (\\theta_{1}^{2}{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} and \\rho{(f_{\\mathbf{p}})} = (\\theta_{1}{(f_{\\mathbf{p}})} \\cos{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} and \\rho{(f_{\\mathbf{p}})} = (\\cos^{2}{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Pow(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Mul(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Mul(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Pow(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\rho')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Mul(Function('\\\\theta_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\rho')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Pow(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(q,\\lambda)} = \\lambda + q, then obtain \\frac{\\sin{(2 \\phi_{2}{(q,\\lambda)})}}{\\lambda + q + \\phi_{2}{(q,\\lambda)}} = \\frac{\\sin{(\\lambda + q + \\phi_{2}{(q,\\lambda)})}}{\\lambda + q + \\phi_{2}{(q,\\lambda)}}", "derivation": "\\phi_{2}{(q,\\lambda)} = \\lambda + q and 2 \\phi_{2}{(q,\\lambda)} = \\lambda + q + \\phi_{2}{(q,\\lambda)} and \\sin{(2 \\phi_{2}{(q,\\lambda)})} = \\sin{(\\lambda + q + \\phi_{2}{(q,\\lambda)})} and \\frac{\\sin{(2 \\phi_{2}{(q,\\lambda)})}}{\\lambda + q + \\phi_{2}{(q,\\lambda)}} = \\frac{\\sin{(\\lambda + q + \\phi_{2}{(q,\\lambda)})}}{\\lambda + q + \\phi_{2}{(q,\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)))"], [["add", 1, "Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)))), sin(Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1)), sin(Mul(Integer(2), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))))), Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1)), sin(Add(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}_0{(\\psi)} = \\log{(e^{\\psi})} and \\operatorname{a^{\\dagger}}{(\\psi)} = e^{\\psi}, then obtain \\hat{p}_0{(\\psi)} \\operatorname{a^{\\dagger}}{(\\psi)} = \\operatorname{a^{\\dagger}}{(\\psi)} \\log{(\\operatorname{a^{\\dagger}}{(\\psi)})}", "derivation": "\\hat{p}_0{(\\psi)} = \\log{(e^{\\psi})} and \\hat{p}_0{(\\psi)} e^{\\psi} = e^{\\psi} \\log{(e^{\\psi})} and \\operatorname{a^{\\dagger}}{(\\psi)} = e^{\\psi} and \\hat{p}_0{(\\psi)} \\operatorname{a^{\\dagger}}{(\\psi)} = \\operatorname{a^{\\dagger}}{(\\psi)} \\log{(\\operatorname{a^{\\dagger}}{(\\psi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), log(exp(Symbol('\\\\psi', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True))), Mul(exp(Symbol('\\\\psi', commutative=True)), log(exp(Symbol('\\\\psi', commutative=True)))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True))), Mul(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), log(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(t)} = t, then derive \\frac{\\hat{x} + t \\mathbf{S}{(t)} + t + \\frac{\\mathbf{S}^{2}{(t)}}{2}}{\\int 2 t d\\mathbf{S}{(t)}} = \\frac{t + \\int 2 t d\\mathbf{S}{(t)}}{\\int 2 t d\\mathbf{S}{(t)}}, then obtain \\frac{\\hat{x} + \\frac{3 t^{2}}{2} + t}{\\int 2 t dt} = \\frac{t + \\int 2 t dt}{\\int 2 t dt}", "derivation": "\\mathbf{S}{(t)} = t and t + \\mathbf{S}{(t)} = 2 t and \\int (t + \\mathbf{S}{(t)}) dt = \\int 2 t dt and \\int (t + \\mathbf{S}{(t)}) d\\mathbf{S}{(t)} = \\int 2 t d\\mathbf{S}{(t)} and t + \\int (t + \\mathbf{S}{(t)}) d\\mathbf{S}{(t)} = t + \\int 2 t d\\mathbf{S}{(t)} and \\frac{t + \\int (t + \\mathbf{S}{(t)}) d\\mathbf{S}{(t)}}{\\int 2 t d\\mathbf{S}{(t)}} = \\frac{t + \\int 2 t d\\mathbf{S}{(t)}}{\\int 2 t d\\mathbf{S}{(t)}} and \\frac{\\hat{x} + t \\mathbf{S}{(t)} + t + \\frac{\\mathbf{S}^{2}{(t)}}{2}}{\\int 2 t d\\mathbf{S}{(t)}} = \\frac{t + \\int 2 t d\\mathbf{S}{(t)}}{\\int 2 t d\\mathbf{S}{(t)}} and \\frac{\\hat{x} + \\frac{3 t^{2}}{2} + t}{\\int 2 t dt} = \\frac{t + \\int 2 t dt}{\\int 2 t dt}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)), Symbol('t', commutative=True))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Mul(Integer(2), Symbol('t', commutative=True)))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Add(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))))"], [["add", 4, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Integral(Add(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))), Add(Symbol('t', commutative=True), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))))"], [["divide", 5, "Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))"], "Equality(Mul(Add(Symbol('t', commutative=True), Integral(Add(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))), Integer(-1))), Mul(Add(Symbol('t', commutative=True), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))), Integer(-1))))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('t', commutative=True), Function('\\\\mathbf{S}')(Symbol('t', commutative=True))), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)), Integer(2)))), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))), Integer(-1))), Mul(Add(Symbol('t', commutative=True), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True))))), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{S}')(Symbol('t', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Rational(3, 2), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('t', commutative=True)), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(-1))), Mul(Add(Symbol('t', commutative=True), Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Pow(Integral(Mul(Integer(2), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{1}{(M_{E},\\mathbf{v})} = e^{- M_{E} + \\mathbf{v}} and \\mathbf{H}{(M_{E},\\mathbf{v})} = - M_{E} + \\mathbf{v}, then obtain \\theta_{1}{(M_{E},\\mathbf{v})} + 1 = e^{\\mathbf{H}{(M_{E},\\mathbf{v})}} + 1", "derivation": "\\theta_{1}{(M_{E},\\mathbf{v})} = e^{- M_{E} + \\mathbf{v}} and \\theta_{1}{(M_{E},\\mathbf{v})} + 1 = e^{- M_{E} + \\mathbf{v}} + 1 and \\mathbf{H}{(M_{E},\\mathbf{v})} = - M_{E} + \\mathbf{v} and \\theta_{1}{(M_{E},\\mathbf{v})} + 1 = e^{\\mathbf{H}{(M_{E},\\mathbf{v})}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta_1')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)), Add(exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\theta_1')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)), Add(exp(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(v_{1},\\Psi_{nl})} = \\frac{v_{1}}{\\Psi_{nl}} and \\operatorname{a^{\\dagger}}{(v_{1},\\Psi_{nl})} = (\\frac{v_{1}}{\\Psi_{nl}})^{- v_{1}}, then obtain \\int \\operatorname{a^{\\dagger}}{(v_{1},\\Psi_{nl})} d\\Psi_{nl} = \\int \\operatorname{E_{n}}^{- v_{1}}{(v_{1},\\Psi_{nl})} d\\Psi_{nl}", "derivation": "\\operatorname{E_{n}}{(v_{1},\\Psi_{nl})} = \\frac{v_{1}}{\\Psi_{nl}} and \\operatorname{E_{n}}^{v_{1}}{(v_{1},\\Psi_{nl})} = (\\frac{v_{1}}{\\Psi_{nl}})^{v_{1}} and (\\frac{v_{1}}{\\Psi_{nl}})^{- v_{1}} = \\operatorname{E_{n}}^{- v_{1}}{(v_{1},\\Psi_{nl})} and \\operatorname{a^{\\dagger}}{(v_{1},\\Psi_{nl})} = (\\frac{v_{1}}{\\Psi_{nl}})^{- v_{1}} and \\operatorname{a^{\\dagger}}{(v_{1},\\Psi_{nl})} = \\operatorname{E_{n}}^{- v_{1}}{(v_{1},\\Psi_{nl})} and \\int \\operatorname{a^{\\dagger}}{(v_{1},\\Psi_{nl})} d\\Psi_{nl} = \\int \\operatorname{E_{n}}^{- v_{1}}{(v_{1},\\Psi_{nl})} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('v_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["divide", 2, "Mul(Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('v_1', commutative=True)))"], "Equality(Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Pow(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["integrate", 5, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Function('E_n')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\delta{(m_{s})} = \\cos{(m_{s})}, then derive \\delta{(m_{s})} \\int \\delta{(m_{s})} dm_{s} = (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})}, then derive (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} = (\\mathbf{S} + \\sin{(m_{s})}) \\delta{(m_{s})}, then obtain \\int (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} d\\mathbf{S} = \\int (\\mathbf{S} + \\sin{(m_{s})}) \\delta{(m_{s})} d\\mathbf{S}", "derivation": "\\delta{(m_{s})} = \\cos{(m_{s})} and \\int \\delta{(m_{s})} dm_{s} = \\int \\cos{(m_{s})} dm_{s} and \\delta{(m_{s})} \\int \\delta{(m_{s})} dm_{s} = \\delta{(m_{s})} \\int \\cos{(m_{s})} dm_{s} and \\delta{(m_{s})} \\int \\delta{(m_{s})} dm_{s} = (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} and (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} = \\delta{(m_{s})} \\int \\cos{(m_{s})} dm_{s} and (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} = (\\mathbf{S} + \\sin{(m_{s})}) \\delta{(m_{s})} and \\int (v_{2} + \\sin{(m_{s})}) \\delta{(m_{s})} d\\mathbf{S} = \\int (\\mathbf{S} + \\sin{(m_{s})}) \\delta{(m_{s})} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('m_s', commutative=True)), cos(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["times", 2, "Function('\\\\delta')(Symbol('m_s', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('m_s', commutative=True)), Integral(Function('\\\\delta')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Function('\\\\delta')(Symbol('m_s', commutative=True)), Integral(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('\\\\delta')(Symbol('m_s', commutative=True)), Integral(Function('\\\\delta')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Add(Symbol('v_2', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('v_2', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))), Mul(Function('\\\\delta')(Symbol('m_s', commutative=True)), Integral(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('v_2', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))))"], [["integrate", 6, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('v_2', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('m_s', commutative=True))), Function('\\\\delta')(Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\Omega)} = e^{\\Omega} and g{(\\Omega)} = (((e^{\\operatorname{z^{*}}{(\\Omega)}})^{\\Omega})^{\\Omega})^{\\Omega}, then obtain \\int g{(\\Omega)} d\\Omega = \\int (((e^{e^{\\Omega}})^{\\Omega})^{\\Omega})^{\\Omega} d\\Omega", "derivation": "\\operatorname{z^{*}}{(\\Omega)} = e^{\\Omega} and e^{\\operatorname{z^{*}}{(\\Omega)}} = e^{e^{\\Omega}} and (e^{\\operatorname{z^{*}}{(\\Omega)}})^{\\Omega} = (e^{e^{\\Omega}})^{\\Omega} and ((e^{\\operatorname{z^{*}}{(\\Omega)}})^{\\Omega})^{\\Omega} = ((e^{e^{\\Omega}})^{\\Omega})^{\\Omega} and (((e^{\\operatorname{z^{*}}{(\\Omega)}})^{\\Omega})^{\\Omega})^{\\Omega} = (((e^{e^{\\Omega}})^{\\Omega})^{\\Omega})^{\\Omega} and g{(\\Omega)} = (((e^{\\operatorname{z^{*}}{(\\Omega)}})^{\\Omega})^{\\Omega})^{\\Omega} and g{(\\Omega)} = (((e^{e^{\\Omega}})^{\\Omega})^{\\Omega})^{\\Omega} and \\int g{(\\Omega)} d\\Omega = \\int (((e^{e^{\\Omega}})^{\\Omega})^{\\Omega})^{\\Omega} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["exp", 1], "Equality(exp(Function('z^*')(Symbol('\\\\Omega', commutative=True))), exp(exp(Symbol('\\\\Omega', commutative=True))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(exp(Function('z^*')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(exp(exp(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(exp(Function('z^*')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(exp(exp(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Pow(exp(Function('z^*')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Pow(exp(exp(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\Omega', commutative=True)), Pow(Pow(Pow(exp(Function('z^*')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('g')(Symbol('\\\\Omega', commutative=True)), Pow(Pow(Pow(exp(exp(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 7, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Pow(Pow(Pow(exp(exp(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\Psi_{\\lambda},\\mathbf{S})} = \\frac{\\mathbf{S}}{\\Psi_{\\lambda}} and \\hat{\\mathbf{r}}{(\\mathbf{S})} = \\mathbf{S}^{2}, then obtain \\frac{\\mathbf{S} \\operatorname{A_{z}}{(\\Psi_{\\lambda},\\mathbf{S})}}{\\Psi_{\\lambda}} = \\frac{\\hat{\\mathbf{r}}{(\\mathbf{S})}}{\\Psi_{\\lambda}^{2}}", "derivation": "\\operatorname{A_{z}}{(\\Psi_{\\lambda},\\mathbf{S})} = \\frac{\\mathbf{S}}{\\Psi_{\\lambda}} and \\frac{\\mathbf{S} \\operatorname{A_{z}}{(\\Psi_{\\lambda},\\mathbf{S})}}{\\Psi_{\\lambda}} = \\frac{\\mathbf{S}^{2}}{\\Psi_{\\lambda}^{2}} and \\hat{\\mathbf{r}}{(\\mathbf{S})} = \\mathbf{S}^{2} and \\frac{\\mathbf{S} \\operatorname{A_{z}}{(\\Psi_{\\lambda},\\mathbf{S})}}{\\Psi_{\\lambda}} = \\frac{\\hat{\\mathbf{r}}{(\\mathbf{S})}}{\\Psi_{\\lambda}^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True), Function('A_z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True), Function('A_z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{s})} = \\mathbf{s}, then obtain \\int \\mathbf{p}^{\\mathbf{s}}{(\\mathbf{s})} d\\mathbf{p}{(\\mathbf{s})} = \\int \\mathbf{s}^{\\mathbf{s}} d\\mathbf{p}{(\\mathbf{s})}", "derivation": "\\mathbf{p}{(\\mathbf{s})} = \\mathbf{s} and \\mathbf{p}^{\\mathbf{s}}{(\\mathbf{s})} = \\mathbf{s}^{\\mathbf{s}} and \\int \\mathbf{p}^{\\mathbf{s}}{(\\mathbf{s})} d\\mathbf{s} = \\int \\mathbf{s}^{\\mathbf{s}} d\\mathbf{s} and \\int \\mathbf{p}^{\\mathbf{s}}{(\\mathbf{s})} d\\mathbf{p}{(\\mathbf{s})} = \\int \\mathbf{s}^{\\mathbf{s}} d\\mathbf{p}{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)))), Integral(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\chi,\\theta_2)} = \\chi - \\theta_2, then obtain \\chi - \\theta_2 + \\int \\Omega^{\\chi}{(\\chi,\\theta_2)} d\\chi = \\chi - \\theta_2 + \\int (\\chi - \\theta_2)^{\\chi} d\\chi", "derivation": "\\Omega{(\\chi,\\theta_2)} = \\chi - \\theta_2 and \\Omega^{\\chi}{(\\chi,\\theta_2)} = (\\chi - \\theta_2)^{\\chi} and \\int \\Omega^{\\chi}{(\\chi,\\theta_2)} d\\chi = \\int (\\chi - \\theta_2)^{\\chi} d\\chi and \\chi - \\theta_2 + \\int \\Omega^{\\chi}{(\\chi,\\theta_2)} d\\chi = \\chi - \\theta_2 + \\int (\\chi - \\theta_2)^{\\chi} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(Pow(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\chi{(F_{c},\\eta^{\\prime})} = F_{c} + \\eta^{\\prime}, then obtain F_{c} + \\eta^{\\prime} + \\chi{(F_{c},\\eta^{\\prime})} = 2 \\chi{(F_{c},\\eta^{\\prime})}", "derivation": "\\chi{(F_{c},\\eta^{\\prime})} = F_{c} + \\eta^{\\prime} and F_{c} + \\eta^{\\prime} + \\chi{(F_{c},\\eta^{\\prime})} = 2 F_{c} + 2 \\eta^{\\prime} and 2 \\chi{(F_{c},\\eta^{\\prime})} = 2 F_{c} + 2 \\eta^{\\prime} and F_{c} + \\eta^{\\prime} + \\chi{(F_{c},\\eta^{\\prime})} = 2 \\chi{(F_{c},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 1, "Add(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given E{(A_{2})} = \\frac{d}{d A_{2}} e^{A_{2}}, then derive E{(A_{2})} = e^{A_{2}}, then obtain \\hat{X} + e^{A_{2}} = \\mathbf{f} + e^{A_{2}}", "derivation": "E{(A_{2})} = \\frac{d}{d A_{2}} e^{A_{2}} and E{(A_{2})} = e^{A_{2}} and e^{A_{2}} = \\frac{d}{d A_{2}} e^{A_{2}} and E{(A_{2})} = \\frac{d}{d A_{2}} E{(A_{2})} and \\int E{(A_{2})} dA_{2} = \\int \\frac{d}{d A_{2}} E{(A_{2})} dA_{2} and \\int e^{A_{2}} dA_{2} = \\int \\frac{d}{d A_{2}} e^{A_{2}} dA_{2} and \\hat{X} + e^{A_{2}} = \\mathbf{f} + e^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('A_2', commutative=True)), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('A_2', commutative=True)), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('E')(Symbol('A_2', commutative=True)), Derivative(Function('E')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('E')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Derivative(Function('E')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('A_2', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\chi{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\hat{\\mathbf{r}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\chi{(y^{\\prime})}, then obtain \\hat{\\mathbf{r}}{(y^{\\prime})} = - \\sin{(y^{\\prime})}", "derivation": "\\chi{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\hat{\\mathbf{r}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\chi{(y^{\\prime})} and \\hat{\\mathbf{r}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\cos{(y^{\\prime})} and \\hat{\\mathbf{r}}{(y^{\\prime})} = - \\sin{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('\\\\chi')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(C)} = \\log{(C)}, then obtain C \\frac{d}{d C} \\int \\mathbf{s}{(C)} dC = C \\frac{d}{d C} \\int \\log{(C)} dC", "derivation": "\\mathbf{s}{(C)} = \\log{(C)} and \\int \\mathbf{s}{(C)} dC = \\int \\log{(C)} dC and \\frac{d}{d C} \\int \\mathbf{s}{(C)} dC = \\frac{d}{d C} \\int \\log{(C)} dC and C \\frac{d}{d C} \\int \\mathbf{s}{(C)} dC = C \\frac{d}{d C} \\int \\log{(C)} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 3, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('C', commutative=True), Derivative(Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{2}{(H,\\mathbf{J})} = H + \\mathbf{J} and \\varphi{(H)} = 2 H, then obtain H - \\mathbf{J} + \\theta_{2}{(H,\\mathbf{J})} = \\varphi{(H)}", "derivation": "\\theta_{2}{(H,\\mathbf{J})} = H + \\mathbf{J} and H + \\theta_{2}{(H,\\mathbf{J})} = 2 H + \\mathbf{J} and \\varphi{(H)} = 2 H and H + \\theta_{2}{(H,\\mathbf{J})} = \\mathbf{J} + \\varphi{(H)} and H - \\mathbf{J} + \\theta_{2}{(H,\\mathbf{J})} = \\varphi{(H)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(2), Symbol('H', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('H', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('H', commutative=True), Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\varphi')(Symbol('H', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Function('\\\\varphi')(Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\tilde{g},h)} = \\tilde{g} + h and \\operatorname{c_{0}}{(\\tilde{g},h)} = (\\tilde{g} + h) \\varepsilon{(\\tilde{g},h)}, then obtain \\frac{\\partial^{2}}{\\partial h\\partial \\tilde{g}} \\operatorname{c_{0}}{(\\tilde{g},h)} = \\frac{\\partial^{2}}{\\partial h\\partial \\tilde{g}} \\varepsilon^{2}{(\\tilde{g},h)}", "derivation": "\\varepsilon{(\\tilde{g},h)} = \\tilde{g} + h and \\varepsilon^{2}{(\\tilde{g},h)} = (\\tilde{g} + h) \\varepsilon{(\\tilde{g},h)} and \\operatorname{c_{0}}{(\\tilde{g},h)} = (\\tilde{g} + h) \\varepsilon{(\\tilde{g},h)} and \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{c_{0}}{(\\tilde{g},h)} = \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + h) \\varepsilon{(\\tilde{g},h)} and \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{c_{0}}{(\\tilde{g},h)} = \\frac{\\partial}{\\partial \\tilde{g}} \\varepsilon^{2}{(\\tilde{g},h)} and \\frac{\\partial^{2}}{\\partial h\\partial \\tilde{g}} \\operatorname{c_{0}}{(\\tilde{g},h)} = \\frac{\\partial^{2}}{\\partial h\\partial \\tilde{g}} \\varepsilon^{2}{(\\tilde{g},h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)))"], [["times", 1, "Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True))"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Integer(2)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Integer(2)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(\\phi_2)} = \\log{(\\phi_2)}, then obtain \\frac{d}{d \\phi_2} (4 \\mathbf{p}{(\\phi_2)} - 2 \\log{(\\phi_2)}) = \\frac{d}{d \\phi_2} 2 \\mathbf{p}{(\\phi_2)}", "derivation": "\\mathbf{p}{(\\phi_2)} = \\log{(\\phi_2)} and 2 \\mathbf{p}{(\\phi_2)} = \\mathbf{p}{(\\phi_2)} + \\log{(\\phi_2)} and 2 \\mathbf{p}{(\\phi_2)} - \\log{(\\phi_2)} = \\mathbf{p}{(\\phi_2)} and \\frac{d}{d \\phi_2} 2 \\mathbf{p}{(\\phi_2)} = \\frac{d}{d \\phi_2} (\\mathbf{p}{(\\phi_2)} + \\log{(\\phi_2)}) and \\frac{d}{d \\phi_2} (4 \\mathbf{p}{(\\phi_2)} - 2 \\log{(\\phi_2)}) = \\frac{d}{d \\phi_2} 2 \\mathbf{p}{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))), Add(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))))"], [["minus", 2, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi_2', commutative=True)))), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(4), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(v_{1})} = \\log{(v_{1})}, then obtain \\frac{d}{d v_{1}} 1 = \\frac{d}{d v_{1}} (\\mathbf{M}{(v_{1})} - \\log{(v_{1})})^{v_{1}}", "derivation": "\\mathbf{M}{(v_{1})} = \\log{(v_{1})} and \\mathbf{M}{(v_{1})} - \\log{(v_{1})} = 0 and (\\mathbf{M}{(v_{1})} - \\log{(v_{1})})^{v_{1}} = 0^{v_{1}} and \\frac{d}{d v_{1}} (\\mathbf{M}{(v_{1})} - \\log{(v_{1})})^{v_{1}} = \\frac{d}{d v_{1}} 0^{v_{1}} and \\frac{d}{d v_{1}} 1 = \\frac{d}{d v_{1}} (\\mathbf{M}{(v_{1})} - \\log{(v_{1})})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["minus", 1, "log(Symbol('v_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Pow(Integer(0), Symbol('v_1', commutative=True)))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Add(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(u)} = \\int \\log{(u)} du, then derive I{(u)} = \\phi + u \\log{(u)} - u, then obtain - u (\\phi + u \\log{(u)} - u) = - u I{(u)}", "derivation": "I{(u)} = \\int \\log{(u)} du and I{(u)} = \\phi + u \\log{(u)} - u and \\phi + u \\log{(u)} - u = \\int \\log{(u)} du and - u (\\phi + u \\log{(u)} - u) = - u \\int \\log{(u)} du and - u (\\phi + u \\log{(u)} - u) = - u I{(u)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('u', commutative=True)), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('I')(Symbol('u', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True), Function('I')(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(A_{y})} = \\sin{(A_{y})} and \\sigma_{p}{(A_{y})} = A_{y} \\operatorname{A_{1}}{(A_{y})}, then obtain \\sin^{A_{y}}{(A_{y} \\operatorname{A_{1}}{(A_{y})})} = \\sin^{A_{y}}{(A_{y} \\sin{(A_{y})})}", "derivation": "\\operatorname{A_{1}}{(A_{y})} = \\sin{(A_{y})} and A_{y} \\operatorname{A_{1}}{(A_{y})} = A_{y} \\sin{(A_{y})} and \\sin{(A_{y} \\operatorname{A_{1}}{(A_{y})})} = \\sin{(A_{y} \\sin{(A_{y})})} and \\sigma_{p}{(A_{y})} = A_{y} \\operatorname{A_{1}}{(A_{y})} and \\sin{(\\sigma_{p}{(A_{y})})} = \\sin{(A_{y} \\sin{(A_{y})})} and \\sin^{A_{y}}{(\\sigma_{p}{(A_{y})})} = \\sin^{A_{y}}{(A_{y} \\sin{(A_{y})})} and \\sin{(A_{y} \\operatorname{A_{1}}{(A_{y})})} = \\sin{(\\sigma_{p}{(A_{y})})} and \\sin^{A_{y}}{(A_{y} \\operatorname{A_{1}}{(A_{y})})} = \\sin^{A_{y}}{(A_{y} \\sin{(A_{y})})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["times", 1, "Symbol('A_y', commutative=True)"], "Equality(Mul(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True))), Mul(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True)))), sin(Mul(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Function('\\\\sigma_p')(Symbol('A_y', commutative=True))), sin(Mul(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True)))))"], [["power", 5, "Symbol('A_y', commutative=True)"], "Equality(Pow(sin(Function('\\\\sigma_p')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Pow(sin(Mul(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(sin(Mul(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True)))), sin(Function('\\\\sigma_p')(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(sin(Mul(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True)), Pow(sin(Mul(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given I{(x^\\prime,A_{2})} = \\frac{A_{2}}{x^\\prime} and \\operatorname{A_{y}}{(x^\\prime,A_{2})} = \\frac{A_{2}}{x^\\prime} + x^\\prime, then obtain \\int (\\frac{A_{2}}{x^\\prime} + x^\\prime + I{(x^\\prime,A_{2})}) dx^\\prime = \\int (\\frac{2 A_{2}}{x^\\prime} + x^\\prime) dx^\\prime", "derivation": "I{(x^\\prime,A_{2})} = \\frac{A_{2}}{x^\\prime} and \\operatorname{A_{y}}{(x^\\prime,A_{2})} = \\frac{A_{2}}{x^\\prime} + x^\\prime and \\operatorname{A_{y}}{(x^\\prime,A_{2})} = x^\\prime + I{(x^\\prime,A_{2})} and \\operatorname{A_{y}}{(x^\\prime,A_{2})} + I{(x^\\prime,A_{2})} = \\frac{A_{2}}{x^\\prime} + x^\\prime + I{(x^\\prime,A_{2})} and \\frac{A_{2}}{x^\\prime} + \\operatorname{A_{y}}{(x^\\prime,A_{2})} = \\frac{2 A_{2}}{x^\\prime} + x^\\prime and \\int (\\frac{A_{2}}{x^\\prime} + \\operatorname{A_{y}}{(x^\\prime,A_{2})}) dx^\\prime = \\int (\\frac{2 A_{2}}{x^\\prime} + x^\\prime) dx^\\prime and \\int (\\frac{A_{2}}{x^\\prime} + x^\\prime + I{(x^\\prime,A_{2})}) dx^\\prime = \\int (\\frac{2 A_{2}}{x^\\prime} + x^\\prime) dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True)), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('x^\\\\prime', commutative=True), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))))"], [["add", 2, "Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Add(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True)), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(2), Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integral(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('A_2', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given I{(l)} = \\sin{(l)}, then obtain - \\sin^{l}{(l)} + \\int (- l + I{(l)}) dl = - \\sin^{l}{(l)} + \\int (- l + \\sin{(l)}) dl", "derivation": "I{(l)} = \\sin{(l)} and - l + I{(l)} = - l + \\sin{(l)} and \\int (- l + I{(l)}) dl = \\int (- l + \\sin{(l)}) dl and - \\sin^{l}{(l)} + \\int (- l + I{(l)}) dl = - \\sin^{l}{(l)} + \\int (- l + \\sin{(l)}) dl", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('I')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('I')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["minus", 3, "Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('I')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\Psi,\\mu)} = \\Psi \\mu, then derive \\frac{\\partial}{\\partial \\mu} \\phi{(\\Psi,\\mu)} = \\Psi, then obtain \\phi{(\\frac{\\partial}{\\partial \\mu} \\Psi \\mu,\\mu)} = \\mu \\frac{\\partial}{\\partial \\mu} \\Psi \\mu", "derivation": "\\phi{(\\Psi,\\mu)} = \\Psi \\mu and \\frac{\\partial}{\\partial \\mu} \\phi{(\\Psi,\\mu)} = \\frac{\\partial}{\\partial \\mu} \\Psi \\mu and \\frac{\\partial}{\\partial \\mu} \\phi{(\\Psi,\\mu)} = \\Psi and \\frac{\\partial}{\\partial \\mu} \\Psi \\mu = \\Psi and \\phi{(\\frac{\\partial}{\\partial \\mu} \\Psi \\mu,\\mu)} = \\mu \\frac{\\partial}{\\partial \\mu} \\Psi \\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\phi')(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(\\Omega,m_{s})} = \\Omega^{m_{s}}, then obtain i{(\\Omega,m_{s})} \\int \\log{(i{(\\Omega,m_{s})})} dm_{s} = i{(\\Omega,m_{s})} \\int \\log{(\\Omega^{m_{s}})} dm_{s}", "derivation": "i{(\\Omega,m_{s})} = \\Omega^{m_{s}} and \\log{(i{(\\Omega,m_{s})})} = \\log{(\\Omega^{m_{s}})} and \\int \\log{(i{(\\Omega,m_{s})})} dm_{s} = \\int \\log{(\\Omega^{m_{s}})} dm_{s} and \\Omega^{m_{s}} \\int \\log{(i{(\\Omega,m_{s})})} dm_{s} = \\Omega^{m_{s}} \\int \\log{(\\Omega^{m_{s}})} dm_{s} and i{(\\Omega,m_{s})} \\int \\log{(i{(\\Omega,m_{s})})} dm_{s} = i{(\\Omega,m_{s})} \\int \\log{(\\Omega^{m_{s}})} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)))"], [["log", 1], "Equality(log(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(log(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integral(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"], [["times", 3, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integral(log(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integral(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integral(log(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True)))), Mul(Function('i')(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integral(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given b{(z^{*},\\varepsilon,n)} = \\frac{- \\varepsilon + z^{*}}{n}, then obtain b{(z^{*},\\varepsilon,n)} - \\frac{- \\varepsilon + z^{*}}{n} = b{(z^{*},\\varepsilon,n)} + \\int \\frac{- \\varepsilon + z^{*}}{n} dn - \\int b{(z^{*},\\varepsilon,n)} dn - \\frac{- \\varepsilon + z^{*}}{n}", "derivation": "b{(z^{*},\\varepsilon,n)} = \\frac{- \\varepsilon + z^{*}}{n} and \\int b{(z^{*},\\varepsilon,n)} dn = \\int \\frac{- \\varepsilon + z^{*}}{n} dn and \\int b{(z^{*},\\varepsilon,n)} dn + \\frac{- \\varepsilon + z^{*}}{n} = \\int \\frac{- \\varepsilon + z^{*}}{n} dn + \\frac{- \\varepsilon + z^{*}}{n} and b{(z^{*},\\varepsilon,n)} + \\int b{(z^{*},\\varepsilon,n)} dn = b{(z^{*},\\varepsilon,n)} + \\int \\frac{- \\varepsilon + z^{*}}{n} dn and b{(z^{*},\\varepsilon,n)} - \\frac{- \\varepsilon + z^{*}}{n} = b{(z^{*},\\varepsilon,n)} + \\int \\frac{- \\varepsilon + z^{*}}{n} dn - \\int b{(z^{*},\\varepsilon,n)} dn - \\frac{- \\varepsilon + z^{*}}{n}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True)))"], "Equality(Add(Integral(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True)))), Add(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))), Tuple(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Integral(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["minus", 4, "Add(Integral(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))))"], "Equality(Add(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True)))), Add(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True))), Tuple(Symbol('n', commutative=True))), Mul(Integer(-1), Integral(Function('b')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(y)} = \\cos{(y)}, then obtain \\operatorname{A_{x}}^{y}{(y)} - \\cos^{y}{(y)} = 0", "derivation": "\\operatorname{A_{x}}{(y)} = \\cos{(y)} and \\operatorname{A_{x}}^{y}{(y)} = \\cos^{y}{(y)} and \\operatorname{A_{x}}^{y}{(y)} + \\cos{(\\log{(f_{\\mathbf{p}})})} = \\cos^{y}{(y)} + \\cos{(\\log{(f_{\\mathbf{p}})})} and \\operatorname{A_{x}}^{y}{(y)} - \\cos^{y}{(y)} = 0", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["add", 2, "cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Add(Pow(Function('A_x')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)), cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["minus", 3, "Add(Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)), cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], "Equality(Add(Pow(Function('A_x')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)))), Integer(0))"]]}, {"prompt": "Given k{(\\mathbf{J},\\pi)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi, then derive 1 = \\frac{\\pi}{k{(\\mathbf{J},\\pi)}}, then obtain 2 = \\frac{\\pi}{\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi} + 1", "derivation": "k{(\\mathbf{J},\\pi)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi and 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi}{k{(\\mathbf{J},\\pi)}} and 1 = \\frac{\\pi}{k{(\\mathbf{J},\\pi)}} and 1 = \\frac{\\pi}{\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi} and 2 = \\frac{\\pi}{\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\pi} + 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 1, "Function('k')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('k')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(1), Mul(Symbol('\\\\pi', commutative=True), Pow(Function('k')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Mul(Symbol('\\\\pi', commutative=True), Pow(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 4, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(v_{x},l)} = l^{v_{x}}, then obtain (l^{v_{x}} + v_{x})^{- l} \\int (v_{x} + \\hat{H}_{\\lambda}{(v_{x},l)})^{l} dl = (l^{v_{x}} + v_{x})^{- l} \\int (l^{v_{x}} + v_{x})^{l} dl", "derivation": "\\hat{H}_{\\lambda}{(v_{x},l)} = l^{v_{x}} and v_{x} + \\hat{H}_{\\lambda}{(v_{x},l)} = l^{v_{x}} + v_{x} and (v_{x} + \\hat{H}_{\\lambda}{(v_{x},l)})^{l} = (l^{v_{x}} + v_{x})^{l} and \\int (v_{x} + \\hat{H}_{\\lambda}{(v_{x},l)})^{l} dl = \\int (l^{v_{x}} + v_{x})^{l} dl and (l^{v_{x}} + v_{x})^{- l} \\int (v_{x} + \\hat{H}_{\\lambda}{(v_{x},l)})^{l} dl = (l^{v_{x}} + v_{x})^{- l} \\int (l^{v_{x}} + v_{x})^{l} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Symbol('v_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('l', commutative=True)))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('v_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["divide", 4, "Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Pow(Add(Symbol('v_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Pow(Add(Pow(Symbol('l', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\mathbf{J},b,\\mathbf{M})} = \\mathbf{J} \\mathbf{M} b, then obtain \\mathbf{J} \\chi{(\\mathbf{J},b,\\mathbf{M})} + \\chi^{b}{(\\mathbf{J},b,\\mathbf{M})} = \\mathbf{J}^{2} \\mathbf{M} b + \\chi^{b}{(\\mathbf{J},b,\\mathbf{M})}", "derivation": "\\chi{(\\mathbf{J},b,\\mathbf{M})} = \\mathbf{J} \\mathbf{M} b and \\chi^{b}{(\\mathbf{J},b,\\mathbf{M})} = (\\mathbf{J} \\mathbf{M} b)^{b} and \\mathbf{J} \\chi{(\\mathbf{J},b,\\mathbf{M})} = \\mathbf{J}^{2} \\mathbf{M} b and \\mathbf{J} \\chi{(\\mathbf{J},b,\\mathbf{M})} + (\\mathbf{J} \\mathbf{M} b)^{b} = \\mathbf{J}^{2} \\mathbf{M} b + (\\mathbf{J} \\mathbf{M} b)^{b} and \\mathbf{J} \\chi{(\\mathbf{J},b,\\mathbf{M})} + \\chi^{b}{(\\mathbf{J},b,\\mathbf{M})} = \\mathbf{J}^{2} \\mathbf{M} b + \\chi^{b}{(\\mathbf{J},b,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('b', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)))"], [["add", 3, "Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Add(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('b', commutative=True))), Add(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('b', commutative=True)), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(V)} = \\sin{(V)} and \\psi{(V)} = \\int \\frac{\\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV}{\\operatorname{F_{H}}{(V)}} dV, then obtain \\psi{(V)} = \\int \\frac{\\int 0 dV}{\\operatorname{F_{H}}{(V)}} dV", "derivation": "\\operatorname{F_{H}}{(V)} = \\sin{(V)} and \\operatorname{F_{H}}{(V)} - \\sin{(V)} = 0 and \\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV = \\int 0 dV and \\frac{\\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV}{\\sin{(V)}} = \\frac{\\int 0 dV}{\\sin{(V)}} and \\frac{\\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV}{\\operatorname{F_{H}}{(V)}} = \\frac{\\int 0 dV}{\\operatorname{F_{H}}{(V)}} and \\int \\frac{\\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV}{\\operatorname{F_{H}}{(V)}} dV = \\int \\frac{\\int 0 dV}{\\operatorname{F_{H}}{(V)}} dV and \\psi{(V)} = \\int \\frac{\\int (\\operatorname{F_{H}}{(V)} - \\sin{(V)}) dV}{\\operatorname{F_{H}}{(V)}} dV and \\psi{(V)} = \\int \\frac{\\int 0 dV}{\\operatorname{F_{H}}{(V)}} dV", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["minus", 1, "sin(Symbol('V', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Integer(0), Tuple(Symbol('V', commutative=True))))"], [["divide", 3, "sin(Symbol('V', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), Integral(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True)))), Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True)))), Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))))"], [["integrate", 5, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('V', commutative=True)), Integral(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Add(Function('F_H')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Function('\\\\psi')(Symbol('V', commutative=True)), Integral(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Q)} = \\log{(\\log{(Q)})}, then obtain Q^{2} (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{4} = Q^{2} (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{2} (\\log{(Q)} + \\log{(\\log{(Q)})})^{2}", "derivation": "\\operatorname{y^{\\prime}}{(Q)} = \\log{(\\log{(Q)})} and \\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)} = \\log{(Q)} + \\log{(\\log{(Q)})} and (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{2} = (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)}) (\\log{(Q)} + \\log{(\\log{(Q)})}) and Q (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{2} = Q (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)}) (\\log{(Q)} + \\log{(\\log{(Q)})}) and Q^{2} (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{4} = Q^{2} (\\operatorname{y^{\\prime}}{(Q)} + \\log{(Q)})^{2} (\\log{(Q)} + \\log{(\\log{(Q)})})^{2}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))"], [["add", 1, "log(Symbol('Q', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True)))))"], [["times", 2, "Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], "Equality(Pow(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Integer(2)), Mul(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))))"], [["times", 3, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Pow(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Integer(2))), Mul(Symbol('Q', commutative=True), Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Integer(4))), Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Integer(2)), Pow(Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\hat{H}_l{(P_{e},\\lambda)} = P_{e} + \\lambda, then obtain (P_{e} + \\lambda)^{2} \\hat{H}_l^{2}{(P_{e},\\lambda)} = (P_{e} + \\lambda)^{3} \\hat{H}_l{(P_{e},\\lambda)}", "derivation": "\\hat{H}_l{(P_{e},\\lambda)} = P_{e} + \\lambda and \\hat{H}_l^{2}{(P_{e},\\lambda)} = (P_{e} + \\lambda) \\hat{H}_l{(P_{e},\\lambda)} and \\hat{H}_l^{4}{(P_{e},\\lambda)} = (P_{e} + \\lambda)^{2} \\hat{H}_l^{2}{(P_{e},\\lambda)} and (P_{e} + \\lambda)^{2} \\hat{H}_l^{2}{(P_{e},\\lambda)} = (P_{e} + \\lambda)^{3} \\hat{H}_l{(P_{e},\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Add(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(4)), Mul(Pow(Add(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Pow(Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Pow(Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(3)), Function('\\\\hat{H}_l')(Symbol('P_e', commutative=True), Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} = \\psi^* + g - v_{x}, then obtain - \\psi^* + g \\int \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} dg - g + v_{x} = - \\psi^* + g (\\phi + \\frac{g^{2}}{2} + g (\\psi^* - v_{x})) - g + v_{x}", "derivation": "\\operatorname{r_{0}}{(\\psi^*,v_{x},g)} = \\psi^* + g - v_{x} and \\int \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} dg = \\int (\\psi^* + g - v_{x}) dg and g \\int \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} dg = g \\int (\\psi^* + g - v_{x}) dg and - \\psi^* + g \\int \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} dg - g + v_{x} = - \\psi^* + g \\int (\\psi^* + g - v_{x}) dg - g + v_{x} and - \\psi^* + g \\int \\operatorname{r_{0}}{(\\psi^*,v_{x},g)} dg - g + v_{x} = - \\psi^* + g (\\phi + \\frac{g^{2}}{2} + g (\\psi^* - v_{x})) - g + v_{x}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('v_x', commutative=True), Symbol('g', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('v_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["times", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('v_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Symbol('g', commutative=True), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["minus", 3, "Add(Symbol('\\\\psi^*', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('g', commutative=True), Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('v_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('g', commutative=True), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_x', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('g', commutative=True), Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('v_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('g', commutative=True), Add(Symbol('\\\\phi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger}, then obtain (\\int \\hat{p}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} d\\hat{p}{(\\Psi^{\\dagger})})^{\\hat{p}{(\\Psi^{\\dagger})}} = (\\int (\\Psi^{\\dagger})^{\\Psi^{\\dagger}} d\\hat{p}{(\\Psi^{\\dagger})})^{\\hat{p}{(\\Psi^{\\dagger})}}", "derivation": "\\hat{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} and \\hat{p}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} = (\\Psi^{\\dagger})^{\\Psi^{\\dagger}} and \\int \\hat{p}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int (\\Psi^{\\dagger})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} and (\\int \\hat{p}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (\\int (\\Psi^{\\dagger})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} and (\\int \\hat{p}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} d\\hat{p}{(\\Psi^{\\dagger})})^{\\hat{p}{(\\Psi^{\\dagger})}} = (\\int (\\Psi^{\\dagger})^{\\Psi^{\\dagger}} d\\hat{p}{(\\Psi^{\\dagger})})^{\\hat{p}{(\\Psi^{\\dagger})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], [["power", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Integral(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(Integral(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Function('\\\\hat{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\chi)} = \\log{(\\chi)} and \\operatorname{P_{e}}{(\\chi)} = (\\int (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} - \\log{(\\chi)}) d\\chi)^{\\chi}, then obtain \\operatorname{P_{e}}^{\\chi}{(\\chi)} = ((\\int 0 d\\chi)^{\\chi})^{\\chi}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\chi)} = \\log{(\\chi)} and \\operatorname{V_{\\mathbf{B}}}{(\\chi)} - \\log{(\\chi)} = 0 and \\int (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} - \\log{(\\chi)}) d\\chi = \\int 0 d\\chi and \\operatorname{P_{e}}{(\\chi)} = (\\int (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} - \\log{(\\chi)}) d\\chi)^{\\chi} and \\operatorname{P_{e}}{(\\chi)} = (\\int 0 d\\chi)^{\\chi} and \\operatorname{P_{e}}^{\\chi}{(\\chi)} = ((\\int 0 d\\chi)^{\\chi})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('P_e')(Symbol('\\\\chi', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["power", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(P_{g},b)} = e^{P_{g} b}, then derive - \\mathbf{B} - e^{P_{g} b} + \\int P_{g} \\Psi^{\\dagger}{(P_{g},b)} db = 0, then obtain - \\mathbf{B} - e^{P_{g} b} + \\int P_{g} e^{P_{g} b} db = 0", "derivation": "\\Psi^{\\dagger}{(P_{g},b)} = e^{P_{g} b} and P_{g} \\Psi^{\\dagger}{(P_{g},b)} = P_{g} e^{P_{g} b} and \\int P_{g} \\Psi^{\\dagger}{(P_{g},b)} db = \\int P_{g} e^{P_{g} b} db and \\int P_{g} \\Psi^{\\dagger}{(P_{g},b)} db - \\int P_{g} e^{P_{g} b} db = 0 and - \\mathbf{B} - e^{P_{g} b} + \\int P_{g} \\Psi^{\\dagger}{(P_{g},b)} db = 0 and - \\mathbf{B} - e^{P_{g} b} + \\int P_{g} e^{P_{g} b} db = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('b', commutative=True)), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True))))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('P_g', commutative=True), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('P_g', commutative=True), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))))"], [["minus", 3, "Integral(Mul(Symbol('P_g', commutative=True), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('P_g', commutative=True), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))))), Integer(0))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Integral(Mul(Symbol('P_g', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Integral(Mul(Symbol('P_g', commutative=True), exp(Mul(Symbol('P_g', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\psi{(I)} = \\cos{(I)} and t{(\\varepsilon,U,\\omega)} = U + \\omega + \\varepsilon, then obtain - t{(\\varepsilon,U,\\omega)} \\sin{(I)} = - (U + \\omega + \\varepsilon) \\sin{(I)}", "derivation": "\\psi{(I)} = \\cos{(I)} and t{(\\varepsilon,U,\\omega)} = U + \\omega + \\varepsilon and t{(\\varepsilon,U,\\omega)} \\frac{d}{d I} \\psi{(I)} = (U + \\omega + \\varepsilon) \\frac{d}{d I} \\psi{(I)} and t{(\\varepsilon,U,\\omega)} \\frac{d}{d I} \\cos{(I)} = (U + \\omega + \\varepsilon) \\frac{d}{d I} \\cos{(I)} and - t{(\\varepsilon,U,\\omega)} \\sin{(I)} = - (U + \\omega + \\varepsilon) \\sin{(I)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], ["get_premise", "Equality(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 2, "Derivative(Function('\\\\psi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Mul(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\psi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\psi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Symbol('I', commutative=True))), Mul(Integer(-1), Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\phi_1)} = e^{\\phi_1} and \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} = J_{\\varepsilon}^{n_{1}}, then obtain (\\hat{x}{(\\phi_1)} \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} e^{\\phi_1})^{\\phi_1} = (J_{\\varepsilon}^{n_{1}} \\hat{x}{(\\phi_1)} e^{\\phi_1})^{\\phi_1}", "derivation": "\\hat{x}{(\\phi_1)} = e^{\\phi_1} and \\hat{x}{(\\phi_1)} e^{\\phi_1} = e^{2 \\phi_1} and \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} = J_{\\varepsilon}^{n_{1}} and \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} e^{2 \\phi_1} = J_{\\varepsilon}^{n_{1}} e^{2 \\phi_1} and \\hat{x}{(\\phi_1)} \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} e^{\\phi_1} = J_{\\varepsilon}^{n_{1}} \\hat{x}{(\\phi_1)} e^{\\phi_1} and (\\hat{x}{(\\phi_1)} \\operatorname{v_{t}}{(n_{1},J_{\\varepsilon})} e^{\\phi_1})^{\\phi_1} = (J_{\\varepsilon}^{n_{1}} \\hat{x}{(\\phi_1)} e^{\\phi_1})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], ["get_premise", "Equality(Function('v_t')(Symbol('n_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('n_1', commutative=True)))"], [["times", 3, "exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Function('v_t')(Symbol('n_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), Function('v_t')(Symbol('n_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))))"], [["power", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), Function('v_t')(Symbol('n_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then obtain \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{v_{x}}{(V_{\\mathbf{B}})} + 2 + \\frac{1}{V_{\\mathbf{B}}} = 2 + \\frac{2}{V_{\\mathbf{B}}}", "derivation": "\\operatorname{v_{x}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and V_{\\mathbf{B}} + \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} + \\log{(V_{\\mathbf{B}})} and 2 V_{\\mathbf{B}} + \\operatorname{v_{x}}{(V_{\\mathbf{B}})} + \\log{(V_{\\mathbf{B}})} = 2 V_{\\mathbf{B}} + 2 \\log{(V_{\\mathbf{B}})} and \\frac{d}{d V_{\\mathbf{B}}} (2 V_{\\mathbf{B}} + \\operatorname{v_{x}}{(V_{\\mathbf{B}})} + \\log{(V_{\\mathbf{B}})}) = \\frac{d}{d V_{\\mathbf{B}}} (2 V_{\\mathbf{B}} + 2 \\log{(V_{\\mathbf{B}})}) and \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{v_{x}}{(V_{\\mathbf{B}})} + 2 + \\frac{1}{V_{\\mathbf{B}}} = 2 + \\frac{2}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 2, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(2), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(2), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(2), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))), Add(Integer(2), Mul(Integer(2), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\sigma_p,z)} = - \\sigma_p + z, then derive (\\frac{\\partial}{\\partial \\sigma_p} \\hat{p}_0{(\\sigma_p,z)})^{\\sigma_p} = (-1)^{\\sigma_p}, then obtain \\frac{(\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z))^{\\sigma_p}}{\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z)} = \\frac{(-1)^{\\sigma_p}}{\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z)}", "derivation": "\\hat{p}_0{(\\sigma_p,z)} = - \\sigma_p + z and \\frac{\\partial}{\\partial \\sigma_p} \\hat{p}_0{(\\sigma_p,z)} = \\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z) and (\\frac{\\partial}{\\partial \\sigma_p} \\hat{p}_0{(\\sigma_p,z)})^{\\sigma_p} = (\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z))^{\\sigma_p} and (\\frac{\\partial}{\\partial \\sigma_p} \\hat{p}_0{(\\sigma_p,z)})^{\\sigma_p} = (-1)^{\\sigma_p} and (\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z))^{\\sigma_p} = (-1)^{\\sigma_p} and \\frac{(\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z))^{\\sigma_p}}{\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z)} = \\frac{(-1)^{\\sigma_p}}{\\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 5, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})} = (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}}, then obtain - (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} (- (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} - \\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})}) = 2 (- \\Psi_{\\lambda} + \\tilde{g})^{2 \\hat{X}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})} = (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} and (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} + \\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})} = 2 (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} and - (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} - \\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})} = - 2 (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} and - (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} (- (- \\Psi_{\\lambda} + \\tilde{g})^{\\hat{X}} - \\operatorname{E_{\\lambda}}{(\\tilde{g},\\hat{X},\\Psi_{\\lambda})}) = 2 (- \\Psi_{\\lambda} + \\tilde{g})^{2 \\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given I{(J)} = \\cos{(J)}, then derive I{(J)} + \\int I{(J)} dJ = a^{\\dagger} + I{(J)} + \\sin{(J)}, then obtain \\frac{d}{d a^{\\dagger}} (I{(J)} + \\int \\cos{(J)} dJ) I{(J)} \\cos{(J)} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} + I{(J)} + \\sin{(J)}) I{(J)} \\cos{(J)}", "derivation": "I{(J)} = \\cos{(J)} and \\int I{(J)} dJ = \\int \\cos{(J)} dJ and I{(J)} + \\int I{(J)} dJ = I{(J)} + \\int \\cos{(J)} dJ and I{(J)} + \\int I{(J)} dJ = a^{\\dagger} + I{(J)} + \\sin{(J)} and (I{(J)} + \\int I{(J)} dJ) \\cos{(J)} = (a^{\\dagger} + I{(J)} + \\sin{(J)}) \\cos{(J)} and (I{(J)} + \\int \\cos{(J)} dJ) \\cos{(J)} = (a^{\\dagger} + I{(J)} + \\sin{(J)}) \\cos{(J)} and (I{(J)} + \\int \\cos{(J)} dJ) I{(J)} \\cos{(J)} = (a^{\\dagger} + I{(J)} + \\sin{(J)}) I{(J)} \\cos{(J)} and \\frac{d}{d a^{\\dagger}} (I{(J)} + \\int \\cos{(J)} dJ) I{(J)} \\cos{(J)} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} + I{(J)} + \\sin{(J)}) I{(J)} \\cos{(J)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('I')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["add", 2, "Function('I')(Symbol('J', commutative=True))"], "Equality(Add(Function('I')(Symbol('J', commutative=True)), Integral(Function('I')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Function('I')(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('I')(Symbol('J', commutative=True)), Integral(Function('I')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('I')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))))"], [["times", 4, "cos(Symbol('J', commutative=True))"], "Equality(Mul(Add(Function('I')(Symbol('J', commutative=True)), Integral(Function('I')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), cos(Symbol('J', commutative=True))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('I')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Function('I')(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), cos(Symbol('J', commutative=True))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('I')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True))))"], [["times", 6, "Function('I')(Symbol('J', commutative=True))"], "Equality(Mul(Add(Function('I')(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Function('I')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('I')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))), Function('I')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))))"], [["differentiate", 7, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Add(Function('I')(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Function('I')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('I')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))), Function('I')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(A_{z},\\sigma_p)} = \\cos{(A_{z} + \\sigma_p)}, then obtain - \\sigma_p \\cos{(A_{z} + \\sigma_p)} + \\hat{x}{(A_{z},\\sigma_p)} = - \\sigma_p \\cos{(A_{z} + \\sigma_p)} + \\cos{(A_{z} + \\sigma_p)}", "derivation": "\\hat{x}{(A_{z},\\sigma_p)} = \\cos{(A_{z} + \\sigma_p)} and \\sigma_p \\hat{x}{(A_{z},\\sigma_p)} = \\sigma_p \\cos{(A_{z} + \\sigma_p)} and - \\sigma_p \\hat{x}{(A_{z},\\sigma_p)} + \\hat{x}{(A_{z},\\sigma_p)} = - \\sigma_p \\hat{x}{(A_{z},\\sigma_p)} + \\cos{(A_{z} + \\sigma_p)} and - \\sigma_p \\cos{(A_{z} + \\sigma_p)} + \\hat{x}{(A_{z},\\sigma_p)} = - \\sigma_p \\cos{(A_{z} + \\sigma_p)} + \\cos{(A_{z} + \\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["minus", 1, "Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Function('\\\\hat{x}')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbb{I})} = \\log{(\\mathbb{I})}, then obtain \\frac{\\log{(\\frac{\\hat{H}_{\\lambda}{(\\mathbb{I})}}{\\log{(\\mathbb{I})}})}}{A_{y}} = 0", "derivation": "\\hat{H}_{\\lambda}{(\\mathbb{I})} = \\log{(\\mathbb{I})} and \\frac{\\hat{H}_{\\lambda}{(\\mathbb{I})}}{\\log{(\\mathbb{I})}} = 1 and \\log{(\\frac{\\hat{H}_{\\lambda}{(\\mathbb{I})}}{\\log{(\\mathbb{I})}})} = 0 and \\frac{\\log{(\\frac{\\hat{H}_{\\lambda}{(\\mathbb{I})}}{\\log{(\\mathbb{I})}})}}{A_{y}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True)), log(Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))), Integer(1))"], [["log", 2], "Equality(log(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)))), Integer(0))"], [["divide", 3, "Symbol('A_y', commutative=True)"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), log(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\varphi)} = \\sin{(\\varphi)}, then obtain \\varphi - \\operatorname{t_{1}}{(\\varphi)} + \\int (- \\varphi + \\operatorname{t_{1}}{(\\varphi)}) d\\varphi = \\varphi - \\operatorname{t_{1}}{(\\varphi)} + \\int (- \\varphi + \\sin{(\\varphi)}) d\\varphi", "derivation": "\\operatorname{t_{1}}{(\\varphi)} = \\sin{(\\varphi)} and - \\varphi + \\operatorname{t_{1}}{(\\varphi)} = - \\varphi + \\sin{(\\varphi)} and \\int (- \\varphi + \\operatorname{t_{1}}{(\\varphi)}) d\\varphi = \\int (- \\varphi + \\sin{(\\varphi)}) d\\varphi and \\varphi - \\operatorname{t_{1}}{(\\varphi)} + \\int (- \\varphi + \\operatorname{t_{1}}{(\\varphi)}) d\\varphi = \\varphi - \\operatorname{t_{1}}{(\\varphi)} + \\int (- \\varphi + \\sin{(\\varphi)}) d\\varphi", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["minus", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('t_1')(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('t_1')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('t_1')(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('t_1')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(i)} = \\sin{(\\log{(i)})} and u{(i)} = \\log{(i)}, then obtain \\int \\log{(\\frac{d}{d i} \\sin{(u{(i)})})} \\int \\sin{(u{(i)})} di di = \\int \\log{(\\frac{d}{d i} \\sin{(\\log{(i)})})} \\int \\sin{(u{(i)})} di di", "derivation": "\\Psi_{nl}{(i)} = \\sin{(\\log{(i)})} and \\frac{d}{d i} \\Psi_{nl}{(i)} = \\frac{d}{d i} \\sin{(\\log{(i)})} and \\log{(\\frac{d}{d i} \\Psi_{nl}{(i)})} = \\log{(\\frac{d}{d i} \\sin{(\\log{(i)})})} and u{(i)} = \\log{(i)} and \\Psi_{nl}{(i)} = \\sin{(u{(i)})} and \\log{(\\frac{d}{d i} \\sin{(u{(i)})})} = \\log{(\\frac{d}{d i} \\sin{(\\log{(i)})})} and \\log{(\\frac{d}{d i} \\sin{(u{(i)})})} \\int \\sin{(u{(i)})} di = \\log{(\\frac{d}{d i} \\sin{(\\log{(i)})})} \\int \\sin{(u{(i)})} di and \\int \\log{(\\frac{d}{d i} \\sin{(u{(i)})})} \\int \\sin{(u{(i)})} di di = \\int \\log{(\\frac{d}{d i} \\sin{(\\log{(i)})})} \\int \\sin{(u{(i)})} di di", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), log(Derivative(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('u')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), sin(Function('u')(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Derivative(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), log(Derivative(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["times", 6, "Integral(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(log(Derivative(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Integral(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Mul(log(Derivative(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Integral(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))))"], [["integrate", 7, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(log(Derivative(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Integral(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integral(Mul(log(Derivative(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Integral(sin(Function('u')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\hat{x}_0)} = \\log{(\\log{(\\hat{x}_0)})}, then derive \\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)} = \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}}, then obtain \\frac{\\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)}}{f^{*}} + \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}} = \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}} + \\frac{1}{\\hat{x}_0 f^{*} \\log{(\\hat{x}_0)}}", "derivation": "\\mathbf{s}{(\\hat{x}_0)} = \\log{(\\log{(\\hat{x}_0)})} and \\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\log{(\\log{(\\hat{x}_0)})} and \\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)} = \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}} and \\frac{\\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)}}{f^{*}} = \\frac{1}{\\hat{x}_0 f^{*} \\log{(\\hat{x}_0)}} and \\frac{\\frac{d}{d \\hat{x}_0} \\mathbf{s}{(\\hat{x}_0)}}{f^{*}} + \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}} = \\frac{1}{\\hat{x}_0 \\log{(\\hat{x}_0)}} + \\frac{1}{\\hat{x}_0 f^{*} \\log{(\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True)), log(log(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))))"], [["times", 3, "Pow(Symbol('f^*', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))))"], [["add", 4, "Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given G{(P_{e},G)} = G + \\cos{(P_{e})}, then obtain (P_{e} + G{(P_{e},G)})^{P_{e}} G{(P_{e},G)} = (G + \\cos{(P_{e})}) (P_{e} + G{(P_{e},G)})^{P_{e}}", "derivation": "G{(P_{e},G)} = G + \\cos{(P_{e})} and P_{e} + G{(P_{e},G)} = G + P_{e} + \\cos{(P_{e})} and (G + P_{e} + \\cos{(P_{e})})^{P_{e}} G{(P_{e},G)} = (G + \\cos{(P_{e})}) (G + P_{e} + \\cos{(P_{e})})^{P_{e}} and (P_{e} + G{(P_{e},G)})^{P_{e}} G{(P_{e},G)} = (G + \\cos{(P_{e})}) (P_{e} + G{(P_{e},G)})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["times", 1, "Pow(Add(Symbol('G', commutative=True), Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Mul(Add(Symbol('G', commutative=True), cos(Symbol('P_e', commutative=True))), Pow(Add(Symbol('G', commutative=True), Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Symbol('P_e', commutative=True)), Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Mul(Add(Symbol('G', commutative=True), cos(Symbol('P_e', commutative=True))), Pow(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\chi{(m,Z)} = \\frac{Z}{m} and \\mathbf{M}{(m,Z)} = \\frac{Z}{m}, then obtain \\frac{\\partial}{\\partial m} \\frac{Z}{m} - \\frac{\\partial}{\\partial m} \\chi{(m,Z)} = 0", "derivation": "\\chi{(m,Z)} = \\frac{Z}{m} and \\mathbf{M}{(m,Z)} = \\frac{Z}{m} and \\frac{\\partial}{\\partial m} \\mathbf{M}{(m,Z)} = \\frac{\\partial}{\\partial m} \\frac{Z}{m} and - \\chi{(m,Z)} + \\frac{\\partial}{\\partial m} \\mathbf{M}{(m,Z)} = - \\chi{(m,Z)} + \\frac{\\partial}{\\partial m} \\frac{Z}{m} and - \\frac{\\partial}{\\partial m} \\frac{Z}{m} + \\frac{\\partial}{\\partial m} \\mathbf{M}{(m,Z)} = 0 and - \\frac{\\partial}{\\partial m} \\chi{(m,Z)} + \\frac{\\partial}{\\partial m} \\mathbf{M}{(m,Z)} = 0 and \\frac{\\partial}{\\partial m} \\frac{Z}{m} - \\frac{\\partial}{\\partial m} \\chi{(m,Z)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 3, "Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["minus", 4, "Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given x{(\\lambda)} = \\sin{(\\log{(\\lambda)})} and U{(a,n,\\mathbf{B})} = \\mathbf{B} a^{n}, then obtain \\sin^{\\lambda}{(\\log{(\\lambda)})} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} U{(a,n,\\mathbf{B})} dn = \\sin^{\\lambda}{(\\log{(\\lambda)})} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} a^{n} dn", "derivation": "x{(\\lambda)} = \\sin{(\\log{(\\lambda)})} and U{(a,n,\\mathbf{B})} = \\mathbf{B} a^{n} and \\frac{\\partial}{\\partial \\mathbf{B}} U{(a,n,\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} a^{n} and x^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\log{(\\lambda)})} and \\int \\frac{\\partial}{\\partial \\mathbf{B}} U{(a,n,\\mathbf{B})} dn = \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} a^{n} dn and x^{\\lambda}{(\\lambda)} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} U{(a,n,\\mathbf{B})} dn = x^{\\lambda}{(\\lambda)} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} a^{n} dn and \\sin^{\\lambda}{(\\log{(\\lambda)})} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} U{(a,n,\\mathbf{B})} dn = \\sin^{\\lambda}{(\\log{(\\lambda)})} + \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} a^{n} dn", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\lambda', commutative=True)), sin(log(Symbol('\\\\lambda', commutative=True))))"], ["get_premise", "Equality(Function('U')(Symbol('a', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('a', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('x')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Function('U')(Symbol('a', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))))"], [["add", 5, "Pow(Function('x')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Pow(Function('x')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Integral(Derivative(Function('U')(Symbol('a', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Add(Pow(Function('x')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Integral(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(sin(log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Integral(Derivative(Function('U')(Symbol('a', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Add(Pow(sin(log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Integral(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given s{(\\eta)} = \\eta, then derive \\lambda + \\frac{s^{2}{(\\eta)}}{2} = \\int \\eta ds{(\\eta)}, then derive \\frac{\\eta^{2}}{2} + \\lambda = E_{\\lambda} + \\frac{\\eta^{2}}{2}, then obtain \\Psi_{\\lambda} + \\frac{\\eta^{2}}{2} = \\frac{(E_{\\lambda} + \\frac{\\eta^{2}}{2}) (\\Psi_{\\lambda} + \\frac{\\eta^{2}}{2})}{\\int \\eta d\\eta}", "derivation": "s{(\\eta)} = \\eta and \\int s{(\\eta)} d\\eta = \\int \\eta d\\eta and \\int s{(\\eta)} ds{(\\eta)} = \\int \\eta ds{(\\eta)} and \\lambda + \\frac{s^{2}{(\\eta)}}{2} = \\int \\eta ds{(\\eta)} and \\frac{\\eta^{2}}{2} + \\lambda = \\int \\eta d\\eta and \\frac{\\eta^{2}}{2} + \\lambda = E_{\\lambda} + \\frac{\\eta^{2}}{2} and \\frac{\\frac{\\eta^{2}}{2} + \\lambda}{\\int \\eta d\\eta} = \\frac{E_{\\lambda} + \\frac{\\eta^{2}}{2}}{\\int \\eta d\\eta} and 1 = \\frac{E_{\\lambda} + \\frac{\\eta^{2}}{2}}{\\int \\eta d\\eta} and \\Psi_{\\lambda} + \\frac{\\eta^{2}}{2} = \\frac{(E_{\\lambda} + \\frac{\\eta^{2}}{2}) (\\Psi_{\\lambda} + \\frac{\\eta^{2}}{2})}{\\int \\eta d\\eta}", "srepr_derivation": [["renaming_premise", "Equality(Function('s')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('s')(Symbol('\\\\eta', commutative=True)), Tuple(Function('s')(Symbol('\\\\eta', commutative=True)))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Function('s')(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Function('s')(Symbol('\\\\eta', commutative=True)), Integer(2)))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Function('s')(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))))"], [["divide", 6, "Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Pow(Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integer(1), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Pow(Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["divide", 8, "Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Integer(-1))"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))), Pow(Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} = (e^{\\mathbf{v}})^{J_{\\varepsilon}}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} (- J_{\\varepsilon} + \\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} (e^{\\mathbf{v}})^{- J_{\\varepsilon}}) = \\frac{d}{d \\mathbf{v}} (1 - J_{\\varepsilon})", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} = (e^{\\mathbf{v}})^{J_{\\varepsilon}} and \\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} (e^{\\mathbf{v}})^{- J_{\\varepsilon}} = 1 and - J_{\\varepsilon} + \\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} (e^{\\mathbf{v}})^{- J_{\\varepsilon}} = 1 - J_{\\varepsilon} and \\frac{\\partial}{\\partial \\mathbf{v}} (- J_{\\varepsilon} + \\hat{\\mathbf{r}}{(\\mathbf{v},J_{\\varepsilon})} (e^{\\mathbf{v}})^{- J_{\\varepsilon}}) = \\frac{d}{d \\mathbf{v}} (1 - J_{\\varepsilon})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(1))"], [["add", 2, "Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(Q,\\Psi)} = \\Psi + \\sin{(Q)}, then obtain (\\Psi + \\frac{(\\Psi + \\frac{(\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}{(Q,\\Psi)}}) (\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}^{2}{(Q,\\Psi)}})^{Q} = (\\Psi + \\sin{(Q)})^{Q}", "derivation": "\\mathbf{P}{(Q,\\Psi)} = \\Psi + \\sin{(Q)} and \\sin{(Q)} = \\frac{(\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}{(Q,\\Psi)}} and \\mathbf{P}^{Q}{(Q,\\Psi)} = (\\Psi + \\sin{(Q)})^{Q} and \\mathbf{P}^{Q}{(Q,\\Psi)} = (\\Psi + \\frac{(\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}{(Q,\\Psi)}})^{Q} and \\mathbf{P}^{Q}{(Q,\\Psi)} = (\\Psi + \\frac{(\\Psi + \\frac{(\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}{(Q,\\Psi)}}) (\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}^{2}{(Q,\\Psi)}})^{Q} and (\\Psi + \\frac{(\\Psi + \\frac{(\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}{(Q,\\Psi)}}) (\\Psi + \\sin{(Q)}) \\sin{(Q)}}{\\mathbf{P}^{2}{(Q,\\Psi)}})^{Q} = (\\Psi + \\sin{(Q)})^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))))"], [["divide", 1, "Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Integer(-1)))"], "Equality(sin(Symbol('Q', commutative=True)), Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('Q', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('Q', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('Q', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Add(Symbol('\\\\Psi', commutative=True), Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('Q', commutative=True)))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-2)), sin(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Add(Symbol('\\\\Psi', commutative=True), Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('Q', commutative=True)))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-2)), sin(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})} = \\log{(F_{g} + a^{\\dagger})}, then obtain (F_{g} + a^{\\dagger}) (- F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})}) = (F_{g} + a^{\\dagger}) (- F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\log{(F_{g} + a^{\\dagger})})", "derivation": "\\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})} = \\log{(F_{g} + a^{\\dagger})} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})} = \\frac{\\partial}{\\partial F_{g}} \\log{(F_{g} + a^{\\dagger})} and - F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})} = - F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\log{(F_{g} + a^{\\dagger})} and (F_{g} + a^{\\dagger}) (- F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{E_{\\lambda}}{(a^{\\dagger},F_{g})}) = (F_{g} + a^{\\dagger}) (- F_{g} - a^{\\dagger} + \\frac{\\partial}{\\partial F_{g}} \\log{(F_{g} + a^{\\dagger})})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_g', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(log(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["minus", 2, "Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(log(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["times", 3, "Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))), Mul(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(log(Add(Symbol('F_g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{M}{(x^\\prime,\\varepsilon_0)} = \\frac{\\varepsilon_0}{x^\\prime}, then obtain \\frac{\\varepsilon_0 (\\frac{\\varepsilon_0}{x^\\prime})^{- \\varepsilon_0} \\mathbf{M}^{\\varepsilon_0}{(x^\\prime,\\varepsilon_0)}}{x^\\prime} = \\frac{\\varepsilon_0}{x^\\prime}", "derivation": "\\mathbf{M}{(x^\\prime,\\varepsilon_0)} = \\frac{\\varepsilon_0}{x^\\prime} and \\mathbf{M}^{\\varepsilon_0}{(x^\\prime,\\varepsilon_0)} = (\\frac{\\varepsilon_0}{x^\\prime})^{\\varepsilon_0} and (\\frac{\\varepsilon_0}{x^\\prime})^{- \\varepsilon_0} \\mathbf{M}^{\\varepsilon_0}{(x^\\prime,\\varepsilon_0)} = 1 and \\varepsilon_0 (\\frac{\\varepsilon_0}{x^\\prime})^{- \\varepsilon_0} \\mathbf{M}^{\\varepsilon_0}{(x^\\prime,\\varepsilon_0)} = \\varepsilon_0 and \\frac{\\varepsilon_0 (\\frac{\\varepsilon_0}{x^\\prime})^{- \\varepsilon_0} \\mathbf{M}^{\\varepsilon_0}{(x^\\prime,\\varepsilon_0)}}{x^\\prime} = \\frac{\\varepsilon_0}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 2, "Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Integer(1))"], [["times", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))"], [["times", 4, "Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"]]}, {"prompt": "Given t{(\\varepsilon)} = \\log{(e^{\\varepsilon})}, then derive \\frac{d^{2}}{d \\varepsilon^{2}} t{(\\varepsilon)} = 0, then obtain 0 = - \\frac{d^{2}}{d \\varepsilon^{2}} \\log{(e^{\\varepsilon})}", "derivation": "t{(\\varepsilon)} = \\log{(e^{\\varepsilon})} and \\frac{d}{d \\varepsilon} t{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\log{(e^{\\varepsilon})} and \\frac{d^{2}}{d \\varepsilon^{2}} t{(\\varepsilon)} = \\frac{d^{2}}{d \\varepsilon^{2}} \\log{(e^{\\varepsilon})} and \\frac{d^{2}}{d \\varepsilon^{2}} t{(\\varepsilon)} = 0 and 0 = - \\frac{d^{2}}{d \\varepsilon^{2}} t{(\\varepsilon)} and 0 = - \\frac{d^{2}}{d \\varepsilon^{2}} \\log{(e^{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\varepsilon', commutative=True)), log(exp(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Derivative(log(exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('t')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Integer(0))"], [["minus", 4, "Derivative(Function('t')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2)))"], "Equality(Integer(0), Mul(Integer(-1), Derivative(Function('t')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\phi_{2}{(a^{\\dagger},\\mathbf{r})} = \\mathbf{r} e^{a^{\\dagger}}, then derive \\frac{\\partial}{\\partial \\mathbf{r}} \\phi_{2}{(a^{\\dagger},\\mathbf{r})} = e^{a^{\\dagger}}, then obtain e^{a^{\\dagger}} (\\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}})^{2} = e^{3 a^{\\dagger}}", "derivation": "\\phi_{2}{(a^{\\dagger},\\mathbf{r})} = \\mathbf{r} e^{a^{\\dagger}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\phi_{2}{(a^{\\dagger},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\phi_{2}{(a^{\\dagger},\\mathbf{r})} = e^{a^{\\dagger}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}} = e^{a^{\\dagger}} and e^{a^{\\dagger}} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}} = e^{2 a^{\\dagger}} and e^{2 a^{\\dagger}} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}} = e^{3 a^{\\dagger}} and e^{a^{\\dagger}} (\\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} e^{a^{\\dagger}})^{2} = e^{3 a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 4, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(exp(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 5, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), exp(Mul(Integer(3), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(exp(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(2))), exp(Mul(Integer(3), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\theta_2,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta_2 and \\operatorname{M_{E}}{(\\mathbf{A})} = \\int \\log{(\\mathbf{A})} d\\mathbf{A}, then obtain - \\mathbf{J}_P \\theta_2 + \\operatorname{M_{E}}^{\\mathbf{A}}{(\\mathbf{A})} = - \\mathbf{J}_P \\theta_2 + (\\int \\log{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}}", "derivation": "\\mathbf{P}{(\\theta_2,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta_2 and \\operatorname{M_{E}}{(\\mathbf{A})} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\operatorname{M_{E}}^{\\mathbf{A}}{(\\mathbf{A})} = (\\int \\log{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} and \\operatorname{M_{E}}^{\\mathbf{A}}{(\\mathbf{A})} - \\mathbf{P}{(\\theta_2,\\mathbf{J}_P)} = - \\mathbf{P}{(\\theta_2,\\mathbf{J}_P)} + (\\int \\log{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} and - \\mathbf{J}_P \\theta_2 + \\operatorname{M_{E}}^{\\mathbf{A}}{(\\mathbf{A})} = - \\mathbf{J}_P \\theta_2 + (\\int \\log{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], ["get_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 3, "Function('\\\\mathbf{P}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Pow(Function('M_E')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('M_E')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then obtain \\int \\sin{(V_{\\mathbf{B}} \\operatorname{v_{t}}{(V_{\\mathbf{B}})})} dV_{\\mathbf{B}} = \\int \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})} dV_{\\mathbf{B}}", "derivation": "\\operatorname{v_{t}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and V_{\\mathbf{B}} \\operatorname{v_{t}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} and \\sin{(V_{\\mathbf{B}} \\operatorname{v_{t}}{(V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})} and \\int \\sin{(V_{\\mathbf{B}} \\operatorname{v_{t}}{(V_{\\mathbf{B}})})} dV_{\\mathbf{B}} = \\int \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_t')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_t')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["integrate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_t')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given l{(u)} = \\cos{(\\sin{(u)})}, then obtain l{(u)} \\int (l{(u)} + \\cos{(\\sin{(u)})}) du = l{(u)} \\int 2 \\cos{(\\sin{(u)})} du", "derivation": "l{(u)} = \\cos{(\\sin{(u)})} and l{(u)} + \\cos{(\\sin{(u)})} = 2 \\cos{(\\sin{(u)})} and \\int (l{(u)} + \\cos{(\\sin{(u)})}) du = \\int 2 \\cos{(\\sin{(u)})} du and \\cos{(\\sin{(u)})} \\int (l{(u)} + \\cos{(\\sin{(u)})}) du = \\cos{(\\sin{(u)})} \\int 2 \\cos{(\\sin{(u)})} du and l{(u)} \\int (l{(u)} + \\cos{(\\sin{(u)})}) du = l{(u)} \\int 2 \\cos{(\\sin{(u)})} du", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True))))"], [["add", 1, "cos(sin(Symbol('u', commutative=True)))"], "Equality(Add(Function('l')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('u', commutative=True)))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Function('l')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(2), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "cos(sin(Symbol('u', commutative=True)))"], "Equality(Mul(cos(sin(Symbol('u', commutative=True))), Integral(Add(Function('l')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), Mul(cos(sin(Symbol('u', commutative=True))), Integral(Mul(Integer(2), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('l')(Symbol('u', commutative=True)), Integral(Add(Function('l')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), Mul(Function('l')(Symbol('u', commutative=True)), Integral(Mul(Integer(2), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(A_{z},m_{s})} = \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s}, then obtain \\int 4 \\operatorname{t_{1}}^{2}{(A_{z},m_{s})} dA_{z} = \\int (\\operatorname{t_{1}}{(A_{z},m_{s})} + \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s})^{2} dA_{z}", "derivation": "\\operatorname{t_{1}}{(A_{z},m_{s})} = \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s} and 2 \\operatorname{t_{1}}{(A_{z},m_{s})} = \\operatorname{t_{1}}{(A_{z},m_{s})} + \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s} and 4 \\operatorname{t_{1}}^{2}{(A_{z},m_{s})} = (\\operatorname{t_{1}}{(A_{z},m_{s})} + \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s})^{2} and \\int 4 \\operatorname{t_{1}}^{2}{(A_{z},m_{s})} dA_{z} = \\int (\\operatorname{t_{1}}{(A_{z},m_{s})} + \\frac{\\partial}{\\partial m_{s}} A_{z} m_{s})^{2} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["add", 1, "Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(2), Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True))), Add(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Integer(2))), Pow(Add(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Integer(2)))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Integer(2))), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(Add(Function('t_1')(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Integer(2)), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\Omega)} = \\log{(\\cos{(\\Omega)})}, then obtain \\rho_b + 2 \\int \\operatorname{F_{c}}{(\\Omega)} d\\Omega + 2 \\int - \\log{(\\cos{(\\Omega)})} d\\Omega = \\int 0 d\\Omega", "derivation": "\\operatorname{F_{c}}{(\\Omega)} = \\log{(\\cos{(\\Omega)})} and \\operatorname{F_{c}}{(\\Omega)} - \\log{(\\cos{(\\Omega)})} = 0 and 2 \\operatorname{F_{c}}{(\\Omega)} - 2 \\log{(\\cos{(\\Omega)})} = \\operatorname{F_{c}}{(\\Omega)} - \\log{(\\cos{(\\Omega)})} and 2 \\operatorname{F_{c}}{(\\Omega)} - 2 \\log{(\\cos{(\\Omega)})} = 0 and \\int (2 \\operatorname{F_{c}}{(\\Omega)} - 2 \\log{(\\cos{(\\Omega)})}) d\\Omega = \\int 0 d\\Omega and \\rho_b + 2 \\int \\operatorname{F_{c}}{(\\Omega)} d\\Omega + 2 \\int - \\log{(\\cos{(\\Omega)})} d\\Omega = \\int 0 d\\Omega", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))))"], [["minus", 1, "log(cos(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\Omega', commutative=True))))), Integer(0))"], [["add", 2, "Add(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\Omega', commutative=True)))))"], "Equality(Add(Mul(Integer(2), Function('F_c')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integer(2), log(cos(Symbol('\\\\Omega', commutative=True))))), Add(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Function('F_c')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integer(2), log(cos(Symbol('\\\\Omega', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('F_c')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integer(2), log(cos(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(2), Add(Integral(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(-1), log(cos(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True)))))), Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(t,\\Omega)} = \\Omega + t, then derive E_{\\lambda} \\theta_{2}^{\\Omega}{(t,\\Omega)} = E_{\\lambda} (\\Omega + t)^{\\Omega}, then obtain E_{\\lambda} \\theta_{2}^{\\Omega}{(t,\\Omega)} - 1 = E_{\\lambda} (\\Omega + t)^{\\Omega} - 1", "derivation": "\\theta_{2}{(t,\\Omega)} = \\Omega + t and \\theta_{2}^{\\Omega}{(t,\\Omega)} = (\\Omega + t)^{\\Omega} and E_{\\lambda} f \\theta_{2}^{\\Omega}{(t,\\Omega)} = E_{\\lambda} f (\\Omega + t)^{\\Omega} and E_{\\lambda} f \\theta_{2}^{\\Omega}{(t,\\Omega)} + \\theta_{2}^{\\Omega}{(t,\\Omega)} = E_{\\lambda} f (\\Omega + t)^{\\Omega} + \\theta_{2}^{\\Omega}{(t,\\Omega)} and \\frac{\\partial}{\\partial f} (E_{\\lambda} f \\theta_{2}^{\\Omega}{(t,\\Omega)} + \\theta_{2}^{\\Omega}{(t,\\Omega)}) = \\frac{\\partial}{\\partial f} (E_{\\lambda} f (\\Omega + t)^{\\Omega} + \\theta_{2}^{\\Omega}{(t,\\Omega)}) and E_{\\lambda} \\theta_{2}^{\\Omega}{(t,\\Omega)} = E_{\\lambda} (\\Omega + t)^{\\Omega} and E_{\\lambda} \\theta_{2}^{\\Omega}{(t,\\Omega)} - 1 = E_{\\lambda} (\\Omega + t)^{\\Omega} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["add", 3, "Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["minus", 6, 1], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(-1)), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_P{(i)} = \\cos{(e^{i})}, then obtain \\cos{(i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\mathbf{J}_P{(i)} \\cos{(e^{i})})^{i})} = \\cos{(i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\cos^{2}{(e^{i})})^{i})}", "derivation": "\\mathbf{J}_P{(i)} = \\cos{(e^{i})} and \\mathbf{J}_P{(i)} \\cos{(e^{i})} = \\cos^{2}{(e^{i})} and i \\mathbf{J}_P{(i)} \\cos{(e^{i})} = i \\cos^{2}{(e^{i})} and (i \\mathbf{J}_P{(i)} \\cos{(e^{i})})^{i} = (i \\cos^{2}{(e^{i})})^{i} and i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\mathbf{J}_P{(i)} \\cos{(e^{i})})^{i} = i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\cos^{2}{(e^{i})})^{i} and \\cos{(i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\mathbf{J}_P{(i)} \\cos{(e^{i})})^{i})} = \\cos{(i \\mathbf{J}_P{(i)} \\cos{(e^{i})} + (i \\cos^{2}{(e^{i})})^{i})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True))))"], [["times", 1, "cos(exp(Symbol('i', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Pow(cos(exp(Symbol('i', commutative=True))), Integer(2)))"], [["times", 2, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Mul(Symbol('i', commutative=True), Pow(cos(exp(Symbol('i', commutative=True))), Integer(2))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Pow(cos(exp(Symbol('i', commutative=True))), Integer(2))), Symbol('i', commutative=True)))"], [["add", 4, "Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True))))"], "Equality(Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Pow(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Symbol('i', commutative=True))), Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Pow(Mul(Symbol('i', commutative=True), Pow(cos(exp(Symbol('i', commutative=True))), Integer(2))), Symbol('i', commutative=True))))"], [["cos", 5], "Equality(cos(Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Pow(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Symbol('i', commutative=True)))), cos(Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True)))), Pow(Mul(Symbol('i', commutative=True), Pow(cos(exp(Symbol('i', commutative=True))), Integer(2))), Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(r_{0},\\hat{p}_0)} = \\hat{p}_0 r_{0}, then derive \\frac{\\partial}{\\partial r_{0}} \\mathbf{M}{(r_{0},\\hat{p}_0)} = \\hat{p}_0, then obtain \\frac{\\partial}{\\partial r_{0}} \\hat{p}_0 r_{0} = \\hat{p}_0", "derivation": "\\mathbf{M}{(r_{0},\\hat{p}_0)} = \\hat{p}_0 r_{0} and \\frac{\\partial}{\\partial r_{0}} \\mathbf{M}{(r_{0},\\hat{p}_0)} = \\frac{\\partial}{\\partial r_{0}} \\hat{p}_0 r_{0} and \\frac{\\partial}{\\partial r_{0}} \\mathbf{M}{(r_{0},\\hat{p}_0)} = \\hat{p}_0 and \\frac{\\partial}{\\partial r_{0}} \\hat{p}_0 r_{0} = \\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))"]]}, {"prompt": "Given m{(E_{\\lambda})} = \\log{(\\log{(E_{\\lambda})})}, then derive \\int m{(E_{\\lambda})} dE_{\\lambda} = E_{\\lambda} \\log{(\\log{(E_{\\lambda})})} + f - \\operatorname{li}{(E_{\\lambda})}, then obtain \\Omega (E_{\\lambda} \\log{(\\log{(E_{\\lambda})})} + f - \\operatorname{li}{(E_{\\lambda})}) = \\Omega \\int \\log{(\\log{(E_{\\lambda})})} dE_{\\lambda}", "derivation": "m{(E_{\\lambda})} = \\log{(\\log{(E_{\\lambda})})} and \\int m{(E_{\\lambda})} dE_{\\lambda} = \\int \\log{(\\log{(E_{\\lambda})})} dE_{\\lambda} and \\Omega \\int m{(E_{\\lambda})} dE_{\\lambda} = \\Omega \\int \\log{(\\log{(E_{\\lambda})})} dE_{\\lambda} and \\int m{(E_{\\lambda})} dE_{\\lambda} = E_{\\lambda} \\log{(\\log{(E_{\\lambda})})} + f - \\operatorname{li}{(E_{\\lambda})} and \\Omega (E_{\\lambda} \\log{(\\log{(E_{\\lambda})})} + f - \\operatorname{li}{(E_{\\lambda})}) = \\Omega \\int \\log{(\\log{(E_{\\lambda})})} dE_{\\lambda}", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('E_{\\\\lambda}', commutative=True)), log(log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('m')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(log(log(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Integral(Function('m')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Integral(log(log(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), log(log(Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), log(log(Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('E_{\\\\lambda}', commutative=True))))), Mul(Symbol('\\\\Omega', commutative=True), Integral(log(log(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(c)} = \\sin{(\\sin{(c)})}, then obtain \\frac{\\mathbf{J}_P{(c)}}{\\sin{(c)}} + \\sin{(c)} + \\frac{1}{\\sin{(c)}} = \\sin{(c)} + \\frac{\\sin{(\\sin{(c)})}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}}", "derivation": "\\mathbf{J}_P{(c)} = \\sin{(\\sin{(c)})} and \\frac{\\mathbf{J}_P{(c)}}{\\sin{(c)}} = \\frac{\\sin{(\\sin{(c)})}}{\\sin{(c)}} and \\frac{\\mathbf{J}_P{(c)}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}} = \\frac{\\sin{(\\sin{(c)})}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}} and \\frac{\\mathbf{J}_P{(c)}}{\\sin{(c)}} + \\sin{(c)} + \\frac{1}{\\sin{(c)}} = \\sin{(c)} + \\frac{\\sin{(\\sin{(c)})}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))))"], [["divide", 1, "sin(Symbol('c', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), sin(sin(Symbol('c', commutative=True)))))"], [["add", 2, "Pow(sin(Symbol('c', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('\\\\mathbf{J}_P')(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Add(Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), sin(sin(Symbol('c', commutative=True)))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"], [["add", 3, "sin(Symbol('c', commutative=True))"], "Equality(Add(Mul(Function('\\\\mathbf{J}_P')(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), sin(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Add(sin(Symbol('c', commutative=True)), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), sin(sin(Symbol('c', commutative=True)))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given W{(a^{\\dagger})} = \\cos{(\\cos{(a^{\\dagger})})}, then obtain \\int (a^{\\dagger} + W{(a^{\\dagger})}) da^{\\dagger} + \\frac{1}{W{(a^{\\dagger})}} = \\int (a^{\\dagger} + \\cos{(\\cos{(a^{\\dagger})})}) da^{\\dagger} + \\frac{1}{W{(a^{\\dagger})}}", "derivation": "W{(a^{\\dagger})} = \\cos{(\\cos{(a^{\\dagger})})} and a^{\\dagger} + W{(a^{\\dagger})} = a^{\\dagger} + \\cos{(\\cos{(a^{\\dagger})})} and \\int (a^{\\dagger} + W{(a^{\\dagger})}) da^{\\dagger} = \\int (a^{\\dagger} + \\cos{(\\cos{(a^{\\dagger})})}) da^{\\dagger} and \\int (a^{\\dagger} + W{(a^{\\dagger})}) da^{\\dagger} + \\frac{1}{\\cos{(\\cos{(a^{\\dagger})})}} = \\int (a^{\\dagger} + \\cos{(\\cos{(a^{\\dagger})})}) da^{\\dagger} + \\frac{1}{\\cos{(\\cos{(a^{\\dagger})})}} and \\int (a^{\\dagger} + W{(a^{\\dagger})}) da^{\\dagger} + \\frac{1}{W{(a^{\\dagger})}} = \\int (a^{\\dagger} + \\cos{(\\cos{(a^{\\dagger})})}) da^{\\dagger} + \\frac{1}{W{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), cos(cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('W')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('W')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 3, "Pow(cos(cos(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('W')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Pow(cos(cos(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))), Add(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Pow(cos(cos(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('W')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Add(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(J,\\varphi)} = - J + \\varphi, then obtain (- J + \\varphi + 1) \\eta^{\\prime}{(J,\\varphi)} + 1 = (- J + \\varphi) (- J + \\varphi + 1) + 1", "derivation": "\\eta^{\\prime}{(J,\\varphi)} = - J + \\varphi and \\eta^{\\prime}{(J,\\varphi)} + 1 = - J + \\varphi + 1 and (\\eta^{\\prime}{(J,\\varphi)} + 1) \\eta^{\\prime}{(J,\\varphi)} = (- J + \\varphi) (\\eta^{\\prime}{(J,\\varphi)} + 1) and (\\eta^{\\prime}{(J,\\varphi)} + 1) \\eta^{\\prime}{(J,\\varphi)} + 1 = (- J + \\varphi) (\\eta^{\\prime}{(J,\\varphi)} + 1) + 1 and (- J + \\varphi + 1) \\eta^{\\prime}{(J,\\varphi)} + 1 = (- J + \\varphi) (- J + \\varphi + 1) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True), Integer(1)))"], [["times", 1, "Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1))"], "Equality(Mul(Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True)), Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(1)), Add(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True)), Add(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(1)), Add(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\eta{(x,\\phi_1)} = \\int (\\phi_1 + x) d\\phi_1, then derive - \\phi_1 + \\eta{(x,\\phi_1)} = A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1, then obtain \\frac{2 (A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1)}{\\phi_1^{2}} = \\frac{2 (- \\phi_1 + \\eta{(x,\\phi_1)})}{\\phi_1^{2}}", "derivation": "\\eta{(x,\\phi_1)} = \\int (\\phi_1 + x) d\\phi_1 and - \\phi_1 + \\eta{(x,\\phi_1)} = - \\phi_1 + \\int (\\phi_1 + x) d\\phi_1 and - \\phi_1 + \\eta{(x,\\phi_1)} = A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1 and A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1 = - \\phi_1 + \\int (\\phi_1 + x) d\\phi_1 and \\frac{2 (A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1)}{\\phi_1^{2}} = \\frac{2 (- \\phi_1 + \\int (\\phi_1 + x) d\\phi_1)}{\\phi_1^{2}} and \\frac{2 (A_{y} + \\frac{\\phi_1^{2}}{2} + \\phi_1 x - \\phi_1)}{\\phi_1^{2}} = \\frac{2 (- \\phi_1 + \\eta{(x,\\phi_1)})}{\\phi_1^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('\\\\eta')(Symbol('x', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('\\\\eta')(Symbol('x', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["divide", 4, "Mul(Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('\\\\eta')(Symbol('x', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(f,\\mu_0)} = \\log{(\\mu_0^{f})}, then obtain \\frac{\\partial}{\\partial f} \\dot{\\mathbf{r}}{(f,\\mu_0)} = \\log{(\\mu_0)}", "derivation": "\\dot{\\mathbf{r}}{(f,\\mu_0)} = \\log{(\\mu_0^{f})} and - \\mu_0 + \\dot{\\mathbf{r}}{(f,\\mu_0)} = - \\mu_0 + \\log{(\\mu_0^{f})} and \\frac{\\partial}{\\partial f} (- \\mu_0 + \\dot{\\mathbf{r}}{(f,\\mu_0)}) = \\frac{\\partial}{\\partial f} (- \\mu_0 + \\log{(\\mu_0^{f})}) and \\frac{\\partial}{\\partial f} \\dot{\\mathbf{r}}{(f,\\mu_0)} = \\log{(\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('f', commutative=True))))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('f', commutative=True)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), log(Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(i,S,\\mathbf{J}_M)} = i (S - \\mathbf{J}_M) and \\operatorname{M_{E}}{(q,f)} = f q, then obtain \\frac{f i q (S - \\mathbf{J}_M) \\ddot{x}{(i,S,\\mathbf{J}_M)}}{S} = \\frac{f i^{2} q (S - \\mathbf{J}_M)^{2}}{S}", "derivation": "\\ddot{x}{(i,S,\\mathbf{J}_M)} = i (S - \\mathbf{J}_M) and \\operatorname{M_{E}}{(q,f)} = f q and \\frac{i (S - \\mathbf{J}_M) \\ddot{x}{(i,S,\\mathbf{J}_M)}}{S} = \\frac{i^{2} (S - \\mathbf{J}_M)^{2}}{S} and \\frac{i (S - \\mathbf{J}_M) \\operatorname{M_{E}}{(q,f)} \\ddot{x}{(i,S,\\mathbf{J}_M)}}{S} = \\frac{i^{2} (S - \\mathbf{J}_M)^{2} \\operatorname{M_{E}}{(q,f)}}{S} and \\frac{f i q (S - \\mathbf{J}_M) \\ddot{x}{(i,S,\\mathbf{J}_M)}}{S} = \\frac{f i^{2} q (S - \\mathbf{J}_M)^{2}}{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('i', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('i', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)))))"], ["get_premise", "Equality(Function('M_E')(Symbol('q', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('i', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('i', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Function('\\\\ddot{x}')(Symbol('i', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2))))"], [["times", 3, "Function('M_E')(Symbol('q', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('i', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Function('M_E')(Symbol('q', commutative=True), Symbol('f', commutative=True)), Function('\\\\ddot{x}')(Symbol('i', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)), Function('M_E')(Symbol('q', commutative=True), Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('q', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Function('\\\\ddot{x}')(Symbol('i', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f', commutative=True), Pow(Symbol('i', commutative=True), Integer(2)), Symbol('q', commutative=True), Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2))))"]]}, {"prompt": "Given A{(\\hat{X},l)} = \\frac{\\partial}{\\partial l} (\\hat{X} + l), then derive A{(\\hat{X},l)} = 1, then obtain - l + \\frac{\\partial}{\\partial l} \\frac{(\\frac{\\partial}{\\partial l} (\\hat{X} + l))^{2}}{l^{2}} = - l + \\frac{\\partial}{\\partial l} \\frac{\\frac{\\partial}{\\partial l} (\\hat{X} + l)}{l^{2}}", "derivation": "A{(\\hat{X},l)} = \\frac{\\partial}{\\partial l} (\\hat{X} + l) and \\frac{A{(\\hat{X},l)}}{\\frac{\\partial}{\\partial l} (\\hat{X} + l)} = 1 and A{(\\hat{X},l)} = 1 and \\frac{\\partial}{\\partial l} (\\hat{X} + l) = 1 and \\frac{\\frac{\\partial}{\\partial l} (\\hat{X} + l)}{l} = \\frac{1}{l} and \\frac{(\\frac{\\partial}{\\partial l} (\\hat{X} + l))^{2}}{l^{2}} = \\frac{\\frac{\\partial}{\\partial l} (\\hat{X} + l)}{l^{2}} and \\frac{\\partial}{\\partial l} \\frac{(\\frac{\\partial}{\\partial l} (\\hat{X} + l))^{2}}{l^{2}} = \\frac{\\partial}{\\partial l} \\frac{\\frac{\\partial}{\\partial l} (\\hat{X} + l)}{l^{2}} and - l + \\frac{\\partial}{\\partial l} \\frac{(\\frac{\\partial}{\\partial l} (\\hat{X} + l))^{2}}{l^{2}} = - l + \\frac{\\partial}{\\partial l} \\frac{\\frac{\\partial}{\\partial l} (\\hat{X} + l)}{l^{2}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Function('A')(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Function('A')(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('l', commutative=True), Integer(-1)))"], [["times", 5, "Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Pow(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Pow(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 7, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Pow(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-2)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(\\mu,t_{2})} = \\mu t_{2} and \\mathbf{g}{(\\mu,t_{2})} = \\mu t_{2}, then obtain - \\frac{\\frac{\\partial}{\\partial \\mu} \\delta{(\\mu,t_{2})}}{\\delta^{2}{(\\mu,t_{2})}} = - \\frac{\\frac{\\partial}{\\partial \\mu} \\mathbf{g}{(\\mu,t_{2})}}{\\mathbf{g}^{2}{(\\mu,t_{2})}}", "derivation": "\\delta{(\\mu,t_{2})} = \\mu t_{2} and \\frac{1}{\\mu t_{2}} = \\frac{1}{\\delta{(\\mu,t_{2})}} and \\mathbf{g}{(\\mu,t_{2})} = \\mu t_{2} and \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\mu t_{2}} = \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\delta{(\\mu,t_{2})}} and \\frac{1}{\\mathbf{g}{(\\mu,t_{2})}} = \\frac{1}{\\delta{(\\mu,t_{2})}} and \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\mu t_{2}} = \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\mathbf{g}{(\\mu,t_{2})}} and \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\delta{(\\mu,t_{2})}} = \\frac{\\partial}{\\partial \\mu} \\frac{1}{\\mathbf{g}{(\\mu,t_{2})}} and - \\frac{\\frac{\\partial}{\\partial \\mu} \\delta{(\\mu,t_{2})}}{\\delta^{2}{(\\mu,t_{2})}} = - \\frac{\\frac{\\partial}{\\partial \\mu} \\mathbf{g}{(\\mu,t_{2})}}{\\mathbf{g}^{2}{(\\mu,t_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True), Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Pow(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Pow(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Pow(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-2)), Derivative(Function('\\\\delta')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Integer(-2)), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(\\dot{\\mathbf{r}},G)} = \\sin^{G}{(\\dot{\\mathbf{r}})} and \\mathbf{f}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})}, then obtain G = G - \\mathbf{f}^{G}{(\\dot{\\mathbf{r}})} + \\sin^{G}{(\\dot{\\mathbf{r}})}", "derivation": "h{(\\dot{\\mathbf{r}},G)} = \\sin^{G}{(\\dot{\\mathbf{r}})} and 0 = - h{(\\dot{\\mathbf{r}},G)} + \\sin^{G}{(\\dot{\\mathbf{r}})} and G = G - h{(\\dot{\\mathbf{r}},G)} + \\sin^{G}{(\\dot{\\mathbf{r}})} and \\mathbf{f}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})} and G = G + \\mathbf{f}^{G}{(\\dot{\\mathbf{r}})} - h{(\\dot{\\mathbf{r}},G)} and G - \\sin^{G}{(\\dot{\\mathbf{r}})} = G + \\mathbf{f}^{G}{(\\dot{\\mathbf{r}})} - h{(\\dot{\\mathbf{r}},G)} - \\sin^{G}{(\\dot{\\mathbf{r}})} and G - \\mathbf{f}^{G}{(\\dot{\\mathbf{r}})} = G - h{(\\dot{\\mathbf{r}},G)} and G = G - \\mathbf{f}^{G}{(\\dot{\\mathbf{r}})} + \\sin^{G}{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)))"], [["minus", 1, "Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True))), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))))"], [["add", 2, "Symbol('G', commutative=True)"], "Equality(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True))), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)))))"], [["minus", 5, "Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)))), Add(Symbol('G', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Integer(-1), Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))), Pow(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))))"]]}, {"prompt": "Given k{(W)} = e^{W}, then derive \\int k{(W)} dW = k + e^{W}, then obtain \\int e^{W} dW = k + k{(W)}", "derivation": "k{(W)} = e^{W} and \\int k{(W)} dW = \\int e^{W} dW and \\int k{(W)} dW = k + e^{W} and \\int e^{W} dW = k + e^{W} and \\int e^{W} dW = k + k{(W)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('k')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('k', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('k', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('k', commutative=True), Function('k')(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(S,\\mathbf{J}_M)} = S + \\mathbf{J}_M, then obtain \\cos{(2 \\eta^{\\prime}{(S,\\mathbf{J}_M)})} = \\cos{(2 S + 2 \\mathbf{J}_M)}", "derivation": "\\eta^{\\prime}{(S,\\mathbf{J}_M)} = S + \\mathbf{J}_M and \\mathbf{J}_M + \\eta^{\\prime}{(S,\\mathbf{J}_M)} = S + 2 \\mathbf{J}_M and 2 \\eta^{\\prime}{(S,\\mathbf{J}_M)} = S + \\mathbf{J}_M + \\eta^{\\prime}{(S,\\mathbf{J}_M)} and 2 \\eta^{\\prime}{(S,\\mathbf{J}_M)} = 2 S + 2 \\mathbf{J}_M and \\cos{(2 \\eta^{\\prime}{(S,\\mathbf{J}_M)})} = \\cos{(2 S + 2 \\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["cos", 4], "Equality(cos(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))), cos(Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given L{(C,q)} = C \\cos{(q)}, then obtain \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC + \\int \\frac{\\partial}{\\partial C} (q + L^{C}{(C,q)}) dC - 2 = 2 \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC - 2", "derivation": "L{(C,q)} = C \\cos{(q)} and L^{C}{(C,q)} = (C \\cos{(q)})^{C} and q + L^{C}{(C,q)} = q + (C \\cos{(q)})^{C} and \\frac{\\partial}{\\partial C} (q + L^{C}{(C,q)}) = \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) and \\int \\frac{\\partial}{\\partial C} (q + L^{C}{(C,q)}) dC = \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC and \\int \\frac{\\partial}{\\partial C} (q + L^{C}{(C,q)}) dC - 1 = \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC - 1 and \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC + \\int \\frac{\\partial}{\\partial C} (q + L^{C}{(C,q)}) dC - 2 = 2 \\int \\frac{\\partial}{\\partial C} (q + (C \\cos{(q)})^{C}) dC - 2", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True)))"], [["add", 2, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True))), Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Symbol('q', commutative=True), Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))))"], [["minus", 5, 1], "Equality(Add(Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integer(-1)), Add(Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integer(-1)))"], [["add", 6, "Add(Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Function('L')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integer(-2)), Add(Mul(Integer(2), Integral(Derivative(Add(Symbol('q', commutative=True), Pow(Mul(Symbol('C', commutative=True), cos(Symbol('q', commutative=True))), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Integer(-2)))"]]}, {"prompt": "Given \\dot{z}{(\\hat{x},A_{y})} = \\hat{x}^{A_{y}} and \\operatorname{r_{0}}{(A_{y})} = A_{y}, then obtain \\int - A_{y} d\\operatorname{r_{0}}{(A_{y})} = \\int (- A_{y} + \\hat{x}^{A_{y}} - \\dot{z}{(\\hat{x},A_{y})}) d\\operatorname{r_{0}}{(A_{y})}", "derivation": "\\dot{z}{(\\hat{x},A_{y})} = \\hat{x}^{A_{y}} and 0 = \\hat{x}^{A_{y}} - \\dot{z}{(\\hat{x},A_{y})} and \\operatorname{r_{0}}{(A_{y})} = A_{y} and - A_{y} = - A_{y} + \\hat{x}^{A_{y}} - \\dot{z}{(\\hat{x},A_{y})} and \\int - A_{y} dA_{y} = \\int (- A_{y} + \\hat{x}^{A_{y}} - \\dot{z}{(\\hat{x},A_{y})}) dA_{y} and \\int - A_{y} d\\operatorname{r_{0}}{(A_{y})} = \\int (- A_{y} + \\hat{x}^{A_{y}} - \\dot{z}{(\\hat{x},A_{y})}) d\\operatorname{r_{0}}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)))"], [["minus", 1, "Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["minus", 2, "Symbol('A_y', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('A_y', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)))))"], [["integrate", 4, "Symbol('A_y', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Mul(Integer(-1), Symbol('A_y', commutative=True)), Tuple(Function('r_0')(Symbol('A_y', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_y', commutative=True)))), Tuple(Function('r_0')(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given H{(\\mu)} = \\sin{(\\log{(\\mu)})}, then obtain 5 \\frac{d}{d \\mu} H{(\\mu)} + \\frac{\\cos{(\\log{(\\mu)})}}{\\mu} = 4 \\frac{d}{d \\mu} H{(\\mu)} + \\frac{2 \\cos{(\\log{(\\mu)})}}{\\mu}", "derivation": "H{(\\mu)} = \\sin{(\\log{(\\mu)})} and 2 H{(\\mu)} = H{(\\mu)} + \\sin{(\\log{(\\mu)})} and 4 H{(\\mu)} = 3 H{(\\mu)} + \\sin{(\\log{(\\mu)})} and 5 H{(\\mu)} + \\sin{(\\log{(\\mu)})} = 4 H{(\\mu)} + 2 \\sin{(\\log{(\\mu)})} and \\frac{d}{d \\mu} (5 H{(\\mu)} + \\sin{(\\log{(\\mu)})}) = \\frac{d}{d \\mu} (4 H{(\\mu)} + 2 \\sin{(\\log{(\\mu)})}) and 5 \\frac{d}{d \\mu} H{(\\mu)} + \\frac{\\cos{(\\log{(\\mu)})}}{\\mu} = 4 \\frac{d}{d \\mu} H{(\\mu)} + \\frac{2 \\cos{(\\log{(\\mu)})}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mu', commutative=True)), sin(log(Symbol('\\\\mu', commutative=True))))"], [["add", 1, "Function('H')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(2), Function('H')(Symbol('\\\\mu', commutative=True))), Add(Function('H')(Symbol('\\\\mu', commutative=True)), sin(log(Symbol('\\\\mu', commutative=True)))))"], [["add", 2, "Mul(Integer(2), Function('H')(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Integer(4), Function('H')(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(3), Function('H')(Symbol('\\\\mu', commutative=True))), sin(log(Symbol('\\\\mu', commutative=True)))))"], [["add", 3, "Add(Function('H')(Symbol('\\\\mu', commutative=True)), sin(log(Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(5), Function('H')(Symbol('\\\\mu', commutative=True))), sin(log(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(4), Function('H')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), sin(log(Symbol('\\\\mu', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(5), Function('H')(Symbol('\\\\mu', commutative=True))), sin(log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(4), Function('H')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), sin(log(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(5), Derivative(Function('H')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(log(Symbol('\\\\mu', commutative=True))))), Add(Mul(Integer(4), Derivative(Function('H')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(log(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} = - V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon}, then obtain - \\frac{\\int J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} d\\sigma_x}{V_{\\mathbf{B}}^{2}} = - \\frac{\\int (- V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon}) d\\sigma_x}{V_{\\mathbf{B}}^{2}}", "derivation": "J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} = - V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon} and \\int J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} d\\sigma_x = \\int (- V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon}) d\\sigma_x and - \\frac{\\int J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} d\\sigma_x}{V_{\\mathbf{B}}} = - \\frac{\\int (- V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon}) d\\sigma_x}{V_{\\mathbf{B}}} and - \\frac{\\int J{(V_{\\mathbf{B}},g^{\\prime}_{\\varepsilon},\\sigma_x)} d\\sigma_x}{V_{\\mathbf{B}}^{2}} = - \\frac{\\int (- V_{\\mathbf{B}} + \\sigma_x + g^{\\prime}_{\\varepsilon}) d\\sigma_x}{V_{\\mathbf{B}}^{2}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Integral(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 3, "Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-2)), Integral(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-2)), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{g},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}}, then obtain (\\frac{\\operatorname{m_{s}}^{\\mathbf{M}}{(\\mathbf{g},\\mathbf{M})}}{\\mathbf{M}})^{\\mathbf{g}} = (\\frac{(\\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}})^{\\mathbf{M}}}{\\mathbf{M}})^{\\mathbf{g}}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{g},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}} and \\operatorname{m_{s}}^{\\mathbf{M}}{(\\mathbf{g},\\mathbf{M})} = (\\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}})^{\\mathbf{M}} and \\frac{\\operatorname{m_{s}}^{\\mathbf{M}}{(\\mathbf{g},\\mathbf{M})}}{\\mathbf{M}} = \\frac{(\\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}})^{\\mathbf{M}}}{\\mathbf{M}} and (\\frac{\\operatorname{m_{s}}^{\\mathbf{M}}{(\\mathbf{g},\\mathbf{M})}}{\\mathbf{M}})^{\\mathbf{g}} = (\\frac{(\\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\mathbf{g}})^{\\mathbf{M}}}{\\mathbf{M}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Function('m_s')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Function('m_s')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\theta{(f_{\\mathbf{p}},L)} = L^{f_{\\mathbf{p}}}, then derive \\eta^{\\prime} + \\theta{(f_{\\mathbf{p}},L)} = L^{f_{\\mathbf{p}}} + T, then obtain L^{f_{\\mathbf{p}}} + T = T + \\theta{(f_{\\mathbf{p}},L)}", "derivation": "\\theta{(f_{\\mathbf{p}},L)} = L^{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}},L)} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} L^{f_{\\mathbf{p}}} and \\int \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}},L)} df_{\\mathbf{p}} = \\int \\frac{\\partial}{\\partial f_{\\mathbf{p}}} L^{f_{\\mathbf{p}}} df_{\\mathbf{p}} and \\eta^{\\prime} + \\theta{(f_{\\mathbf{p}},L)} = L^{f_{\\mathbf{p}}} + T and \\eta^{\\prime} + \\theta{(f_{\\mathbf{p}},L)} = T + \\theta{(f_{\\mathbf{p}},L)} and L^{f_{\\mathbf{p}}} + T = T + \\theta{(f_{\\mathbf{p}},L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Pow(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Derivative(Pow(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True))), Add(Symbol('T', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given v{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}}, then obtain \\hat{\\mathbf{r}}^{4} v^{2}{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}}^{5} v{(\\hat{\\mathbf{r}},F_{x})}", "derivation": "v{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}} and \\hat{\\mathbf{r}} v{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}}^{2} and \\hat{\\mathbf{r}}^{2} v{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}}^{3} and \\hat{\\mathbf{r}}^{4} v^{2}{(\\hat{\\mathbf{r}},F_{x})} = F_{x} \\hat{\\mathbf{r}}^{5} v{(\\hat{\\mathbf{r}},F_{x})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))))"], [["times", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(3))))"], [["times", 3, "Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(4)), Pow(Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True)), Integer(2))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(5)), Function('v')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\delta,\\theta_2)} = \\frac{\\theta_2}{\\delta}, then obtain (\\frac{\\partial}{\\partial \\theta_2} \\mathbf{F}{(\\delta,\\theta_2)})^{\\theta_2} = (\\frac{1}{\\delta})^{\\theta_2}", "derivation": "\\mathbf{F}{(\\delta,\\theta_2)} = \\frac{\\theta_2}{\\delta} and \\frac{\\partial}{\\partial \\theta_2} \\mathbf{F}{(\\delta,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\delta} and (\\frac{\\partial}{\\partial \\theta_2} \\mathbf{F}{(\\delta,\\theta_2)})^{\\theta_2} = (\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\delta})^{\\theta_2} and (\\frac{\\partial}{\\partial \\theta_2} \\mathbf{F}{(\\delta,\\theta_2)})^{\\theta_2} = (\\frac{1}{\\delta})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(G,C_{d})} = G^{C_{d}}, then obtain \\frac{- G - 1 + 2 G^{- C_{d}} \\mathbf{P}{(G,C_{d})}}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}} = \\frac{- G + G^{- C_{d}} \\mathbf{P}{(G,C_{d})}}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}}", "derivation": "\\mathbf{P}{(G,C_{d})} = G^{C_{d}} and G^{- C_{d}} \\mathbf{P}{(G,C_{d})} = 1 and - G + G^{- C_{d}} \\mathbf{P}{(G,C_{d})} = 1 - G and \\frac{- G + G^{- C_{d}} \\mathbf{P}{(G,C_{d})}}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}} = \\frac{1 - G}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}} and \\frac{- G - 1 + 2 G^{- C_{d}} \\mathbf{P}{(G,C_{d})}}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}} = \\frac{- G + G^{- C_{d}} \\mathbf{P}{(G,C_{d})}}{- G^{C_{d}} + \\mathbf{P}{(G,C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True)))"], [["divide", 1, "Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Integer(1))"], [["minus", 2, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('G', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True)))), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Integer(-1))), Mul(Add(Integer(1), Mul(Integer(-1), Symbol('G', commutative=True))), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Integer(-1), Mul(Integer(2), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True)))), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{P}')(Symbol('G', commutative=True), Symbol('C_d', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given z{(v_{y})} = e^{v_{y}}, then derive \\int z{(v_{y})} dv_{y} = A_{x} + e^{v_{y}}, then obtain \\frac{d}{d v_{y}} z{(v_{y})} = \\frac{d}{d v_{y}} \\int e^{v_{y}} dv_{y}", "derivation": "z{(v_{y})} = e^{v_{y}} and \\int z{(v_{y})} dv_{y} = \\int e^{v_{y}} dv_{y} and \\int z{(v_{y})} dv_{y} = A_{x} + e^{v_{y}} and A_{x} + e^{v_{y}} = \\int e^{v_{y}} dv_{y} and \\frac{\\partial}{\\partial v_{y}} (A_{x} + e^{v_{y}}) = \\frac{d}{d v_{y}} \\int e^{v_{y}} dv_{y} and \\frac{\\partial}{\\partial v_{y}} (A_{x} + z{(v_{y})}) = \\frac{d}{d v_{y}} \\int e^{v_{y}} dv_{y} and \\frac{d}{d v_{y}} z{(v_{y})} = \\frac{d}{d v_{y}} \\int e^{v_{y}} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('z')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('A_x', commutative=True), exp(Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A_x', commutative=True), exp(Symbol('v_y', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 4, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Symbol('A_x', commutative=True), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Function('z')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('z')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(\\phi_2,g)} = \\phi_2 g, then obtain (\\frac{\\mathbf{p}{(\\phi_2,g)}}{\\phi_2 g^{2}})^{\\phi_2} = (\\frac{1}{g})^{\\phi_2}", "derivation": "\\mathbf{p}{(\\phi_2,g)} = \\phi_2 g and \\frac{\\mathbf{p}{(\\phi_2,g)}}{\\phi_2 g} = 1 and \\frac{\\mathbf{p}{(\\phi_2,g)}}{g} = \\phi_2 and \\frac{\\mathbf{p}{(\\phi_2,g)}}{\\phi_2 g^{2}} = \\frac{1}{g} and (\\frac{\\mathbf{p}{(\\phi_2,g)}}{\\phi_2 g^{2}})^{\\phi_2} = (\\frac{1}{g})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Integer(1))"], [["divide", 2, "Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], [["divide", 3, "Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-2)), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Integer(-1)))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-2)), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(L)} = \\sin{(e^{L})}, then obtain \\frac{d}{d L} \\frac{\\hat{H}_{\\lambda}{(L)}}{L \\sin{(e^{L})}} = \\frac{d}{d L} \\frac{1}{L}", "derivation": "\\hat{H}_{\\lambda}{(L)} = \\sin{(e^{L})} and \\frac{\\hat{H}_{\\lambda}{(L)}}{\\sin{(e^{L})}} = 1 and \\frac{\\hat{H}_{\\lambda}{(L)}}{L \\sin{(e^{L})}} = \\frac{1}{L} and \\frac{d}{d L} \\frac{\\hat{H}_{\\lambda}{(L)}}{L \\sin{(e^{L})}} = \\frac{d}{d L} \\frac{1}{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('L', commutative=True)), sin(exp(Symbol('L', commutative=True))))"], [["divide", 1, "sin(exp(Symbol('L', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('L', commutative=True)), Pow(sin(exp(Symbol('L', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('L', commutative=True)), Pow(sin(exp(Symbol('L', commutative=True))), Integer(-1))), Pow(Symbol('L', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('L', commutative=True)), Pow(sin(exp(Symbol('L', commutative=True))), Integer(-1))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Symbol('L', commutative=True), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(I,s,\\omega)} = (\\omega^{I})^{s}, then derive \\frac{\\partial}{\\partial \\omega} a{(I,s,\\omega)} = \\frac{I s (\\omega^{I})^{s}}{\\omega}, then obtain \\frac{\\partial^{2}}{\\partial s\\partial \\omega} a{(I,s,\\omega)} = \\frac{\\partial}{\\partial s} \\frac{I s (\\omega^{I})^{s}}{\\omega}", "derivation": "a{(I,s,\\omega)} = (\\omega^{I})^{s} and \\frac{\\partial}{\\partial \\omega} a{(I,s,\\omega)} = \\frac{\\partial}{\\partial \\omega} (\\omega^{I})^{s} and \\frac{\\partial}{\\partial \\omega} a{(I,s,\\omega)} = \\frac{I s (\\omega^{I})^{s}}{\\omega} and \\frac{\\partial^{2}}{\\partial s\\partial \\omega} a{(I,s,\\omega)} = \\frac{\\partial}{\\partial s} \\frac{I s (\\omega^{I})^{s}}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('I', commutative=True), Symbol('s', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('I', commutative=True), Symbol('s', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('I', commutative=True), Symbol('s', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Symbol('s', commutative=True))))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('I', commutative=True), Symbol('s', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(i,x)} = i - x, then derive \\frac{\\partial}{\\partial x} \\Psi_{\\lambda}{(i,x)} = -1, then obtain 2 (-1)^{x} \\cos{(\\Psi_{\\lambda}{(i,x)})} = 2 \\cos{(\\Psi_{\\lambda}{(i,x)})} (\\frac{\\partial}{\\partial x} \\Psi_{\\lambda}{(i,x)})^{x}", "derivation": "\\Psi_{\\lambda}{(i,x)} = i - x and \\frac{\\partial}{\\partial x} \\Psi_{\\lambda}{(i,x)} = \\frac{\\partial}{\\partial x} (i - x) and \\frac{\\partial}{\\partial x} \\Psi_{\\lambda}{(i,x)} = -1 and -1 = \\frac{\\partial}{\\partial x} (i - x) and (-1)^{x} = (\\frac{\\partial}{\\partial x} (i - x))^{x} and 2 (-1)^{x} \\cos{(\\Psi_{\\lambda}{(i,x)})} = 2 \\cos{(\\Psi_{\\lambda}{(i,x)})} (\\frac{\\partial}{\\partial x} (i - x))^{x} and 2 (-1)^{x} \\cos{(\\Psi_{\\lambda}{(i,x)})} = 2 \\cos{(\\Psi_{\\lambda}{(i,x)})} (\\frac{\\partial}{\\partial x} \\Psi_{\\lambda}{(i,x)})^{x}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)), Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["power", 4, "Symbol('x', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('x', commutative=True)), Pow(Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["times", 5, "Mul(Integer(2), cos(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Integer(-1), Symbol('x', commutative=True)), cos(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(2), cos(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True))), Pow(Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(2), Pow(Integer(-1), Symbol('x', commutative=True)), cos(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(2), cos(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True))), Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(Z)} = \\log{(Z)} and \\dot{\\mathbf{r}}{(Z)} = \\int \\mathbf{r}{(Z)} dZ, then obtain \\sin{(\\dot{\\mathbf{r}}{(Z)})} = \\sin{(\\int \\mathbf{r}{(Z)} dZ)}", "derivation": "\\mathbf{r}{(Z)} = \\log{(Z)} and \\int \\mathbf{r}{(Z)} dZ = \\int \\log{(Z)} dZ and \\dot{\\mathbf{r}}{(Z)} = \\int \\mathbf{r}{(Z)} dZ and \\dot{\\mathbf{r}}{(Z)} = \\int \\log{(Z)} dZ and \\sin{(\\dot{\\mathbf{r}}{(Z)})} = \\sin{(\\int \\log{(Z)} dZ)} and \\sin{(\\dot{\\mathbf{r}}{(Z)})} = \\sin{(\\int \\mathbf{r}{(Z)} dZ)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('Z', commutative=True)), Integral(Function('\\\\mathbf{r}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('Z', commutative=True)), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["sin", 4], "Equality(sin(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('Z', commutative=True))), sin(Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(sin(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('Z', commutative=True))), sin(Integral(Function('\\\\mathbf{r}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(\\dot{x},E)} = E - \\dot{x}, then obtain \\int (\\int 1 d\\dot{x})^{\\dot{x}} d\\dot{x} = \\int (\\int \\frac{E - \\dot{x}}{\\psi^{*}{(\\dot{x},E)}} d\\dot{x})^{\\dot{x}} d\\dot{x}", "derivation": "\\psi^{*}{(\\dot{x},E)} = E - \\dot{x} and 1 = \\frac{E - \\dot{x}}{\\psi^{*}{(\\dot{x},E)}} and \\int 1 d\\dot{x} = \\int \\frac{E - \\dot{x}}{\\psi^{*}{(\\dot{x},E)}} d\\dot{x} and (\\int 1 d\\dot{x})^{\\dot{x}} = (\\int \\frac{E - \\dot{x}}{\\psi^{*}{(\\dot{x},E)}} d\\dot{x})^{\\dot{x}} and \\int (\\int 1 d\\dot{x})^{\\dot{x}} d\\dot{x} = \\int (\\int \\frac{E - \\dot{x}}{\\psi^{*}{(\\dot{x},E)}} d\\dot{x})^{\\dot{x}} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 1, "Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(Mul(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Pow(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Pow(Integral(Mul(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Function('\\\\psi^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('E', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(B,M)} = \\sin{(B M)}, then obtain \\dot{x}^{4}{(B,M)} \\cos{(\\phi)} = \\dot{x}^{2}{(B,M)} \\sin^{2}{(B M)} \\cos{(\\phi)}", "derivation": "\\dot{x}{(B,M)} = \\sin{(B M)} and \\dot{x}^{2}{(B,M)} = \\dot{x}{(B,M)} \\sin{(B M)} and \\dot{x}^{4}{(B,M)} = \\dot{x}^{2}{(B,M)} \\sin^{2}{(B M)} and \\dot{x}^{4}{(B,M)} \\cos{(\\phi)} = \\dot{x}^{2}{(B,M)} \\sin^{2}{(B M)} \\cos{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), sin(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True))))"], [["times", 1, "Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True))"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), Integer(2)), Mul(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), sin(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True))), Integer(2))))"], [["times", 3, "cos(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), Integer(4)), cos(Symbol('\\\\phi', commutative=True))), Mul(Pow(Function('\\\\dot{x}')(Symbol('B', commutative=True), Symbol('M', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True))), Integer(2)), cos(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given c{(\\phi_2)} = \\phi_2, then derive \\frac{d}{d \\phi_2} c{(\\phi_2)} - 1 = 0, then obtain - \\phi_2 + \\phi_2^{- \\phi_2} \\int (\\frac{d}{d c{(\\phi_2)}} c{(\\phi_2)} - 1) d\\phi_2 = - \\phi_2 + \\phi_2^{- \\phi_2} \\int 0 d\\phi_2", "derivation": "c{(\\phi_2)} = \\phi_2 and \\frac{d}{d \\phi_2} c{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 and \\frac{d}{d \\phi_2} c{(\\phi_2)} - 1 = \\frac{d}{d \\phi_2} \\phi_2 - 1 and \\frac{d}{d \\phi_2} c{(\\phi_2)} - 1 = 0 and \\frac{d}{d \\phi_2} \\phi_2 - 1 = 0 and \\frac{d}{d c{(\\phi_2)}} c{(\\phi_2)} - 1 = 0 and \\int (\\frac{d}{d c{(\\phi_2)}} c{(\\phi_2)} - 1) d\\phi_2 = \\int 0 d\\phi_2 and \\phi_2^{- \\phi_2} \\int (\\frac{d}{d c{(\\phi_2)}} c{(\\phi_2)} - 1) d\\phi_2 = \\phi_2^{- \\phi_2} \\int 0 d\\phi_2 and - \\phi_2 + \\phi_2^{- \\phi_2} \\int (\\frac{d}{d c{(\\phi_2)}} c{(\\phi_2)} - 1) d\\phi_2 = - \\phi_2 + \\phi_2^{- \\phi_2} \\int 0 d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('c')(Symbol('\\\\phi_2', commutative=True)), Integer(1))), Integer(-1)), Integer(0))"], [["integrate", 6, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('c')(Symbol('\\\\phi_2', commutative=True)), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 7, "Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Integral(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('c')(Symbol('\\\\phi_2', commutative=True)), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 8, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Integral(Add(Derivative(Function('c')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('c')(Symbol('\\\\phi_2', commutative=True)), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(q,v_{t})} = q + v_{t}, then obtain \\frac{\\frac{\\partial}{\\partial v_{t}} \\hat{p}{(q,v_{t})}}{q} = \\frac{1}{q}", "derivation": "\\hat{p}{(q,v_{t})} = q + v_{t} and \\frac{\\partial}{\\partial v_{t}} \\hat{p}{(q,v_{t})} = \\frac{\\partial}{\\partial v_{t}} (q + v_{t}) and \\frac{\\frac{\\partial}{\\partial v_{t}} \\hat{p}{(q,v_{t})}}{\\frac{\\partial}{\\partial v_{t}} (q + v_{t})} = 1 and \\frac{\\frac{\\partial}{\\partial v_{t}} \\hat{p}{(q,v_{t})}}{q \\frac{\\partial}{\\partial v_{t}} (q + v_{t})} = \\frac{1}{q} and \\frac{\\frac{\\partial}{\\partial v_{t}} \\hat{p}{(q,v_{t})}}{q} = \\frac{1}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('q', commutative=True), Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integer(1))"], [["divide", 3, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Derivative(Add(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Pow(Symbol('q', commutative=True), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('q', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Pow(Symbol('q', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} = \\frac{\\sin{(\\theta_1)}}{b}, then obtain (2 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1)^{b} = (-1 + \\frac{2 \\sin{(\\theta_1)}}{b})^{b}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} = \\frac{\\sin{(\\theta_1)}}{b} and \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1 = -1 + \\frac{\\sin{(\\theta_1)}}{b} and 2 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1 = \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1 + \\frac{\\sin{(\\theta_1)}}{b} and (2 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1)^{b} = (\\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1 + \\frac{\\sin{(\\theta_1)}}{b})^{b} and (2 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1,b)} - 1)^{b} = (-1 + \\frac{2 \\sin{(\\theta_1)}}{b})^{b}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 2, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True)), Integer(-1), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Symbol('b', commutative=True)), Pow(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True)), Integer(-1), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Symbol('b', commutative=True)), Pow(Add(Integer(-1), Mul(Integer(2), Pow(Symbol('b', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(a^{\\dagger})} = e^{e^{a^{\\dagger}}}, then derive \\int \\mathbf{A}{(a^{\\dagger})} da^{\\dagger} = \\mathbf{s} + \\operatorname{Ei}{(e^{a^{\\dagger}})}, then derive \\mathbf{s} + \\operatorname{Ei}{(e^{a^{\\dagger}})} = Z + \\operatorname{Ei}{(e^{a^{\\dagger}})}, then derive Z + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\mathbf{E} + \\operatorname{Ei}{(e^{a^{\\dagger}})}, then obtain \\mathbf{E} + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\int e^{e^{a^{\\dagger}}} da^{\\dagger}", "derivation": "\\mathbf{A}{(a^{\\dagger})} = e^{e^{a^{\\dagger}}} and \\int \\mathbf{A}{(a^{\\dagger})} da^{\\dagger} = \\int e^{e^{a^{\\dagger}}} da^{\\dagger} and \\int \\mathbf{A}{(a^{\\dagger})} da^{\\dagger} = \\mathbf{s} + \\operatorname{Ei}{(e^{a^{\\dagger}})} and \\mathbf{s} + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\int e^{e^{a^{\\dagger}}} da^{\\dagger} and \\mathbf{s} + \\operatorname{Ei}{(e^{a^{\\dagger}})} = Z + \\operatorname{Ei}{(e^{a^{\\dagger}})} and Z + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\int e^{e^{a^{\\dagger}}} da^{\\dagger} and Z + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\mathbf{E} + \\operatorname{Ei}{(e^{a^{\\dagger}})} and \\mathbf{E} + \\operatorname{Ei}{(e^{a^{\\dagger}})} = \\int e^{e^{a^{\\dagger}}} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(exp(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Integral(exp(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('Z', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('Z', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Integral(exp(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('Z', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('\\\\mathbf{E}', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Ei(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Integral(exp(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(Z,\\pi)} = \\frac{Z}{\\pi}, then derive \\frac{\\partial}{\\partial Z} \\bar{\\h}{(Z,\\pi)} = \\frac{1}{\\pi}, then obtain \\frac{\\partial^{2}}{\\partial \\pi\\partial Z} \\bar{\\h}{(Z,\\pi)} - \\frac{1}{Z} = \\frac{d}{d \\pi} \\frac{1}{\\pi} - \\frac{1}{Z}", "derivation": "\\bar{\\h}{(Z,\\pi)} = \\frac{Z}{\\pi} and \\frac{\\partial}{\\partial Z} \\bar{\\h}{(Z,\\pi)} = \\frac{\\partial}{\\partial Z} \\frac{Z}{\\pi} and \\frac{\\partial}{\\partial Z} \\bar{\\h}{(Z,\\pi)} = \\frac{1}{\\pi} and \\frac{\\partial^{2}}{\\partial \\pi\\partial Z} \\bar{\\h}{(Z,\\pi)} = \\frac{d}{d \\pi} \\frac{1}{\\pi} and \\frac{\\partial^{2}}{\\partial \\pi\\partial Z} \\bar{\\h}{(Z,\\pi)} - \\frac{1}{Z} = \\frac{d}{d \\pi} \\frac{1}{\\pi} - \\frac{1}{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["minus", 4, "Pow(Symbol('Z', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))), Add(Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(v_{1})} = \\cos{(\\sin{(v_{1})})}, then obtain 2 = 1 + \\frac{\\log{(\\cos{(\\sin{(v_{1})})})}}{\\log{(\\mathbf{J}_f{(v_{1})})}}", "derivation": "\\mathbf{J}_f{(v_{1})} = \\cos{(\\sin{(v_{1})})} and \\log{(\\mathbf{J}_f{(v_{1})})} = \\log{(\\cos{(\\sin{(v_{1})})})} and 1 = \\frac{\\log{(\\cos{(\\sin{(v_{1})})})}}{\\log{(\\mathbf{J}_f{(v_{1})})}} and 2 = 1 + \\frac{\\log{(\\cos{(\\sin{(v_{1})})})}}{\\log{(\\mathbf{J}_f{(v_{1})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('v_1', commutative=True)), cos(sin(Symbol('v_1', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\mathbf{J}_f')(Symbol('v_1', commutative=True))), log(cos(sin(Symbol('v_1', commutative=True)))))"], [["divide", 2, "log(Function('\\\\mathbf{J}_f')(Symbol('v_1', commutative=True)))"], "Equality(Integer(1), Mul(Pow(log(Function('\\\\mathbf{J}_f')(Symbol('v_1', commutative=True))), Integer(-1)), log(cos(sin(Symbol('v_1', commutative=True))))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(log(Function('\\\\mathbf{J}_f')(Symbol('v_1', commutative=True))), Integer(-1)), log(cos(sin(Symbol('v_1', commutative=True)))))))"]]}, {"prompt": "Given \\varphi{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain (\\int (2 \\varphi^{\\rho_f}{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f)^{2} = (\\int (\\varphi^{\\rho_f}{(\\rho_f)} + 2 \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f)^{2}", "derivation": "\\varphi{(\\rho_f)} = \\sin{(\\rho_f)} and \\varphi^{\\rho_f}{(\\rho_f)} = \\sin^{\\rho_f}{(\\rho_f)} and 2 \\varphi^{\\rho_f}{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)} = \\varphi^{\\rho_f}{(\\rho_f)} + 2 \\sin^{\\rho_f}{(\\rho_f)} and \\int (2 \\varphi^{\\rho_f}{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f = \\int (\\varphi^{\\rho_f}{(\\rho_f)} + 2 \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f and (\\int (2 \\varphi^{\\rho_f}{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f)^{2} = (\\int (\\varphi^{\\rho_f}{(\\rho_f)} + 2 \\sin^{\\rho_f}{(\\rho_f)}) d\\rho_f)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["add", 2, "Add(Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Add(Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Mul(Integer(2), Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(2)), Pow(Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\mathbf{A})} = \\mathbf{A} and \\operatorname{n_{1}}{(\\mathbf{A},V)} = V (V + \\mathbf{A}), then obtain \\mathbf{J}_M{(\\mathbf{A})} \\operatorname{n_{1}}{(\\mathbf{A},V)} = \\mathbf{A} \\operatorname{n_{1}}{(\\mathbf{A},V)}", "derivation": "\\mathbf{J}_M{(\\mathbf{A})} = \\mathbf{A} and \\operatorname{n_{1}}{(\\mathbf{A},V)} = V (V + \\mathbf{A}) and V (V + \\mathbf{A}) \\mathbf{J}_M{(\\mathbf{A})} = V \\mathbf{A} (V + \\mathbf{A}) and \\mathbf{J}_M{(\\mathbf{A})} \\operatorname{n_{1}}{(\\mathbf{A},V)} = \\mathbf{A} \\operatorname{n_{1}}{(\\mathbf{A},V)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 1, "Mul(Symbol('V', commutative=True), Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Symbol('V', commutative=True), Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{A}', commutative=True)), Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('V', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('V', commutative=True))))"]]}, {"prompt": "Given G{(\\eta)} = \\log{(\\eta)}, then obtain \\int e^{(G^{\\eta}{(\\eta)})^{\\eta}} d\\eta - 1 = \\int e^{(\\log{(\\eta)}^{\\eta})^{\\eta}} d\\eta - 1", "derivation": "G{(\\eta)} = \\log{(\\eta)} and G^{\\eta}{(\\eta)} = \\log{(\\eta)}^{\\eta} and (G^{\\eta}{(\\eta)})^{\\eta} = (\\log{(\\eta)}^{\\eta})^{\\eta} and e^{(G^{\\eta}{(\\eta)})^{\\eta}} = e^{(\\log{(\\eta)}^{\\eta})^{\\eta}} and \\int e^{(G^{\\eta}{(\\eta)})^{\\eta}} d\\eta = \\int e^{(\\log{(\\eta)}^{\\eta})^{\\eta}} d\\eta and \\int e^{(G^{\\eta}{(\\eta)})^{\\eta}} d\\eta - 1 = \\int e^{(\\log{(\\eta)}^{\\eta})^{\\eta}} d\\eta - 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(log(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Pow(Function('G')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Pow(log(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Pow(Function('G')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), exp(Pow(Pow(log(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(exp(Pow(Pow(Function('G')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(exp(Pow(Pow(log(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Integral(exp(Pow(Pow(Function('G')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1)), Add(Integral(exp(Pow(Pow(log(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given g{(p)} = \\log{(p)}, then derive \\int g{(p)} dp = \\mu_0 + p \\log{(p)} - p, then derive \\sigma_x + p \\log{(p)} - p = \\mu_0 + p g{(p)} - p, then obtain \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + p g{(p)} - p) + \\frac{d}{d p} g{(p)} = \\frac{\\partial}{\\partial \\sigma_x} (\\mu_0 + p g{(p)} - p) + \\frac{d}{d p} g{(p)}", "derivation": "g{(p)} = \\log{(p)} and \\int g{(p)} dp = \\int \\log{(p)} dp and \\int g{(p)} dp = \\mu_0 + p \\log{(p)} - p and \\int g{(p)} dp = \\mu_0 + p g{(p)} - p and \\int \\log{(p)} dp = \\mu_0 + p g{(p)} - p and \\sigma_x + p \\log{(p)} - p = \\mu_0 + p g{(p)} - p and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + p \\log{(p)} - p) = \\frac{\\partial}{\\partial \\sigma_x} (\\mu_0 + p g{(p)} - p) and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + p \\log{(p)} - p) + \\frac{d}{d p} g{(p)} = \\frac{\\partial}{\\partial \\sigma_x} (\\mu_0 + p g{(p)} - p) + \\frac{d}{d p} g{(p)} and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + p g{(p)} - p) + \\frac{d}{d p} g{(p)} = \\frac{\\partial}{\\partial \\sigma_x} (\\mu_0 + p g{(p)} - p) + \\frac{d}{d p} g{(p)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["add", 7, "Derivative(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Add(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\mu_0', commutative=True), Mul(Symbol('p', commutative=True), Function('g')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Function('g')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(Q,\\phi_1)} = Q \\cos{(\\phi_1)} and V{(\\rho)} = e^{\\rho} and \\sigma_{x}{(\\rho)} = e^{\\rho}, then obtain Q V{(\\rho)} \\cos{(\\phi_1)} = Q \\sigma_{x}{(\\rho)} \\cos{(\\phi_1)}", "derivation": "\\operatorname{C_{1}}{(Q,\\phi_1)} = Q \\cos{(\\phi_1)} and V{(\\rho)} = e^{\\rho} and \\operatorname{C_{1}}{(Q,\\phi_1)} V{(\\rho)} = \\operatorname{C_{1}}{(Q,\\phi_1)} e^{\\rho} and \\sigma_{x}{(\\rho)} = e^{\\rho} and Q V{(\\rho)} \\cos{(\\phi_1)} = Q e^{\\rho} \\cos{(\\phi_1)} and Q V{(\\rho)} \\cos{(\\phi_1)} = Q \\sigma_{x}{(\\rho)} \\cos{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Symbol('Q', commutative=True), cos(Symbol('\\\\phi_1', commutative=True))))"], ["get_premise", "Equality(Function('V')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["times", 2, "Function('C_1')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('C_1')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('V')(Symbol('\\\\rho', commutative=True))), Mul(Function('C_1')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('Q', commutative=True), Function('V')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('Q', commutative=True), exp(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('Q', commutative=True), Function('V')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('Q', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and c{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain 2 c{(a^{\\dagger})} = \\mathbf{s}{(a^{\\dagger})} + c{(a^{\\dagger})}", "derivation": "\\mathbf{s}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and c{(a^{\\dagger})} = \\log{(a^{\\dagger})} and c{(a^{\\dagger})} = \\mathbf{s}{(a^{\\dagger})} and c{(a^{\\dagger})} + \\log{(a^{\\dagger})} = \\mathbf{s}{(a^{\\dagger})} + \\log{(a^{\\dagger})} and 2 c{(a^{\\dagger})} = \\mathbf{s}{(a^{\\dagger})} + c{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{s}')(Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 3, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('c')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('a^{\\\\dagger}', commutative=True)), Function('c')(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(q)} = e^{q} and \\tilde{g}^*{(q)} = \\iint (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq dq, then obtain \\tilde{g}^*{(q)} + \\int 0^{q} dq = \\int 0^{q} dq + \\iint (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq dq", "derivation": "\\operatorname{E_{\\lambda}}{(q)} = e^{q} and \\operatorname{E_{\\lambda}}{(q)} - e^{q} = 0 and (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} = 0^{q} and \\int (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq = \\int 0^{q} dq and \\iint (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq dq = \\iint 0^{q} dq dq and \\tilde{g}^*{(q)} = \\iint (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq dq and \\tilde{g}^*{(q)} = \\iint 0^{q} dq dq and \\tilde{g}^*{(q)} + \\int 0^{q} dq = \\int 0^{q} dq + \\iint 0^{q} dq dq and \\tilde{g}^*{(q)} + \\int 0^{q} dq = \\int 0^{q} dq + \\iint (\\operatorname{E_{\\lambda}}{(q)} - e^{q})^{q} dq dq", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["minus", 1, "exp(Symbol('q', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Pow(Integer(0), Symbol('q', commutative=True)))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True)), Integral(Pow(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True)), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["add", 7, "Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True)), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True)), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Add(Function('E_{\\\\lambda}')(Symbol('q', commutative=True)), Mul(Integer(-1), exp(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)}, then obtain (\\cos{(\\tilde{g}^*)} + \\int \\hat{\\mathbf{x}}{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\cos{(\\tilde{g}^*)} + \\int \\cos{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*}", "derivation": "\\hat{\\mathbf{x}}{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)} and \\int \\hat{\\mathbf{x}}{(\\tilde{g}^*)} d\\tilde{g}^* = \\int \\cos{(\\tilde{g}^*)} d\\tilde{g}^* and \\cos{(\\tilde{g}^*)} + \\int \\hat{\\mathbf{x}}{(\\tilde{g}^*)} d\\tilde{g}^* = \\cos{(\\tilde{g}^*)} + \\int \\cos{(\\tilde{g}^*)} d\\tilde{g}^* and (\\cos{(\\tilde{g}^*)} + \\int \\hat{\\mathbf{x}}{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\cos{(\\tilde{g}^*)} + \\int \\cos{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["power", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given c{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})}, then derive \\frac{d}{d a^{\\dagger}} c{(a^{\\dagger})} = - \\sin{(a^{\\dagger})}, then obtain \\frac{d}{d a^{\\dagger}} c{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} \\sin{(a^{\\dagger})}", "derivation": "c{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})} and c{(a^{\\dagger})} + 1 = \\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})} + 1 and \\frac{d}{d a^{\\dagger}} (c{(a^{\\dagger})} + 1) = \\frac{d}{d a^{\\dagger}} (\\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})} + 1) and \\frac{d}{d a^{\\dagger}} c{(a^{\\dagger})} = - \\sin{(a^{\\dagger})} and \\frac{d^{2}}{d (a^{\\dagger})^{2}} \\sin{(a^{\\dagger})} = - \\sin{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} c{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} \\sin{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(g)} = e^{g}, then derive \\frac{d}{d g} \\int \\frac{\\operatorname{x^{{\\}'}}{(g)}}{g} dg = \\frac{\\partial}{\\partial g} (b + \\operatorname{Ei}{(g)}), then obtain \\frac{\\frac{d}{d g} \\int \\frac{e^{g}}{g} dg}{\\operatorname{x^{{\\}'}}{(g)}} = \\frac{\\frac{\\partial}{\\partial g} (b + \\operatorname{Ei}{(g)})}{\\operatorname{x^{{\\}'}}{(g)}}", "derivation": "\\operatorname{x^{{\\}'}}{(g)} = e^{g} and \\frac{\\operatorname{x^{{\\}'}}{(g)}}{g} = \\frac{e^{g}}{g} and \\int \\frac{\\operatorname{x^{{\\}'}}{(g)}}{g} dg = \\int \\frac{e^{g}}{g} dg and \\frac{d}{d g} \\int \\frac{\\operatorname{x^{{\\}'}}{(g)}}{g} dg = \\frac{d}{d g} \\int \\frac{e^{g}}{g} dg and \\frac{d}{d g} \\int \\frac{\\operatorname{x^{{\\}'}}{(g)}}{g} dg = \\frac{\\partial}{\\partial g} (b + \\operatorname{Ei}{(g)}) and \\frac{d}{d g} \\int \\frac{e^{g}}{g} dg = \\frac{\\partial}{\\partial g} (b + \\operatorname{Ei}{(g)}) and \\frac{\\frac{d}{d g} \\int \\frac{e^{g}}{g} dg}{\\operatorname{x^{{\\}'}}{(g)}} = \\frac{\\frac{\\partial}{\\partial g} (b + \\operatorname{Ei}{(g)})}{\\operatorname{x^{{\\}'}}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('g', commutative=True))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), Ei(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), Ei(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["divide", 6, "Function('x^\\\\prime')(Symbol('g', commutative=True))"], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('g', commutative=True)), Integer(-1)), Derivative(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(Function('x^\\\\prime')(Symbol('g', commutative=True)), Integer(-1)), Derivative(Add(Symbol('b', commutative=True), Ei(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given J{(f)} = \\sin{(f)}, then obtain \\int \\frac{d}{d f} \\frac{J{(f)}}{f^{2}} df = \\int \\frac{d}{d f} \\frac{\\sin{(f)}}{f^{2}} df", "derivation": "J{(f)} = \\sin{(f)} and \\frac{J{(f)}}{f} = \\frac{\\sin{(f)}}{f} and \\frac{J{(f)}}{f^{2}} = \\frac{\\sin{(f)}}{f^{2}} and \\frac{d}{d f} \\frac{J{(f)}}{f^{2}} = \\frac{d}{d f} \\frac{\\sin{(f)}}{f^{2}} and \\int \\frac{d}{d f} \\frac{J{(f)}}{f^{2}} df = \\int \\frac{d}{d f} \\frac{\\sin{(f)}}{f^{2}} df", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('J')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), sin(Symbol('f', commutative=True))))"], [["times", 2, "Pow(Symbol('f', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-2)), Function('J')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-2)), sin(Symbol('f', commutative=True))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-2)), Function('J')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-2)), sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-2)), Function('J')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-2)), sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} = b + \\cos{(\\tilde{g}^*)} and U{(\\tilde{g}^*)} = \\sin{(\\cos{(\\tilde{g}^*)} + 1)}, then obtain \\frac{d}{d b} U{(\\tilde{g}^*)} = \\frac{d}{d b} \\sin{(\\cos{(\\tilde{g}^*)} + 1)}", "derivation": "\\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} = b + \\cos{(\\tilde{g}^*)} and - b + \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} = \\cos{(\\tilde{g}^*)} and - b + \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} + 1 = \\cos{(\\tilde{g}^*)} + 1 and \\sin{(- b + \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} + 1)} = \\sin{(\\cos{(\\tilde{g}^*)} + 1)} and \\frac{\\partial}{\\partial b} \\sin{(- b + \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} + 1)} = \\frac{d}{d b} \\sin{(\\cos{(\\tilde{g}^*)} + 1)} and U{(\\tilde{g}^*)} = \\sin{(\\cos{(\\tilde{g}^*)} + 1)} and U{(\\tilde{g}^*)} = \\sin{(- b + \\dot{\\mathbf{r}}{(\\tilde{g}^*,b)} + 1)} and \\frac{d}{d b} U{(\\tilde{g}^*)} = \\frac{d}{d b} \\sin{(\\cos{(\\tilde{g}^*)} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), cos(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True))), cos(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True)), Integer(1)), Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1)))"], [["sin", 3], "Equality(sin(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True)), Integer(1))), sin(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(sin(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True)), Integer(1))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1))), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Function('U')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('b', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Derivative(Function('U')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Add(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(f,A_{x})} = \\log{(\\frac{f}{A_{x}})}, then derive A_{x} + E_{n} = \\int \\frac{\\log{(\\frac{f}{A_{x}})}}{\\operatorname{P_{g}}{(f,A_{x})}} dA_{x}, then derive A_{x} + E_{n} = A_{x} + J_{\\varepsilon}, then obtain \\int 1 dA_{x} = A_{x} + J_{\\varepsilon}", "derivation": "\\operatorname{P_{g}}{(f,A_{x})} = \\log{(\\frac{f}{A_{x}})} and 1 = \\frac{\\log{(\\frac{f}{A_{x}})}}{\\operatorname{P_{g}}{(f,A_{x})}} and \\int 1 dA_{x} = \\int \\frac{\\log{(\\frac{f}{A_{x}})}}{\\operatorname{P_{g}}{(f,A_{x})}} dA_{x} and A_{x} + E_{n} = \\int \\frac{\\log{(\\frac{f}{A_{x}})}}{\\operatorname{P_{g}}{(f,A_{x})}} dA_{x} and A_{x} + E_{n} = \\int 1 dA_{x} and A_{x} + E_{n} = A_{x} + J_{\\varepsilon} and \\int 1 dA_{x} = A_{x} + J_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('f', commutative=True), Symbol('A_x', commutative=True)), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["divide", 1, "Function('P_g')(Symbol('f', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_g')(Symbol('f', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Pow(Function('P_g')(Symbol('f', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_x', commutative=True), Symbol('E_n', commutative=True)), Integral(Mul(Pow(Function('P_g')(Symbol('f', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('A_x', commutative=True), Symbol('E_n', commutative=True)), Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('A_x', commutative=True), Symbol('E_n', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(A_{1},\\hat{p})} = - A_{1} + \\hat{p}, then obtain 2 \\frac{\\partial}{\\partial \\hat{p}} \\tilde{g}^*{(A_{1},\\hat{p})} = 2", "derivation": "\\tilde{g}^*{(A_{1},\\hat{p})} = - A_{1} + \\hat{p} and - A_{1} + \\hat{p} + \\tilde{g}^*{(A_{1},\\hat{p})} = - 2 A_{1} + 2 \\hat{p} and 2 \\tilde{g}^*{(A_{1},\\hat{p})} = - 2 A_{1} + 2 \\hat{p} and \\frac{\\partial}{\\partial \\hat{p}} 2 \\tilde{g}^*{(A_{1},\\hat{p})} = \\frac{\\partial}{\\partial \\hat{p}} (- 2 A_{1} + 2 \\hat{p}) and 2 \\frac{\\partial}{\\partial \\hat{p}} \\tilde{g}^*{(A_{1},\\hat{p})} = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A_1', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\hat{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('A_1', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('A_1', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('A_1', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('\\\\tilde{g}^*')(Symbol('A_1', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(2))"]]}, {"prompt": "Given \\bar{\\h}{(i,p)} = \\frac{\\partial}{\\partial p} p^{i} and \\operatorname{r_{0}}{(i,p)} = \\frac{i p^{i}}{p}, then derive \\bar{\\h}{(i,p)} = \\frac{i p^{i}}{p}, then obtain \\frac{\\bar{\\h}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} + \\int e^{\\frac{i p^{i}}{p}} dp = \\frac{\\operatorname{r_{0}}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} + \\int e^{\\frac{i p^{i}}{p}} dp", "derivation": "\\bar{\\h}{(i,p)} = \\frac{\\partial}{\\partial p} p^{i} and \\bar{\\h}{(i,p)} = \\frac{i p^{i}}{p} and \\operatorname{r_{0}}{(i,p)} = \\frac{i p^{i}}{p} and \\bar{\\h}{(i,p)} = \\operatorname{r_{0}}{(i,p)} and \\int \\bar{\\h}{(i,p)} di = \\int \\operatorname{r_{0}}{(i,p)} di and \\frac{\\bar{\\h}{(i,p)}}{\\int \\operatorname{r_{0}}{(i,p)} di} = \\frac{\\operatorname{r_{0}}{(i,p)}}{\\int \\operatorname{r_{0}}{(i,p)} di} and \\frac{\\bar{\\h}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} = \\frac{\\operatorname{r_{0}}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} and \\frac{\\bar{\\h}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} + \\int e^{\\frac{i p^{i}}{p}} dp = \\frac{\\operatorname{r_{0}}{(i,p)}}{\\int \\bar{\\h}{(i,p)} di} + \\int e^{\\frac{i p^{i}}{p}} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Symbol('p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('i', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('i', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 4, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["divide", 4, "Integral(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Mul(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Mul(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))))"], [["add", 7, "Integral(exp(Mul(Symbol('i', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Mul(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integral(exp(Mul(Symbol('i', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('p', commutative=True)))), Add(Mul(Function('r_0')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integral(exp(Mul(Symbol('i', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given s{(\\hat{p},\\rho_f)} = \\hat{p} - \\rho_f, then obtain \\iint (\\rho_f + s^{\\rho_f}{(\\hat{p},\\rho_f)}) d\\rho_f d\\hat{p} = \\iint (\\rho_f + (\\hat{p} - \\rho_f)^{\\rho_f}) d\\rho_f d\\hat{p}", "derivation": "s{(\\hat{p},\\rho_f)} = \\hat{p} - \\rho_f and s^{\\rho_f}{(\\hat{p},\\rho_f)} = (\\hat{p} - \\rho_f)^{\\rho_f} and \\rho_f + s^{\\rho_f}{(\\hat{p},\\rho_f)} = \\rho_f + (\\hat{p} - \\rho_f)^{\\rho_f} and \\int (\\rho_f + s^{\\rho_f}{(\\hat{p},\\rho_f)}) d\\rho_f = \\int (\\rho_f + (\\hat{p} - \\rho_f)^{\\rho_f}) d\\rho_f and \\iint (\\rho_f + s^{\\rho_f}{(\\hat{p},\\rho_f)}) d\\rho_f d\\hat{p} = \\iint (\\rho_f + (\\hat{p} - \\rho_f)^{\\rho_f}) d\\rho_f d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\rho_f', commutative=True), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Symbol('\\\\rho_f', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\rho_f', commutative=True), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\rho_f', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)}, then derive \\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0 = M_{E} + \\hat{x}_0, then obtain \\frac{e^{\\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0}}{\\log{(\\hat{x}_0)}} = \\frac{e^{M_{E} + \\hat{x}_0}}{\\log{(\\hat{x}_0)}}", "derivation": "\\sigma_{p}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)} and \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} = 1 and \\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0 = \\int 1 d\\hat{x}_0 and \\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0 = M_{E} + \\hat{x}_0 and e^{\\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0} = e^{M_{E} + \\hat{x}_0} and \\frac{e^{\\int \\frac{\\sigma_{p}{(\\hat{x}_0)}}{\\log{(\\hat{x}_0)}} d\\hat{x}_0}}{\\log{(\\hat{x}_0)}} = \\frac{e^{M_{E} + \\hat{x}_0}}{\\log{(\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["exp", 4], "Equality(exp(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), exp(Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 5, "Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))"], "Equality(Mul(exp(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))), Mul(exp(Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{F},\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda} - \\mathbf{F})}, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi^{\\dagger}{(\\mathbf{F},\\Psi_{\\lambda})} = - \\cos{(\\Psi_{\\lambda} - \\mathbf{F})}, then obtain - \\cos{(\\Psi_{\\lambda} - \\mathbf{F})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} - \\sin{(\\Psi_{\\lambda} - \\mathbf{F})}", "derivation": "\\Psi^{\\dagger}{(\\mathbf{F},\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda} - \\mathbf{F})} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi^{\\dagger}{(\\mathbf{F},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} - \\sin{(\\Psi_{\\lambda} - \\mathbf{F})} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi^{\\dagger}{(\\mathbf{F},\\Psi_{\\lambda})} = - \\cos{(\\Psi_{\\lambda} - \\mathbf{F})} and - \\cos{(\\Psi_{\\lambda} - \\mathbf{F})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} - \\sin{(\\Psi_{\\lambda} - \\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(v_{y},\\Psi)} = \\Psi + v_{y} and \\operatorname{v_{t}}{(v_{y},\\Psi)} = \\Psi \\frac{\\partial}{\\partial v_{y}} \\Psi_{\\lambda}{(v_{y},\\Psi)}, then derive \\Psi \\frac{\\partial}{\\partial v_{y}} \\Psi_{\\lambda}{(v_{y},\\Psi)} = \\Psi, then obtain \\operatorname{v_{t}}{(v_{y},\\Psi)} = \\Psi", "derivation": "\\Psi_{\\lambda}{(v_{y},\\Psi)} = \\Psi + v_{y} and \\Psi \\Psi_{\\lambda}{(v_{y},\\Psi)} = \\Psi (\\Psi + v_{y}) and \\frac{\\partial}{\\partial v_{y}} \\Psi \\Psi_{\\lambda}{(v_{y},\\Psi)} = \\frac{\\partial}{\\partial v_{y}} \\Psi (\\Psi + v_{y}) and \\Psi \\frac{\\partial}{\\partial v_{y}} \\Psi_{\\lambda}{(v_{y},\\Psi)} = \\Psi and \\operatorname{v_{t}}{(v_{y},\\Psi)} = \\Psi \\frac{\\partial}{\\partial v_{y}} \\Psi_{\\lambda}{(v_{y},\\Psi)} and \\operatorname{v_{t}}{(v_{y},\\Psi)} = \\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('v_y', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Symbol('\\\\Psi', commutative=True))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('v_t')(Symbol('v_y', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"]]}, {"prompt": "Given \\mathbf{J}_f{(m_{s})} = \\frac{d}{d m_{s}} e^{m_{s}} and v{(m_{s})} = m_{s}, then obtain v{(m_{s})} + \\frac{d}{d m_{s}} e^{m_{s}} = m_{s} + \\frac{d}{d m_{s}} e^{m_{s}}", "derivation": "\\mathbf{J}_f{(m_{s})} = \\frac{d}{d m_{s}} e^{m_{s}} and v{(m_{s})} = m_{s} and \\mathbf{J}_f{(m_{s})} + v{(m_{s})} = m_{s} + \\mathbf{J}_f{(m_{s})} and v{(m_{s})} + \\frac{d}{d m_{s}} e^{m_{s}} = m_{s} + \\frac{d}{d m_{s}} e^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('m_s', commutative=True)), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], [["add", 2, "Function('\\\\mathbf{J}_f')(Symbol('m_s', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('m_s', commutative=True)), Function('v')(Symbol('m_s', commutative=True))), Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('v')(Symbol('m_s', commutative=True)), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Symbol('m_s', commutative=True), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{H}{(a^{\\dagger},\\omega)} = \\sin{(\\frac{a^{\\dagger}}{\\omega})}, then obtain (a^{\\dagger} \\mathbf{H}{(a^{\\dagger},\\omega)} + a^{\\dagger})^{a^{\\dagger}} = (a^{\\dagger} \\sin{(\\frac{a^{\\dagger}}{\\omega})} + a^{\\dagger})^{a^{\\dagger}}", "derivation": "\\mathbf{H}{(a^{\\dagger},\\omega)} = \\sin{(\\frac{a^{\\dagger}}{\\omega})} and a^{\\dagger} \\mathbf{H}{(a^{\\dagger},\\omega)} = a^{\\dagger} \\sin{(\\frac{a^{\\dagger}}{\\omega})} and a^{\\dagger} \\mathbf{H}{(a^{\\dagger},\\omega)} + a^{\\dagger} = a^{\\dagger} \\sin{(\\frac{a^{\\dagger}}{\\omega})} + a^{\\dagger} and (a^{\\dagger} \\mathbf{H}{(a^{\\dagger},\\omega)} + a^{\\dagger})^{a^{\\dagger}} = (a^{\\dagger} \\sin{(\\frac{a^{\\dagger}}{\\omega})} + a^{\\dagger})^{a^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["add", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\chi{(\\mathbf{M})} = \\mathbf{M}, then obtain \\mathbf{M} (\\mathbf{M}^{- \\mathbf{M}} \\operatorname{t_{2}}{(\\mathbf{M})})^{\\mathbf{M}} = \\mathbf{M} (\\mathbf{M}^{- \\mathbf{M}} e^{\\mathbf{M}})^{\\mathbf{M}}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\chi{(\\mathbf{M})} = \\mathbf{M} and \\mathbf{M}^{- \\mathbf{M}} \\operatorname{t_{2}}{(\\mathbf{M})} = \\mathbf{M}^{- \\mathbf{M}} e^{\\mathbf{M}} and (\\mathbf{M}^{- \\mathbf{M}} \\operatorname{t_{2}}{(\\mathbf{M})})^{\\mathbf{M}} = (\\mathbf{M}^{- \\mathbf{M}} e^{\\mathbf{M}})^{\\mathbf{M}} and (\\mathbf{M}^{- \\mathbf{M}} \\operatorname{t_{2}}{(\\mathbf{M})})^{\\mathbf{M}} \\chi{(\\mathbf{M})} = (\\mathbf{M}^{- \\mathbf{M}} e^{\\mathbf{M}})^{\\mathbf{M}} \\chi{(\\mathbf{M})} and \\mathbf{M} (\\mathbf{M}^{- \\mathbf{M}} \\operatorname{t_{2}}{(\\mathbf{M})})^{\\mathbf{M}} = \\mathbf{M} (\\mathbf{M}^{- \\mathbf{M}} e^{\\mathbf{M}})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 4, "Function('\\\\chi')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given x{(\\varepsilon_0)} = \\log{(\\sin{(\\varepsilon_0)})}, then obtain e^{\\sin^{\\varepsilon_0}{(\\frac{x{(\\varepsilon_0)}}{\\sin{(\\varepsilon_0)}})}} = e^{\\sin^{\\varepsilon_0}{(\\frac{\\log{(\\sin{(\\varepsilon_0)})}}{\\sin{(\\varepsilon_0)}})}}", "derivation": "x{(\\varepsilon_0)} = \\log{(\\sin{(\\varepsilon_0)})} and \\frac{x{(\\varepsilon_0)}}{\\sin{(\\varepsilon_0)}} = \\frac{\\log{(\\sin{(\\varepsilon_0)})}}{\\sin{(\\varepsilon_0)}} and \\sin{(\\frac{x{(\\varepsilon_0)}}{\\sin{(\\varepsilon_0)}})} = \\sin{(\\frac{\\log{(\\sin{(\\varepsilon_0)})}}{\\sin{(\\varepsilon_0)}})} and \\sin^{\\varepsilon_0}{(\\frac{x{(\\varepsilon_0)}}{\\sin{(\\varepsilon_0)}})} = \\sin^{\\varepsilon_0}{(\\frac{\\log{(\\sin{(\\varepsilon_0)})}}{\\sin{(\\varepsilon_0)}})} and e^{\\sin^{\\varepsilon_0}{(\\frac{x{(\\varepsilon_0)}}{\\sin{(\\varepsilon_0)}})}} = e^{\\sin^{\\varepsilon_0}{(\\frac{\\log{(\\sin{(\\varepsilon_0)})}}{\\sin{(\\varepsilon_0)}})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), log(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Mul(log(sin(Symbol('\\\\varepsilon_0', commutative=True))), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))))"], [["sin", 2], "Equality(sin(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))), sin(Mul(log(sin(Symbol('\\\\varepsilon_0', commutative=True))), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(sin(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Mul(log(sin(Symbol('\\\\varepsilon_0', commutative=True))), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(sin(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))), Symbol('\\\\varepsilon_0', commutative=True))), exp(Pow(sin(Mul(log(sin(Symbol('\\\\varepsilon_0', commutative=True))), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))), Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(A_{1},t)} = \\sin{(A_{1}^{t})}, then obtain - \\rho_{f}{(A_{1},t)} \\sin{(\\rho_{f}{(A_{1},t)})} = - \\rho_{f}{(A_{1},t)} \\sin{(2 \\rho_{f}{(A_{1},t)} - \\sin{(A_{1}^{t})})}", "derivation": "\\rho_{f}{(A_{1},t)} = \\sin{(A_{1}^{t})} and 0 = - \\rho_{f}{(A_{1},t)} + \\sin{(A_{1}^{t})} and - \\rho_{f}{(A_{1},t)} = - 2 \\rho_{f}{(A_{1},t)} + \\sin{(A_{1}^{t})} and - \\sin{(\\rho_{f}{(A_{1},t)})} = - \\sin{(2 \\rho_{f}{(A_{1},t)} - \\sin{(A_{1}^{t})})} and - \\rho_{f}{(A_{1},t)} \\sin{(\\rho_{f}{(A_{1},t)})} = - \\rho_{f}{(A_{1},t)} \\sin{(2 \\rho_{f}{(A_{1},t)} - \\sin{(A_{1}^{t})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True)), sin(Pow(Symbol('A_1', commutative=True), Symbol('t', commutative=True))))"], [["minus", 1, "Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))), sin(Pow(Symbol('A_1', commutative=True), Symbol('t', commutative=True)))))"], [["minus", 2, "Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))), sin(Pow(Symbol('A_1', commutative=True), Symbol('t', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True)))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), sin(Pow(Symbol('A_1', commutative=True), Symbol('t', commutative=True))))))))"], [["times", 4, "Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True)), sin(Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True)))), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True)), sin(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('A_1', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), sin(Pow(Symbol('A_1', commutative=True), Symbol('t', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})} = \\frac{\\sin{(\\mathbf{s})}}{\\dot{z}}, then obtain \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\dot{z} \\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})}}{\\sin{(\\mathbf{s})}} = \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}}", "derivation": "\\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})} = \\frac{\\sin{(\\mathbf{s})}}{\\dot{z}} and \\dot{z} \\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})} = \\sin{(\\mathbf{s})} and \\frac{\\partial}{\\partial \\dot{z}} \\dot{z} \\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})} = \\frac{d}{d \\dot{z}} \\sin{(\\mathbf{s})} and \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\dot{z} \\operatorname{A_{y}}{(\\dot{z},\\mathbf{s})}}{\\sin{(\\mathbf{s})}} = \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_y')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_y')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["divide", 3, "sin(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_y')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(\\nabla)} = e^{\\cos{(\\nabla)}} and S{(\\nabla)} = e^{\\cos{(\\nabla)}}, then derive \\frac{d}{d \\nabla} H{(\\nabla)} = - e^{\\cos{(\\nabla)}} \\sin{(\\nabla)}, then derive y^{\\prime} + H{(\\nabla)} = \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla, then obtain \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + e^{\\cos{(\\nabla)}}) = \\frac{d}{d y^{\\prime}} \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla", "derivation": "H{(\\nabla)} = e^{\\cos{(\\nabla)}} and \\frac{d}{d \\nabla} H{(\\nabla)} = \\frac{d}{d \\nabla} e^{\\cos{(\\nabla)}} and \\frac{d}{d \\nabla} H{(\\nabla)} = - e^{\\cos{(\\nabla)}} \\sin{(\\nabla)} and S{(\\nabla)} = e^{\\cos{(\\nabla)}} and \\frac{d}{d \\nabla} H{(\\nabla)} = - S{(\\nabla)} \\sin{(\\nabla)} and \\int \\frac{d}{d \\nabla} H{(\\nabla)} d\\nabla = \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla and y^{\\prime} + H{(\\nabla)} = \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla and y^{\\prime} + e^{\\cos{(\\nabla)}} = \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla and \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + e^{\\cos{(\\nabla)}}) = \\frac{d}{d y^{\\prime}} \\int - S{(\\nabla)} \\sin{(\\nabla)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\nabla', commutative=True)), exp(cos(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\nabla', commutative=True))), sin(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\nabla', commutative=True)), exp(cos(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('H')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Function('S')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Derivative(Function('H')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Integer(-1), Function('S')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('H')(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Integer(-1), Function('S')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), exp(cos(Symbol('\\\\nabla', commutative=True)))), Integral(Mul(Integer(-1), Function('S')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 8, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), exp(cos(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), Function('S')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\operatorname{A_{x}}{(f_{\\mathbf{p}})} = \\frac{1}{f_{\\mathbf{p}}}, then obtain \\operatorname{A_{x}}{(f_{\\mathbf{p}})} + \\phi_{1}^{2}{(\\mathbf{J}_P)} = \\phi_{1}^{2}{(\\mathbf{J}_P)} + \\frac{1}{f_{\\mathbf{p}}}", "derivation": "\\phi_{1}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\operatorname{A_{x}}{(f_{\\mathbf{p}})} = \\frac{1}{f_{\\mathbf{p}}} and \\operatorname{A_{x}}{(f_{\\mathbf{p}})} + \\log{(\\mathbf{J}_P)}^{2} = \\log{(\\mathbf{J}_P)}^{2} + \\frac{1}{f_{\\mathbf{p}}} and \\operatorname{A_{x}}{(f_{\\mathbf{p}})} + \\phi_{1}^{2}{(\\mathbf{J}_P)} = \\phi_{1}^{2}{(\\mathbf{J}_P)} + \\frac{1}{f_{\\mathbf{p}}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))"], [["add", 2, "Pow(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))"], "Equality(Add(Function('A_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))), Add(Pow(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))), Add(Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\sigma_p)} = \\cos{(e^{\\sigma_p})}, then obtain \\cos{(\\int \\operatorname{z^{*}}{(\\sigma_p)} e^{\\sigma_p} d\\sigma_p)} = \\cos{(\\int e^{\\sigma_p} \\cos{(e^{\\sigma_p})} d\\sigma_p)}", "derivation": "\\operatorname{z^{*}}{(\\sigma_p)} = \\cos{(e^{\\sigma_p})} and \\operatorname{z^{*}}{(\\sigma_p)} e^{\\sigma_p} = e^{\\sigma_p} \\cos{(e^{\\sigma_p})} and \\int \\operatorname{z^{*}}{(\\sigma_p)} e^{\\sigma_p} d\\sigma_p = \\int e^{\\sigma_p} \\cos{(e^{\\sigma_p})} d\\sigma_p and \\cos{(\\int \\operatorname{z^{*}}{(\\sigma_p)} e^{\\sigma_p} d\\sigma_p)} = \\cos{(\\int e^{\\sigma_p} \\cos{(e^{\\sigma_p})} d\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('z^*')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Function('z^*')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Mul(Function('z^*')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))), cos(Integral(Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given H{(G,v_{t})} = \\sin{(G - v_{t})} and \\operatorname{f^{\\prime}}{(G,v_{t})} = H{(G,v_{t})} \\sin{(G - v_{t})}, then obtain \\int H^{2}{(G,v_{t})} dG = \\frac{G \\sin^{2}{(G - v_{t})}}{2} + \\frac{G \\cos^{2}{(G - v_{t})}}{2} + t_{1} - \\frac{\\sin{(G - v_{t})} \\cos{(G - v_{t})}}{2}", "derivation": "H{(G,v_{t})} = \\sin{(G - v_{t})} and H^{2}{(G,v_{t})} = H{(G,v_{t})} \\sin{(G - v_{t})} and \\operatorname{f^{\\prime}}{(G,v_{t})} = H{(G,v_{t})} \\sin{(G - v_{t})} and \\operatorname{f^{\\prime}}{(G,v_{t})} = \\sin^{2}{(G - v_{t})} and H^{2}{(G,v_{t})} = \\operatorname{f^{\\prime}}{(G,v_{t})} and \\int \\operatorname{f^{\\prime}}{(G,v_{t})} dG = \\int \\sin^{2}{(G - v_{t})} dG and \\int H^{2}{(G,v_{t})} dG = \\int \\sin^{2}{(G - v_{t})} dG and \\int H^{2}{(G,v_{t})} dG = \\frac{G \\sin^{2}{(G - v_{t})}}{2} + \\frac{G \\cos^{2}{(G - v_{t})}}{2} + t_{1} - \\frac{\\sin{(G - v_{t})} \\cos{(G - v_{t})}}{2}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["times", 1, "Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Pow(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Mul(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Integer(2)), Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(2)), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integral(Pow(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Integer(2)), Tuple(Symbol('G', commutative=True))), Integral(Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(2)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Integral(Pow(Function('H')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Integer(2)), Tuple(Symbol('G', commutative=True))), Add(Mul(Rational(1, 2), Symbol('G', commutative=True), Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(2))), Mul(Rational(1, 2), Symbol('G', commutative=True), Pow(cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(2))), Symbol('t_1', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))))"]]}, {"prompt": "Given b{(a^{\\dagger})} = \\int \\log{(a^{\\dagger})} da^{\\dagger}, then derive a^{\\dagger} b{(a^{\\dagger})} = a^{\\dagger} (C_{2} + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}), then obtain (a^{\\dagger})^{2} b^{2}{(a^{\\dagger})} \\log{(a^{\\dagger})} = (a^{\\dagger})^{2} (C_{2} + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}) b{(a^{\\dagger})} \\log{(a^{\\dagger})}", "derivation": "b{(a^{\\dagger})} = \\int \\log{(a^{\\dagger})} da^{\\dagger} and a^{\\dagger} b{(a^{\\dagger})} = a^{\\dagger} \\int \\log{(a^{\\dagger})} da^{\\dagger} and a^{\\dagger} b{(a^{\\dagger})} = a^{\\dagger} (C_{2} + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}) and a^{\\dagger} b{(a^{\\dagger})} \\log{(a^{\\dagger})} = a^{\\dagger} (C_{2} + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}) \\log{(a^{\\dagger})} and (a^{\\dagger})^{2} b^{2}{(a^{\\dagger})} \\log{(a^{\\dagger})} = (a^{\\dagger})^{2} (C_{2} + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}) b{(a^{\\dagger})} \\log{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('b')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('b')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('C_2', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["times", 3, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('C_2', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 4, "Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('b')(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Pow(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Add(Symbol('C_2', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(\\theta_1)} = e^{e^{\\theta_1}}, then obtain -1 + \\frac{\\ddot{x}{(\\theta_1)} e^{\\theta_1}}{\\theta_1} = -1 + \\frac{e^{\\theta_1} e^{e^{\\theta_1}}}{\\theta_1}", "derivation": "\\ddot{x}{(\\theta_1)} = e^{e^{\\theta_1}} and \\frac{\\ddot{x}{(\\theta_1)}}{\\theta_1} = \\frac{e^{e^{\\theta_1}}}{\\theta_1} and \\ddot{x}{(\\theta_1)} - 1 = e^{e^{\\theta_1}} - 1 and \\frac{\\ddot{x}{(\\theta_1)} e^{\\theta_1}}{\\theta_1} = \\frac{e^{\\theta_1} e^{e^{\\theta_1}}}{\\theta_1} and - \\frac{e^{e^{\\theta_1}} - 1}{\\ddot{x}{(\\theta_1)} - 1} + \\frac{\\ddot{x}{(\\theta_1)} e^{\\theta_1}}{\\theta_1} = - \\frac{e^{e^{\\theta_1}} - 1}{\\ddot{x}{(\\theta_1)} - 1} + \\frac{e^{\\theta_1} e^{e^{\\theta_1}}}{\\theta_1} and -1 + \\frac{\\ddot{x}{(\\theta_1)} e^{\\theta_1}}{\\theta_1} = -1 + \\frac{e^{\\theta_1} e^{e^{\\theta_1}}}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), exp(exp(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Add(exp(exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1)))"], [["times", 2, "exp(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)), exp(exp(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 4, "Mul(Pow(Add(Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integer(-1)), Add(exp(exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integer(-1)), Add(exp(exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integer(-1)), Add(exp(exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)), exp(exp(Symbol('\\\\theta_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)), exp(exp(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(t)} = \\sin{(t)}, then derive \\frac{d^{2}}{d t^{2}} \\operatorname{C_{1}}{(t)} = - \\sin{(t)}, then obtain \\frac{d^{2}}{d t^{2}} \\sin{(t)} = - \\sin{(t)}", "derivation": "\\operatorname{C_{1}}{(t)} = \\sin{(t)} and t + \\operatorname{C_{1}}{(t)} = t + \\sin{(t)} and \\frac{d}{d t} (t + \\operatorname{C_{1}}{(t)}) = \\frac{d}{d t} (t + \\sin{(t)}) and \\frac{d^{2}}{d t^{2}} (t + \\operatorname{C_{1}}{(t)}) = \\frac{d^{2}}{d t^{2}} (t + \\sin{(t)}) and \\frac{d^{2}}{d t^{2}} \\operatorname{C_{1}}{(t)} = - \\sin{(t)} and \\frac{d^{2}}{d t^{2}} \\operatorname{C_{1}}{(t)} = - \\operatorname{C_{1}}{(t)} and \\frac{d^{2}}{d t^{2}} \\sin{(t)} = - \\sin{(t)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('C_1')(Symbol('t', commutative=True))), Add(Symbol('t', commutative=True), sin(Symbol('t', commutative=True))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Function('C_1')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('t', commutative=True), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Function('C_1')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(2))), Derivative(Add(Symbol('t', commutative=True), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('C_1')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('C_1')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), Function('C_1')(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\sigma_p)} = \\cos{(\\cos{(\\sigma_p)})} and \\operatorname{A_{2}}{(\\sigma_p)} = - \\operatorname{n_{2}}{(\\sigma_p)} and \\Psi_{nl}{(\\sigma_p)} = \\operatorname{A_{2}}{(\\sigma_p)} - \\frac{d}{d \\sigma_p} 0, then obtain \\Psi_{nl}{(\\sigma_p)} = \\operatorname{A_{2}}{(\\sigma_p)} - \\frac{d}{d \\sigma_p} (\\operatorname{A_{2}}{(\\sigma_p)} + \\cos{(\\cos{(\\sigma_p)})})", "derivation": "\\operatorname{n_{2}}{(\\sigma_p)} = \\cos{(\\cos{(\\sigma_p)})} and 0 = - \\operatorname{n_{2}}{(\\sigma_p)} + \\cos{(\\cos{(\\sigma_p)})} and \\frac{d}{d \\sigma_p} 0 = \\frac{d}{d \\sigma_p} (- \\operatorname{n_{2}}{(\\sigma_p)} + \\cos{(\\cos{(\\sigma_p)})}) and \\operatorname{A_{2}}{(\\sigma_p)} = - \\operatorname{n_{2}}{(\\sigma_p)} and \\Psi_{nl}{(\\sigma_p)} = \\operatorname{A_{2}}{(\\sigma_p)} - \\frac{d}{d \\sigma_p} 0 and \\frac{d}{d \\sigma_p} 0 = \\frac{d}{d \\sigma_p} (\\operatorname{A_{2}}{(\\sigma_p)} + \\cos{(\\cos{(\\sigma_p)})}) and \\Psi_{nl}{(\\sigma_p)} = \\operatorname{A_{2}}{(\\sigma_p)} - \\frac{d}{d \\sigma_p} (\\operatorname{A_{2}}{(\\sigma_p)} + \\cos{(\\cos{(\\sigma_p)})})", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 1, "Function('n_2')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n_2')(Symbol('\\\\sigma_p', commutative=True))), cos(cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('n_2')(Symbol('\\\\sigma_p', commutative=True))), cos(cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\sigma_p', commutative=True)), Add(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\sigma_p', commutative=True)), Add(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(Add(Function('A_2')(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{H}{(A_{2},T)} = \\sin^{A_{2}}{(T)} and Q{(T)} = \\sin{(T)}, then obtain e^{\\hat{H}{(A_{2},T)}} = e^{Q^{A_{2}}{(T)}}", "derivation": "\\hat{H}{(A_{2},T)} = \\sin^{A_{2}}{(T)} and Q{(T)} = \\sin{(T)} and \\hat{H}{(A_{2},T)} = Q^{A_{2}}{(T)} and e^{\\hat{H}{(A_{2},T)}} = e^{Q^{A_{2}}{(T)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('A_2', commutative=True), Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Symbol('A_2', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{H}')(Symbol('A_2', commutative=True), Symbol('T', commutative=True)), Pow(Function('Q')(Symbol('T', commutative=True)), Symbol('A_2', commutative=True)))"], [["exp", 3], "Equality(exp(Function('\\\\hat{H}')(Symbol('A_2', commutative=True), Symbol('T', commutative=True))), exp(Pow(Function('Q')(Symbol('T', commutative=True)), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(f,L)} = \\frac{e^{L}}{f}, then obtain \\mathbf{p}^{L}{(f,L)} \\frac{\\partial}{\\partial f} (\\mathbf{p}{(f,L)} + 1) = (\\frac{e^{L}}{f})^{L} \\frac{\\partial}{\\partial f} (\\mathbf{p}{(f,L)} + 1)", "derivation": "\\mathbf{p}{(f,L)} = \\frac{e^{L}}{f} and \\mathbf{p}{(f,L)} + 1 = 1 + \\frac{e^{L}}{f} and \\frac{\\partial}{\\partial f} (\\mathbf{p}{(f,L)} + 1) = \\frac{\\partial}{\\partial f} (1 + \\frac{e^{L}}{f}) and \\mathbf{p}^{L}{(f,L)} = (\\frac{e^{L}}{f})^{L} and \\mathbf{p}^{L}{(f,L)} \\frac{\\partial}{\\partial f} (1 + \\frac{e^{L}}{f}) = (\\frac{e^{L}}{f})^{L} \\frac{\\partial}{\\partial f} (1 + \\frac{e^{L}}{f}) and \\mathbf{p}^{L}{(f,L)} \\frac{\\partial}{\\partial f} (\\mathbf{p}{(f,L)} + 1) = (\\frac{e^{L}}{f})^{L} \\frac{\\partial}{\\partial f} (\\mathbf{p}{(f,L)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["times", 4, "Derivative(Add(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Derivative(Add(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Derivative(Add(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Derivative(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Derivative(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 \\phi_1, then derive \\frac{\\partial}{\\partial \\phi_1} \\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0, then obtain \\frac{\\frac{\\partial}{\\partial \\phi_1} \\hat{x}_0 \\phi_1}{\\hat{x}_0} = 1", "derivation": "\\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 \\phi_1 and - \\hat{x}_0 + \\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 \\phi_1 - \\hat{x}_0 and \\frac{\\partial}{\\partial \\phi_1} (- \\hat{x}_0 + \\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)}) = \\frac{\\partial}{\\partial \\phi_1} (\\hat{x}_0 \\phi_1 - \\hat{x}_0) and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{z^{*}}{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 and \\frac{\\partial}{\\partial \\phi_1} \\hat{x}_0 \\phi_1 = \\hat{x}_0 and \\phi_1 \\frac{\\partial}{\\partial \\phi_1} \\hat{x}_0 \\phi_1 = \\hat{x}_0 \\phi_1 and \\frac{\\frac{\\partial}{\\partial \\phi_1} \\hat{x}_0 \\phi_1}{\\hat{x}_0} = 1", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('z^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('z^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('z^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\hat{x}_0', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\hat{x}_0', commutative=True))"], [["times", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 6, "Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\varphi{(\\lambda)} = \\int \\log{(\\lambda)} d\\lambda, then obtain (\\frac{\\varphi{(\\lambda)}}{\\int \\log{(\\lambda)} d\\lambda} - 1)^{\\lambda} = 0^{\\lambda}", "derivation": "\\varphi{(\\lambda)} = \\int \\log{(\\lambda)} d\\lambda and \\frac{\\varphi{(\\lambda)}}{\\int \\log{(\\lambda)} d\\lambda} = 1 and \\frac{\\varphi{(\\lambda)}}{\\int \\log{(\\lambda)} d\\lambda} - 1 = 0 and (\\frac{\\varphi{(\\lambda)}}{\\int \\log{(\\lambda)} d\\lambda} - 1)^{\\lambda} = 0^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Mul(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(-1)), Integer(0))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Pow(Integer(0), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\eta,P_{g},c)} = c (P_{g} + \\eta), then obtain - \\frac{- \\eta + \\int \\operatorname{F_{x}}{(\\eta,P_{g},c)} dP_{g}}{\\eta} = - \\frac{- \\eta + \\int c (P_{g} + \\eta) dP_{g}}{\\eta}", "derivation": "\\operatorname{F_{x}}{(\\eta,P_{g},c)} = c (P_{g} + \\eta) and \\int \\operatorname{F_{x}}{(\\eta,P_{g},c)} dP_{g} = \\int c (P_{g} + \\eta) dP_{g} and - \\eta + \\int \\operatorname{F_{x}}{(\\eta,P_{g},c)} dP_{g} = - \\eta + \\int c (P_{g} + \\eta) dP_{g} and - \\frac{- \\eta + \\int \\operatorname{F_{x}}{(\\eta,P_{g},c)} dP_{g}}{\\eta} = - \\frac{- \\eta + \\int c (P_{g} + \\eta) dP_{g}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\eta', commutative=True), Symbol('P_g', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('\\\\eta', commutative=True), Symbol('P_g', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Mul(Symbol('c', commutative=True), Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Function('F_x')(Symbol('\\\\eta', commutative=True), Symbol('P_g', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Mul(Symbol('c', commutative=True), Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('P_g', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Function('F_x')(Symbol('\\\\eta', commutative=True), Symbol('P_g', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('P_g', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Mul(Symbol('c', commutative=True), Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('P_g', commutative=True))))))"]]}, {"prompt": "Given b{(C_{d})} = \\log{(C_{d})}, then obtain -1 - \\frac{1}{C_{d}} = - \\frac{d}{d C_{d}} b{(C_{d})} - 1", "derivation": "b{(C_{d})} = \\log{(C_{d})} and b{(C_{d})} - \\log{(C_{d})} = 0 and b{(C_{d})} - \\log{(C_{d})} + 1 = 1 and - C_{d} + b{(C_{d})} - \\log{(C_{d})} + 1 = 1 - C_{d} and - C_{d} - \\log{(C_{d})} + 1 = - C_{d} - b{(C_{d})} + 1 and \\frac{d}{d C_{d}} (- C_{d} - \\log{(C_{d})} + 1) = \\frac{d}{d C_{d}} (- C_{d} - b{(C_{d})} + 1) and -1 - \\frac{1}{C_{d}} = - \\frac{d}{d C_{d}} b{(C_{d})} - 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["minus", 1, "log(Symbol('C_d', commutative=True))"], "Equality(Add(Function('b')(Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('C_d', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('b')(Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('C_d', commutative=True))), Integer(1)), Integer(1))"], [["minus", 3, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Function('b')(Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('C_d', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Symbol('C_d', commutative=True))))"], [["minus", 4, "Function('b')(Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('C_d', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Function('b')(Symbol('C_d', commutative=True))), Integer(1)))"], [["differentiate", 5, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Symbol('C_d', commutative=True))), Integer(1)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Function('b')(Symbol('C_d', commutative=True))), Integer(1)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Derivative(Function('b')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given S{(i,a^{\\dagger},t)} = a^{\\dagger} i t, then derive \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)} = i t, then obtain - i^{2} + \\frac{\\partial}{\\partial t} (- S{(i,a^{\\dagger},t)} + \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)}) = - i^{2} + \\frac{\\partial}{\\partial t} (i t - S{(i,a^{\\dagger},t)})", "derivation": "S{(i,a^{\\dagger},t)} = a^{\\dagger} i t and \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)} = \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} i t and \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)} = i t and - S{(i,a^{\\dagger},t)} + \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)} = i t - S{(i,a^{\\dagger},t)} and \\frac{\\partial}{\\partial t} (- S{(i,a^{\\dagger},t)} + \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)}) = \\frac{\\partial}{\\partial t} (i t - S{(i,a^{\\dagger},t)}) and - i^{2} + \\frac{\\partial}{\\partial t} (- S{(i,a^{\\dagger},t)} + \\frac{\\partial}{\\partial a^{\\dagger}} S{(i,a^{\\dagger},t)}) = - i^{2} + \\frac{\\partial}{\\partial t} (i t - S{(i,a^{\\dagger},t)})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('i', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('i', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('i', commutative=True), Symbol('t', commutative=True)))"], [["minus", 3, "Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Derivative(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Mul(Symbol('i', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)))))"], [["differentiate", 4, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Derivative(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('i', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["minus", 5, "Pow(Symbol('i', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Derivative(Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('i', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('i', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mu)} = \\sin{(\\cos{(\\mu)})}, then obtain - \\cos{(\\cos{(\\mu)})} + \\frac{d}{d \\mu} \\Psi^{\\dagger}{(\\mu)} - 1 = - \\cos{(\\cos{(\\mu)})} + \\frac{d}{d \\mu} \\sin{(\\cos{(\\mu)})} - 1", "derivation": "\\Psi^{\\dagger}{(\\mu)} = \\sin{(\\cos{(\\mu)})} and \\frac{d}{d \\mu} \\Psi^{\\dagger}{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\cos{(\\mu)})} and \\frac{d}{d \\mu} \\Psi^{\\dagger}{(\\mu)} - 1 = \\frac{d}{d \\mu} \\sin{(\\cos{(\\mu)})} - 1 and - \\cos{(\\cos{(\\mu)})} + \\frac{d}{d \\mu} \\Psi^{\\dagger}{(\\mu)} - 1 = - \\cos{(\\cos{(\\mu)})} + \\frac{d}{d \\mu} \\sin{(\\cos{(\\mu)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), sin(cos(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(cos(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 3, "cos(cos(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(cos(Symbol('\\\\mu', commutative=True)))), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), cos(cos(Symbol('\\\\mu', commutative=True)))), Derivative(sin(cos(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given W{(\\dot{\\mathbf{r}})} = \\cos{(e^{\\dot{\\mathbf{r}}})}, then derive Q + W{(\\dot{\\mathbf{r}})} = \\mathbb{I} + \\cos{(e^{\\dot{\\mathbf{r}}})}, then obtain Q + W{(\\dot{\\mathbf{r}})} = \\mathbb{I} + W{(\\dot{\\mathbf{r}})}", "derivation": "W{(\\dot{\\mathbf{r}})} = \\cos{(e^{\\dot{\\mathbf{r}}})} and \\frac{d}{d \\dot{\\mathbf{r}}} W{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(e^{\\dot{\\mathbf{r}}})} and \\int \\frac{d}{d \\dot{\\mathbf{r}}} W{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(e^{\\dot{\\mathbf{r}}})} d\\dot{\\mathbf{r}} and Q + W{(\\dot{\\mathbf{r}})} = \\mathbb{I} + \\cos{(e^{\\dot{\\mathbf{r}}})} and Q + \\cos{(e^{\\dot{\\mathbf{r}}})} = \\mathbb{I} + \\cos{(e^{\\dot{\\mathbf{r}}})} and Q + W{(\\dot{\\mathbf{r}})} = \\mathbb{I} + W{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('Q', commutative=True), Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('Q', commutative=True), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('\\\\mathbb{I}', commutative=True), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('Q', commutative=True), Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{r})} = \\mathbf{r} and \\operatorname{c_{0}}{(\\mathbf{r})} = \\frac{d}{d \\operatorname{z^{*}}{(\\mathbf{r})}} \\operatorname{z^{*}}{(\\mathbf{r})}, then derive \\frac{d}{d \\mathbf{r}} \\operatorname{z^{*}}{(\\mathbf{r})} = 1, then obtain \\int \\operatorname{c_{0}}{(\\mathbf{r})} d\\mathbf{r} = \\int 1 d\\mathbf{r}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{r})} = \\mathbf{r} and \\frac{d}{d \\mathbf{r}} \\operatorname{z^{*}}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\mathbf{r} and \\frac{d}{d \\mathbf{r}} \\operatorname{z^{*}}{(\\mathbf{r})} = 1 and \\frac{d}{d \\mathbf{r}} \\mathbf{r} = 1 and \\frac{d}{d \\operatorname{z^{*}}{(\\mathbf{r})}} \\operatorname{z^{*}}{(\\mathbf{r})} = 1 and \\operatorname{c_{0}}{(\\mathbf{r})} = \\frac{d}{d \\operatorname{z^{*}}{(\\mathbf{r})}} \\operatorname{z^{*}}{(\\mathbf{r})} and \\operatorname{c_{0}}{(\\mathbf{r})} = 1 and \\int \\operatorname{c_{0}}{(\\mathbf{r})} d\\mathbf{r} = \\int 1 d\\mathbf{r}", "srepr_derivation": [["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Function('z^*')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(1))"], [["integrate", 7, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})} = - \\hat{x} + v_{1}, then obtain b + \\int (\\frac{\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1} = b + \\int (\\frac{- \\hat{x} + v_{1}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1}", "derivation": "\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})} = - \\hat{x} + v_{1} and \\frac{\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})}}{v_{1}} = \\frac{- \\hat{x} + v_{1}}{v_{1}} and \\frac{\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})}}{v_{1}} + \\frac{1}{v_{1}} = \\frac{- \\hat{x} + v_{1}}{v_{1}} + \\frac{1}{v_{1}} and \\int (\\frac{\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1} = \\int (\\frac{- \\hat{x} + v_{1}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1} and b + \\int (\\frac{\\operatorname{E_{\\lambda}}{(\\hat{x},v_{1})}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1} = b + \\int (\\frac{- \\hat{x} + v_{1}}{v_{1}} + \\frac{1}{v_{1}}) dv_{1}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_1', commutative=True)))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_1', commutative=True))))"], [["add", 2, "Pow(Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Tuple(Symbol('v_1', commutative=True))))"], [["add", 4, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Integral(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('b', commutative=True), Integral(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Tuple(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(C_{2})} = e^{C_{2}}, then derive \\frac{d}{d C_{2}} \\operatorname{t_{1}}{(C_{2})} = e^{C_{2}}, then obtain - C_{2} + \\operatorname{t_{1}}{(C_{2})} = - C_{2} + \\frac{d}{d C_{2}} e^{C_{2}}", "derivation": "\\operatorname{t_{1}}{(C_{2})} = e^{C_{2}} and - C_{2} + \\operatorname{t_{1}}{(C_{2})} = - C_{2} + e^{C_{2}} and \\frac{d}{d C_{2}} \\operatorname{t_{1}}{(C_{2})} = \\frac{d}{d C_{2}} e^{C_{2}} and \\frac{d}{d C_{2}} \\operatorname{t_{1}}{(C_{2})} = e^{C_{2}} and e^{C_{2}} = \\frac{d}{d C_{2}} e^{C_{2}} and - C_{2} + \\operatorname{t_{1}}{(C_{2})} = - C_{2} + \\frac{d}{d C_{2}} e^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["minus", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('t_1')(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('t_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), exp(Symbol('C_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Symbol('C_2', commutative=True)), Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('t_1')(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v_{z},z)} = \\log{(\\frac{z}{v_{z}})}, then obtain \\frac{z \\int (v_{z} + \\operatorname{f_{E}}{(v_{z},z)}) dv_{z}}{v_{z}} = \\frac{z (F_{g} + \\frac{v_{z}^{2}}{2} + v_{z} \\log{(\\frac{z}{v_{z}})} + v_{z})}{v_{z}}", "derivation": "\\operatorname{f_{E}}{(v_{z},z)} = \\log{(\\frac{z}{v_{z}})} and v_{z} + \\operatorname{f_{E}}{(v_{z},z)} = v_{z} + \\log{(\\frac{z}{v_{z}})} and \\int (v_{z} + \\operatorname{f_{E}}{(v_{z},z)}) dv_{z} = \\int (v_{z} + \\log{(\\frac{z}{v_{z}})}) dv_{z} and \\frac{z \\int (v_{z} + \\operatorname{f_{E}}{(v_{z},z)}) dv_{z}}{v_{z}} = \\frac{z \\int (v_{z} + \\log{(\\frac{z}{v_{z}})}) dv_{z}}{v_{z}} and \\frac{z \\int (v_{z} + \\operatorname{f_{E}}{(v_{z},z)}) dv_{z}}{v_{z}} = \\frac{z (F_{g} + \\frac{v_{z}^{2}}{2} + v_{z} \\log{(\\frac{z}{v_{z}})} + v_{z})}{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v_z', commutative=True), Symbol('z', commutative=True)), log(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["add", 1, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Function('f_E')(Symbol('v_z', commutative=True), Symbol('z', commutative=True))), Add(Symbol('v_z', commutative=True), log(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True)))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Symbol('v_z', commutative=True), Function('f_E')(Symbol('v_z', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('v_z', commutative=True), log(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Tuple(Symbol('v_z', commutative=True))))"], [["times", 3, "Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Add(Symbol('v_z', commutative=True), Function('f_E')(Symbol('v_z', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Add(Symbol('v_z', commutative=True), log(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Tuple(Symbol('v_z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Add(Symbol('v_z', commutative=True), Function('f_E')(Symbol('v_z', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True), Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2))), Mul(Symbol('v_z', commutative=True), log(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(F_{c},A_{z})} = A_{z} + F_{c}, then obtain (((\\operatorname{F_{N}}^{F_{c}}{(F_{c},A_{z})})^{F_{c}})^{F_{c}})^{A_{z}} = ((((A_{z} + F_{c})^{F_{c}})^{F_{c}})^{F_{c}})^{A_{z}}", "derivation": "\\operatorname{F_{N}}{(F_{c},A_{z})} = A_{z} + F_{c} and \\operatorname{F_{N}}^{F_{c}}{(F_{c},A_{z})} = (A_{z} + F_{c})^{F_{c}} and (\\operatorname{F_{N}}^{F_{c}}{(F_{c},A_{z})})^{F_{c}} = ((A_{z} + F_{c})^{F_{c}})^{F_{c}} and ((\\operatorname{F_{N}}^{F_{c}}{(F_{c},A_{z})})^{F_{c}})^{F_{c}} = (((A_{z} + F_{c})^{F_{c}})^{F_{c}})^{F_{c}} and (((\\operatorname{F_{N}}^{F_{c}}{(F_{c},A_{z})})^{F_{c}})^{F_{c}})^{A_{z}} = ((((A_{z} + F_{c})^{F_{c}})^{F_{c}})^{F_{c}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('F_c', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('F_c', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('F_c', commutative=True), Symbol('A_z', commutative=True)), Symbol('F_c', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Function('F_N')(Symbol('F_c', commutative=True), Symbol('A_z', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(Add(Symbol('A_z', commutative=True), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Pow(Function('F_N')(Symbol('F_c', commutative=True), Symbol('A_z', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(Pow(Add(Symbol('A_z', commutative=True), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 4, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Pow(Pow(Function('F_N')(Symbol('F_c', commutative=True), Symbol('A_z', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(Pow(Pow(Add(Symbol('A_z', commutative=True), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(E_{x})} = \\sin{(\\log{(E_{x})})}, then obtain (\\mu_{0}{(E_{x})} + \\sin{(\\log{(E_{x})})}) \\mu_{0}{(E_{x})} + \\mu_{0}{(E_{x})} = 2 \\mu_{0}{(E_{x})} \\sin{(\\log{(E_{x})})} + \\mu_{0}{(E_{x})}", "derivation": "\\mu_{0}{(E_{x})} = \\sin{(\\log{(E_{x})})} and \\mu_{0}{(E_{x})} + \\sin{(\\log{(E_{x})})} = 2 \\sin{(\\log{(E_{x})})} and (\\mu_{0}{(E_{x})} + \\sin{(\\log{(E_{x})})}) \\mu_{0}{(E_{x})} = 2 \\mu_{0}{(E_{x})} \\sin{(\\log{(E_{x})})} and (\\mu_{0}{(E_{x})} + \\sin{(\\log{(E_{x})})}) \\mu_{0}{(E_{x})} + \\mu_{0}{(E_{x})} = 2 \\mu_{0}{(E_{x})} \\sin{(\\log{(E_{x})})} + \\mu_{0}{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True))))"], [["add", 1, "sin(log(Symbol('E_x', commutative=True)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True)))), Mul(Integer(2), sin(log(Symbol('E_x', commutative=True)))))"], [["times", 2, "Function('\\\\mu_0')(Symbol('E_x', commutative=True))"], "Equality(Mul(Add(Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True)))), Function('\\\\mu_0')(Symbol('E_x', commutative=True))), Mul(Integer(2), Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True)))))"], [["add", 3, "Function('\\\\mu_0')(Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Add(Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True)))), Function('\\\\mu_0')(Symbol('E_x', commutative=True))), Function('\\\\mu_0')(Symbol('E_x', commutative=True))), Add(Mul(Integer(2), Function('\\\\mu_0')(Symbol('E_x', commutative=True)), sin(log(Symbol('E_x', commutative=True)))), Function('\\\\mu_0')(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(v_{y},Z)} = Z v_{y} and g{(v_{y},Z)} = \\frac{\\partial}{\\partial Z} Z v_{y}, then obtain g{(v_{y},Z)} + \\frac{\\partial}{\\partial Z} Z v_{y} = 2 \\frac{\\partial}{\\partial Z} Z v_{y}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(v_{y},Z)} = Z v_{y} and \\frac{\\partial}{\\partial Z} \\operatorname{V_{\\mathbf{E}}}{(v_{y},Z)} = \\frac{\\partial}{\\partial Z} Z v_{y} and g{(v_{y},Z)} = \\frac{\\partial}{\\partial Z} Z v_{y} and g{(v_{y},Z)} = \\frac{\\partial}{\\partial Z} \\operatorname{V_{\\mathbf{E}}}{(v_{y},Z)} and g{(v_{y},Z)} + \\frac{\\partial}{\\partial Z} Z v_{y} = \\frac{\\partial}{\\partial Z} Z v_{y} + \\frac{\\partial}{\\partial Z} \\operatorname{V_{\\mathbf{E}}}{(v_{y},Z)} and g{(v_{y},Z)} + \\frac{\\partial}{\\partial Z} Z v_{y} = 2 \\frac{\\partial}{\\partial Z} Z v_{y}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('g')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Add(Function('g')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('g')(Symbol('v_y', commutative=True), Symbol('Z', commutative=True)), Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Symbol('Z', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mu)} = \\cos{(\\mu)}, then derive \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} = - \\sin{(\\mu)}, then obtain - \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} + \\frac{d}{d \\mu} \\cos{(\\mu)} + (\\frac{d}{d \\mu} \\cos{(\\mu)})^{\\mu} = - \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} + (\\frac{d}{d \\mu} \\mathbf{H}{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)}", "derivation": "\\mathbf{H}{(\\mu)} = \\cos{(\\mu)} and \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} = \\frac{d}{d \\mu} \\cos{(\\mu)} and \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} = - \\sin{(\\mu)} and \\frac{d}{d \\mu} \\cos{(\\mu)} = - \\sin{(\\mu)} and (\\frac{d}{d \\mu} \\cos{(\\mu)})^{\\mu} = (- \\sin{(\\mu)})^{\\mu} and (\\frac{d}{d \\mu} \\cos{(\\mu)})^{\\mu} = (\\frac{d}{d \\mu} \\mathbf{H}{(\\mu)})^{\\mu} and - \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} + \\frac{d}{d \\mu} \\cos{(\\mu)} + (\\frac{d}{d \\mu} \\cos{(\\mu)})^{\\mu} = - \\frac{d}{d \\mu} \\mathbf{H}{(\\mu)} + (\\frac{d}{d \\mu} \\mathbf{H}{(\\mu)})^{\\mu} + \\frac{d}{d \\mu} \\cos{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["add", 6, "Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(x^\\prime)} = e^{x^\\prime} and \\mathbf{B}{(F_{g},\\varphi)} = F_{g} \\log{(\\varphi)}, then obtain \\frac{\\partial}{\\partial \\varphi} (\\mathbf{B}{(F_{g},\\varphi)} + \\cos{(Q{(x^\\prime)})}) = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{B}{(F_{g},\\varphi)} + \\cos{(e^{x^\\prime})})", "derivation": "Q{(x^\\prime)} = e^{x^\\prime} and \\cos{(Q{(x^\\prime)})} = \\cos{(e^{x^\\prime})} and \\mathbf{B}{(F_{g},\\varphi)} = F_{g} \\log{(\\varphi)} and F_{g} \\log{(\\varphi)} + \\cos{(Q{(x^\\prime)})} = F_{g} \\log{(\\varphi)} + \\cos{(e^{x^\\prime})} and \\mathbf{B}{(F_{g},\\varphi)} + \\cos{(Q{(x^\\prime)})} = \\mathbf{B}{(F_{g},\\varphi)} + \\cos{(e^{x^\\prime})} and \\frac{\\partial}{\\partial \\varphi} (\\mathbf{B}{(F_{g},\\varphi)} + \\cos{(Q{(x^\\prime)})}) = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{B}{(F_{g},\\varphi)} + \\cos{(e^{x^\\prime})})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["cos", 1], "Equality(cos(Function('Q')(Symbol('x^\\\\prime', commutative=True))), cos(exp(Symbol('x^\\\\prime', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('F_g', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "Mul(Symbol('F_g', commutative=True), log(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Symbol('F_g', commutative=True), log(Symbol('\\\\varphi', commutative=True))), cos(Function('Q')(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Symbol('F_g', commutative=True), log(Symbol('\\\\varphi', commutative=True))), cos(exp(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Function('Q')(Symbol('x^\\\\prime', commutative=True)))), Add(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(exp(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Function('Q')(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(exp(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(G)} = e^{\\sin{(G)}}, then derive \\frac{d}{d G} \\sigma_{p}{(G)} = e^{\\sin{(G)}} \\cos{(G)}, then obtain \\frac{\\partial}{\\partial G} (W + \\sigma_{p}{(G)}) = \\frac{\\partial}{\\partial G} (P_{g} + e^{\\sin{(G)}})", "derivation": "\\sigma_{p}{(G)} = e^{\\sin{(G)}} and \\frac{d}{d G} \\sigma_{p}{(G)} = \\frac{d}{d G} e^{\\sin{(G)}} and \\frac{d}{d G} \\sigma_{p}{(G)} = e^{\\sin{(G)}} \\cos{(G)} and \\int \\frac{d}{d G} \\sigma_{p}{(G)} dG = \\int e^{\\sin{(G)}} \\cos{(G)} dG and \\frac{d}{d G} \\int \\frac{d}{d G} \\sigma_{p}{(G)} dG = \\frac{d}{d G} \\int e^{\\sin{(G)}} \\cos{(G)} dG and \\frac{\\partial}{\\partial G} (W + \\sigma_{p}{(G)}) = \\frac{\\partial}{\\partial G} (P_{g} + e^{\\sin{(G)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('G', commutative=True)), exp(sin(Symbol('G', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(exp(sin(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integral(Mul(exp(sin(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integral(Mul(exp(sin(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('W', commutative=True), Function('\\\\sigma_p')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('P_g', commutative=True), exp(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(\\hat{H},M)} = M \\hat{H} and \\theta_{2}{(\\hat{H},M)} = \\int \\phi_{2}{(\\hat{H},M)} d\\hat{H}, then obtain - 2 \\hat{x}_0 + \\theta_{2}{(\\hat{H},M)} = - 2 \\hat{x}_0 + \\int M \\hat{H} d\\hat{H}", "derivation": "\\phi_{2}{(\\hat{H},M)} = M \\hat{H} and \\int \\phi_{2}{(\\hat{H},M)} d\\hat{H} = \\int M \\hat{H} d\\hat{H} and - \\hat{x}_0 + \\int \\phi_{2}{(\\hat{H},M)} d\\hat{H} = - \\hat{x}_0 + \\int M \\hat{H} d\\hat{H} and \\theta_{2}{(\\hat{H},M)} = \\int \\phi_{2}{(\\hat{H},M)} d\\hat{H} and - \\hat{x}_0 + \\theta_{2}{(\\hat{H},M)} = - \\hat{x}_0 + \\int M \\hat{H} d\\hat{H} and - 2 \\hat{x}_0 + \\theta_{2}{(\\hat{H},M)} = - 2 \\hat{x}_0 + \\int M \\hat{H} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True)), Integral(Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 5, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} e^{\\hat{X}}, then obtain \\hat{X} \\int \\operatorname{C_{2}}{(\\hat{X})} d\\hat{X} = \\hat{X} (V_{\\mathbf{B}} + e^{\\hat{X}})", "derivation": "\\operatorname{C_{2}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} e^{\\hat{X}} and \\int \\operatorname{C_{2}}{(\\hat{X})} d\\hat{X} = \\int \\frac{d}{d \\hat{X}} e^{\\hat{X}} d\\hat{X} and \\hat{X} \\int \\operatorname{C_{2}}{(\\hat{X})} d\\hat{X} = \\hat{X} \\int \\frac{d}{d \\hat{X}} e^{\\hat{X}} d\\hat{X} and \\hat{X} \\int \\operatorname{C_{2}}{(\\hat{X})} d\\hat{X} = \\hat{X} (V_{\\mathbf{B}} + e^{\\hat{X}})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{X}', commutative=True)), Derivative(exp(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Derivative(exp(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Integral(Function('C_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Mul(Symbol('\\\\hat{X}', commutative=True), Integral(Derivative(exp(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Integral(Function('C_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Mul(Symbol('\\\\hat{X}', commutative=True), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(z^{*},\\phi)} = \\cos{(\\phi - z^{*})}, then obtain \\phi_{1}{(z^{*},\\phi)} - \\int \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} d\\phi = \\cos{(\\phi - z^{*})} - \\int \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} d\\phi", "derivation": "\\phi_{1}{(z^{*},\\phi)} = \\cos{(\\phi - z^{*})} and \\phi \\phi_{1}{(z^{*},\\phi)} = \\phi \\cos{(\\phi - z^{*})} and 1 = \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} and \\int 1 d\\phi = \\int \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} d\\phi and \\phi_{1}{(z^{*},\\phi)} - \\int 1 d\\phi = \\cos{(\\phi - z^{*})} - \\int 1 d\\phi and \\phi_{1}{(z^{*},\\phi)} - \\int \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} d\\phi = \\cos{(\\phi - z^{*})} - \\int \\frac{\\cos{(\\phi - z^{*})}}{\\phi_{1}{(z^{*},\\phi)}} d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["divide", 2, "Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))))), Add(cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('\\\\phi', commutative=True))))), Add(cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Mul(Integer(-1), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('z^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given m{(p)} = \\cos{(p)} and \\eta^{\\prime}{(L_{\\varepsilon})} = \\cos{(\\cos{(L_{\\varepsilon})})}, then obtain (\\eta^{\\prime}{(L_{\\varepsilon})} + 1) \\cos{(p)} = (\\cos{(\\cos{(L_{\\varepsilon})})} + 1) \\cos{(p)}", "derivation": "m{(p)} = \\cos{(p)} and \\eta^{\\prime}{(L_{\\varepsilon})} = \\cos{(\\cos{(L_{\\varepsilon})})} and \\eta^{\\prime}{(L_{\\varepsilon})} + \\frac{\\cos{(p)}}{m{(p)}} = \\cos{(\\cos{(L_{\\varepsilon})})} + \\frac{\\cos{(p)}}{m{(p)}} and \\eta^{\\prime}{(L_{\\varepsilon})} + 1 = \\cos{(\\cos{(L_{\\varepsilon})})} + 1 and (\\eta^{\\prime}{(L_{\\varepsilon})} + 1) m^{\\frac{\\cos{(p)}}{m{(p)}}}{(p)} = (\\cos{(\\cos{(L_{\\varepsilon})})} + 1) m^{\\frac{\\cos{(p)}}{m{(p)}}}{(p)} and (\\eta^{\\prime}{(L_{\\varepsilon})} + 1) \\cos{(p)} = (\\cos{(\\cos{(L_{\\varepsilon})})} + 1) \\cos{(p)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(cos(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["add", 2, "Mul(Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True)))), Add(cos(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(cos(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(1)))"], [["divide", 4, "Pow(Function('m')(Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True))))"], "Equality(Mul(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(1)), Pow(Function('m')(Symbol('p', commutative=True)), Mul(Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True))))), Mul(Add(cos(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(1)), Pow(Function('m')(Symbol('p', commutative=True)), Mul(Pow(Function('m')(Symbol('p', commutative=True)), Integer(-1)), cos(Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(1)), cos(Symbol('p', commutative=True))), Mul(Add(cos(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(1)), cos(Symbol('p', commutative=True))))"]]}, {"prompt": "Given c{(\\eta)} = e^{\\sin{(\\eta)}}, then obtain ((\\frac{c{(\\eta)}}{\\eta})^{\\eta})^{\\eta} = ((\\frac{e^{\\sin{(\\eta)}}}{\\eta})^{\\eta})^{\\eta}", "derivation": "c{(\\eta)} = e^{\\sin{(\\eta)}} and \\frac{c{(\\eta)}}{\\eta} = \\frac{e^{\\sin{(\\eta)}}}{\\eta} and (\\frac{c{(\\eta)}}{\\eta})^{\\eta} = (\\frac{e^{\\sin{(\\eta)}}}{\\eta})^{\\eta} and ((\\frac{c{(\\eta)}}{\\eta})^{\\eta})^{\\eta} = ((\\frac{e^{\\sin{(\\eta)}}}{\\eta})^{\\eta})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\eta', commutative=True)), exp(sin(Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(sin(Symbol('\\\\eta', commutative=True)))))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)))"], [["power", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\varphi{(P_{e})} = \\log{(P_{e})}, then derive 2 \\frac{d}{d P_{e}} \\varphi{(P_{e})} = \\frac{d}{d P_{e}} \\varphi{(P_{e})} + \\frac{1}{P_{e}}, then obtain 2 \\frac{d}{d P_{e}} \\log{(P_{e})} = \\frac{d}{d P_{e}} \\log{(P_{e})} + \\frac{1}{P_{e}}", "derivation": "\\varphi{(P_{e})} = \\log{(P_{e})} and \\frac{d}{d P_{e}} \\varphi{(P_{e})} = \\frac{d}{d P_{e}} \\log{(P_{e})} and 2 \\frac{d}{d P_{e}} \\varphi{(P_{e})} = \\frac{d}{d P_{e}} \\varphi{(P_{e})} + \\frac{d}{d P_{e}} \\log{(P_{e})} and 2 \\frac{d}{d P_{e}} \\varphi{(P_{e})} = \\frac{d}{d P_{e}} \\varphi{(P_{e})} + \\frac{1}{P_{e}} and 2 \\frac{d}{d P_{e}} \\log{(P_{e})} = \\frac{d}{d P_{e}} \\log{(P_{e})} + \\frac{1}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\varphi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Pow(Symbol('P_e', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Pow(Symbol('P_e', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(h)} = \\cos{(h)}, then obtain - h = - h \\varepsilon^{- h}{(h)} \\cos^{h}{(h)}", "derivation": "\\varepsilon{(h)} = \\cos{(h)} and \\varepsilon^{h}{(h)} = \\cos^{h}{(h)} and \\varepsilon^{h}{(h)} \\cos^{- h}{(h)} = 1 and 1 = \\varepsilon^{- h}{(h)} \\cos^{h}{(h)} and - h = - h \\varepsilon^{- h}{(h)} \\cos^{h}{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["divide", 2, "Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varepsilon')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))), Integer(1))"], [["divide", 3, "Mul(Pow(Function('\\\\varepsilon')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given k{(Q)} = \\cos{(Q)}, then derive \\frac{d}{d Q} k{(Q)} = - \\sin{(Q)}, then obtain - \\sin{(Q)} - \\frac{d}{d Q} \\cos{(Q)} = 0", "derivation": "k{(Q)} = \\cos{(Q)} and \\frac{d}{d Q} k{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\frac{d}{d Q} k{(Q)} = - \\sin{(Q)} and \\frac{d}{d Q} k{(Q)} - 1 = - \\sin{(Q)} - 1 and \\sin{(Q)} + \\frac{d}{d Q} k{(Q)} = 0 and \\sin{(Q)} + \\frac{d}{d Q} \\cos{(Q)} = 0 and - \\sin{(Q)} - \\frac{d}{d Q} \\cos{(Q)} = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('k')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Integer(-1)))"], [["minus", 4, "Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Integer(-1))"], "Equality(Add(sin(Symbol('Q', commutative=True)), Derivative(Function('k')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(sin(Symbol('Q', commutative=True)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(0))"], [["divide", 6, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\ddot{x}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime})}, then derive \\int \\ddot{x}{(y^{\\prime})} dy^{\\prime} = A_{z} + \\operatorname{a^{\\dagger}}{(y^{\\prime})}, then obtain \\iiint \\ddot{x}{(y^{\\prime})} dy^{\\prime} dy^{\\prime} dy^{\\prime} = \\iint (A_{z} + \\operatorname{a^{\\dagger}}{(y^{\\prime})}) dy^{\\prime} dy^{\\prime}", "derivation": "\\ddot{x}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime})} and \\int \\ddot{x}{(y^{\\prime})} dy^{\\prime} = \\int \\frac{d}{d y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime})} dy^{\\prime} and \\int \\ddot{x}{(y^{\\prime})} dy^{\\prime} = A_{z} + \\operatorname{a^{\\dagger}}{(y^{\\prime})} and \\iint \\ddot{x}{(y^{\\prime})} dy^{\\prime} dy^{\\prime} = \\int (A_{z} + \\operatorname{a^{\\dagger}}{(y^{\\prime})}) dy^{\\prime} and \\iiint \\ddot{x}{(y^{\\prime})} dy^{\\prime} dy^{\\prime} dy^{\\prime} = \\iint (A_{z} + \\operatorname{a^{\\dagger}}{(y^{\\prime})}) dy^{\\prime} dy^{\\prime}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('A_z', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\theta,A_{y},\\theta_2)} = A_{y} \\theta_2^{\\theta} and \\sigma_{x}{(\\theta,\\theta_2)} = \\theta_2^{\\theta} and n{(\\theta,\\theta_2)} = \\theta_2^{\\theta}, then obtain A_{y} n{(\\theta,\\theta_2)} = A_{y} \\sigma_{x}{(\\theta,\\theta_2)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\theta,A_{y},\\theta_2)} = A_{y} \\theta_2^{\\theta} and \\sigma_{x}{(\\theta,\\theta_2)} = \\theta_2^{\\theta} and \\operatorname{g_{\\varepsilon}}{(\\theta,A_{y},\\theta_2)} = A_{y} \\sigma_{x}{(\\theta,\\theta_2)} and A_{y} \\theta_2^{\\theta} = A_{y} \\sigma_{x}{(\\theta,\\theta_2)} and n{(\\theta,\\theta_2)} = \\theta_2^{\\theta} and A_{y} n{(\\theta,\\theta_2)} = A_{y} \\sigma_{x}{(\\theta,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('A_y', commutative=True), Function('n')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('A_y', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} = \\dot{z} + \\rho_b, then obtain - \\dot{z} + \\operatorname{f_{\\mathbf{v}}}^{2}{(\\dot{z},\\rho_b)} = \\dot{z} \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} - \\dot{z} + \\rho_b \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} = \\dot{z} + \\rho_b and \\operatorname{f_{\\mathbf{v}}}^{2}{(\\dot{z},\\rho_b)} = (\\dot{z} + \\rho_b) \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} and - \\dot{z} + \\operatorname{f_{\\mathbf{v}}}^{2}{(\\dot{z},\\rho_b)} = - \\dot{z} + (\\dot{z} + \\rho_b) \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} and - \\dot{z} + \\operatorname{f_{\\mathbf{v}}}^{2}{(\\dot{z},\\rho_b)} = \\dot{z} \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)} - \\dot{z} + \\rho_b \\operatorname{f_{\\mathbf{v}}}{(\\dot{z},\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["expand", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(2))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given k{(l,W)} = \\sin{(l^{W})}, then obtain (3 ((\\frac{\\partial}{\\partial W} k{(l,W)})^{l})^{W} \\sin{(l^{W})})^{l} = (3 ((\\frac{\\partial}{\\partial W} \\sin{(l^{W})})^{l})^{W} \\sin{(l^{W})})^{l}", "derivation": "k{(l,W)} = \\sin{(l^{W})} and \\frac{\\partial}{\\partial W} k{(l,W)} = \\frac{\\partial}{\\partial W} \\sin{(l^{W})} and (\\frac{\\partial}{\\partial W} k{(l,W)})^{l} = (\\frac{\\partial}{\\partial W} \\sin{(l^{W})})^{l} and ((\\frac{\\partial}{\\partial W} k{(l,W)})^{l})^{W} = ((\\frac{\\partial}{\\partial W} \\sin{(l^{W})})^{l})^{W} and 3 ((\\frac{\\partial}{\\partial W} k{(l,W)})^{l})^{W} \\sin{(l^{W})} = 3 ((\\frac{\\partial}{\\partial W} \\sin{(l^{W})})^{l})^{W} \\sin{(l^{W})} and (3 ((\\frac{\\partial}{\\partial W} k{(l,W)})^{l})^{W} \\sin{(l^{W})})^{l} = (3 ((\\frac{\\partial}{\\partial W} \\sin{(l^{W})})^{l})^{W} \\sin{(l^{W})})^{l}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Derivative(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Pow(Derivative(sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Derivative(sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)))"], [["times", 4, "Mul(Integer(3), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))))"], "Equality(Mul(Integer(3), Pow(Pow(Derivative(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)))), Mul(Integer(3), Pow(Pow(Derivative(sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)))))"], [["power", 5, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(Integer(3), Pow(Pow(Derivative(Function('k')(Symbol('l', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)))), Symbol('l', commutative=True)), Pow(Mul(Integer(3), Pow(Pow(Derivative(sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('l', commutative=True)), Symbol('W', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\Omega{(E)} = \\sin{(E)}, then obtain \\frac{e^{\\Omega{(E)}}}{\\cos{(\\Omega{(E)})}} = \\frac{e^{\\sin{(E)}}}{\\cos{(\\Omega{(E)})}}", "derivation": "\\Omega{(E)} = \\sin{(E)} and \\cos{(\\Omega{(E)})} = \\cos{(\\sin{(E)})} and e^{\\Omega{(E)}} = e^{\\sin{(E)}} and \\frac{e^{\\Omega{(E)}}}{\\cos{(\\sin{(E)})}} = \\frac{e^{\\sin{(E)}}}{\\cos{(\\sin{(E)})}} and \\frac{e^{\\Omega{(E)}}}{\\cos{(\\Omega{(E)})}} = \\frac{e^{\\sin{(E)}}}{\\cos{(\\Omega{(E)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\Omega')(Symbol('E', commutative=True))), cos(sin(Symbol('E', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\Omega')(Symbol('E', commutative=True))), exp(sin(Symbol('E', commutative=True))))"], [["divide", 3, "cos(sin(Symbol('E', commutative=True)))"], "Equality(Mul(exp(Function('\\\\Omega')(Symbol('E', commutative=True))), Pow(cos(sin(Symbol('E', commutative=True))), Integer(-1))), Mul(exp(sin(Symbol('E', commutative=True))), Pow(cos(sin(Symbol('E', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(exp(Function('\\\\Omega')(Symbol('E', commutative=True))), Pow(cos(Function('\\\\Omega')(Symbol('E', commutative=True))), Integer(-1))), Mul(exp(sin(Symbol('E', commutative=True))), Pow(cos(Function('\\\\Omega')(Symbol('E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{M}{(m)} = \\sin{(m)}, then derive \\int \\mathbf{M}{(m)} dm = V_{\\mathbf{E}} - \\cos{(m)}, then obtain - \\cos{(m)} \\frac{d}{d V_{\\mathbf{E}}} \\int \\mathbf{M}{(m)} dm = - \\cos{(m)} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (V_{\\mathbf{E}} - \\cos{(m)})", "derivation": "\\mathbf{M}{(m)} = \\sin{(m)} and \\int \\mathbf{M}{(m)} dm = \\int \\sin{(m)} dm and \\int \\mathbf{M}{(m)} dm = V_{\\mathbf{E}} - \\cos{(m)} and \\frac{d}{d V_{\\mathbf{E}}} \\int \\mathbf{M}{(m)} dm = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (V_{\\mathbf{E}} - \\cos{(m)}) and - \\cos{(m)} \\frac{d}{d V_{\\mathbf{E}}} \\int \\mathbf{M}{(m)} dm = - \\cos{(m)} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (V_{\\mathbf{E}} - \\cos{(m)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('m', commutative=True)))))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('m', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["times", 4, "Mul(Integer(-1), cos(Symbol('m', commutative=True)))"], "Equality(Mul(Integer(-1), cos(Symbol('m', commutative=True)), Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('m', commutative=True)), Derivative(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('m', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given n{(\\mathbf{p},y)} = \\mathbf{p} - y, then derive \\int n{(\\mathbf{p},y)} dy = \\mathbf{p} y + r_{0} - \\frac{y^{2}}{2}, then obtain (\\int n{(\\mathbf{p},y)} dy)^{2 y} = (\\mathbf{p} y + r_{0} - \\frac{y^{2}}{2})^{2 y}", "derivation": "n{(\\mathbf{p},y)} = \\mathbf{p} - y and \\int n{(\\mathbf{p},y)} dy = \\int (\\mathbf{p} - y) dy and \\int n{(\\mathbf{p},y)} dy = \\mathbf{p} y + r_{0} - \\frac{y^{2}}{2} and \\int (\\mathbf{p} - y) dy = \\mathbf{p} y + r_{0} - \\frac{y^{2}}{2} and (\\int n{(\\mathbf{p},y)} dy)^{y} = (\\int (\\mathbf{p} - y) dy)^{y} and (\\int n{(\\mathbf{p},y)} dy)^{y} = (\\mathbf{p} y + r_{0} - \\frac{y^{2}}{2})^{y} and (\\int n{(\\mathbf{p},y)} dy)^{2 y} = (\\mathbf{p} y + r_{0} - \\frac{y^{2}}{2})^{2 y}", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Symbol('y', commutative=True)))"], [["power", 6, 2], "Equality(Pow(Integral(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Mul(Integer(2), Symbol('y', commutative=True))), Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('y', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Mul(Integer(2), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\dot{y},C_{2})} = \\log{(C_{2} - \\dot{y})}, then obtain \\frac{\\int 2 \\theta_{2}{(\\dot{y},C_{2})} d\\dot{y}}{\\log{(C_{2} - \\dot{y})}} = \\frac{\\int (\\theta_{2}{(\\dot{y},C_{2})} + \\log{(C_{2} - \\dot{y})}) d\\dot{y}}{\\log{(C_{2} - \\dot{y})}}", "derivation": "\\theta_{2}{(\\dot{y},C_{2})} = \\log{(C_{2} - \\dot{y})} and 2 \\theta_{2}{(\\dot{y},C_{2})} = \\theta_{2}{(\\dot{y},C_{2})} + \\log{(C_{2} - \\dot{y})} and \\int 2 \\theta_{2}{(\\dot{y},C_{2})} d\\dot{y} = \\int (\\theta_{2}{(\\dot{y},C_{2})} + \\log{(C_{2} - \\dot{y})}) d\\dot{y} and \\frac{\\int 2 \\theta_{2}{(\\dot{y},C_{2})} d\\dot{y}}{\\log{(C_{2} - \\dot{y})}} = \\frac{\\int (\\theta_{2}{(\\dot{y},C_{2})} + \\log{(C_{2} - \\dot{y})}) d\\dot{y}}{\\log{(C_{2} - \\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True)), log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 1, "Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True))), Add(Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True)), log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True)), log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["divide", 3, "log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Mul(Pow(log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)), Integral(Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)), Integral(Add(Function('\\\\theta_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('C_2', commutative=True)), log(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\delta,\\chi)} = \\frac{\\partial}{\\partial \\delta} \\chi \\delta, then derive \\operatorname{n_{2}}{(\\delta,\\chi)} = \\chi, then obtain \\frac{\\partial}{\\partial \\delta} \\frac{(\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{2 \\chi}}{\\chi} = \\frac{\\partial}{\\partial \\delta} \\frac{(\\chi + 1)^{\\chi} (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{\\chi}}{\\chi}", "derivation": "\\operatorname{n_{2}}{(\\delta,\\chi)} = \\frac{\\partial}{\\partial \\delta} \\chi \\delta and \\operatorname{n_{2}}{(\\delta,\\chi)} = \\chi and \\operatorname{n_{2}}{(\\delta,\\chi)} + 1 = \\chi + 1 and (\\operatorname{n_{2}}{(\\delta,\\chi)} + 1)^{\\chi} = (\\chi + 1)^{\\chi} and (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{\\chi} = (\\chi + 1)^{\\chi} and (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{2 \\chi} = (\\chi + 1)^{\\chi} (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{\\chi} and \\frac{(\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{2 \\chi}}{\\chi} = \\frac{(\\chi + 1)^{\\chi} (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{\\chi}}{\\chi} and \\frac{\\partial}{\\partial \\delta} \\frac{(\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{2 \\chi}}{\\chi} = \\frac{\\partial}{\\partial \\delta} \\frac{(\\chi + 1)^{\\chi} (\\frac{\\partial}{\\partial \\delta} \\chi \\delta + 1)^{\\chi}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('n_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["add", 2, 1], "Equality(Add(Function('n_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(1)), Add(Symbol('\\\\chi', commutative=True), Integer(1)))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Function('n_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(1)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Symbol('\\\\chi', commutative=True)))"], [["times", 5, "Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\chi', commutative=True))"], "Equality(Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Mul(Integer(2), Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Symbol('\\\\chi', commutative=True)), Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\chi', commutative=True))))"], [["divide", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Mul(Integer(2), Symbol('\\\\chi', commutative=True)))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Symbol('\\\\chi', commutative=True)), Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Mul(Integer(2), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Symbol('\\\\chi', commutative=True)), Pow(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(W)} = \\log{(W)}, then derive m{(W)} + 2 \\int m{(W)} dW = W \\log{(W)} - W + \\mathbf{p} + m{(W)} + \\int m{(W)} dW, then obtain m{(W)} + 2 \\int m{(W)} dW = W \\log{(W)} - W + \\mathbf{p} + m{(W)} + \\int \\log{(W)} dW", "derivation": "m{(W)} = \\log{(W)} and \\int m{(W)} dW = \\int \\log{(W)} dW and \\log{(W)} + \\int m{(W)} dW = \\log{(W)} + \\int \\log{(W)} dW and m{(W)} + \\int m{(W)} dW = m{(W)} + \\int \\log{(W)} dW and m{(W)} + 2 \\int m{(W)} dW = m{(W)} + \\int m{(W)} dW + \\int \\log{(W)} dW and m{(W)} + 2 \\int m{(W)} dW = W \\log{(W)} - W + \\mathbf{p} + m{(W)} + \\int m{(W)} dW and m{(W)} + 2 \\int m{(W)} dW = W \\log{(W)} - W + \\mathbf{p} + m{(W)} + \\int \\log{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["add", 2, "log(Symbol('W', commutative=True))"], "Equality(Add(log(Symbol('W', commutative=True)), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(log(Symbol('W', commutative=True)), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('m')(Symbol('W', commutative=True)), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Function('m')(Symbol('W', commutative=True)), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["add", 4, "Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Function('m')(Symbol('W', commutative=True)), Mul(Integer(2), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Function('m')(Symbol('W', commutative=True)), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Function('m')(Symbol('W', commutative=True)), Mul(Integer(2), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Function('m')(Symbol('W', commutative=True)), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('m')(Symbol('W', commutative=True)), Mul(Integer(2), Integral(Function('m')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Function('m')(Symbol('W', commutative=True)), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{p})} = \\log{(\\sin{(\\mathbf{p})})}, then obtain \\sin{(\\mathbf{p})} \\int 0 d\\mathbf{p} = \\sin{(\\mathbf{p})} \\int (- \\operatorname{f^{*}}{(\\mathbf{p})} + \\log{(\\sin{(\\mathbf{p})})}) d\\mathbf{p}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{p})} = \\log{(\\sin{(\\mathbf{p})})} and 0 = - \\operatorname{f^{*}}{(\\mathbf{p})} + \\log{(\\sin{(\\mathbf{p})})} and \\int 0 d\\mathbf{p} = \\int (- \\operatorname{f^{*}}{(\\mathbf{p})} + \\log{(\\sin{(\\mathbf{p})})}) d\\mathbf{p} and \\sin{(\\mathbf{p})} \\int 0 d\\mathbf{p} = \\sin{(\\mathbf{p})} \\int (- \\operatorname{f^{*}}{(\\mathbf{p})} + \\log{(\\sin{(\\mathbf{p})})}) d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True)), log(sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 1, "Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True))), log(sin(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True))), log(sin(Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["times", 3, "sin(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Add(Mul(Integer(-1), Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True))), log(sin(Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given C{(g,\\mathbf{r})} = \\mathbf{r} - g, then obtain g + (- \\int \\mathbf{r} dg + \\int (\\mathbf{r} - g) dg)^{\\mathbf{r}} - 1 = g + (\\int - g dg)^{\\mathbf{r}} - 1", "derivation": "C{(g,\\mathbf{r})} = \\mathbf{r} - g and \\int C{(g,\\mathbf{r})} dg = \\int (\\mathbf{r} - g) dg and \\int C{(g,\\mathbf{r})} dg = \\int \\mathbf{r} dg + \\int - g dg and - \\int \\mathbf{r} dg + \\int C{(g,\\mathbf{r})} dg = \\int - g dg and (- \\int \\mathbf{r} dg + \\int C{(g,\\mathbf{r})} dg)^{\\mathbf{r}} = (\\int - g dg)^{\\mathbf{r}} and (- \\int \\mathbf{r} dg + \\int (\\mathbf{r} - g) dg)^{\\mathbf{r}} = (\\int - g dg)^{\\mathbf{r}} and (- \\int \\mathbf{r} dg + \\int (\\mathbf{r} - g) dg)^{\\mathbf{r}} - 1 = (\\int - g dg)^{\\mathbf{r}} - 1 and g + (- \\int \\mathbf{r} dg + \\int (\\mathbf{r} - g) dg)^{\\mathbf{r}} - 1 = g + (\\int - g dg)^{\\mathbf{r}} - 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('C')(Symbol('g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('C')(Symbol('g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["minus", 3, "Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))), Integral(Function('C')(Symbol('g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))), Integral(Function('C')(Symbol('g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 6, 1], "Equality(Add(Pow(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Add(Pow(Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))"], [["add", 7, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Pow(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('g', commutative=True)))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Add(Symbol('g', commutative=True), Pow(Integral(Mul(Integer(-1), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given M{(\\mathbb{I},m_{s})} = \\sin{(\\mathbb{I} + m_{s})}, then obtain - (\\frac{m_{s} M{(\\mathbb{I},m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}} = - (\\frac{m_{s} \\sin{(\\mathbb{I} + m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}}", "derivation": "M{(\\mathbb{I},m_{s})} = \\sin{(\\mathbb{I} + m_{s})} and \\frac{M{(\\mathbb{I},m_{s})}}{\\mathbb{I} + m_{s}} = \\frac{\\sin{(\\mathbb{I} + m_{s})}}{\\mathbb{I} + m_{s}} and \\frac{m_{s} M{(\\mathbb{I},m_{s})}}{\\mathbb{I} + m_{s}} = \\frac{m_{s} \\sin{(\\mathbb{I} + m_{s})}}{\\mathbb{I} + m_{s}} and (\\frac{m_{s} M{(\\mathbb{I},m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}} = (\\frac{m_{s} \\sin{(\\mathbb{I} + m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}} and - (\\frac{m_{s} M{(\\mathbb{I},m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}} = - (\\frac{m_{s} \\sin{(\\mathbb{I} + m_{s})}}{\\mathbb{I} + m_{s}})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('M')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)))))"], [["times", 2, "Symbol('m_s', commutative=True)"], "Equality(Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('M')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)))))"], [["power", 3, "Symbol('m_s', commutative=True)"], "Equality(Pow(Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('M')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Pow(Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)))), Symbol('m_s', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('M')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('m_s', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m_s', commutative=True)))), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\omega{(r)} = \\cos{(r)}, then obtain \\frac{(\\frac{d}{d r} \\omega{(r)})^{r}}{\\frac{d}{d r} \\cos{(r)}} = \\frac{(\\frac{d}{d r} \\cos{(r)})^{r}}{\\frac{d}{d r} \\cos{(r)}}", "derivation": "\\omega{(r)} = \\cos{(r)} and \\frac{d}{d r} \\omega{(r)} = \\frac{d}{d r} \\cos{(r)} and (\\frac{d}{d r} \\omega{(r)})^{r} = (\\frac{d}{d r} \\cos{(r)})^{r} and \\frac{(\\frac{d}{d r} \\omega{(r)})^{r}}{\\frac{d}{d r} \\cos{(r)}} = \\frac{(\\frac{d}{d r} \\cos{(r)})^{r}}{\\frac{d}{d r} \\cos{(r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\omega')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["divide", 3, "Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\omega')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\theta,\\Psi,\\theta_2)} = (\\Psi - \\theta)^{\\theta_2}, then obtain \\Psi (- \\Psi + \\frac{\\hat{H}_l{(\\theta,\\Psi,\\theta_2)}}{\\Psi}) = \\Psi (- \\Psi + \\frac{(\\Psi - \\theta)^{\\theta_2}}{\\Psi})", "derivation": "\\hat{H}_l{(\\theta,\\Psi,\\theta_2)} = (\\Psi - \\theta)^{\\theta_2} and \\frac{\\hat{H}_l{(\\theta,\\Psi,\\theta_2)}}{\\Psi} = \\frac{(\\Psi - \\theta)^{\\theta_2}}{\\Psi} and - \\Psi + \\frac{\\hat{H}_l{(\\theta,\\Psi,\\theta_2)}}{\\Psi} = - \\Psi + \\frac{(\\Psi - \\theta)^{\\theta_2}}{\\Psi} and \\Psi (- \\Psi + \\frac{\\hat{H}_l{(\\theta,\\Psi,\\theta_2)}}{\\Psi}) = \\Psi (- \\Psi + \\frac{(\\Psi - \\theta)^{\\theta_2}}{\\Psi})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"], [["minus", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta_2', commutative=True)))))"], [["times", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True))))), Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given \\rho_{b}{(m_{s})} = \\sin{(m_{s})} and u{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})}, then obtain m_{s} + \\rho_{b}{(m_{s})} + \\sin{(\\varepsilon_0)} + \\cos{(\\sin{(\\varepsilon_0)})} = m_{s} + \\sin{(\\varepsilon_0)} + \\sin{(m_{s})} + \\cos{(\\sin{(\\varepsilon_0)})}", "derivation": "\\rho_{b}{(m_{s})} = \\sin{(m_{s})} and m_{s} + \\rho_{b}{(m_{s})} = m_{s} + \\sin{(m_{s})} and u{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})} and m_{s} + \\rho_{b}{(m_{s})} + \\sin{(\\varepsilon_0)} = m_{s} + \\sin{(\\varepsilon_0)} + \\sin{(m_{s})} and m_{s} + \\rho_{b}{(m_{s})} + u{(\\varepsilon_0)} + \\sin{(\\varepsilon_0)} = m_{s} + u{(\\varepsilon_0)} + \\sin{(\\varepsilon_0)} + \\sin{(m_{s})} and m_{s} + \\rho_{b}{(m_{s})} + \\sin{(\\varepsilon_0)} + \\cos{(\\sin{(\\varepsilon_0)})} = m_{s} + \\sin{(\\varepsilon_0)} + \\sin{(m_{s})} + \\cos{(\\sin{(\\varepsilon_0)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["add", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Symbol('m_s', commutative=True), Function('\\\\rho_b')(Symbol('m_s', commutative=True))), Add(Symbol('m_s', commutative=True), sin(Symbol('m_s', commutative=True))))"], ["get_premise", "Equality(Function('u')(Symbol('\\\\varepsilon_0', commutative=True)), cos(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 2, "sin(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Symbol('m_s', commutative=True), Function('\\\\rho_b')(Symbol('m_s', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('m_s', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["add", 4, "Function('u')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Symbol('m_s', commutative=True), Function('\\\\rho_b')(Symbol('m_s', commutative=True)), Function('u')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('m_s', commutative=True), Function('u')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('m_s', commutative=True), Function('\\\\rho_b')(Symbol('m_s', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)), cos(sin(Symbol('\\\\varepsilon_0', commutative=True)))), Add(Symbol('m_s', commutative=True), sin(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('m_s', commutative=True)), cos(sin(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(f)} = \\log{(f)}, then obtain \\frac{d}{d f} (\\log{(f)} - 1 + \\frac{\\log{(f)}}{\\lambda{(f)}}) = \\frac{d}{d f} (\\frac{(\\log{(f)} + \\frac{\\log{(f)}}{\\lambda{(f)}}) \\log{(f)}}{\\lambda{(f)}} - 1)", "derivation": "\\lambda{(f)} = \\log{(f)} and 1 = \\frac{\\log{(f)}}{\\lambda{(f)}} and \\log{(f)} + \\frac{\\log{(f)}}{\\lambda{(f)}} = \\frac{(\\log{(f)} + \\frac{\\log{(f)}}{\\lambda{(f)}}) \\log{(f)}}{\\lambda{(f)}} and \\log{(f)} - 1 + \\frac{\\log{(f)}}{\\lambda{(f)}} = \\frac{(\\log{(f)} + \\frac{\\log{(f)}}{\\lambda{(f)}}) \\log{(f)}}{\\lambda{(f)}} - 1 and \\frac{d}{d f} (\\log{(f)} - 1 + \\frac{\\log{(f)}}{\\lambda{(f)}}) = \\frac{d}{d f} (\\frac{(\\log{(f)} + \\frac{\\log{(f)}}{\\lambda{(f)}}) \\log{(f)}}{\\lambda{(f)}} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["divide", 1, "Function('\\\\lambda')(Symbol('f', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True))))"], [["times", 2, "Add(log(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True))))"], "Equality(Add(log(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Mul(Add(log(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True))))"], [["minus", 3, 1], "Equality(Add(log(Symbol('f', commutative=True)), Integer(-1), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Add(Mul(Add(log(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True))), Integer(-1)))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(log(Symbol('f', commutative=True)), Integer(-1), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Add(log(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('f', commutative=True)), Integer(-1)), log(Symbol('f', commutative=True))), Integer(-1)), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})}, then derive \\int C{(\\sigma_p)} d\\sigma_p = \\sigma_p \\log{(\\log{(\\sigma_p)})} + v_{2} - \\operatorname{li}{(\\sigma_p)}, then obtain \\hbar + \\sigma_p \\log{(\\log{(\\sigma_p)})} - \\operatorname{li}{(\\sigma_p)} = \\sigma_p \\log{(\\log{(\\sigma_p)})} + v_{2} - \\operatorname{li}{(\\sigma_p)}", "derivation": "C{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})} and \\int C{(\\sigma_p)} d\\sigma_p = \\int \\log{(\\log{(\\sigma_p)})} d\\sigma_p and \\int C{(\\sigma_p)} d\\sigma_p = \\sigma_p \\log{(\\log{(\\sigma_p)})} + v_{2} - \\operatorname{li}{(\\sigma_p)} and \\int \\log{(\\log{(\\sigma_p)})} d\\sigma_p = \\sigma_p \\log{(\\log{(\\sigma_p)})} + v_{2} - \\operatorname{li}{(\\sigma_p)} and \\hbar + \\sigma_p \\log{(\\log{(\\sigma_p)})} - \\operatorname{li}{(\\sigma_p)} = \\sigma_p \\log{(\\log{(\\sigma_p)})} + v_{2} - \\operatorname{li}{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\sigma_p', commutative=True)), log(log(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(log(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Symbol('\\\\sigma_p', commutative=True), log(log(Symbol('\\\\sigma_p', commutative=True)))), Symbol('v_2', commutative=True), Mul(Integer(-1), li(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Symbol('\\\\sigma_p', commutative=True), log(log(Symbol('\\\\sigma_p', commutative=True)))), Symbol('v_2', commutative=True), Mul(Integer(-1), li(Symbol('\\\\sigma_p', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('\\\\sigma_p', commutative=True), log(log(Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), li(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Symbol('\\\\sigma_p', commutative=True), log(log(Symbol('\\\\sigma_p', commutative=True)))), Symbol('v_2', commutative=True), Mul(Integer(-1), li(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(p,s)} = - p + s, then obtain \\iint (p + \\operatorname{f_{\\mathbf{v}}}{(p,s)}) dp dp = \\iint s dp dp", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(p,s)} = - p + s and p + \\operatorname{f_{\\mathbf{v}}}{(p,s)} = s and \\int (p + \\operatorname{f_{\\mathbf{v}}}{(p,s)}) dp = \\int s dp and \\iint (p + \\operatorname{f_{\\mathbf{v}}}{(p,s)}) dp dp = \\iint s dp dp", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('s', commutative=True)))"], [["add", 1, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Symbol('p', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Symbol('s', commutative=True), Tuple(Symbol('p', commutative=True))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Symbol('p', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Symbol('s', commutative=True), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(y,\\varphi)} = y + \\log{(\\varphi)}, then obtain 0 = - \\frac{(y + 2 \\log{(\\varphi)}) \\frac{\\partial}{\\partial y} \\phi_{1}{(y,\\varphi)}}{(\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)})^{2}} + \\frac{1}{\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)}}", "derivation": "\\phi_{1}{(y,\\varphi)} = y + \\log{(\\varphi)} and \\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)} = y + 2 \\log{(\\varphi)} and 1 = \\frac{y + 2 \\log{(\\varphi)}}{\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)}} and \\frac{d}{d y} 1 = \\frac{\\partial}{\\partial y} \\frac{y + 2 \\log{(\\varphi)}}{\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)}} and 0 = - \\frac{(y + 2 \\log{(\\varphi)}) \\frac{\\partial}{\\partial y} \\phi_{1}{(y,\\varphi)}}{(\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)})^{2}} + \\frac{1}{\\phi_{1}{(y,\\varphi)} + \\log{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('y', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "log(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Add(Symbol('y', commutative=True), Mul(Integer(2), log(Symbol('\\\\varphi', commutative=True)))))"], [["divide", 2, "Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('y', commutative=True), Mul(Integer(2), log(Symbol('\\\\varphi', commutative=True)))), Pow(Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('y', commutative=True), Mul(Integer(2), log(Symbol('\\\\varphi', commutative=True)))), Pow(Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(2), log(Symbol('\\\\varphi', commutative=True)))), Pow(Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Integer(-2)), Derivative(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Pow(Add(Function('\\\\phi_1')(Symbol('y', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(r,M)} = e^{M - r}, then obtain \\tilde{\\infty} ((\\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} - \\frac{\\partial}{\\partial M} e^{M - r})^{r} - \\frac{1}{M - r}) = \\tilde{\\infty} (0^{r} - \\frac{1}{M - r})", "derivation": "\\hat{H}_{\\lambda}{(r,M)} = e^{M - r} and \\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} = \\frac{\\partial}{\\partial M} e^{M - r} and \\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} - \\frac{\\partial}{\\partial M} e^{M - r} = 0 and (\\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} - \\frac{\\partial}{\\partial M} e^{M - r})^{r} = 0^{r} and (\\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} - \\frac{\\partial}{\\partial M} e^{M - r})^{r} - \\frac{1}{M - r} = 0^{r} - \\frac{1}{M - r} and \\tilde{\\infty} ((\\frac{\\partial}{\\partial M} \\hat{H}_{\\lambda}{(r,M)} - \\frac{\\partial}{\\partial M} e^{M - r})^{r} - \\frac{1}{M - r}) = \\tilde{\\infty} (0^{r} - \\frac{1}{M - r})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))), Integer(0))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))), Symbol('r', commutative=True)), Pow(Integer(0), Symbol('r', commutative=True)))"], [["minus", 4, "Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1))"], "Equality(Add(Pow(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1)))), Add(Pow(Integer(0), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1)))))"], [["divide", 5, 0], "Equality(Mul(zoo, Add(Pow(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1))))), Mul(zoo, Add(Pow(Integer(0), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(E_{x},r)} = \\cos{(E_{x} - r)}, then obtain (\\int (\\mathbf{J}_f^{r}{(E_{x},r)} + \\cos{(E_{x} - r)}) dr)^{r} = (\\int (\\cos{(E_{x} - r)} + \\cos^{r}{(E_{x} - r)}) dr)^{r}", "derivation": "\\mathbf{J}_f{(E_{x},r)} = \\cos{(E_{x} - r)} and \\mathbf{J}_f^{r}{(E_{x},r)} = \\cos^{r}{(E_{x} - r)} and \\mathbf{J}_f^{r}{(E_{x},r)} + \\cos{(E_{x} - r)} = \\cos{(E_{x} - r)} + \\cos^{r}{(E_{x} - r)} and \\int (\\mathbf{J}_f^{r}{(E_{x},r)} + \\cos{(E_{x} - r)}) dr = \\int (\\cos{(E_{x} - r)} + \\cos^{r}{(E_{x} - r)}) dr and (\\int (\\mathbf{J}_f^{r}{(E_{x},r)} + \\cos{(E_{x} - r)}) dr)^{r} = (\\int (\\cos{(E_{x} - r)} + \\cos^{r}{(E_{x} - r)}) dr)^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["add", 2, "cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], "Equality(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Add(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True))))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True))), Integral(Add(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["power", 4, "Symbol('r', commutative=True)"], "Equality(Pow(Integral(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Integral(Add(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(r)} = \\sin{(r)}, then obtain \\frac{d}{d r} \\int (\\mathbf{B}{(r)} + 3 \\sin{(r)}) dr = \\frac{d}{d r} \\int (2 \\mathbf{B}{(r)} + 2 \\sin{(r)}) dr", "derivation": "\\mathbf{B}{(r)} = \\sin{(r)} and 2 \\mathbf{B}{(r)} = \\mathbf{B}{(r)} + \\sin{(r)} and 3 \\mathbf{B}{(r)} + \\sin{(r)} = 2 \\mathbf{B}{(r)} + 2 \\sin{(r)} and 3 \\mathbf{B}{(r)} + \\sin{(r)} = \\mathbf{B}{(r)} + 3 \\sin{(r)} and \\mathbf{B}{(r)} + 3 \\sin{(r)} = 2 \\mathbf{B}{(r)} + 2 \\sin{(r)} and \\int (\\mathbf{B}{(r)} + 3 \\sin{(r)}) dr = \\int (2 \\mathbf{B}{(r)} + 2 \\sin{(r)}) dr and \\frac{d}{d r} \\int (\\mathbf{B}{(r)} + 3 \\sin{(r)}) dr = \\frac{d}{d r} \\int (2 \\mathbf{B}{(r)} + 2 \\sin{(r)}) dr", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{B}')(Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))))"], [["add", 2, "Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), Mul(Integer(2), sin(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True))), Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), Mul(Integer(3), sin(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), Mul(Integer(3), sin(Symbol('r', commutative=True)))), Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), Mul(Integer(2), sin(Symbol('r', commutative=True)))))"], [["integrate", 5, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), Mul(Integer(3), sin(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), Mul(Integer(2), sin(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 6, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mathbf{B}')(Symbol('r', commutative=True)), Mul(Integer(3), sin(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('r', commutative=True))), Mul(Integer(2), sin(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\varphi)} = \\log{(\\varphi)} and \\operatorname{V_{\\mathbf{E}}}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(\\varphi)}^{- \\varphi} \\frac{d}{d \\varphi} \\log{(\\varphi)}, then derive \\frac{d}{d \\varphi} \\operatorname{n_{1}}{(\\varphi)} = \\frac{1}{\\varphi}, then obtain \\int \\operatorname{V_{\\mathbf{E}}}{(\\varphi)} d\\varphi = \\int \\frac{d}{d \\varphi} \\frac{\\log{(\\varphi)}^{- \\varphi}}{\\varphi} d\\varphi", "derivation": "\\operatorname{n_{1}}{(\\varphi)} = \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\operatorname{n_{1}}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\operatorname{n_{1}}{(\\varphi)} = \\frac{1}{\\varphi} and \\frac{d}{d \\varphi} \\log{(\\varphi)} = \\frac{1}{\\varphi} and \\operatorname{V_{\\mathbf{E}}}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(\\varphi)}^{- \\varphi} \\frac{d}{d \\varphi} \\log{(\\varphi)} and \\operatorname{V_{\\mathbf{E}}}{(\\varphi)} = \\frac{d}{d \\varphi} \\frac{\\log{(\\varphi)}^{- \\varphi}}{\\varphi} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\varphi)} d\\varphi = \\int \\frac{d}{d \\varphi} \\frac{\\log{(\\varphi)}^{- \\varphi}}{\\varphi} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\varphi', commutative=True)), Derivative(Mul(Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\varphi', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given g{(\\eta)} = \\sin{(\\eta)}, then obtain \\int 0 d\\eta = \\int \\eta (g^{2}{(\\eta)} - \\sin^{2}{(\\eta)}) d\\eta", "derivation": "g{(\\eta)} = \\sin{(\\eta)} and 0 = - g{(\\eta)} + \\sin{(\\eta)} and -1 = - \\frac{\\sin{(\\eta)}}{g{(\\eta)}} and 0 = (- g{(\\eta)} + \\sin{(\\eta)})^{2} and 0 = g^{2}{(\\eta)} - 2 g{(\\eta)} \\sin{(\\eta)} + \\sin^{2}{(\\eta)} and 2 g{(\\eta)} \\sin{(\\eta)} = 2 \\sin^{2}{(\\eta)} and 0 = g^{2}{(\\eta)} - \\sin^{2}{(\\eta)} and 0 = \\eta (g^{2}{(\\eta)} - \\sin^{2}{(\\eta)}) and \\int 0 d\\eta = \\int \\eta (g^{2}{(\\eta)} - \\sin^{2}{(\\eta)}) d\\eta", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "Function('g')(Symbol('\\\\eta', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g')(Symbol('\\\\eta', commutative=True))), sin(Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Function('g')(Symbol('\\\\eta', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\eta', commutative=True)), Integer(-1)), sin(Symbol('\\\\eta', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Function('g')(Symbol('\\\\eta', commutative=True))), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\eta', commutative=True))), sin(Symbol('\\\\eta', commutative=True))), Integer(2)))"], [["expand", 4], "Equality(Integer(0), Add(Pow(Function('g')(Symbol('\\\\eta', commutative=True)), Integer(2)), Mul(Integer(-1), Integer(2), Function('g')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))"], [["times", 3, "Mul(Integer(-1), Integer(2), Function('g')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(2), Function('g')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Integer(0), Add(Pow(Function('g')(Symbol('\\\\eta', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)))))"], [["times", 7, "Symbol('\\\\eta', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\eta', commutative=True), Add(Pow(Function('g')(Symbol('\\\\eta', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))))"], [["integrate", 8, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Mul(Symbol('\\\\eta', commutative=True), Add(Pow(Function('g')(Symbol('\\\\eta', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain (\\frac{d^{3}}{d (y^{\\prime})^{3}} \\operatorname{t_{1}}{(y^{\\prime})})^{y^{\\prime}} = (\\frac{d^{3}}{d (y^{\\prime})^{3}} e^{y^{\\prime}})^{y^{\\prime}}", "derivation": "\\operatorname{t_{1}}{(y^{\\prime})} = e^{y^{\\prime}} and \\frac{d}{d y^{\\prime}} \\operatorname{t_{1}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} and \\frac{d^{2}}{d (y^{\\prime})^{2}} \\operatorname{t_{1}}{(y^{\\prime})} = \\frac{d^{2}}{d (y^{\\prime})^{2}} e^{y^{\\prime}} and \\frac{d^{3}}{d (y^{\\prime})^{3}} \\operatorname{t_{1}}{(y^{\\prime})} = \\frac{d^{3}}{d (y^{\\prime})^{3}} e^{y^{\\prime}} and (\\frac{d^{3}}{d (y^{\\prime})^{3}} \\operatorname{t_{1}}{(y^{\\prime})})^{y^{\\prime}} = (\\frac{d^{3}}{d (y^{\\prime})^{3}} e^{y^{\\prime}})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(2))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(3))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(3))))"], [["power", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(Z)} = \\frac{d}{d Z} \\log{(Z)}, then derive \\operatorname{P_{g}}{(Z)} = \\frac{1}{Z}, then obtain (- (1 + \\frac{1}{Z})^{Z})^{Z} = (- (\\frac{d}{d Z} \\log{(Z)} + 1)^{Z})^{Z}", "derivation": "\\operatorname{P_{g}}{(Z)} = \\frac{d}{d Z} \\log{(Z)} and \\operatorname{P_{g}}{(Z)} + 1 = \\frac{d}{d Z} \\log{(Z)} + 1 and \\operatorname{P_{g}}{(Z)} = \\frac{1}{Z} and (\\operatorname{P_{g}}{(Z)} + 1)^{Z} = (\\frac{d}{d Z} \\log{(Z)} + 1)^{Z} and (1 + \\frac{1}{Z})^{Z} = (\\frac{d}{d Z} \\log{(Z)} + 1)^{Z} and - (1 + \\frac{1}{Z})^{Z} = - (\\frac{d}{d Z} \\log{(Z)} + 1)^{Z} and (- (1 + \\frac{1}{Z})^{Z})^{Z} = (- (\\frac{d}{d Z} \\log{(Z)} + 1)^{Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('Z', commutative=True)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('P_g')(Symbol('Z', commutative=True)), Integer(1)), Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('P_g')(Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Integer(-1)))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Function('P_g')(Symbol('Z', commutative=True)), Integer(1)), Symbol('Z', commutative=True)), Pow(Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Integer(1), Pow(Symbol('Z', commutative=True), Integer(-1))), Symbol('Z', commutative=True)), Pow(Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Symbol('Z', commutative=True)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Integer(1), Pow(Symbol('Z', commutative=True), Integer(-1))), Symbol('Z', commutative=True))), Mul(Integer(-1), Pow(Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Symbol('Z', commutative=True))))"], [["power", 6, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Add(Integer(1), Pow(Symbol('Z', commutative=True), Integer(-1))), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Mul(Integer(-1), Pow(Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(a)} = \\log{(\\sin{(a)})}, then obtain 2 a + 2 \\int \\sigma_{x}{(a)} da = 2 a + \\int \\sigma_{x}{(a)} da + \\int \\log{(\\sin{(a)})} da", "derivation": "\\sigma_{x}{(a)} = \\log{(\\sin{(a)})} and \\int \\sigma_{x}{(a)} da = \\int \\log{(\\sin{(a)})} da and a + \\int \\sigma_{x}{(a)} da = a + \\int \\log{(\\sin{(a)})} da and 2 a + 2 \\int \\sigma_{x}{(a)} da = 2 a + \\int \\sigma_{x}{(a)} da + \\int \\log{(\\sin{(a)})} da", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["add", 2, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Integral(log(sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True)))))"], [["add", 3, "Add(Symbol('a', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), Integral(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))), Add(Mul(Integer(2), Symbol('a', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(Q,a)} = \\int \\frac{a}{Q} da and \\operatorname{g_{\\varepsilon}}{(Q,a)} = \\int \\frac{a}{Q} da, then obtain \\frac{Q + \\operatorname{g_{\\varepsilon}}{(Q,a)}}{\\int \\frac{a}{Q} da} = \\frac{Q + \\int \\frac{a}{Q} da}{\\int \\frac{a}{Q} da}", "derivation": "\\operatorname{f_{E}}{(Q,a)} = \\int \\frac{a}{Q} da and \\operatorname{g_{\\varepsilon}}{(Q,a)} = \\int \\frac{a}{Q} da and Q + \\operatorname{g_{\\varepsilon}}{(Q,a)} = Q + \\int \\frac{a}{Q} da and \\frac{Q + \\operatorname{g_{\\varepsilon}}{(Q,a)}}{\\operatorname{f_{E}}{(Q,a)}} = \\frac{Q + \\int \\frac{a}{Q} da}{\\operatorname{f_{E}}{(Q,a)}} and \\frac{Q + \\operatorname{g_{\\varepsilon}}{(Q,a)}}{\\int \\frac{a}{Q} da} = \\frac{Q + \\int \\frac{a}{Q} da}{\\int \\frac{a}{Q} da}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["add", 2, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('a', commutative=True))), Add(Symbol('Q', commutative=True), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["divide", 3, "Function('f_E')(Symbol('Q', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Add(Symbol('Q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('a', commutative=True))), Pow(Function('f_E')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Integer(-1))), Mul(Add(Symbol('Q', commutative=True), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Pow(Function('f_E')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Symbol('Q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('a', commutative=True))), Pow(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))), Mul(Add(Symbol('Q', commutative=True), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Pow(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and I{(\\mathbb{I})} = \\mathbb{I}, then obtain \\mathbb{I} (- \\mathbb{I}^{\\mathbb{I}} + I^{\\mathbb{I}}{(\\mathbb{I})} - \\sin{(\\mathbb{I})}) = - \\mathbb{I} \\sin{(\\mathbb{I})}", "derivation": "\\dot{z}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and I{(\\mathbb{I})} = \\mathbb{I} and I^{\\mathbb{I}}{(\\mathbb{I})} = \\mathbb{I}^{\\mathbb{I}} and - \\mathbb{I}^{\\mathbb{I}} + I^{\\mathbb{I}}{(\\mathbb{I})} = 0 and - \\mathbb{I}^{\\mathbb{I}} + I^{\\mathbb{I}}{(\\mathbb{I})} - \\dot{z}{(\\mathbb{I})} = - \\dot{z}{(\\mathbb{I})} and \\mathbb{I} (- \\mathbb{I}^{\\mathbb{I}} + I^{\\mathbb{I}}{(\\mathbb{I})} - \\dot{z}{(\\mathbb{I})}) = - \\mathbb{I} \\dot{z}{(\\mathbb{I})} and \\mathbb{I} (- \\mathbb{I}^{\\mathbb{I}} + I^{\\mathbb{I}}{(\\mathbb{I})} - \\sin{(\\mathbb{I})}) = - \\mathbb{I} \\sin{(\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 3, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Integer(0))"], [["minus", 4, "Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 5, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbb{I}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\phi)} = \\sin{(\\phi)}, then obtain \\mathbf{J}_M{(\\phi)} \\sin{(\\phi)} + \\cos{(\\mathbf{J}_M^{2}{(\\phi)})} = \\mathbf{J}_M{(\\phi)} \\sin{(\\phi)} + \\cos{(\\mathbf{J}_M{(\\phi)} \\sin{(\\phi)})}", "derivation": "\\mathbf{J}_M{(\\phi)} = \\sin{(\\phi)} and \\mathbf{J}_M^{2}{(\\phi)} = \\mathbf{J}_M{(\\phi)} \\sin{(\\phi)} and \\cos{(\\mathbf{J}_M^{2}{(\\phi)})} = \\cos{(\\mathbf{J}_M{(\\phi)} \\sin{(\\phi)})} and \\mathbf{J}_M{(\\phi)} \\sin{(\\phi)} + \\cos{(\\mathbf{J}_M^{2}{(\\phi)})} = \\mathbf{J}_M{(\\phi)} \\sin{(\\phi)} + \\cos{(\\mathbf{J}_M{(\\phi)} \\sin{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), Integer(2))), cos(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))))"], [["add", 3, "Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), cos(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), Integer(2)))), Add(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), cos(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given n{(S)} = \\cos{(S)} and \\dot{\\mathbf{r}}{(S)} = n^{S}{(S)}, then obtain \\dot{\\mathbf{r}}{(S)} - n^{- S}{(S)} \\cos^{S}{(S)} = \\cos^{S}{(S)} - n^{- S}{(S)} \\cos^{S}{(S)}", "derivation": "n{(S)} = \\cos{(S)} and n^{S}{(S)} = \\cos^{S}{(S)} and \\dot{\\mathbf{r}}{(S)} = n^{S}{(S)} and \\dot{\\mathbf{r}}{(S)} = \\cos^{S}{(S)} and \\dot{\\mathbf{r}}{(S)} - n^{- S}{(S)} \\cos^{S}{(S)} = \\cos^{S}{(S)} - n^{- S}{(S)} \\cos^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('n')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('S', commutative=True)), Pow(Function('n')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["minus", 4, "Mul(Pow(Function('n')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('n')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))), Add(Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('n')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(F_{x},C_{d})} = - \\sin{(C_{d} - F_{x})} and z{(F_{x})} = - F_{x}, then obtain - F_{x} + z{(F_{x})} + \\int - \\sin{(C_{d} - F_{x})} dF_{x} = - 2 F_{x} + \\int - \\sin{(C_{d} - F_{x})} dF_{x}", "derivation": "\\nabla{(F_{x},C_{d})} = - \\sin{(C_{d} - F_{x})} and \\int \\nabla{(F_{x},C_{d})} dF_{x} = \\int - \\sin{(C_{d} - F_{x})} dF_{x} and z{(F_{x})} = - F_{x} and - F_{x} + z{(F_{x})} = - 2 F_{x} and - F_{x} + z{(F_{x})} + \\int \\nabla{(F_{x},C_{d})} dF_{x} = - 2 F_{x} + \\int \\nabla{(F_{x},C_{d})} dF_{x} and - F_{x} + z{(F_{x})} + \\int - \\sin{(C_{d} - F_{x})} dF_{x} = - 2 F_{x} + \\int - \\sin{(C_{d} - F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('F_x', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True))))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('F_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True))))), Tuple(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('z')(Symbol('F_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)))"], [["add", 4, "Integral(Function('\\\\nabla')(Symbol('F_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('z')(Symbol('F_x', commutative=True)), Integral(Function('\\\\nabla')(Symbol('F_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Integral(Function('\\\\nabla')(Symbol('F_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('z')(Symbol('F_x', commutative=True)), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True))))), Tuple(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True))))), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given r{(G)} = \\log{(G)}, then derive \\int r{(G)} dG = G \\log{(G)} - G + \\mathbf{P}, then obtain G r{(G)} - G + \\mathbf{P} = \\int \\log{(G)} dG", "derivation": "r{(G)} = \\log{(G)} and \\int r{(G)} dG = \\int \\log{(G)} dG and \\int r{(G)} dG = G \\log{(G)} - G + \\mathbf{P} and G \\log{(G)} - G + \\mathbf{P} = \\int \\log{(G)} dG and G r{(G)} - G + \\mathbf{P} = \\int \\log{(G)} dG", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('r')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('G', commutative=True), Function('r')(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(C_{d})} = \\cos{(C_{d})} and E{(C_{d})} = \\cos{(\\operatorname{A_{y}}{(C_{d})})}, then obtain C_{d} \\log{(E{(C_{d})})} - C_{d} = C_{d} \\log{(\\cos{(\\operatorname{A_{y}}{(C_{d})})})} - C_{d}", "derivation": "\\operatorname{A_{y}}{(C_{d})} = \\cos{(C_{d})} and E{(C_{d})} = \\cos{(\\operatorname{A_{y}}{(C_{d})})} and E{(C_{d})} = \\cos{(\\cos{(C_{d})})} and \\log{(E{(C_{d})})} = \\log{(\\cos{(\\cos{(C_{d})})})} and C_{d} \\log{(E{(C_{d})})} = C_{d} \\log{(\\cos{(\\cos{(C_{d})})})} and C_{d} \\log{(E{(C_{d})})} - C_{d} = C_{d} \\log{(\\cos{(\\cos{(C_{d})})})} - C_{d} and C_{d} \\log{(E{(C_{d})})} - C_{d} = C_{d} \\log{(\\cos{(\\operatorname{A_{y}}{(C_{d})})})} - C_{d}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('C_d', commutative=True)), cos(Function('A_y')(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True))))"], [["log", 3], "Equality(log(Function('E')(Symbol('C_d', commutative=True))), log(cos(cos(Symbol('C_d', commutative=True)))))"], [["times", 4, "Symbol('C_d', commutative=True)"], "Equality(Mul(Symbol('C_d', commutative=True), log(Function('E')(Symbol('C_d', commutative=True)))), Mul(Symbol('C_d', commutative=True), log(cos(cos(Symbol('C_d', commutative=True))))))"], [["minus", 5, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Function('E')(Symbol('C_d', commutative=True)))), Mul(Integer(-1), Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), log(cos(cos(Symbol('C_d', commutative=True))))), Mul(Integer(-1), Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Function('E')(Symbol('C_d', commutative=True)))), Mul(Integer(-1), Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), log(cos(Function('A_y')(Symbol('C_d', commutative=True))))), Mul(Integer(-1), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given g{(\\mathbf{H},\\hbar)} = \\mathbf{H} \\sin{(\\hbar)} and \\operatorname{v_{x}}{(\\mathbf{H})} = \\mathbf{H}, then derive v_{y} + \\frac{\\partial}{\\partial \\mathbf{H}} g{(\\mathbf{H},\\hbar)} = v_{y} + \\sin{(\\hbar)}, then obtain v_{y} + \\frac{\\partial}{\\partial \\operatorname{v_{x}}{(\\mathbf{H})}} \\operatorname{v_{x}}{(\\mathbf{H})} \\sin{(\\hbar)} = v_{y} + \\sin{(\\hbar)}", "derivation": "g{(\\mathbf{H},\\hbar)} = \\mathbf{H} \\sin{(\\hbar)} and \\operatorname{v_{x}}{(\\mathbf{H})} = \\mathbf{H} and \\frac{\\partial}{\\partial \\mathbf{H}} g{(\\mathbf{H},\\hbar)} = \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} \\sin{(\\hbar)} and v_{y} + \\frac{\\partial}{\\partial \\mathbf{H}} g{(\\mathbf{H},\\hbar)} = v_{y} + \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} \\sin{(\\hbar)} and v_{y} + \\frac{\\partial}{\\partial \\mathbf{H}} g{(\\mathbf{H},\\hbar)} = v_{y} + \\sin{(\\hbar)} and v_{y} + \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} \\sin{(\\hbar)} = v_{y} + \\sin{(\\hbar)} and v_{y} + \\frac{\\partial}{\\partial \\operatorname{v_{x}}{(\\mathbf{H})}} \\operatorname{v_{x}}{(\\mathbf{H})} \\sin{(\\hbar)} = v_{y} + \\sin{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["add", 3, "Symbol('v_y', commutative=True)"], "Equality(Add(Symbol('v_y', commutative=True), Derivative(Function('g')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('v_y', commutative=True), Derivative(Function('g')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Symbol('v_y', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Symbol('v_y', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Symbol('v_y', commutative=True), Derivative(Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)))), Add(Symbol('v_y', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = \\tilde{g}^*, then obtain - (\\tilde{g}^*)^{2} + \\tilde{g}^* \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = 0", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = \\tilde{g}^* and \\tilde{g}^* \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = (\\tilde{g}^*)^{2} and 2 \\tilde{g}^* \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = (\\tilde{g}^*)^{2} + \\tilde{g}^* \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} and - (\\tilde{g}^*)^{2} + \\tilde{g}^* \\operatorname{J_{\\varepsilon}}{(\\tilde{g}^*)} = 0", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], [["times", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2)))"], [["add", 2, "Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["minus", 3, "Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(y)} = \\sin{(y)}, then obtain \\frac{4 \\operatorname{y^{\\prime}}^{2}{(y)}}{\\sin{(y)}} = \\frac{2 (\\operatorname{y^{\\prime}}{(y)} + \\sin{(y)}) \\operatorname{y^{\\prime}}{(y)}}{\\sin{(y)}}", "derivation": "\\operatorname{y^{\\prime}}{(y)} = \\sin{(y)} and \\operatorname{y^{\\prime}}{(y)} + \\sin{(y)} = 2 \\sin{(y)} and 2 \\operatorname{y^{\\prime}}{(y)} = \\operatorname{y^{\\prime}}{(y)} + \\sin{(y)} and 2 (\\operatorname{y^{\\prime}}{(y)} + \\sin{(y)}) \\operatorname{y^{\\prime}}{(y)} = 4 \\operatorname{y^{\\prime}}{(y)} \\sin{(y)} and 4 \\operatorname{y^{\\prime}}^{2}{(y)} = 4 \\operatorname{y^{\\prime}}{(y)} \\sin{(y)} and 4 \\operatorname{y^{\\prime}}^{2}{(y)} = 2 (\\operatorname{y^{\\prime}}{(y)} + \\sin{(y)}) \\operatorname{y^{\\prime}}{(y)} and \\frac{4 \\operatorname{y^{\\prime}}^{2}{(y)}}{\\sin{(y)}} = \\frac{2 (\\operatorname{y^{\\prime}}{(y)} + \\sin{(y)}) \\operatorname{y^{\\prime}}{(y)}}{\\sin{(y)}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["add", 1, "sin(Symbol('y', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))), Mul(Integer(2), sin(Symbol('y', commutative=True))))"], [["add", 1, "Function('y^{\\\\prime}')(Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('y', commutative=True))), Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))), Function('y^{\\\\prime}')(Symbol('y', commutative=True))), Mul(Integer(4), Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(4), Pow(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), Integer(2))), Mul(Integer(4), Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(4), Pow(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))), Function('y^{\\\\prime}')(Symbol('y', commutative=True))))"], [["divide", 6, "sin(Symbol('y', commutative=True))"], "Equality(Mul(Integer(4), Pow(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), Integer(2)), Pow(sin(Symbol('y', commutative=True)), Integer(-1))), Mul(Integer(2), Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True))), Function('y^{\\\\prime}')(Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\ddot{x},F_{g})} = F_{g} + \\log{(\\ddot{x})}, then obtain F_{g} + \\nabla = \\int \\frac{F_{g} + \\log{(\\ddot{x})}}{\\operatorname{n_{2}}{(\\ddot{x},F_{g})}} dF_{g}", "derivation": "\\operatorname{n_{2}}{(\\ddot{x},F_{g})} = F_{g} + \\log{(\\ddot{x})} and 1 = \\frac{F_{g} + \\log{(\\ddot{x})}}{\\operatorname{n_{2}}{(\\ddot{x},F_{g})}} and \\int 1 dF_{g} = \\int \\frac{F_{g} + \\log{(\\ddot{x})}}{\\operatorname{n_{2}}{(\\ddot{x},F_{g})}} dF_{g} and F_{g} + \\nabla = \\int \\frac{F_{g} + \\log{(\\ddot{x})}}{\\operatorname{n_{2}}{(\\ddot{x},F_{g})}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True), Symbol('F_g', commutative=True)), Add(Symbol('F_g', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 1, "Function('n_2')(Symbol('\\\\ddot{x}', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('F_g', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True), Symbol('F_g', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Add(Symbol('F_g', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True), Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Integral(Mul(Add(Symbol('F_g', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True), Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(Z)} = e^{Z} and \\Psi^{\\dagger}{(Z)} = e^{Z} + \\frac{d}{d Z} \\operatorname{f^{*}}{(Z)}, then obtain \\Psi^{\\dagger}{(Z)} = e^{Z} + \\frac{d}{d Z} e^{Z}", "derivation": "\\operatorname{f^{*}}{(Z)} = e^{Z} and \\frac{d}{d Z} \\operatorname{f^{*}}{(Z)} = \\frac{d}{d Z} e^{Z} and e^{Z} + \\frac{d}{d Z} \\operatorname{f^{*}}{(Z)} = e^{Z} + \\frac{d}{d Z} e^{Z} and \\Psi^{\\dagger}{(Z)} = e^{Z} + \\frac{d}{d Z} \\operatorname{f^{*}}{(Z)} and \\Psi^{\\dagger}{(Z)} = e^{Z} + \\frac{d}{d Z} e^{Z}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 2, "exp(Symbol('Z', commutative=True))"], "Equality(Add(exp(Symbol('Z', commutative=True)), Derivative(Function('f^*')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(exp(Symbol('Z', commutative=True)), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Add(exp(Symbol('Z', commutative=True)), Derivative(Function('f^*')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Add(exp(Symbol('Z', commutative=True)), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)} = \\frac{\\theta_1 \\varphi^*}{\\varphi}, then obtain \\int \\frac{1}{\\varphi^*} d\\varphi^* = \\int \\frac{\\theta_1}{\\varphi \\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)}} d\\varphi^*", "derivation": "\\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)} = \\frac{\\theta_1 \\varphi^*}{\\varphi} and \\frac{\\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)}}{\\varphi^*} = \\frac{\\theta_1}{\\varphi} and \\frac{1}{\\varphi^*} = \\frac{\\theta_1}{\\varphi \\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)}} and \\int \\frac{1}{\\varphi^*} d\\varphi^* = \\int \\frac{\\theta_1}{\\varphi \\operatorname{A_{1}}{(\\varphi^*,\\theta_1,\\varphi)}} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('A_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], [["divide", 2, "Function('A_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Function('A_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Function('A_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(g,\\mathbf{F})} = \\int \\mathbf{F} g dg and \\phi{(F_{c})} = \\sin{(F_{c})}, then obtain \\int\\limits^{F_{c} + (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega}} \\phi{(F_{c})} dF_{c} = \\int\\limits^{F_{c} + (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega}} \\sin{(F_{c})} dF_{c}", "derivation": "\\rho_{f}{(g,\\mathbf{F})} = \\int \\mathbf{F} g dg and 0 = - \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg and 0 = (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega} and \\phi{(F_{c})} = \\sin{(F_{c})} and F_{c} = F_{c} + (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega} and \\int \\phi{(F_{c})} dF_{c} = \\int \\sin{(F_{c})} dF_{c} and \\int\\limits^{F_{c} + (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega}} \\phi{(F_{c})} dF_{c} = \\int\\limits^{F_{c} + (- \\rho_{f}{(g,\\mathbf{F})} + \\int \\mathbf{F} g dg) e^{S \\Omega}} \\sin{(F_{c})} dF_{c}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["minus", 1, "Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["times", 2, "exp(Mul(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), exp(Mul(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\phi')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["add", 3, "Symbol('F_c', commutative=True)"], "Equality(Symbol('F_c', commutative=True), Add(Symbol('F_c', commutative=True), Mul(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), exp(Mul(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["integrate", 4, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Function('\\\\phi')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Add(Symbol('F_c', commutative=True), Mul(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), exp(Mul(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True))))))), Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Add(Symbol('F_c', commutative=True), Mul(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), exp(Mul(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True))))))))"]]}, {"prompt": "Given A{(n_{1},h)} = \\frac{n_{1}}{h}, then obtain \\frac{h^{2} A^{h}{(n_{1},h)} \\frac{\\partial}{\\partial n_{1}} A{(n_{1},h)}}{A{(n_{1},h)}} = \\frac{h^{2} (\\frac{n_{1}}{h})^{h}}{n_{1}}", "derivation": "A{(n_{1},h)} = \\frac{n_{1}}{h} and A^{h}{(n_{1},h)} = (\\frac{n_{1}}{h})^{h} and \\frac{\\partial}{\\partial n_{1}} A^{h}{(n_{1},h)} = \\frac{\\partial}{\\partial n_{1}} (\\frac{n_{1}}{h})^{h} and h \\frac{\\partial}{\\partial n_{1}} A^{h}{(n_{1},h)} = h \\frac{\\partial}{\\partial n_{1}} (\\frac{n_{1}}{h})^{h} and \\frac{h^{2} A^{h}{(n_{1},h)} \\frac{\\partial}{\\partial n_{1}} A{(n_{1},h)}}{A{(n_{1},h)}} = \\frac{h^{2} (\\frac{n_{1}}{h})^{h}}{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Symbol('h', commutative=True)))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Pow(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 3, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Derivative(Pow(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Derivative(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Pow(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Derivative(Function('A')(Symbol('n_1', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\mu_0)} = \\int \\sin{(\\mu_0)} d\\mu_0, then derive \\sigma_{x}{(\\mu_0)} \\sin{(\\mu_0)} = (\\mathbf{J}_P - \\cos{(\\mu_0)}) \\sin{(\\mu_0)}, then obtain \\sigma_{x}^{2}{(\\mu_0)} \\sin{(\\mu_0)} = (\\mathbf{J}_P - \\cos{(\\mu_0)}) \\sigma_{x}{(\\mu_0)} \\sin{(\\mu_0)}", "derivation": "\\sigma_{x}{(\\mu_0)} = \\int \\sin{(\\mu_0)} d\\mu_0 and \\sigma_{x}{(\\mu_0)} \\sin{(\\mu_0)} = \\sin{(\\mu_0)} \\int \\sin{(\\mu_0)} d\\mu_0 and \\sigma_{x}{(\\mu_0)} \\sin{(\\mu_0)} = (\\mathbf{J}_P - \\cos{(\\mu_0)}) \\sin{(\\mu_0)} and \\sigma_{x}{(\\mu_0)} = \\mathbf{J}_P - \\cos{(\\mu_0)} and \\sigma_{x}^{2}{(\\mu_0)} \\sin{(\\mu_0)} = (\\mathbf{J}_P - \\cos{(\\mu_0)}) \\sigma_{x}{(\\mu_0)} \\sin{(\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Mul(sin(Symbol('\\\\mu_0', commutative=True)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), sin(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 3, "sin(Symbol('\\\\mu_0', commutative=True))"], "Equality(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))))"], [["times", 4, "Mul(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), Integer(2)), sin(Symbol('\\\\mu_0', commutative=True))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), Function('\\\\sigma_x')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(c)} = \\log{(c)}, then obtain \\operatorname{F_{H}}^{2}{(c)} - \\operatorname{F_{H}}{(c)} - \\log{(c)} = \\operatorname{F_{H}}{(c)} \\log{(c)} - \\operatorname{F_{H}}{(c)} - \\log{(c)}", "derivation": "\\operatorname{F_{H}}{(c)} = \\log{(c)} and \\operatorname{F_{H}}{(c)} + \\log{(c)} = 2 \\log{(c)} and \\operatorname{F_{H}}^{2}{(c)} = \\operatorname{F_{H}}{(c)} \\log{(c)} and \\operatorname{F_{H}}^{2}{(c)} - 2 \\log{(c)} = \\operatorname{F_{H}}{(c)} \\log{(c)} - 2 \\log{(c)} and \\operatorname{F_{H}}^{2}{(c)} - \\operatorname{F_{H}}{(c)} - \\log{(c)} = \\operatorname{F_{H}}{(c)} \\log{(c)} - \\operatorname{F_{H}}{(c)} - \\log{(c)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["add", 1, "log(Symbol('c', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Mul(Integer(2), log(Symbol('c', commutative=True))))"], [["times", 1, "Function('F_H')(Symbol('c', commutative=True))"], "Equality(Pow(Function('F_H')(Symbol('c', commutative=True)), Integer(2)), Mul(Function('F_H')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))))"], [["minus", 3, "Mul(Integer(2), log(Symbol('c', commutative=True)))"], "Equality(Add(Pow(Function('F_H')(Symbol('c', commutative=True)), Integer(2)), Mul(Integer(-1), Integer(2), log(Symbol('c', commutative=True)))), Add(Mul(Function('F_H')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('F_H')(Symbol('c', commutative=True)), Integer(2)), Mul(Integer(-1), Function('F_H')(Symbol('c', commutative=True))), Mul(Integer(-1), log(Symbol('c', commutative=True)))), Add(Mul(Function('F_H')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Mul(Integer(-1), Function('F_H')(Symbol('c', commutative=True))), Mul(Integer(-1), log(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given r{(f_{E})} = \\int \\cos{(f_{E})} df_{E}, then derive r{(f_{E})} - \\sin{(f_{E})} = u, then derive (r{(f_{E})} - \\sin{(f_{E})})^{f_{E}} = G^{f_{E}}, then obtain u^{f_{E}} = G^{f_{E}}", "derivation": "r{(f_{E})} = \\int \\cos{(f_{E})} df_{E} and r{(f_{E})} - \\sin{(f_{E})} = - \\sin{(f_{E})} + \\int \\cos{(f_{E})} df_{E} and r{(f_{E})} - \\sin{(f_{E})} = u and (r{(f_{E})} - \\sin{(f_{E})})^{f_{E}} = (- \\sin{(f_{E})} + \\int \\cos{(f_{E})} df_{E})^{f_{E}} and (r{(f_{E})} - \\sin{(f_{E})})^{f_{E}} = G^{f_{E}} and u^{f_{E}} = G^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('f_E', commutative=True)), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["minus", 1, "sin(Symbol('f_E', commutative=True))"], "Equality(Add(Function('r')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('r')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('u', commutative=True))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Function('r')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Function('r')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Symbol('u', commutative=True), Symbol('f_E', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} = q + \\log{(\\pi)}, then obtain \\frac{\\int \\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} dq}{\\pi} = \\frac{\\int (q + \\log{(\\pi)}) dq}{\\pi}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} = q + \\log{(\\pi)} and \\int \\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} dq = \\int (q + \\log{(\\pi)}) dq and \\frac{\\partial}{\\partial \\pi} (q + \\log{(\\pi)}) \\int \\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} dq = \\frac{\\partial}{\\partial \\pi} (q + \\log{(\\pi)}) \\int (q + \\log{(\\pi)}) dq and \\frac{\\int \\operatorname{V_{\\mathbf{B}}}{(q,\\pi)} dq}{\\pi} = \\frac{\\int (q + \\log{(\\pi)}) dq}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('q', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('q', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["times", 2, "Derivative(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('q', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Derivative(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('q', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(Add(Symbol('q', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)} = \\int (E_{x} + \\theta) d\\theta and x{(E_{x},\\theta)} = ((E_{x} + \\theta) \\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)})^{E_{x}}, then obtain x{(E_{x},\\theta)} = ((E_{x} + \\theta) \\int (E_{x} + \\theta) d\\theta)^{E_{x}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)} = \\int (E_{x} + \\theta) d\\theta and (E_{x} + \\theta) \\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)} = (E_{x} + \\theta) \\int (E_{x} + \\theta) d\\theta and ((E_{x} + \\theta) \\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)})^{E_{x}} = ((E_{x} + \\theta) \\int (E_{x} + \\theta) d\\theta)^{E_{x}} and x{(E_{x},\\theta)} = ((E_{x} + \\theta) \\operatorname{f_{\\mathbf{p}}}{(E_{x},\\theta)})^{E_{x}} and x{(E_{x},\\theta)} = ((E_{x} + \\theta) \\int (E_{x} + \\theta) d\\theta)^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('E_x', commutative=True)), Pow(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Symbol('E_x', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('E_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('x')(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('E_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given S{(\\nabla,c,\\theta_1)} = \\frac{\\nabla}{c} + \\theta_1 and \\operatorname{v_{1}}{(\\theta_1)} = \\theta_1, then obtain \\frac{\\partial}{\\partial \\theta_1} \\iint S{(\\nabla,c,\\theta_1)} d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla = \\frac{\\partial}{\\partial \\theta_1} \\iint (\\frac{\\nabla}{c} + \\theta_1) d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla", "derivation": "S{(\\nabla,c,\\theta_1)} = \\frac{\\nabla}{c} + \\theta_1 and \\int S{(\\nabla,c,\\theta_1)} d\\theta_1 = \\int (\\frac{\\nabla}{c} + \\theta_1) d\\theta_1 and \\operatorname{v_{1}}{(\\theta_1)} = \\theta_1 and \\int S{(\\nabla,c,\\theta_1)} d\\operatorname{v_{1}}{(\\theta_1)} = \\int (\\frac{\\nabla}{c} + \\theta_1) d\\operatorname{v_{1}}{(\\theta_1)} and \\iint S{(\\nabla,c,\\theta_1)} d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla = \\iint (\\frac{\\nabla}{c} + \\theta_1) d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla and \\frac{\\partial}{\\partial \\theta_1} \\iint S{(\\nabla,c,\\theta_1)} d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla = \\frac{\\partial}{\\partial \\theta_1} \\iint (\\frac{\\nabla}{c} + \\theta_1) d\\operatorname{v_{1}}{(\\theta_1)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\nabla', commutative=True), Symbol('c', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\nabla', commutative=True), Symbol('c', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('S')(Symbol('\\\\nabla', commutative=True), Symbol('c', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True)))), Integral(Add(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\nabla', commutative=True), Symbol('c', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('\\\\nabla', commutative=True), Symbol('c', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Tuple(Function('v_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then obtain z^{2 \\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} = z^{\\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} \\cos^{\\mathbf{F}}{(\\mathbf{F})}", "derivation": "z{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and z^{\\mathbf{F}}{(\\mathbf{F})} = \\cos^{\\mathbf{F}}{(\\mathbf{F})} and z^{\\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} = \\cos{(\\mathbf{F})} \\cos^{\\mathbf{F}}{(\\mathbf{F})} and z^{\\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} \\cos^{\\mathbf{F}}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} \\cos^{2 \\mathbf{F}}{(\\mathbf{F})} and z^{2 \\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} = z^{\\mathbf{F}}{(\\mathbf{F})} \\cos{(\\mathbf{F})} \\cos^{\\mathbf{F}}{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 2, "cos(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Pow(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Mul(cos(Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 3, "Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Pow(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Mul(cos(Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True))), cos(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Function('z')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} = - \\Psi^{\\dagger} + \\chi, then derive 1 = \\frac{- \\Psi^{\\dagger} \\chi + \\frac{\\chi^{2}}{2} + t_{1}}{\\int \\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} d\\chi}, then obtain 1 = \\frac{- \\Psi^{\\dagger} \\chi + \\frac{\\chi^{2}}{2} + t_{1}}{\\int (- \\Psi^{\\dagger} + \\chi) d\\chi}", "derivation": "\\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} = - \\Psi^{\\dagger} + \\chi and \\int \\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} d\\chi = \\int (- \\Psi^{\\dagger} + \\chi) d\\chi and 1 = \\frac{\\int (- \\Psi^{\\dagger} + \\chi) d\\chi}{\\int \\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} d\\chi} and 1 = \\frac{- \\Psi^{\\dagger} \\chi + \\frac{\\chi^{2}}{2} + t_{1}}{\\int \\operatorname{P_{e}}{(\\Psi^{\\dagger},\\chi)} d\\chi} and 1 = \\frac{- \\Psi^{\\dagger} \\chi + \\frac{\\chi^{2}}{2} + t_{1}}{\\int (- \\Psi^{\\dagger} + \\chi) d\\chi}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Integral(Function('P_e')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Pow(Integral(Function('P_e')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Pow(Integral(Function('P_e')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given S{(\\hbar)} = \\cos{(\\hbar)} and Q{(\\hbar)} = S^{2}{(\\hbar)}, then obtain 4 Q{(\\hbar)} = 2 (S{(\\hbar)} + \\cos{(\\hbar)}) S{(\\hbar)}", "derivation": "S{(\\hbar)} = \\cos{(\\hbar)} and 2 S{(\\hbar)} = S{(\\hbar)} + \\cos{(\\hbar)} and 4 S^{2}{(\\hbar)} = 2 (S{(\\hbar)} + \\cos{(\\hbar)}) S{(\\hbar)} and Q{(\\hbar)} = S^{2}{(\\hbar)} and 4 Q{(\\hbar)} = 2 (S{(\\hbar)} + \\cos{(\\hbar)}) S{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Function('S')(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('\\\\hbar', commutative=True))), Add(Function('S')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('S')(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('S')(Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('S')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Function('S')(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\hbar', commutative=True)), Pow(Function('S')(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(4), Function('Q')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), Add(Function('S')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Function('S')(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given k{(\\phi_2,\\eta^{\\prime})} = \\frac{\\phi_2}{\\eta^{\\prime}}, then obtain \\phi_2 + \\sin{(\\frac{\\phi_2 k{(\\phi_2,\\eta^{\\prime})}}{\\eta^{\\prime}})} = \\phi_2 + \\sin{(\\frac{\\phi_2^{2}}{(\\eta^{\\prime})^{2}})}", "derivation": "k{(\\phi_2,\\eta^{\\prime})} = \\frac{\\phi_2}{\\eta^{\\prime}} and \\frac{\\phi_2 k{(\\phi_2,\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\phi_2^{2}}{(\\eta^{\\prime})^{2}} and \\sin{(\\frac{\\phi_2 k{(\\phi_2,\\eta^{\\prime})}}{\\eta^{\\prime}})} = \\sin{(\\frac{\\phi_2^{2}}{(\\eta^{\\prime})^{2}})} and \\phi_2 + \\sin{(\\frac{\\phi_2 k{(\\phi_2,\\eta^{\\prime})}}{\\eta^{\\prime}})} = \\phi_2 + \\sin{(\\frac{\\phi_2^{2}}{(\\eta^{\\prime})^{2}})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))))"], [["sin", 2], "Equality(sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"], [["add", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))), Add(Symbol('\\\\phi_2', commutative=True), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))))))"]]}, {"prompt": "Given r{(z^{*})} = \\cos{(\\cos{(z^{*})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(z^{*})} = \\cos{(\\cos{(z^{*})})}, then obtain \\frac{d}{d z^{*}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z^{*})} = \\frac{d}{d z^{*}} r{(z^{*})}", "derivation": "r{(z^{*})} = \\cos{(\\cos{(z^{*})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(z^{*})} = \\cos{(\\cos{(z^{*})})} and \\frac{d}{d z^{*}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(\\cos{(z^{*})})} and \\frac{d}{d z^{*}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z^{*})} = \\frac{d}{d z^{*}} r{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('z^*', commutative=True)), cos(cos(Symbol('z^*', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z^*', commutative=True)), cos(cos(Symbol('z^*', commutative=True))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Function('r')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(x^\\prime,\\varepsilon)} = \\sin{(\\varepsilon + x^\\prime)}, then obtain \\varepsilon \\int (\\mathbf{E}{(x^\\prime,\\varepsilon)} - \\sin{(\\varepsilon + x^\\prime)}) d\\varepsilon = \\varepsilon \\int 0 d\\varepsilon", "derivation": "\\mathbf{E}{(x^\\prime,\\varepsilon)} = \\sin{(\\varepsilon + x^\\prime)} and \\mathbf{E}{(x^\\prime,\\varepsilon)} - \\sin{(\\varepsilon + x^\\prime)} = 0 and \\int (\\mathbf{E}{(x^\\prime,\\varepsilon)} - \\sin{(\\varepsilon + x^\\prime)}) d\\varepsilon = \\int 0 d\\varepsilon and \\varepsilon \\int (\\mathbf{E}{(x^\\prime,\\varepsilon)} - \\sin{(\\varepsilon + x^\\prime)}) d\\varepsilon = \\varepsilon \\int 0 d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 1, "sin(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{E}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Add(Function('\\\\mathbf{E}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})} = \\dot{y} \\log{(\\varphi)}, then obtain ((- \\dot{y} + \\sin{(\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})})})^{\\dot{y}})^{\\dot{y}} = ((- \\dot{y} + \\sin{(\\dot{y} \\log{(\\varphi)})})^{\\dot{y}})^{\\dot{y}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})} = \\dot{y} \\log{(\\varphi)} and \\sin{(\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})})} = \\sin{(\\dot{y} \\log{(\\varphi)})} and - \\dot{y} + \\sin{(\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})})} = - \\dot{y} + \\sin{(\\dot{y} \\log{(\\varphi)})} and (- \\dot{y} + \\sin{(\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})})})^{\\dot{y}} = (- \\dot{y} + \\sin{(\\dot{y} \\log{(\\varphi)})})^{\\dot{y}} and ((- \\dot{y} + \\sin{(\\operatorname{L_{\\varepsilon}}{(\\varphi,\\dot{y})})})^{\\dot{y}})^{\\dot{y}} = ((- \\dot{y} + \\sin{(\\dot{y} \\log{(\\varphi)})})^{\\dot{y}})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["sin", 1], "Equality(sin(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\varphi', commutative=True)))))"], [["minus", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\varphi', commutative=True))))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\varphi', commutative=True))))), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\varphi', commutative=True))))), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} = \\frac{C_{2} \\mathbf{S}}{\\chi}, then obtain (C_{2} + \\int \\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} dC_{2})^{\\chi} = (C_{2} + \\int \\frac{C_{2} \\mathbf{S}}{\\chi} dC_{2})^{\\chi}", "derivation": "\\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} = \\frac{C_{2} \\mathbf{S}}{\\chi} and \\int \\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} dC_{2} = \\int \\frac{C_{2} \\mathbf{S}}{\\chi} dC_{2} and C_{2} + \\int \\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} dC_{2} = C_{2} + \\int \\frac{C_{2} \\mathbf{S}}{\\chi} dC_{2} and (C_{2} + \\int \\mathbf{F}{(\\chi,\\mathbf{S},C_{2})} dC_{2})^{\\chi} = (C_{2} + \\int \\frac{C_{2} \\mathbf{S}}{\\chi} dC_{2})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["add", 2, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given M{(\\pi,\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi, then derive M{(\\pi,\\mathbf{B})} = \\pi, then obtain M^{\\mathbf{B}}{(\\pi,\\mathbf{B})} - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi - (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi)^{\\mathbf{B}} = - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi", "derivation": "M{(\\pi,\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi and M{(\\pi,\\mathbf{B})} = \\pi and \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi = \\pi and (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi)^{\\mathbf{B}} = \\pi^{\\mathbf{B}} and M^{\\mathbf{B}}{(\\pi,\\mathbf{B})} = \\pi^{\\mathbf{B}} and - \\pi^{\\mathbf{B}} + M^{\\mathbf{B}}{(\\pi,\\mathbf{B})} - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi = - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi and M^{\\mathbf{B}}{(\\pi,\\mathbf{B})} - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi - (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi)^{\\mathbf{B}} = - \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\pi", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 5, "Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Pow(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(W,G,T)} = (- G + T)^{W}, then obtain \\operatorname{M_{E}}^{W}{(W,G,T)} = ((- G + T)^{3 W} \\operatorname{M_{E}}{(W,G,T)} - \\operatorname{M_{E}}^{4}{(W,G,T)} + 1) \\operatorname{M_{E}}^{W}{(W,G,T)}", "derivation": "\\operatorname{M_{E}}{(W,G,T)} = (- G + T)^{W} and \\operatorname{M_{E}}^{2}{(W,G,T)} = (- G + T)^{W} \\operatorname{M_{E}}{(W,G,T)} and \\operatorname{M_{E}}^{4}{(W,G,T)} = (- G + T)^{2 W} \\operatorname{M_{E}}^{2}{(W,G,T)} and \\operatorname{M_{E}}^{4}{(W,G,T)} + 1 = (- G + T)^{2 W} \\operatorname{M_{E}}^{2}{(W,G,T)} + 1 and 1 = (- G + T)^{2 W} \\operatorname{M_{E}}^{2}{(W,G,T)} - \\operatorname{M_{E}}^{4}{(W,G,T)} + 1 and 1 = (- G + T)^{3 W} \\operatorname{M_{E}}{(W,G,T)} - (- G + T)^{2 W} \\operatorname{M_{E}}^{2}{(W,G,T)} + 1 and 1 = (- G + T)^{3 W} \\operatorname{M_{E}}{(W,G,T)} - \\operatorname{M_{E}}^{4}{(W,G,T)} + 1 and \\operatorname{M_{E}}^{W}{(W,G,T)} = ((- G + T)^{3 W} \\operatorname{M_{E}}{(W,G,T)} - \\operatorname{M_{E}}^{4}{(W,G,T)} + 1) \\operatorname{M_{E}}^{W}{(W,G,T)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Symbol('W', commutative=True)))"], [["times", 1, "Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(2)), Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Symbol('W', commutative=True)), Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4)), Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True))), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(2))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4)), Integer(1)), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True))), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(2))), Integer(1)))"], [["minus", 4, "Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4))"], "Equality(Integer(1), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True))), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(3), Symbol('W', commutative=True))), Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True))), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(2))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(3), Symbol('W', commutative=True))), Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4))), Integer(1)))"], [["times", 7, "Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Symbol('W', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Symbol('W', commutative=True)), Mul(Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(3), Symbol('W', commutative=True))), Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Integer(4))), Integer(1)), Pow(Function('M_E')(Symbol('W', commutative=True), Symbol('G', commutative=True), Symbol('T', commutative=True)), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)}, then obtain (\\frac{d}{d \\mu} \\mathbf{A}^{\\mu}{(\\mu)} + 1)^{\\mu} = (\\frac{d}{d \\mu} (\\frac{d}{d \\mu} \\sin{(\\mu)})^{\\mu} + 1)^{\\mu}", "derivation": "\\mathbf{A}{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)} and \\mathbf{A}^{\\mu}{(\\mu)} = (\\frac{d}{d \\mu} \\sin{(\\mu)})^{\\mu} and \\frac{d}{d \\mu} \\mathbf{A}^{\\mu}{(\\mu)} = \\frac{d}{d \\mu} (\\frac{d}{d \\mu} \\sin{(\\mu)})^{\\mu} and \\frac{d}{d \\mu} \\mathbf{A}^{\\mu}{(\\mu)} + 1 = \\frac{d}{d \\mu} (\\frac{d}{d \\mu} \\sin{(\\mu)})^{\\mu} + 1 and (\\frac{d}{d \\mu} \\mathbf{A}^{\\mu}{(\\mu)} + 1)^{\\mu} = (\\frac{d}{d \\mu} (\\frac{d}{d \\mu} \\sin{(\\mu)})^{\\mu} + 1)^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\mu', commutative=True)), Pow(Add(Derivative(Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{X},C_{2})} = C_{2} + \\hat{X} and \\sigma_{p}{(\\hat{X},C_{2})} = \\frac{C_{2} + \\hat{X}}{\\operatorname{F_{x}}{(\\hat{X},C_{2})}} - \\operatorname{F_{x}}{(\\hat{X},C_{2})}, then obtain 1 - \\operatorname{F_{x}}{(\\hat{X},C_{2})} = - C_{2} - \\hat{X} + 1", "derivation": "\\operatorname{F_{x}}{(\\hat{X},C_{2})} = C_{2} + \\hat{X} and 1 = \\frac{C_{2} + \\hat{X}}{\\operatorname{F_{x}}{(\\hat{X},C_{2})}} and 1 - \\operatorname{F_{x}}{(\\hat{X},C_{2})} = \\frac{C_{2} + \\hat{X}}{\\operatorname{F_{x}}{(\\hat{X},C_{2})}} - \\operatorname{F_{x}}{(\\hat{X},C_{2})} and \\sigma_{p}{(\\hat{X},C_{2})} = \\frac{C_{2} + \\hat{X}}{\\operatorname{F_{x}}{(\\hat{X},C_{2})}} - \\operatorname{F_{x}}{(\\hat{X},C_{2})} and \\sigma_{p}{(\\hat{X},C_{2})} = - C_{2} - \\hat{X} + 1 and 1 - \\operatorname{F_{x}}{(\\hat{X},C_{2})} = \\sigma_{p}{(\\hat{X},C_{2})} and 1 - \\operatorname{F_{x}}{(\\hat{X},C_{2})} = - C_{2} - \\hat{X} + 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('C_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Integer(-1))))"], [["minus", 2, "Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))), Add(Mul(Add(Symbol('C_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Add(Symbol('C_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))), Function('\\\\sigma_p')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('C_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\omega{(\\ddot{x},\\varphi)} = \\frac{\\ddot{x}}{\\varphi} and V{(\\varphi)} = \\frac{1}{\\varphi}, then obtain \\omega{(\\ddot{x},\\varphi)} + \\int V^{2}{(\\varphi)} d\\varphi = \\frac{\\ddot{x}}{\\varphi} + \\int V^{2}{(\\varphi)} d\\varphi", "derivation": "\\omega{(\\ddot{x},\\varphi)} = \\frac{\\ddot{x}}{\\varphi} and V{(\\varphi)} = \\frac{1}{\\varphi} and V^{2}{(\\varphi)} = \\frac{V{(\\varphi)}}{\\varphi} and \\int V^{2}{(\\varphi)} d\\varphi = \\int \\frac{V{(\\varphi)}}{\\varphi} d\\varphi and \\omega{(\\ddot{x},\\varphi)} + \\int \\frac{V{(\\varphi)}}{\\varphi} d\\varphi = \\frac{\\ddot{x}}{\\varphi} + \\int \\frac{V{(\\varphi)}}{\\varphi} d\\varphi and \\omega{(\\ddot{x},\\varphi)} + \\int V^{2}{(\\varphi)} d\\varphi = \\frac{\\ddot{x}}{\\varphi} + \\int V^{2}{(\\varphi)} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], [["times", 2, "Function('V')(Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('V')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('V')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Pow(Function('V')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Integral(Pow(Function('V')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\mu_0,T)} = \\sin{(\\mu_0^{T})}, then obtain \\frac{\\frac{\\partial}{\\partial \\mu_0} \\dot{y}{(\\mu_0,T)}}{\\dot{y}{(\\mu_0,T)}} = \\frac{T \\mu_0^{T} \\cos{(\\mu_0^{T})}}{\\mu_0 \\dot{y}{(\\mu_0,T)}}", "derivation": "\\dot{y}{(\\mu_0,T)} = \\sin{(\\mu_0^{T})} and \\frac{\\partial}{\\partial \\mu_0} \\dot{y}{(\\mu_0,T)} = \\frac{\\partial}{\\partial \\mu_0} \\sin{(\\mu_0^{T})} and \\frac{\\frac{\\partial}{\\partial \\mu_0} \\dot{y}{(\\mu_0,T)}}{\\dot{y}{(\\mu_0,T)}} = \\frac{\\frac{\\partial}{\\partial \\mu_0} \\sin{(\\mu_0^{T})}}{\\dot{y}{(\\mu_0,T)}} and \\frac{\\frac{\\partial}{\\partial \\mu_0} \\dot{y}{(\\mu_0,T)}}{\\dot{y}{(\\mu_0,T)}} = \\frac{T \\mu_0^{T} \\cos{(\\mu_0^{T})}}{\\mu_0 \\dot{y}{(\\mu_0,T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Derivative(sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(A_{2})} = \\log{(A_{2})}, then obtain 0 = - \\frac{\\sin{(\\frac{\\log{(A_{2})} \\frac{d}{d A_{2}} \\hat{X}{(A_{2})}}{\\hat{X}^{2}{(A_{2})}} - \\frac{1}{A_{2} \\hat{X}{(A_{2})}})}}{\\hat{X}{(A_{2})}}", "derivation": "\\hat{X}{(A_{2})} = \\log{(A_{2})} and 1 = \\frac{\\log{(A_{2})}}{\\hat{X}{(A_{2})}} and \\frac{d}{d A_{2}} 1 = \\frac{d}{d A_{2}} \\frac{\\log{(A_{2})}}{\\hat{X}{(A_{2})}} and \\sin{(\\frac{d}{d A_{2}} 1)} = \\sin{(\\frac{d}{d A_{2}} \\frac{\\log{(A_{2})}}{\\hat{X}{(A_{2})}})} and \\frac{\\sin{(\\frac{d}{d A_{2}} 1)}}{\\hat{X}{(A_{2})}} = \\frac{\\sin{(\\frac{d}{d A_{2}} \\frac{\\log{(A_{2})}}{\\hat{X}{(A_{2})}})}}{\\hat{X}{(A_{2})}} and 0 = - \\frac{\\sin{(\\frac{\\log{(A_{2})} \\frac{d}{d A_{2}} \\hat{X}{(A_{2})}}{\\hat{X}^{2}{(A_{2})}} - \\frac{1}{A_{2} \\hat{X}{(A_{2})}})}}{\\hat{X}{(A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["divide", 1, "Function('\\\\hat{X}')(Symbol('A_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), log(Symbol('A_2', commutative=True))))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1)))), sin(Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["divide", 4, "Function('\\\\hat{X}')(Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), sin(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))))), Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), sin(Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)), sin(Add(Mul(Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-2)), log(Symbol('A_2', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Function('\\\\hat{X}')(Symbol('A_2', commutative=True)), Integer(-1)))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J}, then obtain \\frac{d}{d \\mathbf{J}} \\frac{\\int \\operatorname{E_{n}}{(\\mathbf{J})} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} = 0", "derivation": "\\operatorname{E_{n}}{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and \\int \\operatorname{E_{n}}{(\\mathbf{J})} d\\mathbf{J} = \\iint \\cos{(\\mathbf{J})} d\\mathbf{J} d\\mathbf{J} and \\frac{\\int \\operatorname{E_{n}}{(\\mathbf{J})} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} = \\frac{\\iint \\cos{(\\mathbf{J})} d\\mathbf{J} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} and \\frac{d}{d \\mathbf{J}} \\frac{\\int \\operatorname{E_{n}}{(\\mathbf{J})} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} = \\frac{d}{d \\mathbf{J}} \\frac{\\iint \\cos{(\\mathbf{J})} d\\mathbf{J} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} and \\frac{d}{d \\mathbf{J}} \\frac{\\int \\operatorname{E_{n}}{(\\mathbf{J})} d\\mathbf{J}}{\\cos{(\\mathbf{J})}} = 0", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 2, "cos(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Mul(Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})} = H + g^{\\prime}_{\\varepsilon}, then obtain \\frac{(H + g^{\\prime}_{\\varepsilon}) \\hat{H}_l^{2}{(H,g^{\\prime}_{\\varepsilon})}}{H} = \\frac{(H + g^{\\prime}_{\\varepsilon})^{2} \\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})}}{H}", "derivation": "\\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})} = H + g^{\\prime}_{\\varepsilon} and (H + g^{\\prime}_{\\varepsilon}) \\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})} = (H + g^{\\prime}_{\\varepsilon})^{2} and (H + g^{\\prime}_{\\varepsilon}) \\hat{H}_l^{2}{(H,g^{\\prime}_{\\varepsilon})} = (H + g^{\\prime}_{\\varepsilon})^{2} \\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})} and \\frac{(H + g^{\\prime}_{\\varepsilon}) \\hat{H}_l^{2}{(H,g^{\\prime}_{\\varepsilon})}}{H} = \\frac{(H + g^{\\prime}_{\\varepsilon})^{2} \\hat{H}_l{(H,g^{\\prime}_{\\varepsilon})}}{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["times", 2, "Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)), Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["divide", 3, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Add(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)), Function('\\\\hat{H}_l')(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(a,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{a}, then obtain \\frac{\\mathbf{J}_f}{a} + (- \\tilde{g}{(a,\\mathbf{J}_f)})^{a} = \\frac{\\mathbf{J}_f}{a} + (- \\frac{\\mathbf{J}_f}{a})^{a}", "derivation": "\\tilde{g}{(a,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{a} and - \\tilde{g}{(a,\\mathbf{J}_f)} = - \\frac{\\mathbf{J}_f}{a} and (- \\tilde{g}{(a,\\mathbf{J}_f)})^{a} = (- \\frac{\\mathbf{J}_f}{a})^{a} and \\frac{\\mathbf{J}_f}{a} + (- \\tilde{g}{(a,\\mathbf{J}_f)})^{a} = \\frac{\\mathbf{J}_f}{a} + (- \\frac{\\mathbf{J}_f}{a})^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('a', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Pow(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('a', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given H{(t,\\mathbf{S})} = e^{\\frac{t}{\\mathbf{S}}}, then obtain (H{(t,\\mathbf{S})} - \\frac{t}{\\mathbf{S}}) (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}}) (e^{\\frac{t}{\\mathbf{S}}} + \\frac{t}{\\mathbf{S}}) = (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}})^{2} (e^{\\frac{t}{\\mathbf{S}}} + \\frac{t}{\\mathbf{S}})", "derivation": "H{(t,\\mathbf{S})} = e^{\\frac{t}{\\mathbf{S}}} and H{(t,\\mathbf{S})} - \\frac{t}{\\mathbf{S}} = e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}} and (H{(t,\\mathbf{S})} - \\frac{t}{\\mathbf{S}}) (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}}) = (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}})^{2} and (H{(t,\\mathbf{S})} - \\frac{t}{\\mathbf{S}}) (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}}) (e^{\\frac{t}{\\mathbf{S}}} + \\frac{t}{\\mathbf{S}}) = (e^{\\frac{t}{\\mathbf{S}}} - \\frac{t}{\\mathbf{S}})^{2} (e^{\\frac{t}{\\mathbf{S}}} + \\frac{t}{\\mathbf{S}})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))"], "Equality(Add(Function('H')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["times", 2, "Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True)))"], "Equality(Mul(Add(Function('H')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True)))), Pow(Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Integer(2)))"], [["times", 3, "Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True)))"], "Equality(Mul(Add(Function('H')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True)))), Mul(Pow(Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Integer(2)), Add(exp(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given a{(q,\\mu)} = \\int \\mu q d\\mu, then derive \\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq = \\int 0 dq, then obtain \\frac{(\\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq)^{q}}{\\mu q (\\mu + \\int \\mu q d\\mu)} = \\frac{(\\int 0 dq)^{q}}{\\mu q (\\mu + \\int \\mu q d\\mu)}", "derivation": "a{(q,\\mu)} = \\int \\mu q d\\mu and a{(q,\\mu)} - \\int \\mu q d\\mu = 0 and \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) = \\frac{d}{d \\mu} 0 and \\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq = \\int \\frac{d}{d \\mu} 0 dq and \\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq = \\int 0 dq and (\\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq)^{q} = (\\int 0 dq)^{q} and \\frac{(\\int \\frac{\\partial}{\\partial \\mu} (a{(q,\\mu)} - \\int \\mu q d\\mu) dq)^{q}}{\\mu q (\\mu + \\int \\mu q d\\mu)} = \\frac{(\\int 0 dq)^{q}}{\\mu q (\\mu + \\int \\mu q d\\mu)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Derivative(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integral(Derivative(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Integral(Integer(0), Tuple(Symbol('q', commutative=True))))"], [["power", 5, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Derivative(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["divide", 6, "Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mu', commutative=True), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Integer(-1)), Pow(Integral(Derivative(Add(Function('a')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mu', commutative=True), Integral(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Integer(-1)), Pow(Integral(Integer(0), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\hbar)} = \\hbar, then derive \\hat{H} + \\nabla{(\\hbar)} = \\int \\frac{\\hbar}{\\nabla{(\\hbar)}} d\\nabla{(\\hbar)}, then derive \\hat{H} + \\nabla{(\\hbar)} = \\phi_2 + \\nabla{(\\hbar)}, then obtain (\\hat{H} + \\nabla{(\\hbar)})^{\\hbar} = (\\phi_2 + \\nabla{(\\hbar)})^{\\hbar}", "derivation": "\\nabla{(\\hbar)} = \\hbar and 1 = \\frac{\\hbar}{\\nabla{(\\hbar)}} and \\int 1 d\\hbar = \\int \\frac{\\hbar}{\\nabla{(\\hbar)}} d\\hbar and \\int 1 d\\nabla{(\\hbar)} = \\int \\frac{\\hbar}{\\nabla{(\\hbar)}} d\\nabla{(\\hbar)} and \\hat{H} + \\nabla{(\\hbar)} = \\int \\frac{\\hbar}{\\nabla{(\\hbar)}} d\\nabla{(\\hbar)} and \\hat{H} + \\nabla{(\\hbar)} = \\int 1 d\\nabla{(\\hbar)} and \\hat{H} + \\nabla{(\\hbar)} = \\phi_2 + \\nabla{(\\hbar)} and (\\hat{H} + \\nabla{(\\hbar)})^{\\hbar} = (\\phi_2 + \\nabla{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["divide", 1, "Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Integer(1), Tuple(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)))), Integral(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))), Integral(Integer(1), Tuple(Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))))"], [["power", 7, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\nabla')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(T)} = \\int \\log{(T)} dT and \\operatorname{v_{2}}{(T)} = \\int \\hat{p}_0{(T)} dT, then obtain \\operatorname{v_{2}}{(T)} = \\iint \\log{(T)} dT dT", "derivation": "\\hat{p}_0{(T)} = \\int \\log{(T)} dT and \\int \\hat{p}_0{(T)} dT = \\iint \\log{(T)} dT dT and \\operatorname{v_{2}}{(T)} = \\int \\hat{p}_0{(T)} dT and \\operatorname{v_{2}}{(T)} = \\iint \\log{(T)} dT dT", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('T', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('T', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_2')(Symbol('T', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given s{(S,f^{*})} = S \\sin{(f^{*})}, then obtain \\frac{d}{d S} (- S - \\int 0 dS) = \\frac{\\partial}{\\partial S} (S \\sin{(f^{*})} - S - s{(S,f^{*})} - \\int 0 dS)", "derivation": "s{(S,f^{*})} = S \\sin{(f^{*})} and 0 = S \\sin{(f^{*})} - s{(S,f^{*})} and \\int 0 dS = \\int (S \\sin{(f^{*})} - s{(S,f^{*})}) dS and - \\int (S \\sin{(f^{*})} - s{(S,f^{*})}) dS = S \\sin{(f^{*})} - s{(S,f^{*})} - \\int (S \\sin{(f^{*})} - s{(S,f^{*})}) dS and - \\int 0 dS = S \\sin{(f^{*})} - s{(S,f^{*})} - \\int 0 dS and - S - \\int 0 dS = S \\sin{(f^{*})} - S - s{(S,f^{*})} - \\int 0 dS and \\frac{d}{d S} (- S - \\int 0 dS) = \\frac{\\partial}{\\partial S} (S \\sin{(f^{*})} - S - s{(S,f^{*})} - \\int 0 dS)", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["minus", 1, "Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["minus", 2, "Integral(Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('S', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))))"], [["minus", 5, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))))"], [["differentiate", 6, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(E_{n})} = \\log{(\\log{(E_{n})})}, then obtain \\int (f{(E_{n})} - \\log{(\\log{(E_{n})})}) (f{(E_{n})} - \\log{(\\log{(E_{n})})} - 1)^{E_{n}} dE_{n} = \\int (-1)^{E_{n}} (f{(E_{n})} - \\log{(\\log{(E_{n})})}) dE_{n}", "derivation": "f{(E_{n})} = \\log{(\\log{(E_{n})})} and f{(E_{n})} - \\log{(\\log{(E_{n})})} = 0 and f{(E_{n})} - \\log{(\\log{(E_{n})})} - 1 = -1 and (f{(E_{n})} - \\log{(\\log{(E_{n})})} - 1)^{E_{n}} = (-1)^{E_{n}} and (f{(E_{n})} - \\log{(\\log{(E_{n})})}) (f{(E_{n})} - \\log{(\\log{(E_{n})})} - 1)^{E_{n}} = (-1)^{E_{n}} (f{(E_{n})} - \\log{(\\log{(E_{n})})}) and \\int (f{(E_{n})} - \\log{(\\log{(E_{n})})}) (f{(E_{n})} - \\log{(\\log{(E_{n})})} - 1)^{E_{n}} dE_{n} = \\int (-1)^{E_{n}} (f{(E_{n})} - \\log{(\\log{(E_{n})})}) dE_{n}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('E_n', commutative=True)), log(log(Symbol('E_n', commutative=True))))"], [["minus", 1, "log(log(Symbol('E_n', commutative=True)))"], "Equality(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True))))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))), Integer(-1)), Integer(-1))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))), Integer(-1)), Symbol('E_n', commutative=True)), Pow(Integer(-1), Symbol('E_n', commutative=True)))"], [["times", 4, "Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))))"], "Equality(Mul(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True))))), Pow(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))), Integer(-1)), Symbol('E_n', commutative=True))), Mul(Pow(Integer(-1), Symbol('E_n', commutative=True)), Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))))))"], [["integrate", 5, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True))))), Pow(Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))), Integer(-1)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Pow(Integer(-1), Symbol('E_n', commutative=True)), Add(Function('f')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(log(Symbol('E_n', commutative=True)))))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given f{(r,\\mathbf{S})} = - r + \\sin{(\\mathbf{S})}, then derive (\\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})})^{r} = \\cos^{r}{(\\mathbf{S})}, then obtain \\mathbf{S} + \\cos^{r}{(\\mathbf{S})} = \\mathbf{S} + (\\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})})^{r}", "derivation": "f{(r,\\mathbf{S})} = - r + \\sin{(\\mathbf{S})} and \\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})} = \\frac{\\partial}{\\partial \\mathbf{S}} (- r + \\sin{(\\mathbf{S})}) and (\\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})})^{r} = (\\frac{\\partial}{\\partial \\mathbf{S}} (- r + \\sin{(\\mathbf{S})}))^{r} and (\\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})})^{r} = \\cos^{r}{(\\mathbf{S})} and \\cos^{r}{(\\mathbf{S})} = (\\frac{\\partial}{\\partial \\mathbf{S}} (- r + \\sin{(\\mathbf{S})}))^{r} and \\mathbf{S} + \\cos^{r}{(\\mathbf{S})} = \\mathbf{S} + (\\frac{\\partial}{\\partial \\mathbf{S}} (- r + \\sin{(\\mathbf{S})}))^{r} and \\mathbf{S} + \\cos^{r}{(\\mathbf{S})} = \\mathbf{S} + (\\frac{\\partial}{\\partial \\mathbf{S}} f{(r,\\mathbf{S})})^{r}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('r', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["add", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('r', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Derivative(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('r', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(v_{z},x)} = v_{z} - x, then derive \\frac{\\partial}{\\partial x} \\operatorname{F_{g}}{(v_{z},x)} = -1, then obtain - \\frac{1}{v_{z} - x} = \\frac{\\frac{\\partial}{\\partial x} (v_{z} - x)}{v_{z} - x}", "derivation": "\\operatorname{F_{g}}{(v_{z},x)} = v_{z} - x and \\frac{\\partial}{\\partial x} \\operatorname{F_{g}}{(v_{z},x)} = \\frac{\\partial}{\\partial x} (v_{z} - x) and \\frac{\\partial}{\\partial x} \\operatorname{F_{g}}{(v_{z},x)} = -1 and -1 = \\frac{\\partial}{\\partial x} (v_{z} - x) and - \\frac{1}{v_{z} - x} = \\frac{\\frac{\\partial}{\\partial x} (v_{z} - x)}{v_{z} - x}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('v_z', commutative=True), Symbol('x', commutative=True)), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('v_z', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('v_z', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 4, "Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Derivative(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(q,U)} = U q, then derive \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = q, then obtain q^{U} \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} (\\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)})^{U}", "derivation": "\\operatorname{v_{2}}{(q,U)} = U q and \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = \\frac{\\partial}{\\partial U} U q and \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = q and (\\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)})^{U} = q^{U} and (\\frac{\\partial}{\\partial U} U q)^{U} = q^{U} and (\\frac{\\partial}{\\partial U} U q)^{U} = (\\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)})^{U} and (\\frac{\\partial}{\\partial U} U q)^{U} \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} (\\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)})^{U} and q^{U} \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} = \\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)} (\\frac{\\partial}{\\partial U} \\operatorname{v_{2}}{(q,U)})^{U}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('q', commutative=True))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True)), Pow(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True)))"], [["times", 6, "Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True)), Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Pow(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True))))"], [["evaluate_derivatives", 7], "Equality(Mul(Pow(Symbol('q', commutative=True), Symbol('U', commutative=True)), Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Pow(Derivative(Function('v_2')(Symbol('q', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('U', commutative=True))))"]]}, {"prompt": "Given J{(F_{H})} = \\frac{d}{d F_{H}} e^{F_{H}}, then derive - J{(F_{H})} = - e^{F_{H}}, then obtain \\frac{d}{d F_{H}} \\int - \\frac{d}{d F_{H}} e^{F_{H}} dF_{H} = \\frac{d}{d F_{H}} \\int - e^{F_{H}} dF_{H}", "derivation": "J{(F_{H})} = \\frac{d}{d F_{H}} e^{F_{H}} and - J{(F_{H})} = - \\frac{d}{d F_{H}} e^{F_{H}} and - J{(F_{H})} = - e^{F_{H}} and - \\frac{d}{d F_{H}} e^{F_{H}} = - e^{F_{H}} and \\int - \\frac{d}{d F_{H}} e^{F_{H}} dF_{H} = \\int - e^{F_{H}} dF_{H} and \\frac{d}{d F_{H}} \\int - \\frac{d}{d F_{H}} e^{F_{H}} dF_{H} = \\frac{d}{d F_{H}} \\int - e^{F_{H}} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('F_H', commutative=True)), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('J')(Symbol('F_H', commutative=True))), Mul(Integer(-1), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(-1), Function('J')(Symbol('F_H', commutative=True))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["differentiate", 5, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} = \\log{(e^{\\mathbf{g}})}, then derive \\frac{d}{d \\mathbf{g}} \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} = 1, then obtain \\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})} = 1", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} = \\log{(e^{\\mathbf{g}})} and \\frac{d}{d \\mathbf{g}} \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})} and \\frac{d}{d \\mathbf{g}} \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} (\\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})})^{\\mathbf{g}} = \\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})} (\\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})})^{\\mathbf{g}} and \\frac{d}{d \\mathbf{g}} \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} = 1 and \\frac{d}{d \\mathbf{g}} \\log{(e^{\\mathbf{g}})} = 1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), log(exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["times", 2, "Pow(Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Pow(Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Pow(Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\delta{(\\mathbf{P})} = \\sin{(\\mathbf{P})} and \\hat{H}_l{(\\mathbf{P})} = \\sin^{\\mathbf{P}}{(\\mathbf{P})}, then obtain (\\delta^{\\mathbf{P}}{(\\mathbf{P})} + \\sin{(\\mathbf{P})})^{\\mathbf{P}} = (\\hat{H}_l{(\\mathbf{P})} + \\sin{(\\mathbf{P})})^{\\mathbf{P}}", "derivation": "\\delta{(\\mathbf{P})} = \\sin{(\\mathbf{P})} and \\delta^{\\mathbf{P}}{(\\mathbf{P})} = \\sin^{\\mathbf{P}}{(\\mathbf{P})} and \\delta^{\\mathbf{P}}{(\\mathbf{P})} + \\sin{(\\mathbf{P})} = \\sin{(\\mathbf{P})} + \\sin^{\\mathbf{P}}{(\\mathbf{P})} and \\hat{H}_l{(\\mathbf{P})} = \\sin^{\\mathbf{P}}{(\\mathbf{P})} and \\delta^{\\mathbf{P}}{(\\mathbf{P})} + \\sin{(\\mathbf{P})} = \\hat{H}_l{(\\mathbf{P})} + \\sin{(\\mathbf{P})} and (\\delta^{\\mathbf{P}}{(\\mathbf{P})} + \\sin{(\\mathbf{P})})^{\\mathbf{P}} = (\\hat{H}_l{(\\mathbf{P})} + \\sin{(\\mathbf{P})})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 2, "sin(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))), Add(sin(Symbol('\\\\mathbf{P}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given L{(v,l)} = l^{v}, then derive \\frac{\\partial^{2}}{\\partial v\\partial l} L{(v,l)} = \\frac{l^{v} (v \\log{(l)} + 1)}{l}, then obtain \\frac{\\partial^{2}}{\\partial v\\partial l} L{(v,l)} = \\frac{(v \\log{(l)} + 1) L{(v,l)}}{l}", "derivation": "L{(v,l)} = l^{v} and \\frac{\\partial}{\\partial l} L{(v,l)} = \\frac{\\partial}{\\partial l} l^{v} and \\frac{\\partial^{2}}{\\partial v\\partial l} L{(v,l)} = \\frac{\\partial^{2}}{\\partial v\\partial l} l^{v} and \\frac{\\partial^{2}}{\\partial v\\partial l} L{(v,l)} = \\frac{l^{v} (v \\log{(l)} + 1)}{l} and \\frac{\\partial^{2}}{\\partial v\\partial l} L{(v,l)} = \\frac{(v \\log{(l)} + 1) L{(v,l)}}{l}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Symbol('l', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Pow(Symbol('l', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Symbol('v', commutative=True)), Add(Mul(Symbol('v', commutative=True), log(Symbol('l', commutative=True))), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Symbol('v', commutative=True), log(Symbol('l', commutative=True))), Integer(1)), Function('L')(Symbol('v', commutative=True), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\psi{(E_{n},z^{*},\\mu)} = \\frac{E_{n} - z^{*}}{\\mu}, then derive \\frac{\\partial}{\\partial \\mu} \\psi{(E_{n},z^{*},\\mu)} = - \\frac{E_{n} - z^{*}}{\\mu^{2}}, then obtain \\frac{\\partial}{\\partial \\mu} \\frac{E_{n} - z^{*}}{\\mu} = - \\frac{E_{n} - z^{*}}{\\mu^{2}}", "derivation": "\\psi{(E_{n},z^{*},\\mu)} = \\frac{E_{n} - z^{*}}{\\mu} and \\frac{\\partial}{\\partial \\mu} \\psi{(E_{n},z^{*},\\mu)} = \\frac{\\partial}{\\partial \\mu} \\frac{E_{n} - z^{*}}{\\mu} and \\frac{\\partial}{\\partial \\mu} \\psi{(E_{n},z^{*},\\mu)} = - \\frac{E_{n} - z^{*}}{\\mu^{2}} and \\frac{\\partial}{\\partial \\mu} \\psi{(E_{n},z^{*},\\mu)} = \\frac{- E_{n} + z^{*}}{\\mu^{2}} and - \\frac{E_{n} - z^{*}}{\\mu^{2}} = \\frac{- E_{n} + z^{*}}{\\mu^{2}} and \\frac{\\partial}{\\partial \\mu} \\frac{E_{n} - z^{*}}{\\mu} = \\frac{- E_{n} + z^{*}}{\\mu^{2}} and \\frac{\\partial}{\\partial \\mu} \\frac{E_{n} - z^{*}}{\\mu} = - \\frac{E_{n} - z^{*}}{\\mu^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('E_n', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('E_n', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('E_n', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\psi')(Symbol('E_n', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\delta,k)} = \\delta + k, then obtain e^{- \\int \\operatorname{z^{*}}{(\\delta,k)} d\\delta + \\frac{\\operatorname{z^{*}}{(\\delta,k)}}{\\delta}} = e^{- \\int \\operatorname{z^{*}}{(\\delta,k)} d\\delta + \\frac{\\delta + k}{\\delta}}", "derivation": "\\operatorname{z^{*}}{(\\delta,k)} = \\delta + k and \\int \\operatorname{z^{*}}{(\\delta,k)} d\\delta = \\int (\\delta + k) d\\delta and \\frac{\\operatorname{z^{*}}{(\\delta,k)}}{\\delta} = \\frac{\\delta + k}{\\delta} and - \\int (\\delta + k) d\\delta + \\frac{\\operatorname{z^{*}}{(\\delta,k)}}{\\delta} = - \\int (\\delta + k) d\\delta + \\frac{\\delta + k}{\\delta} and e^{- \\int (\\delta + k) d\\delta + \\frac{\\operatorname{z^{*}}{(\\delta,k)}}{\\delta}} = e^{- \\int (\\delta + k) d\\delta + \\frac{\\delta + k}{\\delta}} and e^{- \\int \\operatorname{z^{*}}{(\\delta,k)} d\\delta + \\frac{\\operatorname{z^{*}}{(\\delta,k)}}{\\delta}} = e^{- \\int \\operatorname{z^{*}}{(\\delta,k)} d\\delta + \\frac{\\delta + k}{\\delta}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))))"], [["minus", 3, "Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)))))"], [["exp", 4], "Equality(exp(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))))), exp(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(exp(Add(Mul(Integer(-1), Integral(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))))), exp(Add(Mul(Integer(-1), Integral(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(S,A,F_{N})} = A + \\frac{S}{F_{N}}, then obtain (A + \\frac{S}{F_{N}}) (\\frac{\\partial}{\\partial S} \\operatorname{F_{H}}{(S,A,F_{N})} - \\frac{1}{F_{N}}) = (A + \\frac{S}{F_{N}}) (\\frac{\\partial}{\\partial S} (A + \\frac{S}{F_{N}}) - \\frac{1}{F_{N}})", "derivation": "\\operatorname{F_{H}}{(S,A,F_{N})} = A + \\frac{S}{F_{N}} and \\frac{\\partial}{\\partial S} \\operatorname{F_{H}}{(S,A,F_{N})} = \\frac{\\partial}{\\partial S} (A + \\frac{S}{F_{N}}) and \\frac{\\partial}{\\partial S} \\operatorname{F_{H}}{(S,A,F_{N})} - \\frac{1}{F_{N}} = \\frac{\\partial}{\\partial S} (A + \\frac{S}{F_{N}}) - \\frac{1}{F_{N}} and (A + \\frac{S}{F_{N}}) (\\frac{\\partial}{\\partial S} \\operatorname{F_{H}}{(S,A,F_{N})} - \\frac{1}{F_{N}}) = (A + \\frac{S}{F_{N}}) (\\frac{\\partial}{\\partial S} (A + \\frac{S}{F_{N}}) - \\frac{1}{F_{N}})", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('S', commutative=True), Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('S', commutative=True), Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Symbol('F_N', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('F_H')(Symbol('S', commutative=True), Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)))), Add(Derivative(Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)))))"], [["times", 3, "Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True)))"], "Equality(Mul(Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))), Add(Derivative(Function('F_H')(Symbol('S', commutative=True), Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1))))), Mul(Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))), Add(Derivative(Add(Symbol('A', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given x{(\\hat{H}_{\\lambda},\\omega)} = \\sin^{\\omega}{(\\hat{H}_{\\lambda})}, then obtain - (\\sin^{\\omega}{(\\hat{H}_{\\lambda})})^{\\omega} + \\log{(x{(\\hat{H}_{\\lambda},\\omega)})} = - (\\sin^{\\omega}{(\\hat{H}_{\\lambda})})^{\\omega} + \\log{(\\sin^{\\omega}{(\\hat{H}_{\\lambda})})}", "derivation": "x{(\\hat{H}_{\\lambda},\\omega)} = \\sin^{\\omega}{(\\hat{H}_{\\lambda})} and x^{\\omega}{(\\hat{H}_{\\lambda},\\omega)} = (\\sin^{\\omega}{(\\hat{H}_{\\lambda})})^{\\omega} and \\log{(x{(\\hat{H}_{\\lambda},\\omega)})} = \\log{(\\sin^{\\omega}{(\\hat{H}_{\\lambda})})} and - x^{\\omega}{(\\hat{H}_{\\lambda},\\omega)} + \\log{(x{(\\hat{H}_{\\lambda},\\omega)})} = - x^{\\omega}{(\\hat{H}_{\\lambda},\\omega)} + \\log{(\\sin^{\\omega}{(\\hat{H}_{\\lambda})})} and - (\\sin^{\\omega}{(\\hat{H}_{\\lambda})})^{\\omega} + \\log{(x{(\\hat{H}_{\\lambda},\\omega)})} = - (\\sin^{\\omega}{(\\hat{H}_{\\lambda})})^{\\omega} + \\log{(\\sin^{\\omega}{(\\hat{H}_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["log", 1], "Equality(log(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True))), log(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Pow(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), log(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), log(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), log(Function('x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), log(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\theta_2,t_{2})} = \\log{(- \\theta_2 + t_{2})}, then obtain (- \\operatorname{v_{1}}^{\\theta_2}{(\\theta_2,t_{2})} + \\log{(- \\theta_2 + t_{2})}) \\operatorname{v_{1}}{(\\theta_2,t_{2})} = (- \\operatorname{v_{1}}^{\\theta_2}{(\\theta_2,t_{2})} + \\log{(- \\theta_2 + t_{2})}) \\log{(- \\theta_2 + t_{2})}", "derivation": "\\operatorname{v_{1}}{(\\theta_2,t_{2})} = \\log{(- \\theta_2 + t_{2})} and \\operatorname{v_{1}}^{\\theta_2}{(\\theta_2,t_{2})} = \\log{(- \\theta_2 + t_{2})}^{\\theta_2} and (\\log{(- \\theta_2 + t_{2})} - \\log{(- \\theta_2 + t_{2})}^{\\theta_2}) \\operatorname{v_{1}}{(\\theta_2,t_{2})} = (\\log{(- \\theta_2 + t_{2})} - \\log{(- \\theta_2 + t_{2})}^{\\theta_2}) \\log{(- \\theta_2 + t_{2})} and (- \\operatorname{v_{1}}^{\\theta_2}{(\\theta_2,t_{2})} + \\log{(- \\theta_2 + t_{2})}) \\operatorname{v_{1}}{(\\theta_2,t_{2})} = (- \\operatorname{v_{1}}^{\\theta_2}{(\\theta_2,t_{2})} + \\log{(- \\theta_2 + t_{2})}) \\log{(- \\theta_2 + t_{2})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Add(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Mul(Integer(-1), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"], "Equality(Mul(Add(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Mul(Integer(-1), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))), Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))), Mul(Add(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Mul(Integer(-1), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True)))), Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))), Mul(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} = A_{2} - \\eta^{\\prime}, then obtain A_{2} - \\eta^{\\prime} + 2 \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2} = A_{2} - \\eta^{\\prime} + \\int (A_{2} - \\eta^{\\prime}) dA_{2} + \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2}", "derivation": "\\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} = A_{2} - \\eta^{\\prime} and \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2} = \\int (A_{2} - \\eta^{\\prime}) dA_{2} and 2 \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2} = \\int (A_{2} - \\eta^{\\prime}) dA_{2} + \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2} and A_{2} - \\eta^{\\prime} + 2 \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2} = A_{2} - \\eta^{\\prime} + \\int (A_{2} - \\eta^{\\prime}) dA_{2} + \\int \\operatorname{x^{{\\}'}}{(A_{2},\\eta^{\\prime})} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('A_2', commutative=True))))"], [["add", 2, "Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["add", 3, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True))))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Integral(Function('x^\\\\prime')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J}_M,y)} = y^{\\mathbf{J}_M} and \\operatorname{z^{*}}{(p,l)} = l^{p}, then obtain - 2 \\mathbf{J}_M + y^{\\mathbf{J}_M} + \\iint \\operatorname{z^{*}}{(p,l)} dp dl = - 2 \\mathbf{J}_M + y^{\\mathbf{J}_M} + \\iint l^{p} dp dl", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J}_M,y)} = y^{\\mathbf{J}_M} and - 2 \\mathbf{J}_M + \\operatorname{F_{H}}{(\\mathbf{J}_M,y)} = - 2 \\mathbf{J}_M + y^{\\mathbf{J}_M} and \\operatorname{z^{*}}{(p,l)} = l^{p} and \\int \\operatorname{z^{*}}{(p,l)} dp = \\int l^{p} dp and \\iint \\operatorname{z^{*}}{(p,l)} dp dl = \\iint l^{p} dp dl and - 2 \\mathbf{J}_M + \\operatorname{F_{H}}{(\\mathbf{J}_M,y)} + \\iint \\operatorname{z^{*}}{(p,l)} dp dl = - 2 \\mathbf{J}_M + \\operatorname{F_{H}}{(\\mathbf{J}_M,y)} + \\iint l^{p} dp dl and - 2 \\mathbf{J}_M + y^{\\mathbf{J}_M} + \\iint \\operatorname{z^{*}}{(p,l)} dp dl = - 2 \\mathbf{J}_M + y^{\\mathbf{J}_M} + \\iint l^{p} dp dl", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["get_premise", "Equality(Function('z^*')(Symbol('p', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('p', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('p', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["add", 5, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y', commutative=True)), Integral(Function('z^*')(Symbol('p', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Function('z^*')(Symbol('p', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})} = \\frac{M_{E}}{\\hat{\\mathbf{x}}}, then obtain (\\frac{\\partial}{\\partial M_{E}} - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})})^{2} = \\frac{\\partial}{\\partial M_{E}} - \\frac{M_{E}}{\\hat{\\mathbf{x}}} \\frac{\\partial}{\\partial M_{E}} - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})}", "derivation": "\\theta_{2}{(M_{E},\\hat{\\mathbf{x}})} = \\frac{M_{E}}{\\hat{\\mathbf{x}}} and - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})} = - \\frac{M_{E}}{\\hat{\\mathbf{x}}} and \\frac{\\partial}{\\partial M_{E}} - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial M_{E}} - \\frac{M_{E}}{\\hat{\\mathbf{x}}} and (\\frac{\\partial}{\\partial M_{E}} - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})})^{2} = \\frac{\\partial}{\\partial M_{E}} - \\frac{M_{E}}{\\hat{\\mathbf{x}}} \\frac{\\partial}{\\partial M_{E}} - \\theta_{2}{(M_{E},\\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('M_E', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Integer(-1), Symbol('M_E', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{x}{(i,z^{*},m)} = \\frac{m - z^{*}}{i} and \\operatorname{f_{E}}{(\\pi,\\mu)} = \\frac{\\pi}{\\mu}, then obtain - m + z^{*} + \\operatorname{f_{E}}{(\\pi,\\mu)} - \\int \\sigma_{x}^{i}{(i,z^{*},m)} dm = - m + z^{*} - \\int \\sigma_{x}^{i}{(i,z^{*},m)} dm + \\frac{\\pi}{\\mu}", "derivation": "\\sigma_{x}{(i,z^{*},m)} = \\frac{m - z^{*}}{i} and \\sigma_{x}^{i}{(i,z^{*},m)} = (\\frac{m - z^{*}}{i})^{i} and \\int \\sigma_{x}^{i}{(i,z^{*},m)} dm = \\int (\\frac{m - z^{*}}{i})^{i} dm and \\operatorname{f_{E}}{(\\pi,\\mu)} = \\frac{\\pi}{\\mu} and - m + z^{*} + \\operatorname{f_{E}}{(\\pi,\\mu)} - \\int (\\frac{m - z^{*}}{i})^{i} dm = - m + z^{*} - \\int (\\frac{m - z^{*}}{i})^{i} dm + \\frac{\\pi}{\\mu} and - m + z^{*} + \\operatorname{f_{E}}{(\\pi,\\mu)} - \\int \\sigma_{x}^{i}{(i,z^{*},m)} dm = - m + z^{*} - \\int \\sigma_{x}^{i}{(i,z^{*},m)} dm + \\frac{\\pi}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('i', commutative=True), Symbol('z^*', commutative=True), Symbol('m', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('i', commutative=True), Symbol('z^*', commutative=True), Symbol('m', commutative=True)), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('i', commutative=True)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Function('\\\\sigma_x')(Symbol('i', commutative=True), Symbol('z^*', commutative=True), Symbol('m', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True))))"], ["get_premise", "Equality(Function('f_E')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)))"], [["minus", 4, "Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(Pow(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('z^*', commutative=True), Function('f_E')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('z^*', commutative=True), Function('f_E')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\sigma_x')(Symbol('i', commutative=True), Symbol('z^*', commutative=True), Symbol('m', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Integral(Pow(Function('\\\\sigma_x')(Symbol('i', commutative=True), Symbol('z^*', commutative=True), Symbol('m', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain (\\mathbf{D}{(\\mathbf{P})} - \\log{(\\mathbf{P})})^{\\mathbf{P}} - 1 = 0^{\\mathbf{P}} - 1", "derivation": "\\mathbf{D}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\mathbf{D}{(\\mathbf{P})} - \\log{(\\mathbf{P})} = 0 and (\\mathbf{D}{(\\mathbf{P})} - \\log{(\\mathbf{P})})^{\\mathbf{P}} = 0^{\\mathbf{P}} and (\\mathbf{D}{(\\mathbf{P})} - \\log{(\\mathbf{P})})^{\\mathbf{P}} - 1 = 0^{\\mathbf{P}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 3, 1], "Equality(Add(Pow(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Add(Pow(Integer(0), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(k,\\hbar)} = e^{k^{\\hbar}}, then obtain (\\int \\frac{\\partial}{\\partial \\hbar} \\operatorname{F_{c}}{(k,\\hbar)} d\\hbar) \\int \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} d\\hbar = (\\int \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} d\\hbar)^{2}", "derivation": "\\operatorname{F_{c}}{(k,\\hbar)} = e^{k^{\\hbar}} and \\frac{\\partial}{\\partial \\hbar} \\operatorname{F_{c}}{(k,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} and \\int \\frac{\\partial}{\\partial \\hbar} \\operatorname{F_{c}}{(k,\\hbar)} d\\hbar = \\int \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} d\\hbar and (\\int \\frac{\\partial}{\\partial \\hbar} \\operatorname{F_{c}}{(k,\\hbar)} d\\hbar) \\int \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} d\\hbar = (\\int \\frac{\\partial}{\\partial \\hbar} e^{k^{\\hbar}} d\\hbar)^{2}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True)), exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["times", 3, "Integral(Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Integral(Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))), Pow(Integral(Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\sigma_{x}{(\\pi)} = e^{\\pi}, then obtain (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} (e^{\\pi})^{\\pi} \\int \\sigma_{x}{(\\pi)} d\\pi = (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} (e^{\\pi})^{\\pi} \\int e^{\\pi} d\\pi", "derivation": "\\sigma_{x}{(\\pi)} = e^{\\pi} and \\int \\sigma_{x}{(\\pi)} d\\pi = \\int e^{\\pi} d\\pi and \\sigma_{x}^{\\pi}{(\\pi)} = (e^{\\pi})^{\\pi} and (e^{\\pi})^{- \\pi} \\int \\sigma_{x}{(\\pi)} d\\pi = (e^{\\pi})^{- \\pi} \\int e^{\\pi} d\\pi and \\frac{d}{d \\pi} \\sigma_{x}^{\\pi}{(\\pi)} = \\frac{d}{d \\pi} (e^{\\pi})^{\\pi} and (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} \\sigma_{x}^{\\pi}{(\\pi)} \\int \\sigma_{x}{(\\pi)} d\\pi = (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} \\sigma_{x}^{\\pi}{(\\pi)} \\int e^{\\pi} d\\pi and (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} (e^{\\pi})^{\\pi} \\int \\sigma_{x}{(\\pi)} d\\pi = (e^{\\pi})^{- \\pi} \\frac{d}{d \\pi} (e^{\\pi})^{\\pi} \\int e^{\\pi} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["divide", 2, "Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Integral(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Derivative(Pow(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Derivative(Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(Function('\\\\sigma_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Derivative(Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A_{z})} = e^{A_{z}}, then obtain - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}} + \\operatorname{E_{n}}{(A_{z})} - 1 = - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}}", "derivation": "\\operatorname{E_{n}}{(A_{z})} = e^{A_{z}} and 0 = - \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}} and 0^{A_{z}} = (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}} and - 0^{A_{z}} + \\operatorname{E_{n}}{(A_{z})} = - 0^{A_{z}} + e^{A_{z}} and - 0^{A_{z}} - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}} + \\operatorname{E_{n}}{(A_{z})} = - 0^{A_{z}} - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}} + e^{A_{z}} and - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}} + \\operatorname{E_{n}}{(A_{z})} - 1 = - (- \\operatorname{E_{n}}{(A_{z})} + e^{A_{z}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["minus", 1, "Function('E_n')(Symbol('A_z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["minus", 1, "Pow(Integer(0), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('A_z', commutative=True))), Function('E_n')(Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Function('E_n')(Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Function('E_n')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('A_z', commutative=True))), exp(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given H{(f,\\mu)} = f^{\\mu}, then derive - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = - f^{\\mu} + (\\frac{\\mu f^{\\mu}}{f})^{\\mu} - H{(f,\\mu)}, then obtain - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = - f^{\\mu} + (\\frac{\\mu H{(f,\\mu)}}{f})^{\\mu} - H{(f,\\mu)}", "derivation": "H{(f,\\mu)} = f^{\\mu} and \\frac{H{(f,\\mu)}}{f} = \\frac{f^{\\mu}}{f} and \\frac{\\partial}{\\partial f} H{(f,\\mu)} = \\frac{\\partial}{\\partial f} f^{\\mu} and (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = (\\frac{\\partial}{\\partial f} f^{\\mu})^{\\mu} and - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} f^{\\mu})^{\\mu} and - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = - f^{\\mu} + (\\frac{\\mu f^{\\mu}}{f})^{\\mu} - H{(f,\\mu)} and - f^{\\mu} - H{(f,\\mu)} + (\\frac{\\partial}{\\partial f} H{(f,\\mu)})^{\\mu} = - f^{\\mu} + (\\frac{\\mu H{(f,\\mu)}}{f})^{\\mu} - H{(f,\\mu)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["minus", 4, "Add(Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Derivative(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Derivative(Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Derivative(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Derivative(Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given G{(x)} = \\int e^{x} dx, then derive G{(x)} = F_{H} + e^{x}, then derive F_{H} = F_{c}, then obtain F_{H}^{F_{c}} = F_{c}^{F_{c}}", "derivation": "G{(x)} = \\int e^{x} dx and G{(x)} - e^{x} = - e^{x} + \\int e^{x} dx and G{(x)} = F_{H} + e^{x} and F_{H} = - e^{x} + \\int e^{x} dx and F_{H} = F_{c} and F_{H}^{F_{c}} = F_{c}^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('x', commutative=True)), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["minus", 1, "exp(Symbol('x', commutative=True))"], "Equality(Add(Function('G')(Symbol('x', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('x', commutative=True))), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('G')(Symbol('x', commutative=True)), Add(Symbol('F_H', commutative=True), exp(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), exp(Symbol('x', commutative=True))), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Symbol('F_H', commutative=True), Symbol('F_c', commutative=True))"], [["power", 5, "Symbol('F_c', commutative=True)"], "Equality(Pow(Symbol('F_H', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\varphi)} = \\log{(\\varphi)}, then obtain (\\int \\hat{H}^{2}{(\\varphi)} d\\varphi)^{\\varphi} = (\\int \\hat{H}{(\\varphi)} \\log{(\\varphi)} d\\varphi)^{\\varphi}", "derivation": "\\hat{H}{(\\varphi)} = \\log{(\\varphi)} and \\hat{H}^{2}{(\\varphi)} = \\hat{H}{(\\varphi)} \\log{(\\varphi)} and \\int \\hat{H}^{2}{(\\varphi)} d\\varphi = \\int \\hat{H}{(\\varphi)} \\log{(\\varphi)} d\\varphi and (\\int \\hat{H}^{2}{(\\varphi)} d\\varphi)^{\\varphi} = (\\int \\hat{H}{(\\varphi)} \\log{(\\varphi)} d\\varphi)^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Mul(Function('\\\\hat{H}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{D},A_{z})} = A_{z} + \\log{(\\mathbf{D})} and u{(\\mathbf{D},A_{z})} = \\frac{\\theta_{1}{(\\mathbf{D},A_{z})}}{\\log{(\\mathbf{D})}}, then obtain u^{\\mathbf{D}}{(\\mathbf{D},A_{z})} = (\\frac{A_{z} + \\log{(\\mathbf{D})}}{\\log{(\\mathbf{D})}})^{\\mathbf{D}}", "derivation": "\\theta_{1}{(\\mathbf{D},A_{z})} = A_{z} + \\log{(\\mathbf{D})} and \\frac{\\theta_{1}{(\\mathbf{D},A_{z})}}{\\log{(\\mathbf{D})}} = \\frac{A_{z} + \\log{(\\mathbf{D})}}{\\log{(\\mathbf{D})}} and (\\frac{\\theta_{1}{(\\mathbf{D},A_{z})}}{\\log{(\\mathbf{D})}})^{\\mathbf{D}} = (\\frac{A_{z} + \\log{(\\mathbf{D})}}{\\log{(\\mathbf{D})}})^{\\mathbf{D}} and u{(\\mathbf{D},A_{z})} = \\frac{\\theta_{1}{(\\mathbf{D},A_{z})}}{\\log{(\\mathbf{D})}} and u^{\\mathbf{D}}{(\\mathbf{D},A_{z})} = (\\frac{A_{z} + \\log{(\\mathbf{D})}}{\\log{(\\mathbf{D})}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 1, "log(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Mul(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Mul(Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('u')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Pow(log(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given v{(E_{x},\\nabla,y)} = - E_{x} + \\frac{\\nabla}{y}, then obtain ((\\frac{\\partial}{\\partial y} v{(E_{x},\\nabla,y)})^{y})^{\\nabla} = ((\\frac{\\partial}{\\partial y} (- E_{x} + \\frac{\\nabla}{y}))^{y})^{\\nabla}", "derivation": "v{(E_{x},\\nabla,y)} = - E_{x} + \\frac{\\nabla}{y} and \\frac{\\partial}{\\partial y} v{(E_{x},\\nabla,y)} = \\frac{\\partial}{\\partial y} (- E_{x} + \\frac{\\nabla}{y}) and (\\frac{\\partial}{\\partial y} v{(E_{x},\\nabla,y)})^{y} = (\\frac{\\partial}{\\partial y} (- E_{x} + \\frac{\\nabla}{y}))^{y} and ((\\frac{\\partial}{\\partial y} v{(E_{x},\\nabla,y)})^{y})^{\\nabla} = ((\\frac{\\partial}{\\partial y} (- E_{x} + \\frac{\\nabla}{y}))^{y})^{\\nabla}", "srepr_derivation": [["get_premise", "Equality(Function('v')(Symbol('E_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('E_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Derivative(Function('v')(Symbol('E_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)))"], [["power", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('v')(Symbol('E_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Pow(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(A,C_{d})} = - C_{d} + \\sin{(A)}, then obtain \\int ((\\int \\mathbf{r}{(A,C_{d})} dA)^{C_{d}} + 1) dC_{d} = \\int ((\\int (- C_{d} + \\sin{(A)}) dA)^{C_{d}} + 1) dC_{d}", "derivation": "\\mathbf{r}{(A,C_{d})} = - C_{d} + \\sin{(A)} and \\int \\mathbf{r}{(A,C_{d})} dA = \\int (- C_{d} + \\sin{(A)}) dA and (\\int \\mathbf{r}{(A,C_{d})} dA)^{C_{d}} = (\\int (- C_{d} + \\sin{(A)}) dA)^{C_{d}} and (\\int \\mathbf{r}{(A,C_{d})} dA)^{C_{d}} + 1 = (\\int (- C_{d} + \\sin{(A)}) dA)^{C_{d}} + 1 and \\int ((\\int \\mathbf{r}{(A,C_{d})} dA)^{C_{d}} + 1) dC_{d} = \\int ((\\int (- C_{d} + \\sin{(A)}) dA)^{C_{d}} + 1) dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)), Integer(1)), Add(Pow(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)), Integer(1)))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)), Integer(1)), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Pow(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Symbol('C_d', commutative=True)), Integer(1)), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(h)} = \\log{(h)}, then obtain (1 - \\log{(h)})^{h} = (- \\log{(h)} - 1 + \\frac{2 \\log{(h)}}{\\mathbf{s}{(h)}})^{h}", "derivation": "\\mathbf{s}{(h)} = \\log{(h)} and h \\mathbf{s}{(h)} = h \\log{(h)} and 1 = \\frac{\\log{(h)}}{\\mathbf{s}{(h)}} and 1 - \\log{(h)} = - \\log{(h)} + \\frac{\\log{(h)}}{\\mathbf{s}{(h)}} and (1 - \\log{(h)})^{h} = (- \\log{(h)} + \\frac{\\log{(h)}}{\\mathbf{s}{(h)}})^{h} and (- \\log{(h)} + \\frac{\\log{(h)}}{\\mathbf{s}{(h)}})^{h} = (- \\log{(h)} - 1 + \\frac{2 \\log{(h)}}{\\mathbf{s}{(h)}})^{h} and (1 - \\log{(h)})^{h} = (- \\log{(h)} - 1 + \\frac{2 \\log{(h)}}{\\mathbf{s}{(h)}})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\mathbf{s}')(Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))))"], [["divide", 2, "Mul(Symbol('h', commutative=True), Function('\\\\mathbf{s}')(Symbol('h', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True))))"], [["minus", 3, "log(Symbol('h', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), log(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('h', commutative=True))), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True)))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('h', commutative=True))), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('h', commutative=True))), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('h', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('h', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('\\\\mathbf{s}')(Symbol('h', commutative=True)), Integer(-1)), log(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} = (E_{x} \\eta^{\\prime})^{\\pi}, then obtain \\int \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int \\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} d\\pi dE_{x} = \\int \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int (E_{x} \\eta^{\\prime})^{\\pi} d\\pi dE_{x}", "derivation": "\\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} = (E_{x} \\eta^{\\prime})^{\\pi} and \\int \\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} d\\pi = \\int (E_{x} \\eta^{\\prime})^{\\pi} d\\pi and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int \\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} d\\pi = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int (E_{x} \\eta^{\\prime})^{\\pi} d\\pi and \\int \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int \\mathbf{P}{(E_{x},\\pi,\\eta^{\\prime})} d\\pi dE_{x} = \\int \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int (E_{x} \\eta^{\\prime})^{\\pi} d\\pi dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('E_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Pow(Mul(Symbol('E_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Symbol('E_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(Integral(Pow(Mul(Symbol('E_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(z)} = \\sin{(z)}, then derive \\int \\operatorname{E_{n}}{(z)} dz = f_{\\mathbf{p}} - \\cos{(z)}, then obtain F_{N} - \\cos{(z)} + 1 = \\int \\operatorname{E_{n}}{(z)} dz + 1", "derivation": "\\operatorname{E_{n}}{(z)} = \\sin{(z)} and \\int \\operatorname{E_{n}}{(z)} dz = \\int \\sin{(z)} dz and \\int \\operatorname{E_{n}}{(z)} dz = f_{\\mathbf{p}} - \\cos{(z)} and \\int \\sin{(z)} dz = f_{\\mathbf{p}} - \\cos{(z)} and \\int \\sin{(z)} dz + 1 = f_{\\mathbf{p}} - \\cos{(z)} + 1 and \\int \\sin{(z)} dz + 1 = \\int \\operatorname{E_{n}}{(z)} dz + 1 and F_{N} - \\cos{(z)} + 1 = \\int \\operatorname{E_{n}}{(z)} dz + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)), Add(Integral(Function('E_n')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))), Integer(1)), Add(Integral(Function('E_n')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)))"]]}, {"prompt": "Given T{(C_{1},g^{\\prime}_{\\varepsilon},f)} = C_{1} + f + g^{\\prime}_{\\varepsilon}, then obtain C_{1} + f + g^{\\prime}_{\\varepsilon} = 2 C_{1} + 2 f + 2 g^{\\prime}_{\\varepsilon} - T{(C_{1},g^{\\prime}_{\\varepsilon},f)}", "derivation": "T{(C_{1},g^{\\prime}_{\\varepsilon},f)} = C_{1} + f + g^{\\prime}_{\\varepsilon} and f + T{(C_{1},g^{\\prime}_{\\varepsilon},f)} = C_{1} + 2 f + g^{\\prime}_{\\varepsilon} and 0 = C_{1} + f + g^{\\prime}_{\\varepsilon} - T{(C_{1},g^{\\prime}_{\\varepsilon},f)} and C_{1} + f + g^{\\prime}_{\\varepsilon} = 2 C_{1} + 2 f + 2 g^{\\prime}_{\\varepsilon} - T{(C_{1},g^{\\prime}_{\\varepsilon},f)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('f', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('T')(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(2), Symbol('f', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 2, "Add(Symbol('f', commutative=True), Function('T')(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)))"], "Equality(Integer(0), Add(Symbol('C_1', commutative=True), Symbol('f', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)))))"], [["add", 3, "Add(Symbol('C_1', commutative=True), Symbol('f', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Symbol('f', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given E{(B,g)} = g^{B}, then obtain \\frac{z ((\\frac{E{(B,g)}}{\\mathbf{F}})^{g} + E{(B,g)})}{\\sin{(\\mathbf{F})}} = \\frac{z ((\\frac{g^{B}}{\\mathbf{F}})^{g} + E{(B,g)})}{\\sin{(\\mathbf{F})}}", "derivation": "E{(B,g)} = g^{B} and \\frac{E{(B,g)}}{\\mathbf{F}} = \\frac{g^{B}}{\\mathbf{F}} and (\\frac{E{(B,g)}}{\\mathbf{F}})^{g} = (\\frac{g^{B}}{\\mathbf{F}})^{g} and (\\frac{E{(B,g)}}{\\mathbf{F}})^{g} + E{(B,g)} = (\\frac{g^{B}}{\\mathbf{F}})^{g} + E{(B,g)} and \\frac{z ((\\frac{E{(B,g)}}{\\mathbf{F}})^{g} + E{(B,g)})}{\\sin{(\\mathbf{F})}} = \\frac{z ((\\frac{g^{B}}{\\mathbf{F}})^{g} + E{(B,g)})}{\\sin{(\\mathbf{F})}}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('B', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)))"], [["add", 3, "Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('z', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Symbol('z', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Symbol('z', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('g', commutative=True))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(f)} = \\sin{(f)}, then obtain 2 \\frac{d^{2}}{d f^{2}} \\varphi{(f)} \\frac{d^{3}}{d f^{3}} \\varphi{(f)} = - \\sin{(f)} \\frac{d^{3}}{d f^{3}} \\varphi{(f)} - \\cos{(f)} \\frac{d^{2}}{d f^{2}} \\varphi{(f)}", "derivation": "\\varphi{(f)} = \\sin{(f)} and \\frac{d}{d f} \\varphi{(f)} = \\frac{d}{d f} \\sin{(f)} and \\frac{d^{2}}{d f^{2}} \\varphi{(f)} = \\frac{d^{2}}{d f^{2}} \\sin{(f)} and (\\frac{d^{2}}{d f^{2}} \\varphi{(f)})^{2} = \\frac{d^{2}}{d f^{2}} \\varphi{(f)} \\frac{d^{2}}{d f^{2}} \\sin{(f)} and \\frac{d}{d f} (\\frac{d^{2}}{d f^{2}} \\varphi{(f)})^{2} = \\frac{d}{d f} \\frac{d^{2}}{d f^{2}} \\varphi{(f)} \\frac{d^{2}}{d f^{2}} \\sin{(f)} and 2 \\frac{d^{2}}{d f^{2}} \\varphi{(f)} \\frac{d^{3}}{d f^{3}} \\varphi{(f)} = - \\sin{(f)} \\frac{d^{3}}{d f^{3}} \\varphi{(f)} - \\cos{(f)} \\frac{d^{2}}{d f^{2}} \\varphi{(f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))))"], [["times", 3, "Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))"], "Equality(Pow(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Integer(2)), Mul(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Integer(2)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(2), Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(3)))), Add(Mul(Integer(-1), sin(Symbol('f', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(3)))), Mul(Integer(-1), cos(Symbol('f', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(v,\\mathbf{g})} = e^{\\mathbf{g} + v}, then derive \\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})} = e^{\\mathbf{g} + v}, then obtain (\\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})})^{v} = (\\frac{\\partial^{2}}{\\partial v^{2}} e^{\\mathbf{g} + v})^{v}", "derivation": "\\hat{\\mathbf{x}}{(v,\\mathbf{g})} = e^{\\mathbf{g} + v} and \\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})} = \\frac{\\partial}{\\partial v} e^{\\mathbf{g} + v} and \\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})} = e^{\\mathbf{g} + v} and \\frac{\\partial}{\\partial v} e^{\\mathbf{g} + v} = e^{\\mathbf{g} + v} and (\\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})})^{v} = (\\frac{\\partial}{\\partial v} e^{\\mathbf{g} + v})^{v} and (\\frac{\\partial}{\\partial v} \\hat{\\mathbf{x}}{(v,\\mathbf{g})})^{v} = (\\frac{\\partial^{2}}{\\partial v^{2}} e^{\\mathbf{g} + v})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)), Pow(Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)), Pow(Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(2))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(C_{1})} = \\sin{(C_{1})}, then obtain \\iint \\operatorname{V_{\\mathbf{B}}}{(C_{1})} dC_{1} dC_{1} - 1 = \\iint \\sin{(C_{1})} dC_{1} dC_{1} - 1", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(C_{1})} = \\sin{(C_{1})} and \\int \\operatorname{V_{\\mathbf{B}}}{(C_{1})} dC_{1} = \\int \\sin{(C_{1})} dC_{1} and \\iint \\operatorname{V_{\\mathbf{B}}}{(C_{1})} dC_{1} dC_{1} = \\iint \\sin{(C_{1})} dC_{1} dC_{1} and \\iint \\operatorname{V_{\\mathbf{B}}}{(C_{1})} dC_{1} dC_{1} - 1 = \\iint \\sin{(C_{1})} dC_{1} dC_{1} - 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integer(-1)), Add(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(M)} = \\sin{(M)}, then derive \\int \\operatorname{E_{\\lambda}}{(M)} dM = S - \\cos{(M)}, then obtain \\frac{\\partial}{\\partial M} \\frac{((S - \\cos{(M)})^{M})^{M}}{t_{1}} = \\frac{\\partial}{\\partial M} \\frac{((\\int \\sin{(M)} dM)^{M})^{M}}{t_{1}}", "derivation": "\\operatorname{E_{\\lambda}}{(M)} = \\sin{(M)} and \\int \\operatorname{E_{\\lambda}}{(M)} dM = \\int \\sin{(M)} dM and (\\int \\operatorname{E_{\\lambda}}{(M)} dM)^{M} = (\\int \\sin{(M)} dM)^{M} and \\int \\operatorname{E_{\\lambda}}{(M)} dM = S - \\cos{(M)} and ((\\int \\operatorname{E_{\\lambda}}{(M)} dM)^{M})^{M} = ((\\int \\sin{(M)} dM)^{M})^{M} and ((S - \\cos{(M)})^{M})^{M} = ((\\int \\sin{(M)} dM)^{M})^{M} and \\frac{((S - \\cos{(M)})^{M})^{M}}{t_{1}} = \\frac{((\\int \\sin{(M)} dM)^{M})^{M}}{t_{1}} and \\frac{\\partial}{\\partial M} \\frac{((S - \\cos{(M)})^{M})^{M}}{t_{1}} = \\frac{\\partial}{\\partial M} \\frac{((\\int \\sin{(M)} dM)^{M})^{M}}{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Integral(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Integral(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["divide", 6, "Symbol('t_1', commutative=True)"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], [["differentiate", 7, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(h,\\rho_b,U)} = (\\rho_b - h)^{U}, then obtain \\int (- U + \\rho_b - h + z{(h,\\rho_b,U)}) dU = \\int (- U + \\rho_b - h + (\\rho_b - h)^{U}) dU", "derivation": "z{(h,\\rho_b,U)} = (\\rho_b - h)^{U} and - U + z{(h,\\rho_b,U)} = - U + (\\rho_b - h)^{U} and - U + \\rho_b - h + z{(h,\\rho_b,U)} = - U + \\rho_b - h + (\\rho_b - h)^{U} and \\int (- U + \\rho_b - h + z{(h,\\rho_b,U)}) dU = \\int (- U + \\rho_b - h + (\\rho_b - h)^{U}) dU", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('U', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('U', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('z')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('U', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)), Function('z')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)), Function('z')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(I)} = \\log{(I)} and \\operatorname{E_{\\lambda}}{(I)} = \\log{(I)}, then obtain \\sin{(\\mathbf{F}{(I)})} \\int \\mathbf{F}{(I)} dI = \\sin{(\\operatorname{E_{\\lambda}}{(I)})} \\int \\mathbf{F}{(I)} dI", "derivation": "\\mathbf{F}{(I)} = \\log{(I)} and \\int \\mathbf{F}{(I)} dI = \\int \\log{(I)} dI and \\operatorname{E_{\\lambda}}{(I)} = \\log{(I)} and \\mathbf{F}{(I)} = \\operatorname{E_{\\lambda}}{(I)} and \\sin{(\\mathbf{F}{(I)})} = \\sin{(\\operatorname{E_{\\lambda}}{(I)})} and \\sin{(\\mathbf{F}{(I)})} \\int \\log{(I)} dI = \\sin{(\\operatorname{E_{\\lambda}}{(I)})} \\int \\log{(I)} dI and \\sin{(\\mathbf{F}{(I)})} \\int \\mathbf{F}{(I)} dI = \\sin{(\\operatorname{E_{\\lambda}}{(I)})} \\int \\mathbf{F}{(I)} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{F}')(Symbol('I', commutative=True)), Function('E_{\\\\lambda}')(Symbol('I', commutative=True)))"], [["sin", 4], "Equality(sin(Function('\\\\mathbf{F}')(Symbol('I', commutative=True))), sin(Function('E_{\\\\lambda}')(Symbol('I', commutative=True))))"], [["times", 5, "Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(sin(Function('\\\\mathbf{F}')(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(sin(Function('E_{\\\\lambda}')(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(sin(Function('\\\\mathbf{F}')(Symbol('I', commutative=True))), Integral(Function('\\\\mathbf{F}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(sin(Function('E_{\\\\lambda}')(Symbol('I', commutative=True))), Integral(Function('\\\\mathbf{F}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\varphi)} = \\cos{(\\varphi)}, then derive \\frac{d}{d \\varphi} \\Psi_{\\lambda}{(\\varphi)} = - \\sin{(\\varphi)}, then obtain - \\sin{(\\varphi)} = \\frac{d}{d \\varphi} \\cos{(\\varphi)}", "derivation": "\\Psi_{\\lambda}{(\\varphi)} = \\cos{(\\varphi)} and \\frac{d}{d \\varphi} \\Psi_{\\lambda}{(\\varphi)} = \\frac{d}{d \\varphi} \\cos{(\\varphi)} and \\frac{d}{d \\varphi} \\Psi_{\\lambda}{(\\varphi)} = - \\sin{(\\varphi)} and - \\sin{(\\varphi)} = \\frac{d}{d \\varphi} \\cos{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True))), Derivative(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})} = - J_{\\varepsilon} + \\frac{y}{\\mu}, then obtain - J_{\\varepsilon} + \\frac{\\partial}{\\partial \\mu} (y + \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})})^{y} + \\frac{y}{\\mu} = - J_{\\varepsilon} + \\frac{\\partial}{\\partial \\mu} (- J_{\\varepsilon} + y + \\frac{y}{\\mu})^{y} + \\frac{y}{\\mu}", "derivation": "\\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})} = - J_{\\varepsilon} + \\frac{y}{\\mu} and y + \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})} = - J_{\\varepsilon} + y + \\frac{y}{\\mu} and (y + \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})})^{y} = (- J_{\\varepsilon} + y + \\frac{y}{\\mu})^{y} and \\frac{\\partial}{\\partial \\mu} (y + \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})})^{y} = \\frac{\\partial}{\\partial \\mu} (- J_{\\varepsilon} + y + \\frac{y}{\\mu})^{y} and - J_{\\varepsilon} + \\frac{\\partial}{\\partial \\mu} (y + \\Psi^{\\dagger}{(y,\\mu,J_{\\varepsilon})})^{y} + \\frac{y}{\\mu} = - J_{\\varepsilon} + \\frac{\\partial}{\\partial \\mu} (- J_{\\varepsilon} + y + \\frac{y}{\\mu})^{y} + \\frac{y}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('y', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('y', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Symbol('y', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('y', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('y', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('y', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Pow(Add(Symbol('y', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('y', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\varphi)} = \\sin{(e^{\\varphi})}, then obtain \\int \\frac{\\sigma_{p}{(\\varphi)}}{\\varphi} d\\varphi + \\int \\frac{\\sin{(e^{\\varphi})}}{\\varphi} d\\varphi = 2 \\int \\frac{\\sin{(e^{\\varphi})}}{\\varphi} d\\varphi", "derivation": "\\sigma_{p}{(\\varphi)} = \\sin{(e^{\\varphi})} and \\frac{\\sigma_{p}{(\\varphi)}}{\\varphi} = \\frac{\\sin{(e^{\\varphi})}}{\\varphi} and \\int \\frac{\\sigma_{p}{(\\varphi)}}{\\varphi} d\\varphi = \\int \\frac{\\sin{(e^{\\varphi})}}{\\varphi} d\\varphi and \\int \\frac{\\sigma_{p}{(\\varphi)}}{\\varphi} d\\varphi + \\int \\frac{\\sin{(e^{\\varphi})}}{\\varphi} d\\varphi = 2 \\int \\frac{\\sin{(e^{\\varphi})}}{\\varphi} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\varphi', commutative=True)), sin(exp(Symbol('\\\\varphi', commutative=True))))"], [["divide", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(exp(Symbol('\\\\varphi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["add", 3, "Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(2), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\delta)} = e^{\\delta}, then derive \\frac{d}{d \\delta} \\mathbf{E}{(\\delta)} = e^{\\delta}, then obtain - \\operatorname{J_{\\varepsilon}}{(f^{*})} - e^{\\delta} = - \\operatorname{J_{\\varepsilon}}{(f^{*})} - \\frac{d}{d \\delta} \\mathbf{E}{(\\delta)}", "derivation": "\\mathbf{E}{(\\delta)} = e^{\\delta} and \\operatorname{J_{\\varepsilon}}{(f^{*})} + \\mathbf{E}{(\\delta)} = \\operatorname{J_{\\varepsilon}}{(f^{*})} + e^{\\delta} and \\frac{\\partial}{\\partial \\delta} (\\operatorname{J_{\\varepsilon}}{(f^{*})} + \\mathbf{E}{(\\delta)}) = \\frac{\\partial}{\\partial \\delta} (\\operatorname{J_{\\varepsilon}}{(f^{*})} + e^{\\delta}) and \\frac{d}{d \\delta} \\mathbf{E}{(\\delta)} = e^{\\delta} and - \\operatorname{J_{\\varepsilon}}{(f^{*})} - e^{\\delta} + \\frac{d}{d \\delta} \\mathbf{E}{(\\delta)} = - \\operatorname{J_{\\varepsilon}}{(f^{*})} and - \\operatorname{J_{\\varepsilon}}{(f^{*})} - e^{\\delta} = - \\operatorname{J_{\\varepsilon}}{(f^{*})} - \\frac{d}{d \\delta} \\mathbf{E}{(\\delta)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True))), Add(Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), exp(Symbol('\\\\delta', commutative=True)))"], [["minus", 4, "Add(Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True))))"], [["minus", 5, "Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f^*', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{S}{(\\theta_2)} = e^{\\theta_2}, then derive \\int \\mathbf{S}{(\\theta_2)} d\\theta_2 = V_{\\mathbf{B}} + e^{\\theta_2}, then derive (H + e^{\\theta_2})^{\\theta_2} = (V_{\\mathbf{B}} + \\mathbf{S}{(\\theta_2)})^{\\theta_2}, then obtain E_{\\lambda} + (H + e^{\\theta_2})^{\\theta_2} = E_{\\lambda} + (V_{\\mathbf{B}} + \\mathbf{S}{(\\theta_2)})^{\\theta_2}", "derivation": "\\mathbf{S}{(\\theta_2)} = e^{\\theta_2} and \\int \\mathbf{S}{(\\theta_2)} d\\theta_2 = \\int e^{\\theta_2} d\\theta_2 and \\int \\mathbf{S}{(\\theta_2)} d\\theta_2 = V_{\\mathbf{B}} + e^{\\theta_2} and \\int e^{\\theta_2} d\\theta_2 = V_{\\mathbf{B}} + e^{\\theta_2} and (\\int e^{\\theta_2} d\\theta_2)^{\\theta_2} = (V_{\\mathbf{B}} + e^{\\theta_2})^{\\theta_2} and (\\int e^{\\theta_2} d\\theta_2)^{\\theta_2} = (V_{\\mathbf{B}} + \\mathbf{S}{(\\theta_2)})^{\\theta_2} and (H + e^{\\theta_2})^{\\theta_2} = (V_{\\mathbf{B}} + \\mathbf{S}{(\\theta_2)})^{\\theta_2} and E_{\\lambda} + (H + e^{\\theta_2})^{\\theta_2} = E_{\\lambda} + (V_{\\mathbf{B}} + \\mathbf{S}{(\\theta_2)})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('H', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["add", 7, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Symbol('H', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(c)} = \\log{(\\cos{(c)})}, then obtain \\frac{e^{\\mathbf{S}{(c)} \\log{(\\cos{(c)})}}}{\\cos{(c)}} = \\frac{e^{\\log{(\\cos{(c)})}^{2}}}{\\cos{(c)}}", "derivation": "\\mathbf{S}{(c)} = \\log{(\\cos{(c)})} and \\mathbf{S}{(c)} \\log{(\\cos{(c)})} = \\log{(\\cos{(c)})}^{2} and \\mathbf{S}{(c)} \\log{(\\cos{(c)})} - \\log{(\\cos{(c)})} = \\log{(\\cos{(c)})}^{2} - \\log{(\\cos{(c)})} and \\frac{e^{\\mathbf{S}{(c)} \\log{(\\cos{(c)})}}}{\\cos{(c)}} = \\frac{e^{\\log{(\\cos{(c)})}^{2}}}{\\cos{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), log(cos(Symbol('c', commutative=True))))"], [["times", 1, "log(cos(Symbol('c', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), log(cos(Symbol('c', commutative=True)))), Pow(log(cos(Symbol('c', commutative=True))), Integer(2)))"], [["minus", 2, "log(cos(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), log(cos(Symbol('c', commutative=True)))), Mul(Integer(-1), log(cos(Symbol('c', commutative=True))))), Add(Pow(log(cos(Symbol('c', commutative=True))), Integer(2)), Mul(Integer(-1), log(cos(Symbol('c', commutative=True))))))"], [["exp", 3], "Equality(Mul(exp(Mul(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), log(cos(Symbol('c', commutative=True))))), Pow(cos(Symbol('c', commutative=True)), Integer(-1))), Mul(exp(Pow(log(cos(Symbol('c', commutative=True))), Integer(2))), Pow(cos(Symbol('c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given Z{(\\theta,\\phi)} = \\sin{(\\phi^{\\theta})}, then derive 2 \\frac{\\partial}{\\partial \\phi} Z{(\\theta,\\phi)} = \\frac{\\partial}{\\partial \\phi} Z{(\\theta,\\phi)} + \\frac{\\phi^{\\theta} \\theta \\cos{(\\phi^{\\theta})}}{\\phi}, then obtain \\frac{\\partial}{\\partial \\theta} 2 \\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} = \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} + \\frac{\\phi^{\\theta} \\theta \\cos{(\\phi^{\\theta})}}{\\phi})", "derivation": "Z{(\\theta,\\phi)} = \\sin{(\\phi^{\\theta})} and 2 Z{(\\theta,\\phi)} = Z{(\\theta,\\phi)} + \\sin{(\\phi^{\\theta})} and \\frac{\\partial}{\\partial \\phi} 2 Z{(\\theta,\\phi)} = \\frac{\\partial}{\\partial \\phi} (Z{(\\theta,\\phi)} + \\sin{(\\phi^{\\theta})}) and 2 \\frac{\\partial}{\\partial \\phi} Z{(\\theta,\\phi)} = \\frac{\\partial}{\\partial \\phi} Z{(\\theta,\\phi)} + \\frac{\\phi^{\\theta} \\theta \\cos{(\\phi^{\\theta})}}{\\phi} and 2 \\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} = \\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} + \\frac{\\phi^{\\theta} \\theta \\cos{(\\phi^{\\theta})}}{\\phi} and \\frac{\\partial}{\\partial \\theta} 2 \\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} = \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial \\phi} \\sin{(\\phi^{\\theta})} + \\frac{\\phi^{\\theta} \\theta \\cos{(\\phi^{\\theta})}}{\\phi})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True)), sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True)), sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True)), sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Derivative(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True), cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Derivative(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True), cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))))"], [["differentiate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Derivative(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Derivative(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True), cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\log{(\\eta^{\\prime})}, then derive \\frac{d}{d \\eta^{\\prime}} \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\frac{1}{\\eta^{\\prime}}, then obtain \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\frac{1}{\\eta^{\\prime}}", "derivation": "\\operatorname{n_{1}}{(\\eta^{\\prime})} = \\log{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\log{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\frac{1}{\\eta^{\\prime}} and \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\frac{1}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} = \\frac{k}{f_{E} m_{s}}, then obtain \\frac{\\partial}{\\partial k} m_{s} \\int \\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} dm_{s} = \\frac{\\partial}{\\partial k} m_{s} \\int \\frac{k}{f_{E} m_{s}} dm_{s}", "derivation": "\\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} = \\frac{k}{f_{E} m_{s}} and \\int \\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} dm_{s} = \\int \\frac{k}{f_{E} m_{s}} dm_{s} and m_{s} \\int \\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} dm_{s} = m_{s} \\int \\frac{k}{f_{E} m_{s}} dm_{s} and \\frac{\\partial}{\\partial k} m_{s} \\int \\operatorname{y^{\\prime}}{(k,m_{s},f_{E})} dm_{s} = \\frac{\\partial}{\\partial k} m_{s} \\int \\frac{k}{f_{E} m_{s}} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('k', commutative=True), Symbol('m_s', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('k', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('k', commutative=True), Symbol('m_s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('k', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True))))"], [["divide", 2, "Pow(Symbol('m_s', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('m_s', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('k', commutative=True), Symbol('m_s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Symbol('m_s', commutative=True), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('k', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True)))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Symbol('m_s', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('k', commutative=True), Symbol('m_s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Symbol('m_s', commutative=True), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('k', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(v_{y})} = \\sin{(v_{y})}, then obtain - \\operatorname{E_{n}}^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\operatorname{E_{n}}{(v_{y})} = - \\operatorname{E_{n}}^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})}", "derivation": "\\operatorname{E_{n}}{(v_{y})} = \\sin{(v_{y})} and \\operatorname{E_{n}}^{v_{y}}{(v_{y})} = \\sin^{v_{y}}{(v_{y})} and \\frac{d}{d v_{y}} \\operatorname{E_{n}}{(v_{y})} = \\frac{d}{d v_{y}} \\sin{(v_{y})} and - \\sin^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\operatorname{E_{n}}{(v_{y})} = - \\sin^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})} and - \\operatorname{E_{n}}^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\operatorname{E_{n}}{(v_{y})} = - \\operatorname{E_{n}}^{v_{y}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["minus", 3, "Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Derivative(Function('E_n')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Derivative(Function('E_n')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{F})} = e^{\\mathbf{F}}, then obtain \\frac{- e^{\\mathbf{F}} - \\frac{\\operatorname{F_{N}}{(\\mathbf{F})}}{\\mathbf{F}}}{\\frac{e^{\\mathbf{F}}}{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}^{2}}} = \\frac{- e^{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}}}{\\frac{e^{\\mathbf{F}}}{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}^{2}}}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{F})} = e^{\\mathbf{F}} and \\frac{\\operatorname{F_{N}}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{e^{\\mathbf{F}}}{\\mathbf{F}} and - \\frac{\\operatorname{F_{N}}{(\\mathbf{F})}}{\\mathbf{F}} = - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}} and - e^{\\mathbf{F}} - \\frac{\\operatorname{F_{N}}{(\\mathbf{F})}}{\\mathbf{F}} = - e^{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}} and \\frac{- e^{\\mathbf{F}} - \\frac{\\operatorname{F_{N}}{(\\mathbf{F})}}{\\mathbf{F}}}{\\frac{e^{\\mathbf{F}}}{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}^{2}}} = \\frac{- e^{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}}}{\\frac{e^{\\mathbf{F}}}{\\mathbf{F}} - \\frac{e^{\\mathbf{F}}}{\\mathbf{F}^{2}}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 3, "exp(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 4, "Add(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mathbf{F}', commutative=True))))), Mul(Pow(Add(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(E,x)} = - x + e^{E} and \\operatorname{f_{E}}{(E,x)} = - \\rho_{f}^{E}{(E,x)} - 1, then obtain \\operatorname{f_{E}}{(E,x)} = - (- x + e^{E})^{E} - 1", "derivation": "\\rho_{f}{(E,x)} = - x + e^{E} and \\rho_{f}^{E}{(E,x)} = (- x + e^{E})^{E} and - \\rho_{f}^{E}{(E,x)} = - (- x + e^{E})^{E} and - \\rho_{f}^{E}{(E,x)} - 1 = - (- x + e^{E})^{E} - 1 and \\operatorname{f_{E}}{(E,x)} = - \\rho_{f}^{E}{(E,x)} - 1 and \\operatorname{f_{E}}{(E,x)} = - (- x + e^{E})^{E} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(Symbol('E', commutative=True))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Symbol('E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Symbol('E', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(Symbol('E', commutative=True))), Symbol('E', commutative=True))), Integer(-1)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Symbol('E', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('f_E')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), exp(Symbol('E', commutative=True))), Symbol('E', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\phi_{2}{(A_{1},\\mathbf{B})} = e^{A_{1}^{\\mathbf{B}}} and q{(A_{1},\\mathbf{B})} = A_{1}^{\\mathbf{B}}, then obtain \\phi_{2}{(A_{1},\\mathbf{B})} + e^{q{(A_{1},\\mathbf{B})}} = 2 e^{q{(A_{1},\\mathbf{B})}}", "derivation": "\\phi_{2}{(A_{1},\\mathbf{B})} = e^{A_{1}^{\\mathbf{B}}} and \\phi_{2}{(A_{1},\\mathbf{B})} + e^{A_{1}^{\\mathbf{B}}} = 2 e^{A_{1}^{\\mathbf{B}}} and q{(A_{1},\\mathbf{B})} = A_{1}^{\\mathbf{B}} and \\phi_{2}{(A_{1},\\mathbf{B})} = e^{q{(A_{1},\\mathbf{B})}} and e^{q{(A_{1},\\mathbf{B})}} = e^{A_{1}^{\\mathbf{B}}} and \\phi_{2}{(A_{1},\\mathbf{B})} + e^{q{(A_{1},\\mathbf{B})}} = 2 e^{q{(A_{1},\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 1, "exp(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Function('\\\\phi_2')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(2), exp(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\phi_2')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Function('q')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(exp(Function('q')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), exp(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Function('\\\\phi_2')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Function('q')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(2), exp(Function('q')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(v_{2},z)} = \\sin{(v_{2} + z)}, then obtain \\frac{\\partial}{\\partial v_{2}} e^{\\int \\mathbf{D}{(v_{2},z)} dv_{2}} = \\frac{\\partial}{\\partial v_{2}} e^{\\int \\sin{(v_{2} + z)} dv_{2}}", "derivation": "\\mathbf{D}{(v_{2},z)} = \\sin{(v_{2} + z)} and \\int \\mathbf{D}{(v_{2},z)} dv_{2} = \\int \\sin{(v_{2} + z)} dv_{2} and e^{\\int \\mathbf{D}{(v_{2},z)} dv_{2}} = e^{\\int \\sin{(v_{2} + z)} dv_{2}} and \\frac{\\partial}{\\partial v_{2}} e^{\\int \\mathbf{D}{(v_{2},z)} dv_{2}} = \\frac{\\partial}{\\partial v_{2}} e^{\\int \\sin{(v_{2} + z)} dv_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), sin(Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(sin(Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), exp(Integral(sin(Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_2', commutative=True)))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(exp(Integral(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(exp(Integral(sin(Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\lambda,\\mathbf{M})} = \\mathbf{M} \\sin{(\\lambda)}, then derive \\int (H{(\\lambda,\\mathbf{M})} + \\sin{(\\lambda)}) d\\mathbf{M} = \\Psi + \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)}, then obtain \\Psi + \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} = \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} + \\mathbf{f}", "derivation": "H{(\\lambda,\\mathbf{M})} = \\mathbf{M} \\sin{(\\lambda)} and H{(\\lambda,\\mathbf{M})} + \\sin{(\\lambda)} = \\mathbf{M} \\sin{(\\lambda)} + \\sin{(\\lambda)} and \\int (H{(\\lambda,\\mathbf{M})} + \\sin{(\\lambda)}) d\\mathbf{M} = \\int (\\mathbf{M} \\sin{(\\lambda)} + \\sin{(\\lambda)}) d\\mathbf{M} and \\int (H{(\\lambda,\\mathbf{M})} + \\sin{(\\lambda)}) d\\mathbf{M} = \\Psi + \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} and \\Psi + \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} = \\int (\\mathbf{M} \\sin{(\\lambda)} + \\sin{(\\lambda)}) d\\mathbf{M} and \\Psi + \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} = \\frac{\\mathbf{M}^{2} \\sin{(\\lambda)}}{2} + \\mathbf{M} \\sin{(\\lambda)} + \\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Add(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))), Integral(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)), sin(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(y^{\\prime},\\rho_f)} = \\rho_f^{y^{\\prime}}, then obtain \\rho_f \\hat{H}^{2}{(y^{\\prime},\\rho_f)} + \\rho_f^{y^{\\prime}} = \\rho_f \\rho_f^{2 y^{\\prime}} + \\rho_f^{y^{\\prime}}", "derivation": "\\hat{H}{(y^{\\prime},\\rho_f)} = \\rho_f^{y^{\\prime}} and \\rho_f \\hat{H}{(y^{\\prime},\\rho_f)} = \\rho_f \\rho_f^{y^{\\prime}} and \\rho_f \\hat{H}^{2}{(y^{\\prime},\\rho_f)} = \\rho_f \\rho_f^{y^{\\prime}} \\hat{H}{(y^{\\prime},\\rho_f)} and \\rho_f \\rho_f^{y^{\\prime}} \\hat{H}{(y^{\\prime},\\rho_f)} = \\rho_f \\rho_f^{2 y^{\\prime}} and \\rho_f \\hat{H}^{2}{(y^{\\prime},\\rho_f)} + \\rho_f^{y^{\\prime}} = \\rho_f \\rho_f^{y^{\\prime}} \\hat{H}{(y^{\\prime},\\rho_f)} + \\rho_f^{y^{\\prime}} and \\rho_f \\hat{H}^{2}{(y^{\\prime},\\rho_f)} + \\rho_f^{y^{\\prime}} = \\rho_f \\rho_f^{2 y^{\\prime}} + \\rho_f^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 2, "Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2))), Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 3, "Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2))), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2))), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\Omega,b)} = \\log{(\\Omega^{b})}, then obtain - \\bar{\\h}^{2}{(\\Omega,b)} = - \\frac{\\log{(\\Omega^{b})}^{8}}{\\bar{\\h}^{6}{(\\Omega,b)}}", "derivation": "\\bar{\\h}{(\\Omega,b)} = \\log{(\\Omega^{b})} and \\bar{\\h}{(\\Omega,b)} \\log{(\\Omega^{b})} = \\log{(\\Omega^{b})}^{2} and \\log{(\\Omega^{b})} = \\frac{\\log{(\\Omega^{b})}^{2}}{\\bar{\\h}{(\\Omega,b)}} and \\bar{\\h}{(\\Omega,b)} = \\frac{\\log{(\\Omega^{b})}^{2}}{\\bar{\\h}{(\\Omega,b)}} and - \\bar{\\h}{(\\Omega,b)} = - \\frac{\\log{(\\Omega^{b})}^{2}}{\\bar{\\h}{(\\Omega,b)}} and \\bar{\\h}^{2}{(\\Omega,b)} = \\frac{\\log{(\\Omega^{b})}^{4}}{\\bar{\\h}^{2}{(\\Omega,b)}} and - \\bar{\\h}^{2}{(\\Omega,b)} = - \\frac{\\log{(\\Omega^{b})}^{4}}{\\bar{\\h}^{2}{(\\Omega,b)}} and - \\bar{\\h}^{2}{(\\Omega,b)} = - \\frac{\\log{(\\Omega^{b})}^{8}}{\\bar{\\h}^{6}{(\\Omega,b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))))"], [["times", 1, "log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)))), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(2)))"], [["divide", 2, "Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))"], "Equality(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Mul(Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(2))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(2))))"], [["power", 5, 2], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(2)), Mul(Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-2)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(4))))"], [["divide", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-2)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(4))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Integer(-6)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True))), Integer(8))))"]]}, {"prompt": "Given \\mathbf{g}{(E,f_{E})} = \\log{(E + f_{E})} and \\delta{(E,f_{E})} = \\mathbf{g}{(E,f_{E})} + \\log{(E + f_{E})}, then obtain 2 \\log{(E + f_{E})} = 2 \\mathbf{g}{(E,f_{E})}", "derivation": "\\mathbf{g}{(E,f_{E})} = \\log{(E + f_{E})} and \\delta{(E,f_{E})} = \\mathbf{g}{(E,f_{E})} + \\log{(E + f_{E})} and \\delta{(E,f_{E})} = 2 \\log{(E + f_{E})} and \\delta{(E,f_{E})} = 2 \\mathbf{g}{(E,f_{E})} and 2 \\log{(E + f_{E})} = 2 \\mathbf{g}{(E,f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), log(Add(Symbol('E', commutative=True), Symbol('f_E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Add(Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), log(Add(Symbol('E', commutative=True), Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(2), log(Add(Symbol('E', commutative=True), Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), log(Add(Symbol('E', commutative=True), Symbol('f_E', commutative=True)))), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given S{(v_{2},a^{\\dagger})} = - a^{\\dagger} + \\cos{(v_{2})}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} S{(v_{2},a^{\\dagger})} = -1, then derive \\int (-1) dv_{2} = \\delta - v_{2}, then obtain - \\frac{d}{d v_{2}} \\int (-1) dv_{2} = - \\frac{\\partial}{\\partial v_{2}} (\\delta - v_{2})", "derivation": "S{(v_{2},a^{\\dagger})} = - a^{\\dagger} + \\cos{(v_{2})} and \\frac{\\partial}{\\partial a^{\\dagger}} S{(v_{2},a^{\\dagger})} = \\frac{\\partial}{\\partial a^{\\dagger}} (- a^{\\dagger} + \\cos{(v_{2})}) and \\frac{\\partial}{\\partial a^{\\dagger}} S{(v_{2},a^{\\dagger})} = -1 and -1 = \\frac{\\partial}{\\partial a^{\\dagger}} (- a^{\\dagger} + \\cos{(v_{2})}) and \\int (-1) dv_{2} = \\int \\frac{\\partial}{\\partial a^{\\dagger}} (- a^{\\dagger} + \\cos{(v_{2})}) dv_{2} and \\int (-1) dv_{2} = \\delta - v_{2} and \\frac{d}{d v_{2}} \\int (-1) dv_{2} = \\frac{\\partial}{\\partial v_{2}} (\\delta - v_{2}) and - \\frac{d}{d v_{2}} \\int (-1) dv_{2} = - \\frac{\\partial}{\\partial v_{2}} (\\delta - v_{2})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('v_2', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('v_2', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('v_2', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('v_2', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Integer(-1), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["differentiate", 6, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Integer(-1), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["divide", 7, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Integral(Integer(-1), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(Z)} = \\frac{d}{d Z} e^{Z}, then derive v{(Z)} = e^{Z}, then obtain \\frac{d}{d Z} e^{- Z} \\frac{d}{d Z} v{(Z)} = \\frac{d}{d Z} 1", "derivation": "v{(Z)} = \\frac{d}{d Z} e^{Z} and v{(Z)} e^{- Z} = e^{- Z} \\frac{d}{d Z} e^{Z} and v{(Z)} = e^{Z} and v{(Z)} = \\frac{d}{d Z} v{(Z)} and e^{Z} = \\frac{d}{d Z} e^{Z} and e^{- Z} \\frac{d}{d Z} v{(Z)} = e^{- Z} \\frac{d}{d Z} e^{Z} and \\frac{d}{d Z} e^{- Z} \\frac{d}{d Z} v{(Z)} = \\frac{d}{d Z} e^{- Z} \\frac{d}{d Z} e^{Z} and \\frac{d}{d Z} e^{- Z} \\frac{d}{d Z} v{(Z)} = \\frac{d}{d Z} 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('Z', commutative=True)), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["divide", 1, "exp(Symbol('Z', commutative=True))"], "Equality(Mul(Function('v')(Symbol('Z', commutative=True)), exp(Mul(Integer(-1), Symbol('Z', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('v')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v')(Symbol('Z', commutative=True)), Derivative(Function('v')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Symbol('Z', commutative=True)), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Function('v')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Function('v')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Derivative(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Function('v')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(x)} = \\frac{d}{d x} \\cos{(x)} and \\psi^{*}{(m)} = \\cos{(m)} and \\operatorname{C_{d}}{(x)} = \\cos{(x)}, then derive \\int \\psi^{*}{(m)} dm = \\mathbf{F} + \\sin{(m)}, then obtain \\mathbf{F} + \\operatorname{m_{s}}{(x)} + \\sin{(m)} = \\mathbf{F} + \\sin{(m)} + \\frac{d}{d x} \\operatorname{C_{d}}{(x)}", "derivation": "\\operatorname{m_{s}}{(x)} = \\frac{d}{d x} \\cos{(x)} and \\psi^{*}{(m)} = \\cos{(m)} and \\int \\psi^{*}{(m)} dm = \\int \\cos{(m)} dm and \\operatorname{C_{d}}{(x)} = \\cos{(x)} and \\operatorname{m_{s}}{(x)} = \\frac{d}{d x} \\operatorname{C_{d}}{(x)} and \\int \\psi^{*}{(m)} dm = \\mathbf{F} + \\sin{(m)} and \\operatorname{m_{s}}{(x)} + \\int \\cos{(m)} dm = \\frac{d}{d x} \\operatorname{C_{d}}{(x)} + \\int \\cos{(m)} dm and \\mathbf{F} + \\sin{(m)} = \\int \\cos{(m)} dm and \\mathbf{F} + \\operatorname{m_{s}}{(x)} + \\sin{(m)} = \\mathbf{F} + \\sin{(m)} + \\frac{d}{d x} \\operatorname{C_{d}}{(x)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('x', commutative=True)), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('m_s')(Symbol('x', commutative=True)), Derivative(Function('C_d')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\psi^*')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('m', commutative=True))))"], [["add", 5, "Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))"], "Equality(Add(Function('m_s')(Symbol('x', commutative=True)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Derivative(Function('C_d')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('m_s')(Symbol('x', commutative=True)), sin(Symbol('m', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('m', commutative=True)), Derivative(Function('C_d')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{2}{(A_{z})} = \\cos{(e^{A_{z}})}, then obtain \\int \\cos{(e^{A_{z}})} \\int \\cos{(\\theta_{2}{(A_{z})})} dA_{z} dA_{z} = \\int \\cos{(e^{A_{z}})} \\int \\cos{(\\cos{(e^{A_{z}})})} dA_{z} dA_{z}", "derivation": "\\theta_{2}{(A_{z})} = \\cos{(e^{A_{z}})} and \\cos{(\\theta_{2}{(A_{z})})} = \\cos{(\\cos{(e^{A_{z}})})} and \\int \\cos{(\\theta_{2}{(A_{z})})} dA_{z} = \\int \\cos{(\\cos{(e^{A_{z}})})} dA_{z} and \\cos{(e^{A_{z}})} \\int \\cos{(\\theta_{2}{(A_{z})})} dA_{z} = \\cos{(e^{A_{z}})} \\int \\cos{(\\cos{(e^{A_{z}})})} dA_{z} and \\int \\cos{(e^{A_{z}})} \\int \\cos{(\\theta_{2}{(A_{z})})} dA_{z} dA_{z} = \\int \\cos{(e^{A_{z}})} \\int \\cos{(\\cos{(e^{A_{z}})})} dA_{z} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('A_z', commutative=True)), cos(exp(Symbol('A_z', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\theta_2')(Symbol('A_z', commutative=True))), cos(cos(exp(Symbol('A_z', commutative=True)))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(cos(Function('\\\\theta_2')(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Integral(cos(cos(exp(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))))"], [["times", 3, "cos(exp(Symbol('A_z', commutative=True)))"], "Equality(Mul(cos(exp(Symbol('A_z', commutative=True))), Integral(cos(Function('\\\\theta_2')(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)))), Mul(cos(exp(Symbol('A_z', commutative=True))), Integral(cos(cos(exp(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True)))))"], [["integrate", 4, "Symbol('A_z', commutative=True)"], "Equality(Integral(Mul(cos(exp(Symbol('A_z', commutative=True))), Integral(cos(Function('\\\\theta_2')(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))), Integral(Mul(cos(exp(Symbol('A_z', commutative=True))), Integral(cos(cos(exp(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\eta)} = \\eta and \\mathbf{J}_M{(\\eta)} = \\eta, then obtain \\frac{\\int \\mathbf{J}_M{(\\eta)} d\\eta}{\\frac{d}{d \\eta} \\eta} = \\frac{\\int \\eta d\\eta}{\\frac{d}{d \\eta} \\eta}", "derivation": "\\dot{\\mathbf{r}}{(\\eta)} = \\eta and \\mathbf{J}_M{(\\eta)} = \\eta and \\int \\mathbf{J}_M{(\\eta)} d\\eta = \\int \\eta d\\eta and \\frac{\\int \\mathbf{J}_M{(\\eta)} d\\eta}{\\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)}} = \\frac{\\int \\eta d\\eta}{\\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)}} and \\frac{\\int \\mathbf{J}_M{(\\eta)} d\\eta}{\\frac{d}{d \\eta} \\eta} = \\frac{\\int \\eta d\\eta}{\\frac{d}{d \\eta} \\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True))))"], [["divide", 3, "Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Pow(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Pow(Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given W{(P_{e})} = \\sin{(P_{e})}, then derive \\frac{\\frac{d}{d P_{e}} W{(P_{e})}}{P_{e}} = \\frac{\\cos{(P_{e})}}{P_{e}}, then obtain \\frac{d}{d P_{e}} \\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{P_{e}} = \\frac{d}{d P_{e}} \\frac{\\cos{(P_{e})}}{P_{e}}", "derivation": "W{(P_{e})} = \\sin{(P_{e})} and \\frac{d}{d P_{e}} W{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(P_{e})} and \\frac{\\frac{d}{d P_{e}} W{(P_{e})}}{P_{e}} = \\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{P_{e}} and \\frac{\\frac{d}{d P_{e}} W{(P_{e})}}{P_{e}} = \\frac{\\cos{(P_{e})}}{P_{e}} and \\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{P_{e}} = \\frac{\\cos{(P_{e})}}{P_{e}} and \\frac{d}{d P_{e}} \\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{P_{e}} = \\frac{d}{d P_{e}} \\frac{\\cos{(P_{e})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))))"], [["differentiate", 5, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(u,t_{1})} = u^{t_{1}} and \\lambda{(u,t_{1})} = t_{1} u^{t_{1}}, then obtain \\frac{- u^{t_{1}} + \\int t_{1} \\mathbf{f}{(u,t_{1})} du}{\\lambda{(u,t_{1})}} = \\frac{- u^{t_{1}} + \\int \\lambda{(u,t_{1})} du}{\\lambda{(u,t_{1})}}", "derivation": "\\mathbf{f}{(u,t_{1})} = u^{t_{1}} and t_{1} \\mathbf{f}{(u,t_{1})} = t_{1} u^{t_{1}} and \\lambda{(u,t_{1})} = t_{1} u^{t_{1}} and t_{1} \\mathbf{f}{(u,t_{1})} = \\lambda{(u,t_{1})} and \\int t_{1} \\mathbf{f}{(u,t_{1})} du = \\int \\lambda{(u,t_{1})} du and - u^{t_{1}} + \\int t_{1} \\mathbf{f}{(u,t_{1})} du = - u^{t_{1}} + \\int \\lambda{(u,t_{1})} du and \\frac{- u^{t_{1}} + \\int t_{1} \\mathbf{f}{(u,t_{1})} du}{\\lambda{(u,t_{1})}} = \\frac{- u^{t_{1}} + \\int \\lambda{(u,t_{1})} du}{\\lambda{(u,t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True)))"], [["times", 1, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Mul(Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('t_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Symbol('t_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["minus", 5, "Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Integral(Mul(Symbol('t_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Integral(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["divide", 6, "Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Integral(Mul(Symbol('t_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('u', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('t_1', commutative=True))), Integral(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('u', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('t_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain e^{(\\operatorname{n_{1}}{(\\mathbf{A})} - \\cos{(\\mathbf{A})})^{\\mathbf{A}}} = e^{0^{\\mathbf{A}}}", "derivation": "\\operatorname{n_{1}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\operatorname{n_{1}}{(\\mathbf{A})} - \\cos{(\\mathbf{A})} = 0 and (\\operatorname{n_{1}}{(\\mathbf{A})} - \\cos{(\\mathbf{A})})^{\\mathbf{A}} = 0^{\\mathbf{A}} and e^{(\\operatorname{n_{1}}{(\\mathbf{A})} - \\cos{(\\mathbf{A})})^{\\mathbf{A}}} = e^{0^{\\mathbf{A}}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Add(Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{A}', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Add(Function('n_1')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True))), exp(Pow(Integer(0), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(E_{x},A_{z})} = \\frac{\\partial}{\\partial E_{x}} (A_{z} + E_{x}) and \\mathbf{J}_P{(E_{x},A_{z})} = \\int \\frac{\\partial}{\\partial E_{x}} (A_{z} + E_{x}) dA_{z}, then derive \\operatorname{C_{d}}{(E_{x},A_{z})} = 1, then obtain \\mathbf{J}_P{(E_{x},A_{z})} = A_{z} + v_{1}", "derivation": "\\operatorname{C_{d}}{(E_{x},A_{z})} = \\frac{\\partial}{\\partial E_{x}} (A_{z} + E_{x}) and \\operatorname{C_{d}}{(E_{x},A_{z})} = 1 and \\int \\operatorname{C_{d}}{(E_{x},A_{z})} dA_{z} = \\int 1 dA_{z} and \\int \\frac{\\partial}{\\partial E_{x}} (A_{z} + E_{x}) dA_{z} = \\int 1 dA_{z} and \\mathbf{J}_P{(E_{x},A_{z})} = \\int \\frac{\\partial}{\\partial E_{x}} (A_{z} + E_{x}) dA_{z} and \\mathbf{J}_P{(E_{x},A_{z})} = \\int 1 dA_{z} and \\mathbf{J}_P{(E_{x},A_{z})} = A_{z} + v_{1}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Derivative(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('C_d')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Integer(1))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Integral(Derivative(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\Omega{(v_{z})} = \\sin{(\\sin{(v_{z})})}, then obtain \\Omega^{v_{z}}{(v_{z})} + \\frac{v_{z} + \\Omega{(v_{z})}}{v_{z}} = \\sin^{v_{z}}{(\\sin{(v_{z})})} + \\frac{v_{z} + \\Omega{(v_{z})}}{v_{z}}", "derivation": "\\Omega{(v_{z})} = \\sin{(\\sin{(v_{z})})} and v_{z} + \\Omega{(v_{z})} = v_{z} + \\sin{(\\sin{(v_{z})})} and \\frac{v_{z} + \\Omega{(v_{z})}}{v_{z}} = \\frac{v_{z} + \\sin{(\\sin{(v_{z})})}}{v_{z}} and \\Omega^{v_{z}}{(v_{z})} = \\sin^{v_{z}}{(\\sin{(v_{z})})} and \\Omega^{v_{z}}{(v_{z})} + \\frac{v_{z} + \\sin{(\\sin{(v_{z})})}}{v_{z}} = \\sin^{v_{z}}{(\\sin{(v_{z})})} + \\frac{v_{z} + \\sin{(\\sin{(v_{z})})}}{v_{z}} and \\Omega^{v_{z}}{(v_{z})} + \\frac{v_{z} + \\Omega{(v_{z})}}{v_{z}} = \\sin^{v_{z}}{(\\sin{(v_{z})})} + \\frac{v_{z} + \\Omega{(v_{z})}}{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('v_z', commutative=True)), sin(sin(Symbol('v_z', commutative=True))))"], [["add", 1, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Function('\\\\Omega')(Symbol('v_z', commutative=True))), Add(Symbol('v_z', commutative=True), sin(sin(Symbol('v_z', commutative=True)))))"], [["divide", 2, "Symbol('v_z', commutative=True)"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), Function('\\\\Omega')(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), sin(sin(Symbol('v_z', commutative=True))))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(sin(sin(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 4, "Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), sin(sin(Symbol('v_z', commutative=True)))))"], "Equality(Add(Pow(Function('\\\\Omega')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), sin(sin(Symbol('v_z', commutative=True)))))), Add(Pow(sin(sin(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), sin(sin(Symbol('v_z', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Function('\\\\Omega')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), Function('\\\\Omega')(Symbol('v_z', commutative=True))))), Add(Pow(sin(sin(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('v_z', commutative=True), Function('\\\\Omega')(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}{(c_{0},\\hat{x}_0)} = e^{\\hat{x}_0 c_{0}}, then obtain \\frac{\\mathbf{J}^{3}{(c_{0},\\hat{x}_0)}}{c_{0}} = \\frac{\\mathbf{J}^{2}{(c_{0},\\hat{x}_0)} e^{\\hat{x}_0 c_{0}}}{c_{0}}", "derivation": "\\mathbf{J}{(c_{0},\\hat{x}_0)} = e^{\\hat{x}_0 c_{0}} and \\frac{\\mathbf{J}{(c_{0},\\hat{x}_0)}}{c_{0}} = \\frac{e^{\\hat{x}_0 c_{0}}}{c_{0}} and \\frac{\\mathbf{J}^{2}{(c_{0},\\hat{x}_0)} e^{\\hat{x}_0 c_{0}}}{c_{0}} = \\frac{\\mathbf{J}{(c_{0},\\hat{x}_0)} e^{2 \\hat{x}_0 c_{0}}}{c_{0}} and \\frac{\\mathbf{J}^{3}{(c_{0},\\hat{x}_0)}}{c_{0}} = \\frac{\\mathbf{J}^{2}{(c_{0},\\hat{x}_0)} e^{\\hat{x}_0 c_{0}}}{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True))))"], [["divide", 1, "Symbol('c_0', commutative=True)"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True)))))"], [["times", 2, "Mul(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True))))"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(2)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True)))), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(3))), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(2)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(C_{d})} = \\int \\cos{(C_{d})} dC_{d}, then derive \\frac{d}{d C_{d}} 2 \\mathbf{P}{(C_{d})} = \\frac{\\partial}{\\partial C_{d}} (g + \\mathbf{P}{(C_{d})} + \\sin{(C_{d})}), then obtain \\log{(\\frac{d}{d C_{d}} 2 \\mathbf{P}{(C_{d})})} = \\log{(\\frac{\\partial}{\\partial C_{d}} (g + \\mathbf{P}{(C_{d})} + \\sin{(C_{d})}))}", "derivation": "\\mathbf{P}{(C_{d})} = \\int \\cos{(C_{d})} dC_{d} and 2 \\mathbf{P}{(C_{d})} = \\mathbf{P}{(C_{d})} + \\int \\cos{(C_{d})} dC_{d} and \\frac{d}{d C_{d}} 2 \\mathbf{P}{(C_{d})} = \\frac{d}{d C_{d}} (\\mathbf{P}{(C_{d})} + \\int \\cos{(C_{d})} dC_{d}) and \\frac{d}{d C_{d}} 2 \\mathbf{P}{(C_{d})} = \\frac{\\partial}{\\partial C_{d}} (g + \\mathbf{P}{(C_{d})} + \\sin{(C_{d})}) and \\log{(\\frac{d}{d C_{d}} 2 \\mathbf{P}{(C_{d})})} = \\log{(\\frac{\\partial}{\\partial C_{d}} (g + \\mathbf{P}{(C_{d})} + \\sin{(C_{d})}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True)), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True)), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True)), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Symbol('g', commutative=True), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["log", 4], "Equality(log(Derivative(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), log(Derivative(Add(Symbol('g', commutative=True), Function('\\\\mathbf{P}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(m,Z)} = Z m, then obtain \\frac{d}{d Z} \\sin{(1)} = \\frac{\\partial}{\\partial Z} \\sin{(\\frac{Z m}{\\dot{y}{(m,Z)}})}", "derivation": "\\dot{y}{(m,Z)} = Z m and 1 = \\frac{Z m}{\\dot{y}{(m,Z)}} and \\sin{(1)} = \\sin{(\\frac{Z m}{\\dot{y}{(m,Z)}})} and \\frac{d}{d Z} \\sin{(1)} = \\frac{\\partial}{\\partial Z} \\sin{(\\frac{Z m}{\\dot{y}{(m,Z)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], [["sin", 2], "Equality(sin(Integer(1)), sin(Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Integer(-1)))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(sin(Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Integer(-1)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(i,\\varepsilon)} = \\varepsilon + i, then obtain \\operatorname{v_{x}}{(i,\\varepsilon)} - \\int (\\varepsilon + i) di = \\varepsilon + i - \\int (\\varepsilon + i) di", "derivation": "\\operatorname{v_{x}}{(i,\\varepsilon)} = \\varepsilon + i and \\int \\operatorname{v_{x}}{(i,\\varepsilon)} di = \\int (\\varepsilon + i) di and \\operatorname{v_{x}}{(i,\\varepsilon)} - \\int \\operatorname{v_{x}}{(i,\\varepsilon)} di = \\varepsilon + i - \\int \\operatorname{v_{x}}{(i,\\varepsilon)} di and \\operatorname{v_{x}}{(i,\\varepsilon)} - \\int (\\varepsilon + i) di = \\varepsilon + i - \\int (\\varepsilon + i) di", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["minus", 1, "Integral(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True)))"], "Equality(Add(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Integral(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True))))), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True), Mul(Integer(-1), Integral(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))))"]]}, {"prompt": "Given b{(u,A_{1})} = u^{A_{1}}, then derive 0 = \\frac{A_{1} u^{A_{1}}}{u b{(u,A_{1})}} - \\frac{u^{A_{1}} \\frac{\\partial}{\\partial u} b{(u,A_{1})}}{b^{2}{(u,A_{1})}}, then obtain 0 = \\frac{A_{1} (\\frac{A_{1}}{u} - u^{- A_{1}} \\frac{\\partial}{\\partial u} u^{A_{1}})}{u}", "derivation": "b{(u,A_{1})} = u^{A_{1}} and 1 = \\frac{u^{A_{1}}}{b{(u,A_{1})}} and \\frac{d}{d u} 1 = \\frac{\\partial}{\\partial u} \\frac{u^{A_{1}}}{b{(u,A_{1})}} and 0 = \\frac{A_{1} u^{A_{1}}}{u b{(u,A_{1})}} - \\frac{u^{A_{1}} \\frac{\\partial}{\\partial u} b{(u,A_{1})}}{b^{2}{(u,A_{1})}} and 0 = \\frac{A_{1}}{u} - \\frac{\\frac{\\partial}{\\partial u} b{(u,A_{1})}}{b{(u,A_{1})}} and 0 = \\frac{A_{1} (\\frac{A_{1}}{u} - \\frac{\\frac{\\partial}{\\partial u} b{(u,A_{1})}}{b{(u,A_{1})}})}{u} and 0 = \\frac{A_{1} (\\frac{A_{1}}{u} - u^{- A_{1}} \\frac{\\partial}{\\partial u} u^{A_{1}})}{u}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)))"], [["divide", 1, "Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-2)), Derivative(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["times", 5, "Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)))"], "Equality(Integer(0), Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(0), Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Derivative(Pow(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{f})} = \\log{(\\mathbf{f})}, then obtain 1 = - \\frac{d}{d \\mathbf{f}} \\cos{(\\hat{\\mathbf{r}}{(\\mathbf{f})})} + \\frac{d}{d \\mathbf{f}} \\cos{(\\log{(\\mathbf{f})})} + 1", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{f})} = \\log{(\\mathbf{f})} and \\cos{(\\hat{\\mathbf{r}}{(\\mathbf{f})})} = \\cos{(\\log{(\\mathbf{f})})} and \\frac{d}{d \\mathbf{f}} \\cos{(\\hat{\\mathbf{r}}{(\\mathbf{f})})} = \\frac{d}{d \\mathbf{f}} \\cos{(\\log{(\\mathbf{f})})} and 0 = - \\frac{d}{d \\mathbf{f}} \\cos{(\\hat{\\mathbf{r}}{(\\mathbf{f})})} + \\frac{d}{d \\mathbf{f}} \\cos{(\\log{(\\mathbf{f})})} and 1 = - \\frac{d}{d \\mathbf{f}} \\cos{(\\hat{\\mathbf{r}}{(\\mathbf{f})})} + \\frac{d}{d \\mathbf{f}} \\cos{(\\log{(\\mathbf{f})})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), cos(log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Derivative(cos(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["add", 4, 1], "Equality(Integer(1), Add(Mul(Integer(-1), Derivative(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Derivative(cos(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\psi{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then derive \\frac{d}{d \\mathbf{A}} \\psi{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}, then obtain \\int \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} d\\mathbf{A} = \\int - \\sin{(\\mathbf{A})} d\\mathbf{A}", "derivation": "\\psi{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\psi{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\psi{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} and \\int \\frac{d}{d \\mathbf{A}} \\psi{(\\mathbf{A})} d\\mathbf{A} = \\int - \\sin{(\\mathbf{A})} d\\mathbf{A} and \\int \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} d\\mathbf{A} = \\int - \\sin{(\\mathbf{A})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given a{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain \\frac{2 e^{\\mathbf{S}} + 2 \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})}}{(- a{(\\mathbf{S})} - e^{\\mathbf{S}})^{3}} = - \\frac{e^{- 2 \\mathbf{S}}}{2}", "derivation": "a{(\\mathbf{S})} = e^{\\mathbf{S}} and a{(\\mathbf{S})} + e^{\\mathbf{S}} = 2 e^{\\mathbf{S}} and - a{(\\mathbf{S})} - e^{\\mathbf{S}} = - 2 e^{\\mathbf{S}} and \\frac{1}{(- a{(\\mathbf{S})} - e^{\\mathbf{S}})^{2}} = \\frac{e^{- 2 \\mathbf{S}}}{4} and \\frac{d}{d \\mathbf{S}} \\frac{1}{(- a{(\\mathbf{S})} - e^{\\mathbf{S}})^{2}} = \\frac{d}{d \\mathbf{S}} \\frac{e^{- 2 \\mathbf{S}}}{4} and \\frac{2 e^{\\mathbf{S}} + 2 \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})}}{(- a{(\\mathbf{S})} - e^{\\mathbf{S}})^{3}} = - \\frac{e^{- 2 \\mathbf{S}}}{2}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(-2)), Mul(Rational(1, 4), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(-2)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 4), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(-3)), Add(Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), Derivative(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))), Mul(Integer(-1), Rational(1, 2), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(q,z)} = \\frac{\\sin{(z)}}{q}, then derive \\frac{\\partial}{\\partial z} \\ddot{x}{(q,z)} = \\frac{\\cos{(z)}}{q}, then obtain \\frac{\\cos{(z)} \\frac{\\partial}{\\partial q} \\ddot{x}{(q,z)}}{q} = \\frac{\\cos{(z)} \\frac{\\partial}{\\partial q} \\frac{\\sin{(z)}}{q}}{q}", "derivation": "\\ddot{x}{(q,z)} = \\frac{\\sin{(z)}}{q} and \\frac{\\partial}{\\partial q} \\ddot{x}{(q,z)} = \\frac{\\partial}{\\partial q} \\frac{\\sin{(z)}}{q} and \\ddot{x}{(q,z)} + 1 = 1 + \\frac{\\sin{(z)}}{q} and \\frac{\\partial}{\\partial z} (\\ddot{x}{(q,z)} + 1) = \\frac{\\partial}{\\partial z} (1 + \\frac{\\sin{(z)}}{q}) and \\frac{\\partial}{\\partial z} \\ddot{x}{(q,z)} = \\frac{\\cos{(z)}}{q} and \\frac{\\partial}{\\partial q} \\ddot{x}{(q,z)} \\frac{\\partial}{\\partial z} \\ddot{x}{(q,z)} = \\frac{\\partial}{\\partial q} \\frac{\\sin{(z)}}{q} \\frac{\\partial}{\\partial z} \\ddot{x}{(q,z)} and \\frac{\\cos{(z)} \\frac{\\partial}{\\partial q} \\ddot{x}{(q,z)}}{q} = \\frac{\\cos{(z)} \\frac{\\partial}{\\partial q} \\frac{\\sin{(z)}}{q}}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True)))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))))"], [["times", 2, "Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('q', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True)), Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), sin(Symbol('z', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mu)} = \\log{(\\mu)}, then derive \\frac{d}{d \\mu} \\operatorname{f^{\\prime}}{(\\mu)} = \\frac{1}{\\mu}, then obtain \\log{(\\mu)} + \\frac{d}{d \\mu} \\log{(\\mu)} = \\log{(\\mu)} + \\frac{1}{\\mu}", "derivation": "\\operatorname{f^{\\prime}}{(\\mu)} = \\log{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{f^{\\prime}}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{f^{\\prime}}{(\\mu)} = \\frac{1}{\\mu} and \\log{(\\mu)} + \\frac{d}{d \\mu} \\operatorname{f^{\\prime}}{(\\mu)} = \\log{(\\mu)} + \\frac{1}{\\mu} and \\log{(\\mu)} + \\frac{d}{d \\mu} \\log{(\\mu)} = \\log{(\\mu)} + \\frac{1}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], [["add", 3, "log(Symbol('\\\\mu', commutative=True))"], "Equality(Add(log(Symbol('\\\\mu', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(log(Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('\\\\mu', commutative=True)), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(log(Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(g,f)} = \\frac{\\partial}{\\partial g} (f + g), then derive - f + \\phi_{1}{(g,f)} = 1 - f, then obtain \\frac{- f + \\phi_{1}{(g,f)}}{g + 1} = \\frac{- f + \\frac{\\partial}{\\partial g} (f + g)}{g + 1}", "derivation": "\\phi_{1}{(g,f)} = \\frac{\\partial}{\\partial g} (f + g) and g + \\phi_{1}{(g,f)} = g + \\frac{\\partial}{\\partial g} (f + g) and - f + \\phi_{1}{(g,f)} = - f + \\frac{\\partial}{\\partial g} (f + g) and - f + \\phi_{1}{(g,f)} = 1 - f and \\frac{- f + \\phi_{1}{(g,f)}}{g + \\phi_{1}{(g,f)}} = \\frac{- f + \\frac{\\partial}{\\partial g} (f + g)}{g + \\phi_{1}{(g,f)}} and g + \\phi_{1}{(g,f)} = g + 1 and \\frac{- f + \\phi_{1}{(g,f)}}{g + 1} = \\frac{- f + \\frac{\\partial}{\\partial g} (f + g)}{g + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True)), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["add", 1, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Add(Symbol('g', commutative=True), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["minus", 2, "Add(Symbol('f', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["divide", 3, "Add(Symbol('g', commutative=True), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Pow(Add(Symbol('g', commutative=True), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Pow(Add(Symbol('g', commutative=True), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Integer(-1))))"], [["add", 4, "Add(Symbol('f', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Symbol('g', commutative=True), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Add(Symbol('g', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('g', commutative=True), Symbol('f', commutative=True))), Pow(Add(Symbol('g', commutative=True), Integer(1)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Pow(Add(Symbol('g', commutative=True), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}_0{(v_{1},g)} = \\frac{g}{v_{1}}, then obtain (- (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g} + \\hat{p}_0^{g}{(v_{1},g)})^{v_{1}} = ((\\frac{g}{v_{1}})^{g} - (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g})^{v_{1}}", "derivation": "\\hat{p}_0{(v_{1},g)} = \\frac{g}{v_{1}} and \\hat{p}_0^{g}{(v_{1},g)} = (\\frac{g}{v_{1}})^{g} and - (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g} + \\hat{p}_0^{g}{(v_{1},g)} = (\\frac{g}{v_{1}})^{g} - (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g} and (- (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g} + \\hat{p}_0^{g}{(v_{1},g)})^{v_{1}} = ((\\frac{g}{v_{1}})^{g} - (\\frac{g}{v_{1}})^{g} ((\\frac{g}{v_{1}})^{g})^{- g})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('v_1', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('v_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)))"], [["minus", 2, "Mul(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('v_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('v_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Symbol('v_1', commutative=True)), Pow(Add(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\omega{(M_{E})} = M_{E}, then derive \\int \\omega{(M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + n, then derive \\frac{t_{1}}{2} + \\frac{\\omega^{2}{(M_{E})}}{4} = \\frac{n}{2} + \\frac{\\omega^{2}{(M_{E})}}{4}, then obtain \\int (\\frac{t_{1}}{2} + \\frac{\\omega^{2}{(M_{E})}}{4}) dn = \\int (\\frac{n}{2} + \\frac{\\omega^{2}{(M_{E})}}{4}) dn", "derivation": "\\omega{(M_{E})} = M_{E} and \\int \\omega{(M_{E})} dM_{E} = \\int M_{E} dM_{E} and \\int \\omega{(M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + n and \\int \\omega{(M_{E})} d\\omega{(M_{E})} = n + \\frac{\\omega^{2}{(M_{E})}}{2} and \\frac{\\int \\omega{(M_{E})} d\\omega{(M_{E})}}{2} = \\frac{n}{2} + \\frac{\\omega^{2}{(M_{E})}}{4} and \\frac{t_{1}}{2} + \\frac{\\omega^{2}{(M_{E})}}{4} = \\frac{n}{2} + \\frac{\\omega^{2}{(M_{E})}}{4} and \\int (\\frac{t_{1}}{2} + \\frac{\\omega^{2}{(M_{E})}}{4}) dn = \\int (\\frac{n}{2} + \\frac{\\omega^{2}{(M_{E})}}{4}) dn", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\omega')(Symbol('M_E', commutative=True)), Tuple(Function('\\\\omega')(Symbol('M_E', commutative=True)))), Add(Symbol('n', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))))"], [["times", 4, "Rational(1, 2)"], "Equality(Mul(Rational(1, 2), Integral(Function('\\\\omega')(Symbol('M_E', commutative=True)), Tuple(Function('\\\\omega')(Symbol('M_E', commutative=True))))), Add(Mul(Rational(1, 2), Symbol('n', commutative=True)), Mul(Rational(1, 4), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Symbol('t_1', commutative=True)), Mul(Rational(1, 4), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))), Add(Mul(Rational(1, 2), Symbol('n', commutative=True)), Mul(Rational(1, 4), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))))"], [["integrate", 6, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Rational(1, 2), Symbol('t_1', commutative=True)), Mul(Rational(1, 4), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Rational(1, 2), Symbol('n', commutative=True)), Mul(Rational(1, 4), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True)), Integer(2)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\theta{(I)} = \\int \\cos{(I)} dI, then obtain \\frac{d}{d I} - \\frac{\\theta{(I)} \\cos{(I)}}{\\int \\cos{(I)} dI} = \\sin{(I)}", "derivation": "\\theta{(I)} = \\int \\cos{(I)} dI and I \\theta{(I)} = I \\int \\cos{(I)} dI and \\frac{\\theta{(I)}}{\\int \\cos{(I)} dI} = 1 and - \\frac{\\theta{(I)}}{\\int \\cos{(I)} dI} = -1 and - \\frac{\\theta{(I)} \\cos{(I)}}{\\int \\cos{(I)} dI} = - \\cos{(I)} and \\frac{d}{d I} - \\frac{\\theta{(I)} \\cos{(I)}}{\\int \\cos{(I)} dI} = \\frac{d}{d I} - \\cos{(I)} and \\frac{d}{d I} - \\frac{\\theta{(I)} \\cos{(I)}}{\\int \\cos{(I)} dI} = \\sin{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('I', commutative=True)), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\theta')(Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["divide", 2, "Mul(Symbol('I', commutative=True), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], "Equality(Mul(Function('\\\\theta')(Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(-1))), Integer(1))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta')(Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(-1))), Integer(-1))"], [["times", 4, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\theta')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(Symbol('I', commutative=True))))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\theta')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Mul(Integer(-1), Function('\\\\theta')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))), sin(Symbol('I', commutative=True)))"]]}, {"prompt": "Given g{(\\mathbf{B})} = e^{\\cos{(\\mathbf{B})}}, then derive \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} = - e^{\\cos{(\\mathbf{B})}} \\sin{(\\mathbf{B})}, then obtain \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} - 1 = - g{(\\mathbf{B})} \\sin{(\\mathbf{B})} - 1", "derivation": "g{(\\mathbf{B})} = e^{\\cos{(\\mathbf{B})}} and \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} e^{\\cos{(\\mathbf{B})}} and \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} = - e^{\\cos{(\\mathbf{B})}} \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} = - g{(\\mathbf{B})} \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} - 1 = \\frac{d}{d \\mathbf{B}} e^{\\cos{(\\mathbf{B})}} - 1 and - g{(\\mathbf{B})} \\sin{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} e^{\\cos{(\\mathbf{B})}} and \\frac{d}{d \\mathbf{B}} g{(\\mathbf{B})} - 1 = - g{(\\mathbf{B})} \\sin{(\\mathbf{B})} - 1", "srepr_derivation": [["get_premise", "Equality(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), exp(cos(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Mul(Integer(-1), Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(exp(cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(exp(cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Derivative(Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Function('g')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\ddot{x},V)} = V + \\ddot{x}, then obtain \\frac{(V + \\ddot{x})^{V}}{F_{x}} = \\frac{(\\frac{(V + \\ddot{x})^{2}}{\\operatorname{A_{x}}{(\\ddot{x},V)}})^{V}}{F_{x}}", "derivation": "\\operatorname{A_{x}}{(\\ddot{x},V)} = V + \\ddot{x} and 1 = \\frac{V + \\ddot{x}}{\\operatorname{A_{x}}{(\\ddot{x},V)}} and V + \\ddot{x} = \\frac{(V + \\ddot{x})^{2}}{\\operatorname{A_{x}}{(\\ddot{x},V)}} and (V + \\ddot{x})^{V} = (\\frac{(V + \\ddot{x})^{2}}{\\operatorname{A_{x}}{(\\ddot{x},V)}})^{V} and \\frac{(V + \\ddot{x})^{V}}{F_{x}} = \\frac{(\\frac{(V + \\ddot{x})^{2}}{\\operatorname{A_{x}}{(\\ddot{x},V)}})^{V}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["divide", 1, "Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Integer(-1))))"], [["divide", 2, "Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))"], "Equality(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('V', commutative=True)), Pow(Mul(Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Symbol('V', commutative=True)))"], [["divide", 4, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('V', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Mul(Pow(Add(Symbol('V', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then derive \\frac{d}{d \\mathbf{J}_P} \\operatorname{v_{y}}{(\\mathbf{J}_P)} = \\frac{1}{\\mathbf{J}_P}, then obtain \\int \\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\frac{1}{\\mathbf{J}_P} d\\mathbf{J}_P", "derivation": "\\operatorname{v_{y}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{v_{y}}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{v_{y}}{(\\mathbf{J}_P)} = \\frac{1}{\\mathbf{J}_P} and \\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)} = \\frac{1}{\\mathbf{J}_P} and \\int \\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\frac{1}{\\mathbf{J}_P} d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Derivative(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\mathbf{E}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} - 1, then obtain 0 = - \\operatorname{F_{N}}{(\\hat{x}_0)} + \\mathbf{E}{(\\hat{x}_0)} + 1", "derivation": "\\operatorname{F_{N}}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\operatorname{F_{N}}{(\\hat{x}_0)} - 1 = \\sin{(\\hat{x}_0)} - 1 and \\mathbf{E}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} - 1 and \\operatorname{F_{N}}{(\\hat{x}_0)} - 1 = \\mathbf{E}{(\\hat{x}_0)} and 0 = - \\operatorname{F_{N}}{(\\hat{x}_0)} + \\mathbf{E}{(\\hat{x}_0)} + 1", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('F_N')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Add(sin(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Add(sin(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('F_N')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 4, "Add(Function('F_N')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}_0', commutative=True))), Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(1)))"]]}, {"prompt": "Given p{(L)} = \\log{(e^{L})}, then obtain 0 = - \\frac{d}{d L} p{(L)} + \\frac{d}{d L} \\log{(e^{L})}", "derivation": "p{(L)} = \\log{(e^{L})} and \\frac{d}{d L} p{(L)} = \\frac{d}{d L} \\log{(e^{L})} and - e^{L} + \\frac{d}{d L} p{(L)} = - e^{L} + \\frac{d}{d L} \\log{(e^{L})} and 0 = - \\frac{d}{d L} p{(L)} + \\frac{d}{d L} \\log{(e^{L})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('L', commutative=True)), log(exp(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('L', commutative=True))), Derivative(Function('p')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('L', commutative=True))), Derivative(log(exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["minus", 3, "Add(Mul(Integer(-1), exp(Symbol('L', commutative=True))), Derivative(Function('p')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('p')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Derivative(log(exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(H,\\hat{X})} = H^{\\hat{X}}, then obtain - H^{\\hat{X}} - \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} 2 Q{(H,\\hat{X})} = - H^{\\hat{X}} - \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} (H^{\\hat{X}} + Q{(H,\\hat{X})})", "derivation": "Q{(H,\\hat{X})} = H^{\\hat{X}} and 2 Q{(H,\\hat{X})} = H^{\\hat{X}} + Q{(H,\\hat{X})} and \\frac{\\partial}{\\partial \\hat{X}} 2 Q{(H,\\hat{X})} = \\frac{\\partial}{\\partial \\hat{X}} (H^{\\hat{X}} + Q{(H,\\hat{X})}) and - H^{\\hat{X}} + \\frac{\\partial}{\\partial \\hat{X}} 2 Q{(H,\\hat{X})} = - H^{\\hat{X}} + \\frac{\\partial}{\\partial \\hat{X}} (H^{\\hat{X}} + Q{(H,\\hat{X})}) and - H^{\\hat{X}} - \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} 2 Q{(H,\\hat{X})} = - H^{\\hat{X}} - \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} (H^{\\hat{X}} + Q{(H,\\hat{X})})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 1, "Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["minus", 3, "Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Derivative(Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["minus", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Derivative(Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('Q')(Symbol('H', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(F_{g},F_{c},P_{e})} = F_{g} (- F_{c} + P_{e}) and n{(F_{c})} = F_{c}, then obtain - F_{c} + \\frac{n{(F_{c})}}{P_{e}} = - F_{c} + \\frac{F_{c}}{P_{e}}", "derivation": "\\varphi^{*}{(F_{g},F_{c},P_{e})} = F_{g} (- F_{c} + P_{e}) and n{(F_{c})} = F_{c} and \\frac{n{(F_{c})}}{P_{e}} = \\frac{F_{c}}{P_{e}} and - F_{c} - F_{g} (- F_{c} + P_{e}) + \\varphi^{*}{(F_{g},F_{c},P_{e})} + \\frac{n{(F_{c})}}{P_{e}} = - F_{c} + \\frac{F_{c}}{P_{e}} - F_{g} (- F_{c} + P_{e}) + \\varphi^{*}{(F_{g},F_{c},P_{e})} and - F_{c} + \\frac{n{(F_{c})}}{P_{e}} = - F_{c} + \\frac{F_{c}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('F_c', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["divide", 2, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('n')(Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1))))"], [["minus", 3, "Add(Symbol('F_c', commutative=True), Mul(Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('P_e', commutative=True))), Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('F_c', commutative=True), Symbol('P_e', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('P_e', commutative=True))), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('F_c', commutative=True), Symbol('P_e', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('n')(Symbol('F_c', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('P_e', commutative=True))), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('F_c', commutative=True), Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('n')(Symbol('F_c', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(M_{E},I)} = - I + M_{E}, then obtain (\\int 1 dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}})^{2} = (\\int 1 dI + 1) (\\int 1 dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}})", "derivation": "\\operatorname{F_{N}}{(M_{E},I)} = - I + M_{E} and \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} = 1 and \\int \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} dI = \\int 1 dI and \\int \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} = \\int \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} dI + 1 and \\int 1 dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}} = \\int 1 dI + 1 and (\\int 1 dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}})^{2} = (\\int 1 dI + 1) (\\int 1 dI + \\frac{\\operatorname{F_{N}}{(M_{E},I)}}{- I + M_{E}})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Integer(1), Tuple(Symbol('I', commutative=True))))"], [["add", 2, "Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)))), Add(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integer(1)))"], [["times", 5, "Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))))"], "Equality(Pow(Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)))), Integer(2)), Mul(Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('M_E', commutative=True)), Integer(-1)), Function('F_N')(Symbol('M_E', commutative=True), Symbol('I', commutative=True))))))"]]}, {"prompt": "Given k{(z^{*},\\varepsilon,n)} = (- \\varepsilon + n)^{z^{*}}, then derive \\frac{\\partial}{\\partial z^{*}} k{(z^{*},\\varepsilon,n)} = (- \\varepsilon + n)^{z^{*}} \\log{(- \\varepsilon + n)}, then obtain \\frac{\\partial}{\\partial z^{*}} k{(z^{*},\\varepsilon,n)} = k{(z^{*},\\varepsilon,n)} \\log{(- \\varepsilon + n)}", "derivation": "k{(z^{*},\\varepsilon,n)} = (- \\varepsilon + n)^{z^{*}} and \\frac{\\partial}{\\partial z^{*}} k{(z^{*},\\varepsilon,n)} = \\frac{\\partial}{\\partial z^{*}} (- \\varepsilon + n)^{z^{*}} and \\frac{\\partial}{\\partial z^{*}} k{(z^{*},\\varepsilon,n)} = (- \\varepsilon + n)^{z^{*}} \\log{(- \\varepsilon + n)} and \\frac{\\partial}{\\partial z^{*}} k{(z^{*},\\varepsilon,n)} = k{(z^{*},\\varepsilon,n)} \\log{(- \\varepsilon + n)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True)), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True)), Symbol('z^*', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('k')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Function('k')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\Psi)} = \\frac{1}{\\Psi}, then obtain (\\frac{1}{\\Psi})^{\\Psi} + \\tilde{g}^*^{\\Psi}{(\\Psi)} + 1 = 2 (\\frac{1}{\\Psi})^{\\Psi} + 1", "derivation": "\\tilde{g}^*{(\\Psi)} = \\frac{1}{\\Psi} and \\tilde{g}^*^{\\Psi}{(\\Psi)} = (\\frac{1}{\\Psi})^{\\Psi} and (\\frac{1}{\\Psi})^{\\Psi} + \\tilde{g}^*^{\\Psi}{(\\Psi)} = 2 (\\frac{1}{\\Psi})^{\\Psi} and (\\frac{1}{\\Psi})^{\\Psi} + \\tilde{g}^*^{\\Psi}{(\\Psi)} + 1 = 2 (\\frac{1}{\\Psi})^{\\Psi} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Integer(1)), Add(Mul(Integer(2), Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(n_{1})} = \\log{(n_{1})}, then derive \\frac{d}{d n_{1}} \\operatorname{P_{e}}{(n_{1})} = \\frac{1}{n_{1}}, then obtain \\frac{\\frac{d}{d n_{1}} \\operatorname{P_{e}}{(n_{1})}}{n_{1}} = \\frac{1}{n_{1}^{2}}", "derivation": "\\operatorname{P_{e}}{(n_{1})} = \\log{(n_{1})} and \\frac{d}{d n_{1}} \\operatorname{P_{e}}{(n_{1})} = \\frac{d}{d n_{1}} \\log{(n_{1})} and \\frac{d}{d n_{1}} \\operatorname{P_{e}}{(n_{1})} = \\frac{1}{n_{1}} and \\frac{d}{d n_{1}} \\log{(n_{1})} = \\frac{1}{n_{1}} and \\frac{\\frac{d}{d n_{1}} \\log{(n_{1})}}{n_{1}} = \\frac{1}{n_{1}^{2}} and \\frac{\\frac{d}{d n_{1}} \\operatorname{P_{e}}{(n_{1})}}{n_{1}} = \\frac{1}{n_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('n_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('n_1', commutative=True), Integer(-1)))"], [["times", 4, "Pow(Symbol('n_1', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Derivative(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Pow(Symbol('n_1', commutative=True), Integer(-2)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Derivative(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Pow(Symbol('n_1', commutative=True), Integer(-2)))"]]}, {"prompt": "Given W{(k,t_{2})} = \\log{(k)}^{t_{2}}, then derive - \\frac{\\frac{\\partial}{\\partial t_{2}} W{(k,t_{2})}}{\\log{(k)}} = - \\frac{\\log{(k)}^{t_{2}} \\log{(\\log{(k)})}}{\\log{(k)}}, then obtain - \\frac{\\frac{\\partial}{\\partial t_{2}} \\log{(k)}^{t_{2}}}{\\log{(k)}} = - \\frac{\\frac{\\partial}{\\partial t_{2}} W{(k,t_{2})}}{\\log{(k)}}", "derivation": "W{(k,t_{2})} = \\log{(k)}^{t_{2}} and - \\frac{W{(k,t_{2})}}{\\log{(k)}} = - \\frac{\\log{(k)}^{t_{2}}}{\\log{(k)}} and \\frac{\\partial}{\\partial t_{2}} - \\frac{W{(k,t_{2})}}{\\log{(k)}} = \\frac{\\partial}{\\partial t_{2}} - \\frac{\\log{(k)}^{t_{2}}}{\\log{(k)}} and - \\frac{\\frac{\\partial}{\\partial t_{2}} W{(k,t_{2})}}{\\log{(k)}} = - \\frac{\\log{(k)}^{t_{2}} \\log{(\\log{(k)})}}{\\log{(k)}} and - \\frac{\\frac{\\partial}{\\partial t_{2}} \\log{(k)}^{t_{2}}}{\\log{(k)}} = - \\frac{\\log{(k)}^{t_{2}} \\log{(\\log{(k)})}}{\\log{(k)}} and - \\frac{\\frac{\\partial}{\\partial t_{2}} \\log{(k)}^{t_{2}}}{\\log{(k)}} = - \\frac{\\frac{\\partial}{\\partial t_{2}} W{(k,t_{2})}}{\\log{(k)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('k', commutative=True), Symbol('t_2', commutative=True)), Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), log(Symbol('k', commutative=True)))"], "Equality(Mul(Integer(-1), Function('W')(Symbol('k', commutative=True), Symbol('t_2', commutative=True)), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('W')(Symbol('k', commutative=True), Symbol('t_2', commutative=True)), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('k', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True)), log(log(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True)), log(log(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Pow(log(Symbol('k', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('k', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(r)} = \\cos{(r)} and \\operatorname{F_{g}}{(r)} = \\frac{d}{d r} \\operatorname{r_{0}}{(r)}, then obtain \\int \\operatorname{F_{g}}{(r)} dr + \\int \\frac{d}{d r} \\cos{(r)} dr = 2 \\int \\frac{d}{d r} \\cos{(r)} dr", "derivation": "\\operatorname{r_{0}}{(r)} = \\cos{(r)} and \\frac{d}{d r} \\operatorname{r_{0}}{(r)} = \\frac{d}{d r} \\cos{(r)} and \\operatorname{F_{g}}{(r)} = \\frac{d}{d r} \\operatorname{r_{0}}{(r)} and \\operatorname{F_{g}}{(r)} = \\frac{d}{d r} \\cos{(r)} and \\int \\operatorname{F_{g}}{(r)} dr = \\int \\frac{d}{d r} \\cos{(r)} dr and \\int \\operatorname{F_{g}}{(r)} dr + \\int \\frac{d}{d r} \\cos{(r)} dr = 2 \\int \\frac{d}{d r} \\cos{(r)} dr", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('r', commutative=True)), Derivative(Function('r_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_g')(Symbol('r', commutative=True)), Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["add", 5, "Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True)))"], "Equality(Add(Integral(Function('F_g')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True)))), Mul(Integer(2), Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\theta)} = \\cos{(e^{\\theta})} and \\hat{x}_0{(q,y)} = q y, then obtain \\theta + 2 \\operatorname{A_{x}}{(\\theta)} + \\hat{x}_0{(q,y)} = \\theta + q y + 2 \\operatorname{A_{x}}{(\\theta)}", "derivation": "\\operatorname{A_{x}}{(\\theta)} = \\cos{(e^{\\theta})} and \\hat{x}_0{(q,y)} = q y and \\operatorname{A_{x}}{(\\theta)} + \\hat{x}_0{(q,y)} = q y + \\operatorname{A_{x}}{(\\theta)} and \\hat{x}_0{(q,y)} + \\cos{(e^{\\theta})} = q y + \\cos{(e^{\\theta})} and \\theta + \\operatorname{A_{x}}{(\\theta)} + \\hat{x}_0{(q,y)} + \\cos{(e^{\\theta})} = \\theta + q y + \\operatorname{A_{x}}{(\\theta)} + \\cos{(e^{\\theta})} and \\theta + \\hat{x}_0{(q,y)} + 2 \\cos{(e^{\\theta})} = \\theta + q y + 2 \\cos{(e^{\\theta})} and \\theta + 2 \\operatorname{A_{x}}{(\\theta)} + \\hat{x}_0{(q,y)} = \\theta + q y + 2 \\operatorname{A_{x}}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\theta', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)))"], [["add", 2, "Function('A_x')(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('A_x')(Symbol('\\\\theta', commutative=True)), Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)), Function('A_x')(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True)))), Add(Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["add", 4, "Add(Symbol('\\\\theta', commutative=True), Function('A_x')(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('A_x')(Symbol('\\\\theta', commutative=True)), Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)), Function('A_x')(Symbol('\\\\theta', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\theta', commutative=True))))), Add(Symbol('\\\\theta', commutative=True), Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\theta', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), Function('A_x')(Symbol('\\\\theta', commutative=True))), Function('\\\\hat{x}_0')(Symbol('q', commutative=True), Symbol('y', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Symbol('q', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(b,W)} = \\frac{\\partial}{\\partial b} (W - b), then derive \\operatorname{n_{2}}{(b,W)} = -1, then derive \\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)} = 0, then obtain \\frac{\\partial}{\\partial b} \\sin{((\\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)})^{b})} = \\frac{d}{d b} \\sin{((\\frac{d}{d b} (-1))^{b})}", "derivation": "\\operatorname{n_{2}}{(b,W)} = \\frac{\\partial}{\\partial b} (W - b) and \\operatorname{n_{2}}{(b,W)} = -1 and \\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)} = \\frac{d}{d b} (-1) and \\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)} = 0 and (\\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)})^{b} = 0^{b} and \\sin{((\\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)})^{b})} = \\sin{(0^{b})} and \\sin{((\\frac{d}{d b} (-1))^{b})} = \\sin{(0^{b})} and \\sin{((\\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)})^{b})} = \\sin{((\\frac{d}{d b} (-1))^{b})} and \\frac{\\partial}{\\partial b} \\sin{((\\frac{\\partial}{\\partial b} \\operatorname{n_{2}}{(b,W)})^{b})} = \\frac{d}{d b} \\sin{((\\frac{d}{d b} (-1))^{b})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Derivative(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Integer(-1))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(0))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True)), Pow(Integer(0), Symbol('b', commutative=True)))"], [["sin", 5], "Equality(sin(Pow(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))), sin(Pow(Integer(0), Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(sin(Pow(Derivative(Integer(-1), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))), sin(Pow(Integer(0), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(sin(Pow(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))), sin(Pow(Derivative(Integer(-1), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))))"], [["differentiate", 8, "Symbol('b', commutative=True)"], "Equality(Derivative(sin(Pow(Derivative(Function('n_2')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Pow(Derivative(Integer(-1), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(l,\\mathbf{v})} = \\frac{\\mathbf{v}}{l} and t{(\\mathbf{v},l)} = 2 \\operatorname{F_{c}}{(l,\\mathbf{v})}, then obtain (t{(\\mathbf{v},l)} - 1)^{\\mathbf{v}} = (\\frac{2 \\mathbf{v}}{l} - 1)^{\\mathbf{v}}", "derivation": "\\operatorname{F_{c}}{(l,\\mathbf{v})} = \\frac{\\mathbf{v}}{l} and \\operatorname{F_{c}}{(l,\\mathbf{v})} - 1 = \\frac{\\mathbf{v}}{l} - 1 and 2 \\operatorname{F_{c}}{(l,\\mathbf{v})} - 1 = \\frac{\\mathbf{v}}{l} + \\operatorname{F_{c}}{(l,\\mathbf{v})} - 1 and t{(\\mathbf{v},l)} = 2 \\operatorname{F_{c}}{(l,\\mathbf{v})} and t{(\\mathbf{v},l)} - 1 = \\frac{\\mathbf{v}}{l} + \\operatorname{F_{c}}{(l,\\mathbf{v})} - 1 and (t{(\\mathbf{v},l)} - 1)^{\\mathbf{v}} = (\\frac{\\mathbf{v}}{l} + \\operatorname{F_{c}}{(l,\\mathbf{v})} - 1)^{\\mathbf{v}} and (t{(\\mathbf{v},l)} - 1)^{\\mathbf{v}} = (\\frac{2 \\mathbf{v}}{l} - 1)^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Integer(-1)))"], [["add", 1, "Add(Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(2), Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('l', commutative=True)), Mul(Integer(2), Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)))"], [["power", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Add(Function('t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('F_c')(Symbol('l', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Add(Function('t')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(v_{t})} = \\cos{(\\sin{(v_{t})})} and E{(v_{t})} = \\cos{(\\sin{(v_{t})})}, then obtain (\\int \\cos{(\\sin{(v_{t})})} dv_{t})^{v_{t}} = (\\int E{(v_{t})} dv_{t})^{v_{t}}", "derivation": "\\operatorname{v_{y}}{(v_{t})} = \\cos{(\\sin{(v_{t})})} and \\int \\operatorname{v_{y}}{(v_{t})} dv_{t} = \\int \\cos{(\\sin{(v_{t})})} dv_{t} and E{(v_{t})} = \\cos{(\\sin{(v_{t})})} and \\int E{(v_{t})} dv_{t} = \\int \\cos{(\\sin{(v_{t})})} dv_{t} and \\int \\operatorname{v_{y}}{(v_{t})} dv_{t} = \\int E{(v_{t})} dv_{t} and (\\int \\operatorname{v_{y}}{(v_{t})} dv_{t})^{v_{t}} = (\\int E{(v_{t})} dv_{t})^{v_{t}} and (\\int \\cos{(\\sin{(v_{t})})} dv_{t})^{v_{t}} = (\\int E{(v_{t})} dv_{t})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('v_t', commutative=True)), cos(sin(Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(cos(sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('v_t', commutative=True)), cos(sin(Symbol('v_t', commutative=True))))"], [["integrate", 3, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('E')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(cos(sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('v_y')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Function('E')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["power", 5, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Function('v_y')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Function('E')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Integral(cos(sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Function('E')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(v,f)} = f - v, then obtain - 2 v (f - v) (2 f - 2 v)^{2} = - v (2 f - 2 v)^{3}", "derivation": "\\operatorname{v_{z}}{(v,f)} = f - v and f - v + \\operatorname{v_{z}}{(v,f)} = 2 f - 2 v and 2 \\operatorname{v_{z}}{(v,f)} = 2 f - 2 v and - 2 v \\operatorname{v_{z}}{(v,f)} = - v (2 f - 2 v) and - 2 v (f - v) = - v (2 f - 2 v) and - 2 v (f - v) (2 f - 2 v)^{2} = - v (2 f - 2 v)^{3}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('v', commutative=True), Symbol('f', commutative=True)), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["add", 1, "Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))"], "Equality(Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Function('v_z')(Symbol('v', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('v_z')(Symbol('v', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('v', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True), Function('v_z')(Symbol('v', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Mul(Integer(-1), Symbol('v', commutative=True), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))))"], [["divide", 5, "Pow(Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True))), Integer(-2))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True))), Integer(2))), Mul(Integer(-1), Symbol('v', commutative=True), Pow(Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True))), Integer(3))))"]]}, {"prompt": "Given Q{(u,C_{d})} = \\log{(u^{C_{d}})} and \\operatorname{A_{x}}{(\\mathbf{f},f_{E})} = \\mathbf{f} + f_{E}, then obtain \\int \\frac{\\operatorname{A_{x}}{(\\mathbf{f},f_{E})}}{Q{(u,C_{d})}} du = \\int \\frac{\\mathbf{f} + f_{E}}{Q{(u,C_{d})}} du", "derivation": "Q{(u,C_{d})} = \\log{(u^{C_{d}})} and \\operatorname{A_{x}}{(\\mathbf{f},f_{E})} = \\mathbf{f} + f_{E} and \\frac{\\operatorname{A_{x}}{(\\mathbf{f},f_{E})}}{\\log{(u^{C_{d}})}} = \\frac{\\mathbf{f} + f_{E}}{\\log{(u^{C_{d}})}} and \\frac{\\operatorname{A_{x}}{(\\mathbf{f},f_{E})}}{Q{(u,C_{d})}} = \\frac{\\mathbf{f} + f_{E}}{Q{(u,C_{d})}} and \\int \\frac{\\operatorname{A_{x}}{(\\mathbf{f},f_{E})}}{Q{(u,C_{d})}} du = \\int \\frac{\\mathbf{f} + f_{E}}{Q{(u,C_{d})}} du", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('u', commutative=True), Symbol('C_d', commutative=True)), log(Pow(Symbol('u', commutative=True), Symbol('C_d', commutative=True))))"], ["get_premise", "Equality(Function('A_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)))"], [["divide", 2, "log(Pow(Symbol('u', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Pow(Symbol('u', commutative=True), Symbol('C_d', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Pow(Symbol('u', commutative=True), Symbol('C_d', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('A_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('Q')(Symbol('u', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('Q')(Symbol('u', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Function('A_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('Q')(Symbol('u', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('Q')(Symbol('u', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(C,\\omega)} = \\sin{(\\frac{\\omega}{C})}, then obtain e^{\\frac{\\operatorname{P_{g}}^{C}{(C,\\omega)}}{\\omega}} = e^{\\frac{\\sin^{C}{(\\frac{\\omega}{C})}}{\\omega}}", "derivation": "\\operatorname{P_{g}}{(C,\\omega)} = \\sin{(\\frac{\\omega}{C})} and \\operatorname{P_{g}}^{C}{(C,\\omega)} = \\sin^{C}{(\\frac{\\omega}{C})} and \\frac{\\operatorname{P_{g}}^{C}{(C,\\omega)}}{\\omega} = \\frac{\\sin^{C}{(\\frac{\\omega}{C})}}{\\omega} and e^{\\frac{\\operatorname{P_{g}}^{C}{(C,\\omega)}}{\\omega}} = e^{\\frac{\\sin^{C}{(\\frac{\\omega}{C})}}{\\omega}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('C', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Symbol('C', commutative=True)))"], [["divide", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('C', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Symbol('C', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('C', commutative=True)))), exp(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} = \\sin{(\\cos{(v_{1})})} and p{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(v_{1})}, then obtain p{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} \\sin{(\\cos{(v_{1})})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} = \\sin{(\\cos{(v_{1})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(v_{1})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} \\sin{(\\cos{(v_{1})})} and \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} \\sin{(\\cos{(v_{1})})} and p{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(v_{1})} and p{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{1})} \\sin{(\\cos{(v_{1})})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(cos(Symbol('v_1', commutative=True))))"], [["times", 1, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True))"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(2)), Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(cos(Symbol('v_1', commutative=True)))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(2)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(cos(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('p')(Symbol('v_1', commutative=True)), Derivative(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(2)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('p')(Symbol('v_1', commutative=True)), Derivative(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(cos(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\dot{x},z)} = \\frac{z}{\\dot{x}}, then obtain \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} = \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} \\sin^{z}{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})}", "derivation": "b{(\\dot{x},z)} = \\frac{z}{\\dot{x}} and b{(\\dot{x},z)} - \\frac{z}{\\dot{x}} = 0 and \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} = 0 and \\sin^{z}{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} = 0^{z} and \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} \\sin^{z}{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} = 0^{z} \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} and \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} = \\sin{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})} \\sin^{z}{(b{(\\dot{x},z)} - \\frac{z}{\\dot{x}})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True))"], "Equality(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Integer(0))"], [["sin", 2], "Equality(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Integer(0), Symbol('z', commutative=True)))"], [["times", 4, "sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], "Equality(Mul(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Pow(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Symbol('z', commutative=True))), Mul(Pow(Integer(0), Symbol('z', commutative=True)), sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Mul(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Pow(sin(Add(Function('b')(Symbol('\\\\dot{x}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(E)} = \\frac{d}{d E} \\log{(E)}, then derive \\operatorname{v_{1}}{(E)} = \\frac{1}{E}, then obtain \\log{(\\frac{d}{d E} \\log{(E)})} = \\log{(\\operatorname{v_{1}}{(\\frac{1}{\\frac{d}{d E} \\log{(E)}})})}", "derivation": "\\operatorname{v_{1}}{(E)} = \\frac{d}{d E} \\log{(E)} and \\operatorname{v_{1}}{(E)} = \\frac{1}{E} and \\log{(\\operatorname{v_{1}}{(E)})} = \\log{(\\frac{d}{d E} \\log{(E)})} and \\log{(\\frac{1}{E})} = \\log{(\\frac{d}{d E} \\log{(E)})} and \\frac{1}{E} = \\frac{d}{d E} \\log{(E)} and \\log{(\\frac{1}{E})} = \\log{(\\operatorname{v_{1}}{(E)})} and \\log{(\\frac{d}{d E} \\log{(E)})} = \\log{(\\operatorname{v_{1}}{(\\frac{1}{\\frac{d}{d E} \\log{(E)}})})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('E', commutative=True)), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_1')(Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1)))"], [["log", 1], "Equality(log(Function('v_1')(Symbol('E', commutative=True))), log(Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(log(Pow(Symbol('E', commutative=True), Integer(-1))), log(Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(log(Pow(Symbol('E', commutative=True), Integer(-1))), log(Function('v_1')(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(log(Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Function('v_1')(Pow(Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(G,k)} = G k, then derive \\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)} = G, then obtain k + \\nabla{(G,k)} + \\int 1 dk + 1 = k + (\\frac{\\frac{\\partial}{\\partial k} G k}{G})^{G} + \\nabla{(G,k)} + \\int 1 dk", "derivation": "\\operatorname{n_{1}}{(G,k)} = G k and \\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)} = \\frac{\\partial}{\\partial k} G k and 1 = \\frac{\\frac{\\partial}{\\partial k} G k}{\\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)}} and \\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)} = G and 1 = (\\frac{\\frac{\\partial}{\\partial k} G k}{\\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)}})^{G} and k + \\nabla{(G,k)} + 1 = k + (\\frac{\\frac{\\partial}{\\partial k} G k}{\\frac{\\partial}{\\partial k} \\operatorname{n_{1}}{(G,k)}})^{G} + \\nabla{(G,k)} and k + \\nabla{(G,k)} + 1 = k + (\\frac{\\frac{\\partial}{\\partial k} G k}{G})^{G} + \\nabla{(G,k)} and k + \\nabla{(G,k)} + \\int 1 dk + 1 = k + (\\frac{\\frac{\\partial}{\\partial k} G k}{G})^{G} + \\nabla{(G,k)} + \\int 1 dk", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Pow(Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('G', commutative=True))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Integer(1), Pow(Mul(Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Pow(Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))), Symbol('G', commutative=True)))"], [["add", 5, "Add(Symbol('k', commutative=True), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True)))"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Integer(1)), Add(Symbol('k', commutative=True), Pow(Mul(Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Pow(Derivative(Function('n_1')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))), Symbol('G', commutative=True)), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Symbol('k', commutative=True), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Integer(1)), Add(Symbol('k', commutative=True), Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Symbol('G', commutative=True)), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True))))"], [["add", 7, "Integral(Integer(1), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Integral(Integer(1), Tuple(Symbol('k', commutative=True))), Integer(1)), Add(Symbol('k', commutative=True), Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(Mul(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Symbol('G', commutative=True)), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Integral(Integer(1), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given s{(E,f^{*})} = (f^{*})^{E}, then obtain - \\frac{\\partial}{\\partial f^{*}} \\frac{s^{f^{*}}{(E,f^{*})}}{s{(E,f^{*})}} = - \\frac{\\partial}{\\partial f^{*}} \\frac{((f^{*})^{E})^{f^{*}}}{s{(E,f^{*})}}", "derivation": "s{(E,f^{*})} = (f^{*})^{E} and s^{f^{*}}{(E,f^{*})} = ((f^{*})^{E})^{f^{*}} and \\frac{s^{f^{*}}{(E,f^{*})}}{s{(E,f^{*})}} = \\frac{((f^{*})^{E})^{f^{*}}}{s{(E,f^{*})}} and \\frac{\\partial}{\\partial f^{*}} \\frac{s^{f^{*}}{(E,f^{*})}}{s{(E,f^{*})}} = \\frac{\\partial}{\\partial f^{*}} \\frac{((f^{*})^{E})^{f^{*}}}{s{(E,f^{*})}} and - \\frac{\\partial}{\\partial f^{*}} \\frac{s^{f^{*}}{(E,f^{*})}}{s{(E,f^{*})}} = - \\frac{\\partial}{\\partial f^{*}} \\frac{((f^{*})^{E})^{f^{*}}}{s{(E,f^{*})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('E', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('E', commutative=True)), Symbol('f^*', commutative=True)))"], [["divide", 2, "Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Mul(Pow(Pow(Symbol('f^*', commutative=True), Symbol('E', commutative=True)), Symbol('f^*', commutative=True)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Pow(Symbol('f^*', commutative=True), Symbol('E', commutative=True)), Symbol('f^*', commutative=True)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Mul(Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Pow(Symbol('f^*', commutative=True), Symbol('E', commutative=True)), Symbol('f^*', commutative=True)), Pow(Function('s')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and V{(\\mathbf{E})} = (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2}, then obtain V{(\\mathbf{E})} + \\sin{(\\mathbf{E})} = \\sin{(\\mathbf{E})} + (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2}", "derivation": "\\mathbf{B}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and \\mathbf{B}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} = (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2} and V{(\\mathbf{E})} = (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2} and V{(\\mathbf{E})} = \\mathbf{B}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and V{(\\mathbf{E})} + \\sin{(\\mathbf{E})} = \\mathbf{B}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} + \\sin{(\\mathbf{E})} and V{(\\mathbf{E})} + \\sin{(\\mathbf{E})} = \\sin{(\\mathbf{E})} + (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2)))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["add", 4, "sin(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), sin(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True))), Add(sin(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{D},\\lambda)} = e^{\\lambda^{\\mathbf{D}}}, then derive \\frac{\\partial}{\\partial \\lambda} \\mathbf{B}{(\\mathbf{D},\\lambda)} = \\frac{\\lambda^{\\mathbf{D}} \\mathbf{D} e^{\\lambda^{\\mathbf{D}}}}{\\lambda}, then obtain \\int \\frac{\\lambda^{\\mathbf{D}} \\mathbf{D} e^{\\lambda^{\\mathbf{D}}}}{\\lambda} d\\lambda = \\int \\frac{\\partial}{\\partial \\lambda} e^{\\lambda^{\\mathbf{D}}} d\\lambda", "derivation": "\\mathbf{B}{(\\mathbf{D},\\lambda)} = e^{\\lambda^{\\mathbf{D}}} and \\frac{\\partial}{\\partial \\lambda} \\mathbf{B}{(\\mathbf{D},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} e^{\\lambda^{\\mathbf{D}}} and \\frac{\\partial}{\\partial \\lambda} \\mathbf{B}{(\\mathbf{D},\\lambda)} = \\frac{\\lambda^{\\mathbf{D}} \\mathbf{D} e^{\\lambda^{\\mathbf{D}}}}{\\lambda} and \\frac{\\lambda^{\\mathbf{D}} \\mathbf{D} e^{\\lambda^{\\mathbf{D}}}}{\\lambda} = \\frac{\\partial}{\\partial \\lambda} e^{\\lambda^{\\mathbf{D}}} and \\int \\frac{\\lambda^{\\mathbf{D}} \\mathbf{D} e^{\\lambda^{\\mathbf{D}}}}{\\lambda} d\\lambda = \\int \\frac{\\partial}{\\partial \\lambda} e^{\\lambda^{\\mathbf{D}}} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\lambda', commutative=True)), exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Derivative(exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Derivative(exp(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(G)} = \\sin{(\\sin{(G)})}, then obtain \\int \\frac{d}{d G} - \\int \\sin{(\\sin{(G)})} dG dG = \\int \\frac{d}{d G} (- \\operatorname{f^{\\prime}}{(G)} + \\sin{(\\sin{(G)})} - \\int \\sin{(\\sin{(G)})} dG) dG", "derivation": "\\operatorname{f^{\\prime}}{(G)} = \\sin{(\\sin{(G)})} and - \\int \\sin{(\\sin{(G)})} dG = - \\operatorname{f^{\\prime}}{(G)} + \\sin{(\\sin{(G)})} - \\int \\sin{(\\sin{(G)})} dG and \\frac{d}{d G} - \\int \\sin{(\\sin{(G)})} dG = \\frac{d}{d G} (- \\operatorname{f^{\\prime}}{(G)} + \\sin{(\\sin{(G)})} - \\int \\sin{(\\sin{(G)})} dG) and \\int \\frac{d}{d G} - \\int \\sin{(\\sin{(G)})} dG dG = \\int \\frac{d}{d G} (- \\operatorname{f^{\\prime}}{(G)} + \\sin{(\\sin{(G)})} - \\int \\sin{(\\sin{(G)})} dG) dG", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], [["minus", 1, "Add(Function('f^{\\\\prime}')(Symbol('G', commutative=True)), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], "Equality(Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True))), Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True))), Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True))), Mul(Integer(-1), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(m)} = \\sin{(m)} and \\operatorname{J_{\\varepsilon}}{(m)} = 1 - \\frac{1}{\\mathbb{I}{(m)}}, then obtain (\\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)})^{2} = \\frac{d}{d m} (1 - \\frac{1}{\\mathbb{I}{(m)}}) \\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)}", "derivation": "\\mathbb{I}{(m)} = \\sin{(m)} and \\operatorname{J_{\\varepsilon}}{(m)} = 1 - \\frac{1}{\\mathbb{I}{(m)}} and \\operatorname{J_{\\varepsilon}}{(m)} = 1 - \\frac{1}{\\sin{(m)}} and \\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)} = \\frac{d}{d m} (1 - \\frac{1}{\\sin{(m)}}) and \\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)} = \\frac{d}{d m} (1 - \\frac{1}{\\mathbb{I}{(m)}}) and (\\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)})^{2} = \\frac{d}{d m} (1 - \\frac{1}{\\mathbb{I}{(m)}}) \\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Integer(-1)))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Integer(-1)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Integer(-1)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["times", 5, "Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Integer(-1)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain \\mathbf{J}_P \\cos{(\\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)})} = \\mathbf{J}_P \\cos{(\\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P})}", "derivation": "\\mu{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} and \\cos{(\\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)})} = \\cos{(\\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P})} and e^{- \\mathbf{J}_P} \\cos{(\\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)})} = e^{- \\mathbf{J}_P} \\cos{(\\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P})} and \\mathbf{J}_P \\cos{(\\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)})} = \\mathbf{J}_P \\cos{(\\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), cos(Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["divide", 3, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))))"], [["times", 4, "Mul(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(t_{1},n)} = \\frac{\\partial}{\\partial n} n t_{1}, then obtain \\frac{\\partial^{2}}{\\partial n^{2}} \\int \\operatorname{P_{g}}^{2}{(t_{1},n)} dt_{1} = \\frac{\\partial^{2}}{\\partial n^{2}} \\int \\operatorname{P_{g}}{(t_{1},n)} \\frac{\\partial}{\\partial n} n t_{1} dt_{1}", "derivation": "\\operatorname{P_{g}}{(t_{1},n)} = \\frac{\\partial}{\\partial n} n t_{1} and \\operatorname{P_{g}}^{2}{(t_{1},n)} = \\operatorname{P_{g}}{(t_{1},n)} \\frac{\\partial}{\\partial n} n t_{1} and \\int \\operatorname{P_{g}}^{2}{(t_{1},n)} dt_{1} = \\int \\operatorname{P_{g}}{(t_{1},n)} \\frac{\\partial}{\\partial n} n t_{1} dt_{1} and \\frac{\\partial}{\\partial n} \\int \\operatorname{P_{g}}^{2}{(t_{1},n)} dt_{1} = \\frac{\\partial}{\\partial n} \\int \\operatorname{P_{g}}{(t_{1},n)} \\frac{\\partial}{\\partial n} n t_{1} dt_{1} and \\frac{\\partial^{2}}{\\partial n^{2}} \\int \\operatorname{P_{g}}^{2}{(t_{1},n)} dt_{1} = \\frac{\\partial^{2}}{\\partial n^{2}} \\int \\operatorname{P_{g}}{(t_{1},n)} \\frac{\\partial}{\\partial n} n t_{1} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 1, "Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True))"], "Equality(Pow(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Mul(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Pow(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('t_1', commutative=True))))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integral(Mul(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(Integral(Mul(Function('P_g')(Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)} = (- A_{1} + \\sigma_p)^{F_{g}}, then obtain 1 = e^{F_{g} (- A_{1} + \\sigma_p)^{F_{g}} - F_{g} \\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)}}", "derivation": "\\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)} = (- A_{1} + \\sigma_p)^{F_{g}} and F_{g} \\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)} = F_{g} (- A_{1} + \\sigma_p)^{F_{g}} and 0 = F_{g} (- A_{1} + \\sigma_p)^{F_{g}} - F_{g} \\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)} and 1 = e^{F_{g} (- A_{1} + \\sigma_p)^{F_{g}} - F_{g} \\operatorname{F_{N}}{(A_{1},F_{g},\\sigma_p)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_g', commutative=True)))"], [["times", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Function('F_N')(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('F_g', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_g', commutative=True))))"], [["minus", 2, "Mul(Symbol('F_g', commutative=True), Function('F_N')(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('F_g', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True), Function('F_N')(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["exp", 3], "Equality(Integer(1), exp(Add(Mul(Symbol('F_g', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True), Function('F_N')(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given y{(H,\\rho_b)} = H e^{\\rho_b}, then derive \\rho_b \\cos{(\\rho_b y{(H,\\rho_b)})} \\frac{\\partial}{\\partial H} y{(H,\\rho_b)} = \\rho_b e^{\\rho_b} \\cos{(H \\rho_b e^{\\rho_b})}, then obtain \\rho_b \\cos{(\\rho_b y{(H,\\rho_b)})} \\frac{\\partial}{\\partial H} y{(H,\\rho_b)} = \\rho_b e^{\\rho_b} \\cos{(\\rho_b y{(H,\\rho_b)})}", "derivation": "y{(H,\\rho_b)} = H e^{\\rho_b} and \\rho_b y{(H,\\rho_b)} = H \\rho_b e^{\\rho_b} and \\sin{(\\rho_b y{(H,\\rho_b)})} = \\sin{(H \\rho_b e^{\\rho_b})} and \\frac{\\partial}{\\partial H} \\sin{(\\rho_b y{(H,\\rho_b)})} = \\frac{\\partial}{\\partial H} \\sin{(H \\rho_b e^{\\rho_b})} and \\rho_b \\cos{(\\rho_b y{(H,\\rho_b)})} \\frac{\\partial}{\\partial H} y{(H,\\rho_b)} = \\rho_b e^{\\rho_b} \\cos{(H \\rho_b e^{\\rho_b})} and \\rho_b \\cos{(\\rho_b y{(H,\\rho_b)})} \\frac{\\partial}{\\partial H} y{(H,\\rho_b)} = \\rho_b e^{\\rho_b} \\cos{(\\rho_b y{(H,\\rho_b)})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('H', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), sin(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(sin(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), cos(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Derivative(Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), cos(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Derivative(Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)), cos(Mul(Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True))))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{x},\\mathbf{F})} = \\hat{x} - \\mathbf{F}, then derive \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} = 1, then obtain \\int \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} d\\hat{x} = \\int \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} - \\mathbf{F}) d\\hat{x}", "derivation": "\\mathbb{I}{(\\hat{x},\\mathbf{F})} = \\hat{x} - \\mathbf{F} and \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} = \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} - \\mathbf{F}) and \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} = 1 and \\int \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} d\\hat{x} = \\int 1 d\\hat{x} and \\int \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} - \\mathbf{F}) d\\hat{x} = \\int 1 d\\hat{x} and \\int \\frac{\\partial}{\\partial \\hat{x}} \\mathbb{I}{(\\hat{x},\\mathbf{F})} d\\hat{x} = \\int \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} - \\mathbf{F}) d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\rho_f,\\mathbf{M})} = e^{\\frac{\\rho_f}{\\mathbf{M}}}, then obtain e^{\\frac{\\rho_f}{\\mathbf{M}}} + \\int (\\mathbf{p}{(\\rho_f,\\mathbf{M})} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M} = e^{\\frac{\\rho_f}{\\mathbf{M}}} + \\int (e^{\\frac{\\rho_f}{\\mathbf{M}}} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M}", "derivation": "\\mathbf{p}{(\\rho_f,\\mathbf{M})} = e^{\\frac{\\rho_f}{\\mathbf{M}}} and \\mathbf{p}{(\\rho_f,\\mathbf{M})} - \\frac{\\rho_f}{\\mathbf{M}} = e^{\\frac{\\rho_f}{\\mathbf{M}}} - \\frac{\\rho_f}{\\mathbf{M}} and \\int (\\mathbf{p}{(\\rho_f,\\mathbf{M})} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M} = \\int (e^{\\frac{\\rho_f}{\\mathbf{M}}} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M} and e^{\\frac{\\rho_f}{\\mathbf{M}}} + \\int (\\mathbf{p}{(\\rho_f,\\mathbf{M})} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M} = e^{\\frac{\\rho_f}{\\mathbf{M}}} + \\int (e^{\\frac{\\rho_f}{\\mathbf{M}}} - \\frac{\\rho_f}{\\mathbf{M}}) d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 3, "exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Integral(Add(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Add(exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Integral(Add(exp(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})}, then obtain (\\frac{d}{d \\theta_2} (\\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} \\cos^{- \\theta_2}{(\\log{(\\theta_2)})})^{\\theta_2})^{\\theta_2} = (\\frac{d}{d \\theta_2} 1)^{\\theta_2}", "derivation": "\\Psi^{\\dagger}{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})} and \\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} = \\cos^{\\theta_2}{(\\log{(\\theta_2)})} and \\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} \\cos^{- \\theta_2}{(\\log{(\\theta_2)})} = 1 and (\\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} \\cos^{- \\theta_2}{(\\log{(\\theta_2)})})^{\\theta_2} = 1 and \\frac{d}{d \\theta_2} (\\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} \\cos^{- \\theta_2}{(\\log{(\\theta_2)})})^{\\theta_2} = \\frac{d}{d \\theta_2} 1 and (\\frac{d}{d \\theta_2} (\\Psi^{\\dagger}^{\\theta_2}{(\\theta_2)} \\cos^{- \\theta_2}{(\\log{(\\theta_2)})})^{\\theta_2})^{\\theta_2} = (\\frac{d}{d \\theta_2} 1)^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 2, "Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Integer(1))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Derivative(Pow(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given r{(y)} = \\cos{(y)}, then obtain \\frac{d}{d y} r^{2 y}{(y)} = \\frac{d}{d y} r^{y}{(y)} \\cos^{y}{(y)}", "derivation": "r{(y)} = \\cos{(y)} and r^{y}{(y)} = \\cos^{y}{(y)} and r^{2 y}{(y)} = r^{y}{(y)} \\cos^{y}{(y)} and \\frac{d}{d y} r^{2 y}{(y)} = \\frac{d}{d y} r^{y}{(y)} \\cos^{y}{(y)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('r')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["times", 2, "Pow(Function('r')(Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Pow(Function('r')(Symbol('y', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Mul(Pow(Function('r')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Function('r')(Symbol('y', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('r')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{x}_0,T)} = T - \\hat{x}_0, then obtain - \\operatorname{t_{2}}{(\\hat{x}_0,T)} - \\frac{\\operatorname{t_{2}}{(\\hat{x}_0,T)}}{\\hat{x}_0} = - \\operatorname{t_{2}}{(\\hat{x}_0,T)} + \\frac{- T + \\hat{x}_0}{\\hat{x}_0}", "derivation": "\\operatorname{t_{2}}{(\\hat{x}_0,T)} = T - \\hat{x}_0 and - \\frac{\\operatorname{t_{2}}{(\\hat{x}_0,T)}}{\\hat{x}_0} = - \\frac{T - \\hat{x}_0}{\\hat{x}_0} and \\frac{\\operatorname{t_{2}}{(\\hat{x}_0,T)}}{\\hat{x}_0} = \\frac{T - \\hat{x}_0}{\\hat{x}_0} and - \\frac{T - \\hat{x}_0}{\\hat{x}_0} = \\frac{- T + \\hat{x}_0}{\\hat{x}_0} and - \\frac{\\operatorname{t_{2}}{(\\hat{x}_0,T)}}{\\hat{x}_0} = \\frac{- T + \\hat{x}_0}{\\hat{x}_0} and - \\operatorname{t_{2}}{(\\hat{x}_0,T)} - \\frac{\\operatorname{t_{2}}{(\\hat{x}_0,T)}}{\\hat{x}_0} = - \\operatorname{t_{2}}{(\\hat{x}_0,T)} + \\frac{- T + \\hat{x}_0}{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))))"], [["times", 1, "Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 5, "Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(B)} = \\sin{(\\sin{(B)})} and \\Psi_{nl}{(Q,t_{1})} = \\frac{t_{1}}{Q}, then obtain \\frac{\\Psi{(B)} \\Psi_{nl}{(Q,t_{1})}}{\\frac{d}{d B} \\sin{(\\sin{(B)})}} = \\frac{t_{1} \\Psi{(B)}}{Q \\frac{d}{d B} \\sin{(\\sin{(B)})}}", "derivation": "\\Psi{(B)} = \\sin{(\\sin{(B)})} and \\frac{d}{d B} \\Psi{(B)} = \\frac{d}{d B} \\sin{(\\sin{(B)})} and \\Psi_{nl}{(Q,t_{1})} = \\frac{t_{1}}{Q} and \\frac{\\Psi{(B)} \\Psi_{nl}{(Q,t_{1})}}{\\frac{d}{d B} \\Psi{(B)}} = \\frac{t_{1} \\Psi{(B)}}{Q \\frac{d}{d B} \\Psi{(B)}} and \\frac{\\Psi{(B)} \\Psi_{nl}{(Q,t_{1})}}{\\frac{d}{d B} \\sin{(\\sin{(B)})}} = \\frac{t_{1} \\Psi{(B)}}{Q \\frac{d}{d B} \\sin{(\\sin{(B)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('B', commutative=True)), sin(sin(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('Q', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["divide", 3, "Mul(Pow(Function('\\\\Psi')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('\\\\Psi')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], "Equality(Mul(Function('\\\\Psi')(Symbol('B', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('Q', commutative=True), Symbol('t_1', commutative=True)), Pow(Derivative(Function('\\\\Psi')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\Psi')(Symbol('B', commutative=True)), Pow(Derivative(Function('\\\\Psi')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\Psi')(Symbol('B', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('Q', commutative=True), Symbol('t_1', commutative=True)), Pow(Derivative(sin(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\Psi')(Symbol('B', commutative=True)), Pow(Derivative(sin(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(v_{2},\\Psi)} = \\log{(\\Psi v_{2})}, then obtain \\frac{1}{\\Psi v_{2} \\log{(\\Psi v_{2})}} = \\frac{\\log{(\\Psi v_{2})}}{\\Psi v_{2} \\operatorname{v_{1}}^{2}{(v_{2},\\Psi)}}", "derivation": "\\operatorname{v_{1}}{(v_{2},\\Psi)} = \\log{(\\Psi v_{2})} and \\Psi v_{2} \\operatorname{v_{1}}{(v_{2},\\Psi)} = \\Psi v_{2} \\log{(\\Psi v_{2})} and \\frac{1}{\\Psi v_{2}} = \\frac{\\log{(\\Psi v_{2})}}{\\Psi v_{2} \\operatorname{v_{1}}{(v_{2},\\Psi)}} and \\frac{1}{\\Psi v_{2} \\operatorname{v_{1}}{(v_{2},\\Psi)}} = \\frac{\\log{(\\Psi v_{2})}}{\\Psi v_{2} \\operatorname{v_{1}}^{2}{(v_{2},\\Psi)}} and \\frac{1}{\\Psi v_{2} \\log{(\\Psi v_{2})}} = \\frac{1}{\\Psi v_{2} \\operatorname{v_{1}}{(v_{2},\\Psi)}} and \\frac{1}{\\Psi v_{2} \\log{(\\Psi v_{2})}} = \\frac{\\log{(\\Psi v_{2})}}{\\Psi v_{2} \\operatorname{v_{1}}^{2}{(v_{2},\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True)))))"], [["divide", 1, "Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True)))))"], [["divide", 3, "Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-2)), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('v_1')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-2)), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(J)} = \\sin{(J)}, then obtain \\frac{\\operatorname{v_{t}}{(J)} \\cos{(J)}}{\\sin^{2}{(J)}} - \\frac{\\frac{d}{d J} \\operatorname{v_{t}}{(J)}}{\\sin{(J)}} = 0", "derivation": "\\operatorname{v_{t}}{(J)} = \\sin{(J)} and \\frac{\\operatorname{v_{t}}{(J)}}{\\sin{(J)}} = 1 and - \\frac{\\operatorname{v_{t}}{(J)}}{\\sin{(J)}} = -1 and \\frac{d}{d J} - \\frac{\\operatorname{v_{t}}{(J)}}{\\sin{(J)}} = \\frac{d}{d J} (-1) and \\frac{\\operatorname{v_{t}}{(J)} \\cos{(J)}}{\\sin^{2}{(J)}} - \\frac{\\frac{d}{d J} \\operatorname{v_{t}}{(J)}}{\\sin{(J)}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["divide", 1, "sin(Symbol('J', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_t')(Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Integer(-1))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('v_t')(Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Function('v_t')(Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Integer(-2)), cos(Symbol('J', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('J', commutative=True)), Integer(-1)), Derivative(Function('v_t')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\pi{(\\mathbf{g},\\rho_f)} = \\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} + \\rho_f), then obtain \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} \\pi{(\\mathbf{g},\\rho_f)} = 0", "derivation": "\\pi{(\\mathbf{g},\\rho_f)} = \\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} + \\rho_f) and \\pi{(\\mathbf{g},\\rho_f)} - 1 = \\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} + \\rho_f) - 1 and \\frac{\\partial}{\\partial \\rho_f} (\\pi{(\\mathbf{g},\\rho_f)} - 1) = \\frac{\\partial}{\\partial \\rho_f} (\\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} + \\rho_f) - 1) and \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (\\pi{(\\mathbf{g},\\rho_f)} - 1) = \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (\\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} + \\rho_f) - 1) and \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} \\pi{(\\mathbf{g},\\rho_f)} = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Add(Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Derivative(Add(Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\bar{\\h}{(k)} = \\sin{(k)}, then derive \\frac{\\bar{\\h}{(k)} \\cos^{2 k}{(k)}}{k} = \\frac{\\sin{(k)} \\cos^{2 k}{(k)}}{k}, then obtain \\frac{\\bar{\\h}{(k)} \\cos^{2 k}{(k)}}{k \\cos{(k)}} = \\frac{\\sin{(k)} \\cos^{2 k}{(k)}}{k \\cos{(k)}}", "derivation": "\\bar{\\h}{(k)} = \\sin{(k)} and \\frac{\\bar{\\h}{(k)}}{k} = \\frac{\\sin{(k)}}{k} and \\frac{\\bar{\\h}{(k)} (\\frac{d}{d k} \\sin{(k)})^{2 k}}{k} = \\frac{\\sin{(k)} (\\frac{d}{d k} \\sin{(k)})^{2 k}}{k} and \\frac{\\bar{\\h}{(k)} \\cos^{2 k}{(k)}}{k} = \\frac{\\sin{(k)} \\cos^{2 k}{(k)}}{k} and \\frac{\\bar{\\h}{(k)} \\cos^{2 k}{(k)}}{k \\cos{(k)}} = \\frac{\\sin{(k)} \\cos^{2 k}{(k)}}{k \\cos{(k)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["divide", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('k', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True))))"], [["times", 2, "Pow(Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(2), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('k', commutative=True)), Pow(Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(2), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)), Pow(Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(2), Symbol('k', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))))"], [["divide", 4, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(P_{e})} = \\cos{(P_{e})}, then obtain \\int (\\operatorname{M_{E}}^{2}{(P_{e})} + \\operatorname{M_{E}}{(P_{e})}) dP_{e} = \\int (\\operatorname{M_{E}}{(P_{e})} \\cos{(P_{e})} + \\operatorname{M_{E}}{(P_{e})}) dP_{e}", "derivation": "\\operatorname{M_{E}}{(P_{e})} = \\cos{(P_{e})} and \\operatorname{M_{E}}^{2}{(P_{e})} = \\operatorname{M_{E}}{(P_{e})} \\cos{(P_{e})} and \\operatorname{M_{E}}^{2}{(P_{e})} + \\operatorname{M_{E}}{(P_{e})} = \\operatorname{M_{E}}{(P_{e})} \\cos{(P_{e})} + \\operatorname{M_{E}}{(P_{e})} and \\int (\\operatorname{M_{E}}^{2}{(P_{e})} + \\operatorname{M_{E}}{(P_{e})}) dP_{e} = \\int (\\operatorname{M_{E}}{(P_{e})} \\cos{(P_{e})} + \\operatorname{M_{E}}{(P_{e})}) dP_{e}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["times", 1, "Function('M_E')(Symbol('P_e', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('M_E')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))))"], [["add", 2, "Function('M_E')(Symbol('P_e', commutative=True))"], "Equality(Add(Pow(Function('M_E')(Symbol('P_e', commutative=True)), Integer(2)), Function('M_E')(Symbol('P_e', commutative=True))), Add(Mul(Function('M_E')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Function('M_E')(Symbol('P_e', commutative=True))))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Pow(Function('M_E')(Symbol('P_e', commutative=True)), Integer(2)), Function('M_E')(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Mul(Function('M_E')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Function('M_E')(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\phi_2)} = \\phi_2, then derive \\int \\mathbf{J}{(\\phi_2)} d\\phi_2 = \\hat{H}_l + \\frac{\\phi_2^{2}}{2}, then derive \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)} = 1, then obtain \\int \\phi_2 d\\phi_2 = \\hat{H}_l + \\frac{\\phi_2^{2} \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)}}{2}", "derivation": "\\mathbf{J}{(\\phi_2)} = \\phi_2 and \\int \\mathbf{J}{(\\phi_2)} d\\phi_2 = \\int \\phi_2 d\\phi_2 and \\int \\mathbf{J}{(\\phi_2)} d\\phi_2 = \\hat{H}_l + \\frac{\\phi_2^{2}}{2} and \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 and \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)} = 1 and \\int \\phi_2 d\\phi_2 = \\hat{H}_l + \\frac{\\phi_2^{2}}{2} and \\frac{\\phi_2^{2} \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)}}{2} = \\frac{\\phi_2^{2}}{2} and \\int \\phi_2 d\\phi_2 = \\hat{H}_l + \\frac{\\phi_2^{2} \\frac{d}{d \\phi_2} \\mathbf{J}{(\\phi_2)}}{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"], [["times", 5, "Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{p}{(\\mu)} = \\sin{(\\mu)}, then obtain (\\mu + (\\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)})^{\\mu})^{\\mu} - \\sin{(\\mu)} = (\\mu + (\\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)})^{\\mu})^{\\mu} - \\sin{(\\mu)}", "derivation": "\\hat{p}{(\\mu)} = \\sin{(\\mu)} and \\hat{p}^{\\mu}{(\\mu)} = \\sin^{\\mu}{(\\mu)} and \\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)} = \\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)} and (\\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)})^{\\mu} = (\\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)})^{\\mu} and \\mu + (\\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)})^{\\mu} = \\mu + (\\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)})^{\\mu} and (\\mu + (\\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)})^{\\mu})^{\\mu} = (\\mu + (\\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)})^{\\mu})^{\\mu} and (\\mu + (\\hat{p}{(\\mu)} + \\hat{p}^{\\mu}{(\\mu)})^{\\mu})^{\\mu} - \\sin{(\\mu)} = (\\mu + (\\hat{p}{(\\mu)} + \\sin^{\\mu}{(\\mu)})^{\\mu})^{\\mu} - \\sin{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["add", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["power", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["minus", 6, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True)))), Add(Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Add(Function('\\\\hat{p}')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given z{(A_{z})} = \\cos{(A_{z})}, then obtain - (\\frac{\\cos{(A_{z})}}{A_{z}})^{A_{z}} + \\int (\\frac{z{(A_{z})}}{A_{z} \\cos{(A_{z})}})^{A_{z}} dA_{z} = - (\\frac{\\cos{(A_{z})}}{A_{z}})^{A_{z}} + \\int (\\frac{1}{A_{z}})^{A_{z}} dA_{z}", "derivation": "z{(A_{z})} = \\cos{(A_{z})} and \\frac{z{(A_{z})}}{A_{z}} = \\frac{\\cos{(A_{z})}}{A_{z}} and \\frac{z{(A_{z})}}{A_{z} \\cos{(A_{z})}} = \\frac{1}{A_{z}} and (\\frac{z{(A_{z})}}{A_{z} \\cos{(A_{z})}})^{A_{z}} = (\\frac{1}{A_{z}})^{A_{z}} and \\int (\\frac{z{(A_{z})}}{A_{z} \\cos{(A_{z})}})^{A_{z}} dA_{z} = \\int (\\frac{1}{A_{z}})^{A_{z}} dA_{z} and - (\\frac{\\cos{(A_{z})}}{A_{z}})^{A_{z}} + \\int (\\frac{z{(A_{z})}}{A_{z} \\cos{(A_{z})}})^{A_{z}} dA_{z} = - (\\frac{\\cos{(A_{z})}}{A_{z}})^{A_{z}} + \\int (\\frac{1}{A_{z}})^{A_{z}} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["divide", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('z')(Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True))))"], [["divide", 2, "cos(Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('z')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Pow(Symbol('A_z', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('z')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Symbol('A_z', commutative=True)), Pow(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('A_z', commutative=True)))"], [["integrate", 4, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('z')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["minus", 5, "Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Integral(Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('z')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Integral(Pow(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(z)} = \\int \\sin{(z)} dz, then derive \\operatorname{F_{c}}{(z)} = B - \\cos{(z)}, then derive B - \\cos{(z)} = M - \\cos{(z)}, then obtain z^{2} (M - \\cos{(z)}) = z^{2} (\\mu_0 - \\cos{(z)})", "derivation": "\\operatorname{F_{c}}{(z)} = \\int \\sin{(z)} dz and z \\operatorname{F_{c}}{(z)} = z \\int \\sin{(z)} dz and \\operatorname{F_{c}}{(z)} = B - \\cos{(z)} and B - \\cos{(z)} = \\int \\sin{(z)} dz and z^{2} \\operatorname{F_{c}}{(z)} = z^{2} \\int \\sin{(z)} dz and B - \\cos{(z)} = M - \\cos{(z)} and z^{2} (B - \\cos{(z)}) = z^{2} \\int \\sin{(z)} dz and z^{2} (M - \\cos{(z)}) = z^{2} \\int \\sin{(z)} dz and z^{2} (M - \\cos{(z)}) = z^{2} (\\mu_0 - \\cos{(z)})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('F_c')(Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('F_c')(Symbol('z', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Symbol('B', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Function('F_c')(Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('B', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Symbol('B', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 8], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given q{(a,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + a), then derive q{(a,C_{2})} = -1, then derive \\int q^{C_{2}}{(a,C_{2})} dC_{2} = - \\frac{(-1)^{C_{2}} i}{\\pi} + \\hat{H}_l, then obtain \\iint q^{C_{2}}{(a,C_{2})} dC_{2} d\\hat{H}_l = \\int (i \\pi^{q{(a,C_{2})}} q{(a,C_{2})} q^{C_{2}}{(a,C_{2})} + \\hat{H}_l) d\\hat{H}_l", "derivation": "q{(a,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + a) and q{(a,C_{2})} = -1 and q^{C_{2}}{(a,C_{2})} = (-1)^{C_{2}} and \\int q^{C_{2}}{(a,C_{2})} dC_{2} = \\int (-1)^{C_{2}} dC_{2} and \\int q^{C_{2}}{(a,C_{2})} dC_{2} = - \\frac{(-1)^{C_{2}} i}{\\pi} + \\hat{H}_l and \\int q^{C_{2}}{(a,C_{2})} dC_{2} = i \\pi^{q{(a,C_{2})}} q{(a,C_{2})} q^{C_{2}}{(a,C_{2})} + \\hat{H}_l and \\iint q^{C_{2}}{(a,C_{2})} dC_{2} d\\hat{H}_l = \\int (i \\pi^{q{(a,C_{2})}} q{(a,C_{2})} q^{C_{2}}{(a,C_{2})} + \\hat{H}_l) d\\hat{H}_l", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Integer(-1))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Integer(-1), Symbol('C_2', commutative=True)))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Pow(Integer(-1), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Pow(Integer(-1), Symbol('C_2', commutative=True)), I, Pow(pi, Integer(-1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(I, Pow(pi, Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True))), Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["integrate", 6, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Add(Mul(I, Pow(pi, Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True))), Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('q')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given M{(z^{*})} = \\int \\sin{(z^{*})} dz^{*}, then derive \\frac{d}{d z^{*}} M{(z^{*})} = \\frac{\\partial}{\\partial z^{*}} (\\phi - \\cos{(z^{*})}), then obtain \\frac{\\partial}{\\partial z^{*}} (\\phi - \\cos{(z^{*})}) = \\frac{d}{d z^{*}} \\int \\sin{(z^{*})} dz^{*}", "derivation": "M{(z^{*})} = \\int \\sin{(z^{*})} dz^{*} and \\frac{d}{d z^{*}} M{(z^{*})} = \\frac{d}{d z^{*}} \\int \\sin{(z^{*})} dz^{*} and \\frac{d}{d z^{*}} M{(z^{*})} = \\frac{\\partial}{\\partial z^{*}} (\\phi - \\cos{(z^{*})}) and \\frac{\\partial}{\\partial z^{*}} (\\phi - \\cos{(z^{*})}) = \\frac{d}{d z^{*}} \\int \\sin{(z^{*})} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('z^*', commutative=True)), Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('M')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} and \\operatorname{P_{g}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} u{(J_{\\varepsilon})} + 1, then derive \\operatorname{P_{g}}{(J_{\\varepsilon})} = 1 - \\sin{(J_{\\varepsilon})}, then obtain t_{1} + \\sin{(\\rho_b)} + \\int \\operatorname{P_{g}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = t_{1} + \\sin{(\\rho_b)} + \\int (1 - \\sin{(J_{\\varepsilon})}) dJ_{\\varepsilon}", "derivation": "u{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} and \\operatorname{P_{g}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} u{(J_{\\varepsilon})} + 1 and \\operatorname{P_{g}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} + 1 and \\operatorname{P_{g}}{(J_{\\varepsilon})} = 1 - \\sin{(J_{\\varepsilon})} and \\int \\operatorname{P_{g}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (1 - \\sin{(J_{\\varepsilon})}) dJ_{\\varepsilon} and t_{1} + \\sin{(\\rho_b)} + \\int \\operatorname{P_{g}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = t_{1} + \\sin{(\\rho_b)} + \\int (1 - \\sin{(J_{\\varepsilon})}) dJ_{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Derivative(Function('u')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_g')(Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Function('P_g')(Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Integer(1), Mul(Integer(-1), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 5, "Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\rho_b', commutative=True)), Integral(Function('P_g')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\rho_b', commutative=True)), Integral(Add(Integer(1), Mul(Integer(-1), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given m{(H)} = \\sin{(H)} and \\operatorname{V_{\\mathbf{E}}}{(H)} = \\int \\sin{(H)} dH, then derive \\operatorname{V_{\\mathbf{E}}}{(H)} = \\hat{\\mathbf{r}} - \\cos{(H)}, then obtain \\int m{(H)} dH = \\hat{\\mathbf{r}} - \\cos{(H)}", "derivation": "m{(H)} = \\sin{(H)} and \\int m{(H)} dH = \\int \\sin{(H)} dH and \\operatorname{V_{\\mathbf{E}}}{(H)} = \\int \\sin{(H)} dH and \\operatorname{V_{\\mathbf{E}}}{(H)} = \\hat{\\mathbf{r}} - \\cos{(H)} and \\hat{\\mathbf{r}} - \\cos{(H)} = \\int \\sin{(H)} dH and \\int m{(H)} dH = \\hat{\\mathbf{r}} - \\cos{(H)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('m')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('H', commutative=True)), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('H', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Integral(Function('m')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given A{(g_{\\varepsilon},\\mathbf{J}_M,c)} = - \\mathbf{J}_M + c g_{\\varepsilon}, then derive \\int \\frac{A{(g_{\\varepsilon},\\mathbf{J}_M,c)}}{- \\mathbf{J}_M + c g_{\\varepsilon}} dg_{\\varepsilon} = g_{\\varepsilon} + v_{z}, then derive F_{c} + g_{\\varepsilon} = g_{\\varepsilon} + v_{z}, then obtain \\sin{(\\int 1 dg_{\\varepsilon})} = \\sin{(F_{c} + g_{\\varepsilon})}", "derivation": "A{(g_{\\varepsilon},\\mathbf{J}_M,c)} = - \\mathbf{J}_M + c g_{\\varepsilon} and \\frac{A{(g_{\\varepsilon},\\mathbf{J}_M,c)}}{- \\mathbf{J}_M + c g_{\\varepsilon}} = 1 and \\int \\frac{A{(g_{\\varepsilon},\\mathbf{J}_M,c)}}{- \\mathbf{J}_M + c g_{\\varepsilon}} dg_{\\varepsilon} = \\int 1 dg_{\\varepsilon} and \\int \\frac{A{(g_{\\varepsilon},\\mathbf{J}_M,c)}}{- \\mathbf{J}_M + c g_{\\varepsilon}} dg_{\\varepsilon} = g_{\\varepsilon} + v_{z} and \\int 1 dg_{\\varepsilon} = g_{\\varepsilon} + v_{z} and F_{c} + g_{\\varepsilon} = g_{\\varepsilon} + v_{z} and \\sin{(\\int 1 dg_{\\varepsilon})} = \\sin{(g_{\\varepsilon} + v_{z})} and \\sin{(\\int 1 dg_{\\varepsilon})} = \\sin{(F_{c} + g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1)), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1)), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1)), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_z', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_z', commutative=True)))"], [["sin", 5], "Equality(sin(Integral(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), sin(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(sin(Integral(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), sin(Add(Symbol('F_c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(E_{n},\\mathbf{J})} = \\mathbf{J}^{E_{n}}, then obtain - \\sin{(\\operatorname{t_{2}}{(E_{n},\\mathbf{J})})} \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{t_{2}}{(E_{n},\\mathbf{J})} = - \\frac{E_{n} \\mathbf{J}^{E_{n}} \\sin{(\\mathbf{J}^{E_{n}})}}{\\mathbf{J}}", "derivation": "\\operatorname{t_{2}}{(E_{n},\\mathbf{J})} = \\mathbf{J}^{E_{n}} and \\cos{(\\operatorname{t_{2}}{(E_{n},\\mathbf{J})})} = \\cos{(\\mathbf{J}^{E_{n}})} and \\frac{\\partial}{\\partial \\mathbf{J}} \\cos{(\\operatorname{t_{2}}{(E_{n},\\mathbf{J})})} = \\frac{\\partial}{\\partial \\mathbf{J}} \\cos{(\\mathbf{J}^{E_{n}})} and - \\sin{(\\operatorname{t_{2}}{(E_{n},\\mathbf{J})})} \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{t_{2}}{(E_{n},\\mathbf{J})} = - \\frac{E_{n} \\mathbf{J}^{E_{n}} \\sin{(\\mathbf{J}^{E_{n}})}}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_n', commutative=True)))"], [["cos", 1], "Equality(cos(Function('t_2')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), cos(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_n', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(cos(Function('t_2')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), sin(Function('t_2')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Derivative(Function('t_2')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('E_n', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_n', commutative=True)), sin(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given v{(\\dot{z})} = \\log{(\\dot{z})}, then derive \\int v{(\\dot{z})} d\\dot{z} = \\dot{z} \\log{(\\dot{z})} - \\dot{z} + \\mathbf{f}, then obtain \\int v{(\\dot{z})} d\\dot{z} = \\dot{z} v{(\\dot{z})} - \\dot{z} + \\mathbf{f}", "derivation": "v{(\\dot{z})} = \\log{(\\dot{z})} and \\int v{(\\dot{z})} d\\dot{z} = \\int \\log{(\\dot{z})} d\\dot{z} and \\int v{(\\dot{z})} d\\dot{z} = \\dot{z} \\log{(\\dot{z})} - \\dot{z} + \\mathbf{f} and \\int v{(\\dot{z})} d\\dot{z} = \\dot{z} v{(\\dot{z})} - \\dot{z} + \\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('v')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\rho{(i,s)} = \\frac{i}{s} and C{(i,s)} = \\frac{i}{s}, then derive \\frac{\\frac{\\partial}{\\partial s} \\rho{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}} = \\frac{\\frac{\\partial}{\\partial s} C{(i,s)}}{s} - \\frac{C{(i,s)}}{s^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial s} \\rho{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}} = \\frac{\\frac{\\partial}{\\partial s} C{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}}", "derivation": "\\rho{(i,s)} = \\frac{i}{s} and \\frac{\\rho{(i,s)}}{s} = \\frac{i}{s^{2}} and C{(i,s)} = \\frac{i}{s} and \\frac{\\rho{(i,s)}}{s} = \\frac{C{(i,s)}}{s} and \\frac{\\partial}{\\partial s} \\frac{\\rho{(i,s)}}{s} = \\frac{\\partial}{\\partial s} \\frac{C{(i,s)}}{s} and \\frac{\\frac{\\partial}{\\partial s} \\rho{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}} = \\frac{\\frac{\\partial}{\\partial s} C{(i,s)}}{s} - \\frac{C{(i,s)}}{s^{2}} and \\frac{\\frac{\\partial}{\\partial s} \\rho{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}} = \\frac{\\frac{\\partial}{\\partial s} C{(i,s)}}{s} - \\frac{\\rho{(i,s)}}{s^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('i', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('s', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('i', commutative=True), Pow(Symbol('s', commutative=True), Integer(-2))))"], ["renaming_premise", "Equality(Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('i', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-2)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-2)), Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-2)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('C')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-2)), Function('\\\\rho')(Symbol('i', commutative=True), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given V{(l)} = e^{\\sin{(l)}}, then derive \\frac{d}{d l} V{(l)} = e^{\\sin{(l)}} \\cos{(l)}, then obtain (V{(l)} + \\frac{d}{d l} V{(l)})^{l} = (V{(l)} + e^{\\sin{(l)}} \\cos{(l)})^{l}", "derivation": "V{(l)} = e^{\\sin{(l)}} and \\frac{d}{d l} V{(l)} = \\frac{d}{d l} e^{\\sin{(l)}} and V{(l)} + \\frac{d}{d l} V{(l)} = V{(l)} + \\frac{d}{d l} e^{\\sin{(l)}} and \\frac{d}{d l} V{(l)} = e^{\\sin{(l)}} \\cos{(l)} and (V{(l)} + \\frac{d}{d l} V{(l)})^{l} = (V{(l)} + \\frac{d}{d l} e^{\\sin{(l)}})^{l} and (V{(l)} + e^{\\sin{(l)}} \\cos{(l)})^{l} = (V{(l)} + \\frac{d}{d l} e^{\\sin{(l)}})^{l} and (V{(l)} + \\frac{d}{d l} V{(l)})^{l} = (V{(l)} + e^{\\sin{(l)}} \\cos{(l)})^{l}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 2, "Function('V')(Symbol('l', commutative=True))"], "Equality(Add(Function('V')(Symbol('l', commutative=True)), Derivative(Function('V')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Function('V')(Symbol('l', commutative=True)), Derivative(exp(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(exp(sin(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Function('V')(Symbol('l', commutative=True)), Derivative(Function('V')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)), Pow(Add(Function('V')(Symbol('l', commutative=True)), Derivative(exp(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Function('V')(Symbol('l', commutative=True)), Mul(exp(sin(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True)))), Symbol('l', commutative=True)), Pow(Add(Function('V')(Symbol('l', commutative=True)), Derivative(exp(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Function('V')(Symbol('l', commutative=True)), Derivative(Function('V')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)), Pow(Add(Function('V')(Symbol('l', commutative=True)), Mul(exp(sin(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True)))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\dot{x},A_{1})} = \\frac{\\dot{x}}{A_{1}}, then obtain \\frac{\\partial}{\\partial \\dot{x}} \\int \\mathbf{P}^{\\dot{x}}{(\\dot{x},A_{1})} d\\dot{x} = \\frac{\\partial}{\\partial \\dot{x}} \\int (\\frac{\\dot{x}}{A_{1}})^{\\dot{x}} d\\dot{x}", "derivation": "\\mathbf{P}{(\\dot{x},A_{1})} = \\frac{\\dot{x}}{A_{1}} and \\mathbf{P}^{\\dot{x}}{(\\dot{x},A_{1})} = (\\frac{\\dot{x}}{A_{1}})^{\\dot{x}} and \\int \\mathbf{P}^{\\dot{x}}{(\\dot{x},A_{1})} d\\dot{x} = \\int (\\frac{\\dot{x}}{A_{1}})^{\\dot{x}} d\\dot{x} and \\frac{\\partial}{\\partial \\dot{x}} \\int \\mathbf{P}^{\\dot{x}}{(\\dot{x},A_{1})} d\\dot{x} = \\frac{\\partial}{\\partial \\dot{x}} \\int (\\frac{\\dot{x}}{A_{1}})^{\\dot{x}} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} = \\frac{c_{0}}{\\varepsilon_0}, then obtain - \\varepsilon_0 \\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} - c_{0} = - 2 c_{0}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} = \\frac{c_{0}}{\\varepsilon_0} and \\varepsilon_0 \\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} = c_{0} and - \\varepsilon_0 \\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} = - c_{0} and - \\varepsilon_0 \\operatorname{g^{\\prime}_{\\varepsilon}}{(c_{0},\\varepsilon_0)} - c_{0} = - 2 c_{0}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('c_0', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(x)} = e^{\\sin{(x)}}, then derive \\frac{d}{d x} \\mathbb{I}{(x)} = e^{\\sin{(x)}} \\cos{(x)}, then obtain \\frac{(\\frac{d}{d x} \\mathbb{I}{(x)})^{x}}{\\frac{d}{d x} \\mathbb{I}{(x)}} = \\frac{(\\mathbb{I}{(x)} \\cos{(x)})^{x}}{\\frac{d}{d x} \\mathbb{I}{(x)}}", "derivation": "\\mathbb{I}{(x)} = e^{\\sin{(x)}} and \\frac{d}{d x} \\mathbb{I}{(x)} = \\frac{d}{d x} e^{\\sin{(x)}} and \\frac{d}{d x} \\mathbb{I}{(x)} = e^{\\sin{(x)}} \\cos{(x)} and (\\frac{d}{d x} \\mathbb{I}{(x)})^{x} = (e^{\\sin{(x)}} \\cos{(x)})^{x} and \\frac{(\\frac{d}{d x} \\mathbb{I}{(x)})^{x}}{\\frac{d}{d x} e^{\\sin{(x)}}} = \\frac{(e^{\\sin{(x)}} \\cos{(x)})^{x}}{\\frac{d}{d x} e^{\\sin{(x)}}} and \\frac{(\\frac{d}{d x} \\mathbb{I}{(x)})^{x}}{\\frac{d}{d x} \\mathbb{I}{(x)}} = \\frac{(\\mathbb{I}{(x)} \\cos{(x)})^{x}}{\\frac{d}{d x} \\mathbb{I}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(exp(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Mul(exp(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["divide", 4, "Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Mul(exp(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True))), Mul(Pow(Mul(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\chi{(F_{c},f^{*})} = F_{c} \\cos{(f^{*})}, then obtain \\frac{\\partial}{\\partial f^{*}} F_{c} \\cos{(f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\frac{F_{c}^{2} \\cos^{2}{(f^{*})}}{\\chi{(F_{c},f^{*})}}", "derivation": "\\chi{(F_{c},f^{*})} = F_{c} \\cos{(f^{*})} and f^{*} \\chi{(F_{c},f^{*})} = F_{c} f^{*} \\cos{(f^{*})} and F_{c} \\cos{(f^{*})} = \\frac{F_{c}^{2} \\cos^{2}{(f^{*})}}{\\chi{(F_{c},f^{*})}} and \\frac{\\partial}{\\partial f^{*}} F_{c} \\cos{(f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\frac{F_{c}^{2} \\cos^{2}{(f^{*})}}{\\chi{(F_{c},f^{*})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('F_c', commutative=True), cos(Symbol('f^*', commutative=True))))"], [["times", 1, "Symbol('f^*', commutative=True)"], "Equality(Mul(Symbol('f^*', commutative=True), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True))), Mul(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True), cos(Symbol('f^*', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('f^*', commutative=True), Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1)))"], "Equality(Mul(Symbol('F_c', commutative=True), cos(Symbol('f^*', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('f^*', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_c', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('F_c', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('f^*', commutative=True)), Integer(2))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(y^{\\prime})} = e^{y^{\\prime}}, then derive \\frac{d}{d y^{\\prime}} \\operatorname{A_{x}}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain \\frac{d^{3}}{d (y^{\\prime})^{3}} \\operatorname{A_{x}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}}", "derivation": "\\operatorname{A_{x}}{(y^{\\prime})} = e^{y^{\\prime}} and \\frac{d}{d y^{\\prime}} \\operatorname{A_{x}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} and \\frac{d}{d y^{\\prime}} \\operatorname{A_{x}}{(y^{\\prime})} = e^{y^{\\prime}} and \\frac{d^{2}}{d (y^{\\prime})^{2}} \\operatorname{A_{x}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} and \\operatorname{A_{x}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\operatorname{A_{x}}{(y^{\\prime})} and \\frac{d^{3}}{d (y^{\\prime})^{3}} \\operatorname{A_{x}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(2))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Function('A_x')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(3))), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(v_{t})} = \\sin{(v_{t})} and \\operatorname{A_{y}}{(v_{t})} = - \\int \\frac{d}{d v_{t}} \\sin{(v_{t})} dv_{t}, then obtain - t - \\sin{(v_{t})} = - \\dot{\\mathbf{r}} - \\mathbf{s}{(v_{t})}", "derivation": "\\mathbf{s}{(v_{t})} = \\sin{(v_{t})} and \\frac{d}{d v_{t}} \\mathbf{s}{(v_{t})} = \\frac{d}{d v_{t}} \\sin{(v_{t})} and \\int \\frac{d}{d v_{t}} \\mathbf{s}{(v_{t})} dv_{t} = \\int \\frac{d}{d v_{t}} \\sin{(v_{t})} dv_{t} and \\operatorname{A_{y}}{(v_{t})} = - \\int \\frac{d}{d v_{t}} \\sin{(v_{t})} dv_{t} and \\operatorname{A_{y}}{(v_{t})} = - \\int \\frac{d}{d v_{t}} \\mathbf{s}{(v_{t})} dv_{t} and - \\int \\frac{d}{d v_{t}} \\sin{(v_{t})} dv_{t} = - \\int \\frac{d}{d v_{t}} \\mathbf{s}{(v_{t})} dv_{t} and - t - \\sin{(v_{t})} = - \\dot{\\mathbf{r}} - \\mathbf{s}{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Integral(Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('A_y')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Integral(Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True)))), Mul(Integer(-1), Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(i,\\psi^*)} = \\psi^* i, then obtain \\int \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} di) di = \\int \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int (\\psi^* i)^{\\psi^*} di) di", "derivation": "\\operatorname{A_{2}}{(i,\\psi^*)} = \\psi^* i and \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} = (\\psi^* i)^{\\psi^*} and \\int \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} di = \\int (\\psi^* i)^{\\psi^*} di and - \\psi^* i + \\int \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} di = - \\psi^* i + \\int (\\psi^* i)^{\\psi^*} di and \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} di) = \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int (\\psi^* i)^{\\psi^*} di) and \\int \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int \\operatorname{A_{2}}^{\\psi^*}{(i,\\psi^*)} di) di = \\int \\frac{\\partial}{\\partial i} (- \\psi^* i + \\int (\\psi^* i)^{\\psi^*} di) di", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Function('A_2')(Symbol('i', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\Omega{(v_{t})} = \\sin{(\\log{(v_{t})})}, then derive - \\sin{(\\log{(v_{t})})} + \\int \\Omega{(v_{t})} dv_{t} = \\theta_1 + \\frac{v_{t} \\sin{(\\log{(v_{t})})}}{2} - \\frac{v_{t} \\cos{(\\log{(v_{t})})}}{2} - \\sin{(\\log{(v_{t})})}, then obtain - A_{1} - \\sin{(\\log{(v_{t})})} + \\int \\Omega{(v_{t})} dv_{t} = - A_{1} + \\theta_1 + \\frac{v_{t} \\sin{(\\log{(v_{t})})}}{2} - \\frac{v_{t} \\cos{(\\log{(v_{t})})}}{2} - \\sin{(\\log{(v_{t})})}", "derivation": "\\Omega{(v_{t})} = \\sin{(\\log{(v_{t})})} and \\int \\Omega{(v_{t})} dv_{t} = \\int \\sin{(\\log{(v_{t})})} dv_{t} and - \\sin{(\\log{(v_{t})})} + \\int \\Omega{(v_{t})} dv_{t} = - \\sin{(\\log{(v_{t})})} + \\int \\sin{(\\log{(v_{t})})} dv_{t} and - \\sin{(\\log{(v_{t})})} + \\int \\Omega{(v_{t})} dv_{t} = \\theta_1 + \\frac{v_{t} \\sin{(\\log{(v_{t})})}}{2} - \\frac{v_{t} \\cos{(\\log{(v_{t})})}}{2} - \\sin{(\\log{(v_{t})})} and - A_{1} - \\sin{(\\log{(v_{t})})} + \\int \\Omega{(v_{t})} dv_{t} = - A_{1} + \\theta_1 + \\frac{v_{t} \\sin{(\\log{(v_{t})})}}{2} - \\frac{v_{t} \\cos{(\\log{(v_{t})})}}{2} - \\sin{(\\log{(v_{t})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('v_t', commutative=True)), sin(log(Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(sin(log(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["minus", 2, "sin(log(Symbol('v_t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True)))), Integral(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True)))), Integral(sin(log(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True)))), Integral(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Rational(1, 2), Symbol('v_t', commutative=True), sin(log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('v_t', commutative=True), cos(log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True))))))"], [["minus", 4, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True)))), Integral(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Rational(1, 2), Symbol('v_t', commutative=True), sin(log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('v_t', commutative=True), cos(log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), sin(log(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given x{(A_{1},\\mathbf{H})} = \\frac{\\mathbf{H}}{A_{1}}, then derive \\int A_{1} x{(A_{1},\\mathbf{H})} d\\mathbf{H} = \\frac{\\mathbf{H}^{2}}{2} + \\varepsilon_0, then obtain x{(A_{1},\\mathbf{H})} + \\int A_{1} x{(A_{1},\\mathbf{H})} d\\mathbf{H} = \\frac{\\mathbf{H}^{2}}{2} + \\varepsilon_0 + x{(A_{1},\\mathbf{H})}", "derivation": "x{(A_{1},\\mathbf{H})} = \\frac{\\mathbf{H}}{A_{1}} and A_{1} x{(A_{1},\\mathbf{H})} = \\mathbf{H} and \\int A_{1} x{(A_{1},\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H} d\\mathbf{H} and \\int A_{1} x{(A_{1},\\mathbf{H})} d\\mathbf{H} = \\frac{\\mathbf{H}^{2}}{2} + \\varepsilon_0 and x{(A_{1},\\mathbf{H})} + \\int A_{1} x{(A_{1},\\mathbf{H})} d\\mathbf{H} = \\frac{\\mathbf{H}^{2}}{2} + \\varepsilon_0 + x{(A_{1},\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Mul(Symbol('A_1', commutative=True), Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('A_1', commutative=True), Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 4, "Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Mul(Symbol('A_1', commutative=True), Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Symbol('\\\\varepsilon_0', commutative=True), Function('x')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(V)} = e^{V} and \\operatorname{F_{x}}{(V)} = \\int 0 dV, then obtain \\frac{\\operatorname{F_{x}}^{V}{(V)}}{\\mathbf{J}_M{(V)}} = \\frac{(\\int (\\mathbf{J}_M{(V)} - e^{V}) dV)^{V}}{\\mathbf{J}_M{(V)}}", "derivation": "\\mathbf{J}_M{(V)} = e^{V} and \\mathbf{J}_M{(V)} - e^{V} = 0 and \\int (\\mathbf{J}_M{(V)} - e^{V}) dV = \\int 0 dV and \\operatorname{F_{x}}{(V)} = \\int 0 dV and \\operatorname{F_{x}}^{V}{(V)} = (\\int 0 dV)^{V} and \\frac{\\operatorname{F_{x}}^{V}{(V)}}{\\mathbf{J}_M{(V)}} = \\frac{(\\int 0 dV)^{V}}{\\mathbf{J}_M{(V)}} and \\frac{\\operatorname{F_{x}}^{V}{(V)}}{\\mathbf{J}_M{(V)}} = \\frac{(\\int (\\mathbf{J}_M{(V)} - e^{V}) dV)^{V}}{\\mathbf{J}_M{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["minus", 1, "exp(Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Integer(0), Tuple(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('V', commutative=True)), Integral(Integer(0), Tuple(Symbol('V', commutative=True))))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["divide", 5, "Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('F_x')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Integer(-1)), Pow(Integral(Integer(0), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Function('F_x')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Integer(-1)), Pow(Integral(Add(Function('\\\\mathbf{J}_M')(Symbol('V', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(x^\\prime,b)} = b + x^\\prime, then derive \\frac{\\partial}{\\partial b} \\mathbf{B}{(x^\\prime,b)} = 1, then obtain \\frac{\\frac{\\partial}{\\partial b} (b + x^\\prime)}{x^\\prime} = \\frac{1}{x^\\prime}", "derivation": "\\mathbf{B}{(x^\\prime,b)} = b + x^\\prime and \\frac{\\partial}{\\partial b} \\mathbf{B}{(x^\\prime,b)} = \\frac{\\partial}{\\partial b} (b + x^\\prime) and \\frac{\\partial}{\\partial b} \\mathbf{B}{(x^\\prime,b)} = 1 and \\frac{\\partial}{\\partial b} (b + x^\\prime) = 1 and \\frac{\\frac{\\partial}{\\partial b} (b + x^\\prime)}{x^\\prime} = \\frac{1}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Add(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"]]}, {"prompt": "Given u{(\\omega)} = \\cos{(\\omega)}, then obtain 0 = \\cos{(\\omega)} + \\frac{d^{2}}{d \\omega^{2}} u{(\\omega)}", "derivation": "u{(\\omega)} = \\cos{(\\omega)} and 0 = - u{(\\omega)} + \\cos{(\\omega)} and \\frac{d}{d \\omega} 0 = \\frac{d}{d \\omega} (- u{(\\omega)} + \\cos{(\\omega)}) and - \\frac{d}{d \\omega} 0 = - \\frac{d}{d \\omega} (- u{(\\omega)} + \\cos{(\\omega)}) and \\frac{d}{d \\omega} - \\frac{d}{d \\omega} 0 = \\frac{d}{d \\omega} - \\frac{d}{d \\omega} (- u{(\\omega)} + \\cos{(\\omega)}) and 0 = \\cos{(\\omega)} + \\frac{d^{2}}{d \\omega^{2}} u{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "Function('u')(Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('u')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('u')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Function('u')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Function('u')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Add(cos(Symbol('\\\\omega', commutative=True)), Derivative(Function('u')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(L)} = e^{L}, then obtain - L (\\operatorname{E_{x}}{(L)} + \\operatorname{E_{x}}{(L)} (e^{L})^{- L} - 1) = - L (\\operatorname{E_{x}}{(L)} (e^{L})^{- L} + e^{L} - 1)", "derivation": "\\operatorname{E_{x}}{(L)} = e^{L} and \\operatorname{E_{x}}{(L)} (e^{L})^{- L} = e^{L} (e^{L})^{- L} and \\operatorname{E_{x}}{(L)} + e^{L} (e^{L})^{- L} = e^{L} + e^{L} (e^{L})^{- L} and \\operatorname{E_{x}}{(L)} + e^{L} (e^{L})^{- L} - 1 = e^{L} + e^{L} (e^{L})^{- L} - 1 and \\operatorname{E_{x}}{(L)} + \\operatorname{E_{x}}{(L)} (e^{L})^{- L} - 1 = \\operatorname{E_{x}}{(L)} (e^{L})^{- L} + e^{L} - 1 and - L (\\operatorname{E_{x}}{(L)} + \\operatorname{E_{x}}{(L)} (e^{L})^{- L} - 1) = - L (\\operatorname{E_{x}}{(L)} (e^{L})^{- L} + e^{L} - 1)", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["divide", 1, "Pow(exp(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Function('E_x')(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))))"], [["add", 1, "Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))))"], "Equality(Add(Function('E_x')(Symbol('L', commutative=True)), Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))))), Add(exp(Symbol('L', commutative=True)), Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('E_x')(Symbol('L', commutative=True)), Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), Integer(-1)), Add(exp(Symbol('L', commutative=True)), Mul(exp(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('E_x')(Symbol('L', commutative=True)), Mul(Function('E_x')(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), Integer(-1)), Add(Mul(Function('E_x')(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), exp(Symbol('L', commutative=True)), Integer(-1)))"], [["times", 5, "Mul(Integer(-1), Symbol('L', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('L', commutative=True), Add(Function('E_x')(Symbol('L', commutative=True)), Mul(Function('E_x')(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), Integer(-1))), Mul(Integer(-1), Symbol('L', commutative=True), Add(Mul(Function('E_x')(Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)))), exp(Symbol('L', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{v}{(v_{z})} = e^{v_{z}}, then derive e^{v_{z}} + \\frac{d}{d v_{z}} \\mathbf{v}{(v_{z})} = 2 e^{v_{z}}, then obtain (e^{v_{z}} + \\frac{d}{d v_{z}} \\mathbf{v}{(v_{z})})^{2} = 4 e^{2 v_{z}}", "derivation": "\\mathbf{v}{(v_{z})} = e^{v_{z}} and \\mathbf{v}{(v_{z})} + e^{v_{z}} = 2 e^{v_{z}} and \\frac{d}{d v_{z}} (\\mathbf{v}{(v_{z})} + e^{v_{z}}) = \\frac{d}{d v_{z}} 2 e^{v_{z}} and e^{v_{z}} + \\frac{d}{d v_{z}} \\mathbf{v}{(v_{z})} = 2 e^{v_{z}} and (e^{v_{z}} + \\frac{d}{d v_{z}} \\mathbf{v}{(v_{z})})^{2} = 4 e^{2 v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["add", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Mul(Integer(2), exp(Symbol('v_z', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{v}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('v_z', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('v_z', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(exp(Symbol('v_z', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\mu{(y^{\\prime})} = \\log{(y^{\\prime})}, then obtain \\frac{d}{d y^{\\prime}} (\\mu{(y^{\\prime})} \\log{(y^{\\prime})})^{y^{\\prime}} (\\log{(y^{\\prime})}^{2})^{- y^{\\prime}} = \\frac{d}{d y^{\\prime}} 1", "derivation": "\\mu{(y^{\\prime})} = \\log{(y^{\\prime})} and \\mu{(y^{\\prime})} \\log{(y^{\\prime})} = \\log{(y^{\\prime})}^{2} and (\\mu{(y^{\\prime})} \\log{(y^{\\prime})})^{y^{\\prime}} = (\\log{(y^{\\prime})}^{2})^{y^{\\prime}} and (\\mu{(y^{\\prime})} \\log{(y^{\\prime})})^{y^{\\prime}} (\\log{(y^{\\prime})}^{2})^{- y^{\\prime}} = 1 and \\frac{d}{d y^{\\prime}} (\\mu{(y^{\\prime})} \\log{(y^{\\prime})})^{y^{\\prime}} (\\log{(y^{\\prime})}^{2})^{- y^{\\prime}} = \\frac{d}{d y^{\\prime}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "log(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mu')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 3, "Pow(Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Mul(Function('\\\\mu')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Integer(1))"], [["differentiate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Pow(Mul(Function('\\\\mu')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{z},\\hat{H}_l)} = \\hat{H}_l^{v_{z}}, then obtain \\frac{d^{2}}{d v_{z}^{2}} 1 = \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\frac{\\hat{H}_l^{v_{z}}}{\\operatorname{t_{2}}{(v_{z},\\hat{H}_l)}}", "derivation": "\\operatorname{t_{2}}{(v_{z},\\hat{H}_l)} = \\hat{H}_l^{v_{z}} and 1 = \\frac{\\hat{H}_l^{v_{z}}}{\\operatorname{t_{2}}{(v_{z},\\hat{H}_l)}} and \\frac{d}{d v_{z}} 1 = \\frac{\\partial}{\\partial v_{z}} \\frac{\\hat{H}_l^{v_{z}}}{\\operatorname{t_{2}}{(v_{z},\\hat{H}_l)}} and \\frac{d^{2}}{d v_{z}^{2}} 1 = \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\frac{\\hat{H}_l^{v_{z}}}{\\operatorname{t_{2}}{(v_{z},\\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True)))"], [["divide", 1, "Function('t_2')(Symbol('v_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('t_2')(Symbol('v_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('t_2')(Symbol('v_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('t_2')(Symbol('v_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(l)} = \\log{(\\cos{(l)})}, then obtain \\iint (l + \\operatorname{E_{n}}{(l)}) dl dl = \\iint (l + \\log{(\\cos{(l)})}) dl dl", "derivation": "\\operatorname{E_{n}}{(l)} = \\log{(\\cos{(l)})} and l + \\operatorname{E_{n}}{(l)} = l + \\log{(\\cos{(l)})} and \\int (l + \\operatorname{E_{n}}{(l)}) dl = \\int (l + \\log{(\\cos{(l)})}) dl and \\iint (l + \\operatorname{E_{n}}{(l)}) dl dl = \\iint (l + \\log{(\\cos{(l)})}) dl dl", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('l', commutative=True)), log(cos(Symbol('l', commutative=True))))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('E_n')(Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), log(cos(Symbol('l', commutative=True)))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Function('E_n')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), log(cos(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Function('E_n')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), log(cos(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(P_{g},k)} = \\frac{\\sin{(k)}}{P_{g}}, then obtain \\frac{\\partial}{\\partial P_{g}} (\\cos{(\\mathbf{J}{(P_{g},k)})} + 1) = \\frac{\\partial}{\\partial P_{g}} (\\cos{(\\frac{\\sin{(k)}}{P_{g}})} + 1)", "derivation": "\\mathbf{J}{(P_{g},k)} = \\frac{\\sin{(k)}}{P_{g}} and \\cos{(\\mathbf{J}{(P_{g},k)})} = \\cos{(\\frac{\\sin{(k)}}{P_{g}})} and \\cos{(\\mathbf{J}{(P_{g},k)})} + 1 = \\cos{(\\frac{\\sin{(k)}}{P_{g}})} + 1 and \\frac{\\partial}{\\partial P_{g}} (\\cos{(\\mathbf{J}{(P_{g},k)})} + 1) = \\frac{\\partial}{\\partial P_{g}} (\\cos{(\\frac{\\sin{(k)}}{P_{g}})} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('k', commutative=True)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('k', commutative=True))), cos(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(cos(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('k', commutative=True))), Integer(1)), Add(cos(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)))), Integer(1)))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Add(cos(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('k', commutative=True))), Integer(1)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(cos(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), sin(Symbol('k', commutative=True)))), Integer(1)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(P_{g},A_{1})} = \\log{(A_{1}^{P_{g}})} and S{(P_{g},A_{1})} = \\frac{Z{(P_{g},A_{1})}}{A_{1}}, then obtain - A_{1}^{P_{g}} + \\sin{(\\frac{\\log{(A_{1}^{P_{g}})}}{A_{1}})} = - A_{1}^{P_{g}} + \\sin{(\\frac{Z{(P_{g},A_{1})}}{A_{1}})}", "derivation": "Z{(P_{g},A_{1})} = \\log{(A_{1}^{P_{g}})} and \\frac{Z{(P_{g},A_{1})}}{A_{1}} = \\frac{\\log{(A_{1}^{P_{g}})}}{A_{1}} and S{(P_{g},A_{1})} = \\frac{Z{(P_{g},A_{1})}}{A_{1}} and S{(P_{g},A_{1})} = \\frac{\\log{(A_{1}^{P_{g}})}}{A_{1}} and \\sin{(S{(P_{g},A_{1})})} = \\sin{(\\frac{Z{(P_{g},A_{1})}}{A_{1}})} and \\sin{(\\frac{\\log{(A_{1}^{P_{g}})}}{A_{1}})} = \\sin{(\\frac{Z{(P_{g},A_{1})}}{A_{1}})} and - A_{1}^{P_{g}} + \\sin{(\\frac{\\log{(A_{1}^{P_{g}})}}{A_{1}})} = - A_{1}^{P_{g}} + \\sin{(\\frac{Z{(P_{g},A_{1})}}{A_{1}})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True)), log(Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True)))))"], ["renaming_premise", "Equality(Function('S')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('S')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True)))))"], [["sin", 3], "Equality(sin(Function('S')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True))), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))))), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True)))))"], [["minus", 6, "Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True)))))), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))), sin(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('Z')(Symbol('P_g', commutative=True), Symbol('A_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{P},\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}} - \\mathbf{P}}, then obtain (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}^{\\hat{\\mathbf{r}}}{(\\mathbf{P},\\hat{\\mathbf{r}})})^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (e^{\\hat{\\mathbf{r}} - \\mathbf{P}})^{\\hat{\\mathbf{r}}})^{\\mathbf{P}}", "derivation": "\\mathbf{r}{(\\mathbf{P},\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}} - \\mathbf{P}} and \\mathbf{r}^{\\hat{\\mathbf{r}}}{(\\mathbf{P},\\hat{\\mathbf{r}})} = (e^{\\hat{\\mathbf{r}} - \\mathbf{P}})^{\\hat{\\mathbf{r}}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}^{\\hat{\\mathbf{r}}}{(\\mathbf{P},\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (e^{\\hat{\\mathbf{r}} - \\mathbf{P}})^{\\hat{\\mathbf{r}}} and (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}^{\\hat{\\mathbf{r}}}{(\\mathbf{P},\\hat{\\mathbf{r}})})^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (e^{\\hat{\\mathbf{r}} - \\mathbf{P}})^{\\hat{\\mathbf{r}}})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Pow(exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(p)} = \\int \\log{(p)} dp, then derive \\hat{x}_0{(p)} + 1 = E_{x} + p \\log{(p)} - p + 1, then obtain \\int (\\eta^{\\prime} + p \\log{(p)} - p + 1) dE_{x} = \\frac{E_{x}^{2}}{2} + E_{x} (p \\log{(p)} - p + 1) + \\mathbf{r}", "derivation": "\\hat{x}_0{(p)} = \\int \\log{(p)} dp and \\hat{x}_0{(p)} + 1 = \\int \\log{(p)} dp + 1 and \\hat{x}_0{(p)} + 1 = E_{x} + p \\log{(p)} - p + 1 and \\int \\log{(p)} dp + 1 = E_{x} + p \\log{(p)} - p + 1 and \\int (\\int \\log{(p)} dp + 1) dE_{x} = \\int (E_{x} + p \\log{(p)} - p + 1) dE_{x} and \\int (\\eta^{\\prime} + p \\log{(p)} - p + 1) dE_{x} = \\frac{E_{x}^{2}}{2} + E_{x} (p \\log{(p)} - p + 1) + \\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Integer(1)), Add(Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(1)))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Integer(1)), Add(Symbol('E_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(1)), Add(Symbol('E_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)), Integer(1)))"], [["integrate", 4, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(1)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)), Integer(1)), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)), Integer(1)), Tuple(Symbol('E_x', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_x', commutative=True), Integer(2))), Mul(Symbol('E_x', commutative=True), Add(Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)), Integer(1))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(C)} = \\sin{(\\log{(C)})}, then derive \\int (\\hat{H}_{\\lambda}{(C)} + \\log{(C)}) dC = C \\log{(C)} + \\frac{C \\sin{(\\log{(C)})}}{2} - \\frac{C \\cos{(\\log{(C)})}}{2} - C + G, then obtain \\int (\\hat{H}_{\\lambda}{(C)} + \\log{(C)}) dC = \\frac{C \\hat{H}_{\\lambda}{(C)}}{2} + C \\log{(C)} - \\frac{C \\cos{(\\log{(C)})}}{2} - C + G", "derivation": "\\hat{H}_{\\lambda}{(C)} = \\sin{(\\log{(C)})} and \\hat{H}_{\\lambda}{(C)} + \\log{(C)} = \\log{(C)} + \\sin{(\\log{(C)})} and \\int (\\hat{H}_{\\lambda}{(C)} + \\log{(C)}) dC = \\int (\\log{(C)} + \\sin{(\\log{(C)})}) dC and \\int (\\hat{H}_{\\lambda}{(C)} + \\log{(C)}) dC = C \\log{(C)} + \\frac{C \\sin{(\\log{(C)})}}{2} - \\frac{C \\cos{(\\log{(C)})}}{2} - C + G and \\int (\\hat{H}_{\\lambda}{(C)} + \\log{(C)}) dC = \\frac{C \\hat{H}_{\\lambda}{(C)}}{2} + C \\log{(C)} - \\frac{C \\cos{(\\log{(C)})}}{2} - C + G", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True)), sin(log(Symbol('C', commutative=True))))"], [["add", 1, "log(Symbol('C', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Add(log(Symbol('C', commutative=True)), sin(log(Symbol('C', commutative=True)))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Add(log(Symbol('C', commutative=True)), sin(log(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Rational(1, 2), Symbol('C', commutative=True), sin(log(Symbol('C', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('C', commutative=True), cos(log(Symbol('C', commutative=True)))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Add(Mul(Rational(1, 2), Symbol('C', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('C', commutative=True), cos(log(Symbol('C', commutative=True)))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} = \\mathbf{E} b, then obtain - \\mathbf{E}^{2} b + (\\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)})^{2} = - \\mathbf{E}^{2} b + \\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} \\frac{\\partial}{\\partial b} \\mathbf{E}^{2} b", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} = \\mathbf{E} b and \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} = \\mathbf{E}^{2} b and \\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} = \\frac{\\partial}{\\partial b} \\mathbf{E}^{2} b and (\\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)})^{2} = \\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} \\frac{\\partial}{\\partial b} \\mathbf{E}^{2} b and - \\mathbf{E}^{2} b + (\\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)})^{2} = - \\mathbf{E}^{2} b + \\frac{\\partial}{\\partial b} \\mathbf{E} \\operatorname{E_{\\lambda}}{(\\mathbf{E},b)} \\frac{\\partial}{\\partial b} \\mathbf{E}^{2} b", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["minus", 4, "Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(2))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)), Mul(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{1},\\hbar)} = \\sin^{\\hbar}{(C_{1})}, then obtain - 2 C_{1} + \\operatorname{g^{\\prime}_{\\varepsilon}}^{2 C_{1}}{(C_{1},\\hbar)} = - 2 C_{1} + (\\sin^{\\hbar}{(C_{1})})^{C_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{1}}{(C_{1},\\hbar)}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{1},\\hbar)} = \\sin^{\\hbar}{(C_{1})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{1}}{(C_{1},\\hbar)} = (\\sin^{\\hbar}{(C_{1})})^{C_{1}} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{2 C_{1}}{(C_{1},\\hbar)} = (\\sin^{\\hbar}{(C_{1})})^{C_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{1}}{(C_{1},\\hbar)} and - 2 C_{1} + \\operatorname{g^{\\prime}_{\\varepsilon}}^{2 C_{1}}{(C_{1},\\hbar)} = - 2 C_{1} + (\\sin^{\\hbar}{(C_{1})})^{C_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{1}}{(C_{1},\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Symbol('C_1', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(sin(Symbol('C_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True)))"], [["times", 2, "Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True))"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('C_1', commutative=True))), Mul(Pow(Pow(sin(Symbol('C_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('C_1', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Mul(Pow(Pow(sin(Symbol('C_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given h{(\\theta_1)} = e^{\\theta_1}, then obtain h^{2 \\theta_1}{(\\theta_1)} (\\int h{(\\theta_1)} d\\theta_1) \\int e^{\\theta_1} d\\theta_1 = h^{\\theta_1}{(\\theta_1)} (e^{\\theta_1})^{\\theta_1} (\\int h{(\\theta_1)} d\\theta_1) \\int e^{\\theta_1} d\\theta_1", "derivation": "h{(\\theta_1)} = e^{\\theta_1} and \\int h{(\\theta_1)} d\\theta_1 = \\int e^{\\theta_1} d\\theta_1 and h^{\\theta_1}{(\\theta_1)} = (e^{\\theta_1})^{\\theta_1} and h^{\\theta_1}{(\\theta_1)} \\int h{(\\theta_1)} d\\theta_1 = (e^{\\theta_1})^{\\theta_1} \\int h{(\\theta_1)} d\\theta_1 and h^{\\theta_1}{(\\theta_1)} \\int e^{\\theta_1} d\\theta_1 = (e^{\\theta_1})^{\\theta_1} \\int e^{\\theta_1} d\\theta_1 and h^{2 \\theta_1}{(\\theta_1)} (\\int h{(\\theta_1)} d\\theta_1) \\int e^{\\theta_1} d\\theta_1 = h^{\\theta_1}{(\\theta_1)} (e^{\\theta_1})^{\\theta_1} (\\int h{(\\theta_1)} d\\theta_1) \\int e^{\\theta_1} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["times", 3, "Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["times", 5, "Mul(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))), Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Function('h')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Integral(Function('h')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(W)} = \\cos{(W)} and \\operatorname{v_{1}}{(W)} = \\tilde{g}{(W)} - \\int \\cos^{W}{(W)} dW, then obtain \\operatorname{v_{1}}{(W)} = \\cos{(W)} - \\int \\cos^{W}{(W)} dW", "derivation": "\\tilde{g}{(W)} = \\cos{(W)} and \\tilde{g}^{W}{(W)} = \\cos^{W}{(W)} and \\int \\tilde{g}^{W}{(W)} dW = \\int \\cos^{W}{(W)} dW and \\tilde{g}{(W)} - \\int \\tilde{g}^{W}{(W)} dW = \\cos{(W)} - \\int \\tilde{g}^{W}{(W)} dW and \\tilde{g}{(W)} - \\int \\cos^{W}{(W)} dW = \\cos{(W)} - \\int \\cos^{W}{(W)} dW and \\operatorname{v_{1}}{(W)} = \\tilde{g}{(W)} - \\int \\cos^{W}{(W)} dW and \\operatorname{v_{1}}{(W)} = \\cos{(W)} - \\int \\cos^{W}{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["minus", 1, "Integral(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(cos(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(cos(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('W', commutative=True)), Add(Function('\\\\tilde{g}')(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('v_1')(Symbol('W', commutative=True)), Add(cos(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{J}_M,v_{t})} = - \\mathbf{J}_M + e^{v_{t}}, then obtain \\int (\\mathbf{J}_M - \\frac{- \\mathbf{J}_M + e^{v_{t}}}{\\Omega{(\\mathbf{J}_M,v_{t})}} + \\Omega{(\\mathbf{J}_M,v_{t})} + 1) d\\mathbf{J}_M = \\int e^{v_{t}} d\\mathbf{J}_M", "derivation": "\\Omega{(\\mathbf{J}_M,v_{t})} = - \\mathbf{J}_M + e^{v_{t}} and 1 = \\frac{- \\mathbf{J}_M + e^{v_{t}}}{\\Omega{(\\mathbf{J}_M,v_{t})}} and \\mathbf{J}_M + \\Omega{(\\mathbf{J}_M,v_{t})} = e^{v_{t}} and - \\frac{- \\mathbf{J}_M + e^{v_{t}}}{\\Omega{(\\mathbf{J}_M,v_{t})}} + \\Omega{(\\mathbf{J}_M,v_{t})} + 1 = \\Omega{(\\mathbf{J}_M,v_{t})} and \\int (\\mathbf{J}_M + \\Omega{(\\mathbf{J}_M,v_{t})}) d\\mathbf{J}_M = \\int e^{v_{t}} d\\mathbf{J}_M and \\int (\\mathbf{J}_M - \\frac{- \\mathbf{J}_M + e^{v_{t}}}{\\Omega{(\\mathbf{J}_M,v_{t})}} + \\Omega{(\\mathbf{J}_M,v_{t})} + 1) d\\mathbf{J}_M = \\int e^{v_{t}} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('v_t', commutative=True))))"], [["divide", 1, "Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('v_t', commutative=True))), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True))), exp(Symbol('v_t', commutative=True)))"], [["minus", 2, "Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('v_t', commutative=True))), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('v_t', commutative=True))), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(1)), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('v_t', commutative=True))), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))), Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_t', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(x^\\prime,l)} = l + x^\\prime and \\operatorname{v_{z}}{(T)} = \\sin{(e^{T})}, then obtain \\mathbf{H}{(x^\\prime,l)} \\int \\mathbf{H}{(x^\\prime,l)} dx^\\prime + \\operatorname{v_{z}}{(T)} - 1 = \\mathbf{H}{(x^\\prime,l)} \\int (l + x^\\prime) dx^\\prime + \\operatorname{v_{z}}{(T)} - 1", "derivation": "\\mathbf{H}{(x^\\prime,l)} = l + x^\\prime and \\int \\mathbf{H}{(x^\\prime,l)} dx^\\prime = \\int (l + x^\\prime) dx^\\prime and \\mathbf{H}{(x^\\prime,l)} \\int \\mathbf{H}{(x^\\prime,l)} dx^\\prime = \\mathbf{H}{(x^\\prime,l)} \\int (l + x^\\prime) dx^\\prime and \\operatorname{v_{z}}{(T)} = \\sin{(e^{T})} and \\operatorname{v_{z}}{(T)} - 1 = \\sin{(e^{T})} - 1 and \\mathbf{H}{(x^\\prime,l)} \\int \\mathbf{H}{(x^\\prime,l)} dx^\\prime + \\sin{(e^{T})} - 1 = \\mathbf{H}{(x^\\prime,l)} \\int (l + x^\\prime) dx^\\prime + \\sin{(e^{T})} - 1 and \\mathbf{H}{(x^\\prime,l)} \\int \\mathbf{H}{(x^\\prime,l)} dx^\\prime + \\operatorname{v_{z}}{(T)} - 1 = \\mathbf{H}{(x^\\prime,l)} \\int (l + x^\\prime) dx^\\prime + \\operatorname{v_{z}}{(T)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Symbol('l', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Add(Symbol('l', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], ["get_premise", "Equality(Function('v_z')(Symbol('T', commutative=True)), sin(exp(Symbol('T', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Function('v_z')(Symbol('T', commutative=True)), Integer(-1)), Add(sin(exp(Symbol('T', commutative=True))), Integer(-1)))"], [["add", 3, "Add(sin(exp(Symbol('T', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), sin(exp(Symbol('T', commutative=True))), Integer(-1)), Add(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Add(Symbol('l', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), sin(exp(Symbol('T', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Function('v_z')(Symbol('T', commutative=True)), Integer(-1)), Add(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Integral(Add(Symbol('l', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Function('v_z')(Symbol('T', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given u{(\\varepsilon,n)} = \\varepsilon - n, then obtain \\iint (u{(\\varepsilon,n)} + \\frac{u{(\\varepsilon,n)}}{n}) d\\varepsilon dn = \\iint (u{(\\varepsilon,n)} + \\frac{\\varepsilon - n}{n}) d\\varepsilon dn", "derivation": "u{(\\varepsilon,n)} = \\varepsilon - n and \\frac{u{(\\varepsilon,n)}}{n} = \\frac{\\varepsilon - n}{n} and u{(\\varepsilon,n)} + \\frac{u{(\\varepsilon,n)}}{n} = u{(\\varepsilon,n)} + \\frac{\\varepsilon - n}{n} and \\int (u{(\\varepsilon,n)} + \\frac{u{(\\varepsilon,n)}}{n}) d\\varepsilon = \\int (u{(\\varepsilon,n)} + \\frac{\\varepsilon - n}{n}) d\\varepsilon and \\iint (u{(\\varepsilon,n)} + \\frac{u{(\\varepsilon,n)}}{n}) d\\varepsilon dn = \\iint (u{(\\varepsilon,n)} + \\frac{\\varepsilon - n}{n}) d\\varepsilon dn", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["divide", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["add", 2, "Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)))), Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Function('u')(Symbol('\\\\varepsilon', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(E_{n},k)} = E_{n} k, then obtain \\int\\limits^{\\frac{\\mathbf{P}{(E_{n},k)}}{E_{n}}} \\frac{\\partial}{\\partial k} (E_{n} + \\mathbf{P}{(E_{n},k)}) dk = \\int\\limits^{\\frac{\\mathbf{P}{(E_{n},k)}}{E_{n}}} \\frac{\\partial}{\\partial k} (E_{n} k + E_{n}) dk", "derivation": "\\mathbf{P}{(E_{n},k)} = E_{n} k and E_{n} + \\mathbf{P}{(E_{n},k)} = E_{n} k + E_{n} and \\frac{\\partial}{\\partial k} (E_{n} + \\mathbf{P}{(E_{n},k)}) = \\frac{\\partial}{\\partial k} (E_{n} k + E_{n}) and \\frac{\\mathbf{P}{(E_{n},k)}}{E_{n}} = k and \\int \\frac{\\partial}{\\partial k} (E_{n} + \\mathbf{P}{(E_{n},k)}) dk = \\int \\frac{\\partial}{\\partial k} (E_{n} k + E_{n}) dk and \\int\\limits^{\\frac{\\mathbf{P}{(E_{n},k)}}{E_{n}}} \\frac{\\partial}{\\partial k} (E_{n} + \\mathbf{P}{(E_{n},k)}) dk = \\int\\limits^{\\frac{\\mathbf{P}{(E_{n},k)}}{E_{n}}} \\frac{\\partial}{\\partial k} (E_{n} k + E_{n}) dk", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('k', commutative=True)))"], [["add", 1, "Symbol('E_n', commutative=True)"], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Symbol('k', commutative=True)), Symbol('E_n', commutative=True)))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Symbol('E_n', commutative=True), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('k', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('E_n', commutative=True)"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('E_n', commutative=True), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))), Integral(Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('k', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Derivative(Add(Symbol('E_n', commutative=True), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))))), Integral(Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('k', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('E_n', commutative=True), Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{2},z)} = z^{v_{2}}, then obtain \\frac{\\partial}{\\partial v_{2}} \\frac{z^{v_{2}}}{v_{2}} + \\frac{\\partial}{\\partial v_{2}} \\frac{\\operatorname{F_{H}}{(v_{2},z)}}{v_{2}} = 2 \\frac{\\partial}{\\partial v_{2}} \\frac{z^{v_{2}}}{v_{2}}", "derivation": "\\operatorname{F_{H}}{(v_{2},z)} = z^{v_{2}} and \\frac{\\operatorname{F_{H}}{(v_{2},z)}}{v_{2}} = \\frac{z^{v_{2}}}{v_{2}} and \\frac{\\partial}{\\partial v_{2}} \\frac{\\operatorname{F_{H}}{(v_{2},z)}}{v_{2}} = \\frac{\\partial}{\\partial v_{2}} \\frac{z^{v_{2}}}{v_{2}} and \\frac{\\partial}{\\partial v_{2}} \\frac{z^{v_{2}}}{v_{2}} + \\frac{\\partial}{\\partial v_{2}} \\frac{\\operatorname{F_{H}}{(v_{2},z)}}{v_{2}} = 2 \\frac{\\partial}{\\partial v_{2}} \\frac{z^{v_{2}}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True)))"], [["divide", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('F_H')(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('F_H')(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('F_H')(Symbol('v_2', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{X},k)} = \\hat{X} + e^{k} and x{(\\hat{X},k)} = \\frac{\\partial}{\\partial \\hat{X}} (\\hat{X} + e^{k}), then obtain (\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)})^{2} = x{(\\hat{X},k)} \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)}", "derivation": "\\operatorname{t_{2}}{(\\hat{X},k)} = \\hat{X} + e^{k} and \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)} = \\frac{\\partial}{\\partial \\hat{X}} (\\hat{X} + e^{k}) and x{(\\hat{X},k)} = \\frac{\\partial}{\\partial \\hat{X}} (\\hat{X} + e^{k}) and (\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)})^{2} = \\frac{\\partial}{\\partial \\hat{X}} (\\hat{X} + e^{k}) \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)} and (\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)})^{2} = x{(\\hat{X},k)} \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{t_{2}}{(\\hat{X},k)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('x')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Derivative(Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(2)), Mul(Function('x')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Omega{(\\hat{H}_l,\\omega)} = \\hat{H}_l^{\\omega}, then obtain \\frac{\\omega (\\omega + \\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l)}{\\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l} = \\frac{\\omega (\\omega + \\int \\hat{H}_l^{\\omega} d\\hat{H}_l)}{\\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l}", "derivation": "\\Omega{(\\hat{H}_l,\\omega)} = \\hat{H}_l^{\\omega} and \\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l = \\int \\hat{H}_l^{\\omega} d\\hat{H}_l and \\omega + \\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l = \\omega + \\int \\hat{H}_l^{\\omega} d\\hat{H}_l and \\omega (\\omega + \\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l) = \\omega (\\omega + \\int \\hat{H}_l^{\\omega} d\\hat{H}_l) and \\frac{\\omega (\\omega + \\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l)}{\\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l} = \\frac{\\omega (\\omega + \\int \\hat{H}_l^{\\omega} d\\hat{H}_l)}{\\int \\Omega{(\\hat{H}_l,\\omega)} d\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Integral(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["times", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Integral(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))))"], [["divide", 4, "Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Integral(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\psi^{*}{(U,C_{d})} = \\sin^{C_{d}}{(U)}, then derive \\frac{\\partial}{\\partial C_{d}} \\psi^{*}{(U,C_{d})} = \\log{(\\sin{(U)})} \\sin^{C_{d}}{(U)}, then obtain \\frac{J_{\\varepsilon} + \\psi^{*}{(U,C_{d})}}{C_{d}} = \\frac{E_{\\lambda} + \\sin^{C_{d}}{(U)}}{C_{d}}", "derivation": "\\psi^{*}{(U,C_{d})} = \\sin^{C_{d}}{(U)} and \\frac{\\partial}{\\partial C_{d}} \\psi^{*}{(U,C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\sin^{C_{d}}{(U)} and \\frac{\\partial}{\\partial C_{d}} \\psi^{*}{(U,C_{d})} = \\log{(\\sin{(U)})} \\sin^{C_{d}}{(U)} and \\int \\frac{\\partial}{\\partial C_{d}} \\psi^{*}{(U,C_{d})} dC_{d} = \\int \\log{(\\sin{(U)})} \\sin^{C_{d}}{(U)} dC_{d} and \\frac{\\int \\frac{\\partial}{\\partial C_{d}} \\psi^{*}{(U,C_{d})} dC_{d}}{C_{d}} = \\frac{\\int \\log{(\\sin{(U)})} \\sin^{C_{d}}{(U)} dC_{d}}{C_{d}} and \\frac{J_{\\varepsilon} + \\psi^{*}{(U,C_{d})}}{C_{d}} = \\frac{E_{\\lambda} + \\sin^{C_{d}}{(U)}}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(log(sin(Symbol('U', commutative=True))), Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(log(sin(Symbol('U', commutative=True))), Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 4, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Derivative(Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Mul(log(sin(Symbol('U', commutative=True))), Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\psi^*')(Symbol('U', commutative=True), Symbol('C_d', commutative=True)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(sin(Symbol('U', commutative=True)), Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{J})} = \\cos{(\\sin{(\\mathbf{J})})}, then obtain (\\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})})^{\\mathbf{J}} \\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})} = 1", "derivation": "\\lambda{(\\mathbf{J})} = \\cos{(\\sin{(\\mathbf{J})})} and \\lambda^{\\mathbf{J}}{(\\mathbf{J})} = \\cos^{\\mathbf{J}}{(\\sin{(\\mathbf{J})})} and \\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})} = 1 and (\\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})})^{\\mathbf{J}} = 1 and (\\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})})^{\\mathbf{J}} \\lambda^{\\mathbf{J}}{(\\mathbf{J})} = \\lambda^{\\mathbf{J}}{(\\mathbf{J})} and (\\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})})^{\\mathbf{J}} \\lambda^{\\mathbf{J}}{(\\mathbf{J})} \\cos^{- \\mathbf{J}}{(\\sin{(\\mathbf{J})})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 2, "Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1))"], [["times", 4, "Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = \\cos{(e^{\\mathbf{F}})}, then obtain 2 \\frac{d}{d \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = - e^{\\mathbf{F}} \\sin{(e^{\\mathbf{F}})} + \\frac{d}{d \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = \\cos{(e^{\\mathbf{F}})} and 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} + \\cos{(e^{\\mathbf{F}})} and \\frac{d}{d \\mathbf{F}} 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} (\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} + \\cos{(e^{\\mathbf{F}})}) and 2 \\frac{d}{d \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})} = - e^{\\mathbf{F}} \\sin{(e^{\\mathbf{F}})} + \\frac{d}{d \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True)), cos(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True)), cos(exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True)), cos(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(P_{e},\\mathbf{F},g)} = g (P_{e} + \\mathbf{F}) and T{(P_{e},\\mathbf{F})} = P_{e} + \\mathbf{F}, then obtain \\int 0 dg = \\int (g T{(P_{e},\\mathbf{F})} - \\hat{p}{(P_{e},\\mathbf{F},g)}) dg", "derivation": "\\hat{p}{(P_{e},\\mathbf{F},g)} = g (P_{e} + \\mathbf{F}) and P_{e} + \\hat{p}{(P_{e},\\mathbf{F},g)} = P_{e} + g (P_{e} + \\mathbf{F}) and 0 = g (P_{e} + \\mathbf{F}) - \\hat{p}{(P_{e},\\mathbf{F},g)} and T{(P_{e},\\mathbf{F})} = P_{e} + \\mathbf{F} and \\int 0 dg = \\int (g (P_{e} + \\mathbf{F}) - \\hat{p}{(P_{e},\\mathbf{F},g)}) dg and \\int 0 dg = \\int (g T{(P_{e},\\mathbf{F})} - \\hat{p}{(P_{e},\\mathbf{F},g)}) dg", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Symbol('g', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["minus", 2, "Add(Symbol('P_e', commutative=True), Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('g', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)))))"], ["renaming_premise", "Equality(Function('T')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Symbol('g', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Symbol('g', commutative=True), Function('T')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given q{(S,\\psi)} = S^{\\psi}, then obtain \\int (\\psi + q{(S,\\psi)}) q{(S,\\psi)} d\\psi = \\int (S^{\\psi} + \\psi) q{(S,\\psi)} d\\psi", "derivation": "q{(S,\\psi)} = S^{\\psi} and \\psi + q{(S,\\psi)} = S^{\\psi} + \\psi and S^{\\psi} (\\psi + q{(S,\\psi)}) = S^{\\psi} (S^{\\psi} + \\psi) and (\\psi + q{(S,\\psi)}) q{(S,\\psi)} = (S^{\\psi} + \\psi) q{(S,\\psi)} and \\int (\\psi + q{(S,\\psi)}) q{(S,\\psi)} d\\psi = \\int (S^{\\psi} + \\psi) q{(S,\\psi)} d\\psi", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Symbol('\\\\psi', commutative=True), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Add(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["times", 2, "Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(Symbol('\\\\psi', commutative=True), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Add(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\psi', commutative=True), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Add(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Function('q')(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{r})} = \\sin{(\\sin{(\\mathbf{r})})}, then obtain \\hat{H}_l{(\\mathbf{r})} + \\frac{1}{\\sin{(\\sin{(\\mathbf{r})})}} = \\sin{(\\sin{(\\mathbf{r})})} + \\frac{1}{\\sin{(\\sin{(\\mathbf{r})})}}", "derivation": "\\hat{H}_l{(\\mathbf{r})} = \\sin{(\\sin{(\\mathbf{r})})} and 1 = \\frac{\\sin{(\\sin{(\\mathbf{r})})}}{\\hat{H}_l{(\\mathbf{r})}} and \\frac{1}{\\sin{(\\sin{(\\mathbf{r})})}} = \\frac{1}{\\hat{H}_l{(\\mathbf{r})}} and \\hat{H}_l{(\\mathbf{r})} + \\frac{1}{\\hat{H}_l{(\\mathbf{r})}} = \\sin{(\\sin{(\\mathbf{r})})} + \\frac{1}{\\hat{H}_l{(\\mathbf{r})}} and \\hat{H}_l{(\\mathbf{r})} + \\frac{1}{\\sin{(\\sin{(\\mathbf{r})})}} = \\sin{(\\sin{(\\mathbf{r})})} + \\frac{1}{\\sin{(\\sin{(\\mathbf{r})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), sin(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["divide", 2, "sin(sin(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))"], [["add", 1, "Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Add(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))), Add(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given W{(E,k)} = E + e^{k} and B{(E,k)} = E + \\frac{3 W{(E,k)}}{2} + e^{k}, then obtain \\int (E + \\frac{3 W{(E,k)}}{2} + e^{k})^{k} dk = \\int (\\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2})^{k} dk", "derivation": "W{(E,k)} = E + e^{k} and E W{(E,k)} = E (E + e^{k}) and \\frac{W{(E,k)}}{2} = \\frac{E}{2} + \\frac{e^{k}}{2} and E + \\frac{3 W{(E,k)}}{2} + e^{k} = \\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2} and B{(E,k)} = E + \\frac{3 W{(E,k)}}{2} + e^{k} and B{(E,k)} = \\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2} and B^{k}{(E,k)} = (\\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2})^{k} and \\int B^{k}{(E,k)} dk = \\int (\\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2})^{k} dk and \\int (E + \\frac{3 W{(E,k)}}{2} + e^{k})^{k} dk = \\int (\\frac{3 E}{2} + W{(E,k)} + \\frac{3 e^{k}}{2})^{k} dk", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Add(Symbol('E', commutative=True), exp(Symbol('k', commutative=True))))"], [["times", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), exp(Symbol('k', commutative=True)))))"], [["divide", 2, "Mul(Integer(2), Symbol('E', commutative=True))"], "Equality(Mul(Rational(1, 2), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True))), Add(Mul(Rational(1, 2), Symbol('E', commutative=True)), Mul(Rational(1, 2), exp(Symbol('k', commutative=True)))))"], [["add", 3, "Add(Symbol('E', commutative=True), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Mul(Rational(3, 2), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))), Add(Mul(Rational(3, 2), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Mul(Rational(3, 2), exp(Symbol('k', commutative=True)))))"], ["renaming_premise", "Equality(Function('B')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Add(Symbol('E', commutative=True), Mul(Rational(3, 2), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('B')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Add(Mul(Rational(3, 2), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Mul(Rational(3, 2), exp(Symbol('k', commutative=True)))))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(Function('B')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Mul(Rational(3, 2), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Mul(Rational(3, 2), exp(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["integrate", 7, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Function('B')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Add(Mul(Rational(3, 2), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Mul(Rational(3, 2), exp(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Integral(Pow(Add(Symbol('E', commutative=True), Mul(Rational(3, 2), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Add(Mul(Rational(3, 2), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Mul(Rational(3, 2), exp(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(c,y)} = \\int (c + y) dy, then derive \\frac{\\rho_{f}{(c,y)}}{S + c y + \\frac{y^{2}}{2}} = 1, then obtain - \\frac{\\rho_{f}{(c,y)}}{S + c y + \\frac{y^{2}}{2}} = -1", "derivation": "\\rho_{f}{(c,y)} = \\int (c + y) dy and \\frac{\\rho_{f}{(c,y)}}{\\int (c + y) dy} = 1 and \\frac{\\rho_{f}{(c,y)}}{S + c y + \\frac{y^{2}}{2}} = 1 and - \\frac{\\rho_{f}{(c,y)}}{S + c y + \\frac{y^{2}}{2}} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('y', commutative=True)), Integral(Add(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["divide", 1, "Integral(Add(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Add(Symbol('c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('S', commutative=True), Mul(Symbol('c', commutative=True), Symbol('y', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Integer(-1)), Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('y', commutative=True))), Integer(1))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('S', commutative=True), Mul(Symbol('c', commutative=True), Symbol('y', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Integer(-1)), Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('y', commutative=True))), Integer(-1))"]]}, {"prompt": "Given \\mathbf{M}{(c,U)} = U c, then derive \\hat{X} + \\frac{\\int \\frac{c^{2}}{U^{2}} dU + \\int \\frac{\\mathbf{M}{(c,U)}}{U} dU}{c^{2}} = \\int (\\frac{1}{c} + \\frac{1}{U^{2}}) dU, then obtain \\hat{X} + \\frac{\\int \\frac{c^{2}}{U^{2}} dU + \\int \\frac{\\mathbf{M}{(c,U)}}{U} dU}{c^{2}} = \\int (\\frac{\\mathbf{M}{(c,U)}}{U c^{2}} + \\frac{1}{U^{2}}) dU", "derivation": "\\mathbf{M}{(c,U)} = U c and \\frac{\\mathbf{M}{(c,U)}}{U c} = 1 and \\frac{\\mathbf{M}{(c,U)}}{U c^{2}} = \\frac{1}{c} and \\frac{\\mathbf{M}{(c,U)}}{U c^{2}} + \\frac{1}{U^{2}} = \\frac{1}{c} + \\frac{1}{U^{2}} and \\int (\\frac{\\mathbf{M}{(c,U)}}{U c^{2}} + \\frac{1}{U^{2}}) dU = \\int (\\frac{1}{c} + \\frac{1}{U^{2}}) dU and \\hat{X} + \\frac{\\int \\frac{c^{2}}{U^{2}} dU + \\int \\frac{\\mathbf{M}{(c,U)}}{U} dU}{c^{2}} = \\int (\\frac{1}{c} + \\frac{1}{U^{2}}) dU and \\hat{X} + \\frac{\\int \\frac{c^{2}}{U^{2}} dU + \\int \\frac{\\mathbf{M}{(c,U)}}{U} dU}{c^{2}} = \\int (\\frac{\\mathbf{M}{(c,U)}}{U c^{2}} + \\frac{1}{U^{2}}) dU", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Mul(Symbol('U', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Integer(1))"], [["times", 2, "Pow(Symbol('c', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-2)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('c', commutative=True), Integer(-1)))"], [["add", 3, "Pow(Symbol('U', commutative=True), Integer(-2))"], "Equality(Add(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-2)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Integer(-2))), Add(Pow(Symbol('c', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-2))))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-2)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Integer(-2))), Tuple(Symbol('U', commutative=True))), Integral(Add(Pow(Symbol('c', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-2))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-2)), Add(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('c', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))), Integral(Add(Pow(Symbol('c', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-2))), Tuple(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-2)), Add(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('c', commutative=True), Integer(2))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))), Integral(Add(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-2)), Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Integer(-2))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain - \\mathbf{H} + \\hat{H}{(\\mathbf{H})} - \\int \\sin{(\\mathbf{H})} d\\mathbf{H} = - \\mathbf{H} + \\sin{(\\mathbf{H})} - \\int \\sin{(\\mathbf{H})} d\\mathbf{H}", "derivation": "\\hat{H}{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\int \\hat{H}{(\\mathbf{H})} d\\mathbf{H} = \\int \\sin{(\\mathbf{H})} d\\mathbf{H} and - \\mathbf{H} + \\hat{H}{(\\mathbf{H})} = - \\mathbf{H} + \\sin{(\\mathbf{H})} and - \\mathbf{H} + \\hat{H}{(\\mathbf{H})} - \\int \\hat{H}{(\\mathbf{H})} d\\mathbf{H} = - \\mathbf{H} + \\sin{(\\mathbf{H})} - \\int \\hat{H}{(\\mathbf{H})} d\\mathbf{H} and - \\mathbf{H} + \\hat{H}{(\\mathbf{H})} - \\int \\sin{(\\mathbf{H})} d\\mathbf{H} = - \\mathbf{H} + \\sin{(\\mathbf{H})} - \\int \\sin{(\\mathbf{H})} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 3, "Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"]]}, {"prompt": "Given z{(\\phi_2,\\mathbf{f})} = \\mathbf{f} + \\log{(\\phi_2)}, then obtain \\frac{2 z{(\\phi_2,\\mathbf{f})} + \\log{(\\phi_2)}}{\\log{(\\phi_2)}} = \\frac{\\mathbf{f} + z{(\\phi_2,\\mathbf{f})} + 2 \\log{(\\phi_2)}}{\\log{(\\phi_2)}}", "derivation": "z{(\\phi_2,\\mathbf{f})} = \\mathbf{f} + \\log{(\\phi_2)} and z{(\\phi_2,\\mathbf{f})} + \\log{(\\phi_2)} = \\mathbf{f} + 2 \\log{(\\phi_2)} and 2 z{(\\phi_2,\\mathbf{f})} + \\log{(\\phi_2)} = \\mathbf{f} + z{(\\phi_2,\\mathbf{f})} + 2 \\log{(\\phi_2)} and \\frac{2 z{(\\phi_2,\\mathbf{f})} + \\log{(\\phi_2)}}{\\log{(\\phi_2)}} = \\frac{\\mathbf{f} + z{(\\phi_2,\\mathbf{f})} + 2 \\log{(\\phi_2)}}{\\log{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), log(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))))"], [["add", 2, "Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))))"], [["divide", 3, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given A{(y^{\\prime},\\mathbf{S})} = (y^{\\prime})^{\\mathbf{S}}, then derive \\frac{\\partial}{\\partial y^{\\prime}} A{(y^{\\prime},\\mathbf{S})} = \\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}}, then obtain 0 = e^{\\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}}} - e^{\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime})^{\\mathbf{S}}}", "derivation": "A{(y^{\\prime},\\mathbf{S})} = (y^{\\prime})^{\\mathbf{S}} and \\frac{\\partial}{\\partial y^{\\prime}} A{(y^{\\prime},\\mathbf{S})} = \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime})^{\\mathbf{S}} and \\frac{\\partial}{\\partial y^{\\prime}} A{(y^{\\prime},\\mathbf{S})} = \\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}} and e^{\\frac{\\partial}{\\partial y^{\\prime}} A{(y^{\\prime},\\mathbf{S})}} = e^{\\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}}} and e^{\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime})^{\\mathbf{S}}} = e^{\\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}}} and 0 = e^{\\frac{\\mathbf{S} (y^{\\prime})^{\\mathbf{S}}}{y^{\\prime}}} - e^{\\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime})^{\\mathbf{S}}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["exp", 3], "Equality(exp(Derivative(Function('A')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), exp(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), exp(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["minus", 5, "exp(Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(exp(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(-1), exp(Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\mathbf{B}{(E_{\\lambda},t_{2})} = \\frac{\\partial}{\\partial t_{2}} (- E_{\\lambda} + t_{2}), then derive \\mathbf{B}{(E_{\\lambda},t_{2})} = 1, then obtain \\int \\frac{\\partial}{\\partial t_{2}} (- E_{\\lambda} + t_{2}) dt_{2} = \\int 1 dt_{2}", "derivation": "\\mathbf{B}{(E_{\\lambda},t_{2})} = \\frac{\\partial}{\\partial t_{2}} (- E_{\\lambda} + t_{2}) and \\mathbf{B}{(E_{\\lambda},t_{2})} = 1 and \\frac{\\partial}{\\partial t_{2}} (- E_{\\lambda} + t_{2}) = 1 and \\int \\frac{\\partial}{\\partial t_{2}} (- E_{\\lambda} + t_{2}) dt_{2} = \\int 1 dt_{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given m{(E_{x},P_{e})} = P_{e} + e^{E_{x}}, then derive \\int m{(E_{x},P_{e})} dP_{e} = \\frac{P_{e}^{2}}{2} + P_{e} e^{E_{x}} + \\varphi, then obtain 0 = - \\frac{P_{e}^{2}}{2} - P_{e} e^{E_{x}} - \\varphi + \\int m{(E_{x},P_{e})} dP_{e}", "derivation": "m{(E_{x},P_{e})} = P_{e} + e^{E_{x}} and \\int m{(E_{x},P_{e})} dP_{e} = \\int (P_{e} + e^{E_{x}}) dP_{e} and \\int m{(E_{x},P_{e})} dP_{e} = \\frac{P_{e}^{2}}{2} + P_{e} e^{E_{x}} + \\varphi and \\frac{P_{e}^{2}}{2} + P_{e} e^{E_{x}} + \\varphi = \\int (P_{e} + e^{E_{x}}) dP_{e} and 0 = - \\frac{P_{e}^{2}}{2} - P_{e} e^{E_{x}} - \\varphi + \\int (P_{e} + e^{E_{x}}) dP_{e} and 0 = - \\frac{P_{e}^{2}}{2} - P_{e} e^{E_{x}} - \\varphi + \\int m{(E_{x},P_{e})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('E_x', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('m')(Symbol('E_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('E_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["minus", 4, "Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('P_e', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integral(Function('m')(Symbol('E_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given A{(\\nabla)} = e^{e^{\\nabla}}, then obtain \\nabla A{(\\nabla)} e^{e^{\\nabla}} = \\nabla e^{2 e^{\\nabla}}", "derivation": "A{(\\nabla)} = e^{e^{\\nabla}} and A^{2}{(\\nabla)} = A{(\\nabla)} e^{e^{\\nabla}} and \\nabla A^{2}{(\\nabla)} = \\nabla A{(\\nabla)} e^{e^{\\nabla}} and \\nabla A{(\\nabla)} = \\nabla e^{e^{\\nabla}} and \\nabla A{(\\nabla)} e^{e^{\\nabla}} = \\nabla e^{2 e^{\\nabla}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\nabla', commutative=True)), exp(exp(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Function('A')(Symbol('\\\\nabla', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('\\\\nabla', commutative=True)), exp(exp(Symbol('\\\\nabla', commutative=True)))))"], [["times", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('A')(Symbol('\\\\nabla', commutative=True)), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Function('A')(Symbol('\\\\nabla', commutative=True)), exp(exp(Symbol('\\\\nabla', commutative=True)))))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('A')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), exp(exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('A')(Symbol('\\\\nabla', commutative=True)), exp(exp(Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), exp(Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)} and M{(\\tilde{g}^*)} = 4 \\log{(\\tilde{g}^*)}^{2}, then obtain 2 (\\mathbf{s}{(\\tilde{g}^*)} + \\log{(\\tilde{g}^*)}) \\log{(\\tilde{g}^*)} = M{(\\tilde{g}^*)}", "derivation": "\\mathbf{s}{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)} and \\mathbf{s}{(\\tilde{g}^*)} + \\log{(\\tilde{g}^*)} = 2 \\log{(\\tilde{g}^*)} and 2 (\\mathbf{s}{(\\tilde{g}^*)} + \\log{(\\tilde{g}^*)}) \\log{(\\tilde{g}^*)} = 4 \\log{(\\tilde{g}^*)}^{2} and M{(\\tilde{g}^*)} = 4 \\log{(\\tilde{g}^*)}^{2} and 2 (\\mathbf{s}{(\\tilde{g}^*)} + \\log{(\\tilde{g}^*)}) \\log{(\\tilde{g}^*)} = M{(\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 1, "log(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(2), log(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 2, "Mul(Integer(2), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('\\\\mathbf{s}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))), log(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(4), Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(4), Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Add(Function('\\\\mathbf{s}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))), log(Symbol('\\\\tilde{g}^*', commutative=True))), Function('M')(Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(C_{d})} = \\frac{d}{d C_{d}} \\sin{(C_{d})} and \\hat{p}{(C_{d})} = \\sin{(C_{d})} \\cos{(C_{d})} + \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})}, then derive \\operatorname{L_{\\varepsilon}}{(C_{d})} = \\cos{(C_{d})}, then derive \\hat{p}{(C_{d})} = 2 \\sin{(C_{d})} \\cos{(C_{d})}, then obtain \\hat{p}{(C_{d})} = 2 \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(C_{d})} = \\frac{d}{d C_{d}} \\sin{(C_{d})} and \\operatorname{L_{\\varepsilon}}{(C_{d})} = \\cos{(C_{d})} and \\operatorname{L_{\\varepsilon}}{(C_{d})} \\sin{(C_{d})} = \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})} and \\sin{(C_{d})} \\cos{(C_{d})} = \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})} and \\hat{p}{(C_{d})} = \\sin{(C_{d})} \\cos{(C_{d})} + \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})} and \\hat{p}{(C_{d})} = 2 \\sin{(C_{d})} \\cos{(C_{d})} and \\hat{p}{(C_{d})} = 2 \\sin{(C_{d})} \\frac{d}{d C_{d}} \\sin{(C_{d})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Derivative(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["times", 1, "sin(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Mul(sin(Symbol('C_d', commutative=True)), Derivative(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(sin(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True))), Mul(sin(Symbol('C_d', commutative=True)), Derivative(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('C_d', commutative=True)), Add(Mul(sin(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True))), Mul(sin(Symbol('C_d', commutative=True)), Derivative(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\hat{p}')(Symbol('C_d', commutative=True)), Mul(Integer(2), sin(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('\\\\hat{p}')(Symbol('C_d', commutative=True)), Mul(Integer(2), sin(Symbol('C_d', commutative=True)), Derivative(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given E{(\\nabla)} = \\log{(\\nabla)}, then derive \\frac{d}{d \\nabla} E{(\\nabla)} = \\frac{1}{\\nabla}, then obtain E{(\\frac{1}{\\frac{d}{d \\nabla} \\log{(\\nabla)}})} - \\frac{d}{d \\nabla} \\log{(\\nabla)} = \\log{(\\frac{1}{\\frac{d}{d \\nabla} \\log{(\\nabla)}})} - \\frac{d}{d \\nabla} \\log{(\\nabla)}", "derivation": "E{(\\nabla)} = \\log{(\\nabla)} and \\frac{d}{d \\nabla} E{(\\nabla)} = \\frac{d}{d \\nabla} \\log{(\\nabla)} and \\frac{d}{d \\nabla} E{(\\nabla)} = \\frac{1}{\\nabla} and \\frac{d}{d \\nabla} \\log{(\\nabla)} = \\frac{1}{\\nabla} and E{(\\nabla)} - \\frac{1}{\\nabla} = \\log{(\\nabla)} - \\frac{1}{\\nabla} and E{(\\frac{1}{\\frac{d}{d \\nabla} \\log{(\\nabla)}})} - \\frac{d}{d \\nabla} \\log{(\\nabla)} = \\log{(\\frac{1}{\\frac{d}{d \\nabla} \\log{(\\nabla)}})} - \\frac{d}{d \\nabla} \\log{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))"], [["minus", 1, "Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))"], "Equality(Add(Function('E')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Add(log(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('E')(Pow(Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))), Add(log(Pow(Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{f}{(A_{1})} = A_{1}, then derive \\int \\mathbf{f}{(A_{1})} dA_{1} = \\frac{A_{1}^{2}}{2} + E_{\\lambda}, then obtain \\int \\mathbf{f}{(A_{1})} d\\mathbf{f}{(A_{1})} = E_{\\lambda} + \\frac{\\mathbf{f}^{2}{(A_{1})}}{2}", "derivation": "\\mathbf{f}{(A_{1})} = A_{1} and \\int \\mathbf{f}{(A_{1})} dA_{1} = \\int A_{1} dA_{1} and \\int \\mathbf{f}{(A_{1})} dA_{1} = \\frac{A_{1}^{2}}{2} + E_{\\lambda} and \\int \\mathbf{f}{(A_{1})} d\\mathbf{f}{(A_{1})} = E_{\\lambda} + \\frac{\\mathbf{f}^{2}{(A_{1})}}{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Symbol('A_1', commutative=True), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Tuple(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\ddot{x}{(B)} = e^{B}, then obtain \\frac{d}{d B} - 2 (e^{B})^{B} = \\frac{d}{d B} (- \\ddot{x}{(B)} + e^{B} - 2 (e^{B})^{B})", "derivation": "\\ddot{x}{(B)} = e^{B} and \\ddot{x}^{B}{(B)} = (e^{B})^{B} and 0 = - \\ddot{x}{(B)} + e^{B} and - \\ddot{x}^{B}{(B)} = - \\ddot{x}{(B)} - \\ddot{x}^{B}{(B)} + e^{B} and - (e^{B})^{B} = - \\ddot{x}{(B)} + e^{B} - (e^{B})^{B} and - 2 (e^{B})^{B} = - \\ddot{x}{(B)} + e^{B} - 2 (e^{B})^{B} and - 2 \\ddot{x}^{B}{(B)} = - \\ddot{x}{(B)} - 2 \\ddot{x}^{B}{(B)} + e^{B} and \\frac{d}{d B} - 2 \\ddot{x}^{B}{(B)} = \\frac{d}{d B} (- \\ddot{x}{(B)} - 2 \\ddot{x}^{B}{(B)} + e^{B}) and \\frac{d}{d B} - 2 (e^{B})^{B} = \\frac{d}{d B} (- \\ddot{x}{(B)} + e^{B} - 2 (e^{B})^{B})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["minus", 1, "Function('\\\\ddot{x}')(Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))))"], [["add", 5, "Mul(Integer(-1), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True)), Mul(Integer(-1), Integer(2), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Integer(-1), Integer(2), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))))"], [["differentiate", 7, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integer(2), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Derivative(Mul(Integer(-1), Integer(2), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True)), Mul(Integer(-1), Integer(2), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}} m_{s}, then derive \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}}, then obtain \\frac{\\partial}{\\partial m_{s}} \\dot{\\mathbf{r}} m_{s} - \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}} - \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})}", "derivation": "\\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}} m_{s} and \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\frac{\\partial}{\\partial m_{s}} \\dot{\\mathbf{r}} m_{s} and \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}} and \\frac{\\partial}{\\partial m_{s}} \\dot{\\mathbf{r}} m_{s} = \\dot{\\mathbf{r}} and \\frac{\\partial}{\\partial m_{s}} \\dot{\\mathbf{r}} m_{s} - \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})} = \\dot{\\mathbf{r}} - \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(\\dot{\\mathbf{r}},m_{s})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["minus", 4, "Derivative(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Derivative(Function('P_e')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})}, then derive \\int \\operatorname{x^{{\\}'}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = E_{\\lambda} + \\sin{(g^{\\prime}_{\\varepsilon})}, then obtain \\int \\cos{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = E_{\\lambda} + \\sin{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\operatorname{x^{{\\}'}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and \\int \\operatorname{x^{{\\}'}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int \\cos{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\int \\operatorname{x^{{\\}'}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = E_{\\lambda} + \\sin{(g^{\\prime}_{\\varepsilon})} and \\int \\cos{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = E_{\\lambda} + \\sin{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x^\\\\prime')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(L,J_{\\varepsilon})} = \\frac{L}{J_{\\varepsilon}}, then obtain \\frac{2 L \\frac{\\partial}{\\partial L} (\\mu_{0}{(L,J_{\\varepsilon})} + \\frac{L}{J_{\\varepsilon}})}{J_{\\varepsilon}} = \\frac{2 L \\frac{\\partial}{\\partial L} \\frac{2 L}{J_{\\varepsilon}}}{J_{\\varepsilon}}", "derivation": "\\mu_{0}{(L,J_{\\varepsilon})} = \\frac{L}{J_{\\varepsilon}} and \\mu_{0}{(L,J_{\\varepsilon})} + \\frac{L}{J_{\\varepsilon}} = \\frac{2 L}{J_{\\varepsilon}} and \\frac{\\partial}{\\partial L} (\\mu_{0}{(L,J_{\\varepsilon})} + \\frac{L}{J_{\\varepsilon}}) = \\frac{\\partial}{\\partial L} \\frac{2 L}{J_{\\varepsilon}} and \\frac{2 L \\frac{\\partial}{\\partial L} (\\mu_{0}{(L,J_{\\varepsilon})} + \\frac{L}{J_{\\varepsilon}})}{J_{\\varepsilon}} = \\frac{2 L \\frac{\\partial}{\\partial L} \\frac{2 L}{J_{\\varepsilon}}}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('L', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('L', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True)))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('L', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True), Derivative(Add(Function('\\\\mu_0')(Symbol('L', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True), Derivative(Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(G,B)} = \\cos{(B G)} and \\operatorname{n_{1}}{(G,B)} = B G, then derive 0 = - G \\sin{(B G)} - \\frac{\\partial}{\\partial B} \\sigma_{p}{(G,B)}, then derive 0 = - B G \\cos{(B G)} - \\sin{(B G)} - \\frac{\\partial^{2}}{\\partial G\\partial B} \\sigma_{p}{(G,B)}, then obtain 0 = - \\operatorname{n_{1}}{(G,B)} \\cos{(\\operatorname{n_{1}}{(G,B)})} - \\sin{(\\operatorname{n_{1}}{(G,B)})} - \\frac{\\partial^{2}}{\\partial G\\partial B} \\cos{(\\operatorname{n_{1}}{(G,B)})}", "derivation": "\\sigma_{p}{(G,B)} = \\cos{(B G)} and 0 = - \\sigma_{p}{(G,B)} + \\cos{(B G)} and \\frac{d}{d B} 0 = \\frac{\\partial}{\\partial B} (- \\sigma_{p}{(G,B)} + \\cos{(B G)}) and 0 = - G \\sin{(B G)} - \\frac{\\partial}{\\partial B} \\sigma_{p}{(G,B)} and \\frac{d}{d G} 0 = \\frac{\\partial}{\\partial G} (- G \\sin{(B G)} - \\frac{\\partial}{\\partial B} \\sigma_{p}{(G,B)}) and 0 = - B G \\cos{(B G)} - \\sin{(B G)} - \\frac{\\partial^{2}}{\\partial G\\partial B} \\sigma_{p}{(G,B)} and 0 = - B G \\cos{(B G)} - \\sin{(B G)} - \\frac{\\partial^{2}}{\\partial G\\partial B} \\cos{(B G)} and \\operatorname{n_{1}}{(G,B)} = B G and 0 = - \\operatorname{n_{1}}{(G,B)} \\cos{(\\operatorname{n_{1}}{(G,B)})} - \\sin{(\\operatorname{n_{1}}{(G,B)})} - \\frac{\\partial^{2}}{\\partial G\\partial B} \\cos{(\\operatorname{n_{1}}{(G,B)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True)), cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True))))"], [["minus", 1, "Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True))), cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True))), cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('G', commutative=True), sin(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True), sin(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('G', commutative=True), cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), sin(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('G', commutative=True), cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), sin(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(cos(Mul(Symbol('B', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n_1')(Symbol('G', commutative=True), Symbol('B', commutative=True)), cos(Function('n_1')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Mul(Integer(-1), sin(Function('n_1')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Mul(Integer(-1), Derivative(cos(Function('n_1')(Symbol('G', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(i,A_{1},\\lambda)} = A_{1}^{\\lambda} i, then obtain A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)} - \\int (A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)}) d\\lambda = 2 A_{1}^{\\lambda} i - \\int (A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)}) d\\lambda", "derivation": "\\operatorname{v_{2}}{(i,A_{1},\\lambda)} = A_{1}^{\\lambda} i and A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)} = 2 A_{1}^{\\lambda} i and \\int (A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)}) d\\lambda = \\int 2 A_{1}^{\\lambda} i d\\lambda and A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)} - \\int 2 A_{1}^{\\lambda} i d\\lambda = 2 A_{1}^{\\lambda} i - \\int 2 A_{1}^{\\lambda} i d\\lambda and A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)} - \\int (A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)}) d\\lambda = 2 A_{1}^{\\lambda} i - \\int (A_{1}^{\\lambda} i + \\operatorname{v_{2}}{(i,A_{1},\\lambda)}) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["minus", 2, "Integral(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}{(f^{*})} = e^{f^{*}} and n{(\\chi)} = e^{\\chi}, then obtain \\int \\tilde{g}{(f^{*})} n{(\\chi)} df^{*} = \\int \\tilde{g}{(f^{*})} e^{\\chi} df^{*}", "derivation": "\\tilde{g}{(f^{*})} = e^{f^{*}} and n{(\\chi)} = e^{\\chi} and n{(\\chi)} e^{f^{*}} = e^{\\chi} e^{f^{*}} and \\tilde{g}{(f^{*})} n{(\\chi)} = \\tilde{g}{(f^{*})} e^{\\chi} and \\int \\tilde{g}{(f^{*})} n{(\\chi)} df^{*} = \\int \\tilde{g}{(f^{*})} e^{\\chi} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], ["get_premise", "Equality(Function('n')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["times", 2, "exp(Symbol('f^*', commutative=True))"], "Equality(Mul(Function('n')(Symbol('\\\\chi', commutative=True)), exp(Symbol('f^*', commutative=True))), Mul(exp(Symbol('\\\\chi', commutative=True)), exp(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True)), Function('n')(Symbol('\\\\chi', commutative=True))), Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True)), exp(Symbol('\\\\chi', commutative=True))))"], [["integrate", 4, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True)), Function('n')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True)), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\hat{H}_l)} = \\cos{(\\cos{(\\hat{H}_l)})}, then obtain \\hat{H}{(\\hat{H}_l)} = \\hat{H}{(\\hat{H}_l)} - \\hat{H}^{\\hat{H}_l}{(\\hat{H}_l)} + \\cos^{\\hat{H}_l}{(\\cos{(\\hat{H}_l)})}", "derivation": "\\hat{H}{(\\hat{H}_l)} = \\cos{(\\cos{(\\hat{H}_l)})} and \\hat{H}^{\\hat{H}_l}{(\\hat{H}_l)} = \\cos^{\\hat{H}_l}{(\\cos{(\\hat{H}_l)})} and \\hat{H}{(\\hat{H}_l)} + \\hat{H}^{\\hat{H}_l}{(\\hat{H}_l)} = \\hat{H}{(\\hat{H}_l)} + \\cos^{\\hat{H}_l}{(\\cos{(\\hat{H}_l)})} and \\hat{H}{(\\hat{H}_l)} = \\hat{H}{(\\hat{H}_l)} - \\hat{H}^{\\hat{H}_l}{(\\hat{H}_l)} + \\cos^{\\hat{H}_l}{(\\cos{(\\hat{H}_l)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(cos(cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 2, "Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Add(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(cos(cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Add(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Pow(cos(cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(u,\\mathbf{r})} = \\mathbf{r} + \\sin{(u)}, then obtain - u (- u + \\hat{H}{(u,\\mathbf{r})}) (- u (\\mathbf{r} - u + \\sin{(u)}) + u) = - u (- u (\\mathbf{r} - u + \\sin{(u)}) + u) (\\mathbf{r} - u + \\sin{(u)})", "derivation": "\\hat{H}{(u,\\mathbf{r})} = \\mathbf{r} + \\sin{(u)} and - u + \\hat{H}{(u,\\mathbf{r})} = \\mathbf{r} - u + \\sin{(u)} and - u (- u + \\hat{H}{(u,\\mathbf{r})}) = - u (\\mathbf{r} - u + \\sin{(u)}) and - u (- u + \\hat{H}{(u,\\mathbf{r})}) + u = - u (\\mathbf{r} - u + \\sin{(u)}) + u and - u (- u + \\hat{H}{(u,\\mathbf{r})}) (- u (- u + \\hat{H}{(u,\\mathbf{r})}) + u) = - u (- u (- u + \\hat{H}{(u,\\mathbf{r})}) + u) (\\mathbf{r} - u + \\sin{(u)}) and - u (- u + \\hat{H}{(u,\\mathbf{r})}) (- u (\\mathbf{r} - u + \\sin{(u)}) + u) = - u (- u (\\mathbf{r} - u + \\sin{(u)}) + u) (\\mathbf{r} - u + \\sin{(u)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('u', commutative=True))))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))))"], [["add", 3, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('u', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{H}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))), Symbol('u', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{H})} = \\cos{(\\hat{H})}, then derive \\int \\mu_{0}{(\\hat{H})} d\\hat{H} = \\hat{\\mathbf{r}} + \\sin{(\\hat{H})}, then obtain - \\hat{\\mathbf{r}} + \\mu_{0}{(\\hat{H})} - \\sin{(\\hat{H})} = - \\hat{\\mathbf{r}} - \\sin{(\\hat{H})} + \\cos{(\\hat{H})}", "derivation": "\\mu_{0}{(\\hat{H})} = \\cos{(\\hat{H})} and \\int \\mu_{0}{(\\hat{H})} d\\hat{H} = \\int \\cos{(\\hat{H})} d\\hat{H} and \\mu_{0}{(\\hat{H})} - \\int \\cos{(\\hat{H})} d\\hat{H} = \\cos{(\\hat{H})} - \\int \\cos{(\\hat{H})} d\\hat{H} and \\int \\mu_{0}{(\\hat{H})} d\\hat{H} = \\hat{\\mathbf{r}} + \\sin{(\\hat{H})} and \\int \\cos{(\\hat{H})} d\\hat{H} = \\hat{\\mathbf{r}} + \\sin{(\\hat{H})} and - \\hat{\\mathbf{r}} + \\mu_{0}{(\\hat{H})} - \\sin{(\\hat{H})} = - \\hat{\\mathbf{r}} - \\sin{(\\hat{H})} + \\cos{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(cos(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True))), cos(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(M,\\mathbf{M})} = - M + \\mathbf{M} and U{(\\delta)} = \\cos{(\\cos{(\\delta)})}, then obtain (U^{\\delta}{(\\delta)} \\cos{(\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})})})^{\\mathbf{M}} = (\\cos{(\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})})} \\cos^{\\delta}{(\\cos{(\\delta)})})^{\\mathbf{M}}", "derivation": "\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})} = - M + \\mathbf{M} and \\cos{(\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})})} = \\cos{(M - \\mathbf{M})} and U{(\\delta)} = \\cos{(\\cos{(\\delta)})} and U^{\\delta}{(\\delta)} = \\cos^{\\delta}{(\\cos{(\\delta)})} and U^{\\delta}{(\\delta)} \\cos{(M - \\mathbf{M})} = \\cos{(M - \\mathbf{M})} \\cos^{\\delta}{(\\cos{(\\delta)})} and (U^{\\delta}{(\\delta)} \\cos{(M - \\mathbf{M})})^{\\mathbf{M}} = (\\cos{(M - \\mathbf{M})} \\cos^{\\delta}{(\\cos{(\\delta)})})^{\\mathbf{M}} and (U^{\\delta}{(\\delta)} \\cos{(\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})})})^{\\mathbf{M}} = (\\cos{(\\operatorname{E_{\\lambda}}{(M,\\mathbf{M})})} \\cos^{\\delta}{(\\cos{(\\delta)})})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('E_{\\\\lambda}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], ["get_premise", "Equality(Function('U')(Symbol('\\\\delta', commutative=True)), cos(cos(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(cos(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["times", 4, "cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Pow(cos(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('U')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(cos(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Pow(cos(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Mul(Pow(Function('U')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), cos(Function('E_{\\\\lambda}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(cos(Function('E_{\\\\lambda}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(cos(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\delta)} = \\cos{(\\delta)}, then obtain \\frac{\\cos^{4}{(\\delta)}}{(\\int \\operatorname{E_{x}}^{3}{(\\delta)} d\\delta)^{2}} = \\frac{\\cos^{4}{(\\delta)}}{(\\int \\operatorname{E_{x}}{(\\delta)} \\cos^{2}{(\\delta)} d\\delta)^{2}}", "derivation": "\\operatorname{E_{x}}{(\\delta)} = \\cos{(\\delta)} and \\operatorname{E_{x}}^{2}{(\\delta)} = \\operatorname{E_{x}}{(\\delta)} \\cos{(\\delta)} and \\operatorname{E_{x}}^{3}{(\\delta)} = \\operatorname{E_{x}}^{2}{(\\delta)} \\cos{(\\delta)} and \\operatorname{E_{x}}^{3}{(\\delta)} = \\operatorname{E_{x}}{(\\delta)} \\cos^{2}{(\\delta)} and \\int \\operatorname{E_{x}}^{3}{(\\delta)} d\\delta = \\int \\operatorname{E_{x}}{(\\delta)} \\cos^{2}{(\\delta)} d\\delta and \\frac{\\int \\operatorname{E_{x}}^{3}{(\\delta)} d\\delta}{\\cos^{2}{(\\delta)}} = \\frac{\\int \\operatorname{E_{x}}{(\\delta)} \\cos^{2}{(\\delta)} d\\delta}{\\cos^{2}{(\\delta)}} and \\frac{\\cos^{4}{(\\delta)}}{(\\int \\operatorname{E_{x}}^{3}{(\\delta)} d\\delta)^{2}} = \\frac{\\cos^{4}{(\\delta)}}{(\\int \\operatorname{E_{x}}{(\\delta)} \\cos^{2}{(\\delta)} d\\delta)^{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Function('E_x')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))))"], [["times", 2, "Function('E_x')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(3)), Mul(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(2)), cos(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(3)), Mul(Function('E_x')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(3)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Function('E_x')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 5, "Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(2))"], "Equality(Mul(Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(-2)), Integral(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(3)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(-2)), Integral(Mul(Function('E_x')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["power", 6, "Integer(-2)"], "Equality(Mul(Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(4)), Pow(Integral(Pow(Function('E_x')(Symbol('\\\\delta', commutative=True)), Integer(3)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-2))), Mul(Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(4)), Pow(Integral(Mul(Function('E_x')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{J}{(\\hat{X},J)} = J \\hat{X}, then derive \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{J}{(\\hat{X},J)} = J, then obtain J \\mathbf{J}{(\\hat{X},J)} = J^{2} \\hat{X}", "derivation": "\\mathbf{J}{(\\hat{X},J)} = J \\hat{X} and \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{J}{(\\hat{X},J)} = \\frac{\\partial}{\\partial \\hat{X}} J \\hat{X} and \\mathbf{J}{(\\hat{X},J)} \\frac{\\partial}{\\partial \\hat{X}} J \\hat{X} = J \\hat{X} \\frac{\\partial}{\\partial \\hat{X}} J \\hat{X} and \\frac{\\partial}{\\partial \\hat{X}} \\mathbf{J}{(\\hat{X},J)} = J and \\frac{\\partial}{\\partial \\hat{X}} J \\hat{X} = J and J \\mathbf{J}{(\\hat{X},J)} = J^{2} \\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('J', commutative=True)), Derivative(Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True), Derivative(Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('J', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('J', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('J', commutative=True))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\rho{(M)} = \\sin{(M)}, then derive \\frac{d}{d M} \\rho{(M)} = \\cos{(M)}, then obtain \\frac{\\cos^{M}{(M)}}{\\cos{(M)}} = \\frac{(\\frac{d}{d M} \\sin{(M)})^{M}}{\\cos{(M)}}", "derivation": "\\rho{(M)} = \\sin{(M)} and \\frac{d}{d M} \\rho{(M)} = \\frac{d}{d M} \\sin{(M)} and \\frac{d}{d M} \\rho{(M)} = \\cos{(M)} and (\\frac{d}{d M} \\rho{(M)})^{M} = \\cos^{M}{(M)} and \\frac{d}{d M} \\sin{(M)} = \\cos{(M)} and (\\frac{d}{d M} \\rho{(M)})^{M} = (\\frac{d}{d M} \\sin{(M)})^{M} and \\frac{(\\frac{d}{d M} \\rho{(M)})^{M}}{\\cos{(M)}} = \\frac{(\\frac{d}{d M} \\sin{(M)})^{M}}{\\cos{(M)}} and \\frac{\\cos^{M}{(M)}}{\\cos{(M)}} = \\frac{(\\frac{d}{d M} \\sin{(M)})^{M}}{\\cos{(M)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), cos(Symbol('M', commutative=True)))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), cos(Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], [["divide", 6, "cos(Symbol('M', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))), Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Pow(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Mul(Pow(cos(Symbol('M', commutative=True)), Integer(-1)), Pow(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True))))"]]}, {"prompt": "Given b{(\\mathbf{J})} = \\log{(\\mathbf{J})}, then derive \\frac{d}{d \\mathbf{J}} b{(\\mathbf{J})} = \\frac{1}{\\mathbf{J}}, then obtain \\frac{\\frac{d^{2}}{d \\mathbf{J}^{2}} \\log{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\frac{1}{\\mathbf{J}}}{\\mathbf{J}}", "derivation": "b{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} b{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} b{(\\mathbf{J})} = \\frac{1}{\\mathbf{J}} and \\frac{d^{2}}{d \\mathbf{J}^{2}} b{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\frac{1}{\\mathbf{J}} and \\frac{\\frac{d^{2}}{d \\mathbf{J}^{2}} b{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\frac{1}{\\mathbf{J}}}{\\mathbf{J}} and \\frac{\\frac{d^{2}}{d \\mathbf{J}^{2}} \\log{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\frac{1}{\\mathbf{J}}}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Function('b')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\chi)} = \\sin{(\\log{(\\chi)})}, then derive \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{\\chi \\sin{(\\log{(\\chi)})}}{2} - \\frac{\\chi \\cos{(\\log{(\\chi)})}}{2} + \\nabla, then obtain - \\frac{d^{2}}{d \\chi^{2}} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi + \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{\\chi \\operatorname{P_{g}}{(\\chi)}}{2} - \\frac{\\chi \\cos{(\\log{(\\chi)})}}{2} + \\nabla - \\frac{d^{2}}{d \\chi^{2}} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi", "derivation": "\\operatorname{P_{g}}{(\\chi)} = \\sin{(\\log{(\\chi)})} and \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\int \\sin{(\\log{(\\chi)})} d\\chi and \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{\\chi \\sin{(\\log{(\\chi)})}}{2} - \\frac{\\chi \\cos{(\\log{(\\chi)})}}{2} + \\nabla and \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{\\chi \\operatorname{P_{g}}{(\\chi)}}{2} - \\frac{\\chi \\cos{(\\log{(\\chi)})}}{2} + \\nabla and - \\frac{d^{2}}{d \\chi^{2}} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi + \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{\\chi \\operatorname{P_{g}}{(\\chi)}}{2} - \\frac{\\chi \\cos{(\\log{(\\chi)})}}{2} + \\nabla - \\frac{d^{2}}{d \\chi^{2}} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\chi', commutative=True)), sin(log(Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(sin(log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\chi', commutative=True), sin(log(Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\chi', commutative=True), cos(log(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\chi', commutative=True), Function('P_g')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\chi', commutative=True), cos(log(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"], [["minus", 4, "Derivative(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))), Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('\\\\chi', commutative=True), Function('P_g')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\chi', commutative=True), cos(log(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(a,r_{0})} = \\frac{\\partial}{\\partial r_{0}} (- a + r_{0}), then derive \\frac{\\partial}{\\partial r_{0}} \\operatorname{A_{1}}{(a,r_{0})} = 0, then obtain \\frac{\\partial}{\\partial a} (\\frac{\\partial^{2}}{\\partial r_{0}^{2}} (- a + r_{0}))^{a} = \\frac{d}{d a} 0^{a}", "derivation": "\\operatorname{A_{1}}{(a,r_{0})} = \\frac{\\partial}{\\partial r_{0}} (- a + r_{0}) and \\frac{\\partial}{\\partial r_{0}} \\operatorname{A_{1}}{(a,r_{0})} = \\frac{\\partial^{2}}{\\partial r_{0}^{2}} (- a + r_{0}) and \\frac{\\partial}{\\partial r_{0}} \\operatorname{A_{1}}{(a,r_{0})} = 0 and \\frac{\\partial^{2}}{\\partial r_{0}^{2}} (- a + r_{0}) = 0 and (\\frac{\\partial^{2}}{\\partial r_{0}^{2}} (- a + r_{0}))^{a} = 0^{a} and \\frac{\\partial}{\\partial a} (\\frac{\\partial^{2}}{\\partial r_{0}^{2}} (- a + r_{0}))^{a} = \\frac{d}{d a} 0^{a}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('a', commutative=True), Symbol('r_0', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('a', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('a', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))), Integer(0))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))), Symbol('a', commutative=True)), Pow(Integer(0), Symbol('a', commutative=True)))"], [["differentiate", 5, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(v_{1},V)} = - V + v_{1} and \\dot{y}{(v_{1},V)} = J^{V}{(v_{1},V)}, then obtain \\int (\\frac{\\partial}{\\partial V} \\dot{y}{(v_{1},V)})^{v_{1}} dv_{1} = \\int (\\frac{\\partial}{\\partial V} J^{V}{(v_{1},V)})^{v_{1}} dv_{1}", "derivation": "J{(v_{1},V)} = - V + v_{1} and J^{V}{(v_{1},V)} = (- V + v_{1})^{V} and \\frac{\\partial}{\\partial V} J^{V}{(v_{1},V)} = \\frac{\\partial}{\\partial V} (- V + v_{1})^{V} and \\dot{y}{(v_{1},V)} = J^{V}{(v_{1},V)} and \\frac{\\partial}{\\partial V} \\dot{y}{(v_{1},V)} = \\frac{\\partial}{\\partial V} (- V + v_{1})^{V} and \\frac{\\partial}{\\partial V} \\dot{y}{(v_{1},V)} = \\frac{\\partial}{\\partial V} J^{V}{(v_{1},V)} and (\\frac{\\partial}{\\partial V} \\dot{y}{(v_{1},V)})^{v_{1}} = (\\frac{\\partial}{\\partial V} J^{V}{(v_{1},V)})^{v_{1}} and \\int (\\frac{\\partial}{\\partial V} \\dot{y}{(v_{1},V)})^{v_{1}} dv_{1} = \\int (\\frac{\\partial}{\\partial V} J^{V}{(v_{1},V)})^{v_{1}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_1', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_1', commutative=True)), Symbol('V', commutative=True)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_1', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('v_1', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 6, "Symbol('v_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"], [["integrate", 7, "Symbol('v_1', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Pow(Derivative(Pow(Function('J')(Symbol('v_1', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(Z,v_{z})} = \\frac{v_{z}}{Z}, then obtain Z \\frac{\\partial}{\\partial v_{z}} \\operatorname{f_{E}}{(Z,v_{z})} = 1", "derivation": "\\operatorname{f_{E}}{(Z,v_{z})} = \\frac{v_{z}}{Z} and \\frac{\\partial}{\\partial v_{z}} \\operatorname{f_{E}}{(Z,v_{z})} = \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{Z} and \\frac{\\frac{\\partial}{\\partial v_{z}} \\operatorname{f_{E}}{(Z,v_{z})}}{\\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{Z}} = 1 and Z \\frac{\\partial}{\\partial v_{z}} \\operatorname{f_{E}}{(Z,v_{z})} = 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('f_E')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('Z', commutative=True), Derivative(Function('f_E')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = g_{\\varepsilon} + y^{\\prime}, then derive \\mu + \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = g_{\\varepsilon} + m, then obtain \\mu + \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = \\mu + g_{\\varepsilon} + y^{\\prime}", "derivation": "\\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = g_{\\varepsilon} + y^{\\prime} and \\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = \\frac{\\partial}{\\partial g_{\\varepsilon}} (g_{\\varepsilon} + y^{\\prime}) and \\int \\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} dg_{\\varepsilon} = \\int \\frac{\\partial}{\\partial g_{\\varepsilon}} (g_{\\varepsilon} + y^{\\prime}) dg_{\\varepsilon} and \\mu + \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = g_{\\varepsilon} + m and \\mu + g_{\\varepsilon} + y^{\\prime} = g_{\\varepsilon} + m and \\mu + \\operatorname{E_{n}}{(g_{\\varepsilon},y^{\\prime})} = \\mu + g_{\\varepsilon} + y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Derivative(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given Z{(x^\\prime,g)} = g x^\\prime, then obtain Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} Z{(x^\\prime,g)} - Z{(x^\\prime,g)} = Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} g x^\\prime - Z{(x^\\prime,g)}", "derivation": "Z{(x^\\prime,g)} = g x^\\prime and \\frac{\\partial}{\\partial g} Z{(x^\\prime,g)} = \\frac{\\partial}{\\partial g} g x^\\prime and Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} Z{(x^\\prime,g)} = Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} g x^\\prime and Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} Z{(x^\\prime,g)} - Z{(x^\\prime,g)} = Z{(x^\\prime,g)} \\frac{\\partial}{\\partial g} g x^\\prime - Z{(x^\\prime,g)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 2, "Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Derivative(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["minus", 3, "Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Mul(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Derivative(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)))), Add(Mul(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\hat{p},h)} = h^{\\hat{p}} and \\hat{H}_l{(\\hat{p},h)} = \\Psi^{h}{(\\hat{p},h)}, then obtain (h^{\\hat{p}})^{h} + \\hat{H}_l{(\\hat{p},h)} = 2 (h^{\\hat{p}})^{h}", "derivation": "\\Psi{(\\hat{p},h)} = h^{\\hat{p}} and \\Psi^{h}{(\\hat{p},h)} = (h^{\\hat{p}})^{h} and \\hat{H}_l{(\\hat{p},h)} = \\Psi^{h}{(\\hat{p},h)} and (h^{\\hat{p}})^{h} + \\Psi^{h}{(\\hat{p},h)} = 2 (h^{\\hat{p}})^{h} and (h^{\\hat{p}})^{h} + \\hat{H}_l{(\\hat{p},h)} = 2 (h^{\\hat{p}})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True)), Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True)), Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('h', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(c,l)} = \\frac{c}{l}, then obtain (\\frac{c}{l} + 2 l + \\operatorname{F_{N}}{(c,l)}) \\operatorname{F_{N}}{(c,l)} = (\\frac{2 c}{l} + 2 l) \\operatorname{F_{N}}{(c,l)}", "derivation": "\\operatorname{F_{N}}{(c,l)} = \\frac{c}{l} and l + \\operatorname{F_{N}}{(c,l)} = \\frac{c}{l} + l and \\frac{c}{l} + 2 l + \\operatorname{F_{N}}{(c,l)} = \\frac{2 c}{l} + 2 l and (\\frac{c}{l} + 2 l + \\operatorname{F_{N}}{(c,l)}) \\operatorname{F_{N}}{(c,l)} = (\\frac{2 c}{l} + 2 l) \\operatorname{F_{N}}{(c,l)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Add(Mul(Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('l', commutative=True)))"], [["add", 2, "Add(Mul(Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('l', commutative=True))"], "Equality(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('l', commutative=True)), Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(2), Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('l', commutative=True))))"], [["times", 3, "Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('l', commutative=True)), Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('c', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('l', commutative=True))), Function('F_N')(Symbol('c', commutative=True), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{J}_P)} = e^{\\sin{(\\mathbf{J}_P)}}, then obtain \\frac{\\lambda^{2}{(\\mathbf{J}_P)}}{\\sin{(\\mathbf{J}_P)}} = \\frac{\\lambda{(\\mathbf{J}_P)} e^{\\sin{(\\mathbf{J}_P)}}}{\\sin{(\\mathbf{J}_P)}}", "derivation": "\\lambda{(\\mathbf{J}_P)} = e^{\\sin{(\\mathbf{J}_P)}} and \\lambda{(\\mathbf{J}_P)} e^{\\sin{(\\mathbf{J}_P)}} = e^{2 \\sin{(\\mathbf{J}_P)}} and \\frac{\\lambda{(\\mathbf{J}_P)}}{\\sin{(\\mathbf{J}_P)}} = \\frac{e^{\\sin{(\\mathbf{J}_P)}}}{\\sin{(\\mathbf{J}_P)}} and \\frac{\\lambda{(\\mathbf{J}_P)} e^{\\sin{(\\mathbf{J}_P)}}}{\\sin{(\\mathbf{J}_P)}} = \\frac{e^{2 \\sin{(\\mathbf{J}_P)}}}{\\sin{(\\mathbf{J}_P)}} and \\frac{\\lambda^{2}{(\\mathbf{J}_P)}}{\\sin{(\\mathbf{J}_P)}} = \\frac{\\lambda{(\\mathbf{J}_P)} e^{\\sin{(\\mathbf{J}_P)}}}{\\sin{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["divide", 1, "sin(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Mul(exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"], [["divide", 2, "sin(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Mul(exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{s}{(h,\\rho)} = \\sin{(\\rho + h)}, then obtain (\\int \\mathbf{s}{(h,\\rho)} d\\rho)^{h} + \\int \\sin{(\\rho + h)} d\\rho = \\int \\sin{(\\rho + h)} d\\rho + (\\int \\sin{(\\rho + h)} d\\rho)^{h}", "derivation": "\\mathbf{s}{(h,\\rho)} = \\sin{(\\rho + h)} and \\int \\mathbf{s}{(h,\\rho)} d\\rho = \\int \\sin{(\\rho + h)} d\\rho and (\\int \\mathbf{s}{(h,\\rho)} d\\rho)^{h} = (\\int \\sin{(\\rho + h)} d\\rho)^{h} and (\\int \\mathbf{s}{(h,\\rho)} d\\rho)^{h} + \\int \\sin{(\\rho + h)} d\\rho = \\int \\sin{(\\rho + h)} d\\rho + (\\int \\sin{(\\rho + h)} d\\rho)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)), sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True)))"], [["add", 3, "Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Pow(Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True)), Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True)))), Add(Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Pow(Integral(sin(Add(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\dot{x},r_{0})} = \\frac{\\dot{x}}{r_{0}}, then obtain \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} \\int \\Omega{(\\dot{x},r_{0})} dr_{0})^{\\dot{x}} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} \\int \\frac{\\dot{x}}{r_{0}} dr_{0})^{\\dot{x}}", "derivation": "\\Omega{(\\dot{x},r_{0})} = \\frac{\\dot{x}}{r_{0}} and \\int \\Omega{(\\dot{x},r_{0})} dr_{0} = \\int \\frac{\\dot{x}}{r_{0}} dr_{0} and \\dot{x} \\int \\Omega{(\\dot{x},r_{0})} dr_{0} = \\dot{x} \\int \\frac{\\dot{x}}{r_{0}} dr_{0} and (\\dot{x} \\int \\Omega{(\\dot{x},r_{0})} dr_{0})^{\\dot{x}} = (\\dot{x} \\int \\frac{\\dot{x}}{r_{0}} dr_{0})^{\\dot{x}} and \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} \\int \\Omega{(\\dot{x},r_{0})} dr_{0})^{\\dot{x}} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} \\int \\frac{\\dot{x}}{r_{0}} dr_{0})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True))))"], [["times", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True)))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True)))), Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True)))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)} = \\mathbf{J}_P^{\\mathbf{s}}, then obtain \\frac{\\mathbf{J}_P + \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)}}{\\mathbf{J}_P + \\mathbf{J}_P^{\\mathbf{s}} \\mathbf{s}} = 1", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)} = \\mathbf{J}_P^{\\mathbf{s}} and \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)} = \\mathbf{J}_P^{\\mathbf{s}} \\mathbf{s} and \\mathbf{J}_P + \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)} = \\mathbf{J}_P + \\mathbf{J}_P^{\\mathbf{s}} \\mathbf{s} and \\frac{\\mathbf{J}_P + \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s},\\mathbf{J}_P)}}{\\mathbf{J}_P + \\mathbf{J}_P^{\\mathbf{s}} \\mathbf{s}} = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))), Integer(1))"]]}, {"prompt": "Given J{(B)} = \\log{(B)} and \\operatorname{f_{\\mathbf{v}}}{(B)} = \\frac{d}{d B} J{(B)}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB = Q + J{(B)}, then obtain \\frac{(\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB)^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB} = \\frac{(Q + J{(B)})^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB}", "derivation": "J{(B)} = \\log{(B)} and \\operatorname{f_{\\mathbf{v}}}{(B)} = \\frac{d}{d B} J{(B)} and \\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB = \\int \\frac{d}{d B} J{(B)} dB and \\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB = Q + J{(B)} and \\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB = Q + \\log{(B)} and (\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB)^{B} = (Q + \\log{(B)})^{B} and \\frac{(\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB)^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB} = \\frac{(Q + \\log{(B)})^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB} and \\frac{(\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB)^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB} = \\frac{(Q + J{(B)})^{B}}{\\int \\operatorname{f_{\\mathbf{v}}}{(B)} dB}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Derivative(Function('J')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Function('J')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('Q', commutative=True), Function('J')(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('Q', commutative=True), log(Symbol('B', commutative=True))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Symbol('Q', commutative=True), log(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["divide", 6, "Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integer(-1)), Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Pow(Add(Symbol('Q', commutative=True), log(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Mul(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integer(-1)), Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Pow(Add(Symbol('Q', commutative=True), Function('J')(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\dot{x},y)} = \\dot{x} \\cos{(y)}, then obtain - \\mathbf{J}_M{(\\dot{x},y)} + \\int (\\mathbf{J}_M^{\\dot{x}}{(\\dot{x},y)})^{\\dot{x}} d\\dot{x} = - \\mathbf{J}_M{(\\dot{x},y)} + \\int ((\\dot{x} \\cos{(y)})^{\\dot{x}})^{\\dot{x}} d\\dot{x}", "derivation": "\\mathbf{J}_M{(\\dot{x},y)} = \\dot{x} \\cos{(y)} and \\mathbf{J}_M^{\\dot{x}}{(\\dot{x},y)} = (\\dot{x} \\cos{(y)})^{\\dot{x}} and (\\mathbf{J}_M^{\\dot{x}}{(\\dot{x},y)})^{\\dot{x}} = ((\\dot{x} \\cos{(y)})^{\\dot{x}})^{\\dot{x}} and \\int (\\mathbf{J}_M^{\\dot{x}}{(\\dot{x},y)})^{\\dot{x}} d\\dot{x} = \\int ((\\dot{x} \\cos{(y)})^{\\dot{x}})^{\\dot{x}} d\\dot{x} and - \\mathbf{J}_M{(\\dot{x},y)} + \\int (\\mathbf{J}_M^{\\dot{x}}{(\\dot{x},y)})^{\\dot{x}} d\\dot{x} = - \\mathbf{J}_M{(\\dot{x},y)} + \\int ((\\dot{x} \\cos{(y)})^{\\dot{x}})^{\\dot{x}} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('y', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('y', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('y', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Pow(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('y', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 4, "Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True))), Integral(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True))), Integral(Pow(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('y', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(J,l)} = \\frac{l}{J}, then derive \\frac{\\frac{\\partial}{\\partial l} \\varphi{(J,l)}}{J} = \\frac{1}{J^{2}}, then obtain - \\frac{\\frac{\\partial}{\\partial l} \\frac{l}{J}}{J} = - \\frac{1}{J^{2}}", "derivation": "\\varphi{(J,l)} = \\frac{l}{J} and - J + \\varphi{(J,l)} = - J + \\frac{l}{J} and \\frac{\\partial}{\\partial l} (- J + \\varphi{(J,l)}) = \\frac{\\partial}{\\partial l} (- J + \\frac{l}{J}) and \\frac{\\frac{\\partial}{\\partial l} (- J + \\varphi{(J,l)})}{J} = \\frac{\\frac{\\partial}{\\partial l} (- J + \\frac{l}{J})}{J} and \\frac{\\frac{\\partial}{\\partial l} \\varphi{(J,l)}}{J} = \\frac{1}{J^{2}} and - \\frac{\\frac{\\partial}{\\partial l} \\varphi{(J,l)}}{J} = - \\frac{1}{J^{2}} and - \\frac{\\frac{\\partial}{\\partial l} \\frac{l}{J}}{J} = - \\frac{1}{J^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('J', commutative=True), Integer(-2)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2))))"]]}, {"prompt": "Given V{(M_{E},G)} = e^{G + M_{E}}, then derive \\int \\frac{\\partial}{\\partial M_{E}} V{(M_{E},G)} dG = \\sigma_p + e^{G + M_{E}}, then obtain - M_{E} + \\sigma_p + V{(M_{E},G)} = - M_{E} + \\int \\frac{\\partial}{\\partial M_{E}} e^{G + M_{E}} dG", "derivation": "V{(M_{E},G)} = e^{G + M_{E}} and \\frac{\\partial}{\\partial M_{E}} V{(M_{E},G)} = \\frac{\\partial}{\\partial M_{E}} e^{G + M_{E}} and \\int \\frac{\\partial}{\\partial M_{E}} V{(M_{E},G)} dG = \\int \\frac{\\partial}{\\partial M_{E}} e^{G + M_{E}} dG and \\int \\frac{\\partial}{\\partial M_{E}} V{(M_{E},G)} dG = \\sigma_p + e^{G + M_{E}} and \\int \\frac{\\partial}{\\partial M_{E}} e^{G + M_{E}} dG = \\sigma_p + e^{G + M_{E}} and \\int \\frac{\\partial}{\\partial M_{E}} V{(M_{E},G)} dG = \\sigma_p + V{(M_{E},G)} and \\sigma_p + V{(M_{E},G)} = \\sigma_p + e^{G + M_{E}} and - M_{E} + \\sigma_p + V{(M_{E},G)} = - M_{E} + \\sigma_p + e^{G + M_{E}} and - M_{E} + \\sigma_p + V{(M_{E},G)} = - M_{E} + \\int \\frac{\\partial}{\\partial M_{E}} e^{G + M_{E}} dG", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True)), exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Derivative(Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integral(Derivative(exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Derivative(exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Derivative(Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True)))))"], [["minus", 7, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\sigma_p', commutative=True), exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Function('V')(Symbol('M_E', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Integral(Derivative(exp(Add(Symbol('G', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(C)} = \\frac{d}{d C} e^{C}, then obtain - \\int \\operatorname{f_{\\mathbf{p}}}{(C)} (\\frac{d}{d C} e^{C})^{- C} dC = - \\int \\frac{d}{d C} e^{C} (\\frac{d}{d C} e^{C})^{- C} dC", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(C)} = \\frac{d}{d C} e^{C} and \\operatorname{f_{\\mathbf{p}}}{(C)} (\\frac{d}{d C} e^{C})^{- C} = \\frac{d}{d C} e^{C} (\\frac{d}{d C} e^{C})^{- C} and \\int \\operatorname{f_{\\mathbf{p}}}{(C)} (\\frac{d}{d C} e^{C})^{- C} dC = \\int \\frac{d}{d C} e^{C} (\\frac{d}{d C} e^{C})^{- C} dC and - \\int \\operatorname{f_{\\mathbf{p}}}{(C)} (\\frac{d}{d C} e^{C})^{- C} dC = - \\int \\frac{d}{d C} e^{C} (\\frac{d}{d C} e^{C})^{- C} dC", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('C', commutative=True)), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["divide", 1, "Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('C', commutative=True)), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))), Mul(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('C', commutative=True)), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('C', commutative=True)), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))), Mul(Integer(-1), Integral(Mul(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c, then derive \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = c, then obtain \\hat{H} c + \\hat{H} + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c + \\hat{H} + c", "derivation": "\\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} c and \\hat{H} c + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c + \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} c and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = c and \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} c = c and \\hat{H} c + \\hat{H} + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c + \\hat{H} + \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} c and \\hat{H} c + \\hat{H} + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{g_{\\varepsilon}}{(c,\\hat{H})} = \\hat{H} c + \\hat{H} + c", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["add", 2, "Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Derivative(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Symbol('c', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Symbol('c', commutative=True))"], [["add", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Derivative(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Derivative(Function('g_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(y)} = \\cos{(\\log{(y)})}, then derive \\int \\Psi_{nl}{(y)} \\log{(y)} dy = \\dot{y} + \\frac{y \\log{(y)} \\sin{(\\log{(y)})}}{2} + \\frac{y \\log{(y)} \\cos{(\\log{(y)})}}{2} - \\frac{y \\sin{(\\log{(y)})}}{2}, then obtain \\int \\log{(y)} \\cos{(\\log{(y)})} dy = \\dot{y} + \\frac{y \\log{(y)} \\sin{(\\log{(y)})}}{2} + \\frac{y \\log{(y)} \\cos{(\\log{(y)})}}{2} - \\frac{y \\sin{(\\log{(y)})}}{2}", "derivation": "\\Psi_{nl}{(y)} = \\cos{(\\log{(y)})} and \\Psi_{nl}{(y)} \\log{(y)} = \\log{(y)} \\cos{(\\log{(y)})} and \\int \\Psi_{nl}{(y)} \\log{(y)} dy = \\int \\log{(y)} \\cos{(\\log{(y)})} dy and \\int \\Psi_{nl}{(y)} \\log{(y)} dy = \\dot{y} + \\frac{y \\log{(y)} \\sin{(\\log{(y)})}}{2} + \\frac{y \\log{(y)} \\cos{(\\log{(y)})}}{2} - \\frac{y \\sin{(\\log{(y)})}}{2} and \\int \\log{(y)} \\cos{(\\log{(y)})} dy = \\dot{y} + \\frac{y \\log{(y)} \\sin{(\\log{(y)})}}{2} + \\frac{y \\log{(y)} \\cos{(\\log{(y)})}}{2} - \\frac{y \\sin{(\\log{(y)})}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True))))"], [["times", 1, "log(Symbol('y', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True))), Mul(log(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True)))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Function('\\\\Psi_{nl}')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Mul(log(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\Psi_{nl}')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Rational(1, 2), Symbol('y', commutative=True), log(Symbol('y', commutative=True)), sin(log(Symbol('y', commutative=True)))), Mul(Rational(1, 2), Symbol('y', commutative=True), log(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('y', commutative=True), sin(log(Symbol('y', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(log(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Rational(1, 2), Symbol('y', commutative=True), log(Symbol('y', commutative=True)), sin(log(Symbol('y', commutative=True)))), Mul(Rational(1, 2), Symbol('y', commutative=True), log(Symbol('y', commutative=True)), cos(log(Symbol('y', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('y', commutative=True), sin(log(Symbol('y', commutative=True))))))"]]}, {"prompt": "Given a{(A_{z},\\phi_1)} = A_{z} + \\phi_1, then obtain 2 a{(A_{z},\\phi_1)} (\\int 2 a{(A_{z},\\phi_1)} d\\phi_1)^{2} = 2 a{(A_{z},\\phi_1)} (\\int (A_{z} + \\phi_1 + a{(A_{z},\\phi_1)}) d\\phi_1) \\int 2 a{(A_{z},\\phi_1)} d\\phi_1", "derivation": "a{(A_{z},\\phi_1)} = A_{z} + \\phi_1 and 2 a{(A_{z},\\phi_1)} = A_{z} + \\phi_1 + a{(A_{z},\\phi_1)} and \\int 2 a{(A_{z},\\phi_1)} d\\phi_1 = \\int (A_{z} + \\phi_1 + a{(A_{z},\\phi_1)}) d\\phi_1 and 2 a{(A_{z},\\phi_1)} \\int 2 a{(A_{z},\\phi_1)} d\\phi_1 = 2 a{(A_{z},\\phi_1)} \\int (A_{z} + \\phi_1 + a{(A_{z},\\phi_1)}) d\\phi_1 and 2 a{(A_{z},\\phi_1)} (\\int 2 a{(A_{z},\\phi_1)} d\\phi_1)^{2} = 2 a{(A_{z},\\phi_1)} (\\int (A_{z} + \\phi_1 + a{(A_{z},\\phi_1)}) d\\phi_1) \\int 2 a{(A_{z},\\phi_1)} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 3, "Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["times", 4, "Integral(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Integral(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(2))), Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Integer(2), Function('a')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\delta{(r,\\phi_2)} = \\frac{\\phi_2}{r}, then obtain ((\\frac{\\phi_2}{r})^{\\phi_2} + \\delta{(r,\\phi_2)} - \\frac{1}{r}) (\\delta{(r,\\phi_2)} + \\delta^{\\phi_2}{(r,\\phi_2)} - \\frac{1}{r}) = ((\\frac{\\phi_2}{r})^{\\phi_2} + \\delta{(r,\\phi_2)} - \\frac{1}{r})^{2}", "derivation": "\\delta{(r,\\phi_2)} = \\frac{\\phi_2}{r} and \\delta^{\\phi_2}{(r,\\phi_2)} = (\\frac{\\phi_2}{r})^{\\phi_2} and \\delta^{\\phi_2}{(r,\\phi_2)} - \\frac{1}{r} = (\\frac{\\phi_2}{r})^{\\phi_2} - \\frac{1}{r} and \\delta{(r,\\phi_2)} + \\delta^{\\phi_2}{(r,\\phi_2)} - \\frac{1}{r} = (\\frac{\\phi_2}{r})^{\\phi_2} + \\delta{(r,\\phi_2)} - \\frac{1}{r} and ((\\frac{\\phi_2}{r})^{\\phi_2} + \\delta{(r,\\phi_2)} - \\frac{1}{r}) (\\delta{(r,\\phi_2)} + \\delta^{\\phi_2}{(r,\\phi_2)} - \\frac{1}{r}) = ((\\frac{\\phi_2}{r})^{\\phi_2} + \\delta{(r,\\phi_2)} - \\frac{1}{r})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 2, "Pow(Symbol('r', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Add(Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))))"], [["add", 3, "Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Add(Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))))"], [["times", 4, "Add(Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Add(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))))), Pow(Add(Pow(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Function('\\\\delta')(Symbol('r', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Integer(2)))"]]}, {"prompt": "Given q{(W)} = \\log{(W)}, then obtain \\frac{q{(W)}}{W + q^{- W}{(W)} \\log{(W)}^{W}} = \\frac{\\log{(W)}}{W + q^{- W}{(W)} \\log{(W)}^{W}}", "derivation": "q{(W)} = \\log{(W)} and q^{W}{(W)} = \\log{(W)}^{W} and 1 = q^{- W}{(W)} \\log{(W)}^{W} and W + 1 = W + q^{- W}{(W)} \\log{(W)}^{W} and \\frac{q{(W)}}{W + 1} = \\frac{\\log{(W)}}{W + 1} and \\frac{q{(W)}}{W + q^{- W}{(W)} \\log{(W)}^{W}} = \\frac{\\log{(W)}}{W + q^{- W}{(W)} \\log{(W)}^{W}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('q')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["divide", 2, "Pow(Function('q')(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('q')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Symbol('W', commutative=True))"], "Equality(Add(Symbol('W', commutative=True), Integer(1)), Add(Symbol('W', commutative=True), Mul(Pow(Function('q')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))))"], [["divide", 1, "Add(Symbol('W', commutative=True), Integer(1))"], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Integer(1)), Integer(-1)), Function('q')(Symbol('W', commutative=True))), Mul(Pow(Add(Symbol('W', commutative=True), Integer(1)), Integer(-1)), log(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Mul(Pow(Function('q')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))), Integer(-1)), Function('q')(Symbol('W', commutative=True))), Mul(Pow(Add(Symbol('W', commutative=True), Mul(Pow(Function('q')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))), Integer(-1)), log(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(r_{0},C_{d})} = \\log{(C_{d}^{r_{0}})} and \\operatorname{A_{1}}{(r_{0},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\log{(C_{d}^{r_{0}})}, then obtain - \\log{(C_{d}^{r_{0}})} + \\frac{\\partial}{\\partial C_{d}} \\hat{H}{(r_{0},C_{d})} = \\operatorname{A_{1}}{(r_{0},C_{d})} - \\log{(C_{d}^{r_{0}})}", "derivation": "\\hat{H}{(r_{0},C_{d})} = \\log{(C_{d}^{r_{0}})} and \\frac{\\partial}{\\partial C_{d}} \\hat{H}{(r_{0},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\log{(C_{d}^{r_{0}})} and - \\log{(C_{d}^{r_{0}})} + \\frac{\\partial}{\\partial C_{d}} \\hat{H}{(r_{0},C_{d})} = - \\log{(C_{d}^{r_{0}})} + \\frac{\\partial}{\\partial C_{d}} \\log{(C_{d}^{r_{0}})} and \\operatorname{A_{1}}{(r_{0},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\log{(C_{d}^{r_{0}})} and - \\log{(C_{d}^{r_{0}})} + \\frac{\\partial}{\\partial C_{d}} \\hat{H}{(r_{0},C_{d})} = \\operatorname{A_{1}}{(r_{0},C_{d})} - \\log{(C_{d}^{r_{0}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 2, "log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True)))), Derivative(Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True)))), Derivative(log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), Derivative(log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True)))), Derivative(Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Function('A_1')(Symbol('r_0', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), log(Pow(Symbol('C_d', commutative=True), Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(L)} = \\int e^{L} dL, then derive \\operatorname{F_{c}}{(L)} + e^{L} = \\Psi^{\\dagger} + 2 e^{L}, then obtain \\frac{d}{d L} (e^{L} + \\int e^{L} dL) = \\frac{\\partial}{\\partial L} (\\Psi^{\\dagger} + 2 e^{L})", "derivation": "\\operatorname{F_{c}}{(L)} = \\int e^{L} dL and \\operatorname{F_{c}}{(L)} + e^{L} = e^{L} + \\int e^{L} dL and \\operatorname{F_{c}}{(L)} + e^{L} = \\Psi^{\\dagger} + 2 e^{L} and e^{L} + \\int e^{L} dL = \\Psi^{\\dagger} + 2 e^{L} and \\frac{d}{d L} (e^{L} + \\int e^{L} dL) = \\frac{\\partial}{\\partial L} (\\Psi^{\\dagger} + 2 e^{L})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('L', commutative=True)), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["add", 1, "exp(Symbol('L', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Add(exp(Symbol('L', commutative=True)), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('F_c')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(2), exp(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(exp(Symbol('L', commutative=True)), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(2), exp(Symbol('L', commutative=True)))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(exp(Symbol('L', commutative=True)), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(2), exp(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(\\dot{y})} = \\cos{(\\log{(\\dot{y})})} and \\operatorname{a^{\\dagger}}{(\\dot{y})} = \\log{(\\dot{y})}, then derive \\frac{d}{d \\dot{y}} L{(\\dot{y})} = - \\frac{\\sin{(\\log{(\\dot{y})})}}{\\dot{y}}, then obtain - \\frac{\\sin{(\\operatorname{a^{\\dagger}}{(\\dot{y})})}}{\\dot{y}} = \\frac{d}{d \\dot{y}} \\cos{(\\log{(\\dot{y})})}", "derivation": "L{(\\dot{y})} = \\cos{(\\log{(\\dot{y})})} and \\frac{d}{d \\dot{y}} L{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\cos{(\\log{(\\dot{y})})} and \\operatorname{a^{\\dagger}}{(\\dot{y})} = \\log{(\\dot{y})} and \\frac{d}{d \\dot{y}} L{(\\dot{y})} = - \\frac{\\sin{(\\log{(\\dot{y})})}}{\\dot{y}} and \\frac{d}{d \\dot{y}} L{(\\dot{y})} = - \\frac{\\sin{(\\operatorname{a^{\\dagger}}{(\\dot{y})})}}{\\dot{y}} and - \\frac{\\sin{(\\operatorname{a^{\\dagger}}{(\\dot{y})})}}{\\dot{y}} = \\frac{d}{d \\dot{y}} \\cos{(\\log{(\\dot{y})})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\dot{y}', commutative=True)), cos(log(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('L')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), sin(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), sin(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)))), Derivative(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(\\mathbf{J}_M,y^{\\prime},T)} = T \\mathbf{J}_M y^{\\prime}, then obtain T \\mathbf{J}_M y^{\\prime} + \\mathbf{J}_M^{2} + (y^{\\prime})^{2} f^{2}{(\\mathbf{J}_M,y^{\\prime},T)} = T^{2} \\mathbf{J}_M^{2} (y^{\\prime})^{4} + T \\mathbf{J}_M y^{\\prime} + \\mathbf{J}_M^{2}", "derivation": "f{(\\mathbf{J}_M,y^{\\prime},T)} = T \\mathbf{J}_M y^{\\prime} and y^{\\prime} f{(\\mathbf{J}_M,y^{\\prime},T)} = T \\mathbf{J}_M (y^{\\prime})^{2} and (y^{\\prime})^{2} f^{2}{(\\mathbf{J}_M,y^{\\prime},T)} = T^{2} \\mathbf{J}_M^{2} (y^{\\prime})^{4} and \\mathbf{J}_M^{2} + (y^{\\prime})^{2} f^{2}{(\\mathbf{J}_M,y^{\\prime},T)} = T^{2} \\mathbf{J}_M^{2} (y^{\\prime})^{4} + \\mathbf{J}_M^{2} and T \\mathbf{J}_M y^{\\prime} + \\mathbf{J}_M^{2} + (y^{\\prime})^{2} f^{2}{(\\mathbf{J}_M,y^{\\prime},T)} = T^{2} \\mathbf{J}_M^{2} (y^{\\prime})^{4} + T \\mathbf{J}_M y^{\\prime} + \\mathbf{J}_M^{2}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True))), Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)), Pow(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Pow(Symbol('T', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(4))))"], [["add", 3, "Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)), Pow(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Add(Mul(Pow(Symbol('T', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(4))), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))))"], [["add", 4, "Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)), Pow(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Add(Mul(Pow(Symbol('T', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(4))), Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))))"]]}, {"prompt": "Given s{(i,n_{1})} = \\log{(i + n_{1})}, then obtain (i + \\frac{\\partial}{\\partial i} i s{(i,n_{1})})^{i} = (i + \\frac{\\partial}{\\partial i} i \\log{(i + n_{1})})^{i}", "derivation": "s{(i,n_{1})} = \\log{(i + n_{1})} and i s{(i,n_{1})} = i \\log{(i + n_{1})} and \\frac{\\partial}{\\partial i} i s{(i,n_{1})} = \\frac{\\partial}{\\partial i} i \\log{(i + n_{1})} and i + \\frac{\\partial}{\\partial i} i s{(i,n_{1})} = i + \\frac{\\partial}{\\partial i} i \\log{(i + n_{1})} and (i + \\frac{\\partial}{\\partial i} i s{(i,n_{1})})^{i} = (i + \\frac{\\partial}{\\partial i} i \\log{(i + n_{1})})^{i}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), log(Add(Symbol('i', commutative=True), Symbol('n_1', commutative=True))))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('s')(Symbol('i', commutative=True), Symbol('n_1', commutative=True))), Mul(Symbol('i', commutative=True), log(Add(Symbol('i', commutative=True), Symbol('n_1', commutative=True)))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Symbol('i', commutative=True), Function('s')(Symbol('i', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Symbol('i', commutative=True), log(Add(Symbol('i', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["add", 3, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('i', commutative=True), Function('s')(Symbol('i', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('i', commutative=True), log(Add(Symbol('i', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('i', commutative=True), Function('s')(Symbol('i', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Symbol('i', commutative=True)), Pow(Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('i', commutative=True), log(Add(Symbol('i', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1)))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(E)} = \\sin{(\\sin{(E)})}, then obtain \\int (\\int (- E + \\int \\mathbf{v}{(E)} dE) dE)^{E} dE = \\int (\\int (- E + \\int \\sin{(\\sin{(E)})} dE) dE)^{E} dE", "derivation": "\\mathbf{v}{(E)} = \\sin{(\\sin{(E)})} and \\int \\mathbf{v}{(E)} dE = \\int \\sin{(\\sin{(E)})} dE and - E + \\int \\mathbf{v}{(E)} dE = - E + \\int \\sin{(\\sin{(E)})} dE and \\int (- E + \\int \\mathbf{v}{(E)} dE) dE = \\int (- E + \\int \\sin{(\\sin{(E)})} dE) dE and (\\int (- E + \\int \\mathbf{v}{(E)} dE) dE)^{E} = (\\int (- E + \\int \\sin{(\\sin{(E)})} dE) dE)^{E} and \\int (\\int (- E + \\int \\mathbf{v}{(E)} dE) dE)^{E} dE = \\int (\\int (- E + \\int \\sin{(\\sin{(E)})} dE) dE)^{E} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), sin(sin(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["minus", 2, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["power", 4, "Symbol('E', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["integrate", 5, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\theta{(v_{t})} = \\log{(\\sin{(v_{t})})} and \\theta_{1}{(v_{t})} = (\\frac{\\log{(\\sin{(v_{t})})}}{\\sin{(v_{t})}})^{v_{t}}, then obtain \\theta_{1}{(v_{t})} + 1 = (\\frac{\\log{(\\sin{(v_{t})})}}{\\sin{(v_{t})}})^{v_{t}} + 1", "derivation": "\\theta{(v_{t})} = \\log{(\\sin{(v_{t})})} and \\frac{\\theta{(v_{t})}}{\\sin{(v_{t})}} = \\frac{\\log{(\\sin{(v_{t})})}}{\\sin{(v_{t})}} and \\theta_{1}{(v_{t})} = (\\frac{\\log{(\\sin{(v_{t})})}}{\\sin{(v_{t})}})^{v_{t}} and \\theta_{1}{(v_{t})} = (\\frac{\\theta{(v_{t})}}{\\sin{(v_{t})}})^{v_{t}} and \\theta_{1}{(v_{t})} + 1 = (\\frac{\\theta{(v_{t})}}{\\sin{(v_{t})}})^{v_{t}} + 1 and \\theta_{1}{(v_{t})} + 1 = (\\frac{\\log{(\\sin{(v_{t})})}}{\\sin{(v_{t})}})^{v_{t}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('v_t', commutative=True)), log(sin(Symbol('v_t', commutative=True))))"], [["divide", 1, "sin(Symbol('v_t', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))), Mul(log(sin(Symbol('v_t', commutative=True))), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('v_t', commutative=True)), Pow(Mul(log(sin(Symbol('v_t', commutative=True))), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))), Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\theta_1')(Symbol('v_t', commutative=True)), Pow(Mul(Function('\\\\theta')(Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))), Symbol('v_t', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('\\\\theta_1')(Symbol('v_t', commutative=True)), Integer(1)), Add(Pow(Mul(Function('\\\\theta')(Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))), Symbol('v_t', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\theta_1')(Symbol('v_t', commutative=True)), Integer(1)), Add(Pow(Mul(log(sin(Symbol('v_t', commutative=True))), Pow(sin(Symbol('v_t', commutative=True)), Integer(-1))), Symbol('v_t', commutative=True)), Integer(1)))"]]}, {"prompt": "Given t{(n)} = \\int e^{n} dn and \\operatorname{t_{2}}{(\\mathbf{D},n)} = \\mathbf{D} + e^{n}, then derive \\frac{t{(n)}}{\\mathbf{D} + e^{n}} = 1, then obtain (\\frac{t{(n)}}{\\mathbf{D} + e^{n}})^{n} = 1", "derivation": "t{(n)} = \\int e^{n} dn and \\frac{t{(n)}}{\\int e^{n} dn} = 1 and \\frac{t{(n)}}{\\mathbf{D} + e^{n}} = 1 and \\operatorname{t_{2}}{(\\mathbf{D},n)} = \\mathbf{D} + e^{n} and \\frac{t{(n)}}{\\operatorname{t_{2}}{(\\mathbf{D},n)}} = 1 and (\\frac{t{(n)}}{\\operatorname{t_{2}}{(\\mathbf{D},n)}})^{n} = 1 and (\\frac{t{(n)}}{\\mathbf{D} + e^{n}})^{n} = 1", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('n', commutative=True)), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["divide", 1, "Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Mul(Function('t')(Symbol('n', commutative=True)), Pow(Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('n', commutative=True))), Integer(-1)), Function('t')(Symbol('n', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('t')(Symbol('n', commutative=True)), Pow(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Integer(1))"], [["power", 5, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Function('t')(Symbol('n', commutative=True)), Pow(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Symbol('n', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('n', commutative=True))), Integer(-1)), Function('t')(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\phi{(u,\\omega)} = \\int \\omega u du and \\eta{(u,\\omega)} = 2 \\int \\omega u du, then obtain u + 2 \\phi{(u,\\omega)} = u + 2 \\int \\omega u du", "derivation": "\\phi{(u,\\omega)} = \\int \\omega u du and \\eta{(u,\\omega)} = 2 \\int \\omega u du and u + \\eta{(u,\\omega)} = u + 2 \\int \\omega u du and \\eta{(u,\\omega)} = 2 \\phi{(u,\\omega)} and u + 2 \\phi{(u,\\omega)} = u + 2 \\int \\omega u du", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["add", 2, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Symbol('u', commutative=True), Mul(Integer(2), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\eta')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('u', commutative=True), Mul(Integer(2), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Symbol('u', commutative=True), Mul(Integer(2), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\phi{(I,\\dot{y})} = \\dot{y}^{I} and \\theta_{2}{(I,\\dot{y})} = \\cos{(\\phi{(I,\\dot{y})})} \\frac{\\partial}{\\partial I} \\phi{(I,\\dot{y})}, then derive \\cos{(\\phi{(I,\\dot{y})})} \\frac{\\partial}{\\partial I} \\phi{(I,\\dot{y})} = \\dot{y}^{I} \\log{(\\dot{y})} \\cos{(\\dot{y}^{I})}, then obtain \\theta_{2}{(I,\\dot{y})} = \\dot{y}^{I} \\log{(\\dot{y})} \\cos{(\\dot{y}^{I})}", "derivation": "\\phi{(I,\\dot{y})} = \\dot{y}^{I} and \\sin{(\\phi{(I,\\dot{y})})} = \\sin{(\\dot{y}^{I})} and \\frac{\\partial}{\\partial I} \\sin{(\\phi{(I,\\dot{y})})} = \\frac{\\partial}{\\partial I} \\sin{(\\dot{y}^{I})} and \\cos{(\\phi{(I,\\dot{y})})} \\frac{\\partial}{\\partial I} \\phi{(I,\\dot{y})} = \\dot{y}^{I} \\log{(\\dot{y})} \\cos{(\\dot{y}^{I})} and \\theta_{2}{(I,\\dot{y})} = \\cos{(\\phi{(I,\\dot{y})})} \\frac{\\partial}{\\partial I} \\phi{(I,\\dot{y})} and \\theta_{2}{(I,\\dot{y})} = \\dot{y}^{I} \\log{(\\dot{y})} \\cos{(\\dot{y}^{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(sin(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Derivative(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)), cos(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(cos(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Derivative(Function('\\\\phi')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)), cos(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(A)} = \\sin{(A)}, then obtain \\frac{\\frac{d^{2}}{d A^{2}} (A + \\dot{y}{(A)})}{\\frac{d^{2}}{d A^{2}} (A + \\sin{(A)})} = 1", "derivation": "\\dot{y}{(A)} = \\sin{(A)} and A + \\dot{y}{(A)} = A + \\sin{(A)} and \\frac{d}{d A} (A + \\dot{y}{(A)}) = \\frac{d}{d A} (A + \\sin{(A)}) and \\frac{d^{2}}{d A^{2}} (A + \\dot{y}{(A)}) = \\frac{d^{2}}{d A^{2}} (A + \\sin{(A)}) and \\frac{\\frac{d^{2}}{d A^{2}} (A + \\dot{y}{(A)})}{\\dot{y}{(A)}} = \\frac{\\frac{d^{2}}{d A^{2}} (A + \\sin{(A)})}{\\dot{y}{(A)}} and \\frac{\\frac{d^{2}}{d A^{2}} (A + \\dot{y}{(A)})}{\\frac{d^{2}}{d A^{2}} (A + \\sin{(A)})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["add", 1, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Function('\\\\dot{y}')(Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Symbol('A', commutative=True), Function('\\\\dot{y}')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Symbol('A', commutative=True), Function('\\\\dot{y}')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))))"], [["divide", 4, "Function('\\\\dot{y}')(Symbol('A', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Integer(-1)), Derivative(Add(Symbol('A', commutative=True), Function('\\\\dot{y}')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2)))), Mul(Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Integer(-1)), Derivative(Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2)))))"], [["divide", 5, "Mul(Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Integer(-1)), Derivative(Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))))"], "Equality(Mul(Derivative(Add(Symbol('A', commutative=True), Function('\\\\dot{y}')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))), Pow(Derivative(Add(Symbol('A', commutative=True), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given H{(y^{\\prime},E_{\\lambda})} = \\log{(E_{\\lambda}^{y^{\\prime}})}, then derive \\frac{\\partial}{\\partial y^{\\prime}} H{(y^{\\prime},E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain (\\frac{\\partial}{\\partial y^{\\prime}} H{(y^{\\prime},E_{\\lambda})})^{y^{\\prime}} = \\log{(E_{\\lambda})}^{y^{\\prime}}", "derivation": "H{(y^{\\prime},E_{\\lambda})} = \\log{(E_{\\lambda}^{y^{\\prime}})} and \\frac{\\partial}{\\partial y^{\\prime}} H{(y^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial y^{\\prime}} \\log{(E_{\\lambda}^{y^{\\prime}})} and \\frac{\\partial}{\\partial y^{\\prime}} H{(y^{\\prime},E_{\\lambda})} = \\log{(E_{\\lambda})} and (\\frac{\\partial}{\\partial y^{\\prime}} H{(y^{\\prime},E_{\\lambda})})^{y^{\\prime}} = \\log{(E_{\\lambda})}^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), log(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(t)} = t, then derive \\int \\mathbf{J}_P{(t)} dt = A_{z} + \\frac{t^{2}}{2}, then obtain \\iint \\mathbf{J}_P{(t)} d\\mathbf{J}_P{(t)} dA_{z} + 1 = \\int (A_{z} + \\frac{\\mathbf{J}_P^{2}{(t)}}{2}) dA_{z} + 1", "derivation": "\\mathbf{J}_P{(t)} = t and \\int \\mathbf{J}_P{(t)} dt = \\int t dt and \\int \\mathbf{J}_P{(t)} dt = A_{z} + \\frac{t^{2}}{2} and \\iint \\mathbf{J}_P{(t)} dt dA_{z} = \\int (A_{z} + \\frac{t^{2}}{2}) dA_{z} and \\iint \\mathbf{J}_P{(t)} dt dA_{z} + 1 = \\int (A_{z} + \\frac{t^{2}}{2}) dA_{z} + 1 and \\iint \\mathbf{J}_P{(t)} d\\mathbf{J}_P{(t)} dA_{z} + 1 = \\int (A_{z} + \\frac{\\mathbf{J}_P^{2}{(t)}}{2}) dA_{z} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Symbol('t', commutative=True))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('A_z', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integer(1)), Add(Integral(Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('A_z', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Integral(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Integer(1)), Add(Integral(Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{J}_P')(Symbol('t', commutative=True)), Integer(2)))), Tuple(Symbol('A_z', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then derive \\frac{d}{d \\mathbf{J}_M} \\Psi_{nl}{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)}, then obtain \\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)}", "derivation": "\\Psi_{nl}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\Psi_{nl}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\Psi_{nl}{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(t,\\eta^{\\prime})} = \\eta^{\\prime} + t, then obtain \\frac{\\rho_{f}{(t,\\eta^{\\prime})}}{t} = \\frac{\\eta^{\\prime} + t}{t}", "derivation": "\\rho_{f}{(t,\\eta^{\\prime})} = \\eta^{\\prime} + t and - \\eta^{\\prime} + \\rho_{f}{(t,\\eta^{\\prime})} = t and \\frac{\\rho_{f}{(t,\\eta^{\\prime})}}{- \\eta^{\\prime} + \\rho_{f}{(t,\\eta^{\\prime})}} = \\frac{\\eta^{\\prime} + t}{- \\eta^{\\prime} + \\rho_{f}{(t,\\eta^{\\prime})}} and \\frac{\\rho_{f}{(t,\\eta^{\\prime})}}{t} = \\frac{\\eta^{\\prime} + t}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)))"], [["minus", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('t', commutative=True))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\eta{(g_{\\varepsilon},M_{E})} = M_{E} \\cos{(g_{\\varepsilon})}, then obtain 2 M_{E} \\cos{(g_{\\varepsilon})} + 1 = 2 M_{E} \\cos{(g_{\\varepsilon})} + \\frac{2 M_{E} \\cos{(g_{\\varepsilon})}}{M_{E} \\cos{(g_{\\varepsilon})} + \\eta{(g_{\\varepsilon},M_{E})}}", "derivation": "\\eta{(g_{\\varepsilon},M_{E})} = M_{E} \\cos{(g_{\\varepsilon})} and M_{E} \\cos{(g_{\\varepsilon})} + \\eta{(g_{\\varepsilon},M_{E})} = 2 M_{E} \\cos{(g_{\\varepsilon})} and 1 = \\frac{2 M_{E} \\cos{(g_{\\varepsilon})}}{M_{E} \\cos{(g_{\\varepsilon})} + \\eta{(g_{\\varepsilon},M_{E})}} and 2 M_{E} \\cos{(g_{\\varepsilon})} + 1 = 2 M_{E} \\cos{(g_{\\varepsilon})} + \\frac{2 M_{E} \\cos{(g_{\\varepsilon})}}{M_{E} \\cos{(g_{\\varepsilon})} + \\eta{(g_{\\varepsilon},M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('\\\\eta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Add(Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('\\\\eta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Integer(1), Mul(Integer(2), Symbol('M_E', commutative=True), Pow(Add(Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('\\\\eta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('M_E', commutative=True), Pow(Add(Mul(Symbol('M_E', commutative=True), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('\\\\eta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\phi_1)} = \\sin{(\\phi_1)} and \\rho{(\\phi_1)} = \\log{(\\phi_1)}, then obtain \\rho{(\\phi_1)} \\sin{(\\phi_1)} - \\sin{(\\phi_1)} = \\log{(\\phi_1)} \\sin{(\\phi_1)} - \\sin{(\\phi_1)}", "derivation": "\\tilde{g}^*{(\\phi_1)} = \\sin{(\\phi_1)} and \\rho{(\\phi_1)} = \\log{(\\phi_1)} and \\rho{(\\phi_1)} \\tilde{g}^*{(\\phi_1)} = \\tilde{g}^*{(\\phi_1)} \\log{(\\phi_1)} and \\rho{(\\phi_1)} \\sin{(\\phi_1)} = \\log{(\\phi_1)} \\sin{(\\phi_1)} and \\rho{(\\phi_1)} \\sin{(\\phi_1)} - \\tilde{g}^*{(\\phi_1)} = - \\tilde{g}^*{(\\phi_1)} + \\log{(\\phi_1)} \\sin{(\\phi_1)} and \\rho{(\\phi_1)} \\sin{(\\phi_1)} - \\sin{(\\phi_1)} = \\log{(\\phi_1)} \\sin{(\\phi_1)} - \\sin{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["times", 2, "Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\phi_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True))), Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(log(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 4, "Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Mul(Function('\\\\rho')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True))), Mul(log(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Function('\\\\rho')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\phi_1', commutative=True)))), Add(Mul(log(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(F_{c})} = \\sin{(\\log{(F_{c})})} and \\operatorname{E_{x}}{(\\dot{x},u)} = \\dot{x} u, then obtain (\\frac{\\partial}{\\partial u} \\operatorname{E_{x}}{(\\dot{x},u)} + 1) \\sin{(\\log{(F_{c})})} = (\\frac{\\partial}{\\partial u} \\dot{x} u + 1) \\sin{(\\log{(F_{c})})}", "derivation": "\\operatorname{F_{g}}{(F_{c})} = \\sin{(\\log{(F_{c})})} and \\operatorname{E_{x}}{(\\dot{x},u)} = \\dot{x} u and \\frac{\\partial}{\\partial u} \\operatorname{E_{x}}{(\\dot{x},u)} = \\frac{\\partial}{\\partial u} \\dot{x} u and \\frac{\\partial}{\\partial u} \\operatorname{E_{x}}{(\\dot{x},u)} + \\frac{\\sin{(\\log{(F_{c})})}}{\\operatorname{F_{g}}{(F_{c})}} = \\frac{\\partial}{\\partial u} \\dot{x} u + \\frac{\\sin{(\\log{(F_{c})})}}{\\operatorname{F_{g}}{(F_{c})}} and \\frac{\\partial}{\\partial u} \\operatorname{E_{x}}{(\\dot{x},u)} + 1 = \\frac{\\partial}{\\partial u} \\dot{x} u + 1 and (\\frac{\\partial}{\\partial u} \\operatorname{E_{x}}{(\\dot{x},u)} + 1) \\sin{(\\log{(F_{c})})} = (\\frac{\\partial}{\\partial u} \\dot{x} u + 1) \\sin{(\\log{(F_{c})})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('F_c', commutative=True)), sin(log(Symbol('F_c', commutative=True))))"], ["get_premise", "Equality(Function('E_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["add", 3, "Mul(Pow(Function('F_g')(Symbol('F_c', commutative=True)), Integer(-1)), sin(log(Symbol('F_c', commutative=True))))"], "Equality(Add(Derivative(Function('E_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Pow(Function('F_g')(Symbol('F_c', commutative=True)), Integer(-1)), sin(log(Symbol('F_c', commutative=True))))), Add(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Pow(Function('F_g')(Symbol('F_c', commutative=True)), Integer(-1)), sin(log(Symbol('F_c', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('E_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1)))"], [["times", 5, "sin(log(Symbol('F_c', commutative=True)))"], "Equality(Mul(Add(Derivative(Function('E_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1)), sin(log(Symbol('F_c', commutative=True)))), Mul(Add(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1)), sin(log(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}, then derive \\pi{(\\mathbf{E})} = - \\sin{(\\mathbf{E})}, then obtain (- \\sin{(\\mathbf{E})} + \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} - (\\cos{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} = 0", "derivation": "\\pi{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and \\pi{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} and - \\sin{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and - \\sin{(\\mathbf{E})} + \\cos{(\\mathbf{E})} = \\cos{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and (- \\sin{(\\mathbf{E})} + \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} = (\\cos{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} and (- \\sin{(\\mathbf{E})} + \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} - (\\cos{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}) \\pi{(\\mathbf{E})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 3, "cos(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), cos(Symbol('\\\\mathbf{E}', commutative=True))), Add(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["times", 4, "Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), cos(Symbol('\\\\mathbf{E}', commutative=True))), Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Add(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 5, "Mul(Add(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), cos(Symbol('\\\\mathbf{E}', commutative=True))), Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Add(cos(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Function('\\\\pi')(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\sigma_{x}{(\\psi^*,f)} = f^{\\psi^*}, then obtain \\frac{\\partial}{\\partial f} \\iint 0^{\\psi^*} df d\\psi^* = \\frac{\\partial}{\\partial f} \\iint (f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}))^{\\psi^*} df d\\psi^*", "derivation": "\\sigma_{x}{(\\psi^*,f)} = f^{\\psi^*} and 0 = f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)} and 0 = f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}) and 0^{\\psi^*} = (f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}))^{\\psi^*} and \\int 0^{\\psi^*} df = \\int (f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}))^{\\psi^*} df and \\iint 0^{\\psi^*} df d\\psi^* = \\iint (f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}))^{\\psi^*} df d\\psi^* and \\frac{\\partial}{\\partial f} \\iint 0^{\\psi^*} df d\\psi^* = \\frac{\\partial}{\\partial f} \\iint (f (f^{\\psi^*} - \\sigma_{x}{(\\psi^*,f)}))^{\\psi^*} df d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)))))"], [["times", 2, "Symbol('f', commutative=True)"], "Equality(Integer(0), Mul(Symbol('f', commutative=True), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))))"], [["power", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Symbol('f', commutative=True), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(Mul(Symbol('f', commutative=True), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["integrate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(Mul(Symbol('f', commutative=True), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 6, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Symbol('f', commutative=True), Add(Pow(Symbol('f', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\mathbf{J}_f,\\mathbf{F})} = \\mathbf{F} + \\mathbf{J}_f, then obtain \\frac{\\mathbf{F} + \\mathbf{J}_f}{\\mathbf{F}} + \\frac{t{(\\mathbf{J}_f,\\mathbf{F})}}{\\mathbf{F}} = \\frac{2 \\mathbf{F} + 2 \\mathbf{J}_f}{\\mathbf{F}}", "derivation": "t{(\\mathbf{J}_f,\\mathbf{F})} = \\mathbf{F} + \\mathbf{J}_f and \\frac{t{(\\mathbf{J}_f,\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\mathbf{F} + \\mathbf{J}_f}{\\mathbf{F}} and \\frac{\\mathbf{F} + \\mathbf{J}_f}{\\mathbf{F}} + \\frac{t{(\\mathbf{J}_f,\\mathbf{F})}}{\\mathbf{F}} = \\frac{2 (\\mathbf{F} + \\mathbf{J}_f)}{\\mathbf{F}} and \\frac{2 t{(\\mathbf{J}_f,\\mathbf{F})}}{\\mathbf{F}} = \\frac{2 \\mathbf{F} + 2 \\mathbf{J}_f}{\\mathbf{F}} and \\frac{2 (\\mathbf{F} + \\mathbf{J}_f)}{\\mathbf{F}} = \\frac{2 \\mathbf{F} + 2 \\mathbf{J}_f}{\\mathbf{F}} and \\frac{\\mathbf{F} + \\mathbf{J}_f}{\\mathbf{F}} + \\frac{t{(\\mathbf{J}_f,\\mathbf{F})}}{\\mathbf{F}} = \\frac{2 \\mathbf{F} + 2 \\mathbf{J}_f}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(c)} = \\int \\log{(c)} dc, then derive \\hat{H}_{\\lambda}{(c)} + \\log{(c)} = \\mathbf{v} + c \\log{(c)} - c + \\log{(c)}, then obtain \\mathbf{v} + c \\log{(c)} - c + 2 \\log{(c)} + \\int \\log{(c)} dc = 2 \\mathbf{v} + 2 c \\log{(c)} - 2 c + 2 \\log{(c)}", "derivation": "\\hat{H}_{\\lambda}{(c)} = \\int \\log{(c)} dc and \\hat{H}_{\\lambda}{(c)} + \\log{(c)} = \\log{(c)} + \\int \\log{(c)} dc and \\hat{H}_{\\lambda}{(c)} + \\log{(c)} = \\mathbf{v} + c \\log{(c)} - c + \\log{(c)} and \\mathbf{v} + c \\log{(c)} - c + \\hat{H}_{\\lambda}{(c)} + 2 \\log{(c)} = 2 \\mathbf{v} + 2 c \\log{(c)} - 2 c + 2 \\log{(c)} and \\mathbf{v} + c \\log{(c)} - c + 2 \\log{(c)} + \\int \\log{(c)} dc = 2 \\mathbf{v} + 2 c \\log{(c)} - 2 c + 2 \\log{(c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 1, "log(Symbol('c', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Add(log(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\theta_2)} = e^{\\theta_2}, then derive - \\theta_2 + \\int \\hat{p}{(\\theta_2)} d\\theta_2 = \\hat{X} - \\theta_2 + e^{\\theta_2}, then obtain - \\theta_2 + \\int \\hat{p}{(\\theta_2)} d\\theta_2 = \\hat{X} - \\theta_2 + \\hat{p}{(\\theta_2)}", "derivation": "\\hat{p}{(\\theta_2)} = e^{\\theta_2} and \\int \\hat{p}{(\\theta_2)} d\\theta_2 = \\int e^{\\theta_2} d\\theta_2 and - \\theta_2 + \\int \\hat{p}{(\\theta_2)} d\\theta_2 = - \\theta_2 + \\int e^{\\theta_2} d\\theta_2 and - \\theta_2 + \\int \\hat{p}{(\\theta_2)} d\\theta_2 = \\hat{X} - \\theta_2 + e^{\\theta_2} and - \\theta_2 + \\int \\hat{p}{(\\theta_2)} d\\theta_2 = \\hat{X} - \\theta_2 + \\hat{p}{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{p},B)} = \\frac{\\partial}{\\partial B} B \\hat{p}, then derive \\operatorname{E_{x}}{(\\hat{p},B)} = \\hat{p}, then obtain \\int 1 dB = \\int \\frac{\\hat{p}}{\\frac{\\partial}{\\partial B} B \\hat{p}} dB", "derivation": "\\operatorname{E_{x}}{(\\hat{p},B)} = \\frac{\\partial}{\\partial B} B \\hat{p} and \\operatorname{E_{x}}{(\\hat{p},B)} = \\hat{p} and \\frac{\\partial}{\\partial B} B \\hat{p} = \\hat{p} and 1 = \\frac{\\hat{p}}{\\frac{\\partial}{\\partial B} B \\hat{p}} and \\int 1 dB = \\int \\frac{\\hat{p}}{\\frac{\\partial}{\\partial B} B \\hat{p}} dB", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\hat{p}', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E_x')(Symbol('\\\\hat{p}', commutative=True), Symbol('B', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True))"], [["divide", 3, "Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 4, "Symbol('B', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('B', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\pi{(F_{H})} = \\log{(F_{H})}, then obtain - \\frac{\\sin{(\\theta_1)}}{\\log{(F_{H})}} = \\frac{F_{H} (- \\pi{(F_{H})} + \\log{(F_{H})}) - \\sin{(\\theta_1)}}{\\log{(F_{H})}}", "derivation": "\\pi{(F_{H})} = \\log{(F_{H})} and 0 = - \\pi{(F_{H})} + \\log{(F_{H})} and 0 = - F_{H} (- \\pi{(F_{H})} + \\log{(F_{H})}) and 0 = F_{H} (- \\pi{(F_{H})} + \\log{(F_{H})}) and - \\sin{(\\theta_1)} = F_{H} (- \\pi{(F_{H})} + \\log{(F_{H})}) - \\sin{(\\theta_1)} and - \\frac{\\sin{(\\theta_1)}}{\\log{(F_{H})}} = \\frac{F_{H} (- \\pi{(F_{H})} + \\log{(F_{H})}) - \\sin{(\\theta_1)}}{\\log{(F_{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["minus", 1, "Function('\\\\pi')(Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('F_H', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Integer(0), Mul(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True)))))"], [["minus", 4, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["divide", 5, "log(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Pow(log(Symbol('F_H', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Add(Mul(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Pow(log(Symbol('F_H', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(M_{E})} = \\log{(M_{E})}, then derive \\dot{z}{(M_{E})} + \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} \\log{(M_{E})} - M_{E} + \\Psi_{nl} + \\dot{z}{(M_{E})}, then obtain \\dot{z}{(M_{E})} + \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} \\dot{z}{(M_{E})} - M_{E} + \\Psi_{nl} + \\dot{z}{(M_{E})}", "derivation": "\\dot{z}{(M_{E})} = \\log{(M_{E})} and M_{E} + \\dot{z}{(M_{E})} = M_{E} + \\log{(M_{E})} and \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\int (M_{E} + \\log{(M_{E})}) dM_{E} and \\dot{z}{(M_{E})} + \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\dot{z}{(M_{E})} + \\int (M_{E} + \\log{(M_{E})}) dM_{E} and \\dot{z}{(M_{E})} + \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} \\log{(M_{E})} - M_{E} + \\Psi_{nl} + \\dot{z}{(M_{E})} and \\dot{z}{(M_{E})} + \\int (M_{E} + \\dot{z}{(M_{E})}) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} \\dot{z}{(M_{E})} - M_{E} + \\Psi_{nl} + \\dot{z}{(M_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["add", 3, "Function('\\\\dot{z}')(Symbol('M_E', commutative=True))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Add(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{z}')(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\tilde{g},\\hat{p}_0)} = \\hat{p}_0 + \\tilde{g} and \\operatorname{A_{x}}{(i,\\mathbf{F})} = \\frac{\\log{(i)}}{\\mathbf{F}}, then obtain - \\operatorname{A_{x}}{(i,\\mathbf{F})} + \\mathbf{f}{(\\tilde{g},\\hat{p}_0)} = \\hat{p}_0 + \\tilde{g} - \\operatorname{A_{x}}{(i,\\mathbf{F})}", "derivation": "\\mathbf{f}{(\\tilde{g},\\hat{p}_0)} = \\hat{p}_0 + \\tilde{g} and \\operatorname{A_{x}}{(i,\\mathbf{F})} = \\frac{\\log{(i)}}{\\mathbf{F}} and \\mathbf{f}{(\\tilde{g},\\hat{p}_0)} - \\frac{\\log{(i)}}{\\mathbf{F}} = \\hat{p}_0 + \\tilde{g} - \\frac{\\log{(i)}}{\\mathbf{F}} and - \\operatorname{A_{x}}{(i,\\mathbf{F})} + \\mathbf{f}{(\\tilde{g},\\hat{p}_0)} = \\hat{p}_0 + \\tilde{g} - \\operatorname{A_{x}}{(i,\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], ["get_premise", "Equality(Function('A_x')(Symbol('i', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), log(Symbol('i', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), log(Symbol('i', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), log(Symbol('i', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), log(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('A_x')(Symbol('i', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Function('\\\\mathbf{f}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Function('A_x')(Symbol('i', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} = - \\hat{H}_{\\lambda} + \\lambda^{m} and \\operatorname{A_{z}}{(r)} = \\sin{(r)}, then obtain \\operatorname{A_{z}}{(r)} \\int \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} d\\lambda = \\sin{(r)} \\int \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} d\\lambda", "derivation": "\\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} = - \\hat{H}_{\\lambda} + \\lambda^{m} and \\int \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} d\\lambda = \\int (- \\hat{H}_{\\lambda} + \\lambda^{m}) d\\lambda and \\operatorname{A_{z}}{(r)} = \\sin{(r)} and \\operatorname{A_{z}}{(r)} \\int (- \\hat{H}_{\\lambda} + \\lambda^{m}) d\\lambda = \\sin{(r)} \\int (- \\hat{H}_{\\lambda} + \\lambda^{m}) d\\lambda and \\operatorname{A_{z}}{(r)} \\int \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} d\\lambda = \\sin{(r)} \\int \\mathbf{f}{(m,\\hat{H}_{\\lambda},\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], ["get_premise", "Equality(Function('A_z')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["times", 3, "Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('A_z')(Symbol('r', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(sin(Symbol('r', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('A_z')(Symbol('r', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(sin(Symbol('r', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{\\nabla}, then obtain - \\frac{\\partial}{\\partial \\nabla} \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})} = - \\Psi_{\\lambda}^{\\nabla} \\log{(\\Psi_{\\lambda})}", "derivation": "\\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{\\nabla} and - \\Psi_{\\lambda} + \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\Psi_{\\lambda}^{\\nabla} and \\frac{\\partial}{\\partial \\nabla} (- \\Psi_{\\lambda} + \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial \\nabla} (- \\Psi_{\\lambda} + \\Psi_{\\lambda}^{\\nabla}) and - \\frac{\\partial}{\\partial \\nabla} (- \\Psi_{\\lambda} + \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})}) = - \\frac{\\partial}{\\partial \\nabla} (- \\Psi_{\\lambda} + \\Psi_{\\lambda}^{\\nabla}) and - \\frac{\\partial}{\\partial \\nabla} \\dot{\\mathbf{r}}{(\\nabla,\\Psi_{\\lambda})} = - \\Psi_{\\lambda}^{\\nabla} \\log{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given y{(A_{2},A)} = A - A_{2}, then obtain \\int 1 dA = \\int ((A - 2 A_{2} + 1)^{A_{2}})^{A_{2}} ((- A_{2} + y{(A_{2},A)} + 1)^{A_{2}})^{- A_{2}} dA", "derivation": "y{(A_{2},A)} = A - A_{2} and y{(A_{2},A)} + 1 = A - A_{2} + 1 and - A_{2} + y{(A_{2},A)} + 1 = A - 2 A_{2} + 1 and (- A_{2} + y{(A_{2},A)} + 1)^{A_{2}} = (A - 2 A_{2} + 1)^{A_{2}} and ((- A_{2} + y{(A_{2},A)} + 1)^{A_{2}})^{A_{2}} = ((A - 2 A_{2} + 1)^{A_{2}})^{A_{2}} and 1 = ((A - 2 A_{2} + 1)^{A_{2}})^{A_{2}} ((- A_{2} + y{(A_{2},A)} + 1)^{A_{2}})^{- A_{2}} and \\int 1 dA = \\int ((A - 2 A_{2} + 1)^{A_{2}})^{A_{2}} ((- A_{2} + y{(A_{2},A)} + 1)^{A_{2}})^{- A_{2}} dA", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True)), Integer(1)))"], [["minus", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Integer(2), Symbol('A_2', commutative=True)), Integer(1)))"], [["power", 3, "Symbol('A_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Integer(2), Symbol('A_2', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)))"], [["power", 4, "Symbol('A_2', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Integer(2), Symbol('A_2', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["divide", 5, "Pow(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Integer(2), Symbol('A_2', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True)))))"], [["integrate", 6, "Symbol('A', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A', commutative=True))), Integral(Mul(Pow(Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Integer(2), Symbol('A_2', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('y')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Integer(1)), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True)))), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})} = \\frac{F_{g} \\mathbf{H}}{J_{\\varepsilon}}, then obtain \\int (- F_{g} + \\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})}) d\\mathbf{H} = - F_{g} \\mathbf{H} + \\frac{F_{g} \\mathbf{H}^{2}}{2 J_{\\varepsilon}} + a", "derivation": "\\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})} = \\frac{F_{g} \\mathbf{H}}{J_{\\varepsilon}} and - F_{g} + \\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})} = - F_{g} + \\frac{F_{g} \\mathbf{H}}{J_{\\varepsilon}} and \\int (- F_{g} + \\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})}) d\\mathbf{H} = \\int (- F_{g} + \\frac{F_{g} \\mathbf{H}}{J_{\\varepsilon}}) d\\mathbf{H} and \\int (- F_{g} + \\operatorname{f_{E}}{(J_{\\varepsilon},F_{g},\\mathbf{H})}) d\\mathbf{H} = - F_{g} \\mathbf{H} + \\frac{F_{g} \\mathbf{H}^{2}}{2 J_{\\varepsilon}} + a", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('f_E')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('f_E')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('f_E')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Rational(1, 2), Symbol('F_g', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(f_{E},\\varphi)} = \\frac{\\varphi}{f_{E}}, then derive \\frac{\\partial}{\\partial f_{E}} \\operatorname{F_{g}}{(f_{E},\\varphi)} = - \\frac{\\varphi}{f_{E}^{2}}, then obtain \\frac{\\partial}{\\partial f_{E}} \\operatorname{F_{g}}{(f_{E},\\varphi)} = - \\frac{\\operatorname{F_{g}}{(f_{E},\\varphi)}}{f_{E}}", "derivation": "\\operatorname{F_{g}}{(f_{E},\\varphi)} = \\frac{\\varphi}{f_{E}} and \\frac{\\partial}{\\partial f_{E}} \\operatorname{F_{g}}{(f_{E},\\varphi)} = \\frac{\\partial}{\\partial f_{E}} \\frac{\\varphi}{f_{E}} and \\frac{\\partial}{\\partial f_{E}} \\operatorname{F_{g}}{(f_{E},\\varphi)} = - \\frac{\\varphi}{f_{E}^{2}} and \\frac{\\partial}{\\partial f_{E}} \\frac{\\varphi}{f_{E}} = - \\frac{\\varphi}{f_{E}^{2}} and \\frac{\\partial}{\\partial f_{E}} \\operatorname{F_{g}}{(f_{E},\\varphi)} = - \\frac{\\operatorname{F_{g}}{(f_{E},\\varphi)}}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('f_E', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('f_E', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('f_E', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('F_g')(Symbol('f_E', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('F_g')(Symbol('f_E', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\mathbf{A},A_{z})} = A_{z} \\mathbf{A} and a{(t)} = \\log{(t)}, then obtain (\\nabla^{A_{z}}{(\\mathbf{A},A_{z})} a{(t)} + a{(t)})^{\\mathbf{A}} = (\\nabla^{A_{z}}{(\\mathbf{A},A_{z})} \\log{(t)} + a{(t)})^{\\mathbf{A}}", "derivation": "\\nabla{(\\mathbf{A},A_{z})} = A_{z} \\mathbf{A} and \\nabla^{A_{z}}{(\\mathbf{A},A_{z})} = (A_{z} \\mathbf{A})^{A_{z}} and a{(t)} = \\log{(t)} and (A_{z} \\mathbf{A})^{A_{z}} a{(t)} = (A_{z} \\mathbf{A})^{A_{z}} \\log{(t)} and (A_{z} \\mathbf{A})^{A_{z}} a{(t)} + a{(t)} = (A_{z} \\mathbf{A})^{A_{z}} \\log{(t)} + a{(t)} and ((A_{z} \\mathbf{A})^{A_{z}} a{(t)} + a{(t)})^{\\mathbf{A}} = ((A_{z} \\mathbf{A})^{A_{z}} \\log{(t)} + a{(t)})^{\\mathbf{A}} and (\\nabla^{A_{z}}{(\\mathbf{A},A_{z})} a{(t)} + a{(t)})^{\\mathbf{A}} = (\\nabla^{A_{z}}{(\\mathbf{A},A_{z})} \\log{(t)} + a{(t)})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)))"], ["get_premise", "Equality(Function('a')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["times", 3, "Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), Function('a')(Symbol('t', commutative=True))), Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), log(Symbol('t', commutative=True))))"], [["add", 4, "Function('a')(Symbol('t', commutative=True))"], "Equality(Add(Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), Function('a')(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))), Add(Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), log(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), Function('a')(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Mul(Pow(Mul(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('A_z', commutative=True)), log(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Add(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Function('a')(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), log(Symbol('t', commutative=True))), Function('a')(Symbol('t', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(G,\\theta_1)} = \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(G,\\theta_1)} d\\theta_1 = T - \\frac{\\theta_1^{2}}{2 G^{2}}, then obtain \\int (T - \\frac{\\theta_1^{2}}{2 G^{2}}) dG = \\iint \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G} d\\theta_1 dG", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(G,\\theta_1)} = \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G} and \\int \\operatorname{f_{\\mathbf{v}}}{(G,\\theta_1)} d\\theta_1 = \\int \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G} d\\theta_1 and \\int \\operatorname{f_{\\mathbf{v}}}{(G,\\theta_1)} d\\theta_1 = T - \\frac{\\theta_1^{2}}{2 G^{2}} and T - \\frac{\\theta_1^{2}}{2 G^{2}} = \\int \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G} d\\theta_1 and \\int (T - \\frac{\\theta_1^{2}}{2 G^{2}}) dG = \\iint \\frac{\\partial}{\\partial G} \\frac{\\theta_1}{G} d\\theta_1 dG", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(-2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(-2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)))), Integral(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(-2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)))), Tuple(Symbol('G', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{B})} = e^{\\sin{(\\mathbf{B})}} and \\varepsilon_{0}{(\\mathbf{B})} = \\mathbf{B}, then obtain \\frac{\\operatorname{M_{E}}{(\\mathbf{B})}}{\\mathbf{B} + e^{\\sin{(\\mathbf{B})}}} = \\frac{e^{\\sin{(\\mathbf{B})}}}{\\mathbf{B} + e^{\\sin{(\\mathbf{B})}}}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{B})} = e^{\\sin{(\\mathbf{B})}} and \\varepsilon_{0}{(\\mathbf{B})} = \\mathbf{B} and \\varepsilon_{0}{(\\mathbf{B})} + e^{\\sin{(\\mathbf{B})}} = \\mathbf{B} + e^{\\sin{(\\mathbf{B})}} and \\frac{\\operatorname{M_{E}}{(\\mathbf{B})}}{\\varepsilon_{0}{(\\mathbf{B})} + e^{\\sin{(\\mathbf{B})}}} = \\frac{e^{\\sin{(\\mathbf{B})}}}{\\varepsilon_{0}{(\\mathbf{B})} + e^{\\sin{(\\mathbf{B})}}} and \\frac{\\operatorname{M_{E}}{(\\mathbf{B})}}{\\mathbf{B} + e^{\\sin{(\\mathbf{B})}}} = \\frac{e^{\\sin{(\\mathbf{B})}}}{\\mathbf{B} + e^{\\sin{(\\mathbf{B})}}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], [["add", 2, "exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{B}', commutative=True)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Symbol('\\\\mathbf{B}', commutative=True), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["divide", 1, "Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{B}', commutative=True)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{B}', commutative=True)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)), Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{B}', commutative=True)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)), Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)), exp(sin(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given T{(\\eta,\\varepsilon)} = \\frac{\\varepsilon}{\\eta}, then derive \\eta \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = - \\frac{\\varepsilon}{\\eta}, then obtain \\frac{\\partial}{\\partial \\varepsilon} - \\frac{\\varepsilon}{\\eta} = \\frac{\\partial}{\\partial \\varepsilon} - T{(\\eta,\\varepsilon)}", "derivation": "T{(\\eta,\\varepsilon)} = \\frac{\\varepsilon}{\\eta} and \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = \\frac{\\partial}{\\partial \\eta} \\frac{\\varepsilon}{\\eta} and \\eta \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = \\eta \\frac{\\partial}{\\partial \\eta} \\frac{\\varepsilon}{\\eta} and \\eta \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = - \\frac{\\varepsilon}{\\eta} and \\eta \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = - T{(\\eta,\\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} \\eta \\frac{\\partial}{\\partial \\eta} T{(\\eta,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} - T{(\\eta,\\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} - \\frac{\\varepsilon}{\\eta} = \\frac{\\partial}{\\partial \\varepsilon} - T{(\\eta,\\varepsilon)}", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Symbol('\\\\eta', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Derivative(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Symbol('\\\\eta', commutative=True), Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\eta', commutative=True), Derivative(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('\\\\eta', commutative=True), Derivative(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\eta', commutative=True), Derivative(Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(F_{H})} = e^{F_{H}} and \\mathbf{M}{(F_{H})} = \\operatorname{f_{\\mathbf{p}}}^{F_{H}}{(F_{H})} (e^{F_{H}})^{F_{H}}, then obtain \\int \\mathbf{M}{(F_{H})} dF_{H} = \\int \\operatorname{f_{\\mathbf{p}}}^{2 F_{H}}{(F_{H})} dF_{H}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(F_{H})} = e^{F_{H}} and \\operatorname{f_{\\mathbf{p}}}^{F_{H}}{(F_{H})} = (e^{F_{H}})^{F_{H}} and \\mathbf{M}{(F_{H})} = \\operatorname{f_{\\mathbf{p}}}^{F_{H}}{(F_{H})} (e^{F_{H}})^{F_{H}} and \\mathbf{M}{(F_{H})} = \\operatorname{f_{\\mathbf{p}}}^{2 F_{H}}{(F_{H})} and \\int \\mathbf{M}{(F_{H})} dF_{H} = \\int \\operatorname{f_{\\mathbf{p}}}^{2 F_{H}}{(F_{H})} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(exp(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('F_H', commutative=True)), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(exp(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{M}')(Symbol('F_H', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('F_H', commutative=True)), Mul(Integer(2), Symbol('F_H', commutative=True))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('F_H', commutative=True)), Mul(Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v_{y},t)} = t + v_{y}, then derive \\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} - 1 = 0, then obtain t + (\\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} - 1)^{t} = 0^{t} + t", "derivation": "\\operatorname{v_{t}}{(v_{y},t)} = t + v_{y} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} = \\frac{\\partial}{\\partial v_{y}} (t + v_{y}) and - \\frac{\\partial}{\\partial v_{y}} (t + v_{y}) + \\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} = 0 and \\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} - 1 = 0 and \\frac{\\partial}{\\partial v_{y}} (t + v_{y}) - 1 = 0 and (\\frac{\\partial}{\\partial v_{y}} (t + v_{y}) - 1)^{t} = 0^{t} and (\\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} - 1)^{t} = 0^{t} and t + (\\frac{\\partial}{\\partial v_{y}} \\operatorname{v_{t}}{(v_{y},t)} - 1)^{t} = 0^{t} + t", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Derivative(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["power", 5, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Derivative(Add(Symbol('t', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Pow(Integer(0), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Derivative(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Pow(Integer(0), Symbol('t', commutative=True)))"], [["add", 7, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Pow(Add(Derivative(Function('v_t')(Symbol('v_y', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True))), Add(Pow(Integer(0), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(v_{1},B,f^{*})} = B f^{*} v_{1}, then obtain \\frac{\\partial}{\\partial B} (\\int \\mathbf{v}{(v_{1},B,f^{*})} dB)^{v_{1}} = \\frac{\\partial}{\\partial B} (\\int B f^{*} v_{1} dB)^{v_{1}}", "derivation": "\\mathbf{v}{(v_{1},B,f^{*})} = B f^{*} v_{1} and \\int \\mathbf{v}{(v_{1},B,f^{*})} dB = \\int B f^{*} v_{1} dB and (\\int \\mathbf{v}{(v_{1},B,f^{*})} dB)^{v_{1}} = (\\int B f^{*} v_{1} dB)^{v_{1}} and \\frac{\\partial}{\\partial B} (\\int \\mathbf{v}{(v_{1},B,f^{*})} dB)^{v_{1}} = \\frac{\\partial}{\\partial B} (\\int B f^{*} v_{1} dB)^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True), Symbol('B', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('f^*', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True), Symbol('B', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Mul(Symbol('B', commutative=True), Symbol('f^*', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True), Symbol('B', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('v_1', commutative=True)), Pow(Integral(Mul(Symbol('B', commutative=True), Symbol('f^*', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('v_1', commutative=True)))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True), Symbol('B', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Symbol('B', commutative=True), Symbol('f^*', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(z^{*},a)} = \\frac{a}{z^{*}} and \\operatorname{E_{x}}{(z^{*},a)} = \\frac{a}{z^{*}}, then obtain z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\Omega{(z^{*},a)} = z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\operatorname{E_{x}}{(z^{*},a)}", "derivation": "\\Omega{(z^{*},a)} = \\frac{a}{z^{*}} and \\frac{\\partial}{\\partial a} \\Omega{(z^{*},a)} = \\frac{\\partial}{\\partial a} \\frac{a}{z^{*}} and \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\Omega{(z^{*},a)} = \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\frac{a}{z^{*}} and \\operatorname{E_{x}}{(z^{*},a)} = \\frac{a}{z^{*}} and z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\Omega{(z^{*},a)} = z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\frac{a}{z^{*}} and z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\Omega{(z^{*},a)} = z^{*} \\frac{\\partial^{2}}{\\partial z^{*}\\partial a} \\operatorname{E_{x}}{(z^{*},a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["times", 3, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Derivative(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Symbol('z^*', commutative=True), Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('z^*', commutative=True), Derivative(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Symbol('z^*', commutative=True), Derivative(Function('E_x')(Symbol('z^*', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)} = \\mu^{\\varepsilon_0}, then obtain e^{\\mu (- \\mu^{\\varepsilon_0} + \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)}) - 1} = e^{-1}", "derivation": "\\operatorname{M_{E}}{(\\varepsilon_0,\\mu)} = \\mu^{\\varepsilon_0} and 0 = \\mu^{\\varepsilon_0} - \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)} and - \\mu^{\\varepsilon_0} + \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)} = 0 and \\mu (- \\mu^{\\varepsilon_0} + \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)}) = 0 and \\mu (- \\mu^{\\varepsilon_0} + \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)}) - 1 = -1 and e^{\\mu (- \\mu^{\\varepsilon_0} + \\operatorname{M_{E}}{(\\varepsilon_0,\\mu)}) - 1} = e^{-1}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["minus", 2, "Add(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["times", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(0))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(-1)), Integer(-1))"], [["exp", 5], "Equality(exp(Add(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Function('M_E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(-1))), exp(Integer(-1)))"]]}, {"prompt": "Given \\tilde{g}^*{(r_{0},\\mathbf{M})} = \\mathbf{M} r_{0}, then obtain r_{0} - 2 \\tilde{g}^*{(r_{0},\\mathbf{M})} + \\frac{\\tilde{g}^*{(r_{0},\\mathbf{M})}}{\\mathbf{M}} = 2 r_{0} - 2 \\tilde{g}^*{(r_{0},\\mathbf{M})}", "derivation": "\\tilde{g}^*{(r_{0},\\mathbf{M})} = \\mathbf{M} r_{0} and \\frac{\\tilde{g}^*{(r_{0},\\mathbf{M})}}{\\mathbf{M}} = r_{0} and - \\tilde{g}^*{(r_{0},\\mathbf{M})} + \\frac{\\tilde{g}^*{(r_{0},\\mathbf{M})}}{\\mathbf{M}} = r_{0} - \\tilde{g}^*{(r_{0},\\mathbf{M})} and r_{0} - 2 \\tilde{g}^*{(r_{0},\\mathbf{M})} + \\frac{\\tilde{g}^*{(r_{0},\\mathbf{M})}}{\\mathbf{M}} = 2 r_{0} - 2 \\tilde{g}^*{(r_{0},\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('r_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('r_0', commutative=True))"], [["minus", 2, "Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 3, "Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\omega)} = \\log{(\\omega)}, then obtain (\\frac{d}{d \\omega} \\operatorname{F_{N}}{(\\omega)})^{\\omega} + \\frac{1}{\\omega} = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} + \\frac{1}{\\omega}", "derivation": "\\operatorname{F_{N}}{(\\omega)} = \\log{(\\omega)} and \\frac{d}{d \\omega} \\operatorname{F_{N}}{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\omega)} and (\\frac{d}{d \\omega} \\operatorname{F_{N}}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} and (\\frac{d}{d \\omega} \\operatorname{F_{N}}{(\\omega)})^{\\omega} + \\frac{1}{\\omega} = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} + \\frac{1}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Function('F_N')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["add", 3, "Pow(Symbol('\\\\omega', commutative=True), Integer(-1))"], "Equality(Add(Pow(Derivative(Function('F_N')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Add(Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"]]}, {"prompt": "Given M{(\\hat{x}_0,M_{E},\\omega)} = M_{E}^{\\hat{x}_0} \\omega and q{(g,\\dot{y})} = e^{\\dot{y} + g}, then obtain (\\omega + g + M{(\\hat{x}_0,M_{E},\\omega)}) e^{- \\dot{y} - g} = (M_{E}^{\\hat{x}_0} \\omega + \\omega + g) e^{- \\dot{y} - g}", "derivation": "M{(\\hat{x}_0,M_{E},\\omega)} = M_{E}^{\\hat{x}_0} \\omega and \\omega + M{(\\hat{x}_0,M_{E},\\omega)} = M_{E}^{\\hat{x}_0} \\omega + \\omega and q{(g,\\dot{y})} = e^{\\dot{y} + g} and \\omega + g + M{(\\hat{x}_0,M_{E},\\omega)} = M_{E}^{\\hat{x}_0} \\omega + \\omega + g and \\frac{\\omega + g + M{(\\hat{x}_0,M_{E},\\omega)}}{q{(g,\\dot{y})}} = \\frac{M_{E}^{\\hat{x}_0} \\omega + \\omega + g}{q{(g,\\dot{y})}} and (\\omega + g + M{(\\hat{x}_0,M_{E},\\omega)}) e^{- \\dot{y} - g} = (M_{E}^{\\hat{x}_0} \\omega + \\omega + g) e^{- \\dot{y} - g}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], ["get_premise", "Equality(Function('q')(Symbol('g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('g', commutative=True))))"], [["add", 2, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True)))"], [["divide", 4, "Function('q')(Symbol('g', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('q')(Symbol('g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Mul(Add(Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True)), Pow(Function('q')(Symbol('g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))), Mul(Add(Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\phi{(J_{\\varepsilon},W)} = W + \\sin{(J_{\\varepsilon})} and \\varphi^{*}{(J_{\\varepsilon},W)} = W + \\sin{(J_{\\varepsilon})}, then obtain (W + \\varphi^{*}{(J_{\\varepsilon},W)} + \\sin{(J_{\\varepsilon})})^{J_{\\varepsilon}} = (2 W + 2 \\sin{(J_{\\varepsilon})})^{J_{\\varepsilon}}", "derivation": "\\phi{(J_{\\varepsilon},W)} = W + \\sin{(J_{\\varepsilon})} and W + \\phi{(J_{\\varepsilon},W)} + \\sin{(J_{\\varepsilon})} = 2 W + 2 \\sin{(J_{\\varepsilon})} and \\varphi^{*}{(J_{\\varepsilon},W)} = W + \\sin{(J_{\\varepsilon})} and \\varphi^{*}{(J_{\\varepsilon},W)} = \\phi{(J_{\\varepsilon},W)} and W + \\varphi^{*}{(J_{\\varepsilon},W)} + \\sin{(J_{\\varepsilon})} = 2 W + 2 \\sin{(J_{\\varepsilon})} and (W + \\varphi^{*}{(J_{\\varepsilon},W)} + \\sin{(J_{\\varepsilon})})^{J_{\\varepsilon}} = (2 W + 2 \\sin{(J_{\\varepsilon})})^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Add(Symbol('W', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Symbol('W', commutative=True), Function('\\\\phi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\varphi^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Function('\\\\phi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('W', commutative=True), Function('\\\\varphi^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["power", 5, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Symbol('W', commutative=True), Function('\\\\varphi^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{f},L)} = L \\mathbf{f} and \\operatorname{v_{x}}{(\\mathbf{f},L)} = L \\mathbf{f}, then obtain \\frac{\\operatorname{v_{x}}{(\\mathbf{f},L)}}{\\tilde{g}^*{(\\mathbf{f},L)}} = 1", "derivation": "\\tilde{g}^*{(\\mathbf{f},L)} = L \\mathbf{f} and \\operatorname{v_{x}}{(\\mathbf{f},L)} = L \\mathbf{f} and \\frac{\\operatorname{v_{x}}{(\\mathbf{f},L)}}{\\tilde{g}^*{(\\mathbf{f},L)}} = \\frac{L \\mathbf{f}}{\\tilde{g}^*{(\\mathbf{f},L)}} and \\frac{\\operatorname{v_{x}}{(\\mathbf{f},L)}}{L \\mathbf{f}} = 1 and \\frac{\\operatorname{v_{x}}{(\\mathbf{f},L)}}{\\tilde{g}^*{(\\mathbf{f},L)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 2, "Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True))), Integer(1))"]]}, {"prompt": "Given I{(t_{2})} = e^{t_{2}} and \\mathbf{f}{(t_{2})} = I{(t_{2})} + e^{t_{2}}, then derive \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})} = 2 e^{t_{2}}, then obtain \\mathbf{f}{(t_{2})} = \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})}", "derivation": "I{(t_{2})} = e^{t_{2}} and I{(t_{2})} + e^{t_{2}} = 2 e^{t_{2}} and \\mathbf{f}{(t_{2})} = I{(t_{2})} + e^{t_{2}} and \\mathbf{f}{(t_{2})} = 2 e^{t_{2}} and \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})} = \\frac{d}{d t_{2}} 2 e^{t_{2}} and \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})} = 2 e^{t_{2}} and \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})} = I{(t_{2})} + e^{t_{2}} and \\mathbf{f}{(t_{2})} = \\frac{d}{d t_{2}} \\mathbf{f}{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["add", 1, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(Function('I')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Mul(Integer(2), exp(Symbol('t_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Add(Function('I')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Mul(Integer(2), exp(Symbol('t_2', commutative=True))))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Integer(2), exp(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Add(Function('I')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(c_{0})} = \\log{(c_{0})}, then obtain (\\int\\limits^{e^{J{(c_{0})}}} e^{J{(c_{0})}} dc_{0})^{2} = (\\int\\limits^{e^{J{(c_{0})}}} c_{0} dc_{0}) \\int\\limits^{e^{J{(c_{0})}}} e^{J{(c_{0})}} dc_{0}", "derivation": "J{(c_{0})} = \\log{(c_{0})} and e^{J{(c_{0})}} = c_{0} and \\int e^{J{(c_{0})}} dc_{0} = \\int c_{0} dc_{0} and \\int\\limits^{e^{J{(c_{0})}}} e^{J{(c_{0})}} dc_{0} = \\int\\limits^{e^{J{(c_{0})}}} c_{0} dc_{0} and (\\int\\limits^{e^{J{(c_{0})}}} e^{J{(c_{0})}} dc_{0})^{2} = (\\int\\limits^{e^{J{(c_{0})}}} c_{0} dc_{0}) \\int\\limits^{e^{J{(c_{0})}}} e^{J{(c_{0})}} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["exp", 1], "Equality(exp(Function('J')(Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True))"], [["integrate", 2, "Symbol('c_0', commutative=True)"], "Equality(Integral(exp(Function('J')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(exp(Function('J')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True))))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True))))))"], [["times", 4, "Integral(exp(Function('J')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True)))))"], "Equality(Pow(Integral(exp(Function('J')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True))))), Integer(2)), Mul(Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True))))), Integral(exp(Function('J')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), exp(Function('J')(Symbol('c_0', commutative=True)))))))"]]}, {"prompt": "Given x{(Z,J)} = \\frac{\\partial}{\\partial J} (J + Z), then derive \\int x{(Z,J)} dZ = Z + \\rho_f, then obtain \\iint x{(Z,J)} dZ dJ = \\int (Z + \\rho_f) dJ", "derivation": "x{(Z,J)} = \\frac{\\partial}{\\partial J} (J + Z) and \\int x{(Z,J)} dZ = \\int \\frac{\\partial}{\\partial J} (J + Z) dZ and \\int x{(Z,J)} dZ = Z + \\rho_f and \\iint x{(Z,J)} dZ dJ = \\int (Z + \\rho_f) dJ", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('Z', commutative=True), Symbol('J', commutative=True)), Derivative(Add(Symbol('J', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Add(Symbol('J', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Function('x')(Symbol('Z', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given Q{(\\dot{z})} = \\log{(\\dot{z})}, then obtain (\\frac{d^{2}}{d \\dot{z}^{2}} Q{(\\dot{z})})^{\\dot{z}} = (\\frac{d^{2}}{d \\dot{z}^{2}} \\log{(\\dot{z})})^{\\dot{z}}", "derivation": "Q{(\\dot{z})} = \\log{(\\dot{z})} and \\frac{d}{d \\dot{z}} Q{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\log{(\\dot{z})} and \\frac{d^{2}}{d \\dot{z}^{2}} Q{(\\dot{z})} = \\frac{d^{2}}{d \\dot{z}^{2}} \\log{(\\dot{z})} and (\\frac{d^{2}}{d \\dot{z}^{2}} Q{(\\dot{z})})^{\\dot{z}} = (\\frac{d^{2}}{d \\dot{z}^{2}} \\log{(\\dot{z})})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Derivative(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(2))))"], [["power", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Derivative(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\dot{z}', commutative=True)), Pow(Derivative(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})}, then obtain (\\int \\frac{\\operatorname{t_{2}}{(\\sigma_x)}}{\\cos{(\\sigma_x)}} d\\sigma_x)^{\\sigma_x} = (\\int \\frac{\\sin{(\\cos{(\\sigma_x)})}}{\\cos{(\\sigma_x)}} d\\sigma_x)^{\\sigma_x}", "derivation": "\\operatorname{t_{2}}{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})} and \\frac{\\operatorname{t_{2}}{(\\sigma_x)}}{\\cos{(\\sigma_x)}} = \\frac{\\sin{(\\cos{(\\sigma_x)})}}{\\cos{(\\sigma_x)}} and \\int \\frac{\\operatorname{t_{2}}{(\\sigma_x)}}{\\cos{(\\sigma_x)}} d\\sigma_x = \\int \\frac{\\sin{(\\cos{(\\sigma_x)})}}{\\cos{(\\sigma_x)}} d\\sigma_x and (\\int \\frac{\\operatorname{t_{2}}{(\\sigma_x)}}{\\cos{(\\sigma_x)}} d\\sigma_x)^{\\sigma_x} = (\\int \\frac{\\sin{(\\cos{(\\sigma_x)})}}{\\cos{(\\sigma_x)}} d\\sigma_x)^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\sigma_x', commutative=True)), sin(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 1, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('t_2')(Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Mul(sin(cos(Symbol('\\\\sigma_x', commutative=True))), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Function('t_2')(Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(sin(cos(Symbol('\\\\sigma_x', commutative=True))), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Integral(Mul(Function('t_2')(Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(sin(cos(Symbol('\\\\sigma_x', commutative=True))), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})} = J_{\\varepsilon} + \\mathbf{F}, then obtain - \\frac{- J_{\\varepsilon} - \\mathbf{F} + 2 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}}{3 J_{\\varepsilon} + 3 \\mathbf{F} - 4 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}} = 1", "derivation": "\\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})} = J_{\\varepsilon} + \\mathbf{F} and 0 = J_{\\varepsilon} + \\mathbf{F} - \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})} and - \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})} = J_{\\varepsilon} + \\mathbf{F} - 2 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})} and - \\frac{\\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}}{J_{\\varepsilon} + \\mathbf{F} - 2 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}} = 1 and - \\frac{- J_{\\varepsilon} - \\mathbf{F} + 2 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}}{3 J_{\\varepsilon} + 3 \\mathbf{F} - 4 \\Psi^{\\dagger}{(\\mathbf{F},J_{\\varepsilon})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["divide", 3, "Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Pow(Add(Mul(Integer(3), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(4), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f_{E},y)} = \\frac{y}{f_{E}} and \\mathbf{r}{(f_{E},y)} = \\frac{y}{f_{E}}, then obtain (\\int y dy)^{y} = (\\int \\frac{y^{2}}{f_{E} \\mathbf{r}{(f_{E},y)}} dy)^{y}", "derivation": "\\operatorname{n_{1}}{(f_{E},y)} = \\frac{y}{f_{E}} and y = \\frac{y^{2}}{f_{E} \\operatorname{n_{1}}{(f_{E},y)}} and \\mathbf{r}{(f_{E},y)} = \\frac{y}{f_{E}} and \\mathbf{r}{(f_{E},y)} = \\operatorname{n_{1}}{(f_{E},y)} and y = \\frac{y^{2}}{f_{E} \\mathbf{r}{(f_{E},y)}} and \\int y dy = \\int \\frac{y^{2}}{f_{E} \\mathbf{r}{(f_{E},y)}} dy and (\\int y dy)^{y} = (\\int \\frac{y^{2}}{f_{E} \\mathbf{r}{(f_{E},y)}} dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('n_1')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)))"], "Equality(Symbol('y', commutative=True), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Function('n_1')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{r}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Function('n_1')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Symbol('y', commutative=True), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{r}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True))), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{r}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1))), Tuple(Symbol('y', commutative=True))))"], [["power", 6, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{r}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1))), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given V{(\\mathbf{B})} = \\cos{(\\mathbf{B})}, then derive \\int V{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{S} + \\sin{(\\mathbf{B})}, then obtain \\mathbf{S} + \\sin{(\\mathbf{B})} = \\int \\cos{(\\mathbf{B})} d\\mathbf{B}", "derivation": "V{(\\mathbf{B})} = \\cos{(\\mathbf{B})} and \\int V{(\\mathbf{B})} d\\mathbf{B} = \\int \\cos{(\\mathbf{B})} d\\mathbf{B} and \\int V{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{S} + \\sin{(\\mathbf{B})} and \\mathbf{S} + \\sin{(\\mathbf{B})} = \\int \\cos{(\\mathbf{B})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('V')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{B}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = - \\Omega + \\hat{H}_l, then derive \\frac{\\partial}{\\partial \\Omega} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = -1, then derive \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = 0, then obtain \\int \\frac{\\partial^{2}}{\\partial \\Omega^{2}} (- \\Omega + \\hat{H}_l) d\\Omega = \\int 0 d\\Omega", "derivation": "\\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = - \\Omega + \\hat{H}_l and \\frac{\\partial}{\\partial \\Omega} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = \\frac{\\partial}{\\partial \\Omega} (- \\Omega + \\hat{H}_l) and \\frac{\\partial}{\\partial \\Omega} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = -1 and \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = \\frac{d}{d \\Omega} (-1) and \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\Psi^{\\dagger}{(\\Omega,\\hat{H}_l)} = 0 and \\frac{\\partial^{2}}{\\partial \\Omega^{2}} (- \\Omega + \\hat{H}_l) = 0 and \\int \\frac{\\partial^{2}}{\\partial \\Omega^{2}} (- \\Omega + \\hat{H}_l) d\\Omega = \\int 0 d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Derivative(Integer(-1), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Integer(0))"], [["integrate", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{M},\\hbar)} = - \\hbar + \\mathbf{M}, then derive \\int \\psi^{*}{(\\mathbf{M},\\hbar)} d\\hbar = \\ddot{x} - \\frac{\\hbar^{2}}{2} + \\hbar \\mathbf{M}, then obtain \\ddot{x} - \\frac{\\hbar^{2}}{2} + \\hbar \\mathbf{M} = \\int (- \\hbar + \\mathbf{M}) d\\hbar", "derivation": "\\psi^{*}{(\\mathbf{M},\\hbar)} = - \\hbar + \\mathbf{M} and \\int \\psi^{*}{(\\mathbf{M},\\hbar)} d\\hbar = \\int (- \\hbar + \\mathbf{M}) d\\hbar and \\int \\psi^{*}{(\\mathbf{M},\\hbar)} d\\hbar = \\ddot{x} - \\frac{\\hbar^{2}}{2} + \\hbar \\mathbf{M} and \\ddot{x} - \\frac{\\hbar^{2}}{2} + \\hbar \\mathbf{M} = \\int (- \\hbar + \\mathbf{M}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})} = \\mathbf{E} e^{\\dot{y}}, then obtain 0 = (\\mathbf{E} e^{\\dot{y}} - \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})}) e^{\\dot{y}}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})} = \\mathbf{E} e^{\\dot{y}} and - \\mathbf{E} e^{\\dot{y}} + \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})} = 0 and - \\mathbf{E} e^{\\dot{y}} + \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})} - 1 = -1 and 0 = \\mathbf{E} e^{\\dot{y}} - \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})} and 0 = (\\mathbf{E} e^{\\dot{y}} - \\hat{\\mathbf{x}}{(\\mathbf{E},\\dot{y})}) e^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Integer(-1))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["times", 4, "exp(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), exp(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(C)} = e^{C}, then obtain - (\\mu_{0}{(C)} - e^{C}) e^{C} (e^{C})^{C} = 0", "derivation": "\\mu_{0}{(C)} = e^{C} and \\mu_{0}{(C)} - e^{C} = 0 and (\\mu_{0}{(C)} - e^{C}) (e^{C})^{C} = 0 and - (\\mu_{0}{(C)} - e^{C}) e^{C} (e^{C})^{C} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["minus", 1, "exp(Symbol('C', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(0))"], [["times", 2, "Pow(exp(Symbol('C', commutative=True)), Symbol('C', commutative=True))"], "Equality(Mul(Add(Function('\\\\mu_0')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Pow(exp(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Integer(0))"], [["times", 3, "Mul(Integer(-1), exp(Symbol('C', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('\\\\mu_0')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), exp(Symbol('C', commutative=True)), Pow(exp(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{t})} = \\sin{(v_{t})}, then obtain (\\operatorname{t_{2}}^{v_{t}}{(v_{t})} - \\sin^{v_{t}}{(v_{t})}) (\\operatorname{t_{2}}^{v_{t}}{(v_{t})})^{v_{t}} = 0", "derivation": "\\operatorname{t_{2}}{(v_{t})} = \\sin{(v_{t})} and \\operatorname{t_{2}}^{v_{t}}{(v_{t})} = \\sin^{v_{t}}{(v_{t})} and \\operatorname{t_{2}}^{v_{t}}{(v_{t})} - \\sin^{v_{t}}{(v_{t})} = 0 and (\\operatorname{t_{2}}^{v_{t}}{(v_{t})})^{v_{t}} = (\\sin^{v_{t}}{(v_{t})})^{v_{t}} and (\\operatorname{t_{2}}^{v_{t}}{(v_{t})} - \\sin^{v_{t}}{(v_{t})}) (\\sin^{v_{t}}{(v_{t})})^{v_{t}} = 0 and (\\operatorname{t_{2}}^{v_{t}}{(v_{t})} - \\sin^{v_{t}}{(v_{t})}) (\\operatorname{t_{2}}^{v_{t}}{(v_{t})})^{v_{t}} = 0", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["minus", 2, "Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Add(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["times", 3, "Pow(Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Mul(Add(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))), Pow(Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))), Pow(Pow(Function('t_2')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))), Integer(0))"]]}, {"prompt": "Given I{(E,\\hat{H})} = E \\hat{H}, then obtain \\hat{H} \\sin{(\\int \\sin{(E \\hat{H} - I{(E,\\hat{H})})} dE)} = 0", "derivation": "I{(E,\\hat{H})} = E \\hat{H} and - E \\hat{H} + I{(E,\\hat{H})} = 0 and E \\hat{H} - I{(E,\\hat{H})} = 0 and \\sin{(E \\hat{H} - I{(E,\\hat{H})})} = 0 and \\int \\sin{(E \\hat{H} - I{(E,\\hat{H})})} dE = \\int 0 dE and \\sin{(\\int \\sin{(E \\hat{H} - I{(E,\\hat{H})})} dE)} = 0 and \\hat{H} \\sin{(\\int \\sin{(E \\hat{H} - I{(E,\\hat{H})})} dE)} = 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, "Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Integer(0))"], [["sin", 3], "Equality(sin(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(sin(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Tuple(Symbol('E', commutative=True))), Integral(Integer(0), Tuple(Symbol('E', commutative=True))))"], [["sin", 5], "Equality(sin(Integral(sin(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Tuple(Symbol('E', commutative=True)))), Integer(0))"], [["times", 6, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), sin(Integral(sin(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Tuple(Symbol('E', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{S}{(\\theta)} = \\log{(\\theta)}, then derive \\int \\mathbf{S}{(\\theta)} d\\theta = \\hbar + \\theta \\log{(\\theta)} - \\theta, then obtain (\\int \\mathbf{S}{(\\theta)} d\\theta)^{\\theta} = (\\hbar + \\theta \\log{(\\theta)} - \\theta)^{\\theta}", "derivation": "\\mathbf{S}{(\\theta)} = \\log{(\\theta)} and \\int \\mathbf{S}{(\\theta)} d\\theta = \\int \\log{(\\theta)} d\\theta and \\int \\mathbf{S}{(\\theta)} d\\theta = \\hbar + \\theta \\log{(\\theta)} - \\theta and \\hbar + \\theta \\log{(\\theta)} - \\theta = \\int \\log{(\\theta)} d\\theta and (\\int \\mathbf{S}{(\\theta)} d\\theta)^{\\theta} = (\\int \\log{(\\theta)} d\\theta)^{\\theta} and (\\int \\mathbf{S}{(\\theta)} d\\theta)^{\\theta} = (\\hbar + \\theta \\log{(\\theta)} - \\theta)^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["power", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given Z{(t,f)} = \\cos{(f + t)}, then obtain \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + Z{(t,f)}) dt)} + \\int (- f - t + \\cos{(f + t)}) dt = \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + \\cos{(f + t)}) dt)} + \\int (- f - t + \\cos{(f + t)}) dt", "derivation": "Z{(t,f)} = \\cos{(f + t)} and - f - t + Z{(t,f)} = - f - t + \\cos{(f + t)} and \\int (- f - t + Z{(t,f)}) dt = \\int (- f - t + \\cos{(f + t)}) dt and \\frac{\\partial}{\\partial t} \\int (- f - t + Z{(t,f)}) dt = \\frac{\\partial}{\\partial t} \\int (- f - t + \\cos{(f + t)}) dt and \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + Z{(t,f)}) dt)} = \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + \\cos{(f + t)}) dt)} and \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + Z{(t,f)}) dt)} + \\int (- f - t + \\cos{(f + t)}) dt = \\log{(\\frac{\\partial}{\\partial t} \\int (- f - t + \\cos{(f + t)}) dt)} + \\int (- f - t + \\cos{(f + t)}) dt", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True))))"], [["minus", 1, "Add(Symbol('f', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["log", 4], "Equality(log(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), log(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["add", 5, "Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(log(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('Z')(Symbol('t', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))), Add(log(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Add(Symbol('f', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{J}_M,C_{d})} = - C_{d} + \\mathbf{J}_M, then obtain \\frac{- 2 \\mathbf{J}_M + 2 \\varphi{(\\mathbf{J}_M,C_{d})}}{\\varphi{(\\mathbf{J}_M,C_{d})}} = \\frac{- C_{d} - \\mathbf{J}_M + \\varphi{(\\mathbf{J}_M,C_{d})}}{\\varphi{(\\mathbf{J}_M,C_{d})}}", "derivation": "\\varphi{(\\mathbf{J}_M,C_{d})} = - C_{d} + \\mathbf{J}_M and - \\mathbf{J}_M + \\varphi{(\\mathbf{J}_M,C_{d})} = - C_{d} and - 2 \\mathbf{J}_M + 2 \\varphi{(\\mathbf{J}_M,C_{d})} = - C_{d} - \\mathbf{J}_M + \\varphi{(\\mathbf{J}_M,C_{d})} and \\frac{- 2 \\mathbf{J}_M + 2 \\varphi{(\\mathbf{J}_M,C_{d})}}{- C_{d} + \\mathbf{J}_M} = \\frac{- C_{d} - \\mathbf{J}_M + \\varphi{(\\mathbf{J}_M,C_{d})}}{- C_{d} + \\mathbf{J}_M} and \\frac{- 2 \\mathbf{J}_M + 2 \\varphi{(\\mathbf{J}_M,C_{d})}}{\\varphi{(\\mathbf{J}_M,C_{d})}} = \\frac{- C_{d} - \\mathbf{J}_M + \\varphi{(\\mathbf{J}_M,C_{d})}}{\\varphi{(\\mathbf{J}_M,C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)))), Pow(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True))), Pow(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('C_d', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\phi_2)} = \\cos{(\\cos{(\\phi_2)})}, then obtain \\frac{d^{2}}{d \\phi_2^{2}} (\\operatorname{c_{0}}{(\\phi_2)} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}}) = \\frac{d^{2}}{d \\phi_2^{2}} (\\cos{(\\cos{(\\phi_2)})} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}})", "derivation": "\\operatorname{c_{0}}{(\\phi_2)} = \\cos{(\\cos{(\\phi_2)})} and \\operatorname{c_{0}}{(\\phi_2)} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}} = \\cos{(\\cos{(\\phi_2)})} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}} and \\frac{d}{d \\phi_2} (\\operatorname{c_{0}}{(\\phi_2)} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}}) = \\frac{d}{d \\phi_2} (\\cos{(\\cos{(\\phi_2)})} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}}) and \\frac{d^{2}}{d \\phi_2^{2}} (\\operatorname{c_{0}}{(\\phi_2)} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}}) = \\frac{d^{2}}{d \\phi_2^{2}} (\\cos{(\\cos{(\\phi_2)})} + \\frac{1}{\\cos{(\\cos{(\\phi_2)})}})", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))"], "Equality(Add(Function('c_0')(Symbol('\\\\phi_2', commutative=True)), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Add(cos(cos(Symbol('\\\\phi_2', commutative=True))), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('\\\\phi_2', commutative=True)), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(cos(cos(Symbol('\\\\phi_2', commutative=True))), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('\\\\phi_2', commutative=True)), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2))), Derivative(Add(cos(cos(Symbol('\\\\phi_2', commutative=True))), Pow(cos(cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2))))"]]}, {"prompt": "Given i{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} = 2 \\sin{(g^{\\prime}_{\\varepsilon})}, then obtain 4 \\sin^{2}{(g^{\\prime}_{\\varepsilon})} = 2 \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} \\sin{(g^{\\prime}_{\\varepsilon})}", "derivation": "i{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and i{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} = 2 \\sin{(g^{\\prime}_{\\varepsilon})} and \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} = 2 \\sin{(g^{\\prime}_{\\varepsilon})} and i{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} = \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} and 2 (i{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})}) \\sin{(g^{\\prime}_{\\varepsilon})} = 2 \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} \\sin{(g^{\\prime}_{\\varepsilon})} and 4 \\sin^{2}{(g^{\\prime}_{\\varepsilon})} = 2 \\mathbf{M}{(g^{\\prime}_{\\varepsilon})} \\sin{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('i')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('i')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('\\\\mathbf{M}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["times", 4, "Mul(Integer(2), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('i')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(4), Pow(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(P_{g},\\mathbf{r})} = - P_{g} + \\mathbf{r}, then obtain I + \\hat{H}{(P_{g},\\mathbf{r})} = \\mathbf{r} + \\omega", "derivation": "\\hat{H}{(P_{g},\\mathbf{r})} = - P_{g} + \\mathbf{r} and \\frac{\\partial}{\\partial \\mathbf{r}} \\hat{H}{(P_{g},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} (- P_{g} + \\mathbf{r}) and \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\hat{H}{(P_{g},\\mathbf{r})} d\\mathbf{r} = \\int \\frac{\\partial}{\\partial \\mathbf{r}} (- P_{g} + \\mathbf{r}) d\\mathbf{r} and I + \\hat{H}{(P_{g},\\mathbf{r})} = \\mathbf{r} + \\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{H}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('I', commutative=True), Function('\\\\hat{H}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(k,g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} - k, then derive \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\hat{X}{(k,g^{\\prime}_{\\varepsilon})} = 1, then obtain \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} (g^{\\prime}_{\\varepsilon} - k) = 1", "derivation": "\\hat{X}{(k,g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} - k and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\hat{X}{(k,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} (g^{\\prime}_{\\varepsilon} - k) and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\hat{X}{(k,g^{\\prime}_{\\varepsilon})} = 1 and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} (g^{\\prime}_{\\varepsilon} - k) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given t{(W)} = \\frac{d}{d W} \\sin{(W)}, then derive \\frac{d}{d W} t{(W)} = - \\sin{(W)}, then obtain - \\dot{y}^{S}{(S)} \\frac{d^{2}}{d W^{2}} t{(W)} = - \\dot{y}^{S}{(S)} \\frac{d^{3}}{d W^{3}} - \\frac{d}{d W} t{(W)}", "derivation": "t{(W)} = \\frac{d}{d W} \\sin{(W)} and \\frac{d}{d W} t{(W)} = \\frac{d^{2}}{d W^{2}} \\sin{(W)} and \\frac{d}{d W} t{(W)} = - \\sin{(W)} and - \\sin{(W)} = \\frac{d^{2}}{d W^{2}} \\sin{(W)} and \\frac{d}{d W} - \\sin{(W)} = \\frac{d^{3}}{d W^{3}} \\sin{(W)} and - \\dot{y}^{S}{(S)} \\frac{d}{d W} - \\sin{(W)} = - \\dot{y}^{S}{(S)} \\frac{d^{3}}{d W^{3}} \\sin{(W)} and - \\dot{y}^{S}{(S)} \\frac{d^{2}}{d W^{2}} t{(W)} = - \\dot{y}^{S}{(S)} \\frac{d^{3}}{d W^{3}} - \\frac{d}{d W} t{(W)}", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(3))))"], [["times", 5, "Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(3)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Derivative(Function('t')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Derivative(Mul(Integer(-1), Derivative(Function('t')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Tuple(Symbol('W', commutative=True), Integer(3)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\phi_2)} = e^{\\sin{(\\phi_2)}} and \\mathbf{J}{(\\phi_2)} = \\phi_2, then obtain \\mathbf{J}{(\\phi_2)} - \\mathbf{S}^{\\phi_2}{(\\phi_2)} + 1 = \\phi_2 - \\mathbf{S}^{\\phi_2}{(\\phi_2)} + 1", "derivation": "\\mathbf{S}{(\\phi_2)} = e^{\\sin{(\\phi_2)}} and \\mathbf{J}{(\\phi_2)} = \\phi_2 and \\mathbf{J}{(\\phi_2)} - (e^{\\sin{(\\phi_2)}})^{\\phi_2} = \\phi_2 - (e^{\\sin{(\\phi_2)}})^{\\phi_2} and \\mathbf{J}{(\\phi_2)} - \\mathbf{S}^{\\phi_2}{(\\phi_2)} = \\phi_2 - \\mathbf{S}^{\\phi_2}{(\\phi_2)} and \\mathbf{J}{(\\phi_2)} - \\mathbf{S}^{\\phi_2}{(\\phi_2)} + 1 = \\phi_2 - \\mathbf{S}^{\\phi_2}{(\\phi_2)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\phi_2', commutative=True)), exp(sin(Symbol('\\\\phi_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["minus", 2, "Pow(exp(sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(exp(sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Integer(1)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(A)} = e^{\\sin{(A)}}, then obtain \\sin{(\\operatorname{v_{2}}^{2}{(A)} e^{2 \\sin{(A)}} + \\operatorname{v_{2}}{(A)})} = \\sin{(\\operatorname{v_{2}}{(A)} e^{3 \\sin{(A)}} + \\operatorname{v_{2}}{(A)})}", "derivation": "\\operatorname{v_{2}}{(A)} = e^{\\sin{(A)}} and \\operatorname{v_{2}}^{2}{(A)} = \\operatorname{v_{2}}{(A)} e^{\\sin{(A)}} and \\operatorname{v_{2}}^{4}{(A)} = \\operatorname{v_{2}}^{2}{(A)} e^{2 \\sin{(A)}} and \\operatorname{v_{2}}^{2}{(A)} e^{2 \\sin{(A)}} = \\operatorname{v_{2}}{(A)} e^{3 \\sin{(A)}} and \\operatorname{v_{2}}^{2}{(A)} e^{2 \\sin{(A)}} + \\operatorname{v_{2}}{(A)} = \\operatorname{v_{2}}{(A)} e^{3 \\sin{(A)}} + \\operatorname{v_{2}}{(A)} and \\sin{(\\operatorname{v_{2}}^{2}{(A)} e^{2 \\sin{(A)}} + \\operatorname{v_{2}}{(A)})} = \\sin{(\\operatorname{v_{2}}{(A)} e^{3 \\sin{(A)}} + \\operatorname{v_{2}}{(A)})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('A', commutative=True)), exp(sin(Symbol('A', commutative=True))))"], [["times", 1, "Function('v_2')(Symbol('A', commutative=True))"], "Equality(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(2)), Mul(Function('v_2')(Symbol('A', commutative=True)), exp(sin(Symbol('A', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(4)), Mul(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('A', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('A', commutative=True))))), Mul(Function('v_2')(Symbol('A', commutative=True)), exp(Mul(Integer(3), sin(Symbol('A', commutative=True))))))"], [["add", 4, "Function('v_2')(Symbol('A', commutative=True))"], "Equality(Add(Mul(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('A', commutative=True))))), Function('v_2')(Symbol('A', commutative=True))), Add(Mul(Function('v_2')(Symbol('A', commutative=True)), exp(Mul(Integer(3), sin(Symbol('A', commutative=True))))), Function('v_2')(Symbol('A', commutative=True))))"], [["sin", 5], "Equality(sin(Add(Mul(Pow(Function('v_2')(Symbol('A', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('A', commutative=True))))), Function('v_2')(Symbol('A', commutative=True)))), sin(Add(Mul(Function('v_2')(Symbol('A', commutative=True)), exp(Mul(Integer(3), sin(Symbol('A', commutative=True))))), Function('v_2')(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(C_{1},p,V_{\\mathbf{E}})} = C_{1} (V_{\\mathbf{E}} + p), then obtain (\\int C_{1}^{2} (V_{\\mathbf{E}} + p) dC_{1})^{p} = (\\int (C_{1}^{2} V_{\\mathbf{E}} + C_{1}^{2} p) dC_{1})^{p}", "derivation": "\\Psi_{\\lambda}{(C_{1},p,V_{\\mathbf{E}})} = C_{1} (V_{\\mathbf{E}} + p) and C_{1} \\Psi_{\\lambda}{(C_{1},p,V_{\\mathbf{E}})} = C_{1}^{2} (V_{\\mathbf{E}} + p) and C_{1} \\Psi_{\\lambda}{(C_{1},p,V_{\\mathbf{E}})} = C_{1}^{2} V_{\\mathbf{E}} + C_{1}^{2} p and C_{1}^{2} (V_{\\mathbf{E}} + p) = C_{1}^{2} V_{\\mathbf{E}} + C_{1}^{2} p and \\int C_{1}^{2} (V_{\\mathbf{E}} + p) dC_{1} = \\int (C_{1}^{2} V_{\\mathbf{E}} + C_{1}^{2} p) dC_{1} and (\\int C_{1}^{2} (V_{\\mathbf{E}} + p) dC_{1})^{p} = (\\int (C_{1}^{2} V_{\\mathbf{E}} + C_{1}^{2} p) dC_{1})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('p', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('C_1', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('p', commutative=True))))"], [["times", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('p', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('p', commutative=True))))"], [["expand", 2], "Equality(Mul(Symbol('C_1', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('p', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('p', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["power", 5, "Symbol('p', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(2)), Symbol('p', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given S{(f_{\\mathbf{p}},U)} = e^{U f_{\\mathbf{p}}}, then obtain (\\int (f_{\\mathbf{p}} + S{(f_{\\mathbf{p}},U)})^{f_{\\mathbf{p}}} dU)^{U} = (\\int (f_{\\mathbf{p}} + e^{U f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} dU)^{U}", "derivation": "S{(f_{\\mathbf{p}},U)} = e^{U f_{\\mathbf{p}}} and f_{\\mathbf{p}} + S{(f_{\\mathbf{p}},U)} = f_{\\mathbf{p}} + e^{U f_{\\mathbf{p}}} and (f_{\\mathbf{p}} + S{(f_{\\mathbf{p}},U)})^{f_{\\mathbf{p}}} = (f_{\\mathbf{p}} + e^{U f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} and \\int (f_{\\mathbf{p}} + S{(f_{\\mathbf{p}},U)})^{f_{\\mathbf{p}}} dU = \\int (f_{\\mathbf{p}} + e^{U f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} dU and (\\int (f_{\\mathbf{p}} + S{(f_{\\mathbf{p}},U)})^{f_{\\mathbf{p}}} dU)^{U} = (\\int (f_{\\mathbf{p}} + e^{U f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} dU)^{U}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('U', commutative=True)), exp(Mul(Symbol('U', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('U', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Mul(Symbol('U', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["power", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('U', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Mul(Symbol('U', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('U', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Mul(Symbol('U', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('U', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Mul(Symbol('U', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(A_{z})} = \\cos{(A_{z})}, then obtain \\int 2 (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}} dA_{z} = \\int (e^{A_{z}} + (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}}) dA_{z}", "derivation": "\\operatorname{A_{2}}{(A_{z})} = \\cos{(A_{z})} and \\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}} = 1 and e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}} = e and (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}} = e^{A_{z}} and 2 (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}} = e^{A_{z}} + (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}} and \\int 2 (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}} dA_{z} = \\int (e^{A_{z}} + (e^{\\frac{\\operatorname{A_{2}}{(A_{z})}}{\\cos{(A_{z})}}})^{A_{z}}) dA_{z}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["divide", 1, "cos(Symbol('A_z', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1))), Integer(1))"], [["exp", 2], "Equality(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), E)"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["add", 4, "Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True))"], "Equality(Mul(Integer(2), Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True))), Add(exp(Symbol('A_z', commutative=True)), Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True))))"], [["integrate", 5, "Symbol('A_z', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Integral(Add(exp(Symbol('A_z', commutative=True)), Pow(exp(Mul(Function('A_2')(Symbol('A_z', commutative=True)), Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)))), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{r})} = e^{\\mathbf{r}} and L{(\\mathbf{r})} = e^{\\mathbf{r}} and \\hat{\\mathbf{x}}{(\\mathbf{r})} = \\mathbf{r} + e^{\\mathbf{r}}, then obtain \\hat{\\mathbf{x}}{(\\mathbf{r})} = \\mathbf{r} + L{(\\mathbf{r})}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{r})} = e^{\\mathbf{r}} and \\mathbf{r} + \\operatorname{f^{\\prime}}{(\\mathbf{r})} = \\mathbf{r} + e^{\\mathbf{r}} and L{(\\mathbf{r})} = e^{\\mathbf{r}} and \\hat{\\mathbf{x}}{(\\mathbf{r})} = \\mathbf{r} + e^{\\mathbf{r}} and \\mathbf{r} + \\operatorname{f^{\\prime}}{(\\mathbf{r})} = \\mathbf{r} + L{(\\mathbf{r})} and \\hat{\\mathbf{x}}{(\\mathbf{r})} = \\mathbf{r} + \\operatorname{f^{\\prime}}{(\\mathbf{r})} and \\hat{\\mathbf{x}}{(\\mathbf{r})} = \\mathbf{r} + L{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Function('L')(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Function('L')(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(F_{H})} = e^{F_{H}}, then derive F_{H} + \\mathbf{s} = \\int \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} dF_{H}, then derive (F_{H} + \\mathbf{s}) T{(\\mathbf{E},t)} = (F_{H} + \\mu) T{(\\mathbf{E},t)}, then obtain (F_{H} + \\mathbf{s}) T{(\\mathbf{E},t)} - \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} = (F_{H} + \\mu) T{(\\mathbf{E},t)} - \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}}", "derivation": "\\operatorname{v_{t}}{(F_{H})} = e^{F_{H}} and 1 = \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} and \\int 1 dF_{H} = \\int \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} dF_{H} and F_{H} + \\mathbf{s} = \\int \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} dF_{H} and F_{H} + \\mathbf{s} = \\int 1 dF_{H} and (F_{H} + \\mathbf{s}) T{(\\mathbf{E},t)} = T{(\\mathbf{E},t)} \\int 1 dF_{H} and (F_{H} + \\mathbf{s}) T{(\\mathbf{E},t)} = (F_{H} + \\mu) T{(\\mathbf{E},t)} and (F_{H} + \\mathbf{s}) T{(\\mathbf{E},t)} - \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}} = (F_{H} + \\mu) T{(\\mathbf{E},t)} - \\frac{e^{F_{H}}}{\\operatorname{v_{t}}{(F_{H})}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["divide", 1, "Function('v_t')(Symbol('F_H', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True))))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Mul(Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_H', commutative=True))))"], [["times", 5, "Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Mul(Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_H', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mu', commutative=True)), Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))))"], [["minus", 7, "Mul(Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True)))), Add(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mu', commutative=True)), Function('T')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(V)} = \\frac{d}{d V} \\cos{(V)}, then derive \\operatorname{M_{E}}{(V)} = - \\sin{(V)}, then obtain - \\operatorname{M_{E}}{(V)} - \\sin{(V)} + \\frac{d}{d V} \\cos{(V)} = - \\sin{(V)}", "derivation": "\\operatorname{M_{E}}{(V)} = \\frac{d}{d V} \\cos{(V)} and \\operatorname{M_{E}}{(V)} = - \\sin{(V)} and \\frac{d}{d V} \\cos{(V)} = - \\operatorname{M_{E}}{(V)} - \\sin{(V)} + \\frac{d}{d V} \\cos{(V)} and \\operatorname{M_{E}}{(V)} = - \\operatorname{M_{E}}{(V)} - \\sin{(V)} + \\frac{d}{d V} \\cos{(V)} and - \\operatorname{M_{E}}{(V)} - \\sin{(V)} + \\frac{d}{d V} \\cos{(V)} = - \\sin{(V)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('V', commutative=True)), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M_E')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True))))"], [["minus", 2, "Add(Function('M_E')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], "Equality(Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Add(Mul(Integer(-1), Function('M_E')(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('M_E')(Symbol('V', commutative=True)), Add(Mul(Integer(-1), Function('M_E')(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\psi^*)} = \\log{(\\log{(\\psi^*)})}, then obtain \\frac{\\int - \\rho_{b}{(\\psi^*)} d\\psi^*}{- \\rho_{b}{(\\psi^*)} + \\log{(\\log{(\\psi^*)})}} = \\frac{\\int - \\log{(\\log{(\\psi^*)})} d\\psi^*}{- \\rho_{b}{(\\psi^*)} + \\log{(\\log{(\\psi^*)})}}", "derivation": "\\rho_{b}{(\\psi^*)} = \\log{(\\log{(\\psi^*)})} and - \\rho_{b}{(\\psi^*)} = - \\log{(\\log{(\\psi^*)})} and \\int - \\rho_{b}{(\\psi^*)} d\\psi^* = \\int - \\log{(\\log{(\\psi^*)})} d\\psi^* and \\frac{\\int - \\rho_{b}{(\\psi^*)} d\\psi^*}{- \\rho_{b}{(\\psi^*)} + \\log{(\\log{(\\psi^*)})}} = \\frac{\\int - \\log{(\\log{(\\psi^*)})} d\\psi^*}{- \\rho_{b}{(\\psi^*)} + \\log{(\\log{(\\psi^*)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True)), log(log(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), log(log(Symbol('\\\\psi^*', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(-1), log(log(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), log(log(Symbol('\\\\psi^*', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), log(log(Symbol('\\\\psi^*', commutative=True)))), Integer(-1)), Integral(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\psi^*', commutative=True))), log(log(Symbol('\\\\psi^*', commutative=True)))), Integer(-1)), Integral(Mul(Integer(-1), log(log(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\hat{x})} = \\sin{(\\hat{x})}, then derive \\int \\operatorname{t_{1}}{(\\hat{x})} d\\hat{x} = \\varphi - \\cos{(\\hat{x})}, then derive M - \\hat{x} - \\cos{(\\hat{x})} = - \\hat{x} + \\varphi - \\cos{(\\hat{x})}, then obtain - \\hat{x} + \\int \\operatorname{t_{1}}{(\\hat{x})} d\\hat{x} = M - \\hat{x} - \\cos{(\\hat{x})}", "derivation": "\\operatorname{t_{1}}{(\\hat{x})} = \\sin{(\\hat{x})} and \\int \\operatorname{t_{1}}{(\\hat{x})} d\\hat{x} = \\int \\sin{(\\hat{x})} d\\hat{x} and \\int \\operatorname{t_{1}}{(\\hat{x})} d\\hat{x} = \\varphi - \\cos{(\\hat{x})} and \\int \\sin{(\\hat{x})} d\\hat{x} = \\varphi - \\cos{(\\hat{x})} and - \\hat{x} + \\int \\sin{(\\hat{x})} d\\hat{x} = - \\hat{x} + \\varphi - \\cos{(\\hat{x})} and M - \\hat{x} - \\cos{(\\hat{x})} = - \\hat{x} + \\varphi - \\cos{(\\hat{x})} and - \\hat{x} + \\int \\sin{(\\hat{x})} d\\hat{x} = M - \\hat{x} - \\cos{(\\hat{x})} and - \\hat{x} + \\int \\operatorname{t_{1}}{(\\hat{x})} d\\hat{x} = M - \\hat{x} - \\cos{(\\hat{x})}", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_1')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["minus", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(Function('t_1')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(L,\\mu)} = \\frac{\\mu}{L}, then obtain (\\frac{\\mu}{L})^{L} \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{(\\tilde{g}^*^{\\mu}{(L,\\mu)})} = (\\frac{\\mu}{L})^{L} \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{((\\frac{\\mu}{L})^{\\mu})}", "derivation": "\\tilde{g}^*{(L,\\mu)} = \\frac{\\mu}{L} and \\tilde{g}^*^{\\mu}{(L,\\mu)} = (\\frac{\\mu}{L})^{\\mu} and - \\tilde{g}^*^{\\mu}{(L,\\mu)} = - (\\frac{\\mu}{L})^{\\mu} and \\cos{(\\tilde{g}^*^{\\mu}{(L,\\mu)})} = \\cos{((\\frac{\\mu}{L})^{\\mu})} and \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{(\\tilde{g}^*^{\\mu}{(L,\\mu)})} = \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{((\\frac{\\mu}{L})^{\\mu})} and (\\frac{\\mu}{L})^{L} \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{(\\tilde{g}^*^{\\mu}{(L,\\mu)})} = (\\frac{\\mu}{L})^{L} \\tilde{g}^*^{\\mu}{(L,\\mu)} \\cos{((\\frac{\\mu}{L})^{\\mu})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["cos", 3], "Equality(cos(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), cos(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["times", 4, "Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), cos(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), cos(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["times", 5, "Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('L', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), cos(Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('L', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('L', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), cos(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(k)} = \\log{(\\sin{(k)})} and \\mathbf{f}{(k)} = k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})}, then obtain (k \\log{(\\sin{(k)})} + \\log{(\\sin{(k)})})^{k} - \\log{(\\sin{(k)})} = (k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})})^{k} - \\log{(\\sin{(k)})}", "derivation": "\\mathbf{s}{(k)} = \\log{(\\sin{(k)})} and k \\mathbf{s}{(k)} = k \\log{(\\sin{(k)})} and k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})} = k \\log{(\\sin{(k)})} + \\log{(\\sin{(k)})} and \\mathbf{f}{(k)} = k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})} and \\mathbf{f}{(k)} = k \\log{(\\sin{(k)})} + \\log{(\\sin{(k)})} and \\mathbf{f}^{k}{(k)} = (k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})})^{k} and \\mathbf{f}^{k}{(k)} - \\log{(\\sin{(k)})} = (k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})})^{k} - \\log{(\\sin{(k)})} and (k \\log{(\\sin{(k)})} + \\log{(\\sin{(k)})})^{k} - \\log{(\\sin{(k)})} = (k \\mathbf{s}{(k)} + \\log{(\\sin{(k)})})^{k} - \\log{(\\sin{(k)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), log(sin(Symbol('k', commutative=True))))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), log(sin(Symbol('k', commutative=True)))))"], [["add", 2, "log(sin(Symbol('k', commutative=True)))"], "Equality(Add(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), log(sin(Symbol('k', commutative=True)))), Add(Mul(Symbol('k', commutative=True), log(sin(Symbol('k', commutative=True)))), log(sin(Symbol('k', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('k', commutative=True)), Add(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), log(sin(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{f}')(Symbol('k', commutative=True)), Add(Mul(Symbol('k', commutative=True), log(sin(Symbol('k', commutative=True)))), log(sin(Symbol('k', commutative=True)))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), log(sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["minus", 6, "log(sin(Symbol('k', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), log(sin(Symbol('k', commutative=True))))), Add(Pow(Add(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), log(sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Mul(Integer(-1), log(sin(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Pow(Add(Mul(Symbol('k', commutative=True), log(sin(Symbol('k', commutative=True)))), log(sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Mul(Integer(-1), log(sin(Symbol('k', commutative=True))))), Add(Pow(Add(Mul(Symbol('k', commutative=True), Function('\\\\mathbf{s}')(Symbol('k', commutative=True))), log(sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Mul(Integer(-1), log(sin(Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})} = \\mathbf{B} e^{\\Psi^{\\dagger}}, then obtain ((- \\Psi^{\\dagger} \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})})^{\\mathbf{B}})^{\\mathbf{B}} = ((- \\Psi^{\\dagger} \\mathbf{B} e^{\\Psi^{\\dagger}})^{\\mathbf{B}})^{\\mathbf{B}}", "derivation": "\\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})} = \\mathbf{B} e^{\\Psi^{\\dagger}} and \\Psi^{\\dagger} \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})} = \\Psi^{\\dagger} \\mathbf{B} e^{\\Psi^{\\dagger}} and - \\Psi^{\\dagger} \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})} = - \\Psi^{\\dagger} \\mathbf{B} e^{\\Psi^{\\dagger}} and (- \\Psi^{\\dagger} \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})})^{\\mathbf{B}} = (- \\Psi^{\\dagger} \\mathbf{B} e^{\\Psi^{\\dagger}})^{\\mathbf{B}} and ((- \\Psi^{\\dagger} \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},\\mathbf{B})})^{\\mathbf{B}})^{\\mathbf{B}} = ((- \\Psi^{\\dagger} \\mathbf{B} e^{\\Psi^{\\dagger}})^{\\mathbf{B}})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\omega{(\\pi,\\theta_1)} = \\log{(\\frac{\\pi}{\\theta_1})}, then derive \\frac{\\partial}{\\partial \\pi} \\omega{(\\pi,\\theta_1)} = \\frac{1}{\\pi}, then obtain \\int - \\frac{\\partial}{\\partial \\pi} \\log{(\\frac{\\pi}{\\theta_1})} d\\pi = \\int - \\frac{1}{\\pi} d\\pi", "derivation": "\\omega{(\\pi,\\theta_1)} = \\log{(\\frac{\\pi}{\\theta_1})} and \\frac{\\partial}{\\partial \\pi} \\omega{(\\pi,\\theta_1)} = \\frac{\\partial}{\\partial \\pi} \\log{(\\frac{\\pi}{\\theta_1})} and \\frac{\\partial}{\\partial \\pi} \\omega{(\\pi,\\theta_1)} = \\frac{1}{\\pi} and \\frac{\\partial}{\\partial \\pi} \\log{(\\frac{\\pi}{\\theta_1})} = \\frac{1}{\\pi} and - \\frac{\\partial}{\\partial \\pi} \\log{(\\frac{\\pi}{\\theta_1})} = - \\frac{1}{\\pi} and - \\frac{\\partial}{\\partial \\pi} \\omega{(\\pi,\\theta_1)} = - \\frac{1}{\\pi} and \\int - \\frac{\\partial}{\\partial \\pi} \\omega{(\\pi,\\theta_1)} d\\pi = \\int - \\frac{1}{\\pi} d\\pi and \\int - \\frac{\\partial}{\\partial \\pi} \\log{(\\frac{\\pi}{\\theta_1})} d\\pi = \\int - \\frac{1}{\\pi} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(log(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["integrate", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integral(Mul(Integer(-1), Derivative(log(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(C_{d})} = \\cos{(C_{d})} and \\dot{y}{(C_{d})} = \\frac{d}{d C_{d}} (C_{d} + \\operatorname{v_{x}}{(C_{d})} - \\cos{(C_{d})}), then obtain \\dot{y}{(C_{d})} = \\frac{d}{d C_{d}} C_{d}", "derivation": "\\operatorname{v_{x}}{(C_{d})} = \\cos{(C_{d})} and \\operatorname{v_{x}}{(C_{d})} - \\cos{(C_{d})} = 0 and C_{d} + \\operatorname{v_{x}}{(C_{d})} - \\cos{(C_{d})} = C_{d} and \\frac{d}{d C_{d}} (C_{d} + \\operatorname{v_{x}}{(C_{d})} - \\cos{(C_{d})}) = \\frac{d}{d C_{d}} C_{d} and \\dot{y}{(C_{d})} = \\frac{d}{d C_{d}} (C_{d} + \\operatorname{v_{x}}{(C_{d})} - \\cos{(C_{d})}) and \\dot{y}{(C_{d})} = \\frac{d}{d C_{d}} C_{d}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["minus", 1, "cos(Symbol('C_d', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('C_d', commutative=True)), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Integer(0))"], [["add", 2, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Function('v_x')(Symbol('C_d', commutative=True)), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Symbol('C_d', commutative=True), Function('v_x')(Symbol('C_d', commutative=True)), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), Derivative(Add(Symbol('C_d', commutative=True), Function('v_x')(Symbol('C_d', commutative=True)), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), Derivative(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(M)} = \\log{(M)}, then derive \\int (- M + \\Psi_{\\lambda}{(M)}) dM = - \\frac{M^{2}}{2} + M \\log{(M)} - M + \\theta_1, then obtain \\int (- M + \\log{(M)}) dM = - \\frac{M^{2}}{2} + M \\log{(M)} - M + \\theta_1", "derivation": "\\Psi_{\\lambda}{(M)} = \\log{(M)} and - M + \\Psi_{\\lambda}{(M)} = - M + \\log{(M)} and \\int (- M + \\Psi_{\\lambda}{(M)}) dM = \\int (- M + \\log{(M)}) dM and \\int (- M + \\Psi_{\\lambda}{(M)}) dM = - \\frac{M^{2}}{2} + M \\log{(M)} - M + \\theta_1 and \\int (- M + \\log{(M)}) dM = - \\frac{M^{2}}{2} + M \\log{(M)} - M + \\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["minus", 1, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), log(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), log(Symbol('M', commutative=True))), Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), log(Symbol('M', commutative=True))), Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{2},Q)} = \\log{(\\frac{Q}{v_{2}})}, then obtain 0 = \\log{(\\frac{v_{2} + \\log{(\\frac{Q}{v_{2}})}}{v_{2} + \\operatorname{F_{c}}{(v_{2},Q)}})}", "derivation": "\\operatorname{F_{c}}{(v_{2},Q)} = \\log{(\\frac{Q}{v_{2}})} and v_{2} + \\operatorname{F_{c}}{(v_{2},Q)} = v_{2} + \\log{(\\frac{Q}{v_{2}})} and (v_{2} + \\operatorname{F_{c}}{(v_{2},Q)}) \\operatorname{F_{c}}{(v_{2},Q)} = (v_{2} + \\log{(\\frac{Q}{v_{2}})}) \\operatorname{F_{c}}{(v_{2},Q)} and 1 = \\frac{v_{2} + \\log{(\\frac{Q}{v_{2}})}}{v_{2} + \\operatorname{F_{c}}{(v_{2},Q)}} and 0 = \\log{(\\frac{v_{2} + \\log{(\\frac{Q}{v_{2}})}}{v_{2} + \\operatorname{F_{c}}{(v_{2},Q)}})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True)), log(Mul(Symbol('Q', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)))))"], [["add", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Symbol('v_2', commutative=True), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('v_2', commutative=True), log(Mul(Symbol('Q', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))))"], [["times", 2, "Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Add(Symbol('v_2', commutative=True), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Mul(Add(Symbol('v_2', commutative=True), log(Mul(Symbol('Q', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))))"], [["divide", 3, "Mul(Add(Symbol('v_2', commutative=True), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('v_2', commutative=True), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Integer(-1)), Add(Symbol('v_2', commutative=True), log(Mul(Symbol('Q', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)))))))"], [["log", 4], "Equality(Integer(0), log(Mul(Pow(Add(Symbol('v_2', commutative=True), Function('F_c')(Symbol('v_2', commutative=True), Symbol('Q', commutative=True))), Integer(-1)), Add(Symbol('v_2', commutative=True), log(Mul(Symbol('Q', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))))))"]]}, {"prompt": "Given u{(h,\\mathbf{M})} = \\frac{\\mathbf{M}}{h} and \\operatorname{x^{{\\}'}}{(S,\\theta_1)} = - \\theta_1 + \\log{(S)} and \\varepsilon_{0}{(S,\\theta_1)} = - \\theta_1 + \\log{(S)}, then obtain \\varepsilon_{0}{(S,\\theta_1)} + \\int u{(h,\\mathbf{M})} dh = \\varepsilon_{0}{(S,\\theta_1)} + \\int \\frac{\\mathbf{M}}{h} dh", "derivation": "u{(h,\\mathbf{M})} = \\frac{\\mathbf{M}}{h} and \\int u{(h,\\mathbf{M})} dh = \\int \\frac{\\mathbf{M}}{h} dh and \\operatorname{x^{{\\}'}}{(S,\\theta_1)} = - \\theta_1 + \\log{(S)} and \\varepsilon_{0}{(S,\\theta_1)} = - \\theta_1 + \\log{(S)} and \\operatorname{x^{{\\}'}}{(S,\\theta_1)} = \\varepsilon_{0}{(S,\\theta_1)} and - \\theta_1 + \\log{(S)} + \\int u{(h,\\mathbf{M})} dh = - \\theta_1 + \\log{(S)} + \\int \\frac{\\mathbf{M}}{h} dh and \\operatorname{x^{{\\}'}}{(S,\\theta_1)} + \\int u{(h,\\mathbf{M})} dh = \\operatorname{x^{{\\}'}}{(S,\\theta_1)} + \\int \\frac{\\mathbf{M}}{h} dh and \\varepsilon_{0}{(S,\\theta_1)} + \\int u{(h,\\mathbf{M})} dh = \\varepsilon_{0}{(S,\\theta_1)} + \\int \\frac{\\mathbf{M}}{h} dh", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('h', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('u')(Symbol('h', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))))"], ["get_premise", "Equality(Function('x^\\\\prime')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('x^\\\\prime')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Symbol('S', commutative=True)), Integral(Function('u')(Symbol('h', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Symbol('S', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Function('x^\\\\prime')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Function('u')(Symbol('h', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Function('x^\\\\prime')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Function('u')(Symbol('h', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}}, then obtain 1 = \\frac{\\int (\\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}})^{\\Psi_{nl}} d\\Psi_{nl}}{\\int \\mathbf{F}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl}}", "derivation": "\\mathbf{F}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}} and \\mathbf{F}^{\\Psi_{nl}}{(\\Psi_{nl})} = (\\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}})^{\\Psi_{nl}} and \\int \\mathbf{F}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl} = \\int (\\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}})^{\\Psi_{nl}} d\\Psi_{nl} and 1 = \\frac{\\int (\\frac{d}{d \\Psi_{nl}} e^{\\Psi_{nl}})^{\\Psi_{nl}} d\\Psi_{nl}}{\\int \\mathbf{F}^{\\Psi_{nl}}{(\\Psi_{nl})} d\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Derivative(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Derivative(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 3, "Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Integral(Pow(Derivative(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{1})} = \\frac{d}{d A_{1}} \\cos{(A_{1})}, then derive 2 + \\frac{\\operatorname{F_{x}}{(A_{1})}}{A_{1}} = 2 - \\frac{\\sin{(A_{1})}}{A_{1}}, then obtain 2 - \\frac{\\sin{(A_{1})}}{A_{1}} = 2 + \\frac{\\frac{d}{d A_{1}} \\cos{(A_{1})}}{A_{1}}", "derivation": "\\operatorname{F_{x}}{(A_{1})} = \\frac{d}{d A_{1}} \\cos{(A_{1})} and \\frac{\\operatorname{F_{x}}{(A_{1})}}{A_{1}} = \\frac{\\frac{d}{d A_{1}} \\cos{(A_{1})}}{A_{1}} and 1 + \\frac{\\operatorname{F_{x}}{(A_{1})}}{A_{1}} = 1 + \\frac{\\frac{d}{d A_{1}} \\cos{(A_{1})}}{A_{1}} and 2 + \\frac{\\operatorname{F_{x}}{(A_{1})}}{A_{1}} = 2 + \\frac{\\frac{d}{d A_{1}} \\cos{(A_{1})}}{A_{1}} and 2 + \\frac{\\operatorname{F_{x}}{(A_{1})}}{A_{1}} = 2 - \\frac{\\sin{(A_{1})}}{A_{1}} and 2 - \\frac{\\sin{(A_{1})}}{A_{1}} = 2 + \\frac{\\frac{d}{d A_{1}} \\cos{(A_{1})}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('A_1', commutative=True)), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('F_x')(Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('F_x')(Symbol('A_1', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(2), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('F_x')(Symbol('A_1', commutative=True)))), Add(Integer(2), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(2), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('F_x')(Symbol('A_1', commutative=True)))), Add(Integer(2), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), sin(Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Integer(2), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), sin(Symbol('A_1', commutative=True)))), Add(Integer(2), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\hbar)} = \\log{(\\hbar)}, then derive (\\log{(\\hbar)} \\frac{d}{d \\hbar} \\operatorname{f^{*}}{(\\hbar)} + \\frac{\\operatorname{f^{*}}{(\\hbar)}}{\\hbar})^{\\hbar} = (\\frac{2 \\log{(\\hbar)}}{\\hbar})^{\\hbar}, then obtain (\\log{(\\hbar)} \\frac{d}{d \\hbar} \\log{(\\hbar)} + \\frac{\\log{(\\hbar)}}{\\hbar})^{\\hbar} = (\\frac{2 \\log{(\\hbar)}}{\\hbar})^{\\hbar}", "derivation": "\\operatorname{f^{*}}{(\\hbar)} = \\log{(\\hbar)} and \\operatorname{f^{*}}{(\\hbar)} \\log{(\\hbar)} = \\log{(\\hbar)}^{2} and \\frac{d}{d \\hbar} \\operatorname{f^{*}}{(\\hbar)} \\log{(\\hbar)} = \\frac{d}{d \\hbar} \\log{(\\hbar)}^{2} and (\\frac{d}{d \\hbar} \\operatorname{f^{*}}{(\\hbar)} \\log{(\\hbar)})^{\\hbar} = (\\frac{d}{d \\hbar} \\log{(\\hbar)}^{2})^{\\hbar} and (\\log{(\\hbar)} \\frac{d}{d \\hbar} \\operatorname{f^{*}}{(\\hbar)} + \\frac{\\operatorname{f^{*}}{(\\hbar)}}{\\hbar})^{\\hbar} = (\\frac{2 \\log{(\\hbar)}}{\\hbar})^{\\hbar} and (\\log{(\\hbar)} \\frac{d}{d \\hbar} \\log{(\\hbar)} + \\frac{\\log{(\\hbar)}}{\\hbar})^{\\hbar} = (\\frac{2 \\log{(\\hbar)}}{\\hbar})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "log(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Mul(Function('f^*')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('f^*')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Mul(log(Symbol('\\\\hbar', commutative=True)), Derivative(Function('f^*')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('f^*')(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Mul(log(Symbol('\\\\hbar', commutative=True)), Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), log(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given W{(\\mathbf{J}_P,\\hat{x}_0)} = \\hat{x}_0 + \\mathbf{J}_P, then obtain \\frac{W^{2}{(\\mathbf{J}_P,\\hat{x}_0)}}{(\\hat{x}_0 + \\mathbf{J}_P)^{2}} = 1", "derivation": "W{(\\mathbf{J}_P,\\hat{x}_0)} = \\hat{x}_0 + \\mathbf{J}_P and \\frac{W{(\\mathbf{J}_P,\\hat{x}_0)}}{\\hat{x}_0 + \\mathbf{J}_P} = 1 and \\frac{W^{2}{(\\mathbf{J}_P,\\hat{x}_0)}}{\\hat{x}_0 + \\mathbf{J}_P} = W{(\\mathbf{J}_P,\\hat{x}_0)} and \\frac{W^{2}{(\\mathbf{J}_P,\\hat{x}_0)}}{(\\hat{x}_0 + \\mathbf{J}_P)^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Integer(1))"], [["times", 2, "Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(2))), Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-2)), Pow(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)} = T + g^{\\prime}_{\\varepsilon}, then obtain \\int \\frac{g^{\\prime}_{\\varepsilon} \\mathbf{D}^{2}{(g^{\\prime}_{\\varepsilon},T)}}{T + g^{\\prime}_{\\varepsilon}} dg^{\\prime}_{\\varepsilon} = \\int g^{\\prime}_{\\varepsilon} \\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)} dg^{\\prime}_{\\varepsilon}", "derivation": "\\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)} = T + g^{\\prime}_{\\varepsilon} and \\frac{\\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)}}{T + g^{\\prime}_{\\varepsilon}} = 1 and \\frac{g^{\\prime}_{\\varepsilon} \\mathbf{D}^{2}{(g^{\\prime}_{\\varepsilon},T)}}{T + g^{\\prime}_{\\varepsilon}} = g^{\\prime}_{\\varepsilon} \\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)} and \\int \\frac{g^{\\prime}_{\\varepsilon} \\mathbf{D}^{2}{(g^{\\prime}_{\\varepsilon},T)}}{T + g^{\\prime}_{\\varepsilon}} dg^{\\prime}_{\\varepsilon} = \\int g^{\\prime}_{\\varepsilon} \\mathbf{D}{(g^{\\prime}_{\\varepsilon},T)} dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Add(Symbol('T', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Integer(1))"], [["times", 2, "Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Pow(Add(Symbol('T', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Pow(Add(Symbol('T', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{D}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\mathbf{J})} = \\sin{(\\mathbf{J})}, then obtain \\frac{d}{d \\mathbf{J}} \\int \\frac{\\mathbf{J}_M{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\int \\frac{\\sin{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J}", "derivation": "\\mathbf{J}_M{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\frac{\\mathbf{J}_M{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\sin{(\\mathbf{J})}}{\\mathbf{J}} and \\int \\frac{\\mathbf{J}_M{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} = \\int \\frac{\\sin{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\int \\frac{\\mathbf{J}_M{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\int \\frac{\\sin{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\Psi_{nl},a)} = \\Psi_{nl} a and U{(\\Psi_{nl},a)} = \\frac{\\partial}{\\partial a} (\\Psi_{nl} a - \\operatorname{F_{g}}{(\\Psi_{nl},a)}), then derive U{(\\Psi_{nl},a)} = 0, then derive U{(\\Psi_{nl},a)} = \\Psi_{nl} - \\frac{\\partial}{\\partial a} \\operatorname{F_{g}}{(\\Psi_{nl},a)}, then derive \\Psi_{nl} - \\frac{\\partial}{\\partial a} \\operatorname{F_{g}}{(\\Psi_{nl},a)} = 0, then obtain \\frac{d}{d a} 0 = 0", "derivation": "\\operatorname{F_{g}}{(\\Psi_{nl},a)} = \\Psi_{nl} a and U{(\\Psi_{nl},a)} = \\frac{\\partial}{\\partial a} (\\Psi_{nl} a - \\operatorname{F_{g}}{(\\Psi_{nl},a)}) and U{(\\Psi_{nl},a)} = \\frac{d}{d a} 0 and U{(\\Psi_{nl},a)} = 0 and U{(\\Psi_{nl},a)} = \\Psi_{nl} - \\frac{\\partial}{\\partial a} \\operatorname{F_{g}}{(\\Psi_{nl},a)} and \\frac{\\partial}{\\partial a} (\\Psi_{nl} a - \\operatorname{F_{g}}{(\\Psi_{nl},a)}) = 0 and \\Psi_{nl} - \\frac{\\partial}{\\partial a} \\operatorname{F_{g}}{(\\Psi_{nl},a)} = 0 and \\Psi_{nl} - \\frac{\\partial}{\\partial a} \\operatorname{F_{g}}{(\\Psi_{nl},a)} = \\frac{d}{d a} 0 and \\frac{d}{d a} 0 = 0", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('U')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('U')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Function('U')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Derivative(Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(0))"], [["evaluate_derivatives", 6], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Derivative(Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Derivative(Function('F_g')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given s{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain \\sin{((- \\mathbf{s} + s{(\\mathbf{s})})^{\\mathbf{s}})} = \\sin{((- \\mathbf{s} + \\cos{(\\mathbf{s})})^{\\mathbf{s}})}", "derivation": "s{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and - \\mathbf{s} + s{(\\mathbf{s})} = - \\mathbf{s} + \\cos{(\\mathbf{s})} and (- \\mathbf{s} + s{(\\mathbf{s})})^{\\mathbf{s}} = (- \\mathbf{s} + \\cos{(\\mathbf{s})})^{\\mathbf{s}} and \\sin{((- \\mathbf{s} + s{(\\mathbf{s})})^{\\mathbf{s}})} = \\sin{((- \\mathbf{s} + \\cos{(\\mathbf{s})})^{\\mathbf{s}})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('s')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('s')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('s')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(H)} = \\log{(\\sin{(H)})}, then obtain \\int (- \\mathbf{E}{(H)} + \\sin{(\\mathbf{E}{(H)})})^{H} dH = \\int (- \\mathbf{E}{(H)} + \\sin{(\\log{(\\sin{(H)})})})^{H} dH", "derivation": "\\mathbf{E}{(H)} = \\log{(\\sin{(H)})} and \\sin{(\\mathbf{E}{(H)})} = \\sin{(\\log{(\\sin{(H)})})} and - \\mathbf{E}{(H)} + \\sin{(\\mathbf{E}{(H)})} = - \\mathbf{E}{(H)} + \\sin{(\\log{(\\sin{(H)})})} and (- \\mathbf{E}{(H)} + \\sin{(\\mathbf{E}{(H)})})^{H} = (- \\mathbf{E}{(H)} + \\sin{(\\log{(\\sin{(H)})})})^{H} and \\int (- \\mathbf{E}{(H)} + \\sin{(\\mathbf{E}{(H)})})^{H} dH = \\int (- \\mathbf{E}{(H)} + \\sin{(\\log{(\\sin{(H)})})})^{H} dH", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(log(sin(Symbol('H', commutative=True)))))"], [["minus", 2, "Function('\\\\mathbf{E}')(Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(Function('\\\\mathbf{E}')(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(log(sin(Symbol('H', commutative=True))))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(Function('\\\\mathbf{E}')(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(log(sin(Symbol('H', commutative=True))))), Symbol('H', commutative=True)))"], [["integrate", 4, "Symbol('H', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(Function('\\\\mathbf{E}')(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), sin(log(sin(Symbol('H', commutative=True))))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given v{(v_{t},s)} = s v_{t}, then obtain 2 (\\int v{(v_{t},s)} ds)^{v_{t}} = (\\int s v_{t} ds)^{v_{t}} + (\\int v{(v_{t},s)} ds)^{v_{t}}", "derivation": "v{(v_{t},s)} = s v_{t} and \\int v{(v_{t},s)} ds = \\int s v_{t} ds and (\\int v{(v_{t},s)} ds)^{v_{t}} = (\\int s v_{t} ds)^{v_{t}} and 2 (\\int v{(v_{t},s)} ds)^{v_{t}} = (\\int s v_{t} ds)^{v_{t}} + (\\int v{(v_{t},s)} ds)^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True)))"], [["add", 3, "Pow(Integral(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True))"], "Equality(Mul(Integer(2), Pow(Integral(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True))), Add(Pow(Integral(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Function('v')(Symbol('v_t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{t})} = e^{v_{t}}, then obtain \\frac{d}{d v_{t}} \\sigma_{p}{(v_{t})} + 1 = e^{v_{t}} + 1", "derivation": "\\sigma_{p}{(v_{t})} = e^{v_{t}} and v_{t} + \\sigma_{p}{(v_{t})} = v_{t} + e^{v_{t}} and \\frac{d}{d v_{t}} (v_{t} + \\sigma_{p}{(v_{t})}) = \\frac{d}{d v_{t}} (v_{t} + e^{v_{t}}) and \\frac{d}{d v_{t}} \\sigma_{p}{(v_{t})} + 1 = e^{v_{t}} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["add", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Symbol('v_t', commutative=True), Function('\\\\sigma_p')(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Symbol('v_t', commutative=True), Function('\\\\sigma_p')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\sigma_p')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('v_t', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\theta_{1}{(h,\\hat{H}_{\\lambda})} = \\frac{h}{\\hat{H}_{\\lambda}} and \\mathbb{I}{(h,\\hat{H}_{\\lambda})} = - \\theta_{1}{(h,\\hat{H}_{\\lambda})}, then obtain - \\frac{e^{\\mathbb{I}{(h,\\hat{H}_{\\lambda})}}}{h} = - \\frac{e^{- \\frac{h}{\\hat{H}_{\\lambda}}}}{h}", "derivation": "\\theta_{1}{(h,\\hat{H}_{\\lambda})} = \\frac{h}{\\hat{H}_{\\lambda}} and - \\theta_{1}{(h,\\hat{H}_{\\lambda})} = - \\frac{h}{\\hat{H}_{\\lambda}} and e^{- \\theta_{1}{(h,\\hat{H}_{\\lambda})}} = e^{- \\frac{h}{\\hat{H}_{\\lambda}}} and - e^{- \\theta_{1}{(h,\\hat{H}_{\\lambda})}} = - e^{- \\frac{h}{\\hat{H}_{\\lambda}}} and - \\frac{e^{- \\theta_{1}{(h,\\hat{H}_{\\lambda})}}}{h} = - \\frac{e^{- \\frac{h}{\\hat{H}_{\\lambda}}}}{h} and \\mathbb{I}{(h,\\hat{H}_{\\lambda})} = - \\theta_{1}{(h,\\hat{H}_{\\lambda})} and - \\frac{e^{\\mathbb{I}{(h,\\hat{H}_{\\lambda})}}}{h} = - \\frac{e^{- \\frac{h}{\\hat{H}_{\\lambda}}}}{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], [["exp", 2], "Equality(exp(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Mul(Integer(-1), exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))))"], [["divide", 4, "Symbol('h', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), exp(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\tilde{g})} = \\log{(\\tilde{g})}, then obtain \\int \\tilde{g} \\hat{p}{(\\tilde{g})} \\log{(\\tilde{g})} d\\tilde{g} = \\lambda + \\frac{\\tilde{g}^{2} \\log{(\\tilde{g})}^{2}}{2} - \\frac{\\tilde{g}^{2} \\log{(\\tilde{g})}}{2} + \\frac{\\tilde{g}^{2}}{4}", "derivation": "\\hat{p}{(\\tilde{g})} = \\log{(\\tilde{g})} and \\tilde{g} \\hat{p}{(\\tilde{g})} = \\tilde{g} \\log{(\\tilde{g})} and \\tilde{g} \\hat{p}{(\\tilde{g})} \\log{(\\tilde{g})} = \\tilde{g} \\log{(\\tilde{g})}^{2} and \\int \\tilde{g} \\hat{p}{(\\tilde{g})} \\log{(\\tilde{g})} d\\tilde{g} = \\int \\tilde{g} \\log{(\\tilde{g})}^{2} d\\tilde{g} and \\int \\tilde{g} \\hat{p}{(\\tilde{g})} \\log{(\\tilde{g})} d\\tilde{g} = \\lambda + \\frac{\\tilde{g}^{2} \\log{(\\tilde{g})}^{2}}{2} - \\frac{\\tilde{g}^{2} \\log{(\\tilde{g})}}{2} + \\frac{\\tilde{g}^{2}}{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "log(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\omega{(v_{2},p)} = e^{v_{2}^{p}}, then obtain \\frac{\\partial}{\\partial p} (\\omega{(v_{2},p)} + e^{v_{2}^{p}})^{p} = \\frac{\\partial}{\\partial p} (2 e^{v_{2}^{p}})^{p}", "derivation": "\\omega{(v_{2},p)} = e^{v_{2}^{p}} and \\omega{(v_{2},p)} + e^{v_{2}^{p}} = 2 e^{v_{2}^{p}} and (\\omega{(v_{2},p)} + e^{v_{2}^{p}})^{p} = (2 e^{v_{2}^{p}})^{p} and \\frac{\\partial}{\\partial p} (\\omega{(v_{2},p)} + e^{v_{2}^{p}})^{p} = \\frac{\\partial}{\\partial p} (2 e^{v_{2}^{p}})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v_2', commutative=True), Symbol('p', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True))))"], [["add", 1, "exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('v_2', commutative=True), Symbol('p', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))), Mul(Integer(2), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Function('\\\\omega')(Symbol('v_2', commutative=True), Symbol('p', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True)), Pow(Mul(Integer(2), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True)))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\omega')(Symbol('v_2', commutative=True), Symbol('p', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(2), exp(Pow(Symbol('v_2', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(\\theta,M,\\theta_1)} = - M - \\theta + \\theta_1, then derive y + \\hat{p}{(\\theta,M,\\theta_1)} = C_{1} - \\theta, then obtain (y + \\hat{p}{(\\theta,M,\\theta_1)}) \\hat{p}{(\\theta,M,\\theta_1)} = (- M - \\theta + \\theta_1 + y) \\hat{p}{(\\theta,M,\\theta_1)}", "derivation": "\\hat{p}{(\\theta,M,\\theta_1)} = - M - \\theta + \\theta_1 and \\frac{\\partial}{\\partial \\theta} \\hat{p}{(\\theta,M,\\theta_1)} = \\frac{\\partial}{\\partial \\theta} (- M - \\theta + \\theta_1) and \\int \\frac{\\partial}{\\partial \\theta} \\hat{p}{(\\theta,M,\\theta_1)} d\\theta = \\int \\frac{\\partial}{\\partial \\theta} (- M - \\theta + \\theta_1) d\\theta and y + \\hat{p}{(\\theta,M,\\theta_1)} = C_{1} - \\theta and - M - \\theta + \\theta_1 + y = C_{1} - \\theta and (y + \\hat{p}{(\\theta,M,\\theta_1)}) \\hat{p}{(\\theta,M,\\theta_1)} = (C_{1} - \\theta) \\hat{p}{(\\theta,M,\\theta_1)} and (y + \\hat{p}{(\\theta,M,\\theta_1)}) \\hat{p}{(\\theta,M,\\theta_1)} = (- M - \\theta + \\theta_1 + y) \\hat{p}{(\\theta,M,\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["times", 4, "Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Add(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mu)} = e^{\\sin{(\\mu)}}, then obtain \\frac{d^{3}}{d \\mu^{3}} \\frac{\\tilde{g}^*{(\\mu)}}{\\sin{(\\mu)}} = \\frac{d^{3}}{d \\mu^{3}} \\frac{e^{\\sin{(\\mu)}}}{\\sin{(\\mu)}}", "derivation": "\\tilde{g}^*{(\\mu)} = e^{\\sin{(\\mu)}} and \\frac{\\tilde{g}^*{(\\mu)}}{\\sin{(\\mu)}} = \\frac{e^{\\sin{(\\mu)}}}{\\sin{(\\mu)}} and \\frac{d}{d \\mu} \\frac{\\tilde{g}^*{(\\mu)}}{\\sin{(\\mu)}} = \\frac{d}{d \\mu} \\frac{e^{\\sin{(\\mu)}}}{\\sin{(\\mu)}} and \\frac{d^{2}}{d \\mu^{2}} \\frac{\\tilde{g}^*{(\\mu)}}{\\sin{(\\mu)}} = \\frac{d^{2}}{d \\mu^{2}} \\frac{e^{\\sin{(\\mu)}}}{\\sin{(\\mu)}} and \\frac{d^{3}}{d \\mu^{3}} \\frac{\\tilde{g}^*{(\\mu)}}{\\sin{(\\mu)}} = \\frac{d^{3}}{d \\mu^{3}} \\frac{e^{\\sin{(\\mu)}}}{\\sin{(\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mu', commutative=True)), exp(sin(Symbol('\\\\mu', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(exp(sin(Symbol('\\\\mu', commutative=True))), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(exp(sin(Symbol('\\\\mu', commutative=True))), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))), Derivative(Mul(exp(sin(Symbol('\\\\mu', commutative=True))), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(3))), Derivative(Mul(exp(sin(Symbol('\\\\mu', commutative=True))), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True), Integer(3))))"]]}, {"prompt": "Given l{(T)} = e^{T}, then derive - T + \\frac{d}{d T} l{(T)} = - T + e^{T}, then obtain - T + \\frac{d}{d T} e^{T} = - T + e^{T}", "derivation": "l{(T)} = e^{T} and \\frac{d}{d T} l{(T)} = \\frac{d}{d T} e^{T} and - T + \\frac{d}{d T} l{(T)} = - T + \\frac{d}{d T} e^{T} and - T + \\frac{d}{d T} l{(T)} = - T + e^{T} and - T + \\frac{d}{d T} e^{T} = - T + e^{T}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"]]}, {"prompt": "Given H{(\\mathbf{J},\\pi)} = \\cos{(\\mathbf{J} \\pi)}, then obtain \\mathbf{J} + ((\\frac{\\int H{(\\mathbf{J},\\pi)} d\\mathbf{J}}{\\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J}})^{\\pi})^{\\pi} - \\cos{(\\mathbf{J} \\pi)} + 1 = \\mathbf{J} - \\cos{(\\mathbf{J} \\pi)} + 2", "derivation": "H{(\\mathbf{J},\\pi)} = \\cos{(\\mathbf{J} \\pi)} and \\int H{(\\mathbf{J},\\pi)} d\\mathbf{J} = \\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J} and \\frac{\\int H{(\\mathbf{J},\\pi)} d\\mathbf{J}}{\\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J}} = 1 and (\\frac{\\int H{(\\mathbf{J},\\pi)} d\\mathbf{J}}{\\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J}})^{\\pi} = 1 and ((\\frac{\\int H{(\\mathbf{J},\\pi)} d\\mathbf{J}}{\\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J}})^{\\pi})^{\\pi} = 1 and \\mathbf{J} + ((\\frac{\\int H{(\\mathbf{J},\\pi)} d\\mathbf{J}}{\\int \\cos{(\\mathbf{J} \\pi)} d\\mathbf{J}})^{\\pi})^{\\pi} - \\cos{(\\mathbf{J} \\pi)} + 1 = \\mathbf{J} - \\cos{(\\mathbf{J} \\pi)} + 2", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 2, "Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Integral(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Mul(Integral(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Pow(Mul(Integral(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["add", 5, "Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)))), Integer(1))"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Pow(Mul(Integral(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Integral(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)))), Integer(1)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\pi', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} = - \\sin{(\\mathbf{E} - \\mathbf{J}_f)}, then obtain - (\\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} - \\sin{(\\mathbf{E} - \\mathbf{J}_f)}) \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} \\sin{(\\mathbf{E} - \\mathbf{J}_f)} = 2 \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} \\sin^{2}{(\\mathbf{E} - \\mathbf{J}_f)}", "derivation": "\\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} = - \\sin{(\\mathbf{E} - \\mathbf{J}_f)} and \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} - \\sin{(\\mathbf{E} - \\mathbf{J}_f)} = - 2 \\sin{(\\mathbf{E} - \\mathbf{J}_f)} and (\\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} - \\sin{(\\mathbf{E} - \\mathbf{J}_f)}) \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} = - 2 \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} \\sin{(\\mathbf{E} - \\mathbf{J}_f)} and - (\\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} - \\sin{(\\mathbf{E} - \\mathbf{J}_f)}) \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} \\sin{(\\mathbf{E} - \\mathbf{J}_f)} = 2 \\varphi^{*}{(\\mathbf{E},\\mathbf{J}_f)} \\sin^{2}{(\\mathbf{E} - \\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["add", 1, "Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["times", 2, "Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Add(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))), Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["times", 3, "Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))), Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\eta{(m_{s})} = m_{s}, then obtain ((\\eta{(m_{s})} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} - 1 = ((m_{s} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} - 1", "derivation": "\\eta{(m_{s})} = m_{s} and \\eta{(m_{s})} + \\frac{d}{d m_{s}} m_{s} = m_{s} + \\frac{d}{d m_{s}} m_{s} and (\\eta{(m_{s})} + \\frac{d}{d m_{s}} m_{s})^{m_{s}} = (m_{s} + \\frac{d}{d m_{s}} m_{s})^{m_{s}} and ((\\eta{(m_{s})} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} = ((m_{s} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} and ((\\eta{(m_{s})} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} - 1 = ((m_{s} + \\frac{d}{d m_{s}} m_{s})^{m_{s}})^{m_{s}} - 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\eta')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], [["add", 1, "Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\eta')(Symbol('m_s', commutative=True)), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Symbol('m_s', commutative=True), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Add(Function('\\\\eta')(Symbol('m_s', commutative=True)), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)))"], [["power", 3, "Symbol('m_s', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\eta')(Symbol('m_s', commutative=True)), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Add(Symbol('m_s', commutative=True), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Pow(Pow(Add(Function('\\\\eta')(Symbol('m_s', commutative=True)), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Integer(-1)), Add(Pow(Pow(Add(Symbol('m_s', commutative=True), Derivative(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{S}{(n_{1})} = e^{n_{1}}, then derive \\int \\mathbf{S}{(n_{1})} dn_{1} = f^{\\prime} + e^{n_{1}}, then obtain (- \\sigma_x + f^{\\prime} + e^{n_{1}})^{2 f^{\\prime}} = (e^{n_{1}})^{2 f^{\\prime}}", "derivation": "\\mathbf{S}{(n_{1})} = e^{n_{1}} and \\int \\mathbf{S}{(n_{1})} dn_{1} = \\int e^{n_{1}} dn_{1} and \\int \\mathbf{S}{(n_{1})} dn_{1} = f^{\\prime} + e^{n_{1}} and f^{\\prime} + e^{n_{1}} = \\int e^{n_{1}} dn_{1} and f^{\\prime} + 2 e^{n_{1}} - \\int e^{n_{1}} dn_{1} = e^{n_{1}} and (f^{\\prime} + 2 e^{n_{1}} - \\int e^{n_{1}} dn_{1})^{f^{\\prime}} = (e^{n_{1}})^{f^{\\prime}} and (f^{\\prime} + 2 e^{n_{1}} - \\int e^{n_{1}} dn_{1})^{2 f^{\\prime}} = (e^{n_{1}})^{2 f^{\\prime}} and (- \\sigma_x + f^{\\prime} + e^{n_{1}})^{2 f^{\\prime}} = (e^{n_{1}})^{2 f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), exp(Symbol('n_1', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))), exp(Symbol('n_1', commutative=True)))"], [["power", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), exp(Symbol('n_1', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))), Symbol('f^{\\\\prime}', commutative=True)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 6, 2], "Equality(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), exp(Symbol('n_1', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))), Pow(exp(Symbol('n_1', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('n_1', commutative=True))), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))), Pow(exp(Symbol('n_1', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\Omega)} = \\sin{(\\Omega)}, then obtain (\\sin^{2}{(\\Omega)})^{\\Omega} \\mathbf{J}^{2}{(\\Omega)} \\sin^{2}{(\\Omega)} = (\\sin^{2}{(\\Omega)})^{\\Omega} \\mathbf{J}^{3}{(\\Omega)} \\sin{(\\Omega)}", "derivation": "\\mathbf{J}{(\\Omega)} = \\sin{(\\Omega)} and \\mathbf{J}{(\\Omega)} \\sin{(\\Omega)} = \\sin^{2}{(\\Omega)} and (\\mathbf{J}{(\\Omega)} \\sin{(\\Omega)})^{\\Omega} = (\\sin^{2}{(\\Omega)})^{\\Omega} and \\mathbf{J}^{2}{(\\Omega)} \\sin^{2}{(\\Omega)} = \\mathbf{J}{(\\Omega)} \\sin^{3}{(\\Omega)} and \\mathbf{J}^{3}{(\\Omega)} \\sin{(\\Omega)} = \\mathbf{J}{(\\Omega)} \\sin^{3}{(\\Omega)} and \\mathbf{J}^{2}{(\\Omega)} \\sin^{2}{(\\Omega)} = \\mathbf{J}^{3}{(\\Omega)} \\sin{(\\Omega)} and (\\mathbf{J}{(\\Omega)} \\sin{(\\Omega)})^{\\Omega} \\mathbf{J}^{2}{(\\Omega)} \\sin^{2}{(\\Omega)} = (\\mathbf{J}{(\\Omega)} \\sin{(\\Omega)})^{\\Omega} \\mathbf{J}^{3}{(\\Omega)} \\sin{(\\Omega)} and (\\sin^{2}{(\\Omega)})^{\\Omega} \\mathbf{J}^{2}{(\\Omega)} \\sin^{2}{(\\Omega)} = (\\sin^{2}{(\\Omega)})^{\\Omega} \\mathbf{J}^{3}{(\\Omega)} \\sin{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2)), Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(3)), sin(Symbol('\\\\Omega', commutative=True))), Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(3)), sin(Symbol('\\\\Omega', commutative=True))))"], [["times", 6, "Pow(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Pow(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(3)), sin(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Pow(Pow(sin(Symbol('\\\\Omega', commutative=True)), Integer(2)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\Omega', commutative=True)), Integer(3)), sin(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\Psi_{\\lambda})} = \\sin{(\\sin{(\\Psi_{\\lambda})})}, then obtain \\frac{d^{2}}{d \\Psi_{\\lambda}^{2}} \\Psi_{\\lambda} \\dot{x}{(\\Psi_{\\lambda})} = \\frac{d^{2}}{d \\Psi_{\\lambda}^{2}} \\Psi_{\\lambda} \\sin{(\\sin{(\\Psi_{\\lambda})})}", "derivation": "\\dot{x}{(\\Psi_{\\lambda})} = \\sin{(\\sin{(\\Psi_{\\lambda})})} and \\Psi_{\\lambda} \\dot{x}{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\sin{(\\sin{(\\Psi_{\\lambda})})} and \\frac{d}{d \\Psi_{\\lambda}} \\Psi_{\\lambda} \\dot{x}{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\Psi_{\\lambda} \\sin{(\\sin{(\\Psi_{\\lambda})})} and \\frac{d^{2}}{d \\Psi_{\\lambda}^{2}} \\Psi_{\\lambda} \\dot{x}{(\\Psi_{\\lambda})} = \\frac{d^{2}}{d \\Psi_{\\lambda}^{2}} \\Psi_{\\lambda} \\sin{(\\sin{(\\Psi_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(2))), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mu{(\\varepsilon,y)} = e^{\\frac{y}{\\varepsilon}}, then obtain 0 = - (- \\mu^{y}{(\\varepsilon,y)} + (e^{\\frac{y}{\\varepsilon}})^{y}) \\mu^{y}{(\\varepsilon,y)}", "derivation": "\\mu{(\\varepsilon,y)} = e^{\\frac{y}{\\varepsilon}} and \\mu^{y}{(\\varepsilon,y)} = (e^{\\frac{y}{\\varepsilon}})^{y} and 0 = - \\mu^{y}{(\\varepsilon,y)} + (e^{\\frac{y}{\\varepsilon}})^{y} and 0 = - (- \\mu^{y}{(\\varepsilon,y)} + (e^{\\frac{y}{\\varepsilon}})^{y}) \\mu^{y}{(\\varepsilon,y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), exp(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Pow(exp(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Pow(exp(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True))), Pow(Function('\\\\mu')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(g)} = \\log{(g)}, then obtain \\log{(g)} + 1 = \\log{(g)} + \\frac{\\iiint \\log{(g)} dg dg dg}{\\iiint \\mathbf{B}{(g)} dg dg dg}", "derivation": "\\mathbf{B}{(g)} = \\log{(g)} and \\int \\mathbf{B}{(g)} dg = \\int \\log{(g)} dg and \\iint \\mathbf{B}{(g)} dg dg = \\iint \\log{(g)} dg dg and \\iiint \\mathbf{B}{(g)} dg dg dg = \\iiint \\log{(g)} dg dg dg and 1 = \\frac{\\iiint \\log{(g)} dg dg dg}{\\iiint \\mathbf{B}{(g)} dg dg dg} and \\log{(g)} + 1 = \\log{(g)} + \\frac{\\iiint \\log{(g)} dg dg dg}{\\iiint \\mathbf{B}{(g)} dg dg dg}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["divide", 4, "Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["add", 5, "log(Symbol('g', commutative=True))"], "Equality(Add(log(Symbol('g', commutative=True)), Integer(1)), Add(log(Symbol('g', commutative=True)), Mul(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given U{(s,\\varphi)} = \\cos{(\\varphi s)} and \\hat{p}{(s,\\varphi)} = - \\int U{(s,\\varphi)} ds, then obtain 0^{s} = (\\hat{p}{(s,\\varphi)} + \\int \\cos{(\\varphi s)} ds)^{s}", "derivation": "U{(s,\\varphi)} = \\cos{(\\varphi s)} and \\int U{(s,\\varphi)} ds = \\int \\cos{(\\varphi s)} ds and 0 = - \\int U{(s,\\varphi)} ds + \\int \\cos{(\\varphi s)} ds and \\hat{p}{(s,\\varphi)} = - \\int U{(s,\\varphi)} ds and 0 = \\hat{p}{(s,\\varphi)} + \\int \\cos{(\\varphi s)} ds and 0^{s} = (\\hat{p}{(s,\\varphi)} + \\int \\cos{(\\varphi s)} ds)^{s}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('U')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["minus", 2, "Integral(Function('U')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('U')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integral(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integral(Function('U')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Function('\\\\hat{p}')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["power", 5, "Symbol('s', commutative=True)"], "Equality(Pow(Integer(0), Symbol('s', commutative=True)), Pow(Add(Function('\\\\hat{p}')(Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Symbol('s', commutative=True)))"]]}, {"prompt": "Given t{(C_{1},\\mathbf{v})} = \\mathbf{v} e^{C_{1}}, then obtain \\int 2 t^{C_{1}}{(C_{1},\\mathbf{v})} d\\mathbf{v} = \\int ((\\mathbf{v} e^{C_{1}})^{C_{1}} + t^{C_{1}}{(C_{1},\\mathbf{v})}) d\\mathbf{v}", "derivation": "t{(C_{1},\\mathbf{v})} = \\mathbf{v} e^{C_{1}} and t^{C_{1}}{(C_{1},\\mathbf{v})} = (\\mathbf{v} e^{C_{1}})^{C_{1}} and 2 t^{C_{1}}{(C_{1},\\mathbf{v})} = (\\mathbf{v} e^{C_{1}})^{C_{1}} + t^{C_{1}}{(C_{1},\\mathbf{v})} and \\int 2 t^{C_{1}}{(C_{1},\\mathbf{v})} d\\mathbf{v} = \\int ((\\mathbf{v} e^{C_{1}})^{C_{1}} + t^{C_{1}}{(C_{1},\\mathbf{v})}) d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], [["add", 2, "Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True))), Add(Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Function('t')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('C_1', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)} = \\frac{\\phi}{\\pi}, then obtain \\int (- \\phi + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)})^{\\pi} d\\phi = \\int (- \\phi + \\frac{\\phi}{\\pi})^{\\pi} d\\phi", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)} = \\frac{\\phi}{\\pi} and - \\phi + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)} = - \\phi + \\frac{\\phi}{\\pi} and (- \\phi + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)})^{\\pi} = (- \\phi + \\frac{\\phi}{\\pi})^{\\pi} and \\int (- \\phi + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,\\pi)})^{\\pi} d\\phi = \\int (- \\phi + \\frac{\\phi}{\\pi})^{\\pi} d\\phi", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Symbol('\\\\pi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(C,\\dot{z},\\rho_f)} = C^{\\rho_f} \\dot{z}, then derive \\frac{\\frac{\\partial}{\\partial C} \\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f} = \\frac{C^{\\rho_f} \\dot{z}}{C}, then obtain \\frac{\\frac{\\partial}{\\partial C} \\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f \\ddot{x}{(C,\\dot{z},\\rho_f)}} = \\frac{C^{\\rho_f} \\dot{z}}{C \\ddot{x}{(C,\\dot{z},\\rho_f)}}", "derivation": "\\ddot{x}{(C,\\dot{z},\\rho_f)} = C^{\\rho_f} \\dot{z} and \\frac{\\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f} = \\frac{C^{\\rho_f} \\dot{z}}{\\rho_f} and \\frac{\\partial}{\\partial C} \\frac{\\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f} = \\frac{\\partial}{\\partial C} \\frac{C^{\\rho_f} \\dot{z}}{\\rho_f} and \\frac{\\frac{\\partial}{\\partial C} \\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f} = \\frac{C^{\\rho_f} \\dot{z}}{C} and \\frac{\\frac{\\partial}{\\partial C} \\ddot{x}{(C,\\dot{z},\\rho_f)}}{\\rho_f \\ddot{x}{(C,\\dot{z},\\rho_f)}} = \\frac{C^{\\rho_f} \\dot{z}}{C \\ddot{x}{(C,\\dot{z},\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('C', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 4, "Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('C', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(C)} = \\sin{(\\cos{(C)})}, then obtain (\\int \\mathbf{B}{(C)} dC)^{C} - \\frac{\\sin{(\\cos{(C)})}}{\\mathbf{B}{(C)}} = (\\int \\sin{(\\cos{(C)})} dC)^{C} - \\frac{\\sin{(\\cos{(C)})}}{\\mathbf{B}{(C)}}", "derivation": "\\mathbf{B}{(C)} = \\sin{(\\cos{(C)})} and \\int \\mathbf{B}{(C)} dC = \\int \\sin{(\\cos{(C)})} dC and (\\int \\mathbf{B}{(C)} dC)^{C} = (\\int \\sin{(\\cos{(C)})} dC)^{C} and (\\int \\mathbf{B}{(C)} dC)^{C} - \\frac{\\sin{(\\cos{(C)})}}{\\mathbf{B}{(C)}} = (\\int \\sin{(\\cos{(C)})} dC)^{C} - \\frac{\\sin{(\\cos{(C)})}}{\\mathbf{B}{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), sin(cos(Symbol('C', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(sin(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(sin(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["minus", 3, "Mul(Pow(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Integer(-1)), sin(cos(Symbol('C', commutative=True))))"], "Equality(Add(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Integer(-1)), sin(cos(Symbol('C', commutative=True))))), Add(Pow(Integral(sin(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('C', commutative=True)), Integer(-1)), sin(cos(Symbol('C', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(P_{e},\\mathbf{M})} = - P_{e} + \\log{(\\mathbf{M})}, then obtain - \\frac{\\rho_{f}^{P_{e}}{(P_{e},\\mathbf{M})} + \\log{(\\mathbf{M})}}{P_{e}} = - \\frac{(- P_{e} + \\log{(\\mathbf{M})})^{P_{e}} + \\log{(\\mathbf{M})}}{P_{e}}", "derivation": "\\rho_{f}{(P_{e},\\mathbf{M})} = - P_{e} + \\log{(\\mathbf{M})} and \\rho_{f}^{P_{e}}{(P_{e},\\mathbf{M})} = (- P_{e} + \\log{(\\mathbf{M})})^{P_{e}} and \\rho_{f}^{P_{e}}{(P_{e},\\mathbf{M})} + \\log{(\\mathbf{M})} = (- P_{e} + \\log{(\\mathbf{M})})^{P_{e}} + \\log{(\\mathbf{M})} and - \\frac{\\rho_{f}^{P_{e}}{(P_{e},\\mathbf{M})} + \\log{(\\mathbf{M})}}{P_{e}} = - \\frac{(- P_{e} + \\log{(\\mathbf{M})})^{P_{e}} + \\log{(\\mathbf{M})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('P_e', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('P_e', commutative=True)))"], [["add", 2, "log(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Pow(Function('\\\\rho_f')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('P_e', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Pow(Function('\\\\rho_f')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('P_e', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\phi_2,\\mathbf{p})} = \\frac{\\phi_2}{\\mathbf{p}}, then obtain \\cos{(\\pi{(\\phi_2,\\mathbf{p})} + \\int \\pi{(\\phi_2,\\mathbf{p})} d\\mathbf{p})} = \\cos{(\\pi{(\\phi_2,\\mathbf{p})} + \\int \\frac{\\phi_2}{\\mathbf{p}} d\\mathbf{p})}", "derivation": "\\pi{(\\phi_2,\\mathbf{p})} = \\frac{\\phi_2}{\\mathbf{p}} and \\int \\pi{(\\phi_2,\\mathbf{p})} d\\mathbf{p} = \\int \\frac{\\phi_2}{\\mathbf{p}} d\\mathbf{p} and \\pi{(\\phi_2,\\mathbf{p})} + \\int \\pi{(\\phi_2,\\mathbf{p})} d\\mathbf{p} = \\pi{(\\phi_2,\\mathbf{p})} + \\int \\frac{\\phi_2}{\\mathbf{p}} d\\mathbf{p} and \\cos{(\\pi{(\\phi_2,\\mathbf{p})} + \\int \\pi{(\\phi_2,\\mathbf{p})} d\\mathbf{p})} = \\cos{(\\pi{(\\phi_2,\\mathbf{p})} + \\int \\frac{\\phi_2}{\\mathbf{p}} d\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 2, "Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))), cos(Add(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given \\Psi{(\\varphi)} = e^{\\varphi} and Z{(\\varphi)} = e^{\\varphi} and Q{(\\varphi)} = e^{\\varphi}, then obtain Q{(\\varphi)} \\frac{d}{d \\varphi} \\Psi{(\\varphi)} = e^{\\varphi} \\frac{d}{d \\varphi} \\Psi{(\\varphi)}", "derivation": "\\Psi{(\\varphi)} = e^{\\varphi} and Z{(\\varphi)} = e^{\\varphi} and \\frac{d}{d \\varphi} Z{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and Z{(\\varphi)} = \\Psi{(\\varphi)} and Q{(\\varphi)} = e^{\\varphi} and \\frac{d}{d \\varphi} \\Psi{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and Q{(\\varphi)} \\frac{d}{d \\varphi} e^{\\varphi} = e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} and Q{(\\varphi)} \\frac{d}{d \\varphi} \\Psi{(\\varphi)} = e^{\\varphi} \\frac{d}{d \\varphi} \\Psi{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('Z')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Z')(Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["times", 5, "Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Mul(Function('Q')(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Function('Q')(Symbol('\\\\varphi', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(t,v_{x})} = \\sin{(t + v_{x})}, then obtain (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + 2 \\sin{(t + v_{x})} + 1 = 2 (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + 2 \\sin{(t + v_{x})}", "derivation": "\\lambda{(t,v_{x})} = \\sin{(t + v_{x})} and \\lambda{(t,v_{x})} - \\sin{(t + v_{x})} = 0 and (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} = 0^{v_{x}} and (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + \\sin{(t + v_{x})} = 0^{v_{x}} + \\sin{(t + v_{x})} and \\sin{(t + v_{x})} + 1 = (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + \\sin{(t + v_{x})} and (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + 2 \\sin{(t + v_{x})} + 1 = 2 (\\lambda{(t,v_{x})} - \\sin{(t + v_{x})})^{v_{x}} + 2 \\sin{(t + v_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))"], [["minus", 1, "sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)), Pow(Integer(0), Symbol('v_x', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))"], "Equality(Add(Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True)))), Add(Pow(Integer(0), Symbol('v_x', commutative=True)), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))), Integer(1)), Add(Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True)))))"], [["add", 5, "Add(Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))"], "Equality(Add(Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)), Mul(Integer(2), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True)))), Integer(1)), Add(Mul(Integer(2), Pow(Add(Function('\\\\lambda')(Symbol('t', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True))), Mul(Integer(2), sin(Add(Symbol('t', commutative=True), Symbol('v_x', commutative=True))))))"]]}, {"prompt": "Given \\dot{z}{(E)} = \\frac{d}{d E} e^{E}, then derive \\dot{z}{(E)} = e^{E}, then obtain - e^{E} + \\frac{d}{d E} e^{E} (\\frac{d}{d E} e^{E})^{E} - \\frac{1}{f} = e^{E} (\\frac{d}{d E} e^{E})^{E} - e^{E} - \\frac{1}{f}", "derivation": "\\dot{z}{(E)} = \\frac{d}{d E} e^{E} and \\dot{z}{(E)} = e^{E} and \\dot{z}^{E}{(E)} = (e^{E})^{E} and \\dot{z}{(E)} (e^{E})^{E} = e^{E} (e^{E})^{E} and \\dot{z}{(E)} \\dot{z}^{E}{(E)} = \\dot{z}^{E}{(E)} e^{E} and \\frac{d}{d E} e^{E} (\\frac{d}{d E} e^{E})^{E} = e^{E} (\\frac{d}{d E} e^{E})^{E} and - e^{E} + \\frac{d}{d E} e^{E} (\\frac{d}{d E} e^{E})^{E} - \\frac{1}{f} = e^{E} (\\frac{d}{d E} e^{E})^{E} - e^{E} - \\frac{1}{f}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{z}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Mul(exp(Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Pow(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Mul(Pow(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Mul(exp(Symbol('E', commutative=True)), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["minus", 6, "Add(exp(Symbol('E', commutative=True)), Pow(Symbol('f', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('E', commutative=True))), Mul(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)))), Add(Mul(exp(Symbol('E', commutative=True)), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Mul(Integer(-1), exp(Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\ddot{x}{(U)} = \\log{(U)}, then obtain \\log{(U)} + 1 - \\frac{1}{\\ddot{x}{(U)}} = 0^{U} + \\log{(U)} - \\frac{1}{\\ddot{x}{(U)}}", "derivation": "\\ddot{x}{(U)} = \\log{(U)} and 0 = - \\ddot{x}{(U)} + \\log{(U)} and \\ddot{x}{(U)} - \\log{(U)} = 0 and (\\ddot{x}{(U)} - \\log{(U)})^{U} = 0^{U} and (\\ddot{x}{(U)} - \\log{(U)})^{U} - \\frac{\\ddot{x}^{U}{(U)}}{\\log{(U)}} = 0^{U} - \\frac{\\ddot{x}^{U}{(U)}}{\\log{(U)}} and 1 - \\frac{1}{\\log{(U)}} = (\\ddot{x}{(U)} - \\log{(U)})^{U} - \\frac{1}{\\log{(U)}} and 1 - \\frac{1}{\\ddot{x}{(U)}} = 0^{U} - \\frac{1}{\\ddot{x}{(U)}} and \\log{(U)} + 1 - \\frac{1}{\\ddot{x}{(U)}} = 0^{U} + \\log{(U)} - \\frac{1}{\\ddot{x}{(U)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["minus", 1, "Function('\\\\ddot{x}')(Symbol('U', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Integer(0), Symbol('U', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(-1)))"], "Equality(Add(Pow(Add(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(-1)))), Add(Pow(Integer(0), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(log(Symbol('U', commutative=True)), Integer(-1)))), Add(Pow(Add(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('U', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Integer(-1)))), Add(Pow(Integer(0), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Integer(-1)))))"], [["minus", 7, "Mul(Integer(-1), log(Symbol('U', commutative=True)))"], "Equality(Add(log(Symbol('U', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Integer(-1)))), Add(Pow(Integer(0), Symbol('U', commutative=True)), log(Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{s}{(h,\\hbar)} = - \\hbar + h, then obtain \\frac{\\partial}{\\partial h} \\mathbf{s}{(h,\\hbar)} + \\iint \\mathbf{s}{(h,\\hbar)} dh dh = \\frac{\\partial}{\\partial h} (- \\hbar + h) + \\iint \\mathbf{s}{(h,\\hbar)} dh dh", "derivation": "\\mathbf{s}{(h,\\hbar)} = - \\hbar + h and \\int \\mathbf{s}{(h,\\hbar)} dh = \\int (- \\hbar + h) dh and \\iint \\mathbf{s}{(h,\\hbar)} dh dh = \\iint (- \\hbar + h) dh dh and \\frac{\\partial}{\\partial h} \\mathbf{s}{(h,\\hbar)} = \\frac{\\partial}{\\partial h} (- \\hbar + h) and \\frac{\\partial}{\\partial h} \\mathbf{s}{(h,\\hbar)} + \\iint (- \\hbar + h) dh dh = \\frac{\\partial}{\\partial h} (- \\hbar + h) + \\iint (- \\hbar + h) dh dh and \\frac{\\partial}{\\partial h} \\mathbf{s}{(h,\\hbar)} + \\iint \\mathbf{s}{(h,\\hbar)} dh dh = \\frac{\\partial}{\\partial h} (- \\hbar + h) + \\iint \\mathbf{s}{(h,\\hbar)} dh dh", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["add", 4, "Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{s}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(M_{E})} = e^{M_{E}}, then obtain \\frac{d}{d M_{E}} M_{E} e^{M_{E}} \\int M_{E} \\operatorname{E_{x}}{(M_{E})} dM_{E} = \\frac{d}{d M_{E}} M_{E} e^{M_{E}} \\int M_{E} e^{M_{E}} dM_{E}", "derivation": "\\operatorname{E_{x}}{(M_{E})} = e^{M_{E}} and M_{E} \\operatorname{E_{x}}{(M_{E})} = M_{E} e^{M_{E}} and \\frac{d}{d M_{E}} M_{E} \\operatorname{E_{x}}{(M_{E})} = \\frac{d}{d M_{E}} M_{E} e^{M_{E}} and \\int M_{E} \\operatorname{E_{x}}{(M_{E})} dM_{E} = \\int M_{E} e^{M_{E}} dM_{E} and \\frac{d}{d M_{E}} M_{E} \\operatorname{E_{x}}{(M_{E})} \\int M_{E} \\operatorname{E_{x}}{(M_{E})} dM_{E} = \\frac{d}{d M_{E}} M_{E} \\operatorname{E_{x}}{(M_{E})} \\int M_{E} e^{M_{E}} dM_{E} and \\frac{d}{d M_{E}} M_{E} e^{M_{E}} \\int M_{E} \\operatorname{E_{x}}{(M_{E})} dM_{E} = \\frac{d}{d M_{E}} M_{E} e^{M_{E}} \\int M_{E} e^{M_{E}} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["times", 4, "Derivative(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integral(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Mul(Derivative(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integral(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Derivative(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integral(Mul(Symbol('M_E', commutative=True), Function('E_x')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Mul(Derivative(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integral(Mul(Symbol('M_E', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})} = - \\dot{\\mathbf{r}} + \\eta^{\\prime} - q, then obtain \\frac{d}{d \\dot{\\mathbf{r}}} \\eta^{\\prime} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\frac{\\eta^{\\prime} (- \\dot{\\mathbf{r}} + \\eta^{\\prime} - q)}{L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})}}", "derivation": "L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})} = - \\dot{\\mathbf{r}} + \\eta^{\\prime} - q and 1 = \\frac{- \\dot{\\mathbf{r}} + \\eta^{\\prime} - q}{L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})}} and \\eta^{\\prime} = \\frac{\\eta^{\\prime} (- \\dot{\\mathbf{r}} + \\eta^{\\prime} - q)}{L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})}} and \\frac{d}{d \\dot{\\mathbf{r}}} \\eta^{\\prime} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\frac{\\eta^{\\prime} (- \\dot{\\mathbf{r}} + \\eta^{\\prime} - q)}{L{(\\dot{\\mathbf{r}},q,\\eta^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["divide", 1, "Function('L')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Function('L')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))))"], [["times", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Function('L')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Symbol('\\\\eta^{\\\\prime}', commutative=True), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Function('L')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})} = \\log{(\\dot{z})} and f{(\\dot{z})} = \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})} + \\log{(\\dot{z})}, then obtain \\frac{f{(\\dot{z})}}{l} = \\frac{2 \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})}}{l}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\dot{z})} = \\log{(\\dot{z})} and f{(\\dot{z})} = \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})} + \\log{(\\dot{z})} and f{(\\dot{z})} = 2 \\log{(\\dot{z})} and f{(\\dot{z})} = 2 \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})} and \\frac{f{(\\dot{z})}}{l} = \\frac{2 \\operatorname{V_{\\mathbf{E}}}{(\\dot{z})}}{l}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 4, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), Pow(Symbol('l', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(z^{*},k)} = k z^{*}, then obtain \\iint \\frac{1}{2} dz^{*} dk = \\iint \\frac{k z^{*}}{2 \\psi^{*}{(z^{*},k)}} dz^{*} dk", "derivation": "\\psi^{*}{(z^{*},k)} = k z^{*} and \\frac{1}{2} = \\frac{k z^{*}}{2 \\psi^{*}{(z^{*},k)}} and \\int \\frac{1}{2} dz^{*} = \\int \\frac{k z^{*}}{2 \\psi^{*}{(z^{*},k)}} dz^{*} and \\iint \\frac{1}{2} dz^{*} dk = \\iint \\frac{k z^{*}}{2 \\psi^{*}{(z^{*},k)}} dz^{*} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('z^*', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('k', commutative=True), Symbol('z^*', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Function('\\\\psi^*')(Symbol('z^*', commutative=True), Symbol('k', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Symbol('k', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\psi^*')(Symbol('z^*', commutative=True), Symbol('k', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Rational(1, 2), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Rational(1, 2), Symbol('k', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\psi^*')(Symbol('z^*', commutative=True), Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('z^*', commutative=True))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Rational(1, 2), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Rational(1, 2), Symbol('k', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\psi^*')(Symbol('z^*', commutative=True), Symbol('k', commutative=True)), Integer(-1))), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} = \\frac{e^{a^{\\dagger}}}{\\hat{x}}, then obtain (\\frac{\\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} e^{a^{\\dagger}}}{\\hat{x}})^{a^{\\dagger}} + \\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} = (\\frac{e^{2 a^{\\dagger}}}{\\hat{x}^{2}})^{a^{\\dagger}} + \\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})}", "derivation": "\\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} = \\frac{e^{a^{\\dagger}}}{\\hat{x}} and \\frac{\\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} e^{a^{\\dagger}}}{\\hat{x}} = \\frac{e^{2 a^{\\dagger}}}{\\hat{x}^{2}} and (\\frac{\\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} e^{a^{\\dagger}}}{\\hat{x}})^{a^{\\dagger}} = (\\frac{e^{2 a^{\\dagger}}}{\\hat{x}^{2}})^{a^{\\dagger}} and (\\frac{\\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} e^{a^{\\dagger}}}{\\hat{x}})^{a^{\\dagger}} + \\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})} = (\\frac{e^{2 a^{\\dagger}}}{\\hat{x}^{2}})^{a^{\\dagger}} + \\operatorname{A_{z}}{(\\hat{x},a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 3, "Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Function('A_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain \\theta_2 = \\frac{\\theta_2 (\\mathbf{r} + \\sin{(\\theta_2)})}{\\int \\pi{(\\theta_2)} d\\theta_2}", "derivation": "\\pi{(\\theta_2)} = \\cos{(\\theta_2)} and \\int \\pi{(\\theta_2)} d\\theta_2 = \\int \\cos{(\\theta_2)} d\\theta_2 and 1 = \\frac{\\int \\cos{(\\theta_2)} d\\theta_2}{\\int \\pi{(\\theta_2)} d\\theta_2} and \\theta_2 = \\frac{\\theta_2 \\int \\cos{(\\theta_2)} d\\theta_2}{\\int \\pi{(\\theta_2)} d\\theta_2} and \\theta_2 = \\frac{\\theta_2 (\\mathbf{r} + \\sin{(\\theta_2)})}{\\int \\pi{(\\theta_2)} d\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["times", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Symbol('\\\\theta_2', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), Pow(Integral(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Symbol('\\\\theta_2', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Pow(Integral(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} = \\log{(\\sin{(\\hat{H}_{\\lambda})})}, then obtain \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} - 2 \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1 = - \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1", "derivation": "\\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} = \\log{(\\sin{(\\hat{H}_{\\lambda})})} and 0 = - \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} + \\log{(\\sin{(\\hat{H}_{\\lambda})})} and -1 = - \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} + \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1 and \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} - \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1 = -1 and \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} - 2 \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1 = - \\log{(\\sin{(\\hat{H}_{\\lambda})})} - 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1)))"], [["minus", 3, "Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], "Equality(Add(Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Integer(-1)), Integer(-1))"], [["minus", 4, "log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), log(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(a)} = \\frac{d}{d a} \\cos{(a)}, then obtain - \\operatorname{A_{2}}{(a)} + \\frac{d}{d a} \\operatorname{A_{2}}{(a)} = - \\operatorname{A_{2}}{(a)} - \\cos{(a)}", "derivation": "\\operatorname{A_{2}}{(a)} = \\frac{d}{d a} \\cos{(a)} and \\frac{d}{d a} \\operatorname{A_{2}}{(a)} = \\frac{d^{2}}{d a^{2}} \\cos{(a)} and - \\operatorname{A_{2}}{(a)} + \\frac{d}{d a} \\operatorname{A_{2}}{(a)} = - \\operatorname{A_{2}}{(a)} + \\frac{d^{2}}{d a^{2}} \\cos{(a)} and - \\operatorname{A_{2}}{(a)} + \\frac{d}{d a} \\operatorname{A_{2}}{(a)} = - \\operatorname{A_{2}}{(a)} - \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('a', commutative=True)), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))))"], [["minus", 2, "Function('A_2')(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_2')(Symbol('a', commutative=True))), Derivative(Function('A_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('A_2')(Symbol('a', commutative=True))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('A_2')(Symbol('a', commutative=True))), Derivative(Function('A_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('A_2')(Symbol('a', commutative=True))), Mul(Integer(-1), cos(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(A_{y})} = e^{A_{y}}, then derive \\frac{d}{d A_{y}} \\mathbf{s}{(A_{y})} + 1 = e^{A_{y}} + 1, then obtain \\mathbf{s}{(A_{y})} + 1 = e^{A_{y}} + 1", "derivation": "\\mathbf{s}{(A_{y})} = e^{A_{y}} and \\frac{d}{d A_{y}} \\mathbf{s}{(A_{y})} = \\frac{d}{d A_{y}} e^{A_{y}} and \\frac{d}{d A_{y}} \\mathbf{s}{(A_{y})} + 1 = \\frac{d}{d A_{y}} e^{A_{y}} + 1 and \\frac{d}{d A_{y}} \\mathbf{s}{(A_{y})} + 1 = e^{A_{y}} + 1 and \\frac{d}{d A_{y}} \\mathbf{s}{(A_{y})} + 1 = \\mathbf{s}{(A_{y})} + 1 and \\mathbf{s}{(A_{y})} + 1 = e^{A_{y}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('A_y', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)), Add(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('A_y', commutative=True)), Integer(1)), Add(exp(Symbol('A_y', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\eta^{\\prime}{(E_{n},W)} = \\frac{E_{n}}{W}, then obtain (- E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} \\eta^{\\prime}^{E_{n}}{(E_{n},W)})^{E_{n}} = (- E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} (\\frac{E_{n}}{W})^{E_{n}})^{E_{n}}", "derivation": "\\eta^{\\prime}{(E_{n},W)} = \\frac{E_{n}}{W} and \\eta^{\\prime}^{E_{n}}{(E_{n},W)} = (\\frac{E_{n}}{W})^{E_{n}} and \\frac{\\partial}{\\partial E_{n}} \\eta^{\\prime}^{E_{n}}{(E_{n},W)} = \\frac{\\partial}{\\partial E_{n}} (\\frac{E_{n}}{W})^{E_{n}} and - E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} \\eta^{\\prime}^{E_{n}}{(E_{n},W)} = - E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} (\\frac{E_{n}}{W})^{E_{n}} and (- E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} \\eta^{\\prime}^{E_{n}}{(E_{n},W)})^{E_{n}} = (- E_{n} - \\frac{E_{n}}{W} + \\frac{\\partial}{\\partial E_{n}} (\\frac{E_{n}}{W})^{E_{n}})^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Symbol('E_n', commutative=True)), Pow(Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Symbol('E_n', commutative=True)))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('E_n', commutative=True), Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Derivative(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Derivative(Pow(Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('E_n', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Derivative(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('E_n', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Derivative(Pow(Mul(Symbol('E_n', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given C{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain - \\frac{\\log{(E_{\\lambda})}}{C{(E_{\\lambda})}} = -1", "derivation": "C{(E_{\\lambda})} = \\log{(E_{\\lambda})} and 0 = - C{(E_{\\lambda})} + \\log{(E_{\\lambda})} and - \\log{(E_{\\lambda})} = - C{(E_{\\lambda})} and - \\frac{\\log{(E_{\\lambda})}}{C{(E_{\\lambda})}} = -1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 1, "Function('C')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C')(Symbol('E_{\\\\lambda}', commutative=True))), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 2, "log(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), log(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Function('C')(Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Function('C')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('C')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))"]]}, {"prompt": "Given i{(t_{2})} = \\frac{d}{d t_{2}} \\cos{(t_{2})}, then obtain - \\frac{t_{2} i{(t_{2})}}{\\sin{(t_{2})}} = t_{2}", "derivation": "i{(t_{2})} = \\frac{d}{d t_{2}} \\cos{(t_{2})} and t_{2} i{(t_{2})} = t_{2} \\frac{d}{d t_{2}} \\cos{(t_{2})} and \\frac{t_{2} i{(t_{2})}}{\\frac{d}{d t_{2}} \\cos{(t_{2})}} = t_{2} and - \\frac{t_{2} i{(t_{2})}}{\\sin{(t_{2})}} = t_{2}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('t_2', commutative=True)), Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["times", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Function('i')(Symbol('t_2', commutative=True))), Mul(Symbol('t_2', commutative=True), Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["divide", 2, "Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('t_2', commutative=True), Function('i')(Symbol('t_2', commutative=True)), Pow(Derivative(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1))), Symbol('t_2', commutative=True))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Symbol('t_2', commutative=True), Function('i')(Symbol('t_2', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Integer(-1))), Symbol('t_2', commutative=True))"]]}, {"prompt": "Given B{(T)} = \\cos{(\\cos{(T)})} and \\omega{(T)} = \\int \\cos{(\\cos{(T)})} dT, then obtain \\frac{d}{d T} \\int B{(T)} dT = \\frac{d}{d T} \\omega{(T)}", "derivation": "B{(T)} = \\cos{(\\cos{(T)})} and \\int B{(T)} dT = \\int \\cos{(\\cos{(T)})} dT and \\frac{d}{d T} \\int B{(T)} dT = \\frac{d}{d T} \\int \\cos{(\\cos{(T)})} dT and \\omega{(T)} = \\int \\cos{(\\cos{(T)})} dT and \\frac{d}{d T} \\int B{(T)} dT = \\frac{d}{d T} \\omega{(T)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('T', commutative=True)), cos(cos(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('B')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(cos(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Integral(Function('B')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(cos(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('T', commutative=True)), Integral(cos(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integral(Function('B')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Function('\\\\omega')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{F})} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F}, then derive - z + \\phi_{1}{(\\mathbf{F})} - \\sin{(\\mathbf{F})} = 0, then obtain \\int (- z - \\sin{(\\mathbf{F})} + \\int \\cos{(\\mathbf{F})} d\\mathbf{F}) dz + \\int \\cos{(\\mathbf{F})} d\\mathbf{F} = \\int 0 dz + \\int \\cos{(\\mathbf{F})} d\\mathbf{F}", "derivation": "\\phi_{1}{(\\mathbf{F})} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and \\phi_{1}{(\\mathbf{F})} - \\int \\cos{(\\mathbf{F})} d\\mathbf{F} = 0 and - z + \\phi_{1}{(\\mathbf{F})} - \\sin{(\\mathbf{F})} = 0 and \\int (- z + \\phi_{1}{(\\mathbf{F})} - \\sin{(\\mathbf{F})}) dz = \\int 0 dz and \\phi_{1}{(\\mathbf{F})} + \\int (- z + \\phi_{1}{(\\mathbf{F})} - \\sin{(\\mathbf{F})}) dz = \\phi_{1}{(\\mathbf{F})} + \\int 0 dz and \\int (- z - \\sin{(\\mathbf{F})} + \\int \\cos{(\\mathbf{F})} d\\mathbf{F}) dz + \\int \\cos{(\\mathbf{F})} d\\mathbf{F} = \\int 0 dz + \\int \\cos{(\\mathbf{F})} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 1, "Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('z', commutative=True))), Integral(Integer(0), Tuple(Symbol('z', commutative=True))))"], [["add", 4, "Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('z', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integral(Integer(0), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('z', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\phi_2,t)} = t e^{\\phi_2}, then obtain ((\\mathbf{D}^{t}{(\\phi_2,t)})^{t})^{\\phi_2} = (((t e^{\\phi_2})^{t})^{t})^{\\phi_2}", "derivation": "\\mathbf{D}{(\\phi_2,t)} = t e^{\\phi_2} and \\mathbf{D}^{t}{(\\phi_2,t)} = (t e^{\\phi_2})^{t} and (\\mathbf{D}^{t}{(\\phi_2,t)})^{t} = ((t e^{\\phi_2})^{t})^{t} and ((\\mathbf{D}^{t}{(\\phi_2,t)})^{t})^{\\phi_2} = (((t e^{\\phi_2})^{t})^{t})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Mul(Symbol('t', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Symbol('t', commutative=True)))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Pow(Mul(Symbol('t', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["power", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Pow(Mul(Symbol('t', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\operatorname{E_{n}}{(\\mathbf{J})} = \\cos{(\\mathbf{J})}, then obtain \\mathbf{J} \\cos{(\\mathbf{J})} = \\mathbf{J} \\operatorname{E_{n}}{(\\mathbf{J})}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\operatorname{E_{n}}{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\mathbf{J} \\operatorname{f^{*}}{(\\mathbf{J})} = \\mathbf{J} \\cos{(\\mathbf{J})} and \\mathbf{J} \\operatorname{f^{*}}{(\\mathbf{J})} = \\mathbf{J} \\operatorname{E_{n}}{(\\mathbf{J})} and \\mathbf{J} \\cos{(\\mathbf{J})} = \\mathbf{J} \\operatorname{E_{n}}{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True)), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True)), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(F_{g})} = \\sin{(F_{g})}, then obtain e^{3 \\mathbf{s}{(F_{g})}} \\int e^{3 \\mathbf{s}{(F_{g})}} dF_{g} = e^{3 \\mathbf{s}{(F_{g})}} \\int e^{2 \\mathbf{s}{(F_{g})} + \\sin{(F_{g})}} dF_{g}", "derivation": "\\mathbf{s}{(F_{g})} = \\sin{(F_{g})} and 2 \\mathbf{s}{(F_{g})} = \\mathbf{s}{(F_{g})} + \\sin{(F_{g})} and 3 \\mathbf{s}{(F_{g})} = 2 \\mathbf{s}{(F_{g})} + \\sin{(F_{g})} and e^{3 \\mathbf{s}{(F_{g})}} = e^{2 \\mathbf{s}{(F_{g})} + \\sin{(F_{g})}} and \\int e^{3 \\mathbf{s}{(F_{g})}} dF_{g} = \\int e^{2 \\mathbf{s}{(F_{g})} + \\sin{(F_{g})}} dF_{g} and e^{3 \\mathbf{s}{(F_{g})}} \\int e^{3 \\mathbf{s}{(F_{g})}} dF_{g} = e^{3 \\mathbf{s}{(F_{g})}} \\int e^{2 \\mathbf{s}{(F_{g})} + \\sin{(F_{g})}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), sin(Symbol('F_g', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)))), exp(Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), sin(Symbol('F_g', commutative=True)))))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))), Integral(exp(Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), sin(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["times", 5, "exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))))"], "Equality(Mul(exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)))), Integral(exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True)))), Mul(exp(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True)))), Integral(exp(Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True))), sin(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{r})} = \\int \\sin{(\\mathbf{r})} d\\mathbf{r}, then derive \\hat{\\mathbf{r}}{(\\mathbf{r})} = J - \\cos{(\\mathbf{r})}, then obtain (J - \\cos{(\\mathbf{r})})^{\\mathbf{r}} = (\\int \\sin{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{r})} = \\int \\sin{(\\mathbf{r})} d\\mathbf{r} and \\hat{\\mathbf{r}}^{\\mathbf{r}}{(\\mathbf{r})} = (\\int \\sin{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} and \\hat{\\mathbf{r}}{(\\mathbf{r})} = J - \\cos{(\\mathbf{r})} and (J - \\cos{(\\mathbf{r})})^{\\mathbf{r}} = (\\int \\sin{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hbar)} = \\sin{(\\hbar)} and r{(\\hbar)} = \\sin^{2}{(\\hbar)}, then obtain (- \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}) (\\operatorname{F_{c}}^{2}{(\\hbar)} + 1) = (- \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}) (\\sin^{2}{(\\hbar)} + 1)", "derivation": "\\operatorname{F_{c}}{(\\hbar)} = \\sin{(\\hbar)} and \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} = \\sin^{2}{(\\hbar)} and r{(\\hbar)} = \\sin^{2}{(\\hbar)} and r{(\\hbar)} = \\operatorname{F_{c}}^{2}{(\\hbar)} and \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} = r{(\\hbar)} and \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} = \\operatorname{F_{c}}^{2}{(\\hbar)} and \\operatorname{F_{c}}^{2}{(\\hbar)} = \\sin^{2}{(\\hbar)} and \\operatorname{F_{c}}^{2}{(\\hbar)} + 1 = \\sin^{2}{(\\hbar)} + 1 and (- \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}) (\\operatorname{F_{c}}^{2}{(\\hbar)} + 1) = (- \\operatorname{F_{c}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}) (\\sin^{2}{(\\hbar)} + 1)", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\hbar', commutative=True)), Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('r')(Symbol('\\\\hbar', commutative=True)), Pow(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Function('r')(Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Pow(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Pow(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["minus", 7, "Integer(-1)"], "Equality(Add(Pow(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Integer(1)), Add(Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(2)), Integer(1)))"], [["times", 8, "Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Add(Pow(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Integer(1))), Mul(Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Add(Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(2)), Integer(1))))"]]}, {"prompt": "Given \\varphi{(A)} = \\log{(\\log{(A)})}, then derive v_{2} + \\varphi{(A)} = x + \\log{(\\log{(A)})}, then obtain \\int (- 2 x + \\varphi{(A)}) dA = \\int (- v_{2} - x + \\varphi{(A)}) dA", "derivation": "\\varphi{(A)} = \\log{(\\log{(A)})} and \\frac{d}{d A} \\varphi{(A)} = \\frac{d}{d A} \\log{(\\log{(A)})} and \\int \\frac{d}{d A} \\varphi{(A)} dA = \\int \\frac{d}{d A} \\log{(\\log{(A)})} dA and v_{2} + \\varphi{(A)} = x + \\log{(\\log{(A)})} and v_{2} + \\log{(\\log{(A)})} = x + \\log{(\\log{(A)})} and v_{2} - 2 x + \\log{(\\log{(A)})} = - x + \\log{(\\log{(A)})} and v_{2} - 2 x + \\varphi{(A)} = - x + \\varphi{(A)} and - 2 x + \\varphi{(A)} = - v_{2} - x + \\varphi{(A)} and \\int (- 2 x + \\varphi{(A)}) dA = \\int (- v_{2} - x + \\varphi{(A)}) dA", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('A', commutative=True)), log(log(Symbol('A', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\varphi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))), Integral(Derivative(log(log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v_2', commutative=True), Function('\\\\varphi')(Symbol('A', commutative=True))), Add(Symbol('x', commutative=True), log(log(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('v_2', commutative=True), log(log(Symbol('A', commutative=True)))), Add(Symbol('x', commutative=True), log(log(Symbol('A', commutative=True)))))"], [["minus", 5, "Mul(Integer(2), Symbol('x', commutative=True))"], "Equality(Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), log(log(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(log(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))))"], [["minus", 7, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))))"], [["integrate", 8, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{J}_f,z)} = \\frac{\\mathbf{J}_f}{z}, then obtain - z + \\frac{\\partial}{\\partial z} \\theta_{2}^{z}{(\\mathbf{J}_f,z)} - \\frac{1}{z} = - z + \\frac{\\partial}{\\partial z} (\\frac{\\mathbf{J}_f}{z})^{z} - \\frac{1}{z}", "derivation": "\\theta_{2}{(\\mathbf{J}_f,z)} = \\frac{\\mathbf{J}_f}{z} and \\theta_{2}^{z}{(\\mathbf{J}_f,z)} = (\\frac{\\mathbf{J}_f}{z})^{z} and \\frac{\\partial}{\\partial z} \\theta_{2}^{z}{(\\mathbf{J}_f,z)} = \\frac{\\partial}{\\partial z} (\\frac{\\mathbf{J}_f}{z})^{z} and \\frac{\\partial}{\\partial z} \\theta_{2}^{z}{(\\mathbf{J}_f,z)} - \\frac{1}{z} = \\frac{\\partial}{\\partial z} (\\frac{\\mathbf{J}_f}{z})^{z} - \\frac{1}{z} and - z + \\frac{\\partial}{\\partial z} \\theta_{2}^{z}{(\\mathbf{J}_f,z)} - \\frac{1}{z} = - z + \\frac{\\partial}{\\partial z} (\\frac{\\mathbf{J}_f}{z})^{z} - \\frac{1}{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('z', commutative=True)))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 3, "Pow(Symbol('z', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))), Add(Derivative(Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))))"], [["minus", 4, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)} = \\mathbf{E} e^{\\lambda}, then obtain (- \\mathbf{E} e^{\\lambda} + \\mathbf{E} + \\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)})^{\\mathbf{E}} = \\mathbf{E}^{\\mathbf{E}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)} = \\mathbf{E} e^{\\lambda} and - \\mathbf{E} e^{\\lambda} + \\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)} = 0 and - \\mathbf{E} e^{\\lambda} + \\mathbf{E} + \\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)} = \\mathbf{E} and (- \\mathbf{E} e^{\\lambda} + \\mathbf{E} + \\operatorname{g_{\\varepsilon}}{(\\mathbf{E},\\lambda)})^{\\mathbf{E}} = \\mathbf{E}^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\lambda', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(0))"], [["add", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))"], [["power", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"]]}, {"prompt": "Given T{(a^{\\dagger},f^{\\prime},p)} = - a^{\\dagger} + p^{f^{\\prime}}, then obtain - \\frac{(a^{\\dagger})^{2} p^{f^{\\prime}} T{(a^{\\dagger},f^{\\prime},p)}}{f^{\\prime}} = - \\frac{(a^{\\dagger})^{2} p^{f^{\\prime}} (- a^{\\dagger} + p^{f^{\\prime}})}{f^{\\prime}}", "derivation": "T{(a^{\\dagger},f^{\\prime},p)} = - a^{\\dagger} + p^{f^{\\prime}} and - p^{f^{\\prime}} T{(a^{\\dagger},f^{\\prime},p)} = - p^{f^{\\prime}} (- a^{\\dagger} + p^{f^{\\prime}}) and - (a^{\\dagger})^{2} p^{f^{\\prime}} T{(a^{\\dagger},f^{\\prime},p)} = - (a^{\\dagger})^{2} p^{f^{\\prime}} (- a^{\\dagger} + p^{f^{\\prime}}) and - \\frac{(a^{\\dagger})^{2} p^{f^{\\prime}} T{(a^{\\dagger},f^{\\prime},p)}}{f^{\\prime}} = - \\frac{(a^{\\dagger})^{2} p^{f^{\\prime}} (- a^{\\dagger} + p^{f^{\\prime}})}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('T')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["times", 2, "Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))"], "Equality(Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('T')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["divide", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('T')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given Q{(\\mathbf{D},\\rho_f)} = \\mathbf{D}^{\\rho_f}, then obtain \\sin^{\\mathbf{D}}{((\\int Q{(\\mathbf{D},\\rho_f)} d\\rho_f)^{\\rho_f})} = \\sin^{\\mathbf{D}}{((\\int \\mathbf{D}^{\\rho_f} d\\rho_f)^{\\rho_f})}", "derivation": "Q{(\\mathbf{D},\\rho_f)} = \\mathbf{D}^{\\rho_f} and \\int Q{(\\mathbf{D},\\rho_f)} d\\rho_f = \\int \\mathbf{D}^{\\rho_f} d\\rho_f and (\\int Q{(\\mathbf{D},\\rho_f)} d\\rho_f)^{\\rho_f} = (\\int \\mathbf{D}^{\\rho_f} d\\rho_f)^{\\rho_f} and \\sin{((\\int Q{(\\mathbf{D},\\rho_f)} d\\rho_f)^{\\rho_f})} = \\sin{((\\int \\mathbf{D}^{\\rho_f} d\\rho_f)^{\\rho_f})} and \\sin^{\\mathbf{D}}{((\\int Q{(\\mathbf{D},\\rho_f)} d\\rho_f)^{\\rho_f})} = \\sin^{\\mathbf{D}}{((\\int \\mathbf{D}^{\\rho_f} d\\rho_f)^{\\rho_f})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), sin(Pow(Integral(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(sin(Pow(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Pow(Integral(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given V{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then derive \\frac{d}{d \\varepsilon_0} V{(\\varepsilon_0)} - 1 = \\cos{(\\varepsilon_0)} - 1, then obtain \\int \\frac{\\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} - 1}{\\cos{(\\varepsilon_0)} - 1} d\\varepsilon_0 = \\int 1 d\\varepsilon_0", "derivation": "V{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and - \\varepsilon_0 + V{(\\varepsilon_0)} = - \\varepsilon_0 + \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} (- \\varepsilon_0 + V{(\\varepsilon_0)}) = \\frac{d}{d \\varepsilon_0} (- \\varepsilon_0 + \\sin{(\\varepsilon_0)}) and \\frac{d}{d \\varepsilon_0} V{(\\varepsilon_0)} - 1 = \\cos{(\\varepsilon_0)} - 1 and \\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} - 1 = \\cos{(\\varepsilon_0)} - 1 and \\frac{\\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} - 1}{\\cos{(\\varepsilon_0)} - 1} = 1 and \\int \\frac{\\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} - 1}{\\cos{(\\varepsilon_0)} - 1} d\\varepsilon_0 = \\int 1 d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('V')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('V')(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)))"], [["divide", 5, "Add(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["integrate", 6, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Mul(Pow(Add(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} = \\frac{g^{\\prime}_{\\varepsilon}}{B V}, then derive \\int \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} dB = \\lambda - \\frac{g^{\\prime}_{\\varepsilon} \\log{(B)}}{V^{2}}, then obtain \\iint \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} dB dg^{\\prime}_{\\varepsilon} = \\int (\\lambda - \\frac{g^{\\prime}_{\\varepsilon} \\log{(B)}}{V^{2}}) dg^{\\prime}_{\\varepsilon}", "derivation": "\\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} = \\frac{g^{\\prime}_{\\varepsilon}}{B V} and \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} = \\frac{\\partial}{\\partial V} \\frac{g^{\\prime}_{\\varepsilon}}{B V} and \\int \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} dB = \\int \\frac{\\partial}{\\partial V} \\frac{g^{\\prime}_{\\varepsilon}}{B V} dB and \\int \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} dB = \\lambda - \\frac{g^{\\prime}_{\\varepsilon} \\log{(B)}}{V^{2}} and \\iint \\frac{\\partial}{\\partial V} \\operatorname{v_{1}}{(B,g^{\\prime}_{\\varepsilon},V)} dB dg^{\\prime}_{\\varepsilon} = \\int (\\lambda - \\frac{g^{\\prime}_{\\varepsilon} \\log{(B)}}{V^{2}}) dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('B', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('B', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Function('v_1')(Symbol('B', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('v_1')(Symbol('B', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-2)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('B', commutative=True)))))"], [["integrate", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(Function('v_1')(Symbol('B', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-2)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('B', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} = L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) and \\operatorname{v_{y}}{(\\hbar)} = \\hbar, then obtain \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} d\\operatorname{v_{y}}{(\\hbar)} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) d\\operatorname{v_{y}}{(\\hbar)}", "derivation": "\\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} = L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) and \\int \\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} d\\hbar = \\int L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) d\\hbar and \\operatorname{v_{y}}{(\\hbar)} = \\hbar and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} d\\hbar = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) d\\hbar and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\hat{x}{(J_{\\varepsilon},\\hbar,L_{\\varepsilon})} d\\operatorname{v_{y}}{(\\hbar)} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int L_{\\varepsilon} (J_{\\varepsilon} + \\hbar) d\\operatorname{v_{y}}{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Integral(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Function('v_y')(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Function('v_y')(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(r,t_{1})} = r - t_{1}, then derive \\frac{\\frac{\\partial}{\\partial r} b{(r,t_{1})}}{b{(r,t_{1})}} = \\frac{1}{b{(r,t_{1})}}, then obtain \\frac{\\frac{\\partial}{\\partial r} (r - t_{1})}{r - t_{1}} = \\frac{1}{r - t_{1}}", "derivation": "b{(r,t_{1})} = r - t_{1} and \\frac{\\partial}{\\partial r} b{(r,t_{1})} = \\frac{\\partial}{\\partial r} (r - t_{1}) and \\frac{\\frac{\\partial}{\\partial r} b{(r,t_{1})}}{b{(r,t_{1})}} = \\frac{\\frac{\\partial}{\\partial r} (r - t_{1})}{b{(r,t_{1})}} and \\frac{\\frac{\\partial}{\\partial r} b{(r,t_{1})}}{b{(r,t_{1})}} = \\frac{1}{b{(r,t_{1})}} and \\frac{\\frac{\\partial}{\\partial r} (r - t_{1})}{r - t_{1}} = \\frac{1}{r - t_{1}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["divide", 2, "Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Mul(Pow(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Pow(Function('b')(Symbol('r', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integer(-1)), Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Pow(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\psi^{*}{(\\theta)} = e^{\\theta} and L{(\\theta)} = e^{- \\theta}, then obtain \\sin{(L{(\\theta)} e^{\\theta})} = \\sin{(1)}", "derivation": "\\psi^{*}{(\\theta)} = e^{\\theta} and \\psi^{*}{(\\theta)} e^{- \\theta} = 1 and L{(\\theta)} = e^{- \\theta} and L{(\\theta)} \\psi^{*}{(\\theta)} = 1 and L{(\\theta)} e^{\\theta} = 1 and \\sin{(L{(\\theta)} e^{\\theta})} = \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('L')(Symbol('\\\\theta', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\theta', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('L')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True))), Integer(1))"], [["sin", 5], "Equality(sin(Mul(Function('L')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))), sin(Integer(1)))"]]}, {"prompt": "Given v{(x,C_{d})} = \\sin^{C_{d}}{(x)} and \\varepsilon{(x)} = \\sin{(x)}, then obtain 1 = (\\varepsilon^{C_{d}}{(x)})^{x} v^{- x}{(x,C_{d})}", "derivation": "v{(x,C_{d})} = \\sin^{C_{d}}{(x)} and \\varepsilon{(x)} = \\sin{(x)} and v^{x}{(x,C_{d})} = (\\sin^{C_{d}}{(x)})^{x} and 1 = (\\sin^{C_{d}}{(x)})^{x} v^{- x}{(x,C_{d})} and 1 = (\\varepsilon^{C_{d}}{(x)})^{x} v^{- x}{(x,C_{d})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('C_d', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('v')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(sin(Symbol('x', commutative=True)), Symbol('C_d', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 3, "Pow(Function('v')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Symbol('x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(sin(Symbol('x', commutative=True)), Symbol('C_d', commutative=True)), Symbol('x', commutative=True)), Pow(Function('v')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Mul(Pow(Pow(Function('\\\\varepsilon')(Symbol('x', commutative=True)), Symbol('C_d', commutative=True)), Symbol('x', commutative=True)), Pow(Function('v')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(f_{E},s)} = \\cos{(\\frac{s}{f_{E}})} and \\eta^{\\prime}{(f_{E},s)} = \\frac{s}{f_{E}}, then obtain (- \\cos{(\\eta^{\\prime}{(f_{E},s)})})^{s} = (- \\operatorname{P_{e}}{(f_{E},s)})^{s}", "derivation": "\\operatorname{P_{e}}{(f_{E},s)} = \\cos{(\\frac{s}{f_{E}})} and \\eta^{\\prime}{(f_{E},s)} = \\frac{s}{f_{E}} and \\operatorname{P_{e}}{(f_{E},s)} = \\cos{(\\eta^{\\prime}{(f_{E},s)})} and - \\operatorname{P_{e}}{(f_{E},s)} = - \\cos{(\\eta^{\\prime}{(f_{E},s)})} and - \\cos{(\\frac{s}{f_{E}})} = - \\cos{(\\eta^{\\prime}{(f_{E},s)})} and (- \\cos{(\\frac{s}{f_{E}})})^{s} = (- \\cos{(\\eta^{\\prime}{(f_{E},s)})})^{s} and (- \\cos{(\\frac{s}{f_{E}})})^{s} = (- \\operatorname{P_{e}}{(f_{E},s)})^{s} and (- \\cos{(\\eta^{\\prime}{(f_{E},s)})})^{s} = (- \\operatorname{P_{e}}{(f_{E},s)})^{s}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)), cos(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('P_e')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)), cos(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), cos(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), cos(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Mul(Integer(-1), cos(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)))))"], [["power", 5, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Integer(-1), cos(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), cos(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)))), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Mul(Integer(-1), cos(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Mul(Integer(-1), cos(Function('\\\\eta^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\eta)} = \\sin{(e^{\\eta})} and \\operatorname{t_{2}}{(\\eta)} = \\operatorname{C_{2}}{(\\eta)} + \\sin{(e^{\\eta})}, then obtain - \\frac{\\operatorname{t_{2}}{(\\eta)}}{3 \\sin{(e^{\\eta})}} = - \\frac{\\operatorname{C_{2}}{(\\eta)} + \\sin{(e^{\\eta})}}{3 \\sin{(e^{\\eta})}}", "derivation": "\\operatorname{C_{2}}{(\\eta)} = \\sin{(e^{\\eta})} and \\operatorname{C_{2}}{(\\eta)} - 2 \\sin{(e^{\\eta})} = - \\sin{(e^{\\eta})} and \\operatorname{t_{2}}{(\\eta)} = \\operatorname{C_{2}}{(\\eta)} + \\sin{(e^{\\eta})} and - 3 \\sin{(e^{\\eta})} = - \\operatorname{C_{2}}{(\\eta)} - 2 \\sin{(e^{\\eta})} and \\frac{\\operatorname{t_{2}}{(\\eta)}}{- \\operatorname{C_{2}}{(\\eta)} - 2 \\sin{(e^{\\eta})}} = \\frac{\\operatorname{C_{2}}{(\\eta)} + \\sin{(e^{\\eta})}}{- \\operatorname{C_{2}}{(\\eta)} - 2 \\sin{(e^{\\eta})}} and - \\frac{\\operatorname{t_{2}}{(\\eta)}}{3 \\sin{(e^{\\eta})}} = - \\frac{\\operatorname{C_{2}}{(\\eta)} + \\sin{(e^{\\eta})}}{3 \\sin{(e^{\\eta})}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True))))"], [["minus", 1, "Mul(Integer(2), sin(exp(Symbol('\\\\eta', commutative=True))))"], "Equality(Add(Function('C_2')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Integer(2), sin(exp(Symbol('\\\\eta', commutative=True))))), Mul(Integer(-1), sin(exp(Symbol('\\\\eta', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\eta', commutative=True)), Add(Function('C_2')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True)))))"], [["minus", 2, "Add(Function('C_2')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True))))"], "Equality(Mul(Integer(-1), Integer(3), sin(exp(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integer(2), sin(exp(Symbol('\\\\eta', commutative=True))))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integer(2), sin(exp(Symbol('\\\\eta', commutative=True)))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integer(2), sin(exp(Symbol('\\\\eta', commutative=True))))), Integer(-1)), Function('t_2')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integer(2), sin(exp(Symbol('\\\\eta', commutative=True))))), Integer(-1)), Add(Function('C_2')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Rational(1, 3), Function('t_2')(Symbol('\\\\eta', commutative=True)), Pow(sin(exp(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Integer(-1), Rational(1, 3), Add(Function('C_2')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True)))), Pow(sin(exp(Symbol('\\\\eta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\dot{y})} = e^{\\dot{y}} and \\varphi{(\\dot{y})} = e^{\\dot{y}} and \\rho_{f}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\operatorname{v_{z}}^{\\dot{y}}{(\\dot{y})}, then obtain \\rho_{f}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\varphi^{\\dot{y}}{(\\dot{y})}", "derivation": "\\operatorname{v_{z}}{(\\dot{y})} = e^{\\dot{y}} and \\operatorname{v_{z}}^{\\dot{y}}{(\\dot{y})} = (e^{\\dot{y}})^{\\dot{y}} and \\varphi{(\\dot{y})} = e^{\\dot{y}} and \\frac{d}{d \\dot{y}} \\operatorname{v_{z}}^{\\dot{y}}{(\\dot{y})} = \\frac{d}{d \\dot{y}} (e^{\\dot{y}})^{\\dot{y}} and \\rho_{f}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\operatorname{v_{z}}^{\\dot{y}}{(\\dot{y})} and \\frac{d}{d \\dot{y}} \\operatorname{v_{z}}^{\\dot{y}}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\varphi^{\\dot{y}}{(\\dot{y})} and \\rho_{f}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\varphi^{\\dot{y}}{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Pow(Function('v_z')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\dot{y}', commutative=True)), Derivative(Pow(Function('v_z')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Pow(Function('v_z')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\varphi')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('\\\\rho_f')(Symbol('\\\\dot{y}', commutative=True)), Derivative(Pow(Function('\\\\varphi')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(G)} = \\cos{(G)}, then obtain \\frac{(4 \\operatorname{v_{t}}^{G}{(G)} - 3 \\cos^{G}{(G)})^{G}}{G} = \\frac{(2 \\operatorname{v_{t}}^{G}{(G)} - \\cos^{G}{(G)})^{G}}{G}", "derivation": "\\operatorname{v_{t}}{(G)} = \\cos{(G)} and \\operatorname{v_{t}}^{G}{(G)} = \\cos^{G}{(G)} and 2 \\operatorname{v_{t}}^{G}{(G)} - \\cos^{G}{(G)} = \\operatorname{v_{t}}^{G}{(G)} and (2 \\operatorname{v_{t}}^{G}{(G)} - \\cos^{G}{(G)})^{G} = (\\operatorname{v_{t}}^{G}{(G)})^{G} and (4 \\operatorname{v_{t}}^{G}{(G)} - 3 \\cos^{G}{(G)})^{G} = (2 \\operatorname{v_{t}}^{G}{(G)} - \\cos^{G}{(G)})^{G} and \\frac{(4 \\operatorname{v_{t}}^{G}{(G)} - 3 \\cos^{G}{(G)})^{G}}{G} = \\frac{(2 \\operatorname{v_{t}}^{G}{(G)} - \\cos^{G}{(G)})^{G}}{G}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["add", 2, "Add(Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True))))"], "Equality(Add(Mul(Integer(2), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Pow(Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Mul(Integer(4), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Integer(3), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Pow(Add(Mul(Integer(2), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Symbol('G', commutative=True)))"], [["divide", 5, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(4), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Integer(3), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(2), Pow(Function('v_t')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Symbol('G', commutative=True))))"]]}, {"prompt": "Given s{(\\varphi^*,\\hbar)} = \\hbar \\varphi^*, then obtain \\frac{\\partial}{\\partial \\hbar} (- \\hbar \\varphi^* + s{(\\varphi^*,\\hbar)} + 1) = \\frac{d}{d \\hbar} 1", "derivation": "s{(\\varphi^*,\\hbar)} = \\hbar \\varphi^* and s{(\\varphi^*,\\hbar)} + 1 = \\hbar \\varphi^* + 1 and - \\hbar \\varphi^* + s{(\\varphi^*,\\hbar)} + 1 = 1 and \\frac{\\partial}{\\partial \\hbar} (- \\hbar \\varphi^* + s{(\\varphi^*,\\hbar)} + 1) = \\frac{d}{d \\hbar} 1", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)))"], [["minus", 2, "Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(H,\\pi)} = \\cos{(\\pi^{H})} and \\mathbf{E}{(H,\\pi)} = \\pi^{H}, then obtain (\\frac{\\partial}{\\partial \\pi} \\frac{I^{H}{(H,\\pi)}}{\\mathbf{E}{(H,\\pi)}})^{H} = (\\frac{\\partial}{\\partial \\pi} \\frac{\\cos^{H}{(\\mathbf{E}{(H,\\pi)})}}{\\mathbf{E}{(H,\\pi)}})^{H}", "derivation": "I{(H,\\pi)} = \\cos{(\\pi^{H})} and I^{H}{(H,\\pi)} = \\cos^{H}{(\\pi^{H})} and \\mathbf{E}{(H,\\pi)} = \\pi^{H} and \\pi^{- H} I^{H}{(H,\\pi)} = \\pi^{- H} \\cos^{H}{(\\pi^{H})} and \\frac{\\partial}{\\partial \\pi} \\pi^{- H} I^{H}{(H,\\pi)} = \\frac{\\partial}{\\partial \\pi} \\pi^{- H} \\cos^{H}{(\\pi^{H})} and \\frac{\\partial}{\\partial \\pi} \\frac{I^{H}{(H,\\pi)}}{\\mathbf{E}{(H,\\pi)}} = \\frac{\\partial}{\\partial \\pi} \\frac{\\cos^{H}{(\\mathbf{E}{(H,\\pi)})}}{\\mathbf{E}{(H,\\pi)}} and (\\frac{\\partial}{\\partial \\pi} \\frac{I^{H}{(H,\\pi)}}{\\mathbf{E}{(H,\\pi)}})^{H} = (\\frac{\\partial}{\\partial \\pi} \\frac{\\cos^{H}{(\\mathbf{E}{(H,\\pi)})}}{\\mathbf{E}{(H,\\pi)}})^{H}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)), Pow(cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)))"], [["divide", 2, "Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))), Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))), Pow(cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))), Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))), Pow(cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Mul(Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(cos(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["power", 6, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Derivative(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(cos(Function('\\\\mathbf{E}')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{E},\\Omega)} = \\cos{(\\mathbf{E}^{\\Omega})}, then obtain \\sin{(2 \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})})} = \\sin{(\\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + 2 \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{E},\\Omega)} = \\cos{(\\mathbf{E}^{\\Omega})} and \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} = \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})} and \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})} = 2 \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})} and 2 \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})} = \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + 2 \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})} and \\sin{(2 \\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})})} = \\sin{(\\operatorname{V_{\\mathbf{E}}}^{\\Omega}{(\\mathbf{E},\\Omega)} + 2 \\cos^{\\Omega}{(\\mathbf{E}^{\\Omega})})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["add", 2, "Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))))"], [["add", 3, "Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))), Add(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Mul(Integer(2), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))), sin(Add(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Pow(cos(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\Psi{(y^{\\prime})} = \\cos{(y^{\\prime})}, then obtain \\frac{\\sin{(\\int \\Psi{(y^{\\prime})} dy^{\\prime})}}{\\Psi{(y^{\\prime})}} = \\frac{\\sin{(\\int \\cos{(y^{\\prime})} dy^{\\prime})}}{\\Psi{(y^{\\prime})}}", "derivation": "\\Psi{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\int \\Psi{(y^{\\prime})} dy^{\\prime} = \\int \\cos{(y^{\\prime})} dy^{\\prime} and \\sin{(\\int \\Psi{(y^{\\prime})} dy^{\\prime})} = \\sin{(\\int \\cos{(y^{\\prime})} dy^{\\prime})} and \\frac{\\sin{(\\int \\Psi{(y^{\\prime})} dy^{\\prime})}}{\\Psi{(y^{\\prime})}} = \\frac{\\sin{(\\int \\cos{(y^{\\prime})} dy^{\\prime})}}{\\Psi{(y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), sin(Integral(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["divide", 3, "Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Integral(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Mul(Pow(Function('\\\\Psi')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Integral(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(f^{*})} = f^{*} and b{(f^{*})} = \\frac{d}{d f^{*}} \\Omega{(f^{*})}, then obtain 0 = - b{(f^{*})} + \\frac{d}{d f^{*}} f^{*}", "derivation": "\\Omega{(f^{*})} = f^{*} and b{(f^{*})} = \\frac{d}{d f^{*}} \\Omega{(f^{*})} and b{(f^{*})} = \\frac{d}{d f^{*}} f^{*} and 0 = - b{(f^{*})} + \\frac{d}{d f^{*}} f^{*}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], ["renaming_premise", "Equality(Function('b')(Symbol('f^*', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('b')(Symbol('f^*', commutative=True)), Derivative(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["minus", 3, "Function('b')(Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('b')(Symbol('f^*', commutative=True))), Derivative(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{X}{(\\eta,x^\\prime)} = \\eta + x^\\prime, then derive \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(\\eta,x^\\prime)} = 1, then obtain (\\frac{\\partial}{\\partial x^\\prime} (\\eta + x^\\prime))^{x^\\prime} = 1", "derivation": "\\hat{X}{(\\eta,x^\\prime)} = \\eta + x^\\prime and - g + \\hat{X}{(\\eta,x^\\prime)} = \\eta - g + x^\\prime and \\frac{\\partial}{\\partial x^\\prime} (- g + \\hat{X}{(\\eta,x^\\prime)}) = \\frac{\\partial}{\\partial x^\\prime} (\\eta - g + x^\\prime) and \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(\\eta,x^\\prime)} = 1 and \\frac{\\partial}{\\partial x^\\prime} (\\eta + x^\\prime) = 1 and (\\frac{\\partial}{\\partial x^\\prime} (\\eta + x^\\prime))^{x^\\prime} = 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1))"], [["power", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\dot{z}{(\\delta,p)} = \\log{(\\delta - p)} and \\phi_{2}{(\\delta)} = \\delta, then obtain \\int (\\frac{\\partial}{\\partial \\delta} \\dot{z}{(\\delta,p)})^{\\delta} d\\phi_{2}{(\\delta)} = \\int (\\frac{\\partial}{\\partial \\delta} \\log{(\\delta - p)})^{\\delta} d\\phi_{2}{(\\delta)}", "derivation": "\\dot{z}{(\\delta,p)} = \\log{(\\delta - p)} and \\frac{\\partial}{\\partial \\delta} \\dot{z}{(\\delta,p)} = \\frac{\\partial}{\\partial \\delta} \\log{(\\delta - p)} and (\\frac{\\partial}{\\partial \\delta} \\dot{z}{(\\delta,p)})^{\\delta} = (\\frac{\\partial}{\\partial \\delta} \\log{(\\delta - p)})^{\\delta} and \\phi_{2}{(\\delta)} = \\delta and \\int (\\frac{\\partial}{\\partial \\delta} \\dot{z}{(\\delta,p)})^{\\delta} d\\delta = \\int (\\frac{\\partial}{\\partial \\delta} \\log{(\\delta - p)})^{\\delta} d\\delta and \\int (\\frac{\\partial}{\\partial \\delta} \\dot{z}{(\\delta,p)})^{\\delta} d\\phi_{2}{(\\delta)} = \\int (\\frac{\\partial}{\\partial \\delta} \\log{(\\delta - p)})^{\\delta} d\\phi_{2}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('p', commutative=True)), log(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(log(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Derivative(log(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Function('\\\\phi_2')(Symbol('\\\\delta', commutative=True)))), Integral(Pow(Derivative(log(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Function('\\\\phi_2')(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{E})} = \\sin{(\\mathbf{E})}, then obtain \\frac{d}{d \\mathbf{E}} (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E})^{2} = \\frac{d}{d \\mathbf{E}} (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E}) \\int \\sin{(\\mathbf{E})} d\\mathbf{E}", "derivation": "\\hat{p}{(\\mathbf{E})} = \\sin{(\\mathbf{E})} and \\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E} = \\int \\sin{(\\mathbf{E})} d\\mathbf{E} and (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E})^{2} = (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E}) \\int \\sin{(\\mathbf{E})} d\\mathbf{E} and \\frac{d}{d \\mathbf{E}} (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E})^{2} = \\frac{d}{d \\mathbf{E}} (\\int \\hat{p}{(\\mathbf{E})} d\\mathbf{E}) \\int \\sin{(\\mathbf{E})} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["times", 2, "Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integer(2)), Mul(Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Integral(Function('\\\\hat{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(a,f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}}}{a}, then obtain \\iint (- a + \\int f{(a,f_{\\mathbf{p}})} da)^{a} df_{\\mathbf{p}} da = \\iint (- a + \\int \\frac{f_{\\mathbf{p}}}{a} da)^{a} df_{\\mathbf{p}} da", "derivation": "f{(a,f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}}}{a} and \\int f{(a,f_{\\mathbf{p}})} da = \\int \\frac{f_{\\mathbf{p}}}{a} da and - a + \\int f{(a,f_{\\mathbf{p}})} da = - a + \\int \\frac{f_{\\mathbf{p}}}{a} da and (- a + \\int f{(a,f_{\\mathbf{p}})} da)^{a} = (- a + \\int \\frac{f_{\\mathbf{p}}}{a} da)^{a} and \\int (- a + \\int f{(a,f_{\\mathbf{p}})} da)^{a} df_{\\mathbf{p}} = \\int (- a + \\int \\frac{f_{\\mathbf{p}}}{a} da)^{a} df_{\\mathbf{p}} and \\iint (- a + \\int f{(a,f_{\\mathbf{p}})} da)^{a} df_{\\mathbf{p}} da = \\iint (- a + \\int \\frac{f_{\\mathbf{p}}}{a} da)^{a} df_{\\mathbf{p}} da", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["minus", 2, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)))"], [["integrate", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 5, "Symbol('a', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Function('f')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Integral(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given a{(v,U)} = U v, then obtain \\int (v + a{(v,U)})^{v} dU = \\int (U v + v)^{v} dU", "derivation": "a{(v,U)} = U v and v + a{(v,U)} = U v + v and (v + a{(v,U)})^{v} = (U v + v)^{v} and \\int (v + a{(v,U)})^{v} dU = \\int (U v + v)^{v} dU", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('v', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('a')(Symbol('v', commutative=True), Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Symbol('v', commutative=True), Function('a')(Symbol('v', commutative=True), Symbol('U', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('v', commutative=True), Function('a')(Symbol('v', commutative=True), Symbol('U', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(C,E,W)} = (C^{E})^{W}, then obtain (0^{W})^{C} = (((C^{E})^{W} - \\dot{x}{(C,E,W)})^{W})^{C}", "derivation": "\\dot{x}{(C,E,W)} = (C^{E})^{W} and 0 = (C^{E})^{W} - \\dot{x}{(C,E,W)} and 0^{W} = ((C^{E})^{W} - \\dot{x}{(C,E,W)})^{W} and (0^{W})^{C} = (((C^{E})^{W} - \\dot{x}{(C,E,W)})^{W})^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('C', commutative=True), Symbol('E', commutative=True), Symbol('W', commutative=True)), Pow(Pow(Symbol('C', commutative=True), Symbol('E', commutative=True)), Symbol('W', commutative=True)))"], [["minus", 1, "Function('\\\\dot{x}')(Symbol('C', commutative=True), Symbol('E', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Pow(Pow(Symbol('C', commutative=True), Symbol('E', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C', commutative=True), Symbol('E', commutative=True), Symbol('W', commutative=True)))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Add(Pow(Pow(Symbol('C', commutative=True), Symbol('E', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C', commutative=True), Symbol('E', commutative=True), Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('C', commutative=True)), Pow(Pow(Add(Pow(Pow(Symbol('C', commutative=True), Symbol('E', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C', commutative=True), Symbol('E', commutative=True), Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} = \\phi_2 - \\varepsilon_0, then derive \\int (\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1) d\\phi_2 = \\eta^{\\prime} + \\frac{\\phi_2^{2}}{2} + \\phi_2 (1 - \\varepsilon_0), then obtain (\\int (\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1) d\\phi_2)^{2} = (\\eta^{\\prime} + \\frac{\\phi_2^{2}}{2} + \\phi_2 (1 - \\varepsilon_0))^{2}", "derivation": "\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} = \\phi_2 - \\varepsilon_0 and \\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1 = \\phi_2 - \\varepsilon_0 + 1 and \\int (\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1) d\\phi_2 = \\int (\\phi_2 - \\varepsilon_0 + 1) d\\phi_2 and \\int (\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1) d\\phi_2 = \\eta^{\\prime} + \\frac{\\phi_2^{2}}{2} + \\phi_2 (1 - \\varepsilon_0) and (\\int (\\operatorname{z^{*}}{(\\phi_2,\\varepsilon_0)} + 1) d\\phi_2)^{2} = (\\eta^{\\prime} + \\frac{\\phi_2^{2}}{2} + \\phi_2 (1 - \\varepsilon_0))^{2}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('z^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Function('z^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('z^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Function('z^*')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(2)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))), Integer(2)))"]]}, {"prompt": "Given \\delta{(v_{1},\\chi)} = e^{\\chi v_{1}}, then obtain - \\chi v_{1} + \\frac{\\partial}{\\partial \\chi} \\delta{(v_{1},\\chi)} = - \\chi v_{1} + v_{1} e^{\\chi v_{1}}", "derivation": "\\delta{(v_{1},\\chi)} = e^{\\chi v_{1}} and \\frac{\\partial}{\\partial \\chi} \\delta{(v_{1},\\chi)} = \\frac{\\partial}{\\partial \\chi} e^{\\chi v_{1}} and - \\chi v_{1} + \\frac{\\partial}{\\partial \\chi} \\delta{(v_{1},\\chi)} = - \\chi v_{1} + \\frac{\\partial}{\\partial \\chi} e^{\\chi v_{1}} and - \\chi v_{1} + \\frac{\\partial}{\\partial \\chi} \\delta{(v_{1},\\chi)} = - \\chi v_{1} + v_{1} e^{\\chi v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('v_1', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('v_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Derivative(Function('\\\\delta')(Symbol('v_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Derivative(exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Derivative(Function('\\\\delta')(Symbol('v_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('v_1', commutative=True), exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given y{(m,\\tilde{g}^*)} = \\sin{(\\tilde{g}^* m)}, then obtain - m + y{(m,\\tilde{g}^*)} + \\sin{(\\tilde{g}^* m)} = - m + 2 \\sin{(\\tilde{g}^* m)}", "derivation": "y{(m,\\tilde{g}^*)} = \\sin{(\\tilde{g}^* m)} and - m + y{(m,\\tilde{g}^*)} = - m + \\sin{(\\tilde{g}^* m)} and - m + 2 y{(m,\\tilde{g}^*)} = - m + y{(m,\\tilde{g}^*)} + \\sin{(\\tilde{g}^* m)} and - m + 2 y{(m,\\tilde{g}^*)} = - m + 2 \\sin{(\\tilde{g}^* m)} and - m + y{(m,\\tilde{g}^*)} + \\sin{(\\tilde{g}^* m)} = - m + 2 \\sin{(\\tilde{g}^* m)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True))))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True)))))"], [["add", 2, "Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('y')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), sin(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(\\hat{H}_l)} = \\log{(\\sin{(\\hat{H}_l)})}, then derive \\int \\frac{\\Omega{(\\hat{H}_l)}}{\\log{(\\sin{(\\hat{H}_l)})}} d\\hat{H}_l = \\hat{H}_l + \\psi^*, then obtain \\frac{d}{d \\psi^*} \\int 1 d\\hat{H}_l = \\frac{\\partial}{\\partial \\psi^*} (\\hat{H}_l + \\psi^*)", "derivation": "\\Omega{(\\hat{H}_l)} = \\log{(\\sin{(\\hat{H}_l)})} and \\frac{\\Omega{(\\hat{H}_l)}}{\\log{(\\sin{(\\hat{H}_l)})}} = 1 and \\int \\frac{\\Omega{(\\hat{H}_l)}}{\\log{(\\sin{(\\hat{H}_l)})}} d\\hat{H}_l = \\int 1 d\\hat{H}_l and \\int \\frac{\\Omega{(\\hat{H}_l)}}{\\log{(\\sin{(\\hat{H}_l)})}} d\\hat{H}_l = \\hat{H}_l + \\psi^* and \\frac{d}{d \\psi^*} \\int \\frac{\\Omega{(\\hat{H}_l)}}{\\log{(\\sin{(\\hat{H}_l)})}} d\\hat{H}_l = \\frac{\\partial}{\\partial \\psi^*} (\\hat{H}_l + \\psi^*) and \\frac{d}{d \\psi^*} \\int 1 d\\hat{H}_l = \\frac{\\partial}{\\partial \\psi^*} (\\hat{H}_l + \\psi^*)", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True)), log(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 1, "log(sin(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Mul(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\Omega')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\theta,\\mathbf{A})} = - \\theta + \\cos{(\\mathbf{A})} and \\eta{(\\theta)} = - \\theta, then derive \\frac{\\partial}{\\partial \\theta} \\operatorname{E_{n}}{(\\theta,\\mathbf{A})} = -1, then obtain \\sin{(\\eta{(\\theta)} - \\cos{(\\mathbf{A})} + 1)} = - \\sin{(\\theta + \\cos{(\\mathbf{A})} - 1)}", "derivation": "\\operatorname{E_{n}}{(\\theta,\\mathbf{A})} = - \\theta + \\cos{(\\mathbf{A})} and \\frac{\\partial}{\\partial \\theta} \\operatorname{E_{n}}{(\\theta,\\mathbf{A})} = \\frac{\\partial}{\\partial \\theta} (- \\theta + \\cos{(\\mathbf{A})}) and \\frac{\\partial}{\\partial \\theta} \\operatorname{E_{n}}{(\\theta,\\mathbf{A})} = -1 and \\frac{\\partial}{\\partial \\theta} (- \\theta + \\cos{(\\mathbf{A})}) = -1 and \\eta{(\\theta)} = - \\theta and \\eta{(\\theta)} - \\cos{(\\mathbf{A})} - \\frac{\\partial}{\\partial \\theta} (- \\theta + \\cos{(\\mathbf{A})}) = - \\theta - \\cos{(\\mathbf{A})} - \\frac{\\partial}{\\partial \\theta} (- \\theta + \\cos{(\\mathbf{A})}) and \\eta{(\\theta)} - \\cos{(\\mathbf{A})} + 1 = - \\theta - \\cos{(\\mathbf{A})} + 1 and \\sin{(\\eta{(\\theta)} - \\cos{(\\mathbf{A})} + 1)} = - \\sin{(\\theta + \\cos{(\\mathbf{A})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))"], [["minus", 5, "Add(cos(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('\\\\eta')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(1)))"], [["sin", 7], "Equality(sin(Add(Function('\\\\eta')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\theta', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\omega{(L)} = L, then derive \\int \\omega{(L)} dL = \\frac{L^{2}}{2} + \\mu_0, then obtain (\\mu_0 - \\frac{\\omega^{2}{(L)}}{2})^{L} - 1 = (- \\omega^{2}{(L)} + \\int L d\\omega{(L)})^{L} - 1", "derivation": "\\omega{(L)} = L and \\int \\omega{(L)} dL = \\int L dL and \\int \\omega{(L)} dL = \\frac{L^{2}}{2} + \\mu_0 and \\frac{L^{2}}{2} + \\mu_0 = \\int L dL and - \\frac{L^{2}}{2} + \\mu_0 = - L^{2} + \\int L dL and \\mu_0 - \\frac{\\omega^{2}{(L)}}{2} = - \\omega^{2}{(L)} + \\int L d\\omega{(L)} and (\\mu_0 - \\frac{\\omega^{2}{(L)}}{2})^{L} = (- \\omega^{2}{(L)} + \\int L d\\omega{(L)})^{L} and (\\mu_0 - \\frac{\\omega^{2}{(L)}}{2})^{L} - 1 = (- \\omega^{2}{(L)} + \\int L d\\omega{(L)})^{L} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('\\\\mu_0', commutative=True)), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True))))"], [["minus", 4, "Pow(Symbol('L', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(2))), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2))), Integral(Symbol('L', commutative=True), Tuple(Function('\\\\omega')(Symbol('L', commutative=True))))))"], [["power", 6, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2)))), Symbol('L', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2))), Integral(Symbol('L', commutative=True), Tuple(Function('\\\\omega')(Symbol('L', commutative=True))))), Symbol('L', commutative=True)))"], [["add", 7, "Integer(-1)"], "Equality(Add(Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2)))), Symbol('L', commutative=True)), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('L', commutative=True)), Integer(2))), Integral(Symbol('L', commutative=True), Tuple(Function('\\\\omega')(Symbol('L', commutative=True))))), Symbol('L', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given n{(r,c)} = \\frac{r}{c}, then obtain \\iint (2 n{(r,c)} + 1) dr dc = \\iint (1 + \\frac{2 r}{c}) dr dc", "derivation": "n{(r,c)} = \\frac{r}{c} and n{(r,c)} + 1 = 1 + \\frac{r}{c} and n{(r,c)} + 1 + \\frac{r}{c} = 1 + \\frac{2 r}{c} and 2 n{(r,c)} + 1 = 1 + \\frac{2 r}{c} and \\int (2 n{(r,c)} + 1) dr = \\int (1 + \\frac{2 r}{c}) dr and \\iint (2 n{(r,c)} + 1) dr dc = \\iint (1 + \\frac{2 r}{c}) dr dc", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))"], "Equality(Add(Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True)), Integer(1), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))), Add(Integer(1), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True))), Integer(1)), Tuple(Symbol('r', commutative=True))), Integral(Add(Integer(1), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["integrate", 5, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('n')(Symbol('r', commutative=True), Symbol('c', commutative=True))), Integer(1)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Add(Integer(1), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given t{(\\delta,v)} = \\log{(v)}^{\\delta}, then derive \\frac{\\partial}{\\partial \\delta} t{(\\delta,v)} - 1 = \\log{(v)}^{\\delta} \\log{(\\log{(v)})} - 1, then obtain \\frac{\\partial}{\\partial \\delta} t{(\\delta,v)} - 1 = \\frac{\\partial}{\\partial \\delta} \\log{(v)}^{\\delta} - 1", "derivation": "t{(\\delta,v)} = \\log{(v)}^{\\delta} and - \\delta + t{(\\delta,v)} = - \\delta + \\log{(v)}^{\\delta} and \\frac{\\partial}{\\partial \\delta} (- \\delta + t{(\\delta,v)}) = \\frac{\\partial}{\\partial \\delta} (- \\delta + \\log{(v)}^{\\delta}) and \\frac{\\partial}{\\partial \\delta} t{(\\delta,v)} - 1 = \\log{(v)}^{\\delta} \\log{(\\log{(v)})} - 1 and \\frac{\\partial}{\\partial \\delta} \\log{(v)}^{\\delta} - 1 = \\log{(v)}^{\\delta} \\log{(\\log{(v)})} - 1 and \\frac{\\partial}{\\partial \\delta} t{(\\delta,v)} - 1 = \\frac{\\partial}{\\partial \\delta} \\log{(v)}^{\\delta} - 1", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('v', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('t')(Symbol('\\\\delta', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('t')(Symbol('\\\\delta', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True)), log(log(Symbol('v', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True)), log(log(Symbol('v', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('t')(Symbol('\\\\delta', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(log(Symbol('v', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\tilde{g}^*{(f,k,F_{N})} = \\frac{F_{N} f}{k}, then obtain \\frac{F_{N} f}{k} + (- \\frac{k (- \\frac{F_{N} f}{k} + \\tilde{g}^*{(f,k,F_{N})})}{F_{N} f})^{f} = 0^{f} + \\frac{F_{N} f}{k}", "derivation": "\\tilde{g}^*{(f,k,F_{N})} = \\frac{F_{N} f}{k} and - \\frac{F_{N} f}{k} + \\tilde{g}^*{(f,k,F_{N})} = 0 and - \\frac{k (- \\frac{F_{N} f}{k} + \\tilde{g}^*{(f,k,F_{N})})}{F_{N} f} = 0 and (- \\frac{k (- \\frac{F_{N} f}{k} + \\tilde{g}^*{(f,k,F_{N})})}{F_{N} f})^{f} = 0^{f} and \\frac{F_{N} f}{k} + (- \\frac{k (- \\frac{F_{N} f}{k} + \\tilde{g}^*{(f,k,F_{N})})}{F_{N} f})^{f} = 0^{f} + \\frac{F_{N} f}{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True), Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["minus", 1, "Mul(Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True), Symbol('k', commutative=True), Symbol('F_N', commutative=True))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('k', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True), Symbol('k', commutative=True), Symbol('F_N', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('k', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True), Symbol('k', commutative=True), Symbol('F_N', commutative=True)))), Symbol('f', commutative=True)), Pow(Integer(0), Symbol('f', commutative=True)))"], [["add", 4, "Mul(Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Pow(Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('k', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True), Symbol('k', commutative=True), Symbol('F_N', commutative=True)))), Symbol('f', commutative=True))), Add(Pow(Integer(0), Symbol('f', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('f', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{S}{(b,C_{d})} = \\log{(C_{d} - b)}, then obtain (C_{d} - b) \\frac{\\partial}{\\partial b} \\mathbf{S}^{C_{d}}{(b,C_{d})} = (C_{d} - b) \\frac{\\partial}{\\partial b} \\log{(C_{d} - b)}^{C_{d}}", "derivation": "\\mathbf{S}{(b,C_{d})} = \\log{(C_{d} - b)} and \\mathbf{S}^{C_{d}}{(b,C_{d})} = \\log{(C_{d} - b)}^{C_{d}} and \\frac{\\partial}{\\partial b} \\mathbf{S}^{C_{d}}{(b,C_{d})} = \\frac{\\partial}{\\partial b} \\log{(C_{d} - b)}^{C_{d}} and (C_{d} - b) \\frac{\\partial}{\\partial b} \\mathbf{S}^{C_{d}}{(b,C_{d})} = (C_{d} - b) \\frac{\\partial}{\\partial b} \\log{(C_{d} - b)}^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('b', commutative=True), Symbol('C_d', commutative=True)), log(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('b', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(log(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Symbol('C_d', commutative=True)))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('b', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(log(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Symbol('C_d', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 3, "Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Mul(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('b', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Derivative(Pow(log(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Symbol('C_d', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\varepsilon_0)} = e^{\\varepsilon_0}, then derive \\frac{d}{d \\varepsilon_0} \\operatorname{v_{z}}{(\\varepsilon_0)} = e^{\\varepsilon_0}, then obtain \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} = e^{\\varepsilon_0}", "derivation": "\\operatorname{v_{z}}{(\\varepsilon_0)} = e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} \\operatorname{v_{z}}{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} \\operatorname{v_{z}}{(\\varepsilon_0)} = e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} = e^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), exp(Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(V)} = \\log{(V)}, then obtain \\frac{\\tilde{g}^*^{V}{(V)} \\sin{(\\tilde{g}^*{(V)})}}{\\tilde{g}^*{(V)}} = \\frac{\\tilde{g}^*^{V}{(V)} \\sin{(\\log{(V)})}}{\\tilde{g}^*{(V)}}", "derivation": "\\tilde{g}^*{(V)} = \\log{(V)} and \\sin{(\\tilde{g}^*{(V)})} = \\sin{(\\log{(V)})} and \\tilde{g}^*^{V}{(V)} = \\log{(V)}^{V} and \\log{(V)}^{V} \\sin{(\\tilde{g}^*{(V)})} = \\log{(V)}^{V} \\sin{(\\log{(V)})} and \\frac{\\log{(V)}^{V} \\sin{(\\tilde{g}^*{(V)})}}{\\tilde{g}^*{(V)}} = \\frac{\\log{(V)}^{V} \\sin{(\\log{(V)})}}{\\tilde{g}^*{(V)}} and \\frac{\\tilde{g}^*^{V}{(V)} \\sin{(\\tilde{g}^*{(V)})}}{\\tilde{g}^*{(V)}} = \\frac{\\tilde{g}^*^{V}{(V)} \\sin{(\\log{(V)})}}{\\tilde{g}^*{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["times", 2, "Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], "Equality(Mul(Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)))), Mul(Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True)))))"], [["divide", 4, "Function('\\\\tilde{g}^*')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Integer(-1)), Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Integer(-1)), Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Integer(-1)), Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Integer(-1)), Pow(Function('\\\\tilde{g}^*')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} = \\hat{p}_0^{x}, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{x} \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} dx = \\frac{\\partial}{\\partial \\hat{p}_0} \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{2 x} dx", "derivation": "\\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} = \\hat{p}_0^{x} and \\hat{p}_0^{x} \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} = \\hat{p}_0^{2 x} and \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{x} \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{2 x} and \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{x} \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} dx = \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{2 x} dx and \\frac{\\partial}{\\partial \\hat{p}_0} \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{x} \\operatorname{y^{\\prime}}{(x,\\hat{p}_0)} dx = \\frac{\\partial}{\\partial \\hat{p}_0} \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0^{2 x} dx", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('x', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Integral(Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('x', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integral(Derivative(Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(g)} = \\log{(g)}, then obtain \\frac{d}{d g} 2 \\iint \\mathbf{J}_f{(g)} dg dg = \\frac{d}{d g} 2 \\iint \\log{(g)} dg dg", "derivation": "\\mathbf{J}_f{(g)} = \\log{(g)} and \\int \\mathbf{J}_f{(g)} dg = \\int \\log{(g)} dg and \\iint \\mathbf{J}_f{(g)} dg dg = \\iint \\log{(g)} dg dg and 2 \\iint \\mathbf{J}_f{(g)} dg dg = \\iint \\mathbf{J}_f{(g)} dg dg + \\iint \\log{(g)} dg dg and \\iint \\mathbf{J}_f{(g)} dg dg + \\iint \\log{(g)} dg dg = 2 \\iint \\log{(g)} dg dg and 2 \\iint \\mathbf{J}_f{(g)} dg dg = 2 \\iint \\log{(g)} dg dg and \\frac{d}{d g} 2 \\iint \\mathbf{J}_f{(g)} dg dg = \\frac{d}{d g} 2 \\iint \\log{(g)} dg dg", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["add", 3, "Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["add", 3, "Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(2), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(2), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["differentiate", 6, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Integral(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(A_{y})} = \\log{(A_{y})}, then obtain - A_{y} + 3 \\pi{(A_{y})} - \\log{(A_{y})} - \\log{(A_{y} - \\pi{(A_{y})} + \\log{(A_{y})})} = - A_{y} + \\log{(A_{y})}", "derivation": "\\pi{(A_{y})} = \\log{(A_{y})} and - A_{y} + \\pi{(A_{y})} = - A_{y} + \\log{(A_{y})} and - A_{y} + \\pi{(A_{y})} - \\log{(A_{y})} = - A_{y} and - A_{y} + 2 \\pi{(A_{y})} - \\log{(A_{y})} = - A_{y} + \\pi{(A_{y})} and - A_{y} + 3 \\pi{(A_{y})} - \\log{(A_{y})} - \\log{(A_{y} - \\pi{(A_{y})} + \\log{(A_{y})})} = - A_{y} + 2 \\pi{(A_{y})} - \\log{(A_{y})} and - A_{y} + 2 \\pi{(A_{y})} - \\log{(A_{y})} = - A_{y} + \\log{(A_{y})} and - A_{y} + 3 \\pi{(A_{y})} - \\log{(A_{y})} - \\log{(A_{y} - \\pi{(A_{y})} + \\log{(A_{y})})} = - A_{y} + \\log{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True)))"], [["minus", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\pi')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True))))"], [["minus", 2, "log(Symbol('A_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\pi')(Symbol('A_y', commutative=True)), Mul(Integer(-1), log(Symbol('A_y', commutative=True)))), Mul(Integer(-1), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Symbol('A_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\pi')(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(3), Function('\\\\pi')(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('A_y', commutative=True))), log(Symbol('A_y', commutative=True)))))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Symbol('A_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Symbol('A_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(3), Function('\\\\pi')(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Symbol('A_y', commutative=True))), Mul(Integer(-1), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('A_y', commutative=True))), log(Symbol('A_y', commutative=True)))))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} = \\theta_2 + \\log{(n)} and \\delta{(\\theta_2)} = 2 \\theta_2, then obtain 2 \\operatorname{g_{\\varepsilon}}^{2}{(\\theta_2,n)} = (\\delta{(\\theta_2)} + 2 \\log{(n)}) \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} = \\theta_2 + \\log{(n)} and 2 \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} = \\theta_2 + \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} + \\log{(n)} and 2 \\operatorname{g_{\\varepsilon}}^{2}{(\\theta_2,n)} = (\\theta_2 + \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} + \\log{(n)}) \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} and 2 (\\theta_2 + \\log{(n)})^{2} = (\\theta_2 + \\log{(n)}) (2 \\theta_2 + 2 \\log{(n)}) and \\delta{(\\theta_2)} = 2 \\theta_2 and 2 \\operatorname{g_{\\varepsilon}}^{2}{(\\theta_2,n)} = (2 \\theta_2 + 2 \\log{(n)}) \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)} and 2 \\operatorname{g_{\\varepsilon}}^{2}{(\\theta_2,n)} = (\\delta{(\\theta_2)} + 2 \\log{(n)}) \\operatorname{g_{\\varepsilon}}{(\\theta_2,n)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('n', commutative=True))))"], [["add", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), log(Symbol('n', commutative=True))))"], [["times", 2, "Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('n', commutative=True))), Integer(2))), Mul(Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), log(Symbol('n', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), log(Symbol('n', commutative=True)))), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(2), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Add(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), log(Symbol('n', commutative=True)))), Function('g_{\\\\varepsilon}')(Symbol('\\\\theta_2', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{J}_f)} = e^{e^{\\mathbf{J}_f}}, then derive \\int \\tilde{g}^*{(\\mathbf{J}_f)} d\\mathbf{J}_f = A + \\operatorname{Ei}{(e^{\\mathbf{J}_f})}, then derive \\hat{H}_l + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} = A + \\operatorname{Ei}{(e^{\\mathbf{J}_f})}, then obtain \\frac{d}{d \\hat{H}_l} \\int e^{e^{\\mathbf{J}_f}} d\\mathbf{J}_f = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\operatorname{Ei}{(e^{\\mathbf{J}_f})})", "derivation": "\\tilde{g}^*{(\\mathbf{J}_f)} = e^{e^{\\mathbf{J}_f}} and \\int \\tilde{g}^*{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int e^{e^{\\mathbf{J}_f}} d\\mathbf{J}_f and \\int \\tilde{g}^*{(\\mathbf{J}_f)} d\\mathbf{J}_f = A + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} and \\int e^{e^{\\mathbf{J}_f}} d\\mathbf{J}_f = A + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} and \\hat{H}_l + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} = A + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} and \\int e^{e^{\\mathbf{J}_f}} d\\mathbf{J}_f = \\hat{H}_l + \\operatorname{Ei}{(e^{\\mathbf{J}_f})} and \\frac{d}{d \\hat{H}_l} \\int e^{e^{\\mathbf{J}_f}} d\\mathbf{J}_f = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\operatorname{Ei}{(e^{\\mathbf{J}_f})})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('A', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('A', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('A', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Integral(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Ei(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(F_{g},\\dot{z})} = \\cos{(F_{g} \\dot{z})}, then derive \\frac{\\partial}{\\partial F_{g}} \\operatorname{f^{\\prime}}{(F_{g},\\dot{z})} = - \\dot{z} \\sin{(F_{g} \\dot{z})}, then obtain (\\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g} \\dot{z})})^{F_{g}} = (- \\dot{z} \\sin{(F_{g} \\dot{z})})^{F_{g}}", "derivation": "\\operatorname{f^{\\prime}}{(F_{g},\\dot{z})} = \\cos{(F_{g} \\dot{z})} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{f^{\\prime}}{(F_{g},\\dot{z})} = \\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g} \\dot{z})} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{f^{\\prime}}{(F_{g},\\dot{z})} = - \\dot{z} \\sin{(F_{g} \\dot{z})} and (\\frac{\\partial}{\\partial F_{g}} \\operatorname{f^{\\prime}}{(F_{g},\\dot{z})})^{F_{g}} = (- \\dot{z} \\sin{(F_{g} \\dot{z})})^{F_{g}} and (\\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g} \\dot{z})})^{F_{g}} = (- \\dot{z} \\sin{(F_{g} \\dot{z})})^{F_{g}}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), cos(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(s)} = s, then obtain - s^{s} + \\mathbf{r}{(s)} = s - s^{s}", "derivation": "\\mathbf{r}{(s)} = s and \\mathbf{r}^{s}{(s)} = s^{s} and \\mathbf{r}{(s)} - \\mathbf{r}^{s}{(s)} = s - \\mathbf{r}^{s}{(s)} and - s^{s} + \\mathbf{r}{(s)} = s - s^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))), Add(Symbol('s', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('s', commutative=True), Symbol('s', commutative=True))), Function('\\\\mathbf{r}')(Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v_{y})} = \\log{(v_{y})}, then obtain - v_{y} + 2 \\operatorname{v_{t}}{(v_{y})} - \\log{(v_{y})} = - v_{y} + \\operatorname{v_{t}}{(v_{y})}", "derivation": "\\operatorname{v_{t}}{(v_{y})} = \\log{(v_{y})} and - v_{y} + \\operatorname{v_{t}}{(v_{y})} = - v_{y} + \\log{(v_{y})} and - v_{y} + \\operatorname{v_{t}}{(v_{y})} - \\log{(v_{y})} = - v_{y} and - v_{y} + 2 \\operatorname{v_{t}}{(v_{y})} - \\log{(v_{y})} = - v_{y} + \\operatorname{v_{t}}{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_t')(Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))))"], [["minus", 2, "log(Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_t')(Symbol('v_y', commutative=True)), Mul(Integer(-1), log(Symbol('v_y', commutative=True)))), Mul(Integer(-1), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Mul(Integer(2), Function('v_t')(Symbol('v_y', commutative=True))), Mul(Integer(-1), log(Symbol('v_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('v_t')(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(n_{1},\\rho_b,A_{x})} = A_{x} n_{1} - \\rho_b, then obtain \\rho_b = \\rho_b + (A_{x} n_{1} - \\rho_b)^{\\rho_b} - \\hat{\\mathbf{x}}^{\\rho_b}{(n_{1},\\rho_b,A_{x})}", "derivation": "\\hat{\\mathbf{x}}{(n_{1},\\rho_b,A_{x})} = A_{x} n_{1} - \\rho_b and \\hat{\\mathbf{x}}^{\\rho_b}{(n_{1},\\rho_b,A_{x})} = (A_{x} n_{1} - \\rho_b)^{\\rho_b} and 0 = (A_{x} n_{1} - \\rho_b)^{\\rho_b} - \\hat{\\mathbf{x}}^{\\rho_b}{(n_{1},\\rho_b,A_{x})} and \\rho_b = \\rho_b + (A_{x} n_{1} - \\rho_b)^{\\rho_b} - \\hat{\\mathbf{x}}^{\\rho_b}{(n_{1},\\rho_b,A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Add(Mul(Symbol('A_x', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)))))"], [["add", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Symbol('\\\\rho_b', commutative=True), Add(Symbol('\\\\rho_b', commutative=True), Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\log{(v_{1})}, then obtain \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{1})}}{\\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})}} = \\frac{\\log{(v_{1})}}{\\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\log{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\frac{d}{d v_{1}} \\log{(v_{1})} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{1})}}{\\frac{d}{d v_{1}} \\log{(v_{1})}} = \\frac{\\log{(v_{1})}}{\\frac{d}{d v_{1}} \\log{(v_{1})}} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(v_{1})}}{\\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})}} = \\frac{\\log{(v_{1})}}{\\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Pow(Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('v_1', commutative=True)), Pow(Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('v_1', commutative=True)), Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given I{(b)} = \\log{(b)}, then obtain b - I{(b)} + \\cos{(\\frac{- b + I{(b)} + 1}{b})} = b - I{(b)} + \\cos{(\\frac{- b + \\log{(b)} + 1}{b})}", "derivation": "I{(b)} = \\log{(b)} and - b + I{(b)} = - b + \\log{(b)} and - b + I{(b)} + 1 = - b + \\log{(b)} + 1 and - \\frac{- b + I{(b)} + 1}{b} = - \\frac{- b + \\log{(b)} + 1}{b} and \\cos{(\\frac{- b + I{(b)} + 1}{b})} = \\cos{(\\frac{- b + \\log{(b)} + 1}{b})} and b - I{(b)} + \\cos{(\\frac{- b + I{(b)} + 1}{b})} = b - I{(b)} + \\cos{(\\frac{- b + \\log{(b)} + 1}{b})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('b', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('b', commutative=True)), Integer(1)))"], [["divide", 3, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True)), Integer(1))), Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('b', commutative=True)), Integer(1))))"], [["cos", 4], "Equality(cos(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True)), Integer(1)))), cos(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('b', commutative=True)), Integer(1)))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True)))"], "Equality(Add(Symbol('b', commutative=True), Mul(Integer(-1), Function('I')(Symbol('b', commutative=True))), cos(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('I')(Symbol('b', commutative=True)), Integer(1))))), Add(Symbol('b', commutative=True), Mul(Integer(-1), Function('I')(Symbol('b', commutative=True))), cos(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('b', commutative=True)), Integer(1))))))"]]}, {"prompt": "Given n{(\\phi,\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda}^{\\phi})}, then obtain - \\frac{n{(\\phi,\\Psi_{\\lambda})} - \\cos{(\\Psi_{\\lambda}^{\\phi})}}{\\cos{(\\Psi_{\\lambda}^{\\phi})}} - n{(\\phi,\\Psi_{\\lambda})} = - n{(\\phi,\\Psi_{\\lambda})}", "derivation": "n{(\\phi,\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda}^{\\phi})} and n{(\\phi,\\Psi_{\\lambda})} - \\cos{(\\Psi_{\\lambda}^{\\phi})} = 0 and - \\frac{n{(\\phi,\\Psi_{\\lambda})} - \\cos{(\\Psi_{\\lambda}^{\\phi})}}{\\cos{(\\Psi_{\\lambda}^{\\phi})}} = 0 and - \\frac{n{(\\phi,\\Psi_{\\lambda})} - \\cos{(\\Psi_{\\lambda}^{\\phi})}}{\\cos{(\\Psi_{\\lambda}^{\\phi})}} - n{(\\phi,\\Psi_{\\lambda})} = - n{(\\phi,\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))), Pow(cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1))), Integer(0))"], [["minus", 3, "Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Add(Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))))), Pow(cos(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1))), Mul(Integer(-1), Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), Function('n')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\hat{p}_0,y)} = e^{\\hat{p}_0 - y}, then derive \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy = \\Psi^{\\dagger} - e^{\\hat{p}_0 - y}, then obtain 0 = - m + e^{\\hat{p}_0 - y} + \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy", "derivation": "\\operatorname{F_{H}}{(\\hat{p}_0,y)} = e^{\\hat{p}_0 - y} and \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy = \\int e^{\\hat{p}_0 - y} dy and \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy = \\Psi^{\\dagger} - e^{\\hat{p}_0 - y} and \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy = \\Psi^{\\dagger} - \\operatorname{F_{H}}{(\\hat{p}_0,y)} and \\int e^{\\hat{p}_0 - y} dy = \\Psi^{\\dagger} - e^{\\hat{p}_0 - y} and 0 = \\Psi^{\\dagger} - e^{\\hat{p}_0 - y} - \\int e^{\\hat{p}_0 - y} dy and 0 = \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy - \\int e^{\\hat{p}_0 - y} dy and 0 = - m + e^{\\hat{p}_0 - y} + \\int \\operatorname{F_{H}}{(\\hat{p}_0,y)} dy", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))))"], [["minus", 5, "Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Mul(Integer(-1), Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(0), Add(Integral(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Mul(Integer(-1), Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))))"], [["evaluate_integrals", 7], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Integral(Function('F_H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(M_{E})} = \\frac{d}{d M_{E}} \\log{(M_{E})}, then derive \\theta_{2}{(M_{E})} = \\frac{1}{M_{E}}, then obtain 1 + \\frac{\\log{(M_{E})}}{M_{E}} = \\theta_{2}{(\\frac{1}{\\frac{d}{d M_{E}} \\log{(M_{E})}})} \\log{(M_{E})} + 1", "derivation": "\\theta_{2}{(M_{E})} = \\frac{d}{d M_{E}} \\log{(M_{E})} and \\theta_{2}{(M_{E})} = \\frac{1}{M_{E}} and \\frac{d}{d M_{E}} \\log{(M_{E})} = \\frac{1}{M_{E}} and \\theta_{2}{(M_{E})} \\log{(M_{E})} = \\log{(M_{E})} \\frac{d}{d M_{E}} \\log{(M_{E})} and \\theta_{2}{(M_{E})} \\log{(M_{E})} + 1 = \\log{(M_{E})} \\frac{d}{d M_{E}} \\log{(M_{E})} + 1 and \\theta_{2}{(\\frac{1}{\\frac{d}{d M_{E}} \\log{(M_{E})}})} = \\frac{d}{d M_{E}} \\log{(M_{E})} and 1 + \\frac{\\log{(M_{E})}}{M_{E}} = \\log{(M_{E})} \\frac{d}{d M_{E}} \\log{(M_{E})} + 1 and 1 + \\frac{\\log{(M_{E})}}{M_{E}} = \\theta_{2}{(\\frac{1}{\\frac{d}{d M_{E}} \\log{(M_{E})}})} \\log{(M_{E})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('M_E', commutative=True)), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta_2')(Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], [["times", 1, "log(Symbol('M_E', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True))), Mul(log(Symbol('M_E', commutative=True)), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["add", 4, 1], "Equality(Add(Mul(Function('\\\\theta_2')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True))), Integer(1)), Add(Mul(log(Symbol('M_E', commutative=True)), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\theta_2')(Pow(Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1))), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Integer(1), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True)))), Add(Mul(log(Symbol('M_E', commutative=True)), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Integer(1), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True)))), Add(Mul(Function('\\\\theta_2')(Pow(Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1))), log(Symbol('M_E', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(S)} = \\sin{(S)}, then derive \\int \\operatorname{v_{1}}{(S)} dS = \\lambda - \\cos{(S)}, then derive \\lambda - \\cos{(S)} = A_{x} - \\cos{(S)}, then obtain (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)}) \\int \\operatorname{v_{1}}{(S)} dS = (A_{x} - \\cos{(S)}) (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)})", "derivation": "\\operatorname{v_{1}}{(S)} = \\sin{(S)} and \\int \\operatorname{v_{1}}{(S)} dS = \\int \\sin{(S)} dS and \\int \\operatorname{v_{1}}{(S)} dS = \\lambda - \\cos{(S)} and \\lambda - \\cos{(S)} = \\int \\sin{(S)} dS and \\lambda - \\cos{(S)} = A_{x} - \\cos{(S)} and (\\lambda - \\cos{(S)}) (2 \\sin{(S)} + 2 \\int \\operatorname{v_{1}}{(S)} dS) = (A_{x} - \\cos{(S)}) (2 \\sin{(S)} + 2 \\int \\operatorname{v_{1}}{(S)} dS) and (\\lambda - \\cos{(S)}) (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)}) = (A_{x} - \\cos{(S)}) (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)}) and (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)}) \\int \\operatorname{v_{1}}{(S)} dS = (A_{x} - \\cos{(S)}) (2 \\lambda + 2 \\sin{(S)} - 2 \\cos{(S)})", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))))"], [["times", 5, "Add(Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(2), Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(2), Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))), Mul(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(2), Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('S', commutative=True))))), Mul(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('S', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('S', commutative=True)))), Integral(Function('v_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('S', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(l,C_{d})} = l^{C_{d}} and \\mathbf{H}{(l,C_{d})} = (l^{C_{d}})^{l}, then obtain \\frac{l \\mathbf{A}^{l}{(l,C_{d})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(l,C_{d})}}{\\mathbf{A}{(l,C_{d})}} = \\frac{\\partial}{\\partial C_{d}} \\mathbf{H}{(l,C_{d})}", "derivation": "\\mathbf{A}{(l,C_{d})} = l^{C_{d}} and \\mathbf{A}^{l}{(l,C_{d})} = (l^{C_{d}})^{l} and \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}^{l}{(l,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (l^{C_{d}})^{l} and \\mathbf{H}{(l,C_{d})} = (l^{C_{d}})^{l} and \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}^{l}{(l,C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\mathbf{H}{(l,C_{d})} and \\frac{l \\mathbf{A}^{l}{(l,C_{d})} \\frac{\\partial}{\\partial C_{d}} \\mathbf{A}{(l,C_{d})}}{\\mathbf{A}{(l,C_{d})}} = \\frac{\\partial}{\\partial C_{d}} \\mathbf{H}{(l,C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('C_d', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Pow(Pow(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('l', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Symbol('l', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(T)} = e^{\\sin{(T)}}, then derive - e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)} = e^{\\sin{(T)}} \\cos{(T)} - e^{\\sin{(T)}}, then obtain \\int \\frac{d}{d T} (- e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)}) dT = \\int \\frac{d}{d T} (e^{\\sin{(T)}} \\cos{(T)} - e^{\\sin{(T)}}) dT", "derivation": "\\operatorname{r_{0}}{(T)} = e^{\\sin{(T)}} and \\frac{d}{d T} \\operatorname{r_{0}}{(T)} = \\frac{d}{d T} e^{\\sin{(T)}} and - e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)} = - e^{\\sin{(T)}} + \\frac{d}{d T} e^{\\sin{(T)}} and - e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)} = e^{\\sin{(T)}} \\cos{(T)} - e^{\\sin{(T)}} and \\frac{d}{d T} (- e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)}) = \\frac{d}{d T} (e^{\\sin{(T)}} \\cos{(T)} - e^{\\sin{(T)}}) and \\int \\frac{d}{d T} (- e^{\\sin{(T)}} + \\frac{d}{d T} \\operatorname{r_{0}}{(T)}) dT = \\int \\frac{d}{d T} (e^{\\sin{(T)}} \\cos{(T)} - e^{\\sin{(T)}}) dT", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('T', commutative=True)), exp(sin(Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "exp(sin(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(sin(Symbol('T', commutative=True)))), Derivative(Function('r_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(sin(Symbol('T', commutative=True)))), Derivative(exp(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(sin(Symbol('T', commutative=True)))), Derivative(Function('r_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(exp(sin(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('T', commutative=True))))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(sin(Symbol('T', commutative=True)))), Derivative(Function('r_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(exp(sin(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('T', commutative=True))))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), exp(sin(Symbol('T', commutative=True)))), Derivative(Function('r_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(Add(Mul(exp(sin(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('T', commutative=True))))), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\Omega{(E_{x},s)} = E_{x} + s, then obtain 1 + \\frac{\\Omega^{2}{(E_{x},s)}}{E_{x}} = 1 + \\frac{(E_{x} + s)^{2}}{E_{x}}", "derivation": "\\Omega{(E_{x},s)} = E_{x} + s and \\frac{\\Omega{(E_{x},s)}}{E_{x}} = \\frac{E_{x} + s}{E_{x}} and \\frac{\\Omega^{2}{(E_{x},s)}}{E_{x}} = \\frac{(E_{x} + s) \\Omega{(E_{x},s)}}{E_{x}} and \\frac{(E_{x} + s) \\Omega{(E_{x},s)}}{E_{x}} = \\frac{(E_{x} + s)^{2}}{E_{x}} and \\frac{\\Omega^{2}{(E_{x},s)}}{E_{x}} = \\frac{(E_{x} + s)^{2}}{E_{x}} and 1 + \\frac{\\Omega^{2}{(E_{x},s)}}{E_{x}} = 1 + \\frac{(E_{x} + s)^{2}}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2)))), Add(Integer(1), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Add(Symbol('E_x', commutative=True), Symbol('s', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given J{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then obtain - \\mathbf{S} - J{(\\mathbf{S})} + 3 \\sin{(\\mathbf{S})} = - \\mathbf{S} + 2 \\sin{(\\mathbf{S})}", "derivation": "J{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and \\mathbf{S} + J{(\\mathbf{S})} = \\mathbf{S} + \\sin{(\\mathbf{S})} and \\mathbf{S} + J{(\\mathbf{S})} - \\sin{(\\mathbf{S})} = \\mathbf{S} and \\mathbf{S} + J{(\\mathbf{S})} - 2 \\sin{(\\mathbf{S})} = \\mathbf{S} - \\sin{(\\mathbf{S})} and \\mathbf{S} + J{(\\mathbf{S})} - 3 \\sin{(\\mathbf{S})} = \\mathbf{S} - 2 \\sin{(\\mathbf{S})} and - \\mathbf{S} - J{(\\mathbf{S})} + 3 \\sin{(\\mathbf{S})} = - \\mathbf{S} + 2 \\sin{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('J')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 2, "sin(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('J')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True))"], [["add", 3, "Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('J')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["minus", 4, "sin(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('J')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integer(3), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["times", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(3), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(a,n)} = \\log{(a - n)}, then derive \\frac{\\partial}{\\partial a} \\operatorname{c_{0}}{(a,n)} = \\frac{1}{a - n}, then derive \\rho_f + \\operatorname{c_{0}}{(a,n)} = f^{\\prime} + \\log{(a - n)}, then obtain (\\rho_f + \\operatorname{c_{0}}{(a,n)})^{a} = (f^{\\prime} + \\operatorname{c_{0}}{(a,n)})^{a}", "derivation": "\\operatorname{c_{0}}{(a,n)} = \\log{(a - n)} and \\frac{\\partial}{\\partial a} \\operatorname{c_{0}}{(a,n)} = \\frac{\\partial}{\\partial a} \\log{(a - n)} and \\frac{\\partial}{\\partial a} \\operatorname{c_{0}}{(a,n)} = \\frac{1}{a - n} and \\int \\frac{\\partial}{\\partial a} \\operatorname{c_{0}}{(a,n)} da = \\int \\frac{1}{a - n} da and \\rho_f + \\operatorname{c_{0}}{(a,n)} = f^{\\prime} + \\log{(a - n)} and (\\rho_f + \\operatorname{c_{0}}{(a,n)})^{a} = (f^{\\prime} + \\log{(a - n)})^{a} and (\\rho_f + \\operatorname{c_{0}}{(a,n)})^{a} = (f^{\\prime} + \\operatorname{c_{0}}{(a,n)})^{a}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True)), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Integer(-1)))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Derivative(Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Integral(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["power", 5, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\rho_f', commutative=True), Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('a', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), log(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Symbol('\\\\rho_f', commutative=True), Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('a', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Function('c_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(r,\\theta_1)} = - \\theta_1 + r, then obtain \\int \\theta_{2}{(r,\\theta_1)} d\\theta_1 + \\int \\theta_{2}{(r,\\theta_1)} dr = \\int (- \\theta_1 + r) dr + \\int \\theta_{2}{(r,\\theta_1)} d\\theta_1", "derivation": "\\theta_{2}{(r,\\theta_1)} = - \\theta_1 + r and \\int \\theta_{2}{(r,\\theta_1)} d\\theta_1 = \\int (- \\theta_1 + r) d\\theta_1 and \\int \\theta_{2}{(r,\\theta_1)} dr = \\int (- \\theta_1 + r) dr and \\int (- \\theta_1 + r) d\\theta_1 + \\int \\theta_{2}{(r,\\theta_1)} dr = \\int (- \\theta_1 + r) d\\theta_1 + \\int (- \\theta_1 + r) dr and \\int \\theta_{2}{(r,\\theta_1)} d\\theta_1 + \\int \\theta_{2}{(r,\\theta_1)} dr = \\int (- \\theta_1 + r) dr + \\int \\theta_{2}{(r,\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(a^{\\dagger})} = \\cos{(\\log{(a^{\\dagger})})}, then obtain 0 = - (\\int (a^{\\dagger} + \\operatorname{F_{c}}{(a^{\\dagger})}) da^{\\dagger})^{a^{\\dagger}} + (\\int (a^{\\dagger} + \\cos{(\\log{(a^{\\dagger})})}) da^{\\dagger})^{a^{\\dagger}}", "derivation": "\\operatorname{F_{c}}{(a^{\\dagger})} = \\cos{(\\log{(a^{\\dagger})})} and a^{\\dagger} + \\operatorname{F_{c}}{(a^{\\dagger})} = a^{\\dagger} + \\cos{(\\log{(a^{\\dagger})})} and \\int (a^{\\dagger} + \\operatorname{F_{c}}{(a^{\\dagger})}) da^{\\dagger} = \\int (a^{\\dagger} + \\cos{(\\log{(a^{\\dagger})})}) da^{\\dagger} and (\\int (a^{\\dagger} + \\operatorname{F_{c}}{(a^{\\dagger})}) da^{\\dagger})^{a^{\\dagger}} = (\\int (a^{\\dagger} + \\cos{(\\log{(a^{\\dagger})})}) da^{\\dagger})^{a^{\\dagger}} and 0 = - (\\int (a^{\\dagger} + \\operatorname{F_{c}}{(a^{\\dagger})}) da^{\\dagger})^{a^{\\dagger}} + (\\int (a^{\\dagger} + \\cos{(\\log{(a^{\\dagger})})}) da^{\\dagger})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), cos(log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(log(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(log(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(log(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 4, "Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(log(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\ddot{x},S)} = S - \\ddot{x}, then obtain \\log{(f^{*})} \\frac{d}{d S} 1 = \\log{(f^{*})} \\frac{\\partial}{\\partial S} \\frac{S - \\ddot{x}}{\\mathbf{J}_f{(\\ddot{x},S)}}", "derivation": "\\mathbf{J}_f{(\\ddot{x},S)} = S - \\ddot{x} and 1 = \\frac{S - \\ddot{x}}{\\mathbf{J}_f{(\\ddot{x},S)}} and \\frac{d}{d S} 1 = \\frac{\\partial}{\\partial S} \\frac{S - \\ddot{x}}{\\mathbf{J}_f{(\\ddot{x},S)}} and \\log{(f^{*})} \\frac{d}{d S} 1 = \\log{(f^{*})} \\frac{\\partial}{\\partial S} \\frac{S - \\ddot{x}}{\\mathbf{J}_f{(\\ddot{x},S)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('S', commutative=True)), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('S', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('S', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('S', commutative=True)), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["divide", 3, "Pow(log(Symbol('f^*', commutative=True)), Integer(-1))"], "Equality(Mul(log(Symbol('f^*', commutative=True)), Derivative(Integer(1), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(log(Symbol('f^*', commutative=True)), Derivative(Mul(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('S', commutative=True)), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(Z)} = \\sin{(Z)}, then derive (\\int (Z + v{(Z)}) dZ)^{Z} = (\\frac{Z^{2}}{2} + v_{z} - \\cos{(Z)})^{Z}, then obtain (\\int (Z + v{(Z)}) dZ)^{Z} + \\iint (Z + v{(Z)}) dZ dZ = (\\frac{Z^{2}}{2} + v_{z} - \\cos{(Z)})^{Z} + \\iint (Z + v{(Z)}) dZ dZ", "derivation": "v{(Z)} = \\sin{(Z)} and Z + v{(Z)} = Z + \\sin{(Z)} and \\int (Z + v{(Z)}) dZ = \\int (Z + \\sin{(Z)}) dZ and \\iint (Z + v{(Z)}) dZ dZ = \\iint (Z + \\sin{(Z)}) dZ dZ and (\\int (Z + v{(Z)}) dZ)^{Z} = (\\int (Z + \\sin{(Z)}) dZ)^{Z} and (\\int (Z + v{(Z)}) dZ)^{Z} = (\\frac{Z^{2}}{2} + v_{z} - \\cos{(Z)})^{Z} and (\\int (Z + v{(Z)}) dZ)^{Z} + \\iint (Z + \\sin{(Z)}) dZ dZ = (\\frac{Z^{2}}{2} + v_{z} - \\cos{(Z)})^{Z} + \\iint (Z + \\sin{(Z)}) dZ dZ and (\\int (Z + v{(Z)}) dZ)^{Z} + \\iint (Z + v{(Z)}) dZ dZ = (\\frac{Z^{2}}{2} + v_{z} - \\cos{(Z)})^{Z} + \\iint (Z + v{(Z)}) dZ dZ", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["add", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)))"], [["add", 6, "Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Add(Pow(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Pow(Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), Function('v')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given m{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} = \\cos{(\\mathbf{v})} - \\frac{d}{d t_{2}} \\sin{(t_{2})}, then derive \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} = \\cos{(\\mathbf{v})} - \\cos{(t_{2})}, then obtain \\int \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} d\\mathbf{v} = \\int (m{(\\mathbf{v})} - \\cos{(t_{2})}) d\\mathbf{v}", "derivation": "m{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} = \\cos{(\\mathbf{v})} - \\frac{d}{d t_{2}} \\sin{(t_{2})} and \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} = \\cos{(\\mathbf{v})} - \\cos{(t_{2})} and \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} = m{(\\mathbf{v})} - \\cos{(t_{2})} and \\int \\dot{\\mathbf{r}}{(\\mathbf{v},t_{2})} d\\mathbf{v} = \\int (m{(\\mathbf{v})} - \\cos{(t_{2})}) d\\mathbf{v}", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('t_2', commutative=True)), Add(cos(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('t_2', commutative=True)), Add(cos(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('t_2', commutative=True)), Add(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\delta{(F_{c})} = e^{F_{c}}, then obtain e^{- 2 F_{c}} \\int \\sin{(\\delta{(F_{c})})} dF_{c} = (\\mathbf{A} + \\operatorname{Si}{(e^{F_{c}})}) e^{- 2 F_{c}}", "derivation": "\\delta{(F_{c})} = e^{F_{c}} and \\sin{(\\delta{(F_{c})})} = \\sin{(e^{F_{c}})} and \\int \\sin{(\\delta{(F_{c})})} dF_{c} = \\int \\sin{(e^{F_{c}})} dF_{c} and e^{- 2 F_{c}} \\int \\sin{(\\delta{(F_{c})})} dF_{c} = e^{- 2 F_{c}} \\int \\sin{(e^{F_{c}})} dF_{c} and e^{- 2 F_{c}} \\int \\sin{(\\delta{(F_{c})})} dF_{c} = (\\mathbf{A} + \\operatorname{Si}{(e^{F_{c}})}) e^{- 2 F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\delta')(Symbol('F_c', commutative=True))), sin(exp(Symbol('F_c', commutative=True))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(sin(Function('\\\\delta')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(sin(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], [["divide", 3, "exp(Mul(Integer(2), Symbol('F_c', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True))), Integral(sin(Function('\\\\delta')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True))), Integral(sin(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(exp(Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True))), Integral(sin(Function('\\\\delta')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Si(exp(Symbol('F_c', commutative=True)))), exp(Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given J{(M_{E})} = \\int \\cos{(M_{E})} dM_{E}, then derive \\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E} = \\sin{(M_{E})}, then obtain (\\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E})^{2} = (\\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E}) \\frac{d}{d M_{E}} \\iint \\cos{(M_{E})} dM_{E} dM_{E}", "derivation": "J{(M_{E})} = \\int \\cos{(M_{E})} dM_{E} and \\int J{(M_{E})} dM_{E} = \\iint \\cos{(M_{E})} dM_{E} dM_{E} and \\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E} = \\frac{d}{d M_{E}} \\iint \\cos{(M_{E})} dM_{E} dM_{E} and \\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E} = \\sin{(M_{E})} and \\sin{(M_{E})} \\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E} = \\sin{(M_{E})} \\frac{d}{d M_{E}} \\iint \\cos{(M_{E})} dM_{E} dM_{E} and (\\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E})^{2} = (\\frac{d}{d M_{E}} \\int J{(M_{E})} dM_{E}) \\frac{d}{d M_{E}} \\iint \\cos{(M_{E})} dM_{E} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('M_E', commutative=True)), Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), sin(Symbol('M_E', commutative=True)))"], [["times", 3, "sin(Symbol('M_E', commutative=True))"], "Equality(Mul(sin(Symbol('M_E', commutative=True)), Derivative(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(sin(Symbol('M_E', commutative=True)), Derivative(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Integral(Function('J')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given q{(\\theta_1)} = \\cos{(\\theta_1)}, then derive \\frac{d}{d \\theta_1} q{(\\theta_1)} = - \\sin{(\\theta_1)}, then obtain \\int \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} d\\theta_1 = \\int - \\sin{(\\theta_1)} d\\theta_1", "derivation": "q{(\\theta_1)} = \\cos{(\\theta_1)} and \\frac{d}{d \\theta_1} q{(\\theta_1)} = \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} and \\frac{d}{d \\theta_1} q{(\\theta_1)} = - \\sin{(\\theta_1)} and \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} = - \\sin{(\\theta_1)} and \\int \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} d\\theta_1 = \\int - \\sin{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{J}_M,L)} = L + \\mathbf{J}_M and \\chi{(\\mathbf{J}_M,L)} = \\frac{\\partial}{\\partial L} (L + \\mathbf{J}_M), then derive (\\frac{\\partial}{\\partial L} \\operatorname{c_{0}}{(\\mathbf{J}_M,L)})^{\\mathbf{J}_M} = 1, then obtain 1 = \\chi^{- \\mathbf{J}_M}{(\\mathbf{J}_M,L)}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{J}_M,L)} = L + \\mathbf{J}_M and \\frac{\\partial}{\\partial L} \\operatorname{c_{0}}{(\\mathbf{J}_M,L)} = \\frac{\\partial}{\\partial L} (L + \\mathbf{J}_M) and (\\frac{\\partial}{\\partial L} \\operatorname{c_{0}}{(\\mathbf{J}_M,L)})^{\\mathbf{J}_M} = (\\frac{\\partial}{\\partial L} (L + \\mathbf{J}_M))^{\\mathbf{J}_M} and (\\frac{\\partial}{\\partial L} \\operatorname{c_{0}}{(\\mathbf{J}_M,L)})^{\\mathbf{J}_M} = 1 and (\\frac{\\partial}{\\partial L} (L + \\mathbf{J}_M))^{\\mathbf{J}_M} = 1 and \\chi{(\\mathbf{J}_M,L)} = \\frac{\\partial}{\\partial L} (L + \\mathbf{J}_M) and \\chi^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} = 1 and 1 = \\chi^{- \\mathbf{J}_M}{(\\mathbf{J}_M,L)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Derivative(Function('c_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('c_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Derivative(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], [["divide", 7, "Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(1), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given G{(m,J)} = \\frac{e^{J}}{m}, then obtain (- (\\frac{e^{J}}{m})^{J} + G{(m,J)})^{J} = (- (\\frac{e^{J}}{m})^{J} + \\frac{e^{J}}{m})^{J}", "derivation": "G{(m,J)} = \\frac{e^{J}}{m} and G^{J}{(m,J)} = (\\frac{e^{J}}{m})^{J} and G{(m,J)} - G^{J}{(m,J)} = - G^{J}{(m,J)} + \\frac{e^{J}}{m} and - (\\frac{e^{J}}{m})^{J} + G{(m,J)} = - (\\frac{e^{J}}{m})^{J} + \\frac{e^{J}}{m} and (- (\\frac{e^{J}}{m})^{J} + G{(m,J)})^{J} = (- (\\frac{e^{J}}{m})^{J} + \\frac{e^{J}}{m})^{J}", "srepr_derivation": [["get_premise", "Equality(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["minus", 1, "Pow(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Add(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True)))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Function('G')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)} = \\dot{\\mathbf{r}}^{\\theta_2}, then obtain \\int 0 d\\theta_2 + 1 = \\int (\\dot{\\mathbf{r}}^{\\theta_2} - \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)}) d\\theta_2 + 1", "derivation": "\\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)} = \\dot{\\mathbf{r}}^{\\theta_2} and t_{1} + \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)} = \\dot{\\mathbf{r}}^{\\theta_2} + t_{1} and 0 = \\dot{\\mathbf{r}}^{\\theta_2} - \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)} and \\int 0 d\\theta_2 = \\int (\\dot{\\mathbf{r}}^{\\theta_2} - \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)}) d\\theta_2 and \\int 0 d\\theta_2 + 1 = \\int (\\dot{\\mathbf{r}}^{\\theta_2} - \\eta^{\\prime}{(\\dot{\\mathbf{r}},\\theta_2)}) d\\theta_2 + 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 2, "Add(Symbol('t_1', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)), Add(Integral(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} = \\frac{\\partial}{\\partial g} (\\hat{H} + g), then derive \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = \\dot{z} + \\hat{H}, then obtain \\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} = \\frac{\\partial}{\\partial g} (\\hat{H} + g) and \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = \\int \\frac{\\partial}{\\partial g} (\\hat{H} + g) d\\hat{H} and \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = \\dot{z} + \\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = \\frac{\\partial}{\\partial \\hat{H}} (\\dot{z} + \\hat{H}) and \\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{f_{\\mathbf{v}}}{(g,\\hat{H})} d\\hat{H} = 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given i{(\\phi_2,\\sigma_p)} = \\sigma_p^{\\phi_2}, then obtain \\frac{\\partial}{\\partial \\phi_2} (- \\dot{y} + \\frac{i{(\\phi_2,\\sigma_p)}}{\\dot{y}}) = \\frac{\\partial}{\\partial \\phi_2} (- \\dot{y} + \\frac{\\sigma_p^{\\phi_2}}{\\dot{y}})", "derivation": "i{(\\phi_2,\\sigma_p)} = \\sigma_p^{\\phi_2} and \\frac{i{(\\phi_2,\\sigma_p)}}{\\dot{y}} = \\frac{\\sigma_p^{\\phi_2}}{\\dot{y}} and - \\dot{y} + \\frac{i{(\\phi_2,\\sigma_p)}}{\\dot{y}} = - \\dot{y} + \\frac{\\sigma_p^{\\phi_2}}{\\dot{y}} and \\frac{\\partial}{\\partial \\phi_2} (- \\dot{y} + \\frac{i{(\\phi_2,\\sigma_p)}}{\\dot{y}}) = \\frac{\\partial}{\\partial \\phi_2} (- \\dot{y} + \\frac{\\sigma_p^{\\phi_2}}{\\dot{y}})", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["minus", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(Q)} = \\cos{(Q)}, then obtain \\mathbf{s}{(Q)} \\cos{(Q)} + \\frac{d}{d Q} \\mathbf{s}{(Q)} = \\cos^{2}{(Q)} + \\frac{d}{d Q} \\mathbf{s}{(Q)}", "derivation": "\\mathbf{s}{(Q)} = \\cos{(Q)} and \\frac{d}{d Q} \\mathbf{s}{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\mathbf{s}{(Q)} \\cos{(Q)} = \\cos^{2}{(Q)} and \\mathbf{s}{(Q)} \\cos{(Q)} + \\frac{d}{d Q} \\cos{(Q)} = \\cos^{2}{(Q)} + \\frac{d}{d Q} \\cos{(Q)} and \\mathbf{s}{(Q)} \\cos{(Q)} + \\frac{d}{d Q} \\mathbf{s}{(Q)} = \\cos^{2}{(Q)} + \\frac{d}{d Q} \\mathbf{s}{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 1, "cos(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Pow(cos(Symbol('Q', commutative=True)), Integer(2)))"], [["add", 3, "Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})} = - B + J_{\\varepsilon}, then obtain - \\frac{2 \\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})}}{B} = - \\frac{\\frac{\\partial}{\\partial B} (- B + J_{\\varepsilon})}{B} - \\frac{\\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})}}{B}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})} = - B + J_{\\varepsilon} and \\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})} = \\frac{\\partial}{\\partial B} (- B + J_{\\varepsilon}) and - \\frac{\\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})}}{B} = - \\frac{\\frac{\\partial}{\\partial B} (- B + J_{\\varepsilon})}{B} and - \\frac{2 \\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})}}{B} = - \\frac{\\frac{\\partial}{\\partial B} (- B + J_{\\varepsilon})}{B} - \\frac{\\frac{\\partial}{\\partial B} \\operatorname{f_{\\mathbf{p}}}{(B,J_{\\varepsilon})}}{B}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Integer(2), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('B', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then obtain ((\\frac{\\operatorname{C_{2}}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})})^{\\mathbf{F}} = ((\\frac{\\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})})^{\\mathbf{F}}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\frac{\\operatorname{C_{2}}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\cos{(\\mathbf{F})}}{\\mathbf{F}} and (\\frac{\\operatorname{C_{2}}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} = (\\frac{\\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} and (\\frac{\\operatorname{C_{2}}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})} = (\\frac{\\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})} and ((\\frac{\\operatorname{C_{2}}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})})^{\\mathbf{F}} = ((\\frac{\\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + \\cos{(\\mathbf{F})})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 3, "cos(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(y)} = \\int \\sin{(y)} dy, then derive \\dot{x}^{2}{(y)} = (A - \\cos{(y)}) \\dot{x}{(y)}, then derive \\dot{x}{(y)} = Z - \\cos{(y)}, then obtain - \\frac{- A + (A - \\cos{(y)}) (Z - \\cos{(y)})}{A} = - \\frac{- A + (Z - \\cos{(y)}) \\int \\sin{(y)} dy}{A}", "derivation": "\\dot{x}{(y)} = \\int \\sin{(y)} dy and \\dot{x}^{2}{(y)} = \\dot{x}{(y)} \\int \\sin{(y)} dy and \\dot{x}^{2}{(y)} = (A - \\cos{(y)}) \\dot{x}{(y)} and (A - \\cos{(y)}) \\dot{x}{(y)} = \\dot{x}{(y)} \\int \\sin{(y)} dy and - A + (A - \\cos{(y)}) \\dot{x}{(y)} = - A + \\dot{x}{(y)} \\int \\sin{(y)} dy and \\dot{x}{(y)} = Z - \\cos{(y)} and - A + (A - \\cos{(y)}) (Z - \\cos{(y)}) = - A + (Z - \\cos{(y)}) \\int \\sin{(y)} dy and - \\frac{- A + (A - \\cos{(y)}) (Z - \\cos{(y)})}{A} = - \\frac{- A + (Z - \\cos{(y)}) \\int \\sin{(y)} dy}{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["times", 1, "Function('\\\\dot{x}')(Symbol('y', commutative=True))"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integer(2)), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Function('\\\\dot{x}')(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Function('\\\\dot{x}')(Symbol('y', commutative=True))), Mul(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["minus", 4, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Function('\\\\dot{x}')(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{x}')(Symbol('y', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Add(Symbol('Z', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"], [["divide", 7, "Mul(Integer(-1), Symbol('A', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True))))))), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Add(Symbol('Z', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(t_{1},L)} = \\cos{(L + t_{1})}, then derive \\sin{(L + t_{1})} + \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)} = 0, then obtain (\\sin{(L + t_{1})} + \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)}) \\frac{\\partial}{\\partial t_{1}} \\cos{(L + t_{1})} = 0", "derivation": "\\operatorname{v_{1}}{(t_{1},L)} = \\cos{(L + t_{1})} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)} = \\frac{\\partial}{\\partial t_{1}} \\cos{(L + t_{1})} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)} - \\frac{\\partial}{\\partial t_{1}} \\cos{(L + t_{1})} = 0 and \\sin{(L + t_{1})} + \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)} = 0 and (\\sin{(L + t_{1})} + \\frac{\\partial}{\\partial t_{1}} \\operatorname{v_{1}}{(t_{1},L)}) \\frac{\\partial}{\\partial t_{1}} \\cos{(L + t_{1})} = 0", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('t_1', commutative=True), Symbol('L', commutative=True)), cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('t_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('v_1')(Symbol('t_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(sin(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Derivative(Function('v_1')(Symbol('t_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Integer(0))"], [["times", 4, "Derivative(cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Mul(Add(sin(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Derivative(Function('v_1')(Symbol('t_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Derivative(cos(Add(Symbol('L', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given q{(\\theta)} = \\sin{(\\theta)} and \\operatorname{L_{\\varepsilon}}{(\\theta)} = \\sin{(\\theta)}, then obtain (\\frac{q{(\\theta)} - \\sin{(\\theta)}}{2 \\sin{(\\theta)}})^{\\theta} = 0^{\\theta}", "derivation": "q{(\\theta)} = \\sin{(\\theta)} and \\operatorname{L_{\\varepsilon}}{(\\theta)} = \\sin{(\\theta)} and q{(\\theta)} = \\operatorname{L_{\\varepsilon}}{(\\theta)} and \\operatorname{L_{\\varepsilon}}{(\\theta)} - \\sin{(\\theta)} = 0 and q{(\\theta)} - \\sin{(\\theta)} = 0 and \\frac{q{(\\theta)} - \\sin{(\\theta)}}{2 \\sin{(\\theta)}} = 0 and (\\frac{q{(\\theta)} - \\sin{(\\theta)}}{2 \\sin{(\\theta)}})^{\\theta} = 0^{\\theta}", "srepr_derivation": [["renaming_premise", "Equality(Function('q')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('q')(Symbol('\\\\theta', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True)))"], [["minus", 2, "sin(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('q')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["divide", 5, "Mul(Integer(2), sin(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('q')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(-1))), Integer(0))"], [["power", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Add(Function('q')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(-1))), Symbol('\\\\theta', commutative=True)), Pow(Integer(0), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given E{(\\mathbf{M})} = \\log{(\\mathbf{M})}, then obtain \\frac{E{(\\mathbf{M})} - \\log{(\\mathbf{M})}}{\\mathbf{M} \\log{(\\mathbf{M})}} = 0", "derivation": "E{(\\mathbf{M})} = \\log{(\\mathbf{M})} and E{(\\mathbf{M})} - \\log{(\\mathbf{M})} = 0 and \\frac{E{(\\mathbf{M})} - \\log{(\\mathbf{M})}}{\\mathbf{M}} = 0 and \\frac{E{(\\mathbf{M})} - \\log{(\\mathbf{M})}}{\\mathbf{M} \\log{(\\mathbf{M})}} = 0", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Function('E')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{M}', commutative=True)))), Integer(0))"], [["divide", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Function('E')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{M}', commutative=True))))), Integer(0))"], [["divide", 3, "log(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Function('E')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{M}', commutative=True)))), Pow(log(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\chi{(T)} = \\sin{(T)} and Q{(T)} = \\chi{(T)} + \\sin{(T)}, then obtain T + (- \\frac{T^{2}}{2} + \\mathbf{F} + \\chi{(T)} + \\sin{(T)} - 2 \\cos{(T)})^{T} = T + (- \\frac{T^{2}}{2} + \\mathbf{F} + 2 \\chi{(T)} - 2 \\cos{(T)})^{T}", "derivation": "\\chi{(T)} = \\sin{(T)} and Q{(T)} = \\chi{(T)} + \\sin{(T)} and Q{(T)} = 2 \\chi{(T)} and \\chi{(T)} + \\sin{(T)} = 2 \\chi{(T)} and \\chi{(T)} + \\sin{(T)} + \\int (- T + 2 \\sin{(T)}) dT = 2 \\chi{(T)} + \\int (- T + 2 \\sin{(T)}) dT and (\\chi{(T)} + \\sin{(T)} + \\int (- T + 2 \\sin{(T)}) dT)^{T} = (2 \\chi{(T)} + \\int (- T + 2 \\sin{(T)}) dT)^{T} and T + (\\chi{(T)} + \\sin{(T)} + \\int (- T + 2 \\sin{(T)}) dT)^{T} = T + (2 \\chi{(T)} + \\int (- T + 2 \\sin{(T)}) dT)^{T} and T + (- \\frac{T^{2}}{2} + \\mathbf{F} + \\chi{(T)} + \\sin{(T)} - 2 \\cos{(T)})^{T} = T + (- \\frac{T^{2}}{2} + \\mathbf{F} + 2 \\chi{(T)} - 2 \\cos{(T)})^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('T', commutative=True)), Add(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Q')(Symbol('T', commutative=True)), Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))), Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))))"], [["add", 4, "Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))))"], [["power", 5, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["minus", 6, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Pow(Add(Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), Pow(Add(Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), sin(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\chi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(2), Function('\\\\chi')(Symbol('T', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(A_{x})} = e^{A_{x}}, then obtain \\frac{d}{d A_{x}} \\int (A_{x} + e^{A_{x}}) (\\operatorname{f_{E}}{(A_{x})} - e^{A_{x}}) dA_{x} = \\frac{d}{d A_{x}} \\int 0 dA_{x}", "derivation": "\\operatorname{f_{E}}{(A_{x})} = e^{A_{x}} and \\operatorname{f_{E}}{(A_{x})} - e^{A_{x}} = 0 and A_{x} + \\operatorname{f_{E}}{(A_{x})} = A_{x} + e^{A_{x}} and (A_{x} + \\operatorname{f_{E}}{(A_{x})}) (\\operatorname{f_{E}}{(A_{x})} - e^{A_{x}}) = 0 and (A_{x} + e^{A_{x}}) (\\operatorname{f_{E}}{(A_{x})} - e^{A_{x}}) = 0 and \\int (A_{x} + e^{A_{x}}) (\\operatorname{f_{E}}{(A_{x})} - e^{A_{x}}) dA_{x} = \\int 0 dA_{x} and \\frac{d}{d A_{x}} \\int (A_{x} + e^{A_{x}}) (\\operatorname{f_{E}}{(A_{x})} - e^{A_{x}}) dA_{x} = \\frac{d}{d A_{x}} \\int 0 dA_{x}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["minus", 1, "exp(Symbol('A_x', commutative=True))"], "Equality(Add(Function('f_E')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))), Integer(0))"], [["add", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Function('f_E')(Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), exp(Symbol('A_x', commutative=True))))"], [["times", 2, "Add(Symbol('A_x', commutative=True), Function('f_E')(Symbol('A_x', commutative=True)))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Function('f_E')(Symbol('A_x', commutative=True))), Add(Function('f_E')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('A_x', commutative=True), exp(Symbol('A_x', commutative=True))), Add(Function('f_E')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True))))), Integer(0))"], [["integrate", 5, "Symbol('A_x', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('A_x', commutative=True), exp(Symbol('A_x', commutative=True))), Add(Function('f_E')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True))))), Tuple(Symbol('A_x', commutative=True))), Integral(Integer(0), Tuple(Symbol('A_x', commutative=True))))"], [["differentiate", 6, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Integral(Mul(Add(Symbol('A_x', commutative=True), exp(Symbol('A_x', commutative=True))), Add(Function('f_E')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True))))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(M)} = \\sin{(M)} and p{(M)} = - \\sin{(M)}, then obtain (- q{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} = (p{(M)} - 1)^{M} - q{(M)} + \\sin{(M)}", "derivation": "q{(M)} = \\sin{(M)} and p{(M)} = - \\sin{(M)} and p{(M)} - 1 = - \\sin{(M)} - 1 and (p{(M)} - 1)^{M} = (- \\sin{(M)} - 1)^{M} and (p{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} = (- \\sin{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} and p{(M)} - 1 = - q{(M)} - 1 and (- q{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} = (- \\sin{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} and (- q{(M)} - 1)^{M} - q{(M)} + \\sin{(M)} = (p{(M)} - 1)^{M} - q{(M)} + \\sin{(M)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('M', commutative=True)), Mul(Integer(-1), sin(Symbol('M', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('p')(Symbol('M', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('M', commutative=True))), Integer(-1)))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Add(Function('p')(Symbol('M', commutative=True)), Integer(-1)), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('M', commutative=True))), Integer(-1)), Symbol('M', commutative=True)))"], [["add", 4, "Add(Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True)))"], "Equality(Add(Pow(Add(Function('p')(Symbol('M', commutative=True)), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))), Add(Pow(Add(Mul(Integer(-1), sin(Symbol('M', commutative=True))), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('p')(Symbol('M', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))), Add(Pow(Add(Mul(Integer(-1), sin(Symbol('M', commutative=True))), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))), Add(Pow(Add(Function('p')(Symbol('M', commutative=True)), Integer(-1)), Symbol('M', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\eta{(J)} = e^{J}, then obtain \\frac{d}{d J} \\eta{(J)} e^{J \\eta{(J)} e^{- J} - 2 J} = \\frac{d}{d J} 1", "derivation": "\\eta{(J)} = e^{J} and \\eta{(J)} e^{- J} = 1 and \\frac{d}{d J} \\eta{(J)} e^{- J} = \\frac{d}{d J} 1 and J \\eta{(J)} e^{- J} = J and J \\eta{(J)} e^{- J} - J = 0 and J \\eta{(J)} e^{- J} - 2 J = - J and \\frac{d}{d J} \\eta{(J)} e^{J \\eta{(J)} e^{- J} - 2 J} = \\frac{d}{d J} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["divide", 1, "exp(Symbol('J', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["times", 2, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Symbol('J', commutative=True))"], [["add", 4, "Mul(Integer(-1), Symbol('J', commutative=True))"], "Equality(Add(Mul(Symbol('J', commutative=True), Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(-1), Symbol('J', commutative=True))), Integer(0))"], [["minus", 5, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Symbol('J', commutative=True), Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('J', commutative=True))), Mul(Integer(-1), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Derivative(Mul(Function('\\\\eta')(Symbol('J', commutative=True)), exp(Add(Mul(Symbol('J', commutative=True), Function('\\\\eta')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('J', commutative=True))))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(u,f)} = f u, then obtain (\\pi{(u,f)} + 1) \\int 0 du = (\\frac{f u}{\\pi{(u,f)}} + \\pi{(u,f)}) \\int 0 du", "derivation": "\\pi{(u,f)} = f u and 1 = \\frac{f u}{\\pi{(u,f)}} and 0 = \\frac{f u}{\\pi{(u,f)}} - 1 and \\int 0 du = \\int (\\frac{f u}{\\pi{(u,f)}} - 1) du and \\pi{(u,f)} + 1 = \\frac{f u}{\\pi{(u,f)}} + \\pi{(u,f)} and (\\pi{(u,f)} + 1) \\int (\\frac{f u}{\\pi{(u,f)}} - 1) du = (\\frac{f u}{\\pi{(u,f)}} + \\pi{(u,f)}) \\int (\\frac{f u}{\\pi{(u,f)}} - 1) du and (\\pi{(u,f)} + 1) \\int 0 du = (\\frac{f u}{\\pi{(u,f)}} + \\pi{(u,f)}) \\int 0 du", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('u', commutative=True)))"], [["divide", 1, "Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(1), Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Integer(-1)))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('u', commutative=True))))"], [["add", 2, "Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(1)), Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True))))"], [["times", 5, "Integral(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Add(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(1)), Integral(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('u', commutative=True)))), Mul(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True))), Integral(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Add(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(1)), Integral(Integer(0), Tuple(Symbol('u', commutative=True)))), Mul(Add(Mul(Symbol('f', commutative=True), Symbol('u', commutative=True), Pow(Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('u', commutative=True), Symbol('f', commutative=True))), Integral(Integer(0), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given b{(q,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q), then derive \\hbar - q + b{(q,\\hbar)} = \\hbar - q - 1, then obtain \\hbar - q + b{(q,\\hbar)} + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q) + 1 = \\hbar - q + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q)", "derivation": "b{(q,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q) and \\hbar - q + b{(q,\\hbar)} = \\hbar - q + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q) and \\hbar - q + b{(q,\\hbar)} = \\hbar - q - 1 and \\hbar - q + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q) = \\hbar - q - 1 and \\hbar - q + b{(q,\\hbar)} + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q) + 1 = \\hbar - q + \\frac{\\partial}{\\partial \\hbar} (- \\hbar + q)", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Function('b')(Symbol('q', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Function('b')(Symbol('q', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Function('b')(Symbol('q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"]]}, {"prompt": "Given A{(u,C_{1},t_{1})} = \\frac{t_{1}}{C_{1} u}, then obtain \\frac{t_{1}}{C_{1} u} + \\frac{A{(u,C_{1},t_{1})}}{C_{1}} + \\frac{1}{C_{1}^{3}} = \\frac{t_{1}}{C_{1} u} + \\frac{t_{1}}{C_{1}^{2} u} + \\frac{1}{C_{1}^{3}}", "derivation": "A{(u,C_{1},t_{1})} = \\frac{t_{1}}{C_{1} u} and \\frac{A{(u,C_{1},t_{1})}}{C_{1}} = \\frac{t_{1}}{C_{1}^{2} u} and \\frac{A{(u,C_{1},t_{1})}}{C_{1}} + \\frac{1}{C_{1}^{3}} = \\frac{t_{1}}{C_{1}^{2} u} + \\frac{1}{C_{1}^{3}} and \\frac{t_{1}}{C_{1} u} + \\frac{A{(u,C_{1},t_{1})}}{C_{1}} + \\frac{1}{C_{1}^{3}} = \\frac{t_{1}}{C_{1} u} + \\frac{t_{1}}{C_{1}^{2} u} + \\frac{1}{C_{1}^{3}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('u', commutative=True), Symbol('C_1', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True), Symbol('C_1', commutative=True), Symbol('t_1', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["add", 2, "Pow(Symbol('C_1', commutative=True), Integer(-3))"], "Equality(Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True), Symbol('C_1', commutative=True), Symbol('t_1', commutative=True))), Pow(Symbol('C_1', commutative=True), Integer(-3))), Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Pow(Symbol('C_1', commutative=True), Integer(-3))))"], [["add", 3, "Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True), Symbol('C_1', commutative=True), Symbol('t_1', commutative=True))), Pow(Symbol('C_1', commutative=True), Integer(-3))), Add(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Symbol('t_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Pow(Symbol('C_1', commutative=True), Integer(-3))))"]]}, {"prompt": "Given U{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} = - F_{x} + \\log{(\\mathbf{p})}, then obtain - F_{x} + \\mathbf{v} \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} + \\log{(\\mathbf{p})} - \\cos{(\\mathbf{v})} = - F_{x} + \\mathbf{v} (- F_{x} + \\log{(\\mathbf{p})}) + \\log{(\\mathbf{p})} - \\cos{(\\mathbf{v})}", "derivation": "U{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} = - F_{x} + \\log{(\\mathbf{p})} and \\mathbf{v} \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} = \\mathbf{v} (- F_{x} + \\log{(\\mathbf{p})}) and - F_{x} + \\mathbf{v} \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} - U{(\\mathbf{v})} + \\log{(\\mathbf{p})} = - F_{x} + \\mathbf{v} (- F_{x} + \\log{(\\mathbf{p})}) - U{(\\mathbf{v})} + \\log{(\\mathbf{p})} and - F_{x} + \\mathbf{v} \\Psi_{\\lambda}{(\\mathbf{p},F_{x})} + \\log{(\\mathbf{p})} - \\cos{(\\mathbf{v})} = - F_{x} + \\mathbf{v} (- F_{x} + \\log{(\\mathbf{p})}) + \\log{(\\mathbf{p})} - \\cos{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{v}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{v}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{v}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))), log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(r_{0})} = \\int e^{r_{0}} dr_{0}, then obtain \\frac{\\sin{(\\operatorname{F_{c}}{(r_{0})})}}{\\iint e^{r_{0}} dr_{0} dr_{0}} = \\frac{\\sin{(\\int e^{r_{0}} dr_{0})}}{\\iint e^{r_{0}} dr_{0} dr_{0}}", "derivation": "\\operatorname{F_{c}}{(r_{0})} = \\int e^{r_{0}} dr_{0} and \\int \\operatorname{F_{c}}{(r_{0})} dr_{0} = \\iint e^{r_{0}} dr_{0} dr_{0} and \\sin{(\\operatorname{F_{c}}{(r_{0})})} = \\sin{(\\int e^{r_{0}} dr_{0})} and \\frac{\\sin{(\\operatorname{F_{c}}{(r_{0})})}}{\\int \\operatorname{F_{c}}{(r_{0})} dr_{0}} = \\frac{\\sin{(\\int e^{r_{0}} dr_{0})}}{\\int \\operatorname{F_{c}}{(r_{0})} dr_{0}} and \\frac{\\sin{(\\operatorname{F_{c}}{(r_{0})})}}{\\iint e^{r_{0}} dr_{0} dr_{0}} = \\frac{\\sin{(\\int e^{r_{0}} dr_{0})}}{\\iint e^{r_{0}} dr_{0} dr_{0}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('r_0', commutative=True)), Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["sin", 1], "Equality(sin(Function('F_c')(Symbol('r_0', commutative=True))), sin(Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))))"], [["divide", 3, "Integral(Function('F_c')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))"], "Equality(Mul(sin(Function('F_c')(Symbol('r_0', commutative=True))), Pow(Integral(Function('F_c')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integer(-1))), Mul(sin(Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Pow(Integral(Function('F_c')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(sin(Function('F_c')(Symbol('r_0', commutative=True))), Pow(Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integer(-1))), Mul(sin(Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Pow(Integral(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}{(H,Z)} = H + Z, then derive H + \\int \\tilde{g}{(H,Z)} dH = \\frac{H^{2}}{2} + H Z + H + x^\\prime, then obtain \\int (H + \\int (H + Z) dH) dZ = \\int (\\frac{H^{2}}{2} + H Z + H + x^\\prime) dZ", "derivation": "\\tilde{g}{(H,Z)} = H + Z and \\int \\tilde{g}{(H,Z)} dH = \\int (H + Z) dH and - Z + \\int \\tilde{g}{(H,Z)} dH = - Z + \\int (H + Z) dH and H + \\int \\tilde{g}{(H,Z)} dH = H + \\int (H + Z) dH and H + \\int \\tilde{g}{(H,Z)} dH = \\frac{H^{2}}{2} + H Z + H + x^\\prime and H + \\int (H + Z) dH = \\frac{H^{2}}{2} + H Z + H + x^\\prime and \\int (H + \\int (H + Z) dH) dZ = \\int (\\frac{H^{2}}{2} + H Z + H + x^\\prime) dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["minus", 2, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Integral(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Integral(Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["add", 3, "Add(Symbol('H', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Symbol('H', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), Integral(Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('H', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Symbol('H', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('H', commutative=True), Integral(Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Symbol('H', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 6, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Integral(Add(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('Z', commutative=True)), Symbol('H', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\theta{(M)} = e^{M} and E{(M)} = e^{M}, then obtain - M - E{(M)} + \\theta^{M}{(M)} e^{M} = - M - E{(M)} + E^{M}{(M)} e^{M}", "derivation": "\\theta{(M)} = e^{M} and E{(M)} = e^{M} and \\theta{(M)} = E{(M)} and \\theta^{M}{(M)} = E^{M}{(M)} and \\theta^{M}{(M)} e^{M} = E^{M}{(M)} e^{M} and - M - E{(M)} + \\theta^{M}{(M)} e^{M} = - M - E{(M)} + E^{M}{(M)} e^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\theta')(Symbol('M', commutative=True)), Function('E')(Symbol('M', commutative=True)))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Function('E')(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["times", 4, "exp(Symbol('M', commutative=True))"], "Equality(Mul(Pow(Function('\\\\theta')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Mul(Pow(Function('E')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))))"], [["minus", 5, "Add(Symbol('M', commutative=True), Function('E')(Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('M', commutative=True))), Mul(Pow(Function('\\\\theta')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('M', commutative=True))), Mul(Pow(Function('E')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(I,U)} = \\cos{(I + U)} and \\sigma_{p}{(I,U)} = \\frac{\\int \\mathbf{r}{(I,U)} dI}{\\int \\cos{(I + U)} dI}, then derive \\int \\mathbf{r}{(I,U)} dI = B + \\sin{(I + U)}, then derive \\sigma_{p}{(I,U)} = \\frac{\\int \\mathbf{r}{(I,U)} dI}{\\varphi + \\sin{(I + U)}}, then obtain \\sigma_{p}{(I,U)} = \\frac{B + \\sin{(I + U)}}{\\varphi + \\sin{(I + U)}}", "derivation": "\\mathbf{r}{(I,U)} = \\cos{(I + U)} and \\int \\mathbf{r}{(I,U)} dI = \\int \\cos{(I + U)} dI and \\int \\mathbf{r}{(I,U)} dI = B + \\sin{(I + U)} and \\sigma_{p}{(I,U)} = \\frac{\\int \\mathbf{r}{(I,U)} dI}{\\int \\cos{(I + U)} dI} and \\sigma_{p}{(I,U)} = \\frac{\\int \\mathbf{r}{(I,U)} dI}{\\varphi + \\sin{(I + U)}} and \\sigma_{p}{(I,U)} = \\frac{B + \\sin{(I + U)}}{\\varphi + \\sin{(I + U)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('U', commutative=True)), cos(Add(Symbol('I', commutative=True), Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(Add(Symbol('I', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('B', commutative=True), sin(Add(Symbol('I', commutative=True), Symbol('U', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Mul(Integral(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('I', commutative=True))), Pow(Integral(cos(Add(Symbol('I', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('I', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Function('\\\\sigma_p')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Add(Symbol('I', commutative=True), Symbol('U', commutative=True)))), Integer(-1)), Integral(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('\\\\sigma_p')(Symbol('I', commutative=True), Symbol('U', commutative=True)), Mul(Add(Symbol('B', commutative=True), sin(Add(Symbol('I', commutative=True), Symbol('U', commutative=True)))), Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Add(Symbol('I', commutative=True), Symbol('U', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(\\eta,\\phi_1,v_{x})} = \\eta + \\phi_1 + v_{x}, then obtain - \\eta - \\phi_1 - v_{x} + \\Omega{(\\eta,\\phi_1,v_{x})} = 0", "derivation": "\\Omega{(\\eta,\\phi_1,v_{x})} = \\eta + \\phi_1 + v_{x} and \\Omega^{v_{x}}{(\\eta,\\phi_1,v_{x})} = (\\eta + \\phi_1 + v_{x})^{v_{x}} and (\\eta + \\phi_1 + v_{x})^{v_{x}} + \\Omega{(\\eta,\\phi_1,v_{x})} = \\eta + \\phi_1 + v_{x} + (\\eta + \\phi_1 + v_{x})^{v_{x}} and 2 (\\eta + \\phi_1 + v_{x})^{v_{x}} + \\Omega{(\\eta,\\phi_1,v_{x})} = \\eta + \\phi_1 + v_{x} + 2 (\\eta + \\phi_1 + v_{x})^{v_{x}} and \\Omega{(\\eta,\\phi_1,v_{x})} + 2 \\Omega^{v_{x}}{(\\eta,\\phi_1,v_{x})} = \\eta + \\phi_1 + v_{x} + 2 \\Omega^{v_{x}}{(\\eta,\\phi_1,v_{x})} and - \\eta - \\phi_1 - v_{x} + \\Omega{(\\eta,\\phi_1,v_{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["add", 1, "Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["add", 3, "Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True), Mul(Integer(2), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(2), Pow(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True), Mul(Integer(2), Pow(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))))"], [["minus", 5, "Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True), Mul(Integer(2), Pow(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Symbol('v_x', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(a,\\tilde{g}^*)} = - a + e^{\\tilde{g}^*}, then derive \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)} = e^{\\tilde{g}^*}, then obtain 0 = \\frac{\\partial}{\\partial \\tilde{g}^*} (- a + \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)}) - \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)}", "derivation": "\\phi_{1}{(a,\\tilde{g}^*)} = - a + e^{\\tilde{g}^*} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} (- a + e^{\\tilde{g}^*}) and \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)} = e^{\\tilde{g}^*} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} (- a + \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)}) and 0 = \\frac{\\partial}{\\partial \\tilde{g}^*} (- a + \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)}) - \\frac{\\partial}{\\partial \\tilde{g}^*} \\phi_{1}{(a,\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{M}{(Q,\\mu)} = Q + \\mu, then obtain \\sin{(Q + \\mu + \\mathbf{M}{(Q,\\mu)})} = \\sin{(2 \\mathbf{M}{(Q,\\mu)})}", "derivation": "\\mathbf{M}{(Q,\\mu)} = Q + \\mu and Q + \\mu + \\mathbf{M}{(Q,\\mu)} = 2 Q + 2 \\mu and \\sin{(Q + \\mu + \\mathbf{M}{(Q,\\mu)})} = \\sin{(2 Q + 2 \\mu)} and \\sin{(2 \\mathbf{M}{(Q,\\mu)})} = \\sin{(2 Q + 2 \\mu)} and \\sin{(Q + \\mu + \\mathbf{M}{(Q,\\mu)})} = \\sin{(2 \\mathbf{M}{(Q,\\mu)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Add(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["sin", 2], "Equality(sin(Add(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)))), sin(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(sin(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)))), sin(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Add(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)))), sin(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given U{(J,s)} = J + s, then obtain (J + s)^{2} U^{2}{(J,s)} = (J + s) U^{3}{(J,s)}", "derivation": "U{(J,s)} = J + s and U^{2}{(J,s)} = (J + s) U{(J,s)} and U^{4}{(J,s)} = (J + s)^{2} U^{2}{(J,s)} and (J + s)^{2} U^{2}{(J,s)} = (J + s)^{3} U{(J,s)} and (J + s) U^{2}{(J,s)} = (J + s)^{2} U{(J,s)} and U^{4}{(J,s)} = (J + s) U^{3}{(J,s)} and (J + s)^{2} U^{2}{(J,s)} = (J + s) U^{3}{(J,s)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Add(Symbol('J', commutative=True), Symbol('s', commutative=True)))"], [["times", 1, "Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True))"], "Equality(Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2)), Mul(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(4)), Mul(Pow(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(3)), Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True))))"], [["divide", 4, "Add(Symbol('J', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2)), Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(4)), Mul(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Add(Symbol('J', commutative=True), Symbol('s', commutative=True)), Pow(Function('U')(Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\dot{x}{(\\varphi^*)} = \\cos{(\\varphi^*)}, then obtain - \\dot{x}{(\\varphi^*)} + \\frac{\\varphi^* \\dot{x}{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}} = - \\dot{x}{(\\varphi^*)} + \\frac{\\varphi^* \\cos{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}}", "derivation": "\\dot{x}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\varphi^* \\dot{x}{(\\varphi^*)} = \\varphi^* \\cos{(\\varphi^*)} and \\varphi^* \\dot{x}{(\\varphi^*)} + \\dot{x}{(\\varphi^*)} = \\varphi^* \\cos{(\\varphi^*)} + \\dot{x}{(\\varphi^*)} and \\frac{\\varphi^* \\dot{x}{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}} = \\frac{\\varphi^* \\cos{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}} and - \\dot{x}{(\\varphi^*)} + \\frac{\\varphi^* \\dot{x}{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}} = - \\dot{x}{(\\varphi^*)} + \\frac{\\varphi^* \\cos{(\\varphi^*)} + \\dot{x}{(\\varphi^*)}}{\\varphi^* \\dot{x}{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 2, "Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 3, "Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["minus", 4, "Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given u{(A_{y})} = e^{A_{y}}, then derive \\int u{(A_{y})} dA_{y} = P_{g} + e^{A_{y}}, then obtain A_{y} + e^{A_{y}} + \\int u{(A_{y})} dA_{y} = A_{y} + P_{g} + 2 e^{A_{y}}", "derivation": "u{(A_{y})} = e^{A_{y}} and \\int u{(A_{y})} dA_{y} = \\int e^{A_{y}} dA_{y} and A_{y} + e^{A_{y}} + \\int u{(A_{y})} dA_{y} = A_{y} + e^{A_{y}} + \\int e^{A_{y}} dA_{y} and \\int u{(A_{y})} dA_{y} = P_{g} + e^{A_{y}} and \\int e^{A_{y}} dA_{y} = P_{g} + e^{A_{y}} and A_{y} + e^{A_{y}} + \\int u{(A_{y})} dA_{y} = A_{y} + P_{g} + 2 e^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["add", 2, "Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True)))"], "Equality(Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True)), Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True)), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('P_g', commutative=True), exp(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('P_g', commutative=True), exp(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True)), Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Symbol('A_y', commutative=True), Symbol('P_g', commutative=True), Mul(Integer(2), exp(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)}, then obtain - \\operatorname{c_{0}}{(\\varepsilon_0)} \\frac{d}{d \\varepsilon_0} \\operatorname{c_{0}}^{\\varepsilon_0}{(\\varepsilon_0)} = - \\operatorname{c_{0}}{(\\varepsilon_0)} \\frac{d}{d \\varepsilon_0} \\cos^{\\varepsilon_0}{(\\varepsilon_0)}", "derivation": "\\operatorname{c_{0}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\operatorname{c_{0}}^{\\varepsilon_0}{(\\varepsilon_0)} = \\cos^{\\varepsilon_0}{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\operatorname{c_{0}}^{\\varepsilon_0}{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\cos^{\\varepsilon_0}{(\\varepsilon_0)} and - \\operatorname{c_{0}}{(\\varepsilon_0)} \\frac{d}{d \\varepsilon_0} \\operatorname{c_{0}}^{\\varepsilon_0}{(\\varepsilon_0)} = - \\operatorname{c_{0}}{(\\varepsilon_0)} \\frac{d}{d \\varepsilon_0} \\cos^{\\varepsilon_0}{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Pow(Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}{(\\rho,\\eta^{\\prime})} = - \\sin{(\\eta^{\\prime} - \\rho)}, then obtain - \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\tilde{g}^{\\eta^{\\prime}}{(\\rho,\\eta^{\\prime})}}{\\sin{(\\eta^{\\prime} - \\rho)}} = - \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} (- \\sin{(\\eta^{\\prime} - \\rho)})^{\\eta^{\\prime}}}{\\sin{(\\eta^{\\prime} - \\rho)}}", "derivation": "\\tilde{g}{(\\rho,\\eta^{\\prime})} = - \\sin{(\\eta^{\\prime} - \\rho)} and \\tilde{g}^{\\eta^{\\prime}}{(\\rho,\\eta^{\\prime})} = (- \\sin{(\\eta^{\\prime} - \\rho)})^{\\eta^{\\prime}} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\tilde{g}^{\\eta^{\\prime}}{(\\rho,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} (- \\sin{(\\eta^{\\prime} - \\rho)})^{\\eta^{\\prime}} and - \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\tilde{g}^{\\eta^{\\prime}}{(\\rho,\\eta^{\\prime})}}{\\sin{(\\eta^{\\prime} - \\rho)}} = - \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} (- \\sin{(\\eta^{\\prime} - \\rho)})^{\\eta^{\\prime}}}{\\sin{(\\eta^{\\prime} - \\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))))"], "Equality(Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Integer(-1)), Derivative(Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Integer(-1)), Derivative(Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(T)} = \\log{(\\cos{(T)})}, then derive \\frac{d}{d T} \\varepsilon{(T)} = - \\frac{\\sin{(T)}}{\\cos{(T)}}, then obtain 1 = - \\frac{\\sin{(T)}}{\\cos{(T)} \\frac{d}{d T} \\varepsilon{(T)}}", "derivation": "\\varepsilon{(T)} = \\log{(\\cos{(T)})} and \\frac{d}{d T} \\varepsilon{(T)} = \\frac{d}{d T} \\log{(\\cos{(T)})} and \\frac{d}{d T} \\varepsilon{(T)} = - \\frac{\\sin{(T)}}{\\cos{(T)}} and \\frac{\\frac{d}{d T} \\varepsilon{(T)}}{\\frac{d}{d T} \\log{(\\cos{(T)})}} = - \\frac{\\sin{(T)}}{\\cos{(T)} \\frac{d}{d T} \\log{(\\cos{(T)})}} and 1 = - \\frac{\\sin{(T)}}{\\cos{(T)} \\frac{d}{d T} \\varepsilon{(T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('T', commutative=True)), log(cos(Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1))))"], [["divide", 3, "Derivative(log(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Derivative(log(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), sin(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Pow(Derivative(log(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(1), Mul(Integer(-1), sin(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(P_{g},F_{N})} = e^{\\frac{F_{N}}{P_{g}}}, then derive \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})} = \\frac{e^{\\frac{F_{N}}{P_{g}}}}{P_{g}}, then obtain (\\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})})^{2} = \\frac{\\operatorname{A_{x}}{(P_{g},F_{N})} \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})}}{P_{g}}", "derivation": "\\operatorname{A_{x}}{(P_{g},F_{N})} = e^{\\frac{F_{N}}{P_{g}}} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})} = \\frac{\\partial}{\\partial F_{N}} e^{\\frac{F_{N}}{P_{g}}} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})} = \\frac{e^{\\frac{F_{N}}{P_{g}}}}{P_{g}} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})} = \\frac{\\operatorname{A_{x}}{(P_{g},F_{N})}}{P_{g}} and (\\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})})^{2} = \\frac{\\operatorname{A_{x}}{(P_{g},F_{N})} \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{x}}{(P_{g},F_{N})}}{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True))))"], [["times", 4, "Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Derivative(Function('A_x')(Symbol('P_g', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(\\Omega)} = e^{\\Omega}, then obtain \\int Q{(\\Omega)} d\\Omega = \\int (Q{(\\Omega)} - e^{\\Omega})^{\\Omega} Q{(\\Omega)} d\\Omega", "derivation": "Q{(\\Omega)} = e^{\\Omega} and Q{(\\Omega)} - e^{\\Omega} = 0 and (Q{(\\Omega)} - e^{\\Omega})^{\\Omega} = 0^{\\Omega} and (Q{(\\Omega)} - e^{\\Omega})^{\\Omega} Q{(\\Omega)} = 0^{\\Omega} Q{(\\Omega)} and Q{(\\Omega)} = (Q{(\\Omega)} - e^{\\Omega})^{\\Omega} Q{(\\Omega)} and \\int Q{(\\Omega)} d\\Omega = \\int (Q{(\\Omega)} - e^{\\Omega})^{\\Omega} Q{(\\Omega)} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Pow(Integer(0), Symbol('\\\\Omega', commutative=True)))"], [["times", 3, "Function('Q')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Add(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Function('Q')(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\Omega', commutative=True)), Function('Q')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Pow(Add(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Function('Q')(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Add(Function('Q')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Function('Q')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\delta)} = e^{\\delta}, then derive \\int \\mathbf{s}{(\\delta)} d\\delta = u + e^{\\delta}, then obtain \\int (\\iint e^{\\delta} d\\delta du)^{\\delta} du = \\int (\\int (u + e^{\\delta}) du)^{\\delta} du", "derivation": "\\mathbf{s}{(\\delta)} = e^{\\delta} and \\int \\mathbf{s}{(\\delta)} d\\delta = \\int e^{\\delta} d\\delta and \\int \\mathbf{s}{(\\delta)} d\\delta = u + e^{\\delta} and \\int e^{\\delta} d\\delta = u + e^{\\delta} and \\iint e^{\\delta} d\\delta du = \\int (u + e^{\\delta}) du and (\\iint e^{\\delta} d\\delta du)^{\\delta} = (\\int (u + e^{\\delta}) du)^{\\delta} and \\int (\\iint e^{\\delta} d\\delta du)^{\\delta} du = \\int (\\int (u + e^{\\delta}) du)^{\\delta} du", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('u', commutative=True), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('u', commutative=True), exp(Symbol('\\\\delta', commutative=True))))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('u', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["power", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Add(Symbol('u', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["integrate", 6, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(Integral(Add(Symbol('u', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(F_{c},\\sigma_p)} = \\sigma_p^{F_{c}} and \\mathbf{J}_f{(F_{c},\\sigma_p)} = - \\sigma_p^{F_{c}}, then obtain \\sin{(- \\sigma_p^{F_{c}} + 2 \\dot{y}{(F_{c},\\sigma_p)} + \\mathbf{J}_f{(F_{c},\\sigma_p)})} = 0", "derivation": "\\dot{y}{(F_{c},\\sigma_p)} = \\sigma_p^{F_{c}} and - \\sigma_p^{F_{c}} + \\dot{y}{(F_{c},\\sigma_p)} = 0 and - \\sigma_p^{F_{c}} + 2 \\dot{y}{(F_{c},\\sigma_p)} = \\dot{y}{(F_{c},\\sigma_p)} and - \\sin{(\\sigma_p^{F_{c}} - \\dot{y}{(F_{c},\\sigma_p)})} = 0 and \\mathbf{J}_f{(F_{c},\\sigma_p)} = - \\sigma_p^{F_{c}} and 2 \\dot{y}{(F_{c},\\sigma_p)} + \\mathbf{J}_f{(F_{c},\\sigma_p)} = \\dot{y}{(F_{c},\\sigma_p)} and \\sin{(- \\sigma_p^{F_{c}} + 2 \\dot{y}{(F_{c},\\sigma_p)} + \\mathbf{J}_f{(F_{c},\\sigma_p)})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(0))"], [["add", 1, "Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(sin(Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mu_{0}{(C,z)} = C^{z}, then obtain \\frac{\\partial}{\\partial C} 4 C^{2 z} \\mu_{0}{(C,z)} = \\frac{\\partial}{\\partial C} 4 C^{3 z}", "derivation": "\\mu_{0}{(C,z)} = C^{z} and C^{z} + \\mu_{0}{(C,z)} = 2 C^{z} and 2 C^{z} (C^{z} + \\mu_{0}{(C,z)}) \\mu_{0}{(C,z)} = 2 C^{2 z} (C^{z} + \\mu_{0}{(C,z)}) and (C^{z} + \\mu_{0}{(C,z)})^{2} \\mu_{0}{(C,z)} = C^{z} (C^{z} + \\mu_{0}{(C,z)})^{2} and \\frac{\\partial}{\\partial C} (C^{z} + \\mu_{0}{(C,z)})^{2} \\mu_{0}{(C,z)} = \\frac{\\partial}{\\partial C} C^{z} (C^{z} + \\mu_{0}{(C,z)})^{2} and \\frac{\\partial}{\\partial C} 4 C^{2 z} \\mu_{0}{(C,z)} = \\frac{\\partial}{\\partial C} 4 C^{3 z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)))"], [["add", 1, "Pow(Symbol('C', commutative=True), Symbol('z', commutative=True))"], "Equality(Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Mul(Integer(2), Pow(Symbol('C', commutative=True), Symbol('z', commutative=True))))"], [["times", 1, "Mul(Integer(2), Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Mul(Integer(2), Pow(Symbol('C', commutative=True), Mul(Integer(2), Symbol('z', commutative=True))), Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Integer(2)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Pow(Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Integer(2))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Integer(2)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Pow(Add(Pow(Symbol('C', commutative=True), Symbol('z', commutative=True)), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Mul(Integer(4), Pow(Symbol('C', commutative=True), Mul(Integer(2), Symbol('z', commutative=True))), Function('\\\\mu_0')(Symbol('C', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(4), Pow(Symbol('C', commutative=True), Mul(Integer(3), Symbol('z', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(f^{\\prime},\\sigma_x)} = \\cos{((f^{\\prime})^{\\sigma_x})}, then derive \\frac{\\partial}{\\partial f^{\\prime}} M{(f^{\\prime},\\sigma_x)} = - \\frac{\\sigma_x (f^{\\prime})^{\\sigma_x} \\sin{((f^{\\prime})^{\\sigma_x})}}{f^{\\prime}}, then obtain - \\frac{\\sigma_x (f^{\\prime})^{\\sigma_x} \\sin{((f^{\\prime})^{\\sigma_x})}}{f^{\\prime}} = \\frac{\\partial}{\\partial f^{\\prime}} \\cos{((f^{\\prime})^{\\sigma_x})}", "derivation": "M{(f^{\\prime},\\sigma_x)} = \\cos{((f^{\\prime})^{\\sigma_x})} and \\frac{\\partial}{\\partial f^{\\prime}} M{(f^{\\prime},\\sigma_x)} = \\frac{\\partial}{\\partial f^{\\prime}} \\cos{((f^{\\prime})^{\\sigma_x})} and \\frac{\\partial}{\\partial f^{\\prime}} M{(f^{\\prime},\\sigma_x)} = - \\frac{\\sigma_x (f^{\\prime})^{\\sigma_x} \\sin{((f^{\\prime})^{\\sigma_x})}}{f^{\\prime}} and - \\frac{\\sigma_x (f^{\\prime})^{\\sigma_x} \\sin{((f^{\\prime})^{\\sigma_x})}}{f^{\\prime}} = \\frac{\\partial}{\\partial f^{\\prime}} \\cos{((f^{\\prime})^{\\sigma_x})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), sin(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), sin(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Derivative(cos(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(v_{y},\\theta_2)} = \\frac{e^{\\theta_2}}{v_{y}}, then obtain ((\\operatorname{A_{y}}{(v_{y},\\theta_2)} + 1) e^{\\theta_2})^{\\theta_2} = ((1 + \\frac{e^{\\theta_2}}{v_{y}}) e^{\\theta_2})^{\\theta_2}", "derivation": "\\operatorname{A_{y}}{(v_{y},\\theta_2)} = \\frac{e^{\\theta_2}}{v_{y}} and \\operatorname{A_{y}}{(v_{y},\\theta_2)} + 1 = 1 + \\frac{e^{\\theta_2}}{v_{y}} and (\\operatorname{A_{y}}{(v_{y},\\theta_2)} + 1) e^{\\theta_2} = (1 + \\frac{e^{\\theta_2}}{v_{y}}) e^{\\theta_2} and ((\\operatorname{A_{y}}{(v_{y},\\theta_2)} + 1) e^{\\theta_2})^{\\theta_2} = ((1 + \\frac{e^{\\theta_2}}{v_{y}}) e^{\\theta_2})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('A_y')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_2', commutative=True)))))"], [["times", 2, "exp(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Add(Function('A_y')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), exp(Symbol('\\\\theta_2', commutative=True))), Mul(Add(Integer(1), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_2', commutative=True)))), exp(Symbol('\\\\theta_2', commutative=True))))"], [["power", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Add(Function('A_y')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Add(Integer(1), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_2', commutative=True)))), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given k{(\\pi)} = \\cos{(\\pi)}, then derive \\frac{d}{d \\pi} k{(\\pi)} = - \\sin{(\\pi)}, then obtain - \\sin{(\\pi)} \\frac{d}{d \\pi} k{(\\pi)} = \\sin^{2}{(\\pi)}", "derivation": "k{(\\pi)} = \\cos{(\\pi)} and \\frac{d}{d \\pi} k{(\\pi)} = \\frac{d}{d \\pi} \\cos{(\\pi)} and \\frac{d}{d \\pi} k{(\\pi)} = - \\sin{(\\pi)} and \\frac{d}{d \\pi} \\cos{(\\pi)} = - \\sin{(\\pi)} and \\frac{d}{d \\pi} k{(\\pi)} \\frac{d}{d \\pi} \\cos{(\\pi)} = - \\sin{(\\pi)} \\frac{d}{d \\pi} \\cos{(\\pi)} and - \\sin{(\\pi)} \\frac{d}{d \\pi} k{(\\pi)} = \\sin^{2}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))))"], [["times", 3, "Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('k')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True)), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True)), Derivative(Function('k')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(v_{2},\\theta_2)} = \\theta_2 - v_{2}, then obtain - \\frac{\\theta_2 - v_{2}}{2 \\theta_2 - v_{2}} = \\frac{- \\theta_2 + v_{2}}{2 \\theta_2 - v_{2}}", "derivation": "\\operatorname{f^{*}}{(v_{2},\\theta_2)} = \\theta_2 - v_{2} and \\frac{\\operatorname{f^{*}}{(v_{2},\\theta_2)}}{2 \\theta_2 - v_{2}} = \\frac{\\theta_2 - v_{2}}{2 \\theta_2 - v_{2}} and - \\frac{\\operatorname{f^{*}}{(v_{2},\\theta_2)}}{2 \\theta_2 - v_{2}} = - \\frac{\\theta_2 - v_{2}}{2 \\theta_2 - v_{2}} and - \\frac{\\theta_2 - v_{2}}{2 \\theta_2 - v_{2}} = \\frac{- \\theta_2 + v_{2}}{2 \\theta_2 - v_{2}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1)), Function('f^*')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1)), Function('f^*')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given G{(\\rho_f)} = \\cos{(\\cos{(\\rho_f)})} and \\hat{p}{(\\rho_f)} = \\cos{(\\rho_f)}, then obtain \\frac{d}{d \\rho_f} (- \\rho_f + G{(\\rho_f)}) = \\frac{d}{d \\rho_f} (- \\rho_f + \\cos{(\\hat{p}{(\\rho_f)})})", "derivation": "G{(\\rho_f)} = \\cos{(\\cos{(\\rho_f)})} and \\hat{p}{(\\rho_f)} = \\cos{(\\rho_f)} and G{(\\rho_f)} = \\cos{(\\hat{p}{(\\rho_f)})} and - \\rho_f + G{(\\rho_f)} = - \\rho_f + \\cos{(\\hat{p}{(\\rho_f)})} and \\frac{d}{d \\rho_f} (- \\rho_f + G{(\\rho_f)}) = \\frac{d}{d \\rho_f} (- \\rho_f + \\cos{(\\hat{p}{(\\rho_f)})})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\rho_f', commutative=True)), cos(cos(Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('G')(Symbol('\\\\rho_f', commutative=True)), cos(Function('\\\\hat{p}')(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('G')(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), cos(Function('\\\\hat{p}')(Symbol('\\\\rho_f', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('G')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), cos(Function('\\\\hat{p}')(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(H)} = e^{H}, then derive \\frac{d}{d H} \\hat{p}_0{(H)} = e^{H}, then obtain \\int \\frac{d}{d H} e^{H} dH = \\int e^{H} dH", "derivation": "\\hat{p}_0{(H)} = e^{H} and \\frac{d}{d H} \\hat{p}_0{(H)} = \\frac{d}{d H} e^{H} and \\frac{d}{d H} \\hat{p}_0{(H)} = e^{H} and \\int \\frac{d}{d H} \\hat{p}_0{(H)} dH = \\int e^{H} dH and \\int \\frac{d}{d H} e^{H} dH = \\int e^{H} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), exp(Symbol('H', commutative=True)))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}_0')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\lambda{(p)} = \\sin{(p)}, then obtain \\sin{(\\frac{\\sin{(p)} \\frac{d}{d p} \\lambda{(p)}}{p})} = \\sin{(\\frac{\\sin{(p)} \\frac{d}{d p} \\sin{(p)}}{p})}", "derivation": "\\lambda{(p)} = \\sin{(p)} and \\frac{d}{d p} \\lambda{(p)} = \\frac{d}{d p} \\sin{(p)} and \\frac{\\sin{(p)} \\frac{d}{d p} \\lambda{(p)}}{p} = \\frac{\\sin{(p)} \\frac{d}{d p} \\sin{(p)}}{p} and \\sin{(\\frac{\\sin{(p)} \\frac{d}{d p} \\lambda{(p)}}{p})} = \\sin{(\\frac{\\sin{(p)} \\frac{d}{d p} \\sin{(p)}}{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["times", 2, "Mul(Pow(Symbol('p', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)), Derivative(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), sin(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)), Derivative(sin(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\phi{(\\hat{H},\\rho_f)} = \\frac{\\rho_f}{\\hat{H}} and U{(\\rho_f)} = \\rho_f^{2}, then obtain \\frac{\\rho_f \\phi{(\\hat{H},\\rho_f)}}{\\hat{H}} = \\frac{U{(\\rho_f)}}{\\hat{H}^{2}}", "derivation": "\\phi{(\\hat{H},\\rho_f)} = \\frac{\\rho_f}{\\hat{H}} and - \\phi{(\\hat{H},\\rho_f)} = - \\frac{\\rho_f}{\\hat{H}} and \\frac{\\rho_f \\phi{(\\hat{H},\\rho_f)}}{\\hat{H}} = \\frac{\\rho_f^{2}}{\\hat{H}^{2}} and U{(\\rho_f)} = \\rho_f^{2} and \\frac{\\rho_f \\phi{(\\hat{H},\\rho_f)}}{\\hat{H}} = \\frac{U{(\\rho_f)}}{\\hat{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), Function('U')(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(c)} = \\int \\log{(c)} dc, then derive \\operatorname{A_{y}}{(c)} = \\mu + c \\log{(c)} - c, then derive \\operatorname{A_{y}}{(c)} = 2 \\mu + 2 c \\log{(c)} - 2 c, then obtain 2 \\mu + 2 c \\log{(c)} - 2 c = \\mathbf{g} + c \\log{(c)} - c", "derivation": "\\operatorname{A_{y}}{(c)} = \\int \\log{(c)} dc and \\operatorname{A_{y}}{(c)} = \\mu + c \\log{(c)} - c and (\\mu + c \\log{(c)} - c) \\operatorname{A_{y}}{(c)} = (\\mu + c \\log{(c)} - c)^{2} and \\frac{\\partial}{\\partial \\mu} (\\mu + c \\log{(c)} - c) \\operatorname{A_{y}}{(c)} = \\frac{\\partial}{\\partial \\mu} (\\mu + c \\log{(c)} - c)^{2} and \\operatorname{A_{y}}{(c)} = 2 \\mu + 2 c \\log{(c)} - 2 c and 2 \\mu + 2 c \\log{(c)} - 2 c = \\int \\log{(c)} dc and 2 \\mu + 2 c \\log{(c)} - 2 c = \\mathbf{g} + c \\log{(c)} - c", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('A_y')(Symbol('c', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Function('A_y')(Symbol('c', commutative=True))), Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Function('A_y')(Symbol('c', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('A_y')(Symbol('c', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} = \\frac{e^{J}}{V_{\\mathbf{E}}}, then derive \\frac{\\partial}{\\partial J} \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} = \\frac{e^{J}}{V_{\\mathbf{E}}}, then obtain \\int \\frac{\\partial}{\\partial J} \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} dJ = \\int \\frac{e^{J}}{V_{\\mathbf{E}}} dJ", "derivation": "\\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} = \\frac{e^{J}}{V_{\\mathbf{E}}} and \\frac{\\partial}{\\partial J} \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial J} \\frac{e^{J}}{V_{\\mathbf{E}}} and \\frac{\\partial}{\\partial J} \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} = \\frac{e^{J}}{V_{\\mathbf{E}}} and \\int \\frac{\\partial}{\\partial J} \\operatorname{A_{2}}{(J,V_{\\mathbf{E}})} dJ = \\int \\frac{e^{J}}{V_{\\mathbf{E}}} dJ", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('A_2')(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given E{(\\psi,C_{2})} = C_{2}^{\\psi}, then obtain \\frac{C_{2}^{4}}{E^{2}{(\\psi,C_{2})}} = C_{2}^{4} C_{2}^{- 2 \\psi}", "derivation": "E{(\\psi,C_{2})} = C_{2}^{\\psi} and \\frac{E{(\\psi,C_{2})}}{C_{2}} = \\frac{C_{2}^{\\psi}}{C_{2}} and \\frac{E{(\\psi,C_{2})}}{C_{2}^{2}} = \\frac{C_{2}^{\\psi}}{C_{2}^{2}} and \\frac{C_{2}^{4}}{E^{2}{(\\psi,C_{2})}} = C_{2}^{4} C_{2}^{- 2 \\psi}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\psi', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\psi', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), Function('E')(Symbol('\\\\psi', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["power", 3, "Integer(-2)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(4)), Pow(Function('E')(Symbol('\\\\psi', commutative=True), Symbol('C_2', commutative=True)), Integer(-2))), Mul(Pow(Symbol('C_2', commutative=True), Integer(4)), Pow(Symbol('C_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(C_{1},n)} = C_{1} n, then obtain - n + \\frac{\\partial}{\\partial C_{1}} \\Psi_{nl}{(C_{1},n)} = 0", "derivation": "\\Psi_{nl}{(C_{1},n)} = C_{1} n and \\frac{\\partial}{\\partial C_{1}} \\Psi_{nl}{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} C_{1} n and - n + \\frac{\\partial}{\\partial C_{1}} \\Psi_{nl}{(C_{1},n)} = - n + \\frac{\\partial}{\\partial C_{1}} C_{1} n and - n + \\frac{\\partial}{\\partial C_{1}} \\Psi_{nl}{(C_{1},n)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(g,A)} = A g, then derive \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} = g, then obtain 0^{A} - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} = (g - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)})^{A} - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)}", "derivation": "\\operatorname{v_{1}}{(g,A)} = A g and \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} = \\frac{\\partial}{\\partial A} A g and \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} = g and 0 = g - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} and 0^{A} = (g - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)})^{A} and 0^{A} - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)} = (g - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)})^{A} - \\frac{\\partial}{\\partial A} \\operatorname{v_{1}}{(g,A)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Symbol('A', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('g', commutative=True))"], [["minus", 3, "Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A', commutative=True)), Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Symbol('A', commutative=True)))"], [["add", 5, "Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Add(Pow(Integer(0), Symbol('A', commutative=True)), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Symbol('A', commutative=True)), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(c_{0})} = \\cos{(c_{0})}, then obtain (c_{0} + \\operatorname{f^{\\prime}}{(c_{0})}) \\operatorname{f^{\\prime}}{(c_{0})} = (c_{0} + \\operatorname{f^{\\prime}}{(c_{0})}) \\cos{(c_{0})}", "derivation": "\\operatorname{f^{\\prime}}{(c_{0})} = \\cos{(c_{0})} and c_{0} + \\operatorname{f^{\\prime}}{(c_{0})} = c_{0} + \\cos{(c_{0})} and (c_{0} + \\cos{(c_{0})}) \\operatorname{f^{\\prime}}{(c_{0})} = (c_{0} + \\cos{(c_{0})}) \\cos{(c_{0})} and (c_{0} + \\operatorname{f^{\\prime}}{(c_{0})}) \\operatorname{f^{\\prime}}{(c_{0})} = (c_{0} + \\operatorname{f^{\\prime}}{(c_{0})}) \\cos{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["add", 1, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Function('f^{\\\\prime}')(Symbol('c_0', commutative=True))), Add(Symbol('c_0', commutative=True), cos(Symbol('c_0', commutative=True))))"], [["times", 1, "Add(Symbol('c_0', commutative=True), cos(Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Symbol('c_0', commutative=True), cos(Symbol('c_0', commutative=True))), Function('f^{\\\\prime}')(Symbol('c_0', commutative=True))), Mul(Add(Symbol('c_0', commutative=True), cos(Symbol('c_0', commutative=True))), cos(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('c_0', commutative=True), Function('f^{\\\\prime}')(Symbol('c_0', commutative=True))), Function('f^{\\\\prime}')(Symbol('c_0', commutative=True))), Mul(Add(Symbol('c_0', commutative=True), Function('f^{\\\\prime}')(Symbol('c_0', commutative=True))), cos(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(v_{y})} = \\sin{(v_{y})}, then obtain ((\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\psi^{*}{(v_{y})} dv_{y} dv_{y})^{\\eta} = ((\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\sin{(v_{y})} dv_{y} dv_{y})^{\\eta}", "derivation": "\\psi^{*}{(v_{y})} = \\sin{(v_{y})} and \\int \\psi^{*}{(v_{y})} dv_{y} = \\int \\sin{(v_{y})} dv_{y} and \\iint \\psi^{*}{(v_{y})} dv_{y} dv_{y} = \\iint \\sin{(v_{y})} dv_{y} dv_{y} and (\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\psi^{*}{(v_{y})} dv_{y} dv_{y} = (\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\sin{(v_{y})} dv_{y} dv_{y} and ((\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\psi^{*}{(v_{y})} dv_{y} dv_{y})^{\\eta} = ((\\int \\mathbf{S}{(\\eta,\\tilde{g})} d\\eta) \\iint \\sin{(v_{y})} dv_{y} dv_{y})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["times", 3, "Integral(Function('\\\\mathbf{S}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Mul(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))))"], [["power", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given W{(y,f^{*})} = (f^{*})^{y}, then obtain (y + W{(y,f^{*})} + \\int W{(y,f^{*})} dy)^{y} = (y + W{(y,f^{*})} + \\int (f^{*})^{y} dy)^{y}", "derivation": "W{(y,f^{*})} = (f^{*})^{y} and y + W{(y,f^{*})} = (f^{*})^{y} + y and \\int W{(y,f^{*})} dy = \\int (f^{*})^{y} dy and (f^{*})^{y} + y + \\int W{(y,f^{*})} dy = (f^{*})^{y} + y + \\int (f^{*})^{y} dy and y + W{(y,f^{*})} + \\int W{(y,f^{*})} dy = y + W{(y,f^{*})} + \\int (f^{*})^{y} dy and (y + W{(y,f^{*})} + \\int W{(y,f^{*})} dy)^{y} = (y + W{(y,f^{*})} + \\int (f^{*})^{y} dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True))), Add(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 3, "Add(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Integral(Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Integral(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('y', commutative=True), Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Integral(Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Integral(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Symbol('y', commutative=True), Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Integral(Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Pow(Add(Symbol('y', commutative=True), Function('W')(Symbol('y', commutative=True), Symbol('f^*', commutative=True)), Integral(Pow(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given J{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then obtain \\frac{J{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}} = \\frac{\\sin{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}}", "derivation": "J{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and J^{\\mathbb{I}}{(\\mathbb{I})} = \\sin^{\\mathbb{I}}{(\\mathbb{I})} and J{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})} and J{(\\mathbb{I})} - J^{\\mathbb{I}}{(\\mathbb{I})} = - J^{\\mathbb{I}}{(\\mathbb{I})} + \\sin{(\\mathbb{I})} and \\frac{J{(\\mathbb{I})} - J^{\\mathbb{I}}{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}} = \\frac{- J^{\\mathbb{I}}{(\\mathbb{I})} + \\sin{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}} and \\frac{J{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}} = \\frac{\\sin{(\\mathbb{I})} - \\sin^{\\mathbb{I}}{(\\mathbb{I})}}{\\sin{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))), Add(sin(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 4, "sin(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), sin(Symbol('\\\\mathbb{I}', commutative=True))), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))), Mul(Add(sin(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))), Pow(sin(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(P_{g})} = \\int \\cos{(P_{g})} dP_{g}, then obtain e^{\\mu + \\mathbb{I}{(P_{g})}} = e^{A_{y} + \\sin{(P_{g})}}", "derivation": "\\mathbb{I}{(P_{g})} = \\int \\cos{(P_{g})} dP_{g} and \\frac{d}{d P_{g}} \\mathbb{I}{(P_{g})} = \\frac{d}{d P_{g}} \\int \\cos{(P_{g})} dP_{g} and \\int \\frac{d}{d P_{g}} \\mathbb{I}{(P_{g})} dP_{g} = \\int \\frac{d}{d P_{g}} \\int \\cos{(P_{g})} dP_{g} dP_{g} and e^{\\int \\frac{d}{d P_{g}} \\mathbb{I}{(P_{g})} dP_{g}} = e^{\\int \\frac{d}{d P_{g}} \\int \\cos{(P_{g})} dP_{g} dP_{g}} and e^{\\mu + \\mathbb{I}{(P_{g})}} = e^{A_{y} + \\sin{(P_{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True)), Integral(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Integral(Derivative(Integral(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Derivative(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True)))), exp(Integral(Derivative(Integral(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(exp(Add(Symbol('\\\\mu', commutative=True), Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True)))), exp(Add(Symbol('A_y', commutative=True), sin(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\omega{(y)} = e^{y}, then derive \\int \\omega{(y)} dy = \\mathbf{v} + e^{y}, then obtain (e^{\\int \\omega{(y)} dy})^{\\mathbf{v}} = (e^{\\int e^{y} dy})^{\\mathbf{v}}", "derivation": "\\omega{(y)} = e^{y} and \\int \\omega{(y)} dy = \\int e^{y} dy and e^{\\int \\omega{(y)} dy} = e^{\\int e^{y} dy} and \\int \\omega{(y)} dy = \\mathbf{v} + e^{y} and e^{\\mathbf{v} + e^{y}} = e^{\\int e^{y} dy} and (e^{\\mathbf{v} + e^{y}})^{\\mathbf{v}} = (e^{\\int e^{y} dy})^{\\mathbf{v}} and (e^{\\int \\omega{(y)} dy})^{\\mathbf{v}} = (e^{\\int e^{y} dy})^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\omega')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('y', commutative=True)))), exp(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('y', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(exp(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(exp(Integral(Function('\\\\omega')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(exp(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(F_{g})} = F_{g}, then derive \\frac{d}{d F_{g}} \\mathbf{A}{(F_{g})} = 1, then obtain (\\frac{d}{d F_{g}} F_{g})^{F_{g}} = 1", "derivation": "\\mathbf{A}{(F_{g})} = F_{g} and \\frac{d}{d F_{g}} \\mathbf{A}{(F_{g})} = \\frac{d}{d F_{g}} F_{g} and \\frac{d}{d F_{g}} \\mathbf{A}{(F_{g})} = 1 and \\frac{d}{d F_{g}} F_{g} = 1 and (\\frac{d}{d F_{g}} F_{g})^{F_{g}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('F_g', commutative=True)"], "Equality(Pow(Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hbar)} = \\cos{(\\hbar)} and \\operatorname{P_{e}}{(\\hbar)} = \\frac{d}{d \\hbar} (- \\hbar + \\cos{(\\hbar)}), then derive - \\operatorname{P_{e}}{(\\hbar)} = 1 - \\frac{d}{d \\hbar} \\operatorname{f_{\\mathbf{p}}}{(\\hbar)}, then obtain (- \\operatorname{P_{e}}{(\\hbar)})^{\\hbar} = (1 - \\frac{d}{d \\hbar} \\operatorname{f_{\\mathbf{p}}}{(\\hbar)})^{\\hbar}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hbar)} = \\cos{(\\hbar)} and - \\hbar + \\operatorname{f_{\\mathbf{p}}}{(\\hbar)} = - \\hbar + \\cos{(\\hbar)} and \\operatorname{P_{e}}{(\\hbar)} = \\frac{d}{d \\hbar} (- \\hbar + \\cos{(\\hbar)}) and - \\operatorname{P_{e}}{(\\hbar)} = - \\frac{d}{d \\hbar} (- \\hbar + \\cos{(\\hbar)}) and - \\operatorname{P_{e}}{(\\hbar)} = - \\frac{d}{d \\hbar} (- \\hbar + \\operatorname{f_{\\mathbf{p}}}{(\\hbar)}) and - \\operatorname{P_{e}}{(\\hbar)} = 1 - \\frac{d}{d \\hbar} \\operatorname{f_{\\mathbf{p}}}{(\\hbar)} and (- \\operatorname{P_{e}}{(\\hbar)})^{\\hbar} = (1 - \\frac{d}{d \\hbar} \\operatorname{f_{\\mathbf{p}}}{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True))), Add(Integer(1), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))))"], [["power", 6, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(n_{1})} = n_{1} and Z{(n_{1})} = n_{1} \\hat{X}{(n_{1})}, then obtain \\int n_{1} Z{(n_{1})} dn_{1} = \\int n_{1}^{3} dn_{1}", "derivation": "\\hat{X}{(n_{1})} = n_{1} and Z{(n_{1})} = n_{1} \\hat{X}{(n_{1})} and \\frac{Z{(n_{1})}}{n_{1}} = \\hat{X}{(n_{1})} and n_{1} Z{(n_{1})} = n_{1}^{2} \\hat{X}{(n_{1})} and \\int n_{1} Z{(n_{1})} dn_{1} = \\int n_{1}^{2} \\hat{X}{(n_{1})} dn_{1} and \\int n_{1} Z{(n_{1})} dn_{1} = \\int n_{1}^{3} dn_{1}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], ["renaming_premise", "Equality(Function('Z')(Symbol('n_1', commutative=True)), Mul(Symbol('n_1', commutative=True), Function('\\\\hat{X}')(Symbol('n_1', commutative=True))))"], [["divide", 2, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('Z')(Symbol('n_1', commutative=True))), Function('\\\\hat{X}')(Symbol('n_1', commutative=True)))"], [["times", 3, "Pow(Symbol('n_1', commutative=True), Integer(2))"], "Equality(Mul(Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(2)), Function('\\\\hat{X}')(Symbol('n_1', commutative=True))))"], [["integrate", 4, "Symbol('n_1', commutative=True)"], "Equality(Integral(Mul(Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(2)), Function('\\\\hat{X}')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Integer(3)), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then derive \\frac{d}{d \\hat{H}_l} \\operatorname{A_{y}}{(\\hat{H}_l)} = \\frac{1}{\\hat{H}_l}, then obtain (\\frac{d}{d \\hat{H}_l} \\log{(\\hat{H}_l)})^{2} = \\frac{\\frac{d}{d \\hat{H}_l} \\log{(\\hat{H}_l)}}{\\hat{H}_l}", "derivation": "\\operatorname{A_{y}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\operatorname{A_{y}}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\log{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\operatorname{A_{y}}{(\\hat{H}_l)} = \\frac{1}{\\hat{H}_l} and (\\frac{d}{d \\hat{H}_l} \\operatorname{A_{y}}{(\\hat{H}_l)})^{2} = \\frac{\\frac{d}{d \\hat{H}_l} \\operatorname{A_{y}}{(\\hat{H}_l)}}{\\hat{H}_l} and (\\frac{d}{d \\hat{H}_l} \\log{(\\hat{H}_l)})^{2} = \\frac{\\frac{d}{d \\hat{H}_l} \\log{(\\hat{H}_l)}}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)))"], [["times", 3, "Derivative(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Function('A_y')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(log(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{1}{(F_{g})} = e^{\\sin{(F_{g})}}, then obtain \\frac{d}{d F_{g}} \\theta_{1}^{2}{(F_{g})} e^{\\sin{(F_{g})}} = \\frac{d}{d F_{g}} \\theta_{1}{(F_{g})} e^{2 \\sin{(F_{g})}}", "derivation": "\\theta_{1}{(F_{g})} = e^{\\sin{(F_{g})}} and \\theta_{1}{(F_{g})} e^{\\sin{(F_{g})}} = e^{2 \\sin{(F_{g})}} and \\theta_{1}^{2}{(F_{g})} e^{\\sin{(F_{g})}} = \\theta_{1}{(F_{g})} e^{2 \\sin{(F_{g})}} and \\frac{d}{d F_{g}} \\theta_{1}^{2}{(F_{g})} e^{\\sin{(F_{g})}} = \\frac{d}{d F_{g}} \\theta_{1}{(F_{g})} e^{2 \\sin{(F_{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), exp(sin(Symbol('F_g', commutative=True))))"], [["times", 1, "exp(sin(Symbol('F_g', commutative=True)))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), exp(sin(Symbol('F_g', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('F_g', commutative=True)))))"], [["times", 2, "Function('\\\\theta_1')(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), Integer(2)), exp(sin(Symbol('F_g', commutative=True)))), Mul(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), exp(Mul(Integer(2), sin(Symbol('F_g', commutative=True))))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), Integer(2)), exp(sin(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\theta_1')(Symbol('F_g', commutative=True)), exp(Mul(Integer(2), sin(Symbol('F_g', commutative=True))))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} = \\Psi + \\mathbf{J}, then derive (\\int \\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} d\\Psi)^{\\Psi} = (\\frac{\\Psi^{2}}{2} + \\Psi \\mathbf{J} + \\phi)^{\\Psi}, then obtain (\\int (\\Psi + \\mathbf{J}) d\\Psi)^{\\Psi} = (\\frac{\\Psi^{2}}{2} + \\Psi \\mathbf{J} + \\phi)^{\\Psi}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} = \\Psi + \\mathbf{J} and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} d\\Psi = \\int (\\Psi + \\mathbf{J}) d\\Psi and (\\int \\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} d\\Psi)^{\\Psi} = (\\int (\\Psi + \\mathbf{J}) d\\Psi)^{\\Psi} and (\\int \\operatorname{E_{\\lambda}}{(\\mathbf{J},\\Psi)} d\\Psi)^{\\Psi} = (\\frac{\\Psi^{2}}{2} + \\Psi \\mathbf{J} + \\phi)^{\\Psi} and (\\int (\\Psi + \\mathbf{J}) d\\Psi)^{\\Psi} = (\\frac{\\Psi^{2}}{2} + \\Psi \\mathbf{J} + \\phi)^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given J{(v_{x},\\varphi)} = \\frac{\\varphi}{v_{x}}, then obtain v_{x} J{(v_{x},\\varphi)} + J{(v_{x},\\varphi)} = \\varphi + J{(v_{x},\\varphi)}", "derivation": "J{(v_{x},\\varphi)} = \\frac{\\varphi}{v_{x}} and v_{x} J{(v_{x},\\varphi)} = \\varphi and \\frac{\\varphi}{v_{x}} + v_{x} J{(v_{x},\\varphi)} = \\varphi + \\frac{\\varphi}{v_{x}} and v_{x} J{(v_{x},\\varphi)} + J{(v_{x},\\varphi)} = \\varphi + J{(v_{x},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('v_x', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v_x', commutative=True), Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))"], [["add", 2, "Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Symbol('v_x', commutative=True), Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Symbol('v_x', commutative=True), Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Function('J')(Symbol('v_x', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given a{(\\phi)} = \\sin{(\\phi)}, then obtain (a{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)}) (a{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = (a{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)}) (\\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi}", "derivation": "a{(\\phi)} = \\sin{(\\phi)} and \\frac{d}{d \\phi} a{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and a{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)} = \\sin{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)} and a{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)} = \\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)} and (a{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = (\\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} and (a{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)}) (a{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = (a{(\\phi)} + \\frac{d}{d \\phi} a{(\\phi)}) (\\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\phi', commutative=True)), Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Symbol('\\\\phi', commutative=True)), Pow(Add(sin(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Symbol('\\\\phi', commutative=True)))"], [["times", 5, "Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], "Equality(Mul(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Pow(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Symbol('\\\\phi', commutative=True))), Mul(Add(Function('a')(Symbol('\\\\phi', commutative=True)), Derivative(Function('a')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Pow(Add(sin(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})} and \\Psi^{\\dagger}{(P_{g},L)} = - L + P_{g}, then obtain \\frac{(\\mathbb{I} + \\mathbf{D}{(\\mathbb{I})}) \\log{(\\mathbb{I})}}{\\Psi^{\\dagger}{(P_{g},L)}} = \\frac{(\\mathbb{I} + \\log{(\\log{(\\mathbb{I})})}) \\log{(\\mathbb{I})}}{\\Psi^{\\dagger}{(P_{g},L)}}", "derivation": "\\mathbf{D}{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})} and \\mathbb{I} + \\mathbf{D}{(\\mathbb{I})} = \\mathbb{I} + \\log{(\\log{(\\mathbb{I})})} and (\\mathbb{I} + \\mathbf{D}{(\\mathbb{I})}) \\log{(\\mathbb{I})} = (\\mathbb{I} + \\log{(\\log{(\\mathbb{I})})}) \\log{(\\mathbb{I})} and \\Psi^{\\dagger}{(P_{g},L)} = - L + P_{g} and \\frac{(\\mathbb{I} + \\mathbf{D}{(\\mathbb{I})}) \\log{(\\mathbb{I})}}{- L + P_{g}} = \\frac{(\\mathbb{I} + \\log{(\\log{(\\mathbb{I})})}) \\log{(\\mathbb{I})}}{- L + P_{g}} and \\frac{(\\mathbb{I} + \\mathbf{D}{(\\mathbb{I})}) \\log{(\\mathbb{I})}}{\\Psi^{\\dagger}{(P_{g},L)}} = \\frac{(\\mathbb{I} + \\log{(\\log{(\\mathbb{I})})}) \\log{(\\mathbb{I})}}{\\Psi^{\\dagger}{(P_{g},L)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True)), log(log(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), log(log(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["times", 2, "log(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True))), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), log(log(Symbol('\\\\mathbb{I}', commutative=True)))), log(Symbol('\\\\mathbb{I}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('P_g', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('P_g', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('P_g', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True))), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('P_g', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), log(log(Symbol('\\\\mathbb{I}', commutative=True)))), log(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), log(log(Symbol('\\\\mathbb{I}', commutative=True)))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{r},t_{2})} = \\log{(\\mathbf{r} - t_{2})}, then obtain \\frac{\\partial}{\\partial t_{2}} y^{\\prime} (t_{2} + \\sin{(\\operatorname{C_{d}}{(\\mathbf{r},t_{2})})}) = \\frac{\\partial}{\\partial t_{2}} y^{\\prime} (t_{2} + \\sin{(\\log{(\\mathbf{r} - t_{2})})})", "derivation": "\\operatorname{C_{d}}{(\\mathbf{r},t_{2})} = \\log{(\\mathbf{r} - t_{2})} and \\sin{(\\operatorname{C_{d}}{(\\mathbf{r},t_{2})})} = \\sin{(\\log{(\\mathbf{r} - t_{2})})} and t_{2} + \\sin{(\\operatorname{C_{d}}{(\\mathbf{r},t_{2})})} = t_{2} + \\sin{(\\log{(\\mathbf{r} - t_{2})})} and y^{\\prime} (t_{2} + \\sin{(\\operatorname{C_{d}}{(\\mathbf{r},t_{2})})}) = y^{\\prime} (t_{2} + \\sin{(\\log{(\\mathbf{r} - t_{2})})}) and \\frac{\\partial}{\\partial t_{2}} y^{\\prime} (t_{2} + \\sin{(\\operatorname{C_{d}}{(\\mathbf{r},t_{2})})}) = \\frac{\\partial}{\\partial t_{2}} y^{\\prime} (t_{2} + \\sin{(\\log{(\\mathbf{r} - t_{2})})})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True)), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)))))"], [["sin", 1], "Equality(sin(Function('C_d')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))), sin(log(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))))"], [["add", 2, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), sin(Function('C_d')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True)))), Add(Symbol('t_2', commutative=True), sin(log(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)))))))"], [["times", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Add(Symbol('t_2', commutative=True), sin(Function('C_d')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))))), Mul(Symbol('y^{\\\\prime}', commutative=True), Add(Symbol('t_2', commutative=True), sin(log(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))))))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Add(Symbol('t_2', commutative=True), sin(Function('C_d')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Add(Symbol('t_2', commutative=True), sin(log(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(A)} = \\log{(A)} and \\varepsilon{(A)} = \\log{(A)}^{A}, then obtain \\phi^{A}{(A)} \\int \\phi^{A}{(A)} dA = \\phi^{A}{(A)} \\int \\log{(A)}^{A} dA", "derivation": "\\phi{(A)} = \\log{(A)} and \\phi^{A}{(A)} = \\log{(A)}^{A} and \\varepsilon{(A)} = \\log{(A)}^{A} and \\int \\phi^{A}{(A)} dA = \\int \\log{(A)}^{A} dA and \\varepsilon{(A)} = \\phi^{A}{(A)} and \\varepsilon{(A)} \\int \\phi^{A}{(A)} dA = \\varepsilon{(A)} \\int \\log{(A)}^{A} dA and \\phi^{A}{(A)} \\int \\phi^{A}{(A)} dA = \\phi^{A}{(A)} \\int \\log{(A)}^{A} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('A', commutative=True)), Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\varepsilon')(Symbol('A', commutative=True)), Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["times", 4, "Function('\\\\varepsilon')(Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('A', commutative=True)), Integral(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Function('\\\\varepsilon')(Symbol('A', commutative=True)), Integral(Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integral(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Function('\\\\phi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integral(Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(A_{1},J_{\\varepsilon})} = - J_{\\varepsilon} + e^{A_{1}} and \\operatorname{x^{{\\}'}}{(A_{1},J_{\\varepsilon})} = (\\frac{\\partial}{\\partial J_{\\varepsilon}} (- J_{\\varepsilon} + e^{A_{1}}))^{A_{1}}, then obtain \\operatorname{x^{{\\}'}}^{J_{\\varepsilon}}{(A_{1},J_{\\varepsilon})} = ((\\frac{\\partial}{\\partial J_{\\varepsilon}} \\operatorname{V_{\\mathbf{B}}}{(A_{1},J_{\\varepsilon})})^{A_{1}})^{J_{\\varepsilon}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(A_{1},J_{\\varepsilon})} = - J_{\\varepsilon} + e^{A_{1}} and \\operatorname{x^{{\\}'}}{(A_{1},J_{\\varepsilon})} = (\\frac{\\partial}{\\partial J_{\\varepsilon}} (- J_{\\varepsilon} + e^{A_{1}}))^{A_{1}} and \\operatorname{x^{{\\}'}}{(A_{1},J_{\\varepsilon})} = (\\frac{\\partial}{\\partial J_{\\varepsilon}} \\operatorname{V_{\\mathbf{B}}}{(A_{1},J_{\\varepsilon})})^{A_{1}} and \\operatorname{x^{{\\}'}}^{J_{\\varepsilon}}{(A_{1},J_{\\varepsilon})} = ((\\frac{\\partial}{\\partial J_{\\varepsilon}} \\operatorname{V_{\\mathbf{B}}}{(A_{1},J_{\\varepsilon})})^{A_{1}})^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('A_1', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('A_1', commutative=True)))"], [["power", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('A_1', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(C_{d})} = \\cos{(\\cos{(C_{d})})}, then obtain (- C_{d} + (\\operatorname{A_{y}}{(C_{d})} + \\cos{(\\cos{(C_{d})})})^{C_{d}})^{2} = (- C_{d} + (2 \\cos{(\\cos{(C_{d})})})^{C_{d}})^{2}", "derivation": "\\operatorname{A_{y}}{(C_{d})} = \\cos{(\\cos{(C_{d})})} and \\operatorname{A_{y}}{(C_{d})} + \\cos{(\\cos{(C_{d})})} = 2 \\cos{(\\cos{(C_{d})})} and (\\operatorname{A_{y}}{(C_{d})} + \\cos{(\\cos{(C_{d})})})^{C_{d}} = (2 \\cos{(\\cos{(C_{d})})})^{C_{d}} and - C_{d} + (\\operatorname{A_{y}}{(C_{d})} + \\cos{(\\cos{(C_{d})})})^{C_{d}} = - C_{d} + (2 \\cos{(\\cos{(C_{d})})})^{C_{d}} and (- C_{d} + (\\operatorname{A_{y}}{(C_{d})} + \\cos{(\\cos{(C_{d})})})^{C_{d}})^{2} = (- C_{d} + (2 \\cos{(\\cos{(C_{d})})})^{C_{d}})^{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True))))"], [["add", 1, "cos(cos(Symbol('C_d', commutative=True)))"], "Equality(Add(Function('A_y')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True)))), Mul(Integer(2), cos(cos(Symbol('C_d', commutative=True)))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Add(Function('A_y')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Pow(Mul(Integer(2), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)))"], [["minus", 3, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Pow(Add(Function('A_y')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Pow(Mul(Integer(2), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Pow(Add(Function('A_y')(Symbol('C_d', commutative=True)), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Pow(Mul(Integer(2), cos(cos(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True))), Integer(2)))"]]}, {"prompt": "Given i{(c)} = e^{c} and \\operatorname{v_{y}}{(c)} = e^{c}, then obtain (i{(c)} e^{- c})^{c} (i{(c)} + 1) e^{- c} = (i{(c)} e^{- c})^{c} (\\operatorname{v_{y}}{(c)} + 1) e^{- c}", "derivation": "i{(c)} = e^{c} and i{(c)} + 1 = e^{c} + 1 and \\operatorname{v_{y}}{(c)} = e^{c} and i{(c)} + 1 = \\operatorname{v_{y}}{(c)} + 1 and (i{(c)} e^{- c})^{c} (i{(c)} + 1) = (i{(c)} e^{- c})^{c} (\\operatorname{v_{y}}{(c)} + 1) and (i{(c)} e^{- c})^{c} (i{(c)} + 1) e^{- c} = (i{(c)} e^{- c})^{c} (\\operatorname{v_{y}}{(c)} + 1) e^{- c}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('i')(Symbol('c', commutative=True)), Integer(1)), Add(exp(Symbol('c', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('i')(Symbol('c', commutative=True)), Integer(1)), Add(Function('v_y')(Symbol('c', commutative=True)), Integer(1)))"], [["times", 4, "Pow(Mul(Function('i')(Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Mul(Function('i')(Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Add(Function('i')(Symbol('c', commutative=True)), Integer(1))), Mul(Pow(Mul(Function('i')(Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Add(Function('v_y')(Symbol('c', commutative=True)), Integer(1))))"], [["times", 5, "exp(Mul(Integer(-1), Symbol('c', commutative=True)))"], "Equality(Mul(Pow(Mul(Function('i')(Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Add(Function('i')(Symbol('c', commutative=True)), Integer(1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Pow(Mul(Function('i')(Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Add(Function('v_y')(Symbol('c', commutative=True)), Integer(1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given c{(\\sigma_p)} = e^{\\sigma_p}, then obtain ((c{(\\sigma_p)} + e^{\\sigma_p}) e^{\\sigma_p} - 2 e^{\\sigma_p})^{\\sigma_p} = (2 e^{2 \\sigma_p} - 2 e^{\\sigma_p})^{\\sigma_p}", "derivation": "c{(\\sigma_p)} = e^{\\sigma_p} and c{(\\sigma_p)} + e^{\\sigma_p} = 2 e^{\\sigma_p} and (c{(\\sigma_p)} + e^{\\sigma_p}) e^{\\sigma_p} = 2 e^{2 \\sigma_p} and (c{(\\sigma_p)} + e^{\\sigma_p}) e^{\\sigma_p} - 2 e^{\\sigma_p} = 2 e^{2 \\sigma_p} - 2 e^{\\sigma_p} and ((c{(\\sigma_p)} + e^{\\sigma_p}) e^{\\sigma_p} - 2 e^{\\sigma_p})^{\\sigma_p} = (2 e^{2 \\sigma_p} - 2 e^{\\sigma_p})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('c')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Add(Function('c')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Add(Function('c')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Add(Mul(Add(Function('c')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\phi{(s)} = \\log{(s)}, then obtain e^{- s + \\int \\phi{(s)} ds} = e^{- s + \\int \\log{(s)} ds}", "derivation": "\\phi{(s)} = \\log{(s)} and \\int \\phi{(s)} ds = \\int \\log{(s)} ds and - s + \\int \\phi{(s)} ds = - s + \\int \\log{(s)} ds and e^{- s + \\int \\phi{(s)} ds} = e^{- s + \\int \\log{(s)} ds}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["minus", 2, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Function('\\\\phi')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Function('\\\\phi')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))), exp(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})} = (\\eta^{\\prime} + \\pi)^{A_{1}}, then obtain A_{1} \\int ((\\eta^{\\prime} + \\pi)^{A_{1}} + \\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})}) dA_{1} = A_{1} \\int 2 (\\eta^{\\prime} + \\pi)^{A_{1}} dA_{1}", "derivation": "\\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})} = (\\eta^{\\prime} + \\pi)^{A_{1}} and (\\eta^{\\prime} + \\pi)^{A_{1}} + \\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})} = 2 (\\eta^{\\prime} + \\pi)^{A_{1}} and \\int ((\\eta^{\\prime} + \\pi)^{A_{1}} + \\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})}) dA_{1} = \\int 2 (\\eta^{\\prime} + \\pi)^{A_{1}} dA_{1} and A_{1} \\int ((\\eta^{\\prime} + \\pi)^{A_{1}} + \\operatorname{A_{z}}{(A_{1},\\pi,\\eta^{\\prime})}) dA_{1} = A_{1} \\int 2 (\\eta^{\\prime} + \\pi)^{A_{1}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)))"], [["add", 1, "Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)), Function('A_z')(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)), Function('A_z')(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Integer(2), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"], [["times", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Integral(Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)), Function('A_z')(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), Mul(Symbol('A_1', commutative=True), Integral(Mul(Integer(2), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(z^{*},E)} = E z^{*}, then obtain \\int (- E z^{*} + E) dz^{*} = \\int (E - \\mathbf{r}{(z^{*},E)}) dz^{*}", "derivation": "\\mathbf{r}{(z^{*},E)} = E z^{*} and 0 = E z^{*} - \\mathbf{r}{(z^{*},E)} and - E z^{*} = - \\mathbf{r}{(z^{*},E)} and - E z^{*} + E = E - \\mathbf{r}{(z^{*},E)} and \\int (- E z^{*} + E) dz^{*} = \\int (E - \\mathbf{r}{(z^{*},E)}) dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('z^*', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('E', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('E', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True))))"], [["add", 3, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('z^*', commutative=True)), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True)))))"], [["integrate", 4, "Symbol('z^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('z^*', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('z^*', commutative=True), Symbol('E', commutative=True)))), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(c)} = \\sin{(c)}, then derive - \\sin{(c)} \\frac{d^{2}}{d c^{2}} \\mathbf{v}{(c)} = \\sin^{2}{(c)}, then obtain - \\sin{(c)} \\frac{d^{2}}{d c^{2}} \\sin{(c)} = \\sin^{2}{(c)}", "derivation": "\\mathbf{v}{(c)} = \\sin{(c)} and \\frac{d}{d c} \\mathbf{v}{(c)} = \\frac{d}{d c} \\sin{(c)} and \\frac{d^{2}}{d c^{2}} \\mathbf{v}{(c)} = \\frac{d^{2}}{d c^{2}} \\sin{(c)} and \\frac{d^{2}}{d c^{2}} \\mathbf{v}{(c)} \\frac{d^{2}}{d c^{2}} \\sin{(c)} = (\\frac{d^{2}}{d c^{2}} \\sin{(c)})^{2} and - \\sin{(c)} \\frac{d^{2}}{d c^{2}} \\mathbf{v}{(c)} = \\sin^{2}{(c)} and - \\sin{(c)} \\frac{d^{2}}{d c^{2}} \\sin{(c)} = \\sin^{2}{(c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))))"], [["times", 3, "Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{v}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Pow(Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Integer(2)))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), sin(Symbol('c', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Pow(sin(Symbol('c', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), sin(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Pow(sin(Symbol('c', commutative=True)), Integer(2)))"]]}, {"prompt": "Given S{(S,F_{N})} = \\frac{\\cos{(F_{N})}}{S}, then derive \\frac{\\partial}{\\partial S} S{(S,F_{N})} = - \\frac{\\cos{(F_{N})}}{S^{2}}, then obtain \\frac{\\partial}{\\partial F_{N}} - \\sin{(\\frac{S{(S,F_{N})}}{S})} = \\frac{\\partial}{\\partial F_{N}} \\sin{(\\frac{\\partial}{\\partial S} S{(S,F_{N})})}", "derivation": "S{(S,F_{N})} = \\frac{\\cos{(F_{N})}}{S} and \\frac{\\partial}{\\partial S} S{(S,F_{N})} = \\frac{\\partial}{\\partial S} \\frac{\\cos{(F_{N})}}{S} and \\frac{\\partial}{\\partial S} S{(S,F_{N})} = - \\frac{\\cos{(F_{N})}}{S^{2}} and - \\frac{\\cos{(F_{N})}}{S^{2}} = \\frac{\\partial}{\\partial S} \\frac{\\cos{(F_{N})}}{S} and - \\frac{S{(S,F_{N})}}{S} = \\frac{\\partial}{\\partial S} S{(S,F_{N})} and - \\sin{(\\frac{S{(S,F_{N})}}{S})} = \\sin{(\\frac{\\partial}{\\partial S} S{(S,F_{N})})} and \\frac{\\partial}{\\partial F_{N}} - \\sin{(\\frac{S{(S,F_{N})}}{S})} = \\frac{\\partial}{\\partial F_{N}} \\sin{(\\frac{\\partial}{\\partial S} S{(S,F_{N})})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), cos(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), cos(Symbol('F_N', commutative=True))), Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True))), Derivative(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["sin", 5], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True))))), sin(Derivative(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(sin(Derivative(Function('S')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(\\mathbf{F},C_{2})} = \\sin{(C_{2} + \\mathbf{F})} and \\Psi_{\\lambda}{(\\mathbf{F},C_{2})} = \\frac{\\partial}{\\partial C_{2}} m{(\\mathbf{F},C_{2})}, then obtain \\Psi_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F},C_{2})} = (\\frac{\\partial}{\\partial C_{2}} \\sin{(C_{2} + \\mathbf{F})})^{\\mathbf{F}}", "derivation": "m{(\\mathbf{F},C_{2})} = \\sin{(C_{2} + \\mathbf{F})} and \\frac{\\partial}{\\partial C_{2}} m{(\\mathbf{F},C_{2})} = \\frac{\\partial}{\\partial C_{2}} \\sin{(C_{2} + \\mathbf{F})} and \\Psi_{\\lambda}{(\\mathbf{F},C_{2})} = \\frac{\\partial}{\\partial C_{2}} m{(\\mathbf{F},C_{2})} and \\Psi_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F},C_{2})} = (\\frac{\\partial}{\\partial C_{2}} m{(\\mathbf{F},C_{2})})^{\\mathbf{F}} and \\Psi_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F},C_{2})} = (\\frac{\\partial}{\\partial C_{2}} \\sin{(C_{2} + \\mathbf{F})})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), sin(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Derivative(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(Function('m')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(sin(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\sigma_p,n)} = \\sin^{\\sigma_p}{(n)}, then obtain \\frac{n + \\operatorname{E_{x}}{(\\sigma_p,n)} + \\sin^{\\sigma_p}{(n)}}{\\sigma_p} = \\frac{n + 2 \\operatorname{E_{x}}{(\\sigma_p,n)}}{\\sigma_p}", "derivation": "\\operatorname{E_{x}}{(\\sigma_p,n)} = \\sin^{\\sigma_p}{(n)} and n + \\operatorname{E_{x}}{(\\sigma_p,n)} = n + \\sin^{\\sigma_p}{(n)} and n + 2 \\operatorname{E_{x}}{(\\sigma_p,n)} = n + \\operatorname{E_{x}}{(\\sigma_p,n)} + \\sin^{\\sigma_p}{(n)} and n + 2 \\operatorname{E_{x}}{(\\sigma_p,n)} = n + 2 \\sin^{\\sigma_p}{(n)} and \\frac{n + 2 \\operatorname{E_{x}}{(\\sigma_p,n)}}{\\sigma_p} = \\frac{n + 2 \\sin^{\\sigma_p}{(n)}}{\\sigma_p} and \\frac{n + \\operatorname{E_{x}}{(\\sigma_p,n)} + \\sin^{\\sigma_p}{(n)}}{\\sigma_p} = \\frac{n + 2 \\sin^{\\sigma_p}{(n)}}{\\sigma_p} and \\frac{n + \\operatorname{E_{x}}{(\\sigma_p,n)} + \\sin^{\\sigma_p}{(n)}}{\\sigma_p} = \\frac{n + 2 \\operatorname{E_{x}}{(\\sigma_p,n)}}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Symbol('n', commutative=True), Mul(Integer(2), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)))), Add(Symbol('n', commutative=True), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('n', commutative=True), Mul(Integer(2), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)))), Add(Symbol('n', commutative=True), Mul(Integer(2), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))))"], [["divide", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(2), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(2), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(2), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(2), Function('E_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))))))"]]}, {"prompt": "Given u{(c_{0})} = \\log{(c_{0})}, then obtain \\cos{(\\int (\\frac{u{(c_{0})}}{\\log{(c_{0})}})^{c_{0}} dc_{0})} = \\cos{(\\int 1 dc_{0})}", "derivation": "u{(c_{0})} = \\log{(c_{0})} and \\frac{u{(c_{0})}}{\\log{(c_{0})}} = 1 and (\\frac{u{(c_{0})}}{\\log{(c_{0})}})^{c_{0}} = 1 and \\int (\\frac{u{(c_{0})}}{\\log{(c_{0})}})^{c_{0}} dc_{0} = \\int 1 dc_{0} and \\cos{(\\int (\\frac{u{(c_{0})}}{\\log{(c_{0})}})^{c_{0}} dc_{0})} = \\cos{(\\int 1 dc_{0})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["divide", 1, "log(Symbol('c_0', commutative=True))"], "Equality(Mul(Function('u')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('c_0', commutative=True)"], "Equality(Pow(Mul(Function('u')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Symbol('c_0', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('c_0', commutative=True)"], "Equality(Integral(Pow(Mul(Function('u')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('c_0', commutative=True))))"], [["cos", 4], "Equality(cos(Integral(Pow(Mul(Function('u')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), cos(Integral(Integer(1), Tuple(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given v{(W)} = \\cos{(W)}, then obtain \\frac{d}{d W} (- 2 W + 2 v{(W)}) = \\frac{d}{d W} (- 2 W + v{(W)} + \\cos{(W)})", "derivation": "v{(W)} = \\cos{(W)} and - W + v{(W)} = - W + \\cos{(W)} and - 2 W + 2 v{(W)} = - 2 W + v{(W)} + \\cos{(W)} and \\frac{d}{d W} (- 2 W + 2 v{(W)}) = \\frac{d}{d W} (- 2 W + v{(W)} + \\cos{(W)})", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["minus", 1, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('v')(Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('v')(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Function('v')(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True)), Function('v')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Function('v')(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True)), Function('v')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\theta_1)} = \\sin{(\\theta_1)} and \\nabla{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain \\Omega + \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)} + \\nabla{(\\theta_1)} \\sin{(\\theta_1)} = \\Omega + 2 \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)}", "derivation": "\\operatorname{F_{H}}{(\\theta_1)} = \\sin{(\\theta_1)} and \\nabla{(\\theta_1)} = \\sin{(\\theta_1)} and \\nabla^{2}{(\\theta_1)} = \\nabla{(\\theta_1)} \\sin{(\\theta_1)} and \\nabla^{2}{(\\theta_1)} = \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)} and \\nabla{(\\theta_1)} \\sin{(\\theta_1)} = \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)} and \\Omega + \\nabla{(\\theta_1)} \\sin{(\\theta_1)} = \\Omega + \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)} and \\Omega + \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)} + \\nabla{(\\theta_1)} \\sin{(\\theta_1)} = \\Omega + 2 \\operatorname{F_{H}}{(\\theta_1)} \\nabla{(\\theta_1)}", "srepr_derivation": [["get_premise", "Equality(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["times", 2, "Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Mul(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Mul(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True))))"], [["add", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 6, "Mul(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True))), Mul(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('F_H')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\delta{(l,b)} = e^{b l}, then obtain \\frac{l^{2} \\delta^{3 b}{(l,b)} - l \\delta^{b}{(l,b)} e^{b l}}{b l} = \\frac{l^{2} \\delta^{2 b}{(l,b)} (e^{b l})^{b} - l \\delta^{b}{(l,b)} e^{b l}}{b l}", "derivation": "\\delta{(l,b)} = e^{b l} and \\delta^{b}{(l,b)} = (e^{b l})^{b} and l \\delta^{b}{(l,b)} = l (e^{b l})^{b} and l \\delta^{2 b}{(l,b)} = l \\delta^{b}{(l,b)} (e^{b l})^{b} and l^{2} \\delta^{3 b}{(l,b)} = l^{2} \\delta^{2 b}{(l,b)} (e^{b l})^{b} and l^{2} \\delta^{3 b}{(l,b)} - l \\delta^{b}{(l,b)} e^{b l} = l^{2} \\delta^{2 b}{(l,b)} (e^{b l})^{b} - l \\delta^{b}{(l,b)} e^{b l} and \\frac{l^{2} \\delta^{3 b}{(l,b)} - l \\delta^{b}{(l,b)} e^{b l}}{b l} = \\frac{l^{2} \\delta^{2 b}{(l,b)} (e^{b l})^{b} - l \\delta^{b}{(l,b)} e^{b l}}{b l}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True)))"], [["times", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Mul(Symbol('l', commutative=True), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True))))"], [["times", 3, "Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))"], "Equality(Mul(Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True)))), Mul(Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True))))"], [["times", 4, "Mul(Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(3), Symbol('b', commutative=True)))), Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True))))"], [["minus", 5, "Mul(Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(3), Symbol('b', commutative=True)))), Mul(Integer(-1), Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))))), Add(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))))))"], [["divide", 6, "Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(3), Symbol('b', commutative=True)))), Mul(Integer(-1), Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True)))))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Pow(exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True), Pow(Function('\\\\delta')(Symbol('l', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), exp(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} = \\mathbf{f} - \\phi_2, then obtain \\int \\phi_2 \\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} d\\mathbf{f} = \\frac{\\mathbf{f}^{2} \\phi_2}{2} - \\mathbf{f} \\phi_2^{2} + \\theta_2", "derivation": "\\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} = \\mathbf{f} - \\phi_2 and \\phi_2 \\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} = \\phi_2 (\\mathbf{f} - \\phi_2) and \\int \\phi_2 \\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} d\\mathbf{f} = \\int \\phi_2 (\\mathbf{f} - \\phi_2) d\\mathbf{f} and \\int \\phi_2 \\operatorname{M_{E}}{(\\phi_2,\\mathbf{f})} d\\mathbf{f} = \\frac{\\mathbf{f}^{2} \\phi_2}{2} - \\mathbf{f} \\phi_2^{2} + \\theta_2", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\phi_1)} = \\sin{(\\phi_1)}, then obtain 0 = - \\sin{(\\frac{\\ddot{x}{(\\phi_1)}}{\\sin{(\\phi_1)}} - 1)}", "derivation": "\\ddot{x}{(\\phi_1)} = \\sin{(\\phi_1)} and \\frac{\\ddot{x}{(\\phi_1)}}{\\sin{(\\phi_1)}} = 1 and \\frac{\\ddot{x}{(\\phi_1)}}{\\sin{(\\phi_1)}} + \\sin{(\\phi_1)} = \\sin{(\\phi_1)} + 1 and 0 = - \\frac{\\ddot{x}{(\\phi_1)}}{\\sin{(\\phi_1)}} + 1 and 0 = - \\sin{(\\frac{\\ddot{x}{(\\phi_1)}}{\\sin{(\\phi_1)}} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Mul(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), sin(Symbol('\\\\phi_1', commutative=True))), Add(sin(Symbol('\\\\phi_1', commutative=True)), Integer(1)))"], [["minus", 3, "Add(Mul(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), sin(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Integer(1)))"], [["sin", 4], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Mul(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Integer(-1)))))"]]}, {"prompt": "Given y{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}}, then derive \\frac{d}{d r_{0}} y{(r_{0})} = e^{r_{0}}, then obtain \\frac{d^{2}}{d r_{0}^{2}} y{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}}", "derivation": "y{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}} and \\frac{d}{d r_{0}} y{(r_{0})} = \\frac{d^{2}}{d r_{0}^{2}} e^{r_{0}} and \\frac{d}{d r_{0}} y{(r_{0})} = e^{r_{0}} and y{(r_{0})} = \\frac{d^{2}}{d r_{0}^{2}} y{(r_{0})} and \\frac{d^{2}}{d r_{0}^{2}} y{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('r_0', commutative=True)), Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), exp(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('y')(Symbol('r_0', commutative=True)), Derivative(Function('y')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Derivative(Function('y')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))), Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(g)} = \\log{(g)}, then obtain (\\frac{I^{g}{(g)} \\int I{(g)} dg}{\\log{(g)}})^{g} = (\\frac{\\log{(g)}^{g} \\int I{(g)} dg}{\\log{(g)}})^{g}", "derivation": "I{(g)} = \\log{(g)} and I^{g}{(g)} = \\log{(g)}^{g} and \\int I{(g)} dg = \\int \\log{(g)} dg and I^{g}{(g)} \\int \\log{(g)} dg = \\log{(g)}^{g} \\int \\log{(g)} dg and \\frac{I^{g}{(g)} \\int \\log{(g)} dg}{\\log{(g)}} = \\frac{\\log{(g)}^{g} \\int \\log{(g)} dg}{\\log{(g)}} and \\frac{I^{g}{(g)} \\int I{(g)} dg}{\\log{(g)}} = \\frac{\\log{(g)}^{g} \\int I{(g)} dg}{\\log{(g)}} and (\\frac{I^{g}{(g)} \\int I{(g)} dg}{\\log{(g)}})^{g} = (\\frac{\\log{(g)}^{g} \\int I{(g)} dg}{\\log{(g)}})^{g}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('I')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('I')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["times", 2, "Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Function('I')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["divide", 4, "log(Symbol('g', commutative=True))"], "Equality(Mul(Pow(Function('I')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(-1)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(log(Symbol('g', commutative=True)), Integer(-1)), Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('I')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(-1)), Integral(Function('I')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(log(Symbol('g', commutative=True)), Integer(-1)), Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Integral(Function('I')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["power", 6, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Pow(Function('I')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(-1)), Integral(Function('I')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Mul(Pow(log(Symbol('g', commutative=True)), Integer(-1)), Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Integral(Function('I')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given E{(F_{g},\\mathbf{f})} = \\int (- F_{g} + \\mathbf{f}) d\\mathbf{f}, then obtain ((- F_{g} + \\mathbf{f}) E{(F_{g},\\mathbf{f})})^{F_{g}} = ((- F_{g} + \\mathbf{f}) (- F_{g} \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\varphi))^{F_{g}}", "derivation": "E{(F_{g},\\mathbf{f})} = \\int (- F_{g} + \\mathbf{f}) d\\mathbf{f} and (- F_{g} + \\mathbf{f}) E{(F_{g},\\mathbf{f})} = (- F_{g} + \\mathbf{f}) \\int (- F_{g} + \\mathbf{f}) d\\mathbf{f} and ((- F_{g} + \\mathbf{f}) E{(F_{g},\\mathbf{f})})^{F_{g}} = ((- F_{g} + \\mathbf{f}) \\int (- F_{g} + \\mathbf{f}) d\\mathbf{f})^{F_{g}} and ((- F_{g} + \\mathbf{f}) E{(F_{g},\\mathbf{f})})^{F_{g}} = ((- F_{g} + \\mathbf{f}) (- F_{g} \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\varphi))^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Function('E')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Function('E')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('F_g', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('F_g', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Function('E')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('F_g', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Symbol('\\\\varphi', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(M_{E})} = M_{E}, then derive \\int \\psi^{*}{(M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + \\dot{x}, then obtain \\frac{d}{d \\dot{x}} \\int M_{E} dM_{E} = \\frac{\\partial}{\\partial \\dot{x}} (\\frac{M_{E}^{2}}{2} + \\dot{x})", "derivation": "\\psi^{*}{(M_{E})} = M_{E} and \\int \\psi^{*}{(M_{E})} dM_{E} = \\int M_{E} dM_{E} and \\int \\psi^{*}{(M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + \\dot{x} and \\int M_{E} dM_{E} = \\frac{M_{E}^{2}}{2} + \\dot{x} and \\frac{d}{d \\dot{x}} \\int M_{E} dM_{E} = \\frac{\\partial}{\\partial \\dot{x}} (\\frac{M_{E}^{2}}{2} + \\dot{x})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and \\mathbf{A}{(\\eta^{\\prime})} = \\int e^{\\frac{\\operatorname{f^{*}}{(\\eta^{\\prime})}}{\\cos{(\\eta^{\\prime})}}} d\\eta^{\\prime}, then obtain \\mathbf{A}{(\\eta^{\\prime})} = \\int e d\\eta^{\\prime}", "derivation": "\\operatorname{f^{*}}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and \\frac{\\operatorname{f^{*}}{(\\eta^{\\prime})}}{\\cos{(\\eta^{\\prime})}} = 1 and e^{\\frac{\\operatorname{f^{*}}{(\\eta^{\\prime})}}{\\cos{(\\eta^{\\prime})}}} = e and \\int e^{\\frac{\\operatorname{f^{*}}{(\\eta^{\\prime})}}{\\cos{(\\eta^{\\prime})}}} d\\eta^{\\prime} = \\int e d\\eta^{\\prime} and \\mathbf{A}{(\\eta^{\\prime})} = \\int e^{\\frac{\\operatorname{f^{*}}{(\\eta^{\\prime})}}{\\cos{(\\eta^{\\prime})}}} d\\eta^{\\prime} and \\mathbf{A}{(\\eta^{\\prime})} = \\int e d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))), Integer(1))"], [["exp", 2], "Equality(exp(Mul(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))), E)"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(exp(Mul(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(E, Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(exp(Mul(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(E, Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(A_{1},Q,A)} = A - A_{1} + Q, then obtain A_{1} (\\cos{(\\hat{H}_l{(A_{1},Q,A)})} + 1) = A_{1} (\\cos{(A - A_{1} + Q)} + 1)", "derivation": "\\hat{H}_l{(A_{1},Q,A)} = A - A_{1} + Q and \\cos{(\\hat{H}_l{(A_{1},Q,A)})} = \\cos{(A - A_{1} + Q)} and \\cos{(\\hat{H}_l{(A_{1},Q,A)})} + 1 = \\cos{(A - A_{1} + Q)} + 1 and A_{1} (\\cos{(\\hat{H}_l{(A_{1},Q,A)})} + 1) = A_{1} (\\cos{(A - A_{1} + Q)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('Q', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True), Symbol('A', commutative=True))), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('Q', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(cos(Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True), Symbol('A', commutative=True))), Integer(1)), Add(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('Q', commutative=True))), Integer(1)))"], [["times", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Add(cos(Function('\\\\hat{H}_l')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True), Symbol('A', commutative=True))), Integer(1))), Mul(Symbol('A_1', commutative=True), Add(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('Q', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} = - p + \\log{(c_{0})}, then obtain \\frac{\\partial}{\\partial p} \\int \\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} dp = \\frac{\\partial}{\\partial p} (n_{2} - \\frac{p^{2}}{2} + p \\log{(c_{0})})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} = - p + \\log{(c_{0})} and \\int \\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} dp = \\int (- p + \\log{(c_{0})}) dp and \\frac{\\partial}{\\partial p} \\int \\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} dp = \\frac{\\partial}{\\partial p} \\int (- p + \\log{(c_{0})}) dp and \\frac{\\partial}{\\partial p} \\int \\operatorname{V_{\\mathbf{B}}}{(c_{0},p)} dp = \\frac{\\partial}{\\partial p} (n_{2} - \\frac{p^{2}}{2} + p \\log{(c_{0})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('c_0', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('c_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('c_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('c_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2))), Mul(Symbol('p', commutative=True), log(Symbol('c_0', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\pi)} = \\int \\sin{(\\pi)} d\\pi, then derive I{(\\pi)} = \\mathbf{g} - \\cos{(\\pi)}, then derive \\mu_0 - \\cos{(\\pi)} = \\mathbf{g} - \\cos{(\\pi)}, then derive 0 = - \\cos{(\\pi)}, then obtain 0^{\\pi} = (- \\cos{(\\pi)})^{\\pi}", "derivation": "I{(\\pi)} = \\int \\sin{(\\pi)} d\\pi and I{(\\pi)} = \\mathbf{g} - \\cos{(\\pi)} and - I{(\\pi)} \\cos{(\\pi)} = - (\\mathbf{g} - \\cos{(\\pi)}) \\cos{(\\pi)} and \\int \\sin{(\\pi)} d\\pi = \\mathbf{g} - \\cos{(\\pi)} and \\mu_0 - \\cos{(\\pi)} = \\mathbf{g} - \\cos{(\\pi)} and - I{(\\pi)} \\cos{(\\pi)} = - (\\mu_0 - \\cos{(\\pi)}) \\cos{(\\pi)} and \\frac{d}{d \\mu_0} - I{(\\pi)} \\cos{(\\pi)} = \\frac{\\partial}{\\partial \\mu_0} - (\\mu_0 - \\cos{(\\pi)}) \\cos{(\\pi)} and 0 = - \\cos{(\\pi)} and 0^{\\pi} = (- \\cos{(\\pi)})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\pi', commutative=True)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('I')(Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Function('I')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), cos(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(-1), Function('I')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), cos(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('I')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))))"], [["power", 8, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\pi', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{E},\\psi^*)} = \\cos{(\\frac{\\mathbf{E}}{\\psi^*})}, then derive \\int \\frac{\\mathbb{I}{(\\mathbf{E},\\psi^*)}}{\\cos{(\\frac{\\mathbf{E}}{\\psi^*})}} d\\mathbf{E} = \\mathbf{E} + \\mathbf{s}, then obtain (\\mathbf{E} + \\mathbf{s})^{\\mathbf{s}} = (\\int 1 d\\mathbf{E})^{\\mathbf{s}}", "derivation": "\\mathbb{I}{(\\mathbf{E},\\psi^*)} = \\cos{(\\frac{\\mathbf{E}}{\\psi^*})} and \\frac{\\mathbb{I}{(\\mathbf{E},\\psi^*)}}{\\cos{(\\frac{\\mathbf{E}}{\\psi^*})}} = 1 and \\int \\frac{\\mathbb{I}{(\\mathbf{E},\\psi^*)}}{\\cos{(\\frac{\\mathbf{E}}{\\psi^*})}} d\\mathbf{E} = \\int 1 d\\mathbf{E} and \\int \\frac{\\mathbb{I}{(\\mathbf{E},\\psi^*)}}{\\cos{(\\frac{\\mathbf{E}}{\\psi^*})}} d\\mathbf{E} = \\mathbf{E} + \\mathbf{s} and \\mathbf{E} + \\mathbf{s} = \\int 1 d\\mathbf{E} and (\\mathbf{E} + \\mathbf{s})^{\\mathbf{s}} = (\\int 1 d\\mathbf{E})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi^*', commutative=True)), cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))))"], [["divide", 1, "cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(x^\\prime)} = x^\\prime and E{(x^\\prime)} = \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}}, then derive \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} = \\frac{\\int x^\\prime dx^\\prime}{\\cos{(x^\\prime)}}, then derive \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} = \\frac{\\sigma_x + \\frac{(x^\\prime)^{2}}{2}}{\\cos{(x^\\prime)}}, then obtain E{(x^\\prime)} + 1 = \\frac{\\sigma_x + \\frac{(x^\\prime)^{2}}{2}}{\\cos{(x^\\prime)}} + 1", "derivation": "\\hat{p}{(x^\\prime)} = x^\\prime and \\int \\hat{p}{(x^\\prime)} dx^\\prime = \\int x^\\prime dx^\\prime and \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\frac{d}{d x^\\prime} \\sin{(x^\\prime)}} = \\frac{\\int x^\\prime dx^\\prime}{\\frac{d}{d x^\\prime} \\sin{(x^\\prime)}} and \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} = \\frac{\\int x^\\prime dx^\\prime}{\\cos{(x^\\prime)}} and \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} = \\frac{\\sigma_x + \\frac{(x^\\prime)^{2}}{2}}{\\cos{(x^\\prime)}} and E{(x^\\prime)} = \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} and E{(x^\\prime)} + 1 = 1 + \\frac{\\int \\hat{p}{(x^\\prime)} dx^\\prime}{\\cos{(x^\\prime)}} and E{(x^\\prime)} + 1 = \\frac{\\sigma_x + \\frac{(x^\\prime)^{2}}{2}}{\\cos{(x^\\prime)}} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 2, "Derivative(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(Derivative(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('x^\\\\prime', commutative=True)), Mul(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["add", 6, 1], "Equality(Add(Function('E')(Symbol('x^\\\\prime', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Function('E')(Symbol('x^\\\\prime', commutative=True)), Integer(1)), Add(Mul(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given E{(B,T)} = \\int (B - T) dT, then derive E{(B,T)} = B T - \\frac{T^{2}}{2} + x, then obtain 2 (T + 2 E^{2}{(B,T)} + \\int (B - T) dT) (B T - \\frac{T^{2}}{2} + x) E{(B,T)} = 2 (T + 2 E{(B,T)} \\int (B - T) dT + \\int (B - T) dT) (B T - \\frac{T^{2}}{2} + x) E{(B,T)}", "derivation": "E{(B,T)} = \\int (B - T) dT and E{(B,T)} = B T - \\frac{T^{2}}{2} + x and \\int (B - T) dT = B T - \\frac{T^{2}}{2} + x and 2 E^{2}{(B,T)} = 2 (B T - \\frac{T^{2}}{2} + x) E{(B,T)} and 2 E^{2}{(B,T)} = 2 E{(B,T)} \\int (B - T) dT and T + 2 E^{2}{(B,T)} + \\int (B - T) dT = T + 2 E{(B,T)} \\int (B - T) dT + \\int (B - T) dT and 2 (T + 2 E^{2}{(B,T)} + \\int (B - T) dT) (B T - \\frac{T^{2}}{2} + x) E{(B,T)} = 2 (T + 2 E{(B,T)} \\int (B - T) dT + \\int (B - T) dT) (B T - \\frac{T^{2}}{2} + x) E{(B,T)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)))"], [["times", 2, "Mul(Integer(2), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Integer(2), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["add", 5, "Add(Symbol('T', commutative=True), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(2), Pow(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integer(2))), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Symbol('T', commutative=True), Mul(Integer(2), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["times", 6, "Mul(Integer(2), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Integer(2), Add(Symbol('T', commutative=True), Mul(Integer(2), Pow(Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integer(2))), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Add(Symbol('T', commutative=True), Mul(Integer(2), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Symbol('B', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('x', commutative=True)), Function('E')(Symbol('B', commutative=True), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi_1)} = \\int \\sin{(\\phi_1)} d\\phi_1, then obtain \\frac{\\frac{d}{d \\phi_1} \\operatorname{t_{2}}^{\\phi_1}{(\\phi_1)}}{\\phi_1 + \\operatorname{t_{2}}{(\\phi_1)}} = \\frac{\\frac{d}{d \\phi_1} (\\int \\sin{(\\phi_1)} d\\phi_1)^{\\phi_1}}{\\phi_1 + \\operatorname{t_{2}}{(\\phi_1)}}", "derivation": "\\operatorname{t_{2}}{(\\phi_1)} = \\int \\sin{(\\phi_1)} d\\phi_1 and \\operatorname{t_{2}}^{\\phi_1}{(\\phi_1)} = (\\int \\sin{(\\phi_1)} d\\phi_1)^{\\phi_1} and \\frac{d}{d \\phi_1} \\operatorname{t_{2}}^{\\phi_1}{(\\phi_1)} = \\frac{d}{d \\phi_1} (\\int \\sin{(\\phi_1)} d\\phi_1)^{\\phi_1} and \\frac{\\frac{d}{d \\phi_1} \\operatorname{t_{2}}^{\\phi_1}{(\\phi_1)}}{\\phi_1 + \\operatorname{t_{2}}{(\\phi_1)}} = \\frac{\\frac{d}{d \\phi_1} (\\int \\sin{(\\phi_1)} d\\phi_1)^{\\phi_1}}{\\phi_1 + \\operatorname{t_{2}}{(\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi_1', commutative=True)), Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Function('t_2')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["divide", 3, "Add(Symbol('\\\\phi_1', commutative=True), Function('t_2')(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\phi_1', commutative=True), Function('t_2')(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Derivative(Pow(Function('t_2')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\phi_1', commutative=True), Function('t_2')(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Derivative(Pow(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given U{(\\lambda)} = \\cos{(\\sin{(\\lambda)})}, then obtain \\frac{d}{d \\lambda} \\frac{1}{U{(\\lambda)}} = \\frac{d}{d \\lambda} \\frac{\\int \\cos{(\\sin{(\\lambda)})} d\\lambda}{U{(\\lambda)} \\int U{(\\lambda)} d\\lambda}", "derivation": "U{(\\lambda)} = \\cos{(\\sin{(\\lambda)})} and \\int U{(\\lambda)} d\\lambda = \\int \\cos{(\\sin{(\\lambda)})} d\\lambda and \\frac{\\int U{(\\lambda)} d\\lambda}{\\lambda U{(\\lambda)}} = \\frac{\\int \\cos{(\\sin{(\\lambda)})} d\\lambda}{\\lambda U{(\\lambda)}} and \\frac{\\int U{(\\lambda)} d\\lambda}{U{(\\lambda)}} = \\frac{\\int \\cos{(\\sin{(\\lambda)})} d\\lambda}{U{(\\lambda)}} and \\frac{1}{U{(\\lambda)}} = \\frac{\\int \\cos{(\\sin{(\\lambda)})} d\\lambda}{U{(\\lambda)} \\int U{(\\lambda)} d\\lambda} and \\frac{d}{d \\lambda} \\frac{1}{U{(\\lambda)}} = \\frac{d}{d \\lambda} \\frac{\\int \\cos{(\\sin{(\\lambda)})} d\\lambda}{U{(\\lambda)} \\int U{(\\lambda)} d\\lambda}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\lambda', commutative=True)), cos(sin(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\lambda', commutative=True), Function('U')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["times", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Integral(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["divide", 4, "Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Mul(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('U')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(H)} = \\cos{(e^{H})}, then obtain 0 = - \\frac{H \\operatorname{C_{1}}{(H)} e^{H} \\sin{(e^{H})}}{\\cos^{2}{(e^{H})}} - \\frac{H \\frac{d}{d H} \\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} - \\frac{\\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} + 1", "derivation": "\\operatorname{C_{1}}{(H)} = \\cos{(e^{H})} and H \\operatorname{C_{1}}{(H)} = H \\cos{(e^{H})} and H \\operatorname{C_{1}}^{2}{(H)} = H \\operatorname{C_{1}}{(H)} \\cos{(e^{H})} and H \\operatorname{C_{1}}{(H)} \\cos{(e^{H})} = H \\cos^{2}{(e^{H})} and \\frac{H \\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} = H and 0 = - \\frac{H \\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} + H and \\frac{d}{d H} 0 = \\frac{d}{d H} (- \\frac{H \\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} + H) and 0 = - \\frac{H \\operatorname{C_{1}}{(H)} e^{H} \\sin{(e^{H})}}{\\cos^{2}{(e^{H})}} - \\frac{H \\frac{d}{d H} \\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} - \\frac{\\operatorname{C_{1}}{(H)}}{\\cos{(e^{H})}} + 1", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True))))"], [["times", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Mul(Symbol('H', commutative=True), cos(exp(Symbol('H', commutative=True)))))"], [["times", 2, "Function('C_1')(Symbol('H', commutative=True))"], "Equality(Mul(Symbol('H', commutative=True), Pow(Function('C_1')(Symbol('H', commutative=True)), Integer(2))), Mul(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True)))), Mul(Symbol('H', commutative=True), Pow(cos(exp(Symbol('H', commutative=True))), Integer(2))))"], [["divide", 4, "Pow(cos(exp(Symbol('H', commutative=True))), Integer(2))"], "Equality(Mul(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1))), Symbol('H', commutative=True))"], [["minus", 5, "Mul(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1))), Symbol('H', commutative=True)))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), sin(exp(Symbol('H', commutative=True))), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-2))), Mul(Integer(-1), Symbol('H', commutative=True), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1)), Derivative(Function('C_1')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Function('C_1')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\dot{y}{(I)} = \\sin{(I)}, then obtain \\frac{d}{d I} 1 = \\frac{d}{d I} 0^{I}", "derivation": "\\dot{y}{(I)} = \\sin{(I)} and \\dot{y}{(I)} - \\sin{(I)} = 0 and (\\dot{y}{(I)} - \\sin{(I)})^{I} = 0^{I} and \\frac{d}{d I} (\\dot{y}{(I)} - \\sin{(I)})^{I} = \\frac{d}{d I} 0^{I} and \\frac{d}{d I} 1 = \\frac{d}{d I} (\\dot{y}{(I)} - \\sin{(I)})^{I} and \\frac{d}{d I} 1 = \\frac{d}{d I} 0^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["minus", 1, "sin(Symbol('I', commutative=True))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{y}')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Pow(Integer(0), Symbol('I', commutative=True)))"], [["differentiate", 3, "Symbol('I', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\dot{y}')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Add(Function('\\\\dot{y}')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integer(1), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}}, then derive \\frac{A + a^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} = \\frac{\\int \\frac{e^{\\sin{(a^{\\dagger})}}}{\\bar{\\h}{(a^{\\dagger})}} da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}}, then obtain \\frac{A + a^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} = \\frac{\\int 1 da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}}", "derivation": "\\bar{\\h}{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}} and 1 = \\frac{e^{\\sin{(a^{\\dagger})}}}{\\bar{\\h}{(a^{\\dagger})}} and \\int 1 da^{\\dagger} = \\int \\frac{e^{\\sin{(a^{\\dagger})}}}{\\bar{\\h}{(a^{\\dagger})}} da^{\\dagger} and \\frac{\\int 1 da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} = \\frac{\\int \\frac{e^{\\sin{(a^{\\dagger})}}}{\\bar{\\h}{(a^{\\dagger})}} da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} and \\frac{A + a^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} = \\frac{\\int \\frac{e^{\\sin{(a^{\\dagger})}}}{\\bar{\\h}{(a^{\\dagger})}} da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} and \\frac{A + a^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}} = \\frac{\\int 1 da^{\\dagger}}{\\bar{\\h}{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 3, "Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('A', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Symbol('A', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\hbar')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(F_{g},V_{\\mathbf{E}})} = \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})}, then obtain (F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{H}_l{(F_{g},V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})})^{V_{\\mathbf{E}}}", "derivation": "\\hat{H}_l{(F_{g},V_{\\mathbf{E}})} = \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{H}_l{(F_{g},V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})} and F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{H}_l{(F_{g},V_{\\mathbf{E}})} = F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})} and (F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{H}_l{(F_{g},V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (F_{g} \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{F_{g}})})^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(Symbol('F_g', commutative=True), Derivative(cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Mul(Symbol('F_g', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Mul(Symbol('F_g', commutative=True), Derivative(cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} and r{(A_{1},\\dot{y})} = \\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} + 1, then obtain - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + r{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} and \\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} + 1 = - A_{1} + \\dot{y} + 1 and r{(A_{1},\\dot{y})} = \\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} + 1 and - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + r{(A_{1},\\dot{y})} = - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + \\operatorname{L_{\\varepsilon}}{(A_{1},\\dot{y})} + 1 and - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + r{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} - (\\mathbf{H}{(A_{1},\\dot{y})} + 1)^{\\dot{y}} + 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True), Integer(1)))"], ["renaming_premise", "Equality(Function('r')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"], [["minus", 3, "Pow(Add(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Symbol('\\\\dot{y}', commutative=True))), Function('r')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Symbol('\\\\dot{y}', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Symbol('\\\\dot{y}', commutative=True))), Function('r')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Symbol('\\\\dot{y}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\psi{(v_{2},\\mathbf{S},A_{x})} = \\frac{A_{x}}{v_{2}} + \\mathbf{S}, then derive \\frac{\\partial}{\\partial v_{2}} \\psi{(v_{2},\\mathbf{S},A_{x})} = - \\frac{A_{x}}{v_{2}^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial v_{2}} \\psi{(v_{2},\\mathbf{S},A_{x})}}{v_{2}} = - \\frac{A_{x}}{v_{2}^{3}}", "derivation": "\\psi{(v_{2},\\mathbf{S},A_{x})} = \\frac{A_{x}}{v_{2}} + \\mathbf{S} and \\frac{\\partial}{\\partial v_{2}} \\psi{(v_{2},\\mathbf{S},A_{x})} = \\frac{\\partial}{\\partial v_{2}} (\\frac{A_{x}}{v_{2}} + \\mathbf{S}) and \\frac{\\partial}{\\partial v_{2}} \\psi{(v_{2},\\mathbf{S},A_{x})} = - \\frac{A_{x}}{v_{2}^{2}} and \\frac{\\partial}{\\partial v_{2}} (\\frac{A_{x}}{v_{2}} + \\mathbf{S}) = - \\frac{A_{x}}{v_{2}^{2}} and \\frac{\\frac{\\partial}{\\partial v_{2}} (\\frac{A_{x}}{v_{2}} + \\mathbf{S})}{v_{2}} = - \\frac{A_{x}}{v_{2}^{3}} and \\frac{\\frac{\\partial}{\\partial v_{2}} \\psi{(v_{2},\\mathbf{S},A_{x})}}{v_{2}} = - \\frac{A_{x}}{v_{2}^{3}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_x', commutative=True)), Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-2))))"], [["times", 4, "Pow(Symbol('v_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Derivative(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-3))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Derivative(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(y)} = \\sin{(\\cos{(y)})}, then obtain \\operatorname{f^{\\prime}}^{y}{(y)} \\sin^{y}{(\\cos{(y)})} \\cos{(y)} = \\sin^{2 y}{(\\cos{(y)})} \\cos{(y)}", "derivation": "\\operatorname{f^{\\prime}}{(y)} = \\sin{(\\cos{(y)})} and \\operatorname{f^{\\prime}}^{y}{(y)} = \\sin^{y}{(\\cos{(y)})} and \\operatorname{f^{\\prime}}^{y}{(y)} \\cos{(y)} = \\sin^{y}{(\\cos{(y)})} \\cos{(y)} and \\operatorname{f^{\\prime}}^{y}{(y)} \\sin^{y}{(\\cos{(y)})} \\cos{(y)} = \\sin^{2 y}{(\\cos{(y)})} \\cos{(y)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('y', commutative=True)), sin(cos(Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(sin(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["times", 2, "cos(Symbol('y', commutative=True))"], "Equality(Mul(Pow(Function('f^{\\\\prime}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Mul(Pow(sin(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))))"], [["times", 3, "Pow(sin(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Function('f^{\\\\prime}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(sin(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Mul(Pow(sin(cos(Symbol('y', commutative=True))), Mul(Integer(2), Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))"]]}, {"prompt": "Given H{(n_{1})} = e^{n_{1}} and \\mathbf{g}{(n_{1})} = e^{n_{1}} and \\mathbf{H}{(n_{1})} = H^{n_{1}}{(n_{1})}, then obtain \\mathbf{H}{(n_{1})} = \\mathbf{g}^{n_{1}}{(n_{1})}", "derivation": "H{(n_{1})} = e^{n_{1}} and \\mathbf{g}{(n_{1})} = e^{n_{1}} and \\mathbf{H}{(n_{1})} = H^{n_{1}}{(n_{1})} and H{(n_{1})} = \\mathbf{g}{(n_{1})} and H^{n_{1}}{(n_{1})} = \\mathbf{g}^{n_{1}}{(n_{1})} and \\mathbf{H}{(n_{1})} = \\mathbf{g}^{n_{1}}{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('H')(Symbol('n_1', commutative=True)), Function('\\\\mathbf{g}')(Symbol('n_1', commutative=True)))"], [["power", 4, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(l)} = e^{\\cos{(l)}}, then obtain (\\int \\rho_{b}{(l)} dl)^{l} - (\\int e^{\\cos{(l)}} dl)^{l} = 0", "derivation": "\\rho_{b}{(l)} = e^{\\cos{(l)}} and \\int \\rho_{b}{(l)} dl = \\int e^{\\cos{(l)}} dl and (\\int \\rho_{b}{(l)} dl)^{l} = (\\int e^{\\cos{(l)}} dl)^{l} and (\\int \\rho_{b}{(l)} dl)^{l} - (\\int e^{\\cos{(l)}} dl)^{l} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('l', commutative=True)), exp(cos(Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(exp(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(exp(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["minus", 3, "Pow(Integral(exp(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True))"], "Equality(Add(Pow(Integral(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Integral(exp(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(A_{x})} = \\sin{(A_{x})}, then obtain \\int \\frac{d}{d A_{x}} (-1) dA_{x} + \\int \\frac{d}{d A_{x}} - \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} dA_{x} = 2 \\int \\frac{d}{d A_{x}} (-1) dA_{x}", "derivation": "\\operatorname{f^{*}}{(A_{x})} = \\sin{(A_{x})} and \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} = 1 and - \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} = -1 and \\frac{d}{d A_{x}} - \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} = \\frac{d}{d A_{x}} (-1) and \\int \\frac{d}{d A_{x}} - \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} dA_{x} = \\int \\frac{d}{d A_{x}} (-1) dA_{x} and \\int \\frac{d}{d A_{x}} (-1) dA_{x} + \\int \\frac{d}{d A_{x}} - \\frac{\\operatorname{f^{*}}{(A_{x})}}{\\sin{(A_{x})}} dA_{x} = 2 \\int \\frac{d}{d A_{x}} (-1) dA_{x}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('A_x', commutative=True)), sin(Symbol('A_x', commutative=True)))"], [["divide", 1, "sin(Symbol('A_x', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('A_x', commutative=True)), Pow(sin(Symbol('A_x', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f^*')(Symbol('A_x', commutative=True)), Pow(sin(Symbol('A_x', commutative=True)), Integer(-1))), Integer(-1))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('f^*')(Symbol('A_x', commutative=True)), Pow(sin(Symbol('A_x', commutative=True)), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('A_x', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), Function('f^*')(Symbol('A_x', commutative=True)), Pow(sin(Symbol('A_x', commutative=True)), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Integral(Derivative(Integer(-1), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))))"], [["add", 5, "Integral(Derivative(Integer(-1), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True)))"], "Equality(Add(Integral(Derivative(Integer(-1), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Integral(Derivative(Mul(Integer(-1), Function('f^*')(Symbol('A_x', commutative=True)), Pow(sin(Symbol('A_x', commutative=True)), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True)))), Mul(Integer(2), Integral(Derivative(Integer(-1), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given h{(\\dot{z})} = e^{\\dot{z}}, then obtain - \\dot{z} h{(\\dot{z})} + 1 = - \\dot{z} h{(\\dot{z})} + \\frac{e^{\\dot{z}}}{h{(\\dot{z})}}", "derivation": "h{(\\dot{z})} = e^{\\dot{z}} and \\dot{z} h{(\\dot{z})} = \\dot{z} e^{\\dot{z}} and 1 = \\frac{e^{\\dot{z}}}{h{(\\dot{z})}} and - \\dot{z} e^{\\dot{z}} + 1 = - \\dot{z} e^{\\dot{z}} + \\frac{e^{\\dot{z}}}{h{(\\dot{z})}} and - \\dot{z} h{(\\dot{z})} + 1 = - \\dot{z} h{(\\dot{z})} + \\frac{e^{\\dot{z}}}{h{(\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('h')(Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 1, "Function('h')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('h')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Function('h')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('h')(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('h')(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Function('h')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given x{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain - (\\frac{e^{E_{\\lambda}}}{E_{\\lambda}})^{E_{\\lambda}} x{(E_{\\lambda})} = - (\\frac{e^{E_{\\lambda}}}{E_{\\lambda}})^{E_{\\lambda}} e^{E_{\\lambda}}", "derivation": "x{(E_{\\lambda})} = e^{E_{\\lambda}} and \\frac{x{(E_{\\lambda})}}{E_{\\lambda}} = \\frac{e^{E_{\\lambda}}}{E_{\\lambda}} and - x{(E_{\\lambda})} = - e^{E_{\\lambda}} and - (\\frac{x{(E_{\\lambda})}}{E_{\\lambda}})^{E_{\\lambda}} x{(E_{\\lambda})} = - (\\frac{x{(E_{\\lambda})}}{E_{\\lambda}})^{E_{\\lambda}} e^{E_{\\lambda}} and - (\\frac{e^{E_{\\lambda}}}{E_{\\lambda}})^{E_{\\lambda}} x{(E_{\\lambda})} = - (\\frac{e^{E_{\\lambda}}}{E_{\\lambda}})^{E_{\\lambda}} e^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 3, "Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Function('x')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\theta_1)} = e^{\\theta_1}, then obtain - \\theta_1 + \\dot{\\mathbf{r}}{(\\theta_1)} = - \\theta_1 - \\dot{\\mathbf{r}}{(\\theta_1)} + 2 e^{\\theta_1}", "derivation": "\\dot{\\mathbf{r}}{(\\theta_1)} = e^{\\theta_1} and - \\theta_1 + \\dot{\\mathbf{r}}{(\\theta_1)} = - \\theta_1 + e^{\\theta_1} and - \\theta_1 = - \\theta_1 - \\dot{\\mathbf{r}}{(\\theta_1)} + e^{\\theta_1} and - \\theta_1 + e^{\\theta_1} = - \\theta_1 - \\dot{\\mathbf{r}}{(\\theta_1)} + 2 e^{\\theta_1} and - \\theta_1 + \\dot{\\mathbf{r}}{(\\theta_1)} = - \\theta_1 - \\dot{\\mathbf{r}}{(\\theta_1)} + 2 e^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 2, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))), exp(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(I,v_{x})} = \\frac{v_{x}}{I} and \\mathbf{p}{(v_{x})} = v_{x}, then derive \\frac{\\partial}{\\partial v_{x}} \\hat{\\mathbf{x}}{(I,v_{x})} = \\frac{1}{I}, then obtain v_{x} e^{\\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{I}} = v_{x} e^{\\frac{1}{I}}", "derivation": "\\hat{\\mathbf{x}}{(I,v_{x})} = \\frac{v_{x}}{I} and \\frac{\\partial}{\\partial v_{x}} \\hat{\\mathbf{x}}{(I,v_{x})} = \\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{I} and \\frac{\\partial}{\\partial v_{x}} \\hat{\\mathbf{x}}{(I,v_{x})} = \\frac{1}{I} and \\mathbf{p}{(v_{x})} = v_{x} and e^{\\frac{\\partial}{\\partial v_{x}} \\hat{\\mathbf{x}}{(I,v_{x})}} = e^{\\frac{1}{I}} and e^{\\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{I}} = e^{\\frac{1}{I}} and \\mathbf{p}{(v_{x})} e^{\\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{I}} = \\mathbf{p}{(v_{x})} e^{\\frac{1}{I}} and v_{x} e^{\\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{I}} = v_{x} e^{\\frac{1}{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Pow(Symbol('I', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], [["exp", 3], "Equality(exp(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), exp(Pow(Symbol('I', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(exp(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), exp(Pow(Symbol('I', commutative=True), Integer(-1))))"], [["times", 6, "Function('\\\\mathbf{p}')(Symbol('v_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('v_x', commutative=True)), exp(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))), Mul(Function('\\\\mathbf{p}')(Symbol('v_x', commutative=True)), exp(Pow(Symbol('I', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Symbol('v_x', commutative=True), exp(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))), Mul(Symbol('v_x', commutative=True), exp(Pow(Symbol('I', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)} = - \\mathbf{H} + \\mathbf{s} + y, then derive \\int (- \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)}) dy = - \\mathbf{H} y + a + \\frac{y^{2}}{2}, then obtain \\iint (- \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)}) dy d\\mathbf{s} = \\int (- \\mathbf{H} y + a + \\frac{y^{2}}{2}) d\\mathbf{s}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)} = - \\mathbf{H} + \\mathbf{s} + y and - \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)} = - \\mathbf{H} + y and \\int (- \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)}) dy = \\int (- \\mathbf{H} + y) dy and \\int (- \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)}) dy = - \\mathbf{H} y + a + \\frac{y^{2}}{2} and \\iint (- \\mathbf{s} + \\hat{\\mathbf{r}}{(\\mathbf{H},\\mathbf{s},y)}) dy d\\mathbf{s} = \\int (- \\mathbf{H} y + a + \\frac{y^{2}}{2}) d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('y', commutative=True)))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["integrate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a)} = e^{a}, then derive \\int \\operatorname{M_{E}}{(a)} da = \\mathbf{f} + e^{a}, then obtain \\mathbf{f} + \\operatorname{M_{E}}{(a)} = \\operatorname{M_{E}}{(a)} - e^{a} + \\int e^{a} da", "derivation": "\\operatorname{M_{E}}{(a)} = e^{a} and \\int \\operatorname{M_{E}}{(a)} da = \\int e^{a} da and \\int \\operatorname{M_{E}}{(a)} da = \\mathbf{f} + e^{a} and \\operatorname{M_{E}}{(a)} - e^{a} = 0 and \\operatorname{M_{E}}{(a)} - e^{a} + \\int e^{a} da = \\int e^{a} da and \\mathbf{f} + e^{a} = \\int e^{a} da and \\mathbf{f} + \\operatorname{M_{E}}{(a)} = \\int e^{a} da and \\mathbf{f} + \\operatorname{M_{E}}{(a)} = \\operatorname{M_{E}}{(a)} - e^{a} + \\int e^{a} da", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('a', commutative=True))))"], [["minus", 1, "exp(Symbol('a', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Integer(0))"], [["add", 4, "Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Add(Function('M_E')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('M_E')(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('M_E')(Symbol('a', commutative=True))), Add(Function('M_E')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(A_{x},\\dot{y})} = \\dot{y}^{A_{x}}, then derive \\frac{\\partial}{\\partial A_{x}} \\varphi^{*}{(A_{x},\\dot{y})} = \\dot{y}^{A_{x}} \\log{(\\dot{y})}, then obtain \\dot{y}^{A_{x}} \\log{(\\dot{y})} = \\varphi^{*}{(A_{x},\\dot{y})} \\log{(\\dot{y})}", "derivation": "\\varphi^{*}{(A_{x},\\dot{y})} = \\dot{y}^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} \\varphi^{*}{(A_{x},\\dot{y})} = \\frac{\\partial}{\\partial A_{x}} \\dot{y}^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} \\varphi^{*}{(A_{x},\\dot{y})} = \\dot{y}^{A_{x}} \\log{(\\dot{y})} and \\dot{y}^{A_{x}} \\log{(\\dot{y})} = \\frac{\\partial}{\\partial A_{x}} \\dot{y}^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} \\varphi^{*}{(A_{x},\\dot{y})} = \\varphi^{*}{(A_{x},\\dot{y})} \\log{(\\dot{y})} and \\varphi^{*}{(A_{x},\\dot{y})} \\log{(\\dot{y})} = \\frac{\\partial}{\\partial A_{x}} \\dot{y}^{A_{x}} and \\dot{y}^{A_{x}} \\log{(\\dot{y})} = \\varphi^{*}{(A_{x},\\dot{y})} \\log{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))), Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))), Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(f)} = \\cos{(e^{f})}, then derive (\\int (- f + \\phi_{1}{(f)}) df)^{f} = (- \\frac{f^{2}}{2} + t_{1} + \\operatorname{Ci}{(e^{f})})^{f}, then obtain (\\int (- f + \\phi_{1}{(f)}) df)^{f} = (t_{1} - \\frac{(f - \\phi_{1}{(f)} + \\cos{(e^{f})})^{2}}{2} + \\operatorname{Ci}{(e^{f})})^{f}", "derivation": "\\phi_{1}{(f)} = \\cos{(e^{f})} and - f + \\phi_{1}{(f)} = - f + \\cos{(e^{f})} and - f + \\phi_{1}{(f)} - \\cos{(e^{f})} = - f and \\int (- f + \\phi_{1}{(f)}) df = \\int (- f + \\cos{(e^{f})}) df and (\\int (- f + \\phi_{1}{(f)}) df)^{f} = (\\int (- f + \\cos{(e^{f})}) df)^{f} and (\\int (- f + \\phi_{1}{(f)}) df)^{f} = (- \\frac{f^{2}}{2} + t_{1} + \\operatorname{Ci}{(e^{f})})^{f} and (\\int (- f + \\phi_{1}{(f)}) df)^{f} = (t_{1} - \\frac{(f - \\phi_{1}{(f)} + \\cos{(e^{f})})^{2}}{2} + \\operatorname{Ci}{(e^{f})})^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('f', commutative=True)), cos(exp(Symbol('f', commutative=True))))"], [["minus", 1, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), cos(exp(Symbol('f', commutative=True)))))"], [["minus", 2, "cos(exp(Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('f', commutative=True))))), Mul(Integer(-1), Symbol('f', commutative=True)))"], [["integrate", 2, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), cos(exp(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), cos(exp(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f', commutative=True), Integer(2))), Symbol('t_1', commutative=True), Ci(exp(Symbol('f', commutative=True)))), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\phi_1')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('f', commutative=True))), cos(exp(Symbol('f', commutative=True)))), Integer(2))), Ci(exp(Symbol('f', commutative=True)))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(V,L)} = e^{\\frac{L}{V}}, then derive \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL - 1 = \\frac{\\partial}{\\partial L} (V e^{\\frac{L}{V}} + \\mathbf{A}) - 1, then obtain \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL - 1 = \\frac{\\partial}{\\partial L} (V \\hat{p}{(V,L)} + \\mathbf{A}) - 1", "derivation": "\\hat{p}{(V,L)} = e^{\\frac{L}{V}} and \\int \\hat{p}{(V,L)} dL = \\int e^{\\frac{L}{V}} dL and \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL = \\frac{\\partial}{\\partial L} \\int e^{\\frac{L}{V}} dL and \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL - 1 = \\frac{\\partial}{\\partial L} \\int e^{\\frac{L}{V}} dL - 1 and \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL - 1 = \\frac{\\partial}{\\partial L} (V e^{\\frac{L}{V}} + \\mathbf{A}) - 1 and \\frac{\\partial}{\\partial L} \\int \\hat{p}{(V,L)} dL - 1 = \\frac{\\partial}{\\partial L} (V \\hat{p}{(V,L)} + \\mathbf{A}) - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), exp(Mul(Symbol('L', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(exp(Mul(Symbol('L', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))), Tuple(Symbol('L', commutative=True))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Integral(exp(Mul(Symbol('L', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(exp(Mul(Symbol('L', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Derivative(Integral(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Symbol('V', commutative=True), exp(Mul(Symbol('L', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Integral(Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Symbol('V', commutative=True), Function('\\\\hat{p}')(Symbol('V', commutative=True), Symbol('L', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}} + y^{\\prime}}, then derive \\int \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} dy^{\\prime} = G + e^{V_{\\mathbf{E}} + y^{\\prime}}, then obtain \\frac{\\int e^{V_{\\mathbf{E}} + y^{\\prime}} dy^{\\prime}}{y^{\\prime}} = \\frac{G + \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})}}{y^{\\prime}}", "derivation": "\\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}} + y^{\\prime}} and \\int \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} dy^{\\prime} = \\int e^{V_{\\mathbf{E}} + y^{\\prime}} dy^{\\prime} and \\int \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} dy^{\\prime} = G + e^{V_{\\mathbf{E}} + y^{\\prime}} and \\int \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} dy^{\\prime} = G + \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} and \\int e^{V_{\\mathbf{E}} + y^{\\prime}} dy^{\\prime} = G + \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})} and \\frac{\\int e^{V_{\\mathbf{E}} + y^{\\prime}} dy^{\\prime}}{y^{\\prime}} = \\frac{G + \\operatorname{E_{\\lambda}}{(y^{\\prime},V_{\\mathbf{E}})}}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('G', commutative=True), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('G', commutative=True), Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('G', commutative=True), Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["divide", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Integral(exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Function('E_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(v_{1})} = e^{e^{v_{1}}} and \\hat{H}{(t_{1})} = t_{1}, then derive \\int (\\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}}) dv_{1} = t_{1} + e^{v_{1}}, then obtain \\int (\\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}}) dv_{1} = \\hat{H}{(t_{1})} + e^{v_{1}}", "derivation": "\\rho_{f}{(v_{1})} = e^{e^{v_{1}}} and \\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}} = e^{v_{1}} and \\int (\\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}}) dv_{1} = \\int e^{v_{1}} dv_{1} and \\int (\\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}}) dv_{1} = t_{1} + e^{v_{1}} and \\hat{H}{(t_{1})} = t_{1} and \\int (\\rho_{f}{(v_{1})} + e^{v_{1}} - e^{e^{v_{1}}}) dv_{1} = \\hat{H}{(t_{1})} + e^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), exp(Symbol('v_1', commutative=True))), exp(exp(Symbol('v_1', commutative=True))))"], "Equality(Add(Function('\\\\rho_f')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('v_1', commutative=True))))), exp(Symbol('v_1', commutative=True)))"], [["integrate", 2, "Symbol('v_1', commutative=True)"], "Equality(Integral(Add(Function('\\\\rho_f')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('v_1', commutative=True))))), Tuple(Symbol('v_1', commutative=True))), Integral(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('\\\\rho_f')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('v_1', commutative=True))))), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('t_1', commutative=True), exp(Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Function('\\\\rho_f')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('v_1', commutative=True))))), Tuple(Symbol('v_1', commutative=True))), Add(Function('\\\\hat{H}')(Symbol('t_1', commutative=True)), exp(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given Q{(k)} = e^{k}, then derive Q{(k)} e^{k} + e^{k} \\frac{d}{d k} Q{(k)} = 2 e^{2 k}, then obtain Q^{2}{(k)} + Q{(k)} \\frac{d}{d k} Q{(k)} = 2 Q^{2}{(k)}", "derivation": "Q{(k)} = e^{k} and Q{(k)} e^{k} = e^{2 k} and \\frac{d}{d k} Q{(k)} e^{k} = \\frac{d}{d k} e^{2 k} and Q{(k)} e^{k} + e^{k} \\frac{d}{d k} Q{(k)} = 2 e^{2 k} and Q^{2}{(k)} + Q{(k)} \\frac{d}{d k} Q{(k)} = 2 Q^{2}{(k)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], [["times", 1, "exp(Symbol('k', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True))), exp(Mul(Integer(2), Symbol('k', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Function('Q')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('Q')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True))), Mul(exp(Symbol('k', commutative=True)), Derivative(Function('Q')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('Q')(Symbol('k', commutative=True)), Integer(2)), Mul(Function('Q')(Symbol('k', commutative=True)), Derivative(Function('Q')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Mul(Integer(2), Pow(Function('Q')(Symbol('k', commutative=True)), Integer(2))))"]]}, {"prompt": "Given A{(\\phi_2,g)} = \\phi_2 - g and B{(\\phi_2)} = - \\phi_2, then obtain (\\phi_2 - g + B{(\\phi_2)})^{\\phi_2} = (- g)^{\\phi_2}", "derivation": "A{(\\phi_2,g)} = \\phi_2 - g and - \\phi_2 + A{(\\phi_2,g)} = - g and B{(\\phi_2)} = - \\phi_2 and A{(\\phi_2,g)} + B{(\\phi_2)} = - g and \\phi_2 - g + B{(\\phi_2)} = - g and (\\phi_2 - g + B{(\\phi_2)})^{\\phi_2} = (- g)^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('A')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('A')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Function('B')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Function('B')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["power", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Function('B')(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(i,\\theta)} = \\theta i and V{(i,\\theta)} = (\\theta i)^{\\theta} and \\mathbf{P}{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} V{(i,\\theta)}, then obtain \\mathbf{P}{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta i)^{\\theta}", "derivation": "\\operatorname{M_{E}}{(i,\\theta)} = \\theta i and \\operatorname{M_{E}}^{\\theta}{(i,\\theta)} = (\\theta i)^{\\theta} and V{(i,\\theta)} = (\\theta i)^{\\theta} and \\frac{\\partial}{\\partial \\theta} \\operatorname{M_{E}}^{\\theta}{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta i)^{\\theta} and \\operatorname{M_{E}}^{\\theta}{(i,\\theta)} = V{(i,\\theta)} and \\frac{\\partial}{\\partial \\theta} V{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta i)^{\\theta} and \\mathbf{P}{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} V{(i,\\theta)} and \\mathbf{P}{(i,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta i)^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Pow(Function('M_E')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('M_E')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Function('V')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Function('V')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Function('V')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('\\\\mathbf{P}')(Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('i', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} = \\log{(\\mathbf{r} + \\theta_2)}, then obtain (- 2 \\log{(\\mathbf{r} + \\theta_2)})^{\\mathbf{r}} = (- \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} - \\log{(\\mathbf{r} + \\theta_2)})^{\\mathbf{r}}", "derivation": "\\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} = \\log{(\\mathbf{r} + \\theta_2)} and 0 = - \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} + \\log{(\\mathbf{r} + \\theta_2)} and - \\log{(\\mathbf{r} + \\theta_2)} = - \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} and - 2 \\log{(\\mathbf{r} + \\theta_2)} = - \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} - \\log{(\\mathbf{r} + \\theta_2)} and (- 2 \\log{(\\mathbf{r} + \\theta_2)})^{\\mathbf{r}} = (- \\operatorname{v_{2}}{(\\theta_2,\\mathbf{r})} - \\log{(\\mathbf{r} + \\theta_2)})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 2, "log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 3, "log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True))))))"], [["power", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integer(2), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True))))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\phi_1)} = \\int \\log{(\\phi_1)} d\\phi_1 and \\mathbf{J}{(\\phi_1)} = \\int \\log{(\\phi_1)} d\\phi_1, then derive \\operatorname{v_{y}}{(\\phi_1)} = \\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z}, then obtain \\mathbf{J}{(\\phi_1)} + \\int (\\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z}) d\\phi_1 = \\mathbf{J}{(\\phi_1)} + \\iint \\log{(\\phi_1)} d\\phi_1 d\\phi_1", "derivation": "\\operatorname{v_{y}}{(\\phi_1)} = \\int \\log{(\\phi_1)} d\\phi_1 and \\mathbf{J}{(\\phi_1)} = \\int \\log{(\\phi_1)} d\\phi_1 and \\operatorname{v_{y}}{(\\phi_1)} = \\mathbf{J}{(\\phi_1)} and \\operatorname{v_{y}}{(\\phi_1)} = \\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z} and \\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z} = \\mathbf{J}{(\\phi_1)} and \\int (\\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z}) d\\phi_1 = \\int \\mathbf{J}{(\\phi_1)} d\\phi_1 and \\int (\\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z}) d\\phi_1 = \\iint \\log{(\\phi_1)} d\\phi_1 d\\phi_1 and \\mathbf{J}{(\\phi_1)} + \\int (\\phi_1 \\log{(\\phi_1)} - \\phi_1 + v_{z}) d\\phi_1 = \\mathbf{J}{(\\phi_1)} + \\iint \\log{(\\phi_1)} d\\phi_1 d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('v_z', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["add", 7, "Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Add(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(b)} = \\log{(\\log{(b)})} and \\varphi^{*}{(b)} = e^{\\operatorname{f^{*}}{(b)}}, then obtain e^{3 \\operatorname{f^{*}}{(b)}} = e^{2 \\operatorname{f^{*}}{(b)}} \\log{(b)}", "derivation": "\\operatorname{f^{*}}{(b)} = \\log{(\\log{(b)})} and 2 \\operatorname{f^{*}}{(b)} = \\operatorname{f^{*}}{(b)} + \\log{(\\log{(b)})} and e^{2 \\operatorname{f^{*}}{(b)}} = e^{\\operatorname{f^{*}}{(b)}} \\log{(b)} and \\varphi^{*}{(b)} = e^{\\operatorname{f^{*}}{(b)}} and \\varphi^{*}{(b)} e^{2 \\operatorname{f^{*}}{(b)}} = \\varphi^{*}{(b)} e^{\\operatorname{f^{*}}{(b)}} \\log{(b)} and e^{3 \\operatorname{f^{*}}{(b)}} = e^{2 \\operatorname{f^{*}}{(b)}} \\log{(b)}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('b', commutative=True)), log(log(Symbol('b', commutative=True))))"], [["add", 1, "Function('f^*')(Symbol('b', commutative=True))"], "Equality(Mul(Integer(2), Function('f^*')(Symbol('b', commutative=True))), Add(Function('f^*')(Symbol('b', commutative=True)), log(log(Symbol('b', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Integer(2), Function('f^*')(Symbol('b', commutative=True)))), Mul(exp(Function('f^*')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('b', commutative=True)), exp(Function('f^*')(Symbol('b', commutative=True))))"], [["times", 3, "Function('\\\\varphi^*')(Symbol('b', commutative=True))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('b', commutative=True)), exp(Mul(Integer(2), Function('f^*')(Symbol('b', commutative=True))))), Mul(Function('\\\\varphi^*')(Symbol('b', commutative=True)), exp(Function('f^*')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(exp(Mul(Integer(3), Function('f^*')(Symbol('b', commutative=True)))), Mul(exp(Mul(Integer(2), Function('f^*')(Symbol('b', commutative=True)))), log(Symbol('b', commutative=True))))"]]}, {"prompt": "Given r{(\\mathbf{J}_M,P_{e})} = \\mathbf{J}_M + \\log{(P_{e})}, then obtain \\frac{\\mathbf{J}_M + \\log{(P_{e})}}{\\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})}} = \\frac{(\\mathbf{J}_M + \\log{(P_{e})})^{2}}{(\\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})}) r{(\\mathbf{J}_M,P_{e})}}", "derivation": "r{(\\mathbf{J}_M,P_{e})} = \\mathbf{J}_M + \\log{(P_{e})} and 2 r{(\\mathbf{J}_M,P_{e})} = \\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})} and \\frac{1}{2} = \\frac{\\mathbf{J}_M + \\log{(P_{e})}}{2 r{(\\mathbf{J}_M,P_{e})}} and \\frac{1}{2} = \\frac{\\mathbf{J}_M + \\log{(P_{e})}}{\\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})}} and \\frac{\\mathbf{J}_M + \\log{(P_{e})}}{\\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})}} = \\frac{(\\mathbf{J}_M + \\log{(P_{e})})^{2}}{(\\mathbf{J}_M + r{(\\mathbf{J}_M,P_{e})} + \\log{(P_{e})}) r{(\\mathbf{J}_M,P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('P_e', commutative=True))))"], [["add", 1, "Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True))"], "Equality(Mul(Integer(2), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('P_e', commutative=True))), Pow(Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('P_e', commutative=True))), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('P_e', commutative=True))), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('P_e', commutative=True))), Integer(2)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Integer(-1)), Pow(Function('r')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mu,L)} = \\cos{(L - \\mu)}, then obtain - \\cos{(L - \\mu)} + \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\mathbf{H}{(\\mu,L)}) + \\cos{(L - \\mu)}) = - \\cos{(L - \\mu)} + \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\cos{(L - \\mu)}) + \\cos{(L - \\mu)})", "derivation": "\\mathbf{H}{(\\mu,L)} = \\cos{(L - \\mu)} and - \\mu + \\mathbf{H}{(\\mu,L)} = - \\mu + \\cos{(L - \\mu)} and L (- \\mu + \\mathbf{H}{(\\mu,L)}) = L (- \\mu + \\cos{(L - \\mu)}) and L (- \\mu + \\mathbf{H}{(\\mu,L)}) + \\cos{(L - \\mu)} = L (- \\mu + \\cos{(L - \\mu)}) + \\cos{(L - \\mu)} and \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\mathbf{H}{(\\mu,L)}) + \\cos{(L - \\mu)}) = \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\cos{(L - \\mu)}) + \\cos{(L - \\mu)}) and - \\cos{(L - \\mu)} + \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\mathbf{H}{(\\mu,L)}) + \\cos{(L - \\mu)}) = - \\cos{(L - \\mu)} + \\frac{\\partial}{\\partial \\mu} (L (- \\mu + \\cos{(L - \\mu)}) + \\cos{(L - \\mu)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True)))), Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))))"], [["add", 3, "cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True)))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True)))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 5, "cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Derivative(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mu', commutative=True), Symbol('L', commutative=True)))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Derivative(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(H,\\mathbf{A})} = \\sin{(H + \\mathbf{A})}, then obtain (\\frac{\\partial}{\\partial \\mathbf{A}} \\frac{r{(H,\\mathbf{A})}}{\\mathbf{A}})^{\\mathbf{A}} = (\\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\sin{(H + \\mathbf{A})}}{\\mathbf{A}})^{\\mathbf{A}}", "derivation": "r{(H,\\mathbf{A})} = \\sin{(H + \\mathbf{A})} and \\frac{r{(H,\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\sin{(H + \\mathbf{A})}}{\\mathbf{A}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{r{(H,\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\sin{(H + \\mathbf{A})}}{\\mathbf{A}} and (\\frac{\\partial}{\\partial \\mathbf{A}} \\frac{r{(H,\\mathbf{A})}}{\\mathbf{A}})^{\\mathbf{A}} = (\\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\sin{(H + \\mathbf{A})}}{\\mathbf{A}})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), sin(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('r')(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('r')(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('r')(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(v_{2})} = \\log{(\\cos{(v_{2})})} and \\dot{x}{(v_{2})} = \\cos{(v_{2})}, then obtain - v_{2} + \\frac{\\operatorname{J_{\\varepsilon}}{(v_{2})}}{\\log{(\\dot{x}{(v_{2})})}} = 1 - v_{2}", "derivation": "\\operatorname{J_{\\varepsilon}}{(v_{2})} = \\log{(\\cos{(v_{2})})} and \\frac{\\operatorname{J_{\\varepsilon}}{(v_{2})}}{\\log{(\\cos{(v_{2})})}} = 1 and - v_{2} + \\frac{\\operatorname{J_{\\varepsilon}}{(v_{2})}}{\\log{(\\cos{(v_{2})})}} = 1 - v_{2} and \\dot{x}{(v_{2})} = \\cos{(v_{2})} and - v_{2} + \\frac{\\operatorname{J_{\\varepsilon}}{(v_{2})}}{\\log{(\\dot{x}{(v_{2})})}} = 1 - v_{2}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('v_2', commutative=True)), log(cos(Symbol('v_2', commutative=True))))"], [["divide", 1, "log(cos(Symbol('v_2', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('v_2', commutative=True)), Pow(log(cos(Symbol('v_2', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Function('J_{\\\\varepsilon}')(Symbol('v_2', commutative=True)), Pow(log(cos(Symbol('v_2', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Function('J_{\\\\varepsilon}')(Symbol('v_2', commutative=True)), Pow(log(Function('\\\\dot{x}')(Symbol('v_2', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(z^{*},\\mathbf{s},\\hat{H})} = z^{*} + m{(z^{*},\\mathbf{s},\\hat{H})} and x{(z^{*},\\mathbf{s},\\hat{H})} = z^{*} + m{(z^{*},\\mathbf{s},\\hat{H})}, then obtain \\log{(- m{(z^{*},\\mathbf{s},\\hat{H})} + x{(z^{*},\\mathbf{s},\\hat{H})})} = \\log{(z^{*})}", "derivation": "\\hat{x}_0{(z^{*},\\mathbf{s},\\hat{H})} = z^{*} + m{(z^{*},\\mathbf{s},\\hat{H})} and \\hat{x}_0{(z^{*},\\mathbf{s},\\hat{H})} - m{(z^{*},\\mathbf{s},\\hat{H})} = z^{*} and \\log{(\\hat{x}_0{(z^{*},\\mathbf{s},\\hat{H})} - m{(z^{*},\\mathbf{s},\\hat{H})})} = \\log{(z^{*})} and x{(z^{*},\\mathbf{s},\\hat{H})} = z^{*} + m{(z^{*},\\mathbf{s},\\hat{H})} and \\hat{x}_0{(z^{*},\\mathbf{s},\\hat{H})} = x{(z^{*},\\mathbf{s},\\hat{H})} and \\log{(- m{(z^{*},\\mathbf{s},\\hat{H})} + x{(z^{*},\\mathbf{s},\\hat{H})})} = \\log{(z^{*})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('z^*', commutative=True), Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Symbol('z^*', commutative=True))"], [["log", 2], "Equality(log(Add(Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), log(Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('z^*', commutative=True), Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Function('x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Add(Mul(Integer(-1), Function('m')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Function('x')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), log(Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given Z{(\\rho_b)} = - \\rho_b + e^{\\rho_b} and h{(\\rho_b)} = - \\rho_b, then obtain (\\frac{d}{d \\rho_b} Z{(\\rho_b)})^{\\rho_b} = (\\frac{d}{d \\rho_b} (h{(\\rho_b)} + e^{\\rho_b}))^{\\rho_b}", "derivation": "Z{(\\rho_b)} = - \\rho_b + e^{\\rho_b} and \\frac{d}{d \\rho_b} Z{(\\rho_b)} = \\frac{d}{d \\rho_b} (- \\rho_b + e^{\\rho_b}) and h{(\\rho_b)} = - \\rho_b and \\frac{d}{d \\rho_b} Z{(\\rho_b)} = \\frac{d}{d \\rho_b} (h{(\\rho_b)} + e^{\\rho_b}) and (\\frac{d}{d \\rho_b} Z{(\\rho_b)})^{\\rho_b} = (\\frac{d}{d \\rho_b} (h{(\\rho_b)} + e^{\\rho_b}))^{\\rho_b}", "srepr_derivation": [["renaming_premise", "Equality(Function('Z')(Symbol('\\\\rho_b', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('Z')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Add(Function('h')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Derivative(Function('Z')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('\\\\rho_b', commutative=True)), Pow(Derivative(Add(Function('h')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given Z{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain \\frac{Z^{\\mathbf{H}}{(\\mathbf{H})}}{\\int (e^{\\mathbf{H}})^{\\mathbf{H}} d\\mathbf{H}} = \\frac{(e^{\\mathbf{H}})^{\\mathbf{H}}}{\\int (e^{\\mathbf{H}})^{\\mathbf{H}} d\\mathbf{H}}", "derivation": "Z{(\\mathbf{H})} = e^{\\mathbf{H}} and Z^{\\mathbf{H}}{(\\mathbf{H})} = (e^{\\mathbf{H}})^{\\mathbf{H}} and \\int Z^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H} = \\int (e^{\\mathbf{H}})^{\\mathbf{H}} d\\mathbf{H} and \\frac{Z^{\\mathbf{H}}{(\\mathbf{H})}}{\\int Z^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H}} = \\frac{(e^{\\mathbf{H}})^{\\mathbf{H}}}{\\int Z^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H}} and \\frac{Z^{\\mathbf{H}}{(\\mathbf{H})}}{\\int (e^{\\mathbf{H}})^{\\mathbf{H}} d\\mathbf{H}} = \\frac{(e^{\\mathbf{H}})^{\\mathbf{H}}}{\\int (e^{\\mathbf{H}})^{\\mathbf{H}} d\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 2, "Integral(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))), Mul(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))), Mul(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Pow(exp(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(\\dot{z},V)} = \\log{(V + \\dot{z})}, then obtain \\frac{\\partial}{\\partial \\dot{z}} \\int \\ddot{x}{(\\dot{z},V)} d\\dot{z} = \\frac{\\partial}{\\partial \\dot{z}} (V \\log{(V + \\dot{z})} + \\dot{z} \\log{(V + \\dot{z})} - \\dot{z} + \\mathbf{f})", "derivation": "\\ddot{x}{(\\dot{z},V)} = \\log{(V + \\dot{z})} and \\int \\ddot{x}{(\\dot{z},V)} d\\dot{z} = \\int \\log{(V + \\dot{z})} d\\dot{z} and \\frac{\\partial}{\\partial \\dot{z}} \\int \\ddot{x}{(\\dot{z},V)} d\\dot{z} = \\frac{\\partial}{\\partial \\dot{z}} \\int \\log{(V + \\dot{z})} d\\dot{z} and \\frac{\\partial}{\\partial \\dot{z}} \\int \\ddot{x}{(\\dot{z},V)} d\\dot{z} = \\frac{\\partial}{\\partial \\dot{z}} (V \\log{(V + \\dot{z})} + \\dot{z} \\log{(V + \\dot{z})} - \\dot{z} + \\mathbf{f})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('V', commutative=True)), log(Add(Symbol('V', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(log(Add(Symbol('V', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Integral(log(Add(Symbol('V', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('V', commutative=True), log(Add(Symbol('V', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Mul(Symbol('\\\\dot{z}', commutative=True), log(Add(Symbol('V', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})} = e^{\\mathbf{D} n_{1}}, then obtain \\frac{\\partial}{\\partial n_{1}} \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + \\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})}) = \\frac{\\partial}{\\partial n_{1}} \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + e^{\\mathbf{D} n_{1}})", "derivation": "\\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})} = e^{\\mathbf{D} n_{1}} and - \\mathbf{D} n_{1} + \\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})} = - \\mathbf{D} n_{1} + e^{\\mathbf{D} n_{1}} and \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + \\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})}) = \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + e^{\\mathbf{D} n_{1}}) and \\frac{\\partial}{\\partial n_{1}} \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + \\operatorname{L_{\\varepsilon}}{(n_{1},\\mathbf{D})}) = \\frac{\\partial}{\\partial n_{1}} \\mathbf{D} n_{1} (- \\mathbf{D} n_{1} + e^{\\mathbf{D} n_{1}})", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)))))"], [["times", 2, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True))))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True))))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} = \\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}}, then obtain \\int (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1)^{2} dV_{\\mathbf{E}} = \\int (\\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}} - 1) (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1) dV_{\\mathbf{E}}", "derivation": "B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} = \\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}} and B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1 = \\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}} - 1 and (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1)^{2} = (\\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}} - 1) (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1) and \\int (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1)^{2} dV_{\\mathbf{E}} = \\int (\\frac{V_{\\mathbf{E}} a^{\\dagger}}{\\hat{H}_{\\lambda}} - 1) (B{(a^{\\dagger},\\hat{H}_{\\lambda},V_{\\mathbf{E}})} - 1) dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))"], [["times", 2, "Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))"], "Equality(Pow(Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Integer(2)), Mul(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Pow(Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Mul(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Function('B')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)}, then obtain \\log{(\\hat{x}_0)} \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\mathbf{E}{(\\hat{x}_0)} d\\hat{x}_0 = \\log{(\\hat{x}_0)} \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\log{(\\hat{x}_0)} d\\hat{x}_0", "derivation": "\\mathbf{E}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)} and \\int \\mathbf{E}{(\\hat{x}_0)} d\\hat{x}_0 = \\int \\log{(\\hat{x}_0)} d\\hat{x}_0 and \\frac{d}{d \\hat{x}_0} \\int \\mathbf{E}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{d}{d \\hat{x}_0} \\int \\log{(\\hat{x}_0)} d\\hat{x}_0 and \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\mathbf{E}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\log{(\\hat{x}_0)} d\\hat{x}_0 and \\log{(\\hat{x}_0)} \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\mathbf{E}{(\\hat{x}_0)} d\\hat{x}_0 = \\log{(\\hat{x}_0)} \\frac{d^{2}}{d \\hat{x}_0^{2}} \\int \\log{(\\hat{x}_0)} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(log(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))), Derivative(Integral(log(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))))"], [["times", 4, "log(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)))), Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Integral(log(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{X})} = \\log{(\\hat{X})} and \\Omega{(\\hat{X})} = \\log{(\\hat{X})}, then derive - \\frac{1}{\\hat{X}^{2}} = \\frac{d^{2}}{d \\hat{X}^{2}} \\Omega{(\\hat{X})}, then obtain \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})} = - \\frac{1}{\\hat{X}^{2}}", "derivation": "\\mathbf{H}{(\\hat{X})} = \\log{(\\hat{X})} and \\Omega{(\\hat{X})} = \\log{(\\hat{X})} and \\frac{d}{d \\hat{X}} \\mathbf{H}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\log{(\\hat{X})} and \\frac{d}{d \\hat{X}} \\mathbf{H}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\Omega{(\\hat{X})} and \\frac{d}{d \\hat{X}} \\log{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\Omega{(\\hat{X})} and \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\Omega{(\\hat{X})} and - \\frac{1}{\\hat{X}^{2}} = \\frac{d^{2}}{d \\hat{X}^{2}} \\Omega{(\\hat{X})} and \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})} = - \\frac{1}{\\hat{X}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Function('\\\\Omega')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Function('\\\\Omega')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Derivative(Function('\\\\Omega')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2))), Derivative(Function('\\\\Omega')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\Psi{(\\psi^*,z)} = \\sin{(\\psi^* z)}, then derive \\frac{\\partial}{\\partial z} \\Psi{(\\psi^*,z)} = \\psi^* \\cos{(\\psi^* z)}, then obtain \\cos{(\\psi^* z)} \\frac{\\partial}{\\partial z} \\Psi{(\\psi^*,z)} = \\psi^* \\cos^{2}{(\\psi^* z)}", "derivation": "\\Psi{(\\psi^*,z)} = \\sin{(\\psi^* z)} and \\frac{\\partial}{\\partial z} \\Psi{(\\psi^*,z)} = \\frac{\\partial}{\\partial z} \\sin{(\\psi^* z)} and \\frac{\\partial}{\\partial z} \\Psi{(\\psi^*,z)} = \\psi^* \\cos{(\\psi^* z)} and \\cos{(\\psi^* z)} \\frac{\\partial}{\\partial z} \\Psi{(\\psi^*,z)} = \\psi^* \\cos^{2}{(\\psi^* z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Symbol('\\\\psi^*', commutative=True), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)))))"], [["times", 3, "cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi^*', commutative=True), Pow(cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{E},\\rho_f)} = \\sin{(\\mathbf{E}^{\\rho_f})} and \\operatorname{E_{x}}{(\\eta)} = \\eta, then obtain - \\sin{(\\mathbf{E}^{\\rho_f})} + \\frac{d}{d \\eta} (- \\eta + \\operatorname{E_{x}}{(\\eta)}) = - \\sin{(\\mathbf{E}^{\\rho_f})} + \\frac{d}{d \\eta} 0", "derivation": "\\operatorname{A_{2}}{(\\mathbf{E},\\rho_f)} = \\sin{(\\mathbf{E}^{\\rho_f})} and \\operatorname{E_{x}}{(\\eta)} = \\eta and \\operatorname{A_{2}}{(\\mathbf{E},\\rho_f)} + \\operatorname{E_{x}}{(\\eta)} = \\eta + \\operatorname{A_{2}}{(\\mathbf{E},\\rho_f)} and \\operatorname{E_{x}}{(\\eta)} + \\sin{(\\mathbf{E}^{\\rho_f})} = \\eta + \\sin{(\\mathbf{E}^{\\rho_f})} and - \\eta + \\operatorname{E_{x}}{(\\eta)} = 0 and \\frac{d}{d \\eta} (- \\eta + \\operatorname{E_{x}}{(\\eta)}) = \\frac{d}{d \\eta} 0 and - \\sin{(\\mathbf{E}^{\\rho_f})} + \\frac{d}{d \\eta} (- \\eta + \\operatorname{E_{x}}{(\\eta)}) = - \\sin{(\\mathbf{E}^{\\rho_f})} + \\frac{d}{d \\eta} 0", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], ["get_premise", "Equality(Function('E_x')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["add", 2, "Function('A_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Function('E_x')(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Function('A_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('E_x')(Symbol('\\\\eta', commutative=True)), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)))))"], [["minus", 4, "Add(Symbol('\\\\eta', commutative=True), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('E_x')(Symbol('\\\\eta', commutative=True))), Integer(0))"], [["differentiate", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('E_x')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 6, "sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('E_x')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(f_{\\mathbf{p}},g)} = \\sin{(f_{\\mathbf{p}} g)} and \\mathbf{D}{(f_{\\mathbf{p}},g)} = \\sin{(f_{\\mathbf{p}} g)}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{p}}} s{(f_{\\mathbf{p}},g)} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\mathbf{D}{(f_{\\mathbf{p}},g)}", "derivation": "s{(f_{\\mathbf{p}},g)} = \\sin{(f_{\\mathbf{p}} g)} and \\mathbf{D}{(f_{\\mathbf{p}},g)} = \\sin{(f_{\\mathbf{p}} g)} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} s{(f_{\\mathbf{p}},g)} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}} g)} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} s{(f_{\\mathbf{p}},g)} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\mathbf{D}{(f_{\\mathbf{p}},g)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True)), sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True)), sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('s')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta f_{E}, then derive \\frac{\\partial}{\\partial f_{E}} \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta, then obtain \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)} + \\frac{\\partial}{\\partial f_{E}} \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta + \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)}", "derivation": "\\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta f_{E} and \\frac{\\partial}{\\partial f_{E}} \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\frac{\\partial}{\\partial f_{E}} \\theta f_{E} and \\frac{\\partial}{\\partial f_{E}} \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta and \\frac{\\partial}{\\partial f_{E}} \\theta f_{E} = \\theta and \\frac{\\partial}{\\partial f_{E}} \\theta f_{E} + \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)} = \\theta + \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)} and \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)} + \\frac{\\partial}{\\partial f_{E}} \\hat{\\mathbf{x}}{(\\theta,f_{E})} = \\theta + \\frac{\\partial}{\\partial x^\\prime} \\hat{X}{(x^\\prime,g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True))"], [["add", 4, "Derivative(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Symbol('\\\\theta', commutative=True), Derivative(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Derivative(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Add(Symbol('\\\\theta', commutative=True), Derivative(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(M)} = \\sin{(M)}, then derive \\int (M + \\operatorname{E_{n}}{(M)}) dM = \\frac{M^{2}}{2} + q - \\cos{(M)}, then obtain 0 = \\frac{M^{2}}{2} + q - \\cos{(M)} - \\int (M + \\sin{(M)}) dM", "derivation": "\\operatorname{E_{n}}{(M)} = \\sin{(M)} and M + \\operatorname{E_{n}}{(M)} = M + \\sin{(M)} and \\int (M + \\operatorname{E_{n}}{(M)}) dM = \\int (M + \\sin{(M)}) dM and \\int (M + \\operatorname{E_{n}}{(M)}) dM = \\frac{M^{2}}{2} + q - \\cos{(M)} and \\int (M + \\sin{(M)}) dM = \\frac{M^{2}}{2} + q - \\cos{(M)} and - \\int (M + \\operatorname{E_{n}}{(M)}) dM + \\int (M + \\sin{(M)}) dM = \\frac{M^{2}}{2} + q - \\cos{(M)} - \\int (M + \\operatorname{E_{n}}{(M)}) dM and 0 = \\frac{M^{2}}{2} + q - \\cos{(M)} - \\int (M + \\operatorname{E_{n}}{(M)}) dM and 0 = \\frac{M^{2}}{2} + q - \\cos{(M)} - \\int (M + \\sin{(M)}) dM", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["add", 1, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Add(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["minus", 5, "Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), Function('E_n')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(m)} = \\sin{(\\cos{(m)})} and \\operatorname{t_{1}}{(m)} = \\frac{\\sin{(\\cos{(m)})}}{m}, then obtain \\int \\frac{\\operatorname{v_{2}}{(m)}}{m} dm = \\int \\frac{\\sin{(\\cos{(m)})}}{m} dm", "derivation": "\\operatorname{v_{2}}{(m)} = \\sin{(\\cos{(m)})} and \\frac{\\operatorname{v_{2}}{(m)}}{m} = \\frac{\\sin{(\\cos{(m)})}}{m} and \\operatorname{t_{1}}{(m)} = \\frac{\\sin{(\\cos{(m)})}}{m} and \\int \\operatorname{t_{1}}{(m)} dm = \\int \\frac{\\sin{(\\cos{(m)})}}{m} dm and \\frac{\\operatorname{v_{2}}{(m)}}{m} = \\operatorname{t_{1}}{(m)} and \\int \\frac{\\operatorname{v_{2}}{(m)}}{m} dm = \\int \\frac{\\sin{(\\cos{(m)})}}{m} dm", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('m', commutative=True)), sin(cos(Symbol('m', commutative=True))))"], [["divide", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('v_2')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), sin(cos(Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('m', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), sin(cos(Symbol('m', commutative=True)))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), sin(cos(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('v_2')(Symbol('m', commutative=True))), Function('t_1')(Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('v_2')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), sin(cos(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(Q)} = \\sin{(Q)}, then derive \\frac{d}{d Q} \\dot{y}{(Q)} = \\cos{(Q)}, then obtain S^{\\psi} (I + \\dot{y}{(Q)}) = S^{\\psi} (\\mathbf{s} + \\sin{(Q)})", "derivation": "\\dot{y}{(Q)} = \\sin{(Q)} and \\frac{d}{d Q} \\dot{y}{(Q)} = \\frac{d}{d Q} \\sin{(Q)} and \\frac{d}{d Q} \\dot{y}{(Q)} = \\cos{(Q)} and \\int \\frac{d}{d Q} \\dot{y}{(Q)} dQ = \\int \\cos{(Q)} dQ and S^{\\psi} \\int \\frac{d}{d Q} \\dot{y}{(Q)} dQ = S^{\\psi} \\int \\cos{(Q)} dQ and S^{\\psi} (I + \\dot{y}{(Q)}) = S^{\\psi} (\\mathbf{s} + \\sin{(Q)})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), cos(Symbol('Q', commutative=True)))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{y}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["times", 4, "Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Integral(Derivative(Function('\\\\dot{y}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('I', commutative=True), Function('\\\\dot{y}')(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given E{(r,\\hat{H})} = - \\sin{(\\hat{H} - r)}, then obtain - \\sin{(\\hat{H} - r)} + \\int (E{(r,\\hat{H})} - \\sin{(\\hat{H} - r)}) dr = - \\sin{(\\hat{H} - r)} + \\int - 2 \\sin{(\\hat{H} - r)} dr", "derivation": "E{(r,\\hat{H})} = - \\sin{(\\hat{H} - r)} and E{(r,\\hat{H})} - \\sin{(\\hat{H} - r)} = - 2 \\sin{(\\hat{H} - r)} and \\int (E{(r,\\hat{H})} - \\sin{(\\hat{H} - r)}) dr = \\int - 2 \\sin{(\\hat{H} - r)} dr and - \\sin{(\\hat{H} - r)} + \\int (E{(r,\\hat{H})} - \\sin{(\\hat{H} - r)}) dr = - \\sin{(\\hat{H} - r)} + \\int - 2 \\sin{(\\hat{H} - r)} dr", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('r', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))))"], [["add", 1, "Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], "Equality(Add(Function('E')(Symbol('r', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Function('E')(Symbol('r', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))), Tuple(Symbol('r', commutative=True))), Integral(Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True))))"], [["add", 3, "Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Integral(Add(Function('E')(Symbol('r', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Integral(Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given f{(E_{x},\\mathbf{F})} = \\log{(E_{x} \\mathbf{F})}, then obtain \\int \\frac{f{(E_{x},\\mathbf{F})} + \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} dE_{x} = \\int \\frac{2 \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} dE_{x}", "derivation": "f{(E_{x},\\mathbf{F})} = \\log{(E_{x} \\mathbf{F})} and f{(E_{x},\\mathbf{F})} + \\log{(E_{x} \\mathbf{F})} = 2 \\log{(E_{x} \\mathbf{F})} and \\frac{f{(E_{x},\\mathbf{F})} + \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} = \\frac{2 \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} and \\int \\frac{f{(E_{x},\\mathbf{F})} + \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} dE_{x} = \\int \\frac{2 \\log{(E_{x} \\mathbf{F})}}{\\frac{\\partial}{\\partial E_{x}} f{(E_{x},\\mathbf{F})}} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(2), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 2, "Derivative(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Pow(Derivative(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(2), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Derivative(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Add(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Pow(Derivative(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Integer(2), log(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Derivative(Function('f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\nabla{(P_{g},\\dot{z})} = \\dot{z} \\sin{(P_{g})} and \\pi{(P_{g})} = P_{g}, then obtain (\\frac{\\pi{(P_{g})}}{\\nabla{(P_{g},\\dot{z})}})^{P_{g}} = (\\frac{P_{g}}{\\nabla{(P_{g},\\dot{z})}})^{P_{g}}", "derivation": "\\nabla{(P_{g},\\dot{z})} = \\dot{z} \\sin{(P_{g})} and \\pi{(P_{g})} = P_{g} and \\frac{\\pi{(P_{g})}}{\\dot{z} \\sin{(P_{g})}} = \\frac{P_{g}}{\\dot{z} \\sin{(P_{g})}} and (\\frac{\\pi{(P_{g})}}{\\dot{z} \\sin{(P_{g})}})^{P_{g}} = (\\frac{P_{g}}{\\dot{z} \\sin{(P_{g})}})^{P_{g}} and (\\frac{\\pi{(P_{g})}}{\\nabla{(P_{g},\\dot{z})}})^{P_{g}} = (\\frac{P_{g}}{\\nabla{(P_{g},\\dot{z})}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), sin(Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["divide", 2, "Mul(Symbol('\\\\dot{z}', commutative=True), sin(Symbol('P_g', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))), Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True)), Pow(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Mul(Pow(Function('\\\\nabla')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Function('\\\\pi')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Mul(Symbol('P_g', commutative=True), Pow(Function('\\\\nabla')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\hat{H})} = \\sin{(\\log{(\\hat{H})})}, then obtain \\Psi + \\psi^{*}{(\\hat{H})} - \\sin{(\\log{(\\hat{H})})} = \\int \\frac{d}{d \\hat{H}} 0 d\\hat{H}", "derivation": "\\psi^{*}{(\\hat{H})} = \\sin{(\\log{(\\hat{H})})} and \\psi^{*}{(\\hat{H})} - \\sin{(\\log{(\\hat{H})})} = 0 and \\frac{d}{d \\hat{H}} (\\psi^{*}{(\\hat{H})} - \\sin{(\\log{(\\hat{H})})}) = \\frac{d}{d \\hat{H}} 0 and \\int \\frac{d}{d \\hat{H}} (\\psi^{*}{(\\hat{H})} - \\sin{(\\log{(\\hat{H})})}) d\\hat{H} = \\int \\frac{d}{d \\hat{H}} 0 d\\hat{H} and \\Psi + \\psi^{*}{(\\hat{H})} - \\sin{(\\log{(\\hat{H})})} = \\int \\frac{d}{d \\hat{H}} 0 d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{H}', commutative=True)), sin(log(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "sin(log(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\hat{H}', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\psi^*')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\hat{H}', commutative=True))))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\psi^*')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\hat{H}', commutative=True))))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\psi^*')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\hat{H}', commutative=True))))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(\\tilde{g},\\mu_0)} = \\sin{(\\mu_0^{\\tilde{g}})}, then obtain - \\mu_0^{\\tilde{g}} + \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} - \\frac{\\partial}{\\partial \\tilde{g}} \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} = - \\mu_0^{\\tilde{g}} + \\sin^{\\tilde{g}}{(\\mu_0^{\\tilde{g}})} - \\frac{\\partial}{\\partial \\tilde{g}} \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)}", "derivation": "\\dot{y}{(\\tilde{g},\\mu_0)} = \\sin{(\\mu_0^{\\tilde{g}})} and \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} = \\sin^{\\tilde{g}}{(\\mu_0^{\\tilde{g}})} and - \\mu_0^{\\tilde{g}} + \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} = - \\mu_0^{\\tilde{g}} + \\sin^{\\tilde{g}}{(\\mu_0^{\\tilde{g}})} and - \\mu_0^{\\tilde{g}} + \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} - \\frac{\\partial}{\\partial \\tilde{g}} \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)} = - \\mu_0^{\\tilde{g}} + \\sin^{\\tilde{g}}{(\\mu_0^{\\tilde{g}})} - \\frac{\\partial}{\\partial \\tilde{g}} \\dot{y}^{\\tilde{g}}{(\\tilde{g},\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 2, "Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 3, "Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(sin(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\hbar)} = \\cos{(\\hbar)}, then obtain e^{\\sin{(\\hbar)} + \\frac{d}{d \\hbar} \\mathbf{A}{(\\hbar)}} = 1", "derivation": "\\mathbf{A}{(\\hbar)} = \\cos{(\\hbar)} and \\mathbf{A}{(\\hbar)} - \\cos{(\\hbar)} = 0 and \\frac{d}{d \\hbar} (\\mathbf{A}{(\\hbar)} - \\cos{(\\hbar)}) = \\frac{d}{d \\hbar} 0 and e^{\\frac{d}{d \\hbar} (\\mathbf{A}{(\\hbar)} - \\cos{(\\hbar)})} = e^{\\frac{d}{d \\hbar} 0} and e^{\\sin{(\\hbar)} + \\frac{d}{d \\hbar} \\mathbf{A}{(\\hbar)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), exp(Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(exp(Add(sin(Symbol('\\\\hbar', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v,\\rho_f)} = \\rho_f + v, then obtain (\\frac{\\operatorname{v_{t}}^{2}{(v,\\rho_f)}}{v^{2}})^{v} = (\\frac{(\\rho_f + v) \\operatorname{v_{t}}{(v,\\rho_f)}}{v^{2}})^{v}", "derivation": "\\operatorname{v_{t}}{(v,\\rho_f)} = \\rho_f + v and \\frac{\\operatorname{v_{t}}{(v,\\rho_f)}}{v} = \\frac{\\rho_f + v}{v} and \\frac{\\operatorname{v_{t}}^{2}{(v,\\rho_f)}}{v^{2}} = \\frac{(\\rho_f + v) \\operatorname{v_{t}}{(v,\\rho_f)}}{v^{2}} and (\\frac{\\operatorname{v_{t}}^{2}{(v,\\rho_f)}}{v^{2}})^{v} = (\\frac{(\\rho_f + v) \\operatorname{v_{t}}{(v,\\rho_f)}}{v^{2}})^{v}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)))"], [["divide", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-2)), Pow(Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2))), Mul(Pow(Symbol('v', commutative=True), Integer(-2)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-2)), Pow(Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2))), Symbol('v', commutative=True)), Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-2)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Function('v_t')(Symbol('v', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\hat{p}_0,l)} = \\hat{p}_0 \\log{(l)}, then obtain e^{- (\\hat{p}_0 \\log{(l)})^{l} + (\\operatorname{x^{{\\}'}}^{l}{(\\hat{p}_0,l)})^{l}} = e^{- (\\hat{p}_0 \\log{(l)})^{l} + ((\\hat{p}_0 \\log{(l)})^{l})^{l}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\hat{p}_0,l)} = \\hat{p}_0 \\log{(l)} and \\operatorname{x^{{\\}'}}^{l}{(\\hat{p}_0,l)} = (\\hat{p}_0 \\log{(l)})^{l} and (\\operatorname{x^{{\\}'}}^{l}{(\\hat{p}_0,l)})^{l} = ((\\hat{p}_0 \\log{(l)})^{l})^{l} and - (\\hat{p}_0 \\log{(l)})^{l} + (\\operatorname{x^{{\\}'}}^{l}{(\\hat{p}_0,l)})^{l} = - (\\hat{p}_0 \\log{(l)})^{l} + ((\\hat{p}_0 \\log{(l)})^{l})^{l} and e^{- (\\hat{p}_0 \\log{(l)})^{l} + (\\operatorname{x^{{\\}'}}^{l}{(\\hat{p}_0,l)})^{l}} = e^{- (\\hat{p}_0 \\log{(l)})^{l} + ((\\hat{p}_0 \\log{(l)})^{l})^{l}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["minus", 3, "Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Pow(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Pow(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["exp", 4], "Equality(exp(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Pow(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)))), exp(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Pow(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(C_{d})} = \\log{(\\log{(C_{d})})} and \\operatorname{v_{y}}{(C_{d})} = \\log{(C_{d})}, then obtain \\log{(\\operatorname{v_{y}}{(C_{d})})} + \\log{(\\log{(C_{d})})} = 2 \\log{(\\log{(C_{d})})}", "derivation": "\\mathbf{r}{(C_{d})} = \\log{(\\log{(C_{d})})} and \\operatorname{v_{y}}{(C_{d})} = \\log{(C_{d})} and \\mathbf{r}{(C_{d})} = \\log{(\\operatorname{v_{y}}{(C_{d})})} and \\mathbf{r}{(C_{d})} + \\log{(\\operatorname{v_{y}}{(C_{d})})} = 2 \\log{(\\operatorname{v_{y}}{(C_{d})})} and \\mathbf{r}{(C_{d})} + \\log{(\\log{(C_{d})})} = 2 \\log{(\\log{(C_{d})})} and \\log{(\\operatorname{v_{y}}{(C_{d})})} + \\log{(\\log{(C_{d})})} = 2 \\log{(\\log{(C_{d})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('C_d', commutative=True)), log(log(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{r}')(Symbol('C_d', commutative=True)), log(Function('v_y')(Symbol('C_d', commutative=True))))"], [["add", 3, "log(Function('v_y')(Symbol('C_d', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('C_d', commutative=True)), log(Function('v_y')(Symbol('C_d', commutative=True)))), Mul(Integer(2), log(Function('v_y')(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('C_d', commutative=True)), log(log(Symbol('C_d', commutative=True)))), Mul(Integer(2), log(log(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(log(Function('v_y')(Symbol('C_d', commutative=True))), log(log(Symbol('C_d', commutative=True)))), Mul(Integer(2), log(log(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given f{(\\sigma_x)} = \\log{(\\log{(\\sigma_x)})} and \\operatorname{f_{E}}{(\\lambda)} = e^{\\lambda} and \\mathbf{M}{(\\eta)} = \\eta, then obtain \\frac{\\mathbf{M}{(\\eta)} \\operatorname{f_{E}}^{- \\lambda}{(\\lambda)}}{\\log{(\\log{(\\sigma_x)})}^{2}} = \\frac{\\eta \\operatorname{f_{E}}^{- \\lambda}{(\\lambda)}}{\\log{(\\log{(\\sigma_x)})}^{2}}", "derivation": "f{(\\sigma_x)} = \\log{(\\log{(\\sigma_x)})} and \\operatorname{f_{E}}{(\\lambda)} = e^{\\lambda} and \\mathbf{M}{(\\eta)} = \\eta and \\frac{\\mathbf{M}{(\\eta)}}{f{(\\sigma_x)}} = \\frac{\\eta}{f{(\\sigma_x)}} and \\frac{\\mathbf{M}{(\\eta)}}{\\log{(\\log{(\\sigma_x)})}} = \\frac{\\eta}{\\log{(\\log{(\\sigma_x)})}} and \\frac{\\mathbf{M}{(\\eta)} (e^{\\lambda})^{- \\lambda}}{\\log{(\\log{(\\sigma_x)})}^{2}} = \\frac{\\eta (e^{\\lambda})^{- \\lambda}}{\\log{(\\log{(\\sigma_x)})}^{2}} and \\frac{\\mathbf{M}{(\\eta)} \\operatorname{f_{E}}^{- \\lambda}{(\\lambda)}}{\\log{(\\log{(\\sigma_x)})}^{2}} = \\frac{\\eta \\operatorname{f_{E}}^{- \\lambda}{(\\lambda)}}{\\log{(\\log{(\\sigma_x)})}^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('f')(Symbol('\\\\sigma_x', commutative=True)), log(log(Symbol('\\\\sigma_x', commutative=True))))"], ["get_premise", "Equality(Function('f_E')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["divide", 3, "Function('f')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\eta', commutative=True)), Pow(Function('f')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Mul(Symbol('\\\\eta', commutative=True), Pow(Function('f')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\eta', commutative=True)), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))), Mul(Symbol('\\\\eta', commutative=True), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))))"], [["divide", 5, "Mul(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), log(log(Symbol('\\\\sigma_x', commutative=True))))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\eta', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-2))), Mul(Symbol('\\\\eta', commutative=True), Pow(exp(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\eta', commutative=True)), Pow(Function('f_E')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-2))), Mul(Symbol('\\\\eta', commutative=True), Pow(Function('f_E')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(log(log(Symbol('\\\\sigma_x', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = p v_{1}, then obtain - 2 p v_{1} + 3 \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = p v_{1} and - p v_{1} + \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = 0 and - 2 p v_{1} + \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = - p v_{1} and - 2 p v_{1} + 2 \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = 0 and - 2 p v_{1} + 3 \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)} = \\operatorname{f_{\\mathbf{p}}}{(v_{1},p)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Symbol('v_1', commutative=True)))"], [["minus", 1, "Mul(Symbol('p', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), Symbol('v_1', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True))), Integer(0))"], [["minus", 2, "Mul(Symbol('p', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('p', commutative=True), Symbol('v_1', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True), Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('p', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True)))), Integer(0))"], [["add", 4, "Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('p', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(3), Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True)))), Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True), Symbol('p', commutative=True)))"]]}, {"prompt": "Given B{(E,J_{\\varepsilon})} = E J_{\\varepsilon}, then derive \\frac{\\partial}{\\partial J_{\\varepsilon}} B{(E,J_{\\varepsilon})} = E, then obtain B{(E,J_{\\varepsilon})} \\frac{\\partial}{\\partial J_{\\varepsilon}} B{(E,J_{\\varepsilon})} = E B{(E,J_{\\varepsilon})}", "derivation": "B{(E,J_{\\varepsilon})} = E J_{\\varepsilon} and \\frac{\\partial}{\\partial J_{\\varepsilon}} B{(E,J_{\\varepsilon})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} E J_{\\varepsilon} and \\frac{\\partial}{\\partial J_{\\varepsilon}} B{(E,J_{\\varepsilon})} = E and B{(E,J_{\\varepsilon})} \\frac{\\partial}{\\partial J_{\\varepsilon}} B{(E,J_{\\varepsilon})} = E B{(E,J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('E', commutative=True))"], [["times", 3, "Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Symbol('E', commutative=True), Function('B')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\mathbf{g},n,\\varphi)} = \\frac{\\varphi^{n}}{\\mathbf{g}}, then obtain \\int 0 dn = \\int (- \\mathbf{g} \\pi{(\\mathbf{g},n,\\varphi)} + \\varphi^{n}) dn", "derivation": "\\pi{(\\mathbf{g},n,\\varphi)} = \\frac{\\varphi^{n}}{\\mathbf{g}} and \\mathbf{g} \\pi{(\\mathbf{g},n,\\varphi)} = \\varphi^{n} and 0 = - \\mathbf{g} \\pi{(\\mathbf{g},n,\\varphi)} + \\varphi^{n} and \\int 0 dn = \\int (- \\mathbf{g} \\pi{(\\mathbf{g},n,\\varphi)} + \\varphi^{n}) dn", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varphi', commutative=True))), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varphi', commutative=True))), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\pi')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varphi', commutative=True))), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\pi)} = e^{\\pi}, then derive \\int \\frac{\\operatorname{C_{2}}{(\\pi)}}{\\pi} d\\pi = t_{1} + \\operatorname{Ei}{(\\pi)}, then obtain t_{1} + \\operatorname{Ei}{(\\pi)} = V + \\operatorname{Ei}{(\\pi)}", "derivation": "\\operatorname{C_{2}}{(\\pi)} = e^{\\pi} and \\frac{\\operatorname{C_{2}}{(\\pi)}}{\\pi} = \\frac{e^{\\pi}}{\\pi} and \\int \\frac{\\operatorname{C_{2}}{(\\pi)}}{\\pi} d\\pi = \\int \\frac{e^{\\pi}}{\\pi} d\\pi and \\int \\frac{\\operatorname{C_{2}}{(\\pi)}}{\\pi} d\\pi = t_{1} + \\operatorname{Ei}{(\\pi)} and t_{1} + \\operatorname{Ei}{(\\pi)} = \\int \\frac{e^{\\pi}}{\\pi} d\\pi and t_{1} + \\operatorname{Ei}{(\\pi)} = V + \\operatorname{Ei}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), exp(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('t_1', commutative=True), Ei(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('t_1', commutative=True), Ei(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('t_1', commutative=True), Ei(Symbol('\\\\pi', commutative=True))), Add(Symbol('V', commutative=True), Ei(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mu)} = e^{\\mu}, then derive - e^{\\mu} + e^{- \\mu} \\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)} = 1 - e^{\\mu}, then obtain e^{- e^{\\mu} + e^{- \\mu} \\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)}} = e^{1 - e^{\\mu}}", "derivation": "\\operatorname{v_{z}}{(\\mu)} = e^{\\mu} and \\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)} = \\frac{d}{d \\mu} e^{\\mu} and \\frac{\\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)}}{\\frac{d}{d \\mu} e^{\\mu}} = 1 and - e^{\\mu} + \\frac{\\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)}}{\\frac{d}{d \\mu} e^{\\mu}} = 1 - e^{\\mu} and - e^{\\mu} + e^{- \\mu} \\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)} = 1 - e^{\\mu} and e^{- e^{\\mu} + e^{- \\mu} \\frac{d}{d \\mu} \\operatorname{v_{z}}{(\\mu)}} = e^{1 - e^{\\mu}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('v_z')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["minus", 3, "exp(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Mul(Derivative(Function('v_z')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Derivative(Function('v_z')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True)))))"], [["exp", 5], "Equality(exp(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Derivative(Function('v_z')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))), exp(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given W{(C)} = \\sin{(C)}, then derive \\cos{(W{(C)})} \\frac{d}{d C} W{(C)} = \\cos{(C)} \\cos{(\\sin{(C)})}, then obtain \\cos{(\\sin{(C)})} \\frac{d}{d C} \\sin{(C)} = \\cos{(C)} \\cos{(\\sin{(C)})}", "derivation": "W{(C)} = \\sin{(C)} and \\sin{(W{(C)})} = \\sin{(\\sin{(C)})} and \\frac{d}{d C} \\sin{(W{(C)})} = \\frac{d}{d C} \\sin{(\\sin{(C)})} and \\cos{(W{(C)})} \\frac{d}{d C} W{(C)} = \\cos{(C)} \\cos{(\\sin{(C)})} and \\cos{(\\sin{(C)})} \\frac{d}{d C} \\sin{(C)} = \\cos{(C)} \\cos{(\\sin{(C)})}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["sin", 1], "Equality(sin(Function('W')(Symbol('C', commutative=True))), sin(sin(Symbol('C', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(sin(Function('W')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('W')(Symbol('C', commutative=True))), Derivative(Function('W')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(cos(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(cos(sin(Symbol('C', commutative=True))), Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(cos(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{P},M)} = \\mathbf{P}^{M}, then obtain (\\frac{\\operatorname{v_{t}}{(\\mathbf{P},M)}}{- M + \\mathbf{P}^{M}})^{\\mathbf{P}} = (\\frac{\\mathbf{P}^{M}}{- M + \\mathbf{P}^{M}})^{\\mathbf{P}}", "derivation": "\\operatorname{v_{t}}{(\\mathbf{P},M)} = \\mathbf{P}^{M} and - M + \\operatorname{v_{t}}{(\\mathbf{P},M)} = - M + \\mathbf{P}^{M} and \\frac{\\operatorname{v_{t}}{(\\mathbf{P},M)}}{- M + \\operatorname{v_{t}}{(\\mathbf{P},M)}} = \\frac{\\mathbf{P}^{M}}{- M + \\operatorname{v_{t}}{(\\mathbf{P},M)}} and (\\frac{\\operatorname{v_{t}}{(\\mathbf{P},M)}}{- M + \\operatorname{v_{t}}{(\\mathbf{P},M)}})^{\\mathbf{P}} = (\\frac{\\mathbf{P}^{M}}{- M + \\operatorname{v_{t}}{(\\mathbf{P},M)}})^{\\mathbf{P}} and (\\frac{\\operatorname{v_{t}}{(\\mathbf{P},M)}}{- M + \\mathbf{P}^{M}})^{\\mathbf{P}} = (\\frac{\\mathbf{P}^{M}}{- M + \\mathbf{P}^{M}})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)))"], [["minus", 1, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(P_{g})} = \\cos{(\\cos{(P_{g})})} and \\theta{(P_{g})} = \\cos{(\\cos{(P_{g})})}, then obtain \\theta^{P_{g}}{(P_{g})} = \\cos^{P_{g}}{(\\cos{(P_{g})})}", "derivation": "\\operatorname{M_{E}}{(P_{g})} = \\cos{(\\cos{(P_{g})})} and \\operatorname{M_{E}}^{P_{g}}{(P_{g})} = \\cos^{P_{g}}{(\\cos{(P_{g})})} and \\theta{(P_{g})} = \\cos{(\\cos{(P_{g})})} and \\theta{(P_{g})} = \\operatorname{M_{E}}{(P_{g})} and \\theta^{P_{g}}{(P_{g})} = \\cos^{P_{g}}{(\\cos{(P_{g})})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('P_g', commutative=True)), cos(cos(Symbol('P_g', commutative=True))))"], [["power", 1, "Symbol('P_g', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(cos(cos(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('P_g', commutative=True)), cos(cos(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\theta')(Symbol('P_g', commutative=True)), Function('M_E')(Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\theta')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(cos(cos(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(I)} = \\sin{(I)}, then obtain \\operatorname{C_{1}}{(I)} + \\frac{\\sin{(I)}}{I} = \\sin{(I)} + \\frac{\\operatorname{C_{1}}{(I)}}{I}", "derivation": "\\operatorname{C_{1}}{(I)} = \\sin{(I)} and \\frac{\\operatorname{C_{1}}{(I)}}{I} = \\frac{\\sin{(I)}}{I} and \\sin{(I)} + \\frac{\\operatorname{C_{1}}{(I)}}{I} = \\sin{(I)} + \\frac{\\sin{(I)}}{I} and \\operatorname{C_{1}}{(I)} + \\frac{\\sin{(I)}}{I} = \\sin{(I)} + \\frac{\\sin{(I)}}{I} and \\operatorname{C_{1}}{(I)} + \\frac{\\sin{(I)}}{I} = \\sin{(I)} + \\frac{\\operatorname{C_{1}}{(I)}}{I}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('C_1')(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True))))"], [["add", 2, "sin(Symbol('I', commutative=True))"], "Equality(Add(sin(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('C_1')(Symbol('I', commutative=True)))), Add(sin(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True)))))"], [["add", 1, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True)))"], "Equality(Add(Function('C_1')(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True)))), Add(sin(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('C_1')(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Symbol('I', commutative=True)))), Add(sin(Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('C_1')(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(f^{\\prime},W)} = (f^{\\prime})^{W}, then derive \\frac{\\partial}{\\partial f^{\\prime}} \\hat{H}_l{(f^{\\prime},W)} = \\frac{W (f^{\\prime})^{W}}{f^{\\prime}}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime})^{W} = \\frac{W (f^{\\prime})^{W}}{f^{\\prime}}", "derivation": "\\hat{H}_l{(f^{\\prime},W)} = (f^{\\prime})^{W} and \\frac{\\partial}{\\partial f^{\\prime}} \\hat{H}_l{(f^{\\prime},W)} = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime})^{W} and \\hat{H}_l{(f^{\\prime},W)} + 1 = (f^{\\prime})^{W} + 1 and \\frac{\\partial}{\\partial f^{\\prime}} (\\hat{H}_l{(f^{\\prime},W)} + 1) = \\frac{\\partial}{\\partial f^{\\prime}} ((f^{\\prime})^{W} + 1) and \\frac{\\partial}{\\partial f^{\\prime}} \\hat{H}_l{(f^{\\prime},W)} = \\frac{W (f^{\\prime})^{W}}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime})^{W} = \\frac{W (f^{\\prime})^{W}}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Integer(1)), Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\psi{(f^{\\prime})} = \\sin{(e^{f^{\\prime}})} and \\operatorname{v_{1}}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain \\int (f^{\\prime} + \\psi{(f^{\\prime})}) df^{\\prime} = \\int (f^{\\prime} + \\sin{(\\operatorname{v_{1}}{(f^{\\prime})})}) df^{\\prime}", "derivation": "\\psi{(f^{\\prime})} = \\sin{(e^{f^{\\prime}})} and f^{\\prime} + \\psi{(f^{\\prime})} = f^{\\prime} + \\sin{(e^{f^{\\prime}})} and \\operatorname{v_{1}}{(f^{\\prime})} = e^{f^{\\prime}} and f^{\\prime} + \\psi{(f^{\\prime})} = f^{\\prime} + \\sin{(\\operatorname{v_{1}}{(f^{\\prime})})} and \\int (f^{\\prime} + \\psi{(f^{\\prime})}) df^{\\prime} = \\int (f^{\\prime} + \\sin{(\\operatorname{v_{1}}{(f^{\\prime})})}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True)), sin(exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(exp(Symbol('f^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(Function('v_1')(Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Function('v_1')(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)} = \\frac{P_{e} \\nabla}{\\mu}, then obtain \\frac{\\partial}{\\partial P_{e}} (\\frac{\\partial}{\\partial P_{e}} \\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)})^{\\mu} = \\frac{\\partial}{\\partial P_{e}} (\\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\nabla}{\\mu})^{\\mu}", "derivation": "\\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)} = \\frac{P_{e} \\nabla}{\\mu} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)} = \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\nabla}{\\mu} and (\\frac{\\partial}{\\partial P_{e}} \\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)})^{\\mu} = (\\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\nabla}{\\mu})^{\\mu} and \\frac{\\partial}{\\partial P_{e}} (\\frac{\\partial}{\\partial P_{e}} \\operatorname{f^{\\prime}}{(\\nabla,P_{e},\\mu)})^{\\mu} = \\frac{\\partial}{\\partial P_{e}} (\\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\nabla}{\\mu})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(z^{*})} = \\sin{(z^{*})}, then obtain \\frac{d}{d z^{*}} (4 z^{*} + \\sin{(z^{*})}) = \\frac{(2 z^{*} + \\sin{(z^{*})}) \\frac{d}{d z^{*}} (4 z^{*} + \\sin{(z^{*})})}{2 z^{*} + g{(z^{*})}}", "derivation": "g{(z^{*})} = \\sin{(z^{*})} and z^{*} + g{(z^{*})} = z^{*} + \\sin{(z^{*})} and 2 z^{*} + g{(z^{*})} = 2 z^{*} + \\sin{(z^{*})} and \\frac{2 z^{*} + g{(z^{*})}}{z^{*}} = \\frac{2 z^{*} + \\sin{(z^{*})}}{z^{*}} and 1 = \\frac{2 z^{*} + \\sin{(z^{*})}}{2 z^{*} + g{(z^{*})}} and \\frac{d}{d z^{*}} (4 z^{*} + \\sin{(z^{*})}) = \\frac{(2 z^{*} + \\sin{(z^{*})}) \\frac{d}{d z^{*}} (4 z^{*} + \\sin{(z^{*})})}{2 z^{*} + g{(z^{*})}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["add", 1, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Function('g')(Symbol('z^*', commutative=True))), Add(Symbol('z^*', commutative=True), sin(Symbol('z^*', commutative=True))))"], [["add", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('z^*', commutative=True)), Function('g')(Symbol('z^*', commutative=True))), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))))"], [["divide", 3, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), Function('g')(Symbol('z^*', commutative=True)))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))))"], [["divide", 4, "Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), Function('g')(Symbol('z^*', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(2), Symbol('z^*', commutative=True)), Function('g')(Symbol('z^*', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))))"], [["times", 5, "Derivative(Add(Mul(Integer(4), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))"], "Equality(Derivative(Add(Mul(Integer(4), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Pow(Add(Mul(Integer(2), Symbol('z^*', commutative=True)), Function('g')(Symbol('z^*', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Derivative(Add(Mul(Integer(4), Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(P_{g},\\mathbf{B})} = - P_{g} + \\mathbf{B}, then derive \\frac{\\partial}{\\partial P_{g}} \\tilde{g}^*{(P_{g},\\mathbf{B})} = -1, then obtain \\iint (- \\omega^{n} + \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B})) d\\omega dn = \\iint (- \\omega^{n} - 1) d\\omega dn", "derivation": "\\tilde{g}^*{(P_{g},\\mathbf{B})} = - P_{g} + \\mathbf{B} and \\frac{\\partial}{\\partial P_{g}} \\tilde{g}^*{(P_{g},\\mathbf{B})} = \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B}) and \\frac{\\partial}{\\partial P_{g}} \\tilde{g}^*{(P_{g},\\mathbf{B})} = -1 and \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B}) = -1 and - \\omega^{n} + \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B}) = - \\omega^{n} - 1 and \\int (- \\omega^{n} + \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B})) d\\omega = \\int (- \\omega^{n} - 1) d\\omega and \\iint (- \\omega^{n} + \\frac{\\partial}{\\partial P_{g}} (- P_{g} + \\mathbf{B})) d\\omega dn = \\iint (- \\omega^{n} - 1) d\\omega dn", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Integer(-1))"], [["minus", 4, "Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 6, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(v_{z},\\varepsilon_0)} = - \\varepsilon_0 + \\sin{(v_{z})}, then derive \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{v_{2}}{(v_{z},\\varepsilon_0)} = -1, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + \\sin{(v_{z})}) = -1", "derivation": "\\operatorname{v_{2}}{(v_{z},\\varepsilon_0)} = - \\varepsilon_0 + \\sin{(v_{z})} and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{v_{2}}{(v_{z},\\varepsilon_0)} = \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + \\sin{(v_{z})}) and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{v_{2}}{(v_{z},\\varepsilon_0)} = -1 and \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + \\sin{(v_{z})}) = -1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('v_z', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(V,n)} = \\frac{V}{n}, then obtain - (\\frac{V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)}) \\operatorname{F_{c}}{(V,n)} = - (\\frac{2 V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)} - 1) \\operatorname{F_{c}}{(V,n)}", "derivation": "\\operatorname{F_{c}}{(V,n)} = \\frac{V}{n} and 1 = \\frac{V}{n \\operatorname{F_{c}}{(V,n)}} and 1 - \\operatorname{F_{c}}{(V,n)} = \\frac{V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)} and - (1 - \\operatorname{F_{c}}{(V,n)}) \\operatorname{F_{c}}{(V,n)} = - (\\frac{V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)}) \\operatorname{F_{c}}{(V,n)} and - (\\frac{V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)}) \\operatorname{F_{c}}{(V,n)} = - (\\frac{2 V}{n \\operatorname{F_{c}}{(V,n)}} - \\operatorname{F_{c}}{(V,n)} - 1) \\operatorname{F_{c}}{(V,n)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["divide", 1, "Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))"], "Equality(Integer(1), Mul(Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Integer(-1))))"], [["minus", 2, "Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Add(Mul(Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Integer(-1)), Function('F_c')(Symbol('V', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)}, then derive \\int \\mathbf{r}{(\\mu)} d\\mu = F_{N} + \\log{(\\mu)}, then obtain \\frac{\\partial}{\\partial \\mu} (F_{N} + \\log{(\\mu)}) = \\frac{d}{d \\mu} \\int \\frac{d}{d \\mu} \\log{(\\mu)} d\\mu", "derivation": "\\mathbf{r}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)} and \\int \\mathbf{r}{(\\mu)} d\\mu = \\int \\frac{d}{d \\mu} \\log{(\\mu)} d\\mu and \\int \\mathbf{r}{(\\mu)} d\\mu = F_{N} + \\log{(\\mu)} and F_{N} + \\log{(\\mu)} = \\int \\frac{d}{d \\mu} \\log{(\\mu)} d\\mu and \\frac{\\partial}{\\partial \\mu} (F_{N} + \\log{(\\mu)}) = \\frac{d}{d \\mu} \\int \\frac{d}{d \\mu} \\log{(\\mu)} d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('F_N', commutative=True), log(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_N', commutative=True), log(Symbol('\\\\mu', commutative=True))), Integral(Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Symbol('F_N', commutative=True), log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integral(Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(v_{1},q)} = q^{v_{1}}, then obtain 4 q^{2} \\rho^{2}{(v_{1},q)} = q^{2} (q^{v_{1}} + \\rho{(v_{1},q)})^{2}", "derivation": "\\rho{(v_{1},q)} = q^{v_{1}} and 2 \\rho{(v_{1},q)} = q^{v_{1}} + \\rho{(v_{1},q)} and 2 q \\rho{(v_{1},q)} = q (q^{v_{1}} + \\rho{(v_{1},q)}) and 4 q^{2} \\rho^{2}{(v_{1},q)} = q^{2} (q^{v_{1}} + \\rho{(v_{1},q)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('v_1', commutative=True)))"], [["add", 1, "Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Add(Pow(Symbol('q', commutative=True), Symbol('v_1', commutative=True)), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))))"], [["times", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Integer(2), Symbol('q', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), Add(Pow(Symbol('q', commutative=True), Symbol('v_1', commutative=True)), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Symbol('q', commutative=True), Integer(2)), Pow(Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Integer(2))), Mul(Pow(Symbol('q', commutative=True), Integer(2)), Pow(Add(Pow(Symbol('q', commutative=True), Symbol('v_1', commutative=True)), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Integer(2))))"]]}, {"prompt": "Given C{(z^{*},\\Omega)} = \\Omega + z^{*}, then obtain \\frac{\\partial}{\\partial \\Omega} (\\frac{\\Omega^{2}}{2} + \\Omega z^{*} + f_{\\mathbf{p}} + C{(z^{*},\\Omega)}) = \\frac{\\partial}{\\partial \\Omega} (\\frac{\\Omega^{2}}{2} + \\Omega z^{*} + \\Omega + f_{\\mathbf{p}} + z^{*})", "derivation": "C{(z^{*},\\Omega)} = \\Omega + z^{*} and \\int C{(z^{*},\\Omega)} d\\Omega = \\int (\\Omega + z^{*}) d\\Omega and C{(z^{*},\\Omega)} + \\int C{(z^{*},\\Omega)} d\\Omega = \\Omega + z^{*} + \\int C{(z^{*},\\Omega)} d\\Omega and \\frac{\\partial}{\\partial \\Omega} (C{(z^{*},\\Omega)} + \\int C{(z^{*},\\Omega)} d\\Omega) = \\frac{\\partial}{\\partial \\Omega} (\\Omega + z^{*} + \\int C{(z^{*},\\Omega)} d\\Omega) and \\frac{\\partial}{\\partial \\Omega} (C{(z^{*},\\Omega)} + \\int (\\Omega + z^{*}) d\\Omega) = \\frac{\\partial}{\\partial \\Omega} (\\Omega + z^{*} + \\int (\\Omega + z^{*}) d\\Omega) and \\frac{\\partial}{\\partial \\Omega} (\\frac{\\Omega^{2}}{2} + \\Omega z^{*} + f_{\\mathbf{p}} + C{(z^{*},\\Omega)}) = \\frac{\\partial}{\\partial \\Omega} (\\frac{\\Omega^{2}}{2} + \\Omega z^{*} + \\Omega + f_{\\mathbf{p}} + z^{*})", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True), Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True), Integral(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('C')(Symbol('z^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\Omega', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(A_{z},\\mu_0)} = \\mu_0^{A_{z}} and y{(A_{z},\\mu_0)} = - A_{z} + \\mu_0^{A_{z}} - g{(A_{z},\\mu_0)}, then obtain y{(A_{z},\\mu_0)} + e^{y{(A_{z},\\mu_0)}} = - A_{z} + e^{y{(A_{z},\\mu_0)}}", "derivation": "g{(A_{z},\\mu_0)} = \\mu_0^{A_{z}} and y{(A_{z},\\mu_0)} = - A_{z} + \\mu_0^{A_{z}} - g{(A_{z},\\mu_0)} and y{(A_{z},\\mu_0)} = - A_{z} and y{(A_{z},\\mu_0)} + e^{y{(A_{z},\\mu_0)}} = - A_{z} + e^{y{(A_{z},\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True)))"], [["add", 3, "exp(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)), exp(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), exp(Function('y')(Symbol('A_z', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} = \\Psi_{nl} \\chi, then obtain \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl} \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} + \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)}) = \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl} \\chi + \\Psi_{nl} \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)})", "derivation": "\\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} = \\Psi_{nl} \\chi and \\Psi_{nl} \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} = \\Psi_{nl}^{2} \\chi and \\Psi_{nl}^{2} \\chi + \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} = \\Psi_{nl}^{2} \\chi + \\Psi_{nl} \\chi and \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl}^{2} \\chi + \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)}) = \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl}^{2} \\chi + \\Psi_{nl} \\chi) and \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl} \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)} + \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)}) = \\frac{\\partial}{\\partial \\chi} (\\Psi_{nl} \\chi + \\Psi_{nl} \\operatorname{f^{*}}{(\\Psi_{nl},\\chi)})", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('f^*')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(u,n)} = - n + u, then derive \\int \\operatorname{P_{e}}{(u,n)} du = I - n u + \\frac{u^{2}}{2}, then derive \\hat{X} - n u + \\frac{u^{2}}{2} = I - n u + \\frac{u^{2}}{2}, then obtain \\frac{\\partial}{\\partial I} (\\hat{X} - n u + \\frac{u^{2}}{2}) = \\frac{\\partial}{\\partial I} (I - n u + \\frac{u^{2}}{2})", "derivation": "\\operatorname{P_{e}}{(u,n)} = - n + u and \\int \\operatorname{P_{e}}{(u,n)} du = \\int (- n + u) du and \\int \\operatorname{P_{e}}{(u,n)} du = I - n u + \\frac{u^{2}}{2} and \\int (- n + u) du = I - n u + \\frac{u^{2}}{2} and \\hat{X} - n u + \\frac{u^{2}}{2} = I - n u + \\frac{u^{2}}{2} and \\frac{\\partial}{\\partial I} (\\hat{X} - n u + \\frac{u^{2}}{2}) = \\frac{\\partial}{\\partial I} (I - n u + \\frac{u^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('u', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('u', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('u', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(f^{\\prime},\\mathbf{S})} = \\mathbf{S} \\cos{(f^{\\prime})}, then obtain \\tilde{g} + \\log{(\\mathbf{S})} = \\int \\frac{\\cos{(f^{\\prime})}}{v{(f^{\\prime},\\mathbf{S})}} d\\mathbf{S}", "derivation": "v{(f^{\\prime},\\mathbf{S})} = \\mathbf{S} \\cos{(f^{\\prime})} and 1 = \\frac{\\mathbf{S} \\cos{(f^{\\prime})}}{v{(f^{\\prime},\\mathbf{S})}} and \\frac{1}{\\mathbf{S}} = \\frac{\\cos{(f^{\\prime})}}{v{(f^{\\prime},\\mathbf{S})}} and \\int \\frac{1}{\\mathbf{S}} d\\mathbf{S} = \\int \\frac{\\cos{(f^{\\prime})}}{v{(f^{\\prime},\\mathbf{S})}} d\\mathbf{S} and \\tilde{g} + \\log{(\\mathbf{S})} = \\int \\frac{\\cos{(f^{\\prime})}}{v{(f^{\\prime},\\mathbf{S})}} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Mul(Pow(Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(Pow(Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(Pow(Function('v')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given S{(J)} = \\cos{(J)}, then derive \\frac{S^{2}{(J)}}{4 (\\frac{d}{d J} S{(J)})^{2}} = \\frac{\\cos^{2}{(J)}}{4 (\\frac{d}{d J} S{(J)})^{2}}, then obtain \\frac{S^{6}{(J)}}{\\cos^{2}{(J)}} = S^{4}{(J)}", "derivation": "S{(J)} = \\cos{(J)} and 2 S{(J)} = S{(J)} + \\cos{(J)} and \\frac{S{(J)}}{\\frac{d}{d J} (S{(J)} + \\cos{(J)})} = \\frac{\\cos{(J)}}{\\frac{d}{d J} (S{(J)} + \\cos{(J)})} and \\frac{S{(J)}}{\\frac{d}{d J} 2 S{(J)}} = \\frac{\\cos{(J)}}{\\frac{d}{d J} 2 S{(J)}} and \\frac{S^{2}{(J)}}{(\\frac{d}{d J} 2 S{(J)})^{2}} = \\frac{\\cos^{2}{(J)}}{(\\frac{d}{d J} 2 S{(J)})^{2}} and \\frac{S^{2}{(J)}}{4 (\\frac{d}{d J} S{(J)})^{2}} = \\frac{\\cos^{2}{(J)}}{4 (\\frac{d}{d J} S{(J)})^{2}} and \\frac{S^{2}{(J)}}{\\cos^{2}{(J)}} = 1 and \\frac{S^{6}{(J)}}{\\cos^{2}{(J)}} = S^{4}{(J)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["add", 1, "Function('S')(Symbol('J', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('J', commutative=True))), Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))))"], [["divide", 1, "Derivative(Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Function('S')(Symbol('J', commutative=True)), Pow(Derivative(Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(cos(Symbol('J', commutative=True)), Pow(Derivative(Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('S')(Symbol('J', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('S')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(cos(Symbol('J', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('S')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))))"], [["power", 4, 2], "Equality(Mul(Pow(Function('S')(Symbol('J', commutative=True)), Integer(2)), Pow(Derivative(Mul(Integer(2), Function('S')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-2))), Mul(Pow(cos(Symbol('J', commutative=True)), Integer(2)), Pow(Derivative(Mul(Integer(2), Function('S')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-2))))"], [["evaluate_derivatives", 5], "Equality(Mul(Rational(1, 4), Pow(Function('S')(Symbol('J', commutative=True)), Integer(2)), Pow(Derivative(Function('S')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-2))), Mul(Rational(1, 4), Pow(cos(Symbol('J', commutative=True)), Integer(2)), Pow(Derivative(Function('S')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-2))))"], [["divide", 6, "Mul(Rational(1, 4), Pow(cos(Symbol('J', commutative=True)), Integer(2)), Pow(Derivative(Function('S')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-2)))"], "Equality(Mul(Pow(Function('S')(Symbol('J', commutative=True)), Integer(2)), Pow(cos(Symbol('J', commutative=True)), Integer(-2))), Integer(1))"], [["times", 7, "Pow(Function('S')(Symbol('J', commutative=True)), Integer(4))"], "Equality(Mul(Pow(Function('S')(Symbol('J', commutative=True)), Integer(6)), Pow(cos(Symbol('J', commutative=True)), Integer(-2))), Pow(Function('S')(Symbol('J', commutative=True)), Integer(4)))"]]}, {"prompt": "Given l{(C_{1})} = \\int \\sin{(C_{1})} dC_{1}, then derive l{(C_{1})} = f - \\cos{(C_{1})}, then obtain - (f - \\cos{(C_{1})})^{C_{1}} \\cos{(C_{1})} = - l^{C_{1}}{(C_{1})} \\cos{(C_{1})}", "derivation": "l{(C_{1})} = \\int \\sin{(C_{1})} dC_{1} and l{(C_{1})} = f - \\cos{(C_{1})} and l^{C_{1}}{(C_{1})} = (\\int \\sin{(C_{1})} dC_{1})^{C_{1}} and (f - \\cos{(C_{1})})^{C_{1}} = (\\int \\sin{(C_{1})} dC_{1})^{C_{1}} and (f - \\cos{(C_{1})})^{C_{1}} = l^{C_{1}}{(C_{1})} and - (f - \\cos{(C_{1})})^{C_{1}} \\cos{(C_{1})} = - l^{C_{1}}{(C_{1})} \\cos{(C_{1})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('C_1', commutative=True)), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('l')(Symbol('C_1', commutative=True)), Add(Symbol('f', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('l')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Add(Symbol('f', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)), Pow(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Symbol('f', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)), Pow(Function('l')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["times", 5, "Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('f', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), Mul(Integer(-1), Pow(Function('l')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})} = \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}}, then obtain (\\frac{v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} - \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}} = (\\frac{V_{\\mathbf{B}} + a^{\\dagger}}{V_{\\mathbf{B}} v_{1}})^{V_{\\mathbf{B}}} - \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}}", "derivation": "v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})} = \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}} and \\frac{v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{V_{\\mathbf{B}} v_{1}} and (\\frac{v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} = (\\frac{V_{\\mathbf{B}} + a^{\\dagger}}{V_{\\mathbf{B}} v_{1}})^{V_{\\mathbf{B}}} and (\\frac{v{(a^{\\dagger},v_{1},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} - \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}} = (\\frac{V_{\\mathbf{B}} + a^{\\dagger}}{V_{\\mathbf{B}} v_{1}})^{V_{\\mathbf{B}}} - \\frac{V_{\\mathbf{B}} + a^{\\dagger}}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('v')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('v')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Pow(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('v')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Add(Pow(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(v)} = \\log{(v)} and k{(v)} = \\log{(v)}^{v}, then obtain \\Psi_{\\lambda}^{v}{(v)} = k{(v)}", "derivation": "\\Psi_{\\lambda}{(v)} = \\log{(v)} and \\Psi_{\\lambda}^{v}{(v)} = \\log{(v)}^{v} and k{(v)} = \\log{(v)}^{v} and \\Psi_{\\lambda}^{v}{(v)} = k{(v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('v', commutative=True)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Function('k')(Symbol('v', commutative=True)))"]]}, {"prompt": "Given x{(\\hat{p},\\pi)} = \\hat{p} + \\pi, then derive \\frac{\\partial}{\\partial \\hat{p}} x{(\\hat{p},\\pi)} = 1, then obtain \\frac{\\partial}{\\partial \\pi} \\int \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\pi) d\\pi = \\frac{d}{d \\pi} \\int 1 d\\pi", "derivation": "x{(\\hat{p},\\pi)} = \\hat{p} + \\pi and \\frac{\\partial}{\\partial \\hat{p}} x{(\\hat{p},\\pi)} = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\pi) and \\frac{\\partial}{\\partial \\hat{p}} x{(\\hat{p},\\pi)} = 1 and \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\pi) = 1 and \\int \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\pi) d\\pi = \\int 1 d\\pi and \\frac{\\partial}{\\partial \\pi} \\int \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\pi) d\\pi = \\frac{d}{d \\pi} \\int 1 d\\pi", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integral(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(\\dot{\\mathbf{r}},v_{x})} = \\dot{\\mathbf{r}}^{v_{x}}, then obtain (e^{- \\dot{\\mathbf{r}}^{v_{x}} + \\eta{(\\dot{\\mathbf{r}},v_{x})}})^{\\dot{\\mathbf{r}}} + 1 = 2", "derivation": "\\eta{(\\dot{\\mathbf{r}},v_{x})} = \\dot{\\mathbf{r}}^{v_{x}} and - \\dot{\\mathbf{r}}^{v_{x}} + \\eta{(\\dot{\\mathbf{r}},v_{x})} = 0 and e^{- \\dot{\\mathbf{r}}^{v_{x}} + \\eta{(\\dot{\\mathbf{r}},v_{x})}} = 1 and (e^{- \\dot{\\mathbf{r}}^{v_{x}} + \\eta{(\\dot{\\mathbf{r}},v_{x})}})^{\\dot{\\mathbf{r}}} = 1 and (e^{- \\dot{\\mathbf{r}}^{v_{x}} + \\eta{(\\dot{\\mathbf{r}},v_{x})}})^{\\dot{\\mathbf{r}}} + 1 = 2", "srepr_derivation": [["get_premise", "Equality(Function('\\\\eta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Function('\\\\eta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Function('\\\\eta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(exp(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Function('\\\\eta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(1))"], [["add", 4, 1], "Equality(Add(Pow(exp(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Function('\\\\eta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(1)), Integer(2))"]]}, {"prompt": "Given \\mu_{0}{(v,C_{2})} = \\log{(\\frac{C_{2}}{v})}, then obtain \\mu_{0}{(v,C_{2})} + \\frac{\\partial^{3}}{\\partial v\\partial C_{2}\\partial v} (\\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})}) = \\mu_{0}{(v,C_{2})} + \\frac{d^{3}}{d vd C_{2}d v} 0", "derivation": "\\mu_{0}{(v,C_{2})} = \\log{(\\frac{C_{2}}{v})} and \\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})} = 0 and \\frac{\\partial}{\\partial v} (\\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})}) = \\frac{d}{d v} 0 and \\frac{\\partial^{2}}{\\partial C_{2}\\partial v} (\\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})}) = \\frac{d^{2}}{d C_{2}d v} 0 and \\frac{\\partial^{3}}{\\partial v\\partial C_{2}\\partial v} (\\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})}) = \\frac{d^{3}}{d vd C_{2}d v} 0 and \\mu_{0}{(v,C_{2})} + \\frac{\\partial^{3}}{\\partial v\\partial C_{2}\\partial v} (\\mu_{0}{(v,C_{2})} - \\log{(\\frac{C_{2}}{v})}) = \\mu_{0}{(v,C_{2})} + \\frac{d^{3}}{d vd C_{2}d v} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))"], [["minus", 1, "log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], "Equality(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Integer(0))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 5, "Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Derivative(Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Function('\\\\mu_0')(Symbol('v', commutative=True), Symbol('C_2', commutative=True)), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} = \\frac{\\omega}{\\mathbf{J}_f}, then obtain - \\int (\\mathbf{F} + \\mathbf{f}) d\\mathbf{f} + \\iint \\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} d\\omega d\\omega = - \\int (\\mathbf{F} + \\mathbf{f}) d\\mathbf{f} + \\iint \\frac{\\omega}{\\mathbf{J}_f} d\\omega d\\omega", "derivation": "\\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} = \\frac{\\omega}{\\mathbf{J}_f} and \\int \\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} d\\omega = \\int \\frac{\\omega}{\\mathbf{J}_f} d\\omega and \\iint \\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} d\\omega d\\omega = \\iint \\frac{\\omega}{\\mathbf{J}_f} d\\omega d\\omega and - \\int (\\mathbf{F} + \\mathbf{f}) d\\mathbf{f} + \\iint \\mathbf{J}_M{(\\mathbf{J}_f,\\omega)} d\\omega d\\omega = - \\int (\\mathbf{F} + \\mathbf{f}) d\\mathbf{f} + \\iint \\frac{\\omega}{\\mathbf{J}_f} d\\omega d\\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(v)} = \\cos{(v)}, then obtain (\\frac{d}{d v} 0^{v})^{v} = (\\frac{d}{d v} 1)^{v}", "derivation": "\\mathbf{E}{(v)} = \\cos{(v)} and 0 = - \\mathbf{E}{(v)} + \\cos{(v)} and 0^{v} = (- \\mathbf{E}{(v)} + \\cos{(v)})^{v} and \\frac{d}{d v} 0^{v} = \\frac{d}{d v} (- \\mathbf{E}{(v)} + \\cos{(v)})^{v} and \\frac{d}{d v} (- \\mathbf{E}{(v)} + \\cos{(v)})^{v} = \\frac{d}{d v} 1 and \\frac{d}{d v} 0^{v} = \\frac{d}{d v} 1 and (\\frac{d}{d v} 0^{v})^{v} = (\\frac{d}{d v} 1)^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{E}')(Symbol('v', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Pow(Integer(0), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["power", 6, "Symbol('v', commutative=True)"], "Equality(Pow(Derivative(Pow(Integer(0), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = (- J_{\\varepsilon} + \\theta)^{\\mu_0}, then derive \\frac{\\partial}{\\partial \\mu_0} \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = (- J_{\\varepsilon} + \\theta)^{\\mu_0} \\log{(- J_{\\varepsilon} + \\theta)}, then obtain \\frac{\\partial}{\\partial \\mu_0} \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} \\log{(- J_{\\varepsilon} + \\theta)}", "derivation": "\\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = (- J_{\\varepsilon} + \\theta)^{\\mu_0} and J_{\\varepsilon} + \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = J_{\\varepsilon} + (- J_{\\varepsilon} + \\theta)^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} (J_{\\varepsilon} + \\psi{(\\theta,\\mu_0,J_{\\varepsilon})}) = \\frac{\\partial}{\\partial \\mu_0} (J_{\\varepsilon} + (- J_{\\varepsilon} + \\theta)^{\\mu_0}) and \\frac{\\partial}{\\partial \\mu_0} \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = (- J_{\\varepsilon} + \\theta)^{\\mu_0} \\log{(- J_{\\varepsilon} + \\theta)} and \\frac{\\partial}{\\partial \\mu_0} \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} = \\psi{(\\theta,\\mu_0,J_{\\varepsilon})} \\log{(- J_{\\varepsilon} + \\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mu_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(E,\\tilde{g}^*)} = \\cos{(E^{\\tilde{g}^*})}, then derive \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} = - \\frac{E^{\\tilde{g}^*} \\tilde{g}^* \\sin{(E^{\\tilde{g}^*})}}{E}, then obtain \\int \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} dE - 1 = \\int - \\frac{E^{\\tilde{g}^*} \\tilde{g}^* \\sin{(E^{\\tilde{g}^*})}}{E} dE - 1", "derivation": "\\operatorname{m_{s}}{(E,\\tilde{g}^*)} = \\cos{(E^{\\tilde{g}^*})} and \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} = \\frac{\\partial}{\\partial E} \\cos{(E^{\\tilde{g}^*})} and \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} = - \\frac{E^{\\tilde{g}^*} \\tilde{g}^* \\sin{(E^{\\tilde{g}^*})}}{E} and \\int \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} dE = \\int - \\frac{E^{\\tilde{g}^*} \\tilde{g}^* \\sin{(E^{\\tilde{g}^*})}}{E} dE and \\int \\frac{\\partial}{\\partial E} \\operatorname{m_{s}}{(E,\\tilde{g}^*)} dE - 1 = \\int - \\frac{E^{\\tilde{g}^*} \\tilde{g}^* \\sin{(E^{\\tilde{g}^*})}}{E} dE - 1", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Derivative(Function('m_s')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Integral(Derivative(Function('m_s')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))), Integer(-1)), Add(Integral(Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), sin(Pow(Symbol('E', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('E', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})}, then derive \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})}, then obtain \\frac{d^{2}}{d \\mathbf{F}^{2}} \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} and \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\sin{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} and \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})} and \\frac{d^{2}}{d \\mathbf{F}^{2}} \\Psi_{\\lambda}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["times", 1, "sin(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Mul(sin(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Mul(sin(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(sin(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Derivative(Mul(sin(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\omega{(\\phi_1,\\mu)} = \\mu - \\phi_1 and \\varepsilon{(\\phi_1,\\mu)} = \\int \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu, then obtain \\iint \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu d\\mu = \\int \\varepsilon{(\\phi_1,\\mu)} d\\mu", "derivation": "\\omega{(\\phi_1,\\mu)} = \\mu - \\phi_1 and \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} = \\frac{\\partial}{\\partial \\mu} (\\mu - \\phi_1) and \\int \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu = \\int \\frac{\\partial}{\\partial \\mu} (\\mu - \\phi_1) d\\mu and \\varepsilon{(\\phi_1,\\mu)} = \\int \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu and \\varepsilon{(\\phi_1,\\mu)} = \\int \\frac{\\partial}{\\partial \\mu} (\\mu - \\phi_1) d\\mu and \\int \\varepsilon{(\\phi_1,\\mu)} d\\mu = \\iint \\frac{\\partial}{\\partial \\mu} (\\mu - \\phi_1) d\\mu d\\mu and \\iint \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu d\\mu = \\iint \\frac{\\partial}{\\partial \\mu} (\\mu - \\phi_1) d\\mu d\\mu and \\iint \\frac{\\partial}{\\partial \\mu} \\omega{(\\phi_1,\\mu)} d\\mu d\\mu = \\int \\varepsilon{(\\phi_1,\\mu)} d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Derivative(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\mathbf{J}_M \\cos{(n_{2})}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\cos{(n_{2})}, then obtain \\frac{\\partial}{\\partial n_{2}} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_M \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\mathbf{J}_M \\cos{(n_{2})} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{J}_M \\cos{(n_{2})} and \\frac{\\partial}{\\partial n_{2}} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_M \\cos{(n_{2})} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\cos{(n_{2})} and \\frac{\\partial}{\\partial n_{2}} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})} = \\frac{\\partial}{\\partial n_{2}} \\mathbf{J}_M \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{t_{2}}{(\\mathbf{J}_M,n_{2})}", "srepr_derivation": [["get_premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), cos(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), cos(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Function('t_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} \\cos{(\\Psi^{\\dagger})}, then derive \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = - \\sin{(\\Psi^{\\dagger})}, then derive \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}, then obtain \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} - \\sin{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}", "derivation": "\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} \\cos{(\\Psi^{\\dagger})} and \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = - \\sin{(\\Psi^{\\dagger})} and \\frac{d}{d \\Psi^{\\dagger}} \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} - \\sin{(\\Psi^{\\dagger})} and \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} - \\sin{(\\Psi^{\\dagger})} and \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})} and \\frac{d^{2}}{d (\\Psi^{\\dagger})^{2}} - \\sin{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbb{I})} = \\cos{(\\sin{(\\mathbb{I})})}, then derive \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} = - \\sin{(\\sin{(\\mathbb{I})})} \\cos{(\\mathbb{I})}, then obtain \\int \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} d\\mathbb{I} = \\int \\frac{d}{d \\mathbb{I}} \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I}", "derivation": "\\phi_{1}{(\\mathbb{I})} = \\cos{(\\sin{(\\mathbb{I})})} and \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\cos{(\\sin{(\\mathbb{I})})} and \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} = - \\sin{(\\sin{(\\mathbb{I})})} \\cos{(\\mathbb{I})} and \\int \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} d\\mathbb{I} = \\int - \\sin{(\\sin{(\\mathbb{I})})} \\cos{(\\mathbb{I})} d\\mathbb{I} and \\int \\frac{d}{d \\mathbb{I}} \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I} = \\int - \\sin{(\\sin{(\\mathbb{I})})} \\cos{(\\mathbb{I})} d\\mathbb{I} and \\int \\frac{d}{d \\mathbb{I}} \\phi_{1}{(\\mathbb{I})} d\\mathbb{I} = \\int \\frac{d}{d \\mathbb{I}} \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbb{I}', commutative=True)), cos(sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\mathbb{I}', commutative=True))), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Integer(-1), sin(sin(Symbol('\\\\mathbb{I}', commutative=True))), cos(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Integer(-1), sin(sin(Symbol('\\\\mathbb{I}', commutative=True))), cos(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Derivative(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(t,f_{\\mathbf{p}})} = t \\sin{(f_{\\mathbf{p}})}, then obtain 2 \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}}) = \\frac{d}{d f_{\\mathbf{p}}} 2 + \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}})", "derivation": "\\mathbf{r}{(t,f_{\\mathbf{p}})} = t \\sin{(f_{\\mathbf{p}})} and \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}} = 1 and 1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}} = 2 and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}}) = \\frac{d}{d f_{\\mathbf{p}}} 2 and 2 \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}}) = \\frac{d}{d f_{\\mathbf{p}}} 2 + \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (1 + \\frac{\\mathbf{r}{(t,f_{\\mathbf{p}})}}{t \\sin{(f_{\\mathbf{p}})}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Symbol('t', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["divide", 1, "Mul(Symbol('t', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))), Integer(2))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Add(Integer(1), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Add(Integer(1), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Add(Derivative(Integer(2), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(E,Q,S)} = (E^{S})^{Q}, then derive \\frac{\\partial}{\\partial Q} G{(E,Q,S)} = (E^{S})^{Q} \\log{(E^{S})}, then obtain \\frac{\\partial}{\\partial Q} (E^{S})^{Q} = (E^{S})^{Q} \\log{(E^{S})}", "derivation": "G{(E,Q,S)} = (E^{S})^{Q} and E^{S} + G{(E,Q,S)} = E^{S} + (E^{S})^{Q} and \\frac{\\partial}{\\partial Q} (E^{S} + G{(E,Q,S)}) = \\frac{\\partial}{\\partial Q} (E^{S} + (E^{S})^{Q}) and \\frac{\\partial}{\\partial Q} G{(E,Q,S)} = (E^{S})^{Q} \\log{(E^{S})} and \\frac{\\partial}{\\partial Q} (E^{S})^{Q} = (E^{S})^{Q} \\log{(E^{S})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('E', commutative=True), Symbol('Q', commutative=True), Symbol('S', commutative=True)), Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True)))"], [["add", 1, "Pow(Symbol('E', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Function('G')(Symbol('E', commutative=True), Symbol('Q', commutative=True), Symbol('S', commutative=True))), Add(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Function('G')(Symbol('E', commutative=True), Symbol('Q', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('G')(Symbol('E', commutative=True), Symbol('Q', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True)), log(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Pow(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True)), log(Pow(Symbol('E', commutative=True), Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(g)} = \\cos{(g)}, then obtain 2 \\eta^{\\prime}{(g)} - \\frac{\\cos{(g)}}{g} = 2 \\cos{(g)} - \\frac{\\cos{(g)}}{g}", "derivation": "\\eta^{\\prime}{(g)} = \\cos{(g)} and \\frac{\\eta^{\\prime}{(g)}}{g} = \\frac{\\cos{(g)}}{g} and \\eta^{\\prime}{(g)} - \\frac{\\eta^{\\prime}{(g)}}{g} = \\cos{(g)} - \\frac{\\eta^{\\prime}{(g)}}{g} and \\eta^{\\prime}{(g)} - \\frac{\\cos{(g)}}{g} = \\cos{(g)} - \\frac{\\cos{(g)}}{g} and 2 \\eta^{\\prime}{(g)} - \\frac{\\eta^{\\prime}{(g)}}{g} = \\eta^{\\prime}{(g)} + \\cos{(g)} - \\frac{\\eta^{\\prime}{(g)}}{g} and 2 \\eta^{\\prime}{(g)} - \\frac{\\cos{(g)}}{g} = \\eta^{\\prime}{(g)} + \\cos{(g)} - \\frac{\\cos{(g)}}{g} and 2 \\eta^{\\prime}{(g)} - \\frac{\\cos{(g)}}{g} = 2 \\cos{(g)} - \\frac{\\cos{(g)}}{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)))), Add(cos(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Add(cos(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))))"], [["add", 1, "Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Add(Mul(Integer(2), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(Z,\\sigma_x)} = Z \\sigma_x, then derive \\frac{\\partial^{2}}{\\partial \\sigma_x\\partial Z} \\operatorname{P_{e}}{(Z,\\sigma_x)} = 1, then obtain 0 = \\frac{\\partial^{3}}{\\partial \\sigma_x\\partial Z^{2}} \\operatorname{P_{e}}{(Z,\\sigma_x)}", "derivation": "\\operatorname{P_{e}}{(Z,\\sigma_x)} = Z \\sigma_x and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{P_{e}}{(Z,\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} Z \\sigma_x and \\frac{\\partial^{2}}{\\partial Z\\partial \\sigma_x} \\operatorname{P_{e}}{(Z,\\sigma_x)} = \\frac{\\partial^{2}}{\\partial Z\\partial \\sigma_x} Z \\sigma_x and \\frac{\\partial^{2}}{\\partial \\sigma_x\\partial Z} \\operatorname{P_{e}}{(Z,\\sigma_x)} = 1 and \\frac{\\partial^{3}}{\\partial Z\\partial \\sigma_x\\partial Z} \\operatorname{P_{e}}{(Z,\\sigma_x)} = \\frac{d}{d Z} 1 and \\frac{\\partial^{3}}{\\partial Z\\partial \\sigma_x\\partial Z} Z \\sigma_x = \\frac{d}{d Z} 1 and \\frac{\\partial^{3}}{\\partial Z\\partial \\sigma_x\\partial Z} Z \\sigma_x = \\frac{\\partial^{3}}{\\partial Z\\partial \\sigma_x\\partial Z} \\operatorname{P_{e}}{(Z,\\sigma_x)} and 0 = \\frac{\\partial^{3}}{\\partial \\sigma_x\\partial Z^{2}} \\operatorname{P_{e}}{(Z,\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Derivative(Function('P_e')(Symbol('Z', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(\\lambda)} = \\cos{(\\lambda)} and \\tilde{g}^*{(\\lambda)} = \\lambda + \\cos{(\\lambda)} - 1, then obtain \\frac{d}{d \\lambda} (\\lambda + \\cos{(\\lambda)})^{\\lambda} = \\frac{d}{d \\lambda} (\\tilde{g}^*{(\\lambda)} + 1)^{\\lambda}", "derivation": "\\varphi^{*}{(\\lambda)} = \\cos{(\\lambda)} and \\lambda + \\varphi^{*}{(\\lambda)} = \\lambda + \\cos{(\\lambda)} and \\lambda + \\varphi^{*}{(\\lambda)} - 1 = \\lambda + \\cos{(\\lambda)} - 1 and \\tilde{g}^*{(\\lambda)} = \\lambda + \\cos{(\\lambda)} - 1 and \\lambda + \\varphi^{*}{(\\lambda)} - 1 = \\tilde{g}^*{(\\lambda)} and \\lambda + \\varphi^{*}{(\\lambda)} = \\tilde{g}^*{(\\lambda)} + 1 and \\lambda + \\cos{(\\lambda)} = \\tilde{g}^*{(\\lambda)} + 1 and (\\lambda + \\cos{(\\lambda)})^{\\lambda} = (\\tilde{g}^*{(\\lambda)} + 1)^{\\lambda} and \\frac{d}{d \\lambda} (\\lambda + \\cos{(\\lambda)})^{\\lambda} = \\frac{d}{d \\lambda} (\\tilde{g}^*{(\\lambda)} + 1)^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)))"], [["add", 5, 1], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)), Integer(1)))"], [["power", 7, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 8, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\lambda', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(M,s)} = \\log{(M + s)}, then obtain \\int \\frac{- \\log{(M + s)} + \\int M \\operatorname{v_{x}}{(M,s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} dM = \\int \\frac{- \\log{(M + s)} + \\int M \\log{(M + s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} dM", "derivation": "\\operatorname{v_{x}}{(M,s)} = \\log{(M + s)} and M \\operatorname{v_{x}}{(M,s)} = M \\log{(M + s)} and \\int M \\operatorname{v_{x}}{(M,s)} dM = \\int M \\log{(M + s)} dM and - \\log{(M + s)} + \\int M \\operatorname{v_{x}}{(M,s)} dM = - \\log{(M + s)} + \\int M \\log{(M + s)} dM and \\frac{- \\log{(M + s)} + \\int M \\operatorname{v_{x}}{(M,s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} = \\frac{- \\log{(M + s)} + \\int M \\log{(M + s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} and \\int \\frac{- \\log{(M + s)} + \\int M \\operatorname{v_{x}}{(M,s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} dM = \\int \\frac{- \\log{(M + s)} + \\int M \\log{(M + s)} dM}{\\int M \\operatorname{v_{x}}{(M,s)} dM} dM", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True)), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True))))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('M', commutative=True), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('M', commutative=True), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["minus", 3, "log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('M', commutative=True)))))"], [["divide", 4, "Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True)))), Pow(Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('M', commutative=True)))), Pow(Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True))), Integer(-1))))"], [["integrate", 5, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True)))), Pow(Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True))), Integer(-1))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Integral(Mul(Symbol('M', commutative=True), log(Add(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('M', commutative=True)))), Pow(Integral(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('M', commutative=True))), Integer(-1))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\log{(\\frac{z^{*}}{\\mathbf{S}})}, then obtain \\frac{\\partial^{3}}{\\partial z^{*}\\partial \\mathbf{S}\\partial z^{*}} \\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\frac{\\partial^{3}}{\\partial z^{*}\\partial \\mathbf{S}\\partial z^{*}} \\log{(\\frac{z^{*}}{\\mathbf{S}})}", "derivation": "\\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\log{(\\frac{z^{*}}{\\mathbf{S}})} and \\frac{\\partial}{\\partial z^{*}} \\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\frac{\\partial}{\\partial z^{*}} \\log{(\\frac{z^{*}}{\\mathbf{S}})} and \\frac{\\partial^{2}}{\\partial \\mathbf{S}\\partial z^{*}} \\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\frac{\\partial^{2}}{\\partial \\mathbf{S}\\partial z^{*}} \\log{(\\frac{z^{*}}{\\mathbf{S}})} and \\frac{\\partial^{3}}{\\partial z^{*}\\partial \\mathbf{S}\\partial z^{*}} \\Psi_{\\lambda}{(z^{*},\\mathbf{S})} = \\frac{\\partial^{3}}{\\partial z^{*}\\partial \\mathbf{S}\\partial z^{*}} \\log{(\\frac{z^{*}}{\\mathbf{S}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(S,p)} = \\frac{e^{p}}{S}, then derive \\int \\Psi_{nl}{(S,p)} e^{- p} dS = \\theta_1 + \\log{(S)}, then obtain \\theta_1 + \\log{(S)} = \\int \\frac{1}{S} dS", "derivation": "\\Psi_{nl}{(S,p)} = \\frac{e^{p}}{S} and \\Psi_{nl}{(S,p)} e^{- p} = \\frac{1}{S} and \\int \\Psi_{nl}{(S,p)} e^{- p} dS = \\int \\frac{1}{S} dS and \\int \\Psi_{nl}{(S,p)} e^{- p} dS = \\theta_1 + \\log{(S)} and \\theta_1 + \\log{(S)} = \\int \\frac{1}{S} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))))"], [["divide", 1, "exp(Symbol('p', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True)))), Pow(Symbol('S', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('S', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Integer(-1)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('p', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('S', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Integer(-1)), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given n{(F_{g},f^{\\prime})} = \\log{(F_{g}^{f^{\\prime}})} and \\delta{(F_{g},f^{\\prime})} = \\frac{(f^{\\prime})^{2} \\log{(F_{g})}}{2}, then derive \\int n{(F_{g},f^{\\prime})} df^{\\prime} = \\mathbf{J}_M + \\frac{(f^{\\prime})^{2} \\log{(F_{g})}}{2}, then obtain \\int \\log{(F_{g}^{f^{\\prime}})} df^{\\prime} = \\mathbf{J}_M + \\delta{(F_{g},f^{\\prime})}", "derivation": "n{(F_{g},f^{\\prime})} = \\log{(F_{g}^{f^{\\prime}})} and \\int n{(F_{g},f^{\\prime})} df^{\\prime} = \\int \\log{(F_{g}^{f^{\\prime}})} df^{\\prime} and \\int n{(F_{g},f^{\\prime})} df^{\\prime} = \\mathbf{J}_M + \\frac{(f^{\\prime})^{2} \\log{(F_{g})}}{2} and \\delta{(F_{g},f^{\\prime})} = \\frac{(f^{\\prime})^{2} \\log{(F_{g})}}{2} and \\int n{(F_{g},f^{\\prime})} df^{\\prime} = \\mathbf{J}_M + \\delta{(F_{g},f^{\\prime})} and \\int \\log{(F_{g}^{f^{\\prime}})} df^{\\prime} = \\mathbf{J}_M + \\delta{(F_{g},f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), log(Pow(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('n')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Pow(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)), log(Symbol('F_g', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)), log(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('n')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(log(Pow(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given J{(C_{1},W)} = C_{1} + e^{W} and \\hat{H}_l{(C_{1},W)} = C_{1} + W + e^{W}, then obtain \\frac{(C_{1} + W + e^{W})^{2}}{C_{1} + e^{W}} = \\frac{(C_{1} + W + e^{W}) \\hat{H}_l{(C_{1},W)}}{C_{1} + e^{W}}", "derivation": "J{(C_{1},W)} = C_{1} + e^{W} and W + J{(C_{1},W)} = C_{1} + W + e^{W} and \\hat{H}_l{(C_{1},W)} = C_{1} + W + e^{W} and W + J{(C_{1},W)} = \\hat{H}_l{(C_{1},W)} and (W + J{(C_{1},W)})^{2} = (W + J{(C_{1},W)}) \\hat{H}_l{(C_{1},W)} and (C_{1} + W + e^{W})^{2} = (C_{1} + W + e^{W}) \\hat{H}_l{(C_{1},W)} and \\frac{(C_{1} + W + e^{W})^{2}}{C_{1} + e^{W}} = \\frac{(C_{1} + W + e^{W}) \\hat{H}_l{(C_{1},W)}}{C_{1} + e^{W}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Add(Symbol('C_1', commutative=True), exp(Symbol('W', commutative=True))))"], [["add", 1, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('W', commutative=True), Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Function('\\\\hat{H}_l')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))"], [["times", 4, "Add(Symbol('W', commutative=True), Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))"], "Equality(Pow(Add(Symbol('W', commutative=True), Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Integer(2)), Mul(Add(Symbol('W', commutative=True), Function('J')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Function('\\\\hat{H}_l')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))), Integer(2)), Mul(Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))), Function('\\\\hat{H}_l')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))))"], [["divide", 6, "Add(Symbol('C_1', commutative=True), exp(Symbol('W', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), exp(Symbol('W', commutative=True))), Integer(-1)), Pow(Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))), Integer(2))), Mul(Pow(Add(Symbol('C_1', commutative=True), exp(Symbol('W', commutative=True))), Integer(-1)), Add(Symbol('C_1', commutative=True), Symbol('W', commutative=True), exp(Symbol('W', commutative=True))), Function('\\\\hat{H}_l')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{g})} = \\log{(\\mathbf{g})} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{g})} = \\operatorname{t_{2}}^{\\mathbf{g}}{(\\mathbf{g})}, then obtain \\operatorname{g_{\\varepsilon}}{(\\mathbf{g})} = \\log{(\\mathbf{g})}^{\\mathbf{g}}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{g})} = \\log{(\\mathbf{g})} and \\operatorname{t_{2}}^{\\mathbf{g}}{(\\mathbf{g})} = \\log{(\\mathbf{g})}^{\\mathbf{g}} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{g})} = \\operatorname{t_{2}}^{\\mathbf{g}}{(\\mathbf{g})} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{g})} = \\log{(\\mathbf{g})}^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(log(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('t_2')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(log(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(f^{*},\\mathbf{M})} = \\mathbf{M} f^{*} and \\phi_{1}{(f^{*},\\mathbf{M})} = \\int (- \\mathbf{M} f^{*} + \\rho_{b}{(f^{*},\\mathbf{M})}) df^{*}, then obtain \\phi_{1}{(f^{*},\\mathbf{M})} = \\int 0 df^{*}", "derivation": "\\rho_{b}{(f^{*},\\mathbf{M})} = \\mathbf{M} f^{*} and - \\mathbf{M} f^{*} + \\rho_{b}{(f^{*},\\mathbf{M})} = 0 and \\int (- \\mathbf{M} f^{*} + \\rho_{b}{(f^{*},\\mathbf{M})}) df^{*} = \\int 0 df^{*} and \\phi_{1}{(f^{*},\\mathbf{M})} = \\int (- \\mathbf{M} f^{*} + \\rho_{b}{(f^{*},\\mathbf{M})}) df^{*} and \\phi_{1}{(f^{*},\\mathbf{M})} = \\int 0 df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f^*', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\rho_b')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\rho_b')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\rho_b')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\phi_1')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(x)} = \\log{(x)}, then obtain \\frac{(\\frac{1}{\\log{(x)}})^{x}}{\\log{(x)}} = \\frac{(\\frac{1}{\\log{(x)}})^{x}}{\\operatorname{F_{c}}{(x)}}", "derivation": "\\operatorname{F_{c}}{(x)} = \\log{(x)} and \\frac{1}{\\log{(x)}} = \\frac{1}{\\operatorname{F_{c}}{(x)}} and (\\frac{1}{\\log{(x)}})^{x} = (\\frac{1}{\\operatorname{F_{c}}{(x)}})^{x} and \\frac{(\\frac{1}{\\operatorname{F_{c}}{(x)}})^{x}}{\\log{(x)}} = \\frac{(\\frac{1}{\\operatorname{F_{c}}{(x)}})^{x}}{\\operatorname{F_{c}}{(x)}} and \\frac{(\\frac{1}{\\log{(x)}})^{x}}{\\log{(x)}} = \\frac{(\\frac{1}{\\log{(x)}})^{x}}{\\operatorname{F_{c}}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["divide", 1, "Mul(Function('F_c')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], "Equality(Pow(log(Symbol('x', commutative=True)), Integer(-1)), Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Pow(log(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)), Pow(Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)))"], [["times", 2, "Pow(Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Mul(Pow(Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)), Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Pow(log(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Mul(Pow(Pow(log(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True)), Pow(Function('F_c')(Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(F_{N})} = \\sin{(F_{N})}, then obtain \\frac{1}{2} = \\frac{\\sin{(F_{N})}}{\\operatorname{g_{\\varepsilon}}{(F_{N})} + \\sin{(F_{N})}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(F_{N})} = \\sin{(F_{N})} and 2 \\operatorname{g_{\\varepsilon}}{(F_{N})} = \\operatorname{g_{\\varepsilon}}{(F_{N})} + \\sin{(F_{N})} and \\frac{1}{2} = \\frac{\\sin{(F_{N})}}{2 \\operatorname{g_{\\varepsilon}}{(F_{N})}} and \\frac{1}{2} = \\frac{\\sin{(F_{N})}}{\\operatorname{g_{\\varepsilon}}{(F_{N})} + \\sin{(F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["add", 1, "Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Add(Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True)), Integer(-1)), sin(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True))), Integer(-1)), sin(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given q{(A_{z})} = \\cos{(A_{z})}, then derive \\frac{d^{2}}{d A_{z}^{2}} q{(A_{z})} = - \\cos{(A_{z})}, then obtain \\frac{d^{2}}{d A_{z}^{2}} q{(A_{z})} = - q{(A_{z})}", "derivation": "q{(A_{z})} = \\cos{(A_{z})} and \\frac{d}{d A_{z}} q{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})} and \\frac{d^{2}}{d A_{z}^{2}} q{(A_{z})} = \\frac{d^{2}}{d A_{z}^{2}} \\cos{(A_{z})} and \\frac{d^{2}}{d A_{z}^{2}} q{(A_{z})} = - \\cos{(A_{z})} and \\frac{d^{2}}{d A_{z}^{2}} q{(A_{z})} = - q{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('q')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('A_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('q')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Mul(Integer(-1), Function('q')(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\theta_2)} = e^{\\sin{(\\theta_2)}}, then obtain e^{\\sin{(\\theta_2)}} \\cos{(\\theta_2)} - \\frac{d}{d \\theta_2} \\sigma_{p}{(\\theta_2)} = 0", "derivation": "\\sigma_{p}{(\\theta_2)} = e^{\\sin{(\\theta_2)}} and - \\sigma_{p}{(\\theta_2)} = - e^{\\sin{(\\theta_2)}} and - \\sigma_{p}{(\\theta_2)} + e^{\\sin{(\\theta_2)}} = 0 and \\frac{d}{d \\theta_2} (- \\sigma_{p}{(\\theta_2)} + e^{\\sin{(\\theta_2)}}) = \\frac{d}{d \\theta_2} 0 and e^{\\sin{(\\theta_2)}} \\cos{(\\theta_2)} - \\frac{d}{d \\theta_2} \\sigma_{p}{(\\theta_2)} = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\theta_2', commutative=True)), exp(sin(Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), exp(sin(Symbol('\\\\theta_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\theta_2', commutative=True))), exp(sin(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\theta_2', commutative=True))), exp(sin(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(exp(sin(Symbol('\\\\theta_2', commutative=True))), cos(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(r_{0})} = e^{\\sin{(r_{0})}}, then obtain \\log{(- \\int \\operatorname{c_{0}}{(r_{0})} dr_{0})} = \\log{(- \\int e^{\\sin{(r_{0})}} dr_{0})}", "derivation": "\\operatorname{c_{0}}{(r_{0})} = e^{\\sin{(r_{0})}} and \\int \\operatorname{c_{0}}{(r_{0})} dr_{0} = \\int e^{\\sin{(r_{0})}} dr_{0} and - \\int \\operatorname{c_{0}}{(r_{0})} dr_{0} = - \\int e^{\\sin{(r_{0})}} dr_{0} and \\log{(- \\int \\operatorname{c_{0}}{(r_{0})} dr_{0})} = \\log{(- \\int e^{\\sin{(r_{0})}} dr_{0})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('r_0', commutative=True)), exp(sin(Symbol('r_0', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(exp(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('c_0')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Mul(Integer(-1), Integral(exp(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))))"], [["log", 3], "Equality(log(Mul(Integer(-1), Integral(Function('c_0')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))), log(Mul(Integer(-1), Integral(exp(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(k,c,V)} = c^{V} - k, then derive \\frac{\\partial}{\\partial k} \\operatorname{f_{E}}{(k,c,V)} = -1, then obtain \\sin{(\\frac{\\partial}{\\partial k} (c^{V} - k))} \\frac{\\partial}{\\partial k} (c^{V} - k) = - \\sin{(\\frac{\\partial}{\\partial k} (c^{V} - k))}", "derivation": "\\operatorname{f_{E}}{(k,c,V)} = c^{V} - k and \\frac{\\partial}{\\partial k} \\operatorname{f_{E}}{(k,c,V)} = \\frac{\\partial}{\\partial k} (c^{V} - k) and \\frac{\\partial}{\\partial k} \\operatorname{f_{E}}{(k,c,V)} = -1 and \\frac{\\partial}{\\partial k} (c^{V} - k) = -1 and \\sin{(\\frac{\\partial}{\\partial k} (c^{V} - k))} \\frac{\\partial}{\\partial k} (c^{V} - k) = - \\sin{(\\frac{\\partial}{\\partial k} (c^{V} - k))}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('k', commutative=True), Symbol('c', commutative=True), Symbol('V', commutative=True)), Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('k', commutative=True), Symbol('c', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('k', commutative=True), Symbol('c', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))"], [["times", 4, "sin(Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(sin(Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Derivative(Add(Pow(Symbol('c', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))))"]]}, {"prompt": "Given v{(P_{g},\\mathbf{p})} = P_{g} + \\mathbf{p}, then derive \\int v{(P_{g},\\mathbf{p})} d\\mathbf{p} = P_{g} \\mathbf{p} + \\hat{p} + \\frac{\\mathbf{p}^{2}}{2}, then obtain \\iint v{(P_{g},\\mathbf{p})} d\\mathbf{p} dP_{g} = \\iint (P_{g} + \\mathbf{p}) d\\mathbf{p} dP_{g}", "derivation": "v{(P_{g},\\mathbf{p})} = P_{g} + \\mathbf{p} and \\int v{(P_{g},\\mathbf{p})} d\\mathbf{p} = \\int (P_{g} + \\mathbf{p}) d\\mathbf{p} and \\int v{(P_{g},\\mathbf{p})} d\\mathbf{p} = P_{g} \\mathbf{p} + \\hat{p} + \\frac{\\mathbf{p}^{2}}{2} and \\iint v{(P_{g},\\mathbf{p})} d\\mathbf{p} dP_{g} = \\int (P_{g} \\mathbf{p} + \\hat{p} + \\frac{\\mathbf{p}^{2}}{2}) dP_{g} and \\iint (P_{g} + \\mathbf{p}) d\\mathbf{p} dP_{g} = \\int (P_{g} \\mathbf{p} + \\hat{p} + \\frac{\\mathbf{p}^{2}}{2}) dP_{g} and \\iint v{(P_{g},\\mathbf{p})} d\\mathbf{p} dP_{g} = \\iint (P_{g} + \\mathbf{p}) d\\mathbf{p} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('v')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)))), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)))), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('v')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(v_{t})} = \\cos{(v_{t})} and C{(\\hat{\\mathbf{x}},\\hat{H})} = \\frac{\\hat{H}}{\\hat{\\mathbf{x}}}, then obtain - \\frac{\\hat{H}}{\\hat{\\mathbf{x}}} + C{(\\hat{\\mathbf{x}},\\hat{H})} + \\tilde{g}{(v_{t})} = \\tilde{g}{(v_{t})}", "derivation": "\\tilde{g}{(v_{t})} = \\cos{(v_{t})} and C{(\\hat{\\mathbf{x}},\\hat{H})} = \\frac{\\hat{H}}{\\hat{\\mathbf{x}}} and - \\frac{\\hat{H}}{\\hat{\\mathbf{x}}} + C{(\\hat{\\mathbf{x}},\\hat{H})} = 0 and - \\frac{\\hat{H}}{\\hat{\\mathbf{x}}} + C{(\\hat{\\mathbf{x}},\\hat{H})} + \\cos{(v_{t})} = \\cos{(v_{t})} and - \\frac{\\hat{H}}{\\hat{\\mathbf{x}}} + C{(\\hat{\\mathbf{x}},\\hat{H})} + \\tilde{g}{(v_{t})} = \\tilde{g}{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], ["get_premise", "Equality(Function('C')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))))"], [["minus", 2, "Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Function('C')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["add", 3, "cos(Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Function('C')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('v_t', commutative=True))), cos(Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Function('C')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\tilde{g}')(Symbol('v_t', commutative=True))), Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(A)} = e^{A}, then obtain \\sin{((2 \\operatorname{E_{x}}{(A)})^{A})} = \\sin{((\\operatorname{E_{x}}{(A)} + e^{A})^{A})}", "derivation": "\\operatorname{E_{x}}{(A)} = e^{A} and 2 \\operatorname{E_{x}}{(A)} = \\operatorname{E_{x}}{(A)} + e^{A} and (2 \\operatorname{E_{x}}{(A)})^{A} = (\\operatorname{E_{x}}{(A)} + e^{A})^{A} and \\sin{((2 \\operatorname{E_{x}}{(A)})^{A})} = \\sin{((\\operatorname{E_{x}}{(A)} + e^{A})^{A})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["add", 1, "Function('E_x')(Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('E_x')(Symbol('A', commutative=True))), Add(Function('E_x')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('E_x')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Add(Function('E_x')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Mul(Integer(2), Function('E_x')(Symbol('A', commutative=True))), Symbol('A', commutative=True))), sin(Pow(Add(Function('E_x')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"]]}, {"prompt": "Given A{(\\Omega,f^{*})} = \\Omega + f^{*} and b{(\\Omega)} = \\Omega, then derive b{(\\Omega)} - \\cos{(A{(\\Omega,f^{*})})} \\frac{\\partial}{\\partial f^{*}} A{(\\Omega,f^{*})} = \\Omega - \\cos{(A{(\\Omega,f^{*})})} \\frac{\\partial}{\\partial f^{*}} A{(\\Omega,f^{*})}, then obtain b{(\\Omega)} - \\cos{(\\Omega + f^{*})} \\frac{\\partial}{\\partial f^{*}} (\\Omega + f^{*}) = \\Omega - \\cos{(\\Omega + f^{*})} \\frac{\\partial}{\\partial f^{*}} (\\Omega + f^{*})", "derivation": "A{(\\Omega,f^{*})} = \\Omega + f^{*} and \\sin{(A{(\\Omega,f^{*})})} = \\sin{(\\Omega + f^{*})} and b{(\\Omega)} = \\Omega and b{(\\Omega)} - \\frac{\\partial}{\\partial f^{*}} \\sin{(\\Omega + f^{*})} = \\Omega - \\frac{\\partial}{\\partial f^{*}} \\sin{(\\Omega + f^{*})} and b{(\\Omega)} - \\frac{\\partial}{\\partial f^{*}} \\sin{(A{(\\Omega,f^{*})})} = \\Omega - \\frac{\\partial}{\\partial f^{*}} \\sin{(A{(\\Omega,f^{*})})} and b{(\\Omega)} - \\cos{(A{(\\Omega,f^{*})})} \\frac{\\partial}{\\partial f^{*}} A{(\\Omega,f^{*})} = \\Omega - \\cos{(A{(\\Omega,f^{*})})} \\frac{\\partial}{\\partial f^{*}} A{(\\Omega,f^{*})} and b{(\\Omega)} - \\cos{(\\Omega + f^{*})} \\frac{\\partial}{\\partial f^{*}} (\\Omega + f^{*}) = \\Omega - \\cos{(\\Omega + f^{*})} \\frac{\\partial}{\\partial f^{*}} (\\Omega + f^{*})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)))"], [["sin", 1], "Equality(sin(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["minus", 3, "Derivative(sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))"], "Equality(Add(Function('b')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Derivative(sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('b')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Derivative(sin(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(sin(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Function('b')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), cos(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Derivative(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Derivative(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Function('b')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{P}{(v_{t},l)} = - l + v_{t} and \\varepsilon_{0}{(v_{t},l)} = - l + v_{t}, then obtain \\mathbf{P}{(v_{t},l)} - \\frac{d}{d l} 0 = \\varepsilon_{0}{(v_{t},l)} - \\frac{d}{d l} 0", "derivation": "\\mathbf{P}{(v_{t},l)} = - l + v_{t} and 0 = - l + v_{t} - \\mathbf{P}{(v_{t},l)} and \\varepsilon_{0}{(v_{t},l)} = - l + v_{t} and 0 = - \\mathbf{P}{(v_{t},l)} + \\varepsilon_{0}{(v_{t},l)} and - \\frac{d}{d l} 0 = - \\mathbf{P}{(v_{t},l)} + \\varepsilon_{0}{(v_{t},l)} - \\frac{d}{d l} 0 and \\mathbf{P}{(v_{t},l)} - \\frac{d}{d l} 0 = \\varepsilon_{0}{(v_{t},l)} - \\frac{d}{d l} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_t', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True))), Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True), Symbol('l', commutative=True))))"], [["minus", 4, "Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True))), Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["minus", 5, "Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\theta_{1}{(\\eta^{\\prime})} = \\cos{(\\log{(\\eta^{\\prime})})}, then derive \\frac{d}{d \\eta^{\\prime}} \\theta_{1}{(\\eta^{\\prime})} = - \\frac{\\sin{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}}, then obtain - \\frac{\\sin{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} = \\frac{d}{d \\eta^{\\prime}} \\cos{(\\log{(\\eta^{\\prime})})}", "derivation": "\\theta_{1}{(\\eta^{\\prime})} = \\cos{(\\log{(\\eta^{\\prime})})} and \\frac{d}{d \\eta^{\\prime}} \\theta_{1}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\cos{(\\log{(\\eta^{\\prime})})} and \\frac{d}{d \\eta^{\\prime}} \\theta_{1}{(\\eta^{\\prime})} = - \\frac{\\sin{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} and - \\frac{\\sin{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} = \\frac{d}{d \\eta^{\\prime}} \\cos{(\\log{(\\eta^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Derivative(cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(a^{\\dagger},V)} = V a^{\\dagger} and n{(a^{\\dagger},V)} = \\frac{\\operatorname{r_{0}}{(a^{\\dagger},V)}}{a^{\\dagger}}, then obtain n{(a^{\\dagger},V)} = V", "derivation": "\\operatorname{r_{0}}{(a^{\\dagger},V)} = V a^{\\dagger} and \\frac{\\operatorname{r_{0}}{(a^{\\dagger},V)}}{a^{\\dagger}} = V and n{(a^{\\dagger},V)} = \\frac{\\operatorname{r_{0}}{(a^{\\dagger},V)}}{a^{\\dagger}} and n{(a^{\\dagger},V)} = V", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V', commutative=True))), Symbol('V', commutative=True))"], ["renaming_premise", "Equality(Function('n')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('n')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True))"]]}, {"prompt": "Given \\mathbf{E}{(E,\\rho_b,k)} = k (E + \\rho_b) and \\operatorname{E_{\\lambda}}{(E,\\rho_b,k)} = k (E + \\rho_b), then obtain - 2 E - 2 \\rho_b + k (E + \\rho_b) + \\mathbf{E}{(E,\\rho_b,k)} = - 2 E - 2 \\rho_b + k (E + \\rho_b) + \\operatorname{E_{\\lambda}}{(E,\\rho_b,k)}", "derivation": "\\mathbf{E}{(E,\\rho_b,k)} = k (E + \\rho_b) and - E - \\rho_b + \\mathbf{E}{(E,\\rho_b,k)} = - E - \\rho_b + k (E + \\rho_b) and \\operatorname{E_{\\lambda}}{(E,\\rho_b,k)} = k (E + \\rho_b) and - E - \\rho_b + \\mathbf{E}{(E,\\rho_b,k)} = - E - \\rho_b + \\operatorname{E_{\\lambda}}{(E,\\rho_b,k)} and - 2 E - 2 \\rho_b + k (E + \\rho_b) + \\mathbf{E}{(E,\\rho_b,k)} = - 2 E - 2 \\rho_b + k (E + \\rho_b) + \\operatorname{E_{\\lambda}}{(E,\\rho_b,k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 1, "Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\mathbf{E}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\mathbf{E}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))), Function('\\\\mathbf{E}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('k', commutative=True), Add(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})} and \\mathbf{g}{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})}, then obtain \\int (- \\tilde{g} + \\mathbf{J}_P{(\\tilde{g})}) d\\tilde{g} = \\int (- \\tilde{g} + \\mathbf{g}{(\\tilde{g})}) d\\tilde{g}", "derivation": "\\mathbf{J}_P{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})} and \\mathbf{g}{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})} and \\mathbf{J}_P{(\\tilde{g})} = \\mathbf{g}{(\\tilde{g})} and - \\tilde{g} + \\mathbf{J}_P{(\\tilde{g})} = - \\tilde{g} + \\mathbf{g}{(\\tilde{g})} and \\int (- \\tilde{g} + \\mathbf{J}_P{(\\tilde{g})}) d\\tilde{g} = \\int (- \\tilde{g} + \\mathbf{g}{(\\tilde{g})}) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True)), log(cos(Symbol('\\\\tilde{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}', commutative=True)), log(cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then derive \\frac{d}{d \\mathbf{E}} \\operatorname{F_{c}}{(\\mathbf{E})} = \\frac{1}{\\mathbf{E}}, then obtain \\operatorname{F_{c}}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\log{(\\mathbf{E})} = \\frac{\\operatorname{F_{c}}{(\\mathbf{E})}}{\\mathbf{E}}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and \\frac{d}{d \\mathbf{E}} \\operatorname{F_{c}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\log{(\\mathbf{E})} and \\frac{d}{d \\mathbf{E}} \\operatorname{F_{c}}{(\\mathbf{E})} = \\frac{1}{\\mathbf{E}} and \\operatorname{F_{c}}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\operatorname{F_{c}}{(\\mathbf{E})} = \\frac{\\operatorname{F_{c}}{(\\mathbf{E})}}{\\mathbf{E}} and \\operatorname{F_{c}}{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\log{(\\mathbf{E})} = \\frac{\\operatorname{F_{c}}{(\\mathbf{E})}}{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))"], [["times", 3, "Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\Omega)} = \\cos{(\\Omega)} and \\varphi^{*}{(\\Omega)} = \\frac{1}{\\operatorname{C_{d}}{(\\Omega)}}, then obtain \\frac{d}{d \\Omega} (\\varphi^{*}{(\\Omega)} - 1) = \\frac{d}{d \\Omega} (- \\operatorname{C_{d}}{(\\Omega)} \\varphi^{*}{(\\Omega)} + \\varphi^{*}{(\\Omega)})", "derivation": "\\operatorname{C_{d}}{(\\Omega)} = \\cos{(\\Omega)} and -1 = - \\frac{\\cos{(\\Omega)}}{\\operatorname{C_{d}}{(\\Omega)}} and -1 + \\frac{1}{\\operatorname{C_{d}}{(\\Omega)}} = - \\frac{\\cos{(\\Omega)}}{\\operatorname{C_{d}}{(\\Omega)}} + \\frac{1}{\\operatorname{C_{d}}{(\\Omega)}} and \\varphi^{*}{(\\Omega)} = \\frac{1}{\\operatorname{C_{d}}{(\\Omega)}} and \\varphi^{*}{(\\Omega)} - 1 = - \\varphi^{*}{(\\Omega)} \\cos{(\\Omega)} + \\varphi^{*}{(\\Omega)} and \\frac{d}{d \\Omega} (\\varphi^{*}{(\\Omega)} - 1) = \\frac{d}{d \\Omega} (- \\varphi^{*}{(\\Omega)} \\cos{(\\Omega)} + \\varphi^{*}{(\\Omega)}) and \\frac{d}{d \\Omega} (\\varphi^{*}{(\\Omega)} - 1) = \\frac{d}{d \\Omega} (- \\operatorname{C_{d}}{(\\Omega)} \\varphi^{*}{(\\Omega)} + \\varphi^{*}{(\\Omega)})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Function('C_d')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1))"], "Equality(Add(Integer(-1), Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), Pow(Function('C_d')(Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\Omega', commutative=True)), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbb{I},z^{*})} = \\sin{(\\mathbb{I} + z^{*})}, then obtain - \\mathbb{I} - z^{*} + 1 = - \\mathbb{I} - z^{*} + \\frac{\\int \\sin{(\\mathbb{I} + z^{*})} dz^{*}}{\\int \\operatorname{t_{1}}{(\\mathbb{I},z^{*})} dz^{*}}", "derivation": "\\operatorname{t_{1}}{(\\mathbb{I},z^{*})} = \\sin{(\\mathbb{I} + z^{*})} and \\int \\operatorname{t_{1}}{(\\mathbb{I},z^{*})} dz^{*} = \\int \\sin{(\\mathbb{I} + z^{*})} dz^{*} and 1 = \\frac{\\int \\sin{(\\mathbb{I} + z^{*})} dz^{*}}{\\int \\operatorname{t_{1}}{(\\mathbb{I},z^{*})} dz^{*}} and - \\mathbb{I} - z^{*} + 1 = - \\mathbb{I} - z^{*} + \\frac{\\int \\sin{(\\mathbb{I} + z^{*})} dz^{*}}{\\int \\operatorname{t_{1}}{(\\mathbb{I},z^{*})} dz^{*}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True)), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"], [["divide", 2, "Integral(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(-1)), Integral(sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))))"], [["minus", 3, "Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Pow(Integral(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(-1)), Integral(sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(f^{*})} = \\frac{d}{d f^{*}} \\sin{(f^{*})} and \\operatorname{x^{{\\}'}}{(f^{*})} = \\cos{(f^{*})}, then derive \\mathbf{F}{(f^{*})} = \\cos{(f^{*})}, then derive \\frac{\\operatorname{x^{{\\}'}}{(f^{*})}}{\\sin{(f^{*})}} = \\frac{\\cos{(f^{*})}}{\\sin{(f^{*})}}, then obtain \\frac{\\frac{d}{d f^{*}} \\sin{(f^{*})}}{\\sin{(f^{*})}} = \\frac{\\cos{(f^{*})}}{\\sin{(f^{*})}}", "derivation": "\\mathbf{F}{(f^{*})} = \\frac{d}{d f^{*}} \\sin{(f^{*})} and \\mathbf{F}{(f^{*})} = \\cos{(f^{*})} and \\operatorname{x^{{\\}'}}{(f^{*})} = \\cos{(f^{*})} and \\operatorname{x^{{\\}'}}{(f^{*})} = \\mathbf{F}{(f^{*})} and \\operatorname{x^{{\\}'}}{(f^{*})} = \\frac{d}{d f^{*}} \\sin{(f^{*})} and \\frac{\\operatorname{x^{{\\}'}}{(f^{*})}}{\\sin{(f^{*})}} = \\frac{\\frac{d}{d f^{*}} \\sin{(f^{*})}}{\\sin{(f^{*})}} and \\frac{\\operatorname{x^{{\\}'}}{(f^{*})}}{\\sin{(f^{*})}} = \\frac{\\cos{(f^{*})}}{\\sin{(f^{*})}} and \\frac{\\frac{d}{d f^{*}} \\sin{(f^{*})}}{\\sin{(f^{*})}} = \\frac{\\cos{(f^{*})}}{\\sin{(f^{*})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f^*', commutative=True)), Derivative(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('x^\\\\prime')(Symbol('f^*', commutative=True)), Function('\\\\mathbf{F}')(Symbol('f^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('x^\\\\prime')(Symbol('f^*', commutative=True)), Derivative(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["divide", 5, "sin(Symbol('f^*', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('f^*', commutative=True)), Pow(sin(Symbol('f^*', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Derivative(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Function('x^\\\\prime')(Symbol('f^*', commutative=True)), Pow(sin(Symbol('f^*', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), cos(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Derivative(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), cos(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A,M)} = \\frac{M}{A} and \\operatorname{v_{t}}{(A,M)} = \\frac{A (- 2 \\Psi_{\\lambda}{(A,M)} + \\frac{2 M}{A})}{2 M}, then obtain \\frac{\\partial}{\\partial A} \\frac{A (- \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A})}{2 M} = \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,M)}", "derivation": "\\Psi_{\\lambda}{(A,M)} = \\frac{M}{A} and 0 = - \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A} and - \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A} = - 2 \\Psi_{\\lambda}{(A,M)} + \\frac{2 M}{A} and \\frac{A (- \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A})}{2 M} = \\frac{A (- 2 \\Psi_{\\lambda}{(A,M)} + \\frac{2 M}{A})}{2 M} and \\operatorname{v_{t}}{(A,M)} = \\frac{A (- 2 \\Psi_{\\lambda}{(A,M)} + \\frac{2 M}{A})}{2 M} and \\frac{A (- \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A})}{2 M} = \\operatorname{v_{t}}{(A,M)} and \\frac{\\partial}{\\partial A} \\frac{A (- \\Psi_{\\lambda}{(A,M)} + \\frac{M}{A})}{2 M} = \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True))"], "Equality(Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('A', commutative=True), Symbol('M', commutative=True)), Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Function('v_t')(Symbol('A', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 6, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Function('v_t')(Symbol('A', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(F_{c})} = \\sin{(\\cos{(F_{c})})}, then obtain (((M^{F_{c}}{(F_{c})})^{F_{c}})^{F_{c}})^{F_{c}} = (((\\sin^{F_{c}}{(\\cos{(F_{c})})})^{F_{c}})^{F_{c}})^{F_{c}}", "derivation": "M{(F_{c})} = \\sin{(\\cos{(F_{c})})} and M^{F_{c}}{(F_{c})} = \\sin^{F_{c}}{(\\cos{(F_{c})})} and (M^{F_{c}}{(F_{c})})^{F_{c}} = (\\sin^{F_{c}}{(\\cos{(F_{c})})})^{F_{c}} and ((M^{F_{c}}{(F_{c})})^{F_{c}})^{F_{c}} = ((\\sin^{F_{c}}{(\\cos{(F_{c})})})^{F_{c}})^{F_{c}} and (((M^{F_{c}}{(F_{c})})^{F_{c}})^{F_{c}})^{F_{c}} = (((\\sin^{F_{c}}{(\\cos{(F_{c})})})^{F_{c}})^{F_{c}})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_c', commutative=True)), sin(cos(Symbol('F_c', commutative=True))))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('M')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(sin(cos(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Function('M')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(sin(cos(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Pow(Function('M')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(Pow(sin(cos(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Pow(Pow(Function('M')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(Pow(Pow(sin(cos(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given G{(M_{E},\\psi^*,c)} = - M_{E} + \\psi^* + c, then obtain (M_{E} - \\psi^* - c + \\frac{G{(M_{E},\\psi^*,c)}}{\\psi^*})^{M_{E}} = (M_{E} - \\psi^* - c + \\frac{- M_{E} + \\psi^* + c}{\\psi^*})^{M_{E}}", "derivation": "G{(M_{E},\\psi^*,c)} = - M_{E} + \\psi^* + c and \\frac{G{(M_{E},\\psi^*,c)}}{\\psi^*} = \\frac{- M_{E} + \\psi^* + c}{\\psi^*} and M_{E} - \\psi^* - c + \\frac{G{(M_{E},\\psi^*,c)}}{\\psi^*} = M_{E} - \\psi^* - c + \\frac{- M_{E} + \\psi^* + c}{\\psi^*} and (M_{E} - \\psi^* - c + \\frac{G{(M_{E},\\psi^*,c)}}{\\psi^*})^{M_{E}} = (M_{E} - \\psi^* - c + \\frac{- M_{E} + \\psi^* + c}{\\psi^*})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('M_E', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('G')(Symbol('M_E', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('G')(Symbol('M_E', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)))), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)))))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('G')(Symbol('M_E', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True)))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\lambda{(z)} = \\log{(z)}, then obtain - \\log{(z)} + (\\frac{d^{2}}{d z^{2}} \\int (\\lambda{(z)} - \\log{(z)}) dz)^{z} = - \\log{(z)} + (\\frac{d^{2}}{d z^{2}} \\int 0 dz)^{z}", "derivation": "\\lambda{(z)} = \\log{(z)} and \\lambda{(z)} - \\log{(z)} = 0 and \\int (\\lambda{(z)} - \\log{(z)}) dz = \\int 0 dz and \\frac{d}{d z} \\int (\\lambda{(z)} - \\log{(z)}) dz = \\frac{d}{d z} \\int 0 dz and \\frac{d^{2}}{d z^{2}} \\int (\\lambda{(z)} - \\log{(z)}) dz = \\frac{d^{2}}{d z^{2}} \\int 0 dz and (\\frac{d^{2}}{d z^{2}} \\int (\\lambda{(z)} - \\log{(z)}) dz)^{z} = (\\frac{d^{2}}{d z^{2}} \\int 0 dz)^{z} and - \\log{(z)} + (\\frac{d^{2}}{d z^{2}} \\int (\\lambda{(z)} - \\log{(z)}) dz)^{z} = - \\log{(z)} + (\\frac{d^{2}}{d z^{2}} \\int 0 dz)^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["minus", 1, "log(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))), Integral(Integer(0), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))), Derivative(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))))"], [["power", 5, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Integral(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))), Symbol('z', commutative=True)), Pow(Derivative(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["add", 6, "Mul(Integer(-1), log(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Pow(Derivative(Integral(Add(Function('\\\\lambda')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))), Symbol('z', commutative=True))), Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Pow(Derivative(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(2))), Symbol('z', commutative=True))))"]]}, {"prompt": "Given L{(\\chi,\\hbar)} = \\frac{\\hbar}{\\chi}, then obtain (\\int \\frac{\\partial}{\\partial \\chi} L{(\\chi,\\hbar)} d\\chi)^{\\chi} = (\\int \\frac{\\partial}{\\partial \\chi} \\frac{\\hbar}{\\chi} d\\chi)^{\\chi}", "derivation": "L{(\\chi,\\hbar)} = \\frac{\\hbar}{\\chi} and \\frac{\\partial}{\\partial \\chi} L{(\\chi,\\hbar)} = \\frac{\\partial}{\\partial \\chi} \\frac{\\hbar}{\\chi} and \\int \\frac{\\partial}{\\partial \\chi} L{(\\chi,\\hbar)} d\\chi = \\int \\frac{\\partial}{\\partial \\chi} \\frac{\\hbar}{\\chi} d\\chi and (\\int \\frac{\\partial}{\\partial \\chi} L{(\\chi,\\hbar)} d\\chi)^{\\chi} = (\\int \\frac{\\partial}{\\partial \\chi} \\frac{\\hbar}{\\chi} d\\chi)^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integral(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\chi)} = \\cos{(\\sin{(\\chi)})} and \\dot{x}{(\\chi)} = \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} and \\mathbf{A}{(\\chi)} = \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} + \\int \\operatorname{f^{\\prime}}{(\\chi)} d\\chi, then obtain \\mathbf{A}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\sin{(\\chi)})} + \\int \\operatorname{f^{\\prime}}{(\\chi)} d\\chi", "derivation": "\\operatorname{f^{\\prime}}{(\\chi)} = \\cos{(\\sin{(\\chi)})} and \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\sin{(\\chi)})} and \\dot{x}{(\\chi)} = \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} and \\dot{x}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\sin{(\\chi)})} and \\mathbf{A}{(\\chi)} = \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} + \\int \\operatorname{f^{\\prime}}{(\\chi)} d\\chi and \\mathbf{A}{(\\chi)} = \\dot{x}{(\\chi)} + \\int \\operatorname{f^{\\prime}}{(\\chi)} d\\chi and \\mathbf{A}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\sin{(\\chi)})} + \\int \\operatorname{f^{\\prime}}{(\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\chi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\dot{x}')(Symbol('\\\\chi', commutative=True)), Derivative(cos(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\chi', commutative=True)), Add(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integral(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\chi', commutative=True)), Add(Function('\\\\dot{x}')(Symbol('\\\\chi', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\chi', commutative=True)), Add(Derivative(cos(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integral(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given v{(\\chi,\\mu)} = \\mu^{\\chi}, then derive (\\frac{\\partial}{\\partial \\chi} v{(\\chi,\\mu)})^{\\mu} = (\\mu^{\\chi} \\log{(\\mu)})^{\\mu}, then obtain (v{(\\chi,\\mu)} \\log{(\\mu)})^{\\mu} = (\\mu^{\\chi} \\log{(\\mu)})^{\\mu}", "derivation": "v{(\\chi,\\mu)} = \\mu^{\\chi} and \\frac{\\partial}{\\partial \\chi} v{(\\chi,\\mu)} = \\frac{\\partial}{\\partial \\chi} \\mu^{\\chi} and (\\frac{\\partial}{\\partial \\chi} v{(\\chi,\\mu)})^{\\mu} = (\\frac{\\partial}{\\partial \\chi} \\mu^{\\chi})^{\\mu} and (\\frac{\\partial}{\\partial \\chi} v{(\\chi,\\mu)})^{\\mu} = (\\mu^{\\chi} \\log{(\\mu)})^{\\mu} and (\\frac{\\partial}{\\partial \\chi} v{(\\chi,\\mu)})^{\\mu} = (v{(\\chi,\\mu)} \\log{(\\mu)})^{\\mu} and (v{(\\chi,\\mu)} \\log{(\\mu)})^{\\mu} = (\\frac{\\partial}{\\partial \\chi} \\mu^{\\chi})^{\\mu} and (v{(\\chi,\\mu)} \\log{(\\mu)})^{\\mu} = (\\mu^{\\chi} \\log{(\\mu)})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Mul(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Mul(Function('v')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\Omega)} = \\Omega and \\operatorname{A_{z}}{(\\Omega)} = - \\Omega + \\mathbf{g}{(\\Omega)}, then obtain - \\Omega \\operatorname{A_{z}}{(\\Omega)} = 0", "derivation": "\\mathbf{g}{(\\Omega)} = \\Omega and - \\Omega + \\mathbf{g}{(\\Omega)} = 0 and - \\Omega (- \\Omega + \\mathbf{g}{(\\Omega)}) = 0 and \\operatorname{A_{z}}{(\\Omega)} = - \\Omega + \\mathbf{g}{(\\Omega)} and - \\Omega \\operatorname{A_{z}}{(\\Omega)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["minus", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True))), Integer(0))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Function('A_z')(Symbol('\\\\Omega', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(l)} = \\sin{(\\cos{(l)})} and \\mathbf{J}_P{(r)} = \\log{(r)}, then obtain \\frac{d}{d r} (\\mathbf{J}_P{(r)} + 1) = \\frac{d}{d r} (\\log{(r)} + 1)", "derivation": "\\operatorname{F_{g}}{(l)} = \\sin{(\\cos{(l)})} and \\mathbf{J}_P{(r)} = \\log{(r)} and \\frac{\\operatorname{F_{g}}{(l)}}{\\sin{(\\cos{(l)})}} + \\mathbf{J}_P{(r)} = \\frac{\\operatorname{F_{g}}{(l)}}{\\sin{(\\cos{(l)})}} + \\log{(r)} and \\mathbf{J}_P{(r)} + 1 = \\log{(r)} + 1 and \\frac{d}{d r} (\\mathbf{J}_P{(r)} + 1) = \\frac{d}{d r} (\\log{(r)} + 1)", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('l', commutative=True)), sin(cos(Symbol('l', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["add", 2, "Mul(Function('F_g')(Symbol('l', commutative=True)), Pow(sin(cos(Symbol('l', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Function('F_g')(Symbol('l', commutative=True)), Pow(sin(cos(Symbol('l', commutative=True))), Integer(-1))), Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True))), Add(Mul(Function('F_g')(Symbol('l', commutative=True)), Pow(sin(cos(Symbol('l', commutative=True))), Integer(-1))), log(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True)), Integer(1)), Add(log(Symbol('r', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(log(Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(g)} = e^{g}, then derive (\\int \\mathbf{A}{(g)} dg)^{g} = (\\mu_0 + e^{g})^{g}, then obtain \\frac{d}{d \\mu_0} ((\\int \\mathbf{A}{(g)} dg)^{g} - \\frac{(\\int e^{g} dg)^{g}}{g}) = \\frac{\\partial}{\\partial \\mu_0} ((\\mu_0 + \\mathbf{A}{(g)})^{g} - \\frac{(\\int e^{g} dg)^{g}}{g})", "derivation": "\\mathbf{A}{(g)} = e^{g} and \\int \\mathbf{A}{(g)} dg = \\int e^{g} dg and (\\int \\mathbf{A}{(g)} dg)^{g} = (\\int e^{g} dg)^{g} and (\\int \\mathbf{A}{(g)} dg)^{g} = (\\mu_0 + e^{g})^{g} and (\\int \\mathbf{A}{(g)} dg)^{g} = (\\mu_0 + \\mathbf{A}{(g)})^{g} and (\\int \\mathbf{A}{(g)} dg)^{g} - \\frac{(\\int e^{g} dg)^{g}}{g} = (\\mu_0 + \\mathbf{A}{(g)})^{g} - \\frac{(\\int e^{g} dg)^{g}}{g} and \\frac{d}{d \\mu_0} ((\\int \\mathbf{A}{(g)} dg)^{g} - \\frac{(\\int e^{g} dg)^{g}}{g}) = \\frac{\\partial}{\\partial \\mu_0} ((\\mu_0 + \\mathbf{A}{(g)})^{g} - \\frac{(\\int e^{g} dg)^{g}}{g})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\mathbf{A}')(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["minus", 5, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], "Equality(Add(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))), Add(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\mathbf{A}')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\mathbf{A}')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(\\dot{x})} = \\sin{(\\sin{(\\dot{x})})}, then obtain ((a^{\\dot{x}}{(\\dot{x})})^{\\dot{x}})^{\\dot{x}} = ((\\sin^{\\dot{x}}{(\\sin{(\\dot{x})})})^{\\dot{x}})^{\\dot{x}}", "derivation": "a{(\\dot{x})} = \\sin{(\\sin{(\\dot{x})})} and a^{\\dot{x}}{(\\dot{x})} = \\sin^{\\dot{x}}{(\\sin{(\\dot{x})})} and (a^{\\dot{x}}{(\\dot{x})})^{\\dot{x}} = (\\sin^{\\dot{x}}{(\\sin{(\\dot{x})})})^{\\dot{x}} and ((a^{\\dot{x}}{(\\dot{x})})^{\\dot{x}})^{\\dot{x}} = ((\\sin^{\\dot{x}}{(\\sin{(\\dot{x})})})^{\\dot{x}})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\dot{x}', commutative=True)), sin(sin(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Pow(Function('a')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Pow(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Pow(Pow(Function('a')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Pow(Pow(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given p{(\\mathbf{A},t_{1},A_{x})} = (A_{x}^{t_{1}})^{\\mathbf{A}}, then obtain 2 = \\frac{\\mathbf{A} (A_{x}^{t_{1}})^{\\mathbf{A}} + \\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})}}{\\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})}}", "derivation": "p{(\\mathbf{A},t_{1},A_{x})} = (A_{x}^{t_{1}})^{\\mathbf{A}} and \\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})} = \\mathbf{A} (A_{x}^{t_{1}})^{\\mathbf{A}} and 2 \\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})} = \\mathbf{A} (A_{x}^{t_{1}})^{\\mathbf{A}} + \\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})} and 2 = \\frac{\\mathbf{A} (A_{x}^{t_{1}})^{\\mathbf{A}} + \\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})}}{\\mathbf{A} p{(\\mathbf{A},t_{1},A_{x})}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)), Pow(Pow(Symbol('A_x', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Pow(Symbol('A_x', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Pow(Symbol('A_x', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)))"], "Equality(Integer(2), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Pow(Symbol('A_x', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)))), Pow(Function('p')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True), Symbol('A_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given J{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}, then obtain J{(a^{\\dagger})} - e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} + 1 = - e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} + \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} + 1", "derivation": "J{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and J{(a^{\\dagger})} + 1 = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} + 1 and e^{J{(a^{\\dagger})}} = e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} and J{(a^{\\dagger})} - e^{J{(a^{\\dagger})}} + 1 = - e^{J{(a^{\\dagger})}} + \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} + 1 and - e^{J{(a^{\\dagger})}} = - e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} and J{(a^{\\dagger})} - e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} + 1 = - e^{\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}} + \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} + 1", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1)), Add(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1)))"], [["exp", 1], "Equality(exp(Function('J')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["minus", 2, "exp(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1)), Add(Mul(Integer(-1), exp(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), exp(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('J')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Integer(1)), Add(Mul(Integer(-1), exp(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and p{(\\mathbf{s})} = \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})}, then obtain \\frac{d}{d \\mathbf{s}} \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} p{(\\mathbf{s})}", "derivation": "\\hat{x}_0{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})} = \\sin^{\\mathbf{s}}{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\sin^{\\mathbf{s}}{(\\mathbf{s})} and p{(\\mathbf{s})} = \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})} and p{(\\mathbf{s})} = \\sin^{\\mathbf{s}}{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} \\hat{x}_0^{\\mathbf{s}}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} p{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('p')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Function('p')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbb{I},x^\\prime)} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime}, then obtain \\int \\frac{\\partial}{\\partial x^\\prime} \\frac{\\mathbf{E}{(\\mathbb{I},x^\\prime)}}{x^\\prime} d\\mathbb{I} = \\int \\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime}}{x^\\prime} d\\mathbb{I}", "derivation": "\\mathbf{E}{(\\mathbb{I},x^\\prime)} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime} and \\frac{\\mathbf{E}{(\\mathbb{I},x^\\prime)}}{x^\\prime} = \\frac{\\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime}}{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\frac{\\mathbf{E}{(\\mathbb{I},x^\\prime)}}{x^\\prime} = \\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime}}{x^\\prime} and \\int \\frac{\\partial}{\\partial x^\\prime} \\frac{\\mathbf{E}{(\\mathbb{I},x^\\prime)}}{x^\\prime} d\\mathbb{I} = \\int \\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{x^\\prime}}{x^\\prime} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(M)} = \\log{(M)}, then derive \\frac{d}{d M} \\mathbf{v}{(M)} = \\frac{1}{M}, then obtain \\frac{d}{d M} \\log{(M)} \\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z} + \\frac{d}{d v_{z}} \\sin{(v_{z})} = \\frac{d}{d v_{z}} \\sin{(v_{z})} + \\frac{\\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z}}{M}", "derivation": "\\mathbf{v}{(M)} = \\log{(M)} and \\frac{d}{d M} \\mathbf{v}{(M)} = \\frac{d}{d M} \\log{(M)} and \\frac{d}{d M} \\mathbf{v}{(M)} = \\frac{1}{M} and \\frac{d}{d M} \\log{(M)} = \\frac{1}{M} and \\frac{d}{d M} \\log{(M)} \\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z} = \\frac{\\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z}}{M} and \\frac{d}{d M} \\log{(M)} \\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z} + \\frac{d}{d v_{z}} \\sin{(v_{z})} = \\frac{d}{d v_{z}} \\sin{(v_{z})} + \\frac{\\int \\frac{d}{d v_{z}} \\sin{(v_{z})} dv_{z}}{M}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["times", 4, "Integral(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Mul(Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integral(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Integral(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True)))))"], [["add", 5, "Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Add(Mul(Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integral(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True)))), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Integral(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(q,h)} = h q, then obtain ((h q \\mathbf{J}_P{(q,h)})^{q})^{q} = ((h^{2} q^{2})^{q})^{q}", "derivation": "\\mathbf{J}_P{(q,h)} = h q and h q \\mathbf{J}_P{(q,h)} = h^{2} q^{2} and (h q \\mathbf{J}_P{(q,h)})^{q} = (h^{2} q^{2})^{q} and ((h q \\mathbf{J}_P{(q,h)})^{q})^{q} = ((h^{2} q^{2})^{q})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Mul(Symbol('h', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Symbol('h', commutative=True), Symbol('q', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Symbol('h', commutative=True), Symbol('q', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('h', commutative=True))), Symbol('q', commutative=True)), Pow(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('q', commutative=True)))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('h', commutative=True), Symbol('q', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('h', commutative=True))), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\theta_2)} = \\int \\sin{(\\theta_2)} d\\theta_2, then derive \\operatorname{c_{0}}{(\\theta_2)} = U - \\cos{(\\theta_2)}, then obtain \\operatorname{c_{0}}^{\\theta_2}{(\\theta_2)} = (a^{\\dagger} - \\cos{(\\theta_2)})^{\\theta_2}", "derivation": "\\operatorname{c_{0}}{(\\theta_2)} = \\int \\sin{(\\theta_2)} d\\theta_2 and \\operatorname{c_{0}}{(\\theta_2)} = U - \\cos{(\\theta_2)} and \\operatorname{c_{0}}^{\\theta_2}{(\\theta_2)} = (U - \\cos{(\\theta_2)})^{\\theta_2} and \\int \\sin{(\\theta_2)} d\\theta_2 = U - \\cos{(\\theta_2)} and \\operatorname{c_{0}}^{\\theta_2}{(\\theta_2)} = (\\int \\sin{(\\theta_2)} d\\theta_2)^{\\theta_2} and \\operatorname{c_{0}}^{\\theta_2}{(\\theta_2)} = (a^{\\dagger} - \\cos{(\\theta_2)})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)} = \\mathbf{J}_f v_{2} and t{(v_{2},\\mathbf{J}_f)} = - \\mathbf{J}_f v_{2} + \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = 0, then obtain - \\mathbf{J}_f v_{2} (\\mathbf{J}_f v_{2} - \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)}) \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = 0", "derivation": "\\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)} = \\mathbf{J}_f v_{2} and t{(v_{2},\\mathbf{J}_f)} = - \\mathbf{J}_f v_{2} + \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)} and t{(v_{2},\\mathbf{J}_f)} = 0 and \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} 0 and \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = 0 and (\\mathbf{J}_f v_{2} - \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)}) \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = 0 and - \\mathbf{J}_f v_{2} (\\mathbf{J}_f v_{2} - \\operatorname{A_{1}}{(v_{2},\\mathbf{J}_f)}) \\frac{\\partial}{\\partial \\mathbf{J}_f} t{(v_{2},\\mathbf{J}_f)} = 0", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Function('A_1')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(0))"], [["times", 5, "Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], "Equality(Mul(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Derivative(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Integer(0))"], [["times", 6, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Derivative(Function('t')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given n{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then derive \\frac{\\frac{d}{d \\hat{H}_l} n{(\\hat{H}_l)} - 1}{\\cos{(\\hat{H}_l)} - 1} = 1, then obtain \\frac{d}{d \\hat{H}_l} \\frac{\\frac{d}{d \\hat{H}_l} n{(\\hat{H}_l)} - 1}{\\cos{(\\hat{H}_l)} - 1} = \\frac{d}{d \\hat{H}_l} 1", "derivation": "n{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and - \\hat{H}_l + n{(\\hat{H}_l)} = - \\hat{H}_l + \\sin{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} (- \\hat{H}_l + n{(\\hat{H}_l)}) = \\frac{d}{d \\hat{H}_l} (- \\hat{H}_l + \\sin{(\\hat{H}_l)}) and \\frac{\\frac{d}{d \\hat{H}_l} (- \\hat{H}_l + n{(\\hat{H}_l)})}{\\frac{d}{d \\hat{H}_l} (- \\hat{H}_l + \\sin{(\\hat{H}_l)})} = 1 and \\frac{\\frac{d}{d \\hat{H}_l} n{(\\hat{H}_l)} - 1}{\\cos{(\\hat{H}_l)} - 1} = 1 and \\frac{d}{d \\hat{H}_l} \\frac{\\frac{d}{d \\hat{H}_l} n{(\\hat{H}_l)} - 1}{\\cos{(\\hat{H}_l)} - 1} = \\frac{d}{d \\hat{H}_l} 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('n')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('n')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('n')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(cos(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(Function('n')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["differentiate", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(cos(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(Function('n')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(n_{1})} = e^{n_{1}}, then obtain - e^{n_{1}} = - n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} e^{n_{1}} + n_{1} e^{2 n_{1}} - e^{n_{1}}", "derivation": "\\operatorname{a^{\\dagger}}{(n_{1})} = e^{n_{1}} and n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} = n_{1} e^{n_{1}} and n_{1} \\operatorname{a^{\\dagger}}^{2}{(n_{1})} = n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} e^{n_{1}} and 0 = - n_{1} \\operatorname{a^{\\dagger}}^{2}{(n_{1})} + n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} e^{n_{1}} and 0 = - n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} e^{n_{1}} + n_{1} e^{2 n_{1}} and - e^{n_{1}} = - n_{1} \\operatorname{a^{\\dagger}}{(n_{1})} e^{n_{1}} + n_{1} e^{2 n_{1}} - e^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["times", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), exp(Symbol('n_1', commutative=True))))"], [["times", 2, "Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True))"], "Equality(Mul(Symbol('n_1', commutative=True), Pow(Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), Integer(2))), Mul(Symbol('n_1', commutative=True), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))))"], [["minus", 3, "Mul(Symbol('n_1', commutative=True), Pow(Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), Integer(2)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), Integer(2))), Mul(Symbol('n_1', commutative=True), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('n_1', commutative=True), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), exp(Mul(Integer(2), Symbol('n_1', commutative=True))))))"], [["minus", 5, "exp(Symbol('n_1', commutative=True))"], "Equality(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), exp(Mul(Integer(2), Symbol('n_1', commutative=True)))), Mul(Integer(-1), exp(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\Psi)} = \\cos{(\\Psi)}, then obtain 0 = - 2 \\operatorname{A_{z}}{(\\Psi)} + 2 \\cos{(\\Psi)}", "derivation": "\\operatorname{A_{z}}{(\\Psi)} = \\cos{(\\Psi)} and 0 = - \\operatorname{A_{z}}{(\\Psi)} + \\cos{(\\Psi)} and - \\operatorname{A_{z}}{(\\Psi)} = - 2 \\operatorname{A_{z}}{(\\Psi)} + \\cos{(\\Psi)} and 0 = - 2 \\operatorname{A_{z}}{(\\Psi)} + 2 \\cos{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('A_z')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["minus", 2, "Function('A_z')(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Integer(-1), Function('A_z')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('A_z')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('A_z')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\theta_2)} = e^{e^{\\theta_2}}, then obtain (\\rho^{\\theta_2}{(\\theta_2)})^{\\theta_2} = ((e^{\\rho^{- \\theta_2}{(\\theta_2)} e^{\\theta_2} (e^{e^{\\theta_2}})^{\\theta_2}})^{\\theta_2})^{\\theta_2}", "derivation": "\\rho{(\\theta_2)} = e^{e^{\\theta_2}} and \\rho^{\\theta_2}{(\\theta_2)} = (e^{e^{\\theta_2}})^{\\theta_2} and 1 = \\rho^{- \\theta_2}{(\\theta_2)} (e^{e^{\\theta_2}})^{\\theta_2} and e^{\\theta_2} = \\rho^{- \\theta_2}{(\\theta_2)} e^{\\theta_2} (e^{e^{\\theta_2}})^{\\theta_2} and (\\rho^{\\theta_2}{(\\theta_2)})^{\\theta_2} = ((e^{e^{\\theta_2}})^{\\theta_2})^{\\theta_2} and (\\rho^{\\theta_2}{(\\theta_2)})^{\\theta_2} = ((e^{\\rho^{- \\theta_2}{(\\theta_2)} e^{\\theta_2} (e^{e^{\\theta_2}})^{\\theta_2}})^{\\theta_2})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), exp(exp(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(exp(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Pow(exp(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"], [["times", 3, "exp(Symbol('\\\\theta_2', commutative=True))"], "Equality(exp(Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), exp(Symbol('\\\\theta_2', commutative=True)), Pow(exp(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(exp(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(exp(Mul(Pow(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), exp(Symbol('\\\\theta_2', commutative=True)), Pow(exp(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given g{(I,A_{2})} = I^{A_{2}} and \\dot{y}{(I,A_{2})} = 2 I^{A_{2}} g{(I,A_{2})}, then obtain \\dot{y}{(I,A_{2})} = I^{2 A_{2}} + I^{A_{2}} g{(I,A_{2})}", "derivation": "g{(I,A_{2})} = I^{A_{2}} and I^{A_{2}} g{(I,A_{2})} = I^{2 A_{2}} and A_{2} + I^{A_{2}} g{(I,A_{2})} = A_{2} + I^{2 A_{2}} and A_{2} + 2 I^{A_{2}} g{(I,A_{2})} = A_{2} + I^{2 A_{2}} + I^{A_{2}} g{(I,A_{2})} and 2 I^{A_{2}} g{(I,A_{2})} = I^{2 A_{2}} + I^{A_{2}} g{(I,A_{2})} and \\dot{y}{(I,A_{2})} = 2 I^{A_{2}} g{(I,A_{2})} and \\dot{y}{(I,A_{2})} = I^{2 A_{2}} + I^{A_{2}} g{(I,A_{2})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))"], [["times", 1, "Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('A_2', commutative=True))))"], [["add", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))), Add(Symbol('A_2', commutative=True), Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('A_2', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))), Add(Symbol('A_2', commutative=True), Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('A_2', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))))"], [["minus", 4, "Symbol('A_2', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Add(Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('A_2', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\dot{y}')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Add(Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('A_2', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), Function('g')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(Z)} = \\log{(Z)} and \\mathbf{J}{(Z)} = \\log{(Z)} - 1, then obtain (\\frac{d}{d Z} (\\operatorname{c_{0}}{(Z)} - 1))^{Z} = (\\frac{d}{d Z} \\mathbf{J}{(Z)})^{Z}", "derivation": "\\operatorname{c_{0}}{(Z)} = \\log{(Z)} and \\operatorname{c_{0}}{(Z)} - 1 = \\log{(Z)} - 1 and \\mathbf{J}{(Z)} = \\log{(Z)} - 1 and \\frac{d}{d Z} (\\operatorname{c_{0}}{(Z)} - 1) = \\frac{d}{d Z} (\\log{(Z)} - 1) and \\frac{d}{d Z} (\\operatorname{c_{0}}{(Z)} - 1) = \\frac{d}{d Z} \\mathbf{J}{(Z)} and (\\frac{d}{d Z} (\\operatorname{c_{0}}{(Z)} - 1))^{Z} = (\\frac{d}{d Z} \\mathbf{J}{(Z)})^{Z}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('c_0')(Symbol('Z', commutative=True)), Integer(-1)), Add(log(Symbol('Z', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('Z', commutative=True)), Add(log(Symbol('Z', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('Z', commutative=True)), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(log(Symbol('Z', commutative=True)), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Function('c_0')(Symbol('Z', commutative=True)), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["power", 5, "Symbol('Z', commutative=True)"], "Equality(Pow(Derivative(Add(Function('c_0')(Symbol('Z', commutative=True)), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given W{(r,C_{d})} = \\frac{r}{C_{d}} and \\operatorname{F_{x}}{(r,C_{d})} = \\frac{r}{C_{d}}, then obtain 0 = \\frac{\\partial}{\\partial C_{d}} \\frac{r}{C_{d}} - \\frac{\\partial}{\\partial C_{d}} W{(r,C_{d})}", "derivation": "W{(r,C_{d})} = \\frac{r}{C_{d}} and \\operatorname{F_{x}}{(r,C_{d})} = \\frac{r}{C_{d}} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{F_{x}}{(r,C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\frac{r}{C_{d}} and C_{d} + \\frac{\\partial}{\\partial C_{d}} \\operatorname{F_{x}}{(r,C_{d})} = C_{d} + \\frac{\\partial}{\\partial C_{d}} \\frac{r}{C_{d}} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{F_{x}}{(r,C_{d})} = \\frac{\\partial}{\\partial C_{d}} W{(r,C_{d})} and 0 = \\frac{\\partial}{\\partial C_{d}} \\frac{r}{C_{d}} - \\frac{\\partial}{\\partial C_{d}} \\operatorname{F_{x}}{(r,C_{d})} and 0 = \\frac{\\partial}{\\partial C_{d}} \\frac{r}{C_{d}} - \\frac{\\partial}{\\partial C_{d}} W{(r,C_{d})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 3, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Derivative(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Symbol('C_d', commutative=True), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Function('W')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 4, "Add(Symbol('C_d', commutative=True), Derivative(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('F_x')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integer(0), Add(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('W')(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))))"]]}, {"prompt": "Given u{(I)} = \\int \\sin{(I)} dI, then obtain - \\mathbf{r} + \\cos{(I)} + \\frac{1}{I} = - \\mathbf{r} + \\cos{(I)} - \\frac{u{(I)}}{I} + \\frac{\\int \\sin{(I)} dI}{I} + \\frac{1}{I}", "derivation": "u{(I)} = \\int \\sin{(I)} dI and \\frac{u{(I)}}{I} = \\frac{\\int \\sin{(I)} dI}{I} and \\frac{u{(I)}}{I} + \\frac{1}{I} = \\frac{\\int \\sin{(I)} dI}{I} + \\frac{1}{I} and \\frac{1}{I} = - \\frac{u{(I)}}{I} + \\frac{\\int \\sin{(I)} dI}{I} + \\frac{1}{I} and - \\mathbf{r} + \\cos{(I)} + \\frac{1}{I} = - \\mathbf{r} + \\cos{(I)} - \\frac{u{(I)}}{I} + \\frac{\\int \\sin{(I)} dI}{I} + \\frac{1}{I}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('I', commutative=True)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('u')(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["add", 2, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('u')(Symbol('I', commutative=True))), Pow(Symbol('I', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('u')(Symbol('I', commutative=True)))"], "Equality(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('u')(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["minus", 4, "Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('u')(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Pow(Symbol('I', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(G,\\mathbf{r})} = \\frac{\\partial}{\\partial G} (- G + \\mathbf{r}), then derive \\operatorname{v_{y}}{(G,\\mathbf{r})} = -1, then obtain \\frac{d^{2}}{d \\mathbf{r}^{2}} (-1) = \\frac{\\partial^{3}}{\\partial \\mathbf{r}^{2}\\partial G} (- G + \\mathbf{r})", "derivation": "\\operatorname{v_{y}}{(G,\\mathbf{r})} = \\frac{\\partial}{\\partial G} (- G + \\mathbf{r}) and \\operatorname{v_{y}}{(G,\\mathbf{r})} = -1 and -1 = \\frac{\\partial}{\\partial G} (- G + \\mathbf{r}) and \\frac{d}{d \\mathbf{r}} (-1) = \\frac{\\partial^{2}}{\\partial \\mathbf{r}\\partial G} (- G + \\mathbf{r}) and \\frac{d^{2}}{d \\mathbf{r}^{2}} (-1) = \\frac{\\partial^{3}}{\\partial \\mathbf{r}^{2}\\partial G} (- G + \\mathbf{r})", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_y')(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(f_{E})} = e^{\\sin{(f_{E})}}, then derive \\frac{d}{d f_{E}} \\operatorname{v_{x}}{(f_{E})} = e^{\\sin{(f_{E})}} \\cos{(f_{E})}, then obtain (\\frac{d}{d f_{E}} e^{\\sin{(f_{E})}})^{f_{E}} = (e^{\\sin{(f_{E})}} \\cos{(f_{E})})^{f_{E}}", "derivation": "\\operatorname{v_{x}}{(f_{E})} = e^{\\sin{(f_{E})}} and \\frac{d}{d f_{E}} \\operatorname{v_{x}}{(f_{E})} = \\frac{d}{d f_{E}} e^{\\sin{(f_{E})}} and \\frac{d}{d f_{E}} \\operatorname{v_{x}}{(f_{E})} = e^{\\sin{(f_{E})}} \\cos{(f_{E})} and \\frac{d}{d f_{E}} e^{\\sin{(f_{E})}} = e^{\\sin{(f_{E})}} \\cos{(f_{E})} and (\\frac{d}{d f_{E}} e^{\\sin{(f_{E})}})^{f_{E}} = (e^{\\sin{(f_{E})}} \\cos{(f_{E})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('f_E', commutative=True)), exp(sin(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(exp(sin(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(exp(sin(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True))))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(Derivative(exp(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('f_E', commutative=True)), Pow(Mul(exp(sin(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(E)} = \\cos{(E)}, then obtain (- \\mathbf{B}{(E)} \\cos{(E)} + \\mathbf{B}{(E)}) (- \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)}) = (- \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)})^{2}", "derivation": "\\mathbf{B}{(E)} = \\cos{(E)} and \\mathbf{B}{(E)} \\cos{(E)} = \\cos^{2}{(E)} and \\mathbf{B}{(E)} - \\cos^{2}{(E)} = - \\cos^{2}{(E)} + \\cos{(E)} and - \\mathbf{B}{(E)} \\cos{(E)} + \\mathbf{B}{(E)} = - \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)} and (- \\mathbf{B}{(E)} \\cos{(E)} + \\mathbf{B}{(E)}) (- \\cos^{2}{(E)} + \\cos{(E)}) = (- \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)}) (- \\cos^{2}{(E)} + \\cos{(E)}) and (- \\mathbf{B}{(E)} \\cos{(E)} + \\mathbf{B}{(E)}) (- \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)}) = (- \\mathbf{B}{(E)} \\cos{(E)} + \\cos{(E)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["times", 1, "cos(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Pow(cos(Symbol('E', commutative=True)), Integer(2)))"], [["minus", 1, "Pow(cos(Symbol('E', commutative=True)), Integer(2))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('E', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(cos(Symbol('E', commutative=True)), Integer(2))), cos(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Function('\\\\mathbf{B}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Pow(cos(Symbol('E', commutative=True)), Integer(2))), cos(Symbol('E', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Function('\\\\mathbf{B}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Pow(cos(Symbol('E', commutative=True)), Integer(2))), cos(Symbol('E', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Pow(cos(Symbol('E', commutative=True)), Integer(2))), cos(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Function('\\\\mathbf{B}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True)))), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(J)} = \\cos{(J)}, then derive \\int \\operatorname{A_{2}}{(J)} dJ = \\Psi_{nl} + \\sin{(J)}, then obtain (\\int \\cos{(J)} dJ) (\\int \\cos{(J)} dJ)^{J} = (\\int \\operatorname{A_{2}}{(J)} dJ) (\\int \\cos{(J)} dJ)^{J}", "derivation": "\\operatorname{A_{2}}{(J)} = \\cos{(J)} and \\int \\operatorname{A_{2}}{(J)} dJ = \\int \\cos{(J)} dJ and \\int \\operatorname{A_{2}}{(J)} dJ = \\Psi_{nl} + \\sin{(J)} and (\\int \\operatorname{A_{2}}{(J)} dJ)^{J} = (\\int \\cos{(J)} dJ)^{J} and (\\int \\operatorname{A_{2}}{(J)} dJ) (\\int \\cos{(J)} dJ)^{J} = (\\Psi_{nl} + \\sin{(J)}) (\\int \\cos{(J)} dJ)^{J} and (\\int \\operatorname{A_{2}}{(J)} dJ) (\\int \\operatorname{A_{2}}{(J)} dJ)^{J} = (\\Psi_{nl} + \\sin{(J)}) (\\int \\operatorname{A_{2}}{(J)} dJ)^{J} and (\\int \\cos{(J)} dJ) (\\int \\cos{(J)} dJ)^{J} = (\\Psi_{nl} + \\sin{(J)}) (\\int \\cos{(J)} dJ)^{J} and (\\int \\cos{(J)} dJ) (\\int \\cos{(J)} dJ)^{J} = (\\int \\operatorname{A_{2}}{(J)} dJ) (\\int \\cos{(J)} dJ)^{J}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["times", 3, "Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))"], "Equality(Mul(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('J', commutative=True))), Pow(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Integral(Function('A_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} = V_{\\mathbf{E}} - \\hat{\\mathbf{r}}, then obtain - \\sin{(\\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} - \\mu^{V_{\\mathbf{E}}}{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})})} = \\sin{((V_{\\mathbf{E}} - \\hat{\\mathbf{r}})^{V_{\\mathbf{E}}} - \\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})})}", "derivation": "\\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} = V_{\\mathbf{E}} - \\hat{\\mathbf{r}} and \\mu^{V_{\\mathbf{E}}}{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} = (V_{\\mathbf{E}} - \\hat{\\mathbf{r}})^{V_{\\mathbf{E}}} and - \\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} + \\mu^{V_{\\mathbf{E}}}{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} = (V_{\\mathbf{E}} - \\hat{\\mathbf{r}})^{V_{\\mathbf{E}}} - \\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} and - \\sin{(\\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})} - \\mu^{V_{\\mathbf{E}}}{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})})} = \\sin{((V_{\\mathbf{E}} - \\hat{\\mathbf{r}})^{V_{\\mathbf{E}}} - \\mu{(\\hat{\\mathbf{r}},V_{\\mathbf{E}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["minus", 2, "Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Add(Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))), sin(Add(Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{F})} = \\log{(\\sin{(\\mathbf{F})})}, then obtain \\frac{d^{2}}{d \\mathbf{F}^{2}} \\phi_{1}{(\\mathbf{F})} = -1 - \\frac{\\cos^{2}{(\\mathbf{F})}}{\\sin^{2}{(\\mathbf{F})}}", "derivation": "\\phi_{1}{(\\mathbf{F})} = \\log{(\\sin{(\\mathbf{F})})} and \\frac{d}{d \\mathbf{F}} \\phi_{1}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\sin{(\\mathbf{F})})} and \\frac{d^{2}}{d \\mathbf{F}^{2}} \\phi_{1}{(\\mathbf{F})} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\sin{(\\mathbf{F})})} and \\frac{d^{2}}{d \\mathbf{F}^{2}} \\phi_{1}{(\\mathbf{F})} = -1 - \\frac{\\cos^{2}{(\\mathbf{F})}}{\\sin^{2}{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), log(sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Derivative(log(sin(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Mul(Integer(-1), Add(Integer(1), Mul(Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))))))"]]}, {"prompt": "Given u{(A_{2})} = \\log{(A_{2})}, then derive \\frac{d}{d A_{2}} u{(A_{2})} = \\frac{1}{A_{2}}, then obtain \\frac{1}{A_{2}} = \\frac{d}{d A_{2}} \\log{(A_{2})}", "derivation": "u{(A_{2})} = \\log{(A_{2})} and \\frac{d}{d A_{2}} u{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and \\frac{d}{d A_{2}} u{(A_{2})} = \\frac{1}{A_{2}} and \\frac{1}{A_{2}} = \\frac{d}{d A_{2}} \\log{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Pow(Symbol('A_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('A_2', commutative=True), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(T)} = \\sin{(\\log{(T)})}, then obtain \\frac{d}{d T} 2 (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + 1 = \\frac{d}{d T} ((- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + (- T + \\sin{(\\log{(T)})})^{2}) + 1", "derivation": "\\lambda{(T)} = \\sin{(\\log{(T)})} and - T + \\lambda{(T)} = - T + \\sin{(\\log{(T)})} and (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) = (- T + \\sin{(\\log{(T)})})^{2} and 2 (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) = (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + (- T + \\sin{(\\log{(T)})})^{2} and \\frac{d}{d T} 2 (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) = \\frac{d}{d T} ((- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + (- T + \\sin{(\\log{(T)})})^{2}) and \\frac{d}{d T} 2 (- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + 1 = \\frac{d}{d T} ((- T + \\lambda{(T)}) (- T + \\sin{(\\log{(T)})}) + (- T + \\sin{(\\log{(T)})})^{2}) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))), Integer(2)))"], [["add", 3, "Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Add(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))), Integer(2))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))), Integer(2))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Derivative(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\lambda')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(log(Symbol('T', commutative=True)))), Integer(2))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} = V_{\\mathbf{B}} - i, then obtain (V_{\\mathbf{B}} - i)^{2} \\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} - 1 = (V_{\\mathbf{B}} - i)^{3} - 1", "derivation": "\\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} = V_{\\mathbf{B}} - i and (V_{\\mathbf{B}} - i) \\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} = (V_{\\mathbf{B}} - i)^{2} and (V_{\\mathbf{B}} - i)^{2} \\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} = (V_{\\mathbf{B}} - i)^{3} and (V_{\\mathbf{B}} - i)^{2} \\operatorname{M_{E}}{(i,V_{\\mathbf{B}})} - 1 = (V_{\\mathbf{B}} - i)^{3} - 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('i', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["times", 1, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))"], "Equality(Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Function('M_E')(Symbol('i', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(2)))"], [["times", 2, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(2)), Function('M_E')(Symbol('i', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(3)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(2)), Function('M_E')(Symbol('i', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(-1)), Add(Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(3)), Integer(-1)))"]]}, {"prompt": "Given B{(C_{1},z)} = - z + e^{C_{1}}, then obtain - 2 e^{C_{1}} + \\frac{B{(C_{1},z)}}{z} = - 2 e^{C_{1}} + \\frac{- z + e^{C_{1}}}{z}", "derivation": "B{(C_{1},z)} = - z + e^{C_{1}} and C_{1} B{(C_{1},z)} = C_{1} (- z + e^{C_{1}}) and \\frac{C_{1} B{(C_{1},z)}}{z} = \\frac{C_{1} (- z + e^{C_{1}})}{z} and \\frac{B{(C_{1},z)}}{z} = \\frac{- z + e^{C_{1}}}{z} and - e^{C_{1}} + \\frac{B{(C_{1},z)}}{z} = - e^{C_{1}} + \\frac{- z + e^{C_{1}}}{z} and - 2 e^{C_{1}} + \\frac{B{(C_{1},z)}}{z} = - 2 e^{C_{1}} + \\frac{- z + e^{C_{1}}}{z}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True))))"], [["times", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True)))))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('C_1', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True)))))"], [["divide", 3, "Symbol('C_1', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True)))))"], [["minus", 4, "exp(Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('C_1', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('C_1', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True))))))"], [["add", 5, "Mul(Integer(-1), exp(Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), exp(Symbol('C_1', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('B')(Symbol('C_1', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Integer(2), exp(Symbol('C_1', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given B{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})}, then derive \\Psi^{\\dagger} + \\dot{y} = \\int \\frac{\\cos{(\\Psi^{\\dagger})}}{B{(\\Psi^{\\dagger})}} d\\Psi^{\\dagger}, then obtain \\int 1 d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\dot{y}", "derivation": "B{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})} and 1 = \\frac{\\cos{(\\Psi^{\\dagger})}}{B{(\\Psi^{\\dagger})}} and \\int 1 d\\Psi^{\\dagger} = \\int \\frac{\\cos{(\\Psi^{\\dagger})}}{B{(\\Psi^{\\dagger})}} d\\Psi^{\\dagger} and \\Psi^{\\dagger} + \\dot{y} = \\int \\frac{\\cos{(\\Psi^{\\dagger})}}{B{(\\Psi^{\\dagger})}} d\\Psi^{\\dagger} and \\int 1 d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\dot{y}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integral(Mul(Pow(Function('B')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\omega)} = e^{\\sin{(\\omega)}}, then obtain \\frac{\\ddot{x}^{\\omega}{(\\omega)}}{- \\omega + e^{\\sin{(\\omega)}}} = \\frac{(e^{\\sin{(\\omega)}})^{\\omega}}{- \\omega + e^{\\sin{(\\omega)}}}", "derivation": "\\ddot{x}{(\\omega)} = e^{\\sin{(\\omega)}} and - \\omega + \\ddot{x}{(\\omega)} = - \\omega + e^{\\sin{(\\omega)}} and \\ddot{x}^{\\omega}{(\\omega)} = (e^{\\sin{(\\omega)}})^{\\omega} and \\frac{\\ddot{x}^{\\omega}{(\\omega)}}{- \\omega + \\ddot{x}{(\\omega)}} = \\frac{(e^{\\sin{(\\omega)}})^{\\omega}}{- \\omega + \\ddot{x}{(\\omega)}} and \\frac{\\ddot{x}^{\\omega}{(\\omega)}}{- \\omega + e^{\\sin{(\\omega)}}} = \\frac{(e^{\\sin{(\\omega)}})^{\\omega}}{- \\omega + e^{\\sin{(\\omega)}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True))), Integer(-1)), Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True))), Integer(-1)), Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Integer(-1)), Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Integer(-1)), Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given b{(v)} = \\cos{(v)} and \\mathbf{B}{(v)} = \\cos{(v)}, then obtain \\frac{\\cos{(v)}}{\\mathbf{B}{(v)}} = 1", "derivation": "b{(v)} = \\cos{(v)} and \\frac{b{(v)}}{\\cos{(v)}} = 1 and \\mathbf{B}{(v)} = \\cos{(v)} and \\frac{b{(v)}}{\\mathbf{B}{(v)}} = 1 and \\frac{\\cos{(v)}}{\\mathbf{B}{(v)}} = 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["divide", 1, "cos(Symbol('v', commutative=True))"], "Equality(Mul(Function('b')(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Integer(-1)), Function('b')(Symbol('v', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} = (e^{\\sigma_p})^{\\mathbf{B}} and J{(\\sigma_p)} = \\sigma_p, then obtain J^{\\sigma_p}{(\\sigma_p)} - \\int \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} d\\mathbf{B} = \\sigma_p^{\\sigma_p} - \\int \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} d\\mathbf{B}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} = (e^{\\sigma_p})^{\\mathbf{B}} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} d\\mathbf{B} = \\int (e^{\\sigma_p})^{\\mathbf{B}} d\\mathbf{B} and J{(\\sigma_p)} = \\sigma_p and J^{\\sigma_p}{(\\sigma_p)} = \\sigma_p^{\\sigma_p} and J^{\\sigma_p}{(\\sigma_p)} - \\int (e^{\\sigma_p})^{\\mathbf{B}} d\\mathbf{B} = \\sigma_p^{\\sigma_p} - \\int (e^{\\sigma_p})^{\\mathbf{B}} d\\mathbf{B} and J^{\\sigma_p}{(\\sigma_p)} - \\int \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} d\\mathbf{B} = \\sigma_p^{\\sigma_p} - \\int \\operatorname{V_{\\mathbf{E}}}{(\\sigma_p,\\mathbf{B})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 4, "Integral(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Pow(Function('J')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integral(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integral(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('J')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(f^{\\prime},y)} = \\frac{y}{f^{\\prime}}, then obtain \\int (- 2 f^{\\prime} + 2 \\hat{\\mathbf{x}}{(f^{\\prime},y)}) df^{\\prime} = \\int (- 2 f^{\\prime} + \\frac{2 y}{f^{\\prime}}) df^{\\prime}", "derivation": "\\hat{\\mathbf{x}}{(f^{\\prime},y)} = \\frac{y}{f^{\\prime}} and - f^{\\prime} + \\hat{\\mathbf{x}}{(f^{\\prime},y)} = - f^{\\prime} + \\frac{y}{f^{\\prime}} and \\hat{\\mathbf{x}}{(f^{\\prime},y)} + \\frac{y}{f^{\\prime}} = \\frac{2 y}{f^{\\prime}} and - 2 f^{\\prime} + 2 \\hat{\\mathbf{x}}{(f^{\\prime},y)} = - 2 f^{\\prime} + \\hat{\\mathbf{x}}{(f^{\\prime},y)} + \\frac{y}{f^{\\prime}} and - 2 f^{\\prime} + 2 \\hat{\\mathbf{x}}{(f^{\\prime},y)} = - 2 f^{\\prime} + \\frac{2 y}{f^{\\prime}} and \\int (- 2 f^{\\prime} + 2 \\hat{\\mathbf{x}}{(f^{\\prime},y)}) df^{\\prime} = \\int (- 2 f^{\\prime} + \\frac{2 y}{f^{\\prime}}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Mul(Integer(2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["integrate", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\Psi{(C)} = \\cos{(\\log{(C)})} and \\operatorname{f^{*}}{(C)} = \\log{(C)}, then obtain \\int e^{\\Psi^{C}{(C)}} \\cos{(\\log{(C)})} dC = \\int e^{\\cos^{C}{(\\operatorname{f^{*}}{(C)})}} \\cos{(\\log{(C)})} dC", "derivation": "\\Psi{(C)} = \\cos{(\\log{(C)})} and \\Psi^{C}{(C)} = \\cos^{C}{(\\log{(C)})} and \\operatorname{f^{*}}{(C)} = \\log{(C)} and \\Psi^{C}{(C)} = \\cos^{C}{(\\operatorname{f^{*}}{(C)})} and e^{\\Psi^{C}{(C)}} = e^{\\cos^{C}{(\\operatorname{f^{*}}{(C)})}} and e^{\\Psi^{C}{(C)}} \\cos{(\\log{(C)})} = e^{\\cos^{C}{(\\operatorname{f^{*}}{(C)})}} \\cos{(\\log{(C)})} and \\int e^{\\Psi^{C}{(C)}} \\cos{(\\log{(C)})} dC = \\int e^{\\cos^{C}{(\\operatorname{f^{*}}{(C)})}} \\cos{(\\log{(C)})} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('C', commutative=True)), cos(log(Symbol('C', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(cos(log(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Psi')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(cos(Function('f^*')(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Function('\\\\Psi')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), exp(Pow(cos(Function('f^*')(Symbol('C', commutative=True))), Symbol('C', commutative=True))))"], [["times", 5, "cos(log(Symbol('C', commutative=True)))"], "Equality(Mul(exp(Pow(Function('\\\\Psi')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), cos(log(Symbol('C', commutative=True)))), Mul(exp(Pow(cos(Function('f^*')(Symbol('C', commutative=True))), Symbol('C', commutative=True))), cos(log(Symbol('C', commutative=True)))))"], [["integrate", 6, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(exp(Pow(Function('\\\\Psi')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), cos(log(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Mul(exp(Pow(cos(Function('f^*')(Symbol('C', commutative=True))), Symbol('C', commutative=True))), cos(log(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(r)} = \\cos{(\\sin{(r)})}, then obtain - \\sin{(r)} + \\cos{(\\sin{(r)})} = - \\Psi_{\\lambda}{(r)} - \\sin{(r)} + 2 \\cos{(\\sin{(r)})}", "derivation": "\\Psi_{\\lambda}{(r)} = \\cos{(\\sin{(r)})} and \\Psi_{\\lambda}{(r)} - \\sin{(r)} = - \\sin{(r)} + \\cos{(\\sin{(r)})} and - \\sin{(r)} = - \\Psi_{\\lambda}{(r)} - \\sin{(r)} + \\cos{(\\sin{(r)})} and - \\sin{(r)} + \\cos{(\\sin{(r)})} = - \\Psi_{\\lambda}{(r)} - \\sin{(r)} + 2 \\cos{(\\sin{(r)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('r', commutative=True)), cos(sin(Symbol('r', commutative=True))))"], [["minus", 1, "sin(Symbol('r', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('r', commutative=True)), Mul(Integer(-1), sin(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('r', commutative=True))), cos(sin(Symbol('r', commutative=True)))))"], [["minus", 2, "Function('\\\\Psi_{\\\\lambda}')(Symbol('r', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('r', commutative=True))), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True))), cos(sin(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('r', commutative=True))), cos(sin(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True))), Mul(Integer(2), cos(sin(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\chi{(\\pi)} = \\cos{(\\pi)} and G{(\\pi)} = \\cos{(\\pi)}, then obtain - \\frac{G{(\\pi)} - 2 \\chi{(\\pi)}}{G{(\\pi)} \\chi{(\\pi)}} = \\frac{1}{G{(\\pi)}}", "derivation": "\\chi{(\\pi)} = \\cos{(\\pi)} and G{(\\pi)} = \\cos{(\\pi)} and G{(\\pi)} - \\cos{(\\pi)} = 0 and G{(\\pi)} - \\chi{(\\pi)} - \\cos{(\\pi)} = - \\chi{(\\pi)} and G{(\\pi)} - 2 \\chi{(\\pi)} = - \\chi{(\\pi)} and - \\frac{G{(\\pi)} - 2 \\chi{(\\pi)}}{\\chi{(\\pi)}} = 1 and - \\frac{G{(\\pi)} - 2 \\chi{(\\pi)}}{G{(\\pi)} \\chi{(\\pi)}} = \\frac{1}{G{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('G')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Integer(0))"], [["minus", 3, "Function('\\\\chi')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('G')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('G')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\chi')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\pi', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('G')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\chi')(Symbol('\\\\pi', commutative=True)))), Pow(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 6, "Function('G')(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), Add(Function('G')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\chi')(Symbol('\\\\pi', commutative=True)))), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(-1)), Pow(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Integer(-1))), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given W{(n_{2})} = \\sin{(n_{2})}, then derive \\int W{(n_{2})} dn_{2} = g^{\\prime}_{\\varepsilon} - \\cos{(n_{2})}, then obtain (\\int W{(n_{2})} dn_{2})^{n_{2}} = (g^{\\prime}_{\\varepsilon} - \\cos{(n_{2})})^{n_{2}}", "derivation": "W{(n_{2})} = \\sin{(n_{2})} and \\int W{(n_{2})} dn_{2} = \\int \\sin{(n_{2})} dn_{2} and \\int W{(n_{2})} dn_{2} = g^{\\prime}_{\\varepsilon} - \\cos{(n_{2})} and (\\int W{(n_{2})} dn_{2})^{n_{2}} = (g^{\\prime}_{\\varepsilon} - \\cos{(n_{2})})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('W')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(E,\\psi)} = \\frac{\\partial}{\\partial E} (E + \\psi), then derive - E - \\psi + \\mathbf{F}{(E,\\psi)} = - E - \\psi + 1, then obtain \\frac{\\partial}{\\partial E} (- E - \\psi + 2 \\mathbf{F}{(E,\\psi)} - 1) = \\frac{\\partial}{\\partial E} (- E - \\psi + \\mathbf{F}{(E,\\psi)})", "derivation": "\\mathbf{F}{(E,\\psi)} = \\frac{\\partial}{\\partial E} (E + \\psi) and - E - \\psi + \\mathbf{F}{(E,\\psi)} = - E - \\psi + \\frac{\\partial}{\\partial E} (E + \\psi) and - E - \\psi + \\mathbf{F}{(E,\\psi)} = - E - \\psi + 1 and \\frac{\\partial}{\\partial E} (- E - \\psi + \\mathbf{F}{(E,\\psi)}) = \\frac{\\partial}{\\partial E} (- E - \\psi + 1) and \\frac{\\partial}{\\partial E} (- E - \\psi + 2 \\mathbf{F}{(E,\\psi)} - 1) = \\frac{\\partial}{\\partial E} (- E - \\psi + \\mathbf{F}{(E,\\psi)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Add(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 1, "Add(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Derivative(Add(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))), Integer(-1)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(n_{1},c)} = c^{n_{1}} and \\mathbf{p}{(n_{1},c)} = (c + c^{n_{1}})^{c}, then obtain 0 = - (c + c^{n_{1}})^{c} + \\mathbf{p}{(n_{1},c)}", "derivation": "\\hat{x}_0{(n_{1},c)} = c^{n_{1}} and c + \\hat{x}_0{(n_{1},c)} = c + c^{n_{1}} and (c + \\hat{x}_0{(n_{1},c)})^{c} = (c + c^{n_{1}})^{c} and 0 = (c + c^{n_{1}})^{c} - (c + \\hat{x}_0{(n_{1},c)})^{c} and \\mathbf{p}{(n_{1},c)} = (c + c^{n_{1}})^{c} and 0 = - (c + \\hat{x}_0{(n_{1},c)})^{c} + \\mathbf{p}{(n_{1},c)} and 0 = - (c + c^{n_{1}})^{c} + \\mathbf{p}{(n_{1},c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Add(Symbol('c', commutative=True), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Add(Symbol('c', commutative=True), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True))), Symbol('c', commutative=True)))"], [["minus", 3, "Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('c', commutative=True), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True))), Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Pow(Add(Symbol('c', commutative=True), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True))), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{x}_0')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))), Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Add(Symbol('c', commutative=True), Pow(Symbol('c', commutative=True), Symbol('n_1', commutative=True))), Symbol('c', commutative=True))), Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\varepsilon,\\mathbf{F})} = \\mathbf{F} \\varepsilon, then obtain \\frac{\\partial}{\\partial \\varepsilon} (\\mathbf{F} \\varepsilon \\int \\psi^{*}{(\\varepsilon,\\mathbf{F})} d\\mathbf{F} + \\mathbf{F} \\varepsilon) = \\frac{\\partial}{\\partial \\varepsilon} (\\mathbf{F} \\varepsilon \\int \\mathbf{F} \\varepsilon d\\mathbf{F} + \\mathbf{F} \\varepsilon)", "derivation": "\\psi^{*}{(\\varepsilon,\\mathbf{F})} = \\mathbf{F} \\varepsilon and \\int \\psi^{*}{(\\varepsilon,\\mathbf{F})} d\\mathbf{F} = \\int \\mathbf{F} \\varepsilon d\\mathbf{F} and \\mathbf{F} \\varepsilon \\int \\psi^{*}{(\\varepsilon,\\mathbf{F})} d\\mathbf{F} = \\mathbf{F} \\varepsilon \\int \\mathbf{F} \\varepsilon d\\mathbf{F} and \\mathbf{F} \\varepsilon \\int \\psi^{*}{(\\varepsilon,\\mathbf{F})} d\\mathbf{F} + \\mathbf{F} \\varepsilon = \\mathbf{F} \\varepsilon \\int \\mathbf{F} \\varepsilon d\\mathbf{F} + \\mathbf{F} \\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} (\\mathbf{F} \\varepsilon \\int \\psi^{*}{(\\varepsilon,\\mathbf{F})} d\\mathbf{F} + \\mathbf{F} \\varepsilon) = \\frac{\\partial}{\\partial \\varepsilon} (\\mathbf{F} \\varepsilon \\int \\mathbf{F} \\varepsilon d\\mathbf{F} + \\mathbf{F} \\varepsilon)", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 2, "Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["add", 3, "Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Function('\\\\psi^*')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(F_{g},\\Omega)} = F_{g}^{\\Omega}, then obtain F_{g} + F_{g}^{\\Omega} H{(F_{g},\\Omega)} + H{(F_{g},\\Omega)} = F_{g} + F_{g}^{\\Omega} H{(F_{g},\\Omega)} + F_{g}^{\\Omega}", "derivation": "H{(F_{g},\\Omega)} = F_{g}^{\\Omega} and F_{g}^{\\Omega} H{(F_{g},\\Omega)} = F_{g}^{2 \\Omega} and F_{g} + F_{g}^{2 \\Omega} + H{(F_{g},\\Omega)} = F_{g} + F_{g}^{2 \\Omega} + F_{g}^{\\Omega} and F_{g} + F_{g}^{\\Omega} H{(F_{g},\\Omega)} + H{(F_{g},\\Omega)} = F_{g} + F_{g}^{\\Omega} H{(F_{g},\\Omega)} + F_{g}^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Add(Symbol('F_g', commutative=True), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))))"], "Equality(Add(Symbol('F_g', commutative=True), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('F_g', commutative=True), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('H')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\hat{X},\\theta_2)} = \\hat{X} + \\theta_2 and c{(\\hat{X},\\theta_2)} = \\hat{X} + \\theta_2, then derive \\int \\bar{\\h}{(\\hat{X},\\theta_2)} d\\theta_2 = \\hat{X} \\theta_2 + \\mathbf{A} + \\frac{\\theta_2^{2}}{2}, then obtain \\int c{(\\hat{X},\\theta_2)} d\\theta_2 = \\hat{X} \\theta_2 + \\mathbf{A} + \\frac{\\theta_2^{2}}{2}", "derivation": "\\bar{\\h}{(\\hat{X},\\theta_2)} = \\hat{X} + \\theta_2 and c{(\\hat{X},\\theta_2)} = \\hat{X} + \\theta_2 and \\int \\bar{\\h}{(\\hat{X},\\theta_2)} d\\theta_2 = \\int (\\hat{X} + \\theta_2) d\\theta_2 and \\int \\bar{\\h}{(\\hat{X},\\theta_2)} d\\theta_2 = \\hat{X} \\theta_2 + \\mathbf{A} + \\frac{\\theta_2^{2}}{2} and c{(\\hat{X},\\theta_2)} = \\bar{\\h}{(\\hat{X},\\theta_2)} and \\int c{(\\hat{X},\\theta_2)} d\\theta_2 = \\hat{X} \\theta_2 + \\mathbf{A} + \\frac{\\theta_2^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('c')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hbar')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('c')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given y{(\\tilde{g},\\sigma_p)} = e^{\\frac{\\sigma_p}{\\tilde{g}}}, then obtain \\frac{\\partial}{\\partial \\tilde{g}} (- y{(\\tilde{g},\\sigma_p)} + y^{\\sigma_p}{(\\tilde{g},\\sigma_p)}) = \\frac{\\partial}{\\partial \\tilde{g}} (- y{(\\tilde{g},\\sigma_p)} + (e^{\\frac{\\sigma_p}{\\tilde{g}}})^{\\sigma_p})", "derivation": "y{(\\tilde{g},\\sigma_p)} = e^{\\frac{\\sigma_p}{\\tilde{g}}} and y^{\\sigma_p}{(\\tilde{g},\\sigma_p)} = (e^{\\frac{\\sigma_p}{\\tilde{g}}})^{\\sigma_p} and - y{(\\tilde{g},\\sigma_p)} + y^{\\sigma_p}{(\\tilde{g},\\sigma_p)} = - y{(\\tilde{g},\\sigma_p)} + (e^{\\frac{\\sigma_p}{\\tilde{g}}})^{\\sigma_p} and \\frac{\\partial}{\\partial \\tilde{g}} (- y{(\\tilde{g},\\sigma_p)} + y^{\\sigma_p}{(\\tilde{g},\\sigma_p)}) = \\frac{\\partial}{\\partial \\tilde{g}} (- y{(\\tilde{g},\\sigma_p)} + (e^{\\frac{\\sigma_p}{\\tilde{g}}})^{\\sigma_p})", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), exp(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 2, "Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(exp(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))), Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(exp(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(s,T)} = \\frac{T}{s} and n{(s,T)} = \\frac{2 T}{s}, then obtain - \\hat{\\mathbf{x}}{(s,T)} + n{(s,T)} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT = \\frac{T}{s} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT", "derivation": "\\hat{\\mathbf{x}}{(s,T)} = \\frac{T}{s} and \\frac{T}{s} + \\hat{\\mathbf{x}}{(s,T)} = \\frac{2 T}{s} and n{(s,T)} = \\frac{2 T}{s} and - \\hat{\\mathbf{x}}{(s,T)} + n{(s,T)} = \\frac{2 T}{s} - \\hat{\\mathbf{x}}{(s,T)} and - \\hat{\\mathbf{x}}{(s,T)} + n{(s,T)} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT = \\frac{2 T}{s} - \\hat{\\mathbf{x}}{(s,T)} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT and - \\hat{\\mathbf{x}}{(s,T)} + n{(s,T)} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT = \\frac{T}{s} + \\int 2 \\hat{\\mathbf{x}}{(s,T)} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('s', commutative=True), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["minus", 3, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Function('n')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Add(Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True)))))"], [["add", 4, "Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Function('n')(Symbol('s', commutative=True), Symbol('T', commutative=True)), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Function('n')(Symbol('s', commutative=True), Symbol('T', commutative=True)), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Symbol('T', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(E,T)} = \\cos{(E + T)}, then derive \\int \\hat{x}{(E,T)} dT = \\hat{p}_0 + \\sin{(E + T)}, then obtain V + \\sin{(E + T)} = \\hat{p}_0 + \\sin{(E + T)}", "derivation": "\\hat{x}{(E,T)} = \\cos{(E + T)} and \\int \\hat{x}{(E,T)} dT = \\int \\cos{(E + T)} dT and \\int \\hat{x}{(E,T)} dT = \\hat{p}_0 + \\sin{(E + T)} and \\int \\cos{(E + T)} dT = \\hat{p}_0 + \\sin{(E + T)} and V + \\sin{(E + T)} = \\hat{p}_0 + \\sin{(E + T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('E', commutative=True), Symbol('T', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(cos(Add(Symbol('E', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('E', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('V', commutative=True), sin(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(J,B)} = B^{J} and \\rho{(J,B)} = \\frac{\\partial}{\\partial B} (B B^{J} + \\frac{1}{B}), then derive \\rho{(J,B)} = B^{J} J + B^{J} - \\frac{1}{B^{2}}, then obtain - B B^{J} + \\rho{(J,B)} - \\frac{1}{B} = - B B^{J} + J \\lambda{(J,B)} + \\lambda{(J,B)} - \\frac{1}{B} - \\frac{1}{B^{2}}", "derivation": "\\lambda{(J,B)} = B^{J} and B \\lambda{(J,B)} = B B^{J} and \\rho{(J,B)} = \\frac{\\partial}{\\partial B} (B B^{J} + \\frac{1}{B}) and \\rho{(J,B)} = B^{J} J + B^{J} - \\frac{1}{B^{2}} and - B \\lambda{(J,B)} + \\rho{(J,B)} - \\frac{1}{B} = - B \\lambda{(J,B)} + B^{J} J + B^{J} - \\frac{1}{B} - \\frac{1}{B^{2}} and - B \\lambda{(J,B)} + \\rho{(J,B)} - \\frac{1}{B} = - B \\lambda{(J,B)} + J \\lambda{(J,B)} + \\lambda{(J,B)} - \\frac{1}{B} - \\frac{1}{B^{2}} and - B B^{J} + \\rho{(J,B)} - \\frac{1}{B} = - B B^{J} + J \\lambda{(J,B)} + \\lambda{(J,B)} - \\frac{1}{B} - \\frac{1}{B^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True)))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Derivative(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True))), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\rho')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Add(Mul(Pow(Symbol('B', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)))))"], [["minus", 4, "Add(Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Pow(Symbol('B', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Function('\\\\rho')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Function('\\\\rho')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('J', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True))), Function('\\\\rho')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given I{(\\psi^*,z)} = e^{z^{\\psi^*}} and c{(\\psi^*,z)} = z e^{z^{\\psi^*}}, then obtain g z I{(\\psi^*,z)} = g z e^{z^{\\psi^*}}", "derivation": "I{(\\psi^*,z)} = e^{z^{\\psi^*}} and c{(\\psi^*,z)} = z e^{z^{\\psi^*}} and c{(\\psi^*,z)} = z I{(\\psi^*,z)} and g c{(\\psi^*,z)} = g z e^{z^{\\psi^*}} and g z I{(\\psi^*,z)} = g z e^{z^{\\psi^*}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), exp(Pow(Symbol('z', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), exp(Pow(Symbol('z', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('c')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))))"], [["times", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('c')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('g', commutative=True), Symbol('z', commutative=True), exp(Pow(Symbol('z', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('g', commutative=True), Symbol('z', commutative=True), Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('g', commutative=True), Symbol('z', commutative=True), exp(Pow(Symbol('z', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{J}_f,G)} = \\frac{\\log{(G)}}{\\mathbf{J}_f}, then derive \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = E - \\frac{G \\log{(G)}}{\\mathbf{J}_f} + \\frac{G}{\\mathbf{J}_f}, then obtain G + \\hat{x}{(\\mathbf{J}_f,G)} + \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = E + G - \\frac{G \\log{(G)}}{\\mathbf{J}_f} + \\frac{G}{\\mathbf{J}_f} + \\hat{x}{(\\mathbf{J}_f,G)}", "derivation": "\\hat{x}{(\\mathbf{J}_f,G)} = \\frac{\\log{(G)}}{\\mathbf{J}_f} and - \\hat{x}{(\\mathbf{J}_f,G)} = - \\frac{\\log{(G)}}{\\mathbf{J}_f} and \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = \\int - \\frac{\\log{(G)}}{\\mathbf{J}_f} dG and \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = E - \\frac{G \\log{(G)}}{\\mathbf{J}_f} + \\frac{G}{\\mathbf{J}_f} and G + \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = E + G - \\frac{G \\log{(G)}}{\\mathbf{J}_f} + \\frac{G}{\\mathbf{J}_f} and G + \\hat{x}{(\\mathbf{J}_f,G)} + \\int - \\hat{x}{(\\mathbf{J}_f,G)} dG = E + G - \\frac{G \\log{(G)}}{\\mathbf{J}_f} + \\frac{G}{\\mathbf{J}_f} + \\hat{x}{(\\mathbf{J}_f,G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)))))"], [["add", 4, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Integral(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), Add(Symbol('E', commutative=True), Symbol('G', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)))))"], [["add", 5, "Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True)), Integral(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))), Add(Symbol('E', commutative=True), Symbol('G', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), log(Symbol('G', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Function('\\\\hat{x}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})} = \\mathbf{B} + \\phi_2 and \\chi{(\\phi_2,\\mathbf{B})} = 2 \\mathbf{B} + \\phi_2, then obtain \\frac{\\chi{(\\phi_2,\\mathbf{B})}}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}} = \\frac{\\mathbf{B} + \\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}}", "derivation": "\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})} = \\mathbf{B} + \\phi_2 and \\mathbf{B} + \\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})} = 2 \\mathbf{B} + \\phi_2 and \\chi{(\\phi_2,\\mathbf{B})} = 2 \\mathbf{B} + \\phi_2 and \\frac{\\chi{(\\phi_2,\\mathbf{B})}}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}} = \\frac{2 \\mathbf{B} + \\phi_2}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}} and \\frac{\\chi{(\\phi_2,\\mathbf{B})}}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}} = \\frac{\\mathbf{B} + \\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}}{\\operatorname{v_{z}}{(\\phi_2,\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 3, "Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Function('v_z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(y^{\\prime})} = \\cos{(y^{\\prime})}, then obtain \\log{(\\hat{\\mathbf{x}}^{2}{(y^{\\prime})})}^{2} = \\log{(\\hat{\\mathbf{x}}{(y^{\\prime})} \\cos{(y^{\\prime})})} \\log{(\\hat{\\mathbf{x}}^{2}{(y^{\\prime})})}", "derivation": "\\hat{\\mathbf{x}}{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\hat{\\mathbf{x}}^{2}{(y^{\\prime})} = \\hat{\\mathbf{x}}{(y^{\\prime})} \\cos{(y^{\\prime})} and \\log{(\\hat{\\mathbf{x}}^{2}{(y^{\\prime})})} = \\log{(\\hat{\\mathbf{x}}{(y^{\\prime})} \\cos{(y^{\\prime})})} and \\log{(\\hat{\\mathbf{x}}^{2}{(y^{\\prime})})}^{2} = \\log{(\\hat{\\mathbf{x}}{(y^{\\prime})} \\cos{(y^{\\prime})})} \\log{(\\hat{\\mathbf{x}}^{2}{(y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["log", 2], "Equality(log(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2))), log(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["times", 3, "log(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))"], "Equality(Pow(log(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2))), Integer(2)), Mul(log(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))), log(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\nabla{(\\nabla,\\phi_2)} = \\frac{e^{\\phi_2}}{\\nabla}, then derive \\frac{\\partial}{\\partial \\phi_2} \\nabla{(\\nabla,\\phi_2)} = \\frac{e^{\\phi_2}}{\\nabla}, then obtain (\\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\nabla{(\\nabla,\\phi_2)})^{2} = \\frac{\\partial}{\\partial \\phi_2} \\nabla{(\\nabla,\\phi_2)} \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\nabla{(\\nabla,\\phi_2)}", "derivation": "\\nabla{(\\nabla,\\phi_2)} = \\frac{e^{\\phi_2}}{\\nabla} and \\frac{\\partial}{\\partial \\phi_2} \\nabla{(\\nabla,\\phi_2)} = \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{\\nabla} and \\frac{\\partial}{\\partial \\phi_2} \\nabla{(\\nabla,\\phi_2)} = \\frac{e^{\\phi_2}}{\\nabla} and \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{\\nabla} = \\frac{e^{\\phi_2}}{\\nabla} and (\\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{\\nabla})^{2} = \\frac{e^{\\phi_2} \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{\\nabla}}{\\nabla} and (\\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\nabla{(\\nabla,\\phi_2)})^{2} = \\frac{\\partial}{\\partial \\phi_2} \\nabla{(\\nabla,\\phi_2)} \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\nabla{(\\nabla,\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["times", 4, "Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2))), Integer(2)), Mul(Derivative(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Function('\\\\nabla')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given t{(Q)} = \\log{(Q)} and B{(Q)} = - t{(Q)}, then obtain (\\int 0 dQ)^{Q} = (\\int Q (B{(Q)} + t{(Q)}) dQ)^{Q}", "derivation": "t{(Q)} = \\log{(Q)} and 0 = - t{(Q)} + \\log{(Q)} and 0 = Q (- t{(Q)} + \\log{(Q)}) and B{(Q)} = - t{(Q)} and 0 = Q (B{(Q)} + \\log{(Q)}) and 0 = Q (B{(Q)} + t{(Q)}) and \\int 0 dQ = \\int Q (B{(Q)} + t{(Q)}) dQ and (\\int 0 dQ)^{Q} = (\\int Q (B{(Q)} + t{(Q)}) dQ)^{Q}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["minus", 1, "Function('t')(Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t')(Symbol('Q', commutative=True))), log(Symbol('Q', commutative=True))))"], [["times", 2, "Symbol('Q', commutative=True)"], "Equality(Integer(0), Mul(Symbol('Q', commutative=True), Add(Mul(Integer(-1), Function('t')(Symbol('Q', commutative=True))), log(Symbol('Q', commutative=True)))))"], ["renaming_premise", "Equality(Function('B')(Symbol('Q', commutative=True)), Mul(Integer(-1), Function('t')(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Mul(Symbol('Q', commutative=True), Add(Function('B')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Mul(Symbol('Q', commutative=True), Add(Function('B')(Symbol('Q', commutative=True)), Function('t')(Symbol('Q', commutative=True)))))"], [["integrate", 6, "Symbol('Q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Add(Function('B')(Symbol('Q', commutative=True)), Function('t')(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["power", 7, "Symbol('Q', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Integral(Mul(Symbol('Q', commutative=True), Add(Function('B')(Symbol('Q', commutative=True)), Function('t')(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(I,f^{*})} = I^{f^{*}}, then obtain I^{3 f^{*}} \\mathbf{g}^{3}{(I,f^{*})} - \\mathbf{g}{(I,f^{*})} = I^{6 f^{*}} - \\mathbf{g}{(I,f^{*})}", "derivation": "\\mathbf{g}{(I,f^{*})} = I^{f^{*}} and I^{f^{*}} \\mathbf{g}{(I,f^{*})} = I^{2 f^{*}} and I^{f^{*}} \\mathbf{g}^{2}{(I,f^{*})} = I^{2 f^{*}} \\mathbf{g}{(I,f^{*})} and I^{2 f^{*}} \\mathbf{g}^{4}{(I,f^{*})} = I^{4 f^{*}} \\mathbf{g}^{2}{(I,f^{*})} and I^{2 f^{*}} \\mathbf{g}{(I,f^{*})} = I^{3 f^{*}} and I^{3 f^{*}} \\mathbf{g}^{3}{(I,f^{*})} = I^{6 f^{*}} and I^{3 f^{*}} \\mathbf{g}^{3}{(I,f^{*})} - \\mathbf{g}{(I,f^{*})} = I^{6 f^{*}} - \\mathbf{g}{(I,f^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('f^*', commutative=True)))"], [["times", 1, "Pow(Symbol('I', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True))), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Integer(4))), Mul(Pow(Symbol('I', commutative=True), Mul(Integer(4), Symbol('f^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('I', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True))), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('I', commutative=True), Mul(Integer(3), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Symbol('I', commutative=True), Mul(Integer(3), Symbol('f^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Integer(3))), Pow(Symbol('I', commutative=True), Mul(Integer(6), Symbol('f^*', commutative=True))))"], [["minus", 6, "Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('I', commutative=True), Mul(Integer(3), Symbol('f^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)), Integer(3))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)))), Add(Pow(Symbol('I', commutative=True), Mul(Integer(6), Symbol('f^*', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given g{(H)} = e^{H}, then obtain \\cos{(\\frac{g^{2}{(H)}}{\\int \\frac{\\operatorname{v_{1}}{(H)} e^{2 H}}{g{(H)}} dH})} = \\cos{(\\frac{g{(H)} e^{H}}{\\int \\frac{\\operatorname{v_{1}}{(H)} e^{2 H}}{g{(H)}} dH})}", "derivation": "g{(H)} = e^{H} and g^{2}{(H)} = g{(H)} e^{H} and g{(H)} \\operatorname{v_{1}}{(H)} = \\operatorname{v_{1}}{(H)} e^{H} and \\operatorname{v_{1}}{(H)} = \\frac{\\operatorname{v_{1}}{(H)} e^{H}}{g{(H)}} and \\frac{g^{2}{(H)}}{\\int \\operatorname{v_{1}}{(H)} e^{H} dH} = \\frac{g{(H)} e^{H}}{\\int \\operatorname{v_{1}}{(H)} e^{H} dH} and \\cos{(\\frac{g^{2}{(H)}}{\\int \\operatorname{v_{1}}{(H)} e^{H} dH})} = \\cos{(\\frac{g{(H)} e^{H}}{\\int \\operatorname{v_{1}}{(H)} e^{H} dH})} and \\cos{(\\frac{g^{2}{(H)}}{\\int \\frac{\\operatorname{v_{1}}{(H)} e^{2 H}}{g{(H)}} dH})} = \\cos{(\\frac{g{(H)} e^{H}}{\\int \\frac{\\operatorname{v_{1}}{(H)} e^{2 H}}{g{(H)}} dH})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["times", 1, "Function('g')(Symbol('H', commutative=True))"], "Equality(Pow(Function('g')(Symbol('H', commutative=True)), Integer(2)), Mul(Function('g')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))))"], [["times", 1, "Function('v_1')(Symbol('H', commutative=True))"], "Equality(Mul(Function('g')(Symbol('H', commutative=True)), Function('v_1')(Symbol('H', commutative=True))), Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))))"], [["divide", 3, "Function('g')(Symbol('H', commutative=True))"], "Equality(Function('v_1')(Symbol('H', commutative=True)), Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(-1)), Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))))"], [["divide", 2, "Integral(Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(2)), Pow(Integral(Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(Function('g')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Pow(Integral(Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["cos", 5], "Equality(cos(Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(2)), Pow(Integral(Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1)))), cos(Mul(Function('g')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Pow(Integral(Mul(Function('v_1')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(cos(Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(2)), Pow(Integral(Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(-1)), Function('v_1')(Symbol('H', commutative=True)), exp(Mul(Integer(2), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integer(-1)))), cos(Mul(Function('g')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Pow(Integral(Mul(Pow(Function('g')(Symbol('H', commutative=True)), Integer(-1)), Function('v_1')(Symbol('H', commutative=True)), exp(Mul(Integer(2), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} = \\frac{\\phi_2}{\\nabla r_{0}}, then obtain \\frac{\\partial}{\\partial r_{0}} \\frac{\\int \\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} d\\phi_2}{\\int \\frac{\\phi_2}{\\nabla r_{0}} d\\phi_2} = \\frac{d}{d r_{0}} 1", "derivation": "\\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} = \\frac{\\phi_2}{\\nabla r_{0}} and \\int \\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} d\\phi_2 = \\int \\frac{\\phi_2}{\\nabla r_{0}} d\\phi_2 and \\frac{\\int \\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} d\\phi_2}{\\int \\frac{\\phi_2}{\\nabla r_{0}} d\\phi_2} = 1 and \\frac{\\partial}{\\partial r_{0}} \\frac{\\int \\operatorname{m_{s}}{(\\phi_2,r_{0},\\nabla)} d\\phi_2}{\\int \\frac{\\phi_2}{\\nabla r_{0}} d\\phi_2} = \\frac{d}{d r_{0}} 1", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 2, "Integral(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Integral(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integer(1))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Integral(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(P_{g})} = \\log{(P_{g})}, then obtain \\cos{((- P_{g} + 2 \\mathbf{F}{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}})} = \\cos{((- P_{g} + \\mathbf{F}{(P_{g})} + \\log{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}})}", "derivation": "\\mathbf{F}{(P_{g})} = \\log{(P_{g})} and - P_{g} + \\mathbf{F}{(P_{g})} = - P_{g} + \\log{(P_{g})} and - P_{g} + 2 \\mathbf{F}{(P_{g})} = - P_{g} + \\mathbf{F}{(P_{g})} + \\log{(P_{g})} and (- P_{g} + 2 \\mathbf{F}{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}} = (- P_{g} + \\mathbf{F}{(P_{g})} + \\log{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}} and \\cos{((- P_{g} + 2 \\mathbf{F}{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}})} = \\cos{((- P_{g} + \\mathbf{F}{(P_{g})} + \\log{(P_{g})}) \\frac{d}{d P_{g}} \\log{(P_{g})}^{P_{g}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True)))"], [["minus", 1, "Symbol('P_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True))))"], [["times", 3, "Derivative(Pow(log(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)))), Derivative(Pow(log(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True))), Derivative(Pow(log(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["cos", 4], "Equality(cos(Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)))), Derivative(Pow(log(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))), cos(Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True))), Derivative(Pow(log(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\omega)} = \\sin{(\\omega)}, then obtain \\dot{\\mathbf{r}}{(\\omega)} + 4 \\sin^{2}{(\\omega)} = 4 \\sin^{2}{(\\omega)} + \\sin{(\\omega)}", "derivation": "\\dot{\\mathbf{r}}{(\\omega)} = \\sin{(\\omega)} and \\dot{\\mathbf{r}}{(\\omega)} + \\sin{(\\omega)} = 2 \\sin{(\\omega)} and 2 (\\dot{\\mathbf{r}}{(\\omega)} + \\sin{(\\omega)}) \\sin{(\\omega)} = 4 \\sin^{2}{(\\omega)} and 2 (\\dot{\\mathbf{r}}{(\\omega)} + \\sin{(\\omega)}) \\sin{(\\omega)} + \\dot{\\mathbf{r}}{(\\omega)} = 2 (\\dot{\\mathbf{r}}{(\\omega)} + \\sin{(\\omega)}) \\sin{(\\omega)} + \\sin{(\\omega)} and \\dot{\\mathbf{r}}{(\\omega)} + 4 \\sin^{2}{(\\omega)} = 4 \\sin^{2}{(\\omega)} + \\sin{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True))))"], [["times", 2, "Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(4), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2))))"], [["add", 1, "Mul(Integer(2), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), sin(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(2), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), sin(Symbol('\\\\omega', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(2), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), sin(Symbol('\\\\omega', commutative=True))), sin(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(4), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2)))), Add(Mul(Integer(4), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2))), sin(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given n{(f_{E})} = e^{\\cos{(f_{E})}}, then obtain \\frac{d}{d f_{E}} \\cos^{f_{E}}{(f_{E})} = \\frac{d}{d f_{E}} ((- n{(f_{E})} + e^{\\cos{(f_{E})}})^{2} + \\cos{(f_{E})})^{f_{E}}", "derivation": "n{(f_{E})} = e^{\\cos{(f_{E})}} and 0 = - n{(f_{E})} + e^{\\cos{(f_{E})}} and 0 = (- n{(f_{E})} + e^{\\cos{(f_{E})}})^{2} and \\cos{(f_{E})} = (- n{(f_{E})} + e^{\\cos{(f_{E})}})^{2} + \\cos{(f_{E})} and \\cos^{f_{E}}{(f_{E})} = ((- n{(f_{E})} + e^{\\cos{(f_{E})}})^{2} + \\cos{(f_{E})})^{f_{E}} and \\frac{d}{d f_{E}} \\cos^{f_{E}}{(f_{E})} = \\frac{d}{d f_{E}} ((- n{(f_{E})} + e^{\\cos{(f_{E})}})^{2} + \\cos{(f_{E})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f_E', commutative=True)), exp(cos(Symbol('f_E', commutative=True))))"], [["minus", 1, "Function('n')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True))))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True)))), Integer(2)))"], [["add", 3, "cos(Symbol('f_E', commutative=True))"], "Equality(cos(Symbol('f_E', commutative=True)), Add(Pow(Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True)))), Integer(2)), cos(Symbol('f_E', commutative=True))))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(cos(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(Add(Pow(Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True)))), Integer(2)), cos(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"], [["differentiate", 5, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Pow(cos(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Pow(Add(Pow(Add(Mul(Integer(-1), Function('n')(Symbol('f_E', commutative=True))), exp(cos(Symbol('f_E', commutative=True)))), Integer(2)), cos(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(a,g)} = \\log{(a)}^{g}, then obtain \\frac{a + 2 g + J{(a,g)} + \\log{(a)}}{\\sin{(g + J{(a,g)} + \\log{(a)})}} = \\frac{a + 2 g + \\log{(a)} + \\log{(a)}^{g}}{\\sin{(g + J{(a,g)} + \\log{(a)})}}", "derivation": "J{(a,g)} = \\log{(a)}^{g} and J{(a,g)} + \\log{(a)} = \\log{(a)} + \\log{(a)}^{g} and g + J{(a,g)} + \\log{(a)} = g + \\log{(a)} + \\log{(a)}^{g} and a + g + J{(a,g)} + \\log{(a)} = a + g + \\log{(a)} + \\log{(a)}^{g} and a + 2 g + J{(a,g)} + \\log{(a)} = a + 2 g + \\log{(a)} + \\log{(a)}^{g} and \\frac{a + 2 g + J{(a,g)} + \\log{(a)}}{\\sin{(g + \\log{(a)} + \\log{(a)}^{g})}} = \\frac{a + 2 g + \\log{(a)} + \\log{(a)}^{g}}{\\sin{(g + \\log{(a)} + \\log{(a)}^{g})}} and \\frac{a + 2 g + J{(a,g)} + \\log{(a)}}{\\sin{(g + J{(a,g)} + \\log{(a)})}} = \\frac{a + 2 g + \\log{(a)} + \\log{(a)}^{g}}{\\sin{(g + J{(a,g)} + \\log{(a)})}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True)))"], [["add", 1, "log(Symbol('a', commutative=True))"], "Equality(Add(Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Add(log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))))"], [["add", 2, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Add(Symbol('g', commutative=True), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))))"], [["add", 3, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Symbol('g', commutative=True), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), Symbol('g', commutative=True), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))))"], [["add", 4, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))))"], [["divide", 5, "sin(Add(Symbol('g', commutative=True), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))))"], "Equality(Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Pow(sin(Add(Symbol('g', commutative=True), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True)))), Integer(-1))), Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))), Pow(sin(Add(Symbol('g', commutative=True), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True))), Pow(sin(Add(Symbol('g', commutative=True), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True)))), Integer(-1))), Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('g', commutative=True))), Pow(sin(Add(Symbol('g', commutative=True), Function('J')(Symbol('a', commutative=True), Symbol('g', commutative=True)), log(Symbol('a', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{A}{(A,T)} = A \\log{(T)} and \\eta^{\\prime}{(A,T)} = (A \\log{(T)})^{A}, then obtain \\eta^{\\prime}^{2 T}{(A,T)} = ((A \\log{(T)})^{A})^{T} \\eta^{\\prime}^{T}{(A,T)}", "derivation": "\\mathbf{A}{(A,T)} = A \\log{(T)} and \\mathbf{A}^{A}{(A,T)} = (A \\log{(T)})^{A} and \\eta^{\\prime}{(A,T)} = (A \\log{(T)})^{A} and \\eta^{\\prime}{(A,T)} = \\mathbf{A}^{A}{(A,T)} and \\eta^{\\prime}^{T}{(A,T)} = (\\mathbf{A}^{A}{(A,T)})^{T} and \\eta^{\\prime}^{T}{(A,T)} = ((A \\log{(T)})^{A})^{T} and \\eta^{\\prime}^{2 T}{(A,T)} = ((A \\log{(T)})^{A})^{T} \\eta^{\\prime}^{T}{(A,T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('A', commutative=True), log(Symbol('T', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Symbol('A', commutative=True), log(Symbol('T', commutative=True))), Symbol('A', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Pow(Mul(Symbol('A', commutative=True), log(Symbol('T', commutative=True))), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('A', commutative=True)))"], [["power", 4, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('A', commutative=True)), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Mul(Symbol('A', commutative=True), log(Symbol('T', commutative=True))), Symbol('A', commutative=True)), Symbol('T', commutative=True)))"], [["times", 6, "Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True))), Mul(Pow(Pow(Mul(Symbol('A', commutative=True), log(Symbol('T', commutative=True))), Symbol('A', commutative=True)), Symbol('T', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta}, then derive \\mathbb{I}{(\\eta)} = e^{\\eta}, then obtain \\int e^{\\eta} d\\eta = \\int \\frac{d}{d \\eta} e^{\\eta} d\\eta", "derivation": "\\mathbb{I}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\mathbb{I}{(\\eta)} = e^{\\eta} and e^{\\eta} = \\frac{d}{d \\eta} e^{\\eta} and \\int e^{\\eta} d\\eta = \\int \\frac{d}{d \\eta} e^{\\eta} d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(C)} = \\sin{(C)}, then obtain \\operatorname{x^{{\\}'}}{(C)} \\operatorname{x^{{\\}'}}^{C}{(C)} + 2 \\sin^{C}{(C)} = \\operatorname{x^{{\\}'}}{(C)} \\sin^{C}{(C)} + 2 \\sin^{C}{(C)}", "derivation": "\\operatorname{x^{{\\}'}}{(C)} = \\sin{(C)} and \\operatorname{x^{{\\}'}}^{C}{(C)} = \\sin^{C}{(C)} and \\operatorname{x^{{\\}'}}{(C)} \\operatorname{x^{{\\}'}}^{C}{(C)} = \\operatorname{x^{{\\}'}}{(C)} \\sin^{C}{(C)} and \\operatorname{x^{{\\}'}}^{C}{(C)} + \\sin^{C}{(C)} = 2 \\sin^{C}{(C)} and \\operatorname{x^{{\\}'}}{(C)} \\operatorname{x^{{\\}'}}^{C}{(C)} + \\operatorname{x^{{\\}'}}^{C}{(C)} + \\sin^{C}{(C)} = \\operatorname{x^{{\\}'}}{(C)} \\sin^{C}{(C)} + \\operatorname{x^{{\\}'}}^{C}{(C)} + \\sin^{C}{(C)} and \\operatorname{x^{{\\}'}}{(C)} \\operatorname{x^{{\\}'}}^{C}{(C)} + 2 \\sin^{C}{(C)} = \\operatorname{x^{{\\}'}}{(C)} \\sin^{C}{(C)} + 2 \\sin^{C}{(C)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["times", 2, "Function('x^\\\\prime')(Symbol('C', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["add", 2, "Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))"], "Equality(Add(Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["add", 3, "Add(Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], "Equality(Add(Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Add(Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True)))), Add(Mul(Function('x^\\\\prime')(Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\Psi)} = \\cos{(\\Psi)}, then obtain \\iint \\operatorname{F_{g}}^{3}{(\\Psi)} \\cos{(\\Psi)} d\\Psi d\\Psi = \\iint \\operatorname{F_{g}}^{2}{(\\Psi)} \\cos^{2}{(\\Psi)} d\\Psi d\\Psi", "derivation": "\\operatorname{F_{g}}{(\\Psi)} = \\cos{(\\Psi)} and \\operatorname{F_{g}}{(\\Psi)} \\cos{(\\Psi)} = \\cos^{2}{(\\Psi)} and \\operatorname{F_{g}}^{2}{(\\Psi)} \\cos^{2}{(\\Psi)} = \\cos^{4}{(\\Psi)} and \\operatorname{F_{g}}^{3}{(\\Psi)} \\cos{(\\Psi)} = \\operatorname{F_{g}}^{2}{(\\Psi)} \\cos^{2}{(\\Psi)} and \\int \\operatorname{F_{g}}^{3}{(\\Psi)} \\cos{(\\Psi)} d\\Psi = \\int \\operatorname{F_{g}}^{2}{(\\Psi)} \\cos^{2}{(\\Psi)} d\\Psi and \\iint \\operatorname{F_{g}}^{3}{(\\Psi)} \\cos{(\\Psi)} d\\Psi d\\Psi = \\iint \\operatorname{F_{g}}^{2}{(\\Psi)} \\cos^{2}{(\\Psi)} d\\Psi d\\Psi", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(3)), cos(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(3)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(3)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Function('F_g')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given A{(C)} = \\log{(C)}, then derive \\frac{d}{d C} A{(C)} = \\frac{1}{C}, then obtain \\frac{d^{2}}{d C^{2}} \\frac{1}{C} = \\frac{d^{3}}{d C^{3}} \\log{(C)}", "derivation": "A{(C)} = \\log{(C)} and \\frac{d}{d C} A{(C)} = \\frac{d}{d C} \\log{(C)} and \\frac{d}{d C} A{(C)} = \\frac{1}{C} and \\frac{1}{C} = \\frac{d}{d C} \\log{(C)} and \\frac{d}{d C} \\frac{1}{C} = \\frac{d^{2}}{d C^{2}} \\log{(C)} and \\frac{d^{2}}{d C^{2}} \\frac{1}{C} = \\frac{d^{3}}{d C^{3}} \\log{(C)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Symbol('C', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Pow(Symbol('C', commutative=True), Integer(-1)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["differentiate", 5, "Symbol('C', commutative=True)"], "Equality(Derivative(Pow(Symbol('C', commutative=True), Integer(-1)), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\theta_{2}{(\\eta,y)} = \\frac{y}{\\eta}, then obtain \\tilde{\\infty} \\eta \\theta_{2}{(\\eta,y)} = \\tilde{\\infty} y", "derivation": "\\theta_{2}{(\\eta,y)} = \\frac{y}{\\eta} and \\eta \\theta_{2}{(\\eta,y)} = y and \\frac{\\eta \\theta_{2}{(\\eta,y)}}{\\theta_{2}{(\\eta,y)} - \\frac{y}{\\eta}} = \\frac{y}{\\theta_{2}{(\\eta,y)} - \\frac{y}{\\eta}} and \\tilde{\\infty} \\eta \\theta_{2}{(\\eta,y)} = \\tilde{\\infty} y", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\eta', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], [["divide", 2, "Add(Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Pow(Add(Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), Pow(Add(Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(zoo, Symbol('\\\\eta', commutative=True), Function('\\\\theta_2')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Mul(zoo, Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\rho)} = \\sin{(\\rho)}, then obtain \\sin{(\\rho)} + \\sin^{2}{(\\rho - \\Psi^{\\dagger}^{2}{(\\rho)})} = \\sin{(\\rho)} + \\sin^{2}{(\\rho - \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)})}", "derivation": "\\Psi^{\\dagger}{(\\rho)} = \\sin{(\\rho)} and \\Psi^{\\dagger}^{2}{(\\rho)} = \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)} and - \\rho + \\Psi^{\\dagger}^{2}{(\\rho)} = - \\rho + \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)} and - \\sin{(\\rho - \\Psi^{\\dagger}^{2}{(\\rho)})} = - \\sin{(\\rho - \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)})} and \\sin^{2}{(\\rho - \\Psi^{\\dagger}^{2}{(\\rho)})} = \\sin^{2}{(\\rho - \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)})} and \\sin{(\\rho)} + \\sin^{2}{(\\rho - \\Psi^{\\dagger}^{2}{(\\rho)})} = \\sin{(\\rho)} + \\sin^{2}{(\\rho - \\Psi^{\\dagger}{(\\rho)} \\sin{(\\rho)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True)))"], [["times", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True))"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), Integer(2)), Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True))))"], [["minus", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), Integer(2)))))), Mul(Integer(-1), sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True)))))))"], [["power", 4, 2], "Equality(Pow(sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), Integer(2))))), Integer(2)), Pow(sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True))))), Integer(2)))"], [["add", 5, "sin(Symbol('\\\\rho', commutative=True))"], "Equality(Add(sin(Symbol('\\\\rho', commutative=True)), Pow(sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), Integer(2))))), Integer(2))), Add(sin(Symbol('\\\\rho', commutative=True)), Pow(sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), sin(Symbol('\\\\rho', commutative=True))))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\lambda)} = \\sin{(\\lambda)}, then derive (\\phi - \\cos{(\\lambda)}) \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = (\\phi - \\cos{(\\lambda)})^{2}, then obtain (\\phi - \\cos{(\\lambda)}) \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = (\\phi - \\cos{(\\lambda)}) \\int \\sin{(\\lambda)} d\\lambda", "derivation": "\\operatorname{f^{*}}{(\\lambda)} = \\sin{(\\lambda)} and \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = \\int \\sin{(\\lambda)} d\\lambda and (\\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda) \\int \\sin{(\\lambda)} d\\lambda = (\\int \\sin{(\\lambda)} d\\lambda)^{2} and (\\phi - \\cos{(\\lambda)}) \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = (\\phi - \\cos{(\\lambda)})^{2} and (\\phi - \\cos{(\\lambda)}) \\int \\sin{(\\lambda)} d\\lambda = (\\phi - \\cos{(\\lambda)})^{2} and (\\phi - \\cos{(\\lambda)}) \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = (\\phi - \\cos{(\\lambda)}) \\int \\sin{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Pow(Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Pow(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Pow(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given f{(J,f^{\\prime})} = \\int J f^{\\prime} dJ, then obtain \\frac{2 f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} - 1 + \\frac{1}{\\int J f^{\\prime} dJ} = 1 + \\frac{1}{\\int J f^{\\prime} dJ}", "derivation": "f{(J,f^{\\prime})} = \\int J f^{\\prime} dJ and \\frac{f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} = 1 and \\frac{f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} + \\frac{1}{\\int J f^{\\prime} dJ} = 1 + \\frac{1}{\\int J f^{\\prime} dJ} and \\frac{2 f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} = \\frac{f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} + 1 and \\frac{2 f{(J,f^{\\prime})}}{\\int J f^{\\prime} dJ} - 1 + \\frac{1}{\\int J f^{\\prime} dJ} = 1 + \\frac{1}{\\int J f^{\\prime} dJ}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["divide", 1, "Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Add(Integer(1), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"], [["add", 2, "Mul(Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(2), Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Add(Mul(Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(2), Function('f')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Integer(-1), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Add(Integer(1), Pow(Integral(Mul(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given m{(x^\\prime,P_{e})} = x^\\prime e^{P_{e}} and \\mathbf{J}_f{(P_{e})} = e^{P_{e}}, then obtain m{(x^\\prime,P_{e})} - e^{P_{e}} = x^\\prime e^{P_{e}} - e^{P_{e}}", "derivation": "m{(x^\\prime,P_{e})} = x^\\prime e^{P_{e}} and \\mathbf{J}_f{(P_{e})} = e^{P_{e}} and m{(x^\\prime,P_{e})} = x^\\prime \\mathbf{J}_f{(P_{e})} and x^\\prime e^{P_{e}} = x^\\prime \\mathbf{J}_f{(P_{e})} and m{(x^\\prime,P_{e})} - e^{P_{e}} = x^\\prime \\mathbf{J}_f{(P_{e})} - e^{P_{e}} and m{(x^\\prime,P_{e})} - e^{P_{e}} = x^\\prime e^{P_{e}} - e^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('x^\\\\prime', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), exp(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('m')(Symbol('x^\\\\prime', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), exp(Symbol('P_e', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('P_e', commutative=True))))"], [["minus", 3, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Function('m')(Symbol('x^\\\\prime', commutative=True), Symbol('P_e', commutative=True)), Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('P_e', commutative=True))), Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('m')(Symbol('x^\\\\prime', commutative=True), Symbol('P_e', commutative=True)), Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))), Add(Mul(Symbol('x^\\\\prime', commutative=True), exp(Symbol('P_e', commutative=True))), Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\omega{(y^{\\prime})} = y^{\\prime} e^{y^{\\prime}}, then obtain \\int \\log{(1 - y^{\\prime})} dy^{\\prime} = \\int \\log{(- y^{\\prime} + (\\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}})^{y^{\\prime}})} dy^{\\prime}", "derivation": "\\omega{(y^{\\prime})} = y^{\\prime} e^{y^{\\prime}} and 1 = \\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}} and 1 = (\\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}})^{y^{\\prime}} and 1 - y^{\\prime} = - y^{\\prime} + (\\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}})^{y^{\\prime}} and \\log{(1 - y^{\\prime})} = \\log{(- y^{\\prime} + (\\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}})^{y^{\\prime}})} and \\int \\log{(1 - y^{\\prime})} dy^{\\prime} = \\int \\log{(- y^{\\prime} + (\\frac{y^{\\prime} e^{y^{\\prime}}}{\\omega{(y^{\\prime})}})^{y^{\\prime}})} dy^{\\prime}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))))"], [["log", 4], "Equality(log(Add(Integer(1), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))))"], [["integrate", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(log(Add(Integer(1), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('\\\\omega')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given k{(P_{e},g^{\\prime}_{\\varepsilon})} = \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})} and \\dot{x}{(P_{e},g^{\\prime}_{\\varepsilon})} = k^{P_{e}}{(P_{e},g^{\\prime}_{\\varepsilon})} + \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}}, then obtain \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\dot{x}{(P_{e},g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} 2 \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}}", "derivation": "k{(P_{e},g^{\\prime}_{\\varepsilon})} = \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})} and k^{P_{e}}{(P_{e},g^{\\prime}_{\\varepsilon})} = \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}} and \\dot{x}{(P_{e},g^{\\prime}_{\\varepsilon})} = k^{P_{e}}{(P_{e},g^{\\prime}_{\\varepsilon})} + \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}} and \\dot{x}{(P_{e},g^{\\prime}_{\\varepsilon})} = 2 \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}} and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\dot{x}{(P_{e},g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} 2 \\log{(P_{e}^{g^{\\prime}_{\\varepsilon}})}^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Pow(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('k')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('P_e', commutative=True)), Pow(log(Pow(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('P_e', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Pow(Function('k')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('P_e', commutative=True)), Pow(log(Pow(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Pow(log(Pow(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('P_e', commutative=True))))"], [["differentiate", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(log(Pow(Symbol('P_e', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('P_e', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{g})} = \\log{(\\log{(\\mathbf{g})})}, then obtain \\frac{\\operatorname{A_{y}}^{4}{(\\mathbf{g})}}{\\log{(\\log{(\\mathbf{g})})}} = \\operatorname{A_{y}}^{3}{(\\mathbf{g})}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{g})} = \\log{(\\log{(\\mathbf{g})})} and \\frac{\\operatorname{A_{y}}{(\\mathbf{g})}}{\\log{(\\log{(\\mathbf{g})})}} = 1 and \\frac{\\operatorname{A_{y}}^{2}{(\\mathbf{g})}}{\\log{(\\log{(\\mathbf{g})})}} = \\operatorname{A_{y}}{(\\mathbf{g})} and \\frac{\\operatorname{A_{y}}^{4}{(\\mathbf{g})}}{\\log{(\\log{(\\mathbf{g})})}} = \\operatorname{A_{y}}^{3}{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), log(log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 1, "log(log(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(log(log(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Pow(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Pow(log(log(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 3, "Pow(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(4)), Pow(log(log(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Pow(Function('A_y')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} = Z \\hat{H}, then derive \\int \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} dZ = \\frac{Z^{2}}{2} + q, then obtain 0 = \\frac{Z^{2}}{2} + q - \\int \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} dZ", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} = Z \\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} = \\frac{\\partial}{\\partial \\hat{H}} Z \\hat{H} and \\int \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} dZ = \\int \\frac{\\partial}{\\partial \\hat{H}} Z \\hat{H} dZ and \\int \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} dZ = \\frac{Z^{2}}{2} + q and 0 = \\frac{Z^{2}}{2} + q - \\int \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{V_{\\mathbf{B}}}{(Z,\\hat{H})} dZ", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('q', commutative=True)))"], [["minus", 4, "Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True)))"], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('Z', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))))"]]}, {"prompt": "Given V{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\chi)}, then derive V{(\\chi)} = - \\sin{(\\chi)}, then derive - V{(\\chi)} = \\sin{(\\chi)}, then derive V{(\\chi)} + \\sin^{2}{(\\chi)} = \\sin^{2}{(\\chi)} - \\sin{(\\chi)}, then obtain V{(\\chi)} + (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} = (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} + \\frac{d}{d \\chi} \\cos{(\\chi)}", "derivation": "V{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\chi)} and V{(\\chi)} = - \\sin{(\\chi)} and - V{(\\chi)} = - \\frac{d}{d \\chi} \\cos{(\\chi)} and - V{(\\chi)} = \\sin{(\\chi)} and - \\frac{d}{d \\chi} \\cos{(\\chi)} = \\sin{(\\chi)} and V{(\\chi)} + (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} = - \\sin{(\\chi)} + (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} and V{(\\chi)} + \\sin^{2}{(\\chi)} = \\sin^{2}{(\\chi)} - \\sin{(\\chi)} and V{(\\chi)} + (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} = (\\frac{d}{d \\chi} \\cos{(\\chi)})^{2} + \\frac{d}{d \\chi} \\cos{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\chi', commutative=True)), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('V')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('V')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Function('V')(Symbol('\\\\chi', commutative=True))), sin(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), sin(Symbol('\\\\chi', commutative=True)))"], [["add", 2, "Pow(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))"], "Equality(Add(Function('V')(Symbol('\\\\chi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))), Add(Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))), Pow(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Add(Function('V')(Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(2))), Add(Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Function('V')(Symbol('\\\\chi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))), Add(Pow(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2)), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given n{(\\Psi_{\\lambda},\\mu_0)} = \\sin{(\\Psi_{\\lambda} - \\mu_0)}, then derive - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)} = - \\cos{(\\Psi_{\\lambda} - \\mu_0)}, then obtain (- \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)})^{\\Psi_{\\lambda}} = (- \\cos{(\\Psi_{\\lambda} - \\mu_0)})^{\\Psi_{\\lambda}}", "derivation": "n{(\\Psi_{\\lambda},\\mu_0)} = \\sin{(\\Psi_{\\lambda} - \\mu_0)} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\sin{(\\Psi_{\\lambda} - \\mu_0)} and - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)} = - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\sin{(\\Psi_{\\lambda} - \\mu_0)} and - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)} = - \\cos{(\\Psi_{\\lambda} - \\mu_0)} and (- \\frac{\\partial}{\\partial \\Psi_{\\lambda}} n{(\\Psi_{\\lambda},\\mu_0)})^{\\Psi_{\\lambda}} = (- \\cos{(\\Psi_{\\lambda} - \\mu_0)})^{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('n')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('n')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))))"], [["power", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Derivative(Function('n')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(A_{2},\\sigma_p)} = - A_{2} + \\cos{(\\sigma_p)} and \\mathbf{D}{(A_{2},\\sigma_p)} = - A_{2} + \\cos{(\\sigma_p)}, then obtain \\varepsilon^{\\sigma_p}{(A_{2},\\sigma_p)} = (- A_{2} + \\cos{(\\sigma_p)})^{\\sigma_p}", "derivation": "\\varepsilon{(A_{2},\\sigma_p)} = - A_{2} + \\cos{(\\sigma_p)} and \\mathbf{D}{(A_{2},\\sigma_p)} = - A_{2} + \\cos{(\\sigma_p)} and \\mathbf{D}^{\\sigma_p}{(A_{2},\\sigma_p)} = (- A_{2} + \\cos{(\\sigma_p)})^{\\sigma_p} and \\varepsilon{(A_{2},\\sigma_p)} = \\mathbf{D}{(A_{2},\\sigma_p)} and \\varepsilon^{\\sigma_p}{(A_{2},\\sigma_p)} = (- A_{2} + \\cos{(\\sigma_p)})^{\\sigma_p}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varepsilon')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\varepsilon')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} = \\log{(\\sigma_p + v_{x})}, then obtain \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} \\log{(\\sigma_p + v_{x})} = \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} \\log{(\\sigma_p + v_{x})}^{2}", "derivation": "\\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} = \\log{(\\sigma_p + v_{x})} and \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} \\log{(\\sigma_p + v_{x})} = \\log{(\\sigma_p + v_{x})}^{2} and \\frac{\\partial}{\\partial \\sigma_p} \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} \\log{(\\sigma_p + v_{x})} = \\frac{\\partial}{\\partial \\sigma_p} \\log{(\\sigma_p + v_{x})}^{2} and \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} \\log{(\\sigma_p + v_{x})} = \\hat{\\mathbf{x}}{(v_{x},\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} \\log{(\\sigma_p + v_{x})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True))))"], [["times", 1, "log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True)))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True)))), Pow(log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True))), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Pow(log(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('v_x', commutative=True))), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(\\hat{H})} = e^{\\hat{H}}, then obtain \\hat{H} C^{2}{(\\hat{H})} = \\hat{H} e^{2 \\hat{H}}", "derivation": "C{(\\hat{H})} = e^{\\hat{H}} and \\hat{H} C{(\\hat{H})} = \\hat{H} e^{\\hat{H}} and \\hat{H} C^{2}{(\\hat{H})} = \\hat{H} C{(\\hat{H})} e^{\\hat{H}} and \\hat{H} C{(\\hat{H})} e^{\\hat{H}} = \\hat{H} e^{2 \\hat{H}} and \\hat{H} C^{2}{(\\hat{H})} = \\hat{H} e^{2 \\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('C')(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\hat{H}', commutative=True), Function('C')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Function('C')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('C')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given l{(P_{g},z)} = P_{g} + z, then obtain ((\\int l{(P_{g},z)} dz)^{2})^{P_{g}} = ((\\int (P_{g} + z) dz) \\int l{(P_{g},z)} dz)^{P_{g}}", "derivation": "l{(P_{g},z)} = P_{g} + z and \\int l{(P_{g},z)} dz = \\int (P_{g} + z) dz and (\\int l{(P_{g},z)} dz)^{2} = (\\int (P_{g} + z) dz) \\int l{(P_{g},z)} dz and ((\\int l{(P_{g},z)} dz)^{2})^{P_{g}} = ((\\int (P_{g} + z) dz) \\int l{(P_{g},z)} dz)^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 2, "Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Pow(Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(2)), Mul(Integral(Add(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Pow(Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(2)), Symbol('P_g', commutative=True)), Pow(Mul(Integral(Add(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Function('l')(Symbol('P_g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\psi{(u,L)} = u^{L}, then obtain - u u^{2 L} + \\cos{(u u^{L} \\psi^{3}{(u,L)})} = - u u^{2 L} + \\cos{(u u^{2 L} \\psi^{2}{(u,L)})}", "derivation": "\\psi{(u,L)} = u^{L} and u \\psi{(u,L)} = u u^{L} and \\psi^{2}{(u,L)} = u^{L} \\psi{(u,L)} and u \\psi^{2}{(u,L)} = u u^{L} \\psi{(u,L)} and u u^{L} \\psi{(u,L)} = u u^{2 L} and u u^{2 L} \\psi^{2}{(u,L)} = u u^{3 L} \\psi{(u,L)} and \\cos{(u u^{2 L} \\psi^{2}{(u,L)})} = \\cos{(u u^{3 L} \\psi{(u,L)})} and \\cos{(u u^{L} \\psi^{3}{(u,L)})} = \\cos{(u u^{2 L} \\psi^{2}{(u,L)})} and - u u^{2 L} + \\cos{(u u^{L} \\psi^{3}{(u,L)})} = - u u^{2 L} + \\cos{(u u^{2 L} \\psi^{2}{(u,L)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True))))"], [["times", 1, "Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))"], "Equality(Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))))"], [["times", 3, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2))), Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True)))))"], [["times", 4, "Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True)))"], "Equality(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2))), Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(3), Symbol('L', commutative=True))), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True))))"], [["cos", 6], "Equality(cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2)))), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(3), Symbol('L', commutative=True))), Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(3)))), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2)))))"], [["minus", 8, "Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True)))), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Symbol('L', commutative=True)), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(3))))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True)))), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('u', commutative=True), Mul(Integer(2), Symbol('L', commutative=True))), Pow(Function('\\\\psi')(Symbol('u', commutative=True), Symbol('L', commutative=True)), Integer(2))))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(F_{g})} = \\sin{(F_{g})}, then derive \\sin{(F_{g})} \\int \\operatorname{t_{2}}{(F_{g})} dF_{g} = (J - \\cos{(F_{g})}) \\sin{(F_{g})}, then obtain \\log{(\\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g})} = \\log{((J - \\cos{(F_{g})}) \\sin{(F_{g})})}", "derivation": "\\operatorname{t_{2}}{(F_{g})} = \\sin{(F_{g})} and \\int \\operatorname{t_{2}}{(F_{g})} dF_{g} = \\int \\sin{(F_{g})} dF_{g} and \\sin{(F_{g})} \\int \\operatorname{t_{2}}{(F_{g})} dF_{g} = \\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g} and \\sin{(F_{g})} \\int \\operatorname{t_{2}}{(F_{g})} dF_{g} = (J - \\cos{(F_{g})}) \\sin{(F_{g})} and \\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g} = (J - \\cos{(F_{g})}) \\sin{(F_{g})} and \\log{(\\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g})} = \\log{((J - \\cos{(F_{g})}) \\sin{(F_{g})})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["times", 2, "sin(Symbol('F_g', commutative=True))"], "Equality(Mul(sin(Symbol('F_g', commutative=True)), Integral(Function('t_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(sin(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(sin(Symbol('F_g', commutative=True)), Integral(Function('t_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('F_g', commutative=True)))), sin(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(sin(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('F_g', commutative=True)))), sin(Symbol('F_g', commutative=True))))"], [["log", 5], "Equality(log(Mul(sin(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))), log(Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('F_g', commutative=True)))), sin(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(E)} = e^{e^{E}}, then derive \\int \\lambda{(E)} dE = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})}, then derive C_{d} + \\operatorname{Ei}{(e^{E})} + 1 = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})} + 1, then obtain C_{d} + \\operatorname{Ei}{(e^{E})} + 1 = \\int e^{e^{E}} dE + 1", "derivation": "\\lambda{(E)} = e^{e^{E}} and \\int \\lambda{(E)} dE = \\int e^{e^{E}} dE and \\int \\lambda{(E)} dE = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})} and \\int e^{e^{E}} dE = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})} and \\int e^{e^{E}} dE + 1 = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})} + 1 and C_{d} + \\operatorname{Ei}{(e^{E})} + 1 = f_{\\mathbf{p}} + \\operatorname{Ei}{(e^{E})} + 1 and C_{d} + \\operatorname{Ei}{(e^{E})} + 1 = \\int e^{e^{E}} dE + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('E', commutative=True)), exp(exp(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(exp(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Ei(exp(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Ei(exp(Symbol('E', commutative=True)))))"], [["add", 4, 1], "Equality(Add(Integral(exp(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integer(1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Ei(exp(Symbol('E', commutative=True))), Integer(1)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('C_d', commutative=True), Ei(exp(Symbol('E', commutative=True))), Integer(1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Ei(exp(Symbol('E', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('C_d', commutative=True), Ei(exp(Symbol('E', commutative=True))), Integer(1)), Add(Integral(exp(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\sigma_{x}{(V,\\mathbf{B})} = V \\mathbf{B} and \\mathbf{s}{(V,\\mathbf{B})} = \\int (V \\frac{\\partial}{\\partial V} V \\mathbf{B})^{V} d\\mathbf{B}, then derive \\frac{\\partial}{\\partial V} \\sigma_{x}{(V,\\mathbf{B})} = \\mathbf{B}, then obtain \\int \\sigma_{x}^{V}{(V,\\frac{\\partial}{\\partial V} V \\mathbf{B})} d\\mathbf{B} = \\mathbf{s}{(V,\\mathbf{B})}", "derivation": "\\sigma_{x}{(V,\\mathbf{B})} = V \\mathbf{B} and \\sigma_{x}^{V}{(V,\\mathbf{B})} = (V \\mathbf{B})^{V} and \\frac{\\partial}{\\partial V} \\sigma_{x}{(V,\\mathbf{B})} = \\frac{\\partial}{\\partial V} V \\mathbf{B} and \\frac{\\partial}{\\partial V} \\sigma_{x}{(V,\\mathbf{B})} = \\mathbf{B} and \\frac{\\partial}{\\partial V} V \\mathbf{B} = \\mathbf{B} and \\sigma_{x}^{V}{(V,\\frac{\\partial}{\\partial V} V \\mathbf{B})} = (V \\frac{\\partial}{\\partial V} V \\mathbf{B})^{V} and \\int \\sigma_{x}^{V}{(V,\\frac{\\partial}{\\partial V} V \\mathbf{B})} d\\mathbf{B} = \\int (V \\frac{\\partial}{\\partial V} V \\mathbf{B})^{V} d\\mathbf{B} and \\mathbf{s}{(V,\\mathbf{B})} = \\int (V \\frac{\\partial}{\\partial V} V \\mathbf{B})^{V} d\\mathbf{B} and \\int \\sigma_{x}^{V}{(V,\\frac{\\partial}{\\partial V} V \\mathbf{B})} d\\mathbf{B} = \\mathbf{s}{(V,\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('V', commutative=True)), Pow(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('\\\\sigma_x')(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)), Pow(Mul(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)))"], [["integrate", 6, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\sigma_x')(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Pow(Mul(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Pow(Mul(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Integral(Pow(Function('\\\\sigma_x')(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Symbol('V', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Function('\\\\mathbf{s}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(s)} = \\sin{(\\sin{(s)})}, then obtain \\cos{(\\dot{y}{(s)})} \\frac{d}{d s} \\dot{y}{(s)} = \\cos{(s)} \\cos{(\\sin{(s)})} \\cos{(\\sin{(\\sin{(s)})})}", "derivation": "\\dot{y}{(s)} = \\sin{(\\sin{(s)})} and \\sin{(\\dot{y}{(s)})} = \\sin{(\\sin{(\\sin{(s)})})} and \\frac{d}{d s} \\sin{(\\dot{y}{(s)})} = \\frac{d}{d s} \\sin{(\\sin{(\\sin{(s)})})} and \\cos{(\\dot{y}{(s)})} \\frac{d}{d s} \\dot{y}{(s)} = \\cos{(s)} \\cos{(\\sin{(s)})} \\cos{(\\sin{(\\sin{(s)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('s', commutative=True)), sin(sin(Symbol('s', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\dot{y}')(Symbol('s', commutative=True))), sin(sin(sin(Symbol('s', commutative=True)))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(sin(Function('\\\\dot{y}')(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(sin(sin(sin(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('\\\\dot{y}')(Symbol('s', commutative=True))), Derivative(Function('\\\\dot{y}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(cos(Symbol('s', commutative=True)), cos(sin(Symbol('s', commutative=True))), cos(sin(sin(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = E - \\mathbf{J}_M, then obtain (- \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M} + \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)}) \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = (E - 2 \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M}) \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)}", "derivation": "\\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = E - \\mathbf{J}_M and - \\mathbf{J}_M + \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = E - 2 \\mathbf{J}_M and - \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M} + \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = E - 2 \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M} and (- \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M} + \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)}) \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)} = (E - 2 \\mathbf{J}_M + (E - \\mathbf{J}_M)^{\\mathbf{J}_M}) \\operatorname{y^{\\prime}}{(E,\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 2, "Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 3, "Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Add(Symbol('E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))), Function('y^{\\\\prime}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given s{(M,A_{x})} = A_{x} M and \\operatorname{f_{\\mathbf{v}}}{(A_{x})} = A_{x}, then derive \\frac{\\partial^{2}}{\\partial M\\partial A_{x}} s{(M,A_{x})} = 1, then obtain \\frac{\\partial^{2}}{\\partial M\\partial \\operatorname{f_{\\mathbf{v}}}{(A_{x})}} M \\operatorname{f_{\\mathbf{v}}}{(A_{x})} = 1", "derivation": "s{(M,A_{x})} = A_{x} M and \\frac{\\partial}{\\partial A_{x}} s{(M,A_{x})} = \\frac{\\partial}{\\partial A_{x}} A_{x} M and \\operatorname{f_{\\mathbf{v}}}{(A_{x})} = A_{x} and \\frac{\\partial^{2}}{\\partial M\\partial A_{x}} s{(M,A_{x})} = \\frac{\\partial^{2}}{\\partial M\\partial A_{x}} A_{x} M and \\frac{\\partial^{2}}{\\partial M\\partial A_{x}} s{(M,A_{x})} = 1 and \\frac{\\partial^{2}}{\\partial M\\partial A_{x}} A_{x} M = 1 and \\frac{\\partial^{2}}{\\partial M\\partial \\operatorname{f_{\\mathbf{v}}}{(A_{x})}} M \\operatorname{f_{\\mathbf{v}}}{(A_{x})} = 1", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('s')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Symbol('A_x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Derivative(Mul(Symbol('M', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('A_x', commutative=True))), Tuple(Function('f_{\\\\mathbf{v}}')(Symbol('A_x', commutative=True)), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given y{(\\mathbf{s},\\theta_1)} = \\sin{(\\mathbf{s} \\theta_1)}, then obtain y{(\\mathbf{s},\\theta_1)} - \\int y{(\\mathbf{s},\\theta_1)} d\\mathbf{s} = \\sin{(\\mathbf{s} \\theta_1)} - \\int y{(\\mathbf{s},\\theta_1)} d\\mathbf{s}", "derivation": "y{(\\mathbf{s},\\theta_1)} = \\sin{(\\mathbf{s} \\theta_1)} and \\int y{(\\mathbf{s},\\theta_1)} d\\mathbf{s} = \\int \\sin{(\\mathbf{s} \\theta_1)} d\\mathbf{s} and y{(\\mathbf{s},\\theta_1)} - \\int \\sin{(\\mathbf{s} \\theta_1)} d\\mathbf{s} = \\sin{(\\mathbf{s} \\theta_1)} - \\int \\sin{(\\mathbf{s} \\theta_1)} d\\mathbf{s} and y{(\\mathbf{s},\\theta_1)} - \\int y{(\\mathbf{s},\\theta_1)} d\\mathbf{s} = \\sin{(\\mathbf{s} \\theta_1)} - \\int y{(\\mathbf{s},\\theta_1)} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Integral(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integral(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integral(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integral(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(sin(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integral(Function('y')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given \\pi{(B,A_{z},y)} = A_{z} + B + y, then obtain \\pi^{A_{z}}{(B,A_{z},y)} + \\sin{(\\pi^{A_{z}}{(B,A_{z},y)})} = \\pi^{A_{z}}{(B,A_{z},y)} + \\sin{((A_{z} + B + y)^{A_{z}})}", "derivation": "\\pi{(B,A_{z},y)} = A_{z} + B + y and \\pi^{A_{z}}{(B,A_{z},y)} = (A_{z} + B + y)^{A_{z}} and \\sin{(\\pi^{A_{z}}{(B,A_{z},y)})} = \\sin{((A_{z} + B + y)^{A_{z}})} and \\pi^{A_{z}}{(B,A_{z},y)} + \\sin{(\\pi^{A_{z}}{(B,A_{z},y)})} = \\pi^{A_{z}}{(B,A_{z},y)} + \\sin{((A_{z} + B + y)^{A_{z}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('B', commutative=True), Symbol('y', commutative=True)))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('B', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True))), sin(Pow(Add(Symbol('A_z', commutative=True), Symbol('B', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True))))"], [["add", 3, "Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True))"], "Equality(Add(Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)), sin(Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)))), Add(Pow(Function('\\\\pi')(Symbol('B', commutative=True), Symbol('A_z', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)), sin(Pow(Add(Symbol('A_z', commutative=True), Symbol('B', commutative=True), Symbol('y', commutative=True)), Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given r{(y^{\\prime})} = \\sin{(y^{\\prime})}, then obtain \\frac{d}{d y^{\\prime}} (r{(y^{\\prime})} + \\int (r{(y^{\\prime})} - \\sin{(y^{\\prime})}) dy^{\\prime}) = \\frac{d}{d y^{\\prime}} (r{(y^{\\prime})} + \\int 0 dy^{\\prime})", "derivation": "r{(y^{\\prime})} = \\sin{(y^{\\prime})} and r{(y^{\\prime})} - \\sin{(y^{\\prime})} = 0 and \\int (r{(y^{\\prime})} - \\sin{(y^{\\prime})}) dy^{\\prime} = \\int 0 dy^{\\prime} and \\sin{(y^{\\prime})} + \\int (r{(y^{\\prime})} - \\sin{(y^{\\prime})}) dy^{\\prime} = \\sin{(y^{\\prime})} + \\int 0 dy^{\\prime} and \\frac{d}{d y^{\\prime}} (\\sin{(y^{\\prime})} + \\int (r{(y^{\\prime})} - \\sin{(y^{\\prime})}) dy^{\\prime}) = \\frac{d}{d y^{\\prime}} (\\sin{(y^{\\prime})} + \\int 0 dy^{\\prime}) and \\frac{d}{d y^{\\prime}} (r{(y^{\\prime})} + \\int (r{(y^{\\prime})} - \\sin{(y^{\\prime})}) dy^{\\prime}) = \\frac{d}{d y^{\\prime}} (r{(y^{\\prime})} + \\int 0 dy^{\\prime})", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "sin(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 3, "sin(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integral(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["differentiate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integral(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Integral(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Function('r')(Symbol('y^{\\\\prime}', commutative=True)), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} and \\psi{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} - F_{x}, then obtain - F_{x} - \\psi{(E_{x},r_{0},F_{x})} + \\rho{(E_{x},r_{0},F_{x})} = 0", "derivation": "\\rho{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} and - F_{x} + \\rho{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} - F_{x} and - \\frac{E_{x} r_{0}}{F_{x}} + \\rho{(E_{x},r_{0},F_{x})} = 0 and \\psi{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} - F_{x} and F_{x} + \\psi{(E_{x},r_{0},F_{x})} = \\frac{E_{x} r_{0}}{F_{x}} and - F_{x} - \\psi{(E_{x},r_{0},F_{x})} + \\rho{(E_{x},r_{0},F_{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\rho')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True))))"], [["minus", 2, "Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)), Function('\\\\rho')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True))))"], [["add", 4, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\psi')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('E_x', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\psi')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True))), Function('\\\\rho')(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True), Symbol('F_x', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(Z,C_{1})} = \\log{(C_{1} Z)}, then derive (Z + \\frac{\\partial}{\\partial C_{1}} \\operatorname{a^{\\dagger}}{(Z,C_{1})})^{Z} = (Z + \\frac{1}{C_{1}})^{Z}, then obtain (Z + \\frac{\\partial}{\\partial C_{1}} \\log{(C_{1} Z)})^{Z} = (Z + \\frac{1}{C_{1}})^{Z}", "derivation": "\\operatorname{a^{\\dagger}}{(Z,C_{1})} = \\log{(C_{1} Z)} and C_{1} Z + \\operatorname{a^{\\dagger}}{(Z,C_{1})} = C_{1} Z + \\log{(C_{1} Z)} and \\frac{\\partial}{\\partial C_{1}} (C_{1} Z + \\operatorname{a^{\\dagger}}{(Z,C_{1})}) = \\frac{\\partial}{\\partial C_{1}} (C_{1} Z + \\log{(C_{1} Z)}) and (\\frac{\\partial}{\\partial C_{1}} (C_{1} Z + \\operatorname{a^{\\dagger}}{(Z,C_{1})}))^{Z} = (\\frac{\\partial}{\\partial C_{1}} (C_{1} Z + \\log{(C_{1} Z)}))^{Z} and (Z + \\frac{\\partial}{\\partial C_{1}} \\operatorname{a^{\\dagger}}{(Z,C_{1})})^{Z} = (Z + \\frac{1}{C_{1}})^{Z} and (Z + \\frac{\\partial}{\\partial C_{1}} \\log{(C_{1} Z)})^{Z} = (Z + \\frac{1}{C_{1}})^{Z}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True))))"], [["add", 1, "Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Derivative(Add(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Symbol('Z', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Symbol('Z', commutative=True)), Pow(Add(Symbol('Z', commutative=True), Pow(Symbol('C_1', commutative=True), Integer(-1))), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Symbol('Z', commutative=True), Derivative(log(Mul(Symbol('C_1', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Symbol('Z', commutative=True)), Pow(Add(Symbol('Z', commutative=True), Pow(Symbol('C_1', commutative=True), Integer(-1))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\delta{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then obtain \\delta{(a^{\\dagger})} \\cos^{2}{(a^{\\dagger})} = \\delta^{2}{(a^{\\dagger})} \\cos{(a^{\\dagger})}", "derivation": "\\delta{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\delta^{2}{(a^{\\dagger})} = \\delta{(a^{\\dagger})} \\cos{(a^{\\dagger})} and \\delta^{3}{(a^{\\dagger})} = \\delta^{2}{(a^{\\dagger})} \\cos{(a^{\\dagger})} and \\delta^{3}{(a^{\\dagger})} = \\delta{(a^{\\dagger})} \\cos^{2}{(a^{\\dagger})} and \\delta{(a^{\\dagger})} \\cos^{2}{(a^{\\dagger})} = \\delta^{2}{(a^{\\dagger})} \\cos{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 1, "Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), Mul(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 2, "Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(3)), Mul(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\rho_b)} = \\rho_b, then derive \\varphi + \\frac{\\operatorname{F_{c}}^{2}{(\\rho_b)}}{2} = \\int \\rho_b d\\operatorname{F_{c}}{(\\rho_b)}, then obtain \\varphi + \\frac{\\operatorname{F_{c}}^{2}{(\\rho_b)}}{2} = \\int \\operatorname{F_{c}}{(\\rho_b)} d\\operatorname{F_{c}}{(\\rho_b)}", "derivation": "\\operatorname{F_{c}}{(\\rho_b)} = \\rho_b and \\int \\operatorname{F_{c}}{(\\rho_b)} d\\rho_b = \\int \\rho_b d\\rho_b and \\int \\operatorname{F_{c}}{(\\rho_b)} d\\operatorname{F_{c}}{(\\rho_b)} = \\int \\rho_b d\\operatorname{F_{c}}{(\\rho_b)} and \\varphi + \\frac{\\operatorname{F_{c}}^{2}{(\\rho_b)}}{2} = \\int \\rho_b d\\operatorname{F_{c}}{(\\rho_b)} and \\varphi + \\frac{\\operatorname{F_{c}}^{2}{(\\rho_b)}}{2} = \\int \\operatorname{F_{c}}{(\\rho_b)} d\\operatorname{F_{c}}{(\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Symbol('\\\\rho_b', commutative=True), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Tuple(Function('F_c')(Symbol('\\\\rho_b', commutative=True)))), Integral(Symbol('\\\\rho_b', commutative=True), Tuple(Function('F_c')(Symbol('\\\\rho_b', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Rational(1, 2), Pow(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Integer(2)))), Integral(Symbol('\\\\rho_b', commutative=True), Tuple(Function('F_c')(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Rational(1, 2), Pow(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Integer(2)))), Integral(Function('F_c')(Symbol('\\\\rho_b', commutative=True)), Tuple(Function('F_c')(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given V{(Z,Q)} = \\frac{\\partial}{\\partial Z} Q Z, then derive V{(Z,Q)} = Q, then derive \\int (Q + \\int Q dZ) dQ = Q^{2} (\\frac{Z}{2} + \\frac{1}{2}) + S, then obtain \\int (Q + \\int Q dZ) dV{(Z,Q)} = S + (\\frac{Z}{2} + \\frac{1}{2}) V^{2}{(Z,Q)}", "derivation": "V{(Z,Q)} = \\frac{\\partial}{\\partial Z} Q Z and V{(Z,Q)} = Q and Q = \\frac{\\partial}{\\partial Z} Q Z and Q + \\int Q dZ = \\frac{\\partial}{\\partial Z} Q Z + \\int Q dZ and \\int (Q + \\int Q dZ) dQ = \\int (\\frac{\\partial}{\\partial Z} Q Z + \\int Q dZ) dQ and \\int (Q + \\int Q dZ) dQ = Q^{2} (\\frac{Z}{2} + \\frac{1}{2}) + S and \\int (Q + \\int Q dZ) dV{(Z,Q)} = S + (\\frac{Z}{2} + \\frac{1}{2}) V^{2}{(Z,Q)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('V')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Symbol('Q', commutative=True), Derivative(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 3, "Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))"], "Equality(Add(Symbol('Q', commutative=True), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))), Add(Derivative(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Derivative(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Symbol('Q', commutative=True), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Add(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Add(Mul(Rational(1, 2), Symbol('Z', commutative=True)), Rational(1, 2))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integral(Add(Symbol('Q', commutative=True), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Z', commutative=True)))), Tuple(Function('V')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)))), Add(Symbol('S', commutative=True), Mul(Add(Mul(Rational(1, 2), Symbol('Z', commutative=True)), Rational(1, 2)), Pow(Function('V')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given J{(F_{N})} = \\log{(F_{N})}, then derive 0 = - \\frac{d}{d F_{N}} J{(F_{N})} + \\frac{1}{F_{N}}, then obtain 0 = \\sin{(\\int (- \\frac{d}{d F_{N}} \\log{(F_{N})} + \\frac{1}{F_{N}}) dF_{N})}", "derivation": "J{(F_{N})} = \\log{(F_{N})} and 0 = - J{(F_{N})} + \\log{(F_{N})} and \\frac{d}{d F_{N}} 0 = \\frac{d}{d F_{N}} (- J{(F_{N})} + \\log{(F_{N})}) and 0 = - \\frac{d}{d F_{N}} J{(F_{N})} + \\frac{1}{F_{N}} and \\int 0 dF_{N} = \\int (- \\frac{d}{d F_{N}} J{(F_{N})} + \\frac{1}{F_{N}}) dF_{N} and 0 = \\sin{(\\int (- \\frac{d}{d F_{N}} J{(F_{N})} + \\frac{1}{F_{N}}) dF_{N})} and 0 = \\sin{(\\int (- \\frac{d}{d F_{N}} \\log{(F_{N})} + \\frac{1}{F_{N}}) dF_{N})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["minus", 1, "Function('J')(Symbol('F_N', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('J')(Symbol('F_N', commutative=True))), log(Symbol('F_N', commutative=True))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('J')(Symbol('F_N', commutative=True))), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('J')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Symbol('F_N', commutative=True), Integer(-1))))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Function('J')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Symbol('F_N', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True))))"], [["sin", 5], "Equality(Integer(0), sin(Integral(Add(Mul(Integer(-1), Derivative(Function('J')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Symbol('F_N', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(0), sin(Integral(Add(Mul(Integer(-1), Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Symbol('F_N', commutative=True), Integer(-1))), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(z)} = \\cos{(z)}, then obtain \\frac{1}{\\cos{(z)}} = \\frac{1}{\\operatorname{n_{2}}{(z)}}", "derivation": "\\operatorname{n_{2}}{(z)} = \\cos{(z)} and 1 = \\frac{\\cos{(z)}}{\\operatorname{n_{2}}{(z)}} and \\frac{\\operatorname{n_{2}}{(z)}}{\\cos{(z)}} = 1 and \\frac{1}{\\cos{(z)}} = \\frac{1}{\\operatorname{n_{2}}{(z)}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["divide", 1, "Function('n_2')(Symbol('z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('n_2')(Symbol('z', commutative=True)), Integer(-1)), cos(Symbol('z', commutative=True))))"], [["divide", 2, "Mul(Pow(Function('n_2')(Symbol('z', commutative=True)), Integer(-1)), cos(Symbol('z', commutative=True)))"], "Equality(Mul(Function('n_2')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 3, "Function('n_2')(Symbol('z', commutative=True))"], "Equality(Pow(cos(Symbol('z', commutative=True)), Integer(-1)), Pow(Function('n_2')(Symbol('z', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given I{(c_{0})} = \\log{(c_{0})}, then obtain - c_{0} + (- c_{0} + I{(c_{0})}) I{(c_{0})} + I{(c_{0})} = - c_{0} + (- c_{0} + I{(c_{0})}) \\log{(c_{0})} + I{(c_{0})}", "derivation": "I{(c_{0})} = \\log{(c_{0})} and - c_{0} + I{(c_{0})} = - c_{0} + \\log{(c_{0})} and (- c_{0} + \\log{(c_{0})}) I{(c_{0})} = (- c_{0} + \\log{(c_{0})}) \\log{(c_{0})} and - c_{0} + (- c_{0} + \\log{(c_{0})}) I{(c_{0})} + \\log{(c_{0})} = - c_{0} + (- c_{0} + \\log{(c_{0})}) \\log{(c_{0})} + \\log{(c_{0})} and - c_{0} + (- c_{0} + I{(c_{0})}) I{(c_{0})} + I{(c_{0})} = - c_{0} + (- c_{0} + I{(c_{0})}) \\log{(c_{0})} + I{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["minus", 1, "Symbol('c_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('I')(Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), Function('I')(Symbol('c_0', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), log(Symbol('c_0', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), Function('I')(Symbol('c_0', commutative=True))), log(Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), log(Symbol('c_0', commutative=True))), log(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('I')(Symbol('c_0', commutative=True))), Function('I')(Symbol('c_0', commutative=True))), Function('I')(Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('I')(Symbol('c_0', commutative=True))), log(Symbol('c_0', commutative=True))), Function('I')(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\pi{(v_{y},\\theta_2)} = - \\theta_2 + v_{y} and L{(v_{y},\\theta_2)} = - \\theta_2 + v_{y}, then obtain \\int \\frac{\\partial}{\\partial v_{y}} L{(v_{y},\\theta_2)} dv_{y} = \\int \\frac{\\partial}{\\partial v_{y}} (- \\theta_2 + v_{y}) dv_{y}", "derivation": "\\pi{(v_{y},\\theta_2)} = - \\theta_2 + v_{y} and L{(v_{y},\\theta_2)} = - \\theta_2 + v_{y} and - v_{y} + \\pi{(v_{y},\\theta_2)} = - \\theta_2 and L{(v_{y},\\theta_2)} = \\pi{(v_{y},\\theta_2)} and \\frac{\\partial}{\\partial v_{y}} L{(v_{y},\\theta_2)} = \\frac{\\partial}{\\partial v_{y}} \\pi{(v_{y},\\theta_2)} and \\frac{\\partial}{\\partial v_{y}} L{(v_{y},\\theta_2)} = \\frac{\\partial}{\\partial v_{y}} (- \\theta_2 + v_{y}) and \\int \\frac{\\partial}{\\partial v_{y}} L{(v_{y},\\theta_2)} dv_{y} = \\int \\frac{\\partial}{\\partial v_{y}} (- \\theta_2 + v_{y}) dv_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\pi')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('L')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\pi')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 4, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Function('\\\\pi')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Function('L')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('v_y', commutative=True)"], "Equality(Integral(Derivative(Function('L')(Symbol('v_y', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Tuple(Symbol('v_y', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(E,\\mathbf{B})} = E \\mathbf{B}, then obtain \\mathbf{B} = \\mathbf{B} ((\\int E \\mathbf{B} d\\mathbf{B})^{\\mathbf{B}}) (\\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B})^{- \\mathbf{B}}", "derivation": "\\mathbf{J}_M{(E,\\mathbf{B})} = E \\mathbf{B} and \\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B} = \\int E \\mathbf{B} d\\mathbf{B} and (\\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B})^{\\mathbf{B}} = (\\int E \\mathbf{B} d\\mathbf{B})^{\\mathbf{B}} and 1 = ((\\int E \\mathbf{B} d\\mathbf{B})^{\\mathbf{B}}) (\\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B})^{- \\mathbf{B}} and - \\mathbf{B} = - \\mathbf{B} ((\\int E \\mathbf{B} d\\mathbf{B})^{\\mathbf{B}}) (\\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B})^{- \\mathbf{B}} and \\mathbf{B} = \\mathbf{B} ((\\int E \\mathbf{B} d\\mathbf{B})^{\\mathbf{B}}) (\\int \\mathbf{J}_M{(E,\\mathbf{B})} d\\mathbf{B})^{- \\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 3, "Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Pow(Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 5, "Integer(-1)"], "Equality(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given M{(E_{n},m)} = E_{n} m, then obtain \\frac{\\partial}{\\partial E_{n}} E_{n} m \\int M^{m}{(E_{n},m)} dm = \\frac{\\partial}{\\partial E_{n}} E_{n} m \\int (E_{n} m)^{m} dm", "derivation": "M{(E_{n},m)} = E_{n} m and M^{m}{(E_{n},m)} = (E_{n} m)^{m} and \\int M^{m}{(E_{n},m)} dm = \\int (E_{n} m)^{m} dm and \\frac{\\partial}{\\partial E_{n}} E_{n} m \\int M^{m}{(E_{n},m)} dm = \\frac{\\partial}{\\partial E_{n}} E_{n} m \\int (E_{n} m)^{m} dm", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('M')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Function('M')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["times", 3, "Derivative(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Integral(Pow(Function('M')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Derivative(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Integral(Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\hat{X})}, then derive \\operatorname{A_{1}}^{\\hat{X}}{(\\hat{X})} = (- \\sin{(\\hat{X})})^{\\hat{X}}, then obtain (\\frac{d}{d \\hat{X}} \\cos{(\\hat{X})})^{\\hat{X}} = (- \\sin{(\\hat{X})})^{\\hat{X}}", "derivation": "\\operatorname{A_{1}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\hat{X})} and \\operatorname{A_{1}}^{\\hat{X}}{(\\hat{X})} = (\\frac{d}{d \\hat{X}} \\cos{(\\hat{X})})^{\\hat{X}} and \\operatorname{A_{1}}^{\\hat{X}}{(\\hat{X})} = (- \\sin{(\\hat{X})})^{\\hat{X}} and (\\frac{d}{d \\hat{X}} \\cos{(\\hat{X})})^{\\hat{X}} = (- \\sin{(\\hat{X})})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\hat{X}', commutative=True)), Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('A_1')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(t,\\hbar)} = \\int (\\hbar + t) d\\hbar, then derive - \\frac{\\hbar^{2}}{2} - \\hbar t - s + \\ddot{x}{(t,\\hbar)} = 0, then obtain (- \\frac{\\hbar^{2}}{2} - \\hbar t - s + \\int (\\hbar + t) d\\hbar)^{\\hbar} = 0^{\\hbar}", "derivation": "\\ddot{x}{(t,\\hbar)} = \\int (\\hbar + t) d\\hbar and \\ddot{x}{(t,\\hbar)} - \\int (\\hbar + t) d\\hbar = 0 and - \\frac{\\hbar^{2}}{2} - \\hbar t - s + \\ddot{x}{(t,\\hbar)} = 0 and (- \\frac{\\hbar^{2}}{2} - \\hbar t - s + \\ddot{x}{(t,\\hbar)})^{\\hbar} = 0^{\\hbar} and (- \\frac{\\hbar^{2}}{2} - \\hbar t - s + \\int (\\hbar + t) d\\hbar)^{\\hbar} = 0^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(0))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\phi,l)} = \\phi l, then obtain - \\operatorname{F_{N}}{(\\phi,l)} - e^{\\operatorname{F_{N}}{(\\phi,l)}} = - \\operatorname{F_{N}}{(\\phi,l)} - e^{\\phi l}", "derivation": "\\operatorname{F_{N}}{(\\phi,l)} = \\phi l and e^{\\operatorname{F_{N}}{(\\phi,l)}} = e^{\\phi l} and \\operatorname{F_{N}}{(\\phi,l)} + e^{\\operatorname{F_{N}}{(\\phi,l)}} = \\operatorname{F_{N}}{(\\phi,l)} + e^{\\phi l} and - \\operatorname{F_{N}}{(\\phi,l)} - e^{\\operatorname{F_{N}}{(\\phi,l)}} = - \\operatorname{F_{N}}{(\\phi,l)} - e^{\\phi l}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))"], [["exp", 1], "Equality(exp(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))), exp(Mul(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))))"], [["add", 2, "Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))), Add(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)), exp(Mul(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), exp(Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))))), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), exp(Mul(Symbol('\\\\phi', commutative=True), Symbol('l', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(Q)} = \\sin{(Q)}, then derive \\frac{d}{d Q} \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\frac{\\partial}{\\partial Q} (\\frac{Q}{2} + \\delta - \\frac{\\sin{(Q)} \\cos{(Q)}}{2}), then obtain \\frac{d}{d Q} \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\frac{\\sin^{2}{(Q)}}{2} - \\frac{\\cos^{2}{(Q)}}{2} + \\frac{1}{2}", "derivation": "\\mathbf{A}{(Q)} = \\sin{(Q)} and \\mathbf{A}{(Q)} \\sin{(Q)} = \\sin^{2}{(Q)} and \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\int \\sin^{2}{(Q)} dQ and \\frac{d}{d Q} \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\frac{d}{d Q} \\int \\sin^{2}{(Q)} dQ and \\frac{d}{d Q} \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\frac{\\partial}{\\partial Q} (\\frac{Q}{2} + \\delta - \\frac{\\sin{(Q)} \\cos{(Q)}}{2}) and \\frac{d}{d Q} \\int \\mathbf{A}{(Q)} \\sin{(Q)} dQ = \\frac{\\sin^{2}{(Q)}}{2} - \\frac{\\cos^{2}{(Q)}}{2} + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["times", 1, "sin(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Pow(sin(Symbol('Q', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Pow(sin(Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Symbol('Q', commutative=True)), Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Add(Mul(Rational(1, 2), Pow(sin(Symbol('Q', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(cos(Symbol('Q', commutative=True)), Integer(2))), Rational(1, 2)))"]]}, {"prompt": "Given \\hat{p}_0{(E_{n})} = \\cos{(E_{n})}, then obtain (- \\hat{p}_0{(E_{n})} + \\cos{(E_{n})})^{E_{n}} - 1 = 0", "derivation": "\\hat{p}_0{(E_{n})} = \\cos{(E_{n})} and 0 = - \\hat{p}_0{(E_{n})} + \\cos{(E_{n})} and 0^{E_{n}} = (- \\hat{p}_0{(E_{n})} + \\cos{(E_{n})})^{E_{n}} and 0^{E_{n}} - (- \\hat{p}_0{(E_{n})} + \\cos{(E_{n})})^{E_{n}} = 0 and (- \\hat{p}_0{(E_{n})} + \\cos{(E_{n})})^{E_{n}} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["minus", 1, "Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))), cos(Symbol('E_n', commutative=True))))"], [["power", 2, "Symbol('E_n', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E_n', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('E_n', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('E_n', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{A}{(\\phi,p)} = \\phi p, then obtain (\\frac{\\mathbf{A}{(\\phi,p)}}{\\phi p})^{\\phi} + (p + (p + \\mathbf{A}{(\\phi,p)})^{p} + \\mathbf{A}{(\\phi,p)})^{p} = (\\frac{\\mathbf{A}{(\\phi,p)}}{\\phi p})^{\\phi} + (p + (\\phi p + p)^{p} + \\mathbf{A}{(\\phi,p)})^{p}", "derivation": "\\mathbf{A}{(\\phi,p)} = \\phi p and p + \\mathbf{A}{(\\phi,p)} = \\phi p + p and (p + \\mathbf{A}{(\\phi,p)})^{p} = (\\phi p + p)^{p} and p + (p + \\mathbf{A}{(\\phi,p)})^{p} + \\mathbf{A}{(\\phi,p)} = p + (\\phi p + p)^{p} + \\mathbf{A}{(\\phi,p)} and (p + (p + \\mathbf{A}{(\\phi,p)})^{p} + \\mathbf{A}{(\\phi,p)})^{p} = (p + (\\phi p + p)^{p} + \\mathbf{A}{(\\phi,p)})^{p} and (\\frac{\\mathbf{A}{(\\phi,p)}}{\\phi p})^{\\phi} + (p + (p + \\mathbf{A}{(\\phi,p)})^{p} + \\mathbf{A}{(\\phi,p)})^{p} = (\\frac{\\mathbf{A}{(\\phi,p)}}{\\phi p})^{\\phi} + (p + (\\phi p + p)^{p} + \\mathbf{A}{(\\phi,p)})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)))"], [["add", 1, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["add", 3, "Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Symbol('p', commutative=True), Pow(Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Add(Symbol('p', commutative=True), Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Symbol('p', commutative=True), Pow(Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Symbol('p', commutative=True), Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["add", 5, "Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('p', commutative=True), Pow(Add(Symbol('p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('p', commutative=True), Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hbar)} = \\int \\log{(\\hbar)} d\\hbar and C{(\\hbar)} = \\hbar \\log{(\\hbar)}, then derive \\mathbf{J}_f{(\\hbar)} = \\hbar \\log{(\\hbar)} - \\hbar + c_{0}, then obtain \\frac{d}{d c_{0}} \\mathbf{J}_f^{\\hbar}{(\\hbar)} = \\frac{\\partial}{\\partial c_{0}} (- \\hbar + c_{0} + C{(\\hbar)})^{\\hbar}", "derivation": "\\mathbf{J}_f{(\\hbar)} = \\int \\log{(\\hbar)} d\\hbar and \\mathbf{J}_f{(\\hbar)} = \\hbar \\log{(\\hbar)} - \\hbar + c_{0} and \\mathbf{J}_f^{\\hbar}{(\\hbar)} = (\\hbar \\log{(\\hbar)} - \\hbar + c_{0})^{\\hbar} and \\frac{d}{d c_{0}} \\mathbf{J}_f^{\\hbar}{(\\hbar)} = \\frac{\\partial}{\\partial c_{0}} (\\hbar \\log{(\\hbar)} - \\hbar + c_{0})^{\\hbar} and C{(\\hbar)} = \\hbar \\log{(\\hbar)} and \\frac{d}{d c_{0}} \\mathbf{J}_f^{\\hbar}{(\\hbar)} = \\frac{\\partial}{\\partial c_{0}} (- \\hbar + c_{0} + C{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('c_0', commutative=True)))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('c_0', commutative=True), Function('C')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(r,\\rho_b)} = \\log{(\\rho_b + r)} and \\operatorname{v_{1}}{(L)} = \\cos{(L)}, then obtain \\frac{\\operatorname{v_{1}}{(L)}}{2 \\log{(\\rho_b + r)}} - \\int \\log{(\\rho_b + r)} dr = - \\int \\log{(\\rho_b + r)} dr + \\frac{\\cos{(L)}}{2 \\log{(\\rho_b + r)}}", "derivation": "\\operatorname{M_{E}}{(r,\\rho_b)} = \\log{(\\rho_b + r)} and \\operatorname{v_{1}}{(L)} = \\cos{(L)} and \\frac{\\operatorname{v_{1}}{(L)}}{2 \\operatorname{M_{E}}{(r,\\rho_b)}} = \\frac{\\cos{(L)}}{2 \\operatorname{M_{E}}{(r,\\rho_b)}} and - \\int \\operatorname{M_{E}}{(r,\\rho_b)} dr + \\frac{\\operatorname{v_{1}}{(L)}}{2 \\operatorname{M_{E}}{(r,\\rho_b)}} = - \\int \\operatorname{M_{E}}{(r,\\rho_b)} dr + \\frac{\\cos{(L)}}{2 \\operatorname{M_{E}}{(r,\\rho_b)}} and \\frac{\\operatorname{v_{1}}{(L)}}{2 \\log{(\\rho_b + r)}} - \\int \\log{(\\rho_b + r)} dr = - \\int \\log{(\\rho_b + r)} dr + \\frac{\\cos{(L)}}{2 \\log{(\\rho_b + r)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), log(Add(Symbol('\\\\rho_b', commutative=True), Symbol('r', commutative=True))))"], ["get_premise", "Equality(Function('v_1')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["divide", 2, "Mul(Integer(2), Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Function('v_1')(Symbol('L', commutative=True))), Mul(Rational(1, 2), Pow(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), cos(Symbol('L', commutative=True))))"], [["minus", 3, "Integral(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Rational(1, 2), Pow(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Function('v_1')(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Integral(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Rational(1, 2), Pow(Function('M_E')(Symbol('r', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), cos(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Function('v_1')(Symbol('L', commutative=True)), Pow(log(Add(Symbol('\\\\rho_b', commutative=True), Symbol('r', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(log(Add(Symbol('\\\\rho_b', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))), Add(Mul(Integer(-1), Integral(log(Add(Symbol('\\\\rho_b', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))), Mul(Rational(1, 2), Pow(log(Add(Symbol('\\\\rho_b', commutative=True), Symbol('r', commutative=True))), Integer(-1)), cos(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(k,\\sigma_p)} = \\int (\\sigma_p + k) dk, then obtain \\sigma_p + k + \\theta_{2}^{k}{(k,\\sigma_p)} - 1 = \\sigma_p + k + (\\int (\\sigma_p + k) dk)^{k} - 1", "derivation": "\\theta_{2}{(k,\\sigma_p)} = \\int (\\sigma_p + k) dk and \\theta_{2}^{k}{(k,\\sigma_p)} = (\\int (\\sigma_p + k) dk)^{k} and \\theta_{2}^{k}{(k,\\sigma_p)} - 1 = (\\int (\\sigma_p + k) dk)^{k} - 1 and \\sigma_p + k + \\theta_{2}^{k}{(k,\\sigma_p)} - 1 = \\sigma_p + k + (\\int (\\sigma_p + k) dk)^{k} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('k', commutative=True)), Pow(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["minus", 2, 1], "Equality(Add(Pow(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('k', commutative=True)), Integer(-1)), Add(Pow(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Integer(-1)))"], [["add", 3, "Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True), Pow(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('k', commutative=True)), Integer(-1)), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True), Pow(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given i{(P_{e},\\varphi)} = \\cos{(\\frac{\\varphi}{P_{e}})}, then obtain (\\int \\frac{\\partial}{\\partial P_{e}} i^{P_{e}}{(P_{e},\\varphi)} d\\varphi)^{P_{e}} = (\\int \\frac{\\partial}{\\partial P_{e}} \\cos^{P_{e}}{(\\frac{\\varphi}{P_{e}})} d\\varphi)^{P_{e}}", "derivation": "i{(P_{e},\\varphi)} = \\cos{(\\frac{\\varphi}{P_{e}})} and i^{P_{e}}{(P_{e},\\varphi)} = \\cos^{P_{e}}{(\\frac{\\varphi}{P_{e}})} and \\frac{\\partial}{\\partial P_{e}} i^{P_{e}}{(P_{e},\\varphi)} = \\frac{\\partial}{\\partial P_{e}} \\cos^{P_{e}}{(\\frac{\\varphi}{P_{e}})} and \\int \\frac{\\partial}{\\partial P_{e}} i^{P_{e}}{(P_{e},\\varphi)} d\\varphi = \\int \\frac{\\partial}{\\partial P_{e}} \\cos^{P_{e}}{(\\frac{\\varphi}{P_{e}})} d\\varphi and (\\int \\frac{\\partial}{\\partial P_{e}} i^{P_{e}}{(P_{e},\\varphi)} d\\varphi)^{P_{e}} = (\\int \\frac{\\partial}{\\partial P_{e}} \\cos^{P_{e}}{(\\frac{\\varphi}{P_{e}})} d\\varphi)^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('i')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('P_e', commutative=True)), Pow(cos(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Pow(Function('i')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Pow(cos(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('i')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Pow(cos(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 4, "Symbol('P_e', commutative=True)"], "Equality(Pow(Integral(Derivative(Pow(Function('i')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)), Pow(Integral(Derivative(Pow(cos(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given g{(\\tilde{g})} = e^{\\tilde{g}}, then obtain \\tilde{\\infty} e^{2 \\tilde{g}} = \\frac{\\tilde{\\infty} (g{(\\tilde{g})} + e^{\\tilde{g}}) e^{2 \\tilde{g}}}{g{(\\tilde{g})}}", "derivation": "g{(\\tilde{g})} = e^{\\tilde{g}} and 2 g{(\\tilde{g})} = g{(\\tilde{g})} + e^{\\tilde{g}} and \\tilde{\\infty} g{(\\tilde{g})} = \\tilde{\\infty} (g{(\\tilde{g})} + e^{\\tilde{g}}) and \\tilde{\\infty} e^{2 \\tilde{g}} = \\frac{\\tilde{\\infty} (g{(\\tilde{g})} + e^{\\tilde{g}}) e^{2 \\tilde{g}}}{g{(\\tilde{g})}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "Function('g')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('g')(Symbol('\\\\tilde{g}', commutative=True))), Add(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 2, 0], "Equality(Mul(zoo, Function('g')(Symbol('\\\\tilde{g}', commutative=True))), Mul(zoo, Add(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True))))"], "Equality(Mul(zoo, exp(Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)))), Mul(zoo, Add(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Pow(Function('g')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(M_{E})} = \\int \\cos{(M_{E})} dM_{E}, then derive \\mathbf{H}{(M_{E})} = \\mathbf{P} + \\sin{(M_{E})}, then obtain \\frac{\\mathbf{P} + \\sin{(M_{E})}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\cos{(M_{E})} dM_{E}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}}", "derivation": "\\mathbf{H}{(M_{E})} = \\int \\cos{(M_{E})} dM_{E} and \\int \\mathbf{H}{(M_{E})} dM_{E} = \\iint \\cos{(M_{E})} dM_{E} dM_{E} and \\mathbf{H}{(M_{E})} = \\mathbf{P} + \\sin{(M_{E})} and \\frac{\\mathbf{H}{(M_{E})}}{\\int \\mathbf{H}{(M_{E})} dM_{E}} = \\frac{\\int \\cos{(M_{E})} dM_{E}}{\\int \\mathbf{H}{(M_{E})} dM_{E}} and \\frac{\\mathbf{H}{(M_{E})}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\cos{(M_{E})} dM_{E}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}} and \\frac{\\mathbf{P} + \\sin{(M_{E})}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\cos{(M_{E})} dM_{E}}{\\iint \\cos{(M_{E})} dM_{E} dM_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('M_E', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Pow(Integral(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Pow(Integral(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1)), Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('M_E', commutative=True)), Pow(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('M_E', commutative=True))), Pow(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given G{(\\rho_b,n,\\eta^{\\prime})} = (\\eta^{\\prime} \\rho_b)^{n} and \\hat{H}_l{(\\rho_b,n,\\eta^{\\prime})} = (\\eta^{\\prime} \\rho_b)^{n}, then obtain \\frac{\\partial}{\\partial \\eta^{\\prime}} \\hat{H}_l{(\\rho_b,n,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} G{(\\rho_b,n,\\eta^{\\prime})}", "derivation": "G{(\\rho_b,n,\\eta^{\\prime})} = (\\eta^{\\prime} \\rho_b)^{n} and \\hat{H}_l{(\\rho_b,n,\\eta^{\\prime})} = (\\eta^{\\prime} \\rho_b)^{n} and \\hat{H}_l{(\\rho_b,n,\\eta^{\\prime})} = G{(\\rho_b,n,\\eta^{\\prime})} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\hat{H}_l{(\\rho_b,n,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} G{(\\rho_b,n,\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('G')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('G')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\delta,t_{1})} = \\frac{\\delta}{t_{1}} and \\operatorname{v_{t}}{(t_{1})} = t_{1}, then obtain (\\frac{\\operatorname{v_{t}}{(t_{1})}}{\\hat{p}_0{(\\delta,t_{1})} - 1})^{\\delta} = (\\frac{t_{1}}{\\hat{p}_0{(\\delta,t_{1})} - 1})^{\\delta}", "derivation": "\\hat{p}_0{(\\delta,t_{1})} = \\frac{\\delta}{t_{1}} and \\operatorname{v_{t}}{(t_{1})} = t_{1} and \\frac{\\operatorname{v_{t}}{(t_{1})}}{\\frac{\\delta}{t_{1}} - 1} = \\frac{t_{1}}{\\frac{\\delta}{t_{1}} - 1} and (\\frac{\\operatorname{v_{t}}{(t_{1})}}{\\frac{\\delta}{t_{1}} - 1})^{\\delta} = (\\frac{t_{1}}{\\frac{\\delta}{t_{1}} - 1})^{\\delta} and (\\frac{\\operatorname{v_{t}}{(t_{1})}}{\\hat{p}_0{(\\delta,t_{1})} - 1})^{\\delta} = (\\frac{t_{1}}{\\hat{p}_0{(\\delta,t_{1})} - 1})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["divide", 2, "Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integer(-1)), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True))), Mul(Symbol('t_1', commutative=True), Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integer(-1)), Integer(-1))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integer(-1)), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Symbol('t_1', commutative=True), Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integer(-1)), Integer(-1))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Mul(Pow(Add(Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Symbol('t_1', commutative=True), Pow(Add(Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Integer(-1))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} = e^{- \\Psi_{\\lambda} + \\mathbf{D}}, then obtain \\frac{(\\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} - e^{- \\Psi_{\\lambda} + \\mathbf{D}})^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda}} = \\frac{0^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda}}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} = e^{- \\Psi_{\\lambda} + \\mathbf{D}} and \\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} - e^{- \\Psi_{\\lambda} + \\mathbf{D}} = 0 and (\\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} - e^{- \\Psi_{\\lambda} + \\mathbf{D}})^{\\Psi_{\\lambda}} = 0^{\\Psi_{\\lambda}} and \\frac{(\\dot{\\mathbf{r}}{(\\mathbf{D},\\Psi_{\\lambda})} - e^{- \\Psi_{\\lambda} + \\mathbf{D}})^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda}} = \\frac{0^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 1, "exp(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Integer(0), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(a,f^{*})} = a + f^{*}, then obtain (a + f^{*}) \\frac{\\partial}{\\partial a} \\phi_{1}{(a,f^{*})} + \\phi_{1}{(a,f^{*})} = 2 a + 2 f^{*}", "derivation": "\\phi_{1}{(a,f^{*})} = a + f^{*} and (a + f^{*}) \\phi_{1}{(a,f^{*})} = (a + f^{*})^{2} and \\frac{\\partial}{\\partial a} (a + f^{*}) \\phi_{1}{(a,f^{*})} = \\frac{\\partial}{\\partial a} (a + f^{*})^{2} and (a + f^{*}) \\frac{\\partial}{\\partial a} \\phi_{1}{(a,f^{*})} + \\phi_{1}{(a,f^{*})} = 2 a + 2 f^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)))"], [["times", 1, "Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('f^*', commutative=True))), Pow(Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Add(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Function('\\\\phi_1')(Symbol('a', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\lambda)} = \\cos{(\\lambda)}, then obtain 2 \\mathbb{I}{(\\lambda)} = 2 \\cos{(\\lambda)}", "derivation": "\\mathbb{I}{(\\lambda)} = \\cos{(\\lambda)} and \\mathbb{I}{(\\lambda)} + \\cos{(\\lambda)} = 2 \\cos{(\\lambda)} and 2 \\mathbb{I}{(\\lambda)} = \\mathbb{I}{(\\lambda)} + \\cos{(\\lambda)} and 2 \\mathbb{I}{(\\lambda)} = 2 \\cos{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True))), Add(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(G,\\mathbf{F})} = \\log{(\\mathbf{F})}^{G}, then obtain \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\operatorname{f^{*}}^{\\mathbf{F}}{(G,\\mathbf{F})}}{\\mathbf{F} \\log{(\\mathbf{F})}} = \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{(\\log{(\\mathbf{F})}^{G})^{\\mathbf{F}}}{\\mathbf{F} \\log{(\\mathbf{F})}}", "derivation": "\\operatorname{f^{*}}{(G,\\mathbf{F})} = \\log{(\\mathbf{F})}^{G} and \\operatorname{f^{*}}^{\\mathbf{F}}{(G,\\mathbf{F})} = (\\log{(\\mathbf{F})}^{G})^{\\mathbf{F}} and \\frac{\\operatorname{f^{*}}^{\\mathbf{F}}{(G,\\mathbf{F})}}{\\log{(\\mathbf{F})}} = \\frac{(\\log{(\\mathbf{F})}^{G})^{\\mathbf{F}}}{\\log{(\\mathbf{F})}} and \\frac{\\operatorname{f^{*}}^{\\mathbf{F}}{(G,\\mathbf{F})}}{\\mathbf{F} \\log{(\\mathbf{F})}} = \\frac{(\\log{(\\mathbf{F})}^{G})^{\\mathbf{F}}}{\\mathbf{F} \\log{(\\mathbf{F})}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\operatorname{f^{*}}^{\\mathbf{F}}{(G,\\mathbf{F})}}{\\mathbf{F} \\log{(\\mathbf{F})}} = \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{(\\log{(\\mathbf{F})}^{G})^{\\mathbf{F}}}{\\mathbf{F} \\log{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('G', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('G', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 2, "log(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Pow(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Pow(Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('G', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"], [["divide", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('G', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('G', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(v,\\hbar)} = - v + e^{\\hbar}, then obtain \\log{(k{(v,\\hbar)} + \\int ((- v + e^{\\hbar})^{\\hbar})^{- v} (k^{\\hbar}{(v,\\hbar)})^{v} d\\hbar)} = \\log{(k{(v,\\hbar)} + \\int 1 d\\hbar)}", "derivation": "k{(v,\\hbar)} = - v + e^{\\hbar} and k^{\\hbar}{(v,\\hbar)} = (- v + e^{\\hbar})^{\\hbar} and (k^{\\hbar}{(v,\\hbar)})^{v} = ((- v + e^{\\hbar})^{\\hbar})^{v} and ((- v + e^{\\hbar})^{\\hbar})^{- v} (k^{\\hbar}{(v,\\hbar)})^{v} = 1 and \\int ((- v + e^{\\hbar})^{\\hbar})^{- v} (k^{\\hbar}{(v,\\hbar)})^{v} d\\hbar = \\int 1 d\\hbar and k{(v,\\hbar)} + \\int ((- v + e^{\\hbar})^{\\hbar})^{- v} (k^{\\hbar}{(v,\\hbar)})^{v} d\\hbar = k{(v,\\hbar)} + \\int 1 d\\hbar and \\log{(k{(v,\\hbar)} + \\int ((- v + e^{\\hbar})^{\\hbar})^{- v} (k^{\\hbar}{(v,\\hbar)})^{v} d\\hbar)} = \\log{(k{(v,\\hbar)} + \\int 1 d\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True)))"], [["divide", 3, "Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Pow(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True))), Integer(1))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Pow(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 5, "Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Mul(Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Pow(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["log", 6], "Equality(log(Add(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Mul(Pow(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Pow(Pow(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))), log(Add(Function('k')(Symbol('v', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}_0{(\\psi^*)} = e^{\\psi^*}, then derive \\int \\psi^* \\hat{p}_0{(\\psi^*)} d\\psi^* = A_{x} + (\\psi^* - 1) e^{\\psi^*}, then obtain \\int \\psi^* e^{\\psi^*} d\\psi^* = A_{x} + (\\psi^* - 1) e^{\\psi^*}", "derivation": "\\hat{p}_0{(\\psi^*)} = e^{\\psi^*} and \\psi^* \\hat{p}_0{(\\psi^*)} = \\psi^* e^{\\psi^*} and \\int \\psi^* \\hat{p}_0{(\\psi^*)} d\\psi^* = \\int \\psi^* e^{\\psi^*} d\\psi^* and \\int \\psi^* \\hat{p}_0{(\\psi^*)} d\\psi^* = A_{x} + (\\psi^* - 1) e^{\\psi^*} and \\int \\psi^* e^{\\psi^*} d\\psi^* = A_{x} + (\\psi^* - 1) e^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Add(Symbol('\\\\psi^*', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Add(Symbol('\\\\psi^*', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\sigma_x)} = e^{\\sigma_x}, then obtain \\cos{(\\lambda{(\\sigma_x)})} - \\int \\lambda{(\\sigma_x)} d\\sigma_x = \\cos{(e^{\\sigma_x})} - \\int \\lambda{(\\sigma_x)} d\\sigma_x", "derivation": "\\lambda{(\\sigma_x)} = e^{\\sigma_x} and \\int \\lambda{(\\sigma_x)} d\\sigma_x = \\int e^{\\sigma_x} d\\sigma_x and \\cos{(\\lambda{(\\sigma_x)})} = \\cos{(e^{\\sigma_x})} and \\cos{(\\lambda{(\\sigma_x)})} - \\int e^{\\sigma_x} d\\sigma_x = \\cos{(e^{\\sigma_x})} - \\int e^{\\sigma_x} d\\sigma_x and \\cos{(\\lambda{(\\sigma_x)})} - \\int \\lambda{(\\sigma_x)} d\\sigma_x = \\cos{(e^{\\sigma_x})} - \\int \\lambda{(\\sigma_x)} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True))), cos(exp(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 3, "Integral(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(cos(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))), Add(cos(exp(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(cos(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))), Add(cos(exp(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))))"]]}, {"prompt": "Given M{(\\Psi_{nl},\\sigma_p)} = \\Psi_{nl} + \\cos{(\\sigma_p)}, then derive \\int \\sigma_p M{(\\Psi_{nl},\\sigma_p)} d\\sigma_p = \\frac{\\Psi_{nl} \\sigma_p^{2}}{2} + \\sigma_p \\sin{(\\sigma_p)} + r + \\cos{(\\sigma_p)}, then obtain \\frac{\\Psi_{nl} \\sigma_p^{2}}{2} + \\sigma_p \\sin{(\\sigma_p)} + r + \\cos{(\\sigma_p)} = \\int \\sigma_p (\\Psi_{nl} + \\cos{(\\sigma_p)}) d\\sigma_p", "derivation": "M{(\\Psi_{nl},\\sigma_p)} = \\Psi_{nl} + \\cos{(\\sigma_p)} and \\sigma_p M{(\\Psi_{nl},\\sigma_p)} = \\sigma_p (\\Psi_{nl} + \\cos{(\\sigma_p)}) and \\int \\sigma_p M{(\\Psi_{nl},\\sigma_p)} d\\sigma_p = \\int \\sigma_p (\\Psi_{nl} + \\cos{(\\sigma_p)}) d\\sigma_p and \\int \\sigma_p M{(\\Psi_{nl},\\sigma_p)} d\\sigma_p = \\frac{\\Psi_{nl} \\sigma_p^{2}}{2} + \\sigma_p \\sin{(\\sigma_p)} + r + \\cos{(\\sigma_p)} and \\frac{\\Psi_{nl} \\sigma_p^{2}}{2} + \\sigma_p \\sin{(\\sigma_p)} + r + \\cos{(\\sigma_p)} = \\int \\sigma_p (\\Psi_{nl} + \\cos{(\\sigma_p)}) d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('M')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('M')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('M')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\sigma_p', commutative=True))), Symbol('r', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Rational(1, 2), Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\sigma_p', commutative=True))), Symbol('r', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\dot{z},\\mathbf{r})} = \\cos{(\\frac{\\dot{z}}{\\mathbf{r}})}, then derive \\frac{\\partial}{\\partial \\mathbf{r}} \\varepsilon_{0}{(\\dot{z},\\mathbf{r})} = \\frac{\\dot{z} \\sin{(\\frac{\\dot{z}}{\\mathbf{r}})}}{\\mathbf{r}^{2}}, then obtain \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\dot{z}}{\\mathbf{r}})} = \\frac{\\dot{z} \\sin{(\\frac{\\dot{z}}{\\mathbf{r}})}}{\\mathbf{r}^{2}}", "derivation": "\\varepsilon_{0}{(\\dot{z},\\mathbf{r})} = \\cos{(\\frac{\\dot{z}}{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\mathbf{r}} \\varepsilon_{0}{(\\dot{z},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\dot{z}}{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\mathbf{r}} \\varepsilon_{0}{(\\dot{z},\\mathbf{r})} = \\frac{\\dot{z} \\sin{(\\frac{\\dot{z}}{\\mathbf{r}})}}{\\mathbf{r}^{2}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\dot{z}}{\\mathbf{r}})} = \\frac{\\dot{z} \\sin{(\\frac{\\dot{z}}{\\mathbf{r}})}}{\\mathbf{r}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-2)), sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-2)), sin(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\phi_{1}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then derive \\int \\phi_{1}{(L_{\\varepsilon})} dL_{\\varepsilon} = f + \\sin{(L_{\\varepsilon})}, then obtain \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} = f + \\sin{(L_{\\varepsilon})}", "derivation": "\\phi_{1}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\int \\phi_{1}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and \\int \\phi_{1}{(L_{\\varepsilon})} dL_{\\varepsilon} = f + \\sin{(L_{\\varepsilon})} and \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} = f + \\sin{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(i,\\varepsilon_0)} = - \\varepsilon_0 + i, then derive \\frac{\\partial}{\\partial \\varepsilon_0} \\dot{y}{(i,\\varepsilon_0)} = -1, then obtain (\\int \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + i) di)^{\\varepsilon_0} = (\\int (-1) di)^{\\varepsilon_0}", "derivation": "\\dot{y}{(i,\\varepsilon_0)} = - \\varepsilon_0 + i and \\frac{\\partial}{\\partial \\varepsilon_0} \\dot{y}{(i,\\varepsilon_0)} = \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + i) and \\frac{\\partial}{\\partial \\varepsilon_0} \\dot{y}{(i,\\varepsilon_0)} = -1 and \\int \\frac{\\partial}{\\partial \\varepsilon_0} \\dot{y}{(i,\\varepsilon_0)} di = \\int (-1) di and \\int \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + i) di = \\int (-1) di and (\\int \\frac{\\partial}{\\partial \\varepsilon_0} (- \\varepsilon_0 + i) di)^{\\varepsilon_0} = (\\int (-1) di)^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{y}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Integer(-1), Tuple(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Integer(-1), Tuple(Symbol('i', commutative=True))))"], [["power", 5, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(Integer(-1), Tuple(Symbol('i', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\operatorname{m_{s}}{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then obtain \\frac{\\operatorname{m_{s}}{(\\mathbf{F})} \\sin^{- \\mathbf{F}}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\sin{(\\mathbf{F})} \\sin^{- \\mathbf{F}}{(\\mathbf{F})}}{\\mathbf{F}}", "derivation": "\\rho{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\operatorname{m_{s}}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\frac{\\rho^{- \\mathbf{F}}{(\\mathbf{F})} \\operatorname{m_{s}}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\rho^{- \\mathbf{F}}{(\\mathbf{F})} \\sin{(\\mathbf{F})}}{\\mathbf{F}} and \\frac{\\operatorname{m_{s}}{(\\mathbf{F})} \\sin^{- \\mathbf{F}}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\sin{(\\mathbf{F})} \\sin^{- \\mathbf{F}}{(\\mathbf{F})}}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Function('m_s')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{\\mathbf{x}},V)} = V + \\hat{\\mathbf{x}}, then derive \\int \\varphi^{*}{(\\hat{\\mathbf{x}},V)} dV = \\frac{V^{2}}{2} + V \\hat{\\mathbf{x}} + \\Psi_{\\lambda}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\int \\varphi^{*}{(\\hat{\\mathbf{x}},V)} dV = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\frac{V^{2}}{2} + V \\hat{\\mathbf{x}} + \\Psi_{\\lambda})", "derivation": "\\varphi^{*}{(\\hat{\\mathbf{x}},V)} = V + \\hat{\\mathbf{x}} and \\int \\varphi^{*}{(\\hat{\\mathbf{x}},V)} dV = \\int (V + \\hat{\\mathbf{x}}) dV and \\int \\varphi^{*}{(\\hat{\\mathbf{x}},V)} dV = \\frac{V^{2}}{2} + V \\hat{\\mathbf{x}} + \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\int \\varphi^{*}{(\\hat{\\mathbf{x}},V)} dV = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\frac{V^{2}}{2} + V \\hat{\\mathbf{x}} + \\Psi_{\\lambda})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Symbol('V', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Symbol('V', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Symbol('V', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(t_{1})} = e^{\\sin{(t_{1})}} and \\operatorname{v_{t}}{(t_{1})} = 2 k{(t_{1})}, then obtain \\frac{\\operatorname{v_{t}}{(t_{1})} e^{- \\sin{(t_{1})}}}{2} + \\sin{(t_{1})} = \\sin{(t_{1})} + 1", "derivation": "k{(t_{1})} = e^{\\sin{(t_{1})}} and 2 k{(t_{1})} = k{(t_{1})} + e^{\\sin{(t_{1})}} and \\operatorname{v_{t}}{(t_{1})} = 2 k{(t_{1})} and \\frac{\\operatorname{v_{t}}{(t_{1})}}{k{(t_{1})} + e^{\\sin{(t_{1})}}} = \\frac{2 k{(t_{1})}}{k{(t_{1})} + e^{\\sin{(t_{1})}}} and \\frac{\\operatorname{v_{t}}{(t_{1})}}{k{(t_{1})} + e^{\\sin{(t_{1})}}} = 1 and \\sin{(t_{1})} + \\frac{\\operatorname{v_{t}}{(t_{1})}}{k{(t_{1})} + e^{\\sin{(t_{1})}}} = \\sin{(t_{1})} + 1 and \\frac{\\operatorname{v_{t}}{(t_{1})} e^{- \\sin{(t_{1})}}}{2} + \\sin{(t_{1})} = \\sin{(t_{1})} + 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True))))"], [["add", 1, "Function('k')(Symbol('t_1', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('t_1', commutative=True))), Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('t_1', commutative=True)), Mul(Integer(2), Function('k')(Symbol('t_1', commutative=True))))"], [["divide", 3, "Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True))))"], "Equality(Mul(Pow(Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True)))), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True))), Mul(Integer(2), Pow(Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True)))), Integer(-1)), Function('k')(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True)))), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True))), Integer(1))"], [["add", 5, "sin(Symbol('t_1', commutative=True))"], "Equality(Add(sin(Symbol('t_1', commutative=True)), Mul(Pow(Add(Function('k')(Symbol('t_1', commutative=True)), exp(sin(Symbol('t_1', commutative=True)))), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True)))), Add(sin(Symbol('t_1', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Rational(1, 2), Function('v_t')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))))), sin(Symbol('t_1', commutative=True))), Add(sin(Symbol('t_1', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\sigma_{x}{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})} and l{(\\rho_f)} = - \\sigma_{x}{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} + \\sin^{2}{(\\sin{(\\rho_f)})}, then obtain 0 = l{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})}", "derivation": "\\sigma_{x}{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})} and \\sigma_{x}{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} = \\sin^{2}{(\\sin{(\\rho_f)})} and 0 = - \\sigma_{x}{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} + \\sin^{2}{(\\sin{(\\rho_f)})} and l{(\\rho_f)} = - \\sigma_{x}{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} + \\sin^{2}{(\\sin{(\\rho_f)})} and 0 = l{(\\rho_f)} and 0 = l{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "sin(sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Pow(sin(sin(Symbol('\\\\rho_f', commutative=True))), Integer(2)))"], [["minus", 2, "Mul(Function('\\\\sigma_x')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Pow(sin(sin(Symbol('\\\\rho_f', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Pow(sin(sin(Symbol('\\\\rho_f', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Function('l')(Symbol('\\\\rho_f', commutative=True)))"], [["times", 5, "sin(sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Integer(0), Mul(Function('l')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{p},g)} = \\log{(- \\mathbf{p} + g)} and \\operatorname{v_{z}}{(\\mathbf{p},g)} = - \\mathbf{p} + g, then obtain \\log{(- \\mathbf{p} + g)} = \\log{(\\operatorname{v_{z}}{(\\mathbf{p},g)})}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{p},g)} = \\log{(- \\mathbf{p} + g)} and \\operatorname{v_{z}}{(\\mathbf{p},g)} = - \\mathbf{p} + g and \\operatorname{A_{1}}{(\\mathbf{p},g)} = \\log{(\\operatorname{v_{z}}{(\\mathbf{p},g)})} and \\log{(- \\mathbf{p} + g)} = \\log{(\\operatorname{v_{z}}{(\\mathbf{p},g)})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('g', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_1')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('g', commutative=True)), log(Function('v_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('g', commutative=True))), log(Function('v_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(P_{g})} = \\log{(P_{g})} and \\mathbf{F}{(P_{g})} = \\log{(P_{g})}, then obtain \\mathbf{F}{(P_{g})} + 2 = \\log{(P_{g})} + 2", "derivation": "\\rho_{b}{(P_{g})} = \\log{(P_{g})} and \\rho_{b}{(P_{g})} + 1 = \\log{(P_{g})} + 1 and \\rho_{b}{(P_{g})} + 2 = \\log{(P_{g})} + 2 and \\mathbf{F}{(P_{g})} = \\log{(P_{g})} and \\rho_{b}{(P_{g})} + 2 = \\mathbf{F}{(P_{g})} + 2 and \\mathbf{F}{(P_{g})} + 2 = \\log{(P_{g})} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\rho_b')(Symbol('P_g', commutative=True)), Integer(1)), Add(log(Symbol('P_g', commutative=True)), Integer(1)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\rho_b')(Symbol('P_g', commutative=True)), Integer(2)), Add(log(Symbol('P_g', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\rho_b')(Symbol('P_g', commutative=True)), Integer(2)), Add(Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('P_g', commutative=True)), Integer(2)), Add(log(Symbol('P_g', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{H},J_{\\varepsilon})} = - J_{\\varepsilon} + \\mathbf{H} and \\operatorname{v_{z}}{(\\eta,g)} = \\frac{\\eta}{g}, then obtain - J_{\\varepsilon} + \\operatorname{v_{z}}{(\\eta,g)} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}} = - J_{\\varepsilon} + \\frac{\\eta}{g} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}}", "derivation": "\\hat{p}{(\\mathbf{H},J_{\\varepsilon})} = - J_{\\varepsilon} + \\mathbf{H} and \\operatorname{v_{z}}{(\\eta,g)} = \\frac{\\eta}{g} and \\operatorname{v_{z}}{(\\eta,g)} + \\frac{\\hat{p}{(\\mathbf{H},J_{\\varepsilon})}}{\\mathbf{H}} = \\frac{\\eta}{g} + \\frac{\\hat{p}{(\\mathbf{H},J_{\\varepsilon})}}{\\mathbf{H}} and \\operatorname{v_{z}}{(\\eta,g)} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}} = \\frac{\\eta}{g} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}} and - J_{\\varepsilon} + \\operatorname{v_{z}}{(\\eta,g)} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}} = - J_{\\varepsilon} + \\frac{\\eta}{g} + \\frac{- J_{\\varepsilon} + \\mathbf{H}}{\\mathbf{H}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], ["get_premise", "Equality(Function('v_z')(Symbol('\\\\eta', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["add", 2, "Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Function('v_z')(Symbol('\\\\eta', commutative=True), Symbol('g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('v_z')(Symbol('\\\\eta', commutative=True), Symbol('g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["minus", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('v_z')(Symbol('\\\\eta', commutative=True), Symbol('g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f^{*},b)} = \\frac{b}{f^{*}}, then obtain \\int \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\operatorname{F_{H}}{(f^{*},b)} db = \\int \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\frac{b}{f^{*}} db", "derivation": "\\operatorname{F_{H}}{(f^{*},b)} = \\frac{b}{f^{*}} and \\frac{\\partial}{\\partial f^{*}} \\operatorname{F_{H}}{(f^{*},b)} = \\frac{\\partial}{\\partial f^{*}} \\frac{b}{f^{*}} and \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\operatorname{F_{H}}{(f^{*},b)} = \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\frac{b}{f^{*}} and \\int \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\operatorname{F_{H}}{(f^{*},b)} db = \\int \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} \\frac{b}{f^{*}} db", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('b', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(2))), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Derivative(Function('F_H')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(2))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(2))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given U{(y,n)} = \\frac{n}{y}, then obtain 0^{n} (n + y + U{(y,n)}) = (\\frac{\\frac{n}{y} - U{(y,n)}}{n})^{n} (n + y + U{(y,n)})", "derivation": "U{(y,n)} = \\frac{n}{y} and y + U{(y,n)} = \\frac{n}{y} + y and 0 = \\frac{n}{y} - U{(y,n)} and n + y + U{(y,n)} = n + \\frac{n}{y} + y and 0 = \\frac{\\frac{n}{y} - U{(y,n)}}{n} and 0^{n} = (\\frac{\\frac{n}{y} - U{(y,n)}}{n})^{n} and 0^{n} (n + \\frac{n}{y} + y) = (\\frac{\\frac{n}{y} - U{(y,n)}}{n})^{n} (n + \\frac{n}{y} + y) and 0^{n} (n + y + U{(y,n)}) = (\\frac{\\frac{n}{y} - U{(y,n)}}{n})^{n} (n + y + U{(y,n)})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True)))"], [["minus", 2, "Add(Symbol('y', commutative=True), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True)))))"], [["add", 2, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Symbol('y', commutative=True), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True)))"], [["divide", 3, "Symbol('n', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))))))"], [["power", 5, "Symbol('n', commutative=True)"], "Equality(Pow(Integer(0), Symbol('n', commutative=True)), Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))))), Symbol('n', commutative=True)))"], [["times", 6, "Add(Symbol('n', commutative=True), Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))), Mul(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))))), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Pow(Integer(0), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Symbol('n', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True))))), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True), Function('U')(Symbol('y', commutative=True), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(h,f^{*})} = f^{*} h, then obtain \\frac{\\partial}{\\partial f^{*}} (- f^{*} h + 2 \\sigma_{p}{(h,f^{*})}) = \\frac{\\partial}{\\partial f^{*}} f^{*} h", "derivation": "\\sigma_{p}{(h,f^{*})} = f^{*} h and f^{*} + \\sigma_{p}{(h,f^{*})} = f^{*} h + f^{*} and - f^{*} h + \\sigma_{p}{(h,f^{*})} = 0 and \\frac{\\partial}{\\partial f^{*}} \\sigma_{p}{(h,f^{*})} = \\frac{\\partial}{\\partial f^{*}} f^{*} h and - f^{*} h + 2 \\sigma_{p}{(h,f^{*})} = \\sigma_{p}{(h,f^{*})} and \\frac{\\partial}{\\partial f^{*}} (- f^{*} h + 2 \\sigma_{p}{(h,f^{*})}) = \\frac{\\partial}{\\partial f^{*}} \\sigma_{p}{(h,f^{*})} and \\frac{\\partial}{\\partial f^{*}} (- f^{*} h + 2 \\sigma_{p}{(h,f^{*})}) = \\frac{\\partial}{\\partial f^{*}} f^{*} h", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('f^*', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Symbol('f^*', commutative=True)))"], [["minus", 2, "Add(Mul(Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True))), Integer(0))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)))), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)))"], [["differentiate", 5, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('h', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('f^*', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(v_{z},\\mathbf{f})} = \\mathbf{f} v_{z} and J{(v_{z})} = v_{z}, then obtain v_{z} \\dot{y}^{v_{z}}{(v_{z},\\mathbf{f})} = v_{z} (\\mathbf{f} v_{z})^{v_{z}}", "derivation": "\\dot{y}{(v_{z},\\mathbf{f})} = \\mathbf{f} v_{z} and \\dot{y}^{v_{z}}{(v_{z},\\mathbf{f})} = (\\mathbf{f} v_{z})^{v_{z}} and J{(v_{z})} = v_{z} and J{(v_{z})} \\dot{y}^{v_{z}}{(v_{z},\\mathbf{f})} = (\\mathbf{f} v_{z})^{v_{z}} J{(v_{z})} and v_{z} \\dot{y}^{v_{z}}{(v_{z},\\mathbf{f})} = v_{z} (\\mathbf{f} v_{z})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('v_z', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], [["times", 2, "Function('J')(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('J')(Symbol('v_z', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('v_z', commutative=True))), Mul(Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Function('J')(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('v_z', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('v_z', commutative=True))), Mul(Symbol('v_z', commutative=True), Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\psi{(A_{1},v_{z})} = \\cos{(\\frac{v_{z}}{A_{1}})}, then obtain \\frac{\\partial}{\\partial A_{1}} ((- A_{1} + \\psi{(A_{1},v_{z})})^{A_{1}})^{A_{1}} = \\frac{\\partial}{\\partial A_{1}} ((- A_{1} + \\cos{(\\frac{v_{z}}{A_{1}})})^{A_{1}})^{A_{1}}", "derivation": "\\psi{(A_{1},v_{z})} = \\cos{(\\frac{v_{z}}{A_{1}})} and - A_{1} + \\psi{(A_{1},v_{z})} = - A_{1} + \\cos{(\\frac{v_{z}}{A_{1}})} and (- A_{1} + \\psi{(A_{1},v_{z})})^{A_{1}} = (- A_{1} + \\cos{(\\frac{v_{z}}{A_{1}})})^{A_{1}} and ((- A_{1} + \\psi{(A_{1},v_{z})})^{A_{1}})^{A_{1}} = ((- A_{1} + \\cos{(\\frac{v_{z}}{A_{1}})})^{A_{1}})^{A_{1}} and \\frac{\\partial}{\\partial A_{1}} ((- A_{1} + \\psi{(A_{1},v_{z})})^{A_{1}})^{A_{1}} = \\frac{\\partial}{\\partial A_{1}} ((- A_{1} + \\cos{(\\frac{v_{z}}{A_{1}})})^{A_{1}})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True)), cos(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["minus", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\psi')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), cos(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\psi')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), cos(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))), Symbol('A_1', commutative=True)))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\psi')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), cos(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\psi')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), cos(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(J)} = e^{e^{J}}, then derive \\frac{d}{d J} \\pi{(J)} = e^{J} e^{e^{J}}, then obtain \\int \\frac{d}{d J} e^{e^{J}} dJ = \\int e^{J} e^{e^{J}} dJ", "derivation": "\\pi{(J)} = e^{e^{J}} and \\frac{d}{d J} \\pi{(J)} = \\frac{d}{d J} e^{e^{J}} and \\frac{d}{d J} \\pi{(J)} = e^{J} e^{e^{J}} and \\frac{d}{d J} \\pi{(J)} = \\pi{(J)} e^{J} and \\int \\frac{d}{d J} \\pi{(J)} dJ = \\int \\pi{(J)} e^{J} dJ and \\int \\frac{d}{d J} e^{e^{J}} dJ = \\int e^{J} e^{e^{J}} dJ", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('J', commutative=True)), exp(exp(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(exp(Symbol('J', commutative=True)), exp(exp(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\pi')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Function('\\\\pi')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True))))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\pi')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Function('\\\\pi')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(exp(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(exp(Symbol('J', commutative=True)), exp(exp(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\Omega)} = e^{\\Omega}, then obtain M + \\operatorname{f_{\\mathbf{v}}}{(\\Omega)} = c_{0} + e^{\\Omega}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\Omega)} = e^{\\Omega} and \\operatorname{f_{\\mathbf{v}}}{(\\Omega)} - 1 = e^{\\Omega} - 1 and \\frac{d}{d \\Omega} (\\operatorname{f_{\\mathbf{v}}}{(\\Omega)} - 1) = \\frac{d}{d \\Omega} (e^{\\Omega} - 1) and \\int \\frac{d}{d \\Omega} (\\operatorname{f_{\\mathbf{v}}}{(\\Omega)} - 1) d\\Omega = \\int \\frac{d}{d \\Omega} (e^{\\Omega} - 1) d\\Omega and M + \\operatorname{f_{\\mathbf{v}}}{(\\Omega)} = c_{0} + e^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Add(exp(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('M', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(r_{0},z)} = r_{0} \\sin{(z)}, then derive (\\frac{\\partial}{\\partial r_{0}} \\hat{\\mathbf{x}}{(r_{0},z)})^{z} = \\sin^{z}{(z)}, then obtain r_{0} \\sin{(z)} (\\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(z)})^{z} = r_{0} \\sin{(z)} \\sin^{z}{(z)}", "derivation": "\\hat{\\mathbf{x}}{(r_{0},z)} = r_{0} \\sin{(z)} and \\frac{\\partial}{\\partial r_{0}} \\hat{\\mathbf{x}}{(r_{0},z)} = \\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(z)} and (\\frac{\\partial}{\\partial r_{0}} \\hat{\\mathbf{x}}{(r_{0},z)})^{z} = (\\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(z)})^{z} and (\\frac{\\partial}{\\partial r_{0}} \\hat{\\mathbf{x}}{(r_{0},z)})^{z} = \\sin^{z}{(z)} and r_{0} \\sin{(z)} (\\frac{\\partial}{\\partial r_{0}} \\hat{\\mathbf{x}}{(r_{0},z)})^{z} = r_{0} \\sin{(z)} \\sin^{z}{(z)} and r_{0} \\sin{(z)} (\\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(z)})^{z} = r_{0} \\sin{(z)} \\sin^{z}{(z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r_0', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["times", 4, "Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True)))"], "Equality(Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True)), Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('z', commutative=True))), Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True)), Pow(Derivative(Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('z', commutative=True))), Mul(Symbol('r_0', commutative=True), sin(Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True))))"]]}, {"prompt": "Given C{(\\dot{x},L_{\\varepsilon})} = L_{\\varepsilon}^{\\dot{x}}, then obtain - \\frac{1}{C{(\\dot{x},L_{\\varepsilon})}} + \\frac{L_{\\varepsilon} + 1}{L_{\\varepsilon}} = - \\frac{1}{C{(\\dot{x},L_{\\varepsilon})}} + \\frac{L_{\\varepsilon} + \\frac{L_{\\varepsilon}^{\\dot{x}}}{C{(\\dot{x},L_{\\varepsilon})}}}{L_{\\varepsilon}}", "derivation": "C{(\\dot{x},L_{\\varepsilon})} = L_{\\varepsilon}^{\\dot{x}} and 1 = \\frac{L_{\\varepsilon}^{\\dot{x}}}{C{(\\dot{x},L_{\\varepsilon})}} and L_{\\varepsilon} + 1 = L_{\\varepsilon} + \\frac{L_{\\varepsilon}^{\\dot{x}}}{C{(\\dot{x},L_{\\varepsilon})}} and \\frac{L_{\\varepsilon} + 1}{L_{\\varepsilon}} = \\frac{L_{\\varepsilon} + \\frac{L_{\\varepsilon}^{\\dot{x}}}{C{(\\dot{x},L_{\\varepsilon})}}}{L_{\\varepsilon}} and - \\frac{1}{C{(\\dot{x},L_{\\varepsilon})}} + \\frac{L_{\\varepsilon} + 1}{L_{\\varepsilon}} = - \\frac{1}{C{(\\dot{x},L_{\\varepsilon})}} + \\frac{L_{\\varepsilon} + \\frac{L_{\\varepsilon}^{\\dot{x}}}{C{(\\dot{x},L_{\\varepsilon})}}}{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)))))"], [["divide", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))))))"], [["minus", 4, "Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('C')(Symbol('\\\\dot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(T)} = \\log{(T)}, then derive \\frac{d}{d T} \\operatorname{F_{x}}{(T)} = \\frac{1}{T}, then obtain (- \\log{(T)} + \\cos{(\\frac{d}{d T} \\log{(T)})})^{T} - 1 = (- \\log{(T)} + \\cos{(\\frac{1}{T})})^{T} - 1", "derivation": "\\operatorname{F_{x}}{(T)} = \\log{(T)} and \\frac{d}{d T} \\operatorname{F_{x}}{(T)} = \\frac{d}{d T} \\log{(T)} and \\frac{d}{d T} \\operatorname{F_{x}}{(T)} = \\frac{1}{T} and \\cos{(\\frac{d}{d T} \\operatorname{F_{x}}{(T)})} = \\cos{(\\frac{1}{T})} and \\cos{(\\frac{d}{d T} \\log{(T)})} = \\cos{(\\frac{1}{T})} and - \\log{(T)} + \\cos{(\\frac{d}{d T} \\log{(T)})} = - \\log{(T)} + \\cos{(\\frac{1}{T})} and (- \\log{(T)} + \\cos{(\\frac{d}{d T} \\log{(T)})})^{T} = (- \\log{(T)} + \\cos{(\\frac{1}{T})})^{T} and (- \\log{(T)} + \\cos{(\\frac{d}{d T} \\log{(T)})})^{T} - 1 = (- \\log{(T)} + \\cos{(\\frac{1}{T})})^{T} - 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Symbol('T', commutative=True), Integer(-1)))"], [["cos", 3], "Equality(cos(Derivative(Function('F_x')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), cos(Pow(Symbol('T', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), cos(Pow(Symbol('T', commutative=True), Integer(-1))))"], [["minus", 5, "log(Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Pow(Symbol('T', commutative=True), Integer(-1)))))"], [["power", 6, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Pow(Symbol('T', commutative=True), Integer(-1)))), Symbol('T', commutative=True)))"], [["add", 7, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Symbol('T', commutative=True)), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), cos(Pow(Symbol('T', commutative=True), Integer(-1)))), Symbol('T', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{x},\\mathbf{J}_M)} = \\frac{\\hat{x}}{\\mathbf{J}_M}, then obtain F_{N}^{- 2 v_{2}} \\operatorname{F_{g}}^{2}{(\\hat{x},\\mathbf{J}_M)} = \\frac{F_{N}^{- 2 v_{2}} \\hat{x}^{2}}{\\mathbf{J}_M^{2}}", "derivation": "\\operatorname{F_{g}}{(\\hat{x},\\mathbf{J}_M)} = \\frac{\\hat{x}}{\\mathbf{J}_M} and \\operatorname{F_{g}}^{2}{(\\hat{x},\\mathbf{J}_M)} = \\frac{\\hat{x} \\operatorname{F_{g}}{(\\hat{x},\\mathbf{J}_M)}}{\\mathbf{J}_M} and F_{N}^{- v_{2}} \\operatorname{F_{g}}{(\\hat{x},\\mathbf{J}_M)} = \\frac{F_{N}^{- v_{2}} \\hat{x}}{\\mathbf{J}_M} and \\frac{F_{N}^{- 2 v_{2}} \\hat{x} \\operatorname{F_{g}}{(\\hat{x},\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{F_{N}^{- 2 v_{2}} \\hat{x}^{2}}{\\mathbf{J}_M^{2}} and F_{N}^{- 2 v_{2}} \\operatorname{F_{g}}^{2}{(\\hat{x},\\mathbf{J}_M)} = \\frac{F_{N}^{- 2 v_{2}} \\hat{x}^{2}}{\\mathbf{J}_M^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))))"], [["times", 1, "Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 1, "Pow(Symbol('F_N', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))))"], [["times", 3, "Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))), Pow(Function('F_g')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Pow(Symbol('F_N', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-2))))"]]}, {"prompt": "Given f{(n_{1},x)} = n_{1} e^{x}, then obtain 2 \\frac{\\partial}{\\partial x} f{(n_{1},x)} - 1 = n_{1} e^{x} + \\frac{\\partial}{\\partial x} f{(n_{1},x)} - 1", "derivation": "f{(n_{1},x)} = n_{1} e^{x} and 2 f{(n_{1},x)} = n_{1} e^{x} + f{(n_{1},x)} and - x + 2 f{(n_{1},x)} = n_{1} e^{x} - x + f{(n_{1},x)} and \\frac{\\partial}{\\partial x} (- x + 2 f{(n_{1},x)}) = \\frac{\\partial}{\\partial x} (n_{1} e^{x} - x + f{(n_{1},x)}) and 2 \\frac{\\partial}{\\partial x} f{(n_{1},x)} - 1 = n_{1} e^{x} + \\frac{\\partial}{\\partial x} f{(n_{1},x)} - 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('n_1', commutative=True), exp(Symbol('x', commutative=True))))"], [["add", 1, "Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))), Add(Mul(Symbol('n_1', commutative=True), exp(Symbol('x', commutative=True))), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))))"], [["minus", 2, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(2), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)))), Add(Mul(Symbol('n_1', commutative=True), exp(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True)), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(2), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('n_1', commutative=True), exp(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True)), Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Derivative(Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('n_1', commutative=True), exp(Symbol('x', commutative=True))), Derivative(Function('f')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(A_{z},l)} = \\cos{(\\frac{A_{z}}{l})} and \\omega{(A_{z},l)} = \\int - \\cos{(\\frac{A_{z}}{l})} dl, then obtain \\cos{(\\omega{(A_{z},l)} - \\int - \\operatorname{v_{2}}{(A_{z},l)} dl)} = 1", "derivation": "\\operatorname{v_{2}}{(A_{z},l)} = \\cos{(\\frac{A_{z}}{l})} and - \\operatorname{v_{2}}{(A_{z},l)} = - \\cos{(\\frac{A_{z}}{l})} and \\int - \\operatorname{v_{2}}{(A_{z},l)} dl = \\int - \\cos{(\\frac{A_{z}}{l})} dl and \\omega{(A_{z},l)} = \\int - \\cos{(\\frac{A_{z}}{l})} dl and - \\omega{(A_{z},l)} = - \\int - \\cos{(\\frac{A_{z}}{l})} dl and - \\omega{(A_{z},l)} + \\int - \\cos{(\\frac{A_{z}}{l})} dl = 0 and \\cos{(\\omega{(A_{z},l)} - \\int - \\cos{(\\frac{A_{z}}{l})} dl)} = 1 and \\cos{(\\omega{(A_{z},l)} - \\int - \\operatorname{v_{2}}{(A_{z},l)} dl)} = 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('A_z', commutative=True), Symbol('l', commutative=True)), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_2')(Symbol('A_z', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('v_2')(Symbol('A_z', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('A_z', commutative=True), Symbol('l', commutative=True)), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\omega')(Symbol('A_z', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('A_z', commutative=True), Symbol('l', commutative=True))), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True)))), Integer(0))"], [["cos", 6], "Equality(cos(Add(Function('\\\\omega')(Symbol('A_z', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(-1), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))), Tuple(Symbol('l', commutative=True)))))), Integer(1))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(cos(Add(Function('\\\\omega')(Symbol('A_z', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(-1), Function('v_2')(Symbol('A_z', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))))), Integer(1))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{J})} = \\mathbf{J}, then derive - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\operatorname{a^{\\dagger}}{(\\mathbf{J})} = 1 - \\eta{(\\mathbf{J},t_{2})}, then obtain - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\mathbf{J} = 1 - \\eta{(\\mathbf{J},t_{2})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{J})} = \\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\operatorname{a^{\\dagger}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\mathbf{J} and - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\operatorname{a^{\\dagger}}{(\\mathbf{J})} = - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\mathbf{J} and - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\operatorname{a^{\\dagger}}{(\\mathbf{J})} = 1 - \\eta{(\\mathbf{J},t_{2})} and - \\eta{(\\mathbf{J},t_{2})} + \\frac{d}{d \\mathbf{J}} \\mathbf{J} = 1 - \\eta{(\\mathbf{J},t_{2})}", "srepr_derivation": [["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{J}', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True))), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True))), Derivative(Symbol('\\\\mathbf{J}', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True))), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True))), Derivative(Symbol('\\\\mathbf{J}', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given M{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})}, then derive - \\sin{(V_{\\mathbf{B}})} + \\frac{d}{d V_{\\mathbf{B}}} M{(V_{\\mathbf{B}})} = - 2 \\sin{(V_{\\mathbf{B}})}, then obtain - \\sin{(V_{\\mathbf{B}})} + \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} = - 2 \\sin{(V_{\\mathbf{B}})}", "derivation": "M{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\frac{d}{d V_{\\mathbf{B}}} M{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} and \\frac{d}{d V_{\\mathbf{B}}} M{(V_{\\mathbf{B}})} + \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} = 2 \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} and - \\sin{(V_{\\mathbf{B}})} + \\frac{d}{d V_{\\mathbf{B}}} M{(V_{\\mathbf{B}})} = - 2 \\sin{(V_{\\mathbf{B}})} and - \\sin{(V_{\\mathbf{B}})} + \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} = - 2 \\sin{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Derivative(Function('M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Derivative(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\theta_1)} = \\log{(\\log{(\\theta_1)})} and v{(\\theta_1)} = \\log{(\\log{(\\theta_1)})}, then obtain \\nabla{(\\theta_1)} + \\int v{(\\theta_1)} d\\theta_1 = \\nabla{(\\theta_1)} + \\int \\nabla{(\\theta_1)} d\\theta_1", "derivation": "\\nabla{(\\theta_1)} = \\log{(\\log{(\\theta_1)})} and v{(\\theta_1)} = \\log{(\\log{(\\theta_1)})} and v{(\\theta_1)} = \\nabla{(\\theta_1)} and \\int v{(\\theta_1)} d\\theta_1 = \\int \\nabla{(\\theta_1)} d\\theta_1 and \\nabla{(\\theta_1)} + \\int v{(\\theta_1)} d\\theta_1 = \\nabla{(\\theta_1)} + \\int \\nabla{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), log(log(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\theta_1', commutative=True)), log(log(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["add", 4, "Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Integral(Function('v')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Add(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\nabla')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\rho_f)} = e^{\\rho_f}, then obtain - \\operatorname{M_{E}}{(\\rho_f)} + \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f = - \\operatorname{M_{E}}{(\\rho_f)} + (e^{\\rho_f})^{\\rho_f} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f", "derivation": "\\operatorname{M_{E}}{(\\rho_f)} = e^{\\rho_f} and \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} = (e^{\\rho_f})^{\\rho_f} and \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f = (e^{\\rho_f})^{\\rho_f} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f and - \\operatorname{M_{E}}{(\\rho_f)} + \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f = - \\operatorname{M_{E}}{(\\rho_f)} + (e^{\\rho_f})^{\\rho_f} \\int \\operatorname{M_{E}}^{\\rho_f}{(\\rho_f)} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(exp(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["times", 2, "Integral(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integral(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Pow(exp(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integral(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["minus", 3, "Function('M_E')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integral(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(exp(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integral(Pow(Function('M_E')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\psi,m_{s})} = \\sin{(\\psi^{m_{s}})} and \\theta{(\\psi,m_{s})} = \\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})}, then obtain \\sin{(\\theta{(\\psi,m_{s})})} = \\sin{(\\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})})}", "derivation": "\\mathbf{s}{(\\psi,m_{s})} = \\sin{(\\psi^{m_{s}})} and \\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})} = \\frac{\\partial}{\\partial \\psi} \\sin{(\\psi^{m_{s}})} and \\theta{(\\psi,m_{s})} = \\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})} and \\sin{(\\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})})} = \\sin{(\\frac{\\partial}{\\partial \\psi} \\sin{(\\psi^{m_{s}})})} and \\sin{(\\theta{(\\psi,m_{s})})} = \\sin{(\\frac{\\partial}{\\partial \\psi} \\sin{(\\psi^{m_{s}})})} and \\sin{(\\theta{(\\psi,m_{s})})} = \\sin{(\\frac{\\partial}{\\partial \\psi} \\mathbf{s}{(\\psi,m_{s})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), sin(Derivative(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(sin(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))), sin(Derivative(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(sin(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True))), sin(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\dot{z})} = \\sin{(\\dot{z})} and m{(\\dot{z})} = \\log{(\\int \\sin{(\\dot{z})} d\\dot{z})}, then obtain \\dot{z} + \\log{(\\int \\sin{(\\dot{z})} d\\dot{z})} = \\dot{z} + \\log{(\\int \\eta^{\\prime}{(\\dot{z})} d\\dot{z})}", "derivation": "\\eta^{\\prime}{(\\dot{z})} = \\sin{(\\dot{z})} and \\int \\eta^{\\prime}{(\\dot{z})} d\\dot{z} = \\int \\sin{(\\dot{z})} d\\dot{z} and m{(\\dot{z})} = \\log{(\\int \\sin{(\\dot{z})} d\\dot{z})} and \\dot{z} + m{(\\dot{z})} = \\dot{z} + \\log{(\\int \\sin{(\\dot{z})} d\\dot{z})} and \\dot{z} + m{(\\dot{z})} = \\dot{z} + \\log{(\\int \\eta^{\\prime}{(\\dot{z})} d\\dot{z})} and \\dot{z} + \\log{(\\int \\sin{(\\dot{z})} d\\dot{z})} = \\dot{z} + \\log{(\\int \\eta^{\\prime}{(\\dot{z})} d\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\dot{z}', commutative=True)), log(Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), log(Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), log(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), log(Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))), Add(Symbol('\\\\dot{z}', commutative=True), log(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\mathbf{J}{(\\mathbf{A})} = \\mathbf{A} - e^{\\hat{p}_0{(\\mathbf{A})}}, then obtain \\frac{d^{2}}{d \\mathbf{A}^{2}} \\mathbf{J}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} 0", "derivation": "\\hat{p}_0{(\\mathbf{A})} = \\log{(\\mathbf{A})} and e^{\\hat{p}_0{(\\mathbf{A})}} = \\mathbf{A} and 0 = \\mathbf{A} - e^{\\hat{p}_0{(\\mathbf{A})}} and \\frac{d}{d \\mathbf{A}} 0 = \\frac{d}{d \\mathbf{A}} (\\mathbf{A} - e^{\\hat{p}_0{(\\mathbf{A})}}) and \\mathbf{J}{(\\mathbf{A})} = \\mathbf{A} - e^{\\hat{p}_0{(\\mathbf{A})}} and \\frac{d}{d \\mathbf{A}} \\mathbf{J}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} (\\mathbf{A} - e^{\\hat{p}_0{(\\mathbf{A})}}) and \\frac{d}{d \\mathbf{A}} \\mathbf{J}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} 0 and \\frac{d^{2}}{d \\mathbf{A}^{2}} \\mathbf{J}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))"], [["minus", 2, "exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True))))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["differentiate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), exp(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True))))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))))"]]}, {"prompt": "Given H{(v_{z},A_{2})} = A_{2} + v_{z}, then obtain \\frac{\\partial^{3}}{\\partial A_{2}\\partial v_{z}\\partial A_{2}} (A_{2} + H{(v_{z},A_{2})}) = \\frac{\\partial^{3}}{\\partial A_{2}\\partial v_{z}\\partial A_{2}} (2 A_{2} + v_{z})", "derivation": "H{(v_{z},A_{2})} = A_{2} + v_{z} and A_{2} + H{(v_{z},A_{2})} = 2 A_{2} + v_{z} and \\frac{\\partial}{\\partial A_{2}} (A_{2} + H{(v_{z},A_{2})}) = \\frac{\\partial}{\\partial A_{2}} (2 A_{2} + v_{z}) and \\frac{\\partial^{2}}{\\partial v_{z}\\partial A_{2}} (A_{2} + H{(v_{z},A_{2})}) = \\frac{\\partial^{2}}{\\partial v_{z}\\partial A_{2}} (2 A_{2} + v_{z}) and \\frac{\\partial^{3}}{\\partial A_{2}\\partial v_{z}\\partial A_{2}} (A_{2} + H{(v_{z},A_{2})}) = \\frac{\\partial^{3}}{\\partial A_{2}\\partial v_{z}\\partial A_{2}} (2 A_{2} + v_{z})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True)))"], [["add", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Function('H')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('v_z', commutative=True)))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Symbol('A_2', commutative=True), Function('H')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Symbol('A_2', commutative=True), Function('H')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Symbol('A_2', commutative=True), Function('H')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)}, then obtain - (\\operatorname{C_{2}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}) \\sin{(\\varepsilon_0)} + (\\sin{(\\varepsilon_0)} + \\frac{d}{d \\varepsilon_0} \\operatorname{C_{2}}{(\\varepsilon_0)}) \\cos{(\\varepsilon_0)} = 0", "derivation": "\\operatorname{C_{2}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\operatorname{C_{2}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)} = 0 and (\\operatorname{C_{2}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}) \\cos{(\\varepsilon_0)} = 0 and \\frac{d}{d \\varepsilon_0} (\\operatorname{C_{2}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}) \\cos{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} 0 and - (\\operatorname{C_{2}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}) \\sin{(\\varepsilon_0)} + (\\sin{(\\varepsilon_0)} + \\frac{d}{d \\varepsilon_0} \\operatorname{C_{2}}{(\\varepsilon_0)}) \\cos{(\\varepsilon_0)} = 0", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(0))"], [["divide", 2, "Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), cos(Symbol('\\\\varepsilon_0', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Mul(Add(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), cos(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Add(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), sin(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Add(sin(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Function('C_2')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\phi{(g)} = \\log{(\\sin{(g)})} and \\operatorname{c_{0}}{(g)} = (\\int \\log{(\\sin{(g)})} dg)^{g}, then obtain ((\\int \\log{(\\sin{(g)})} dg)^{g})^{g} + \\operatorname{c_{0}}^{g}{(g)} = 2 ((\\int \\log{(\\sin{(g)})} dg)^{g})^{g}", "derivation": "\\phi{(g)} = \\log{(\\sin{(g)})} and \\int \\phi{(g)} dg = \\int \\log{(\\sin{(g)})} dg and (\\int \\phi{(g)} dg)^{g} = (\\int \\log{(\\sin{(g)})} dg)^{g} and \\operatorname{c_{0}}{(g)} = (\\int \\log{(\\sin{(g)})} dg)^{g} and \\operatorname{c_{0}}{(g)} = (\\int \\phi{(g)} dg)^{g} and \\operatorname{c_{0}}^{g}{(g)} = ((\\int \\phi{(g)} dg)^{g})^{g} and \\operatorname{c_{0}}^{g}{(g)} = ((\\int \\log{(\\sin{(g)})} dg)^{g})^{g} and ((\\int \\log{(\\sin{(g)})} dg)^{g})^{g} + \\operatorname{c_{0}}^{g}{(g)} = 2 ((\\int \\log{(\\sin{(g)})} dg)^{g})^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('g', commutative=True)), log(sin(Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Integral(Function('\\\\phi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('g', commutative=True)), Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('c_0')(Symbol('g', commutative=True)), Pow(Integral(Function('\\\\phi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Integral(Function('\\\\phi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Function('c_0')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["add", 7, "Pow(Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))"], "Equality(Add(Pow(Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Function('c_0')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Integer(2), Pow(Pow(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\chi)} = e^{\\cos{(\\chi)}}, then obtain \\int \\frac{d}{d \\chi} (\\hat{H}{(\\chi)} \\cos{(\\chi)})^{\\chi} d\\chi = \\int \\frac{d}{d \\chi} (e^{\\cos{(\\chi)}} \\cos{(\\chi)})^{\\chi} d\\chi", "derivation": "\\hat{H}{(\\chi)} = e^{\\cos{(\\chi)}} and \\hat{H}{(\\chi)} \\cos{(\\chi)} = e^{\\cos{(\\chi)}} \\cos{(\\chi)} and (\\hat{H}{(\\chi)} \\cos{(\\chi)})^{\\chi} = (e^{\\cos{(\\chi)}} \\cos{(\\chi)})^{\\chi} and \\frac{d}{d \\chi} (\\hat{H}{(\\chi)} \\cos{(\\chi)})^{\\chi} = \\frac{d}{d \\chi} (e^{\\cos{(\\chi)}} \\cos{(\\chi)})^{\\chi} and \\int \\frac{d}{d \\chi} (\\hat{H}{(\\chi)} \\cos{(\\chi)})^{\\chi} d\\chi = \\int \\frac{d}{d \\chi} (e^{\\cos{(\\chi)}} \\cos{(\\chi)})^{\\chi} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), exp(cos(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Mul(exp(cos(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Mul(exp(cos(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(Mul(exp(cos(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Pow(Mul(exp(cos(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}} and T{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}}, then obtain V_{\\mathbf{E}} (- T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} = V_{\\mathbf{E}}", "derivation": "\\mathbf{F}{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}} and T{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}} and \\mathbf{F}{(V_{\\mathbf{E}})} = T{(V_{\\mathbf{E}})} and - T{(V_{\\mathbf{E}})} + \\mathbf{F}{(V_{\\mathbf{E}})} = - T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}} and 0 = - T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}} and 0^{V_{\\mathbf{E}}} = (- T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} and 0^{V_{\\mathbf{E}}} V_{\\mathbf{E}} = V_{\\mathbf{E}} (- T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} and V_{\\mathbf{E}} (- T{(V_{\\mathbf{E}})} + e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} = V_{\\mathbf{E}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["minus", 1, "Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["power", 5, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["times", 6, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True))"]]}, {"prompt": "Given \\mathbf{g}{(v_{x},l)} = v_{x}^{l}, then derive l (l \\frac{\\partial}{\\partial l} \\mathbf{g}{(v_{x},l)} + \\mathbf{g}{(v_{x},l)}) = l (l v_{x}^{l} \\log{(v_{x})} + v_{x}^{l}), then obtain l (l \\frac{\\partial}{\\partial l} \\mathbf{g}{(v_{x},l)} + \\mathbf{g}{(v_{x},l)}) = l (l \\mathbf{g}{(v_{x},l)} \\log{(v_{x})} + \\mathbf{g}{(v_{x},l)})", "derivation": "\\mathbf{g}{(v_{x},l)} = v_{x}^{l} and l \\mathbf{g}{(v_{x},l)} = l v_{x}^{l} and \\frac{\\partial}{\\partial l} l \\mathbf{g}{(v_{x},l)} = \\frac{\\partial}{\\partial l} l v_{x}^{l} and l \\frac{\\partial}{\\partial l} l \\mathbf{g}{(v_{x},l)} = l \\frac{\\partial}{\\partial l} l v_{x}^{l} and l (l \\frac{\\partial}{\\partial l} \\mathbf{g}{(v_{x},l)} + \\mathbf{g}{(v_{x},l)}) = l (l v_{x}^{l} \\log{(v_{x})} + v_{x}^{l}) and l (l \\frac{\\partial}{\\partial l} \\mathbf{g}{(v_{x},l)} + \\mathbf{g}{(v_{x},l)}) = l (l \\mathbf{g}{(v_{x},l)} \\log{(v_{x})} + \\mathbf{g}{(v_{x},l)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 3, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Derivative(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Symbol('l', commutative=True), Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('l', commutative=True), Add(Mul(Symbol('l', commutative=True), Derivative(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Add(Mul(Symbol('l', commutative=True), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), log(Symbol('v_x', commutative=True))), Pow(Symbol('v_x', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('l', commutative=True), Add(Mul(Symbol('l', commutative=True), Derivative(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Add(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)), log(Symbol('v_x', commutative=True))), Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{1})} = e^{\\cos{(v_{1})}}, then derive \\sin{(v_{1})} + \\frac{d}{d v_{1}} \\mathbf{J}_P{(v_{1})} = - e^{\\cos{(v_{1})}} \\sin{(v_{1})} + \\sin{(v_{1})}, then obtain \\sin{(v_{1})} + \\frac{d}{d v_{1}} e^{\\cos{(v_{1})}} = - e^{\\cos{(v_{1})}} \\sin{(v_{1})} + \\sin{(v_{1})}", "derivation": "\\mathbf{J}_P{(v_{1})} = e^{\\cos{(v_{1})}} and \\mathbf{J}_P{(v_{1})} - \\cos{(v_{1})} = e^{\\cos{(v_{1})}} - \\cos{(v_{1})} and \\frac{d}{d v_{1}} (\\mathbf{J}_P{(v_{1})} - \\cos{(v_{1})}) = \\frac{d}{d v_{1}} (e^{\\cos{(v_{1})}} - \\cos{(v_{1})}) and \\sin{(v_{1})} + \\frac{d}{d v_{1}} \\mathbf{J}_P{(v_{1})} = - e^{\\cos{(v_{1})}} \\sin{(v_{1})} + \\sin{(v_{1})} and \\sin{(v_{1})} + \\frac{d}{d v_{1}} \\mathbf{J}_P{(v_{1})} = - \\mathbf{J}_P{(v_{1})} \\sin{(v_{1})} + \\sin{(v_{1})} and \\sin{(v_{1})} + \\frac{d}{d v_{1}} e^{\\cos{(v_{1})}} = - e^{\\cos{(v_{1})}} \\sin{(v_{1})} + \\sin{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), exp(cos(Symbol('v_1', commutative=True))))"], [["minus", 1, "cos(Symbol('v_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Add(exp(cos(Symbol('v_1', commutative=True))), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(exp(cos(Symbol('v_1', commutative=True))), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(sin(Symbol('v_1', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(cos(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(sin(Symbol('v_1', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(sin(Symbol('v_1', commutative=True)), Derivative(exp(cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(cos(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain (2 \\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}})^{\\mathbf{M}} = (\\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}} + 1)^{\\mathbf{M}}", "derivation": "\\mathbf{E}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}} = 1 and 2 \\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}} = \\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}} + 1 and (2 \\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}})^{\\mathbf{M}} = (\\mathbf{E}{(\\mathbf{M})} e^{- \\mathbf{M}} + 1)^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1))"], [["add", 2, "Mul(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1)))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1)), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(F_{c})} = \\log{(\\cos{(F_{c})})}, then obtain \\cos{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{A_{x}}{(F_{c})} - 1 = - \\frac{\\sin{(F_{c})}}{\\cos{(F_{c})}} + \\cos{(F_{c})} - 1", "derivation": "\\operatorname{A_{x}}{(F_{c})} = \\log{(\\cos{(F_{c})})} and - F_{c} + \\operatorname{A_{x}}{(F_{c})} = - F_{c} + \\log{(\\cos{(F_{c})})} and \\frac{d}{d F_{c}} (- F_{c} + \\operatorname{A_{x}}{(F_{c})}) = \\frac{d}{d F_{c}} (- F_{c} + \\log{(\\cos{(F_{c})})}) and \\cos{(F_{c})} + \\frac{d}{d F_{c}} (- F_{c} + \\operatorname{A_{x}}{(F_{c})}) = \\cos{(F_{c})} + \\frac{d}{d F_{c}} (- F_{c} + \\log{(\\cos{(F_{c})})}) and \\cos{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{A_{x}}{(F_{c})} - 1 = - \\frac{\\sin{(F_{c})}}{\\cos{(F_{c})}} + \\cos{(F_{c})} - 1", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('F_c', commutative=True)), log(cos(Symbol('F_c', commutative=True))))"], [["minus", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), log(cos(Symbol('F_c', commutative=True)))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), log(cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["add", 3, "cos(Symbol('F_c', commutative=True))"], "Equality(Add(cos(Symbol('F_c', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(cos(Symbol('F_c', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), log(cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(cos(Symbol('F_c', commutative=True)), Derivative(Function('A_x')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), cos(Symbol('F_c', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given n{(\\theta)} = \\log{(e^{\\theta})} and \\delta{(t,g)} = \\frac{g}{t}, then obtain (- \\theta + \\log{(e^{\\theta})})^{\\theta} (- t + \\delta{(t,g)}) = (- \\theta + \\log{(e^{\\theta})})^{\\theta} (\\frac{g}{t} - t)", "derivation": "n{(\\theta)} = \\log{(e^{\\theta})} and - \\theta + n{(\\theta)} = - \\theta + \\log{(e^{\\theta})} and (- \\theta + n{(\\theta)})^{\\theta} = (- \\theta + \\log{(e^{\\theta})})^{\\theta} and \\delta{(t,g)} = \\frac{g}{t} and - t + \\delta{(t,g)} = \\frac{g}{t} - t and (- \\theta + n{(\\theta)})^{\\theta} (- t + \\delta{(t,g)}) = (- \\theta + n{(\\theta)})^{\\theta} (\\frac{g}{t} - t) and (- \\theta + \\log{(e^{\\theta})})^{\\theta} (- t + \\delta{(t,g)}) = (- \\theta + \\log{(e^{\\theta})})^{\\theta} (\\frac{g}{t} - t)", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True))))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('n')(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True)))))"], [["power", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('n')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('t', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["minus", 4, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\delta')(Symbol('t', commutative=True), Symbol('g', commutative=True))), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["times", 5, "Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('n')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('n')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\delta')(Symbol('t', commutative=True), Symbol('g', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('n')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\delta')(Symbol('t', commutative=True), Symbol('g', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{J}_M)} = \\sin{(\\sin{(\\mathbf{J}_M)})} and \\operatorname{r_{0}}{(f^{*})} = \\log{(f^{*})}, then obtain 1 = \\frac{\\log{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})}}{\\operatorname{r_{0}}{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})}}", "derivation": "\\mathbf{P}{(\\mathbf{J}_M)} = \\sin{(\\sin{(\\mathbf{J}_M)})} and \\log{(\\mathbf{P}{(\\mathbf{J}_M)})} = \\log{(\\sin{(\\sin{(\\mathbf{J}_M)})})} and \\operatorname{r_{0}}{(f^{*})} = \\log{(f^{*})} and \\operatorname{r_{0}}{(f^{*})} + \\log{(\\sin{(\\sin{(\\mathbf{J}_M)})})} = \\log{(f^{*})} + \\log{(\\sin{(\\sin{(\\mathbf{J}_M)})})} and \\operatorname{r_{0}}{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})} = \\log{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})} and 1 = \\frac{\\log{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})}}{\\operatorname{r_{0}}{(f^{*})} + \\log{(\\mathbf{P}{(\\mathbf{J}_M)})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True))), log(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], ["get_premise", "Equality(Function('r_0')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], [["add", 3, "log(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Add(Function('r_0')(Symbol('f^*', commutative=True)), log(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))), Add(log(Symbol('f^*', commutative=True)), log(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('r_0')(Symbol('f^*', commutative=True)), log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(log(Symbol('f^*', commutative=True)), log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["divide", 5, "Add(Function('r_0')(Symbol('f^*', commutative=True)), log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Function('r_0')(Symbol('f^*', commutative=True)), log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(-1)), Add(log(Symbol('f^*', commutative=True)), log(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{J}_M', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{x}{(C,q)} = C + \\sin{(q)}, then obtain \\int (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{2} (\\frac{\\partial}{\\partial q} \\sigma_{x}{(C,q)})^{2} dC = \\int (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{4} dC", "derivation": "\\sigma_{x}{(C,q)} = C + \\sin{(q)} and \\frac{\\partial}{\\partial q} \\sigma_{x}{(C,q)} = \\frac{\\partial}{\\partial q} (C + \\sin{(q)}) and \\frac{\\partial}{\\partial q} (C + \\sin{(q)}) \\frac{\\partial}{\\partial q} \\sigma_{x}{(C,q)} = (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{2} and (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{2} (\\frac{\\partial}{\\partial q} \\sigma_{x}{(C,q)})^{2} = (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{4} and \\int (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{2} (\\frac{\\partial}{\\partial q} \\sigma_{x}{(C,q)})^{2} dC = \\int (\\frac{\\partial}{\\partial q} (C + \\sin{(q)}))^{4} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\sigma_x')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(2)))"], [["power", 3, 2], "Equality(Mul(Pow(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Function('\\\\sigma_x')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(4)))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Pow(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Function('\\\\sigma_x')(Symbol('C', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(2))), Tuple(Symbol('C', commutative=True))), Integral(Pow(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(4)), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given Q{(\\mathbf{D},\\mu_0)} = \\frac{\\mathbf{D}}{\\mu_0}, then obtain \\frac{(\\int Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D}) \\iint Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} d\\mu_0}{\\int \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D}} = \\iint Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} d\\mu_0", "derivation": "Q{(\\mathbf{D},\\mu_0)} = \\frac{\\mathbf{D}}{\\mu_0} and \\int Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} = \\int \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D} and \\iint Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} d\\mu_0 = \\iint \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D} d\\mu_0 and \\frac{(\\int Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D}) \\iint \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D} d\\mu_0}{\\int \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D}} = \\iint \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D} d\\mu_0 and \\frac{(\\int Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D}) \\iint Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} d\\mu_0}{\\int \\frac{\\mathbf{D}}{\\mu_0} d\\mathbf{D}} = \\iint Q{(\\mathbf{D},\\mu_0)} d\\mathbf{D} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 2, "Mul(Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Pow(Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Integral(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = - \\sigma_p + \\sin{(\\hbar)} and \\pi{(\\sigma_p,\\hbar)} = \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)}, then derive \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = -1, then obtain \\frac{\\partial^{- \\pi{(\\sigma_p,\\hbar)}}}{\\partial \\sigma_p^{- \\pi{(\\sigma_p,\\hbar)}}} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = \\pi{(\\sigma_p,\\hbar)}", "derivation": "\\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = - \\sigma_p + \\sin{(\\hbar)} and \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = \\frac{\\partial}{\\partial \\sigma_p} (- \\sigma_p + \\sin{(\\hbar)}) and \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = -1 and \\pi{(\\sigma_p,\\hbar)} = \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} and \\pi{(\\sigma_p,\\hbar)} = -1 and \\frac{\\partial^{- \\pi{(\\sigma_p,\\hbar)}}}{\\partial \\sigma_p^{- \\pi{(\\sigma_p,\\hbar)}}} \\operatorname{E_{n}}{(\\sigma_p,\\hbar)} = \\pi{(\\sigma_p,\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('E_n')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\pi')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('E_n')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True))))), Function('\\\\pi')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\chi,l)} = - \\chi + l, then obtain \\frac{\\partial}{\\partial l} (\\chi - l) \\frac{\\partial}{\\partial l} \\sin{(\\mathbf{J}_P{(\\chi,l)})} = \\frac{\\partial}{\\partial l} (\\chi - l) \\frac{\\partial}{\\partial l} - \\sin{(\\chi - l)}", "derivation": "\\mathbf{J}_P{(\\chi,l)} = - \\chi + l and \\sin{(\\mathbf{J}_P{(\\chi,l)})} = - \\sin{(\\chi - l)} and \\frac{\\partial}{\\partial l} \\sin{(\\mathbf{J}_P{(\\chi,l)})} = \\frac{\\partial}{\\partial l} - \\sin{(\\chi - l)} and (\\chi - l) \\frac{\\partial}{\\partial l} \\sin{(\\mathbf{J}_P{(\\chi,l)})} = (\\chi - l) \\frac{\\partial}{\\partial l} - \\sin{(\\chi - l)} and \\frac{\\partial}{\\partial l} (\\chi - l) \\frac{\\partial}{\\partial l} \\sin{(\\mathbf{J}_P{(\\chi,l)})} = \\frac{\\partial}{\\partial l} (\\chi - l) \\frac{\\partial}{\\partial l} - \\sin{(\\chi - l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('l', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{J}_P')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(sin(Function('\\\\mathbf{J}_P')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 3, "Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Derivative(sin(Function('\\\\mathbf{J}_P')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Derivative(sin(Function('\\\\mathbf{J}_P')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(U)} = \\sin{(\\log{(U)})} and \\hat{p}{(U)} = \\log{(U)}, then obtain \\frac{d}{d U} \\sin{(\\log{(U)})} + \\int \\sin{(\\log{(U)})} dU = \\frac{d}{d U} x{(U)} + \\int \\sin{(\\log{(U)})} dU", "derivation": "x{(U)} = \\sin{(\\log{(U)})} and \\int x{(U)} dU = \\int \\sin{(\\log{(U)})} dU and \\hat{p}{(U)} = \\log{(U)} and x{(U)} = \\sin{(\\hat{p}{(U)})} and \\frac{d}{d U} x{(U)} = \\frac{d}{d U} \\sin{(\\hat{p}{(U)})} and \\frac{d}{d U} \\sin{(\\log{(U)})} = \\frac{d}{d U} \\sin{(\\hat{p}{(U)})} and \\frac{d}{d U} \\sin{(\\log{(U)})} = \\frac{d}{d U} x{(U)} and \\frac{d}{d U} \\sin{(\\log{(U)})} + \\int x{(U)} dU = \\frac{d}{d U} x{(U)} + \\int x{(U)} dU and \\frac{d}{d U} \\sin{(\\log{(U)})} + \\int \\sin{(\\log{(U)})} dU = \\frac{d}{d U} x{(U)} + \\int \\sin{(\\log{(U)})} dU", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('U', commutative=True)), sin(log(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('x')(Symbol('U', commutative=True)), sin(Function('\\\\hat{p}')(Symbol('U', commutative=True))))"], [["differentiate", 4, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Function('\\\\hat{p}')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Function('\\\\hat{p}')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["add", 7, "Integral(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Derivative(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Derivative(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Add(Derivative(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))), Add(Derivative(Function('x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given M{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\log{(g^{\\prime}_{\\varepsilon})}, then derive 0 = - M{(g^{\\prime}_{\\varepsilon})} + \\frac{1}{g^{\\prime}_{\\varepsilon}}, then obtain M{(g^{\\prime}_{\\varepsilon})} = \\frac{1}{g^{\\prime}_{\\varepsilon}}", "derivation": "M{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\log{(g^{\\prime}_{\\varepsilon})} and 0 = - M{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\log{(g^{\\prime}_{\\varepsilon})} and 0 = - M{(g^{\\prime}_{\\varepsilon})} + \\frac{1}{g^{\\prime}_{\\varepsilon}} and M{(g^{\\prime}_{\\varepsilon})} = \\frac{1}{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Derivative(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 1, "Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Derivative(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Integer(-1), Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Function('M')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon{(i,\\rho_b)} = \\rho_b i, then obtain \\sigma_p + (i + \\varepsilon{(i,\\rho_b)})^{\\rho_b} + (i + \\varepsilon{(i,\\rho_b)})^{i} = \\sigma_p + (i + \\varepsilon{(i,\\rho_b)})^{i} + (\\rho_b i + i)^{\\rho_b}", "derivation": "\\varepsilon{(i,\\rho_b)} = \\rho_b i and i + \\varepsilon{(i,\\rho_b)} = \\rho_b i + i and (i + \\varepsilon{(i,\\rho_b)})^{i} = (\\rho_b i + i)^{i} and (i + \\varepsilon{(i,\\rho_b)})^{\\rho_b} = (\\rho_b i + i)^{\\rho_b} and \\sigma_p + (i + \\varepsilon{(i,\\rho_b)})^{\\rho_b} + (\\rho_b i + i)^{i} = \\sigma_p + (\\rho_b i + i)^{\\rho_b} + (\\rho_b i + i)^{i} and \\sigma_p + (i + \\varepsilon{(i,\\rho_b)})^{\\rho_b} + (i + \\varepsilon{(i,\\rho_b)})^{i} = \\sigma_p + (i + \\varepsilon{(i,\\rho_b)})^{i} + (\\rho_b i + i)^{\\rho_b}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)))"], [["add", 1, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 4, "Add(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('i', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Symbol('i', commutative=True), Function('\\\\varepsilon')(Symbol('i', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given m{(b,T)} = e^{T - b}, then obtain - (T - b) e^{T - b} + \\iint m{(b,T)} dT db = - (T - b) e^{T - b} + \\iint e^{T - b} dT db", "derivation": "m{(b,T)} = e^{T - b} and \\int m{(b,T)} dT = \\int e^{T - b} dT and \\iint m{(b,T)} dT db = \\iint e^{T - b} dT db and - (T - b) e^{T - b} + \\iint m{(b,T)} dT db = - (T - b) e^{T - b} + \\iint e^{T - b} dT db", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('b', commutative=True), Symbol('T', commutative=True)), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('m')(Symbol('b', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Function('m')(Symbol('b', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 3, "Mul(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))), Integral(Function('m')(Symbol('b', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))), Integral(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(F_{N})} = \\cos{(F_{N})}, then obtain (\\psi^{*}{(F_{N})} - \\cos^{F_{N}}{(F_{N})}) \\cos^{F_{N}}{(F_{N})} = (- \\psi^{*}{(F_{N})} + 2 \\cos{(F_{N})} - \\cos^{F_{N}}{(F_{N})}) \\cos^{F_{N}}{(F_{N})}", "derivation": "\\psi^{*}{(F_{N})} = \\cos{(F_{N})} and \\cos{(F_{N})} = - \\psi^{*}{(F_{N})} + 2 \\cos{(F_{N})} and \\psi^{*}{(F_{N})} = - \\psi^{*}{(F_{N})} + 2 \\cos{(F_{N})} and \\psi^{*}{(F_{N})} - \\cos^{F_{N}}{(F_{N})} = - \\psi^{*}{(F_{N})} + 2 \\cos{(F_{N})} - \\cos^{F_{N}}{(F_{N})} and (\\psi^{*}{(F_{N})} - \\cos^{F_{N}}{(F_{N})}) \\cos^{F_{N}}{(F_{N})} = (- \\psi^{*}{(F_{N})} + 2 \\cos{(F_{N})} - \\cos^{F_{N}}{(F_{N})}) \\cos^{F_{N}}{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["minus", 1, "Add(Function('\\\\psi^*')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True))))"], "Equality(cos(Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('F_N', commutative=True))), Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('\\\\psi^*')(Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('F_N', commutative=True))), Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('F_N', commutative=True))), Mul(Integer(2), cos(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))))"], [["times", 4, "Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], "Equality(Mul(Add(Function('\\\\psi^*')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('F_N', commutative=True))), Mul(Integer(2), cos(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\delta{(\\eta,z)} = \\frac{\\partial}{\\partial \\eta} \\eta^{z}, then derive \\delta{(\\eta,z)} = \\frac{\\eta^{z} z}{\\eta}, then obtain 0^{z} = (- \\delta{(\\eta,z)} + \\frac{\\partial}{\\partial \\eta} \\eta^{z})^{z}", "derivation": "\\delta{(\\eta,z)} = \\frac{\\partial}{\\partial \\eta} \\eta^{z} and \\delta{(\\eta,z)} = \\frac{\\eta^{z} z}{\\eta} and \\frac{\\partial}{\\partial \\eta} \\eta^{z} = \\frac{\\eta^{z} z}{\\eta} and 0 = - \\frac{\\partial}{\\partial \\eta} \\eta^{z} + \\frac{\\eta^{z} z}{\\eta} and 0 = \\delta{(\\eta,z)} - \\frac{\\partial}{\\partial \\eta} \\eta^{z} and 0^{z} = (\\delta{(\\eta,z)} - \\frac{\\partial}{\\partial \\eta} \\eta^{z})^{z} and 0^{z} = (- \\frac{\\partial}{\\partial \\eta} \\eta^{z} + \\frac{\\eta^{z} z}{\\eta})^{z} and 0^{z} = (- \\delta{(\\eta,z)} + \\frac{\\eta^{z} z}{\\eta})^{z} and 0^{z} = (- \\delta{(\\eta,z)} + \\frac{\\partial}{\\partial \\eta} \\eta^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["minus", 3, "Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Add(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))))"], [["power", 5, "Symbol('z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('z', commutative=True)), Pow(Add(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Integer(0), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Pow(Integer(0), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Pow(Integer(0), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True))), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given C{(Z,v)} = Z + v and \\mathbf{M}{(Z,v)} = Z + v - C{(Z,v)} - 1 and z{(A_{x})} = e^{\\cos{(A_{x})}}, then obtain (z{(A_{x})} + 1) \\mathbf{M}{(Z,v)} = (e^{\\cos{(A_{x})}} + 1) \\mathbf{M}{(Z,v)}", "derivation": "C{(Z,v)} = Z + v and 0 = Z + v - C{(Z,v)} and -1 = Z + v - C{(Z,v)} - 1 and \\mathbf{M}{(Z,v)} = Z + v - C{(Z,v)} - 1 and -1 = \\mathbf{M}{(Z,v)} and z{(A_{x})} = e^{\\cos{(A_{x})}} and - \\mathbf{M}{(Z,v)} + z{(A_{x})} = - \\mathbf{M}{(Z,v)} + e^{\\cos{(A_{x})}} and z{(A_{x})} + 1 = e^{\\cos{(A_{x})}} + 1 and (z{(A_{x})} + 1) \\mathbf{M}{(Z,v)} = (e^{\\cos{(A_{x})}} + 1) \\mathbf{M}{(Z,v)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('Z', commutative=True), Symbol('v', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('v', commutative=True)))"], [["minus", 1, "Function('C')(Symbol('Z', commutative=True), Symbol('v', commutative=True))"], "Equality(Integer(0), Add(Symbol('Z', commutative=True), Symbol('v', commutative=True), Mul(Integer(-1), Function('C')(Symbol('Z', commutative=True), Symbol('v', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Symbol('Z', commutative=True), Symbol('v', commutative=True), Mul(Integer(-1), Function('C')(Symbol('Z', commutative=True), Symbol('v', commutative=True))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('v', commutative=True), Mul(Integer(-1), Function('C')(Symbol('Z', commutative=True), Symbol('v', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(-1), Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True)))"], ["get_premise", "Equality(Function('z')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True))))"], [["minus", 6, "Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))), Function('z')(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))), exp(cos(Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Function('z')(Symbol('A_x', commutative=True)), Integer(1)), Add(exp(cos(Symbol('A_x', commutative=True))), Integer(1)))"], [["times", 8, "Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Add(Function('z')(Symbol('A_x', commutative=True)), Integer(1)), Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))), Mul(Add(exp(cos(Symbol('A_x', commutative=True))), Integer(1)), Function('\\\\mathbf{M}')(Symbol('Z', commutative=True), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(s)} = \\log{(s)} and b{(F_{g})} = \\int \\log{(F_{g})} dF_{g}, then derive \\int \\dot{z}{(s)} ds = \\mathbf{E} + s \\log{(s)} - s, then obtain \\frac{b{(F_{g})}}{\\int \\dot{z}{(s)} ds} = \\frac{F_{g} \\log{(F_{g})} - F_{g} + \\hat{H}}{\\int \\dot{z}{(s)} ds}", "derivation": "\\dot{z}{(s)} = \\log{(s)} and \\int \\dot{z}{(s)} ds = \\int \\log{(s)} ds and b{(F_{g})} = \\int \\log{(F_{g})} dF_{g} and \\int \\dot{z}{(s)} ds = \\mathbf{E} + s \\log{(s)} - s and \\frac{b{(F_{g})}}{\\mathbf{E} + s \\log{(s)} - s} = \\frac{\\int \\log{(F_{g})} dF_{g}}{\\mathbf{E} + s \\log{(s)} - s} and \\frac{b{(F_{g})}}{\\int \\dot{z}{(s)} ds} = \\frac{\\int \\log{(F_{g})} dF_{g}}{\\int \\dot{z}{(s)} ds} and \\frac{b{(F_{g})}}{\\int \\dot{z}{(s)} ds} = \\frac{F_{g} \\log{(F_{g})} - F_{g} + \\hat{H}}{\\int \\dot{z}{(s)} ds}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], ["get_premise", "Equality(Function('b')(Symbol('F_g', commutative=True)), Integral(log(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Integer(-1)), Function('b')(Symbol('F_g', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Integer(-1)), Integral(log(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('b')(Symbol('F_g', commutative=True)), Pow(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(-1))), Mul(Pow(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(-1)), Integral(log(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Function('b')(Symbol('F_g', commutative=True)), Pow(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(-1))), Mul(Add(Mul(Symbol('F_g', commutative=True), log(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\psi{(Q)} = \\log{(Q)}, then obtain \\int (\\psi{(Q)} \\log{(Q)})^{Q} dQ = \\int (\\log{(Q)}^{2})^{Q} dQ", "derivation": "\\psi{(Q)} = \\log{(Q)} and \\psi{(Q)} \\log{(Q)} = \\log{(Q)}^{2} and (\\psi{(Q)} \\log{(Q)})^{Q} = (\\log{(Q)}^{2})^{Q} and \\int (\\psi{(Q)} \\log{(Q)})^{Q} dQ = \\int (\\log{(Q)}^{2})^{Q} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["times", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\psi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Pow(log(Symbol('Q', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Mul(Function('\\\\psi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Pow(log(Symbol('Q', commutative=True)), Integer(2)), Symbol('Q', commutative=True)))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\psi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Pow(log(Symbol('Q', commutative=True)), Integer(2)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given v{(E_{n},g^{\\prime}_{\\varepsilon})} = \\cos{(E_{n} - g^{\\prime}_{\\varepsilon})} and M{(E_{n},g^{\\prime}_{\\varepsilon})} = E_{n} - g^{\\prime}_{\\varepsilon}, then obtain 1 = \\frac{\\cos{(M{(E_{n},g^{\\prime}_{\\varepsilon})})}}{\\cos{(E_{n} - g^{\\prime}_{\\varepsilon})}}", "derivation": "v{(E_{n},g^{\\prime}_{\\varepsilon})} = \\cos{(E_{n} - g^{\\prime}_{\\varepsilon})} and 1 = \\frac{\\cos{(E_{n} - g^{\\prime}_{\\varepsilon})}}{v{(E_{n},g^{\\prime}_{\\varepsilon})}} and M{(E_{n},g^{\\prime}_{\\varepsilon})} = E_{n} - g^{\\prime}_{\\varepsilon} and 1 = \\frac{\\cos{(M{(E_{n},g^{\\prime}_{\\varepsilon})})}}{v{(E_{n},g^{\\prime}_{\\varepsilon})}} and 1 = \\frac{\\cos{(M{(E_{n},g^{\\prime}_{\\varepsilon})})}}{\\cos{(E_{n} - g^{\\prime}_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["divide", 1, "Function('v')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"], ["renaming_premise", "Equality(Function('M')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('v')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Function('M')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Pow(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Integer(-1)), cos(Function('M')(Symbol('E_n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\hat{H},V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\hat{H}, then obtain - V_{\\mathbf{B}} \\hat{H} + \\frac{\\lambda{(\\hat{H},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} + \\frac{1}{V_{\\mathbf{B}}} = - V_{\\mathbf{B}} \\hat{H} + \\hat{H} + \\frac{1}{V_{\\mathbf{B}}}", "derivation": "\\lambda{(\\hat{H},V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\hat{H} and \\frac{\\lambda{(\\hat{H},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\hat{H} and \\frac{\\lambda{(\\hat{H},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} + \\frac{1}{V_{\\mathbf{B}}} = \\hat{H} + \\frac{1}{V_{\\mathbf{B}}} and - V_{\\mathbf{B}} \\hat{H} + \\frac{\\lambda{(\\hat{H},V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} + \\frac{1}{V_{\\mathbf{B}}} = - V_{\\mathbf{B}} \\hat{H} + \\hat{H} + \\frac{1}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))"], [["add", 2, "Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))), Add(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given i{(U,\\phi_1)} = \\sin{(U + \\phi_1)}, then obtain \\frac{2 \\phi_1 i{(U,\\phi_1)}}{U} = \\frac{\\phi_1 i{(U,\\phi_1)}}{U} + \\frac{\\phi_1 \\sin{(U + \\phi_1)}}{U}", "derivation": "i{(U,\\phi_1)} = \\sin{(U + \\phi_1)} and \\phi_1 i{(U,\\phi_1)} = \\phi_1 \\sin{(U + \\phi_1)} and \\frac{\\phi_1 i{(U,\\phi_1)}}{U} = \\frac{\\phi_1 \\sin{(U + \\phi_1)}}{U} and \\frac{2 \\phi_1 i{(U,\\phi_1)}}{U} = \\frac{\\phi_1 i{(U,\\phi_1)}}{U} + \\frac{\\phi_1 \\sin{(U + \\phi_1)}}{U}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True)), sin(Add(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), sin(Add(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["divide", 2, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), sin(Add(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), sin(Add(Symbol('U', commutative=True), Symbol('\\\\phi_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} = e^{\\cos{(\\mathbf{J})}}, then obtain \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})}}{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} - \\cos{(\\mathbf{J})}} = \\frac{e^{\\cos{(\\mathbf{J})}}}{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} - \\cos{(\\mathbf{J})}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} = e^{\\cos{(\\mathbf{J})}} and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} - \\cos{(\\mathbf{J})} = e^{\\cos{(\\mathbf{J})}} - \\cos{(\\mathbf{J})} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})}}{e^{\\cos{(\\mathbf{J})}} - \\cos{(\\mathbf{J})}} = \\frac{e^{\\cos{(\\mathbf{J})}}}{e^{\\cos{(\\mathbf{J})}} - \\cos{(\\mathbf{J})}} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})}}{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} - \\cos{(\\mathbf{J})}} = \\frac{e^{\\cos{(\\mathbf{J})}}}{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{J})} - \\cos{(\\mathbf{J})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True)), exp(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 1, "cos(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Add(exp(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 1, "Add(exp(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Mul(Pow(Add(exp(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Add(exp(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1)), exp(cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1)), exp(cos(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{1},W)} = v_{1}^{W}, then obtain \\frac{(W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)})^{v_{1}}}{W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)}} = \\frac{(W + v_{1} v_{1}^{W})^{v_{1}}}{W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)}}", "derivation": "\\operatorname{y^{\\prime}}{(v_{1},W)} = v_{1}^{W} and v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)} = v_{1} v_{1}^{W} and W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)} = W + v_{1} v_{1}^{W} and (W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)})^{v_{1}} = (W + v_{1} v_{1}^{W})^{v_{1}} and \\frac{(W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)})^{v_{1}}}{W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)}} = \\frac{(W + v_{1} v_{1}^{W})^{v_{1}}}{W + v_{1} \\operatorname{y^{\\prime}}{(v_{1},W)}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))"], [["times", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('W', commutative=True))))"], [["add", 2, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Symbol('v_1', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Symbol('v_1', commutative=True)))"], [["divide", 4, "Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Integer(-1)), Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Symbol('v_1', commutative=True))), Mul(Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Symbol('v_1', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Symbol('v_1', commutative=True), Function('y^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('W', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{s}{(S,g)} = \\cos{(\\frac{S}{g})}, then obtain \\int \\mathbf{s}{(S,g)} dg = - \\int \\mathbf{s}{(S,g)} dS + \\int \\cos{(\\frac{S}{g})} dS + \\int \\cos{(\\frac{S}{g})} dg", "derivation": "\\mathbf{s}{(S,g)} = \\cos{(\\frac{S}{g})} and \\int \\mathbf{s}{(S,g)} dg = \\int \\cos{(\\frac{S}{g})} dg and \\int \\mathbf{s}{(S,g)} dS = \\int \\cos{(\\frac{S}{g})} dS and - \\frac{\\partial}{\\partial g} \\mathbf{s}{(S,g)} + \\int \\mathbf{s}{(S,g)} dS = - \\frac{\\partial}{\\partial g} \\mathbf{s}{(S,g)} + \\int \\cos{(\\frac{S}{g})} dS and 0 = - \\int \\mathbf{s}{(S,g)} dS + \\int \\cos{(\\frac{S}{g})} dS and \\int \\cos{(\\frac{S}{g})} dg = - \\int \\mathbf{s}{(S,g)} dS + \\int \\cos{(\\frac{S}{g})} dS + \\int \\cos{(\\frac{S}{g})} dg and \\int \\mathbf{s}{(S,g)} dg = - \\int \\mathbf{s}{(S,g)} dS + \\int \\cos{(\\frac{S}{g})} dS + \\int \\cos{(\\frac{S}{g})} dg", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('S', commutative=True))))"], [["minus", 3, "Derivative(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('S', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True)))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('S', commutative=True)))))"], [["add", 5, "Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True)))"], "Equality(Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True)))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('S', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{s}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True)))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('S', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{H},U)} = e^{U + \\mathbf{H}}, then obtain \\mathbf{g}{(\\mathbf{H},U)} - \\frac{\\partial}{\\partial U} \\mathbf{g}{(\\mathbf{H},U)} = e^{U + \\mathbf{H}} - \\frac{\\partial}{\\partial U} \\mathbf{g}{(\\mathbf{H},U)}", "derivation": "\\mathbf{g}{(\\mathbf{H},U)} = e^{U + \\mathbf{H}} and \\frac{\\partial}{\\partial U} \\mathbf{g}{(\\mathbf{H},U)} = \\frac{\\partial}{\\partial U} e^{U + \\mathbf{H}} and \\mathbf{g}{(\\mathbf{H},U)} - \\frac{\\partial}{\\partial U} e^{U + \\mathbf{H}} = e^{U + \\mathbf{H}} - \\frac{\\partial}{\\partial U} e^{U + \\mathbf{H}} and \\mathbf{g}{(\\mathbf{H},U)} - \\frac{\\partial}{\\partial U} \\mathbf{g}{(\\mathbf{H},U)} = e^{U + \\mathbf{H}} - \\frac{\\partial}{\\partial U} \\mathbf{g}{(\\mathbf{H},U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))), Add(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Derivative(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))), Add(exp(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\theta{(m)} = \\log{(e^{m})}, then obtain (\\int (\\frac{\\theta{(m)}}{m} + \\frac{\\log{(e^{m})}}{m}) dm)^{2} = (\\int \\frac{2 \\log{(e^{m})}}{m} dm)^{2}", "derivation": "\\theta{(m)} = \\log{(e^{m})} and \\frac{\\theta{(m)}}{m} = \\frac{\\log{(e^{m})}}{m} and \\frac{\\theta{(m)}}{m} + \\frac{\\log{(e^{m})}}{m} = \\frac{2 \\log{(e^{m})}}{m} and \\int (\\frac{\\theta{(m)}}{m} + \\frac{\\log{(e^{m})}}{m}) dm = \\int \\frac{2 \\log{(e^{m})}}{m} dm and (\\int (\\frac{\\theta{(m)}}{m} + \\frac{\\log{(e^{m})}}{m}) dm)^{2} = (\\int \\frac{2 \\log{(e^{m})}}{m} dm)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True))))"], [["divide", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True))))), Mul(Integer(2), Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True)))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), Pow(Symbol('m', commutative=True), Integer(-1)), log(exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{S}{(F_{g},L)} = F_{g}^{L}, then obtain e^{\\frac{\\partial}{\\partial F_{g}} \\cos{(\\mathbf{S}{(F_{g},L)})}} = e^{\\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g}^{L})}}", "derivation": "\\mathbf{S}{(F_{g},L)} = F_{g}^{L} and \\cos{(\\mathbf{S}{(F_{g},L)})} = \\cos{(F_{g}^{L})} and \\frac{\\partial}{\\partial F_{g}} \\cos{(\\mathbf{S}{(F_{g},L)})} = \\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g}^{L})} and e^{\\frac{\\partial}{\\partial F_{g}} \\cos{(\\mathbf{S}{(F_{g},L)})}} = e^{\\frac{\\partial}{\\partial F_{g}} \\cos{(F_{g}^{L})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('F_g', commutative=True), Symbol('L', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), cos(Pow(Symbol('F_g', commutative=True), Symbol('L', commutative=True))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(cos(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(cos(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), exp(Derivative(cos(Pow(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{M})} = \\cos{(\\mathbf{M})}, then obtain \\frac{\\dot{x}{(\\mathbf{M})} \\sin^{\\mathbf{M}}{(\\frac{\\dot{x}{(\\mathbf{M})}}{\\cos{(\\mathbf{M})}})}}{\\cos{(\\mathbf{M})}} = \\frac{\\dot{x}{(\\mathbf{M})} \\sin^{\\mathbf{M}}{(1)}}{\\cos{(\\mathbf{M})}}", "derivation": "\\dot{x}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\frac{\\dot{x}{(\\mathbf{M})}}{\\cos{(\\mathbf{M})}} = 1 and \\sin{(\\frac{\\dot{x}{(\\mathbf{M})}}{\\cos{(\\mathbf{M})}})} = \\sin{(1)} and \\sin^{\\mathbf{M}}{(\\frac{\\dot{x}{(\\mathbf{M})}}{\\cos{(\\mathbf{M})}})} = \\sin^{\\mathbf{M}}{(1)} and \\frac{\\dot{x}{(\\mathbf{M})} \\sin^{\\mathbf{M}}{(\\frac{\\dot{x}{(\\mathbf{M})}}{\\cos{(\\mathbf{M})}})}}{\\cos{(\\mathbf{M})}} = \\frac{\\dot{x}{(\\mathbf{M})} \\sin^{\\mathbf{M}}{(1)}}{\\cos{(\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Integer(1))"], [["sin", 2], "Equality(sin(Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)))), sin(Integer(1)))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(sin(Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Integer(1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 4, "Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Integer(1)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given U{(E_{x},\\mathbf{g})} = \\frac{e^{\\mathbf{g}}}{E_{x}}, then obtain \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g}^{2} + \\mathbf{g} U{(E_{x},\\mathbf{g})}) = \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g}^{2} + \\frac{\\mathbf{g} e^{\\mathbf{g}}}{E_{x}})", "derivation": "U{(E_{x},\\mathbf{g})} = \\frac{e^{\\mathbf{g}}}{E_{x}} and - \\mathbf{g} + U{(E_{x},\\mathbf{g})} = - \\mathbf{g} + \\frac{e^{\\mathbf{g}}}{E_{x}} and \\mathbf{g} (- \\mathbf{g} + U{(E_{x},\\mathbf{g})}) = \\mathbf{g} (- \\mathbf{g} + \\frac{e^{\\mathbf{g}}}{E_{x}}) and - \\mathbf{g}^{2} + \\mathbf{g} U{(E_{x},\\mathbf{g})} = - \\mathbf{g}^{2} + \\frac{\\mathbf{g} e^{\\mathbf{g}}}{E_{x}} and \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g}^{2} + \\mathbf{g} U{(E_{x},\\mathbf{g})}) = \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g}^{2} + \\frac{\\mathbf{g} e^{\\mathbf{g}}}{E_{x}})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["times", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{g}', commutative=True))))))"], [["expand", 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('U')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('U')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and \\rho_{f}{(J_{\\varepsilon})} = J_{\\varepsilon}, then obtain - J_{\\varepsilon} \\log{(J_{\\varepsilon})} + \\rho_{f}{(J_{\\varepsilon})} = - J_{\\varepsilon} \\log{(J_{\\varepsilon})} + J_{\\varepsilon}", "derivation": "\\operatorname{A_{z}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and J_{\\varepsilon} \\operatorname{A_{z}}{(J_{\\varepsilon})} = J_{\\varepsilon} \\log{(J_{\\varepsilon})} and \\rho_{f}{(J_{\\varepsilon})} = J_{\\varepsilon} and - J_{\\varepsilon} \\operatorname{A_{z}}{(J_{\\varepsilon})} + \\rho_{f}{(J_{\\varepsilon})} = - J_{\\varepsilon} \\operatorname{A_{z}}{(J_{\\varepsilon})} + J_{\\varepsilon} and - J_{\\varepsilon} \\log{(J_{\\varepsilon})} + \\rho_{f}{(J_{\\varepsilon})} = - J_{\\varepsilon} \\log{(J_{\\varepsilon})} + J_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["minus", 3, "Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True))), Function('\\\\rho_f')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Function('\\\\rho_f')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given t{(p,\\hat{x}_0)} = \\hat{x}_0 + \\log{(p)}, then derive \\frac{\\partial^{3}}{\\partial p^{3}} t{(p,\\hat{x}_0)} = \\frac{2}{p^{3}}, then obtain \\frac{\\partial^{3}}{\\partial p^{3}} (\\hat{x}_0 + \\log{(p)}) = \\frac{2}{p^{3}}", "derivation": "t{(p,\\hat{x}_0)} = \\hat{x}_0 + \\log{(p)} and \\frac{\\partial}{\\partial p} t{(p,\\hat{x}_0)} = \\frac{\\partial}{\\partial p} (\\hat{x}_0 + \\log{(p)}) and \\frac{\\partial^{2}}{\\partial p^{2}} t{(p,\\hat{x}_0)} = \\frac{\\partial^{2}}{\\partial p^{2}} (\\hat{x}_0 + \\log{(p)}) and \\frac{\\partial^{3}}{\\partial p^{3}} t{(p,\\hat{x}_0)} = \\frac{\\partial^{3}}{\\partial p^{3}} (\\hat{x}_0 + \\log{(p)}) and \\frac{\\partial^{3}}{\\partial p^{3}} t{(p,\\hat{x}_0)} = \\frac{2}{p^{3}} and \\frac{\\partial^{3}}{\\partial p^{3}} (\\hat{x}_0 + \\log{(p)}) = \\frac{2}{p^{3}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('p', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('p', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('p', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('p', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(3))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(3))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('t')(Symbol('p', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(3))), Mul(Integer(2), Pow(Symbol('p', commutative=True), Integer(-3))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(3))), Mul(Integer(2), Pow(Symbol('p', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\mathbf{g}{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})}, then obtain \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} \\mathbf{g}{(f_{\\mathbf{v}})} = - \\sin{(f_{\\mathbf{v}})}", "derivation": "\\mathbf{g}{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})} and \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{g}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} and \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} \\mathbf{g}{(f_{\\mathbf{v}})} = \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} \\sin{(f_{\\mathbf{v}})} and \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} \\mathbf{g}{(f_{\\mathbf{v}})} = - \\sin{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))), Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(n_{2})} = \\log{(n_{2})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} (- e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\mathbf{r}{(n_{2})})}) = \\frac{\\partial}{\\partial \\mathbf{J}} (- e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\log{(n_{2})})})", "derivation": "\\mathbf{r}{(n_{2})} = \\log{(n_{2})} and \\cos{(\\mathbf{r}{(n_{2})})} = \\cos{(\\log{(n_{2})})} and \\frac{d}{d n_{2}} \\cos{(\\mathbf{r}{(n_{2})})} = \\frac{d}{d n_{2}} \\cos{(\\log{(n_{2})})} and - e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\mathbf{r}{(n_{2})})} = - e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\log{(n_{2})})} and \\frac{\\partial}{\\partial \\mathbf{J}} (- e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\mathbf{r}{(n_{2})})}) = \\frac{\\partial}{\\partial \\mathbf{J}} (- e^{- \\mathbf{A} + \\mathbf{J}} + \\frac{d}{d n_{2}} \\cos{(\\log{(n_{2})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{r}')(Symbol('n_2', commutative=True))), cos(log(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(cos(Function('\\\\mathbf{r}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["minus", 3, "exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))), Derivative(cos(Function('\\\\mathbf{r}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))), Derivative(cos(log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))), Derivative(cos(Function('\\\\mathbf{r}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))), Derivative(cos(log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(A)} = \\log{(A)}, then derive \\int I{(A)} dA = A \\log{(A)} - A + l, then obtain \\frac{\\partial}{\\partial A} (A \\log{(A)} - A + l)^{A} = \\frac{d}{d A} (\\int \\log{(A)} dA)^{A}", "derivation": "I{(A)} = \\log{(A)} and \\int I{(A)} dA = \\int \\log{(A)} dA and (\\int I{(A)} dA)^{A} = (\\int \\log{(A)} dA)^{A} and \\frac{d}{d A} (\\int I{(A)} dA)^{A} = \\frac{d}{d A} (\\int \\log{(A)} dA)^{A} and \\int I{(A)} dA = A \\log{(A)} - A + l and A \\log{(A)} - A + l = \\int \\log{(A)} dA and (\\int I{(A)} dA)^{A} = (A \\log{(A)} - A + l)^{A} and \\frac{\\partial}{\\partial A} (A \\log{(A)} - A + l)^{A} = \\frac{d}{d A} (\\int \\log{(A)} dA)^{A}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('I')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Function('I')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('I')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('l', commutative=True)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Integral(Function('I')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Add(Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('l', commutative=True)), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Derivative(Pow(Add(Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('l', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)} = - \\mathbf{J}_f + e^{\\dot{\\mathbf{r}}}, then obtain (- \\frac{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)}}{\\mathbf{J}_f - e^{\\dot{\\mathbf{r}}}})^{\\dot{\\mathbf{r}}} = 1", "derivation": "\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)} = - \\mathbf{J}_f + e^{\\dot{\\mathbf{r}}} and - \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)} = \\mathbf{J}_f - e^{\\dot{\\mathbf{r}}} and - \\frac{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)}}{\\mathbf{J}_f - e^{\\dot{\\mathbf{r}}}} = 1 and (- \\frac{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},\\mathbf{J}_f)}}{\\mathbf{J}_f - e^{\\dot{\\mathbf{r}}}})^{\\dot{\\mathbf{r}}} = 1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["divide", 2, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integer(-1)), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(1))"], [["power", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integer(-1)), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbb{I},g)} = - \\mathbb{I} + g, then obtain 0 = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + g) - \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{F_{N}}{(\\mathbb{I},g)}", "derivation": "\\operatorname{F_{N}}{(\\mathbb{I},g)} = - \\mathbb{I} + g and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{F_{N}}{(\\mathbb{I},g)} = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + g) and - g + \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{F_{N}}{(\\mathbb{I},g)} = - g + \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + g) and 0 = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + g) - \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{F_{N}}{(\\mathbb{I},g)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Function('F_N')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Function('F_N')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('F_N')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given T{(\\dot{z},l)} = - \\dot{z} + l, then obtain - \\dot{z} (- 3 \\dot{z} + 2 l) = - \\dot{z} (- 2 \\dot{z} + l + T{(\\dot{z},l)})", "derivation": "T{(\\dot{z},l)} = - \\dot{z} + l and - \\dot{z} + T{(\\dot{z},l)} = - 2 \\dot{z} + l and - 2 \\dot{z} + l + T{(\\dot{z},l)} = - 3 \\dot{z} + 2 l and - \\dot{z} + 2 T{(\\dot{z},l)} = - 3 \\dot{z} + 2 l and - \\dot{z} (- \\dot{z} + 2 T{(\\dot{z},l)}) = - \\dot{z} (- 3 \\dot{z} + 2 l) and - \\dot{z} (- \\dot{z} + 2 T{(\\dot{z},l)}) = - \\dot{z} (- 2 \\dot{z} + l + T{(\\dot{z},l)}) and - \\dot{z} (- 3 \\dot{z} + 2 l) = - \\dot{z} (- 2 \\dot{z} + l + T{(\\dot{z},l)})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True))))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True))))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True)))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)), Symbol('l', commutative=True), Function('T')(Symbol('\\\\dot{z}', commutative=True), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given J{(q,\\phi_1)} = - \\phi_1 + q, then derive J{(q,\\phi_1)} + \\log{(2 \\frac{\\partial}{\\partial \\phi_1} J{(q,\\phi_1)})} = J{(q,\\phi_1)} + \\log{(\\frac{\\partial}{\\partial \\phi_1} J{(q,\\phi_1)} - 1)}, then obtain - \\phi_1 + q + \\log{(2 \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q))} = - \\phi_1 + q + \\log{(\\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q) - 1)}", "derivation": "J{(q,\\phi_1)} = - \\phi_1 + q and 2 J{(q,\\phi_1)} = - \\phi_1 + q + J{(q,\\phi_1)} and \\frac{\\partial}{\\partial \\phi_1} 2 J{(q,\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q + J{(q,\\phi_1)}) and \\log{(\\frac{\\partial}{\\partial \\phi_1} 2 J{(q,\\phi_1)})} = \\log{(\\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q + J{(q,\\phi_1)}))} and J{(q,\\phi_1)} + \\log{(\\frac{\\partial}{\\partial \\phi_1} 2 J{(q,\\phi_1)})} = J{(q,\\phi_1)} + \\log{(\\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q + J{(q,\\phi_1)}))} and J{(q,\\phi_1)} + \\log{(2 \\frac{\\partial}{\\partial \\phi_1} J{(q,\\phi_1)})} = J{(q,\\phi_1)} + \\log{(\\frac{\\partial}{\\partial \\phi_1} J{(q,\\phi_1)} - 1)} and - \\phi_1 + q + \\log{(2 \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q))} = - \\phi_1 + q + \\log{(\\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + q) - 1)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True)))"], [["add", 1, "Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(2), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Integer(2), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), log(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["add", 4, "Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Derivative(Mul(Integer(2), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))), Add(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Mul(Integer(2), Derivative(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))), Add(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Add(Derivative(Function('J')(Symbol('q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), log(Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True), log(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\Psi{(u)} = u and \\theta{(u)} = - u, then obtain (u^{2} - u) \\Psi{(u)} \\theta{(u)} = - u (u^{2} - u) \\Psi{(u)}", "derivation": "\\Psi{(u)} = u and u \\Psi{(u)} = u^{2} and u \\Psi{(u)} - u = u^{2} - u and (u \\Psi{(u)} - u) \\Psi{(u)} = (u^{2} - u) \\Psi{(u)} and \\theta{(u)} = - u and (u \\Psi{(u)} - u) \\Psi{(u)} \\theta{(u)} = - u (u \\Psi{(u)} - u) \\Psi{(u)} and (u^{2} - u) \\Psi{(u)} \\theta{(u)} = - u (u^{2} - u) \\Psi{(u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Pow(Symbol('u', commutative=True), Integer(2)))"], [["add", 2, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Add(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Add(Pow(Symbol('u', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["times", 3, "Function('\\\\Psi')(Symbol('u', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Add(Pow(Symbol('u', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)))"], [["times", 5, "Mul(Add(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Mul(Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Pow(Symbol('u', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Pow(Symbol('u', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\Psi')(Symbol('u', commutative=True))))"]]}, {"prompt": "Given Q{(\\varepsilon)} = \\varepsilon and E{(\\varepsilon)} = \\int Q{(\\varepsilon)} d\\varepsilon, then derive \\frac{d}{d \\varepsilon} E{(\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} (F_{H} + \\frac{\\varepsilon^{2}}{2}), then obtain a^{2}{(\\varepsilon)} (\\frac{d}{d \\varepsilon} E{(\\varepsilon)})^{2} = a^{2}{(\\varepsilon)} (\\frac{\\partial}{\\partial \\varepsilon} (F_{H} + \\frac{\\varepsilon^{2}}{2}))^{2}", "derivation": "Q{(\\varepsilon)} = \\varepsilon and E{(\\varepsilon)} = \\int Q{(\\varepsilon)} d\\varepsilon and E{(\\varepsilon)} = \\int \\varepsilon d\\varepsilon and \\frac{d}{d \\varepsilon} E{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\int \\varepsilon d\\varepsilon and \\frac{d}{d \\varepsilon} E{(\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} (F_{H} + \\frac{\\varepsilon^{2}}{2}) and a{(\\varepsilon)} \\frac{d}{d \\varepsilon} E{(\\varepsilon)} = a{(\\varepsilon)} \\frac{\\partial}{\\partial \\varepsilon} (F_{H} + \\frac{\\varepsilon^{2}}{2}) and a^{2}{(\\varepsilon)} (\\frac{d}{d \\varepsilon} E{(\\varepsilon)})^{2} = a^{2}{(\\varepsilon)} (\\frac{\\partial}{\\partial \\varepsilon} (F_{H} + \\frac{\\varepsilon^{2}}{2}))^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('Q')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Integral(Function('Q')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["divide", 5, "Pow(Function('a')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))"], "Equality(Mul(Function('a')(Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(Function('a')(Symbol('\\\\varepsilon', commutative=True)), Derivative(Add(Symbol('F_H', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["power", 6, 2], "Equality(Mul(Pow(Function('a')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Pow(Derivative(Function('E')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Function('a')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Pow(Derivative(Add(Symbol('F_H', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\phi{(v_{x},C_{2})} = C_{2} + v_{x}, then obtain C_{2} \\frac{\\partial}{\\partial v_{x}} \\phi{(v_{x},C_{2})} = C_{2}", "derivation": "\\phi{(v_{x},C_{2})} = C_{2} + v_{x} and C_{2} \\phi{(v_{x},C_{2})} = C_{2} (C_{2} + v_{x}) and \\frac{\\partial}{\\partial v_{x}} C_{2} \\phi{(v_{x},C_{2})} = \\frac{\\partial}{\\partial v_{x}} C_{2} (C_{2} + v_{x}) and C_{2} \\frac{\\partial}{\\partial v_{x}} \\phi{(v_{x},C_{2})} = C_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('v_x', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('v_x', commutative=True)))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('\\\\phi')(Symbol('v_x', commutative=True), Symbol('C_2', commutative=True))), Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Symbol('v_x', commutative=True))))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('C_2', commutative=True), Function('\\\\phi')(Symbol('v_x', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('C_2', commutative=True), Derivative(Function('\\\\phi')(Symbol('v_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Symbol('C_2', commutative=True))"]]}, {"prompt": "Given k{(v_{2},\\mathbf{B})} = \\sin{(\\mathbf{B} - v_{2})}, then derive \\int k{(v_{2},\\mathbf{B})} d\\mathbf{B} = P_{g} - \\cos{(\\mathbf{B} - v_{2})}, then obtain - \\cos{(\\mathbf{B} - v_{2})} \\int \\sin{(\\mathbf{B} - v_{2})} d\\mathbf{B} = - (P_{g} - \\cos{(\\mathbf{B} - v_{2})}) \\cos{(\\mathbf{B} - v_{2})}", "derivation": "k{(v_{2},\\mathbf{B})} = \\sin{(\\mathbf{B} - v_{2})} and \\int k{(v_{2},\\mathbf{B})} d\\mathbf{B} = \\int \\sin{(\\mathbf{B} - v_{2})} d\\mathbf{B} and \\int k{(v_{2},\\mathbf{B})} d\\mathbf{B} = P_{g} - \\cos{(\\mathbf{B} - v_{2})} and \\int \\sin{(\\mathbf{B} - v_{2})} d\\mathbf{B} = P_{g} - \\cos{(\\mathbf{B} - v_{2})} and - \\cos{(\\mathbf{B} - v_{2})} \\int \\sin{(\\mathbf{B} - v_{2})} d\\mathbf{B} = - (P_{g} - \\cos{(\\mathbf{B} - v_{2})}) \\cos{(\\mathbf{B} - v_{2})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), sin(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(sin(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('P_g', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('P_g', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))))"], [["times", 4, "Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], "Equality(Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Integral(sin(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(-1), Add(Symbol('P_g', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))), cos(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"]]}, {"prompt": "Given \\eta{(f^{\\prime},F_{x})} = \\log{(F_{x} - f^{\\prime})} and \\mu_{0}{(f^{\\prime},F_{x})} = \\log{(F_{x} - f^{\\prime})}, then obtain - f^{\\prime} - \\log{(F_{x} - f^{\\prime})} = - f^{\\prime} - \\mu_{0}{(f^{\\prime},F_{x})}", "derivation": "\\eta{(f^{\\prime},F_{x})} = \\log{(F_{x} - f^{\\prime})} and - \\eta{(f^{\\prime},F_{x})} = - \\log{(F_{x} - f^{\\prime})} and - f^{\\prime} - \\eta{(f^{\\prime},F_{x})} = - f^{\\prime} - \\log{(F_{x} - f^{\\prime})} and \\mu_{0}{(f^{\\prime},F_{x})} = \\log{(F_{x} - f^{\\prime})} and - f^{\\prime} - \\eta{(f^{\\prime},F_{x})} = - f^{\\prime} - \\mu_{0}{(f^{\\prime},F_{x})} and - f^{\\prime} - \\log{(F_{x} - f^{\\prime})} = - f^{\\prime} - \\mu_{0}{(f^{\\prime},F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), log(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\eta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), log(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))))"], [["minus", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), log(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(A_{2})} = \\log{(\\cos{(A_{2})})} and \\psi^{*}{(A_{2})} = \\frac{\\operatorname{t_{2}}{(A_{2})} \\log{(\\cos{(A_{2})})}}{A_{2}^{2} \\cos^{2}{(A_{2})}}, then obtain \\psi^{*}{(A_{2})} = \\frac{\\operatorname{t_{2}}^{2}{(A_{2})}}{A_{2}^{2} \\cos^{2}{(A_{2})}}", "derivation": "\\operatorname{t_{2}}{(A_{2})} = \\log{(\\cos{(A_{2})})} and \\frac{\\operatorname{t_{2}}{(A_{2})}}{A_{2}} = \\frac{\\log{(\\cos{(A_{2})})}}{A_{2}} and \\frac{\\operatorname{t_{2}}{(A_{2})}}{A_{2} \\cos{(A_{2})}} = \\frac{\\log{(\\cos{(A_{2})})}}{A_{2} \\cos{(A_{2})}} and \\frac{\\operatorname{t_{2}}^{2}{(A_{2})}}{A_{2}^{2} \\cos^{2}{(A_{2})}} = \\frac{\\operatorname{t_{2}}{(A_{2})} \\log{(\\cos{(A_{2})})}}{A_{2}^{2} \\cos^{2}{(A_{2})}} and \\psi^{*}{(A_{2})} = \\frac{\\operatorname{t_{2}}{(A_{2})} \\log{(\\cos{(A_{2})})}}{A_{2}^{2} \\cos^{2}{(A_{2})}} and \\psi^{*}{(A_{2})} = \\frac{\\operatorname{t_{2}}^{2}{(A_{2})}}{A_{2}^{2} \\cos^{2}{(A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True))))"], [["divide", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('t_2')(Symbol('A_2', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), log(cos(Symbol('A_2', commutative=True)))))"], [["divide", 2, "cos(Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('t_2')(Symbol('A_2', commutative=True)), Pow(cos(Symbol('A_2', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), log(cos(Symbol('A_2', commutative=True))), Pow(cos(Symbol('A_2', commutative=True)), Integer(-1))))"], [["times", 3, "Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('t_2')(Symbol('A_2', commutative=True)), Pow(cos(Symbol('A_2', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('t_2')(Symbol('A_2', commutative=True)), Integer(2)), Pow(cos(Symbol('A_2', commutative=True)), Integer(-2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Function('t_2')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True))), Pow(cos(Symbol('A_2', commutative=True)), Integer(-2))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Function('t_2')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True))), Pow(cos(Symbol('A_2', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('t_2')(Symbol('A_2', commutative=True)), Integer(2)), Pow(cos(Symbol('A_2', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{J}{(z)} = \\cos{(z)} and m{(F_{g},s)} = F_{g} s, then obtain \\frac{- \\sigma_x - \\mathbf{J}{(z)}}{F_{g} s} = \\frac{- \\sigma_x - \\cos{(z)}}{F_{g} s}", "derivation": "\\mathbf{J}{(z)} = \\cos{(z)} and - \\mathbf{J}{(z)} = - \\cos{(z)} and - \\sigma_x - \\mathbf{J}{(z)} = - \\sigma_x - \\cos{(z)} and m{(F_{g},s)} = F_{g} s and \\frac{- \\sigma_x - \\mathbf{J}{(z)}}{m{(F_{g},s)}} = \\frac{- \\sigma_x - \\cos{(z)}}{m{(F_{g},s)}} and \\frac{- \\sigma_x - \\mathbf{J}{(z)}}{F_{g} s} = \\frac{- \\sigma_x - \\cos{(z)}}{F_{g} s}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('z', commutative=True))), Mul(Integer(-1), cos(Symbol('z', commutative=True))))"], [["minus", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], ["get_premise", "Equality(Function('m')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('s', commutative=True)))"], [["divide", 3, "Function('m')(Symbol('F_g', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('z', commutative=True)))), Pow(Function('m')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Pow(Function('m')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('z', commutative=True))))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given f{(\\hbar)} = \\log{(\\log{(\\hbar)})}, then derive \\frac{d}{d \\hbar} f{(\\hbar)} = \\frac{1}{\\hbar \\log{(\\hbar)}}, then obtain \\int (\\int \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} d\\hbar)^{\\hbar} d\\hbar = \\int (\\int \\frac{1}{\\hbar \\log{(\\hbar)}} d\\hbar)^{\\hbar} d\\hbar", "derivation": "f{(\\hbar)} = \\log{(\\log{(\\hbar)})} and \\frac{d}{d \\hbar} f{(\\hbar)} = \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} and \\frac{d}{d \\hbar} f{(\\hbar)} = \\frac{1}{\\hbar \\log{(\\hbar)}} and \\int \\frac{d}{d \\hbar} f{(\\hbar)} d\\hbar = \\int \\frac{1}{\\hbar \\log{(\\hbar)}} d\\hbar and (\\int \\frac{d}{d \\hbar} f{(\\hbar)} d\\hbar)^{\\hbar} = (\\int \\frac{1}{\\hbar \\log{(\\hbar)}} d\\hbar)^{\\hbar} and \\int (\\int \\frac{d}{d \\hbar} f{(\\hbar)} d\\hbar)^{\\hbar} d\\hbar = \\int (\\int \\frac{1}{\\hbar \\log{(\\hbar)}} d\\hbar)^{\\hbar} d\\hbar and \\int (\\int \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} d\\hbar)^{\\hbar} d\\hbar = \\int (\\int \\frac{1}{\\hbar \\log{(\\hbar)}} d\\hbar)^{\\hbar} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\hbar', commutative=True)), log(log(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Integral(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Pow(Integral(Derivative(log(log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain \\theta_2 - \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} = \\theta_2 - \\frac{2 \\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} + 1", "derivation": "\\hat{H}_{\\lambda}{(\\theta_2)} = \\cos{(\\theta_2)} and \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} = 1 and \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} - 1 = 0 and \\theta_2 + \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} - 1 = \\theta_2 and \\theta_2 = \\theta_2 - \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} + 1 and \\theta_2 - \\frac{\\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} = \\theta_2 - \\frac{2 \\hat{H}_{\\lambda}{(\\theta_2)}}{\\cos{(\\theta_2)}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(-1)), Integer(0))"], [["add", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(-1)), Symbol('\\\\theta_2', commutative=True))"], [["minus", 4, "Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(-1))"], "Equality(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(1)))"], [["add", 5, "Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\dot{x})} = \\log{(\\dot{x})}, then derive \\cos{(\\frac{d}{d \\dot{x}} \\hat{\\mathbf{r}}{(\\dot{x})} + \\frac{1}{\\dot{x}})} = \\cos{(\\frac{2}{\\dot{x}})}, then obtain \\cos{(\\frac{d}{d \\dot{x}} \\log{(\\dot{x})} + \\frac{1}{\\dot{x}})} = \\cos{(\\frac{2}{\\dot{x}})}", "derivation": "\\hat{\\mathbf{r}}{(\\dot{x})} = \\log{(\\dot{x})} and \\hat{\\mathbf{r}}{(\\dot{x})} + \\log{(\\dot{x})} = 2 \\log{(\\dot{x})} and \\frac{d}{d \\dot{x}} (\\hat{\\mathbf{r}}{(\\dot{x})} + \\log{(\\dot{x})}) = \\frac{d}{d \\dot{x}} 2 \\log{(\\dot{x})} and \\cos{(\\frac{d}{d \\dot{x}} (\\hat{\\mathbf{r}}{(\\dot{x})} + \\log{(\\dot{x})}))} = \\cos{(\\frac{d}{d \\dot{x}} 2 \\log{(\\dot{x})})} and \\cos{(\\frac{d}{d \\dot{x}} \\hat{\\mathbf{r}}{(\\dot{x})} + \\frac{1}{\\dot{x}})} = \\cos{(\\frac{2}{\\dot{x}})} and \\cos{(\\frac{d}{d \\dot{x}} \\log{(\\dot{x})} + \\frac{1}{\\dot{x}})} = \\cos{(\\frac{2}{\\dot{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), cos(Derivative(Mul(Integer(2), log(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(cos(Add(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))), cos(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(cos(Add(Derivative(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))), cos(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(z)} = \\log{(z)} and \\ddot{x}{(\\varphi^*,V)} = e^{V + \\varphi^*}, then obtain \\operatorname{A_{z}}{(z)} + \\ddot{x}{(\\varphi^*,V)} = \\operatorname{A_{z}}{(z)} + e^{V + \\varphi^*}", "derivation": "\\operatorname{A_{z}}{(z)} = \\log{(z)} and \\ddot{x}{(\\varphi^*,V)} = e^{V + \\varphi^*} and \\ddot{x}{(\\varphi^*,V)} + \\log{(z)} = e^{V + \\varphi^*} + \\log{(z)} and \\operatorname{A_{z}}{(z)} + \\ddot{x}{(\\varphi^*,V)} = \\operatorname{A_{z}}{(z)} + e^{V + \\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\varphi^*', commutative=True), Symbol('V', commutative=True)), exp(Add(Symbol('V', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["add", 2, "log(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\varphi^*', commutative=True), Symbol('V', commutative=True)), log(Symbol('z', commutative=True))), Add(exp(Add(Symbol('V', commutative=True), Symbol('\\\\varphi^*', commutative=True))), log(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A_z')(Symbol('z', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\varphi^*', commutative=True), Symbol('V', commutative=True))), Add(Function('A_z')(Symbol('z', commutative=True)), exp(Add(Symbol('V', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(F_{g})} = \\log{(F_{g})} and z{(F_{g})} = \\frac{\\log{(F_{g})}^{2}}{2}, then derive \\int \\frac{\\hat{p}{(F_{g})}}{F_{g}} dF_{g} = \\hat{H}_{\\lambda} + \\frac{\\log{(F_{g})}^{2}}{2}, then obtain \\hat{H}_{\\lambda} + z{(F_{g})} = f + \\frac{\\log{(F_{g})}^{2}}{2}", "derivation": "\\hat{p}{(F_{g})} = \\log{(F_{g})} and \\frac{\\hat{p}{(F_{g})}}{F_{g}} = \\frac{\\log{(F_{g})}}{F_{g}} and \\int \\frac{\\hat{p}{(F_{g})}}{F_{g}} dF_{g} = \\int \\frac{\\log{(F_{g})}}{F_{g}} dF_{g} and \\int \\frac{\\hat{p}{(F_{g})}}{F_{g}} dF_{g} = \\hat{H}_{\\lambda} + \\frac{\\log{(F_{g})}^{2}}{2} and \\hat{H}_{\\lambda} + \\frac{\\log{(F_{g})}^{2}}{2} = \\int \\frac{\\log{(F_{g})}}{F_{g}} dF_{g} and z{(F_{g})} = \\frac{\\log{(F_{g})}^{2}}{2} and \\hat{H}_{\\lambda} + z{(F_{g})} = \\int \\frac{\\log{(F_{g})}}{F_{g}} dF_{g} and \\hat{H}_{\\lambda} + z{(F_{g})} = f + \\frac{\\log{(F_{g})}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))"], [["divide", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('F_g', commutative=True))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('F_g', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('F_g', commutative=True)), Integer(2)))), Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('F_g', commutative=True)), Mul(Rational(1, 2), Pow(log(Symbol('F_g', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('z')(Symbol('F_g', commutative=True))), Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('z')(Symbol('F_g', commutative=True))), Add(Symbol('f', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('F_g', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given S{(v_{1},\\lambda)} = \\lambda + v_{1}, then obtain \\frac{(\\lambda + \\frac{\\partial}{\\partial v_{1}} S{(v_{1},\\lambda)})^{v_{1}}}{v_{1}} = \\frac{(\\lambda + \\frac{\\partial}{\\partial v_{1}} (\\lambda + v_{1}))^{v_{1}}}{v_{1}}", "derivation": "S{(v_{1},\\lambda)} = \\lambda + v_{1} and \\frac{\\partial}{\\partial v_{1}} S{(v_{1},\\lambda)} = \\frac{\\partial}{\\partial v_{1}} (\\lambda + v_{1}) and \\lambda + \\frac{\\partial}{\\partial v_{1}} S{(v_{1},\\lambda)} = \\lambda + \\frac{\\partial}{\\partial v_{1}} (\\lambda + v_{1}) and (\\lambda + \\frac{\\partial}{\\partial v_{1}} S{(v_{1},\\lambda)})^{v_{1}} = (\\lambda + \\frac{\\partial}{\\partial v_{1}} (\\lambda + v_{1}))^{v_{1}} and \\frac{(\\lambda + \\frac{\\partial}{\\partial v_{1}} S{(v_{1},\\lambda)})^{v_{1}}}{v_{1}} = \\frac{(\\lambda + \\frac{\\partial}{\\partial v_{1}} (\\lambda + v_{1}))^{v_{1}}}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('S')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Symbol('\\\\lambda', commutative=True), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('S')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)), Pow(Add(Symbol('\\\\lambda', commutative=True), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)))"], [["divide", 4, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('S')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\lambda', commutative=True), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(n_{2})} = \\sin{(n_{2})} and \\sigma_{x}{(n_{2})} = \\mathbf{H}^{2}{(n_{2})}, then obtain \\sin^{2}{(n_{2})} = \\mathbf{H}{(n_{2})} \\sin{(n_{2})}", "derivation": "\\mathbf{H}{(n_{2})} = \\sin{(n_{2})} and \\mathbf{H}^{2}{(n_{2})} = \\mathbf{H}{(n_{2})} \\sin{(n_{2})} and \\sigma_{x}{(n_{2})} = \\mathbf{H}^{2}{(n_{2})} and \\sigma_{x}{(n_{2})} = \\sin^{2}{(n_{2})} and \\sigma_{x}{(n_{2})} = \\mathbf{H}{(n_{2})} \\sin{(n_{2})} and \\sin^{2}{(n_{2})} = \\mathbf{H}{(n_{2})} \\sin{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('n_2', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\sigma_x')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\sigma_x')(Symbol('n_2', commutative=True)), Mul(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(sin(Symbol('n_2', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(u,\\theta_2)} = \\theta_2 u, then obtain (\\frac{\\theta_2^{2} u^{3}}{\\operatorname{v_{z}}{(u,\\theta_2)}})^{\\theta_2} = (\\theta_2 u^{2})^{\\theta_2}", "derivation": "\\operatorname{v_{z}}{(u,\\theta_2)} = \\theta_2 u and u \\operatorname{v_{z}}{(u,\\theta_2)} = \\theta_2 u^{2} and u^{2} \\operatorname{v_{z}}^{2}{(u,\\theta_2)} = \\theta_2^{2} u^{4} and (u \\operatorname{v_{z}}{(u,\\theta_2)})^{\\theta_2} = (\\theta_2 u^{2})^{\\theta_2} and u \\operatorname{v_{z}}{(u,\\theta_2)} = \\frac{\\theta_2^{2} u^{3}}{\\operatorname{v_{z}}{(u,\\theta_2)}} and (\\frac{\\theta_2^{2} u^{3}}{\\operatorname{v_{z}}{(u,\\theta_2)}})^{\\theta_2} = (\\theta_2 u^{2})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('u', commutative=True)))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Pow(Symbol('u', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(2)), Pow(Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)), Pow(Symbol('u', commutative=True), Integer(4))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Symbol('u', commutative=True), Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Pow(Symbol('u', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 3, "Mul(Symbol('u', commutative=True), Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Symbol('u', commutative=True), Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)), Pow(Symbol('u', commutative=True), Integer(3)), Pow(Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)), Pow(Symbol('u', commutative=True), Integer(3)), Pow(Function('v_z')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Pow(Symbol('u', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\mathbf{r})} = \\mathbf{r}, then obtain \\int \\frac{d}{d \\mathbf{r}} \\varepsilon_{0}^{\\mathbf{r}}{(\\mathbf{r})} d\\mathbf{r} = \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r}^{\\mathbf{r}} d\\mathbf{r}", "derivation": "\\varepsilon_{0}{(\\mathbf{r})} = \\mathbf{r} and \\varepsilon_{0}^{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r}^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\varepsilon_{0}^{\\mathbf{r}}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\mathbf{r}^{\\mathbf{r}} and \\int \\frac{d}{d \\mathbf{r}} \\varepsilon_{0}^{\\mathbf{r}}{(\\mathbf{r})} d\\mathbf{r} = \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r}^{\\mathbf{r}} d\\mathbf{r}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Q,\\phi_1)} = Q + \\phi_1 and \\hat{x}_0{(Q,\\phi_1)} = Q + 2 \\phi_1, then obtain \\frac{\\partial}{\\partial Q} 2 \\phi_1 (Q + 2 \\phi_1) (\\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)}) = \\frac{\\partial}{\\partial Q} 2 \\phi_1 (Q + 2 \\phi_1)^{2}", "derivation": "\\operatorname{y^{\\prime}}{(Q,\\phi_1)} = Q + \\phi_1 and \\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)} = Q + 2 \\phi_1 and \\hat{x}_0{(Q,\\phi_1)} = Q + 2 \\phi_1 and 2 \\phi_1 (\\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)}) = 2 \\phi_1 (Q + 2 \\phi_1) and 2 \\phi_1 (\\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)}) \\hat{x}_0{(Q,\\phi_1)} = 2 \\phi_1 (Q + 2 \\phi_1) \\hat{x}_0{(Q,\\phi_1)} and 2 \\phi_1 (Q + 2 \\phi_1) (\\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)}) = 2 \\phi_1 (Q + 2 \\phi_1)^{2} and \\frac{\\partial}{\\partial Q} 2 \\phi_1 (Q + 2 \\phi_1) (\\phi_1 + \\operatorname{y^{\\prime}}{(Q,\\phi_1)}) = \\frac{\\partial}{\\partial Q} 2 \\phi_1 (Q + 2 \\phi_1)^{2}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('\\\\phi_1', commutative=True), Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))))"], [["times", 4, "Function('\\\\hat{x}_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('\\\\phi_1', commutative=True), Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Function('\\\\hat{x}_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Function('\\\\hat{x}_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Integer(2))))"], [["differentiate", 6, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Function('y^{\\\\prime}')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Integer(2))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(i,F_{N})} = \\frac{F_{N}}{i} and Z{(i,F_{N})} = \\frac{F_{N}}{i} + \\omega{(i,F_{N})}, then obtain Z{(i,F_{N})} = 2 \\omega{(i,F_{N})}", "derivation": "\\omega{(i,F_{N})} = \\frac{F_{N}}{i} and 2 \\omega{(i,F_{N})} = \\frac{F_{N}}{i} + \\omega{(i,F_{N})} and Z{(i,F_{N})} = \\frac{F_{N}}{i} + \\omega{(i,F_{N})} and Z{(i,F_{N})} = 2 \\omega{(i,F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["add", 1, "Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Symbol('F_N', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('Z')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('i', commutative=True), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(a)} = e^{\\cos{(a)}}, then obtain (a + \\rho_{f}{(a)} + e^{\\cos{(a)}}) e^{\\cos{(a)}} = (a + 2 e^{\\cos{(a)}}) e^{\\cos{(a)}}", "derivation": "\\rho_{f}{(a)} = e^{\\cos{(a)}} and a + \\rho_{f}{(a)} = a + e^{\\cos{(a)}} and a + 2 \\rho_{f}{(a)} = a + \\rho_{f}{(a)} + e^{\\cos{(a)}} and a + 2 \\rho_{f}{(a)} = a + 2 e^{\\cos{(a)}} and (a + 2 \\rho_{f}{(a)}) e^{\\cos{(a)}} = (a + 2 e^{\\cos{(a)}}) e^{\\cos{(a)}} and (a + \\rho_{f}{(a)} + e^{\\cos{(a)}}) e^{\\cos{(a)}} = (a + 2 e^{\\cos{(a)}}) e^{\\cos{(a)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('a', commutative=True)), exp(cos(Symbol('a', commutative=True))))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('\\\\rho_f')(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), exp(cos(Symbol('a', commutative=True)))))"], [["add", 1, "Add(Symbol('a', commutative=True), Function('\\\\rho_f')(Symbol('a', commutative=True)))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(2), Function('\\\\rho_f')(Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Function('\\\\rho_f')(Symbol('a', commutative=True)), exp(cos(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(2), Function('\\\\rho_f')(Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(2), exp(cos(Symbol('a', commutative=True))))))"], [["times", 4, "exp(cos(Symbol('a', commutative=True)))"], "Equality(Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), Function('\\\\rho_f')(Symbol('a', commutative=True)))), exp(cos(Symbol('a', commutative=True)))), Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), exp(cos(Symbol('a', commutative=True))))), exp(cos(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('a', commutative=True), Function('\\\\rho_f')(Symbol('a', commutative=True)), exp(cos(Symbol('a', commutative=True)))), exp(cos(Symbol('a', commutative=True)))), Mul(Add(Symbol('a', commutative=True), Mul(Integer(2), exp(cos(Symbol('a', commutative=True))))), exp(cos(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given V{(\\mathbf{g})} = \\log{(\\sin{(\\mathbf{g})})}, then obtain \\int V{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} d\\mathbf{g} + \\frac{1}{V{(\\mathbf{g})}} = \\int \\log{(\\sin{(\\mathbf{g})})}^{2} d\\mathbf{g} + \\frac{1}{V{(\\mathbf{g})}}", "derivation": "V{(\\mathbf{g})} = \\log{(\\sin{(\\mathbf{g})})} and V{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} = \\log{(\\sin{(\\mathbf{g})})}^{2} and \\int V{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} d\\mathbf{g} = \\int \\log{(\\sin{(\\mathbf{g})})}^{2} d\\mathbf{g} and \\int V{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} d\\mathbf{g} + \\frac{1}{\\log{(\\sin{(\\mathbf{g})})}} = \\int \\log{(\\sin{(\\mathbf{g})})}^{2} d\\mathbf{g} + \\frac{1}{\\log{(\\sin{(\\mathbf{g})})}} and \\int V{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} d\\mathbf{g} + \\frac{1}{V{(\\mathbf{g})}} = \\int \\log{(\\sin{(\\mathbf{g})})}^{2} d\\mathbf{g} + \\frac{1}{V{(\\mathbf{g})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 1, "Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))"], "Equality(Mul(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))), Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 3, "Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Mul(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Add(Integral(Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Mul(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Add(Integral(Pow(log(sin(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given I{(\\mathbf{f},L_{\\varepsilon})} = L_{\\varepsilon}^{\\mathbf{f}}, then obtain \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{I{(\\mathbf{f},L_{\\varepsilon})}}{\\mathbf{f}} d\\mathbf{f} = \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{L_{\\varepsilon}^{\\mathbf{f}}}{\\mathbf{f}} d\\mathbf{f}", "derivation": "I{(\\mathbf{f},L_{\\varepsilon})} = L_{\\varepsilon}^{\\mathbf{f}} and \\frac{I{(\\mathbf{f},L_{\\varepsilon})}}{\\mathbf{f}} = \\frac{L_{\\varepsilon}^{\\mathbf{f}}}{\\mathbf{f}} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{I{(\\mathbf{f},L_{\\varepsilon})}}{\\mathbf{f}} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{L_{\\varepsilon}^{\\mathbf{f}}}{\\mathbf{f}} and \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{I{(\\mathbf{f},L_{\\varepsilon})}}{\\mathbf{f}} d\\mathbf{f} = \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{L_{\\varepsilon}^{\\mathbf{f}}}{\\mathbf{f}} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given x{(\\mu,\\theta_1)} = \\cos{(\\theta_1^{\\mu})} and \\hat{\\mathbf{x}}{(\\theta_1,\\mu)} = \\frac{\\partial}{\\partial \\theta_1} x{(\\mu,\\theta_1)}, then derive \\frac{\\partial}{\\partial \\theta_1} x{(\\mu,\\theta_1)} = - \\frac{\\mu \\theta_1^{\\mu} \\sin{(\\theta_1^{\\mu})}}{\\theta_1}, then obtain (- x{(\\mu,\\theta_1)} + \\sin{(\\eta)}) \\hat{\\mathbf{x}}{(\\theta_1,\\mu)} = - \\frac{\\mu \\theta_1^{\\mu} (- x{(\\mu,\\theta_1)} + \\sin{(\\eta)}) \\sin{(\\theta_1^{\\mu})}}{\\theta_1}", "derivation": "x{(\\mu,\\theta_1)} = \\cos{(\\theta_1^{\\mu})} and \\frac{\\partial}{\\partial \\theta_1} x{(\\mu,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\cos{(\\theta_1^{\\mu})} and \\frac{\\partial}{\\partial \\theta_1} x{(\\mu,\\theta_1)} = - \\frac{\\mu \\theta_1^{\\mu} \\sin{(\\theta_1^{\\mu})}}{\\theta_1} and \\hat{\\mathbf{x}}{(\\theta_1,\\mu)} = \\frac{\\partial}{\\partial \\theta_1} x{(\\mu,\\theta_1)} and \\hat{\\mathbf{x}}{(\\theta_1,\\mu)} = - \\frac{\\mu \\theta_1^{\\mu} \\sin{(\\theta_1^{\\mu})}}{\\theta_1} and (- x{(\\mu,\\theta_1)} + \\sin{(\\eta)}) \\hat{\\mathbf{x}}{(\\theta_1,\\mu)} = - \\frac{\\mu \\theta_1^{\\mu} (- x{(\\mu,\\theta_1)} + \\sin{(\\eta)}) \\sin{(\\theta_1^{\\mu})}}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)), cos(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["times", 5, "Add(Mul(Integer(-1), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\eta', commutative=True))), sin(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(H,t_{2})} = \\sin{(\\frac{t_{2}}{H})}, then obtain \\int \\frac{\\tilde{g}{(H,t_{2})} + \\sin{(\\frac{t_{2}}{H})}}{H} dH = \\int \\frac{2 \\sin{(\\frac{t_{2}}{H})}}{H} dH", "derivation": "\\tilde{g}{(H,t_{2})} = \\sin{(\\frac{t_{2}}{H})} and \\tilde{g}{(H,t_{2})} + \\sin{(\\frac{t_{2}}{H})} = 2 \\sin{(\\frac{t_{2}}{H})} and \\frac{\\tilde{g}{(H,t_{2})} + \\sin{(\\frac{t_{2}}{H})}}{H} = \\frac{2 \\sin{(\\frac{t_{2}}{H})}}{H} and \\int \\frac{\\tilde{g}{(H,t_{2})} + \\sin{(\\frac{t_{2}}{H})}}{H} dH = \\int \\frac{2 \\sin{(\\frac{t_{2}}{H})}}{H} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('t_2', commutative=True)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["add", 1, "sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('t_2', commutative=True)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))), Mul(Integer(2), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))))"], [["times", 2, "Pow(Symbol('H', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('t_2', commutative=True)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))), Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('t_2', commutative=True)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(z^{*},t_{2})} = \\frac{\\partial}{\\partial z^{*}} t_{2} z^{*}, then derive \\mathbf{J}_P{(z^{*},t_{2})} = t_{2}, then obtain z^{*} + \\mathbf{J}_P{(z^{*},\\frac{\\partial}{\\partial z^{*}} t_{2} z^{*})} = z^{*} + \\frac{\\partial}{\\partial z^{*}} t_{2} z^{*}", "derivation": "\\mathbf{J}_P{(z^{*},t_{2})} = \\frac{\\partial}{\\partial z^{*}} t_{2} z^{*} and \\mathbf{J}_P{(z^{*},t_{2})} = t_{2} and z^{*} + \\mathbf{J}_P{(z^{*},t_{2})} = t_{2} + z^{*} and t_{2} = \\frac{\\partial}{\\partial z^{*}} t_{2} z^{*} and z^{*} + \\mathbf{J}_P{(z^{*},\\frac{\\partial}{\\partial z^{*}} t_{2} z^{*})} = z^{*} + \\frac{\\partial}{\\partial z^{*}} t_{2} z^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True)), Derivative(Mul(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["add", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Symbol('t_2', commutative=True), Derivative(Mul(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('z^*', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('z^*', commutative=True), Derivative(Mul(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), Add(Symbol('z^*', commutative=True), Derivative(Mul(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\mathbf{B})} = \\cos{(\\mathbf{B})} and \\rho_{b}{(\\rho)} = \\cos{(\\log{(\\rho)})}, then obtain \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\rho_{b}{(\\rho)} + \\cos{(S{(\\mathbf{B})})} = \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\rho_{b}{(\\rho)} + \\cos{(\\cos{(\\mathbf{B})})}", "derivation": "S{(\\mathbf{B})} = \\cos{(\\mathbf{B})} and \\rho_{b}{(\\rho)} = \\cos{(\\log{(\\rho)})} and \\cos{(S{(\\mathbf{B})})} = \\cos{(\\cos{(\\mathbf{B})})} and \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\cos{(S{(\\mathbf{B})})} + \\cos{(\\log{(\\rho)})} = \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\cos{(\\log{(\\rho)})} + \\cos{(\\cos{(\\mathbf{B})})} and \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\rho_{b}{(\\rho)} + \\cos{(S{(\\mathbf{B})})} = \\hat{H}_{\\lambda}^{2}{(v_{z})} + \\rho_{b}{(\\rho)} + \\cos{(\\cos{(\\mathbf{B})})}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True))))"], [["cos", 1], "Equality(cos(Function('S')(Symbol('\\\\mathbf{B}', commutative=True))), cos(cos(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 3, "Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Integer(2)), cos(log(Symbol('\\\\rho', commutative=True))))"], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Integer(2)), cos(Function('S')(Symbol('\\\\mathbf{B}', commutative=True))), cos(log(Symbol('\\\\rho', commutative=True)))), Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Integer(2)), cos(log(Symbol('\\\\rho', commutative=True))), cos(cos(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Integer(2)), Function('\\\\rho_b')(Symbol('\\\\rho', commutative=True)), cos(Function('S')(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Integer(2)), Function('\\\\rho_b')(Symbol('\\\\rho', commutative=True)), cos(cos(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\mu{(F_{N})} = \\log{(\\cos{(F_{N})})} and U{(F_{N})} = \\cos{(F_{N})}, then obtain (- \\log{(U{(F_{N})})})^{F_{N}} = (- \\mu{(F_{N})})^{F_{N}}", "derivation": "\\mu{(F_{N})} = \\log{(\\cos{(F_{N})})} and U{(F_{N})} = \\cos{(F_{N})} and - \\log{(\\cos{(F_{N})})} = - \\mu{(F_{N})} and - \\log{(U{(F_{N})})} = - \\mu{(F_{N})} and (- \\log{(U{(F_{N})})})^{F_{N}} = (- \\mu{(F_{N})})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('F_N', commutative=True)), log(cos(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["minus", 1, "Add(Function('\\\\mu')(Symbol('F_N', commutative=True)), log(cos(Symbol('F_N', commutative=True))))"], "Equality(Mul(Integer(-1), log(cos(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(-1), log(Function('U')(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('F_N', commutative=True))))"], [["power", 4, "Symbol('F_N', commutative=True)"], "Equality(Pow(Mul(Integer(-1), log(Function('U')(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\mu')(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\nabla{(J,A)} = A + J and \\theta{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive 0 = - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}), then obtain - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}) + e^{\\mathbf{p}} = e^{\\mathbf{p}}", "derivation": "\\nabla{(J,A)} = A + J and 0 = A + J - \\nabla{(J,A)} and \\frac{d}{d A} 0 = \\frac{\\partial}{\\partial A} (A + J - \\nabla{(J,A)}) and - A \\frac{d}{d A} 0 = - A \\frac{\\partial}{\\partial A} (A + J - \\nabla{(J,A)}) and \\theta{(\\mathbf{p})} = e^{\\mathbf{p}} and 0 = - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}) and \\theta{(\\mathbf{p})} = - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}) + \\theta{(\\mathbf{p})} and - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}) + \\theta{(\\mathbf{p})} = e^{\\mathbf{p}} and - A (1 - \\frac{\\partial}{\\partial A} \\nabla{(J,A)}) + e^{\\mathbf{p}} = e^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('J', commutative=True)))"], [["minus", 1, "Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True))"], "Equality(Integer(0), Add(Symbol('A', commutative=True), Symbol('J', commutative=True), Mul(Integer(-1), Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), Symbol('J', commutative=True), Mul(Integer(-1), Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Symbol('A', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('A', commutative=True), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A', commutative=True), Derivative(Add(Symbol('A', commutative=True), Symbol('J', commutative=True), Mul(Integer(-1), Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)))))"], ["get_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Integer(-1), Symbol('A', commutative=True), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))))"], [["add", 6, "Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))), Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))), Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))), exp(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given z{(r,l)} = - l + r and \\operatorname{C_{2}}{(r,l)} = \\int r dl, then obtain \\frac{\\operatorname{C_{2}}{(r,l)}}{l - 2 r - 1} = \\frac{\\int r dl}{l - 2 r - 1}", "derivation": "z{(r,l)} = - l + r and - z{(r,l)} = l - r and - r - z{(r,l)} = l - 2 r and \\operatorname{C_{2}}{(r,l)} = \\int r dl and \\frac{\\operatorname{C_{2}}{(r,l)}}{- r - z{(r,l)} - 1} = \\frac{\\int r dl}{- r - z{(r,l)} - 1} and \\frac{\\operatorname{C_{2}}{(r,l)}}{l - 2 r - 1} = \\frac{\\int r dl}{l - 2 r - 1}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('r', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('r', commutative=True), Symbol('l', commutative=True)), Integral(Symbol('r', commutative=True), Tuple(Symbol('l', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Integer(-1)), Integer(-1)), Function('C_2')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Integer(-1)), Integer(-1)), Integral(Symbol('r', commutative=True), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Integer(-1)), Integer(-1)), Function('C_2')(Symbol('r', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Integer(-1)), Integer(-1)), Integral(Symbol('r', commutative=True), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{v})} = \\sin{(e^{\\mathbf{v}})} and \\chi{(\\mathbf{v})} = e^{\\mathbf{v}}, then obtain \\mathbf{v} \\operatorname{v_{2}}^{\\mathbf{v}}{(\\mathbf{v})} = \\mathbf{v} \\sin^{\\mathbf{v}}{(\\chi{(\\mathbf{v})})}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{v})} = \\sin{(e^{\\mathbf{v}})} and \\operatorname{v_{2}}^{\\mathbf{v}}{(\\mathbf{v})} = \\sin^{\\mathbf{v}}{(e^{\\mathbf{v}})} and \\chi{(\\mathbf{v})} = e^{\\mathbf{v}} and \\operatorname{v_{2}}^{\\mathbf{v}}{(\\mathbf{v})} = \\sin^{\\mathbf{v}}{(\\chi{(\\mathbf{v})})} and \\mathbf{v} \\operatorname{v_{2}}^{\\mathbf{v}}{(\\mathbf{v})} = \\mathbf{v} \\sin^{\\mathbf{v}}{(\\chi{(\\mathbf{v})})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{v}', commutative=True)), sin(exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('v_2')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('v_2')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(sin(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{f})} = \\sin{(\\mathbf{f})}, then derive 0 = \\cos{(\\mathbf{f})} - \\frac{d}{d \\mathbf{f}} \\dot{\\mathbf{r}}{(\\mathbf{f})}, then obtain 0 = \\cos{(\\mathbf{f})} - \\frac{d}{d \\mathbf{f}} \\sin{(\\mathbf{f})}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and 0 = - \\dot{\\mathbf{r}}{(\\mathbf{f})} + \\sin{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} 0 = \\frac{d}{d \\mathbf{f}} (- \\dot{\\mathbf{r}}{(\\mathbf{f})} + \\sin{(\\mathbf{f})}) and 0 = \\cos{(\\mathbf{f})} - \\frac{d}{d \\mathbf{f}} \\dot{\\mathbf{r}}{(\\mathbf{f})} and 0 = \\cos{(\\mathbf{f})} - \\frac{d}{d \\mathbf{f}} \\sin{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True))), sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(cos(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(cos(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given Q{(u,T)} = T u, then obtain (\\int (- T u + Q{(u,T)} - \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du)^{u} = (\\int (- \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du)^{u}", "derivation": "Q{(u,T)} = T u and - T u + Q{(u,T)} = 0 and \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) = \\frac{d}{d T} 0 and - T u + Q{(u,T)} - \\frac{d}{d T} 0 - 1 = - \\frac{d}{d T} 0 - 1 and - T u + Q{(u,T)} - \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1 = - \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1 and \\int (- T u + Q{(u,T)} - \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du = \\int (- \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du and (\\int (- T u + Q{(u,T)} - \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du)^{u} = (\\int (- \\frac{\\partial}{\\partial T} (- T u + Q{(u,T)}) - 1) du)^{u}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Mul(Symbol('T', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "Add(Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('u', commutative=True))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('u', commutative=True)), Function('Q')(Symbol('u', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\mathbf{p})} = \\sin{(\\cos{(\\mathbf{p})})} and \\Psi^{\\dagger}{(\\mathbf{p})} = \\int \\sin{(\\cos{(\\mathbf{p})})} d\\mathbf{p} and \\operatorname{f^{*}}{(\\mathbf{p})} = \\int \\rho{(\\mathbf{p})} d\\mathbf{p}, then obtain \\operatorname{f^{*}}{(\\mathbf{p})} \\cos{(\\mathbf{p})} = \\cos{(\\mathbf{p})} \\int \\rho{(\\mathbf{p})} d\\mathbf{p}", "derivation": "\\rho{(\\mathbf{p})} = \\sin{(\\cos{(\\mathbf{p})})} and \\int \\rho{(\\mathbf{p})} d\\mathbf{p} = \\int \\sin{(\\cos{(\\mathbf{p})})} d\\mathbf{p} and \\Psi^{\\dagger}{(\\mathbf{p})} = \\int \\sin{(\\cos{(\\mathbf{p})})} d\\mathbf{p} and \\operatorname{f^{*}}{(\\mathbf{p})} = \\int \\rho{(\\mathbf{p})} d\\mathbf{p} and \\Psi^{\\dagger}{(\\mathbf{p})} = \\int \\rho{(\\mathbf{p})} d\\mathbf{p} and \\Psi^{\\dagger}{(\\mathbf{p})} = \\operatorname{f^{*}}{(\\mathbf{p})} and \\Psi^{\\dagger}{(\\mathbf{p})} \\cos{(\\mathbf{p})} = \\cos{(\\mathbf{p})} \\int \\rho{(\\mathbf{p})} d\\mathbf{p} and \\operatorname{f^{*}}{(\\mathbf{p})} \\cos{(\\mathbf{p})} = \\cos{(\\mathbf{p})} \\int \\rho{(\\mathbf{p})} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), sin(cos(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(cos(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Integral(sin(cos(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 5, "cos(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True))), Mul(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Function('f^*')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True))), Mul(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(A)} = \\sin{(e^{A})} and \\dot{z}{(B,b)} = - B + b, then obtain \\frac{(e^{A} + \\sin{(e^{A})}) (- B + b + \\dot{z}{(B,b)})}{\\sin{(e^{A})}} = \\frac{(- 2 B + 2 b) (e^{A} + \\sin{(e^{A})})}{\\sin{(e^{A})}}", "derivation": "\\varepsilon_{0}{(A)} = \\sin{(e^{A})} and \\dot{z}{(B,b)} = - B + b and - B + b + \\dot{z}{(B,b)} = - 2 B + 2 b and \\frac{(e^{A} + \\sin{(e^{A})}) (- B + b + \\dot{z}{(B,b)})}{\\varepsilon_{0}{(A)}} = \\frac{(- 2 B + 2 b) (e^{A} + \\sin{(e^{A})})}{\\varepsilon_{0}{(A)}} and \\frac{(e^{A} + \\sin{(e^{A})}) (- B + b + \\dot{z}{(B,b)})}{\\sin{(e^{A})}} = \\frac{(- 2 B + 2 b) (e^{A} + \\sin{(e^{A})})}{\\sin{(e^{A})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('B', commutative=True), Symbol('b', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('b', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('b', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["times", 3, "Mul(Add(exp(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True)))), Pow(Function('\\\\varepsilon_0')(Symbol('A', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(exp(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('b', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True), Symbol('b', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('A', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Add(exp(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True)))), Pow(Function('\\\\varepsilon_0')(Symbol('A', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(exp(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('b', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True), Symbol('b', commutative=True))), Pow(sin(exp(Symbol('A', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Add(exp(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True)))), Pow(sin(exp(Symbol('A', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{1}{(c_{0},\\mathbf{S})} = \\log{(\\mathbf{S} - c_{0})} and v{(\\mathbf{S})} = - \\mathbf{S}, then obtain (- \\mathbf{S} + \\theta_{1}{(c_{0},\\mathbf{S})})^{\\mathbf{S}} = (- \\mathbf{S} + \\log{(\\mathbf{S} - c_{0})})^{\\mathbf{S}}", "derivation": "\\theta_{1}{(c_{0},\\mathbf{S})} = \\log{(\\mathbf{S} - c_{0})} and - \\mathbf{S} + \\theta_{1}{(c_{0},\\mathbf{S})} = - \\mathbf{S} + \\log{(\\mathbf{S} - c_{0})} and v{(\\mathbf{S})} = - \\mathbf{S} and \\theta_{1}{(c_{0},\\mathbf{S})} + v{(\\mathbf{S})} = v{(\\mathbf{S})} + \\log{(\\mathbf{S} - c_{0})} and (\\theta_{1}{(c_{0},\\mathbf{S})} + v{(\\mathbf{S})})^{\\mathbf{S}} = (v{(\\mathbf{S})} + \\log{(\\mathbf{S} - c_{0})})^{\\mathbf{S}} and (- \\mathbf{S} + \\theta_{1}{(c_{0},\\mathbf{S})})^{\\mathbf{S}} = (- \\mathbf{S} + \\log{(\\mathbf{S} - c_{0})})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))"], [["minus", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\theta_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\theta_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('v')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Function('v')(Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Function('\\\\theta_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('v')(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Function('v')(Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\theta_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(q)} = \\log{(q)}, then derive q \\frac{d}{d q} \\Psi^{\\dagger}{(q)} + \\Psi^{\\dagger}{(q)} - 1 = \\log{(q)}, then obtain q \\frac{d}{d q} \\log{(q)} + \\log{(q)} - 1 = \\log{(q)}", "derivation": "\\Psi^{\\dagger}{(q)} = \\log{(q)} and q \\Psi^{\\dagger}{(q)} = q \\log{(q)} and \\frac{d}{d q} q \\Psi^{\\dagger}{(q)} = \\frac{d}{d q} q \\log{(q)} and \\frac{d}{d q} q \\Psi^{\\dagger}{(q)} - 1 = \\frac{d}{d q} q \\log{(q)} - 1 and q \\frac{d}{d q} \\Psi^{\\dagger}{(q)} + \\Psi^{\\dagger}{(q)} - 1 = \\log{(q)} and q \\frac{d}{d q} \\log{(q)} + \\log{(q)} - 1 = \\log{(q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Symbol('q', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Mul(Symbol('q', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('q', commutative=True), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('q', commutative=True)), Integer(-1)), log(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('q', commutative=True), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), log(Symbol('q', commutative=True)), Integer(-1)), log(Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(v_{x},p)} = - p + v_{x} and \\operatorname{v_{1}}{(p)} = - p, then obtain \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} \\bar{\\h}^{p}{(v_{x},p)} = \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} (v_{x} + \\operatorname{v_{1}}{(p)})^{p}", "derivation": "\\bar{\\h}{(v_{x},p)} = - p + v_{x} and \\bar{\\h}^{p}{(v_{x},p)} = (- p + v_{x})^{p} and \\frac{\\partial}{\\partial p} \\bar{\\h}^{p}{(v_{x},p)} = \\frac{\\partial}{\\partial p} (- p + v_{x})^{p} and \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} \\bar{\\h}^{p}{(v_{x},p)} = \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} (- p + v_{x})^{p} and \\operatorname{v_{1}}{(p)} = - p and \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} \\bar{\\h}^{p}{(v_{x},p)} = \\bar{\\h}{(v_{x},p)} \\frac{\\partial}{\\partial p} (v_{x} + \\operatorname{v_{1}}{(p)})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["times", 3, "Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Add(Symbol('v_x', commutative=True), Function('v_1')(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(E_{x},\\mathbf{J}_P)} = - E_{x} + \\mathbf{J}_P, then obtain \\frac{\\frac{\\partial}{\\partial E_{x}} \\frac{z{(E_{x},\\mathbf{J}_P)}}{E_{x}}}{\\mathbf{J}_P} = \\frac{\\frac{\\partial}{\\partial E_{x}} \\frac{- E_{x} + \\mathbf{J}_P}{E_{x}}}{\\mathbf{J}_P}", "derivation": "z{(E_{x},\\mathbf{J}_P)} = - E_{x} + \\mathbf{J}_P and \\frac{z{(E_{x},\\mathbf{J}_P)}}{E_{x}} = \\frac{- E_{x} + \\mathbf{J}_P}{E_{x}} and \\frac{\\partial}{\\partial E_{x}} \\frac{z{(E_{x},\\mathbf{J}_P)}}{E_{x}} = \\frac{\\partial}{\\partial E_{x}} \\frac{- E_{x} + \\mathbf{J}_P}{E_{x}} and \\frac{\\frac{\\partial}{\\partial E_{x}} \\frac{z{(E_{x},\\mathbf{J}_P)}}{E_{x}}}{\\mathbf{J}_P} = \\frac{\\frac{\\partial}{\\partial E_{x}} \\frac{- E_{x} + \\mathbf{J}_P}{E_{x}}}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('z')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 2, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('z')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('z')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(E_{n},\\dot{z})} = - E_{n} + \\dot{z}, then derive \\int - V{(E_{n},\\dot{z})} dE_{n} = \\frac{E_{n}^{2}}{2} - E_{n} \\dot{z} + n_{2}, then obtain - E_{n} = \\frac{\\partial}{\\partial \\dot{z}} \\int (E_{n} - \\dot{z}) dE_{n}", "derivation": "V{(E_{n},\\dot{z})} = - E_{n} + \\dot{z} and - V{(E_{n},\\dot{z})} = E_{n} - \\dot{z} and \\int - V{(E_{n},\\dot{z})} dE_{n} = \\int (E_{n} - \\dot{z}) dE_{n} and \\frac{\\partial}{\\partial \\dot{z}} \\int - V{(E_{n},\\dot{z})} dE_{n} = \\frac{\\partial}{\\partial \\dot{z}} \\int (E_{n} - \\dot{z}) dE_{n} and \\int - V{(E_{n},\\dot{z})} dE_{n} = \\frac{E_{n}^{2}}{2} - E_{n} \\dot{z} + n_{2} and \\frac{\\partial}{\\partial \\dot{z}} (\\frac{E_{n}^{2}}{2} - E_{n} \\dot{z} + n_{2}) = \\frac{\\partial}{\\partial \\dot{z}} \\int (E_{n} - \\dot{z}) dE_{n} and - E_{n} = \\frac{\\partial}{\\partial \\dot{z}} \\int (E_{n} - \\dot{z}) dE_{n}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('V')(Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('V')(Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Function('V')(Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Function('V')(Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('E_n', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(-1), Symbol('E_n', commutative=True)), Derivative(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(U,y)} = U + y, then derive U + y + \\frac{\\partial}{\\partial y} \\hat{x}_0{(U,y)} = U + y + 1, then derive \\varphi + \\frac{y^{2}}{2} + y (U + 1) = W + \\frac{y^{2}}{2} + y (U + 1), then obtain \\frac{(\\varphi + \\frac{y^{2}}{2} + y (U + 1)) \\hat{\\mathbf{x}}{(r,\\varphi^*)}}{g_{\\varepsilon}} = \\frac{(W + \\frac{y^{2}}{2} + y (U + 1)) \\hat{\\mathbf{x}}{(r,\\varphi^*)}}{g_{\\varepsilon}}", "derivation": "\\hat{x}_0{(U,y)} = U + y and \\frac{\\partial}{\\partial y} \\hat{x}_0{(U,y)} = \\frac{\\partial}{\\partial y} (U + y) and U + y + \\frac{\\partial}{\\partial y} \\hat{x}_0{(U,y)} = U + y + \\frac{\\partial}{\\partial y} (U + y) and U + y + \\frac{\\partial}{\\partial y} \\hat{x}_0{(U,y)} = U + y + 1 and U + y + 1 = U + y + \\frac{\\partial}{\\partial y} (U + y) and \\int (U + y + 1) dy = \\int (U + y + \\frac{\\partial}{\\partial y} (U + y)) dy and \\varphi + \\frac{y^{2}}{2} + y (U + 1) = W + \\frac{y^{2}}{2} + y (U + 1) and \\frac{(\\varphi + \\frac{y^{2}}{2} + y (U + 1)) \\hat{\\mathbf{x}}{(r,\\varphi^*)}}{g_{\\varepsilon}} = \\frac{(W + \\frac{y^{2}}{2} + y (U + 1)) \\hat{\\mathbf{x}}{(r,\\varphi^*)}}{g_{\\varepsilon}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Add(Symbol('U', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["add", 2, "Add(Symbol('U', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Derivative(Add(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Integer(1)), Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Derivative(Add(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('U', commutative=True), Symbol('y', commutative=True), Derivative(Add(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))), Add(Symbol('W', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))))"], [["times", 7, "Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('r', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\chi)} = e^{\\chi}, then derive 0 = \\frac{e^{\\chi}}{\\varepsilon{(\\chi)}} - \\frac{e^{\\chi} \\frac{d}{d \\chi} \\varepsilon{(\\chi)}}{\\varepsilon^{2}{(\\chi)}}, then obtain 0 = 1 - \\frac{\\frac{d}{d \\chi} \\varepsilon{(\\chi)}}{\\varepsilon{(\\chi)}}", "derivation": "\\varepsilon{(\\chi)} = e^{\\chi} and \\frac{\\varepsilon{(\\chi)}}{\\chi} = \\frac{e^{\\chi}}{\\chi} and - \\frac{\\varepsilon{(\\chi)}}{\\chi} = - \\frac{e^{\\chi}}{\\chi} and - \\frac{\\varepsilon{(\\chi)} e^{- \\chi}}{\\chi} = - \\frac{1}{\\chi} and 1 = \\frac{e^{\\chi}}{\\varepsilon{(\\chi)}} and \\frac{d}{d \\chi} 1 = \\frac{d}{d \\chi} \\frac{e^{\\chi}}{\\varepsilon{(\\chi)}} and 0 = \\frac{e^{\\chi}}{\\varepsilon{(\\chi)}} - \\frac{e^{\\chi} \\frac{d}{d \\chi} \\varepsilon{(\\chi)}}{\\varepsilon^{2}{(\\chi)}} and 0 = 1 - e^{- \\chi} \\frac{d}{d \\chi} e^{\\chi} and 0 = 1 - \\frac{\\frac{d}{d \\chi} \\varepsilon{(\\chi)}}{\\varepsilon{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))))"], [["divide", 3, "exp(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Add(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Integer(-2)), exp(Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)}, then obtain (\\int (2 \\mathbf{J}_M + \\operatorname{P_{e}}{(\\mathbf{J}_M)}) d\\mathbf{J}_M)^{2} = (\\int (2 \\mathbf{J}_M + \\log{(\\mathbf{J}_M)}) d\\mathbf{J}_M)^{2}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)} and \\mathbf{J}_M + \\operatorname{P_{e}}{(\\mathbf{J}_M)} = \\mathbf{J}_M + \\log{(\\mathbf{J}_M)} and 2 \\mathbf{J}_M + \\operatorname{P_{e}}{(\\mathbf{J}_M)} = 2 \\mathbf{J}_M + \\log{(\\mathbf{J}_M)} and \\int (2 \\mathbf{J}_M + \\operatorname{P_{e}}{(\\mathbf{J}_M)}) d\\mathbf{J}_M = \\int (2 \\mathbf{J}_M + \\log{(\\mathbf{J}_M)}) d\\mathbf{J}_M and (\\int (2 \\mathbf{J}_M + \\operatorname{P_{e}}{(\\mathbf{J}_M)}) d\\mathbf{J}_M)^{2} = (\\int (2 \\mathbf{J}_M + \\log{(\\mathbf{J}_M)}) d\\mathbf{J}_M)^{2}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('P_e')(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('P_e')(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('P_e')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('P_e')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)), Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\omega{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\operatorname{E_{n}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})}, then obtain \\operatorname{E_{n}}{(E_{\\lambda})} = (\\operatorname{E_{n}}{(E_{\\lambda})} - \\cos{(E_{\\lambda})}) \\operatorname{E_{n}}^{- E_{\\lambda}}{(E_{\\lambda})} \\cos{(E_{\\lambda})} + \\operatorname{E_{n}}{(E_{\\lambda})}", "derivation": "\\omega{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\operatorname{E_{n}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\omega{(E_{\\lambda})} = \\operatorname{E_{n}}{(E_{\\lambda})} and 0 = \\operatorname{E_{n}}{(E_{\\lambda})} - \\omega{(E_{\\lambda})} and 0 = \\operatorname{E_{n}}{(E_{\\lambda})} - \\cos{(E_{\\lambda})} and 0 = (\\operatorname{E_{n}}{(E_{\\lambda})} - \\cos{(E_{\\lambda})}) \\operatorname{E_{n}}^{- E_{\\lambda}}{(E_{\\lambda})} \\cos{(E_{\\lambda})} and \\operatorname{E_{n}}{(E_{\\lambda})} = (\\operatorname{E_{n}}{(E_{\\lambda})} - \\cos{(E_{\\lambda})}) \\operatorname{E_{n}}^{- E_{\\lambda}}{(E_{\\lambda})} \\cos{(E_{\\lambda})} + \\operatorname{E_{n}}{(E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\omega')(Symbol('E_{\\\\lambda}', commutative=True)), Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 3, "Function('\\\\omega')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('\\\\omega')(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["divide", 5, "Mul(Pow(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Mul(Add(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Symbol('E_{\\\\lambda}', commutative=True)))), Pow(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), cos(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 6, "Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Add(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Symbol('E_{\\\\lambda}', commutative=True)))), Pow(Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), cos(Symbol('E_{\\\\lambda}', commutative=True))), Function('E_n')(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\rho,F_{N})} = F_{N} \\rho, then derive \\frac{\\partial}{\\partial \\rho} \\operatorname{t_{1}}{(\\rho,F_{N})} = F_{N}, then obtain 0 = F_{N} \\sin{(F_{N} \\rho - \\operatorname{t_{1}}{(\\rho,F_{N})})}", "derivation": "\\operatorname{t_{1}}{(\\rho,F_{N})} = F_{N} \\rho and 0 = F_{N} \\rho - \\operatorname{t_{1}}{(\\rho,F_{N})} and \\frac{\\partial}{\\partial \\rho} \\operatorname{t_{1}}{(\\rho,F_{N})} = \\frac{\\partial}{\\partial \\rho} F_{N} \\rho and \\frac{\\partial}{\\partial \\rho} \\operatorname{t_{1}}{(\\rho,F_{N})} = F_{N} and 0 = \\sin{(F_{N} \\rho - \\operatorname{t_{1}}{(\\rho,F_{N})})} and \\frac{\\partial}{\\partial \\rho} F_{N} \\rho = F_{N} and 0 = \\sin{(F_{N} \\rho - \\operatorname{t_{1}}{(\\rho,F_{N})})} \\frac{\\partial}{\\partial \\rho} F_{N} \\rho and 0 = F_{N} \\sin{(F_{N} \\rho - \\operatorname{t_{1}}{(\\rho,F_{N})})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["minus", 1, "Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('F_N', commutative=True))"], [["sin", 2], "Equality(Integer(0), sin(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('F_N', commutative=True))"], [["times", 5, "Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(sin(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True))))), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integer(0), Mul(Symbol('F_N', commutative=True), sin(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\rho', commutative=True), Symbol('F_N', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(E)} = \\log{(E)} and \\Psi_{nl}{(E)} = (\\log{(E)}^{E})^{E}, then obtain \\Psi_{nl}{(E)} - \\operatorname{n_{2}}^{E}{(E)} + \\log{(E)}^{E} = (\\operatorname{n_{2}}^{E}{(E)})^{E} - \\operatorname{n_{2}}^{E}{(E)} + \\log{(E)}^{E}", "derivation": "\\operatorname{n_{2}}{(E)} = \\log{(E)} and \\operatorname{n_{2}}^{E}{(E)} = \\log{(E)}^{E} and (\\operatorname{n_{2}}^{E}{(E)})^{E} = (\\log{(E)}^{E})^{E} and \\Psi_{nl}{(E)} = (\\log{(E)}^{E})^{E} and \\Psi_{nl}{(E)} = (\\operatorname{n_{2}}^{E}{(E)})^{E} and \\Psi_{nl}{(E)} - \\operatorname{n_{2}}^{E}{(E)} + \\log{(E)}^{E} = (\\operatorname{n_{2}}^{E}{(E)})^{E} - \\operatorname{n_{2}}^{E}{(E)} + \\log{(E)}^{E}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Pow(Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Pow(Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["minus", 5, "Add(Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True))))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Add(Pow(Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('n_2')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(q,\\mu)} = \\log{(\\mu q)}, then derive \\int \\eta^{\\prime}{(q,\\mu)} dq = \\varepsilon + q \\log{(\\mu q)} - q, then derive \\varepsilon + q \\log{(\\mu q)} - q = \\mathbf{B} + q \\log{(\\mu q)} - q, then obtain \\mathbf{B} + q \\log{(\\mu q)} - q = \\tilde{g}^* + q \\log{(\\mu q)} - q", "derivation": "\\eta^{\\prime}{(q,\\mu)} = \\log{(\\mu q)} and \\int \\eta^{\\prime}{(q,\\mu)} dq = \\int \\log{(\\mu q)} dq and \\int \\eta^{\\prime}{(q,\\mu)} dq = \\varepsilon + q \\log{(\\mu q)} - q and \\varepsilon + q \\log{(\\mu q)} - q = \\int \\log{(\\mu q)} dq and \\varepsilon + q \\log{(\\mu q)} - q = \\mathbf{B} + q \\log{(\\mu q)} - q and \\mathbf{B} + q \\log{(\\mu q)} - q = \\int \\log{(\\mu q)} dq and \\mathbf{B} + q \\log{(\\mu q)} - q = \\tilde{g}^* + q \\log{(\\mu q)} - q", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))), Integral(log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))), Integral(log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Symbol('q', commutative=True), log(Mul(Symbol('\\\\mu', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(-1), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(A_{1})} = \\cos{(A_{1})} and \\operatorname{v_{1}}{(A_{1})} = \\frac{1}{\\mathbf{E}{(A_{1})}}, then obtain \\int \\frac{1}{\\cos{(A_{1})}} dA_{1} = \\int \\frac{1}{\\mathbf{E}{(A_{1})}} dA_{1}", "derivation": "\\mathbf{E}{(A_{1})} = \\cos{(A_{1})} and \\operatorname{v_{1}}{(A_{1})} = \\frac{1}{\\mathbf{E}{(A_{1})}} and \\operatorname{v_{1}}{(A_{1})} = \\frac{1}{\\cos{(A_{1})}} and \\frac{1}{\\cos{(A_{1})}} = \\frac{1}{\\mathbf{E}{(A_{1})}} and \\int \\frac{1}{\\cos{(A_{1})}} dA_{1} = \\int \\frac{1}{\\mathbf{E}{(A_{1})}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('A_1', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('A_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('v_1')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(cos(Symbol('A_1', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('A_1', commutative=True)), Integer(-1)))"], [["integrate", 4, "Symbol('A_1', commutative=True)"], "Equality(Integral(Pow(cos(Symbol('A_1', commutative=True)), Integer(-1)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Function('\\\\mathbf{E}')(Symbol('A_1', commutative=True)), Integer(-1)), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(t_{1},b)} = \\int b t_{1} db, then obtain \\frac{d}{d t_{1}} 0 = \\frac{\\partial}{\\partial t_{1}} \\frac{- \\mathbf{J}^{b}{(t_{1},b)} + (\\int b t_{1} db)^{b}}{b}", "derivation": "\\mathbf{J}{(t_{1},b)} = \\int b t_{1} db and \\mathbf{J}^{b}{(t_{1},b)} = (\\int b t_{1} db)^{b} and 0 = - \\mathbf{J}^{b}{(t_{1},b)} + (\\int b t_{1} db)^{b} and 0 = \\frac{- \\mathbf{J}^{b}{(t_{1},b)} + (\\int b t_{1} db)^{b}}{b} and \\frac{d}{d t_{1}} 0 = \\frac{\\partial}{\\partial t_{1}} \\frac{- \\mathbf{J}^{b}{(t_{1},b)} + (\\int b t_{1} db)^{b}}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Integral(Mul(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Integral(Mul(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Integral(Mul(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"], [["divide", 3, "Symbol('b', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Integral(Mul(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))))"], [["differentiate", 4, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Integral(Mul(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(C,\\mathbf{E})} = - C + \\mathbf{E}, then obtain (\\int \\operatorname{v_{y}}^{\\mathbf{E}}{(C,\\mathbf{E})} d\\mathbf{E})^{C} = (\\int (- C + \\mathbf{E})^{\\mathbf{E}} d\\mathbf{E})^{C}", "derivation": "\\operatorname{v_{y}}{(C,\\mathbf{E})} = - C + \\mathbf{E} and \\operatorname{v_{y}}^{\\mathbf{E}}{(C,\\mathbf{E})} = (- C + \\mathbf{E})^{\\mathbf{E}} and \\int \\operatorname{v_{y}}^{\\mathbf{E}}{(C,\\mathbf{E})} d\\mathbf{E} = \\int (- C + \\mathbf{E})^{\\mathbf{E}} d\\mathbf{E} and (\\int \\operatorname{v_{y}}^{\\mathbf{E}}{(C,\\mathbf{E})} d\\mathbf{E})^{C} = (\\int (- C + \\mathbf{E})^{\\mathbf{E}} d\\mathbf{E})^{C}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given i{(B)} = \\log{(\\sin{(B)})}, then obtain \\int i{(B)} dB + \\iint i{(B)} dB dB = \\int i{(B)} dB + \\iint \\log{(\\sin{(B)})} dB dB", "derivation": "i{(B)} = \\log{(\\sin{(B)})} and \\int i{(B)} dB = \\int \\log{(\\sin{(B)})} dB and \\iint i{(B)} dB dB = \\iint \\log{(\\sin{(B)})} dB dB and \\int \\log{(\\sin{(B)})} dB + \\iint i{(B)} dB dB = \\int \\log{(\\sin{(B)})} dB + \\iint \\log{(\\sin{(B)})} dB dB and \\int i{(B)} dB + \\iint i{(B)} dB dB = \\int i{(B)} dB + \\iint \\log{(\\sin{(B)})} dB dB", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('B', commutative=True)), log(sin(Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["add", 3, "Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))"], "Equality(Add(Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Integral(Function('i')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(log(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given t{(F_{x})} = \\cos{(e^{F_{x}})}, then obtain - (2 t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})}) \\cos^{F_{x}}{(e^{F_{x}})} = - (t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})} + \\cos^{F_{x}}{(e^{F_{x}})}) \\cos^{F_{x}}{(e^{F_{x}})}", "derivation": "t{(F_{x})} = \\cos{(e^{F_{x}})} and t^{F_{x}}{(F_{x})} = \\cos^{F_{x}}{(e^{F_{x}})} and 2 t^{F_{x}}{(F_{x})} = t^{F_{x}}{(F_{x})} + \\cos^{F_{x}}{(e^{F_{x}})} and 2 t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})} = t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})} + \\cos^{F_{x}}{(e^{F_{x}})} and - (2 t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})}) \\cos^{F_{x}}{(e^{F_{x}})} = - (t^{F_{x}}{(F_{x})} - \\cos{(e^{F_{x}})} + \\cos^{F_{x}}{(e^{F_{x}})}) \\cos^{F_{x}}{(e^{F_{x}})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('F_x', commutative=True)), cos(exp(Symbol('F_x', commutative=True))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["add", 2, "Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Add(Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["minus", 3, "cos(exp(Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('F_x', commutative=True))))), Add(Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('F_x', commutative=True)))), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(2), Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('F_x', commutative=True))))), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))), Mul(Integer(-1), Add(Pow(Function('t')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('F_x', commutative=True)))), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))), Pow(cos(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given U{(P_{e},\\eta^{\\prime})} = P_{e} + \\sin{(\\eta^{\\prime})}, then obtain \\frac{\\partial}{\\partial \\eta^{\\prime}} \\sin{((- \\eta^{\\prime} + U{(P_{e},\\eta^{\\prime})})^{P_{e}})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\sin{((P_{e} - \\eta^{\\prime} + \\sin{(\\eta^{\\prime})})^{P_{e}})}", "derivation": "U{(P_{e},\\eta^{\\prime})} = P_{e} + \\sin{(\\eta^{\\prime})} and - \\eta^{\\prime} + U{(P_{e},\\eta^{\\prime})} = P_{e} - \\eta^{\\prime} + \\sin{(\\eta^{\\prime})} and (- \\eta^{\\prime} + U{(P_{e},\\eta^{\\prime})})^{P_{e}} = (P_{e} - \\eta^{\\prime} + \\sin{(\\eta^{\\prime})})^{P_{e}} and \\sin{((- \\eta^{\\prime} + U{(P_{e},\\eta^{\\prime})})^{P_{e}})} = \\sin{((P_{e} - \\eta^{\\prime} + \\sin{(\\eta^{\\prime})})^{P_{e}})} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\sin{((- \\eta^{\\prime} + U{(P_{e},\\eta^{\\prime})})^{P_{e}})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\sin{((P_{e} - \\eta^{\\prime} + \\sin{(\\eta^{\\prime})})^{P_{e}})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('P_e', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('P_e', commutative=True), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["minus", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('U')(Symbol('P_e', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('U')(Symbol('P_e', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True)), Pow(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('U')(Symbol('P_e', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True))), sin(Pow(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('U')(Symbol('P_e', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Pow(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} = e^{\\mathbf{D}}, then obtain \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} + 1 = \\frac{d^{2}}{d \\mathbf{D}^{2}} e^{\\mathbf{D}} + 1", "derivation": "\\phi{(\\mathbf{D})} = e^{\\mathbf{D}} and \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} e^{\\mathbf{D}} and \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} = e^{\\mathbf{D}} and \\frac{d}{d \\mathbf{D}} e^{\\mathbf{D}} = e^{\\mathbf{D}} and \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} = \\frac{d^{2}}{d \\mathbf{D}^{2}} e^{\\mathbf{D}} and \\frac{d}{d \\mathbf{D}} \\phi{(\\mathbf{D})} + 1 = \\frac{d^{2}}{d \\mathbf{D}^{2}} e^{\\mathbf{D}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))))"], [["add", 5, 1], "Equality(Add(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Integer(1)))"]]}, {"prompt": "Given \\rho_{b}{(V)} = \\sin{(\\log{(V)})}, then derive U + \\rho_{b}{(V)} = a + \\sin{(\\log{(V)})}, then obtain \\frac{\\partial}{\\partial V} (U + \\rho_{b}{(V)}) = \\frac{\\partial}{\\partial V} (a + \\sin{(\\log{(V)})})", "derivation": "\\rho_{b}{(V)} = \\sin{(\\log{(V)})} and \\frac{d}{d V} \\rho_{b}{(V)} = \\frac{d}{d V} \\sin{(\\log{(V)})} and \\int \\frac{d}{d V} \\rho_{b}{(V)} dV = \\int \\frac{d}{d V} \\sin{(\\log{(V)})} dV and U + \\rho_{b}{(V)} = a + \\sin{(\\log{(V)})} and \\frac{\\partial}{\\partial V} (U + \\rho_{b}{(V)}) = \\frac{\\partial}{\\partial V} (a + \\sin{(\\log{(V)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(sin(log(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\rho_b')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(sin(log(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('U', commutative=True), Function('\\\\rho_b')(Symbol('V', commutative=True))), Add(Symbol('a', commutative=True), sin(log(Symbol('V', commutative=True)))))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Symbol('U', commutative=True), Function('\\\\rho_b')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), sin(log(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(n_{2})} = \\sin{(n_{2})} and \\operatorname{a^{\\dagger}}{(r,\\hbar)} = r + e^{\\hbar}, then obtain - \\eta^{n_{2}}{(n_{2})} + \\iint \\operatorname{a^{\\dagger}}{(r,\\hbar)} dr d\\hbar = - \\eta^{n_{2}}{(n_{2})} + \\iint (r + e^{\\hbar}) dr d\\hbar", "derivation": "\\eta{(n_{2})} = \\sin{(n_{2})} and \\operatorname{a^{\\dagger}}{(r,\\hbar)} = r + e^{\\hbar} and \\int \\operatorname{a^{\\dagger}}{(r,\\hbar)} dr = \\int (r + e^{\\hbar}) dr and \\iint \\operatorname{a^{\\dagger}}{(r,\\hbar)} dr d\\hbar = \\iint (r + e^{\\hbar}) dr d\\hbar and - \\sin^{n_{2}}{(n_{2})} + \\iint \\operatorname{a^{\\dagger}}{(r,\\hbar)} dr d\\hbar = - \\sin^{n_{2}}{(n_{2})} + \\iint (r + e^{\\hbar}) dr d\\hbar and - \\eta^{n_{2}}{(n_{2})} + \\iint \\operatorname{a^{\\dagger}}{(r,\\hbar)} dr d\\hbar = - \\eta^{n_{2}}{(n_{2})} + \\iint (r + e^{\\hbar}) dr d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], ["get_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('r', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 4, "Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Integral(Function('a^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Integral(Function('a^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\mathbf{r})} = \\log{(\\cos{(\\mathbf{r})})} and \\dot{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r}, then obtain \\dot{\\mathbf{r}}{(\\mathbf{r})} \\log{(\\cos{(\\mathbf{r})})}^{\\mathbf{r}} = \\mathbf{r} \\log{(\\cos{(\\mathbf{r})})}^{\\mathbf{r}}", "derivation": "\\chi{(\\mathbf{r})} = \\log{(\\cos{(\\mathbf{r})})} and \\chi^{\\mathbf{r}}{(\\mathbf{r})} = \\log{(\\cos{(\\mathbf{r})})}^{\\mathbf{r}} and \\dot{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r} and \\chi^{\\mathbf{r}}{(\\mathbf{r})} \\dot{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r} \\chi^{\\mathbf{r}}{(\\mathbf{r})} and \\dot{\\mathbf{r}}{(\\mathbf{r})} \\log{(\\cos{(\\mathbf{r})})}^{\\mathbf{r}} = \\mathbf{r} \\log{(\\cos{(\\mathbf{r})})}^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{r}', commutative=True)), log(cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["times", 3, "Pow(Function('\\\\chi')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('\\\\chi')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(log(cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P}, then derive - \\dot{x}{(\\mathbf{J}_P)} = - 2 \\dot{x}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P}, then obtain - 2 \\dot{x}{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} = - 2 \\dot{x}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P}", "derivation": "\\dot{x}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} and 2 \\dot{x}{(\\mathbf{J}_P)} = \\dot{x}{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} and - \\dot{x}{(\\mathbf{J}_P)} = - 2 \\dot{x}{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} and - \\dot{x}{(\\mathbf{J}_P)} = - 2 \\dot{x}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} and - \\dot{x}{(\\mathbf{J}_P)} = - \\dot{x}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} - \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} and - 2 \\dot{x}{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} e^{\\mathbf{J}_P} = - 2 \\dot{x}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["add", 1, "Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["minus", 1, "Mul(Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(f^{*})} = \\cos{(f^{*})}, then obtain \\frac{4 \\operatorname{g_{\\varepsilon}}^{2}{(f^{*})}}{(\\operatorname{g_{\\varepsilon}}{(f^{*})} + \\cos{(f^{*})})^{2}} = 1", "derivation": "\\operatorname{g_{\\varepsilon}}{(f^{*})} = \\cos{(f^{*})} and 2 \\operatorname{g_{\\varepsilon}}{(f^{*})} = \\operatorname{g_{\\varepsilon}}{(f^{*})} + \\cos{(f^{*})} and 4 \\operatorname{g_{\\varepsilon}}^{2}{(f^{*})} = (\\operatorname{g_{\\varepsilon}}{(f^{*})} + \\cos{(f^{*})})^{2} and \\frac{4 \\operatorname{g_{\\varepsilon}}^{2}{(f^{*})}}{(\\operatorname{g_{\\varepsilon}}{(f^{*})} + \\cos{(f^{*})})^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["add", 1, "Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True))), Add(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), Integer(2))), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Integer(2)))"], [["divide", 3, "Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Integer(2))"], "Equality(Mul(Integer(4), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Integer(-2)), Pow(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(s)} = e^{s}, then derive \\int \\operatorname{A_{x}}{(s)} ds = \\eta^{\\prime} + e^{s}, then obtain \\int e^{s} ds = \\eta^{\\prime} + e^{s}", "derivation": "\\operatorname{A_{x}}{(s)} = e^{s} and \\int \\operatorname{A_{x}}{(s)} ds = \\int e^{s} ds and \\int \\operatorname{A_{x}}{(s)} ds = \\eta^{\\prime} + e^{s} and \\int e^{s} ds = \\eta^{\\prime} + e^{s}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_x')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(k)} = \\log{(k)} and \\Psi_{\\lambda}{(k)} = \\log{(k)}, then derive \\int \\operatorname{F_{N}}{(k)} dk = \\pi + k \\log{(k)} - k, then obtain \\frac{\\pi + k \\Psi_{\\lambda}{(k)} - k}{\\pi} = \\frac{\\int \\log{(k)} dk}{\\pi}", "derivation": "\\operatorname{F_{N}}{(k)} = \\log{(k)} and \\Psi_{\\lambda}{(k)} = \\log{(k)} and \\int \\operatorname{F_{N}}{(k)} dk = \\int \\log{(k)} dk and \\int \\operatorname{F_{N}}{(k)} dk = \\pi + k \\log{(k)} - k and \\pi + k \\log{(k)} - k = \\int \\log{(k)} dk and \\pi + k \\Psi_{\\lambda}{(k)} - k = \\int \\log{(k)} dk and \\frac{\\pi + k \\Psi_{\\lambda}{(k)} - k}{\\pi} = \\frac{\\int \\log{(k)} dk}{\\pi}", "srepr_derivation": [["get_premise", "Equality(Function('F_N')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('F_N')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('k', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["divide", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('k', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(a)} = \\cos{(e^{a})}, then derive \\theta_1 + \\int \\frac{\\tilde{g}{(a)} - \\cos{(e^{a})}}{\\cos{(e^{a})}} da = \\int 0 da, then obtain \\int (\\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} - 1) da = \\theta_1 + \\int 0 da", "derivation": "\\tilde{g}{(a)} = \\cos{(e^{a})} and \\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} = 1 and \\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} - 1 = 0 and \\int (\\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} - 1) da = \\int 0 da and \\theta_1 + \\int \\frac{\\tilde{g}{(a)} - \\cos{(e^{a})}}{\\cos{(e^{a})}} da = \\int 0 da and \\int (\\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} - 1) da = \\theta_1 + \\int \\frac{\\tilde{g}{(a)} - \\cos{(e^{a})}}{\\cos{(e^{a})}} da and \\int 0 da = \\theta_1 + \\int 0 da and \\int (\\frac{\\tilde{g}{(a)}}{\\cos{(e^{a})}} - 1) da = \\theta_1 + \\int 0 da", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), cos(exp(Symbol('a', commutative=True))))"], [["divide", 1, "cos(exp(Symbol('a', commutative=True)))"], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Integer(-1)), Integer(0))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Mul(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Integer(-1)), Tuple(Symbol('a', commutative=True))), Integral(Integer(0), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Integral(Mul(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('a', commutative=True))))), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Tuple(Symbol('a', commutative=True)))), Integral(Integer(0), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Mul(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Integer(-1)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Integral(Mul(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('a', commutative=True))))), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Tuple(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Integer(0), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Integral(Integer(0), Tuple(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Integral(Add(Mul(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Integer(-1))), Integer(-1)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Integral(Integer(0), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given V{(\\tilde{g})} = \\int \\log{(\\tilde{g})} d\\tilde{g}, then derive V{(\\tilde{g})} = \\Psi^{\\dagger} + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g}, then obtain \\hat{p}_0 \\int \\log{(\\tilde{g})} d\\tilde{g} = \\hat{p}_0 (\\Psi^{\\dagger} + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g})", "derivation": "V{(\\tilde{g})} = \\int \\log{(\\tilde{g})} d\\tilde{g} and V{(\\tilde{g})} = \\Psi^{\\dagger} + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g} and \\int \\log{(\\tilde{g})} d\\tilde{g} = \\Psi^{\\dagger} + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g} and \\hat{p}_0 \\int \\log{(\\tilde{g})} d\\tilde{g} = \\hat{p}_0 (\\Psi^{\\dagger} + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g})", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\tilde{g}', commutative=True)), Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('V')(Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Symbol('\\\\hat{p}_0', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given U{(h)} = \\int \\log{(h)} dh, then derive U{(h)} = h \\log{(h)} - h + v_{1}, then obtain \\hat{x} k \\frac{\\partial}{\\partial h} (h \\log{(h)} - h + v_{1}) = \\hat{x} k \\frac{d}{d h} \\int \\log{(h)} dh", "derivation": "U{(h)} = \\int \\log{(h)} dh and U{(h)} = h \\log{(h)} - h + v_{1} and \\frac{d}{d h} U{(h)} = \\frac{d}{d h} \\int \\log{(h)} dh and \\hat{x} k \\frac{d}{d h} U{(h)} = \\hat{x} k \\frac{d}{d h} \\int \\log{(h)} dh and \\hat{x} k \\frac{\\partial}{\\partial h} (h \\log{(h)} - h + v_{1}) = \\hat{x} k \\frac{d}{d h} \\int \\log{(h)} dh", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('h', commutative=True)), Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('U')(Symbol('h', commutative=True)), Add(Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 3, "Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('k', commutative=True), Derivative(Function('U')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('k', commutative=True), Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('k', commutative=True), Derivative(Add(Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('k', commutative=True), Derivative(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(\\hbar)} = \\frac{d}{d \\hbar} e^{\\hbar}, then derive \\hbar + \\int \\hat{H}{(\\hbar)} d\\hbar = P_{e} + \\hbar + e^{\\hbar}, then obtain e^{P_{e} + \\hbar + e^{\\hbar}} = e^{\\hbar + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar}", "derivation": "\\hat{H}{(\\hbar)} = \\frac{d}{d \\hbar} e^{\\hbar} and \\int \\hat{H}{(\\hbar)} d\\hbar = \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar and \\hbar + \\int \\hat{H}{(\\hbar)} d\\hbar = \\hbar + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar and \\hbar + \\int \\hat{H}{(\\hbar)} d\\hbar = P_{e} + \\hbar + e^{\\hbar} and P_{e} + \\hbar + e^{\\hbar} = \\hbar + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar and e^{P_{e} + \\hbar + e^{\\hbar}} = e^{\\hbar + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\hbar', commutative=True), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hbar', commutative=True), Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["exp", 5], "Equality(exp(Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), exp(Symbol('\\\\hbar', commutative=True)))), exp(Add(Symbol('\\\\hbar', commutative=True), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{v}{(\\psi^*)} = \\cos{(\\psi^*)}, then derive \\int \\mathbf{v}{(\\psi^*)} d\\psi^* = \\theta_2 + \\sin{(\\psi^*)}, then derive \\theta_2 + \\sin{(\\psi^*)} = \\pi + \\sin{(\\psi^*)}, then obtain \\psi^* (\\theta_2 + \\sin{(\\psi^*)}) = \\psi^* (\\pi + \\sin{(\\psi^*)})", "derivation": "\\mathbf{v}{(\\psi^*)} = \\cos{(\\psi^*)} and \\int \\mathbf{v}{(\\psi^*)} d\\psi^* = \\int \\cos{(\\psi^*)} d\\psi^* and \\int \\mathbf{v}{(\\psi^*)} d\\psi^* = \\theta_2 + \\sin{(\\psi^*)} and \\theta_2 + \\sin{(\\psi^*)} = \\int \\cos{(\\psi^*)} d\\psi^* and \\theta_2 + \\sin{(\\psi^*)} = \\pi + \\sin{(\\psi^*)} and \\psi^* (\\theta_2 + \\sin{(\\psi^*)}) = \\psi^* (\\pi + \\sin{(\\psi^*)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\psi^*', commutative=True))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi^*', commutative=True))))"], [["times", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\psi^*', commutative=True)))), Mul(Symbol('\\\\psi^*', commutative=True), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given L{(\\mathbf{g})} = e^{\\mathbf{g}}, then obtain - 2 L{(\\mathbf{g})} + 2 L^{\\mathbf{g}}{(\\mathbf{g})} - 2 e^{\\mathbf{g}} = - 2 L{(\\mathbf{g})} + L^{\\mathbf{g}}{(\\mathbf{g})} - 2 e^{\\mathbf{g}} + (e^{\\mathbf{g}})^{\\mathbf{g}}", "derivation": "L{(\\mathbf{g})} = e^{\\mathbf{g}} and 2 L{(\\mathbf{g})} = L{(\\mathbf{g})} + e^{\\mathbf{g}} and L^{\\mathbf{g}}{(\\mathbf{g})} = (e^{\\mathbf{g}})^{\\mathbf{g}} and - 2 L{(\\mathbf{g})} + L^{\\mathbf{g}}{(\\mathbf{g})} = - 2 L{(\\mathbf{g})} + (e^{\\mathbf{g}})^{\\mathbf{g}} and - 4 L{(\\mathbf{g})} + 2 L^{\\mathbf{g}}{(\\mathbf{g})} = - 4 L{(\\mathbf{g})} + L^{\\mathbf{g}}{(\\mathbf{g})} + (e^{\\mathbf{g}})^{\\mathbf{g}} and - 2 L{(\\mathbf{g})} + 2 L^{\\mathbf{g}}{(\\mathbf{g})} - 2 e^{\\mathbf{g}} = - 2 L{(\\mathbf{g})} + L^{\\mathbf{g}}{(\\mathbf{g})} - 2 e^{\\mathbf{g}} + (e^{\\mathbf{g}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\mathbf{g}', commutative=True)))"], [["add", 1, "Function('L')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(4), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(2), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Integer(-1), Integer(4), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(2), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\mathbf{g}', commutative=True))), Pow(Function('L')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mathbf{g}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(F_{c})} = e^{F_{c}}, then obtain F_{c} \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})} + \\operatorname{E_{n}}{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})} = F_{c} e^{F_{c}} + e^{F_{c}} + \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})}", "derivation": "\\operatorname{E_{n}}{(F_{c})} = e^{F_{c}} and F_{c} \\operatorname{E_{n}}{(F_{c})} = F_{c} e^{F_{c}} and F_{c} \\operatorname{E_{n}}{(F_{c})} + \\operatorname{E_{n}}{(F_{c})} = F_{c} e^{F_{c}} + \\operatorname{E_{n}}{(F_{c})} and \\frac{d}{d F_{c}} (F_{c} \\operatorname{E_{n}}{(F_{c})} + \\operatorname{E_{n}}{(F_{c})}) = \\frac{d}{d F_{c}} (F_{c} e^{F_{c}} + \\operatorname{E_{n}}{(F_{c})}) and F_{c} \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})} + \\operatorname{E_{n}}{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})} = F_{c} e^{F_{c}} + e^{F_{c}} + \\frac{d}{d F_{c}} \\operatorname{E_{n}}{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["times", 1, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Function('E_n')(Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), exp(Symbol('F_c', commutative=True))))"], [["add", 2, "Function('E_n')(Symbol('F_c', commutative=True))"], "Equality(Add(Mul(Symbol('F_c', commutative=True), Function('E_n')(Symbol('F_c', commutative=True))), Function('E_n')(Symbol('F_c', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), exp(Symbol('F_c', commutative=True))), Function('E_n')(Symbol('F_c', commutative=True))))"], [["differentiate", 3, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_c', commutative=True), Function('E_n')(Symbol('F_c', commutative=True))), Function('E_n')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_c', commutative=True), exp(Symbol('F_c', commutative=True))), Function('E_n')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('F_c', commutative=True), Derivative(Function('E_n')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Function('E_n')(Symbol('F_c', commutative=True)), Derivative(Function('E_n')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Mul(Symbol('F_c', commutative=True), exp(Symbol('F_c', commutative=True))), exp(Symbol('F_c', commutative=True)), Derivative(Function('E_n')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi{(L)} = \\int \\log{(L)} dL, then derive \\Psi{(L)} = L \\log{(L)} - L + \\eta^{\\prime}, then obtain \\cos{(L)} = \\cos{(L \\log{(L)} + \\eta^{\\prime} - \\int \\log{(L)} dL)}", "derivation": "\\Psi{(L)} = \\int \\log{(L)} dL and 0 = - \\Psi{(L)} + \\int \\log{(L)} dL and \\Psi{(L)} = L \\log{(L)} - L + \\eta^{\\prime} and 0 = - L \\log{(L)} + L - \\eta^{\\prime} + \\int \\log{(L)} dL and - L = - L \\log{(L)} - \\eta^{\\prime} + \\int \\log{(L)} dL and - L = - L \\log{(L)} - \\eta^{\\prime} + \\Psi{(L)} and \\cos{(L)} = \\cos{(L \\log{(L)} + \\eta^{\\prime} - \\Psi{(L)})} and \\cos{(L)} = \\cos{(L \\log{(L)} + \\eta^{\\prime} - \\int \\log{(L)} dL)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('L', commutative=True)), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["minus", 1, "Function('\\\\Psi')(Symbol('L', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('L', commutative=True))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\Psi')(Symbol('L', commutative=True)), Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('L', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\Psi')(Symbol('L', commutative=True))))"], [["cos", 6], "Equality(cos(Symbol('L', commutative=True)), cos(Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('\\\\Psi')(Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(cos(Symbol('L', commutative=True)), cos(Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))))"]]}, {"prompt": "Given \\rho{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)}, then obtain 0^{\\hat{p}_0} \\tilde{\\infty}^{2 \\hat{p}_0} (\\rho{(\\hat{p}_0)} - \\sin{(\\hat{p}_0)})^{\\hat{p}_0} = 0^{\\hat{p}_0} \\tilde{\\infty}^{2 \\hat{p}_0}", "derivation": "\\rho{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\rho{(\\hat{p}_0)} - \\sin{(\\hat{p}_0)} = 0 and (\\rho{(\\hat{p}_0)} - \\sin{(\\hat{p}_0)})^{\\hat{p}_0} = 0^{\\hat{p}_0} and \\tilde{\\infty}^{\\hat{p}_0} (\\rho{(\\hat{p}_0)} - \\sin{(\\hat{p}_0)})^{\\hat{p}_0} = 0^{\\hat{p}_0} \\tilde{\\infty}^{\\hat{p}_0} and 0^{\\hat{p}_0} \\tilde{\\infty}^{2 \\hat{p}_0} (\\rho{(\\hat{p}_0)} - \\sin{(\\hat{p}_0)})^{\\hat{p}_0} = 0^{\\hat{p}_0} \\tilde{\\infty}^{2 \\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 3, "Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True)), Pow(zoo, Symbol('\\\\hat{p}_0', commutative=True))))"], [["times", 4, "Mul(Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True)), Pow(zoo, Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True)), Pow(zoo, Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True))), Pow(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\hat{p}_0', commutative=True)), Pow(zoo, Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(i,E,\\hat{H})} = E \\hat{H} i, then obtain \\frac{\\hat{H}^{4} \\phi_{1}^{2}{(i,E,\\hat{H})}}{E^{2}} = \\frac{\\hat{H}^{5} i \\phi_{1}{(i,E,\\hat{H})}}{E}", "derivation": "\\phi_{1}{(i,E,\\hat{H})} = E \\hat{H} i and E \\hat{H} i \\phi_{1}{(i,E,\\hat{H})} = E^{2} \\hat{H}^{2} i^{2} and E \\hat{H}^{2} i \\phi_{1}{(i,E,\\hat{H})} = E^{2} \\hat{H}^{3} i^{2} and \\frac{E \\hat{H}^{2} \\phi_{1}{(i,E,\\hat{H})}}{i} = E^{2} \\hat{H}^{3} and \\frac{\\hat{H}^{2} \\phi_{1}{(i,E,\\hat{H})}}{E i} = \\hat{H}^{3} and \\frac{\\hat{H}^{4} \\phi_{1}^{2}{(i,E,\\hat{H})}}{E^{2}} = \\frac{\\hat{H}^{5} i \\phi_{1}{(i,E,\\hat{H})}}{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)))"], [["times", 1, "Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["times", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('i', commutative=True), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(3)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["divide", 3, "Pow(Symbol('i', commutative=True), Integer(2))"], "Equality(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(3))))"], [["divide", 4, "Pow(Symbol('E', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(3)))"], [["times", 5, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('i', commutative=True), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(4)), Pow(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(5)), Symbol('i', commutative=True), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('E', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given J{(\\psi)} = \\sin{(\\log{(\\psi)})} and \\Psi^{\\dagger}{(\\psi)} = \\frac{J{(\\psi)}}{\\sin{(\\log{(\\psi)})}}, then obtain \\int \\Psi^{\\dagger}{(\\psi)} d\\psi = \\int 1 d\\psi", "derivation": "J{(\\psi)} = \\sin{(\\log{(\\psi)})} and \\Psi^{\\dagger}{(\\psi)} = \\frac{J{(\\psi)}}{\\sin{(\\log{(\\psi)})}} and \\Psi^{\\dagger}{(\\psi)} = 1 and \\int \\Psi^{\\dagger}{(\\psi)} d\\psi = \\int 1 d\\psi", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\psi', commutative=True)), sin(log(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Mul(Function('J')(Symbol('\\\\psi', commutative=True)), Pow(sin(log(Symbol('\\\\psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given A{(v_{y})} = \\sin{(\\cos{(v_{y})})} and \\mathbf{r}{(v_{y})} = \\cos{(v_{y})}, then obtain - v_{y} + A{(v_{y})} = - v_{y} + \\sin{(\\mathbf{r}{(v_{y})})}", "derivation": "A{(v_{y})} = \\sin{(\\cos{(v_{y})})} and - v_{y} + A{(v_{y})} = - v_{y} + \\sin{(\\cos{(v_{y})})} and \\mathbf{r}{(v_{y})} = \\cos{(v_{y})} and - v_{y} + A{(v_{y})} = - v_{y} + \\sin{(\\mathbf{r}{(v_{y})})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('v_y', commutative=True)), sin(cos(Symbol('v_y', commutative=True))))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('A')(Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), sin(cos(Symbol('v_y', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('A')(Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), sin(Function('\\\\mathbf{r}')(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\nabla,M)} = \\cos{(M + \\nabla)}, then derive \\int \\operatorname{E_{n}}{(\\nabla,M)} d\\nabla = n + \\sin{(M + \\nabla)}, then obtain \\int (n + \\sin{(M + \\nabla)}) d\\nabla = \\iint \\cos{(M + \\nabla)} d\\nabla d\\nabla", "derivation": "\\operatorname{E_{n}}{(\\nabla,M)} = \\cos{(M + \\nabla)} and \\int \\operatorname{E_{n}}{(\\nabla,M)} d\\nabla = \\int \\cos{(M + \\nabla)} d\\nabla and \\int \\operatorname{E_{n}}{(\\nabla,M)} d\\nabla = n + \\sin{(M + \\nabla)} and n + \\sin{(M + \\nabla)} = \\int \\cos{(M + \\nabla)} d\\nabla and \\int (n + \\sin{(M + \\nabla)}) d\\nabla = \\iint \\cos{(M + \\nabla)} d\\nabla d\\nabla", "srepr_derivation": [["get_premise", "Equality(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), cos(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('n', commutative=True), sin(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('n', commutative=True), sin(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integral(cos(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Add(Symbol('n', commutative=True), sin(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Add(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given f{(\\mathbf{P},C_{1})} = \\mathbf{P}^{C_{1}} and \\operatorname{r_{0}}{(\\mathbf{P},C_{1},\\eta^{\\prime})} = \\eta^{\\prime} + f{(\\mathbf{P},C_{1})}, then obtain \\operatorname{r_{0}}{(\\mathbf{P},C_{1},\\eta^{\\prime})} = \\eta^{\\prime} + \\mathbf{P}^{C_{1}}", "derivation": "f{(\\mathbf{P},C_{1})} = \\mathbf{P}^{C_{1}} and \\eta^{\\prime} + f{(\\mathbf{P},C_{1})} = \\eta^{\\prime} + \\mathbf{P}^{C_{1}} and \\operatorname{r_{0}}{(\\mathbf{P},C_{1},\\eta^{\\prime})} = \\eta^{\\prime} + f{(\\mathbf{P},C_{1})} and \\operatorname{r_{0}}{(\\mathbf{P},C_{1},\\eta^{\\prime})} = \\eta^{\\prime} + \\mathbf{P}^{C_{1}}", "srepr_derivation": [["renaming_premise", "Equality(Function('f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True)))"], [["add", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('r_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(E)} = \\log{(E)}, then derive 0 = - \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{1}{E}, then obtain 0 = \\frac{- \\hat{x}_0{(E)} + \\log{(E)}}{\\hat{x}_0{(E)}} - \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{1}{E}", "derivation": "\\hat{x}_0{(E)} = \\log{(E)} and 0 = - \\hat{x}_0{(E)} + \\log{(E)} and \\frac{d}{d E} 0 = \\frac{d}{d E} (- \\hat{x}_0{(E)} + \\log{(E)}) and 0 = \\frac{- \\hat{x}_0{(E)} + \\log{(E)}}{\\hat{x}_0{(E)}} and 0 = - \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{1}{E} and - \\frac{d}{d E} \\hat{x}_0{(E)} = \\frac{- \\hat{x}_0{(E)} + \\log{(E)}}{\\hat{x}_0{(E)}} - \\frac{d}{d E} \\hat{x}_0{(E)} and 0 = \\frac{- \\hat{x}_0{(E)} + \\log{(E)}}{\\hat{x}_0{(E)}} - \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{1}{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["minus", 1, "Function('\\\\hat{x}_0')(Symbol('E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('E', commutative=True))), log(Symbol('E', commutative=True))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('E', commutative=True))), log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\hat{x}_0')(Symbol('E', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('E', commutative=True))), log(Symbol('E', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Pow(Symbol('E', commutative=True), Integer(-1))))"], [["add", 4, "Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('E', commutative=True))), log(Symbol('E', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Integer(-1))), Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Integer(0), Add(Mul(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('E', commutative=True))), log(Symbol('E', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Integer(-1))), Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Pow(Symbol('E', commutative=True), Integer(-1))))"]]}, {"prompt": "Given k{(\\pi,F_{x})} = F_{x} - \\pi, then obtain \\frac{\\partial}{\\partial F_{x}} - \\frac{k{(\\pi,F_{x})}}{\\pi} = \\frac{\\partial}{\\partial F_{x}} \\frac{- F_{x} + \\pi}{\\pi}", "derivation": "k{(\\pi,F_{x})} = F_{x} - \\pi and - \\frac{k{(\\pi,F_{x})}}{\\pi} = - \\frac{F_{x} - \\pi}{\\pi} and \\frac{\\partial}{\\partial F_{x}} - \\frac{k{(\\pi,F_{x})}}{\\pi} = \\frac{\\partial}{\\partial F_{x}} - \\frac{F_{x} - \\pi}{\\pi} and \\frac{\\partial}{\\partial F_{x}} - \\frac{F_{x} - \\pi}{\\pi} = \\frac{\\partial}{\\partial F_{x}} \\frac{- F_{x} + \\pi}{\\pi} and \\frac{\\partial}{\\partial F_{x}} - \\frac{k{(\\pi,F_{x})}}{\\pi} = \\frac{\\partial}{\\partial F_{x}} \\frac{- F_{x} + \\pi}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(F_{H})} = \\cos{(F_{H})}, then obtain (E{(F_{H})} + \\cos{(F_{H})}) E^{F_{H}}{(F_{H})} = 2 E^{F_{H}}{(F_{H})} \\cos{(F_{H})}", "derivation": "E{(F_{H})} = \\cos{(F_{H})} and E{(F_{H})} + \\cos{(F_{H})} = 2 \\cos{(F_{H})} and E^{F_{H}}{(F_{H})} = \\cos^{F_{H}}{(F_{H})} and (E{(F_{H})} + \\cos{(F_{H})}) \\cos^{F_{H}}{(F_{H})} = 2 \\cos{(F_{H})} \\cos^{F_{H}}{(F_{H})} and (E{(F_{H})} + \\cos{(F_{H})}) E^{F_{H}}{(F_{H})} = 2 E^{F_{H}}{(F_{H})} \\cos{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["add", 1, "cos(Symbol('F_H', commutative=True))"], "Equality(Add(Function('E')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Mul(Integer(2), cos(Symbol('F_H', commutative=True))))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('E')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["times", 2, "Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], "Equality(Mul(Add(Function('E')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Integer(2), cos(Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Function('E')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Pow(Function('E')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Integer(2), Pow(Function('E')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\chi)} = \\int e^{\\chi} d\\chi, then derive (c + e^{\\chi}) \\theta{(\\chi)} = (c + e^{\\chi})^{2}, then obtain (c + e^{\\chi})^{2} (c + e^{\\chi})^{\\chi} \\theta{(\\chi)} = \\frac{(c + e^{\\chi})^{6} (c + e^{\\chi})^{\\chi}}{\\theta^{3}{(\\chi)}}", "derivation": "\\theta{(\\chi)} = \\int e^{\\chi} d\\chi and \\theta{(\\chi)} \\int e^{\\chi} d\\chi = (\\int e^{\\chi} d\\chi)^{2} and (c + e^{\\chi}) \\theta{(\\chi)} = (c + e^{\\chi})^{2} and c + e^{\\chi} = \\frac{(c + e^{\\chi})^{2}}{\\theta{(\\chi)}} and (c + e^{\\chi})^{2} \\theta^{2}{(\\chi)} = (c + e^{\\chi})^{4} and (c + e^{\\chi})^{2} = \\frac{(c + e^{\\chi})^{4}}{\\theta^{2}{(\\chi)}} and \\frac{(c + e^{\\chi})^{4} (c + e^{\\chi})^{\\chi}}{\\theta{(\\chi)}} = \\frac{(c + e^{\\chi})^{6} (c + e^{\\chi})^{\\chi}}{\\theta^{3}{(\\chi)}} and (c + e^{\\chi})^{2} (c + e^{\\chi})^{\\chi} \\theta{(\\chi)} = \\frac{(c + e^{\\chi})^{6} (c + e^{\\chi})^{\\chi}}{\\theta^{3}{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Function('\\\\theta')(Symbol('\\\\chi', commutative=True))), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)))"], [["divide", 3, "Function('\\\\theta')(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["power", 3, 2], "Equality(Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(2))), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(4)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)), Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(4)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-2))))"], [["times", 6, "Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(4)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(6)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-3))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('\\\\theta')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Integer(6)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\chi', commutative=True)), Integer(-3))))"]]}, {"prompt": "Given \\sigma_{x}{(a,G,l)} = a^{G} - l and \\delta{(a,G)} = - a^{G}, then obtain l + 2 \\delta{(a,G)} = - a^{G} + l + \\delta{(a,G)}", "derivation": "\\sigma_{x}{(a,G,l)} = a^{G} - l and 0 = a^{G} - l - \\sigma_{x}{(a,G,l)} and - 2 a^{G} + l = - a^{G} - \\sigma_{x}{(a,G,l)} and \\delta{(a,G)} = - a^{G} and l + 2 \\delta{(a,G)} = \\delta{(a,G)} - \\sigma_{x}{(a,G,l)} and l + 2 \\delta{(a,G)} = - a^{G} + l + \\delta{(a,G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('G', commutative=True), Symbol('l', commutative=True)), Add(Pow(Symbol('a', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["minus", 1, "Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('G', commutative=True), Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('a', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('G', commutative=True), Symbol('l', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(2), Pow(Symbol('a', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('a', commutative=True), Symbol('G', commutative=True))), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('G', commutative=True), Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('a', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('G', commutative=True)))), Add(Function('\\\\delta')(Symbol('a', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('G', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('G', commutative=True))), Symbol('l', commutative=True), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('G', commutative=True))))"]]}, {"prompt": "Given U{(C,g)} = C + g, then obtain \\frac{(C U{(C,g)})^{4 C}}{U^{2}{(C,g)}} = \\frac{(C (C + g))^{2 C} (C U{(C,g)})^{2 C}}{U^{2}{(C,g)}}", "derivation": "U{(C,g)} = C + g and C U{(C,g)} = C (C + g) and (C U{(C,g)})^{C} = (C (C + g))^{C} and \\frac{(C U{(C,g)})^{C}}{U{(C,g)}} = \\frac{(C (C + g))^{C}}{U{(C,g)}} and \\frac{(C U{(C,g)})^{2 C}}{U{(C,g)}} = \\frac{(C (C + g))^{C} (C U{(C,g)})^{C}}{U{(C,g)}} and \\frac{(C U{(C,g)})^{4 C}}{U^{2}{(C,g)}} = \\frac{(C (C + g))^{2 C} (C U{(C,g)})^{2 C}}{U^{2}{(C,g)}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Add(Symbol('C', commutative=True), Symbol('g', commutative=True)))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Symbol('g', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)))"], [["times", 3, "Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-1))), Mul(Pow(Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-1))))"], [["times", 4, "Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Mul(Integer(2), Symbol('C', commutative=True))), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-1))), Mul(Pow(Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Symbol('C', commutative=True)), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-1))))"], [["power", 5, 2], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Mul(Integer(4), Symbol('C', commutative=True))), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-2))), Mul(Pow(Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Symbol('g', commutative=True))), Mul(Integer(2), Symbol('C', commutative=True))), Pow(Mul(Symbol('C', commutative=True), Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True))), Mul(Integer(2), Symbol('C', commutative=True))), Pow(Function('U')(Symbol('C', commutative=True), Symbol('g', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})}, then obtain \\int 0 d\\mu_0 = \\int (- \\log{(\\int \\hat{x}_0^{\\mu_0}{(\\mu_0)} d\\mu_0)} + \\log{(\\int \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} d\\mu_0)}) d\\mu_0", "derivation": "\\hat{x}_0{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})} and \\hat{x}_0^{\\mu_0}{(\\mu_0)} = \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} and \\int \\hat{x}_0^{\\mu_0}{(\\mu_0)} d\\mu_0 = \\int \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} d\\mu_0 and \\log{(\\int \\hat{x}_0^{\\mu_0}{(\\mu_0)} d\\mu_0)} = \\log{(\\int \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} d\\mu_0)} and 0 = - \\log{(\\int \\hat{x}_0^{\\mu_0}{(\\mu_0)} d\\mu_0)} + \\log{(\\int \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} d\\mu_0)} and \\int 0 d\\mu_0 = \\int (- \\log{(\\int \\hat{x}_0^{\\mu_0}{(\\mu_0)} d\\mu_0)} + \\log{(\\int \\sin^{\\mu_0}{(\\cos{(\\mu_0)})} d\\mu_0)}) d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), sin(cos(Symbol('\\\\mu_0', commutative=True))))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["log", 3], "Equality(log(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), log(Integral(Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 4, "log(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), log(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), log(Integral(Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))))"], [["integrate", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), log(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), log(Integral(Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(i)} = \\sin{(i)}, then obtain \\frac{\\int \\operatorname{V_{\\mathbf{B}}}{(i)} di}{\\delta - \\cos{(i)}} = 1", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(i)} = \\sin{(i)} and \\int \\operatorname{V_{\\mathbf{B}}}{(i)} di = \\int \\sin{(i)} di and \\frac{\\int \\operatorname{V_{\\mathbf{B}}}{(i)} di}{\\int \\sin{(i)} di} = 1 and \\frac{\\int \\operatorname{V_{\\mathbf{B}}}{(i)} di}{\\delta - \\cos{(i)}} = 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Pow(Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('i', commutative=True)))), Integer(-1)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Integer(1))"]]}, {"prompt": "Given U{(I,y^{\\prime})} = I + y^{\\prime} and \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} = 2 y^{\\prime}, then obtain \\cos{(I + 2 y^{\\prime})} = \\cos{(y^{\\prime} + U{(I,y^{\\prime})})}", "derivation": "U{(I,y^{\\prime})} = I + y^{\\prime} and y^{\\prime} + U{(I,y^{\\prime})} = I + 2 y^{\\prime} and \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} = 2 y^{\\prime} and y^{\\prime} + U{(I,y^{\\prime})} = I + \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} and I + 2 y^{\\prime} = I + \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} and \\cos{(I + 2 y^{\\prime})} = \\cos{(I + \\operatorname{g_{\\varepsilon}}{(y^{\\prime})})} and \\cos{(I + 2 y^{\\prime})} = \\cos{(y^{\\prime} + U{(I,y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('U')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('U')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('I', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('I', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('I', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True))))"], [["cos", 5], "Equality(cos(Add(Symbol('I', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), cos(Add(Symbol('I', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(cos(Add(Symbol('I', commutative=True), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Function('U')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given u{(E,J)} = \\log{(E + J)}, then obtain \\frac{\\frac{d^{2}}{d Ed J} 1}{u{(E,J)}} = \\frac{\\frac{\\partial^{2}}{\\partial E\\partial J} u^{- J}{(E,J)} \\log{(E + J)}^{J}}{u{(E,J)}}", "derivation": "u{(E,J)} = \\log{(E + J)} and u^{J}{(E,J)} = \\log{(E + J)}^{J} and 1 = u^{- J}{(E,J)} \\log{(E + J)}^{J} and \\frac{d}{d J} 1 = \\frac{\\partial}{\\partial J} u^{- J}{(E,J)} \\log{(E + J)}^{J} and \\frac{d^{2}}{d Ed J} 1 = \\frac{\\partial^{2}}{\\partial E\\partial J} u^{- J}{(E,J)} \\log{(E + J)}^{J} and \\frac{\\frac{d^{2}}{d Ed J} 1}{u{(E,J)}} = \\frac{\\frac{\\partial^{2}}{\\partial E\\partial J} u^{- J}{(E,J)} \\log{(E + J)}^{J}}{u{(E,J)}}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["divide", 2, "Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["divide", 5, "Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True))"], "Equality(Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Derivative(Mul(Pow(Function('u')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(log(Add(Symbol('E', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(\\mathbf{H},\\Omega)} = \\int \\Omega^{\\mathbf{H}} d\\mathbf{H}, then obtain (\\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} M{(\\mathbf{H},\\Omega)}}{M{(\\mathbf{H},\\Omega)}})^{\\mathbf{H}} = (\\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} \\int \\Omega^{\\mathbf{H}} d\\mathbf{H}}{M{(\\mathbf{H},\\Omega)}})^{\\mathbf{H}}", "derivation": "M{(\\mathbf{H},\\Omega)} = \\int \\Omega^{\\mathbf{H}} d\\mathbf{H} and \\frac{\\partial}{\\partial \\Omega} M{(\\mathbf{H},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\int \\Omega^{\\mathbf{H}} d\\mathbf{H} and \\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} M{(\\mathbf{H},\\Omega)} = \\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} \\int \\Omega^{\\mathbf{H}} d\\mathbf{H} and \\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} M{(\\mathbf{H},\\Omega)}}{M{(\\mathbf{H},\\Omega)}} = \\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} \\int \\Omega^{\\mathbf{H}} d\\mathbf{H}}{M{(\\mathbf{H},\\Omega)}} and (\\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} M{(\\mathbf{H},\\Omega)}}{M{(\\mathbf{H},\\Omega)}})^{\\mathbf{H}} = (\\frac{\\mathbf{H} + \\frac{\\partial}{\\partial \\Omega} \\int \\Omega^{\\mathbf{H}} d\\mathbf{H}}{M{(\\mathbf{H},\\Omega)}})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["divide", 3, "Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given q{(f^{\\prime},\\hat{H})} = \\hat{H} f^{\\prime}, then obtain 1 - \\frac{q{(f^{\\prime},\\hat{H})}}{f^{\\prime}} = 1 - \\hat{H}", "derivation": "q{(f^{\\prime},\\hat{H})} = \\hat{H} f^{\\prime} and \\frac{q{(f^{\\prime},\\hat{H})}}{f^{\\prime}} = \\hat{H} and -1 + \\frac{q{(f^{\\prime},\\hat{H})}}{f^{\\prime}} = \\hat{H} - 1 and 1 - \\frac{q{(f^{\\prime},\\hat{H})}}{f^{\\prime}} = 1 - \\hat{H}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], [["times", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(A_{2},V)} = - A_{2} + V, then derive \\frac{\\partial}{\\partial V} \\tilde{g}^*{(A_{2},V)} = 1, then obtain \\frac{(\\frac{\\partial^{2}}{\\partial A_{2}\\partial V} (- A_{2} + V))^{V}}{- A_{2} + \\frac{\\partial}{\\partial V} (- A_{2} + V)} = \\frac{(\\frac{d}{d A_{2}} 1)^{V}}{- A_{2} + \\frac{\\partial}{\\partial V} (- A_{2} + V)}", "derivation": "\\tilde{g}^*{(A_{2},V)} = - A_{2} + V and \\frac{\\partial}{\\partial V} \\tilde{g}^*{(A_{2},V)} = \\frac{\\partial}{\\partial V} (- A_{2} + V) and \\frac{\\partial}{\\partial V} \\tilde{g}^*{(A_{2},V)} = 1 and \\frac{\\partial^{2}}{\\partial A_{2}\\partial V} \\tilde{g}^*{(A_{2},V)} = \\frac{d}{d A_{2}} 1 and (\\frac{\\partial^{2}}{\\partial A_{2}\\partial V} \\tilde{g}^*{(A_{2},V)})^{V} = (\\frac{d}{d A_{2}} 1)^{V} and (\\frac{\\partial^{2}}{\\partial A_{2}\\partial V} (- A_{2} + V))^{V} = (\\frac{d}{d A_{2}} 1)^{V} and \\frac{(\\frac{\\partial^{2}}{\\partial A_{2}\\partial V} (- A_{2} + V))^{V}}{- A_{2} + \\frac{\\partial}{\\partial V} (- A_{2} + V)} = \\frac{(\\frac{d}{d A_{2}} 1)^{V}}{- A_{2} + \\frac{\\partial}{\\partial V} (- A_{2} + V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('A_2', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["divide", 6, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(-1)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(-1)), Pow(Derivative(Integer(1), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\psi{(H)} = e^{H}, then derive (\\sigma_p + \\psi{(H)}) e^{\\frac{d}{d H} e^{H}} = (y^{\\prime} + e^{H}) e^{\\frac{d}{d H} e^{H}}, then obtain (\\sigma_p + \\psi{(H)}) e^{\\frac{d}{d H} \\psi{(H)}} = (y^{\\prime} + e^{H}) e^{\\frac{d}{d H} \\psi{(H)}}", "derivation": "\\psi{(H)} = e^{H} and \\frac{d}{d H} \\psi{(H)} = \\frac{d}{d H} e^{H} and e^{\\frac{d}{d H} \\psi{(H)}} = e^{\\frac{d}{d H} e^{H}} and \\int \\frac{d}{d H} \\psi{(H)} dH = \\int \\frac{d}{d H} e^{H} dH and e^{\\frac{d}{d H} e^{H}} \\int \\frac{d}{d H} \\psi{(H)} dH = e^{\\frac{d}{d H} e^{H}} \\int \\frac{d}{d H} e^{H} dH and (\\sigma_p + \\psi{(H)}) e^{\\frac{d}{d H} e^{H}} = (y^{\\prime} + e^{H}) e^{\\frac{d}{d H} e^{H}} and (\\sigma_p + \\psi{(H)}) e^{\\frac{d}{d H} \\psi{(H)}} = (y^{\\prime} + e^{H}) e^{\\frac{d}{d H} \\psi{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"], [["times", 4, "exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], "Equality(Mul(exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integral(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)))), Mul(exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integral(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\psi')(Symbol('H', commutative=True))), exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('H', commutative=True))), exp(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\psi')(Symbol('H', commutative=True))), exp(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('H', commutative=True))), exp(Derivative(Function('\\\\psi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\phi_{2}{(\\rho_f)} = \\log{(\\rho_f)} and \\Psi^{\\dagger}{(\\rho_f)} = \\log{(\\rho_f)}, then obtain \\rho_f \\Psi^{\\dagger}{(\\rho_f)} = \\rho_f \\log{(\\rho_f)}", "derivation": "\\phi_{2}{(\\rho_f)} = \\log{(\\rho_f)} and \\rho_f \\phi_{2}{(\\rho_f)} = \\rho_f \\log{(\\rho_f)} and \\Psi^{\\dagger}{(\\rho_f)} = \\log{(\\rho_f)} and \\phi_{2}{(\\rho_f)} = \\Psi^{\\dagger}{(\\rho_f)} and \\rho_f \\Psi^{\\dagger}{(\\rho_f)} = \\rho_f \\log{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Function('\\\\phi_2')(Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\phi_2')(Symbol('\\\\rho_f', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(v_{z})} = \\cos{(v_{z})}, then obtain 2 \\dot{\\mathbf{r}}^{2}{(v_{z})} = \\dot{\\mathbf{r}}^{2}{(v_{z})} + \\cos^{2}{(v_{z})}", "derivation": "\\dot{\\mathbf{r}}{(v_{z})} = \\cos{(v_{z})} and \\dot{\\mathbf{r}}{(v_{z})} \\cos{(v_{z})} = \\cos^{2}{(v_{z})} and \\dot{\\mathbf{r}}^{2}{(v_{z})} = \\dot{\\mathbf{r}}{(v_{z})} \\cos{(v_{z})} and \\dot{\\mathbf{r}}^{2}{(v_{z})} = \\cos^{2}{(v_{z})} and 2 \\dot{\\mathbf{r}}^{2}{(v_{z})} = \\dot{\\mathbf{r}}^{2}{(v_{z})} + \\cos^{2}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True)))"], [["times", 1, "cos(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True))), Pow(cos(Symbol('v_z', commutative=True)), Integer(2)))"], [["times", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), Integer(2)), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), Integer(2)), Pow(cos(Symbol('v_z', commutative=True)), Integer(2)))"], [["add", 4, "Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), Integer(2))), Add(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True)), Integer(2)), Pow(cos(Symbol('v_z', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{H}{(a^{\\dagger},U)} = e^{\\frac{a^{\\dagger}}{U}}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\mathbf{H}{(a^{\\dagger},U)} = \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U}, then obtain \\int \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{a^{\\dagger}}{U}}}{\\mathbf{A}} da^{\\dagger} = \\int \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U \\mathbf{A}} da^{\\dagger}", "derivation": "\\mathbf{H}{(a^{\\dagger},U)} = e^{\\frac{a^{\\dagger}}{U}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\mathbf{H}{(a^{\\dagger},U)} = \\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{a^{\\dagger}}{U}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\mathbf{H}{(a^{\\dagger},U)} = \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U} and \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} \\mathbf{H}{(a^{\\dagger},U)}}{\\mathbf{A}} = \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U \\mathbf{A}} and \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{a^{\\dagger}}{U}}}{\\mathbf{A}} = \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U \\mathbf{A}} and \\int \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{a^{\\dagger}}{U}}}{\\mathbf{A}} da^{\\dagger} = \\int \\frac{e^{\\frac{a^{\\dagger}}{U}}}{U \\mathbf{A}} da^{\\dagger}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["divide", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Derivative(exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Derivative(exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mu{(B,F_{x})} = \\int F_{x}^{B} dB, then obtain \\sin{(\\frac{\\mu{(B,F_{x})} \\log{(\\mu{(B,F_{x})})}}{F_{x}})} = \\sin{(\\frac{\\mu{(B,F_{x})} \\log{(\\int F_{x}^{B} dB)}}{F_{x}})}", "derivation": "\\mu{(B,F_{x})} = \\int F_{x}^{B} dB and \\log{(\\mu{(B,F_{x})})} = \\log{(\\int F_{x}^{B} dB)} and \\frac{\\mu{(B,F_{x})} \\log{(\\mu{(B,F_{x})})}}{F_{x}} = \\frac{\\mu{(B,F_{x})} \\log{(\\int F_{x}^{B} dB)}}{F_{x}} and \\sin{(\\frac{\\mu{(B,F_{x})} \\log{(\\mu{(B,F_{x})})}}{F_{x}})} = \\sin{(\\frac{\\mu{(B,F_{x})} \\log{(\\int F_{x}^{B} dB)}}{F_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)), Integral(Pow(Symbol('F_x', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True))), log(Integral(Pow(Symbol('F_x', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["times", 2, "Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)), log(Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)), log(Integral(Pow(Symbol('F_x', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)), log(Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True))))), sin(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('B', commutative=True), Symbol('F_x', commutative=True)), log(Integral(Pow(Symbol('F_x', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))))"]]}, {"prompt": "Given \\eta{(x^\\prime,\\hat{H},t_{2})} = \\hat{H} - t_{2} + x^\\prime and \\operatorname{F_{x}}{(\\hat{H})} = \\hat{H}, then obtain (- \\hat{H} + \\eta{(x^\\prime,\\hat{H},t_{2})})^{x^\\prime} = (- t_{2} + x^\\prime)^{x^\\prime}", "derivation": "\\eta{(x^\\prime,\\hat{H},t_{2})} = \\hat{H} - t_{2} + x^\\prime and \\operatorname{F_{x}}{(\\hat{H})} = \\hat{H} and - \\operatorname{F_{x}}{(\\hat{H})} + \\eta{(x^\\prime,\\hat{H},t_{2})} = \\hat{H} - t_{2} + x^\\prime - \\operatorname{F_{x}}{(\\hat{H})} and - \\hat{H} + \\eta{(x^\\prime,\\hat{H},t_{2})} = - t_{2} + x^\\prime and (- \\hat{H} + \\eta{(x^\\prime,\\hat{H},t_{2})})^{x^\\prime} = (- t_{2} + x^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], [["minus", 1, "Function('F_x')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{H}', commutative=True))), Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('F_x')(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('t_2', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{A_{x}}{\\Psi^{\\dagger}}, then obtain \\frac{A_{x}}{\\Psi^{\\dagger}} + 2 \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{2 A_{x}}{\\Psi^{\\dagger}} + \\lambda{(\\Psi^{\\dagger},A_{x})}", "derivation": "\\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{A_{x}}{\\Psi^{\\dagger}} and \\frac{A_{x}}{\\Psi^{\\dagger}} + \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{2 A_{x}}{\\Psi^{\\dagger}} and \\frac{2 A_{x}}{\\Psi^{\\dagger}} + \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{3 A_{x}}{\\Psi^{\\dagger}} and \\frac{A_{x}}{\\Psi^{\\dagger}} + 2 \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{3 A_{x}}{\\Psi^{\\dagger}} and \\frac{A_{x}}{\\Psi^{\\dagger}} + 2 \\lambda{(\\Psi^{\\dagger},A_{x})} = \\frac{2 A_{x}}{\\Psi^{\\dagger}} + \\lambda{(\\Psi^{\\dagger},A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(2), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(2), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(3), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)))), Mul(Integer(3), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)))), Add(Mul(Integer(2), Symbol('A_x', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\tilde{g}^*)} = e^{\\tilde{g}^*}, then obtain \\int \\frac{d}{d \\tilde{g}^*} 0 d\\tilde{g}^* = \\int \\frac{d}{d \\tilde{g}^*} (\\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*})^{3} d\\tilde{g}^*", "derivation": "\\Psi^{\\dagger}{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and 0 = - \\Psi^{\\dagger}{(\\tilde{g}^*)} + e^{\\tilde{g}^*} and 0 = \\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*} and 0 = (\\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*})^{2} and 0 = (\\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*})^{3} and \\frac{d}{d \\tilde{g}^*} 0 = \\frac{d}{d \\tilde{g}^*} (\\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*})^{3} and \\int \\frac{d}{d \\tilde{g}^*} 0 d\\tilde{g}^* = \\int \\frac{d}{d \\tilde{g}^*} (\\Psi^{\\dagger}{(\\tilde{g}^*)} - e^{\\tilde{g}^*})^{3} d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True))), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["times", 3, "Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Integer(0), Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(2)))"], [["times", 4, "Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Integer(0), Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(3)))"], [["differentiate", 5, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(3)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Derivative(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(3)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{P} + \\tilde{g}^*), then derive \\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} = -1, then obtain \\frac{\\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} - 2 \\mathbf{v}{(\\tilde{g}^*)}}{\\mathbf{v}{(\\tilde{g}^*)}} = \\frac{- 2 \\mathbf{v}{(\\tilde{g}^*)} - 1}{\\mathbf{v}{(\\tilde{g}^*)}}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{P} + \\tilde{g}^*) and \\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} = -1 and \\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} - \\mathbf{v}{(\\tilde{g}^*)} = - \\mathbf{v}{(\\tilde{g}^*)} - 1 and \\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} - 2 \\mathbf{v}{(\\tilde{g}^*)} = - 2 \\mathbf{v}{(\\tilde{g}^*)} - 1 and \\frac{\\operatorname{M_{E}}{(\\mathbf{P},\\tilde{g}^*)} - 2 \\mathbf{v}{(\\tilde{g}^*)}}{\\mathbf{v}{(\\tilde{g}^*)}} = \\frac{- 2 \\mathbf{v}{(\\tilde{g}^*)} - 1}{\\mathbf{v}{(\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M_E')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))"], [["minus", 2, "Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"], [["minus", 3, "Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"], [["divide", 4, "Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Add(Function('M_E')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Pow(Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Pow(Function('\\\\mathbf{v}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(c_{0},\\mathbf{M})} = \\mathbf{M} c_{0} and \\mathbf{J}_P{(\\varepsilon,\\sigma_x)} = \\varepsilon \\cos{(\\sigma_x)}, then obtain \\mathbf{J}_P{(\\varepsilon,\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\mathbf{M} c_{0} = \\varepsilon \\cos{(\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\mathbf{M} c_{0}", "derivation": "\\operatorname{E_{\\lambda}}{(c_{0},\\mathbf{M})} = \\mathbf{M} c_{0} and \\frac{\\partial}{\\partial c_{0}} \\operatorname{E_{\\lambda}}{(c_{0},\\mathbf{M})} = \\frac{\\partial}{\\partial c_{0}} \\mathbf{M} c_{0} and \\mathbf{J}_P{(\\varepsilon,\\sigma_x)} = \\varepsilon \\cos{(\\sigma_x)} and \\mathbf{J}_P{(\\varepsilon,\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\operatorname{E_{\\lambda}}{(c_{0},\\mathbf{M})} = \\varepsilon \\cos{(\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\operatorname{E_{\\lambda}}{(c_{0},\\mathbf{M})} and \\mathbf{J}_P{(\\varepsilon,\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\mathbf{M} c_{0} = \\varepsilon \\cos{(\\sigma_x)} + \\frac{\\partial}{\\partial c_{0}} \\mathbf{M} c_{0}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 3, "Derivative(Function('E_{\\\\lambda}')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('E_{\\\\lambda}')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{1})} = \\cos{(v_{1})} and \\dot{y}{(v_{1})} = \\cos{(v_{1})}, then obtain (\\frac{d}{d v_{1}} \\dot{y}{(v_{1})})^{v_{1}} = (\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}}", "derivation": "\\operatorname{n_{1}}{(v_{1})} = \\cos{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and (\\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})})^{v_{1}} = (\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}} and \\dot{y}{(v_{1})} = \\cos{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} = \\frac{d}{d v_{1}} \\dot{y}{(v_{1})} and (\\frac{d}{d v_{1}} \\dot{y}{(v_{1})})^{v_{1}} = (\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(g)} = e^{g}, then derive \\frac{d}{d g} \\hat{x}{(g)} = e^{g}, then obtain (y{(r_{0})} + \\frac{d}{d g} e^{g})^{r_{0}} + \\cos{(e^{r_{0}})} = (y{(r_{0})} + e^{g})^{r_{0}} + \\cos{(e^{r_{0}})}", "derivation": "\\hat{x}{(g)} = e^{g} and \\frac{d}{d g} \\hat{x}{(g)} = \\frac{d}{d g} e^{g} and \\frac{d}{d g} \\hat{x}{(g)} = e^{g} and \\frac{d}{d g} e^{g} = e^{g} and y{(r_{0})} + \\frac{d}{d g} e^{g} = y{(r_{0})} + e^{g} and (y{(r_{0})} + \\frac{d}{d g} e^{g})^{r_{0}} = (y{(r_{0})} + e^{g})^{r_{0}} and (y{(r_{0})} + \\frac{d}{d g} e^{g})^{r_{0}} + \\cos{(e^{r_{0}})} = (y{(r_{0})} + e^{g})^{r_{0}} + \\cos{(e^{r_{0}})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), exp(Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), exp(Symbol('g', commutative=True)))"], [["add", 4, "Function('y')(Symbol('r_0', commutative=True))"], "Equality(Add(Function('y')(Symbol('r_0', commutative=True)), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Function('y')(Symbol('r_0', commutative=True)), exp(Symbol('g', commutative=True))))"], [["power", 5, "Symbol('r_0', commutative=True)"], "Equality(Pow(Add(Function('y')(Symbol('r_0', commutative=True)), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('r_0', commutative=True)), Pow(Add(Function('y')(Symbol('r_0', commutative=True)), exp(Symbol('g', commutative=True))), Symbol('r_0', commutative=True)))"], [["add", 6, "cos(exp(Symbol('r_0', commutative=True)))"], "Equality(Add(Pow(Add(Function('y')(Symbol('r_0', commutative=True)), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('r_0', commutative=True)), cos(exp(Symbol('r_0', commutative=True)))), Add(Pow(Add(Function('y')(Symbol('r_0', commutative=True)), exp(Symbol('g', commutative=True))), Symbol('r_0', commutative=True)), cos(exp(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given u{(\\mu_0)} = e^{\\sin{(\\mu_0)}}, then derive \\mu_0 \\frac{d}{d \\mu_0} u{(\\mu_0)} + u{(\\mu_0)} = \\mu_0 e^{\\sin{(\\mu_0)}} \\cos{(\\mu_0)} + e^{\\sin{(\\mu_0)}}, then obtain \\mu_0 \\frac{d}{d \\mu_0} e^{\\sin{(\\mu_0)}} + e^{\\sin{(\\mu_0)}} = \\mu_0 e^{\\sin{(\\mu_0)}} \\cos{(\\mu_0)} + e^{\\sin{(\\mu_0)}}", "derivation": "u{(\\mu_0)} = e^{\\sin{(\\mu_0)}} and \\mu_0 u{(\\mu_0)} = \\mu_0 e^{\\sin{(\\mu_0)}} and \\frac{d}{d \\mu_0} \\mu_0 u{(\\mu_0)} = \\frac{d}{d \\mu_0} \\mu_0 e^{\\sin{(\\mu_0)}} and \\mu_0 \\frac{d}{d \\mu_0} u{(\\mu_0)} + u{(\\mu_0)} = \\mu_0 e^{\\sin{(\\mu_0)}} \\cos{(\\mu_0)} + e^{\\sin{(\\mu_0)}} and \\mu_0 \\frac{d}{d \\mu_0} e^{\\sin{(\\mu_0)}} + e^{\\sin{(\\mu_0)}} = \\mu_0 e^{\\sin{(\\mu_0)}} \\cos{(\\mu_0)} + e^{\\sin{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(sin(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Function('u')(Symbol('\\\\mu_0', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Function('u')(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), exp(sin(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Derivative(Function('u')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Function('u')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), exp(sin(Symbol('\\\\mu_0', commutative=True))), cos(Symbol('\\\\mu_0', commutative=True))), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Derivative(exp(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), exp(sin(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Symbol('\\\\mu_0', commutative=True), exp(sin(Symbol('\\\\mu_0', commutative=True))), cos(Symbol('\\\\mu_0', commutative=True))), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(\\ddot{x},r_{0})} = \\ddot{x} + r_{0}, then derive \\int \\rho_{b}{(\\ddot{x},r_{0})} d\\ddot{x} = \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} r_{0} + \\mathbf{J}_M, then obtain \\int (\\ddot{x} + r_{0}) d\\ddot{x} = \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} r_{0} + \\mathbf{J}_M", "derivation": "\\rho_{b}{(\\ddot{x},r_{0})} = \\ddot{x} + r_{0} and \\int \\rho_{b}{(\\ddot{x},r_{0})} d\\ddot{x} = \\int (\\ddot{x} + r_{0}) d\\ddot{x} and \\int \\rho_{b}{(\\ddot{x},r_{0})} d\\ddot{x} = \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} r_{0} + \\mathbf{J}_M and \\int (\\ddot{x} + r_{0}) d\\ddot{x} = \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} r_{0} + \\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given r{(\\sigma_p)} = \\sin{(e^{\\sigma_p})}, then derive \\frac{d}{d \\sigma_p} r{(\\sigma_p)} = e^{\\sigma_p} \\cos{(e^{\\sigma_p})}, then obtain 0 = e^{\\sigma_p} \\cos{(e^{\\sigma_p})} - \\frac{d}{d \\sigma_p} \\sin{(e^{\\sigma_p})}", "derivation": "r{(\\sigma_p)} = \\sin{(e^{\\sigma_p})} and \\frac{d}{d \\sigma_p} r{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\sin{(e^{\\sigma_p})} and \\frac{d}{d \\sigma_p} r{(\\sigma_p)} = e^{\\sigma_p} \\cos{(e^{\\sigma_p})} and \\frac{d}{d \\sigma_p} \\sin{(e^{\\sigma_p})} = e^{\\sigma_p} \\cos{(e^{\\sigma_p})} and 0 = e^{\\sigma_p} \\cos{(e^{\\sigma_p})} - \\frac{d}{d \\sigma_p} \\sin{(e^{\\sigma_p})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\sigma_p', commutative=True)), sin(exp(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["minus", 4, "Derivative(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(exp(Symbol('\\\\sigma_p', commutative=True)), cos(exp(Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Derivative(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given t{(\\pi,T)} = \\frac{\\partial}{\\partial T} \\frac{T}{\\pi}, then derive t{(\\pi,T)} = \\frac{1}{\\pi}, then obtain - \\frac{d}{d T} \\frac{1}{\\pi} = - \\frac{\\partial}{\\partial T} t{(\\pi,T)}", "derivation": "t{(\\pi,T)} = \\frac{\\partial}{\\partial T} \\frac{T}{\\pi} and t{(\\pi,T)} = \\frac{1}{\\pi} and \\frac{\\partial}{\\partial T} t{(\\pi,T)} = \\frac{d}{d T} \\frac{1}{\\pi} and - \\frac{d}{d T} \\frac{1}{\\pi} + \\frac{\\partial}{\\partial T} t{(\\pi,T)} = 0 and - \\frac{d}{d T} \\frac{1}{\\pi} = - \\frac{\\partial}{\\partial T} t{(\\pi,T)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Derivative(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('T', commutative=True), Integer(1)))), Derivative(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(0))"], [["minus", 4, "Derivative(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('t')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{v}{(a)} = \\cos{(a)}, then derive - \\frac{d}{d a} \\mathbf{v}{(a)} = \\sin{(a)}, then obtain - \\frac{d}{d a} \\cos{(a)} - 1 = \\sin{(a)} - 1", "derivation": "\\mathbf{v}{(a)} = \\cos{(a)} and \\frac{d}{d a} \\mathbf{v}{(a)} = \\frac{d}{d a} \\cos{(a)} and - \\frac{d}{d a} \\mathbf{v}{(a)} = - \\frac{d}{d a} \\cos{(a)} and - \\frac{d}{d a} \\mathbf{v}{(a)} = \\sin{(a)} and - \\frac{d}{d a} \\cos{(a)} = \\sin{(a)} and - \\frac{d}{d a} \\cos{(a)} - 1 = \\sin{(a)} - 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{v}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{v}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), sin(Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), sin(Symbol('a', commutative=True)))"], [["minus", 5, 1], "Equality(Add(Mul(Integer(-1), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(-1)), Add(sin(Symbol('a', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = e^{\\sin{(V_{\\mathbf{E}})}}, then derive - \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = - e^{\\sin{(V_{\\mathbf{E}})}} \\cos{(V_{\\mathbf{E}})}, then obtain - \\frac{d}{d V_{\\mathbf{E}}} e^{\\sin{(V_{\\mathbf{E}})}} = - e^{\\sin{(V_{\\mathbf{E}})}} \\cos{(V_{\\mathbf{E}})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = e^{\\sin{(V_{\\mathbf{E}})}} and - \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = - e^{\\sin{(V_{\\mathbf{E}})}} and \\frac{d}{d V_{\\mathbf{E}}} - \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} - e^{\\sin{(V_{\\mathbf{E}})}} and - \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{E}})} = - e^{\\sin{(V_{\\mathbf{E}})}} \\cos{(V_{\\mathbf{E}})} and - \\frac{d}{d V_{\\mathbf{E}}} e^{\\sin{(V_{\\mathbf{E}})}} = - e^{\\sin{(V_{\\mathbf{E}})}} \\cos{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(Integer(-1), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(Integer(-1), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given L{(\\theta)} = \\int \\log{(\\theta)} d\\theta, then derive L{(\\theta)} = \\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}}, then obtain (\\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}})^{\\theta} + L^{\\theta}{(\\theta)} = 2 L^{\\theta}{(\\theta)}", "derivation": "L{(\\theta)} = \\int \\log{(\\theta)} d\\theta and L{(\\theta)} = \\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}} and \\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}} = \\int \\log{(\\theta)} d\\theta and (\\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}})^{\\theta} = (\\int \\log{(\\theta)} d\\theta)^{\\theta} and (\\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}})^{\\theta} + (\\int \\log{(\\theta)} d\\theta)^{\\theta} = 2 (\\int \\log{(\\theta)} d\\theta)^{\\theta} and (\\theta \\log{(\\theta)} - \\theta + f_{\\mathbf{p}})^{\\theta} + L^{\\theta}{(\\theta)} = 2 L^{\\theta}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta', commutative=True)), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('L')(Symbol('\\\\theta', commutative=True)), Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["add", 4, "Pow(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Pow(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Pow(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Pow(Add(Mul(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Function('L')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Pow(Function('L')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})}, then obtain \\frac{\\operatorname{C_{2}}^{2}{(\\tilde{g})}}{\\sin^{2}{(\\tilde{g})}} = 1", "derivation": "\\operatorname{C_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\frac{\\operatorname{C_{2}}{(\\tilde{g})}}{\\sin{(\\tilde{g})}} = 1 and \\frac{\\operatorname{C_{2}}^{2}{(\\tilde{g})}}{\\sin{(\\tilde{g})}} = \\operatorname{C_{2}}{(\\tilde{g})} and \\frac{\\operatorname{C_{2}}^{2}{(\\tilde{g})}}{\\sin^{2}{(\\tilde{g})}} = 1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Function('C_2')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Pow(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(-2))), Integer(1))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(u,c,L)} = \\frac{L u}{c}, then obtain \\int (- \\hat{H}_{\\lambda}{(u,c,L)} + \\hat{H}_{\\lambda}^{L}{(u,c,L)}) dL = \\int ((\\frac{L u}{c})^{L} - \\hat{H}_{\\lambda}{(u,c,L)}) dL", "derivation": "\\hat{H}_{\\lambda}{(u,c,L)} = \\frac{L u}{c} and \\hat{H}_{\\lambda}^{L}{(u,c,L)} = (\\frac{L u}{c})^{L} and - \\hat{H}_{\\lambda}{(u,c,L)} + \\hat{H}_{\\lambda}^{L}{(u,c,L)} = (\\frac{L u}{c})^{L} - \\hat{H}_{\\lambda}{(u,c,L)} and \\int (- \\hat{H}_{\\lambda}{(u,c,L)} + \\hat{H}_{\\lambda}^{L}{(u,c,L)}) dL = \\int ((\\frac{L u}{c})^{L} - \\hat{H}_{\\lambda}{(u,c,L)}) dL", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('L', commutative=True)))"], [["minus", 2, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Add(Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('L', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('L', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True), Symbol('c', commutative=True), Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + u, then obtain - \\log{(\\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} + u))} + \\int \\frac{\\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + u} du = - \\log{(\\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} + u))} + \\int 1 du", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + u and \\hat{\\mathbf{r}} \\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} + u) and \\frac{\\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + u} = 1 and \\int \\frac{\\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + u} du = \\int 1 du and - \\log{(\\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} + u))} + \\int \\frac{\\operatorname{f_{\\mathbf{v}}}{(u,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + u} du = - \\log{(\\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} + u))} + \\int 1 du", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('u', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('u', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('u', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(1))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('u', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"], [["minus", 4, "log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))))"], "Equality(Add(Mul(Integer(-1), log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))))), Integral(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('u', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('u', commutative=True)))), Add(Mul(Integer(-1), log(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))))), Integral(Integer(1), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given c{(f_{\\mathbf{p}})} = \\cos{(\\log{(f_{\\mathbf{p}})})} and \\varepsilon_{0}{(B)} = e^{B}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{\\varepsilon_{0}{(B)}}{\\cos{(\\log{(f_{\\mathbf{p}})})}} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{e^{B}}{\\cos{(\\log{(f_{\\mathbf{p}})})}}", "derivation": "c{(f_{\\mathbf{p}})} = \\cos{(\\log{(f_{\\mathbf{p}})})} and \\varepsilon_{0}{(B)} = e^{B} and \\frac{\\varepsilon_{0}{(B)}}{c{(f_{\\mathbf{p}})}} = \\frac{e^{B}}{c{(f_{\\mathbf{p}})}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{\\varepsilon_{0}{(B)}}{c{(f_{\\mathbf{p}})}} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{e^{B}}{c{(f_{\\mathbf{p}})}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{\\varepsilon_{0}{(B)}}{\\cos{(\\log{(f_{\\mathbf{p}})})}} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{e^{B}}{\\cos{(\\log{(f_{\\mathbf{p}})})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["divide", 2, "Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('B', commutative=True)), Pow(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Pow(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\varepsilon_0')(Symbol('B', commutative=True)), Pow(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Function('\\\\varepsilon_0')(Symbol('B', commutative=True)), Pow(cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('B', commutative=True)), Pow(cos(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(\\varepsilon,\\hat{p}_0)} = - \\varepsilon + \\sin{(\\hat{p}_0)} and \\theta_{2}{(F_{g},\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} F_{g} \\varepsilon, then obtain \\theta_{2}{(F_{g},\\varepsilon)} + \\frac{\\varepsilon + s{(\\varepsilon,\\hat{p}_0)}}{\\varepsilon} = \\frac{\\partial}{\\partial \\varepsilon} F_{g} \\varepsilon + \\frac{\\varepsilon + s{(\\varepsilon,\\hat{p}_0)}}{\\varepsilon}", "derivation": "s{(\\varepsilon,\\hat{p}_0)} = - \\varepsilon + \\sin{(\\hat{p}_0)} and \\varepsilon + s{(\\varepsilon,\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\theta_{2}{(F_{g},\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} F_{g} \\varepsilon and \\theta_{2}{(F_{g},\\varepsilon)} + \\frac{\\sin{(\\hat{p}_0)}}{\\varepsilon} = \\frac{\\partial}{\\partial \\varepsilon} F_{g} \\varepsilon + \\frac{\\sin{(\\hat{p}_0)}}{\\varepsilon} and \\theta_{2}{(F_{g},\\varepsilon)} + \\frac{\\varepsilon + s{(\\varepsilon,\\hat{p}_0)}}{\\varepsilon} = \\frac{\\partial}{\\partial \\varepsilon} F_{g} \\varepsilon + \\frac{\\varepsilon + s{(\\varepsilon,\\hat{p}_0)}}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Function('s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))), Add(Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Function('s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(\\delta)} = \\sin{(\\sin{(\\delta)})}, then obtain \\nabla^{2}{(\\delta)} \\sin{(\\sin{(\\delta)})} = \\sin^{3}{(\\sin{(\\delta)})}", "derivation": "\\nabla{(\\delta)} = \\sin{(\\sin{(\\delta)})} and \\nabla{(\\delta)} \\sin{(\\sin{(\\delta)})} = \\sin^{2}{(\\sin{(\\delta)})} and \\nabla{(\\delta)} \\sin^{2}{(\\sin{(\\delta)})} = \\sin^{3}{(\\sin{(\\delta)})} and \\nabla^{2}{(\\delta)} \\sin{(\\sin{(\\delta)})} = \\sin^{3}{(\\sin{(\\delta)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), sin(sin(Symbol('\\\\delta', commutative=True))))"], [["times", 1, "sin(sin(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), sin(sin(Symbol('\\\\delta', commutative=True)))), Pow(sin(sin(Symbol('\\\\delta', commutative=True))), Integer(2)))"], [["times", 1, "Pow(sin(sin(Symbol('\\\\delta', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Pow(sin(sin(Symbol('\\\\delta', commutative=True))), Integer(2))), Pow(sin(sin(Symbol('\\\\delta', commutative=True))), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Integer(2)), sin(sin(Symbol('\\\\delta', commutative=True)))), Pow(sin(sin(Symbol('\\\\delta', commutative=True))), Integer(3)))"]]}, {"prompt": "Given T{(C,t_{2})} = \\log{(C + t_{2})}, then obtain \\frac{\\partial}{\\partial C} \\iint \\frac{\\partial}{\\partial C} (t_{2} + T{(C,t_{2})}) dC dC = \\frac{\\partial}{\\partial C} \\iint \\frac{\\partial}{\\partial C} (t_{2} + \\log{(C + t_{2})}) dC dC", "derivation": "T{(C,t_{2})} = \\log{(C + t_{2})} and t_{2} + T{(C,t_{2})} = t_{2} + \\log{(C + t_{2})} and \\frac{\\partial}{\\partial C} (t_{2} + T{(C,t_{2})}) = \\frac{\\partial}{\\partial C} (t_{2} + \\log{(C + t_{2})}) and \\int \\frac{\\partial}{\\partial C} (t_{2} + T{(C,t_{2})}) dC = \\int \\frac{\\partial}{\\partial C} (t_{2} + \\log{(C + t_{2})}) dC and \\iint \\frac{\\partial}{\\partial C} (t_{2} + T{(C,t_{2})}) dC dC = \\iint \\frac{\\partial}{\\partial C} (t_{2} + \\log{(C + t_{2})}) dC dC and \\frac{\\partial}{\\partial C} \\iint \\frac{\\partial}{\\partial C} (t_{2} + T{(C,t_{2})}) dC dC = \\frac{\\partial}{\\partial C} \\iint \\frac{\\partial}{\\partial C} (t_{2} + \\log{(C + t_{2})}) dC dC", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True))))"], [["add", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('t_2', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Symbol('t_2', commutative=True), Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('t_2', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('t_2', commutative=True), Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Add(Symbol('t_2', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('t_2', commutative=True), Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Add(Symbol('t_2', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 5, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Derivative(Add(Symbol('t_2', commutative=True), Function('T')(Symbol('C', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Symbol('t_2', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}}, then derive \\delta{(F_{x})} e^{F_{x}} = e^{2 F_{x}}, then obtain \\hat{H}{(f_{E},\\mu_0)} + e^{F_{x}} \\frac{d}{d F_{x}} e^{F_{x}} = \\hat{H}{(f_{E},\\mu_0)} + e^{2 F_{x}}", "derivation": "\\delta{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}} and \\delta{(F_{x})} e^{F_{x}} = e^{F_{x}} \\frac{d}{d F_{x}} e^{F_{x}} and \\delta{(F_{x})} e^{F_{x}} = e^{2 F_{x}} and e^{F_{x}} \\frac{d}{d F_{x}} e^{F_{x}} = e^{2 F_{x}} and \\hat{H}{(f_{E},\\mu_0)} + e^{F_{x}} \\frac{d}{d F_{x}} e^{F_{x}} = \\hat{H}{(f_{E},\\mu_0)} + e^{2 F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 1, "exp(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True))), Mul(exp(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('\\\\delta')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True))), exp(Mul(Integer(2), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(exp(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('F_x', commutative=True))))"], [["add", 4, "Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(exp(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), exp(Mul(Integer(2), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(x^\\prime,a)} = - a + x^\\prime and C{(x^\\prime,a)} = 2 \\operatorname{C_{1}}{(x^\\prime,a)}, then obtain - C{(x^\\prime,a)} + \\operatorname{C_{1}}^{a}{(x^\\prime,a)} = (- a + x^\\prime)^{a} - C{(x^\\prime,a)}", "derivation": "\\operatorname{C_{1}}{(x^\\prime,a)} = - a + x^\\prime and C{(x^\\prime,a)} = 2 \\operatorname{C_{1}}{(x^\\prime,a)} and \\operatorname{C_{1}}^{a}{(x^\\prime,a)} = (- a + x^\\prime)^{a} and - 2 \\operatorname{C_{1}}{(x^\\prime,a)} + \\operatorname{C_{1}}^{a}{(x^\\prime,a)} = (- a + x^\\prime)^{a} - 2 \\operatorname{C_{1}}{(x^\\prime,a)} and - C{(x^\\prime,a)} + \\operatorname{C_{1}}^{a}{(x^\\prime,a)} = (- a + x^\\prime)^{a} - C{(x^\\prime,a)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Mul(Integer(2), Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('a', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True))), Pow(Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('a', commutative=True)), Mul(Integer(-1), Integer(2), Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('C')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True))), Pow(Function('C_1')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('a', commutative=True)), Mul(Integer(-1), Function('C')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\Omega,l)} = \\Omega - l, then derive \\frac{\\frac{\\partial}{\\partial \\Omega} \\omega{(\\Omega,l)}}{\\Omega} = \\frac{1}{\\Omega}, then obtain (\\frac{\\frac{\\partial}{\\partial \\Omega} (\\Omega - l)}{\\Omega})^{\\Omega} = (\\frac{1}{\\Omega})^{\\Omega}", "derivation": "\\omega{(\\Omega,l)} = \\Omega - l and \\frac{\\partial}{\\partial \\Omega} \\omega{(\\Omega,l)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega - l) and \\frac{\\frac{\\partial}{\\partial \\Omega} \\omega{(\\Omega,l)}}{\\Omega} = \\frac{\\frac{\\partial}{\\partial \\Omega} (\\Omega - l)}{\\Omega} and \\frac{\\frac{\\partial}{\\partial \\Omega} \\omega{(\\Omega,l)}}{\\Omega} = \\frac{1}{\\Omega} and \\frac{\\frac{\\partial}{\\partial \\Omega} (\\Omega - l)}{\\Omega} = \\frac{1}{\\Omega} and (\\frac{\\frac{\\partial}{\\partial \\Omega} (\\Omega - l)}{\\Omega})^{\\Omega} = (\\frac{1}{\\Omega})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Function('\\\\omega')(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Function('\\\\omega')(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(v_{x})} = \\sin{(\\cos{(v_{x})})} and L{(v_{x})} = \\cos{(v_{x})}, then obtain \\frac{d}{d v_{x}} 2 \\sin{(\\cos{(v_{x})})} = \\frac{d}{d v_{x}} (\\mathbf{A}{(v_{x})} + \\sin{(\\cos{(v_{x})})})", "derivation": "\\mathbf{A}{(v_{x})} = \\sin{(\\cos{(v_{x})})} and L{(v_{x})} = \\cos{(v_{x})} and \\mathbf{A}{(v_{x})} = \\sin{(L{(v_{x})})} and 2 \\mathbf{A}{(v_{x})} = \\mathbf{A}{(v_{x})} + \\sin{(L{(v_{x})})} and 2 \\sin{(\\cos{(v_{x})})} = \\sin{(L{(v_{x})})} + \\sin{(\\cos{(v_{x})})} and 2 \\sin{(\\cos{(v_{x})})} = \\mathbf{A}{(v_{x})} + \\sin{(\\cos{(v_{x})})} and \\frac{d}{d v_{x}} 2 \\sin{(\\cos{(v_{x})})} = \\frac{d}{d v_{x}} (\\mathbf{A}{(v_{x})} + \\sin{(\\cos{(v_{x})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), sin(cos(Symbol('v_x', commutative=True))))"], ["get_premise", "Equality(Function('L')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), sin(Function('L')(Symbol('v_x', commutative=True))))"], [["add", 3, "Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Add(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), sin(Function('L')(Symbol('v_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), sin(cos(Symbol('v_x', commutative=True)))), Add(sin(Function('L')(Symbol('v_x', commutative=True))), sin(cos(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), sin(cos(Symbol('v_x', commutative=True)))), Add(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), sin(cos(Symbol('v_x', commutative=True)))))"], [["differentiate", 6, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Mul(Integer(2), sin(cos(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), sin(cos(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(E_{n})} = \\sin{(\\cos{(E_{n})})}, then obtain (3 \\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})}) \\rho_{f}{(E_{n})} = (\\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})})^{2}", "derivation": "\\rho_{f}{(E_{n})} = \\sin{(\\cos{(E_{n})})} and 2 \\rho_{f}{(E_{n})} = \\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})} and 4 \\rho_{f}{(E_{n})} = 3 \\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})} and 4 \\rho_{f}^{2}{(E_{n})} = (\\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})})^{2} and (3 \\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})}) \\rho_{f}{(E_{n})} = (\\rho_{f}{(E_{n})} + \\sin{(\\cos{(E_{n})})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('E_n', commutative=True)), sin(cos(Symbol('E_n', commutative=True))))"], [["add", 1, "Function('\\\\rho_f')(Symbol('E_n', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('E_n', commutative=True))), Add(Function('\\\\rho_f')(Symbol('E_n', commutative=True)), sin(cos(Symbol('E_n', commutative=True)))))"], [["add", 2, "Mul(Integer(2), Function('\\\\rho_f')(Symbol('E_n', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\rho_f')(Symbol('E_n', commutative=True))), Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('E_n', commutative=True))), sin(cos(Symbol('E_n', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\rho_f')(Symbol('E_n', commutative=True)), Integer(2))), Pow(Add(Function('\\\\rho_f')(Symbol('E_n', commutative=True)), sin(cos(Symbol('E_n', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('E_n', commutative=True))), sin(cos(Symbol('E_n', commutative=True)))), Function('\\\\rho_f')(Symbol('E_n', commutative=True))), Pow(Add(Function('\\\\rho_f')(Symbol('E_n', commutative=True)), sin(cos(Symbol('E_n', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\chi{(m,E)} = E + m, then obtain E (\\int E^{2} \\chi{(m,E)} dm + \\int E m \\chi{(m,E)} dm + \\int \\chi{(m,E)} dm) + c_{0} = E (\\int E dm + \\int m dm + \\int E^{2} \\chi{(m,E)} dm + \\int E m \\chi{(m,E)} dm) + F_{H}", "derivation": "\\chi{(m,E)} = E + m and E \\chi{(m,E)} = E (E + m) and E^{2} \\chi^{2}{(m,E)} = E^{2} (E + m) \\chi{(m,E)} and E^{2} \\chi^{2}{(m,E)} + E \\chi{(m,E)} = E^{2} \\chi^{2}{(m,E)} + E (E + m) and E^{2} (E + m) \\chi{(m,E)} + E \\chi{(m,E)} = E^{2} (E + m) \\chi{(m,E)} + E (E + m) and \\int (E^{2} (E + m) \\chi{(m,E)} + E \\chi{(m,E)}) dm = \\int (E^{2} (E + m) \\chi{(m,E)} + E (E + m)) dm and E (\\int E^{2} \\chi{(m,E)} dm + \\int E m \\chi{(m,E)} dm + \\int \\chi{(m,E)} dm) + c_{0} = E (\\int E dm + \\int m dm + \\int E^{2} \\chi{(m,E)} dm + \\int E m \\chi{(m,E)} dm) + F_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)))"], [["times", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Symbol('m', commutative=True))))"], [["times", 2, "Mul(Symbol('E', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Integer(2))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Integer(2))), Mul(Symbol('E', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)))), Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Integer(2))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)))), Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Symbol('E', commutative=True), Add(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('m', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('m', commutative=True))))), Symbol('c_0', commutative=True)), Add(Mul(Symbol('E', commutative=True), Add(Integral(Symbol('E', commutative=True), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True))), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('m', commutative=True), Function('\\\\chi')(Symbol('m', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('m', commutative=True))))), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(f^{\\prime})} = f^{\\prime}, then derive \\int \\mathbf{J}_M{(f^{\\prime})} df^{\\prime} = \\frac{(f^{\\prime})^{2}}{2} + t, then derive \\sigma_p + \\frac{(f^{\\prime})^{2}}{2} = \\frac{(f^{\\prime})^{2}}{2} + t, then obtain (\\sigma_p + \\frac{(f^{\\prime})^{2}}{2}) (\\frac{(f^{\\prime})^{2}}{2} + t) = (\\frac{(f^{\\prime})^{2}}{2} + t)^{2}", "derivation": "\\mathbf{J}_M{(f^{\\prime})} = f^{\\prime} and \\int \\mathbf{J}_M{(f^{\\prime})} df^{\\prime} = \\int f^{\\prime} df^{\\prime} and \\int \\mathbf{J}_M{(f^{\\prime})} df^{\\prime} = \\frac{(f^{\\prime})^{2}}{2} + t and \\int f^{\\prime} df^{\\prime} = \\frac{(f^{\\prime})^{2}}{2} + t and \\sigma_p + \\frac{(f^{\\prime})^{2}}{2} = \\frac{(f^{\\prime})^{2}}{2} + t and (\\sigma_p + \\frac{(f^{\\prime})^{2}}{2}) \\int f^{\\prime} df^{\\prime} = (\\frac{(f^{\\prime})^{2}}{2} + t) \\int f^{\\prime} df^{\\prime} and (\\sigma_p + \\frac{(f^{\\prime})^{2}}{2}) (\\frac{(f^{\\prime})^{2}}{2} + t) = (\\frac{(f^{\\prime})^{2}}{2} + t)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))), Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True)))"], [["times", 5, "Integral(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))), Integral(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True)), Integral(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))), Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True))), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Symbol('t', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{v}{(\\phi_1,t_{2})} = \\phi_1 + \\cos{(t_{2})}, then obtain 3 \\phi_1 - 2 \\mathbf{v}{(\\phi_1,t_{2})} + 3 \\cos{(t_{2})} = \\phi_1 + \\cos{(t_{2})}", "derivation": "\\mathbf{v}{(\\phi_1,t_{2})} = \\phi_1 + \\cos{(t_{2})} and \\mathbf{v}{(\\phi_1,t_{2})} - \\cos{(t_{2})} = \\phi_1 and \\mathbf{v}{(\\phi_1,t_{2})} + \\cos{(t_{2})} = \\phi_1 + 2 \\cos{(t_{2})} and \\cos{(t_{2})} = \\phi_1 - \\mathbf{v}{(\\phi_1,t_{2})} + 2 \\cos{(t_{2})} and - \\phi_1 + 2 \\mathbf{v}{(\\phi_1,t_{2})} - 2 \\cos{(t_{2})} = \\phi_1 and \\mathbf{v}{(\\phi_1,t_{2})} = 2 \\phi_1 - \\mathbf{v}{(\\phi_1,t_{2})} + 2 \\cos{(t_{2})} and - \\phi_1 + 2 \\mathbf{v}{(\\phi_1,t_{2})} - \\cos{(t_{2})} = \\phi_1 + \\cos{(t_{2})} and 3 \\phi_1 - 2 \\mathbf{v}{(\\phi_1,t_{2})} + 3 \\cos{(t_{2})} = \\phi_1 + \\cos{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('t_2', commutative=True))))"], [["minus", 1, "cos(Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Symbol('\\\\phi_1', commutative=True))"], [["minus", 1, "Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(2), cos(Symbol('t_2', commutative=True)))))"], [["minus", 3, "Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))"], "Equality(cos(Symbol('t_2', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(2), cos(Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('t_2', commutative=True)))), Symbol('\\\\phi_1', commutative=True))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(2), cos(Symbol('t_2', commutative=True)))))"], [["add", 5, "cos(Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Mul(Integer(3), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(3), cos(Symbol('t_2', commutative=True)))), Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given s{(\\mu_0,T)} = T \\mu_0, then obtain \\frac{\\log{(\\mu_0 + s{(\\mu_0,T)})} - \\int s{(\\mu_0,T)} dT}{\\log{(T \\mu_0 + \\mu_0)}} = \\frac{\\log{(T \\mu_0 + \\mu_0)} - \\int s{(\\mu_0,T)} dT}{\\log{(T \\mu_0 + \\mu_0)}}", "derivation": "s{(\\mu_0,T)} = T \\mu_0 and \\mu_0 + s{(\\mu_0,T)} = T \\mu_0 + \\mu_0 and \\int s{(\\mu_0,T)} dT = \\int T \\mu_0 dT and \\log{(\\mu_0 + s{(\\mu_0,T)})} = \\log{(T \\mu_0 + \\mu_0)} and \\log{(\\mu_0 + s{(\\mu_0,T)})} - \\int T \\mu_0 dT = \\log{(T \\mu_0 + \\mu_0)} - \\int T \\mu_0 dT and \\frac{\\log{(\\mu_0 + s{(\\mu_0,T)})} - \\int T \\mu_0 dT}{\\log{(T \\mu_0 + \\mu_0)}} = \\frac{\\log{(T \\mu_0 + \\mu_0)} - \\int T \\mu_0 dT}{\\log{(T \\mu_0 + \\mu_0)}} and \\frac{\\log{(\\mu_0 + s{(\\mu_0,T)})} - \\int s{(\\mu_0,T)} dT}{\\log{(T \\mu_0 + \\mu_0)}} = \\frac{\\log{(T \\mu_0 + \\mu_0)} - \\int s{(\\mu_0,T)} dT}{\\log{(T \\mu_0 + \\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True))), Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["log", 2], "Equality(log(Add(Symbol('\\\\mu_0', commutative=True), Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)))), log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["minus", 4, "Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(log(Add(Symbol('\\\\mu_0', commutative=True), Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True))))))"], [["divide", 5, "log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Add(log(Add(Symbol('\\\\mu_0', commutative=True), Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True))))), Pow(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Add(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('T', commutative=True))))), Pow(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(log(Add(Symbol('\\\\mu_0', commutative=True), Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Pow(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Add(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integral(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Pow(log(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} = \\varphi (n_{2} - s), then derive \\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} = \\varphi, then obtain - \\varphi (n_{2} - s) + (\\frac{\\partial}{\\partial n_{2}} \\varphi (n_{2} - s) - 1)^{s} = - \\varphi (n_{2} - s) + (\\varphi - 1)^{s}", "derivation": "\\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} = \\varphi (n_{2} - s) and \\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} + 1 = \\varphi (n_{2} - s) + 1 and \\frac{\\partial}{\\partial n_{2}} (\\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} + 1) = \\frac{\\partial}{\\partial n_{2}} (\\varphi (n_{2} - s) + 1) and \\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{\\prime}}{(n_{2},s,\\varphi)} = \\varphi and \\frac{\\partial}{\\partial n_{2}} \\varphi (n_{2} - s) = \\varphi and \\frac{\\partial}{\\partial n_{2}} \\varphi (n_{2} - s) - 1 = \\varphi - 1 and (\\frac{\\partial}{\\partial n_{2}} \\varphi (n_{2} - s) - 1)^{s} = (\\varphi - 1)^{s} and - \\varphi (n_{2} - s) + (\\frac{\\partial}{\\partial n_{2}} \\varphi (n_{2} - s) - 1)^{s} = - \\varphi (n_{2} - s) + (\\varphi - 1)^{s}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Integer(1)))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Add(Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Integer(1)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('s', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))"], [["minus", 5, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], [["power", 6, "Symbol('s', commutative=True)"], "Equality(Pow(Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1)), Symbol('s', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["minus", 7, "Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Pow(Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1)), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Pow(Add(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\psi,Q,E_{\\lambda})} = E_{\\lambda} Q \\psi, then derive \\frac{\\partial}{\\partial E_{\\lambda}} \\phi_{2}{(\\psi,Q,E_{\\lambda})} = Q \\psi, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda} Q \\psi = Q \\psi", "derivation": "\\phi_{2}{(\\psi,Q,E_{\\lambda})} = E_{\\lambda} Q \\psi and \\frac{\\partial}{\\partial E_{\\lambda}} \\phi_{2}{(\\psi,Q,E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda} Q \\psi and \\frac{\\partial}{\\partial E_{\\lambda}} \\phi_{2}{(\\psi,Q,E_{\\lambda})} = Q \\psi and \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda} Q \\psi = Q \\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\phi,A_{x})} = \\phi \\log{(A_{x})} and \\hat{x}_0{(\\tilde{g})} = e^{\\sin{(\\tilde{g})}}, then obtain (- \\phi + \\frac{\\partial}{\\partial \\phi} \\sigma_{x}{(\\phi,A_{x})}) \\hat{x}_0{(\\tilde{g})} = (- \\phi + \\frac{\\partial}{\\partial \\phi} \\phi \\log{(A_{x})}) \\hat{x}_0{(\\tilde{g})}", "derivation": "\\sigma_{x}{(\\phi,A_{x})} = \\phi \\log{(A_{x})} and \\frac{\\partial}{\\partial \\phi} \\sigma_{x}{(\\phi,A_{x})} = \\frac{\\partial}{\\partial \\phi} \\phi \\log{(A_{x})} and - \\phi + \\frac{\\partial}{\\partial \\phi} \\sigma_{x}{(\\phi,A_{x})} = - \\phi + \\frac{\\partial}{\\partial \\phi} \\phi \\log{(A_{x})} and \\hat{x}_0{(\\tilde{g})} = e^{\\sin{(\\tilde{g})}} and (- \\phi + \\frac{\\partial}{\\partial \\phi} \\sigma_{x}{(\\phi,A_{x})}) e^{\\sin{(\\tilde{g})}} = (- \\phi + \\frac{\\partial}{\\partial \\phi} \\phi \\log{(A_{x})}) e^{\\sin{(\\tilde{g})}} and (- \\phi + \\frac{\\partial}{\\partial \\phi} \\sigma_{x}{(\\phi,A_{x})}) \\hat{x}_0{(\\tilde{g})} = (- \\phi + \\frac{\\partial}{\\partial \\phi} \\phi \\log{(A_{x})}) \\hat{x}_0{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\phi', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('A_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\phi', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\phi', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], ["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(sin(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 3, "exp(sin(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\phi', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), exp(sin(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), exp(sin(Symbol('\\\\tilde{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\phi', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Function('\\\\hat{x}_0')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Function('\\\\hat{x}_0')(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = \\sin{(\\frac{\\mathbf{v}}{\\mathbf{S}})} and V{(\\mathbf{S},\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\mathbf{v}}{\\mathbf{S}})}, then derive \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = \\frac{\\cos{(\\frac{\\mathbf{v}}{\\mathbf{S}})}}{\\mathbf{S}}, then derive V{(\\mathbf{S},\\mathbf{v})} = \\frac{\\cos{(\\frac{\\mathbf{v}}{\\mathbf{S}})}}{\\mathbf{S}}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = V{(\\mathbf{S},\\mathbf{v})}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = \\sin{(\\frac{\\mathbf{v}}{\\mathbf{S}})} and \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\mathbf{v}}{\\mathbf{S}})} and \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = \\frac{\\cos{(\\frac{\\mathbf{v}}{\\mathbf{S}})}}{\\mathbf{S}} and V{(\\mathbf{S},\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\mathbf{v}}{\\mathbf{S}})} and V{(\\mathbf{S},\\mathbf{v})} = \\frac{\\cos{(\\frac{\\mathbf{v}}{\\mathbf{S}})}}{\\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{E_{x}}{(\\mathbf{S},\\mathbf{v})} = V{(\\mathbf{S},\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), cos(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(sin(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('V')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), cos(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Function('V')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given m{(t_{1},s)} = s - t_{1}, then obtain (1 - t_{1}) \\int m{(t_{1},s)} ds = (1 - t_{1}) \\int (s - t_{1}) ds", "derivation": "m{(t_{1},s)} = s - t_{1} and t_{1} + m{(t_{1},s)} = s and \\int m{(t_{1},s)} ds = \\int (s - t_{1}) ds and (\\frac{s}{t_{1} + m{(t_{1},s)}} - t_{1}) \\int m{(t_{1},s)} ds = (\\frac{s}{t_{1} + m{(t_{1},s)}} - t_{1}) \\int (s - t_{1}) ds and (1 - t_{1}) \\int m{(t_{1},s)} ds = (1 - t_{1}) \\int (s - t_{1}) ds", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True)), Add(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["add", 1, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["times", 3, "Add(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('t_1', commutative=True), Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('t_1', commutative=True), Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integral(Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('t_1', commutative=True), Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integral(Add(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integral(Function('m')(Symbol('t_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Integer(1), Mul(Integer(-1), Symbol('t_1', commutative=True))), Integral(Add(Symbol('s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(m,f^{\\prime})} = \\sin{(f^{\\prime} m)} and \\Psi_{nl}{(t_{1})} = e^{t_{1}}, then derive \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})} = m \\cos{(f^{\\prime} m)}, then obtain \\Psi_{nl}{(t_{1})} - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})} = e^{t_{1}} - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})}", "derivation": "\\operatorname{v_{x}}{(m,f^{\\prime})} = \\sin{(f^{\\prime} m)} and \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})} = \\frac{\\partial}{\\partial f^{\\prime}} \\sin{(f^{\\prime} m)} and \\Psi_{nl}{(t_{1})} = e^{t_{1}} and \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})} = m \\cos{(f^{\\prime} m)} and - m \\cos{(f^{\\prime} m)} + \\Psi_{nl}{(t_{1})} = - m \\cos{(f^{\\prime} m)} + e^{t_{1}} and \\Psi_{nl}{(t_{1})} - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})} = e^{t_{1}} - \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{v_{x}}{(m,f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('m', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True))))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('m', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('m', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('m', commutative=True), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))))"], [["minus", 3, "Mul(Symbol('m', commutative=True), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))), Function('\\\\Psi_{nl}')(Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))), exp(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Derivative(Function('v_x')(Symbol('m', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))), Add(exp(Symbol('t_1', commutative=True)), Mul(Integer(-1), Derivative(Function('v_x')(Symbol('m', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Psi{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})}, then derive \\frac{d}{d x^\\prime} \\Psi{(x^\\prime)} = - \\frac{\\sin{(x^\\prime)}}{\\cos{(x^\\prime)}}, then obtain \\cos{(x^\\prime)} \\frac{d}{d x^\\prime} \\Psi{(x^\\prime)} = - \\sin{(x^\\prime)}", "derivation": "\\Psi{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\Psi{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(\\cos{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\Psi{(x^\\prime)} = - \\frac{\\sin{(x^\\prime)}}{\\cos{(x^\\prime)}} and \\cos{(x^\\prime)} \\frac{d}{d x^\\prime} \\Psi{(x^\\prime)} = - \\sin{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('x^\\\\prime', commutative=True)), log(cos(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1))))"], [["divide", 3, "Pow(cos(Symbol('x^\\\\prime', commutative=True)), Integer(-1))"], "Equality(Mul(cos(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{y})} = \\log{(\\log{(A_{y})})}, then obtain 2 (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} = (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} + 1", "derivation": "\\operatorname{F_{x}}{(A_{y})} = \\log{(\\log{(A_{y})})} and 0 = - \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})} and 0^{A_{y}} = (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} and 2 \\cdot 0^{A_{y}} = 0^{A_{y}} + (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} and 2 (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} = (- \\operatorname{F_{x}}{(A_{y})} + \\log{(\\log{(A_{y})})})^{A_{y}} + 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('A_y', commutative=True)), log(log(Symbol('A_y', commutative=True))))"], [["minus", 1, "Function('F_x')(Symbol('A_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_x')(Symbol('A_y', commutative=True))), log(log(Symbol('A_y', commutative=True)))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('A_y', commutative=True))), log(log(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True)))"], [["add", 3, "Pow(Integer(0), Symbol('A_y', commutative=True))"], "Equality(Mul(Integer(2), Pow(Integer(0), Symbol('A_y', commutative=True))), Add(Pow(Integer(0), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('A_y', commutative=True))), log(log(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('A_y', commutative=True))), log(log(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('A_y', commutative=True))), log(log(Symbol('A_y', commutative=True)))), Symbol('A_y', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbb{I}{(\\ddot{x},\\Psi,v)} = \\ddot{x} + \\frac{v}{\\Psi}, then obtain \\frac{\\partial^{2}}{\\partial \\ddot{x}^{2}} \\mathbb{I}^{v}{(\\ddot{x},\\Psi,v)} = \\frac{\\partial^{2}}{\\partial \\ddot{x}^{2}} (\\ddot{x} + \\frac{v}{\\Psi})^{v}", "derivation": "\\mathbb{I}{(\\ddot{x},\\Psi,v)} = \\ddot{x} + \\frac{v}{\\Psi} and \\mathbb{I}^{v}{(\\ddot{x},\\Psi,v)} = (\\ddot{x} + \\frac{v}{\\Psi})^{v} and \\frac{\\partial}{\\partial \\ddot{x}} \\mathbb{I}^{v}{(\\ddot{x},\\Psi,v)} = \\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\frac{v}{\\Psi})^{v} and \\frac{\\partial^{2}}{\\partial \\ddot{x}^{2}} \\mathbb{I}^{v}{(\\ddot{x},\\Psi,v)} = \\frac{\\partial^{2}}{\\partial \\ddot{x}^{2}} (\\ddot{x} + \\frac{v}{\\Psi})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)} = \\nabla g_{\\varepsilon}, then obtain (- 3 \\nabla g_{\\varepsilon} + 4 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)})^{2} = \\operatorname{A_{1}}^{2}{(g_{\\varepsilon},\\nabla)}", "derivation": "\\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)} = \\nabla g_{\\varepsilon} and - \\nabla g_{\\varepsilon} + 2 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)} = \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)} and (- \\nabla g_{\\varepsilon} + 2 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)})^{2} = \\operatorname{A_{1}}^{2}{(g_{\\varepsilon},\\nabla)} and (- 3 \\nabla g_{\\varepsilon} + 4 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)})^{2} = (- \\nabla g_{\\varepsilon} + 2 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)})^{2} and (- 3 \\nabla g_{\\varepsilon} + 4 \\operatorname{A_{1}}{(g_{\\varepsilon},\\nabla)})^{2} = \\operatorname{A_{1}}^{2}{(g_{\\varepsilon},\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(2)), Pow(Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Add(Mul(Integer(-1), Integer(3), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(4), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Mul(Integer(-1), Integer(3), Symbol('\\\\nabla', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(4), Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(2)), Pow(Function('A_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(v_{z},A_{2})} = \\frac{e^{A_{2}}}{v_{z}} and \\operatorname{f_{E}}{(v_{z},A_{2})} = - A_{2} + \\operatorname{m_{s}}{(v_{z},A_{2})}, then obtain (- \\frac{\\operatorname{f_{E}}{(v_{z},A_{2})}}{A_{2}})^{v_{z}} = (- \\frac{- A_{2} + \\frac{e^{A_{2}}}{v_{z}}}{A_{2}})^{v_{z}}", "derivation": "\\operatorname{m_{s}}{(v_{z},A_{2})} = \\frac{e^{A_{2}}}{v_{z}} and - A_{2} + \\operatorname{m_{s}}{(v_{z},A_{2})} = - A_{2} + \\frac{e^{A_{2}}}{v_{z}} and \\operatorname{f_{E}}{(v_{z},A_{2})} = - A_{2} + \\operatorname{m_{s}}{(v_{z},A_{2})} and \\operatorname{f_{E}}{(v_{z},A_{2})} = - A_{2} + \\frac{e^{A_{2}}}{v_{z}} and - \\frac{\\operatorname{f_{E}}{(v_{z},A_{2})}}{A_{2}} = - \\frac{- A_{2} + \\frac{e^{A_{2}}}{v_{z}}}{A_{2}} and (- \\frac{\\operatorname{f_{E}}{(v_{z},A_{2})}}{A_{2}})^{v_{z}} = (- \\frac{- A_{2} + \\frac{e^{A_{2}}}{v_{z}}}{A_{2}})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), exp(Symbol('A_2', commutative=True))))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('m_s')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), exp(Symbol('A_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('m_s')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('f_E')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), exp(Symbol('A_2', commutative=True)))))"], [["divide", 4, "Mul(Integer(-1), Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f_E')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), exp(Symbol('A_2', commutative=True))))))"], [["power", 5, "Symbol('v_z', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f_E')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Symbol('v_z', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), exp(Symbol('A_2', commutative=True))))), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given u{(E_{n},i)} = E_{n} + i, then derive \\int u{(E_{n},i)} dE_{n} = \\frac{E_{n}^{2}}{2} + E_{n} i + t_{2}, then obtain A_{y} + \\frac{E_{n}^{2}}{2} + E_{n} i = \\frac{E_{n}^{2}}{2} + E_{n} i + t_{2}", "derivation": "u{(E_{n},i)} = E_{n} + i and \\int u{(E_{n},i)} dE_{n} = \\int (E_{n} + i) dE_{n} and \\int u{(E_{n},i)} dE_{n} = \\frac{E_{n}^{2}}{2} + E_{n} i + t_{2} and \\int (E_{n} + i) dE_{n} = \\frac{E_{n}^{2}}{2} + E_{n} i + t_{2} and A_{y} + \\frac{E_{n}^{2}}{2} + E_{n} i = \\frac{E_{n}^{2}}{2} + E_{n} i + t_{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('u')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Symbol('t_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = e^{\\mathbf{r}}, then derive \\frac{d}{d \\mathbf{r}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = e^{\\mathbf{r}}, then obtain \\frac{d}{d \\mathbf{r}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = \\frac{d^{2}}{d \\mathbf{r}^{2}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})} = \\frac{d^{2}}{d \\mathbf{r}^{2}} \\operatorname{L_{\\varepsilon}}{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))))"]]}, {"prompt": "Given r{(\\Omega,\\theta_2)} = \\log{(\\Omega^{\\theta_2})}, then obtain 1 = \\frac{\\int (- r^{\\Omega}{(\\Omega,\\theta_2)} + \\log{(\\Omega^{\\theta_2})}^{\\Omega}) d\\Omega}{\\int 0 d\\Omega}", "derivation": "r{(\\Omega,\\theta_2)} = \\log{(\\Omega^{\\theta_2})} and r^{\\Omega}{(\\Omega,\\theta_2)} = \\log{(\\Omega^{\\theta_2})}^{\\Omega} and \\Omega^{\\theta_2} + r^{\\Omega}{(\\Omega,\\theta_2)} = \\Omega^{\\theta_2} + \\log{(\\Omega^{\\theta_2})}^{\\Omega} and 0 = - r^{\\Omega}{(\\Omega,\\theta_2)} + \\log{(\\Omega^{\\theta_2})}^{\\Omega} and \\int 0 d\\Omega = \\int (- r^{\\Omega}{(\\Omega,\\theta_2)} + \\log{(\\Omega^{\\theta_2})}^{\\Omega}) d\\Omega and 1 = \\frac{\\int (- r^{\\Omega}{(\\Omega,\\theta_2)} + \\log{(\\Omega^{\\theta_2})}^{\\Omega}) d\\Omega}{\\int 0 d\\Omega}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\Omega', commutative=True))))"], [["minus", 3, "Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 5, "Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Function('r')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\Omega', commutative=True))), Pow(log(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\omega)} = \\log{(\\sin{(\\omega)})} and \\operatorname{E_{x}}{(\\omega)} = \\cos{(\\omega)}, then derive \\frac{d}{d \\omega} \\operatorname{f_{\\mathbf{p}}}{(\\omega)} = \\frac{\\cos{(\\omega)}}{\\sin{(\\omega)}}, then obtain \\log{(\\frac{d}{d \\omega} \\log{(\\sin{(\\omega)})})} = \\log{(\\frac{\\operatorname{E_{x}}{(\\omega)}}{\\sin{(\\omega)}})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\omega)} = \\log{(\\sin{(\\omega)})} and \\frac{d}{d \\omega} \\operatorname{f_{\\mathbf{p}}}{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\sin{(\\omega)})} and \\frac{d}{d \\omega} \\operatorname{f_{\\mathbf{p}}}{(\\omega)} = \\frac{\\cos{(\\omega)}}{\\sin{(\\omega)}} and \\log{(\\frac{d}{d \\omega} \\operatorname{f_{\\mathbf{p}}}{(\\omega)})} = \\log{(\\frac{\\cos{(\\omega)}}{\\sin{(\\omega)}})} and \\operatorname{E_{x}}{(\\omega)} = \\cos{(\\omega)} and \\log{(\\frac{d}{d \\omega} \\log{(\\sin{(\\omega)})})} = \\log{(\\frac{\\cos{(\\omega)}}{\\sin{(\\omega)}})} and \\log{(\\frac{d}{d \\omega} \\log{(\\sin{(\\omega)})})} = \\log{(\\frac{\\operatorname{E_{x}}{(\\omega)}}{\\sin{(\\omega)}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\omega', commutative=True)), log(sin(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\omega', commutative=True))))"], [["log", 3], "Equality(log(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), log(Mul(Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(log(Derivative(log(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), log(Mul(Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(log(Derivative(log(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), log(Mul(Function('E_x')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\rho{(\\varepsilon)} = \\sin{(\\varepsilon)}, then derive \\frac{d}{d \\varepsilon} \\rho{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain (\\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)})^{\\varepsilon} = \\cos^{\\varepsilon}{(\\varepsilon)}", "derivation": "\\rho{(\\varepsilon)} = \\sin{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\rho{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\rho{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)} = \\cos{(\\varepsilon)} and (\\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)})^{\\varepsilon} = \\cos^{\\varepsilon}{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Derivative(sin(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} = \\sin{(\\frac{\\hat{x}}{U})}, then obtain \\frac{\\int (\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} - \\sin{(\\frac{\\hat{x}}{U})}) d\\hat{x}}{\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)}} = \\frac{\\int 0 d\\hat{x}}{\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} = \\sin{(\\frac{\\hat{x}}{U})} and \\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} - \\sin{(\\frac{\\hat{x}}{U})} = 0 and \\int (\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} - \\sin{(\\frac{\\hat{x}}{U})}) d\\hat{x} = \\int 0 d\\hat{x} and \\frac{\\int (\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} - \\sin{(\\frac{\\hat{x}}{U})}) d\\hat{x}}{\\sin{(\\frac{\\hat{x}}{U})}} = \\frac{\\int 0 d\\hat{x}}{\\sin{(\\frac{\\hat{x}}{U})}} and \\frac{\\int (\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)} - \\sin{(\\frac{\\hat{x}}{U})}) d\\hat{x}}{\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)}} = \\frac{\\int 0 d\\hat{x}}{\\operatorname{J_{\\varepsilon}}{(\\hat{x},U)}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 3, "sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))), Integer(-1)), Integral(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Pow(sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Integral(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{s})} = \\mathbf{s}, then derive e^{\\frac{d}{d \\mathbf{s}} \\operatorname{v_{x}}{(\\mathbf{s})}} = e, then obtain e = e^{\\frac{d}{d \\mathbf{s}} \\mathbf{s}}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{s})} = \\mathbf{s} and \\frac{d}{d \\mathbf{s}} \\operatorname{v_{x}}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\mathbf{s} and e^{\\frac{d}{d \\mathbf{s}} \\operatorname{v_{x}}{(\\mathbf{s})}} = e^{\\frac{d}{d \\mathbf{s}} \\mathbf{s}} and e^{\\frac{d}{d \\mathbf{s}} \\operatorname{v_{x}}{(\\mathbf{s})}} = e and e = e^{\\frac{d}{d \\mathbf{s}} \\mathbf{s}}", "srepr_derivation": [["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('v_x')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), exp(Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(exp(Derivative(Function('v_x')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), E)"], [["substitute_LHS_for_RHS", 3, 4], "Equality(E, exp(Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(\\phi_2,\\mathbf{M})} = \\phi_2^{\\mathbf{M}}, then obtain 2 \\mathbf{M} M{(\\phi_2,\\mathbf{M})} - \\phi_2 = 2 \\mathbf{M} \\phi_2^{\\mathbf{M}} - \\phi_2", "derivation": "M{(\\phi_2,\\mathbf{M})} = \\phi_2^{\\mathbf{M}} and \\mathbf{M} M{(\\phi_2,\\mathbf{M})} = \\mathbf{M} \\phi_2^{\\mathbf{M}} and \\mathbf{M} M{(\\phi_2,\\mathbf{M})} - \\phi_2 = \\mathbf{M} \\phi_2^{\\mathbf{M}} - \\phi_2 and 2 \\mathbf{M} M{(\\phi_2,\\mathbf{M})} - \\phi_2 = \\mathbf{M} \\phi_2^{\\mathbf{M}} + \\mathbf{M} M{(\\phi_2,\\mathbf{M})} - \\phi_2 and 2 \\mathbf{M} M{(\\phi_2,\\mathbf{M})} - \\phi_2 = 2 \\mathbf{M} \\phi_2^{\\mathbf{M}} - \\phi_2", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(H,\\dot{y})} = H - \\dot{y}, then obtain \\frac{\\partial}{\\partial H} \\int (\\dot{y} + \\mathbf{D}{(H,\\dot{y})}) d\\dot{y} = \\frac{\\partial}{\\partial H} \\int H d\\dot{y}", "derivation": "\\mathbf{D}{(H,\\dot{y})} = H - \\dot{y} and \\dot{y} + \\mathbf{D}{(H,\\dot{y})} = H and \\int (\\dot{y} + \\mathbf{D}{(H,\\dot{y})}) d\\dot{y} = \\int H d\\dot{y} and \\frac{\\partial}{\\partial H} \\int (\\dot{y} + \\mathbf{D}{(H,\\dot{y})}) d\\dot{y} = \\frac{\\partial}{\\partial H} \\int H d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('H', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{D}')(Symbol('H', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Symbol('H', commutative=True))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{D}')(Symbol('H', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Symbol('H', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{D}')(Symbol('H', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(Symbol('H', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(\\theta_2)} = \\sin{(\\theta_2)}, then derive A_{2} + W{(\\theta_2)} = \\chi + \\sin{(\\theta_2)}, then obtain \\frac{A_{2} + W{(\\theta_2)}}{\\theta_2 + \\sin{(\\theta_2)}} = \\frac{\\chi + \\sin{(\\theta_2)}}{\\theta_2 + \\sin{(\\theta_2)}}", "derivation": "W{(\\theta_2)} = \\sin{(\\theta_2)} and \\theta_2 + W{(\\theta_2)} = \\theta_2 + \\sin{(\\theta_2)} and \\frac{d}{d \\theta_2} W{(\\theta_2)} = \\frac{d}{d \\theta_2} \\sin{(\\theta_2)} and \\int \\frac{d}{d \\theta_2} W{(\\theta_2)} d\\theta_2 = \\int \\frac{d}{d \\theta_2} \\sin{(\\theta_2)} d\\theta_2 and A_{2} + W{(\\theta_2)} = \\chi + \\sin{(\\theta_2)} and \\frac{A_{2} + W{(\\theta_2)}}{\\theta_2 + W{(\\theta_2)}} = \\frac{\\chi + \\sin{(\\theta_2)}}{\\theta_2 + W{(\\theta_2)}} and \\frac{A_{2} + W{(\\theta_2)}}{\\theta_2 + \\sin{(\\theta_2)}} = \\frac{\\chi + \\sin{(\\theta_2)}}{\\theta_2 + \\sin{(\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Function('W')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 5, "Add(Symbol('\\\\theta_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Add(Symbol('A_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\chi', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Add(Symbol('A_2', commutative=True), Function('W')(Symbol('\\\\theta_2', commutative=True))), Pow(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\chi', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Pow(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\varphi^*,L_{\\varepsilon})} = - L_{\\varepsilon} + \\log{(\\varphi^*)}, then derive \\int \\operatorname{v_{y}}{(\\varphi^*,L_{\\varepsilon})} d\\varphi^* = \\varphi^* (- L_{\\varepsilon} - 1) + \\varphi^* \\log{(\\varphi^*)} + t_{1}, then obtain \\varphi^* (- L_{\\varepsilon} - 1) + \\varphi^* \\log{(\\varphi^*)} + t_{1} = \\int (- L_{\\varepsilon} + \\log{(\\varphi^*)}) d\\varphi^*", "derivation": "\\operatorname{v_{y}}{(\\varphi^*,L_{\\varepsilon})} = - L_{\\varepsilon} + \\log{(\\varphi^*)} and \\int \\operatorname{v_{y}}{(\\varphi^*,L_{\\varepsilon})} d\\varphi^* = \\int (- L_{\\varepsilon} + \\log{(\\varphi^*)}) d\\varphi^* and \\int \\operatorname{v_{y}}{(\\varphi^*,L_{\\varepsilon})} d\\varphi^* = \\varphi^* (- L_{\\varepsilon} - 1) + \\varphi^* \\log{(\\varphi^*)} + t_{1} and \\varphi^* (- L_{\\varepsilon} - 1) + \\varphi^* \\log{(\\varphi^*)} + t_{1} = \\int (- L_{\\varepsilon} + \\log{(\\varphi^*)}) d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\varphi^*', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\varphi^*', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_y')(Symbol('\\\\varphi^*', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('t_1', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\pi{(c)} = \\sin{(c)}, then derive \\int \\pi{(c)} dc = \\sigma_x - \\cos{(c)}, then derive \\sigma_x - \\cos{(c)} = \\mathbf{J}_f - \\cos{(c)}, then obtain \\frac{(\\int \\sin{(c)} dc)^{\\mathbf{J}_f}}{\\frac{d}{d F_{x}} (-1)} = \\frac{(\\mathbf{J}_f - \\cos{(c)})^{\\mathbf{J}_f}}{\\frac{d}{d F_{x}} (-1)}", "derivation": "\\pi{(c)} = \\sin{(c)} and \\int \\pi{(c)} dc = \\int \\sin{(c)} dc and \\int \\pi{(c)} dc = \\sigma_x - \\cos{(c)} and \\sigma_x - \\cos{(c)} = \\int \\sin{(c)} dc and \\sigma_x - \\cos{(c)} = \\mathbf{J}_f - \\cos{(c)} and \\int \\sin{(c)} dc = \\mathbf{J}_f - \\cos{(c)} and (\\int \\sin{(c)} dc)^{\\mathbf{J}_f} = (\\mathbf{J}_f - \\cos{(c)})^{\\mathbf{J}_f} and \\frac{(\\int \\sin{(c)} dc)^{\\mathbf{J}_f}}{\\frac{d}{d F_{x}} (-1)} = \\frac{(\\mathbf{J}_f - \\cos{(c)})^{\\mathbf{J}_f}}{\\frac{d}{d F_{x}} (-1)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))))"], [["power", 6, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 7, "Derivative(Integer(-1), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Integer(-1), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Pow(Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Integer(-1), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(x,\\mathbf{D})} = \\frac{\\mathbf{D}}{x} and \\rho_{f}{(x)} = \\frac{1}{x}, then obtain - (- \\frac{\\mathbf{D}}{x} + \\rho_{f}{(x)}) \\int \\frac{\\mathbf{D}}{x} dx = - (- \\frac{\\mathbf{D}}{x} + \\frac{1}{x}) \\int \\frac{\\mathbf{D}}{x} dx", "derivation": "\\hat{\\mathbf{x}}{(x,\\mathbf{D})} = \\frac{\\mathbf{D}}{x} and \\int \\hat{\\mathbf{x}}{(x,\\mathbf{D})} dx = \\int \\frac{\\mathbf{D}}{x} dx and \\rho_{f}{(x)} = \\frac{1}{x} and - \\frac{\\mathbf{D}}{x} + \\rho_{f}{(x)} = - \\frac{\\mathbf{D}}{x} + \\frac{1}{x} and - (- \\frac{\\mathbf{D}}{x} + \\rho_{f}{(x)}) \\int \\hat{\\mathbf{x}}{(x,\\mathbf{D})} dx = - (- \\frac{\\mathbf{D}}{x} + \\frac{1}{x}) \\int \\hat{\\mathbf{x}}{(x,\\mathbf{D})} dx and - (- \\frac{\\mathbf{D}}{x} + \\rho_{f}{(x)}) \\int \\frac{\\mathbf{D}}{x} dx = - (- \\frac{\\mathbf{D}}{x} + \\frac{1}{x}) \\int \\frac{\\mathbf{D}}{x} dx", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('x', commutative=True)), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["minus", 3, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\rho_f')(Symbol('x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["times", 4, "Mul(Integer(-1), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('x', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\rho_f')(Symbol('x', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Pow(Symbol('x', commutative=True), Integer(-1))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\rho_f')(Symbol('x', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Pow(Symbol('x', commutative=True), Integer(-1))), Integral(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given C{(g^{\\prime}_{\\varepsilon},\\chi)} = \\cos{(\\chi - g^{\\prime}_{\\varepsilon})} and \\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)} = \\chi - g^{\\prime}_{\\varepsilon}, then obtain - g^{\\prime}_{\\varepsilon} + \\cos{(\\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)})} = - g^{\\prime}_{\\varepsilon} + \\cos{(\\chi - g^{\\prime}_{\\varepsilon})}", "derivation": "C{(g^{\\prime}_{\\varepsilon},\\chi)} = \\cos{(\\chi - g^{\\prime}_{\\varepsilon})} and \\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)} = \\chi - g^{\\prime}_{\\varepsilon} and C{(g^{\\prime}_{\\varepsilon},\\chi)} = \\cos{(\\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)})} and \\cos{(\\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)})} = \\cos{(\\chi - g^{\\prime}_{\\varepsilon})} and - g^{\\prime}_{\\varepsilon} + \\cos{(\\operatorname{f^{*}}{(g^{\\prime}_{\\varepsilon},\\chi)})} = - g^{\\prime}_{\\varepsilon} + \\cos{(\\chi - g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), cos(Function('f^*')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(cos(Function('f^*')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True))), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["minus", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Function('f^*')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)}, then derive - \\operatorname{A_{z}}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} \\frac{d}{d \\hat{H}_l} \\operatorname{A_{z}}{(\\hat{H}_l)} = - 2 \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)}, then obtain - \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)} = - 2 \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)}", "derivation": "\\operatorname{A_{z}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\operatorname{A_{z}}{(\\hat{H}_l)} \\cos{(\\hat{H}_l)} = \\cos^{2}{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\operatorname{A_{z}}{(\\hat{H}_l)} \\cos{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\cos^{2}{(\\hat{H}_l)} and - \\operatorname{A_{z}}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} \\frac{d}{d \\hat{H}_l} \\operatorname{A_{z}}{(\\hat{H}_l)} = - 2 \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)} and - \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)} = - 2 \\sin{(\\hat{H}_l)} \\cos{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Pow(cos(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Mul(cos(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Mul(cos(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(cos(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(a,J)} = - J + a, then derive \\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)} = 1, then obtain (- J + \\frac{\\partial}{\\partial a} (- J + a)) (- \\cos{(\\frac{\\partial}{\\partial a} (- J + a))} + \\cos{(\\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)})}) = 0", "derivation": "\\operatorname{a^{\\dagger}}{(a,J)} = - J + a and \\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)} = \\frac{\\partial}{\\partial a} (- J + a) and \\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)} = 1 and \\frac{\\partial}{\\partial a} (- J + a) = 1 and \\cos{(\\frac{\\partial}{\\partial a} (- J + a))} = \\cos{(1)} and \\cos{(\\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)})} = \\cos{(1)} and \\cos{(\\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)})} = \\cos{(\\frac{\\partial}{\\partial a} (- J + a))} and - \\cos{(\\frac{\\partial}{\\partial a} (- J + a))} + \\cos{(\\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)})} = 0 and (- J + \\frac{\\partial}{\\partial a} (- J + a)) (- \\cos{(\\frac{\\partial}{\\partial a} (- J + a))} + \\cos{(\\frac{\\partial}{\\partial a} \\operatorname{a^{\\dagger}}{(a,J)})}) = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1))"], [["cos", 4], "Equality(cos(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Integer(1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(cos(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(cos(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["minus", 7, "cos(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), cos(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), cos(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Integer(0))"], [["times", 8, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), cos(Derivative(Function('a^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(I)} = \\sin{(I)}, then obtain 2 (I + \\operatorname{A_{y}}{(I)}) \\operatorname{A_{y}}{(I)} \\sin{(I)} = (I + \\operatorname{A_{y}}{(I)}) (\\operatorname{A_{y}}{(I)} + \\sin{(I)}) \\sin{(I)}", "derivation": "\\operatorname{A_{y}}{(I)} = \\sin{(I)} and 2 \\operatorname{A_{y}}{(I)} = \\operatorname{A_{y}}{(I)} + \\sin{(I)} and I + \\operatorname{A_{y}}{(I)} = I + \\sin{(I)} and 2 \\operatorname{A_{y}}{(I)} \\sin{(I)} = (\\operatorname{A_{y}}{(I)} + \\sin{(I)}) \\sin{(I)} and 2 (I + \\sin{(I)}) \\operatorname{A_{y}}{(I)} \\sin{(I)} = (I + \\sin{(I)}) (\\operatorname{A_{y}}{(I)} + \\sin{(I)}) \\sin{(I)} and 2 (I + \\operatorname{A_{y}}{(I)}) \\operatorname{A_{y}}{(I)} \\sin{(I)} = (I + \\operatorname{A_{y}}{(I)}) (\\operatorname{A_{y}}{(I)} + \\sin{(I)}) \\sin{(I)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["add", 1, "Function('A_y')(Symbol('I', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('I', commutative=True))), Add(Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('A_y')(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))))"], [["times", 2, "sin(Symbol('I', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), Mul(Add(Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))))"], [["times", 4, "Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True)))"], "Equality(Mul(Integer(2), Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), Mul(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Add(Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Add(Symbol('I', commutative=True), Function('A_y')(Symbol('I', commutative=True))), Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), Mul(Add(Symbol('I', commutative=True), Function('A_y')(Symbol('I', commutative=True))), Add(Function('A_y')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))))"]]}, {"prompt": "Given r{(\\psi,\\chi,\\sigma_x)} = \\chi + \\psi - \\sigma_x, then derive \\frac{\\partial}{\\partial \\sigma_x} r{(\\psi,\\chi,\\sigma_x)} = -1, then obtain \\frac{\\partial}{\\partial \\sigma_x} (\\chi + \\psi - \\sigma_x) = -1", "derivation": "r{(\\psi,\\chi,\\sigma_x)} = \\chi + \\psi - \\sigma_x and \\frac{\\partial}{\\partial \\sigma_x} r{(\\psi,\\chi,\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} (\\chi + \\psi - \\sigma_x) and \\frac{\\partial}{\\partial \\sigma_x} r{(\\psi,\\chi,\\sigma_x)} = -1 and \\frac{\\partial}{\\partial \\sigma_x} (\\chi + \\psi - \\sigma_x) = -1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\sigma_{x}{(f^{\\prime})} = e^{f^{\\prime}}, then derive e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} \\sigma_{x}{(f^{\\prime})} = 2 e^{f^{\\prime}}, then obtain e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} \\sigma_{x}{(f^{\\prime})} = \\sigma_{x}{(f^{\\prime})} + e^{f^{\\prime}}", "derivation": "\\sigma_{x}{(f^{\\prime})} = e^{f^{\\prime}} and \\sigma_{x}{(f^{\\prime})} + e^{f^{\\prime}} = 2 e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} (\\sigma_{x}{(f^{\\prime})} + e^{f^{\\prime}}) = \\frac{d}{d f^{\\prime}} 2 e^{f^{\\prime}} and e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} \\sigma_{x}{(f^{\\prime})} = 2 e^{f^{\\prime}} and e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} \\sigma_{x}{(f^{\\prime})} = \\sigma_{x}{(f^{\\prime})} + e^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "exp(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(exp(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\psi,F_{N},\\sigma_x)} = F_{N} (\\psi - \\sigma_x), then obtain \\int \\frac{\\partial}{\\partial \\psi} \\sigma_x \\operatorname{M_{E}}^{F_{N}}{(\\psi,F_{N},\\sigma_x)} dF_{N} = \\int \\frac{\\partial}{\\partial \\psi} \\sigma_x (F_{N} (\\psi - \\sigma_x))^{F_{N}} dF_{N}", "derivation": "\\operatorname{M_{E}}{(\\psi,F_{N},\\sigma_x)} = F_{N} (\\psi - \\sigma_x) and \\operatorname{M_{E}}^{F_{N}}{(\\psi,F_{N},\\sigma_x)} = (F_{N} (\\psi - \\sigma_x))^{F_{N}} and \\sigma_x \\operatorname{M_{E}}^{F_{N}}{(\\psi,F_{N},\\sigma_x)} = \\sigma_x (F_{N} (\\psi - \\sigma_x))^{F_{N}} and \\frac{\\partial}{\\partial \\psi} \\sigma_x \\operatorname{M_{E}}^{F_{N}}{(\\psi,F_{N},\\sigma_x)} = \\frac{\\partial}{\\partial \\psi} \\sigma_x (F_{N} (\\psi - \\sigma_x))^{F_{N}} and \\int \\frac{\\partial}{\\partial \\psi} \\sigma_x \\operatorname{M_{E}}^{F_{N}}{(\\psi,F_{N},\\sigma_x)} dF_{N} = \\int \\frac{\\partial}{\\partial \\psi} \\sigma_x (F_{N} (\\psi - \\sigma_x))^{F_{N}} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('F_N', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('F_N', commutative=True)), Pow(Mul(Symbol('F_N', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('F_N', commutative=True)))"], [["times", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('F_N', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Mul(Symbol('F_N', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('F_N', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Mul(Symbol('F_N', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Function('M_E')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('F_N', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Mul(Symbol('F_N', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\rho)} = e^{e^{\\rho}}, then obtain e^{e^{\\rho}} + \\frac{\\operatorname{E_{n}}^{\\rho}{(\\rho)}}{\\rho} = e^{e^{\\rho}} + \\frac{(e^{e^{\\rho}})^{\\rho}}{\\rho}", "derivation": "\\operatorname{E_{n}}{(\\rho)} = e^{e^{\\rho}} and \\operatorname{E_{n}}^{\\rho}{(\\rho)} = (e^{e^{\\rho}})^{\\rho} and \\frac{\\operatorname{E_{n}}^{\\rho}{(\\rho)}}{\\rho} = \\frac{(e^{e^{\\rho}})^{\\rho}}{\\rho} and e^{e^{\\rho}} + \\frac{\\operatorname{E_{n}}^{\\rho}{(\\rho)}}{\\rho} = e^{e^{\\rho}} + \\frac{(e^{e^{\\rho}})^{\\rho}}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\rho', commutative=True)), exp(exp(Symbol('\\\\rho', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(exp(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["divide", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(exp(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))))"], [["add", 3, "exp(exp(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(exp(exp(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))), Add(exp(exp(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(exp(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given V{(S,y)} = S y and \\operatorname{V_{\\mathbf{B}}}{(S,y)} = - S y, then derive - \\frac{- S y + y + V{(S,y)}}{y} = -1, then obtain \\frac{y + V{(S,y)} + \\operatorname{V_{\\mathbf{B}}}{(S,y)}}{S y^{2}} = \\frac{1}{S y}", "derivation": "V{(S,y)} = S y and y + V{(S,y)} = S y + y and - S y + y + V{(S,y)} = y and \\operatorname{V_{\\mathbf{B}}}{(S,y)} = - S y and \\frac{- S y + y + V{(S,y)}}{\\frac{\\partial}{\\partial S} - S y} = \\frac{y}{\\frac{\\partial}{\\partial S} - S y} and - \\frac{- S y + y + V{(S,y)}}{y} = -1 and \\frac{- S y + y + V{(S,y)}}{S y^{2}} = \\frac{1}{S y} and \\frac{y + V{(S,y)} + \\operatorname{V_{\\mathbf{B}}}{(S,y)}}{S y^{2}} = \\frac{1}{S y}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)))"], [["divide", 3, "Derivative(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True))), Pow(Derivative(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('y', commutative=True), Pow(Derivative(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True)))), Integer(-1))"], [["divide", 6, "Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-2)), Add(Symbol('y', commutative=True), Function('V')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('S', commutative=True), Symbol('y', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1))))"]]}, {"prompt": "Given i{(r)} = r, then derive (\\frac{d^{2}}{d r^{2}} i{(r)})^{2} = 0, then obtain - i{(r)} + (\\frac{d^{2}}{d r^{2}} r)^{2} = - i{(r)}", "derivation": "i{(r)} = r and \\frac{d}{d r} i{(r)} = \\frac{d}{d r} r and \\frac{d^{2}}{d r^{2}} i{(r)} = \\frac{d^{2}}{d r^{2}} r and (\\frac{d^{2}}{d r^{2}} i{(r)})^{2} = (\\frac{d^{2}}{d r^{2}} r)^{2} and (\\frac{d^{2}}{d r^{2}} i{(r)})^{2} = 0 and (\\frac{d^{2}}{d r^{2}} r)^{2} = 0 and - i{(r)} + (\\frac{d^{2}}{d r^{2}} r)^{2} = - i{(r)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('r', commutative=True)), Symbol('r', commutative=True))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(2)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(2)), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(2)), Integer(0))"], [["minus", 6, "Function('i')(Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('r', commutative=True))), Pow(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(2))), Mul(Integer(-1), Function('i')(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(i,I)} = \\frac{I}{i} and \\mathbf{s}{(i,I)} = - \\operatorname{F_{x}}{(i,I)}, then obtain \\frac{\\frac{\\partial}{\\partial I} - \\operatorname{F_{x}}{(i,I)}}{\\frac{\\partial}{\\partial I} \\mathbf{s}{(i,I)}} = 1", "derivation": "\\operatorname{F_{x}}{(i,I)} = \\frac{I}{i} and \\mathbf{s}{(i,I)} = - \\operatorname{F_{x}}{(i,I)} and \\mathbf{s}{(i,I)} = - \\frac{I}{i} and - \\operatorname{F_{x}}{(i,I)} = - \\frac{I}{i} and \\frac{\\partial}{\\partial I} - \\operatorname{F_{x}}{(i,I)} = \\frac{\\partial}{\\partial I} - \\frac{I}{i} and \\frac{\\partial}{\\partial I} - \\operatorname{F_{x}}{(i,I)} = \\frac{\\partial}{\\partial I} \\mathbf{s}{(i,I)} and \\frac{\\frac{\\partial}{\\partial I} - \\operatorname{F_{x}}{(i,I)}}{\\frac{\\partial}{\\partial I} \\mathbf{s}{(i,I)}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{s}')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Mul(Integer(-1), Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{s}')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["divide", 6, "Derivative(Function('\\\\mathbf{s}')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Integer(-1), Function('F_x')(Symbol('i', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{s}')(Symbol('i', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given n{(\\mu_0)} = \\log{(e^{\\mu_0})}, then obtain 1 = ((\\frac{e^{\\mu_0} \\log{(e^{\\mu_0})} + \\log{(e^{\\mu_0})}}{n{(\\mu_0)} + e^{\\mu_0} \\log{(e^{\\mu_0})}})^{\\mu_0})^{\\mu_0}", "derivation": "n{(\\mu_0)} = \\log{(e^{\\mu_0})} and n{(\\mu_0)} e^{\\mu_0} = e^{\\mu_0} \\log{(e^{\\mu_0})} and n{(\\mu_0)} e^{\\mu_0} + n{(\\mu_0)} = n{(\\mu_0)} e^{\\mu_0} + \\log{(e^{\\mu_0})} and 1 = \\frac{n{(\\mu_0)} e^{\\mu_0} + \\log{(e^{\\mu_0})}}{n{(\\mu_0)} e^{\\mu_0} + n{(\\mu_0)}} and 1 = (\\frac{n{(\\mu_0)} e^{\\mu_0} + \\log{(e^{\\mu_0})}}{n{(\\mu_0)} e^{\\mu_0} + n{(\\mu_0)}})^{\\mu_0} and 1 = (\\frac{e^{\\mu_0} \\log{(e^{\\mu_0})} + \\log{(e^{\\mu_0})}}{n{(\\mu_0)} + e^{\\mu_0} \\log{(e^{\\mu_0})}})^{\\mu_0} and 1 = ((\\frac{e^{\\mu_0} \\log{(e^{\\mu_0})} + \\log{(e^{\\mu_0})}}{n{(\\mu_0)} + e^{\\mu_0} \\log{(e^{\\mu_0})}})^{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Mul(exp(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))))"], [["add", 1, "Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Function('n')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True)))))"], [["divide", 3, "Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Function('n')(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Function('n')(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True))))))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Function('n')(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Add(Mul(Function('n')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Pow(Mul(Add(Mul(exp(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))), log(exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(exp(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True))))), Integer(-1))), Symbol('\\\\mu_0', commutative=True)))"], [["power", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Add(Mul(exp(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))), log(exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(exp(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True))))), Integer(-1))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{J},t)} = \\mathbf{J}^{t}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} (((- \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)})^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} = \\frac{d}{d \\mathbf{J}} ((0^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}}", "derivation": "\\theta_{1}{(\\mathbf{J},t)} = \\mathbf{J}^{t} and - \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)} = 0 and (- \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)})^{\\mathbf{J}} = 0^{\\mathbf{J}} and ((- \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)})^{\\mathbf{J}})^{\\mathbf{J}} = (0^{\\mathbf{J}})^{\\mathbf{J}} and (((- \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)})^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} = ((0^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mathbf{J}} (((- \\mathbf{J}^{t} + \\theta_{1}{(\\mathbf{J},t)})^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} = \\frac{d}{d \\mathbf{J}} ((0^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Pow(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Pow(Pow(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Pow(Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\hat{x}_0,z,L_{\\varepsilon})} = L_{\\varepsilon}^{\\hat{x}_0} - z, then obtain L_{\\varepsilon}^{\\hat{x}_0} - z + L_{\\varepsilon}^{- \\hat{x}_0} = L_{\\varepsilon}^{\\hat{x}_0} - z + \\frac{L_{\\varepsilon}^{- \\hat{x}_0} (L_{\\varepsilon}^{\\hat{x}_0} - z)}{t{(\\hat{x}_0,z,L_{\\varepsilon})}}", "derivation": "t{(\\hat{x}_0,z,L_{\\varepsilon})} = L_{\\varepsilon}^{\\hat{x}_0} - z and L_{\\varepsilon}^{- \\hat{x}_0} t{(\\hat{x}_0,z,L_{\\varepsilon})} = L_{\\varepsilon}^{- \\hat{x}_0} (L_{\\varepsilon}^{\\hat{x}_0} - z) and L_{\\varepsilon}^{- \\hat{x}_0} = \\frac{L_{\\varepsilon}^{- \\hat{x}_0} (L_{\\varepsilon}^{\\hat{x}_0} - z)}{t{(\\hat{x}_0,z,L_{\\varepsilon})}} and L_{\\varepsilon}^{\\hat{x}_0} - z + L_{\\varepsilon}^{- \\hat{x}_0} = L_{\\varepsilon}^{\\hat{x}_0} - z + \\frac{L_{\\varepsilon}^{- \\hat{x}_0} (L_{\\varepsilon}^{\\hat{x}_0} - z)}{t{(\\hat{x}_0,z,L_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["divide", 1, "Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Function('t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["divide", 2, "Function('t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Pow(Function('t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["add", 3, "Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)))"], "Equality(Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))), Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Add(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Pow(Function('t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('z', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{B}{(l)} = \\cos{(l)}, then derive \\Omega + \\mathbf{B}{(l)} = \\Omega + \\cos{(l)}, then obtain - \\Omega (\\Omega + \\cos{(l)}) + \\Omega + \\mathbf{B}{(l)} = - \\Omega (\\Omega + \\cos{(l)}) + \\Omega + \\cos{(l)}", "derivation": "\\mathbf{B}{(l)} = \\cos{(l)} and \\frac{d}{d l} \\mathbf{B}{(l)} = \\frac{d}{d l} \\cos{(l)} and \\int \\frac{d}{d l} \\mathbf{B}{(l)} dl = \\int \\frac{d}{d l} \\cos{(l)} dl and \\Omega + \\mathbf{B}{(l)} = \\Omega + \\cos{(l)} and \\Omega (\\Omega + \\mathbf{B}{(l)}) = \\Omega (\\Omega + \\cos{(l)}) and - \\Omega (\\Omega + \\mathbf{B}{(l)}) + \\Omega + \\mathbf{B}{(l)} = - \\Omega (\\Omega + \\mathbf{B}{(l)}) + \\Omega + \\cos{(l)} and - \\Omega (\\Omega + \\cos{(l)}) + \\Omega + \\mathbf{B}{(l)} = - \\Omega (\\Omega + \\cos{(l)}) + \\Omega + \\cos{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True))))"], [["times", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True)))), Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True)))), Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True)))), Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{B}')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True)))), Symbol('\\\\Omega', commutative=True), cos(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{E},M)} = \\log{(M + \\mathbf{E})}, then derive \\frac{\\partial}{\\partial M} \\mathbf{J}_f{(\\mathbf{E},M)} = \\frac{1}{M + \\mathbf{E}}, then obtain \\frac{\\frac{\\partial}{\\partial M} \\log{(M + \\mathbf{E})}}{M} = \\frac{1}{M (M + \\mathbf{E})}", "derivation": "\\mathbf{J}_f{(\\mathbf{E},M)} = \\log{(M + \\mathbf{E})} and \\frac{\\partial}{\\partial M} \\mathbf{J}_f{(\\mathbf{E},M)} = \\frac{\\partial}{\\partial M} \\log{(M + \\mathbf{E})} and \\frac{\\partial}{\\partial M} \\mathbf{J}_f{(\\mathbf{E},M)} = \\frac{1}{M + \\mathbf{E}} and \\frac{\\partial}{\\partial M} \\log{(M + \\mathbf{E})} = \\frac{1}{M + \\mathbf{E}} and \\frac{\\frac{\\partial}{\\partial M} \\log{(M + \\mathbf{E})}}{M} = \\frac{1}{M (M + \\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True)), log(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(log(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["divide", 4, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(log(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{P},\\theta)} = \\cos{(\\frac{\\mathbf{P}}{\\theta})} and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)} = \\frac{\\mathbf{P}}{\\theta}, then obtain \\log{(\\rho_{b}{(\\mathbf{P},\\theta)} \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)})})} = \\log{(\\cos^{2}{(\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)})})}", "derivation": "\\rho_{b}{(\\mathbf{P},\\theta)} = \\cos{(\\frac{\\mathbf{P}}{\\theta})} and \\rho_{b}{(\\mathbf{P},\\theta)} \\cos{(\\frac{\\mathbf{P}}{\\theta})} = \\cos^{2}{(\\frac{\\mathbf{P}}{\\theta})} and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)} = \\frac{\\mathbf{P}}{\\theta} and \\log{(\\rho_{b}{(\\mathbf{P},\\theta)} \\cos{(\\frac{\\mathbf{P}}{\\theta})})} = \\log{(\\cos^{2}{(\\frac{\\mathbf{P}}{\\theta})})} and \\log{(\\rho_{b}{(\\mathbf{P},\\theta)} \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)})})} = \\log{(\\cos^{2}{(\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},\\theta)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))))"], [["times", 1, "cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))), Pow(cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Integer(2)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["log", 2], "Equality(log(Mul(Function('\\\\rho_b')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))))), log(Pow(cos(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(log(Mul(Function('\\\\rho_b')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True))))), log(Pow(cos(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(2))))"]]}, {"prompt": "Given J{(\\mathbf{J},U)} = \\sin{(U \\mathbf{J})}, then obtain U + \\sin^{\\mathbf{J}}{(U \\mathbf{J})} + 1 = U + \\sin^{\\mathbf{J}}{(U \\mathbf{J})} + J^{- \\mathbf{J}}{(\\mathbf{J},U)} \\sin^{\\mathbf{J}}{(U \\mathbf{J})}", "derivation": "J{(\\mathbf{J},U)} = \\sin{(U \\mathbf{J})} and J^{\\mathbf{J}}{(\\mathbf{J},U)} = \\sin^{\\mathbf{J}}{(U \\mathbf{J})} and U + J^{\\mathbf{J}}{(\\mathbf{J},U)} = U + \\sin^{\\mathbf{J}}{(U \\mathbf{J})} and 1 = J^{- \\mathbf{J}}{(\\mathbf{J},U)} \\sin^{\\mathbf{J}}{(U \\mathbf{J})} and U + J^{\\mathbf{J}}{(\\mathbf{J},U)} + 1 = U + J^{\\mathbf{J}}{(\\mathbf{J},U)} + J^{- \\mathbf{J}}{(\\mathbf{J},U)} \\sin^{\\mathbf{J}}{(U \\mathbf{J})} and U + \\sin^{\\mathbf{J}}{(U \\mathbf{J})} + 1 = U + \\sin^{\\mathbf{J}}{(U \\mathbf{J})} + J^{- \\mathbf{J}}{(\\mathbf{J},U)} \\sin^{\\mathbf{J}}{(U \\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 2, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('U', commutative=True), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 2, "Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 4, "Add(Symbol('U', commutative=True), Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1)), Add(Symbol('U', commutative=True), Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('U', commutative=True), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1)), Add(Symbol('U', commutative=True), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Function('J')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(sin(Mul(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given T{(i)} = \\cos{(\\log{(i)})}, then obtain - \\frac{i (\\frac{d}{d i} T{(i)} - \\frac{\\sin{(\\log{(i)})}}{i})}{\\sin{(\\log{(i)})}} = 2", "derivation": "T{(i)} = \\cos{(\\log{(i)})} and \\frac{d}{d i} T{(i)} = \\frac{d}{d i} \\cos{(\\log{(i)})} and \\frac{d}{d i} T{(i)} + \\frac{d}{d i} \\cos{(\\log{(i)})} = 2 \\frac{d}{d i} \\cos{(\\log{(i)})} and \\frac{\\frac{d}{d i} T{(i)} + \\frac{d}{d i} \\cos{(\\log{(i)})}}{\\frac{d}{d i} \\cos{(\\log{(i)})}} = 2 and - \\frac{i (\\frac{d}{d i} T{(i)} - \\frac{\\sin{(\\log{(i)})}}{i})}{\\sin{(\\log{(i)})}} = 2", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('i', commutative=True)), cos(log(Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('T')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["divide", 3, "Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('T')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Pow(Derivative(cos(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(2))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Add(Derivative(Function('T')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), sin(log(Symbol('i', commutative=True))))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))), Integer(2))"]]}, {"prompt": "Given \\theta_{2}{(z^{*},G)} = G z^{*} and W{(z^{*},G)} = \\frac{\\partial}{\\partial z^{*}} \\theta_{2}{(z^{*},G)}, then derive W{(z^{*},G)} = G, then obtain \\frac{\\partial}{\\partial z^{*}} G z^{*} = G", "derivation": "\\theta_{2}{(z^{*},G)} = G z^{*} and \\frac{\\partial}{\\partial z^{*}} \\theta_{2}{(z^{*},G)} = \\frac{\\partial}{\\partial z^{*}} G z^{*} and W{(z^{*},G)} = \\frac{\\partial}{\\partial z^{*}} \\theta_{2}{(z^{*},G)} and W{(z^{*},G)} = \\frac{\\partial}{\\partial z^{*}} G z^{*} and W{(z^{*},G)} = G and \\frac{\\partial}{\\partial z^{*}} G z^{*} = G", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Derivative(Mul(Symbol('G', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('G', commutative=True))"]]}, {"prompt": "Given E{(v_{2})} = \\cos{(v_{2})} and \\hat{x}{(v_{2})} = \\int E{(v_{2})} \\cos{(v_{2})} dv_{2}, then obtain \\frac{d}{d v_{2}} \\int \\cos^{2}{(v_{2})} dv_{2} = \\frac{d}{d v_{2}} \\hat{x}{(v_{2})}", "derivation": "E{(v_{2})} = \\cos{(v_{2})} and E^{2}{(v_{2})} = E{(v_{2})} \\cos{(v_{2})} and \\int E^{2}{(v_{2})} dv_{2} = \\int E{(v_{2})} \\cos{(v_{2})} dv_{2} and \\hat{x}{(v_{2})} = \\int E{(v_{2})} \\cos{(v_{2})} dv_{2} and \\int E^{2}{(v_{2})} dv_{2} = \\hat{x}{(v_{2})} and \\int \\cos^{2}{(v_{2})} dv_{2} = \\hat{x}{(v_{2})} and \\frac{d}{d v_{2}} \\int \\cos^{2}{(v_{2})} dv_{2} = \\frac{d}{d v_{2}} \\hat{x}{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["times", 1, "Function('E')(Symbol('v_2', commutative=True))"], "Equality(Pow(Function('E')(Symbol('v_2', commutative=True)), Integer(2)), Mul(Function('E')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["integrate", 2, "Symbol('v_2', commutative=True)"], "Equality(Integral(Pow(Function('E')(Symbol('v_2', commutative=True)), Integer(2)), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Function('E')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('v_2', commutative=True)), Integral(Mul(Function('E')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Pow(Function('E')(Symbol('v_2', commutative=True)), Integer(2)), Tuple(Symbol('v_2', commutative=True))), Function('\\\\hat{x}')(Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Pow(cos(Symbol('v_2', commutative=True)), Integer(2)), Tuple(Symbol('v_2', commutative=True))), Function('\\\\hat{x}')(Symbol('v_2', commutative=True)))"], [["differentiate", 6, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Pow(cos(Symbol('v_2', commutative=True)), Integer(2)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Function('\\\\hat{x}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(m,F_{c})} = \\frac{F_{c}}{m}, then obtain F_{c} - \\operatorname{y^{\\prime}}^{2}{(m,F_{c})} = F_{c} - \\frac{F_{c} \\operatorname{y^{\\prime}}{(m,F_{c})}}{m}", "derivation": "\\operatorname{y^{\\prime}}{(m,F_{c})} = \\frac{F_{c}}{m} and \\operatorname{y^{\\prime}}^{2}{(m,F_{c})} = \\frac{F_{c} \\operatorname{y^{\\prime}}{(m,F_{c})}}{m} and - F_{c} + \\operatorname{y^{\\prime}}^{2}{(m,F_{c})} = - F_{c} + \\frac{F_{c} \\operatorname{y^{\\prime}}{(m,F_{c})}}{m} and F_{c} - \\operatorname{y^{\\prime}}^{2}{(m,F_{c})} = F_{c} - \\frac{F_{c} \\operatorname{y^{\\prime}}{(m,F_{c})}}{m}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)), Integer(2)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True))))"], [["minus", 2, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)), Integer(2)))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(A_{y})} = \\log{(e^{A_{y}})} and Q{(P_{g},\\mathbf{H},A)} = \\frac{A^{P_{g}}}{\\mathbf{H}}, then obtain \\frac{\\partial}{\\partial P_{g}} Q{(P_{g},\\mathbf{H},A)} = \\frac{A^{P_{g}} \\log{(A)}}{\\mathbf{H}}", "derivation": "\\hat{X}{(A_{y})} = \\log{(e^{A_{y}})} and Q{(P_{g},\\mathbf{H},A)} = \\frac{A^{P_{g}}}{\\mathbf{H}} and Q{(P_{g},\\mathbf{H},A)} + \\hat{X}{(A_{y})} = \\frac{A^{P_{g}}}{\\mathbf{H}} + \\hat{X}{(A_{y})} and Q{(P_{g},\\mathbf{H},A)} + \\log{(e^{A_{y}})} = \\frac{A^{P_{g}}}{\\mathbf{H}} + \\log{(e^{A_{y}})} and \\frac{\\partial}{\\partial P_{g}} (Q{(P_{g},\\mathbf{H},A)} + \\log{(e^{A_{y}})}) = \\frac{\\partial}{\\partial P_{g}} (\\frac{A^{P_{g}}}{\\mathbf{H}} + \\log{(e^{A_{y}})}) and \\frac{\\partial}{\\partial P_{g}} Q{(P_{g},\\mathbf{H},A)} = \\frac{A^{P_{g}} \\log{(A)}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('A_y', commutative=True)), log(exp(Symbol('A_y', commutative=True))))"], ["get_premise", "Equality(Function('Q')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["add", 2, "Function('\\\\hat{X}')(Symbol('A_y', commutative=True))"], "Equality(Add(Function('Q')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A', commutative=True)), Function('\\\\hat{X}')(Symbol('A_y', commutative=True))), Add(Mul(Pow(Symbol('A', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Function('\\\\hat{X}')(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('Q')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A', commutative=True)), log(exp(Symbol('A_y', commutative=True)))), Add(Mul(Pow(Symbol('A', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), log(exp(Symbol('A_y', commutative=True)))))"], [["differentiate", 4, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Add(Function('Q')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A', commutative=True)), log(exp(Symbol('A_y', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('A', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), log(exp(Symbol('A_y', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('Q')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Mul(Pow(Symbol('A', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), log(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(a)} = e^{e^{a}} and S{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)}, then obtain \\frac{S{(\\hat{p}_0)}}{\\operatorname{t_{2}}{(a)} - e^{a}} = \\frac{\\sin{(\\hat{p}_0)}}{\\operatorname{t_{2}}{(a)} - e^{a}}", "derivation": "\\operatorname{t_{2}}{(a)} = e^{e^{a}} and \\operatorname{t_{2}}{(a)} - e^{a} = - e^{a} + e^{e^{a}} and S{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\frac{S{(\\hat{p}_0)}}{- e^{a} + e^{e^{a}}} = \\frac{\\sin{(\\hat{p}_0)}}{- e^{a} + e^{e^{a}}} and \\frac{S{(\\hat{p}_0)}}{\\operatorname{t_{2}}{(a)} - e^{a}} = \\frac{\\sin{(\\hat{p}_0)}}{\\operatorname{t_{2}}{(a)} - e^{a}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('a', commutative=True)), exp(exp(Symbol('a', commutative=True))))"], [["minus", 1, "exp(Symbol('a', commutative=True))"], "Equality(Add(Function('t_2')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), exp(exp(Symbol('a', commutative=True)))))"], ["get_premise", "Equality(Function('S')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), exp(exp(Symbol('a', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), exp(exp(Symbol('a', commutative=True)))), Integer(-1)), Function('S')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), exp(exp(Symbol('a', commutative=True)))), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Function('t_2')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Integer(-1)), Function('S')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Add(Function('t_2')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\psi^*)} = \\log{(\\psi^*)}, then obtain \\frac{d}{d \\psi^*} \\frac{e^{\\operatorname{F_{g}}{(\\psi^*)}}}{2 \\log{(\\psi^*)}} = \\frac{d}{d \\psi^*} \\frac{\\psi^*}{2 \\log{(\\psi^*)}}", "derivation": "\\operatorname{F_{g}}{(\\psi^*)} = \\log{(\\psi^*)} and \\operatorname{F_{g}}{(\\psi^*)} + \\log{(\\psi^*)} = 2 \\log{(\\psi^*)} and e^{\\operatorname{F_{g}}{(\\psi^*)}} = \\psi^* and \\frac{e^{\\operatorname{F_{g}}{(\\psi^*)}}}{\\operatorname{F_{g}}{(\\psi^*)} + \\log{(\\psi^*)}} = \\frac{\\psi^*}{\\operatorname{F_{g}}{(\\psi^*)} + \\log{(\\psi^*)}} and \\frac{e^{\\operatorname{F_{g}}{(\\psi^*)}}}{2 \\log{(\\psi^*)}} = \\frac{\\psi^*}{2 \\log{(\\psi^*)}} and \\frac{d}{d \\psi^*} \\frac{e^{\\operatorname{F_{g}}{(\\psi^*)}}}{2 \\log{(\\psi^*)}} = \\frac{d}{d \\psi^*} \\frac{\\psi^*}{2 \\log{(\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "log(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), log(Symbol('\\\\psi^*', commutative=True))))"], [["exp", 1], "Equality(exp(Function('F_g')(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))"], [["divide", 3, "Add(Function('F_g')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Pow(Add(Function('F_g')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Integer(-1)), exp(Function('F_g')(Symbol('\\\\psi^*', commutative=True)))), Mul(Symbol('\\\\psi^*', commutative=True), Pow(Add(Function('F_g')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Rational(1, 2), exp(Function('F_g')(Symbol('\\\\psi^*', commutative=True))), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Symbol('\\\\psi^*', commutative=True), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["differentiate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), exp(Function('F_g')(Symbol('\\\\psi^*', commutative=True))), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Symbol('\\\\psi^*', commutative=True), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(l,\\mathbf{r})} = \\mathbf{r} + l and Z{(l,\\mathbf{r})} = \\mathbf{r} + l, then obtain \\frac{T{(l,\\mathbf{r})}}{\\mathbf{r} l + a + \\frac{l^{2}}{2}} = \\frac{Z{(l,\\mathbf{r})}}{\\mathbf{r} l + a + \\frac{l^{2}}{2}}", "derivation": "T{(l,\\mathbf{r})} = \\mathbf{r} + l and Z{(l,\\mathbf{r})} = \\mathbf{r} + l and T{(l,\\mathbf{r})} = Z{(l,\\mathbf{r})} and \\int Z{(l,\\mathbf{r})} dl = \\int (\\mathbf{r} + l) dl and \\frac{T{(l,\\mathbf{r})}}{\\int Z{(l,\\mathbf{r})} dl} = \\frac{Z{(l,\\mathbf{r})}}{\\int Z{(l,\\mathbf{r})} dl} and \\frac{T{(l,\\mathbf{r})}}{\\int (\\mathbf{r} + l) dl} = \\frac{Z{(l,\\mathbf{r})}}{\\int (\\mathbf{r} + l) dl} and \\frac{T{(l,\\mathbf{r})}}{\\mathbf{r} l + a + \\frac{l^{2}}{2}} = \\frac{Z{(l,\\mathbf{r})}}{\\mathbf{r} l + a + \\frac{l^{2}}{2}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)))"], ["renaming_premise", "Equality(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('T')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["divide", 3, "Integral(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Function('T')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(-1))), Mul(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('T')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(-1))), Mul(Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))), Integer(-1)), Function('T')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Symbol('a', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))), Integer(-1)), Function('Z')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(Q)} = \\sin{(Q)}, then obtain \\psi^{*}{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}{(Q)} + \\psi^{*}^{Q}{(Q)}) dQ = \\psi^{*}{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}^{Q}{(Q)} + \\sin{(Q)}) dQ", "derivation": "\\psi^{*}{(Q)} = \\sin{(Q)} and \\psi^{*}^{Q}{(Q)} = \\sin^{Q}{(Q)} and \\psi^{*}{(Q)} + \\psi^{*}^{Q}{(Q)} = \\psi^{*}^{Q}{(Q)} + \\sin{(Q)} and \\psi^{*}{(Q)} + \\sin^{Q}{(Q)} = \\sin{(Q)} + \\sin^{Q}{(Q)} and \\int (\\psi^{*}{(Q)} + \\psi^{*}^{Q}{(Q)}) dQ = \\int (\\psi^{*}^{Q}{(Q)} + \\sin{(Q)}) dQ and \\sin{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}{(Q)} + \\psi^{*}^{Q}{(Q)}) dQ = \\sin{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}^{Q}{(Q)} + \\sin{(Q)}) dQ and \\psi^{*}{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}{(Q)} + \\psi^{*}^{Q}{(Q)}) dQ = \\psi^{*}{(Q)} + \\sin^{Q}{(Q)} + \\int (\\psi^{*}^{Q}{(Q)} + \\sin{(Q)}) dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["add", 1, "Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Add(Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Add(sin(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["add", 5, "Add(sin(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Add(sin(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(sin(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Add(Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Add(Pow(Function('\\\\psi^*')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(H,n_{1})} = H + n_{1}, then obtain \\int (2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} \\mathbf{M}{(H,n_{1})}) dn_{1} = \\int (2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} (H + n_{1})) dn_{1}", "derivation": "\\mathbf{M}{(H,n_{1})} = H + n_{1} and \\frac{\\partial}{\\partial n_{1}} \\mathbf{M}{(H,n_{1})} = \\frac{\\partial}{\\partial n_{1}} (H + n_{1}) and \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} \\mathbf{M}{(H,n_{1})} = \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} (H + n_{1}) and 2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} \\mathbf{M}{(H,n_{1})} = 2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} (H + n_{1}) and \\int (2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} \\mathbf{M}{(H,n_{1})}) dn_{1} = \\int (2 \\mathbf{M}{(H,n_{1})} + \\frac{\\partial}{\\partial n_{1}} (H + n_{1})) dn_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('H', commutative=True), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Derivative(Add(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["add", 3, "Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))), Derivative(Add(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('n_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))), Integral(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('H', commutative=True), Symbol('n_1', commutative=True))), Derivative(Add(Symbol('H', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given c{(\\dot{\\mathbf{r}},r,z)} = \\frac{r - z}{\\dot{\\mathbf{r}}} and \\hat{X}{(r,z)} = - r + z, then obtain c{(\\dot{\\mathbf{r}},r,z)} + \\frac{\\hat{X}{(r,z)}}{2 r - 2 z} = \\frac{- r + z}{2 r - 2 z} + c{(\\dot{\\mathbf{r}},r,z)}", "derivation": "c{(\\dot{\\mathbf{r}},r,z)} = \\frac{r - z}{\\dot{\\mathbf{r}}} and \\hat{X}{(r,z)} = - r + z and \\frac{\\hat{X}{(r,z)}}{2 r - 2 z} = \\frac{- r + z}{2 r - 2 z} and \\frac{\\hat{X}{(r,z)}}{2 r - 2 z} + \\frac{r - z}{\\dot{\\mathbf{r}}} = \\frac{- r + z}{2 r - 2 z} + \\frac{r - z}{\\dot{\\mathbf{r}}} and c{(\\dot{\\mathbf{r}},r,z)} + \\frac{\\hat{X}{(r,z)}}{2 r - 2 z} = \\frac{- r + z}{2 r - 2 z} + c{(\\dot{\\mathbf{r}},r,z)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('r', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('z', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\hat{X}')(Symbol('r', commutative=True), Symbol('z', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1))))"], [["add", 3, "Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], "Equality(Add(Mul(Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\hat{X}')(Symbol('r', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))), Add(Mul(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\hat{X}')(Symbol('r', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Integer(-1))), Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(F_{c})} = \\sin{(F_{c})} and \\chi{(F_{c})} = - F_{c} + \\sin{(F_{c})}, then obtain \\iint \\sin^{2}{(\\chi{(F_{c})})} dF_{c} dF_{c} = \\iint - \\sin{(F_{c} - \\hat{x}_0{(F_{c})})} \\sin{(\\chi{(F_{c})})} dF_{c} dF_{c}", "derivation": "\\hat{x}_0{(F_{c})} = \\sin{(F_{c})} and \\chi{(F_{c})} = - F_{c} + \\sin{(F_{c})} and \\chi{(F_{c})} = - F_{c} + \\hat{x}_0{(F_{c})} and \\sin{(\\chi{(F_{c})})} = - \\sin{(F_{c} - \\hat{x}_0{(F_{c})})} and \\sin^{2}{(\\chi{(F_{c})})} = - \\sin{(F_{c} - \\hat{x}_0{(F_{c})})} \\sin{(\\chi{(F_{c})})} and \\int \\sin^{2}{(\\chi{(F_{c})})} dF_{c} = \\int - \\sin{(F_{c} - \\hat{x}_0{(F_{c})})} \\sin{(\\chi{(F_{c})})} dF_{c} and \\iint \\sin^{2}{(\\chi{(F_{c})})} dF_{c} dF_{c} = \\iint - \\sin{(F_{c} - \\hat{x}_0{(F_{c})})} \\sin{(\\chi{(F_{c})})} dF_{c} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('F_c', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('F_c', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))))"], [["sin", 3], "Equality(sin(Function('\\\\chi')(Symbol('F_c', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)))))))"], [["times", 4, "sin(Function('\\\\chi')(Symbol('F_c', commutative=True)))"], "Equality(Pow(sin(Function('\\\\chi')(Symbol('F_c', commutative=True))), Integer(2)), Mul(Integer(-1), sin(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))))), sin(Function('\\\\chi')(Symbol('F_c', commutative=True)))))"], [["integrate", 5, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(sin(Function('\\\\chi')(Symbol('F_c', commutative=True))), Integer(2)), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))))), sin(Function('\\\\chi')(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True))))"], [["integrate", 6, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(sin(Function('\\\\chi')(Symbol('F_c', commutative=True))), Integer(2)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))))), sin(Function('\\\\chi')(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(H,z)} = - H + \\sin{(z)} and \\dot{\\mathbf{r}}{(H,z)} = - H + \\sin{(z)}, then obtain \\sin^{H}{(\\hat{H}_{\\lambda}{(H,z)} - \\sin{(z)})} = \\sin^{H}{(\\dot{\\mathbf{r}}{(H,z)} - \\sin{(z)})}", "derivation": "\\hat{H}_{\\lambda}{(H,z)} = - H + \\sin{(z)} and \\dot{\\mathbf{r}}{(H,z)} = - H + \\sin{(z)} and \\hat{H}_{\\lambda}{(H,z)} = \\dot{\\mathbf{r}}{(H,z)} and \\hat{H}_{\\lambda}{(H,z)} - \\sin{(z)} = \\dot{\\mathbf{r}}{(H,z)} - \\sin{(z)} and \\sin{(\\hat{H}_{\\lambda}{(H,z)} - \\sin{(z)})} = \\sin{(\\dot{\\mathbf{r}}{(H,z)} - \\sin{(z)})} and \\sin^{H}{(\\hat{H}_{\\lambda}{(H,z)} - \\sin{(z)})} = \\sin^{H}{(\\dot{\\mathbf{r}}{(H,z)} - \\sin{(z)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('z', commutative=True)))"], [["minus", 3, "sin(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))), sin(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(sin(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))), Symbol('H', commutative=True)), Pow(sin(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(z^{*})} = e^{z^{*}} and L{(\\mathbf{E},v_{x})} = e^{\\mathbf{E}^{v_{x}}}, then obtain \\operatorname{F_{H}}{(z^{*})} + \\frac{L{(\\mathbf{E},v_{x})}}{z^{*}} = \\operatorname{F_{H}}{(z^{*})} + \\frac{e^{\\mathbf{E}^{v_{x}}}}{z^{*}}", "derivation": "\\operatorname{F_{H}}{(z^{*})} = e^{z^{*}} and L{(\\mathbf{E},v_{x})} = e^{\\mathbf{E}^{v_{x}}} and \\frac{L{(\\mathbf{E},v_{x})}}{z^{*}} = \\frac{e^{\\mathbf{E}^{v_{x}}}}{z^{*}} and e^{z^{*}} + \\frac{L{(\\mathbf{E},v_{x})}}{z^{*}} = e^{z^{*}} + \\frac{e^{\\mathbf{E}^{v_{x}}}}{z^{*}} and \\operatorname{F_{H}}{(z^{*})} + \\frac{L{(\\mathbf{E},v_{x})}}{z^{*}} = \\operatorname{F_{H}}{(z^{*})} + \\frac{e^{\\mathbf{E}^{v_{x}}}}{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('z^*', commutative=True)), exp(Symbol('z^*', commutative=True)))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True)), exp(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True))))"], [["divide", 2, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True)))))"], [["add", 3, "exp(Symbol('z^*', commutative=True))"], "Equality(Add(exp(Symbol('z^*', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True)))), Add(exp(Symbol('z^*', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('F_H')(Symbol('z^*', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True)))), Add(Function('F_H')(Symbol('z^*', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('v_x', commutative=True))))))"]]}, {"prompt": "Given b{(r_{0},\\sigma_p)} = \\sigma_p r_{0}, then derive \\frac{\\partial}{\\partial \\sigma_p} b{(r_{0},\\sigma_p)} = r_{0}, then obtain \\frac{\\frac{\\partial}{\\partial \\sigma_p} \\sigma_p r_{0}}{m} = \\frac{r_{0}}{m}", "derivation": "b{(r_{0},\\sigma_p)} = \\sigma_p r_{0} and \\frac{\\partial}{\\partial \\sigma_p} b{(r_{0},\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p r_{0} and \\frac{\\partial}{\\partial \\sigma_p} b{(r_{0},\\sigma_p)} = r_{0} and \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p r_{0} = r_{0} and \\frac{\\frac{\\partial}{\\partial \\sigma_p} \\sigma_p r_{0}}{m} = \\frac{r_{0}}{m}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('r_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('r_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('r_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["divide", 4, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(C_{2})} = \\sin{(\\cos{(C_{2})})} and \\phi_{2}{(C_{2})} = \\varepsilon_{0}{(C_{2})} + \\int \\sin{(\\cos{(C_{2})})} dC_{2}, then obtain \\varepsilon_{0}{(C_{2})} + \\int \\varepsilon_{0}{(C_{2})} dC_{2} = \\phi_{2}{(C_{2})}", "derivation": "\\varepsilon_{0}{(C_{2})} = \\sin{(\\cos{(C_{2})})} and \\int \\varepsilon_{0}{(C_{2})} dC_{2} = \\int \\sin{(\\cos{(C_{2})})} dC_{2} and \\varepsilon_{0}{(C_{2})} + \\int \\varepsilon_{0}{(C_{2})} dC_{2} = \\varepsilon_{0}{(C_{2})} + \\int \\sin{(\\cos{(C_{2})})} dC_{2} and \\phi_{2}{(C_{2})} = \\varepsilon_{0}{(C_{2})} + \\int \\sin{(\\cos{(C_{2})})} dC_{2} and \\varepsilon_{0}{(C_{2})} + \\int \\varepsilon_{0}{(C_{2})} dC_{2} = \\phi_{2}{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), sin(cos(Symbol('C_2', commutative=True))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(sin(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["add", 2, "Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Integral(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Integral(sin(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('C_2', commutative=True)), Add(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Integral(sin(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Integral(Function('\\\\varepsilon_0')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Function('\\\\phi_2')(Symbol('C_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} = C_{1} + \\sin{(V)} and \\operatorname{v_{x}}{(C_{1},V)} = C_{1} + \\sin{(V)}, then obtain (\\int \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} dC_{1})^{V} = (\\int \\operatorname{v_{x}}{(C_{1},V)} dC_{1})^{V}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} = C_{1} + \\sin{(V)} and \\operatorname{v_{x}}{(C_{1},V)} = C_{1} + \\sin{(V)} and \\operatorname{v_{x}}{(C_{1},V)} = \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} and \\int \\operatorname{v_{x}}{(C_{1},V)} dC_{1} = \\int (C_{1} + \\sin{(V)}) dC_{1} and \\int \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} dC_{1} = \\int (C_{1} + \\sin{(V)}) dC_{1} and (\\int \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} dC_{1})^{V} = (\\int (C_{1} + \\sin{(V)}) dC_{1})^{V} and (\\int \\operatorname{V_{\\mathbf{B}}}{(C_{1},V)} dC_{1})^{V} = (\\int \\operatorname{v_{x}}{(C_{1},V)} dC_{1})^{V}", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Add(Symbol('C_1', commutative=True), sin(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Add(Symbol('C_1', commutative=True), sin(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_x')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Symbol('C_1', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Symbol('C_1', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["power", 5, "Symbol('V', commutative=True)"], "Equality(Pow(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('V', commutative=True)), Pow(Integral(Add(Symbol('C_1', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('V', commutative=True)), Pow(Integral(Function('v_x')(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\Psi^{\\dagger},B)} = \\log{(B^{\\Psi^{\\dagger}})} and \\hat{H}{(\\Psi^{\\dagger},B)} = B^{\\Psi^{\\dagger}}, then obtain \\frac{\\partial}{\\partial B} \\log{(\\hat{H}{(\\Psi^{\\dagger},B)})} = \\frac{\\partial}{\\partial B} \\log{(B^{\\Psi^{\\dagger}})}", "derivation": "\\operatorname{C_{2}}{(\\Psi^{\\dagger},B)} = \\log{(B^{\\Psi^{\\dagger}})} and \\frac{\\partial}{\\partial B} \\operatorname{C_{2}}{(\\Psi^{\\dagger},B)} = \\frac{\\partial}{\\partial B} \\log{(B^{\\Psi^{\\dagger}})} and \\hat{H}{(\\Psi^{\\dagger},B)} = B^{\\Psi^{\\dagger}} and \\operatorname{C_{2}}{(\\Psi^{\\dagger},B)} = \\log{(\\hat{H}{(\\Psi^{\\dagger},B)})} and \\frac{\\partial}{\\partial B} \\log{(\\hat{H}{(\\Psi^{\\dagger},B)})} = \\frac{\\partial}{\\partial B} \\log{(B^{\\Psi^{\\dagger}})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('B', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('B', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('C_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True)), log(Function('\\\\hat{H}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(log(Function('\\\\hat{H}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('B', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(s)} = e^{s}, then obtain (- s (Z{(s)} e^{- s})^{s})^{s} = (- s)^{s}", "derivation": "Z{(s)} = e^{s} and Z{(s)} e^{- s} = 1 and - s Z{(s)} e^{- s} = - s and (Z{(s)} e^{- s})^{s} = 1 and (Z{(s)} e^{- s Z{(s)} e^{- s}})^{s} = 1 and (Z{(s)} e^{- s Z{(s)} e^{- s Z{(s)} e^{- s}}})^{s} = 1 and - s (Z{(s)} e^{- s Z{(s)} e^{- s Z{(s)} e^{- s}}})^{s} = - s and - s (Z{(s)} e^{- s})^{s} = - s and (- s (Z{(s)} e^{- s})^{s})^{s} = (- s)^{s}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["divide", 1, "exp(Symbol('s', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Integer(1))"], [["times", 2, "Mul(Integer(-1), Symbol('s', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Mul(Integer(-1), Symbol('s', commutative=True)))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))))), Symbol('s', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))))))), Symbol('s', commutative=True)), Integer(1))"], [["times", 6, "Mul(Integer(-1), Symbol('s', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('s', commutative=True), Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True), Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))))))), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Integer(-1), Symbol('s', commutative=True), Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))"], [["power", 8, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Pow(Mul(Function('Z')(Symbol('s', commutative=True)), exp(Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\sigma_x)} = e^{\\sin{(\\sigma_x)}}, then obtain \\frac{\\int \\frac{\\hat{p}{(\\sigma_x)}}{\\sin{(\\sigma_x)}} d\\sigma_x}{\\sigma_x} = \\frac{\\int \\frac{e^{\\sin{(\\sigma_x)}}}{\\sin{(\\sigma_x)}} d\\sigma_x}{\\sigma_x}", "derivation": "\\hat{p}{(\\sigma_x)} = e^{\\sin{(\\sigma_x)}} and \\frac{\\hat{p}{(\\sigma_x)}}{\\sin{(\\sigma_x)}} = \\frac{e^{\\sin{(\\sigma_x)}}}{\\sin{(\\sigma_x)}} and \\int \\frac{\\hat{p}{(\\sigma_x)}}{\\sin{(\\sigma_x)}} d\\sigma_x = \\int \\frac{e^{\\sin{(\\sigma_x)}}}{\\sin{(\\sigma_x)}} d\\sigma_x and \\frac{\\int \\frac{\\hat{p}{(\\sigma_x)}}{\\sin{(\\sigma_x)}} d\\sigma_x}{\\sigma_x} = \\frac{\\int \\frac{e^{\\sin{(\\sigma_x)}}}{\\sin{(\\sigma_x)}} d\\sigma_x}{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\sigma_x', commutative=True)), exp(sin(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Mul(exp(sin(Symbol('\\\\sigma_x', commutative=True))), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{p}')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(exp(sin(Symbol('\\\\sigma_x', commutative=True))), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Integral(Mul(Function('\\\\hat{p}')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Integral(Mul(exp(sin(Symbol('\\\\sigma_x', commutative=True))), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given a{(\\omega)} = e^{\\omega}, then obtain ((a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega})^{\\omega} a^{- \\omega}{(\\omega)} e^{\\omega})^{\\omega} e^{- \\omega} = e^{- \\omega} (e^{\\omega})^{\\omega}", "derivation": "a{(\\omega)} = e^{\\omega} and a^{\\omega}{(\\omega)} = (e^{\\omega})^{\\omega} and a^{\\omega}{(\\omega)} e^{- \\omega} = e^{- \\omega} (e^{\\omega})^{\\omega} and e^{\\omega} = a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega} and a^{\\omega}{(\\omega)} = (a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega})^{\\omega} and (a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega})^{\\omega} = (e^{\\omega})^{\\omega} and a^{\\omega}{(\\omega)} = ((a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega})^{\\omega} a^{- \\omega}{(\\omega)} e^{\\omega})^{\\omega} and ((a^{- \\omega}{(\\omega)} e^{\\omega} (e^{\\omega})^{\\omega})^{\\omega} a^{- \\omega}{(\\omega)} e^{\\omega})^{\\omega} e^{- \\omega} = e^{- \\omega} (e^{\\omega})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["divide", 2, "exp(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["divide", 2, "Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], "Equality(exp(Symbol('\\\\omega', commutative=True)), Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Pow(Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Mul(Pow(Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 7], "Equality(Mul(Pow(Mul(Pow(Mul(Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Function('a')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given m{(x)} = \\cos{(x)}, then obtain 0 = - \\frac{1}{\\cos{(x)}} + \\frac{1}{m{(x)}}", "derivation": "m{(x)} = \\cos{(x)} and \\frac{m{(x)}}{x} = \\frac{\\cos{(x)}}{x} and \\frac{m{(x)}}{\\cos{(x)}} = 1 and \\frac{1}{\\cos{(x)}} = \\frac{1}{m{(x)}} and 0 = - \\frac{1}{\\cos{(x)}} + \\frac{1}{m{(x)}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["divide", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('m')(Symbol('x', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), cos(Symbol('x', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), cos(Symbol('x', commutative=True)))"], "Equality(Mul(Function('m')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 3, "Function('m')(Symbol('x', commutative=True))"], "Equality(Pow(cos(Symbol('x', commutative=True)), Integer(-1)), Pow(Function('m')(Symbol('x', commutative=True)), Integer(-1)))"], [["minus", 4, "Pow(cos(Symbol('x', commutative=True)), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Pow(Function('m')(Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(c)} = \\cos{(c)} and \\operatorname{v_{x}}{(c)} = \\frac{d}{d c} \\cos{(c)}, then obtain (\\frac{d}{d c} \\operatorname{F_{N}}{(c)})^{c} = \\operatorname{v_{x}}^{c}{(c)}", "derivation": "\\operatorname{F_{N}}{(c)} = \\cos{(c)} and \\frac{d}{d c} \\operatorname{F_{N}}{(c)} = \\frac{d}{d c} \\cos{(c)} and (\\frac{d}{d c} \\operatorname{F_{N}}{(c)})^{c} = (\\frac{d}{d c} \\cos{(c)})^{c} and \\operatorname{v_{x}}{(c)} = \\frac{d}{d c} \\cos{(c)} and (\\frac{d}{d c} \\operatorname{F_{N}}{(c)})^{c} = \\operatorname{v_{x}}^{c}{(c)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Function('F_N')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('c', commutative=True)), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Derivative(Function('F_N')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Function('v_x')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\omega{(c)} = e^{c}, then derive \\frac{d}{d c} \\omega{(c)} = e^{c}, then obtain (- e^{c} + \\cos{(\\frac{d}{d c} e^{c})})^{c} = (- e^{c} + \\cos{(e^{c})})^{c}", "derivation": "\\omega{(c)} = e^{c} and \\frac{d}{d c} \\omega{(c)} = \\frac{d}{d c} e^{c} and \\frac{d}{d c} \\omega{(c)} = e^{c} and \\frac{d}{d c} e^{c} = e^{c} and \\cos{(\\frac{d}{d c} e^{c})} = \\cos{(e^{c})} and \\cos{(\\frac{d}{d c} e^{c})} - \\frac{d}{d c} \\omega{(c)} = \\cos{(e^{c})} - \\frac{d}{d c} \\omega{(c)} and - e^{c} + \\cos{(\\frac{d}{d c} e^{c})} = - e^{c} + \\cos{(e^{c})} and (- e^{c} + \\cos{(\\frac{d}{d c} e^{c})})^{c} = (- e^{c} + \\cos{(e^{c})})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), exp(Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), exp(Symbol('c', commutative=True)))"], [["cos", 4], "Equality(cos(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), cos(exp(Symbol('c', commutative=True))))"], [["minus", 5, "Derivative(Function('\\\\omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Add(cos(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))), Add(cos(exp(Symbol('c', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('c', commutative=True))), cos(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('c', commutative=True))), cos(exp(Symbol('c', commutative=True)))))"], [["power", 7, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), exp(Symbol('c', commutative=True))), cos(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))), Symbol('c', commutative=True)), Pow(Add(Mul(Integer(-1), exp(Symbol('c', commutative=True))), cos(exp(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{P})} = \\log{(e^{\\mathbf{P}})}, then obtain e^{- \\mathbf{P}} \\int (\\mathbf{P} + \\mathbf{r}{(\\mathbf{P})}) d\\mathbf{P} = e^{- \\mathbf{P}} \\int (\\mathbf{P} + \\log{(e^{\\mathbf{P}})}) d\\mathbf{P}", "derivation": "\\mathbf{r}{(\\mathbf{P})} = \\log{(e^{\\mathbf{P}})} and \\mathbf{P} + \\mathbf{r}{(\\mathbf{P})} = \\mathbf{P} + \\log{(e^{\\mathbf{P}})} and \\int (\\mathbf{P} + \\mathbf{r}{(\\mathbf{P})}) d\\mathbf{P} = \\int (\\mathbf{P} + \\log{(e^{\\mathbf{P}})}) d\\mathbf{P} and e^{- \\mathbf{P}} \\int (\\mathbf{P} + \\mathbf{r}{(\\mathbf{P})}) d\\mathbf{P} = e^{- \\mathbf{P}} \\int (\\mathbf{P} + \\log{(e^{\\mathbf{P}})}) d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True)), log(exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), log(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), log(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 3, "exp(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), log(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(U)} = \\log{(U)}, then derive \\int \\operatorname{L_{\\varepsilon}}{(U)} dU = U \\log{(U)} - U + \\mathbf{v}, then obtain U \\log{(U)} - U + V_{\\mathbf{B}} + \\tilde{g} = U \\log{(U)} - U + V_{\\mathbf{B}} + \\mathbf{v}", "derivation": "\\operatorname{L_{\\varepsilon}}{(U)} = \\log{(U)} and \\int \\operatorname{L_{\\varepsilon}}{(U)} dU = \\int \\log{(U)} dU and \\int \\operatorname{L_{\\varepsilon}}{(U)} dU = U \\log{(U)} - U + \\mathbf{v} and \\int \\log{(U)} dU = U \\log{(U)} - U + \\mathbf{v} and V_{\\mathbf{B}} + \\int \\log{(U)} dU = U \\log{(U)} - U + V_{\\mathbf{B}} + \\mathbf{v} and U \\log{(U)} - U + V_{\\mathbf{B}} + \\tilde{g} = U \\log{(U)} - U + V_{\\mathbf{B}} + \\mathbf{v}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)} = \\mathbf{S}^{\\theta_2}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)}}{\\mathbf{S}} d\\theta_2 d\\mathbf{S} = \\int \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\mathbf{S}^{\\theta_2}}{\\mathbf{S}} d\\theta_2 d\\mathbf{S}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)} = \\mathbf{S}^{\\theta_2} and \\frac{\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)}}{\\mathbf{S}} = \\frac{\\mathbf{S}^{\\theta_2}}{\\mathbf{S}} and \\int \\frac{\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)}}{\\mathbf{S}} d\\theta_2 = \\int \\frac{\\mathbf{S}^{\\theta_2}}{\\mathbf{S}} d\\theta_2 and \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)}}{\\mathbf{S}} d\\theta_2 = \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\mathbf{S}^{\\theta_2}}{\\mathbf{S}} d\\theta_2 and \\int \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\operatorname{v_{2}}{(\\mathbf{S},\\theta_2)}}{\\mathbf{S}} d\\theta_2 d\\mathbf{S} = \\int \\frac{\\partial}{\\partial \\mathbf{S}} \\int \\frac{\\mathbf{S}^{\\theta_2}}{\\mathbf{S}} d\\theta_2 d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given G{(b)} = e^{b}, then obtain e^{\\frac{G{(b)} - e^{b}}{\\int (G{(b)} - e^{b}) db}} = 1", "derivation": "G{(b)} = e^{b} and G{(b)} - e^{b} = 0 and \\int (G{(b)} - e^{b}) db = \\int 0 db and \\frac{G{(b)} - e^{b}}{\\int (G{(b)} - e^{b}) db} = 0 and \\frac{G{(b)} - e^{b}}{\\int 0 db} = 0 and e^{\\frac{G{(b)} - e^{b}}{\\int 0 db}} = 1 and e^{\\frac{G{(b)} - e^{b}}{\\int (G{(b)} - e^{b}) db}} = 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["minus", 1, "exp(Symbol('b', commutative=True))"], "Equality(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integral(Integer(0), Tuple(Symbol('b', commutative=True))))"], [["divide", 2, "Integral(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True)))"], "Equality(Mul(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Pow(Integral(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Pow(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Integer(-1))), Integer(0))"], [["exp", 5], "Equality(exp(Mul(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Pow(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Integer(-1)))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(exp(Mul(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Pow(Integral(Add(Function('G')(Symbol('b', commutative=True)), Mul(Integer(-1), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integer(-1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(B)} = e^{B}, then obtain (\\frac{- B + \\operatorname{F_{g}}{(B)}}{\\operatorname{F_{g}}{(B)}})^{B} = (\\frac{- B + e^{B}}{\\operatorname{F_{g}}{(B)}})^{B}", "derivation": "\\operatorname{F_{g}}{(B)} = e^{B} and - B + \\operatorname{F_{g}}{(B)} = - B + e^{B} and \\frac{- B + \\operatorname{F_{g}}{(B)}}{\\operatorname{F_{g}}{(B)}} = \\frac{- B + e^{B}}{\\operatorname{F_{g}}{(B)}} and (\\frac{- B + \\operatorname{F_{g}}{(B)}}{\\operatorname{F_{g}}{(B)}})^{B} = (\\frac{- B + e^{B}}{\\operatorname{F_{g}}{(B)}})^{B}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('F_g')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))))"], [["divide", 2, "Function('F_g')(Symbol('B', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('F_g')(Symbol('B', commutative=True))), Pow(Function('F_g')(Symbol('B', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Pow(Function('F_g')(Symbol('B', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('F_g')(Symbol('B', commutative=True))), Pow(Function('F_g')(Symbol('B', commutative=True)), Integer(-1))), Symbol('B', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Pow(Function('F_g')(Symbol('B', commutative=True)), Integer(-1))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(C)} = e^{C}, then derive \\frac{d}{d C} \\mathbf{J}{(C)} = e^{C}, then obtain e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + \\frac{d^{2}}{d C^{2}} \\mathbf{J}{(C)} = e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + e^{C}", "derivation": "\\mathbf{J}{(C)} = e^{C} and \\frac{d}{d C} \\mathbf{J}{(C)} = \\frac{d}{d C} e^{C} and \\mathbf{J}{(C)} e^{C} + \\frac{d}{d C} \\mathbf{J}{(C)} = \\mathbf{J}{(C)} e^{C} + \\frac{d}{d C} e^{C} and \\frac{d}{d C} \\mathbf{J}{(C)} = e^{C} and \\mathbf{J}{(C)} = \\frac{d}{d C} \\mathbf{J}{(C)} and e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + \\frac{d^{2}}{d C^{2}} \\mathbf{J}{(C)} = e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + \\frac{d}{d C} e^{C} and e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + \\frac{d^{2}}{d C^{2}} \\mathbf{J}{(C)} = e^{C} \\frac{d}{d C} \\mathbf{J}{(C)} + e^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 2, "Mul(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Add(Mul(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), exp(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(exp(Symbol('C', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2)))), Add(Mul(exp(Symbol('C', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(exp(Symbol('C', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2)))), Add(Mul(exp(Symbol('C', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), exp(Symbol('C', commutative=True))))"]]}, {"prompt": "Given r{(L,\\omega)} = \\sin{(L + \\omega)}, then derive \\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} = \\cos{(L + \\omega)}, then obtain \\cos{(\\frac{\\partial}{\\partial L} (\\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} + 1))} = \\cos{(\\frac{\\partial}{\\partial L} (\\cos{(L + \\omega)} + 1))}", "derivation": "r{(L,\\omega)} = \\sin{(L + \\omega)} and \\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\sin{(L + \\omega)} and \\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} = \\cos{(L + \\omega)} and \\frac{\\partial}{\\partial \\omega} \\sin{(L + \\omega)} = \\cos{(L + \\omega)} and \\frac{\\partial}{\\partial \\omega} \\sin{(L + \\omega)} + 1 = \\cos{(L + \\omega)} + 1 and \\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} + 1 = \\cos{(L + \\omega)} + 1 and \\frac{\\partial}{\\partial L} (\\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} + 1) = \\frac{\\partial}{\\partial L} (\\cos{(L + \\omega)} + 1) and \\cos{(\\frac{\\partial}{\\partial L} (\\frac{\\partial}{\\partial \\omega} r{(L,\\omega)} + 1))} = \\cos{(\\frac{\\partial}{\\partial L} (\\cos{(L + \\omega)} + 1))}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(sin(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Add(cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Add(cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(1)))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["cos", 7], "Equality(cos(Derivative(Add(Derivative(Function('r')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1)))), cos(Derivative(Add(cos(Add(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{f},t_{2})} = \\frac{\\mathbf{f}}{t_{2}}, then obtain \\frac{\\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\operatorname{F_{H}}{(\\mathbf{f},t_{2})} dt_{2}}{\\mathbf{f}} = \\frac{\\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\frac{\\mathbf{f}}{t_{2}} dt_{2}}{\\mathbf{f}}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{f},t_{2})} = \\frac{\\mathbf{f}}{t_{2}} and \\int \\operatorname{F_{H}}{(\\mathbf{f},t_{2})} dt_{2} = \\int \\frac{\\mathbf{f}}{t_{2}} dt_{2} and \\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\operatorname{F_{H}}{(\\mathbf{f},t_{2})} dt_{2} = \\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\frac{\\mathbf{f}}{t_{2}} dt_{2} and \\frac{\\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\operatorname{F_{H}}{(\\mathbf{f},t_{2})} dt_{2}}{\\mathbf{f}} = \\frac{\\phi_{1}{(q,\\mathbf{f},t_{2})} + \\int \\frac{\\mathbf{f}}{t_{2}} dt_{2}}{\\mathbf{f}}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True))))"], [["add", 2, "Function('\\\\phi_1')(Symbol('q', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('q', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Integral(Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('q', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True)))))"], [["divide", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Function('\\\\phi_1')(Symbol('q', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Integral(Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Function('\\\\phi_1')(Symbol('q', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}_0{(C,I,\\lambda)} = \\frac{C + \\lambda}{I}, then obtain \\frac{d}{d C} 0 = \\frac{\\partial}{\\partial C} ((- C - \\lambda + \\frac{C + \\lambda}{I}) \\hat{p}_0{(C,I,\\lambda)} - (- C - \\lambda + \\hat{p}_0{(C,I,\\lambda)}) \\hat{p}_0{(C,I,\\lambda)})", "derivation": "\\hat{p}_0{(C,I,\\lambda)} = \\frac{C + \\lambda}{I} and - C - \\lambda + \\hat{p}_0{(C,I,\\lambda)} = - C - \\lambda + \\frac{C + \\lambda}{I} and (- C - \\lambda + \\hat{p}_0{(C,I,\\lambda)}) \\hat{p}_0{(C,I,\\lambda)} = (- C - \\lambda + \\frac{C + \\lambda}{I}) \\hat{p}_0{(C,I,\\lambda)} and 0 = (- C - \\lambda + \\frac{C + \\lambda}{I}) \\hat{p}_0{(C,I,\\lambda)} - (- C - \\lambda + \\hat{p}_0{(C,I,\\lambda)}) \\hat{p}_0{(C,I,\\lambda)} and \\frac{d}{d C} 0 = \\frac{\\partial}{\\partial C} ((- C - \\lambda + \\frac{C + \\lambda}{I}) \\hat{p}_0{(C,I,\\lambda)} - (- C - \\lambda + \\hat{p}_0{(C,I,\\lambda)}) \\hat{p}_0{(C,I,\\lambda)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["minus", 1, "Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["times", 2, "Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["minus", 3, "Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Integer(0), Add(Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\hat{p}_0')(Symbol('C', commutative=True), Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = \\delta^{V_{\\mathbf{B}}}, then obtain (- 4 \\delta + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})}) (- 2 \\delta^{V_{\\mathbf{B}}} + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})}) = 0", "derivation": "\\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = \\delta^{V_{\\mathbf{B}}} and - \\delta - \\delta^{V_{\\mathbf{B}}} + \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = - \\delta and - \\delta + \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = - \\delta + \\delta^{V_{\\mathbf{B}}} and - \\delta - \\delta^{V_{\\mathbf{B}}} + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = - \\delta + \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} and - \\delta - \\delta^{V_{\\mathbf{B}}} + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = - \\delta + \\delta^{V_{\\mathbf{B}}} and - 2 \\delta^{V_{\\mathbf{B}}} + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})} = 0 and (- 4 \\delta + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})}) (- 2 \\delta^{V_{\\mathbf{B}}} + 2 \\operatorname{M_{E}}{(\\delta,V_{\\mathbf{B}})}) = 0", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(0))"], [["times", 6, "Add(Mul(Integer(-1), Integer(4), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(4), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given E{(\\varphi)} = \\cos{(\\sin{(\\varphi)})} and \\operatorname{F_{N}}{(C_{d})} = \\cos{(\\sin{(C_{d})})}, then obtain - E^{\\varphi}{(\\varphi)} - \\operatorname{F_{N}}{(C_{d})} = - E^{\\varphi}{(\\varphi)} - \\cos{(\\sin{(C_{d})})}", "derivation": "E{(\\varphi)} = \\cos{(\\sin{(\\varphi)})} and \\operatorname{F_{N}}{(C_{d})} = \\cos{(\\sin{(C_{d})})} and \\operatorname{F_{N}}{(C_{d})} + \\cos^{\\varphi}{(\\sin{(\\varphi)})} = \\cos{(\\sin{(C_{d})})} + \\cos^{\\varphi}{(\\sin{(\\varphi)})} and E^{\\varphi}{(\\varphi)} + \\operatorname{F_{N}}{(C_{d})} = E^{\\varphi}{(\\varphi)} + \\cos{(\\sin{(C_{d})})} and - E^{\\varphi}{(\\varphi)} - \\operatorname{F_{N}}{(C_{d})} = - E^{\\varphi}{(\\varphi)} - \\cos{(\\sin{(C_{d})})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\varphi', commutative=True)), cos(sin(Symbol('\\\\varphi', commutative=True))))"], ["get_premise", "Equality(Function('F_N')(Symbol('C_d', commutative=True)), cos(sin(Symbol('C_d', commutative=True))))"], [["add", 2, "Pow(cos(sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('C_d', commutative=True)), Pow(cos(sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), Add(cos(sin(Symbol('C_d', commutative=True))), Pow(cos(sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Pow(Function('E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Function('F_N')(Symbol('C_d', commutative=True))), Add(Pow(Function('E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), cos(sin(Symbol('C_d', commutative=True)))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Function('F_N')(Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), cos(sin(Symbol('C_d', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(u,q)} = \\frac{u}{q}, then derive \\frac{\\partial}{\\partial q} \\mathbf{r}{(u,q)} - \\frac{u}{q} + \\frac{u}{q^{2}} = - \\frac{u}{q}, then obtain - \\frac{\\frac{\\partial}{\\partial q} \\frac{u}{q} - \\frac{u}{q} + \\frac{u}{q^{2}}}{\\frac{\\partial}{\\partial q} \\frac{u}{q}} = \\frac{u}{q \\frac{\\partial}{\\partial q} \\frac{u}{q}}", "derivation": "\\mathbf{r}{(u,q)} = \\frac{u}{q} and \\frac{\\partial}{\\partial q} \\mathbf{r}{(u,q)} = \\frac{\\partial}{\\partial q} \\frac{u}{q} and - \\frac{\\partial}{\\partial q} \\frac{u}{q} + \\frac{\\partial}{\\partial q} \\mathbf{r}{(u,q)} - \\frac{u}{q} = - \\frac{u}{q} and \\frac{\\partial}{\\partial q} \\mathbf{r}{(u,q)} - \\frac{u}{q} + \\frac{u}{q^{2}} = - \\frac{u}{q} and - \\frac{\\frac{\\partial}{\\partial q} \\mathbf{r}{(u,q)} - \\frac{u}{q} + \\frac{u}{q^{2}}}{\\frac{\\partial}{\\partial q} \\frac{u}{q}} = \\frac{u}{q \\frac{\\partial}{\\partial q} \\frac{u}{q}} and - \\frac{\\frac{\\partial}{\\partial q} \\frac{u}{q} - \\frac{u}{q} + \\frac{u}{q^{2}}}{\\frac{\\partial}{\\partial q} \\frac{u}{q}} = \\frac{u}{q \\frac{\\partial}{\\partial q} \\frac{u}{q}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 2, "Add(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{r}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{r}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Add(Derivative(Function('\\\\mathbf{r}')(Symbol('u', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Symbol('u', commutative=True))), Pow(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True), Pow(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Add(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Symbol('u', commutative=True))), Pow(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True), Pow(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(F_{c})} = \\cos{(F_{c})}, then obtain 1 = \\operatorname{a^{\\dagger}}^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(F_{c})}", "derivation": "\\operatorname{a^{\\dagger}}{(F_{c})} = \\cos{(F_{c})} and \\operatorname{a^{\\dagger}}^{F_{c}}{(F_{c})} = \\cos^{F_{c}}{(F_{c})} and \\frac{\\operatorname{a^{\\dagger}}^{F_{c}}{(F_{c})}}{F_{c}} = \\frac{\\cos^{F_{c}}{(F_{c})}}{F_{c}} and 1 = \\operatorname{a^{\\dagger}}^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["divide", 2, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\nabla{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} and \\psi{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}}, then obtain 2 \\nabla{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} + 2 \\nabla{(t_{1},\\dot{y})} - \\psi{(t_{1},\\dot{y})}", "derivation": "\\nabla{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} and 2 \\nabla{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} + \\nabla{(t_{1},\\dot{y})} and \\psi{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} and \\psi{(t_{1},\\dot{y})} = \\nabla{(t_{1},\\dot{y})} and 2 \\psi{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} + \\psi{(t_{1},\\dot{y})} and 0 = t_{1}^{\\dot{y}} - \\psi{(t_{1},\\dot{y})} and 2 \\nabla{(t_{1},\\dot{y})} = t_{1}^{\\dot{y}} + 2 \\nabla{(t_{1},\\dot{y})} - \\psi{(t_{1},\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Integer(2), Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 5, "Mul(Integer(2), Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Integer(0), Add(Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 6, "Mul(Integer(2), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Function('\\\\psi')(Symbol('t_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given t{(\\sigma_x,s)} = \\sigma_x^{s}, then derive (\\frac{\\sigma_x \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,s)}}{t{(\\sigma_x,s)}} + \\log{(t{(\\sigma_x,s)})}) t^{\\sigma_x}{(\\sigma_x,s)} = (s + \\log{(\\sigma_x^{s})}) (\\sigma_x^{s})^{\\sigma_x}, then obtain (\\frac{\\sigma_x \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,s)}}{t{(\\sigma_x,s)}} + \\log{(t{(\\sigma_x,s)})}) (\\sigma_x^{s})^{\\sigma_x} = (s + \\log{(\\sigma_x^{s})}) (\\sigma_x^{s})^{\\sigma_x}", "derivation": "t{(\\sigma_x,s)} = \\sigma_x^{s} and t^{\\sigma_x}{(\\sigma_x,s)} = (\\sigma_x^{s})^{\\sigma_x} and \\frac{\\partial}{\\partial \\sigma_x} t^{\\sigma_x}{(\\sigma_x,s)} = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x^{s})^{\\sigma_x} and (\\frac{\\sigma_x \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,s)}}{t{(\\sigma_x,s)}} + \\log{(t{(\\sigma_x,s)})}) t^{\\sigma_x}{(\\sigma_x,s)} = (s + \\log{(\\sigma_x^{s})}) (\\sigma_x^{s})^{\\sigma_x} and (\\frac{\\sigma_x \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,s)}}{t{(\\sigma_x,s)}} + \\log{(t{(\\sigma_x,s)})}) (\\sigma_x^{s})^{\\sigma_x} = (s + \\log{(\\sigma_x^{s})}) (\\sigma_x^{s})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Pow(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Derivative(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), log(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)))), Pow(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Mul(Add(Symbol('s', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)))), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Derivative(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), log(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)))), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Mul(Add(Symbol('s', commutative=True), log(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)))), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\theta{(L)} = e^{L}, then obtain L \\sin{(4 e^{2 \\theta{(L)}})} = L \\sin{((e^{\\theta{(L)}} + e^{e^{L}})^{2})}", "derivation": "\\theta{(L)} = e^{L} and e^{\\theta{(L)}} = e^{e^{L}} and 2 e^{\\theta{(L)}} = e^{\\theta{(L)}} + e^{e^{L}} and 4 e^{2 \\theta{(L)}} = (e^{\\theta{(L)}} + e^{e^{L}})^{2} and \\sin{(4 e^{2 \\theta{(L)}})} = \\sin{((e^{\\theta{(L)}} + e^{e^{L}})^{2})} and L \\sin{(4 e^{2 \\theta{(L)}})} = L \\sin{((e^{\\theta{(L)}} + e^{e^{L}})^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\theta')(Symbol('L', commutative=True))), exp(exp(Symbol('L', commutative=True))))"], [["add", 2, "exp(Function('\\\\theta')(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(2), exp(Function('\\\\theta')(Symbol('L', commutative=True)))), Add(exp(Function('\\\\theta')(Symbol('L', commutative=True))), exp(exp(Symbol('L', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), exp(Mul(Integer(2), Function('\\\\theta')(Symbol('L', commutative=True))))), Pow(Add(exp(Function('\\\\theta')(Symbol('L', commutative=True))), exp(exp(Symbol('L', commutative=True)))), Integer(2)))"], [["sin", 4], "Equality(sin(Mul(Integer(4), exp(Mul(Integer(2), Function('\\\\theta')(Symbol('L', commutative=True)))))), sin(Pow(Add(exp(Function('\\\\theta')(Symbol('L', commutative=True))), exp(exp(Symbol('L', commutative=True)))), Integer(2))))"], [["times", 5, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), sin(Mul(Integer(4), exp(Mul(Integer(2), Function('\\\\theta')(Symbol('L', commutative=True))))))), Mul(Symbol('L', commutative=True), sin(Pow(Add(exp(Function('\\\\theta')(Symbol('L', commutative=True))), exp(exp(Symbol('L', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given \\theta_{1}{(\\tilde{g}^*)} = \\sin{(\\cos{(\\tilde{g}^*)})}, then obtain 3 \\theta_{1}{(\\tilde{g}^*)} - \\sin{(\\cos{(\\tilde{g}^*)})} - 1 = 2 \\sin{(\\cos{(\\tilde{g}^*)})} - 1", "derivation": "\\theta_{1}{(\\tilde{g}^*)} = \\sin{(\\cos{(\\tilde{g}^*)})} and \\theta_{1}{(\\tilde{g}^*)} - 1 = \\sin{(\\cos{(\\tilde{g}^*)})} - 1 and 2 \\theta_{1}{(\\tilde{g}^*)} - 1 = \\theta_{1}{(\\tilde{g}^*)} + \\sin{(\\cos{(\\tilde{g}^*)})} - 1 and 2 \\theta_{1}{(\\tilde{g}^*)} - 1 = 2 \\sin{(\\cos{(\\tilde{g}^*)})} - 1 and 2 \\theta_{1}{(\\tilde{g}^*)} - \\sin{(\\cos{(\\tilde{g}^*)})} - 1 = \\sin{(\\cos{(\\tilde{g}^*)})} - 1 and 2 \\theta_{1}{(\\tilde{g}^*)} - 1 = 3 \\theta_{1}{(\\tilde{g}^*)} - \\sin{(\\cos{(\\tilde{g}^*)})} - 1 and 3 \\theta_{1}{(\\tilde{g}^*)} - \\sin{(\\cos{(\\tilde{g}^*)})} - 1 = 2 \\sin{(\\cos{(\\tilde{g}^*)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Add(sin(cos(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"], [["add", 2, "Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Add(Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Add(Mul(Integer(2), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"], [["add", 4, "Mul(Integer(-1), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)), Add(sin(cos(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Add(Mul(Integer(3), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(3), Function('\\\\theta_1')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)), Add(Mul(Integer(2), sin(cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\hat{p}{(\\phi_1,\\omega)} = - \\omega + \\phi_1 and \\operatorname{C_{2}}{(\\phi_1,\\omega)} = (- 2 \\omega + 3 \\phi_1 + \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1}, then obtain (- 3 \\omega + 4 \\phi_1)^{\\phi_1} = (- \\omega + 2 \\phi_1 + 2 \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1}", "derivation": "\\hat{p}{(\\phi_1,\\omega)} = - \\omega + \\phi_1 and - \\omega + 2 \\phi_1 + 2 \\hat{p}{(\\phi_1,\\omega)} = - 2 \\omega + 3 \\phi_1 + \\hat{p}{(\\phi_1,\\omega)} and (- \\omega + 2 \\phi_1 + 2 \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1} = (- 2 \\omega + 3 \\phi_1 + \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1} and \\operatorname{C_{2}}{(\\phi_1,\\omega)} = (- 2 \\omega + 3 \\phi_1 + \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1} and \\operatorname{C_{2}}{(\\phi_1,\\omega)} = (- 3 \\omega + 4 \\phi_1)^{\\phi_1} and \\operatorname{C_{2}}{(\\phi_1,\\omega)} = (- \\omega + 2 \\phi_1 + 2 \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1} and (- 3 \\omega + 4 \\phi_1)^{\\phi_1} = (- \\omega + 2 \\phi_1 + 2 \\hat{p}{(\\phi_1,\\omega)})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(3), Symbol('\\\\phi_1', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(3), Symbol('\\\\phi_1', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(3), Symbol('\\\\phi_1', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('C_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(3), Symbol('\\\\omega', commutative=True)), Mul(Integer(4), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('C_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Add(Mul(Integer(-1), Integer(3), Symbol('\\\\omega', commutative=True)), Mul(Integer(4), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(H,\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{H}{\\mathbf{B}}, then derive 1 - \\hat{H}{(H,\\mathbf{B})} = - \\hat{H}{(H,\\mathbf{B})} - \\frac{1}{\\mathbf{B}^{2} \\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}}, then obtain 1 = - \\frac{1}{\\mathbf{B}^{2} \\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}}", "derivation": "\\hat{H}{(H,\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{H}{\\mathbf{B}} and \\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})} = \\frac{\\partial^{2}}{\\partial H\\partial \\mathbf{B}} \\frac{H}{\\mathbf{B}} and 1 = \\frac{\\frac{\\partial^{2}}{\\partial H\\partial \\mathbf{B}} \\frac{H}{\\mathbf{B}}}{\\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}} and 1 - \\hat{H}{(H,\\mathbf{B})} = - \\hat{H}{(H,\\mathbf{B})} + \\frac{\\frac{\\partial^{2}}{\\partial H\\partial \\mathbf{B}} \\frac{H}{\\mathbf{B}}}{\\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}} and 1 - \\hat{H}{(H,\\mathbf{B})} = - \\hat{H}{(H,\\mathbf{B})} - \\frac{1}{\\mathbf{B}^{2} \\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}} and 1 = - \\frac{1}{\\mathbf{B}^{2} \\frac{\\partial}{\\partial H} \\hat{H}{(H,\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["minus", 3, "Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)))))"], [["minus", 5, "Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\pi{(\\dot{x},V)} = \\frac{\\partial}{\\partial \\dot{x}} (V + \\dot{x}), then derive \\pi{(\\dot{x},V)} = 1, then obtain \\cos{(\\frac{\\partial^{2}}{\\partial V\\partial \\dot{x}} (V + \\dot{x}))} = \\cos{(\\frac{d}{d V} 1)}", "derivation": "\\pi{(\\dot{x},V)} = \\frac{\\partial}{\\partial \\dot{x}} (V + \\dot{x}) and \\pi{(\\dot{x},V)} = 1 and \\frac{\\partial}{\\partial \\dot{x}} (V + \\dot{x}) = 1 and \\frac{\\partial^{2}}{\\partial V\\partial \\dot{x}} (V + \\dot{x}) = \\frac{d}{d V} 1 and \\cos{(\\frac{\\partial^{2}}{\\partial V\\partial \\dot{x}} (V + \\dot{x}))} = \\cos{(\\frac{d}{d V} 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\dot{x}', commutative=True), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\pi')(Symbol('\\\\dot{x}', commutative=True), Symbol('V', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True), Integer(1)))), cos(Derivative(Integer(1), Tuple(Symbol('V', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(Z,U)} = U + Z, then derive \\frac{\\partial}{\\partial Z} h{(Z,U)} - 1 = 0, then obtain - U - Z + \\int (\\frac{\\partial}{\\partial Z} (U + Z) - 1) dU = - U - Z + \\int 0 dU", "derivation": "h{(Z,U)} = U + Z and - U - Z + h{(Z,U)} = 0 and - 2 U - Z + h{(Z,U)} = - U and \\frac{\\partial}{\\partial Z} (- 2 U - Z + h{(Z,U)}) = \\frac{d}{d Z} - U and \\frac{\\partial}{\\partial Z} h{(Z,U)} - 1 = 0 and \\int (\\frac{\\partial}{\\partial Z} h{(Z,U)} - 1) dU = \\int 0 dU and \\int (\\frac{\\partial}{\\partial Z} (U + Z) - 1) dU = \\int 0 dU and - U - Z + \\int (\\frac{\\partial}{\\partial Z} (U + Z) - 1) dU = - U - Z + \\int 0 dU", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('Z', commutative=True)))"], [["minus", 1, "Add(Symbol('U', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('U', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Derivative(Function('h')(Symbol('Z', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Add(Derivative(Add(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["minus", 7, "Add(Symbol('U', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integral(Add(Derivative(Add(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integral(Integer(0), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given A{(u)} = \\log{(\\log{(u)})}, then obtain 2 A^{u}{(u)} - \\log{(\\log{(u)})} = - \\log{(\\log{(u)})} + 2 \\log{(\\log{(u)})}^{u}", "derivation": "A{(u)} = \\log{(\\log{(u)})} and A^{u}{(u)} = \\log{(\\log{(u)})}^{u} and A^{u}{(u)} - \\log{(\\log{(u)})} = - \\log{(\\log{(u)})} + \\log{(\\log{(u)})}^{u} and 2 A^{u}{(u)} - \\log{(\\log{(u)})} = A^{u}{(u)} - \\log{(\\log{(u)})} + \\log{(\\log{(u)})}^{u} and 2 A^{u}{(u)} - \\log{(\\log{(u)})} = - \\log{(\\log{(u)})} + 2 \\log{(\\log{(u)})}^{u}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('u', commutative=True)), log(log(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(log(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["minus", 2, "log(log(Symbol('u', commutative=True)))"], "Equality(Add(Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), log(log(Symbol('u', commutative=True))))), Add(Mul(Integer(-1), log(log(Symbol('u', commutative=True)))), Pow(log(log(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["add", 2, "Add(Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), log(log(Symbol('u', commutative=True)))))"], "Equality(Add(Mul(Integer(2), Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(-1), log(log(Symbol('u', commutative=True))))), Add(Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), log(log(Symbol('u', commutative=True)))), Pow(log(log(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Pow(Function('A')(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(-1), log(log(Symbol('u', commutative=True))))), Add(Mul(Integer(-1), log(log(Symbol('u', commutative=True)))), Mul(Integer(2), Pow(log(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given J{(S)} = \\cos{(\\sin{(S)})}, then obtain (\\int (J{(S)} - \\sin{(S)}) dS)^{S} = (\\int (- \\sin{(S)} + \\cos{(\\sin{(S)})}) dS)^{S}", "derivation": "J{(S)} = \\cos{(\\sin{(S)})} and J{(S)} - \\sin{(S)} = - \\sin{(S)} + \\cos{(\\sin{(S)})} and \\int (J{(S)} - \\sin{(S)}) dS = \\int (- \\sin{(S)} + \\cos{(\\sin{(S)})}) dS and (\\int (J{(S)} - \\sin{(S)}) dS)^{S} = (\\int (- \\sin{(S)} + \\cos{(\\sin{(S)})}) dS)^{S}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('S', commutative=True)), cos(sin(Symbol('S', commutative=True))))"], [["minus", 1, "sin(Symbol('S', commutative=True))"], "Equality(Add(Function('J')(Symbol('S', commutative=True)), Mul(Integer(-1), sin(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('S', commutative=True))), cos(sin(Symbol('S', commutative=True)))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Function('J')(Symbol('S', commutative=True)), Mul(Integer(-1), sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('S', commutative=True))), cos(sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Integral(Add(Function('J')(Symbol('S', commutative=True)), Mul(Integer(-1), sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), sin(Symbol('S', commutative=True))), cos(sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\rho_b,b)} = \\rho_b e^{b}, then obtain \\rho_b + \\int \\frac{\\hat{H}{(\\rho_b,b)}}{b} d\\rho_b - \\frac{1}{b} = \\rho_b + \\int \\frac{\\rho_b e^{b}}{b} d\\rho_b - \\frac{1}{b}", "derivation": "\\hat{H}{(\\rho_b,b)} = \\rho_b e^{b} and \\frac{\\hat{H}{(\\rho_b,b)}}{b} = \\frac{\\rho_b e^{b}}{b} and \\int \\frac{\\hat{H}{(\\rho_b,b)}}{b} d\\rho_b = \\int \\frac{\\rho_b e^{b}}{b} d\\rho_b and \\int \\frac{\\hat{H}{(\\rho_b,b)}}{b} d\\rho_b - \\frac{1}{b} = \\int \\frac{\\rho_b e^{b}}{b} d\\rho_b - \\frac{1}{b} and \\rho_b + \\int \\frac{\\hat{H}{(\\rho_b,b)}}{b} d\\rho_b - \\frac{1}{b} = \\rho_b + \\int \\frac{\\rho_b e^{b}}{b} d\\rho_b - \\frac{1}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('b', commutative=True))))"], [["divide", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), exp(Symbol('b', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), exp(Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Pow(Symbol('b', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), exp(Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)))))"], [["add", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Integral(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)))), Add(Symbol('\\\\rho_b', commutative=True), Integral(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), exp(Symbol('b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{M}{(c_{0},J,s)} = J - c_{0} + s, then obtain \\int (c_{0} + \\mathbf{M}{(c_{0},J,s)} + 1) dJ = \\int (J + s + 1) dJ", "derivation": "\\mathbf{M}{(c_{0},J,s)} = J - c_{0} + s and \\mathbf{M}{(c_{0},J,s)} + 1 = J - c_{0} + s + 1 and c_{0} + \\mathbf{M}{(c_{0},J,s)} + 1 = J + s + 1 and \\int (c_{0} + \\mathbf{M}{(c_{0},J,s)} + 1) dJ = \\int (J + s + 1) dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True), Symbol('J', commutative=True), Symbol('s', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('s', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True), Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(1)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('s', commutative=True), Integer(1)))"], [["minus", 2, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True), Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(1)), Add(Symbol('J', commutative=True), Symbol('s', commutative=True), Integer(1)))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True), Symbol('J', commutative=True), Symbol('s', commutative=True)), Integer(1)), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), Symbol('s', commutative=True), Integer(1)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(q,\\Omega)} = \\int \\Omega^{q} d\\Omega and \\operatorname{z^{*}}{(q,\\Omega)} = (\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)}) (\\Omega^{q} + \\int \\Omega^{q} d\\Omega) - \\int \\Omega^{q} d\\Omega, then obtain - q \\operatorname{z^{*}}{(q,\\Omega)} = - q ((\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)})^{2} - \\int \\Omega^{q} d\\Omega)", "derivation": "\\operatorname{g_{\\varepsilon}}{(q,\\Omega)} = \\int \\Omega^{q} d\\Omega and \\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)} = \\Omega^{q} + \\int \\Omega^{q} d\\Omega and (\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)})^{2} = (\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)}) (\\Omega^{q} + \\int \\Omega^{q} d\\Omega) and \\operatorname{z^{*}}{(q,\\Omega)} = (\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)}) (\\Omega^{q} + \\int \\Omega^{q} d\\Omega) - \\int \\Omega^{q} d\\Omega and \\operatorname{z^{*}}{(q,\\Omega)} = (\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)})^{2} - \\int \\Omega^{q} d\\Omega and - q \\operatorname{z^{*}}{(q,\\Omega)} = - q ((\\Omega^{q} + \\operatorname{g_{\\varepsilon}}{(q,\\Omega)})^{2} - \\int \\Omega^{q} d\\Omega)", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["times", 2, "Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2)), Mul(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Mul(Integer(-1), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('z^*')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2)), Mul(Integer(-1), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["times", 5, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Function('z^*')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True), Add(Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2)), Mul(Integer(-1), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))))"]]}, {"prompt": "Given \\psi{(\\varphi^*)} = \\log{(\\varphi^*)}, then obtain - \\frac{\\varphi^* \\psi^{2}{(\\varphi^*)}}{\\log{(\\varphi^*)}} + \\frac{\\psi{(\\varphi^*)}}{\\log{(\\varphi^*)}} = - \\frac{\\varphi^* \\psi^{2}{(\\varphi^*)}}{\\log{(\\varphi^*)}} + 1", "derivation": "\\psi{(\\varphi^*)} = \\log{(\\varphi^*)} and \\varphi^* \\psi{(\\varphi^*)} = \\varphi^* \\log{(\\varphi^*)} and \\frac{\\psi{(\\varphi^*)}}{\\log{(\\varphi^*)}} = 1 and - \\varphi^* \\log{(\\varphi^*)} + \\frac{\\psi{(\\varphi^*)}}{\\log{(\\varphi^*)}} = - \\varphi^* \\log{(\\varphi^*)} + 1 and - \\varphi^* \\psi{(\\varphi^*)} + \\frac{\\psi{(\\varphi^*)}}{\\log{(\\varphi^*)}} = - \\varphi^* \\psi{(\\varphi^*)} + 1 and \\frac{\\varphi^* \\psi^{2}{(\\varphi^*)}}{\\log{(\\varphi^*)}} = \\varphi^* \\psi{(\\varphi^*)} and - \\frac{\\varphi^* \\psi^{2}{(\\varphi^*)}}{\\log{(\\varphi^*)}} + \\frac{\\psi{(\\varphi^*)}}{\\log{(\\varphi^*)}} = - \\frac{\\varphi^* \\psi^{2}{(\\varphi^*)}}{\\log{(\\varphi^*)}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 1, "log(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 3, "Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True))), Mul(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True))), Integer(1)))"], [["times", 3, "Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} = \\hat{x}_0 \\mathbf{E}, then obtain \\hat{x}_0 \\mathbf{E} + \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} + \\int \\hat{x}_0 \\mathbf{E} d\\hat{x}_0 = 2 \\hat{x}_0 \\mathbf{E} + \\int \\hat{x}_0 \\mathbf{E} d\\hat{x}_0", "derivation": "\\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} = \\hat{x}_0 \\mathbf{E} and \\hat{x}_0 \\mathbf{E} + \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} = 2 \\hat{x}_0 \\mathbf{E} and \\int \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} d\\hat{x}_0 = \\int \\hat{x}_0 \\mathbf{E} d\\hat{x}_0 and \\hat{x}_0 \\mathbf{E} + \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} + \\int \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} d\\hat{x}_0 = 2 \\hat{x}_0 \\mathbf{E} + \\int \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} d\\hat{x}_0 and \\hat{x}_0 \\mathbf{E} + \\mathbb{I}{(\\mathbf{E},\\hat{x}_0)} + \\int \\hat{x}_0 \\mathbf{E} d\\hat{x}_0 = 2 \\hat{x}_0 \\mathbf{E} + \\int \\hat{x}_0 \\mathbf{E} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["add", 2, "Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given c{(\\Psi)} = \\sin{(\\log{(\\Psi)})}, then obtain - \\Psi (c{(\\Psi)} - \\log{(\\Psi)}) \\log{(\\Psi)} - \\Psi = - \\Psi (- \\log{(\\Psi)} + \\sin{(\\log{(\\Psi)})}) \\log{(\\Psi)} - \\Psi", "derivation": "c{(\\Psi)} = \\sin{(\\log{(\\Psi)})} and c{(\\Psi)} - \\log{(\\Psi)} = - \\log{(\\Psi)} + \\sin{(\\log{(\\Psi)})} and - (c{(\\Psi)} - \\log{(\\Psi)}) \\log{(\\Psi)} = - (- \\log{(\\Psi)} + \\sin{(\\log{(\\Psi)})}) \\log{(\\Psi)} and - \\Psi (c{(\\Psi)} - \\log{(\\Psi)}) \\log{(\\Psi)} = - \\Psi (- \\log{(\\Psi)} + \\sin{(\\log{(\\Psi)})}) \\log{(\\Psi)} and - \\Psi (c{(\\Psi)} - \\log{(\\Psi)}) \\log{(\\Psi)} - \\Psi = - \\Psi (- \\log{(\\Psi)} + \\sin{(\\log{(\\Psi)})}) \\log{(\\Psi)} - \\Psi", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Psi', commutative=True)), sin(log(Symbol('\\\\Psi', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('c')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))), sin(log(Symbol('\\\\Psi', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('c')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))), sin(log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))))"], [["times", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Function('c')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))), sin(log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))))"], [["minus", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Function('c')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))), sin(log(Symbol('\\\\Psi', commutative=True)))), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(n_{1})} = \\cos{(n_{1})} and \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = \\cos{(n_{1})}, then obtain \\cos{(n_{1})} = 2 \\Psi_{\\lambda}{(n_{1})} - \\cos{(n_{1})}", "derivation": "\\Psi_{\\lambda}{(n_{1})} = \\cos{(n_{1})} and \\Psi_{\\lambda}{(n_{1})} - \\cos{(n_{1})} = 0 and \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = \\cos{(n_{1})} and \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = \\Psi_{\\lambda}{(n_{1})} and \\Psi_{\\lambda}{(n_{1})} - \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = 0 and 2 \\Psi_{\\lambda}{(n_{1})} - \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = \\Psi_{\\lambda}{(n_{1})} and \\operatorname{f_{\\mathbf{p}}}{(n_{1})} = 2 \\Psi_{\\lambda}{(n_{1})} - \\operatorname{f_{\\mathbf{p}}}{(n_{1})} and \\cos{(n_{1})} = 2 \\Psi_{\\lambda}{(n_{1})} - \\cos{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["minus", 1, "cos(Symbol('n_1', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True)), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)))), Integer(0))"], [["add", 5, "Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True))), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)))), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)), Add(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True))), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(cos(Symbol('n_1', commutative=True)), Add(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True))), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(A_{1},P_{e})} = \\frac{e^{P_{e}}}{A_{1}}, then derive \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} + \\frac{e^{P_{e}}}{A_{1}} = \\frac{e^{P_{e}}}{A_{1}} - \\frac{e^{P_{e}}}{A_{1}^{2}}, then obtain 0 = - \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} - \\frac{e^{P_{e}}}{A_{1}^{2}}", "derivation": "\\operatorname{m_{s}}{(A_{1},P_{e})} = \\frac{e^{P_{e}}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} = \\frac{\\partial}{\\partial A_{1}} \\frac{e^{P_{e}}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} + \\frac{e^{P_{e}}}{A_{1}} = \\frac{\\partial}{\\partial A_{1}} \\frac{e^{P_{e}}}{A_{1}} + \\frac{e^{P_{e}}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} + \\frac{e^{P_{e}}}{A_{1}} = \\frac{e^{P_{e}}}{A_{1}} - \\frac{e^{P_{e}}}{A_{1}^{2}} and 0 = - \\frac{\\partial}{\\partial A_{1}} \\operatorname{m_{s}}{(A_{1},P_{e})} - \\frac{e^{P_{e}}}{A_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["add", 2, "Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True)))"], "Equality(Add(Derivative(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True)))), Add(Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True)))), Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), exp(Symbol('P_e', commutative=True)))))"], [["minus", 4, "Add(Derivative(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), exp(Symbol('P_e', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('m_s')(Symbol('A_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), exp(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)} = \\hat{\\mathbf{r}} + \\mathbf{J}_P, then derive \\hat{\\mathbf{r}} + r_{0} = \\int \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} d\\hat{\\mathbf{r}}, then obtain \\log{(\\hat{\\mathbf{r}} + r_{0})} = \\log{(\\int \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} d\\hat{\\mathbf{r}})}", "derivation": "\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)} = \\hat{\\mathbf{r}} + \\mathbf{J}_P and 1 = \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} and \\int 1 d\\hat{\\mathbf{r}} = \\int \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} d\\hat{\\mathbf{r}} and \\hat{\\mathbf{r}} + r_{0} = \\int \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} d\\hat{\\mathbf{r}} and \\log{(\\hat{\\mathbf{r}} + r_{0})} = \\log{(\\int \\frac{\\hat{\\mathbf{r}} + \\mathbf{J}_P}{\\hat{p}_0{(\\hat{\\mathbf{r}},\\mathbf{J}_P)}} d\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r_0', commutative=True)), Integral(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["log", 4], "Equality(log(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r_0', commutative=True))), log(Integral(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(S)} = \\cos{(\\cos{(S)})} and \\Psi_{\\lambda}{(S)} = \\sin{(\\cos^{S}{(\\cos{(S)})})}, then obtain \\sin^{S}{(\\tilde{g}^{S}{(S)})} = \\Psi_{\\lambda}^{S}{(S)}", "derivation": "\\tilde{g}{(S)} = \\cos{(\\cos{(S)})} and \\tilde{g}^{S}{(S)} = \\cos^{S}{(\\cos{(S)})} and \\sin{(\\tilde{g}^{S}{(S)})} = \\sin{(\\cos^{S}{(\\cos{(S)})})} and \\sin^{S}{(\\tilde{g}^{S}{(S)})} = \\sin^{S}{(\\cos^{S}{(\\cos{(S)})})} and \\Psi_{\\lambda}{(S)} = \\sin{(\\cos^{S}{(\\cos{(S)})})} and \\sin^{S}{(\\tilde{g}^{S}{(S)})} = \\Psi_{\\lambda}^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), cos(cos(Symbol('S', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(cos(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), sin(Pow(cos(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(sin(Pow(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(sin(Pow(cos(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True)), sin(Pow(cos(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(sin(Pow(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(E)} = \\log{(E)}, then derive \\frac{d}{d E} \\mathbf{r}{(E)} = \\frac{1}{E}, then obtain \\frac{\\Omega \\frac{d}{d E} \\log{(E)}}{J{(F_{g},v_{t},\\Omega)}} + \\log{(E)} = \\log{(E)} + \\frac{\\Omega}{E J{(F_{g},v_{t},\\Omega)}}", "derivation": "\\mathbf{r}{(E)} = \\log{(E)} and \\frac{d}{d E} \\mathbf{r}{(E)} = \\frac{d}{d E} \\log{(E)} and \\frac{d}{d E} \\mathbf{r}{(E)} = \\frac{1}{E} and \\frac{\\Omega \\frac{d}{d E} \\mathbf{r}{(E)}}{J{(F_{g},v_{t},\\Omega)}} = \\frac{\\Omega}{E J{(F_{g},v_{t},\\Omega)}} and \\frac{\\Omega \\frac{d}{d E} \\mathbf{r}{(E)}}{J{(F_{g},v_{t},\\Omega)}} + \\mathbf{r}{(E)} = \\mathbf{r}{(E)} + \\frac{\\Omega}{E J{(F_{g},v_{t},\\Omega)}} and \\frac{\\Omega \\frac{d}{d E} \\log{(E)}}{J{(F_{g},v_{t},\\Omega)}} + \\log{(E)} = \\log{(E)} + \\frac{\\Omega}{E J{(F_{g},v_{t},\\Omega)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Symbol('E', commutative=True), Integer(-1)))"], [["divide", 3, "Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], [["add", 4, "Function('\\\\mathbf{r}')(Symbol('E', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Function('\\\\mathbf{r}')(Symbol('E', commutative=True))), Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Symbol('E', commutative=True))), Add(log(Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True), Pow(Function('J')(Symbol('F_g', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{D}{(E,\\sigma_x)} = E - \\sigma_x and \\operatorname{F_{c}}{(\\sigma_x)} = \\sigma_x, then derive \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\sigma_x)} = 1, then obtain \\frac{\\frac{\\partial}{\\partial E} (E - \\sigma_x)}{\\sigma_x} = \\frac{1}{\\sigma_x}", "derivation": "\\mathbf{D}{(E,\\sigma_x)} = E - \\sigma_x and \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\sigma_x)} = \\frac{\\partial}{\\partial E} (E - \\sigma_x) and \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\sigma_x)} = 1 and \\frac{\\partial}{\\partial E} (E - \\sigma_x) = 1 and \\operatorname{F_{c}}{(\\sigma_x)} = \\sigma_x and \\frac{\\partial}{\\partial E} (E - \\operatorname{F_{c}}{(\\sigma_x)}) = 1 and \\frac{\\frac{\\partial}{\\partial E} (E - \\operatorname{F_{c}}{(\\sigma_x)})}{\\operatorname{F_{c}}{(\\sigma_x)}} = \\frac{1}{\\operatorname{F_{c}}{(\\sigma_x)}} and \\frac{\\frac{\\partial}{\\partial E} (E - \\sigma_x)}{\\sigma_x} = \\frac{1}{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Function('F_c')(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(1))"], [["divide", 6, "Function('F_c')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Pow(Function('F_c')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Function('F_c')(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1)))), Pow(Function('F_c')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(G)} = e^{G} and q{(G)} = e^{G}, then obtain 0 = \\operatorname{n_{1}}{(G)} - q{(G)}", "derivation": "\\operatorname{n_{1}}{(G)} = e^{G} and q{(G)} = e^{G} and q{(G)} = \\operatorname{n_{1}}{(G)} and 0 = \\operatorname{n_{1}}{(G)} - q{(G)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('q')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('q')(Symbol('G', commutative=True)), Function('n_1')(Symbol('G', commutative=True)))"], [["minus", 3, "Function('q')(Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Function('n_1')(Symbol('G', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain 0 = (- \\bar{\\h}{(\\mathbf{A})} + \\cos{(\\mathbf{A})})^{2} \\cos{(\\mathbf{A})}", "derivation": "\\bar{\\h}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and 0 = - \\bar{\\h}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and 0 = (- \\bar{\\h}{(\\mathbf{A})} + \\cos{(\\mathbf{A})})^{2} and 0 = (- \\bar{\\h}{(\\mathbf{A})} + \\cos{(\\mathbf{A})})^{2} \\cos{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))"], [["times", 3, "cos(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)), cos(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} = E_{n}^{v_{z}}, then obtain \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\frac{\\int \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} dE_{n}}{\\int E_{n}^{v_{z}} dE_{n}} = \\frac{d^{2}}{d v_{z}^{2}} 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} = E_{n}^{v_{z}} and \\int \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} dE_{n} = \\int E_{n}^{v_{z}} dE_{n} and \\frac{\\int \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} dE_{n}}{\\int E_{n}^{v_{z}} dE_{n}} = 1 and \\frac{\\partial}{\\partial v_{z}} \\frac{\\int \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} dE_{n}}{\\int E_{n}^{v_{z}} dE_{n}} = \\frac{d}{d v_{z}} 1 and \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\frac{\\int \\operatorname{L_{\\varepsilon}}{(E_{n},v_{z})} dE_{n}}{\\int E_{n}^{v_{z}} dE_{n}} = \\frac{d^{2}}{d v_{z}^{2}} 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["divide", 2, "Integral(Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Mul(Pow(Integral(Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(-1)), Integral(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Integer(1))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Integral(Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(-1)), Integral(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Integral(Pow(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(-1)), Integral(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(2))))"]]}, {"prompt": "Given p{(q,E_{\\lambda})} = - \\sin{(E_{\\lambda} - q)} and \\hat{\\mathbf{x}}{(q,E_{\\lambda})} = \\frac{\\partial}{\\partial q} p{(q,E_{\\lambda})}, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} - \\frac{\\hat{\\mathbf{x}}{(q,E_{\\lambda})}}{q} = \\frac{\\partial}{\\partial E_{\\lambda}} - \\frac{\\frac{\\partial}{\\partial q} - \\sin{(E_{\\lambda} - q)}}{q}", "derivation": "p{(q,E_{\\lambda})} = - \\sin{(E_{\\lambda} - q)} and \\frac{\\partial}{\\partial q} p{(q,E_{\\lambda})} = \\frac{\\partial}{\\partial q} - \\sin{(E_{\\lambda} - q)} and \\hat{\\mathbf{x}}{(q,E_{\\lambda})} = \\frac{\\partial}{\\partial q} p{(q,E_{\\lambda})} and - \\frac{\\hat{\\mathbf{x}}{(q,E_{\\lambda})}}{q} = - \\frac{\\frac{\\partial}{\\partial q} p{(q,E_{\\lambda})}}{q} and - \\frac{\\hat{\\mathbf{x}}{(q,E_{\\lambda})}}{q} = - \\frac{\\frac{\\partial}{\\partial q} - \\sin{(E_{\\lambda} - q)}}{q} and \\frac{\\partial}{\\partial E_{\\lambda}} - \\frac{\\hat{\\mathbf{x}}{(q,E_{\\lambda})}}{q} = \\frac{\\partial}{\\partial E_{\\lambda}} - \\frac{\\frac{\\partial}{\\partial q} - \\sin{(E_{\\lambda} - q)}}{q}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Function('p')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Function('p')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Mul(Integer(-1), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('q', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Mul(Integer(-1), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain (\\bar{\\h}{(\\hat{H}_{\\lambda})} \\log{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} = (\\log{(\\hat{H}_{\\lambda})}^{2} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}}", "derivation": "\\bar{\\h}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\bar{\\h}{(\\hat{H}_{\\lambda})} \\log{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}^{2} and \\bar{\\h}{(\\hat{H}_{\\lambda})} \\log{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}^{2} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} and (\\bar{\\h}{(\\hat{H}_{\\lambda})} \\log{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} = (\\log{(\\hat{H}_{\\lambda})}^{2} - \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["times", 1, "log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)))"], [["minus", 2, "Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('\\\\hbar')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))), Add(Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))))"], [["power", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\hbar')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Add(Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(F_{H},b)} = - \\sin{(F_{H} - b)}, then obtain \\frac{\\partial}{\\partial F_{H}} (\\int \\hat{X}{(F_{H},b)} dF_{H})^{b} = \\frac{\\partial}{\\partial F_{H}} (\\int - \\sin{(F_{H} - b)} dF_{H})^{b}", "derivation": "\\hat{X}{(F_{H},b)} = - \\sin{(F_{H} - b)} and \\int \\hat{X}{(F_{H},b)} dF_{H} = \\int - \\sin{(F_{H} - b)} dF_{H} and (\\int \\hat{X}{(F_{H},b)} dF_{H})^{b} = (\\int - \\sin{(F_{H} - b)} dF_{H})^{b} and \\frac{\\partial}{\\partial F_{H}} (\\int \\hat{X}{(F_{H},b)} dF_{H})^{b} = \\frac{\\partial}{\\partial F_{H}} (\\int - \\sin{(F_{H} - b)} dF_{H})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))), Tuple(Symbol('F_H', commutative=True))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{X}')(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))), Tuple(Symbol('F_H', commutative=True))), Symbol('b', commutative=True)))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\hat{X}')(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))), Tuple(Symbol('F_H', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\phi_2,v,b)} = \\phi_2 b^{v}, then obtain (\\phi_2 b^{v} + \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)}) (2 \\phi_2 b^{v} + 2 \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)}) = (3 \\phi_2 b^{v} + 2 \\phi_2) (\\phi_2 b^{v} + \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)})", "derivation": "\\bar{\\h}{(\\phi_2,v,b)} = \\phi_2 b^{v} and \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)} = \\phi_2 b^{v} + \\phi_2 and \\phi_2 b^{v} + \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)} = 2 \\phi_2 b^{v} + \\phi_2 and 2 \\phi_2 b^{v} + 2 \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)} = 3 \\phi_2 b^{v} + 2 \\phi_2 and (2 \\phi_2 b^{v} + \\phi_2) (2 \\phi_2 b^{v} + 2 \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)}) = (2 \\phi_2 b^{v} + \\phi_2) (3 \\phi_2 b^{v} + 2 \\phi_2) and (\\phi_2 b^{v} + \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)}) (2 \\phi_2 b^{v} + 2 \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)}) = (3 \\phi_2 b^{v} + 2 \\phi_2) (\\phi_2 b^{v} + \\phi_2 + \\bar{\\h}{(\\phi_2,v,b)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))))"], [["add", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True))), Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["add", 2, "Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True))))"], [["times", 4, "Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True)))), Mul(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(3), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True)))), Mul(Add(Mul(Integer(3), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('b', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\phi_2', commutative=True), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given S{(\\phi_1)} = \\cos{(\\phi_1)}, then obtain \\frac{d^{2}}{d \\phi_1^{2}} ((- \\phi_1 + S{(\\phi_1)})^{\\phi_1})^{\\phi_1} = \\frac{d^{2}}{d \\phi_1^{2}} ((- \\phi_1 + \\cos{(\\phi_1)})^{\\phi_1})^{\\phi_1}", "derivation": "S{(\\phi_1)} = \\cos{(\\phi_1)} and - \\phi_1 + S{(\\phi_1)} = - \\phi_1 + \\cos{(\\phi_1)} and (- \\phi_1 + S{(\\phi_1)})^{\\phi_1} = (- \\phi_1 + \\cos{(\\phi_1)})^{\\phi_1} and ((- \\phi_1 + S{(\\phi_1)})^{\\phi_1})^{\\phi_1} = ((- \\phi_1 + \\cos{(\\phi_1)})^{\\phi_1})^{\\phi_1} and \\frac{d}{d \\phi_1} ((- \\phi_1 + S{(\\phi_1)})^{\\phi_1})^{\\phi_1} = \\frac{d}{d \\phi_1} ((- \\phi_1 + \\cos{(\\phi_1)})^{\\phi_1})^{\\phi_1} and \\frac{d^{2}}{d \\phi_1^{2}} ((- \\phi_1 + S{(\\phi_1)})^{\\phi_1})^{\\phi_1} = \\frac{d^{2}}{d \\phi_1^{2}} ((- \\phi_1 + \\cos{(\\phi_1)})^{\\phi_1})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('S')(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('S')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('S')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('S')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('S')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\eta{(\\psi,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} \\psi, then obtain \\int (\\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\eta{(\\psi,\\hat{\\mathbf{x}})}) d\\psi = \\int (\\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\psi) d\\psi", "derivation": "\\eta{(\\psi,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} \\psi and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\eta{(\\psi,\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\psi and \\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\eta{(\\psi,\\hat{\\mathbf{x}})} = \\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\psi and \\int (\\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\eta{(\\psi,\\hat{\\mathbf{x}})}) d\\psi = \\int (\\psi + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\psi) d\\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Symbol('\\\\psi', commutative=True), Derivative(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Add(Symbol('\\\\psi', commutative=True), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\psi', commutative=True), Derivative(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Symbol('\\\\psi', commutative=True), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given r{(T)} = e^{T}, then obtain - T + 2 e^{T} = - T + r{(T)} + e^{T}", "derivation": "r{(T)} = e^{T} and - T + r{(T)} = - T + e^{T} and - T + 2 r{(T)} = - T + r{(T)} + e^{T} and - T + 2 r{(T)} = - T + 2 e^{T} and - T + 2 e^{T} = - T + r{(T)} + e^{T}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('r')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('r')(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), Function('r')(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('r')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), Function('r')(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), exp(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(2), exp(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('r')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"]]}, {"prompt": "Given A{(E)} = \\log{(E)}, then obtain \\frac{d}{d E} A^{3}{(E)} \\log{(E)} = \\frac{d}{d E} A^{2}{(E)} \\log{(E)}^{2}", "derivation": "A{(E)} = \\log{(E)} and A{(E)} \\log{(E)} = \\log{(E)}^{2} and A^{2}{(E)} \\log{(E)}^{2} = \\log{(E)}^{4} and A^{3}{(E)} \\log{(E)} = A^{2}{(E)} \\log{(E)}^{2} and \\frac{d}{d E} A^{3}{(E)} \\log{(E)} = \\frac{d}{d E} A^{2}{(E)} \\log{(E)}^{2}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["times", 1, "log(Symbol('E', commutative=True))"], "Equality(Mul(Function('A')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(log(Symbol('E', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('A')(Symbol('E', commutative=True)), Integer(2)), Pow(log(Symbol('E', commutative=True)), Integer(2))), Pow(log(Symbol('E', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('A')(Symbol('E', commutative=True)), Integer(3)), log(Symbol('E', commutative=True))), Mul(Pow(Function('A')(Symbol('E', commutative=True)), Integer(2)), Pow(log(Symbol('E', commutative=True)), Integer(2))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('A')(Symbol('E', commutative=True)), Integer(3)), log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('A')(Symbol('E', commutative=True)), Integer(2)), Pow(log(Symbol('E', commutative=True)), Integer(2))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(P_{e},b)} = P_{e} b, then obtain (\\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} - b) \\operatorname{f_{\\mathbf{v}}}{(P_{e},b)} = P_{e} b (\\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} - b)", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)} = P_{e} b and 1 = \\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} and 1 - b = \\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} - b and (1 - b) \\operatorname{f_{\\mathbf{v}}}{(P_{e},b)} = P_{e} b (1 - b) and (\\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} - b) \\operatorname{f_{\\mathbf{v}}}{(P_{e},b)} = P_{e} b (\\frac{P_{e} b}{\\operatorname{f_{\\mathbf{v}}}{(P_{e},b)}} - b)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True)))"], [["divide", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))"], "Equality(Integer(1), Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Integer(-1))))"], [["minus", 2, "Symbol('b', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('b', commutative=True))), Add(Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["times", 1, "Add(Integer(1), Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Symbol('b', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('b', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Add(Mul(Symbol('P_e', commutative=True), Symbol('b', commutative=True), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given n{(f_{\\mathbf{v}},f)} = \\sin^{f_{\\mathbf{v}}}{(f)}, then obtain \\frac{d}{d f} 0 - 1 = \\frac{\\partial}{\\partial f} (- n{(f_{\\mathbf{v}},f)} + \\sin^{f_{\\mathbf{v}}}{(f)}) - 1", "derivation": "n{(f_{\\mathbf{v}},f)} = \\sin^{f_{\\mathbf{v}}}{(f)} and 0 = - n{(f_{\\mathbf{v}},f)} + \\sin^{f_{\\mathbf{v}}}{(f)} and \\frac{d}{d f} 0 = \\frac{\\partial}{\\partial f} (- n{(f_{\\mathbf{v}},f)} + \\sin^{f_{\\mathbf{v}}}{(f)}) and \\frac{d}{d f} 0 - 1 = \\frac{\\partial}{\\partial f} (- n{(f_{\\mathbf{v}},f)} + \\sin^{f_{\\mathbf{v}}}{(f)}) - 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Function('n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Function('n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\delta)} = \\log{(\\delta)}, then obtain (\\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\operatorname{f^{*}}{(\\delta)}))^{\\delta} = (\\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\log{(\\delta)}))^{\\delta}", "derivation": "\\operatorname{f^{*}}{(\\delta)} = \\log{(\\delta)} and \\delta + \\operatorname{f^{*}}{(\\delta)} = \\delta + \\log{(\\delta)} and \\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\operatorname{f^{*}}{(\\delta)} = \\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\log{(\\delta)} and \\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\operatorname{f^{*}}{(\\delta)}) = \\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\log{(\\delta)}) and (\\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\operatorname{f^{*}}{(\\delta)}))^{\\delta} = (\\frac{d}{d \\delta} (\\delta - (\\delta + \\log{(\\delta)})^{\\delta} + \\log{(\\delta)}))^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('f^*')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Function('f^*')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), log(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Function('f^*')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Function('f^*')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{P})} = \\sin{(\\sin{(\\mathbf{P})})}, then obtain \\int \\frac{d}{d \\mathbf{P}} \\operatorname{f_{E}}{(\\mathbf{P})} d\\mathbf{P} + \\frac{\\sin{(\\sin{(\\mathbf{P})})}}{\\mathbf{P}} = \\int \\frac{d}{d \\mathbf{P}} \\sin{(\\sin{(\\mathbf{P})})} d\\mathbf{P} + \\frac{\\sin{(\\sin{(\\mathbf{P})})}}{\\mathbf{P}}", "derivation": "\\operatorname{f_{E}}{(\\mathbf{P})} = \\sin{(\\sin{(\\mathbf{P})})} and \\frac{d}{d \\mathbf{P}} \\operatorname{f_{E}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\sin{(\\sin{(\\mathbf{P})})} and \\int \\frac{d}{d \\mathbf{P}} \\operatorname{f_{E}}{(\\mathbf{P})} d\\mathbf{P} = \\int \\frac{d}{d \\mathbf{P}} \\sin{(\\sin{(\\mathbf{P})})} d\\mathbf{P} and \\int \\frac{d}{d \\mathbf{P}} \\operatorname{f_{E}}{(\\mathbf{P})} d\\mathbf{P} + \\frac{\\sin{(\\sin{(\\mathbf{P})})}}{\\mathbf{P}} = \\int \\frac{d}{d \\mathbf{P}} \\sin{(\\sin{(\\mathbf{P})})} d\\mathbf{P} + \\frac{\\sin{(\\sin{(\\mathbf{P})})}}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{P}', commutative=True)), sin(sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Function('f_E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(sin(sin(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\mathbf{P}', commutative=True))))"], "Equality(Add(Integral(Derivative(Function('f_E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\mathbf{P}', commutative=True))))), Add(Integral(Derivative(sin(sin(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain (\\mathbf{J}{(L_{\\varepsilon})} \\mathbf{J}^{L_{\\varepsilon}}{(L_{\\varepsilon})})^{L_{\\varepsilon}} = (\\mathbf{J}{(L_{\\varepsilon})} (e^{L_{\\varepsilon}})^{L_{\\varepsilon}})^{L_{\\varepsilon}}", "derivation": "\\mathbf{J}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\mathbf{J}^{L_{\\varepsilon}}{(L_{\\varepsilon})} = (e^{L_{\\varepsilon}})^{L_{\\varepsilon}} and \\mathbf{J}{(L_{\\varepsilon})} \\mathbf{J}^{L_{\\varepsilon}}{(L_{\\varepsilon})} = \\mathbf{J}{(L_{\\varepsilon})} (e^{L_{\\varepsilon}})^{L_{\\varepsilon}} and (\\mathbf{J}{(L_{\\varepsilon})} \\mathbf{J}^{L_{\\varepsilon}}{(L_{\\varepsilon})})^{L_{\\varepsilon}} = (\\mathbf{J}{(L_{\\varepsilon})} (e^{L_{\\varepsilon}})^{L_{\\varepsilon}})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}, then derive \\frac{d}{d \\hat{H}_{\\lambda}} \\mathbf{M}{(\\hat{H}_{\\lambda})} = 1, then obtain U (\\frac{d}{d \\mathbf{M}{(\\hat{H}_{\\lambda})}} \\mathbf{M}{(\\hat{H}_{\\lambda})})^{\\mathbf{M}{(\\hat{H}_{\\lambda})}} = U", "derivation": "\\mathbf{M}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and \\frac{d}{d \\hat{H}_{\\lambda}} \\mathbf{M}{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} and \\frac{d}{d \\hat{H}_{\\lambda}} \\mathbf{M}{(\\hat{H}_{\\lambda})} = 1 and \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} = 1 and (\\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda})^{\\hat{H}_{\\lambda}} = 1 and (\\frac{d}{d \\mathbf{M}{(\\hat{H}_{\\lambda})}} \\mathbf{M}{(\\hat{H}_{\\lambda})})^{\\mathbf{M}{(\\hat{H}_{\\lambda})}} = 1 and U (\\frac{d}{d \\mathbf{M}{(\\hat{H}_{\\lambda})}} \\mathbf{M}{(\\hat{H}_{\\lambda})})^{\\mathbf{M}{(\\hat{H}_{\\lambda})}} = U", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))), Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(1))"], [["times", 6, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Pow(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))), Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Symbol('U', commutative=True))"]]}, {"prompt": "Given \\Psi{(E_{n},S,\\tilde{g}^*)} = E_{n} S - \\tilde{g}^* and \\operatorname{A_{1}}{(t,t_{2})} = t + t_{2} and I{(t,\\tilde{g}^*,t_{2})} = \\tilde{g}^* + t + t_{2}, then obtain E_{n} S + \\operatorname{A_{1}}{(t,t_{2})} = E_{n} S - \\tilde{g}^* + I{(t,\\tilde{g}^*,t_{2})}", "derivation": "\\Psi{(E_{n},S,\\tilde{g}^*)} = E_{n} S - \\tilde{g}^* and \\operatorname{A_{1}}{(t,t_{2})} = t + t_{2} and \\tilde{g}^* + \\operatorname{A_{1}}{(t,t_{2})} = \\tilde{g}^* + t + t_{2} and \\tilde{g}^* + \\operatorname{A_{1}}{(t,t_{2})} + \\Psi{(E_{n},S,\\tilde{g}^*)} = \\tilde{g}^* + t + t_{2} + \\Psi{(E_{n},S,\\tilde{g}^*)} and I{(t,\\tilde{g}^*,t_{2})} = \\tilde{g}^* + t + t_{2} and \\tilde{g}^* + \\operatorname{A_{1}}{(t,t_{2})} + \\Psi{(E_{n},S,\\tilde{g}^*)} = I{(t,\\tilde{g}^*,t_{2})} + \\Psi{(E_{n},S,\\tilde{g}^*)} and E_{n} S + \\operatorname{A_{1}}{(t,t_{2})} = E_{n} S - \\tilde{g}^* + I{(t,\\tilde{g}^*,t_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Symbol('E_n', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], ["get_premise", "Equality(Function('A_1')(Symbol('t', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('t', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('A_1')(Symbol('t', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 3, "Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('A_1')(Symbol('t', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t', commutative=True), Symbol('t_2', commutative=True), Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t', commutative=True), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('A_1')(Symbol('t', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Function('I')(Symbol('t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\Psi')(Symbol('E_n', commutative=True), Symbol('S', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Symbol('E_n', commutative=True), Symbol('S', commutative=True)), Function('A_1')(Symbol('t', commutative=True), Symbol('t_2', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('I')(Symbol('t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\varphi^*)} = \\varphi^*, then obtain \\frac{\\int 1 d\\varphi^*}{\\int \\frac{\\varphi^* + \\operatorname{t_{2}}{(\\varphi^*)}}{2 \\operatorname{t_{2}}{(\\varphi^*)}} d\\varphi^*} = 1", "derivation": "\\operatorname{t_{2}}{(\\varphi^*)} = \\varphi^* and 2 \\operatorname{t_{2}}{(\\varphi^*)} = \\varphi^* + \\operatorname{t_{2}}{(\\varphi^*)} and 1 = \\frac{\\varphi^* + \\operatorname{t_{2}}{(\\varphi^*)}}{2 \\operatorname{t_{2}}{(\\varphi^*)}} and \\int 1 d\\varphi^* = \\int \\frac{\\varphi^* + \\operatorname{t_{2}}{(\\varphi^*)}}{2 \\operatorname{t_{2}}{(\\varphi^*)}} d\\varphi^* and \\frac{\\int 1 d\\varphi^*}{\\int \\frac{\\varphi^* + \\operatorname{t_{2}}{(\\varphi^*)}}{2 \\operatorname{t_{2}}{(\\varphi^*)}} d\\varphi^*} = 1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], [["add", 1, "Function('t_2')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(2), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('t_2')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Symbol('\\\\varphi^*', commutative=True), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('t_2')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Rational(1, 2), Add(Symbol('\\\\varphi^*', commutative=True), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('t_2')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 4, "Integral(Mul(Rational(1, 2), Add(Symbol('\\\\varphi^*', commutative=True), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('t_2')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Pow(Integral(Mul(Rational(1, 2), Add(Symbol('\\\\varphi^*', commutative=True), Function('t_2')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('t_2')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\hat{x}_0{(G)} = \\log{(G)}, then derive \\int \\log{(\\hat{x}_0{(G)} + \\log{(G)})} dG = G \\log{(2 \\log{(G)})} + \\mathbf{M} - \\operatorname{li}{(G)}, then obtain \\frac{d}{d G} \\int \\log{(\\hat{x}_0{(G)} + \\log{(G)})} dG = \\frac{\\partial}{\\partial G} (G \\log{(2 \\log{(G)})} + \\mathbf{M} - \\operatorname{li}{(G)})", "derivation": "\\hat{x}_0{(G)} = \\log{(G)} and \\hat{x}_0{(G)} + \\log{(G)} = 2 \\log{(G)} and \\log{(\\hat{x}_0{(G)} + \\log{(G)})} = \\log{(2 \\log{(G)})} and \\int \\log{(\\hat{x}_0{(G)} + \\log{(G)})} dG = \\int \\log{(2 \\log{(G)})} dG and \\int \\log{(\\hat{x}_0{(G)} + \\log{(G)})} dG = G \\log{(2 \\log{(G)})} + \\mathbf{M} - \\operatorname{li}{(G)} and \\frac{d}{d G} \\int \\log{(\\hat{x}_0{(G)} + \\log{(G)})} dG = \\frac{\\partial}{\\partial G} (G \\log{(2 \\log{(G)})} + \\mathbf{M} - \\operatorname{li}{(G)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["add", 1, "log(Symbol('G', commutative=True))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True))), Mul(Integer(2), log(Symbol('G', commutative=True))))"], [["log", 2], "Equality(log(Add(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))), log(Mul(Integer(2), log(Symbol('G', commutative=True)))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(log(Add(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Integral(log(Mul(Integer(2), log(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(log(Add(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), log(Mul(Integer(2), log(Symbol('G', commutative=True))))), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), li(Symbol('G', commutative=True)))))"], [["differentiate", 5, "Symbol('G', commutative=True)"], "Equality(Derivative(Integral(log(Add(Function('\\\\hat{x}_0')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('G', commutative=True), log(Mul(Integer(2), log(Symbol('G', commutative=True))))), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), li(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(\\theta)} = \\int \\cos{(\\theta)} d\\theta, then derive (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} = (\\mathbf{v} + \\sin{(\\theta)})^{2}, then obtain \\frac{\\partial}{\\partial \\theta} \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} d\\theta = \\frac{\\partial}{\\partial \\theta} \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)})^{2} d\\theta", "derivation": "\\delta{(\\theta)} = \\int \\cos{(\\theta)} d\\theta and \\delta{(\\theta)} \\int \\cos{(\\theta)} d\\theta = (\\int \\cos{(\\theta)} d\\theta)^{2} and (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} = (\\mathbf{v} + \\sin{(\\theta)})^{2} and \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} = \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)})^{2} and \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} d\\theta = \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)})^{2} d\\theta and \\frac{\\partial}{\\partial \\theta} \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)}) \\delta{(\\theta)} d\\theta = \\frac{\\partial}{\\partial \\theta} \\int \\frac{\\partial}{\\partial \\mathbf{v}} (\\mathbf{v} + \\sin{(\\theta)})^{2} d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\theta', commutative=True)), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Function('\\\\delta')(Symbol('\\\\theta', commutative=True)), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Pow(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Function('\\\\delta')(Symbol('\\\\theta', commutative=True))), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Function('\\\\delta')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Function('\\\\delta')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Integral(Derivative(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Function('\\\\delta')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Integral(Derivative(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)}, then derive \\Psi^{\\dagger}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain - \\sin{(\\varepsilon)} \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\Psi^{\\dagger}{(\\varepsilon)}) d\\sigma_x = - \\sin{(\\varepsilon)} \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\cos{(\\varepsilon)}) d\\sigma_x", "derivation": "\\Psi^{\\dagger}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\sin{(\\varepsilon)} and \\Psi^{\\dagger}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\Psi^{\\dagger}{(\\varepsilon)} = \\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\cos{(\\varepsilon)} and \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\Psi^{\\dagger}{(\\varepsilon)}) d\\sigma_x = \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\cos{(\\varepsilon)}) d\\sigma_x and - \\sin{(\\varepsilon)} \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\Psi^{\\dagger}{(\\varepsilon)}) d\\sigma_x = - \\sin{(\\varepsilon)} \\int (\\Omega{(\\psi^*,v_{t},\\sigma_x)} + \\cos{(\\varepsilon)}) d\\sigma_x", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True)), Derivative(sin(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["add", 2, "Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True))), Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 4, "Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True)), Integral(Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True)), Integral(Add(Function('\\\\Omega')(Symbol('\\\\psi^*', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given Q{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\Omega{(\\tilde{g},g^{\\prime}_{\\varepsilon})} = Q{(\\tilde{g})} - \\frac{1}{g^{\\prime}_{\\varepsilon}}, then obtain - \\sin{(\\tilde{g})} + \\frac{1}{g^{\\prime}_{\\varepsilon}} = - Q{(\\tilde{g})} + \\frac{1}{g^{\\prime}_{\\varepsilon}}", "derivation": "Q{(\\tilde{g})} = \\sin{(\\tilde{g})} and Q{(\\tilde{g})} - \\frac{1}{g^{\\prime}_{\\varepsilon}} = \\sin{(\\tilde{g})} - \\frac{1}{g^{\\prime}_{\\varepsilon}} and \\Omega{(\\tilde{g},g^{\\prime}_{\\varepsilon})} = Q{(\\tilde{g})} - \\frac{1}{g^{\\prime}_{\\varepsilon}} and \\Omega{(\\tilde{g},g^{\\prime}_{\\varepsilon})} = \\sin{(\\tilde{g})} - \\frac{1}{g^{\\prime}_{\\varepsilon}} and - \\Omega{(\\tilde{g},g^{\\prime}_{\\varepsilon})} = - Q{(\\tilde{g})} + \\frac{1}{g^{\\prime}_{\\varepsilon}} and - \\sin{(\\tilde{g})} + \\frac{1}{g^{\\prime}_{\\varepsilon}} = - Q{(\\tilde{g})} + \\frac{1}{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 1, "Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Add(Function('Q')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))), Add(sin(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Function('Q')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(sin(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Function('Q')(Symbol('\\\\tilde{g}', commutative=True))), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True))), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Function('Q')(Symbol('\\\\tilde{g}', commutative=True))), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = - B + \\log{(V_{\\mathbf{B}})}, then derive \\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = -1, then obtain \\cos{(\\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})})} \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = 0", "derivation": "\\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = - B + \\log{(V_{\\mathbf{B}})} and \\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial B} (- B + \\log{(V_{\\mathbf{B}})}) and \\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = -1 and \\sin{(\\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})})} = - \\sin{(1)} and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\sin{(\\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})})} = \\frac{d}{d V_{\\mathbf{B}}} - \\sin{(1)} and \\cos{(\\frac{\\partial}{\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})})} \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}\\partial B} \\operatorname{g_{\\varepsilon}}{(B,V_{\\mathbf{B}})} = 0", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))"], [["sin", 3], "Equality(sin(Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Integer(1))))"], [["differentiate", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(sin(Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Integer(1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(cos(Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Derivative(Function('g_{\\\\varepsilon}')(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\rho_{f}{(\\Omega,C_{d})} = C_{d} \\Omega and \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} = \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial \\Omega} (C_{d} \\Omega)^{\\Omega} \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} = \\frac{\\partial}{\\partial \\Omega} (C_{d} \\Omega)^{\\Omega} \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon}", "derivation": "\\rho_{f}{(\\Omega,C_{d})} = C_{d} \\Omega and \\rho_{f}^{\\Omega}{(\\Omega,C_{d})} = (C_{d} \\Omega)^{\\Omega} and \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} = \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon} and \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} \\rho_{f}^{\\Omega}{(\\Omega,C_{d})} = \\rho_{f}^{\\Omega}{(\\Omega,C_{d})} \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon} and (C_{d} \\Omega)^{\\Omega} \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} = (C_{d} \\Omega)^{\\Omega} \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon} and \\frac{\\partial}{\\partial \\Omega} (C_{d} \\Omega)^{\\Omega} \\hat{\\mathbf{x}}{(g_{\\varepsilon},\\lambda)} = \\frac{\\partial}{\\partial \\Omega} (C_{d} \\Omega)^{\\Omega} \\int \\lambda g_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 3, "Pow(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}{(U)} = \\frac{d}{d U} \\log{(U)}, then derive \\mathbf{J}{(U)} = \\frac{1}{U}, then obtain \\cos{(U)} = \\cos{(U + \\mathbf{J}{(U)} - \\frac{1}{U})}", "derivation": "\\mathbf{J}{(U)} = \\frac{d}{d U} \\log{(U)} and \\mathbf{J}{(U)} = \\frac{1}{U} and - U + \\mathbf{J}{(U)} = - U + \\frac{1}{U} and - U = - U - \\mathbf{J}{(U)} + \\frac{1}{U} and \\cos{(U)} = \\cos{(U + \\mathbf{J}{(U)} - \\frac{1}{U})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('U', commutative=True)), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}')(Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Integer(-1)))"], [["minus", 2, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{J}')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Integer(-1))))"], [["minus", 3, "Function('\\\\mathbf{J}')(Symbol('U', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Integer(-1))))"], [["cos", 4], "Equality(cos(Symbol('U', commutative=True)), cos(Add(Symbol('U', commutative=True), Function('\\\\mathbf{J}')(Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\hat{X}{(n)} = \\sin{(\\cos{(n)})}, then derive \\frac{d}{d n} \\hat{X}{(n)} = - \\sin{(n)} \\cos{(\\cos{(n)})}, then obtain \\frac{d}{d n} \\sin{(\\cos{(n)})} = - \\sin{(n)} \\cos{(\\cos{(n)})}", "derivation": "\\hat{X}{(n)} = \\sin{(\\cos{(n)})} and \\frac{d}{d n} \\hat{X}{(n)} = \\frac{d}{d n} \\sin{(\\cos{(n)})} and \\frac{d}{d n} \\hat{X}{(n)} = - \\sin{(n)} \\cos{(\\cos{(n)})} and \\frac{d}{d n} \\sin{(\\cos{(n)})} = - \\sin{(n)} \\cos{(\\cos{(n)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('n', commutative=True)), sin(cos(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n', commutative=True)), cos(cos(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n', commutative=True)), cos(cos(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(x^\\prime,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{x^\\prime}, then obtain \\Psi_{\\lambda} (- 8 \\Psi_{\\lambda}^{x^\\prime} + 8 \\Omega{(x^\\prime,\\Psi_{\\lambda})}) = 0", "derivation": "\\Omega{(x^\\prime,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{x^\\prime} and - \\Psi_{\\lambda}^{x^\\prime} + \\Omega{(x^\\prime,\\Psi_{\\lambda})} = 0 and - \\Psi_{\\lambda}^{x^\\prime} + 2 \\Omega{(x^\\prime,\\Psi_{\\lambda})} = \\Omega{(x^\\prime,\\Psi_{\\lambda})} and \\Psi_{\\lambda} (- \\Psi_{\\lambda}^{x^\\prime} + \\Omega{(x^\\prime,\\Psi_{\\lambda})}) = 0 and \\Psi_{\\lambda} (- 2 \\Psi_{\\lambda}^{x^\\prime} + 2 \\Omega{(x^\\prime,\\Psi_{\\lambda})}) = 0 and \\Psi_{\\lambda} (- 4 \\Psi_{\\lambda}^{x^\\prime} + 4 \\Omega{(x^\\prime,\\Psi_{\\lambda})}) = 0 and \\Psi_{\\lambda} (- 8 \\Psi_{\\lambda}^{x^\\prime} + 8 \\Omega{(x^\\prime,\\Psi_{\\lambda})}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(0))"], [["add", 2, "Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Integer(4), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(4), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Integer(8), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(8), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})}, then derive \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{F_{x}}{(\\hat{\\mathbf{r}})} = \\frac{1}{\\hat{\\mathbf{r}}}, then obtain \\hat{\\mathbf{r}} (\\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}}) = 2", "derivation": "\\operatorname{F_{x}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{F_{x}}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{F_{x}}{(\\hat{\\mathbf{r}})} = \\frac{1}{\\hat{\\mathbf{r}}} and \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} = \\frac{1}{\\hat{\\mathbf{r}}} and \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}} = \\frac{2}{\\hat{\\mathbf{r}}} and \\hat{\\mathbf{r}} (\\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}}) = 2", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)))"], [["add", 4, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Add(Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["divide", 5, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)))), Integer(2))"]]}, {"prompt": "Given h{(\\lambda,n)} = n^{\\lambda} and \\mu{(\\lambda,n)} = n^{\\lambda}, then obtain \\int \\mu{(\\lambda,n)} dn = \\int h{(\\lambda,n)} dn", "derivation": "h{(\\lambda,n)} = n^{\\lambda} and \\int h{(\\lambda,n)} dn = \\int n^{\\lambda} dn and \\mu{(\\lambda,n)} = n^{\\lambda} and h{(\\lambda,n)} = \\mu{(\\lambda,n)} and \\int \\mu{(\\lambda,n)} dn = \\int n^{\\lambda} dn and \\int \\mu{(\\lambda,n)} dn = \\int h{(\\lambda,n)} dn", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Symbol('n', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('h')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Function('\\\\mu')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Symbol('n', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Function('h')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(Q,g)} = g \\log{(Q)}, then obtain - \\hat{\\mathbf{r}} (\\varepsilon^{Q}{(Q,g)})^{Q} = - \\hat{\\mathbf{r}} ((g \\log{(Q)})^{Q})^{Q}", "derivation": "\\varepsilon{(Q,g)} = g \\log{(Q)} and \\varepsilon^{Q}{(Q,g)} = (g \\log{(Q)})^{Q} and (\\varepsilon^{Q}{(Q,g)})^{Q} = ((g \\log{(Q)})^{Q})^{Q} and - \\hat{\\mathbf{r}} (\\varepsilon^{Q}{(Q,g)})^{Q} = - \\hat{\\mathbf{r}} ((g \\log{(Q)})^{Q})^{Q}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('Q', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), log(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('Q', commutative=True), Symbol('g', commutative=True)), Symbol('Q', commutative=True)), Pow(Mul(Symbol('g', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(Function('\\\\varepsilon')(Symbol('Q', commutative=True), Symbol('g', commutative=True)), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Pow(Function('\\\\varepsilon')(Symbol('Q', commutative=True), Symbol('g', commutative=True)), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Pow(Mul(Symbol('g', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})}, then obtain \\iint \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} dx^\\prime dx^\\prime = \\iint \\log{(\\cos{(x^\\prime)})}^{x^\\prime} dx^\\prime dx^\\prime", "derivation": "\\operatorname{v_{2}}{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})} and \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})}^{x^\\prime} and \\int \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} dx^\\prime = \\int \\log{(\\cos{(x^\\prime)})}^{x^\\prime} dx^\\prime and \\iint \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} dx^\\prime dx^\\prime = \\iint \\log{(\\cos{(x^\\prime)})}^{x^\\prime} dx^\\prime dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), log(cos(Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\theta_1,\\mathbf{B})} = \\frac{\\partial}{\\partial \\theta_1} \\mathbf{B} \\theta_1, then derive \\hat{X}{(\\theta_1,\\mathbf{B})} = \\mathbf{B}, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{B} \\theta_1}{\\mathbf{B}}", "derivation": "\\hat{X}{(\\theta_1,\\mathbf{B})} = \\frac{\\partial}{\\partial \\theta_1} \\mathbf{B} \\theta_1 and \\hat{X}{(\\theta_1,\\mathbf{B})} = \\mathbf{B} and 1 = \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{B} \\theta_1}{\\hat{X}{(\\theta_1,\\mathbf{B})}} and 1 = \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{B} \\theta_1}{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], [["divide", 1, "Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{p})} = e^{e^{\\hat{p}}}, then derive \\int \\mathbf{J}_M{(\\hat{p})} d\\hat{p} = \\mathbf{S} + \\operatorname{Ei}{(e^{\\hat{p}})}, then obtain \\frac{d}{d \\mathbf{S}} \\int e^{e^{\\hat{p}}} d\\hat{p} = \\frac{\\partial}{\\partial \\mathbf{S}} (\\mathbf{S} + \\operatorname{Ei}{(e^{\\hat{p}})})", "derivation": "\\mathbf{J}_M{(\\hat{p})} = e^{e^{\\hat{p}}} and \\int \\mathbf{J}_M{(\\hat{p})} d\\hat{p} = \\int e^{e^{\\hat{p}}} d\\hat{p} and \\int \\mathbf{J}_M{(\\hat{p})} d\\hat{p} = \\mathbf{S} + \\operatorname{Ei}{(e^{\\hat{p}})} and \\int e^{e^{\\hat{p}}} d\\hat{p} = \\mathbf{S} + \\operatorname{Ei}{(e^{\\hat{p}})} and \\frac{d}{d \\mathbf{S}} \\int e^{e^{\\hat{p}}} d\\hat{p} = \\frac{\\partial}{\\partial \\mathbf{S}} (\\mathbf{S} + \\operatorname{Ei}{(e^{\\hat{p}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Ei(exp(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Ei(exp(Symbol('\\\\hat{p}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Integral(exp(exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Ei(exp(Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(x)} = \\log{(e^{x})}, then obtain \\hat{p}_0{(x)} \\log{(e^{x})} + \\hat{p}_0{(x)} = \\log{(e^{x})}^{2} + \\log{(e^{x})}", "derivation": "\\hat{p}_0{(x)} = \\log{(e^{x})} and \\hat{p}_0{(x)} \\log{(e^{x})} = \\log{(e^{x})}^{2} and \\hat{p}_0{(x)} + \\log{(e^{x})}^{2} = \\log{(e^{x})}^{2} + \\log{(e^{x})} and \\hat{p}_0{(x)} \\log{(e^{x})} + \\hat{p}_0{(x)} = \\hat{p}_0{(x)} + \\log{(e^{x})}^{2} and \\hat{p}_0{(x)} \\log{(e^{x})} + \\hat{p}_0{(x)} = \\log{(e^{x})}^{2} + \\log{(e^{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), log(exp(Symbol('x', commutative=True))))"], [["times", 1, "log(exp(Symbol('x', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), log(exp(Symbol('x', commutative=True)))), Pow(log(exp(Symbol('x', commutative=True))), Integer(2)))"], [["add", 1, "Pow(log(exp(Symbol('x', commutative=True))), Integer(2))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), Pow(log(exp(Symbol('x', commutative=True))), Integer(2))), Add(Pow(log(exp(Symbol('x', commutative=True))), Integer(2)), log(exp(Symbol('x', commutative=True)))))"], [["add", 2, "Function('\\\\hat{p}_0')(Symbol('x', commutative=True))"], "Equality(Add(Mul(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), log(exp(Symbol('x', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('x', commutative=True))), Add(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), Pow(log(exp(Symbol('x', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Function('\\\\hat{p}_0')(Symbol('x', commutative=True)), log(exp(Symbol('x', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('x', commutative=True))), Add(Pow(log(exp(Symbol('x', commutative=True))), Integer(2)), log(exp(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\pi,b)} = \\frac{\\partial}{\\partial \\pi} \\pi b, then derive \\operatorname{F_{c}}{(\\pi,b)} + 1 = b + 1, then obtain \\pi (\\frac{b^{2}}{2} + b) + f = \\pi (\\frac{b^{2}}{2} + b) + t", "derivation": "\\operatorname{F_{c}}{(\\pi,b)} = \\frac{\\partial}{\\partial \\pi} \\pi b and \\operatorname{F_{c}}{(\\pi,b)} + 1 = \\frac{\\partial}{\\partial \\pi} \\pi b + 1 and \\operatorname{F_{c}}{(\\pi,b)} + 1 = b + 1 and \\frac{\\partial}{\\partial \\pi} \\pi b + 1 = b + 1 and \\int (\\frac{\\partial}{\\partial \\pi} \\pi b + 1) db = \\int (b + 1) db and \\int 1 db + \\int \\frac{\\partial}{\\partial \\pi} \\pi b db = \\int 1 db + \\int b db and \\int (\\int 1 db + \\int \\frac{\\partial}{\\partial \\pi} \\pi b db) d\\pi = \\int (\\int 1 db + \\int b db) d\\pi and \\pi (\\frac{b^{2}}{2} + b) + f = \\pi (\\frac{b^{2}}{2} + b) + t", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('F_c')(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Integer(1)), Add(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('F_c')(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Integer(1)), Add(Symbol('b', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(Symbol('b', commutative=True), Integer(1)))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('b', commutative=True))), Integral(Add(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True))))"], [["expand", 5], "Equality(Add(Integral(Integer(1), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('b', commutative=True))), Integral(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Integral(Integer(1), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Integral(Integer(1), Tuple(Symbol('b', commutative=True))), Integral(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + i and \\operatorname{P_{g}}{(v_{t})} = \\log{(v_{t})}, then obtain - \\frac{(- i + \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})}) \\operatorname{P_{g}}{(v_{t})}}{\\delta} = \\frac{f_{\\mathbf{v}} \\operatorname{P_{g}}{(v_{t})}}{\\delta}", "derivation": "\\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + i and - i + \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} and \\operatorname{P_{g}}{(v_{t})} = \\log{(v_{t})} and (- i + \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})}) \\log{(v_{t})} = - f_{\\mathbf{v}} \\log{(v_{t})} and (- i + \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})}) \\operatorname{P_{g}}{(v_{t})} = - f_{\\mathbf{v}} \\operatorname{P_{g}}{(v_{t})} and - \\frac{(- i + \\operatorname{g_{\\varepsilon}}{(i,f_{\\mathbf{v}})}) \\operatorname{P_{g}}{(v_{t})}}{\\delta} = \\frac{f_{\\mathbf{v}} \\operatorname{P_{g}}{(v_{t})}}{\\delta}", "srepr_derivation": [["get_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["times", 2, "log(Symbol('v_t', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True), log(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('P_g')(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('P_g')(Symbol('v_t', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('P_g')(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('P_g')(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(z)} = \\cos{(\\sin{(z)})}, then obtain \\frac{d}{d z} (\\frac{\\operatorname{f_{\\mathbf{v}}}{(z)}}{\\sin{(z)}} + \\operatorname{f_{\\mathbf{v}}}^{z}{(z)}) = \\frac{d}{d z} (\\operatorname{f_{\\mathbf{v}}}^{z}{(z)} + \\frac{\\cos{(\\sin{(z)})}}{\\sin{(z)}})", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(z)} = \\cos{(\\sin{(z)})} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(z)}}{\\sin{(z)}} = \\frac{\\cos{(\\sin{(z)})}}{\\sin{(z)}} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(z)}}{\\sin{(z)}} + \\operatorname{f_{\\mathbf{v}}}^{z}{(z)} = \\operatorname{f_{\\mathbf{v}}}^{z}{(z)} + \\frac{\\cos{(\\sin{(z)})}}{\\sin{(z)}} and \\frac{d}{d z} (\\frac{\\operatorname{f_{\\mathbf{v}}}{(z)}}{\\sin{(z)}} + \\operatorname{f_{\\mathbf{v}}}^{z}{(z)}) = \\frac{d}{d z} (\\operatorname{f_{\\mathbf{v}}}^{z}{(z)} + \\frac{\\cos{(\\sin{(z)})}}{\\sin{(z)}})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), cos(sin(Symbol('z', commutative=True))))"], [["divide", 1, "sin(Symbol('z', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('z', commutative=True)), Integer(-1)), cos(sin(Symbol('z', commutative=True)))))"], [["add", 2, "Pow(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Symbol('z', commutative=True))"], "Equality(Add(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Add(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Mul(Pow(sin(Symbol('z', commutative=True)), Integer(-1)), cos(sin(Symbol('z', commutative=True))))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Mul(Pow(sin(Symbol('z', commutative=True)), Integer(-1)), cos(sin(Symbol('z', commutative=True))))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(h)} = \\cos{(h)}, then obtain e^{\\operatorname{v_{z}}{(h)} - \\cos^{h}{(h)}} = e^{\\cos{(h)} - \\cos^{h}{(h)}}", "derivation": "\\operatorname{v_{z}}{(h)} = \\cos{(h)} and \\operatorname{v_{z}}^{h}{(h)} = \\cos^{h}{(h)} and \\operatorname{v_{z}}{(h)} - \\operatorname{v_{z}}^{h}{(h)} = - \\operatorname{v_{z}}^{h}{(h)} + \\cos{(h)} and \\operatorname{v_{z}}{(h)} - \\cos^{h}{(h)} = \\cos{(h)} - \\cos^{h}{(h)} and e^{\\operatorname{v_{z}}{(h)} - \\cos^{h}{(h)}} = e^{\\cos{(h)} - \\cos^{h}{(h)}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 1, "Pow(Function('v_z')(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Function('v_z')(Symbol('h', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('v_z')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('v_z')(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))), Add(cos(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))))"], [["exp", 4], "Equality(exp(Add(Function('v_z')(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))), exp(Add(cos(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\hat{x}_0)} = e^{e^{\\hat{x}_0}}, then obtain 0 = - (\\int \\operatorname{A_{x}}{(\\hat{x}_0)} d\\hat{x}_0)^{\\hat{x}_0} + (\\int e^{e^{\\hat{x}_0}} d\\hat{x}_0)^{\\hat{x}_0}", "derivation": "\\operatorname{A_{x}}{(\\hat{x}_0)} = e^{e^{\\hat{x}_0}} and \\int \\operatorname{A_{x}}{(\\hat{x}_0)} d\\hat{x}_0 = \\int e^{e^{\\hat{x}_0}} d\\hat{x}_0 and (\\int \\operatorname{A_{x}}{(\\hat{x}_0)} d\\hat{x}_0)^{\\hat{x}_0} = (\\int e^{e^{\\hat{x}_0}} d\\hat{x}_0)^{\\hat{x}_0} and 0 = - (\\int \\operatorname{A_{x}}{(\\hat{x}_0)} d\\hat{x}_0)^{\\hat{x}_0} + (\\int e^{e^{\\hat{x}_0}} d\\hat{x}_0)^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\hat{x}_0', commutative=True)), exp(exp(Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(exp(exp(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Integral(Function('A_x')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Integral(exp(exp(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 3, "Pow(Integral(Function('A_x')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Integral(Function('A_x')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Pow(Integral(exp(exp(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\nabla)} = e^{\\nabla}, then derive e^{\\nabla} + \\int \\mathbf{v}{(\\nabla)} d\\nabla = n_{1} + 2 e^{\\nabla}, then obtain n_{1} + 2 \\mathbf{v}{(\\nabla)} = \\mathbf{v}{(\\nabla)} + \\int e^{\\nabla} d\\nabla", "derivation": "\\mathbf{v}{(\\nabla)} = e^{\\nabla} and \\int \\mathbf{v}{(\\nabla)} d\\nabla = \\int e^{\\nabla} d\\nabla and e^{\\nabla} + \\int \\mathbf{v}{(\\nabla)} d\\nabla = e^{\\nabla} + \\int e^{\\nabla} d\\nabla and e^{\\nabla} + \\int \\mathbf{v}{(\\nabla)} d\\nabla = n_{1} + 2 e^{\\nabla} and \\mathbf{v}{(\\nabla)} + \\int \\mathbf{v}{(\\nabla)} d\\nabla = \\mathbf{v}{(\\nabla)} + \\int e^{\\nabla} d\\nabla and \\mathbf{v}{(\\nabla)} + \\int \\mathbf{v}{(\\nabla)} d\\nabla = n_{1} + 2 \\mathbf{v}{(\\nabla)} and n_{1} + 2 \\mathbf{v}{(\\nabla)} = \\mathbf{v}{(\\nabla)} + \\int e^{\\nabla} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(exp(Symbol('\\\\nabla', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(exp(Symbol('\\\\nabla', commutative=True)), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(exp(Symbol('\\\\nabla', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('n_1', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('n_1', commutative=True), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('n_1', commutative=True), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)))), Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(v_{2},C_{d})} = C_{d} v_{2}, then obtain \\frac{\\partial}{\\partial v_{2}} - (C_{d} v_{2})^{C_{d}} \\lambda{(v_{2},C_{d})} = \\frac{\\partial}{\\partial v_{2}} - (2 C_{d} v_{2} - \\lambda{(v_{2},C_{d})})^{C_{d}} \\lambda{(v_{2},C_{d})}", "derivation": "\\lambda{(v_{2},C_{d})} = C_{d} v_{2} and C_{d} v_{2} = 2 C_{d} v_{2} - \\lambda{(v_{2},C_{d})} and (C_{d} v_{2})^{C_{d}} = (2 C_{d} v_{2} - \\lambda{(v_{2},C_{d})})^{C_{d}} and - (C_{d} v_{2})^{C_{d}} \\lambda{(v_{2},C_{d})} = - (2 C_{d} v_{2} - \\lambda{(v_{2},C_{d})})^{C_{d}} \\lambda{(v_{2},C_{d})} and \\frac{\\partial}{\\partial v_{2}} - (C_{d} v_{2})^{C_{d}} \\lambda{(v_{2},C_{d})} = \\frac{\\partial}{\\partial v_{2}} - (2 C_{d} v_{2} - \\lambda{(v_{2},C_{d})})^{C_{d}} \\lambda{(v_{2},C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Add(Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True))))"], "Equality(Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Symbol('C_d', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Symbol('C_d', commutative=True)), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True))))"], [["differentiate", 4, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Mul(Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Symbol('C_d', commutative=True)), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(C_{1},\\omega)} = \\sin{(C_{1}^{\\omega})}, then obtain h{(C_{1},\\omega)} \\sin^{\\omega}{(C_{1}^{\\omega})} = \\sin{(C_{1}^{\\omega})} \\sin^{\\omega}{(C_{1}^{\\omega})}", "derivation": "h{(C_{1},\\omega)} = \\sin{(C_{1}^{\\omega})} and h^{\\omega}{(C_{1},\\omega)} = \\sin^{\\omega}{(C_{1}^{\\omega})} and h{(C_{1},\\omega)} h^{\\omega}{(C_{1},\\omega)} = h^{\\omega}{(C_{1},\\omega)} \\sin{(C_{1}^{\\omega})} and h{(C_{1},\\omega)} \\sin^{\\omega}{(C_{1}^{\\omega})} = \\sin{(C_{1}^{\\omega})} \\sin^{\\omega}{(C_{1}^{\\omega})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Pow(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('h')(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))), Mul(sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\dot{z},b)} = \\dot{z} b, then obtain \\frac{\\partial}{\\partial b} (- \\mathbf{B}{(\\dot{z},b)} + \\int \\mathbf{B}{(\\dot{z},b)} d\\dot{z}) = \\frac{\\partial}{\\partial b} (- \\mathbf{B}{(\\dot{z},b)} + \\int \\dot{z} b d\\dot{z})", "derivation": "\\mathbf{B}{(\\dot{z},b)} = \\dot{z} b and \\int \\mathbf{B}{(\\dot{z},b)} d\\dot{z} = \\int \\dot{z} b d\\dot{z} and - \\mathbf{B}{(\\dot{z},b)} + \\int \\mathbf{B}{(\\dot{z},b)} d\\dot{z} = - \\mathbf{B}{(\\dot{z},b)} + \\int \\dot{z} b d\\dot{z} and \\frac{\\partial}{\\partial b} (- \\mathbf{B}{(\\dot{z},b)} + \\int \\mathbf{B}{(\\dot{z},b)} d\\dot{z}) = \\frac{\\partial}{\\partial b} (- \\mathbf{B}{(\\dot{z},b)} + \\int \\dot{z} b d\\dot{z})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain - \\theta_1 + e^{\\int \\theta_1 d\\theta_1} = - \\theta_1 + e^{\\int \\frac{\\theta_1 \\sin{(\\theta_1)}}{\\mathbf{M}{(\\theta_1)}} d\\theta_1}", "derivation": "\\mathbf{M}{(\\theta_1)} = \\sin{(\\theta_1)} and \\theta_1 \\mathbf{M}{(\\theta_1)} = \\theta_1 \\sin{(\\theta_1)} and \\theta_1 = \\frac{\\theta_1 \\sin{(\\theta_1)}}{\\mathbf{M}{(\\theta_1)}} and \\int \\theta_1 d\\theta_1 = \\int \\frac{\\theta_1 \\sin{(\\theta_1)}}{\\mathbf{M}{(\\theta_1)}} d\\theta_1 and e^{\\int \\theta_1 d\\theta_1} = e^{\\int \\frac{\\theta_1 \\sin{(\\theta_1)}}{\\mathbf{M}{(\\theta_1)}} d\\theta_1} and - \\theta_1 + e^{\\int \\theta_1 d\\theta_1} = - \\theta_1 + e^{\\int \\frac{\\theta_1 \\sin{(\\theta_1)}}{\\mathbf{M}{(\\theta_1)}} d\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Symbol('\\\\theta_1', commutative=True), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Symbol('\\\\theta_1', commutative=True), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(Integral(Symbol('\\\\theta_1', commutative=True), Tuple(Symbol('\\\\theta_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{x}_0)} = \\log{(\\hat{x}_0)}, then obtain (\\hat{x}_0 \\tilde{g}^*{(\\hat{x}_0)})^{\\hat{x}_0} - \\tilde{g}^*{(\\hat{x}_0)} = (\\hat{x}_0 \\log{(\\hat{x}_0)})^{\\hat{x}_0} - \\tilde{g}^*{(\\hat{x}_0)}", "derivation": "\\tilde{g}^*{(\\hat{x}_0)} = \\log{(\\hat{x}_0)} and \\hat{x}_0 \\tilde{g}^*{(\\hat{x}_0)} = \\hat{x}_0 \\log{(\\hat{x}_0)} and (\\hat{x}_0 \\tilde{g}^*{(\\hat{x}_0)})^{\\hat{x}_0} = (\\hat{x}_0 \\log{(\\hat{x}_0)})^{\\hat{x}_0} and (\\hat{x}_0 \\tilde{g}^*{(\\hat{x}_0)})^{\\hat{x}_0} - \\tilde{g}^*{(\\hat{x}_0)} = (\\hat{x}_0 \\log{(\\hat{x}_0)})^{\\hat{x}_0} - \\tilde{g}^*{(\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\hat{x}_0', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Mul(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 3, "Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Pow(Mul(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given s{(C)} = \\log{(C)}, then obtain \\frac{s^{2}{(C)}}{\\log{(C)} \\frac{d}{d C} \\int s{(C)} dC} = \\frac{s{(C)}}{\\frac{d}{d C} \\int s{(C)} dC}", "derivation": "s{(C)} = \\log{(C)} and s^{2}{(C)} = s{(C)} \\log{(C)} and \\frac{s^{2}{(C)}}{\\log{(C)}} = s{(C)} and \\int \\frac{s^{2}{(C)}}{\\log{(C)}} dC = \\int s{(C)} dC and \\frac{d}{d C} \\int \\frac{s^{2}{(C)}}{\\log{(C)}} dC = \\frac{d}{d C} \\int s{(C)} dC and \\frac{s^{2}{(C)}}{\\log{(C)} \\frac{d}{d C} \\int \\frac{s^{2}{(C)}}{\\log{(C)}} dC} = \\frac{s{(C)}}{\\frac{d}{d C} \\int \\frac{s^{2}{(C)}}{\\log{(C)}} dC} and \\frac{s^{2}{(C)}}{\\log{(C)} \\frac{d}{d C} \\int s{(C)} dC} = \\frac{s{(C)}}{\\frac{d}{d C} \\int s{(C)} dC}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["times", 1, "Function('s')(Symbol('C', commutative=True))"], "Equality(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))))"], [["divide", 2, "log(Symbol('C', commutative=True))"], "Equality(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Function('s')(Symbol('C', commutative=True)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True))), Integral(Function('s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Function('s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Integral(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1)), Pow(Derivative(Integral(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))), Mul(Function('s')(Symbol('C', commutative=True)), Pow(Derivative(Integral(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Function('s')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-1)), Pow(Derivative(Integral(Function('s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))), Mul(Function('s')(Symbol('C', commutative=True)), Pow(Derivative(Integral(Function('s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(M_{E})} = \\log{(M_{E})}, then obtain \\log{(M_{E})} (\\int \\frac{1}{\\dot{z}{(M_{E})}} dM_{E})^{M_{E}} = \\log{(M_{E})} (\\int \\frac{\\log{(M_{E})}}{\\dot{z}^{2}{(M_{E})}} dM_{E})^{M_{E}}", "derivation": "\\dot{z}{(M_{E})} = \\log{(M_{E})} and 1 = \\frac{\\log{(M_{E})}}{\\dot{z}{(M_{E})}} and \\frac{1}{\\dot{z}{(M_{E})}} = \\frac{\\log{(M_{E})}}{\\dot{z}^{2}{(M_{E})}} and \\int \\frac{1}{\\dot{z}{(M_{E})}} dM_{E} = \\int \\frac{\\log{(M_{E})}}{\\dot{z}^{2}{(M_{E})}} dM_{E} and (\\int \\frac{1}{\\dot{z}{(M_{E})}} dM_{E})^{M_{E}} = (\\int \\frac{\\log{(M_{E})}}{\\dot{z}^{2}{(M_{E})}} dM_{E})^{M_{E}} and \\log{(M_{E})} (\\int \\frac{1}{\\dot{z}{(M_{E})}} dM_{E})^{M_{E}} = \\log{(M_{E})} (\\int \\frac{\\log{(M_{E})}}{\\dot{z}^{2}{(M_{E})}} dM_{E})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["divide", 1, "Function('\\\\dot{z}')(Symbol('M_E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-1)), log(Symbol('M_E', commutative=True))))"], [["divide", 2, "Function('\\\\dot{z}')(Symbol('M_E', commutative=True))"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-2)), log(Symbol('M_E', commutative=True))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-1)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-2)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-1)), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Integral(Mul(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-2)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"], [["divide", 5, "Pow(log(Symbol('M_E', commutative=True)), Integer(-1))"], "Equality(Mul(log(Symbol('M_E', commutative=True)), Pow(Integral(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-1)), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))), Mul(log(Symbol('M_E', commutative=True)), Pow(Integral(Mul(Pow(Function('\\\\dot{z}')(Symbol('M_E', commutative=True)), Integer(-2)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{x},\\mathbf{H})} = \\hat{x} \\mathbf{H} and \\hat{\\mathbf{x}}{(\\mathbf{H})} = \\mathbf{H}, then obtain (\\hat{x} \\hat{\\mathbf{x}}{(\\mathbf{H})})^{\\mathbf{H}} = \\operatorname{a^{\\dagger}}^{\\mathbf{H}}{(\\hat{x},\\mathbf{H})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{x},\\mathbf{H})} = \\hat{x} \\mathbf{H} and \\hat{\\mathbf{x}}{(\\mathbf{H})} = \\mathbf{H} and \\hat{x} \\hat{\\mathbf{x}}{(\\mathbf{H})} = \\hat{x} \\mathbf{H} and (\\hat{x} \\hat{\\mathbf{x}}{(\\mathbf{H})})^{\\mathbf{H}} = (\\hat{x} \\mathbf{H})^{\\mathbf{H}} and (\\hat{x} \\hat{\\mathbf{x}}{(\\mathbf{H})})^{\\mathbf{H}} = \\operatorname{a^{\\dagger}}^{\\mathbf{H}}{(\\hat{x},\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["times", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(Q,g,n)} = g^{Q} n, then obtain n + \\mathbf{s}{(Q,g,n)} \\int \\mathbf{s}{(Q,g,n)} dQ = n + \\mathbf{s}{(Q,g,n)} \\int g^{Q} n dQ", "derivation": "\\mathbf{s}{(Q,g,n)} = g^{Q} n and \\int \\mathbf{s}{(Q,g,n)} dQ = \\int g^{Q} n dQ and g^{Q} n \\int \\mathbf{s}{(Q,g,n)} dQ = g^{Q} n \\int g^{Q} n dQ and g^{Q} n \\int \\mathbf{s}{(Q,g,n)} dQ + n = g^{Q} n \\int g^{Q} n dQ + n and n + \\mathbf{s}{(Q,g,n)} \\int \\mathbf{s}{(Q,g,n)} dQ = n + \\mathbf{s}{(Q,g,n)} \\int g^{Q} n dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True), Integral(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True), Integral(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["add", 3, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True), Integral(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('n', commutative=True)), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True), Integral(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('n', commutative=True), Mul(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True))))), Add(Symbol('n', commutative=True), Mul(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('g', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('Q', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p} - x^\\prime and \\mathbf{p}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p} - x^\\prime, then obtain \\frac{\\partial}{\\partial x^\\prime} \\operatorname{m_{s}}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\frac{\\partial}{\\partial x^\\prime} (\\eta^{\\prime} + \\hat{p} - x^\\prime)", "derivation": "\\operatorname{m_{s}}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p} - x^\\prime and \\mathbf{p}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p} - x^\\prime and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\frac{\\partial}{\\partial x^\\prime} (\\eta^{\\prime} + \\hat{p} - x^\\prime) and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\frac{\\partial}{\\partial x^\\prime} \\operatorname{m_{s}}{(x^\\prime,\\hat{p},\\eta^{\\prime})} and \\frac{\\partial}{\\partial x^\\prime} \\operatorname{m_{s}}{(x^\\prime,\\hat{p},\\eta^{\\prime})} = \\frac{\\partial}{\\partial x^\\prime} (\\eta^{\\prime} + \\hat{p} - x^\\prime)", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Function('m_s')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('m_s')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\dot{y})} = \\dot{y}, then derive \\frac{d}{d \\dot{y}} \\operatorname{t_{1}}{(\\dot{y})} = 1, then obtain \\frac{d}{d \\dot{y}} \\dot{y} = 1", "derivation": "\\operatorname{t_{1}}{(\\dot{y})} = \\dot{y} and \\frac{d}{d \\dot{y}} \\operatorname{t_{1}}{(\\dot{y})} = \\frac{d}{d \\dot{y}} \\dot{y} and \\frac{d}{d \\dot{y}} \\operatorname{t_{1}}{(\\dot{y})} = 1 and \\frac{d}{d \\dot{y}} \\dot{y} = 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Symbol('\\\\dot{y}', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('\\\\dot{y}', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\nabla{(\\hat{X})} = \\log{(\\hat{X})}, then obtain \\nabla{(\\hat{X})} + \\int \\log{(\\hat{X})} d\\hat{X} = \\log{(\\hat{X})} + \\int \\log{(\\hat{X})} d\\hat{X}", "derivation": "\\nabla{(\\hat{X})} = \\log{(\\hat{X})} and \\int \\nabla{(\\hat{X})} d\\hat{X} = \\int \\log{(\\hat{X})} d\\hat{X} and \\nabla{(\\hat{X})} + \\int \\nabla{(\\hat{X})} d\\hat{X} = \\log{(\\hat{X})} + \\int \\nabla{(\\hat{X})} d\\hat{X} and \\nabla{(\\hat{X})} + \\int \\log{(\\hat{X})} d\\hat{X} = \\log{(\\hat{X})} + \\int \\log{(\\hat{X})} d\\hat{X}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["add", 1, "Integral(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Integral(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Add(log(Symbol('\\\\hat{X}', commutative=True)), Integral(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\hat{X}', commutative=True)), Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Add(log(Symbol('\\\\hat{X}', commutative=True)), Integral(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{x})} = \\log{(\\cos{(v_{x})})}, then obtain e^{\\int (- v_{x} + \\operatorname{F_{H}}{(v_{x})}) dv_{x}} = e^{\\int (- v_{x} + \\log{(\\cos{(v_{x})})}) dv_{x}}", "derivation": "\\operatorname{F_{H}}{(v_{x})} = \\log{(\\cos{(v_{x})})} and - v_{x} + \\operatorname{F_{H}}{(v_{x})} = - v_{x} + \\log{(\\cos{(v_{x})})} and \\int (- v_{x} + \\operatorname{F_{H}}{(v_{x})}) dv_{x} = \\int (- v_{x} + \\log{(\\cos{(v_{x})})}) dv_{x} and e^{\\int (- v_{x} + \\operatorname{F_{H}}{(v_{x})}) dv_{x}} = e^{\\int (- v_{x} + \\log{(\\cos{(v_{x})})}) dv_{x}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True))))"], [["minus", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('F_H')(Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True)))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('F_H')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('F_H')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), exp(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\Psi_{\\lambda})} = \\sin{(\\log{(\\Psi_{\\lambda})})} and \\tilde{g}{(\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})}, then obtain - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})} + 1 = \\tilde{g}{(\\Psi_{\\lambda})} + 1", "derivation": "\\theta{(\\Psi_{\\lambda})} = \\sin{(\\log{(\\Psi_{\\lambda})})} and - \\Psi_{\\lambda} + \\theta{(\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})} and - \\Psi_{\\lambda} + \\theta{(\\Psi_{\\lambda})} + 1 = - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})} + 1 and \\tilde{g}{(\\Psi_{\\lambda})} = - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})} and - \\Psi_{\\lambda} + \\theta{(\\Psi_{\\lambda})} + 1 = \\tilde{g}{(\\Psi_{\\lambda})} + 1 and - \\Psi_{\\lambda} + \\sin{(\\log{(\\Psi_{\\lambda})})} + 1 = \\tilde{g}{(\\Psi_{\\lambda})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1)), Add(Function('\\\\tilde{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)), Add(Function('\\\\tilde{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + \\int \\sin{(\\mathbf{v}{(\\mathbf{J})})} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + \\int \\sin{(e^{\\mathbf{J}})} d\\mathbf{J}", "derivation": "\\mathbf{v}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\sin{(\\mathbf{v}{(\\mathbf{J})})} = \\sin{(e^{\\mathbf{J}})} and \\frac{d}{d \\mathbf{J}} \\mathbf{v}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} and \\int \\sin{(\\mathbf{v}{(\\mathbf{J})})} d\\mathbf{J} = \\int \\sin{(e^{\\mathbf{J}})} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\mathbf{v}{(\\mathbf{J})} + \\int \\sin{(\\mathbf{v}{(\\mathbf{J})})} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\mathbf{v}{(\\mathbf{J})} + \\int \\sin{(e^{\\mathbf{J}})} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + \\int \\sin{(\\mathbf{v}{(\\mathbf{J})})} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + \\int \\sin{(e^{\\mathbf{J}})} d\\mathbf{J}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True))), sin(exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(sin(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(sin(exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 4, "Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integral(sin(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Add(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integral(sin(exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integral(sin(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integral(sin(exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(C_{1})} = C_{1}, then derive \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = \\frac{C_{1}^{2}}{2} + k, then obtain e^{C_{1}} + \\int C_{1} dC_{1} = \\frac{C_{1}^{2}}{2} + k + e^{C_{1}}", "derivation": "\\operatorname{r_{0}}{(C_{1})} = C_{1} and \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = \\int C_{1} dC_{1} and \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = \\frac{C_{1}^{2}}{2} + k and e^{C_{1}} + \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = \\frac{C_{1}^{2}}{2} + k + e^{C_{1}} and e^{C_{1}} + \\int C_{1} dC_{1} = \\frac{C_{1}^{2}}{2} + k + e^{C_{1}}", "srepr_derivation": [["renaming_premise", "Equality(Function('r_0')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_1', commutative=True), Integer(2))), Symbol('k', commutative=True)))"], [["add", 3, "exp(Symbol('C_1', commutative=True))"], "Equality(Add(exp(Symbol('C_1', commutative=True)), Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('C_1', commutative=True), Integer(2))), Symbol('k', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(exp(Symbol('C_1', commutative=True)), Integral(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('C_1', commutative=True), Integer(2))), Symbol('k', commutative=True), exp(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})} = H^{\\hat{\\mathbf{r}}} and \\operatorname{E_{x}}{(H,\\hat{\\mathbf{r}})} = H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})}, then obtain \\operatorname{E_{x}}^{\\hat{\\mathbf{r}}}{(H,\\hat{\\mathbf{r}})} = (H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}}", "derivation": "\\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})} = H^{\\hat{\\mathbf{r}}} and \\operatorname{E_{x}}{(H,\\hat{\\mathbf{r}})} = H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})} and \\operatorname{E_{x}}{(H,\\hat{\\mathbf{r}})} = 2 H^{\\hat{\\mathbf{r}}} and 2 H^{\\hat{\\mathbf{r}}} = H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})} and (2 H^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} = (H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} and \\operatorname{E_{x}}^{\\hat{\\mathbf{r}}}{(H,\\hat{\\mathbf{r}})} = (H^{\\hat{\\mathbf{r}}} + \\Psi_{\\lambda}{(H,\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E_x')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(2), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('E_x')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Pow(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('H', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)} = \\Omega \\mathbf{J}_P, then obtain \\int \\frac{d}{d \\mathbf{J}_P} 0 d\\mathbf{J}_P = M + \\Omega \\mathbf{J}_P - \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)}", "derivation": "\\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)} = \\Omega \\mathbf{J}_P and 0 = \\Omega \\mathbf{J}_P - \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} 0 = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\Omega \\mathbf{J}_P - \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)}) and \\int \\frac{d}{d \\mathbf{J}_P} 0 d\\mathbf{J}_P = \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\Omega \\mathbf{J}_P - \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)}) d\\mathbf{J}_P and \\int \\frac{d}{d \\mathbf{J}_P} 0 d\\mathbf{J}_P = M + \\Omega \\mathbf{J}_P - \\operatorname{A_{x}}{(\\Omega,\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Derivative(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given l{(\\tilde{g},\\rho_b,\\mu_0)} = (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}, then derive \\frac{\\partial}{\\partial \\tilde{g}} l{(\\tilde{g},\\rho_b,\\mu_0)} = \\frac{\\rho_b (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}}{\\tilde{g}}, then obtain \\int \\frac{\\partial}{\\partial \\tilde{g}} l{(\\tilde{g},\\rho_b,\\mu_0)} d\\tilde{g} = \\int \\frac{\\rho_b (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}}{\\tilde{g}} d\\tilde{g}", "derivation": "l{(\\tilde{g},\\rho_b,\\mu_0)} = (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b} and \\mu_0 + l{(\\tilde{g},\\rho_b,\\mu_0)} = \\mu_0 + (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b} and \\frac{\\partial}{\\partial \\tilde{g}} (\\mu_0 + l{(\\tilde{g},\\rho_b,\\mu_0)}) = \\frac{\\partial}{\\partial \\tilde{g}} (\\mu_0 + (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}) and \\frac{\\partial}{\\partial \\tilde{g}} l{(\\tilde{g},\\rho_b,\\mu_0)} = \\frac{\\rho_b (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}}{\\tilde{g}} and \\int \\frac{\\partial}{\\partial \\tilde{g}} l{(\\tilde{g},\\rho_b,\\mu_0)} d\\tilde{g} = \\int \\frac{\\rho_b (\\frac{\\tilde{g}}{\\mu_0})^{\\rho_b}}{\\tilde{g}} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mu_0', commutative=True), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu_0', commutative=True), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Derivative(Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\omega{(F_{N})} = \\log{(\\log{(F_{N})})}, then derive - \\int \\omega{(F_{N})} dF_{N} = - F_{N} \\log{(\\log{(F_{N})})} - \\hbar + \\operatorname{li}{(F_{N})}, then obtain \\log{(- \\int \\log{(\\log{(F_{N})})} dF_{N})} = \\log{(- F_{N} \\log{(\\log{(F_{N})})} - \\hbar + \\operatorname{li}{(F_{N})})}", "derivation": "\\omega{(F_{N})} = \\log{(\\log{(F_{N})})} and \\int \\omega{(F_{N})} dF_{N} = \\int \\log{(\\log{(F_{N})})} dF_{N} and - \\int \\omega{(F_{N})} dF_{N} = - \\int \\log{(\\log{(F_{N})})} dF_{N} and - \\int \\omega{(F_{N})} dF_{N} = - F_{N} \\log{(\\log{(F_{N})})} - \\hbar + \\operatorname{li}{(F_{N})} and - \\int \\log{(\\log{(F_{N})})} dF_{N} = - F_{N} \\log{(\\log{(F_{N})})} - \\hbar + \\operatorname{li}{(F_{N})} and \\log{(- \\int \\log{(\\log{(F_{N})})} dF_{N})} = \\log{(- F_{N} \\log{(\\log{(F_{N})})} - \\hbar + \\operatorname{li}{(F_{N})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('F_N', commutative=True)), log(log(Symbol('F_N', commutative=True))))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(log(log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\omega')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Integral(log(log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\omega')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), log(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), li(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Integral(log(log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), log(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), li(Symbol('F_N', commutative=True))))"], [["log", 5], "Equality(log(Mul(Integer(-1), Integral(log(log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))), log(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), log(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), li(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} = \\log{(A_{y})}, then derive \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + C_{d}, then obtain \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} dA_{y} = A_{y} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} - A_{y} + C_{d}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} = \\log{(A_{y})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} dA_{y} = \\int \\log{(A_{y})} dA_{y} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + C_{d} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} dA_{y} = A_{y} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{y})} - A_{y} + C_{d}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), log(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\mathbf{S},\\dot{y})} = \\dot{y} + \\mathbf{S}, then obtain - \\mathbf{S} \\phi{(\\mathbf{S},\\dot{y})} + \\phi{(\\mathbf{S},\\dot{y})} = \\dot{y} - \\mathbf{S} \\phi{(\\mathbf{S},\\dot{y})} + \\mathbf{S}", "derivation": "\\phi{(\\mathbf{S},\\dot{y})} = \\dot{y} + \\mathbf{S} and \\mathbf{S} \\phi{(\\mathbf{S},\\dot{y})} = \\mathbf{S} (\\dot{y} + \\mathbf{S}) and - \\mathbf{S} (\\dot{y} + \\mathbf{S}) + \\phi{(\\mathbf{S},\\dot{y})} = \\dot{y} - \\mathbf{S} (\\dot{y} + \\mathbf{S}) + \\mathbf{S} and - \\mathbf{S} \\phi{(\\mathbf{S},\\dot{y})} + \\phi{(\\mathbf{S},\\dot{y})} = \\dot{y} - \\mathbf{S} \\phi{(\\mathbf{S},\\dot{y})} + \\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(\\eta,t)} = \\eta^{t}, then obtain \\frac{\\partial}{\\partial t} (t + 2 \\mathbf{D}{(\\eta,t)}) = \\frac{\\partial}{\\partial t} (\\eta^{t} + t + \\mathbf{D}{(\\eta,t)})", "derivation": "\\mathbf{D}{(\\eta,t)} = \\eta^{t} and 2 \\mathbf{D}{(\\eta,t)} = \\eta^{t} + \\mathbf{D}{(\\eta,t)} and t + 2 \\mathbf{D}{(\\eta,t)} = \\eta^{t} + t + \\mathbf{D}{(\\eta,t)} and \\frac{\\partial}{\\partial t} (t + 2 \\mathbf{D}{(\\eta,t)}) = \\frac{\\partial}{\\partial t} (\\eta^{t} + t + \\mathbf{D}{(\\eta,t)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True))), Add(Pow(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True))))"], [["add", 2, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)))), Add(Pow(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\eta', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(H,I,Q)} = H Q - I, then obtain H Q + \\frac{Q J^{Q}{(H,I,Q)} \\frac{\\partial}{\\partial I} J{(H,I,Q)}}{J{(H,I,Q)}} = H Q - \\frac{Q (H Q - I)^{Q}}{H Q - I}", "derivation": "J{(H,I,Q)} = H Q - I and J^{Q}{(H,I,Q)} = (H Q - I)^{Q} and \\frac{\\partial}{\\partial I} J^{Q}{(H,I,Q)} = \\frac{\\partial}{\\partial I} (H Q - I)^{Q} and H Q + \\frac{\\partial}{\\partial I} J^{Q}{(H,I,Q)} = H Q + \\frac{\\partial}{\\partial I} (H Q - I)^{Q} and H Q + \\frac{Q J^{Q}{(H,I,Q)} \\frac{\\partial}{\\partial I} J{(H,I,Q)}}{J{(H,I,Q)}} = H Q - \\frac{Q (H Q - I)^{Q}}{H Q - I}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))), Symbol('Q', commutative=True)))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Pow(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Derivative(Pow(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Derivative(Pow(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), Pow(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Derivative(Function('J')(Symbol('H', commutative=True), Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))), Integer(-1)), Pow(Add(Mul(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\theta{(t_{1},k)} = k - t_{1}, then derive \\int \\theta{(t_{1},k)} dt_{1} = k t_{1} - \\frac{t_{1}^{2}}{2} + z, then derive k t_{1} - \\frac{t_{1}^{2}}{2} + z = \\theta + k t_{1} - \\frac{t_{1}^{2}}{2}, then obtain \\theta + k t_{1} - \\frac{t_{1}^{2}}{2} = \\int (k - t_{1}) dt_{1}", "derivation": "\\theta{(t_{1},k)} = k - t_{1} and \\int \\theta{(t_{1},k)} dt_{1} = \\int (k - t_{1}) dt_{1} and \\int \\theta{(t_{1},k)} dt_{1} = k t_{1} - \\frac{t_{1}^{2}}{2} + z and k t_{1} - \\frac{t_{1}^{2}}{2} + z = \\int (k - t_{1}) dt_{1} and k t_{1} - \\frac{t_{1}^{2}}{2} + z = \\theta + k t_{1} - \\frac{t_{1}^{2}}{2} and \\theta + k t_{1} - \\frac{t_{1}^{2}}{2} = \\int (k - t_{1}) dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('t_1', commutative=True), Symbol('k', commutative=True)), Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('t_1', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta')(Symbol('t_1', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Mul(Symbol('k', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('k', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Symbol('z', commutative=True)), Integral(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('k', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Symbol('z', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Mul(Symbol('k', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Symbol('k', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))), Integral(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(P_{e})} = \\sin{(\\cos{(P_{e})})}, then derive \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})} = - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})}, then obtain (1 - \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})})^{P_{e}} = (\\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} + 1)^{P_{e}}", "derivation": "\\operatorname{r_{0}}{(P_{e})} = \\sin{(\\cos{(P_{e})})} and \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(\\cos{(P_{e})})} and \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})} = - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} and \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})} - 1 = - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} - 1 and 1 - \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})} = \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} + 1 and (1 - \\frac{d}{d P_{e}} \\operatorname{r_{0}}{(P_{e})})^{P_{e}} = (\\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} + 1)^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('P_e', commutative=True)), sin(cos(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('r_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))), Integer(-1)))"], [["times", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Function('r_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))), Add(Mul(sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))), Integer(1)))"], [["power", 5, "Symbol('P_e', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Function('r_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))), Symbol('P_e', commutative=True)), Pow(Add(Mul(sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))), Integer(1)), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(E_{n},F_{x})} = E_{n} F_{x} and c{(E_{n},F_{x})} = \\int E_{n} F_{x} dF_{x}, then obtain - E_{n} c{(E_{n},F_{x})} + F_{x} + \\int \\sigma_{x}{(E_{n},F_{x})} dF_{x} = - E_{n} c{(E_{n},F_{x})} + F_{x} + c{(E_{n},F_{x})}", "derivation": "\\sigma_{x}{(E_{n},F_{x})} = E_{n} F_{x} and \\int \\sigma_{x}{(E_{n},F_{x})} dF_{x} = \\int E_{n} F_{x} dF_{x} and F_{x} + \\int \\sigma_{x}{(E_{n},F_{x})} dF_{x} = F_{x} + \\int E_{n} F_{x} dF_{x} and c{(E_{n},F_{x})} = \\int E_{n} F_{x} dF_{x} and F_{x} + \\int \\sigma_{x}{(E_{n},F_{x})} dF_{x} = F_{x} + c{(E_{n},F_{x})} and - E_{n} c{(E_{n},F_{x})} + F_{x} + \\int \\sigma_{x}{(E_{n},F_{x})} dF_{x} = - E_{n} c{(E_{n},F_{x})} + F_{x} + c{(E_{n},F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["add", 2, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Symbol('F_x', commutative=True), Integral(Mul(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Integral(Mul(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('F_x', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Symbol('F_x', commutative=True), Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True))))"], [["minus", 5, "Mul(Symbol('E_n', commutative=True), Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True), Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True), Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True), Function('c')(Symbol('E_n', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}} \\theta_2, then obtain \\hat{\\mathbf{r}}^{2} + 2 \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}}^{2} \\theta_2 + \\hat{\\mathbf{r}}^{2} + \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)}", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}} \\theta_2 and \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}}^{2} \\theta_2 and 2 \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}}^{2} \\theta_2 + \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} and \\hat{\\mathbf{r}}^{2} + 2 \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)} = \\hat{\\mathbf{r}}^{2} \\theta_2 + \\hat{\\mathbf{r}}^{2} + \\hat{\\mathbf{r}} \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\theta_2', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["add", 3, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(H)} = e^{H}, then obtain \\iint \\mathbf{J}_M^{H}{(H)} (e^{H})^{H} dH dH = \\iint (e^{H})^{2 H} dH dH", "derivation": "\\mathbf{J}_M{(H)} = e^{H} and \\mathbf{J}_M^{H}{(H)} = (e^{H})^{H} and \\mathbf{J}_M^{H}{(H)} (e^{H})^{H} = (e^{H})^{2 H} and \\int \\mathbf{J}_M^{H}{(H)} (e^{H})^{H} dH = \\int (e^{H})^{2 H} dH and \\iint \\mathbf{J}_M^{H}{(H)} (e^{H})^{H} dH dH = \\iint (e^{H})^{2 H} dH dH", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Pow(exp(Symbol('H', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Pow(exp(Symbol('H', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["integrate", 4, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Pow(exp(Symbol('H', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given U{(F_{x})} = \\int \\cos{(F_{x})} dF_{x}, then derive U{(F_{x})} = \\hat{p}_0 + \\sin{(F_{x})}, then derive \\frac{\\partial}{\\partial F_{x}} (\\hat{p}_0 + \\sin{(F_{x})}) = \\frac{\\partial}{\\partial F_{x}} (x^\\prime + \\sin{(F_{x})}), then obtain \\frac{\\partial}{\\partial F_{x}} (x^\\prime + \\sin{(F_{x})}) = \\frac{d}{d F_{x}} \\int \\cos{(F_{x})} dF_{x}", "derivation": "U{(F_{x})} = \\int \\cos{(F_{x})} dF_{x} and U{(F_{x})} = \\hat{p}_0 + \\sin{(F_{x})} and \\hat{p}_0 + \\sin{(F_{x})} = \\int \\cos{(F_{x})} dF_{x} and \\frac{\\partial}{\\partial F_{x}} (\\hat{p}_0 + \\sin{(F_{x})}) = \\frac{d}{d F_{x}} \\int \\cos{(F_{x})} dF_{x} and \\frac{\\partial}{\\partial F_{x}} (\\hat{p}_0 + \\sin{(F_{x})}) = \\frac{\\partial}{\\partial F_{x}} (x^\\prime + \\sin{(F_{x})}) and \\frac{\\partial}{\\partial F_{x}} (x^\\prime + \\sin{(F_{x})}) = \\frac{d}{d F_{x}} \\int \\cos{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('F_x', commutative=True)), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('U')(Symbol('F_x', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('F_x', commutative=True))), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(z)} = \\frac{d}{d z} e^{z}, then derive \\mathbf{D}{(z)} - e^{z} = 0, then obtain \\frac{d^{2}}{d z^{2}} (- e^{z} + \\frac{d}{d z} e^{z}) = \\frac{d^{2}}{d z^{2}} 0", "derivation": "\\mathbf{D}{(z)} = \\frac{d}{d z} e^{z} and \\mathbf{D}{(z)} - e^{z} = - e^{z} + \\frac{d}{d z} e^{z} and \\mathbf{D}{(z)} - e^{z} = 0 and - e^{z} + \\frac{d}{d z} e^{z} = 0 and \\frac{d}{d z} (- e^{z} + \\frac{d}{d z} e^{z}) = \\frac{d}{d z} 0 and \\frac{d^{2}}{d z^{2}} (- e^{z} + \\frac{d}{d z} e^{z}) = \\frac{d^{2}}{d z^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 1, "exp(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('z', commutative=True))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('z', commutative=True))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(Symbol('z', commutative=True))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(Symbol('z', commutative=True))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\hat{x}_0{(r_{0},\\phi)} = r_{0}^{\\phi}, then obtain (\\phi r_{0}^{\\phi} \\hat{x}_0{(r_{0},\\phi)} + r_{0}^{2 \\phi})^{2} = (\\phi r_{0}^{2 \\phi} + r_{0}^{2 \\phi})^{2}", "derivation": "\\hat{x}_0{(r_{0},\\phi)} = r_{0}^{\\phi} and r_{0}^{\\phi} \\hat{x}_0{(r_{0},\\phi)} = r_{0}^{2 \\phi} and \\phi r_{0}^{\\phi} \\hat{x}_0{(r_{0},\\phi)} = \\phi r_{0}^{2 \\phi} and \\phi r_{0}^{\\phi} \\hat{x}_0{(r_{0},\\phi)} + r_{0}^{2 \\phi} = \\phi r_{0}^{2 \\phi} + r_{0}^{2 \\phi} and (\\phi r_{0}^{\\phi} \\hat{x}_0{(r_{0},\\phi)} + r_{0}^{2 \\phi})^{2} = (\\phi r_{0}^{2 \\phi} + r_{0}^{2 \\phi})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["times", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))))"], [["add", 3, "Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Add(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Integer(2)), Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\rho{(\\hat{X},\\hat{p}_0)} = \\hat{p}_0^{\\hat{X}} and f{(Q)} = \\cos{(Q)} and l{(\\hat{X},Q,\\hat{p}_0)} = \\cos{(Q)} - \\frac{1}{\\rho{(\\hat{X},\\hat{p}_0)}}, then obtain \\frac{\\partial}{\\partial Q} l{(\\hat{X},Q,\\hat{p}_0)} = - \\sin{(Q)}", "derivation": "\\rho{(\\hat{X},\\hat{p}_0)} = \\hat{p}_0^{\\hat{X}} and f{(Q)} = \\cos{(Q)} and l{(\\hat{X},Q,\\hat{p}_0)} = \\cos{(Q)} - \\frac{1}{\\rho{(\\hat{X},\\hat{p}_0)}} and l{(\\hat{X},Q,\\hat{p}_0)} = \\cos{(Q)} - \\hat{p}_0^{- \\hat{X}} and l{(\\hat{X},Q,\\hat{p}_0)} = f{(Q)} - \\hat{p}_0^{- \\hat{X}} and \\frac{\\partial}{\\partial Q} l{(\\hat{X},Q,\\hat{p}_0)} = \\frac{\\partial}{\\partial Q} (f{(Q)} - \\hat{p}_0^{- \\hat{X}}) and \\frac{\\partial}{\\partial Q} l{(\\hat{X},Q,\\hat{p}_0)} = \\frac{\\partial}{\\partial Q} (\\cos{(Q)} - \\hat{p}_0^{- \\hat{X}}) and \\frac{\\partial}{\\partial Q} l{(\\hat{X},Q,\\hat{p}_0)} = - \\sin{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["get_premise", "Equality(Function('f')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(cos(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(cos(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Function('f')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Function('f')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(F_{g},\\rho_b)} = \\rho_b \\log{(F_{g})}, then obtain \\log{(\\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)})} + \\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)} = \\log{(\\frac{\\partial}{\\partial \\rho_b} \\rho_b \\log{(F_{g})})} + \\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)}", "derivation": "\\mathbf{J}_P{(F_{g},\\rho_b)} = \\rho_b \\log{(F_{g})} and \\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)} = \\frac{\\partial}{\\partial \\rho_b} \\rho_b \\log{(F_{g})} and \\log{(\\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)})} = \\log{(\\frac{\\partial}{\\partial \\rho_b} \\rho_b \\log{(F_{g})})} and \\log{(\\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)})} + \\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)} = \\log{(\\frac{\\partial}{\\partial \\rho_b} \\rho_b \\log{(F_{g})})} + \\frac{\\partial}{\\partial \\rho_b} \\mathbf{J}_P{(F_{g},\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('F_g', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), log(Derivative(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('F_g', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Add(log(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(log(Derivative(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('F_g', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{A}{(n)} = e^{n}, then obtain \\mathbf{A}{(n)} \\iint \\mathbf{A}{(n)} e^{n} dn dn = \\mathbf{A}{(n)} \\iint e^{2 n} dn dn", "derivation": "\\mathbf{A}{(n)} = e^{n} and \\mathbf{A}{(n)} e^{n} = e^{2 n} and \\int \\mathbf{A}{(n)} e^{n} dn = \\int e^{2 n} dn and \\iint \\mathbf{A}{(n)} e^{n} dn dn = \\iint e^{2 n} dn dn and e^{n} \\iint \\mathbf{A}{(n)} e^{n} dn dn = e^{n} \\iint e^{2 n} dn dn and \\mathbf{A}{(n)} \\iint \\mathbf{A}{(n)} e^{n} dn dn = \\mathbf{A}{(n)} \\iint e^{2 n} dn dn", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["times", 1, "exp(Symbol('n', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), exp(Mul(Integer(2), Symbol('n', commutative=True))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["times", 4, "exp(Symbol('n', commutative=True))"], "Equality(Mul(exp(Symbol('n', commutative=True)), Integral(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(exp(Symbol('n', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), Integral(Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Function('\\\\mathbf{A}')(Symbol('n', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\mathbf{J}_f,\\chi,L)} = \\frac{L}{\\mathbf{J}_f} - \\chi and \\operatorname{E_{n}}{(\\mathbf{J}_f,\\chi,L)} = \\frac{L}{\\mathbf{J}_f} - \\chi - 1, then obtain \\operatorname{E_{n}}{(\\mathbf{J}_f,\\chi,L)} = \\theta{(\\mathbf{J}_f,\\chi,L)} - 1", "derivation": "\\theta{(\\mathbf{J}_f,\\chi,L)} = \\frac{L}{\\mathbf{J}_f} - \\chi and \\theta{(\\mathbf{J}_f,\\chi,L)} - 1 = \\frac{L}{\\mathbf{J}_f} - \\chi - 1 and \\operatorname{E_{n}}{(\\mathbf{J}_f,\\chi,L)} = \\frac{L}{\\mathbf{J}_f} - \\chi - 1 and \\operatorname{E_{n}}{(\\mathbf{J}_f,\\chi,L)} = \\theta{(\\mathbf{J}_f,\\chi,L)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('L', commutative=True)), Add(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Add(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('L', commutative=True)), Add(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('E_n')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('L', commutative=True)), Add(Function('\\\\theta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('L', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(F_{N})} = e^{\\cos{(F_{N})}}, then obtain (\\frac{\\operatorname{M_{E}}^{F_{N}}{(F_{N})}}{F_{N}})^{F_{N}} = (\\frac{(e^{\\cos{(F_{N})}})^{F_{N}}}{F_{N}})^{F_{N}}", "derivation": "\\operatorname{M_{E}}{(F_{N})} = e^{\\cos{(F_{N})}} and \\operatorname{M_{E}}^{F_{N}}{(F_{N})} = (e^{\\cos{(F_{N})}})^{F_{N}} and \\frac{\\operatorname{M_{E}}^{F_{N}}{(F_{N})}}{F_{N}} = \\frac{(e^{\\cos{(F_{N})}})^{F_{N}}}{F_{N}} and (\\frac{\\operatorname{M_{E}}^{F_{N}}{(F_{N})}}{F_{N}})^{F_{N}} = (\\frac{(e^{\\cos{(F_{N})}})^{F_{N}}}{F_{N}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(exp(cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"], [["divide", 2, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Function('M_E')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(exp(cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Function('M_E')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(exp(cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\psi^*)} = \\cos{(\\psi^*)}, then obtain \\frac{d^{2}}{d (\\psi^*)^{2}} 0 = \\frac{d^{2}}{d (\\psi^*)^{2}} (- \\operatorname{y^{\\prime}}^{\\psi^*}{(\\psi^*)} + \\cos^{\\psi^*}{(\\psi^*)})", "derivation": "\\operatorname{y^{\\prime}}{(\\psi^*)} = \\cos{(\\psi^*)} and \\operatorname{y^{\\prime}}^{\\psi^*}{(\\psi^*)} = \\cos^{\\psi^*}{(\\psi^*)} and 0 = - \\operatorname{y^{\\prime}}^{\\psi^*}{(\\psi^*)} + \\cos^{\\psi^*}{(\\psi^*)} and \\frac{d}{d \\psi^*} 0 = \\frac{d}{d \\psi^*} (- \\operatorname{y^{\\prime}}^{\\psi^*}{(\\psi^*)} + \\cos^{\\psi^*}{(\\psi^*)}) and \\frac{d^{2}}{d (\\psi^*)^{2}} 0 = \\frac{d^{2}}{d (\\psi^*)^{2}} (- \\operatorname{y^{\\prime}}^{\\psi^*}{(\\psi^*)} + \\cos^{\\psi^*}{(\\psi^*)})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "Pow(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given S{(E)} = \\sin{(\\log{(E)})}, then derive (\\int \\frac{S{(E)}}{E} dE)^{E} = (\\mathbf{E} - \\cos{(\\log{(E)})})^{E}, then derive (\\theta_2 - \\cos{(\\log{(E)})})^{E} = (\\mathbf{E} - \\cos{(\\log{(E)})})^{E}, then obtain \\int (\\theta_2 - \\cos{(\\log{(E)})})^{E} dE = \\int (\\mathbf{E} - \\cos{(\\log{(E)})})^{E} dE", "derivation": "S{(E)} = \\sin{(\\log{(E)})} and \\frac{S{(E)}}{E} = \\frac{\\sin{(\\log{(E)})}}{E} and \\int \\frac{S{(E)}}{E} dE = \\int \\frac{\\sin{(\\log{(E)})}}{E} dE and (\\int \\frac{S{(E)}}{E} dE)^{E} = (\\int \\frac{\\sin{(\\log{(E)})}}{E} dE)^{E} and (\\int \\frac{S{(E)}}{E} dE)^{E} = (\\mathbf{E} - \\cos{(\\log{(E)})})^{E} and (\\int \\frac{\\sin{(\\log{(E)})}}{E} dE)^{E} = (\\mathbf{E} - \\cos{(\\log{(E)})})^{E} and (\\theta_2 - \\cos{(\\log{(E)})})^{E} = (\\mathbf{E} - \\cos{(\\log{(E)})})^{E} and \\int (\\theta_2 - \\cos{(\\log{(E)})})^{E} dE = \\int (\\mathbf{E} - \\cos{(\\log{(E)})})^{E} dE", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('E', commutative=True)), sin(log(Symbol('E', commutative=True))))"], [["divide", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)))"], [["integrate", 7, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(log(Symbol('E', commutative=True))))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given z{(H,J,M)} = - H + M^{J}, then obtain \\iint (- H + M^{J} + z{(H,J,M)}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM dM = \\iint (- 2 H + 2 M^{J}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM dM", "derivation": "z{(H,J,M)} = - H + M^{J} and - H + M^{J} + z{(H,J,M)} = - 2 H + 2 M^{J} and (- H + M^{J} + z{(H,J,M)}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} = (- 2 H + 2 M^{J}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} and \\int (- H + M^{J} + z{(H,J,M)}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM = \\int (- 2 H + 2 M^{J}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM and \\iint (- H + M^{J} + z{(H,J,M)}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM dM = \\iint (- 2 H + 2 M^{J}) \\frac{\\partial}{\\partial M} z^{H}{(H,J,M)} dM dM", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)))))"], [["times", 2, "Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Pow(Symbol('M', commutative=True), Symbol('J', commutative=True)))), Derivative(Pow(Function('z')(Symbol('H', commutative=True), Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(b)} = \\log{(b)}, then derive \\log{(\\int \\mathbf{J}{(b)} db)} = \\log{(A_{z} + b \\log{(b)} - b)}, then obtain \\frac{d}{d A_{z}} \\log{(\\int \\log{(b)} db)} = \\frac{\\partial}{\\partial A_{z}} \\log{(A_{z} + b \\mathbf{J}{(b)} - b)}", "derivation": "\\mathbf{J}{(b)} = \\log{(b)} and \\int \\mathbf{J}{(b)} db = \\int \\log{(b)} db and \\log{(\\int \\mathbf{J}{(b)} db)} = \\log{(\\int \\log{(b)} db)} and \\log{(\\int \\mathbf{J}{(b)} db)} = \\log{(A_{z} + b \\log{(b)} - b)} and \\log{(\\int \\mathbf{J}{(b)} db)} = \\log{(A_{z} + b \\mathbf{J}{(b)} - b)} and \\log{(\\int \\log{(b)} db)} = \\log{(A_{z} + b \\mathbf{J}{(b)} - b)} and \\frac{d}{d A_{z}} \\log{(\\int \\log{(b)} db)} = \\frac{\\partial}{\\partial A_{z}} \\log{(A_{z} + b \\mathbf{J}{(b)} - b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\mathbf{J}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('\\\\mathbf{J}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Add(Symbol('A_z', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(log(Integral(Function('\\\\mathbf{J}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Add(Symbol('A_z', commutative=True), Mul(Symbol('b', commutative=True), Function('\\\\mathbf{J}')(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(log(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Add(Symbol('A_z', commutative=True), Mul(Symbol('b', commutative=True), Function('\\\\mathbf{J}')(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["differentiate", 6, "Symbol('A_z', commutative=True)"], "Equality(Derivative(log(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(log(Add(Symbol('A_z', commutative=True), Mul(Symbol('b', commutative=True), Function('\\\\mathbf{J}')(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} = \\Psi + \\hat{X} and \\mathbf{D}{(\\hat{X},\\Psi)} = \\int \\Psi \\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} d\\Psi, then obtain \\mathbf{D}{(\\hat{X},\\Psi)} = \\int \\Psi (\\Psi + \\hat{X}) d\\Psi", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} = \\Psi + \\hat{X} and \\Psi \\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} = \\Psi (\\Psi + \\hat{X}) and \\int \\Psi \\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} d\\Psi = \\int \\Psi (\\Psi + \\hat{X}) d\\Psi and \\mathbf{D}{(\\hat{X},\\Psi)} = \\int \\Psi \\operatorname{V_{\\mathbf{B}}}{(\\hat{X},\\Psi)} d\\Psi and \\mathbf{D}{(\\hat{X},\\Psi)} = \\int \\Psi (\\Psi + \\hat{X}) d\\Psi", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given S{(\\rho_b)} = \\rho_b and \\operatorname{V_{\\mathbf{E}}}{(m,\\rho_b)} = \\rho_b - \\frac{1}{m} and \\omega{(m,\\rho_b)} = S{(\\rho_b)} - \\frac{1}{m}, then obtain \\frac{\\omega{(m,\\rho_b)}}{\\int \\sin{(\\dot{z})} d\\dot{z}} = \\frac{\\operatorname{V_{\\mathbf{E}}}{(m,\\rho_b)}}{\\int \\sin{(\\dot{z})} d\\dot{z}}", "derivation": "S{(\\rho_b)} = \\rho_b and S{(\\rho_b)} - \\frac{1}{m} = \\rho_b - \\frac{1}{m} and \\operatorname{V_{\\mathbf{E}}}{(m,\\rho_b)} = \\rho_b - \\frac{1}{m} and \\omega{(m,\\rho_b)} = S{(\\rho_b)} - \\frac{1}{m} and \\omega{(m,\\rho_b)} = \\rho_b - \\frac{1}{m} and \\omega{(m,\\rho_b)} = \\operatorname{V_{\\mathbf{E}}}{(m,\\rho_b)} and \\frac{\\omega{(m,\\rho_b)}}{\\int \\sin{(\\dot{z})} d\\dot{z}} = \\frac{\\operatorname{V_{\\mathbf{E}}}{(m,\\rho_b)}}{\\int \\sin{(\\dot{z})} d\\dot{z}}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], [["minus", 1, "Pow(Symbol('m', commutative=True), Integer(-1))"], "Equality(Add(Function('S')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Function('S')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 6, "Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))), Mul(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given n{(r_{0})} = \\sin{(r_{0})}, then derive \\int n{(r_{0})} dr_{0} = L - \\cos{(r_{0})}, then obtain - L + n{(r_{0})} + \\cos{(r_{0})} = - L + \\sin{(r_{0})} + \\cos{(r_{0})}", "derivation": "n{(r_{0})} = \\sin{(r_{0})} and \\int n{(r_{0})} dr_{0} = \\int \\sin{(r_{0})} dr_{0} and \\int n{(r_{0})} dr_{0} = L - \\cos{(r_{0})} and n{(r_{0})} - \\int \\sin{(r_{0})} dr_{0} = \\sin{(r_{0})} - \\int \\sin{(r_{0})} dr_{0} and n{(r_{0})} - \\int n{(r_{0})} dr_{0} = \\sin{(r_{0})} - \\int n{(r_{0})} dr_{0} and - L + n{(r_{0})} + \\cos{(r_{0})} = - L + \\sin{(r_{0})} + \\cos{(r_{0})}", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('n')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Add(Symbol('L', commutative=True), Mul(Integer(-1), cos(Symbol('r_0', commutative=True)))))"], [["minus", 1, "Integral(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))), Add(sin(Symbol('r_0', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Integral(Function('n')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))), Add(sin(Symbol('r_0', commutative=True)), Mul(Integer(-1), Integral(Function('n')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('n')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\psi{(Z)} = \\sin{(\\sin{(Z)})} and \\operatorname{v_{z}}{(Z)} = \\sin{(\\sin{(Z)})}, then obtain \\int (Z + \\frac{\\psi{(Z)}}{\\sin{(Z)}} - \\operatorname{v_{t}}{(\\hbar)}) d\\hbar = \\int (Z - \\operatorname{v_{t}}{(\\hbar)} + \\frac{\\operatorname{v_{z}}{(Z)}}{\\sin{(Z)}}) d\\hbar", "derivation": "\\psi{(Z)} = \\sin{(\\sin{(Z)})} and \\frac{\\psi{(Z)}}{\\sin{(Z)}} = \\frac{\\sin{(\\sin{(Z)})}}{\\sin{(Z)}} and \\operatorname{v_{z}}{(Z)} = \\sin{(\\sin{(Z)})} and \\operatorname{v_{z}}{(Z)} = \\psi{(Z)} and \\frac{\\operatorname{v_{z}}{(Z)}}{\\sin{(Z)}} = \\frac{\\sin{(\\sin{(Z)})}}{\\sin{(Z)}} and \\frac{\\psi{(Z)}}{\\sin{(Z)}} = \\frac{\\operatorname{v_{z}}{(Z)}}{\\sin{(Z)}} and Z + \\frac{\\psi{(Z)}}{\\sin{(Z)}} - \\operatorname{v_{t}}{(\\hbar)} = Z - \\operatorname{v_{t}}{(\\hbar)} + \\frac{\\operatorname{v_{z}}{(Z)}}{\\sin{(Z)}} and \\int (Z + \\frac{\\psi{(Z)}}{\\sin{(Z)}} - \\operatorname{v_{t}}{(\\hbar)}) d\\hbar = \\int (Z - \\operatorname{v_{t}}{(\\hbar)} + \\frac{\\operatorname{v_{z}}{(Z)}}{\\sin{(Z)}}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('Z', commutative=True)), sin(sin(Symbol('Z', commutative=True))))"], [["divide", 1, "sin(Symbol('Z', commutative=True))"], "Equality(Mul(Function('\\\\psi')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), sin(sin(Symbol('Z', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('Z', commutative=True)), sin(sin(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('v_z')(Symbol('Z', commutative=True)), Function('\\\\psi')(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Function('v_z')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), sin(sin(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Function('\\\\psi')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Function('v_z')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Symbol('Z', commutative=True), Mul(Function('\\\\psi')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('v_t')(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('v_t')(Symbol('\\\\hbar', commutative=True))), Mul(Function('v_z')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)))))"], [["integrate", 7, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Symbol('Z', commutative=True), Mul(Function('\\\\psi')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('v_t')(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('v_t')(Symbol('\\\\hbar', commutative=True))), Mul(Function('v_z')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given g{(x^\\prime,\\varepsilon_0)} = \\varepsilon_0 x^\\prime, then obtain \\frac{\\partial}{\\partial E_{x}} \\frac{\\varepsilon_0 x^\\prime + g^{x^\\prime}{(x^\\prime,\\varepsilon_0)}}{\\operatorname{F_{g}}{(E_{x},c)}} = \\frac{\\partial}{\\partial E_{x}} \\frac{\\varepsilon_0 x^\\prime + (\\varepsilon_0 x^\\prime)^{x^\\prime}}{\\operatorname{F_{g}}{(E_{x},c)}}", "derivation": "g{(x^\\prime,\\varepsilon_0)} = \\varepsilon_0 x^\\prime and g^{x^\\prime}{(x^\\prime,\\varepsilon_0)} = (\\varepsilon_0 x^\\prime)^{x^\\prime} and \\varepsilon_0 x^\\prime + g^{x^\\prime}{(x^\\prime,\\varepsilon_0)} = \\varepsilon_0 x^\\prime + (\\varepsilon_0 x^\\prime)^{x^\\prime} and \\frac{\\varepsilon_0 x^\\prime + g^{x^\\prime}{(x^\\prime,\\varepsilon_0)}}{\\operatorname{F_{g}}{(E_{x},c)}} = \\frac{\\varepsilon_0 x^\\prime + (\\varepsilon_0 x^\\prime)^{x^\\prime}}{\\operatorname{F_{g}}{(E_{x},c)}} and \\frac{\\partial}{\\partial E_{x}} \\frac{\\varepsilon_0 x^\\prime + g^{x^\\prime}{(x^\\prime,\\varepsilon_0)}}{\\operatorname{F_{g}}{(E_{x},c)}} = \\frac{\\partial}{\\partial E_{x}} \\frac{\\varepsilon_0 x^\\prime + (\\varepsilon_0 x^\\prime)^{x^\\prime}}{\\operatorname{F_{g}}{(E_{x},c)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["divide", 3, "Function('F_g')(Symbol('E_x', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('F_g')(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('F_g')(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('F_g')(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('F_g')(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(E_{x},\\theta_2)} = - E_{x} + \\theta_2, then derive 0 = - E_{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + y - \\int L{(E_{x},\\theta_2)} d\\theta_2, then obtain E_{x} \\theta_2 - \\frac{\\theta_2^{2}}{2} - y + \\int L{(E_{x},\\theta_2)} d\\theta_2 = 0", "derivation": "L{(E_{x},\\theta_2)} = - E_{x} + \\theta_2 and \\int L{(E_{x},\\theta_2)} d\\theta_2 = \\int (- E_{x} + \\theta_2) d\\theta_2 and 0 = \\int (- E_{x} + \\theta_2) d\\theta_2 - \\int L{(E_{x},\\theta_2)} d\\theta_2 and 0 = - E_{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + y - \\int L{(E_{x},\\theta_2)} d\\theta_2 and 0 = - E_{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + y - \\int (- E_{x} + \\theta_2) d\\theta_2 and E_{x} \\theta_2 - \\frac{\\theta_2^{2}}{2} - y + \\int (- E_{x} + \\theta_2) d\\theta_2 = 0 and E_{x} \\theta_2 - \\frac{\\theta_2^{2}}{2} - y + \\int L{(E_{x},\\theta_2)} d\\theta_2 = 0", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 2, "Integral(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Integral(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('y', commutative=True), Mul(Integer(-1), Integral(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('y', commutative=True), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('y', commutative=True), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], "Equality(Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('y', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('y', commutative=True)), Integral(Function('L')(Symbol('E_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"]]}, {"prompt": "Given t{(b)} = \\log{(b)}, then obtain \\frac{((b + t{(b)}) \\log{(b)})^{b} ((b + \\log{(b)}) \\log{(b)})^{- b}}{(b + \\log{(b)}) \\log{(b)}} = \\frac{1}{(b + \\log{(b)}) \\log{(b)}}", "derivation": "t{(b)} = \\log{(b)} and b + t{(b)} = b + \\log{(b)} and (b + t{(b)}) \\log{(b)} = (b + \\log{(b)}) \\log{(b)} and ((b + t{(b)}) \\log{(b)})^{b} = ((b + \\log{(b)}) \\log{(b)})^{b} and \\frac{((b + t{(b)}) \\log{(b)})^{b}}{(b + \\log{(b)}) \\log{(b)}} = \\frac{((b + \\log{(b)}) \\log{(b)})^{b}}{(b + \\log{(b)}) \\log{(b)}} and \\frac{((b + t{(b)}) \\log{(b)})^{b} ((b + \\log{(b)}) \\log{(b)})^{- b}}{(b + \\log{(b)}) \\log{(b)}} = \\frac{1}{(b + \\log{(b)}) \\log{(b)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('t')(Symbol('b', commutative=True))), Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))))"], [["times", 2, "log(Symbol('b', commutative=True))"], "Equality(Mul(Add(Symbol('b', commutative=True), Function('t')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('b', commutative=True), Function('t')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["divide", 4, "Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Mul(Add(Symbol('b', commutative=True), Function('t')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Integer(-1)), Pow(log(Symbol('b', commutative=True)), Integer(-1))), Mul(Pow(Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Integer(-1)), Pow(log(Symbol('b', commutative=True)), Integer(-1))))"], [["divide", 5, "Pow(Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True))"], "Equality(Mul(Pow(Mul(Add(Symbol('b', commutative=True), Function('t')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Mul(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Pow(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Integer(-1)), Pow(log(Symbol('b', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Integer(-1)), Pow(log(Symbol('b', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\theta{(J,F_{c})} = F_{c}^{J}, then derive J \\frac{\\partial}{\\partial F_{c}} \\theta{(J,F_{c})} = \\frac{F_{c}^{J} J^{2}}{F_{c}}, then obtain J \\frac{\\partial}{\\partial F_{c}} J \\theta{(J,F_{c})} \\frac{\\partial}{\\partial F_{c}} \\theta{(J,F_{c})} = \\frac{J^{2} \\theta{(J,F_{c})} \\frac{\\partial}{\\partial F_{c}} J \\theta{(J,F_{c})}}{F_{c}}", "derivation": "\\theta{(J,F_{c})} = F_{c}^{J} and J \\theta{(J,F_{c})} = F_{c}^{J} J and \\frac{\\partial}{\\partial F_{c}} J \\theta{(J,F_{c})} = \\frac{\\partial}{\\partial F_{c}} F_{c}^{J} J and J \\frac{\\partial}{\\partial F_{c}} \\theta{(J,F_{c})} = \\frac{F_{c}^{J} J^{2}}{F_{c}} and J \\frac{\\partial}{\\partial F_{c}} F_{c}^{J} J \\frac{\\partial}{\\partial F_{c}} \\theta{(J,F_{c})} = \\frac{F_{c}^{J} J^{2} \\frac{\\partial}{\\partial F_{c}} F_{c}^{J} J}{F_{c}} and J \\frac{\\partial}{\\partial F_{c}} J \\theta{(J,F_{c})} \\frac{\\partial}{\\partial F_{c}} \\theta{(J,F_{c})} = \\frac{J^{2} \\theta{(J,F_{c})} \\frac{\\partial}{\\partial F_{c}} J \\theta{(J,F_{c})}}{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Mul(Symbol('J', commutative=True), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('J', commutative=True), Derivative(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(2))))"], [["times", 4, "Derivative(Mul(Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('J', commutative=True), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(2)), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('J', commutative=True), Derivative(Mul(Symbol('J', commutative=True), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('J', commutative=True), Integer(2)), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True)), Derivative(Mul(Symbol('J', commutative=True), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(v_{t},g)} = - g + v_{t} and a{(v_{t},g)} = 2 \\mathbf{J}_M{(v_{t},g)}, then obtain (- v_{t} + \\mathbf{J}_M{(v_{t},g)}) a{(v_{t},g)} = - g a{(v_{t},g)}", "derivation": "\\mathbf{J}_M{(v_{t},g)} = - g + v_{t} and - v_{t} + \\mathbf{J}_M{(v_{t},g)} = - g and 2 \\mathbf{J}_M{(v_{t},g)} = - g + v_{t} + \\mathbf{J}_M{(v_{t},g)} and (- v_{t} + \\mathbf{J}_M{(v_{t},g)}) (- g + v_{t} + \\mathbf{J}_M{(v_{t},g)}) = - g (- g + v_{t} + \\mathbf{J}_M{(v_{t},g)}) and a{(v_{t},g)} = 2 \\mathbf{J}_M{(v_{t},g)} and a{(v_{t},g)} = - g + v_{t} + \\mathbf{J}_M{(v_{t},g)} and (- v_{t} + \\mathbf{J}_M{(v_{t},g)}) a{(v_{t},g)} = - g a{(v_{t},g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True)))"], [["minus", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["add", 2, "Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)))))"], ["renaming_premise", "Equality(Function('a')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('a')(Symbol('v_t', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_t', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))), Function('a')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True), Function('a')(Symbol('v_t', commutative=True), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\mu{(p)} = \\cos{(e^{p})}, then obtain - \\mu{(p)} + 2 \\mu^{p}{(p)} = - \\mu{(p)} + \\mu^{p}{(p)} + \\cos^{p}{(e^{p})}", "derivation": "\\mu{(p)} = \\cos{(e^{p})} and \\mu^{p}{(p)} = \\cos^{p}{(e^{p})} and 2 \\mu^{p}{(p)} = \\mu^{p}{(p)} + \\cos^{p}{(e^{p})} and - \\mu{(p)} + 2 \\mu^{p}{(p)} = - \\mu{(p)} + \\mu^{p}{(p)} + \\cos^{p}{(e^{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["add", 2, "Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(exp(Symbol('p', commutative=True))), Symbol('p', commutative=True))))"], [["minus", 3, "Function('\\\\mu')(Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('p', commutative=True))), Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('p', commutative=True))), Pow(Function('\\\\mu')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(exp(Symbol('p', commutative=True))), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(E_{x},C_{d})} = C_{d} E_{x} and \\theta{(E_{x},C_{d})} = \\sin{(E_{x} + \\operatorname{F_{g}}{(E_{x},C_{d})})}, then obtain \\frac{\\theta^{C_{d}}{(E_{x},C_{d})}}{E_{x}} = \\frac{\\sin^{C_{d}}{(C_{d} E_{x} + E_{x})}}{E_{x}}", "derivation": "\\operatorname{F_{g}}{(E_{x},C_{d})} = C_{d} E_{x} and E_{x} + \\operatorname{F_{g}}{(E_{x},C_{d})} = C_{d} E_{x} + E_{x} and \\theta{(E_{x},C_{d})} = \\sin{(E_{x} + \\operatorname{F_{g}}{(E_{x},C_{d})})} and \\theta{(E_{x},C_{d})} = \\sin{(C_{d} E_{x} + E_{x})} and \\theta^{C_{d}}{(E_{x},C_{d})} = \\sin^{C_{d}}{(C_{d} E_{x} + E_{x})} and \\frac{\\theta^{C_{d}}{(E_{x},C_{d})}}{E_{x}} = \\frac{\\sin^{C_{d}}{(C_{d} E_{x} + E_{x})}}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('E_x', commutative=True)))"], [["add", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Function('F_g')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)), sin(Add(Symbol('E_x', commutative=True), Function('F_g')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\theta')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)), sin(Add(Mul(Symbol('C_d', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))))"], [["power", 4, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(sin(Add(Mul(Symbol('C_d', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Symbol('C_d', commutative=True)))"], [["divide", 5, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Function('\\\\theta')(Symbol('E_x', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(sin(Add(Mul(Symbol('C_d', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(W)} = \\cos{(W)} and \\operatorname{C_{2}}{(W)} = \\frac{\\cos{(W)}}{W}, then obtain W + \\frac{1}{2} + \\frac{2 \\operatorname{n_{2}}{(W)}}{W} = W + 2 \\operatorname{C_{2}}{(W)} + \\frac{1}{2}", "derivation": "\\operatorname{n_{2}}{(W)} = \\cos{(W)} and \\frac{\\operatorname{n_{2}}{(W)}}{W} = \\frac{\\cos{(W)}}{W} and W + \\frac{\\operatorname{n_{2}}{(W)}}{W} = W + \\frac{\\cos{(W)}}{W} and W + \\frac{\\operatorname{n_{2}}{(W)}}{W} + \\frac{\\cos{(W)}}{W} = W + \\frac{2 \\cos{(W)}}{W} and W + \\frac{2 \\operatorname{n_{2}}{(W)}}{W} = W + \\frac{2 \\cos{(W)}}{W} and \\operatorname{C_{2}}{(W)} = \\frac{\\cos{(W)}}{W} and W + \\frac{2 \\operatorname{n_{2}}{(W)}}{W} = W + 2 \\operatorname{C_{2}}{(W)} and W + \\frac{1}{2} + \\frac{2 \\operatorname{n_{2}}{(W)}}{W} = W + 2 \\operatorname{C_{2}}{(W)} + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["divide", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))))"], [["add", 2, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))"], "Equality(Add(Symbol('W', commutative=True), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Integer(2), Function('C_2')(Symbol('W', commutative=True)))))"], [["add", 7, "Rational(1, 2)"], "Equality(Add(Symbol('W', commutative=True), Rational(1, 2), Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-1)), Function('n_2')(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Integer(2), Function('C_2')(Symbol('W', commutative=True))), Rational(1, 2)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})} = e^{\\mathbf{A} + \\mathbf{E}} and s{(\\mathbf{E},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}}}, then obtain 1 = \\frac{\\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}}}{\\mathbf{A} + \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})} = e^{\\mathbf{A} + \\mathbf{E}} and \\mathbf{A} + \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})} = \\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}} and s{(\\mathbf{E},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}}} and s{(\\mathbf{E},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})}} and \\frac{1}{\\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}}} = \\frac{1}{\\mathbf{A} + \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})}} and 1 = \\frac{\\mathbf{A} + e^{\\mathbf{A} + \\mathbf{E}}}{\\mathbf{A} + \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"], [["divide", 5, "Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Integer(-1))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(i)} = e^{i}, then obtain \\frac{d}{d i} \\cos{(\\operatorname{M_{E}}^{3}{(i)})} = \\frac{d}{d i} \\cos{(\\operatorname{M_{E}}{(i)} e^{2 i})}", "derivation": "\\operatorname{M_{E}}{(i)} = e^{i} and \\operatorname{M_{E}}^{2}{(i)} = \\operatorname{M_{E}}{(i)} e^{i} and \\operatorname{M_{E}}^{4}{(i)} = \\operatorname{M_{E}}^{2}{(i)} e^{2 i} and \\operatorname{M_{E}}^{3}{(i)} = \\operatorname{M_{E}}{(i)} e^{2 i} and \\cos{(\\operatorname{M_{E}}^{3}{(i)})} = \\cos{(\\operatorname{M_{E}}{(i)} e^{2 i})} and \\frac{d}{d i} \\cos{(\\operatorname{M_{E}}^{3}{(i)})} = \\frac{d}{d i} \\cos{(\\operatorname{M_{E}}{(i)} e^{2 i})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["times", 1, "Function('M_E')(Symbol('i', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(2)), Mul(Function('M_E')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(4)), Mul(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('i', commutative=True)))))"], [["divide", 3, "Function('M_E')(Symbol('i', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(3)), Mul(Function('M_E')(Symbol('i', commutative=True)), exp(Mul(Integer(2), Symbol('i', commutative=True)))))"], [["cos", 4], "Equality(cos(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(3))), cos(Mul(Function('M_E')(Symbol('i', commutative=True)), exp(Mul(Integer(2), Symbol('i', commutative=True))))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(cos(Pow(Function('M_E')(Symbol('i', commutative=True)), Integer(3))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(Mul(Function('M_E')(Symbol('i', commutative=True)), exp(Mul(Integer(2), Symbol('i', commutative=True))))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(Q,C_{d})} = - C_{d} + \\sin{(Q)}, then derive \\frac{\\partial}{\\partial Q} \\phi_{2}{(Q,C_{d})} = \\cos{(Q)}, then obtain - \\frac{\\partial}{\\partial Q} (- C_{d} + \\sin{(Q)}) = - \\cos{(Q)}", "derivation": "\\phi_{2}{(Q,C_{d})} = - C_{d} + \\sin{(Q)} and - C_{d} + \\phi_{2}{(Q,C_{d})} = - 2 C_{d} + \\sin{(Q)} and \\frac{\\partial}{\\partial Q} (- C_{d} + \\phi_{2}{(Q,C_{d})}) = \\frac{\\partial}{\\partial Q} (- 2 C_{d} + \\sin{(Q)}) and \\frac{\\partial}{\\partial Q} \\phi_{2}{(Q,C_{d})} = \\cos{(Q)} and - \\frac{\\partial}{\\partial Q} \\phi_{2}{(Q,C_{d})} = - \\cos{(Q)} and - \\frac{\\partial}{\\partial Q} (- C_{d} + \\sin{(Q)}) = - \\cos{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('Q', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["minus", 1, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Function('\\\\phi_2')(Symbol('Q', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_d', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Function('\\\\phi_2')(Symbol('Q', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('C_d', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi_2')(Symbol('Q', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), cos(Symbol('Q', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('Q', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given r{(\\tilde{g})} = \\sin{(\\tilde{g})}, then obtain r^{3}{(\\tilde{g})} \\sin{(\\tilde{g})} = \\sin^{4}{(\\tilde{g})}", "derivation": "r{(\\tilde{g})} = \\sin{(\\tilde{g})} and r{(\\tilde{g})} \\sin{(\\tilde{g})} = \\sin^{2}{(\\tilde{g})} and r^{2}{(\\tilde{g})} \\sin^{2}{(\\tilde{g})} = \\sin^{4}{(\\tilde{g})} and r^{3}{(\\tilde{g})} \\sin{(\\tilde{g})} = r^{2}{(\\tilde{g})} \\sin^{2}{(\\tilde{g})} and r^{3}{(\\tilde{g})} \\sin{(\\tilde{g})} = \\sin^{4}{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True))), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Integer(3)), sin(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Integer(3)), sin(Symbol('\\\\tilde{g}', commutative=True))), Pow(sin(Symbol('\\\\tilde{g}', commutative=True)), Integer(4)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)} = (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} and \\operatorname{f_{E}}{(a^{\\dagger})} = 0^{a^{\\dagger}}, then obtain \\operatorname{f_{E}}{(a^{\\dagger})} = (- (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)})^{a^{\\dagger}}", "derivation": "\\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)} = (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} and - (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)} = 0 and (- (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)})^{a^{\\dagger}} = 0^{a^{\\dagger}} and \\operatorname{f_{E}}{(a^{\\dagger})} = 0^{a^{\\dagger}} and \\operatorname{f_{E}}{(a^{\\dagger})} = (- (\\frac{a^{\\dagger}}{L})^{\\dot{\\mathbf{r}}} + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},a^{\\dagger},L)})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["minus", 1, "Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('L', commutative=True))), Integer(0))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('L', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('L', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\chi{(Q,M)} = M Q and \\eta^{\\prime}{(M)} = M^{2}, then obtain (M Q \\chi{(Q,M)})^{M} = (Q^{2} \\eta^{\\prime}{(M)})^{M}", "derivation": "\\chi{(Q,M)} = M Q and M Q \\chi{(Q,M)} = M^{2} Q^{2} and \\eta^{\\prime}{(M)} = M^{2} and (M Q \\chi{(Q,M)})^{M} = (M^{2} Q^{2})^{M} and (M Q \\chi{(Q,M)})^{M} = (Q^{2} \\eta^{\\prime}{(M)})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('Q', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('Q', commutative=True)))"], [["times", 1, "Mul(Symbol('M', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Symbol('M', commutative=True), Symbol('Q', commutative=True), Function('\\\\chi')(Symbol('Q', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('Q', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('M', commutative=True)), Pow(Symbol('M', commutative=True), Integer(2)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Symbol('M', commutative=True), Symbol('Q', commutative=True), Function('\\\\chi')(Symbol('Q', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Symbol('M', commutative=True), Symbol('Q', commutative=True), Function('\\\\chi')(Symbol('Q', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Function('\\\\eta^{\\\\prime}')(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},\\rho)} = \\int A_{2} \\rho dA_{2}, then obtain (\\frac{\\partial^{2}}{\\partial \\rho^{2}} \\mathbf{S}^{\\rho}{(A_{2},\\rho)})^{A_{2}} = (\\frac{\\partial^{2}}{\\partial \\rho^{2}} (\\int A_{2} \\rho dA_{2})^{\\rho})^{A_{2}}", "derivation": "\\mathbf{S}{(A_{2},\\rho)} = \\int A_{2} \\rho dA_{2} and \\mathbf{S}^{\\rho}{(A_{2},\\rho)} = (\\int A_{2} \\rho dA_{2})^{\\rho} and \\frac{\\partial}{\\partial \\rho} \\mathbf{S}^{\\rho}{(A_{2},\\rho)} = \\frac{\\partial}{\\partial \\rho} (\\int A_{2} \\rho dA_{2})^{\\rho} and \\frac{\\partial^{2}}{\\partial \\rho^{2}} \\mathbf{S}^{\\rho}{(A_{2},\\rho)} = \\frac{\\partial^{2}}{\\partial \\rho^{2}} (\\int A_{2} \\rho dA_{2})^{\\rho} and (\\frac{\\partial^{2}}{\\partial \\rho^{2}} \\mathbf{S}^{\\rho}{(A_{2},\\rho)})^{A_{2}} = (\\frac{\\partial^{2}}{\\partial \\rho^{2}} (\\int A_{2} \\rho dA_{2})^{\\rho})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))), Derivative(Pow(Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))))"], [["power", 4, "Symbol('A_2', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))), Symbol('A_2', commutative=True)), Pow(Derivative(Pow(Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))), Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\theta)} = \\sin{(\\theta)}, then derive \\int \\sigma_{x}{(\\theta)} d\\theta = E_{n} - \\cos{(\\theta)}, then obtain - E_{n} + \\cos{(\\theta)} + \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = - E_{n} + 2 \\cos{(\\theta)}", "derivation": "\\sigma_{x}{(\\theta)} = \\sin{(\\theta)} and \\int \\sigma_{x}{(\\theta)} d\\theta = \\int \\sin{(\\theta)} d\\theta and \\int \\sigma_{x}{(\\theta)} d\\theta = E_{n} - \\cos{(\\theta)} and \\int \\sin{(\\theta)} d\\theta = E_{n} - \\cos{(\\theta)} and \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\theta)} and \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} - \\int \\sin{(\\theta)} d\\theta = \\frac{d}{d \\theta} \\sin{(\\theta)} - \\int \\sin{(\\theta)} d\\theta and - E_{n} + \\cos{(\\theta)} + \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = - E_{n} + \\cos{(\\theta)} + \\frac{d}{d \\theta} \\sin{(\\theta)} and - E_{n} + \\cos{(\\theta)} + \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} = - E_{n} + 2 \\cos{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 5, "Integral(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))), Add(Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), cos(Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), cos(Symbol('\\\\theta', commutative=True)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 7], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), cos(Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(A_{y},W)} = \\int (A_{y} + W) dW and \\hat{H}{(A_{y})} = A_{y}, then derive \\varepsilon_{0}{(A_{y},W)} = A_{y} W + \\frac{W^{2}}{2} + x^\\prime, then derive V_{\\mathbf{B}} + \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})} = \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})} + x^\\prime, then obtain \\int (W + \\hat{H}{(A_{y})}) dW = V_{\\mathbf{B}} + \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})}", "derivation": "\\varepsilon_{0}{(A_{y},W)} = \\int (A_{y} + W) dW and \\hat{H}{(A_{y})} = A_{y} and \\varepsilon_{0}{(A_{y},W)} = A_{y} W + \\frac{W^{2}}{2} + x^\\prime and \\int (A_{y} + W) dW = A_{y} W + \\frac{W^{2}}{2} + x^\\prime and \\int (W + \\hat{H}{(A_{y})}) dW = \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})} + x^\\prime and V_{\\mathbf{B}} + \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})} = \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})} + x^\\prime and \\int (W + \\hat{H}{(A_{y})}) dW = V_{\\mathbf{B}} + \\frac{W^{2}}{2} + W \\hat{H}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Integral(Add(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["evaluate_integrals", 1], "Equality(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Add(Mul(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Symbol('W', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Add(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True))), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Add(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True))), Tuple(Symbol('W', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Function('\\\\hat{H}')(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given n{(\\rho_f)} = \\sin{(e^{\\rho_f})}, then derive \\int n{(\\rho_f)} d\\rho_f = T + \\operatorname{Si}{(e^{\\rho_f})}, then obtain \\frac{d}{d T} \\int \\sin{(e^{\\rho_f})} d\\rho_f = 1", "derivation": "n{(\\rho_f)} = \\sin{(e^{\\rho_f})} and \\int n{(\\rho_f)} d\\rho_f = \\int \\sin{(e^{\\rho_f})} d\\rho_f and \\int n{(\\rho_f)} d\\rho_f = T + \\operatorname{Si}{(e^{\\rho_f})} and \\int \\sin{(e^{\\rho_f})} d\\rho_f = T + \\operatorname{Si}{(e^{\\rho_f})} and \\frac{d}{d T} \\int \\sin{(e^{\\rho_f})} d\\rho_f = \\frac{\\partial}{\\partial T} (T + \\operatorname{Si}{(e^{\\rho_f})}) and \\frac{d}{d T} \\int \\sin{(e^{\\rho_f})} d\\rho_f = 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\rho_f', commutative=True)), sin(exp(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(sin(exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('T', commutative=True), Si(exp(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('T', commutative=True), Si(exp(Symbol('\\\\rho_f', commutative=True)))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Integral(sin(exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Si(exp(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(sin(exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given E{(Z)} = \\cos{(Z)}, then obtain \\frac{A_{y}^{2} \\frac{d}{d Z} E{(Z)}}{A_{y}^{2} E{(Z)} + 1} = - \\frac{A_{y}^{2} \\sin{(Z)}}{A_{y}^{2} \\cos{(Z)} + 1}", "derivation": "E{(Z)} = \\cos{(Z)} and A_{y}^{2} E{(Z)} = A_{y}^{2} \\cos{(Z)} and A_{y}^{2} E{(Z)} + 1 = A_{y}^{2} \\cos{(Z)} + 1 and \\log{(A_{y}^{2} E{(Z)} + 1)} = \\log{(A_{y}^{2} \\cos{(Z)} + 1)} and \\frac{\\partial}{\\partial Z} \\log{(A_{y}^{2} E{(Z)} + 1)} = \\frac{\\partial}{\\partial Z} \\log{(A_{y}^{2} \\cos{(Z)} + 1)} and \\frac{A_{y}^{2} \\frac{d}{d Z} E{(Z)}}{A_{y}^{2} E{(Z)} + 1} = - \\frac{A_{y}^{2} \\sin{(Z)}}{A_{y}^{2} \\cos{(Z)} + 1}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["divide", 1, "Pow(Symbol('A_y', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('E')(Symbol('Z', commutative=True))), Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), cos(Symbol('Z', commutative=True))))"], [["add", 2, 1], "Equality(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('E')(Symbol('Z', commutative=True))), Integer(1)), Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), cos(Symbol('Z', commutative=True))), Integer(1)))"], [["log", 3], "Equality(log(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('E')(Symbol('Z', commutative=True))), Integer(1))), log(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), cos(Symbol('Z', commutative=True))), Integer(1))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(log(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('E')(Symbol('Z', commutative=True))), Integer(1))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(log(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), cos(Symbol('Z', commutative=True))), Integer(1))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('E')(Symbol('Z', commutative=True))), Integer(1)), Integer(-1)), Derivative(Function('E')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), cos(Symbol('Z', commutative=True))), Integer(1)), Integer(-1)), sin(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(l)} = \\log{(l)}, then derive \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = \\frac{\\rho_{b}{(l)}}{l}, then obtain \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} - \\frac{d}{d l} \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = - \\frac{d}{d l} \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} + \\frac{\\log{(l)}}{l}", "derivation": "\\rho_{b}{(l)} = \\log{(l)} and \\frac{\\rho_{b}{(l)}}{l} = \\frac{\\log{(l)}}{l} and \\frac{d}{d l} \\rho_{b}{(l)} = \\frac{d}{d l} \\log{(l)} and \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = \\rho_{b}{(l)} \\frac{d}{d l} \\log{(l)} and \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = \\frac{\\rho_{b}{(l)}}{l} and \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = \\frac{\\log{(l)}}{l} and \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} - \\frac{d}{d l} \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} = - \\frac{d}{d l} \\rho_{b}{(l)} \\frac{d}{d l} \\rho_{b}{(l)} + \\frac{\\log{(l)}}{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["divide", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('l', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 3, "Function('\\\\rho_b')(Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('l', commutative=True))))"], [["minus", 6, "Derivative(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Mul(Function('\\\\rho_b')(Symbol('l', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(q,\\mathbf{r})} = e^{\\mathbf{r} - q}, then derive \\int \\tilde{g}^*{(q,\\mathbf{r})} dq = \\dot{y} - e^{\\mathbf{r} - q}, then obtain \\int \\tilde{g}^*{(q,\\mathbf{r})} dq = \\dot{y} - \\tilde{g}^*{(q,\\mathbf{r})}", "derivation": "\\tilde{g}^*{(q,\\mathbf{r})} = e^{\\mathbf{r} - q} and \\int \\tilde{g}^*{(q,\\mathbf{r})} dq = \\int e^{\\mathbf{r} - q} dq and \\int \\tilde{g}^*{(q,\\mathbf{r})} dq = \\dot{y} - e^{\\mathbf{r} - q} and \\int \\tilde{g}^*{(q,\\mathbf{r})} dq = \\dot{y} - \\tilde{g}^*{(q,\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('q', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('q', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\nabla)} = e^{e^{\\nabla}}, then obtain \\int \\operatorname{J_{\\varepsilon}}{(\\nabla)} d\\nabla = \\operatorname{Ei}{(e^{\\nabla})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\nabla)} = e^{e^{\\nabla}} and \\int \\operatorname{J_{\\varepsilon}}{(\\nabla)} d\\nabla = \\int e^{e^{\\nabla}} d\\nabla and \\iint \\operatorname{J_{\\varepsilon}}{(\\nabla)} d\\nabla d\\nabla = \\iint e^{e^{\\nabla}} d\\nabla d\\nabla and \\frac{d}{d \\nabla} \\iint \\operatorname{J_{\\varepsilon}}{(\\nabla)} d\\nabla d\\nabla = \\frac{d}{d \\nabla} \\iint e^{e^{\\nabla}} d\\nabla d\\nabla and \\int \\operatorname{J_{\\varepsilon}}{(\\nabla)} d\\nabla = \\operatorname{Ei}{(e^{\\nabla})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), exp(exp(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(exp(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(exp(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Integral(exp(exp(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Ei(exp(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbf{f})} = \\log{(\\log{(\\mathbf{f})})} and \\varepsilon{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\log{(\\mathbf{f})})}, then obtain \\log{(\\varepsilon{(\\mathbf{f})})} = \\log{(\\frac{d}{d \\mathbf{f}} \\mathbf{J}_P{(\\mathbf{f})})}", "derivation": "\\mathbf{J}_P{(\\mathbf{f})} = \\log{(\\log{(\\mathbf{f})})} and \\frac{d}{d \\mathbf{f}} \\mathbf{J}_P{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\log{(\\mathbf{f})})} and \\varepsilon{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\log{(\\mathbf{f})})} and \\log{(\\varepsilon{(\\mathbf{f})})} = \\log{(\\frac{d}{d \\mathbf{f}} \\log{(\\log{(\\mathbf{f})})})} and \\log{(\\varepsilon{(\\mathbf{f})})} = \\log{(\\frac{d}{d \\mathbf{f}} \\mathbf{J}_P{(\\mathbf{f})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{f}', commutative=True)), log(log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{f}', commutative=True)), Derivative(log(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Function('\\\\varepsilon')(Symbol('\\\\mathbf{f}', commutative=True))), log(Derivative(log(log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(log(Function('\\\\varepsilon')(Symbol('\\\\mathbf{f}', commutative=True))), log(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})} = \\mathbf{A} + x^\\prime, then obtain \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})} - 1 = \\frac{(\\mathbf{A} + x^\\prime - 1) \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})}}{\\mathbf{A} + x^\\prime}", "derivation": "\\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})} = \\mathbf{A} + x^\\prime and \\frac{\\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})}}{\\mathbf{A} + x^\\prime} = 1 and \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})} - 1 = \\mathbf{A} + x^\\prime - 1 and \\frac{(\\mathbf{A} + x^\\prime - 1) \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})}}{\\mathbf{A} + x^\\prime} = \\mathbf{A} + x^\\prime - 1 and \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})} - 1 = \\frac{(\\mathbf{A} + x^\\prime - 1) \\operatorname{A_{2}}{(x^\\prime,\\mathbf{A})}}{\\mathbf{A} + x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["times", 2, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('A_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\hat{p},A_{y})} = \\hat{p} e^{A_{y}} and V{(F_{g},y)} = \\cos{(\\frac{y}{F_{g}})}, then obtain ((- \\hat{p} + \\hat{H}{(\\hat{p},A_{y})}) V{(F_{g},y)})^{y} = ((- \\hat{p} + \\hat{H}{(\\hat{p},A_{y})}) \\cos{(\\frac{y}{F_{g}})})^{y}", "derivation": "\\hat{H}{(\\hat{p},A_{y})} = \\hat{p} e^{A_{y}} and V{(F_{g},y)} = \\cos{(\\frac{y}{F_{g}})} and (\\hat{p} e^{A_{y}} - \\hat{p}) V{(F_{g},y)} = (\\hat{p} e^{A_{y}} - \\hat{p}) \\cos{(\\frac{y}{F_{g}})} and ((\\hat{p} e^{A_{y}} - \\hat{p}) V{(F_{g},y)})^{y} = ((\\hat{p} e^{A_{y}} - \\hat{p}) \\cos{(\\frac{y}{F_{g}})})^{y} and ((- \\hat{p} + \\hat{H}{(\\hat{p},A_{y})}) V{(F_{g},y)})^{y} = ((- \\hat{p} + \\hat{H}{(\\hat{p},A_{y})}) \\cos{(\\frac{y}{F_{g}})})^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))))"], ["get_premise", "Equality(Function('V')(Symbol('F_g', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["times", 2, "Add(Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Function('V')(Symbol('F_g', commutative=True), Symbol('y', commutative=True))), Mul(Add(Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)))))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Function('V')(Symbol('F_g', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Add(Mul(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True))), Function('V')(Symbol('F_g', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True))), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{y})} = \\sin{(A_{y})}, then obtain \\operatorname{A_{x}}^{2}{(A_{y})} + \\operatorname{A_{x}}^{A_{y}}{(A_{y})} - \\sin{(A_{y})} = \\operatorname{A_{x}}^{2}{(A_{y})} - \\sin{(A_{y})} + \\sin^{A_{y}}{(A_{y})}", "derivation": "\\operatorname{A_{x}}{(A_{y})} = \\sin{(A_{y})} and \\operatorname{A_{x}}^{2}{(A_{y})} = \\operatorname{A_{x}}{(A_{y})} \\sin{(A_{y})} and \\operatorname{A_{x}}^{A_{y}}{(A_{y})} = \\sin^{A_{y}}{(A_{y})} and \\operatorname{A_{x}}{(A_{y})} \\sin{(A_{y})} + \\operatorname{A_{x}}^{A_{y}}{(A_{y})} - \\sin{(A_{y})} = \\operatorname{A_{x}}{(A_{y})} \\sin{(A_{y})} - \\sin{(A_{y})} + \\sin^{A_{y}}{(A_{y})} and \\operatorname{A_{x}}^{2}{(A_{y})} + \\operatorname{A_{x}}^{A_{y}}{(A_{y})} - \\sin{(A_{y})} = \\operatorname{A_{x}}^{2}{(A_{y})} - \\sin{(A_{y})} + \\sin^{A_{y}}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["times", 1, "Function('A_x')(Symbol('A_y', commutative=True))"], "Equality(Pow(Function('A_x')(Symbol('A_y', commutative=True)), Integer(2)), Mul(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(sin(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["add", 3, "Add(Mul(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], "Equality(Add(Mul(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))), Pow(Function('A_x')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), Add(Mul(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))), Pow(sin(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('A_x')(Symbol('A_y', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), Add(Pow(Function('A_x')(Symbol('A_y', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))), Pow(sin(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(\\lambda)} = e^{\\sin{(\\lambda)}}, then obtain \\frac{\\int (\\lambda + \\mu_{0}{(\\lambda)}) d\\lambda}{\\lambda + \\mu_{0}{(\\lambda)}} = \\frac{\\int (\\lambda + e^{\\sin{(\\lambda)}}) d\\lambda}{\\lambda + \\mu_{0}{(\\lambda)}}", "derivation": "\\mu_{0}{(\\lambda)} = e^{\\sin{(\\lambda)}} and \\lambda + \\mu_{0}{(\\lambda)} = \\lambda + e^{\\sin{(\\lambda)}} and \\int (\\lambda + \\mu_{0}{(\\lambda)}) d\\lambda = \\int (\\lambda + e^{\\sin{(\\lambda)}}) d\\lambda and \\frac{\\int (\\lambda + \\mu_{0}{(\\lambda)}) d\\lambda}{\\lambda + \\mu_{0}{(\\lambda)}} = \\frac{\\int (\\lambda + e^{\\sin{(\\lambda)}}) d\\lambda}{\\lambda + \\mu_{0}{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True)), exp(sin(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), exp(sin(Symbol('\\\\lambda', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), exp(sin(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mu_0')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\lambda', commutative=True), exp(sin(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given S{(r,F_{g})} = e^{r^{F_{g}}}, then derive F_{g} \\frac{\\partial}{\\partial F_{g}} S{(r,F_{g})} = F_{g} r^{F_{g}} e^{r^{F_{g}}} \\log{(r)}, then obtain F_{g} r^{F_{g}} e^{r^{F_{g}}} \\log{(r)} = F_{g} r^{F_{g}} S{(r,F_{g})} \\log{(r)}", "derivation": "S{(r,F_{g})} = e^{r^{F_{g}}} and \\frac{\\partial}{\\partial F_{g}} S{(r,F_{g})} = \\frac{\\partial}{\\partial F_{g}} e^{r^{F_{g}}} and F_{g} \\frac{\\partial}{\\partial F_{g}} S{(r,F_{g})} = F_{g} \\frac{\\partial}{\\partial F_{g}} e^{r^{F_{g}}} and F_{g} \\frac{\\partial}{\\partial F_{g}} S{(r,F_{g})} = F_{g} r^{F_{g}} e^{r^{F_{g}}} \\log{(r)} and F_{g} \\frac{\\partial}{\\partial F_{g}} S{(r,F_{g})} = F_{g} r^{F_{g}} S{(r,F_{g})} \\log{(r)} and F_{g} \\frac{\\partial}{\\partial F_{g}} e^{r^{F_{g}}} = F_{g} r^{F_{g}} S{(r,F_{g})} \\log{(r)} and F_{g} r^{F_{g}} e^{r^{F_{g}}} \\log{(r)} = F_{g} r^{F_{g}} S{(r,F_{g})} \\log{(r)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["times", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Derivative(Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Symbol('F_g', commutative=True), Derivative(exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('F_g', commutative=True), Derivative(Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))), log(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('F_g', commutative=True), Derivative(Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('F_g', commutative=True), Derivative(exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('r', commutative=True))))"], [["evaluate_derivatives", 6], "Equality(Mul(Symbol('F_g', commutative=True), Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), exp(Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True))), log(Symbol('r', commutative=True))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\delta,C_{2})} = \\frac{e^{\\delta}}{C_{2}}, then obtain \\frac{\\frac{\\operatorname{P_{g}}{(\\delta,C_{2})}}{C_{2}} - \\frac{1}{C_{2}^{2}}}{\\frac{e^{\\delta}}{C_{2}^{2}} - \\frac{1}{C_{2}^{2}}} = 1", "derivation": "\\operatorname{P_{g}}{(\\delta,C_{2})} = \\frac{e^{\\delta}}{C_{2}} and \\frac{\\operatorname{P_{g}}{(\\delta,C_{2})}}{C_{2}} = \\frac{e^{\\delta}}{C_{2}^{2}} and \\frac{\\operatorname{P_{g}}{(\\delta,C_{2})}}{C_{2}} - \\frac{1}{C_{2}^{2}} = \\frac{e^{\\delta}}{C_{2}^{2}} - \\frac{1}{C_{2}^{2}} and \\frac{\\frac{\\operatorname{P_{g}}{(\\delta,C_{2})}}{C_{2}} - \\frac{1}{C_{2}^{2}}}{\\frac{e^{\\delta}}{C_{2}^{2}} - \\frac{1}{C_{2}^{2}}} = 1", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), exp(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Pow(Symbol('C_2', commutative=True), Integer(-2))"], "Equality(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('C_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-2)))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-2)))))"], [["divide", 3, "Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-2))))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-2)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-2)))), Integer(-1)), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('C_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-2))))), Integer(1))"]]}, {"prompt": "Given C{(H)} = e^{H}, then obtain 1 = \\frac{e^{H}}{C{(H)} - 1 + \\frac{e^{H}}{C{(H)}}}", "derivation": "C{(H)} = e^{H} and 1 = \\frac{e^{H}}{C{(H)}} and C{(H)} + 1 = C{(H)} + \\frac{e^{H}}{C{(H)}} and C{(H)} = C{(H)} - 1 + \\frac{e^{H}}{C{(H)}} and 1 = \\frac{e^{H}}{C{(H)} - 1 + \\frac{e^{H}}{C{(H)}}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["divide", 1, "Function('C')(Symbol('H', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C')(Symbol('H', commutative=True)), Integer(-1)), exp(Symbol('H', commutative=True))))"], [["add", 2, "Function('C')(Symbol('H', commutative=True))"], "Equality(Add(Function('C')(Symbol('H', commutative=True)), Integer(1)), Add(Function('C')(Symbol('H', commutative=True)), Mul(Pow(Function('C')(Symbol('H', commutative=True)), Integer(-1)), exp(Symbol('H', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Function('C')(Symbol('H', commutative=True)), Add(Function('C')(Symbol('H', commutative=True)), Integer(-1), Mul(Pow(Function('C')(Symbol('H', commutative=True)), Integer(-1)), exp(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(1), Mul(Pow(Add(Function('C')(Symbol('H', commutative=True)), Integer(-1), Mul(Pow(Function('C')(Symbol('H', commutative=True)), Integer(-1)), exp(Symbol('H', commutative=True)))), Integer(-1)), exp(Symbol('H', commutative=True))))"]]}, {"prompt": "Given m{(A_{1},A)} = A A_{1}, then obtain (\\int (- A_{1} + m{(A_{1},A)}) dA_{1})^{A} = (\\int (A A_{1} - A_{1}) dA_{1})^{A}", "derivation": "m{(A_{1},A)} = A A_{1} and - A_{1} + m{(A_{1},A)} = A A_{1} - A_{1} and \\int (- A_{1} + m{(A_{1},A)}) dA_{1} = \\int (A A_{1} - A_{1}) dA_{1} and (\\int (- A_{1} + m{(A_{1},A)}) dA_{1})^{A} = (\\int (A A_{1} - A_{1}) dA_{1})^{A}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('A_1', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('A_1', commutative=True)))"], [["minus", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('m')(Symbol('A_1', commutative=True), Symbol('A', commutative=True))), Add(Mul(Symbol('A', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Symbol('A_1', commutative=True))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('m')(Symbol('A_1', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Mul(Symbol('A', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('m')(Symbol('A_1', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(Add(Mul(Symbol('A', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given v{(A_{y},\\hat{H}_l)} = \\frac{A_{y}}{\\hat{H}_l} and \\rho_{b}{(\\hat{H}_l)} = \\hat{H}_l, then obtain 2 v{(A_{y},\\hat{H}_l)} + 2 = \\frac{\\hat{H}_l}{\\rho_{b}{(\\hat{H}_l)}} + 2 v{(A_{y},\\hat{H}_l)} + 1", "derivation": "v{(A_{y},\\hat{H}_l)} = \\frac{A_{y}}{\\hat{H}_l} and 2 v{(A_{y},\\hat{H}_l)} = \\frac{A_{y}}{\\hat{H}_l} + v{(A_{y},\\hat{H}_l)} and \\rho_{b}{(\\hat{H}_l)} = \\hat{H}_l and 1 = \\frac{\\hat{H}_l}{\\rho_{b}{(\\hat{H}_l)}} and 2 = \\frac{\\hat{H}_l}{\\rho_{b}{(\\hat{H}_l)}} + 1 and \\frac{A_{y}}{\\hat{H}_l} + v{(A_{y},\\hat{H}_l)} + 2 = \\frac{A_{y}}{\\hat{H}_l} + \\frac{\\hat{H}_l}{\\rho_{b}{(\\hat{H}_l)}} + v{(A_{y},\\hat{H}_l)} + 1 and 2 v{(A_{y},\\hat{H}_l)} + 2 = \\frac{\\hat{H}_l}{\\rho_{b}{(\\hat{H}_l)}} + 2 v{(A_{y},\\hat{H}_l)} + 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"], [["add", 1, "Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Integer(2), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], [["divide", 3, "Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))))"], [["add", 4, 1], "Equality(Integer(2), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Integer(1)))"], [["add", 5, "Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Mul(Integer(2), Function('v')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(1)))"]]}, {"prompt": "Given b{(M_{E},I)} = \\int \\frac{M_{E}}{I} dI, then obtain \\cos^{I}{(\\frac{M_{E} \\int b{(M_{E},I)} dM_{E}}{I})} = \\cos^{I}{(\\frac{M_{E} \\iint \\frac{M_{E}}{I} dI dM_{E}}{I})}", "derivation": "b{(M_{E},I)} = \\int \\frac{M_{E}}{I} dI and \\int b{(M_{E},I)} dM_{E} = \\iint \\frac{M_{E}}{I} dI dM_{E} and - \\frac{M_{E} \\int b{(M_{E},I)} dM_{E}}{I} = - \\frac{M_{E} \\iint \\frac{M_{E}}{I} dI dM_{E}}{I} and \\cos{(\\frac{M_{E} \\int b{(M_{E},I)} dM_{E}}{I})} = \\cos{(\\frac{M_{E} \\iint \\frac{M_{E}}{I} dI dM_{E}}{I})} and \\cos^{I}{(\\frac{M_{E} \\int b{(M_{E},I)} dM_{E}}{I})} = \\cos^{I}{(\\frac{M_{E} \\iint \\frac{M_{E}}{I} dI dM_{E}}{I})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('b')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Function('b')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["cos", 3], "Equality(cos(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Function('b')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), cos(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(cos(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Function('b')(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), Symbol('I', commutative=True)), Pow(cos(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(G,\\sigma_p)} = \\sigma_p + \\cos{(G)}, then obtain \\frac{\\partial}{\\partial \\sigma_p} \\int \\operatorname{f^{\\prime}}^{G}{(G,\\sigma_p)} dG = \\frac{\\partial}{\\partial \\sigma_p} \\int (\\sigma_p + \\cos{(G)})^{G} dG", "derivation": "\\operatorname{f^{\\prime}}{(G,\\sigma_p)} = \\sigma_p + \\cos{(G)} and \\operatorname{f^{\\prime}}^{G}{(G,\\sigma_p)} = (\\sigma_p + \\cos{(G)})^{G} and \\int \\operatorname{f^{\\prime}}^{G}{(G,\\sigma_p)} dG = \\int (\\sigma_p + \\cos{(G)})^{G} dG and \\frac{\\partial}{\\partial \\sigma_p} \\int \\operatorname{f^{\\prime}}^{G}{(G,\\sigma_p)} dG = \\frac{\\partial}{\\partial \\sigma_p} \\int (\\sigma_p + \\cos{(G)})^{G} dG", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), cos(Symbol('G', commutative=True))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('G', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), cos(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Pow(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Pow(Add(Symbol('\\\\sigma_p', commutative=True), cos(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('f^{\\\\prime}')(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Symbol('\\\\sigma_p', commutative=True), cos(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(f_{E})} = \\log{(\\sin{(f_{E})})}, then derive \\frac{d}{d f_{E}} \\mathbf{A}{(f_{E})} = \\frac{\\cos{(f_{E})}}{\\sin{(f_{E})}}, then obtain \\int \\frac{d}{d f_{E}} \\log{(\\sin{(f_{E})})} df_{E} = \\int \\frac{\\cos{(f_{E})}}{\\sin{(f_{E})}} df_{E}", "derivation": "\\mathbf{A}{(f_{E})} = \\log{(\\sin{(f_{E})})} and \\frac{d}{d f_{E}} \\mathbf{A}{(f_{E})} = \\frac{d}{d f_{E}} \\log{(\\sin{(f_{E})})} and \\frac{d}{d f_{E}} \\mathbf{A}{(f_{E})} = \\frac{\\cos{(f_{E})}}{\\sin{(f_{E})}} and \\frac{d}{d f_{E}} \\log{(\\sin{(f_{E})})} = \\frac{\\cos{(f_{E})}}{\\sin{(f_{E})}} and \\int \\frac{d}{d f_{E}} \\log{(\\sin{(f_{E})})} df_{E} = \\int \\frac{\\cos{(f_{E})}}{\\sin{(f_{E})}} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('f_E', commutative=True)), log(sin(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(log(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('f_E', commutative=True)), Integer(-1)), cos(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('f_E', commutative=True)), Integer(-1)), cos(Symbol('f_E', commutative=True))))"], [["integrate", 4, "Symbol('f_E', commutative=True)"], "Equality(Integral(Derivative(log(sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Pow(sin(Symbol('f_E', commutative=True)), Integer(-1)), cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0} and \\phi_{2}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0}, then obtain Z^{\\hat{x}_0} \\mathbf{p}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0} \\phi_{2}{(Z,\\hat{x}_0)}", "derivation": "\\mathbf{p}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0} and \\phi_{2}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0} and \\mathbf{p}{(Z,\\hat{x}_0)} = \\phi_{2}{(Z,\\hat{x}_0)} and Z^{\\hat{x}_0} \\mathbf{p}{(Z,\\hat{x}_0)} = Z^{\\hat{x}_0} \\phi_{2}{(Z,\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{p}')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\phi_2')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 3, "Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\phi_2')(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\omega{(M_{E},\\Psi_{\\lambda})} = - M_{E} + \\Psi_{\\lambda}, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\omega{(M_{E},\\Psi_{\\lambda})} = 1, then obtain \\frac{\\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- M_{E} + \\Psi_{\\lambda})}{\\omega{(M_{E},\\Psi_{\\lambda})}} = \\frac{1}{\\omega{(M_{E},\\Psi_{\\lambda})}}", "derivation": "\\omega{(M_{E},\\Psi_{\\lambda})} = - M_{E} + \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\omega{(M_{E},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- M_{E} + \\Psi_{\\lambda}) and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\omega{(M_{E},\\Psi_{\\lambda})} = 1 and \\frac{\\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\omega{(M_{E},\\Psi_{\\lambda})}}{\\omega{(M_{E},\\Psi_{\\lambda})}} = \\frac{1}{\\omega{(M_{E},\\Psi_{\\lambda})}} and \\frac{\\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- M_{E} + \\Psi_{\\lambda})}{\\omega{(M_{E},\\Psi_{\\lambda})}} = \\frac{1}{\\omega{(M_{E},\\Psi_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Derivative(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Pow(Function('\\\\omega')(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\hat{H},b)} = e^{\\hat{H}^{b}}, then derive - \\hat{H}^{b} e^{\\hat{H}^{b}} \\log{(\\hat{H})} + \\frac{\\partial}{\\partial b} \\operatorname{v_{1}}{(\\hat{H},b)} = 0, then obtain - \\hat{H}^{b} e^{\\hat{H}^{b}} \\log{(\\hat{H})} + \\frac{\\partial}{\\partial b} e^{\\hat{H}^{b}} = 0", "derivation": "\\operatorname{v_{1}}{(\\hat{H},b)} = e^{\\hat{H}^{b}} and - b + \\operatorname{v_{1}}{(\\hat{H},b)} = - b + e^{\\hat{H}^{b}} and \\operatorname{v_{1}}{(\\hat{H},b)} - e^{\\hat{H}^{b}} = 0 and \\frac{\\partial}{\\partial b} (\\operatorname{v_{1}}{(\\hat{H},b)} - e^{\\hat{H}^{b}}) = \\frac{d}{d b} 0 and - \\hat{H}^{b} e^{\\hat{H}^{b}} \\log{(\\hat{H})} + \\frac{\\partial}{\\partial b} \\operatorname{v_{1}}{(\\hat{H},b)} = 0 and - \\hat{H}^{b} e^{\\hat{H}^{b}} \\log{(\\hat{H})} + \\frac{\\partial}{\\partial b} e^{\\hat{H}^{b}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('v_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('b', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))))"], "Equality(Add(Function('v_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Function('v_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Derivative(Function('v_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Derivative(exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given b{(E_{x})} = \\cos{(E_{x})}, then obtain \\frac{d}{d E_{x}} \\int b^{E_{x}}{(E_{x})} dE_{x} = \\frac{d}{d E_{x}} \\int \\cos^{E_{x}}{(E_{x})} dE_{x}", "derivation": "b{(E_{x})} = \\cos{(E_{x})} and b^{E_{x}}{(E_{x})} = \\cos^{E_{x}}{(E_{x})} and \\int b^{E_{x}}{(E_{x})} dE_{x} = \\int \\cos^{E_{x}}{(E_{x})} dE_{x} and \\frac{d}{d E_{x}} \\int b^{E_{x}}{(E_{x})} dE_{x} = \\frac{d}{d E_{x}} \\int \\cos^{E_{x}}{(E_{x})} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('b')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(cos(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Pow(Function('b')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(cos(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('b')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(t)} = \\cos{(t)}, then derive - \\cos{(t)} + \\frac{d}{d t} n{(t)} = - \\sin{(t)} - \\cos{(t)}, then obtain \\frac{d}{d t} \\cos{(t)} + \\int (- \\cos{(t)} + \\frac{d}{d t} \\cos{(t)}) dt = \\frac{d}{d t} \\cos{(t)} + \\int (- \\sin{(t)} - \\cos{(t)}) dt", "derivation": "n{(t)} = \\cos{(t)} and \\frac{d}{d t} n{(t)} = \\frac{d}{d t} \\cos{(t)} and - \\cos{(t)} + \\frac{d}{d t} n{(t)} = - \\cos{(t)} + \\frac{d}{d t} \\cos{(t)} and - \\cos{(t)} + \\frac{d}{d t} n{(t)} = - \\sin{(t)} - \\cos{(t)} and - \\cos{(t)} + \\frac{d}{d t} \\cos{(t)} = - \\sin{(t)} - \\cos{(t)} and \\int (- \\cos{(t)} + \\frac{d}{d t} \\cos{(t)}) dt = \\int (- \\sin{(t)} - \\cos{(t)}) dt and \\frac{d}{d t} \\cos{(t)} + \\int (- \\cos{(t)} + \\frac{d}{d t} \\cos{(t)}) dt = \\frac{d}{d t} \\cos{(t)} + \\int (- \\sin{(t)} - \\cos{(t)}) dt", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["minus", 2, "cos(Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(Function('n')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(Function('n')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"], [["integrate", 5, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))))"], [["add", 6, "Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integral(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True)))), Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integral(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(E,Z)} = E - Z, then derive \\frac{\\partial}{\\partial Z} \\theta_{2}{(E,Z)} = -1, then obtain \\frac{\\partial}{\\partial Z} (E - Z) + 1 = 0", "derivation": "\\theta_{2}{(E,Z)} = E - Z and \\frac{\\partial}{\\partial Z} \\theta_{2}{(E,Z)} = \\frac{\\partial}{\\partial Z} (E - Z) and \\frac{\\partial}{\\partial Z} \\theta_{2}{(E,Z)} = -1 and \\frac{\\partial}{\\partial Z} \\theta_{2}{(E,Z)} + 1 = 0 and \\frac{\\partial}{\\partial Z} (E - Z) + 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1)), Integer(0))"]]}, {"prompt": "Given E{(\\hat{\\mathbf{r}},A_{x})} = A_{x} + \\hat{\\mathbf{r}}, then obtain \\frac{\\frac{\\partial}{\\partial A_{x}} E{(\\hat{\\mathbf{r}},A_{x})}}{\\hat{\\mathbf{r}}} = \\frac{1}{\\hat{\\mathbf{r}}}", "derivation": "E{(\\hat{\\mathbf{r}},A_{x})} = A_{x} + \\hat{\\mathbf{r}} and \\frac{\\partial}{\\partial A_{x}} E{(\\hat{\\mathbf{r}},A_{x})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\hat{\\mathbf{r}}) and \\frac{\\frac{\\partial}{\\partial A_{x}} E{(\\hat{\\mathbf{r}},A_{x})}}{\\hat{\\mathbf{r}}} = \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} + \\hat{\\mathbf{r}})}{\\hat{\\mathbf{r}}} and \\frac{\\frac{\\partial}{\\partial A_{x}} E{(\\hat{\\mathbf{r}},A_{x})}}{\\hat{\\mathbf{r}}} = \\frac{1}{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given G{(\\varepsilon_0)} = \\log{(\\varepsilon_0)}, then derive (\\frac{d}{d \\varepsilon_0} G{(\\varepsilon_0)})^{\\varepsilon_0} = (\\frac{1}{\\varepsilon_0})^{\\varepsilon_0}, then obtain \\frac{d}{d \\varepsilon_0} (\\frac{1}{\\varepsilon_0})^{\\varepsilon_0} = \\frac{d}{d \\varepsilon_0} (\\frac{d}{d \\varepsilon_0} \\log{(\\varepsilon_0)})^{\\varepsilon_0}", "derivation": "G{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} G{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\log{(\\varepsilon_0)} and (\\frac{d}{d \\varepsilon_0} G{(\\varepsilon_0)})^{\\varepsilon_0} = (\\frac{d}{d \\varepsilon_0} \\log{(\\varepsilon_0)})^{\\varepsilon_0} and (\\frac{d}{d \\varepsilon_0} G{(\\varepsilon_0)})^{\\varepsilon_0} = (\\frac{1}{\\varepsilon_0})^{\\varepsilon_0} and (\\frac{1}{\\varepsilon_0})^{\\varepsilon_0} = (\\frac{d}{d \\varepsilon_0} \\log{(\\varepsilon_0)})^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} (\\frac{1}{\\varepsilon_0})^{\\varepsilon_0} = \\frac{d}{d \\varepsilon_0} (\\frac{d}{d \\varepsilon_0} \\log{(\\varepsilon_0)})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Derivative(Function('G')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Derivative(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('G')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Derivative(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(Derivative(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(b)} = \\cos{(b)} and \\operatorname{C_{1}}{(b)} = \\frac{d}{d b} A{(b)}, then obtain \\operatorname{C_{1}}{(b)} + \\frac{d}{d b} A{(b)} = \\frac{d}{d b} A{(b)} + \\frac{d}{d b} \\cos{(b)}", "derivation": "A{(b)} = \\cos{(b)} and \\operatorname{C_{1}}{(b)} = \\frac{d}{d b} A{(b)} and \\operatorname{C_{1}}{(b)} = \\frac{d}{d b} \\cos{(b)} and \\operatorname{C_{1}}{(b)} + \\frac{d}{d b} A{(b)} = \\frac{d}{d b} A{(b)} + \\frac{d}{d b} \\cos{(b)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('b', commutative=True)), Derivative(Function('A')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_1')(Symbol('b', commutative=True)), Derivative(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Function('A')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))"], "Equality(Add(Function('C_1')(Symbol('b', commutative=True)), Derivative(Function('A')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Derivative(Function('A')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(t_{1},\\lambda)} = \\lambda^{t_{1}}, then obtain \\frac{\\partial}{\\partial \\lambda} (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)})^{2} = \\frac{\\partial}{\\partial \\lambda} (\\lambda^{t_{1}} - 2 t_{1}) (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)})", "derivation": "\\operatorname{A_{x}}{(t_{1},\\lambda)} = \\lambda^{t_{1}} and - 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)} = \\lambda^{t_{1}} - 2 t_{1} and (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)})^{2} = (\\lambda^{t_{1}} - 2 t_{1}) (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)}) and \\frac{\\partial}{\\partial \\lambda} (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)})^{2} = \\frac{\\partial}{\\partial \\lambda} (\\lambda^{t_{1}} - 2 t_{1}) (- 2 t_{1} + \\operatorname{A_{x}}{(t_{1},\\lambda)})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Pow(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(2)), Mul(Add(Pow(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(2)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Add(Pow(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('t_1', commutative=True)), Function('A_x')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(P_{g},y^{\\prime})} = \\sin{(P_{g} y^{\\prime})}, then obtain \\int U{(P_{g},y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} dP_{g} = \\int \\sin{(P_{g} y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} dP_{g}", "derivation": "U{(P_{g},y^{\\prime})} = \\sin{(P_{g} y^{\\prime})} and \\frac{\\partial}{\\partial P_{g}} U{(P_{g},y^{\\prime})} = \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} and U{(P_{g},y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} U{(P_{g},y^{\\prime})} = \\sin{(P_{g} y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} U{(P_{g},y^{\\prime})} and U{(P_{g},y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} = \\sin{(P_{g} y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} and \\int U{(P_{g},y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} dP_{g} = \\int \\sin{(P_{g} y^{\\prime})} \\frac{\\partial}{\\partial P_{g}} \\sin{(P_{g} y^{\\prime})} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))"], "Equality(Mul(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Mul(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Derivative(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Mul(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Derivative(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('P_g', commutative=True)"], "Equality(Integral(Mul(Function('U')(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Tuple(Symbol('P_g', commutative=True))), Integral(Mul(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Derivative(sin(Mul(Symbol('P_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Tuple(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(g_{\\varepsilon},A_{1})} = \\cos^{g_{\\varepsilon}}{(A_{1})}, then obtain \\frac{\\partial}{\\partial A_{1}} \\int (\\bar{\\h}{(g_{\\varepsilon},A_{1})} - \\cos^{g_{\\varepsilon}}{(A_{1})}) dg_{\\varepsilon} = \\frac{d}{d A_{1}} \\int 0 dg_{\\varepsilon}", "derivation": "\\bar{\\h}{(g_{\\varepsilon},A_{1})} = \\cos^{g_{\\varepsilon}}{(A_{1})} and \\bar{\\h}{(g_{\\varepsilon},A_{1})} - \\cos^{g_{\\varepsilon}}{(A_{1})} = 0 and \\int (\\bar{\\h}{(g_{\\varepsilon},A_{1})} - \\cos^{g_{\\varepsilon}}{(A_{1})}) dg_{\\varepsilon} = \\int 0 dg_{\\varepsilon} and \\frac{\\partial}{\\partial A_{1}} \\int (\\bar{\\h}{(g_{\\varepsilon},A_{1})} - \\cos^{g_{\\varepsilon}}{(A_{1})}) dg_{\\varepsilon} = \\frac{d}{d A_{1}} \\int 0 dg_{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hbar')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Pow(cos(Symbol('A_1', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\hbar')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_1', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Function('\\\\hbar')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_1', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 3, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\hbar')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_1', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(u)} = - u, then obtain \\int (\\iint \\mathbf{J}_f{(u)} du du - \\frac{1}{I - \\mathbf{J}_f{(u)}}) dI = \\int (\\iint - u du du - \\frac{1}{I - \\mathbf{J}_f{(u)}}) dI", "derivation": "\\mathbf{J}_f{(u)} = - u and \\int \\mathbf{J}_f{(u)} du = \\int - u du and \\iint \\mathbf{J}_f{(u)} du du = \\iint - u du du and \\iint \\mathbf{J}_f{(u)} du du - \\frac{1}{I + u} = \\iint - u du du - \\frac{1}{I + u} and \\iint \\mathbf{J}_f{(u)} du du - \\frac{1}{I - \\mathbf{J}_f{(u)}} = \\iint - u du du - \\frac{1}{I - \\mathbf{J}_f{(u)}} and \\int (\\iint \\mathbf{J}_f{(u)} du du - \\frac{1}{I - \\mathbf{J}_f{(u)}}) dI = \\int (\\iint - u du du - \\frac{1}{I - \\mathbf{J}_f{(u)}}) dI", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(-1), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(-1), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["minus", 3, "Pow(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1))"], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)))), Add(Integral(Mul(Integer(-1), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)))), Integer(-1)))), Add(Integral(Mul(Integer(-1), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)))), Integer(-1)))))"], [["integrate", 5, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)))), Integer(-1)))), Tuple(Symbol('I', commutative=True))), Integral(Add(Integral(Mul(Integer(-1), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('u', commutative=True)))), Integer(-1)))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given a{(f_{\\mathbf{p}})} = \\cos{(\\cos{(f_{\\mathbf{p}})})}, then obtain - \\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} + \\log{(\\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}})} = - \\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} + \\log{(\\log{(\\cos{(\\cos{(f_{\\mathbf{p}})})})}^{f_{\\mathbf{p}}})}", "derivation": "a{(f_{\\mathbf{p}})} = \\cos{(\\cos{(f_{\\mathbf{p}})})} and \\log{(a{(f_{\\mathbf{p}})})} = \\log{(\\cos{(\\cos{(f_{\\mathbf{p}})})})} and \\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} = \\log{(\\cos{(\\cos{(f_{\\mathbf{p}})})})}^{f_{\\mathbf{p}}} and \\log{(\\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}})} = \\log{(\\log{(\\cos{(\\cos{(f_{\\mathbf{p}})})})}^{f_{\\mathbf{p}}})} and - \\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} + \\log{(\\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}})} = - \\log{(a{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} + \\log{(\\log{(\\cos{(\\cos{(f_{\\mathbf{p}})})})}^{f_{\\mathbf{p}}})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["log", 1], "Equality(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), log(cos(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["power", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(log(cos(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["log", 3], "Equality(log(Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))), log(Pow(log(cos(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 4, "Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))), log(Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Mul(Integer(-1), Pow(log(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))), log(Pow(log(cos(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\rho_f)} = e^{\\rho_f}, then obtain \\frac{\\operatorname{E_{\\lambda}}{(\\rho_f)} - \\int \\operatorname{E_{\\lambda}}{(\\rho_f)} d\\rho_f}{e^{\\rho_f} - \\int \\operatorname{E_{\\lambda}}{(\\rho_f)} d\\rho_f} = 1", "derivation": "\\operatorname{E_{\\lambda}}{(\\rho_f)} = e^{\\rho_f} and \\int \\operatorname{E_{\\lambda}}{(\\rho_f)} d\\rho_f = \\int e^{\\rho_f} d\\rho_f and \\operatorname{E_{\\lambda}}{(\\rho_f)} - \\int e^{\\rho_f} d\\rho_f = e^{\\rho_f} - \\int e^{\\rho_f} d\\rho_f and \\frac{\\operatorname{E_{\\lambda}}{(\\rho_f)} - \\int e^{\\rho_f} d\\rho_f}{e^{\\rho_f} - \\int e^{\\rho_f} d\\rho_f} = 1 and \\frac{\\operatorname{E_{\\lambda}}{(\\rho_f)} - \\int \\operatorname{E_{\\lambda}}{(\\rho_f)} d\\rho_f}{e^{\\rho_f} - \\int \\operatorname{E_{\\lambda}}{(\\rho_f)} d\\rho_f} = 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Add(exp(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))))"], [["divide", 3, "Add(exp(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], "Equality(Mul(Add(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Pow(Add(exp(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Pow(Add(exp(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given Q{(\\mathbf{D},\\sigma_x)} = \\mathbf{D} + \\sigma_x, then derive \\frac{\\partial}{\\partial \\mathbf{D}} Q{(\\mathbf{D},\\sigma_x)} = 1, then obtain \\cos{(\\frac{\\partial}{\\partial \\mathbf{D}} Q{(\\mathbf{D},\\sigma_x)})} = \\cos{(1)}", "derivation": "Q{(\\mathbf{D},\\sigma_x)} = \\mathbf{D} + \\sigma_x and \\frac{\\partial}{\\partial \\mathbf{D}} Q{(\\mathbf{D},\\sigma_x)} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + \\sigma_x) and \\frac{\\partial}{\\partial \\mathbf{D}} Q{(\\mathbf{D},\\sigma_x)} = 1 and \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + \\sigma_x) = 1 and \\cos{(\\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + \\sigma_x))} = \\cos{(1)} and \\cos{(\\frac{\\partial}{\\partial \\mathbf{D}} Q{(\\mathbf{D},\\sigma_x)})} = \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))"], [["cos", 4], "Equality(cos(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), cos(Integer(1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(cos(Derivative(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), cos(Integer(1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{p}_0)} = \\cos{(\\sin{(\\hat{p}_0)})}, then obtain \\frac{d}{d \\hat{p}_0} \\operatorname{n_{2}}{(\\hat{p}_0)} + \\frac{d^{2}}{d \\hat{p}_0^{2}} \\operatorname{n_{2}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\operatorname{n_{2}}{(\\hat{p}_0)} + \\frac{d^{2}}{d \\hat{p}_0^{2}} \\cos{(\\sin{(\\hat{p}_0)})}", "derivation": "\\operatorname{n_{2}}{(\\hat{p}_0)} = \\cos{(\\sin{(\\hat{p}_0)})} and \\frac{d}{d \\hat{p}_0} \\operatorname{n_{2}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\cos{(\\sin{(\\hat{p}_0)})} and \\frac{d^{2}}{d \\hat{p}_0^{2}} \\operatorname{n_{2}}{(\\hat{p}_0)} = \\frac{d^{2}}{d \\hat{p}_0^{2}} \\cos{(\\sin{(\\hat{p}_0)})} and \\frac{d}{d \\hat{p}_0} \\operatorname{n_{2}}{(\\hat{p}_0)} + \\frac{d^{2}}{d \\hat{p}_0^{2}} \\operatorname{n_{2}}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\operatorname{n_{2}}{(\\hat{p}_0)} + \\frac{d^{2}}{d \\hat{p}_0^{2}} \\cos{(\\sin{(\\hat{p}_0)})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), cos(sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Derivative(cos(sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))))"], [["add", 3, "Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2)))), Add(Derivative(Function('n_2')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\pi{(G)} = \\log{(G)}, then derive \\frac{d}{d G} \\pi{(G)} = \\frac{1}{G}, then obtain \\frac{d}{d G} \\pi{(G)} + 1 + \\frac{1}{G} = 1 + \\frac{2}{G}", "derivation": "\\pi{(G)} = \\log{(G)} and \\frac{d}{d G} \\pi{(G)} = \\frac{d}{d G} \\log{(G)} and \\frac{d}{d G} \\pi{(G)} = \\frac{1}{G} and \\frac{d}{d G} \\pi{(G)} + 1 = 1 + \\frac{1}{G} and \\frac{d}{d G} \\log{(G)} + 1 = 1 + \\frac{1}{G} and \\frac{1}{G} = \\frac{d}{d G} \\log{(G)} and \\frac{d}{d G} \\pi{(G)} + 1 = \\frac{d}{d G} \\log{(G)} + 1 and \\frac{d}{d G} \\pi{(G)} + 1 + \\frac{1}{G} = \\frac{d}{d G} \\log{(G)} + 1 + \\frac{1}{G} and \\frac{d}{d G} \\pi{(G)} + 1 + \\frac{1}{G} = 1 + \\frac{2}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Add(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)))"], [["add", 7, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Add(Derivative(Function('\\\\pi')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Integer(1), Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\phi{(\\nabla,T)} = \\frac{\\partial}{\\partial \\nabla} T \\nabla, then obtain \\phi{(\\nabla,T)} - \\frac{\\int \\phi{(\\nabla,T)} d\\nabla}{\\nabla} = \\frac{\\partial}{\\partial \\nabla} T \\nabla - \\frac{\\int \\phi{(\\nabla,T)} d\\nabla}{\\nabla}", "derivation": "\\phi{(\\nabla,T)} = \\frac{\\partial}{\\partial \\nabla} T \\nabla and \\int \\phi{(\\nabla,T)} d\\nabla = \\int \\frac{\\partial}{\\partial \\nabla} T \\nabla d\\nabla and \\frac{\\int \\phi{(\\nabla,T)} d\\nabla}{\\nabla} = \\frac{\\int \\frac{\\partial}{\\partial \\nabla} T \\nabla d\\nabla}{\\nabla} and \\phi{(\\nabla,T)} - \\frac{\\int \\frac{\\partial}{\\partial \\nabla} T \\nabla d\\nabla}{\\nabla} = \\frac{\\partial}{\\partial \\nabla} T \\nabla - \\frac{\\int \\frac{\\partial}{\\partial \\nabla} T \\nabla d\\nabla}{\\nabla} and \\phi{(\\nabla,T)} - \\frac{\\int \\phi{(\\nabla,T)} d\\nabla}{\\nabla} = \\frac{\\partial}{\\partial \\nabla} T \\nabla - \\frac{\\int \\phi{(\\nabla,T)} d\\nabla}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], "Equality(Add(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))), Add(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))), Add(Derivative(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(I)} = \\cos{(I)}, then derive \\frac{d}{d I} \\operatorname{t_{2}}{(I)} = - \\sin{(I)}, then obtain e^{\\frac{d}{d I} \\cos{(I)}} = e^{- \\sin{(I)}}", "derivation": "\\operatorname{t_{2}}{(I)} = \\cos{(I)} and \\frac{d}{d I} \\operatorname{t_{2}}{(I)} = \\frac{d}{d I} \\cos{(I)} and \\frac{d}{d I} \\operatorname{t_{2}}{(I)} = - \\sin{(I)} and \\frac{d}{d I} \\cos{(I)} = - \\sin{(I)} and e^{\\frac{d}{d I} \\cos{(I)}} = e^{- \\sin{(I)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('I', commutative=True))))"], [["exp", 4], "Equality(exp(Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), exp(Mul(Integer(-1), sin(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)} = - \\mathbf{s} + \\rho_f, then obtain \\frac{\\rho_f^{2}}{2} + \\int 1 d\\rho_f = \\frac{\\rho_f^{2}}{2} + \\int \\rho_f^{\\mathbf{s}} (\\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)})^{- \\mathbf{s}} d\\rho_f", "derivation": "\\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)} = - \\mathbf{s} + \\rho_f and \\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)} = \\rho_f and (\\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)})^{\\mathbf{s}} = \\rho_f^{\\mathbf{s}} and 1 = \\rho_f^{\\mathbf{s}} (\\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)})^{- \\mathbf{s}} and \\int 1 d\\rho_f = \\int \\rho_f^{\\mathbf{s}} (\\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)})^{- \\mathbf{s}} d\\rho_f and \\frac{\\rho_f^{2}}{2} + \\int 1 d\\rho_f = \\frac{\\rho_f^{2}}{2} + \\int \\rho_f^{\\mathbf{s}} (\\mathbf{s} + \\operatorname{A_{2}}{(\\mathbf{s},\\rho_f)})^{- \\mathbf{s}} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 3, "Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["add", 5, "Mul(Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))), Integral(Integer(1), Tuple(Symbol('\\\\rho_f', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('A_2')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given k{(\\phi_1)} = e^{\\phi_1}, then obtain k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1} + \\frac{d}{d \\phi_1} (k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1}) k{(\\phi_1)} = k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1} + \\frac{d}{d \\phi_1} (e^{\\phi_1} + (e^{\\phi_1})^{\\phi_1}) k{(\\phi_1)}", "derivation": "k{(\\phi_1)} = e^{\\phi_1} and k^{\\phi_1}{(\\phi_1)} = (e^{\\phi_1})^{\\phi_1} and k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1} = e^{\\phi_1} + (e^{\\phi_1})^{\\phi_1} and (k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1}) k{(\\phi_1)} = (e^{\\phi_1} + (e^{\\phi_1})^{\\phi_1}) k{(\\phi_1)} and \\frac{d}{d \\phi_1} (k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1}) k{(\\phi_1)} = \\frac{d}{d \\phi_1} (e^{\\phi_1} + (e^{\\phi_1})^{\\phi_1}) k{(\\phi_1)} and k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1} + \\frac{d}{d \\phi_1} (k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1}) k{(\\phi_1)} = k^{\\phi_1}{(\\phi_1)} + e^{\\phi_1} + \\frac{d}{d \\phi_1} (e^{\\phi_1} + (e^{\\phi_1})^{\\phi_1}) k{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["add", 2, "exp(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Add(exp(Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"], [["times", 3, "Function('k')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))), Mul(Add(exp(Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Mul(Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Add(exp(Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["add", 5, "Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)), Derivative(Mul(Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Add(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)), Derivative(Mul(Add(exp(Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Function('k')(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{f},T)} = \\mathbf{f}^{T}, then obtain \\int 2 \\eta^{\\prime}^{T}{(\\mathbf{f},T)} d\\mathbf{f} = \\int ((\\mathbf{f}^{T})^{T} + \\eta^{\\prime}^{T}{(\\mathbf{f},T)}) d\\mathbf{f}", "derivation": "\\eta^{\\prime}{(\\mathbf{f},T)} = \\mathbf{f}^{T} and \\eta^{\\prime}^{T}{(\\mathbf{f},T)} = (\\mathbf{f}^{T})^{T} and 2 \\eta^{\\prime}^{T}{(\\mathbf{f},T)} = (\\mathbf{f}^{T})^{T} + \\eta^{\\prime}^{T}{(\\mathbf{f},T)} and \\int 2 \\eta^{\\prime}^{T}{(\\mathbf{f},T)} d\\mathbf{f} = \\int ((\\mathbf{f}^{T})^{T} + \\eta^{\\prime}^{T}{(\\mathbf{f},T)}) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["add", 2, "Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Add(Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} = e^{\\lambda}, then obtain (\\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda})^{\\lambda} = 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\lambda)} = e^{\\lambda} and \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda} = 1 and - \\lambda \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda} = - \\lambda and \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda}} = 1 and (\\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda \\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda}})^{\\lambda} = 1 and (\\operatorname{f_{\\mathbf{v}}}{(\\lambda)} e^{- \\lambda})^{\\lambda} = 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Integer(1))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))), Integer(1))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))), Symbol('\\\\lambda', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Integer(1))"]]}, {"prompt": "Given x{(v_{1})} = \\log{(v_{1})}, then obtain (-1 + \\frac{x{(v_{1})}}{v_{1}})^{v_{1}} = (-1 + \\frac{\\log{(v_{1})}}{v_{1}})^{v_{1}}", "derivation": "x{(v_{1})} = \\log{(v_{1})} and \\frac{x{(v_{1})}}{v_{1}} = \\frac{\\log{(v_{1})}}{v_{1}} and -1 + \\frac{x{(v_{1})}}{v_{1}} = -1 + \\frac{\\log{(v_{1})}}{v_{1}} and (-1 + \\frac{x{(v_{1})}}{v_{1}})^{v_{1}} = (-1 + \\frac{\\log{(v_{1})}}{v_{1}})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('x')(Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), log(Symbol('v_1', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('x')(Symbol('v_1', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), log(Symbol('v_1', commutative=True)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Integer(-1), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('x')(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), log(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\omega,\\mathbf{B})} = \\cos{(\\frac{\\mathbf{B}}{\\omega})} and \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = - \\mathbf{J}_P + s, then obtain - \\cos{(\\frac{\\mathbf{B}}{\\omega})} \\frac{\\partial}{\\partial s} \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = - \\cos{(\\frac{\\mathbf{B}}{\\omega})} \\frac{\\partial}{\\partial s} (- \\mathbf{J}_P + s)", "derivation": "\\dot{y}{(\\omega,\\mathbf{B})} = \\cos{(\\frac{\\mathbf{B}}{\\omega})} and \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = - \\mathbf{J}_P + s and \\frac{\\partial}{\\partial s} \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = \\frac{\\partial}{\\partial s} (- \\mathbf{J}_P + s) and - \\dot{y}{(\\omega,\\mathbf{B})} \\frac{\\partial}{\\partial s} \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = - \\dot{y}{(\\omega,\\mathbf{B})} \\frac{\\partial}{\\partial s} (- \\mathbf{J}_P + s) and - \\cos{(\\frac{\\mathbf{B}}{\\omega})} \\frac{\\partial}{\\partial s} \\operatorname{C_{1}}{(\\mathbf{J}_P,s)} = - \\cos{(\\frac{\\mathbf{B}}{\\omega})} \\frac{\\partial}{\\partial s} (- \\mathbf{J}_P + s)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))))"], ["get_premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('s', commutative=True)))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('C_1')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Derivative(Function('C_1')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\mathbf{D})} = \\cos{(\\log{(\\mathbf{D})})}, then obtain 2 \\sigma_{x}{(\\mathbf{D})} + \\log{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = \\log{(\\mathbf{D})} + \\cos{(\\log{(\\mathbf{D})})}", "derivation": "\\sigma_{x}{(\\mathbf{D})} = \\cos{(\\log{(\\mathbf{D})})} and \\sigma_{x}{(\\mathbf{D})} + \\log{(\\mathbf{D})} = \\log{(\\mathbf{D})} + \\cos{(\\log{(\\mathbf{D})})} and 2 \\sigma_{x}{(\\mathbf{D})} + 2 \\log{(\\mathbf{D})} = \\sigma_{x}{(\\mathbf{D})} + 2 \\log{(\\mathbf{D})} + \\cos{(\\log{(\\mathbf{D})})} and 2 \\sigma_{x}{(\\mathbf{D})} + \\log{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = \\sigma_{x}{(\\mathbf{D})} + \\log{(\\mathbf{D})} and 2 \\sigma_{x}{(\\mathbf{D})} + \\log{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = \\log{(\\mathbf{D})} + \\cos{(\\log{(\\mathbf{D})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 1, "log(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Add(log(Symbol('\\\\mathbf{D}', commutative=True)), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["add", 2, "Add(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{D}', commutative=True))), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["minus", 3, "Add(log(Symbol('\\\\mathbf{D}', commutative=True)), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True))), log(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True))), log(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))), Add(log(Symbol('\\\\mathbf{D}', commutative=True)), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(f,E_{\\lambda})} = - E_{\\lambda} + f, then obtain (f \\theta_{1}{(f,E_{\\lambda})} + f) \\frac{\\partial}{\\partial E_{\\lambda}} f \\theta_{1}{(f,E_{\\lambda})} = (f \\theta_{1}{(f,E_{\\lambda})} + f) \\frac{\\partial}{\\partial E_{\\lambda}} f (- E_{\\lambda} + f)", "derivation": "\\theta_{1}{(f,E_{\\lambda})} = - E_{\\lambda} + f and f \\theta_{1}{(f,E_{\\lambda})} = f (- E_{\\lambda} + f) and f \\theta_{1}{(f,E_{\\lambda})} + f = f (- E_{\\lambda} + f) + f and \\frac{\\partial}{\\partial E_{\\lambda}} f \\theta_{1}{(f,E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} f (- E_{\\lambda} + f) and (f (- E_{\\lambda} + f) + f) \\frac{\\partial}{\\partial E_{\\lambda}} f \\theta_{1}{(f,E_{\\lambda})} = (f (- E_{\\lambda} + f) + f) \\frac{\\partial}{\\partial E_{\\lambda}} f (- E_{\\lambda} + f) and (f \\theta_{1}{(f,E_{\\lambda})} + f) \\frac{\\partial}{\\partial E_{\\lambda}} f \\theta_{1}{(f,E_{\\lambda})} = (f \\theta_{1}{(f,E_{\\lambda})} + f) \\frac{\\partial}{\\partial E_{\\lambda}} f (- E_{\\lambda} + f)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True)))"], [["times", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))))"], [["add", 2, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 4, "Add(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('f', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('f', commutative=True), Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('f', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(G)} = e^{G} and \\hat{H}{(G)} = \\frac{\\varphi^{*}{(G)}}{G}, then obtain \\hat{H}^{2}{(G)} = \\frac{\\hat{H}{(G)} e^{G}}{G}", "derivation": "\\varphi^{*}{(G)} = e^{G} and \\frac{\\varphi^{*}{(G)}}{G} = \\frac{e^{G}}{G} and \\frac{\\varphi^{*}^{2}{(G)}}{G^{2}} = \\frac{\\varphi^{*}{(G)} e^{G}}{G^{2}} and \\hat{H}{(G)} = \\frac{\\varphi^{*}{(G)}}{G} and \\hat{H}{(G)} = \\frac{e^{G}}{G} and \\frac{\\varphi^{*}^{2}{(G)}}{G^{2}} = \\frac{\\hat{H}{(G)} \\varphi^{*}{(G)}}{G} and \\frac{\\hat{H}{(G)} \\varphi^{*}{(G)}}{G} = \\frac{\\varphi^{*}{(G)} e^{G}}{G^{2}} and \\hat{H}^{2}{(G)} = \\frac{\\hat{H}{(G)} e^{G}}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["divide", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('G', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('G', commutative=True)))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Pow(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Integer(2))), Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Function('\\\\varphi^*')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\hat{H}')(Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Pow(Function('\\\\varphi^*')(Symbol('G', commutative=True)), Integer(2))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('G', commutative=True)), Function('\\\\varphi^*')(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('G', commutative=True)), Function('\\\\varphi^*')(Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Function('\\\\varphi^*')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Function('\\\\hat{H}')(Symbol('G', commutative=True)), Integer(2)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(l)} = \\int \\cos{(l)} dl and \\psi^{*}{(l)} = l \\int \\cos{(l)} dl, then obtain \\psi^{*}{(l)} + \\int \\cos{(l)} dl = l \\int \\cos{(l)} dl + \\int \\cos{(l)} dl", "derivation": "\\phi_{2}{(l)} = \\int \\cos{(l)} dl and l \\phi_{2}{(l)} = l \\int \\cos{(l)} dl and \\psi^{*}{(l)} = l \\int \\cos{(l)} dl and \\psi^{*}{(l)} = l \\phi_{2}{(l)} and l \\phi_{2}{(l)} + \\int \\cos{(l)} dl = l \\int \\cos{(l)} dl + \\int \\cos{(l)} dl and \\psi^{*}{(l)} + \\int \\cos{(l)} dl = l \\int \\cos{(l)} dl + \\int \\cos{(l)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\phi_2')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\psi^*')(Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), Function('\\\\phi_2')(Symbol('l', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Add(Mul(Symbol('l', commutative=True), Function('\\\\phi_2')(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Symbol('l', commutative=True), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('\\\\psi^*')(Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Symbol('l', commutative=True), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} and \\pi{(\\Psi^{\\dagger})} = \\chi{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})}, then obtain \\chi^{2}{(\\Psi^{\\dagger})} \\cos{(\\log{(\\Psi^{\\dagger})}^{2})} = \\log{(\\Psi^{\\dagger})}^{2} \\cos{(\\log{(\\Psi^{\\dagger})}^{2})}", "derivation": "\\chi{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} and \\chi{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})}^{2} and \\pi{(\\Psi^{\\dagger})} = \\chi{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})} and \\pi{(\\Psi^{\\dagger})} = \\chi^{2}{(\\Psi^{\\dagger})} and \\chi{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})} = \\chi^{2}{(\\Psi^{\\dagger})} and \\chi^{2}{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})}^{2} and \\chi^{2}{(\\Psi^{\\dagger})} \\cos{(\\log{(\\Psi^{\\dagger})}^{2})} = \\log{(\\Psi^{\\dagger})}^{2} \\cos{(\\log{(\\Psi^{\\dagger})}^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["times", 1, "log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\pi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))"], [["times", 6, "cos(Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)), cos(Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))), Mul(Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)), cos(Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given G{(y^{\\prime},f)} = f + e^{y^{\\prime}}, then derive \\int \\frac{G{(y^{\\prime},f)}}{f + e^{y^{\\prime}}} dy^{\\prime} = s + y^{\\prime}, then obtain \\iint 1 dy^{\\prime} dy^{\\prime} = \\int (s + y^{\\prime}) dy^{\\prime}", "derivation": "G{(y^{\\prime},f)} = f + e^{y^{\\prime}} and G{(y^{\\prime},f)} e^{- y^{\\prime}} = (f + e^{y^{\\prime}}) e^{- y^{\\prime}} and \\frac{G{(y^{\\prime},f)}}{f + e^{y^{\\prime}}} = 1 and \\int \\frac{G{(y^{\\prime},f)}}{f + e^{y^{\\prime}}} dy^{\\prime} = \\int 1 dy^{\\prime} and \\int \\frac{G{(y^{\\prime},f)}}{f + e^{y^{\\prime}}} dy^{\\prime} = s + y^{\\prime} and \\iint \\frac{G{(y^{\\prime},f)}}{f + e^{y^{\\prime}}} dy^{\\prime} dy^{\\prime} = \\int (s + y^{\\prime}) dy^{\\prime} and \\iint 1 dy^{\\prime} dy^{\\prime} = \\int (s + y^{\\prime}) dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Mul(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), exp(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["divide", 2, "Mul(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), exp(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Integer(1))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Integer(1), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('s', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('f', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Function('G')(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('s', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Integer(1), Tuple(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('s', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta), then obtain (- 2 \\theta + Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})}) \\dot{x}{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = (- 2 \\theta + f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta)) \\dot{x}{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})}", "derivation": "Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta) and - \\theta + Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = - \\theta + f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta) and - 2 \\theta + Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = - 2 \\theta + f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta) and (- 2 \\theta + Z{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})}) \\dot{x}{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})} = (- 2 \\theta + f_{\\mathbf{v}} (V_{\\mathbf{E}} - \\theta)) \\dot{x}{(\\theta,f_{\\mathbf{v}},V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))))"], [["minus", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))))"], [["times", 3, "Function('\\\\dot{x}')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Function('Z')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('\\\\dot{x}')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))), Function('\\\\dot{x}')(Symbol('\\\\theta', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\rho_b)} = \\sin{(\\rho_b)}, then obtain 2 \\frac{d}{d \\rho_b} \\Omega{(\\rho_b)} \\sin{(\\rho_b)} = \\frac{d}{d \\rho_b} \\Omega{(\\rho_b)} \\sin{(\\rho_b)} + \\frac{d}{d \\rho_b} \\sin^{2}{(\\rho_b)}", "derivation": "\\Omega{(\\rho_b)} = \\sin{(\\rho_b)} and \\Omega{(\\rho_b)} \\sin{(\\rho_b)} = \\sin^{2}{(\\rho_b)} and \\frac{d}{d \\rho_b} \\Omega{(\\rho_b)} \\sin{(\\rho_b)} = \\frac{d}{d \\rho_b} \\sin^{2}{(\\rho_b)} and 2 \\frac{d}{d \\rho_b} \\Omega{(\\rho_b)} \\sin{(\\rho_b)} = \\frac{d}{d \\rho_b} \\Omega{(\\rho_b)} \\sin{(\\rho_b)} + \\frac{d}{d \\rho_b} \\sin^{2}{(\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True))), Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Mul(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(Derivative(Mul(Function('\\\\Omega')(Symbol('\\\\rho_b', commutative=True)), sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\nabla,f)} = \\log{(\\nabla f)}, then obtain \\int 0 d\\nabla = \\int (1 - (\\frac{\\hat{\\mathbf{r}}{(\\nabla,f)}}{\\log{(\\nabla f)}})^{\\nabla}) d\\nabla", "derivation": "\\hat{\\mathbf{r}}{(\\nabla,f)} = \\log{(\\nabla f)} and \\frac{\\hat{\\mathbf{r}}{(\\nabla,f)}}{\\log{(\\nabla f)}} = 1 and (\\frac{\\hat{\\mathbf{r}}{(\\nabla,f)}}{\\log{(\\nabla f)}})^{\\nabla} = 1 and 0 = 1 - (\\frac{\\hat{\\mathbf{r}}{(\\nabla,f)}}{\\log{(\\nabla f)}})^{\\nabla} and \\int 0 d\\nabla = \\int (1 - (\\frac{\\hat{\\mathbf{r}}{(\\nabla,f)}}{\\log{(\\nabla f)}})^{\\nabla}) d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))))"], [["divide", 1, "log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Symbol('\\\\nabla', commutative=True)), Integer(1))"], [["minus", 3, "Pow(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Pow(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\phi_2)} = \\phi_2, then derive \\phi^{\\varphi^*} + \\operatorname{F_{g}}{(\\phi_2)} = \\phi^{\\varphi^*} + \\phi_2, then obtain \\frac{\\cos{(\\phi^{\\varphi^*} + \\operatorname{F_{g}}{(\\phi_2)})}}{n_{1}} = \\frac{\\cos{(\\phi^{\\varphi^*} + \\phi_2)}}{n_{1}}", "derivation": "\\operatorname{F_{g}}{(\\phi_2)} = \\phi_2 and \\phi \\operatorname{F_{g}}{(\\phi_2)} = \\phi \\phi_2 and \\frac{\\partial}{\\partial \\phi} \\phi \\operatorname{F_{g}}{(\\phi_2)} = \\frac{\\partial}{\\partial \\phi} \\phi \\phi_2 and \\phi^{\\varphi^*} + \\frac{\\partial}{\\partial \\phi} \\phi \\operatorname{F_{g}}{(\\phi_2)} = \\phi^{\\varphi^*} + \\frac{\\partial}{\\partial \\phi} \\phi \\phi_2 and \\phi^{\\varphi^*} + \\operatorname{F_{g}}{(\\phi_2)} = \\phi^{\\varphi^*} + \\phi_2 and \\cos{(\\phi^{\\varphi^*} + \\operatorname{F_{g}}{(\\phi_2)})} = \\cos{(\\phi^{\\varphi^*} + \\phi_2)} and \\frac{\\cos{(\\phi^{\\varphi^*} + \\operatorname{F_{g}}{(\\phi_2)})}}{n_{1}} = \\frac{\\cos{(\\phi^{\\varphi^*} + \\phi_2)}}{n_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('F_g')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["add", 3, "Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('F_g')(Symbol('\\\\phi_2', commutative=True))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["cos", 5], "Equality(cos(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('F_g')(Symbol('\\\\phi_2', commutative=True)))), cos(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["times", 6, "Pow(Symbol('n_1', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('F_g')(Symbol('\\\\phi_2', commutative=True))))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\eta,\\mathbb{I})} = \\sin{(\\eta \\mathbb{I})}, then derive \\frac{\\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}} = \\frac{\\mathbb{I} \\cos{(\\eta \\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}}, then obtain (\\frac{\\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}})^{\\mathbb{I}} = (\\frac{\\mathbb{I} \\cos{(\\eta \\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}})^{\\mathbb{I}}", "derivation": "\\delta{(\\eta,\\mathbb{I})} = \\sin{(\\eta \\mathbb{I})} and \\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})} = \\frac{\\partial}{\\partial \\eta} \\sin{(\\eta \\mathbb{I})} and \\frac{\\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}} = \\frac{\\frac{\\partial}{\\partial \\eta} \\sin{(\\eta \\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}} and \\frac{\\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}} = \\frac{\\mathbb{I} \\cos{(\\eta \\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}} and (\\frac{\\frac{\\partial}{\\partial \\eta} \\delta{(\\eta,\\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}})^{\\mathbb{I}} = (\\frac{\\mathbb{I} \\cos{(\\eta \\mathbb{I})}}{\\sin{(\\eta \\mathbb{I})}})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 2, "sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Derivative(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), cos(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["power", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Mul(Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(sin(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), cos(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t,\\mathbf{M})} = \\mathbf{M} t, then derive F_{H} + \\operatorname{F_{g}}{(t,\\mathbf{M})} = \\mathbf{M} t + \\mu, then obtain F_{H} + \\mathbf{M} t = \\mu + \\operatorname{F_{g}}{(t,\\mathbf{M})}", "derivation": "\\operatorname{F_{g}}{(t,\\mathbf{M})} = \\mathbf{M} t and \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(t,\\mathbf{M})} = \\frac{\\partial}{\\partial t} \\mathbf{M} t and \\int \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(t,\\mathbf{M})} dt = \\int \\frac{\\partial}{\\partial t} \\mathbf{M} t dt and F_{H} + \\operatorname{F_{g}}{(t,\\mathbf{M})} = \\mathbf{M} t + \\mu and F_{H} + \\mathbf{M} t = \\mathbf{M} t + \\mu and F_{H} + \\operatorname{F_{g}}{(t,\\mathbf{M})} = \\mu + \\operatorname{F_{g}}{(t,\\mathbf{M})} and \\mathbf{M} t + \\mu = \\mu + \\operatorname{F_{g}}{(t,\\mathbf{M})} and F_{H} + \\mathbf{M} t = \\mu + \\operatorname{F_{g}}{(t,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_H', commutative=True), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('F_H', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('F_H', commutative=True), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Symbol('F_H', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('t', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\mu{(E,J,A)} = A J + E, then derive - \\frac{A^{2} J}{2} - A E - \\hat{x} + \\int \\mu{(E,J,A)} dA = 0, then obtain (- \\frac{A^{2} J}{2} - A E - \\hat{x} + \\int \\mu{(E,J,A)} dA - 1)^{A} = (-1)^{A}", "derivation": "\\mu{(E,J,A)} = A J + E and \\int \\mu{(E,J,A)} dA = \\int (A J + E) dA and - \\int (A J + E) dA + \\int \\mu{(E,J,A)} dA = 0 and - \\frac{A^{2} J}{2} - A E - \\hat{x} + \\int \\mu{(E,J,A)} dA = 0 and - \\frac{A^{2} J}{2} - A E - \\hat{x} + \\int \\mu{(E,J,A)} dA - 1 = -1 and (- \\frac{A^{2} J}{2} - A E - \\hat{x} + \\int \\mu{(E,J,A)} dA - 1)^{A} = (-1)^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Add(Mul(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Mul(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["minus", 2, "Integral(Add(Mul(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Mul(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('A', commutative=True)))), Integral(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2)), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Integer(0))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2)), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1)), Integer(-1))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2)), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Integral(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1)), Symbol('A', commutative=True)), Pow(Integer(-1), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(r,c_{0})} = c_{0}^{r}, then obtain r \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} \\mathbf{B}{(r,c_{0})} = r \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} c_{0}^{r}", "derivation": "\\mathbf{B}{(r,c_{0})} = c_{0}^{r} and \\frac{\\partial}{\\partial c_{0}} \\mathbf{B}{(r,c_{0})} = \\frac{\\partial}{\\partial c_{0}} c_{0}^{r} and \\frac{\\partial^{2}}{\\partial r\\partial c_{0}} \\mathbf{B}{(r,c_{0})} = \\frac{\\partial^{2}}{\\partial r\\partial c_{0}} c_{0}^{r} and \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} \\mathbf{B}{(r,c_{0})} = \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} c_{0}^{r} and r \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} \\mathbf{B}{(r,c_{0})} = r \\frac{\\partial^{3}}{\\partial r^{2}\\partial c_{0}} c_{0}^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('r', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('r', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('c_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('r', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Symbol('c_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('r', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(Pow(Symbol('c_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["times", 4, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Derivative(Function('\\\\mathbf{B}')(Symbol('r', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(2)))), Mul(Symbol('r', commutative=True), Derivative(Pow(Symbol('c_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(2)))))"]]}, {"prompt": "Given t{(M)} = e^{M}, then obtain \\frac{(t{(M)} t^{- M}{(M)})^{M} (t^{- M}{(M)} e^{M})^{- M} t^{M}{(M)}}{t{(M)}} = \\frac{t^{M}{(M)}}{t{(M)}}", "derivation": "t{(M)} = e^{M} and t^{M}{(M)} = (e^{M})^{M} and t{(M)} (e^{M})^{- M} = e^{M} (e^{M})^{- M} and (t{(M)} (e^{M})^{- M})^{M} = (e^{M} (e^{M})^{- M})^{M} and (t{(M)} (e^{M})^{- M})^{M} (e^{M} (e^{M})^{- M})^{- M} = 1 and \\frac{(t{(M)} (e^{M})^{- M})^{M} (e^{M} (e^{M})^{- M})^{- M} (e^{M})^{M}}{t{(M)}} = \\frac{(e^{M})^{M}}{t{(M)}} and \\frac{(t{(M)} t^{- M}{(M)})^{M} (t^{- M}{(M)} e^{M})^{- M} t^{M}{(M)}}{t{(M)}} = \\frac{t^{M}{(M)}}{t{(M)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('t')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["divide", 1, "Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))"], "Equality(Mul(Function('t')(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Mul(exp(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Function('t')(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(exp(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True)))"], [["divide", 4, "Pow(Mul(exp(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True))"], "Equality(Mul(Pow(Mul(Function('t')(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(exp(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(1))"], [["divide", 5, "Mul(Function('t')(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True))))"], "Equality(Mul(Pow(Mul(Function('t')(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(exp(Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Function('t')(Symbol('M', commutative=True)), Integer(-1)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Mul(Pow(Function('t')(Symbol('M', commutative=True)), Integer(-1)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(Mul(Function('t')(Symbol('M', commutative=True)), Pow(Function('t')(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(Pow(Function('t')(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True))), exp(Symbol('M', commutative=True))), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Function('t')(Symbol('M', commutative=True)), Integer(-1)), Pow(Function('t')(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Mul(Pow(Function('t')(Symbol('M', commutative=True)), Integer(-1)), Pow(Function('t')(Symbol('M', commutative=True)), Symbol('M', commutative=True))))"]]}, {"prompt": "Given x{(\\lambda)} = \\log{(\\lambda)} and \\varepsilon_{0}{(\\lambda)} = \\frac{d}{d \\lambda} \\int \\log{(\\lambda)} d\\lambda, then obtain (\\lambda \\log{(\\lambda)} - \\lambda) \\varepsilon_{0}{(\\lambda)} = (\\lambda \\log{(\\lambda)} - \\lambda) \\frac{d}{d \\lambda} \\int x{(\\lambda)} d\\lambda", "derivation": "x{(\\lambda)} = \\log{(\\lambda)} and \\int x{(\\lambda)} d\\lambda = \\int \\log{(\\lambda)} d\\lambda and \\varepsilon_{0}{(\\lambda)} = \\frac{d}{d \\lambda} \\int \\log{(\\lambda)} d\\lambda and \\varepsilon_{0}{(\\lambda)} = \\frac{d}{d \\lambda} \\int x{(\\lambda)} d\\lambda and \\varepsilon_{0}{(\\lambda)} \\frac{d}{d \\lambda} (\\iint \\log{(\\lambda)} d\\lambda d\\lambda - 1) = (\\frac{d}{d \\lambda} (\\iint \\log{(\\lambda)} d\\lambda d\\lambda - 1)) \\frac{d}{d \\lambda} \\int x{(\\lambda)} d\\lambda and (\\lambda \\log{(\\lambda)} - \\lambda) \\varepsilon_{0}{(\\lambda)} = (\\lambda \\log{(\\lambda)} - \\lambda) \\frac{d}{d \\lambda} \\int x{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True)), Derivative(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True)), Derivative(Integral(Function('x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Add(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True)), Derivative(Add(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Derivative(Add(Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Function('x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True))), Mul(Add(Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Derivative(Integral(Function('x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(\\sigma_p,V)} = V - \\sigma_p, then obtain \\frac{\\partial^{2}}{\\partial \\sigma_p\\partial V} \\frac{L{(\\sigma_p,V)}}{\\sigma_p} = \\frac{\\partial^{2}}{\\partial \\sigma_p\\partial V} \\frac{V - \\sigma_p}{\\sigma_p}", "derivation": "L{(\\sigma_p,V)} = V - \\sigma_p and \\frac{L{(\\sigma_p,V)}}{\\sigma_p} = \\frac{V - \\sigma_p}{\\sigma_p} and \\frac{\\partial}{\\partial V} \\frac{L{(\\sigma_p,V)}}{\\sigma_p} = \\frac{\\partial}{\\partial V} \\frac{V - \\sigma_p}{\\sigma_p} and \\frac{\\partial^{2}}{\\partial \\sigma_p\\partial V} \\frac{L{(\\sigma_p,V)}}{\\sigma_p} = \\frac{\\partial^{2}}{\\partial \\sigma_p\\partial V} \\frac{V - \\sigma_p}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\sigma_p', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\sigma_p', commutative=True), Symbol('V', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\sigma_p', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\sigma_p', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain H{(\\mathbf{s})} (e^{\\mathbf{s}})^{\\mathbf{s}} + H^{\\mathbf{s}}{(\\mathbf{s})} = H{(\\mathbf{s})} (e^{\\mathbf{s}})^{\\mathbf{s}} + (e^{\\mathbf{s}})^{\\mathbf{s}}", "derivation": "H{(\\mathbf{s})} = e^{\\mathbf{s}} and H^{\\mathbf{s}}{(\\mathbf{s})} = (e^{\\mathbf{s}})^{\\mathbf{s}} and H{(\\mathbf{s})} H^{\\mathbf{s}}{(\\mathbf{s})} = H{(\\mathbf{s})} (e^{\\mathbf{s}})^{\\mathbf{s}} and H{(\\mathbf{s})} H^{\\mathbf{s}}{(\\mathbf{s})} + H^{\\mathbf{s}}{(\\mathbf{s})} = H{(\\mathbf{s})} H^{\\mathbf{s}}{(\\mathbf{s})} + (e^{\\mathbf{s}})^{\\mathbf{s}} and H{(\\mathbf{s})} (e^{\\mathbf{s}})^{\\mathbf{s}} + H^{\\mathbf{s}}{(\\mathbf{s})} = H{(\\mathbf{s})} (e^{\\mathbf{s}})^{\\mathbf{s}} + (e^{\\mathbf{s}})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 2, "Function('H')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 2, "Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Pow(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Function('H')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{y})} = \\cos{(v_{y})} and J{(v_{y})} = \\cos{(v_{y})}, then obtain J{(v_{y})} \\cos{(v_{y})} + J{(v_{y})} = J{(v_{y})} \\cos{(v_{y})} + \\cos{(v_{y})}", "derivation": "\\operatorname{y^{\\prime}}{(v_{y})} = \\cos{(v_{y})} and \\operatorname{y^{\\prime}}{(v_{y})} \\cos{(v_{y})} = \\cos^{2}{(v_{y})} and J{(v_{y})} = \\cos{(v_{y})} and J{(v_{y})} = \\operatorname{y^{\\prime}}{(v_{y})} and J{(v_{y})} \\cos{(v_{y})} = \\cos^{2}{(v_{y})} and J{(v_{y})} + \\cos^{2}{(v_{y})} = \\cos^{2}{(v_{y})} + \\cos{(v_{y})} and J{(v_{y})} \\cos{(v_{y})} + J{(v_{y})} = J{(v_{y})} \\cos{(v_{y})} + \\cos{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["times", 1, "cos(Symbol('v_y', commutative=True))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Pow(cos(Symbol('v_y', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('J')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('J')(Symbol('v_y', commutative=True)), Function('y^{\\\\prime}')(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Function('J')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Pow(cos(Symbol('v_y', commutative=True)), Integer(2)))"], [["add", 3, "Pow(cos(Symbol('v_y', commutative=True)), Integer(2))"], "Equality(Add(Function('J')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(2))), Add(Pow(cos(Symbol('v_y', commutative=True)), Integer(2)), cos(Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Function('J')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Function('J')(Symbol('v_y', commutative=True))), Add(Mul(Function('J')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\eta,\\dot{z})} = \\dot{z} \\eta, then derive \\frac{\\partial}{\\partial \\eta} \\Psi_{\\lambda}{(\\eta,\\dot{z})} = \\dot{z}, then obtain \\int (- \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta + \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta) d\\dot{z} = \\int (\\dot{z} - \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta) d\\dot{z}", "derivation": "\\Psi_{\\lambda}{(\\eta,\\dot{z})} = \\dot{z} \\eta and \\frac{\\partial}{\\partial \\eta} \\Psi_{\\lambda}{(\\eta,\\dot{z})} = \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta and \\frac{\\partial}{\\partial \\eta} \\Psi_{\\lambda}{(\\eta,\\dot{z})} = \\dot{z} and \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta = \\dot{z} and - \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta + \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta = \\dot{z} - \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta and \\int (- \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta + \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta) d\\dot{z} = \\int (\\dot{z} - \\eta \\frac{\\partial}{\\partial \\eta} \\dot{z} \\eta) d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\dot{z}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\dot{z}', commutative=True))"], [["minus", 4, "Mul(Symbol('\\\\eta', commutative=True), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))))"], [["integrate", 5, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given g{(v_{1},\\mu_0)} = \\cos{(\\frac{v_{1}}{\\mu_0})}, then derive \\frac{\\partial}{\\partial v_{1}} g{(v_{1},\\mu_0)} = - \\frac{\\sin{(\\frac{v_{1}}{\\mu_0})}}{\\mu_0}, then obtain \\frac{\\frac{\\partial}{\\partial v_{1}} g{(v_{1},\\mu_0)}}{\\sin{(\\frac{v_{1}}{\\mu_0})}} = - \\frac{1}{\\mu_0}", "derivation": "g{(v_{1},\\mu_0)} = \\cos{(\\frac{v_{1}}{\\mu_0})} and \\frac{\\partial}{\\partial v_{1}} g{(v_{1},\\mu_0)} = \\frac{\\partial}{\\partial v_{1}} \\cos{(\\frac{v_{1}}{\\mu_0})} and \\frac{\\partial}{\\partial v_{1}} g{(v_{1},\\mu_0)} = - \\frac{\\sin{(\\frac{v_{1}}{\\mu_0})}}{\\mu_0} and \\frac{\\frac{\\partial}{\\partial v_{1}} g{(v_{1},\\mu_0)}}{\\sin{(\\frac{v_{1}}{\\mu_0})}} = - \\frac{1}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('v_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), cos(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('v_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('v_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))))"], [["divide", 3, "sin(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))), Integer(-1)), Derivative(Function('g')(Symbol('v_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(E,V)} = E V, then obtain \\int \\frac{E V + \\ddot{x}{(E,V)}}{2 E V} dV = \\int 1 dV", "derivation": "\\ddot{x}{(E,V)} = E V and E V + \\ddot{x}{(E,V)} = 2 E V and E (E V + \\ddot{x}{(E,V)}) = 2 E^{2} V and \\frac{E V + \\ddot{x}{(E,V)}}{2 E V} = 1 and \\int \\frac{E V + \\ddot{x}{(E,V)}}{2 E V} dV = \\int 1 dV", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('E', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('V', commutative=True)))"], [["add", 1, "Mul(Symbol('E', commutative=True), Symbol('V', commutative=True))"], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('V', commutative=True)), Function('\\\\ddot{x}')(Symbol('E', commutative=True), Symbol('V', commutative=True))), Mul(Integer(2), Symbol('E', commutative=True), Symbol('V', commutative=True)))"], [["times", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Add(Mul(Symbol('E', commutative=True), Symbol('V', commutative=True)), Function('\\\\ddot{x}')(Symbol('E', commutative=True), Symbol('V', commutative=True)))), Mul(Integer(2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('V', commutative=True)))"], [["divide", 3, "Mul(Integer(2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('V', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Symbol('E', commutative=True), Symbol('V', commutative=True)), Function('\\\\ddot{x}')(Symbol('E', commutative=True), Symbol('V', commutative=True)))), Integer(1))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Symbol('E', commutative=True), Symbol('V', commutative=True)), Function('\\\\ddot{x}')(Symbol('E', commutative=True), Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Integer(1), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain \\sin{((\\frac{d}{d \\mathbf{P}} \\chi{(\\mathbf{P})})^{\\mathbf{P}})} = \\sin{((\\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}})^{\\mathbf{P}})}", "derivation": "\\chi{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\chi{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}} and (\\frac{d}{d \\mathbf{P}} \\chi{(\\mathbf{P})})^{\\mathbf{P}} = (\\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}})^{\\mathbf{P}} and \\sin{((\\frac{d}{d \\mathbf{P}} \\chi{(\\mathbf{P})})^{\\mathbf{P}})} = \\sin{((\\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}})^{\\mathbf{P}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True))), sin(Pow(Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(U,W)} = \\cos{(U + W)}, then derive \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\tilde{g}^*{(U,W)} = - \\sin{(U + W)} \\cos{(U + W)}, then obtain - \\sin{(U + W)} \\cos{(U + W)} = \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\cos{(U + W)}", "derivation": "\\tilde{g}^*{(U,W)} = \\cos{(U + W)} and \\frac{\\partial}{\\partial W} \\tilde{g}^*{(U,W)} = \\frac{\\partial}{\\partial W} \\cos{(U + W)} and \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\tilde{g}^*{(U,W)} = \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\cos{(U + W)} and \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\tilde{g}^*{(U,W)} = - \\sin{(U + W)} \\cos{(U + W)} and - \\sin{(U + W)} \\cos{(U + W)} = \\cos{(U + W)} \\frac{\\partial}{\\partial W} \\cos{(U + W)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('W', commutative=True)), cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["times", 2, "cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Derivative(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True)))), Mul(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Derivative(cos(Add(Symbol('U', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(m,\\psi)} = (e^{m})^{\\psi}, then obtain 0^{\\psi} = (- (- \\mathbf{J}{(m,\\psi)} + (e^{m})^{\\psi}) \\mathbf{J}{(m,\\psi)})^{\\psi}", "derivation": "\\mathbf{J}{(m,\\psi)} = (e^{m})^{\\psi} and 0 = - \\mathbf{J}{(m,\\psi)} + (e^{m})^{\\psi} and 0 = - (- \\mathbf{J}{(m,\\psi)} + (e^{m})^{\\psi}) \\mathbf{J}{(m,\\psi)} and 0^{\\psi} = (- (- \\mathbf{J}{(m,\\psi)} + (e^{m})^{\\psi}) \\mathbf{J}{(m,\\psi)})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(exp(Symbol('m', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Pow(exp(Symbol('m', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Pow(exp(Symbol('m', commutative=True)), Symbol('\\\\psi', commutative=True))), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Pow(exp(Symbol('m', commutative=True)), Symbol('\\\\psi', commutative=True))), Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(F_{g},f)} = F_{g}^{f}, then obtain \\frac{\\partial}{\\partial f} \\int \\frac{F_{g}^{f} \\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} df = \\frac{\\partial}{\\partial f} \\int \\frac{F_{g}^{2 f}}{F_{g}} df", "derivation": "\\operatorname{P_{e}}{(F_{g},f)} = F_{g}^{f} and \\frac{\\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} = \\frac{F_{g}^{f}}{F_{g}} and \\frac{\\operatorname{P_{e}}^{2}{(F_{g},f)}}{F_{g}} = \\frac{F_{g}^{f} \\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} and \\frac{F_{g}^{f} \\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} = \\frac{F_{g}^{2 f}}{F_{g}} and \\int \\frac{F_{g}^{f} \\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} df = \\int \\frac{F_{g}^{2 f}}{F_{g}} df and \\frac{\\partial}{\\partial f} \\int \\frac{F_{g}^{f} \\operatorname{P_{e}}{(F_{g},f)}}{F_{g}} df = \\frac{\\partial}{\\partial f} \\int \\frac{F_{g}^{2 f}}{F_{g}} df", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True))))"], [["times", 2, "Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('f', commutative=True)))))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))))"], [["differentiate", 5, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(z^{*})} = \\sin{(z^{*})} and \\mathbf{J}_M{(z^{*})} = \\sin{(z^{*})}, then obtain \\frac{d}{d z^{*}} (\\varepsilon^{z^{*}}{(z^{*})})^{z^{*}} = \\frac{d}{d z^{*}} (\\mathbf{J}_M^{z^{*}}{(z^{*})})^{z^{*}}", "derivation": "\\varepsilon{(z^{*})} = \\sin{(z^{*})} and \\varepsilon^{z^{*}}{(z^{*})} = \\sin^{z^{*}}{(z^{*})} and \\mathbf{J}_M{(z^{*})} = \\sin{(z^{*})} and (\\varepsilon^{z^{*}}{(z^{*})})^{z^{*}} = (\\sin^{z^{*}}{(z^{*})})^{z^{*}} and (\\varepsilon^{z^{*}}{(z^{*})})^{z^{*}} = (\\mathbf{J}_M^{z^{*}}{(z^{*})})^{z^{*}} and \\frac{d}{d z^{*}} (\\varepsilon^{z^{*}}{(z^{*})})^{z^{*}} = \\frac{d}{d z^{*}} (\\mathbf{J}_M^{z^{*}}{(z^{*})})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["power", 1, "Symbol('z^*', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(sin(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Pow(Function('\\\\varepsilon')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(Pow(sin(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Pow(Function('\\\\varepsilon')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\varepsilon')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(\\hat{x},C_{1})} = - C_{1} + \\hat{x}, then obtain \\frac{\\partial^{2}}{\\partial C_{1}^{2}} m{(\\hat{x},C_{1})} = 2 \\frac{\\partial^{2}}{\\partial C_{1}^{2}} m{(\\hat{x},C_{1})}", "derivation": "m{(\\hat{x},C_{1})} = - C_{1} + \\hat{x} and - C_{1} + \\hat{x} + m{(\\hat{x},C_{1})} = - 2 C_{1} + 2 \\hat{x} and 2 m{(\\hat{x},C_{1})} = - 2 C_{1} + 2 \\hat{x} and - C_{1} + \\hat{x} + m{(\\hat{x},C_{1})} = 2 m{(\\hat{x},C_{1})} and \\frac{\\partial}{\\partial C_{1}} (- C_{1} + \\hat{x} + m{(\\hat{x},C_{1})}) = \\frac{\\partial}{\\partial C_{1}} 2 m{(\\hat{x},C_{1})} and \\frac{\\partial^{2}}{\\partial C_{1}^{2}} (- C_{1} + \\hat{x} + m{(\\hat{x},C_{1})}) = \\frac{\\partial^{2}}{\\partial C_{1}^{2}} 2 m{(\\hat{x},C_{1})} and \\frac{\\partial^{2}}{\\partial C_{1}^{2}} m{(\\hat{x},C_{1})} = 2 \\frac{\\partial^{2}}{\\partial C_{1}^{2}} m{(\\hat{x},C_{1})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Mul(Integer(2), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))))"], [["differentiate", 4, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(2))), Derivative(Mul(Integer(2), Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2))), Mul(Integer(2), Derivative(Function('m')(Symbol('\\\\hat{x}', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(x,L)} = \\cos^{L}{(x)}, then obtain \\frac{L \\operatorname{M_{E}}^{x}{(x,L)}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}} = \\frac{L (\\cos^{L}{(x)})^{x}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}}", "derivation": "\\operatorname{M_{E}}{(x,L)} = \\cos^{L}{(x)} and \\operatorname{M_{E}}^{x}{(x,L)} = (\\cos^{L}{(x)})^{x} and \\frac{\\operatorname{M_{E}}^{x}{(x,L)}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}} = \\frac{(\\cos^{L}{(x)})^{x}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}} and \\frac{L \\operatorname{M_{E}}^{x}{(x,L)}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}} = \\frac{L (\\cos^{L}{(x)})^{x}}{(\\cos^{L}{(x)})^{x} - \\operatorname{M_{E}}^{x}{(x,L)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 2, "Add(Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True))))"], "Equality(Mul(Pow(Add(Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True)))), Integer(-1)), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Add(Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True)))), Integer(-1)), Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True))))"], [["times", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Pow(Add(Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True)))), Integer(-1)), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True))), Mul(Symbol('L', commutative=True), Pow(Add(Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True), Symbol('L', commutative=True)), Symbol('x', commutative=True)))), Integer(-1)), Pow(Pow(cos(Symbol('x', commutative=True)), Symbol('L', commutative=True)), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\varepsilon,y)} = \\varepsilon y, then derive \\frac{\\partial}{\\partial y} \\phi_{2}{(\\varepsilon,y)} = \\varepsilon, then obtain \\frac{\\partial}{\\partial y} (\\varepsilon + y) = \\frac{\\partial}{\\partial y} (y + \\frac{\\partial}{\\partial y} \\varepsilon y)", "derivation": "\\phi_{2}{(\\varepsilon,y)} = \\varepsilon y and \\frac{\\partial}{\\partial y} \\phi_{2}{(\\varepsilon,y)} = \\frac{\\partial}{\\partial y} \\varepsilon y and \\frac{\\partial}{\\partial y} \\phi_{2}{(\\varepsilon,y)} = \\varepsilon and \\varepsilon = \\frac{\\partial}{\\partial y} \\varepsilon y and \\varepsilon + y = y + \\frac{\\partial}{\\partial y} \\varepsilon y and \\frac{\\partial}{\\partial y} (\\varepsilon + y) = \\frac{\\partial}{\\partial y} (y + \\frac{\\partial}{\\partial y} \\varepsilon y)", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('\\\\varepsilon', commutative=True), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["add", 4, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('y', commutative=True), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(H,C_{2})} = C_{2} + H, then obtain \\int \\cos{(\\cos{(J{(H,C_{2})})})} dH = \\int \\cos{(\\cos{(C_{2} + H)})} dH", "derivation": "J{(H,C_{2})} = C_{2} + H and \\cos{(J{(H,C_{2})})} = \\cos{(C_{2} + H)} and \\cos{(\\cos{(J{(H,C_{2})})})} = \\cos{(\\cos{(C_{2} + H)})} and \\int \\cos{(\\cos{(J{(H,C_{2})})})} dH = \\int \\cos{(\\cos{(C_{2} + H)})} dH", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('H', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('H', commutative=True)))"], [["cos", 1], "Equality(cos(Function('J')(Symbol('H', commutative=True), Symbol('C_2', commutative=True))), cos(Add(Symbol('C_2', commutative=True), Symbol('H', commutative=True))))"], [["cos", 2], "Equality(cos(cos(Function('J')(Symbol('H', commutative=True), Symbol('C_2', commutative=True)))), cos(cos(Add(Symbol('C_2', commutative=True), Symbol('H', commutative=True)))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(cos(cos(Function('J')(Symbol('H', commutative=True), Symbol('C_2', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integral(cos(cos(Add(Symbol('C_2', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given J{(t_{2},C_{2})} = C_{2} - t_{2}, then derive \\frac{\\partial}{\\partial t_{2}} J{(t_{2},C_{2})} = -1, then obtain C_{2} J{(t_{2},C_{2})} \\frac{\\partial}{\\partial t_{2}} (C_{2} - t_{2}) = - C_{2} J{(t_{2},C_{2})}", "derivation": "J{(t_{2},C_{2})} = C_{2} - t_{2} and \\frac{\\partial}{\\partial t_{2}} J{(t_{2},C_{2})} = \\frac{\\partial}{\\partial t_{2}} (C_{2} - t_{2}) and \\frac{\\partial}{\\partial t_{2}} J{(t_{2},C_{2})} = -1 and C_{2} J{(t_{2},C_{2})} \\frac{\\partial}{\\partial t_{2}} J{(t_{2},C_{2})} = - C_{2} J{(t_{2},C_{2})} and C_{2} J{(t_{2},C_{2})} \\frac{\\partial}{\\partial t_{2}} (C_{2} - t_{2}) = - C_{2} J{(t_{2},C_{2})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1))"], [["times", 3, "Mul(Symbol('C_2', commutative=True), Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)))"], "Equality(Mul(Symbol('C_2', commutative=True), Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Derivative(Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('C_2', commutative=True), Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('C_2', commutative=True), Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True)), Derivative(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('C_2', commutative=True), Function('J')(Symbol('t_2', commutative=True), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(E_{n})} = e^{E_{n}}, then derive \\frac{d}{d E_{n}} \\mathbf{D}{(E_{n})} = e^{E_{n}}, then obtain \\frac{d^{2}}{d E_{n}^{2}} e^{E_{n}} = \\frac{d}{d E_{n}} e^{E_{n}}", "derivation": "\\mathbf{D}{(E_{n})} = e^{E_{n}} and \\frac{d}{d E_{n}} \\mathbf{D}{(E_{n})} = \\frac{d}{d E_{n}} e^{E_{n}} and \\frac{d}{d E_{n}} \\mathbf{D}{(E_{n})} = e^{E_{n}} and \\frac{d}{d E_{n}} e^{E_{n}} = e^{E_{n}} and \\frac{d^{2}}{d E_{n}^{2}} e^{E_{n}} = \\frac{d}{d E_{n}} e^{E_{n}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), exp(Symbol('E_n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), exp(Symbol('E_n', commutative=True)))"], [["differentiate", 4, "Symbol('E_n', commutative=True)"], "Equality(Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2))), Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(v_{z},\\phi_1)} = \\phi_1^{v_{z}} and \\mathbf{H}{(v_{z},\\phi_1)} = (\\phi_1^{v_{z}} v_{z})^{v_{z}} (v_{z} \\rho_{f}{(v_{z},\\phi_1)})^{- v_{z}}, then obtain - \\mathbf{H}{(v_{z},\\phi_1)} \\rho_{f}{(v_{z},\\phi_1)} + 1 = 1 - \\rho_{f}{(v_{z},\\phi_1)}", "derivation": "\\rho_{f}{(v_{z},\\phi_1)} = \\phi_1^{v_{z}} and \\mathbf{H}{(v_{z},\\phi_1)} = (\\phi_1^{v_{z}} v_{z})^{v_{z}} (v_{z} \\rho_{f}{(v_{z},\\phi_1)})^{- v_{z}} and \\mathbf{H}{(v_{z},\\phi_1)} = 1 and \\mathbf{H}{(v_{z},\\phi_1)} \\rho_{f}{(v_{z},\\phi_1)} = \\rho_{f}{(v_{z},\\phi_1)} and - \\mathbf{H}{(v_{z},\\phi_1)} \\rho_{f}{(v_{z},\\phi_1)} = - \\rho_{f}{(v_{z},\\phi_1)} and - \\mathbf{H}{(v_{z},\\phi_1)} \\rho_{f}{(v_{z},\\phi_1)} + 1 = 1 - \\rho_{f}{(v_{z},\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Mul(Symbol('v_z', commutative=True), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(1))"], [["times", 3, "Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('v_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(h,\\rho_b)} = \\rho_b h and \\Omega{(h,\\rho_b)} = h + \\operatorname{v_{1}}{(h,\\rho_b)}, then obtain \\frac{\\rho_b h - \\operatorname{v_{1}}{(h,\\rho_b)}}{2 \\operatorname{v_{1}}{(h,\\rho_b)}} = \\frac{h - \\Omega{(h,\\rho_b)} + \\operatorname{v_{1}}{(h,\\rho_b)}}{2 \\operatorname{v_{1}}{(h,\\rho_b)}}", "derivation": "\\operatorname{v_{1}}{(h,\\rho_b)} = \\rho_b h and h + \\operatorname{v_{1}}{(h,\\rho_b)} = \\rho_b h + h and \\Omega{(h,\\rho_b)} = h + \\operatorname{v_{1}}{(h,\\rho_b)} and \\rho_b h + \\Omega{(h,\\rho_b)} = \\rho_b h + h + \\operatorname{v_{1}}{(h,\\rho_b)} and \\rho_b h + \\Omega{(h,\\rho_b)} = h + 2 \\operatorname{v_{1}}{(h,\\rho_b)} and \\rho_b h - \\operatorname{v_{1}}{(h,\\rho_b)} = h - \\Omega{(h,\\rho_b)} + \\operatorname{v_{1}}{(h,\\rho_b)} and \\frac{\\rho_b h - \\operatorname{v_{1}}{(h,\\rho_b)}}{2 \\operatorname{v_{1}}{(h,\\rho_b)}} = \\frac{h - \\Omega{(h,\\rho_b)} + \\operatorname{v_{1}}{(h,\\rho_b)}}{2 \\operatorname{v_{1}}{(h,\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('h', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Symbol('h', commutative=True), Mul(Integer(2), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["minus", 5, "Add(Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["divide", 6, "Mul(Integer(2), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Pow(Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True))), Pow(Function('v_1')(Symbol('h', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(q)} = \\log{(q)}, then obtain \\log{(q + \\frac{- q + \\delta{(q)}}{- q + \\log{(q)}})} = \\log{(q + 1)}", "derivation": "\\delta{(q)} = \\log{(q)} and - q + \\delta{(q)} = - q + \\log{(q)} and \\frac{- q + \\delta{(q)}}{- q + \\log{(q)}} = 1 and q + \\frac{- q + \\delta{(q)}}{- q + \\log{(q)}} = q + 1 and \\log{(q + \\frac{- q + \\delta{(q)}}{- q + \\log{(q)}})} = \\log{(q + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\delta')(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('q', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\delta')(Symbol('q', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('q', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Add(Symbol('q', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\delta')(Symbol('q', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('q', commutative=True))), Integer(-1)))), Add(Symbol('q', commutative=True), Integer(1)))"], [["log", 4], "Equality(log(Add(Symbol('q', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\delta')(Symbol('q', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('q', commutative=True))), Integer(-1))))), log(Add(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(\\sigma_p,A_{x})} = A_{x} + \\sigma_p, then derive \\frac{\\partial}{\\partial A_{x}} G{(\\sigma_p,A_{x})} = 1, then obtain (\\tilde{g}^*{(i)} \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p))^{i} - \\tilde{g}^*{(i)} = - \\tilde{g}^*{(i)} + \\tilde{g}^*^{i}{(i)}", "derivation": "G{(\\sigma_p,A_{x})} = A_{x} + \\sigma_p and \\frac{\\partial}{\\partial A_{x}} G{(\\sigma_p,A_{x})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p) and \\frac{\\partial}{\\partial A_{x}} G{(\\sigma_p,A_{x})} = 1 and \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p) = 1 and \\tilde{g}^*{(i)} \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p) = \\tilde{g}^*{(i)} and (\\tilde{g}^*{(i)} \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p))^{i} = \\tilde{g}^*^{i}{(i)} and (\\tilde{g}^*{(i)} \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sigma_p))^{i} - \\tilde{g}^*{(i)} = - \\tilde{g}^*{(i)} + \\tilde{g}^*^{i}{(i)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\sigma_p', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\sigma_p', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('\\\\sigma_p', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Function('\\\\tilde{g}^*')(Symbol('i', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)))"], [["power", 5, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Symbol('i', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 6, "Function('\\\\tilde{g}^*')(Symbol('i', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Symbol('i', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('i', commutative=True))), Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\lambda{(f_{E},\\Psi_{\\lambda},\\sigma_p)} = \\frac{f_{E}^{\\sigma_p}}{\\Psi_{\\lambda}}, then derive \\frac{\\partial}{\\partial \\sigma_p} \\lambda{(f_{E},\\Psi_{\\lambda},\\sigma_p)} = \\frac{f_{E}^{\\sigma_p} \\log{(f_{E})}}{\\Psi_{\\lambda}}, then obtain \\frac{\\partial}{\\partial \\sigma_p} \\frac{f_{E}^{\\sigma_p}}{\\Psi_{\\lambda}} = \\frac{f_{E}^{\\sigma_p} \\log{(f_{E})}}{\\Psi_{\\lambda}}", "derivation": "\\lambda{(f_{E},\\Psi_{\\lambda},\\sigma_p)} = \\frac{f_{E}^{\\sigma_p}}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\sigma_p} \\lambda{(f_{E},\\Psi_{\\lambda},\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\frac{f_{E}^{\\sigma_p}}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\sigma_p} \\lambda{(f_{E},\\Psi_{\\lambda},\\sigma_p)} = \\frac{f_{E}^{\\sigma_p} \\log{(f_{E})}}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\sigma_p} \\frac{f_{E}^{\\sigma_p}}{\\Psi_{\\lambda}} = \\frac{f_{E}^{\\sigma_p} \\log{(f_{E})}}{\\Psi_{\\lambda}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\lambda')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given v{(v_{z})} = \\frac{d}{d v_{z}} \\log{(v_{z})}, then derive \\int v{(v_{z})} dv_{z} = f^{\\prime} + \\log{(v_{z})}, then obtain \\frac{d}{d v_{z}} (\\int v{(v_{z})} dv_{z} + 1) = \\frac{\\partial}{\\partial v_{z}} (f^{\\prime} + \\log{(v_{z})} + 1)", "derivation": "v{(v_{z})} = \\frac{d}{d v_{z}} \\log{(v_{z})} and \\int v{(v_{z})} dv_{z} = \\int \\frac{d}{d v_{z}} \\log{(v_{z})} dv_{z} and \\int v{(v_{z})} dv_{z} = f^{\\prime} + \\log{(v_{z})} and \\int v{(v_{z})} dv_{z} + 1 = f^{\\prime} + \\log{(v_{z})} + 1 and \\frac{d}{d v_{z}} (\\int v{(v_{z})} dv_{z} + 1) = \\frac{\\partial}{\\partial v_{z}} (f^{\\prime} + \\log{(v_{z})} + 1)", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('v_z', commutative=True)), Derivative(log(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('v')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Derivative(log(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('v_z', commutative=True))))"], [["add", 3, 1], "Equality(Add(Integral(Function('v')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(1)), Add(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('v_z', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Integral(Function('v')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('v_z', commutative=True)), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} = e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}}, then obtain 1 = \\frac{\\hat{\\mathbf{x}}^{2} e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} - \\varepsilon_0}{\\hat{\\mathbf{x}}^{2} \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} - \\varepsilon_0}", "derivation": "\\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} = e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} and \\hat{\\mathbf{x}} \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} = \\hat{\\mathbf{x}} e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} and \\hat{\\mathbf{x}}^{2} \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} = \\hat{\\mathbf{x}}^{2} e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} and \\hat{\\mathbf{x}}^{2} \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} - \\varepsilon_0 = \\hat{\\mathbf{x}}^{2} e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} - \\varepsilon_0 and 1 = \\frac{\\hat{\\mathbf{x}}^{2} e^{\\frac{\\varepsilon_0}{\\hat{\\mathbf{x}}}} - \\varepsilon_0}{\\hat{\\mathbf{x}}^{2} \\delta{(\\hat{\\mathbf{x}},\\varepsilon_0)} - \\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["divide", 2, "Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 4, "Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1)), Add(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and k{(\\mathbf{S})} = \\mathbf{J}{(\\mathbf{S})} - \\cos{(\\mathbf{S})}, then obtain \\frac{d}{d \\mathbf{S}} (\\mathbf{J}{(\\mathbf{S})} - \\cos{(\\mathbf{S})}) + \\frac{d}{d \\mathbf{S}} k{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} 0 + \\frac{d}{d \\mathbf{S}} k{(\\mathbf{S})}", "derivation": "\\mathbf{J}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and k{(\\mathbf{S})} = \\mathbf{J}{(\\mathbf{S})} - \\cos{(\\mathbf{S})} and k{(\\mathbf{S})} = 0 and \\frac{d}{d \\mathbf{S}} k{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} 0 and \\frac{d}{d \\mathbf{S}} (\\mathbf{J}{(\\mathbf{S})} - \\cos{(\\mathbf{S})}) = \\frac{d}{d \\mathbf{S}} 0 and \\frac{d}{d \\mathbf{S}} (\\mathbf{J}{(\\mathbf{S})} - \\cos{(\\mathbf{S})}) + \\frac{d}{d \\mathbf{S}} k{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} 0 + \\frac{d}{d \\mathbf{S}} k{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["add", 5, "Derivative(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Add(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Function('k')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\bar{\\h}{(H,k)} = H k, then obtain - (\\frac{\\partial}{\\partial H} \\bar{\\h}{(H,k)})^{H} \\frac{\\partial^{2}}{\\partial H^{2}} \\bar{\\h}{(H,k)} = 0", "derivation": "\\bar{\\h}{(H,k)} = H k and \\frac{\\partial}{\\partial H} \\bar{\\h}{(H,k)} = \\frac{\\partial}{\\partial H} H k and \\frac{\\partial^{2}}{\\partial H^{2}} \\bar{\\h}{(H,k)} = \\frac{\\partial^{2}}{\\partial H^{2}} H k and - (\\frac{\\partial}{\\partial H} \\bar{\\h}{(H,k)})^{H} \\frac{\\partial^{2}}{\\partial H^{2}} \\bar{\\h}{(H,k)} = - \\frac{\\partial^{2}}{\\partial H^{2}} H k (\\frac{\\partial}{\\partial H} \\bar{\\h}{(H,k)})^{H} and - (\\frac{\\partial}{\\partial H} \\bar{\\h}{(H,k)})^{H} \\frac{\\partial^{2}}{\\partial H^{2}} \\bar{\\h}{(H,k)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Mul(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["times", 3, "Mul(Integer(-1), Pow(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2)))), Mul(Integer(-1), Derivative(Mul(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Pow(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Derivative(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2)))), Integer(0))"]]}, {"prompt": "Given \\mu_{0}{(V)} = \\sin{(\\log{(V)})}, then obtain - \\sin^{V}{(\\log{(V)})} + (\\frac{d}{d V} \\mu_{0}^{V}{(V)})^{V} = - \\sin^{V}{(\\log{(V)})} + (\\frac{d}{d V} \\sin^{V}{(\\log{(V)})})^{V}", "derivation": "\\mu_{0}{(V)} = \\sin{(\\log{(V)})} and \\mu_{0}^{V}{(V)} = \\sin^{V}{(\\log{(V)})} and \\frac{d}{d V} \\mu_{0}^{V}{(V)} = \\frac{d}{d V} \\sin^{V}{(\\log{(V)})} and (\\frac{d}{d V} \\mu_{0}^{V}{(V)})^{V} = (\\frac{d}{d V} \\sin^{V}{(\\log{(V)})})^{V} and - \\sin^{V}{(\\log{(V)})} + (\\frac{d}{d V} \\mu_{0}^{V}{(V)})^{V} = - \\sin^{V}{(\\log{(V)})} + (\\frac{d}{d V} \\sin^{V}{(\\log{(V)})})^{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu_0')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mu_0')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["minus", 4, "Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Pow(Derivative(Pow(Function('\\\\mu_0')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True))), Add(Mul(Integer(-1), Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Pow(Derivative(Pow(sin(log(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given u{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then derive \\int \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} = \\hat{H}_l + \\mathbf{B} + \\frac{\\log{(\\mathbf{B})}^{2}}{2}, then obtain \\int \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} = \\hat{H}_l + \\mathbf{B} + \\frac{u^{2}{(\\mathbf{B})}}{2}", "derivation": "u{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\mathbf{B} + u{(\\mathbf{B})} = \\mathbf{B} + \\log{(\\mathbf{B})} and \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} = \\frac{\\mathbf{B} + \\log{(\\mathbf{B})}}{\\mathbf{B}} and \\int \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} = \\int \\frac{\\mathbf{B} + \\log{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} and \\int \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} = \\hat{H}_l + \\mathbf{B} + \\frac{\\log{(\\mathbf{B})}^{2}}{2} and \\int \\frac{\\mathbf{B} + u{(\\mathbf{B})}}{\\mathbf{B}} d\\mathbf{B} = \\hat{H}_l + \\mathbf{B} + \\frac{u^{2}{(\\mathbf{B})}}{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Function('u')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given x{(q)} = \\frac{d}{d q} \\log{(q)}, then obtain (q \\frac{d}{d q} x{(q)})^{q} + \\int (q \\frac{d}{d q} x{(q)})^{q} dq = (q \\frac{d}{d q} x{(q)})^{q} + \\int (q \\frac{d^{2}}{d q^{2}} \\log{(q)})^{q} dq", "derivation": "x{(q)} = \\frac{d}{d q} \\log{(q)} and \\frac{d}{d q} x{(q)} = \\frac{d^{2}}{d q^{2}} \\log{(q)} and q \\frac{d}{d q} x{(q)} = q \\frac{d^{2}}{d q^{2}} \\log{(q)} and (q \\frac{d}{d q} x{(q)})^{q} = (q \\frac{d^{2}}{d q^{2}} \\log{(q)})^{q} and \\int (q \\frac{d}{d q} x{(q)})^{q} dq = \\int (q \\frac{d^{2}}{d q^{2}} \\log{(q)})^{q} dq and (q \\frac{d}{d q} x{(q)})^{q} + \\int (q \\frac{d}{d q} x{(q)})^{q} dq = (q \\frac{d}{d q} x{(q)})^{q} + \\int (q \\frac{d^{2}}{d q^{2}} \\log{(q)})^{q} dq", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('q', commutative=True)), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(2))))"], [["times", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Symbol('q', commutative=True), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(2)))))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True)), Pow(Mul(Symbol('q', commutative=True), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(2)))), Symbol('q', commutative=True)))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Mul(Symbol('q', commutative=True), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(2)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["add", 5, "Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True)), Integral(Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Pow(Mul(Symbol('q', commutative=True), Derivative(Function('x')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Symbol('q', commutative=True)), Integral(Pow(Mul(Symbol('q', commutative=True), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(2)))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(v_{1})} = \\log{(v_{1})}, then obtain - 2 v_{1} \\mathbf{P}{(v_{1})} \\mathbf{P}^{v_{1}}{(v_{1})} = - 2 v_{1} \\mathbf{P}{(v_{1})} \\log{(v_{1})}^{v_{1}}", "derivation": "\\mathbf{P}{(v_{1})} = \\log{(v_{1})} and 2 \\mathbf{P}{(v_{1})} = \\mathbf{P}{(v_{1})} + \\log{(v_{1})} and \\mathbf{P}^{v_{1}}{(v_{1})} = \\log{(v_{1})}^{v_{1}} and (\\mathbf{P}{(v_{1})} + \\log{(v_{1})}) \\mathbf{P}^{v_{1}}{(v_{1})} = (\\mathbf{P}{(v_{1})} + \\log{(v_{1})}) \\log{(v_{1})}^{v_{1}} and - v_{1} (\\mathbf{P}{(v_{1})} + \\log{(v_{1})}) \\mathbf{P}^{v_{1}}{(v_{1})} = - v_{1} (\\mathbf{P}{(v_{1})} + \\log{(v_{1})}) \\log{(v_{1})}^{v_{1}} and - 2 v_{1} \\mathbf{P}{(v_{1})} \\mathbf{P}^{v_{1}}{(v_{1})} = - 2 v_{1} \\mathbf{P}{(v_{1})} \\log{(v_{1})}^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True))))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["times", 3, "Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True))), Pow(log(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_1', commutative=True), Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True), Add(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True))), Pow(log(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('v_1', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(i)} = \\log{(i)}, then derive \\int \\operatorname{C_{1}}{(i)} di = \\varepsilon + i \\log{(i)} - i, then obtain (\\int \\frac{d}{d i} \\int \\operatorname{C_{1}}{(i)} di d\\varepsilon)^{\\varepsilon} = (\\int \\frac{\\partial}{\\partial i} (\\varepsilon + i \\operatorname{C_{1}}{(i)} - i) d\\varepsilon)^{\\varepsilon}", "derivation": "\\operatorname{C_{1}}{(i)} = \\log{(i)} and \\int \\operatorname{C_{1}}{(i)} di = \\int \\log{(i)} di and \\int \\operatorname{C_{1}}{(i)} di = \\varepsilon + i \\log{(i)} - i and \\int \\operatorname{C_{1}}{(i)} di = \\varepsilon + i \\operatorname{C_{1}}{(i)} - i and \\frac{d}{d i} \\int \\operatorname{C_{1}}{(i)} di = \\frac{\\partial}{\\partial i} (\\varepsilon + i \\operatorname{C_{1}}{(i)} - i) and \\int \\frac{d}{d i} \\int \\operatorname{C_{1}}{(i)} di d\\varepsilon = \\int \\frac{\\partial}{\\partial i} (\\varepsilon + i \\operatorname{C_{1}}{(i)} - i) d\\varepsilon and (\\int \\frac{d}{d i} \\int \\operatorname{C_{1}}{(i)} di d\\varepsilon)^{\\varepsilon} = (\\int \\frac{\\partial}{\\partial i} (\\varepsilon + i \\operatorname{C_{1}}{(i)} - i) d\\varepsilon)^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('i', commutative=True), log(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('i', commutative=True), Function('C_1')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('i', commutative=True), Function('C_1')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('i', commutative=True), Function('C_1')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 6, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Integral(Derivative(Integral(Function('C_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Integral(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Symbol('i', commutative=True), Function('C_1')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{P},\\hat{H})} = \\log{(\\hat{H} + \\mathbf{P})}, then obtain \\frac{\\partial}{\\partial \\hat{H}} (\\tilde{g}^{\\hat{H}}{(\\mathbf{P},\\hat{H})} + \\log{(\\hat{H} + \\mathbf{P})}) = \\frac{\\partial}{\\partial \\hat{H}} (\\log{(\\hat{H} + \\mathbf{P})} + \\log{(\\hat{H} + \\mathbf{P})}^{\\hat{H}})", "derivation": "\\tilde{g}{(\\mathbf{P},\\hat{H})} = \\log{(\\hat{H} + \\mathbf{P})} and \\tilde{g}^{\\hat{H}}{(\\mathbf{P},\\hat{H})} = \\log{(\\hat{H} + \\mathbf{P})}^{\\hat{H}} and \\tilde{g}^{\\hat{H}}{(\\mathbf{P},\\hat{H})} + \\log{(\\hat{H} + \\mathbf{P})} = \\log{(\\hat{H} + \\mathbf{P})} + \\log{(\\hat{H} + \\mathbf{P})}^{\\hat{H}} and \\frac{\\partial}{\\partial \\hat{H}} (\\tilde{g}^{\\hat{H}}{(\\mathbf{P},\\hat{H})} + \\log{(\\hat{H} + \\mathbf{P})}) = \\frac{\\partial}{\\partial \\hat{H}} (\\log{(\\hat{H} + \\mathbf{P})} + \\log{(\\hat{H} + \\mathbf{P})}^{\\hat{H}})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["add", 2, "log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Add(log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\psi,\\rho,\\delta)} = \\frac{\\psi \\rho}{\\delta}, then obtain \\frac{d}{d \\psi} 2 = \\frac{\\partial}{\\partial \\psi} (1 + \\frac{\\psi \\rho}{\\delta \\varepsilon{(\\psi,\\rho,\\delta)}})", "derivation": "\\varepsilon{(\\psi,\\rho,\\delta)} = \\frac{\\psi \\rho}{\\delta} and 1 = \\frac{\\psi \\rho}{\\delta \\varepsilon{(\\psi,\\rho,\\delta)}} and 2 = 1 + \\frac{\\psi \\rho}{\\delta \\varepsilon{(\\psi,\\rho,\\delta)}} and \\frac{d}{d \\psi} 2 = \\frac{\\partial}{\\partial \\psi} (1 + \\frac{\\psi \\rho}{\\delta \\varepsilon{(\\psi,\\rho,\\delta)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["add", 2, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Integer(2), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(W)} = \\cos{(W)} and z{(W)} = \\cos{(W)}, then obtain (\\varepsilon_{0}^{W}{(W)})^{W} \\frac{d}{d W} z{(W)} = (\\cos^{W}{(W)})^{W} \\frac{d}{d W} z{(W)}", "derivation": "\\varepsilon_{0}{(W)} = \\cos{(W)} and z{(W)} = \\cos{(W)} and z^{W}{(W)} = \\cos^{W}{(W)} and (z^{W}{(W)})^{W} = (\\cos^{W}{(W)})^{W} and z^{W}{(W)} = \\varepsilon_{0}^{W}{(W)} and (\\varepsilon_{0}^{W}{(W)})^{W} = (\\cos^{W}{(W)})^{W} and (\\varepsilon_{0}^{W}{(W)})^{W} \\frac{d}{d W} z{(W)} = (\\cos^{W}{(W)})^{W} \\frac{d}{d W} z{(W)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Function('z')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Pow(Function('z')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('z')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["times", 6, "Derivative(Function('z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Pow(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Derivative(Function('z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Derivative(Function('z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given t{(\\chi)} = e^{\\cos{(\\chi)}}, then obtain \\int \\cos{(\\chi - t^{\\chi}{(\\chi)})} d\\chi = \\int \\cos{(\\chi - (e^{\\cos{(\\chi)}})^{\\chi})} d\\chi", "derivation": "t{(\\chi)} = e^{\\cos{(\\chi)}} and t^{\\chi}{(\\chi)} = (e^{\\cos{(\\chi)}})^{\\chi} and - \\chi + t^{\\chi}{(\\chi)} = - \\chi + (e^{\\cos{(\\chi)}})^{\\chi} and \\cos{(\\chi - t^{\\chi}{(\\chi)})} = \\cos{(\\chi - (e^{\\cos{(\\chi)}})^{\\chi})} and \\int \\cos{(\\chi - t^{\\chi}{(\\chi)})} d\\chi = \\int \\cos{(\\chi - (e^{\\cos{(\\chi)}})^{\\chi})} d\\chi", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\chi', commutative=True)), exp(cos(Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(exp(cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["minus", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Function('t')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(exp(cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"], [["cos", 3], "Equality(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Pow(exp(cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Pow(exp(cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given U{(r_{0},\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}} - r_{0})}, then obtain (r_{0} U{(r_{0},\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} (r_{0} U{(r_{0},\\hat{\\mathbf{x}})} + r_{0}) = (r_{0} U{(r_{0},\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} (r_{0} \\cos{(\\hat{\\mathbf{x}} - r_{0})} + r_{0})", "derivation": "U{(r_{0},\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}} - r_{0})} and r_{0} U{(r_{0},\\hat{\\mathbf{x}})} = r_{0} \\cos{(\\hat{\\mathbf{x}} - r_{0})} and r_{0} U{(r_{0},\\hat{\\mathbf{x}})} + r_{0} = r_{0} \\cos{(\\hat{\\mathbf{x}} - r_{0})} + r_{0} and (r_{0} U{(r_{0},\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} (r_{0} U{(r_{0},\\hat{\\mathbf{x}})} + r_{0}) = (r_{0} U{(r_{0},\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} (r_{0} \\cos{(\\hat{\\mathbf{x}} - r_{0})} + r_{0})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"], [["minus", 2, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('r_0', commutative=True)), Add(Mul(Symbol('r_0', commutative=True), cos(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))), Symbol('r_0', commutative=True)))"], [["times", 3, "Pow(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('r_0', commutative=True))), Mul(Pow(Mul(Symbol('r_0', commutative=True), Function('U')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Symbol('r_0', commutative=True), cos(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\delta,\\mathbf{S})} = \\mathbf{S} \\log{(\\delta)}, then obtain ((- \\mathbf{S} + \\operatorname{v_{y}}{(\\delta,\\mathbf{S})})^{\\delta})^{\\mathbf{S}} - ((\\mathbf{S} \\log{(\\delta)} - \\mathbf{S})^{\\delta})^{\\mathbf{S}} = 0", "derivation": "\\operatorname{v_{y}}{(\\delta,\\mathbf{S})} = \\mathbf{S} \\log{(\\delta)} and - \\mathbf{S} + \\operatorname{v_{y}}{(\\delta,\\mathbf{S})} = \\mathbf{S} \\log{(\\delta)} - \\mathbf{S} and (- \\mathbf{S} + \\operatorname{v_{y}}{(\\delta,\\mathbf{S})})^{\\delta} = (\\mathbf{S} \\log{(\\delta)} - \\mathbf{S})^{\\delta} and ((- \\mathbf{S} + \\operatorname{v_{y}}{(\\delta,\\mathbf{S})})^{\\delta})^{\\mathbf{S}} = ((\\mathbf{S} \\log{(\\delta)} - \\mathbf{S})^{\\delta})^{\\mathbf{S}} and ((- \\mathbf{S} + \\operatorname{v_{y}}{(\\delta,\\mathbf{S})})^{\\delta})^{\\mathbf{S}} - ((\\mathbf{S} \\log{(\\delta)} - \\mathbf{S})^{\\delta})^{\\mathbf{S}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 4, "Pow(Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbb{I}{(B,\\Psi_{\\lambda})} = - B + \\cos{(\\Psi_{\\lambda})}, then obtain \\int \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbb{I}^{B}{(B,\\Psi_{\\lambda})} dB = \\int \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- B + \\cos{(\\Psi_{\\lambda})})^{B} dB", "derivation": "\\mathbb{I}{(B,\\Psi_{\\lambda})} = - B + \\cos{(\\Psi_{\\lambda})} and \\mathbb{I}^{B}{(B,\\Psi_{\\lambda})} = (- B + \\cos{(\\Psi_{\\lambda})})^{B} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbb{I}^{B}{(B,\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- B + \\cos{(\\Psi_{\\lambda})})^{B} and \\int \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbb{I}^{B}{(B,\\Psi_{\\lambda})} dB = \\int \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- B + \\cos{(\\Psi_{\\lambda})})^{B} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('B', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('B', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('B', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbb{I}')(Symbol('B', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\mathbb{I}')(Symbol('B', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\theta,n_{2})} = \\theta n_{2}, then obtain - \\frac{- \\theta n_{2} - n_{2} + 2 \\theta_{2}{(\\theta,n_{2})}}{\\theta_{2}{(\\theta,n_{2})}} = - \\frac{\\theta n_{2} - n_{2}}{\\theta_{2}{(\\theta,n_{2})}}", "derivation": "\\theta_{2}{(\\theta,n_{2})} = \\theta n_{2} and - n_{2} + \\theta_{2}{(\\theta,n_{2})} = \\theta n_{2} - n_{2} and - \\theta n_{2} - n_{2} + \\theta_{2}{(\\theta,n_{2})} = - n_{2} and - \\theta n_{2} - n_{2} + 2 \\theta_{2}{(\\theta,n_{2})} = - n_{2} + \\theta_{2}{(\\theta,n_{2})} and - \\theta n_{2} - n_{2} + 2 \\theta_{2}{(\\theta,n_{2})} = \\theta n_{2} - n_{2} and - \\frac{- \\theta n_{2} - n_{2} + 2 \\theta_{2}{(\\theta,n_{2})}}{\\theta_{2}{(\\theta,n_{2})}} = - \\frac{\\theta n_{2} - n_{2}}{\\theta_{2}{(\\theta,n_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))), Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} = \\frac{\\cos{(\\hat{X})}}{z}, then obtain \\int \\cos{(\\operatorname{g_{\\varepsilon}}{(\\hat{X},z)})} d\\hat{X} = \\int \\cos{(2 \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} - \\frac{\\cos{(\\hat{X})}}{z})} d\\hat{X}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} = \\frac{\\cos{(\\hat{X})}}{z} and 0 = - \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} + \\frac{\\cos{(\\hat{X})}}{z} and - \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} = - 2 \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} + \\frac{\\cos{(\\hat{X})}}{z} and \\cos{(\\operatorname{g_{\\varepsilon}}{(\\hat{X},z)})} = \\cos{(2 \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} - \\frac{\\cos{(\\hat{X})}}{z})} and \\int \\cos{(\\operatorname{g_{\\varepsilon}}{(\\hat{X},z)})} d\\hat{X} = \\int \\cos{(2 \\operatorname{g_{\\varepsilon}}{(\\hat{X},z)} - \\frac{\\cos{(\\hat{X})}}{z})} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True)))))"], [["minus", 2, "Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True)))))"], [["cos", 3], "Equality(cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), cos(Add(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(cos(Add(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{X}', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})} = (f^{*})^{\\mathbf{J}} - v_{1}, then obtain \\frac{((f^{*})^{\\mathbf{J}} - v_{1}) \\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})}}{v_{1}} = \\frac{((f^{*})^{\\mathbf{J}} - v_{1})^{2}}{v_{1}}", "derivation": "\\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})} = (f^{*})^{\\mathbf{J}} - v_{1} and \\frac{\\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})}}{v_{1}} = \\frac{(f^{*})^{\\mathbf{J}} - v_{1}}{v_{1}} and \\frac{\\operatorname{E_{\\lambda}}^{2}{(v_{1},\\mathbf{J},f^{*})}}{v_{1}} = \\frac{((f^{*})^{\\mathbf{J}} - v_{1}) \\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})}}{v_{1}} and \\frac{((f^{*})^{\\mathbf{J}} - v_{1}) \\operatorname{E_{\\lambda}}{(v_{1},\\mathbf{J},f^{*})}}{v_{1}} = \\frac{((f^{*})^{\\mathbf{J}} - v_{1})^{2}}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True)), Add(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"], [["times", 2, "Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Function('E_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Add(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)} = \\Psi^{\\dagger} \\psi^*, then obtain \\int (\\frac{\\partial}{\\partial \\psi^*} \\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} = \\int (\\frac{\\partial}{\\partial \\psi^*} \\Psi^{\\dagger} \\psi^*)^{\\Psi^{\\dagger}} d\\Psi^{\\dagger}", "derivation": "\\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)} = \\Psi^{\\dagger} \\psi^* and \\frac{\\partial}{\\partial \\psi^*} \\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)} = \\frac{\\partial}{\\partial \\psi^*} \\Psi^{\\dagger} \\psi^* and (\\frac{\\partial}{\\partial \\psi^*} \\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)})^{\\Psi^{\\dagger}} = (\\frac{\\partial}{\\partial \\psi^*} \\Psi^{\\dagger} \\psi^*)^{\\Psi^{\\dagger}} and \\int (\\frac{\\partial}{\\partial \\psi^*} \\operatorname{E_{x}}{(\\Psi^{\\dagger},\\psi^*)})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} = \\int (\\frac{\\partial}{\\partial \\psi^*} \\Psi^{\\dagger} \\psi^*)^{\\Psi^{\\dagger}} d\\Psi^{\\dagger}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Derivative(Function('E_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('E_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given z{(v_{1})} = \\cos{(v_{1})}, then obtain 1 = - 2 z{(v_{1})} + 2 \\cos{(v_{1})} + 1", "derivation": "z{(v_{1})} = \\cos{(v_{1})} and 0 = - z{(v_{1})} + \\cos{(v_{1})} and 1 = - z{(v_{1})} + \\cos{(v_{1})} + 1 and \\cos{(v_{1})} = - z{(v_{1})} + 2 \\cos{(v_{1})} and 1 = - 2 z{(v_{1})} + 2 \\cos{(v_{1})} + 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["minus", 1, "Function('z')(Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z')(Symbol('v_1', commutative=True))), cos(Symbol('v_1', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('z')(Symbol('v_1', commutative=True))), cos(Symbol('v_1', commutative=True)), Integer(1)))"], [["add", 1, "Add(Mul(Integer(-1), Function('z')(Symbol('v_1', commutative=True))), cos(Symbol('v_1', commutative=True)))"], "Equality(cos(Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Function('z')(Symbol('v_1', commutative=True))), Mul(Integer(2), cos(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Add(Mul(Integer(-1), Integer(2), Function('z')(Symbol('v_1', commutative=True))), Mul(Integer(2), cos(Symbol('v_1', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\tilde{g}^*{(m,t)} = t^{m}, then derive \\frac{\\partial}{\\partial m} \\tilde{g}^*{(m,t)} + 1 = t^{m} \\log{(t)} + 1, then obtain (\\tilde{g}^*{(m,t)} \\log{(t)} + 1)^{m} = (t^{m} \\log{(t)} + 1)^{m}", "derivation": "\\tilde{g}^*{(m,t)} = t^{m} and m + \\tilde{g}^*{(m,t)} = m + t^{m} and \\frac{\\partial}{\\partial m} (m + \\tilde{g}^*{(m,t)}) = \\frac{\\partial}{\\partial m} (m + t^{m}) and \\frac{\\partial}{\\partial m} \\tilde{g}^*{(m,t)} + 1 = t^{m} \\log{(t)} + 1 and \\frac{\\partial}{\\partial m} \\tilde{g}^*{(m,t)} + 1 = \\tilde{g}^*{(m,t)} \\log{(t)} + 1 and (\\frac{\\partial}{\\partial m} \\tilde{g}^*{(m,t)} + 1)^{m} = (t^{m} \\log{(t)} + 1)^{m} and (\\tilde{g}^*{(m,t)} \\log{(t)} + 1)^{m} = (t^{m} \\log{(t)} + 1)^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('m', commutative=True)))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True))), Add(Symbol('m', commutative=True), Pow(Symbol('t', commutative=True), Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), Pow(Symbol('t', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Add(Mul(Pow(Symbol('t', commutative=True), Symbol('m', commutative=True)), log(Symbol('t', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Add(Mul(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Integer(1)))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Symbol('m', commutative=True)), Pow(Add(Mul(Pow(Symbol('t', commutative=True), Symbol('m', commutative=True)), log(Symbol('t', commutative=True))), Integer(1)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Add(Mul(Function('\\\\tilde{g}^*')(Symbol('m', commutative=True), Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Integer(1)), Symbol('m', commutative=True)), Pow(Add(Mul(Pow(Symbol('t', commutative=True), Symbol('m', commutative=True)), log(Symbol('t', commutative=True))), Integer(1)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\mu_0)} = \\cos{(e^{\\mu_0})} and B{(\\mu_0)} = \\cos{(e^{\\mu_0})}, then obtain \\frac{d}{d \\mu_0} (-1 + \\frac{\\mathbf{P}{(\\mu_0)}}{\\mu_0}) = \\frac{d}{d \\mu_0} (-1 + \\frac{B{(\\mu_0)}}{\\mu_0})", "derivation": "\\mathbf{P}{(\\mu_0)} = \\cos{(e^{\\mu_0})} and \\frac{\\mathbf{P}{(\\mu_0)}}{\\mu_0} = \\frac{\\cos{(e^{\\mu_0})}}{\\mu_0} and -1 + \\frac{\\mathbf{P}{(\\mu_0)}}{\\mu_0} = -1 + \\frac{\\cos{(e^{\\mu_0})}}{\\mu_0} and B{(\\mu_0)} = \\cos{(e^{\\mu_0})} and \\frac{d}{d \\mu_0} (-1 + \\frac{\\mathbf{P}{(\\mu_0)}}{\\mu_0}) = \\frac{d}{d \\mu_0} (-1 + \\frac{\\cos{(e^{\\mu_0})}}{\\mu_0}) and \\frac{d}{d \\mu_0} (-1 + \\frac{\\mathbf{P}{(\\mu_0)}}{\\mu_0}) = \\frac{d}{d \\mu_0} (-1 + \\frac{B{(\\mu_0)}}{\\mu_0})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mu_0', commutative=True)), cos(exp(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mu_0', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mu_0', commutative=True))))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\mu_0', commutative=True)), cos(exp(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(a,\\mu)} = a + \\sin{(\\mu)}, then obtain \\cos{(\\frac{a + \\operatorname{F_{g}}{(a,\\mu)} + \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}})} = \\cos{(\\frac{2 a + 2 \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}})}", "derivation": "\\operatorname{F_{g}}{(a,\\mu)} = a + \\sin{(\\mu)} and a + \\operatorname{F_{g}}{(a,\\mu)} + \\sin{(\\mu)} = 2 a + 2 \\sin{(\\mu)} and \\frac{a + \\operatorname{F_{g}}{(a,\\mu)} + \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}} = \\frac{2 a + 2 \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}} and \\cos{(\\frac{a + \\operatorname{F_{g}}{(a,\\mu)} + \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}})} = \\cos{(\\frac{2 a + 2 \\sin{(\\mu)}}{\\operatorname{F_{g}}{(a,\\mu)}})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('a', commutative=True), sin(Symbol('\\\\mu', commutative=True))))"], [["add", 1, "Add(Symbol('a', commutative=True), sin(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Symbol('a', commutative=True), Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mu', commutative=True)))))"], [["divide", 2, "Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Add(Symbol('a', commutative=True), Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Pow(Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mu', commutative=True)))), Pow(Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["cos", 3], "Equality(cos(Mul(Add(Symbol('a', commutative=True), Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Pow(Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)))), cos(Mul(Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mu', commutative=True)))), Pow(Function('F_g')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given i{(H,C_{2})} = \\frac{\\partial}{\\partial H} C_{2} H and \\nabla{(x^\\prime)} = \\sin{(\\cos{(x^\\prime)})}, then derive i{(H,C_{2})} = C_{2}, then obtain x^\\prime i{(H,\\frac{\\partial}{\\partial H} C_{2} H)} + \\sin{(\\cos{(x^\\prime)})} = x^\\prime \\frac{\\partial}{\\partial H} C_{2} H + \\sin{(\\cos{(x^\\prime)})}", "derivation": "i{(H,C_{2})} = \\frac{\\partial}{\\partial H} C_{2} H and i{(H,C_{2})} = C_{2} and \\frac{\\partial}{\\partial H} C_{2} H = C_{2} and i{(H,\\frac{\\partial}{\\partial H} C_{2} H)} = \\frac{\\partial}{\\partial H} C_{2} H and \\nabla{(x^\\prime)} = \\sin{(\\cos{(x^\\prime)})} and x^\\prime i{(H,\\frac{\\partial}{\\partial H} C_{2} H)} = x^\\prime \\frac{\\partial}{\\partial H} C_{2} H and x^\\prime i{(H,\\frac{\\partial}{\\partial H} C_{2} H)} + \\nabla{(x^\\prime)} = x^\\prime \\frac{\\partial}{\\partial H} C_{2} H + \\nabla{(x^\\prime)} and x^\\prime i{(H,\\frac{\\partial}{\\partial H} C_{2} H)} + \\sin{(\\cos{(x^\\prime)})} = x^\\prime \\frac{\\partial}{\\partial H} C_{2} H + \\sin{(\\cos{(x^\\prime)})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('H', commutative=True), Symbol('C_2', commutative=True)), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('i')(Symbol('H', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('C_2', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('i')(Symbol('H', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True)), sin(cos(Symbol('x^\\\\prime', commutative=True))))"], [["times", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('i')(Symbol('H', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Mul(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["add", 6, "Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('x^\\\\prime', commutative=True), Function('i')(Symbol('H', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Symbol('x^\\\\prime', commutative=True), Function('i')(Symbol('H', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), sin(cos(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), sin(cos(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given J{(m,\\mathbf{v})} = \\frac{m}{\\mathbf{v}}, then obtain (- m + \\frac{m}{\\mathbf{v}}) (- m + J{(m,\\mathbf{v})}) + J{(m,\\mathbf{v})} = (- m + \\frac{m}{\\mathbf{v}}) (- m + J{(m,\\mathbf{v})}) + \\frac{m}{\\mathbf{v}}", "derivation": "J{(m,\\mathbf{v})} = \\frac{m}{\\mathbf{v}} and - m + J{(m,\\mathbf{v})} = - m + \\frac{m}{\\mathbf{v}} and (- m + \\frac{m}{\\mathbf{v}}) (- m + J{(m,\\mathbf{v})}) = (- m + \\frac{m}{\\mathbf{v}})^{2} and (- m + \\frac{m}{\\mathbf{v}})^{2} + J{(m,\\mathbf{v})} = (- m + \\frac{m}{\\mathbf{v}})^{2} + \\frac{m}{\\mathbf{v}} and (- m + \\frac{m}{\\mathbf{v}}) (- m + J{(m,\\mathbf{v})}) + J{(m,\\mathbf{v})} = (- m + \\frac{m}{\\mathbf{v}}) (- m + J{(m,\\mathbf{v})}) + \\frac{m}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Integer(2)))"], [["add", 1, "Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Integer(2))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Integer(2)), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Integer(2)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('J')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"]]}, {"prompt": "Given u{(J,L_{\\varepsilon})} = J^{L_{\\varepsilon}}, then obtain 0 = J^{L_{\\varepsilon}} \\log{(J)} - \\frac{\\partial}{\\partial L_{\\varepsilon}} u{(J,L_{\\varepsilon})}", "derivation": "u{(J,L_{\\varepsilon})} = J^{L_{\\varepsilon}} and \\frac{\\partial}{\\partial L_{\\varepsilon}} u{(J,L_{\\varepsilon})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} J^{L_{\\varepsilon}} and 0 = \\frac{\\partial}{\\partial L_{\\varepsilon}} J^{L_{\\varepsilon}} - \\frac{\\partial}{\\partial L_{\\varepsilon}} u{(J,L_{\\varepsilon})} and 0 = J^{L_{\\varepsilon}} \\log{(J)} - \\frac{\\partial}{\\partial L_{\\varepsilon}} u{(J,L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('u')(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('u')(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('J', commutative=True))), Mul(Integer(-1), Derivative(Function('u')(Symbol('J', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given p{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then obtain \\int (- \\hat{\\mathbf{x}} + 2 p{(\\hat{\\mathbf{x}})} - 2 \\log{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}} = \\int - \\hat{\\mathbf{x}} d\\hat{\\mathbf{x}}", "derivation": "p{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and - \\hat{\\mathbf{x}} + p{(\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + \\log{(\\hat{\\mathbf{x}})} and - \\hat{\\mathbf{x}} + p{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} and - \\hat{\\mathbf{x}} + 2 p{(\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + p{(\\hat{\\mathbf{x}})} and - \\hat{\\mathbf{x}} + 2 p{(\\hat{\\mathbf{x}})} - 2 \\log{(\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} and \\int (- \\hat{\\mathbf{x}} + 2 p{(\\hat{\\mathbf{x}})} - 2 \\log{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}} = \\int - \\hat{\\mathbf{x}} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["minus", 2, "log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Function('p')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})} = \\sin{(\\mathbb{I} \\mathbf{M})} and \\operatorname{F_{g}}{(\\mathbb{I},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\sin{(\\mathbb{I} \\mathbf{M})}}{\\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})}}, then obtain \\operatorname{F_{g}}{(\\mathbb{I},\\mathbf{M})} = \\frac{d}{d \\mathbb{I}} 1", "derivation": "\\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})} = \\sin{(\\mathbb{I} \\mathbf{M})} and 1 = \\frac{\\sin{(\\mathbb{I} \\mathbf{M})}}{\\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})}} and \\frac{d}{d \\mathbb{I}} 1 = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\sin{(\\mathbb{I} \\mathbf{M})}}{\\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})}} and \\operatorname{F_{g}}{(\\mathbb{I},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\sin{(\\mathbb{I} \\mathbf{M})}}{\\eta^{\\prime}{(\\mathbb{I},\\mathbf{M})}} and \\operatorname{F_{g}}{(\\mathbb{I},\\mathbf{M})} = \\frac{d}{d \\mathbb{I}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(a,\\mathbf{B})} = \\mathbf{B} + a, then derive (\\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B})^{a} = (\\mathbf{B} + \\mathbf{f})^{a}, then obtain ((\\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B})^{a})^{\\mathbf{f}} = ((\\mathbf{B} + \\mathbf{f})^{a})^{\\mathbf{f}}", "derivation": "\\dot{z}{(a,\\mathbf{B})} = \\mathbf{B} + a and \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} = 1 and \\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B} = \\int 1 d\\mathbf{B} and (\\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B})^{a} = (\\int 1 d\\mathbf{B})^{a} and (\\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B})^{a} = (\\mathbf{B} + \\mathbf{f})^{a} and ((\\int \\frac{\\dot{z}{(a,\\mathbf{B})}}{\\mathbf{B} + a} d\\mathbf{B})^{a})^{\\mathbf{f}} = ((\\mathbf{B} + \\mathbf{f})^{a})^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('a', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('a', commutative=True)))"], [["power", 5, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Pow(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('a', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('a', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given b{(I,t_{1})} = \\cos^{t_{1}}{(I)}, then derive \\frac{\\frac{\\partial}{\\partial t_{1}} b{(I,t_{1})}}{\\cos{(I)}} = \\frac{\\log{(\\cos{(I)})} \\cos^{t_{1}}{(I)}}{\\cos{(I)}}, then obtain (\\frac{\\frac{\\partial}{\\partial t_{1}} \\cos^{t_{1}}{(I)}}{\\cos{(I)}})^{I} = (\\frac{\\log{(\\cos{(I)})} \\cos^{t_{1}}{(I)}}{\\cos{(I)}})^{I}", "derivation": "b{(I,t_{1})} = \\cos^{t_{1}}{(I)} and \\frac{\\partial}{\\partial t_{1}} b{(I,t_{1})} = \\frac{\\partial}{\\partial t_{1}} \\cos^{t_{1}}{(I)} and \\frac{\\frac{\\partial}{\\partial t_{1}} b{(I,t_{1})}}{\\cos{(I)}} = \\frac{\\frac{\\partial}{\\partial t_{1}} \\cos^{t_{1}}{(I)}}{\\cos{(I)}} and \\frac{\\frac{\\partial}{\\partial t_{1}} b{(I,t_{1})}}{\\cos{(I)}} = \\frac{\\log{(\\cos{(I)})} \\cos^{t_{1}}{(I)}}{\\cos{(I)}} and \\frac{\\frac{\\partial}{\\partial t_{1}} \\cos^{t_{1}}{(I)}}{\\cos{(I)}} = \\frac{\\log{(\\cos{(I)})} \\cos^{t_{1}}{(I)}}{\\cos{(I)}} and (\\frac{\\frac{\\partial}{\\partial t_{1}} \\cos^{t_{1}}{(I)}}{\\cos{(I)}})^{I} = (\\frac{\\log{(\\cos{(I)})} \\cos^{t_{1}}{(I)}}{\\cos{(I)}})^{I}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('I', commutative=True), Symbol('t_1', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('I', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["divide", 2, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('I', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('I', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(log(cos(Symbol('I', commutative=True))), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(log(cos(Symbol('I', commutative=True))), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True))))"], [["power", 5, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Symbol('I', commutative=True)), Pow(Mul(log(cos(Symbol('I', commutative=True))), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('t_1', commutative=True))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(C_{2})} = e^{e^{C_{2}}}, then derive \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = e^{C_{2}} e^{e^{C_{2}}}, then obtain e^{C_{2}} e^{e^{C_{2}}} + \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = \\mathbf{A}{(C_{2})} e^{C_{2}} + e^{C_{2}} e^{e^{C_{2}}}", "derivation": "\\mathbf{A}{(C_{2})} = e^{e^{C_{2}}} and \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = \\frac{d}{d C_{2}} e^{e^{C_{2}}} and \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = e^{C_{2}} e^{e^{C_{2}}} and \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = \\mathbf{A}{(C_{2})} e^{C_{2}} and e^{C_{2}} e^{e^{C_{2}}} + \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = \\mathbf{A}{(C_{2})} e^{C_{2}} + e^{C_{2}} e^{e^{C_{2}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), exp(exp(Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(exp(Symbol('C_2', commutative=True)), exp(exp(Symbol('C_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True))))"], [["add", 4, "Mul(exp(Symbol('C_2', commutative=True)), exp(exp(Symbol('C_2', commutative=True))))"], "Equality(Add(Mul(exp(Symbol('C_2', commutative=True)), exp(exp(Symbol('C_2', commutative=True)))), Derivative(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Mul(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True))), Mul(exp(Symbol('C_2', commutative=True)), exp(exp(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}_0{(A_{y},\\dot{y})} = \\cos^{\\dot{y}}{(A_{y})}, then derive \\frac{\\dot{y}^{2}}{2} + \\mathbf{s} = \\int (\\dot{y} - \\hat{x}_0{(A_{y},\\dot{y})} + \\cos^{\\dot{y}}{(A_{y})}) d\\dot{y}, then obtain \\frac{\\dot{y}^{2}}{2} + \\mathbf{s} = \\int \\dot{y} d\\dot{y}", "derivation": "\\hat{x}_0{(A_{y},\\dot{y})} = \\cos^{\\dot{y}}{(A_{y})} and \\dot{y} + \\hat{x}_0{(A_{y},\\dot{y})} = \\dot{y} + \\cos^{\\dot{y}}{(A_{y})} and \\dot{y} = \\dot{y} - \\hat{x}_0{(A_{y},\\dot{y})} + \\cos^{\\dot{y}}{(A_{y})} and \\int \\dot{y} d\\dot{y} = \\int (\\dot{y} - \\hat{x}_0{(A_{y},\\dot{y})} + \\cos^{\\dot{y}}{(A_{y})}) d\\dot{y} and \\frac{\\dot{y}^{2}}{2} + \\mathbf{s} = \\int (\\dot{y} - \\hat{x}_0{(A_{y},\\dot{y})} + \\cos^{\\dot{y}}{(A_{y})}) d\\dot{y} and \\frac{\\dot{y}^{2}}{2} + \\mathbf{s} = \\int \\dot{y} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Pow(cos(Symbol('A_y', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 2, "Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(cos(Symbol('A_y', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Symbol('\\\\dot{y}', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(cos(Symbol('A_y', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(cos(Symbol('A_y', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Symbol('\\\\dot{y}', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} and \\operatorname{v_{x}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\operatorname{t_{2}}{(\\hat{\\mathbf{r}})}, then obtain (\\hat{p} + z^{*}) \\operatorname{v_{x}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} (\\hat{p} + z^{*}) \\log{(\\hat{\\mathbf{r}})}", "derivation": "\\operatorname{t_{2}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} and \\tilde{\\infty} \\operatorname{t_{2}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\log{(\\hat{\\mathbf{r}})} and \\operatorname{v_{x}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\operatorname{t_{2}}{(\\hat{\\mathbf{r}})} and \\operatorname{v_{x}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\log{(\\hat{\\mathbf{r}})} and (\\hat{p} + z^{*}) \\operatorname{v_{x}}{(\\hat{\\mathbf{r}})} = \\tilde{\\infty} (\\hat{p} + z^{*}) \\log{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, 0], "Equality(Mul(zoo, Function('t_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(zoo, log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(zoo, Function('t_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(zoo, log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 4, "Add(Symbol('\\\\hat{p}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('z^*', commutative=True)), Function('v_x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(zoo, Add(Symbol('\\\\hat{p}', commutative=True), Symbol('z^*', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given t{(i,H)} = - H + \\log{(i)}, then obtain \\frac{(\\int t{(i,H)} dH)^{H}}{(- H + \\log{(i)}) t{(i,H)}} = \\frac{(\\int (- H + \\log{(i)}) dH)^{H}}{(- H + \\log{(i)}) t{(i,H)}}", "derivation": "t{(i,H)} = - H + \\log{(i)} and \\int t{(i,H)} dH = \\int (- H + \\log{(i)}) dH and (\\int t{(i,H)} dH)^{H} = (\\int (- H + \\log{(i)}) dH)^{H} and \\frac{(\\int t{(i,H)} dH)^{H}}{(- H + \\log{(i)}) t{(i,H)}} = \\frac{(\\int (- H + \\log{(i)}) dH)^{H}}{(- H + \\log{(i)}) t{(i,H)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["divide", 3, "Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Integer(-1)), Pow(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(Integral(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Integer(-1)), Pow(Function('t')(Symbol('i', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('i', commutative=True))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(J,C_{d})} = e^{C_{d}^{J}} and \\mathbf{H}{(J,C_{d})} = e^{C_{d}^{J}}, then obtain - \\mathbf{H}{(J,C_{d})} + e^{C_{d}^{J}} = 0", "derivation": "\\operatorname{F_{g}}{(J,C_{d})} = e^{C_{d}^{J}} and \\mathbf{H}{(J,C_{d})} = e^{C_{d}^{J}} and \\mathbf{H}{(J,C_{d})} = \\operatorname{F_{g}}{(J,C_{d})} and - \\operatorname{F_{g}}{(J,C_{d})} + \\mathbf{H}{(J,C_{d})} = 0 and \\mathbf{H}{(J,C_{d})} - e^{C_{d}^{J}} = 0 and - \\mathbf{H}{(J,C_{d})} + e^{C_{d}^{J}} = 0", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('J', commutative=True), Symbol('C_d', commutative=True)), exp(Pow(Symbol('C_d', commutative=True), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('J', commutative=True), Symbol('C_d', commutative=True)), exp(Pow(Symbol('C_d', commutative=True), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{H}')(Symbol('J', commutative=True), Symbol('C_d', commutative=True)), Function('F_g')(Symbol('J', commutative=True), Symbol('C_d', commutative=True)))"], [["minus", 3, "Function('F_g')(Symbol('J', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_g')(Symbol('J', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\mathbf{H}')(Symbol('J', commutative=True), Symbol('C_d', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('J', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('C_d', commutative=True), Symbol('J', commutative=True))))), Integer(0))"], [["times", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('J', commutative=True), Symbol('C_d', commutative=True))), exp(Pow(Symbol('C_d', commutative=True), Symbol('J', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\Omega{(v_{y},I)} = \\frac{\\cos{(I)}}{v_{y}}, then obtain \\frac{\\partial}{\\partial v_{y}} (- v_{y} + \\Omega^{v_{y}}{(v_{y},I)}) = \\frac{\\partial}{\\partial v_{y}} (- v_{y} + (\\frac{\\cos{(I)}}{v_{y}})^{v_{y}})", "derivation": "\\Omega{(v_{y},I)} = \\frac{\\cos{(I)}}{v_{y}} and \\Omega^{v_{y}}{(v_{y},I)} = (\\frac{\\cos{(I)}}{v_{y}})^{v_{y}} and - v_{y} + \\Omega^{v_{y}}{(v_{y},I)} = - v_{y} + (\\frac{\\cos{(I)}}{v_{y}})^{v_{y}} and \\frac{\\partial}{\\partial v_{y}} (- v_{y} + \\Omega^{v_{y}}{(v_{y},I)}) = \\frac{\\partial}{\\partial v_{y}} (- v_{y} + (\\frac{\\cos{(I)}}{v_{y}})^{v_{y}})", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('v_y', commutative=True), Symbol('I', commutative=True)), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), cos(Symbol('I', commutative=True))))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('v_y', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True)), Pow(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), cos(Symbol('I', commutative=True))), Symbol('v_y', commutative=True)))"], [["minus", 2, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Pow(Function('\\\\Omega')(Symbol('v_y', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Pow(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), cos(Symbol('I', commutative=True))), Symbol('v_y', commutative=True))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Pow(Function('\\\\Omega')(Symbol('v_y', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Pow(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), cos(Symbol('I', commutative=True))), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{x},x)} = \\sin^{\\dot{x}}{(x)}, then obtain x^{2} \\operatorname{A_{y}}{(\\dot{x},x)} \\sin^{3 \\dot{x}}{(x)} = x^{2} \\sin^{4 \\dot{x}}{(x)}", "derivation": "\\operatorname{A_{y}}{(\\dot{x},x)} = \\sin^{\\dot{x}}{(x)} and \\operatorname{A_{y}}{(\\dot{x},x)} \\sin^{\\dot{x}}{(x)} = \\sin^{2 \\dot{x}}{(x)} and x \\operatorname{A_{y}}{(\\dot{x},x)} \\sin^{\\dot{x}}{(x)} = x \\sin^{2 \\dot{x}}{(x)} and x^{2} \\operatorname{A_{y}}{(\\dot{x},x)} \\sin^{3 \\dot{x}}{(x)} = x^{2} \\sin^{4 \\dot{x}}{(x)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "Pow(sin(Symbol('x', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Pow(sin(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Mul(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True)))))"], [["times", 3, "Mul(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(2)), Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Mul(Integer(3), Symbol('\\\\dot{x}', commutative=True)))), Mul(Pow(Symbol('x', commutative=True), Integer(2)), Pow(sin(Symbol('x', commutative=True)), Mul(Integer(4), Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(E,J)} = E J, then obtain (- E + \\frac{\\iiint \\hat{X}{(E,J)} dJ dJ dJ}{E})^{E} = (- E + \\frac{\\iiint E J dJ dJ dJ}{E})^{E}", "derivation": "\\hat{X}{(E,J)} = E J and \\int \\hat{X}{(E,J)} dJ = \\int E J dJ and \\iint \\hat{X}{(E,J)} dJ dJ = \\iint E J dJ dJ and \\iiint \\hat{X}{(E,J)} dJ dJ dJ = \\iiint E J dJ dJ dJ and \\frac{\\iiint \\hat{X}{(E,J)} dJ dJ dJ}{E} = \\frac{\\iiint E J dJ dJ dJ}{E} and - E + \\frac{\\iiint \\hat{X}{(E,J)} dJ dJ dJ}{E} = - E + \\frac{\\iiint E J dJ dJ dJ}{E} and (- E + \\frac{\\iiint \\hat{X}{(E,J)} dJ dJ dJ}{E})^{E} = (- E + \\frac{\\iiint E J dJ dJ dJ}{E})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["times", 4, "Pow(Symbol('E', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["add", 5, "Mul(Integer(-1), Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))))"], [["power", 6, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Function('\\\\hat{X}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))), Symbol('E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Integral(Mul(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(F_{N})} = F_{N}, then obtain 0^{\\operatorname{c_{0}}{(F_{N})}} = (\\int F_{N} d\\operatorname{c_{0}}{(F_{N})} - \\int \\operatorname{c_{0}}{(F_{N})} d\\operatorname{c_{0}}{(F_{N})})^{\\operatorname{c_{0}}{(F_{N})}}", "derivation": "\\operatorname{c_{0}}{(F_{N})} = F_{N} and \\int \\operatorname{c_{0}}{(F_{N})} dF_{N} = \\int F_{N} dF_{N} and \\int \\operatorname{c_{0}}{(F_{N})} d\\operatorname{c_{0}}{(F_{N})} = \\int F_{N} d\\operatorname{c_{0}}{(F_{N})} and 0 = \\int F_{N} d\\operatorname{c_{0}}{(F_{N})} - \\int \\operatorname{c_{0}}{(F_{N})} d\\operatorname{c_{0}}{(F_{N})} and 0^{F_{N}} = (\\int F_{N} d\\operatorname{c_{0}}{(F_{N})} - \\int \\operatorname{c_{0}}{(F_{N})} d\\operatorname{c_{0}}{(F_{N})})^{F_{N}} and 0^{\\operatorname{c_{0}}{(F_{N})}} = (\\int F_{N} d\\operatorname{c_{0}}{(F_{N})} - \\int \\operatorname{c_{0}}{(F_{N})} d\\operatorname{c_{0}}{(F_{N})})^{\\operatorname{c_{0}}{(F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))), Integral(Symbol('F_N', commutative=True), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Function('c_0')(Symbol('F_N', commutative=True))))"], "Equality(Integer(0), Add(Integral(Symbol('F_N', commutative=True), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))))))"], [["power", 4, "Symbol('F_N', commutative=True)"], "Equality(Pow(Integer(0), Symbol('F_N', commutative=True)), Pow(Add(Integral(Symbol('F_N', commutative=True), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))))), Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integer(0), Function('c_0')(Symbol('F_N', commutative=True))), Pow(Add(Integral(Symbol('F_N', commutative=True), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Integral(Function('c_0')(Symbol('F_N', commutative=True)), Tuple(Function('c_0')(Symbol('F_N', commutative=True)))))), Function('c_0')(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(C_{2})} = \\frac{d}{d C_{2}} \\log{(C_{2})}, then derive \\operatorname{A_{2}}{(C_{2})} = \\frac{1}{C_{2}}, then obtain \\int \\frac{1}{C_{2}} dC_{2} = \\int \\frac{d}{d C_{2}} \\log{(C_{2})} dC_{2}", "derivation": "\\operatorname{A_{2}}{(C_{2})} = \\frac{d}{d C_{2}} \\log{(C_{2})} and \\int \\operatorname{A_{2}}{(C_{2})} dC_{2} = \\int \\frac{d}{d C_{2}} \\log{(C_{2})} dC_{2} and \\operatorname{A_{2}}{(C_{2})} = \\frac{1}{C_{2}} and \\int \\frac{1}{C_{2}} dC_{2} = \\int \\frac{d}{d C_{2}} \\log{(C_{2})} dC_{2}", "srepr_derivation": [["get_premise", "Equality(Function('A_2')(Symbol('C_2', commutative=True)), Derivative(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('A_2')(Symbol('C_2', commutative=True)), Pow(Symbol('C_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Pow(Symbol('C_2', commutative=True), Integer(-1)), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given I{(S,\\sigma_x)} = S - \\sigma_x, then derive I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = I{(S,\\sigma_x)} + 1, then obtain I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} - 1", "derivation": "I{(S,\\sigma_x)} = S - \\sigma_x and \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = \\frac{\\partial}{\\partial S} (S - \\sigma_x) and I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) and I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = I{(S,\\sigma_x)} + 1 and I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) = I{(S,\\sigma_x)} + 1 and I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} - 1 = I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) and I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} = I{(S,\\sigma_x)} + \\frac{\\partial}{\\partial S} (S - \\sigma_x) + \\frac{\\partial}{\\partial S} I{(S,\\sigma_x)} - 1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 2, "Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('I')(Symbol('S', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(A_{1},\\Omega)} = \\cos{(A_{1} + \\Omega)}, then derive \\Omega \\int \\operatorname{C_{d}}{(A_{1},\\Omega)} d\\Omega = \\Omega (\\mathbf{F} + \\sin{(A_{1} + \\Omega)}), then obtain \\frac{\\partial}{\\partial \\Omega} \\Omega \\int \\cos{(A_{1} + \\Omega)} d\\Omega = \\frac{\\partial}{\\partial \\Omega} \\Omega (\\mathbf{F} + \\sin{(A_{1} + \\Omega)})", "derivation": "\\operatorname{C_{d}}{(A_{1},\\Omega)} = \\cos{(A_{1} + \\Omega)} and \\int \\operatorname{C_{d}}{(A_{1},\\Omega)} d\\Omega = \\int \\cos{(A_{1} + \\Omega)} d\\Omega and \\Omega \\int \\operatorname{C_{d}}{(A_{1},\\Omega)} d\\Omega = \\Omega \\int \\cos{(A_{1} + \\Omega)} d\\Omega and \\Omega \\int \\operatorname{C_{d}}{(A_{1},\\Omega)} d\\Omega = \\Omega (\\mathbf{F} + \\sin{(A_{1} + \\Omega)}) and \\frac{\\partial}{\\partial \\Omega} \\Omega \\int \\operatorname{C_{d}}{(A_{1},\\Omega)} d\\Omega = \\frac{\\partial}{\\partial \\Omega} \\Omega (\\mathbf{F} + \\sin{(A_{1} + \\Omega)}) and \\frac{\\partial}{\\partial \\Omega} \\Omega \\int \\cos{(A_{1} + \\Omega)} d\\Omega = \\frac{\\partial}{\\partial \\Omega} \\Omega (\\mathbf{F} + \\sin{(A_{1} + \\Omega)})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(cos(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["times", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Integral(Function('C_d')(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Integral(cos(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Integral(Function('C_d')(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Integral(Function('C_d')(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Integral(cos(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Add(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(C_{1})} = \\sin{(\\sin{(C_{1})})} and \\varepsilon_{0}{(C_{1})} = \\frac{\\mathbf{A}{(C_{1})}}{C_{1}}, then obtain \\varepsilon_{0}{(C_{1})} = \\frac{\\sin{(\\sin{(C_{1})})}}{C_{1}}", "derivation": "\\mathbf{A}{(C_{1})} = \\sin{(\\sin{(C_{1})})} and \\frac{\\mathbf{A}{(C_{1})}}{C_{1}} = \\frac{\\sin{(\\sin{(C_{1})})}}{C_{1}} and \\varepsilon_{0}{(C_{1})} = \\frac{\\mathbf{A}{(C_{1})}}{C_{1}} and \\varepsilon_{0}{(C_{1})} = \\frac{\\sin{(\\sin{(C_{1})})}}{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('C_1', commutative=True)), sin(sin(Symbol('C_1', commutative=True))))"], [["divide", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('C_1', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), sin(sin(Symbol('C_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('C_1', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varepsilon_0')(Symbol('C_1', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), sin(sin(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} = - \\Psi + \\mathbf{f}, then obtain \\Psi - \\mathbf{f} + 4 (- \\Psi + \\mathbf{f})^{2} - \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} = \\Psi - \\mathbf{f} + 2 (- 2 \\Psi + 2 \\mathbf{f}) (- \\Psi + \\mathbf{f}) - \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})}", "derivation": "\\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} = - \\Psi + \\mathbf{f} and 2 \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} = - \\Psi + \\mathbf{f} + \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} and 4 \\hat{\\mathbf{x}}^{2}{(\\Psi,\\mathbf{f})} = 2 (- \\Psi + \\mathbf{f} + \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})}) \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} and 4 (- \\Psi + \\mathbf{f})^{2} = 2 (- 2 \\Psi + 2 \\mathbf{f}) (- \\Psi + \\mathbf{f}) and \\Psi - \\mathbf{f} + 4 (- \\Psi + \\mathbf{f})^{2} - \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})} = \\Psi - \\mathbf{f} + 2 (- 2 \\Psi + 2 \\mathbf{f}) (- \\Psi + \\mathbf{f}) - \\hat{\\mathbf{x}}{(\\Psi,\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given C{(H,Q)} = \\frac{\\cos{(H)}}{Q}, then obtain \\frac{d}{d Q} \\frac{1}{Q} = \\frac{\\partial}{\\partial Q} (\\int \\frac{\\cos{(H)}}{Q} dH - \\int C{(H,Q)} dH + \\frac{1}{Q})", "derivation": "C{(H,Q)} = \\frac{\\cos{(H)}}{Q} and \\int C{(H,Q)} dH = \\int \\frac{\\cos{(H)}}{Q} dH and 0 = \\int \\frac{\\cos{(H)}}{Q} dH - \\int C{(H,Q)} dH and \\frac{1}{Q} = \\int \\frac{\\cos{(H)}}{Q} dH - \\int C{(H,Q)} dH + \\frac{1}{Q} and \\frac{d}{d Q} \\frac{1}{Q} = \\frac{\\partial}{\\partial Q} (\\int \\frac{\\cos{(H)}}{Q} dH - \\int C{(H,Q)} dH + \\frac{1}{Q})", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), cos(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["minus", 2, "Integral(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["add", 3, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('H', commutative=True)))), Pow(Symbol('Q', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Pow(Symbol('Q', commutative=True), Integer(-1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('C')(Symbol('H', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('H', commutative=True)))), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(Q)} = e^{Q} and \\operatorname{f_{\\mathbf{v}}}{(Q)} = - \\mathbf{P}{(Q)} e^{- Q}, then obtain - l{(P_{e},\\varphi)} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ = \\operatorname{f_{\\mathbf{v}}}{(Q)} - l{(P_{e},\\varphi)} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ + 1", "derivation": "\\mathbf{P}{(Q)} = e^{Q} and \\mathbf{P}{(Q)} e^{- Q} = 1 and 0 = - \\mathbf{P}{(Q)} e^{- Q} + 1 and \\int \\mathbf{P}{(Q)} e^{- Q} dQ = \\int 1 dQ and - \\int 1 dQ = - \\mathbf{P}{(Q)} e^{- Q} - \\int 1 dQ + 1 and \\operatorname{f_{\\mathbf{v}}}{(Q)} = - \\mathbf{P}{(Q)} e^{- Q} and - \\int \\mathbf{P}{(Q)} e^{- Q} dQ = - \\mathbf{P}{(Q)} e^{- Q} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ + 1 and - \\int \\mathbf{P}{(Q)} e^{- Q} dQ = \\operatorname{f_{\\mathbf{v}}}{(Q)} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ + 1 and - l{(P_{e},\\varphi)} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ = \\operatorname{f_{\\mathbf{v}}}{(Q)} - l{(P_{e},\\varphi)} - \\int \\mathbf{P}{(Q)} e^{- Q} dQ + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["divide", 1, "exp(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Integer(1))"], [["minus", 2, "Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Integer(1)))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True))))"], [["minus", 3, "Integral(Integer(1), Tuple(Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('Q', commutative=True)))), Integer(1)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True)), Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Integer(1)))"], [["minus", 8, "Function('l')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True)), Mul(Integer(-1), Function('l')(Symbol('P_e', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Integral(Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given G{(s)} = \\log{(s)}, then obtain (1 + \\frac{1}{s}) e^{G{(s)}} + (s + \\log{(s)}) e^{G{(s)}} \\frac{d}{d s} G{(s)} = s (1 + \\frac{1}{s}) + s + \\log{(s)}", "derivation": "G{(s)} = \\log{(s)} and e^{G{(s)}} = s and (s + \\log{(s)}) e^{G{(s)}} = s (s + \\log{(s)}) and \\frac{d}{d s} (s + \\log{(s)}) e^{G{(s)}} = \\frac{d}{d s} s (s + \\log{(s)}) and (1 + \\frac{1}{s}) e^{G{(s)}} + (s + \\log{(s)}) e^{G{(s)}} \\frac{d}{d s} G{(s)} = s (1 + \\frac{1}{s}) + s + \\log{(s)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["exp", 1], "Equality(exp(Function('G')(Symbol('s', commutative=True))), Symbol('s', commutative=True))"], [["times", 2, "Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True)))"], "Equality(Mul(Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), exp(Function('G')(Symbol('s', commutative=True)))), Mul(Symbol('s', commutative=True), Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True)))))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), exp(Function('G')(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('s', commutative=True), Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Integer(1), Pow(Symbol('s', commutative=True), Integer(-1))), exp(Function('G')(Symbol('s', commutative=True)))), Mul(Add(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), exp(Function('G')(Symbol('s', commutative=True))), Derivative(Function('G')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Mul(Symbol('s', commutative=True), Add(Integer(1), Pow(Symbol('s', commutative=True), Integer(-1)))), Symbol('s', commutative=True), log(Symbol('s', commutative=True))))"]]}, {"prompt": "Given u{(x,b)} = x^{b}, then obtain \\frac{u{(x,b)} + \\int u{(x,b)} dx}{x} = \\frac{x^{b} + \\int u{(x,b)} dx}{x}", "derivation": "u{(x,b)} = x^{b} and \\int u{(x,b)} dx = \\int x^{b} dx and u{(x,b)} + \\int x^{b} dx = x^{b} + \\int x^{b} dx and \\frac{u{(x,b)} + \\int x^{b} dx}{x} = \\frac{x^{b} + \\int x^{b} dx}{x} and \\frac{u{(x,b)} + \\int u{(x,b)} dx}{x} = \\frac{x^{b} + \\int u{(x,b)} dx}{x}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["add", 1, "Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Add(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["divide", 3, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Pow(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integral(Function('u')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given x{(a,f^{\\prime})} = - a + f^{\\prime}, then obtain - f^{\\prime} = a - 2 f^{\\prime} + x{(a,f^{\\prime})}", "derivation": "x{(a,f^{\\prime})} = - a + f^{\\prime} and 0 = - a + f^{\\prime} - x{(a,f^{\\prime})} and 0 = a - f^{\\prime} + x{(a,f^{\\prime})} and - f^{\\prime} = a - 2 f^{\\prime} + x{(a,f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('x')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('x')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('x')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Function('x')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\phi{(r,Q)} = \\cos{(Q^{r})}, then derive (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{r} = (- Q^{r} \\log{(Q)} \\sin{(Q^{r})})^{r}, then obtain - (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{Q} + (\\frac{\\partial}{\\partial r} \\cos{(Q^{r})})^{r} = (- Q^{r} \\log{(Q)} \\sin{(Q^{r})})^{r} - (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{Q}", "derivation": "\\phi{(r,Q)} = \\cos{(Q^{r})} and \\frac{\\partial}{\\partial r} \\phi{(r,Q)} = \\frac{\\partial}{\\partial r} \\cos{(Q^{r})} and (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{r} = (\\frac{\\partial}{\\partial r} \\cos{(Q^{r})})^{r} and (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{r} = (- Q^{r} \\log{(Q)} \\sin{(Q^{r})})^{r} and (\\frac{\\partial}{\\partial r} \\cos{(Q^{r})})^{r} = (- Q^{r} \\log{(Q)} \\sin{(Q^{r})})^{r} and - (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{Q} + (\\frac{\\partial}{\\partial r} \\cos{(Q^{r})})^{r} = (- Q^{r} \\log{(Q)} \\sin{(Q^{r})})^{r} - (\\frac{\\partial}{\\partial r} \\phi{(r,Q)})^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), cos(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(cos(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)), log(Symbol('Q', commutative=True)), sin(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)), log(Symbol('Q', commutative=True)), sin(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["minus", 5, "Pow(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('Q', commutative=True))), Pow(Derivative(cos(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True))), Add(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)), log(Symbol('Q', commutative=True)), sin(Pow(Symbol('Q', commutative=True), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Derivative(Function('\\\\phi')(Symbol('r', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(h)} = \\sin{(\\sin{(h)})} and M{(h)} = \\frac{\\operatorname{A_{z}}{(h)}}{\\sin{(\\sin{(h)})}}, then obtain ((\\frac{M{(h)}}{\\sin{(h)}})^{h})^{h} = ((\\frac{1}{\\sin{(h)}})^{h})^{h}", "derivation": "\\operatorname{A_{z}}{(h)} = \\sin{(\\sin{(h)})} and M{(h)} = \\frac{\\operatorname{A_{z}}{(h)}}{\\sin{(\\sin{(h)})}} and \\frac{M{(h)}}{\\sin{(h)}} = \\frac{\\operatorname{A_{z}}{(h)}}{\\sin{(h)} \\sin{(\\sin{(h)})}} and (\\frac{M{(h)}}{\\sin{(h)}})^{h} = (\\frac{\\operatorname{A_{z}}{(h)}}{\\sin{(h)} \\sin{(\\sin{(h)})}})^{h} and ((\\frac{M{(h)}}{\\sin{(h)}})^{h})^{h} = ((\\frac{\\operatorname{A_{z}}{(h)}}{\\sin{(h)} \\sin{(\\sin{(h)})}})^{h})^{h} and ((\\frac{M{(h)}}{\\sin{(h)}})^{h})^{h} = ((\\frac{1}{\\sin{(h)}})^{h})^{h}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('h', commutative=True)), sin(sin(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('h', commutative=True)), Mul(Function('A_z')(Symbol('h', commutative=True)), Pow(sin(sin(Symbol('h', commutative=True))), Integer(-1))))"], [["divide", 2, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Function('M')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Mul(Function('A_z')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(sin(Symbol('h', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Function('M')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Symbol('h', commutative=True)), Pow(Mul(Function('A_z')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(sin(Symbol('h', commutative=True))), Integer(-1))), Symbol('h', commutative=True)))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Mul(Function('M')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Mul(Function('A_z')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(sin(Symbol('h', commutative=True))), Integer(-1))), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Pow(Mul(Function('M')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(c_{0})} = e^{c_{0}}, then obtain (T \\mathbf{g}{(c_{0})} + t_{1})^{t_{1}} = (T e^{c_{0}} + t_{1})^{t_{1}}", "derivation": "\\mathbf{g}{(c_{0})} = e^{c_{0}} and T \\mathbf{g}{(c_{0})} = T e^{c_{0}} and T \\mathbf{g}{(c_{0})} + t_{1} = T e^{c_{0}} + t_{1} and (T \\mathbf{g}{(c_{0})} + t_{1})^{t_{1}} = (T e^{c_{0}} + t_{1})^{t_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], [["times", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Function('\\\\mathbf{g}')(Symbol('c_0', commutative=True))), Mul(Symbol('T', commutative=True), exp(Symbol('c_0', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Symbol('T', commutative=True), Function('\\\\mathbf{g}')(Symbol('c_0', commutative=True))), Symbol('t_1', commutative=True)), Add(Mul(Symbol('T', commutative=True), exp(Symbol('c_0', commutative=True))), Symbol('t_1', commutative=True)))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('T', commutative=True), Function('\\\\mathbf{g}')(Symbol('c_0', commutative=True))), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Add(Mul(Symbol('T', commutative=True), exp(Symbol('c_0', commutative=True))), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given Z{(\\nabla,\\Omega,M)} = (\\Omega^{M})^{\\nabla}, then obtain ((\\Omega + (\\Omega^{M})^{\\nabla}) (\\Omega + Z{(\\nabla,\\Omega,M)})^{2})^{M} = ((\\Omega + (\\Omega^{M})^{\\nabla})^{2} (\\Omega + Z{(\\nabla,\\Omega,M)}))^{M}", "derivation": "Z{(\\nabla,\\Omega,M)} = (\\Omega^{M})^{\\nabla} and \\Omega + Z{(\\nabla,\\Omega,M)} = \\Omega + (\\Omega^{M})^{\\nabla} and (\\Omega + (\\Omega^{M})^{\\nabla}) (\\Omega + Z{(\\nabla,\\Omega,M)}) = (\\Omega + (\\Omega^{M})^{\\nabla})^{2} and (\\Omega + (\\Omega^{M})^{\\nabla}) (\\Omega + Z{(\\nabla,\\Omega,M)})^{2} = (\\Omega + (\\Omega^{M})^{\\nabla})^{2} (\\Omega + Z{(\\nabla,\\Omega,M)}) and ((\\Omega + (\\Omega^{M})^{\\nabla}) (\\Omega + Z{(\\nabla,\\Omega,M)})^{2})^{M} = ((\\Omega + (\\Omega^{M})^{\\nabla})^{2} (\\Omega + Z{(\\nabla,\\Omega,M)}))^{M}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)))), Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Integer(2)))"], [["times", 3, "Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Pow(Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True))), Integer(2))), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Integer(2)), Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)))))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Pow(Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True))), Integer(2))), Symbol('M', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\nabla', commutative=True))), Integer(2)), Add(Symbol('\\\\Omega', commutative=True), Function('Z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('M', commutative=True)))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given t{(C,m_{s})} = - m_{s} + \\sin{(C)}, then derive \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})} = -1, then obtain (m_{s} + \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})})^{C} = (m_{s} - 1)^{C}", "derivation": "t{(C,m_{s})} = - m_{s} + \\sin{(C)} and \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (- m_{s} + \\sin{(C)}) and m_{s} + \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})} = m_{s} + \\frac{\\partial}{\\partial m_{s}} (- m_{s} + \\sin{(C)}) and (m_{s} + \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})})^{C} = (m_{s} + \\frac{\\partial}{\\partial m_{s}} (- m_{s} + \\sin{(C)}))^{C} and \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})} = -1 and m_{s} - 1 = m_{s} + \\frac{\\partial}{\\partial m_{s}} (- m_{s} + \\sin{(C)}) and (m_{s} + \\frac{\\partial}{\\partial m_{s}} t{(C,m_{s})})^{C} = (m_{s} - 1)^{C}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Add(Symbol('m_s', commutative=True), Derivative(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Symbol('m_s', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Add(Symbol('m_s', commutative=True), Derivative(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('C', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('C', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('m_s', commutative=True), Integer(-1)), Add(Symbol('m_s', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Pow(Add(Symbol('m_s', commutative=True), Derivative(Function('t')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('C', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Integer(-1)), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\delta,\\theta_1,V)} = \\delta + \\theta_1^{V}, then derive \\int (- \\theta_1^{V} + \\operatorname{C_{1}}{(\\delta,\\theta_1,V)}) d\\delta = \\frac{\\delta^{2}}{2} + f_{E}, then obtain \\int \\delta d\\delta = \\frac{\\delta^{2}}{2} + f_{E}", "derivation": "\\operatorname{C_{1}}{(\\delta,\\theta_1,V)} = \\delta + \\theta_1^{V} and - \\theta_1^{V} + \\operatorname{C_{1}}{(\\delta,\\theta_1,V)} = \\delta and \\int (- \\theta_1^{V} + \\operatorname{C_{1}}{(\\delta,\\theta_1,V)}) d\\delta = \\int \\delta d\\delta and \\int (- \\theta_1^{V} + \\operatorname{C_{1}}{(\\delta,\\theta_1,V)}) d\\delta = \\frac{\\delta^{2}}{2} + f_{E} and \\int \\delta d\\delta = \\frac{\\delta^{2}}{2} + f_{E}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Function('C_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Symbol('\\\\delta', commutative=True))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Function('C_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Function('C_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\varphi^*,A_{x})} = \\frac{A_{x}}{\\varphi^*}, then obtain \\frac{((\\int \\frac{A_{x}}{\\varphi^*} dA_{x})^{A_{x}}) (\\int \\Omega{(\\varphi^*,A_{x})} dA_{x})^{A_{x}}}{\\int \\frac{A_{x}}{\\varphi^*} dA_{x}} = \\frac{(\\int \\frac{A_{x}}{\\varphi^*} dA_{x})^{2 A_{x}}}{\\int \\frac{A_{x}}{\\varphi^*} dA_{x}}", "derivation": "\\Omega{(\\varphi^*,A_{x})} = \\frac{A_{x}}{\\varphi^*} and \\int \\Omega{(\\varphi^*,A_{x})} dA_{x} = \\int \\frac{A_{x}}{\\varphi^*} dA_{x} and (\\int \\Omega{(\\varphi^*,A_{x})} dA_{x})^{A_{x}} = (\\int \\frac{A_{x}}{\\varphi^*} dA_{x})^{A_{x}} and \\frac{((\\int \\frac{A_{x}}{\\varphi^*} dA_{x})^{A_{x}}) (\\int \\Omega{(\\varphi^*,A_{x})} dA_{x})^{A_{x}}}{\\int \\frac{A_{x}}{\\varphi^*} dA_{x}} = \\frac{(\\int \\frac{A_{x}}{\\varphi^*} dA_{x})^{2 A_{x}}}{\\int \\frac{A_{x}}{\\varphi^*} dA_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["divide", 3, "Mul(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True))))"], "Equality(Mul(Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Integer(-1)), Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integral(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Mul(Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Integer(-1)), Pow(Integral(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Mul(Integer(2), Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\delta{(k)} = \\cos{(k)}, then derive \\mu_0 - \\frac{k^{2}}{2} + \\tilde{\\infty} \\delta{(k)} \\log{(\\delta{(k)})} + \\tilde{\\infty} \\delta{(k)} = \\int (- k + \\log{(\\cos{(k)})}) dk, then obtain \\int (\\mu_0 - \\frac{k^{2}}{2} + \\tilde{\\infty} \\delta{(k)} \\log{(\\cos{(k)})} + \\tilde{\\infty} \\delta{(k)}) d\\mu_0 = \\iint (- k + \\log{(\\cos{(k)})}) dk d\\mu_0", "derivation": "\\delta{(k)} = \\cos{(k)} and \\log{(\\delta{(k)})} = \\log{(\\cos{(k)})} and - k + \\log{(\\delta{(k)})} = - k + \\log{(\\cos{(k)})} and \\int (- k + \\log{(\\delta{(k)})}) dk = \\int (- k + \\log{(\\cos{(k)})}) dk and \\mu_0 - \\frac{k^{2}}{2} + \\tilde{\\infty} \\delta{(k)} \\log{(\\delta{(k)})} + \\tilde{\\infty} \\delta{(k)} = \\int (- k + \\log{(\\cos{(k)})}) dk and \\mu_0 - \\frac{k^{2}}{2} + \\tilde{\\infty} \\delta{(k)} \\log{(\\cos{(k)})} + \\tilde{\\infty} \\delta{(k)} = \\int (- k + \\log{(\\cos{(k)})}) dk and \\int (\\mu_0 - \\frac{k^{2}}{2} + \\tilde{\\infty} \\delta{(k)} \\log{(\\cos{(k)})} + \\tilde{\\infty} \\delta{(k)}) d\\mu_0 = \\iint (- k + \\log{(\\cos{(k)})}) dk d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\delta')(Symbol('k', commutative=True))), log(cos(Symbol('k', commutative=True))))"], [["minus", 2, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Function('\\\\delta')(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Function('\\\\delta')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)), log(Function('\\\\delta')(Symbol('k', commutative=True)))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["integrate", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Mul(zoo, Function('\\\\delta')(Symbol('k', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(cos(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\hat{p},A_{x})} = - A_{x} + e^{\\hat{p}}, then obtain - A_{x} - \\operatorname{v_{1}}{(\\hat{p},A_{x})} = - e^{\\hat{p}}", "derivation": "\\operatorname{v_{1}}{(\\hat{p},A_{x})} = - A_{x} + e^{\\hat{p}} and - \\operatorname{v_{1}}{(\\hat{p},A_{x})} = A_{x} - e^{\\hat{p}} and - A_{x} - \\operatorname{v_{1}}{(\\hat{p},A_{x})} + e^{\\hat{p}} = 0 and - A_{x} - \\operatorname{v_{1}}{(\\hat{p},A_{x})} = - e^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_x', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True)))))"], [["minus", 2, "Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_x', commutative=True))), exp(Symbol('\\\\hat{p}', commutative=True))), Integer(0))"], [["minus", 3, "exp(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_x', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given H{(r,P_{g})} = - P_{g} + r, then obtain \\frac{d}{d r} 1 = \\frac{\\partial}{\\partial r} \\frac{\\int (- P_{g} + r) dP_{g}}{\\int H{(r,P_{g})} dP_{g}}", "derivation": "H{(r,P_{g})} = - P_{g} + r and \\int H{(r,P_{g})} dP_{g} = \\int (- P_{g} + r) dP_{g} and 1 = \\frac{\\int (- P_{g} + r) dP_{g}}{\\int H{(r,P_{g})} dP_{g}} and \\frac{d}{d r} 1 = \\frac{\\partial}{\\partial r} \\frac{\\int (- P_{g} + r) dP_{g}}{\\int H{(r,P_{g})} dP_{g}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('r', commutative=True), Symbol('P_g', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('H')(Symbol('r', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["divide", 2, "Integral(Function('H')(Symbol('r', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Pow(Integral(Function('H')(Symbol('r', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integer(-1))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Pow(Integral(Function('H')(Symbol('r', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} = \\frac{E - M}{\\theta}, then obtain \\frac{\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1}{\\theta + 1} = \\frac{(E - M) (\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1)}{\\theta (\\theta + 1) \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} = \\frac{E - M}{\\theta} and 1 = \\frac{E - M}{\\theta \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)}} and \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1 = \\frac{(E - M) (\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1)}{\\theta \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)}} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1}{\\theta + 1} = \\frac{(E - M) (\\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)} - 1)}{\\theta (\\theta + 1) \\operatorname{f_{\\mathbf{p}}}{(E,\\theta,M)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))))"], [["divide", 1, "Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1))))"], [["times", 2, "Add(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Add(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1))))"], [["divide", 3, "Add(Symbol('\\\\theta', commutative=True), Integer(1))"], "Equality(Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Integer(1)), Integer(-1)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Add(Symbol('\\\\theta', commutative=True), Integer(1)), Integer(-1)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('M', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given h{(a)} = \\cos{(\\cos{(a)})}, then obtain \\iint e^{h{(a)}} da da = \\iint e^{\\cos{(\\cos{(a)})}} da da", "derivation": "h{(a)} = \\cos{(\\cos{(a)})} and e^{h{(a)}} = e^{\\cos{(\\cos{(a)})}} and \\int e^{h{(a)}} da = \\int e^{\\cos{(\\cos{(a)})}} da and \\iint e^{h{(a)}} da da = \\iint e^{\\cos{(\\cos{(a)})}} da da", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('a', commutative=True)), cos(cos(Symbol('a', commutative=True))))"], [["exp", 1], "Equality(exp(Function('h')(Symbol('a', commutative=True))), exp(cos(cos(Symbol('a', commutative=True)))))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(exp(Function('h')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(exp(cos(cos(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(exp(Function('h')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(exp(cos(cos(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given m{(\\theta_1,V)} = V + \\theta_1 and \\Psi^{\\dagger}{(\\theta_1,V)} = (V + \\theta_1) m{(\\theta_1,V)}, then obtain \\int (V + \\theta_1)^{2} dV = \\int (V + \\theta_1) m{(\\theta_1,V)} dV", "derivation": "m{(\\theta_1,V)} = V + \\theta_1 and (V + \\theta_1) m{(\\theta_1,V)} = (V + \\theta_1)^{2} and \\Psi^{\\dagger}{(\\theta_1,V)} = (V + \\theta_1) m{(\\theta_1,V)} and \\Psi^{\\dagger}{(\\theta_1,V)} = (V + \\theta_1)^{2} and \\int \\Psi^{\\dagger}{(\\theta_1,V)} dV = \\int (V + \\theta_1) m{(\\theta_1,V)} dV and \\int (V + \\theta_1)^{2} dV = \\int (V + \\theta_1) m{(\\theta_1,V)} dV", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Pow(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True)), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True)), Pow(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Pow(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = J_{\\varepsilon} + \\hat{H}, then derive \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = 1, then obtain \\frac{\\partial}{\\partial \\hat{H}} (J_{\\varepsilon} + \\hat{H}) + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} + 1", "derivation": "\\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = J_{\\varepsilon} + \\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\hat{H}} (J_{\\varepsilon} + \\hat{H}) and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = 1 and \\frac{\\partial}{\\partial \\hat{H}} (J_{\\varepsilon} + \\hat{H}) = 1 and \\frac{\\partial}{\\partial \\hat{H}} (J_{\\varepsilon} + \\hat{H}) + \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{E_{\\lambda}}{(\\hat{H},J_{\\varepsilon})} + 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{J},\\Psi)} = \\mathbf{J} \\sin{(\\Psi)}, then obtain \\frac{1}{\\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}}} = \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)} \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}}}", "derivation": "\\theta_{2}{(\\mathbf{J},\\Psi)} = \\mathbf{J} \\sin{(\\Psi)} and 1 = \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}} and \\frac{d}{d \\mathbf{J}} 1 = \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}} and \\frac{1}{\\frac{d}{d \\mathbf{J}} 1} = \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)} \\frac{d}{d \\mathbf{J}} 1} and \\frac{1}{\\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}}} = \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)} \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J} \\sin{(\\Psi)}}{\\theta_{2}{(\\mathbf{J},\\Psi)}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\chi,\\varphi)} = \\sin{(\\chi + \\varphi)}, then derive \\int \\operatorname{t_{1}}{(\\chi,\\varphi)} d\\varphi = z^{*} - \\cos{(\\chi + \\varphi)}, then derive \\mathbf{J}_M - \\cos{(\\chi + \\varphi)} = z^{*} - \\cos{(\\chi + \\varphi)}, then obtain \\frac{\\mathbf{J}_M - \\cos{(\\chi + \\varphi)}}{\\operatorname{t_{1}}{(\\chi,\\varphi)}} = \\frac{z^{*} - \\cos{(\\chi + \\varphi)}}{\\operatorname{t_{1}}{(\\chi,\\varphi)}}", "derivation": "\\operatorname{t_{1}}{(\\chi,\\varphi)} = \\sin{(\\chi + \\varphi)} and \\int \\operatorname{t_{1}}{(\\chi,\\varphi)} d\\varphi = \\int \\sin{(\\chi + \\varphi)} d\\varphi and \\int \\operatorname{t_{1}}{(\\chi,\\varphi)} d\\varphi = z^{*} - \\cos{(\\chi + \\varphi)} and \\int \\sin{(\\chi + \\varphi)} d\\varphi = z^{*} - \\cos{(\\chi + \\varphi)} and \\mathbf{J}_M - \\cos{(\\chi + \\varphi)} = z^{*} - \\cos{(\\chi + \\varphi)} and \\frac{\\mathbf{J}_M - \\cos{(\\chi + \\varphi)}}{\\operatorname{t_{1}}{(\\chi,\\varphi)}} = \\frac{z^{*} - \\cos{(\\chi + \\varphi)}}{\\operatorname{t_{1}}{(\\chi,\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), sin(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))))"], [["divide", 5, "Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))), Pow(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))), Mul(Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))), Pow(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho{(B)} = \\int \\cos{(B)} dB, then derive \\rho{(B)} = v + \\sin{(B)}, then obtain - \\cos{(B)} + \\int \\rho{(B)} dB + \\iint \\cos{(B)} dB dB = - \\cos{(B)} + \\int (v + \\sin{(B)}) dB + \\iint \\cos{(B)} dB dB", "derivation": "\\rho{(B)} = \\int \\cos{(B)} dB and \\int \\rho{(B)} dB = \\iint \\cos{(B)} dB dB and \\int \\rho{(B)} dB + \\iint \\cos{(B)} dB dB = 2 \\iint \\cos{(B)} dB dB and - \\cos{(B)} + \\int \\rho{(B)} dB + \\iint \\cos{(B)} dB dB = - \\cos{(B)} + 2 \\iint \\cos{(B)} dB dB and \\rho{(B)} = v + \\sin{(B)} and \\int (v + \\sin{(B)}) dB + \\iint \\cos{(B)} dB dB = 2 \\iint \\cos{(B)} dB dB and - \\cos{(B)} + \\int \\rho{(B)} dB + \\iint \\cos{(B)} dB dB = - \\cos{(B)} + \\int (v + \\sin{(B)}) dB + \\iint \\cos{(B)} dB dB", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('B', commutative=True)), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))"], "Equality(Add(Integral(Function('\\\\rho')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["minus", 3, "cos(Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('B', commutative=True))), Integral(Function('\\\\rho')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('B', commutative=True))), Mul(Integer(2), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\rho')(Symbol('B', commutative=True)), Add(Symbol('v', commutative=True), sin(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Integral(Add(Symbol('v', commutative=True), sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Mul(Integer(-1), cos(Symbol('B', commutative=True))), Integral(Function('\\\\rho')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('B', commutative=True))), Integral(Add(Symbol('v', commutative=True), sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(t_{1},v)} = \\frac{\\log{(v)}}{t_{1}}, then derive \\frac{\\partial}{\\partial t_{1}} \\int \\lambda{(t_{1},v)} dv = \\frac{\\partial}{\\partial t_{1}} (\\mathbf{F} + \\frac{v \\log{(v)}}{t_{1}} - \\frac{v}{t_{1}}), then obtain \\frac{\\partial}{\\partial t_{1}} (\\mathbf{F} + \\frac{v \\log{(v)}}{t_{1}} - \\frac{v}{t_{1}}) = \\frac{\\partial}{\\partial t_{1}} \\int \\frac{\\log{(v)}}{t_{1}} dv", "derivation": "\\lambda{(t_{1},v)} = \\frac{\\log{(v)}}{t_{1}} and \\int \\lambda{(t_{1},v)} dv = \\int \\frac{\\log{(v)}}{t_{1}} dv and \\frac{\\partial}{\\partial t_{1}} \\int \\lambda{(t_{1},v)} dv = \\frac{\\partial}{\\partial t_{1}} \\int \\frac{\\log{(v)}}{t_{1}} dv and \\frac{\\partial}{\\partial t_{1}} \\int \\lambda{(t_{1},v)} dv = \\frac{\\partial}{\\partial t_{1}} (\\mathbf{F} + \\frac{v \\log{(v)}}{t_{1}} - \\frac{v}{t_{1}}) and \\frac{\\partial}{\\partial t_{1}} (\\mathbf{F} + \\frac{v \\log{(v)}}{t_{1}} - \\frac{v}{t_{1}}) = \\frac{\\partial}{\\partial t_{1}} \\int \\frac{\\log{(v)}}{t_{1}} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), log(Symbol('v', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\lambda')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\lambda')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('v', commutative=True), log(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('v', commutative=True), log(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(T)} = e^{e^{T}}, then obtain - \\frac{1}{J{(T)}} = - \\frac{- T + (- J{(T)} + e^{e^{T}})^{T}}{(0^{T} - T) J{(T)}}", "derivation": "J{(T)} = e^{e^{T}} and 0 = - J{(T)} + e^{e^{T}} and 0^{T} = (- J{(T)} + e^{e^{T}})^{T} and 0^{T} - T = - T + (- J{(T)} + e^{e^{T}})^{T} and - \\frac{0^{T} - T}{J{(T)}} = - \\frac{- T + (- J{(T)} + e^{e^{T}})^{T}}{J{(T)}} and - \\frac{1}{J{(T)}} = - \\frac{- T + (- J{(T)} + e^{e^{T}})^{T}}{(0^{T} - T) J{(T)}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('T', commutative=True)), exp(exp(Symbol('T', commutative=True))))"], [["minus", 1, "Function('J')(Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), exp(exp(Symbol('T', commutative=True)))))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Integer(0), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), exp(exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["minus", 3, "Symbol('T', commutative=True)"], "Equality(Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), exp(exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Function('J')(Symbol('T', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Pow(Function('J')(Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), exp(exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Pow(Function('J')(Symbol('T', commutative=True)), Integer(-1))))"], [["divide", 5, "Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('J')(Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Function('J')(Symbol('T', commutative=True))), exp(exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Pow(Function('J')(Symbol('T', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*,c,W)} = W \\tilde{g}^* + c and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g}^*,c,W)} = W \\tilde{g}^* + c, then obtain \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g}^*,c,W)}}{(W \\tilde{g}^* + c)^{2}} = \\frac{1}{W \\tilde{g}^* + c}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*,c,W)} = W \\tilde{g}^* + c and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*,c,W)}}{W \\tilde{g}^* + c} = 1 and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*,c,W)}}{(W \\tilde{g}^* + c)^{2}} = \\frac{1}{W \\tilde{g}^* + c} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g}^*,c,W)} = W \\tilde{g}^* + c and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g}^*,c,W)} = \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*,c,W)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\tilde{g}^*,c,W)}}{(W \\tilde{g}^* + c)^{2}} = \\frac{1}{W \\tilde{g}^* + c}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True)), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)))"], [["divide", 1, "Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True))), Integer(1))"], [["times", 2, "Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-2)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True))), Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True)), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-2)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('c', commutative=True), Symbol('W', commutative=True))), Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('c', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given G{(v_{x})} = \\log{(v_{x})}, then derive \\frac{d}{d v_{x}} G{(v_{x})} = \\frac{1}{v_{x}}, then obtain 0 = - \\frac{d}{d v_{x}} G{(v_{x})} + \\frac{1}{v_{x}}", "derivation": "G{(v_{x})} = \\log{(v_{x})} and \\frac{d}{d v_{x}} G{(v_{x})} = \\frac{d}{d v_{x}} \\log{(v_{x})} and \\frac{d}{d v_{x}} G{(v_{x})} = \\frac{1}{v_{x}} and \\frac{d}{d v_{x}} G{(v_{x})} - \\frac{d}{d v_{x}} \\log{(v_{x})} = - \\frac{d}{d v_{x}} \\log{(v_{x})} + \\frac{1}{v_{x}} and 0 = - \\frac{d}{d v_{x}} G{(v_{x})} + \\frac{1}{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Pow(Symbol('v_x', commutative=True), Integer(-1)))"], [["minus", 3, "Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('G')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('G')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Pow(Symbol('v_x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(n_{1},f^{*},\\mathbf{S})} = \\frac{f^{*} + n_{1}}{\\mathbf{S}}, then obtain \\frac{(f^{*} + n_{1}) (\\frac{- f^{*} - n_{1}}{\\mathbf{S}} + \\frac{f^{*} + n_{1}}{\\mathbf{S}})}{\\mathbf{S}} = 0", "derivation": "\\dot{z}{(n_{1},f^{*},\\mathbf{S})} = \\frac{f^{*} + n_{1}}{\\mathbf{S}} and \\dot{z}{(n_{1},f^{*},\\mathbf{S})} - \\frac{f^{*} + n_{1}}{\\mathbf{S}} = 0 and \\frac{(f^{*} + n_{1}) (\\dot{z}{(n_{1},f^{*},\\mathbf{S})} - \\frac{f^{*} + n_{1}}{\\mathbf{S}})}{\\mathbf{S}} = 0 and (\\dot{z}{(n_{1},f^{*},\\mathbf{S})} + \\frac{- f^{*} - n_{1}}{\\mathbf{S}}) \\dot{z}{(n_{1},f^{*},\\mathbf{S})} = 0 and \\frac{(f^{*} + n_{1}) (\\frac{- f^{*} - n_{1}}{\\mathbf{S}} + \\frac{f^{*} + n_{1}}{\\mathbf{S}})}{\\mathbf{S}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('f^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('f^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True)))), Integer(0))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True)), Add(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('f^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('f^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True))))), Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('f^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True)), Add(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), Symbol('n_1', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\hat{x}_0{(i,M)} = \\cos{(M + i)}, then obtain \\hat{x}_0{(i,M)} \\sin{(\\cos{(M + i)})} = \\sin{(\\cos{(M + i)})} \\cos{(M + i)}", "derivation": "\\hat{x}_0{(i,M)} = \\cos{(M + i)} and \\sin{(\\hat{x}_0{(i,M)})} = \\sin{(\\cos{(M + i)})} and \\hat{x}_0{(i,M)} \\sin{(\\hat{x}_0{(i,M)})} = \\sin{(\\hat{x}_0{(i,M)})} \\cos{(M + i)} and \\hat{x}_0{(i,M)} \\sin{(\\cos{(M + i)})} = \\sin{(\\cos{(M + i)})} \\cos{(M + i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True)), cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True))), sin(cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True)))))"], [["times", 1, "sin(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True)), sin(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True)))), Mul(sin(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True))), cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('M', commutative=True)), sin(cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True))))), Mul(sin(cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True)))), cos(Add(Symbol('M', commutative=True), Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})} = \\mathbb{I} \\mathbf{S} and \\operatorname{t_{2}}{(\\mathbb{I},\\mathbf{S})} = \\mathbb{I} \\mathbf{S}, then obtain \\int 0 d\\mathbf{S} = \\int (\\mathbb{I} \\mathbf{S} - \\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})}) d\\mathbf{S}", "derivation": "\\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})} = \\mathbb{I} \\mathbf{S} and 0 = \\mathbb{I} \\mathbf{S} - \\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})} and \\operatorname{t_{2}}{(\\mathbb{I},\\mathbf{S})} = \\mathbb{I} \\mathbf{S} and \\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})} = \\operatorname{t_{2}}{(\\mathbb{I},\\mathbf{S})} and 0 = \\mathbb{I} \\mathbf{S} - \\operatorname{t_{2}}{(\\mathbb{I},\\mathbf{S})} and \\int 0 d\\mathbf{S} = \\int (\\mathbb{I} \\mathbf{S} - \\operatorname{t_{2}}{(\\mathbb{I},\\mathbf{S})}) d\\mathbf{S} and \\int 0 d\\mathbf{S} = \\int (\\mathbb{I} \\mathbf{S} - \\operatorname{C_{1}}{(\\mathbb{I},\\mathbf{S})}) d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('t_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(A_{z},\\mathbf{f})} = A_{z} \\cos{(\\mathbf{f})}, then obtain 0 = \\sin{((A_{z} \\cos{(\\mathbf{f})})^{- A_{z}} \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})})} - \\sin{(1)}", "derivation": "\\operatorname{y^{\\prime}}{(A_{z},\\mathbf{f})} = A_{z} \\cos{(\\mathbf{f})} and \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})} = (A_{z} \\cos{(\\mathbf{f})})^{A_{z}} and (A_{z} \\cos{(\\mathbf{f})})^{- A_{z}} \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})} = 1 and \\sin{((A_{z} \\cos{(\\mathbf{f})})^{- A_{z}} \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})})} = \\sin{(1)} and 0 = - \\sin{((A_{z} \\cos{(\\mathbf{f})})^{- A_{z}} \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})})} + \\sin{(1)} and 0 = \\sin{((A_{z} \\cos{(\\mathbf{f})})^{- A_{z}} \\operatorname{y^{\\prime}}^{A_{z}}{(A_{z},\\mathbf{f})})} - \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('A_z', commutative=True)))"], [["divide", 2, "Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True))), Integer(1))"], [["sin", 3], "Equality(sin(Mul(Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True)))), sin(Integer(1)))"], [["minus", 4, "sin(Mul(Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Mul(Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True))))), sin(Integer(1))))"], [["divide", 5, "Integer(-1)"], "Equality(Integer(0), Add(sin(Mul(Pow(Mul(Symbol('A_z', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_z', commutative=True)))), Mul(Integer(-1), sin(Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(k)} = k, then derive \\int \\hat{H}_l{(k)} dk = c + \\frac{k^{2}}{2}, then obtain \\frac{\\partial}{\\partial c} (- c + \\int k d\\hat{H}_l{(k)}) = \\frac{d}{d c} \\frac{\\hat{H}_l^{2}{(k)}}{2}", "derivation": "\\hat{H}_l{(k)} = k and \\int \\hat{H}_l{(k)} dk = \\int k dk and \\int \\hat{H}_l{(k)} dk = c + \\frac{k^{2}}{2} and \\int k dk = c + \\frac{k^{2}}{2} and - c + \\int k dk = \\frac{k^{2}}{2} and - c + \\int k d\\hat{H}_l{(k)} = \\frac{\\hat{H}_l^{2}{(k)}}{2} and \\frac{\\partial}{\\partial c} (- c + \\int k d\\hat{H}_l{(k)}) = \\frac{d}{d c} \\frac{\\hat{H}_l^{2}{(k)}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Symbol('c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True))), Add(Symbol('c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))))"], [["minus", 4, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Symbol('k', commutative=True), Tuple(Function('\\\\hat{H}_l')(Symbol('k', commutative=True))))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Integer(2))))"], [["differentiate", 6, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Symbol('k', commutative=True), Tuple(Function('\\\\hat{H}_l')(Symbol('k', commutative=True))))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Integer(2))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})}, then derive \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = - \\sin{(v_{1})}, then derive - \\cos{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})}, then obtain - \\cos{(v_{1})} = \\frac{d}{d v_{1}} - \\sin{(v_{1})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = - \\sin{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} = \\frac{d}{d v_{1}} - \\sin{(v_{1})} and \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} = \\frac{d}{d v_{1}} - \\sin{(v_{1})} and \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} and - \\cos{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{f_{\\mathbf{p}}}{(v_{1})} and \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} = - \\cos{(v_{1})} and - \\cos{(v_{1})} = \\frac{d}{d v_{1}} - \\sin{(v_{1})}", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), sin(Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Derivative(Mul(Integer(-1), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(k,\\mu)} = \\int \\frac{k}{\\mu} d\\mu, then derive \\hat{p} + k = \\int \\frac{\\int \\frac{k}{\\mu} d\\mu}{M{(k,\\mu)}} dk, then obtain \\cos{(\\hat{p} + k + (\\hat{p} + k)^{\\hat{p}})} = \\cos{((\\hat{p} + k)^{\\hat{p}} + \\int 1 dk)}", "derivation": "M{(k,\\mu)} = \\int \\frac{k}{\\mu} d\\mu and \\frac{M{(k,\\mu)}}{\\mu} = \\frac{\\int \\frac{k}{\\mu} d\\mu}{\\mu} and 1 = \\frac{\\int \\frac{k}{\\mu} d\\mu}{M{(k,\\mu)}} and \\int 1 dk = \\int \\frac{\\int \\frac{k}{\\mu} d\\mu}{M{(k,\\mu)}} dk and \\hat{p} + k = \\int \\frac{\\int \\frac{k}{\\mu} d\\mu}{M{(k,\\mu)}} dk and \\hat{p} + k = \\int 1 dk and \\hat{p} + k + (\\hat{p} + k)^{\\hat{p}} = (\\hat{p} + k)^{\\hat{p}} + \\int 1 dk and \\cos{(\\hat{p} + k + (\\hat{p} + k)^{\\hat{p}})} = \\cos{((\\hat{p} + k)^{\\hat{p}} + \\int 1 dk)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\mu', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('k', commutative=True))), Integral(Mul(Pow(Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Integral(Mul(Pow(Function('M')(Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Integral(Integer(1), Tuple(Symbol('k', commutative=True))))"], [["add", 6, "Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Add(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integral(Integer(1), Tuple(Symbol('k', commutative=True)))))"], [["cos", 7], "Equality(cos(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))), cos(Add(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integral(Integer(1), Tuple(Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(k,\\mu_0)} = \\log{(\\mu_0 - k)}, then derive \\int \\dot{y}{(k,\\mu_0)} dk = - \\mu_0 \\log{(- \\mu_0 + k)} + \\nabla + k \\log{(\\mu_0 - k)} - k, then obtain \\frac{\\partial}{\\partial k} \\int \\dot{y}{(k,\\mu_0)} dk = \\frac{\\partial}{\\partial k} (- \\mu_0 \\log{(- \\mu_0 + k)} + \\nabla + k \\dot{y}{(k,\\mu_0)} - k)", "derivation": "\\dot{y}{(k,\\mu_0)} = \\log{(\\mu_0 - k)} and \\int \\dot{y}{(k,\\mu_0)} dk = \\int \\log{(\\mu_0 - k)} dk and \\int \\dot{y}{(k,\\mu_0)} dk = - \\mu_0 \\log{(- \\mu_0 + k)} + \\nabla + k \\log{(\\mu_0 - k)} - k and \\int \\dot{y}{(k,\\mu_0)} dk = - \\mu_0 \\log{(- \\mu_0 + k)} + \\nabla + k \\dot{y}{(k,\\mu_0)} - k and \\frac{\\partial}{\\partial k} \\int \\dot{y}{(k,\\mu_0)} dk = \\frac{\\partial}{\\partial k} (- \\mu_0 \\log{(- \\mu_0 + k)} + \\nabla + k \\dot{y}{(k,\\mu_0)} - k)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(log(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('k', commutative=True)))), Symbol('\\\\nabla', commutative=True), Mul(Symbol('k', commutative=True), log(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('k', commutative=True)))), Symbol('\\\\nabla', commutative=True), Mul(Symbol('k', commutative=True), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('k', commutative=True)))), Symbol('\\\\nabla', commutative=True), Mul(Symbol('k', commutative=True), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\dot{y})} = \\sin{(\\dot{y})} and M{(\\dot{y})} = \\hat{\\mathbf{x}}^{2}{(\\dot{y})}, then obtain M^{\\dot{y}}{(\\dot{y})} \\sin{(\\dot{y})} = (\\hat{\\mathbf{x}}^{2}{(\\dot{y})})^{\\dot{y}} \\sin{(\\dot{y})}", "derivation": "\\hat{\\mathbf{x}}{(\\dot{y})} = \\sin{(\\dot{y})} and \\hat{\\mathbf{x}}^{2}{(\\dot{y})} = \\hat{\\mathbf{x}}{(\\dot{y})} \\sin{(\\dot{y})} and M{(\\dot{y})} = \\hat{\\mathbf{x}}^{2}{(\\dot{y})} and M{(\\dot{y})} = \\hat{\\mathbf{x}}{(\\dot{y})} \\sin{(\\dot{y})} and M^{\\dot{y}}{(\\dot{y})} = (\\hat{\\mathbf{x}}{(\\dot{y})} \\sin{(\\dot{y})})^{\\dot{y}} and M^{\\dot{y}}{(\\dot{y})} = (\\hat{\\mathbf{x}}^{2}{(\\dot{y})})^{\\dot{y}} and M^{\\dot{y}}{(\\dot{y})} \\sin{(\\dot{y})} = (\\hat{\\mathbf{x}}^{2}{(\\dot{y})})^{\\dot{y}} \\sin{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\dot{y}', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('M')(Symbol('\\\\dot{y}', commutative=True)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))))"], [["power", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('M')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('M')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 6, "Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('M')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(v_{2},\\phi_1)} = e^{\\frac{v_{2}}{\\phi_1}} and \\psi^{*}{(v_{2},\\phi_1)} = \\frac{v_{2}}{\\phi_1}, then obtain - \\phi_1 + \\theta_{1}{(v_{2},\\phi_1)} = - \\phi_1 + e^{\\psi^{*}{(v_{2},\\phi_1)}}", "derivation": "\\theta_{1}{(v_{2},\\phi_1)} = e^{\\frac{v_{2}}{\\phi_1}} and \\psi^{*}{(v_{2},\\phi_1)} = \\frac{v_{2}}{\\phi_1} and \\theta_{1}{(v_{2},\\phi_1)} = e^{\\psi^{*}{(v_{2},\\phi_1)}} and - \\phi_1 + \\theta_{1}{(v_{2},\\phi_1)} = - \\phi_1 + e^{\\psi^{*}{(v_{2},\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), exp(Function('\\\\psi^*')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), exp(Function('\\\\psi^*')(Symbol('v_2', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{H})} = e^{\\hat{H}}, then obtain - e^{\\hat{H}} + \\int \\dot{\\mathbf{r}}{(\\hat{H})} e^{- \\hat{H}} d\\hat{H} = \\hat{H} + \\mathbf{s} - e^{\\hat{H}}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{H})} = e^{\\hat{H}} and \\dot{\\mathbf{r}}{(\\hat{H})} e^{- \\hat{H}} = 1 and \\int \\dot{\\mathbf{r}}{(\\hat{H})} e^{- \\hat{H}} d\\hat{H} = \\int 1 d\\hat{H} and - e^{\\hat{H}} + \\int \\dot{\\mathbf{r}}{(\\hat{H})} e^{- \\hat{H}} d\\hat{H} = - e^{\\hat{H}} + \\int 1 d\\hat{H} and - e^{\\hat{H}} + \\int \\dot{\\mathbf{r}}{(\\hat{H})} e^{- \\hat{H}} d\\hat{H} = \\hat{H} + \\mathbf{s} - e^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 3, "exp(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given t{(\\theta,f)} = \\frac{\\partial}{\\partial f} (\\theta - f), then derive t{(\\theta,f)} = -1, then obtain e^{t{(\\theta,f)}} = e^{\\frac{\\partial}{\\partial f} (\\theta - f)}", "derivation": "t{(\\theta,f)} = \\frac{\\partial}{\\partial f} (\\theta - f) and t{(\\theta,f)} = -1 and \\frac{\\partial}{\\partial f} (\\theta - f) = -1 and e^{\\frac{\\partial}{\\partial f} (\\theta - f)} = e^{-1} and e^{t{(\\theta,f)}} = e^{-1} and e^{t{(\\theta,f)}} = e^{\\frac{\\partial}{\\partial f} (\\theta - f)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t')(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))"], [["exp", 3], "Equality(exp(Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), exp(Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(exp(Function('t')(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True))), exp(Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(exp(Function('t')(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True))), exp(Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(F_{g},\\mathbf{E})} = \\int (F_{g} + \\mathbf{E}) dF_{g}, then derive 0 = \\frac{\\partial}{\\partial \\mathbf{E}} (- F_{g} - f{(F_{g},\\mathbf{E})} + \\int (F_{g} + \\mathbf{E}) dF_{g}), then obtain 0 = \\frac{d}{d \\mathbf{E}} - F_{g}", "derivation": "f{(F_{g},\\mathbf{E})} = \\int (F_{g} + \\mathbf{E}) dF_{g} and - F_{g} + f{(F_{g},\\mathbf{E})} = - F_{g} + \\int (F_{g} + \\mathbf{E}) dF_{g} and - F_{g} = - F_{g} - f{(F_{g},\\mathbf{E})} + \\int (F_{g} + \\mathbf{E}) dF_{g} and \\frac{d}{d \\mathbf{E}} - F_{g} = \\frac{\\partial}{\\partial \\mathbf{E}} (- F_{g} - f{(F_{g},\\mathbf{E})} + \\int (F_{g} + \\mathbf{E}) dF_{g}) and 0 = \\frac{\\partial}{\\partial \\mathbf{E}} (- F_{g} - f{(F_{g},\\mathbf{E})} + \\int (F_{g} + \\mathbf{E}) dF_{g}) and 0 = \\frac{d}{d \\mathbf{E}} - F_{g}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["minus", 2, "Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_g', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(0), Derivative(Mul(Integer(-1), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\Omega,y)} = \\frac{y}{\\Omega}, then obtain \\frac{4 y^{2} \\mathbf{J}_P^{2}{(\\Omega,y)}}{\\Omega^{2}} = \\frac{4 y^{4}}{\\Omega^{4}}", "derivation": "\\mathbf{J}_P{(\\Omega,y)} = \\frac{y}{\\Omega} and 2 \\mathbf{J}_P{(\\Omega,y)} = \\mathbf{J}_P{(\\Omega,y)} + \\frac{y}{\\Omega} and \\frac{2 y \\mathbf{J}_P{(\\Omega,y)}}{\\Omega} = \\frac{y (\\mathbf{J}_P{(\\Omega,y)} + \\frac{y}{\\Omega})}{\\Omega} and y \\mathbf{J}_P{(\\Omega,y)} = \\frac{y^{2}}{\\Omega} and \\frac{2 y^{2}}{\\Omega^{2}} = \\frac{y (\\mathbf{J}_P{(\\Omega,y)} + \\frac{y}{\\Omega})}{\\Omega} and \\frac{2 y \\mathbf{J}_P{(\\Omega,y)}}{\\Omega} = \\frac{2 y^{2}}{\\Omega^{2}} and \\frac{4 y^{2} \\mathbf{J}_P^{2}{(\\Omega,y)}}{\\Omega^{2}} = \\frac{4 y^{4}}{\\Omega^{4}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True))), Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True), Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True)))))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True), Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Symbol('y', commutative=True), Integer(2))))"], [["power", 6, 2], "Equality(Mul(Integer(4), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Symbol('y', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Integer(4), Pow(Symbol('\\\\Omega', commutative=True), Integer(-4)), Pow(Symbol('y', commutative=True), Integer(4))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(i)} = \\cos{(i)} and \\mathbf{B}{(i)} = \\operatorname{v_{t}}{(i)} - \\cos{(i)}, then obtain \\frac{\\mathbf{B}{(i)}}{\\operatorname{v_{t}}^{2}{(i)}} = 0", "derivation": "\\operatorname{v_{t}}{(i)} = \\cos{(i)} and \\mathbf{B}{(i)} = \\operatorname{v_{t}}{(i)} - \\cos{(i)} and \\frac{\\mathbf{B}{(i)}}{\\operatorname{v_{t}}{(i)} \\cos{(i)}} = \\frac{\\operatorname{v_{t}}{(i)} - \\cos{(i)}}{\\operatorname{v_{t}}{(i)} \\cos{(i)}} and \\frac{\\mathbf{B}{(i)}}{\\operatorname{v_{t}}^{2}{(i)}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('i', commutative=True)), Add(Function('v_t')(Symbol('i', commutative=True)), Mul(Integer(-1), cos(Symbol('i', commutative=True)))))"], [["divide", 2, "Mul(Function('v_t')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('i', commutative=True)), Pow(Function('v_t')(Symbol('i', commutative=True)), Integer(-1)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Mul(Add(Function('v_t')(Symbol('i', commutative=True)), Mul(Integer(-1), cos(Symbol('i', commutative=True)))), Pow(Function('v_t')(Symbol('i', commutative=True)), Integer(-1)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('i', commutative=True)), Pow(Function('v_t')(Symbol('i', commutative=True)), Integer(-2))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(I,\\varepsilon_0)} = \\sin{(\\frac{I}{\\varepsilon_0})}, then obtain \\log{(\\operatorname{A_{y}}{(I,\\varepsilon_0)})} - \\frac{\\sin{(\\frac{I}{\\varepsilon_0})}}{I} = \\log{(\\sin{(\\frac{I}{\\varepsilon_0})})} - \\frac{\\sin{(\\frac{I}{\\varepsilon_0})}}{I}", "derivation": "\\operatorname{A_{y}}{(I,\\varepsilon_0)} = \\sin{(\\frac{I}{\\varepsilon_0})} and \\log{(\\operatorname{A_{y}}{(I,\\varepsilon_0)})} = \\log{(\\sin{(\\frac{I}{\\varepsilon_0})})} and \\frac{\\operatorname{A_{y}}{(I,\\varepsilon_0)}}{I} = \\frac{\\sin{(\\frac{I}{\\varepsilon_0})}}{I} and \\log{(\\operatorname{A_{y}}{(I,\\varepsilon_0)})} - \\frac{\\operatorname{A_{y}}{(I,\\varepsilon_0)}}{I} = \\log{(\\sin{(\\frac{I}{\\varepsilon_0})})} - \\frac{\\operatorname{A_{y}}{(I,\\varepsilon_0)}}{I} and \\log{(\\operatorname{A_{y}}{(I,\\varepsilon_0)})} - \\frac{\\sin{(\\frac{I}{\\varepsilon_0})}}{I} = \\log{(\\sin{(\\frac{I}{\\varepsilon_0})})} - \\frac{\\sin{(\\frac{I}{\\varepsilon_0})}}{I}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))))"], [["log", 1], "Equality(log(Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), log(sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))))"], [["minus", 2, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Add(log(Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Add(log(sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(log(Function('A_y')(Symbol('I', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))))), Add(log(sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), sin(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given p{(W)} = \\sin{(W)} and \\operatorname{C_{2}}{(W)} = \\sin{(W)}, then obtain \\int p{(W)} dW = \\int \\sin{(W)} dW", "derivation": "p{(W)} = \\sin{(W)} and \\operatorname{C_{2}}{(W)} = \\sin{(W)} and \\operatorname{C_{2}}{(W)} = p{(W)} and \\int \\operatorname{C_{2}}{(W)} dW = \\int \\sin{(W)} dW and \\int p{(W)} dW = \\int \\sin{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('C_2')(Symbol('W', commutative=True)), Function('p')(Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('p')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\dot{x},V_{\\mathbf{E}})} = - \\dot{x} + \\cos{(V_{\\mathbf{E}})}, then obtain \\int \\frac{\\partial}{\\partial \\dot{x}} \\theta{(\\dot{x},V_{\\mathbf{E}})} dV_{\\mathbf{E}} = - V_{\\mathbf{E}} + \\mathbf{D}", "derivation": "\\theta{(\\dot{x},V_{\\mathbf{E}})} = - \\dot{x} + \\cos{(V_{\\mathbf{E}})} and \\frac{\\partial}{\\partial \\dot{x}} \\theta{(\\dot{x},V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\dot{x}} (- \\dot{x} + \\cos{(V_{\\mathbf{E}})}) and \\int \\frac{\\partial}{\\partial \\dot{x}} \\theta{(\\dot{x},V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\frac{\\partial}{\\partial \\dot{x}} (- \\dot{x} + \\cos{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}} and \\int \\frac{\\partial}{\\partial \\dot{x}} \\theta{(\\dot{x},V_{\\mathbf{E}})} dV_{\\mathbf{E}} = - V_{\\mathbf{E}} + \\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})} = (\\frac{t_{2}}{\\mathbf{J}})^{\\hat{p}}, then obtain - \\frac{(\\frac{t_{2}}{\\mathbf{J}})^{- \\hat{p}} \\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})}}{\\frac{d}{d \\hat{p}} 1} = - \\frac{1}{\\frac{d}{d \\hat{p}} 1}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})} = (\\frac{t_{2}}{\\mathbf{J}})^{\\hat{p}} and (\\frac{t_{2}}{\\mathbf{J}})^{- \\hat{p}} \\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})} = 1 and - (\\frac{t_{2}}{\\mathbf{J}})^{- \\hat{p}} \\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})} = -1 and - \\frac{(\\frac{t_{2}}{\\mathbf{J}})^{- \\hat{p}} \\operatorname{M_{E}}{(\\mathbf{J},\\hat{p},t_{2})}}{\\frac{d}{d \\hat{p}} 1} = - \\frac{1}{\\frac{d}{d \\hat{p}} 1}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "Pow(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Function('M_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Function('M_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Integer(-1))"], [["divide", 3, "Derivative(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Function('M_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Derivative(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given h{(m,i,n_{2})} = - m + \\frac{n_{2}}{i}, then derive \\frac{\\partial}{\\partial i} h{(m,i,n_{2})} + \\frac{n_{2}}{i^{2}} = 0, then obtain \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} h{(m,i,n_{2})} + \\frac{n_{2}}{i^{2}}) = \\frac{d}{d i} 0", "derivation": "h{(m,i,n_{2})} = - m + \\frac{n_{2}}{i} and m + h{(m,i,n_{2})} - \\frac{n_{2}}{i} = 0 and \\frac{\\partial}{\\partial i} (m + h{(m,i,n_{2})} - \\frac{n_{2}}{i}) = \\frac{d}{d i} 0 and \\frac{\\partial}{\\partial i} h{(m,i,n_{2})} + \\frac{n_{2}}{i^{2}} = 0 and \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} h{(m,i,n_{2})} + \\frac{n_{2}}{i^{2}}) = \\frac{d}{d i} 0", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], "Equality(Add(Symbol('m', commutative=True), Function('h')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('h')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('h')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Symbol('n_2', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('h')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Symbol('n_2', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(Z,\\tilde{g}^*)} = \\cos{(Z^{\\tilde{g}^*})}, then derive \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)} = - Z^{\\tilde{g}^*} \\log{(Z)} \\sin{(Z^{\\tilde{g}^*})}, then obtain ((\\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)})^{Z})^{Z} = ((- Z^{\\tilde{g}^*} \\log{(Z)} \\sin{(Z^{\\tilde{g}^*})})^{Z})^{Z}", "derivation": "\\operatorname{t_{2}}{(Z,\\tilde{g}^*)} = \\cos{(Z^{\\tilde{g}^*})} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\cos{(Z^{\\tilde{g}^*})} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)} = - Z^{\\tilde{g}^*} \\log{(Z)} \\sin{(Z^{\\tilde{g}^*})} and (\\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)})^{Z} = (- Z^{\\tilde{g}^*} \\log{(Z)} \\sin{(Z^{\\tilde{g}^*})})^{Z} and ((\\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(Z,\\tilde{g}^*)})^{Z})^{Z} = ((- Z^{\\tilde{g}^*} \\log{(Z)} \\sin{(Z^{\\tilde{g}^*})})^{Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('Z', commutative=True)), sin(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('Z', commutative=True)), sin(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('Z', commutative=True)))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Pow(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('Z', commutative=True)), sin(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{x},\\mathbf{f})} = \\cos{(\\hat{x} - \\mathbf{f})}, then obtain \\int (\\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{p}{(\\hat{x},\\mathbf{f})}) d\\mathbf{f} = \\int (\\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\hat{x} - \\mathbf{f})}) d\\mathbf{f}", "derivation": "\\mathbf{p}{(\\hat{x},\\mathbf{f})} = \\cos{(\\hat{x} - \\mathbf{f})} and \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{p}{(\\hat{x},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\hat{x} - \\mathbf{f})} and \\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{p}{(\\hat{x},\\mathbf{f})} = \\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\hat{x} - \\mathbf{f})} and \\int (\\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{p}{(\\hat{x},\\mathbf{f})}) d\\mathbf{f} = \\int (\\cos{(\\hat{x} - \\mathbf{f})} + \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\hat{x} - \\mathbf{f})}) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["add", 2, "cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], "Equality(Add(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Add(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Derivative(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Derivative(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given W{(\\mu_0)} = \\sin{(\\mu_0)}, then derive \\int W{(\\mu_0)} d\\mu_0 = z - \\cos{(\\mu_0)}, then obtain \\mu_0 (z - \\cos{(\\mu_0)}) \\sin{(\\mu_0)} = \\mu_0 (m_{s} - \\cos{(\\mu_0)}) \\sin{(\\mu_0)}", "derivation": "W{(\\mu_0)} = \\sin{(\\mu_0)} and \\int W{(\\mu_0)} d\\mu_0 = \\int \\sin{(\\mu_0)} d\\mu_0 and \\mu_0 \\int W{(\\mu_0)} d\\mu_0 = \\mu_0 \\int \\sin{(\\mu_0)} d\\mu_0 and \\mu_0 \\sin{(\\mu_0)} \\int W{(\\mu_0)} d\\mu_0 = \\mu_0 \\sin{(\\mu_0)} \\int \\sin{(\\mu_0)} d\\mu_0 and \\int W{(\\mu_0)} d\\mu_0 = z - \\cos{(\\mu_0)} and \\mu_0 (z - \\cos{(\\mu_0)}) \\sin{(\\mu_0)} = \\mu_0 \\sin{(\\mu_0)} \\int \\sin{(\\mu_0)} d\\mu_0 and \\mu_0 (z - \\cos{(\\mu_0)}) \\sin{(\\mu_0)} = \\mu_0 (m_{s} - \\cos{(\\mu_0)}) \\sin{(\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["times", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Function('W')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["times", 3, "sin(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\mu_0', commutative=True)), Integral(Function('W')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\mu_0', commutative=True)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), sin(Symbol('\\\\mu_0', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\mu_0', commutative=True)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), sin(Symbol('\\\\mu_0', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), sin(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given I{(\\pi,l)} = \\pi + l, then derive \\int I{(\\pi,l)} dl = \\pi l + \\varepsilon_0 + \\frac{l^{2}}{2}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} (\\int (\\pi + l) dl)^{\\pi} = \\frac{\\partial}{\\partial \\varepsilon_0} (\\pi l + \\varepsilon_0 + \\frac{l^{2}}{2})^{\\pi}", "derivation": "I{(\\pi,l)} = \\pi + l and \\int I{(\\pi,l)} dl = \\int (\\pi + l) dl and \\int I{(\\pi,l)} dl = \\pi l + \\varepsilon_0 + \\frac{l^{2}}{2} and \\int (\\pi + l) dl = \\pi l + \\varepsilon_0 + \\frac{l^{2}}{2} and (\\int (\\pi + l) dl)^{\\pi} = (\\pi l + \\varepsilon_0 + \\frac{l^{2}}{2})^{\\pi} and \\frac{\\partial}{\\partial \\varepsilon_0} (\\int (\\pi + l) dl)^{\\pi} = \\frac{\\partial}{\\partial \\varepsilon_0} (\\pi l + \\varepsilon_0 + \\frac{l^{2}}{2})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(C)} = e^{\\sin{(C)}} and c{(C)} = e^{\\sin{(C)}}, then obtain \\theta{(C)} - e^{\\sin{(C)}} = c{(C)} - e^{\\sin{(C)}}", "derivation": "\\theta{(C)} = e^{\\sin{(C)}} and - C + \\theta{(C)} = - C + e^{\\sin{(C)}} and c{(C)} = e^{\\sin{(C)}} and - C + \\theta{(C)} = - C + c{(C)} and \\theta{(C)} - e^{\\sin{(C)}} = c{(C)} - e^{\\sin{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('C', commutative=True)), exp(sin(Symbol('C', commutative=True))))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('\\\\theta')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), exp(sin(Symbol('C', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('C', commutative=True)), exp(sin(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('\\\\theta')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('c')(Symbol('C', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('C', commutative=True)), exp(sin(Symbol('C', commutative=True))))"], "Equality(Add(Function('\\\\theta')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('C', commutative=True))))), Add(Function('c')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('C', commutative=True))))))"]]}, {"prompt": "Given u{(x^\\prime,h)} = h x^\\prime, then derive \\frac{\\partial}{\\partial A} \\int (2 A + u{(x^\\prime,h)}) dh = \\frac{\\partial}{\\partial A} (2 A h + \\dot{z} + \\frac{h^{2} x^\\prime}{2}), then obtain \\frac{\\partial^{2}}{\\partial x^\\prime\\partial A} (2 A h + \\dot{z} + \\frac{h u{(x^\\prime,h)}}{2}) = \\frac{\\partial^{2}}{\\partial x^\\prime\\partial A} \\int (2 A + h x^\\prime) dh", "derivation": "u{(x^\\prime,h)} = h x^\\prime and 2 A + u{(x^\\prime,h)} = 2 A + h x^\\prime and \\int (2 A + u{(x^\\prime,h)}) dh = \\int (2 A + h x^\\prime) dh and \\frac{\\partial}{\\partial A} \\int (2 A + u{(x^\\prime,h)}) dh = \\frac{\\partial}{\\partial A} \\int (2 A + h x^\\prime) dh and \\frac{\\partial}{\\partial A} \\int (2 A + u{(x^\\prime,h)}) dh = \\frac{\\partial}{\\partial A} (2 A h + \\dot{z} + \\frac{h^{2} x^\\prime}{2}) and \\frac{\\partial}{\\partial A} (2 A h + \\dot{z} + \\frac{h^{2} x^\\prime}{2}) = \\frac{\\partial}{\\partial A} \\int (2 A + h x^\\prime) dh and \\frac{\\partial}{\\partial A} (2 A h + \\dot{z} + \\frac{h u{(x^\\prime,h)}}{2}) = \\frac{\\partial}{\\partial A} \\int (2 A + h x^\\prime) dh and \\frac{\\partial^{2}}{\\partial x^\\prime\\partial A} (2 A h + \\dot{z} + \\frac{h u{(x^\\prime,h)}}{2}) = \\frac{\\partial^{2}}{\\partial x^\\prime\\partial A} \\int (2 A + h x^\\prime) dh", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Mul(Integer(2), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('A', commutative=True)), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Symbol('h', commutative=True), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Symbol('h', commutative=True), Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\phi_1,\\Omega)} = \\frac{e^{\\Omega}}{\\phi_1}, then obtain \\iint \\phi_1 (\\lambda{(\\phi_1,\\Omega)} - e^{\\Omega}) e^{- \\Omega} d\\phi_1 d\\Omega = \\iint \\phi_1 (- e^{\\Omega} + \\frac{e^{\\Omega}}{\\phi_1}) e^{- \\Omega} d\\phi_1 d\\Omega", "derivation": "\\lambda{(\\phi_1,\\Omega)} = \\frac{e^{\\Omega}}{\\phi_1} and \\lambda{(\\phi_1,\\Omega)} - e^{\\Omega} = - e^{\\Omega} + \\frac{e^{\\Omega}}{\\phi_1} and \\phi_1 (\\lambda{(\\phi_1,\\Omega)} - e^{\\Omega}) e^{- \\Omega} = \\phi_1 (- e^{\\Omega} + \\frac{e^{\\Omega}}{\\phi_1}) e^{- \\Omega} and \\int \\phi_1 (\\lambda{(\\phi_1,\\Omega)} - e^{\\Omega}) e^{- \\Omega} d\\phi_1 = \\int \\phi_1 (- e^{\\Omega} + \\frac{e^{\\Omega}}{\\phi_1}) e^{- \\Omega} d\\phi_1 and \\iint \\phi_1 (\\lambda{(\\phi_1,\\Omega)} - e^{\\Omega}) e^{- \\Omega} d\\phi_1 d\\Omega = \\iint \\phi_1 (- e^{\\Omega} + \\frac{e^{\\Omega}}{\\phi_1}) e^{- \\Omega} d\\phi_1 d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Add(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_1', commutative=True), Add(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_1', commutative=True), Add(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(n_{2})} = \\cos{(n_{2})} and t{(n_{2})} = - n_{2}, then obtain \\int (- n_{2} + \\operatorname{V_{\\mathbf{E}}}{(n_{2})}) dn_{2} = \\int (- n_{2} + \\cos{(n_{2})}) dn_{2}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(n_{2})} = \\cos{(n_{2})} and - n_{2} + \\operatorname{V_{\\mathbf{E}}}{(n_{2})} = - n_{2} + \\cos{(n_{2})} and t{(n_{2})} = - n_{2} and \\operatorname{V_{\\mathbf{E}}}{(n_{2})} + t{(n_{2})} = t{(n_{2})} + \\cos{(n_{2})} and \\int (\\operatorname{V_{\\mathbf{E}}}{(n_{2})} + t{(n_{2})}) dn_{2} = \\int (t{(n_{2})} + \\cos{(n_{2})}) dn_{2} and \\int (- n_{2} + \\operatorname{V_{\\mathbf{E}}}{(n_{2})}) dn_{2} = \\int (- n_{2} + \\cos{(n_{2})}) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)), Function('t')(Symbol('n_2', commutative=True))), Add(Function('t')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))))"], [["integrate", 4, "Symbol('n_2', commutative=True)"], "Equality(Integral(Add(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True)), Function('t')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Function('t')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(r,v_{t})} = \\cos{(r + v_{t})}, then derive \\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\phi_{1}{(r,v_{t})} = 0, then obtain (\\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\phi_{1}{(r,v_{t})}) (\\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\cos{(r + v_{t})}) = 0", "derivation": "\\phi_{1}{(r,v_{t})} = \\cos{(r + v_{t})} and \\phi_{1}{(r,v_{t})} - \\cos{(r + v_{t})} = 0 and \\frac{\\partial}{\\partial r} (\\phi_{1}{(r,v_{t})} - \\cos{(r + v_{t})}) = \\frac{d}{d r} 0 and \\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\phi_{1}{(r,v_{t})} = 0 and \\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\cos{(r + v_{t})} = 0 and (\\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\phi_{1}{(r,v_{t})}) (\\sin{(r + v_{t})} + \\frac{\\partial}{\\partial r} \\cos{(r + v_{t})}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))))"], [["minus", 1, "cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(sin(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Derivative(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(sin(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Derivative(cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(0))"], [["times", 5, "Add(sin(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Derivative(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], "Equality(Mul(Add(sin(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Derivative(Function('\\\\phi_1')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(sin(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Derivative(cos(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given b{(y)} = \\frac{1}{y}, then obtain (\\frac{d}{d y} \\log{(b{(y)})})^{y} = (\\frac{d}{d y} \\log{(\\frac{1}{y})})^{y}", "derivation": "b{(y)} = \\frac{1}{y} and \\log{(b{(y)})} = \\log{(\\frac{1}{y})} and \\frac{d}{d y} \\log{(b{(y)})} = \\frac{d}{d y} \\log{(\\frac{1}{y})} and (\\frac{d}{d y} \\log{(b{(y)})})^{y} = (\\frac{d}{d y} \\log{(\\frac{1}{y})})^{y}", "srepr_derivation": [["renaming_premise", "Equality(Function('b')(Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1)))"], [["log", 1], "Equality(log(Function('b')(Symbol('y', commutative=True))), log(Pow(Symbol('y', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(log(Function('b')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Derivative(log(Function('b')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Pow(Derivative(log(Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(r_{0},B)} = r_{0}^{B}, then derive \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)} = r_{0}^{B} \\log{(r_{0})}, then obtain r_{0}^{B} v_{x} \\log{(r_{0})} = v_{x} \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)}", "derivation": "\\operatorname{P_{g}}{(r_{0},B)} = r_{0}^{B} and \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)} = \\frac{\\partial}{\\partial B} r_{0}^{B} and \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)} = r_{0}^{B} \\log{(r_{0})} and \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)} = \\operatorname{P_{g}}{(r_{0},B)} \\log{(r_{0})} and v_{x} \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)} = v_{x} \\operatorname{P_{g}}{(r_{0},B)} \\log{(r_{0})} and r_{0}^{B} v_{x} \\log{(r_{0})} = v_{x} \\operatorname{P_{g}}{(r_{0},B)} \\log{(r_{0})} and r_{0}^{B} v_{x} \\log{(r_{0})} = v_{x} \\frac{\\partial}{\\partial B} \\operatorname{P_{g}}{(r_{0},B)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Pow(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), log(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), log(Symbol('r_0', commutative=True))))"], [["times", 4, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Derivative(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Symbol('v_x', commutative=True), Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), log(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Symbol('v_x', commutative=True), log(Symbol('r_0', commutative=True))), Mul(Symbol('v_x', commutative=True), Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), log(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Symbol('v_x', commutative=True), log(Symbol('r_0', commutative=True))), Mul(Symbol('v_x', commutative=True), Derivative(Function('P_g')(Symbol('r_0', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(x,f,W)} = (W + x)^{f}, then obtain \\frac{\\partial}{\\partial f} (\\int \\operatorname{F_{x}}{(x,f,W)} df)^{f} + 1 = \\frac{\\partial}{\\partial f} (\\int (W + x)^{f} df)^{f} + 1", "derivation": "\\operatorname{F_{x}}{(x,f,W)} = (W + x)^{f} and \\int \\operatorname{F_{x}}{(x,f,W)} df = \\int (W + x)^{f} df and (\\int \\operatorname{F_{x}}{(x,f,W)} df)^{f} = (\\int (W + x)^{f} df)^{f} and \\frac{\\partial}{\\partial f} (\\int \\operatorname{F_{x}}{(x,f,W)} df)^{f} = \\frac{\\partial}{\\partial f} (\\int (W + x)^{f} df)^{f} and \\frac{\\partial}{\\partial f} (\\int \\operatorname{F_{x}}{(x,f,W)} df)^{f} + 1 = \\frac{\\partial}{\\partial f} (\\int (W + x)^{f} df)^{f} + 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('x', commutative=True), Symbol('f', commutative=True), Symbol('W', commutative=True)), Pow(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)), Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('x', commutative=True), Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Function('F_x')(Symbol('x', commutative=True), Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(Pow(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('F_x')(Symbol('x', commutative=True), Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Pow(Integral(Function('F_x')(Symbol('x', commutative=True), Symbol('f', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Integral(Pow(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\hat{X}{(P_{e},b)} = P_{e} + b, then obtain \\int b \\hat{X}{(P_{e},b)} dP_{e} = \\frac{P_{e}^{2} b}{2} + P_{e} b^{2} + \\mu", "derivation": "\\hat{X}{(P_{e},b)} = P_{e} + b and b \\hat{X}{(P_{e},b)} = b (P_{e} + b) and \\int b \\hat{X}{(P_{e},b)} dP_{e} = \\int b (P_{e} + b) dP_{e} and \\int b \\hat{X}{(P_{e},b)} dP_{e} = \\frac{P_{e}^{2} b}{2} + P_{e} b^{2} + \\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('b', commutative=True))))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Mul(Symbol('b', commutative=True), Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('b', commutative=True), Function('\\\\hat{X}')(Symbol('P_e', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2)), Symbol('b', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(I)} = \\log{(\\log{(I)})}, then obtain \\operatorname{A_{2}}^{4}{(I)} = \\operatorname{A_{2}}^{3}{(I)} \\log{(\\log{(I)})}", "derivation": "\\operatorname{A_{2}}{(I)} = \\log{(\\log{(I)})} and \\operatorname{A_{2}}^{2}{(I)} = \\operatorname{A_{2}}{(I)} \\log{(\\log{(I)})} and \\operatorname{A_{2}}^{3}{(I)} \\log{(\\log{(I)})} = \\operatorname{A_{2}}^{2}{(I)} \\log{(\\log{(I)})}^{2} and \\operatorname{A_{2}}^{3}{(I)} \\log{(\\log{(I)})} = \\operatorname{A_{2}}{(I)} \\log{(\\log{(I)})}^{3} and \\operatorname{A_{2}}^{4}{(I)} = \\operatorname{A_{2}}^{2}{(I)} \\log{(\\log{(I)})}^{2} and \\operatorname{A_{2}}^{4}{(I)} = \\operatorname{A_{2}}^{3}{(I)} \\log{(\\log{(I)})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('I', commutative=True)), log(log(Symbol('I', commutative=True))))"], [["times", 1, "Function('A_2')(Symbol('I', commutative=True))"], "Equality(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(2)), Mul(Function('A_2')(Symbol('I', commutative=True)), log(log(Symbol('I', commutative=True)))))"], [["times", 2, "Mul(Function('A_2')(Symbol('I', commutative=True)), log(log(Symbol('I', commutative=True))))"], "Equality(Mul(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(3)), log(log(Symbol('I', commutative=True)))), Mul(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(2)), Pow(log(log(Symbol('I', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(3)), log(log(Symbol('I', commutative=True)))), Mul(Function('A_2')(Symbol('I', commutative=True)), Pow(log(log(Symbol('I', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(4)), Mul(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(2)), Pow(log(log(Symbol('I', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(4)), Mul(Pow(Function('A_2')(Symbol('I', commutative=True)), Integer(3)), log(log(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(T)} = \\log{(T)}, then derive \\frac{\\int \\mathbf{J}_f{(T)} dT}{G + T \\log{(T)} - T} = 1, then obtain \\frac{\\partial}{\\partial T} \\frac{\\int \\log{(T)} dT}{G + T \\log{(T)} - T} = \\frac{d}{d T} 1", "derivation": "\\mathbf{J}_f{(T)} = \\log{(T)} and \\int \\mathbf{J}_f{(T)} dT = \\int \\log{(T)} dT and \\frac{\\int \\mathbf{J}_f{(T)} dT}{\\int \\log{(T)} dT} = 1 and \\frac{\\int \\mathbf{J}_f{(T)} dT}{G + T \\log{(T)} - T} = 1 and \\frac{\\int \\mathbf{J}_f{(T)} dT}{G + T \\mathbf{J}_f{(T)} - T} = 1 and \\frac{\\int \\log{(T)} dT}{G + T \\log{(T)} - T} = 1 and \\frac{\\partial}{\\partial T} \\frac{\\int \\log{(T)} dT}{G + T \\log{(T)} - T} = \\frac{d}{d T} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["divide", 2, "Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Pow(Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Symbol('T', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{J}_f')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True))), Integer(-1)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Integer(1))"], [["differentiate", 6, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True))), Integer(-1)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(v)} = \\log{(\\log{(v)})}, then obtain \\int (v \\cos{(v + \\mu{(v)})} + \\frac{v \\cos{(v + \\mu{(v)})}}{\\cos{(v + \\log{(\\log{(v)})})}}) dv = \\int (v \\cos{(v + \\mu{(v)})} + v) dv", "derivation": "\\mu{(v)} = \\log{(\\log{(v)})} and v + \\mu{(v)} = v + \\log{(\\log{(v)})} and \\cos{(v + \\mu{(v)})} = \\cos{(v + \\log{(\\log{(v)})})} and \\frac{\\cos{(v + \\mu{(v)})}}{\\cos{(v + \\log{(\\log{(v)})})}} = 1 and \\frac{v \\cos{(v + \\mu{(v)})}}{\\cos{(v + \\log{(\\log{(v)})})}} = v and v \\cos{(v + \\mu{(v)})} + \\frac{v \\cos{(v + \\mu{(v)})}}{\\cos{(v + \\log{(\\log{(v)})})}} = v \\cos{(v + \\mu{(v)})} + v and \\int (v \\cos{(v + \\mu{(v)})} + \\frac{v \\cos{(v + \\mu{(v)})}}{\\cos{(v + \\log{(\\log{(v)})})}}) dv = \\int (v \\cos{(v + \\mu{(v)})} + v) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('v', commutative=True)), log(log(Symbol('v', commutative=True))))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))), cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True))))))"], [["divide", 3, "cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True)))))"], "Equality(Mul(cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))), Pow(cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True))))), Integer(-1))), Integer(1))"], [["times", 4, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))), Pow(cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True))))), Integer(-1))), Symbol('v', commutative=True))"], [["add", 5, "Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))))"], "Equality(Add(Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True))))), Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))), Pow(cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True))))), Integer(-1)))), Add(Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True))))), Symbol('v', commutative=True)))"], [["integrate", 6, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True))))), Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True)))), Pow(cos(Add(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True))))), Integer(-1)))), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Symbol('v', commutative=True), cos(Add(Symbol('v', commutative=True), Function('\\\\mu')(Symbol('v', commutative=True))))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given b{(F_{g},q)} = \\sin{(F_{g} - q)}, then derive \\frac{\\partial}{\\partial F_{g}} b{(F_{g},q)} = \\cos{(F_{g} - q)}, then obtain \\sin{(F_{g} - q)} \\cos{(F_{g} - q)} + \\sin{(F_{g} - q)} \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)} = 2 \\sin{(F_{g} - q)} \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)}", "derivation": "b{(F_{g},q)} = \\sin{(F_{g} - q)} and \\frac{\\partial}{\\partial F_{g}} b{(F_{g},q)} = \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)} and \\frac{\\partial}{\\partial F_{g}} b{(F_{g},q)} = \\cos{(F_{g} - q)} and \\cos{(F_{g} - q)} = \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)} and \\sin{(F_{g} - q)} \\cos{(F_{g} - q)} = \\sin{(F_{g} - q)} \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)} and \\sin{(F_{g} - q)} \\cos{(F_{g} - q)} + \\sin{(F_{g} - q)} \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)} = 2 \\sin{(F_{g} - q)} \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} - q)}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('F_g', commutative=True), Symbol('q', commutative=True)), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('F_g', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('F_g', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["times", 4, "sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], "Equality(Mul(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Mul(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["add", 5, "Mul(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], "Equality(Add(Mul(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), cos(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Mul(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))), Mul(Integer(2), sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Derivative(sin(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi^{*}{(\\hat{X})} = \\log{(\\hat{X})}, then obtain (- \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})})^{\\hat{X}} + \\log{(\\hat{X})} = \\log{(\\hat{X})} + 1", "derivation": "\\psi^{*}{(\\hat{X})} = \\log{(\\hat{X})} and 2 \\psi^{*}{(\\hat{X})} = \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})} and 0 = - \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})} and 0^{\\hat{X}} = (- \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})})^{\\hat{X}} and 0^{\\hat{X}} + \\log{(\\hat{X})} = (- \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})})^{\\hat{X}} + \\log{(\\hat{X})} and (- \\psi^{*}{(\\hat{X})} + \\log{(\\hat{X})})^{\\hat{X}} + \\log{(\\hat{X})} = \\log{(\\hat{X})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["add", 1, "Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))), Add(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))), log(Symbol('\\\\hat{X}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))), log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 4, "log(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))), log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True))), log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True))), Add(log(Symbol('\\\\hat{X}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{f}{(\\ddot{x})} = \\log{(\\ddot{x})}, then obtain \\frac{d^{2}}{d \\ddot{x}^{2}} \\mathbf{f}{(\\ddot{x})} - \\frac{1}{\\ddot{x}^{2}} = - \\frac{2}{\\ddot{x}^{2}}", "derivation": "\\mathbf{f}{(\\ddot{x})} = \\log{(\\ddot{x})} and \\mathbf{f}{(\\ddot{x})} + \\log{(\\ddot{x})} = 2 \\log{(\\ddot{x})} and \\frac{d}{d \\ddot{x}} (\\mathbf{f}{(\\ddot{x})} + \\log{(\\ddot{x})}) = \\frac{d}{d \\ddot{x}} 2 \\log{(\\ddot{x})} and \\frac{d^{2}}{d \\ddot{x}^{2}} (\\mathbf{f}{(\\ddot{x})} + \\log{(\\ddot{x})}) = \\frac{d^{2}}{d \\ddot{x}^{2}} 2 \\log{(\\ddot{x})} and \\frac{d^{2}}{d \\ddot{x}^{2}} \\mathbf{f}{(\\ddot{x})} - \\frac{1}{\\ddot{x}^{2}} = - \\frac{2}{\\ddot{x}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{f}')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{f}')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Derivative(Mul(Integer(2), log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-2)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\varepsilon,\\theta_2)} = \\frac{\\log{(\\theta_2)}}{\\varepsilon} and t{(\\varepsilon,\\theta_2)} = \\frac{\\log{(\\theta_2)}}{\\varepsilon}, then obtain t{(\\varepsilon,\\theta_2)} - 1 + \\frac{1}{\\log{(\\theta_2)}} = -1 + \\frac{1}{\\log{(\\theta_2)}} + \\frac{\\log{(\\theta_2)}}{\\varepsilon}", "derivation": "\\operatorname{m_{s}}{(\\varepsilon,\\theta_2)} = \\frac{\\log{(\\theta_2)}}{\\varepsilon} and t{(\\varepsilon,\\theta_2)} = \\frac{\\log{(\\theta_2)}}{\\varepsilon} and t{(\\varepsilon,\\theta_2)} - 1 = -1 + \\frac{\\log{(\\theta_2)}}{\\varepsilon} and t{(\\varepsilon,\\theta_2)} - 1 = \\operatorname{m_{s}}{(\\varepsilon,\\theta_2)} - 1 and t{(\\varepsilon,\\theta_2)} - 1 + \\frac{1}{\\log{(\\theta_2)}} = \\operatorname{m_{s}}{(\\varepsilon,\\theta_2)} - 1 + \\frac{1}{\\log{(\\theta_2)}} and \\operatorname{m_{s}}{(\\varepsilon,\\theta_2)} - 1 = -1 + \\frac{\\log{(\\theta_2)}}{\\varepsilon} and t{(\\varepsilon,\\theta_2)} - 1 + \\frac{1}{\\log{(\\theta_2)}} = -1 + \\frac{1}{\\log{(\\theta_2)}} + \\frac{\\log{(\\theta_2)}}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Add(Function('m_s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], [["add", 4, "Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))"], "Equality(Add(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Add(Function('m_s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('m_s')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Function('t')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Add(Integer(-1), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(F_{H})} = F_{H}, then derive \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} = 1, then obtain \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} = (\\frac{d}{d F_{H}} F_{H})^{2}", "derivation": "\\mathbf{J}_f{(F_{H})} = F_{H} and \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} = \\frac{d}{d F_{H}} F_{H} and \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} = 1 and (\\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})})^{2} = \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} and (\\frac{d}{d F_{H}} F_{H})^{2} = \\frac{d}{d F_{H}} F_{H} and \\frac{d}{d F_{H}} \\mathbf{J}_f{(F_{H})} = (\\frac{d}{d F_{H}} F_{H})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(2)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(2)), Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Pow(Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given p{(\\mathbf{g})} = \\mathbf{g}, then obtain \\sin{((\\cos^{\\mathbf{g}}{(p{(\\mathbf{g})})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})})} = \\sin{((\\cos^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})})}", "derivation": "p{(\\mathbf{g})} = \\mathbf{g} and \\cos{(p{(\\mathbf{g})})} = \\cos{(\\mathbf{g})} and \\cos^{\\mathbf{g}}{(p{(\\mathbf{g})})} = \\cos^{\\mathbf{g}}{(\\mathbf{g})} and (\\cos^{\\mathbf{g}}{(p{(\\mathbf{g})})})^{\\mathbf{g}} = (\\cos^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} and (\\cos^{\\mathbf{g}}{(p{(\\mathbf{g})})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})} = (\\cos^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})} and \\sin{((\\cos^{\\mathbf{g}}{(p{(\\mathbf{g})})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})})} = \\sin{((\\cos^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} - \\cos{(p{(\\mathbf{g})})})}", "srepr_derivation": [["renaming_premise", "Equality(Function('p')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))"], [["cos", 1], "Equality(cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Pow(cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 4, "cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Pow(Pow(cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))))), Add(Pow(Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))))))"], [["sin", 5], "Equality(sin(Add(Pow(Pow(cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True)))))), sin(Add(Pow(Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), cos(Function('p')(Symbol('\\\\mathbf{g}', commutative=True)))))))"]]}, {"prompt": "Given \\delta{(\\mathbf{s},k)} = \\int k^{\\mathbf{s}} dk and \\psi^{*}{(r)} = \\sin{(r)}, then obtain (\\frac{\\cos{(\\psi^{*}^{r}{(r)})}}{\\int k^{\\mathbf{s}} dk})^{r} = (\\frac{\\cos{(\\sin^{r}{(r)})}}{\\int k^{\\mathbf{s}} dk})^{r}", "derivation": "\\delta{(\\mathbf{s},k)} = \\int k^{\\mathbf{s}} dk and \\psi^{*}{(r)} = \\sin{(r)} and \\psi^{*}^{r}{(r)} = \\sin^{r}{(r)} and \\cos{(\\psi^{*}^{r}{(r)})} = \\cos{(\\sin^{r}{(r)})} and \\frac{\\cos{(\\psi^{*}^{r}{(r)})}}{\\delta{(\\mathbf{s},k)}} = \\frac{\\cos{(\\sin^{r}{(r)})}}{\\delta{(\\mathbf{s},k)}} and (\\frac{\\cos{(\\psi^{*}^{r}{(r)})}}{\\delta{(\\mathbf{s},k)}})^{r} = (\\frac{\\cos{(\\sin^{r}{(r)})}}{\\delta{(\\mathbf{s},k)}})^{r} and (\\frac{\\cos{(\\psi^{*}^{r}{(r)})}}{\\int k^{\\mathbf{s}} dk})^{r} = (\\frac{\\cos{(\\sin^{r}{(r)})}}{\\int k^{\\mathbf{s}} dk})^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True)), Integral(Pow(Symbol('k', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('k', commutative=True))))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), cos(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], [["divide", 4, "Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), cos(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)))), Mul(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), cos(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)))))"], [["power", 5, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), cos(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Mul(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), cos(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Mul(cos(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Pow(Integral(Pow(Symbol('k', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Symbol('r', commutative=True)), Pow(Mul(cos(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Pow(Integral(Pow(Symbol('k', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(l,\\hbar)} = \\hbar l, then derive l + \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} = \\hbar + l, then obtain \\int (\\hbar + 1) dl = \\int (\\hbar + e^{\\hbar - \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)}}) dl", "derivation": "\\operatorname{c_{0}}{(l,\\hbar)} = \\hbar l and \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} = \\frac{\\partial}{\\partial l} \\hbar l and l + \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} = l + \\frac{\\partial}{\\partial l} \\hbar l and l + \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} = \\hbar + l and - \\frac{\\partial}{\\partial l} \\hbar l + \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} = \\hbar - \\frac{\\partial}{\\partial l} \\hbar l and 0 = \\hbar - \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)} and 1 = e^{\\hbar - \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)}} and \\hbar + 1 = \\hbar + e^{\\hbar - \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)}} and \\int (\\hbar + 1) dl = \\int (\\hbar + e^{\\hbar - \\frac{\\partial}{\\partial l} \\operatorname{c_{0}}{(l,\\hbar)}}) dl", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 2, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('l', commutative=True), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('l', commutative=True), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)))"], [["minus", 4, "Add(Symbol('l', commutative=True), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["exp", 6], "Equality(Integer(1), exp(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))))"], [["minus", 7, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Integer(1)), Add(Symbol('\\\\hbar', commutative=True), exp(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))))"], [["integrate", 8, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\hbar', commutative=True), exp(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Function('c_0')(Symbol('l', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(t,\\dot{z})} = t^{\\dot{z}}, then obtain \\frac{\\iint \\hat{p}_0{(t,\\dot{z})} d\\dot{z} dt}{\\int \\hat{p}_0{(t,\\dot{z})} d\\dot{z}} = \\frac{\\iint t^{\\dot{z}} d\\dot{z} dt}{\\int \\hat{p}_0{(t,\\dot{z})} d\\dot{z}}", "derivation": "\\hat{p}_0{(t,\\dot{z})} = t^{\\dot{z}} and \\int \\hat{p}_0{(t,\\dot{z})} d\\dot{z} = \\int t^{\\dot{z}} d\\dot{z} and \\iint \\hat{p}_0{(t,\\dot{z})} d\\dot{z} dt = \\iint t^{\\dot{z}} d\\dot{z} dt and \\frac{\\iint \\hat{p}_0{(t,\\dot{z})} d\\dot{z} dt}{\\int t^{\\dot{z}} d\\dot{z}} = \\frac{\\iint t^{\\dot{z}} d\\dot{z} dt}{\\int t^{\\dot{z}} d\\dot{z}} and \\frac{\\iint \\hat{p}_0{(t,\\dot{z})} d\\dot{z} dt}{\\int \\hat{p}_0{(t,\\dot{z})} d\\dot{z}} = \\frac{\\iint t^{\\dot{z}} d\\dot{z} dt}{\\int \\hat{p}_0{(t,\\dot{z})} d\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["divide", 3, "Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Pow(Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Integral(Pow(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})}, then obtain 2 \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} + \\cos{(\\Psi_{\\lambda})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} + \\operatorname{f_{\\mathbf{p}}}{(\\Psi_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} + \\cos{(\\Psi_{\\lambda})} and \\operatorname{f_{\\mathbf{p}}}{(\\Psi_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} and 2 \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(\\Psi_{\\lambda})} + \\cos{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given c{(\\Psi_{\\lambda},\\dot{y})} = \\sin{(\\Psi_{\\lambda} + \\dot{y})}, then derive \\int c{(\\Psi_{\\lambda},\\dot{y})} d\\dot{y} = S - \\cos{(\\Psi_{\\lambda} + \\dot{y})}, then obtain \\int \\sin{(\\Psi_{\\lambda} + \\dot{y})} d\\dot{y} = S - \\cos{(\\Psi_{\\lambda} + \\dot{y})}", "derivation": "c{(\\Psi_{\\lambda},\\dot{y})} = \\sin{(\\Psi_{\\lambda} + \\dot{y})} and \\int c{(\\Psi_{\\lambda},\\dot{y})} d\\dot{y} = \\int \\sin{(\\Psi_{\\lambda} + \\dot{y})} d\\dot{y} and \\int c{(\\Psi_{\\lambda},\\dot{y})} d\\dot{y} = S - \\cos{(\\Psi_{\\lambda} + \\dot{y})} and \\int \\sin{(\\Psi_{\\lambda} + \\dot{y})} d\\dot{y} = S - \\cos{(\\Psi_{\\lambda} + \\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} = (y^{\\prime})^{\\mathbf{S}} and U{(v)} = \\log{(v)} and y{(v,y^{\\prime},\\mathbf{S})} = \\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} \\log{(v)}, then obtain (y^{\\prime})^{\\mathbf{S}} U{(v)} = y{(v,y^{\\prime},\\mathbf{S})}", "derivation": "\\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} = (y^{\\prime})^{\\mathbf{S}} and U{(v)} = \\log{(v)} and \\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} U{(v)} = \\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} \\log{(v)} and (y^{\\prime})^{\\mathbf{S}} U{(v)} = (y^{\\prime})^{\\mathbf{S}} \\log{(v)} and y{(v,y^{\\prime},\\mathbf{S})} = \\operatorname{C_{1}}{(y^{\\prime},\\mathbf{S})} \\log{(v)} and y{(v,y^{\\prime},\\mathbf{S})} = (y^{\\prime})^{\\mathbf{S}} \\log{(v)} and (y^{\\prime})^{\\mathbf{S}} U{(v)} = y{(v,y^{\\prime},\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], ["get_premise", "Equality(Function('U')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["times", 2, "Function('C_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('C_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('U')(Symbol('v', commutative=True))), Mul(Function('C_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('U')(Symbol('v', commutative=True))), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('v', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Function('C_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('y')(Symbol('v', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('U')(Symbol('v', commutative=True))), Function('y')(Symbol('v', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(S,G)} = S^{G} and \\operatorname{r_{0}}{(v_{2})} = e^{v_{2}}, then obtain \\frac{\\partial}{\\partial G} \\frac{\\operatorname{r_{0}}{(v_{2})}}{\\cos{(S^{G})}} = \\frac{\\partial}{\\partial G} \\frac{e^{v_{2}}}{\\cos{(S^{G})}}", "derivation": "\\operatorname{P_{g}}{(S,G)} = S^{G} and \\operatorname{r_{0}}{(v_{2})} = e^{v_{2}} and \\frac{\\operatorname{r_{0}}{(v_{2})}}{\\cos{(\\operatorname{P_{g}}{(S,G)})}} = \\frac{e^{v_{2}}}{\\cos{(\\operatorname{P_{g}}{(S,G)})}} and \\frac{\\partial}{\\partial G} \\frac{\\operatorname{r_{0}}{(v_{2})}}{\\cos{(\\operatorname{P_{g}}{(S,G)})}} = \\frac{\\partial}{\\partial G} \\frac{e^{v_{2}}}{\\cos{(\\operatorname{P_{g}}{(S,G)})}} and \\frac{\\partial}{\\partial G} \\frac{\\operatorname{r_{0}}{(v_{2})}}{\\cos{(S^{G})}} = \\frac{\\partial}{\\partial G} \\frac{e^{v_{2}}}{\\cos{(S^{G})}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('G', commutative=True)))"], ["get_premise", "Equality(Function('r_0')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["divide", 2, "cos(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True)))"], "Equality(Mul(Function('r_0')(Symbol('v_2', commutative=True)), Pow(cos(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Mul(exp(Symbol('v_2', commutative=True)), Pow(cos(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Function('r_0')(Symbol('v_2', commutative=True)), Pow(cos(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('v_2', commutative=True)), Pow(cos(Function('P_g')(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Function('r_0')(Symbol('v_2', commutative=True)), Pow(cos(Pow(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('v_2', commutative=True)), Pow(cos(Pow(Symbol('S', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(I,A)} = A I and \\psi{(I,A)} = I (\\int A I dI)^{I}, then obtain \\psi{(I,A)} = I (\\int \\operatorname{v_{2}}{(I,A)} dI)^{I}", "derivation": "\\operatorname{v_{2}}{(I,A)} = A I and \\int \\operatorname{v_{2}}{(I,A)} dI = \\int A I dI and (\\int \\operatorname{v_{2}}{(I,A)} dI)^{I} = (\\int A I dI)^{I} and I (\\int \\operatorname{v_{2}}{(I,A)} dI)^{I} = I (\\int A I dI)^{I} and \\psi{(I,A)} = I (\\int A I dI)^{I} and \\psi{(I,A)} = I (\\int \\operatorname{v_{2}}{(I,A)} dI)^{I}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Integral(Function('v_2')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Integral(Mul(Symbol('A', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["times", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Pow(Integral(Function('v_2')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), Pow(Integral(Mul(Symbol('A', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Integral(Mul(Symbol('A', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\psi')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Integral(Function('v_2')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(a^{\\dagger},\\varphi)} = \\varphi + a^{\\dagger}, then derive \\frac{\\partial}{\\partial \\varphi} \\hat{X}{(a^{\\dagger},\\varphi)} = 1, then obtain (\\frac{\\partial}{\\partial \\varphi} (\\varphi + a^{\\dagger}))^{\\varphi} = 1", "derivation": "\\hat{X}{(a^{\\dagger},\\varphi)} = \\varphi + a^{\\dagger} and \\frac{\\partial}{\\partial \\varphi} \\hat{X}{(a^{\\dagger},\\varphi)} = \\frac{\\partial}{\\partial \\varphi} (\\varphi + a^{\\dagger}) and \\frac{\\partial}{\\partial \\varphi} \\hat{X}{(a^{\\dagger},\\varphi)} = 1 and \\frac{\\partial}{\\partial \\varphi} (\\varphi + a^{\\dagger}) = 1 and (\\frac{\\partial}{\\partial \\varphi} (\\varphi + a^{\\dagger}))^{\\varphi} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Integer(1))"]]}, {"prompt": "Given Z{(\\varphi^*,E_{\\lambda})} = E_{\\lambda} \\varphi^*, then derive (\\varphi^* + v_{x})^{E_{\\lambda}} = (\\int \\frac{E_{\\lambda} \\varphi^*}{Z{(\\varphi^*,E_{\\lambda})}} d\\varphi^*)^{E_{\\lambda}}, then obtain \\frac{(\\varphi^* + v_{x})^{E_{\\lambda}}}{\\varphi^*} = \\frac{(\\int 1 d\\varphi^*)^{E_{\\lambda}}}{\\varphi^*}", "derivation": "Z{(\\varphi^*,E_{\\lambda})} = E_{\\lambda} \\varphi^* and 1 = \\frac{E_{\\lambda} \\varphi^*}{Z{(\\varphi^*,E_{\\lambda})}} and \\int 1 d\\varphi^* = \\int \\frac{E_{\\lambda} \\varphi^*}{Z{(\\varphi^*,E_{\\lambda})}} d\\varphi^* and (\\int 1 d\\varphi^*)^{E_{\\lambda}} = (\\int \\frac{E_{\\lambda} \\varphi^*}{Z{(\\varphi^*,E_{\\lambda})}} d\\varphi^*)^{E_{\\lambda}} and (\\varphi^* + v_{x})^{E_{\\lambda}} = (\\int \\frac{E_{\\lambda} \\varphi^*}{Z{(\\varphi^*,E_{\\lambda})}} d\\varphi^*)^{E_{\\lambda}} and (\\varphi^* + v_{x})^{E_{\\lambda}} = (\\int 1 d\\varphi^*)^{E_{\\lambda}} and \\frac{(\\varphi^* + v_{x})^{E_{\\lambda}}}{\\varphi^*} = \\frac{(\\int 1 d\\varphi^*)^{E_{\\lambda}}}{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 1, "Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Pow(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Pow(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Pow(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('v_x', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Pow(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('v_x', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 6, "Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('v_x', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{P})} = e^{e^{\\mathbf{P}}}, then obtain e^{e^{\\mathbf{P}}} \\frac{d}{d \\mathbf{P}} (\\mathbf{D}{(\\mathbf{P})} + e^{e^{\\mathbf{P}}}) = e^{e^{\\mathbf{P}}} \\frac{d}{d \\mathbf{P}} 2 e^{e^{\\mathbf{P}}}", "derivation": "\\mathbf{D}{(\\mathbf{P})} = e^{e^{\\mathbf{P}}} and \\mathbf{D}{(\\mathbf{P})} + e^{e^{\\mathbf{P}}} = 2 e^{e^{\\mathbf{P}}} and 2 \\mathbf{D}{(\\mathbf{P})} = \\mathbf{D}{(\\mathbf{P})} + e^{e^{\\mathbf{P}}} and 2 \\mathbf{D}{(\\mathbf{P})} = 2 e^{e^{\\mathbf{P}}} and \\frac{d}{d \\mathbf{P}} 2 \\mathbf{D}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} 2 e^{e^{\\mathbf{P}}} and \\frac{d}{d \\mathbf{P}} (\\mathbf{D}{(\\mathbf{P})} + e^{e^{\\mathbf{P}}}) = \\frac{d}{d \\mathbf{P}} 2 e^{e^{\\mathbf{P}}} and e^{e^{\\mathbf{P}}} \\frac{d}{d \\mathbf{P}} (\\mathbf{D}{(\\mathbf{P})} + e^{e^{\\mathbf{P}}}) = e^{e^{\\mathbf{P}}} \\frac{d}{d \\mathbf{P}} 2 e^{e^{\\mathbf{P}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 1, "exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Integer(2), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(2), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["times", 6, "exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(exp(exp(Symbol('\\\\mathbf{P}', commutative=True))), Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(exp(exp(Symbol('\\\\mathbf{P}', commutative=True))), Derivative(Mul(Integer(2), exp(exp(Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{S},y)} = y^{\\mathbf{S}} and \\mathbf{s}{(\\mathbf{S},y)} = y^{\\mathbf{S}}, then obtain \\pi y^{\\mathbf{S}} + \\cos{(\\operatorname{J_{\\varepsilon}}^{\\mathbf{F}}{(\\pi,\\mathbf{F})})} = \\pi \\rho_{b}{(\\mathbf{S},y)} + \\cos{(\\operatorname{J_{\\varepsilon}}^{\\mathbf{F}}{(\\pi,\\mathbf{F})})}", "derivation": "\\rho_{b}{(\\mathbf{S},y)} = y^{\\mathbf{S}} and \\mathbf{s}{(\\mathbf{S},y)} = y^{\\mathbf{S}} and \\pi \\mathbf{s}{(\\mathbf{S},y)} = \\pi y^{\\mathbf{S}} and \\pi \\mathbf{s}{(\\mathbf{S},y)} = \\pi \\rho_{b}{(\\mathbf{S},y)} and \\pi y^{\\mathbf{S}} = \\pi \\rho_{b}{(\\mathbf{S},y)} and \\pi y^{\\mathbf{S}} + \\cos{(\\operatorname{J_{\\varepsilon}}^{\\mathbf{F}}{(\\pi,\\mathbf{F})})} = \\pi \\rho_{b}{(\\mathbf{S},y)} + \\cos{(\\operatorname{J_{\\varepsilon}}^{\\mathbf{F}}{(\\pi,\\mathbf{F})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))))"], [["add", 5, "cos(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), cos(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))), cos(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given J{(\\hat{X})} = e^{\\hat{X}}, then obtain 0 = - \\int J^{\\hat{X}}{(\\hat{X})} d\\hat{X} + \\int (e^{\\hat{X}})^{\\hat{X}} d\\hat{X}", "derivation": "J{(\\hat{X})} = e^{\\hat{X}} and J^{\\hat{X}}{(\\hat{X})} = (e^{\\hat{X}})^{\\hat{X}} and \\int J^{\\hat{X}}{(\\hat{X})} d\\hat{X} = \\int (e^{\\hat{X}})^{\\hat{X}} d\\hat{X} and 0 = - \\int J^{\\hat{X}}{(\\hat{X})} d\\hat{X} + \\int (e^{\\hat{X}})^{\\hat{X}} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\hat{X}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Pow(Function('J')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 3, "Integral(Pow(Function('J')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Pow(Function('J')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Integral(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi and \\chi{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi, then obtain \\frac{(C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\mathbb{I}^{\\Psi}{(\\Psi)})^{\\Psi}}{\\Psi} = \\frac{(C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\chi^{\\Psi}{(\\Psi)})^{\\Psi}}{\\Psi}", "derivation": "\\mathbb{I}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi and \\mathbb{I}^{\\Psi}{(\\Psi)} = (\\int \\log{(\\Psi)} d\\Psi)^{\\Psi} and (\\mathbb{I}^{\\Psi}{(\\Psi)})^{\\Psi} = ((\\int \\log{(\\Psi)} d\\Psi)^{\\Psi})^{\\Psi} and \\chi{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi and (\\mathbb{I}^{\\Psi}{(\\Psi)})^{\\Psi} = (\\chi^{\\Psi}{(\\Psi)})^{\\Psi} and (C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\mathbb{I}^{\\Psi}{(\\Psi)})^{\\Psi} = (C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\chi^{\\Psi}{(\\Psi)})^{\\Psi} and \\frac{(C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\mathbb{I}^{\\Psi}{(\\Psi)})^{\\Psi}}{\\Psi} = \\frac{(C_{1} + \\Psi \\log{(\\Psi)} - \\Psi) (\\chi^{\\Psi}{(\\Psi)})^{\\Psi}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\Psi', commutative=True)), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Function('\\\\chi')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["times", 5, "Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(Pow(Function('\\\\chi')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))))"], [["divide", 6, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(Pow(Function('\\\\chi')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given B{(M_{E})} = \\log{(M_{E})} and \\operatorname{f_{E}}{(M_{E})} = \\frac{d}{d M_{E}} \\frac{B{(M_{E})}}{M_{E}}, then obtain 0 = - \\operatorname{f_{E}}{(M_{E})} + \\frac{d}{d M_{E}} \\frac{\\log{(M_{E})}}{M_{E}}", "derivation": "B{(M_{E})} = \\log{(M_{E})} and \\frac{B{(M_{E})}}{M_{E}} = \\frac{\\log{(M_{E})}}{M_{E}} and \\frac{d}{d M_{E}} \\frac{B{(M_{E})}}{M_{E}} = \\frac{d}{d M_{E}} \\frac{\\log{(M_{E})}}{M_{E}} and \\operatorname{f_{E}}{(M_{E})} = \\frac{d}{d M_{E}} \\frac{B{(M_{E})}}{M_{E}} and \\operatorname{f_{E}}{(M_{E})} = \\frac{d}{d M_{E}} \\frac{\\log{(M_{E})}}{M_{E}} and 0 = - \\operatorname{f_{E}}{(M_{E})} + \\frac{d}{d M_{E}} \\frac{\\log{(M_{E})}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["divide", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('B')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('B')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('M_E', commutative=True)), Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('B')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('f_E')(Symbol('M_E', commutative=True)), Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["minus", 5, "Function('f_E')(Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_E')(Symbol('M_E', commutative=True))), Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(n_{1},x)} = \\frac{\\partial}{\\partial n_{1}} n_{1} x, then obtain \\log{(\\operatorname{f^{*}}{(n_{1},x)})} \\frac{\\partial^{2}}{\\partial n_{1}^{2}} n_{1} x = \\log{(\\frac{\\partial}{\\partial n_{1}} n_{1} x)} \\frac{\\partial^{2}}{\\partial n_{1}^{2}} n_{1} x", "derivation": "\\operatorname{f^{*}}{(n_{1},x)} = \\frac{\\partial}{\\partial n_{1}} n_{1} x and \\frac{\\partial}{\\partial n_{1}} \\operatorname{f^{*}}{(n_{1},x)} = \\frac{\\partial^{2}}{\\partial n_{1}^{2}} n_{1} x and \\log{(\\operatorname{f^{*}}{(n_{1},x)})} = \\log{(\\frac{\\partial}{\\partial n_{1}} n_{1} x)} and \\log{(\\operatorname{f^{*}}{(n_{1},x)})} \\frac{\\partial}{\\partial n_{1}} \\operatorname{f^{*}}{(n_{1},x)} = \\log{(\\frac{\\partial}{\\partial n_{1}} n_{1} x)} \\frac{\\partial}{\\partial n_{1}} \\operatorname{f^{*}}{(n_{1},x)} and \\log{(\\operatorname{f^{*}}{(n_{1},x)})} \\frac{\\partial^{2}}{\\partial n_{1}^{2}} n_{1} x = \\log{(\\frac{\\partial}{\\partial n_{1}} n_{1} x)} \\frac{\\partial^{2}}{\\partial n_{1}^{2}} n_{1} x", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))))"], [["log", 1], "Equality(log(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))), log(Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))"], "Equality(Mul(log(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))), Derivative(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(log(Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Derivative(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(log(Function('f^*')(Symbol('n_1', commutative=True), Symbol('x', commutative=True))), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2)))), Mul(log(Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\Omega{(H,r_{0})} = \\int (H - r_{0}) dH, then derive \\Omega{(H,r_{0})} = \\frac{H^{2}}{2} - H r_{0} + n_{1}, then obtain \\int \\Omega{(H,r_{0})} dr_{0} = \\int (\\frac{H^{2}}{2} - H r_{0} + n_{1}) dr_{0}", "derivation": "\\Omega{(H,r_{0})} = \\int (H - r_{0}) dH and \\int \\Omega{(H,r_{0})} dr_{0} = \\iint (H - r_{0}) dH dr_{0} and \\Omega{(H,r_{0})} = \\frac{H^{2}}{2} - H r_{0} + n_{1} and \\int (\\frac{H^{2}}{2} - H r_{0} + n_{1}) dr_{0} = \\iint (H - r_{0}) dH dr_{0} and \\int \\Omega{(H,r_{0})} dr_{0} = \\int (\\frac{H^{2}}{2} - H r_{0} + n_{1}) dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\Omega')(Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('\\\\Omega')(Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('H', commutative=True), Symbol('r_0', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\hat{\\mathbf{r}})} = \\log{(\\sin{(\\hat{\\mathbf{r}})})} and \\delta{(V)} = \\cos{(V)}, then obtain \\delta{(V)} \\log{(\\sin{(\\hat{\\mathbf{r}})})} = \\log{(\\sin{(\\hat{\\mathbf{r}})})} \\cos{(V)}", "derivation": "\\pi{(\\hat{\\mathbf{r}})} = \\log{(\\sin{(\\hat{\\mathbf{r}})})} and \\delta{(V)} = \\cos{(V)} and \\delta{(V)} \\pi{(\\hat{\\mathbf{r}})} = \\pi{(\\hat{\\mathbf{r}})} \\cos{(V)} and \\delta{(V)} \\log{(\\sin{(\\hat{\\mathbf{r}})})} = \\log{(\\sin{(\\hat{\\mathbf{r}})})} \\cos{(V)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["times", 2, "Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('V', commutative=True)), Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\delta')(Symbol('V', commutative=True)), log(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(log(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), cos(Symbol('V', commutative=True))))"]]}, {"prompt": "Given T{(\\mathbf{J})} = - \\mathbf{J} and f{(\\mathbf{J})} = - \\mathbf{J}, then derive \\int e^{T{(\\mathbf{J})}} d\\mathbf{J} = x^\\prime - e^{- \\mathbf{J}}, then obtain - \\frac{\\int e^{- \\mathbf{J}} d\\mathbf{J}}{\\mathbf{J}} = - \\frac{x^\\prime - e^{T{(\\mathbf{J})}}}{\\mathbf{J}}", "derivation": "T{(\\mathbf{J})} = - \\mathbf{J} and e^{T{(\\mathbf{J})}} = e^{- \\mathbf{J}} and \\int e^{T{(\\mathbf{J})}} d\\mathbf{J} = \\int e^{- \\mathbf{J}} d\\mathbf{J} and f{(\\mathbf{J})} = - \\mathbf{J} and \\int e^{T{(\\mathbf{J})}} d\\mathbf{J} = x^\\prime - e^{- \\mathbf{J}} and \\int e^{- \\mathbf{J}} d\\mathbf{J} = x^\\prime - e^{- \\mathbf{J}} and \\int e^{- \\mathbf{J}} d\\mathbf{J} = x^\\prime - e^{T{(\\mathbf{J})}} and \\frac{\\int e^{- \\mathbf{J}} d\\mathbf{J}}{f{(\\mathbf{J})}} = \\frac{x^\\prime - e^{T{(\\mathbf{J})}}}{f{(\\mathbf{J})}} and - \\frac{\\int e^{- \\mathbf{J}} d\\mathbf{J}}{\\mathbf{J}} = - \\frac{x^\\prime - e^{T{(\\mathbf{J})}}}{\\mathbf{J}}", "srepr_derivation": [["renaming_premise", "Equality(Function('T')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integral(exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integral(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True))))))"], [["divide", 7, "Function('f')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Function('f')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True))))), Pow(Function('f')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Integral(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), exp(Function('T')(Symbol('\\\\mathbf{J}', commutative=True)))))))"]]}, {"prompt": "Given \\mathbb{I}{(T)} = \\sin{(T)} and J{(i)} = \\log{(i)}, then obtain T - \\sin{(T)} + \\frac{- T + \\mathbb{I}{(T)}}{T \\log{(i)}} = T - \\sin{(T)} + \\frac{- T + \\sin{(T)}}{T \\log{(i)}}", "derivation": "\\mathbb{I}{(T)} = \\sin{(T)} and - T + \\mathbb{I}{(T)} = - T + \\sin{(T)} and J{(i)} = \\log{(i)} and \\frac{- T + \\mathbb{I}{(T)}}{T} = \\frac{- T + \\sin{(T)}}{T} and \\frac{- T + \\mathbb{I}{(T)}}{T J{(i)}} = \\frac{- T + \\sin{(T)}}{T J{(i)}} and \\frac{- T + \\mathbb{I}{(T)}}{T \\log{(i)}} = \\frac{- T + \\sin{(T)}}{T \\log{(i)}} and T - \\sin{(T)} + \\frac{- T + \\mathbb{I}{(T)}}{T \\log{(i)}} = T - \\sin{(T)} + \\frac{- T + \\sin{(T)}}{T \\log{(i)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbb{I}')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))))"], ["get_premise", "Equality(Function('J')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["divide", 2, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbb{I}')(Symbol('T', commutative=True)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))))"], [["divide", 4, "Function('J')(Symbol('i', commutative=True))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbb{I}')(Symbol('T', commutative=True))), Pow(Function('J')(Symbol('i', commutative=True)), Integer(-1))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))), Pow(Function('J')(Symbol('i', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbb{I}')(Symbol('T', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1))))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), sin(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbb{I}')(Symbol('T', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1)))), Add(Symbol('T', commutative=True), Mul(Integer(-1), sin(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given G{(\\mu)} = \\log{(\\mu)} and \\rho_{f}{(\\mu)} = \\log{(\\mu)}, then derive \\frac{d}{d \\mu} \\rho_{f}{(\\mu)} = \\frac{1}{\\mu}, then obtain \\frac{d}{d \\mu} G{(\\mu)} = \\frac{1}{\\mu}", "derivation": "G{(\\mu)} = \\log{(\\mu)} and \\frac{d}{d \\mu} G{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)} and \\rho_{f}{(\\mu)} = \\log{(\\mu)} and \\frac{d}{d \\mu} \\rho_{f}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)} and \\frac{d}{d \\mu} \\rho_{f}{(\\mu)} = \\frac{1}{\\mu} and \\frac{d}{d \\mu} \\log{(\\mu)} = \\frac{1}{\\mu} and \\frac{d}{d \\mu} G{(\\mu)} = \\frac{1}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Derivative(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"]]}, {"prompt": "Given k{(q,y)} = q + y, then obtain (\\iint (q + y) dy dy + \\iint k{(q,y)} dy dy + 1)^{y} = (2 \\iint (q + y) dy dy + 1)^{y}", "derivation": "k{(q,y)} = q + y and \\int k{(q,y)} dy = \\int (q + y) dy and \\iint k{(q,y)} dy dy = \\iint (q + y) dy dy and \\iint (q + y) dy dy + \\iint k{(q,y)} dy dy = 2 \\iint (q + y) dy dy and \\iint (q + y) dy dy + \\iint k{(q,y)} dy dy + 1 = 2 \\iint (q + y) dy dy + 1 and (\\iint (q + y) dy dy + \\iint k{(q,y)} dy dy + 1)^{y} = (2 \\iint (q + y) dy dy + 1)^{y}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Add(Symbol('q', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 3, "Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(Integer(2), Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["add", 4, 1], "Equality(Add(Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(1)), Add(Mul(Integer(2), Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Integer(1)))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Function('k')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(1)), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(2), Integral(Add(Symbol('q', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Integer(1)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given A{(t_{2})} = \\sin{(t_{2})}, then derive t_{2} + \\frac{d}{d t_{2}} A{(t_{2})} - 1 = t_{2} + \\cos{(t_{2})} - 1, then obtain t_{2} + \\frac{d}{d t_{2}} \\sin{(t_{2})} - 1 = t_{2} + \\cos{(t_{2})} - 1", "derivation": "A{(t_{2})} = \\sin{(t_{2})} and \\frac{d}{d t_{2}} A{(t_{2})} = \\frac{d}{d t_{2}} \\sin{(t_{2})} and t_{2} + \\frac{d}{d t_{2}} A{(t_{2})} = t_{2} + \\frac{d}{d t_{2}} \\sin{(t_{2})} and t_{2} + \\frac{d}{d t_{2}} A{(t_{2})} - 1 = t_{2} + \\frac{d}{d t_{2}} \\sin{(t_{2})} - 1 and t_{2} + \\frac{d}{d t_{2}} A{(t_{2})} - 1 = t_{2} + \\cos{(t_{2})} - 1 and t_{2} + \\frac{d}{d t_{2}} \\sin{(t_{2})} - 1 = t_{2} + \\cos{(t_{2})} - 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["add", 2, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Derivative(Function('A')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Symbol('t_2', commutative=True), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["minus", 3, 1], "Equality(Add(Symbol('t_2', commutative=True), Derivative(Function('A')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('t_2', commutative=True), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('t_2', commutative=True), Derivative(Function('A')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('t_2', commutative=True), cos(Symbol('t_2', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('t_2', commutative=True), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('t_2', commutative=True), cos(Symbol('t_2', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\chi{(P_{e})} = \\sin{(\\sin{(P_{e})})}, then obtain e^{e^{\\int \\chi{(P_{e})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e}} = e^{e^{\\int \\sin{(\\sin{(P_{e})})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e}}", "derivation": "\\chi{(P_{e})} = \\sin{(\\sin{(P_{e})})} and \\int \\chi{(P_{e})} dP_{e} = \\int \\sin{(\\sin{(P_{e})})} dP_{e} and e^{\\int \\chi{(P_{e})} dP_{e}} = e^{\\int \\sin{(\\sin{(P_{e})})} dP_{e}} and e^{\\int \\chi{(P_{e})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e} = e^{\\int \\sin{(\\sin{(P_{e})})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e} and e^{e^{\\int \\chi{(P_{e})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e}} = e^{e^{\\int \\sin{(\\sin{(P_{e})})} dP_{e}} + \\int \\sin{(\\sin{(P_{e})})} dP_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('P_e', commutative=True)), sin(sin(Symbol('P_e', commutative=True))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), exp(Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["add", 3, "Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))"], "Equality(Add(exp(Integral(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Add(exp(Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["exp", 4], "Equality(exp(Add(exp(Integral(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))), exp(Add(exp(Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Integral(sin(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))))"]]}, {"prompt": "Given r{(\\lambda,\\mu)} = \\frac{\\mu}{\\lambda}, then derive \\frac{\\partial}{\\partial \\lambda} r{(\\lambda,\\mu)} - \\frac{1}{\\lambda^{2}} = - \\frac{\\mu}{\\lambda^{2}} - \\frac{1}{\\lambda^{2}}, then obtain \\int (\\frac{\\partial}{\\partial \\lambda} \\frac{\\mu}{\\lambda} - \\frac{1}{\\lambda^{2}}) d\\mu = \\int (- \\frac{\\mu}{\\lambda^{2}} - \\frac{1}{\\lambda^{2}}) d\\mu", "derivation": "r{(\\lambda,\\mu)} = \\frac{\\mu}{\\lambda} and r{(\\lambda,\\mu)} + \\frac{1}{\\lambda} = \\frac{\\mu}{\\lambda} + \\frac{1}{\\lambda} and \\frac{\\partial}{\\partial \\lambda} (r{(\\lambda,\\mu)} + \\frac{1}{\\lambda}) = \\frac{\\partial}{\\partial \\lambda} (\\frac{\\mu}{\\lambda} + \\frac{1}{\\lambda}) and \\frac{\\partial}{\\partial \\lambda} r{(\\lambda,\\mu)} - \\frac{1}{\\lambda^{2}} = - \\frac{\\mu}{\\lambda^{2}} - \\frac{1}{\\lambda^{2}} and \\frac{\\partial}{\\partial \\lambda} \\frac{\\mu}{\\lambda} - \\frac{1}{\\lambda^{2}} = - \\frac{\\mu}{\\lambda^{2}} - \\frac{1}{\\lambda^{2}} and \\int (\\frac{\\partial}{\\partial \\lambda} \\frac{\\mu}{\\lambda} - \\frac{1}{\\lambda^{2}}) d\\mu = \\int (- \\frac{\\mu}{\\lambda^{2}} - \\frac{1}{\\lambda^{2}}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))"], "Equality(Add(Function('r')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('r')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))))"], [["integrate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Derivative(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given v{(S)} = e^{S}, then derive S \\frac{d}{d S} v{(S)} = S e^{S}, then obtain S \\frac{d}{d S} e^{S} = S e^{S}", "derivation": "v{(S)} = e^{S} and \\frac{d}{d S} v{(S)} = \\frac{d}{d S} e^{S} and S \\frac{d}{d S} v{(S)} = S \\frac{d}{d S} e^{S} and S \\frac{d}{d S} v{(S)} = S e^{S} and S \\frac{d}{d S} e^{S} = S e^{S}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 2, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Derivative(Function('v')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Symbol('S', commutative=True), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('S', commutative=True), Derivative(Function('v')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Symbol('S', commutative=True), exp(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('S', commutative=True), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Symbol('S', commutative=True), exp(Symbol('S', commutative=True))))"]]}, {"prompt": "Given L{(\\psi,v)} = \\cos{(\\psi + v)} and u{(\\psi,v)} = \\cos{(\\psi + v)}, then obtain (L{(\\psi,v)} + u{(\\psi,v)})^{2} = 2 (L{(\\psi,v)} + u{(\\psi,v)}) u{(\\psi,v)}", "derivation": "L{(\\psi,v)} = \\cos{(\\psi + v)} and L{(\\psi,v)} + \\cos{(\\psi + v)} = 2 \\cos{(\\psi + v)} and (L{(\\psi,v)} + \\cos{(\\psi + v)})^{2} = 2 (L{(\\psi,v)} + \\cos{(\\psi + v)}) \\cos{(\\psi + v)} and u{(\\psi,v)} = \\cos{(\\psi + v)} and (L{(\\psi,v)} + u{(\\psi,v)})^{2} = 2 (L{(\\psi,v)} + u{(\\psi,v)}) u{(\\psi,v)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))))"], [["add", 1, "cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))))"], [["times", 2, "Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))))"], "Equality(Pow(Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))), Integer(2)), Mul(Integer(2), Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), cos(Add(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), Function('u')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))), Integer(2)), Mul(Integer(2), Add(Function('L')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True)), Function('u')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))), Function('u')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(L)} = \\sin{(L)}, then derive 2 \\frac{d}{d L} \\tilde{g}{(L)} = \\cos{(L)} + \\frac{d}{d L} \\tilde{g}{(L)}, then obtain \\cos{(L)} + 3 \\frac{d}{d L} \\sin{(L)} = 3 \\cos{(L)} + \\frac{d}{d L} \\sin{(L)}", "derivation": "\\tilde{g}{(L)} = \\sin{(L)} and 2 \\tilde{g}{(L)} = \\tilde{g}{(L)} + \\sin{(L)} and \\frac{d}{d L} 2 \\tilde{g}{(L)} = \\frac{d}{d L} (\\tilde{g}{(L)} + \\sin{(L)}) and 2 \\frac{d}{d L} \\tilde{g}{(L)} = \\cos{(L)} + \\frac{d}{d L} \\tilde{g}{(L)} and 2 \\frac{d}{d L} \\sin{(L)} = \\cos{(L)} + \\frac{d}{d L} \\sin{(L)} and \\cos{(L)} + 3 \\frac{d}{d L} \\tilde{g}{(L)} = 2 \\cos{(L)} + 2 \\frac{d}{d L} \\tilde{g}{(L)} and \\cos{(L)} + 3 \\frac{d}{d L} \\sin{(L)} = 2 \\cos{(L)} + 2 \\frac{d}{d L} \\sin{(L)} and \\cos{(L)} + 3 \\frac{d}{d L} \\sin{(L)} = 3 \\cos{(L)} + \\frac{d}{d L} \\sin{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('L', commutative=True))), Add(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(cos(Symbol('L', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(cos(Symbol('L', commutative=True)), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["add", 4, "Add(cos(Symbol('L', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], "Equality(Add(cos(Symbol('L', commutative=True)), Mul(Integer(3), Derivative(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(Mul(Integer(2), cos(Symbol('L', commutative=True))), Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(cos(Symbol('L', commutative=True)), Mul(Integer(3), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(Mul(Integer(2), cos(Symbol('L', commutative=True))), Mul(Integer(2), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(cos(Symbol('L', commutative=True)), Mul(Integer(3), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(Mul(Integer(3), cos(Symbol('L', commutative=True))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbb{I},Z)} = \\frac{\\mathbb{I}}{Z}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\dot{\\mathbf{r}}{(\\mathbb{I},Z)} = \\frac{1}{Z}, then obtain 1 + \\frac{1}{Z} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Z} + 1", "derivation": "\\dot{\\mathbf{r}}{(\\mathbb{I},Z)} = \\frac{\\mathbb{I}}{Z} and \\frac{\\partial}{\\partial \\mathbb{I}} \\dot{\\mathbf{r}}{(\\mathbb{I},Z)} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Z} and \\frac{\\partial}{\\partial \\mathbb{I}} \\dot{\\mathbf{r}}{(\\mathbb{I},Z)} + 1 = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Z} + 1 and \\frac{\\partial}{\\partial \\mathbb{I}} \\dot{\\mathbf{r}}{(\\mathbb{I},Z)} = \\frac{1}{Z} and 1 + \\frac{1}{Z} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Z} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Pow(Symbol('Z', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Integer(1), Pow(Symbol('Z', commutative=True), Integer(-1))), Add(Derivative(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + v_{t}, then obtain - 2 \\hat{\\mathbf{x}} - 3 v_{t} + 2 \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = - v_{t}", "derivation": "\\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + v_{t} and - \\hat{\\mathbf{x}} - v_{t} + \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = 0 and - 2 \\hat{\\mathbf{x}} - v_{t} + \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} and - 2 \\hat{\\mathbf{x}} - 2 v_{t} + 2 \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = 0 and - 2 \\hat{\\mathbf{x}} - 3 v_{t} + 2 \\operatorname{v_{2}}{(v_{t},\\hat{\\mathbf{x}})} = - v_{t}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('v_t', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_2')(Symbol('v_t', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_2')(Symbol('v_t', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('v_t', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(0))"], [["add", 4, "Mul(Integer(-1), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('v_t', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('v_t', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Integer(-1), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(b)} = \\sin{(e^{b})}, then derive - e^{b} = (b (e^{b} \\cos{(e^{b})} - \\frac{d}{d b} \\mathbf{f}{(b)}) - \\mathbf{f}{(b)} + \\sin{(e^{b})}) \\cos{(b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}))} - e^{b}, then obtain - e^{b} = b (e^{b} \\cos{(e^{b})} - \\frac{d}{d b} \\mathbf{f}{(b)}) - e^{b}", "derivation": "\\mathbf{f}{(b)} = \\sin{(e^{b})} and 0 = - \\mathbf{f}{(b)} + \\sin{(e^{b})} and 0 = b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}) and 0 = \\sin{(b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}))} and \\frac{d}{d b} 0 = \\frac{d}{d b} \\sin{(b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}))} and - e^{b} + \\frac{d}{d b} 0 = - e^{b} + \\frac{d}{d b} \\sin{(b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}))} and - e^{b} = (b (e^{b} \\cos{(e^{b})} - \\frac{d}{d b} \\mathbf{f}{(b)}) - \\mathbf{f}{(b)} + \\sin{(e^{b})}) \\cos{(b (- \\mathbf{f}{(b)} + \\sin{(e^{b})}))} - e^{b} and - e^{b} = b (e^{b} \\cos{(e^{b})} - \\frac{d}{d b} \\mathbf{f}{(b)}) - e^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('b', commutative=True)), sin(exp(Symbol('b', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{f}')(Symbol('b', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True)))))"], [["times", 2, "Symbol('b', commutative=True)"], "Equality(Integer(0), Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True))))))"], [["sin", 3], "Equality(Integer(0), sin(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True)))))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True)))))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["minus", 5, "exp(Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('b', commutative=True))), Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('b', commutative=True))), Derivative(sin(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True)))))), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(-1), exp(Symbol('b', commutative=True))), Add(Mul(Add(Mul(Symbol('b', commutative=True), Add(Mul(exp(Symbol('b', commutative=True)), cos(exp(Symbol('b', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{f}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True)))), cos(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('b', commutative=True))), sin(exp(Symbol('b', commutative=True))))))), Mul(Integer(-1), exp(Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Mul(Integer(-1), exp(Symbol('b', commutative=True))), Add(Mul(Symbol('b', commutative=True), Add(Mul(exp(Symbol('b', commutative=True)), cos(exp(Symbol('b', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{f}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))), Mul(Integer(-1), exp(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(p)} = e^{p}, then derive \\int \\hat{x}_0{(p)} dp = n + e^{p}, then derive \\mathbf{f} + e^{p} = n + e^{p}, then obtain - \\mathbf{B}^{\\psi} \\frac{d}{d p} \\int e^{p} dp = - \\mathbf{B}^{\\psi} \\frac{\\partial}{\\partial p} (\\mathbf{f} + e^{p})", "derivation": "\\hat{x}_0{(p)} = e^{p} and \\int \\hat{x}_0{(p)} dp = \\int e^{p} dp and \\int \\hat{x}_0{(p)} dp = n + e^{p} and \\int e^{p} dp = n + e^{p} and \\frac{d}{d p} \\int \\hat{x}_0{(p)} dp = \\frac{\\partial}{\\partial p} (n + e^{p}) and \\mathbf{f} + e^{p} = n + e^{p} and - \\mathbf{B}^{\\psi} \\frac{d}{d p} \\int \\hat{x}_0{(p)} dp = - \\mathbf{B}^{\\psi} \\frac{\\partial}{\\partial p} (n + e^{p}) and - \\mathbf{B}^{\\psi} \\frac{d}{d p} \\int \\hat{x}_0{(p)} dp = - \\mathbf{B}^{\\psi} \\frac{\\partial}{\\partial p} (\\mathbf{f} + e^{p}) and - \\mathbf{B}^{\\psi} \\frac{d}{d p} \\int e^{p} dp = - \\mathbf{B}^{\\psi} \\frac{\\partial}{\\partial p} (\\mathbf{f} + e^{p})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('n', commutative=True), exp(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('n', commutative=True), exp(Symbol('p', commutative=True))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('p', commutative=True))), Add(Symbol('n', commutative=True), exp(Symbol('p', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Integral(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Add(Symbol('n', commutative=True), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Integral(Function('\\\\hat{x}_0')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(\\nabla,A_{1},y)} = A_{1} y + \\nabla, then obtain \\int \\frac{1}{4 A_{1}^{2} y^{2} - 4 A_{1} y V{(\\nabla,A_{1},y)} + V^{2}{(\\nabla,A_{1},y)}} dA_{1} = \\int \\frac{1}{A_{1}^{2} y^{2} - 2 A_{1} \\nabla y + \\nabla^{2}} dA_{1}", "derivation": "V{(\\nabla,A_{1},y)} = A_{1} y + \\nabla and - A_{1} y + V{(\\nabla,A_{1},y)} = \\nabla and - 2 A_{1} y + V{(\\nabla,A_{1},y)} = - A_{1} y + \\nabla and \\frac{1}{(- 2 A_{1} y + V{(\\nabla,A_{1},y)})^{2}} = \\frac{1}{(- A_{1} y + \\nabla)^{2}} and \\frac{1}{4 A_{1}^{2} y^{2} - 4 A_{1} y V{(\\nabla,A_{1},y)} + V^{2}{(\\nabla,A_{1},y)}} = \\frac{1}{A_{1}^{2} y^{2} - 2 A_{1} \\nabla y + \\nabla^{2}} and \\int \\frac{1}{4 A_{1}^{2} y^{2} - 4 A_{1} y V{(\\nabla,A_{1},y)} + V^{2}{(\\nabla,A_{1},y)}} dA_{1} = \\int \\frac{1}{A_{1}^{2} y^{2} - 2 A_{1} \\nabla y + \\nabla^{2}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Add(Mul(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Mul(Symbol('A_1', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Symbol('\\\\nabla', commutative=True))"], [["minus", 2, "Mul(Symbol('A_1', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\nabla', commutative=True)), Integer(-2)))"], [["expand", 4], "Equality(Pow(Add(Mul(Integer(4), Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Integer(-1), Integer(4), Symbol('A_1', commutative=True), Symbol('y', commutative=True), Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Pow(Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Integer(2))), Integer(-1)), Pow(Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Integer(-1)))"], [["integrate", 5, "Symbol('A_1', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(4), Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Integer(-1), Integer(4), Symbol('A_1', commutative=True), Symbol('y', commutative=True), Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Pow(Function('V')(Symbol('\\\\nabla', commutative=True), Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Integer(2))), Integer(-1)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Integer(-1)), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given E{(\\varphi^*,\\Omega)} = \\frac{\\Omega}{\\varphi^*}, then obtain (- \\varphi^* + E{(\\varphi^*,\\Omega)} + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega)^{\\Omega} = (\\frac{\\Omega}{\\varphi^*} - \\varphi^* + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega)^{\\Omega}", "derivation": "E{(\\varphi^*,\\Omega)} = \\frac{\\Omega}{\\varphi^*} and - \\varphi^* + E{(\\varphi^*,\\Omega)} = \\frac{\\Omega}{\\varphi^*} - \\varphi^* and \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega = \\int (\\frac{\\Omega}{\\varphi^*} - \\varphi^*) d\\Omega and - \\varphi^* + E{(\\varphi^*,\\Omega)} + \\int (\\frac{\\Omega}{\\varphi^*} - \\varphi^*) d\\Omega = \\frac{\\Omega}{\\varphi^*} - \\varphi^* + \\int (\\frac{\\Omega}{\\varphi^*} - \\varphi^*) d\\Omega and - \\varphi^* + E{(\\varphi^*,\\Omega)} + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega = \\frac{\\Omega}{\\varphi^*} - \\varphi^* + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega and (- \\varphi^* + E{(\\varphi^*,\\Omega)} + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega)^{\\Omega} = (\\frac{\\Omega}{\\varphi^*} - \\varphi^* + \\int (- \\varphi^* + E{(\\varphi^*,\\Omega)}) d\\Omega)^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('E')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\hat{H},s)} = \\int \\frac{\\hat{H}}{s} d\\hat{H}, then obtain A_{y} + \\frac{\\int \\frac{\\hat{H}^{2}}{s} ds}{2} + \\frac{\\int 4 \\phi{(\\hat{H},s)} ds}{2} = \\int 3 \\int \\frac{\\hat{H}}{s} d\\hat{H} ds", "derivation": "\\phi{(\\hat{H},s)} = \\int \\frac{\\hat{H}}{s} d\\hat{H} and \\phi{(\\hat{H},s)} + \\int \\frac{\\hat{H}}{s} d\\hat{H} = 2 \\int \\frac{\\hat{H}}{s} d\\hat{H} and \\phi{(\\hat{H},s)} + 2 \\int \\frac{\\hat{H}}{s} d\\hat{H} = 3 \\int \\frac{\\hat{H}}{s} d\\hat{H} and 2 \\phi{(\\hat{H},s)} + \\int \\frac{\\hat{H}}{s} d\\hat{H} = 3 \\int \\frac{\\hat{H}}{s} d\\hat{H} and \\int (2 \\phi{(\\hat{H},s)} + \\int \\frac{\\hat{H}}{s} d\\hat{H}) ds = \\int 3 \\int \\frac{\\hat{H}}{s} d\\hat{H} ds and A_{y} + \\frac{\\int \\frac{\\hat{H}^{2}}{s} ds}{2} + \\frac{\\int 4 \\phi{(\\hat{H},s)} ds}{2} = \\int 3 \\int \\frac{\\hat{H}}{s} d\\hat{H} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 1, "Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["add", 2, "Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Mul(Integer(3), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True))), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(3), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True))), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Integer(3), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Add(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Integer(4), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))), Integral(Mul(Integer(3), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(A_{x},L_{\\varepsilon})} = A_{x}^{L_{\\varepsilon}}, then obtain (\\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x}) \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x} = (\\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int A_{x}^{L_{\\varepsilon}} dA_{x}) \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x}", "derivation": "\\mu_{0}{(A_{x},L_{\\varepsilon})} = A_{x}^{L_{\\varepsilon}} and \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x} = \\int A_{x}^{L_{\\varepsilon}} dA_{x} and \\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x} = \\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int A_{x}^{L_{\\varepsilon}} dA_{x} and (\\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x}) \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x} = (\\mu_{0}{(A_{x},L_{\\varepsilon})} + \\int A_{x}^{L_{\\varepsilon}} dA_{x}) \\int \\mu_{0}{(A_{x},L_{\\varepsilon})} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["add", 2, "Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Pow(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["times", 3, "Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Mul(Add(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Pow(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Integral(Function('\\\\mu_0')(Symbol('A_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(n)} = \\sin{(n)}, then obtain (\\frac{\\hat{p}_0^{n}{(n)}}{n})^{n} = (\\frac{\\sin^{n}{(n)}}{n})^{n}", "derivation": "\\hat{p}_0{(n)} = \\sin{(n)} and \\hat{p}_0^{n}{(n)} = \\sin^{n}{(n)} and \\frac{\\hat{p}_0^{n}{(n)}}{n} = \\frac{\\sin^{n}{(n)}}{n} and (\\frac{\\hat{p}_0^{n}{(n)}}{n})^{n} = (\\frac{\\sin^{n}{(n)}}{n})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["divide", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(c_{0},g)} = \\frac{\\partial}{\\partial c_{0}} c_{0}^{g}, then derive (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1)^{2} = (1 + \\frac{c_{0}^{g} g}{c_{0}}) (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1), then obtain (\\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1)^{2} = (1 + \\frac{c_{0}^{g} g}{c_{0}}) (\\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1)", "derivation": "\\operatorname{y^{\\prime}}{(c_{0},g)} = \\frac{\\partial}{\\partial c_{0}} c_{0}^{g} and \\operatorname{y^{\\prime}}{(c_{0},g)} + 1 = \\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1 and (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1)^{2} = (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1) (\\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1) and (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1)^{2} = (1 + \\frac{c_{0}^{g} g}{c_{0}}) (\\operatorname{y^{\\prime}}{(c_{0},g)} + 1) and (\\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1)^{2} = (1 + \\frac{c_{0}^{g} g}{c_{0}}) (\\frac{\\partial}{\\partial c_{0}} c_{0}^{g} + 1)", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Derivative(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1)), Add(Derivative(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(1)))"], [["times", 2, "Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1))"], "Equality(Pow(Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1)), Integer(2)), Mul(Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1)), Add(Derivative(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Pow(Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1)), Integer(2)), Mul(Add(Integer(1), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Add(Derivative(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(1)), Integer(2)), Mul(Add(Integer(1), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Derivative(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(1))))"]]}, {"prompt": "Given q{(u,\\rho_f)} = \\rho_f u, then obtain \\int \\frac{\\partial}{\\partial u} \\frac{u (- q{(u,\\rho_f)})^{u}}{q{(u,\\rho_f)}} du = \\int \\frac{\\partial}{\\partial u} \\frac{u (- \\rho_f u)^{u}}{q{(u,\\rho_f)}} du", "derivation": "q{(u,\\rho_f)} = \\rho_f u and - q{(u,\\rho_f)} = - \\rho_f u and (- q{(u,\\rho_f)})^{u} = (- \\rho_f u)^{u} and \\frac{u (- q{(u,\\rho_f)})^{u}}{q{(u,\\rho_f)}} = \\frac{u (- \\rho_f u)^{u}}{q{(u,\\rho_f)}} and \\frac{\\partial}{\\partial u} \\frac{u (- q{(u,\\rho_f)})^{u}}{q{(u,\\rho_f)}} = \\frac{\\partial}{\\partial u} \\frac{u (- \\rho_f u)^{u}}{q{(u,\\rho_f)}} and \\int \\frac{\\partial}{\\partial u} \\frac{u (- q{(u,\\rho_f)})^{u}}{q{(u,\\rho_f)}} du = \\int \\frac{\\partial}{\\partial u} \\frac{u (- \\rho_f u)^{u}}{q{(u,\\rho_f)}} du", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('u', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Symbol('u', commutative=True), Pow(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Function('q')(Symbol('u', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\nabla,t)} = \\frac{\\nabla}{t} and q{(\\nabla,t)} = 2 \\sigma_{x}{(\\nabla,t)}, then obtain \\int (2 \\sigma_{x}{(\\nabla,t)})^{\\nabla} dt + (\\int q{(\\nabla,t)} d\\nabla)^{2} = \\int (2 \\sigma_{x}{(\\nabla,t)})^{\\nabla} dt + (\\int \\frac{2 \\nabla}{t} d\\nabla)^{2}", "derivation": "\\sigma_{x}{(\\nabla,t)} = \\frac{\\nabla}{t} and q{(\\nabla,t)} = 2 \\sigma_{x}{(\\nabla,t)} and \\int q{(\\nabla,t)} d\\nabla = \\int 2 \\sigma_{x}{(\\nabla,t)} d\\nabla and \\int q{(\\nabla,t)} d\\nabla = \\int \\frac{2 \\nabla}{t} d\\nabla and (\\int q{(\\nabla,t)} d\\nabla)^{2} = (\\int \\frac{2 \\nabla}{t} d\\nabla)^{2} and \\int (2 \\sigma_{x}{(\\nabla,t)})^{\\nabla} dt + (\\int q{(\\nabla,t)} d\\nabla)^{2} = \\int (2 \\sigma_{x}{(\\nabla,t)})^{\\nabla} dt + (\\int \\frac{2 \\nabla}{t} d\\nabla)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2)))"], [["add", 5, "Integral(Pow(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Integral(Pow(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2))), Add(Integral(Pow(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\dot{z})} = \\dot{z}, then derive \\sigma_x + \\frac{n^{2}}{2} + n \\frac{d}{d \\dot{z}} \\operatorname{v_{2}}{(\\dot{z})} = C_{1} + \\frac{n^{2}}{2} + n, then obtain \\sigma_x + \\frac{n^{2}}{2} + n \\frac{d}{d \\dot{z}} \\dot{z} = C_{1} + \\frac{n^{2}}{2} + n", "derivation": "\\operatorname{v_{2}}{(\\dot{z})} = \\dot{z} and \\frac{d}{d \\dot{z}} \\operatorname{v_{2}}{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\dot{z} and n + \\frac{d}{d \\dot{z}} \\operatorname{v_{2}}{(\\dot{z})} = n + \\frac{d}{d \\dot{z}} \\dot{z} and \\int (n + \\frac{d}{d \\dot{z}} \\operatorname{v_{2}}{(\\dot{z})}) dn = \\int (n + \\frac{d}{d \\dot{z}} \\dot{z}) dn and \\sigma_x + \\frac{n^{2}}{2} + n \\frac{d}{d \\dot{z}} \\operatorname{v_{2}}{(\\dot{z})} = C_{1} + \\frac{n^{2}}{2} + n and \\sigma_x + \\frac{n^{2}}{2} + n \\frac{d}{d \\dot{z}} \\dot{z} = C_{1} + \\frac{n^{2}}{2} + n", "srepr_derivation": [["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["add", 2, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Derivative(Function('v_2')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Symbol('n', commutative=True), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Symbol('n', commutative=True), Derivative(Function('v_2')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('n', commutative=True), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Mul(Symbol('n', commutative=True), Derivative(Function('v_2')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Mul(Symbol('n', commutative=True), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(E,A_{x})} = E + e^{A_{x}} and z{(E,A_{x})} = \\frac{\\eta^{\\prime}{(E,A_{x})}}{A_{x}}, then obtain z^{A_{x}}{(E,A_{x})} = (\\frac{E + e^{A_{x}}}{A_{x}})^{A_{x}}", "derivation": "\\eta^{\\prime}{(E,A_{x})} = E + e^{A_{x}} and \\frac{\\eta^{\\prime}{(E,A_{x})}}{A_{x}} = \\frac{E + e^{A_{x}}}{A_{x}} and z{(E,A_{x})} = \\frac{\\eta^{\\prime}{(E,A_{x})}}{A_{x}} and z{(E,A_{x})} = \\frac{E + e^{A_{x}}}{A_{x}} and z^{A_{x}}{(E,A_{x})} = (\\frac{E + e^{A_{x}}}{A_{x}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('E', commutative=True), exp(Symbol('A_x', commutative=True))))"], [["divide", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), exp(Symbol('A_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('E', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('z')(Symbol('E', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), exp(Symbol('A_x', commutative=True)))))"], [["power", 4, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('z')(Symbol('E', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), exp(Symbol('A_x', commutative=True)))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(v_{y})} = \\cos{(v_{y})}, then obtain \\frac{0^{v_{y}}}{- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})}} = \\frac{(- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})})^{v_{y}}}{- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})}}", "derivation": "\\bar{\\h}{(v_{y})} = \\cos{(v_{y})} and 0 = - \\bar{\\h}{(v_{y})} + \\cos{(v_{y})} and 0^{v_{y}} = (- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})})^{v_{y}} and \\frac{0^{v_{y}}}{- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})}} = \\frac{(- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})})^{v_{y}}}{- \\bar{\\h}{(v_{y})} + \\cos{(v_{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('v_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_y', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True)))"], "Equality(Mul(Pow(Integer(0), Symbol('v_y', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{1},y^{\\prime})} = \\frac{\\sin{(C_{1})}}{y^{\\prime}}, then obtain \\int (-1) dC_{1} = \\int - \\frac{\\sin{(C_{1})}}{y^{\\prime} \\operatorname{E_{n}}{(C_{1},y^{\\prime})}} dC_{1}", "derivation": "\\operatorname{E_{n}}{(C_{1},y^{\\prime})} = \\frac{\\sin{(C_{1})}}{y^{\\prime}} and - \\operatorname{E_{n}}{(C_{1},y^{\\prime})} = - \\frac{\\sin{(C_{1})}}{y^{\\prime}} and -1 = - \\frac{\\sin{(C_{1})}}{y^{\\prime} \\operatorname{E_{n}}{(C_{1},y^{\\prime})}} and \\int (-1) dC_{1} = \\int - \\frac{\\sin{(C_{1})}}{y^{\\prime} \\operatorname{E_{n}}{(C_{1},y^{\\prime})}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('C_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), sin(Symbol('C_1', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E_n')(Symbol('C_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), sin(Symbol('C_1', commutative=True))))"], [["divide", 2, "Function('E_n')(Symbol('C_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('C_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Symbol('C_1', commutative=True))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('C_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(P_{e})} = \\cos{(P_{e})}, then obtain \\frac{2}{P_{e} \\cos{(P_{e})}} = \\frac{\\theta_{1}{(P_{e})} + \\cos{(P_{e})}}{P_{e} \\theta_{1}{(P_{e})} \\cos{(P_{e})}}", "derivation": "\\theta_{1}{(P_{e})} = \\cos{(P_{e})} and P_{e} \\theta_{1}{(P_{e})} = P_{e} \\cos{(P_{e})} and 2 \\theta_{1}{(P_{e})} = \\theta_{1}{(P_{e})} + \\cos{(P_{e})} and 2 = \\frac{\\theta_{1}{(P_{e})} + \\cos{(P_{e})}}{\\theta_{1}{(P_{e})}} and \\frac{2}{\\theta_{1}{(P_{e})}} = \\frac{\\theta_{1}{(P_{e})} + \\cos{(P_{e})}}{\\theta_{1}^{2}{(P_{e})}} and \\frac{2}{P_{e} \\theta_{1}{(P_{e})}} = \\frac{\\theta_{1}{(P_{e})} + \\cos{(P_{e})}}{P_{e} \\theta_{1}^{2}{(P_{e})}} and \\frac{2}{P_{e} \\cos{(P_{e})}} = \\frac{\\theta_{1}{(P_{e})} + \\cos{(P_{e})}}{P_{e} \\theta_{1}{(P_{e})} \\cos{(P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["times", 1, "Symbol('P_e', commutative=True)"], "Equality(Mul(Symbol('P_e', commutative=True), Function('\\\\theta_1')(Symbol('P_e', commutative=True))), Mul(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["add", 1, "Function('\\\\theta_1')(Symbol('P_e', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta_1')(Symbol('P_e', commutative=True))), Add(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))))"], [["divide", 3, "Function('\\\\theta_1')(Symbol('P_e', commutative=True))"], "Equality(Integer(2), Mul(Add(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-1))))"], [["times", 4, "Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-2))))"], [["divide", 5, "Symbol('P_e', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(2), Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('P_e', commutative=True)), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{p}{(\\rho_f)} = \\log{(\\log{(\\rho_f)})}, then derive (\\int \\sigma_{p}{(\\rho_f)} d\\rho_f)^{\\rho_f} = (\\rho_f \\log{(\\log{(\\rho_f)})} + u - \\operatorname{li}{(\\rho_f)})^{\\rho_f}, then obtain (\\int \\log{(\\log{(\\rho_f)})} d\\rho_f)^{\\rho_f} = (\\rho_f \\log{(\\log{(\\rho_f)})} + u - \\operatorname{li}{(\\rho_f)})^{\\rho_f}", "derivation": "\\sigma_{p}{(\\rho_f)} = \\log{(\\log{(\\rho_f)})} and \\int \\sigma_{p}{(\\rho_f)} d\\rho_f = \\int \\log{(\\log{(\\rho_f)})} d\\rho_f and (\\int \\sigma_{p}{(\\rho_f)} d\\rho_f)^{\\rho_f} = (\\int \\log{(\\log{(\\rho_f)})} d\\rho_f)^{\\rho_f} and (\\int \\sigma_{p}{(\\rho_f)} d\\rho_f)^{\\rho_f} = (\\rho_f \\log{(\\log{(\\rho_f)})} + u - \\operatorname{li}{(\\rho_f)})^{\\rho_f} and (\\int \\log{(\\log{(\\rho_f)})} d\\rho_f)^{\\rho_f} = (\\rho_f \\log{(\\log{(\\rho_f)})} + u - \\operatorname{li}{(\\rho_f)})^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\rho_f', commutative=True)), log(log(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(log(log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(log(log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_f', commutative=True), log(log(Symbol('\\\\rho_f', commutative=True)))), Symbol('u', commutative=True), Mul(Integer(-1), li(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(log(log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_f', commutative=True), log(log(Symbol('\\\\rho_f', commutative=True)))), Symbol('u', commutative=True), Mul(Integer(-1), li(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given k{(a^{\\dagger},\\varepsilon_0)} = - a^{\\dagger} + e^{\\varepsilon_0} and \\mathbf{p}{(\\varepsilon_0)} = 2 e^{\\varepsilon_0}, then obtain - \\varepsilon_0 - a^{\\dagger} + k{(a^{\\dagger},\\varepsilon_0)} + e^{\\varepsilon_0} = - \\varepsilon_0 - 2 a^{\\dagger} + \\mathbf{p}{(\\varepsilon_0)}", "derivation": "k{(a^{\\dagger},\\varepsilon_0)} = - a^{\\dagger} + e^{\\varepsilon_0} and - a^{\\dagger} + k{(a^{\\dagger},\\varepsilon_0)} + e^{\\varepsilon_0} = - 2 a^{\\dagger} + 2 e^{\\varepsilon_0} and - \\varepsilon_0 - a^{\\dagger} + k{(a^{\\dagger},\\varepsilon_0)} + e^{\\varepsilon_0} = - \\varepsilon_0 - 2 a^{\\dagger} + 2 e^{\\varepsilon_0} and \\mathbf{p}{(\\varepsilon_0)} = 2 e^{\\varepsilon_0} and - \\varepsilon_0 - a^{\\dagger} + k{(a^{\\dagger},\\varepsilon_0)} + e^{\\varepsilon_0} = - \\varepsilon_0 - 2 a^{\\dagger} + \\mathbf{p}{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\varepsilon_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given m{(s,\\hbar)} = \\cos{(\\hbar + s)}, then derive x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} m{(s,\\hbar)} = - x{(\\varepsilon_0,E_{\\lambda})} \\sin{(\\hbar + s)}, then obtain x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} m{(s,\\hbar)} - \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)} = - x{(\\varepsilon_0,E_{\\lambda})} \\sin{(\\hbar + s)} - \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)}", "derivation": "m{(s,\\hbar)} = \\cos{(\\hbar + s)} and \\frac{\\partial}{\\partial s} m{(s,\\hbar)} = \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)} and x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} m{(s,\\hbar)} = x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)} and x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} m{(s,\\hbar)} = - x{(\\varepsilon_0,E_{\\lambda})} \\sin{(\\hbar + s)} and x{(\\varepsilon_0,E_{\\lambda})} \\frac{\\partial}{\\partial s} m{(s,\\hbar)} - \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)} = - x{(\\varepsilon_0,E_{\\lambda})} \\sin{(\\hbar + s)} - \\frac{\\partial}{\\partial s} \\cos{(\\hbar + s)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 2, "Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True)))))"], [["minus", 4, "Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True)))), Mul(Integer(-1), Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))))"]]}, {"prompt": "Given g{(\\theta)} = e^{e^{\\theta}}, then obtain - g{(\\theta)} e^{- \\theta} + e^{- \\theta} \\frac{d}{d \\theta} g{(\\theta)} = e^{e^{\\theta}} - e^{- \\theta} e^{e^{\\theta}}", "derivation": "g{(\\theta)} = e^{e^{\\theta}} and g{(\\theta)} e^{- \\theta} = e^{- \\theta} e^{e^{\\theta}} and \\frac{d}{d \\theta} g{(\\theta)} e^{- \\theta} = \\frac{d}{d \\theta} e^{- \\theta} e^{e^{\\theta}} and - g{(\\theta)} e^{- \\theta} + e^{- \\theta} \\frac{d}{d \\theta} g{(\\theta)} = e^{e^{\\theta}} - e^{- \\theta} e^{e^{\\theta}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\theta', commutative=True)), exp(exp(Symbol('\\\\theta', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('g')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), exp(exp(Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Function('g')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), exp(exp(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\theta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Derivative(Function('g')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Add(exp(exp(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), exp(exp(Symbol('\\\\theta', commutative=True))))))"]]}, {"prompt": "Given A{(r_{0})} = \\frac{d}{d r_{0}} \\sin{(r_{0})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(r_{0})} = A^{r_{0}}{(r_{0})}, then derive A{(r_{0})} = \\cos{(r_{0})}, then obtain (\\cos^{r_{0}}{(r_{0})})^{r_{0}} = \\operatorname{g^{\\prime}_{\\varepsilon}}^{r_{0}}{(r_{0})}", "derivation": "A{(r_{0})} = \\frac{d}{d r_{0}} \\sin{(r_{0})} and A{(r_{0})} = \\cos{(r_{0})} and A^{r_{0}}{(r_{0})} = (\\frac{d}{d r_{0}} \\sin{(r_{0})})^{r_{0}} and \\cos^{r_{0}}{(r_{0})} = (\\frac{d}{d r_{0}} \\sin{(r_{0})})^{r_{0}} and \\cos^{r_{0}}{(r_{0})} = A^{r_{0}}{(r_{0})} and (\\cos^{r_{0}}{(r_{0})})^{r_{0}} = (A^{r_{0}}{(r_{0})})^{r_{0}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(r_{0})} = A^{r_{0}}{(r_{0})} and (\\cos^{r_{0}}{(r_{0})})^{r_{0}} = \\operatorname{g^{\\prime}_{\\varepsilon}}^{r_{0}}{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('r_0', commutative=True)), Derivative(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('A')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Derivative(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Derivative(sin(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('A')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["power", 5, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Function('A')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Pow(Function('A')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\chi{(P_{e},n)} = (e^{P_{e}})^{n}, then derive \\frac{\\partial}{\\partial P_{e}} \\chi{(P_{e},n)} + 1 = n (e^{P_{e}})^{n} + 1, then obtain n (\\frac{\\partial}{\\partial P_{e}} \\chi{(P_{e},n)} + 1) (e^{P_{e}})^{n} = n (\\frac{\\partial}{\\partial P_{e}} (e^{P_{e}})^{n} + 1) (e^{P_{e}})^{n}", "derivation": "\\chi{(P_{e},n)} = (e^{P_{e}})^{n} and P_{e} + \\chi{(P_{e},n)} = P_{e} + (e^{P_{e}})^{n} and \\frac{\\partial}{\\partial P_{e}} (P_{e} + \\chi{(P_{e},n)}) = \\frac{\\partial}{\\partial P_{e}} (P_{e} + (e^{P_{e}})^{n}) and \\frac{\\partial}{\\partial P_{e}} \\chi{(P_{e},n)} + 1 = n (e^{P_{e}})^{n} + 1 and \\frac{\\partial}{\\partial P_{e}} (e^{P_{e}})^{n} + 1 = n (e^{P_{e}})^{n} + 1 and \\frac{\\partial}{\\partial P_{e}} \\chi{(P_{e},n)} + 1 = \\frac{\\partial}{\\partial P_{e}} (e^{P_{e}})^{n} + 1 and n (\\frac{\\partial}{\\partial P_{e}} \\chi{(P_{e},n)} + 1) (e^{P_{e}})^{n} = n (\\frac{\\partial}{\\partial P_{e}} (e^{P_{e}})^{n} + 1) (e^{P_{e}})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True)))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True))), Add(Symbol('P_e', commutative=True), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Add(Symbol('P_e', commutative=True), Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)), Add(Mul(Symbol('n', commutative=True), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)), Add(Mul(Symbol('n', commutative=True), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)))"], [["times", 6, "Mul(Symbol('n', commutative=True), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True)))"], "Equality(Mul(Symbol('n', commutative=True), Add(Derivative(Function('\\\\chi')(Symbol('P_e', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))), Mul(Symbol('n', commutative=True), Add(Derivative(Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(f)} = \\frac{d}{d f} \\log{(f)}, then derive 0 = - \\mathbf{P}{(f)} + \\frac{1}{f}, then obtain (\\int 0 df)^{f} = (\\int - \\frac{- \\frac{d}{d f} \\log{(f)} + \\frac{1}{f}}{\\frac{d}{d f} \\log{(f)}} df)^{f}", "derivation": "\\mathbf{P}{(f)} = \\frac{d}{d f} \\log{(f)} and 0 = - \\mathbf{P}{(f)} + \\frac{d}{d f} \\log{(f)} and 0 = - \\mathbf{P}{(f)} + \\frac{1}{f} and 0 = - \\frac{d}{d f} \\log{(f)} + \\frac{1}{f} and 0 = - \\frac{- \\frac{d}{d f} \\log{(f)} + \\frac{1}{f}}{\\frac{d}{d f} \\log{(f)}} and \\int 0 df = \\int - \\frac{- \\frac{d}{d f} \\log{(f)} + \\frac{1}{f}}{\\frac{d}{d f} \\log{(f)}} df and (\\int 0 df)^{f} = (\\int - \\frac{- \\frac{d}{d f} \\log{(f)} + \\frac{1}{f}}{\\frac{d}{d f} \\log{(f)}} df)^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('f', commutative=True)), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\mathbf{P}')(Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('f', commutative=True))), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('f', commutative=True))), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('f', commutative=True), Integer(-1))), Pow(Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 5, "Symbol('f', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('f', commutative=True))), Integral(Mul(Integer(-1), Add(Mul(Integer(-1), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('f', commutative=True), Integer(-1))), Pow(Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('f', commutative=True))))"], [["power", 6, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(Mul(Integer(-1), Add(Mul(Integer(-1), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('f', commutative=True), Integer(-1))), Pow(Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given U{(t,n)} = \\frac{\\partial}{\\partial t} (- n + t), then derive U{(t,n)} = 1, then obtain \\frac{\\partial}{\\partial t} \\int \\frac{\\partial}{\\partial t} (- n + t) dt = \\frac{d}{d t} \\int 1 dt", "derivation": "U{(t,n)} = \\frac{\\partial}{\\partial t} (- n + t) and U{(t,n)} = 1 and \\frac{\\partial}{\\partial t} (- n + t) = 1 and \\int \\frac{\\partial}{\\partial t} (- n + t) dt = \\int 1 dt and \\frac{\\partial}{\\partial t} \\int \\frac{\\partial}{\\partial t} (- n + t) dt = \\frac{d}{d t} \\int 1 dt", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('t', commutative=True), Symbol('n', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('U')(Symbol('t', commutative=True), Symbol('n', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(Integer(1), Tuple(Symbol('t', commutative=True))))"], [["differentiate", 4, "Symbol('t', commutative=True)"], "Equality(Derivative(Integral(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(M,n)} = M n and C{(M,n)} = M n, then obtain \\frac{d}{d n} 0 = \\frac{\\partial}{\\partial n} (0^{n} - (- C{(M,n)} + Q{(M,n)})^{n})", "derivation": "Q{(M,n)} = M n and - M n + Q{(M,n)} = 0 and C{(M,n)} = M n and (- M n + Q{(M,n)})^{n} = 0^{n} and 0 = 0^{n} - (- M n + Q{(M,n)})^{n} and \\frac{d}{d n} 0 = \\frac{\\partial}{\\partial n} (0^{n} - (- M n + Q{(M,n)})^{n}) and \\frac{d}{d n} 0 = \\frac{\\partial}{\\partial n} (0^{n} - (- C{(M,n)} + Q{(M,n)})^{n})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('n', commutative=True)))"], [["minus", 1, "Mul(Symbol('M', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('n', commutative=True)), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('C')(Symbol('M', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('n', commutative=True)))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('n', commutative=True)), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Integer(0), Symbol('n', commutative=True)))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('n', commutative=True)), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True))"], "Equality(Integer(0), Add(Pow(Integer(0), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('n', commutative=True)), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)))))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Pow(Integer(0), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('n', commutative=True)), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Pow(Integer(0), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('C')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Function('Q')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(f_{\\mathbf{v}},s)} = f_{\\mathbf{v}} - s, then derive 0 = 1 - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{v_{1}}{(f_{\\mathbf{v}},s)}, then obtain 0 = 1 - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} - s)", "derivation": "\\operatorname{v_{1}}{(f_{\\mathbf{v}},s)} = f_{\\mathbf{v}} - s and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{v_{1}}{(f_{\\mathbf{v}},s)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} - s) and 0 = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} - s) - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{v_{1}}{(f_{\\mathbf{v}},s)} and 0 = 1 - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{v_{1}}{(f_{\\mathbf{v}},s)} and 0 = 1 - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} - s)", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('v_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\ddot{x}{(B)} = \\int \\log{(B)} dB, then derive \\ddot{x}^{B}{(B)} = (B \\log{(B)} - B + \\mathbf{J})^{B}, then obtain (B \\log{(B)} - B + \\mathbf{J})^{B} = (\\int \\log{(B)} dB)^{B}", "derivation": "\\ddot{x}{(B)} = \\int \\log{(B)} dB and \\ddot{x}^{B}{(B)} = (\\int \\log{(B)} dB)^{B} and \\ddot{x}^{B}{(B)} = (B \\log{(B)} - B + \\mathbf{J})^{B} and (B \\log{(B)} - B + \\mathbf{J})^{B} = (\\int \\log{(B)} dB)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Mul(Symbol('B', commutative=True), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('B', commutative=True)), Pow(Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(v_{1})} = e^{v_{1}}, then obtain v_{1} = v_{1} + \\mathbf{f}^{2}{(v_{1})} e^{v_{1}} - e^{3 v_{1}}", "derivation": "\\mathbf{f}{(v_{1})} = e^{v_{1}} and \\mathbf{f}{(v_{1})} e^{v_{1}} = e^{2 v_{1}} and \\mathbf{f}{(v_{1})} e^{2 v_{1}} = e^{3 v_{1}} and \\mathbf{f}^{2}{(v_{1})} e^{v_{1}} = e^{3 v_{1}} and - 2 v_{1} + \\mathbf{f}{(v_{1})} e^{2 v_{1}} = - 2 v_{1} + e^{3 v_{1}} and - 2 v_{1} + \\mathbf{f}{(v_{1})} e^{2 v_{1}} = - 2 v_{1} + \\mathbf{f}^{2}{(v_{1})} e^{v_{1}} and - 2 v_{1} + e^{3 v_{1}} = - 2 v_{1} + \\mathbf{f}^{2}{(v_{1})} e^{v_{1}} and v_{1} = v_{1} + \\mathbf{f}^{2}{(v_{1})} e^{v_{1}} - e^{3 v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["times", 1, "exp(Symbol('v_1', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True))), exp(Mul(Integer(2), Symbol('v_1', commutative=True))))"], [["times", 1, "exp(Mul(Integer(2), Symbol('v_1', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), exp(Mul(Integer(2), Symbol('v_1', commutative=True)))), exp(Mul(Integer(3), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), Integer(2)), exp(Symbol('v_1', commutative=True))), exp(Mul(Integer(3), Symbol('v_1', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), exp(Mul(Integer(2), Symbol('v_1', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), exp(Mul(Integer(3), Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), exp(Mul(Integer(2), Symbol('v_1', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Pow(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), Integer(2)), exp(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), exp(Mul(Integer(3), Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Pow(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), Integer(2)), exp(Symbol('v_1', commutative=True)))))"], [["minus", 7, "Add(Mul(Integer(-1), Integer(3), Symbol('v_1', commutative=True)), exp(Mul(Integer(3), Symbol('v_1', commutative=True))))"], "Equality(Symbol('v_1', commutative=True), Add(Symbol('v_1', commutative=True), Mul(Pow(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), Integer(2)), exp(Symbol('v_1', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(3), Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(v_{2},v_{t})} = \\log{(v_{2} v_{t})}, then obtain 8 \\phi_{2}^{2}{(v_{2},v_{t})} = 4 (\\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})}) \\phi_{2}{(v_{2},v_{t})}", "derivation": "\\phi_{2}{(v_{2},v_{t})} = \\log{(v_{2} v_{t})} and 2 \\phi_{2}{(v_{2},v_{t})} = \\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})} and 4 \\phi_{2}{(v_{2},v_{t})} = 3 \\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})} and 2 (3 \\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})}) \\phi_{2}{(v_{2},v_{t})} = (\\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})}) (3 \\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})}) and 8 \\phi_{2}^{2}{(v_{2},v_{t})} = 4 (\\phi_{2}{(v_{2},v_{t})} + \\log{(v_{2} v_{t})}) \\phi_{2}{(v_{2},v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))"], [["add", 1, "Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Add(Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))))"], [["add", 2, "Mul(Integer(2), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(3), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(3), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))"], "Equality(Mul(Integer(2), Add(Mul(Integer(3), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Mul(Add(Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))), Add(Mul(Integer(3), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(8), Pow(Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), Integer(2))), Mul(Integer(4), Add(Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), log(Mul(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))), Function('\\\\phi_2')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(C_{2})} = \\sin{(\\sin{(C_{2})})}, then obtain (\\mathbf{D}{(C_{2})} - \\sin{(C_{2})}) \\int \\mathbf{D}{(C_{2})} dC_{2} = (- \\sin{(C_{2})} + \\sin{(\\sin{(C_{2})})}) \\int \\mathbf{D}{(C_{2})} dC_{2}", "derivation": "\\mathbf{D}{(C_{2})} = \\sin{(\\sin{(C_{2})})} and \\int \\mathbf{D}{(C_{2})} dC_{2} = \\int \\sin{(\\sin{(C_{2})})} dC_{2} and \\mathbf{D}{(C_{2})} - \\sin{(C_{2})} = - \\sin{(C_{2})} + \\sin{(\\sin{(C_{2})})} and (\\mathbf{D}{(C_{2})} - \\sin{(C_{2})}) \\int \\sin{(\\sin{(C_{2})})} dC_{2} = (- \\sin{(C_{2})} + \\sin{(\\sin{(C_{2})})}) \\int \\sin{(\\sin{(C_{2})})} dC_{2} and (\\mathbf{D}{(C_{2})} - \\sin{(C_{2})}) \\int \\mathbf{D}{(C_{2})} dC_{2} = (- \\sin{(C_{2})} + \\sin{(\\sin{(C_{2})})}) \\int \\mathbf{D}{(C_{2})} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), sin(sin(Symbol('C_2', commutative=True))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(sin(sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["minus", 1, "sin(Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), sin(Symbol('C_2', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), sin(sin(Symbol('C_2', commutative=True)))))"], [["times", 3, "Integral(sin(sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), sin(Symbol('C_2', commutative=True)))), Integral(sin(sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Mul(Add(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), sin(sin(Symbol('C_2', commutative=True)))), Integral(sin(sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), sin(Symbol('C_2', commutative=True)))), Integral(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Mul(Add(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), sin(sin(Symbol('C_2', commutative=True)))), Integral(Function('\\\\mathbf{D}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{y})} = \\cos{(v_{y})}, then derive \\int \\operatorname{F_{H}}{(v_{y})} dv_{y} = E_{\\lambda} + \\sin{(v_{y})}, then obtain \\iiiint \\operatorname{F_{H}}{(v_{y})} dv_{y} dE_{\\lambda} dv_{y} dv_{y} = \\iiint (E_{\\lambda} + \\sin{(v_{y})}) dE_{\\lambda} dv_{y} dv_{y}", "derivation": "\\operatorname{F_{H}}{(v_{y})} = \\cos{(v_{y})} and \\int \\operatorname{F_{H}}{(v_{y})} dv_{y} = \\int \\cos{(v_{y})} dv_{y} and \\int \\operatorname{F_{H}}{(v_{y})} dv_{y} = E_{\\lambda} + \\sin{(v_{y})} and \\iint \\operatorname{F_{H}}{(v_{y})} dv_{y} dE_{\\lambda} = \\int (E_{\\lambda} + \\sin{(v_{y})}) dE_{\\lambda} and \\iiint \\operatorname{F_{H}}{(v_{y})} dv_{y} dE_{\\lambda} dv_{y} = \\iint (E_{\\lambda} + \\sin{(v_{y})}) dE_{\\lambda} dv_{y} and \\iiiint \\operatorname{F_{H}}{(v_{y})} dv_{y} dE_{\\lambda} dv_{y} dv_{y} = \\iiint (E_{\\lambda} + \\sin{(v_{y})}) dE_{\\lambda} dv_{y} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(cos(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('v_y', commutative=True))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 4, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["integrate", 5, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\dot{x},\\pi)} = \\dot{x} + \\pi, then obtain \\frac{(\\dot{x} + \\Omega{(\\dot{x},\\pi)})^{\\dot{x}}}{\\dot{x} + \\Omega{(\\dot{x},\\pi)}} = \\frac{(2 \\dot{x} + \\pi)^{\\dot{x}}}{\\dot{x} + \\Omega{(\\dot{x},\\pi)}}", "derivation": "\\Omega{(\\dot{x},\\pi)} = \\dot{x} + \\pi and \\dot{x} + \\Omega{(\\dot{x},\\pi)} = 2 \\dot{x} + \\pi and (\\dot{x} + \\Omega{(\\dot{x},\\pi)})^{\\dot{x}} = (2 \\dot{x} + \\pi)^{\\dot{x}} and \\frac{(\\dot{x} + \\Omega{(\\dot{x},\\pi)})^{\\dot{x}}}{\\dot{x} + \\Omega{(\\dot{x},\\pi)}} = \\frac{(2 \\dot{x} + \\pi)^{\\dot{x}}}{\\dot{x} + \\Omega{(\\dot{x},\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 3, "Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\Omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(A_{1},\\chi)} = \\sin{(A_{1} \\chi)}, then derive \\frac{\\partial}{\\partial A_{1}} \\Psi^{\\dagger}{(A_{1},\\chi)} = \\chi \\cos{(A_{1} \\chi)}, then obtain \\frac{\\partial}{\\partial A_{1}} \\chi \\cos{(A_{1} \\chi)} = \\frac{\\partial^{2}}{\\partial A_{1}^{2}} \\sin{(A_{1} \\chi)}", "derivation": "\\Psi^{\\dagger}{(A_{1},\\chi)} = \\sin{(A_{1} \\chi)} and \\frac{\\partial}{\\partial A_{1}} \\Psi^{\\dagger}{(A_{1},\\chi)} = \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} \\chi)} and \\frac{\\partial}{\\partial A_{1}} \\Psi^{\\dagger}{(A_{1},\\chi)} = \\chi \\cos{(A_{1} \\chi)} and \\chi \\cos{(A_{1} \\chi)} = \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} \\chi)} and \\frac{\\partial}{\\partial A_{1}} \\chi \\cos{(A_{1} \\chi)} = \\frac{\\partial^{2}}{\\partial A_{1}^{2}} \\sin{(A_{1} \\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Symbol('\\\\chi', commutative=True), cos(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('\\\\chi', commutative=True), cos(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)))), Derivative(sin(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\chi', commutative=True), cos(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('A_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi_{2}{(\\dot{z})} = \\log{(\\dot{z})}, then obtain - \\phi_{2}{(\\dot{z})} + 3 \\log{(\\dot{z})} - 1 = 2 \\log{(\\dot{z})} - 1", "derivation": "\\phi_{2}{(\\dot{z})} = \\log{(\\dot{z})} and \\phi_{2}{(\\dot{z})} - \\log{(\\dot{z})} = 0 and \\phi_{2}{(\\dot{z})} - \\log{(\\dot{z})} + 1 = 1 and \\phi_{2}{(\\dot{z})} - 2 \\log{(\\dot{z})} + 1 = 1 - \\log{(\\dot{z})} and - \\phi_{2}{(\\dot{z})} + 2 \\log{(\\dot{z})} - 1 = \\log{(\\dot{z})} - 1 and - \\phi_{2}{(\\dot{z})} + 3 \\log{(\\dot{z})} - 1 = 2 \\log{(\\dot{z})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\dot{z}', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Integer(1))"], [["add", 3, "Mul(Integer(-1), log(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(-1), log(Symbol('\\\\dot{z}', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Add(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)))"], [["add", 5, "log(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(3), log(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Add(Mul(Integer(2), log(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given C{(G,T)} = G^{T}, then obtain (G + G^{T} e^{C{(G,T)}}) (G + C{(G,T)} e^{C{(G,T)}}) = (G + G^{T} e^{C{(G,T)}})^{2}", "derivation": "C{(G,T)} = G^{T} and e^{C{(G,T)}} = e^{G^{T}} and C{(G,T)} e^{C{(G,T)}} = G^{T} e^{C{(G,T)}} and C{(G,T)} e^{G^{T}} = G^{T} e^{G^{T}} and G + C{(G,T)} e^{G^{T}} = G + G^{T} e^{G^{T}} and (G + G^{T} e^{G^{T}}) (G + C{(G,T)} e^{G^{T}}) = (G + G^{T} e^{G^{T}})^{2} and (G + G^{T} e^{C{(G,T)}}) (G + C{(G,T)} e^{C{(G,T)}}) = (G + G^{T} e^{C{(G,T)}})^{2}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)))"], [["exp", 1], "Equality(exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True))), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True))))"], [["times", 1, "exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)))))"], [["add", 4, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Mul(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True))))), Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True))))))"], [["times", 5, "Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)))))"], "Equality(Mul(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True))))), Add(Symbol('G', commutative=True), Mul(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)))))), Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True))))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True))))), Add(Symbol('G', commutative=True), Mul(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True)))))), Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('G', commutative=True), Symbol('T', commutative=True)), exp(Function('C')(Symbol('G', commutative=True), Symbol('T', commutative=True))))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{F}{(z^{*},y^{\\prime})} = \\frac{\\partial}{\\partial z^{*}} (y^{\\prime} + z^{*}), then derive \\mathbf{F}{(z^{*},y^{\\prime})} = 1, then obtain \\int \\mathbf{F}{(z^{*},y^{\\prime})} dz^{*} - 1 = \\int 1 dz^{*} - 1", "derivation": "\\mathbf{F}{(z^{*},y^{\\prime})} = \\frac{\\partial}{\\partial z^{*}} (y^{\\prime} + z^{*}) and \\mathbf{F}{(z^{*},y^{\\prime})} = 1 and \\int \\mathbf{F}{(z^{*},y^{\\prime})} dz^{*} = \\int 1 dz^{*} and \\int \\mathbf{F}{(z^{*},y^{\\prime})} dz^{*} - 1 = \\int 1 dz^{*} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('z^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('z^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('z^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Integer(1), Tuple(Symbol('z^*', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Integral(Function('\\\\mathbf{F}')(Symbol('z^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('z^*', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbb{I})} = e^{\\cos{(\\mathbb{I})}}, then obtain 0 = \\log{(\\frac{\\int (- \\operatorname{F_{g}}{(\\mathbb{I})} + e^{\\cos{(\\mathbb{I})}}) d\\mathbb{I}}{\\int 0 d\\mathbb{I}})}", "derivation": "\\operatorname{F_{g}}{(\\mathbb{I})} = e^{\\cos{(\\mathbb{I})}} and 0 = - \\operatorname{F_{g}}{(\\mathbb{I})} + e^{\\cos{(\\mathbb{I})}} and \\int 0 d\\mathbb{I} = \\int (- \\operatorname{F_{g}}{(\\mathbb{I})} + e^{\\cos{(\\mathbb{I})}}) d\\mathbb{I} and 1 = \\frac{\\int (- \\operatorname{F_{g}}{(\\mathbb{I})} + e^{\\cos{(\\mathbb{I})}}) d\\mathbb{I}}{\\int 0 d\\mathbb{I}} and 0 = \\log{(\\frac{\\int (- \\operatorname{F_{g}}{(\\mathbb{I})} + e^{\\cos{(\\mathbb{I})}}) d\\mathbb{I}}{\\int 0 d\\mathbb{I}})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True)), exp(cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 1, "Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True))), exp(cos(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True))), exp(cos(Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 3, "Integral(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True))), exp(cos(Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["log", 4], "Equality(Integer(0), log(Mul(Pow(Integral(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True))), exp(cos(Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(k)} = \\log{(k)}, then obtain - \\frac{- \\log{(k)} + \\log{(k)}^{k}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} = \\frac{\\log{(k)} - \\log{(k)}^{k}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}", "derivation": "\\mathbf{P}{(k)} = \\log{(k)} and \\mathbf{P}^{k}{(k)} = \\log{(k)}^{k} and \\mathbf{P}^{k}{(k)} - \\log{(k)} = - \\log{(k)} + \\log{(k)}^{k} and - \\frac{\\mathbf{P}^{k}{(k)} - \\log{(k)}}{\\log{(k)}} = - \\frac{- \\log{(k)} + \\log{(k)}^{k}}{\\log{(k)}} and - \\frac{\\mathbf{P}^{k}{(k)} - \\log{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} = - \\frac{- \\log{(k)} + \\log{(k)}^{k}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} and - \\frac{- \\log{(k)} + \\log{(k)}^{k}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} = \\frac{\\log{(k)} - \\log{(k)}^{k}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 2, "log(Symbol('k', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{P}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), log(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), log(Symbol('k', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Pow(Function('\\\\mathbf{P}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), log(Symbol('k', commutative=True)))), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Integer(-1))))"], [["minus", 4, "Pow(log(Symbol('k', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Add(Pow(Function('\\\\mathbf{P}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), log(Symbol('k', commutative=True)))), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))), Add(Mul(Add(log(Symbol('k', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(C_{d})} = \\frac{d}{d C_{d}} \\log{(C_{d})} and \\mathbf{D}{(C_{d})} = C_{d}, then obtain \\cos{(\\int \\operatorname{g_{\\varepsilon}}{(C_{d})} dC_{d})} = \\cos{(\\int \\frac{d}{d C_{d}} \\log{(C_{d})} dC_{d})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(C_{d})} = \\frac{d}{d C_{d}} \\log{(C_{d})} and \\int \\operatorname{g_{\\varepsilon}}{(C_{d})} dC_{d} = \\int \\frac{d}{d C_{d}} \\log{(C_{d})} dC_{d} and \\mathbf{D}{(C_{d})} = C_{d} and \\int \\operatorname{g_{\\varepsilon}}{(C_{d})} d\\mathbf{D}{(C_{d})} = \\int \\frac{d}{d C_{d}} \\log{(C_{d})} d\\mathbf{D}{(C_{d})} and \\cos{(\\int \\operatorname{g_{\\varepsilon}}{(C_{d})} d\\mathbf{D}{(C_{d})})} = \\cos{(\\int \\frac{d}{d C_{d}} \\log{(C_{d})} d\\mathbf{D}{(C_{d})})} and \\cos{(\\int \\operatorname{g_{\\varepsilon}}{(C_{d})} dC_{d})} = \\cos{(\\int \\frac{d}{d C_{d}} \\log{(C_{d})} dC_{d})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True)))), Integral(Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True)))))"], [["cos", 4], "Equality(cos(Integral(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True))))), cos(Integral(Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(cos(Integral(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), cos(Integral(Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given S{(z)} = \\sin{(z)} and U{(z)} = S{(z)} + \\sin{(z)}, then obtain (\\frac{\\log{(2 S{(z)})}}{S{(z)}})^{z} = (\\frac{\\log{(2 \\sin{(z)})}}{S{(z)}})^{z}", "derivation": "S{(z)} = \\sin{(z)} and 2 S{(z)} = S{(z)} + \\sin{(z)} and U{(z)} = S{(z)} + \\sin{(z)} and \\log{(U{(z)})} = \\log{(S{(z)} + \\sin{(z)})} and \\log{(U{(z)})} = \\log{(2 S{(z)})} and \\log{(U{(z)})} = \\log{(2 \\sin{(z)})} and \\log{(2 S{(z)})} = \\log{(2 \\sin{(z)})} and \\frac{\\log{(2 S{(z)})}}{S{(z)}} = \\frac{\\log{(2 \\sin{(z)})}}{S{(z)}} and (\\frac{\\log{(2 S{(z)})}}{S{(z)}})^{z} = (\\frac{\\log{(2 \\sin{(z)})}}{S{(z)}})^{z}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["add", 1, "Function('S')(Symbol('z', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('z', commutative=True))), Add(Function('S')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('z', commutative=True)), Add(Function('S')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))))"], [["log", 3], "Equality(log(Function('U')(Symbol('z', commutative=True))), log(Add(Function('S')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(log(Function('U')(Symbol('z', commutative=True))), log(Mul(Integer(2), Function('S')(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Function('U')(Symbol('z', commutative=True))), log(Mul(Integer(2), sin(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(log(Mul(Integer(2), Function('S')(Symbol('z', commutative=True)))), log(Mul(Integer(2), sin(Symbol('z', commutative=True)))))"], [["divide", 7, "Function('S')(Symbol('z', commutative=True))"], "Equality(Mul(Pow(Function('S')(Symbol('z', commutative=True)), Integer(-1)), log(Mul(Integer(2), Function('S')(Symbol('z', commutative=True))))), Mul(Pow(Function('S')(Symbol('z', commutative=True)), Integer(-1)), log(Mul(Integer(2), sin(Symbol('z', commutative=True))))))"], [["power", 8, "Symbol('z', commutative=True)"], "Equality(Pow(Mul(Pow(Function('S')(Symbol('z', commutative=True)), Integer(-1)), log(Mul(Integer(2), Function('S')(Symbol('z', commutative=True))))), Symbol('z', commutative=True)), Pow(Mul(Pow(Function('S')(Symbol('z', commutative=True)), Integer(-1)), log(Mul(Integer(2), sin(Symbol('z', commutative=True))))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\hat{X},k)} = e^{\\frac{\\hat{X}}{k}} and \\operatorname{r_{0}}{(\\hat{x},\\phi)} = \\hat{x} \\phi, then obtain \\frac{\\int \\hat{p}^{k}{(\\hat{X},k)} dk}{\\phi + \\operatorname{r_{0}}{(\\hat{x},\\phi)}} = \\frac{\\int (e^{\\frac{\\hat{X}}{k}})^{k} dk}{\\phi + \\operatorname{r_{0}}{(\\hat{x},\\phi)}}", "derivation": "\\hat{p}{(\\hat{X},k)} = e^{\\frac{\\hat{X}}{k}} and \\operatorname{r_{0}}{(\\hat{x},\\phi)} = \\hat{x} \\phi and \\hat{p}^{k}{(\\hat{X},k)} = (e^{\\frac{\\hat{X}}{k}})^{k} and \\int \\hat{p}^{k}{(\\hat{X},k)} dk = \\int (e^{\\frac{\\hat{X}}{k}})^{k} dk and \\phi + \\operatorname{r_{0}}{(\\hat{x},\\phi)} = \\hat{x} \\phi + \\phi and \\frac{\\int \\hat{p}^{k}{(\\hat{X},k)} dk}{\\hat{x} \\phi + \\phi} = \\frac{\\int (e^{\\frac{\\hat{X}}{k}})^{k} dk}{\\hat{x} \\phi + \\phi} and \\frac{\\int \\hat{p}^{k}{(\\hat{X},k)} dk}{\\phi + \\operatorname{r_{0}}{(\\hat{x},\\phi)}} = \\frac{\\int (e^{\\frac{\\hat{X}}{k}})^{k} dk}{\\phi + \\operatorname{r_{0}}{(\\hat{x},\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))))"], ["get_premise", "Equality(Function('r_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))), Symbol('k', commutative=True)))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["add", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('r_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["divide", 4, "Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Add(Symbol('\\\\phi', commutative=True), Function('r_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)), Integral(Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Add(Symbol('\\\\phi', commutative=True), Function('r_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)), Integral(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(f_{E})} = \\sin{(f_{E})}, then obtain - \\mathbf{s}^{2}{(f_{E})} + \\mathbf{s}^{f_{E}}{(f_{E})} = - \\mathbf{s}^{2}{(f_{E})} + \\sin^{f_{E}}{(f_{E})}", "derivation": "\\mathbf{s}{(f_{E})} = \\sin{(f_{E})} and \\mathbf{s}^{2}{(f_{E})} = \\mathbf{s}{(f_{E})} \\sin{(f_{E})} and \\mathbf{s}^{f_{E}}{(f_{E})} = \\sin^{f_{E}}{(f_{E})} and - \\mathbf{s}{(f_{E})} \\sin{(f_{E})} + \\mathbf{s}^{f_{E}}{(f_{E})} = - \\mathbf{s}{(f_{E})} \\sin{(f_{E})} + \\sin^{f_{E}}{(f_{E})} and - \\mathbf{s}^{2}{(f_{E})} + \\mathbf{s}^{f_{E}}{(f_{E})} = - \\mathbf{s}^{2}{(f_{E})} + \\sin^{f_{E}}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True))))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["minus", 3, "Mul(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True))), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('f_E', commutative=True)), Integer(2))), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(v_{2})} = \\int \\cos{(v_{2})} dv_{2}, then derive \\psi^{*}{(v_{2})} = \\hat{H}_{\\lambda} + \\sin{(v_{2})}, then obtain \\psi^{*}{(v_{2})} \\int \\cos{(v_{2})} dv_{2} = (\\int \\cos{(v_{2})} dv_{2})^{2}", "derivation": "\\psi^{*}{(v_{2})} = \\int \\cos{(v_{2})} dv_{2} and \\psi^{*}{(v_{2})} = \\hat{H}_{\\lambda} + \\sin{(v_{2})} and (\\hat{H}_{\\lambda} + \\sin{(v_{2})}) \\psi^{*}{(v_{2})} = (\\hat{H}_{\\lambda} + \\sin{(v_{2})})^{2} and \\hat{H}_{\\lambda} + \\sin{(v_{2})} = \\int \\cos{(v_{2})} dv_{2} and \\psi^{*}{(v_{2})} \\int \\cos{(v_{2})} dv_{2} = (\\int \\cos{(v_{2})} dv_{2})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('v_2', commutative=True)), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\psi^*')(Symbol('v_2', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True))), Function('\\\\psi^*')(Symbol('v_2', commutative=True))), Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True))), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Function('\\\\psi^*')(Symbol('v_2', commutative=True)), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Pow(Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\omega{(\\theta_2)} = \\sin{(\\theta_2)}, then derive \\int \\omega{(\\theta_2)} d\\theta_2 = v - \\cos{(\\theta_2)}, then derive t - \\cos{(\\theta_2)} = v - \\cos{(\\theta_2)}, then obtain (t - \\cos{(\\theta_2)})^{\\theta_2} - 1 = (\\int \\sin{(\\theta_2)} d\\theta_2)^{\\theta_2} - 1", "derivation": "\\omega{(\\theta_2)} = \\sin{(\\theta_2)} and \\int \\omega{(\\theta_2)} d\\theta_2 = \\int \\sin{(\\theta_2)} d\\theta_2 and \\int \\omega{(\\theta_2)} d\\theta_2 = v - \\cos{(\\theta_2)} and \\int \\sin{(\\theta_2)} d\\theta_2 = v - \\cos{(\\theta_2)} and t - \\cos{(\\theta_2)} = v - \\cos{(\\theta_2)} and (t - \\cos{(\\theta_2)})^{\\theta_2} = (v - \\cos{(\\theta_2)})^{\\theta_2} and (t - \\cos{(\\theta_2)})^{\\theta_2} = (\\int \\omega{(\\theta_2)} d\\theta_2)^{\\theta_2} and (t - \\cos{(\\theta_2)})^{\\theta_2} - 1 = (\\int \\omega{(\\theta_2)} d\\theta_2)^{\\theta_2} - 1 and (t - \\cos{(\\theta_2)})^{\\theta_2} - 1 = (\\int \\sin{(\\theta_2)} d\\theta_2)^{\\theta_2} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 5, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["add", 7, "Integer(-1)"], "Equality(Add(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Add(Pow(Integral(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Add(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Add(Pow(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\eta,\\rho_b)} = (e^{\\rho_b})^{\\eta}, then obtain \\frac{\\partial}{\\partial \\rho_b} (e^{\\rho_b} - \\int \\operatorname{c_{0}}{(\\eta,\\rho_b)} d\\rho_b) = \\frac{\\partial}{\\partial \\rho_b} (e^{\\rho_b} - \\int (e^{\\rho_b})^{\\eta} d\\rho_b)", "derivation": "\\operatorname{c_{0}}{(\\eta,\\rho_b)} = (e^{\\rho_b})^{\\eta} and \\int \\operatorname{c_{0}}{(\\eta,\\rho_b)} d\\rho_b = \\int (e^{\\rho_b})^{\\eta} d\\rho_b and - e^{\\rho_b} + \\int \\operatorname{c_{0}}{(\\eta,\\rho_b)} d\\rho_b = - e^{\\rho_b} + \\int (e^{\\rho_b})^{\\eta} d\\rho_b and e^{\\rho_b} - \\int \\operatorname{c_{0}}{(\\eta,\\rho_b)} d\\rho_b = e^{\\rho_b} - \\int (e^{\\rho_b})^{\\eta} d\\rho_b and \\frac{\\partial}{\\partial \\rho_b} (e^{\\rho_b} - \\int \\operatorname{c_{0}}{(\\eta,\\rho_b)} d\\rho_b) = \\frac{\\partial}{\\partial \\rho_b} (e^{\\rho_b} - \\int (e^{\\rho_b})^{\\eta} d\\rho_b)", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(exp(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Pow(exp(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\rho_b', commutative=True))), Integral(Function('c_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\rho_b', commutative=True))), Integral(Pow(exp(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(exp(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Integral(Function('c_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))), Add(exp(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Integral(Pow(exp(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Add(exp(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Integral(Function('c_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Integral(Pow(exp(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\delta)} = \\log{(\\delta)}, then obtain \\frac{\\operatorname{A_{y}}^{\\delta}{(\\delta)}}{\\int \\log{(\\delta)}^{\\delta} d\\delta} = \\frac{\\log{(\\delta)}^{\\delta}}{\\int \\log{(\\delta)}^{\\delta} d\\delta}", "derivation": "\\operatorname{A_{y}}{(\\delta)} = \\log{(\\delta)} and \\operatorname{A_{y}}^{\\delta}{(\\delta)} = \\log{(\\delta)}^{\\delta} and \\int \\operatorname{A_{y}}^{\\delta}{(\\delta)} d\\delta = \\int \\log{(\\delta)}^{\\delta} d\\delta and \\frac{\\operatorname{A_{y}}^{\\delta}{(\\delta)}}{\\int \\operatorname{A_{y}}^{\\delta}{(\\delta)} d\\delta} = \\frac{\\log{(\\delta)}^{\\delta}}{\\int \\operatorname{A_{y}}^{\\delta}{(\\delta)} d\\delta} and \\frac{\\operatorname{A_{y}}^{\\delta}{(\\delta)}}{\\int \\log{(\\delta)}^{\\delta} d\\delta} = \\frac{\\log{(\\delta)}^{\\delta}}{\\int \\log{(\\delta)}^{\\delta} d\\delta}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 2, "Integral(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1))), Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('A_y')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1))), Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(Pow(log(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(\\phi_1,n_{2})} = \\phi_1 n_{2} and J{(\\phi_1)} = \\phi_1, then obtain \\frac{\\partial}{\\partial n_{2}} (- \\phi_1 n_{2} + J{(\\phi_1)} \\int 2 \\phi_1 n_{2} dn_{2}) = \\frac{\\partial}{\\partial n_{2}} (- \\phi_1 n_{2} + \\phi_1 \\int 2 \\phi_1 n_{2} dn_{2})", "derivation": "k{(\\phi_1,n_{2})} = \\phi_1 n_{2} and J{(\\phi_1)} = \\phi_1 and J{(\\phi_1)} \\int (\\phi_1 n_{2} + k{(\\phi_1,n_{2})}) dn_{2} = \\phi_1 \\int (\\phi_1 n_{2} + k{(\\phi_1,n_{2})}) dn_{2} and J{(\\phi_1)} \\int 2 \\phi_1 n_{2} dn_{2} = \\phi_1 \\int 2 \\phi_1 n_{2} dn_{2} and - \\phi_1 n_{2} + J{(\\phi_1)} \\int 2 \\phi_1 n_{2} dn_{2} = - \\phi_1 n_{2} + \\phi_1 \\int 2 \\phi_1 n_{2} dn_{2} and \\frac{\\partial}{\\partial n_{2}} (- \\phi_1 n_{2} + J{(\\phi_1)} \\int 2 \\phi_1 n_{2} dn_{2}) = \\frac{\\partial}{\\partial n_{2}} (- \\phi_1 n_{2} + \\phi_1 \\int 2 \\phi_1 n_{2} dn_{2})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], [["times", 2, "Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Function('J')(Symbol('\\\\phi_1', commutative=True)), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('J')(Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Function('J')(Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))))"], [["differentiate", 5, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Function('J')(Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(L)} = \\sin{(\\cos{(L)})}, then obtain \\log{(q{(L)})} = \\log{(\\sin{(\\cos{(L)})})}", "derivation": "q{(L)} = \\sin{(\\cos{(L)})} and 0 = - q{(L)} + \\sin{(\\cos{(L)})} and \\sin{(\\cos{(L)})} = - q{(L)} + 2 \\sin{(\\cos{(L)})} and q{(L)} = - q{(L)} + 2 \\sin{(\\cos{(L)})} and \\log{(q{(L)})} = \\log{(- q{(L)} + 2 \\sin{(\\cos{(L)})})} and \\log{(q{(L)})} = \\log{(\\sin{(\\cos{(L)})})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('L', commutative=True)), sin(cos(Symbol('L', commutative=True))))"], [["minus", 1, "Function('q')(Symbol('L', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('q')(Symbol('L', commutative=True))), sin(cos(Symbol('L', commutative=True)))))"], [["add", 2, "sin(cos(Symbol('L', commutative=True)))"], "Equality(sin(cos(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Function('q')(Symbol('L', commutative=True))), Mul(Integer(2), sin(cos(Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('q')(Symbol('L', commutative=True)), Add(Mul(Integer(-1), Function('q')(Symbol('L', commutative=True))), Mul(Integer(2), sin(cos(Symbol('L', commutative=True))))))"], [["log", 4], "Equality(log(Function('q')(Symbol('L', commutative=True))), log(Add(Mul(Integer(-1), Function('q')(Symbol('L', commutative=True))), Mul(Integer(2), sin(cos(Symbol('L', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(log(Function('q')(Symbol('L', commutative=True))), log(sin(cos(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{f})} = \\log{(\\cos{(\\mathbf{f})})} and \\mu_{0}{(\\mathbf{f})} = \\cos{(\\mathbf{f})}, then obtain \\frac{d}{d \\mathbf{f}} \\int 0 d\\mathbf{f} = \\frac{d}{d \\mathbf{f}} \\int (- \\log{(\\mu_{0}{(\\mathbf{f})})} + \\log{(\\cos{(\\mathbf{f})})}) d\\mathbf{f}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{f})} = \\log{(\\cos{(\\mathbf{f})})} and 0 = - \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{f})} + \\log{(\\cos{(\\mathbf{f})})} and \\mu_{0}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\int 0 d\\mathbf{f} = \\int (- \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{f})} + \\log{(\\cos{(\\mathbf{f})})}) d\\mathbf{f} and \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{f})} = \\log{(\\mu_{0}{(\\mathbf{f})})} and \\int 0 d\\mathbf{f} = \\int (- \\log{(\\mu_{0}{(\\mathbf{f})})} + \\log{(\\cos{(\\mathbf{f})})}) d\\mathbf{f} and \\frac{d}{d \\mathbf{f}} \\int 0 d\\mathbf{f} = \\frac{d}{d \\mathbf{f}} \\int (- \\log{(\\mu_{0}{(\\mathbf{f})})} + \\log{(\\cos{(\\mathbf{f})})}) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{f}', commutative=True)), log(cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{f}', commutative=True))), log(cos(Symbol('\\\\mathbf{f}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{f}', commutative=True))), log(cos(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{f}', commutative=True)), log(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Mul(Integer(-1), log(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True)))), log(cos(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), log(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True)))), log(cos(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(q,\\mathbf{M})} = \\mathbf{M}^{q}, then derive \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = \\mathbf{M}^{q} \\log{(\\mathbf{M})}, then obtain - \\mathbf{M}^{q} \\log{(\\mathbf{M})} + \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = - \\mathbf{M}^{q} \\log{(\\mathbf{M})} + n{(q,\\mathbf{M})} \\log{(\\mathbf{M})}", "derivation": "n{(q,\\mathbf{M})} = \\mathbf{M}^{q} and \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = \\frac{\\partial}{\\partial q} \\mathbf{M}^{q} and \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = \\mathbf{M}^{q} \\log{(\\mathbf{M})} and \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = n{(q,\\mathbf{M})} \\log{(\\mathbf{M})} and - \\frac{\\partial}{\\partial q} \\mathbf{M}^{q} + \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = n{(q,\\mathbf{M})} \\log{(\\mathbf{M})} - \\frac{\\partial}{\\partial q} \\mathbf{M}^{q} and - \\mathbf{M}^{q} \\log{(\\mathbf{M})} + \\frac{\\partial}{\\partial q} n{(q,\\mathbf{M})} = - \\mathbf{M}^{q} \\log{(\\mathbf{M})} + n{(q,\\mathbf{M})} \\log{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 4, "Derivative(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Function('n')(Symbol('q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} = \\varepsilon_0 (\\mathbf{P} - x^\\prime), then obtain - \\varepsilon_0 + \\int \\mathbf{P} \\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} d\\mathbf{P} = - \\varepsilon_0 + \\int \\mathbf{P} \\varepsilon_0 (\\mathbf{P} - x^\\prime) d\\mathbf{P}", "derivation": "\\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} = \\varepsilon_0 (\\mathbf{P} - x^\\prime) and \\mathbf{P} \\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} = \\mathbf{P} \\varepsilon_0 (\\mathbf{P} - x^\\prime) and \\int \\mathbf{P} \\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} d\\mathbf{P} = \\int \\mathbf{P} \\varepsilon_0 (\\mathbf{P} - x^\\prime) d\\mathbf{P} and - \\varepsilon_0 + \\int \\mathbf{P} \\eta^{\\prime}{(\\mathbf{P},x^\\prime,\\varepsilon_0)} d\\mathbf{P} = - \\varepsilon_0 + \\int \\mathbf{P} \\varepsilon_0 (\\mathbf{P} - x^\\prime) d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["times", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(r,\\hat{\\mathbf{x}})} = \\frac{r}{\\hat{\\mathbf{x}}} and L{(r,\\hat{\\mathbf{x}})} = \\operatorname{v_{1}}^{\\hat{\\mathbf{x}}}{(r,\\hat{\\mathbf{x}})}, then obtain \\int L{(r,\\hat{\\mathbf{x}})} dr = \\int (\\frac{r}{\\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} dr", "derivation": "\\operatorname{v_{1}}{(r,\\hat{\\mathbf{x}})} = \\frac{r}{\\hat{\\mathbf{x}}} and \\operatorname{v_{1}}^{\\hat{\\mathbf{x}}}{(r,\\hat{\\mathbf{x}})} = (\\frac{r}{\\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} and L{(r,\\hat{\\mathbf{x}})} = \\operatorname{v_{1}}^{\\hat{\\mathbf{x}}}{(r,\\hat{\\mathbf{x}})} and L{(r,\\hat{\\mathbf{x}})} = (\\frac{r}{\\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} and \\int L{(r,\\hat{\\mathbf{x}})} dr = \\int (\\frac{r}{\\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} dr", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Function('v_1')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('L')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Function('L')(Symbol('r', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{H},\\Omega,\\rho)} = \\rho (- \\Omega + \\hat{H}), then obtain - \\frac{\\mathbf{J}_M^{\\rho}{(\\hat{H},\\Omega,\\rho)}}{\\rho^{2} (- \\Omega + \\hat{H})} = - \\frac{(\\rho (- \\Omega + \\hat{H}))^{\\rho}}{\\rho^{2} (- \\Omega + \\hat{H})}", "derivation": "\\mathbf{J}_M{(\\hat{H},\\Omega,\\rho)} = \\rho (- \\Omega + \\hat{H}) and \\mathbf{J}_M^{\\rho}{(\\hat{H},\\Omega,\\rho)} = (\\rho (- \\Omega + \\hat{H}))^{\\rho} and - \\frac{\\mathbf{J}_M^{\\rho}{(\\hat{H},\\Omega,\\rho)}}{\\rho (- \\Omega + \\hat{H})} = - \\frac{(\\rho (- \\Omega + \\hat{H}))^{\\rho}}{\\rho (- \\Omega + \\hat{H})} and - \\frac{\\mathbf{J}_M^{\\rho}{(\\hat{H},\\Omega,\\rho)}}{\\rho^{2} (- \\Omega + \\hat{H})} = - \\frac{(\\rho (- \\Omega + \\hat{H}))^{\\rho}}{\\rho^{2} (- \\Omega + \\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Mul(Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"], [["divide", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Mul(Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given m{(T)} = \\log{(\\sin{(T)})} and \\operatorname{C_{1}}{(T)} = \\int m{(T)} dT, then obtain \\log{(\\sin{(T)})} + \\int \\operatorname{C_{1}}{(T)} dT = \\log{(\\sin{(T)})} + \\iint \\log{(\\sin{(T)})} dT dT", "derivation": "m{(T)} = \\log{(\\sin{(T)})} and \\int m{(T)} dT = \\int \\log{(\\sin{(T)})} dT and \\operatorname{C_{1}}{(T)} = \\int m{(T)} dT and \\operatorname{C_{1}}{(T)} = \\int \\log{(\\sin{(T)})} dT and \\int \\operatorname{C_{1}}{(T)} dT = \\iint \\log{(\\sin{(T)})} dT dT and \\log{(\\sin{(T)})} + \\int \\operatorname{C_{1}}{(T)} dT = \\log{(\\sin{(T)})} + \\iint \\log{(\\sin{(T)})} dT dT", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('T', commutative=True)), log(sin(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('m')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('T', commutative=True)), Integral(Function('m')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('C_1')(Symbol('T', commutative=True)), Integral(log(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 5, "log(sin(Symbol('T', commutative=True)))"], "Equality(Add(log(sin(Symbol('T', commutative=True))), Integral(Function('C_1')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(log(sin(Symbol('T', commutative=True))), Integral(log(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(C,A_{1})} = e^{A_{1} C}, then obtain 2 \\mathbf{M}{(C,A_{1})} \\frac{\\partial}{\\partial A_{1}} \\mathbf{M}{(C,A_{1})} = 2 C \\mathbf{M}{(C,A_{1})} e^{A_{1} C}", "derivation": "\\mathbf{M}{(C,A_{1})} = e^{A_{1} C} and 2 \\mathbf{M}{(C,A_{1})} = \\mathbf{M}{(C,A_{1})} + e^{A_{1} C} and \\frac{\\partial}{\\partial A_{1}} \\mathbf{M}{(C,A_{1})} = \\frac{\\partial}{\\partial A_{1}} e^{A_{1} C} and (\\mathbf{M}{(C,A_{1})} + e^{A_{1} C}) \\frac{\\partial}{\\partial A_{1}} \\mathbf{M}{(C,A_{1})} = (\\mathbf{M}{(C,A_{1})} + e^{A_{1} C}) \\frac{\\partial}{\\partial A_{1}} e^{A_{1} C} and 2 \\mathbf{M}{(C,A_{1})} \\frac{\\partial}{\\partial A_{1}} \\mathbf{M}{(C,A_{1})} = 2 \\mathbf{M}{(C,A_{1})} \\frac{\\partial}{\\partial A_{1}} e^{A_{1} C} and 2 \\mathbf{M}{(C,A_{1})} \\frac{\\partial}{\\partial A_{1}} \\mathbf{M}{(C,A_{1})} = 2 C \\mathbf{M}{(C,A_{1})} e^{A_{1} C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True))), Add(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True)))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["times", 3, "Add(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True)))), Derivative(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Add(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True)))), Derivative(exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Derivative(exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('C', commutative=True), Function('\\\\mathbf{M}')(Symbol('C', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(f_{\\mathbf{p}},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}}, then derive \\mu_{0}{(f_{\\mathbf{p}},C_{d})} = - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}}, then obtain \\int (-1 - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}}) dC_{d} = \\int (\\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}} - 1) dC_{d}", "derivation": "\\mu_{0}{(f_{\\mathbf{p}},C_{d})} = \\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}} and \\mu_{0}{(f_{\\mathbf{p}},C_{d})} = - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}} and - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}} = \\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}} and -1 - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}} = \\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}} - 1 and \\int (-1 - \\frac{f_{\\mathbf{p}}}{C_{d}^{2}}) dC_{d} = \\int (\\frac{\\partial}{\\partial C_{d}} \\frac{f_{\\mathbf{p}}}{C_{d}} - 1) dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_d', commutative=True)), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu_0')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-2)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-2)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-2)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-2)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\lambda{(u,\\tilde{g}^*)} = \\tilde{g}^* + u, then derive \\int (\\tilde{g}^* + \\lambda{(u,\\tilde{g}^*)}) du = J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2}, then obtain \\int (J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2}) d\\tilde{g}^* - 1 = \\iint (2 \\tilde{g}^* + u) du d\\tilde{g}^* - 1", "derivation": "\\lambda{(u,\\tilde{g}^*)} = \\tilde{g}^* + u and \\tilde{g}^* + \\lambda{(u,\\tilde{g}^*)} = 2 \\tilde{g}^* + u and \\int (\\tilde{g}^* + \\lambda{(u,\\tilde{g}^*)}) du = \\int (2 \\tilde{g}^* + u) du and \\int (\\tilde{g}^* + \\lambda{(u,\\tilde{g}^*)}) du = J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2} and J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2} = \\int (2 \\tilde{g}^* + u) du and \\int (J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2}) d\\tilde{g}^* = \\iint (2 \\tilde{g}^* + u) du d\\tilde{g}^* and \\int (J_{\\varepsilon} + 2 \\tilde{g}^* u + \\frac{u^{2}}{2}) d\\tilde{g}^* - 1 = \\iint (2 \\tilde{g}^* + u) du d\\tilde{g}^* - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True)))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Integral(Add(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["integrate", 5, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('u', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Add(Integral(Add(Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given I{(\\mathbb{I},\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{\\mathbb{I}}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} \\int (I{(\\mathbb{I},\\mathbf{J}_f)} - \\frac{\\mathbf{J}_f}{\\mathbb{I}})^{\\mathbf{J}_f} d\\mathbb{I} = \\frac{\\partial}{\\partial \\mathbb{I}} \\int 0^{\\mathbf{J}_f} d\\mathbb{I}", "derivation": "I{(\\mathbb{I},\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{\\mathbb{I}} and I{(\\mathbb{I},\\mathbf{J}_f)} - \\frac{\\mathbf{J}_f}{\\mathbb{I}} = 0 and (I{(\\mathbb{I},\\mathbf{J}_f)} - \\frac{\\mathbf{J}_f}{\\mathbb{I}})^{\\mathbf{J}_f} = 0^{\\mathbf{J}_f} and \\int (I{(\\mathbb{I},\\mathbf{J}_f)} - \\frac{\\mathbf{J}_f}{\\mathbb{I}})^{\\mathbf{J}_f} d\\mathbb{I} = \\int 0^{\\mathbf{J}_f} d\\mathbb{I} and \\frac{\\partial}{\\partial \\mathbb{I}} \\int (I{(\\mathbb{I},\\mathbf{J}_f)} - \\frac{\\mathbf{J}_f}{\\mathbb{I}})^{\\mathbf{J}_f} d\\mathbb{I} = \\frac{\\partial}{\\partial \\mathbb{I}} \\int 0^{\\mathbf{J}_f} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Pow(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Integral(Pow(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integral(Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\rho_b,V)} = e^{\\frac{V}{\\rho_b}}, then obtain \\frac{\\dot{y}{(\\rho_b,V)} - e^{\\frac{V}{\\rho_b}}}{\\rho_b} + \\frac{1}{\\rho_b} = \\frac{1}{\\rho_b}", "derivation": "\\dot{y}{(\\rho_b,V)} = e^{\\frac{V}{\\rho_b}} and \\dot{y}{(\\rho_b,V)} - e^{\\frac{V}{\\rho_b}} = 0 and \\frac{\\dot{y}{(\\rho_b,V)} - e^{\\frac{V}{\\rho_b}}}{\\rho_b} = 0 and \\frac{\\dot{y}{(\\rho_b,V)} - e^{\\frac{V}{\\rho_b}}}{\\rho_b} + \\frac{1}{\\rho_b} = \\frac{1}{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))))"], [["minus", 1, "exp(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))))), Integer(0))"], [["times", 2, "Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))))))), Integer(0))"], [["add", 3, "Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))))))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\nabla,\\lambda)} = \\lambda \\nabla and \\mathbf{J}_f{(\\nabla,\\lambda)} = \\lambda \\nabla, then obtain \\int \\nabla \\mathbf{J}_f^{2 \\nabla}{(\\nabla,\\lambda)} d\\lambda = \\int \\nabla \\Psi_{nl}^{\\nabla}{(\\nabla,\\lambda)} \\mathbf{J}_f^{\\nabla}{(\\nabla,\\lambda)} d\\lambda", "derivation": "\\Psi_{nl}{(\\nabla,\\lambda)} = \\lambda \\nabla and \\mathbf{J}_f{(\\nabla,\\lambda)} = \\lambda \\nabla and \\mathbf{J}_f^{\\nabla}{(\\nabla,\\lambda)} = (\\lambda \\nabla)^{\\nabla} and \\mathbf{J}_f^{\\nabla}{(\\nabla,\\lambda)} = \\Psi_{nl}^{\\nabla}{(\\nabla,\\lambda)} and \\nabla \\mathbf{J}_f^{2 \\nabla}{(\\nabla,\\lambda)} = \\nabla \\Psi_{nl}^{\\nabla}{(\\nabla,\\lambda)} \\mathbf{J}_f^{\\nabla}{(\\nabla,\\lambda)} and \\int \\nabla \\mathbf{J}_f^{2 \\nabla}{(\\nabla,\\lambda)} d\\lambda = \\int \\nabla \\Psi_{nl}^{\\nabla}{(\\nabla,\\lambda)} \\mathbf{J}_f^{\\nabla}{(\\nabla,\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["times", 4, "Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["integrate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\theta_2,F_{H})} = - F_{H} + \\theta_2, then derive \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} = -1, then obtain F_{H} + \\frac{\\partial}{\\partial F_{H}} (- F_{H} + \\theta_2 - \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} - 1) + \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} + 1 = F_{H} + \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})}", "derivation": "\\hat{H}_l{(\\theta_2,F_{H})} = - F_{H} + \\theta_2 and \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} = \\frac{\\partial}{\\partial F_{H}} (- F_{H} + \\theta_2) and \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} = -1 and \\frac{\\partial}{\\partial F_{H}} (- F_{H} + \\theta_2) = -1 and F_{H} + \\frac{\\partial}{\\partial F_{H}} (- F_{H} + \\theta_2) = F_{H} - 1 and F_{H} + \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} = F_{H} - 1 and F_{H} + \\frac{\\partial}{\\partial F_{H}} (- F_{H} + \\theta_2 - \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} - 1) + \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})} + 1 = F_{H} + \\frac{\\partial}{\\partial F_{H}} \\hat{H}_l{(\\theta_2,F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(-1))"], [["add", 4, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('F_H', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('F_H', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Add(Symbol('F_H', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{r})} = \\int \\cos{(\\mathbf{r})} d\\mathbf{r}, then obtain \\frac{d^{2}}{d \\mathbf{r}^{2}} \\operatorname{f^{*}}{(\\mathbf{r})} = \\frac{\\partial^{2}}{\\partial \\mathbf{r}^{2}} (A_{y} + \\sin{(\\mathbf{r})})", "derivation": "\\operatorname{f^{*}}{(\\mathbf{r})} = \\int \\cos{(\\mathbf{r})} d\\mathbf{r} and \\frac{d}{d \\mathbf{r}} \\operatorname{f^{*}}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\int \\cos{(\\mathbf{r})} d\\mathbf{r} and \\frac{d^{2}}{d \\mathbf{r}^{2}} \\operatorname{f^{*}}{(\\mathbf{r})} = \\frac{d^{2}}{d \\mathbf{r}^{2}} \\int \\cos{(\\mathbf{r})} d\\mathbf{r} and \\frac{d^{2}}{d \\mathbf{r}^{2}} \\operatorname{f^{*}}{(\\mathbf{r})} = \\frac{\\partial^{2}}{\\partial \\mathbf{r}^{2}} (A_{y} + \\sin{(\\mathbf{r})})", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{r}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(Integral(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))))"], [["evaluate_integrals", 3], "Equality(Derivative(Function('f^*')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))))"]]}, {"prompt": "Given V{(\\psi,\\phi)} = \\cos{(\\frac{\\psi}{\\phi})} and \\omega{(\\psi,\\phi)} = V{(\\psi,\\phi)} - \\cos{(\\frac{\\psi}{\\phi})}, then obtain - \\cos{(\\frac{\\psi}{\\phi})} = - \\cos{(\\frac{\\psi}{\\phi})} + \\frac{\\omega{(\\psi,\\phi)}}{V{(\\psi,\\phi)} + 1}", "derivation": "V{(\\psi,\\phi)} = \\cos{(\\frac{\\psi}{\\phi})} and V{(\\psi,\\phi)} + 1 = \\cos{(\\frac{\\psi}{\\phi})} + 1 and - V{(\\psi,\\phi)} - 1 = - \\cos{(\\frac{\\psi}{\\phi})} - 1 and 0 = V{(\\psi,\\phi)} - \\cos{(\\frac{\\psi}{\\phi})} and 0 = \\frac{V{(\\psi,\\phi)} - \\cos{(\\frac{\\psi}{\\phi})}}{V{(\\psi,\\phi)} + 1} and - \\cos{(\\frac{\\psi}{\\phi})} = - \\cos{(\\frac{\\psi}{\\phi})} + \\frac{V{(\\psi,\\phi)} - \\cos{(\\frac{\\psi}{\\phi})}}{V{(\\psi,\\phi)} + 1} and \\omega{(\\psi,\\phi)} = V{(\\psi,\\phi)} - \\cos{(\\frac{\\psi}{\\phi})} and - \\cos{(\\frac{\\psi}{\\phi})} = - \\cos{(\\frac{\\psi}{\\phi})} + \\frac{\\omega{(\\psi,\\phi)}}{V{(\\psi,\\phi)} + 1}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1)), Add(cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Integer(1)))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Integer(-1)))"], [["add", 3, "Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1))"], "Equality(Integer(0), Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))))"], [["divide", 4, "Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1))"], "Equality(Integer(0), Mul(Pow(Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1)), Integer(-1)), Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))))))"], [["add", 5, "Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))"], "Equality(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Mul(Pow(Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1)), Integer(-1)), Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Mul(Pow(Add(Function('V')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1)), Integer(-1)), Function('\\\\omega')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given k{(\\sigma_p)} = \\sin{(\\sigma_p)} and a{(\\sigma_p)} = 2 \\sin{(\\sigma_p)}, then obtain - \\sigma_p + 2 k{(\\sigma_p)} \\sin{(\\sigma_p)} = - \\sigma_p + 2 \\sin^{2}{(\\sigma_p)}", "derivation": "k{(\\sigma_p)} = \\sin{(\\sigma_p)} and a{(\\sigma_p)} = 2 \\sin{(\\sigma_p)} and a{(\\sigma_p)} k{(\\sigma_p)} = a{(\\sigma_p)} \\sin{(\\sigma_p)} and 2 k{(\\sigma_p)} \\sin{(\\sigma_p)} = 2 \\sin^{2}{(\\sigma_p)} and - \\sigma_p + 2 k{(\\sigma_p)} \\sin{(\\sigma_p)} = - \\sigma_p + 2 \\sin^{2}{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 1, "Function('a')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('a')(Symbol('\\\\sigma_p', commutative=True)), Function('k')(Symbol('\\\\sigma_p', commutative=True))), Mul(Function('a')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(2))))"], [["minus", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Function('k')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{s},y^{\\prime})} = \\mathbf{s} + y^{\\prime} and \\psi^{*}{(\\mathbf{s},y^{\\prime})} = (\\mathbf{s} + y^{\\prime})^{2}, then obtain \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{s} + y^{\\prime})^{2} = \\frac{\\partial}{\\partial y^{\\prime}} \\psi^{*}{(\\mathbf{s},y^{\\prime})}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{s},y^{\\prime})} = \\mathbf{s} + y^{\\prime} and (\\mathbf{s} + y^{\\prime}) \\operatorname{P_{g}}{(\\mathbf{s},y^{\\prime})} = (\\mathbf{s} + y^{\\prime})^{2} and \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{s} + y^{\\prime}) \\operatorname{P_{g}}{(\\mathbf{s},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{s} + y^{\\prime})^{2} and \\psi^{*}{(\\mathbf{s},y^{\\prime})} = (\\mathbf{s} + y^{\\prime})^{2} and \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{s} + y^{\\prime}) \\operatorname{P_{g}}{(\\mathbf{s},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\psi^{*}{(\\mathbf{s},y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{s} + y^{\\prime})^{2} = \\frac{\\partial}{\\partial y^{\\prime}} \\psi^{*}{(\\mathbf{s},y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('P_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('P_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Function('P_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(\\hbar)} = \\log{(\\hbar)}, then obtain \\frac{\\frac{d}{d \\hbar} G{(\\hbar)}}{\\hbar} - \\frac{G{(\\hbar)}}{\\hbar^{2}} = - \\frac{\\log{(\\hbar)}}{\\hbar^{2}} + \\frac{1}{\\hbar^{2}}", "derivation": "G{(\\hbar)} = \\log{(\\hbar)} and \\frac{G{(\\hbar)}}{\\hbar} = \\frac{\\log{(\\hbar)}}{\\hbar} and \\frac{d}{d \\hbar} \\frac{G{(\\hbar)}}{\\hbar} = \\frac{d}{d \\hbar} \\frac{\\log{(\\hbar)}}{\\hbar} and \\frac{\\frac{d}{d \\hbar} G{(\\hbar)}}{\\hbar} - \\frac{G{(\\hbar)}}{\\hbar^{2}} = - \\frac{\\log{(\\hbar)}}{\\hbar^{2}} + \\frac{1}{\\hbar^{2}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('G')(Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), log(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('G')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Derivative(Function('G')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Function('G')(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), log(Symbol('\\\\hbar', commutative=True))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2))))"]]}, {"prompt": "Given U{(r,A_{1})} = \\sin{(A_{1}^{r})}, then obtain \\iint (- r + U{(r,A_{1})}) dr dr = \\iint (- r + \\sin{(A_{1}^{r})}) dr dr", "derivation": "U{(r,A_{1})} = \\sin{(A_{1}^{r})} and - r + U{(r,A_{1})} = - r + \\sin{(A_{1}^{r})} and \\int (- r + U{(r,A_{1})}) dr = \\int (- r + \\sin{(A_{1}^{r})}) dr and \\iint (- r + U{(r,A_{1})}) dr dr = \\iint (- r + \\sin{(A_{1}^{r})}) dr dr", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), sin(Pow(Symbol('A_1', commutative=True), Symbol('r', commutative=True))))"], [["minus", 1, "Symbol('r', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('U')(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Pow(Symbol('A_1', commutative=True), Symbol('r', commutative=True)))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('U')(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Pow(Symbol('A_1', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('U')(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Pow(Symbol('A_1', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(V)} = \\frac{d}{d V} \\sin{(V)}, then derive \\operatorname{E_{\\lambda}}^{V}{(V)} = \\cos^{V}{(V)}, then obtain (\\cos^{V}{(V)} - (\\frac{d}{d V} \\sin{(V)})^{V})^{V} = 0^{V}", "derivation": "\\operatorname{E_{\\lambda}}{(V)} = \\frac{d}{d V} \\sin{(V)} and \\operatorname{E_{\\lambda}}^{V}{(V)} = (\\frac{d}{d V} \\sin{(V)})^{V} and \\operatorname{E_{\\lambda}}^{V}{(V)} = \\cos^{V}{(V)} and \\operatorname{E_{\\lambda}}^{V}{(V)} - \\cos^{V}{(V)} = 0 and (\\frac{d}{d V} \\sin{(V)})^{V} = \\cos^{V}{(V)} and \\operatorname{E_{\\lambda}}^{V}{(V)} - (\\frac{d}{d V} \\sin{(V)})^{V} = 0 and (\\operatorname{E_{\\lambda}}^{V}{(V)} - (\\frac{d}{d V} \\sin{(V)})^{V})^{V} = 0^{V} and (\\cos^{V}{(V)} - (\\frac{d}{d V} \\sin{(V)})^{V})^{V} = 0^{V}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(cos(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["minus", 3, "Pow(cos(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('V', commutative=True)), Symbol('V', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(cos(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))), Integer(0))"], [["power", 6, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Integer(0), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Pow(Add(Pow(cos(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Integer(0), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(v_{z})} = e^{v_{z}} and \\mathbf{g}{(v_{z})} = \\frac{e^{v_{z}}}{\\frac{d}{d v_{z}} \\phi_{2}{(v_{z})}}, then derive \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})} = e^{v_{z}}, then obtain \\frac{d^{2}}{d v_{z}^{2}} \\phi_{2}{(v_{z})} = \\mathbf{g}{(v_{z})} \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})}", "derivation": "\\phi_{2}{(v_{z})} = e^{v_{z}} and \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})} = \\frac{d}{d v_{z}} e^{v_{z}} and \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})} = e^{v_{z}} and \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})} = \\phi_{2}{(v_{z})} and \\mathbf{g}{(v_{z})} = \\frac{e^{v_{z}}}{\\frac{d}{d v_{z}} \\phi_{2}{(v_{z})}} and \\mathbf{g}{(v_{z})} \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})} = e^{v_{z}} and \\frac{d^{2}}{d v_{z}^{2}} \\phi_{2}{(v_{z})} = e^{v_{z}} and \\frac{d^{2}}{d v_{z}^{2}} \\phi_{2}{(v_{z})} = \\mathbf{g}{(v_{z})} \\frac{d}{d v_{z}} \\phi_{2}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), exp(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Function('\\\\phi_2')(Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_z', commutative=True)), Mul(exp(Symbol('v_z', commutative=True)), Pow(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(-1))))"], [["times", 5, "Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('v_z', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), exp(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2))), exp(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2))), Mul(Function('\\\\mathbf{g}')(Symbol('v_z', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{E}{(t,L)} = (e^{L})^{t}, then derive \\frac{\\partial}{\\partial L} \\mathbf{E}{(t,L)} = t (e^{L})^{t}, then obtain (\\frac{\\partial}{\\partial b} (t \\mathbf{E}{(t,L)} + \\cos{(b)}))^{t} = (\\frac{\\partial}{\\partial b} (t (e^{L})^{t} + \\cos{(b)}))^{t}", "derivation": "\\mathbf{E}{(t,L)} = (e^{L})^{t} and \\frac{\\partial}{\\partial L} \\mathbf{E}{(t,L)} = \\frac{\\partial}{\\partial L} (e^{L})^{t} and \\frac{\\partial}{\\partial L} \\mathbf{E}{(t,L)} = t (e^{L})^{t} and \\cos{(b)} + \\frac{\\partial}{\\partial L} \\mathbf{E}{(t,L)} = t (e^{L})^{t} + \\cos{(b)} and \\frac{\\partial}{\\partial L} (e^{L})^{t} = t (e^{L})^{t} and \\frac{\\partial}{\\partial L} \\mathbf{E}{(t,L)} = t \\mathbf{E}{(t,L)} and t \\mathbf{E}{(t,L)} + \\cos{(b)} = t (e^{L})^{t} + \\cos{(b)} and \\frac{\\partial}{\\partial b} (t \\mathbf{E}{(t,L)} + \\cos{(b)}) = \\frac{\\partial}{\\partial b} (t (e^{L})^{t} + \\cos{(b)}) and (\\frac{\\partial}{\\partial b} (t \\mathbf{E}{(t,L)} + \\cos{(b)}))^{t} = (\\frac{\\partial}{\\partial b} (t (e^{L})^{t} + \\cos{(b)}))^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))))"], [["add", 3, "cos(Symbol('b', commutative=True))"], "Equality(Add(cos(Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))), cos(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Symbol('t', commutative=True), Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Symbol('t', commutative=True), Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True))), cos(Symbol('b', commutative=True))), Add(Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))), cos(Symbol('b', commutative=True))))"], [["differentiate", 7, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('t', commutative=True), Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True))), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["power", 8, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Symbol('t', commutative=True), Function('\\\\mathbf{E}')(Symbol('t', commutative=True), Symbol('L', commutative=True))), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Add(Mul(Symbol('t', commutative=True), Pow(exp(Symbol('L', commutative=True)), Symbol('t', commutative=True))), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(A_{1})} = \\log{(A_{1})}, then obtain \\rho_{b}^{- A_{1}}{(A_{1})} \\frac{d}{d A_{1}} \\rho_{b}^{A_{1}}{(A_{1})} = \\rho_{b}^{- A_{1}}{(A_{1})} \\frac{d}{d A_{1}} \\log{(A_{1})}^{A_{1}}", "derivation": "\\rho_{b}{(A_{1})} = \\log{(A_{1})} and \\rho_{b}^{A_{1}}{(A_{1})} = \\log{(A_{1})}^{A_{1}} and \\frac{d}{d A_{1}} \\rho_{b}^{A_{1}}{(A_{1})} = \\frac{d}{d A_{1}} \\log{(A_{1})}^{A_{1}} and \\rho_{b}^{- A_{1}}{(A_{1})} \\frac{d}{d A_{1}} \\rho_{b}^{A_{1}}{(A_{1})} = \\rho_{b}^{- A_{1}}{(A_{1})} \\frac{d}{d A_{1}} \\log{(A_{1})}^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["divide", 3, "Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Mul(Integer(-1), Symbol('A_1', commutative=True))), Derivative(Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), Mul(Integer(-1), Symbol('A_1', commutative=True))), Derivative(Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(f_{E},E_{\\lambda})} = \\frac{f_{E}}{E_{\\lambda}} and J{(f_{E},E_{\\lambda})} = \\cos{(Z{(f_{E},E_{\\lambda})})}, then obtain (\\int \\cos{(\\frac{f_{E}}{E_{\\lambda}})} dE_{\\lambda})^{E_{\\lambda}} = (\\int \\cos{(Z{(f_{E},E_{\\lambda})})} dE_{\\lambda})^{E_{\\lambda}}", "derivation": "Z{(f_{E},E_{\\lambda})} = \\frac{f_{E}}{E_{\\lambda}} and \\cos{(Z{(f_{E},E_{\\lambda})})} = \\cos{(\\frac{f_{E}}{E_{\\lambda}})} and J{(f_{E},E_{\\lambda})} = \\cos{(Z{(f_{E},E_{\\lambda})})} and \\int J{(f_{E},E_{\\lambda})} dE_{\\lambda} = \\int \\cos{(Z{(f_{E},E_{\\lambda})})} dE_{\\lambda} and J{(f_{E},E_{\\lambda})} = \\cos{(\\frac{f_{E}}{E_{\\lambda}})} and \\int \\cos{(\\frac{f_{E}}{E_{\\lambda}})} dE_{\\lambda} = \\int \\cos{(Z{(f_{E},E_{\\lambda})})} dE_{\\lambda} and (\\int \\cos{(\\frac{f_{E}}{E_{\\lambda}})} dE_{\\lambda})^{E_{\\lambda}} = (\\int \\cos{(Z{(f_{E},E_{\\lambda})})} dE_{\\lambda})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))"], [["cos", 1], "Equality(cos(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), cos(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('J')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('J')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(cos(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 6, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integral(cos(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(cos(Function('Z')(Symbol('f_E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given g{(F_{N})} = \\sin{(F_{N})}, then obtain \\frac{\\frac{d}{d F_{N}} (g{(F_{N})} - \\sin{(F_{N})})^{3}}{\\int 0 dF_{N}} = \\frac{\\frac{d}{d F_{N}} 0}{\\int 0 dF_{N}}", "derivation": "g{(F_{N})} = \\sin{(F_{N})} and g{(F_{N})} - \\sin{(F_{N})} = 0 and (g{(F_{N})} - \\sin{(F_{N})})^{2} = 0 and (g{(F_{N})} - \\sin{(F_{N})})^{3} = 0 and \\frac{d}{d F_{N}} (g{(F_{N})} - \\sin{(F_{N})})^{3} = \\frac{d}{d F_{N}} 0 and \\frac{\\frac{d}{d F_{N}} (g{(F_{N})} - \\sin{(F_{N})})^{3}}{\\int 0 dF_{N}} = \\frac{\\frac{d}{d F_{N}} 0}{\\int 0 dF_{N}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["minus", 1, "sin(Symbol('F_N', commutative=True))"], "Equality(Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(0))"], [["times", 2, "Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True))))"], "Equality(Pow(Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(2)), Integer(0))"], [["times", 3, "Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True))))"], "Equality(Pow(Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(3)), Integer(0))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Pow(Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(3)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["divide", 5, "Integral(Integer(0), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Mul(Derivative(Pow(Add(Function('g')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(3)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Integral(Integer(0), Tuple(Symbol('F_N', commutative=True))), Integer(-1))), Mul(Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Integral(Integer(0), Tuple(Symbol('F_N', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(P_{g},B)} = P_{g} e^{B}, then derive \\frac{\\partial}{\\partial B} \\theta_{2}{(P_{g},B)} = P_{g} e^{B}, then obtain (P_{g} e^{B} - P_{g})^{B} = (- P_{g} + \\frac{\\partial}{\\partial B} P_{g} e^{B})^{B}", "derivation": "\\theta_{2}{(P_{g},B)} = P_{g} e^{B} and \\frac{\\partial}{\\partial B} \\theta_{2}{(P_{g},B)} = \\frac{\\partial}{\\partial B} P_{g} e^{B} and \\frac{\\partial}{\\partial B} \\theta_{2}{(P_{g},B)} = P_{g} e^{B} and - P_{g} + \\frac{\\partial}{\\partial B} \\theta_{2}{(P_{g},B)} = - P_{g} + \\frac{\\partial}{\\partial B} P_{g} e^{B} and (- P_{g} + \\frac{\\partial}{\\partial B} \\theta_{2}{(P_{g},B)})^{B} = (- P_{g} + \\frac{\\partial}{\\partial B} P_{g} e^{B})^{B} and (P_{g} e^{B} - P_{g})^{B} = (- P_{g} + \\frac{\\partial}{\\partial B} P_{g} e^{B})^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('P_g', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('P_g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('P_g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))))"], [["minus", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('P_g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('P_g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\dot{x},v_{z})} = \\dot{x} + v_{z}, then obtain \\frac{\\operatorname{C_{2}}^{2}{(\\dot{x},v_{z})}}{\\dot{x} (\\dot{x} + v_{z})^{2}} = \\frac{1}{\\dot{x}}", "derivation": "\\operatorname{C_{2}}{(\\dot{x},v_{z})} = \\dot{x} + v_{z} and \\frac{\\operatorname{C_{2}}{(\\dot{x},v_{z})}}{\\dot{x} + v_{z}} = 1 and \\frac{\\operatorname{C_{2}}{(\\dot{x},v_{z})}}{\\dot{x} (\\dot{x} + v_{z})} = \\frac{1}{\\dot{x}} and \\dot{x} (\\dot{x} + v_{z}) = \\frac{\\dot{x} (\\dot{x} + v_{z})^{2}}{\\operatorname{C_{2}}{(\\dot{x},v_{z})}} and \\frac{\\operatorname{C_{2}}^{2}{(\\dot{x},v_{z})}}{\\dot{x} (\\dot{x} + v_{z})^{2}} = \\frac{1}{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True))), Integer(1))"], [["divide", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True))), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))"], [["divide", 1, "Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True))), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(2)), Pow(Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(-2)), Pow(Function('C_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(2))), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{nl}{(a,M)} = \\frac{\\cos{(M)}}{a}, then derive \\varepsilon_0 + \\int (\\frac{\\partial}{\\partial a} \\Psi_{nl}{(a,M)} - 1) dM = - M + T - \\frac{\\sin{(M)}}{a^{2}}, then obtain \\int (\\varepsilon_0 + \\int (\\frac{\\partial}{\\partial a} \\Psi_{nl}{(a,M)} - 1) dM) da = \\int (- M + T - \\frac{\\sin{(M)}}{a^{2}}) da", "derivation": "\\Psi_{nl}{(a,M)} = \\frac{\\cos{(M)}}{a} and - a + \\Psi_{nl}{(a,M)} = - a + \\frac{\\cos{(M)}}{a} and \\frac{\\partial}{\\partial a} (- a + \\Psi_{nl}{(a,M)}) = \\frac{\\partial}{\\partial a} (- a + \\frac{\\cos{(M)}}{a}) and \\int \\frac{\\partial}{\\partial a} (- a + \\Psi_{nl}{(a,M)}) dM = \\int \\frac{\\partial}{\\partial a} (- a + \\frac{\\cos{(M)}}{a}) dM and \\varepsilon_0 + \\int (\\frac{\\partial}{\\partial a} \\Psi_{nl}{(a,M)} - 1) dM = - M + T - \\frac{\\sin{(M)}}{a^{2}} and \\int (\\varepsilon_0 + \\int (\\frac{\\partial}{\\partial a} \\Psi_{nl}{(a,M)} - 1) dM) da = \\int (- M + T - \\frac{\\sin{(M)}}{a^{2}}) da", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(Symbol('M', commutative=True))))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(Symbol('M', commutative=True)))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(Symbol('M', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(Symbol('M', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('T', commutative=True), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-2)), sin(Symbol('M', commutative=True)))))"], [["integrate", 5, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('a', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('T', commutative=True), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-2)), sin(Symbol('M', commutative=True)))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(t_{1},M)} = M t_{1}, then obtain \\frac{\\partial}{\\partial t_{1}} M^{2} \\operatorname{f_{\\mathbf{v}}}^{2}{(t_{1},M)} = \\frac{\\partial}{\\partial t_{1}} M^{3} t_{1} \\operatorname{f_{\\mathbf{v}}}{(t_{1},M)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(t_{1},M)} = M t_{1} and M \\operatorname{f_{\\mathbf{v}}}{(t_{1},M)} = M^{2} t_{1} and M^{2} \\operatorname{f_{\\mathbf{v}}}^{2}{(t_{1},M)} = M^{3} t_{1} \\operatorname{f_{\\mathbf{v}}}{(t_{1},M)} and \\frac{\\partial}{\\partial t_{1}} M^{2} \\operatorname{f_{\\mathbf{v}}}^{2}{(t_{1},M)} = \\frac{\\partial}{\\partial t_{1}} M^{3} t_{1} \\operatorname{f_{\\mathbf{v}}}{(t_{1},M)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('t_1', commutative=True)))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(2)), Symbol('t_1', commutative=True)))"], [["times", 2, "Mul(Symbol('M', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True)), Integer(2))), Mul(Pow(Symbol('M', commutative=True), Integer(3)), Symbol('t_1', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True))))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True)), Integer(2))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(3)), Symbol('t_1', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t_1', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(C_{1},t_{2})} = C_{1} + t_{2} and L{(C_{1},t_{2})} = t_{2} v{(C_{1},t_{2})}, then obtain C_{1} + \\int L^{2}{(C_{1},t_{2})} dC_{1} = C_{1} + \\int t_{2} (C_{1} + t_{2}) L{(C_{1},t_{2})} dC_{1}", "derivation": "v{(C_{1},t_{2})} = C_{1} + t_{2} and t_{2} v{(C_{1},t_{2})} = t_{2} (C_{1} + t_{2}) and L{(C_{1},t_{2})} = t_{2} v{(C_{1},t_{2})} and L{(C_{1},t_{2})} = t_{2} (C_{1} + t_{2}) and L^{2}{(C_{1},t_{2})} = t_{2} (C_{1} + t_{2}) L{(C_{1},t_{2})} and \\int L^{2}{(C_{1},t_{2})} dC_{1} = \\int t_{2} (C_{1} + t_{2}) L{(C_{1},t_{2})} dC_{1} and C_{1} + \\int L^{2}{(C_{1},t_{2})} dC_{1} = C_{1} + \\int t_{2} (C_{1} + t_{2}) L{(C_{1},t_{2})} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)))"], [["times", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Function('v')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Symbol('t_2', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('t_2', commutative=True), Function('v')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('t_2', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))))"], [["times", 4, "Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Pow(Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Integer(2)), Mul(Symbol('t_2', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Pow(Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Integer(2)), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Symbol('t_2', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["add", 6, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Integral(Pow(Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Integer(2)), Tuple(Symbol('C_1', commutative=True)))), Add(Symbol('C_1', commutative=True), Integral(Mul(Symbol('t_2', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True)), Function('L')(Symbol('C_1', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given B{(\\mathbf{f},\\dot{y})} = \\dot{y} + \\mathbf{f}, then obtain \\int (\\int \\log{(B{(\\mathbf{f},\\dot{y})})} d\\mathbf{f})^{\\dot{y}} d\\dot{y} = \\int (\\int \\log{(\\dot{y} + \\mathbf{f})} d\\mathbf{f})^{\\dot{y}} d\\dot{y}", "derivation": "B{(\\mathbf{f},\\dot{y})} = \\dot{y} + \\mathbf{f} and \\log{(B{(\\mathbf{f},\\dot{y})})} = \\log{(\\dot{y} + \\mathbf{f})} and \\int \\log{(B{(\\mathbf{f},\\dot{y})})} d\\mathbf{f} = \\int \\log{(\\dot{y} + \\mathbf{f})} d\\mathbf{f} and (\\int \\log{(B{(\\mathbf{f},\\dot{y})})} d\\mathbf{f})^{\\dot{y}} = (\\int \\log{(\\dot{y} + \\mathbf{f})} d\\mathbf{f})^{\\dot{y}} and \\int (\\int \\log{(B{(\\mathbf{f},\\dot{y})})} d\\mathbf{f})^{\\dot{y}} d\\dot{y} = \\int (\\int \\log{(\\dot{y} + \\mathbf{f})} d\\mathbf{f})^{\\dot{y}} d\\dot{y}", "srepr_derivation": [["get_premise", "Equality(Function('B')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["log", 1], "Equality(log(Function('B')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(log(Function('B')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Integral(log(Function('B')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integral(log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Pow(Integral(log(Function('B')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Pow(Integral(log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given M{(F_{x},\\mathbf{r})} = F_{x}^{\\mathbf{r}}, then derive \\frac{\\partial}{\\partial \\mathbf{r}} M{(F_{x},\\mathbf{r})} = F_{x}^{\\mathbf{r}} \\log{(F_{x})}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{r}^{2}} M{(F_{x},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} F_{x}^{\\mathbf{r}} \\log{(F_{x})}", "derivation": "M{(F_{x},\\mathbf{r})} = F_{x}^{\\mathbf{r}} and \\frac{\\partial}{\\partial \\mathbf{r}} M{(F_{x},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} F_{x}^{\\mathbf{r}} and \\frac{\\partial}{\\partial \\mathbf{r}} M{(F_{x},\\mathbf{r})} = F_{x}^{\\mathbf{r}} \\log{(F_{x})} and \\frac{\\partial^{2}}{\\partial \\mathbf{r}^{2}} M{(F_{x},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} F_{x}^{\\mathbf{r}} \\log{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Mul(Pow(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{1}{(\\Psi^{\\dagger},t_{1})} = \\Psi^{\\dagger} + \\log{(t_{1})}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta_{1}{(\\Psi^{\\dagger},t_{1})} = 1, then obtain t_{1} e^{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\log{(t_{1})})} = e t_{1}", "derivation": "\\theta_{1}{(\\Psi^{\\dagger},t_{1})} = \\Psi^{\\dagger} + \\log{(t_{1})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta_{1}{(\\Psi^{\\dagger},t_{1})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\log{(t_{1})}) and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta_{1}{(\\Psi^{\\dagger},t_{1})} = 1 and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\log{(t_{1})}) = 1 and e^{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\log{(t_{1})})} = e and t_{1} e^{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\log{(t_{1})})} = e t_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"], [["exp", 4], "Equality(exp(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), E)"], [["times", 5, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), exp(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))), Mul(E, Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given i{(\\theta,\\pi,\\mathbf{B})} = (- \\pi + \\theta)^{\\mathbf{B}} and \\hat{p}_0{(\\theta,\\pi,\\mathbf{B})} = \\pi + i{(\\theta,\\pi,\\mathbf{B})}, then obtain \\hat{p}_0^{4 \\theta}{(\\theta,\\pi,\\mathbf{B})} = (\\pi + (- \\pi + \\theta)^{\\mathbf{B}})^{2 \\theta} \\hat{p}_0^{2 \\theta}{(\\theta,\\pi,\\mathbf{B})}", "derivation": "i{(\\theta,\\pi,\\mathbf{B})} = (- \\pi + \\theta)^{\\mathbf{B}} and \\pi + i{(\\theta,\\pi,\\mathbf{B})} = \\pi + (- \\pi + \\theta)^{\\mathbf{B}} and \\hat{p}_0{(\\theta,\\pi,\\mathbf{B})} = \\pi + i{(\\theta,\\pi,\\mathbf{B})} and \\hat{p}_0{(\\theta,\\pi,\\mathbf{B})} = \\pi + (- \\pi + \\theta)^{\\mathbf{B}} and \\hat{p}_0^{\\theta}{(\\theta,\\pi,\\mathbf{B})} = (\\pi + (- \\pi + \\theta)^{\\mathbf{B}})^{\\theta} and \\hat{p}_0^{2 \\theta}{(\\theta,\\pi,\\mathbf{B})} = (\\pi + (- \\pi + \\theta)^{\\mathbf{B}})^{\\theta} \\hat{p}_0^{\\theta}{(\\theta,\\pi,\\mathbf{B})} and \\hat{p}_0^{4 \\theta}{(\\theta,\\pi,\\mathbf{B})} = (\\pi + (- \\pi + \\theta)^{\\mathbf{B}})^{2 \\theta} \\hat{p}_0^{2 \\theta}{(\\theta,\\pi,\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('i')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Function('i')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["times", 5, "Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\theta', commutative=True))"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["power", 6, 2], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(4), Symbol('\\\\theta', commutative=True))), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given T{(I)} = \\cos{(\\cos{(I)})} and \\Psi{(I)} = \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI, then obtain \\Psi{(I)} + 1 = 1", "derivation": "T{(I)} = \\cos{(\\cos{(I)})} and \\int T{(I)} dI = \\int \\cos{(\\cos{(I)})} dI and \\Psi{(I)} = \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI and \\Psi{(I)} = 0 and \\Psi{(I)} + \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI = \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI and \\Psi{(I)} + \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI + 1 = \\int T{(I)} dI - \\int \\cos{(\\cos{(I)})} dI + 1 and \\Psi{(I)} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('I', commutative=True)), cos(cos(Symbol('I', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('I', commutative=True)), Add(Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\Psi')(Symbol('I', commutative=True)), Integer(0))"], [["add", 4, "Add(Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], "Equality(Add(Function('\\\\Psi')(Symbol('I', commutative=True)), Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))), Add(Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Function('\\\\Psi')(Symbol('I', commutative=True)), Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Integer(1)), Add(Integral(Function('T')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Function('\\\\Psi')(Symbol('I', commutative=True)), Integer(1)), Integer(1))"]]}, {"prompt": "Given r{(S,g)} = \\frac{\\partial}{\\partial g} \\frac{S}{g}, then derive \\int r{(S,g)} dS = - \\frac{S^{2}}{2 g^{2}} + \\psi, then obtain \\frac{g^{3} \\int \\frac{\\partial}{\\partial g} \\frac{S}{g} dS}{2 (\\frac{\\partial}{\\partial g} \\frac{r{(S,g)}}{S} + 1)} = \\frac{g^{3} (- \\frac{S^{2}}{2 g^{2}} + \\psi)}{2 (\\frac{\\partial}{\\partial g} \\frac{r{(S,g)}}{S} + 1)}", "derivation": "r{(S,g)} = \\frac{\\partial}{\\partial g} \\frac{S}{g} and \\int r{(S,g)} dS = \\int \\frac{\\partial}{\\partial g} \\frac{S}{g} dS and \\int r{(S,g)} dS = - \\frac{S^{2}}{2 g^{2}} + \\psi and \\frac{g^{3} \\int r{(S,g)} dS}{2} = \\frac{g^{3} (- \\frac{S^{2}}{2 g^{2}} + \\psi)}{2} and \\frac{g^{3} \\int \\frac{\\partial}{\\partial g} \\frac{S}{g} dS}{2} = \\frac{g^{3} (- \\frac{S^{2}}{2 g^{2}} + \\psi)}{2} and \\frac{g^{3} \\int \\frac{\\partial}{\\partial g} \\frac{S}{g} dS}{2 (\\frac{\\partial}{\\partial g} \\frac{r{(S,g)}}{S} + 1)} = \\frac{g^{3} (- \\frac{S^{2}}{2 g^{2}} + \\psi)}{2 (\\frac{\\partial}{\\partial g} \\frac{r{(S,g)}}{S} + 1)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('g', commutative=True), Integer(-2))), Symbol('\\\\psi', commutative=True)))"], [["divide", 3, "Mul(Integer(2), Pow(Symbol('g', commutative=True), Integer(-3)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Integral(Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('g', commutative=True), Integer(-2))), Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Integral(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('g', commutative=True), Integer(-2))), Symbol('\\\\psi', commutative=True))))"], [["divide", 5, "Add(Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Pow(Add(Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Integral(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(3)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('g', commutative=True), Integer(-2))), Symbol('\\\\psi', commutative=True)), Pow(Add(Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\dot{z})} = \\log{(\\dot{z})}, then obtain \\tilde{\\infty} (\\dot{z} + \\frac{\\operatorname{F_{g}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}}) = \\tilde{\\infty} (\\dot{z} + \\frac{\\log{(\\dot{z})}^{\\dot{z}}}{\\dot{z}})", "derivation": "\\operatorname{F_{g}}{(\\dot{z})} = \\log{(\\dot{z})} and \\operatorname{F_{g}}^{\\dot{z}}{(\\dot{z})} = \\log{(\\dot{z})}^{\\dot{z}} and \\frac{\\operatorname{F_{g}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = \\frac{\\log{(\\dot{z})}^{\\dot{z}}}{\\dot{z}} and \\dot{z} + \\frac{\\operatorname{F_{g}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = \\dot{z} + \\frac{\\log{(\\dot{z})}^{\\dot{z}}}{\\dot{z}} and \\tilde{\\infty} (\\dot{z} + \\frac{\\operatorname{F_{g}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}}) = \\tilde{\\infty} (\\dot{z} + \\frac{\\log{(\\dot{z})}^{\\dot{z}}}{\\dot{z}})", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["add", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))))"], [["divide", 4, 0], "Equality(Mul(zoo, Add(Symbol('\\\\dot{z}', commutative=True), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))), Mul(zoo, Add(Symbol('\\\\dot{z}', commutative=True), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(I)} = \\cos{(\\log{(I)})} and \\mathbf{v}{(I)} = \\log{(I)} + \\int \\operatorname{A_{1}}{(I)} dI, then obtain - \\operatorname{A_{1}}{(I)} + \\mathbf{v}{(I)} - \\log{(I)} - \\int \\cos{(\\log{(I)})} dI = - \\operatorname{A_{1}}{(I)}", "derivation": "\\operatorname{A_{1}}{(I)} = \\cos{(\\log{(I)})} and \\int \\operatorname{A_{1}}{(I)} dI = \\int \\cos{(\\log{(I)})} dI and \\log{(I)} + \\int \\operatorname{A_{1}}{(I)} dI = \\log{(I)} + \\int \\cos{(\\log{(I)})} dI and \\mathbf{v}{(I)} = \\log{(I)} + \\int \\operatorname{A_{1}}{(I)} dI and \\mathbf{v}{(I)} = \\log{(I)} + \\int \\cos{(\\log{(I)})} dI and - \\operatorname{A_{1}}{(I)} + \\mathbf{v}{(I)} - \\log{(I)} - \\int \\cos{(\\log{(I)})} dI = - \\operatorname{A_{1}}{(I)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('I', commutative=True)), cos(log(Symbol('I', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["add", 2, "log(Symbol('I', commutative=True))"], "Equality(Add(log(Symbol('I', commutative=True)), Integral(Function('A_1')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(log(Symbol('I', commutative=True)), Integral(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('I', commutative=True)), Add(log(Symbol('I', commutative=True)), Integral(Function('A_1')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('I', commutative=True)), Add(log(Symbol('I', commutative=True)), Integral(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["minus", 5, "Add(Function('A_1')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)), Integral(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('A_1')(Symbol('I', commutative=True))), Function('\\\\mathbf{v}')(Symbol('I', commutative=True)), Mul(Integer(-1), log(Symbol('I', commutative=True))), Mul(Integer(-1), Integral(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))), Mul(Integer(-1), Function('A_1')(Symbol('I', commutative=True))))"]]}, {"prompt": "Given b{(a,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}), then derive \\int b{(a,y^{\\prime})} da = \\dot{\\mathbf{r}} + a, then derive \\delta + a = \\dot{\\mathbf{r}} + a, then obtain \\delta + a + \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da = \\dot{\\mathbf{r}} + a + \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da", "derivation": "b{(a,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) and \\int b{(a,y^{\\prime})} da = \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da and \\int b{(a,y^{\\prime})} da = \\dot{\\mathbf{r}} + a and \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da = \\dot{\\mathbf{r}} + a and \\delta + a = \\dot{\\mathbf{r}} + a and \\delta + a + \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da = \\dot{\\mathbf{r}} + a + \\int \\frac{\\partial}{\\partial y^{\\prime}} (a + y^{\\prime}) da", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('b')(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a', commutative=True)))"], [["add", 5, "Integral(Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True)))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Symbol('a', commutative=True), Integral(Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True)))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('a', commutative=True), Integral(Derivative(Add(Symbol('a', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given L{(m)} = \\sin{(m)}, then derive \\frac{d}{d m} L{(m)} = \\cos{(m)}, then obtain L{(m)} + \\frac{d}{d m} (\\frac{d}{d m} L{(m)})^{m} = L{(m)} + \\frac{d}{d m} (\\frac{d}{d m} \\sin{(m)})^{m}", "derivation": "L{(m)} = \\sin{(m)} and \\frac{d}{d m} L{(m)} = \\frac{d}{d m} \\sin{(m)} and \\frac{d}{d m} L{(m)} = \\cos{(m)} and \\cos{(m)} = \\frac{d}{d m} \\sin{(m)} and (\\frac{d}{d m} L{(m)})^{m} = \\cos^{m}{(m)} and (\\frac{d}{d m} L{(m)})^{m} = (\\frac{d}{d m} \\sin{(m)})^{m} and \\frac{d}{d m} (\\frac{d}{d m} L{(m)})^{m} = \\frac{d}{d m} (\\frac{d}{d m} \\sin{(m)})^{m} and L{(m)} + \\frac{d}{d m} (\\frac{d}{d m} L{(m)})^{m} = L{(m)} + \\frac{d}{d m} (\\frac{d}{d m} \\sin{(m)})^{m}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), cos(Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('m', commutative=True)), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)))"], [["differentiate", 6, "Symbol('m', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["add", 7, "Function('L')(Symbol('m', commutative=True))"], "Equality(Add(Function('L')(Symbol('m', commutative=True)), Derivative(Pow(Derivative(Function('L')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Function('L')(Symbol('m', commutative=True)), Derivative(Pow(Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\omega)} = e^{\\omega}, then obtain \\operatorname{P_{e}}^{2 \\omega}{(\\omega)} (e^{\\omega})^{2 \\omega} = (e^{\\omega})^{4 \\omega}", "derivation": "\\operatorname{P_{e}}{(\\omega)} = e^{\\omega} and \\operatorname{P_{e}}^{\\omega}{(\\omega)} = (e^{\\omega})^{\\omega} and \\operatorname{P_{e}}^{\\omega}{(\\omega)} (e^{\\omega})^{\\omega} = (e^{\\omega})^{2 \\omega} and \\operatorname{P_{e}}^{2 \\omega}{(\\omega)} (e^{\\omega})^{2 \\omega} = (e^{\\omega})^{4 \\omega}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Pow(exp(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True))), Pow(exp(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('\\\\omega', commutative=True)))), Pow(exp(Symbol('\\\\omega', commutative=True)), Mul(Integer(4), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given h{(r_{0})} = \\cos{(\\cos{(r_{0})})}, then obtain h{(r_{0})} \\cos^{2}{(\\cos{(r_{0})})} - h{(r_{0})} = h^{3}{(r_{0})} - h{(r_{0})}", "derivation": "h{(r_{0})} = \\cos{(\\cos{(r_{0})})} and h^{2}{(r_{0})} = h{(r_{0})} \\cos{(\\cos{(r_{0})})} and h^{3}{(r_{0})} = h^{2}{(r_{0})} \\cos{(\\cos{(r_{0})})} and h^{3}{(r_{0})} = h{(r_{0})} \\cos^{2}{(\\cos{(r_{0})})} and h{(r_{0})} \\cos^{2}{(\\cos{(r_{0})})} = h^{2}{(r_{0})} \\cos{(\\cos{(r_{0})})} and h{(r_{0})} \\cos^{2}{(\\cos{(r_{0})})} - h{(r_{0})} = h^{2}{(r_{0})} \\cos{(\\cos{(r_{0})})} - h{(r_{0})} and h{(r_{0})} \\cos^{2}{(\\cos{(r_{0})})} - h{(r_{0})} = h^{3}{(r_{0})} - h{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('r_0', commutative=True)), cos(cos(Symbol('r_0', commutative=True))))"], [["times", 1, "Function('h')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('h')(Symbol('r_0', commutative=True)), cos(cos(Symbol('r_0', commutative=True)))))"], [["times", 2, "Function('h')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(3)), Mul(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(2)), cos(cos(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(3)), Mul(Function('h')(Symbol('r_0', commutative=True)), Pow(cos(cos(Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Function('h')(Symbol('r_0', commutative=True)), Pow(cos(cos(Symbol('r_0', commutative=True))), Integer(2))), Mul(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(2)), cos(cos(Symbol('r_0', commutative=True)))))"], [["minus", 5, "Function('h')(Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Function('h')(Symbol('r_0', commutative=True)), Pow(cos(cos(Symbol('r_0', commutative=True))), Integer(2))), Mul(Integer(-1), Function('h')(Symbol('r_0', commutative=True)))), Add(Mul(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(2)), cos(cos(Symbol('r_0', commutative=True)))), Mul(Integer(-1), Function('h')(Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Function('h')(Symbol('r_0', commutative=True)), Pow(cos(cos(Symbol('r_0', commutative=True))), Integer(2))), Mul(Integer(-1), Function('h')(Symbol('r_0', commutative=True)))), Add(Pow(Function('h')(Symbol('r_0', commutative=True)), Integer(3)), Mul(Integer(-1), Function('h')(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\theta_2)} = \\log{(\\theta_2)}, then derive \\frac{\\frac{d}{d \\theta_2} \\operatorname{C_{1}}{(\\theta_2)}}{\\log{(\\theta_2)}} = \\frac{1}{\\theta_2 \\log{(\\theta_2)}}, then obtain \\frac{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}}{\\log{(\\theta_2)}} = \\frac{1}{\\theta_2 \\log{(\\theta_2)}}", "derivation": "\\operatorname{C_{1}}{(\\theta_2)} = \\log{(\\theta_2)} and \\frac{d}{d \\theta_2} \\operatorname{C_{1}}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and \\frac{\\frac{d}{d \\theta_2} \\operatorname{C_{1}}{(\\theta_2)}}{\\log{(\\theta_2)}} = \\frac{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}}{\\log{(\\theta_2)}} and \\frac{\\frac{d}{d \\theta_2} \\operatorname{C_{1}}{(\\theta_2)}}{\\log{(\\theta_2)}} = \\frac{1}{\\theta_2 \\log{(\\theta_2)}} and \\frac{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}}{\\log{(\\theta_2)}} = \\frac{1}{\\theta_2 \\log{(\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Derivative(Function('C_1')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Derivative(Function('C_1')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{M}{(f^{\\prime})} = e^{f^{\\prime}} and \\rho_{f}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}}, then obtain \\rho_{f}{(f^{\\prime})} - e^{f^{\\prime}} = - e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} e^{f^{\\prime}}", "derivation": "\\mathbf{M}{(f^{\\prime})} = e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\mathbf{M}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\rho_{f}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and - e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} \\mathbf{M}{(f^{\\prime})} = - e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\rho_{f}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\mathbf{M}{(f^{\\prime})} and \\rho_{f}{(f^{\\prime})} - e^{f^{\\prime}} = - e^{f^{\\prime}} + \\frac{d}{d f^{\\prime}} e^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('f^{\\\\prime}', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('f^{\\\\prime}', commutative=True))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), exp(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('f^{\\\\prime}', commutative=True))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})}, then derive \\dot{\\mathbf{r}} + \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = W + \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})}, then obtain \\dot{\\mathbf{r}} + \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = W + \\dot{\\mathbf{r}} x{(\\dot{\\mathbf{r}})}", "derivation": "x{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})} and \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\dot{\\mathbf{r}} + \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\dot{\\mathbf{r}} + \\int \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\dot{\\mathbf{r}} + \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = W + \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})} and \\dot{\\mathbf{r}} + \\int x{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = W + \\dot{\\mathbf{r}} x{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["add", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integral(Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integral(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integral(Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integral(Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given S{(\\hat{p}_0)} = e^{\\sin{(\\hat{p}_0)}} and \\mathbf{B}{(\\hat{p}_0)} = - \\sin{(\\hat{p}_0)}, then obtain S^{\\hat{p}_0}{(\\hat{p}_0)} e^{\\mathbf{B}{(\\hat{p}_0)}} = e^{\\mathbf{B}{(\\hat{p}_0)}} (e^{\\sin{(\\hat{p}_0)}})^{\\hat{p}_0}", "derivation": "S{(\\hat{p}_0)} = e^{\\sin{(\\hat{p}_0)}} and S^{\\hat{p}_0}{(\\hat{p}_0)} = (e^{\\sin{(\\hat{p}_0)}})^{\\hat{p}_0} and S^{\\hat{p}_0}{(\\hat{p}_0)} e^{- \\sin{(\\hat{p}_0)}} = e^{- \\sin{(\\hat{p}_0)}} (e^{\\sin{(\\hat{p}_0)}})^{\\hat{p}_0} and \\mathbf{B}{(\\hat{p}_0)} = - \\sin{(\\hat{p}_0)} and S^{\\hat{p}_0}{(\\hat{p}_0)} e^{\\mathbf{B}{(\\hat{p}_0)}} = e^{\\mathbf{B}{(\\hat{p}_0)}} (e^{\\sin{(\\hat{p}_0)}})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hat{p}_0', commutative=True)), exp(sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(exp(sin(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 2, "exp(sin(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Pow(Function('S')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True))))), Mul(exp(Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True)))), Pow(exp(sin(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('S')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), exp(Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(exp(Function('\\\\mathbf{B}')(Symbol('\\\\hat{p}_0', commutative=True))), Pow(exp(sin(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\psi{(V)} = \\cos{(V)} and \\operatorname{n_{1}}{(V)} = \\int \\psi{(V)} dV, then derive \\int \\psi{(V)} dV = \\hat{\\mathbf{x}} + \\sin{(V)}, then obtain \\operatorname{n_{1}}{(V)} = \\hat{\\mathbf{x}} + \\sin{(V)}", "derivation": "\\psi{(V)} = \\cos{(V)} and \\int \\psi{(V)} dV = \\int \\cos{(V)} dV and \\int \\psi{(V)} dV = \\hat{\\mathbf{x}} + \\sin{(V)} and \\operatorname{n_{1}}{(V)} = \\int \\psi{(V)} dV and \\operatorname{n_{1}}{(V)} = \\hat{\\mathbf{x}} + \\sin{(V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('V', commutative=True)), Integral(Function('\\\\psi')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('n_1')(Symbol('V', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(t)} = \\log{(t)} and \\sigma_{p}{(t)} = \\mathbf{g}{(t)} \\log{(t)}, then obtain \\frac{\\log{(t)}^{2}}{A_{2}} = \\frac{\\mathbf{g}{(t)} \\log{(t)}}{A_{2}}", "derivation": "\\mathbf{g}{(t)} = \\log{(t)} and \\mathbf{g}{(t)} \\log{(t)} = \\log{(t)}^{2} and \\sigma_{p}{(t)} = \\mathbf{g}{(t)} \\log{(t)} and \\sigma_{p}{(t)} = \\log{(t)}^{2} and \\frac{\\sigma_{p}{(t)}}{A_{2}} = \\frac{\\log{(t)}^{2}}{A_{2}} and \\frac{\\sigma_{p}{(t)}}{A_{2}} = \\frac{\\mathbf{g}^{2}{(t)}}{A_{2}} and \\frac{\\mathbf{g}{(t)} \\log{(t)}}{A_{2}} = \\frac{\\mathbf{g}^{2}{(t)}}{A_{2}} and \\frac{\\log{(t)}^{2}}{A_{2}} = \\frac{\\mathbf{g}^{2}{(t)}}{A_{2}} and \\frac{\\log{(t)}^{2}}{A_{2}} = \\frac{\\mathbf{g}{(t)} \\log{(t)}}{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["times", 1, "log(Symbol('t', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('t', commutative=True)), Mul(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\sigma_p')(Symbol('t', commutative=True)), Pow(log(Symbol('t', commutative=True)), Integer(2)))"], [["times", 4, "Pow(Symbol('A_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('t', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(log(Symbol('t', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('t', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(log(Symbol('t', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(log(Symbol('t', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\varphi{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\mathbb{I}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})}, then derive \\mathbb{I}{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})}, then obtain \\frac{d}{d J_{\\varepsilon}} \\varphi{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})}", "derivation": "\\varphi{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} \\varphi{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and \\mathbb{I}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and \\mathbb{I}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\varphi{(J_{\\varepsilon})} and \\mathbb{I}{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} \\varphi{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbb{I}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\mathbb{I}')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Function('\\\\varphi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)} = \\sigma_x^{\\mathbf{J}} and g{(\\mathbf{J},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)}, then obtain g{(\\mathbf{J},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\sigma_x^{\\mathbf{J}}", "derivation": "\\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)} = \\sigma_x^{\\mathbf{J}} and \\mathbf{J} \\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)} = \\mathbf{J} \\sigma_x^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\sigma_x^{\\mathbf{J}} and g{(\\mathbf{J},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\operatorname{t_{1}}{(\\mathbf{J},\\sigma_x)} and g{(\\mathbf{J},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} \\mathbf{J} \\sigma_x^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('g')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(a)} = \\cos{(a)} and U{(i)} = \\cos{(i)}, then obtain \\frac{\\partial}{\\partial a} ((U{(i)} + 1) \\cos{(a)} + 1) = \\frac{\\partial}{\\partial a} ((\\cos{(i)} + 1) \\cos{(a)} + 1)", "derivation": "\\mathbf{J}_f{(a)} = \\cos{(a)} and U{(i)} = \\cos{(i)} and U{(i)} + 1 = \\cos{(i)} + 1 and (U{(i)} + 1) \\cos{(a)} = (\\cos{(i)} + 1) \\cos{(a)} and (U{(i)} + 1) \\cos{(a)} + \\frac{\\mathbf{J}_f{(a)}}{\\cos{(a)}} = (\\cos{(i)} + 1) \\cos{(a)} + \\frac{\\mathbf{J}_f{(a)}}{\\cos{(a)}} and (U{(i)} + 1) \\mathbf{J}_f{(a)} + 1 = (\\cos{(i)} + 1) \\mathbf{J}_f{(a)} + 1 and \\frac{\\partial}{\\partial a} ((U{(i)} + 1) \\mathbf{J}_f{(a)} + 1) = \\frac{\\partial}{\\partial a} ((\\cos{(i)} + 1) \\mathbf{J}_f{(a)} + 1) and \\frac{\\partial}{\\partial a} ((U{(i)} + 1) \\cos{(a)} + 1) = \\frac{\\partial}{\\partial a} ((\\cos{(i)} + 1) \\cos{(a)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], ["get_premise", "Equality(Function('U')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["add", 2, 1], "Equality(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), Add(cos(Symbol('i', commutative=True)), Integer(1)))"], [["divide", 3, "Pow(cos(Symbol('a', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))), Mul(Add(cos(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))))"], [["add", 4, "Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))), Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(-1)))), Add(Mul(Add(cos(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))), Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True))), Integer(1)), Add(Mul(Add(cos(Symbol('i', commutative=True)), Integer(1)), Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True))), Integer(1)))"], [["differentiate", 6, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Add(cos(Symbol('i', commutative=True)), Integer(1)), Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Derivative(Add(Mul(Add(Function('U')(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Add(cos(Symbol('i', commutative=True)), Integer(1)), cos(Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(C)} = e^{\\cos{(C)}}, then derive \\frac{\\frac{d}{d C} \\ddot{x}{(C)}}{C} = - \\frac{e^{\\cos{(C)}} \\sin{(C)}}{C}, then obtain - \\frac{e^{\\cos{(C)}}}{C} = - \\frac{\\ddot{x}{(C)}}{C}", "derivation": "\\ddot{x}{(C)} = e^{\\cos{(C)}} and \\frac{d}{d C} \\ddot{x}{(C)} = \\frac{d}{d C} e^{\\cos{(C)}} and \\frac{\\frac{d}{d C} \\ddot{x}{(C)}}{C} = \\frac{\\frac{d}{d C} e^{\\cos{(C)}}}{C} and \\frac{\\frac{d}{d C} \\ddot{x}{(C)}}{C} = - \\frac{e^{\\cos{(C)}} \\sin{(C)}}{C} and \\frac{\\frac{d}{d C} \\ddot{x}{(C)}}{C} = - \\frac{\\ddot{x}{(C)} \\sin{(C)}}{C} and - \\frac{e^{\\cos{(C)}} \\sin{(C)}}{C} = - \\frac{\\ddot{x}{(C)} \\sin{(C)}}{C} and - \\frac{e^{\\cos{(C)}}}{C} = - \\frac{\\ddot{x}{(C)}}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), exp(cos(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(exp(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["divide", 6, "sin(Symbol('C', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C', commutative=True))))"]]}, {"prompt": "Given A{(\\eta,k,W)} = - W + \\eta - k, then derive \\int A{(\\eta,k,W)} d\\eta = A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k), then obtain e^{\\frac{\\eta^{2}}{2} + \\eta (- W - k) + u} = e^{A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k)}", "derivation": "A{(\\eta,k,W)} = - W + \\eta - k and \\int A{(\\eta,k,W)} d\\eta = \\int (- W + \\eta - k) d\\eta and \\int A{(\\eta,k,W)} d\\eta = A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k) and \\int (- W + \\eta - k) d\\eta = A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k) and e^{\\int (- W + \\eta - k) d\\eta} = e^{A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k)} and e^{\\frac{\\eta^{2}}{2} + \\eta (- W - k) + u} = e^{A_{z} + \\frac{\\eta^{2}}{2} + \\eta (- W - k)}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('\\\\eta', commutative=True), Symbol('k', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\eta', commutative=True), Symbol('k', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A')(Symbol('\\\\eta', commutative=True), Symbol('k', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))))))"], [["exp", 4], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))), exp(Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)))))))"], [["evaluate_integrals", 5], "Equality(exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)))), Symbol('u', commutative=True))), exp(Add(Symbol('A_z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)))))))"]]}, {"prompt": "Given A{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\phi{(g_{\\varepsilon})} = - A{(g_{\\varepsilon})} + \\cos{(g_{\\varepsilon})}, then obtain \\cos^{g_{\\varepsilon}}{(g_{\\varepsilon})} = A^{g_{\\varepsilon}}{(g_{\\varepsilon})}", "derivation": "A{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\phi{(g_{\\varepsilon})} = - A{(g_{\\varepsilon})} + \\cos{(g_{\\varepsilon})} and A{(g_{\\varepsilon})} - \\phi{(g_{\\varepsilon})} = - \\phi{(g_{\\varepsilon})} + \\cos{(g_{\\varepsilon})} and 2 A{(g_{\\varepsilon})} - \\cos{(g_{\\varepsilon})} = A{(g_{\\varepsilon})} and (2 A{(g_{\\varepsilon})} - \\cos{(g_{\\varepsilon})})^{g_{\\varepsilon}} = A^{g_{\\varepsilon}}{(g_{\\varepsilon})} and 2 A{(g_{\\varepsilon})} - \\cos{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\cos^{g_{\\varepsilon}}{(g_{\\varepsilon})} = A^{g_{\\varepsilon}}{(g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True))), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Function('\\\\phi')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('g_{\\\\varepsilon}', commutative=True))), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["power", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('A')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A)} = e^{A}, then obtain \\frac{d}{d A} \\cos{(1)} = \\frac{d}{d A} \\cos{((\\frac{e^{A}}{\\operatorname{P_{e}}{(A)}})^{A})}", "derivation": "\\operatorname{P_{e}}{(A)} = e^{A} and 1 = \\frac{e^{A}}{\\operatorname{P_{e}}{(A)}} and 1 = (\\frac{e^{A}}{\\operatorname{P_{e}}{(A)}})^{A} and \\cos{(1)} = \\cos{((\\frac{e^{A}}{\\operatorname{P_{e}}{(A)}})^{A})} and \\frac{d}{d A} \\cos{(1)} = \\frac{d}{d A} \\cos{((\\frac{e^{A}}{\\operatorname{P_{e}}{(A)}})^{A})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["divide", 1, "Function('P_e')(Symbol('A', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_e')(Symbol('A', commutative=True)), Integer(-1)), exp(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('P_e')(Symbol('A', commutative=True)), Integer(-1)), exp(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["cos", 3], "Equality(cos(Integer(1)), cos(Pow(Mul(Pow(Function('P_e')(Symbol('A', commutative=True)), Integer(-1)), exp(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(cos(Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(Pow(Mul(Pow(Function('P_e')(Symbol('A', commutative=True)), Integer(-1)), exp(Symbol('A', commutative=True))), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(A_{y})} = \\cos{(A_{y})}, then obtain \\int \\frac{d}{d A_{y}} h{(A_{y})} \\cos{(A_{y})} dA_{y} = \\dot{\\mathbf{r}} - \\sin^{2}{(A_{y})}", "derivation": "h{(A_{y})} = \\cos{(A_{y})} and h{(A_{y})} \\cos{(A_{y})} = \\cos^{2}{(A_{y})} and \\frac{d}{d A_{y}} h{(A_{y})} \\cos{(A_{y})} = \\frac{d}{d A_{y}} \\cos^{2}{(A_{y})} and \\int \\frac{d}{d A_{y}} h{(A_{y})} \\cos{(A_{y})} dA_{y} = \\int \\frac{d}{d A_{y}} \\cos^{2}{(A_{y})} dA_{y} and \\int \\frac{d}{d A_{y}} h{(A_{y})} \\cos{(A_{y})} dA_{y} = \\dot{\\mathbf{r}} - \\sin^{2}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["times", 1, "cos(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('h')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Pow(cos(Symbol('A_y', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Function('h')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('A_y', commutative=True)), Integer(2)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('A_y', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('h')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('A_y', commutative=True))), Integral(Derivative(Pow(cos(Symbol('A_y', commutative=True)), Integer(2)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Mul(Function('h')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Pow(sin(Symbol('A_y', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given W{(\\pi,\\mathbf{M})} = \\mathbf{M} + \\pi, then obtain - \\mathbf{M} \\frac{\\partial^{2}}{\\partial \\mathbf{M}^{2}} \\frac{- \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})}}{W{(\\pi,\\mathbf{M})}} = - \\mathbf{M} \\frac{d^{2}}{d \\mathbf{M}^{2}} 0", "derivation": "W{(\\pi,\\mathbf{M})} = \\mathbf{M} + \\pi and - \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})} = 0 and \\frac{- \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})}}{W{(\\pi,\\mathbf{M})}} = 0 and \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{- \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})}}{W{(\\pi,\\mathbf{M})}} = \\frac{d}{d \\mathbf{M}} 0 and \\frac{\\partial^{2}}{\\partial \\mathbf{M}^{2}} \\frac{- \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})}}{W{(\\pi,\\mathbf{M})}} = \\frac{d^{2}}{d \\mathbf{M}^{2}} 0 and - \\mathbf{M} \\frac{\\partial^{2}}{\\partial \\mathbf{M}^{2}} \\frac{- \\mathbf{M} - \\pi + W{(\\pi,\\mathbf{M})}}{W{(\\pi,\\mathbf{M})}} = - \\mathbf{M} \\frac{d^{2}}{d \\mathbf{M}^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(0))"], [["divide", 2, "Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))))"], [["times", 5, "Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{X}{(z)} = e^{z}, then derive 0 = \\frac{e^{z}}{\\hat{X}{(z)}} - \\frac{e^{z} \\frac{d}{d z} \\hat{X}{(z)}}{\\hat{X}^{2}{(z)}}, then obtain \\iint 0 dz dz = \\iint (1 - e^{- z} \\frac{d}{d z} e^{z}) dz dz", "derivation": "\\hat{X}{(z)} = e^{z} and 1 = \\frac{e^{z}}{\\hat{X}{(z)}} and \\frac{d}{d z} 1 = \\frac{d}{d z} \\frac{e^{z}}{\\hat{X}{(z)}} and 0 = \\frac{e^{z}}{\\hat{X}{(z)}} - \\frac{e^{z} \\frac{d}{d z} \\hat{X}{(z)}}{\\hat{X}^{2}{(z)}} and 0 = 1 - \\frac{\\frac{d}{d z} \\hat{X}{(z)}}{\\hat{X}{(z)}} and \\int 0 dz = \\int (1 - \\frac{\\frac{d}{d z} \\hat{X}{(z)}}{\\hat{X}{(z)}}) dz and \\iint 0 dz dz = \\iint (1 - \\frac{\\frac{d}{d z} \\hat{X}{(z)}}{\\hat{X}{(z)}}) dz dz and \\iint 0 dz dz = \\iint (1 - e^{- z} \\frac{d}{d z} e^{z}) dz dz", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{X}')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["divide", 1, "Function('\\\\hat{X}')(Symbol('z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), exp(Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), exp(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-2)), exp(Symbol('z', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["integrate", 5, "Symbol('z', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('z', commutative=True))))"], [["integrate", 6, "Symbol('z', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integral(Integer(0), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('z', commutative=True))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(t_{2})} = \\int \\sin{(t_{2})} dt_{2}, then derive (a - p) (- a + \\operatorname{A_{z}}{(t_{2})} + \\cos{(t_{2})}) = 0, then obtain a (a - p) (- a + \\cos{(t_{2})} + \\int \\sin{(t_{2})} dt_{2}) = 0", "derivation": "\\operatorname{A_{z}}{(t_{2})} = \\int \\sin{(t_{2})} dt_{2} and \\operatorname{A_{z}}{(t_{2})} - \\int \\sin{(t_{2})} dt_{2} = 0 and (\\operatorname{A_{z}}{(t_{2})} - \\int \\sin{(t_{2})} dt_{2}) (- p + \\cos{(t_{2})} + \\int \\sin{(t_{2})} dt_{2}) = 0 and (a - p) (- a + \\operatorname{A_{z}}{(t_{2})} + \\cos{(t_{2})}) = 0 and a (a - p) (- a + \\operatorname{A_{z}}{(t_{2})} + \\cos{(t_{2})}) = 0 and a (a - p) (- a + \\cos{(t_{2})} + \\int \\sin{(t_{2})} dt_{2}) = 0", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('t_2', commutative=True)), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["minus", 1, "Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Add(Function('A_z')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Integer(0))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('p', commutative=True)), cos(Symbol('t_2', commutative=True)), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], "Equality(Mul(Add(Function('A_z')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), cos(Symbol('t_2', commutative=True)), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('A_z')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True)))), Integer(0))"], [["times", 4, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('A_z')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('a', commutative=True), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('t_2', commutative=True)), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\eta{(\\Omega,\\mathbf{J}_P)} = - \\Omega + e^{\\mathbf{J}_P} and \\operatorname{x^{{\\}'}}{(E)} = \\cos{(e^{E})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} \\eta{(\\Omega,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\Omega - \\operatorname{x^{{\\}'}}{(E)} + e^{\\mathbf{J}_P} + \\cos{(e^{E})})", "derivation": "\\eta{(\\Omega,\\mathbf{J}_P)} = - \\Omega + e^{\\mathbf{J}_P} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\eta{(\\Omega,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\Omega + e^{\\mathbf{J}_P}) and \\operatorname{x^{{\\}'}}{(E)} = \\cos{(e^{E})} and - \\Omega + \\operatorname{x^{{\\}'}}{(E)} = - \\Omega + \\cos{(e^{E})} and - \\Omega = - \\Omega - \\operatorname{x^{{\\}'}}{(E)} + \\cos{(e^{E})} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\eta{(\\Omega,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\Omega - \\operatorname{x^{{\\}'}}{(E)} + e^{\\mathbf{J}_P} + \\cos{(e^{E})})", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('x^\\\\prime')(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True))))"], [["minus", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('x^\\\\prime')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('E', commutative=True)))))"], [["minus", 4, "Function('x^\\\\prime')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('E', commutative=True))), cos(exp(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('E', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(exp(Symbol('E', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(\\rho_f)} = \\log{(\\rho_f)}, then obtain - \\frac{\\frac{d}{d \\rho_f} M{(\\rho_f)}}{\\log{(\\rho_f)}} = - \\frac{\\frac{d}{d \\rho_f} \\log{(\\rho_f)}}{\\log{(\\rho_f)}}", "derivation": "M{(\\rho_f)} = \\log{(\\rho_f)} and \\frac{d}{d \\rho_f} M{(\\rho_f)} = \\frac{d}{d \\rho_f} \\log{(\\rho_f)} and \\frac{\\frac{d}{d \\rho_f} M{(\\rho_f)}}{\\log{(\\rho_f)}} = \\frac{\\frac{d}{d \\rho_f} \\log{(\\rho_f)}}{\\log{(\\rho_f)}} and - \\frac{\\frac{d}{d \\rho_f} M{(\\rho_f)}}{\\log{(\\rho_f)}} = - \\frac{\\frac{d}{d \\rho_f} \\log{(\\rho_f)}}{\\log{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(\\mathbf{P},c)} = - \\mathbf{P} + c, then derive (- \\mathbf{P} + c) \\frac{\\partial}{\\partial c} W{(\\mathbf{P},c)} + W{(\\mathbf{P},c)} = - 2 \\mathbf{P} + 2 c, then obtain W{(\\mathbf{P},c)} \\frac{\\partial}{\\partial c} W{(\\mathbf{P},c)} + W{(\\mathbf{P},c)} = - 2 \\mathbf{P} + 2 c", "derivation": "W{(\\mathbf{P},c)} = - \\mathbf{P} + c and (- \\mathbf{P} + c) W{(\\mathbf{P},c)} = (- \\mathbf{P} + c)^{2} and \\frac{\\partial}{\\partial c} (- \\mathbf{P} + c) W{(\\mathbf{P},c)} = \\frac{\\partial}{\\partial c} (- \\mathbf{P} + c)^{2} and (- \\mathbf{P} + c) \\frac{\\partial}{\\partial c} W{(\\mathbf{P},c)} + W{(\\mathbf{P},c)} = - 2 \\mathbf{P} + 2 c and W{(\\mathbf{P},c)} \\frac{\\partial}{\\partial c} W{(\\mathbf{P},c)} + W{(\\mathbf{P},c)} = - 2 \\mathbf{P} + 2 c", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)), Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)), Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)), Integer(2)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True)), Derivative(Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Function('W')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(m,A_{z})} = \\log{(\\frac{A_{z}}{m})}, then obtain \\frac{A_{z} \\operatorname{E_{n}}{(m,A_{z})}}{m \\frac{\\partial}{\\partial A_{z}} \\operatorname{E_{n}}{(m,A_{z})}} = \\frac{A_{z} \\log{(\\frac{A_{z}}{m})}}{m \\frac{\\partial}{\\partial A_{z}} \\operatorname{E_{n}}{(m,A_{z})}}", "derivation": "\\operatorname{E_{n}}{(m,A_{z})} = \\log{(\\frac{A_{z}}{m})} and \\frac{\\partial}{\\partial A_{z}} \\operatorname{E_{n}}{(m,A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\log{(\\frac{A_{z}}{m})} and \\frac{A_{z} \\operatorname{E_{n}}{(m,A_{z})}}{m \\frac{\\partial}{\\partial A_{z}} \\log{(\\frac{A_{z}}{m})}} = \\frac{A_{z} \\log{(\\frac{A_{z}}{m})}}{m \\frac{\\partial}{\\partial A_{z}} \\log{(\\frac{A_{z}}{m})}} and \\frac{A_{z} \\operatorname{E_{n}}{(m,A_{z})}}{m \\frac{\\partial}{\\partial A_{z}} \\operatorname{E_{n}}{(m,A_{z})}} = \\frac{A_{z} \\log{(\\frac{A_{z}}{m})}}{m \\frac{\\partial}{\\partial A_{z}} \\operatorname{E_{n}}{(m,A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["divide", 1, "Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True), Derivative(log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), Pow(Derivative(log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Pow(Derivative(log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), Pow(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), log(Mul(Symbol('A_z', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))), Pow(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given U{(Z)} = \\log{(\\log{(Z)})} and \\operatorname{n_{2}}{(Z)} = \\frac{1}{\\log{(Z)}}, then obtain (U{(Z)} \\operatorname{n_{2}}{(Z)})^{Z} - \\operatorname{n_{2}}{(Z)} = (\\operatorname{n_{2}}{(Z)} \\log{(\\log{(Z)})})^{Z} - \\operatorname{n_{2}}{(Z)}", "derivation": "U{(Z)} = \\log{(\\log{(Z)})} and \\frac{U{(Z)}}{\\log{(Z)}} = \\frac{\\log{(\\log{(Z)})}}{\\log{(Z)}} and (\\frac{U{(Z)}}{\\log{(Z)}})^{Z} = (\\frac{\\log{(\\log{(Z)})}}{\\log{(Z)}})^{Z} and (\\frac{U{(Z)}}{\\log{(Z)}})^{Z} - \\frac{1}{\\log{(Z)}} = (\\frac{\\log{(\\log{(Z)})}}{\\log{(Z)}})^{Z} - \\frac{1}{\\log{(Z)}} and \\operatorname{n_{2}}{(Z)} = \\frac{1}{\\log{(Z)}} and (U{(Z)} \\operatorname{n_{2}}{(Z)})^{Z} - \\operatorname{n_{2}}{(Z)} = (\\operatorname{n_{2}}{(Z)} \\log{(\\log{(Z)})})^{Z} - \\operatorname{n_{2}}{(Z)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True))))"], [["divide", 1, "log(Symbol('Z', commutative=True))"], "Equality(Mul(Function('U')(Symbol('Z', commutative=True)), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), log(log(Symbol('Z', commutative=True)))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Function('U')(Symbol('Z', commutative=True)), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)), Pow(Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), log(log(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)))"], [["minus", 3, "Pow(log(Symbol('Z', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Mul(Function('U')(Symbol('Z', commutative=True)), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Z', commutative=True)), Integer(-1)))), Add(Pow(Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), log(log(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Z', commutative=True)), Integer(-1)))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('Z', commutative=True)), Pow(log(Symbol('Z', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Mul(Function('U')(Symbol('Z', commutative=True)), Function('n_2')(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True)))), Add(Pow(Mul(Function('n_2')(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(r,v)} = \\frac{r}{v}, then derive 1 = - \\frac{r}{v^{2} \\frac{\\partial}{\\partial v} \\operatorname{C_{1}}{(r,v)}}, then obtain 1 = - \\frac{r}{v^{2} \\frac{\\partial}{\\partial v} \\frac{r}{v}}", "derivation": "\\operatorname{C_{1}}{(r,v)} = \\frac{r}{v} and \\frac{\\partial}{\\partial v} \\operatorname{C_{1}}{(r,v)} = \\frac{\\partial}{\\partial v} \\frac{r}{v} and 1 = \\frac{\\frac{\\partial}{\\partial v} \\frac{r}{v}}{\\frac{\\partial}{\\partial v} \\operatorname{C_{1}}{(r,v)}} and 1 = - \\frac{r}{v^{2} \\frac{\\partial}{\\partial v} \\operatorname{C_{1}}{(r,v)}} and 1 = - \\frac{r}{v^{2} \\frac{\\partial}{\\partial v} \\frac{r}{v}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('r', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('r', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('C_1')(Symbol('r', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Mul(Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(Function('C_1')(Symbol('r', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(Integer(-1), Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-2)), Pow(Derivative(Function('C_1')(Symbol('r', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Integer(-1), Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-2)), Pow(Derivative(Mul(Symbol('r', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(E_{x},g)} = \\cos{(E_{x} + g)}, then derive \\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)} = - \\sin{(E_{x} + g)}, then obtain (\\frac{\\partial}{\\partial E_{x}} \\cos{(E_{x} + g)})^{E_{x}} = (\\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)})^{E_{x}}", "derivation": "\\operatorname{A_{y}}{(E_{x},g)} = \\cos{(E_{x} + g)} and \\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)} = \\frac{\\partial}{\\partial E_{x}} \\cos{(E_{x} + g)} and \\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)} = - \\sin{(E_{x} + g)} and (\\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)})^{E_{x}} = (- \\sin{(E_{x} + g)})^{E_{x}} and (\\frac{\\partial}{\\partial E_{x}} \\cos{(E_{x} + g)})^{E_{x}} = (- \\sin{(E_{x} + g)})^{E_{x}} and (\\frac{\\partial}{\\partial E_{x}} \\cos{(E_{x} + g)})^{E_{x}} = (\\frac{\\partial}{\\partial E_{x}} \\operatorname{A_{y}}{(E_{x},g)})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('E_x', commutative=True), Symbol('g', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('E_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('E_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True)))))"], [["power", 3, "Symbol('E_x', commutative=True)"], "Equality(Pow(Derivative(Function('A_y')(Symbol('E_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True)))), Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(cos(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True)))), Symbol('E_x', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(cos(Add(Symbol('E_x', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)), Pow(Derivative(Function('A_y')(Symbol('E_x', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(S)} = \\cos{(S)} and \\lambda{(S)} = \\frac{\\cos{(S)}}{S}, then obtain C_{d}^{\\hat{x}_0} + (\\frac{\\operatorname{E_{\\lambda}}{(S)}}{S})^{S} = C_{d}^{\\hat{x}_0} + \\lambda^{S}{(S)}", "derivation": "\\operatorname{E_{\\lambda}}{(S)} = \\cos{(S)} and \\frac{\\operatorname{E_{\\lambda}}{(S)}}{S} = \\frac{\\cos{(S)}}{S} and \\lambda{(S)} = \\frac{\\cos{(S)}}{S} and (\\frac{\\operatorname{E_{\\lambda}}{(S)}}{S})^{S} = (\\frac{\\cos{(S)}}{S})^{S} and (\\frac{\\operatorname{E_{\\lambda}}{(S)}}{S})^{S} = \\lambda^{S}{(S)} and C_{d}^{\\hat{x}_0} + (\\frac{\\operatorname{E_{\\lambda}}{(S)}}{S})^{S} = C_{d}^{\\hat{x}_0} + \\lambda^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('S', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Function('\\\\lambda')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["add", 5, "Pow(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('S', commutative=True))), Symbol('S', commutative=True))), Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Function('\\\\lambda')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\mathbf{p},G)} = \\frac{G}{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial G} \\pi{(\\mathbf{p},G)} = \\frac{1}{\\mathbf{p}}, then obtain \\frac{\\partial}{\\partial G} \\frac{G}{\\mathbf{p}} = \\frac{1}{\\mathbf{p}}", "derivation": "\\pi{(\\mathbf{p},G)} = \\frac{G}{\\mathbf{p}} and \\pi{(\\mathbf{p},G)} - \\frac{1}{\\mathbf{p}} = \\frac{G}{\\mathbf{p}} - \\frac{1}{\\mathbf{p}} and \\frac{\\partial}{\\partial G} (\\pi{(\\mathbf{p},G)} - \\frac{1}{\\mathbf{p}}) = \\frac{\\partial}{\\partial G} (\\frac{G}{\\mathbf{p}} - \\frac{1}{\\mathbf{p}}) and \\frac{\\partial}{\\partial G} \\pi{(\\mathbf{p},G)} = \\frac{1}{\\mathbf{p}} and \\frac{\\partial}{\\partial G} \\frac{G}{\\mathbf{p}} = \\frac{1}{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\pi,A_{2},s)} = \\frac{A_{2}}{\\pi} - s, then obtain \\frac{\\partial}{\\partial A_{2}} \\int (- s \\eta^{\\prime}{(\\pi,A_{2},s)})^{s} ds = \\frac{\\partial}{\\partial A_{2}} \\int (- s (\\frac{A_{2}}{\\pi} - s))^{s} ds", "derivation": "\\eta^{\\prime}{(\\pi,A_{2},s)} = \\frac{A_{2}}{\\pi} - s and - s \\eta^{\\prime}{(\\pi,A_{2},s)} = - s (\\frac{A_{2}}{\\pi} - s) and (- s \\eta^{\\prime}{(\\pi,A_{2},s)})^{s} = (- s (\\frac{A_{2}}{\\pi} - s))^{s} and \\int (- s \\eta^{\\prime}{(\\pi,A_{2},s)})^{s} ds = \\int (- s (\\frac{A_{2}}{\\pi} - s))^{s} ds and \\frac{\\partial}{\\partial A_{2}} \\int (- s \\eta^{\\prime}{(\\pi,A_{2},s)})^{s} ds = \\frac{\\partial}{\\partial A_{2}} \\int (- s (\\frac{A_{2}}{\\pi} - s))^{s} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('s', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('s', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), Symbol('s', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True)))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 4, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Integral(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Integer(-1), Symbol('s', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{B})} = \\log{(e^{\\mathbf{B}})}, then obtain \\frac{d}{d \\mathbf{B}} (2 \\operatorname{n_{2}}{(\\mathbf{B})} + \\log{(e^{\\mathbf{B}})}) = \\frac{d}{d \\mathbf{B}} (\\operatorname{n_{2}}{(\\mathbf{B})} + 2 \\log{(e^{\\mathbf{B}})})", "derivation": "\\operatorname{n_{2}}{(\\mathbf{B})} = \\log{(e^{\\mathbf{B}})} and 2 \\operatorname{n_{2}}{(\\mathbf{B})} = \\operatorname{n_{2}}{(\\mathbf{B})} + \\log{(e^{\\mathbf{B}})} and 2 \\operatorname{n_{2}}{(\\mathbf{B})} + \\log{(e^{\\mathbf{B}})} = \\operatorname{n_{2}}{(\\mathbf{B})} + 2 \\log{(e^{\\mathbf{B}})} and \\frac{d}{d \\mathbf{B}} (2 \\operatorname{n_{2}}{(\\mathbf{B})} + \\log{(e^{\\mathbf{B}})}) = \\frac{d}{d \\mathbf{B}} (\\operatorname{n_{2}}{(\\mathbf{B})} + 2 \\log{(e^{\\mathbf{B}})})", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), log(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 1, "Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), log(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["add", 2, "log(exp(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), log(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), log(exp(Symbol('\\\\mathbf{B}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), log(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), log(exp(Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(f^{\\prime})} = \\int \\cos{(f^{\\prime})} df^{\\prime}, then derive \\log{(\\dot{y}{(f^{\\prime})})} = \\log{(f + \\sin{(f^{\\prime})})}, then derive \\log{(\\pi + \\sin{(f^{\\prime})})} = \\log{(f + \\sin{(f^{\\prime})})}, then obtain \\sin{(\\log{(\\pi + \\sin{(f^{\\prime})})})} = \\sin{(\\log{(\\int \\cos{(f^{\\prime})} df^{\\prime})})}", "derivation": "\\dot{y}{(f^{\\prime})} = \\int \\cos{(f^{\\prime})} df^{\\prime} and \\log{(\\dot{y}{(f^{\\prime})})} = \\log{(\\int \\cos{(f^{\\prime})} df^{\\prime})} and \\log{(\\dot{y}{(f^{\\prime})})} = \\log{(f + \\sin{(f^{\\prime})})} and \\log{(\\int \\cos{(f^{\\prime})} df^{\\prime})} = \\log{(f + \\sin{(f^{\\prime})})} and \\log{(\\pi + \\sin{(f^{\\prime})})} = \\log{(f + \\sin{(f^{\\prime})})} and \\log{(\\pi + \\sin{(f^{\\prime})})} = \\log{(\\int \\cos{(f^{\\prime})} df^{\\prime})} and \\sin{(\\log{(\\pi + \\sin{(f^{\\prime})})})} = \\sin{(\\log{(\\int \\cos{(f^{\\prime})} df^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('f^{\\\\prime}', commutative=True)), Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\dot{y}')(Symbol('f^{\\\\prime}', commutative=True))), log(Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(log(Function('\\\\dot{y}')(Symbol('f^{\\\\prime}', commutative=True))), log(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(log(Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), log(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(log(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))), log(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(log(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))), log(Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["sin", 6], "Equality(sin(log(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True))))), sin(log(Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\rho)} = e^{\\rho}, then obtain e^{- \\rho} = \\operatorname{A_{x}}^{-1 - \\frac{e^{\\rho}}{\\operatorname{A_{x}}{(\\rho)}}}{(\\rho)} e^{\\rho}", "derivation": "\\operatorname{A_{x}}{(\\rho)} = e^{\\rho} and 1 = \\frac{e^{\\rho}}{\\operatorname{A_{x}}{(\\rho)}} and 2 = 1 + \\frac{e^{\\rho}}{\\operatorname{A_{x}}{(\\rho)}} and e^{- \\rho} = \\frac{1}{\\operatorname{A_{x}}{(\\rho)}} and \\frac{1}{\\operatorname{A_{x}}{(\\rho)}} = \\frac{e^{\\rho}}{\\operatorname{A_{x}}^{2}{(\\rho)}} and \\frac{1}{\\operatorname{A_{x}}{(\\rho)}} = \\operatorname{A_{x}}^{-1 - \\frac{e^{\\rho}}{\\operatorname{A_{x}}{(\\rho)}}}{(\\rho)} e^{\\rho} and e^{- \\rho} = \\operatorname{A_{x}}^{-1 - \\frac{e^{\\rho}}{\\operatorname{A_{x}}{(\\rho)}}}{(\\rho)} e^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Function('A_x')(Symbol('\\\\rho', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))))"], [["add", 2, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True)))))"], [["divide", 2, "exp(Symbol('\\\\rho', commutative=True))"], "Equality(exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)))"], [["divide", 2, "Function('A_x')(Symbol('\\\\rho', commutative=True))"], "Equality(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Mul(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-2)), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Mul(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Add(Integer(-1), Mul(Integer(-1), Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))))), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Mul(Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Add(Integer(-1), Mul(Integer(-1), Pow(Function('A_x')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))))), exp(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\omega{(J_{\\varepsilon})} = \\cos{(\\log{(J_{\\varepsilon})})}, then obtain \\frac{\\int - \\omega^{2}{(J_{\\varepsilon})} dJ_{\\varepsilon}}{\\log{(J_{\\varepsilon})}} = \\frac{\\int - \\omega{(J_{\\varepsilon})} \\cos{(\\log{(J_{\\varepsilon})})} dJ_{\\varepsilon}}{\\log{(J_{\\varepsilon})}}", "derivation": "\\omega{(J_{\\varepsilon})} = \\cos{(\\log{(J_{\\varepsilon})})} and - \\omega^{2}{(J_{\\varepsilon})} = - \\omega{(J_{\\varepsilon})} \\cos{(\\log{(J_{\\varepsilon})})} and \\int - \\omega^{2}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int - \\omega{(J_{\\varepsilon})} \\cos{(\\log{(J_{\\varepsilon})})} dJ_{\\varepsilon} and \\frac{\\int - \\omega^{2}{(J_{\\varepsilon})} dJ_{\\varepsilon}}{\\log{(J_{\\varepsilon})}} = \\frac{\\int - \\omega{(J_{\\varepsilon})} \\cos{(\\log{(J_{\\varepsilon})})} dJ_{\\varepsilon}}{\\log{(J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["divide", 3, "log(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Pow(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Function('\\\\omega')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{p},t_{1})} = t_{1} + \\log{(\\mathbf{p})}, then obtain - t_{1} - \\mathbf{P}{(\\mathbf{p},t_{1})} - \\log{(\\mathbf{p})} + \\log{(\\int \\mathbf{P}{(\\mathbf{p},t_{1})} dt_{1})} = - t_{1} - \\mathbf{P}{(\\mathbf{p},t_{1})} - \\log{(\\mathbf{p})} + \\log{(\\int (t_{1} + \\log{(\\mathbf{p})}) dt_{1})}", "derivation": "\\mathbf{P}{(\\mathbf{p},t_{1})} = t_{1} + \\log{(\\mathbf{p})} and \\int \\mathbf{P}{(\\mathbf{p},t_{1})} dt_{1} = \\int (t_{1} + \\log{(\\mathbf{p})}) dt_{1} and \\log{(\\int \\mathbf{P}{(\\mathbf{p},t_{1})} dt_{1})} = \\log{(\\int (t_{1} + \\log{(\\mathbf{p})}) dt_{1})} and - t_{1} - \\mathbf{P}{(\\mathbf{p},t_{1})} - \\log{(\\mathbf{p})} + \\log{(\\int \\mathbf{P}{(\\mathbf{p},t_{1})} dt_{1})} = - t_{1} - \\mathbf{P}{(\\mathbf{p},t_{1})} - \\log{(\\mathbf{p})} + \\log{(\\int (t_{1} + \\log{(\\mathbf{p})}) dt_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('t_1', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), log(Integral(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('t_1', commutative=True)))))"], [["minus", 3, "Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\mathbf{p}', commutative=True))), log(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\mathbf{p}', commutative=True))), log(Integral(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('t_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbb{I},m)} = \\mathbb{I}^{m} and i{(\\mathbb{I},m)} = \\mathbb{I}^{m}, then obtain \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(i{(\\mathbb{I},m)})} = \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(\\mathbb{I}^{m})}", "derivation": "\\operatorname{v_{x}}{(\\mathbb{I},m)} = \\mathbb{I}^{m} and \\log{(\\operatorname{v_{x}}{(\\mathbb{I},m)})} = \\log{(\\mathbb{I}^{m})} and i{(\\mathbb{I},m)} = \\mathbb{I}^{m} and \\operatorname{v_{x}}{(\\mathbb{I},m)} = i{(\\mathbb{I},m)} and \\frac{\\partial}{\\partial m} \\log{(\\operatorname{v_{x}}{(\\mathbb{I},m)})} = \\frac{\\partial}{\\partial m} \\log{(\\mathbb{I}^{m})} and \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(\\operatorname{v_{x}}{(\\mathbb{I},m)})} = \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(\\mathbb{I}^{m})} and \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(i{(\\mathbb{I},m)})} = \\sin^{m}{(i{(\\mathbb{I},m)})} + \\frac{\\partial}{\\partial m} \\log{(\\mathbb{I}^{m})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)))"], [["log", 1], "Equality(log(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), log(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)), Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(log(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["add", 5, "Pow(sin(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))"], "Equality(Add(Pow(sin(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Derivative(log(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Pow(sin(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Derivative(log(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(sin(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Derivative(log(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Pow(sin(Function('i')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Derivative(log(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} = \\dot{x} + \\sin{(\\Psi)} and \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} = U - \\lambda, then obtain - \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} + \\int \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} d\\lambda = - \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} + \\int (U - \\lambda) d\\lambda", "derivation": "\\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} = \\dot{x} + \\sin{(\\Psi)} and - \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} = - \\dot{x} - \\sin{(\\Psi)} and \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} = U - \\lambda and \\int \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} d\\lambda = \\int (U - \\lambda) d\\lambda and - \\dot{x} - \\sin{(\\Psi)} + \\int \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} d\\lambda = - \\dot{x} - \\sin{(\\Psi)} + \\int (U - \\lambda) d\\lambda and - \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} + \\int \\operatorname{V_{\\mathbf{E}}}{(U,\\lambda)} d\\lambda = - \\operatorname{f^{\\prime}}{(\\dot{x},\\Psi)} + \\int (U - \\lambda) d\\lambda", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["minus", 4, "Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given q{(f_{E},\\sigma_x)} = (e^{\\sigma_x})^{f_{E}}, then derive \\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)} = f_{E} (e^{\\sigma_x})^{f_{E}}, then obtain \\frac{(e^{\\sigma_x})^{- f_{E}} \\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)}}{f_{E}} = 1", "derivation": "q{(f_{E},\\sigma_x)} = (e^{\\sigma_x})^{f_{E}} and \\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} (e^{\\sigma_x})^{f_{E}} and \\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)} = f_{E} (e^{\\sigma_x})^{f_{E}} and \\frac{\\partial}{\\partial \\sigma_x} (e^{\\sigma_x})^{f_{E}} = f_{E} (e^{\\sigma_x})^{f_{E}} and \\frac{\\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)}}{\\frac{\\partial}{\\partial \\sigma_x} (e^{\\sigma_x})^{f_{E}}} = \\frac{f_{E} (e^{\\sigma_x})^{f_{E}}}{\\frac{\\partial}{\\partial \\sigma_x} (e^{\\sigma_x})^{f_{E}}} and \\frac{(e^{\\sigma_x})^{- f_{E}} \\frac{\\partial}{\\partial \\sigma_x} q{(f_{E},\\sigma_x)}}{f_{E}} = 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True))))"], [["divide", 3, "Derivative(Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('q')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Pow(Derivative(Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Pow(Derivative(Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(exp(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))), Derivative(Function('q')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\hat{p}_0{(r)} = \\sin{(r)}, then derive \\frac{d}{d r} \\hat{p}_0{(r)} = \\cos{(r)}, then obtain \\sin{(\\sin{(\\frac{d}{d r} \\hat{p}_0{(r)})})} = \\sin{(\\sin{(\\cos{(r)})})}", "derivation": "\\hat{p}_0{(r)} = \\sin{(r)} and \\frac{d}{d r} \\hat{p}_0{(r)} = \\frac{d}{d r} \\sin{(r)} and \\frac{d}{d r} \\hat{p}_0{(r)} = \\cos{(r)} and \\sin{(\\frac{d}{d r} \\hat{p}_0{(r)})} = \\sin{(\\frac{d}{d r} \\sin{(r)})} and \\frac{d}{d r} \\sin{(r)} = \\cos{(r)} and \\sin{(\\frac{d}{d r} \\hat{p}_0{(r)})} = \\sin{(\\cos{(r)})} and \\sin{(\\sin{(\\frac{d}{d r} \\hat{p}_0{(r)})})} = \\sin{(\\sin{(\\cos{(r)})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(sin(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), cos(Symbol('r', commutative=True)))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), sin(Derivative(sin(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), cos(Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(sin(Derivative(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), sin(cos(Symbol('r', commutative=True))))"], [["sin", 6], "Equality(sin(sin(Derivative(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), sin(sin(cos(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\mu_0)} = \\mu_0, then derive \\frac{d}{d \\mu_0} \\eta{(\\mu_0)} + 1 = 2, then obtain - \\eta{(\\mu_0)} + \\frac{d}{d \\mu_0} \\mu_0 + 1 = - \\eta{(\\mu_0)} + \\frac{d}{d \\mu_0} \\eta{(\\mu_0)} + 1", "derivation": "\\eta{(\\mu_0)} = \\mu_0 and \\mu_0 - i + \\eta{(\\mu_0)} = 2 \\mu_0 - i and \\frac{\\partial}{\\partial \\mu_0} (\\mu_0 - i + \\eta{(\\mu_0)}) = \\frac{\\partial}{\\partial \\mu_0} (2 \\mu_0 - i) and \\frac{d}{d \\mu_0} \\eta{(\\mu_0)} + 1 = 2 and \\frac{d}{d \\mu_0} \\mu_0 + 1 = 2 and \\frac{d}{d \\mu_0} \\mu_0 + 1 = \\frac{d}{d \\mu_0} \\eta{(\\mu_0)} + 1 and - \\eta{(\\mu_0)} + \\frac{d}{d \\mu_0} \\mu_0 + 1 = - \\eta{(\\mu_0)} + \\frac{d}{d \\mu_0} \\eta{(\\mu_0)} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["add", 1, "Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Symbol('\\\\mu_0', commutative=True), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Symbol('\\\\mu_0', commutative=True), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)))"], [["minus", 6, "Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True))), Derivative(Symbol('\\\\mu_0', commutative=True), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True))), Derivative(Function('\\\\eta')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})}, then derive \\int \\dot{\\mathbf{r}}{(\\Psi_{nl})} d\\Psi_{nl} = \\pi + \\sin{(\\Psi_{nl})}, then obtain \\iiint \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} d\\Psi_{nl} d\\pi d\\pi = \\iint (\\pi + \\sin{(\\Psi_{nl})}) d\\pi d\\pi", "derivation": "\\dot{\\mathbf{r}}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} and \\int \\dot{\\mathbf{r}}{(\\Psi_{nl})} d\\Psi_{nl} = \\int \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} d\\Psi_{nl} and \\int \\dot{\\mathbf{r}}{(\\Psi_{nl})} d\\Psi_{nl} = \\pi + \\sin{(\\Psi_{nl})} and \\iint \\dot{\\mathbf{r}}{(\\Psi_{nl})} d\\Psi_{nl} d\\pi = \\int (\\pi + \\sin{(\\Psi_{nl})}) d\\pi and \\iint \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} d\\Psi_{nl} d\\pi = \\int (\\pi + \\sin{(\\Psi_{nl})}) d\\pi and \\iiint \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} d\\Psi_{nl} d\\pi d\\pi = \\iint (\\pi + \\sin{(\\Psi_{nl})}) d\\pi d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b^{\\mathbf{f}} and t{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b^{\\mathbf{f}}, then obtain b + t{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b + b^{\\mathbf{f}}", "derivation": "\\hat{p}_0{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b^{\\mathbf{f}} and b + \\hat{p}_0{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b + b^{\\mathbf{f}} and t{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b^{\\mathbf{f}} and t{(\\mathbf{f},b,\\theta_1)} = \\hat{p}_0{(\\mathbf{f},b,\\theta_1)} and b + t{(\\mathbf{f},b,\\theta_1)} = \\theta_1 + b + b^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True), Pow(Symbol('b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('t')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('b', commutative=True), Function('t')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('b', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Symbol('b', commutative=True), Pow(Symbol('b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(P_{e},G)} = G^{P_{e}}, then obtain \\frac{\\partial^{2}}{\\partial P_{e}^{2}} 2 \\hat{\\mathbf{r}}{(P_{e},G)} = \\frac{\\partial^{2}}{\\partial P_{e}^{2}} (G^{P_{e}} + \\hat{\\mathbf{r}}{(P_{e},G)})", "derivation": "\\hat{\\mathbf{r}}{(P_{e},G)} = G^{P_{e}} and 2 \\hat{\\mathbf{r}}{(P_{e},G)} = G^{P_{e}} + \\hat{\\mathbf{r}}{(P_{e},G)} and \\frac{\\partial}{\\partial P_{e}} 2 \\hat{\\mathbf{r}}{(P_{e},G)} = \\frac{\\partial}{\\partial P_{e}} (G^{P_{e}} + \\hat{\\mathbf{r}}{(P_{e},G)}) and \\frac{\\partial^{2}}{\\partial P_{e}^{2}} 2 \\hat{\\mathbf{r}}{(P_{e},G)} = \\frac{\\partial^{2}}{\\partial P_{e}^{2}} (G^{P_{e}} + \\hat{\\mathbf{r}}{(P_{e},G)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('P_e', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('G', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(2))), Derivative(Add(Pow(Symbol('G', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(2))))"]]}, {"prompt": "Given f{(u,G)} = \\frac{u}{G} and \\eta{(u,G)} = u + \\frac{u}{G}, then obtain \\log{(u + f{(u,G)})}^{G} = \\log{(\\eta{(u,G)})}^{G}", "derivation": "f{(u,G)} = \\frac{u}{G} and u + f{(u,G)} = u + \\frac{u}{G} and \\eta{(u,G)} = u + \\frac{u}{G} and u + f{(u,G)} = \\eta{(u,G)} and \\log{(u + f{(u,G)})} = \\log{(\\eta{(u,G)})} and \\log{(u + f{(u,G)})} = \\log{(u + \\frac{u}{G})} and \\log{(u + f{(u,G)})}^{G} = \\log{(u + \\frac{u}{G})}^{G} and \\log{(u + f{(u,G)})}^{G} = \\log{(\\eta{(u,G)})}^{G}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True))), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True), Symbol('G', commutative=True)), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True))), Function('\\\\eta')(Symbol('u', commutative=True), Symbol('G', commutative=True)))"], [["log", 4], "Equality(log(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True)))), log(Function('\\\\eta')(Symbol('u', commutative=True), Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(log(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True)))), log(Add(Symbol('u', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('u', commutative=True)))))"], [["power", 6, "Symbol('G', commutative=True)"], "Equality(Pow(log(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Pow(log(Add(Symbol('u', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('u', commutative=True)))), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Pow(log(Add(Symbol('u', commutative=True), Function('f')(Symbol('u', commutative=True), Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Pow(log(Function('\\\\eta')(Symbol('u', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(r)} = \\sin{(r)}, then obtain r \\operatorname{f_{\\mathbf{v}}}{(r)} + \\operatorname{f_{\\mathbf{v}}}{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)} = r \\operatorname{f_{\\mathbf{v}}}{(r)} + \\sin{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(r)} = \\sin{(r)} and r \\operatorname{f_{\\mathbf{v}}}{(r)} = r \\sin{(r)} and r \\sin{(r)} + \\operatorname{f_{\\mathbf{v}}}{(r)} = r \\sin{(r)} + \\sin{(r)} and r \\sin{(r)} + \\operatorname{f_{\\mathbf{v}}}{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)} = r \\sin{(r)} + \\sin{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)} and r \\operatorname{f_{\\mathbf{v}}}{(r)} + \\operatorname{f_{\\mathbf{v}}}{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)} = r \\operatorname{f_{\\mathbf{v}}}{(r)} + \\sin{(r)} - \\frac{d}{d r} \\operatorname{f_{\\mathbf{v}}}{(r)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True))))"], [["add", 1, "Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True))), Add(Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True))))"], [["minus", 3, "Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Add(Mul(Symbol('r', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('r', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Add(Mul(Symbol('r', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\psi^{*}{(\\psi)} = \\cos{(\\psi)}, then obtain \\frac{\\partial}{\\partial \\ddot{x}} \\int - \\ddot{x} n_{2} (\\psi^{*}{(\\psi)} + 1) d\\psi = \\frac{\\partial}{\\partial \\ddot{x}} \\int - \\ddot{x} n_{2} (\\cos{(\\psi)} + 1) d\\psi", "derivation": "\\psi^{*}{(\\psi)} = \\cos{(\\psi)} and \\psi^{*}{(\\psi)} + 1 = \\cos{(\\psi)} + 1 and - \\ddot{x} n_{2} (\\psi^{*}{(\\psi)} + 1) = - \\ddot{x} n_{2} (\\cos{(\\psi)} + 1) and \\int - \\ddot{x} n_{2} (\\psi^{*}{(\\psi)} + 1) d\\psi = \\int - \\ddot{x} n_{2} (\\cos{(\\psi)} + 1) d\\psi and \\frac{\\partial}{\\partial \\ddot{x}} \\int - \\ddot{x} n_{2} (\\psi^{*}{(\\psi)} + 1) d\\psi = \\frac{\\partial}{\\partial \\ddot{x}} \\int - \\ddot{x} n_{2} (\\cos{(\\psi)} + 1) d\\psi", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\psi', commutative=True)), Integer(1)), Add(cos(Symbol('\\\\psi', commutative=True)), Integer(1)))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(Function('\\\\psi^*')(Symbol('\\\\psi', commutative=True)), Integer(1))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(cos(Symbol('\\\\psi', commutative=True)), Integer(1))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(Function('\\\\psi^*')(Symbol('\\\\psi', commutative=True)), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(cos(Symbol('\\\\psi', commutative=True)), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(Function('\\\\psi^*')(Symbol('\\\\psi', commutative=True)), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True), Add(cos(Symbol('\\\\psi', commutative=True)), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then obtain (\\iiint c{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r})^{\\mathbf{r}} = (\\iiint \\log{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r})^{\\mathbf{r}}", "derivation": "c{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\int c{(\\mathbf{r})} d\\mathbf{r} = \\int \\log{(\\mathbf{r})} d\\mathbf{r} and \\iint c{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} = \\iint \\log{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} and \\iiint c{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r} = \\iiint \\log{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r} and (\\iiint c{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r})^{\\mathbf{r}} = (\\iiint \\log{(\\mathbf{r})} d\\mathbf{r} d\\mathbf{r} d\\mathbf{r})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Integral(Function('c')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\chi)} = \\log{(\\sin{(\\chi)})}, then obtain 3 (2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})}) \\operatorname{A_{z}}{(\\chi)} = (2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})})^{2}", "derivation": "\\operatorname{A_{z}}{(\\chi)} = \\log{(\\sin{(\\chi)})} and 2 \\operatorname{A_{z}}{(\\chi)} = \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})} and 3 \\operatorname{A_{z}}{(\\chi)} = 2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})} and 3 \\operatorname{A_{z}}{(\\chi)} = \\operatorname{A_{z}}{(\\chi)} + 2 \\log{(\\sin{(\\chi)})} and 3 (\\operatorname{A_{z}}{(\\chi)} + 2 \\log{(\\sin{(\\chi)})}) \\operatorname{A_{z}}{(\\chi)} = (\\operatorname{A_{z}}{(\\chi)} + 2 \\log{(\\sin{(\\chi)})})^{2} and 2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})} = \\operatorname{A_{z}}{(\\chi)} + 2 \\log{(\\sin{(\\chi)})} and 3 (2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})}) \\operatorname{A_{z}}{(\\chi)} = (2 \\operatorname{A_{z}}{(\\chi)} + \\log{(\\sin{(\\chi)})})^{2}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\chi', commutative=True)), log(sin(Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Function('A_z')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(2), Function('A_z')(Symbol('\\\\chi', commutative=True))), Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), log(sin(Symbol('\\\\chi', commutative=True)))))"], [["add", 2, "Function('A_z')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(3), Function('A_z')(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Function('A_z')(Symbol('\\\\chi', commutative=True))), log(sin(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('A_z')(Symbol('\\\\chi', commutative=True))), Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), log(sin(Symbol('\\\\chi', commutative=True))))))"], [["times", 4, "Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), log(sin(Symbol('\\\\chi', commutative=True)))))"], "Equality(Mul(Integer(3), Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), log(sin(Symbol('\\\\chi', commutative=True))))), Function('A_z')(Symbol('\\\\chi', commutative=True))), Pow(Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), log(sin(Symbol('\\\\chi', commutative=True))))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('A_z')(Symbol('\\\\chi', commutative=True))), log(sin(Symbol('\\\\chi', commutative=True)))), Add(Function('A_z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), log(sin(Symbol('\\\\chi', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Integer(3), Add(Mul(Integer(2), Function('A_z')(Symbol('\\\\chi', commutative=True))), log(sin(Symbol('\\\\chi', commutative=True)))), Function('A_z')(Symbol('\\\\chi', commutative=True))), Pow(Add(Mul(Integer(2), Function('A_z')(Symbol('\\\\chi', commutative=True))), log(sin(Symbol('\\\\chi', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\phi{(\\Psi_{nl},T)} = \\int \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} and C{(\\Psi_{nl},T)} = \\int \\phi{(\\Psi_{nl},T)} d\\Psi_{nl}, then obtain - T + \\frac{\\partial}{\\partial T} C{(\\Psi_{nl},T)} = - T + \\frac{\\partial}{\\partial T} \\iint \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl}", "derivation": "\\phi{(\\Psi_{nl},T)} = \\int \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} and \\int \\phi{(\\Psi_{nl},T)} d\\Psi_{nl} = \\iint \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl} and C{(\\Psi_{nl},T)} = \\int \\phi{(\\Psi_{nl},T)} d\\Psi_{nl} and C{(\\Psi_{nl},T)} = \\iint \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl} and \\frac{\\partial}{\\partial T} C{(\\Psi_{nl},T)} = \\frac{\\partial}{\\partial T} \\iint \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl} and - T + \\frac{\\partial}{\\partial T} C{(\\Psi_{nl},T)} = - T + \\frac{\\partial}{\\partial T} \\iint \\frac{T}{\\Psi_{nl}} d\\Psi_{nl} d\\Psi_{nl}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Integral(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 5, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Derivative(Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(u,m_{s},n)} = \\frac{n}{m_{s} u} and \\hat{\\mathbf{x}}{(u,m_{s},n)} = \\operatorname{P_{g}}{(u,m_{s},n)} - \\frac{1}{m_{s}}, then obtain \\hat{\\mathbf{x}}{(u,m_{s},n)} = \\frac{n}{m_{s} u} - \\frac{1}{m_{s}}", "derivation": "\\operatorname{P_{g}}{(u,m_{s},n)} = \\frac{n}{m_{s} u} and \\operatorname{P_{g}}{(u,m_{s},n)} - \\frac{1}{m_{s}} = \\frac{n}{m_{s} u} - \\frac{1}{m_{s}} and \\hat{\\mathbf{x}}{(u,m_{s},n)} = \\operatorname{P_{g}}{(u,m_{s},n)} - \\frac{1}{m_{s}} and \\hat{\\mathbf{x}}{(u,m_{s},n)} = \\frac{n}{m_{s} u} - \\frac{1}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('u', commutative=True), Symbol('m_s', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('m_s', commutative=True), Integer(-1))"], "Equality(Add(Function('P_g')(Symbol('u', commutative=True), Symbol('m_s', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('u', commutative=True), Symbol('m_s', commutative=True), Symbol('n', commutative=True)), Add(Function('P_g')(Symbol('u', commutative=True), Symbol('m_s', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('u', commutative=True), Symbol('m_s', commutative=True), Symbol('n', commutative=True)), Add(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given y{(V)} = \\cos{(\\cos{(V)})}, then obtain V + y{(V)} \\cos{(V)} - y{(V)} = V - y{(V)} + \\cos{(V)} \\cos{(\\cos{(V)})}", "derivation": "y{(V)} = \\cos{(\\cos{(V)})} and y{(V)} \\cos{(V)} = \\cos{(V)} \\cos{(\\cos{(V)})} and V + y{(V)} \\cos{(V)} = V + \\cos{(V)} \\cos{(\\cos{(V)})} and V + y{(V)} \\cos{(V)} - y{(V)} = V - y{(V)} + \\cos{(V)} \\cos{(\\cos{(V)})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))))"], [["times", 1, "cos(Symbol('V', commutative=True))"], "Equality(Mul(Function('y')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Mul(cos(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True)))))"], [["add", 2, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Mul(Function('y')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))), Add(Symbol('V', commutative=True), Mul(cos(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))))))"], [["minus", 3, "Function('y')(Symbol('V', commutative=True))"], "Equality(Add(Symbol('V', commutative=True), Mul(Function('y')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Mul(Integer(-1), Function('y')(Symbol('V', commutative=True)))), Add(Symbol('V', commutative=True), Mul(Integer(-1), Function('y')(Symbol('V', commutative=True))), Mul(cos(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))))))"]]}, {"prompt": "Given u{(v_{1},s)} = \\cos{(\\frac{s}{v_{1}})}, then obtain v_{1} u^{2}{(v_{1},s)} (\\frac{\\partial}{\\partial v_{1}} u{(v_{1},s)})^{- v_{1}} = v_{1} \\cos^{2}{(\\frac{s}{v_{1}})} (\\frac{\\partial}{\\partial v_{1}} u{(v_{1},s)})^{- v_{1}}", "derivation": "u{(v_{1},s)} = \\cos{(\\frac{s}{v_{1}})} and v_{1} u{(v_{1},s)} = v_{1} \\cos{(\\frac{s}{v_{1}})} and v_{1} u^{2}{(v_{1},s)} = v_{1} u{(v_{1},s)} \\cos{(\\frac{s}{v_{1}})} and v_{1} u{(v_{1},s)} \\cos{(\\frac{s}{v_{1}})} = v_{1} \\cos^{2}{(\\frac{s}{v_{1}})} and v_{1} u^{2}{(v_{1},s)} = v_{1} \\cos^{2}{(\\frac{s}{v_{1}})} and v_{1} u^{2}{(v_{1},s)} (\\frac{\\partial}{\\partial v_{1}} u{(v_{1},s)})^{- v_{1}} = v_{1} \\cos^{2}{(\\frac{s}{v_{1}})} (\\frac{\\partial}{\\partial v_{1}} u{(v_{1},s)})^{- v_{1}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('v_1', commutative=True), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))))"], [["times", 1, "Mul(Symbol('v_1', commutative=True), Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Symbol('v_1', commutative=True), Pow(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Symbol('v_1', commutative=True), Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('v_1', commutative=True), Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))), Mul(Symbol('v_1', commutative=True), Pow(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('v_1', commutative=True), Pow(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Symbol('v_1', commutative=True), Pow(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Integer(2))))"], [["divide", 5, "Pow(Derivative(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True))"], "Equality(Mul(Symbol('v_1', commutative=True), Pow(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Integer(2)), Pow(Derivative(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Mul(Symbol('v_1', commutative=True), Pow(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Integer(2)), Pow(Derivative(Function('u')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(v_{z},\\theta_1)} = \\frac{v_{z}}{\\theta_1}, then derive \\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}_M{(v_{z},\\theta_1)} = - \\frac{v_{z}}{\\theta_1^{2}}, then obtain 2 \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} = \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} - \\frac{v_{z}}{\\theta_1^{2}}", "derivation": "\\mathbf{J}_M{(v_{z},\\theta_1)} = \\frac{v_{z}}{\\theta_1} and \\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}_M{(v_{z},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} and \\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}_M{(v_{z},\\theta_1)} = - \\frac{v_{z}}{\\theta_1^{2}} and \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} + \\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}_M{(v_{z},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} - \\frac{v_{z}}{\\theta_1^{2}} and 2 \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} = \\frac{\\partial}{\\partial \\theta_1} \\frac{v_{z}}{\\theta_1} - \\frac{v_{z}}{\\theta_1^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('v_z', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('v_z', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('v_z', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2)), Symbol('v_z', commutative=True)))"], [["add", 3, "Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_M')(Symbol('v_z', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2)), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given t{(F_{c})} = \\log{(F_{c})}, then obtain \\tilde{\\infty}^{F_{c}} (t{(F_{c})} - \\log{(F_{c})})^{F_{c}} = 0^{F_{c}} \\tilde{\\infty}^{F_{c}}", "derivation": "t{(F_{c})} = \\log{(F_{c})} and t{(F_{c})} - \\log{(F_{c})} = 0 and (t{(F_{c})} - \\log{(F_{c})})^{F_{c}} = 0^{F_{c}} and \\tilde{\\infty}^{F_{c}} (t{(F_{c})} - \\log{(F_{c})})^{F_{c}} = 0^{F_{c}} \\tilde{\\infty}^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["minus", 1, "log(Symbol('F_c', commutative=True))"], "Equality(Add(Function('t')(Symbol('F_c', commutative=True)), Mul(Integer(-1), log(Symbol('F_c', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Function('t')(Symbol('F_c', commutative=True)), Mul(Integer(-1), log(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)), Pow(Integer(0), Symbol('F_c', commutative=True)))"], [["divide", 3, "Pow(Integer(0), Symbol('F_c', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('F_c', commutative=True)), Pow(Add(Function('t')(Symbol('F_c', commutative=True)), Mul(Integer(-1), log(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True))), Mul(Pow(Integer(0), Symbol('F_c', commutative=True)), Pow(zoo, Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(m,a)} = \\log{(a + m)}, then derive \\int \\mathbf{F}{(m,a)} dm = C_{d} + a \\log{(a + m)} + m \\log{(a + m)} - m, then obtain \\frac{m \\log{(a + m)} + \\int \\mathbf{F}{(m,a)} dm}{\\mathbf{F}{(m,a)}} = \\frac{C_{d} + a \\mathbf{F}{(m,a)} + m \\mathbf{F}{(m,a)} + m \\log{(a + m)} - m}{\\mathbf{F}{(m,a)}}", "derivation": "\\mathbf{F}{(m,a)} = \\log{(a + m)} and \\int \\mathbf{F}{(m,a)} dm = \\int \\log{(a + m)} dm and \\int \\mathbf{F}{(m,a)} dm = C_{d} + a \\log{(a + m)} + m \\log{(a + m)} - m and \\int \\mathbf{F}{(m,a)} dm = C_{d} + a \\mathbf{F}{(m,a)} + m \\mathbf{F}{(m,a)} - m and m \\log{(a + m)} + \\int \\mathbf{F}{(m,a)} dm = C_{d} + a \\mathbf{F}{(m,a)} + m \\mathbf{F}{(m,a)} + m \\log{(a + m)} - m and \\frac{m \\log{(a + m)} + \\int \\mathbf{F}{(m,a)} dm}{\\mathbf{F}{(m,a)}} = \\frac{C_{d} + a \\mathbf{F}{(m,a)} + m \\mathbf{F}{(m,a)} + m \\log{(a + m)} - m}{\\mathbf{F}{(m,a)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('a', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('m', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["add", 4, "Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True))))"], "Equality(Add(Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Integral(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Symbol('C_d', commutative=True), Mul(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('m', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["divide", 5, "Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Integral(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True)))), Pow(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Integer(-1))), Mul(Add(Symbol('C_d', commutative=True), Mul(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('m', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('m', commutative=True), log(Add(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(-1), Symbol('m', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given L{(s)} = \\log{(s)}, then derive L{(s)} \\int L{(s)} ds = (P_{g} + s \\log{(s)} - s) L{(s)}, then obtain \\frac{d}{d s} L{(s)} \\int L{(s)} ds = \\frac{\\partial}{\\partial s} (P_{g} + s \\log{(s)} - s) L{(s)}", "derivation": "L{(s)} = \\log{(s)} and \\int L{(s)} ds = \\int \\log{(s)} ds and \\log{(s)} \\int L{(s)} ds = \\log{(s)} \\int \\log{(s)} ds and L{(s)} \\int L{(s)} ds = L{(s)} \\int \\log{(s)} ds and L{(s)} \\int L{(s)} ds = (P_{g} + s \\log{(s)} - s) L{(s)} and \\frac{d}{d s} L{(s)} \\int L{(s)} ds = \\frac{\\partial}{\\partial s} (P_{g} + s \\log{(s)} - s) L{(s)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('L')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["times", 2, "log(Symbol('s', commutative=True))"], "Equality(Mul(log(Symbol('s', commutative=True)), Integral(Function('L')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(log(Symbol('s', commutative=True)), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('L')(Symbol('s', commutative=True)), Integral(Function('L')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Function('L')(Symbol('s', commutative=True)), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Function('L')(Symbol('s', commutative=True)), Integral(Function('L')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Symbol('P_g', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Function('L')(Symbol('s', commutative=True))))"], [["differentiate", 5, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Function('L')(Symbol('s', commutative=True)), Integral(Function('L')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('P_g', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Function('L')(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{F_{g}}{(\\mathbf{P})} = \\frac{\\Psi_{\\lambda}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}}, then obtain \\int (\\operatorname{F_{g}}{(\\mathbf{P})} + \\log{(\\mathbf{P})}) d\\mathbf{P} = \\int (\\log{(\\mathbf{P})} + 1) d\\mathbf{P}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{F_{g}}{(\\mathbf{P})} = \\frac{\\Psi_{\\lambda}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} and \\operatorname{F_{g}}{(\\mathbf{P})} + \\log{(\\mathbf{P})} = \\frac{\\Psi_{\\lambda}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} + \\log{(\\mathbf{P})} and \\int (\\operatorname{F_{g}}{(\\mathbf{P})} + \\log{(\\mathbf{P})}) d\\mathbf{P} = \\int (\\frac{\\Psi_{\\lambda}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} + \\log{(\\mathbf{P})}) d\\mathbf{P} and \\int (\\operatorname{F_{g}}{(\\mathbf{P})} + \\log{(\\mathbf{P})}) d\\mathbf{P} = \\int (\\log{(\\mathbf{P})} + 1) d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"], [["add", 2, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Add(Function('F_g')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Function('F_g')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then derive \\int \\operatorname{F_{g}}{(L_{\\varepsilon})} dL_{\\varepsilon} = v_{1} + \\sin{(L_{\\varepsilon})}, then obtain v_{1} + \\sin{(L_{\\varepsilon})} = F_{g} + \\sin{(L_{\\varepsilon})}", "derivation": "\\operatorname{F_{g}}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\int \\operatorname{F_{g}}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and \\int \\operatorname{F_{g}}{(L_{\\varepsilon})} dL_{\\varepsilon} = v_{1} + \\sin{(L_{\\varepsilon})} and v_{1} + \\sin{(L_{\\varepsilon})} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and v_{1} + \\sin{(L_{\\varepsilon})} = F_{g} + \\sin{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('v_1', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('v_1', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('v_1', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('F_g', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\hat{p},\\mathbf{S})} = \\mathbf{S} + e^{\\hat{p}} and \\operatorname{v_{2}}{(\\hat{p})} = e^{\\hat{p}}, then obtain \\int (\\mathbf{S} + \\operatorname{v_{2}}{(\\hat{p})}) d\\hat{p} = \\int (\\mathbf{S} + e^{\\hat{p}}) d\\hat{p}", "derivation": "\\mathbf{v}{(\\hat{p},\\mathbf{S})} = \\mathbf{S} + e^{\\hat{p}} and \\int \\mathbf{v}{(\\hat{p},\\mathbf{S})} d\\hat{p} = \\int (\\mathbf{S} + e^{\\hat{p}}) d\\hat{p} and \\operatorname{v_{2}}{(\\hat{p})} = e^{\\hat{p}} and \\mathbf{v}{(\\hat{p},\\mathbf{S})} = \\mathbf{S} + \\operatorname{v_{2}}{(\\hat{p})} and \\int (\\mathbf{S} + \\operatorname{v_{2}}{(\\hat{p})}) d\\hat{p} = \\int (\\mathbf{S} + e^{\\hat{p}}) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Function('v_2')(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('v_2')(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(A_{x})} = \\log{(A_{x})}, then obtain (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{- A_{x}} (\\frac{\\log{(A_{x})}}{A_{x}})^{A_{x}} = (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{- 2 A_{x}} (\\frac{\\log{(A_{x})}}{A_{x}})^{2 A_{x}}", "derivation": "\\mathbf{M}{(A_{x})} = \\log{(A_{x})} and \\frac{\\mathbf{M}{(A_{x})}}{A_{x}} = \\frac{\\log{(A_{x})}}{A_{x}} and (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{A_{x}} = (\\frac{\\log{(A_{x})}}{A_{x}})^{A_{x}} and 1 = (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{- A_{x}} (\\frac{\\log{(A_{x})}}{A_{x}})^{A_{x}} and (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{- A_{x}} (\\frac{\\log{(A_{x})}}{A_{x}})^{A_{x}} = (\\frac{\\mathbf{M}{(A_{x})}}{A_{x}})^{- 2 A_{x}} (\\frac{\\log{(A_{x})}}{A_{x}})^{2 A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["divide", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["divide", 3, "Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], [["times", 4, "Mul(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], "Equality(Mul(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Mul(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('A_x', commutative=True))), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))), Mul(Integer(2), Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(c_{0},z)} = \\log{(\\frac{z}{c_{0}})} and W{(c_{0},z)} = \\frac{1}{\\log{(\\frac{z}{c_{0}})}}, then obtain \\mathbf{P}{(c_{0},z)} - \\frac{1}{\\mathbf{P}{(c_{0},z)}} = \\mathbf{P}{(c_{0},z)} - \\frac{1}{\\log{(\\frac{z}{c_{0}})}}", "derivation": "\\mathbf{P}{(c_{0},z)} = \\log{(\\frac{z}{c_{0}})} and W{(c_{0},z)} = \\frac{1}{\\log{(\\frac{z}{c_{0}})}} and W{(c_{0},z)} = \\frac{1}{\\mathbf{P}{(c_{0},z)}} and W{(c_{0},z)} - \\mathbf{P}{(c_{0},z)} = - \\mathbf{P}{(c_{0},z)} + \\frac{1}{\\log{(\\frac{z}{c_{0}})}} and - \\mathbf{P}{(c_{0},z)} + \\frac{1}{\\mathbf{P}{(c_{0},z)}} = - \\mathbf{P}{(c_{0},z)} + \\frac{1}{\\log{(\\frac{z}{c_{0}})}} and \\mathbf{P}{(c_{0},z)} - \\frac{1}{\\mathbf{P}{(c_{0},z)}} = \\mathbf{P}{(c_{0},z)} - \\frac{1}{\\log{(\\frac{z}{c_{0}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('W')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Pow(log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('W')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"], [["minus", 2, "Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True))"], "Equality(Add(Function('W')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True))), Pow(log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True))), Pow(log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Integer(-1))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Integer(-1)))), Add(Function('\\\\mathbf{P}')(Symbol('c_0', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Pow(log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A}, then obtain \\frac{d}{d \\mathbf{A}} \\frac{\\operatorname{n_{2}}^{\\mathbf{A}}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\frac{(\\int e^{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}}}{\\mathbf{A}}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A} and \\operatorname{n_{2}}^{\\mathbf{A}}{(\\mathbf{A})} = (\\int e^{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}} and \\frac{\\operatorname{n_{2}}^{\\mathbf{A}}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{(\\int e^{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}}}{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\frac{\\operatorname{n_{2}}^{\\mathbf{A}}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\frac{(\\int e^{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Function('n_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Function('n_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(s)} = \\cos{(s)}, then obtain \\mathbf{r}^{2 s}{(s)} \\cos^{- 2 s}{(s)} = 1", "derivation": "\\mathbf{r}{(s)} = \\cos{(s)} and \\mathbf{r}^{s}{(s)} = \\cos^{s}{(s)} and \\mathbf{r}^{s}{(s)} \\cos^{- s}{(s)} = 1 and \\mathbf{r}^{s}{(s)} \\cos^{- 2 s}{(s)} = \\cos^{- s}{(s)} and \\mathbf{r}^{2 s}{(s)} \\cos^{- 2 s}{(s)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["divide", 2, "Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)))), Integer(1))"], [["divide", 3, "Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('s', commutative=True)))), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('s', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('s', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{r}{(C_{1},x)} = C_{1} - x, then obtain - 2 C_{1} + 2 x + \\frac{C_{1} - x}{\\mathbf{r}{(C_{1},x)}} + 1 = - 2 C_{1} + 2 x + \\frac{2 (C_{1} - x)}{\\mathbf{r}{(C_{1},x)}}", "derivation": "\\mathbf{r}{(C_{1},x)} = C_{1} - x and 1 = \\frac{C_{1} - x}{\\mathbf{r}{(C_{1},x)}} and - C_{1} + x + 1 = - C_{1} + x + \\frac{C_{1} - x}{\\mathbf{r}{(C_{1},x)}} and - 2 C_{1} + 2 x + \\frac{C_{1} - x}{\\mathbf{r}{(C_{1},x)}} + 1 = - 2 C_{1} + 2 x + \\frac{2 (C_{1} - x)}{\\mathbf{r}{(C_{1},x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Integer(-1))))"], [["minus", 2, "Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('x', commutative=True), Integer(1)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('x', commutative=True), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Integer(-1)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('x', commutative=True), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given G{(F_{N},V)} = F_{N} + V, then obtain (\\frac{F_{N} + V}{G{(F_{N},V)}} + 2) G{(F_{N},V)} = (\\frac{2 (F_{N} + V)}{G{(F_{N},V)}} + 1) G{(F_{N},V)}", "derivation": "G{(F_{N},V)} = F_{N} + V and 1 = \\frac{F_{N} + V}{G{(F_{N},V)}} and \\frac{F_{N} + V}{G{(F_{N},V)}} + 1 = \\frac{2 (F_{N} + V)}{G{(F_{N},V)}} and \\frac{F_{N} + V}{G{(F_{N},V)}} + 2 = \\frac{2 (F_{N} + V)}{G{(F_{N},V)}} + 1 and (\\frac{F_{N} + V}{G{(F_{N},V)}} + 2) G{(F_{N},V)} = (\\frac{2 (F_{N} + V)}{G{(F_{N},V)}} + 1) G{(F_{N},V)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)))"], [["divide", 1, "Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))))"], [["add", 2, "Mul(Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Integer(1)), Mul(Integer(2), Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Integer(2)), Add(Mul(Integer(2), Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Integer(1)))"], [["times", 4, "Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Add(Mul(Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Integer(2)), Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True))), Mul(Add(Mul(Integer(2), Add(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Pow(Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True)), Integer(-1))), Integer(1)), Function('G')(Symbol('F_N', commutative=True), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(I,f)} = \\sin{(\\frac{f}{I})} and \\mathbf{p}{(I,f)} = - I + \\sin{(\\frac{f}{I})}, then obtain \\mathbf{p}^{2}{(I,f)} + \\sin{(\\frac{f}{I})} = (- I + \\sin{(\\frac{f}{I})}) \\mathbf{p}{(I,f)} + \\sin{(\\frac{f}{I})}", "derivation": "\\operatorname{r_{0}}{(I,f)} = \\sin{(\\frac{f}{I})} and - I + \\operatorname{r_{0}}{(I,f)} = - I + \\sin{(\\frac{f}{I})} and (- I + \\operatorname{r_{0}}{(I,f)})^{2} = (- I + \\operatorname{r_{0}}{(I,f)}) (- I + \\sin{(\\frac{f}{I})}) and \\mathbf{p}{(I,f)} = - I + \\sin{(\\frac{f}{I})} and (- I + \\operatorname{r_{0}}{(I,f)})^{2} + \\sin{(\\frac{f}{I})} = (- I + \\operatorname{r_{0}}{(I,f)}) (- I + \\sin{(\\frac{f}{I})}) + \\sin{(\\frac{f}{I})} and \\mathbf{p}{(I,f)} = - I + \\operatorname{r_{0}}{(I,f)} and \\mathbf{p}^{2}{(I,f)} + \\sin{(\\frac{f}{I})} = (- I + \\sin{(\\frac{f}{I})}) \\mathbf{p}{(I,f)} + \\sin{(\\frac{f}{I})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))), Integer(2)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True))))), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('r_0')(Symbol('I', commutative=True), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Pow(Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(2)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('f', commutative=True))), sin(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\delta{(F_{H},\\eta^{\\prime})} = F_{H} \\eta^{\\prime}, then derive \\frac{\\frac{\\partial}{\\partial F_{H}} \\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} - \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}^{2}} = 0, then obtain \\eta^{\\prime} (\\frac{\\frac{\\partial}{\\partial F_{H}} \\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} - \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}^{2}}) = 0", "derivation": "\\delta{(F_{H},\\eta^{\\prime})} = F_{H} \\eta^{\\prime} and \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} = \\eta^{\\prime} and \\frac{\\partial}{\\partial F_{H}} \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} = \\frac{d}{d F_{H}} \\eta^{\\prime} and \\frac{\\frac{\\partial}{\\partial F_{H}} \\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} - \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}^{2}} = 0 and \\eta^{\\prime} (\\frac{\\frac{\\partial}{\\partial F_{H}} \\delta{(F_{H},\\eta^{\\prime})}}{F_{H}} - \\frac{\\delta{(F_{H},\\eta^{\\prime})}}{F_{H}^{2}}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 1, "Symbol('F_H', commutative=True)"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Symbol('\\\\eta^{\\\\prime}', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-2)), Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integer(0))"], [["times", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-2)), Function('\\\\delta')(Symbol('F_H', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(H,M_{E})} = H + \\sin{(M_{E})}, then obtain - \\int M_{E} (H + \\sin{(M_{E})}) dH + \\int M_{E} \\operatorname{F_{H}}{(H,M_{E})} dH = 0", "derivation": "\\operatorname{F_{H}}{(H,M_{E})} = H + \\sin{(M_{E})} and M_{E} \\operatorname{F_{H}}{(H,M_{E})} = M_{E} (H + \\sin{(M_{E})}) and \\int M_{E} \\operatorname{F_{H}}{(H,M_{E})} dH = \\int M_{E} (H + \\sin{(M_{E})}) dH and - \\int M_{E} (H + \\sin{(M_{E})}) dH + \\int M_{E} \\operatorname{F_{H}}{(H,M_{E})} dH = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('H', commutative=True), sin(Symbol('M_E', commutative=True))))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('F_H')(Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), Add(Symbol('H', commutative=True), sin(Symbol('M_E', commutative=True)))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Symbol('M_E', commutative=True), Function('F_H')(Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Symbol('M_E', commutative=True), Add(Symbol('H', commutative=True), sin(Symbol('M_E', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], [["minus", 3, "Integral(Mul(Symbol('M_E', commutative=True), Add(Symbol('H', commutative=True), sin(Symbol('M_E', commutative=True)))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('M_E', commutative=True), Add(Symbol('H', commutative=True), sin(Symbol('M_E', commutative=True)))), Tuple(Symbol('H', commutative=True)))), Integral(Mul(Symbol('M_E', commutative=True), Function('F_H')(Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('H', commutative=True)))), Integer(0))"]]}, {"prompt": "Given v{(J,v_{t},\\pi)} = - J + \\frac{v_{t}}{\\pi}, then obtain (\\int - v{(J,v_{t},\\pi)} d\\pi)^{J} = (\\int (J - \\frac{v_{t}}{\\pi}) d\\pi)^{J}", "derivation": "v{(J,v_{t},\\pi)} = - J + \\frac{v_{t}}{\\pi} and - v{(J,v_{t},\\pi)} = J - \\frac{v_{t}}{\\pi} and \\int - v{(J,v_{t},\\pi)} d\\pi = \\int (J - \\frac{v_{t}}{\\pi}) d\\pi and (\\int - v{(J,v_{t},\\pi)} d\\pi)^{J} = (\\int (J - \\frac{v_{t}}{\\pi}) d\\pi)^{J}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v')(Symbol('J', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('v')(Symbol('J', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Function('v')(Symbol('J', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given c{(x,b)} = - x + \\cos{(b)} and \\mathbf{f}{(x,b)} = ((- x + \\cos{(b)})^{b})^{b}, then obtain (b (\\frac{b \\frac{\\partial}{\\partial b} c{(x,b)}}{c{(x,b)}} + \\log{(c{(x,b)})}) + \\log{(c^{b}{(x,b)})}) (c^{b}{(x,b)})^{b} = \\frac{\\partial}{\\partial b} \\mathbf{f}{(x,b)}", "derivation": "c{(x,b)} = - x + \\cos{(b)} and c^{b}{(x,b)} = (- x + \\cos{(b)})^{b} and (c^{b}{(x,b)})^{b} = ((- x + \\cos{(b)})^{b})^{b} and \\frac{\\partial}{\\partial b} (c^{b}{(x,b)})^{b} = \\frac{\\partial}{\\partial b} ((- x + \\cos{(b)})^{b})^{b} and \\mathbf{f}{(x,b)} = ((- x + \\cos{(b)})^{b})^{b} and \\frac{\\partial}{\\partial b} (c^{b}{(x,b)})^{b} = \\frac{\\partial}{\\partial b} \\mathbf{f}{(x,b)} and (b (\\frac{b \\frac{\\partial}{\\partial b} c{(x,b)}}{c{(x,b)}} + \\log{(c{(x,b)})}) + \\log{(c^{b}{(x,b)})}) (c^{b}{(x,b)})^{b} = \\frac{\\partial}{\\partial b} \\mathbf{f}{(x,b)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), cos(Symbol('b', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Pow(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{f}')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Add(Mul(Symbol('b', commutative=True), Add(Mul(Symbol('b', commutative=True), Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), log(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True))))), log(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))), Pow(Pow(Function('c')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Derivative(Function('\\\\mathbf{f}')(Symbol('x', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(C,b)} = \\sin{(C^{b})} and B{(C,b)} = \\log{(\\sin{(C^{b})})}, then obtain \\log{(\\operatorname{v_{1}}{(C,b)})}^{C} = B^{C}{(C,b)}", "derivation": "\\operatorname{v_{1}}{(C,b)} = \\sin{(C^{b})} and \\log{(\\operatorname{v_{1}}{(C,b)})} = \\log{(\\sin{(C^{b})})} and B{(C,b)} = \\log{(\\sin{(C^{b})})} and \\log{(\\operatorname{v_{1}}{(C,b)})} = B{(C,b)} and \\log{(\\operatorname{v_{1}}{(C,b)})}^{C} = B^{C}{(C,b)}", "srepr_derivation": [["get_premise", "Equality(Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)), sin(Pow(Symbol('C', commutative=True), Symbol('b', commutative=True))))"], [["log", 1], "Equality(log(Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), log(sin(Pow(Symbol('C', commutative=True), Symbol('b', commutative=True)))))"], ["renaming_premise", "Equality(Function('B')(Symbol('C', commutative=True), Symbol('b', commutative=True)), log(sin(Pow(Symbol('C', commutative=True), Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(log(Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Function('B')(Symbol('C', commutative=True), Symbol('b', commutative=True)))"], [["power", 4, "Symbol('C', commutative=True)"], "Equality(Pow(log(Function('v_1')(Symbol('C', commutative=True), Symbol('b', commutative=True))), Symbol('C', commutative=True)), Pow(Function('B')(Symbol('C', commutative=True), Symbol('b', commutative=True)), Symbol('C', commutative=True)))"]]}, {"prompt": "Given I{(\\Psi_{\\lambda},z)} = (e^{z})^{\\Psi_{\\lambda}}, then derive \\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)} = \\Psi_{\\lambda} (e^{z})^{\\Psi_{\\lambda}}, then obtain 2 = \\frac{\\Psi_{\\lambda} I{(\\Psi_{\\lambda},z)}}{\\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)}} + 1", "derivation": "I{(\\Psi_{\\lambda},z)} = (e^{z})^{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)} = \\frac{\\partial}{\\partial z} (e^{z})^{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)} = \\Psi_{\\lambda} (e^{z})^{\\Psi_{\\lambda}} and 1 = \\frac{\\Psi_{\\lambda} (e^{z})^{\\Psi_{\\lambda}}}{\\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)}} and 1 = \\frac{\\Psi_{\\lambda} (e^{z})^{\\Psi_{\\lambda}}}{\\frac{\\partial}{\\partial z} (e^{z})^{\\Psi_{\\lambda}}} and 2 = \\frac{\\Psi_{\\lambda} (e^{z})^{\\Psi_{\\lambda}}}{\\frac{\\partial}{\\partial z} (e^{z})^{\\Psi_{\\lambda}}} + 1 and 2 = \\frac{\\Psi_{\\lambda} I{(\\Psi_{\\lambda},z)}}{\\frac{\\partial}{\\partial z} I{(\\Psi_{\\lambda},z)}} + 1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Derivative(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Derivative(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Derivative(Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 5, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Derivative(Pow(exp(Symbol('z', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(2), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Pow(Derivative(Function('I')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given k{(\\mathbf{F})} = \\log{(\\mathbf{F})}, then obtain \\int \\frac{2 k{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} d\\mathbf{F} = \\int \\frac{2}{3} d\\mathbf{F}", "derivation": "k{(\\mathbf{F})} = \\log{(\\mathbf{F})} and 2 k{(\\mathbf{F})} = k{(\\mathbf{F})} + \\log{(\\mathbf{F})} and k{(\\mathbf{F})} + \\log{(\\mathbf{F})} = 2 \\log{(\\mathbf{F})} and 2 k{(\\mathbf{F})} = 2 \\log{(\\mathbf{F})} and \\frac{2 k{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} = \\frac{k{(\\mathbf{F})} + \\log{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} and \\frac{2}{3} = \\frac{k{(\\mathbf{F})} + \\log{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} and \\frac{2 k{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} = \\frac{2}{3} and \\int \\frac{2 k{(\\mathbf{F})}}{3 \\log{(\\mathbf{F})}} d\\mathbf{F} = \\int \\frac{2}{3} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 1, "Function('k')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "log(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 2, "Mul(Integer(3), log(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Rational(2, 3), Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Rational(1, 3), Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Rational(2, 3), Mul(Rational(1, 3), Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Rational(2, 3), Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Rational(2, 3))"], [["integrate", 7, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Mul(Rational(2, 3), Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Rational(2, 3), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(I,\\sigma_x)} = \\log{(I \\sigma_x)} and \\dot{\\mathbf{r}}{(I,\\sigma_x)} = I \\sigma_x, then obtain \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\mathbf{A}{(I,\\sigma_x)} \\log{(\\dot{\\mathbf{r}}{(I,\\sigma_x)})} = \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\log{(\\dot{\\mathbf{r}}{(I,\\sigma_x)})}^{2}", "derivation": "\\mathbf{A}{(I,\\sigma_x)} = \\log{(I \\sigma_x)} and I \\sigma_x \\mathbf{A}{(I,\\sigma_x)} = I \\sigma_x \\log{(I \\sigma_x)} and \\dot{\\mathbf{r}}{(I,\\sigma_x)} = I \\sigma_x and \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\mathbf{A}{(I,\\sigma_x)} = \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\log{(\\dot{\\mathbf{r}}{(I,\\sigma_x)})} and \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\mathbf{A}{(I,\\sigma_x)} \\log{(\\dot{\\mathbf{r}}{(I,\\sigma_x)})} = \\dot{\\mathbf{r}}{(I,\\sigma_x)} \\log{(\\dot{\\mathbf{r}}{(I,\\sigma_x)})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), log(Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 1, "Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True), log(Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), log(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 4, "log(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), log(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(log(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\dot{z}{(S,\\dot{y})} = \\sin{(\\frac{\\dot{y}}{S})} and \\operatorname{E_{n}}{(S,\\dot{y})} = \\dot{z}^{2}{(S,\\dot{y})}, then obtain \\int - \\operatorname{E_{n}}{(S,\\dot{y})} dS = \\int - \\sin^{2}{(\\frac{\\dot{y}}{S})} dS", "derivation": "\\dot{z}{(S,\\dot{y})} = \\sin{(\\frac{\\dot{y}}{S})} and - \\dot{z}{(S,\\dot{y})} = - \\sin{(\\frac{\\dot{y}}{S})} and - \\dot{z}^{2}{(S,\\dot{y})} = - \\dot{z}{(S,\\dot{y})} \\sin{(\\frac{\\dot{y}}{S})} and \\operatorname{E_{n}}{(S,\\dot{y})} = \\dot{z}^{2}{(S,\\dot{y})} and - \\operatorname{E_{n}}{(S,\\dot{y})} = - \\dot{z}{(S,\\dot{y})} \\sin{(\\frac{\\dot{y}}{S})} and - \\operatorname{E_{n}}{(S,\\dot{y})} = - \\sin^{2}{(\\frac{\\dot{y}}{S})} and \\int - \\operatorname{E_{n}}{(S,\\dot{y})} dS = \\int - \\sin^{2}{(\\frac{\\dot{y}}{S})} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)))))"], [["times", 1, "Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Function('E_n')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Function('E_n')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Pow(sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True))), Integer(2))))"], [["integrate", 6, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('E_n')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Integer(-1), Pow(sin(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True))), Integer(2))), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given V{(f)} = \\log{(\\sin{(f)})}, then obtain \\int \\frac{(\\log{(\\sin{(f)})} + \\sin{(f)}) V{(f)}}{V{(f)} + \\sin{(f)}} df - 1 = \\int \\log{(\\sin{(f)})} df - 1", "derivation": "V{(f)} = \\log{(\\sin{(f)})} and V{(f)} + \\sin{(f)} = \\log{(\\sin{(f)})} + \\sin{(f)} and \\frac{V{(f)} + \\sin{(f)}}{\\log{(\\sin{(f)})}} = \\frac{\\log{(\\sin{(f)})} + \\sin{(f)}}{\\log{(\\sin{(f)})}} and \\int V{(f)} df = \\int \\log{(\\sin{(f)})} df and \\int V{(f)} df - 1 = \\int \\log{(\\sin{(f)})} df - 1 and \\frac{(V{(f)} + \\sin{(f)}) V{(f)}}{\\log{(\\sin{(f)})}} = \\frac{(\\log{(\\sin{(f)})} + \\sin{(f)}) V{(f)}}{\\log{(\\sin{(f)})}} and V{(f)} = \\frac{(\\log{(\\sin{(f)})} + \\sin{(f)}) V{(f)}}{V{(f)} + \\sin{(f)}} and \\int \\frac{(\\log{(\\sin{(f)})} + \\sin{(f)}) V{(f)}}{V{(f)} + \\sin{(f)}} df - 1 = \\int \\log{(\\sin{(f)})} df - 1", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('f', commutative=True)), log(sin(Symbol('f', commutative=True))))"], [["add", 1, "sin(Symbol('f', commutative=True))"], "Equality(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Add(log(sin(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))))"], [["divide", 2, "log(sin(Symbol('f', commutative=True)))"], "Equality(Mul(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Pow(log(sin(Symbol('f', commutative=True))), Integer(-1))), Mul(Add(log(sin(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Pow(log(sin(Symbol('f', commutative=True))), Integer(-1))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('V')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(log(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integral(Function('V')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)), Add(Integral(log(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integer(-1)))"], [["times", 3, "Function('V')(Symbol('f', commutative=True))"], "Equality(Mul(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Function('V')(Symbol('f', commutative=True)), Pow(log(sin(Symbol('f', commutative=True))), Integer(-1))), Mul(Add(log(sin(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Function('V')(Symbol('f', commutative=True)), Pow(log(sin(Symbol('f', commutative=True))), Integer(-1))))"], [["divide", 6, "Mul(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Pow(log(sin(Symbol('f', commutative=True))), Integer(-1)))"], "Equality(Function('V')(Symbol('f', commutative=True)), Mul(Pow(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Integer(-1)), Add(log(sin(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Function('V')(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Integral(Mul(Pow(Add(Function('V')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Integer(-1)), Add(log(sin(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))), Function('V')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integer(-1)), Add(Integral(log(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\delta{(v)} = \\log{(\\cos{(v)})}, then obtain \\int 0 dv = \\int (- \\int (v + \\delta{(v)}) dv + \\int (v + \\log{(\\cos{(v)})}) dv) dv", "derivation": "\\delta{(v)} = \\log{(\\cos{(v)})} and v + \\delta{(v)} = v + \\log{(\\cos{(v)})} and \\int (v + \\delta{(v)}) dv = \\int (v + \\log{(\\cos{(v)})}) dv and v + \\delta{(v)} + \\int (v + \\delta{(v)}) dv = v + \\delta{(v)} + \\int (v + \\log{(\\cos{(v)})}) dv and 0 = - \\int (v + \\delta{(v)}) dv + \\int (v + \\log{(\\cos{(v)})}) dv and \\int 0 dv = \\int (- \\int (v + \\delta{(v)}) dv + \\int (v + \\log{(\\cos{(v)})}) dv) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('v', commutative=True)), log(cos(Symbol('v', commutative=True))))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), log(cos(Symbol('v', commutative=True)))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Add(Symbol('v', commutative=True), log(cos(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["add", 3, "Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True)))"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True)), Integral(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True)), Integral(Add(Symbol('v', commutative=True), log(cos(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)))))"], [["minus", 4, "Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True)), Integral(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Integral(Add(Symbol('v', commutative=True), log(cos(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)))))"], [["integrate", 5, "Symbol('v', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Integer(-1), Integral(Add(Symbol('v', commutative=True), Function('\\\\delta')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Integral(Add(Symbol('v', commutative=True), log(cos(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given a{(Q,\\theta_1)} = \\frac{\\theta_1}{Q}, then obtain - \\frac{(\\frac{\\theta_1}{Q})^{Q} (- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} a{(Q,\\theta_1)})}{v_{t}} = - \\frac{(\\frac{\\theta_1}{Q})^{Q} (- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q})}{v_{t}}", "derivation": "a{(Q,\\theta_1)} = \\frac{\\theta_1}{Q} and \\frac{\\partial}{\\partial Q} a{(Q,\\theta_1)} = \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q} and - (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} a{(Q,\\theta_1)} = - (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q} and \\frac{- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} a{(Q,\\theta_1)}}{v_{t}} = \\frac{- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q}}{v_{t}} and - \\frac{(\\frac{\\theta_1}{Q})^{Q} (- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} a{(Q,\\theta_1)})}{v_{t}} = - \\frac{(\\frac{\\theta_1}{Q})^{Q} (- (\\frac{\\theta_1}{Q})^{Q} + \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q})}{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('Q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('Q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Function('a')(Symbol('Q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["divide", 3, "Symbol('v_t', commutative=True)"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Function('a')(Symbol('Q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["times", 4, "Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Function('a')(Symbol('Q', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('Q', commutative=True))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{v}{(E_{n},\\sigma_p)} = \\frac{\\sigma_p}{E_{n}}, then obtain \\cos{(\\frac{\\sigma_p \\int \\frac{\\mathbf{v}{(E_{n},\\sigma_p)}}{\\sigma_p} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}})} = \\cos{(\\frac{\\sigma_p \\int \\frac{1}{E_{n}} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}})}", "derivation": "\\mathbf{v}{(E_{n},\\sigma_p)} = \\frac{\\sigma_p}{E_{n}} and \\frac{\\mathbf{v}{(E_{n},\\sigma_p)}}{\\sigma_p} = \\frac{1}{E_{n}} and \\int \\frac{\\mathbf{v}{(E_{n},\\sigma_p)}}{\\sigma_p} dE_{n} = \\int \\frac{1}{E_{n}} dE_{n} and \\frac{\\sigma_p \\int \\frac{\\mathbf{v}{(E_{n},\\sigma_p)}}{\\sigma_p} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}} = \\frac{\\sigma_p \\int \\frac{1}{E_{n}} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}} and \\cos{(\\frac{\\sigma_p \\int \\frac{\\mathbf{v}{(E_{n},\\sigma_p)}}{\\sigma_p} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}})} = \\cos{(\\frac{\\sigma_p \\int \\frac{1}{E_{n}} dE_{n}}{\\mathbf{v}{(E_{n},\\sigma_p)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('E_n', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integral(Pow(Symbol('E_n', commutative=True), Integer(-1)), Tuple(Symbol('E_n', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('E_n', commutative=True)))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Pow(Symbol('E_n', commutative=True), Integer(-1)), Tuple(Symbol('E_n', commutative=True)))))"], [["cos", 4], "Equality(cos(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('E_n', commutative=True))))), cos(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Pow(Symbol('E_n', commutative=True), Integer(-1)), Tuple(Symbol('E_n', commutative=True))))))"]]}, {"prompt": "Given \\psi{(b,n)} = \\int (- b + n) db, then derive \\psi{(b,n)} = - \\frac{b^{2}}{2} + b n + v_{2}, then obtain \\frac{(\\int (- b + n) db)^{n}}{a^{\\dagger}} = \\frac{(- \\frac{b^{2}}{2} + b n + v_{2})^{n}}{a^{\\dagger}}", "derivation": "\\psi{(b,n)} = \\int (- b + n) db and \\psi^{n}{(b,n)} = (\\int (- b + n) db)^{n} and \\psi{(b,n)} = - \\frac{b^{2}}{2} + b n + v_{2} and (- \\frac{b^{2}}{2} + b n + v_{2})^{n} = (\\int (- b + n) db)^{n} and \\frac{\\psi^{n}{(b,n)}}{a^{\\dagger}} = \\frac{(\\int (- b + n) db)^{n}}{a^{\\dagger}} and \\frac{\\psi^{n}{(b,n)}}{a^{\\dagger}} = \\frac{(- \\frac{b^{2}}{2} + b n + v_{2})^{n}}{a^{\\dagger}} and \\frac{(\\int (- b + n) db)^{n}}{a^{\\dagger}} = \\frac{(- \\frac{b^{2}}{2} + b n + v_{2})^{n}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('n', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('n', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('v_2', commutative=True)), Symbol('n', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('n', commutative=True)))"], [["divide", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('v_2', commutative=True)), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('n', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('n', commutative=True)), Symbol('v_2', commutative=True)), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\Psi_{\\lambda})} = e^{e^{\\Psi_{\\lambda}}}, then obtain \\frac{\\int \\chi^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}}{\\Psi_{\\lambda}} = \\frac{\\int (e^{e^{\\Psi_{\\lambda}}})^{\\Psi_{\\lambda}} d\\Psi_{\\lambda}}{\\Psi_{\\lambda}}", "derivation": "\\chi{(\\Psi_{\\lambda})} = e^{e^{\\Psi_{\\lambda}}} and \\chi^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})} = (e^{e^{\\Psi_{\\lambda}}})^{\\Psi_{\\lambda}} and \\int \\chi^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int (e^{e^{\\Psi_{\\lambda}}})^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} and \\frac{\\int \\chi^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}}{\\Psi_{\\lambda}} = \\frac{\\int (e^{e^{\\Psi_{\\lambda}}})^{\\Psi_{\\lambda}} d\\Psi_{\\lambda}}{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(exp(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Pow(exp(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Pow(exp(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(y)} = \\frac{d}{d y} \\sin{(y)}, then derive \\hat{x}_0{(y)} = \\cos{(y)}, then obtain (y + \\int \\cos^{y}{(y)} dy)^{y} = (y + \\int \\hat{x}_0^{y}{(y)} dy)^{y}", "derivation": "\\hat{x}_0{(y)} = \\frac{d}{d y} \\sin{(y)} and \\hat{x}_0^{y}{(y)} = (\\frac{d}{d y} \\sin{(y)})^{y} and \\hat{x}_0{(y)} = \\cos{(y)} and \\int \\hat{x}_0^{y}{(y)} dy = \\int (\\frac{d}{d y} \\sin{(y)})^{y} dy and y + \\int \\hat{x}_0^{y}{(y)} dy = y + \\int (\\frac{d}{d y} \\sin{(y)})^{y} dy and y + \\int \\cos^{y}{(y)} dy = y + \\int (\\frac{d}{d y} \\sin{(y)})^{y} dy and y + \\int \\cos^{y}{(y)} dy = y + \\int \\hat{x}_0^{y}{(y)} dy and (y + \\int \\cos^{y}{(y)} dy)^{y} = (y + \\int \\hat{x}_0^{y}{(y)} dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Derivative(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Derivative(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Pow(Derivative(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 4, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Integral(Pow(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Integral(Pow(Derivative(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('y', commutative=True), Integral(Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Integral(Pow(Derivative(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('y', commutative=True), Integral(Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Integral(Pow(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["power", 7, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Symbol('y', commutative=True), Integral(Pow(cos(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Pow(Add(Symbol('y', commutative=True), Integral(Pow(Function('\\\\hat{x}_0')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(J,W)} = - J + W, then derive \\int \\operatorname{C_{2}}{(J,W)} dJ = - \\frac{J^{2}}{2} + J W + \\ddot{x}, then obtain \\frac{\\partial}{\\partial W} - \\frac{J^{2} \\int (- J + W) dJ}{2} = \\frac{\\partial}{\\partial W} - \\frac{J^{2} (- \\frac{J^{2}}{2} + J W + \\ddot{x})}{2}", "derivation": "\\operatorname{C_{2}}{(J,W)} = - J + W and \\int \\operatorname{C_{2}}{(J,W)} dJ = \\int (- J + W) dJ and \\int \\operatorname{C_{2}}{(J,W)} dJ = - \\frac{J^{2}}{2} + J W + \\ddot{x} and \\int (- J + W) dJ = - \\frac{J^{2}}{2} + J W + \\ddot{x} and - \\frac{J^{2} \\int (- J + W) dJ}{2} = - \\frac{J^{2} (- \\frac{J^{2}}{2} + J W + \\ddot{x})}{2} and \\frac{\\partial}{\\partial W} - \\frac{J^{2} \\int (- J + W) dJ}{2} = \\frac{\\partial}{\\partial W} - \\frac{J^{2} (- \\frac{J^{2}}{2} + J W + \\ddot{x})}{2}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 4, "Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2)))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2)), Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2)), Integral(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(f^{\\prime})} = \\sin{(\\log{(f^{\\prime})})}, then derive \\int q{(f^{\\prime})} df^{\\prime} = \\pi + \\frac{f^{\\prime} \\sin{(\\log{(f^{\\prime})})}}{2} - \\frac{f^{\\prime} \\cos{(\\log{(f^{\\prime})})}}{2}, then obtain \\int \\sin{(\\log{(f^{\\prime})})} df^{\\prime} = \\pi + \\frac{f^{\\prime} q{(f^{\\prime})}}{2} - \\frac{f^{\\prime} \\cos{(\\log{(f^{\\prime})})}}{2}", "derivation": "q{(f^{\\prime})} = \\sin{(\\log{(f^{\\prime})})} and \\int q{(f^{\\prime})} df^{\\prime} = \\int \\sin{(\\log{(f^{\\prime})})} df^{\\prime} and \\int q{(f^{\\prime})} df^{\\prime} = \\pi + \\frac{f^{\\prime} \\sin{(\\log{(f^{\\prime})})}}{2} - \\frac{f^{\\prime} \\cos{(\\log{(f^{\\prime})})}}{2} and \\int \\sin{(\\log{(f^{\\prime})})} df^{\\prime} = \\pi + \\frac{f^{\\prime} \\sin{(\\log{(f^{\\prime})})}}{2} - \\frac{f^{\\prime} \\cos{(\\log{(f^{\\prime})})}}{2} and \\int \\sin{(\\log{(f^{\\prime})})} df^{\\prime} = \\pi + \\frac{f^{\\prime} q{(f^{\\prime})}}{2} - \\frac{f^{\\prime} \\cos{(\\log{(f^{\\prime})})}}{2}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('f^{\\\\prime}', commutative=True)), sin(log(Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('q')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(sin(log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('q')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), sin(log(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), cos(log(Symbol('f^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), sin(log(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), cos(log(Symbol('f^{\\\\prime}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(sin(log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), Function('q')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('f^{\\\\prime}', commutative=True), cos(log(Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(v_{2},L)} = L + v_{2} and \\mathbf{M}{(\\phi_1,t_{2})} = \\frac{t_{2}}{\\phi_1}, then obtain \\frac{\\mathbf{M}{(\\phi_1,t_{2})}}{L + v_{2} + (L + v_{2})^{2}} = \\frac{t_{2}}{\\phi_1 (L + v_{2} + (L + v_{2})^{2})}", "derivation": "\\mu_{0}{(v_{2},L)} = L + v_{2} and (L + v_{2}) \\mu_{0}{(v_{2},L)} = (L + v_{2})^{2} and L + v_{2} + (L + v_{2}) \\mu_{0}{(v_{2},L)} = L + v_{2} + (L + v_{2})^{2} and \\mathbf{M}{(\\phi_1,t_{2})} = \\frac{t_{2}}{\\phi_1} and \\frac{\\mathbf{M}{(\\phi_1,t_{2})}}{L + v_{2} + (L + v_{2}) \\mu_{0}{(v_{2},L)}} = \\frac{t_{2}}{\\phi_1 (L + v_{2} + (L + v_{2}) \\mu_{0}{(v_{2},L)})} and \\frac{\\mathbf{M}{(\\phi_1,t_{2})}}{L + v_{2} + (L + v_{2})^{2}} = \\frac{t_{2}}{\\phi_1 (L + v_{2} + (L + v_{2})^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Integer(2)))"], [["add", 2, "Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)))), Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Integer(2))))"], ["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["divide", 4, "Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)))), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\phi_1', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Symbol('L', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then obtain (\\mathbf{J}_P + \\mathbf{J}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} + \\log{(\\mathbf{J}_P)} = (\\mathbf{J}_P + \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P} + \\log{(\\mathbf{J}_P)}", "derivation": "\\mathbf{J}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\mathbf{J}_P + \\mathbf{J}{(\\mathbf{J}_P)} = \\mathbf{J}_P + \\log{(\\mathbf{J}_P)} and (\\mathbf{J}_P + \\mathbf{J}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\mathbf{J}_P + \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P} and (\\mathbf{J}_P + \\mathbf{J}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} + \\log{(\\mathbf{J}_P)} = (\\mathbf{J}_P + \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P} + \\log{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 3, "log(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(L,\\sigma_x)} = \\sin{(\\sigma_x^{L})}, then obtain \\int \\frac{\\partial}{\\partial \\sigma_x} \\int \\operatorname{A_{x}}{(L,\\sigma_x)} d\\sigma_x dL = \\int \\frac{\\partial}{\\partial \\sigma_x} \\int \\sin{(\\sigma_x^{L})} d\\sigma_x dL", "derivation": "\\operatorname{A_{x}}{(L,\\sigma_x)} = \\sin{(\\sigma_x^{L})} and \\int \\operatorname{A_{x}}{(L,\\sigma_x)} d\\sigma_x = \\int \\sin{(\\sigma_x^{L})} d\\sigma_x and \\frac{\\partial}{\\partial \\sigma_x} \\int \\operatorname{A_{x}}{(L,\\sigma_x)} d\\sigma_x = \\frac{\\partial}{\\partial \\sigma_x} \\int \\sin{(\\sigma_x^{L})} d\\sigma_x and \\int \\frac{\\partial}{\\partial \\sigma_x} \\int \\operatorname{A_{x}}{(L,\\sigma_x)} d\\sigma_x dL = \\int \\frac{\\partial}{\\partial \\sigma_x} \\int \\sin{(\\sigma_x^{L})} d\\sigma_x dL", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('L', commutative=True), Symbol('\\\\sigma_x', commutative=True)), sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('L', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('L', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integral(Function('A_x')(Symbol('L', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('A_x')(Symbol('L', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Integral(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given u{(F_{x})} = \\log{(\\cos{(F_{x})})}, then obtain (\\log{(\\cos{(F_{x})})} + \\cos{(F_{x})}) \\int u{(F_{x})} dF_{x} = (\\log{(\\cos{(F_{x})})} + \\cos{(F_{x})}) \\int \\log{(\\cos{(F_{x})})} dF_{x}", "derivation": "u{(F_{x})} = \\log{(\\cos{(F_{x})})} and \\int u{(F_{x})} dF_{x} = \\int \\log{(\\cos{(F_{x})})} dF_{x} and u{(F_{x})} + \\cos{(F_{x})} = \\log{(\\cos{(F_{x})})} + \\cos{(F_{x})} and (u{(F_{x})} + \\cos{(F_{x})}) \\int u{(F_{x})} dF_{x} = (u{(F_{x})} + \\cos{(F_{x})}) \\int \\log{(\\cos{(F_{x})})} dF_{x} and (\\log{(\\cos{(F_{x})})} + \\cos{(F_{x})}) \\int u{(F_{x})} dF_{x} = (\\log{(\\cos{(F_{x})})} + \\cos{(F_{x})}) \\int \\log{(\\cos{(F_{x})})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('F_x', commutative=True)), log(cos(Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('u')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(log(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["add", 1, "cos(Symbol('F_x', commutative=True))"], "Equality(Add(Function('u')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Add(log(cos(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))))"], [["times", 2, "Add(Function('u')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], "Equality(Mul(Add(Function('u')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Integral(Function('u')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Add(Function('u')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Integral(log(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(log(cos(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))), Integral(Function('u')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Add(log(cos(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))), Integral(log(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\theta,v,v_{x})} = \\theta - v - v_{x}, then obtain (- \\frac{2 v_{x}}{(- v_{x} + \\operatorname{v_{x}}{(\\theta,v,v_{x})})^{2}})^{v} = (- \\frac{2 v_{x}}{(\\theta - v - 2 v_{x})^{2}})^{v}", "derivation": "\\operatorname{v_{x}}{(\\theta,v,v_{x})} = \\theta - v - v_{x} and - v_{x} + \\operatorname{v_{x}}{(\\theta,v,v_{x})} = \\theta - v - 2 v_{x} and \\frac{1}{(- v_{x} + \\operatorname{v_{x}}{(\\theta,v,v_{x})})^{2}} = \\frac{1}{(\\theta - v - 2 v_{x})^{2}} and - \\frac{2 v_{x}}{(- v_{x} + \\operatorname{v_{x}}{(\\theta,v,v_{x})})^{2}} = - \\frac{2 v_{x}}{(\\theta - v - 2 v_{x})^{2}} and (- \\frac{2 v_{x}}{(- v_{x} + \\operatorname{v_{x}}{(\\theta,v,v_{x})})^{2}})^{v} = (- \\frac{2 v_{x}}{(\\theta - v - 2 v_{x})^{2}})^{v}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\theta', commutative=True), Symbol('v', commutative=True), Symbol('v_x', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["minus", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('v_x')(Symbol('\\\\theta', commutative=True), Symbol('v', commutative=True), Symbol('v_x', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True))))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('v_x')(Symbol('\\\\theta', commutative=True), Symbol('v', commutative=True), Symbol('v_x', commutative=True))), Integer(-2)), Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True))), Integer(-2)))"], [["times", 3, "Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('v_x')(Symbol('\\\\theta', commutative=True), Symbol('v', commutative=True), Symbol('v_x', commutative=True))), Integer(-2))), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True), Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True))), Integer(-2))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('v_x')(Symbol('\\\\theta', commutative=True), Symbol('v', commutative=True), Symbol('v_x', commutative=True))), Integer(-2))), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True), Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True))), Integer(-2))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\pi{(L)} = \\cos{(L)}, then derive \\frac{d^{2}}{d L^{2}} \\pi{(L)} = - \\cos{(L)}, then obtain \\frac{d^{2}}{d L^{2}} \\pi{(L)} = - \\pi{(L)}", "derivation": "\\pi{(L)} = \\cos{(L)} and \\frac{d}{d L} \\pi{(L)} = \\frac{d}{d L} \\cos{(L)} and \\frac{d^{2}}{d L^{2}} \\pi{(L)} = \\frac{d^{2}}{d L^{2}} \\cos{(L)} and \\frac{d^{2}}{d L^{2}} \\pi{(L)} = - \\cos{(L)} and \\frac{d^{2}}{d L^{2}} \\pi{(L)} = - \\pi{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(cos(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Derivative(cos(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Mul(Integer(-1), Function('\\\\pi')(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(u)} = \\cos{(u)}, then obtain (\\int \\frac{\\operatorname{A_{1}}{(u)}}{u} du)^{u} = (\\int \\frac{\\cos{(u)}}{u} du)^{u}", "derivation": "\\operatorname{A_{1}}{(u)} = \\cos{(u)} and \\frac{\\operatorname{A_{1}}{(u)}}{u} = \\frac{\\cos{(u)}}{u} and \\int \\frac{\\operatorname{A_{1}}{(u)}}{u} du = \\int \\frac{\\cos{(u)}}{u} du and (\\int \\frac{\\operatorname{A_{1}}{(u)}}{u} du)^{u} = (\\int \\frac{\\cos{(u)}}{u} du)^{u}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["divide", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A_1')(Symbol('u', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A_1')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A_1')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(n)} = \\sin{(n)} and \\operatorname{g_{\\varepsilon}}{(\\pi)} = \\sin{(\\sin{(\\pi)})}, then obtain \\operatorname{g_{\\varepsilon}}{(\\pi)} + \\int \\operatorname{E_{x}}{(n)} dn = \\operatorname{g_{\\varepsilon}}{(\\pi)} + \\int \\sin{(n)} dn", "derivation": "\\operatorname{E_{x}}{(n)} = \\sin{(n)} and \\int \\operatorname{E_{x}}{(n)} dn = \\int \\sin{(n)} dn and \\operatorname{g_{\\varepsilon}}{(\\pi)} = \\sin{(\\sin{(\\pi)})} and \\sin{(\\sin{(\\pi)})} + \\int \\operatorname{E_{x}}{(n)} dn = \\sin{(\\sin{(\\pi)})} + \\int \\sin{(n)} dn and \\operatorname{g_{\\varepsilon}}{(\\pi)} + \\int \\operatorname{E_{x}}{(n)} dn = \\operatorname{g_{\\varepsilon}}{(\\pi)} + \\int \\sin{(n)} dn", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], ["get_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "sin(sin(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(sin(sin(Symbol('\\\\pi', commutative=True))), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(sin(sin(Symbol('\\\\pi', commutative=True))), Integral(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(M_{E},\\mathbf{A})} = M_{E} + \\mathbf{A}, then obtain \\iiint \\hat{p}^{2}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} dM_{E} = \\iiint (M_{E} + \\mathbf{A}) \\hat{p}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} dM_{E}", "derivation": "\\hat{p}{(M_{E},\\mathbf{A})} = M_{E} + \\mathbf{A} and \\hat{p}^{2}{(M_{E},\\mathbf{A})} = (M_{E} + \\mathbf{A}) \\hat{p}{(M_{E},\\mathbf{A})} and \\int \\hat{p}^{2}{(M_{E},\\mathbf{A})} d\\mathbf{A} = \\int (M_{E} + \\mathbf{A}) \\hat{p}{(M_{E},\\mathbf{A})} d\\mathbf{A} and \\iint \\hat{p}^{2}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} = \\iint (M_{E} + \\mathbf{A}) \\hat{p}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} and \\iiint \\hat{p}^{2}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} dM_{E} = \\iiint (M_{E} + \\mathbf{A}) \\hat{p}{(M_{E},\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{p}')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(P_{e})} = \\cos{(P_{e})}, then obtain \\int (4 \\operatorname{E_{n}}{(P_{e})} - 3 \\cos{(P_{e})}) dP_{e} = \\int \\cos{(P_{e})} dP_{e}", "derivation": "\\operatorname{E_{n}}{(P_{e})} = \\cos{(P_{e})} and \\operatorname{E_{n}}{(P_{e})} - \\cos{(P_{e})} = 0 and 2 \\operatorname{E_{n}}{(P_{e})} - \\cos{(P_{e})} = \\operatorname{E_{n}}{(P_{e})} and 2 \\operatorname{E_{n}}{(P_{e})} - \\cos{(P_{e})} = \\cos{(P_{e})} and \\int (2 \\operatorname{E_{n}}{(P_{e})} - \\cos{(P_{e})}) dP_{e} = \\int \\cos{(P_{e})} dP_{e} and \\int (4 \\operatorname{E_{n}}{(P_{e})} - 3 \\cos{(P_{e})}) dP_{e} = \\int \\cos{(P_{e})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["minus", 1, "cos(Symbol('P_e', commutative=True))"], "Equality(Add(Function('E_n')(Symbol('P_e', commutative=True)), Mul(Integer(-1), cos(Symbol('P_e', commutative=True)))), Integer(0))"], [["add", 2, "Function('E_n')(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('E_n')(Symbol('P_e', commutative=True))), Mul(Integer(-1), cos(Symbol('P_e', commutative=True)))), Function('E_n')(Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Add(Mul(Integer(2), Function('E_n')(Symbol('P_e', commutative=True))), Mul(Integer(-1), cos(Symbol('P_e', commutative=True)))), cos(Symbol('P_e', commutative=True)))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('E_n')(Symbol('P_e', commutative=True))), Mul(Integer(-1), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True))), Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Add(Mul(Integer(4), Function('E_n')(Symbol('P_e', commutative=True))), Mul(Integer(-1), Integer(3), cos(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True))), Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain \\frac{d}{d \\theta_1} \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\operatorname{C_{1}}{(\\theta_1)} d\\theta_1 = \\frac{d}{d \\theta_1} \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\sin{(\\theta_1)} d\\theta_1", "derivation": "\\operatorname{C_{1}}{(\\theta_1)} = \\sin{(\\theta_1)} and \\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)} = 2 \\sin{(\\theta_1)} and 2 \\operatorname{C_{1}}{(\\theta_1)} \\sin{(\\theta_1)} = 2 \\sin^{2}{(\\theta_1)} and (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\operatorname{C_{1}}{(\\theta_1)} = (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\sin{(\\theta_1)} and \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\operatorname{C_{1}}{(\\theta_1)} d\\theta_1 = \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\sin{(\\theta_1)} d\\theta_1 and \\frac{d}{d \\theta_1} \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\operatorname{C_{1}}{(\\theta_1)} d\\theta_1 = \\frac{d}{d \\theta_1} \\int (\\operatorname{C_{1}}{(\\theta_1)} + \\sin{(\\theta_1)}) \\sin{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Mul(Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Integer(2), Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Function('C_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Function('C_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integral(Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Function('C_1')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Mul(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(A_{z})} = \\sin{(A_{z})}, then derive \\frac{d}{d A_{z}} \\rho{(A_{z})} = \\cos{(A_{z})}, then obtain \\cos{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})}", "derivation": "\\rho{(A_{z})} = \\sin{(A_{z})} and \\frac{d}{d A_{z}} \\rho{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})} and \\frac{d}{d A_{z}} \\rho{(A_{z})} = \\cos{(A_{z})} and \\cos{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), cos(Symbol('A_z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('A_z', commutative=True)), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\operatorname{A_{y}}{(J_{\\varepsilon})} \\int \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} dJ_{\\varepsilon} = e^{J_{\\varepsilon}} \\int \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} dJ_{\\varepsilon}", "derivation": "\\operatorname{A_{y}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} = 1 and \\int \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} dJ_{\\varepsilon} = \\int 1 dJ_{\\varepsilon} and \\operatorname{A_{y}}{(J_{\\varepsilon})} \\int 1 dJ_{\\varepsilon} = e^{J_{\\varepsilon}} \\int 1 dJ_{\\varepsilon} and \\operatorname{A_{y}}{(J_{\\varepsilon})} \\int \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} dJ_{\\varepsilon} = e^{J_{\\varepsilon}} \\int \\operatorname{A_{y}}{(J_{\\varepsilon})} e^{- J_{\\varepsilon}} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "exp(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Mul(Function('A_y')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and \\operatorname{m_{s}}{(\\hat{\\mathbf{x}})} = e^{2 \\hat{\\mathbf{x}}}, then obtain \\operatorname{E_{\\lambda}}^{2}{(\\hat{\\mathbf{x}})} \\operatorname{m_{s}}{(\\hat{\\mathbf{x}})} = \\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} e^{3 \\hat{\\mathbf{x}}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and \\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} e^{\\hat{\\mathbf{x}}} = e^{2 \\hat{\\mathbf{x}}} and \\operatorname{m_{s}}{(\\hat{\\mathbf{x}})} = e^{2 \\hat{\\mathbf{x}}} and \\operatorname{E_{\\lambda}}^{2}{(\\hat{\\mathbf{x}})} e^{2 \\hat{\\mathbf{x}}} = \\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} e^{3 \\hat{\\mathbf{x}}} and \\operatorname{E_{\\lambda}}^{2}{(\\hat{\\mathbf{x}})} \\operatorname{m_{s}}{(\\hat{\\mathbf{x}})} = \\operatorname{E_{\\lambda}}{(\\hat{\\mathbf{x}})} e^{3 \\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 2, "Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Mul(Integer(3), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(2)), Function('m_s')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Mul(Integer(3), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given y{(Q)} = e^{Q}, then obtain (2 y{(Q)} \\frac{d}{d Q} y{(Q)})^{Q} = (y{(Q)} e^{Q} + e^{Q} \\frac{d}{d Q} y{(Q)})^{Q}", "derivation": "y{(Q)} = e^{Q} and y^{2}{(Q)} = y{(Q)} e^{Q} and \\frac{d}{d Q} y^{2}{(Q)} = \\frac{d}{d Q} y{(Q)} e^{Q} and (\\frac{d}{d Q} y^{2}{(Q)})^{Q} = (\\frac{d}{d Q} y{(Q)} e^{Q})^{Q} and (2 y{(Q)} \\frac{d}{d Q} y{(Q)})^{Q} = (y{(Q)} e^{Q} + e^{Q} \\frac{d}{d Q} y{(Q)})^{Q}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["times", 1, "Function('y')(Symbol('Q', commutative=True))"], "Equality(Pow(Function('y')(Symbol('Q', commutative=True)), Integer(2)), Mul(Function('y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Pow(Function('y')(Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Function('y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('y')(Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Mul(Function('y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(Integer(2), Function('y')(Symbol('Q', commutative=True)), Derivative(Function('y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Symbol('Q', commutative=True)), Pow(Add(Mul(Function('y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Mul(exp(Symbol('Q', commutative=True)), Derivative(Function('y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(y)} = \\int \\cos{(y)} dy and L{(y)} = - \\frac{\\operatorname{L_{\\varepsilon}}{(y)}}{\\sin{(y)}}, then derive \\operatorname{L_{\\varepsilon}}{(y)} = Q + \\sin{(y)}, then obtain L{(y)} = - \\frac{Q + \\sin{(y)}}{\\sin{(y)}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(y)} = \\int \\cos{(y)} dy and \\operatorname{L_{\\varepsilon}}{(y)} = Q + \\sin{(y)} and \\frac{\\operatorname{L_{\\varepsilon}}{(y)}}{\\sin{(y)}} = \\frac{Q + \\sin{(y)}}{\\sin{(y)}} and - \\frac{\\operatorname{L_{\\varepsilon}}{(y)}}{\\sin{(y)}} = - \\frac{Q + \\sin{(y)}}{\\sin{(y)}} and L{(y)} = - \\frac{\\operatorname{L_{\\varepsilon}}{(y)}}{\\sin{(y)}} and L{(y)} = - \\frac{Q + \\sin{(y)}}{\\sin{(y)}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True)), Add(Symbol('Q', commutative=True), sin(Symbol('y', commutative=True))))"], [["divide", 2, "sin(Symbol('y', commutative=True))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Integer(-1))), Mul(Add(Symbol('Q', commutative=True), sin(Symbol('y', commutative=True))), Pow(sin(Symbol('y', commutative=True)), Integer(-1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('Q', commutative=True), sin(Symbol('y', commutative=True))), Pow(sin(Symbol('y', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('L')(Symbol('y', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('L')(Symbol('y', commutative=True)), Mul(Integer(-1), Add(Symbol('Q', commutative=True), sin(Symbol('y', commutative=True))), Pow(sin(Symbol('y', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{F}{(A_{2})} = e^{A_{2}}, then obtain (\\frac{- A_{2} + \\mathbf{F}{(A_{2})}}{- A_{2} + e^{A_{2}}})^{A_{2}} = 1", "derivation": "\\mathbf{F}{(A_{2})} = e^{A_{2}} and - A_{2} + \\mathbf{F}{(A_{2})} = - A_{2} + e^{A_{2}} and \\frac{- A_{2} + \\mathbf{F}{(A_{2})}}{- A_{2} + e^{A_{2}}} = 1 and (\\frac{- A_{2} + \\mathbf{F}{(A_{2})}}{- A_{2} + e^{A_{2}}})^{A_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\mathbf{F}')(Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\mathbf{F}')(Symbol('A_2', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('A_2', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\mathbf{F}')(Symbol('A_2', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True))), Integer(-1))), Symbol('A_2', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(Q)} = \\log{(\\sin{(Q)})}, then obtain (\\cos^{Q}{(\\operatorname{n_{1}}^{Q}{(Q)})})^{Q} = (\\cos^{Q}{(\\log{(\\sin{(Q)})}^{Q})})^{Q}", "derivation": "\\operatorname{n_{1}}{(Q)} = \\log{(\\sin{(Q)})} and \\operatorname{n_{1}}^{Q}{(Q)} = \\log{(\\sin{(Q)})}^{Q} and \\cos{(\\operatorname{n_{1}}^{Q}{(Q)})} = \\cos{(\\log{(\\sin{(Q)})}^{Q})} and \\cos^{Q}{(\\operatorname{n_{1}}^{Q}{(Q)})} = \\cos^{Q}{(\\log{(\\sin{(Q)})}^{Q})} and (\\cos^{Q}{(\\operatorname{n_{1}}^{Q}{(Q)})})^{Q} = (\\cos^{Q}{(\\log{(\\sin{(Q)})}^{Q})})^{Q}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('Q', commutative=True)), log(sin(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(log(sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('n_1')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), cos(Pow(log(sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))))"], [["power", 3, "Symbol('Q', commutative=True)"], "Equality(Pow(cos(Pow(Function('n_1')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(cos(Pow(log(sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(cos(Pow(Function('n_1')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(cos(Pow(log(sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given T{(\\mathbf{p},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mathbf{p} \\mu_0, then derive T{(\\mathbf{p},\\mu_0)} = \\mathbf{p}, then obtain \\int \\frac{\\int \\mathbf{p} d\\mathbf{p}}{\\mu_0} d\\mathbf{p} = \\int \\frac{\\int T{(\\mathbf{p},\\mu_0)} d\\mathbf{p}}{\\mu_0} d\\mathbf{p}", "derivation": "T{(\\mathbf{p},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mathbf{p} \\mu_0 and \\int T{(\\mathbf{p},\\mu_0)} d\\mathbf{p} = \\int \\frac{\\partial}{\\partial \\mu_0} \\mathbf{p} \\mu_0 d\\mathbf{p} and T{(\\mathbf{p},\\mu_0)} = \\mathbf{p} and \\frac{\\int T{(\\mathbf{p},\\mu_0)} d\\mathbf{p}}{\\mu_0} = \\frac{\\int \\frac{\\partial}{\\partial \\mu_0} \\mathbf{p} \\mu_0 d\\mathbf{p}}{\\mu_0} and \\frac{\\int \\mathbf{p} d\\mathbf{p}}{\\mu_0} = \\frac{\\int \\frac{\\partial}{\\partial \\mu_0} \\mathbf{p} \\mu_0 d\\mathbf{p}}{\\mu_0} and \\frac{\\int \\mathbf{p} d\\mathbf{p}}{\\mu_0} = \\frac{\\int T{(\\mathbf{p},\\mu_0)} d\\mathbf{p}}{\\mu_0} and \\int \\frac{\\int \\mathbf{p} d\\mathbf{p}}{\\mu_0} d\\mathbf{p} = \\int \\frac{\\int T{(\\mathbf{p},\\mu_0)} d\\mathbf{p}}{\\mu_0} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], [["divide", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('T')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given i{(G,\\hbar)} = - G + \\hbar, then obtain \\int (2 i{(G,\\hbar)})^{\\hbar} i{(G,\\hbar)} dG = \\int (- 2 G + 2 \\hbar)^{\\hbar} i{(G,\\hbar)} dG", "derivation": "i{(G,\\hbar)} = - G + \\hbar and - G + \\hbar + i{(G,\\hbar)} = - 2 G + 2 \\hbar and 2 i{(G,\\hbar)} = - 2 G + 2 \\hbar and 2 i{(G,\\hbar)} = - G + \\hbar + i{(G,\\hbar)} and (2 i{(G,\\hbar)})^{\\hbar} = (- G + \\hbar + i{(G,\\hbar)})^{\\hbar} and (2 i{(G,\\hbar)})^{\\hbar} i{(G,\\hbar)} = (- G + \\hbar + i{(G,\\hbar)})^{\\hbar} i{(G,\\hbar)} and (2 i{(G,\\hbar)})^{\\hbar} i{(G,\\hbar)} = (- 2 G + 2 \\hbar)^{\\hbar} i{(G,\\hbar)} and \\int (2 i{(G,\\hbar)})^{\\hbar} i{(G,\\hbar)} dG = \\int (- 2 G + 2 \\hbar)^{\\hbar} i{(G,\\hbar)} dG", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["times", 5, "Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 7, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(Mul(Integer(2), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Function('i')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given h{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger}}{\\mu}, then obtain \\frac{\\Psi^{\\dagger}}{\\mu} + (\\frac{\\Psi^{\\dagger}}{\\mu})^{\\mu} h{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger} (\\frac{\\Psi^{\\dagger}}{\\mu})^{\\mu}}{\\mu} + \\frac{\\Psi^{\\dagger}}{\\mu}", "derivation": "h{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger}}{\\mu} and h^{\\mu}{(\\Psi^{\\dagger},\\mu)} = (\\frac{\\Psi^{\\dagger}}{\\mu})^{\\mu} and h{(\\Psi^{\\dagger},\\mu)} h^{\\mu}{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger} h^{\\mu}{(\\Psi^{\\dagger},\\mu)}}{\\mu} and \\frac{\\Psi^{\\dagger}}{\\mu} + h{(\\Psi^{\\dagger},\\mu)} h^{\\mu}{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger} h^{\\mu}{(\\Psi^{\\dagger},\\mu)}}{\\mu} + \\frac{\\Psi^{\\dagger}}{\\mu} and \\frac{\\Psi^{\\dagger}}{\\mu} + (\\frac{\\Psi^{\\dagger}}{\\mu})^{\\mu} h{(\\Psi^{\\dagger},\\mu)} = \\frac{\\Psi^{\\dagger} (\\frac{\\Psi^{\\dagger}}{\\mu})^{\\mu}}{\\mu} + \\frac{\\Psi^{\\dagger}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Mul(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Mul(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True)), Function('h')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(A)} = \\log{(A)}, then derive \\frac{d}{d A} \\hat{H}_{\\lambda}{(A)} = \\frac{1}{A}, then obtain 1 = \\frac{1}{A \\frac{d}{d A} \\log{(A)}}", "derivation": "\\hat{H}_{\\lambda}{(A)} = \\log{(A)} and \\frac{d}{d A} \\hat{H}_{\\lambda}{(A)} = \\frac{d}{d A} \\log{(A)} and \\frac{d}{d A} \\hat{H}_{\\lambda}{(A)} = \\frac{1}{A} and \\frac{d}{d A} \\log{(A)} = \\frac{1}{A} and 1 = \\frac{1}{A \\frac{d}{d A} \\log{(A)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Pow(Symbol('A', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Pow(Symbol('A', commutative=True), Integer(-1)))"], [["divide", 4, "Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(m)} = \\log{(m)}, then derive \\frac{d}{d m} \\operatorname{c_{0}}{(m)} - \\frac{1}{m} = 0, then obtain \\frac{d}{d m} \\log{(m)} - \\frac{1}{m} = 0", "derivation": "\\operatorname{c_{0}}{(m)} = \\log{(m)} and \\operatorname{c_{0}}{(m)} - \\log{(m)} = 0 and \\frac{d}{d m} \\operatorname{c_{0}}{(m)} = \\frac{d}{d m} \\log{(m)} and \\frac{d}{d m} (\\operatorname{c_{0}}{(m)} - \\log{(m)}) = \\frac{d}{d m} 0 and \\frac{d}{d m} \\operatorname{c_{0}}{(m)} - \\frac{1}{m} = 0 and \\frac{d}{d m} \\log{(m)} - \\frac{1}{m} = 0", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["minus", 1, "log(Symbol('m', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('m', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Integer(0))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('m', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('c_0')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(u)} = \\cos{(\\log{(u)})} and \\operatorname{F_{N}}{(u)} = \\cos^{u}{(\\log{(u)})}, then obtain - \\frac{\\cos^{u}{(\\log{(u)})}}{u} = - \\frac{\\operatorname{t_{1}}^{u}{(u)}}{u}", "derivation": "\\operatorname{t_{1}}{(u)} = \\cos{(\\log{(u)})} and \\operatorname{F_{N}}{(u)} = \\cos^{u}{(\\log{(u)})} and \\operatorname{F_{N}}{(u)} = \\operatorname{t_{1}}^{u}{(u)} and - \\frac{\\operatorname{F_{N}}{(u)}}{u} = - \\frac{\\operatorname{t_{1}}^{u}{(u)}}{u} and - \\frac{\\cos^{u}{(\\log{(u)})}}{u} = - \\frac{\\operatorname{t_{1}}^{u}{(u)}}{u}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('u', commutative=True)), cos(log(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('u', commutative=True)), Pow(cos(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_N')(Symbol('u', commutative=True)), Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Function('F_N')(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(cos(log(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\varphi{(t,A)} = - A + t and \\operatorname{n_{2}}{(t,A)} = (- A + t)^{t}, then obtain \\frac{- (- A + t)^{t} + \\varphi{(t,A)}}{\\cos{((- A + t)^{t})}} = \\frac{- A + t - (- A + t)^{t}}{\\cos{((- A + t)^{t})}}", "derivation": "\\varphi{(t,A)} = - A + t and \\varphi^{t}{(t,A)} = (- A + t)^{t} and \\varphi{(t,A)} - \\varphi^{t}{(t,A)} = - A + t - \\varphi^{t}{(t,A)} and \\operatorname{n_{2}}{(t,A)} = (- A + t)^{t} and - (- A + t)^{t} + \\varphi{(t,A)} = - A + t - (- A + t)^{t} and \\frac{- (- A + t)^{t} + \\varphi{(t,A)}}{\\cos{(\\operatorname{n_{2}}{(t,A)})}} = \\frac{- A + t - (- A + t)^{t}}{\\cos{(\\operatorname{n_{2}}{(t,A)})}} and \\frac{- (- A + t)^{t} + \\varphi{(t,A)}}{\\cos{((- A + t)^{t})}} = \\frac{- A + t - (- A + t)^{t}}{\\cos{((- A + t)^{t})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Symbol('t', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Symbol('t', commutative=True)))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('t', commutative=True), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)))))"], [["divide", 5, "cos(Function('n_2')(Symbol('t', commutative=True), Symbol('A', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True))), Pow(cos(Function('n_2')(Symbol('t', commutative=True), Symbol('A', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)))), Pow(cos(Function('n_2')(Symbol('t', commutative=True), Symbol('A', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('\\\\varphi')(Symbol('t', commutative=True), Symbol('A', commutative=True))), Pow(cos(Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True)))), Pow(cos(Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A)} = \\log{(A)}, then obtain - \\int \\frac{\\log{(A)}}{\\operatorname{E_{n}}{(A)}} dA + \\frac{1}{\\operatorname{E_{n}}{(A)}} = - \\int \\frac{\\log{(A)}}{\\operatorname{E_{n}}{(A)}} dA + \\frac{\\log{(A)}}{\\operatorname{E_{n}}^{2}{(A)}}", "derivation": "\\operatorname{E_{n}}{(A)} = \\log{(A)} and 1 = \\frac{\\log{(A)}}{\\operatorname{E_{n}}{(A)}} and \\frac{1}{\\operatorname{E_{n}}{(A)}} = \\frac{\\log{(A)}}{\\operatorname{E_{n}}^{2}{(A)}} and - \\int \\frac{\\log{(A)}}{\\operatorname{E_{n}}{(A)}} dA + \\frac{1}{\\operatorname{E_{n}}{(A)}} = - \\int \\frac{\\log{(A)}}{\\operatorname{E_{n}}{(A)}} dA + \\frac{\\log{(A)}}{\\operatorname{E_{n}}^{2}{(A)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["divide", 1, "Function('E_n')(Symbol('A', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1)), log(Symbol('A', commutative=True))))"], [["divide", 2, "Function('E_n')(Symbol('A', commutative=True))"], "Equality(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1)), Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-2)), log(Symbol('A', commutative=True))))"], [["minus", 3, "Integral(Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Integral(Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-1)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Function('E_n')(Symbol('A', commutative=True)), Integer(-2)), log(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(z)} = \\sin{(z)}, then derive \\int \\dot{x}{(z)} dz = \\pi - \\cos{(z)}, then obtain \\frac{\\log{(\\int \\dot{x}{(z)} dz)}}{\\int \\dot{x}{(z)} dz} = \\frac{\\log{(\\int \\sin{(z)} dz)}}{\\int \\dot{x}{(z)} dz}", "derivation": "\\dot{x}{(z)} = \\sin{(z)} and \\int \\dot{x}{(z)} dz = \\int \\sin{(z)} dz and \\log{(\\int \\dot{x}{(z)} dz)} = \\log{(\\int \\sin{(z)} dz)} and \\int \\dot{x}{(z)} dz = \\pi - \\cos{(z)} and \\frac{\\log{(\\int \\dot{x}{(z)} dz)}}{\\pi - \\cos{(z)}} = \\frac{\\log{(\\int \\sin{(z)} dz)}}{\\pi - \\cos{(z)}} and \\frac{\\log{(\\int \\dot{x}{(z)} dz)}}{\\int \\dot{x}{(z)} dz} = \\frac{\\log{(\\int \\sin{(z)} dz)}}{\\int \\dot{x}{(z)} dz}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), log(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), log(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), log(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(log(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Pow(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))), Mul(log(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Pow(Integral(Function('\\\\dot{x}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(t,G)} = \\frac{t}{G}, then derive \\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)} = 0, then obtain e^{(\\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)})^{2}} = 1", "derivation": "\\mathbf{r}{(t,G)} = \\frac{t}{G} and \\frac{\\partial}{\\partial t} \\mathbf{r}{(t,G)} = \\frac{\\partial}{\\partial t} \\frac{t}{G} and \\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)} = \\frac{\\partial^{2}}{\\partial t^{2}} \\frac{t}{G} and \\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)} = 0 and (\\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)})^{2} = 0 and e^{(\\frac{\\partial^{2}}{\\partial t^{2}} \\mathbf{r}{(t,G)})^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Integer(0))"], [["power", 4, 2], "Equality(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Integer(2)), Integer(0))"], [["exp", 5], "Equality(exp(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Integer(2))), Integer(1))"]]}, {"prompt": "Given k{(\\hat{x},s,\\mathbf{B})} = \\frac{\\mathbf{B}}{\\hat{x} s}, then obtain \\int \\hat{x} d\\hat{x} = \\int \\frac{\\mathbf{B}}{s k{(\\hat{x},s,\\mathbf{B})}} d\\hat{x}", "derivation": "k{(\\hat{x},s,\\mathbf{B})} = \\frac{\\mathbf{B}}{\\hat{x} s} and \\hat{x} k{(\\hat{x},s,\\mathbf{B})} = \\frac{\\mathbf{B}}{s} and \\hat{x} = \\frac{\\mathbf{B}}{s k{(\\hat{x},s,\\mathbf{B})}} and \\int \\hat{x} d\\hat{x} = \\int \\frac{\\mathbf{B}}{s k{(\\hat{x},s,\\mathbf{B})}} d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["divide", 2, "Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Symbol('\\\\hat{x}', commutative=True), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(P_{g})} = e^{P_{g}}, then obtain P_{g} \\frac{d}{d P_{g}} \\psi^{*}{(P_{g})} + \\psi^{*}{(P_{g})} = P_{g} e^{P_{g}} + e^{P_{g}}", "derivation": "\\psi^{*}{(P_{g})} = e^{P_{g}} and P_{g} \\psi^{*}{(P_{g})} = P_{g} e^{P_{g}} and \\frac{d}{d P_{g}} P_{g} \\psi^{*}{(P_{g})} = \\frac{d}{d P_{g}} P_{g} e^{P_{g}} and P_{g} \\frac{d}{d P_{g}} \\psi^{*}{(P_{g})} + \\psi^{*}{(P_{g})} = P_{g} e^{P_{g}} + e^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\psi^*')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["differentiate", 2, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Mul(Symbol('P_g', commutative=True), Function('\\\\psi^*')(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('P_g', commutative=True), Derivative(Function('\\\\psi^*')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Function('\\\\psi^*')(Symbol('P_g', commutative=True))), Add(Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), exp(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\phi_2,f_{\\mathbf{v}})} = f_{\\mathbf{v}}^{\\phi_2} and \\operatorname{A_{2}}{(\\phi_2,f_{\\mathbf{v}})} = f_{\\mathbf{v}}^{\\phi_2}, then obtain - f_{\\mathbf{v}}^{\\phi_2} + \\operatorname{A_{2}}{(\\phi_2,f_{\\mathbf{v}})} = 0", "derivation": "\\mathbb{I}{(\\phi_2,f_{\\mathbf{v}})} = f_{\\mathbf{v}}^{\\phi_2} and - f_{\\mathbf{v}}^{\\phi_2} + \\mathbb{I}{(\\phi_2,f_{\\mathbf{v}})} = 0 and \\operatorname{A_{2}}{(\\phi_2,f_{\\mathbf{v}})} = f_{\\mathbf{v}}^{\\phi_2} and \\operatorname{A_{2}}{(\\phi_2,f_{\\mathbf{v}})} = \\mathbb{I}{(\\phi_2,f_{\\mathbf{v}})} and - f_{\\mathbf{v}}^{\\phi_2} + \\operatorname{A_{2}}{(\\phi_2,f_{\\mathbf{v}})} = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_2')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Function('A_2')(Symbol('\\\\phi_2', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{A}{(v_{t})} = \\int \\log{(v_{t})} dv_{t}, then derive \\mathbf{A}{(v_{t})} = g_{\\varepsilon} + v_{t} \\log{(v_{t})} - v_{t}, then obtain v_{t} \\log{(v_{t})} \\int \\log{(v_{t})} dv_{t} = v_{t} (g_{\\varepsilon} + v_{t} \\log{(v_{t})} - v_{t}) \\log{(v_{t})}", "derivation": "\\mathbf{A}{(v_{t})} = \\int \\log{(v_{t})} dv_{t} and \\mathbf{A}{(v_{t})} = g_{\\varepsilon} + v_{t} \\log{(v_{t})} - v_{t} and \\int \\log{(v_{t})} dv_{t} = g_{\\varepsilon} + v_{t} \\log{(v_{t})} - v_{t} and v_{t} \\log{(v_{t})} \\int \\log{(v_{t})} dv_{t} = v_{t} (g_{\\varepsilon} + v_{t} \\log{(v_{t})} - v_{t}) \\log{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_t', commutative=True)), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{A}')(Symbol('v_t', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["times", 3, "Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True)))"], "Equality(Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True)), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Mul(Symbol('v_t', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True))), log(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} = \\frac{\\mathbb{I} y^{\\prime}}{E}, then obtain \\frac{\\mathbb{I} (\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} + 1)}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}} = \\frac{\\mathbb{I} (1 + \\frac{\\mathbb{I} y^{\\prime}}{E})}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}}", "derivation": "\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} = \\frac{\\mathbb{I} y^{\\prime}}{E} and \\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} + 1 = 1 + \\frac{\\mathbb{I} y^{\\prime}}{E} and \\frac{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} + 1}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}} = \\frac{1 + \\frac{\\mathbb{I} y^{\\prime}}{E}}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}} and \\frac{\\mathbb{I} (\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})} + 1)}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}} = \\frac{\\mathbb{I} (1 + \\frac{\\mathbb{I} y^{\\prime}}{E})}{\\operatorname{v_{1}}{(\\mathbb{I},E,y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 2, "Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Pow(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Mul(Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"], [["times", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Pow(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('v_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(C_{1})} = e^{\\sin{(C_{1})}}, then obtain \\iint \\varepsilon{(C_{1})} e^{- \\sin{(C_{1})}} dC_{1} dC_{1} = \\iint 1 dC_{1} dC_{1}", "derivation": "\\varepsilon{(C_{1})} = e^{\\sin{(C_{1})}} and \\varepsilon{(C_{1})} \\sin{(C_{1})} = e^{\\sin{(C_{1})}} \\sin{(C_{1})} and 1 = \\frac{e^{\\sin{(C_{1})}}}{\\varepsilon{(C_{1})}} and e^{\\sin{(C_{1})}} \\sin{(C_{1})} = \\frac{e^{2 \\sin{(C_{1})}} \\sin{(C_{1})}}{\\varepsilon{(C_{1})}} and \\varepsilon{(C_{1})} e^{- \\sin{(C_{1})}} = 1 and \\int \\varepsilon{(C_{1})} e^{- \\sin{(C_{1})}} dC_{1} = \\int 1 dC_{1} and \\iint \\varepsilon{(C_{1})} e^{- \\sin{(C_{1})}} dC_{1} dC_{1} = \\iint 1 dC_{1} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), exp(sin(Symbol('C_1', commutative=True))))"], [["times", 1, "sin(Symbol('C_1', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True))), Mul(exp(sin(Symbol('C_1', commutative=True))), sin(Symbol('C_1', commutative=True))))"], [["divide", 2, "Mul(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), Integer(-1)), exp(sin(Symbol('C_1', commutative=True)))))"], [["times", 3, "Mul(exp(sin(Symbol('C_1', commutative=True))), sin(Symbol('C_1', commutative=True)))"], "Equality(Mul(exp(sin(Symbol('C_1', commutative=True))), sin(Symbol('C_1', commutative=True))), Mul(Pow(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Mul(Integer(2), sin(Symbol('C_1', commutative=True)))), sin(Symbol('C_1', commutative=True))))"], [["divide", 4, "Mul(Pow(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Mul(Integer(2), sin(Symbol('C_1', commutative=True)))), sin(Symbol('C_1', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))))), Integer(1))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))))), Tuple(Symbol('C_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('C_1', commutative=True))))"], [["integrate", 6, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varepsilon')(Symbol('C_1', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))))), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given k{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain k{(E_{\\lambda})} + \\int \\log{(E_{\\lambda})} dE_{\\lambda} = \\log{(E_{\\lambda})} + \\int \\log{(E_{\\lambda})} dE_{\\lambda}", "derivation": "k{(E_{\\lambda})} = \\log{(E_{\\lambda})} and \\int k{(E_{\\lambda})} dE_{\\lambda} = \\int \\log{(E_{\\lambda})} dE_{\\lambda} and k{(E_{\\lambda})} + \\int k{(E_{\\lambda})} dE_{\\lambda} = \\log{(E_{\\lambda})} + \\int k{(E_{\\lambda})} dE_{\\lambda} and k{(E_{\\lambda})} + \\int \\log{(E_{\\lambda})} dE_{\\lambda} = \\log{(E_{\\lambda})} + \\int \\log{(E_{\\lambda})} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 1, "Integral(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Add(log(Symbol('E_{\\\\lambda}', commutative=True)), Integral(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('k')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Add(log(Symbol('E_{\\\\lambda}', commutative=True)), Integral(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given I{(x^\\prime,f)} = f \\log{(x^\\prime)}, then obtain \\int (I^{f}{(x^\\prime,f)} + I^{x^\\prime}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} dx^\\prime = \\int ((f \\log{(x^\\prime)})^{x^\\prime} + I^{f}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} dx^\\prime", "derivation": "I{(x^\\prime,f)} = f \\log{(x^\\prime)} and I^{x^\\prime}{(x^\\prime,f)} = (f \\log{(x^\\prime)})^{x^\\prime} and I^{f}{(x^\\prime,f)} + I^{x^\\prime}{(x^\\prime,f)} = (f \\log{(x^\\prime)})^{x^\\prime} + I^{f}{(x^\\prime,f)} and (I^{f}{(x^\\prime,f)} + I^{x^\\prime}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} = ((f \\log{(x^\\prime)})^{x^\\prime} + I^{f}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} and \\int (I^{f}{(x^\\prime,f)} + I^{x^\\prime}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} dx^\\prime = \\int ((f \\log{(x^\\prime)})^{x^\\prime} + I^{f}{(x^\\prime,f)}) I^{- x^\\prime}{(x^\\prime,f)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), log(Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('f', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 2, "Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(Pow(Mul(Symbol('f', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))))"], [["divide", 3, "Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Add(Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Mul(Add(Pow(Mul(Symbol('f', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Add(Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Add(Pow(Mul(Symbol('f', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Pow(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C)} = \\log{(\\cos{(C)})} and \\chi{(C)} = \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC}, then obtain \\frac{d}{d C} \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC} + \\int \\log{(\\cos{(C)})} dC = \\int \\log{(\\cos{(C)})} dC", "derivation": "\\Psi^{\\dagger}{(C)} = \\log{(\\cos{(C)})} and \\int \\Psi^{\\dagger}{(C)} dC = \\int \\log{(\\cos{(C)})} dC and \\chi{(C)} = \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC} and \\chi{(C)} = 1 and \\frac{d}{d C} \\chi{(C)} = \\frac{d}{d C} 1 and \\frac{d}{d C} \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC} = \\frac{d}{d C} 1 and \\frac{d}{d C} \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC} + \\int \\log{(\\cos{(C)})} dC = \\frac{d}{d C} 1 + \\int \\log{(\\cos{(C)})} dC and \\frac{d}{d C} \\frac{\\int \\log{(\\cos{(C)})} dC}{\\int \\Psi^{\\dagger}{(C)} dC} + \\int \\log{(\\cos{(C)})} dC = \\int \\log{(\\cos{(C)})} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), log(cos(Symbol('C', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('C', commutative=True)), Mul(Pow(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1)), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\chi')(Symbol('C', commutative=True)), Integer(1))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Mul(Pow(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1)), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 6, "Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1)), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Add(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"], [["evaluate_derivatives", 7], "Equality(Add(Derivative(Mul(Pow(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1)), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Integral(log(cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} = \\sin{(\\dot{y})}, then derive \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = \\hat{H}_l - \\cos{(\\dot{y})}, then obtain - \\sin^{\\dot{y}}{(\\dot{y})} - \\cos{(\\dot{y})} \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = - (\\hat{H}_l - \\cos{(\\dot{y})}) \\cos{(\\dot{y})} - \\sin^{\\dot{y}}{(\\dot{y})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} = \\sin{(\\dot{y})} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = \\int \\sin{(\\dot{y})} d\\dot{y} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = \\hat{H}_l - \\cos{(\\dot{y})} and - \\cos{(\\dot{y})} \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = - (\\hat{H}_l - \\cos{(\\dot{y})}) \\cos{(\\dot{y})} and - \\sin^{\\dot{y}}{(\\dot{y})} - \\cos{(\\dot{y})} \\int \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} d\\dot{y} = - (\\hat{H}_l - \\cos{(\\dot{y})}) \\cos{(\\dot{y})} - \\sin^{\\dot{y}}{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(sin(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(-1), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))), cos(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 4, "Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))), Add(Mul(Integer(-1), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))), cos(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mathbf{g},n_{2})} = \\mathbf{g} + n_{2}, then obtain (n_{2} \\operatorname{g_{\\varepsilon}}^{n_{2}}{(\\mathbf{g},n_{2})})^{n_{2}} = (n_{2} (\\mathbf{g} + n_{2})^{n_{2}})^{n_{2}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mathbf{g},n_{2})} = \\mathbf{g} + n_{2} and \\operatorname{g_{\\varepsilon}}^{n_{2}}{(\\mathbf{g},n_{2})} = (\\mathbf{g} + n_{2})^{n_{2}} and n_{2} \\operatorname{g_{\\varepsilon}}^{n_{2}}{(\\mathbf{g},n_{2})} = n_{2} (\\mathbf{g} + n_{2})^{n_{2}} and (n_{2} \\operatorname{g_{\\varepsilon}}^{n_{2}}{(\\mathbf{g},n_{2})})^{n_{2}} = (n_{2} (\\mathbf{g} + n_{2})^{n_{2}})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["times", 2, "Symbol('n_2', commutative=True)"], "Equality(Mul(Symbol('n_2', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Mul(Symbol('n_2', commutative=True), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Symbol('n_2', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Mul(Symbol('n_2', commutative=True), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\Psi{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then derive \\int \\Psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = P_{e} + \\hat{\\mathbf{x}} \\log{(\\hat{\\mathbf{x}})} - \\hat{\\mathbf{x}}, then obtain \\int \\Psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = P_{e} + \\hat{\\mathbf{x}} \\Psi{(\\hat{\\mathbf{x}})} - \\hat{\\mathbf{x}}", "derivation": "\\Psi{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and \\int \\Psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\log{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} and \\int \\Psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = P_{e} + \\hat{\\mathbf{x}} \\log{(\\hat{\\mathbf{x}})} - \\hat{\\mathbf{x}} and \\int \\Psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = P_{e} + \\hat{\\mathbf{x}} \\Psi{(\\hat{\\mathbf{x}})} - \\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(P_{e},\\mu_0)} = (e^{P_{e}})^{\\mu_0}, then obtain 1 = ((\\frac{(e^{P_{e}})^{\\mu_0}}{\\mathbf{B}{(P_{e},\\mu_0)}})^{\\mu_0})^{\\mu_0}", "derivation": "\\mathbf{B}{(P_{e},\\mu_0)} = (e^{P_{e}})^{\\mu_0} and 1 = \\frac{(e^{P_{e}})^{\\mu_0}}{\\mathbf{B}{(P_{e},\\mu_0)}} and 1 = (\\frac{(e^{P_{e}})^{\\mu_0}}{\\mathbf{B}{(P_{e},\\mu_0)}})^{\\mu_0} and 1 = ((\\frac{(e^{P_{e}})^{\\mu_0}}{\\mathbf{B}{(P_{e},\\mu_0)}})^{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('P_e', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{B}')(Symbol('P_e', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('P_e', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('P_e', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["power", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('P_e', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(v_{y},\\theta_1)} = \\theta_1 + v_{y}, then obtain 3 \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} + 2", "derivation": "\\bar{\\h}{(v_{y},\\theta_1)} = \\theta_1 + v_{y} and \\theta_1 + v_{y} + \\bar{\\h}{(v_{y},\\theta_1)} = 2 \\theta_1 + 2 v_{y} and 2 \\bar{\\h}{(v_{y},\\theta_1)} = 2 \\theta_1 + 2 v_{y} and \\frac{\\partial}{\\partial \\theta_1} 2 \\bar{\\h}{(v_{y},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (2 \\theta_1 + 2 v_{y}) and \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} 2 \\bar{\\h}{(v_{y},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (2 \\theta_1 + 2 v_{y}) + \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} and 3 \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\bar{\\h}{(v_{y},\\theta_1)} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True), Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Derivative(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(3), Derivative(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\hbar')(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given l{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain (\\frac{d}{d \\dot{x}} (l{(\\dot{x})} + l^{\\dot{x}}{(\\dot{x})}))^{\\dot{x}} = (\\frac{d}{d \\dot{x}} (l{(\\dot{x})} + \\sin^{\\dot{x}}{(\\dot{x})}))^{\\dot{x}}", "derivation": "l{(\\dot{x})} = \\sin{(\\dot{x})} and l^{\\dot{x}}{(\\dot{x})} = \\sin^{\\dot{x}}{(\\dot{x})} and l{(\\dot{x})} + l^{\\dot{x}}{(\\dot{x})} = l{(\\dot{x})} + \\sin^{\\dot{x}}{(\\dot{x})} and \\frac{d}{d \\dot{x}} (l{(\\dot{x})} + l^{\\dot{x}}{(\\dot{x})}) = \\frac{d}{d \\dot{x}} (l{(\\dot{x})} + \\sin^{\\dot{x}}{(\\dot{x})}) and (\\frac{d}{d \\dot{x}} (l{(\\dot{x})} + l^{\\dot{x}}{(\\dot{x})}))^{\\dot{x}} = (\\frac{d}{d \\dot{x}} (l{(\\dot{x})} + \\sin^{\\dot{x}}{(\\dot{x})}))^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(sin(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["add", 2, "Function('l')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(sin(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(sin(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Derivative(Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)), Pow(Derivative(Add(Function('l')(Symbol('\\\\dot{x}', commutative=True)), Pow(sin(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(f^{\\prime})} = e^{f^{\\prime}}, then derive \\frac{d}{d f^{\\prime}} \\mathbf{F}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain (\\frac{d^{2}}{d (f^{\\prime})^{2}} e^{f^{\\prime}})^{2} = e^{2 f^{\\prime}}", "derivation": "\\mathbf{F}{(f^{\\prime})} = e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\mathbf{F}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\mathbf{F}{(f^{\\prime})} = e^{f^{\\prime}} and e^{f^{\\prime}} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\mathbf{F}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\frac{d^{2}}{d (f^{\\prime})^{2}} e^{f^{\\prime}} = e^{f^{\\prime}} and (\\frac{d^{2}}{d (f^{\\prime})^{2}} e^{f^{\\prime}})^{2} = e^{2 f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('f^{\\\\prime}', commutative=True)), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('\\\\mathbf{F}')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 6, 2], "Equality(Pow(Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Integer(2)), exp(Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given S{(\\Psi)} = \\cos{(\\Psi)} and \\nabla{(\\Psi)} = \\frac{S{(\\Psi)}}{\\Psi}, then obtain \\iint \\nabla{(\\Psi)} d\\Psi d\\Psi = \\iint \\frac{\\cos{(\\Psi)}}{\\Psi} d\\Psi d\\Psi", "derivation": "S{(\\Psi)} = \\cos{(\\Psi)} and \\frac{S{(\\Psi)}}{\\Psi} = \\frac{\\cos{(\\Psi)}}{\\Psi} and \\nabla{(\\Psi)} = \\frac{S{(\\Psi)}}{\\Psi} and \\nabla{(\\Psi)} = \\frac{\\cos{(\\Psi)}}{\\Psi} and \\int \\nabla{(\\Psi)} d\\Psi = \\int \\frac{\\cos{(\\Psi)}}{\\Psi} d\\Psi and \\iint \\nabla{(\\Psi)} d\\Psi d\\Psi = \\iint \\frac{\\cos{(\\Psi)}}{\\Psi} d\\Psi d\\Psi", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\nabla')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(n_{1})} = e^{n_{1}}, then derive \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})} = e^{n_{1}}, then obtain \\sigma_{p}{(n_{1})} + \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})} = 2 \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})}", "derivation": "\\sigma_{p}{(n_{1})} = e^{n_{1}} and \\sigma_{p}{(n_{1})} + e^{n_{1}} = 2 e^{n_{1}} and \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})} = \\frac{d}{d n_{1}} e^{n_{1}} and \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})} = e^{n_{1}} and \\sigma_{p}{(n_{1})} + \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})} = 2 \\frac{d}{d n_{1}} \\sigma_{p}{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["add", 1, "exp(Symbol('n_1', commutative=True))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Mul(Integer(2), exp(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), exp(Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('\\\\sigma_p')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(l)} = \\sin{(l)}, then derive l \\frac{d}{d l} \\operatorname{A_{1}}{(l)} + \\operatorname{A_{1}}{(l)} = l \\cos{(l)} + \\sin{(l)}, then obtain e^{l \\frac{d}{d l} \\operatorname{A_{1}}{(l)} + \\operatorname{A_{1}}{(l)}} = e^{l \\cos{(l)} + \\operatorname{A_{1}}{(l)}}", "derivation": "\\operatorname{A_{1}}{(l)} = \\sin{(l)} and l \\operatorname{A_{1}}{(l)} = l \\sin{(l)} and \\frac{d}{d l} l \\operatorname{A_{1}}{(l)} = \\frac{d}{d l} l \\sin{(l)} and l \\frac{d}{d l} \\operatorname{A_{1}}{(l)} + \\operatorname{A_{1}}{(l)} = l \\cos{(l)} + \\sin{(l)} and e^{l \\frac{d}{d l} \\operatorname{A_{1}}{(l)} + \\operatorname{A_{1}}{(l)}} = e^{l \\cos{(l)} + \\sin{(l)}} and e^{l \\frac{d}{d l} \\operatorname{A_{1}}{(l)} + \\operatorname{A_{1}}{(l)}} = e^{l \\cos{(l)} + \\operatorname{A_{1}}{(l)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('A_1')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), sin(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Symbol('l', commutative=True), Function('A_1')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('l', commutative=True), Derivative(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('A_1')(Symbol('l', commutative=True))), Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), sin(Symbol('l', commutative=True))))"], [["exp", 4], "Equality(exp(Add(Mul(Symbol('l', commutative=True), Derivative(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('A_1')(Symbol('l', commutative=True)))), exp(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), sin(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(exp(Add(Mul(Symbol('l', commutative=True), Derivative(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('A_1')(Symbol('l', commutative=True)))), exp(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Function('A_1')(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given E{(\\varphi)} = \\sin{(e^{\\varphi})}, then obtain \\frac{\\frac{d}{d \\varphi} E{(\\varphi)}}{z} = \\frac{e^{\\varphi} \\cos{(e^{\\varphi})}}{z}", "derivation": "E{(\\varphi)} = \\sin{(e^{\\varphi})} and E{(\\varphi)} + 1 = \\sin{(e^{\\varphi})} + 1 and \\frac{E{(\\varphi)} + 1}{z} = \\frac{\\sin{(e^{\\varphi})} + 1}{z} and \\frac{\\partial}{\\partial \\varphi} \\frac{E{(\\varphi)} + 1}{z} = \\frac{\\partial}{\\partial \\varphi} \\frac{\\sin{(e^{\\varphi})} + 1}{z} and \\frac{\\frac{d}{d \\varphi} E{(\\varphi)}}{z} = \\frac{e^{\\varphi} \\cos{(e^{\\varphi})}}{z}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\varphi', commutative=True)), sin(exp(Symbol('\\\\varphi', commutative=True))))"], [["add", 1, 1], "Equality(Add(Function('E')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(sin(exp(Symbol('\\\\varphi', commutative=True))), Integer(1)))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Function('E')(Symbol('\\\\varphi', commutative=True)), Integer(1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(sin(exp(Symbol('\\\\varphi', commutative=True))), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Function('E')(Symbol('\\\\varphi', commutative=True)), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(sin(exp(Symbol('\\\\varphi', commutative=True))), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Symbol('\\\\varphi', commutative=True)), cos(exp(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{X})} = \\cos{(\\hat{X})}, then obtain Q + \\frac{\\operatorname{A_{2}}{(\\hat{X})}}{2 \\cos{(\\hat{X})}} = Q + \\frac{1}{2}", "derivation": "\\operatorname{A_{2}}{(\\hat{X})} = \\cos{(\\hat{X})} and \\operatorname{A_{2}}{(\\hat{X})} + \\cos{(\\hat{X})} = 2 \\cos{(\\hat{X})} and \\frac{\\operatorname{A_{2}}{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})} + \\cos{(\\hat{X})}} = \\frac{\\cos{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})} + \\cos{(\\hat{X})}} and Q + \\frac{\\operatorname{A_{2}}{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})} + \\cos{(\\hat{X})}} = Q + \\frac{\\cos{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})} + \\cos{(\\hat{X})}} and Q + \\frac{\\operatorname{A_{2}}{(\\hat{X})}}{2 \\cos{(\\hat{X})}} = Q + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 1, "Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Pow(Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), Function('A_2')(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True))))"], [["add", 3, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Mul(Pow(Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), Function('A_2')(Symbol('\\\\hat{X}', commutative=True)))), Add(Symbol('Q', commutative=True), Mul(Pow(Add(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), cos(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('Q', commutative=True), Mul(Rational(1, 2), Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))), Add(Symbol('Q', commutative=True), Rational(1, 2)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} \\log{(V_{\\mathbf{E}})}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})}, then obtain \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})}", "derivation": "\\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} \\log{(V_{\\mathbf{E}})} and \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}})} = \\mathbb{I} \\log{(V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}})} and \\frac{\\partial}{\\partial \\mathbb{I}} (\\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}})}) = \\frac{\\partial}{\\partial \\mathbb{I}} (\\mathbb{I} \\log{(V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}})}) and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})} and \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{P_{g}}{(\\mathbb{I},V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 1, "log(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Derivative(Function('P_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(T)} = \\sin{(\\cos{(T)})}, then derive E (\\frac{d}{d T} \\operatorname{F_{H}}{(T)} + 1) = E (- \\sin{(T)} \\cos{(\\cos{(T)})} + 1), then obtain \\int E (\\frac{d}{d T} \\sin{(\\cos{(T)})} + 1) dE = \\int E (\\frac{d}{d T} \\operatorname{F_{H}}{(T)} + 1) dE", "derivation": "\\operatorname{F_{H}}{(T)} = \\sin{(\\cos{(T)})} and T + \\operatorname{F_{H}}{(T)} = T + \\sin{(\\cos{(T)})} and \\frac{d}{d T} (T + \\operatorname{F_{H}}{(T)}) = \\frac{d}{d T} (T + \\sin{(\\cos{(T)})}) and E \\frac{d}{d T} (T + \\operatorname{F_{H}}{(T)}) = E \\frac{d}{d T} (T + \\sin{(\\cos{(T)})}) and E (\\frac{d}{d T} \\operatorname{F_{H}}{(T)} + 1) = E (- \\sin{(T)} \\cos{(\\cos{(T)})} + 1) and \\int E (\\frac{d}{d T} \\operatorname{F_{H}}{(T)} + 1) dE = \\int E (- \\sin{(T)} \\cos{(\\cos{(T)})} + 1) dE and \\int E (\\frac{d}{d T} \\sin{(\\cos{(T)})} + 1) dE = \\int E (- \\sin{(T)} \\cos{(\\cos{(T)})} + 1) dE and \\int E (\\frac{d}{d T} \\sin{(\\cos{(T)})} + 1) dE = \\int E (\\frac{d}{d T} \\operatorname{F_{H}}{(T)} + 1) dE", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('T', commutative=True)), sin(cos(Symbol('T', commutative=True))))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), sin(cos(Symbol('T', commutative=True)))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), sin(cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["times", 3, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Derivative(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Symbol('E', commutative=True), Derivative(Add(Symbol('T', commutative=True), sin(cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('E', commutative=True), Add(Derivative(Function('F_H')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('E', commutative=True), Add(Mul(Integer(-1), sin(Symbol('T', commutative=True)), cos(cos(Symbol('T', commutative=True)))), Integer(1))))"], [["integrate", 5, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Symbol('E', commutative=True), Add(Derivative(Function('F_H')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Add(Mul(Integer(-1), sin(Symbol('T', commutative=True)), cos(cos(Symbol('T', commutative=True)))), Integer(1))), Tuple(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Mul(Symbol('E', commutative=True), Add(Derivative(sin(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Add(Mul(Integer(-1), sin(Symbol('T', commutative=True)), cos(cos(Symbol('T', commutative=True)))), Integer(1))), Tuple(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integral(Mul(Symbol('E', commutative=True), Add(Derivative(sin(cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Add(Derivative(Function('F_H')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}}, then obtain \\frac{d}{d \\eta^{\\prime}} (\\frac{\\hat{x}{(\\eta^{\\prime})}}{\\eta^{\\prime}})^{\\eta^{\\prime}} = \\frac{d}{d \\eta^{\\prime}} (\\frac{e^{e^{\\eta^{\\prime}}}}{\\eta^{\\prime}})^{\\eta^{\\prime}}", "derivation": "\\hat{x}{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}} and \\frac{\\hat{x}{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{e^{e^{\\eta^{\\prime}}}}{\\eta^{\\prime}} and (\\frac{\\hat{x}{(\\eta^{\\prime})}}{\\eta^{\\prime}})^{\\eta^{\\prime}} = (\\frac{e^{e^{\\eta^{\\prime}}}}{\\eta^{\\prime}})^{\\eta^{\\prime}} and \\frac{d}{d \\eta^{\\prime}} (\\frac{\\hat{x}{(\\eta^{\\prime})}}{\\eta^{\\prime}})^{\\eta^{\\prime}} = \\frac{d}{d \\eta^{\\prime}} (\\frac{e^{e^{\\eta^{\\prime}}}}{\\eta^{\\prime}})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["power", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(f^{\\prime},C_{2})} = C_{2} - f^{\\prime}, then obtain - e^{C_{2} - f^{\\prime} + (f^{\\prime} + L{(f^{\\prime},C_{2})})^{C_{2}}} = - e^{C_{2} + C_{2}^{C_{2}} - f^{\\prime}}", "derivation": "L{(f^{\\prime},C_{2})} = C_{2} - f^{\\prime} and f^{\\prime} + L{(f^{\\prime},C_{2})} = C_{2} and (f^{\\prime} + L{(f^{\\prime},C_{2})})^{C_{2}} = C_{2}^{C_{2}} and C_{2} - f^{\\prime} + (f^{\\prime} + L{(f^{\\prime},C_{2})})^{C_{2}} = C_{2} + C_{2}^{C_{2}} - f^{\\prime} and e^{C_{2} - f^{\\prime} + (f^{\\prime} + L{(f^{\\prime},C_{2})})^{C_{2}}} = e^{C_{2} + C_{2}^{C_{2}} - f^{\\prime}} and - e^{C_{2} - f^{\\prime} + (f^{\\prime} + L{(f^{\\prime},C_{2})})^{C_{2}}} = - e^{C_{2} + C_{2}^{C_{2}} - f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)))"], [["add", 3, "Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))), Add(Symbol('C_2', commutative=True), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["exp", 4], "Equality(exp(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)))), exp(Add(Symbol('C_2', commutative=True), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Function('L')(Symbol('f^{\\\\prime}', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))))), Mul(Integer(-1), exp(Add(Symbol('C_2', commutative=True), Pow(Symbol('C_2', commutative=True), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(\\psi^*)} = e^{\\psi^*}, then obtain - \\frac{d}{d \\psi^*} (\\psi^* + \\mathbf{f}{(\\psi^*)} e^{\\psi^*}) = - \\frac{d}{d \\psi^*} (\\psi^* + e^{2 \\psi^*})", "derivation": "\\mathbf{f}{(\\psi^*)} = e^{\\psi^*} and \\mathbf{f}{(\\psi^*)} e^{\\psi^*} = e^{2 \\psi^*} and \\psi^* + \\mathbf{f}{(\\psi^*)} e^{\\psi^*} = \\psi^* + e^{2 \\psi^*} and \\frac{d}{d \\psi^*} (\\psi^* + \\mathbf{f}{(\\psi^*)} e^{\\psi^*}) = \\frac{d}{d \\psi^*} (\\psi^* + e^{2 \\psi^*}) and - \\frac{d}{d \\psi^*} (\\psi^* + \\mathbf{f}{(\\psi^*)} e^{\\psi^*}) = - \\frac{d}{d \\psi^*} (\\psi^* + e^{2 \\psi^*})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Mul(Function('\\\\mathbf{f}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), exp(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Function('\\\\mathbf{f}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Function('\\\\mathbf{f}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(U,\\lambda,C_{2})} = C_{2} \\lambda^{U}, then obtain (U l{(U,\\lambda,C_{2})})^{C_{2}} (U + (U l{(U,\\lambda,C_{2})})^{C_{2}}) = (U l{(U,\\lambda,C_{2})})^{C_{2}} (U + (C_{2} U \\lambda^{U})^{C_{2}})", "derivation": "l{(U,\\lambda,C_{2})} = C_{2} \\lambda^{U} and U l{(U,\\lambda,C_{2})} = C_{2} U \\lambda^{U} and (U l{(U,\\lambda,C_{2})})^{C_{2}} = (C_{2} U \\lambda^{U})^{C_{2}} and U + (U l{(U,\\lambda,C_{2})})^{C_{2}} = U + (C_{2} U \\lambda^{U})^{C_{2}} and (U l{(U,\\lambda,C_{2})})^{C_{2}} (U + (U l{(U,\\lambda,C_{2})})^{C_{2}}) = (U l{(U,\\lambda,C_{2})})^{C_{2}} (U + (C_{2} U \\lambda^{U})^{C_{2}})", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Symbol('U', commutative=True))))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Mul(Symbol('C_2', commutative=True), Symbol('U', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Symbol('U', commutative=True))))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Symbol('U', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Symbol('U', commutative=True))), Symbol('C_2', commutative=True)))"], [["add", 3, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))), Add(Symbol('U', commutative=True), Pow(Mul(Symbol('C_2', commutative=True), Symbol('U', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Symbol('U', commutative=True))), Symbol('C_2', commutative=True))))"], [["times", 4, "Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Add(Symbol('U', commutative=True), Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)))), Mul(Pow(Mul(Symbol('U', commutative=True), Function('l')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Add(Symbol('U', commutative=True), Pow(Mul(Symbol('C_2', commutative=True), Symbol('U', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Symbol('U', commutative=True))), Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given l{(Z,\\varepsilon)} = \\frac{Z}{\\varepsilon}, then derive \\int \\varepsilon l{(Z,\\varepsilon)} dZ = I + \\frac{Z^{2}}{2}, then obtain \\int Z dZ = I + \\frac{Z^{2}}{2}", "derivation": "l{(Z,\\varepsilon)} = \\frac{Z}{\\varepsilon} and \\varepsilon l{(Z,\\varepsilon)} = Z and \\int \\varepsilon l{(Z,\\varepsilon)} dZ = \\int Z dZ and \\int \\varepsilon l{(Z,\\varepsilon)} dZ = I + \\frac{Z^{2}}{2} and \\int Z dZ = I + \\frac{Z^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('Z', commutative=True))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('Z', commutative=True))), Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True))), Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\omega{(S,s)} = S s and \\hat{X}{(S,s)} = S s + s, then obtain \\frac{\\int (S + ((s + \\omega{(S,s)}) \\hat{X}{(S,s)})^{s}) dS}{s} = \\frac{\\int (S + (\\hat{X}^{2}{(S,s)})^{s}) dS}{s}", "derivation": "\\omega{(S,s)} = S s and s + \\omega{(S,s)} = S s + s and (s + \\omega{(S,s)}) (S s + s) = (S s + s)^{2} and ((s + \\omega{(S,s)}) (S s + s))^{s} = ((S s + s)^{2})^{s} and S + ((s + \\omega{(S,s)}) (S s + s))^{s} = S + ((S s + s)^{2})^{s} and \\hat{X}{(S,s)} = S s + s and S + ((s + \\omega{(S,s)}) \\hat{X}{(S,s)})^{s} = S + (\\hat{X}^{2}{(S,s)})^{s} and \\int (S + ((s + \\omega{(S,s)}) \\hat{X}{(S,s)})^{s}) dS = \\int (S + (\\hat{X}^{2}{(S,s)})^{s}) dS and \\frac{\\int (S + ((s + \\omega{(S,s)}) \\hat{X}{(S,s)})^{s}) dS}{s} = \\frac{\\int (S + (\\hat{X}^{2}{(S,s)})^{s}) dS}{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["times", 2, "Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Pow(Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(2)))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Pow(Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(2)), Symbol('s', commutative=True)))"], [["add", 4, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Add(Symbol('S', commutative=True), Pow(Pow(Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(2)), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True)), Add(Mul(Symbol('S', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('S', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Add(Symbol('S', commutative=True), Pow(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True)), Integer(2)), Symbol('s', commutative=True))))"], [["integrate", 7, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Symbol('S', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Pow(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True)), Integer(2)), Symbol('s', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["divide", 8, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Add(Symbol('S', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), Function('\\\\omega')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))), Tuple(Symbol('S', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Add(Symbol('S', commutative=True), Pow(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True), Symbol('s', commutative=True)), Integer(2)), Symbol('s', commutative=True))), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{J}_M,n)} = \\sin{(\\mathbf{J}_M n)}, then obtain \\log{(\\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\operatorname{n_{2}}{(\\mathbf{J}_M,n)}))} = \\log{(\\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\sin{(\\mathbf{J}_M n)}))}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{J}_M,n)} = \\sin{(\\mathbf{J}_M n)} and \\mathbf{J}_M + \\operatorname{n_{2}}{(\\mathbf{J}_M,n)} = \\mathbf{J}_M + \\sin{(\\mathbf{J}_M n)} and \\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\operatorname{n_{2}}{(\\mathbf{J}_M,n)}) = \\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\sin{(\\mathbf{J}_M n)}) and \\log{(\\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\operatorname{n_{2}}{(\\mathbf{J}_M,n)}))} = \\log{(\\frac{\\partial}{\\partial n} (\\mathbf{J}_M + \\sin{(\\mathbf{J}_M n)}))}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), sin(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), log(Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(W)} = \\frac{d}{d W} \\cos{(W)}, then derive \\mathbf{J}{(W)} = - \\sin{(W)}, then obtain ((\\phi_2 + \\mathbf{J}{(W)}) \\mathbf{J}{(W)})^{W} = ((\\pi - \\sin{(W)}) \\mathbf{J}{(W)})^{W}", "derivation": "\\mathbf{J}{(W)} = \\frac{d}{d W} \\cos{(W)} and \\mathbf{J}{(W)} = - \\sin{(W)} and \\frac{d}{d W} \\mathbf{J}{(W)} = \\frac{d}{d W} - \\sin{(W)} and \\int \\frac{d}{d W} \\mathbf{J}{(W)} dW = \\int \\frac{d}{d W} - \\sin{(W)} dW and \\mathbf{J}{(W)} \\int \\frac{d}{d W} \\mathbf{J}{(W)} dW = \\mathbf{J}{(W)} \\int \\frac{d}{d W} - \\sin{(W)} dW and (\\mathbf{J}{(W)} \\int \\frac{d}{d W} \\mathbf{J}{(W)} dW)^{W} = (\\mathbf{J}{(W)} \\int \\frac{d}{d W} - \\sin{(W)} dW)^{W} and ((\\phi_2 + \\mathbf{J}{(W)}) \\mathbf{J}{(W)})^{W} = ((\\pi - \\sin{(W)}) \\mathbf{J}{(W)})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Derivative(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Mul(Integer(-1), sin(Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integral(Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))))"], [["times", 4, "Function('\\\\mathbf{J}')(Symbol('W', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Integral(Derivative(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)))), Mul(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Integral(Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)))))"], [["power", 5, "Symbol('W', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Integral(Derivative(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Pow(Mul(Function('\\\\mathbf{J}')(Symbol('W', commutative=True)), Integral(Derivative(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Mul(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{J}')(Symbol('W', commutative=True))), Function('\\\\mathbf{J}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Mul(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), sin(Symbol('W', commutative=True)))), Function('\\\\mathbf{J}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given U{(c)} = \\sin{(c)} and \\operatorname{F_{N}}{(c)} = \\frac{\\sin^{c}{(c)}}{\\sin{(c)}}, then obtain \\int (\\operatorname{F_{N}}{(c)} + \\frac{1}{U{(c)}}) dc = \\int (\\frac{U^{c}{(c)}}{U{(c)}} + \\frac{1}{U{(c)}}) dc", "derivation": "U{(c)} = \\sin{(c)} and U^{c}{(c)} = \\sin^{c}{(c)} and \\frac{U^{c}{(c)}}{\\sin{(c)}} = \\frac{\\sin^{c}{(c)}}{\\sin{(c)}} and \\operatorname{F_{N}}{(c)} = \\frac{\\sin^{c}{(c)}}{\\sin{(c)}} and \\operatorname{F_{N}}{(c)} + \\frac{1}{\\sin{(c)}} = \\frac{\\sin^{c}{(c)}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}} and \\operatorname{F_{N}}{(c)} + \\frac{1}{\\sin{(c)}} = \\frac{U^{c}{(c)}}{\\sin{(c)}} + \\frac{1}{\\sin{(c)}} and \\operatorname{F_{N}}{(c)} + \\frac{1}{U{(c)}} = \\frac{U^{c}{(c)}}{U{(c)}} + \\frac{1}{U{(c)}} and \\int (\\operatorname{F_{N}}{(c)} + \\frac{1}{U{(c)}}) dc = \\int (\\frac{U^{c}{(c)}}{U{(c)}} + \\frac{1}{U{(c)}}) dc", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('U')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["divide", 2, "sin(Symbol('c', commutative=True))"], "Equality(Mul(Pow(Function('U')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('c', commutative=True)), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["add", 4, "Pow(sin(Symbol('c', commutative=True)), Integer(-1))"], "Equality(Add(Function('F_N')(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Add(Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('F_N')(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('U')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Function('F_N')(Symbol('c', commutative=True)), Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1))))"], [["integrate", 7, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Function('F_N')(Symbol('c', commutative=True)), Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Pow(Function('U')(Symbol('c', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\theta{(F_{N})} = e^{F_{N}}, then obtain \\frac{d}{d F_{N}} \\iint \\theta^{F_{N}}{(F_{N})} dF_{N} dF_{N} = \\frac{d}{d F_{N}} \\iint (e^{F_{N}})^{F_{N}} dF_{N} dF_{N}", "derivation": "\\theta{(F_{N})} = e^{F_{N}} and \\theta^{F_{N}}{(F_{N})} = (e^{F_{N}})^{F_{N}} and \\int \\theta^{F_{N}}{(F_{N})} dF_{N} = \\int (e^{F_{N}})^{F_{N}} dF_{N} and \\iint \\theta^{F_{N}}{(F_{N})} dF_{N} dF_{N} = \\iint (e^{F_{N}})^{F_{N}} dF_{N} dF_{N} and \\frac{d}{d F_{N}} \\iint \\theta^{F_{N}}{(F_{N})} dF_{N} dF_{N} = \\frac{d}{d F_{N}} \\iint (e^{F_{N}})^{F_{N}} dF_{N} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["integrate", 3, "Symbol('F_N', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\theta')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Integral(Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(S,M,C_{d})} = - C_{d} + \\frac{M}{S} and M{(C_{d})} = - C_{d}, then obtain - \\frac{\\frac{\\partial}{\\partial C_{d}} (- \\frac{M}{S} - M{(C_{d})})}{C_{d}} = - \\frac{\\frac{\\partial}{\\partial C_{d}} (C_{d} - \\frac{M}{S})}{C_{d}}", "derivation": "\\tilde{g}^*{(S,M,C_{d})} = - C_{d} + \\frac{M}{S} and - \\tilde{g}^*{(S,M,C_{d})} = C_{d} - \\frac{M}{S} and \\frac{\\partial}{\\partial C_{d}} - \\tilde{g}^*{(S,M,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (C_{d} - \\frac{M}{S}) and - \\frac{\\frac{\\partial}{\\partial C_{d}} - \\tilde{g}^*{(S,M,C_{d})}}{C_{d}} = - \\frac{\\frac{\\partial}{\\partial C_{d}} (C_{d} - \\frac{M}{S})}{C_{d}} and M{(C_{d})} = - C_{d} and \\tilde{g}^*{(S,M,C_{d})} = \\frac{M}{S} + M{(C_{d})} and - \\frac{\\frac{\\partial}{\\partial C_{d}} (- \\frac{M}{S} - M{(C_{d})})}{C_{d}} = - \\frac{\\frac{\\partial}{\\partial C_{d}} (C_{d} - \\frac{M}{S})}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True), Symbol('M', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Mul(Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('S', commutative=True), Symbol('M', commutative=True), Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('S', commutative=True), Symbol('M', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), Symbol('C_d', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('S', commutative=True), Symbol('M', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('M')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True), Symbol('M', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Function('M')(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Function('M')(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(E_{n},W)} = \\frac{\\log{(E_{n})}}{W} and \\dot{z}{(E_{n},W)} = - \\eta^{\\prime}{(E_{n},W)}, then obtain \\frac{\\partial}{\\partial W} - \\frac{\\log{(E_{n})}}{W} = \\frac{\\partial}{\\partial W} \\dot{z}{(E_{n},W)}", "derivation": "\\eta^{\\prime}{(E_{n},W)} = \\frac{\\log{(E_{n})}}{W} and - \\eta^{\\prime}{(E_{n},W)} = - \\frac{\\log{(E_{n})}}{W} and \\dot{z}{(E_{n},W)} = - \\eta^{\\prime}{(E_{n},W)} and \\frac{\\partial}{\\partial W} - \\eta^{\\prime}{(E_{n},W)} = \\frac{\\partial}{\\partial W} - \\frac{\\log{(E_{n})}}{W} and \\dot{z}{(E_{n},W)} = - \\frac{\\log{(E_{n})}}{W} and \\frac{\\partial}{\\partial W} - \\eta^{\\prime}{(E_{n},W)} = \\frac{\\partial}{\\partial W} \\dot{z}{(E_{n},W)} and \\frac{\\partial}{\\partial W} - \\frac{\\log{(E_{n})}}{W} = \\frac{\\partial}{\\partial W} \\dot{z}{(E_{n},W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('E_n', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('E_n', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\dot{z}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Function('\\\\dot{z}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('E_n', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Function('\\\\dot{z}')(Symbol('E_n', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(m_{s},F_{x})} = \\sin^{F_{x}}{(m_{s})}, then obtain ((\\frac{\\bar{\\h}{(m_{s},F_{x})}}{F_{x}})^{m_{s}})^{m_{s}} = ((\\frac{\\sin^{F_{x}}{(m_{s})}}{F_{x}})^{m_{s}})^{m_{s}}", "derivation": "\\bar{\\h}{(m_{s},F_{x})} = \\sin^{F_{x}}{(m_{s})} and \\frac{\\bar{\\h}{(m_{s},F_{x})}}{F_{x}} = \\frac{\\sin^{F_{x}}{(m_{s})}}{F_{x}} and (\\frac{\\bar{\\h}{(m_{s},F_{x})}}{F_{x}})^{m_{s}} = (\\frac{\\sin^{F_{x}}{(m_{s})}}{F_{x}})^{m_{s}} and ((\\frac{\\bar{\\h}{(m_{s},F_{x})}}{F_{x}})^{m_{s}})^{m_{s}} = ((\\frac{\\sin^{F_{x}}{(m_{s})}}{F_{x}})^{m_{s}})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('m_s', commutative=True), Symbol('F_x', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('F_x', commutative=True)))"], [["divide", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('m_s', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('F_x', commutative=True))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('m_s', commutative=True), Symbol('F_x', commutative=True))), Symbol('m_s', commutative=True)), Pow(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('F_x', commutative=True))), Symbol('m_s', commutative=True)))"], [["power", 3, "Symbol('m_s', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('m_s', commutative=True), Symbol('F_x', commutative=True))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('F_x', commutative=True))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given r{(\\ddot{x})} = \\cos{(\\ddot{x})} and \\theta_{2}{(\\ddot{x})} = \\ddot{x} r{(\\ddot{x})}, then obtain \\int (\\theta_{2}{(\\ddot{x})} - \\cos{(\\ddot{x})}) d\\ddot{x} = \\int (\\ddot{x} \\cos{(\\ddot{x})} - \\cos{(\\ddot{x})}) d\\ddot{x}", "derivation": "r{(\\ddot{x})} = \\cos{(\\ddot{x})} and \\theta_{2}{(\\ddot{x})} = \\ddot{x} r{(\\ddot{x})} and \\theta_{2}{(\\ddot{x})} = \\ddot{x} \\cos{(\\ddot{x})} and \\theta_{2}{(\\ddot{x})} - \\cos{(\\ddot{x})} = \\ddot{x} \\cos{(\\ddot{x})} - \\cos{(\\ddot{x})} and \\int (\\theta_{2}{(\\ddot{x})} - \\cos{(\\ddot{x})}) d\\ddot{x} = \\int (\\ddot{x} \\cos{(\\ddot{x})} - \\cos{(\\ddot{x})}) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Function('r')(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 3, "cos(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Add(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})} and Q{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}}, then obtain \\frac{d}{d \\Psi_{\\lambda}} Q{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\ddot{x}^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})}", "derivation": "\\ddot{x}{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})} and \\ddot{x}^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} and Q{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} and \\frac{d}{d \\Psi_{\\lambda}} Q{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\log{(\\log{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} and \\frac{d}{d \\Psi_{\\lambda}} Q{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\ddot{x}^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('Q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)} = \\tilde{g}^* g_{\\varepsilon} and \\operatorname{A_{y}}{(\\dot{x},n_{2})} = \\dot{x} + n_{2}, then obtain (\\frac{\\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)}}{\\operatorname{A_{y}}{(\\dot{x},n_{2})}})^{\\dot{x}} = (\\frac{\\tilde{g}^* g_{\\varepsilon}}{\\operatorname{A_{y}}{(\\dot{x},n_{2})}})^{\\dot{x}}", "derivation": "\\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)} = \\tilde{g}^* g_{\\varepsilon} and \\operatorname{A_{y}}{(\\dot{x},n_{2})} = \\dot{x} + n_{2} and \\frac{\\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)}}{\\dot{x} + n_{2}} = \\frac{\\tilde{g}^* g_{\\varepsilon}}{\\dot{x} + n_{2}} and (\\frac{\\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)}}{\\dot{x} + n_{2}})^{\\dot{x}} = (\\frac{\\tilde{g}^* g_{\\varepsilon}}{\\dot{x} + n_{2}})^{\\dot{x}} and (\\frac{\\mathbf{M}{(g_{\\varepsilon},\\tilde{g}^*)}}{\\operatorname{A_{y}}{(\\dot{x},n_{2})}})^{\\dot{x}} = (\\frac{\\tilde{g}^* g_{\\varepsilon}}{\\operatorname{A_{y}}{(\\dot{x},n_{2})}})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["get_premise", "Equality(Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Mul(Pow(Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Function('A_y')(Symbol('\\\\dot{x}', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given r{(\\hat{X},F_{g})} = \\hat{X} + \\cos{(F_{g})} and C{(F_{g})} = \\cos{(F_{g})}, then obtain ((\\hat{X} + \\cos{(F_{g})})^{F_{g}})^{\\hat{X}} = ((\\hat{X} + C{(F_{g})})^{F_{g}})^{\\hat{X}}", "derivation": "r{(\\hat{X},F_{g})} = \\hat{X} + \\cos{(F_{g})} and C{(F_{g})} = \\cos{(F_{g})} and r{(\\hat{X},F_{g})} = \\hat{X} + C{(F_{g})} and r^{F_{g}}{(\\hat{X},F_{g})} = (\\hat{X} + C{(F_{g})})^{F_{g}} and (r^{F_{g}}{(\\hat{X},F_{g})})^{\\hat{X}} = ((\\hat{X} + C{(F_{g})})^{F_{g}})^{\\hat{X}} and ((\\hat{X} + \\cos{(F_{g})})^{F_{g}})^{\\hat{X}} = ((\\hat{X} + C{(F_{g})})^{F_{g}})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_g', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), cos(Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('r')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_g', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Function('C')(Symbol('F_g', commutative=True))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('r')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Function('C')(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"], [["power", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Pow(Function('r')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Function('C')(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), cos(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Function('C')(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given B{(\\mathbf{J}_P)} = \\int \\sin{(\\mathbf{J}_P)} d\\mathbf{J}_P, then derive \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (C_{d} - \\cos{(\\mathbf{J}_P)}), then obtain - B{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} = - B{(\\mathbf{J}_P)} + \\frac{\\partial}{\\partial \\mathbf{J}_P} (C_{d} - \\cos{(\\mathbf{J}_P)})", "derivation": "B{(\\mathbf{J}_P)} = \\int \\sin{(\\mathbf{J}_P)} d\\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\int \\sin{(\\mathbf{J}_P)} d\\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (C_{d} - \\cos{(\\mathbf{J}_P)}) and \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} - \\int \\sin{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\frac{\\partial}{\\partial \\mathbf{J}_P} (C_{d} - \\cos{(\\mathbf{J}_P)}) - \\int \\sin{(\\mathbf{J}_P)} d\\mathbf{J}_P and - B{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} B{(\\mathbf{J}_P)} = - B{(\\mathbf{J}_P)} + \\frac{\\partial}{\\partial \\mathbf{J}_P} (C_{d} - \\cos{(\\mathbf{J}_P)})", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["minus", 3, "Integral(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Derivative(Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))), Add(Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(z)} = \\log{(z)}, then obtain - \\log{(z)}^{z} + \\frac{C{(z)}}{z} - \\frac{\\log{(z)}}{z} = - \\log{(z)}^{z}", "derivation": "C{(z)} = \\log{(z)} and \\frac{C{(z)}}{z} = \\frac{\\log{(z)}}{z} and - \\log{(z)}^{z} + \\frac{C{(z)}}{z} = - \\log{(z)}^{z} + \\frac{\\log{(z)}}{z} and - \\log{(z)}^{z} + \\frac{C{(z)}}{z} - \\frac{\\log{(z)}}{z} = - \\log{(z)}^{z}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["divide", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C')(Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True))))"], [["minus", 2, "Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C')(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))))"], [["minus", 3, "Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C')(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))), Mul(Integer(-1), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True))))"]]}, {"prompt": "Given p{(x)} = \\cos{(\\cos{(x)})}, then obtain (\\frac{d}{d x} (2 p{(x)} - \\cos{(x)}))^{2} = (\\frac{d}{d x} (- \\cos{(x)} + 2 \\cos{(\\cos{(x)})}))^{2}", "derivation": "p{(x)} = \\cos{(\\cos{(x)})} and p{(x)} - \\cos{(x)} = - \\cos{(x)} + \\cos{(\\cos{(x)})} and 2 p{(x)} - \\cos{(x)} = p{(x)} - \\cos{(x)} + \\cos{(\\cos{(x)})} and 2 p{(x)} - \\cos{(x)} = - \\cos{(x)} + 2 \\cos{(\\cos{(x)})} and \\frac{d}{d x} (2 p{(x)} - \\cos{(x)}) = \\frac{d}{d x} (- \\cos{(x)} + 2 \\cos{(\\cos{(x)})}) and (\\frac{d}{d x} (2 p{(x)} - \\cos{(x)}))^{2} = (\\frac{d}{d x} (- \\cos{(x)} + 2 \\cos{(\\cos{(x)})}))^{2}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True))))"], [["minus", 1, "cos(Symbol('x', commutative=True))"], "Equality(Add(Function('p')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), cos(cos(Symbol('x', commutative=True)))))"], [["add", 2, "Function('p')(Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('p')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Add(Function('p')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True))), cos(cos(Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('p')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Mul(Integer(2), cos(cos(Symbol('x', commutative=True))))))"], [["differentiate", 4, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('p')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Mul(Integer(2), cos(cos(Symbol('x', commutative=True))))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["power", 5, 2], "Equality(Pow(Derivative(Add(Mul(Integer(2), Function('p')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Mul(Integer(2), cos(cos(Symbol('x', commutative=True))))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given v{(A_{1})} = \\sin{(e^{A_{1}})}, then derive \\frac{d}{d A_{1}} v{(A_{1})} = e^{A_{1}} \\cos{(e^{A_{1}})}, then obtain (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + \\frac{d}{d A_{1}} \\sin{(e^{A_{1}})} = (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + e^{A_{1}} \\cos{(e^{A_{1}})}", "derivation": "v{(A_{1})} = \\sin{(e^{A_{1}})} and \\frac{d}{d A_{1}} v{(A_{1})} = \\frac{d}{d A_{1}} \\sin{(e^{A_{1}})} and \\frac{d}{d A_{1}} v{(A_{1})} = e^{A_{1}} \\cos{(e^{A_{1}})} and (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + \\frac{d}{d A_{1}} v{(A_{1})} = (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + e^{A_{1}} \\cos{(e^{A_{1}})} and (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + \\frac{d}{d A_{1}} \\sin{(e^{A_{1}})} = (v{(A_{1})} + e^{A_{1}}) \\sin{(e^{A_{1}})} + e^{A_{1}} \\cos{(e^{A_{1}})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('A_1', commutative=True)), sin(exp(Symbol('A_1', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(exp(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True)))))"], [["add", 3, "Mul(Add(Function('v')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))), sin(exp(Symbol('A_1', commutative=True))))"], "Equality(Add(Mul(Add(Function('v')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))), sin(exp(Symbol('A_1', commutative=True)))), Derivative(Function('v')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Mul(Add(Function('v')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))), sin(exp(Symbol('A_1', commutative=True)))), Mul(exp(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Add(Function('v')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))), sin(exp(Symbol('A_1', commutative=True)))), Derivative(sin(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Mul(Add(Function('v')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))), sin(exp(Symbol('A_1', commutative=True)))), Mul(exp(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = \\dot{z} + \\log{(\\mathbf{D})} and \\mathbf{J}_M{(\\theta)} = e^{\\theta}, then obtain - e^{\\theta} + \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = - e^{\\theta} + \\frac{\\partial}{\\partial \\mathbf{D}} (\\dot{z} + \\log{(\\mathbf{D})})", "derivation": "\\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = \\dot{z} + \\log{(\\mathbf{D})} and \\mathbf{J}_M{(\\theta)} = e^{\\theta} and \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\dot{z} + \\log{(\\mathbf{D})}) and - \\mathbf{J}_M{(\\theta)} + \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = - \\mathbf{J}_M{(\\theta)} + \\frac{\\partial}{\\partial \\mathbf{D}} (\\dot{z} + \\log{(\\mathbf{D})}) and - e^{\\theta} + \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{E_{\\lambda}}{(\\dot{z},\\mathbf{D})} = - e^{\\theta} + \\frac{\\partial}{\\partial \\mathbf{D}} (\\dot{z} + \\log{(\\mathbf{D})})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["minus", 3, "Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True))), Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True))), Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given c{(C_{1})} = e^{C_{1}}, then obtain ((c^{C_{1}}{(C_{1})})^{C_{1}})^{C_{1}} + c{(C_{1})} = (((e^{C_{1}})^{C_{1}})^{C_{1}})^{C_{1}} + c{(C_{1})}", "derivation": "c{(C_{1})} = e^{C_{1}} and c^{C_{1}}{(C_{1})} = (e^{C_{1}})^{C_{1}} and (c^{C_{1}}{(C_{1})})^{C_{1}} = ((e^{C_{1}})^{C_{1}})^{C_{1}} and ((c^{C_{1}}{(C_{1})})^{C_{1}})^{C_{1}} = (((e^{C_{1}})^{C_{1}})^{C_{1}})^{C_{1}} and ((c^{C_{1}}{(C_{1})})^{C_{1}})^{C_{1}} + c{(C_{1})} = (((e^{C_{1}})^{C_{1}})^{C_{1}})^{C_{1}} + c{(C_{1})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('c')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Pow(Function('c')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Pow(Pow(Function('c')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["add", 4, "Function('c')(Symbol('C_1', commutative=True))"], "Equality(Add(Pow(Pow(Pow(Function('c')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Function('c')(Symbol('C_1', commutative=True))), Add(Pow(Pow(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Function('c')(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given V{(I,\\mathbf{s})} = I^{\\mathbf{s}}, then obtain \\int I^{\\mathbf{s}} dI + \\int V{(I,\\mathbf{s})} dI + \\frac{2 \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I V{(I,\\mathbf{s})}} = 2 \\int I^{\\mathbf{s}} dI + \\frac{2 \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I V{(I,\\mathbf{s})}}", "derivation": "V{(I,\\mathbf{s})} = I^{\\mathbf{s}} and \\int V{(I,\\mathbf{s})} dI = \\int I^{\\mathbf{s}} dI and \\int I^{\\mathbf{s}} dI + \\int V{(I,\\mathbf{s})} dI = 2 \\int I^{\\mathbf{s}} dI and \\int I^{\\mathbf{s}} dI + \\int V{(I,\\mathbf{s})} dI + \\frac{2 I^{- \\mathbf{s}} \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I} = 2 \\int I^{\\mathbf{s}} dI + \\frac{2 I^{- \\mathbf{s}} \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I} and \\int I^{\\mathbf{s}} dI + \\int V{(I,\\mathbf{s})} dI + \\frac{2 \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I V{(I,\\mathbf{s})}} = 2 \\int I^{\\mathbf{s}} dI + \\frac{2 \\mathbf{s} \\int I^{\\mathbf{s}} dI}{I V{(I,\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 2, "Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["add", 3, "Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))"], "Equality(Add(Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))), Add(Mul(Integer(2), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))), Add(Mul(Integer(2), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Function('V')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain 2 \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} = \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and 2 \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} = \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} and 2 \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} - \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\cos{(g^{\\prime}_{\\varepsilon})} = \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} - \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\cos{(g^{\\prime}_{\\varepsilon})} and 2 \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} = \\operatorname{F_{H}}{(g^{\\prime}_{\\varepsilon})} + \\sin{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Derivative(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))), Add(Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Function('F_H')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given E{(C_{d})} = \\cos{(e^{C_{d}})} and \\operatorname{a^{\\dagger}}{(C_{d})} = e^{C_{d}} and \\operatorname{P_{g}}{(C_{d})} = \\cos^{C_{d}}{(e^{C_{d}})}, then obtain \\int \\cos^{C_{d}}{(\\operatorname{a^{\\dagger}}{(C_{d})})} dC_{d} = \\int \\operatorname{P_{g}}{(C_{d})} dC_{d}", "derivation": "E{(C_{d})} = \\cos{(e^{C_{d}})} and E^{C_{d}}{(C_{d})} = \\cos^{C_{d}}{(e^{C_{d}})} and \\operatorname{a^{\\dagger}}{(C_{d})} = e^{C_{d}} and E{(C_{d})} = \\cos{(\\operatorname{a^{\\dagger}}{(C_{d})})} and \\cos^{C_{d}}{(\\operatorname{a^{\\dagger}}{(C_{d})})} = \\cos^{C_{d}}{(e^{C_{d}})} and \\operatorname{P_{g}}{(C_{d})} = \\cos^{C_{d}}{(e^{C_{d}})} and \\cos^{C_{d}}{(\\operatorname{a^{\\dagger}}{(C_{d})})} = \\operatorname{P_{g}}{(C_{d})} and \\int \\cos^{C_{d}}{(\\operatorname{a^{\\dagger}}{(C_{d})})} dC_{d} = \\int \\operatorname{P_{g}}{(C_{d})} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True))))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('E')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(cos(exp(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('E')(Symbol('C_d', commutative=True)), cos(Function('a^{\\\\dagger}')(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(cos(Function('a^{\\\\dagger}')(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Pow(cos(exp(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('C_d', commutative=True)), Pow(cos(exp(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(cos(Function('a^{\\\\dagger}')(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Function('P_g')(Symbol('C_d', commutative=True)))"], [["integrate", 7, "Symbol('C_d', commutative=True)"], "Equality(Integral(Pow(cos(Function('a^{\\\\dagger}')(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Function('P_g')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given L{(v)} = e^{v} and \\mathbf{J}_f{(v)} = (v - e^{v} - 1)^{v}, then obtain - v + \\frac{d}{d v} \\mathbf{J}_f{(v)} = - v + \\frac{d}{d v} (v - L{(v)} - 1)^{v}", "derivation": "L{(v)} = e^{v} and - v + L{(v)} = - v + e^{v} and v - L{(v)} = v - e^{v} and v - L{(v)} - 1 = v - e^{v} - 1 and \\mathbf{J}_f{(v)} = (v - e^{v} - 1)^{v} and \\mathbf{J}_f{(v)} = (v - L{(v)} - 1)^{v} and \\frac{d}{d v} \\mathbf{J}_f{(v)} = \\frac{d}{d v} (v - L{(v)} - 1)^{v} and - v + \\frac{d}{d v} \\mathbf{J}_f{(v)} = - v + \\frac{d}{d v} (v - L{(v)} - 1)^{v}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('L')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('L')(Symbol('v', commutative=True)))), Add(Symbol('v', commutative=True), Mul(Integer(-1), exp(Symbol('v', commutative=True)))))"], [["minus", 3, 1], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('L')(Symbol('v', commutative=True))), Integer(-1)), Add(Symbol('v', commutative=True), Mul(Integer(-1), exp(Symbol('v', commutative=True))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), exp(Symbol('v', commutative=True))), Integer(-1)), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('L')(Symbol('v', commutative=True))), Integer(-1)), Symbol('v', commutative=True)))"], [["differentiate", 6, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('L')(Symbol('v', commutative=True))), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 7, "Mul(Integer(-1), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Derivative(Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('L')(Symbol('v', commutative=True))), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\bar{\\h}{(P_{g},\\mathbf{f})} = e^{\\mathbf{f}^{P_{g}}}, then obtain 1 - e^{\\mathbf{f}^{P_{g}}} = - e^{\\mathbf{f}^{P_{g}}} + \\frac{e^{\\mathbf{f}^{P_{g}}}}{\\bar{\\h}{(P_{g},\\mathbf{f})}}", "derivation": "\\bar{\\h}{(P_{g},\\mathbf{f})} = e^{\\mathbf{f}^{P_{g}}} and \\mathbf{f} \\bar{\\h}{(P_{g},\\mathbf{f})} = \\mathbf{f} e^{\\mathbf{f}^{P_{g}}} and 1 = \\frac{e^{\\mathbf{f}^{P_{g}}}}{\\bar{\\h}{(P_{g},\\mathbf{f})}} and 1 - e^{\\mathbf{f}^{P_{g}}} = - e^{\\mathbf{f}^{P_{g}}} + \\frac{e^{\\mathbf{f}^{P_{g}}}}{\\bar{\\h}{(P_{g},\\mathbf{f})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\hbar')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('\\\\mathbf{f}', commutative=True), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True)))))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\hbar')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True)))))"], [["minus", 3, "exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True))))), Add(Mul(Integer(-1), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True)))), Mul(Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('P_g', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})} = \\frac{\\cos{(\\mathbf{M})}}{v_{1}} and v{(v_{1},\\mathbf{M})} = v_{1} \\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})}, then obtain v{(v_{1},\\mathbf{M})} = \\cos{(\\mathbf{M})}", "derivation": "\\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})} = \\frac{\\cos{(\\mathbf{M})}}{v_{1}} and \\frac{v_{1} \\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})}}{\\cos{(\\mathbf{M})}} = 1 and v_{1} \\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})} = \\cos{(\\mathbf{M})} and v{(v_{1},\\mathbf{M})} = v_{1} \\operatorname{f^{\\prime}}{(v_{1},\\mathbf{M})} and v{(v_{1},\\mathbf{M})} = \\cos{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Symbol('v_1', commutative=True), Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Pow(cos(Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))"], "Equality(Mul(Symbol('v_1', commutative=True), Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('v_1', commutative=True), Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('v')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(x)} = \\log{(x)}, then obtain \\operatorname{E_{x}}^{4}{(x)} \\log{(x)}^{2} = \\operatorname{E_{x}}{(x)} \\log{(x)}^{5}", "derivation": "\\operatorname{E_{x}}{(x)} = \\log{(x)} and \\operatorname{E_{x}}^{2}{(x)} = \\operatorname{E_{x}}{(x)} \\log{(x)} and \\operatorname{E_{x}}^{2}{(x)} \\log{(x)} = \\operatorname{E_{x}}{(x)} \\log{(x)}^{2} and \\operatorname{E_{x}}^{3}{(x)} = \\operatorname{E_{x}}^{2}{(x)} \\log{(x)} and \\operatorname{E_{x}}^{4}{(x)} \\log{(x)}^{2} = \\operatorname{E_{x}}^{3}{(x)} \\log{(x)}^{3} and \\operatorname{E_{x}}^{3}{(x)} = \\operatorname{E_{x}}{(x)} \\log{(x)}^{2} and \\operatorname{E_{x}}^{4}{(x)} \\log{(x)}^{2} = \\operatorname{E_{x}}{(x)} \\log{(x)}^{5}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('x', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(2)), Mul(Function('E_x')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))))"], [["times", 1, "Mul(Function('E_x')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(2)), log(Symbol('x', commutative=True))), Mul(Function('E_x')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(3)), Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(2)), log(Symbol('x', commutative=True))))"], [["times", 4, "Mul(Function('E_x')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(4)), Pow(log(Symbol('x', commutative=True)), Integer(2))), Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(3)), Pow(log(Symbol('x', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(3)), Mul(Function('E_x')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(4)), Pow(log(Symbol('x', commutative=True)), Integer(2))), Mul(Function('E_x')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(5))))"]]}, {"prompt": "Given \\rho{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then obtain \\rho{(\\mathbb{I})} + (\\int \\rho{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}} = \\rho{(\\mathbb{I})} + (\\int \\sin{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}}", "derivation": "\\rho{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\int \\rho{(\\mathbb{I})} d\\mathbb{I} = \\int \\sin{(\\mathbb{I})} d\\mathbb{I} and (\\int \\rho{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}} = (\\int \\sin{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}} and \\rho{(\\mathbb{I})} + (\\int \\rho{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}} = \\rho{(\\mathbb{I})} + (\\int \\sin{(\\mathbb{I})} d\\mathbb{I})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 3, "Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))), Add(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\phi_2)} = e^{\\phi_2}, then obtain \\frac{d}{d \\phi_2} (2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + \\operatorname{C_{d}}{(\\phi_2)} + e^{\\phi_2}) = \\frac{d}{d \\phi_2} (2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + 2 e^{\\phi_2})", "derivation": "\\operatorname{C_{d}}{(\\phi_2)} = e^{\\phi_2} and \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} = \\phi_2 e^{\\phi_2} and \\phi_2 e^{\\phi_2} + \\operatorname{C_{d}}{(\\phi_2)} = \\phi_2 e^{\\phi_2} + e^{\\phi_2} and 2 \\phi_2 e^{\\phi_2} + \\operatorname{C_{d}}{(\\phi_2)} + e^{\\phi_2} = 2 \\phi_2 e^{\\phi_2} + 2 e^{\\phi_2} and 2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + \\operatorname{C_{d}}{(\\phi_2)} + e^{\\phi_2} = 2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + 2 e^{\\phi_2} and \\frac{d}{d \\phi_2} (2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + \\operatorname{C_{d}}{(\\phi_2)} + e^{\\phi_2}) = \\frac{d}{d \\phi_2} (2 \\phi_2 \\operatorname{C_{d}}{(\\phi_2)} + 2 e^{\\phi_2})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))))"], [["add", 3, "Add(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Function('C_d')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Function('C_d')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\phi_2', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Function('C_d')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True), Function('C_d')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(A,b)} = A - b, then obtain \\frac{\\partial}{\\partial b} 2 x{(A,b)} = \\frac{\\partial}{\\partial b} (2 A - 2 b)", "derivation": "x{(A,b)} = A - b and A - b + x{(A,b)} = 2 A - 2 b and 2 x{(A,b)} = 2 A - 2 b and \\frac{\\partial}{\\partial b} 2 x{(A,b)} = \\frac{\\partial}{\\partial b} (2 A - 2 b)", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('A', commutative=True), Symbol('b', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["add", 1, "Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), Function('x')(Symbol('A', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('x')(Symbol('A', commutative=True), Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('x')(Symbol('A', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(v_{x})} = e^{v_{x}}, then obtain \\frac{(\\iint \\tilde{g}{(v_{x})} dv_{x} dv_{x})^{v_{x}}}{\\int \\tilde{g}{(v_{x})} dv_{x}} = \\frac{(\\iint e^{v_{x}} dv_{x} dv_{x})^{v_{x}}}{\\int \\tilde{g}{(v_{x})} dv_{x}}", "derivation": "\\tilde{g}{(v_{x})} = e^{v_{x}} and \\int \\tilde{g}{(v_{x})} dv_{x} = \\int e^{v_{x}} dv_{x} and \\iint \\tilde{g}{(v_{x})} dv_{x} dv_{x} = \\iint e^{v_{x}} dv_{x} dv_{x} and (\\iint \\tilde{g}{(v_{x})} dv_{x} dv_{x})^{v_{x}} = (\\iint e^{v_{x}} dv_{x} dv_{x})^{v_{x}} and \\frac{(\\iint \\tilde{g}{(v_{x})} dv_{x} dv_{x})^{v_{x}}}{\\int \\tilde{g}{(v_{x})} dv_{x}} = \\frac{(\\iint e^{v_{x}} dv_{x} dv_{x})^{v_{x}}}{\\int \\tilde{g}{(v_{x})} dv_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["divide", 4, "Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(-1)), Pow(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))), Mul(Pow(Integral(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(-1)), Pow(Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(m,G)} = \\frac{\\partial}{\\partial m} G^{m}, then derive \\operatorname{F_{H}}{(m,G)} = G^{m} \\log{(G)}, then obtain 2 G^{m} \\log{(G)} + \\operatorname{F_{H}}{(m,G)} = 3 G^{m} \\log{(G)}", "derivation": "\\operatorname{F_{H}}{(m,G)} = \\frac{\\partial}{\\partial m} G^{m} and \\operatorname{F_{H}}{(m,G)} = G^{m} \\log{(G)} and G^{m} \\log{(G)} + \\operatorname{F_{H}}{(m,G)} = 2 G^{m} \\log{(G)} and 2 G^{m} \\log{(G)} + \\operatorname{F_{H}}{(m,G)} = 3 G^{m} \\log{(G)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('m', commutative=True), Symbol('G', commutative=True)), Derivative(Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_H')(Symbol('m', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True))), Function('F_H')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Mul(Integer(2), Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True))), Function('F_H')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Mul(Integer(3), Pow(Symbol('G', commutative=True), Symbol('m', commutative=True)), log(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)}, then obtain \\frac{\\cos{(\\hat{p}_0)}}{\\lambda{(\\hat{p}_0)}} = \\frac{- \\lambda{(\\hat{p}_0)} + 2 \\cos{(\\hat{p}_0)}}{\\lambda{(\\hat{p}_0)}}", "derivation": "\\lambda{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)} and \\lambda{(\\hat{p}_0)} + \\cos{(\\hat{p}_0)} = 2 \\cos{(\\hat{p}_0)} and \\cos{(\\hat{p}_0)} = - \\lambda{(\\hat{p}_0)} + 2 \\cos{(\\hat{p}_0)} and \\frac{\\cos{(\\hat{p}_0)}}{\\lambda{(\\hat{p}_0)}} = \\frac{- \\lambda{(\\hat{p}_0)} + 2 \\cos{(\\hat{p}_0)}}{\\lambda{(\\hat{p}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 2, "Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(cos(Symbol('\\\\hat{p}_0', commutative=True)), Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["divide", 3, "Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hat{p}_0', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given h{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain \\frac{d}{d E_{\\lambda}} \\frac{d}{d E_{\\lambda}} E_{\\lambda} h{(E_{\\lambda})} \\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}} = \\frac{d}{d E_{\\lambda}} (\\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}})^{2}", "derivation": "h{(E_{\\lambda})} = e^{E_{\\lambda}} and E_{\\lambda} h{(E_{\\lambda})} = E_{\\lambda} e^{E_{\\lambda}} and \\frac{d}{d E_{\\lambda}} E_{\\lambda} h{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}} and \\frac{d}{d E_{\\lambda}} E_{\\lambda} h{(E_{\\lambda})} \\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}} = (\\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}})^{2} and \\frac{d}{d E_{\\lambda}} \\frac{d}{d E_{\\lambda}} E_{\\lambda} h{(E_{\\lambda})} \\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}} = \\frac{d}{d E_{\\lambda}} (\\frac{d}{d E_{\\lambda}} E_{\\lambda} e^{E_{\\lambda}})^{2}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('h')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('h')(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('h')(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('h')(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_l{(f^{*})} = \\log{(f^{*})}, then obtain 2 (\\hat{H}_l{(f^{*})} + 2 \\log{(f^{*})}) \\log{(f^{*})} = 6 \\log{(f^{*})}^{2}", "derivation": "\\hat{H}_l{(f^{*})} = \\log{(f^{*})} and \\hat{H}_l{(f^{*})} + \\log{(f^{*})} = 2 \\log{(f^{*})} and \\hat{H}_l{(f^{*})} + 2 \\log{(f^{*})} = 3 \\log{(f^{*})} and (\\hat{H}_l{(f^{*})} + \\log{(f^{*})}) (\\hat{H}_l{(f^{*})} + 2 \\log{(f^{*})}) = 3 (\\hat{H}_l{(f^{*})} + \\log{(f^{*})}) \\log{(f^{*})} and 2 (\\hat{H}_l{(f^{*})} + 2 \\log{(f^{*})}) \\log{(f^{*})} = 6 \\log{(f^{*})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], [["add", 1, "log(Symbol('f^*', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True))), Mul(Integer(2), log(Symbol('f^*', commutative=True))))"], [["add", 1, "Mul(Integer(2), log(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), Mul(Integer(2), log(Symbol('f^*', commutative=True)))), Mul(Integer(3), log(Symbol('f^*', commutative=True))))"], [["times", 3, "Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True))), Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), Mul(Integer(2), log(Symbol('f^*', commutative=True))))), Mul(Integer(3), Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True))), log(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Add(Function('\\\\hat{H}_l')(Symbol('f^*', commutative=True)), Mul(Integer(2), log(Symbol('f^*', commutative=True)))), log(Symbol('f^*', commutative=True))), Mul(Integer(6), Pow(log(Symbol('f^*', commutative=True)), Integer(2))))"]]}, {"prompt": "Given S{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)}, then derive \\frac{d}{d \\tilde{g}^*} S{(\\tilde{g}^*)} = \\frac{1}{\\tilde{g}^*}, then obtain \\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\log{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\frac{1}{\\tilde{g}^*}", "derivation": "S{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)} and \\frac{d}{d \\tilde{g}^*} S{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\log{(\\tilde{g}^*)} and \\frac{d}{d \\tilde{g}^*} S{(\\tilde{g}^*)} = \\frac{1}{\\tilde{g}^*} and \\frac{d}{d \\tilde{g}^*} \\log{(\\tilde{g}^*)} = \\frac{1}{\\tilde{g}^*} and \\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\log{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\frac{1}{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(log(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(f)} = \\log{(f)}, then obtain \\frac{- f \\log{(f)} + (f \\operatorname{C_{d}}{(f)})^{f}}{\\log{(f)}} = \\frac{- f \\log{(f)} + (f \\log{(f)})^{f}}{\\log{(f)}}", "derivation": "\\operatorname{C_{d}}{(f)} = \\log{(f)} and f \\operatorname{C_{d}}{(f)} = f \\log{(f)} and (f \\operatorname{C_{d}}{(f)})^{f} = (f \\log{(f)})^{f} and - f \\log{(f)} + (f \\operatorname{C_{d}}{(f)})^{f} = - f \\log{(f)} + (f \\log{(f)})^{f} and \\frac{- f \\log{(f)} + (f \\operatorname{C_{d}}{(f)})^{f}}{\\log{(f)}} = \\frac{- f \\log{(f)} + (f \\log{(f)})^{f}}{\\log{(f)}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["times", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Function('C_d')(Symbol('f', commutative=True))), Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Symbol('f', commutative=True), Function('C_d')(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["minus", 3, "Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Pow(Mul(Symbol('f', commutative=True), Function('C_d')(Symbol('f', commutative=True))), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Pow(Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Symbol('f', commutative=True))))"], [["divide", 4, "log(Symbol('f', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Pow(Mul(Symbol('f', commutative=True), Function('C_d')(Symbol('f', commutative=True))), Symbol('f', commutative=True))), Pow(log(Symbol('f', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Pow(Mul(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Symbol('f', commutative=True))), Pow(log(Symbol('f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{2}{(P_{g})} = e^{P_{g}}, then derive \\int \\phi_{2}{(P_{g})} dP_{g} = v_{1} + e^{P_{g}}, then obtain \\iint e^{P_{g}} dP_{g} dv_{1} = \\int (v_{1} + e^{P_{g}}) dv_{1}", "derivation": "\\phi_{2}{(P_{g})} = e^{P_{g}} and \\int \\phi_{2}{(P_{g})} dP_{g} = \\int e^{P_{g}} dP_{g} and \\int \\phi_{2}{(P_{g})} dP_{g} = v_{1} + e^{P_{g}} and \\int \\phi_{2}{(P_{g})} dP_{g} = v_{1} + \\phi_{2}{(P_{g})} and \\int e^{P_{g}} dP_{g} = v_{1} + \\phi_{2}{(P_{g})} and \\iint e^{P_{g}} dP_{g} dv_{1} = \\int (v_{1} + \\phi_{2}{(P_{g})}) dv_{1} and \\iint e^{P_{g}} dP_{g} dv_{1} = \\int (v_{1} + e^{P_{g}}) dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('v_1', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('v_1', commutative=True), Function('\\\\phi_2')(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('v_1', commutative=True), Function('\\\\phi_2')(Symbol('P_g', commutative=True))))"], [["integrate", 5, "Symbol('v_1', commutative=True)"], "Equality(Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('v_1', commutative=True), Function('\\\\phi_2')(Symbol('P_g', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('v_1', commutative=True), exp(Symbol('P_g', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(g)} = \\sin{(\\cos{(g)})}, then obtain \\frac{\\cos{(g)} + \\iint \\psi^{*}{(g)} \\sin{(\\cos{(g)})} dg dg}{\\psi^{*}{(g)}} = \\frac{\\cos{(g)} + \\iint \\sin^{2}{(\\cos{(g)})} dg dg}{\\psi^{*}{(g)}}", "derivation": "\\psi^{*}{(g)} = \\sin{(\\cos{(g)})} and \\psi^{*}{(g)} \\sin{(\\cos{(g)})} = \\sin^{2}{(\\cos{(g)})} and \\int \\psi^{*}{(g)} \\sin{(\\cos{(g)})} dg = \\int \\sin^{2}{(\\cos{(g)})} dg and \\iint \\psi^{*}{(g)} \\sin{(\\cos{(g)})} dg dg = \\iint \\sin^{2}{(\\cos{(g)})} dg dg and \\cos{(g)} + \\iint \\psi^{*}{(g)} \\sin{(\\cos{(g)})} dg dg = \\cos{(g)} + \\iint \\sin^{2}{(\\cos{(g)})} dg dg and \\frac{\\cos{(g)} + \\iint \\psi^{*}{(g)} \\sin{(\\cos{(g)})} dg dg}{\\psi^{*}{(g)}} = \\frac{\\cos{(g)} + \\iint \\sin^{2}{(\\cos{(g)})} dg dg}{\\psi^{*}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True))))"], [["times", 1, "sin(cos(Symbol('g', commutative=True)))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))), Pow(sin(cos(Symbol('g', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Pow(sin(cos(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(sin(cos(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["add", 4, "cos(Symbol('g', commutative=True))"], "Equality(Add(cos(Symbol('g', commutative=True)), Integral(Mul(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(cos(Symbol('g', commutative=True)), Integral(Pow(sin(cos(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["divide", 5, "Function('\\\\psi^*')(Symbol('g', commutative=True))"], "Equality(Mul(Add(cos(Symbol('g', commutative=True)), Integral(Mul(Function('\\\\psi^*')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Function('\\\\psi^*')(Symbol('g', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('g', commutative=True)), Integral(Pow(sin(cos(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Function('\\\\psi^*')(Symbol('g', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(f^{*})} = e^{f^{*}}, then obtain \\operatorname{v_{1}}^{f^{*}}{(f^{*})} + (e^{f^{*}})^{f^{*}} = \\frac{\\operatorname{v_{1}}^{f^{*}}{(f^{*})}}{2} + \\frac{3 (e^{f^{*}})^{f^{*}}}{2}", "derivation": "\\operatorname{v_{1}}{(f^{*})} = e^{f^{*}} and \\operatorname{v_{1}}^{f^{*}}{(f^{*})} = (e^{f^{*}})^{f^{*}} and 2 \\operatorname{v_{1}}^{f^{*}}{(f^{*})} = \\operatorname{v_{1}}^{f^{*}}{(f^{*})} + (e^{f^{*}})^{f^{*}} and 1 = \\frac{(\\operatorname{v_{1}}^{f^{*}}{(f^{*})} + (e^{f^{*}})^{f^{*}}) \\operatorname{v_{1}}^{- f^{*}}{(f^{*})}}{2} and \\operatorname{v_{1}}^{f^{*}}{(f^{*})} = \\frac{\\operatorname{v_{1}}^{f^{*}}{(f^{*})}}{2} + \\frac{(e^{f^{*}})^{f^{*}}}{2} and \\operatorname{v_{1}}^{f^{*}}{(f^{*})} + (e^{f^{*}})^{f^{*}} = \\frac{\\operatorname{v_{1}}^{f^{*}}{(f^{*})}}{2} + \\frac{3 (e^{f^{*}})^{f^{*}}}{2}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["add", 2, "Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Add(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Pow(Function('v_1')(Symbol('f^*', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["times", 4, "Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Add(Mul(Rational(1, 2), Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Mul(Rational(1, 2), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Add(Mul(Rational(1, 2), Pow(Function('v_1')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Mul(Rational(3, 2), Pow(exp(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given G{(Q,a)} = \\frac{\\partial}{\\partial Q} \\frac{Q}{a}, then derive G{(Q,a)} = \\frac{1}{a}, then obtain \\frac{Q}{a} + \\frac{\\partial}{\\partial a} G{(Q,a)} = \\frac{Q}{a} + \\frac{d}{d a} \\frac{1}{a}", "derivation": "G{(Q,a)} = \\frac{\\partial}{\\partial Q} \\frac{Q}{a} and G{(Q,a)} = \\frac{1}{a} and \\frac{\\partial}{\\partial a} G{(Q,a)} = \\frac{d}{d a} \\frac{1}{a} and \\frac{Q}{a} + \\frac{\\partial}{\\partial a} G{(Q,a)} = \\frac{Q}{a} + \\frac{d}{d a} \\frac{1}{a}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('G')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Pow(Symbol('a', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Symbol('a', commutative=True), Integer(-1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('Q', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Derivative(Function('G')(Symbol('Q', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Symbol('Q', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Derivative(Pow(Symbol('a', commutative=True), Integer(-1)), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain (- \\psi^* + \\sin{(\\psi^*)})^{2} = (- \\psi^* - Q{(\\psi^*)} + 2 \\sin{(\\psi^*)})^{2}", "derivation": "Q{(\\psi^*)} = \\sin{(\\psi^*)} and - \\psi^* + Q{(\\psi^*)} = - \\psi^* + \\sin{(\\psi^*)} and - \\psi^* + Q{(\\psi^*)} + 1 = - \\psi^* + \\sin{(\\psi^*)} + 1 and - 2 \\psi^* + Q{(\\psi^*)} + 1 = - 2 \\psi^* + \\sin{(\\psi^*)} + 1 and - \\psi^* = - \\psi^* - Q{(\\psi^*)} + \\sin{(\\psi^*)} and - \\psi^* + \\sin{(\\psi^*)} = - \\psi^* - Q{(\\psi^*)} + 2 \\sin{(\\psi^*)} and (- \\psi^* + \\sin{(\\psi^*)})^{2} = (- \\psi^* - Q{(\\psi^*)} + 2 \\sin{(\\psi^*)})^{2}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('Q')(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('Q')(Symbol('\\\\psi^*', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)), Integer(1)))"], [["minus", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Function('Q')(Symbol('\\\\psi^*', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)), Integer(1)))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('Q')(Symbol('\\\\psi^*', commutative=True)), Integer(1))"], "Equality(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('\\\\psi^*', commutative=True))), sin(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True)))))"], [["power", 6, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given E{(A_{z},F_{x})} = A_{z} F_{x}, then obtain \\frac{\\partial^{2}}{\\partial A_{z}\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} E{(A_{z},F_{x})}}{A_{z} F_{x}} = \\frac{\\partial^{2}}{\\partial A_{z}\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} A_{z} F_{x}}{A_{z} F_{x}}", "derivation": "E{(A_{z},F_{x})} = A_{z} F_{x} and \\frac{\\partial}{\\partial F_{x}} E{(A_{z},F_{x})} = \\frac{\\partial}{\\partial F_{x}} A_{z} F_{x} and \\frac{\\frac{\\partial}{\\partial F_{x}} E{(A_{z},F_{x})}}{A_{z} F_{x}} = \\frac{\\frac{\\partial}{\\partial F_{x}} A_{z} F_{x}}{A_{z} F_{x}} and \\frac{\\partial}{\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} E{(A_{z},F_{x})}}{A_{z} F_{x}} = \\frac{\\partial}{\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} A_{z} F_{x}}{A_{z} F_{x}} and \\frac{\\partial^{2}}{\\partial A_{z}\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} E{(A_{z},F_{x})}}{A_{z} F_{x}} = \\frac{\\partial^{2}}{\\partial A_{z}\\partial F_{x}} \\frac{\\frac{\\partial}{\\partial F_{x}} A_{z} F_{x}}{A_{z} F_{x}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\sigma_p)} = \\log{(\\sigma_p)} and u{(\\sigma_p)} = (\\sigma_p - \\bar{\\h}{(\\sigma_p)}) \\log{(\\sigma_p)}, then obtain \\sigma_p \\log{(\\sigma_p)} - \\bar{\\h}{(\\sigma_p)} \\log{(\\sigma_p)} = (\\sigma_p - \\log{(\\sigma_p)}) \\log{(\\sigma_p)}", "derivation": "\\bar{\\h}{(\\sigma_p)} = \\log{(\\sigma_p)} and - \\sigma_p + \\bar{\\h}{(\\sigma_p)} = - \\sigma_p + \\log{(\\sigma_p)} and \\sigma_p - \\bar{\\h}{(\\sigma_p)} = \\sigma_p - \\log{(\\sigma_p)} and (\\sigma_p - \\bar{\\h}{(\\sigma_p)}) \\log{(\\sigma_p)} = (\\sigma_p - \\log{(\\sigma_p)}) \\log{(\\sigma_p)} and u{(\\sigma_p)} = (\\sigma_p - \\bar{\\h}{(\\sigma_p)}) \\log{(\\sigma_p)} and u{(\\sigma_p)} = (\\sigma_p - \\log{(\\sigma_p)}) \\log{(\\sigma_p)} and u{(\\sigma_p)} = \\sigma_p \\log{(\\sigma_p)} - \\bar{\\h}{(\\sigma_p)} \\log{(\\sigma_p)} and \\sigma_p \\log{(\\sigma_p)} - \\bar{\\h}{(\\sigma_p)} \\log{(\\sigma_p)} = (\\sigma_p - \\log{(\\sigma_p)}) \\log{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["times", 3, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)))), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), log(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)))), log(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), log(Symbol('\\\\sigma_p', commutative=True))))"], [["expand", 5], "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Symbol('\\\\sigma_p', commutative=True), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Mul(Symbol('\\\\sigma_p', commutative=True), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), log(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} = \\frac{z^{*}}{L}, then obtain - z^{*} + 3 \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L} - \\frac{z^{*} - \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} + \\frac{z^{*}}{L}}{L} = - z^{*} + 2 \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} = \\frac{z^{*}}{L} and \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L} = 0 and - z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L} = - z^{*} and - z^{*} + 2 \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L} = - z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} and - z^{*} + 3 \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L} - \\frac{z^{*} - \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} + \\frac{z^{*}}{L}}{L} = - z^{*} + 2 \\operatorname{f_{\\mathbf{p}}}{(z^{*},L)} - \\frac{z^{*}}{L}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Mul(Integer(-1), Symbol('z^*', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(3), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hbar)} = \\log{(\\sin{(\\hbar)})}, then obtain \\int (\\hbar + (\\operatorname{n_{2}}^{2}{(\\hbar)})^{\\hbar}) d\\hbar = \\int (\\hbar + (\\operatorname{n_{2}}{(\\hbar)} \\log{(\\sin{(\\hbar)})})^{\\hbar}) d\\hbar", "derivation": "\\operatorname{n_{2}}{(\\hbar)} = \\log{(\\sin{(\\hbar)})} and \\operatorname{n_{2}}^{2}{(\\hbar)} = \\operatorname{n_{2}}{(\\hbar)} \\log{(\\sin{(\\hbar)})} and (\\operatorname{n_{2}}^{2}{(\\hbar)})^{\\hbar} = (\\operatorname{n_{2}}{(\\hbar)} \\log{(\\sin{(\\hbar)})})^{\\hbar} and \\hbar + (\\operatorname{n_{2}}^{2}{(\\hbar)})^{\\hbar} = \\hbar + (\\operatorname{n_{2}}{(\\hbar)} \\log{(\\sin{(\\hbar)})})^{\\hbar} and \\int (\\hbar + (\\operatorname{n_{2}}^{2}{(\\hbar)})^{\\hbar}) d\\hbar = \\int (\\hbar + (\\operatorname{n_{2}}{(\\hbar)} \\log{(\\sin{(\\hbar)})})^{\\hbar}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True))))"], [["times", 1, "Function('n_2')(Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('n_2')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Function('n_2')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True)))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Function('n_2')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Function('n_2')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)))"], [["add", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Pow(Pow(Function('n_2')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Pow(Mul(Function('n_2')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hbar', commutative=True), Pow(Pow(Function('n_2')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('\\\\hbar', commutative=True), Pow(Mul(Function('n_2')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(F_{H})} = \\sin{(F_{H})}, then obtain - F_{H} - \\frac{\\operatorname{n_{2}}^{2}{(F_{H})}}{\\sin{(F_{H})}} + \\operatorname{n_{2}}{(F_{H})} = - F_{H}", "derivation": "\\operatorname{n_{2}}{(F_{H})} = \\sin{(F_{H})} and \\operatorname{n_{2}}^{2}{(F_{H})} = \\operatorname{n_{2}}{(F_{H})} \\sin{(F_{H})} and - \\operatorname{n_{2}}^{2}{(F_{H})} = - \\operatorname{n_{2}}{(F_{H})} \\sin{(F_{H})} and - \\frac{\\operatorname{n_{2}}^{2}{(F_{H})}}{\\sin{(F_{H})}} = - \\operatorname{n_{2}}{(F_{H})} and - F_{H} - \\frac{\\operatorname{n_{2}}^{2}{(F_{H})}}{\\sin{(F_{H})}} + \\operatorname{n_{2}}{(F_{H})} = - F_{H}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["times", 1, "Function('n_2')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('n_2')(Symbol('F_H', commutative=True)), Integer(2)), Mul(Function('n_2')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('n_2')(Symbol('F_H', commutative=True)), Integer(2))), Mul(Integer(-1), Function('n_2')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))))"], [["divide", 3, "sin(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('n_2')(Symbol('F_H', commutative=True)), Integer(2)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('n_2')(Symbol('F_H', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('n_2')(Symbol('F_H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Pow(Function('n_2')(Symbol('F_H', commutative=True)), Integer(2)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Function('n_2')(Symbol('F_H', commutative=True))), Mul(Integer(-1), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given L{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain 2 L{(J_{\\varepsilon})} - e^{J_{\\varepsilon}} = e^{J_{\\varepsilon}}", "derivation": "L{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and 2 L{(J_{\\varepsilon})} = L{(J_{\\varepsilon})} + e^{J_{\\varepsilon}} and 2 L{(J_{\\varepsilon})} - e^{J_{\\varepsilon}} = L{(J_{\\varepsilon})} and 2 L{(J_{\\varepsilon})} - e^{J_{\\varepsilon}} = e^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "exp(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Add(Mul(Integer(2), Function('L')(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} = \\frac{a}{\\psi^*}, then obtain (\\log{(\\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} + 1)}^{a})^{a} = (\\log{(1 + \\frac{a}{\\psi^*})}^{a})^{a}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} = \\frac{a}{\\psi^*} and \\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} + 1 = 1 + \\frac{a}{\\psi^*} and \\log{(\\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} + 1)} = \\log{(1 + \\frac{a}{\\psi^*})} and \\log{(\\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} + 1)}^{a} = \\log{(1 + \\frac{a}{\\psi^*})}^{a} and (\\log{(\\operatorname{V_{\\mathbf{E}}}{(a,\\psi^*)} + 1)}^{a})^{a} = (\\log{(1 + \\frac{a}{\\psi^*})}^{a})^{a}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('a', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["log", 2], "Equality(log(Add(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1))), log(Add(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(log(Add(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1))), Symbol('a', commutative=True)), Pow(log(Add(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('a', commutative=True)))), Symbol('a', commutative=True)))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Pow(log(Add(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1))), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Pow(log(Add(Integer(1), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Symbol('a', commutative=True)))"]]}, {"prompt": "Given x{(c_{0},\\phi_1)} = \\phi_1^{c_{0}}, then derive \\frac{\\partial}{\\partial \\phi_1} x{(c_{0},\\phi_1)} = \\frac{\\phi_1^{c_{0}} c_{0}}{\\phi_1}, then obtain \\frac{\\partial}{\\partial \\phi_1} \\phi_1^{c_{0}} = \\frac{\\phi_1^{c_{0}} c_{0}}{\\phi_1}", "derivation": "x{(c_{0},\\phi_1)} = \\phi_1^{c_{0}} and - c_{0} + x{(c_{0},\\phi_1)} = \\phi_1^{c_{0}} - c_{0} and \\frac{\\partial}{\\partial \\phi_1} (- c_{0} + x{(c_{0},\\phi_1)}) = \\frac{\\partial}{\\partial \\phi_1} (\\phi_1^{c_{0}} - c_{0}) and \\frac{\\partial}{\\partial \\phi_1} x{(c_{0},\\phi_1)} = \\frac{\\phi_1^{c_{0}} c_{0}}{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} \\phi_1^{c_{0}} = \\frac{\\phi_1^{c_{0}} c_{0}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('c_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)))"], [["minus", 1, "Symbol('c_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('x')(Symbol('c_0', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('x')(Symbol('c_0', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x')(Symbol('c_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(B)} = e^{B}, then derive \\frac{d}{d B} \\hat{\\mathbf{x}}{(B)} = e^{B}, then obtain \\hat{\\mathbf{x}}^{2}{(B)} \\int \\hat{\\mathbf{x}}^{2}{(B)} dB + \\frac{d^{3}}{d B^{3}} e^{B} = \\hat{\\mathbf{x}}^{2}{(B)} \\int \\hat{\\mathbf{x}}^{2}{(B)} dB + \\frac{d^{4}}{d B^{4}} e^{B}", "derivation": "\\hat{\\mathbf{x}}{(B)} = e^{B} and \\frac{d}{d B} \\hat{\\mathbf{x}}{(B)} = \\frac{d}{d B} e^{B} and \\frac{d}{d B} \\hat{\\mathbf{x}}{(B)} = e^{B} and \\frac{d}{d B} e^{B} = e^{B} and \\frac{d}{d B} \\hat{\\mathbf{x}}{(B)} = \\frac{d^{2}}{d B^{2}} e^{B} and \\frac{d^{2}}{d B^{2}} e^{B} = e^{B} and \\frac{d}{d B} e^{B} = \\frac{d^{2}}{d B^{2}} e^{B} and \\frac{d^{3}}{d B^{3}} e^{B} = \\frac{d^{4}}{d B^{4}} e^{B} and \\hat{\\mathbf{x}}^{2}{(B)} \\int \\hat{\\mathbf{x}}^{2}{(B)} dB + \\frac{d^{3}}{d B^{3}} e^{B} = \\hat{\\mathbf{x}}^{2}{(B)} \\int \\hat{\\mathbf{x}}^{2}{(B)} dB + \\frac{d^{4}}{d B^{4}} e^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), exp(Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), exp(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), exp(Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(3))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(4))))"], [["add", 8, "Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Tuple(Symbol('B', commutative=True))))"], "Equality(Add(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Tuple(Symbol('B', commutative=True)))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(3)))), Add(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Integer(2)), Tuple(Symbol('B', commutative=True)))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(4)))))"]]}, {"prompt": "Given a{(\\Omega)} = e^{\\Omega}, then derive - e^{\\Omega} + \\frac{d}{d \\Omega} a{(\\Omega)} = 0, then obtain (- e^{\\Omega} + \\frac{d}{d \\Omega} a{(\\Omega)})^{\\Omega} = 0^{\\Omega}", "derivation": "a{(\\Omega)} = e^{\\Omega} and \\frac{d}{d \\Omega} a{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\Omega} and \\frac{d}{d \\Omega} a{(\\Omega)} - \\frac{d}{d \\Omega} e^{\\Omega} = 0 and - e^{\\Omega} + \\frac{d}{d \\Omega} a{(\\Omega)} = 0 and (- e^{\\Omega} + \\frac{d}{d \\Omega} a{(\\Omega)})^{\\Omega} = 0^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('a')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Derivative(Function('a')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Integer(0))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Derivative(Function('a')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Integer(0), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given B{(\\Omega,W)} = \\sin^{\\Omega}{(W)}, then derive \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} = \\log{(\\sin{(W)})} \\sin^{\\Omega}{(W)}, then obtain \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega = \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega", "derivation": "B{(\\Omega,W)} = \\sin^{\\Omega}{(W)} and \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} = \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} and \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} = \\log{(\\sin{(W)})} \\sin^{\\Omega}{(W)} and \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} \\iint \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} dW d\\Omega = \\log{(\\sin{(W)})} \\sin^{\\Omega}{(W)} \\iint \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} dW d\\Omega and \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega = \\log{(\\sin{(W)})} \\sin^{\\Omega}{(W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega and \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega = \\frac{\\partial}{\\partial \\Omega} B{(\\Omega,W)} \\iint \\frac{\\partial}{\\partial \\Omega} \\sin^{\\Omega}{(W)} dW d\\Omega", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(log(sin(Symbol('W', commutative=True))), Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["times", 3, "Integral(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(log(sin(Symbol('W', commutative=True))), Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(log(sin(Symbol('W', commutative=True))), Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Derivative(Function('B')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Derivative(Pow(sin(Symbol('W', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\mathbf{A}{(\\mathbf{r})} = \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})}, then obtain \\mathbf{r} \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r} \\mathbf{A}{(\\mathbf{r})}", "derivation": "\\hat{H}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})} = \\log{(\\mathbf{r})}^{\\mathbf{r}} and \\mathbf{r} \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r} \\log{(\\mathbf{r})}^{\\mathbf{r}} and \\mathbf{A}{(\\mathbf{r})} = \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})} and \\mathbf{A}{(\\mathbf{r})} = \\log{(\\mathbf{r})}^{\\mathbf{r}} and \\mathbf{r} \\hat{H}^{\\mathbf{r}}{(\\mathbf{r})} = \\mathbf{r} \\mathbf{A}{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given M{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})}, then derive \\int \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} dL_{\\varepsilon} = B + L_{\\varepsilon}, then obtain L_{\\varepsilon} + \\int \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} dL_{\\varepsilon} = B + 2 L_{\\varepsilon}", "derivation": "M{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})} and \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} = 1 and \\int \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} dL_{\\varepsilon} = \\int 1 dL_{\\varepsilon} and \\int \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} dL_{\\varepsilon} = B + L_{\\varepsilon} and L_{\\varepsilon} + \\int \\frac{M{(L_{\\varepsilon})}}{\\log{(L_{\\varepsilon})}} dL_{\\varepsilon} = B + 2 L_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "log(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Function('M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('B', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["add", 4, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Mul(Function('M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Symbol('B', commutative=True), Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\delta,y)} = \\delta y, then obtain \\frac{\\partial}{\\partial \\delta} e^{\\frac{(- y + \\operatorname{x^{{\\}'}}{(\\delta,y)}) (\\delta y - y)}{\\int \\delta y dy}} = \\frac{\\partial}{\\partial \\delta} e^{\\frac{(\\delta y - y)^{2}}{\\int \\delta y dy}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\delta,y)} = \\delta y and - y + \\operatorname{x^{{\\}'}}{(\\delta,y)} = \\delta y - y and \\frac{- y + \\operatorname{x^{{\\}'}}{(\\delta,y)}}{\\int \\delta y dy} = \\frac{\\delta y - y}{\\int \\delta y dy} and \\frac{(- y + \\operatorname{x^{{\\}'}}{(\\delta,y)}) (\\delta y - y)}{\\int \\delta y dy} = \\frac{(\\delta y - y)^{2}}{\\int \\delta y dy} and e^{\\frac{(- y + \\operatorname{x^{{\\}'}}{(\\delta,y)}) (\\delta y - y)}{\\int \\delta y dy}} = e^{\\frac{(\\delta y - y)^{2}}{\\int \\delta y dy}} and \\frac{\\partial}{\\partial \\delta} e^{\\frac{(- y + \\operatorname{x^{{\\}'}}{(\\delta,y)}) (\\delta y - y)}{\\int \\delta y dy}} = \\frac{\\partial}{\\partial \\delta} e^{\\frac{(\\delta y - y)^{2}}{\\int \\delta y dy}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["divide", 2, "Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"], [["times", 3, "Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Integer(2)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"], [["exp", 4], "Equality(exp(Mul(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)))), exp(Mul(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Integer(2)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)))))"], [["differentiate", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(exp(Mul(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Integer(2)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(p)} = p and \\operatorname{P_{e}}{(p)} = p \\mathbf{D}{(p)} - \\frac{\\mathbf{D}{(p)}}{p}, then obtain p \\mathbf{D}{(p)} + \\operatorname{P_{e}}{(p)} + \\mathbf{D}^{2}{(p)} - \\frac{2 \\mathbf{D}{(p)}}{p} = p \\mathbf{D}{(p)} + 2 \\operatorname{P_{e}}{(p)} - \\frac{\\mathbf{D}{(p)}}{p}", "derivation": "\\mathbf{D}{(p)} = p and \\mathbf{D}^{2}{(p)} = p \\mathbf{D}{(p)} and \\mathbf{D}^{2}{(p)} - \\frac{\\mathbf{D}{(p)}}{p} = p \\mathbf{D}{(p)} - \\frac{\\mathbf{D}{(p)}}{p} and \\operatorname{P_{e}}{(p)} = p \\mathbf{D}{(p)} - \\frac{\\mathbf{D}{(p)}}{p} and \\mathbf{D}^{2}{(p)} - \\frac{\\mathbf{D}{(p)}}{p} = \\operatorname{P_{e}}{(p)} and p \\mathbf{D}{(p)} + \\operatorname{P_{e}}{(p)} + \\mathbf{D}^{2}{(p)} - \\frac{2 \\mathbf{D}{(p)}}{p} = p \\mathbf{D}{(p)} + 2 \\operatorname{P_{e}}{(p)} - \\frac{\\mathbf{D}{(p)}}{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], [["times", 1, "Function('\\\\mathbf{D}')(Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('p', commutative=True)), Integer(2)), Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('p', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))), Add(Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('p', commutative=True)), Add(Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('p', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))), Function('P_e')(Symbol('p', commutative=True)))"], [["add", 5, "Add(Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))), Function('P_e')(Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))))"], "Equality(Add(Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))), Function('P_e')(Symbol('p', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('p', commutative=True)), Integer(2)), Mul(Integer(-1), Integer(2), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))), Add(Mul(Symbol('p', commutative=True), Function('\\\\mathbf{D}')(Symbol('p', commutative=True))), Mul(Integer(2), Function('P_e')(Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given W{(\\mathbf{E},J,\\mu)} = (\\mu^{\\mathbf{E}})^{J} and \\phi_{1}{(\\mu)} = \\mu, then obtain - \\mu (J + W{(\\mathbf{E},J,\\mu)}) + \\int \\mu (J + W{(\\mathbf{E},J,\\mu)}) d\\mathbf{E} = - \\mu (J + W{(\\mathbf{E},J,\\mu)}) + \\int \\mu (J + (\\mu^{\\mathbf{E}})^{J}) d\\mathbf{E}", "derivation": "W{(\\mathbf{E},J,\\mu)} = (\\mu^{\\mathbf{E}})^{J} and J + W{(\\mathbf{E},J,\\mu)} = J + (\\mu^{\\mathbf{E}})^{J} and \\phi_{1}{(\\mu)} = \\mu and (J + W{(\\mathbf{E},J,\\mu)}) \\phi_{1}{(\\mu)} = (J + (\\mu^{\\mathbf{E}})^{J}) \\phi_{1}{(\\mu)} and \\mu (J + W{(\\mathbf{E},J,\\mu)}) = \\mu (J + (\\mu^{\\mathbf{E}})^{J}) and \\int \\mu (J + W{(\\mathbf{E},J,\\mu)}) d\\mathbf{E} = \\int \\mu (J + (\\mu^{\\mathbf{E}})^{J}) d\\mathbf{E} and - \\mu (J + W{(\\mathbf{E},J,\\mu)}) + \\int \\mu (J + W{(\\mathbf{E},J,\\mu)}) d\\mathbf{E} = - \\mu (J + W{(\\mathbf{E},J,\\mu)}) + \\int \\mu (J + (\\mu^{\\mathbf{E}})^{J}) d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True)))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('J', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["times", 2, "Function('\\\\phi_1')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mu', commutative=True))), Mul(Add(Symbol('J', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 6, "Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('J', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('J', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(H,\\mu)} = - H + \\sin{(\\mu)}, then obtain (H + \\Psi{(H,\\mu)})^{\\mu} e^{\\int \\Psi{(H,\\mu)} d\\mu} = (e^{\\int \\Psi{(H,\\mu)} d\\mu}) \\sin^{\\mu}{(\\mu)}", "derivation": "\\Psi{(H,\\mu)} = - H + \\sin{(\\mu)} and H + \\Psi{(H,\\mu)} = \\sin{(\\mu)} and \\int \\Psi{(H,\\mu)} d\\mu = \\int (- H + \\sin{(\\mu)}) d\\mu and e^{\\int \\Psi{(H,\\mu)} d\\mu} = e^{\\int (- H + \\sin{(\\mu)}) d\\mu} and (H + \\Psi{(H,\\mu)})^{\\mu} = \\sin^{\\mu}{(\\mu)} and (H + \\Psi{(H,\\mu)})^{\\mu} e^{\\int (- H + \\sin{(\\mu)}) d\\mu} = (e^{\\int (- H + \\sin{(\\mu)}) d\\mu}) \\sin^{\\mu}{(\\mu)} and (H + \\Psi{(H,\\mu)})^{\\mu} e^{\\int \\Psi{(H,\\mu)} d\\mu} = (e^{\\int \\Psi{(H,\\mu)} d\\mu}) \\sin^{\\mu}{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Add(Symbol('H', commutative=True), Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True))), sin(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), exp(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('H', commutative=True), Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["times", 5, "exp(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), exp(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))), Mul(exp(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), exp(Integral(Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Mul(exp(Integral(Function('\\\\Psi')(Symbol('H', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given E{(\\mathbf{B})} = \\int \\log{(\\mathbf{B})} d\\mathbf{B}, then obtain \\frac{\\int \\mathbf{B} (- \\mathbf{B} + E{(\\mathbf{B})}) d\\mathbf{B}}{- \\mathbf{B} + E{(\\mathbf{B})}} = \\frac{\\int \\mathbf{B} (- \\mathbf{B} + \\int \\log{(\\mathbf{B})} d\\mathbf{B}) d\\mathbf{B}}{- \\mathbf{B} + E{(\\mathbf{B})}}", "derivation": "E{(\\mathbf{B})} = \\int \\log{(\\mathbf{B})} d\\mathbf{B} and - \\mathbf{B} + E{(\\mathbf{B})} = - \\mathbf{B} + \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\mathbf{B} (- \\mathbf{B} + E{(\\mathbf{B})}) = \\mathbf{B} (- \\mathbf{B} + \\int \\log{(\\mathbf{B})} d\\mathbf{B}) and \\int \\mathbf{B} (- \\mathbf{B} + E{(\\mathbf{B})}) d\\mathbf{B} = \\int \\mathbf{B} (- \\mathbf{B} + \\int \\log{(\\mathbf{B})} d\\mathbf{B}) d\\mathbf{B} and \\frac{\\int \\mathbf{B} (- \\mathbf{B} + E{(\\mathbf{B})}) d\\mathbf{B}}{- \\mathbf{B} + E{(\\mathbf{B})}} = \\frac{\\int \\mathbf{B} (- \\mathbf{B} + \\int \\log{(\\mathbf{B})} d\\mathbf{B}) d\\mathbf{B}}{- \\mathbf{B} + E{(\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True))), Integer(-1)), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('E')(Symbol('\\\\mathbf{B}', commutative=True))), Integer(-1)), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\dot{\\mathbf{r}}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then derive \\int \\hat{x}_0{(L_{\\varepsilon})} dL_{\\varepsilon} = \\mathbf{F} - \\cos{(L_{\\varepsilon})}, then obtain \\mathbf{F} - \\cos{(L_{\\varepsilon})} = \\mathbf{F} - \\dot{\\mathbf{r}}{(L_{\\varepsilon})}", "derivation": "\\hat{x}_0{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\int \\hat{x}_0{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\sin{(L_{\\varepsilon})} dL_{\\varepsilon} and \\int \\hat{x}_0{(L_{\\varepsilon})} dL_{\\varepsilon} = \\mathbf{F} - \\cos{(L_{\\varepsilon})} and \\dot{\\mathbf{r}}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\int \\hat{x}_0{(L_{\\varepsilon})} dL_{\\varepsilon} = \\mathbf{F} - \\dot{\\mathbf{r}}{(L_{\\varepsilon})} and \\mathbf{F} - \\cos{(L_{\\varepsilon})} = \\mathbf{F} - \\dot{\\mathbf{r}}{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} = \\cos{(\\theta_2 + t_{2})}, then obtain ((\\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} - \\cos{(\\theta_2 + t_{2})})^{t_{2}})^{\\theta_2} = (0^{t_{2}})^{\\theta_2}", "derivation": "\\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} = \\cos{(\\theta_2 + t_{2})} and \\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} - \\cos{(\\theta_2 + t_{2})} = 0 and (\\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} - \\cos{(\\theta_2 + t_{2})})^{t_{2}} = 0^{t_{2}} and ((\\operatorname{x^{{\\}'}}{(\\theta_2,t_{2})} - \\cos{(\\theta_2 + t_{2})})^{t_{2}})^{\\theta_2} = (0^{t_{2}})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))))"], [["minus", 1, "cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Function('x^\\\\prime')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))))), Symbol('t_2', commutative=True)), Pow(Integer(0), Symbol('t_2', commutative=True)))"], [["power", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Pow(Add(Function('x^\\\\prime')(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('t_2', commutative=True))))), Symbol('t_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(Integer(0), Symbol('t_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(u,V_{\\mathbf{E}})} = - V_{\\mathbf{E}} + u, then obtain \\sin{(2 \\varepsilon{(u,V_{\\mathbf{E}})} + 1)} = \\sin{(- V_{\\mathbf{E}} + u + \\varepsilon{(u,V_{\\mathbf{E}})} + 1)}", "derivation": "\\varepsilon{(u,V_{\\mathbf{E}})} = - V_{\\mathbf{E}} + u and - V_{\\mathbf{E}} + u + \\varepsilon{(u,V_{\\mathbf{E}})} + 1 = - 2 V_{\\mathbf{E}} + 2 u + 1 and 2 \\varepsilon{(u,V_{\\mathbf{E}})} + 1 = - 2 V_{\\mathbf{E}} + 2 u + 1 and 2 \\varepsilon{(u,V_{\\mathbf{E}})} + 1 = - V_{\\mathbf{E}} + u + \\varepsilon{(u,V_{\\mathbf{E}})} + 1 and \\sin{(2 \\varepsilon{(u,V_{\\mathbf{E}})} + 1)} = \\sin{(- V_{\\mathbf{E}} + u + \\varepsilon{(u,V_{\\mathbf{E}})} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('u', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('u', commutative=True), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)))"], [["sin", 4], "Equality(sin(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(1))), sin(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('u', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(y)} = \\sin{(y)}, then obtain - y = - y \\hat{\\mathbf{r}}{(y)} + y \\sin{(y)} - y", "derivation": "\\hat{\\mathbf{r}}{(y)} = \\sin{(y)} and y \\hat{\\mathbf{r}}{(y)} = y \\sin{(y)} and y \\hat{\\mathbf{r}}{(y)} - y = y \\sin{(y)} - y and - y = - y \\hat{\\mathbf{r}}{(y)} + y \\sin{(y)} - y", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), sin(Symbol('y', commutative=True))))"], [["minus", 2, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Symbol('y', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))), Add(Mul(Symbol('y', commutative=True), sin(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["minus", 3, "Mul(Symbol('y', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), sin(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(g)} = \\cos{(g)} and U{(g)} = \\cos{(g)} and \\hat{x}_0{(g)} = U{(g)} - \\cos{(g)}, then obtain \\int \\hat{x}_0{(g)} dg = \\int (- \\operatorname{t_{2}}{(g)} + \\cos{(g)}) dg", "derivation": "\\operatorname{t_{2}}{(g)} = \\cos{(g)} and U{(g)} = \\cos{(g)} and U{(g)} - \\operatorname{t_{2}}{(g)} = - \\operatorname{t_{2}}{(g)} + \\cos{(g)} and \\hat{x}_0{(g)} = U{(g)} - \\cos{(g)} and \\hat{x}_0{(g)} = U{(g)} - \\operatorname{t_{2}}{(g)} and \\hat{x}_0{(g)} = - \\operatorname{t_{2}}{(g)} + \\cos{(g)} and \\int \\hat{x}_0{(g)} dg = \\int (- \\operatorname{t_{2}}{(g)} + \\cos{(g)}) dg", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('U')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["minus", 2, "Function('t_2')(Symbol('g', commutative=True))"], "Equality(Add(Function('U')(Symbol('g', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Function('t_2')(Symbol('g', commutative=True))), cos(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)), Add(Function('U')(Symbol('g', commutative=True)), Mul(Integer(-1), cos(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)), Add(Function('U')(Symbol('g', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)), Add(Mul(Integer(-1), Function('t_2')(Symbol('g', commutative=True))), cos(Symbol('g', commutative=True))))"], [["integrate", 6, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Function('t_2')(Symbol('g', commutative=True))), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\rho_f,\\theta_2)} = \\rho_f \\theta_2 and \\chi{(\\rho_f)} = \\rho_f, then obtain \\rho_f + \\iint \\tilde{g}{(\\rho_f,\\theta_2)} d\\theta_2 d\\chi{(\\rho_f)} = \\rho_f + \\iint \\rho_f \\theta_2 d\\theta_2 d\\chi{(\\rho_f)}", "derivation": "\\tilde{g}{(\\rho_f,\\theta_2)} = \\rho_f \\theta_2 and \\int \\tilde{g}{(\\rho_f,\\theta_2)} d\\theta_2 = \\int \\rho_f \\theta_2 d\\theta_2 and \\chi{(\\rho_f)} = \\rho_f and \\iint \\tilde{g}{(\\rho_f,\\theta_2)} d\\theta_2 d\\rho_f = \\iint \\rho_f \\theta_2 d\\theta_2 d\\rho_f and \\iint \\tilde{g}{(\\rho_f,\\theta_2)} d\\theta_2 d\\chi{(\\rho_f)} = \\iint \\rho_f \\theta_2 d\\theta_2 d\\chi{(\\rho_f)} and \\rho_f + \\iint \\tilde{g}{(\\rho_f,\\theta_2)} d\\theta_2 d\\chi{(\\rho_f)} = \\rho_f + \\iint \\rho_f \\theta_2 d\\theta_2 d\\chi{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["integrate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True)))), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True)))))"], [["add", 5, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True))))), Add(Symbol('\\\\rho_f', commutative=True), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True))))))"]]}, {"prompt": "Given \\psi{(r,f)} = f r, then obtain f r \\psi^{3}{(r,f)} + f r \\psi{(r,f)} = f^{2} r^{2} \\psi^{2}{(r,f)} + f r \\psi{(r,f)}", "derivation": "\\psi{(r,f)} = f r and f r \\psi{(r,f)} = f^{2} r^{2} and f^{2} r^{2} \\psi^{2}{(r,f)} = f^{4} r^{4} and f r \\psi^{3}{(r,f)} = f^{2} r^{2} \\psi^{2}{(r,f)} and f r \\psi^{3}{(r,f)} + f r \\psi{(r,f)} = f^{2} r^{2} \\psi^{2}{(r,f)} + f r \\psi{(r,f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('r', commutative=True)))"], [["times", 1, "Mul(Symbol('f', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2)), Pow(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(4)), Pow(Symbol('r', commutative=True), Integer(4))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Pow(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(3))), Mul(Pow(Symbol('f', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2)), Pow(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Pow(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(3))), Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Pow(Symbol('f', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2)), Pow(Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Symbol('f', commutative=True), Symbol('r', commutative=True), Function('\\\\psi')(Symbol('r', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(f^{*})} = e^{\\cos{(f^{*})}} and Q{(f^{*})} = f^{*}, then derive \\int Q{(f^{*})} df^{*} = \\hat{X} + \\frac{(f^{*})^{2}}{2}, then obtain e^{\\cos{(f^{*})}} \\int f^{*} df^{*} + e^{\\cos{(f^{*})}} \\int Q{(f^{*})} df^{*} = (\\hat{X} + \\frac{(f^{*})^{2}}{2}) e^{\\cos{(f^{*})}} + e^{\\cos{(f^{*})}} \\int f^{*} df^{*}", "derivation": "\\operatorname{A_{y}}{(f^{*})} = e^{\\cos{(f^{*})}} and Q{(f^{*})} = f^{*} and \\int Q{(f^{*})} df^{*} = \\int f^{*} df^{*} and \\int Q{(f^{*})} df^{*} = \\hat{X} + \\frac{(f^{*})^{2}}{2} and \\operatorname{A_{y}}{(f^{*})} \\int Q{(f^{*})} df^{*} = (\\hat{X} + \\frac{(f^{*})^{2}}{2}) \\operatorname{A_{y}}{(f^{*})} and e^{\\cos{(f^{*})}} \\int Q{(f^{*})} df^{*} = (\\hat{X} + \\frac{(f^{*})^{2}}{2}) e^{\\cos{(f^{*})}} and e^{\\cos{(f^{*})}} \\int f^{*} df^{*} + e^{\\cos{(f^{*})}} \\int Q{(f^{*})} df^{*} = (\\hat{X} + \\frac{(f^{*})^{2}}{2}) e^{\\cos{(f^{*})}} + e^{\\cos{(f^{*})}} \\int f^{*} df^{*}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('f^*', commutative=True)), exp(cos(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('Q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2)))))"], [["times", 4, "Function('A_y')(Symbol('f^*', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('f^*', commutative=True)), Integral(Function('Q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2)))), Function('A_y')(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(exp(cos(Symbol('f^*', commutative=True))), Integral(Function('Q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2)))), exp(cos(Symbol('f^*', commutative=True)))))"], [["add", 6, "Mul(exp(cos(Symbol('f^*', commutative=True))), Integral(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True))))"], "Equality(Add(Mul(exp(cos(Symbol('f^*', commutative=True))), Integral(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True)))), Mul(exp(cos(Symbol('f^*', commutative=True))), Integral(Function('Q')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))), Add(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2)))), exp(cos(Symbol('f^*', commutative=True)))), Mul(exp(cos(Symbol('f^*', commutative=True))), Integral(Symbol('f^*', commutative=True), Tuple(Symbol('f^*', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(y)} = \\sin{(y)}, then obtain \\cos{(\\mathbf{J}_M{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y})} = \\cos{(\\sin{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y})}", "derivation": "\\mathbf{J}_M{(y)} = \\sin{(y)} and \\frac{\\mathbf{J}_M{(y)}}{y} = \\frac{\\sin{(y)}}{y} and \\mathbf{J}_M{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y} = \\sin{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y} and \\mathbf{J}_M{(y)} - \\frac{\\sin{(y)}}{y} = \\sin{(y)} - \\frac{\\sin{(y)}}{y} and \\cos{(\\mathbf{J}_M{(y)} - \\frac{\\sin{(y)}}{y})} = \\cos{(\\sin{(y)} - \\frac{\\sin{(y)}}{y})} and \\cos{(\\mathbf{J}_M{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y})} = \\cos{(\\sin{(y)} - \\frac{\\mathbf{J}_M{(y)}}{y})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)))), Add(sin(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True)))), Add(sin(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True)))))"], [["cos", 4], "Equality(cos(Add(Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))))), cos(Add(sin(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(cos(Add(Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True))))), cos(Add(sin(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{t},\\phi)} = \\phi^{v_{t}} and \\Omega{(\\phi)} = - \\phi, then obtain \\cos{(\\Omega{(\\phi)} + \\sigma_{p}{(v_{t},\\phi)})} = \\cos{(\\phi^{v_{t}} + \\Omega{(\\phi)})}", "derivation": "\\sigma_{p}{(v_{t},\\phi)} = \\phi^{v_{t}} and - \\phi + \\sigma_{p}{(v_{t},\\phi)} = - \\phi + \\phi^{v_{t}} and \\Omega{(\\phi)} = - \\phi and \\Omega{(\\phi)} + \\sigma_{p}{(v_{t},\\phi)} = \\phi^{v_{t}} + \\Omega{(\\phi)} and \\cos{(\\Omega{(\\phi)} + \\sigma_{p}{(v_{t},\\phi)})} = \\cos{(\\phi^{v_{t}} + \\Omega{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('v_t', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\sigma_p')(Symbol('v_t', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('v_t', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\phi', commutative=True)), Function('\\\\sigma_p')(Symbol('v_t', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\Omega')(Symbol('\\\\phi', commutative=True))))"], [["cos", 4], "Equality(cos(Add(Function('\\\\Omega')(Symbol('\\\\phi', commutative=True)), Function('\\\\sigma_p')(Symbol('v_t', commutative=True), Symbol('\\\\phi', commutative=True)))), cos(Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\Omega')(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(v_{1},\\psi)} = \\psi v_{1}, then obtain \\psi \\frac{\\partial}{\\partial v_{1}} \\Psi_{nl}{(v_{1},\\psi)} = \\psi^{2}", "derivation": "\\Psi_{nl}{(v_{1},\\psi)} = \\psi v_{1} and \\frac{\\partial}{\\partial v_{1}} \\Psi_{nl}{(v_{1},\\psi)} = \\frac{\\partial}{\\partial v_{1}} \\psi v_{1} and \\frac{\\partial}{\\partial v_{1}} \\psi v_{1} \\frac{\\partial}{\\partial v_{1}} \\Psi_{nl}{(v_{1},\\psi)} = (\\frac{\\partial}{\\partial v_{1}} \\psi v_{1})^{2} and \\psi \\frac{\\partial}{\\partial v_{1}} \\Psi_{nl}{(v_{1},\\psi)} = \\psi^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v_1', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('v_1', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Function('\\\\Psi_{nl}')(Symbol('v_1', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\psi', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('v_1', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Symbol('\\\\psi', commutative=True), Integer(2)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)} = \\log{(\\mathbf{s} + \\rho)}, then obtain \\frac{\\log{(\\mathbf{s} + \\rho)} - 1}{\\mathbf{s}} = \\frac{\\log{(\\mathbf{s} + \\rho)} - \\frac{\\log{(\\mathbf{s} + \\rho)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)}}}{\\mathbf{s}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)} = \\log{(\\mathbf{s} + \\rho)} and 1 = \\frac{\\log{(\\mathbf{s} + \\rho)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)}} and 1 - \\log{(\\mathbf{s} + \\rho)} = - \\log{(\\mathbf{s} + \\rho)} + \\frac{\\log{(\\mathbf{s} + \\rho)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)}} and \\log{(\\mathbf{s} + \\rho)} - 1 = \\log{(\\mathbf{s} + \\rho)} - \\frac{\\log{(\\mathbf{s} + \\rho)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)}} and \\frac{\\log{(\\mathbf{s} + \\rho)} - 1}{\\mathbf{s}} = \\frac{\\log{(\\mathbf{s} + \\rho)} - \\frac{\\log{(\\mathbf{s} + \\rho)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{s},\\rho)}}}{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["divide", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["minus", 2, "log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))))), Add(Mul(Integer(-1), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)))), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1)), Add(log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))))))"], [["divide", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\rho', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\rho_f)} = \\log{(\\rho_f)}, then obtain \\log{(\\rho_f)}^{2} = \\frac{\\log{(\\rho_f)}^{3}}{\\operatorname{z^{*}}{(\\rho_f)}}", "derivation": "\\operatorname{z^{*}}{(\\rho_f)} = \\log{(\\rho_f)} and \\operatorname{z^{*}}{(\\rho_f)} \\log{(\\rho_f)} = \\log{(\\rho_f)}^{2} and \\operatorname{z^{*}}^{2}{(\\rho_f)} \\log{(\\rho_f)}^{2} = \\log{(\\rho_f)}^{4} and \\operatorname{z^{*}}^{3}{(\\rho_f)} \\log{(\\rho_f)} = \\operatorname{z^{*}}^{2}{(\\rho_f)} \\log{(\\rho_f)}^{2} and \\operatorname{z^{*}}^{3}{(\\rho_f)} \\log{(\\rho_f)} = \\log{(\\rho_f)}^{4} and \\operatorname{z^{*}}^{2}{(\\rho_f)} = \\log{(\\rho_f)}^{2} and \\operatorname{z^{*}}^{2}{(\\rho_f)} = \\frac{\\log{(\\rho_f)}^{3}}{\\operatorname{z^{*}}{(\\rho_f)}} and \\log{(\\rho_f)}^{2} = \\frac{\\log{(\\rho_f)}^{3}}{\\operatorname{z^{*}}{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "log(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(3)), log(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(3)), log(Symbol('\\\\rho_f', commutative=True))), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(4)))"], [["divide", 3, "Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2))"], "Equality(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2)))"], [["divide", 5, "Mul(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Mul(Pow(Function('z^*')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(G,n)} = G - n, then derive G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + \\hat{\\mathbf{r}}{(G,n)} = 2 G - n, then obtain G \\frac{\\partial}{\\partial G} (G - n) + G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + G - n + \\hat{\\mathbf{r}}{(G,n)} = G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + 2 G - n + \\hat{\\mathbf{r}}{(G,n)}", "derivation": "\\hat{\\mathbf{r}}{(G,n)} = G - n and G \\hat{\\mathbf{r}}{(G,n)} = G (G - n) and \\frac{\\partial}{\\partial G} G \\hat{\\mathbf{r}}{(G,n)} = \\frac{\\partial}{\\partial G} G (G - n) and G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + \\hat{\\mathbf{r}}{(G,n)} = 2 G - n and G \\frac{\\partial}{\\partial G} (G - n) + G - n = 2 G - n and G \\frac{\\partial}{\\partial G} (G - n) + G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + G - n + \\hat{\\mathbf{r}}{(G,n)} = G \\frac{\\partial}{\\partial G} \\hat{\\mathbf{r}}{(G,n)} + 2 G - n + \\hat{\\mathbf{r}}{(G,n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True))), Mul(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Symbol('G', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('G', commutative=True), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('G', commutative=True), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["add", 5, "Add(Mul(Symbol('G', commutative=True), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(Mul(Symbol('G', commutative=True), Derivative(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Symbol('G', commutative=True), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True))), Add(Mul(Symbol('G', commutative=True), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)} = - \\Psi + \\frac{\\dot{x}}{\\lambda} and \\mu{(\\lambda)} = \\frac{1}{\\lambda}, then obtain - \\mu{(\\lambda)} - \\frac{\\lambda + \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)}}{\\lambda} = - \\frac{\\lambda + \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)}}{\\lambda} - \\frac{1}{\\lambda}", "derivation": "\\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)} = - \\Psi + \\frac{\\dot{x}}{\\lambda} and \\lambda + \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)} = - \\Psi + \\frac{\\dot{x}}{\\lambda} + \\lambda and \\mu{(\\lambda)} = \\frac{1}{\\lambda} and - \\mu{(\\lambda)} = - \\frac{1}{\\lambda} and - \\mu{(\\lambda)} - \\frac{- \\Psi + \\frac{\\dot{x}}{\\lambda} + \\lambda}{\\lambda} = - \\frac{- \\Psi + \\frac{\\dot{x}}{\\lambda} + \\lambda}{\\lambda} - \\frac{1}{\\lambda} and - \\mu{(\\lambda)} - \\frac{\\lambda + \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)}}{\\lambda} = - \\frac{\\lambda + \\operatorname{P_{e}}{(\\dot{x},\\Psi,\\lambda)}}{\\lambda} - \\frac{1}{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('P_e')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))"], [["minus", 4, "Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Symbol('\\\\lambda', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Function('P_e')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Function('P_e')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\delta{(T)} = \\sin{(T)}, then obtain - 0^{T} = - (- \\delta^{T}{(T)} + \\sin^{T}{(T)})^{T}", "derivation": "\\delta{(T)} = \\sin{(T)} and \\delta^{T}{(T)} = \\sin^{T}{(T)} and 0 = - \\delta^{T}{(T)} + \\sin^{T}{(T)} and 0^{T} = (- \\delta^{T}{(T)} + \\sin^{T}{(T)})^{T} and - 0^{T} = - (- \\delta^{T}{(T)} + \\sin^{T}{(T)})^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\delta')(Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Pow(sin(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["power", 3, "Symbol('T', commutative=True)"], "Equality(Pow(Integer(0), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Pow(sin(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Symbol('T', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Pow(sin(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Symbol('T', commutative=True))))"]]}, {"prompt": "Given g{(E_{x},\\mathbf{v})} = e^{\\frac{E_{x}}{\\mathbf{v}}}, then obtain g{(E_{x},\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} g{(E_{x},\\mathbf{v})} = - \\frac{E_{x} g{(E_{x},\\mathbf{v})} e^{\\frac{E_{x}}{\\mathbf{v}}}}{\\mathbf{v}^{2}}", "derivation": "g{(E_{x},\\mathbf{v})} = e^{\\frac{E_{x}}{\\mathbf{v}}} and \\frac{\\partial}{\\partial \\mathbf{v}} g{(E_{x},\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} e^{\\frac{E_{x}}{\\mathbf{v}}} and g{(E_{x},\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} g{(E_{x},\\mathbf{v})} = g{(E_{x},\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} e^{\\frac{E_{x}}{\\mathbf{v}}} and g{(E_{x},\\mathbf{v})} \\frac{\\partial}{\\partial \\mathbf{v}} g{(E_{x},\\mathbf{v})} = - \\frac{E_{x} g{(E_{x},\\mathbf{v})} e^{\\frac{E_{x}}{\\mathbf{v}}}}{\\mathbf{v}^{2}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["times", 2, "Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-2)), Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\varepsilon)} = e^{\\varepsilon}, then obtain \\cos{(\\frac{d}{d \\varepsilon} (\\operatorname{f^{\\prime}}{(\\varepsilon)} - 2))} = \\cos{(\\frac{d}{d \\varepsilon} (e^{\\varepsilon} - 2))}", "derivation": "\\operatorname{f^{\\prime}}{(\\varepsilon)} = e^{\\varepsilon} and \\operatorname{f^{\\prime}}{(\\varepsilon)} - 1 = e^{\\varepsilon} - 1 and \\operatorname{f^{\\prime}}{(\\varepsilon)} - 2 = e^{\\varepsilon} - 2 and \\frac{d}{d \\varepsilon} (\\operatorname{f^{\\prime}}{(\\varepsilon)} - 2) = \\frac{d}{d \\varepsilon} (e^{\\varepsilon} - 2) and \\cos{(\\frac{d}{d \\varepsilon} (\\operatorname{f^{\\prime}}{(\\varepsilon)} - 2))} = \\cos{(\\frac{d}{d \\varepsilon} (e^{\\varepsilon} - 2))}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Add(exp(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), cos(Derivative(Add(exp(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(J)} = \\log{(\\cos{(J)})} and \\phi_{1}{(J)} = h{(J)} + 1, then obtain (h{(J)} + \\frac{2 h{(J)}}{\\log{(\\cos{(J)})}} - 1)^{J} = (h{(J)} + \\frac{h{(J)}}{\\log{(\\cos{(J)})}})^{J}", "derivation": "h{(J)} = \\log{(\\cos{(J)})} and \\frac{h{(J)}}{\\log{(\\cos{(J)})}} = 1 and h{(J)} + \\frac{h{(J)}}{\\log{(\\cos{(J)})}} = h{(J)} + 1 and \\phi_{1}{(J)} = h{(J)} + 1 and \\phi_{1}^{J}{(J)} = (h{(J)} + 1)^{J} and \\phi_{1}{(J)} = h{(J)} + \\frac{h{(J)}}{\\log{(\\cos{(J)})}} and (h{(J)} + \\frac{h{(J)}}{\\log{(\\cos{(J)})}})^{J} = (h{(J)} + 1)^{J} and (h{(J)} + \\frac{2 h{(J)}}{\\log{(\\cos{(J)})}} - 1)^{J} = (h{(J)} + \\frac{h{(J)}}{\\log{(\\cos{(J)})}})^{J}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('J', commutative=True)), log(cos(Symbol('J', commutative=True))))"], [["divide", 1, "log(cos(Symbol('J', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Function('h')(Symbol('J', commutative=True))"], "Equality(Add(Function('h')(Symbol('J', commutative=True)), Mul(Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1)))), Add(Function('h')(Symbol('J', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('J', commutative=True)), Add(Function('h')(Symbol('J', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Add(Function('h')(Symbol('J', commutative=True)), Integer(1)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\phi_1')(Symbol('J', commutative=True)), Add(Function('h')(Symbol('J', commutative=True)), Mul(Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Add(Function('h')(Symbol('J', commutative=True)), Mul(Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1)))), Symbol('J', commutative=True)), Pow(Add(Function('h')(Symbol('J', commutative=True)), Integer(1)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Pow(Add(Function('h')(Symbol('J', commutative=True)), Mul(Integer(2), Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1))), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(Function('h')(Symbol('J', commutative=True)), Mul(Function('h')(Symbol('J', commutative=True)), Pow(log(cos(Symbol('J', commutative=True))), Integer(-1)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(Z,l)} = Z + l, then derive \\frac{\\partial}{\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = 1, then obtain Z + l - \\operatorname{x^{{\\}'}}^{2}{(Z,l)} + \\frac{\\partial^{2}}{\\partial l\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = Z + l - \\operatorname{x^{{\\}'}}^{2}{(Z,l)} + \\frac{d}{d l} 1", "derivation": "\\operatorname{x^{{\\}'}}{(Z,l)} = Z + l and \\operatorname{x^{{\\}'}}^{2}{(Z,l)} = (Z + l) \\operatorname{x^{{\\}'}}{(Z,l)} and \\frac{\\partial}{\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = \\frac{\\partial}{\\partial Z} (Z + l) and \\frac{\\partial}{\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = 1 and \\frac{\\partial^{2}}{\\partial l\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = \\frac{d}{d l} 1 and Z + l - (Z + l) \\operatorname{x^{{\\}'}}{(Z,l)} + \\frac{\\partial^{2}}{\\partial l\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = Z + l - (Z + l) \\operatorname{x^{{\\}'}}{(Z,l)} + \\frac{d}{d l} 1 and Z + l - \\operatorname{x^{{\\}'}}^{2}{(Z,l)} + \\frac{\\partial^{2}}{\\partial l\\partial Z} \\operatorname{x^{{\\}'}}{(Z,l)} = Z + l - \\operatorname{x^{{\\}'}}^{2}{(Z,l)} + \\frac{d}{d l} 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True))"], "Equality(Pow(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Integer(2)), Mul(Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True))))"], "Equality(Add(Symbol('Z', commutative=True), Symbol('l', commutative=True), Mul(Integer(-1), Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('Z', commutative=True), Symbol('l', commutative=True), Mul(Integer(-1), Add(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True))), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Symbol('Z', commutative=True), Symbol('l', commutative=True), Mul(Integer(-1), Pow(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Integer(2))), Derivative(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('Z', commutative=True), Symbol('l', commutative=True), Mul(Integer(-1), Pow(Function('x^\\\\prime')(Symbol('Z', commutative=True), Symbol('l', commutative=True)), Integer(2))), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(v_{2},I)} = - v_{2} + \\log{(I)}, then derive \\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1 = -2, then obtain \\frac{\\partial}{\\partial v_{2}} \\sin^{v_{2}}{(\\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1)} = \\frac{d}{d v_{2}} (- \\sin{(2)})^{v_{2}}", "derivation": "\\mathbf{P}{(v_{2},I)} = - v_{2} + \\log{(I)} and - v_{2} + \\mathbf{P}{(v_{2},I)} = - 2 v_{2} + \\log{(I)} and \\frac{\\partial}{\\partial v_{2}} (- v_{2} + \\mathbf{P}{(v_{2},I)}) = \\frac{\\partial}{\\partial v_{2}} (- 2 v_{2} + \\log{(I)}) and \\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1 = -2 and \\sin{(\\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1)} = - \\sin{(2)} and \\sin^{v_{2}}{(\\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1)} = (- \\sin{(2)})^{v_{2}} and \\frac{\\partial}{\\partial v_{2}} \\sin^{v_{2}}{(\\frac{\\partial}{\\partial v_{2}} \\mathbf{P}{(v_{2},I)} - 1)} = \\frac{d}{d v_{2}} (- \\sin{(2)})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), log(Symbol('I', commutative=True))))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), log(Symbol('I', commutative=True))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"], [["sin", 4], "Equality(sin(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), sin(Integer(2))))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(sin(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Symbol('v_2', commutative=True)), Pow(Mul(Integer(-1), sin(Integer(2))), Symbol('v_2', commutative=True)))"], [["differentiate", 6, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Pow(sin(Add(Derivative(Function('\\\\mathbf{P}')(Symbol('v_2', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), sin(Integer(2))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} = \\frac{a^{\\dagger}}{\\mathbf{E}}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E} + \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} d\\mathbf{E} = 2 \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E}", "derivation": "\\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} = \\frac{a^{\\dagger}}{\\mathbf{E}} and \\int \\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} d\\mathbf{E} = \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E} and \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} d\\mathbf{E} = \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E} and \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E} + \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\operatorname{r_{0}}{(a^{\\dagger},\\mathbf{E})} d\\mathbf{E} = 2 \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{a^{\\dagger}}{\\mathbf{E}} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integral(Function('r_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(B,y)} = \\frac{y}{B}, then obtain \\frac{B ((\\int \\frac{y}{B} dy)^{B} + (\\int \\omega{(B,y)} dy)^{y})}{y} = \\frac{B ((\\int \\frac{y}{B} dy)^{B} + (\\int \\frac{y}{B} dy)^{y})}{y}", "derivation": "\\omega{(B,y)} = \\frac{y}{B} and \\int \\omega{(B,y)} dy = \\int \\frac{y}{B} dy and (\\int \\omega{(B,y)} dy)^{B} = (\\int \\frac{y}{B} dy)^{B} and (\\int \\omega{(B,y)} dy)^{y} = (\\int \\frac{y}{B} dy)^{y} and (\\int \\omega{(B,y)} dy)^{B} + (\\int \\omega{(B,y)} dy)^{y} = (\\int \\frac{y}{B} dy)^{y} + (\\int \\omega{(B,y)} dy)^{B} and (\\int \\frac{y}{B} dy)^{B} + (\\int \\omega{(B,y)} dy)^{y} = (\\int \\frac{y}{B} dy)^{B} + (\\int \\frac{y}{B} dy)^{y} and \\frac{B ((\\int \\frac{y}{B} dy)^{B} + (\\int \\omega{(B,y)} dy)^{y})}{y} = \\frac{B ((\\int \\frac{y}{B} dy)^{B} + (\\int \\frac{y}{B} dy)^{y})}{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["add", 4, "Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True))"], "Equality(Add(Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True))), Add(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True))), Add(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True))))"], [["divide", 6, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True))"], "Equality(Mul(Symbol('B', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Mul(Symbol('B', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})}, then obtain \\hat{X}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} 0 = \\hat{X}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} (- \\hat{X}^{\\mathbf{B}}{(\\mathbf{B})} + (\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})})^{\\mathbf{B}})", "derivation": "\\hat{X}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} and \\hat{X}^{\\mathbf{B}}{(\\mathbf{B})} = (\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})})^{\\mathbf{B}} and 0 = - \\hat{X}^{\\mathbf{B}}{(\\mathbf{B})} + (\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})})^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} 0 = \\frac{d}{d \\mathbf{B}} (- \\hat{X}^{\\mathbf{B}}{(\\mathbf{B})} + (\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})})^{\\mathbf{B}}) and \\hat{X}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} 0 = \\hat{X}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} (- \\hat{X}^{\\mathbf{B}}{(\\mathbf{B})} + (\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})})^{\\mathbf{B}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["add", 4, "Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(E_{n},\\chi)} = \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\chi}, then derive \\frac{\\partial}{\\partial E_{n}} \\hat{p}{(E_{n},\\chi)} = 0, then obtain \\frac{\\partial^{2}}{\\partial E_{n}^{2}} \\frac{E_{n}}{\\chi} = 0", "derivation": "\\hat{p}{(E_{n},\\chi)} = \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\chi} and \\frac{\\partial}{\\partial E_{n}} \\hat{p}{(E_{n},\\chi)} = \\frac{\\partial^{2}}{\\partial E_{n}^{2}} \\frac{E_{n}}{\\chi} and \\frac{\\partial}{\\partial E_{n}} \\hat{p}{(E_{n},\\chi)} = 0 and \\frac{\\partial^{2}}{\\partial E_{n}^{2}} \\frac{E_{n}}{\\chi} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('E_n', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\phi_{2}{(\\rho)} = \\cos{(e^{\\rho})}, then obtain \\cos^{- \\rho}{(e^{\\rho})} \\int \\phi_{2}^{\\rho}{(\\rho)} d\\rho = \\cos^{- \\rho}{(e^{\\rho})} \\int \\cos^{\\rho}{(e^{\\rho})} d\\rho", "derivation": "\\phi_{2}{(\\rho)} = \\cos{(e^{\\rho})} and \\phi_{2}^{\\rho}{(\\rho)} = \\cos^{\\rho}{(e^{\\rho})} and \\int \\phi_{2}^{\\rho}{(\\rho)} d\\rho = \\int \\cos^{\\rho}{(e^{\\rho})} d\\rho and \\cos^{- \\rho}{(e^{\\rho})} \\int \\phi_{2}^{\\rho}{(\\rho)} d\\rho = \\cos^{- \\rho}{(e^{\\rho})} \\int \\cos^{\\rho}{(e^{\\rho})} d\\rho", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\rho', commutative=True)), cos(exp(Symbol('\\\\rho', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["integrate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Pow(Function('\\\\phi_2')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["divide", 3, "Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Integral(Pow(Function('\\\\phi_2')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Mul(Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Integral(Pow(cos(exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(L)} = \\sin{(\\cos{(L)})}, then obtain - \\sin{(\\cos{(L)})} = - \\frac{\\hat{H}{(L)}}{\\sin{(\\cos{(L)})}} - \\sin{(\\cos{(L)})} + 1", "derivation": "\\hat{H}{(L)} = \\sin{(\\cos{(L)})} and \\frac{\\hat{H}{(L)}}{\\sin{(\\cos{(L)})}} = 1 and \\frac{\\hat{H}{(L)}}{\\sin{(\\cos{(L)})}} - \\sin{(\\cos{(L)})} = 1 - \\sin{(\\cos{(L)})} and - \\sin{(\\cos{(L)})} = - \\frac{\\hat{H}{(L)}}{\\sin{(\\cos{(L)})}} - \\sin{(\\cos{(L)})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('L', commutative=True)), sin(cos(Symbol('L', commutative=True))))"], [["divide", 1, "sin(cos(Symbol('L', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Pow(sin(cos(Symbol('L', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "sin(cos(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Pow(sin(cos(Symbol('L', commutative=True))), Integer(-1))), Mul(Integer(-1), sin(cos(Symbol('L', commutative=True))))), Add(Integer(1), Mul(Integer(-1), sin(cos(Symbol('L', commutative=True))))))"], [["minus", 3, "Mul(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Pow(sin(cos(Symbol('L', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(-1), sin(cos(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('L', commutative=True)), Pow(sin(cos(Symbol('L', commutative=True))), Integer(-1))), Mul(Integer(-1), sin(cos(Symbol('L', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\eta{(I,A_{y},J_{\\varepsilon})} = I (- A_{y} + J_{\\varepsilon}), then obtain (I \\int \\eta^{J_{\\varepsilon}}{(I,A_{y},J_{\\varepsilon})} dA_{y})^{J_{\\varepsilon}} = (I \\int (I (- A_{y} + J_{\\varepsilon}))^{J_{\\varepsilon}} dA_{y})^{J_{\\varepsilon}}", "derivation": "\\eta{(I,A_{y},J_{\\varepsilon})} = I (- A_{y} + J_{\\varepsilon}) and \\eta^{J_{\\varepsilon}}{(I,A_{y},J_{\\varepsilon})} = (I (- A_{y} + J_{\\varepsilon}))^{J_{\\varepsilon}} and \\int \\eta^{J_{\\varepsilon}}{(I,A_{y},J_{\\varepsilon})} dA_{y} = \\int (I (- A_{y} + J_{\\varepsilon}))^{J_{\\varepsilon}} dA_{y} and I \\int \\eta^{J_{\\varepsilon}}{(I,A_{y},J_{\\varepsilon})} dA_{y} = I \\int (I (- A_{y} + J_{\\varepsilon}))^{J_{\\varepsilon}} dA_{y} and (I \\int \\eta^{J_{\\varepsilon}}{(I,A_{y},J_{\\varepsilon})} dA_{y})^{J_{\\varepsilon}} = (I \\int (I (- A_{y} + J_{\\varepsilon}))^{J_{\\varepsilon}} dA_{y})^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('I', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Mul(Symbol('I', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(Pow(Mul(Symbol('I', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["times", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Integral(Pow(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Symbol('I', commutative=True), Integral(Pow(Mul(Symbol('I', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"], [["power", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Symbol('I', commutative=True), Integral(Pow(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Mul(Symbol('I', commutative=True), Integral(Pow(Mul(Symbol('I', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\psi,F_{c})} = F_{c} \\psi, then derive \\frac{\\partial}{\\partial \\psi} \\operatorname{E_{x}}{(\\psi,F_{c})} = F_{c}, then derive \\eta + \\operatorname{E_{x}}{(\\psi,F_{c})} = \\int F_{c} d\\psi, then obtain \\frac{\\partial}{\\partial F_{c}} (F_{c} \\psi + \\eta) = \\frac{\\partial}{\\partial F_{c}} \\int F_{c} d\\psi", "derivation": "\\operatorname{E_{x}}{(\\psi,F_{c})} = F_{c} \\psi and \\frac{\\partial}{\\partial \\psi} \\operatorname{E_{x}}{(\\psi,F_{c})} = \\frac{\\partial}{\\partial \\psi} F_{c} \\psi and \\frac{\\partial}{\\partial \\psi} \\operatorname{E_{x}}{(\\psi,F_{c})} = F_{c} and \\int \\frac{\\partial}{\\partial \\psi} \\operatorname{E_{x}}{(\\psi,F_{c})} d\\psi = \\int F_{c} d\\psi and \\eta + \\operatorname{E_{x}}{(\\psi,F_{c})} = \\int F_{c} d\\psi and F_{c} \\psi + \\eta = \\int F_{c} d\\psi and \\frac{\\partial}{\\partial F_{c}} (F_{c} \\psi + \\eta) = \\frac{\\partial}{\\partial F_{c}} \\int F_{c} d\\psi", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('F_c', commutative=True))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Derivative(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\eta', commutative=True)), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))))"], [["differentiate", 6, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Integral(Symbol('F_c', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(t_{2})} = e^{t_{2}}, then obtain (\\hat{p}_0{(t_{2})} + e^{t_{2}} + 1)^{t_{2}} = (2 e^{t_{2}} + 1)^{t_{2}}", "derivation": "\\hat{p}_0{(t_{2})} = e^{t_{2}} and \\hat{p}_0{(t_{2})} + 1 = e^{t_{2}} + 1 and \\hat{p}_0{(t_{2})} + e^{t_{2}} + 1 = 2 e^{t_{2}} + 1 and 2 \\hat{p}_0{(t_{2})} + 1 = 2 e^{t_{2}} + 1 and \\hat{p}_0{(t_{2})} + e^{t_{2}} + 1 = 2 \\hat{p}_0{(t_{2})} + 1 and (2 \\hat{p}_0{(t_{2})} + 1)^{t_{2}} = (2 e^{t_{2}} + 1)^{t_{2}} and (\\hat{p}_0{(t_{2})} + e^{t_{2}} + 1)^{t_{2}} = (2 e^{t_{2}} + 1)^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True)), Integer(1)), Add(exp(Symbol('t_2', commutative=True)), Integer(1)))"], [["add", 1, "Add(exp(Symbol('t_2', commutative=True)), Integer(1))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(1)), Add(Mul(Integer(2), exp(Symbol('t_2', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True))), Integer(1)), Add(Mul(Integer(2), exp(Symbol('t_2', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(1)), Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True))), Integer(1)))"], [["power", 4, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True))), Integer(1)), Symbol('t_2', commutative=True)), Pow(Add(Mul(Integer(2), exp(Symbol('t_2', commutative=True))), Integer(1)), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Add(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)), Integer(1)), Symbol('t_2', commutative=True)), Pow(Add(Mul(Integer(2), exp(Symbol('t_2', commutative=True))), Integer(1)), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(A_{y},\\mu_0)} = \\frac{A_{y}}{\\mu_0}, then obtain \\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0 \\int \\operatorname{C_{2}}{(A_{y},\\mu_0)} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}} = \\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0 \\int \\frac{A_{y}}{\\mu_0} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}}", "derivation": "\\operatorname{C_{2}}{(A_{y},\\mu_0)} = \\frac{A_{y}}{\\mu_0} and \\int \\operatorname{C_{2}}{(A_{y},\\mu_0)} dA_{y} = \\int \\frac{A_{y}}{\\mu_0} dA_{y} and \\mu_0 \\int \\operatorname{C_{2}}{(A_{y},\\mu_0)} dA_{y} = \\mu_0 \\int \\frac{A_{y}}{\\mu_0} dA_{y} and \\frac{\\mu_0 \\int \\operatorname{C_{2}}{(A_{y},\\mu_0)} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}} = \\frac{\\mu_0 \\int \\frac{A_{y}}{\\mu_0} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}} and \\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0 \\int \\operatorname{C_{2}}{(A_{y},\\mu_0)} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}} = \\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0 \\int \\frac{A_{y}}{\\mu_0} dA_{y}}{\\operatorname{C_{2}}{(A_{y},\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True)))))"], [["divide", 3, "Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('C_2')(Symbol('A_y', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Mul(Symbol('A_y', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(H)} = e^{\\sin{(H)}} and \\mathbf{g}{(H)} = \\sin{(H)}, then obtain - H + \\hat{x}_0{(H)} + \\frac{d^{2}}{d H^{2}} e^{\\sin{(H)}} = - H + \\hat{x}_0{(H)} + \\frac{d^{2}}{d H^{2}} e^{\\mathbf{g}{(H)}}", "derivation": "\\hat{x}_0{(H)} = e^{\\sin{(H)}} and - H + \\hat{x}_0{(H)} = - H + e^{\\sin{(H)}} and \\mathbf{g}{(H)} = \\sin{(H)} and \\hat{x}_0{(H)} = e^{\\mathbf{g}{(H)}} and e^{\\sin{(H)}} = e^{\\mathbf{g}{(H)}} and \\frac{d}{d H} e^{\\sin{(H)}} = \\frac{d}{d H} e^{\\mathbf{g}{(H)}} and \\frac{d^{2}}{d H^{2}} e^{\\sin{(H)}} = \\frac{d^{2}}{d H^{2}} e^{\\mathbf{g}{(H)}} and - H + e^{\\sin{(H)}} + \\frac{d^{2}}{d H^{2}} e^{\\sin{(H)}} = - H + e^{\\sin{(H)}} + \\frac{d^{2}}{d H^{2}} e^{\\mathbf{g}{(H)}} and - H + \\hat{x}_0{(H)} + \\frac{d^{2}}{d H^{2}} e^{\\sin{(H)}} = - H + \\hat{x}_0{(H)} + \\frac{d^{2}}{d H^{2}} e^{\\mathbf{g}{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{x}_0')(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(sin(Symbol('H', commutative=True))), exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(exp(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(exp(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["add", 7, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True))), Derivative(exp(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True))), Derivative(exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 8, 2], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), Derivative(exp(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), Derivative(exp(Function('\\\\mathbf{g}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2)))))"]]}, {"prompt": "Given L{(m,\\psi)} = \\psi m, then obtain \\frac{\\partial}{\\partial m} (\\frac{L{(m,\\psi)}}{\\psi m})^{\\psi} + \\frac{1}{m} = \\frac{d}{d m} 1 + \\frac{1}{m}", "derivation": "L{(m,\\psi)} = \\psi m and \\frac{L{(m,\\psi)}}{\\psi m} = 1 and (\\frac{L{(m,\\psi)}}{\\psi m})^{\\psi} = 1 and \\frac{\\partial}{\\partial m} (\\frac{L{(m,\\psi)}}{\\psi m})^{\\psi} = \\frac{d}{d m} 1 and \\frac{\\partial}{\\partial m} (\\frac{L{(m,\\psi)}}{\\psi m})^{\\psi} + \\frac{1}{m} = \\frac{d}{d m} 1 + \\frac{1}{m}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('L')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('L')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('L')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["add", 4, "Pow(Symbol('m', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Pow(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('L')(Symbol('m', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1))), Add(Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(f^{*})} = \\cos{(f^{*})}, then derive \\int \\operatorname{m_{s}}{(f^{*})} df^{*} = \\mathbf{J}_M + \\sin{(f^{*})}, then obtain \\int (- \\mathbf{J}_M + \\int \\cos{(f^{*})} df^{*}) df^{*} = \\int \\sin{(f^{*})} df^{*}", "derivation": "\\operatorname{m_{s}}{(f^{*})} = \\cos{(f^{*})} and \\int \\operatorname{m_{s}}{(f^{*})} df^{*} = \\int \\cos{(f^{*})} df^{*} and \\int \\operatorname{m_{s}}{(f^{*})} df^{*} = \\mathbf{J}_M + \\sin{(f^{*})} and \\int \\cos{(f^{*})} df^{*} = \\mathbf{J}_M + \\sin{(f^{*})} and - \\mathbf{J}_M + \\int \\cos{(f^{*})} df^{*} = \\sin{(f^{*})} and \\int (- \\mathbf{J}_M + \\int \\cos{(f^{*})} df^{*}) df^{*} = \\int \\sin{(f^{*})} df^{*}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m_s')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), sin(Symbol('f^*', commutative=True)))"], [["integrate", 5, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given I{(\\theta_1)} = \\cos{(\\log{(\\theta_1)})}, then obtain \\theta_1 + \\log{(\\frac{d}{d \\theta_1} I{(\\theta_1)})} = \\theta_1 + \\log{(- \\frac{\\sin{(\\log{(\\theta_1)})}}{\\theta_1})}", "derivation": "I{(\\theta_1)} = \\cos{(\\log{(\\theta_1)})} and \\frac{d}{d \\theta_1} I{(\\theta_1)} = \\frac{d}{d \\theta_1} \\cos{(\\log{(\\theta_1)})} and \\log{(\\frac{d}{d \\theta_1} I{(\\theta_1)})} = \\log{(\\frac{d}{d \\theta_1} \\cos{(\\log{(\\theta_1)})})} and \\theta_1 + \\log{(\\frac{d}{d \\theta_1} I{(\\theta_1)})} = \\theta_1 + \\log{(\\frac{d}{d \\theta_1} \\cos{(\\log{(\\theta_1)})})} and \\theta_1 + \\log{(\\frac{d}{d \\theta_1} I{(\\theta_1)})} = \\theta_1 + \\log{(- \\frac{\\sin{(\\log{(\\theta_1)})}}{\\theta_1})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\theta_1', commutative=True)), cos(log(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('I')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), log(Derivative(cos(log(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["add", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Derivative(Function('I')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Add(Symbol('\\\\theta_1', commutative=True), log(Derivative(cos(log(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Derivative(Function('I')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Add(Symbol('\\\\theta_1', commutative=True), log(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(log(Symbol('\\\\theta_1', commutative=True)))))))"]]}, {"prompt": "Given z{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})}, then derive \\int z{(J_{\\varepsilon})} dJ_{\\varepsilon} = f^{\\prime} + \\sin{(J_{\\varepsilon})}, then obtain (\\int \\cos{(J_{\\varepsilon})} dJ_{\\varepsilon})^{f^{\\prime}} = (f^{\\prime} + \\sin{(J_{\\varepsilon})})^{f^{\\prime}}", "derivation": "z{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} and \\int z{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\cos{(J_{\\varepsilon})} dJ_{\\varepsilon} and \\int z{(J_{\\varepsilon})} dJ_{\\varepsilon} = f^{\\prime} + \\sin{(J_{\\varepsilon})} and (\\int z{(J_{\\varepsilon})} dJ_{\\varepsilon})^{f^{\\prime}} = (f^{\\prime} + \\sin{(J_{\\varepsilon})})^{f^{\\prime}} and (\\int \\cos{(J_{\\varepsilon})} dJ_{\\varepsilon})^{f^{\\prime}} = (f^{\\prime} + \\sin{(J_{\\varepsilon})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(Function('z')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given S{(H,t)} = \\cos^{H}{(t)} and \\operatorname{P_{g}}{(H,t)} = 2 H + \\cos^{H}{(t)}, then obtain \\operatorname{P_{g}}{(H,t)} = 2 H + S{(H,t)}", "derivation": "S{(H,t)} = \\cos^{H}{(t)} and H + S{(H,t)} = H + \\cos^{H}{(t)} and 2 H + S{(H,t)} = 2 H + \\cos^{H}{(t)} and \\operatorname{P_{g}}{(H,t)} = 2 H + \\cos^{H}{(t)} and \\operatorname{P_{g}}{(H,t)} = 2 H + S{(H,t)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('H', commutative=True), Symbol('t', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Symbol('H', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('S')(Symbol('H', commutative=True), Symbol('t', commutative=True))), Add(Symbol('H', commutative=True), Pow(cos(Symbol('t', commutative=True)), Symbol('H', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('H', commutative=True)), Function('S')(Symbol('H', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(2), Symbol('H', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('H', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(2), Symbol('H', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('P_g')(Symbol('H', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(2), Symbol('H', commutative=True)), Function('S')(Symbol('H', commutative=True), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(F_{g})} = \\frac{d}{d F_{g}} e^{F_{g}}, then derive \\frac{d}{d F_{g}} \\phi_{2}{(F_{g})} = e^{F_{g}}, then obtain (\\frac{d^{4}}{d F_{g}^{4}} e^{F_{g}})^{2} = e^{F_{g}} \\frac{d^{4}}{d F_{g}^{4}} e^{F_{g}}", "derivation": "\\phi_{2}{(F_{g})} = \\frac{d}{d F_{g}} e^{F_{g}} and \\frac{d}{d F_{g}} \\phi_{2}{(F_{g})} = \\frac{d^{2}}{d F_{g}^{2}} e^{F_{g}} and \\frac{d}{d F_{g}} \\phi_{2}{(F_{g})} = e^{F_{g}} and (\\frac{d}{d F_{g}} \\phi_{2}{(F_{g})})^{2} = e^{F_{g}} \\frac{d}{d F_{g}} \\phi_{2}{(F_{g})} and \\frac{d^{2}}{d F_{g}^{2}} e^{F_{g}} = e^{F_{g}} and \\frac{d}{d F_{g}} \\phi_{2}{(F_{g})} = \\frac{d^{4}}{d F_{g}^{4}} e^{F_{g}} and (\\frac{d^{4}}{d F_{g}^{4}} e^{F_{g}})^{2} = e^{F_{g}} \\frac{d^{4}}{d F_{g}^{4}} e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), exp(Symbol('F_g', commutative=True)))"], [["times", 3, "Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(2)), Mul(exp(Symbol('F_g', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))), exp(Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Function('\\\\phi_2')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(4))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(4))), Integer(2)), Mul(exp(Symbol('F_g', commutative=True)), Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(4)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(F_{N},z)} = - F_{N} + z, then derive - \\frac{\\partial}{\\partial z} \\mathbf{J}_P{(F_{N},z)} = -1, then obtain - \\frac{\\partial}{\\partial z} (- F_{N} + z) = -1", "derivation": "\\mathbf{J}_P{(F_{N},z)} = - F_{N} + z and - \\mathbf{J}_P{(F_{N},z)} = F_{N} - z and \\frac{\\partial}{\\partial z} - \\mathbf{J}_P{(F_{N},z)} = \\frac{\\partial}{\\partial z} (F_{N} - z) and - \\frac{\\partial}{\\partial z} \\mathbf{J}_P{(F_{N},z)} = -1 and - \\frac{\\partial}{\\partial z} (- F_{N} + z) = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('F_N', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('z', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('F_N', commutative=True), Symbol('z', commutative=True))), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('F_N', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_N', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(-1))"]]}, {"prompt": "Given H{(E,f_{E})} = \\frac{\\log{(f_{E})}}{E}, then obtain \\int 1 dE = \\int \\frac{\\frac{\\partial}{\\partial f_{E}} \\int (1 + \\frac{\\log{(f_{E})}}{E}) dE}{\\frac{\\partial}{\\partial f_{E}} \\int (H{(E,f_{E})} + 1) dE} dE", "derivation": "H{(E,f_{E})} = \\frac{\\log{(f_{E})}}{E} and H{(E,f_{E})} + 1 = 1 + \\frac{\\log{(f_{E})}}{E} and \\int (H{(E,f_{E})} + 1) dE = \\int (1 + \\frac{\\log{(f_{E})}}{E}) dE and \\frac{\\partial}{\\partial f_{E}} \\int (H{(E,f_{E})} + 1) dE = \\frac{\\partial}{\\partial f_{E}} \\int (1 + \\frac{\\log{(f_{E})}}{E}) dE and 1 = \\frac{\\frac{\\partial}{\\partial f_{E}} \\int (1 + \\frac{\\log{(f_{E})}}{E}) dE}{\\frac{\\partial}{\\partial f_{E}} \\int (H{(E,f_{E})} + 1) dE} and \\int 1 dE = \\int \\frac{\\frac{\\partial}{\\partial f_{E}} \\int (1 + \\frac{\\log{(f_{E})}}{E}) dE}{\\frac{\\partial}{\\partial f_{E}} \\int (H{(E,f_{E})} + 1) dE} dE", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True)))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True))), Integral(Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Integral(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Integral(Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Integral(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Integral(Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Pow(Derivative(Integral(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 5, "Symbol('E', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E', commutative=True))), Integral(Mul(Derivative(Integral(Add(Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Pow(Derivative(Integral(Add(Function('H')(Symbol('E', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{D},F_{N})} = \\frac{e^{\\mathbf{D}}}{F_{N}}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{f}{(\\mathbf{D},F_{N})} = \\frac{e^{\\mathbf{D}}}{F_{N}}, then obtain - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{e^{\\mathbf{D}}}{F_{N}} = - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{e^{\\mathbf{D}}}{F_{N}}", "derivation": "\\mathbf{f}{(\\mathbf{D},F_{N})} = \\frac{e^{\\mathbf{D}}}{F_{N}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{f}{(\\mathbf{D},F_{N})} = \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{e^{\\mathbf{D}}}{F_{N}} and - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{f}{(\\mathbf{D},F_{N})} = - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{e^{\\mathbf{D}}}{F_{N}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{f}{(\\mathbf{D},F_{N})} = \\frac{e^{\\mathbf{D}}}{F_{N}} and - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{e^{\\mathbf{D}}}{F_{N}} = - \\mathbf{f}{(\\mathbf{D},F_{N})} + \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{e^{\\mathbf{D}}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True))), Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True))), Derivative(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_N', commutative=True))), Derivative(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{S}{(E_{x},q,\\Psi_{nl})} = (q^{E_{x}})^{\\Psi_{nl}}, then obtain \\frac{\\int \\mathbf{S}{(E_{x},q,\\Psi_{nl})} d\\Psi_{nl}}{E_{x} \\int (q^{E_{x}})^{\\Psi_{nl}} d\\Psi_{nl}} = \\frac{1}{E_{x}}", "derivation": "\\mathbf{S}{(E_{x},q,\\Psi_{nl})} = (q^{E_{x}})^{\\Psi_{nl}} and \\int \\mathbf{S}{(E_{x},q,\\Psi_{nl})} d\\Psi_{nl} = \\int (q^{E_{x}})^{\\Psi_{nl}} d\\Psi_{nl} and \\frac{\\int \\mathbf{S}{(E_{x},q,\\Psi_{nl})} d\\Psi_{nl}}{\\int (q^{E_{x}})^{\\Psi_{nl}} d\\Psi_{nl}} = 1 and \\frac{\\int \\mathbf{S}{(E_{x},q,\\Psi_{nl})} d\\Psi_{nl}}{E_{x} \\int (q^{E_{x}})^{\\Psi_{nl}} d\\Psi_{nl}} = \\frac{1}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('E_x', commutative=True), Symbol('q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Pow(Symbol('q', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('E_x', commutative=True), Symbol('q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Pow(Symbol('q', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 2, "Integral(Pow(Pow(Symbol('q', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Pow(Integral(Pow(Pow(Symbol('q', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{S}')(Symbol('E_x', commutative=True), Symbol('q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(1))"], [["divide", 3, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Integral(Pow(Pow(Symbol('q', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{S}')(Symbol('E_x', commutative=True), Symbol('q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Pow(Symbol('E_x', commutative=True), Integer(-1)))"]]}, {"prompt": "Given i{(b)} = \\log{(b)}, then obtain (- i{(b)} + \\int i{(b)} db) i{(b)} = (- i{(b)} + \\int \\log{(b)} db) i{(b)}", "derivation": "i{(b)} = \\log{(b)} and \\int i{(b)} db = \\int \\log{(b)} db and - i{(b)} + \\int i{(b)} db = - i{(b)} + \\int \\log{(b)} db and (- i{(b)} + \\int i{(b)} db) \\log{(b)} = (- i{(b)} + \\int \\log{(b)} db) \\log{(b)} and (- i{(b)} + \\int i{(b)} db) i{(b)} = (- i{(b)} + \\int \\log{(b)} db) i{(b)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('i')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 2, "Function('i')(Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(Function('i')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["times", 3, "log(Symbol('b', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(Function('i')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Symbol('b', commutative=True))), Mul(Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), log(Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(Function('i')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Function('i')(Symbol('b', commutative=True))), Mul(Add(Mul(Integer(-1), Function('i')(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Function('i')(Symbol('b', commutative=True))))"]]}, {"prompt": "Given T{(f)} = \\cos{(f)}, then derive \\frac{d}{d f} T{(f)} = - \\sin{(f)}, then obtain (\\frac{d}{d f} \\cos{(f)})^{2} = - \\sin{(f)} \\frac{d}{d f} \\cos{(f)}", "derivation": "T{(f)} = \\cos{(f)} and \\frac{d}{d f} T{(f)} = \\frac{d}{d f} \\cos{(f)} and \\frac{d}{d f} T{(f)} = - \\sin{(f)} and (\\frac{d}{d f} T{(f)})^{2} = - \\sin{(f)} \\frac{d}{d f} T{(f)} and (\\frac{d}{d f} \\cos{(f)})^{2} = - \\sin{(f)} \\frac{d}{d f} \\cos{(f)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('T')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('f', commutative=True))))"], [["times", 3, "Derivative(Function('T')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('T')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Symbol('f', commutative=True)), Derivative(Function('T')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Symbol('f', commutative=True)), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu_{0}{(\\Omega,\\hbar)} = \\Omega + \\hbar, then obtain \\Omega + 2 \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\hbar) = \\Omega + 2 (\\Omega + \\hbar)^{\\Omega} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\hbar)", "derivation": "\\mu_{0}{(\\Omega,\\hbar)} = \\Omega + \\hbar and \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} = (\\Omega + \\hbar)^{\\Omega} and \\Omega + \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} = \\Omega + (\\Omega + \\hbar)^{\\Omega} and \\Omega + (\\Omega + \\hbar)^{\\Omega} + \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} = \\Omega + 2 (\\Omega + \\hbar)^{\\Omega} and \\Omega + 2 \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} = \\Omega + 2 (\\Omega + \\hbar)^{\\Omega} and \\Omega + 2 \\mu_{0}^{\\Omega}{(\\Omega,\\hbar)} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\hbar) = \\Omega + 2 (\\Omega + \\hbar)^{\\Omega} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\hbar)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["add", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["add", 3, "Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True)))))"], [["minus", 5, "Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(v_{1},\\psi^*)} = \\sin{(\\psi^* + v_{1})}, then obtain - v_{1} + \\operatorname{c_{0}}{(v_{1},\\psi^*)} = - v_{1} + \\sin{(\\psi^* + v_{1})}", "derivation": "\\operatorname{c_{0}}{(v_{1},\\psi^*)} = \\sin{(\\psi^* + v_{1})} and t_{2} \\operatorname{c_{0}}{(v_{1},\\psi^*)} = t_{2} \\sin{(\\psi^* + v_{1})} and \\frac{\\partial}{\\partial t_{2}} t_{2} \\operatorname{c_{0}}{(v_{1},\\psi^*)} = \\frac{\\partial}{\\partial t_{2}} t_{2} \\sin{(\\psi^* + v_{1})} and - v_{1} + \\frac{\\partial}{\\partial t_{2}} t_{2} \\operatorname{c_{0}}{(v_{1},\\psi^*)} = - v_{1} + \\frac{\\partial}{\\partial t_{2}} t_{2} \\sin{(\\psi^* + v_{1})} and - v_{1} + \\operatorname{c_{0}}{(v_{1},\\psi^*)} = - v_{1} + \\sin{(\\psi^* + v_{1})}", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('v_1', commutative=True), Symbol('\\\\psi^*', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_1', commutative=True))))"], [["times", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Function('c_0')(Symbol('v_1', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('t_2', commutative=True), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_1', commutative=True)))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('t_2', commutative=True), Function('c_0')(Symbol('v_1', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('t_2', commutative=True), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Mul(Symbol('t_2', commutative=True), Function('c_0')(Symbol('v_1', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Mul(Symbol('t_2', commutative=True), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('c_0')(Symbol('v_1', commutative=True), Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(f,F_{H})} = F_{H} + \\frac{f}{F_{H}}, then obtain \\int (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}}) (\\int \\operatorname{v_{2}}{(f,F_{H})} df - \\frac{1}{F_{H}}) dF_{H} = \\int (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}})^{2} dF_{H}", "derivation": "\\operatorname{v_{2}}{(f,F_{H})} = F_{H} + \\frac{f}{F_{H}} and \\int \\operatorname{v_{2}}{(f,F_{H})} df = \\int (F_{H} + \\frac{f}{F_{H}}) df and \\int \\operatorname{v_{2}}{(f,F_{H})} df - \\frac{1}{F_{H}} = \\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}} and (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}}) (\\int \\operatorname{v_{2}}{(f,F_{H})} df - \\frac{1}{F_{H}}) = (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}})^{2} and \\int (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}}) (\\int \\operatorname{v_{2}}{(f,F_{H})} df - \\frac{1}{F_{H}}) dF_{H} = \\int (\\int (F_{H} + \\frac{f}{F_{H}}) df - \\frac{1}{F_{H}})^{2} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('f', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["minus", 2, "Pow(Symbol('F_H', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('v_2')(Symbol('f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))))"], [["times", 3, "Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Add(Integral(Function('v_2')(Symbol('f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1))))), Pow(Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Integer(2)))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Add(Integral(Function('v_2')(Symbol('f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1))))), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Add(Integral(Add(Symbol('F_H', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Integer(2)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\theta,\\dot{y})} = \\dot{y} + \\theta, then derive \\int \\operatorname{f_{E}}{(\\theta,\\dot{y})} d\\dot{y} = \\Omega + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta, then derive \\Omega + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta = C_{d} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta, then obtain \\int \\operatorname{f_{E}}{(\\theta,\\dot{y})} d\\dot{y} = C_{d} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta", "derivation": "\\operatorname{f_{E}}{(\\theta,\\dot{y})} = \\dot{y} + \\theta and \\int \\operatorname{f_{E}}{(\\theta,\\dot{y})} d\\dot{y} = \\int (\\dot{y} + \\theta) d\\dot{y} and \\int \\operatorname{f_{E}}{(\\theta,\\dot{y})} d\\dot{y} = \\Omega + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta and \\Omega + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta = \\int (\\dot{y} + \\theta) d\\dot{y} and \\Omega + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta = C_{d} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta and \\int \\operatorname{f_{E}}{(\\theta,\\dot{y})} d\\dot{y} = C_{d} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} \\theta", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Function('f_E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(G,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon}), then obtain \\cos{(\\frac{\\partial}{\\partial G} \\frac{\\dot{z}{(G,g^{\\prime}_{\\varepsilon})}}{G + g^{\\prime}_{\\varepsilon}})} = \\cos{(\\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon})}{G + g^{\\prime}_{\\varepsilon}})}", "derivation": "\\dot{z}{(G,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon}) and \\frac{\\dot{z}{(G,g^{\\prime}_{\\varepsilon})}}{G + g^{\\prime}_{\\varepsilon}} = \\frac{\\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon})}{G + g^{\\prime}_{\\varepsilon}} and \\frac{\\partial}{\\partial G} \\frac{\\dot{z}{(G,g^{\\prime}_{\\varepsilon})}}{G + g^{\\prime}_{\\varepsilon}} = \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon})}{G + g^{\\prime}_{\\varepsilon}} and \\cos{(\\frac{\\partial}{\\partial G} \\frac{\\dot{z}{(G,g^{\\prime}_{\\varepsilon})}}{G + g^{\\prime}_{\\varepsilon}})} = \\cos{(\\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} (G + g^{\\prime}_{\\varepsilon})}{G + g^{\\prime}_{\\varepsilon}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["divide", 1, "Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), cos(Derivative(Mul(Pow(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(g)} = e^{e^{g}}, then obtain \\operatorname{F_{g}}^{g}{(g)} \\log{(\\operatorname{F_{g}}{(g)} + e^{e^{g}})} = \\operatorname{F_{g}}^{g}{(g)} \\log{(2 e^{e^{g}})}", "derivation": "\\operatorname{F_{g}}{(g)} = e^{e^{g}} and \\operatorname{F_{g}}{(g)} + e^{e^{g}} = 2 e^{e^{g}} and \\log{(\\operatorname{F_{g}}{(g)} + e^{e^{g}})} = \\log{(2 e^{e^{g}})} and \\operatorname{F_{g}}^{g}{(g)} = (e^{e^{g}})^{g} and (e^{e^{g}})^{g} \\log{(\\operatorname{F_{g}}{(g)} + e^{e^{g}})} = (e^{e^{g}})^{g} \\log{(2 e^{e^{g}})} and \\operatorname{F_{g}}^{g}{(g)} \\log{(\\operatorname{F_{g}}{(g)} + e^{e^{g}})} = \\operatorname{F_{g}}^{g}{(g)} \\log{(2 e^{e^{g}})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('g', commutative=True)), exp(exp(Symbol('g', commutative=True))))"], [["add", 1, "exp(exp(Symbol('g', commutative=True)))"], "Equality(Add(Function('F_g')(Symbol('g', commutative=True)), exp(exp(Symbol('g', commutative=True)))), Mul(Integer(2), exp(exp(Symbol('g', commutative=True)))))"], [["log", 2], "Equality(log(Add(Function('F_g')(Symbol('g', commutative=True)), exp(exp(Symbol('g', commutative=True))))), log(Mul(Integer(2), exp(exp(Symbol('g', commutative=True))))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(exp(exp(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["times", 3, "Pow(exp(exp(Symbol('g', commutative=True))), Symbol('g', commutative=True))"], "Equality(Mul(Pow(exp(exp(Symbol('g', commutative=True))), Symbol('g', commutative=True)), log(Add(Function('F_g')(Symbol('g', commutative=True)), exp(exp(Symbol('g', commutative=True)))))), Mul(Pow(exp(exp(Symbol('g', commutative=True))), Symbol('g', commutative=True)), log(Mul(Integer(2), exp(exp(Symbol('g', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('F_g')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), log(Add(Function('F_g')(Symbol('g', commutative=True)), exp(exp(Symbol('g', commutative=True)))))), Mul(Pow(Function('F_g')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), log(Mul(Integer(2), exp(exp(Symbol('g', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{B})} = \\log{(\\sin{(\\mathbf{B})})}, then obtain \\operatorname{F_{H}}{(\\mathbf{B})} + \\int \\operatorname{F_{H}}^{\\mathbf{B}}{(\\mathbf{B})} d\\mathbf{B} - \\int \\log{(\\sin{(\\mathbf{B})})}^{\\mathbf{B}} d\\mathbf{B} = \\operatorname{F_{H}}{(\\mathbf{B})}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{B})} = \\log{(\\sin{(\\mathbf{B})})} and \\operatorname{F_{H}}^{\\mathbf{B}}{(\\mathbf{B})} = \\log{(\\sin{(\\mathbf{B})})}^{\\mathbf{B}} and \\int \\operatorname{F_{H}}^{\\mathbf{B}}{(\\mathbf{B})} d\\mathbf{B} = \\int \\log{(\\sin{(\\mathbf{B})})}^{\\mathbf{B}} d\\mathbf{B} and \\operatorname{F_{H}}{(\\mathbf{B})} + \\int \\operatorname{F_{H}}^{\\mathbf{B}}{(\\mathbf{B})} d\\mathbf{B} = \\operatorname{F_{H}}{(\\mathbf{B})} + \\int \\log{(\\sin{(\\mathbf{B})})}^{\\mathbf{B}} d\\mathbf{B} and \\operatorname{F_{H}}{(\\mathbf{B})} + \\int \\operatorname{F_{H}}^{\\mathbf{B}}{(\\mathbf{B})} d\\mathbf{B} - \\int \\log{(\\sin{(\\mathbf{B})})}^{\\mathbf{B}} d\\mathbf{B} = \\operatorname{F_{H}}{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), log(sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Pow(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Pow(log(sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 3, "Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Integral(Pow(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Integral(Pow(log(sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["minus", 4, "Integral(Pow(log(sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Integral(Pow(Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Integral(Pow(log(sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Function('F_H')(Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given C{(U,\\hat{\\mathbf{r}})} = U + \\hat{\\mathbf{r}}, then obtain (\\frac{d}{d \\hat{\\mathbf{r}}} 1)^{2} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\frac{2 U + 2 \\hat{\\mathbf{r}}}{2 (U + \\hat{\\mathbf{r}})})^{2}", "derivation": "C{(U,\\hat{\\mathbf{r}})} = U + \\hat{\\mathbf{r}} and U + \\hat{\\mathbf{r}} + C{(U,\\hat{\\mathbf{r}})} = 2 U + 2 \\hat{\\mathbf{r}} and 1 = \\frac{2 U + 2 \\hat{\\mathbf{r}}}{U + \\hat{\\mathbf{r}} + C{(U,\\hat{\\mathbf{r}})}} and 1 = \\frac{2 U + 2 \\hat{\\mathbf{r}}}{2 C{(U,\\hat{\\mathbf{r}})}} and 1 = \\frac{2 U + 2 \\hat{\\mathbf{r}}}{2 (U + \\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} 1 = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\frac{2 U + 2 \\hat{\\mathbf{r}}}{2 (U + \\hat{\\mathbf{r}})} and (\\frac{d}{d \\hat{\\mathbf{r}}} 1)^{2} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\frac{2 U + 2 \\hat{\\mathbf{r}}}{2 (U + \\hat{\\mathbf{r}})})^{2}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["add", 1, "Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('C')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 2, "Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('C')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Integer(1), Mul(Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('C')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Rational(1, 2), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Function('C')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Rational(1, 2), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["power", 6, 2], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Rational(1, 2), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain \\mathbb{I} + \\operatorname{F_{c}}^{2}{(\\mathbb{I})} + \\cos{(\\mathbb{I})} = \\mathbb{I} + \\operatorname{F_{c}}{(\\mathbb{I})} \\cos{(\\mathbb{I})} + \\cos{(\\mathbb{I})}", "derivation": "\\operatorname{F_{c}}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\mathbb{I} + \\operatorname{F_{c}}{(\\mathbb{I})} = \\mathbb{I} + \\cos{(\\mathbb{I})} and \\operatorname{F_{c}}^{2}{(\\mathbb{I})} = \\operatorname{F_{c}}{(\\mathbb{I})} \\cos{(\\mathbb{I})} and \\mathbb{I} + \\operatorname{F_{c}}^{2}{(\\mathbb{I})} + \\operatorname{F_{c}}{(\\mathbb{I})} = \\mathbb{I} + \\operatorname{F_{c}}{(\\mathbb{I})} \\cos{(\\mathbb{I})} + \\operatorname{F_{c}}{(\\mathbb{I})} and \\mathbb{I} + \\operatorname{F_{c}}^{2}{(\\mathbb{I})} + \\cos{(\\mathbb{I})} = \\mathbb{I} + \\operatorname{F_{c}}{(\\mathbb{I})} \\cos{(\\mathbb{I})} + \\cos{(\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 1, "Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Pow(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Mul(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\mathbb{I}', commutative=True), Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Pow(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Pow(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Function('F_c')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), cos(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{2})} = \\cos{(v_{2})}, then obtain (\\iint \\log{(v_{2} + \\mathbf{E}{(v_{2})})} dv_{2} dv_{2})^{v_{2}} = (\\iint \\log{(v_{2} + \\cos{(v_{2})})} dv_{2} dv_{2})^{v_{2}}", "derivation": "\\mathbf{E}{(v_{2})} = \\cos{(v_{2})} and v_{2} + \\mathbf{E}{(v_{2})} = v_{2} + \\cos{(v_{2})} and \\log{(v_{2} + \\mathbf{E}{(v_{2})})} = \\log{(v_{2} + \\cos{(v_{2})})} and \\int \\log{(v_{2} + \\mathbf{E}{(v_{2})})} dv_{2} = \\int \\log{(v_{2} + \\cos{(v_{2})})} dv_{2} and \\iint \\log{(v_{2} + \\mathbf{E}{(v_{2})})} dv_{2} dv_{2} = \\iint \\log{(v_{2} + \\cos{(v_{2})})} dv_{2} dv_{2} and (\\iint \\log{(v_{2} + \\mathbf{E}{(v_{2})})} dv_{2} dv_{2})^{v_{2}} = (\\iint \\log{(v_{2} + \\cos{(v_{2})})} dv_{2} dv_{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["add", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Symbol('v_2', commutative=True), Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True))), Add(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))))"], [["log", 2], "Equality(log(Add(Symbol('v_2', commutative=True), Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)))), log(Add(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(log(Add(Symbol('v_2', commutative=True), Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True))), Integral(log(Add(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(log(Add(Symbol('v_2', commutative=True), Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(log(Add(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(Integral(log(Add(Symbol('v_2', commutative=True), Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Integral(log(Add(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given u{(r_{0})} = \\cos{(r_{0})} and \\theta{(r_{0})} = \\cos{(r_{0})}, then derive \\frac{\\frac{d}{d r_{0}} \\theta{(r_{0})}}{r_{0}} = - \\frac{\\sin{(r_{0})}}{r_{0}}, then obtain (\\frac{\\frac{d}{d r_{0}} \\theta{(r_{0})}}{r_{0}})^{r_{0}} = (- \\frac{\\sin{(r_{0})}}{r_{0}})^{r_{0}}", "derivation": "u{(r_{0})} = \\cos{(r_{0})} and \\frac{d}{d r_{0}} u{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})} and \\theta{(r_{0})} = \\cos{(r_{0})} and \\theta{(r_{0})} = u{(r_{0})} and \\frac{d}{d r_{0}} \\theta{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})} and \\frac{\\frac{d}{d r_{0}} \\theta{(r_{0})}}{r_{0}} = \\frac{\\frac{d}{d r_{0}} \\cos{(r_{0})}}{r_{0}} and \\frac{\\frac{d}{d r_{0}} \\theta{(r_{0})}}{r_{0}} = - \\frac{\\sin{(r_{0})}}{r_{0}} and (\\frac{\\frac{d}{d r_{0}} \\theta{(r_{0})}}{r_{0}})^{r_{0}} = (- \\frac{\\sin{(r_{0})}}{r_{0}})^{r_{0}}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\theta')(Symbol('r_0', commutative=True)), Function('u')(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\theta')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["divide", 5, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), sin(Symbol('r_0', commutative=True))))"], [["power", 7, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Symbol('r_0', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), sin(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{B})} = \\log{(\\log{(\\mathbf{B})})}, then obtain - \\mathbf{B} + \\theta_{2}^{3}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})} = - \\mathbf{B} + \\log{(\\log{(\\mathbf{B})})}^{4}", "derivation": "\\theta_{2}{(\\mathbf{B})} = \\log{(\\log{(\\mathbf{B})})} and \\theta_{2}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})} = \\log{(\\log{(\\mathbf{B})})}^{2} and \\theta_{2}^{2}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})}^{2} = \\log{(\\log{(\\mathbf{B})})}^{4} and \\theta_{2}^{3}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})} = \\theta_{2}^{2}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})}^{2} and \\theta_{2}^{3}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})} = \\log{(\\log{(\\mathbf{B})})}^{4} and - \\mathbf{B} + \\theta_{2}^{3}{(\\mathbf{B})} \\log{(\\log{(\\mathbf{B})})} = - \\mathbf{B} + \\log{(\\log{(\\mathbf{B})})}^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), log(log(Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 1, "log(log(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), log(log(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2))), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(3)), log(log(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(3)), log(log(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(4)))"], [["minus", 5, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(3)), log(log(Symbol('\\\\mathbf{B}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(log(log(Symbol('\\\\mathbf{B}', commutative=True))), Integer(4))))"]]}, {"prompt": "Given \\tilde{g}{(A,F_{x})} = A + F_{x}, then obtain F_{x} + \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}} = 2 A + 3 F_{x} - \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}}", "derivation": "\\tilde{g}{(A,F_{x})} = A + F_{x} and - A - F_{x} + \\tilde{g}{(A,F_{x})} = 0 and - A + \\tilde{g}{(A,F_{x})} = F_{x} and F_{x} + \\tilde{g}{(A,F_{x})} = A + 2 F_{x} and F_{x} + \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}} = A + 2 F_{x} - \\frac{d}{d F_{x}} 0^{F_{x}} and A + 2 F_{x} - \\frac{d}{d F_{x}} 0^{F_{x}} = 2 A + 3 F_{x} - \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}} and F_{x} + \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}} = 2 A + 3 F_{x} - \\tilde{g}{(A,F_{x})} - \\frac{d}{d F_{x}} 0^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('A', commutative=True), Symbol('F_x', commutative=True)))"], [["minus", 1, "Add(Symbol('A', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Integer(0))"], [["minus", 2, "Mul(Integer(-1), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))"], [["add", 3, "Add(Symbol('A', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('A', commutative=True), Mul(Integer(2), Symbol('F_x', commutative=True))))"], [["minus", 4, "Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Symbol('A', commutative=True), Mul(Integer(2), Symbol('F_x', commutative=True)), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(2), Symbol('F_x', commutative=True)), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(3), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(3), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Derivative(Pow(Integer(0), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{f}{(t,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(t)}, then obtain \\frac{d}{d \\Psi_{\\lambda}} 0 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- t (\\Psi_{\\lambda} + \\cos{(t)}) + t \\mathbf{f}{(t,\\Psi_{\\lambda})})", "derivation": "\\mathbf{f}{(t,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(t)} and t \\mathbf{f}{(t,\\Psi_{\\lambda})} = t (\\Psi_{\\lambda} + \\cos{(t)}) and 0 = t (\\Psi_{\\lambda} + \\cos{(t)}) - t \\mathbf{f}{(t,\\Psi_{\\lambda})} and 0 = - t (\\Psi_{\\lambda} + \\cos{(t)}) + t \\mathbf{f}{(t,\\Psi_{\\lambda})} and \\frac{d}{d \\Psi_{\\lambda}} 0 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- t (\\Psi_{\\lambda} + \\cos{(t)}) + t \\mathbf{f}{(t,\\Psi_{\\lambda})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('t', commutative=True))))"], [["times", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('t', commutative=True)))))"], [["minus", 2, "Mul(Symbol('t', commutative=True), Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('t', commutative=True)))), Mul(Integer(-1), Symbol('t', commutative=True), Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(c)} = \\log{(\\log{(c)})}, then derive \\int \\hat{X}{(c)} dc = \\hat{H}_l + c \\log{(\\log{(c)})} - \\operatorname{li}{(c)}, then obtain \\frac{\\int \\log{(\\log{(c)})} dc}{c} = \\frac{\\hat{H}_l + c \\log{(\\log{(c)})} - \\operatorname{li}{(c)}}{c}", "derivation": "\\hat{X}{(c)} = \\log{(\\log{(c)})} and \\int \\hat{X}{(c)} dc = \\int \\log{(\\log{(c)})} dc and \\int \\hat{X}{(c)} dc = \\hat{H}_l + c \\log{(\\log{(c)})} - \\operatorname{li}{(c)} and \\int \\log{(\\log{(c)})} dc = \\hat{H}_l + c \\log{(\\log{(c)})} - \\operatorname{li}{(c)} and \\frac{\\int \\log{(\\log{(c)})} dc}{c} = \\frac{\\hat{H}_l + c \\log{(\\log{(c)})} - \\operatorname{li}{(c)}}{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('c', commutative=True)), log(log(Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(log(log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('c', commutative=True), log(log(Symbol('c', commutative=True)))), Mul(Integer(-1), li(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('c', commutative=True), log(log(Symbol('c', commutative=True)))), Mul(Integer(-1), li(Symbol('c', commutative=True)))))"], [["divide", 4, "Symbol('c', commutative=True)"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(log(log(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('c', commutative=True), log(log(Symbol('c', commutative=True)))), Mul(Integer(-1), li(Symbol('c', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\varphi^*)} = \\varphi^*, then derive g^{\\prime}_{\\varepsilon} \\frac{d}{d \\varphi^*} \\Psi_{nl}{(\\varphi^*)} = g^{\\prime}_{\\varepsilon}, then obtain (g^{\\prime}_{\\varepsilon} \\frac{d}{d \\varphi^*} \\Psi_{nl}{(\\varphi^*)})^{g^{\\prime}_{\\varepsilon}} = (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}}", "derivation": "\\Psi_{nl}{(\\varphi^*)} = \\varphi^* and g^{\\prime}_{\\varepsilon} \\Psi_{nl}{(\\varphi^*)} = \\varphi^* g^{\\prime}_{\\varepsilon} and \\frac{\\partial}{\\partial \\varphi^*} g^{\\prime}_{\\varepsilon} \\Psi_{nl}{(\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\varphi^* g^{\\prime}_{\\varepsilon} and g^{\\prime}_{\\varepsilon} \\frac{d}{d \\varphi^*} \\Psi_{nl}{(\\varphi^*)} = g^{\\prime}_{\\varepsilon} and (g^{\\prime}_{\\varepsilon} \\frac{d}{d \\varphi^*} \\Psi_{nl}{(\\varphi^*)})^{g^{\\prime}_{\\varepsilon}} = (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], [["times", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], [["power", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(u,B)} = e^{B u} and \\operatorname{v_{2}}{(u,B)} = B u, then obtain \\frac{\\partial}{\\partial u} (B u - \\mathbf{D}{(u,B)} + e^{\\operatorname{v_{2}}{(u,B)}}) = \\frac{\\partial}{\\partial u} (B u - \\mathbf{D}{(u,B)} + e^{B u})", "derivation": "\\operatorname{g_{\\varepsilon}}{(u,B)} = e^{B u} and B u + \\operatorname{g_{\\varepsilon}}{(u,B)} = B u + e^{B u} and \\operatorname{v_{2}}{(u,B)} = B u and \\operatorname{g_{\\varepsilon}}{(u,B)} = e^{\\operatorname{v_{2}}{(u,B)}} and B u + e^{\\operatorname{v_{2}}{(u,B)}} = B u + e^{B u} and B u - \\mathbf{D}{(u,B)} + e^{\\operatorname{v_{2}}{(u,B)}} = B u - \\mathbf{D}{(u,B)} + e^{B u} and \\frac{\\partial}{\\partial u} (B u - \\mathbf{D}{(u,B)} + e^{\\operatorname{v_{2}}{(u,B)}}) = \\frac{\\partial}{\\partial u} (B u - \\mathbf{D}{(u,B)} + e^{B u})", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True))))"], [["add", 1, "Mul(Symbol('B', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), exp(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('u', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('B', commutative=True)), exp(Function('v_2')(Symbol('u', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), exp(Function('v_2')(Symbol('u', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), exp(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)))))"], [["minus", 5, "Function('\\\\mathbf{D}')(Symbol('u', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('u', commutative=True), Symbol('B', commutative=True))), exp(Function('v_2')(Symbol('u', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('u', commutative=True), Symbol('B', commutative=True))), exp(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)))))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('u', commutative=True), Symbol('B', commutative=True))), exp(Function('v_2')(Symbol('u', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('u', commutative=True), Symbol('B', commutative=True))), exp(Mul(Symbol('B', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} = e^{\\varphi^{\\mathbf{J}_P}}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} + 1 = \\varphi^{\\mathbf{J}_P} e^{\\varphi^{\\mathbf{J}_P}} \\log{(\\varphi)} + 1, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} + 1 = \\varphi^{\\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} \\log{(\\varphi)} + 1", "derivation": "\\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} = e^{\\varphi^{\\mathbf{J}_P}} and \\mathbf{J}_P + \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} = \\mathbf{J}_P + e^{\\varphi^{\\mathbf{J}_P}} and \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)}) = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{\\varphi^{\\mathbf{J}_P}}) and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} + 1 = \\varphi^{\\mathbf{J}_P} e^{\\varphi^{\\mathbf{J}_P}} \\log{(\\varphi)} + 1 and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} + 1 = \\varphi^{\\mathbf{J}_P} \\operatorname{f_{E}}{(\\mathbf{J}_P,\\varphi)} \\log{(\\varphi)} + 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(1)), Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), log(Symbol('\\\\varphi', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(1)), Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Integer(1)))"]]}, {"prompt": "Given a{(G)} = \\cos{(G)}, then obtain - \\frac{d^{2}}{d G^{2}} a{(G)} = - \\frac{d^{2}}{d G^{2}} \\cos{(G)}", "derivation": "a{(G)} = \\cos{(G)} and \\frac{d}{d G} a{(G)} = \\frac{d}{d G} \\cos{(G)} and \\frac{d^{2}}{d G^{2}} a{(G)} = \\frac{d^{2}}{d G^{2}} \\cos{(G)} and - \\frac{d^{2}}{d G^{2}} a{(G)} = - \\frac{d^{2}}{d G^{2}} \\cos{(G)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('a')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))), Mul(Integer(-1), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(x^\\prime)} = \\sin{(e^{x^\\prime})}, then derive \\frac{d}{d x^\\prime} \\hat{H}_{\\lambda}{(x^\\prime)} = e^{x^\\prime} \\cos{(e^{x^\\prime})}, then obtain \\log{(\\frac{d^{2}}{d (x^\\prime)^{2}} \\hat{H}_{\\lambda}{(x^\\prime)})} = \\log{(\\frac{d}{d x^\\prime} e^{x^\\prime} \\cos{(e^{x^\\prime})})}", "derivation": "\\hat{H}_{\\lambda}{(x^\\prime)} = \\sin{(e^{x^\\prime})} and \\frac{d}{d x^\\prime} \\hat{H}_{\\lambda}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})} and \\frac{d}{d x^\\prime} \\hat{H}_{\\lambda}{(x^\\prime)} = e^{x^\\prime} \\cos{(e^{x^\\prime})} and \\frac{d^{2}}{d (x^\\prime)^{2}} \\hat{H}_{\\lambda}{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} \\cos{(e^{x^\\prime})} and \\log{(\\frac{d^{2}}{d (x^\\prime)^{2}} \\hat{H}_{\\lambda}{(x^\\prime)})} = \\log{(\\frac{d}{d x^\\prime} e^{x^\\prime} \\cos{(e^{x^\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), sin(exp(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(exp(Symbol('x^\\\\prime', commutative=True)), cos(exp(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Mul(exp(Symbol('x^\\\\prime', commutative=True)), cos(exp(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["log", 4], "Equality(log(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2)))), log(Derivative(Mul(exp(Symbol('x^\\\\prime', commutative=True)), cos(exp(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(C)} = \\cos{(C)}, then obtain \\int (\\dot{x}{(C)} - 1) \\dot{x}{(C)} dC = \\int (\\cos{(C)} - 1) \\dot{x}{(C)} dC", "derivation": "\\dot{x}{(C)} = \\cos{(C)} and \\dot{x}{(C)} - 1 = \\cos{(C)} - 1 and (\\dot{x}{(C)} - 1) \\dot{x}{(C)} = (\\cos{(C)} - 1) \\dot{x}{(C)} and \\int (\\dot{x}{(C)} - 1) \\dot{x}{(C)} dC = \\int (\\cos{(C)} - 1) \\dot{x}{(C)} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Integer(-1)), Add(cos(Symbol('C', commutative=True)), Integer(-1)))"], [["times", 2, "Function('\\\\dot{x}')(Symbol('C', commutative=True))"], "Equality(Mul(Add(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Integer(-1)), Function('\\\\dot{x}')(Symbol('C', commutative=True))), Mul(Add(cos(Symbol('C', commutative=True)), Integer(-1)), Function('\\\\dot{x}')(Symbol('C', commutative=True))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Integer(-1)), Function('\\\\dot{x}')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Add(cos(Symbol('C', commutative=True)), Integer(-1)), Function('\\\\dot{x}')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given W{(A_{2},\\mathbf{J}_P)} = A_{2} + \\mathbf{J}_P, then derive (\\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P)^{A_{2}} = (A_{2} \\mathbf{J}_P + S + \\frac{\\mathbf{J}_P^{2}}{2})^{A_{2}}, then obtain \\int (\\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P)^{A_{2}} dA_{2} = \\int (A_{2} \\mathbf{J}_P + S + \\frac{\\mathbf{J}_P^{2}}{2})^{A_{2}} dA_{2}", "derivation": "W{(A_{2},\\mathbf{J}_P)} = A_{2} + \\mathbf{J}_P and \\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P = \\int (A_{2} + \\mathbf{J}_P) d\\mathbf{J}_P and (\\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P)^{A_{2}} = (\\int (A_{2} + \\mathbf{J}_P) d\\mathbf{J}_P)^{A_{2}} and (\\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P)^{A_{2}} = (A_{2} \\mathbf{J}_P + S + \\frac{\\mathbf{J}_P^{2}}{2})^{A_{2}} and \\int (\\int W{(A_{2},\\mathbf{J}_P)} d\\mathbf{J}_P)^{A_{2}} dA_{2} = \\int (A_{2} \\mathbf{J}_P + S + \\frac{\\mathbf{J}_P^{2}}{2})^{A_{2}} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('W')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('A_2', commutative=True)), Pow(Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('A_2', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('W')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('A_2', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('S', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2)))), Symbol('A_2', commutative=True)))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Pow(Integral(Function('W')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Pow(Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('S', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2)))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} = \\int (- \\dot{\\mathbf{r}} + \\dot{z}) d\\dot{z}, then derive \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} = - \\dot{\\mathbf{r}} \\dot{z} + \\frac{\\dot{z}^{2}}{2} + \\rho, then obtain - \\dot{\\mathbf{r}} \\dot{z} + \\frac{\\dot{z}^{2}}{2} + \\rho - 1 = \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} - 1", "derivation": "\\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} = \\int (- \\dot{\\mathbf{r}} + \\dot{z}) d\\dot{z} and \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} - 1 = \\int (- \\dot{\\mathbf{r}} + \\dot{z}) d\\dot{z} - 1 and \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} = - \\dot{\\mathbf{r}} \\dot{z} + \\frac{\\dot{z}^{2}}{2} + \\rho and - \\dot{\\mathbf{r}} \\dot{z} + \\frac{\\dot{z}^{2}}{2} + \\rho - 1 = \\int (- \\dot{\\mathbf{r}} + \\dot{z}) d\\dot{z} - 1 and - \\dot{\\mathbf{r}} \\dot{z} + \\frac{\\dot{z}^{2}}{2} + \\rho - 1 = \\operatorname{z^{*}}{(\\dot{\\mathbf{r}},\\dot{z})} - 1", "srepr_derivation": [["get_premise", "Equality(Function('z^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 1, 1], "Equality(Add(Function('z^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 1], "Equality(Function('z^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Function('z^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}{(\\mu_0)} = e^{\\mu_0}, then derive \\int \\hat{x}{(\\mu_0)} d\\mu_0 = \\phi_2 + e^{\\mu_0}, then obtain (\\phi_2 - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}} + e^{\\mu_0})^{2} = (\\phi_2 + \\hat{x}{(\\mu_0)} - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}})^{2}", "derivation": "\\hat{x}{(\\mu_0)} = e^{\\mu_0} and \\int \\hat{x}{(\\mu_0)} d\\mu_0 = \\int e^{\\mu_0} d\\mu_0 and \\int \\hat{x}{(\\mu_0)} d\\mu_0 = \\phi_2 + e^{\\mu_0} and \\int \\hat{x}{(\\mu_0)} d\\mu_0 = \\phi_2 + \\hat{x}{(\\mu_0)} and \\phi_2 + e^{\\mu_0} = \\phi_2 + \\hat{x}{(\\mu_0)} and \\phi_2 - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}} + e^{\\mu_0} = \\phi_2 + \\hat{x}{(\\mu_0)} - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}} and (\\phi_2 - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}} + e^{\\mu_0})^{2} = (\\phi_2 + \\hat{x}{(\\mu_0)} - (e^{V_{\\mathbf{E}}^{J_{\\varepsilon}}})^{2 J_{\\varepsilon}})^{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 5, "Pow(exp(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(exp(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)))), exp(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(exp(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True))))))"], [["power", 6, 2], "Equality(Pow(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(exp(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)))), exp(Symbol('\\\\mu_0', commutative=True))), Integer(2)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(exp(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True))))), Integer(2)))"]]}, {"prompt": "Given \\rho{(\\eta^{\\prime})} = \\frac{1}{\\eta^{\\prime}}, then obtain \\frac{d}{d \\eta^{\\prime}} \\rho^{16}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\frac{\\rho^{8}{(\\eta^{\\prime})}}{(\\eta^{\\prime})^{8}}", "derivation": "\\rho{(\\eta^{\\prime})} = \\frac{1}{\\eta^{\\prime}} and \\rho^{2}{(\\eta^{\\prime})} = \\frac{\\rho{(\\eta^{\\prime})}}{\\eta^{\\prime}} and \\rho^{4}{(\\eta^{\\prime})} = \\frac{\\rho^{2}{(\\eta^{\\prime})}}{(\\eta^{\\prime})^{2}} and \\rho^{16}{(\\eta^{\\prime})} = \\frac{\\rho^{8}{(\\eta^{\\prime})}}{(\\eta^{\\prime})^{8}} and \\frac{d}{d \\eta^{\\prime}} \\rho^{16}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\frac{\\rho^{8}{(\\eta^{\\prime})}}{(\\eta^{\\prime})^{8}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)))"], [["times", 1, "Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(4)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2)), Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))))"], [["power", 3, 4], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(16)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-8)), Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(8))))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(16)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-8)), Pow(Function('\\\\rho')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(8))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{D},A_{2})} = \\sin{(A_{2} + \\mathbf{D})}, then derive \\frac{\\partial}{\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\cos{(A_{2} + \\mathbf{D})}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial A_{2}} \\sin{(A_{2} + \\mathbf{D})}", "derivation": "\\hat{x}{(\\mathbf{D},A_{2})} = \\sin{(A_{2} + \\mathbf{D})} and \\frac{\\partial}{\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\frac{\\partial}{\\partial A_{2}} \\sin{(A_{2} + \\mathbf{D})} and \\frac{\\partial}{\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\cos{(A_{2} + \\mathbf{D})} and \\frac{\\partial}{\\partial A_{2}} \\sin{(A_{2} + \\mathbf{D})} = \\cos{(A_{2} + \\mathbf{D})} and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\frac{\\partial}{\\partial \\mathbf{D}} \\cos{(A_{2} + \\mathbf{D})} and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial A_{2}} \\hat{x}{(\\mathbf{D},A_{2})} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial A_{2}} \\sin{(A_{2} + \\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{H}_l,\\theta_2)} = - \\hat{H}_l + \\sin{(\\theta_2)}, then obtain - \\sin{(\\mathbb{I}{(\\hat{H}_l,\\theta_2)})} = \\sin{(\\hat{H}_l - \\sin{(\\theta_2)})}", "derivation": "\\mathbb{I}{(\\hat{H}_l,\\theta_2)} = - \\hat{H}_l + \\sin{(\\theta_2)} and - \\hat{H}_l \\mathbb{I}{(\\hat{H}_l,\\theta_2)} = - \\hat{H}_l (- \\hat{H}_l + \\sin{(\\theta_2)}) and - \\mathbb{I}{(\\hat{H}_l,\\theta_2)} = \\hat{H}_l - \\sin{(\\theta_2)} and - \\sin{(\\mathbb{I}{(\\hat{H}_l,\\theta_2)})} = \\sin{(\\hat{H}_l - \\sin{(\\theta_2)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))))"], [["divide", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\theta_2', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)))), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(P_{e},M)} = M + P_{e}, then obtain - \\operatorname{v_{z}}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} \\operatorname{v_{z}}{(P_{e},M)} = 1 - \\operatorname{v_{z}}{(P_{e},M)}", "derivation": "\\operatorname{v_{z}}{(P_{e},M)} = M + P_{e} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{v_{z}}{(P_{e},M)} = \\frac{\\partial}{\\partial P_{e}} (M + P_{e}) and - \\operatorname{v_{z}}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} \\operatorname{v_{z}}{(P_{e},M)} = - \\operatorname{v_{z}}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} (M + P_{e}) and - \\operatorname{v_{z}}{(P_{e},M)} + \\frac{\\partial}{\\partial P_{e}} \\operatorname{v_{z}}{(P_{e},M)} = 1 - \\operatorname{v_{z}}{(P_{e},M)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Add(Symbol('M', commutative=True), Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["minus", 2, "Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True))), Derivative(Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True))), Derivative(Add(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True))), Derivative(Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('v_z')(Symbol('P_e', commutative=True), Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(Z,C_{1},P_{g})} = C_{1} P_{g} + Z, then obtain \\frac{1}{C_{1} P_{g} (C_{1} P_{g} + P_{g} + Z - 1)} = \\frac{1}{C_{1} P_{g} (P_{g} + \\operatorname{A_{x}}{(Z,C_{1},P_{g})} - 1)}", "derivation": "\\operatorname{A_{x}}{(Z,C_{1},P_{g})} = C_{1} P_{g} + Z and P_{g} + \\operatorname{A_{x}}{(Z,C_{1},P_{g})} = C_{1} P_{g} + P_{g} + Z and P_{g} + \\operatorname{A_{x}}{(Z,C_{1},P_{g})} - 1 = C_{1} P_{g} + P_{g} + Z - 1 and \\frac{P_{g} + \\operatorname{A_{x}}{(Z,C_{1},P_{g})} - 1}{C_{1} P_{g} + P_{g} + Z - 1} = 1 and \\frac{1}{C_{1} P_{g} (C_{1} P_{g} + P_{g} + Z - 1)} = \\frac{1}{C_{1} P_{g} (P_{g} + \\operatorname{A_{x}}{(Z,C_{1},P_{g})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('Z', commutative=True)))"], [["add", 1, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True))), Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('P_g', commutative=True), Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Integer(-1)), Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True), Symbol('Z', commutative=True), Integer(-1)))"], [["divide", 3, "Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True), Symbol('Z', commutative=True), Integer(-1))"], "Equality(Mul(Add(Symbol('P_g', commutative=True), Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True), Symbol('Z', commutative=True), Integer(-1)), Integer(-1))), Integer(1))"], [["divide", 4, "Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True), Add(Symbol('P_g', commutative=True), Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True), Symbol('Z', commutative=True), Integer(-1)), Integer(-1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Add(Symbol('P_g', commutative=True), Function('A_x')(Symbol('Z', commutative=True), Symbol('C_1', commutative=True), Symbol('P_g', commutative=True)), Integer(-1)), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(Z,\\phi_2)} = - Z + \\cos{(\\phi_2)}, then derive \\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)} = - \\sin{(\\phi_2)}, then obtain - Z - \\cos{(\\phi_2)} + \\frac{\\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)}}{- Z + \\cos{(\\phi_2)}} = - Z - \\cos{(\\phi_2)} - \\frac{\\sin{(\\phi_2)}}{- Z + \\cos{(\\phi_2)}}", "derivation": "\\Psi_{\\lambda}{(Z,\\phi_2)} = - Z + \\cos{(\\phi_2)} and \\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)} = \\frac{\\partial}{\\partial \\phi_2} (- Z + \\cos{(\\phi_2)}) and \\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)} = - \\sin{(\\phi_2)} and \\frac{\\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)}}{- Z + \\cos{(\\phi_2)}} = - \\frac{\\sin{(\\phi_2)}}{- Z + \\cos{(\\phi_2)}} and - Z + \\frac{\\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)}}{- Z + \\cos{(\\phi_2)}} = - Z - \\frac{\\sin{(\\phi_2)}}{- Z + \\cos{(\\phi_2)}} and - Z - \\cos{(\\phi_2)} + \\frac{\\frac{\\partial}{\\partial \\phi_2} \\Psi_{\\lambda}{(Z,\\phi_2)}}{- Z + \\cos{(\\phi_2)}} = - Z - \\cos{(\\phi_2)} - \\frac{\\sin{(\\phi_2)}}{- Z + \\cos{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 5, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given l{(y,F_{N})} = \\log{(y)}^{F_{N}}, then obtain - l^{y}{(y,F_{N})} + l^{y}{(y,F_{N})} \\log{(y)}^{- F_{N}} = (\\log{(y)}^{F_{N}})^{y} \\log{(y)}^{- F_{N}} - l^{y}{(y,F_{N})}", "derivation": "l{(y,F_{N})} = \\log{(y)}^{F_{N}} and l^{y}{(y,F_{N})} = (\\log{(y)}^{F_{N}})^{y} and - l^{y}{(y,F_{N})} = - (\\log{(y)}^{F_{N}})^{y} and l^{y}{(y,F_{N})} \\log{(y)}^{- F_{N}} = (\\log{(y)}^{F_{N}})^{y} \\log{(y)}^{- F_{N}} and - l^{y}{(y,F_{N})} + l^{y}{(y,F_{N})} \\log{(y)}^{- F_{N}} = (\\log{(y)}^{F_{N}})^{y} \\log{(y)}^{- F_{N}} - l^{y}{(y,F_{N})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)), Symbol('y', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)))"], "Equality(Mul(Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Pow(Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))))"], [["minus", 4, "Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True))), Mul(Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True))))), Add(Mul(Pow(Pow(log(Symbol('y', commutative=True)), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Integer(-1), Pow(Function('l')(Symbol('y', commutative=True), Symbol('F_N', commutative=True)), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given S{(v_{x},\\mathbf{B})} = \\mathbf{B} v_{x}, then obtain \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\mathbf{B}} \\iint S{(v_{x},\\mathbf{B})} dv_{x} dv_{x} = \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\mathbf{B}} \\iint \\mathbf{B} v_{x} dv_{x} dv_{x}", "derivation": "S{(v_{x},\\mathbf{B})} = \\mathbf{B} v_{x} and \\int S{(v_{x},\\mathbf{B})} dv_{x} = \\int \\mathbf{B} v_{x} dv_{x} and \\iint S{(v_{x},\\mathbf{B})} dv_{x} dv_{x} = \\iint \\mathbf{B} v_{x} dv_{x} dv_{x} and \\frac{\\partial}{\\partial \\mathbf{B}} \\iint S{(v_{x},\\mathbf{B})} dv_{x} dv_{x} = \\frac{\\partial}{\\partial \\mathbf{B}} \\iint \\mathbf{B} v_{x} dv_{x} dv_{x} and \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\mathbf{B}} \\iint S{(v_{x},\\mathbf{B})} dv_{x} dv_{x} = \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\mathbf{B}} \\iint \\mathbf{B} v_{x} dv_{x} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('S')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('S')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(V)} = \\sin{(V)}, then obtain \\mathbf{A}^{V}{(V)} \\sin^{- V}{(V)} + \\sin{(V)} + 1 = \\sin{(V)} + 2", "derivation": "\\mathbf{A}{(V)} = \\sin{(V)} and \\mathbf{A}^{V}{(V)} = \\sin^{V}{(V)} and \\mathbf{A}^{V}{(V)} \\sin^{- V}{(V)} = 1 and \\mathbf{A}^{V}{(V)} \\sin^{- V}{(V)} + \\sin{(V)} = \\sin{(V)} + 1 and \\mathbf{A}^{V}{(V)} \\sin^{- V}{(V)} + \\sin{(V)} + 1 = \\sin{(V)} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["divide", 2, "Pow(sin(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), Integer(1))"], [["add", 3, "sin(Symbol('V', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), sin(Symbol('V', commutative=True))), Add(sin(Symbol('V', commutative=True)), Integer(1)))"], [["add", 4, 1], "Equality(Add(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), sin(Symbol('V', commutative=True)), Integer(1)), Add(sin(Symbol('V', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A_{x})} = \\cos{(A_{x})}, then obtain - \\Psi_{\\lambda}^{A_{x}}{(A_{x})} + \\cos^{A_{x}}{(A_{x})} = 0", "derivation": "\\Psi_{\\lambda}{(A_{x})} = \\cos{(A_{x})} and \\Psi_{\\lambda}^{A_{x}}{(A_{x})} = \\cos^{A_{x}}{(A_{x})} and \\Psi_{\\lambda}^{A_{x}}{(A_{x})} - \\cos^{A_{x}}{(A_{x})} = 0 and - \\Psi_{\\lambda}^{A_{x}}{(A_{x})} + \\cos^{A_{x}}{(A_{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(cos(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["minus", 2, "Pow(cos(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], "Equality(Add(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))), Integer(0))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Pow(cos(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\chi)} = \\log{(\\chi)}, then obtain (\\frac{\\int \\operatorname{C_{1}}{(\\chi)} d\\chi}{\\chi + \\log{(\\chi)}})^{\\chi} = (\\frac{\\int \\log{(\\chi)} d\\chi}{\\chi + \\log{(\\chi)}})^{\\chi}", "derivation": "\\operatorname{C_{1}}{(\\chi)} = \\log{(\\chi)} and \\chi + \\operatorname{C_{1}}{(\\chi)} = \\chi + \\log{(\\chi)} and \\int \\operatorname{C_{1}}{(\\chi)} d\\chi = \\int \\log{(\\chi)} d\\chi and \\frac{\\int \\operatorname{C_{1}}{(\\chi)} d\\chi}{\\chi + \\operatorname{C_{1}}{(\\chi)}} = \\frac{\\int \\log{(\\chi)} d\\chi}{\\chi + \\operatorname{C_{1}}{(\\chi)}} and (\\frac{\\int \\operatorname{C_{1}}{(\\chi)} d\\chi}{\\chi + \\operatorname{C_{1}}{(\\chi)}})^{\\chi} = (\\frac{\\int \\log{(\\chi)} d\\chi}{\\chi + \\operatorname{C_{1}}{(\\chi)}})^{\\chi} and (\\frac{\\int \\operatorname{C_{1}}{(\\chi)} d\\chi}{\\chi + \\log{(\\chi)}})^{\\chi} = (\\frac{\\int \\log{(\\chi)} d\\chi}{\\chi + \\log{(\\chi)}})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('C_1')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('C_1')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Function('C_1')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('C_1')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given x{(\\mathbf{p})} = \\sin{(\\log{(\\mathbf{p})})}, then obtain - (\\mathbf{p} + x{(\\mathbf{p})})^{\\mathbf{p}} + x{(\\mathbf{p})} + \\log{(\\mathbf{p})} = - (\\mathbf{p} + x{(\\mathbf{p})})^{\\mathbf{p}} + \\log{(\\mathbf{p})} + \\sin{(\\log{(\\mathbf{p})})}", "derivation": "x{(\\mathbf{p})} = \\sin{(\\log{(\\mathbf{p})})} and x{(\\mathbf{p})} + \\log{(\\mathbf{p})} = \\log{(\\mathbf{p})} + \\sin{(\\log{(\\mathbf{p})})} and \\mathbf{p} + x{(\\mathbf{p})} = \\mathbf{p} + \\sin{(\\log{(\\mathbf{p})})} and - (\\mathbf{p} + \\sin{(\\log{(\\mathbf{p})})})^{\\mathbf{p}} + x{(\\mathbf{p})} + \\log{(\\mathbf{p})} = - (\\mathbf{p} + \\sin{(\\log{(\\mathbf{p})})})^{\\mathbf{p}} + \\log{(\\mathbf{p})} + \\sin{(\\log{(\\mathbf{p})})} and - (\\mathbf{p} + x{(\\mathbf{p})})^{\\mathbf{p}} + x{(\\mathbf{p})} + \\log{(\\mathbf{p})} = - (\\mathbf{p} + x{(\\mathbf{p})})^{\\mathbf{p}} + \\log{(\\mathbf{p})} + \\sin{(\\log{(\\mathbf{p})})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{p}', commutative=True)), sin(log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 1, "log(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Add(log(Symbol('\\\\mathbf{p}', commutative=True)), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["add", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('x')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\mathbf{p}', commutative=True), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 2, "Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True))), Function('x')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True)), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('x')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))), Function('x')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('x')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True)), sin(log(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(u,z)} = \\sin{(\\frac{u}{z})}, then obtain (\\int \\operatorname{y^{\\prime}}{(u,z)} du) \\iint \\operatorname{y^{\\prime}}{(u,z)} du du = (\\int \\operatorname{y^{\\prime}}{(u,z)} du) \\iint \\sin{(\\frac{u}{z})} du du", "derivation": "\\operatorname{y^{\\prime}}{(u,z)} = \\sin{(\\frac{u}{z})} and \\int \\operatorname{y^{\\prime}}{(u,z)} du = \\int \\sin{(\\frac{u}{z})} du and \\iint \\operatorname{y^{\\prime}}{(u,z)} du du = \\iint \\sin{(\\frac{u}{z})} du du and (\\int \\operatorname{y^{\\prime}}{(u,z)} du) \\iint \\operatorname{y^{\\prime}}{(u,z)} du du = (\\int \\operatorname{y^{\\prime}}{(u,z)} du) \\iint \\sin{(\\frac{u}{z})} du du", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), sin(Mul(Symbol('u', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(sin(Mul(Symbol('u', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))), Tuple(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(sin(Mul(Symbol('u', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Mul(Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(sin(Mul(Symbol('u', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(n_{1},\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}} n_{1}}, then derive \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{B}{(n_{1},\\hat{\\mathbf{r}})} = n_{1} e^{\\hat{\\mathbf{r}} n_{1}}, then obtain \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} e^{\\hat{\\mathbf{r}} n_{1}} = n_{1} e^{\\hat{\\mathbf{r}} n_{1}}", "derivation": "\\mathbf{B}{(n_{1},\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}} n_{1}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{B}{(n_{1},\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} e^{\\hat{\\mathbf{r}} n_{1}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{B}{(n_{1},\\hat{\\mathbf{r}})} = n_{1} e^{\\hat{\\mathbf{r}} n_{1}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} e^{\\hat{\\mathbf{r}} n_{1}} = n_{1} e^{\\hat{\\mathbf{r}} n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('n_1', commutative=True), exp(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(r,\\phi)} = \\phi r, then derive \\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)} = \\phi, then derive \\phi = \\frac{\\phi^{2}}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}}, then obtain \\phi + \\frac{\\phi}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} = \\frac{\\phi^{2}}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} + \\frac{\\phi}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}}", "derivation": "\\mathbf{r}{(r,\\phi)} = \\phi r and \\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)} = \\frac{\\partial}{\\partial r} \\phi r and \\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)} = \\phi and 1 = \\frac{\\phi}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} and \\frac{\\partial}{\\partial r} \\phi r = \\frac{\\phi \\frac{\\partial}{\\partial r} \\phi r}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} and \\phi = \\frac{\\phi^{2}}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} and \\phi + \\frac{\\phi}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} = \\frac{\\phi^{2}}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}} + \\frac{\\phi}{\\frac{\\partial}{\\partial r} \\mathbf{r}{(r,\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], [["divide", 3, "Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Symbol('\\\\phi', commutative=True), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))))"], [["times", 4, "Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Symbol('\\\\phi', commutative=True), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Symbol('\\\\phi', commutative=True), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))))"], [["add", 6, "Mul(Symbol('\\\\phi', commutative=True), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('\\\\phi', commutative=True), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\varepsilon_0)} = \\int \\log{(\\varepsilon_0)} d\\varepsilon_0, then derive \\mathbf{J}^{\\varepsilon_0}{(\\varepsilon_0)} = (\\Psi^{\\dagger} + \\varepsilon_0 \\log{(\\varepsilon_0)} - \\varepsilon_0)^{\\varepsilon_0}, then obtain (\\int \\log{(\\varepsilon_0)} d\\varepsilon_0)^{\\varepsilon_0} = (\\Psi^{\\dagger} + \\varepsilon_0 \\log{(\\varepsilon_0)} - \\varepsilon_0)^{\\varepsilon_0}", "derivation": "\\mathbf{J}{(\\varepsilon_0)} = \\int \\log{(\\varepsilon_0)} d\\varepsilon_0 and \\mathbf{J}^{\\varepsilon_0}{(\\varepsilon_0)} = (\\int \\log{(\\varepsilon_0)} d\\varepsilon_0)^{\\varepsilon_0} and \\mathbf{J}^{\\varepsilon_0}{(\\varepsilon_0)} = (\\Psi^{\\dagger} + \\varepsilon_0 \\log{(\\varepsilon_0)} - \\varepsilon_0)^{\\varepsilon_0} and (\\int \\log{(\\varepsilon_0)} d\\varepsilon_0)^{\\varepsilon_0} = (\\Psi^{\\dagger} + \\varepsilon_0 \\log{(\\varepsilon_0)} - \\varepsilon_0)^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\varepsilon_0', commutative=True)), Integral(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(log(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{s},A_{x})} = \\sin{(\\frac{\\mathbf{s}}{A_{x}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{s}} - \\rho_{b}^{A_{x}}{(\\mathbf{s},A_{x})} = \\frac{\\partial}{\\partial \\mathbf{s}} - \\sin^{A_{x}}{(\\frac{\\mathbf{s}}{A_{x}})}", "derivation": "\\rho_{b}{(\\mathbf{s},A_{x})} = \\sin{(\\frac{\\mathbf{s}}{A_{x}})} and \\rho_{b}^{A_{x}}{(\\mathbf{s},A_{x})} = \\sin^{A_{x}}{(\\frac{\\mathbf{s}}{A_{x}})} and - \\rho_{b}^{A_{x}}{(\\mathbf{s},A_{x})} = - \\sin^{A_{x}}{(\\frac{\\mathbf{s}}{A_{x}})} and \\frac{\\partial}{\\partial \\mathbf{s}} - \\rho_{b}^{A_{x}}{(\\mathbf{s},A_{x})} = \\frac{\\partial}{\\partial \\mathbf{s}} - \\sin^{A_{x}}{(\\frac{\\mathbf{s}}{A_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('A_x', commutative=True)), sin(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(sin(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('A_x', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Mul(Integer(-1), Pow(sin(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('A_x', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(sin(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('A_x', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(F_{g},\\hat{H}_{\\lambda})} = F_{g}^{\\hat{H}_{\\lambda}} and \\operatorname{t_{1}}{(F_{g},\\hat{H}_{\\lambda})} = - \\rho{(F_{g},\\hat{H}_{\\lambda})}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} - \\rho{(F_{g},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} - F_{g}^{\\hat{H}_{\\lambda}}", "derivation": "\\rho{(F_{g},\\hat{H}_{\\lambda})} = F_{g}^{\\hat{H}_{\\lambda}} and \\operatorname{t_{1}}{(F_{g},\\hat{H}_{\\lambda})} = - \\rho{(F_{g},\\hat{H}_{\\lambda})} and \\operatorname{t_{1}}{(F_{g},\\hat{H}_{\\lambda})} = - F_{g}^{\\hat{H}_{\\lambda}} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\operatorname{t_{1}}{(F_{g},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} - F_{g}^{\\hat{H}_{\\lambda}} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} - \\rho{(F_{g},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} - F_{g}^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('\\\\rho')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('t_1')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Integer(-1), Function('\\\\rho')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\hbar)} = \\log{(\\hbar)}, then derive \\hbar \\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar = \\hbar \\int \\log{(\\hbar)} d\\hbar, then derive \\hbar \\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar = \\hbar (E_{x} + \\hbar \\log{(\\hbar)} - \\hbar), then obtain \\hbar \\int \\log{(\\hbar)} d\\hbar = \\hbar (E_{x} + \\hbar \\log{(\\hbar)} - \\hbar)", "derivation": "\\operatorname{A_{x}}{(\\hbar)} = \\log{(\\hbar)} and \\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar = \\int \\log{(\\hbar)} d\\hbar and \\frac{\\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar}{\\frac{d}{d \\hbar} \\log{(\\hbar)}} = \\frac{\\int \\log{(\\hbar)} d\\hbar}{\\frac{d}{d \\hbar} \\log{(\\hbar)}} and \\hbar \\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar = \\hbar \\int \\log{(\\hbar)} d\\hbar and \\hbar \\int \\operatorname{A_{x}}{(\\hbar)} d\\hbar = \\hbar (E_{x} + \\hbar \\log{(\\hbar)} - \\hbar) and \\hbar \\int \\log{(\\hbar)} d\\hbar = \\hbar (E_{x} + \\hbar \\log{(\\hbar)} - \\hbar)", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["divide", 2, "Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Integral(Function('A_x')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Integral(Function('A_x')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('\\\\hbar', commutative=True), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Integral(Function('A_x')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('\\\\hbar', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('\\\\hbar', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(m)} = \\log{(m)}, then obtain \\frac{\\int \\operatorname{v_{z}}{(m)} \\log{(m)} dm}{\\log{(m)}^{2}} = \\frac{\\int \\log{(m)}^{2} dm}{\\log{(m)}^{2}}", "derivation": "\\operatorname{v_{z}}{(m)} = \\log{(m)} and \\operatorname{v_{z}}{(m)} \\log{(m)} = \\log{(m)}^{2} and \\int \\operatorname{v_{z}}{(m)} \\log{(m)} dm = \\int \\log{(m)}^{2} dm and \\frac{\\int \\operatorname{v_{z}}{(m)} \\log{(m)} dm}{\\operatorname{v_{z}}{(m)} \\log{(m)}} = \\frac{\\int \\log{(m)}^{2} dm}{\\operatorname{v_{z}}{(m)} \\log{(m)}} and \\frac{\\int \\operatorname{v_{z}}{(m)} \\log{(m)} dm}{\\log{(m)}^{2}} = \\frac{\\int \\log{(m)}^{2} dm}{\\log{(m)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["times", 1, "log(Symbol('m', commutative=True))"], "Equality(Mul(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Pow(log(Symbol('m', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Pow(log(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True))))"], [["divide", 3, "Mul(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Function('v_z')(Symbol('m', commutative=True)), Integer(-1)), Pow(log(Symbol('m', commutative=True)), Integer(-1)), Integral(Mul(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Function('v_z')(Symbol('m', commutative=True)), Integer(-1)), Pow(log(Symbol('m', commutative=True)), Integer(-1)), Integral(Pow(log(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(log(Symbol('m', commutative=True)), Integer(-2)), Integral(Mul(Function('v_z')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Pow(log(Symbol('m', commutative=True)), Integer(-2)), Integral(Pow(log(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(E,J)} = E e^{J} and L{(E,J)} = \\int \\mathbf{H}{(E,J)} dE, then obtain (L{(E,J)} + \\mathbf{H}{(E,J)} - \\cos{(l)}) \\int \\mathbf{H}{(E,J)} dE = (L{(E,J)} + \\mathbf{H}{(E,J)} - \\cos{(l)}) L{(E,J)}", "derivation": "\\mathbf{H}{(E,J)} = E e^{J} and \\int \\mathbf{H}{(E,J)} dE = \\int E e^{J} dE and (\\mathbf{H}{(E,J)} - \\cos{(l)} + \\int E e^{J} dE) \\int \\mathbf{H}{(E,J)} dE = (\\mathbf{H}{(E,J)} - \\cos{(l)} + \\int E e^{J} dE) \\int E e^{J} dE and L{(E,J)} = \\int \\mathbf{H}{(E,J)} dE and L{(E,J)} = \\int E e^{J} dE and (L{(E,J)} + \\mathbf{H}{(E,J)} - \\cos{(l)}) \\int \\mathbf{H}{(E,J)} dE = (L{(E,J)} + \\mathbf{H}{(E,J)} - \\cos{(l)}) L{(E,J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["times", 2, "Add(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True))), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True))), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True)))), Integral(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Add(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True))), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True)))), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], ["renaming_premise", "Equality(Function('L')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('L')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integral(Mul(Symbol('E', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Add(Function('L')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)))), Integral(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Add(Function('L')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)))), Function('L')(Symbol('E', commutative=True), Symbol('J', commutative=True))))"]]}, {"prompt": "Given u{(\\nabla,k)} = \\frac{k}{\\nabla}, then obtain (k + u{(\\nabla,k)} - \\int (k + u{(\\nabla,k)}) dk)^{\\nabla} = (k - \\int (k + u{(\\nabla,k)}) dk + \\frac{k}{\\nabla})^{\\nabla}", "derivation": "u{(\\nabla,k)} = \\frac{k}{\\nabla} and k + u{(\\nabla,k)} = k + \\frac{k}{\\nabla} and \\int (k + u{(\\nabla,k)}) dk = \\int (k + \\frac{k}{\\nabla}) dk and k + u{(\\nabla,k)} - \\int (k + \\frac{k}{\\nabla}) dk = k - \\int (k + \\frac{k}{\\nabla}) dk + \\frac{k}{\\nabla} and (k + u{(\\nabla,k)} - \\int (k + \\frac{k}{\\nabla}) dk)^{\\nabla} = (k - \\int (k + \\frac{k}{\\nabla}) dk + \\frac{k}{\\nabla})^{\\nabla} and (k + u{(\\nabla,k)} - \\int (k + u{(\\nabla,k)}) dk)^{\\nabla} = (k - \\int (k + u{(\\nabla,k)}) dk + \\frac{k}{\\nabla})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True)))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True))), Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))), Add(Symbol('k', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))))"], [["power", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given E{(\\hat{p})} = \\cos{(\\hat{p})}, then derive \\frac{d}{d \\hat{p}} E{(\\hat{p})} = - \\sin{(\\hat{p})}, then obtain (\\frac{d}{d \\hat{p}} E{(\\hat{p})})^{\\hat{p}} = (- \\sin{(\\hat{p})})^{\\hat{p}}", "derivation": "E{(\\hat{p})} = \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} E{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} E{(\\hat{p})} = - \\sin{(\\hat{p})} and (\\frac{d}{d \\hat{p}} E{(\\hat{p})})^{\\hat{p}} = (- \\sin{(\\hat{p})})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Derivative(Function('E')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(z^{*})} = e^{z^{*}} and L{(z^{*})} = - \\operatorname{f^{*}}{(z^{*})}, then obtain \\int - \\operatorname{f^{*}}{(z^{*})} dz^{*} = \\int - e^{z^{*}} dz^{*}", "derivation": "\\operatorname{f^{*}}{(z^{*})} = e^{z^{*}} and - \\operatorname{f^{*}}{(z^{*})} = - e^{z^{*}} and L{(z^{*})} = - \\operatorname{f^{*}}{(z^{*})} and L{(z^{*})} = - e^{z^{*}} and \\int L{(z^{*})} dz^{*} = \\int - e^{z^{*}} dz^{*} and \\int - \\operatorname{f^{*}}{(z^{*})} dz^{*} = \\int - e^{z^{*}} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('z^*', commutative=True)), exp(Symbol('z^*', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f^*')(Symbol('z^*', commutative=True))), Mul(Integer(-1), exp(Symbol('z^*', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('f^*')(Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('L')(Symbol('z^*', commutative=True)), Mul(Integer(-1), exp(Symbol('z^*', commutative=True))))"], [["integrate", 4, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('L')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Mul(Integer(-1), Function('f^*')(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\tilde{g}^*)} = \\log{(e^{\\tilde{g}^*})}, then obtain \\frac{2 (\\sigma_{x}{(\\tilde{g}^*)} + \\log{(e^{\\tilde{g}^*})}) \\log{(e^{\\tilde{g}^*})}}{\\sigma_{x}{(\\tilde{g}^*)}} = \\frac{4 \\log{(e^{\\tilde{g}^*})}^{2}}{\\sigma_{x}{(\\tilde{g}^*)}}", "derivation": "\\sigma_{x}{(\\tilde{g}^*)} = \\log{(e^{\\tilde{g}^*})} and \\sigma_{x}{(\\tilde{g}^*)} + \\log{(e^{\\tilde{g}^*})} = 2 \\log{(e^{\\tilde{g}^*})} and 2 (\\sigma_{x}{(\\tilde{g}^*)} + \\log{(e^{\\tilde{g}^*})}) \\log{(e^{\\tilde{g}^*})} = 4 \\log{(e^{\\tilde{g}^*})}^{2} and \\frac{2 (\\sigma_{x}{(\\tilde{g}^*)} + \\log{(e^{\\tilde{g}^*})}) \\log{(e^{\\tilde{g}^*})}}{\\sigma_{x}{(\\tilde{g}^*)}} = \\frac{4 \\log{(e^{\\tilde{g}^*})}^{2}}{\\sigma_{x}{(\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 1, "log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Integer(2), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["times", 2, "Mul(Integer(2), log(exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Mul(Integer(2), Add(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Integer(4), Pow(log(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2))))"], [["divide", 3, "Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Integer(2), Add(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Pow(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Integer(4), Pow(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Pow(log(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\eta)} = \\cos{(\\log{(\\eta)})}, then obtain \\frac{d}{d \\eta} (\\eta + \\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)}) = \\frac{d}{d \\eta} (\\eta + \\frac{d}{d \\eta} \\cos{(\\log{(\\eta)})})", "derivation": "\\dot{\\mathbf{r}}{(\\eta)} = \\cos{(\\log{(\\eta)})} and \\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\log{(\\eta)})} and \\eta + \\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)} = \\eta + \\frac{d}{d \\eta} \\cos{(\\log{(\\eta)})} and \\frac{d}{d \\eta} (\\eta + \\frac{d}{d \\eta} \\dot{\\mathbf{r}}{(\\eta)}) = \\frac{d}{d \\eta} (\\eta + \\frac{d}{d \\eta} \\cos{(\\log{(\\eta)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), cos(log(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Symbol('\\\\eta', commutative=True), Derivative(cos(log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\eta', commutative=True), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Derivative(cos(log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(\\pi)} = \\log{(\\pi)}, then derive \\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi = \\mathbf{r} - \\frac{3 \\pi^{2}}{4} - \\pi + (\\frac{\\pi^{2}}{2} + \\pi) \\log{(\\pi)}, then obtain 0 = (\\mathbf{r} - \\frac{3 \\pi^{2}}{4} - \\pi + (\\frac{\\pi^{2}}{2} + \\pi) \\log{(\\pi)})^{\\mathbf{r}} - (\\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi)^{\\mathbf{r}}", "derivation": "\\chi{(\\pi)} = \\log{(\\pi)} and \\int \\chi{(\\pi)} d\\pi = \\int \\log{(\\pi)} d\\pi and \\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi = \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi and \\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi = \\int (\\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi) d\\pi and \\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi = \\mathbf{r} - \\frac{3 \\pi^{2}}{4} - \\pi + (\\frac{\\pi^{2}}{2} + \\pi) \\log{(\\pi)} and (\\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi)^{\\mathbf{r}} = (\\mathbf{r} - \\frac{3 \\pi^{2}}{4} - \\pi + (\\frac{\\pi^{2}}{2} + \\pi) \\log{(\\pi)})^{\\mathbf{r}} and 0 = (\\mathbf{r} - \\frac{3 \\pi^{2}}{4} - \\pi + (\\frac{\\pi^{2}}{2} + \\pi) \\log{(\\pi)})^{\\mathbf{r}} - (\\int (\\log{(\\pi)} + \\int \\chi{(\\pi)} d\\pi) d\\pi)^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "log(Symbol('\\\\pi', commutative=True))"], "Equality(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(log(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Rational(3, 4), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Rational(3, 4), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 6, "Pow(Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Rational(3, 4), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(log(Symbol('\\\\pi', commutative=True)), Integral(Function('\\\\chi')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{A})} = \\mathbf{A}, then obtain 1 = (\\frac{\\log{((- \\mathbf{A})^{\\mathbf{A}})}}{\\log{((- \\operatorname{v_{z}}{(\\mathbf{A})})^{\\mathbf{A}})}})^{\\mathbf{A}}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{A})} = \\mathbf{A} and - \\operatorname{v_{z}}{(\\mathbf{A})} = - \\mathbf{A} and (- \\operatorname{v_{z}}{(\\mathbf{A})})^{\\mathbf{A}} = (- \\mathbf{A})^{\\mathbf{A}} and \\log{((- \\operatorname{v_{z}}{(\\mathbf{A})})^{\\mathbf{A}})} = \\log{((- \\mathbf{A})^{\\mathbf{A}})} and 1 = \\frac{\\log{((- \\mathbf{A})^{\\mathbf{A}})}}{\\log{((- \\operatorname{v_{z}}{(\\mathbf{A})})^{\\mathbf{A}})}} and 1 = (\\frac{\\log{((- \\mathbf{A})^{\\mathbf{A}})}}{\\log{((- \\operatorname{v_{z}}{(\\mathbf{A})})^{\\mathbf{A}})}})^{\\mathbf{A}}", "srepr_derivation": [["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["log", 3], "Equality(log(Pow(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))), log(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 4, "log(Pow(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(1), Mul(log(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Pow(log(Pow(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"], [["power", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integer(1), Pow(Mul(log(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Pow(log(Pow(Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\phi_2,\\theta_1)} = - \\theta_1 + \\sin{(\\phi_2)}, then obtain \\int \\frac{(- \\theta_1 + \\varepsilon_{0}{(\\phi_2,\\theta_1)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} d\\phi_2 = \\int \\frac{(- 2 \\theta_1 + \\sin{(\\phi_2)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} d\\phi_2", "derivation": "\\varepsilon_{0}{(\\phi_2,\\theta_1)} = - \\theta_1 + \\sin{(\\phi_2)} and - \\theta_1 + \\varepsilon_{0}{(\\phi_2,\\theta_1)} = - 2 \\theta_1 + \\sin{(\\phi_2)} and (- \\theta_1 + \\varepsilon_{0}{(\\phi_2,\\theta_1)}) \\sin{(\\phi_2)} = (- 2 \\theta_1 + \\sin{(\\phi_2)}) \\sin{(\\phi_2)} and \\frac{(- \\theta_1 + \\varepsilon_{0}{(\\phi_2,\\theta_1)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} = \\frac{(- 2 \\theta_1 + \\sin{(\\phi_2)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} and \\int \\frac{(- \\theta_1 + \\varepsilon_{0}{(\\phi_2,\\theta_1)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} d\\phi_2 = \\int \\frac{(- 2 \\theta_1 + \\sin{(\\phi_2)}) \\sin{(\\phi_2)}}{\\varepsilon_{0}{(\\phi_2,\\theta_1)}} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))))"], [["times", 2, "sin(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 3, "Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\mu{(C_{1})} = e^{C_{1}}, then derive \\frac{d}{d C_{1}} \\int \\mu{(C_{1})} dC_{1} = \\frac{\\partial}{\\partial C_{1}} (J + e^{C_{1}}), then obtain \\frac{\\partial}{\\partial C_{1}} (J + e^{C_{1}}) = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1}", "derivation": "\\mu{(C_{1})} = e^{C_{1}} and \\int \\mu{(C_{1})} dC_{1} = \\int e^{C_{1}} dC_{1} and \\frac{d}{d C_{1}} \\int \\mu{(C_{1})} dC_{1} = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1} and \\frac{d}{d C_{1}} \\int \\mu{(C_{1})} dC_{1} = \\frac{\\partial}{\\partial C_{1}} (J + e^{C_{1}}) and \\frac{\\partial}{\\partial C_{1}} (J + e^{C_{1}}) = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mu')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mu')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('J', commutative=True), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\cos{(\\hat{H})} and J{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega}, then derive V{(\\hat{H})} = - \\sin{(\\hat{H})}, then obtain \\frac{\\Omega (\\frac{\\Omega V{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})}) V{(\\hat{H})}}{\\mathbf{H}} = \\frac{\\Omega (- \\frac{\\Omega \\sin{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})}) V{(\\hat{H})}}{\\mathbf{H}}", "derivation": "V{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\cos{(\\hat{H})} and J{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega} and V{(\\hat{H})} = - \\sin{(\\hat{H})} and \\frac{V{(\\hat{H})}}{J{(\\mathbf{H},\\Omega)}} = - \\frac{\\sin{(\\hat{H})}}{J{(\\mathbf{H},\\Omega)}} and \\frac{\\Omega V{(\\hat{H})}}{\\mathbf{H}} = - \\frac{\\Omega \\sin{(\\hat{H})}}{\\mathbf{H}} and \\frac{\\Omega V{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})} = - \\frac{\\Omega \\sin{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})} and \\frac{\\Omega (\\frac{\\Omega V{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})}) V{(\\hat{H})}}{\\mathbf{H}} = \\frac{\\Omega (- \\frac{\\Omega \\sin{(\\hat{H})}}{\\mathbf{H}} + \\cos{(\\hat{H})}) V{(\\hat{H})}}{\\mathbf{H}}", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\hat{H}', commutative=True)), Derivative(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('J')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('V')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 3, "Function('J')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Function('J')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Function('V')(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 5, "cos(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{H}', commutative=True))), cos(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))), cos(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 6, "Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{H}', commutative=True))), cos(Symbol('\\\\hat{H}', commutative=True))), Function('V')(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))), cos(Symbol('\\\\hat{H}', commutative=True))), Function('V')(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(f)} = \\sin{(f)} and \\operatorname{g_{\\varepsilon}}{(f)} = \\frac{\\int \\mathbf{F}{(f)} df}{\\mathbf{F}{(f)}}, then obtain \\frac{\\int \\mathbf{F}{(f)} df}{\\mathbf{F}{(f)}} = \\frac{\\int \\mathbf{F}{(f)} df}{\\sin{(f)}}", "derivation": "\\mathbf{F}{(f)} = \\sin{(f)} and \\int \\mathbf{F}{(f)} df = \\int \\sin{(f)} df and \\frac{\\int \\mathbf{F}{(f)} df}{\\sin{(f)}} = \\frac{\\int \\sin{(f)} df}{\\sin{(f)}} and \\operatorname{g_{\\varepsilon}}{(f)} = \\frac{\\int \\mathbf{F}{(f)} df}{\\mathbf{F}{(f)}} and \\operatorname{g_{\\varepsilon}}{(f)} = \\frac{\\int \\sin{(f)} df}{\\sin{(f)}} and \\operatorname{g_{\\varepsilon}}{(f)} = \\frac{\\int \\mathbf{F}{(f)} df}{\\sin{(f)}} and \\frac{\\int \\mathbf{F}{(f)} df}{\\mathbf{F}{(f)}} = \\frac{\\int \\mathbf{F}{(f)} df}{\\sin{(f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "sin(Symbol('f', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f', commutative=True)), Mul(Pow(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('f', commutative=True)), Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('f', commutative=True)), Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given v{(\\hat{p})} = \\cos{(\\hat{p})}, then derive 2 \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = - \\sin{(\\hat{p})} + \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1, then obtain 2 \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = - \\sin{(\\hat{p})} + \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} - 1", "derivation": "v{(\\hat{p})} = \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} v{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} - 1 and 2 \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = \\frac{d}{d \\hat{p}} v{(\\hat{p})} + \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} - 1 and 2 \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = - \\sin{(\\hat{p})} + \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 and 2 \\frac{d}{d \\hat{p}} v{(\\hat{p})} - 1 = - \\sin{(\\hat{p})} + \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} - 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)))"], [["add", 2, "Add(Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Integer(2), Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(-1)), Add(Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Derivative(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given C{(\\phi)} = \\sin{(\\phi)}, then derive \\frac{\\int C{(\\phi)} d\\phi}{C^{2}{(\\phi)}} = \\frac{\\rho - \\cos{(\\phi)}}{C^{2}{(\\phi)}}, then obtain \\frac{\\rho - \\cos{(\\phi)}}{C^{2}{(\\phi)}} = \\frac{\\int \\sin{(\\phi)} d\\phi}{C^{2}{(\\phi)}}", "derivation": "C{(\\phi)} = \\sin{(\\phi)} and \\int C{(\\phi)} d\\phi = \\int \\sin{(\\phi)} d\\phi and \\frac{\\int C{(\\phi)} d\\phi}{\\sin{(\\phi)}} = \\frac{\\int \\sin{(\\phi)} d\\phi}{\\sin{(\\phi)}} and \\frac{\\int C{(\\phi)} d\\phi}{C{(\\phi)} \\sin{(\\phi)}} = \\frac{\\int \\sin{(\\phi)} d\\phi}{C{(\\phi)} \\sin{(\\phi)}} and \\frac{\\int C{(\\phi)} d\\phi}{C^{2}{(\\phi)}} = \\frac{\\int \\sin{(\\phi)} d\\phi}{C^{2}{(\\phi)}} and \\frac{\\int C{(\\phi)} d\\phi}{C^{2}{(\\phi)}} = \\frac{\\rho - \\cos{(\\phi)}}{C^{2}{(\\phi)}} and \\frac{\\rho - \\cos{(\\phi)}}{C^{2}{(\\phi)}} = \\frac{\\int \\sin{(\\phi)} d\\phi}{C^{2}{(\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Function('C')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["divide", 3, "Function('C')(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Function('C')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2)), Integral(Function('C')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2)), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2)), Integral(Function('C')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2))), Mul(Pow(Function('C')(Symbol('\\\\phi', commutative=True)), Integer(-2)), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(W)} = \\log{(W)}, then obtain 2 \\tilde{g}^*{(W)} - 3 \\log{(W)} - 1 = - \\log{(W)} - 1", "derivation": "\\tilde{g}^*{(W)} = \\log{(W)} and \\tilde{g}^*{(W)} + \\log{(W)} = 2 \\log{(W)} and \\tilde{g}^*{(W)} - \\log{(W)} = 0 and \\tilde{g}^*{(W)} - 2 \\log{(W)} = - \\log{(W)} and 2 \\tilde{g}^*{(W)} - 3 \\log{(W)} = \\tilde{g}^*{(W)} - 2 \\log{(W)} and 2 \\tilde{g}^*{(W)} - 3 \\log{(W)} = - \\log{(W)} and 2 \\tilde{g}^*{(W)} - 3 \\log{(W)} - 1 = - \\log{(W)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["add", 1, "log(Symbol('W', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Mul(Integer(2), log(Symbol('W', commutative=True))))"], [["minus", 2, "Mul(Integer(2), log(Symbol('W', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('W', commutative=True)))), Integer(0))"], [["add", 3, "Mul(Integer(-1), log(Symbol('W', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('W', commutative=True)))), Mul(Integer(-1), log(Symbol('W', commutative=True))))"], [["add", 4, "Add(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('W', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('W', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('W', commutative=True)))), Add(Function('\\\\tilde{g}^*')(Symbol('W', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('W', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('W', commutative=True)))), Mul(Integer(-1), log(Symbol('W', commutative=True))))"], [["minus", 6, 1], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('W', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('W', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\dot{z})} = \\cos{(\\cos{(\\dot{z})})}, then obtain \\frac{d}{d \\dot{z}} (\\int \\operatorname{v_{y}}{(\\dot{z})} d\\dot{z})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (\\int \\cos{(\\cos{(\\dot{z})})} d\\dot{z})^{\\dot{z}}", "derivation": "\\operatorname{v_{y}}{(\\dot{z})} = \\cos{(\\cos{(\\dot{z})})} and \\int \\operatorname{v_{y}}{(\\dot{z})} d\\dot{z} = \\int \\cos{(\\cos{(\\dot{z})})} d\\dot{z} and (\\int \\operatorname{v_{y}}{(\\dot{z})} d\\dot{z})^{\\dot{z}} = (\\int \\cos{(\\cos{(\\dot{z})})} d\\dot{z})^{\\dot{z}} and \\frac{d}{d \\dot{z}} (\\int \\operatorname{v_{y}}{(\\dot{z})} d\\dot{z})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (\\int \\cos{(\\cos{(\\dot{z})})} d\\dot{z})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), cos(cos(Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Integral(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Integral(cos(cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Pow(Integral(cos(cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\sigma_p,W)} = \\int \\sigma_p^{W} d\\sigma_p, then obtain \\frac{\\partial}{\\partial \\sigma_p} (\\int \\frac{\\partial}{\\partial \\sigma_p} \\psi{(\\sigma_p,W)} d\\sigma_p)^{W} = \\frac{\\partial}{\\partial \\sigma_p} (\\int \\frac{\\partial}{\\partial \\sigma_p} \\int \\sigma_p^{W} d\\sigma_p d\\sigma_p)^{W}", "derivation": "\\psi{(\\sigma_p,W)} = \\int \\sigma_p^{W} d\\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} \\psi{(\\sigma_p,W)} = \\frac{\\partial}{\\partial \\sigma_p} \\int \\sigma_p^{W} d\\sigma_p and \\int \\frac{\\partial}{\\partial \\sigma_p} \\psi{(\\sigma_p,W)} d\\sigma_p = \\int \\frac{\\partial}{\\partial \\sigma_p} \\int \\sigma_p^{W} d\\sigma_p d\\sigma_p and (\\int \\frac{\\partial}{\\partial \\sigma_p} \\psi{(\\sigma_p,W)} d\\sigma_p)^{W} = (\\int \\frac{\\partial}{\\partial \\sigma_p} \\int \\sigma_p^{W} d\\sigma_p d\\sigma_p)^{W} and \\frac{\\partial}{\\partial \\sigma_p} (\\int \\frac{\\partial}{\\partial \\sigma_p} \\psi{(\\sigma_p,W)} d\\sigma_p)^{W} = \\frac{\\partial}{\\partial \\sigma_p} (\\int \\frac{\\partial}{\\partial \\sigma_p} \\int \\sigma_p^{W} d\\sigma_p d\\sigma_p)^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Derivative(Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Derivative(Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('W', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Pow(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Integral(Derivative(Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(\\mathbf{f},\\hat{H})} = \\log{(\\hat{H} \\mathbf{f})} and J{(\\mathbf{f})} = \\mathbf{f}, then obtain \\int (- \\hat{H} \\mathbf{f} + \\theta{(\\mathbf{f},\\hat{H})}) dJ{(\\mathbf{f})} = \\int (- \\hat{H} \\mathbf{f} + \\log{(\\hat{H} \\mathbf{f})}) dJ{(\\mathbf{f})}", "derivation": "\\theta{(\\mathbf{f},\\hat{H})} = \\log{(\\hat{H} \\mathbf{f})} and - \\hat{H} \\mathbf{f} + \\theta{(\\mathbf{f},\\hat{H})} = - \\hat{H} \\mathbf{f} + \\log{(\\hat{H} \\mathbf{f})} and \\int (- \\hat{H} \\mathbf{f} + \\theta{(\\mathbf{f},\\hat{H})}) d\\mathbf{f} = \\int (- \\hat{H} \\mathbf{f} + \\log{(\\hat{H} \\mathbf{f})}) d\\mathbf{f} and J{(\\mathbf{f})} = \\mathbf{f} and \\int (- \\hat{H} \\mathbf{f} + \\theta{(\\mathbf{f},\\hat{H})}) dJ{(\\mathbf{f})} = \\int (- \\hat{H} \\mathbf{f} + \\log{(\\hat{H} \\mathbf{f})}) dJ{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), log(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\theta')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\theta')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\theta')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Function('J')(Symbol('\\\\mathbf{f}', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Function('J')(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given E{(v_{z},\\Omega)} = \\frac{v_{z}}{\\Omega}, then derive \\frac{\\partial}{\\partial \\Omega} E{(v_{z},\\Omega)} = - \\frac{v_{z}}{\\Omega^{2}}, then obtain - \\frac{\\frac{\\partial}{\\partial \\Omega} E{(v_{z},\\Omega)}}{\\Omega} = \\frac{v_{z}}{\\Omega^{3}}", "derivation": "E{(v_{z},\\Omega)} = \\frac{v_{z}}{\\Omega} and \\frac{\\partial}{\\partial \\Omega} E{(v_{z},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\frac{v_{z}}{\\Omega} and \\frac{\\partial}{\\partial \\Omega} E{(v_{z},\\Omega)} = - \\frac{v_{z}}{\\Omega^{2}} and - \\frac{\\frac{\\partial}{\\partial \\Omega} E{(v_{z},\\Omega)}}{\\Omega} = \\frac{v_{z}}{\\Omega^{3}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Symbol('v_z', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-3)), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given c{(\\chi)} = e^{\\cos{(\\chi)}}, then derive \\frac{d}{d \\chi} c{(\\chi)} = - e^{\\cos{(\\chi)}} \\sin{(\\chi)}, then obtain - \\frac{e^{\\cos{(\\chi)}} \\sin{(\\chi)}}{c{(\\chi)}} = \\frac{\\frac{d}{d \\chi} e^{\\cos{(\\chi)}}}{c{(\\chi)}}", "derivation": "c{(\\chi)} = e^{\\cos{(\\chi)}} and \\frac{d}{d \\chi} c{(\\chi)} = \\frac{d}{d \\chi} e^{\\cos{(\\chi)}} and \\frac{d}{d \\chi} c{(\\chi)} = - e^{\\cos{(\\chi)}} \\sin{(\\chi)} and - e^{\\cos{(\\chi)}} \\sin{(\\chi)} = \\frac{d}{d \\chi} e^{\\cos{(\\chi)}} and - \\frac{e^{\\cos{(\\chi)}} \\sin{(\\chi)}}{c{(\\chi)}} = \\frac{\\frac{d}{d \\chi} e^{\\cos{(\\chi)}}}{c{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\chi', commutative=True)), exp(cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\chi', commutative=True))), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('\\\\chi', commutative=True))), sin(Symbol('\\\\chi', commutative=True))), Derivative(exp(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["divide", 4, "Function('c')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\chi', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\chi', commutative=True))), sin(Symbol('\\\\chi', commutative=True))), Mul(Pow(Function('c')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Derivative(exp(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(\\delta)} = e^{\\delta}, then derive \\int (- \\delta + \\omega{(\\delta)}) d\\delta = - \\frac{\\delta^{2}}{2} + \\mathbf{s} + e^{\\delta}, then obtain - \\frac{\\delta^{2}}{2} + \\mathbf{s} + e^{\\delta} = \\int (- \\delta + e^{\\delta}) d\\delta", "derivation": "\\omega{(\\delta)} = e^{\\delta} and - \\delta + \\omega{(\\delta)} = - \\delta + e^{\\delta} and \\int (- \\delta + \\omega{(\\delta)}) d\\delta = \\int (- \\delta + e^{\\delta}) d\\delta and \\int (- \\delta + \\omega{(\\delta)}) d\\delta = - \\frac{\\delta^{2}}{2} + \\mathbf{s} + e^{\\delta} and - \\frac{\\delta^{2}}{2} + \\mathbf{s} + e^{\\delta} = \\int (- \\delta + e^{\\delta}) d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(A_{1})} = \\log{(A_{1})}, then derive \\frac{d}{d A_{1}} \\operatorname{t_{2}}{(A_{1})} = \\frac{1}{A_{1}}, then obtain \\frac{d}{d A_{1}} \\log{(A_{1})} = \\frac{1}{A_{1}}", "derivation": "\\operatorname{t_{2}}{(A_{1})} = \\log{(A_{1})} and \\frac{d}{d A_{1}} \\operatorname{t_{2}}{(A_{1})} = \\frac{d}{d A_{1}} \\log{(A_{1})} and \\frac{d}{d A_{1}} \\operatorname{t_{2}}{(A_{1})} = \\frac{1}{A_{1}} and \\frac{d}{d A_{1}} \\log{(A_{1})} = \\frac{1}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Pow(Symbol('A_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Pow(Symbol('A_1', commutative=True), Integer(-1)))"]]}, {"prompt": "Given E{(\\rho)} = e^{\\rho}, then obtain \\frac{- \\rho + \\sin{(\\int E{(\\rho)} d\\rho)} + 1}{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)}} = \\frac{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)} + 1}{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)}}", "derivation": "E{(\\rho)} = e^{\\rho} and \\int E{(\\rho)} d\\rho = \\int e^{\\rho} d\\rho and \\sin{(\\int E{(\\rho)} d\\rho)} = \\sin{(\\int e^{\\rho} d\\rho)} and - \\rho + \\sin{(\\int E{(\\rho)} d\\rho)} = - \\rho + \\sin{(\\int e^{\\rho} d\\rho)} and - \\rho + \\sin{(\\int E{(\\rho)} d\\rho)} + 1 = - \\rho + \\sin{(\\int e^{\\rho} d\\rho)} + 1 and \\frac{- \\rho + \\sin{(\\int E{(\\rho)} d\\rho)} + 1}{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)}} = \\frac{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)} + 1}{- \\rho + \\sin{(\\int e^{\\rho} d\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('E')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))))"], [["minus", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(Function('E')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(Function('E')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1)))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(Function('E')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), sin(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(F_{N},M,n)} = (M^{F_{N}})^{n} and g{(F_{N},M,n)} = (M^{F_{N}})^{n}, then obtain \\frac{\\partial}{\\partial M} \\frac{(M^{F_{N}})^{n}}{M} = \\frac{\\partial}{\\partial M} \\frac{g{(F_{N},M,n)}}{M}", "derivation": "\\operatorname{A_{2}}{(F_{N},M,n)} = (M^{F_{N}})^{n} and \\frac{\\operatorname{A_{2}}{(F_{N},M,n)}}{M} = \\frac{(M^{F_{N}})^{n}}{M} and \\frac{\\partial}{\\partial M} \\frac{\\operatorname{A_{2}}{(F_{N},M,n)}}{M} = \\frac{\\partial}{\\partial M} \\frac{(M^{F_{N}})^{n}}{M} and g{(F_{N},M,n)} = (M^{F_{N}})^{n} and \\frac{\\partial}{\\partial M} \\frac{\\operatorname{A_{2}}{(F_{N},M,n)}}{M} = \\frac{\\partial}{\\partial M} \\frac{g{(F_{N},M,n)}}{M} and \\frac{\\partial}{\\partial M} \\frac{(M^{F_{N}})^{n}}{M} = \\frac{\\partial}{\\partial M} \\frac{g{(F_{N},M,n)}}{M}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True)), Pow(Pow(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Symbol('n', commutative=True)))"], [["divide", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('A_2')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Pow(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('A_2')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Pow(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True)), Pow(Pow(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('A_2')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('g')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Pow(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('g')(Symbol('F_N', commutative=True), Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{r})} = \\sin{(e^{\\mathbf{r}})}, then derive e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\operatorname{A_{2}}{(\\mathbf{r})} = \\cos{(e^{\\mathbf{r}})}, then obtain e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\sin{(e^{\\mathbf{r}})} = \\cos{(e^{\\mathbf{r}})}", "derivation": "\\operatorname{A_{2}}{(\\mathbf{r})} = \\sin{(e^{\\mathbf{r}})} and \\frac{d}{d \\mathbf{r}} \\operatorname{A_{2}}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\sin{(e^{\\mathbf{r}})} and e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\operatorname{A_{2}}{(\\mathbf{r})} = e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\sin{(e^{\\mathbf{r}})} and e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\operatorname{A_{2}}{(\\mathbf{r})} = \\cos{(e^{\\mathbf{r}})} and e^{- \\mathbf{r}} \\frac{d}{d \\mathbf{r}} \\sin{(e^{\\mathbf{r}})} = \\cos{(e^{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), sin(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["divide", 2, "exp(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Derivative(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Derivative(sin(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Derivative(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), cos(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Derivative(sin(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), cos(exp(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}}, then obtain \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} \\mathbf{g}{(\\dot{\\mathbf{r}})}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}} = \\frac{e^{\\dot{\\mathbf{r}}}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}}", "derivation": "\\mathbf{g}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\frac{d}{d \\dot{\\mathbf{r}}} \\mathbf{g}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} e^{\\dot{\\mathbf{r}}} and \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} \\mathbf{g}{(\\dot{\\mathbf{r}})}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}} = \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} e^{\\dot{\\mathbf{r}}}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}} and \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} \\mathbf{g}{(\\dot{\\mathbf{r}})}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}} = \\frac{e^{\\dot{\\mathbf{r}}}}{\\mathbf{g}{(\\dot{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given y{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then obtain - y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} + \\int y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = - y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} + \\int \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}}", "derivation": "y{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} = \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} and \\int y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and - y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} + \\int y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = - y^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} + \\int \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["power", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 3, "Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('y')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(C,v_{2})} = \\frac{e^{v_{2}}}{C}, then obtain - C (- C + \\operatorname{P_{g}}^{v_{2}}{(C,v_{2})})^{C} = - C (- C + (\\frac{e^{v_{2}}}{C})^{v_{2}})^{C}", "derivation": "\\operatorname{P_{g}}{(C,v_{2})} = \\frac{e^{v_{2}}}{C} and \\operatorname{P_{g}}^{v_{2}}{(C,v_{2})} = (\\frac{e^{v_{2}}}{C})^{v_{2}} and - C + \\operatorname{P_{g}}^{v_{2}}{(C,v_{2})} = - C + (\\frac{e^{v_{2}}}{C})^{v_{2}} and (- C + \\operatorname{P_{g}}^{v_{2}}{(C,v_{2})})^{C} = (- C + (\\frac{e^{v_{2}}}{C})^{v_{2}})^{C} and - C (- C + \\operatorname{P_{g}}^{v_{2}}{(C,v_{2})})^{C} = - C (- C + (\\frac{e^{v_{2}}}{C})^{v_{2}})^{C}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Symbol('v_2', commutative=True))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"], [["minus", 2, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))), Symbol('C', commutative=True)))"], [["times", 4, "Mul(Integer(-1), Symbol('C', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('C', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('P_g')(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))), Symbol('C', commutative=True))))"]]}, {"prompt": "Given t{(\\theta_2)} = e^{\\theta_2}, then derive \\int t{(\\theta_2)} d\\theta_2 = \\mathbb{I} + e^{\\theta_2}, then derive t{(\\theta_2)} \\int t{(\\theta_2)} d\\theta_2 = (\\Omega + e^{\\theta_2}) t{(\\theta_2)}, then obtain (\\mathbb{I} + e^{\\theta_2}) t{(\\theta_2)} = (\\Omega + e^{\\theta_2}) t{(\\theta_2)}", "derivation": "t{(\\theta_2)} = e^{\\theta_2} and \\int t{(\\theta_2)} d\\theta_2 = \\int e^{\\theta_2} d\\theta_2 and t{(\\theta_2)} \\int t{(\\theta_2)} d\\theta_2 = t{(\\theta_2)} \\int e^{\\theta_2} d\\theta_2 and \\int t{(\\theta_2)} d\\theta_2 = \\mathbb{I} + e^{\\theta_2} and t{(\\theta_2)} \\int t{(\\theta_2)} d\\theta_2 = (\\Omega + e^{\\theta_2}) t{(\\theta_2)} and (\\mathbb{I} + e^{\\theta_2}) t{(\\theta_2)} = (\\Omega + e^{\\theta_2}) t{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["times", 2, "Function('t')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\theta_2', commutative=True)), Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Function('t')(Symbol('\\\\theta_2', commutative=True)), Integral(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('t')(Symbol('\\\\theta_2', commutative=True)), Integral(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Function('t')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Function('t')(Symbol('\\\\theta_2', commutative=True))), Mul(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Function('t')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given y{(l)} = \\cos{(l)}, then obtain \\frac{(l y{(l)} + l)^{l} e^{(l y{(l)} + l)^{l}}}{l y{(l)} + l} = \\frac{(l \\cos{(l)} + l)^{l} e^{(l y{(l)} + l)^{l}}}{l y{(l)} + l}", "derivation": "y{(l)} = \\cos{(l)} and l y{(l)} = l \\cos{(l)} and l y{(l)} + l = l \\cos{(l)} + l and (l y{(l)} + l)^{l} = (l \\cos{(l)} + l)^{l} and e^{(l y{(l)} + l)^{l}} = e^{(l \\cos{(l)} + l)^{l}} and \\frac{(l y{(l)} + l)^{l}}{l y{(l)} + l} = \\frac{(l \\cos{(l)} + l)^{l}}{l y{(l)} + l} and \\frac{(l y{(l)} + l)^{l} e^{(l \\cos{(l)} + l)^{l}}}{l y{(l)} + l} = \\frac{(l \\cos{(l)} + l)^{l} e^{(l \\cos{(l)} + l)^{l}}}{l y{(l)} + l} and \\frac{(l y{(l)} + l)^{l} e^{(l y{(l)} + l)^{l}}}{l y{(l)} + l} = \\frac{(l \\cos{(l)} + l)^{l} e^{(l y{(l)} + l)^{l}}}{l y{(l)} + l}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))))"], [["add", 2, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True))), exp(Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["divide", 4, "Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True))), Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["times", 6, "exp(Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), exp(Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), exp(Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), exp(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Mul(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integer(-1)), Pow(Add(Mul(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), exp(Pow(Add(Mul(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then derive \\int \\theta_{1}{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + \\mu_0, then obtain (\\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + \\mu_0)^{\\mathbf{r}} = (\\int \\log{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "derivation": "\\theta_{1}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\int \\theta_{1}{(\\mathbf{r})} d\\mathbf{r} = \\int \\log{(\\mathbf{r})} d\\mathbf{r} and (\\int \\theta_{1}{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = (\\int \\log{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} and \\int \\theta_{1}{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + \\mu_0 and (\\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + \\mu_0)^{\\mathbf{r}} = (\\int \\log{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\delta{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\hat{p}_0{(Q)} = \\cos{(Q)}, then derive \\delta{(Q)} = - \\sin{(Q)}, then obtain \\frac{d}{d Q} \\hat{p}_0{(Q)} = - \\sin{(Q)}", "derivation": "\\delta{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\hat{p}_0{(Q)} = \\cos{(Q)} and \\delta{(Q)} = - \\sin{(Q)} and \\delta{(Q)} = \\frac{d}{d Q} \\hat{p}_0{(Q)} and \\frac{d}{d Q} \\hat{p}_0{(Q)} = - \\sin{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('Q', commutative=True)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\delta')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\delta')(Symbol('Q', commutative=True)), Derivative(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Q)} = e^{Q}, then derive - (- \\operatorname{y^{\\prime}}{(Q)} + e^{Q}) e^{- Q} + (e^{Q} - \\frac{d}{d Q} \\operatorname{y^{\\prime}}{(Q)}) e^{- Q} = 0, then obtain \\frac{(e^{Q} - \\frac{d}{d Q} e^{Q}) e^{- Q}}{e^{Q} - \\frac{d}{d Q} \\operatorname{y^{\\prime}}{(Q)}} = 0", "derivation": "\\operatorname{y^{\\prime}}{(Q)} = e^{Q} and \\operatorname{y^{\\prime}}{(Q)} - e^{Q} = 0 and - (\\operatorname{y^{\\prime}}{(Q)} - e^{Q}) e^{- Q} = 0 and \\frac{d}{d Q} - (\\operatorname{y^{\\prime}}{(Q)} - e^{Q}) e^{- Q} = \\frac{d}{d Q} 0 and - (- \\operatorname{y^{\\prime}}{(Q)} + e^{Q}) e^{- Q} + (e^{Q} - \\frac{d}{d Q} \\operatorname{y^{\\prime}}{(Q)}) e^{- Q} = 0 and (e^{Q} - \\frac{d}{d Q} e^{Q}) e^{- Q} = 0 and \\frac{(e^{Q} - \\frac{d}{d Q} e^{Q}) e^{- Q}}{e^{Q} - \\frac{d}{d Q} \\operatorname{y^{\\prime}}{(Q)}} = 0", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["minus", 1, "exp(Symbol('Q', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Mul(Integer(-1), exp(Symbol('Q', commutative=True)))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), exp(Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Mul(Integer(-1), exp(Symbol('Q', commutative=True)))), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Mul(Integer(-1), exp(Symbol('Q', commutative=True)))), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('Q', commutative=True))), exp(Symbol('Q', commutative=True))), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Mul(Add(exp(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), exp(Mul(Integer(-1), Symbol('Q', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(exp(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Integer(0))"], [["divide", 6, "Add(exp(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], "Equality(Mul(Pow(Add(exp(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(-1)), Add(exp(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{B}{(C,L)} = \\sin^{L}{(C)} and \\sigma_{p}{(C,L)} = \\sin^{L}{(C)}, then derive \\frac{\\partial}{\\partial L} \\mathbf{B}{(C,L)} = \\log{(\\sin{(C)})} \\sin^{L}{(C)}, then obtain \\frac{\\frac{\\partial}{\\partial L} \\sin^{L}{(C)}}{\\sigma_{p}{(C,L)}} = \\log{(\\sin{(C)})}", "derivation": "\\mathbf{B}{(C,L)} = \\sin^{L}{(C)} and \\frac{\\partial}{\\partial L} \\mathbf{B}{(C,L)} = \\frac{\\partial}{\\partial L} \\sin^{L}{(C)} and \\frac{\\partial}{\\partial L} \\mathbf{B}{(C,L)} = \\log{(\\sin{(C)})} \\sin^{L}{(C)} and \\sigma_{p}{(C,L)} = \\sin^{L}{(C)} and \\frac{\\partial}{\\partial L} \\mathbf{B}{(C,L)} = \\sigma_{p}{(C,L)} \\log{(\\sin{(C)})} and \\frac{\\partial}{\\partial L} \\sin^{L}{(C)} = \\sigma_{p}{(C,L)} \\log{(\\sin{(C)})} and \\frac{\\frac{\\partial}{\\partial L} \\sin^{L}{(C)}}{\\sigma_{p}{(C,L)}} = \\log{(\\sin{(C)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(log(sin(Symbol('C', commutative=True))), Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Function('\\\\sigma_p')(Symbol('C', commutative=True), Symbol('L', commutative=True)), log(sin(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Function('\\\\sigma_p')(Symbol('C', commutative=True), Symbol('L', commutative=True)), log(sin(Symbol('C', commutative=True)))))"], [["divide", 6, "Function('\\\\sigma_p')(Symbol('C', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('C', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Derivative(Pow(sin(Symbol('C', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(sin(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\varphi^*)} = e^{\\sin{(\\varphi^*)}} and \\operatorname{A_{z}}{(\\varphi^*)} = \\varphi^*, then obtain \\cos{(\\frac{e^{\\sin{(\\varphi^*)}}}{\\sin{(\\varphi^*)}})} = \\varphi^* - \\operatorname{A_{z}}{(\\varphi^*)} + \\cos{(\\frac{e^{\\sin{(\\varphi^*)}}}{\\sin{(\\varphi^*)}})}", "derivation": "\\operatorname{z^{*}}{(\\varphi^*)} = e^{\\sin{(\\varphi^*)}} and \\operatorname{A_{z}}{(\\varphi^*)} = \\varphi^* and 0 = \\varphi^* - \\operatorname{A_{z}}{(\\varphi^*)} and \\cos{(\\frac{\\operatorname{z^{*}}{(\\varphi^*)}}{\\sin{(\\varphi^*)}})} = \\varphi^* - \\operatorname{A_{z}}{(\\varphi^*)} + \\cos{(\\frac{\\operatorname{z^{*}}{(\\varphi^*)}}{\\sin{(\\varphi^*)}})} and \\cos{(\\frac{e^{\\sin{(\\varphi^*)}}}{\\sin{(\\varphi^*)}})} = \\varphi^* - \\operatorname{A_{z}}{(\\varphi^*)} + \\cos{(\\frac{e^{\\sin{(\\varphi^*)}}}{\\sin{(\\varphi^*)}})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\varphi^*', commutative=True)), exp(sin(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], [["minus", 2, "Function('A_z')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Function('A_z')(Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 3, "cos(Mul(Function('z^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], "Equality(cos(Mul(Function('z^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Function('A_z')(Symbol('\\\\varphi^*', commutative=True))), cos(Mul(Function('z^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Mul(exp(sin(Symbol('\\\\varphi^*', commutative=True))), Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Function('A_z')(Symbol('\\\\varphi^*', commutative=True))), cos(Mul(exp(sin(Symbol('\\\\varphi^*', commutative=True))), Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given g{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\int g{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + \\mathbf{M}, then obtain \\frac{\\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + \\mathbf{M}}{\\mathbf{H}} = \\frac{\\int \\log{(\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H}}", "derivation": "g{(\\mathbf{H})} = \\log{(\\mathbf{H})} and \\int g{(\\mathbf{H})} d\\mathbf{H} = \\int \\log{(\\mathbf{H})} d\\mathbf{H} and \\int g{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + \\mathbf{M} and \\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + \\mathbf{M} = \\int \\log{(\\mathbf{H})} d\\mathbf{H} and \\frac{\\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + \\mathbf{M}}{\\mathbf{H}} = \\frac{\\int \\log{(\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given L{(\\lambda)} = e^{\\lambda}, then obtain \\log{(L{(\\lambda)})} \\cos^{\\lambda}{(\\cos{(L{(\\lambda)})})} = \\log{(L{(\\lambda)})} \\cos^{\\lambda}{(\\cos{(e^{\\lambda})})}", "derivation": "L{(\\lambda)} = e^{\\lambda} and \\cos{(L{(\\lambda)})} = \\cos{(e^{\\lambda})} and \\cos{(\\cos{(L{(\\lambda)})})} = \\cos{(\\cos{(e^{\\lambda})})} and \\cos^{\\lambda}{(\\cos{(L{(\\lambda)})})} = \\cos^{\\lambda}{(\\cos{(e^{\\lambda})})} and \\log{(L{(\\lambda)})} \\cos^{\\lambda}{(\\cos{(L{(\\lambda)})})} = \\log{(L{(\\lambda)})} \\cos^{\\lambda}{(\\cos{(e^{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["cos", 1], "Equality(cos(Function('L')(Symbol('\\\\lambda', commutative=True))), cos(exp(Symbol('\\\\lambda', commutative=True))))"], [["cos", 2], "Equality(cos(cos(Function('L')(Symbol('\\\\lambda', commutative=True)))), cos(cos(exp(Symbol('\\\\lambda', commutative=True)))))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(cos(cos(Function('L')(Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Pow(cos(cos(exp(Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)))"], [["times", 4, "log(Function('L')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(log(Function('L')(Symbol('\\\\lambda', commutative=True))), Pow(cos(cos(Function('L')(Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True))), Mul(log(Function('L')(Symbol('\\\\lambda', commutative=True))), Pow(cos(cos(exp(Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given H{(v,\\theta_1)} = \\theta_1 + v, then obtain (\\frac{- v + H{(v,\\theta_1)}}{v})^{\\theta_1} - \\frac{(\\theta_1 + v)^{\\theta_1} (- v + H{(v,\\theta_1)})}{v} = (\\frac{\\theta_1}{v})^{\\theta_1} - \\frac{(\\theta_1 + v)^{\\theta_1} (- v + H{(v,\\theta_1)})}{v}", "derivation": "H{(v,\\theta_1)} = \\theta_1 + v and - v + H{(v,\\theta_1)} = \\theta_1 and \\frac{- v + H{(v,\\theta_1)}}{v} = \\frac{\\theta_1}{v} and (\\frac{- v + H{(v,\\theta_1)}}{v})^{\\theta_1} = (\\frac{\\theta_1}{v})^{\\theta_1} and (\\frac{- v + H{(v,\\theta_1)}}{v})^{\\theta_1} - \\frac{(\\theta_1 + v)^{\\theta_1} (- v + H{(v,\\theta_1)})}{v} = (\\frac{\\theta_1}{v})^{\\theta_1} - \\frac{(\\theta_1 + v)^{\\theta_1} (- v + H{(v,\\theta_1)})}{v}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))"], [["divide", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 4, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Add(Pow(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('H')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given Z{(\\omega,W)} = \\cos{(W \\omega)} and \\bar{\\h}{(W,\\omega)} = - \\frac{1}{Z^{2}{(\\omega,W)}}, then obtain \\bar{\\h}{(W,\\omega)} = \\bar{\\h}{(W,\\omega)} - 1 + \\frac{\\cos{(W \\omega)}}{Z{(\\omega,W)}}", "derivation": "Z{(\\omega,W)} = \\cos{(W \\omega)} and 1 = \\frac{\\cos{(W \\omega)}}{Z{(\\omega,W)}} and \\cos{(W \\omega)} = \\frac{\\cos^{2}{(W \\omega)}}{Z{(\\omega,W)}} and 1 = \\frac{\\cos^{2}{(W \\omega)}}{Z^{2}{(\\omega,W)}} and 1 - \\frac{1}{Z^{2}{(\\omega,W)}} = \\frac{\\cos^{2}{(W \\omega)}}{Z^{2}{(\\omega,W)}} - \\frac{1}{Z^{2}{(\\omega,W)}} and - \\frac{1}{Z^{2}{(\\omega,W)}} = -1 + \\frac{\\cos^{2}{(W \\omega)}}{Z^{2}{(\\omega,W)}} - \\frac{1}{Z^{2}{(\\omega,W)}} and \\bar{\\h}{(W,\\omega)} = - \\frac{1}{Z^{2}{(\\omega,W)}} and \\bar{\\h}{(W,\\omega)} = \\bar{\\h}{(W,\\omega)} - 1 + \\frac{\\cos^{2}{(W \\omega)}}{Z^{2}{(\\omega,W)}} and \\bar{\\h}{(W,\\omega)} = \\bar{\\h}{(W,\\omega)} - 1 + \\frac{\\cos{(W \\omega)}}{Z{(\\omega,W)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["divide", 1, "Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-1)), cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["times", 2, "cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-1)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2))))"], [["minus", 4, "Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)))), Add(Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)))))"], [["add", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2))), Add(Integer(-1), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('\\\\hbar')(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Function('\\\\hbar')(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-2)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2)))))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Function('\\\\hbar')(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Function('\\\\hbar')(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1), Mul(Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('W', commutative=True)), Integer(-1)), cos(Mul(Symbol('W', commutative=True), Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(z)} = \\cos{(\\cos{(z)})}, then obtain (\\int \\mathbf{J}_f{(z)} dz - 1)^{z} = (\\int \\cos{(\\cos{(z)})} dz - 1)^{z}", "derivation": "\\mathbf{J}_f{(z)} = \\cos{(\\cos{(z)})} and \\int \\mathbf{J}_f{(z)} dz = \\int \\cos{(\\cos{(z)})} dz and \\int \\mathbf{J}_f{(z)} dz - \\frac{\\cos{(\\cos{(z)})}}{\\mathbf{J}_f{(z)}} = \\int \\cos{(\\cos{(z)})} dz - \\frac{\\cos{(\\cos{(z)})}}{\\mathbf{J}_f{(z)}} and \\int \\mathbf{J}_f{(z)} dz - 1 = \\int \\cos{(\\cos{(z)})} dz - 1 and (\\int \\mathbf{J}_f{(z)} dz - 1)^{z} = (\\int \\cos{(\\cos{(z)})} dz - 1)^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), cos(cos(Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(cos(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Integer(-1)), cos(cos(Symbol('z', commutative=True))))"], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Integer(-1)), cos(cos(Symbol('z', commutative=True))))), Add(Integral(cos(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Integer(-1)), cos(cos(Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1)), Add(Integral(cos(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1)), Symbol('z', commutative=True)), Pow(Add(Integral(cos(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integer(-1)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varphi^*,z)} = \\varphi^* \\cos{(z)}, then derive \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)} = \\cos{(z)}, then obtain (\\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)})^{\\varphi^*} = \\cos^{\\varphi^*}{(z)}", "derivation": "\\operatorname{A_{2}}{(\\varphi^*,z)} = \\varphi^* \\cos{(z)} and \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)} = \\frac{\\partial}{\\partial \\varphi^*} \\varphi^* \\cos{(z)} and \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)} = \\cos{(z)} and (\\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)})^{\\varphi^*} = (\\frac{\\partial}{\\partial \\varphi^*} \\varphi^* \\cos{(z)})^{\\varphi^*} and \\cos^{\\varphi^*}{(z)} = (\\frac{\\partial}{\\partial \\varphi^*} \\varphi^* \\cos{(z)})^{\\varphi^*} and (\\frac{\\partial}{\\partial \\varphi^*} \\operatorname{A_{2}}{(\\varphi^*,z)})^{\\varphi^*} = \\cos^{\\varphi^*}{(z)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('z', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), cos(Symbol('z', commutative=True)))"], [["power", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Derivative(Function('A_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('z', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(cos(Symbol('z', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\varphi^*', commutative=True), cos(Symbol('z', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Function('A_2')(Symbol('\\\\varphi^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(i,F_{c})} = F_{c} i and \\eta{(i,F_{c})} = F_{c}^{2} i^{2} and \\rho_{b}{(i,F_{c})} = F_{c}^{2} + F_{c} i \\operatorname{n_{1}}{(i,F_{c})}, then obtain F_{c}^{2} i^{2} + F_{c}^{2} = F_{c}^{2} + \\eta{(i,F_{c})}", "derivation": "\\operatorname{n_{1}}{(i,F_{c})} = F_{c} i and F_{c} i \\operatorname{n_{1}}{(i,F_{c})} = F_{c}^{2} i^{2} and \\eta{(i,F_{c})} = F_{c}^{2} i^{2} and F_{c}^{2} + F_{c} i \\operatorname{n_{1}}{(i,F_{c})} = F_{c}^{2} i^{2} + F_{c}^{2} and F_{c} i \\operatorname{n_{1}}{(i,F_{c})} = \\eta{(i,F_{c})} and \\rho_{b}{(i,F_{c})} = F_{c}^{2} + F_{c} i \\operatorname{n_{1}}{(i,F_{c})} and \\rho_{b}{(i,F_{c})} = F_{c}^{2} i^{2} + F_{c}^{2} and \\rho_{b}{(i,F_{c})} = F_{c}^{2} + \\eta{(i,F_{c})} and F_{c}^{2} i^{2} + F_{c}^{2} = F_{c}^{2} + \\eta{(i,F_{c})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True)))"], [["times", 1, "Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('F_c', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["add", 2, "Pow(Symbol('F_c', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)))), Add(Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))), Pow(Symbol('F_c', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('F_c', commutative=True))), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)), Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Mul(Symbol('F_c', commutative=True), Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('\\\\rho_b')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)), Add(Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))), Pow(Symbol('F_c', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\rho_b')(Symbol('i', commutative=True), Symbol('F_c', commutative=True)), Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Add(Mul(Pow(Symbol('F_c', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))), Pow(Symbol('F_c', commutative=True), Integer(2))), Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\delta)} = \\cos{(\\delta)}, then derive \\int \\operatorname{r_{0}}{(\\delta)} d\\delta = a + \\sin{(\\delta)}, then obtain \\int (\\cos{(\\delta)} \\int \\operatorname{r_{0}}{(\\delta)} d\\delta)^{\\delta} da = \\int ((a + \\sin{(\\delta)}) \\cos{(\\delta)})^{\\delta} da", "derivation": "\\operatorname{r_{0}}{(\\delta)} = \\cos{(\\delta)} and \\int \\operatorname{r_{0}}{(\\delta)} d\\delta = \\int \\cos{(\\delta)} d\\delta and \\cos{(\\delta)} \\int \\operatorname{r_{0}}{(\\delta)} d\\delta = \\cos{(\\delta)} \\int \\cos{(\\delta)} d\\delta and \\int \\operatorname{r_{0}}{(\\delta)} d\\delta = a + \\sin{(\\delta)} and (a + \\sin{(\\delta)}) \\cos{(\\delta)} = \\cos{(\\delta)} \\int \\cos{(\\delta)} d\\delta and (\\cos{(\\delta)} \\int \\operatorname{r_{0}}{(\\delta)} d\\delta)^{\\delta} = (\\cos{(\\delta)} \\int \\cos{(\\delta)} d\\delta)^{\\delta} and (\\cos{(\\delta)} \\int \\operatorname{r_{0}}{(\\delta)} d\\delta)^{\\delta} = ((a + \\sin{(\\delta)}) \\cos{(\\delta)})^{\\delta} and \\int (\\cos{(\\delta)} \\int \\operatorname{r_{0}}{(\\delta)} d\\delta)^{\\delta} da = \\int ((a + \\sin{(\\delta)}) \\cos{(\\delta)})^{\\delta} da", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 2, "cos(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('a', commutative=True), sin(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('a', commutative=True), sin(Symbol('\\\\delta', commutative=True))), cos(Symbol('\\\\delta', commutative=True))), Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Add(Symbol('a', commutative=True), sin(Symbol('\\\\delta', commutative=True))), cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["integrate", 7, "Symbol('a', commutative=True)"], "Equality(Integral(Pow(Mul(cos(Symbol('\\\\delta', commutative=True)), Integral(Function('r_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), sin(Symbol('\\\\delta', commutative=True))), cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(n_{1},v_{x})} = \\frac{v_{x}}{n_{1}}, then obtain \\int (\\int \\dot{z}{(n_{1},v_{x})} dv_{x})^{2} dv_{x} = \\int (\\int \\frac{v_{x}}{n_{1}} dv_{x}) \\int \\dot{z}{(n_{1},v_{x})} dv_{x} dv_{x}", "derivation": "\\dot{z}{(n_{1},v_{x})} = \\frac{v_{x}}{n_{1}} and \\int \\dot{z}{(n_{1},v_{x})} dv_{x} = \\int \\frac{v_{x}}{n_{1}} dv_{x} and (\\int \\dot{z}{(n_{1},v_{x})} dv_{x})^{2} = (\\int \\frac{v_{x}}{n_{1}} dv_{x}) \\int \\dot{z}{(n_{1},v_{x})} dv_{x} and \\int (\\int \\dot{z}{(n_{1},v_{x})} dv_{x})^{2} dv_{x} = \\int (\\int \\frac{v_{x}}{n_{1}} dv_{x}) \\int \\dot{z}{(n_{1},v_{x})} dv_{x} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["times", 2, "Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(2)), Mul(Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["integrate", 3, "Symbol('v_x', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(2)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Function('\\\\dot{z}')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\eta{(\\hbar)} = \\int \\cos{(\\hbar)} d\\hbar, then derive \\eta{(\\hbar)} = \\varepsilon_0 + \\sin{(\\hbar)}, then derive \\mathbf{J}_f + \\eta{(\\hbar)} = x + \\sin{(\\hbar)}, then obtain \\mathbf{J}_f + h + \\sin{(\\hbar)} = x + \\sin{(\\hbar)}", "derivation": "\\eta{(\\hbar)} = \\int \\cos{(\\hbar)} d\\hbar and \\eta{(\\hbar)} = \\varepsilon_0 + \\sin{(\\hbar)} and \\frac{d}{d \\hbar} \\eta{(\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (\\varepsilon_0 + \\sin{(\\hbar)}) and \\int \\frac{d}{d \\hbar} \\eta{(\\hbar)} d\\hbar = \\int \\frac{\\partial}{\\partial \\hbar} (\\varepsilon_0 + \\sin{(\\hbar)}) d\\hbar and \\mathbf{J}_f + \\eta{(\\hbar)} = x + \\sin{(\\hbar)} and \\mathbf{J}_f + \\int \\cos{(\\hbar)} d\\hbar = x + \\sin{(\\hbar)} and \\mathbf{J}_f + h + \\sin{(\\hbar)} = x + \\sin{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True)), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\eta')(Symbol('\\\\hbar', commutative=True))), Add(Symbol('x', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('x', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('h', commutative=True), sin(Symbol('\\\\hbar', commutative=True))), Add(Symbol('x', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(U,\\pi)} = U - \\pi, then derive (\\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU)^{\\pi} = (- \\frac{U^{2}}{2 \\dot{\\mathbf{r}}} + \\frac{U \\pi}{\\dot{\\mathbf{r}}} + v_{1})^{\\pi}, then obtain e^{(\\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU)^{\\pi}} = e^{(- \\frac{U^{2}}{2 \\dot{\\mathbf{r}}} + \\frac{U \\pi}{\\dot{\\mathbf{r}}} + v_{1})^{\\pi}}", "derivation": "\\mu_{0}{(U,\\pi)} = U - \\pi and - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} = - \\frac{U - \\pi}{\\dot{\\mathbf{r}}} and \\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU = \\int - \\frac{U - \\pi}{\\dot{\\mathbf{r}}} dU and (\\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU)^{\\pi} = (\\int - \\frac{U - \\pi}{\\dot{\\mathbf{r}}} dU)^{\\pi} and (\\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU)^{\\pi} = (- \\frac{U^{2}}{2 \\dot{\\mathbf{r}}} + \\frac{U \\pi}{\\dot{\\mathbf{r}}} + v_{1})^{\\pi} and e^{(\\int - \\frac{\\mu_{0}{(U,\\pi)}}{\\dot{\\mathbf{r}}} dU)^{\\pi}} = e^{(- \\frac{U^{2}}{2 \\dot{\\mathbf{r}}} + \\frac{U \\pi}{\\dot{\\mathbf{r}}} + v_{1})^{\\pi}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('U', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('\\\\pi', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given E{(m)} = \\log{(m)} and \\hat{x}_0{(m)} = E^{m}{(m)}, then obtain 1 - E^{- m}{(m)} = E^{m}{(m)} \\log{(m)}^{- m} - E^{- m}{(m)}", "derivation": "E{(m)} = \\log{(m)} and E^{m}{(m)} = \\log{(m)}^{m} and \\hat{x}_0{(m)} = E^{m}{(m)} and \\hat{x}_0{(m)} \\log{(m)}^{- m} = E^{m}{(m)} \\log{(m)}^{- m} and \\hat{x}_0{(m)} = \\log{(m)}^{m} and \\hat{x}_0{(m)} \\log{(m)}^{- m} - E^{- m}{(m)} = E^{m}{(m)} \\log{(m)}^{- m} - E^{- m}{(m)} and 1 - E^{- m}{(m)} = E^{m}{(m)} \\log{(m)}^{- m} - E^{- m}{(m)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('E')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('m', commutative=True)), Pow(Function('E')(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["divide", 3, "Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Pow(Function('E')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["minus", 4, "Pow(Function('E')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))"], "Equality(Add(Mul(Function('\\\\hat{x}_0')(Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Integer(-1), Pow(Function('E')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))), Add(Mul(Pow(Function('E')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Integer(-1), Pow(Function('E')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('E')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))), Add(Mul(Pow(Function('E')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Integer(-1), Pow(Function('E')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))))"]]}, {"prompt": "Given p{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x}{\\phi_1}, then obtain (\\frac{\\sigma_x}{\\phi_1})^{\\phi_1} p{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x (\\frac{\\sigma_x}{\\phi_1})^{\\phi_1}}{\\phi_1}", "derivation": "p{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x}{\\phi_1} and p^{\\phi_1}{(\\phi_1,\\sigma_x)} = (\\frac{\\sigma_x}{\\phi_1})^{\\phi_1} and p{(\\phi_1,\\sigma_x)} p^{\\phi_1}{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x p^{\\phi_1}{(\\phi_1,\\sigma_x)}}{\\phi_1} and (\\frac{\\sigma_x}{\\phi_1})^{\\phi_1} p{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x (\\frac{\\sigma_x}{\\phi_1})^{\\phi_1}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Pow(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True), Pow(Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Function('p')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True), Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(P_{g},t_{2})} = \\cos{(t_{2}^{P_{g}})}, then obtain t_{2} \\frac{\\partial}{\\partial P_{g}} \\ddot{x}{(P_{g},t_{2})} + 1 = - t_{2} t_{2}^{P_{g}} \\log{(t_{2})} \\sin{(t_{2}^{P_{g}})} + 1", "derivation": "\\ddot{x}{(P_{g},t_{2})} = \\cos{(t_{2}^{P_{g}})} and t_{2} \\ddot{x}{(P_{g},t_{2})} = t_{2} \\cos{(t_{2}^{P_{g}})} and P_{g} + t_{2} \\ddot{x}{(P_{g},t_{2})} = P_{g} + t_{2} \\cos{(t_{2}^{P_{g}})} and \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{2} \\ddot{x}{(P_{g},t_{2})}) = \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{2} \\cos{(t_{2}^{P_{g}})}) and t_{2} \\frac{\\partial}{\\partial P_{g}} \\ddot{x}{(P_{g},t_{2})} + 1 = - t_{2} t_{2}^{P_{g}} \\log{(t_{2})} \\sin{(t_{2}^{P_{g}})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('P_g', commutative=True), Symbol('t_2', commutative=True)), cos(Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True))))"], [["times", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Function('\\\\ddot{x}')(Symbol('P_g', commutative=True), Symbol('t_2', commutative=True))), Mul(Symbol('t_2', commutative=True), cos(Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True)))))"], [["add", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Mul(Symbol('t_2', commutative=True), Function('\\\\ddot{x}')(Symbol('P_g', commutative=True), Symbol('t_2', commutative=True)))), Add(Symbol('P_g', commutative=True), Mul(Symbol('t_2', commutative=True), cos(Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True))))))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Add(Symbol('P_g', commutative=True), Mul(Symbol('t_2', commutative=True), Function('\\\\ddot{x}')(Symbol('P_g', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Symbol('P_g', commutative=True), Mul(Symbol('t_2', commutative=True), cos(Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True))))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('t_2', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('P_g', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True), Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True)), log(Symbol('t_2', commutative=True)), sin(Pow(Symbol('t_2', commutative=True), Symbol('P_g', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(n)} = \\cos{(\\log{(n)})}, then obtain \\frac{- \\phi_1^{\\hat{H}} + \\operatorname{v_{2}}{(n)} + \\log{(n)} + \\cos{(\\log{(n)})}}{\\phi_1} = \\frac{- \\phi_1^{\\hat{H}} + 2 \\operatorname{v_{2}}{(n)} + \\log{(n)}}{\\phi_1}", "derivation": "\\operatorname{v_{2}}{(n)} = \\cos{(\\log{(n)})} and \\operatorname{v_{2}}{(n)} + \\log{(n)} = \\log{(n)} + \\cos{(\\log{(n)})} and \\operatorname{v_{2}}{(n)} + \\log{(n)} + \\cos{(\\log{(n)})} = \\log{(n)} + 2 \\cos{(\\log{(n)})} and 2 \\operatorname{v_{2}}{(n)} + \\log{(n)} = \\log{(n)} + 2 \\cos{(\\log{(n)})} and \\operatorname{v_{2}}{(n)} + \\log{(n)} + \\cos{(\\log{(n)})} = 2 \\operatorname{v_{2}}{(n)} + \\log{(n)} and - \\phi_1^{\\hat{H}} + \\operatorname{v_{2}}{(n)} + \\log{(n)} + \\cos{(\\log{(n)})} = - \\phi_1^{\\hat{H}} + 2 \\operatorname{v_{2}}{(n)} + \\log{(n)} and \\frac{- \\phi_1^{\\hat{H}} + \\operatorname{v_{2}}{(n)} + \\log{(n)} + \\cos{(\\log{(n)})}}{\\phi_1} = \\frac{- \\phi_1^{\\hat{H}} + 2 \\operatorname{v_{2}}{(n)} + \\log{(n)}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True))))"], [["add", 1, "log(Symbol('n', commutative=True))"], "Equality(Add(Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Add(log(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), cos(log(Symbol('n', commutative=True))))"], "Equality(Add(Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True)))), Add(log(Symbol('n', commutative=True)), Mul(Integer(2), cos(log(Symbol('n', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('v_2')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Add(log(Symbol('n', commutative=True)), Mul(Integer(2), cos(log(Symbol('n', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True)))), Add(Mul(Integer(2), Function('v_2')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["minus", 5, "Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(2), Function('v_2')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["divide", 6, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)), cos(log(Symbol('n', commutative=True))))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(2), Function('v_2')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain \\mathbf{H} \\cos^{2}{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\mathbf{J}_M{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\mathbf{H} \\cos^{2}{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\cos^{2}{(\\mathbf{H})}", "derivation": "\\mathbf{J}_M{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\mathbf{J}_M{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\cos^{2}{(\\mathbf{H})} and \\mathbf{H} \\mathbf{J}_M{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\mathbf{H} \\cos^{2}{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\mathbf{J}_M{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\cos^{2}{(\\mathbf{H})} and \\mathbf{H} \\cos^{2}{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\mathbf{J}_M{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\mathbf{H} \\cos^{2}{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\mathbf{H} \\cos^{2}{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["add", 4, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\rho_b,g)} = \\rho_b + g, then obtain \\int 1 dg - \\frac{(- 2 g + \\hat{H}_l{(\\rho_b,g)}) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)} = \\int 1 dg - \\frac{(\\rho_b - g) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)}", "derivation": "\\hat{H}_l{(\\rho_b,g)} = \\rho_b + g and - 2 g + \\hat{H}_l{(\\rho_b,g)} = \\rho_b - g and - \\frac{- 2 g + \\hat{H}_l{(\\rho_b,g)}}{g} = - \\frac{\\rho_b - g}{g} and - \\frac{(- 2 g + \\hat{H}_l{(\\rho_b,g)}) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)} = - \\frac{(\\rho_b - g) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)} and \\int 1 dg - \\frac{(- 2 g + \\hat{H}_l{(\\rho_b,g)}) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)} = \\int 1 dg - \\frac{(\\rho_b - g) (g + \\hat{H}_l{(\\rho_b,g)})}{g (\\rho_b + g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))"], [["times", 3, "Mul(Pow(Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Integer(-1)), Add(Symbol('g', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Integer(-1)), Add(Symbol('g', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)))))"], [["add", 4, "Integral(Integer(1), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))))), Add(Integral(Integer(1), Tuple(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Add(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True)), Integer(-1)), Add(Symbol('g', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\rho_b', commutative=True), Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{1})} = \\sin{(e^{v_{1}})}, then obtain \\sin{((\\frac{\\mathbf{E}{(v_{1})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}})} = \\sin{((\\frac{\\sin{(e^{v_{1}})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}})}", "derivation": "\\mathbf{E}{(v_{1})} = \\sin{(e^{v_{1}})} and \\frac{\\mathbf{E}{(v_{1})}}{v_{1}} = \\frac{\\sin{(e^{v_{1}})}}{v_{1}} and (\\frac{\\mathbf{E}{(v_{1})}}{v_{1}})^{v_{1}} = (\\frac{\\sin{(e^{v_{1}})}}{v_{1}})^{v_{1}} and (\\frac{\\mathbf{E}{(v_{1})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}} = (\\frac{\\sin{(e^{v_{1}})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}} and \\sin{((\\frac{\\mathbf{E}{(v_{1})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}})} = \\sin{((\\frac{\\sin{(e^{v_{1}})}}{v_{1}})^{v_{1}} - \\frac{\\sin{(e^{v_{1}})}}{v_{1}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), sin(exp(Symbol('v_1', commutative=True))))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True))))"], "Equality(Add(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True))))), Add(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True))))))"], [["sin", 4], "Equality(sin(Add(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))))), sin(Add(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(exp(Symbol('v_1', commutative=True)))))))"]]}, {"prompt": "Given \\psi{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain (\\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 1) \\log{(\\sigma_p)} = 2 \\log{(\\sigma_p)}", "derivation": "\\psi{(\\sigma_p)} = \\log{(\\sigma_p)} and \\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} = 1 and \\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 1 = 2 and 2 = - \\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 3 and 2 \\log{(\\sigma_p)} = (- \\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 3) \\log{(\\sigma_p)} and (\\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 1) \\log{(\\sigma_p)} = (- \\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 3) \\log{(\\sigma_p)} and (\\frac{\\psi{(\\sigma_p)}}{\\log{(\\sigma_p)}} + 1) \\log{(\\sigma_p)} = 2 \\log{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, 1], "Equality(Add(Mul(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1)), Integer(2))"], [["add", 3, "Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1))"], "Equality(Integer(2), Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(3)))"], [["times", 4, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(3)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Mul(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1)), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(3)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Add(Mul(Function('\\\\psi')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1)), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(a)} = \\log{(a)}, then obtain \\frac{\\frac{d}{d a} (a \\operatorname{v_{1}}{(a)} - a \\log{(a)})}{a \\log{(a)} - \\log{(a)}} = \\frac{\\frac{d}{d a} 0}{a \\log{(a)} - \\log{(a)}}", "derivation": "\\operatorname{v_{1}}{(a)} = \\log{(a)} and a \\operatorname{v_{1}}{(a)} = a \\log{(a)} and a \\operatorname{v_{1}}{(a)} - \\log{(a)} = a \\log{(a)} - \\log{(a)} and a \\operatorname{v_{1}}{(a)} - a \\log{(a)} = 0 and \\frac{d}{d a} (a \\operatorname{v_{1}}{(a)} - a \\log{(a)}) = \\frac{d}{d a} 0 and \\frac{\\frac{d}{d a} (a \\operatorname{v_{1}}{(a)} - a \\log{(a)})}{a \\operatorname{v_{1}}{(a)} - \\log{(a)}} = \\frac{\\frac{d}{d a} 0}{a \\operatorname{v_{1}}{(a)} - \\log{(a)}} and \\frac{\\frac{d}{d a} (a \\operatorname{v_{1}}{(a)} - a \\log{(a)})}{a \\log{(a)} - \\log{(a)}} = \\frac{\\frac{d}{d a} 0}{a \\log{(a)} - \\log{(a)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))))"], [["minus", 2, "log(Symbol('a', commutative=True))"], "Equality(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))))"], [["minus", 2, "Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["divide", 5, "Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Symbol('a', commutative=True), Function('v_1')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\cos{(g^{\\prime}_{\\varepsilon})}, then derive \\chi{(g^{\\prime}_{\\varepsilon})} = - \\sin{(g^{\\prime}_{\\varepsilon})}, then obtain \\cos{(\\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\cos{(g^{\\prime}_{\\varepsilon})})} = \\cos{(\\frac{d}{d g^{\\prime}_{\\varepsilon}} - \\sin{(g^{\\prime}_{\\varepsilon})})}", "derivation": "\\chi{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\cos{(g^{\\prime}_{\\varepsilon})} and \\chi{(g^{\\prime}_{\\varepsilon})} = - \\sin{(g^{\\prime}_{\\varepsilon})} and \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\chi{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} - \\sin{(g^{\\prime}_{\\varepsilon})} and \\cos{(\\frac{d}{d g^{\\prime}_{\\varepsilon}} \\chi{(g^{\\prime}_{\\varepsilon})})} = \\cos{(\\frac{d}{d g^{\\prime}_{\\varepsilon}} - \\sin{(g^{\\prime}_{\\varepsilon})})} and \\cos{(\\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\cos{(g^{\\prime}_{\\varepsilon})})} = \\cos{(\\frac{d}{d g^{\\prime}_{\\varepsilon}} - \\sin{(g^{\\prime}_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Derivative(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))), cos(Derivative(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Derivative(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2)))), cos(Derivative(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(m)} = \\cos{(m)}, then derive \\frac{d}{d m} \\mathbf{F}{(m)} = - \\sin{(m)}, then obtain B + \\mathbf{F}{(m)} = \\pi + \\cos{(m)}", "derivation": "\\mathbf{F}{(m)} = \\cos{(m)} and \\frac{d}{d m} \\mathbf{F}{(m)} = \\frac{d}{d m} \\cos{(m)} and \\frac{d}{d m} \\mathbf{F}{(m)} = - \\sin{(m)} and \\int \\frac{d}{d m} \\mathbf{F}{(m)} dm = \\int - \\sin{(m)} dm and \\int \\frac{d}{d m} \\cos{(m)} dm = \\int - \\sin{(m)} dm and \\int \\frac{d}{d m} \\mathbf{F}{(m)} dm = \\int \\frac{d}{d m} \\cos{(m)} dm and B + \\mathbf{F}{(m)} = \\pi + \\cos{(m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('m', commutative=True))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{F}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Function('\\\\mathbf{F}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))), Integral(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('B', commutative=True), Function('\\\\mathbf{F}')(Symbol('m', commutative=True))), Add(Symbol('\\\\pi', commutative=True), cos(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})} = \\frac{V_{\\mathbf{E}} + \\mathbf{A}}{\\psi^*}, then obtain \\int \\frac{(\\psi^*)^{2}}{\\operatorname{x^{{\\}'}}^{2}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})}} d\\mathbf{A} = \\int \\frac{(\\psi^*)^{4}}{(V_{\\mathbf{E}} + \\mathbf{A})^{2}} d\\mathbf{A}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})} = \\frac{V_{\\mathbf{E}} + \\mathbf{A}}{\\psi^*} and \\frac{\\operatorname{x^{{\\}'}}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})}}{\\psi^*} = \\frac{V_{\\mathbf{E}} + \\mathbf{A}}{(\\psi^*)^{2}} and \\frac{(\\psi^*)^{2}}{\\operatorname{x^{{\\}'}}^{2}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})}} = \\frac{(\\psi^*)^{4}}{(V_{\\mathbf{E}} + \\mathbf{A})^{2}} and \\int \\frac{(\\psi^*)^{2}}{\\operatorname{x^{{\\}'}}^{2}{(\\mathbf{A},\\psi^*,V_{\\mathbf{E}})}} d\\mathbf{A} = \\int \\frac{(\\psi^*)^{4}}{(V_{\\mathbf{E}} + \\mathbf{A})^{2}} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 2, "Integer(-2)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(4)), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-2))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(4)), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{v},\\mu_0)} = \\mathbf{v} \\log{(\\mu_0)}, then obtain \\frac{\\tilde{g}^*{(\\mathbf{v},\\mu_0)}}{\\int 0 d\\mu_0} = \\frac{\\mathbf{v} \\log{(\\mu_0)}}{\\int 0 d\\mu_0}", "derivation": "\\tilde{g}^*{(\\mathbf{v},\\mu_0)} = \\mathbf{v} \\log{(\\mu_0)} and 0 = \\mathbf{v} \\log{(\\mu_0)} - \\tilde{g}^*{(\\mathbf{v},\\mu_0)} and \\int 0 d\\mu_0 = \\int (\\mathbf{v} \\log{(\\mu_0)} - \\tilde{g}^*{(\\mathbf{v},\\mu_0)}) d\\mu_0 and \\frac{\\tilde{g}^*{(\\mathbf{v},\\mu_0)}}{\\int (\\mathbf{v} \\log{(\\mu_0)} - \\tilde{g}^*{(\\mathbf{v},\\mu_0)}) d\\mu_0} = \\frac{\\mathbf{v} \\log{(\\mu_0)}}{\\int (\\mathbf{v} \\log{(\\mu_0)} - \\tilde{g}^*{(\\mathbf{v},\\mu_0)}) d\\mu_0} and \\frac{\\tilde{g}^*{(\\mathbf{v},\\mu_0)}}{\\int 0 d\\mu_0} = \\frac{\\mathbf{v} \\log{(\\mu_0)}}{\\int 0 d\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 1, "Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{S}{(v_{t},\\hbar)} = \\frac{\\hbar}{v_{t}}, then obtain 1 = (\\frac{\\frac{\\hbar}{v_{t}} - v_{t}}{- v_{t} + \\mathbf{S}{(v_{t},\\hbar)}})^{\\hbar}", "derivation": "\\mathbf{S}{(v_{t},\\hbar)} = \\frac{\\hbar}{v_{t}} and - v_{t} + \\mathbf{S}{(v_{t},\\hbar)} = \\frac{\\hbar}{v_{t}} - v_{t} and 1 = \\frac{\\frac{\\hbar}{v_{t}} - v_{t}}{- v_{t} + \\mathbf{S}{(v_{t},\\hbar)}} and 1 = (\\frac{\\frac{\\hbar}{v_{t}} - v_{t}}{- v_{t} + \\mathbf{S}{(v_{t},\\hbar)}})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('v_t', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{S}')(Symbol('v_t', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{S}')(Symbol('v_t', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{S}')(Symbol('v_t', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\mathbf{S}')(Symbol('v_t', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(A)} = \\sin{(A)}, then derive \\frac{d}{d A} \\operatorname{v_{x}}{(A)} = \\cos{(A)}, then obtain \\frac{d}{d A} \\operatorname{v_{x}}{(A)} - 1 = \\frac{d}{d A} \\sin{(A)} - 1", "derivation": "\\operatorname{v_{x}}{(A)} = \\sin{(A)} and \\frac{d}{d A} \\operatorname{v_{x}}{(A)} = \\frac{d}{d A} \\sin{(A)} and \\frac{d}{d A} \\operatorname{v_{x}}{(A)} = \\cos{(A)} and \\frac{d}{d A} \\operatorname{v_{x}}{(A)} - 1 = \\cos{(A)} - 1 and \\frac{d}{d A} \\sin{(A)} - 1 = \\cos{(A)} - 1 and \\frac{d}{d A} \\operatorname{v_{x}}{(A)} - 1 = \\frac{d}{d A} \\sin{(A)} - 1", "srepr_derivation": [["get_premise", "Equality(Function('v_x')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), cos(Symbol('A', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('v_x')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('A', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('A', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('v_x')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} and \\operatorname{C_{1}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})}, then obtain 3 f_{\\mathbf{p}} = 2 f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})}", "derivation": "\\operatorname{v_{x}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} and 2 \\operatorname{v_{x}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})} and f_{\\mathbf{p}} + 2 \\operatorname{v_{x}}{(f_{\\mathbf{p}})} = 2 f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})} and \\operatorname{C_{1}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})} and \\operatorname{C_{1}}{(f_{\\mathbf{p}})} = 2 f_{\\mathbf{p}} and 2 f_{\\mathbf{p}} = f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})} and 2 f_{\\mathbf{p}} = 2 \\operatorname{v_{x}}{(f_{\\mathbf{p}})} and 3 f_{\\mathbf{p}} = 2 f_{\\mathbf{p}} + \\operatorname{v_{x}}{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["add", 1, "Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Integer(2), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(2), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('C_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(2), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Mul(Integer(3), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\lambda{(v_{1},A_{y})} = A_{y} v_{1}, then derive \\frac{\\frac{\\partial}{\\partial v_{1}} \\lambda{(v_{1},A_{y})}}{\\lambda{(v_{1},A_{y})}} = \\frac{1}{v_{1}}, then obtain \\frac{\\partial}{\\partial v_{1}} \\frac{\\frac{\\partial}{\\partial v_{1}} \\lambda{(v_{1},A_{y})}}{\\lambda{(v_{1},A_{y})}} = \\frac{d}{d v_{1}} \\frac{1}{v_{1}}", "derivation": "\\lambda{(v_{1},A_{y})} = A_{y} v_{1} and \\log{(\\lambda{(v_{1},A_{y})})} = \\log{(A_{y} v_{1})} and \\frac{\\partial}{\\partial v_{1}} \\log{(\\lambda{(v_{1},A_{y})})} = \\frac{\\partial}{\\partial v_{1}} \\log{(A_{y} v_{1})} and \\frac{\\frac{\\partial}{\\partial v_{1}} \\lambda{(v_{1},A_{y})}}{\\lambda{(v_{1},A_{y})}} = \\frac{1}{v_{1}} and \\frac{\\partial}{\\partial v_{1}} \\frac{\\frac{\\partial}{\\partial v_{1}} \\lambda{(v_{1},A_{y})}}{\\lambda{(v_{1},A_{y})}} = \\frac{d}{d v_{1}} \\frac{1}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True))), log(Mul(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(log(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Symbol('v_1', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('v_1', commutative=True), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(F_{H},t_{2})} = F_{H} t_{2}, then obtain t_{2} (F_{H} + 2 \\mathbf{F}{(F_{H},t_{2})}) = t_{2} (2 F_{H} t_{2} + F_{H})", "derivation": "\\mathbf{F}{(F_{H},t_{2})} = F_{H} t_{2} and F_{H} + \\mathbf{F}{(F_{H},t_{2})} = F_{H} t_{2} + F_{H} and F_{H} t_{2} + F_{H} + \\mathbf{F}{(F_{H},t_{2})} = 2 F_{H} t_{2} + F_{H} and F_{H} + 2 \\mathbf{F}{(F_{H},t_{2})} = 2 F_{H} t_{2} + F_{H} and t_{2} (F_{H} + 2 \\mathbf{F}{(F_{H},t_{2})}) = t_{2} (2 F_{H} t_{2} + F_{H})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))), Add(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True)))"], [["add", 1, "Add(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('F_H', commutative=True), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)))), Add(Mul(Integer(2), Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True)))"], [["times", 4, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Add(Symbol('F_H', commutative=True), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))))), Mul(Symbol('t_2', commutative=True), Add(Mul(Integer(2), Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(x^\\prime)} = \\cos{(x^\\prime)} and \\operatorname{v_{2}}{(x^\\prime)} = \\cos^{x^\\prime}{(x^\\prime)}, then obtain \\frac{d}{d x^\\prime} (\\log{(\\mathbf{s}^{x^\\prime}{(x^\\prime)})} - 1) = \\frac{d}{d x^\\prime} (\\log{(\\operatorname{v_{2}}{(x^\\prime)})} - 1)", "derivation": "\\mathbf{s}{(x^\\prime)} = \\cos{(x^\\prime)} and \\mathbf{s}^{x^\\prime}{(x^\\prime)} = \\cos^{x^\\prime}{(x^\\prime)} and \\log{(\\mathbf{s}^{x^\\prime}{(x^\\prime)})} = \\log{(\\cos^{x^\\prime}{(x^\\prime)})} and \\operatorname{v_{2}}{(x^\\prime)} = \\cos^{x^\\prime}{(x^\\prime)} and \\log{(\\mathbf{s}^{x^\\prime}{(x^\\prime)})} = \\log{(\\operatorname{v_{2}}{(x^\\prime)})} and \\log{(\\mathbf{s}^{x^\\prime}{(x^\\prime)})} - 1 = \\log{(\\operatorname{v_{2}}{(x^\\prime)})} - 1 and \\frac{d}{d x^\\prime} (\\log{(\\mathbf{s}^{x^\\prime}{(x^\\prime)})} - 1) = \\frac{d}{d x^\\prime} (\\log{(\\operatorname{v_{2}}{(x^\\prime)})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), log(Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), log(Function('v_2')(Symbol('x^\\\\prime', commutative=True))))"], [["add", 5, "Integer(-1)"], "Equality(Add(log(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Add(log(Function('v_2')(Symbol('x^\\\\prime', commutative=True))), Integer(-1)))"], [["differentiate", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(log(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(log(Function('v_2')(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\varepsilon_0,C_{2})} = C_{2}^{\\varepsilon_0} and \\operatorname{t_{1}}{(\\varepsilon_0,C_{2})} = \\int V{(\\varepsilon_0,C_{2})} d\\varepsilon_0, then obtain \\frac{\\operatorname{t_{1}}{(\\varepsilon_0,C_{2})}}{V{(\\varepsilon_0,C_{2})}} = \\frac{\\int C_{2}^{\\varepsilon_0} d\\varepsilon_0}{V{(\\varepsilon_0,C_{2})}}", "derivation": "V{(\\varepsilon_0,C_{2})} = C_{2}^{\\varepsilon_0} and \\int V{(\\varepsilon_0,C_{2})} d\\varepsilon_0 = \\int C_{2}^{\\varepsilon_0} d\\varepsilon_0 and \\frac{\\int V{(\\varepsilon_0,C_{2})} d\\varepsilon_0}{V{(\\varepsilon_0,C_{2})}} = \\frac{\\int C_{2}^{\\varepsilon_0} d\\varepsilon_0}{V{(\\varepsilon_0,C_{2})}} and \\operatorname{t_{1}}{(\\varepsilon_0,C_{2})} = \\int V{(\\varepsilon_0,C_{2})} d\\varepsilon_0 and \\frac{\\operatorname{t_{1}}{(\\varepsilon_0,C_{2})}}{V{(\\varepsilon_0,C_{2})}} = \\frac{\\int C_{2}^{\\varepsilon_0} d\\varepsilon_0}{V{(\\varepsilon_0,C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Pow(Symbol('C_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 2, "Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Integral(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Pow(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Integral(Pow(Symbol('C_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Integral(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Function('t_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Function('V')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Integral(Pow(Symbol('C_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t,\\mathbf{g})} = \\int (\\mathbf{g} - t) dt, then obtain \\cos{(\\int (- t + \\operatorname{F_{g}}{(t,\\mathbf{g})}) d\\mathbf{g})} = \\cos{(\\int (- t + \\int (\\mathbf{g} - t) dt) d\\mathbf{g})}", "derivation": "\\operatorname{F_{g}}{(t,\\mathbf{g})} = \\int (\\mathbf{g} - t) dt and - t + \\operatorname{F_{g}}{(t,\\mathbf{g})} = - t + \\int (\\mathbf{g} - t) dt and \\int (- t + \\operatorname{F_{g}}{(t,\\mathbf{g})}) d\\mathbf{g} = \\int (- t + \\int (\\mathbf{g} - t) dt) d\\mathbf{g} and \\cos{(\\int (- t + \\operatorname{F_{g}}{(t,\\mathbf{g})}) d\\mathbf{g})} = \\cos{(\\int (- t + \\int (\\mathbf{g} - t) dt) d\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["minus", 1, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integral(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integral(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('F_g')(Symbol('t', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), cos(Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integral(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\chi{(z^{*},H)} = \\frac{z^{*}}{H}, then obtain - (\\frac{z^{*}}{H})^{z^{*}} + \\frac{\\partial^{2}}{\\partial H^{2}} \\chi^{z^{*}}{(z^{*},H)} = - (\\frac{z^{*}}{H})^{z^{*}} + \\frac{\\partial^{2}}{\\partial H^{2}} (\\frac{z^{*}}{H})^{z^{*}}", "derivation": "\\chi{(z^{*},H)} = \\frac{z^{*}}{H} and \\chi^{z^{*}}{(z^{*},H)} = (\\frac{z^{*}}{H})^{z^{*}} and \\frac{\\partial}{\\partial H} \\chi^{z^{*}}{(z^{*},H)} = \\frac{\\partial}{\\partial H} (\\frac{z^{*}}{H})^{z^{*}} and \\frac{\\partial^{2}}{\\partial H^{2}} \\chi^{z^{*}}{(z^{*},H)} = \\frac{\\partial^{2}}{\\partial H^{2}} (\\frac{z^{*}}{H})^{z^{*}} and - (\\frac{z^{*}}{H})^{z^{*}} + \\frac{\\partial^{2}}{\\partial H^{2}} \\chi^{z^{*}}{(z^{*},H)} = - (\\frac{z^{*}}{H})^{z^{*}} + \\frac{\\partial^{2}}{\\partial H^{2}} (\\frac{z^{*}}{H})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], [["power", 1, "Symbol('z^*', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Symbol('z^*', commutative=True)), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["minus", 4, "Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))), Derivative(Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))), Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2)))))"]]}, {"prompt": "Given b{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})} and f{(\\mathbf{g},t_{2})} = b{(\\mathbf{g},t_{2})} - \\frac{\\partial}{\\partial \\mathbf{g}} b{(\\mathbf{g},t_{2})}, then derive f{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})} - t_{2} \\cos{(\\mathbf{g})}, then obtain t_{2} \\cos{(\\mathbf{g})} + f{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})}", "derivation": "b{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})} and f{(\\mathbf{g},t_{2})} = b{(\\mathbf{g},t_{2})} - \\frac{\\partial}{\\partial \\mathbf{g}} b{(\\mathbf{g},t_{2})} and f{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})} - \\frac{\\partial}{\\partial \\mathbf{g}} t_{2} \\sin{(\\mathbf{g})} and f{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})} - t_{2} \\cos{(\\mathbf{g})} and t_{2} \\cos{(\\mathbf{g})} + f{(\\mathbf{g},t_{2})} = t_{2} \\sin{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Add(Function('b')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), Derivative(Function('b')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Function('f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True), cos(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["minus", 4, "Mul(Integer(-1), Symbol('t_2', commutative=True), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Mul(Symbol('t_2', commutative=True), cos(Symbol('\\\\mathbf{g}', commutative=True))), Function('f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('t_2', commutative=True))), Mul(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(C_{d})} = C_{d}, then derive \\frac{d}{d C_{d}} \\hat{X}{(C_{d})} + 1 = 2, then obtain \\frac{d}{d C_{d}} C_{d} + 1 = 2", "derivation": "\\hat{X}{(C_{d})} = C_{d} and C_{d} + \\hat{X}{(C_{d})} = 2 C_{d} and \\frac{d}{d C_{d}} (C_{d} + \\hat{X}{(C_{d})}) = \\frac{d}{d C_{d}} 2 C_{d} and \\frac{d}{d C_{d}} \\hat{X}{(C_{d})} + 1 = 2 and \\frac{d}{d C_{d}} C_{d} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["add", 1, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Function('\\\\hat{X}')(Symbol('C_d', commutative=True))), Mul(Integer(2), Symbol('C_d', commutative=True)))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Symbol('C_d', commutative=True), Function('\\\\hat{X}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\hat{X}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(1)), Integer(2))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{v})} = \\cos{(\\log{(\\mathbf{v})})}, then obtain \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + \\int \\operatorname{v_{y}}{(\\mathbf{v})} d\\mathbf{v}) = \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + \\int \\cos{(\\log{(\\mathbf{v})})} d\\mathbf{v})", "derivation": "\\operatorname{v_{y}}{(\\mathbf{v})} = \\cos{(\\log{(\\mathbf{v})})} and \\int \\operatorname{v_{y}}{(\\mathbf{v})} d\\mathbf{v} = \\int \\cos{(\\log{(\\mathbf{v})})} d\\mathbf{v} and \\mathbf{v} + \\int \\operatorname{v_{y}}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} + \\int \\cos{(\\log{(\\mathbf{v})})} d\\mathbf{v} and \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + \\int \\operatorname{v_{y}}{(\\mathbf{v})} d\\mathbf{v}) = \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + \\int \\cos{(\\log{(\\mathbf{v})})} d\\mathbf{v})", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), cos(log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(cos(log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Integral(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Symbol('\\\\mathbf{v}', commutative=True), Integral(cos(log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Integral(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Integral(cos(log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(f_{E})} = \\frac{d}{d f_{E}} \\log{(f_{E})}, then derive \\mathbf{v}{(f_{E})} = \\frac{1}{f_{E}}, then obtain \\mathbf{v}{(\\frac{1}{\\frac{d}{d f_{E}} \\log{(f_{E})}})} + 1 = \\frac{d}{d f_{E}} \\log{(f_{E})} + 1", "derivation": "\\mathbf{v}{(f_{E})} = \\frac{d}{d f_{E}} \\log{(f_{E})} and \\mathbf{v}{(f_{E})} = \\frac{1}{f_{E}} and \\mathbf{v}{(f_{E})} + 1 = 1 + \\frac{1}{f_{E}} and \\frac{d}{d f_{E}} \\log{(f_{E})} = \\frac{1}{f_{E}} and \\mathbf{v}{(\\frac{1}{\\frac{d}{d f_{E}} \\log{(f_{E})}})} + 1 = \\frac{d}{d f_{E}} \\log{(f_{E})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('f_E', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{v}')(Symbol('f_E', commutative=True)), Pow(Symbol('f_E', commutative=True), Integer(-1)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('f_E', commutative=True)), Integer(1)), Add(Integer(1), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Pow(Symbol('f_E', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\mathbf{v}')(Pow(Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1))), Integer(1)), Add(Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{S},W)} = - \\mathbf{S} + e^{W}, then obtain \\operatorname{F_{N}}{(\\mathbf{S},W)} + \\operatorname{F_{N}}^{W}{(\\mathbf{S},W)} = - \\mathbf{S} + \\operatorname{F_{N}}^{W}{(\\mathbf{S},W)} + e^{W}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{S},W)} = - \\mathbf{S} + e^{W} and \\operatorname{F_{N}}^{W}{(\\mathbf{S},W)} = (- \\mathbf{S} + e^{W})^{W} and (- \\mathbf{S} + e^{W})^{W} + \\operatorname{F_{N}}{(\\mathbf{S},W)} = - \\mathbf{S} + (- \\mathbf{S} + e^{W})^{W} + e^{W} and \\operatorname{F_{N}}{(\\mathbf{S},W)} + \\operatorname{F_{N}}^{W}{(\\mathbf{S},W)} = - \\mathbf{S} + \\operatorname{F_{N}}^{W}{(\\mathbf{S},W)} + e^{W}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('W', commutative=True))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["add", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True)), Pow(Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('F_N')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(G)} = \\log{(e^{G})}, then obtain - 2 G (- 2 G + \\hat{H}_{\\lambda}{(G)}) - 2 G = - 2 G (- 2 G + \\log{(e^{G})}) - 2 G", "derivation": "\\hat{H}_{\\lambda}{(G)} = \\log{(e^{G})} and - G + \\hat{H}_{\\lambda}{(G)} = - G + \\log{(e^{G})} and - 2 G + \\hat{H}_{\\lambda}{(G)} = - 2 G + \\log{(e^{G})} and - 2 G (- 2 G + \\hat{H}_{\\lambda}{(G)}) = - 2 G (- 2 G + \\log{(e^{G})}) and - 2 G (- 2 G + \\hat{H}_{\\lambda}{(G)}) - 2 G = - 2 G (- 2 G + \\log{(e^{G})}) - 2 G", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True))))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Integer(2), Symbol('G', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True))))))"], [["add", 4, "Mul(Integer(-1), Integer(2), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True))))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\mathbf{M})} = \\sin{(\\mathbf{M})}, then obtain (\\frac{d}{d \\mathbf{M}} \\omega^{\\mathbf{M}}{(\\mathbf{M})})^{\\mathbf{M}} = (\\frac{d}{d \\mathbf{M}} \\sin^{\\mathbf{M}}{(\\mathbf{M})})^{\\mathbf{M}}", "derivation": "\\omega{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\omega^{\\mathbf{M}}{(\\mathbf{M})} = \\sin^{\\mathbf{M}}{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} \\omega^{\\mathbf{M}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin^{\\mathbf{M}}{(\\mathbf{M})} and (\\frac{d}{d \\mathbf{M}} \\omega^{\\mathbf{M}}{(\\mathbf{M})})^{\\mathbf{M}} = (\\frac{d}{d \\mathbf{M}} \\sin^{\\mathbf{M}}{(\\mathbf{M})})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Derivative(Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(\\tilde{g},B)} = \\log{(B - \\tilde{g})}, then derive \\frac{\\partial}{\\partial B} \\mathbf{p}{(\\tilde{g},B)} - 1 = -1 + \\frac{1}{B - \\tilde{g}}, then obtain - \\tilde{g} + \\frac{\\partial}{\\partial B} \\log{(B - \\tilde{g})} - 1 = - \\tilde{g} - 1 + \\frac{1}{B - \\tilde{g}}", "derivation": "\\mathbf{p}{(\\tilde{g},B)} = \\log{(B - \\tilde{g})} and - B + \\tilde{g} + \\mathbf{p}{(\\tilde{g},B)} = - B + \\tilde{g} + \\log{(B - \\tilde{g})} and \\frac{\\partial}{\\partial B} (- B + \\tilde{g} + \\mathbf{p}{(\\tilde{g},B)}) = \\frac{\\partial}{\\partial B} (- B + \\tilde{g} + \\log{(B - \\tilde{g})}) and \\frac{\\partial}{\\partial B} \\mathbf{p}{(\\tilde{g},B)} - 1 = -1 + \\frac{1}{B - \\tilde{g}} and - \\tilde{g} + \\frac{\\partial}{\\partial B} \\mathbf{p}{(\\tilde{g},B)} - 1 = - \\tilde{g} - 1 + \\frac{1}{B - \\tilde{g}} and - \\tilde{g} + \\frac{\\partial}{\\partial B} \\log{(B - \\tilde{g})} - 1 = - \\tilde{g} - 1 + \\frac{1}{B - \\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), log(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))))"], [["minus", 1, "Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), log(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), log(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1))))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1), Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Derivative(log(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1), Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(B,F_{H})} = B - F_{H} and \\operatorname{C_{d}}{(B,F_{H})} = \\frac{\\partial}{\\partial B} (B - F_{H}), then obtain (\\frac{\\partial}{\\partial B} \\hat{x}{(B,F_{H})})^{B} = (\\frac{\\partial}{\\partial B} (B - F_{H}))^{B}", "derivation": "\\hat{x}{(B,F_{H})} = B - F_{H} and \\frac{\\partial}{\\partial B} \\hat{x}{(B,F_{H})} = \\frac{\\partial}{\\partial B} (B - F_{H}) and \\operatorname{C_{d}}{(B,F_{H})} = \\frac{\\partial}{\\partial B} (B - F_{H}) and \\operatorname{C_{d}}^{B}{(B,F_{H})} = (\\frac{\\partial}{\\partial B} (B - F_{H}))^{B} and \\operatorname{C_{d}}{(B,F_{H})} = \\frac{\\partial}{\\partial B} \\hat{x}{(B,F_{H})} and (\\frac{\\partial}{\\partial B} \\hat{x}{(B,F_{H})})^{B} = (\\frac{\\partial}{\\partial B} (B - F_{H}))^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('C_d')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\hat{x}')(Symbol('B', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\Psi,\\mathbf{r})} = \\sin{(\\Psi^{\\mathbf{r}})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} = \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})}, then obtain \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})} + \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} = 2 \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})}", "derivation": "\\operatorname{P_{e}}{(\\Psi,\\mathbf{r})} = \\sin{(\\Psi^{\\mathbf{r}})} and \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})} = \\sin^{\\Psi}{(\\Psi^{\\mathbf{r}})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} = \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} = \\sin^{\\Psi}{(\\Psi^{\\mathbf{r}})} and \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} + \\sin^{\\Psi}{(\\Psi^{\\mathbf{r}})} = 2 \\sin^{\\Psi}{(\\Psi^{\\mathbf{r}})} and \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})} + \\operatorname{V_{\\mathbf{E}}}{(\\Psi,\\mathbf{r})} = 2 \\operatorname{P_{e}}^{\\Psi}{(\\Psi,\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('P_e')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["add", 4, "Pow(sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), Pow(sin(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Pow(Function('P_e')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(2), Pow(Function('P_e')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given u{(m,\\mathbf{H})} = \\sin{(\\frac{\\mathbf{H}}{m})}, then obtain \\frac{-1 + \\frac{u{(m,\\mathbf{H})}}{\\mathbf{H}}}{-1 + \\frac{\\sin{(\\frac{\\mathbf{H}}{m})}}{\\mathbf{H}}} = 1", "derivation": "u{(m,\\mathbf{H})} = \\sin{(\\frac{\\mathbf{H}}{m})} and \\frac{u{(m,\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\sin{(\\frac{\\mathbf{H}}{m})}}{\\mathbf{H}} and -1 + \\frac{u{(m,\\mathbf{H})}}{\\mathbf{H}} = -1 + \\frac{\\sin{(\\frac{\\mathbf{H}}{m})}}{\\mathbf{H}} and \\frac{-1 + \\frac{u{(m,\\mathbf{H})}}{\\mathbf{H}}}{-1 + \\frac{\\sin{(\\frac{\\mathbf{H}}{m})}}{\\mathbf{H}}} = 1", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["divide", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('u')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('u')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))))"], [["divide", 3, "Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))))"], "Equality(Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('u')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{M}{(r_{0},f^{\\prime})} = f^{\\prime} + r_{0}, then obtain - f^{\\prime} + r_{0} + 2 (- f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})})^{r_{0}} + \\mathbf{M}{(r_{0},f^{\\prime})} = 2 r_{0} + 2 (- f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})})^{r_{0}}", "derivation": "\\mathbf{M}{(r_{0},f^{\\prime})} = f^{\\prime} + r_{0} and - f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})} = r_{0} and (- f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})})^{r_{0}} = r_{0}^{r_{0}} and - f^{\\prime} + r_{0}^{r_{0}} + \\mathbf{M}{(r_{0},f^{\\prime})} = r_{0} + r_{0}^{r_{0}} and - f^{\\prime} + r_{0} + 2 r_{0}^{r_{0}} + \\mathbf{M}{(r_{0},f^{\\prime})} = 2 r_{0} + 2 r_{0}^{r_{0}} and - f^{\\prime} + r_{0} + 2 (- f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})})^{r_{0}} + \\mathbf{M}{(r_{0},f^{\\prime})} = 2 r_{0} + 2 (- f^{\\prime} + \\mathbf{M}{(r_{0},f^{\\prime})})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('r_0', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('r_0', commutative=True))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 2, "Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True)), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('r_0', commutative=True), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 4, "Add(Symbol('r_0', commutative=True), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True))), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('r_0', commutative=True), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('r_0', commutative=True))), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\mathbf{M}{(\\mathbf{A})} = (e^{\\mathbf{A}})^{\\mathbf{A}}, then obtain \\mathbf{M}{(\\mathbf{A})} - (e^{\\mathbf{A}})^{\\mathbf{A}} = 0", "derivation": "\\operatorname{A_{2}}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\operatorname{A_{2}}^{\\mathbf{A}}{(\\mathbf{A})} = (e^{\\mathbf{A}})^{\\mathbf{A}} and \\operatorname{A_{2}}^{\\mathbf{A}}{(\\mathbf{A})} - (e^{\\mathbf{A}})^{\\mathbf{A}} = 0 and \\mathbf{M}{(\\mathbf{A})} = (e^{\\mathbf{A}})^{\\mathbf{A}} and \\mathbf{M}{(\\mathbf{A})} = \\operatorname{A_{2}}^{\\mathbf{A}}{(\\mathbf{A})} and \\mathbf{M}{(\\mathbf{A})} - (e^{\\mathbf{A}})^{\\mathbf{A}} = 0", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 2, "Pow(exp(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Pow(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(q,f_{E})} = \\frac{q}{f_{E}} and \\operatorname{v_{1}}{(q,f_{E})} = \\frac{2 q}{f_{E}}, then obtain \\operatorname{v_{1}}^{q}{(q,f_{E})} = (\\dot{\\mathbf{r}}{(q,f_{E})} + \\frac{q}{f_{E}})^{q}", "derivation": "\\dot{\\mathbf{r}}{(q,f_{E})} = \\frac{q}{f_{E}} and \\dot{\\mathbf{r}}{(q,f_{E})} + \\frac{q}{f_{E}} = \\frac{2 q}{f_{E}} and \\operatorname{v_{1}}{(q,f_{E})} = \\frac{2 q}{f_{E}} and \\operatorname{v_{1}}{(q,f_{E})} = \\dot{\\mathbf{r}}{(q,f_{E})} + \\frac{q}{f_{E}} and \\operatorname{v_{1}}^{q}{(q,f_{E})} = (\\dot{\\mathbf{r}}{(q,f_{E})} + \\frac{q}{f_{E}})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Mul(Integer(2), Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(2), Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('v_1')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True))))"], [["power", 4, "Symbol('q', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Symbol('q', commutative=True)), Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}, then obtain \\frac{d^{2}}{d \\hat{x}^{2}} (\\operatorname{F_{H}}^{\\hat{x}}{(\\hat{x})} + \\log{(\\sin{(\\hat{x})})}^{\\hat{x}}) = \\frac{d^{2}}{d \\hat{x}^{2}} 2 \\log{(\\sin{(\\hat{x})})}^{\\hat{x}}", "derivation": "\\operatorname{F_{H}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})} and \\operatorname{F_{H}}^{\\hat{x}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} and \\operatorname{F_{H}}^{\\hat{x}}{(\\hat{x})} + \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} = 2 \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} and \\frac{d}{d \\hat{x}} (\\operatorname{F_{H}}^{\\hat{x}}{(\\hat{x})} + \\log{(\\sin{(\\hat{x})})}^{\\hat{x}}) = \\frac{d}{d \\hat{x}} 2 \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} and \\frac{d^{2}}{d \\hat{x}^{2}} (\\operatorname{F_{H}}^{\\hat{x}}{(\\hat{x})} + \\log{(\\sin{(\\hat{x})})}^{\\hat{x}}) = \\frac{d^{2}}{d \\hat{x}^{2}} 2 \\log{(\\sin{(\\hat{x})})}^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["add", 2, "Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Pow(Function('F_H')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(2), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Pow(Function('F_H')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Pow(Function('F_H')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Derivative(Mul(Integer(2), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\Psi{(m,\\mathbb{I})} = \\log{(m^{\\mathbb{I}})} and \\operatorname{f^{*}}{(m,\\mathbb{I})} = 2 \\log{(m^{\\mathbb{I}})}, then obtain \\int \\operatorname{f^{*}}{(m,\\mathbb{I})} d\\mathbb{I} = \\int 2 \\log{(m^{\\mathbb{I}})} d\\mathbb{I}", "derivation": "\\Psi{(m,\\mathbb{I})} = \\log{(m^{\\mathbb{I}})} and \\Psi{(m,\\mathbb{I})} + \\log{(m^{\\mathbb{I}})} = 2 \\log{(m^{\\mathbb{I}})} and \\operatorname{f^{*}}{(m,\\mathbb{I})} = 2 \\log{(m^{\\mathbb{I}})} and \\operatorname{f^{*}}{(m,\\mathbb{I})} = \\Psi{(m,\\mathbb{I})} + \\log{(m^{\\mathbb{I}})} and \\int \\operatorname{f^{*}}{(m,\\mathbb{I})} d\\mathbb{I} = \\int (\\Psi{(m,\\mathbb{I})} + \\log{(m^{\\mathbb{I}})}) d\\mathbb{I} and \\int \\operatorname{f^{*}}{(m,\\mathbb{I})} d\\mathbb{I} = \\int 2 \\log{(m^{\\mathbb{I}})} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 1, "log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Integer(2), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('f^*')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Function('f^*')(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Integer(2), log(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given l{(v_{2},\\psi^*)} = (e^{\\psi^*})^{v_{2}}, then derive v_{2} + z^{*} = \\int (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} dv_{2}, then obtain \\frac{\\partial}{\\partial z^{*}} (v_{2} + z^{*}) = \\frac{\\partial}{\\partial z^{*}} \\int (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} dv_{2}", "derivation": "l{(v_{2},\\psi^*)} = (e^{\\psi^*})^{v_{2}} and 1 = \\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}} and 1 = (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} and \\int 1 dv_{2} = \\int (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} dv_{2} and v_{2} + z^{*} = \\int (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} dv_{2} and \\frac{\\partial}{\\partial z^{*}} (v_{2} + z^{*}) = \\frac{\\partial}{\\partial z^{*}} \\int (\\frac{(e^{\\psi^*})^{v_{2}}}{l{(v_{2},\\psi^*)}})^{v_{2}} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True)))"], [["divide", 1, "Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True))))"], [["power", 2, "Symbol('v_2', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))), Integral(Pow(Mul(Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('v_2', commutative=True), Symbol('z^*', commutative=True)), Integral(Pow(Mul(Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Add(Symbol('v_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(I,\\mathbf{g})} = \\frac{I}{\\mathbf{g}}, then derive \\frac{\\partial}{\\partial \\mathbf{g}} \\phi_{1}{(I,\\mathbf{g})} = - \\frac{I}{\\mathbf{g}^{2}}, then obtain - \\frac{I}{\\mathbf{g}^{2}} = \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{I}{\\mathbf{g}}", "derivation": "\\phi_{1}{(I,\\mathbf{g})} = \\frac{I}{\\mathbf{g}} and \\frac{\\partial}{\\partial \\mathbf{g}} \\phi_{1}{(I,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{I}{\\mathbf{g}} and \\frac{\\partial}{\\partial \\mathbf{g}} \\phi_{1}{(I,\\mathbf{g})} = - \\frac{I}{\\mathbf{g}^{2}} and - \\frac{I}{\\mathbf{g}^{2}} = \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{I}{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2))), Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = \\hbar m_{s}, then derive \\frac{\\partial}{\\partial \\hbar} \\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = m_{s}, then obtain \\hbar + \\operatorname{g_{\\varepsilon}}{(\\frac{\\partial}{\\partial \\hbar} \\hbar m_{s},\\hbar)} = \\hbar \\frac{\\partial}{\\partial \\hbar} \\hbar m_{s} + \\hbar", "derivation": "\\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = \\hbar m_{s} and \\hbar + \\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = \\hbar m_{s} + \\hbar and \\frac{\\partial}{\\partial \\hbar} \\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\hbar m_{s} and \\frac{\\partial}{\\partial \\hbar} \\operatorname{g_{\\varepsilon}}{(m_{s},\\hbar)} = m_{s} and \\frac{\\partial}{\\partial \\hbar} \\hbar m_{s} = m_{s} and \\hbar + \\operatorname{g_{\\varepsilon}}{(\\frac{\\partial}{\\partial \\hbar} \\hbar m_{s},\\hbar)} = \\hbar \\frac{\\partial}{\\partial \\hbar} \\hbar m_{s} + \\hbar", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('m_s', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)))"], [["add", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('m_s', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('m_s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('m_s', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('m_s', commutative=True))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('m_s', commutative=True))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('g_{\\\\varepsilon}')(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given z{(E_{x},G)} = E_{x} + \\sin{(G)}, then obtain \\iint (- E_{x} + z{(E_{x},G)} - \\sin{(G)}) dG dG = \\iint 0 dG dG", "derivation": "z{(E_{x},G)} = E_{x} + \\sin{(G)} and - E_{x} + z{(E_{x},G)} - \\sin{(G)} = 0 and \\int (- E_{x} + z{(E_{x},G)} - \\sin{(G)}) dG = \\int 0 dG and \\iint (- E_{x} + z{(E_{x},G)} - \\sin{(G)}) dG dG = \\iint 0 dG dG", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E_x', commutative=True), Symbol('G', commutative=True)), Add(Symbol('E_x', commutative=True), sin(Symbol('G', commutative=True))))"], [["minus", 1, "Add(Symbol('E_x', commutative=True), sin(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('z')(Symbol('E_x', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('z')(Symbol('E_x', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Integral(Integer(0), Tuple(Symbol('G', commutative=True))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('z')(Symbol('E_x', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Integer(0), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(z^{*})} = \\cos{(z^{*})}, then derive \\int \\operatorname{E_{\\lambda}}{(z^{*})} dz^{*} = \\mathbf{J}_f + \\sin{(z^{*})}, then obtain \\mathbf{J}_f - \\omega = 0", "derivation": "\\operatorname{E_{\\lambda}}{(z^{*})} = \\cos{(z^{*})} and \\int \\operatorname{E_{\\lambda}}{(z^{*})} dz^{*} = \\int \\cos{(z^{*})} dz^{*} and \\int \\operatorname{E_{\\lambda}}{(z^{*})} dz^{*} = \\mathbf{J}_f + \\sin{(z^{*})} and \\int \\operatorname{E_{\\lambda}}{(z^{*})} dz^{*} - \\int \\cos{(z^{*})} dz^{*} = 0 and \\mathbf{J}_f + \\sin{(z^{*})} - \\int \\cos{(z^{*})} dz^{*} = 0 and \\mathbf{J}_f - \\omega = 0", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('z^*', commutative=True))))"], [["minus", 2, "Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))"], "Equality(Add(Integral(Function('E_{\\\\lambda}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('z^*', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))), Integer(0))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\theta_1)} = e^{\\theta_1}, then derive \\frac{d}{d \\theta_1} \\int \\operatorname{v_{z}}{(\\theta_1)} d\\theta_1 = \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + e^{\\theta_1}), then obtain \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + e^{\\theta_1}) = \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + \\operatorname{v_{z}}{(\\theta_1)})", "derivation": "\\operatorname{v_{z}}{(\\theta_1)} = e^{\\theta_1} and \\int \\operatorname{v_{z}}{(\\theta_1)} d\\theta_1 = \\int e^{\\theta_1} d\\theta_1 and \\frac{d}{d \\theta_1} \\int \\operatorname{v_{z}}{(\\theta_1)} d\\theta_1 = \\frac{d}{d \\theta_1} \\int e^{\\theta_1} d\\theta_1 and \\frac{d}{d \\theta_1} \\int \\operatorname{v_{z}}{(\\theta_1)} d\\theta_1 = \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + e^{\\theta_1}) and \\frac{d}{d \\theta_1} \\int \\operatorname{v_{z}}{(\\theta_1)} d\\theta_1 = \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + \\operatorname{v_{z}}{(\\theta_1)}) and \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + e^{\\theta_1}) = \\frac{\\partial}{\\partial \\theta_1} (y^{\\prime} + \\operatorname{v_{z}}{(\\theta_1)})", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integral(Function('v_z')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('v_z')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integral(Function('v_z')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), Function('v_z')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), Function('v_z')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\nabla,\\mu)} = \\nabla^{\\mu}, then obtain - \\nabla + (\\frac{\\partial}{\\partial \\nabla} 2 \\nabla \\mathbf{g}{(\\nabla,\\mu)})^{\\mu} = - \\nabla + (\\frac{\\partial}{\\partial \\nabla} \\nabla (\\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)}))^{\\mu}", "derivation": "\\mathbf{g}{(\\nabla,\\mu)} = \\nabla^{\\mu} and 2 \\mathbf{g}{(\\nabla,\\mu)} = \\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)} and 2 \\nabla \\mathbf{g}{(\\nabla,\\mu)} = \\nabla (\\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)}) and \\frac{\\partial}{\\partial \\nabla} 2 \\nabla \\mathbf{g}{(\\nabla,\\mu)} = \\frac{\\partial}{\\partial \\nabla} \\nabla (\\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)}) and (\\frac{\\partial}{\\partial \\nabla} 2 \\nabla \\mathbf{g}{(\\nabla,\\mu)})^{\\mu} = (\\frac{\\partial}{\\partial \\nabla} \\nabla (\\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)}))^{\\mu} and - \\nabla + (\\frac{\\partial}{\\partial \\nabla} 2 \\nabla \\mathbf{g}{(\\nabla,\\mu)})^{\\mu} = - \\nabla + (\\frac{\\partial}{\\partial \\nabla} \\nabla (\\nabla^{\\mu} + \\mathbf{g}{(\\nabla,\\mu)}))^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["times", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["minus", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Pow(Derivative(Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given k{(\\pi,\\mathbf{M})} = \\mathbf{M}^{\\pi}, then obtain \\frac{\\pi (- 2 k{(\\pi,\\mathbf{M})} + \\frac{k{(\\pi,\\mathbf{M})}}{\\pi})}{k{(\\pi,\\mathbf{M})}} = \\frac{\\pi (\\frac{\\mathbf{M}^{\\pi}}{\\pi} - 2 k{(\\pi,\\mathbf{M})})}{k{(\\pi,\\mathbf{M})}}", "derivation": "k{(\\pi,\\mathbf{M})} = \\mathbf{M}^{\\pi} and \\frac{k{(\\pi,\\mathbf{M})}}{\\pi} = \\frac{\\mathbf{M}^{\\pi}}{\\pi} and - k{(\\pi,\\mathbf{M})} + \\frac{k{(\\pi,\\mathbf{M})}}{\\pi} = \\frac{\\mathbf{M}^{\\pi}}{\\pi} - k{(\\pi,\\mathbf{M})} and - 2 k{(\\pi,\\mathbf{M})} + \\frac{k{(\\pi,\\mathbf{M})}}{\\pi} = \\frac{\\mathbf{M}^{\\pi}}{\\pi} - 2 k{(\\pi,\\mathbf{M})} and \\frac{\\pi (- 2 k{(\\pi,\\mathbf{M})} + \\frac{k{(\\pi,\\mathbf{M})}}{\\pi})}{k{(\\pi,\\mathbf{M})}} = \\frac{\\pi (\\frac{\\mathbf{M}^{\\pi}}{\\pi} - 2 k{(\\pi,\\mathbf{M})})}{k{(\\pi,\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["minus", 2, "Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["minus", 3, "Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["divide", 4, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Integer(2), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Pow(Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Pow(Function('k')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\phi_1,M)} = M \\phi_1, then obtain M^{2} + 1 = M^{2} + \\frac{2 M \\phi_1}{M \\phi_1 + \\operatorname{C_{1}}{(\\phi_1,M)}}", "derivation": "\\operatorname{C_{1}}{(\\phi_1,M)} = M \\phi_1 and M \\phi_1 + \\operatorname{C_{1}}{(\\phi_1,M)} = 2 M \\phi_1 and 1 = \\frac{2 M \\phi_1}{M \\phi_1 + \\operatorname{C_{1}}{(\\phi_1,M)}} and M^{2} + 1 = M^{2} + \\frac{2 M \\phi_1}{M \\phi_1 + \\operatorname{C_{1}}{(\\phi_1,M)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M', commutative=True))), Mul(Integer(2), Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 2, "Add(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M', commutative=True)))"], "Equality(Integer(1), Mul(Integer(2), Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True), Pow(Add(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M', commutative=True))), Integer(-1))))"], [["add", 3, "Pow(Symbol('M', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('M', commutative=True), Integer(2)), Integer(1)), Add(Pow(Symbol('M', commutative=True), Integer(2)), Mul(Integer(2), Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True), Pow(Add(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(E_{n},\\mathbf{J}_M)} = - E_{n} + \\mathbf{J}_M, then obtain - E_{n} (-1 + \\frac{(E_{n} - \\mathbf{J}_M + 1) \\eta^{\\prime}{(E_{n},\\mathbf{J}_M)}}{- E_{n} + \\mathbf{J}_M}) = - E_{n} (E_{n} - \\mathbf{J}_M)", "derivation": "\\eta^{\\prime}{(E_{n},\\mathbf{J}_M)} = - E_{n} + \\mathbf{J}_M and \\frac{\\eta^{\\prime}{(E_{n},\\mathbf{J}_M)}}{- E_{n} + \\mathbf{J}_M} = 1 and \\frac{(E_{n} - \\mathbf{J}_M + 1) \\eta^{\\prime}{(E_{n},\\mathbf{J}_M)}}{- E_{n} + \\mathbf{J}_M} = E_{n} - \\mathbf{J}_M + 1 and -1 + \\frac{(E_{n} - \\mathbf{J}_M + 1) \\eta^{\\prime}{(E_{n},\\mathbf{J}_M)}}{- E_{n} + \\mathbf{J}_M} = E_{n} - \\mathbf{J}_M and - E_{n} (-1 + \\frac{(E_{n} - \\mathbf{J}_M + 1) \\eta^{\\prime}{(E_{n},\\mathbf{J}_M)}}{- E_{n} + \\mathbf{J}_M}) = - E_{n} (E_{n} - \\mathbf{J}_M)", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1))"], [["times", 2, "Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('E_n', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E_n', commutative=True), Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))), Mul(Integer(-1), Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{v},E_{\\lambda})} = E_{\\lambda} \\mathbf{v}, then obtain \\frac{d}{d E_{\\lambda}} \\int 0 dE_{\\lambda} = \\frac{\\partial}{\\partial E_{\\lambda}} \\int (1 - \\frac{\\theta_{1}{(\\mathbf{v},E_{\\lambda})}}{E_{\\lambda} \\mathbf{v}}) dE_{\\lambda}", "derivation": "\\theta_{1}{(\\mathbf{v},E_{\\lambda})} = E_{\\lambda} \\mathbf{v} and \\frac{\\theta_{1}{(\\mathbf{v},E_{\\lambda})}}{E_{\\lambda} \\mathbf{v}} = 1 and 0 = 1 - \\frac{\\theta_{1}{(\\mathbf{v},E_{\\lambda})}}{E_{\\lambda} \\mathbf{v}} and \\int 0 dE_{\\lambda} = \\int (1 - \\frac{\\theta_{1}{(\\mathbf{v},E_{\\lambda})}}{E_{\\lambda} \\mathbf{v}}) dE_{\\lambda} and \\frac{d}{d E_{\\lambda}} \\int 0 dE_{\\lambda} = \\frac{\\partial}{\\partial E_{\\lambda}} \\int (1 - \\frac{\\theta_{1}{(\\mathbf{v},E_{\\lambda})}}{E_{\\lambda} \\mathbf{v}}) dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Integer(1))"], [["minus", 2, "Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["differentiate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integral(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(F_{g},W)} = - W + e^{F_{g}} and g{(W)} = - W, then obtain W + m{(F_{g},W)} e^{F_{g}} = W + (g{(W)} + e^{F_{g}}) e^{F_{g}}", "derivation": "m{(F_{g},W)} = - W + e^{F_{g}} and g{(W)} = - W and m{(F_{g},W)} = g{(W)} + e^{F_{g}} and m{(F_{g},W)} e^{F_{g}} = (g{(W)} + e^{F_{g}}) e^{F_{g}} and W + m{(F_{g},W)} e^{F_{g}} = W + (g{(W)} + e^{F_{g}}) e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('g')(Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('m')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), Add(Function('g')(Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True))))"], [["times", 3, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('m')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True))), Mul(Add(Function('g')(Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True))))"], [["add", 4, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Mul(Function('m')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True)))), Add(Symbol('W', commutative=True), Mul(Add(Function('g')(Symbol('W', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given J{(\\sigma_x,J_{\\varepsilon})} = e^{- J_{\\varepsilon} + \\sigma_x}, then obtain E_{x} J_{\\varepsilon} \\frac{\\partial}{\\partial \\sigma_x} J{(\\sigma_x,J_{\\varepsilon})} = E_{x} J_{\\varepsilon} e^{- J_{\\varepsilon} + \\sigma_x}", "derivation": "J{(\\sigma_x,J_{\\varepsilon})} = e^{- J_{\\varepsilon} + \\sigma_x} and J_{\\varepsilon} J{(\\sigma_x,J_{\\varepsilon})} = J_{\\varepsilon} e^{- J_{\\varepsilon} + \\sigma_x} and E_{x} J_{\\varepsilon} J{(\\sigma_x,J_{\\varepsilon})} = E_{x} J_{\\varepsilon} e^{- J_{\\varepsilon} + \\sigma_x} and \\frac{\\partial}{\\partial \\sigma_x} E_{x} J_{\\varepsilon} J{(\\sigma_x,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\sigma_x} E_{x} J_{\\varepsilon} e^{- J_{\\varepsilon} + \\sigma_x} and E_{x} J_{\\varepsilon} \\frac{\\partial}{\\partial \\sigma_x} J{(\\sigma_x,J_{\\varepsilon})} = E_{x} J_{\\varepsilon} e^{- J_{\\varepsilon} + \\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 2, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Derivative(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Mul(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given y{(\\chi,T)} = \\log{(T + \\chi)}, then obtain - \\frac{\\chi (\\cos{(\\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}})} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}} = - \\frac{\\chi (\\cos{(\\chi)} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}}", "derivation": "y{(\\chi,T)} = \\log{(T + \\chi)} and \\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}} = \\chi and \\cos{(\\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}})} = \\cos{(\\chi)} and \\cos{(\\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}})} - \\frac{1}{\\chi} = \\cos{(\\chi)} - \\frac{1}{\\chi} and \\frac{\\chi (\\cos{(\\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}})} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}} = \\frac{\\chi (\\cos{(\\chi)} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}} and - \\frac{\\chi (\\cos{(\\frac{\\chi y{(\\chi,T)}}{\\log{(T + \\chi)}})} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}} = - \\frac{\\chi (\\cos{(\\chi)} - \\frac{1}{\\chi})}{\\log{(T + \\chi)}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))), Symbol('\\\\chi', commutative=True))"], [["cos", 2], "Equality(cos(Mul(Symbol('\\\\chi', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)))), cos(Symbol('\\\\chi', commutative=True)))"], [["minus", 3, "Pow(Symbol('\\\\chi', commutative=True), Integer(-1))"], "Equality(Add(cos(Mul(Symbol('\\\\chi', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Add(cos(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))))"], [["divide", 4, "Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Add(cos(Mul(Symbol('\\\\chi', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))), Mul(Symbol('\\\\chi', commutative=True), Add(cos(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Add(cos(Mul(Symbol('\\\\chi', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('T', commutative=True)), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Add(cos(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Pow(log(Add(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\chi{(\\Omega)} = e^{\\Omega}, then obtain \\frac{d}{d \\Omega} (\\Omega + e^{\\chi{(\\Omega)}}) \\frac{d}{d \\Omega} (e^{\\Omega})^{\\Omega} = \\frac{d}{d \\Omega} (\\Omega + e^{e^{\\Omega}}) \\frac{d}{d \\Omega} (e^{\\Omega})^{\\Omega}", "derivation": "\\chi{(\\Omega)} = e^{\\Omega} and e^{\\chi{(\\Omega)}} = e^{e^{\\Omega}} and \\Omega + e^{\\chi{(\\Omega)}} = \\Omega + e^{e^{\\Omega}} and \\chi^{\\Omega}{(\\Omega)} = (e^{\\Omega})^{\\Omega} and \\frac{d}{d \\Omega} \\chi^{\\Omega}{(\\Omega)} = \\frac{d}{d \\Omega} (e^{\\Omega})^{\\Omega} and \\frac{d}{d \\Omega} (\\Omega + e^{\\chi{(\\Omega)}}) = \\frac{d}{d \\Omega} (\\Omega + e^{e^{\\Omega}}) and \\frac{d}{d \\Omega} (\\Omega + e^{\\chi{(\\Omega)}}) \\frac{d}{d \\Omega} \\chi^{\\Omega}{(\\Omega)} = \\frac{d}{d \\Omega} (\\Omega + e^{e^{\\Omega}}) \\frac{d}{d \\Omega} \\chi^{\\Omega}{(\\Omega)} and \\frac{d}{d \\Omega} (\\Omega + e^{\\chi{(\\Omega)}}) \\frac{d}{d \\Omega} (e^{\\Omega})^{\\Omega} = \\frac{d}{d \\Omega} (\\Omega + e^{e^{\\Omega}}) \\frac{d}{d \\Omega} (e^{\\Omega})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True))), exp(exp(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), exp(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), exp(exp(Symbol('\\\\Omega', commutative=True)))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["times", 6, "Derivative(Pow(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(Function('\\\\chi')(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), exp(exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given A{(B)} = e^{\\sin{(B)}}, then obtain 2 A^{B}{(B)} + \\frac{(e^{\\sin{(B)}})^{B}}{B} = 2 (e^{\\sin{(B)}})^{B} + \\frac{(e^{\\sin{(B)}})^{B}}{B}", "derivation": "A{(B)} = e^{\\sin{(B)}} and A^{B}{(B)} = (e^{\\sin{(B)}})^{B} and A^{B}{(B)} + \\frac{(e^{\\sin{(B)}})^{B}}{B} = (e^{\\sin{(B)}})^{B} + \\frac{(e^{\\sin{(B)}})^{B}}{B} and 2 A^{B}{(B)} + \\frac{(e^{\\sin{(B)}})^{B}}{B} = A^{B}{(B)} + (e^{\\sin{(B)}})^{B} + \\frac{(e^{\\sin{(B)}})^{B}}{B} and 2 A^{B}{(B)} + \\frac{(e^{\\sin{(B)}})^{B}}{B} = 2 (e^{\\sin{(B)}})^{B} + \\frac{(e^{\\sin{(B)}})^{B}}{B}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('B', commutative=True)), exp(sin(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], "Equality(Add(Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Add(Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))))"], [["add", 2, "Add(Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(2), Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Add(Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Pow(Function('A')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Add(Mul(Integer(2), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(sin(Symbol('B', commutative=True))), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\theta_1)} = \\cos{(\\theta_1)}, then obtain (\\frac{d}{d \\theta_1} \\mathbf{E}{(\\theta_1)} + 1)^{\\theta_1} = (1 - \\sin{(\\theta_1)})^{\\theta_1}", "derivation": "\\mathbf{E}{(\\theta_1)} = \\cos{(\\theta_1)} and \\theta_1 + \\mathbf{E}{(\\theta_1)} = \\theta_1 + \\cos{(\\theta_1)} and \\frac{d}{d \\theta_1} (\\theta_1 + \\mathbf{E}{(\\theta_1)}) = \\frac{d}{d \\theta_1} (\\theta_1 + \\cos{(\\theta_1)}) and (\\frac{d}{d \\theta_1} (\\theta_1 + \\mathbf{E}{(\\theta_1)}))^{\\theta_1} = (\\frac{d}{d \\theta_1} (\\theta_1 + \\cos{(\\theta_1)}))^{\\theta_1} and (\\frac{d}{d \\theta_1} \\mathbf{E}{(\\theta_1)} + 1)^{\\theta_1} = (1 - \\sin{(\\theta_1)})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_1', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(Add(Symbol('\\\\theta_1', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given h{(v_{1})} = \\sin{(v_{1})}, then obtain \\frac{h^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{\\sin^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}}", "derivation": "h{(v_{1})} = \\sin{(v_{1})} and h^{v_{1}}{(v_{1})} = \\sin^{v_{1}}{(v_{1})} and \\frac{h^{v_{1}}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{\\sin^{v_{1}}{(v_{1})}}{\\sin{(v_{1})}} and \\frac{h^{v_{1}}{(v_{1})} \\sin^{v_{1}}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{\\sin^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}} and \\frac{h^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{h^{v_{1}}{(v_{1})} \\sin^{v_{1}}{(v_{1})}}{\\sin{(v_{1})}} and \\frac{h^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{\\sin^{2 v_{1}}{(v_{1})}}{\\sin{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('h')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["divide", 2, "sin(Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Function('h')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["times", 3, "Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Function('h')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('h')(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True))), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(Function('h')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Function('h')(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True))), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(A_{1},v_{z})} = - A_{1} + \\log{(v_{z})}, then obtain 0^{v_{z}} + \\log{(v_{z})} = (- A_{1} - \\mathbf{P}{(A_{1},v_{z})} + \\log{(v_{z})})^{v_{z}} + \\log{(v_{z})}", "derivation": "\\mathbf{P}{(A_{1},v_{z})} = - A_{1} + \\log{(v_{z})} and 0 = - A_{1} - \\mathbf{P}{(A_{1},v_{z})} + \\log{(v_{z})} and 0^{v_{z}} = (- A_{1} - \\mathbf{P}{(A_{1},v_{z})} + \\log{(v_{z})})^{v_{z}} and 0^{v_{z}} + \\log{(v_{z})} = (- A_{1} - \\mathbf{P}{(A_{1},v_{z})} + \\log{(v_{z})})^{v_{z}} + \\log{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('v_z', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{P}')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 3, "log(Symbol('v_z', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('A_1', commutative=True), Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\Psi_{nl})} = \\Psi_{nl}, then obtain (\\Psi_{nl} + \\hat{x}{(\\Psi_{nl})}) (2 \\Psi_{nl} + \\hat{x}{(\\Psi_{nl})}) = 3 \\Psi_{nl} (\\Psi_{nl} + \\hat{x}{(\\Psi_{nl})})", "derivation": "\\hat{x}{(\\Psi_{nl})} = \\Psi_{nl} and \\Psi_{nl} + \\hat{x}{(\\Psi_{nl})} = 2 \\Psi_{nl} and 2 \\Psi_{nl} + \\hat{x}{(\\Psi_{nl})} = 3 \\Psi_{nl} and (\\Psi_{nl} + \\hat{x}{(\\Psi_{nl})}) (2 \\Psi_{nl} + \\hat{x}{(\\Psi_{nl})}) = 3 \\Psi_{nl} (\\Psi_{nl} + \\hat{x}{(\\Psi_{nl})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], [["add", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["add", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(3), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 3, "Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Integer(3), Symbol('\\\\Psi_{nl}', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(a^{\\dagger},n)} = a^{\\dagger} + n, then derive \\hat{x}{(a^{\\dagger},n)} + \\frac{\\partial}{\\partial n} \\hat{x}{(a^{\\dagger},n)} = \\hat{x}{(a^{\\dagger},n)} + 1, then obtain (a^{\\dagger} + n + 1)^{a^{\\dagger}} = (a^{\\dagger} + n + \\frac{\\partial}{\\partial n} (a^{\\dagger} + n))^{a^{\\dagger}}", "derivation": "\\hat{x}{(a^{\\dagger},n)} = a^{\\dagger} + n and \\frac{\\partial}{\\partial n} \\hat{x}{(a^{\\dagger},n)} = \\frac{\\partial}{\\partial n} (a^{\\dagger} + n) and \\hat{x}{(a^{\\dagger},n)} + \\frac{\\partial}{\\partial n} \\hat{x}{(a^{\\dagger},n)} = \\hat{x}{(a^{\\dagger},n)} + \\frac{\\partial}{\\partial n} (a^{\\dagger} + n) and \\hat{x}{(a^{\\dagger},n)} + \\frac{\\partial}{\\partial n} \\hat{x}{(a^{\\dagger},n)} = \\hat{x}{(a^{\\dagger},n)} + 1 and \\hat{x}{(a^{\\dagger},n)} + 1 = \\hat{x}{(a^{\\dagger},n)} + \\frac{\\partial}{\\partial n} (a^{\\dagger} + n) and a^{\\dagger} + n + 1 = a^{\\dagger} + n + \\frac{\\partial}{\\partial n} (a^{\\dagger} + n) and (a^{\\dagger} + n + 1)^{a^{\\dagger}} = (a^{\\dagger} + n + \\frac{\\partial}{\\partial n} (a^{\\dagger} + n))^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(Function('\\\\hat{x}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True), Integer(1)), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True), Integer(1)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\omega)} = \\log{(\\omega)}, then obtain \\operatorname{C_{2}}^{\\omega}{(\\omega)} = (\\operatorname{C_{2}}^{\\omega}{(\\omega)} + \\log{(\\omega)} - \\log{(\\omega)}^{\\omega})^{\\omega}", "derivation": "\\operatorname{C_{2}}{(\\omega)} = \\log{(\\omega)} and \\operatorname{C_{2}}^{\\omega}{(\\omega)} = \\log{(\\omega)}^{\\omega} and \\operatorname{C_{2}}^{\\omega}{(\\omega)} - \\log{(\\omega)}^{\\omega} = 0 and \\operatorname{C_{2}}^{\\omega}{(\\omega)} + \\log{(\\omega)} - \\log{(\\omega)}^{\\omega} = \\log{(\\omega)} and \\operatorname{C_{2}}^{\\omega}{(\\omega)} = (\\operatorname{C_{2}}^{\\omega}{(\\omega)} + \\log{(\\omega)} - \\log{(\\omega)}^{\\omega})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["minus", 2, "Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Pow(Function('C_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["add", 3, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Pow(Function('C_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), log(Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('C_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Pow(Function('C_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given h{(A_{x},T)} = \\frac{T}{A_{x}}, then obtain (\\int h{(A_{x},T)} dA_{x})^{A_{x}} + 1 + \\frac{1}{A_{x}} = (\\int \\frac{T}{A_{x}} dA_{x})^{A_{x}} + 1 + \\frac{1}{A_{x}}", "derivation": "h{(A_{x},T)} = \\frac{T}{A_{x}} and \\int h{(A_{x},T)} dA_{x} = \\int \\frac{T}{A_{x}} dA_{x} and (\\int h{(A_{x},T)} dA_{x})^{A_{x}} = (\\int \\frac{T}{A_{x}} dA_{x})^{A_{x}} and (\\int h{(A_{x},T)} dA_{x})^{A_{x}} + \\frac{1}{A_{x}} = (\\int \\frac{T}{A_{x}} dA_{x})^{A_{x}} + \\frac{1}{A_{x}} and (\\int h{(A_{x},T)} dA_{x})^{A_{x}} + 1 + \\frac{1}{A_{x}} = (\\int \\frac{T}{A_{x}} dA_{x})^{A_{x}} + 1 + \\frac{1}{A_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('h')(Symbol('A_x', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('h')(Symbol('A_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integral(Function('h')(Symbol('A_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["add", 3, "Pow(Symbol('A_x', commutative=True), Integer(-1))"], "Equality(Add(Pow(Integral(Function('h')(Symbol('A_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Symbol('A_x', commutative=True), Integer(-1))), Add(Pow(Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Symbol('A_x', commutative=True), Integer(-1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Pow(Integral(Function('h')(Symbol('A_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))), Add(Pow(Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('T', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given x{(\\mathbf{J}_f,U,f^{*})} = (U + \\mathbf{J}_f)^{f^{*}}, then obtain \\frac{\\partial}{\\partial f^{*}} \\sin{(\\frac{\\partial}{\\partial f^{*}} x{(\\mathbf{J}_f,U,f^{*})})} = \\frac{\\partial}{\\partial f^{*}} \\sin{(\\frac{\\partial}{\\partial f^{*}} (U + \\mathbf{J}_f)^{f^{*}})}", "derivation": "x{(\\mathbf{J}_f,U,f^{*})} = (U + \\mathbf{J}_f)^{f^{*}} and \\frac{\\partial}{\\partial f^{*}} x{(\\mathbf{J}_f,U,f^{*})} = \\frac{\\partial}{\\partial f^{*}} (U + \\mathbf{J}_f)^{f^{*}} and \\sin{(\\frac{\\partial}{\\partial f^{*}} x{(\\mathbf{J}_f,U,f^{*})})} = \\sin{(\\frac{\\partial}{\\partial f^{*}} (U + \\mathbf{J}_f)^{f^{*}})} and \\frac{\\partial}{\\partial f^{*}} \\sin{(\\frac{\\partial}{\\partial f^{*}} x{(\\mathbf{J}_f,U,f^{*})})} = \\frac{\\partial}{\\partial f^{*}} \\sin{(\\frac{\\partial}{\\partial f^{*}} (U + \\mathbf{J}_f)^{f^{*}})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('U', commutative=True), Symbol('f^*', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('U', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('U', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), sin(Derivative(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(sin(Derivative(Function('x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('U', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(sin(Derivative(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbb{I},z^{*},A)} = A + \\mathbb{I} - z^{*} and \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},z^{*},A)} = A + \\mathbb{I} - z^{*}, then obtain \\int \\operatorname{a^{\\dagger}}{(\\mathbb{I},z^{*},A)} dA = \\int (A + \\mathbb{I} - z^{*}) dA", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbb{I},z^{*},A)} = A + \\mathbb{I} - z^{*} and \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},z^{*},A)} = A + \\mathbb{I} - z^{*} and \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},z^{*},A)} = \\operatorname{a^{\\dagger}}{(\\mathbb{I},z^{*},A)} and \\int \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},z^{*},A)} dA = \\int (A + \\mathbb{I} - z^{*}) dA and \\int \\operatorname{a^{\\dagger}}{(\\mathbb{I},z^{*},A)} dA = \\int (A + \\mathbb{I} - z^{*}) dA", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z^*', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(E)} = \\int e^{E} dE, then derive \\frac{\\mathbf{r}{(E)}}{C + e^{E}} = 1, then obtain \\frac{1}{C_{d} + e^{E}} + \\frac{\\mathbf{r}{(E)}}{C + e^{E}} = 1 + \\frac{1}{C_{d} + e^{E}}", "derivation": "\\mathbf{r}{(E)} = \\int e^{E} dE and \\frac{\\mathbf{r}{(E)}}{\\int e^{E} dE} = 1 and \\frac{\\mathbf{r}{(E)}}{C + e^{E}} = 1 and \\frac{1}{\\int e^{E} dE} + \\frac{\\mathbf{r}{(E)}}{C + e^{E}} = 1 + \\frac{1}{\\int e^{E} dE} and \\frac{1}{C_{d} + e^{E}} + \\frac{\\mathbf{r}{(E)}}{C + e^{E}} = 1 + \\frac{1}{C_{d} + e^{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["divide", 1, "Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Pow(Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), exp(Symbol('E', commutative=True))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True))), Integer(1))"], [["add", 3, "Pow(Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(-1))"], "Equality(Add(Pow(Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(-1)), Mul(Pow(Add(Symbol('C', commutative=True), exp(Symbol('E', commutative=True))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True)))), Add(Integer(1), Pow(Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Add(Pow(Add(Symbol('C_d', commutative=True), exp(Symbol('E', commutative=True))), Integer(-1)), Mul(Pow(Add(Symbol('C', commutative=True), exp(Symbol('E', commutative=True))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('E', commutative=True)))), Add(Integer(1), Pow(Add(Symbol('C_d', commutative=True), exp(Symbol('E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given L{(F_{N})} = e^{F_{N}}, then obtain \\sin{(L{(F_{N})} e^{- F_{N}})} = \\sin{(1)}", "derivation": "L{(F_{N})} = e^{F_{N}} and 1 = \\frac{e^{F_{N}}}{L{(F_{N})}} and L{(F_{N})} e^{- F_{N}} = 1 and \\sin{(L{(F_{N})} e^{- F_{N}})} = \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["divide", 1, "Function('L')(Symbol('F_N', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L')(Symbol('F_N', commutative=True)), Integer(-1)), exp(Symbol('F_N', commutative=True))))"], [["divide", 2, "Mul(Pow(Function('L')(Symbol('F_N', commutative=True)), Integer(-1)), exp(Symbol('F_N', commutative=True)))"], "Equality(Mul(Function('L')(Symbol('F_N', commutative=True)), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))), Integer(1))"], [["sin", 3], "Equality(sin(Mul(Function('L')(Symbol('F_N', commutative=True)), exp(Mul(Integer(-1), Symbol('F_N', commutative=True))))), sin(Integer(1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(W)} = \\int e^{W} dW, then obtain 0 = (- \\operatorname{A_{z}}{(W)} + \\int e^{W} dW) (\\operatorname{A_{z}}{(W)} + \\frac{d}{d W} \\int e^{W} dW)^{- W}", "derivation": "\\operatorname{A_{z}}{(W)} = \\int e^{W} dW and \\frac{d}{d W} \\operatorname{A_{z}}{(W)} = \\frac{d}{d W} \\int e^{W} dW and \\operatorname{A_{z}}{(W)} + \\frac{d}{d W} \\operatorname{A_{z}}{(W)} = \\frac{d}{d W} \\operatorname{A_{z}}{(W)} + \\int e^{W} dW and 0 = - \\operatorname{A_{z}}{(W)} + \\int e^{W} dW and 0 = (- \\operatorname{A_{z}}{(W)} + \\int e^{W} dW) (\\operatorname{A_{z}}{(W)} + \\frac{d}{d W} \\operatorname{A_{z}}{(W)})^{- W} and 0 = (- \\operatorname{A_{z}}{(W)} + \\int e^{W} dW) (\\operatorname{A_{z}}{(W)} + \\frac{d}{d W} \\int e^{W} dW)^{- W}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('W', commutative=True)), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Add(Function('A_z')(Symbol('W', commutative=True)), Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Add(Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["minus", 3, "Add(Function('A_z')(Symbol('W', commutative=True)), Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_z')(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["divide", 4, "Pow(Add(Function('A_z')(Symbol('W', commutative=True)), Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Symbol('W', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('A_z')(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Pow(Add(Function('A_z')(Symbol('W', commutative=True)), Derivative(Function('A_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('A_z')(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Pow(Add(Function('A_z')(Symbol('W', commutative=True)), Derivative(Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\theta{(y)} = \\cos{(y)}, then derive \\frac{d}{d y} \\theta{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy = - \\sin{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy, then obtain \\frac{d}{d y} \\cos{(y)} \\int \\frac{d}{d y} \\int \\cos{(y)} dy dy = - \\sin{(y)} \\int \\frac{d}{d y} \\int \\cos{(y)} dy dy", "derivation": "\\theta{(y)} = \\cos{(y)} and \\frac{d}{d y} \\theta{(y)} = \\frac{d}{d y} \\cos{(y)} and \\frac{d}{d y} \\theta{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy = \\frac{d}{d y} \\cos{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy and \\frac{d}{d y} \\theta{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy = - \\sin{(y)} \\int \\frac{d}{d y} \\int \\theta{(y)} dy dy and \\frac{d}{d y} \\cos{(y)} \\int \\frac{d}{d y} \\int \\cos{(y)} dy dy = - \\sin{(y)} \\int \\frac{d}{d y} \\int \\cos{(y)} dy dy", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["times", 2, "Integral(Derivative(Integral(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Derivative(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integral(Derivative(Integral(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))), Mul(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integral(Derivative(Integral(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Derivative(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integral(Derivative(Integral(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))), Mul(Integer(-1), sin(Symbol('y', commutative=True)), Integral(Derivative(Integral(Function('\\\\theta')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integral(Derivative(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))), Mul(Integer(-1), sin(Symbol('y', commutative=True)), Integral(Derivative(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(J,F_{H})} = \\cos{(J^{F_{H}})}, then obtain J^{F_{H}} (- J + \\dot{z}{(J,F_{H})})^{J} + \\cos{(J^{F_{H}})} = J^{F_{H}} (- J + \\cos{(J^{F_{H}})})^{J} + \\cos{(J^{F_{H}})}", "derivation": "\\dot{z}{(J,F_{H})} = \\cos{(J^{F_{H}})} and - J + \\dot{z}{(J,F_{H})} = - J + \\cos{(J^{F_{H}})} and (- J + \\dot{z}{(J,F_{H})})^{J} = (- J + \\cos{(J^{F_{H}})})^{J} and J^{F_{H}} (- J + \\dot{z}{(J,F_{H})})^{J} = J^{F_{H}} (- J + \\cos{(J^{F_{H}})})^{J} and J^{F_{H}} (- J + \\dot{z}{(J,F_{H})})^{J} + \\cos{(J^{F_{H}})} = J^{F_{H}} (- J + \\cos{(J^{F_{H}})})^{J} + \\cos{(J^{F_{H}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True))))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\dot{z}')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\dot{z}')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Symbol('J', commutative=True)))"], [["times", 3, "Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\dot{z}')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Symbol('J', commutative=True))))"], [["add", 4, "cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\dot{z}')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))), Symbol('J', commutative=True))), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Add(Mul(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Symbol('J', commutative=True))), cos(Pow(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(F_{N})} = \\cos{(F_{N})}, then derive \\int (F_{N} + \\operatorname{V_{\\mathbf{B}}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + \\mathbf{v} + \\sin{(F_{N})}, then obtain \\int (F_{N} + \\cos{(F_{N})}) dF_{N} + \\frac{1}{2} = \\frac{F_{N}^{2}}{2} + \\mathbf{v} + \\sin{(F_{N})} + \\frac{1}{2}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(F_{N})} = \\cos{(F_{N})} and F_{N} + \\operatorname{V_{\\mathbf{B}}}{(F_{N})} = F_{N} + \\cos{(F_{N})} and \\int (F_{N} + \\operatorname{V_{\\mathbf{B}}}{(F_{N})}) dF_{N} = \\int (F_{N} + \\cos{(F_{N})}) dF_{N} and \\int (F_{N} + \\operatorname{V_{\\mathbf{B}}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + \\mathbf{v} + \\sin{(F_{N})} and \\int (F_{N} + \\cos{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + \\mathbf{v} + \\sin{(F_{N})} and \\int (F_{N} + \\cos{(F_{N})}) dF_{N} + \\frac{1}{2} = \\frac{F_{N}^{2}}{2} + \\mathbf{v} + \\sin{(F_{N})} + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Symbol('F_N', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('F_N', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["add", 5, "Rational(1, 2)"], "Equality(Add(Integral(Add(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Rational(1, 2)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('F_N', commutative=True)), Rational(1, 2)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(x)} = \\cos{(\\cos{(x)})} and \\operatorname{g_{\\varepsilon}}{(x)} = \\int x \\cos{(\\cos{(x)})} dx, then obtain \\frac{d}{d x} \\int x \\operatorname{f_{E}}{(x)} dx = \\frac{d}{d x} \\operatorname{g_{\\varepsilon}}{(x)}", "derivation": "\\operatorname{f_{E}}{(x)} = \\cos{(\\cos{(x)})} and x \\operatorname{f_{E}}{(x)} = x \\cos{(\\cos{(x)})} and \\int x \\operatorname{f_{E}}{(x)} dx = \\int x \\cos{(\\cos{(x)})} dx and \\frac{d}{d x} \\int x \\operatorname{f_{E}}{(x)} dx = \\frac{d}{d x} \\int x \\cos{(\\cos{(x)})} dx and \\operatorname{g_{\\varepsilon}}{(x)} = \\int x \\cos{(\\cos{(x)})} dx and \\frac{d}{d x} \\int x \\operatorname{f_{E}}{(x)} dx = \\frac{d}{d x} \\operatorname{g_{\\varepsilon}}{(x)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('f_E')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('f_E')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('x', commutative=True), Function('f_E')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('x', commutative=True)), Integral(Mul(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Integral(Mul(Symbol('x', commutative=True), Function('f_E')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Function('g_{\\\\varepsilon}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(E_{n})} = \\cos{(E_{n})} and \\operatorname{V_{\\mathbf{B}}}{(E_{n})} = E_{n}, then derive \\frac{d}{d E_{n}} \\hat{H}{(E_{n})} = - \\sin{(E_{n})}, then obtain \\frac{d}{d \\operatorname{V_{\\mathbf{B}}}{(E_{n})}} \\cos{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} - 1 = - \\sin{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} - 1", "derivation": "\\hat{H}{(E_{n})} = \\cos{(E_{n})} and \\frac{d}{d E_{n}} \\hat{H}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})} and \\operatorname{V_{\\mathbf{B}}}{(E_{n})} = E_{n} and \\frac{d}{d E_{n}} \\hat{H}{(E_{n})} = - \\sin{(E_{n})} and \\frac{d}{d E_{n}} \\cos{(E_{n})} = - \\sin{(E_{n})} and \\frac{d}{d \\operatorname{V_{\\mathbf{B}}}{(E_{n})}} \\cos{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} = - \\sin{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} and \\frac{d}{d \\operatorname{V_{\\mathbf{B}}}{(E_{n})}} \\cos{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} - 1 = - \\sin{(\\operatorname{V_{\\mathbf{B}}}{(E_{n})})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(cos(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True))), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True)), Integer(1))), Mul(Integer(-1), sin(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True)))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Derivative(cos(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True))), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True)), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given s{(H,A_{x})} = A_{x} + \\log{(H)} and u{(H)} = \\log{(H)}, then obtain \\log{(s^{2}{(H,A_{x})} - s{(H,A_{x})} u{(H)})} = \\log{((A_{x} + u{(H)}) s{(H,A_{x})} - s{(H,A_{x})} u{(H)})}", "derivation": "s{(H,A_{x})} = A_{x} + \\log{(H)} and u{(H)} = \\log{(H)} and s^{2}{(H,A_{x})} = (A_{x} + \\log{(H)}) s{(H,A_{x})} and s^{2}{(H,A_{x})} = (A_{x} + u{(H)}) s{(H,A_{x})} and s^{2}{(H,A_{x})} - s{(H,A_{x})} \\log{(H)} = (A_{x} + u{(H)}) s{(H,A_{x})} - s{(H,A_{x})} \\log{(H)} and s^{2}{(H,A_{x})} - s{(H,A_{x})} u{(H)} = (A_{x} + u{(H)}) s{(H,A_{x})} - s{(H,A_{x})} u{(H)} and \\log{(s^{2}{(H,A_{x})} - s{(H,A_{x})} u{(H)})} = \\log{((A_{x} + u{(H)}) s{(H,A_{x})} - s{(H,A_{x})} u{(H)})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), log(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["times", 1, "Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Pow(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Integer(2)), Mul(Add(Symbol('A_x', commutative=True), log(Symbol('H', commutative=True))), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Integer(2)), Mul(Add(Symbol('A_x', commutative=True), Function('u')(Symbol('H', commutative=True))), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))))"], [["minus", 4, "Mul(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('H', commutative=True)))"], "Equality(Add(Pow(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('H', commutative=True)))), Add(Mul(Add(Symbol('A_x', commutative=True), Function('u')(Symbol('H', commutative=True))), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), log(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Function('u')(Symbol('H', commutative=True)))), Add(Mul(Add(Symbol('A_x', commutative=True), Function('u')(Symbol('H', commutative=True))), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Function('u')(Symbol('H', commutative=True)))))"], [["log", 6], "Equality(log(Add(Pow(Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Function('u')(Symbol('H', commutative=True))))), log(Add(Mul(Add(Symbol('A_x', commutative=True), Function('u')(Symbol('H', commutative=True))), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('H', commutative=True), Symbol('A_x', commutative=True)), Function('u')(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(f^{*},n_{2})} = f^{*} n_{2} and \\mathbb{I}{(f^{*},n_{2})} = (f^{*})^{2} n_{2}^{2}, then obtain \\mathbb{I}{(f^{*},n_{2})} = f^{*} n_{2} \\rho_{f}{(f^{*},n_{2})}", "derivation": "\\rho_{f}{(f^{*},n_{2})} = f^{*} n_{2} and f^{*} n_{2} \\rho_{f}{(f^{*},n_{2})} = (f^{*})^{2} n_{2}^{2} and \\mathbb{I}{(f^{*},n_{2})} = (f^{*})^{2} n_{2}^{2} and \\mathbb{I}{(f^{*},n_{2})} = f^{*} n_{2} \\rho_{f}{(f^{*},n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True)))"], [["times", 1, "Mul(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True), Function('\\\\rho_f')(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Pow(Symbol('n_2', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Pow(Symbol('n_2', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True), Function('\\\\rho_f')(Symbol('f^*', commutative=True), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given E{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} \\frac{V_{\\mathbf{E}}}{\\theta} and \\Psi_{nl}{(\\theta,V_{\\mathbf{E}})} = - E{(\\theta,V_{\\mathbf{E}})}, then obtain \\Psi_{nl}{(\\theta,V_{\\mathbf{E}})} = - \\frac{\\partial}{\\partial \\theta} \\frac{V_{\\mathbf{E}}}{\\theta}", "derivation": "E{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} \\frac{V_{\\mathbf{E}}}{\\theta} and - E{(\\theta,V_{\\mathbf{E}})} = - \\frac{\\partial}{\\partial \\theta} \\frac{V_{\\mathbf{E}}}{\\theta} and \\Psi_{nl}{(\\theta,V_{\\mathbf{E}})} = - E{(\\theta,V_{\\mathbf{E}})} and \\Psi_{nl}{(\\theta,V_{\\mathbf{E}})} = - \\frac{\\partial}{\\partial \\theta} \\frac{V_{\\mathbf{E}}}{\\theta}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{H},z)} = \\frac{z}{\\hat{H}}, then obtain \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{a^{\\dagger}}{(\\hat{H},z)} d\\hat{H} - 1) = \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\hat{H}} \\int \\frac{z}{\\hat{H}} d\\hat{H} - 1)", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{H},z)} = \\frac{z}{\\hat{H}} and \\int \\operatorname{a^{\\dagger}}{(\\hat{H},z)} d\\hat{H} = \\int \\frac{z}{\\hat{H}} d\\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{a^{\\dagger}}{(\\hat{H},z)} d\\hat{H} = \\frac{\\partial}{\\partial \\hat{H}} \\int \\frac{z}{\\hat{H}} d\\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{a^{\\dagger}}{(\\hat{H},z)} d\\hat{H} - 1 = \\frac{\\partial}{\\partial \\hat{H}} \\int \\frac{z}{\\hat{H}} d\\hat{H} - 1 and \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\hat{H}} \\int \\operatorname{a^{\\dagger}}{(\\hat{H},z)} d\\hat{H} - 1) = \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\hat{H}} \\int \\frac{z}{\\hat{H}} d\\hat{H} - 1)", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Derivative(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(t,C_{d},\\Psi_{nl})} = t (C_{d} - \\Psi_{nl}), then derive \\int \\nabla{(t,C_{d},\\Psi_{nl})} d\\Psi_{nl} = C_{d} \\Psi_{nl} t - \\frac{\\Psi_{nl}^{2} t}{2} + f^{\\prime}, then obtain \\iint \\nabla{(t,C_{d},\\Psi_{nl})} d\\Psi_{nl} dt = \\int (C_{d} \\Psi_{nl} t - \\frac{\\Psi_{nl}^{2} t}{2} + f^{\\prime}) dt", "derivation": "\\nabla{(t,C_{d},\\Psi_{nl})} = t (C_{d} - \\Psi_{nl}) and \\int \\nabla{(t,C_{d},\\Psi_{nl})} d\\Psi_{nl} = \\int t (C_{d} - \\Psi_{nl}) d\\Psi_{nl} and \\int \\nabla{(t,C_{d},\\Psi_{nl})} d\\Psi_{nl} = C_{d} \\Psi_{nl} t - \\frac{\\Psi_{nl}^{2} t}{2} + f^{\\prime} and \\iint \\nabla{(t,C_{d},\\Psi_{nl})} d\\Psi_{nl} dt = \\int (C_{d} \\Psi_{nl} t - \\frac{\\Psi_{nl}^{2} t}{2} + f^{\\prime}) dt", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('t', commutative=True), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Symbol('t', commutative=True), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('t', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Symbol('t', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(P_{g},\\dot{z})} = \\sin{(P_{g} + \\dot{z})} and \\operatorname{f^{*}}{(P_{g},\\dot{z})} = \\dot{z} \\sin{(P_{g} + \\dot{z})}, then obtain \\frac{\\partial}{\\partial P_{g}} \\int \\dot{z} \\operatorname{A_{1}}{(P_{g},\\dot{z})} dP_{g} = \\frac{\\partial}{\\partial P_{g}} \\int \\operatorname{f^{*}}{(P_{g},\\dot{z})} dP_{g}", "derivation": "\\operatorname{A_{1}}{(P_{g},\\dot{z})} = \\sin{(P_{g} + \\dot{z})} and \\dot{z} \\operatorname{A_{1}}{(P_{g},\\dot{z})} = \\dot{z} \\sin{(P_{g} + \\dot{z})} and \\operatorname{f^{*}}{(P_{g},\\dot{z})} = \\dot{z} \\sin{(P_{g} + \\dot{z})} and \\dot{z} \\operatorname{A_{1}}{(P_{g},\\dot{z})} = \\operatorname{f^{*}}{(P_{g},\\dot{z})} and \\int \\dot{z} \\operatorname{A_{1}}{(P_{g},\\dot{z})} dP_{g} = \\int \\operatorname{f^{*}}{(P_{g},\\dot{z})} dP_{g} and \\frac{\\partial}{\\partial P_{g}} \\int \\dot{z} \\operatorname{A_{1}}{(P_{g},\\dot{z})} dP_{g} = \\frac{\\partial}{\\partial P_{g}} \\int \\operatorname{f^{*}}{(P_{g},\\dot{z})} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), sin(Add(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["times", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 4, "Symbol('P_g', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integral(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["differentiate", 5, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Function('A_1')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Integral(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\omega)} = \\cos{(\\log{(\\omega)})}, then derive \\frac{d}{d \\omega} \\rho_{b}{(\\omega)} = - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega}, then obtain -1 - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} = \\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})} - 1", "derivation": "\\rho_{b}{(\\omega)} = \\cos{(\\log{(\\omega)})} and \\frac{d}{d \\omega} \\rho_{b}{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})} and \\frac{d}{d \\omega} \\rho_{b}{(\\omega)} = - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} and \\frac{d}{d \\omega} \\rho_{b}{(\\omega)} - 1 = \\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})} - 1 and -1 - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} = \\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(log(Symbol('\\\\omega', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(log(Symbol('\\\\omega', commutative=True))))), Add(Derivative(cos(log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(M_{E},J)} = e^{J M_{E}}, then obtain (\\iint \\operatorname{A_{1}}{(M_{E},J)} dJ dM_{E})^{M_{E}} = (\\iint e^{J M_{E}} dJ dM_{E})^{M_{E}}", "derivation": "\\operatorname{A_{1}}{(M_{E},J)} = e^{J M_{E}} and \\int \\operatorname{A_{1}}{(M_{E},J)} dJ = \\int e^{J M_{E}} dJ and \\iint \\operatorname{A_{1}}{(M_{E},J)} dJ dM_{E} = \\iint e^{J M_{E}} dJ dM_{E} and (\\iint \\operatorname{A_{1}}{(M_{E},J)} dJ dM_{E})^{M_{E}} = (\\iint e^{J M_{E}} dJ dM_{E})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), exp(Mul(Symbol('J', commutative=True), Symbol('M_E', commutative=True))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(exp(Mul(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(exp(Mul(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Integral(Function('A_1')(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Integral(exp(Mul(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given u{(\\theta,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\theta, then obtain - \\hat{\\mathbf{r}} \\theta + \\int \\frac{2 u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}} = - \\hat{\\mathbf{r}} \\theta + \\int \\frac{\\hat{\\mathbf{r}} \\theta + u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}}", "derivation": "u{(\\theta,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\theta and 2 u{(\\theta,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\theta + u{(\\theta,\\hat{\\mathbf{r}})} and \\frac{2 u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} = \\frac{\\hat{\\mathbf{r}} \\theta + u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} and \\int \\frac{2 u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}} = \\int \\frac{\\hat{\\mathbf{r}} \\theta + u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}} and - \\hat{\\mathbf{r}} \\theta + \\int \\frac{2 u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}} = - \\hat{\\mathbf{r}} \\theta + \\int \\frac{\\hat{\\mathbf{r}} \\theta + u{(\\theta,\\hat{\\mathbf{r}})}}{\\theta} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(2), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["minus", 4, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Mul(Integer(2), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Function('u')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given q{(r)} = e^{r}, then obtain - U^{3}{(r)} q^{3}{(r)} + 1 = - U^{3}{(r)} q^{3}{(r)} + \\frac{\\cos{((q{(r)} e^{2 r})^{2 r})}}{\\cos{((q^{2}{(r)} e^{r})^{2 r})}}", "derivation": "q{(r)} = e^{r} and q^{2}{(r)} e^{r} = q{(r)} e^{2 r} and (q^{2}{(r)} e^{r})^{r} = (q{(r)} e^{2 r})^{r} and (q^{2}{(r)} e^{r})^{2 r} = (q{(r)} e^{2 r})^{2 r} and \\cos{((q^{2}{(r)} e^{r})^{2 r})} = \\cos{((q{(r)} e^{2 r})^{2 r})} and 1 = \\frac{\\cos{((q{(r)} e^{2 r})^{2 r})}}{\\cos{((q^{2}{(r)} e^{r})^{2 r})}} and - U^{3}{(r)} q^{3}{(r)} + 1 = - U^{3}{(r)} q^{3}{(r)} + \\frac{\\cos{((q{(r)} e^{2 r})^{2 r})}}{\\cos{((q^{2}{(r)} e^{r})^{2 r})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["times", 1, "Mul(Function('q')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))), Pow(Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))), Mul(Integer(2), Symbol('r', commutative=True))))"], [["cos", 4], "Equality(cos(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True)))), cos(Pow(Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))), Mul(Integer(2), Symbol('r', commutative=True)))))"], [["divide", 5, "cos(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))))"], "Equality(Integer(1), Mul(cos(Pow(Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))), Mul(Integer(2), Symbol('r', commutative=True)))), Pow(cos(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True)))), Integer(-1))))"], [["minus", 6, "Mul(Pow(Function('U')(Symbol('r', commutative=True)), Integer(3)), Pow(Function('q')(Symbol('r', commutative=True)), Integer(3)))"], "Equality(Add(Mul(Integer(-1), Pow(Function('U')(Symbol('r', commutative=True)), Integer(3)), Pow(Function('q')(Symbol('r', commutative=True)), Integer(3))), Integer(1)), Add(Mul(Integer(-1), Pow(Function('U')(Symbol('r', commutative=True)), Integer(3)), Pow(Function('q')(Symbol('r', commutative=True)), Integer(3))), Mul(cos(Pow(Mul(Function('q')(Symbol('r', commutative=True)), exp(Mul(Integer(2), Symbol('r', commutative=True)))), Mul(Integer(2), Symbol('r', commutative=True)))), Pow(cos(Pow(Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(2)), exp(Symbol('r', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True)))), Integer(-1)))))"]]}, {"prompt": "Given \\Omega{(M_{E},\\lambda)} = - M_{E} + e^{\\lambda}, then derive \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)} = e^{\\lambda}, then obtain \\frac{\\lambda + e^{\\lambda}}{M_{E}} - \\frac{\\lambda + \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)}}{M_{E}} = 0", "derivation": "\\Omega{(M_{E},\\lambda)} = - M_{E} + e^{\\lambda} and \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} (- M_{E} + e^{\\lambda}) and \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)} = e^{\\lambda} and \\lambda + \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)} = \\lambda + e^{\\lambda} and - \\frac{\\lambda + \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)}}{M_{E}} = - \\frac{\\lambda + e^{\\lambda}}{M_{E}} and \\frac{\\lambda + e^{\\lambda}}{M_{E}} - \\frac{\\lambda + \\frac{\\partial}{\\partial \\lambda} \\Omega{(M_{E},\\lambda)}}{M_{E}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), exp(Symbol('\\\\lambda', commutative=True)))"], [["add", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Symbol('\\\\lambda', commutative=True), exp(Symbol('\\\\lambda', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('M_E', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), exp(Symbol('\\\\lambda', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), exp(Symbol('\\\\lambda', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), exp(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Derivative(Function('\\\\Omega')(Symbol('M_E', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x)} = \\cos{(x)}, then obtain - \\operatorname{A_{y}}{(x)} \\sin{(x)} + \\cos{(x)} \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - 1 = - 2 \\sin{(x)} \\cos{(x)} - \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - 1", "derivation": "\\operatorname{A_{y}}{(x)} = \\cos{(x)} and \\operatorname{A_{y}}{(x)} \\cos{(x)} = \\cos^{2}{(x)} and \\operatorname{A_{y}}{(x)} \\cos{(x)} - \\operatorname{A_{y}}{(x)} = - \\operatorname{A_{y}}{(x)} + \\cos^{2}{(x)} and \\frac{d}{d x} (\\operatorname{A_{y}}{(x)} \\cos{(x)} - \\operatorname{A_{y}}{(x)}) = \\frac{d}{d x} (- \\operatorname{A_{y}}{(x)} + \\cos^{2}{(x)}) and \\frac{d}{d x} (\\operatorname{A_{y}}{(x)} \\cos{(x)} - \\operatorname{A_{y}}{(x)}) - 1 = \\frac{d}{d x} (- \\operatorname{A_{y}}{(x)} + \\cos^{2}{(x)}) - 1 and - \\operatorname{A_{y}}{(x)} \\sin{(x)} + \\cos{(x)} \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - 1 = - 2 \\sin{(x)} \\cos{(x)} - \\frac{d}{d x} \\operatorname{A_{y}}{(x)} - 1", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["times", 1, "cos(Symbol('x', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(2)))"], [["minus", 2, "Function('A_y')(Symbol('x', commutative=True))"], "Equality(Add(Mul(Function('A_y')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Mul(Function('A_y')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(2))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Derivative(Add(Mul(Function('A_y')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(2))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Function('A_y')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(cos(Symbol('x', commutative=True)), Derivative(Function('A_y')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('A_y')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), sin(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Mul(Integer(-1), Derivative(Function('A_y')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A_{2},\\varphi)} = \\cos{(\\frac{A_{2}}{\\varphi})}, then obtain \\int \\operatorname{E_{n}}^{2}{(A_{2},\\varphi)} d\\varphi + 1 = \\int \\operatorname{E_{n}}{(A_{2},\\varphi)} \\cos{(\\frac{A_{2}}{\\varphi})} d\\varphi + 1", "derivation": "\\operatorname{E_{n}}{(A_{2},\\varphi)} = \\cos{(\\frac{A_{2}}{\\varphi})} and \\operatorname{E_{n}}^{2}{(A_{2},\\varphi)} = \\operatorname{E_{n}}{(A_{2},\\varphi)} \\cos{(\\frac{A_{2}}{\\varphi})} and \\int \\operatorname{E_{n}}^{2}{(A_{2},\\varphi)} d\\varphi = \\int \\operatorname{E_{n}}{(A_{2},\\varphi)} \\cos{(\\frac{A_{2}}{\\varphi})} d\\varphi and \\int \\operatorname{E_{n}}^{2}{(A_{2},\\varphi)} d\\varphi + 1 = \\int \\operatorname{E_{n}}{(A_{2},\\varphi)} \\cos{(\\frac{A_{2}}{\\varphi})} d\\varphi + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))))"], [["times", 1, "Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Pow(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Mul(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Pow(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(1)), Add(Integral(Mul(Function('E_n')(Symbol('A_2', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)} = e^{\\varepsilon_0}, then obtain \\mathbf{H} + \\varepsilon_0 = \\int \\frac{e^{\\varepsilon_0}}{\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)}} d\\varepsilon_0", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)} = e^{\\varepsilon_0} and 1 = \\frac{e^{\\varepsilon_0}}{\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)}} and \\int 1 d\\varepsilon_0 = \\int \\frac{e^{\\varepsilon_0}}{\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)}} d\\varepsilon_0 and \\mathbf{H} + \\varepsilon_0 = \\int \\frac{e^{\\varepsilon_0}}{\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0)}} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{y},f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}}, then derive \\operatorname{P_{g}}{(A_{y},f^{*})} = \\frac{1}{A_{y}}, then obtain 2 \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} - 2 + \\frac{2}{A_{y}} = \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} - 2 + \\frac{3}{A_{y}}", "derivation": "\\operatorname{P_{g}}{(A_{y},f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} and \\operatorname{P_{g}}{(A_{y},f^{*})} = \\frac{1}{A_{y}} and \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} = \\frac{1}{A_{y}} and \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} + \\frac{1}{A_{y}} = \\frac{2}{A_{y}} and \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} - 1 + \\frac{1}{A_{y}} = -1 + \\frac{2}{A_{y}} and 2 \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} - 2 + \\frac{2}{A_{y}} = \\frac{\\partial}{\\partial f^{*}} \\frac{f^{*}}{A_{y}} - 2 + \\frac{3}{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_y', commutative=True), Symbol('f^*', commutative=True)), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('P_g')(Symbol('A_y', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('A_y', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Symbol('A_y', commutative=True), Integer(-1)))"], [["add", 3, "Pow(Symbol('A_y', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Symbol('A_y', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('A_y', commutative=True), Integer(-1))))"], [["minus", 4, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-1))), Add(Integer(-1), Mul(Integer(2), Pow(Symbol('A_y', commutative=True), Integer(-1)))))"], [["add", 5, "Add(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(2), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(-2), Mul(Integer(2), Pow(Symbol('A_y', commutative=True), Integer(-1)))), Add(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-2), Mul(Integer(3), Pow(Symbol('A_y', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given n{(\\rho,E_{\\lambda})} = \\rho + \\sin{(E_{\\lambda})} and \\mathbf{P}{(\\rho,E_{\\lambda})} = \\frac{n{(\\rho,E_{\\lambda})}}{\\rho + \\sin{(E_{\\lambda})}}, then obtain \\sin{(\\frac{\\partial}{\\partial \\rho} \\mathbf{P}^{2}{(\\rho,E_{\\lambda})})} = \\sin{(\\frac{\\partial}{\\partial \\rho} \\mathbf{P}{(\\rho,E_{\\lambda})})}", "derivation": "n{(\\rho,E_{\\lambda})} = \\rho + \\sin{(E_{\\lambda})} and \\mathbf{P}{(\\rho,E_{\\lambda})} = \\frac{n{(\\rho,E_{\\lambda})}}{\\rho + \\sin{(E_{\\lambda})}} and \\mathbf{P}{(\\rho,E_{\\lambda})} = 1 and \\mathbf{P}^{2}{(\\rho,E_{\\lambda})} = \\mathbf{P}{(\\rho,E_{\\lambda})} and \\frac{\\partial}{\\partial \\rho} \\mathbf{P}^{2}{(\\rho,E_{\\lambda})} = \\frac{\\partial}{\\partial \\rho} \\mathbf{P}{(\\rho,E_{\\lambda})} and \\sin{(\\frac{\\partial}{\\partial \\rho} \\mathbf{P}^{2}{(\\rho,E_{\\lambda})})} = \\sin{(\\frac{\\partial}{\\partial \\rho} \\mathbf{P}{(\\rho,E_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\rho', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1)), Function('n')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1))"], [["times", 3, "Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["sin", 5], "Equality(sin(Derivative(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), sin(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(E_{x},V)} = e^{\\frac{E_{x}}{V}}, then obtain - \\frac{E_{x} \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}}}{V} + V e^{\\frac{E_{x}}{V}} \\frac{\\partial}{\\partial V} \\mathbf{s}{(E_{x},V)} + \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}} = - \\frac{2 E_{x} e^{\\frac{2 E_{x}}{V}}}{V} + e^{\\frac{2 E_{x}}{V}}", "derivation": "\\mathbf{s}{(E_{x},V)} = e^{\\frac{E_{x}}{V}} and V \\mathbf{s}{(E_{x},V)} = V e^{\\frac{E_{x}}{V}} and V \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}} = V e^{\\frac{2 E_{x}}{V}} and \\frac{\\partial}{\\partial V} V \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}} = \\frac{\\partial}{\\partial V} V e^{\\frac{2 E_{x}}{V}} and - \\frac{E_{x} \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}}}{V} + V e^{\\frac{E_{x}}{V}} \\frac{\\partial}{\\partial V} \\mathbf{s}{(E_{x},V)} + \\mathbf{s}{(E_{x},V)} e^{\\frac{E_{x}}{V}} = - \\frac{2 E_{x} e^{\\frac{2 E_{x}}{V}}}{V} + e^{\\frac{2 E_{x}}{V}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True))), Mul(Symbol('V', commutative=True), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))"], [["times", 2, "exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))"], "Equality(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), Mul(Symbol('V', commutative=True), exp(Mul(Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), exp(Mul(Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), Mul(Symbol('V', commutative=True), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))), Derivative(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))))), Add(Mul(Integer(-1), Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))), exp(Mul(Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(v_{1})} = \\sin{(v_{1})}, then obtain \\operatorname{A_{x}}{(v_{1})} \\sin{(v_{1})} + (\\int \\operatorname{A_{x}}{(v_{1})} dv_{1})^{v_{1}} = \\operatorname{A_{x}}{(v_{1})} \\sin{(v_{1})} + (\\int \\sin{(v_{1})} dv_{1})^{v_{1}}", "derivation": "\\operatorname{A_{x}}{(v_{1})} = \\sin{(v_{1})} and \\int \\operatorname{A_{x}}{(v_{1})} dv_{1} = \\int \\sin{(v_{1})} dv_{1} and (\\int \\operatorname{A_{x}}{(v_{1})} dv_{1})^{v_{1}} = (\\int \\sin{(v_{1})} dv_{1})^{v_{1}} and \\operatorname{A_{x}}{(v_{1})} \\sin{(v_{1})} + (\\int \\operatorname{A_{x}}{(v_{1})} dv_{1})^{v_{1}} = \\operatorname{A_{x}}{(v_{1})} \\sin{(v_{1})} + (\\int \\sin{(v_{1})} dv_{1})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Integral(Function('A_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], [["add", 3, "Mul(Function('A_x')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Function('A_x')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Pow(Integral(Function('A_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))), Add(Mul(Function('A_x')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\pi,\\mu)} = \\mu \\pi, then derive (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} = \\pi^{\\pi}, then obtain \\frac{\\pi (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} \\frac{\\partial^{2}}{\\partial \\mu^{2}} \\mathbf{r}{(\\pi,\\mu)}}{\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)}} = 0", "derivation": "\\mathbf{r}{(\\pi,\\mu)} = \\mu \\pi and \\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)} = \\frac{\\partial}{\\partial \\mu} \\mu \\pi and (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} = (\\frac{\\partial}{\\partial \\mu} \\mu \\pi)^{\\pi} and (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} = \\pi^{\\pi} and \\frac{\\partial}{\\partial \\mu} (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} = \\frac{d}{d \\mu} \\pi^{\\pi} and \\frac{\\pi (\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)})^{\\pi} \\frac{\\partial^{2}}{\\partial \\mu^{2}} \\mathbf{r}{(\\pi,\\mu)}}{\\frac{\\partial}{\\partial \\mu} \\mathbf{r}{(\\pi,\\mu)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('\\\\pi', commutative=True), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2)))), Integer(0))"]]}, {"prompt": "Given p{(C)} = e^{C}, then obtain \\frac{d^{2}}{d C^{2}} 0 = \\frac{d^{2}}{d C^{2}} (- 2 p{(C)} + 2 e^{C})", "derivation": "p{(C)} = e^{C} and 0 = - p{(C)} + e^{C} and - p{(C)} = - 2 p{(C)} + e^{C} and \\frac{d}{d C} 0 = \\frac{d}{d C} (- p{(C)} + e^{C}) and \\frac{d^{2}}{d C^{2}} 0 = \\frac{d^{2}}{d C^{2}} (- p{(C)} + e^{C}) and \\frac{d^{2}}{d C^{2}} 0 = \\frac{d^{2}}{d C^{2}} (- 2 p{(C)} + 2 e^{C})", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["minus", 1, "Function('p')(Symbol('C', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('p')(Symbol('C', commutative=True))), exp(Symbol('C', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('p')(Symbol('C', commutative=True)))"], "Equality(Mul(Integer(-1), Function('p')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('p')(Symbol('C', commutative=True))), exp(Symbol('C', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('p')(Symbol('C', commutative=True))), exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Function('p')(Symbol('C', commutative=True))), exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('p')(Symbol('C', commutative=True))), Mul(Integer(2), exp(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(M_{E},n_{1})} = M_{E} + n_{1}, then obtain M_{E} + n_{1} = M_{E} + n_{1} - \\sin{(\\sin{(M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})})})}", "derivation": "\\operatorname{v_{1}}{(M_{E},n_{1})} = M_{E} + n_{1} and 0 = M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})} and 0 = \\sin{(M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})})} and 0 = \\sin{(\\sin{(M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})})})} and 0 = - \\sin{(\\sin{(M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})})})} and M_{E} + n_{1} = M_{E} + n_{1} - \\sin{(\\sin{(M_{E} + n_{1} - \\operatorname{v_{1}}{(M_{E},n_{1})})})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)))"], [["minus", 1, "Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Integer(0), Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)))))"], [["sin", 2], "Equality(Integer(0), sin(Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True))))))"], [["sin", 3], "Equality(Integer(0), sin(sin(Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)))))))"], [["times", 4, "Integer(-1)"], "Equality(Integer(0), Mul(Integer(-1), sin(sin(Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True))))))))"], [["add", 5, "Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), sin(sin(Add(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(-1), Function('v_1')(Symbol('M_E', commutative=True), Symbol('n_1', commutative=True)))))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} = \\tilde{g}^{\\mathbf{P}} and \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} = \\int \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} d\\mathbf{P}, then obtain 2 \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} = \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} + \\int \\tilde{g}^{\\mathbf{P}} d\\mathbf{P}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} = \\tilde{g}^{\\mathbf{P}} and \\int \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} d\\mathbf{P} = \\int \\tilde{g}^{\\mathbf{P}} d\\mathbf{P} and 2 \\int \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} d\\mathbf{P} = \\int \\tilde{g}^{\\mathbf{P}} d\\mathbf{P} + \\int \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} d\\mathbf{P} and \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} = \\int \\operatorname{F_{c}}{(\\mathbf{P},\\tilde{g})} d\\mathbf{P} and 2 \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} = \\tilde{g}^*{(\\mathbf{P},\\tilde{g})} + \\int \\tilde{g}^{\\mathbf{P}} d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 2, "Integral(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Integral(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('F_c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\dot{z},t,l)} = (\\frac{\\dot{z}}{l})^{t}, then obtain \\cos{(\\int \\lambda^{l}{(\\dot{z},t,l)} d\\dot{z})} = \\cos{(\\int ((\\frac{\\dot{z}}{l})^{t})^{l} d\\dot{z})}", "derivation": "\\lambda{(\\dot{z},t,l)} = (\\frac{\\dot{z}}{l})^{t} and \\lambda^{l}{(\\dot{z},t,l)} = ((\\frac{\\dot{z}}{l})^{t})^{l} and \\int \\lambda^{l}{(\\dot{z},t,l)} d\\dot{z} = \\int ((\\frac{\\dot{z}}{l})^{t})^{l} d\\dot{z} and \\cos{(\\int \\lambda^{l}{(\\dot{z},t,l)} d\\dot{z})} = \\cos{(\\int ((\\frac{\\dot{z}}{l})^{t})^{l} d\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True), Symbol('l', commutative=True)), Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('t', commutative=True)), Symbol('l', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('t', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), cos(Integral(Pow(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('t', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})}, then obtain \\frac{\\mathbf{J}^{2}{(L_{\\varepsilon})}}{\\sin{(L_{\\varepsilon})}} = \\mathbf{J}{(L_{\\varepsilon})}", "derivation": "\\mathbf{J}{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\mathbf{J}^{2}{(L_{\\varepsilon})} = \\mathbf{J}{(L_{\\varepsilon})} \\sin{(L_{\\varepsilon})} and \\frac{\\mathbf{J}{(L_{\\varepsilon})}}{\\sin{(L_{\\varepsilon})}} = 1 and \\frac{\\mathbf{J}^{2}{(L_{\\varepsilon})}}{\\sin{(L_{\\varepsilon})}} = \\mathbf{J}{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1))"], [["times", 3, "Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)} = \\ddot{x} \\cos{(\\hat{H}_l)}, then obtain \\int \\frac{\\partial}{\\partial \\hat{H}_l} (\\ddot{x} \\cos{(\\hat{H}_l)} + \\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)}) d\\ddot{x} = \\int \\frac{\\partial}{\\partial \\hat{H}_l} 2 \\ddot{x} \\cos{(\\hat{H}_l)} d\\ddot{x}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)} = \\ddot{x} \\cos{(\\hat{H}_l)} and \\ddot{x} \\cos{(\\hat{H}_l)} + \\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)} = 2 \\ddot{x} \\cos{(\\hat{H}_l)} and \\frac{\\partial}{\\partial \\hat{H}_l} (\\ddot{x} \\cos{(\\hat{H}_l)} + \\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)}) = \\frac{\\partial}{\\partial \\hat{H}_l} 2 \\ddot{x} \\cos{(\\hat{H}_l)} and \\int \\frac{\\partial}{\\partial \\hat{H}_l} (\\ddot{x} \\cos{(\\hat{H}_l)} + \\operatorname{V_{\\mathbf{E}}}{(\\ddot{x},\\hat{H}_l)}) d\\ddot{x} = \\int \\frac{\\partial}{\\partial \\hat{H}_l} 2 \\ddot{x} \\cos{(\\hat{H}_l)} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Derivative(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(c_{0},\\mathbf{J}_f)} = c_{0} \\sin{(\\mathbf{J}_f)}, then derive \\frac{\\frac{\\partial}{\\partial c_{0}} \\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} = 1, then obtain \\frac{\\frac{\\partial}{\\partial c_{0}} \\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} - \\frac{1}{\\Omega{(c_{0},\\mathbf{J}_f)}} = 1 - \\frac{1}{\\Omega{(c_{0},\\mathbf{J}_f)}}", "derivation": "\\Omega{(c_{0},\\mathbf{J}_f)} = c_{0} \\sin{(\\mathbf{J}_f)} and \\frac{\\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} = c_{0} and \\frac{\\partial}{\\partial c_{0}} \\frac{\\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} = \\frac{d}{d c_{0}} c_{0} and \\frac{\\frac{\\partial}{\\partial c_{0}} \\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} = 1 and \\frac{\\frac{\\partial}{\\partial c_{0}} \\Omega{(c_{0},\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} - \\frac{1}{\\Omega{(c_{0},\\mathbf{J}_f)}} = 1 - \\frac{1}{\\Omega{(c_{0},\\mathbf{J}_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('c_0', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Symbol('c_0', commutative=True))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(1))"], [["minus", 4, "Pow(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given Q{(\\mathbf{D},v,\\mathbf{r})} = \\frac{\\mathbf{r}}{\\mathbf{D} v}, then obtain \\frac{\\mathbf{r} \\frac{\\partial}{\\partial v} Q^{\\mathbf{D}}{(\\mathbf{D},v,\\mathbf{r})}}{\\mathbf{D} v} = \\frac{\\mathbf{r} \\frac{\\partial}{\\partial v} (\\frac{\\mathbf{r}}{\\mathbf{D} v})^{\\mathbf{D}}}{\\mathbf{D} v}", "derivation": "Q{(\\mathbf{D},v,\\mathbf{r})} = \\frac{\\mathbf{r}}{\\mathbf{D} v} and Q^{\\mathbf{D}}{(\\mathbf{D},v,\\mathbf{r})} = (\\frac{\\mathbf{r}}{\\mathbf{D} v})^{\\mathbf{D}} and \\frac{\\partial}{\\partial v} Q^{\\mathbf{D}}{(\\mathbf{D},v,\\mathbf{r})} = \\frac{\\partial}{\\partial v} (\\frac{\\mathbf{r}}{\\mathbf{D} v})^{\\mathbf{D}} and \\frac{\\mathbf{r} \\frac{\\partial}{\\partial v} Q^{\\mathbf{D}}{(\\mathbf{D},v,\\mathbf{r})}}{\\mathbf{D} v} = \\frac{\\mathbf{r} \\frac{\\partial}{\\partial v} (\\frac{\\mathbf{r}}{\\mathbf{D} v})^{\\mathbf{D}}}{\\mathbf{D} v}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Pow(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 3, "Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)), Derivative(Pow(Function('Q')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)), Derivative(Pow(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{F})} = e^{e^{\\mathbf{F}}}, then obtain - \\operatorname{v_{1}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{- e^{\\mathbf{F}}} + e^{- e^{\\mathbf{F}}} \\frac{d}{d \\mathbf{F}} \\operatorname{v_{1}}{(\\mathbf{F})} = 0", "derivation": "\\operatorname{v_{1}}{(\\mathbf{F})} = e^{e^{\\mathbf{F}}} and \\operatorname{v_{1}}{(\\mathbf{F})} e^{- e^{\\mathbf{F}}} = 1 and \\frac{d}{d \\mathbf{F}} \\operatorname{v_{1}}{(\\mathbf{F})} e^{- e^{\\mathbf{F}}} = \\frac{d}{d \\mathbf{F}} 1 and - \\operatorname{v_{1}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{- e^{\\mathbf{F}}} + e^{- e^{\\mathbf{F}}} \\frac{d}{d \\mathbf{F}} \\operatorname{v_{1}}{(\\mathbf{F})} = 0", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True))))), Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Derivative(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given m{(n)} = \\sin{(n)}, then obtain m{(n)} - \\sin{(m^{n}{(n)})} + \\int (- n + 2 m{(n)}) dn = m{(n)} - \\sin{(m^{n}{(n)})} + \\int (- n + m{(n)} + \\sin{(n)}) dn", "derivation": "m{(n)} = \\sin{(n)} and 2 m{(n)} = m{(n)} + \\sin{(n)} and m^{n}{(n)} = \\sin^{n}{(n)} and - n + 2 m{(n)} = - n + m{(n)} + \\sin{(n)} and \\int (- n + 2 m{(n)}) dn = \\int (- n + m{(n)} + \\sin{(n)}) dn and - \\sin{(\\sin^{n}{(n)})} + \\int (- n + 2 m{(n)}) dn = - \\sin{(\\sin^{n}{(n)})} + \\int (- n + m{(n)} + \\sin{(n)}) dn and m{(n)} - \\sin{(\\sin^{n}{(n)})} + \\int (- n + 2 m{(n)}) dn = m{(n)} - \\sin{(\\sin^{n}{(n)})} + \\int (- n + m{(n)} + \\sin{(n)}) dn and m{(n)} - \\sin{(m^{n}{(n)})} + \\int (- n + 2 m{(n)}) dn = m{(n)} - \\sin{(m^{n}{(n)})} + \\int (- n + m{(n)} + \\sin{(n)}) dn", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["add", 1, "Function('m')(Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('m')(Symbol('n', commutative=True))), Add(Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('m')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["minus", 2, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), Function('m')(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), Function('m')(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["minus", 5, "sin(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), Function('m')(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), sin(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["add", 6, "Function('m')(Symbol('n', commutative=True))"], "Equality(Add(Function('m')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), Function('m')(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))), Add(Function('m')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Function('m')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(Pow(Function('m')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), Function('m')(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))), Add(Function('m')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(Pow(Function('m')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('m')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(q)} = \\cos{(q)}, then obtain q + (e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}})^{q} \\sin^{q}{(\\phi_{1}{(q)})} = q + e^{q} \\sin^{q}{(\\phi_{1}{(q)})}", "derivation": "\\phi_{1}{(q)} = \\cos{(q)} and \\frac{\\phi_{1}{(q)}}{\\cos{(q)}} = 1 and \\sin{(\\phi_{1}{(q)})} = \\sin{(\\cos{(q)})} and e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}} = e and (e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}})^{q} = e^{q} and (e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}})^{q} \\sin^{q}{(\\cos{(q)})} = e^{q} \\sin^{q}{(\\cos{(q)})} and (e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}})^{q} \\sin^{q}{(\\phi_{1}{(q)})} = e^{q} \\sin^{q}{(\\phi_{1}{(q)})} and q + (e^{\\frac{\\phi_{1}{(q)}}{\\cos{(q)}}})^{q} \\sin^{q}{(\\phi_{1}{(q)})} = q + e^{q} \\sin^{q}{(\\phi_{1}{(q)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["divide", 1, "cos(Symbol('q', commutative=True))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1))), Integer(1))"], [["sin", 1], "Equality(sin(Function('\\\\phi_1')(Symbol('q', commutative=True))), sin(cos(Symbol('q', commutative=True))))"], [["exp", 2], "Equality(exp(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1)))), E)"], [["power", 4, "Symbol('q', commutative=True)"], "Equality(Pow(exp(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1)))), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["times", 5, "Pow(sin(cos(Symbol('q', commutative=True))), Symbol('q', commutative=True))"], "Equality(Mul(Pow(exp(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1)))), Symbol('q', commutative=True)), Pow(sin(cos(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Pow(sin(cos(Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(exp(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1)))), Symbol('q', commutative=True)), Pow(sin(Function('\\\\phi_1')(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Pow(sin(Function('\\\\phi_1')(Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["add", 7, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Mul(Pow(exp(Mul(Function('\\\\phi_1')(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1)))), Symbol('q', commutative=True)), Pow(sin(Function('\\\\phi_1')(Symbol('q', commutative=True))), Symbol('q', commutative=True)))), Add(Symbol('q', commutative=True), Mul(exp(Symbol('q', commutative=True)), Pow(sin(Function('\\\\phi_1')(Symbol('q', commutative=True))), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(f_{E})} = e^{f_{E}} and \\nabla{(f_{E})} = e^{f_{E}} and H{(f_{E})} = - f_{E}, then obtain \\nabla{(f_{E})} - (H{(f_{E})} + e^{- H{(f_{E})}})^{- H{(f_{E})}} = e^{f_{E}} - (H{(f_{E})} + e^{- H{(f_{E})}})^{- H{(f_{E})}}", "derivation": "\\Omega{(f_{E})} = e^{f_{E}} and - f_{E} + \\Omega{(f_{E})} = - f_{E} + e^{f_{E}} and (- f_{E} + \\Omega{(f_{E})})^{f_{E}} = (- f_{E} + e^{f_{E}})^{f_{E}} and \\nabla{(f_{E})} = e^{f_{E}} and H{(f_{E})} = - f_{E} and - (- f_{E} + \\Omega{(f_{E})})^{f_{E}} + \\nabla{(f_{E})} = - (- f_{E} + \\Omega{(f_{E})})^{f_{E}} + e^{f_{E}} and - (- f_{E} + e^{f_{E}})^{f_{E}} + \\nabla{(f_{E})} = - (- f_{E} + e^{f_{E}})^{f_{E}} + e^{f_{E}} and \\nabla{(f_{E})} - (H{(f_{E})} + e^{- H{(f_{E})}})^{- H{(f_{E})}} = e^{f_{E}} - (H{(f_{E})} + e^{- H{(f_{E})}})^{- H{(f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], [["minus", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True))))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True)))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))), Function('\\\\nabla')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\Omega')(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))), Function('\\\\nabla')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Function('\\\\nabla')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Add(Function('H')(Symbol('f_E', commutative=True)), exp(Mul(Integer(-1), Function('H')(Symbol('f_E', commutative=True))))), Mul(Integer(-1), Function('H')(Symbol('f_E', commutative=True)))))), Add(exp(Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Add(Function('H')(Symbol('f_E', commutative=True)), exp(Mul(Integer(-1), Function('H')(Symbol('f_E', commutative=True))))), Mul(Integer(-1), Function('H')(Symbol('f_E', commutative=True)))))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain \\frac{d}{d \\mathbf{A}} \\hat{x}{(\\mathbf{A})} - \\frac{\\hat{x}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} - \\frac{\\hat{x}{(\\mathbf{A})}}{\\mathbf{A}}", "derivation": "\\hat{x}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\hat{x}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and \\frac{\\hat{x}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\sin{(\\mathbf{A})}}{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\hat{x}{(\\mathbf{A})} - \\frac{\\sin{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} - \\frac{\\sin{(\\mathbf{A})}}{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\hat{x}{(\\mathbf{A})} - \\frac{\\hat{x}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} - \\frac{\\hat{x}{(\\mathbf{A})}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given c{(i,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - i and \\operatorname{f^{*}}{(i,\\hat{H}_{\\lambda})} = - i c{(i,\\hat{H}_{\\lambda})} - i, then obtain (- i (\\hat{H}_{\\lambda} - i) - i)^{i} = (- i c{(i,\\hat{H}_{\\lambda})} - i)^{i}", "derivation": "c{(i,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - i and - i c{(i,\\hat{H}_{\\lambda})} = - i (\\hat{H}_{\\lambda} - i) and - i c{(i,\\hat{H}_{\\lambda})} - i = - i (\\hat{H}_{\\lambda} - i) - i and \\operatorname{f^{*}}{(i,\\hat{H}_{\\lambda})} = - i c{(i,\\hat{H}_{\\lambda})} - i and \\operatorname{f^{*}}{(i,\\hat{H}_{\\lambda})} = - i (\\hat{H}_{\\lambda} - i) - i and \\operatorname{f^{*}}^{i}{(i,\\hat{H}_{\\lambda})} = (- i (\\hat{H}_{\\lambda} - i) - i)^{i} and \\operatorname{f^{*}}^{i}{(i,\\hat{H}_{\\lambda})} = (- i c{(i,\\hat{H}_{\\lambda})} - i)^{i} and (- i (\\hat{H}_{\\lambda} - i) - i)^{i} = (- i c{(i,\\hat{H}_{\\lambda})} - i)^{i}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True), Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True), Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["power", 5, "Symbol('i', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True), Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True), Function('c')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\nabla)} = \\log{(\\sin{(\\nabla)})} and x{(\\nabla)} = \\nabla, then obtain \\frac{\\hat{\\mathbf{x}}{(\\nabla)}}{\\nabla \\log{(\\sin{(\\nabla)})}^{2}} = \\frac{1}{\\nabla \\log{(\\sin{(\\nabla)})}}", "derivation": "\\hat{\\mathbf{x}}{(\\nabla)} = \\log{(\\sin{(\\nabla)})} and x{(\\nabla)} = \\nabla and \\hat{\\mathbf{x}}{(\\nabla)} x{(\\nabla)} = x{(\\nabla)} \\log{(\\sin{(\\nabla)})} and \\nabla \\hat{\\mathbf{x}}{(\\nabla)} = \\nabla \\log{(\\sin{(\\nabla)})} and \\frac{\\nabla \\hat{\\mathbf{x}}{(\\nabla)}}{x{(\\nabla)} \\log{(\\sin{(\\nabla)})}} = \\frac{\\nabla}{x{(\\nabla)}} and \\frac{\\nabla \\hat{\\mathbf{x}}{(\\nabla)}}{x^{2}{(\\nabla)} \\log{(\\sin{(\\nabla)})}^{2}} = \\frac{\\nabla}{x^{2}{(\\nabla)} \\log{(\\sin{(\\nabla)})}} and \\frac{\\hat{\\mathbf{x}}{(\\nabla)}}{\\nabla \\log{(\\sin{(\\nabla)})}^{2}} = \\frac{1}{\\nabla \\log{(\\sin{(\\nabla)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True)), log(sin(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["times", 1, "Function('x')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True)), Function('x')(Symbol('\\\\nabla', commutative=True))), Mul(Function('x')(Symbol('\\\\nabla', commutative=True)), log(sin(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), log(sin(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 4, "Mul(Function('x')(Symbol('\\\\nabla', commutative=True)), log(sin(Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True)), Pow(Function('x')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Pow(log(sin(Symbol('\\\\nabla', commutative=True))), Integer(-1))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('x')(Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["divide", 5, "Mul(Function('x')(Symbol('\\\\nabla', commutative=True)), log(sin(Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True)), Pow(Function('x')(Symbol('\\\\nabla', commutative=True)), Integer(-2)), Pow(log(sin(Symbol('\\\\nabla', commutative=True))), Integer(-2))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Function('x')(Symbol('\\\\nabla', commutative=True)), Integer(-2)), Pow(log(sin(Symbol('\\\\nabla', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\nabla', commutative=True)), Pow(log(sin(Symbol('\\\\nabla', commutative=True))), Integer(-2))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(log(sin(Symbol('\\\\nabla', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(z^{*},\\mathbf{A})} = \\mathbf{A} z^{*}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{F_{c}}{(z^{*},\\mathbf{A})} = z^{*}, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{z^{*}}{\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} z^{*}} = \\frac{d}{d \\mathbf{A}} 1", "derivation": "\\operatorname{F_{c}}{(z^{*},\\mathbf{A})} = \\mathbf{A} z^{*} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{F_{c}}{(z^{*},\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} z^{*} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{F_{c}}{(z^{*},\\mathbf{A})} = z^{*} and z^{*} = \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} z^{*} and \\frac{z^{*}}{\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} z^{*}} = 1 and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{z^{*}}{\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} z^{*}} = \\frac{d}{d \\mathbf{A}} 1", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Symbol('z^*', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('z^*', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('z^*', commutative=True), Pow(Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["differentiate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Symbol('z^*', commutative=True), Pow(Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(v_{1},\\lambda)} = v_{1} + \\sin{(\\lambda)}, then obtain 2^{\\lambda} = (\\frac{2 v_{1} + 2 \\sin{(\\lambda)}}{c{(v_{1},\\lambda)}})^{\\lambda}", "derivation": "c{(v_{1},\\lambda)} = v_{1} + \\sin{(\\lambda)} and 2 c{(v_{1},\\lambda)} = v_{1} + c{(v_{1},\\lambda)} + \\sin{(\\lambda)} and 2 = \\frac{v_{1} + c{(v_{1},\\lambda)} + \\sin{(\\lambda)}}{c{(v_{1},\\lambda)}} and 2^{\\lambda} = (\\frac{v_{1} + c{(v_{1},\\lambda)} + \\sin{(\\lambda)}}{c{(v_{1},\\lambda)}})^{\\lambda} and 2^{\\lambda} = (\\frac{2 v_{1} + 2 \\sin{(\\lambda)}}{v_{1} + \\sin{(\\lambda)}})^{\\lambda} and (\\frac{v_{1} + c{(v_{1},\\lambda)} + \\sin{(\\lambda)}}{c{(v_{1},\\lambda)}})^{\\lambda} = (\\frac{2 v_{1} + 2 \\sin{(\\lambda)}}{v_{1} + \\sin{(\\lambda)}})^{\\lambda} and 2^{\\lambda} = (\\frac{2 v_{1} + 2 \\sin{(\\lambda)}}{c{(v_{1},\\lambda)}})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('v_1', commutative=True), sin(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(2), Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Symbol('v_1', commutative=True), Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))))"], [["divide", 2, "Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Integer(2), Mul(Add(Symbol('v_1', commutative=True), Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Add(Symbol('v_1', commutative=True), Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Pow(Add(Symbol('v_1', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\lambda', commutative=True))))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Mul(Add(Symbol('v_1', commutative=True), Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Pow(Add(Symbol('v_1', commutative=True), sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\lambda', commutative=True))))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Add(Mul(Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\lambda', commutative=True)))), Pow(Function('c')(Symbol('v_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given C{(P_{e},F_{N},b)} = F_{N} - P_{e} + b and \\theta_{1}{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain 1 - \\theta_{1}{(\\mathbf{H})} = - \\theta_{1}{(\\mathbf{H})} + \\frac{\\int (F_{N} - P_{e} + b + 1) dF_{N}}{\\int (C{(P_{e},F_{N},b)} + 1) dF_{N}}", "derivation": "C{(P_{e},F_{N},b)} = F_{N} - P_{e} + b and C{(P_{e},F_{N},b)} + 1 = F_{N} - P_{e} + b + 1 and \\theta_{1}{(\\mathbf{H})} = e^{\\mathbf{H}} and \\int (C{(P_{e},F_{N},b)} + 1) dF_{N} = \\int (F_{N} - P_{e} + b + 1) dF_{N} and 1 = \\frac{\\int (F_{N} - P_{e} + b + 1) dF_{N}}{\\int (C{(P_{e},F_{N},b)} + 1) dF_{N}} and 1 - e^{\\mathbf{H}} = - e^{\\mathbf{H}} + \\frac{\\int (F_{N} - P_{e} + b + 1) dF_{N}}{\\int (C{(P_{e},F_{N},b)} + 1) dF_{N}} and 1 - \\theta_{1}{(\\mathbf{H})} = - \\theta_{1}{(\\mathbf{H})} + \\frac{\\int (F_{N} - P_{e} + b + 1) dF_{N}}{\\int (C{(P_{e},F_{N},b)} + 1) dF_{N}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True), Integer(1)))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('F_N', commutative=True))))"], [["divide", 4, "Integral(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('F_N', commutative=True)))))"], [["minus", 5, "exp(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Integral(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('F_N', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Integral(Add(Function('C')(Symbol('P_e', commutative=True), Symbol('F_N', commutative=True), Symbol('b', commutative=True)), Integer(1)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('F_N', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(E,v)} = v^{E}, then obtain \\int (v^{E} + \\operatorname{E_{\\lambda}}{(E,v)})^{4} dv = \\int 16 v^{4 E} dv", "derivation": "\\operatorname{E_{\\lambda}}{(E,v)} = v^{E} and v^{E} + \\operatorname{E_{\\lambda}}{(E,v)} = 2 v^{E} and (v^{E} + \\operatorname{E_{\\lambda}}{(E,v)})^{2} = 4 v^{2 E} and (v^{E} + \\operatorname{E_{\\lambda}}{(E,v)})^{4} = 16 v^{4 E} and \\int (v^{E} + \\operatorname{E_{\\lambda}}{(E,v)})^{4} dv = \\int 16 v^{4 E} dv", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('E', commutative=True)))"], [["add", 1, "Pow(Symbol('v', commutative=True), Symbol('E', commutative=True))"], "Equality(Add(Pow(Symbol('v', commutative=True), Symbol('E', commutative=True)), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Pow(Symbol('v', commutative=True), Symbol('E', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Pow(Symbol('v', commutative=True), Symbol('E', commutative=True)), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Symbol('v', commutative=True), Mul(Integer(2), Symbol('E', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Symbol('v', commutative=True), Symbol('E', commutative=True)), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Integer(4)), Mul(Integer(16), Pow(Symbol('v', commutative=True), Mul(Integer(4), Symbol('E', commutative=True)))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Add(Pow(Symbol('v', commutative=True), Symbol('E', commutative=True)), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Integer(4)), Tuple(Symbol('v', commutative=True))), Integral(Mul(Integer(16), Pow(Symbol('v', commutative=True), Mul(Integer(4), Symbol('E', commutative=True)))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given b{(\\mathbf{H},\\phi)} = \\int (\\mathbf{H} + \\phi) d\\mathbf{H}, then derive \\phi b{(\\mathbf{H},\\phi)} = \\phi (\\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} \\phi + a), then obtain \\phi (\\int \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} d\\mathbf{H}) \\int (\\mathbf{H} + \\phi) d\\mathbf{H} = \\phi (\\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} \\phi + a) \\int \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} d\\mathbf{H}", "derivation": "b{(\\mathbf{H},\\phi)} = \\int (\\mathbf{H} + \\phi) d\\mathbf{H} and \\phi b{(\\mathbf{H},\\phi)} = \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} and \\phi b{(\\mathbf{H},\\phi)} = \\phi (\\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} \\phi + a) and \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} = \\phi (\\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} \\phi + a) and \\phi (\\int \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} d\\mathbf{H}) \\int (\\mathbf{H} + \\phi) d\\mathbf{H} = \\phi (\\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} \\phi + a) \\int \\phi \\int (\\mathbf{H} + \\phi) d\\mathbf{H} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\phi', commutative=True), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('a', commutative=True))))"], [["times", 4, "Integral(Mul(Symbol('\\\\phi', commutative=True), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Integral(Mul(Symbol('\\\\phi', commutative=True), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('a', commutative=True)), Integral(Mul(Symbol('\\\\phi', commutative=True), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{x},q)} = \\hat{x}^{q}, then derive \\frac{\\partial}{\\partial \\hat{x}} \\phi_{1}{(\\hat{x},q)} = \\frac{\\hat{x}^{q} q}{\\hat{x}}, then obtain \\frac{\\partial}{\\partial q} \\frac{\\hat{x}^{q} q}{\\hat{x}} = \\frac{\\partial^{2}}{\\partial q\\partial \\hat{x}} \\hat{x}^{q}", "derivation": "\\phi_{1}{(\\hat{x},q)} = \\hat{x}^{q} and \\frac{\\partial}{\\partial \\hat{x}} \\phi_{1}{(\\hat{x},q)} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{x}^{q} and \\frac{\\partial}{\\partial \\hat{x}} \\phi_{1}{(\\hat{x},q)} = \\frac{\\hat{x}^{q} q}{\\hat{x}} and \\frac{\\hat{x}^{q} q}{\\hat{x}} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{x}^{q} and \\frac{\\partial}{\\partial q} \\frac{\\hat{x}^{q} q}{\\hat{x}} = \\frac{\\partial^{2}}{\\partial q\\partial \\hat{x}} \\hat{x}^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Derivative(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\rho,\\sigma_p)} = - \\rho + \\sigma_p, then obtain \\frac{\\int 0 d\\rho}{\\sigma_p} = \\frac{\\int (\\int (- \\rho + \\sigma_p) d\\rho - \\int \\operatorname{n_{1}}{(\\rho,\\sigma_p)} d\\rho) d\\rho}{\\sigma_p}", "derivation": "\\operatorname{n_{1}}{(\\rho,\\sigma_p)} = - \\rho + \\sigma_p and \\int \\operatorname{n_{1}}{(\\rho,\\sigma_p)} d\\rho = \\int (- \\rho + \\sigma_p) d\\rho and 0 = \\int (- \\rho + \\sigma_p) d\\rho - \\int \\operatorname{n_{1}}{(\\rho,\\sigma_p)} d\\rho and \\int 0 d\\rho = \\int (\\int (- \\rho + \\sigma_p) d\\rho - \\int \\operatorname{n_{1}}{(\\rho,\\sigma_p)} d\\rho) d\\rho and \\frac{\\int 0 d\\rho}{\\sigma_p} = \\frac{\\int (\\int (- \\rho + \\sigma_p) d\\rho - \\int \\operatorname{n_{1}}{(\\rho,\\sigma_p)} d\\rho) d\\rho}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["minus", 2, "Integral(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Integral(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Integral(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["divide", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Integral(Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Integral(Function('n_1')(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))), Tuple(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given c{(A_{2},\\dot{y})} = A_{2} + \\dot{y} and \\mathbf{H}{(x^\\prime,E_{\\lambda})} = \\sin{(E_{\\lambda} x^\\prime)}, then obtain \\frac{\\frac{\\partial}{\\partial x^\\prime} \\mathbf{H}{(x^\\prime,E_{\\lambda})}}{A_{2} + \\dot{y}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} \\sin{(E_{\\lambda} x^\\prime)}}{A_{2} + \\dot{y}}", "derivation": "c{(A_{2},\\dot{y})} = A_{2} + \\dot{y} and \\mathbf{H}{(x^\\prime,E_{\\lambda})} = \\sin{(E_{\\lambda} x^\\prime)} and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{H}{(x^\\prime,E_{\\lambda})} = \\frac{\\partial}{\\partial x^\\prime} \\sin{(E_{\\lambda} x^\\prime)} and \\frac{\\frac{\\partial}{\\partial x^\\prime} \\mathbf{H}{(x^\\prime,E_{\\lambda})}}{c{(A_{2},\\dot{y})}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} \\sin{(E_{\\lambda} x^\\prime)}}{c{(A_{2},\\dot{y})}} and \\frac{\\frac{\\partial}{\\partial x^\\prime} \\mathbf{H}{(x^\\prime,E_{\\lambda})}}{A_{2} + \\dot{y}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} \\sin{(E_{\\lambda} x^\\prime)}}{A_{2} + \\dot{y}}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 3, "Function('c')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Function('c')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Function('c')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}, then obtain \\frac{\\int \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})}} = \\frac{\\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})}}", "derivation": "\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})} and \\int \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\frac{\\int \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\sin{(\\Psi^{\\dagger})}} = \\frac{\\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\sin{(\\Psi^{\\dagger})}} and \\frac{\\int \\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})}} = \\frac{\\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}}{\\operatorname{y^{\\prime}}{(\\Psi^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Pow(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(P_{g},\\omega)} = P_{g} \\omega, then obtain (P_{g} + 1) (- P_{g} \\omega - 1) = (P_{g} + 1) (- P_{g} \\omega - \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}})", "derivation": "\\operatorname{C_{d}}{(P_{g},\\omega)} = P_{g} \\omega and 1 = \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} and P_{g} + 1 = \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} + P_{g} and -1 = - \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} and - P_{g} \\omega - 1 = - P_{g} \\omega - \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} and (- P_{g} \\omega - 1) (\\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} + P_{g}) = (- P_{g} \\omega - \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}}) (\\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}} + P_{g}) and (P_{g} + 1) (- P_{g} \\omega - 1) = (P_{g} + 1) (- P_{g} \\omega - \\frac{P_{g} \\omega}{\\operatorname{C_{d}}{(P_{g},\\omega)}})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Integer(1), Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Integer(1)), Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))"], [["minus", 4, "Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))))"], [["times", 5, "Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Symbol('P_g', commutative=True), Integer(1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Add(Symbol('P_g', commutative=True), Integer(1)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True), Pow(Function('C_d')(Symbol('P_g', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given h{(J)} = \\sin{(J)}, then obtain \\frac{(\\int h^{J}{(J)} dJ) (\\int h^{J}{(J)} dJ)^{J}}{J} = \\frac{((\\int h^{J}{(J)} dJ)^{J}) \\int \\sin^{J}{(J)} dJ}{J}", "derivation": "h{(J)} = \\sin{(J)} and h^{J}{(J)} = \\sin^{J}{(J)} and \\int h^{J}{(J)} dJ = \\int \\sin^{J}{(J)} dJ and (\\int h^{J}{(J)} dJ)^{J} = (\\int \\sin^{J}{(J)} dJ)^{J} and \\frac{\\int h^{J}{(J)} dJ}{J} = \\frac{\\int \\sin^{J}{(J)} dJ}{J} and \\frac{(\\int h^{J}{(J)} dJ) (\\int \\sin^{J}{(J)} dJ)^{J}}{J} = \\frac{(\\int \\sin^{J}{(J)} dJ) (\\int \\sin^{J}{(J)} dJ)^{J}}{J} and \\frac{(\\int h^{J}{(J)} dJ) (\\int h^{J}{(J)} dJ)^{J}}{J} = \\frac{((\\int h^{J}{(J)} dJ)^{J}) \\int \\sin^{J}{(J)} dJ}{J}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["divide", 3, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["times", 5, "Pow(Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Integral(Pow(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Integral(Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given p{(m,F_{N})} = - F_{N} + m and Q{(F_{N})} = - F_{N}, then obtain (- F_{N} + m + \\pi{(r)})^{F_{N}} = (m + Q{(F_{N})} + \\pi{(r)})^{F_{N}}", "derivation": "p{(m,F_{N})} = - F_{N} + m and Q{(F_{N})} = - F_{N} and p{(m,F_{N})} = m + Q{(F_{N})} and \\pi{(r)} + p{(m,F_{N})} = m + Q{(F_{N})} + \\pi{(r)} and (\\pi{(r)} + p{(m,F_{N})})^{F_{N}} = (m + Q{(F_{N})} + \\pi{(r)})^{F_{N}} and \\pi{(r)} + p{(m,F_{N})} = - F_{N} + m + \\pi{(r)} and (- F_{N} + m + \\pi{(r)})^{F_{N}} = (m + Q{(F_{N})} + \\pi{(r)})^{F_{N}}", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('m', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('p')(Symbol('m', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('m', commutative=True), Function('Q')(Symbol('F_N', commutative=True))))"], [["add", 3, "Function('\\\\pi')(Symbol('r', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('r', commutative=True)), Function('p')(Symbol('m', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('m', commutative=True), Function('Q')(Symbol('F_N', commutative=True)), Function('\\\\pi')(Symbol('r', commutative=True))))"], [["power", 4, "Symbol('F_N', commutative=True)"], "Equality(Pow(Add(Function('\\\\pi')(Symbol('r', commutative=True)), Function('p')(Symbol('m', commutative=True), Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Add(Symbol('m', commutative=True), Function('Q')(Symbol('F_N', commutative=True)), Function('\\\\pi')(Symbol('r', commutative=True))), Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\pi')(Symbol('r', commutative=True)), Function('p')(Symbol('m', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('m', commutative=True), Function('\\\\pi')(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('m', commutative=True), Function('\\\\pi')(Symbol('r', commutative=True))), Symbol('F_N', commutative=True)), Pow(Add(Symbol('m', commutative=True), Function('Q')(Symbol('F_N', commutative=True)), Function('\\\\pi')(Symbol('r', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(z^{*},M,t_{2})} = M + t_{2} + z^{*} and \\tilde{g}{(\\rho_b)} = \\cos{(\\rho_b)}, then obtain \\frac{\\operatorname{F_{c}}^{M}{(z^{*},M,t_{2})} \\tilde{g}{(\\rho_b)}}{\\cos{(\\rho_b)}} = \\frac{(M + t_{2} + z^{*})^{M} \\tilde{g}{(\\rho_b)}}{\\cos{(\\rho_b)}}", "derivation": "\\operatorname{F_{c}}{(z^{*},M,t_{2})} = M + t_{2} + z^{*} and \\tilde{g}{(\\rho_b)} = \\cos{(\\rho_b)} and \\operatorname{F_{c}}^{M}{(z^{*},M,t_{2})} = (M + t_{2} + z^{*})^{M} and \\frac{\\operatorname{F_{c}}^{M}{(z^{*},M,t_{2})}}{\\tilde{g}{(\\rho_b)}} = \\frac{(M + t_{2} + z^{*})^{M}}{\\tilde{g}{(\\rho_b)}} and \\frac{\\operatorname{F_{c}}^{M}{(z^{*},M,t_{2})}}{\\cos{(\\rho_b)}} = \\frac{(M + t_{2} + z^{*})^{M}}{\\cos{(\\rho_b)}} and \\frac{\\operatorname{F_{c}}^{M}{(z^{*},M,t_{2})} \\tilde{g}{(\\rho_b)}}{\\cos{(\\rho_b)}} = \\frac{(M + t_{2} + z^{*})^{M} \\tilde{g}{(\\rho_b)}}{\\cos{(\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('z^*', commutative=True), Symbol('M', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('M', commutative=True), Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('z^*', commutative=True), Symbol('M', commutative=True), Symbol('t_2', commutative=True)), Symbol('M', commutative=True)), Pow(Add(Symbol('M', commutative=True), Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Symbol('M', commutative=True)))"], [["divide", 3, "Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Pow(Function('F_c')(Symbol('z^*', commutative=True), Symbol('M', commutative=True), Symbol('t_2', commutative=True)), Symbol('M', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Symbol('M', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('F_c')(Symbol('z^*', commutative=True), Symbol('M', commutative=True), Symbol('t_2', commutative=True)), Symbol('M', commutative=True)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Symbol('M', commutative=True)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"], [["divide", 5, "Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('F_c')(Symbol('z^*', commutative=True), Symbol('M', commutative=True), Symbol('t_2', commutative=True)), Symbol('M', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Symbol('M', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\rho_b', commutative=True)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(\\phi_2,M)} = \\frac{M}{\\phi_2}, then obtain (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} ((\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 1) + (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 1 = (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 3", "derivation": "\\mathbf{p}{(\\phi_2,M)} = \\frac{M}{\\phi_2} and \\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M} = 1 and (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} = 1 and (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 1 = 2 and 2 (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 2 = (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 3 and (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} ((\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 1) + (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 1 = (\\frac{\\phi_2 \\mathbf{p}{(\\phi_2,M)}}{M})^{M} + 3", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Integer(1))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(1))"], [["add", 3, 1], "Equality(Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(1)), Integer(2))"], [["add", 4, "Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(2), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True))), Integer(2)), Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(1))), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(1)), Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Integer(3)))"]]}, {"prompt": "Given B{(g,s)} = g^{s} - s, then obtain \\frac{(\\int B{(g,s)} dg)^{4}}{(\\int (g^{s} - s) dg)^{2}} = (\\int B{(g,s)} dg)^{2}", "derivation": "B{(g,s)} = g^{s} - s and \\int B{(g,s)} dg = \\int (g^{s} - s) dg and \\frac{\\int B{(g,s)} dg}{\\int (g^{s} - s) dg} = 1 and \\frac{(\\int B{(g,s)} dg)^{2}}{\\int (g^{s} - s) dg} = \\int B{(g,s)} dg and \\frac{(\\int B{(g,s)} dg)^{4}}{(\\int (g^{s} - s) dg)^{2}} = (\\int B{(g,s)} dg)^{2}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["divide", 2, "Integral(Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('g', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integer(1))"], [["times", 3, "Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('g', commutative=True))), Integer(-1)), Pow(Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2))), Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Integral(Add(Pow(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('g', commutative=True))), Integer(-2)), Pow(Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(4))), Pow(Integral(Function('B')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2)))"]]}, {"prompt": "Given z{(A_{x})} = e^{\\cos{(A_{x})}} and \\operatorname{n_{2}}{(A_{x})} = \\cos{(A_{x})}, then obtain 1 = \\frac{\\frac{d}{d A_{x}} (e^{\\cos{(A_{x})}})^{A_{x}}}{\\frac{d}{d A_{x}} z^{A_{x}}{(A_{x})}}", "derivation": "z{(A_{x})} = e^{\\cos{(A_{x})}} and \\operatorname{n_{2}}{(A_{x})} = \\cos{(A_{x})} and z{(A_{x})} = e^{\\operatorname{n_{2}}{(A_{x})}} and z^{A_{x}}{(A_{x})} = (e^{\\operatorname{n_{2}}{(A_{x})}})^{A_{x}} and z^{A_{x}}{(A_{x})} = (e^{\\cos{(A_{x})}})^{A_{x}} and \\frac{d}{d A_{x}} z^{A_{x}}{(A_{x})} = \\frac{d}{d A_{x}} (e^{\\cos{(A_{x})}})^{A_{x}} and 1 = \\frac{\\frac{d}{d A_{x}} (e^{\\cos{(A_{x})}})^{A_{x}}}{\\frac{d}{d A_{x}} z^{A_{x}}{(A_{x})}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('z')(Symbol('A_x', commutative=True)), exp(Function('n_2')(Symbol('A_x', commutative=True))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('z')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(exp(Function('n_2')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('z')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(exp(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["differentiate", 5, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Pow(Function('z')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Pow(exp(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 6, "Derivative(Pow(Function('z')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Pow(Function('z')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Pow(exp(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\psi)} = \\sin{(\\psi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\operatorname{t_{2}}^{\\psi}{(\\psi)} + \\sin{(\\mathbf{H})} + \\sin{(\\psi)} = \\sin{(\\mathbf{H})} + \\sin{(\\psi)} + \\sin^{\\psi}{(\\psi)}", "derivation": "\\operatorname{t_{2}}{(\\psi)} = \\sin{(\\psi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\operatorname{t_{2}}^{\\psi}{(\\psi)} = \\sin^{\\psi}{(\\psi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} + \\operatorname{t_{2}}^{\\psi}{(\\psi)} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} + \\sin^{\\psi}{(\\psi)} and \\operatorname{t_{2}}^{\\psi}{(\\psi)} + \\sin{(\\mathbf{H})} = \\sin{(\\mathbf{H})} + \\sin^{\\psi}{(\\psi)} and \\operatorname{t_{2}}^{\\psi}{(\\psi)} + \\sin{(\\mathbf{H})} + \\sin{(\\psi)} = \\sin{(\\mathbf{H})} + \\sin{(\\psi)} + \\sin^{\\psi}{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)))"], ["get_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["add", 3, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('t_2')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))), Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('t_2')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Add(sin(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["add", 5, "sin(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Pow(Function('t_2')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\psi', commutative=True))), Add(sin(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{J})} = \\log{(\\mathbf{J})}, then derive \\frac{d}{d \\mathbf{J}} \\operatorname{C_{d}}{(\\mathbf{J})} = \\frac{1}{\\mathbf{J}}, then obtain - \\frac{\\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}}{\\mathbf{J}^{2}} = - \\frac{1}{\\mathbf{J}^{3}}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\operatorname{C_{d}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\operatorname{C_{d}}{(\\mathbf{J})} = \\frac{1}{\\mathbf{J}} and \\frac{\\frac{d}{d \\mathbf{J}} \\operatorname{C_{d}}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{1}{\\mathbf{J}^{2}} and \\frac{\\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{1}{\\mathbf{J}^{2}} and \\frac{\\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}}{\\mathbf{J}^{2}} = \\frac{1}{\\mathbf{J}^{3}} and - \\frac{\\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}}{\\mathbf{J}^{2}} = - \\frac{1}{\\mathbf{J}^{3}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))"], [["times", 3, "Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)))"], [["divide", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-3)))"], [["divide", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\varphi^*,\\hbar)} = \\log{(\\hbar)}^{\\varphi^*} and \\operatorname{A_{2}}{(\\hbar)} = \\hbar, then obtain \\int \\varepsilon_{0}{(\\varphi^*,\\hbar)} d\\operatorname{A_{2}}{(\\hbar)} = \\int \\log{(\\hbar)}^{\\varphi^*} d\\operatorname{A_{2}}{(\\hbar)}", "derivation": "\\varepsilon_{0}{(\\varphi^*,\\hbar)} = \\log{(\\hbar)}^{\\varphi^*} and \\int \\varepsilon_{0}{(\\varphi^*,\\hbar)} d\\hbar = \\int \\log{(\\hbar)}^{\\varphi^*} d\\hbar and \\operatorname{A_{2}}{(\\hbar)} = \\hbar and \\int \\varepsilon_{0}{(\\varphi^*,\\hbar)} d\\operatorname{A_{2}}{(\\hbar)} = \\int \\log{(\\hbar)}^{\\varphi^*} d\\operatorname{A_{2}}{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Function('A_2')(Symbol('\\\\hbar', commutative=True)))), Integral(Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Function('A_2')(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)} = \\frac{\\mathbf{J}_f y}{\\Psi^{\\dagger}}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)} = - \\frac{\\mathbf{J}_f y}{(\\Psi^{\\dagger})^{2}}, then obtain \\frac{1}{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\mathbf{J}_f y}{\\Psi^{\\dagger}})^{2}} = \\frac{(\\Psi^{\\dagger})^{4}}{\\mathbf{J}_f^{2} y^{2}}", "derivation": "\\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)} = \\frac{\\mathbf{J}_f y}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\mathbf{J}_f y}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)} = - \\frac{\\mathbf{J}_f y}{(\\Psi^{\\dagger})^{2}} and \\frac{1}{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\dot{z}{(\\Psi^{\\dagger},\\mathbf{J}_f,y)})^{2}} = \\frac{(\\Psi^{\\dagger})^{4}}{\\mathbf{J}_f^{2} y^{2}} and \\frac{1}{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\mathbf{J}_f y}{\\Psi^{\\dagger}})^{2}} = \\frac{(\\Psi^{\\dagger})^{4}}{\\mathbf{J}_f^{2} y^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-2)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-2)), Pow(Symbol('y', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-2)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-2)), Pow(Symbol('y', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\hat{p}{(U,C_{2})} = \\log{(\\frac{C_{2}}{U})}, then obtain S + \\hat{p}{(U,C_{2})} = f + \\log{(C_{2})}", "derivation": "\\hat{p}{(U,C_{2})} = \\log{(\\frac{C_{2}}{U})} and \\frac{\\partial}{\\partial C_{2}} \\hat{p}{(U,C_{2})} = \\frac{\\partial}{\\partial C_{2}} \\log{(\\frac{C_{2}}{U})} and \\int \\frac{\\partial}{\\partial C_{2}} \\hat{p}{(U,C_{2})} dC_{2} = \\int \\frac{\\partial}{\\partial C_{2}} \\log{(\\frac{C_{2}}{U})} dC_{2} and S + \\hat{p}{(U,C_{2})} = f + \\log{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('C_2', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(log(Mul(Symbol('C_2', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('S', commutative=True), Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('C_2', commutative=True))), Add(Symbol('f', commutative=True), log(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given Q{(\\sigma_x,b)} = \\log{(b)}^{\\sigma_x}, then obtain (\\int (b^{2} Q^{2}{(\\sigma_x,b)} + b \\log{(b)}^{\\sigma_x}) db)^{b} = (\\int (b^{2} Q{(\\sigma_x,b)} \\log{(b)}^{\\sigma_x} + b \\log{(b)}^{\\sigma_x}) db)^{b}", "derivation": "Q{(\\sigma_x,b)} = \\log{(b)}^{\\sigma_x} and b Q{(\\sigma_x,b)} = b \\log{(b)}^{\\sigma_x} and b^{2} Q^{2}{(\\sigma_x,b)} = b^{2} Q{(\\sigma_x,b)} \\log{(b)}^{\\sigma_x} and b^{2} Q^{2}{(\\sigma_x,b)} + b \\log{(b)}^{\\sigma_x} = b^{2} Q{(\\sigma_x,b)} \\log{(b)}^{\\sigma_x} + b \\log{(b)}^{\\sigma_x} and \\int (b^{2} Q^{2}{(\\sigma_x,b)} + b \\log{(b)}^{\\sigma_x}) db = \\int (b^{2} Q{(\\sigma_x,b)} \\log{(b)}^{\\sigma_x} + b \\log{(b)}^{\\sigma_x}) db and (\\int (b^{2} Q^{2}{(\\sigma_x,b)} + b \\log{(b)}^{\\sigma_x}) db)^{b} = (\\int (b^{2} Q{(\\sigma_x,b)} \\log{(b)}^{\\sigma_x} + b \\log{(b)}^{\\sigma_x}) db)^{b}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 2, "Mul(Symbol('b', commutative=True), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["add", 3, "Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integral(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('b', commutative=True))))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Integer(2))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('Q')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True)), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('b', commutative=True), Pow(log(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given u{(M_{E},\\mu_0)} = M_{E} + \\log{(\\mu_0)}, then obtain \\int \\frac{\\mu_0 (-1 + \\frac{u{(M_{E},\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} d\\mu_0 = \\int \\frac{\\mu_0 (-1 + \\frac{M_{E} + \\log{(\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} d\\mu_0", "derivation": "u{(M_{E},\\mu_0)} = M_{E} + \\log{(\\mu_0)} and \\frac{u{(M_{E},\\mu_0)}}{\\mu_0} = \\frac{M_{E} + \\log{(\\mu_0)}}{\\mu_0} and -1 + \\frac{u{(M_{E},\\mu_0)}}{\\mu_0} = -1 + \\frac{M_{E} + \\log{(\\mu_0)}}{\\mu_0} and \\frac{\\mu_0 (-1 + \\frac{u{(M_{E},\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} = \\frac{\\mu_0 (-1 + \\frac{M_{E} + \\log{(\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} and \\int \\frac{\\mu_0 (-1 + \\frac{u{(M_{E},\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} d\\mu_0 = \\int \\frac{\\mu_0 (-1 + \\frac{M_{E} + \\log{(\\mu_0)}}{\\mu_0})}{M_{E} + \\log{(\\mu_0)}} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('u')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('u')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('u')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Symbol('\\\\mu_0', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))), Pow(Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mu_0', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('u')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))), Pow(Add(Symbol('M_E', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(a)} = \\log{(\\sin{(a)})}, then obtain \\frac{d}{d a} ((- a + \\hat{X}{(a)})^{a} - \\int \\hat{X}{(a)} da) = \\frac{d}{d a} ((- a + \\log{(\\sin{(a)})})^{a} - \\int \\hat{X}{(a)} da)", "derivation": "\\hat{X}{(a)} = \\log{(\\sin{(a)})} and - a + \\hat{X}{(a)} = - a + \\log{(\\sin{(a)})} and (- a + \\hat{X}{(a)})^{a} = (- a + \\log{(\\sin{(a)})})^{a} and (- a + \\hat{X}{(a)})^{a} - \\int \\hat{X}{(a)} da = (- a + \\log{(\\sin{(a)})})^{a} - \\int \\hat{X}{(a)} da and \\frac{d}{d a} ((- a + \\hat{X}{(a)})^{a} - \\int \\hat{X}{(a)} da) = \\frac{d}{d a} ((- a + \\log{(\\sin{(a)})})^{a} - \\int \\hat{X}{(a)} da)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True))))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True)))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True)))), Symbol('a', commutative=True)))"], [["minus", 3, "Integral(Function('\\\\hat{X}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{X}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))), Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{X}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{X}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(sin(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{X}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(n_{1},k,\\Omega)} = (\\Omega + k)^{n_{1}}, then obtain \\frac{\\partial}{\\partial n_{1}} \\int (- \\Omega - k + \\Psi_{\\lambda}{(n_{1},k,\\Omega)}) dn_{1} = \\frac{\\partial}{\\partial n_{1}} \\int (- \\Omega - k + (\\Omega + k)^{n_{1}}) dn_{1}", "derivation": "\\Psi_{\\lambda}{(n_{1},k,\\Omega)} = (\\Omega + k)^{n_{1}} and - \\Omega - k + \\Psi_{\\lambda}{(n_{1},k,\\Omega)} = - \\Omega - k + (\\Omega + k)^{n_{1}} and \\int (- \\Omega - k + \\Psi_{\\lambda}{(n_{1},k,\\Omega)}) dn_{1} = \\int (- \\Omega - k + (\\Omega + k)^{n_{1}}) dn_{1} and \\frac{\\partial}{\\partial n_{1}} \\int (- \\Omega - k + \\Psi_{\\lambda}{(n_{1},k,\\Omega)}) dn_{1} = \\frac{\\partial}{\\partial n_{1}} \\int (- \\Omega - k + (\\Omega + k)^{n_{1}}) dn_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('k', commutative=True)), Symbol('n_1', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\Omega', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('k', commutative=True)), Symbol('n_1', commutative=True))))"], [["integrate", 2, "Symbol('n_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('k', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('n_1', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('k', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(m,\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}}^{m})}, then obtain \\frac{\\rho_{f}^{m}{(m,\\hat{\\mathbf{r}})} + \\sin^{m}{(\\hat{\\mathbf{r}}^{m})}}{\\rho_{f}{(m,\\hat{\\mathbf{r}})}} = \\frac{2 \\sin^{m}{(\\hat{\\mathbf{r}}^{m})}}{\\rho_{f}{(m,\\hat{\\mathbf{r}})}}", "derivation": "\\rho_{f}{(m,\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}}^{m})} and \\rho_{f}^{m}{(m,\\hat{\\mathbf{r}})} = \\sin^{m}{(\\hat{\\mathbf{r}}^{m})} and \\rho_{f}^{m}{(m,\\hat{\\mathbf{r}})} + \\sin^{m}{(\\hat{\\mathbf{r}}^{m})} = 2 \\sin^{m}{(\\hat{\\mathbf{r}}^{m})} and \\frac{\\rho_{f}^{m}{(m,\\hat{\\mathbf{r}})} + \\sin^{m}{(\\hat{\\mathbf{r}}^{m})}}{\\rho_{f}{(m,\\hat{\\mathbf{r}})}} = \\frac{2 \\sin^{m}{(\\hat{\\mathbf{r}}^{m})}}{\\rho_{f}{(m,\\hat{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["add", 2, "Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))"], "Equality(Add(Pow(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))), Mul(Integer(2), Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["divide", 3, "Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))), Pow(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Pow(sin(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given H{(c)} = e^{c} and \\operatorname{A_{2}}{(c)} = (H{(c)} + 1) e^{c}, then obtain (H{(c)} + 1) e^{c} = (e^{c} + 1) H{(c)}", "derivation": "H{(c)} = e^{c} and H{(c)} + 1 = e^{c} + 1 and \\operatorname{A_{2}}{(c)} = (H{(c)} + 1) e^{c} and \\operatorname{A_{2}}{(c)} = (H{(c)} + 1) H{(c)} and \\operatorname{A_{2}}{(c)} = (e^{c} + 1) H{(c)} and (H{(c)} + 1) e^{c} = (e^{c} + 1) H{(c)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('H')(Symbol('c', commutative=True)), Integer(1)), Add(exp(Symbol('c', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('c', commutative=True)), Mul(Add(Function('H')(Symbol('c', commutative=True)), Integer(1)), exp(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_2')(Symbol('c', commutative=True)), Mul(Add(Function('H')(Symbol('c', commutative=True)), Integer(1)), Function('H')(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('A_2')(Symbol('c', commutative=True)), Mul(Add(exp(Symbol('c', commutative=True)), Integer(1)), Function('H')(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Function('H')(Symbol('c', commutative=True)), Integer(1)), exp(Symbol('c', commutative=True))), Mul(Add(exp(Symbol('c', commutative=True)), Integer(1)), Function('H')(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(u)} = \\sin{(u)}, then derive \\int (u + \\mu_{0}{(u)}) du = m + \\frac{u^{2}}{2} - \\cos{(u)}, then obtain \\int (u + \\sin{(u)}) du = m + \\frac{u^{2}}{2} - \\cos{(u)}", "derivation": "\\mu_{0}{(u)} = \\sin{(u)} and u + \\mu_{0}{(u)} = u + \\sin{(u)} and \\int (u + \\mu_{0}{(u)}) du = \\int (u + \\sin{(u)}) du and \\int (u + \\mu_{0}{(u)}) du = m + \\frac{u^{2}}{2} - \\cos{(u)} and \\int (u + \\sin{(u)}) du = m + \\frac{u^{2}}{2} - \\cos{(u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\mu_0')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Symbol('u', commutative=True), Function('\\\\mu_0')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('u', commutative=True), Function('\\\\mu_0')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\phi_2,C_{d})} = C_{d} \\cos{(\\phi_2)}, then derive \\frac{\\partial}{\\partial C_{d}} \\theta_{1}{(\\phi_2,C_{d})} = \\cos{(\\phi_2)}, then obtain \\theta_{1}{(\\phi_2,C_{d})} = C_{d} \\frac{\\partial}{\\partial C_{d}} C_{d} \\cos{(\\phi_2)}", "derivation": "\\theta_{1}{(\\phi_2,C_{d})} = C_{d} \\cos{(\\phi_2)} and \\frac{\\partial}{\\partial C_{d}} \\theta_{1}{(\\phi_2,C_{d})} = \\frac{\\partial}{\\partial C_{d}} C_{d} \\cos{(\\phi_2)} and \\frac{\\partial}{\\partial C_{d}} \\theta_{1}{(\\phi_2,C_{d})} = \\cos{(\\phi_2)} and \\frac{\\partial}{\\partial C_{d}} C_{d} \\cos{(\\phi_2)} = \\cos{(\\phi_2)} and \\theta_{1}{(\\phi_2,C_{d})} = C_{d} \\frac{\\partial}{\\partial C_{d}} C_{d} \\cos{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), cos(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_d', commutative=True), cos(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), cos(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('C_d', commutative=True), cos(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), cos(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), cos(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\dot{z})} = \\sin{(\\sin{(\\dot{z})})}, then obtain e^{- \\frac{\\hat{\\mathbf{x}}^{2}{(\\dot{z})}}{\\dot{z} \\sin{(\\sin{(\\dot{z})})}}} = e^{- \\frac{\\sin{(\\sin{(\\dot{z})})}}{\\dot{z}}}", "derivation": "\\hat{\\mathbf{x}}{(\\dot{z})} = \\sin{(\\sin{(\\dot{z})})} and - \\frac{\\hat{\\mathbf{x}}{(\\dot{z})}}{\\dot{z}} = - \\frac{\\sin{(\\sin{(\\dot{z})})}}{\\dot{z}} and \\frac{\\hat{\\mathbf{x}}{(\\dot{z})}}{\\sin{(\\sin{(\\dot{z})})}} = 1 and \\frac{\\hat{\\mathbf{x}}^{2}{(\\dot{z})}}{\\sin{(\\sin{(\\dot{z})})}} = \\hat{\\mathbf{x}}{(\\dot{z})} and - \\frac{\\hat{\\mathbf{x}}^{2}{(\\dot{z})}}{\\dot{z} \\sin{(\\sin{(\\dot{z})})}} = - \\frac{\\sin{(\\sin{(\\dot{z})})}}{\\dot{z}} and e^{- \\frac{\\hat{\\mathbf{x}}^{2}{(\\dot{z})}}{\\dot{z} \\sin{(\\sin{(\\dot{z})})}}} = e^{- \\frac{\\sin{(\\sin{(\\dot{z})})}}{\\dot{z}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)), sin(sin(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\dot{z}', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\dot{z}', commutative=True))))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))), Integer(1))"], [["times", 3, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), Pow(sin(sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), Pow(sin(sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\dot{z}', commutative=True)))))"], [["exp", 5], "Equality(exp(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), Pow(sin(sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)))), exp(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\dot{z}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} = \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}}, then obtain \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\iint \\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} d\\psi d\\dot{\\mathbf{r}} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\iint \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}} d\\psi d\\dot{\\mathbf{r}}", "derivation": "\\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} = \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}} and \\int \\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} d\\psi = \\int \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}} d\\psi and \\iint \\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} d\\psi d\\dot{\\mathbf{r}} = \\iint \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}} d\\psi d\\dot{\\mathbf{r}} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\iint \\operatorname{n_{1}}{(\\dot{\\mathbf{r}},\\psi)} d\\psi d\\dot{\\mathbf{r}} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\iint \\frac{\\cos{(\\psi)}}{\\dot{\\mathbf{r}}} d\\psi d\\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integral(Function('n_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(C,m_{s})} = m_{s}^{C}, then obtain \\int \\frac{\\frac{\\partial}{\\partial C} Q{(C,m_{s})}}{C} dm_{s} = \\int \\frac{\\frac{\\partial}{\\partial C} m_{s}^{C}}{C} dm_{s}", "derivation": "Q{(C,m_{s})} = m_{s}^{C} and \\frac{\\partial}{\\partial C} Q{(C,m_{s})} = \\frac{\\partial}{\\partial C} m_{s}^{C} and Q{(C,m_{s})} \\frac{\\partial}{\\partial C} Q{(C,m_{s})} = Q{(C,m_{s})} \\frac{\\partial}{\\partial C} m_{s}^{C} and \\frac{\\frac{\\partial}{\\partial C} Q{(C,m_{s})}}{C} = \\frac{\\frac{\\partial}{\\partial C} m_{s}^{C}}{C} and \\int \\frac{\\frac{\\partial}{\\partial C} Q{(C,m_{s})}}{C} dm_{s} = \\int \\frac{\\frac{\\partial}{\\partial C} m_{s}^{C}}{C} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Symbol('m_s', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 2, "Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Derivative(Pow(Symbol('m_s', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Pow(Symbol('m_s', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('m_s', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Pow(Symbol('m_s', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(v_{x})} = \\cos{(v_{x})}, then obtain \\mathbf{g}^{2 v_{x}}{(v_{x})} + \\mathbf{g}^{v_{x}}{(v_{x})} \\cos^{v_{x}}{(v_{x})} = 2 \\mathbf{g}^{v_{x}}{(v_{x})} \\cos^{v_{x}}{(v_{x})}", "derivation": "\\mathbf{g}{(v_{x})} = \\cos{(v_{x})} and \\mathbf{g}^{v_{x}}{(v_{x})} = \\cos^{v_{x}}{(v_{x})} and \\mathbf{g}^{2 v_{x}}{(v_{x})} = \\mathbf{g}^{v_{x}}{(v_{x})} \\cos^{v_{x}}{(v_{x})} and \\mathbf{g}^{2 v_{x}}{(v_{x})} + \\mathbf{g}^{v_{x}}{(v_{x})} \\cos^{v_{x}}{(v_{x})} = 2 \\mathbf{g}^{v_{x}}{(v_{x})} \\cos^{v_{x}}{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["times", 2, "Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["add", 3, "Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))), Mul(Integer(2), Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\phi_2,\\sigma_p)} = \\log{(\\phi_2 \\sigma_p)}, then obtain ((\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{C_{d}}^{\\sigma_p}{(\\phi_2,\\sigma_p)})^{\\phi_2})^{\\phi_2} = ((\\frac{\\partial}{\\partial \\sigma_p} \\log{(\\phi_2 \\sigma_p)}^{\\sigma_p})^{\\phi_2})^{\\phi_2}", "derivation": "\\operatorname{C_{d}}{(\\phi_2,\\sigma_p)} = \\log{(\\phi_2 \\sigma_p)} and \\operatorname{C_{d}}^{\\sigma_p}{(\\phi_2,\\sigma_p)} = \\log{(\\phi_2 \\sigma_p)}^{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{C_{d}}^{\\sigma_p}{(\\phi_2,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\log{(\\phi_2 \\sigma_p)}^{\\sigma_p} and (\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{C_{d}}^{\\sigma_p}{(\\phi_2,\\sigma_p)})^{\\phi_2} = (\\frac{\\partial}{\\partial \\sigma_p} \\log{(\\phi_2 \\sigma_p)}^{\\sigma_p})^{\\phi_2} and ((\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{C_{d}}^{\\sigma_p}{(\\phi_2,\\sigma_p)})^{\\phi_2})^{\\phi_2} = ((\\frac{\\partial}{\\partial \\sigma_p} \\log{(\\phi_2 \\sigma_p)}^{\\sigma_p})^{\\phi_2})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Pow(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)), Pow(Derivative(Pow(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Pow(Derivative(Pow(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Derivative(Pow(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given m{(g)} = e^{g}, then derive e^{g} + \\frac{d}{d g} m{(g)} = 2 e^{g}, then obtain (m{(g)} + \\frac{d}{d g} m{(g)})^{g} = (2 m{(g)})^{g}", "derivation": "m{(g)} = e^{g} and m{(g)} + e^{g} = 2 e^{g} and \\frac{d}{d g} (m{(g)} + e^{g}) = \\frac{d}{d g} 2 e^{g} and e^{g} + \\frac{d}{d g} m{(g)} = 2 e^{g} and m{(g)} + \\frac{d}{d g} m{(g)} = 2 m{(g)} and (m{(g)} + \\frac{d}{d g} m{(g)})^{g} = (2 m{(g)})^{g}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["add", 1, "exp(Symbol('g', commutative=True))"], "Equality(Add(Function('m')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), Mul(Integer(2), exp(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Function('m')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('g', commutative=True)), Derivative(Function('m')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('m')(Symbol('g', commutative=True)), Derivative(Function('m')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(2), Function('m')(Symbol('g', commutative=True))))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Function('m')(Symbol('g', commutative=True)), Derivative(Function('m')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('g', commutative=True)), Pow(Mul(Integer(2), Function('m')(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(E_{x},C_{1},\\mu)} = (\\frac{E_{x}}{\\mu})^{C_{1}}, then obtain \\int \\frac{\\partial}{\\partial \\mu} \\operatorname{A_{2}}^{E_{x}}{(E_{x},C_{1},\\mu)} dC_{1} = \\int \\frac{\\partial}{\\partial \\mu} ((\\frac{E_{x}}{\\mu})^{C_{1}})^{E_{x}} dC_{1}", "derivation": "\\operatorname{A_{2}}{(E_{x},C_{1},\\mu)} = (\\frac{E_{x}}{\\mu})^{C_{1}} and \\operatorname{A_{2}}^{E_{x}}{(E_{x},C_{1},\\mu)} = ((\\frac{E_{x}}{\\mu})^{C_{1}})^{E_{x}} and \\frac{\\partial}{\\partial \\mu} \\operatorname{A_{2}}^{E_{x}}{(E_{x},C_{1},\\mu)} = \\frac{\\partial}{\\partial \\mu} ((\\frac{E_{x}}{\\mu})^{C_{1}})^{E_{x}} and \\int \\frac{\\partial}{\\partial \\mu} \\operatorname{A_{2}}^{E_{x}}{(E_{x},C_{1},\\mu)} dC_{1} = \\int \\frac{\\partial}{\\partial \\mu} ((\\frac{E_{x}}{\\mu})^{C_{1}})^{E_{x}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('E_x', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('C_1', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('E_x', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('C_1', commutative=True)), Symbol('E_x', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('A_2')(Symbol('E_x', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Pow(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('C_1', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('A_2')(Symbol('E_x', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Derivative(Pow(Pow(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Symbol('C_1', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given v{(\\pi,\\sigma_x)} = \\sin^{\\sigma_x}{(\\pi)} and l{(\\pi)} = \\pi, then obtain \\frac{1}{(- 2 \\pi + l{(\\pi)} + v{(\\pi,\\sigma_x)})^{2}} = \\frac{1}{(- \\pi + \\sin^{\\sigma_x}{(\\pi)})^{2}}", "derivation": "v{(\\pi,\\sigma_x)} = \\sin^{\\sigma_x}{(\\pi)} and l{(\\pi)} = \\pi and - \\pi + v{(\\pi,\\sigma_x)} = - \\pi + \\sin^{\\sigma_x}{(\\pi)} and - \\pi + l{(\\pi)} + v{(\\pi,\\sigma_x)} = v{(\\pi,\\sigma_x)} and - \\pi + v{(\\pi,\\sigma_x)} = - l{(\\pi)} + v{(\\pi,\\sigma_x)} and - 2 \\pi + l{(\\pi)} + v{(\\pi,\\sigma_x)} = - \\pi + v{(\\pi,\\sigma_x)} and \\frac{1}{(- 2 \\pi + l{(\\pi)} + v{(\\pi,\\sigma_x)})^{2}} = \\frac{1}{(- \\pi + v{(\\pi,\\sigma_x)})^{2}} and \\frac{1}{(- 2 \\pi + l{(\\pi)} + v{(\\pi,\\sigma_x)})^{2}} = \\frac{1}{(- \\pi + \\sin^{\\sigma_x}{(\\pi)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["minus", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('l')(Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 4, "Function('l')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Function('l')(Symbol('\\\\pi', commutative=True))), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True)), Function('l')(Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["power", 6, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True)), Function('l')(Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Integer(-2)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True)), Function('l')(Symbol('\\\\pi', commutative=True)), Function('v')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(m,\\mathbf{s})} = \\cos^{\\mathbf{s}}{(m)}, then obtain \\frac{\\partial}{\\partial \\mathbf{s}} e^{\\operatorname{y^{\\prime}}^{m}{(m,\\mathbf{s})}} = \\frac{\\partial}{\\partial \\mathbf{s}} e^{(\\cos^{\\mathbf{s}}{(m)})^{m}}", "derivation": "\\operatorname{y^{\\prime}}{(m,\\mathbf{s})} = \\cos^{\\mathbf{s}}{(m)} and \\operatorname{y^{\\prime}}^{m}{(m,\\mathbf{s})} = (\\cos^{\\mathbf{s}}{(m)})^{m} and e^{\\operatorname{y^{\\prime}}^{m}{(m,\\mathbf{s})}} = e^{(\\cos^{\\mathbf{s}}{(m)})^{m}} and \\frac{\\partial}{\\partial \\mathbf{s}} e^{\\operatorname{y^{\\prime}}^{m}{(m,\\mathbf{s})}} = \\frac{\\partial}{\\partial \\mathbf{s}} e^{(\\cos^{\\mathbf{s}}{(m)})^{m}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True))), exp(Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(exp(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(exp(Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(\\dot{x},G)} = \\frac{\\sin{(\\dot{x})}}{G} and \\mathbf{H}{(\\dot{x},G)} = \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)}, then derive \\delta + \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} = \\delta - \\frac{\\sin{(\\dot{x})}}{G^{2}}, then obtain \\delta + \\mathbf{H}{(\\dot{x},G)} = \\delta - \\frac{\\sin{(\\dot{x})}}{G^{2}}", "derivation": "\\hat{H}{(\\dot{x},G)} = \\frac{\\sin{(\\dot{x})}}{G} and \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} = \\frac{\\partial}{\\partial G} \\frac{\\sin{(\\dot{x})}}{G} and \\delta + \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} = \\delta + \\frac{\\partial}{\\partial G} \\frac{\\sin{(\\dot{x})}}{G} and \\mathbf{H}{(\\dot{x},G)} = \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} and \\delta + \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} = \\delta - \\frac{\\sin{(\\dot{x})}}{G^{2}} and \\delta + \\mathbf{H}{(\\dot{x},G)} = \\delta + \\frac{\\partial}{\\partial G} \\frac{\\sin{(\\dot{x})}}{G} and \\delta + \\mathbf{H}{(\\dot{x},G)} = \\delta + \\frac{\\partial}{\\partial G} \\hat{H}{(\\dot{x},G)} and \\delta + \\mathbf{H}{(\\dot{x},G)} = \\delta - \\frac{\\sin{(\\dot{x})}}{G^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-2)), sin(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\dot{x}', commutative=True), Symbol('G', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-2)), sin(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(J)} = \\log{(J)}, then obtain - \\frac{2 \\mathbf{s}{(J)} - 2 \\log{(J)}}{\\log{(J)}} = - \\frac{\\mathbf{s}{(J)} - \\log{(J)}}{\\log{(J)}}", "derivation": "\\mathbf{s}{(J)} = \\log{(J)} and \\mathbf{s}{(J)} - \\log{(J)} = 0 and 2 \\mathbf{s}{(J)} - 2 \\log{(J)} = \\mathbf{s}{(J)} - \\log{(J)} and - \\frac{2 \\mathbf{s}{(J)} - 2 \\log{(J)}}{\\log{(J)}} = - \\frac{\\mathbf{s}{(J)} - \\log{(J)}}{\\log{(J)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["minus", 1, "log(Symbol('J', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('J', commutative=True)))), Integer(0))"], [["add", 2, "Add(Function('\\\\mathbf{s}')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('J', commutative=True)))), Add(Function('\\\\mathbf{s}')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('J', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), log(Symbol('J', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('J', commutative=True)))), Pow(log(Symbol('J', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Function('\\\\mathbf{s}')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('J', commutative=True)))), Pow(log(Symbol('J', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(E_{x},\\mathbf{p})} = E_{x} \\mathbf{p} and g{(E_{x},\\mathbf{p})} = 2 E_{x} \\mathbf{p} + (- E_{x} \\mathbf{p} + \\rho_{f}{(E_{x},\\mathbf{p})}) \\rho_{f}{(E_{x},\\mathbf{p})}, then obtain \\sin{(g{(E_{x},\\mathbf{p})})} = \\sin{(2 \\rho_{f}{(E_{x},\\mathbf{p})})}", "derivation": "\\rho_{f}{(E_{x},\\mathbf{p})} = E_{x} \\mathbf{p} and g{(E_{x},\\mathbf{p})} = 2 E_{x} \\mathbf{p} + (- E_{x} \\mathbf{p} + \\rho_{f}{(E_{x},\\mathbf{p})}) \\rho_{f}{(E_{x},\\mathbf{p})} and g{(E_{x},\\mathbf{p})} = 2 E_{x} \\mathbf{p} and \\sin{(g{(E_{x},\\mathbf{p})})} = \\sin{(2 E_{x} \\mathbf{p})} and \\sin{(g{(E_{x},\\mathbf{p})})} = \\sin{(2 \\rho_{f}{(E_{x},\\mathbf{p})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\rho_f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\rho_f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["sin", 3], "Equality(sin(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), sin(Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(sin(Function('g')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), sin(Mul(Integer(2), Function('\\\\rho_f')(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{E},k)} = \\frac{\\partial}{\\partial k} \\mathbf{E} k and \\varphi^{*}{(\\mathbf{E},k)} = \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}^*{(\\mathbf{E},k)}, then obtain \\tilde{g}^*{(\\mathbf{E},k)} + \\varphi^{*}{(\\mathbf{E},k)} = \\tilde{g}^*{(\\mathbf{E},k)} + \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial k} \\mathbf{E} k", "derivation": "\\tilde{g}^*{(\\mathbf{E},k)} = \\frac{\\partial}{\\partial k} \\mathbf{E} k and \\varphi^{*}{(\\mathbf{E},k)} = \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}^*{(\\mathbf{E},k)} and \\varphi^{*}{(\\mathbf{E},k)} = \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial k} \\mathbf{E} k and \\tilde{g}^*{(\\mathbf{E},k)} + \\varphi^{*}{(\\mathbf{E},k)} = \\tilde{g}^*{(\\mathbf{E},k)} + \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial k} \\mathbf{E} k", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Function('\\\\varphi^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(\\phi)} = e^{\\phi}, then derive \\int Z{(\\phi)} d\\phi = \\mathbb{I} + e^{\\phi}, then derive - \\mathbb{I} - Z{(\\phi)} = - \\mathbf{J}_M - e^{\\phi}, then obtain - \\sin{(\\mathbb{I} + Z{(\\phi)})} = - \\sin{(\\mathbf{J}_M + e^{\\phi})}", "derivation": "Z{(\\phi)} = e^{\\phi} and \\int Z{(\\phi)} d\\phi = \\int e^{\\phi} d\\phi and \\int Z{(\\phi)} d\\phi = \\mathbb{I} + e^{\\phi} and \\int Z{(\\phi)} d\\phi = \\mathbb{I} + Z{(\\phi)} and \\mathbb{I} + Z{(\\phi)} = \\int e^{\\phi} d\\phi and - \\mathbb{I} - Z{(\\phi)} = - \\int e^{\\phi} d\\phi and - \\mathbb{I} - Z{(\\phi)} = - \\mathbf{J}_M - e^{\\phi} and - \\sin{(\\mathbb{I} + Z{(\\phi)})} = - \\sin{(\\mathbf{J}_M + e^{\\phi})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('Z')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Function('Z')(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('Z')(Symbol('\\\\phi', commutative=True))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["times", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))))"], [["sin", 7], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('Z')(Symbol('\\\\phi', commutative=True))))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{B}{(E)} = \\sin{(E)} and \\theta_{2}{(E)} = \\int (\\mathbf{B}{(E)} \\sin{(E)})^{E} dE, then obtain \\theta_{2}{(E)} + e^{\\mathbf{B}^{2}{(E)}} = e^{\\mathbf{B}^{2}{(E)}} + \\int (\\sin^{2}{(E)})^{E} dE", "derivation": "\\mathbf{B}{(E)} = \\sin{(E)} and \\mathbf{B}{(E)} \\sin{(E)} = \\sin^{2}{(E)} and (\\mathbf{B}{(E)} \\sin{(E)})^{E} = (\\sin^{2}{(E)})^{E} and \\int (\\mathbf{B}{(E)} \\sin{(E)})^{E} dE = \\int (\\sin^{2}{(E)})^{E} dE and e^{\\mathbf{B}{(E)} \\sin{(E)}} + \\int (\\mathbf{B}{(E)} \\sin{(E)})^{E} dE = e^{\\mathbf{B}{(E)} \\sin{(E)}} + \\int (\\sin^{2}{(E)})^{E} dE and e^{\\mathbf{B}^{2}{(E)}} + \\int (\\mathbf{B}{(E)} \\sin{(E)})^{E} dE = e^{\\mathbf{B}^{2}{(E)}} + \\int (\\sin^{2}{(E)})^{E} dE and \\theta_{2}{(E)} = \\int (\\mathbf{B}{(E)} \\sin{(E)})^{E} dE and \\theta_{2}{(E)} + e^{\\mathbf{B}^{2}{(E)}} = e^{\\mathbf{B}^{2}{(E)}} + \\int (\\sin^{2}{(E)})^{E} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["times", 1, "sin(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Pow(sin(Symbol('E', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Pow(sin(Symbol('E', commutative=True)), Integer(2)), Symbol('E', commutative=True)))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Pow(sin(Symbol('E', commutative=True)), Integer(2)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 4, "exp(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))))"], "Equality(Add(exp(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))), Integral(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(exp(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))), Integral(Pow(Pow(sin(Symbol('E', commutative=True)), Integer(2)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(exp(Pow(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), Integer(2))), Integral(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(exp(Pow(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), Integer(2))), Integral(Pow(Pow(sin(Symbol('E', commutative=True)), Integer(2)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('E', commutative=True)), Integral(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Function('\\\\theta_2')(Symbol('E', commutative=True)), exp(Pow(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), Integer(2)))), Add(exp(Pow(Function('\\\\mathbf{B}')(Symbol('E', commutative=True)), Integer(2))), Integral(Pow(Pow(sin(Symbol('E', commutative=True)), Integer(2)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\dot{x},Z,F_{g})} = Z (F_{g} + \\dot{x}) and \\operatorname{m_{s}}{(\\dot{x})} = \\dot{x}, then obtain Z (F_{g} + \\dot{x}) \\operatorname{m_{s}}{(\\dot{x})} \\int Z (F_{g} + \\dot{x}) dF_{g} = Z \\dot{x} (F_{g} + \\dot{x}) \\int Z (F_{g} + \\dot{x}) dF_{g}", "derivation": "\\dot{z}{(\\dot{x},Z,F_{g})} = Z (F_{g} + \\dot{x}) and \\operatorname{m_{s}}{(\\dot{x})} = \\dot{x} and \\int \\dot{z}{(\\dot{x},Z,F_{g})} dF_{g} = \\int Z (F_{g} + \\dot{x}) dF_{g} and Z (F_{g} + \\dot{x}) \\operatorname{m_{s}}{(\\dot{x})} = Z \\dot{x} (F_{g} + \\dot{x}) and Z (F_{g} + \\dot{x}) \\operatorname{m_{s}}{(\\dot{x})} \\int \\dot{z}{(\\dot{x},Z,F_{g})} dF_{g} = Z \\dot{x} (F_{g} + \\dot{x}) \\int \\dot{z}{(\\dot{x},Z,F_{g})} dF_{g} and Z (F_{g} + \\dot{x}) \\operatorname{m_{s}}{(\\dot{x})} \\int Z (F_{g} + \\dot{x}) dF_{g} = Z \\dot{x} (F_{g} + \\dot{x}) \\int Z (F_{g} + \\dot{x}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('F_g', commutative=True)), Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["times", 2, "Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('m_s')(Symbol('\\\\dot{x}', commutative=True))), Mul(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["times", 4, "Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('m_s')(Symbol('\\\\dot{x}', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('m_s')(Symbol('\\\\dot{x}', commutative=True)), Integral(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('F_g', commutative=True)))), Mul(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integral(Mul(Symbol('Z', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\omega)} = \\log{(\\omega)}, then obtain \\omega + \\log{(\\omega)} + \\int (\\omega + \\operatorname{F_{N}}{(\\omega)}) d\\omega = \\omega + \\log{(\\omega)} + \\int (\\omega + \\log{(\\omega)}) d\\omega", "derivation": "\\operatorname{F_{N}}{(\\omega)} = \\log{(\\omega)} and \\omega + \\operatorname{F_{N}}{(\\omega)} = \\omega + \\log{(\\omega)} and \\int (\\omega + \\operatorname{F_{N}}{(\\omega)}) d\\omega = \\int (\\omega + \\log{(\\omega)}) d\\omega and \\omega + \\log{(\\omega)} + \\int (\\omega + \\operatorname{F_{N}}{(\\omega)}) d\\omega = \\omega + \\log{(\\omega)} + \\int (\\omega + \\log{(\\omega)}) d\\omega", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('F_N')(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\omega', commutative=True), Function('F_N')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Function('F_N')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(r_{0},f_{\\mathbf{v}})} = f_{\\mathbf{v}} r_{0}, then derive \\frac{\\partial}{\\partial r_{0}} \\operatorname{n_{2}}{(r_{0},f_{\\mathbf{v}})} = f_{\\mathbf{v}}, then obtain - r_{0} + \\operatorname{n_{2}}{(r_{0},\\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0})} = r_{0} \\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0} - r_{0}", "derivation": "\\operatorname{n_{2}}{(r_{0},f_{\\mathbf{v}})} = f_{\\mathbf{v}} r_{0} and \\frac{\\partial}{\\partial r_{0}} \\operatorname{n_{2}}{(r_{0},f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0} and \\frac{\\partial}{\\partial r_{0}} \\operatorname{n_{2}}{(r_{0},f_{\\mathbf{v}})} = f_{\\mathbf{v}} and \\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0} = f_{\\mathbf{v}} and \\operatorname{n_{2}}{(r_{0},\\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0})} = r_{0} \\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0} and - r_{0} + \\operatorname{n_{2}}{(r_{0},\\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0})} = r_{0} \\frac{\\partial}{\\partial r_{0}} f_{\\mathbf{v}} r_{0} - r_{0}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('r_0', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('r_0', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('r_0', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('n_2')(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["minus", 5, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('n_2')(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))), Add(Mul(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(M_{E})} = \\cos{(M_{E})}, then obtain (M_{E} \\mathbf{M}{(M_{E})} + M_{E} \\cos{(M_{E})})^{M_{E}} = (2 M_{E} \\cos{(M_{E})})^{M_{E}}", "derivation": "\\mathbf{M}{(M_{E})} = \\cos{(M_{E})} and M_{E} \\mathbf{M}{(M_{E})} = M_{E} \\cos{(M_{E})} and M_{E} \\mathbf{M}{(M_{E})} + M_{E} \\cos{(M_{E})} = 2 M_{E} \\cos{(M_{E})} and (M_{E} \\mathbf{M}{(M_{E})} + M_{E} \\cos{(M_{E})})^{M_{E}} = (2 M_{E} \\cos{(M_{E})})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True)))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True))))"], [["add", 2, "Mul(Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True)))"], "Equality(Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True)))), Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True))))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Pow(Mul(Integer(2), Symbol('M_E', commutative=True), cos(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(F_{g},A_{2})} = A_{2} \\sin{(F_{g})}, then obtain - A_{2} \\cos{(F_{g})} + 2 \\frac{\\partial}{\\partial F_{g}} \\mathbb{I}{(F_{g},A_{2})} - 1 = \\frac{\\partial}{\\partial F_{g}} \\mathbb{I}{(F_{g},A_{2})} - 1", "derivation": "\\mathbb{I}{(F_{g},A_{2})} = A_{2} \\sin{(F_{g})} and - F_{g} + \\mathbb{I}{(F_{g},A_{2})} = A_{2} \\sin{(F_{g})} - F_{g} and - A_{2} \\sin{(F_{g})} - F_{g} + \\mathbb{I}{(F_{g},A_{2})} = - F_{g} and - A_{2} \\sin{(F_{g})} - F_{g} + 2 \\mathbb{I}{(F_{g},A_{2})} = - F_{g} + \\mathbb{I}{(F_{g},A_{2})} and \\frac{\\partial}{\\partial F_{g}} (- A_{2} \\sin{(F_{g})} - F_{g} + 2 \\mathbb{I}{(F_{g},A_{2})}) = \\frac{\\partial}{\\partial F_{g}} (- F_{g} + \\mathbb{I}{(F_{g},A_{2})}) and - A_{2} \\cos{(F_{g})} + 2 \\frac{\\partial}{\\partial F_{g}} \\mathbb{I}{(F_{g},A_{2})} - 1 = \\frac{\\partial}{\\partial F_{g}} \\mathbb{I}{(F_{g},A_{2})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True))))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True))))"], [["minus", 2, "Mul(Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True))))"], [["differentiate", 4, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), sin(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), cos(Symbol('F_g', commutative=True))), Mul(Integer(2), Derivative(Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(-1)), Add(Derivative(Function('\\\\mathbb{I}')(Symbol('F_g', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(A_{z})} = \\sin{(A_{z})}, then obtain \\frac{\\frac{d}{d A_{z}} 0}{\\sin{(A_{z})}} = \\frac{\\frac{d}{d A_{z}} (- \\frac{\\operatorname{M_{E}}{(A_{z})}}{\\sin{(A_{z})}} + 1)}{\\sin{(A_{z})}}", "derivation": "\\operatorname{M_{E}}{(A_{z})} = \\sin{(A_{z})} and \\frac{\\operatorname{M_{E}}{(A_{z})}}{\\sin{(A_{z})}} = 1 and 0 = - \\frac{\\operatorname{M_{E}}{(A_{z})}}{\\sin{(A_{z})}} + 1 and \\frac{d}{d A_{z}} 0 = \\frac{d}{d A_{z}} (- \\frac{\\operatorname{M_{E}}{(A_{z})}}{\\sin{(A_{z})}} + 1) and \\frac{\\frac{d}{d A_{z}} 0}{\\sin{(A_{z})}} = \\frac{\\frac{d}{d A_{z}} (- \\frac{\\operatorname{M_{E}}{(A_{z})}}{\\sin{(A_{z})}} + 1)}{\\sin{(A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["divide", 1, "sin(Symbol('A_z', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Function('M_E')(Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M_E')(Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Integer(1)))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('M_E')(Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["divide", 4, "sin(Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('A_z', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('A_z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Function('M_E')(Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(h)} = e^{h}, then obtain \\frac{h + \\operatorname{E_{x}}{(h)}}{\\int e^{h} dh} - \\int e^{h} dh = \\frac{h + e^{h}}{\\int e^{h} dh} - \\int e^{h} dh", "derivation": "\\operatorname{E_{x}}{(h)} = e^{h} and h + \\operatorname{E_{x}}{(h)} = h + e^{h} and \\int \\operatorname{E_{x}}{(h)} dh = \\int e^{h} dh and \\frac{h + \\operatorname{E_{x}}{(h)}}{\\int \\operatorname{E_{x}}{(h)} dh} = \\frac{h + e^{h}}{\\int \\operatorname{E_{x}}{(h)} dh} and \\frac{h + \\operatorname{E_{x}}{(h)}}{\\int \\operatorname{E_{x}}{(h)} dh} - \\int \\operatorname{E_{x}}{(h)} dh = \\frac{h + e^{h}}{\\int \\operatorname{E_{x}}{(h)} dh} - \\int \\operatorname{E_{x}}{(h)} dh and \\frac{h + \\operatorname{E_{x}}{(h)}}{\\int e^{h} dh} - \\int e^{h} dh = \\frac{h + e^{h}}{\\int e^{h} dh} - \\int e^{h} dh", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('E_x')(Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), exp(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["divide", 2, "Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Mul(Add(Symbol('h', commutative=True), Function('E_x')(Symbol('h', commutative=True))), Pow(Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Add(Symbol('h', commutative=True), exp(Symbol('h', commutative=True))), Pow(Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))))"], [["minus", 4, "Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('h', commutative=True), Function('E_x')(Symbol('h', commutative=True))), Pow(Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))), Add(Mul(Add(Symbol('h', commutative=True), exp(Symbol('h', commutative=True))), Pow(Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Function('E_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Add(Symbol('h', commutative=True), Function('E_x')(Symbol('h', commutative=True))), Pow(Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))), Add(Mul(Add(Symbol('h', commutative=True), exp(Symbol('h', commutative=True))), Pow(Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))))"]]}, {"prompt": "Given k{(\\rho,n)} = \\frac{\\rho}{n}, then derive \\frac{\\partial}{\\partial \\rho} k{(\\rho,n)} = \\frac{1}{n}, then obtain (\\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{n})^{n} = (\\frac{\\partial}{\\partial \\rho} k{(\\rho,n)})^{n}", "derivation": "k{(\\rho,n)} = \\frac{\\rho}{n} and \\frac{\\partial}{\\partial \\rho} k{(\\rho,n)} = \\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{n} and \\frac{\\partial}{\\partial \\rho} k{(\\rho,n)} = \\frac{1}{n} and (\\frac{\\partial}{\\partial \\rho} k{(\\rho,n)})^{n} = (\\frac{1}{n})^{n} and (\\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{n})^{n} = (\\frac{1}{n})^{n} and (\\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{n})^{n} = (\\frac{\\partial}{\\partial \\rho} k{(\\rho,n)})^{n}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Pow(Symbol('n', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(\\phi_2)} = e^{\\phi_2}, then obtain \\frac{\\iint \\mathbf{r}^{2}{(\\phi_2)} d\\phi_2 d\\phi_2}{\\mathbf{r}{(\\phi_2)}} = \\frac{\\iint \\mathbf{r}{(\\phi_2)} e^{\\phi_2} d\\phi_2 d\\phi_2}{\\mathbf{r}{(\\phi_2)}}", "derivation": "\\mathbf{r}{(\\phi_2)} = e^{\\phi_2} and \\mathbf{r}^{2}{(\\phi_2)} = \\mathbf{r}{(\\phi_2)} e^{\\phi_2} and \\int \\mathbf{r}^{2}{(\\phi_2)} d\\phi_2 = \\int \\mathbf{r}{(\\phi_2)} e^{\\phi_2} d\\phi_2 and \\iint \\mathbf{r}^{2}{(\\phi_2)} d\\phi_2 d\\phi_2 = \\iint \\mathbf{r}{(\\phi_2)} e^{\\phi_2} d\\phi_2 d\\phi_2 and \\frac{\\iint \\mathbf{r}^{2}{(\\phi_2)} d\\phi_2 d\\phi_2}{\\mathbf{r}{(\\phi_2)}} = \\frac{\\iint \\mathbf{r}{(\\phi_2)} e^{\\phi_2} d\\phi_2 d\\phi_2}{\\mathbf{r}{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 4, "Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\tilde{g}^*)} = \\tilde{g}^*, then obtain \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{x^{{\\}'}}^{\\tilde{g}^*}{(\\tilde{g}^*)}}{\\frac{d}{d \\tilde{g}^*} (\\tilde{g}^*)^{\\tilde{g}^*}} = 1", "derivation": "\\operatorname{x^{{\\}'}}{(\\tilde{g}^*)} = \\tilde{g}^* and \\operatorname{x^{{\\}'}}^{\\tilde{g}^*}{(\\tilde{g}^*)} = (\\tilde{g}^*)^{\\tilde{g}^*} and \\frac{d}{d \\tilde{g}^*} \\operatorname{x^{{\\}'}}^{\\tilde{g}^*}{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} (\\tilde{g}^*)^{\\tilde{g}^*} and \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{x^{{\\}'}}^{\\tilde{g}^*}{(\\tilde{g}^*)}}{\\frac{d}{d \\tilde{g}^*} (\\tilde{g}^*)^{\\tilde{g}^*}} = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], [["power", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Pow(Function('x^\\\\prime')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Integer(-1)), Derivative(Pow(Function('x^\\\\prime')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{s}{(\\psi^*)} = \\log{(\\psi^*)}, then obtain \\int (\\mathbf{s}{(\\psi^*)} + 2) d\\psi^* = \\int (\\log{(\\psi^*)} + 2) d\\psi^*", "derivation": "\\mathbf{s}{(\\psi^*)} = \\log{(\\psi^*)} and \\mathbf{s}{(\\psi^*)} + 1 = \\log{(\\psi^*)} + 1 and \\mathbf{s}{(\\psi^*)} + 2 = \\log{(\\psi^*)} + 2 and \\int (\\mathbf{s}{(\\psi^*)} + 2) d\\psi^* = \\int (\\log{(\\psi^*)} + 2) d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True)), Integer(1)), Add(log(Symbol('\\\\psi^*', commutative=True)), Integer(1)))"], [["add", 2, 1], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Add(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(n_{2})} = \\sin{(n_{2})} and f{(n_{2})} = - \\sin{(n_{2})}, then obtain \\frac{\\operatorname{n_{1}}{(n_{2})} - \\frac{\\operatorname{n_{1}}{(n_{2})}}{\\sin{(n_{2})}}}{\\int - \\sin{(n_{2})} dn_{2}} = \\frac{\\operatorname{n_{1}}{(n_{2})} - 1}{\\int - \\sin{(n_{2})} dn_{2}}", "derivation": "\\operatorname{n_{1}}{(n_{2})} = \\sin{(n_{2})} and f{(n_{2})} = - \\sin{(n_{2})} and \\frac{f{(n_{2})}}{\\sin{(n_{2})}} = -1 and f{(n_{2})} = - \\operatorname{n_{1}}{(n_{2})} and - \\frac{\\operatorname{n_{1}}{(n_{2})}}{\\sin{(n_{2})}} = -1 and \\operatorname{n_{1}}{(n_{2})} - \\frac{\\operatorname{n_{1}}{(n_{2})}}{\\sin{(n_{2})}} = \\operatorname{n_{1}}{(n_{2})} - 1 and \\frac{\\operatorname{n_{1}}{(n_{2})} - \\frac{\\operatorname{n_{1}}{(n_{2})}}{\\sin{(n_{2})}}}{\\int - \\sin{(n_{2})} dn_{2}} = \\frac{\\operatorname{n_{1}}{(n_{2})} - 1}{\\int - \\sin{(n_{2})} dn_{2}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))))"], [["divide", 2, "sin(Symbol('n_2', commutative=True))"], "Equality(Mul(Function('f')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1))), Integer(-1))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Function('n_1')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1))), Integer(-1))"], [["minus", 5, "Mul(Integer(-1), Function('n_1')(Symbol('n_2', commutative=True)))"], "Equality(Add(Function('n_1')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)))), Add(Function('n_1')(Symbol('n_2', commutative=True)), Integer(-1)))"], [["divide", 6, "Integral(Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Add(Function('n_1')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)))), Pow(Integral(Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Mul(Add(Function('n_1')(Symbol('n_2', commutative=True)), Integer(-1)), Pow(Integral(Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(n_{1},y^{\\prime})} = \\int \\frac{y^{\\prime}}{n_{1}} dy^{\\prime} and \\varphi^{*}{(n_{1})} = n_{1}, then obtain - \\ddot{x}{(n_{1},y^{\\prime})} + \\varphi^{*}{(n_{1})} = n_{1} - \\ddot{x}{(n_{1},y^{\\prime})}", "derivation": "\\ddot{x}{(n_{1},y^{\\prime})} = \\int \\frac{y^{\\prime}}{n_{1}} dy^{\\prime} and \\varphi^{*}{(n_{1})} = n_{1} and \\varphi^{*}{(n_{1})} - \\int \\frac{y^{\\prime}}{n_{1}} dy^{\\prime} = n_{1} - \\int \\frac{y^{\\prime}}{n_{1}} dy^{\\prime} and - \\ddot{x}{(n_{1},y^{\\prime})} + \\varphi^{*}{(n_{1})} = n_{1} - \\ddot{x}{(n_{1},y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('n_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], [["minus", 2, "Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('n_1', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Add(Symbol('n_1', commutative=True), Mul(Integer(-1), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('n_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\varphi^*')(Symbol('n_1', commutative=True))), Add(Symbol('n_1', commutative=True), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('n_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given H{(Q)} = \\log{(Q)}, then obtain \\int (Q \\log{(Q)} - \\frac{H{(Q)} \\log{(Q)}}{Q^{2}}) dQ = \\int (Q \\log{(Q)} - \\frac{\\log{(Q)}^{2}}{Q^{2}}) dQ", "derivation": "H{(Q)} = \\log{(Q)} and H{(Q)} \\log{(Q)} = \\log{(Q)}^{2} and - \\frac{H{(Q)} \\log{(Q)}}{Q} = - \\frac{\\log{(Q)}^{2}}{Q} and - \\frac{H{(Q)} \\log{(Q)}}{Q^{2}} = - \\frac{\\log{(Q)}^{2}}{Q^{2}} and Q \\log{(Q)} - \\frac{H{(Q)} \\log{(Q)}}{Q^{2}} = Q \\log{(Q)} - \\frac{\\log{(Q)}^{2}}{Q^{2}} and \\int (Q \\log{(Q)} - \\frac{H{(Q)} \\log{(Q)}}{Q^{2}}) dQ = \\int (Q \\log{(Q)} - \\frac{\\log{(Q)}^{2}}{Q^{2}}) dQ", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["times", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Pow(log(Symbol('Q', commutative=True)), Integer(2)))"], [["divide", 2, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(log(Symbol('Q', commutative=True)), Integer(2))))"], [["times", 3, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(log(Symbol('Q', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(log(Symbol('Q', commutative=True)), Integer(2)))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Function('H')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(log(Symbol('Q', commutative=True)), Integer(2)))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given f{(\\rho_b,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\rho_b})}, then obtain f{(\\rho_b,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} f{(\\rho_b,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} \\frac{\\partial}{\\partial \\mathbf{f}} f{(\\rho_b,\\mathbf{f})}", "derivation": "f{(\\rho_b,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} and \\frac{\\partial}{\\partial \\mathbf{f}} f{(\\rho_b,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} and f{(\\rho_b,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} = \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} and f{(\\rho_b,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} f{(\\rho_b,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\rho_b})} \\frac{\\partial}{\\partial \\mathbf{f}} f{(\\rho_b,\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))"], "Equality(Mul(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Mul(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Derivative(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Mul(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)))), Derivative(Function('f')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(\\Psi_{nl},A)} = A \\Psi_{nl}, then obtain \\frac{A \\Psi_{nl} \\int A \\Psi_{nl} dA + A \\Psi_{nl} \\int \\phi_{1}{(\\Psi_{nl},A)} dA}{A \\Psi_{nl}} = 2 \\int A \\Psi_{nl} dA", "derivation": "\\phi_{1}{(\\Psi_{nl},A)} = A \\Psi_{nl} and \\int \\phi_{1}{(\\Psi_{nl},A)} dA = \\int A \\Psi_{nl} dA and A \\Psi_{nl} \\int \\phi_{1}{(\\Psi_{nl},A)} dA = A \\Psi_{nl} \\int A \\Psi_{nl} dA and A \\Psi_{nl} \\int A \\Psi_{nl} dA + A \\Psi_{nl} \\int \\phi_{1}{(\\Psi_{nl},A)} dA = 2 A \\Psi_{nl} \\int A \\Psi_{nl} dA and \\frac{A \\Psi_{nl} \\int A \\Psi_{nl} dA + A \\Psi_{nl} \\int \\phi_{1}{(\\Psi_{nl},A)} dA}{A \\Psi_{nl}} = 2 \\int A \\Psi_{nl} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 2, "Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Function('\\\\phi_1')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["add", 3, "Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True))))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Function('\\\\phi_1')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["divide", 4, "Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Integral(Function('\\\\phi_1')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))), Mul(Integer(2), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(G,f)} = \\int G^{f} df, then obtain 1 + G^{- f} (G + \\hat{p}{(G,f)}) \\hat{p}{(G,f)} = \\frac{G + \\int G^{f} df}{G + \\hat{p}{(G,f)}} + G^{- f} (G + \\hat{p}{(G,f)}) \\hat{p}{(G,f)}", "derivation": "\\hat{p}{(G,f)} = \\int G^{f} df and G + \\hat{p}{(G,f)} = G + \\int G^{f} df and 1 = \\frac{G + \\int G^{f} df}{G + \\hat{p}{(G,f)}} and 1 + G^{- f} (G + \\hat{p}{(G,f)}) \\hat{p}{(G,f)} = \\frac{G + \\int G^{f} df}{G + \\hat{p}{(G,f)}} + G^{- f} (G + \\hat{p}{(G,f)}) \\hat{p}{(G,f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True)), Integral(Pow(Symbol('G', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Add(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["divide", 2, "Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Add(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"], [["add", 3, "Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Integer(1), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Pow(Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Add(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Add(Symbol('G', commutative=True), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True))), Function('\\\\hat{p}')(Symbol('G', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} = \\mathbf{J}_M + \\mathbf{v} and \\varepsilon_{0}{(\\mathbf{J}_M,\\mathbf{v})} = \\mathbf{J}_M + \\mathbf{v}, then obtain (\\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} - 1)^{\\mathbf{J}_M} = (\\mathbf{J}_M + \\mathbf{v} - 1)^{\\mathbf{J}_M}", "derivation": "\\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} = \\mathbf{J}_M + \\mathbf{v} and \\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} - 1 = \\mathbf{J}_M + \\mathbf{v} - 1 and \\varepsilon_{0}{(\\mathbf{J}_M,\\mathbf{v})} = \\mathbf{J}_M + \\mathbf{v} and \\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} - 1 = \\varepsilon_{0}{(\\mathbf{J}_M,\\mathbf{v})} - 1 and (\\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} - 1)^{\\mathbf{J}_M} = (\\varepsilon_{0}{(\\mathbf{J}_M,\\mathbf{v})} - 1)^{\\mathbf{J}_M} and (\\varepsilon{(\\mathbf{J}_M,\\mathbf{v})} - 1)^{\\mathbf{J}_M} = (\\mathbf{J}_M + \\mathbf{v} - 1)^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)))"], [["power", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(f^{*})} = e^{\\cos{(f^{*})}} and \\theta_{1}{(f^{*})} = \\cos{(f^{*})}, then obtain \\int (e^{\\theta_{1}{(f^{*})}} - e^{\\cos{(f^{*})}}) df^{*} = \\int 0 df^{*}", "derivation": "\\mathbf{f}{(f^{*})} = e^{\\cos{(f^{*})}} and \\mathbf{f}{(f^{*})} - e^{\\cos{(f^{*})}} = 0 and \\theta_{1}{(f^{*})} = \\cos{(f^{*})} and \\mathbf{f}{(f^{*})} = e^{\\theta_{1}{(f^{*})}} and e^{\\theta_{1}{(f^{*})}} - e^{\\cos{(f^{*})}} = 0 and \\int (e^{\\theta_{1}{(f^{*})}} - e^{\\cos{(f^{*})}}) df^{*} = \\int 0 df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), exp(cos(Symbol('f^*', commutative=True))))"], [["minus", 1, "exp(cos(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('f^*', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), exp(Function('\\\\theta_1')(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(exp(Function('\\\\theta_1')(Symbol('f^*', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('f^*', commutative=True))))), Integer(0))"], [["integrate", 5, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(exp(Function('\\\\theta_1')(Symbol('f^*', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('f^*', commutative=True))))), Tuple(Symbol('f^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\varphi^*,\\omega)} = \\omega^{\\varphi^*} and \\operatorname{t_{1}}{(\\varphi^*,\\omega)} = \\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)}, then obtain (\\int ((\\omega^{\\varphi^*})^{\\omega})^{\\omega} d\\omega)^{\\varphi^*} = (\\int (\\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)})^{\\omega} d\\omega)^{\\varphi^*}", "derivation": "\\operatorname{r_{0}}{(\\varphi^*,\\omega)} = \\omega^{\\varphi^*} and \\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)} = (\\omega^{\\varphi^*})^{\\omega} and \\operatorname{t_{1}}{(\\varphi^*,\\omega)} = \\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)} and \\operatorname{t_{1}}{(\\varphi^*,\\omega)} = (\\omega^{\\varphi^*})^{\\omega} and \\operatorname{t_{1}}^{\\omega}{(\\varphi^*,\\omega)} = (\\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)})^{\\omega} and ((\\omega^{\\varphi^*})^{\\omega})^{\\omega} = (\\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)})^{\\omega} and \\int ((\\omega^{\\varphi^*})^{\\omega})^{\\omega} d\\omega = \\int (\\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)})^{\\omega} d\\omega and (\\int ((\\omega^{\\varphi^*})^{\\omega})^{\\omega} d\\omega)^{\\varphi^*} = (\\int (\\operatorname{r_{0}}^{\\omega}{(\\varphi^*,\\omega)})^{\\omega} d\\omega)^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('t_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["integrate", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 7, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Integral(Pow(Pow(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Integral(Pow(Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(G,\\theta_2)} = - \\theta_2 + \\log{(G)} and \\eta{(G,\\theta_2)} = (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\operatorname{x^{{\\}'}}{(G,\\theta_2)})})^{2}, then obtain (\\frac{\\partial}{\\partial G} \\eta{(G,\\theta_2)})^{G} = (\\frac{\\partial}{\\partial G} (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\theta_2 - \\log{(G)})})^{2})^{G}", "derivation": "\\operatorname{x^{{\\}'}}{(G,\\theta_2)} = - \\theta_2 + \\log{(G)} and \\cos{(\\operatorname{x^{{\\}'}}{(G,\\theta_2)})} = \\cos{(\\theta_2 - \\log{(G)})} and \\eta{(G,\\theta_2)} = (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\operatorname{x^{{\\}'}}{(G,\\theta_2)})})^{2} and \\eta{(G,\\theta_2)} = (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\theta_2 - \\log{(G)})})^{2} and \\frac{\\partial}{\\partial G} \\eta{(G,\\theta_2)} = \\frac{\\partial}{\\partial G} (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\theta_2 - \\log{(G)})})^{2} and (\\frac{\\partial}{\\partial G} \\eta{(G,\\theta_2)})^{G} = (\\frac{\\partial}{\\partial G} (- \\theta_2 + \\log{(G)} + 2 \\cos{(\\theta_2 - \\log{(G)})})^{2})^{G}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(Symbol('G', commutative=True))))"], [["cos", 1], "Equality(cos(Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True))), cos(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), log(Symbol('G', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(Symbol('G', commutative=True)), Mul(Integer(2), cos(Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True))))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(Symbol('G', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), log(Symbol('G', commutative=True))))))), Integer(2)))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(Symbol('G', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), log(Symbol('G', commutative=True))))))), Integer(2)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["power", 5, "Symbol('G', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(Symbol('G', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), log(Symbol('G', commutative=True))))))), Integer(2)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\Omega{(a,\\hat{H}_l,E_{\\lambda})} = (\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}, then obtain \\hat{H}_l + \\Omega{(a,\\hat{H}_l,E_{\\lambda})} = \\hat{H}_l (\\frac{(\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}}{\\Omega{(a,\\hat{H}_l,E_{\\lambda})}})^{a} + \\Omega{(a,\\hat{H}_l,E_{\\lambda})}", "derivation": "\\Omega{(a,\\hat{H}_l,E_{\\lambda})} = (\\frac{E_{\\lambda}}{\\hat{H}_l})^{a} and 1 = \\frac{(\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}}{\\Omega{(a,\\hat{H}_l,E_{\\lambda})}} and 1 = (\\frac{(\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}}{\\Omega{(a,\\hat{H}_l,E_{\\lambda})}})^{a} and \\hat{H}_l = \\hat{H}_l (\\frac{(\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}}{\\Omega{(a,\\hat{H}_l,E_{\\lambda})}})^{a} and \\hat{H}_l + \\Omega{(a,\\hat{H}_l,E_{\\lambda})} = \\hat{H}_l (\\frac{(\\frac{E_{\\lambda}}{\\hat{H}_l})^{a}}{\\Omega{(a,\\hat{H}_l,E_{\\lambda})}})^{a} + \\Omega{(a,\\hat{H}_l,E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('a', commutative=True)))"], [["divide", 1, "Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('a', commutative=True)), Pow(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('a', commutative=True)), Pow(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Symbol('a', commutative=True)))"], [["times", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Mul(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('a', commutative=True)), Pow(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Symbol('a', commutative=True))))"], [["add", 4, "Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Mul(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Symbol('a', commutative=True)), Pow(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Symbol('a', commutative=True))), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\pi{(V_{\\mathbf{B}},\\mu)} = V_{\\mathbf{B}} - \\mu, then obtain \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}^{2}} (- V_{\\mathbf{B}} + \\mu + \\pi{(V_{\\mathbf{B}},\\mu)}) = \\frac{d^{2}}{d V_{\\mathbf{B}}^{2}} 0", "derivation": "\\pi{(V_{\\mathbf{B}},\\mu)} = V_{\\mathbf{B}} - \\mu and - V_{\\mathbf{B}} + \\mu + \\pi{(V_{\\mathbf{B}},\\mu)} = 0 and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (- V_{\\mathbf{B}} + \\mu + \\pi{(V_{\\mathbf{B}},\\mu)}) = \\frac{d}{d V_{\\mathbf{B}}} 0 and \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}^{2}} (- V_{\\mathbf{B}} + \\mu + \\pi{(V_{\\mathbf{B}},\\mu)}) = \\frac{d^{2}}{d V_{\\mathbf{B}}^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mu', commutative=True), Function('\\\\pi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mu', commutative=True), Function('\\\\pi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mu', commutative=True), Function('\\\\pi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\delta,U)} = \\log{(\\frac{U}{\\delta})} and \\mathbf{A}{(\\delta,U)} = \\frac{U}{\\delta}, then obtain \\cos{(\\varepsilon_{0}^{\\delta}{(\\delta,U)})} = \\cos{(\\log{(\\mathbf{A}{(\\delta,U)})}^{\\delta})}", "derivation": "\\varepsilon_{0}{(\\delta,U)} = \\log{(\\frac{U}{\\delta})} and \\mathbf{A}{(\\delta,U)} = \\frac{U}{\\delta} and \\varepsilon_{0}^{\\delta}{(\\delta,U)} = \\log{(\\frac{U}{\\delta})}^{\\delta} and \\cos{(\\varepsilon_{0}^{\\delta}{(\\delta,U)})} = \\cos{(\\log{(\\frac{U}{\\delta})}^{\\delta})} and \\varepsilon_{0}^{\\delta}{(\\delta,U)} = \\log{(\\mathbf{A}{(\\delta,U)})}^{\\delta} and \\log{(\\mathbf{A}{(\\delta,U)})}^{\\delta} = \\log{(\\frac{U}{\\delta})}^{\\delta} and \\cos{(\\varepsilon_{0}^{\\delta}{(\\delta,U)})} = \\cos{(\\log{(\\mathbf{A}{(\\delta,U)})}^{\\delta})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), log(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\delta', commutative=True))), cos(Pow(log(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Function('\\\\mathbf{A}')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(log(Function('\\\\mathbf{A}')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(log(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(cos(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\delta', commutative=True))), cos(Pow(log(Function('\\\\mathbf{A}')(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(F_{x},C_{2})} = - C_{2} + F_{x}, then obtain C_{2} - F_{x} + \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(F_{x},C_{2})} - 1 = C_{2} - F_{x} - 2", "derivation": "\\dot{y}{(F_{x},C_{2})} = - C_{2} + F_{x} and \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(F_{x},C_{2})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + F_{x}) and C_{2} - F_{x} + \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(F_{x},C_{2})} = C_{2} - F_{x} + \\frac{\\partial}{\\partial C_{2}} (- C_{2} + F_{x}) and C_{2} - F_{x} + \\frac{\\partial}{\\partial C_{2}} (- C_{2} + F_{x}) + \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(F_{x},C_{2})} = C_{2} - F_{x} + 2 \\frac{\\partial}{\\partial C_{2}} (- C_{2} + F_{x}) and C_{2} - F_{x} + \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(F_{x},C_{2})} - 1 = C_{2} - F_{x} - 2", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('F_x', commutative=True), Symbol('C_2', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('F_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('F_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))"], "Equality(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Function('\\\\dot{y}')(Symbol('F_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('F_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True)), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{H},L)} = e^{L^{\\mathbf{H}}}, then obtain \\frac{\\partial}{\\partial L} L \\operatorname{v_{2}}^{\\mathbf{H}}{(\\mathbf{H},L)} e^{- L^{\\mathbf{H}}} = \\frac{\\partial}{\\partial L} L e^{- L^{\\mathbf{H}}} (e^{L^{\\mathbf{H}}})^{\\mathbf{H}}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{H},L)} = e^{L^{\\mathbf{H}}} and \\operatorname{v_{2}}^{\\mathbf{H}}{(\\mathbf{H},L)} = (e^{L^{\\mathbf{H}}})^{\\mathbf{H}} and \\operatorname{v_{2}}^{\\mathbf{H}}{(\\mathbf{H},L)} e^{- L^{\\mathbf{H}}} = e^{- L^{\\mathbf{H}}} (e^{L^{\\mathbf{H}}})^{\\mathbf{H}} and L \\operatorname{v_{2}}^{\\mathbf{H}}{(\\mathbf{H},L)} e^{- L^{\\mathbf{H}}} = L e^{- L^{\\mathbf{H}}} (e^{L^{\\mathbf{H}}})^{\\mathbf{H}} and \\frac{\\partial}{\\partial L} L \\operatorname{v_{2}}^{\\mathbf{H}}{(\\mathbf{H},L)} e^{- L^{\\mathbf{H}}} = \\frac{\\partial}{\\partial L} L e^{- L^{\\mathbf{H}}} (e^{L^{\\mathbf{H}}})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 2, "exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Pow(Function('v_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))), Mul(exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Pow(Function('v_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))), Mul(Symbol('L', commutative=True), exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Symbol('L', commutative=True), Pow(Function('v_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), exp(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Pow(exp(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\dot{x},C_{1})} = C_{1} - \\dot{x}, then obtain 3 \\varepsilon_{0}{(\\dot{x},C_{1})} = 2 C_{1} - 2 \\dot{x} + \\varepsilon_{0}{(\\dot{x},C_{1})}", "derivation": "\\varepsilon_{0}{(\\dot{x},C_{1})} = C_{1} - \\dot{x} and C_{1} - \\dot{x} + \\varepsilon_{0}{(\\dot{x},C_{1})} = 2 C_{1} - 2 \\dot{x} and 2 \\varepsilon_{0}{(\\dot{x},C_{1})} = 2 C_{1} - 2 \\dot{x} and 3 \\varepsilon_{0}{(\\dot{x},C_{1})} = 2 C_{1} - 2 \\dot{x} + \\varepsilon_{0}{(\\dot{x},C_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))))"], [["add", 1, "Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True))))"], [["add", 3, "Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\dot{x}', commutative=True), Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(h)} = \\sin{(\\log{(h)})}, then obtain e^{(\\int \\operatorname{A_{1}}{(h)} dh)^{h}} = e^{(\\int \\sin{(\\log{(h)})} dh)^{h}}", "derivation": "\\operatorname{A_{1}}{(h)} = \\sin{(\\log{(h)})} and \\int \\operatorname{A_{1}}{(h)} dh = \\int \\sin{(\\log{(h)})} dh and (\\int \\operatorname{A_{1}}{(h)} dh)^{h} = (\\int \\sin{(\\log{(h)})} dh)^{h} and e^{(\\int \\operatorname{A_{1}}{(h)} dh)^{h}} = e^{(\\int \\sin{(\\log{(h)})} dh)^{h}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('h', commutative=True)), sin(log(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(sin(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(sin(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Integral(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), exp(Pow(Integral(sin(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(S)} = \\log{(S)}, then obtain 4 \\log{(S)}^{6} = 2 (\\hat{X}{(S)} + \\log{(S)}) \\log{(S)}^{5}", "derivation": "\\hat{X}{(S)} = \\log{(S)} and \\hat{X}{(S)} + \\log{(S)} = 2 \\log{(S)} and (\\hat{X}{(S)} + \\log{(S)})^{2} = 4 \\log{(S)}^{2} and (\\hat{X}{(S)} + \\log{(S)}) \\log{(S)}^{2} = 2 \\log{(S)}^{3} and (\\hat{X}{(S)} + \\log{(S)})^{2} \\log{(S)}^{4} = 2 (\\hat{X}{(S)} + \\log{(S)}) \\log{(S)}^{5} and 4 \\log{(S)}^{6} = 2 (\\hat{X}{(S)} + \\log{(S)}) \\log{(S)}^{5}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["add", 1, "log(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Mul(Integer(2), log(Symbol('S', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Integer(2)), Mul(Integer(4), Pow(log(Symbol('S', commutative=True)), Integer(2))))"], [["times", 2, "Pow(log(Symbol('S', commutative=True)), Integer(2))"], "Equality(Mul(Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Pow(log(Symbol('S', commutative=True)), Integer(2))), Mul(Integer(2), Pow(log(Symbol('S', commutative=True)), Integer(3))))"], [["times", 4, "Mul(Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Pow(log(Symbol('S', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Integer(2)), Pow(log(Symbol('S', commutative=True)), Integer(4))), Mul(Integer(2), Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Pow(log(Symbol('S', commutative=True)), Integer(5))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(4), Pow(log(Symbol('S', commutative=True)), Integer(6))), Mul(Integer(2), Add(Function('\\\\hat{X}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Pow(log(Symbol('S', commutative=True)), Integer(5))))"]]}, {"prompt": "Given \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = E_{\\lambda} \\dot{z} + y^{\\prime}, then obtain \\frac{\\partial^{3}}{\\partial \\dot{z}^{2}\\partial E_{\\lambda}} \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = 0", "derivation": "\\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = E_{\\lambda} \\dot{z} + y^{\\prime} and \\frac{\\partial}{\\partial \\dot{z}} \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial \\dot{z}} (E_{\\lambda} \\dot{z} + y^{\\prime}) and \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\dot{z}} \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\dot{z}} (E_{\\lambda} \\dot{z} + y^{\\prime}) and \\frac{\\partial^{3}}{\\partial \\dot{z}\\partial E_{\\lambda}\\partial \\dot{z}} \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = \\frac{\\partial^{3}}{\\partial \\dot{z}\\partial E_{\\lambda}\\partial \\dot{z}} (E_{\\lambda} \\dot{z} + y^{\\prime}) and \\frac{\\partial^{3}}{\\partial \\dot{z}^{2}\\partial E_{\\lambda}} \\rho{(\\dot{z},y^{\\prime},E_{\\lambda})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\dot{z}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\delta{(a^{\\dagger},E)} = E - a^{\\dagger}, then derive \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE = - \\frac{E^{2} a^{\\dagger}}{2} + E (a^{\\dagger})^{2} + \\mathbf{D}, then obtain A_{y} + \\int - a^{\\dagger} (E - a^{\\dagger}) dE = A_{y} + \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE", "derivation": "\\delta{(a^{\\dagger},E)} = E - a^{\\dagger} and - a^{\\dagger} \\delta{(a^{\\dagger},E)} = - a^{\\dagger} (E - a^{\\dagger}) and \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE = \\int - a^{\\dagger} (E - a^{\\dagger}) dE and \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE = - \\frac{E^{2} a^{\\dagger}}{2} + E (a^{\\dagger})^{2} + \\mathbf{D} and A_{y} + \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE = A_{y} - \\frac{E^{2} a^{\\dagger}}{2} + E (a^{\\dagger})^{2} + \\mathbf{D} and A_{y} + \\int - a^{\\dagger} (E - a^{\\dagger}) dE = A_{y} - \\frac{E^{2} a^{\\dagger}}{2} + E (a^{\\dagger})^{2} + \\mathbf{D} and A_{y} + \\int - a^{\\dagger} (E - a^{\\dagger}) dE = A_{y} + \\int - a^{\\dagger} \\delta{(a^{\\dagger},E)} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 4, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('A_y', commutative=True), Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('A_y', commutative=True), Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('A_y', commutative=True), Integral(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\delta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given l{(f)} = \\cos{(f)}, then derive \\int l{(f)} df = J + \\sin{(f)}, then derive \\mathbf{r} + \\sin{(f)} = J + \\sin{(f)}, then obtain \\rho_f + \\sin{(f)} = \\mathbf{r} + \\sin{(f)}", "derivation": "l{(f)} = \\cos{(f)} and \\int l{(f)} df = \\int \\cos{(f)} df and \\int l{(f)} df = J + \\sin{(f)} and \\int \\cos{(f)} df = J + \\sin{(f)} and \\mathbf{r} + \\sin{(f)} = J + \\sin{(f)} and \\int \\cos{(f)} df = \\mathbf{r} + \\sin{(f)} and \\rho_f + \\sin{(f)} = \\mathbf{r} + \\sin{(f)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('f', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('f', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('f', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\rho_f', commutative=True), sin(Symbol('f', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{J})} = \\mathbf{J}, then obtain \\frac{\\frac{d}{d \\mathbf{J}} \\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}} - \\frac{\\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}^{2}} = 0", "derivation": "\\mathbb{I}{(\\mathbf{J})} = \\mathbf{J} and \\frac{\\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}} = 1 and \\frac{d}{d \\mathbf{J}} \\frac{\\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{d}{d \\mathbf{J}} 1 and \\frac{\\frac{d}{d \\mathbf{J}} \\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}} - \\frac{\\mathbb{I}{(\\mathbf{J})}}{\\mathbf{J}^{2}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], [["divide", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial a} a^{\\hat{H}_{\\lambda}}, then derive \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})} = \\frac{\\hat{H}_{\\lambda} a^{\\hat{H}_{\\lambda}}}{a}, then obtain \\frac{\\hat{H}_{\\lambda} a^{\\hat{H}_{\\lambda}}}{a} - a = - a + \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})}", "derivation": "\\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial a} a^{\\hat{H}_{\\lambda}} and - a + \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})} = - a + \\frac{\\partial}{\\partial a} a^{\\hat{H}_{\\lambda}} and \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})} = \\frac{\\hat{H}_{\\lambda} a^{\\hat{H}_{\\lambda}}}{a} and \\frac{\\hat{H}_{\\lambda} a^{\\hat{H}_{\\lambda}}}{a} - a = - a + \\frac{\\partial}{\\partial a} a^{\\hat{H}_{\\lambda}} and \\frac{\\hat{H}_{\\lambda} a^{\\hat{H}_{\\lambda}}}{a} - a = - a + \\operatorname{A_{z}}{(a,\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('A_z')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('A_z')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('A_z')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\hat{x})} = e^{\\hat{x}}, then derive - e^{\\hat{x}} + \\frac{d}{d \\hat{x}} \\Psi_{\\lambda}{(\\hat{x})} = 0, then obtain - \\Psi_{\\lambda}{(\\hat{x})} + \\frac{d}{d \\hat{x}} \\Psi_{\\lambda}{(\\hat{x})} = 0", "derivation": "\\Psi_{\\lambda}{(\\hat{x})} = e^{\\hat{x}} and \\Psi_{\\lambda}{(\\hat{x})} - e^{\\hat{x}} = 0 and \\frac{d}{d \\hat{x}} (\\Psi_{\\lambda}{(\\hat{x})} - e^{\\hat{x}}) = \\frac{d}{d \\hat{x}} 0 and - e^{\\hat{x}} + \\frac{d}{d \\hat{x}} \\Psi_{\\lambda}{(\\hat{x})} = 0 and - e^{\\hat{x}} + \\frac{d}{d \\hat{x}} e^{\\hat{x}} = 0 and - \\Psi_{\\lambda}{(\\hat{x})} + \\frac{d}{d \\hat{x}} \\Psi_{\\lambda}{(\\hat{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))), Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{f}{(t_{1})} = \\log{(t_{1})}, then obtain \\int 2 t_{1} \\log{(\\mathbf{f}^{t_{1}}{(t_{1})})} dt_{1} = \\int 2 t_{1} \\log{(\\log{(t_{1})}^{t_{1}})} dt_{1}", "derivation": "\\mathbf{f}{(t_{1})} = \\log{(t_{1})} and \\mathbf{f}^{t_{1}}{(t_{1})} = \\log{(t_{1})}^{t_{1}} and \\log{(\\mathbf{f}^{t_{1}}{(t_{1})})} = \\log{(\\log{(t_{1})}^{t_{1}})} and 2 t_{1} \\log{(\\mathbf{f}^{t_{1}}{(t_{1})})} = 2 t_{1} \\log{(\\log{(t_{1})}^{t_{1}})} and \\int 2 t_{1} \\log{(\\mathbf{f}^{t_{1}}{(t_{1})})} dt_{1} = \\int 2 t_{1} \\log{(\\log{(t_{1})}^{t_{1}})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(log(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))), log(Pow(log(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))))"], [["times", 3, "Mul(Integer(2), Symbol('t_1', commutative=True))"], "Equality(Mul(Integer(2), Symbol('t_1', commutative=True), log(Pow(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))), Mul(Integer(2), Symbol('t_1', commutative=True), log(Pow(log(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))))"], [["integrate", 4, "Symbol('t_1', commutative=True)"], "Equality(Integral(Mul(Integer(2), Symbol('t_1', commutative=True), log(Pow(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Integer(2), Symbol('t_1', commutative=True), log(Pow(log(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\hat{x},E)} = \\frac{\\log{(E)}}{\\hat{x}}, then obtain \\int 0 d\\hat{x} = \\int (\\log{(1 - \\frac{\\log{(E)}}{\\hat{x}^{2}})} - \\log{(1 - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}})}) d\\hat{x}", "derivation": "\\varepsilon{(\\hat{x},E)} = \\frac{\\log{(E)}}{\\hat{x}} and - \\varepsilon{(\\hat{x},E)} = - \\frac{\\log{(E)}}{\\hat{x}} and - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}} = - \\frac{\\log{(E)}}{\\hat{x}^{2}} and 1 - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}} = 1 - \\frac{\\log{(E)}}{\\hat{x}^{2}} and \\log{(1 - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}})} = \\log{(1 - \\frac{\\log{(E)}}{\\hat{x}^{2}})} and 0 = \\log{(1 - \\frac{\\log{(E)}}{\\hat{x}^{2}})} - \\log{(1 - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}})} and \\int 0 d\\hat{x} = \\int (\\log{(1 - \\frac{\\log{(E)}}{\\hat{x}^{2}})} - \\log{(1 - \\frac{\\varepsilon{(\\hat{x},E)}}{\\hat{x}})}) d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E', commutative=True))))"], [["times", 2, "Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), log(Symbol('E', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), log(Symbol('E', commutative=True)))))"], [["log", 4], "Equality(log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True))))), log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), log(Symbol('E', commutative=True))))))"], [["minus", 5, "log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True)))))"], "Equality(Integer(0), Add(log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), log(Symbol('E', commutative=True))))), Mul(Integer(-1), log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True))))))))"], [["integrate", 6, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-2)), log(Symbol('E', commutative=True))))), Mul(Integer(-1), log(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('E', commutative=True))))))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\eta{(t_{1},\\Omega)} = \\Omega + t_{1}, then obtain \\frac{2 \\eta{(t_{1},\\Omega)}}{\\int \\eta{(t_{1},\\Omega)} dt_{1}} = \\frac{2 (\\Omega + t_{1})}{\\int \\eta{(t_{1},\\Omega)} dt_{1}}", "derivation": "\\eta{(t_{1},\\Omega)} = \\Omega + t_{1} and \\int \\eta{(t_{1},\\Omega)} dt_{1} = \\int (\\Omega + t_{1}) dt_{1} and \\frac{\\eta{(t_{1},\\Omega)}}{\\int (\\Omega + t_{1}) dt_{1}} = \\frac{\\Omega + t_{1}}{\\int (\\Omega + t_{1}) dt_{1}} and \\frac{\\eta{(t_{1},\\Omega)}}{\\int \\eta{(t_{1},\\Omega)} dt_{1}} = \\frac{\\Omega + t_{1}}{\\int \\eta{(t_{1},\\Omega)} dt_{1}} and \\frac{2 \\eta{(t_{1},\\Omega)}}{\\int \\eta{(t_{1},\\Omega)} dt_{1}} = \\frac{2 (\\Omega + t_{1})}{\\int \\eta{(t_{1},\\Omega)} dt_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 1, "Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Pow(Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Integral(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Pow(Integral(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))))"], [["divide", 4, "Rational(1, 2)"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Integral(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Mul(Integer(2), Add(Symbol('\\\\Omega', commutative=True), Symbol('t_1', commutative=True)), Pow(Integral(Function('\\\\eta')(Symbol('t_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(A_{y},\\mathbf{g})} = A_{y} + \\mathbf{g}, then obtain (A_{y} + \\mathbf{g}) \\operatorname{E_{x}}^{3}{(A_{y},\\mathbf{g})} = (A_{y} + \\mathbf{g})^{3} \\operatorname{E_{x}}{(A_{y},\\mathbf{g})}", "derivation": "\\operatorname{E_{x}}{(A_{y},\\mathbf{g})} = A_{y} + \\mathbf{g} and \\operatorname{E_{x}}^{2}{(A_{y},\\mathbf{g})} = (A_{y} + \\mathbf{g}) \\operatorname{E_{x}}{(A_{y},\\mathbf{g})} and (A_{y} + \\mathbf{g}) \\operatorname{E_{x}}^{3}{(A_{y},\\mathbf{g})} = (A_{y} + \\mathbf{g})^{2} \\operatorname{E_{x}}^{2}{(A_{y},\\mathbf{g})} and (A_{y} + \\mathbf{g}) \\operatorname{E_{x}}^{3}{(A_{y},\\mathbf{g})} = (A_{y} + \\mathbf{g})^{3} \\operatorname{E_{x}}{(A_{y},\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Mul(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 2, "Mul(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(3))), Mul(Pow(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Pow(Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(3))), Mul(Pow(Add(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(3)), Function('E_x')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})} and \\operatorname{n_{1}}{(\\varepsilon_0)} = \\log{(\\frac{\\mathbf{J}_P{(\\varepsilon_0)}}{\\cos{(\\sin{(\\varepsilon_0)})}})}, then obtain \\operatorname{n_{1}}^{\\varepsilon_0}{(\\varepsilon_0)} = 0^{\\varepsilon_0}", "derivation": "\\mathbf{J}_P{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})} and \\frac{\\mathbf{J}_P{(\\varepsilon_0)}}{\\cos{(\\sin{(\\varepsilon_0)})}} = 1 and \\log{(\\frac{\\mathbf{J}_P{(\\varepsilon_0)}}{\\cos{(\\sin{(\\varepsilon_0)})}})} = 0 and \\operatorname{n_{1}}{(\\varepsilon_0)} = \\log{(\\frac{\\mathbf{J}_P{(\\varepsilon_0)}}{\\cos{(\\sin{(\\varepsilon_0)})}})} and \\operatorname{n_{1}}{(\\varepsilon_0)} = 0 and \\operatorname{n_{1}}^{\\varepsilon_0}{(\\varepsilon_0)} = 0^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon_0', commutative=True)), cos(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "cos(sin(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(sin(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1))), Integer(1))"], [["log", 2], "Equality(log(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(sin(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1)))), Integer(0))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True)), log(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(sin(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(0))"], [["power", 5, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integer(0), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z} C_{1}, then derive \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z}, then obtain \\sin{(\\log{(p)})} \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z} \\sin{(\\log{(p)})}", "derivation": "\\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z} C_{1} and \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{2}}{(C_{1},A_{z})} = \\frac{\\partial}{\\partial C_{1}} A_{z} C_{1} and \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z} and \\sin{(\\log{(p)})} \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{2}}{(C_{1},A_{z})} = A_{z} \\sin{(\\log{(p)})}", "srepr_derivation": [["get_premise", "Equality(Function('A_2')(Symbol('C_1', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('C_1', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('C_1', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('A_z', commutative=True))"], [["times", 3, "sin(log(Symbol('p', commutative=True)))"], "Equality(Mul(sin(log(Symbol('p', commutative=True))), Derivative(Function('A_2')(Symbol('C_1', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Mul(Symbol('A_z', commutative=True), sin(log(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} = \\phi f_{\\mathbf{p}} and r{(\\phi,f_{\\mathbf{p}})} = 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)}, then obtain \\phi + 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} = 2 \\phi f_{\\mathbf{p}} + \\phi", "derivation": "\\mathbf{E}{(f_{\\mathbf{p}},\\phi)} = \\phi f_{\\mathbf{p}} and 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} = \\phi f_{\\mathbf{p}} + \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} and r{(\\phi,f_{\\mathbf{p}})} = 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} and r{(\\phi,f_{\\mathbf{p}})} = \\phi f_{\\mathbf{p}} + \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} and r{(\\phi,f_{\\mathbf{p}})} = 2 \\phi f_{\\mathbf{p}} and \\phi + r{(\\phi,f_{\\mathbf{p}})} = 2 \\phi f_{\\mathbf{p}} + \\phi and \\phi + r{(\\phi,f_{\\mathbf{p}})} = \\phi + 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} and \\phi + 2 \\mathbf{E}{(f_{\\mathbf{p}},\\phi)} = 2 \\phi f_{\\mathbf{p}} + \\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('r')(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('r')(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('r')(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('r')(Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{f})} = \\mathbf{f} - a^{\\dagger} and \\hat{p}_0{(a^{\\dagger},\\mathbf{f})} = \\operatorname{C_{2}}^{a^{\\dagger}}{(a^{\\dagger},\\mathbf{f})}, then obtain a^{\\dagger} \\hat{p}_0{(a^{\\dagger},\\mathbf{f})} = a^{\\dagger} (\\mathbf{f} - a^{\\dagger})^{a^{\\dagger}}", "derivation": "\\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{f})} = \\mathbf{f} - a^{\\dagger} and \\operatorname{C_{2}}^{a^{\\dagger}}{(a^{\\dagger},\\mathbf{f})} = (\\mathbf{f} - a^{\\dagger})^{a^{\\dagger}} and \\hat{p}_0{(a^{\\dagger},\\mathbf{f})} = \\operatorname{C_{2}}^{a^{\\dagger}}{(a^{\\dagger},\\mathbf{f})} and \\hat{p}_0{(a^{\\dagger},\\mathbf{f})} = (\\mathbf{f} - a^{\\dagger})^{a^{\\dagger}} and a^{\\dagger} \\hat{p}_0{(a^{\\dagger},\\mathbf{f})} = a^{\\dagger} (\\mathbf{f} - a^{\\dagger})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{p}_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\hat{p}_0')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} = \\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho}, then obtain 0 = (\\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} + \\frac{- \\hat{x}_0 - \\mathbf{g}}{\\rho}) \\operatorname{F_{H}}^{- \\mathbf{g}}{(\\mathbf{g},\\rho,\\hat{x}_0)}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} = \\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho} and \\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} + 1 = 1 + \\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho} and 0 = - \\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} + \\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho} and 0 = - (\\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho})^{- \\mathbf{g}} (- \\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} + \\frac{\\hat{x}_0 + \\mathbf{g}}{\\rho}) and 0 = (\\operatorname{F_{H}}{(\\mathbf{g},\\rho,\\hat{x}_0)} + \\frac{- \\hat{x}_0 - \\mathbf{g}}{\\rho}) \\operatorname{F_{H}}^{- \\mathbf{g}}{(\\mathbf{g},\\rho,\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["minus", 2, "Add(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Mul(Add(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))), Pow(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(M_{E})} = e^{M_{E}} and \\hat{x}_0{(M_{E})} = \\frac{d}{d M_{E}} \\mathbb{I}{(M_{E})} + \\frac{d}{d M_{E}} e^{M_{E}}, then obtain \\int \\hat{x}_0{(M_{E})} dM_{E} = \\int 2 \\frac{d}{d M_{E}} e^{M_{E}} dM_{E}", "derivation": "\\mathbb{I}{(M_{E})} = e^{M_{E}} and \\frac{d}{d M_{E}} \\mathbb{I}{(M_{E})} = \\frac{d}{d M_{E}} e^{M_{E}} and \\hat{x}_0{(M_{E})} = \\frac{d}{d M_{E}} \\mathbb{I}{(M_{E})} + \\frac{d}{d M_{E}} e^{M_{E}} and \\hat{x}_0{(M_{E})} = 2 \\frac{d}{d M_{E}} e^{M_{E}} and \\int \\hat{x}_0{(M_{E})} dM_{E} = \\int 2 \\frac{d}{d M_{E}} e^{M_{E}} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('M_E', commutative=True)), Add(Derivative(Function('\\\\mathbb{I}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{x}_0')(Symbol('M_E', commutative=True)), Mul(Integer(2), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Integer(2), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given V{(v_{2},C_{1})} = v_{2}^{C_{1}}, then obtain - C_{1} - v_{2}^{- C_{1}} V{(v_{2},C_{1})} - v_{2}^{- C_{1}} = - C_{1} - 1 - v_{2}^{- C_{1}}", "derivation": "V{(v_{2},C_{1})} = v_{2}^{C_{1}} and - v_{2}^{- C_{1}} V{(v_{2},C_{1})} = -1 and - C_{1} - v_{2}^{- C_{1}} V{(v_{2},C_{1})} = - C_{1} - 1 and - C_{1} - v_{2}^{- C_{1}} V{(v_{2},C_{1})} - v_{2}^{- C_{1}} = - C_{1} - 1 - v_{2}^{- C_{1}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('V')(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True))), Integer(-1))"], [["minus", 2, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('V')(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Integer(-1)))"], [["minus", 3, "Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('V')(Symbol('v_2', commutative=True), Symbol('C_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(G,\\chi)} = \\frac{\\partial}{\\partial G} (G + \\chi), then derive \\mathbf{s}{(G,\\chi)} = 1, then obtain - \\chi + ((\\frac{\\partial}{\\partial G} (G + \\chi))^{\\chi})^{G} = 1 - \\chi", "derivation": "\\mathbf{s}{(G,\\chi)} = \\frac{\\partial}{\\partial G} (G + \\chi) and \\mathbf{s}{(G,\\chi)} = 1 and \\frac{\\partial}{\\partial G} (G + \\chi) = 1 and (\\frac{\\partial}{\\partial G} (G + \\chi))^{\\chi} = 1 and ((\\frac{\\partial}{\\partial G} (G + \\chi))^{\\chi})^{G} = 1 and - \\chi + ((\\frac{\\partial}{\\partial G} (G + \\chi))^{\\chi})^{G} = 1 - \\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Integer(1))"], [["power", 4, "Symbol('G', commutative=True)"], "Equality(Pow(Pow(Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Symbol('G', commutative=True)), Integer(1))"], [["minus", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Pow(Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Symbol('G', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(v_{2})} = e^{v_{2}}, then derive \\int \\varepsilon{(v_{2})} dv_{2} = C + e^{v_{2}}, then obtain (-1)^{v_{2}} = (- C - e^{v_{2}} + \\int \\varepsilon{(v_{2})} dv_{2} - 1)^{v_{2}}", "derivation": "\\varepsilon{(v_{2})} = e^{v_{2}} and \\int \\varepsilon{(v_{2})} dv_{2} = \\int e^{v_{2}} dv_{2} and 0 = - \\int \\varepsilon{(v_{2})} dv_{2} + \\int e^{v_{2}} dv_{2} and \\int \\varepsilon{(v_{2})} dv_{2} = C + e^{v_{2}} and 0 = - C - e^{v_{2}} + \\int e^{v_{2}} dv_{2} and -1 = - C - e^{v_{2}} + \\int e^{v_{2}} dv_{2} - 1 and (-1)^{v_{2}} = (- C - e^{v_{2}} + \\int e^{v_{2}} dv_{2} - 1)^{v_{2}} and (-1)^{v_{2}} = (- C - e^{v_{2}} + \\int \\varepsilon{(v_{2})} dv_{2} - 1)^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('C', commutative=True), exp(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))))"], [["add", 5, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integer(-1)))"], [["power", 6, "Symbol('v_2', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integer(-1)), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Pow(Integer(-1), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('v_2', commutative=True))), Integral(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integer(-1)), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(g)} = e^{g}, then obtain - 2 g + \\cos^{2}{(\\operatorname{E_{\\lambda}}{(g)} e^{g})} = - 2 g + \\cos^{2}{(e^{2 g})}", "derivation": "\\operatorname{E_{\\lambda}}{(g)} = e^{g} and \\operatorname{E_{\\lambda}}{(g)} e^{g} = e^{2 g} and \\cos{(\\operatorname{E_{\\lambda}}{(g)} e^{g})} = \\cos{(e^{2 g})} and \\cos^{2}{(\\operatorname{E_{\\lambda}}{(g)} e^{g})} = \\cos^{2}{(e^{2 g})} and - 2 g + \\cos^{2}{(\\operatorname{E_{\\lambda}}{(g)} e^{g})} = - 2 g + \\cos^{2}{(e^{2 g})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["times", 1, "exp(Symbol('g', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), exp(Mul(Integer(2), Symbol('g', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Function('E_{\\\\lambda}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))), cos(exp(Mul(Integer(2), Symbol('g', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(cos(Mul(Function('E_{\\\\lambda}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))), Integer(2)), Pow(cos(exp(Mul(Integer(2), Symbol('g', commutative=True)))), Integer(2)))"], [["minus", 4, "Mul(Integer(2), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Pow(cos(Mul(Function('E_{\\\\lambda}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))), Integer(2))), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Pow(cos(exp(Mul(Integer(2), Symbol('g', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\eta^{\\prime}{(S)} = \\sin{(S)}, then obtain S \\sin{(S)} + \\int \\eta^{\\prime}{(S)} dS = S \\sin{(S)} + c_{0} - \\cos{(S)}", "derivation": "\\eta^{\\prime}{(S)} = \\sin{(S)} and S \\eta^{\\prime}{(S)} = S \\sin{(S)} and \\int \\eta^{\\prime}{(S)} dS = \\int \\sin{(S)} dS and S \\eta^{\\prime}{(S)} + \\int \\eta^{\\prime}{(S)} dS = S \\eta^{\\prime}{(S)} + \\int \\sin{(S)} dS and S \\sin{(S)} + \\int \\eta^{\\prime}{(S)} dS = S \\sin{(S)} + \\int \\sin{(S)} dS and S \\sin{(S)} + \\int \\eta^{\\prime}{(S)} dS = S \\sin{(S)} + c_{0} - \\cos{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), sin(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["add", 3, "Mul(Symbol('S', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Symbol('S', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\phi{(J,\\mu)} = \\frac{J}{\\mu}, then obtain \\frac{\\partial}{\\partial \\mu} \\log{(\\frac{\\mu \\phi{(J,\\mu)}}{J})} = \\frac{d}{d \\mu} 0", "derivation": "\\phi{(J,\\mu)} = \\frac{J}{\\mu} and \\frac{\\mu \\phi{(J,\\mu)}}{J} = 1 and \\log{(\\frac{\\mu \\phi{(J,\\mu)}}{J})} = 0 and \\frac{\\partial}{\\partial \\mu} \\log{(\\frac{\\mu \\phi{(J,\\mu)}}{J})} = \\frac{d}{d \\mu} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(1))"], [["log", 2], "Equality(log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(u,\\varepsilon)} = \\varepsilon + u, then obtain -1 = -2 + \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u}", "derivation": "L{(u,\\varepsilon)} = \\varepsilon + u and \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u} = 1 and - \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u} = -1 and 0 = -1 + \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u} and 0 = (\\varepsilon + u)^{- \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u}} L{(u,\\varepsilon)} - \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u} and - \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u} = (\\varepsilon + u)^{- \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u}} L{(u,\\varepsilon)} - \\frac{2 L{(u,\\varepsilon)}}{\\varepsilon + u} and -1 = -2 + \\frac{L{(u,\\varepsilon)}}{\\varepsilon + u}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integer(-1))"], [["minus", 3, "Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["minus", 5, "Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(-1), Add(Integer(-2), Mul(Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('u', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(i,\\varphi)} = \\varphi - i, then obtain 4 (\\varphi - i)^{2} = (2 \\varphi - 2 i)^{2}", "derivation": "\\operatorname{A_{z}}{(i,\\varphi)} = \\varphi - i and 2 \\operatorname{A_{z}}{(i,\\varphi)} = \\varphi - i + \\operatorname{A_{z}}{(i,\\varphi)} and 4 \\operatorname{A_{z}}^{2}{(i,\\varphi)} = (\\varphi - i + \\operatorname{A_{z}}{(i,\\varphi)})^{2} and 4 (\\varphi - i)^{2} = (2 \\varphi - 2 i)^{2}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["add", 1, "Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(2), Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Function('A_z')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('i', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{r}{(\\rho_b,\\mathbf{S})} = \\mathbf{S} - \\rho_b, then obtain \\frac{\\partial}{\\partial \\mathbf{S}} 2 \\mathbf{r}{(\\rho_b,\\mathbf{S})} = \\frac{\\partial}{\\partial \\mathbf{S}} (2 \\mathbf{S} - 2 \\rho_b)", "derivation": "\\mathbf{r}{(\\rho_b,\\mathbf{S})} = \\mathbf{S} - \\rho_b and \\mathbf{S} - \\rho_b + \\mathbf{r}{(\\rho_b,\\mathbf{S})} = 2 \\mathbf{S} - 2 \\rho_b and 2 \\mathbf{r}{(\\rho_b,\\mathbf{S})} = 2 \\mathbf{S} - 2 \\rho_b and \\frac{\\partial}{\\partial \\mathbf{S}} 2 \\mathbf{r}{(\\rho_b,\\mathbf{S})} = \\frac{\\partial}{\\partial \\mathbf{S}} (2 \\mathbf{S} - 2 \\rho_b)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{B})} = e^{\\mathbf{B}}, then derive \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain 0 = - (- e^{\\mathbf{B}} + \\frac{d}{d \\mathbf{B}} e^{\\mathbf{B}}) e^{- \\mathbf{B}}", "derivation": "\\dot{y}{(\\mathbf{B})} = e^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} e^{\\mathbf{B}} and 0 = - \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} e^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})} = e^{\\mathbf{B}} and 0 = - e^{\\mathbf{B}} + \\frac{d}{d \\mathbf{B}} e^{\\mathbf{B}} and 0 = - e^{\\mathbf{B}} + \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})} and 0 = - (- e^{\\mathbf{B}} + \\frac{d}{d \\mathbf{B}} \\dot{y}{(\\mathbf{B})}) e^{- \\mathbf{B}} and 0 = - (- e^{\\mathbf{B}} + \\frac{d}{d \\mathbf{B}} e^{\\mathbf{B}}) e^{- \\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["divide", 6, "Mul(Integer(-1), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\ddot{x})} = \\cos{(\\ddot{x})}, then derive C_{d} + \\ddot{x} = \\int \\frac{\\cos{(\\ddot{x})}}{\\operatorname{a^{\\dagger}}{(\\ddot{x})}} d\\ddot{x}, then obtain (C_{d} + \\ddot{x})^{C_{d}} = (\\ddot{x} + \\hat{p}_0)^{C_{d}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\ddot{x})} = \\cos{(\\ddot{x})} and 1 = \\frac{\\cos{(\\ddot{x})}}{\\operatorname{a^{\\dagger}}{(\\ddot{x})}} and \\int 1 d\\ddot{x} = \\int \\frac{\\cos{(\\ddot{x})}}{\\operatorname{a^{\\dagger}}{(\\ddot{x})}} d\\ddot{x} and C_{d} + \\ddot{x} = \\int \\frac{\\cos{(\\ddot{x})}}{\\operatorname{a^{\\dagger}}{(\\ddot{x})}} d\\ddot{x} and (C_{d} + \\ddot{x})^{C_{d}} = (\\int \\frac{\\cos{(\\ddot{x})}}{\\operatorname{a^{\\dagger}}{(\\ddot{x})}} d\\ddot{x})^{C_{d}} and (C_{d} + \\ddot{x})^{C_{d}} = (\\int 1 d\\ddot{x})^{C_{d}} and (C_{d} + \\ddot{x})^{C_{d}} = (\\ddot{x} + \\hat{p}_0)^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True)))"], [["divide", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["power", 4, "Symbol('C_d', commutative=True)"], "Equality(Pow(Add(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('C_d', commutative=True)), Pow(Integral(Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('C_d', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('C_d', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('C_d', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\mu{(q,\\varphi)} = - \\varphi + q, then derive \\hat{p} + \\mu{(q,\\varphi)} = \\phi + q, then obtain \\int (\\hat{p} - \\varphi + q) dq = \\int (\\hat{p} + \\mu{(q,\\varphi)}) dq", "derivation": "\\mu{(q,\\varphi)} = - \\varphi + q and \\frac{\\partial}{\\partial q} \\mu{(q,\\varphi)} = \\frac{\\partial}{\\partial q} (- \\varphi + q) and \\int \\frac{\\partial}{\\partial q} \\mu{(q,\\varphi)} dq = \\int \\frac{\\partial}{\\partial q} (- \\varphi + q) dq and \\hat{p} + \\mu{(q,\\varphi)} = \\phi + q and \\hat{p} - \\varphi + q = \\phi + q and \\hat{p} - \\varphi + q = \\hat{p} + \\mu{(q,\\varphi)} and \\int (\\hat{p} - \\varphi + q) dq = \\int (\\hat{p} + \\mu{(q,\\varphi)}) dq", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 6, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v)} = \\sin{(v)}, then obtain - \\operatorname{n_{1}}{(v)} + \\int \\operatorname{n_{1}}{(v)} dv + \\int \\sin{(v)} dv = - \\operatorname{n_{1}}{(v)} + 2 \\int \\sin{(v)} dv", "derivation": "\\operatorname{n_{1}}{(v)} = \\sin{(v)} and \\int \\operatorname{n_{1}}{(v)} dv = \\int \\sin{(v)} dv and \\int \\operatorname{n_{1}}{(v)} dv + \\int \\sin{(v)} dv = 2 \\int \\sin{(v)} dv and - \\sin{(v)} + \\int \\operatorname{n_{1}}{(v)} dv + \\int \\sin{(v)} dv = - \\sin{(v)} + 2 \\int \\sin{(v)} dv and - \\operatorname{n_{1}}{(v)} + \\int \\operatorname{n_{1}}{(v)} dv + \\int \\sin{(v)} dv = - \\operatorname{n_{1}}{(v)} + 2 \\int \\sin{(v)} dv", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 2, "Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Add(Integral(Function('n_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["minus", 3, "sin(Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('v', commutative=True))), Integral(Function('n_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('v', commutative=True))), Mul(Integer(2), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('n_1')(Symbol('v', commutative=True))), Integral(Function('n_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('n_1')(Symbol('v', commutative=True))), Mul(Integer(2), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(v_{1},A_{2})} = A_{2} + v_{1}, then derive \\int \\mathbf{s}{(v_{1},A_{2})} dv_{1} = A_{2} v_{1} + t + \\frac{v_{1}^{2}}{2}, then obtain \\frac{\\int \\mathbf{s}{(v_{1},A_{2})} dv_{1}}{A_{2} (A_{2} + v_{1})} = \\frac{A_{2} v_{1} + t + \\frac{v_{1}^{2}}{2}}{A_{2} (A_{2} + v_{1})}", "derivation": "\\mathbf{s}{(v_{1},A_{2})} = A_{2} + v_{1} and \\int \\mathbf{s}{(v_{1},A_{2})} dv_{1} = \\int (A_{2} + v_{1}) dv_{1} and \\int \\mathbf{s}{(v_{1},A_{2})} dv_{1} = A_{2} v_{1} + t + \\frac{v_{1}^{2}}{2} and \\frac{\\int \\mathbf{s}{(v_{1},A_{2})} dv_{1}}{A_{2} (A_{2} + v_{1})} = \\frac{A_{2} v_{1} + t + \\frac{v_{1}^{2}}{2}}{A_{2} (A_{2} + v_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('v_1', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('v_1', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('v_1', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2)))))"], [["divide", 3, "Mul(Symbol('A_2', commutative=True), Add(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Add(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{s}')(Symbol('v_1', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Add(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Add(Mul(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Symbol('t', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))))))"]]}, {"prompt": "Given Z{(A_{1},C_{1},W)} = C_{1} (- A_{1} + W), then derive \\frac{\\partial}{\\partial A_{1}} Z{(A_{1},C_{1},W)} = - C_{1}, then obtain (- \\frac{Z{(A_{1},C_{1},W)}}{C_{1}})^{C_{1}} (A_{1} - W)^{C_{1}} = (A_{1} - W)^{2 C_{1}}", "derivation": "Z{(A_{1},C_{1},W)} = C_{1} (- A_{1} + W) and \\frac{\\partial}{\\partial A_{1}} Z{(A_{1},C_{1},W)} = \\frac{\\partial}{\\partial A_{1}} C_{1} (- A_{1} + W) and \\frac{Z{(A_{1},C_{1},W)}}{\\frac{\\partial}{\\partial A_{1}} C_{1} (- A_{1} + W)} = \\frac{C_{1} (- A_{1} + W)}{\\frac{\\partial}{\\partial A_{1}} C_{1} (- A_{1} + W)} and \\frac{\\partial}{\\partial A_{1}} Z{(A_{1},C_{1},W)} = - C_{1} and \\frac{\\partial}{\\partial A_{1}} C_{1} (- A_{1} + W) = - C_{1} and - \\frac{Z{(A_{1},C_{1},W)}}{C_{1}} = A_{1} - W and (- \\frac{Z{(A_{1},C_{1},W)}}{C_{1}})^{C_{1}} = (A_{1} - W)^{C_{1}} and (- \\frac{Z{(A_{1},C_{1},W)}}{C_{1}})^{C_{1}} (A_{1} - W)^{C_{1}} = (A_{1} - W)^{2 C_{1}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))"], "Equality(Mul(Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Pow(Derivative(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True)), Pow(Derivative(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["power", 6, "Symbol('C_1', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Symbol('C_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Symbol('C_1', commutative=True)))"], [["times", 7, "Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('Z')(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Symbol('C_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Symbol('C_1', commutative=True))), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Mul(Integer(2), Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\psi^*,\\mu)} = \\mu + \\psi^*, then obtain (\\int \\dot{\\mathbf{r}}{(\\psi^*,\\mu)} d\\psi^*)^{\\psi^*} = (\\mu \\psi^* + \\frac{(\\psi^*)^{2}}{2} + \\sigma_p)^{\\psi^*}", "derivation": "\\dot{\\mathbf{r}}{(\\psi^*,\\mu)} = \\mu + \\psi^* and \\int \\dot{\\mathbf{r}}{(\\psi^*,\\mu)} d\\psi^* = \\int (\\mu + \\psi^*) d\\psi^* and (\\int \\dot{\\mathbf{r}}{(\\psi^*,\\mu)} d\\psi^*)^{\\psi^*} = (\\int (\\mu + \\psi^*) d\\psi^*)^{\\psi^*} and (\\int \\dot{\\mathbf{r}}{(\\psi^*,\\mu)} d\\psi^*)^{\\psi^*} = (\\mu \\psi^* + \\frac{(\\psi^*)^{2}}{2} + \\sigma_p)^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given m{(\\nabla)} = e^{\\nabla} and \\theta_{1}{(\\nabla)} = - m{(\\nabla)}, then obtain 0 = \\theta_{1}{(\\nabla)} + m{(\\nabla)}", "derivation": "m{(\\nabla)} = e^{\\nabla} and \\nabla + m{(\\nabla)} = \\nabla + e^{\\nabla} and 0 = - m{(\\nabla)} + e^{\\nabla} and \\theta_{1}{(\\nabla)} = - m{(\\nabla)} and 0 = \\theta_{1}{(\\nabla)} + e^{\\nabla} and 0 = \\theta_{1}{(\\nabla)} + m{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Function('\\\\theta_1')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Function('\\\\theta_1')(Symbol('\\\\nabla', commutative=True)), Function('m')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{f},\\hbar)} = \\frac{\\mathbf{f}}{\\hbar}, then derive \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)} = \\frac{1}{\\hbar}, then obtain (\\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)})^{\\hbar} = (\\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\mathbf{f}}{\\hbar})^{\\hbar}", "derivation": "\\mathbf{D}{(\\mathbf{f},\\hbar)} = \\frac{\\mathbf{f}}{\\hbar} and \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)} = \\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\mathbf{f}}{\\hbar} and \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)} = \\frac{1}{\\hbar} and \\frac{1}{\\hbar} = \\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\mathbf{f}}{\\hbar} and (\\frac{1}{\\hbar})^{\\hbar} = (\\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\mathbf{f}}{\\hbar})^{\\hbar} and (\\frac{1}{\\hbar})^{\\hbar} = (\\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)})^{\\hbar} and (\\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{D}{(\\mathbf{f},\\hbar)})^{\\hbar} = (\\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\mathbf{f}}{\\hbar})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(H,\\mathbf{D})} = H^{\\mathbf{D}} and I{(H,\\mathbf{D})} = H^{- \\mathbf{D}} \\operatorname{f_{E}}{(H,\\mathbf{D})}, then obtain 1 = I^{- H}{(H,\\mathbf{D})}", "derivation": "\\operatorname{f_{E}}{(H,\\mathbf{D})} = H^{\\mathbf{D}} and I{(H,\\mathbf{D})} = H^{- \\mathbf{D}} \\operatorname{f_{E}}{(H,\\mathbf{D})} and I^{H}{(H,\\mathbf{D})} = (H^{- \\mathbf{D}} \\operatorname{f_{E}}{(H,\\mathbf{D})})^{H} and 1 = (H^{- \\mathbf{D}} \\operatorname{f_{E}}{(H,\\mathbf{D})})^{H} I^{- H}{(H,\\mathbf{D})} and 1 = I^{- H}{(H,\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('f_E')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Pow(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('f_E')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('H', commutative=True)))"], [["divide", 3, "Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('H', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Pow(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('f_E')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('H', commutative=True)), Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Pow(Function('I')(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\hat{\\mathbf{x}})}, then derive \\frac{d}{d \\hat{\\mathbf{x}}} \\mathbf{J}_M{(\\hat{\\mathbf{x}})} = - \\cos{(\\hat{\\mathbf{x}})}, then obtain \\int - \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\frac{d^{2}}{d \\hat{\\mathbf{x}}^{2}} \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}}", "derivation": "\\mathbf{J}_M{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\hat{\\mathbf{x}})} and \\frac{d}{d \\hat{\\mathbf{x}}} \\mathbf{J}_M{(\\hat{\\mathbf{x}})} = \\frac{d^{2}}{d \\hat{\\mathbf{x}}^{2}} \\cos{(\\hat{\\mathbf{x}})} and \\frac{d}{d \\hat{\\mathbf{x}}} \\mathbf{J}_M{(\\hat{\\mathbf{x}})} = - \\cos{(\\hat{\\mathbf{x}})} and - \\cos{(\\hat{\\mathbf{x}})} = \\frac{d^{2}}{d \\hat{\\mathbf{x}}^{2}} \\cos{(\\hat{\\mathbf{x}})} and \\int - \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\frac{d^{2}}{d \\hat{\\mathbf{x}}^{2}} \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Derivative(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given n{(\\dot{x},W)} = \\dot{x}^{W} and \\tilde{g}^*{(\\dot{x},W)} = W n^{2}{(\\dot{x},W)}, then obtain \\frac{\\partial}{\\partial \\dot{x}} W n^{2}{(\\dot{x},W)} = \\frac{\\partial}{\\partial \\dot{x}} \\tilde{g}^*{(\\dot{x},W)}", "derivation": "n{(\\dot{x},W)} = \\dot{x}^{W} and W n{(\\dot{x},W)} = W \\dot{x}^{W} and W \\dot{x}^{W} n{(\\dot{x},W)} = W \\dot{x}^{2 W} and W n^{2}{(\\dot{x},W)} = W \\dot{x}^{W} n{(\\dot{x},W)} and W n^{2}{(\\dot{x},W)} = W \\dot{x}^{2 W} and \\tilde{g}^*{(\\dot{x},W)} = W n^{2}{(\\dot{x},W)} and \\frac{\\partial}{\\partial \\dot{x}} W n^{2}{(\\dot{x},W)} = \\frac{\\partial}{\\partial \\dot{x}} W \\dot{x}^{2 W} and \\tilde{g}^*{(\\dot{x},W)} = W \\dot{x}^{2 W} and \\frac{\\partial}{\\partial \\dot{x}} W n^{2}{(\\dot{x},W)} = \\frac{\\partial}{\\partial \\dot{x}} \\tilde{g}^*{(\\dot{x},W)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True))))"], [["times", 1, "Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(2), Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('W', commutative=True), Pow(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Integer(2))), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('W', commutative=True), Pow(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Integer(2))), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(2), Symbol('W', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Pow(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Symbol('W', commutative=True), Pow(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Integer(2))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(2), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(2), Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Derivative(Mul(Symbol('W', commutative=True), Pow(Function('n')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Integer(2))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\dot{x})} = \\log{(\\dot{x})} and \\rho{(\\dot{x})} = - \\dot{x}, then obtain (\\frac{\\hat{p}_0{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} + \\rho{(\\dot{x})} = (\\frac{\\log{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} + \\rho{(\\dot{x})}", "derivation": "\\hat{p}_0{(\\dot{x})} = \\log{(\\dot{x})} and \\frac{\\hat{p}_0{(\\dot{x})}}{\\dot{x}} = \\frac{\\log{(\\dot{x})}}{\\dot{x}} and (\\frac{\\hat{p}_0{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} = (\\frac{\\log{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} and - \\dot{x} + (\\frac{\\hat{p}_0{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} = - \\dot{x} + (\\frac{\\log{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} and \\rho{(\\dot{x})} = - \\dot{x} and (\\frac{\\hat{p}_0{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} + \\rho{(\\dot{x})} = (\\frac{\\log{(\\dot{x})}}{\\dot{x}})^{\\dot{x}} + \\rho{(\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), log(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{x}', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), log(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = \\frac{F_{g}}{\\tilde{g}^*}, then derive \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = - \\frac{F_{g}}{(\\tilde{g}^*)^{2}}, then obtain - \\frac{\\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*}", "derivation": "\\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = \\frac{F_{g}}{\\tilde{g}^*} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = - \\frac{F_{g}}{(\\tilde{g}^*)^{2}} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)} = - \\frac{\\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)}}{\\tilde{g}^*} and - \\frac{\\operatorname{t_{2}}{(F_{g},\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('t_2')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\nabla,F_{N})} = F_{N} + e^{\\nabla}, then obtain \\cos{((F_{N} + 2 e^{\\nabla}) \\operatorname{C_{1}}{(\\nabla,F_{N})})} = \\cos{((F_{N} + e^{\\nabla}) (F_{N} + 2 e^{\\nabla}))}", "derivation": "\\operatorname{C_{1}}{(\\nabla,F_{N})} = F_{N} + e^{\\nabla} and \\operatorname{C_{1}}{(\\nabla,F_{N})} + e^{\\nabla} = F_{N} + 2 e^{\\nabla} and (\\operatorname{C_{1}}{(\\nabla,F_{N})} + e^{\\nabla}) \\operatorname{C_{1}}{(\\nabla,F_{N})} = (F_{N} + e^{\\nabla}) (\\operatorname{C_{1}}{(\\nabla,F_{N})} + e^{\\nabla}) and (F_{N} + 2 e^{\\nabla}) \\operatorname{C_{1}}{(\\nabla,F_{N})} = (F_{N} + e^{\\nabla}) (F_{N} + 2 e^{\\nabla}) and \\cos{((F_{N} + 2 e^{\\nabla}) \\operatorname{C_{1}}{(\\nabla,F_{N})})} = \\cos{((F_{N} + e^{\\nabla}) (F_{N} + 2 e^{\\nabla}))}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["add", 1, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))), Add(Symbol('F_N', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))))"], [["times", 1, "Add(Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Add(Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))), Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Mul(Add(Symbol('F_N', commutative=True), exp(Symbol('\\\\nabla', commutative=True))), Add(Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('F_N', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))), Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True))), Mul(Add(Symbol('F_N', commutative=True), exp(Symbol('\\\\nabla', commutative=True))), Add(Symbol('F_N', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))))"], [["cos", 4], "Equality(cos(Mul(Add(Symbol('F_N', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))), Function('C_1')(Symbol('\\\\nabla', commutative=True), Symbol('F_N', commutative=True)))), cos(Mul(Add(Symbol('F_N', commutative=True), exp(Symbol('\\\\nabla', commutative=True))), Add(Symbol('F_N', commutative=True), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))))))"]]}, {"prompt": "Given W{(A_{2},l)} = A_{2} l and \\mathbf{r}{(A_{2},l)} = A_{2} l, then obtain \\log{(l \\mathbf{r}{(A_{2},l)})} = \\log{(A_{2} l^{2})}", "derivation": "W{(A_{2},l)} = A_{2} l and l W{(A_{2},l)} = A_{2} l^{2} and \\log{(l W{(A_{2},l)})} = \\log{(A_{2} l^{2})} and \\mathbf{r}{(A_{2},l)} = A_{2} l and l W{(A_{2},l)} = l \\mathbf{r}{(A_{2},l)} and \\log{(l \\mathbf{r}{(A_{2},l)})} = \\log{(A_{2} l^{2})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('A_2', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('W')(Symbol('A_2', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('l', commutative=True), Integer(2))))"], [["log", 2], "Equality(log(Mul(Symbol('l', commutative=True), Function('W')(Symbol('A_2', commutative=True), Symbol('l', commutative=True)))), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Symbol('l', commutative=True), Function('W')(Symbol('A_2', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True), Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True), Symbol('l', commutative=True)))), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)))))"]]}, {"prompt": "Given S{(A_{2},\\varepsilon_0)} = \\log{(\\varepsilon_0^{A_{2}})} and U{(A_{2},\\varepsilon_0)} = \\log{(\\varepsilon_0^{A_{2}})}, then obtain \\sin{(\\sin{(S{(A_{2},\\varepsilon_0)})})} - 1 = \\sin{(\\sin{(U{(A_{2},\\varepsilon_0)})})} - 1", "derivation": "S{(A_{2},\\varepsilon_0)} = \\log{(\\varepsilon_0^{A_{2}})} and \\sin{(S{(A_{2},\\varepsilon_0)})} = \\sin{(\\log{(\\varepsilon_0^{A_{2}})})} and U{(A_{2},\\varepsilon_0)} = \\log{(\\varepsilon_0^{A_{2}})} and \\sin{(\\sin{(S{(A_{2},\\varepsilon_0)})})} = \\sin{(\\sin{(\\log{(\\varepsilon_0^{A_{2}})})})} and \\sin{(\\sin{(S{(A_{2},\\varepsilon_0)})})} = \\sin{(\\sin{(U{(A_{2},\\varepsilon_0)})})} and \\sin{(\\sin{(S{(A_{2},\\varepsilon_0)})})} - 1 = \\sin{(\\sin{(U{(A_{2},\\varepsilon_0)})})} - 1", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_2', commutative=True))))"], [["sin", 1], "Equality(sin(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), sin(log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('U')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_2', commutative=True))))"], [["sin", 2], "Equality(sin(sin(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), sin(sin(log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(sin(sin(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), sin(sin(Function('U')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 5, 1], "Equality(Add(sin(sin(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Integer(-1)), Add(sin(sin(Function('U')(Symbol('A_2', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\eta{(\\dot{x},x^\\prime)} = \\frac{\\dot{x}}{x^\\prime}, then derive \\frac{\\partial}{\\partial \\dot{x}} \\eta{(\\dot{x},x^\\prime)} = \\frac{1}{x^\\prime}, then obtain \\frac{\\frac{\\partial}{\\partial \\dot{x}} (x^\\prime + \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime})}{\\frac{d}{d \\dot{x}} (x^\\prime + \\frac{1}{x^\\prime})} = 1", "derivation": "\\eta{(\\dot{x},x^\\prime)} = \\frac{\\dot{x}}{x^\\prime} and \\frac{\\partial}{\\partial \\dot{x}} \\eta{(\\dot{x},x^\\prime)} = \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime} and \\frac{\\partial}{\\partial \\dot{x}} \\eta{(\\dot{x},x^\\prime)} = \\frac{1}{x^\\prime} and \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime} = \\frac{1}{x^\\prime} and x^\\prime + \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime} = x^\\prime + \\frac{1}{x^\\prime} and \\frac{\\partial}{\\partial \\dot{x}} (x^\\prime + \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime}) = \\frac{d}{d \\dot{x}} (x^\\prime + \\frac{1}{x^\\prime}) and \\frac{\\frac{\\partial}{\\partial \\dot{x}} (x^\\prime + \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x}}{x^\\prime})}{\\frac{d}{d \\dot{x}} (x^\\prime + \\frac{1}{x^\\prime})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\dot{x}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\dot{x}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\dot{x}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["add", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Add(Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Add(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["divide", 6, "Derivative(Add(Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Symbol('x^\\\\prime', commutative=True), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbb{I},\\chi)} = \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I}, then derive \\mathbb{I} \\mathbf{v}{(\\mathbb{I},\\chi)} = \\mathbb{I}^{2}, then obtain \\mathbb{I} \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I} - 1 = \\mathbb{I}^{2} - 1", "derivation": "\\mathbf{v}{(\\mathbb{I},\\chi)} = \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I} and \\mathbb{I} \\mathbf{v}{(\\mathbb{I},\\chi)} = \\mathbb{I} \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I} and \\mathbb{I} \\mathbf{v}{(\\mathbb{I},\\chi)} = \\mathbb{I}^{2} and \\mathbb{I} \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I} = \\mathbb{I}^{2} and \\mathbb{I} \\frac{\\partial}{\\partial \\chi} \\chi \\mathbb{I} - 1 = \\mathbb{I}^{2} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))"], [["minus", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Integer(-1)), Add(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)), Integer(-1)))"]]}, {"prompt": "Given H{(v_{y},\\varphi)} = \\varphi + v_{y}, then derive - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} H{(v_{y},\\varphi)} = 1 - H{(v_{y},\\varphi)}, then obtain - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} (\\varphi + v_{y}) = 1 - H{(v_{y},\\varphi)}", "derivation": "H{(v_{y},\\varphi)} = \\varphi + v_{y} and \\frac{\\partial}{\\partial v_{y}} H{(v_{y},\\varphi)} = \\frac{\\partial}{\\partial v_{y}} (\\varphi + v_{y}) and - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} H{(v_{y},\\varphi)} = - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} (\\varphi + v_{y}) and - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} H{(v_{y},\\varphi)} = 1 - H{(v_{y},\\varphi)} and - H{(v_{y},\\varphi)} + \\frac{\\partial}{\\partial v_{y}} (\\varphi + v_{y}) = 1 - H{(v_{y},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["minus", 2, "Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True))), Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True))), Derivative(Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('H')(Symbol('v_y', commutative=True), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\delta)} = \\cos{(\\delta)}, then obtain \\frac{d}{d \\delta} \\delta (\\mathbf{D}^{\\delta}{(\\delta)} - \\cos{(\\delta)}) = \\frac{d}{d \\delta} \\delta (- \\cos{(\\delta)} + \\cos^{\\delta}{(\\delta)})", "derivation": "\\mathbf{D}{(\\delta)} = \\cos{(\\delta)} and \\mathbf{D}^{\\delta}{(\\delta)} = \\cos^{\\delta}{(\\delta)} and - \\mathbf{D}{(\\delta)} + \\mathbf{D}^{\\delta}{(\\delta)} = - \\mathbf{D}{(\\delta)} + \\cos^{\\delta}{(\\delta)} and - \\mathbf{D}{(\\delta)} = - \\cos{(\\delta)} and \\mathbf{D}^{\\delta}{(\\delta)} - \\cos{(\\delta)} = - \\cos{(\\delta)} + \\cos^{\\delta}{(\\delta)} and \\delta (\\mathbf{D}^{\\delta}{(\\delta)} - \\cos{(\\delta)}) = \\delta (- \\cos{(\\delta)} + \\cos^{\\delta}{(\\delta)}) and \\frac{d}{d \\delta} \\delta (\\mathbf{D}^{\\delta}{(\\delta)} - \\cos{(\\delta)}) = \\frac{d}{d \\delta} \\delta (- \\cos{(\\delta)} + \\cos^{\\delta}{(\\delta)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["minus", 2, "Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True))), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["times", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))))), Mul(Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\delta', commutative=True), Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True))), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(b)} = \\cos{(b)}, then obtain \\frac{(\\mathbf{J}_f{(b)} - \\cos{(b)}) \\int 0 db}{\\cos{(b)}} = 0", "derivation": "\\mathbf{J}_f{(b)} = \\cos{(b)} and \\mathbf{J}_f{(b)} - \\cos{(b)} = 0 and \\frac{\\mathbf{J}_f{(b)} - \\cos{(b)}}{\\cos{(b)}} = 0 and \\int \\frac{\\mathbf{J}_f{(b)} - \\cos{(b)}}{\\cos{(b)}} db = \\int 0 db and \\frac{(\\mathbf{J}_f{(b)} - \\cos{(b)}) \\int \\frac{\\mathbf{J}_f{(b)} - \\cos{(b)}}{\\cos{(b)}} db}{\\cos{(b)}} = 0 and \\frac{(\\mathbf{J}_f{(b)} - \\cos{(b)}) \\int 0 db}{\\cos{(b)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["minus", 1, "cos(Symbol('b', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Integer(0))"], [["divide", 2, "cos(Symbol('b', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Integer(0))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('b', commutative=True))), Integral(Integer(0), Tuple(Symbol('b', commutative=True))))"], [["times", 3, "Integral(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('b', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1)), Integral(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('b', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Pow(cos(Symbol('b', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('b', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain \\frac{\\dot{z}{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\dot{z}{(\\mathbf{A})}} = \\frac{\\sin{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\dot{z}{(\\mathbf{A})}}", "derivation": "\\dot{z}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\dot{z}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and \\frac{d^{2}}{d \\mathbf{A}^{2}} \\dot{z}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} \\sin{(\\mathbf{A})} and \\frac{\\dot{z}{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\sin{(\\mathbf{A})}} = \\frac{\\sin{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\sin{(\\mathbf{A})}} and \\frac{\\dot{z}{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\dot{z}{(\\mathbf{A})}} = \\frac{\\sin{(\\mathbf{A})}}{\\frac{d^{2}}{d \\mathbf{A}^{2}} \\dot{z}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))))"], [["divide", 1, "Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Integer(-1))), Mul(sin(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Integer(-1))), Mul(sin(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}{(E_{x})} = \\log{(E_{x})}, then obtain E_{x}^{2} (\\tilde{g}{(E_{x})} + \\log{(E_{x})})^{2} = 4 E_{x}^{2} \\log{(E_{x})}^{2}", "derivation": "\\tilde{g}{(E_{x})} = \\log{(E_{x})} and \\tilde{g}{(E_{x})} + \\log{(E_{x})} = 2 \\log{(E_{x})} and E_{x} (\\tilde{g}{(E_{x})} + \\log{(E_{x})}) = 2 E_{x} \\log{(E_{x})} and E_{x}^{2} (\\tilde{g}{(E_{x})} + \\log{(E_{x})})^{2} = 4 E_{x}^{2} \\log{(E_{x})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["add", 1, "log(Symbol('E_x', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True))), Mul(Integer(2), log(Symbol('E_x', commutative=True))))"], [["times", 2, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Add(Function('\\\\tilde{g}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))), Mul(Integer(2), Symbol('E_x', commutative=True), log(Symbol('E_x', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Add(Function('\\\\tilde{g}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True))), Integer(2))), Mul(Integer(4), Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(log(Symbol('E_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\lambda{(M_{E},\\hat{p},z)} = (\\hat{p} - z)^{M_{E}}, then obtain \\int \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} d\\hat{p} + \\int \\frac{\\lambda{(M_{E},\\hat{p},z)}}{\\hat{p}} d\\hat{p} = 2 \\int \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} d\\hat{p}", "derivation": "\\lambda{(M_{E},\\hat{p},z)} = (\\hat{p} - z)^{M_{E}} and \\frac{\\lambda{(M_{E},\\hat{p},z)}}{\\hat{p}} = \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} and \\int \\frac{\\lambda{(M_{E},\\hat{p},z)}}{\\hat{p}} d\\hat{p} = \\int \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} d\\hat{p} and \\int \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} d\\hat{p} + \\int \\frac{\\lambda{(M_{E},\\hat{p},z)}}{\\hat{p}} d\\hat{p} = 2 \\int \\frac{(\\hat{p} - z)^{M_{E}}}{\\hat{p}} d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('M_E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('z', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('M_E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('M_E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 3, "Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('M_E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Integer(2), Integral(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given n{(\\mathbf{g},F_{H})} = F_{H} + \\mathbf{g}, then obtain (- \\mathbf{g} + n{(\\mathbf{g},F_{H})}) n{(\\mathbf{g},F_{H})} = F_{H} n{(\\mathbf{g},F_{H})}", "derivation": "n{(\\mathbf{g},F_{H})} = F_{H} + \\mathbf{g} and - \\mathbf{g} + n{(\\mathbf{g},F_{H})} = F_{H} and (F_{H} + \\mathbf{g}) (- \\mathbf{g} + n{(\\mathbf{g},F_{H})}) = F_{H} (F_{H} + \\mathbf{g}) and (- \\mathbf{g} + n{(\\mathbf{g},F_{H})}) n{(\\mathbf{g},F_{H})} = F_{H} n{(\\mathbf{g},F_{H})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))"], [["times", 2, "Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True)))), Mul(Symbol('F_H', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True))), Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Function('n')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(P_{e},z)} = - z + e^{P_{e}}, then derive \\frac{\\partial}{\\partial z} \\dot{x}{(P_{e},z)} = -1, then derive (\\frac{\\partial^{2}}{\\partial z\\partial P_{e}} \\dot{x}{(P_{e},z)})^{z} = 0^{z}, then obtain - z + e^{P_{e}} + (\\frac{\\partial^{2}}{\\partial z\\partial P_{e}} \\dot{x}{(P_{e},z)})^{z} - 1 = 0^{z} - z + e^{P_{e}} - 1", "derivation": "\\dot{x}{(P_{e},z)} = - z + e^{P_{e}} and \\frac{\\partial}{\\partial z} \\dot{x}{(P_{e},z)} = \\frac{\\partial}{\\partial z} (- z + e^{P_{e}}) and \\frac{\\partial}{\\partial z} \\dot{x}{(P_{e},z)} = -1 and \\frac{\\partial^{2}}{\\partial P_{e}\\partial z} \\dot{x}{(P_{e},z)} = \\frac{d}{d P_{e}} (-1) and (\\frac{\\partial^{2}}{\\partial P_{e}\\partial z} \\dot{x}{(P_{e},z)})^{z} = (\\frac{d}{d P_{e}} (-1))^{z} and (\\frac{\\partial^{2}}{\\partial z\\partial P_{e}} \\dot{x}{(P_{e},z)})^{z} = 0^{z} and - z + e^{P_{e}} + (\\frac{\\partial^{2}}{\\partial z\\partial P_{e}} \\dot{x}{(P_{e},z)})^{z} - 1 = 0^{z} - z + e^{P_{e}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Integer(-1), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Integer(0), Symbol('z', commutative=True)))"], [["add", 6, "Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('P_e', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('P_e', commutative=True)), Pow(Derivative(Function('\\\\dot{x}')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Integer(-1)), Add(Pow(Integer(0), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('P_e', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\tilde{g}{(I)} = \\cos{(I)}, then obtain \\sin{(\\frac{d}{d I} (I \\tilde{g}{(I)})^{I})} = \\sin{(\\frac{d}{d I} (I \\cos{(I)})^{I})}", "derivation": "\\tilde{g}{(I)} = \\cos{(I)} and I \\tilde{g}{(I)} = I \\cos{(I)} and (I \\tilde{g}{(I)})^{I} = (I \\cos{(I)})^{I} and \\frac{d}{d I} (I \\tilde{g}{(I)})^{I} = \\frac{d}{d I} (I \\cos{(I)})^{I} and \\sin{(\\frac{d}{d I} (I \\tilde{g}{(I)})^{I})} = \\sin{(\\frac{d}{d I} (I \\cos{(I)})^{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\tilde{g}')(Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), cos(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Symbol('I', commutative=True), Function('\\\\tilde{g}')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Mul(Symbol('I', commutative=True), cos(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["differentiate", 3, "Symbol('I', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('I', commutative=True), Function('\\\\tilde{g}')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('I', commutative=True), cos(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["sin", 4], "Equality(sin(Derivative(Pow(Mul(Symbol('I', commutative=True), Function('\\\\tilde{g}')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), sin(Derivative(Pow(Mul(Symbol('I', commutative=True), cos(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(C_{2})} = \\log{(C_{2})}, then derive \\int \\operatorname{E_{x}}{(C_{2})} dC_{2} = C + C_{2} \\log{(C_{2})} - C_{2}, then obtain \\frac{d}{d C_{2}} \\frac{\\int \\log{(C_{2})} dC_{2}}{C_{2}} = \\frac{\\partial}{\\partial C_{2}} \\frac{C + C_{2} \\operatorname{E_{x}}{(C_{2})} - C_{2}}{C_{2}}", "derivation": "\\operatorname{E_{x}}{(C_{2})} = \\log{(C_{2})} and \\int \\operatorname{E_{x}}{(C_{2})} dC_{2} = \\int \\log{(C_{2})} dC_{2} and \\int \\operatorname{E_{x}}{(C_{2})} dC_{2} = C + C_{2} \\log{(C_{2})} - C_{2} and \\int \\log{(C_{2})} dC_{2} = C + C_{2} \\log{(C_{2})} - C_{2} and \\int \\log{(C_{2})} dC_{2} = C + C_{2} \\operatorname{E_{x}}{(C_{2})} - C_{2} and \\frac{\\int \\log{(C_{2})} dC_{2}}{C_{2}} = \\frac{C + C_{2} \\operatorname{E_{x}}{(C_{2})} - C_{2}}{C_{2}} and \\frac{d}{d C_{2}} \\frac{\\int \\log{(C_{2})} dC_{2}}{C_{2}} = \\frac{\\partial}{\\partial C_{2}} \\frac{C + C_{2} \\operatorname{E_{x}}{(C_{2})} - C_{2}}{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('C', commutative=True), Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('C', commutative=True), Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('C', commutative=True), Mul(Symbol('C_2', commutative=True), Function('E_x')(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True))))"], [["divide", 5, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Symbol('C_2', commutative=True), Function('E_x')(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True)))))"], [["differentiate", 6, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Symbol('C_2', commutative=True), Function('E_x')(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(S)} = e^{S}, then derive \\frac{d}{d S} Z{(S)} = e^{S}, then obtain (\\frac{d^{2}}{d S^{2}} Z{(S)} + \\frac{d^{3}}{d S^{3}} Z{(S)})^{S} = (2 \\frac{d^{3}}{d S^{3}} Z{(S)})^{S}", "derivation": "Z{(S)} = e^{S} and \\frac{d}{d S} Z{(S)} = \\frac{d}{d S} e^{S} and \\frac{d}{d S} Z{(S)} + \\frac{d}{d S} e^{S} = 2 \\frac{d}{d S} e^{S} and \\frac{d}{d S} Z{(S)} = e^{S} and \\frac{d}{d S} Z{(S)} + \\frac{d^{2}}{d S^{2}} Z{(S)} = 2 \\frac{d^{2}}{d S^{2}} Z{(S)} and \\frac{d}{d S} e^{S} + \\frac{d^{2}}{d S^{2}} e^{S} = 2 \\frac{d^{2}}{d S^{2}} e^{S} and \\frac{d^{2}}{d S^{2}} Z{(S)} + \\frac{d^{3}}{d S^{3}} Z{(S)} = 2 \\frac{d^{3}}{d S^{3}} Z{(S)} and (\\frac{d^{2}}{d S^{2}} Z{(S)} + \\frac{d^{3}}{d S^{3}} Z{(S)})^{S} = (2 \\frac{d^{3}}{d S^{3}} Z{(S)})^{S}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 2, "Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), exp(Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(3)))), Mul(Integer(2), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(3)))))"], [["power", 7, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(3)))), Symbol('S', commutative=True)), Pow(Mul(Integer(2), Derivative(Function('Z')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(3)))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})} = g_{\\varepsilon}^{c_{0}}, then obtain g_{\\varepsilon} (\\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})})^{g_{\\varepsilon}} = g_{\\varepsilon} (\\frac{\\partial}{\\partial g_{\\varepsilon}} g_{\\varepsilon}^{c_{0}})^{g_{\\varepsilon}}", "derivation": "\\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})} = g_{\\varepsilon}^{c_{0}} and \\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})} = \\frac{\\partial}{\\partial g_{\\varepsilon}} g_{\\varepsilon}^{c_{0}} and (\\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})})^{g_{\\varepsilon}} = (\\frac{\\partial}{\\partial g_{\\varepsilon}} g_{\\varepsilon}^{c_{0}})^{g_{\\varepsilon}} and g_{\\varepsilon} (\\frac{\\partial}{\\partial g_{\\varepsilon}} \\operatorname{f^{\\prime}}{(g_{\\varepsilon},c_{0})})^{g_{\\varepsilon}} = g_{\\varepsilon} (\\frac{\\partial}{\\partial g_{\\varepsilon}} g_{\\varepsilon}^{c_{0}})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 3, "Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Derivative(Function('f^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Derivative(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given B{(t_{1})} = t_{1}, then derive \\int B{(t_{1})} dt_{1} = \\hat{H}_l + \\frac{t_{1}^{2}}{2}, then obtain \\int t_{1} dt_{1} = \\hat{H}_l + \\frac{t_{1}^{2}}{2}", "derivation": "B{(t_{1})} = t_{1} and \\int B{(t_{1})} dt_{1} = \\int t_{1} dt_{1} and \\int B{(t_{1})} dt_{1} = \\hat{H}_l + \\frac{t_{1}^{2}}{2} and \\int t_{1} dt_{1} = \\hat{H}_l + \\frac{t_{1}^{2}}{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('B')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('B')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\varphi^*,\\theta_1)} = \\theta_1^{\\varphi^*}, then obtain \\int \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}^{\\varphi^*}{(\\varphi^*,\\theta_1)}}{\\theta_1} d\\varphi^* = \\int \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\varphi^*})^{\\varphi^*}}{\\theta_1} d\\varphi^*", "derivation": "\\mathbf{J}{(\\varphi^*,\\theta_1)} = \\theta_1^{\\varphi^*} and \\mathbf{J}^{\\varphi^*}{(\\varphi^*,\\theta_1)} = (\\theta_1^{\\varphi^*})^{\\varphi^*} and \\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}^{\\varphi^*}{(\\varphi^*,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\varphi^*})^{\\varphi^*} and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}^{\\varphi^*}{(\\varphi^*,\\theta_1)}}{\\theta_1} = \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\varphi^*})^{\\varphi^*}}{\\theta_1} and \\int \\frac{\\frac{\\partial}{\\partial \\theta_1} \\mathbf{J}^{\\varphi^*}{(\\varphi^*,\\theta_1)}}{\\theta_1} d\\varphi^* = \\int \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\varphi^*})^{\\varphi^*}}{\\theta_1} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and \\operatorname{E_{x}}{(\\mathbf{E})} = - \\cos{(\\mathbf{E})}, then obtain (\\mathbf{p}{(\\mathbf{E})} - \\cos{(\\mathbf{E})})^{\\mathbf{E}} - \\cos{(\\mathbf{E})} - 1 = 0^{\\mathbf{E}} - \\cos{(\\mathbf{E})} - 1", "derivation": "\\mathbf{p}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and \\mathbf{p}{(\\mathbf{E})} - \\cos{(\\mathbf{E})} = 0 and \\operatorname{E_{x}}{(\\mathbf{E})} = - \\cos{(\\mathbf{E})} and \\operatorname{E_{x}}{(\\mathbf{E})} - 1 = - \\cos{(\\mathbf{E})} - 1 and (\\mathbf{p}{(\\mathbf{E})} - \\cos{(\\mathbf{E})})^{\\mathbf{E}} = 0^{\\mathbf{E}} and (\\mathbf{p}{(\\mathbf{E})} - \\cos{(\\mathbf{E})})^{\\mathbf{E}} + \\operatorname{E_{x}}{(\\mathbf{E})} - 1 = 0^{\\mathbf{E}} + \\operatorname{E_{x}}{(\\mathbf{E})} - 1 and (\\mathbf{p}{(\\mathbf{E})} - \\cos{(\\mathbf{E})})^{\\mathbf{E}} - \\cos{(\\mathbf{E})} - 1 = 0^{\\mathbf{E}} - \\cos{(\\mathbf{E})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))), Integer(-1)))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 5, "Add(Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)), Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Add(Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)), Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))), Integer(-1)), Add(Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(t)} = e^{\\sin{(t)}}, then obtain (\\operatorname{z^{*}}^{2}{(t)} + \\operatorname{z^{*}}{(t)}) e^{- \\sin{(t)}} = (\\operatorname{z^{*}}^{2}{(t)} + e^{\\sin{(t)}}) e^{- \\sin{(t)}}", "derivation": "\\operatorname{z^{*}}{(t)} = e^{\\sin{(t)}} and \\operatorname{z^{*}}^{2}{(t)} = \\operatorname{z^{*}}{(t)} e^{\\sin{(t)}} and \\operatorname{z^{*}}^{2}{(t)} + \\operatorname{z^{*}}{(t)} = \\operatorname{z^{*}}^{2}{(t)} + e^{\\sin{(t)}} and \\operatorname{z^{*}}{(t)} e^{\\sin{(t)}} + \\operatorname{z^{*}}{(t)} = \\operatorname{z^{*}}{(t)} e^{\\sin{(t)}} + e^{\\sin{(t)}} and (\\operatorname{z^{*}}{(t)} e^{\\sin{(t)}} + \\operatorname{z^{*}}{(t)}) e^{- \\sin{(t)}} = (\\operatorname{z^{*}}{(t)} e^{\\sin{(t)}} + e^{\\sin{(t)}}) e^{- \\sin{(t)}} and (\\operatorname{z^{*}}^{2}{(t)} + \\operatorname{z^{*}}{(t)}) e^{- \\sin{(t)}} = (\\operatorname{z^{*}}^{2}{(t)} + e^{\\sin{(t)}}) e^{- \\sin{(t)}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True))))"], [["times", 1, "Function('z^*')(Symbol('t', commutative=True))"], "Equality(Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2)), Mul(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True)))))"], [["add", 1, "Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2)), Function('z^*')(Symbol('t', commutative=True))), Add(Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2)), exp(sin(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True)))), Function('z^*')(Symbol('t', commutative=True))), Add(Mul(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True)))), exp(sin(Symbol('t', commutative=True)))))"], [["divide", 4, "exp(sin(Symbol('t', commutative=True)))"], "Equality(Mul(Add(Mul(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True)))), Function('z^*')(Symbol('t', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True))))), Mul(Add(Mul(Function('z^*')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True)))), exp(sin(Symbol('t', commutative=True)))), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2)), Function('z^*')(Symbol('t', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True))))), Mul(Add(Pow(Function('z^*')(Symbol('t', commutative=True)), Integer(2)), exp(sin(Symbol('t', commutative=True)))), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\rho_{b}{(E_{x},\\Psi_{nl})} = \\frac{\\cos{(E_{x})}}{\\Psi_{nl}}, then obtain \\frac{\\partial}{\\partial E_{x}} - \\rho_{b}^{E_{x}}{(E_{x},\\Psi_{nl})} = \\frac{\\partial}{\\partial E_{x}} - (\\frac{\\cos{(E_{x})}}{\\Psi_{nl}})^{E_{x}}", "derivation": "\\rho_{b}{(E_{x},\\Psi_{nl})} = \\frac{\\cos{(E_{x})}}{\\Psi_{nl}} and \\rho_{b}^{E_{x}}{(E_{x},\\Psi_{nl})} = (\\frac{\\cos{(E_{x})}}{\\Psi_{nl}})^{E_{x}} and - \\rho_{b}^{E_{x}}{(E_{x},\\Psi_{nl})} = - (\\frac{\\cos{(E_{x})}}{\\Psi_{nl}})^{E_{x}} and \\frac{\\partial}{\\partial E_{x}} - \\rho_{b}^{E_{x}}{(E_{x},\\Psi_{nl})} = \\frac{\\partial}{\\partial E_{x}} - (\\frac{\\cos{(E_{x})}}{\\Psi_{nl}})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('E_x', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('E_x', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('E_x', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\rho_b')(Symbol('E_x', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('E_x', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('\\\\rho_b')(Symbol('E_x', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain (\\Psi{(V_{\\mathbf{E}})} - 1) (\\Psi{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} = (\\Psi{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} (\\sin{(V_{\\mathbf{E}})} - 1)", "derivation": "\\Psi{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\Psi{(V_{\\mathbf{E}})} - 1 = \\sin{(V_{\\mathbf{E}})} - 1 and (\\Psi{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} = (\\sin{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} and (\\Psi{(V_{\\mathbf{E}})} - 1) (\\sin{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} = (\\sin{(V_{\\mathbf{E}})} - 1) (\\sin{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} and (\\Psi{(V_{\\mathbf{E}})} - 1) (\\Psi{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} = (\\Psi{(V_{\\mathbf{E}})} - 1)^{V_{\\mathbf{E}}} (\\sin{(V_{\\mathbf{E}})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["times", 2, "Pow(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Pow(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Pow(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Pow(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Pow(Add(Function('\\\\Psi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}}, then derive \\operatorname{m_{s}}{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain \\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d^{2}}{d \\mathbf{P}^{2}} e^{\\mathbf{P}}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d^{2}}{d \\mathbf{P}^{2}} e^{\\mathbf{P}} and \\operatorname{m_{s}}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\operatorname{m_{s}}{(\\mathbf{P})} and \\operatorname{m_{s}}{(\\mathbf{P})} = \\frac{d^{2}}{d \\mathbf{P}^{2}} e^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 1], "Equality(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))))"]]}, {"prompt": "Given k{(\\lambda,\\theta,\\hat{H})} = (\\hat{H} + \\lambda)^{\\theta}, then obtain \\int (- \\theta + k{(\\lambda,\\theta,\\hat{H})}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} d\\theta = \\int (- \\theta + (\\hat{H} + \\lambda)^{\\theta}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} d\\theta", "derivation": "k{(\\lambda,\\theta,\\hat{H})} = (\\hat{H} + \\lambda)^{\\theta} and - \\theta + k{(\\lambda,\\theta,\\hat{H})} = - \\theta + (\\hat{H} + \\lambda)^{\\theta} and (- \\theta + k{(\\lambda,\\theta,\\hat{H})}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} = (- \\theta + (\\hat{H} + \\lambda)^{\\theta}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} and \\int (- \\theta + k{(\\lambda,\\theta,\\hat{H})}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} d\\theta = \\int (- \\theta + (\\hat{H} + \\lambda)^{\\theta}) (- \\theta + k{(\\lambda,\\theta,\\hat{H})})^{\\lambda} d\\theta", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["times", 2, "Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\lambda', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('k')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given g{(\\tilde{g}^*,\\Psi)} = \\Psi + \\tilde{g}^*, then derive \\int g{(\\tilde{g}^*,\\Psi)} d\\Psi = C + \\frac{\\Psi^{2}}{2} + \\Psi \\tilde{g}^*, then obtain - \\frac{d}{d r} e^{r} + \\iint g{(\\tilde{g}^*,\\Psi)} d\\Psi d\\Psi = - \\frac{d}{d r} e^{r} + \\int (C + \\frac{\\Psi^{2}}{2} + \\Psi \\tilde{g}^*) d\\Psi", "derivation": "g{(\\tilde{g}^*,\\Psi)} = \\Psi + \\tilde{g}^* and \\int g{(\\tilde{g}^*,\\Psi)} d\\Psi = \\int (\\Psi + \\tilde{g}^*) d\\Psi and \\int g{(\\tilde{g}^*,\\Psi)} d\\Psi = C + \\frac{\\Psi^{2}}{2} + \\Psi \\tilde{g}^* and \\iint g{(\\tilde{g}^*,\\Psi)} d\\Psi d\\Psi = \\int (C + \\frac{\\Psi^{2}}{2} + \\Psi \\tilde{g}^*) d\\Psi and - \\frac{d}{d r} e^{r} + \\iint g{(\\tilde{g}^*,\\Psi)} d\\Psi d\\Psi = - \\frac{d}{d r} e^{r} + \\int (C + \\frac{\\Psi^{2}}{2} + \\Psi \\tilde{g}^*) d\\Psi", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["minus", 4, "Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integral(Function('g')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integral(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\Psi_{nl},\\hbar)} = \\Psi_{nl} \\hbar, then obtain \\frac{\\hbar^{2} \\operatorname{f_{E}}^{4}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}^{4}} = \\hbar^{6}", "derivation": "\\operatorname{f_{E}}{(\\Psi_{nl},\\hbar)} = \\Psi_{nl} \\hbar and \\frac{\\operatorname{f_{E}}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}} = \\hbar and \\frac{\\hbar \\operatorname{f_{E}}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}} = \\hbar^{2} and \\frac{\\hbar^{2} \\operatorname{f_{E}}^{2}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}^{2}} = \\hbar^{4} and \\frac{\\hbar^{4} \\operatorname{f_{E}}^{2}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}^{2}} = \\hbar^{6} and \\frac{\\hbar^{2} \\operatorname{f_{E}}^{4}{(\\Psi_{nl},\\hbar)}}{\\Psi_{nl}^{4}} = \\hbar^{6}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))"], [["times", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True), Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Pow(Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Pow(Symbol('\\\\hbar', commutative=True), Integer(4)))"], [["times", 4, "Pow(Symbol('\\\\hbar', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Pow(Symbol('\\\\hbar', commutative=True), Integer(4)), Pow(Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Pow(Symbol('\\\\hbar', commutative=True), Integer(6)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-4)), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Pow(Function('f_E')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(4))), Pow(Symbol('\\\\hbar', commutative=True), Integer(6)))"]]}, {"prompt": "Given \\mathbf{P}{(P_{g},\\mathbf{J}_P)} = \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P, then obtain \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P - 1 + \\frac{\\mathbf{J}_P \\mathbf{P}{(P_{g},\\mathbf{J}_P)}}{P_{g}} = \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P - 1 + \\frac{\\mathbf{J}_P \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P}{P_{g}}", "derivation": "\\mathbf{P}{(P_{g},\\mathbf{J}_P)} = \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P and \\frac{\\mathbf{J}_P \\mathbf{P}{(P_{g},\\mathbf{J}_P)}}{P_{g}} = \\frac{\\mathbf{J}_P \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P}{P_{g}} and -1 + \\frac{\\mathbf{J}_P \\mathbf{P}{(P_{g},\\mathbf{J}_P)}}{P_{g}} = -1 + \\frac{\\mathbf{J}_P \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P}{P_{g}} and \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P - 1 + \\frac{\\mathbf{J}_P \\mathbf{P}{(P_{g},\\mathbf{J}_P)}}{P_{g}} = \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P - 1 + \\frac{\\mathbf{J}_P \\int \\frac{P_{g}}{\\mathbf{J}_P} d\\mathbf{J}_P}{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 1, "Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{P}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{P}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["add", 3, "Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{P}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True), Integral(Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(\\omega)} = \\frac{d}{d \\omega} e^{\\omega} and L{(\\omega)} = e^{\\omega}, then obtain \\int \\log{(\\frac{d}{d \\omega} L{(\\omega)})} d\\omega = \\int \\log{(\\frac{d}{d \\omega} e^{\\omega})} d\\omega", "derivation": "\\mu_{0}{(\\omega)} = \\frac{d}{d \\omega} e^{\\omega} and \\log{(\\mu_{0}{(\\omega)})} = \\log{(\\frac{d}{d \\omega} e^{\\omega})} and L{(\\omega)} = e^{\\omega} and \\int \\log{(\\mu_{0}{(\\omega)})} d\\omega = \\int \\log{(\\frac{d}{d \\omega} e^{\\omega})} d\\omega and \\mu_{0}{(\\omega)} = \\frac{d}{d \\omega} L{(\\omega)} and \\int \\log{(\\frac{d}{d \\omega} L{(\\omega)})} d\\omega = \\int \\log{(\\frac{d}{d \\omega} e^{\\omega})} d\\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True))), log(Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(log(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(log(Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True)), Derivative(Function('L')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(log(Derivative(Function('L')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(log(Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\chi)} = \\sin{(e^{\\chi})}, then derive \\int \\operatorname{z^{*}}{(\\chi)} d\\chi = \\mathbf{F} + \\operatorname{Si}{(e^{\\chi})}, then derive \\mathbf{v} + \\operatorname{Si}{(e^{\\chi})} = \\mathbf{F} + \\operatorname{Si}{(e^{\\chi})}, then obtain - 2 \\mathbf{F} (\\mathbf{v} + \\operatorname{Si}{(e^{\\chi})}) = - 2 \\mathbf{F} \\int \\operatorname{z^{*}}{(\\chi)} d\\chi", "derivation": "\\operatorname{z^{*}}{(\\chi)} = \\sin{(e^{\\chi})} and \\int \\operatorname{z^{*}}{(\\chi)} d\\chi = \\int \\sin{(e^{\\chi})} d\\chi and \\int \\operatorname{z^{*}}{(\\chi)} d\\chi = \\mathbf{F} + \\operatorname{Si}{(e^{\\chi})} and \\int \\sin{(e^{\\chi})} d\\chi = \\mathbf{F} + \\operatorname{Si}{(e^{\\chi})} and \\mathbf{v} + \\operatorname{Si}{(e^{\\chi})} = \\mathbf{F} + \\operatorname{Si}{(e^{\\chi})} and - 2 \\mathbf{F} (\\mathbf{v} + \\operatorname{Si}{(e^{\\chi})}) = - 2 \\mathbf{F} (\\mathbf{F} + \\operatorname{Si}{(e^{\\chi})}) and - 2 \\mathbf{F} (\\mathbf{v} + \\operatorname{Si}{(e^{\\chi})}) = - 2 \\mathbf{F} \\int \\operatorname{z^{*}}{(\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\chi', commutative=True)), sin(exp(Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(sin(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True)))))"], [["times", 5, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True), Add(Symbol('\\\\mathbf{v}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True))))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True), Add(Symbol('\\\\mathbf{v}', commutative=True), Si(exp(Symbol('\\\\chi', commutative=True))))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True), Integral(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given p{(\\chi)} = \\cos{(\\chi)} and i{(\\chi)} = \\chi p{(\\chi)}, then obtain \\chi p{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi + \\int i{(\\chi)} d\\chi = \\chi \\cos{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi + \\int i{(\\chi)} d\\chi", "derivation": "p{(\\chi)} = \\cos{(\\chi)} and \\chi p{(\\chi)} = \\chi \\cos{(\\chi)} and i{(\\chi)} = \\chi p{(\\chi)} and i{(\\chi)} = \\chi \\cos{(\\chi)} and \\int i{(\\chi)} d\\chi = \\int \\chi \\cos{(\\chi)} d\\chi and \\chi p{(\\chi)} \\int i{(\\chi)} d\\chi = \\chi \\cos{(\\chi)} \\int i{(\\chi)} d\\chi and \\chi p{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi = \\chi \\cos{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi and \\chi p{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi + \\int i{(\\chi)} d\\chi = \\chi \\cos{(\\chi)} \\int \\chi \\cos{(\\chi)} d\\chi + \\int i{(\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('p')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Function('p')(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('i')(Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('p')(Symbol('\\\\chi', commutative=True)), Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True)), Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('p')(Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["add", 7, "Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Function('p')(Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Integral(Function('i')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(E_{x},A_{1})} = \\log{(\\frac{E_{x}}{A_{1}})}, then derive \\frac{\\partial}{\\partial A_{1}} \\hat{x}{(E_{x},A_{1})} = - \\frac{1}{A_{1}}, then obtain \\frac{\\partial}{\\partial A_{1}} \\log{(\\frac{E_{x}}{A_{1}})} = - \\frac{1}{A_{1}}", "derivation": "\\hat{x}{(E_{x},A_{1})} = \\log{(\\frac{E_{x}}{A_{1}})} and \\frac{\\partial}{\\partial A_{1}} \\hat{x}{(E_{x},A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\log{(\\frac{E_{x}}{A_{1}})} and \\frac{\\partial}{\\partial A_{1}} \\hat{x}{(E_{x},A_{1})} = - \\frac{1}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\log{(\\frac{E_{x}}{A_{1}})} = - \\frac{1}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), log(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_x', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_x', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_x', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1))))"]]}, {"prompt": "Given E{(\\Psi,\\mathbf{M})} = \\Psi - \\mathbf{M}, then obtain - \\Psi (\\Psi - \\mathbf{M}) + E{(\\Psi,\\mathbf{M})} = - \\Psi (\\Psi - \\mathbf{M}) + \\Psi - \\mathbf{M}", "derivation": "E{(\\Psi,\\mathbf{M})} = \\Psi - \\mathbf{M} and \\Psi E{(\\Psi,\\mathbf{M})} = \\Psi (\\Psi - \\mathbf{M}) and - \\Psi E{(\\Psi,\\mathbf{M})} + E{(\\Psi,\\mathbf{M})} = - \\Psi E{(\\Psi,\\mathbf{M})} + \\Psi - \\mathbf{M} and - \\Psi (\\Psi - \\mathbf{M}) + E{(\\Psi,\\mathbf{M})} = - \\Psi (\\Psi - \\mathbf{M}) + \\Psi - \\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["minus", 1, "Mul(Symbol('\\\\Psi', commutative=True), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Function('E')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and S{(F_{c})} = \\cos{(F_{c})}, then obtain \\frac{2 \\mathbf{p}{(\\hat{H}_l)}}{S{(F_{c})}} = \\frac{\\mathbf{p}{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}}{S{(F_{c})}}", "derivation": "\\mathbf{p}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and 2 \\mathbf{p}{(\\hat{H}_l)} = \\mathbf{p}{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} and S{(F_{c})} = \\cos{(F_{c})} and \\frac{2 \\mathbf{p}{(\\hat{H}_l)}}{\\cos{(F_{c})}} = \\frac{\\mathbf{p}{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}}{\\cos{(F_{c})}} and \\frac{2 \\mathbf{p}{(\\hat{H}_l)}}{S{(F_{c})}} = \\frac{\\mathbf{p}{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}}{S{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))))"], ["get_premise", "Equality(Function('S')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["divide", 2, "cos(Symbol('F_c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('S')(Symbol('F_c', commutative=True)), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Add(Function('\\\\mathbf{p}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Pow(Function('S')(Symbol('F_c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda})}, then derive \\int \\varepsilon_{0}{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = C_{1} + \\sin{(\\hat{H}_{\\lambda})}, then obtain - C_{1} + \\varepsilon_{0}{(\\hat{H}_{\\lambda})} - \\sin{(\\hat{H}_{\\lambda})} = - C_{1} - \\sin{(\\hat{H}_{\\lambda})} + \\cos{(\\hat{H}_{\\lambda})}", "derivation": "\\varepsilon_{0}{(\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda})} and \\int \\varepsilon_{0}{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = \\int \\cos{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and \\int \\varepsilon_{0}{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = C_{1} + \\sin{(\\hat{H}_{\\lambda})} and \\varepsilon_{0}{(\\hat{H}_{\\lambda})} - \\int \\varepsilon_{0}{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = \\cos{(\\hat{H}_{\\lambda})} - \\int \\varepsilon_{0}{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and - C_{1} + \\varepsilon_{0}{(\\hat{H}_{\\lambda})} - \\sin{(\\hat{H}_{\\lambda})} = - C_{1} - \\sin{(\\hat{H}_{\\lambda})} + \\cos{(\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Integral(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Add(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(t)} = \\cos{(t)} and B{(t)} = \\cos{(t)}, then obtain \\dot{\\mathbf{r}}{(t)} - 2 \\cos{(t)} = - \\cos{(t)}", "derivation": "\\dot{\\mathbf{r}}{(t)} = \\cos{(t)} and f + \\dot{\\mathbf{r}}{(t)} = f + \\cos{(t)} and B{(t)} = \\cos{(t)} and f - B{(t)} + \\dot{\\mathbf{r}}{(t)} = f - B{(t)} + \\cos{(t)} and - B{(t)} + \\dot{\\mathbf{r}}{(t)} - \\cos{(t)} = - B{(t)} and \\dot{\\mathbf{r}}{(t)} - 2 \\cos{(t)} = - \\cos{(t)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t', commutative=True))), Add(Symbol('f', commutative=True), cos(Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["minus", 2, "Function('B')(Symbol('t', commutative=True))"], "Equality(Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('B')(Symbol('t', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t', commutative=True))), Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('B')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True))))"], [["minus", 4, "Add(Symbol('f', commutative=True), cos(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('B')(Symbol('t', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Mul(Integer(-1), Function('B')(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('t', commutative=True)))), Mul(Integer(-1), cos(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\phi_1)} = \\sin{(\\phi_1)}, then obtain \\frac{\\mathbf{J}^{2}{(\\phi_1)}}{\\sin^{2}{(\\phi_1)}} = 1", "derivation": "\\mathbf{J}{(\\phi_1)} = \\sin{(\\phi_1)} and \\frac{\\mathbf{J}{(\\phi_1)}}{\\sin{(\\phi_1)}} = 1 and \\sin{(\\phi_1)} = \\frac{\\sin^{2}{(\\phi_1)}}{\\mathbf{J}{(\\phi_1)}} and \\frac{\\mathbf{J}^{2}{(\\phi_1)}}{\\sin^{2}{(\\phi_1)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 1, "Mul(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1)))"], "Equality(sin(Symbol('\\\\phi_1', commutative=True)), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-2))), Integer(1))"]]}, {"prompt": "Given \\mathbf{J}_M{(c_{0},F_{x})} = \\frac{F_{x}}{c_{0}} and \\phi_{2}{(q,F_{c})} = \\frac{e^{F_{c}}}{q}, then obtain \\mathbf{J}_M{(c_{0},F_{x})} e^{F_{c}} = \\frac{F_{x} e^{F_{c}}}{c_{0}}", "derivation": "\\mathbf{J}_M{(c_{0},F_{x})} = \\frac{F_{x}}{c_{0}} and \\phi_{2}{(q,F_{c})} = \\frac{e^{F_{c}}}{q} and q \\mathbf{J}_M{(c_{0},F_{x})} = \\frac{F_{x} q}{c_{0}} and q \\mathbf{J}_M{(c_{0},F_{x})} \\phi_{2}{(q,F_{c})} = \\frac{F_{x} q \\phi_{2}{(q,F_{c})}}{c_{0}} and \\mathbf{J}_M{(c_{0},F_{x})} e^{F_{c}} = \\frac{F_{x} e^{F_{c}}}{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))))"], [["divide", 1, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('q', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["times", 3, "Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('F_c', commutative=True))"], "Equality(Mul(Symbol('q', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('F_x', commutative=True)), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('F_c', commutative=True))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('q', commutative=True), Function('\\\\phi_2')(Symbol('q', commutative=True), Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('F_x', commutative=True)), exp(Symbol('F_c', commutative=True))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{H})} = e^{\\sin{(\\mathbf{H})}}, then obtain (C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(\\operatorname{P_{g}}{(\\mathbf{H})})})^{C_{1}} = (C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(e^{\\sin{(\\mathbf{H})}})})^{C_{1}}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{H})} = e^{\\sin{(\\mathbf{H})}} and \\cos{(\\operatorname{P_{g}}{(\\mathbf{H})})} = \\cos{(e^{\\sin{(\\mathbf{H})}})} and C_{1} + \\cos{(\\operatorname{P_{g}}{(\\mathbf{H})})} = C_{1} + \\cos{(e^{\\sin{(\\mathbf{H})}})} and C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(\\operatorname{P_{g}}{(\\mathbf{H})})} = C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(e^{\\sin{(\\mathbf{H})}})} and (C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(\\operatorname{P_{g}}{(\\mathbf{H})})})^{C_{1}} = (C_{1} + r{(\\mathbf{P},C_{1},F_{H})} + \\cos{(e^{\\sin{(\\mathbf{H})}})})^{C_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{H}', commutative=True)), exp(sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["cos", 1], "Equality(cos(Function('P_g')(Symbol('\\\\mathbf{H}', commutative=True))), cos(exp(sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["add", 2, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), cos(Function('P_g')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('C_1', commutative=True), cos(exp(sin(Symbol('\\\\mathbf{H}', commutative=True))))))"], [["add", 3, "Function('r')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Function('r')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('F_H', commutative=True)), cos(Function('P_g')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('C_1', commutative=True), Function('r')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('F_H', commutative=True)), cos(exp(sin(Symbol('\\\\mathbf{H}', commutative=True))))))"], [["power", 4, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Symbol('C_1', commutative=True), Function('r')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('F_H', commutative=True)), cos(Function('P_g')(Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('C_1', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), Function('r')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('C_1', commutative=True), Symbol('F_H', commutative=True)), cos(exp(sin(Symbol('\\\\mathbf{H}', commutative=True))))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\Psi{(B,\\varphi)} = B + \\varphi, then obtain \\frac{\\partial}{\\partial \\varphi} (\\int 3 \\Psi{(B,\\varphi)} dB)^{B} = \\frac{\\partial}{\\partial \\varphi} (\\int (2 B + 2 \\varphi + \\Psi{(B,\\varphi)}) dB)^{B}", "derivation": "\\Psi{(B,\\varphi)} = B + \\varphi and B + \\varphi + 2 \\Psi{(B,\\varphi)} = 2 B + 2 \\varphi + \\Psi{(B,\\varphi)} and 3 \\Psi{(B,\\varphi)} = 2 B + 2 \\varphi + \\Psi{(B,\\varphi)} and \\int 3 \\Psi{(B,\\varphi)} dB = \\int (2 B + 2 \\varphi + \\Psi{(B,\\varphi)}) dB and (\\int 3 \\Psi{(B,\\varphi)} dB)^{B} = (\\int (2 B + 2 \\varphi + \\Psi{(B,\\varphi)}) dB)^{B} and \\frac{\\partial}{\\partial \\varphi} (\\int 3 \\Psi{(B,\\varphi)} dB)^{B} = \\frac{\\partial}{\\partial \\varphi} (\\int (2 B + 2 \\varphi + \\Psi{(B,\\varphi)}) dB)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Add(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(3), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Integer(3), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["power", 4, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(3), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Integral(Mul(Integer(3), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Integral(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\Psi')(Symbol('B', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(m_{s})} = \\sin{(m_{s})}, then derive \\int i{(m_{s})} dm_{s} = \\mathbf{f} - \\cos{(m_{s})}, then derive E_{\\lambda} - \\cos{(m_{s})} = \\mathbf{f} - \\cos{(m_{s})}, then obtain \\int \\sin{(m_{s})} dm_{s} = E_{\\lambda} - \\cos{(m_{s})}", "derivation": "i{(m_{s})} = \\sin{(m_{s})} and \\int i{(m_{s})} dm_{s} = \\int \\sin{(m_{s})} dm_{s} and \\int i{(m_{s})} dm_{s} = \\mathbf{f} - \\cos{(m_{s})} and \\int \\sin{(m_{s})} dm_{s} = \\mathbf{f} - \\cos{(m_{s})} and E_{\\lambda} - \\cos{(m_{s})} = \\mathbf{f} - \\cos{(m_{s})} and \\int \\sin{(m_{s})} dm_{s} = E_{\\lambda} - \\cos{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('i')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('i')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given b{(C_{1},\\varepsilon_0)} = e^{C_{1} + \\varepsilon_0}, then derive \\int b{(C_{1},\\varepsilon_0)} d\\varepsilon_0 = v_{y} + e^{C_{1} + \\varepsilon_0}, then obtain \\mathbf{J} + e^{C_{1} + \\varepsilon_0} = v_{y} + b{(C_{1},\\varepsilon_0)}", "derivation": "b{(C_{1},\\varepsilon_0)} = e^{C_{1} + \\varepsilon_0} and \\int b{(C_{1},\\varepsilon_0)} d\\varepsilon_0 = \\int e^{C_{1} + \\varepsilon_0} d\\varepsilon_0 and \\int b{(C_{1},\\varepsilon_0)} d\\varepsilon_0 = v_{y} + e^{C_{1} + \\varepsilon_0} and \\int e^{C_{1} + \\varepsilon_0} d\\varepsilon_0 = v_{y} + e^{C_{1} + \\varepsilon_0} and \\int e^{C_{1} + \\varepsilon_0} d\\varepsilon_0 = v_{y} + b{(C_{1},\\varepsilon_0)} and \\mathbf{J} + e^{C_{1} + \\varepsilon_0} = v_{y} + b{(C_{1},\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('b')(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('v_y', commutative=True), exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('v_y', commutative=True), exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('v_y', commutative=True), Function('b')(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), exp(Add(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Add(Symbol('v_y', commutative=True), Function('b')(Symbol('C_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(h,T)} = T - h and \\psi{(h,T)} = \\frac{T - h}{T}, then obtain - \\frac{T - h}{T} = - \\frac{\\rho_{b}{(h,T)}}{T}", "derivation": "\\rho_{b}{(h,T)} = T - h and \\frac{\\rho_{b}{(h,T)}}{T} = \\frac{T - h}{T} and \\psi{(h,T)} = \\frac{T - h}{T} and \\psi{(h,T)} = \\frac{\\rho_{b}{(h,T)}}{T} and - \\psi{(h,T)} = - \\frac{\\rho_{b}{(h,T)}}{T} and - \\frac{T - h}{T} = - \\frac{\\rho_{b}{(h,T)}}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\psi')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('h', commutative=True), Symbol('T', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\psi')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('h', commutative=True), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('h', commutative=True), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\varphi^*,\\pi)} = \\pi \\varphi^*, then obtain \\int \\frac{2 \\varphi^* + 2 \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} d\\varphi^* = \\int \\frac{\\pi \\varphi^* + 2 \\varphi^* + \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} d\\varphi^*", "derivation": "\\mathbf{s}{(\\varphi^*,\\pi)} = \\pi \\varphi^* and \\varphi^* + \\mathbf{s}{(\\varphi^*,\\pi)} = \\pi \\varphi^* + \\varphi^* and 2 \\varphi^* + 2 \\mathbf{s}{(\\varphi^*,\\pi)} = \\pi \\varphi^* + 2 \\varphi^* + \\mathbf{s}{(\\varphi^*,\\pi)} and \\frac{2 \\varphi^* + 2 \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} = \\frac{\\pi \\varphi^* + 2 \\varphi^* + \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} and \\int \\frac{2 \\varphi^* + 2 \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} d\\varphi^* = \\int \\frac{\\pi \\varphi^* + 2 \\varphi^* + \\mathbf{s}{(\\varphi^*,\\pi)}}{(\\pi \\varphi^* + \\varphi^*) \\mathbf{s}{(\\varphi^*,\\pi)}} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 2, "Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["divide", 3, "Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)))), Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)))), Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Pow(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(F_{g},n)} = e^{\\frac{n}{F_{g}}} and \\operatorname{A_{y}}{(F_{g},n)} = \\int (\\operatorname{A_{2}}{(F_{g},n)} + 1) dF_{g}, then obtain \\operatorname{A_{y}}^{F_{g}}{(F_{g},n)} = (\\int (e^{\\frac{n}{F_{g}}} + 1) dF_{g})^{F_{g}}", "derivation": "\\operatorname{A_{2}}{(F_{g},n)} = e^{\\frac{n}{F_{g}}} and \\operatorname{A_{y}}{(F_{g},n)} = \\int (\\operatorname{A_{2}}{(F_{g},n)} + 1) dF_{g} and \\operatorname{A_{y}}{(F_{g},n)} = \\int (e^{\\frac{n}{F_{g}}} + 1) dF_{g} and \\operatorname{A_{y}}^{F_{g}}{(F_{g},n)} = (\\int (e^{\\frac{n}{F_{g}}} + 1) dF_{g})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), exp(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Integral(Add(Function('A_2')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_y')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Integral(Add(exp(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Integer(1)), Tuple(Symbol('F_g', commutative=True))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('F_g', commutative=True), Symbol('n', commutative=True)), Symbol('F_g', commutative=True)), Pow(Integral(Add(exp(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Integer(1)), Tuple(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given C{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) and \\rho_{f}{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = \\int C{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} dF_{H}, then obtain - V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) + \\rho_{f}{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = - V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) + \\int V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) dF_{H}", "derivation": "C{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) and \\int C{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} dF_{H} = \\int V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) dF_{H} and \\rho_{f}{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = \\int C{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} dF_{H} and \\rho_{f}{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = \\int V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) dF_{H} and - V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) + \\rho_{f}{(\\Psi_{nl},F_{H},V_{\\mathbf{E}})} = - V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) + \\int V_{\\mathbf{E}} (F_{H} + \\Psi_{nl}) dF_{H}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Function('C')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\rho_f')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["minus", 4, "Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(v_{x},\\mathbf{J}_P)} = \\mathbf{J}_P + v_{x} and \\mathbf{E}{(\\mathbf{J}_P)} = - \\mathbf{J}_P, then obtain (v_{x} (\\mathbf{J}_P + \\mathbf{E}{(\\mathbf{J}_P)}))^{v_{x}} = 0^{v_{x}}", "derivation": "\\mathbf{P}{(v_{x},\\mathbf{J}_P)} = \\mathbf{J}_P + v_{x} and - v_{x} + \\mathbf{P}{(v_{x},\\mathbf{J}_P)} = \\mathbf{J}_P and - \\mathbf{J}_P - v_{x} + \\mathbf{P}{(v_{x},\\mathbf{J}_P)} = 0 and v_{x} (- \\mathbf{J}_P - v_{x} + \\mathbf{P}{(v_{x},\\mathbf{J}_P)}) = 0 and \\mathbf{E}{(\\mathbf{J}_P)} = - \\mathbf{J}_P and (v_{x} (- \\mathbf{J}_P - v_{x} + \\mathbf{P}{(v_{x},\\mathbf{J}_P)}))^{v_{x}} = 0^{v_{x}} and (v_{x} (- v_{x} + \\mathbf{E}{(\\mathbf{J}_P)} + \\mathbf{P}{(v_{x},\\mathbf{J}_P)}))^{v_{x}} = 0^{v_{x}} and (v_{x} (\\mathbf{J}_P + \\mathbf{E}{(\\mathbf{J}_P)}))^{v_{x}} = 0^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v_x', commutative=True)))"], [["minus", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["minus", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(0))"], [["times", 3, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 4, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Integer(0), Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\mathbf{P}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Integer(0), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Pow(Mul(Symbol('v_x', commutative=True), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Integer(0), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(A_{y},M)} = \\frac{A_{y}}{M} and \\operatorname{v_{z}}{(A_{y},M)} = \\frac{A_{y}}{M}, then obtain 0 = \\operatorname{m_{s}}{(A_{y},M)} - \\operatorname{v_{z}}{(A_{y},M)}", "derivation": "\\operatorname{m_{s}}{(A_{y},M)} = \\frac{A_{y}}{M} and \\operatorname{v_{z}}{(A_{y},M)} = \\frac{A_{y}}{M} and 0 = \\frac{A_{y}}{M} - \\operatorname{v_{z}}{(A_{y},M)} and 0 = \\operatorname{m_{s}}{(A_{y},M)} - \\operatorname{v_{z}}{(A_{y},M)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["minus", 2, "Function('v_z')(Symbol('A_y', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Mul(Integer(-1), Function('v_z')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(0), Add(Function('m_s')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\nabla,q)} = \\frac{\\nabla}{q}, then obtain \\operatorname{C_{2}}{(\\nabla,q)} (\\int \\frac{\\nabla}{q} d\\nabla)^{q} = \\frac{\\nabla (\\int \\frac{\\nabla}{q} d\\nabla)^{q}}{q}", "derivation": "\\operatorname{C_{2}}{(\\nabla,q)} = \\frac{\\nabla}{q} and \\int \\operatorname{C_{2}}{(\\nabla,q)} d\\nabla = \\int \\frac{\\nabla}{q} d\\nabla and (\\int \\operatorname{C_{2}}{(\\nabla,q)} d\\nabla)^{q} = (\\int \\frac{\\nabla}{q} d\\nabla)^{q} and \\operatorname{C_{2}}{(\\nabla,q)} (\\int \\operatorname{C_{2}}{(\\nabla,q)} d\\nabla)^{q} = \\frac{\\nabla (\\int \\operatorname{C_{2}}{(\\nabla,q)} d\\nabla)^{q}}{q} and \\operatorname{C_{2}}{(\\nabla,q)} (\\int \\frac{\\nabla}{q} d\\nabla)^{q} = \\frac{\\nabla (\\int \\frac{\\nabla}{q} d\\nabla)^{q}}{q}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True)), Pow(Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True)))"], [["times", 1, "Pow(Integral(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Pow(Integral(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Integral(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('C_2')(Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Pow(Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(t)} = \\cos{(t)}, then obtain \\frac{1}{2} - 2 \\operatorname{t_{1}}{(t)} = - 2 \\operatorname{t_{1}}{(t)} + \\frac{\\cos{(t)}}{2 \\operatorname{t_{1}}{(t)}}", "derivation": "\\operatorname{t_{1}}{(t)} = \\cos{(t)} and 2 \\operatorname{t_{1}}{(t)} = \\operatorname{t_{1}}{(t)} + \\cos{(t)} and \\frac{\\operatorname{t_{1}}{(t)}}{\\operatorname{t_{1}}{(t)} + \\cos{(t)}} = \\frac{\\cos{(t)}}{\\operatorname{t_{1}}{(t)} + \\cos{(t)}} and - \\operatorname{t_{1}}{(t)} - \\cos{(t)} + \\frac{\\operatorname{t_{1}}{(t)}}{\\operatorname{t_{1}}{(t)} + \\cos{(t)}} = - \\operatorname{t_{1}}{(t)} - \\cos{(t)} + \\frac{\\cos{(t)}}{\\operatorname{t_{1}}{(t)} + \\cos{(t)}} and \\frac{1}{2} - 2 \\operatorname{t_{1}}{(t)} = - 2 \\operatorname{t_{1}}{(t)} + \\frac{\\cos{(t)}}{2 \\operatorname{t_{1}}{(t)}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["add", 1, "Function('t_1')(Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('t_1')(Symbol('t', commutative=True))), Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))))"], [["divide", 1, "Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], "Equality(Mul(Pow(Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Integer(-1)), Function('t_1')(Symbol('t', commutative=True))), Mul(Pow(Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Integer(-1)), cos(Symbol('t', commutative=True))))"], [["minus", 3, "Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('t_1')(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(Pow(Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Integer(-1)), Function('t_1')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('t_1')(Symbol('t', commutative=True))), Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(Pow(Add(Function('t_1')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Integer(-1)), cos(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Rational(1, 2), Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('t', commutative=True))), Mul(Rational(1, 2), Pow(Function('t_1')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given Q{(h,\\mathbf{r})} = \\mathbf{r} h and u{(\\mathbf{r})} = \\mathbf{r}, then obtain - \\frac{- \\mathbf{r} h + u{(\\mathbf{r})}}{Q{(h,\\mathbf{r})}} = - \\frac{- \\mathbf{r} h + \\mathbf{r}}{Q{(h,\\mathbf{r})}}", "derivation": "Q{(h,\\mathbf{r})} = \\mathbf{r} h and u{(\\mathbf{r})} = \\mathbf{r} and - Q{(h,\\mathbf{r})} + u{(\\mathbf{r})} = \\mathbf{r} - Q{(h,\\mathbf{r})} and - \\mathbf{r} h + u{(\\mathbf{r})} = - \\mathbf{r} h + \\mathbf{r} and - \\frac{- \\mathbf{r} h + u{(\\mathbf{r})}}{Q{(h,\\mathbf{r})}} = - \\frac{- \\mathbf{r} h + \\mathbf{r}}{Q{(h,\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["minus", 2, "Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Function('u')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Function('u')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Function('u')(Symbol('\\\\mathbf{r}', commutative=True))), Pow(Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('Q')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given L{(F_{N})} = e^{F_{N}}, then obtain - F_{N} L^{F_{N}}{(F_{N})} + L^{F_{N}}{(F_{N})} = - F_{N} L^{F_{N}}{(F_{N})} + (e^{F_{N}})^{F_{N}}", "derivation": "L{(F_{N})} = e^{F_{N}} and L^{F_{N}}{(F_{N})} = (e^{F_{N}})^{F_{N}} and F_{N} L^{F_{N}}{(F_{N})} = F_{N} (e^{F_{N}})^{F_{N}} and - F_{N} (e^{F_{N}})^{F_{N}} + L^{F_{N}}{(F_{N})} = - F_{N} (e^{F_{N}})^{F_{N}} + (e^{F_{N}})^{F_{N}} and - F_{N} L^{F_{N}}{(F_{N})} + L^{F_{N}}{(F_{N})} = - F_{N} L^{F_{N}}{(F_{N})} + (e^{F_{N}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["times", 2, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"], [["minus", 2, "Mul(Symbol('F_N', commutative=True), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), Pow(Function('L')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Pow(exp(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\rho,\\varphi)} = - \\varphi + \\log{(\\rho)} and \\mu{(\\rho)} = \\log{(\\rho)}, then derive \\int \\operatorname{A_{x}}{(\\rho,\\varphi)} d\\varphi = - \\frac{\\varphi^{2}}{2} + \\varphi \\log{(\\rho)} + f^{\\prime}, then obtain \\int (- \\varphi + \\mu{(\\rho)}) d\\varphi = - \\frac{\\varphi^{2}}{2} + \\varphi \\log{(\\rho)} + f^{\\prime}", "derivation": "\\operatorname{A_{x}}{(\\rho,\\varphi)} = - \\varphi + \\log{(\\rho)} and \\mu{(\\rho)} = \\log{(\\rho)} and \\int \\operatorname{A_{x}}{(\\rho,\\varphi)} d\\varphi = \\int (- \\varphi + \\log{(\\rho)}) d\\varphi and \\int \\operatorname{A_{x}}{(\\rho,\\varphi)} d\\varphi = - \\frac{\\varphi^{2}}{2} + \\varphi \\log{(\\rho)} + f^{\\prime} and \\int \\operatorname{A_{x}}{(\\rho,\\varphi)} d\\varphi = \\int (- \\varphi + \\mu{(\\rho)}) d\\varphi and \\int (- \\varphi + \\mu{(\\rho)}) d\\varphi = - \\frac{\\varphi^{2}}{2} + \\varphi \\log{(\\rho)} + f^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\rho', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('A_x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\rho', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Function('A_x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\mu')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\mu')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\rho', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\phi_2,\\mathbf{J})} = \\mathbf{J} + \\phi_2, then derive \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,\\mathbf{J})} = 1, then obtain \\frac{\\partial}{\\partial \\phi_2} (\\mathbf{J} + \\phi_2) = 1", "derivation": "\\mathbf{g}{(\\phi_2,\\mathbf{J})} = \\mathbf{J} + \\phi_2 and \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,\\mathbf{J})} = \\frac{\\partial}{\\partial \\phi_2} (\\mathbf{J} + \\phi_2) and \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,\\mathbf{J})} = 1 and \\frac{\\partial}{\\partial \\phi_2} (\\mathbf{J} + \\phi_2) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\rho_{b}{(H,f_{E})} = \\sin{(H + f_{E})} and Z{(H,f_{E})} = - H + \\cos{(\\rho_{b}{(H,f_{E})})}, then obtain \\frac{\\partial^{2}}{\\partial H^{2}} Z{(H,f_{E})} = \\frac{\\partial^{2}}{\\partial H^{2}} (- H + \\cos{(\\rho_{b}{(H,f_{E})})})", "derivation": "\\rho_{b}{(H,f_{E})} = \\sin{(H + f_{E})} and \\cos{(\\rho_{b}{(H,f_{E})})} = \\cos{(\\sin{(H + f_{E})})} and - H + \\cos{(\\rho_{b}{(H,f_{E})})} = - H + \\cos{(\\sin{(H + f_{E})})} and Z{(H,f_{E})} = - H + \\cos{(\\rho_{b}{(H,f_{E})})} and Z{(H,f_{E})} = - H + \\cos{(\\sin{(H + f_{E})})} and \\frac{\\partial}{\\partial H} Z{(H,f_{E})} = \\frac{\\partial}{\\partial H} (- H + \\cos{(\\sin{(H + f_{E})})}) and \\frac{\\partial^{2}}{\\partial H^{2}} Z{(H,f_{E})} = \\frac{\\partial^{2}}{\\partial H^{2}} (- H + \\cos{(\\sin{(H + f_{E})})}) and \\frac{\\partial^{2}}{\\partial H^{2}} Z{(H,f_{E})} = \\frac{\\partial^{2}}{\\partial H^{2}} (- H + \\cos{(\\rho_{b}{(H,f_{E})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\rho_b')(Symbol('H', commutative=True), Symbol('f_E', commutative=True))), cos(sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True)))))"], [["minus", 2, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(Function('\\\\rho_b')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True))))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(Function('\\\\rho_b')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('Z')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True))))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(sin(Add(Symbol('H', commutative=True), Symbol('f_E', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(Function('Z')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(Function('\\\\rho_b')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} = e^{g^{\\prime}_{\\varepsilon}}, then derive \\int \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = L_{\\varepsilon} + e^{g^{\\prime}_{\\varepsilon}}, then obtain \\int \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = L_{\\varepsilon} + \\mathbf{v}{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\mathbf{v}{(g^{\\prime}_{\\varepsilon})} = e^{g^{\\prime}_{\\varepsilon}} and \\int \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int e^{g^{\\prime}_{\\varepsilon}} dg^{\\prime}_{\\varepsilon} and \\int \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = L_{\\varepsilon} + e^{g^{\\prime}_{\\varepsilon}} and \\int \\mathbf{v}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = L_{\\varepsilon} + \\mathbf{v}{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(s)} = \\sin{(s)}, then obtain (\\frac{d}{d s} \\Psi_{nl}{(s)})^{s} (\\frac{d}{d s} \\sin{(s)})^{s} = (\\frac{d}{d s} \\sin{(s)})^{2 s}", "derivation": "\\Psi_{nl}{(s)} = \\sin{(s)} and \\frac{d}{d s} \\Psi_{nl}{(s)} = \\frac{d}{d s} \\sin{(s)} and (\\frac{d}{d s} \\Psi_{nl}{(s)})^{s} = (\\frac{d}{d s} \\sin{(s)})^{s} and (\\frac{d}{d s} \\Psi_{nl}{(s)})^{s} (\\frac{d}{d s} \\sin{(s)})^{s} = (\\frac{d}{d s} \\sin{(s)})^{2 s}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["times", 3, "Pow(Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))), Pow(Derivative(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Integer(2), Symbol('s', commutative=True))))"]]}, {"prompt": "Given b{(J,\\mathbf{M})} = \\mathbf{M}^{J}, then obtain (- \\int \\mathbf{M} dJ)^{\\mathbf{M}} = (- \\int \\frac{\\mathbf{M} \\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} dJ)^{\\mathbf{M}}", "derivation": "b{(J,\\mathbf{M})} = \\mathbf{M}^{J} and 1 = \\frac{\\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} and \\mathbf{M} = \\frac{\\mathbf{M} \\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} and \\int \\mathbf{M} dJ = \\int \\frac{\\mathbf{M} \\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} dJ and - \\int \\mathbf{M} dJ = - \\int \\frac{\\mathbf{M} \\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} dJ and (- \\int \\mathbf{M} dJ)^{\\mathbf{M}} = (- \\int \\frac{\\mathbf{M} \\mathbf{M}^{J}}{b{(J,\\mathbf{M})}} dJ)^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)))"], [["divide", 1, "Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)), Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"], [["times", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Symbol('\\\\mathbf{M}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)), Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)), Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('J', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)), Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('J', commutative=True)), Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(f^{*},\\lambda)} = \\cos{(\\lambda^{f^{*}})} and \\mathbf{F}{(v_{2},\\tilde{g}^*)} = \\cos{(\\tilde{g}^* - v_{2})}, then obtain \\frac{\\mathbf{F}{(v_{2},\\tilde{g}^*)}}{\\mathbf{J}{(f^{*},\\lambda)}} = \\frac{\\cos{(\\tilde{g}^* - v_{2})}}{\\mathbf{J}{(f^{*},\\lambda)}}", "derivation": "\\mathbf{J}{(f^{*},\\lambda)} = \\cos{(\\lambda^{f^{*}})} and \\mathbf{F}{(v_{2},\\tilde{g}^*)} = \\cos{(\\tilde{g}^* - v_{2})} and \\frac{\\mathbf{F}{(v_{2},\\tilde{g}^*)}}{\\cos{(\\lambda^{f^{*}})}} = \\frac{\\cos{(\\tilde{g}^* - v_{2})}}{\\cos{(\\lambda^{f^{*}})}} and \\frac{\\mathbf{F}{(v_{2},\\tilde{g}^*)}}{\\mathbf{J}{(f^{*},\\lambda)}} = \\frac{\\cos{(\\tilde{g}^* - v_{2})}}{\\mathbf{J}{(f^{*},\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('f^*', commutative=True), Symbol('\\\\lambda', commutative=True)), cos(Pow(Symbol('\\\\lambda', commutative=True), Symbol('f^*', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["divide", 2, "cos(Pow(Symbol('\\\\lambda', commutative=True), Symbol('f^*', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(cos(Pow(Symbol('\\\\lambda', commutative=True), Symbol('f^*', commutative=True))), Integer(-1))), Mul(Pow(cos(Pow(Symbol('\\\\lambda', commutative=True), Symbol('f^*', commutative=True))), Integer(-1)), cos(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('f^*', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('f^*', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\hat{x}_0{(\\sigma_x)} = \\frac{d}{d \\sigma_x} - \\sigma_x, then obtain \\hat{x}_0{(\\sigma_x)} = \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{r}{(\\sigma_x)} - \\log{(\\sigma_x)})", "derivation": "\\mathbf{r}{(\\sigma_x)} = \\log{(\\sigma_x)} and - \\sigma_x + \\mathbf{r}{(\\sigma_x)} - \\log{(\\sigma_x)} = - \\sigma_x and \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{r}{(\\sigma_x)} - \\log{(\\sigma_x)}) = \\frac{d}{d \\sigma_x} - \\sigma_x and \\hat{x}_0{(\\sigma_x)} = \\frac{d}{d \\sigma_x} - \\sigma_x and \\hat{x}_0{(\\sigma_x)} = \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{r}{(\\sigma_x)} - \\log{(\\sigma_x)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{r}')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{r}')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{r}')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(\\mathbf{p},P_{g})} = \\mathbf{p}^{P_{g}}, then obtain (- \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})})^{P_{g}} + 2 J{(\\mathbf{p},P_{g})} = 0^{P_{g}} + 2 J{(\\mathbf{p},P_{g})}", "derivation": "J{(\\mathbf{p},P_{g})} = \\mathbf{p}^{P_{g}} and - \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})} = 0 and 2 J{(\\mathbf{p},P_{g})} = \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})} and (- \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})})^{P_{g}} = 0^{P_{g}} and \\mathbf{p}^{P_{g}} + (- \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})})^{P_{g}} + J{(\\mathbf{p},P_{g})} = 0^{P_{g}} + \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})} and (- \\mathbf{p}^{P_{g}} + J{(\\mathbf{p},P_{g})})^{P_{g}} + 2 J{(\\mathbf{p},P_{g})} = 0^{P_{g}} + 2 J{(\\mathbf{p},P_{g})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Integer(0))"], [["add", 1, "Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(2), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Add(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Integer(0), Symbol('P_g', commutative=True)))"], [["add", 4, "Add(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)))"], "Equality(Add(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Add(Pow(Integer(0), Symbol('P_g', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Mul(Integer(2), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)))), Add(Pow(Integer(0), Symbol('P_g', commutative=True)), Mul(Integer(2), Function('J')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(n_{1})} = e^{n_{1}}, then derive e^{n_{1}} \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = (\\mathbf{J}_P + e^{n_{1}}) e^{n_{1}}, then obtain e^{n_{1}} \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} - \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = (\\mathbf{J}_P + e^{n_{1}}) e^{n_{1}} - \\int \\operatorname{P_{e}}{(n_{1})} dn_{1}", "derivation": "\\operatorname{P_{e}}{(n_{1})} = e^{n_{1}} and \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = \\int e^{n_{1}} dn_{1} and e^{n_{1}} \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = e^{n_{1}} \\int e^{n_{1}} dn_{1} and e^{n_{1}} \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = (\\mathbf{J}_P + e^{n_{1}}) e^{n_{1}} and e^{n_{1}} \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} - \\int \\operatorname{P_{e}}{(n_{1})} dn_{1} = (\\mathbf{J}_P + e^{n_{1}}) e^{n_{1}} - \\int \\operatorname{P_{e}}{(n_{1})} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["times", 2, "exp(Symbol('n_1', commutative=True))"], "Equality(Mul(exp(Symbol('n_1', commutative=True)), Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(exp(Symbol('n_1', commutative=True)), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(exp(Symbol('n_1', commutative=True)), Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('n_1', commutative=True))), exp(Symbol('n_1', commutative=True))))"], [["minus", 4, "Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))"], "Equality(Add(Mul(exp(Symbol('n_1', commutative=True)), Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(Integer(-1), Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))), Add(Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('n_1', commutative=True))), exp(Symbol('n_1', commutative=True))), Mul(Integer(-1), Integral(Function('P_e')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))))"]]}, {"prompt": "Given \\theta{(\\rho_b,E_{\\lambda})} = E_{\\lambda} + \\rho_b, then obtain (E_{\\lambda} + 7 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})}) \\theta{(\\rho_b,E_{\\lambda})} = (2 E_{\\lambda} + 8 \\rho_b) \\theta{(\\rho_b,E_{\\lambda})}", "derivation": "\\theta{(\\rho_b,E_{\\lambda})} = E_{\\lambda} + \\rho_b and \\rho_b + \\theta{(\\rho_b,E_{\\lambda})} = E_{\\lambda} + 2 \\rho_b and E_{\\lambda} + 2 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})} = 2 E_{\\lambda} + 3 \\rho_b and E_{\\lambda} + 5 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})} = 2 E_{\\lambda} + 6 \\rho_b and E_{\\lambda} + 7 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})} = 2 E_{\\lambda} + 8 \\rho_b and (E_{\\lambda} + \\rho_b) (E_{\\lambda} + 7 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})}) = (E_{\\lambda} + \\rho_b) (2 E_{\\lambda} + 8 \\rho_b) and (E_{\\lambda} + 7 \\rho_b + \\theta{(\\rho_b,E_{\\lambda})}) \\theta{(\\rho_b,E_{\\lambda})} = (2 E_{\\lambda} + 8 \\rho_b) \\theta{(\\rho_b,E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))))"], [["add", 2, "Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(3), Symbol('\\\\rho_b', commutative=True))))"], [["add", 3, "Mul(Integer(3), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(5), Symbol('\\\\rho_b', commutative=True)), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(6), Symbol('\\\\rho_b', commutative=True))))"], [["add", 4, "Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(7), Symbol('\\\\rho_b', commutative=True)), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(8), Symbol('\\\\rho_b', commutative=True))))"], [["times", 5, "Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(7), Symbol('\\\\rho_b', commutative=True)), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(8), Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(7), Symbol('\\\\rho_b', commutative=True)), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(8), Symbol('\\\\rho_b', commutative=True))), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\varphi^*,Q)} = (\\varphi^*)^{Q}, then obtain \\hat{x}_0{(\\varphi^*,Q)} + \\int (- \\varphi^* + (\\varphi^*)^{Q}) d\\varphi^* = (\\varphi^*)^{Q} + \\int (- \\varphi^* + (\\varphi^*)^{Q}) d\\varphi^*", "derivation": "\\hat{x}_0{(\\varphi^*,Q)} = (\\varphi^*)^{Q} and - \\varphi^* + \\hat{x}_0{(\\varphi^*,Q)} = - \\varphi^* + (\\varphi^*)^{Q} and \\int (- \\varphi^* + \\hat{x}_0{(\\varphi^*,Q)}) d\\varphi^* = \\int (- \\varphi^* + (\\varphi^*)^{Q}) d\\varphi^* and \\hat{x}_0{(\\varphi^*,Q)} + \\int (- \\varphi^* + \\hat{x}_0{(\\varphi^*,Q)}) d\\varphi^* = (\\varphi^*)^{Q} + \\int (- \\varphi^* + \\hat{x}_0{(\\varphi^*,Q)}) d\\varphi^* and \\hat{x}_0{(\\varphi^*,Q)} + \\int (- \\varphi^* + (\\varphi^*)^{Q}) d\\varphi^* = (\\varphi^*)^{Q} + \\int (- \\varphi^* + (\\varphi^*)^{Q}) d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(H,E)} = \\log{(\\frac{H}{E})}, then obtain \\int \\frac{\\psi^{*}{(H,E)} - \\log{(\\frac{H}{E})}}{\\log{(\\frac{H}{E})} - \\frac{1}{E}} dE = \\int 0 dE", "derivation": "\\psi^{*}{(H,E)} = \\log{(\\frac{H}{E})} and \\psi^{*}{(H,E)} - \\frac{1}{E} = \\log{(\\frac{H}{E})} - \\frac{1}{E} and \\psi^{*}{(H,E)} - \\log{(\\frac{H}{E})} = 0 and \\frac{\\psi^{*}{(H,E)} - \\log{(\\frac{H}{E})}}{\\log{(\\frac{H}{E})} - \\frac{1}{E}} = 0 and \\int \\frac{\\psi^{*}{(H,E)} - \\log{(\\frac{H}{E})}}{\\log{(\\frac{H}{E})} - \\frac{1}{E}} dE = \\int 0 dE", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('H', commutative=True), Symbol('E', commutative=True)), log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))))"], [["minus", 1, "Pow(Symbol('E', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\psi^*')(Symbol('H', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))), Add(log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))))"], [["minus", 2, "Add(log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1))))"], "Equality(Add(Function('\\\\psi^*')(Symbol('H', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))))), Integer(0))"], [["divide", 3, "Add(log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Function('\\\\psi^*')(Symbol('H', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))))), Pow(Add(log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))), Integer(-1))), Integer(0))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\psi^*')(Symbol('H', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))))), Pow(Add(log(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('E', commutative=True))), Integral(Integer(0), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given U{(C_{1},x)} = x^{C_{1}} and \\mathbf{B}{(C_{1},x)} = \\frac{\\partial}{\\partial C_{1}} U{(C_{1},x)}, then obtain C_{1} + \\mathbf{B}{(C_{1},x)} = C_{1} + x^{C_{1}} \\log{(x)}", "derivation": "U{(C_{1},x)} = x^{C_{1}} and \\frac{\\partial}{\\partial C_{1}} U{(C_{1},x)} = \\frac{\\partial}{\\partial C_{1}} x^{C_{1}} and \\mathbf{B}{(C_{1},x)} = \\frac{\\partial}{\\partial C_{1}} U{(C_{1},x)} and \\mathbf{B}{(C_{1},x)} = \\frac{\\partial}{\\partial C_{1}} x^{C_{1}} and C_{1} + \\mathbf{B}{(C_{1},x)} = C_{1} + \\frac{\\partial}{\\partial C_{1}} x^{C_{1}} and C_{1} + \\mathbf{B}{(C_{1},x)} = C_{1} + x^{C_{1}} \\log{(x)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('x', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Derivative(Function('U')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True)), Derivative(Pow(Symbol('x', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Integer(-1), Symbol('C_1', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True))), Add(Symbol('C_1', commutative=True), Derivative(Pow(Symbol('x', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True), Symbol('x', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Pow(Symbol('x', commutative=True), Symbol('C_1', commutative=True)), log(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given g{(\\hat{X},f^{*})} = \\log{(\\hat{X} - f^{*})}, then obtain g{(\\hat{X},f^{*})} - g^{f^{*}}{(\\hat{X},f^{*})} = g{(\\hat{X},f^{*})} - \\log{(\\hat{X} - f^{*})}^{f^{*}}", "derivation": "g{(\\hat{X},f^{*})} = \\log{(\\hat{X} - f^{*})} and g^{f^{*}}{(\\hat{X},f^{*})} = \\log{(\\hat{X} - f^{*})}^{f^{*}} and - g^{f^{*}}{(\\hat{X},f^{*})} = - \\log{(\\hat{X} - f^{*})}^{f^{*}} and g{(\\hat{X},f^{*})} - g^{f^{*}}{(\\hat{X},f^{*})} = g{(\\hat{X},f^{*})} - \\log{(\\hat{X} - f^{*})}^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), log(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(log(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Mul(Integer(-1), Pow(log(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True))))"], [["add", 3, "Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))), Add(Function('g')(Symbol('\\\\hat{X}', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(log(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(y^{\\prime},M)} = (y^{\\prime})^{M}, then derive \\int 0 dy^{\\prime} = \\int \\frac{\\partial}{\\partial M} ((y^{\\prime})^{M} - \\operatorname{A_{1}}{(y^{\\prime},M)}) dy^{\\prime}, then obtain - \\frac{\\int \\frac{d}{d M} 0 dy^{\\prime}}{\\operatorname{A_{1}}{(y^{\\prime},M)}} = - \\frac{\\int 0 dy^{\\prime}}{\\operatorname{A_{1}}{(y^{\\prime},M)}}", "derivation": "\\operatorname{A_{1}}{(y^{\\prime},M)} = (y^{\\prime})^{M} and 0 = (y^{\\prime})^{M} - \\operatorname{A_{1}}{(y^{\\prime},M)} and \\frac{d}{d M} 0 = \\frac{\\partial}{\\partial M} ((y^{\\prime})^{M} - \\operatorname{A_{1}}{(y^{\\prime},M)}) and \\int \\frac{d}{d M} 0 dy^{\\prime} = \\int \\frac{\\partial}{\\partial M} ((y^{\\prime})^{M} - \\operatorname{A_{1}}{(y^{\\prime},M)}) dy^{\\prime} and \\int 0 dy^{\\prime} = \\int \\frac{\\partial}{\\partial M} ((y^{\\prime})^{M} - \\operatorname{A_{1}}{(y^{\\prime},M)}) dy^{\\prime} and \\int \\frac{d}{d M} 0 dy^{\\prime} = \\int 0 dy^{\\prime} and - \\frac{\\int \\frac{d}{d M} 0 dy^{\\prime}}{\\operatorname{A_{1}}{(y^{\\prime},M)}} = - \\frac{\\int 0 dy^{\\prime}}{\\operatorname{A_{1}}{(y^{\\prime},M)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))"], [["minus", 1, "Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Add(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Add(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 6, "Mul(Integer(-1), Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Integral(Derivative(Integer(0), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Pow(Function('A_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given u{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\mathbf{B}{(\\hat{H}_l)} = (\\frac{d}{d \\hat{H}_l} \\hat{H}_l u{(\\hat{H}_l)})^{\\hat{H}_l}, then obtain \\mathbf{B}{(\\hat{H}_l)} = (\\frac{d}{d \\hat{H}_l} \\hat{H}_l \\cos{(\\hat{H}_l)})^{\\hat{H}_l}", "derivation": "u{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\hat{H}_l u{(\\hat{H}_l)} = \\hat{H}_l \\cos{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\hat{H}_l u{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\hat{H}_l \\cos{(\\hat{H}_l)} and \\mathbf{B}{(\\hat{H}_l)} = (\\frac{d}{d \\hat{H}_l} \\hat{H}_l u{(\\hat{H}_l)})^{\\hat{H}_l} and \\mathbf{B}{(\\hat{H}_l)} = (\\frac{d}{d \\hat{H}_l} \\hat{H}_l \\cos{(\\hat{H}_l)})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('u')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('u')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('u')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})}, then derive \\frac{d}{d \\hat{\\mathbf{r}}} \\tilde{g}{(\\hat{\\mathbf{r}})} = - \\sin{(\\hat{\\mathbf{r}})}, then obtain - \\sin{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})}", "derivation": "\\tilde{g}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\tilde{g}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\tilde{g}{(\\hat{\\mathbf{r}})} = - \\sin{(\\hat{\\mathbf{r}})} and - \\sin{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(A_{y},n)} = A_{y} n, then obtain A_{y} + n \\frac{\\partial}{\\partial n} q{(A_{y},n)} = A_{y} n + A_{y}", "derivation": "q{(A_{y},n)} = A_{y} n and \\frac{\\partial}{\\partial n} q{(A_{y},n)} = \\frac{\\partial}{\\partial n} A_{y} n and n \\frac{\\partial}{\\partial n} q{(A_{y},n)} = n \\frac{\\partial}{\\partial n} A_{y} n and n \\frac{\\partial}{\\partial n} q{(A_{y},n)} + \\frac{\\partial}{\\partial n} A_{y} n = n \\frac{\\partial}{\\partial n} A_{y} n + \\frac{\\partial}{\\partial n} A_{y} n and A_{y} + n \\frac{\\partial}{\\partial n} q{(A_{y},n)} = A_{y} n + A_{y}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Derivative(Function('q')(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Symbol('n', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('n', commutative=True), Derivative(Function('q')(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Mul(Symbol('n', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('A_y', commutative=True), Mul(Symbol('n', commutative=True), Derivative(Function('q')(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Add(Mul(Symbol('A_y', commutative=True), Symbol('n', commutative=True)), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(J,M)} = \\frac{J}{M}, then obtain \\frac{- J + \\varepsilon_{0}^{J}{(J,M)}}{M} = \\frac{- J + (\\frac{J}{M})^{J}}{M}", "derivation": "\\varepsilon_{0}{(J,M)} = \\frac{J}{M} and \\varepsilon_{0}^{J}{(J,M)} = (\\frac{J}{M})^{J} and - J + \\varepsilon_{0}^{J}{(J,M)} = - J + (\\frac{J}{M})^{J} and \\frac{- J + \\varepsilon_{0}^{J}{(J,M)}}{M} = \\frac{- J + (\\frac{J}{M})^{J}}{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('J', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('J', commutative=True)), Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Symbol('J', commutative=True)))"], [["minus", 2, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Symbol('J', commutative=True))))"], [["divide", 3, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True), Symbol('M', commutative=True)), Symbol('J', commutative=True)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(H)} = \\log{(H)}, then derive \\int \\bar{\\h}{(H)} dH = A_{1} + H \\log{(H)} - H, then obtain \\int \\frac{\\sin{(\\int \\log{(H)} dH)}}{H} dA_{1} = \\int \\frac{\\sin{(A_{1} + H \\log{(H)} - H)}}{H} dA_{1}", "derivation": "\\bar{\\h}{(H)} = \\log{(H)} and \\int \\bar{\\h}{(H)} dH = \\int \\log{(H)} dH and \\int \\bar{\\h}{(H)} dH = A_{1} + H \\log{(H)} - H and \\sin{(\\int \\bar{\\h}{(H)} dH)} = \\sin{(A_{1} + H \\log{(H)} - H)} and \\frac{\\sin{(\\int \\bar{\\h}{(H)} dH)}}{H} = \\frac{\\sin{(A_{1} + H \\log{(H)} - H)}}{H} and \\int \\frac{\\sin{(\\int \\bar{\\h}{(H)} dH)}}{H} dA_{1} = \\int \\frac{\\sin{(A_{1} + H \\log{(H)} - H)}}{H} dA_{1} and \\int \\frac{\\sin{(\\int \\log{(H)} dH)}}{H} dA_{1} = \\int \\frac{\\sin{(A_{1} + H \\log{(H)} - H)}}{H} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hbar')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Function('\\\\hbar')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), sin(Add(Symbol('A_1', commutative=True), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)))))"], [["times", 4, "Pow(Symbol('H', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Integral(Function('\\\\hbar')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Add(Symbol('A_1', commutative=True), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))))))"], [["integrate", 5, "Symbol('A_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Integral(Function('\\\\hbar')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Add(Symbol('A_1', commutative=True), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))))), Tuple(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), sin(Add(Symbol('A_1', commutative=True), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))))), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)}, then obtain \\frac{\\iint \\operatorname{t_{1}}{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0}{\\iint \\cos{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0} = 1", "derivation": "\\operatorname{t_{1}}{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)} and \\int \\operatorname{t_{1}}{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\cos{(\\hat{p}_0)} d\\hat{p}_0 and \\iint \\operatorname{t_{1}}{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0 = \\iint \\cos{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0 and \\frac{\\iint \\operatorname{t_{1}}{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0}{\\iint \\cos{(\\hat{p}_0)} d\\hat{p}_0 d\\hat{p}_0} = 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["divide", 3, "Integral(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Integral(Function('t_1')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Pow(Integral(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given U{(S,V_{\\mathbf{E}})} = S + V_{\\mathbf{E}}, then obtain - S - V_{\\mathbf{E}} + 2 U{(S,V_{\\mathbf{E}})} = S + V_{\\mathbf{E}}", "derivation": "U{(S,V_{\\mathbf{E}})} = S + V_{\\mathbf{E}} and S + V_{\\mathbf{E}} + U{(S,V_{\\mathbf{E}})} = 2 S + 2 V_{\\mathbf{E}} and 2 U{(S,V_{\\mathbf{E}})} = 2 S + 2 V_{\\mathbf{E}} and - S - V_{\\mathbf{E}} + 2 U{(S,V_{\\mathbf{E}})} = S + V_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, "Add(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('U')(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('U')(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 3, "Add(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(2), Function('U')(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Symbol('S', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(M)} = \\cos{(M)}, then obtain \\frac{\\mathbf{E}{(M)} \\int \\mathbf{E}{(M)} dM}{C} = \\frac{\\mathbf{E}{(M)} \\int \\cos{(M)} dM}{C}", "derivation": "\\mathbf{E}{(M)} = \\cos{(M)} and \\int \\mathbf{E}{(M)} dM = \\int \\cos{(M)} dM and \\frac{\\int \\mathbf{E}{(M)} dM}{C} = \\frac{\\int \\cos{(M)} dM}{C} and \\frac{\\mathbf{E}{(M)} \\int \\mathbf{E}{(M)} dM}{C} = \\frac{\\mathbf{E}{(M)} \\int \\cos{(M)} dM}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["times", 3, "Function('\\\\mathbf{E}')(Symbol('M', commutative=True))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), Integral(Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('M', commutative=True)), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given W{(\\mathbf{J}_f,v,t_{2})} = \\mathbf{J}_f - t_{2} + v, then obtain \\cos{(\\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} W{(\\mathbf{J}_f,v,t_{2})}}{t_{2}})} = \\cos{(\\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - t_{2} + v)}{t_{2}})}", "derivation": "W{(\\mathbf{J}_f,v,t_{2})} = \\mathbf{J}_f - t_{2} + v and \\frac{\\partial}{\\partial \\mathbf{J}_f} W{(\\mathbf{J}_f,v,t_{2})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - t_{2} + v) and - \\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} W{(\\mathbf{J}_f,v,t_{2})}}{t_{2}} = - \\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - t_{2} + v)}{t_{2}} and \\cos{(\\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} W{(\\mathbf{J}_f,v,t_{2})}}{t_{2}})} = \\cos{(\\frac{\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - t_{2} + v)}{t_{2}})}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('t_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))), cos(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))))"]]}, {"prompt": "Given q{(\\nabla,\\lambda)} = \\int \\lambda \\nabla d\\lambda and z{(\\nabla,\\lambda)} = \\lambda \\int (\\lambda + \\int \\lambda \\nabla d\\lambda) d\\nabla, then obtain \\int z{(\\nabla,\\lambda)} d\\lambda = \\int \\lambda \\int (\\lambda + q{(\\nabla,\\lambda)}) d\\nabla d\\lambda", "derivation": "q{(\\nabla,\\lambda)} = \\int \\lambda \\nabla d\\lambda and \\lambda + q{(\\nabla,\\lambda)} = \\lambda + \\int \\lambda \\nabla d\\lambda and \\int (\\lambda + q{(\\nabla,\\lambda)}) d\\nabla = \\int (\\lambda + \\int \\lambda \\nabla d\\lambda) d\\nabla and \\lambda \\int (\\lambda + q{(\\nabla,\\lambda)}) d\\nabla = \\lambda \\int (\\lambda + \\int \\lambda \\nabla d\\lambda) d\\nabla and z{(\\nabla,\\lambda)} = \\lambda \\int (\\lambda + \\int \\lambda \\nabla d\\lambda) d\\nabla and z{(\\nabla,\\lambda)} = \\lambda \\int (\\lambda + q{(\\nabla,\\lambda)}) d\\nabla and \\int z{(\\nabla,\\lambda)} d\\lambda = \\int \\lambda \\int (\\lambda + q{(\\nabla,\\lambda)}) d\\nabla d\\lambda", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\lambda', commutative=True), Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["times", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('q')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\Omega)} = \\cos{(\\Omega)}, then obtain \\frac{\\partial^{2}}{\\partial \\Omega\\partial f_{E}} (f_{E} \\rho_{b}{(\\Omega)} + 1) = \\frac{\\partial^{2}}{\\partial \\Omega\\partial f_{E}} (f_{E} \\cos{(\\Omega)} + 1)", "derivation": "\\rho_{b}{(\\Omega)} = \\cos{(\\Omega)} and f_{E} \\rho_{b}{(\\Omega)} = f_{E} \\cos{(\\Omega)} and f_{E} \\rho_{b}{(\\Omega)} + 1 = f_{E} \\cos{(\\Omega)} + 1 and \\frac{\\partial}{\\partial f_{E}} (f_{E} \\rho_{b}{(\\Omega)} + 1) = \\frac{\\partial}{\\partial f_{E}} (f_{E} \\cos{(\\Omega)} + 1) and \\frac{\\partial^{2}}{\\partial \\Omega\\partial f_{E}} (f_{E} \\rho_{b}{(\\Omega)} + 1) = \\frac{\\partial^{2}}{\\partial \\Omega\\partial f_{E}} (f_{E} \\cos{(\\Omega)} + 1)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "Pow(Symbol('f_E', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\rho_b')(Symbol('\\\\Omega', commutative=True))), Mul(Symbol('f_E', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('f_E', commutative=True), Function('\\\\rho_b')(Symbol('\\\\Omega', commutative=True))), Integer(1)), Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Integer(1)))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('f_E', commutative=True), Function('\\\\rho_b')(Symbol('\\\\Omega', commutative=True))), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('f_E', commutative=True), Function('\\\\rho_b')(Symbol('\\\\Omega', commutative=True))), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\hat{X})} = e^{\\sin{(\\hat{X})}} and \\chi{(\\hat{X})} = B{(\\hat{X})} \\sin{(\\hat{X})}, then obtain (e^{\\sin{(\\hat{X})}} \\sin{(\\hat{X})})^{\\hat{X}} = (B{(\\hat{X})} \\sin{(\\hat{X})})^{\\hat{X}}", "derivation": "B{(\\hat{X})} = e^{\\sin{(\\hat{X})}} and B{(\\hat{X})} \\sin{(\\hat{X})} = e^{\\sin{(\\hat{X})}} \\sin{(\\hat{X})} and \\chi{(\\hat{X})} = B{(\\hat{X})} \\sin{(\\hat{X})} and \\chi^{\\hat{X}}{(\\hat{X})} = (B{(\\hat{X})} \\sin{(\\hat{X})})^{\\hat{X}} and \\chi{(\\hat{X})} = e^{\\sin{(\\hat{X})}} \\sin{(\\hat{X})} and \\chi^{\\hat{X}}{(\\hat{X})} = (e^{\\sin{(\\hat{X})}} \\sin{(\\hat{X})})^{\\hat{X}} and (e^{\\sin{(\\hat{X})}} \\sin{(\\hat{X})})^{\\hat{X}} = (B{(\\hat{X})} \\sin{(\\hat{X})})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{X}', commutative=True)), exp(sin(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Function('B')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True))), Mul(exp(sin(Symbol('\\\\hat{X}', commutative=True))), sin(Symbol('\\\\hat{X}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True)), Mul(Function('B')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Function('B')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True)), Mul(exp(sin(Symbol('\\\\hat{X}', commutative=True))), sin(Symbol('\\\\hat{X}', commutative=True))))"], [["power", 5, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(exp(sin(Symbol('\\\\hat{X}', commutative=True))), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Mul(exp(sin(Symbol('\\\\hat{X}', commutative=True))), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Function('B')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\hat{H})} = \\log{(\\hat{H})}, then obtain 1 = (\\frac{\\log{(\\hat{H})}^{2}}{\\operatorname{v_{z}}^{2}{(\\hat{H})}})^{\\hat{H}}", "derivation": "\\operatorname{v_{z}}{(\\hat{H})} = \\log{(\\hat{H})} and 1 = \\frac{\\log{(\\hat{H})}}{\\operatorname{v_{z}}{(\\hat{H})}} and \\log{(\\hat{H})} = \\frac{\\log{(\\hat{H})}^{2}}{\\operatorname{v_{z}}{(\\hat{H})}} and 1 = (\\frac{\\log{(\\hat{H})}}{\\operatorname{v_{z}}{(\\hat{H})}})^{\\hat{H}} and 1 = (\\frac{\\log{(\\hat{H})}^{2}}{\\operatorname{v_{z}}^{2}{(\\hat{H})}})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Function('v_z')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 1, "Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(log(Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\hat{H}', commutative=True)), Integer(2))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(1), Pow(Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-2)), Pow(log(Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and t{(B,v)} = - B + v, then obtain (- B + v)^{- v} \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} t^{B}{(B,v)} = (- B + v)^{- v} \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} (- B + v)^{B}", "derivation": "\\mathbf{F}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and t{(B,v)} = - B + v and t^{B}{(B,v)} = (- B + v)^{B} and \\frac{\\partial}{\\partial B} t^{B}{(B,v)} = \\frac{\\partial}{\\partial B} (- B + v)^{B} and e^{\\mathbf{J}_f} \\frac{\\partial}{\\partial B} t^{B}{(B,v)} = e^{\\mathbf{J}_f} \\frac{\\partial}{\\partial B} (- B + v)^{B} and \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} t^{B}{(B,v)} = \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} (- B + v)^{B} and (- B + v)^{- v} \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} t^{B}{(B,v)} = (- B + v)^{- v} \\mathbf{F}{(\\mathbf{J}_f)} \\frac{\\partial}{\\partial B} (- B + v)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\mathbf{J}_f', commutative=True)))"], ["get_premise", "Equality(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('B', commutative=True)))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Pow(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["times", 4, "exp(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["divide", 6, "Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Function('t')(Symbol('B', commutative=True), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('v', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\sigma_x)} = \\cos{(\\sigma_x)}, then obtain - \\sin{(\\sigma_x)} + \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{J}_P{(\\sigma_x)}) d\\sigma_x = - \\sin{(\\sigma_x)} + \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\cos{(\\sigma_x)}) d\\sigma_x", "derivation": "\\mathbf{J}_P{(\\sigma_x)} = \\cos{(\\sigma_x)} and - \\sigma_x + \\mathbf{J}_P{(\\sigma_x)} = - \\sigma_x + \\cos{(\\sigma_x)} and \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{J}_P{(\\sigma_x)}) = \\frac{d}{d \\sigma_x} (- \\sigma_x + \\cos{(\\sigma_x)}) and \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{J}_P{(\\sigma_x)}) d\\sigma_x = \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\cos{(\\sigma_x)}) d\\sigma_x and - \\sin{(\\sigma_x)} + \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\mathbf{J}_P{(\\sigma_x)}) d\\sigma_x = - \\sin{(\\sigma_x)} + \\int \\frac{d}{d \\sigma_x} (- \\sigma_x + \\cos{(\\sigma_x)}) d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 4, "Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given S{(E_{x},\\theta_1)} = \\theta_1^{E_{x}} and \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{t_{1}}, then obtain \\tilde{\\infty} S{(E_{x},\\theta_1)} \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\hat{\\mathbf{r}}^{t_{1}} S{(E_{x},\\theta_1)}", "derivation": "S{(E_{x},\\theta_1)} = \\theta_1^{E_{x}} and \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{t_{1}} and \\theta_1^{E_{x}} \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{t_{1}} \\theta_1^{E_{x}} and S{(E_{x},\\theta_1)} \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{t_{1}} S{(E_{x},\\theta_1)} and \\tilde{\\infty} S{(E_{x},\\theta_1)} \\operatorname{V_{\\mathbf{E}}}{(t_{1},\\hat{\\mathbf{r}})} = \\tilde{\\infty} \\hat{\\mathbf{r}}^{t_{1}} S{(E_{x},\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('E_x', commutative=True)))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('t_1', commutative=True)))"], [["times", 2, "Pow(Symbol('\\\\theta_1', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('E_x', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('t_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('S')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('t_1', commutative=True)), Function('S')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["times", 4, "zoo"], "Equality(Mul(zoo, Function('S')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(zoo, Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('t_1', commutative=True)), Function('S')(Symbol('E_x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(v,\\varphi)} = \\cos{(\\frac{\\varphi}{v})}, then obtain \\frac{\\varphi \\int 1 dv}{v} = \\frac{\\varphi \\int \\frac{v + \\cos{(\\frac{\\varphi}{v})}}{v + \\ddot{x}{(v,\\varphi)}} dv}{v}", "derivation": "\\ddot{x}{(v,\\varphi)} = \\cos{(\\frac{\\varphi}{v})} and v + \\ddot{x}{(v,\\varphi)} = v + \\cos{(\\frac{\\varphi}{v})} and 1 = \\frac{v + \\cos{(\\frac{\\varphi}{v})}}{v + \\ddot{x}{(v,\\varphi)}} and \\int 1 dv = \\int \\frac{v + \\cos{(\\frac{\\varphi}{v})}}{v + \\ddot{x}{(v,\\varphi)}} dv and \\frac{\\varphi \\int 1 dv}{v} = \\frac{\\varphi \\int \\frac{v + \\cos{(\\frac{\\varphi}{v})}}{v + \\ddot{x}{(v,\\varphi)}} dv}{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('v', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))))"], [["divide", 2, "Add(Symbol('v', commutative=True), Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('v', commutative=True), Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('v', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(Add(Symbol('v', commutative=True), Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('v', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True))))"], [["times", 4, "Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Mul(Pow(Add(Symbol('v', commutative=True), Function('\\\\ddot{x}')(Symbol('v', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('v', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given s{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}, then derive s{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0, then obtain - \\frac{\\frac{d}{d \\Psi_{nl}} (s{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}})}{\\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}} = - \\frac{\\frac{d}{d \\Psi_{nl}} 0}{\\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}}", "derivation": "s{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and - \\Psi_{nl} + s{(\\Psi_{nl})} = - \\Psi_{nl} + \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and s{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} = 0 and s{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0 and \\frac{d}{d \\Psi_{nl}} (s{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}}) = \\frac{d}{d \\Psi_{nl}} 0 and - \\frac{\\frac{d}{d \\Psi_{nl}} (s{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}})}{\\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}} = - \\frac{\\frac{d}{d \\Psi_{nl}} 0}{\\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('s')(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], "Equality(Add(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Add(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["divide", 5, "Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(Add(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given x{(P_{e},z)} = \\frac{\\sin{(P_{e})}}{z} and \\operatorname{A_{z}}{(P_{e},z)} = \\frac{1}{x{(P_{e},z)}}, then obtain (\\int \\frac{z}{\\sin{(P_{e})}} dP_{e})^{z} = (\\int \\frac{1}{x{(P_{e},z)}} dP_{e})^{z}", "derivation": "x{(P_{e},z)} = \\frac{\\sin{(P_{e})}}{z} and \\operatorname{A_{z}}{(P_{e},z)} = \\frac{1}{x{(P_{e},z)}} and \\operatorname{A_{z}}{(P_{e},z)} = \\frac{z}{\\sin{(P_{e})}} and \\frac{z}{\\sin{(P_{e})}} = \\frac{1}{x{(P_{e},z)}} and \\int \\frac{z}{\\sin{(P_{e})}} dP_{e} = \\int \\frac{1}{x{(P_{e},z)}} dP_{e} and (\\int \\frac{z}{\\sin{(P_{e})}} dP_{e})^{z} = (\\int \\frac{1}{x{(P_{e},z)}} dP_{e})^{z}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), sin(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Pow(Function('x')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_z')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('z', commutative=True), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))), Pow(Function('x')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Mul(Symbol('z', commutative=True), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))), Tuple(Symbol('P_e', commutative=True))), Integral(Pow(Function('x')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Tuple(Symbol('P_e', commutative=True))))"], [["power", 5, "Symbol('z', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('z', commutative=True), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))), Tuple(Symbol('P_e', commutative=True))), Symbol('z', commutative=True)), Pow(Integral(Pow(Function('x')(Symbol('P_e', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Tuple(Symbol('P_e', commutative=True))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\theta_2)} = \\cos{(\\theta_2)} and \\theta_{1}{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain 2 \\theta_{1}{(\\theta_2)} = \\theta_2 (\\frac{\\theta_{1}{(\\theta_2)}}{\\theta_2} + \\frac{\\operatorname{a^{\\dagger}}{(\\theta_2)}}{\\theta_2})", "derivation": "\\operatorname{a^{\\dagger}}{(\\theta_2)} = \\cos{(\\theta_2)} and \\theta_{1}{(\\theta_2)} = \\cos{(\\theta_2)} and \\frac{\\theta_{1}{(\\theta_2)}}{\\theta_2} = \\frac{\\cos{(\\theta_2)}}{\\theta_2} and \\frac{2 \\theta_{1}{(\\theta_2)}}{\\theta_2} = \\frac{\\theta_{1}{(\\theta_2)}}{\\theta_2} + \\frac{\\cos{(\\theta_2)}}{\\theta_2} and 2 \\theta_{1}{(\\theta_2)} = \\theta_2 (\\frac{\\theta_{1}{(\\theta_2)}}{\\theta_2} + \\frac{\\cos{(\\theta_2)}}{\\theta_2}) and 2 \\theta_{1}{(\\theta_2)} = \\theta_2 (\\frac{\\theta_{1}{(\\theta_2)}}{\\theta_2} + \\frac{\\operatorname{a^{\\dagger}}{(\\theta_2)}}{\\theta_2})", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["divide", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_2', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_2', commutative=True)))))"], [["divide", 4, "Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))"], "Equality(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given B{(h,\\hbar)} = \\cos{(\\hbar - h)}, then derive \\frac{\\partial}{\\partial h} B{(h,\\hbar)} - 1 = \\sin{(\\hbar - h)} - 1, then obtain \\frac{\\partial}{\\partial h} \\cos{(\\hbar - h)} - 1 = \\frac{\\partial}{\\partial h} B{(h,\\hbar)} - 1", "derivation": "B{(h,\\hbar)} = \\cos{(\\hbar - h)} and - h + B{(h,\\hbar)} = - h + \\cos{(\\hbar - h)} and \\frac{\\partial}{\\partial h} (- h + B{(h,\\hbar)}) = \\frac{\\partial}{\\partial h} (- h + \\cos{(\\hbar - h)}) and \\frac{\\partial}{\\partial h} B{(h,\\hbar)} - 1 = \\sin{(\\hbar - h)} - 1 and \\frac{\\partial}{\\partial h} \\cos{(\\hbar - h)} - 1 = \\sin{(\\hbar - h)} - 1 and \\frac{\\partial}{\\partial h} \\cos{(\\hbar - h)} - 1 = \\frac{\\partial}{\\partial h} B{(h,\\hbar)} - 1", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('B')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('B')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('B')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)), Add(sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)), Add(sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('B')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given p{(\\Omega)} = \\log{(\\cos{(\\Omega)})}, then obtain p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + 2 \\int \\log{(\\cos{(\\Omega)})} d\\Omega = p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + \\int p{(\\Omega)} d\\Omega + \\int \\log{(\\cos{(\\Omega)})} d\\Omega", "derivation": "p{(\\Omega)} = \\log{(\\cos{(\\Omega)})} and \\int p{(\\Omega)} d\\Omega = \\int \\log{(\\cos{(\\Omega)})} d\\Omega and \\log{(\\cos{(\\Omega)})} + \\int p{(\\Omega)} d\\Omega = \\log{(\\cos{(\\Omega)})} + \\int \\log{(\\cos{(\\Omega)})} d\\Omega and p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + 2 \\int p{(\\Omega)} d\\Omega = p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + \\int p{(\\Omega)} d\\Omega + \\int \\log{(\\cos{(\\Omega)})} d\\Omega and p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + 2 \\int p{(\\Omega)} d\\Omega = p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + 2 \\int \\log{(\\cos{(\\Omega)})} d\\Omega and p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + 2 \\int \\log{(\\cos{(\\Omega)})} d\\Omega = p{(\\Omega)} + \\log{(\\cos{(\\Omega)})} + \\int p{(\\Omega)} d\\Omega + \\int \\log{(\\cos{(\\Omega)})} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "log(cos(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(log(cos(Symbol('\\\\Omega', commutative=True))), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(log(cos(Symbol('\\\\Omega', commutative=True))), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["add", 3, "Add(Function('p')(Symbol('\\\\Omega', commutative=True)), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], "Equality(Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Function('p')(Symbol('\\\\Omega', commutative=True)), log(cos(Symbol('\\\\Omega', commutative=True))), Integral(Function('p')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\sigma_x)} = \\sin{(\\sigma_x)}, then obtain \\operatorname{c_{0}}^{2}{(\\sigma_x)} \\sin{(\\sigma_x)} = \\operatorname{c_{0}}{(\\sigma_x)} \\sin^{2}{(\\sigma_x)}", "derivation": "\\operatorname{c_{0}}{(\\sigma_x)} = \\sin{(\\sigma_x)} and \\operatorname{c_{0}}^{2}{(\\sigma_x)} = \\operatorname{c_{0}}{(\\sigma_x)} \\sin{(\\sigma_x)} and \\operatorname{c_{0}}^{3}{(\\sigma_x)} = \\operatorname{c_{0}}^{2}{(\\sigma_x)} \\sin{(\\sigma_x)} and \\operatorname{c_{0}}^{3}{(\\sigma_x)} = \\operatorname{c_{0}}{(\\sigma_x)} \\sin^{2}{(\\sigma_x)} and \\operatorname{c_{0}}^{2}{(\\sigma_x)} \\sin{(\\sigma_x)} = \\operatorname{c_{0}}{(\\sigma_x)} \\sin^{2}{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Function('c_0')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 2, "Function('c_0')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Integer(3)), Mul(Pow(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Integer(3)), Mul(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), sin(Symbol('\\\\sigma_x', commutative=True))), Mul(Function('c_0')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\dot{y}{(\\Psi_{nl},\\mathbf{S})} = \\Psi_{nl} \\sin{(\\mathbf{S})}, then obtain ((\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})})^{2})^{\\mathbf{S}} = ((\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})}) (\\Psi_{nl} \\sin{(\\mathbf{S})} + \\Psi_{nl}))^{\\mathbf{S}}", "derivation": "\\dot{y}{(\\Psi_{nl},\\mathbf{S})} = \\Psi_{nl} \\sin{(\\mathbf{S})} and \\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})} = \\Psi_{nl} \\sin{(\\mathbf{S})} + \\Psi_{nl} and (\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})})^{2} = (\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})}) (\\Psi_{nl} \\sin{(\\mathbf{S})} + \\Psi_{nl}) and ((\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})})^{2})^{\\mathbf{S}} = ((\\Psi_{nl} + \\dot{y}{(\\Psi_{nl},\\mathbf{S})}) (\\Psi_{nl} \\sin{(\\mathbf{S})} + \\Psi_{nl}))^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(2)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given i{(\\hat{\\mathbf{r}},f)} = \\log{(- \\hat{\\mathbf{r}} + f)}, then derive \\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} = \\frac{1}{- \\hat{\\mathbf{r}} + f}, then obtain \\frac{\\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} - 1}{- \\hat{\\mathbf{r}} + f} = \\frac{-1 + \\frac{1}{- \\hat{\\mathbf{r}} + f}}{- \\hat{\\mathbf{r}} + f}", "derivation": "i{(\\hat{\\mathbf{r}},f)} = \\log{(- \\hat{\\mathbf{r}} + f)} and \\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} = \\frac{\\partial}{\\partial f} \\log{(- \\hat{\\mathbf{r}} + f)} and \\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} = \\frac{1}{- \\hat{\\mathbf{r}} + f} and \\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} - 1 = -1 + \\frac{1}{- \\hat{\\mathbf{r}} + f} and \\frac{\\frac{\\partial}{\\partial f} i{(\\hat{\\mathbf{r}},f)} - 1}{- \\hat{\\mathbf{r}} + f} = \\frac{-1 + \\frac{1}{- \\hat{\\mathbf{r}} + f}}{- \\hat{\\mathbf{r}} + f}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1))))"], [["times", 4, "Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Add(Derivative(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given T{(C_{2})} = \\int \\log{(C_{2})} dC_{2}, then derive T{(C_{2})} + 1 = C_{2} \\log{(C_{2})} - C_{2} + b + 1, then obtain (\\int \\log{(C_{2})} dC_{2} + 1)^{b} = (C_{2} \\log{(C_{2})} - C_{2} + b + 1)^{b}", "derivation": "T{(C_{2})} = \\int \\log{(C_{2})} dC_{2} and T{(C_{2})} + 1 = \\int \\log{(C_{2})} dC_{2} + 1 and T{(C_{2})} + 1 = C_{2} \\log{(C_{2})} - C_{2} + b + 1 and \\int \\log{(C_{2})} dC_{2} + 1 = C_{2} \\log{(C_{2})} - C_{2} + b + 1 and (\\int \\log{(C_{2})} dC_{2} + 1)^{b} = (C_{2} \\log{(C_{2})} - C_{2} + b + 1)^{b}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('C_2', commutative=True)), Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('T')(Symbol('C_2', commutative=True)), Integer(1)), Add(Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(1)))"], [["evaluate_integrals", 2], "Equality(Add(Function('T')(Symbol('C_2', commutative=True)), Integer(1)), Add(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('b', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(1)), Add(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('b', commutative=True), Integer(1)))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Integral(log(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(1)), Symbol('b', commutative=True)), Pow(Add(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('b', commutative=True), Integer(1)), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(\\eta,V)} = \\frac{V}{\\eta}, then obtain (\\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} \\mathbf{p}{(\\eta,V)} = (\\frac{2 V}{\\eta} - \\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} \\mathbf{p}{(\\eta,V)}", "derivation": "\\mathbf{p}{(\\eta,V)} = \\frac{V}{\\eta} and \\mathbf{p}{(\\eta,V)} + 1 = \\frac{V}{\\eta} + 1 and \\frac{V}{\\eta} + \\mathbf{p}{(\\eta,V)} + 1 = \\frac{2 V}{\\eta} + 1 and 2 \\mathbf{p}{(\\eta,V)} + 1 = \\frac{2 V}{\\eta} + 1 and \\mathbf{p}{(\\eta,V)} + 1 = \\frac{2 V}{\\eta} - \\mathbf{p}{(\\eta,V)} + 1 and (\\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} (\\mathbf{p}{(\\eta,V)} + 1) = (\\frac{2 V}{\\eta} - \\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} (\\mathbf{p}{(\\eta,V)} + 1) and (\\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} \\mathbf{p}{(\\eta,V)} = (\\frac{2 V}{\\eta} - \\mathbf{p}{(\\eta,V)} + 1) \\frac{\\partial}{\\partial \\eta} \\mathbf{p}{(\\eta,V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Add(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Integer(1)))"], [["add", 2, "Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Integer(1)))"], [["minus", 4, "Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True))), Integer(1)))"], [["times", 5, "Derivative(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Derivative(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True))), Integer(1)), Derivative(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Integer(1)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True))), Integer(1)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\eta', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\mathbf{J})} = \\log{(\\mathbf{J})}, then obtain \\int \\frac{\\sigma_{x}{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{2}{\\mathbf{J}} = \\int \\frac{\\log{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{2}{\\mathbf{J}}", "derivation": "\\sigma_{x}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{\\sigma_{x}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\log{(\\mathbf{J})}}{\\mathbf{J}} and \\int \\frac{\\sigma_{x}{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} = \\int \\frac{\\log{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} and \\int \\frac{\\sigma_{x}{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{1}{\\mathbf{J}} = \\int \\frac{\\log{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{1}{\\mathbf{J}} and \\int \\frac{\\sigma_{x}{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{2}{\\mathbf{J}} = \\int \\frac{\\log{(\\mathbf{J})}}{\\mathbf{J}} d\\mathbf{J} + \\frac{2}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 3, "Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))), Add(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))))"], [["add", 4, "Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Add(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{X}{(\\hat{H})} = \\log{(\\hat{H})}, then obtain \\int \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\hat{X}{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} d\\hat{H} = \\int \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\log{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} d\\hat{H}", "derivation": "\\hat{X}{(\\hat{H})} = \\log{(\\hat{H})} and \\frac{d}{d \\hat{H}} \\hat{X}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\log{(\\hat{H})} and \\frac{d^{2}}{d \\hat{H}^{2}} \\hat{X}{(\\hat{H})} = \\frac{d^{2}}{d \\hat{H}^{2}} \\log{(\\hat{H})} and \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\hat{X}{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} = \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\log{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} and \\int \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\hat{X}{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} d\\hat{H} = \\int \\frac{\\frac{d^{2}}{d \\hat{H}^{2}} \\log{(\\hat{H})}}{\\psi{(\\rho_b,\\mathbf{v})}} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\hat{H}', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))))"], [["divide", 3, "Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Mul(Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2)))))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given T{(h,\\mu)} = e^{h^{\\mu}} and A{(\\Psi_{nl},\\pi)} = \\frac{\\Psi_{nl}}{\\pi}, then obtain \\int (A{(\\Psi_{nl},\\pi)} + T{(h,\\mu)} - 1) d\\mu = a + \\frac{\\int \\Psi_{nl} d\\mu + \\int - \\pi d\\mu + \\int \\pi T{(h,\\mu)} d\\mu}{\\pi}", "derivation": "T{(h,\\mu)} = e^{h^{\\mu}} and A{(\\Psi_{nl},\\pi)} = \\frac{\\Psi_{nl}}{\\pi} and A{(\\Psi_{nl},\\pi)} + e^{h^{\\mu}} = \\frac{\\Psi_{nl}}{\\pi} + e^{h^{\\mu}} and A{(\\Psi_{nl},\\pi)} + T{(h,\\mu)} = \\frac{\\Psi_{nl}}{\\pi} + T{(h,\\mu)} and A{(\\Psi_{nl},\\pi)} + T{(h,\\mu)} - 1 = \\frac{\\Psi_{nl}}{\\pi} + T{(h,\\mu)} - 1 and \\int (A{(\\Psi_{nl},\\pi)} + T{(h,\\mu)} - 1) d\\mu = \\int (\\frac{\\Psi_{nl}}{\\pi} + T{(h,\\mu)} - 1) d\\mu and \\int (A{(\\Psi_{nl},\\pi)} + T{(h,\\mu)} - 1) d\\mu = a + \\frac{\\int \\Psi_{nl} d\\mu + \\int - \\pi d\\mu + \\int \\pi T{(h,\\mu)} d\\mu}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), exp(Pow(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["get_premise", "Equality(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["add", 2, "exp(Pow(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), exp(Pow(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), exp(Pow(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Add(Function('A')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('a', commutative=True), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Integral(Symbol('\\\\Psi_{nl}', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Symbol('\\\\pi', commutative=True), Function('T')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}} and A{(r_{0})} = (\\frac{d}{d r_{0}} e^{r_{0}})^{r_{0}}, then obtain r_{0} + \\operatorname{n_{2}}^{r_{0}}{(r_{0})} = r_{0} + A{(r_{0})}", "derivation": "\\operatorname{n_{2}}{(r_{0})} = \\frac{d}{d r_{0}} e^{r_{0}} and \\operatorname{n_{2}}^{r_{0}}{(r_{0})} = (\\frac{d}{d r_{0}} e^{r_{0}})^{r_{0}} and r_{0} + \\operatorname{n_{2}}^{r_{0}}{(r_{0})} = r_{0} + (\\frac{d}{d r_{0}} e^{r_{0}})^{r_{0}} and A{(r_{0})} = (\\frac{d}{d r_{0}} e^{r_{0}})^{r_{0}} and r_{0} + \\operatorname{n_{2}}^{r_{0}}{(r_{0})} = r_{0} + A{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('r_0', commutative=True)), Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)))"], [["add", 2, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Pow(Function('n_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Pow(Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('r_0', commutative=True)), Pow(Derivative(exp(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('r_0', commutative=True), Pow(Function('n_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Function('A')(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given E{(P_{e},\\Psi_{\\lambda})} = \\frac{P_{e}}{\\Psi_{\\lambda}}, then obtain \\int (- E{(P_{e},\\Psi_{\\lambda})})^{\\Psi_{\\lambda}} dP_{e} = \\int (- \\frac{P_{e}}{\\Psi_{\\lambda}})^{\\Psi_{\\lambda}} dP_{e}", "derivation": "E{(P_{e},\\Psi_{\\lambda})} = \\frac{P_{e}}{\\Psi_{\\lambda}} and - E{(P_{e},\\Psi_{\\lambda})} = - \\frac{P_{e}}{\\Psi_{\\lambda}} and (- E{(P_{e},\\Psi_{\\lambda})})^{\\Psi_{\\lambda}} = (- \\frac{P_{e}}{\\Psi_{\\lambda}})^{\\Psi_{\\lambda}} and \\int (- E{(P_{e},\\Psi_{\\lambda})})^{\\Psi_{\\lambda}} dP_{e} = \\int (- \\frac{P_{e}}{\\Psi_{\\lambda}})^{\\Psi_{\\lambda}} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Function('E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Pow(Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} = \\mathbb{I}^{\\phi} and \\varphi{(\\mathbb{I},\\phi)} = \\int (\\mathbb{I} \\mathbb{I}^{\\phi} + \\mathbb{I}^{\\phi}) d\\mathbb{I}, then obtain \\int (\\mathbb{I} \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} + \\mathbb{I}^{\\phi}) d\\mathbb{I} = \\varphi{(\\mathbb{I},\\phi)}", "derivation": "\\operatorname{C_{d}}{(\\mathbb{I},\\phi)} = \\mathbb{I}^{\\phi} and \\mathbb{I} \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} = \\mathbb{I} \\mathbb{I}^{\\phi} and \\mathbb{I} \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} + \\mathbb{I}^{\\phi} = \\mathbb{I} \\mathbb{I}^{\\phi} + \\mathbb{I}^{\\phi} and \\int (\\mathbb{I} \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} + \\mathbb{I}^{\\phi}) d\\mathbb{I} = \\int (\\mathbb{I} \\mathbb{I}^{\\phi} + \\mathbb{I}^{\\phi}) d\\mathbb{I} and \\varphi{(\\mathbb{I},\\phi)} = \\int (\\mathbb{I} \\mathbb{I}^{\\phi} + \\mathbb{I}^{\\phi}) d\\mathbb{I} and \\int (\\mathbb{I} \\operatorname{C_{d}}{(\\mathbb{I},\\phi)} + \\mathbb{I}^{\\phi}) d\\mathbb{I} = \\varphi{(\\mathbb{I},\\phi)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True)), Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Function('\\\\varphi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\theta_2)} = \\sin{(\\theta_2)}, then derive \\frac{d}{d \\theta_2} \\lambda{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain (\\frac{d^{2}}{d \\theta_2^{2}} \\lambda{(\\theta_2)})^{2} = (\\frac{d^{2}}{d \\theta_2^{2}} \\sin{(\\theta_2)})^{2}", "derivation": "\\lambda{(\\theta_2)} = \\sin{(\\theta_2)} and \\frac{d}{d \\theta_2} \\lambda{(\\theta_2)} = \\frac{d}{d \\theta_2} \\sin{(\\theta_2)} and \\frac{d}{d \\theta_2} \\lambda{(\\theta_2)} = \\cos{(\\theta_2)} and \\frac{d^{2}}{d \\theta_2^{2}} \\lambda{(\\theta_2)} = \\frac{d}{d \\theta_2} \\cos{(\\theta_2)} and \\cos{(\\theta_2)} = \\frac{d}{d \\theta_2} \\sin{(\\theta_2)} and \\frac{d^{2}}{d \\theta_2^{2}} \\lambda{(\\theta_2)} = \\frac{d^{2}}{d \\theta_2^{2}} \\sin{(\\theta_2)} and (\\frac{d^{2}}{d \\theta_2^{2}} \\lambda{(\\theta_2)})^{2} = (\\frac{d^{2}}{d \\theta_2^{2}} \\sin{(\\theta_2)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), cos(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\theta_2', commutative=True)), Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))))"], [["power", 6, 2], "Equality(Pow(Derivative(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\rho_b)} = e^{\\rho_b} and i{(\\rho_b)} = \\int \\operatorname{M_{E}}{(\\rho_b)} d\\rho_b, then obtain \\operatorname{M_{E}}{(\\rho_b)} \\int e^{\\rho_b} d\\rho_b - e^{\\rho_b} \\int e^{\\rho_b} d\\rho_b = 0", "derivation": "\\operatorname{M_{E}}{(\\rho_b)} = e^{\\rho_b} and i{(\\rho_b)} = \\int \\operatorname{M_{E}}{(\\rho_b)} d\\rho_b and i{(\\rho_b)} = \\int e^{\\rho_b} d\\rho_b and \\operatorname{M_{E}}{(\\rho_b)} i{(\\rho_b)} = i{(\\rho_b)} e^{\\rho_b} and \\rho_b + \\operatorname{M_{E}}{(\\rho_b)} i{(\\rho_b)} = \\rho_b + i{(\\rho_b)} e^{\\rho_b} and \\operatorname{M_{E}}{(\\rho_b)} i{(\\rho_b)} - i{(\\rho_b)} e^{\\rho_b} = 0 and \\operatorname{M_{E}}{(\\rho_b)} \\int e^{\\rho_b} d\\rho_b - e^{\\rho_b} \\int e^{\\rho_b} d\\rho_b = 0", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\rho_b', commutative=True)), Integral(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('i')(Symbol('\\\\rho_b', commutative=True)), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "Function('i')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), Function('i')(Symbol('\\\\rho_b', commutative=True))), Mul(Function('i')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["add", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Mul(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), Function('i')(Symbol('\\\\rho_b', commutative=True)))), Add(Symbol('\\\\rho_b', commutative=True), Mul(Function('i')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))))"], [["minus", 5, "Add(Symbol('\\\\rho_b', commutative=True), Mul(Function('i')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], "Equality(Add(Mul(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), Function('i')(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Function('i')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Function('M_E')(Symbol('\\\\rho_b', commutative=True)), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\rho_b', commutative=True)), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{F}{(H)} = \\cos{(H)} and \\operatorname{x^{{\\}'}}{(H)} = - \\frac{\\frac{d}{d H} (\\mathbf{F}{(H)} - \\cos{(H)})}{\\cos{(H)}}, then obtain \\operatorname{x^{{\\}'}}{(H)} = - \\frac{\\frac{d}{d H} 0}{\\cos{(H)}}", "derivation": "\\mathbf{F}{(H)} = \\cos{(H)} and \\mathbf{F}{(H)} - \\cos{(H)} = 0 and \\frac{d}{d H} (\\mathbf{F}{(H)} - \\cos{(H)}) = \\frac{d}{d H} 0 and - \\frac{\\frac{d}{d H} (\\mathbf{F}{(H)} - \\cos{(H)})}{\\cos{(H)}} = - \\frac{\\frac{d}{d H} 0}{\\cos{(H)}} and \\operatorname{x^{{\\}'}}{(H)} = - \\frac{\\frac{d}{d H} (\\mathbf{F}{(H)} - \\cos{(H)})}{\\cos{(H)}} and \\operatorname{x^{{\\}'}}{(H)} = - \\frac{\\frac{d}{d H} 0}{\\cos{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["minus", 1, "cos(Symbol('H', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{F}')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), cos(Symbol('H', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(cos(Symbol('H', commutative=True)), Integer(-1)), Derivative(Add(Function('\\\\mathbf{F}')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(cos(Symbol('H', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('H', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('H', commutative=True)), Integer(-1)), Derivative(Add(Function('\\\\mathbf{F}')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('x^\\\\prime')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('H', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{1}{(v_{2},\\Omega)} = \\frac{v_{2}}{\\Omega} and \\varphi{(v_{2},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\theta_{1}{(v_{2},\\Omega)}, then obtain \\varphi{(v_{2},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\frac{v_{2}}{\\Omega}", "derivation": "\\theta_{1}{(v_{2},\\Omega)} = \\frac{v_{2}}{\\Omega} and \\frac{\\partial}{\\partial \\Omega} \\theta_{1}{(v_{2},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\frac{v_{2}}{\\Omega} and \\varphi{(v_{2},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\theta_{1}{(v_{2},\\Omega)} and \\varphi{(v_{2},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\frac{v_{2}}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(\\varphi)} = e^{\\varphi}, then obtain \\frac{\\mathbf{v}^{\\varphi}{(\\varphi)}}{\\int (e^{\\varphi})^{\\varphi} d\\varphi} = \\frac{(e^{\\varphi})^{\\varphi}}{\\int (e^{\\varphi})^{\\varphi} d\\varphi}", "derivation": "\\mathbf{v}{(\\varphi)} = e^{\\varphi} and \\mathbf{v}^{\\varphi}{(\\varphi)} = (e^{\\varphi})^{\\varphi} and \\int \\mathbf{v}^{\\varphi}{(\\varphi)} d\\varphi = \\int (e^{\\varphi})^{\\varphi} d\\varphi and \\frac{\\mathbf{v}^{\\varphi}{(\\varphi)}}{\\int \\mathbf{v}^{\\varphi}{(\\varphi)} d\\varphi} = \\frac{(e^{\\varphi})^{\\varphi}}{\\int \\mathbf{v}^{\\varphi}{(\\varphi)} d\\varphi} and \\frac{\\mathbf{v}^{\\varphi}{(\\varphi)}}{\\int (e^{\\varphi})^{\\varphi} d\\varphi} = \\frac{(e^{\\varphi})^{\\varphi}}{\\int (e^{\\varphi})^{\\varphi} d\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["divide", 2, "Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Mul(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Mul(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(H)} = \\cos{(H)}, then obtain \\frac{\\frac{d}{d H} \\hat{x}{(H)}}{- \\hat{x}{(H)} + \\cos{(H)}} = \\frac{\\frac{d}{d H} (- \\hat{x}{(H)} + 2 \\cos{(H)})}{- \\hat{x}{(H)} + \\cos{(H)}}", "derivation": "\\hat{x}{(H)} = \\cos{(H)} and \\cos{(H)} = - \\hat{x}{(H)} + 2 \\cos{(H)} and \\hat{x}{(H)} = - \\hat{x}{(H)} + 2 \\cos{(H)} and \\frac{d}{d H} \\hat{x}{(H)} = \\frac{d}{d H} (- \\hat{x}{(H)} + 2 \\cos{(H)}) and \\frac{\\frac{d}{d H} \\hat{x}{(H)}}{- \\hat{x}{(H)} + \\cos{(H)}} = \\frac{\\frac{d}{d H} (- \\hat{x}{(H)} + 2 \\cos{(H)})}{- \\hat{x}{(H)} + \\cos{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True)))"], "Equality(cos(Symbol('H', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('\\\\hat{x}')(Symbol('H', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 4, "Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(\\hat{X},\\nabla)} = \\frac{\\hat{X}}{\\nabla} and \\phi{(\\hat{X},\\nabla)} = e^{L{(\\hat{X},\\nabla)}}, then obtain ((e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla})^{\\hat{X}} + \\phi^{\\nabla}{(\\hat{X},\\nabla)} = ((e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla})^{\\hat{X}} + (e^{L{(\\hat{X},\\nabla)}})^{\\nabla}", "derivation": "L{(\\hat{X},\\nabla)} = \\frac{\\hat{X}}{\\nabla} and e^{L{(\\hat{X},\\nabla)}} = e^{\\frac{\\hat{X}}{\\nabla}} and (e^{L{(\\hat{X},\\nabla)}})^{\\nabla} = (e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla} and \\phi{(\\hat{X},\\nabla)} = e^{L{(\\hat{X},\\nabla)}} and \\phi^{\\nabla}{(\\hat{X},\\nabla)} = (e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla} and \\phi^{\\nabla}{(\\hat{X},\\nabla)} = (e^{L{(\\hat{X},\\nabla)}})^{\\nabla} and ((e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla})^{\\hat{X}} + \\phi^{\\nabla}{(\\hat{X},\\nabla)} = ((e^{\\frac{\\hat{X}}{\\nabla}})^{\\nabla})^{\\hat{X}} + (e^{L{(\\hat{X},\\nabla)}})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["exp", 1], "Equality(exp(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True))), exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(exp(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('\\\\nabla', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(exp(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["add", 6, "Pow(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Add(Pow(Pow(exp(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(exp(Function('L')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(l)} = \\log{(l)} and \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} = (e^{f_{\\mathbf{p}}})^{\\hat{x}_0}, then obtain - \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} - \\frac{\\int \\mathbf{g}{(l)} dl}{l} = - (e^{f_{\\mathbf{p}}})^{\\hat{x}_0} - \\frac{\\int \\mathbf{g}{(l)} dl}{l}", "derivation": "\\mathbf{g}{(l)} = \\log{(l)} and \\int \\mathbf{g}{(l)} dl = \\int \\log{(l)} dl and \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} = (e^{f_{\\mathbf{p}}})^{\\hat{x}_0} and - \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} = - (e^{f_{\\mathbf{p}}})^{\\hat{x}_0} and - \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} - \\frac{\\int \\log{(l)} dl}{l} = - (e^{f_{\\mathbf{p}}})^{\\hat{x}_0} - \\frac{\\int \\log{(l)} dl}{l} and - \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},\\hat{x}_0)} - \\frac{\\int \\mathbf{g}{(l)} dl}{l} = - (e^{f_{\\mathbf{p}}})^{\\hat{x}_0} - \\frac{\\int \\mathbf{g}{(l)} dl}{l}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Pow(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 4, "Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Integral(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Integral(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Add(Mul(Integer(-1), Pow(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Integral(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{g}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Add(Mul(Integer(-1), Pow(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{g}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger}) and \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},A_{x})} = \\int \\frac{\\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})}}{A_{x}} dA_{x}, then obtain \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},A_{x})} = \\int \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger})}{A_{x}} dA_{x}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger}) and \\frac{\\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})}}{A_{x}} = \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger})}{A_{x}} and \\int \\frac{\\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})}}{A_{x}} dA_{x} = \\int \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger})}{A_{x}} dA_{x} and \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},A_{x})} = \\int \\frac{\\operatorname{L_{\\varepsilon}}{(\\Psi^{\\dagger},A_{x})}}{A_{x}} dA_{x} and \\dot{\\mathbf{r}}{(\\Psi^{\\dagger},A_{x})} = \\int \\frac{\\frac{\\partial}{\\partial A_{x}} (A_{x} - \\Psi^{\\dagger})}{A_{x}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Tuple(Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)), Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('A_x', commutative=True)), Integral(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Tuple(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})} = J_{\\varepsilon} - r_{0} and \\mathbf{p}{(r_{0},J_{\\varepsilon})} = J_{\\varepsilon} - r_{0}, then obtain \\int \\frac{\\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})}}{\\mathbf{p}{(r_{0},J_{\\varepsilon})}} dr_{0} = \\int 1 dr_{0}", "derivation": "\\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})} = J_{\\varepsilon} - r_{0} and \\frac{\\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})}}{J_{\\varepsilon} - r_{0}} = 1 and \\mathbf{p}{(r_{0},J_{\\varepsilon})} = J_{\\varepsilon} - r_{0} and \\frac{\\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})}}{\\mathbf{p}{(r_{0},J_{\\varepsilon})}} = 1 and \\int \\frac{\\operatorname{A_{x}}{(r_{0},J_{\\varepsilon})}}{\\mathbf{p}{(r_{0},J_{\\varepsilon})}} dr_{0} = \\int 1 dr_{0}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["divide", 1, "Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(-1)), Function('A_x')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('A_x')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 4, "Symbol('r_0', commutative=True)"], "Equality(Integral(Mul(Function('A_x')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('r_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(W)} = \\cos{(W)}, then derive \\int \\Psi_{\\lambda}{(W)} dW = \\pi + \\sin{(W)}, then obtain - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\int \\cos{(W)} dW = \\pi - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\sin{(W)}", "derivation": "\\Psi_{\\lambda}{(W)} = \\cos{(W)} and \\int \\Psi_{\\lambda}{(W)} dW = \\int \\cos{(W)} dW and \\int \\Psi_{\\lambda}{(W)} dW = \\pi + \\sin{(W)} and - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\int \\Psi_{\\lambda}{(W)} dW = \\pi - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\sin{(W)} and - \\tilde{g} - \\cos{(W)} + \\int \\cos{(W)} dW = \\pi - \\tilde{g} + \\sin{(W)} - \\cos{(W)} and - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\int \\cos{(W)} dW = \\pi - \\tilde{g} - \\Psi_{\\lambda}{(W)} + \\sin{(W)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('W', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True))), sin(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True))), sin(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{S})} = \\cos{(\\cos{(\\mathbf{S})})}, then derive \\varphi + \\mathbf{H}{(\\mathbf{S})} = \\hat{H} + \\cos{(\\cos{(\\mathbf{S})})}, then obtain \\varphi + \\cos{(\\cos{(\\mathbf{S})})} = \\varphi + \\mathbf{H}{(\\mathbf{S})}", "derivation": "\\mathbf{H}{(\\mathbf{S})} = \\cos{(\\cos{(\\mathbf{S})})} and \\frac{d}{d \\mathbf{S}} \\mathbf{H}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\cos{(\\cos{(\\mathbf{S})})} and \\int \\frac{d}{d \\mathbf{S}} \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S} = \\int \\frac{d}{d \\mathbf{S}} \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S} and \\varphi + \\mathbf{H}{(\\mathbf{S})} = \\hat{H} + \\cos{(\\cos{(\\mathbf{S})})} and \\varphi + \\mathbf{H}{(\\mathbf{S})} = \\hat{H} + \\mathbf{H}{(\\mathbf{S})} and \\varphi + \\cos{(\\cos{(\\mathbf{S})})} = \\hat{H} + \\cos{(\\cos{(\\mathbf{S})})} and \\varphi + \\cos{(\\cos{(\\mathbf{S})})} = \\varphi + \\mathbf{H}{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Derivative(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), cos(cos(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\varphi', commutative=True), cos(cos(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), cos(cos(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('\\\\varphi', commutative=True), cos(cos(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(i)} = \\frac{\\log{(i)}}{i}, then derive \\int \\mathbf{F}{(i)} di = \\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2}, then derive \\frac{d}{d i} \\int \\mathbf{F}{(i)} di = \\frac{\\log{(i)}}{i}, then obtain - \\log{(i)} + \\frac{\\partial}{\\partial i} (\\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2}) = - \\log{(i)} + \\frac{\\log{(i)}}{i}", "derivation": "\\mathbf{F}{(i)} = \\frac{\\log{(i)}}{i} and \\int \\mathbf{F}{(i)} di = \\int \\frac{\\log{(i)}}{i} di and \\int \\mathbf{F}{(i)} di = \\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2} and \\frac{d}{d i} \\int \\mathbf{F}{(i)} di = \\frac{\\partial}{\\partial i} (\\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2}) and \\frac{d}{d i} \\int \\mathbf{F}{(i)} di = \\frac{\\log{(i)}}{i} and \\frac{\\partial}{\\partial i} (\\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2}) = \\frac{\\log{(i)}}{i} and - \\log{(i)} + \\frac{\\partial}{\\partial i} (\\tilde{g}^* + \\frac{\\log{(i)}^{2}}{2}) = - \\log{(i)} + \\frac{\\log{(i)}}{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('i', commutative=True)), Integer(2)))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('i', commutative=True)), Integer(2)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('i', commutative=True)), Integer(2)))), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(Symbol('i', commutative=True))))"], [["minus", 6, "log(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('i', commutative=True))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('i', commutative=True)), Integer(2)))), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\phi_2,\\hat{H}_l)} = \\hat{H}_l + \\phi_2, then obtain \\cos^{\\hat{H}_l}{(\\tilde{g}^{\\hat{H}_l}{(\\phi_2,\\hat{H}_l)})} = \\cos^{\\hat{H}_l}{((\\hat{H}_l + \\phi_2)^{\\hat{H}_l})}", "derivation": "\\tilde{g}{(\\phi_2,\\hat{H}_l)} = \\hat{H}_l + \\phi_2 and \\tilde{g}^{\\hat{H}_l}{(\\phi_2,\\hat{H}_l)} = (\\hat{H}_l + \\phi_2)^{\\hat{H}_l} and \\cos{(\\tilde{g}^{\\hat{H}_l}{(\\phi_2,\\hat{H}_l)})} = \\cos{((\\hat{H}_l + \\phi_2)^{\\hat{H}_l})} and \\cos^{\\hat{H}_l}{(\\tilde{g}^{\\hat{H}_l}{(\\phi_2,\\hat{H}_l)})} = \\cos^{\\hat{H}_l}{((\\hat{H}_l + \\phi_2)^{\\hat{H}_l})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), cos(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(cos(Pow(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(cos(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\omega{(\\hat{H})} = \\sin{(\\hat{H})} and \\hat{H}_l{(\\hat{H})} = \\int \\omega{(\\hat{H})} d\\hat{H} and \\Omega{(\\hat{H})} = \\int \\omega{(\\hat{H})} d\\hat{H}, then obtain \\int \\Omega{(\\hat{H})} d\\hat{H} = \\int \\hat{H}_l{(\\hat{H})} d\\hat{H}", "derivation": "\\omega{(\\hat{H})} = \\sin{(\\hat{H})} and \\hat{H}_l{(\\hat{H})} = \\int \\omega{(\\hat{H})} d\\hat{H} and \\hat{H}_l{(\\hat{H})} = \\int \\sin{(\\hat{H})} d\\hat{H} and \\Omega{(\\hat{H})} = \\int \\omega{(\\hat{H})} d\\hat{H} and \\Omega{(\\hat{H})} = \\int \\sin{(\\hat{H})} d\\hat{H} and \\Omega{(\\hat{H})} = \\hat{H}_l{(\\hat{H})} and \\int \\Omega{(\\hat{H})} d\\hat{H} = \\int \\hat{H}_l{(\\hat{H})} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{H}', commutative=True)), Integral(Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{H}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{H}', commutative=True)), Integral(Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\Omega')(Symbol('\\\\hat{H}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\Omega')(Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 6, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given L{(\\theta,g)} = \\sin{(g^{\\theta})} and \\tilde{g}{(Z,\\Omega)} = \\frac{\\Omega}{Z}, then obtain g^{\\theta} (\\tilde{g}{(Z,\\Omega)} + \\sin{(\\frac{\\Omega}{Z})}) - L^{\\theta}{(\\theta,g)} = g^{\\theta} (\\sin{(\\frac{\\Omega}{Z})} + \\frac{\\Omega}{Z}) - L^{\\theta}{(\\theta,g)}", "derivation": "L{(\\theta,g)} = \\sin{(g^{\\theta})} and \\tilde{g}{(Z,\\Omega)} = \\frac{\\Omega}{Z} and \\tilde{g}{(Z,\\Omega)} + \\sin{(\\frac{\\Omega}{Z})} = \\sin{(\\frac{\\Omega}{Z})} + \\frac{\\Omega}{Z} and g^{\\theta} (\\tilde{g}{(Z,\\Omega)} + \\sin{(\\frac{\\Omega}{Z})}) = g^{\\theta} (\\sin{(\\frac{\\Omega}{Z})} + \\frac{\\Omega}{Z}) and g^{\\theta} (\\tilde{g}{(Z,\\Omega)} + \\sin{(\\frac{\\Omega}{Z})}) - \\sin^{\\theta}{(g^{\\theta})} = g^{\\theta} (\\sin{(\\frac{\\Omega}{Z})} + \\frac{\\Omega}{Z}) - \\sin^{\\theta}{(g^{\\theta})} and g^{\\theta} (\\tilde{g}{(Z,\\Omega)} + \\sin{(\\frac{\\Omega}{Z})}) - L^{\\theta}{(\\theta,g)} = g^{\\theta} (\\sin{(\\frac{\\Omega}{Z})} + \\frac{\\Omega}{Z}) - L^{\\theta}{(\\theta,g)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta', commutative=True), Symbol('g', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], [["add", 2, "sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))), Add(sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))))"], [["times", 3, "Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))))), Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))))"], [["minus", 4, "Pow(sin(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))))), Mul(Integer(-1), Pow(sin(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Pow(sin(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))))), Mul(Integer(-1), Pow(Function('L')(Symbol('\\\\theta', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\theta', commutative=True)))), Add(Mul(Pow(Symbol('g', commutative=True), Symbol('\\\\theta', commutative=True)), Add(sin(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Pow(Function('L')(Symbol('\\\\theta', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\Psi)} = \\cos{(\\Psi)}, then obtain (\\int \\frac{d}{d \\Psi} \\Psi_{nl}{(\\Psi)} d\\Psi)^{\\Psi} = (\\int \\frac{d}{d \\Psi} \\cos{(\\Psi)} d\\Psi)^{\\Psi}", "derivation": "\\Psi_{nl}{(\\Psi)} = \\cos{(\\Psi)} and \\frac{d}{d \\Psi} \\Psi_{nl}{(\\Psi)} = \\frac{d}{d \\Psi} \\cos{(\\Psi)} and \\int \\frac{d}{d \\Psi} \\Psi_{nl}{(\\Psi)} d\\Psi = \\int \\frac{d}{d \\Psi} \\cos{(\\Psi)} d\\Psi and (\\int \\frac{d}{d \\Psi} \\Psi_{nl}{(\\Psi)} d\\Psi)^{\\Psi} = (\\int \\frac{d}{d \\Psi} \\cos{(\\Psi)} d\\Psi)^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Integral(Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(C,\\psi^*)} = \\cos{(C - \\psi^*)} and c{(C,\\psi^*)} = \\int \\theta_{2}{(C,\\psi^*)} dC, then obtain c{(C,\\psi^*)} - 1 = \\int \\cos{(C - \\psi^*)} dC - 1", "derivation": "\\theta_{2}{(C,\\psi^*)} = \\cos{(C - \\psi^*)} and c{(C,\\psi^*)} = \\int \\theta_{2}{(C,\\psi^*)} dC and c{(C,\\psi^*)} = \\int \\cos{(C - \\psi^*)} dC and c{(C,\\psi^*)} - 1 = \\int \\cos{(C - \\psi^*)} dC - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('C', commutative=True), Symbol('\\\\psi^*', commutative=True)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('C', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('C', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('c')(Symbol('C', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('c')(Symbol('C', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Add(Integral(cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A_{2},c)} = \\int (- A_{2} + c) dc, then derive \\frac{\\partial}{\\partial c} (- A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2}) = - \\frac{A_{2}^{2}}{2} + A_{2} c + 1, then obtain - c + \\frac{\\partial}{\\partial c} (- A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2}) = - \\frac{A_{2}^{2}}{2} + A_{2} c - c + 1", "derivation": "\\operatorname{E_{n}}{(A_{2},c)} = \\int (- A_{2} + c) dc and \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2} = \\iint (- A_{2} + c) dc dA_{2} and - A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2} = - A_{2} + c + \\iint (- A_{2} + c) dc dA_{2} and \\frac{\\partial}{\\partial c} (- A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2}) = \\frac{\\partial}{\\partial c} (- A_{2} + c + \\iint (- A_{2} + c) dc dA_{2}) and \\frac{\\partial}{\\partial c} (- A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2}) = - \\frac{A_{2}^{2}}{2} + A_{2} c + 1 and - c + \\frac{\\partial}{\\partial c} (- A_{2} + c + \\int \\operatorname{E_{n}}{(A_{2},c)} dA_{2}) = - \\frac{A_{2}^{2}}{2} + A_{2} c - c + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Integer(1)))"], [["minus", 5, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('c', commutative=True), Integral(Function('E_n')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(a)} = \\cos{(a)}, then obtain - \\operatorname{g_{\\varepsilon}}{(a)} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da = \\omega - \\operatorname{g_{\\varepsilon}}{(a)} + \\sin{(a)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(a)} = \\cos{(a)} and \\int \\operatorname{g_{\\varepsilon}}{(a)} da = \\int \\cos{(a)} da and - \\operatorname{g_{\\varepsilon}}{(a)} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da = - \\operatorname{g_{\\varepsilon}}{(a)} + \\int \\cos{(a)} da and - \\operatorname{g_{\\varepsilon}}{(a)} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da = \\omega - \\operatorname{g_{\\varepsilon}}{(a)} + \\sin{(a)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["minus", 2, "Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True))), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True))), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True))), sin(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\mathbb{I},Q)} = \\mathbb{I} \\log{(Q)}, then obtain (- 2 \\mathbb{I} + 2 \\nabla{(\\mathbb{I},Q)})^{2} = (- 2 \\mathbb{I} + 2 \\nabla{(\\mathbb{I},Q)}) (\\mathbb{I} \\log{(Q)} - 2 \\mathbb{I} + \\nabla{(\\mathbb{I},Q)})", "derivation": "\\nabla{(\\mathbb{I},Q)} = \\mathbb{I} \\log{(Q)} and - \\mathbb{I} + \\nabla{(\\mathbb{I},Q)} = \\mathbb{I} \\log{(Q)} - \\mathbb{I} and - 2 \\mathbb{I} + 2 \\nabla{(\\mathbb{I},Q)} = \\mathbb{I} \\log{(Q)} - 2 \\mathbb{I} + \\nabla{(\\mathbb{I},Q)} and (- 2 \\mathbb{I} + 2 \\nabla{(\\mathbb{I},Q)})^{2} = (- 2 \\mathbb{I} + 2 \\nabla{(\\mathbb{I},Q)}) (\\mathbb{I} \\log{(Q)} - 2 \\mathbb{I} + \\nabla{(\\mathbb{I},Q)})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('Q', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))))"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))), Integer(2)), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(S)} = e^{S}, then obtain \\log{((- \\operatorname{a^{\\dagger}}{(S)} + e^{S})^{S})} = 0", "derivation": "\\operatorname{a^{\\dagger}}{(S)} = e^{S} and 0 = - \\operatorname{a^{\\dagger}}{(S)} + e^{S} and 0^{S} = (- \\operatorname{a^{\\dagger}}{(S)} + e^{S})^{S} and \\log{(0^{S})} = \\log{((- \\operatorname{a^{\\dagger}}{(S)} + e^{S})^{S})} and \\log{((- \\operatorname{a^{\\dagger}}{(S)} + e^{S})^{S})} = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["minus", 1, "Function('a^{\\\\dagger}')(Symbol('S', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Integer(0), Symbol('S', commutative=True)), Pow(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["log", 3], "Equality(log(Pow(Integer(0), Symbol('S', commutative=True))), log(Pow(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(Pow(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), Symbol('S', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(\\pi,\\mathbf{M})} = \\frac{\\cos{(\\pi)}}{\\mathbf{M}}, then derive \\frac{\\partial}{\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})} = - \\frac{\\sin{(\\pi)}}{\\mathbf{M}}, then obtain \\cos{(e^{\\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})}})} = \\cos{(e^{\\frac{\\partial}{\\partial \\mathbf{M}} - \\frac{\\sin{(\\pi)}}{\\mathbf{M}}})}", "derivation": "\\phi_{1}{(\\pi,\\mathbf{M})} = \\frac{\\cos{(\\pi)}}{\\mathbf{M}} and \\frac{\\partial}{\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})} = \\frac{\\partial}{\\partial \\pi} \\frac{\\cos{(\\pi)}}{\\mathbf{M}} and \\frac{\\partial}{\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})} = - \\frac{\\sin{(\\pi)}}{\\mathbf{M}} and \\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} - \\frac{\\sin{(\\pi)}}{\\mathbf{M}} and e^{\\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})}} = e^{\\frac{\\partial}{\\partial \\mathbf{M}} - \\frac{\\sin{(\\pi)}}{\\mathbf{M}}} and \\cos{(e^{\\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\pi} \\phi_{1}{(\\pi,\\mathbf{M})}})} = \\cos{(e^{\\frac{\\partial}{\\partial \\mathbf{M}} - \\frac{\\sin{(\\pi)}}{\\mathbf{M}}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(Derivative(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), exp(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))"], [["cos", 5], "Equality(cos(exp(Derivative(Function('\\\\phi_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))), cos(exp(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given S{(\\delta,M_{E})} = e^{\\frac{M_{E}}{\\delta}}, then obtain \\int \\delta S^{\\delta}{(\\delta,M_{E})} d\\delta - 1 = \\int \\delta (e^{\\frac{M_{E}}{\\delta}})^{\\delta} d\\delta - 1", "derivation": "S{(\\delta,M_{E})} = e^{\\frac{M_{E}}{\\delta}} and S^{\\delta}{(\\delta,M_{E})} = (e^{\\frac{M_{E}}{\\delta}})^{\\delta} and \\delta S^{\\delta}{(\\delta,M_{E})} = \\delta (e^{\\frac{M_{E}}{\\delta}})^{\\delta} and \\int \\delta S^{\\delta}{(\\delta,M_{E})} d\\delta = \\int \\delta (e^{\\frac{M_{E}}{\\delta}})^{\\delta} d\\delta and \\int \\delta S^{\\delta}{(\\delta,M_{E})} d\\delta - 1 = \\int \\delta (e^{\\frac{M_{E}}{\\delta}})^{\\delta} d\\delta - 1", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(exp(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True)))"], [["times", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(exp(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(exp(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Integral(Mul(Symbol('\\\\delta', commutative=True), Pow(exp(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain \\frac{\\mathbf{r} \\varepsilon{(\\mathbf{r})}}{\\cos{(\\mathbf{r})}} - 1 = \\mathbf{r} - 1", "derivation": "\\varepsilon{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\mathbf{r} \\varepsilon{(\\mathbf{r})} = \\mathbf{r} \\cos{(\\mathbf{r})} and \\mathbf{r}^{2} \\varepsilon{(\\mathbf{r})} = \\mathbf{r}^{2} \\cos{(\\mathbf{r})} and \\frac{\\mathbf{r} \\varepsilon{(\\mathbf{r})}}{\\cos{(\\mathbf{r})}} = \\mathbf{r} and \\frac{\\mathbf{r} \\varepsilon{(\\mathbf{r})}}{\\cos{(\\mathbf{r})}} - 1 = \\mathbf{r} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)), Function('\\\\varepsilon')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{r}', commutative=True), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{r}', commutative=True))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Integer(-1)), Add(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{r}{(A_{y})} = \\cos{(\\cos{(A_{y})})} and m{(A_{y})} = \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}}, then obtain \\frac{d}{d A_{y}} m^{A_{y}}{(A_{y})} = \\frac{d}{d A_{y}} \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}}", "derivation": "\\mathbf{r}{(A_{y})} = \\cos{(\\cos{(A_{y})})} and 1 = \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}} and \\frac{d}{d A_{y}} 1 = \\frac{d}{d A_{y}} \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}} and m{(A_{y})} = \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}} and m{(A_{y})} = 1 and m^{A_{y}}{(A_{y})} = 1 and \\frac{d}{d A_{y}} m^{A_{y}}{(A_{y})} = \\frac{d}{d A_{y}} 1 and \\frac{d}{d A_{y}} m^{A_{y}}{(A_{y})} = \\frac{d}{d A_{y}} \\frac{\\cos{(\\cos{(A_{y})})}}{\\mathbf{r}{(A_{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), cos(cos(Symbol('A_y', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Integer(-1)), cos(cos(Symbol('A_y', commutative=True)))))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Integer(-1)), cos(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('m')(Symbol('A_y', commutative=True)), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Integer(-1)), cos(cos(Symbol('A_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('m')(Symbol('A_y', commutative=True)), Integer(1))"], [["power", 5, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('m')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Integer(1))"], [["differentiate", 6, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Pow(Function('m')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Derivative(Pow(Function('m')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Integer(-1)), cos(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(A_{y})} = e^{A_{y}}, then derive \\int u{(A_{y})} dA_{y} = \\mathbf{F} + e^{A_{y}}, then obtain \\int \\frac{d}{d \\mathbf{F}} \\int u{(A_{y})} dA_{y} d\\mathbf{F} = V_{\\mathbf{B}} + \\mathbf{F}", "derivation": "u{(A_{y})} = e^{A_{y}} and \\int u{(A_{y})} dA_{y} = \\int e^{A_{y}} dA_{y} and \\int u{(A_{y})} dA_{y} = \\mathbf{F} + e^{A_{y}} and \\frac{d}{d \\mathbf{F}} \\int u{(A_{y})} dA_{y} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} + e^{A_{y}}) and \\int \\frac{d}{d \\mathbf{F}} \\int u{(A_{y})} dA_{y} d\\mathbf{F} = \\int \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} + e^{A_{y}}) d\\mathbf{F} and \\int \\frac{d}{d \\mathbf{F}} \\int u{(A_{y})} dA_{y} d\\mathbf{F} = \\int \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} + u{(A_{y})}) d\\mathbf{F} and \\int \\frac{d}{d \\mathbf{F}} \\int u{(A_{y})} dA_{y} d\\mathbf{F} = V_{\\mathbf{B}} + \\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('A_y', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Derivative(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('u')(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Derivative(Integral(Function('u')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given J{(\\tilde{g})} = \\sin{(\\sin{(\\tilde{g})})} and \\nabla{(\\tilde{g})} = \\sin{(\\tilde{g})}, then obtain \\frac{J{(\\tilde{g})} + \\sin{(\\nabla{(\\tilde{g})})}}{2 J{(\\tilde{g})}} = \\frac{J{(\\tilde{g})} + \\sin{(\\sin{(\\tilde{g})})}}{2 J{(\\tilde{g})}}", "derivation": "J{(\\tilde{g})} = \\sin{(\\sin{(\\tilde{g})})} and 2 J{(\\tilde{g})} = J{(\\tilde{g})} + \\sin{(\\sin{(\\tilde{g})})} and \\nabla{(\\tilde{g})} = \\sin{(\\tilde{g})} and 2 J{(\\tilde{g})} = J{(\\tilde{g})} + \\sin{(\\nabla{(\\tilde{g})})} and J{(\\tilde{g})} + \\sin{(\\nabla{(\\tilde{g})})} = J{(\\tilde{g})} + \\sin{(\\sin{(\\tilde{g})})} and \\frac{J{(\\tilde{g})} + \\sin{(\\nabla{(\\tilde{g})})}}{2 J{(\\tilde{g})}} = \\frac{J{(\\tilde{g})} + \\sin{(\\sin{(\\tilde{g})})}}{2 J{(\\tilde{g})}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(sin(Symbol('\\\\tilde{g}', commutative=True))))"], [["add", 1, "Function('J')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('J')(Symbol('\\\\tilde{g}', commutative=True))), Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(sin(Symbol('\\\\tilde{g}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('J')(Symbol('\\\\tilde{g}', commutative=True))), Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)))), Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(sin(Symbol('\\\\tilde{g}', commutative=True)))))"], [["divide", 5, "Mul(Integer(2), Function('J')(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)))), Pow(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), sin(sin(Symbol('\\\\tilde{g}', commutative=True)))), Pow(Function('J')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = \\frac{A_{2}}{\\mathbf{r} t_{2}}, then obtain t_{2} \\frac{\\partial}{\\partial t_{2}} \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} + \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = 0", "derivation": "\\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = \\frac{A_{2}}{\\mathbf{r} t_{2}} and t_{2} \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = \\frac{A_{2}}{\\mathbf{r}} and \\frac{\\partial}{\\partial t_{2}} t_{2} \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = \\frac{\\partial}{\\partial t_{2}} \\frac{A_{2}}{\\mathbf{r}} and t_{2} \\frac{\\partial}{\\partial t_{2}} \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} + \\mathbf{J}_M{(A_{2},\\mathbf{r},t_{2})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('t_2', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('t_2', commutative=True), Derivative(Function('\\\\mathbf{J}_M')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Function('\\\\mathbf{J}_M')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('t_2', commutative=True))), Integer(0))"]]}, {"prompt": "Given v{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}, then derive v{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}}, then obtain \\frac{v{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{1}{\\mathbf{F}^{2}}", "derivation": "v{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and \\frac{v{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}}{\\mathbf{F}} and v{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}} and \\frac{1}{\\mathbf{F}} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and \\frac{v{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{1}{\\mathbf{F}^{2}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('v')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('v')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('v')(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)}, then derive \\int \\operatorname{J_{\\varepsilon}}{(\\delta)} d\\delta = k + \\cos{(\\delta)}, then derive k + \\cos{(\\delta)} = \\mathbf{P} + \\cos{(\\delta)}, then obtain \\frac{(\\mathbf{P} + \\cos{(\\delta)}) \\cos{(\\delta)}}{k + \\cos{(\\delta)}} = \\frac{\\cos{(\\delta)} \\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta}{k + \\cos{(\\delta)}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and \\int \\operatorname{J_{\\varepsilon}}{(\\delta)} d\\delta = \\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta and \\int \\operatorname{J_{\\varepsilon}}{(\\delta)} d\\delta = k + \\cos{(\\delta)} and k + \\cos{(\\delta)} = \\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta and k + \\cos{(\\delta)} = \\mathbf{P} + \\cos{(\\delta)} and \\mathbf{P} + \\cos{(\\delta)} = \\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta and \\frac{\\mathbf{P} + \\cos{(\\delta)}}{k + \\cos{(\\delta)}} = \\frac{\\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta}{k + \\cos{(\\delta)}} and \\frac{(\\mathbf{P} + \\cos{(\\delta)}) \\cos{(\\delta)}}{k + \\cos{(\\delta)}} = \\frac{\\cos{(\\delta)} \\int \\frac{d}{d \\delta} \\cos{(\\delta)} d\\delta}{k + \\cos{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integral(Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), cos(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integral(Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 6, "Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Pow(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integer(-1)), Integral(Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["times", 7, "cos(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Pow(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integer(-1)), cos(Symbol('\\\\delta', commutative=True))), Mul(Pow(Add(Symbol('k', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Integer(-1)), cos(Symbol('\\\\delta', commutative=True)), Integral(Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{s},V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}}^{\\mathbf{s}})} and \\psi{(\\mathbf{s})} = \\mathbf{s}, then obtain \\operatorname{A_{1}}^{\\mathbf{s}}{(\\mathbf{s},V_{\\mathbf{E}})} + \\frac{d}{d \\mathbf{s}} \\psi{(\\mathbf{s})} = \\operatorname{A_{1}}^{\\mathbf{s}}{(\\mathbf{s},V_{\\mathbf{E}})} + \\frac{d}{d \\mathbf{s}} \\mathbf{s}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{s},V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}}^{\\mathbf{s}})} and \\psi{(\\mathbf{s})} = \\mathbf{s} and \\frac{d}{d \\mathbf{s}} \\psi{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\mathbf{s} and \\cos^{\\mathbf{s}}{(V_{\\mathbf{E}}^{\\mathbf{s}})} + \\frac{d}{d \\mathbf{s}} \\psi{(\\mathbf{s})} = \\cos^{\\mathbf{s}}{(V_{\\mathbf{E}}^{\\mathbf{s}})} + \\frac{d}{d \\mathbf{s}} \\mathbf{s} and \\operatorname{A_{1}}^{\\mathbf{s}}{(\\mathbf{s},V_{\\mathbf{E}})} + \\frac{d}{d \\mathbf{s}} \\psi{(\\mathbf{s})} = \\operatorname{A_{1}}^{\\mathbf{s}}{(\\mathbf{s},V_{\\mathbf{E}})} + \\frac{d}{d \\mathbf{s}} \\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["add", 3, "Pow(cos(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Pow(cos(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Pow(cos(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('A_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Pow(Function('A_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(L)} = \\sin{(\\cos{(L)})}, then obtain \\operatorname{J_{\\varepsilon}}^{L}{(L)} \\log{(\\operatorname{J_{\\varepsilon}}^{L}{(L)})} = \\operatorname{J_{\\varepsilon}}^{L}{(L)} \\log{(\\sin^{L}{(\\cos{(L)})})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(L)} = \\sin{(\\cos{(L)})} and \\operatorname{J_{\\varepsilon}}^{L}{(L)} = \\sin^{L}{(\\cos{(L)})} and \\log{(\\operatorname{J_{\\varepsilon}}^{L}{(L)})} = \\log{(\\sin^{L}{(\\cos{(L)})})} and \\operatorname{J_{\\varepsilon}}^{L}{(L)} \\log{(\\operatorname{J_{\\varepsilon}}^{L}{(L)})} = \\operatorname{J_{\\varepsilon}}^{L}{(L)} \\log{(\\sin^{L}{(\\cos{(L)})})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), sin(cos(Symbol('L', commutative=True))))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(sin(cos(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), log(Pow(sin(cos(Symbol('L', commutative=True))), Symbol('L', commutative=True))))"], [["times", 3, "Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), log(Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), log(Pow(sin(cos(Symbol('L', commutative=True))), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given g{(\\hat{H}_{\\lambda})} = e^{\\hat{H}_{\\lambda}}, then derive - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} = 0, then obtain - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} + \\frac{\\cos{(v_{2})}}{f_{E}} = \\frac{\\cos{(v_{2})}}{f_{E}}", "derivation": "g{(\\hat{H}_{\\lambda})} = e^{\\hat{H}_{\\lambda}} and \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} e^{\\hat{H}_{\\lambda}} and - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} = - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} e^{\\hat{H}_{\\lambda}} and - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} = 0 and - e^{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} g{(\\hat{H}_{\\lambda})} + \\frac{\\cos{(v_{2})}}{f_{E}} = \\frac{\\cos{(v_{2})}}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Derivative(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Derivative(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Derivative(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Integer(0))"], [["add", 4, "Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Derivative(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True)))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(A_{z},\\lambda)} = - A_{z} + \\lambda, then derive \\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)} = 1, then obtain ((\\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)})^{A_{z}})^{\\lambda} = 1", "derivation": "\\operatorname{n_{1}}{(A_{z},\\lambda)} = - A_{z} + \\lambda and \\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)} = \\frac{\\partial}{\\partial \\lambda} (- A_{z} + \\lambda) and \\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)} = 1 and (\\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)})^{A_{z}} = 1 and ((\\frac{\\partial}{\\partial \\lambda} \\operatorname{n_{1}}{(A_{z},\\lambda)})^{A_{z}})^{\\lambda} = 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('A_z', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('A_z', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('A_z', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Derivative(Function('n_1')(Symbol('A_z', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('A_z', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('n_1')(Symbol('A_z', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('A_z', commutative=True)), Symbol('\\\\lambda', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(G,k)} = k^{G} and \\operatorname{P_{g}}{(G,k)} = \\frac{\\partial}{\\partial k} k^{G} and \\theta_{2}{(k,\\varepsilon,G)} = \\operatorname{P_{g}}{(G,k)} - e^{\\varepsilon}, then obtain \\theta_{2}{(k,\\varepsilon,G)} - \\frac{1}{k} = - e^{\\varepsilon} + \\frac{\\partial}{\\partial k} \\operatorname{F_{H}}{(G,k)} - \\frac{1}{k}", "derivation": "\\operatorname{F_{H}}{(G,k)} = k^{G} and \\operatorname{P_{g}}{(G,k)} = \\frac{\\partial}{\\partial k} k^{G} and \\operatorname{P_{g}}{(G,k)} - e^{\\varepsilon} = - e^{\\varepsilon} + \\frac{\\partial}{\\partial k} k^{G} and \\theta_{2}{(k,\\varepsilon,G)} = \\operatorname{P_{g}}{(G,k)} - e^{\\varepsilon} and \\operatorname{P_{g}}{(G,k)} - e^{\\varepsilon} = - e^{\\varepsilon} + \\frac{\\partial}{\\partial k} \\operatorname{F_{H}}{(G,k)} and \\operatorname{P_{g}}{(G,k)} - e^{\\varepsilon} - \\frac{1}{k} = - e^{\\varepsilon} + \\frac{\\partial}{\\partial k} \\operatorname{F_{H}}{(G,k)} - \\frac{1}{k} and \\theta_{2}{(k,\\varepsilon,G)} - \\frac{1}{k} = - e^{\\varepsilon} + \\frac{\\partial}{\\partial k} \\operatorname{F_{H}}{(G,k)} - \\frac{1}{k}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Derivative(Pow(Symbol('k', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True))), Derivative(Pow(Symbol('k', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('G', commutative=True)), Add(Function('P_g')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('P_g')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True))), Derivative(Function('F_H')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["minus", 5, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Add(Function('P_g')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True))), Derivative(Function('F_H')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('\\\\theta_2')(Symbol('k', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon', commutative=True))), Derivative(Function('F_H')(Symbol('G', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\theta_{2}{(E,\\tilde{g})} = E + \\tilde{g}, then obtain \\theta_{2}{(E,\\tilde{g})} \\int \\theta_{2}{(E,\\tilde{g})} d\\tilde{g} = \\theta_{2}{(E,\\tilde{g})} \\int (E + \\tilde{g}) d\\tilde{g}", "derivation": "\\theta_{2}{(E,\\tilde{g})} = E + \\tilde{g} and \\int \\theta_{2}{(E,\\tilde{g})} d\\tilde{g} = \\int (E + \\tilde{g}) d\\tilde{g} and (E + \\tilde{g}) \\int \\theta_{2}{(E,\\tilde{g})} d\\tilde{g} = (E + \\tilde{g}) \\int (E + \\tilde{g}) d\\tilde{g} and \\theta_{2}{(E,\\tilde{g})} \\int \\theta_{2}{(E,\\tilde{g})} d\\tilde{g} = \\theta_{2}{(E,\\tilde{g})} \\int (E + \\tilde{g}) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(Q,\\eta^{\\prime})} = Q - \\eta^{\\prime}, then obtain (\\int 2 \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} dQ)^{2} = (\\int (\\log{(Q - \\eta^{\\prime})} + \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})}) dQ)^{2}", "derivation": "\\varphi^{*}{(Q,\\eta^{\\prime})} = Q - \\eta^{\\prime} and \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} = \\log{(Q - \\eta^{\\prime})} and 2 \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} = \\log{(Q - \\eta^{\\prime})} + \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} and \\int 2 \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} dQ = \\int (\\log{(Q - \\eta^{\\prime})} + \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})}) dQ and (\\int 2 \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})} dQ)^{2} = (\\int (\\log{(Q - \\eta^{\\prime})} + \\log{(\\varphi^{*}{(Q,\\eta^{\\prime})})}) dQ)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), log(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["add", 2, "log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Integer(2), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(log(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Integer(2), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(log(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Mul(Integer(2), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integer(2)), Pow(Integral(Add(log(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), log(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integer(2)))"]]}, {"prompt": "Given U{(A_{1})} = \\cos{(A_{1})}, then derive \\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1} = A_{1} + \\hat{H}_{\\lambda}, then obtain (\\frac{\\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1}}{A_{1} + \\hat{H}_{\\lambda}})^{A_{1}} = 1", "derivation": "U{(A_{1})} = \\cos{(A_{1})} and \\frac{U{(A_{1})}}{\\cos{(A_{1})}} = 1 and \\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1} = \\int 1 dA_{1} and \\frac{\\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1}}{\\int 1 dA_{1}} = 1 and (\\frac{\\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1}}{\\int 1 dA_{1}})^{A_{1}} = 1 and \\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1} = A_{1} + \\hat{H}_{\\lambda} and A_{1} + \\hat{H}_{\\lambda} = \\int 1 dA_{1} and (\\frac{\\int \\frac{U{(A_{1})}}{\\cos{(A_{1})}} dA_{1}}{A_{1} + \\hat{H}_{\\lambda}})^{A_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))"], [["divide", 1, "cos(Symbol('A_1', commutative=True))"], "Equality(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))))"], [["divide", 3, "Integral(Integer(1), Tuple(Symbol('A_1', commutative=True)))"], "Equality(Mul(Pow(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Integer(-1)), Integral(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True)))), Integer(1))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Pow(Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))), Integer(-1)), Integral(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(1))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Add(Symbol('A_1', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Pow(Mul(Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Integral(Mul(Function('U')(Symbol('A_1', commutative=True)), Pow(cos(Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\dot{x}{(g,\\theta_1)} = \\theta_1 + g and \\operatorname{A_{2}}{(\\theta_1)} = \\theta_1^{\\theta_1}, then obtain \\operatorname{A_{2}}{(\\theta_1)} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})} = \\theta_1^{\\theta_1} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})}", "derivation": "\\dot{x}{(g,\\theta_1)} = \\theta_1 + g and - g + \\dot{x}{(g,\\theta_1)} = \\theta_1 and (- g + \\dot{x}{(g,\\theta_1)})^{\\theta_1} = \\theta_1^{\\theta_1} and \\operatorname{A_{2}}{(\\theta_1)} = \\theta_1^{\\theta_1} and \\operatorname{A_{2}}{(\\theta_1)} = (- g + \\dot{x}{(g,\\theta_1)})^{\\theta_1} and \\operatorname{A_{2}}{(\\theta_1)} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})} = (- g + \\dot{x}{(g,\\theta_1)})^{\\theta_1} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})} and \\operatorname{A_{2}}{(\\theta_1)} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})} = \\theta_1^{\\theta_1} - \\frac{d}{d \\mathbf{s}} \\sin{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('A_2')(Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 5, "Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))"], "Equality(Add(Function('A_2')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Add(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Function('A_2')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta{(F_{c},p)} = F_{c} + \\cos{(p)}, then obtain (F_{c} + \\cos{(p)})^{p} \\int \\eta^{p}{(F_{c},p)} dF_{c} = (F_{c} + \\cos{(p)})^{p} \\int (F_{c} + \\cos{(p)})^{p} dF_{c}", "derivation": "\\eta{(F_{c},p)} = F_{c} + \\cos{(p)} and \\eta^{p}{(F_{c},p)} = (F_{c} + \\cos{(p)})^{p} and \\int \\eta^{p}{(F_{c},p)} dF_{c} = \\int (F_{c} + \\cos{(p)})^{p} dF_{c} and \\eta^{p}{(F_{c},p)} \\int \\eta^{p}{(F_{c},p)} dF_{c} = \\eta^{p}{(F_{c},p)} \\int (F_{c} + \\cos{(p)})^{p} dF_{c} and (F_{c} + \\cos{(p)})^{p} \\int \\eta^{p}{(F_{c},p)} dF_{c} = (F_{c} + \\cos{(p)})^{p} \\int (F_{c} + \\cos{(p)})^{p} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["times", 3, "Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integral(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integral(Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Integral(Pow(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Mul(Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Integral(Pow(Add(Symbol('F_c', commutative=True), cos(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(F_{c},g,\\hbar)} = (F_{c} + g)^{\\hbar}, then obtain - 2 (F_{c} + g)^{\\hbar} + \\varphi^{*}^{F_{c}}{(F_{c},g,\\hbar)} = - 2 (F_{c} + g)^{\\hbar} + ((F_{c} + g)^{\\hbar})^{F_{c}}", "derivation": "\\varphi^{*}{(F_{c},g,\\hbar)} = (F_{c} + g)^{\\hbar} and \\varphi^{*}^{F_{c}}{(F_{c},g,\\hbar)} = ((F_{c} + g)^{\\hbar})^{F_{c}} and - (F_{c} + g)^{\\hbar} + \\varphi^{*}^{F_{c}}{(F_{c},g,\\hbar)} = - (F_{c} + g)^{\\hbar} + ((F_{c} + g)^{\\hbar})^{F_{c}} and - 2 (F_{c} + g)^{\\hbar} + \\varphi^{*}^{F_{c}}{(F_{c},g,\\hbar)} = - 2 (F_{c} + g)^{\\hbar} + ((F_{c} + g)^{\\hbar})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('F_c', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('F_c', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True)))"], [["minus", 2, "Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('F_c', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))), Pow(Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True))))"], [["minus", 3, "Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('F_c', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Integer(2), Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True))), Pow(Pow(Add(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi^{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi^{\\mathbf{p}} \\log{(\\pi)}, then obtain \\pi \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi \\pi^{\\mathbf{p}} \\log{(\\pi)}", "derivation": "\\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi^{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{H}_l{(\\mathbf{p},\\pi)} = \\frac{\\partial}{\\partial \\mathbf{p}} \\pi^{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi^{\\mathbf{p}} \\log{(\\pi)} and \\pi \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{H}_l{(\\mathbf{p},\\pi)} = \\pi \\pi^{\\mathbf{p}} \\log{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\pi', commutative=True))))"], [["times", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\dot{y})} = \\log{(\\dot{y})}, then derive \\int \\mathbf{D}{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + t_{2}, then derive \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\mathbf{F} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + t_{2}, then obtain \\Psi_{\\lambda} + \\dot{y} \\log{(\\dot{y})} - \\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\mathbf{F}", "derivation": "\\mathbf{D}{(\\dot{y})} = \\log{(\\dot{y})} and \\int \\mathbf{D}{(\\dot{y})} d\\dot{y} = \\int \\log{(\\dot{y})} d\\dot{y} and \\int \\mathbf{D}{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + t_{2} and \\int \\log{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + t_{2} and \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\mathbf{F} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + t_{2} and \\int \\log{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\mathbf{F} and \\Psi_{\\lambda} + \\dot{y} \\log{(\\dot{y})} - \\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(log(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('t_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(log(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given h{(A_{x})} = \\log{(A_{x})}, then derive \\int h{(A_{x})} dA_{x} = A_{x} \\log{(A_{x})} - A_{x} + \\mathbf{B}, then obtain \\int \\log{(A_{x})} dA_{x} = A_{x} \\log{(A_{x})} - A_{x} + \\mathbf{B}", "derivation": "h{(A_{x})} = \\log{(A_{x})} and \\int h{(A_{x})} dA_{x} = \\int \\log{(A_{x})} dA_{x} and \\int h{(A_{x})} dA_{x} = A_{x} \\log{(A_{x})} - A_{x} + \\mathbf{B} and \\int \\log{(A_{x})} dA_{x} = A_{x} \\log{(A_{x})} - A_{x} + \\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('h')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Mul(Symbol('A_x', commutative=True), log(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Mul(Symbol('A_x', commutative=True), log(Symbol('A_x', commutative=True))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(P_{g})} = \\frac{d}{d P_{g}} \\cos{(P_{g})}, then derive \\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})} = - \\cos{(P_{g})}, then obtain (\\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})})^{P_{g}} = (\\frac{d^{2}}{d P_{g}^{2}} \\cos{(P_{g})})^{P_{g}}", "derivation": "\\operatorname{t_{2}}{(P_{g})} = \\frac{d}{d P_{g}} \\cos{(P_{g})} and \\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})} = \\frac{d^{2}}{d P_{g}^{2}} \\cos{(P_{g})} and \\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})} = - \\cos{(P_{g})} and \\frac{d^{2}}{d P_{g}^{2}} \\cos{(P_{g})} = - \\cos{(P_{g})} and (\\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})})^{P_{g}} = (- \\cos{(P_{g})})^{P_{g}} and (\\frac{d}{d P_{g}} \\operatorname{t_{2}}{(P_{g})})^{P_{g}} = (\\frac{d^{2}}{d P_{g}^{2}} \\cos{(P_{g})})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('P_g', commutative=True)), Derivative(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('P_g', commutative=True))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Derivative(Function('t_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('P_g', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Function('t_2')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('P_g', commutative=True)), Pow(Derivative(cos(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(F_{g},\\lambda,C_{2})} = \\frac{C_{2} + \\lambda}{F_{g}}, then derive \\frac{\\partial}{\\partial C_{2}} \\mathbf{A}{(F_{g},\\lambda,C_{2})} = \\frac{1}{F_{g}}, then obtain \\frac{\\partial}{\\partial C_{2}} \\frac{C_{2} + \\lambda}{F_{g}} = \\frac{1}{F_{g}}", "derivation": "\\mathbf{A}{(F_{g},\\lambda,C_{2})} = \\frac{C_{2} + \\lambda}{F_{g}} and \\frac{\\partial}{\\partial C_{2}} \\mathbf{A}{(F_{g},\\lambda,C_{2})} = \\frac{\\partial}{\\partial C_{2}} \\frac{C_{2} + \\lambda}{F_{g}} and \\frac{\\partial}{\\partial C_{2}} \\mathbf{A}{(F_{g},\\lambda,C_{2})} = \\frac{1}{F_{g}} and \\frac{\\partial}{\\partial C_{2}} \\frac{C_{2} + \\lambda}{F_{g}} = \\frac{1}{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1)))"]]}, {"prompt": "Given s{(a)} = e^{a} and \\mathbf{J}_f{(a)} = e^{a}, then obtain \\mathbf{J}_f^{2}{(a)} - s{(a)} = \\mathbf{J}_f{(a)} s{(a)} - s{(a)}", "derivation": "s{(a)} = e^{a} and \\mathbf{J}_f{(a)} = e^{a} and \\mathbf{J}_f^{2}{(a)} = \\mathbf{J}_f{(a)} e^{a} and \\mathbf{J}_f^{2}{(a)} - e^{a} = \\mathbf{J}_f{(a)} e^{a} - e^{a} and \\mathbf{J}_f^{2}{(a)} - s{(a)} = \\mathbf{J}_f{(a)} s{(a)} - s{(a)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))))"], [["minus", 3, "exp(Symbol('a', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Integer(2)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))), Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))), Mul(Integer(-1), exp(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Integer(2)), Mul(Integer(-1), Function('s')(Symbol('a', commutative=True)))), Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True)), Function('s')(Symbol('a', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(v_{y})} = e^{v_{y}}, then derive \\int \\lambda{(v_{y})} dv_{y} = e^{v_{y}}, then obtain 0 = - \\lambda{(v_{y})} + \\int \\lambda{(v_{y})} dv_{y}", "derivation": "\\lambda{(v_{y})} = e^{v_{y}} and 0 = - \\lambda{(v_{y})} + e^{v_{y}} and \\int \\lambda{(v_{y})} dv_{y} = \\int e^{v_{y}} dv_{y} and \\iint \\lambda{(v_{y})} dv_{y} dv_{y} = \\iint e^{v_{y}} dv_{y} dv_{y} and \\frac{d}{d v_{y}} \\iint \\lambda{(v_{y})} dv_{y} dv_{y} = \\frac{d}{d v_{y}} \\iint e^{v_{y}} dv_{y} dv_{y} and \\int \\lambda{(v_{y})} dv_{y} = e^{v_{y}} and 0 = - \\lambda{(v_{y})} + \\int \\lambda{(v_{y})} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["minus", 1, "Function('\\\\lambda')(Symbol('v_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_y', commutative=True))), exp(Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["integrate", 3, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 4, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integral(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), exp(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_y', commutative=True))), Integral(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given y{(\\pi,\\sigma_x)} = \\pi - \\sigma_x and \\mu_{0}{(v_{1},r_{0})} = \\sin{(r_{0} + v_{1})}, then obtain \\mu_{0}{(v_{1},r_{0})} \\int \\frac{\\pi - \\sigma_x}{\\pi} d\\sigma_x = \\sin{(r_{0} + v_{1})} \\int \\frac{\\pi - \\sigma_x}{\\pi} d\\sigma_x", "derivation": "y{(\\pi,\\sigma_x)} = \\pi - \\sigma_x and \\frac{y{(\\pi,\\sigma_x)}}{\\pi} = \\frac{\\pi - \\sigma_x}{\\pi} and \\int \\frac{y{(\\pi,\\sigma_x)}}{\\pi} d\\sigma_x = \\int \\frac{\\pi - \\sigma_x}{\\pi} d\\sigma_x and \\mu_{0}{(v_{1},r_{0})} = \\sin{(r_{0} + v_{1})} and \\mu_{0}{(v_{1},r_{0})} \\int \\frac{y{(\\pi,\\sigma_x)}}{\\pi} d\\sigma_x = \\sin{(r_{0} + v_{1})} \\int \\frac{y{(\\pi,\\sigma_x)}}{\\pi} d\\sigma_x and \\mu_{0}{(v_{1},r_{0})} \\int \\frac{\\pi - \\sigma_x}{\\pi} d\\sigma_x = \\sin{(r_{0} + v_{1})} \\int \\frac{\\pi - \\sigma_x}{\\pi} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mu_0')(Symbol('v_1', commutative=True), Symbol('r_0', commutative=True)), sin(Add(Symbol('r_0', commutative=True), Symbol('v_1', commutative=True))))"], [["times", 4, "Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('v_1', commutative=True), Symbol('r_0', commutative=True)), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(sin(Add(Symbol('r_0', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Function('\\\\mu_0')(Symbol('v_1', commutative=True), Symbol('r_0', commutative=True)), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(sin(Add(Symbol('r_0', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(r,r_{0})} = r_{0}^{r} and \\mathbf{g}{(r,r_{0})} = 2 r + r_{0}^{r} \\lambda{(r,r_{0})}, then obtain - \\frac{2 r + r_{0}^{2 r}}{r} + \\frac{\\mathbf{g}{(r,r_{0})}}{r} = 0", "derivation": "\\lambda{(r,r_{0})} = r_{0}^{r} and r_{0}^{r} \\lambda{(r,r_{0})} = r_{0}^{2 r} and 2 r + r_{0}^{r} \\lambda{(r,r_{0})} = 2 r + r_{0}^{2 r} and \\mathbf{g}{(r,r_{0})} = 2 r + r_{0}^{r} \\lambda{(r,r_{0})} and \\frac{2 r + r_{0}^{r} \\lambda{(r,r_{0})}}{r} = \\frac{2 r + r_{0}^{2 r}}{r} and - \\frac{2 r + r_{0}^{2 r}}{r} + \\frac{2 r + r_{0}^{r} \\lambda{(r,r_{0})}}{r} = 0 and - \\frac{2 r + r_{0}^{2 r}}{r} + \\frac{\\mathbf{g}{(r,r_{0})}}{r} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)))"], [["times", 1, "Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True))), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))))"], [["add", 2, "Mul(Integer(2), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)))), Add(Mul(Integer(2), Symbol('r', commutative=True)), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)))))"], [["divide", 3, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True))))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))))))"], [["minus", 5, "Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('r', commutative=True)), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('r_0', commutative=True)))), Integer(0))"]]}, {"prompt": "Given u{(h,B,z)} = B + h + z, then derive \\Psi + \\frac{\\partial}{\\partial z} u{(h,B,z)} = \\Psi + 1, then obtain \\Psi + \\frac{\\partial}{\\partial z} (B + h + z) = \\Psi + 1", "derivation": "u{(h,B,z)} = B + h + z and \\frac{\\partial}{\\partial z} u{(h,B,z)} = \\frac{\\partial}{\\partial z} (B + h + z) and \\Psi + \\frac{\\partial}{\\partial z} u{(h,B,z)} = \\Psi + \\frac{\\partial}{\\partial z} (B + h + z) and \\Psi + \\frac{\\partial}{\\partial z} u{(h,B,z)} = \\Psi + 1 and \\Psi + \\frac{\\partial}{\\partial z} (B + h + z) = \\Psi + 1", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('h', commutative=True), Symbol('B', commutative=True), Symbol('z', commutative=True)), Add(Symbol('B', commutative=True), Symbol('h', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('h', commutative=True), Symbol('B', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Symbol('h', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Derivative(Function('u')(Symbol('h', commutative=True), Symbol('B', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Symbol('\\\\Psi', commutative=True), Derivative(Add(Symbol('B', commutative=True), Symbol('h', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\Psi', commutative=True), Derivative(Function('u')(Symbol('h', commutative=True), Symbol('B', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Symbol('\\\\Psi', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\Psi', commutative=True), Derivative(Add(Symbol('B', commutative=True), Symbol('h', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Symbol('\\\\Psi', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\hat{H}{(\\varphi^*,\\phi)} = \\phi + \\varphi^*, then derive \\frac{\\partial}{\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} = 1, then obtain \\int \\frac{\\partial^{2}}{\\partial \\phi\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} d\\varphi^* = \\int \\frac{d}{d \\phi} 1 d\\varphi^*", "derivation": "\\hat{H}{(\\varphi^*,\\phi)} = \\phi + \\varphi^* and \\frac{\\partial}{\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} = \\frac{\\partial}{\\partial \\varphi^*} (\\phi + \\varphi^*) and \\frac{\\partial}{\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} = 1 and \\frac{\\partial^{2}}{\\partial \\phi\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} = \\frac{d}{d \\phi} 1 and \\int \\frac{\\partial^{2}}{\\partial \\phi\\partial \\varphi^*} \\hat{H}{(\\varphi^*,\\phi)} d\\varphi^* = \\int \\frac{d}{d \\phi} 1 d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(F_{H})} = e^{F_{H}} and V{(F_{H})} = \\frac{d}{d F_{H}} \\operatorname{E_{\\lambda}}{(F_{H})}, then obtain \\frac{d}{d F_{H}} V{(F_{H})} = e^{F_{H}}", "derivation": "\\operatorname{E_{\\lambda}}{(F_{H})} = e^{F_{H}} and V{(F_{H})} = \\frac{d}{d F_{H}} \\operatorname{E_{\\lambda}}{(F_{H})} and \\frac{d}{d F_{H}} V{(F_{H})} = \\frac{d^{2}}{d F_{H}^{2}} \\operatorname{E_{\\lambda}}{(F_{H})} and \\frac{d}{d F_{H}} V{(F_{H})} = \\frac{d^{2}}{d F_{H}^{2}} e^{F_{H}} and \\frac{d}{d F_{H}} V{(F_{H})} = e^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('F_H', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Function('E_{\\\\lambda}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('V')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('V')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), exp(Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(H,\\hat{p}_0)} = \\frac{H}{\\hat{p}_0} and \\delta{(H)} = - 2 H, then obtain \\delta{(H)} + 2 \\hat{p}{(H,\\hat{p}_0)} = \\frac{2 H}{\\hat{p}_0} + \\delta{(H)}", "derivation": "\\hat{p}{(H,\\hat{p}_0)} = \\frac{H}{\\hat{p}_0} and - H + \\hat{p}{(H,\\hat{p}_0)} = - H + \\frac{H}{\\hat{p}_0} and - 2 H + \\hat{p}{(H,\\hat{p}_0)} = - 2 H + \\frac{H}{\\hat{p}_0} and - 2 H + 2 \\hat{p}{(H,\\hat{p}_0)} = - 2 H + \\frac{H}{\\hat{p}_0} + \\hat{p}{(H,\\hat{p}_0)} and \\delta{(H)} = - 2 H and \\delta{(H)} + 2 \\hat{p}{(H,\\hat{p}_0)} = \\frac{H}{\\hat{p}_0} + \\delta{(H)} + \\hat{p}{(H,\\hat{p}_0)} and \\delta{(H)} + \\hat{p}{(H,\\hat{p}_0)} = \\frac{H}{\\hat{p}_0} + \\delta{(H)} and \\delta{(H)} + 2 \\hat{p}{(H,\\hat{p}_0)} = \\frac{2 H}{\\hat{p}_0} + \\delta{(H)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)))))"], [["minus", 2, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\delta')(Symbol('H', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))), Function('\\\\delta')(Symbol('H', commutative=True)), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Function('\\\\delta')(Symbol('H', commutative=True)), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))), Function('\\\\delta')(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Function('\\\\delta')(Symbol('H', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(2), Symbol('H', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))), Function('\\\\delta')(Symbol('H', commutative=True))))"]]}, {"prompt": "Given v{(J,x)} = x^{J} and n{(J,x)} = x^{J}, then obtain \\int 2 x^{J} dJ = \\int (x^{J} + n{(J,x)}) dJ", "derivation": "v{(J,x)} = x^{J} and n{(J,x)} = x^{J} and v{(J,x)} = n{(J,x)} and x^{J} + v{(J,x)} = x^{J} + n{(J,x)} and \\int (x^{J} + v{(J,x)}) dJ = \\int (x^{J} + n{(J,x)}) dJ and \\int 2 x^{J} dJ = \\int (x^{J} + n{(J,x)}) dJ", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('J', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v')(Symbol('J', commutative=True), Symbol('x', commutative=True)), Function('n')(Symbol('J', commutative=True), Symbol('x', commutative=True)))"], [["add", 3, "Pow(Symbol('x', commutative=True), Symbol('J', commutative=True))"], "Equality(Add(Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)), Function('v')(Symbol('J', commutative=True), Symbol('x', commutative=True))), Add(Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)), Function('n')(Symbol('J', commutative=True), Symbol('x', commutative=True))))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)), Function('v')(Symbol('J', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Add(Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)), Function('n')(Symbol('J', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Integer(2), Pow(Symbol('x', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Add(Pow(Symbol('x', commutative=True), Symbol('J', commutative=True)), Function('n')(Symbol('J', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})} = \\mathbf{s}^{\\psi}, then obtain -1 = - (- \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})})^{\\psi}", "derivation": "\\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})} = \\mathbf{s}^{\\psi} and \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})} = 2 \\mathbf{s}^{\\psi} and - \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})} = 0 and (- \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})})^{\\psi} = 0^{\\psi} and - (- \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})})^{\\psi} = - 0^{\\psi} and -1 = - (- \\mathbf{s}^{\\psi} + \\hat{\\mathbf{r}}{(\\psi,\\mathbf{s})})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(0))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Integer(0), Symbol('\\\\psi', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(-1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\psi', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given J{(U)} = \\log{(U)}, then obtain (\\int \\cos{((\\iint J{(U)} dU dU)^{U})} dU)^{U} - (\\int \\cos{((\\iint \\log{(U)} dU dU)^{U})} dU)^{U} = 0", "derivation": "J{(U)} = \\log{(U)} and \\int J{(U)} dU = \\int \\log{(U)} dU and \\iint J{(U)} dU dU = \\iint \\log{(U)} dU dU and (\\iint J{(U)} dU dU)^{U} = (\\iint \\log{(U)} dU dU)^{U} and \\cos{((\\iint J{(U)} dU dU)^{U})} = \\cos{((\\iint \\log{(U)} dU dU)^{U})} and \\int \\cos{((\\iint J{(U)} dU dU)^{U})} dU = \\int \\cos{((\\iint \\log{(U)} dU dU)^{U})} dU and (\\int \\cos{((\\iint J{(U)} dU dU)^{U})} dU)^{U} = (\\int \\cos{((\\iint \\log{(U)} dU dU)^{U})} dU)^{U} and (\\int \\cos{((\\iint J{(U)} dU dU)^{U})} dU)^{U} - (\\int \\cos{((\\iint \\log{(U)} dU dU)^{U})} dU)^{U} = 0", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["cos", 4], "Equality(cos(Pow(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), cos(Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(cos(Pow(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(cos(Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["power", 6, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(cos(Pow(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(cos(Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["minus", 7, "Pow(Integral(cos(Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))"], "Equality(Add(Pow(Integral(cos(Pow(Integral(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Integral(cos(Pow(Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))), Integer(0))"]]}, {"prompt": "Given x{(t,C_{2})} = C_{2} + t, then derive \\int x{(t,C_{2})} dt = C_{2} t + G + \\frac{t^{2}}{2}, then obtain \\frac{t^{2} (C_{2} t + H + \\frac{t^{2}}{2})}{2} = - \\frac{t^{2} (- C_{2} t - G - \\frac{t^{2}}{2})}{2}", "derivation": "x{(t,C_{2})} = C_{2} + t and \\int x{(t,C_{2})} dt = \\int (C_{2} + t) dt and \\int x{(t,C_{2})} dt = C_{2} t + G + \\frac{t^{2}}{2} and - \\int x{(t,C_{2})} dt = - C_{2} t - G - \\frac{t^{2}}{2} and - \\int (C_{2} + t) dt = - C_{2} t - G - \\frac{t^{2}}{2} and \\frac{t^{2} \\int (C_{2} + t) dt}{2} = - \\frac{t^{2} (- C_{2} t - G - \\frac{t^{2}}{2})}{2} and \\frac{t^{2} (C_{2} t + H + \\frac{t^{2}}{2})}{2} = - \\frac{t^{2} (- C_{2} t - G - \\frac{t^{2}}{2})}{2}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('x')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Symbol('G', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('x')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Integral(Add(Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["times", 5, "Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)), Integral(Add(Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"], [["evaluate_integrals", 6], "Equality(Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)), Add(Mul(Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Symbol('H', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\varepsilon_{0}{(P_{e})} = \\cos{(P_{e})}, then derive \\int \\varepsilon_{0}{(P_{e})} dP_{e} = \\mathbf{f} + \\sin{(P_{e})}, then obtain (\\int \\cos{(P_{e})} dP_{e})^{\\mathbf{f}} = (\\mathbf{f} + \\sin{(P_{e})})^{\\mathbf{f}}", "derivation": "\\varepsilon_{0}{(P_{e})} = \\cos{(P_{e})} and \\int \\varepsilon_{0}{(P_{e})} dP_{e} = \\int \\cos{(P_{e})} dP_{e} and \\int \\varepsilon_{0}{(P_{e})} dP_{e} = \\mathbf{f} + \\sin{(P_{e})} and \\int \\cos{(P_{e})} dP_{e} = \\mathbf{f} + \\sin{(P_{e})} and (\\int \\cos{(P_{e})} dP_{e})^{\\mathbf{f}} = (\\mathbf{f} + \\sin{(P_{e})})^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('P_e', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('P_e', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}})}, then obtain \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\phi^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\cos^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}", "derivation": "\\phi{(\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}})} and \\phi^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} = \\cos^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} and \\frac{d}{d \\dot{\\mathbf{r}}} \\phi^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\cos^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} and \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\phi^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\cos^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(Pow(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(F_{c})} = \\log{(F_{c})}, then obtain (F_{c} + \\frac{\\operatorname{t_{1}}{(F_{c})}}{\\log{(F_{c})}})^{F_{c}} = (F_{c} + 1)^{F_{c}}", "derivation": "\\operatorname{t_{1}}{(F_{c})} = \\log{(F_{c})} and \\frac{\\operatorname{t_{1}}{(F_{c})}}{\\log{(F_{c})}} = 1 and F_{c} + \\frac{\\operatorname{t_{1}}{(F_{c})}}{\\log{(F_{c})}} = F_{c} + 1 and (F_{c} + \\frac{\\operatorname{t_{1}}{(F_{c})}}{\\log{(F_{c})}})^{F_{c}} = (F_{c} + 1)^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["divide", 1, "log(Symbol('F_c', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Mul(Function('t_1')(Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1)))), Add(Symbol('F_c', commutative=True), Integer(1)))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Symbol('F_c', commutative=True), Mul(Function('t_1')(Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1)))), Symbol('F_c', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Integer(1)), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})} = E_{x} \\dot{\\mathbf{r}}, then obtain \\int (1 - \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})}) d\\dot{\\mathbf{r}} = - \\frac{E_{x} \\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} + \\rho", "derivation": "\\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})} = E_{x} \\dot{\\mathbf{r}} and \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})} - 1 = E_{x} \\dot{\\mathbf{r}} - 1 and 1 - \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})} = - E_{x} \\dot{\\mathbf{r}} + 1 and \\int (1 - \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})}) d\\dot{\\mathbf{r}} = \\int (- E_{x} \\dot{\\mathbf{r}} + 1) d\\dot{\\mathbf{r}} and \\int (1 - \\mathbf{J}_f{(E_{x},\\dot{\\mathbf{r}})}) d\\dot{\\mathbf{r}} = - \\frac{E_{x} \\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} + \\rho", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], [["times", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(1)))"], [["integrate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Symbol('E_x', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\Psi{(\\hat{p}_0,\\lambda)} = \\frac{\\lambda}{\\hat{p}_0}, then obtain \\cos{(\\frac{\\partial}{\\partial \\hat{p}_0} \\Psi^{\\hat{p}_0}{(\\hat{p}_0,\\lambda)})} = \\cos{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\frac{\\lambda}{\\hat{p}_0})^{\\hat{p}_0})}", "derivation": "\\Psi{(\\hat{p}_0,\\lambda)} = \\frac{\\lambda}{\\hat{p}_0} and \\Psi^{\\hat{p}_0}{(\\hat{p}_0,\\lambda)} = (\\frac{\\lambda}{\\hat{p}_0})^{\\hat{p}_0} and \\frac{\\partial}{\\partial \\hat{p}_0} \\Psi^{\\hat{p}_0}{(\\hat{p}_0,\\lambda)} = \\frac{\\partial}{\\partial \\hat{p}_0} (\\frac{\\lambda}{\\hat{p}_0})^{\\hat{p}_0} and \\cos{(\\frac{\\partial}{\\partial \\hat{p}_0} \\Psi^{\\hat{p}_0}{(\\hat{p}_0,\\lambda)})} = \\cos{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\frac{\\lambda}{\\hat{p}_0})^{\\hat{p}_0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), cos(Derivative(Pow(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(E_{n})} = \\cos{(E_{n})}, then derive \\int \\phi_{1}{(E_{n})} dE_{n} = \\hat{p} + \\sin{(E_{n})}, then obtain 2 \\cos{(E_{n})} + \\cos^{- E_{n}}{(E_{n})} \\int \\phi_{1}{(E_{n})} dE_{n} = (\\hat{p} + \\sin{(E_{n})}) \\cos^{- E_{n}}{(E_{n})} + 2 \\cos{(E_{n})}", "derivation": "\\phi_{1}{(E_{n})} = \\cos{(E_{n})} and \\int \\phi_{1}{(E_{n})} dE_{n} = \\int \\cos{(E_{n})} dE_{n} and \\int \\phi_{1}{(E_{n})} dE_{n} = \\hat{p} + \\sin{(E_{n})} and \\cos^{- E_{n}}{(E_{n})} \\int \\phi_{1}{(E_{n})} dE_{n} = (\\hat{p} + \\sin{(E_{n})}) \\cos^{- E_{n}}{(E_{n})} and 2 \\cos{(E_{n})} + \\cos^{- E_{n}}{(E_{n})} \\int \\phi_{1}{(E_{n})} dE_{n} = (\\hat{p} + \\sin{(E_{n})}) \\cos^{- E_{n}}{(E_{n})} + 2 \\cos{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('E_n', commutative=True))))"], [["divide", 3, "Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Mul(Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('E_n', commutative=True))), Pow(cos(Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True)))))"], [["add", 4, "Mul(Integer(2), cos(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(2), cos(Symbol('E_n', commutative=True))), Mul(Pow(cos(Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))), Add(Mul(Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('E_n', commutative=True))), Pow(cos(Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('E_n', commutative=True)))), Mul(Integer(2), cos(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given y{(v_{z},h)} = \\int h^{v_{z}} dh, then obtain \\frac{\\frac{\\partial^{2}}{\\partial v_{z}^{2}} y{(v_{z},h)}}{h} = \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial v_{z}} \\int h^{v_{z}} dh}{h}", "derivation": "y{(v_{z},h)} = \\int h^{v_{z}} dh and \\frac{\\partial}{\\partial v_{z}} y{(v_{z},h)} = \\frac{\\partial}{\\partial v_{z}} \\int h^{v_{z}} dh and \\frac{\\frac{\\partial}{\\partial v_{z}} y{(v_{z},h)}}{h} = \\frac{\\frac{\\partial}{\\partial v_{z}} \\int h^{v_{z}} dh}{h} and \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial v_{z}} y{(v_{z},h)}}{h} = \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial v_{z}} \\int h^{v_{z}} dh}{h} and \\frac{\\frac{\\partial^{2}}{\\partial v_{z}^{2}} y{(v_{z},h)}}{h} = \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial v_{z}} \\int h^{v_{z}} dh}{h}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_z', commutative=True), Symbol('h', commutative=True)), Integral(Pow(Symbol('h', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('v_z', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('h', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Function('y')(Symbol('v_z', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Integral(Pow(Symbol('h', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Function('y')(Symbol('v_z', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Integral(Pow(Symbol('h', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Function('y')(Symbol('v_z', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2)))), Derivative(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Derivative(Integral(Pow(Symbol('h', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} = \\mathbf{H}^{S}, then obtain S \\mathbf{H}^{S} \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})} = S \\mathbf{H}^{2 S} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} = \\mathbf{H}^{S} and S \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} = S \\mathbf{H}^{S} and \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})} = \\cos{(\\mathbf{H}^{S})} and S \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} \\cos{(\\mathbf{H}^{S})} = S \\mathbf{H}^{S} \\cos{(\\mathbf{H}^{S})} and S \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})} = S \\mathbf{H}^{S} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})} and S \\mathbf{H}^{S} \\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})} = S \\mathbf{H}^{2 S} \\cos{(\\operatorname{J_{\\varepsilon}}{(S,\\mathbf{H})})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True))))"], [["cos", 1], "Equality(cos(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True))))"], [["times", 2, "cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)))"], "Equality(Mul(Symbol('S', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)))), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)), cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('S', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)), cos(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["times", 5, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True))"], "Equality(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('S', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Symbol('S', commutative=True))), cos(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)} = \\hbar + \\cos{(\\varepsilon_0)}, then obtain e^{2 \\hbar (\\hbar + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)})} = e^{2 \\hbar (2 \\hbar + \\cos{(\\varepsilon_0)})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)} = \\hbar + \\cos{(\\varepsilon_0)} and \\hbar + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)} = 2 \\hbar + \\cos{(\\varepsilon_0)} and 2 \\hbar (\\hbar + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)}) = 2 \\hbar (2 \\hbar + \\cos{(\\varepsilon_0)}) and e^{2 \\hbar (\\hbar + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon_0,\\hbar)})} = e^{2 \\hbar (2 \\hbar + \\cos{(\\varepsilon_0)})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["times", 2, "Mul(Integer(2), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["exp", 3], "Equality(exp(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hbar', commutative=True))))), exp(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"]]}, {"prompt": "Given U{(A)} = \\cos{(\\log{(A)})} and \\hat{X}{(A)} = - \\log{(A)} + \\cos{(\\log{(A)})}, then obtain \\frac{\\hat{X}{(A)}}{A} = \\frac{U{(A)} - \\log{(A)}}{A}", "derivation": "U{(A)} = \\cos{(\\log{(A)})} and U{(A)} - \\log{(A)} = - \\log{(A)} + \\cos{(\\log{(A)})} and \\hat{X}{(A)} = - \\log{(A)} + \\cos{(\\log{(A)})} and \\hat{X}{(A)} = U{(A)} - \\log{(A)} and \\frac{\\hat{X}{(A)}}{A} = \\frac{U{(A)} - \\log{(A)}}{A}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('A', commutative=True)), cos(log(Symbol('A', commutative=True))))"], [["minus", 1, "log(Symbol('A', commutative=True))"], "Equality(Add(Function('U')(Symbol('A', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('A', commutative=True))), cos(log(Symbol('A', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Add(Mul(Integer(-1), log(Symbol('A', commutative=True))), cos(log(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{X}')(Symbol('A', commutative=True)), Add(Function('U')(Symbol('A', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True)))))"], [["divide", 4, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('U')(Symbol('A', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(\\hat{H})} = \\cos{(\\sin{(\\hat{H})})} and \\operatorname{v_{x}}{(\\hat{H})} = \\mathbf{f}{(\\hat{H})} + \\cos{(\\sin{(\\hat{H})})}, then obtain \\operatorname{v_{x}}{(\\hat{H})} + 2 \\cos{(\\sin{(\\hat{H})})} = 4 \\cos{(\\sin{(\\hat{H})})}", "derivation": "\\mathbf{f}{(\\hat{H})} = \\cos{(\\sin{(\\hat{H})})} and \\operatorname{v_{x}}{(\\hat{H})} = \\mathbf{f}{(\\hat{H})} + \\cos{(\\sin{(\\hat{H})})} and 2 \\mathbf{f}{(\\hat{H})} + \\operatorname{v_{x}}{(\\hat{H})} = 3 \\mathbf{f}{(\\hat{H})} + \\cos{(\\sin{(\\hat{H})})} and \\operatorname{v_{x}}{(\\hat{H})} + 2 \\cos{(\\sin{(\\hat{H})})} = 4 \\cos{(\\sin{(\\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\hat{H}', commutative=True)), cos(sin(Symbol('\\\\hat{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\hat{H}', commutative=True)), Add(Function('\\\\mathbf{f}')(Symbol('\\\\hat{H}', commutative=True)), cos(sin(Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Function('\\\\mathbf{f}')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('\\\\hat{H}', commutative=True))), Function('v_x')(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(3), Function('\\\\mathbf{f}')(Symbol('\\\\hat{H}', commutative=True))), cos(sin(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('v_x')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), cos(sin(Symbol('\\\\hat{H}', commutative=True))))), Mul(Integer(4), cos(sin(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given n{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)}, then derive \\int n{(\\mathbf{J}_P)} d\\mathbf{J}_P = v + \\sin{(\\mathbf{J}_P)}, then obtain F_{H} + \\sin{(\\mathbf{J}_P)} = v + \\sin{(\\mathbf{J}_P)}", "derivation": "n{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\int n{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\cos{(\\mathbf{J}_P)} d\\mathbf{J}_P and \\int n{(\\mathbf{J}_P)} d\\mathbf{J}_P = v + \\sin{(\\mathbf{J}_P)} and \\int \\cos{(\\mathbf{J}_P)} d\\mathbf{J}_P = v + \\sin{(\\mathbf{J}_P)} and F_{H} + \\sin{(\\mathbf{J}_P)} = v + \\sin{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('v', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('v', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_H', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('v', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given v{(M_{E},F_{H})} = \\sin{(\\frac{M_{E}}{F_{H}})}, then derive \\cos{(\\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})})} = \\cos{(\\frac{\\cos{(\\frac{M_{E}}{F_{H}})}}{F_{H}})}, then obtain \\cos{(\\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})})} + 1 = \\cos{(\\frac{\\cos{(\\frac{M_{E}}{F_{H}})}}{F_{H}})} + 1", "derivation": "v{(M_{E},F_{H})} = \\sin{(\\frac{M_{E}}{F_{H}})} and \\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})} = \\frac{\\partial}{\\partial M_{E}} \\sin{(\\frac{M_{E}}{F_{H}})} and \\cos{(\\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})})} = \\cos{(\\frac{\\partial}{\\partial M_{E}} \\sin{(\\frac{M_{E}}{F_{H}})})} and \\cos{(\\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})})} = \\cos{(\\frac{\\cos{(\\frac{M_{E}}{F_{H}})}}{F_{H}})} and \\cos{(\\frac{\\partial}{\\partial M_{E}} v{(M_{E},F_{H})})} + 1 = \\cos{(\\frac{\\cos{(\\frac{M_{E}}{F_{H}})}}{F_{H}})} + 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('M_E', commutative=True), Symbol('F_H', commutative=True)), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('M_E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('v')(Symbol('M_E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), cos(Derivative(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('v')(Symbol('M_E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(cos(Derivative(Function('v')(Symbol('M_E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Integer(1)), Add(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))))), Integer(1)))"]]}, {"prompt": "Given \\delta{(\\dot{\\mathbf{r}},y^{\\prime})} = - \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\delta{(\\dot{\\mathbf{r}},y^{\\prime})} = \\cos{(\\dot{\\mathbf{r}} - y^{\\prime})}, then obtain \\frac{\\partial}{\\partial y^{\\prime}} - \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})} = \\cos{(\\dot{\\mathbf{r}} - y^{\\prime})}", "derivation": "\\delta{(\\dot{\\mathbf{r}},y^{\\prime})} = - \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})} and \\delta{(\\dot{\\mathbf{r}},y^{\\prime})} - 1 = - \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})} - 1 and \\frac{\\partial}{\\partial y^{\\prime}} (\\delta{(\\dot{\\mathbf{r}},y^{\\prime})} - 1) = \\frac{\\partial}{\\partial y^{\\prime}} (- \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})} - 1) and \\frac{\\partial}{\\partial y^{\\prime}} \\delta{(\\dot{\\mathbf{r}},y^{\\prime})} = \\cos{(\\dot{\\mathbf{r}} - y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} - \\sin{(\\dot{\\mathbf{r}} - y^{\\prime})} = \\cos{(\\dot{\\mathbf{r}} - y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\delta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))), Integer(-1)))"], [["differentiate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\delta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))), Integer(-1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), cos(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), cos(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given c{(i,J)} = \\frac{\\sin{(i)}}{J}, then obtain \\log{(c{(i,J)})}^{J} - \\int \\frac{\\sin{(i)}}{J} dJ = \\log{(\\frac{\\sin{(i)}}{J})}^{J} - \\int \\frac{\\sin{(i)}}{J} dJ", "derivation": "c{(i,J)} = \\frac{\\sin{(i)}}{J} and \\log{(c{(i,J)})} = \\log{(\\frac{\\sin{(i)}}{J})} and \\int c{(i,J)} dJ = \\int \\frac{\\sin{(i)}}{J} dJ and \\log{(c{(i,J)})}^{J} = \\log{(\\frac{\\sin{(i)}}{J})}^{J} and \\log{(c{(i,J)})}^{J} - \\int c{(i,J)} dJ = \\log{(\\frac{\\sin{(i)}}{J})}^{J} - \\int c{(i,J)} dJ and \\log{(c{(i,J)})}^{J} - \\int \\frac{\\sin{(i)}}{J} dJ = \\log{(\\frac{\\sin{(i)}}{J})}^{J} - \\int \\frac{\\sin{(i)}}{J} dJ", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True))))"], [["log", 1], "Equality(log(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True))), log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True)))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(log(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True)))), Symbol('J', commutative=True)))"], [["minus", 4, "Integral(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Pow(log(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))), Add(Pow(log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True)))), Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(log(Function('c')(Symbol('i', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True))), Tuple(Symbol('J', commutative=True))))), Add(Pow(log(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True)))), Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('i', commutative=True))), Tuple(Symbol('J', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{M}{(P_{e})} = \\cos{(\\sin{(P_{e})})}, then derive \\frac{d}{d P_{e}} \\mathbf{M}{(P_{e})} = - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})}, then obtain - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})}", "derivation": "\\mathbf{M}{(P_{e})} = \\cos{(\\sin{(P_{e})})} and \\frac{d}{d P_{e}} \\mathbf{M}{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})} and \\frac{d}{d P_{e}} \\mathbf{M}{(P_{e})} = - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})} and - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True)), cos(sin(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))), Derivative(cos(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(c,m_{s})} = \\cos{(m_{s}^{c})}, then obtain (\\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(c,m_{s})} + 1)^{c} = (\\frac{\\partial}{\\partial m_{s}} \\cos{(m_{s}^{c})} + 1)^{c}", "derivation": "\\operatorname{P_{e}}{(c,m_{s})} = \\cos{(m_{s}^{c})} and \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(c,m_{s})} = \\frac{\\partial}{\\partial m_{s}} \\cos{(m_{s}^{c})} and \\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(c,m_{s})} + 1 = \\frac{\\partial}{\\partial m_{s}} \\cos{(m_{s}^{c})} + 1 and (\\frac{\\partial}{\\partial m_{s}} \\operatorname{P_{e}}{(c,m_{s})} + 1)^{c} = (\\frac{\\partial}{\\partial m_{s}} \\cos{(m_{s}^{c})} + 1)^{c}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('c', commutative=True), Symbol('m_s', commutative=True)), cos(Pow(Symbol('m_s', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('c', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('m_s', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('P_e')(Symbol('c', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(Pow(Symbol('m_s', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1)))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Derivative(Function('P_e')(Symbol('c', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1)), Symbol('c', commutative=True)), Pow(Add(Derivative(cos(Pow(Symbol('m_s', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1)), Symbol('c', commutative=True)))"]]}, {"prompt": "Given L{(\\hat{H},c,\\rho)} = - \\rho + \\frac{c}{\\hat{H}}, then derive \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} = - \\frac{1}{\\hat{H}^{2}}, then obtain - \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} - \\frac{1}{\\hat{H}^{2}} = 0", "derivation": "L{(\\hat{H},c,\\rho)} = - \\rho + \\frac{c}{\\hat{H}} and \\frac{\\partial}{\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} = \\frac{\\partial}{\\partial \\hat{H}} (- \\rho + \\frac{c}{\\hat{H}}) and \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} = \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} (- \\rho + \\frac{c}{\\hat{H}}) and \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} = - \\frac{1}{\\hat{H}^{2}} and - \\frac{1}{\\hat{H}^{2}} = \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} (- \\rho + \\frac{c}{\\hat{H}}) and - \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} (- \\rho + \\frac{c}{\\hat{H}}) - \\frac{1}{\\hat{H}^{2}} = 0 and - \\frac{\\partial^{2}}{\\partial c\\partial \\hat{H}} L{(\\hat{H},c,\\rho)} - \\frac{1}{\\hat{H}^{2}} = 0", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('L')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive \\int 0 d\\mathbf{D} = \\int \\frac{d}{d \\mathbf{D}} \\mathbf{D} (- \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}}) d\\mathbf{D}, then obtain \\int 0 d\\mathbf{D} = \\int \\frac{d}{d \\mathbf{D}} 0 d\\mathbf{D}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{D})} = e^{\\mathbf{D}} and 0 = - \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}} and 0 = \\mathbf{D} (- \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}}) and \\frac{d}{d \\mathbf{D}} 0 = \\frac{d}{d \\mathbf{D}} \\mathbf{D} (- \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}}) and \\int \\frac{d}{d \\mathbf{D}} 0 d\\mathbf{D} = \\int \\frac{d}{d \\mathbf{D}} \\mathbf{D} (- \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}}) d\\mathbf{D} and \\int 0 d\\mathbf{D} = \\int \\frac{d}{d \\mathbf{D}} \\mathbf{D} (- \\operatorname{f^{\\prime}}{(\\mathbf{D})} + e^{\\mathbf{D}}) d\\mathbf{D} and \\int 0 d\\mathbf{D} = \\int \\frac{d}{d \\mathbf{D}} 0 d\\mathbf{D}", "srepr_derivation": [["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(a)} = \\cos{(a)} and \\lambda{(G,A_{y})} = \\frac{A_{y}}{G}, then obtain \\lambda{(G,A_{y})} - \\frac{d}{d a} \\operatorname{F_{x}}{(a)} = \\frac{A_{y}}{G} - \\frac{d}{d a} \\operatorname{F_{x}}{(a)}", "derivation": "\\operatorname{F_{x}}{(a)} = \\cos{(a)} and \\frac{d}{d a} \\operatorname{F_{x}}{(a)} = \\frac{d}{d a} \\cos{(a)} and \\lambda{(G,A_{y})} = \\frac{A_{y}}{G} and \\lambda{(G,A_{y})} - \\frac{d}{d a} \\cos{(a)} = \\frac{A_{y}}{G} - \\frac{d}{d a} \\cos{(a)} and \\lambda{(G,A_{y})} - \\frac{d}{d a} \\operatorname{F_{x}}{(a)} = \\frac{A_{y}}{G} - \\frac{d}{d a} \\operatorname{F_{x}}{(a)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\lambda')(Symbol('G', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["minus", 3, "Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\lambda')(Symbol('G', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('G', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\lambda')(Symbol('G', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Derivative(Function('F_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('G', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Function('F_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(c)} = \\cos{(\\log{(c)})}, then obtain \\eta^{\\prime}{(c)} - \\int \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} dc = \\cos{(\\log{(c)})} - \\int \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} dc", "derivation": "\\eta^{\\prime}{(c)} = \\cos{(\\log{(c)})} and \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} = 1 and \\int \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} dc = \\int 1 dc and \\eta^{\\prime}{(c)} - \\int 1 dc = \\cos{(\\log{(c)})} - \\int 1 dc and \\eta^{\\prime}{(c)} - \\int \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} dc = \\cos{(\\log{(c)})} - \\int \\frac{\\eta^{\\prime}{(c)}}{\\cos{(\\log{(c)})}} dc", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True))))"], [["divide", 1, "cos(log(Symbol('c', commutative=True)))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Pow(cos(log(Symbol('c', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Pow(cos(log(Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integral(Integer(1), Tuple(Symbol('c', commutative=True))))"], [["minus", 1, "Integral(Integer(1), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('c', commutative=True))))), Add(cos(log(Symbol('c', commutative=True))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('c', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Mul(Integer(-1), Integral(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Pow(cos(log(Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))))), Add(cos(log(Symbol('c', commutative=True))), Mul(Integer(-1), Integral(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True)), Pow(cos(log(Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(\\hbar)} = \\sin{(e^{\\hbar})}, then derive \\frac{d}{d \\hbar} \\sigma_{p}{(\\hbar)} + 1 = e^{\\hbar} \\cos{(e^{\\hbar})} + 1, then obtain \\frac{d}{d \\hbar} \\sin{(e^{\\hbar})} + 1 = e^{\\hbar} \\cos{(e^{\\hbar})} + 1", "derivation": "\\sigma_{p}{(\\hbar)} = \\sin{(e^{\\hbar})} and \\hbar + \\sigma_{p}{(\\hbar)} = \\hbar + \\sin{(e^{\\hbar})} and \\frac{d}{d \\hbar} (\\hbar + \\sigma_{p}{(\\hbar)}) = \\frac{d}{d \\hbar} (\\hbar + \\sin{(e^{\\hbar})}) and \\frac{d}{d \\hbar} \\sigma_{p}{(\\hbar)} + 1 = e^{\\hbar} \\cos{(e^{\\hbar})} + 1 and \\frac{d}{d \\hbar} \\sin{(e^{\\hbar})} + 1 = e^{\\hbar} \\cos{(e^{\\hbar})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True)), sin(exp(Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hbar', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(1)), Add(Mul(exp(Symbol('\\\\hbar', commutative=True)), cos(exp(Symbol('\\\\hbar', commutative=True)))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(sin(exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(1)), Add(Mul(exp(Symbol('\\\\hbar', commutative=True)), cos(exp(Symbol('\\\\hbar', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given v{(\\theta)} = \\log{(\\theta)}, then obtain \\int (\\theta + 2 v{(\\theta)}) d\\theta = \\int (\\theta + 2 \\log{(\\theta)}) d\\theta", "derivation": "v{(\\theta)} = \\log{(\\theta)} and \\theta + v{(\\theta)} = \\theta + \\log{(\\theta)} and \\theta + 2 v{(\\theta)} = \\theta + v{(\\theta)} + \\log{(\\theta)} and \\theta + 2 v{(\\theta)} = \\theta + 2 \\log{(\\theta)} and \\int (\\theta + 2 v{(\\theta)}) d\\theta = \\int (\\theta + 2 \\log{(\\theta)}) d\\theta", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('v')(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))))"], [["add", 2, "Function('v')(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), Function('v')(Symbol('\\\\theta', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Function('v')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), Function('v')(Symbol('\\\\theta', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), log(Symbol('\\\\theta', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), Function('v')(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(2), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varphi^*)} = \\log{(\\varphi^*)}, then obtain -1 = \\varphi^* (- \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)}) - \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)} - 1", "derivation": "\\operatorname{A_{x}}{(\\varphi^*)} = \\log{(\\varphi^*)} and 0 = - \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)} and -1 = - \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)} - 1 and 0 = \\varphi^* (- \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)}) and - \\operatorname{A_{x}}{(\\varphi^*)} = \\varphi^* (- \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)}) - \\operatorname{A_{x}}{(\\varphi^*)} and -1 = \\varphi^* (- \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)}) - \\operatorname{A_{x}}{(\\varphi^*)} + \\log{(\\varphi^*)} - 1", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 1, "Function('A_x')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))"], [["times", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(-1), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Function('A_x')(Symbol('\\\\varphi^*', commutative=True))), log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\operatorname{v_{z}}{(\\mathbf{A})} = (\\int \\operatorname{A_{z}}{(\\mathbf{A})} \\sin{(\\mathbf{A})} d\\mathbf{A})^{2}, then obtain \\operatorname{v_{z}}{(\\mathbf{A})} = (\\int \\operatorname{A_{z}}^{2}{(\\mathbf{A})} d\\mathbf{A})^{2}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\operatorname{A_{z}}^{2}{(\\mathbf{A})} = \\operatorname{A_{z}}{(\\mathbf{A})} \\sin{(\\mathbf{A})} and \\int \\operatorname{A_{z}}^{2}{(\\mathbf{A})} d\\mathbf{A} = \\int \\operatorname{A_{z}}{(\\mathbf{A})} \\sin{(\\mathbf{A})} d\\mathbf{A} and \\operatorname{v_{z}}{(\\mathbf{A})} = (\\int \\operatorname{A_{z}}{(\\mathbf{A})} \\sin{(\\mathbf{A})} d\\mathbf{A})^{2} and \\operatorname{v_{z}}{(\\mathbf{A})} = (\\int \\operatorname{A_{z}}^{2}{(\\mathbf{A})} d\\mathbf{A})^{2}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Pow(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(Mul(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(Pow(Function('A_z')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\theta_{1}{(\\phi_1,\\chi)} = \\log{(\\chi + \\phi_1)} and \\hat{\\mathbf{r}}{(\\phi_1,\\chi)} = \\chi + \\phi_1, then obtain \\frac{\\frac{\\partial}{\\partial \\chi} \\log{(\\chi + \\phi_1)}}{\\phi_1} = \\frac{\\frac{\\partial}{\\partial \\chi} \\log{(\\hat{\\mathbf{r}}{(\\phi_1,\\chi)})}}{\\phi_1}", "derivation": "\\theta_{1}{(\\phi_1,\\chi)} = \\log{(\\chi + \\phi_1)} and \\frac{\\partial}{\\partial \\chi} \\theta_{1}{(\\phi_1,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\chi + \\phi_1)} and \\hat{\\mathbf{r}}{(\\phi_1,\\chi)} = \\chi + \\phi_1 and \\frac{\\partial}{\\partial \\chi} \\theta_{1}{(\\phi_1,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\hat{\\mathbf{r}}{(\\phi_1,\\chi)})} and \\frac{\\partial}{\\partial \\chi} \\log{(\\chi + \\phi_1)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\hat{\\mathbf{r}}{(\\phi_1,\\chi)})} and \\frac{\\frac{\\partial}{\\partial \\chi} \\log{(\\chi + \\phi_1)}}{\\phi_1} = \\frac{\\frac{\\partial}{\\partial \\chi} \\log{(\\hat{\\mathbf{r}}{(\\phi_1,\\chi)})}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["divide", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\theta)} = \\sin{(\\sin{(\\theta)})}, then derive \\frac{d}{d \\theta} \\operatorname{A_{y}}{(\\theta)} = \\cos{(\\theta)} \\cos{(\\sin{(\\theta)})}, then obtain \\cos{(\\theta)} \\frac{d}{d \\theta} \\sin{(\\sin{(\\theta)})} = \\cos^{2}{(\\theta)} \\cos{(\\sin{(\\theta)})}", "derivation": "\\operatorname{A_{y}}{(\\theta)} = \\sin{(\\sin{(\\theta)})} and \\frac{d}{d \\theta} \\operatorname{A_{y}}{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\sin{(\\theta)})} and \\frac{d}{d \\theta} \\operatorname{A_{y}}{(\\theta)} = \\cos{(\\theta)} \\cos{(\\sin{(\\theta)})} and \\frac{d}{d \\theta} \\sin{(\\sin{(\\theta)})} = \\cos{(\\theta)} \\cos{(\\sin{(\\theta)})} and \\cos{(\\theta)} \\frac{d}{d \\theta} \\sin{(\\sin{(\\theta)})} = \\cos^{2}{(\\theta)} \\cos{(\\sin{(\\theta)})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\theta', commutative=True)), sin(sin(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\theta', commutative=True)), cos(sin(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\theta', commutative=True)), cos(sin(Symbol('\\\\theta', commutative=True)))))"], [["times", 4, "cos(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\theta', commutative=True)), Derivative(sin(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(2)), cos(sin(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(g,b)} = b \\log{(g)} and E{(g,b)} = \\frac{\\partial}{\\partial g} b \\log{(g)}, then derive g + E{(g,b)} + \\log{(g)} = \\frac{b}{g} + g + \\log{(g)}, then obtain \\frac{g + E{(g,b)} + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} b \\log{(g)})} = \\frac{\\frac{b}{g} + g + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} b \\log{(g)})}", "derivation": "\\operatorname{r_{0}}{(g,b)} = b \\log{(g)} and E{(g,b)} = \\frac{\\partial}{\\partial g} b \\log{(g)} and g + E{(g,b)} + \\frac{\\partial}{\\partial b} b \\log{(g)} = g + \\frac{\\partial}{\\partial b} b \\log{(g)} + \\frac{\\partial}{\\partial g} b \\log{(g)} and g + E{(g,b)} + \\log{(g)} = \\frac{b}{g} + g + \\log{(g)} and \\frac{g + E{(g,b)} + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} \\operatorname{r_{0}}{(g,b)})} = \\frac{\\frac{b}{g} + g + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} \\operatorname{r_{0}}{(g,b)})} and \\frac{g + E{(g,b)} + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} b \\log{(g)})} = \\frac{\\frac{b}{g} + g + \\log{(g)}}{b (g + \\frac{\\partial}{\\partial b} b \\log{(g)})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["add", 2, "Add(Symbol('g', commutative=True), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], "Equality(Add(Symbol('g', commutative=True), Function('E')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Symbol('g', commutative=True), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('g', commutative=True), Function('E')(Symbol('g', commutative=True), Symbol('b', commutative=True)), log(Symbol('g', commutative=True))), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('g', commutative=True), log(Symbol('g', commutative=True))))"], [["divide", 4, "Mul(Symbol('b', commutative=True), Add(Symbol('g', commutative=True), Derivative(Function('r_0')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Derivative(Function('r_0')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(-1)), Add(Symbol('g', commutative=True), Function('E')(Symbol('g', commutative=True), Symbol('b', commutative=True)), log(Symbol('g', commutative=True)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Derivative(Function('r_0')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('g', commutative=True), log(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(-1)), Add(Symbol('g', commutative=True), Function('E')(Symbol('g', commutative=True), Symbol('b', commutative=True)), log(Symbol('g', commutative=True)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Derivative(Mul(Symbol('b', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('g', commutative=True), log(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\Omega)} = \\log{(\\Omega)}, then derive \\int \\operatorname{a^{\\dagger}}{(\\Omega)} d\\Omega = \\Omega \\log{(\\Omega)} - \\Omega + y, then obtain \\int \\log{(\\Omega)} d\\Omega = \\Omega \\log{(\\Omega)} - \\Omega + y", "derivation": "\\operatorname{a^{\\dagger}}{(\\Omega)} = \\log{(\\Omega)} and \\int \\operatorname{a^{\\dagger}}{(\\Omega)} d\\Omega = \\int \\log{(\\Omega)} d\\Omega and \\int \\operatorname{a^{\\dagger}}{(\\Omega)} d\\Omega = \\Omega \\log{(\\Omega)} - \\Omega + y and \\int \\operatorname{a^{\\dagger}}{(\\Omega)} d\\Omega = \\Omega \\operatorname{a^{\\dagger}}{(\\Omega)} - \\Omega + y and \\int \\log{(\\Omega)} d\\Omega = \\Omega \\log{(\\Omega)} - \\Omega + y", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{g})} = \\log{(e^{\\mathbf{g}})}, then obtain \\frac{\\mathbf{J}^{16}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{14}} = \\frac{\\mathbf{J}^{8}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{6}}", "derivation": "\\mathbf{J}{(\\mathbf{g})} = \\log{(e^{\\mathbf{g}})} and \\mathbf{J}^{2}{(\\mathbf{g})} = \\mathbf{J}{(\\mathbf{g})} \\log{(e^{\\mathbf{g}})} and \\frac{\\mathbf{J}^{2}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}} = \\mathbf{J}{(\\mathbf{g})} and \\frac{\\mathbf{J}^{4}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{2}} = \\mathbf{J}^{2}{(\\mathbf{g})} and \\frac{\\mathbf{J}^{8}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{6}} = \\frac{\\mathbf{J}^{4}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{2}} and \\frac{\\mathbf{J}^{16}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{14}} = \\frac{\\mathbf{J}^{8}{(\\mathbf{g})}}{\\log{(e^{\\mathbf{g}})}^{6}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), log(exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), log(exp(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 2, "log(exp(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(4)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-2))), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(8)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-6))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(4)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(16)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-14))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(8)), Pow(log(exp(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-6))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} = \\log{(U^{\\hat{\\mathbf{r}}})}, then obtain \\iint \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} - \\log{(U^{\\hat{\\mathbf{r}}})}) dU dU = \\iint \\frac{d}{d \\hat{\\mathbf{r}}} 0 dU dU", "derivation": "\\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} = \\log{(U^{\\hat{\\mathbf{r}}})} and \\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} - \\log{(U^{\\hat{\\mathbf{r}}})} = 0 and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} - \\log{(U^{\\hat{\\mathbf{r}}})}) = \\frac{d}{d \\hat{\\mathbf{r}}} 0 and \\int \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} - \\log{(U^{\\hat{\\mathbf{r}}})}) dU = \\int \\frac{d}{d \\hat{\\mathbf{r}}} 0 dU and \\iint \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\operatorname{v_{x}}{(U,\\hat{\\mathbf{r}})} - \\log{(U^{\\hat{\\mathbf{r}}})}) dU dU = \\iint \\frac{d}{d \\hat{\\mathbf{r}}} 0 dU dU", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["minus", 1, "log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Add(Function('v_x')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Add(Function('v_x')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Add(Function('v_x')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Add(Function('v_x')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), log(Pow(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f}, then obtain \\iint \\frac{\\int \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} df}{\\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} df} df d\\dot{\\mathbf{r}} = \\iint 1 df d\\dot{\\mathbf{r}}", "derivation": "\\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} and \\int \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} df = \\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} df and \\frac{\\int \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} df}{\\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} df} = 1 and \\int \\frac{\\int \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} df}{\\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} df} df = \\int 1 df and \\iint \\frac{\\int \\operatorname{v_{t}}{(\\dot{\\mathbf{r}},f)} df}{\\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\dot{\\mathbf{r}}^{f} df} df d\\dot{\\mathbf{r}} = \\iint 1 df d\\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True)))"], "Equality(Mul(Integral(Function('v_t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Pow(Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Integral(Function('v_t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Pow(Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True))), Integral(Integer(1), Tuple(Symbol('f', commutative=True))))"], [["integrate", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Mul(Integral(Function('v_t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Pow(Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(\\dot{\\mathbf{r}})}, then derive \\hat{x}{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})} = - \\sin{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})}, then obtain \\cos{(\\dot{\\mathbf{r}})} + \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(\\dot{\\mathbf{r}})} = - \\sin{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})}", "derivation": "\\hat{x}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(\\dot{\\mathbf{r}})} and \\hat{x}{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}})} + \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(\\dot{\\mathbf{r}})} and \\hat{x}{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})} = - \\sin{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})} and \\cos{(\\dot{\\mathbf{r}})} + \\frac{d}{d \\dot{\\mathbf{r}}} \\cos{(\\dot{\\mathbf{r}})} = - \\sin{(\\dot{\\mathbf{r}})} + \\cos{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["add", 1, "cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(-1), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given l{(\\mathbf{S},C_{d})} = C_{d} \\mathbf{S}, then obtain - (C_{d} \\mathbf{S} l^{2}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} + (l^{3}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} = 0", "derivation": "l{(\\mathbf{S},C_{d})} = C_{d} \\mathbf{S} and l^{2}{(\\mathbf{S},C_{d})} = C_{d} \\mathbf{S} l{(\\mathbf{S},C_{d})} and C_{d} \\mathbf{S} l^{2}{(\\mathbf{S},C_{d})} = C_{d}^{2} \\mathbf{S}^{2} l{(\\mathbf{S},C_{d})} and l^{3}{(\\mathbf{S},C_{d})} = C_{d} \\mathbf{S} l^{2}{(\\mathbf{S},C_{d})} and (l^{3}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} = (C_{d} \\mathbf{S} l^{2}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} and - (C_{d} \\mathbf{S} l^{2}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} + (l^{3}{(\\mathbf{S},C_{d})})^{\\mathbf{S}} = 0", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True))))"], [["times", 2, "Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2))), Mul(Pow(Symbol('C_d', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(3)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2))))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(3)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 5, "Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Pow(Function('l')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('C_d', commutative=True)), Integer(3)), Symbol('\\\\mathbf{S}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(v_{z},l,M)} = \\frac{M l}{v_{z}}, then obtain \\frac{\\partial}{\\partial l} \\int (\\cos{(F_{x})} + \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl + 1) dl = \\frac{\\partial}{\\partial l} \\int (\\cos{(F_{x})} + \\int \\frac{M l}{v_{z}} dl + 1) dl", "derivation": "\\hat{\\mathbf{r}}{(v_{z},l,M)} = \\frac{M l}{v_{z}} and \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl = \\int \\frac{M l}{v_{z}} dl and \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl + 1 = \\int \\frac{M l}{v_{z}} dl + 1 and \\cos{(F_{x})} + \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl + 1 = \\cos{(F_{x})} + \\int \\frac{M l}{v_{z}} dl + 1 and \\int (\\cos{(F_{x})} + \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl + 1) dl = \\int (\\cos{(F_{x})} + \\int \\frac{M l}{v_{z}} dl + 1) dl and \\frac{\\partial}{\\partial l} \\int (\\cos{(F_{x})} + \\int \\hat{\\mathbf{r}}{(v_{z},l,M)} dl + 1) dl = \\frac{\\partial}{\\partial l} \\int (\\cos{(F_{x})} + \\int \\frac{M l}{v_{z}} dl + 1) dl", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Tuple(Symbol('l', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Add(Integral(Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Tuple(Symbol('l', commutative=True))), Integer(1)))"], [["add", 3, "cos(Symbol('F_x', commutative=True))"], "Equality(Add(cos(Symbol('F_x', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Add(cos(Symbol('F_x', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Tuple(Symbol('l', commutative=True))), Integer(1)))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Add(cos(Symbol('F_x', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True))), Integral(Add(cos(Symbol('F_x', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True))))"], [["differentiate", 5, "Symbol('l', commutative=True)"], "Equality(Derivative(Integral(Add(cos(Symbol('F_x', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('l', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(Add(cos(Symbol('F_x', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Symbol('l', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(s)} = \\sin{(s)} and \\operatorname{g_{\\varepsilon}}{(s)} = \\sin{(s)}, then obtain \\frac{s + \\operatorname{g_{\\varepsilon}}{(s)}}{\\sin{(s)}} = \\frac{s + \\operatorname{A_{1}}{(s)}}{\\sin{(s)}}", "derivation": "\\operatorname{A_{1}}{(s)} = \\sin{(s)} and \\operatorname{g_{\\varepsilon}}{(s)} = \\sin{(s)} and \\operatorname{g_{\\varepsilon}}{(s)} = \\operatorname{A_{1}}{(s)} and s + \\operatorname{g_{\\varepsilon}}{(s)} = s + \\operatorname{A_{1}}{(s)} and \\frac{s + \\operatorname{g_{\\varepsilon}}{(s)}}{\\sin{(s)}} = \\frac{s + \\operatorname{A_{1}}{(s)}}{\\sin{(s)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('s', commutative=True)), Function('A_1')(Symbol('s', commutative=True)))"], [["add", 3, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), Function('A_1')(Symbol('s', commutative=True))))"], [["divide", 4, "sin(Symbol('s', commutative=True))"], "Equality(Mul(Add(Symbol('s', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1))), Mul(Add(Symbol('s', commutative=True), Function('A_1')(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(H,\\pi)} = H \\pi and u{(H,\\pi)} = (H \\pi)^{H} and \\operatorname{z^{*}}{(H,\\pi)} = H \\pi, then obtain (H \\pi - H) \\tilde{g}^*{(H,\\pi)} + \\operatorname{z^{*}}{(H,\\pi)} \\sin{(u{(H,\\pi)})} - \\frac{1}{\\pi} = (H \\pi - H) \\tilde{g}^*{(H,\\pi)} + \\operatorname{z^{*}}{(H,\\pi)} \\sin{(\\tilde{g}^*^{H}{(H,\\pi)})} - \\frac{1}{\\pi}", "derivation": "\\tilde{g}^*{(H,\\pi)} = H \\pi and \\tilde{g}^*^{H}{(H,\\pi)} = (H \\pi)^{H} and u{(H,\\pi)} = (H \\pi)^{H} and \\sin{(u{(H,\\pi)})} = \\sin{((H \\pi)^{H})} and H \\pi \\sin{(u{(H,\\pi)})} = H \\pi \\sin{((H \\pi)^{H})} and H \\pi \\sin{(u{(H,\\pi)})} = H \\pi \\sin{(\\tilde{g}^*^{H}{(H,\\pi)})} and \\operatorname{z^{*}}{(H,\\pi)} = H \\pi and \\operatorname{z^{*}}{(H,\\pi)} \\sin{(u{(H,\\pi)})} = \\operatorname{z^{*}}{(H,\\pi)} \\sin{(\\tilde{g}^*^{H}{(H,\\pi)})} and (H \\pi - H) \\tilde{g}^*{(H,\\pi)} + \\operatorname{z^{*}}{(H,\\pi)} \\sin{(u{(H,\\pi)})} - \\frac{1}{\\pi} = (H \\pi - H) \\tilde{g}^*{(H,\\pi)} + \\operatorname{z^{*}}{(H,\\pi)} \\sin{(\\tilde{g}^*^{H}{(H,\\pi)})} - \\frac{1}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))"], [["sin", 3], "Equality(sin(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), sin(Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True))))"], [["times", 4, "Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True), sin(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True), sin(Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True), sin(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True), sin(Pow(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Function('z^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Function('z^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))))"], [["add", 8, "Add(Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Function('z^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Function('u')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Add(Mul(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Function('z^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('H', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbb{I}{(m_{s})} = \\cos{(\\sin{(m_{s})})}, then obtain 2 \\mathbb{I}{(m_{s})} = 2 \\cos{(\\sin{(m_{s})})}", "derivation": "\\mathbb{I}{(m_{s})} = \\cos{(\\sin{(m_{s})})} and 2 \\mathbb{I}{(m_{s})} = \\mathbb{I}{(m_{s})} + \\cos{(\\sin{(m_{s})})} and \\mathbb{I}{(m_{s})} + \\cos{(\\sin{(m_{s})})} = 2 \\cos{(\\sin{(m_{s})})} and 2 \\mathbb{I}{(m_{s})} = 2 \\cos{(\\sin{(m_{s})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True)), cos(sin(Symbol('m_s', commutative=True))))"], [["add", 1, "Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True))), Add(Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True)), cos(sin(Symbol('m_s', commutative=True)))))"], [["add", 1, "cos(sin(Symbol('m_s', commutative=True)))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True)), cos(sin(Symbol('m_s', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('m_s', commutative=True))), Mul(Integer(2), cos(sin(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given s{(\\varphi^*,x,\\varepsilon_0)} = \\varepsilon_0 \\varphi^* x, then derive \\frac{\\partial}{\\partial \\varepsilon_0} s{(\\varphi^*,x,\\varepsilon_0)} = \\varphi^* x, then obtain \\varepsilon_0 \\varphi^* x + n = \\int \\varphi^* x d\\varepsilon_0", "derivation": "s{(\\varphi^*,x,\\varepsilon_0)} = \\varepsilon_0 \\varphi^* x and \\frac{\\partial}{\\partial \\varepsilon_0} s{(\\varphi^*,x,\\varepsilon_0)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 \\varphi^* x and \\frac{\\partial}{\\partial \\varepsilon_0} s{(\\varphi^*,x,\\varepsilon_0)} = \\varphi^* x and \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 \\varphi^* x = \\varphi^* x and \\int \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 \\varphi^* x d\\varepsilon_0 = \\int \\varphi^* x d\\varepsilon_0 and \\varepsilon_0 \\varphi^* x + n = \\int \\varphi^* x d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)))"], [["integrate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Symbol('n', commutative=True)), Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})} = \\sin{(\\frac{\\hat{x}}{t_{1}})}, then obtain \\frac{\\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})}}{- \\hat{x} + \\sin{(\\frac{\\hat{x}}{t_{1}})}} = \\frac{\\sin{(\\frac{\\hat{x}}{t_{1}})}}{- \\hat{x} + \\sin{(\\frac{\\hat{x}}{t_{1}})}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})} = \\sin{(\\frac{\\hat{x}}{t_{1}})} and - \\hat{x} + \\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})} = - \\hat{x} + \\sin{(\\frac{\\hat{x}}{t_{1}})} and \\frac{\\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})}}{- \\hat{x} + \\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})}} = \\frac{\\sin{(\\frac{\\hat{x}}{t_{1}})}}{- \\hat{x} + \\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})}} and \\frac{\\operatorname{x^{{\\}'}}{(\\hat{x},t_{1})}}{- \\hat{x} + \\sin{(\\frac{\\hat{x}}{t_{1}})}} = \\frac{\\sin{(\\frac{\\hat{x}}{t_{1}})}}{- \\hat{x} + \\sin{(\\frac{\\hat{x}}{t_{1}})}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Integer(-1)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))), Integer(-1)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{H})} = \\int \\log{(\\mathbf{H})} d\\mathbf{H} and \\mathbf{v}{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\operatorname{E_{n}}^{\\mathbf{H}}{(\\mathbf{H})} = (\\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + c_{0})^{\\mathbf{H}}, then obtain (\\int \\log{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\mathbf{H} \\mathbf{v}{(\\mathbf{H})} - \\mathbf{H} + c_{0})^{\\mathbf{H}}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{H})} = \\int \\log{(\\mathbf{H})} d\\mathbf{H} and \\operatorname{E_{n}}^{\\mathbf{H}}{(\\mathbf{H})} = (\\int \\log{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} and \\operatorname{E_{n}}^{\\mathbf{H}}{(\\mathbf{H})} = (\\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + c_{0})^{\\mathbf{H}} and (\\int \\log{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\mathbf{H} \\log{(\\mathbf{H})} - \\mathbf{H} + c_{0})^{\\mathbf{H}} and \\mathbf{v}{(\\mathbf{H})} = \\log{(\\mathbf{H})} and (\\int \\log{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\mathbf{H} \\mathbf{v}{(\\mathbf{H})} - \\mathbf{H} + c_{0})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{H}', commutative=True)), Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('E_n')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given a{(\\phi,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi}, then obtain a{(\\phi,\\sigma_p)} + (\\frac{\\partial}{\\partial \\phi} \\int a{(\\phi,\\sigma_p)} d\\sigma_p)^{\\phi} = a{(\\phi,\\sigma_p)} + (\\frac{\\partial}{\\partial \\phi} \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} d\\sigma_p)^{\\phi}", "derivation": "a{(\\phi,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} and \\int a{(\\phi,\\sigma_p)} d\\sigma_p = \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} d\\sigma_p and \\frac{\\partial}{\\partial \\phi} \\int a{(\\phi,\\sigma_p)} d\\sigma_p = \\frac{\\partial}{\\partial \\phi} \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} d\\sigma_p and (\\frac{\\partial}{\\partial \\phi} \\int a{(\\phi,\\sigma_p)} d\\sigma_p)^{\\phi} = (\\frac{\\partial}{\\partial \\phi} \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} d\\sigma_p)^{\\phi} and a{(\\phi,\\sigma_p)} + (\\frac{\\partial}{\\partial \\phi} \\int a{(\\phi,\\sigma_p)} d\\sigma_p)^{\\phi} = a{(\\phi,\\sigma_p)} + (\\frac{\\partial}{\\partial \\phi} \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\sigma_p}{\\phi} d\\sigma_p)^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(Integral(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)))"], [["add", 4, "Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integral(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))), Add(Function('a')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integral(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\phi)} = \\log{(\\phi)}, then derive \\frac{\\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi}{\\log{(\\phi)}} = \\frac{\\frac{\\partial}{\\partial \\phi} (B + \\phi \\log{(\\phi)} - \\phi)}{\\log{(\\phi)}}, then obtain \\frac{\\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi}{\\log{(\\phi)}} = 1", "derivation": "\\mathbf{B}{(\\phi)} = \\log{(\\phi)} and \\int \\mathbf{B}{(\\phi)} d\\phi = \\int \\log{(\\phi)} d\\phi and \\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi = \\frac{d}{d \\phi} \\int \\log{(\\phi)} d\\phi and \\frac{\\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi}{\\log{(\\phi)}} = \\frac{\\frac{d}{d \\phi} \\int \\log{(\\phi)} d\\phi}{\\log{(\\phi)}} and \\frac{\\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi}{\\log{(\\phi)}} = \\frac{\\frac{\\partial}{\\partial \\phi} (B + \\phi \\log{(\\phi)} - \\phi)}{\\log{(\\phi)}} and \\frac{\\frac{d}{d \\phi} \\int \\mathbf{B}{(\\phi)} d\\phi}{\\log{(\\phi)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["divide", 3, "log(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1)), Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1)), Derivative(Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1)), Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1)), Derivative(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1)), Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given T{(\\mathbf{A})} = e^{\\mathbf{A}} and A{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}, then derive \\frac{d}{d \\mathbf{A}} T{(\\mathbf{A})} = e^{\\mathbf{A}}, then derive A{(\\mathbf{A})} = e^{\\mathbf{A}}, then obtain \\frac{d}{d \\mathbf{A}} T{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} T{(\\mathbf{A})}", "derivation": "T{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} T{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and A{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} T{(\\mathbf{A})} = e^{\\mathbf{A}} and A{(\\mathbf{A})} = e^{\\mathbf{A}} and e^{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} T{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} T{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('T')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Function('A')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(exp(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Function('T')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Function('T')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi_{1}{(V_{\\mathbf{E}})} = e^{\\cos{(V_{\\mathbf{E}})}}, then derive \\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})} = - e^{\\cos{(V_{\\mathbf{E}})}} \\sin{(V_{\\mathbf{E}})}, then obtain \\frac{d}{d V_{\\mathbf{E}}} e^{\\cos{(V_{\\mathbf{E}})}} = - e^{\\cos{(V_{\\mathbf{E}})}} \\sin{(V_{\\mathbf{E}})}", "derivation": "\\phi_{1}{(V_{\\mathbf{E}})} = e^{\\cos{(V_{\\mathbf{E}})}} and \\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} e^{\\cos{(V_{\\mathbf{E}})}} and \\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})} = - e^{\\cos{(V_{\\mathbf{E}})}} \\sin{(V_{\\mathbf{E}})} and \\frac{d}{d V_{\\mathbf{E}}} e^{\\cos{(V_{\\mathbf{E}})}} = - e^{\\cos{(V_{\\mathbf{E}})}} \\sin{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given m{(b)} = \\cos{(\\sin{(b)})} and \\operatorname{A_{z}}{(\\dot{\\mathbf{r}},\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime}), then obtain 2 \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime}) = \\frac{(m{(b)} + \\cos{(\\sin{(b)})}) \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime})}{m{(b)}}", "derivation": "m{(b)} = \\cos{(\\sin{(b)})} and 2 m{(b)} = m{(b)} + \\cos{(\\sin{(b)})} and 2 = \\frac{m{(b)} + \\cos{(\\sin{(b)})}}{m{(b)}} and \\operatorname{A_{z}}{(\\dot{\\mathbf{r}},\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime}) and 2 \\operatorname{A_{z}}{(\\dot{\\mathbf{r}},\\eta^{\\prime})} = \\frac{(m{(b)} + \\cos{(\\sin{(b)})}) \\operatorname{A_{z}}{(\\dot{\\mathbf{r}},\\eta^{\\prime})}}{m{(b)}} and 2 \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime}) = \\frac{(m{(b)} + \\cos{(\\sin{(b)})}) \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\eta^{\\prime})}{m{(b)}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True))))"], [["add", 1, "Function('m')(Symbol('b', commutative=True))"], "Equality(Mul(Integer(2), Function('m')(Symbol('b', commutative=True))), Add(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True)))))"], [["divide", 2, "Function('m')(Symbol('b', commutative=True))"], "Equality(Integer(2), Mul(Add(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True)))), Pow(Function('m')(Symbol('b', commutative=True)), Integer(-1))))"], ["get_premise", "Equality(Function('A_z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["times", 3, "Function('A_z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(2), Function('A_z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Add(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True)))), Function('A_z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('m')(Symbol('b', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Mul(Add(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True)))), Pow(Function('m')(Symbol('b', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{M})} = e^{\\mathbf{M}}, then derive \\frac{d}{d \\mathbf{M}} \\operatorname{v_{1}}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain e^{\\mathbf{M}} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\operatorname{v_{1}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\operatorname{v_{1}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\operatorname{v_{1}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\operatorname{v_{1}}{(\\mathbf{M})} and e^{\\mathbf{M}} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_1')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Function('v_1')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(J)} = \\log{(\\log{(J)})}, then obtain J + \\int (\\operatorname{c_{0}}{(J)} + \\log{(\\log{(J)})}) dJ = J + \\int 2 \\log{(\\log{(J)})} dJ", "derivation": "\\operatorname{c_{0}}{(J)} = \\log{(\\log{(J)})} and \\operatorname{c_{0}}{(J)} + \\log{(\\log{(J)})} = 2 \\log{(\\log{(J)})} and \\int (\\operatorname{c_{0}}{(J)} + \\log{(\\log{(J)})}) dJ = \\int 2 \\log{(\\log{(J)})} dJ and J + \\int (\\operatorname{c_{0}}{(J)} + \\log{(\\log{(J)})}) dJ = J + \\int 2 \\log{(\\log{(J)})} dJ", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True))))"], [["add", 1, "log(log(Symbol('J', commutative=True)))"], "Equality(Add(Function('c_0')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Mul(Integer(2), log(log(Symbol('J', commutative=True)))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Function('c_0')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Integer(2), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))))"], [["add", 3, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Integral(Add(Function('c_0')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)))), Add(Symbol('J', commutative=True), Integral(Mul(Integer(2), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(u)} = \\log{(u)}, then derive \\int \\theta_{1}{(u)} du = \\hat{p}_0 + u \\log{(u)} - u, then obtain C_{2} + u \\log{(u)} - u = \\hat{p}_0 + u \\log{(u)} - u", "derivation": "\\theta_{1}{(u)} = \\log{(u)} and \\int \\theta_{1}{(u)} du = \\int \\log{(u)} du and \\int \\theta_{1}{(u)} du = \\hat{p}_0 + u \\log{(u)} - u and \\int \\log{(u)} du = \\hat{p}_0 + u \\log{(u)} - u and C_{2} + u \\log{(u)} - u = \\hat{p}_0 + u \\log{(u)} - u", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then derive \\int 0 d\\mathbf{A} = \\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}, then obtain \\frac{\\int 0 d\\mathbf{A}}{\\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}} = \\frac{\\mathbf{g}}{\\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}}", "derivation": "\\mathbf{E}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\mathbf{E}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and 0 = - \\frac{d}{d \\mathbf{A}} \\mathbf{E}{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\int 0 d\\mathbf{A} = \\int (- \\frac{d}{d \\mathbf{A}} \\mathbf{E}{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})}) d\\mathbf{A} and \\int 0 d\\mathbf{A} = \\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and \\int 0 d\\mathbf{A} = \\mathbf{g} and \\frac{\\int 0 d\\mathbf{A}}{\\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}} = \\frac{\\mathbf{g}}{\\mathbf{g} - \\mathbf{E}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))"], [["divide", 6, "Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(a^{\\dagger})} = e^{a^{\\dagger}} and W{(a^{\\dagger})} = (e^{a^{\\dagger}})^{a^{\\dagger}}, then obtain \\frac{d}{d a^{\\dagger}} (e^{a^{\\dagger}})^{a^{\\dagger}} = \\frac{d}{d a^{\\dagger}} W{(a^{\\dagger})}", "derivation": "\\operatorname{P_{e}}{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{P_{e}}^{a^{\\dagger}}{(a^{\\dagger})} = (e^{a^{\\dagger}})^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} \\operatorname{P_{e}}^{a^{\\dagger}}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} (e^{a^{\\dagger}})^{a^{\\dagger}} and W{(a^{\\dagger})} = (e^{a^{\\dagger}})^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} \\operatorname{P_{e}}^{a^{\\dagger}}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} W{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} (e^{a^{\\dagger}})^{a^{\\dagger}} = \\frac{d}{d a^{\\dagger}} W{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Pow(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(exp(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(v_{x},V)} = \\frac{V}{v_{x}} and \\dot{\\mathbf{r}}{(\\nabla,\\lambda)} = e^{\\lambda + \\nabla}, then obtain \\int (\\frac{V}{v_{x}} + \\operatorname{P_{g}}{(v_{x},V)} - e^{\\lambda + \\nabla} + 1) dV = \\int (\\frac{2 V}{v_{x}} - e^{\\lambda + \\nabla} + 1) dV", "derivation": "\\operatorname{P_{g}}{(v_{x},V)} = \\frac{V}{v_{x}} and \\operatorname{P_{g}}{(v_{x},V)} + 1 = \\frac{V}{v_{x}} + 1 and \\dot{\\mathbf{r}}{(\\nabla,\\lambda)} = e^{\\lambda + \\nabla} and \\frac{V}{v_{x}} + \\operatorname{P_{g}}{(v_{x},V)} - \\dot{\\mathbf{r}}{(\\nabla,\\lambda)} + 1 = \\frac{2 V}{v_{x}} - \\dot{\\mathbf{r}}{(\\nabla,\\lambda)} + 1 and \\frac{V}{v_{x}} + \\operatorname{P_{g}}{(v_{x},V)} - e^{\\lambda + \\nabla} + 1 = \\frac{2 V}{v_{x}} - e^{\\lambda + \\nabla} + 1 and \\int (\\frac{V}{v_{x}} + \\operatorname{P_{g}}{(v_{x},V)} - e^{\\lambda + \\nabla} + 1) dV = \\int (\\frac{2 V}{v_{x}} - e^{\\lambda + \\nabla} + 1) dV", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('P_g')(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Integer(1)), Add(Mul(Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Integer(1)))"], ["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)), exp(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Function('P_g')(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Function('P_g')(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(1)), Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), exp(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(1)))"], [["integrate", 5, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Function('P_g')(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(1)), Tuple(Symbol('V', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('V', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), exp(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integer(1)), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given Q{(L,W)} = L - W, then obtain \\int 0 dL = \\int - (L - W - Q{(L,W)}) Q{(L,W)} dL", "derivation": "Q{(L,W)} = L - W and 0 = L - W - Q{(L,W)} and 0 = - (L - W - Q{(L,W)}) Q{(L,W)} and \\int 0 dL = \\int - (L - W - Q{(L,W)}) Q{(L,W)} dL", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["minus", 1, "Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('L', commutative=True))), Integral(Mul(Integer(-1), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Function('Q')(Symbol('L', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{\\ddot{x}}, then obtain \\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)} \\log{(\\frac{\\mathbf{J}_M}{\\ddot{x}})} = \\frac{\\mathbf{J}_M \\log{(\\frac{\\mathbf{J}_M}{\\ddot{x}})}}{\\ddot{x}}", "derivation": "\\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{\\ddot{x}} and \\log{(\\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)})} = \\log{(\\frac{\\mathbf{J}_M}{\\ddot{x}})} and \\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)} \\log{(\\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)})} = \\frac{\\mathbf{J}_M \\log{(\\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)})}}{\\ddot{x}} and \\operatorname{v_{x}}{(\\ddot{x},\\mathbf{J}_M)} \\log{(\\frac{\\mathbf{J}_M}{\\ddot{x}})} = \\frac{\\mathbf{J}_M \\log{(\\frac{\\mathbf{J}_M}{\\ddot{x}})}}{\\ddot{x}}", "srepr_derivation": [["get_premise", "Equality(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["log", 1], "Equality(log(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 1, "log(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True), log(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('v_x')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True), log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\Omega)} = \\sin{(\\sin{(\\Omega)})}, then obtain -1 + \\frac{\\Psi_{\\lambda}{(\\Omega)}}{\\Omega \\sin{(\\sin{(\\Omega)})}} = -1 + \\frac{1}{\\Omega}", "derivation": "\\Psi_{\\lambda}{(\\Omega)} = \\sin{(\\sin{(\\Omega)})} and \\frac{\\Psi_{\\lambda}{(\\Omega)}}{\\sin{(\\sin{(\\Omega)})}} = 1 and \\frac{\\Psi_{\\lambda}{(\\Omega)}}{\\Omega \\sin{(\\sin{(\\Omega)})}} = \\frac{1}{\\Omega} and -1 + \\frac{\\Psi_{\\lambda}{(\\Omega)}}{\\Omega \\sin{(\\sin{(\\Omega)})}} = -1 + \\frac{1}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), sin(sin(Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1)))), Add(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))"]]}, {"prompt": "Given V{(x^\\prime,E_{\\lambda})} = \\frac{\\log{(E_{\\lambda})}}{x^\\prime}, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} V{(x^\\prime,E_{\\lambda})} + E_{\\lambda}) = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\frac{E_{\\lambda} \\log{(E_{\\lambda})}}{x^\\prime})", "derivation": "V{(x^\\prime,E_{\\lambda})} = \\frac{\\log{(E_{\\lambda})}}{x^\\prime} and E_{\\lambda} V{(x^\\prime,E_{\\lambda})} = \\frac{E_{\\lambda} \\log{(E_{\\lambda})}}{x^\\prime} and E_{\\lambda} V{(x^\\prime,E_{\\lambda})} + E_{\\lambda} = E_{\\lambda} + \\frac{E_{\\lambda} \\log{(E_{\\lambda})}}{x^\\prime} and \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} V{(x^\\prime,E_{\\lambda})} + E_{\\lambda}) = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\frac{E_{\\lambda} \\log{(E_{\\lambda})}}{x^\\prime})", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('V')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('V')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["differentiate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('V')(Symbol('x^\\\\prime', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} = \\int (\\omega + \\theta_1) d\\omega, then derive \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} = \\frac{\\omega^{2}}{2} + \\omega \\theta_1 + i, then obtain (\\omega + \\theta_1) (E_{\\lambda} + \\frac{\\omega^{2}}{2} + \\omega \\theta_1) = (\\omega + \\theta_1) \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} = \\int (\\omega + \\theta_1) d\\omega and \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} = \\frac{\\omega^{2}}{2} + \\omega \\theta_1 + i and (\\omega + \\theta_1) \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} = (\\omega + \\theta_1) (\\frac{\\omega^{2}}{2} + \\omega \\theta_1 + i) and (\\omega + \\theta_1) \\int (\\omega + \\theta_1) d\\omega = (\\omega + \\theta_1) (\\frac{\\omega^{2}}{2} + \\omega \\theta_1 + i) and (\\omega + \\theta_1) \\int (\\omega + \\theta_1) d\\omega = (\\omega + \\theta_1) \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)} and (\\omega + \\theta_1) (E_{\\lambda} + \\frac{\\omega^{2}}{2} + \\omega \\theta_1) = (\\omega + \\theta_1) \\operatorname{V_{\\mathbf{B}}}{(\\theta_1,\\omega)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('i', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given p{(E,k)} = \\cos{(E k)} and T{(E,k)} = \\cos{(E k)}, then obtain \\sin{(2 T{(E,k)})} = \\sin{(T{(E,k)} + \\cos{(E k)})}", "derivation": "p{(E,k)} = \\cos{(E k)} and T{(E,k)} = \\cos{(E k)} and T{(E,k)} = p{(E,k)} and 2 p{(E,k)} = p{(E,k)} + \\cos{(E k)} and 2 T{(E,k)} = T{(E,k)} + \\cos{(E k)} and \\sin{(2 T{(E,k)})} = \\sin{(T{(E,k)} + \\cos{(E k)})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('E', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True)), Function('p')(Symbol('E', commutative=True), Symbol('k', commutative=True)))"], [["add", 1, "Function('p')(Symbol('E', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Integer(2), Function('p')(Symbol('E', commutative=True), Symbol('k', commutative=True))), Add(Function('p')(Symbol('E', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True))), Add(Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('k', commutative=True)))))"], [["sin", 5], "Equality(sin(Mul(Integer(2), Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True)))), sin(Add(Function('T')(Symbol('E', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('k', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{H}{(H)} = e^{H}, then obtain \\frac{(e^{H} - \\frac{\\mathbf{H}{(H)}}{H}) \\mathbf{H}{(H)}}{H} = \\frac{(e^{H} - \\frac{\\mathbf{H}{(H)}}{H}) e^{H}}{H}", "derivation": "\\mathbf{H}{(H)} = e^{H} and \\frac{\\mathbf{H}{(H)}}{H} = \\frac{e^{H}}{H} and \\mathbf{H}{(H)} - \\frac{e^{H}}{H} = e^{H} - \\frac{e^{H}}{H} and \\mathbf{H}{(H)} - \\frac{\\mathbf{H}{(H)}}{H} = e^{H} - \\frac{\\mathbf{H}{(H)}}{H} and \\frac{(\\mathbf{H}{(H)} - \\frac{\\mathbf{H}{(H)}}{H}) \\mathbf{H}{(H)}}{H} = \\frac{(\\mathbf{H}{(H)} - \\frac{\\mathbf{H}{(H)}}{H}) e^{H}}{H} and \\frac{(e^{H} - \\frac{\\mathbf{H}{(H)}}{H}) \\mathbf{H}{(H)}}{H} = \\frac{(e^{H} - \\frac{\\mathbf{H}{(H)}}{H}) e^{H}}{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), exp(Symbol('H', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('H', commutative=True), Integer(-1)), exp(Symbol('H', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), exp(Symbol('H', commutative=True)))), Add(exp(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), exp(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))), Add(exp(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))))"], [["times", 2, "Add(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True))))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))), Function('\\\\mathbf{H}')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{H}')(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))), exp(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(exp(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))), Function('\\\\mathbf{H}')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(exp(Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('H', commutative=True)))), exp(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\phi_2)} = \\phi_2, then obtain \\frac{\\log{(- 2 \\phi_2 + \\hat{H}_l{(\\phi_2)})}}{\\phi_2} = \\frac{\\log{(- \\phi_2)}}{\\phi_2}", "derivation": "\\hat{H}_l{(\\phi_2)} = \\phi_2 and - \\phi_2 + \\hat{H}_l{(\\phi_2)} = 0 and - 2 \\phi_2 + \\hat{H}_l{(\\phi_2)} = - \\phi_2 and \\log{(- 2 \\phi_2 + \\hat{H}_l{(\\phi_2)})} = \\log{(- \\phi_2)} and \\frac{\\log{(- 2 \\phi_2 + \\hat{H}_l{(\\phi_2)})}}{\\phi_2} = \\frac{\\log{(- \\phi_2)}}{\\phi_2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True))), Integer(0))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))"], [["log", 3], "Equality(log(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True)))), log(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["divide", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\phi_2', commutative=True))))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given c{(\\mathbf{P},x^\\prime)} = \\mathbf{P} + x^\\prime, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (\\frac{\\partial}{\\partial x^\\prime} c{(\\mathbf{P},x^\\prime)})^{\\mathbf{P}} = \\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{P} + x^\\prime))^{\\mathbf{P}}", "derivation": "c{(\\mathbf{P},x^\\prime)} = \\mathbf{P} + x^\\prime and \\frac{\\partial}{\\partial x^\\prime} c{(\\mathbf{P},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} (\\mathbf{P} + x^\\prime) and (\\frac{\\partial}{\\partial x^\\prime} c{(\\mathbf{P},x^\\prime)})^{\\mathbf{P}} = (\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{P} + x^\\prime))^{\\mathbf{P}} and \\frac{\\partial}{\\partial \\mathbf{P}} (\\frac{\\partial}{\\partial x^\\prime} c{(\\mathbf{P},x^\\prime)})^{\\mathbf{P}} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{P} + x^\\prime))^{\\mathbf{P}} and \\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (\\frac{\\partial}{\\partial x^\\prime} c{(\\mathbf{P},x^\\prime)})^{\\mathbf{P}} = \\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{P} + x^\\prime))^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Derivative(Pow(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))))"]]}, {"prompt": "Given Z{(\\chi)} = \\sin{(\\chi)}, then derive (\\cos{(\\chi)} \\frac{d}{d \\chi} Z{(\\chi)})^{\\chi} = (\\cos^{2}{(\\chi)})^{\\chi}, then obtain (\\cos{(\\chi)} \\frac{d}{d \\chi} Z{(\\chi)})^{\\chi} = (\\cos{(\\chi)} \\frac{d}{d \\chi} \\sin{(\\chi)})^{\\chi}", "derivation": "Z{(\\chi)} = \\sin{(\\chi)} and \\frac{d}{d \\chi} Z{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\chi)} and \\frac{d}{d \\chi} Z{(\\chi)} \\frac{d}{d \\chi} \\sin{(\\chi)} = (\\frac{d}{d \\chi} \\sin{(\\chi)})^{2} and (\\frac{d}{d \\chi} Z{(\\chi)} \\frac{d}{d \\chi} \\sin{(\\chi)})^{\\chi} = ((\\frac{d}{d \\chi} \\sin{(\\chi)})^{2})^{\\chi} and (\\cos{(\\chi)} \\frac{d}{d \\chi} Z{(\\chi)})^{\\chi} = (\\cos^{2}{(\\chi)})^{\\chi} and (\\cos{(\\chi)} \\frac{d}{d \\chi} \\sin{(\\chi)})^{\\chi} = (\\cos^{2}{(\\chi)})^{\\chi} and (\\cos{(\\chi)} \\frac{d}{d \\chi} Z{(\\chi)})^{\\chi} = (\\cos{(\\chi)} \\frac{d}{d \\chi} \\sin{(\\chi)})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 2, "Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('Z')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2)))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Derivative(Function('Z')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Pow(Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2)), Symbol('\\\\chi', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(cos(Symbol('\\\\chi', commutative=True)), Derivative(Function('Z')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(2)), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(cos(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(2)), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Mul(cos(Symbol('\\\\chi', commutative=True)), Derivative(Function('Z')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(cos(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)} = \\hat{\\mathbf{x}} \\mu, then obtain (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu)^{\\hat{\\mathbf{x}}} (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)})^{\\hat{\\mathbf{x}}} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu)^{2 \\hat{\\mathbf{x}}}", "derivation": "\\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)} = \\hat{\\mathbf{x}} \\mu and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)} = \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu and (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)})^{\\hat{\\mathbf{x}}} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu)^{\\hat{\\mathbf{x}}} and (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu)^{\\hat{\\mathbf{x}}} (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\operatorname{v_{1}}{(\\hat{\\mathbf{x}},\\mu)})^{\\hat{\\mathbf{x}}} = (\\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} \\mu)^{2 \\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Derivative(Function('v_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["times", 3, "Pow(Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Pow(Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Derivative(Function('v_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(c_{0},\\varphi^*)} = \\frac{c_{0}}{\\varphi^*}, then obtain \\frac{\\partial}{\\partial c_{0}} \\cos{((\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1) \\operatorname{M_{E}}{(c_{0},\\varphi^*)})} = \\frac{\\partial}{\\partial c_{0}} \\cos{(\\frac{c_{0} (\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1)}{\\varphi^*})}", "derivation": "\\operatorname{M_{E}}{(c_{0},\\varphi^*)} = \\frac{c_{0}}{\\varphi^*} and (\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1) \\operatorname{M_{E}}{(c_{0},\\varphi^*)} = \\frac{c_{0} (\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1)}{\\varphi^*} and \\cos{((\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1) \\operatorname{M_{E}}{(c_{0},\\varphi^*)})} = \\cos{(\\frac{c_{0} (\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1)}{\\varphi^*})} and \\frac{\\partial}{\\partial c_{0}} \\cos{((\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1) \\operatorname{M_{E}}{(c_{0},\\varphi^*)})} = \\frac{\\partial}{\\partial c_{0}} \\cos{(\\frac{c_{0} (\\operatorname{M_{E}}{(c_{0},\\varphi^*)} + 1)}{\\varphi^*})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], [["divide", 1, "Pow(Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)), Integer(-1))"], "Equality(Mul(Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)), Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c_0', commutative=True), Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1))))"], [["cos", 2], "Equality(cos(Mul(Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)), Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), cos(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c_0', commutative=True), Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)))))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(cos(Mul(Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)), Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c_0', commutative=True), Add(Function('M_E')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1)))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(h)} = e^{h}, then obtain ((- \\operatorname{r_{0}}{(h)} + \\operatorname{r_{0}}^{h}{(h)} + e^{h})^{h})^{h} = ((- \\operatorname{r_{0}}{(h)} + e^{h} + (e^{h})^{h})^{h})^{h}", "derivation": "\\operatorname{r_{0}}{(h)} = e^{h} and \\operatorname{r_{0}}^{h}{(h)} = (e^{h})^{h} and - \\operatorname{r_{0}}{(h)} + \\operatorname{r_{0}}^{h}{(h)} = - \\operatorname{r_{0}}{(h)} + (e^{h})^{h} and - \\operatorname{r_{0}}{(h)} + \\operatorname{r_{0}}^{h}{(h)} + e^{h} = - \\operatorname{r_{0}}{(h)} + e^{h} + (e^{h})^{h} and (- \\operatorname{r_{0}}{(h)} + \\operatorname{r_{0}}^{h}{(h)} + e^{h})^{h} = (- \\operatorname{r_{0}}{(h)} + e^{h} + (e^{h})^{h})^{h} and ((- \\operatorname{r_{0}}{(h)} + \\operatorname{r_{0}}^{h}{(h)} + e^{h})^{h})^{h} = ((- \\operatorname{r_{0}}{(h)} + e^{h} + (e^{h})^{h})^{h})^{h}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 2, "Function('r_0')(Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), Pow(Function('r_0')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["add", 3, "exp(Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), Pow(Function('r_0')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), exp(Symbol('h', commutative=True)), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), Pow(Function('r_0')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), exp(Symbol('h', commutative=True)), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["power", 5, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), Pow(Function('r_0')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('r_0')(Symbol('h', commutative=True))), exp(Symbol('h', commutative=True)), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given Z{(h)} = \\cos{(\\log{(h)})}, then obtain \\frac{d}{d h} (\\frac{2 Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})} - 1) = \\frac{d}{d h} (1 - \\cos{(\\log{(h)})})", "derivation": "Z{(h)} = \\cos{(\\log{(h)})} and \\frac{Z{(h)}}{\\cos{(\\log{(h)})}} = 1 and \\frac{Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})} = 1 - \\cos{(\\log{(h)})} and \\frac{d}{d h} (\\frac{Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})}) = \\frac{d}{d h} (1 - \\cos{(\\log{(h)})}) and \\frac{d}{d h} (\\frac{2 Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})} - 1) = \\frac{d}{d h} (\\frac{Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})}) and \\frac{d}{d h} (\\frac{2 Z{(h)}}{\\cos{(\\log{(h)})}} - \\cos{(\\log{(h)})} - 1) = \\frac{d}{d h} (1 - \\cos{(\\log{(h)})})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('h', commutative=True)), cos(log(Symbol('h', commutative=True))))"], [["divide", 1, "cos(log(Symbol('h', commutative=True)))"], "Equality(Mul(Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "cos(log(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))), Add(Integer(1), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(2), Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(log(Symbol('h', commutative=True)))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(2), Function('Z')(Symbol('h', commutative=True)), Pow(cos(log(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(log(Symbol('h', commutative=True)))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), cos(log(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(T)} = \\frac{d}{d T} \\sin{(T)}, then obtain (\\frac{B{(T)} \\frac{d}{d T} \\sin{(T)}}{\\sin^{2}{(T)}} + \\frac{d}{d T} \\sin{(T)})^{2} = (\\frac{d}{d T} \\sin{(T)} + \\frac{(\\frac{d}{d T} \\sin{(T)})^{2}}{\\sin^{2}{(T)}})^{2}", "derivation": "B{(T)} = \\frac{d}{d T} \\sin{(T)} and \\frac{B{(T)}}{\\sin{(T)}} = \\frac{\\frac{d}{d T} \\sin{(T)}}{\\sin{(T)}} and \\frac{B{(T)} \\frac{d}{d T} \\sin{(T)}}{\\sin^{2}{(T)}} = \\frac{(\\frac{d}{d T} \\sin{(T)})^{2}}{\\sin^{2}{(T)}} and \\frac{B{(T)} \\frac{d}{d T} \\sin{(T)}}{\\sin^{2}{(T)}} + \\frac{d}{d T} \\sin{(T)} = \\frac{d}{d T} \\sin{(T)} + \\frac{(\\frac{d}{d T} \\sin{(T)})^{2}}{\\sin^{2}{(T)}} and (\\frac{B{(T)} \\frac{d}{d T} \\sin{(T)}}{\\sin^{2}{(T)}} + \\frac{d}{d T} \\sin{(T)})^{2} = (\\frac{d}{d T} \\sin{(T)} + \\frac{(\\frac{d}{d T} \\sin{(T)})^{2}}{\\sin^{2}{(T)}})^{2}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('T', commutative=True)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 1, "sin(Symbol('T', commutative=True))"], "Equality(Mul(Function('B')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["times", 2, "Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], "Equality(Mul(Function('B')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Pow(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2))))"], [["add", 3, "Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('B')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Pow(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Function('B')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(2)), Pow(Add(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-2)), Pow(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)))), Integer(2)))"]]}, {"prompt": "Given \\hat{x}{(q,y)} = y^{q}, then obtain 2 y^{2 q} (- y + y^{q}) (- y + \\hat{x}{(q,y)}) = 2 y^{2 q} (- y + y^{q})^{2}", "derivation": "\\hat{x}{(q,y)} = y^{q} and - y + \\hat{x}{(q,y)} = - y + y^{q} and y^{q} (- y + \\hat{x}{(q,y)}) = y^{q} (- y + y^{q}) and y^{q} (- y + y^{q}) + y^{q} (- y + \\hat{x}{(q,y)}) = 2 y^{q} (- y + y^{q}) and y^{q} (- y + \\hat{x}{(q,y)}) (y^{q} (- y + y^{q}) + y^{q} (- y + \\hat{x}{(q,y)})) = y^{q} (- y + y^{q}) (y^{q} (- y + y^{q}) + y^{q} (- y + \\hat{x}{(q,y)})) and 2 y^{2 q} (- y + y^{q}) (- y + \\hat{x}{(q,y)}) = 2 y^{2 q} (- y + y^{q})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))))"], [["times", 2, "Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True))))), Mul(Integer(2), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))))"], [["times", 3, "Add(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)))))"], "Equality(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True))), Add(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)))))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))), Add(Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Pow(Symbol('y', commutative=True), Mul(Integer(2), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}')(Symbol('q', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Pow(Symbol('y', commutative=True), Mul(Integer(2), Symbol('q', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('q', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(h,\\hat{p}_0)} = \\hat{p}_0 h, then derive \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{J}{(h,\\hat{p}_0)} = h, then obtain - \\mathbf{J}{(h,\\hat{p}_0)} + \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{J}{(h,\\hat{p}_0)} = h - \\mathbf{J}{(h,\\hat{p}_0)}", "derivation": "\\mathbf{J}{(h,\\hat{p}_0)} = \\hat{p}_0 h and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{J}{(h,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{p}_0 h and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{J}{(h,\\hat{p}_0)} = h and - \\mathbf{J}{(h,\\hat{p}_0)} + \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{J}{(h,\\hat{p}_0)} = h - \\mathbf{J}{(h,\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('h', commutative=True))"], [["minus", 3, "Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Derivative(Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(A)} = e^{A} and W{(A)} = \\log{(A)}, then obtain \\int \\operatorname{A_{1}}{(A)} dA - \\int W{(A)} dA = - \\int W{(A)} dA + \\int e^{A} dA", "derivation": "\\operatorname{A_{1}}{(A)} = e^{A} and W{(A)} = \\log{(A)} and \\int W{(A)} dA = \\int \\log{(A)} dA and \\int \\operatorname{A_{1}}{(A)} dA = \\int e^{A} dA and \\int \\operatorname{A_{1}}{(A)} dA - \\int \\log{(A)} dA = \\int e^{A} dA - \\int \\log{(A)} dA and \\int \\operatorname{A_{1}}{(A)} dA - \\int W{(A)} dA = - \\int W{(A)} dA + \\int e^{A} dA", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], ["get_premise", "Equality(Function('W')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('W')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["minus", 4, "Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Integral(Function('A_1')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Add(Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(Function('A_1')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Function('W')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Add(Mul(Integer(-1), Integral(Function('W')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(f)} = \\cos{(\\log{(f)})}, then obtain \\int \\tilde{g}^*^{4}{(f)} df = \\int \\tilde{g}^*^{2}{(f)} \\cos^{2}{(\\log{(f)})} df", "derivation": "\\tilde{g}^*{(f)} = \\cos{(\\log{(f)})} and \\tilde{g}^*^{2}{(f)} = \\tilde{g}^*{(f)} \\cos{(\\log{(f)})} and \\tilde{g}^*^{4}{(f)} = \\tilde{g}^*^{2}{(f)} \\cos^{2}{(\\log{(f)})} and \\int \\tilde{g}^*^{4}{(f)} df = \\int \\tilde{g}^*^{2}{(f)} \\cos^{2}{(\\log{(f)})} df", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), cos(log(Symbol('f', commutative=True))))"], [["times", 1, "Function('\\\\tilde{g}^*')(Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), cos(log(Symbol('f', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Integer(2)), Pow(cos(log(Symbol('f', commutative=True))), Integer(2))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Integer(4)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Integer(2)), Pow(cos(log(Symbol('f', commutative=True))), Integer(2))), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(k)} = \\cos{(k)}, then obtain \\operatorname{C_{d}}{(k)} \\cos{(k)} (\\int \\operatorname{C_{d}}{(k)} dk)^{2} = \\cos^{2}{(k)} (\\int \\operatorname{C_{d}}{(k)} dk)^{2}", "derivation": "\\operatorname{C_{d}}{(k)} = \\cos{(k)} and \\operatorname{C_{d}}{(k)} \\cos{(k)} = \\cos^{2}{(k)} and \\int \\operatorname{C_{d}}{(k)} dk = \\int \\cos{(k)} dk and (\\int \\operatorname{C_{d}}{(k)} dk)^{2} = (\\int \\operatorname{C_{d}}{(k)} dk) \\int \\cos{(k)} dk and \\operatorname{C_{d}}{(k)} \\cos{(k)} (\\int \\operatorname{C_{d}}{(k)} dk) \\int \\cos{(k)} dk = \\cos^{2}{(k)} (\\int \\operatorname{C_{d}}{(k)} dk) \\int \\cos{(k)} dk and \\operatorname{C_{d}}{(k)} \\cos{(k)} (\\int \\operatorname{C_{d}}{(k)} dk)^{2} = \\cos^{2}{(k)} (\\int \\operatorname{C_{d}}{(k)} dk)^{2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["times", 1, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Function('C_d')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Pow(cos(Symbol('k', commutative=True)), Integer(2)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["times", 3, "Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Pow(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(2)), Mul(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["times", 2, "Mul(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], "Equality(Mul(Function('C_d')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)), Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('C_d')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)), Pow(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(2))), Mul(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Pow(Integral(Function('C_d')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\theta{(V_{\\mathbf{E}})} = \\sin{(e^{V_{\\mathbf{E}}})}, then derive \\int \\theta{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\eta + \\operatorname{Si}{(e^{V_{\\mathbf{E}}})}, then obtain e^{V_{\\mathbf{E}}} \\int \\sin{(e^{V_{\\mathbf{E}}})} dV_{\\mathbf{E}} = (\\eta + \\operatorname{Si}{(e^{V_{\\mathbf{E}}})}) e^{V_{\\mathbf{E}}}", "derivation": "\\theta{(V_{\\mathbf{E}})} = \\sin{(e^{V_{\\mathbf{E}}})} and \\int \\theta{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\sin{(e^{V_{\\mathbf{E}}})} dV_{\\mathbf{E}} and \\int \\theta{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\eta + \\operatorname{Si}{(e^{V_{\\mathbf{E}}})} and e^{V_{\\mathbf{E}}} \\int \\theta{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = (\\eta + \\operatorname{Si}{(e^{V_{\\mathbf{E}}})}) e^{V_{\\mathbf{E}}} and e^{V_{\\mathbf{E}}} \\int \\sin{(e^{V_{\\mathbf{E}}})} dV_{\\mathbf{E}} = (\\eta + \\operatorname{Si}{(e^{V_{\\mathbf{E}}})}) e^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(sin(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Si(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["times", 3, "exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Function('\\\\theta')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Symbol('\\\\eta', commutative=True), Si(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(sin(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Symbol('\\\\eta', commutative=True), Si(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(G)} = e^{G} and \\bar{\\h}{(G)} = e^{e^{G}}, then obtain \\bar{\\h}^{G}{(G)} = (e^{e^{G}})^{G}", "derivation": "\\mu_{0}{(G)} = e^{G} and \\bar{\\h}{(G)} = e^{e^{G}} and \\bar{\\h}{(G)} = e^{\\mu_{0}{(G)}} and \\bar{\\h}^{G}{(G)} = (e^{\\mu_{0}{(G)}})^{G} and \\bar{\\h}^{G}{(G)} = (e^{e^{G}})^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('G', commutative=True)), exp(exp(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hbar')(Symbol('G', commutative=True)), exp(Function('\\\\mu_0')(Symbol('G', commutative=True))))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(exp(Function('\\\\mu_0')(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Function('\\\\hbar')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(exp(exp(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(L_{\\varepsilon},\\omega)} = \\frac{\\omega}{L_{\\varepsilon}}, then obtain \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\rho_{f}{(L_{\\varepsilon},\\omega)} + \\frac{\\omega}{L_{\\varepsilon}} = \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\frac{\\omega}{L_{\\varepsilon}} + \\frac{\\omega}{L_{\\varepsilon}}", "derivation": "\\rho_{f}{(L_{\\varepsilon},\\omega)} = \\frac{\\omega}{L_{\\varepsilon}} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\rho_{f}{(L_{\\varepsilon},\\omega)} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\frac{\\omega}{L_{\\varepsilon}} and \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\rho_{f}{(L_{\\varepsilon},\\omega)} = \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\frac{\\omega}{L_{\\varepsilon}} and \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\rho_{f}{(L_{\\varepsilon},\\omega)} + \\frac{\\omega}{L_{\\varepsilon}} = \\frac{\\partial^{2}}{\\partial \\omega\\partial L_{\\varepsilon}} \\frac{\\omega}{L_{\\varepsilon}} + \\frac{\\omega}{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 3, "Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Derivative(Function('\\\\rho_f')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Add(Derivative(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\sigma_p)} = e^{\\sigma_p}, then derive \\frac{d}{d \\sigma_p} \\mathbf{M}{(\\sigma_p)} = e^{\\sigma_p}, then obtain \\frac{d^{2}}{d \\sigma_p^{2}} \\mathbf{M}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} e^{\\sigma_p}", "derivation": "\\mathbf{M}{(\\sigma_p)} = e^{\\sigma_p} and \\frac{d}{d \\sigma_p} \\mathbf{M}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} e^{\\sigma_p} and \\frac{d}{d \\sigma_p} \\mathbf{M}{(\\sigma_p)} = e^{\\sigma_p} and e^{\\sigma_p} = \\frac{d}{d \\sigma_p} e^{\\sigma_p} and \\frac{d}{d \\sigma_p} \\mathbf{M}{(\\sigma_p)} = \\frac{d^{2}}{d \\sigma_p^{2}} \\mathbf{M}{(\\sigma_p)} and \\frac{d^{2}}{d \\sigma_p^{2}} \\mathbf{M}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} e^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\sigma_p', commutative=True)), Derivative(exp(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(\\hat{H}_l,\\theta_1,\\hbar)} = (\\hat{H}_l^{\\theta_1})^{\\hbar}, then obtain \\hat{H}_l \\cos{((\\hat{H}_l^{\\theta_1})^{\\hbar} - i{(\\hat{H}_l,\\theta_1,\\hbar)})} - 1 = \\hat{H}_l - 1", "derivation": "i{(\\hat{H}_l,\\theta_1,\\hbar)} = (\\hat{H}_l^{\\theta_1})^{\\hbar} and - (\\hat{H}_l^{\\theta_1})^{\\hbar} + i{(\\hat{H}_l,\\theta_1,\\hbar)} = 0 and \\cos{((\\hat{H}_l^{\\theta_1})^{\\hbar} - i{(\\hat{H}_l,\\theta_1,\\hbar)})} = 1 and \\hat{H}_l \\cos{((\\hat{H}_l^{\\theta_1})^{\\hbar} - i{(\\hat{H}_l,\\theta_1,\\hbar)})} = \\hat{H}_l and \\hat{H}_l \\cos{((\\hat{H}_l^{\\theta_1})^{\\hbar} - i{(\\hat{H}_l,\\theta_1,\\hbar)})} - 1 = \\hat{H}_l - 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True))), Function('i')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(0))"], [["cos", 2], "Equality(cos(Add(Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hbar', commutative=True))))), Integer(1))"], [["times", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Add(Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hbar', commutative=True)))))), Symbol('\\\\hat{H}_l', commutative=True))"], [["minus", 4, 1], "Equality(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Add(Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hbar', commutative=True)))))), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)))"]]}, {"prompt": "Given E{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)}, then derive E{(\\theta_2)} = \\frac{1}{\\theta_2}, then obtain - \\frac{1}{\\theta_2} = - E{(\\theta_2)}", "derivation": "E{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and E{(\\theta_2)} = \\frac{1}{\\theta_2} and \\frac{1}{\\theta_2} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and - \\frac{1}{\\theta_2} = - \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and - \\frac{1}{\\theta_2} = - E{(\\theta_2)}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Mul(Integer(-1), Function('E')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})} = \\sin{(C_{d} - f^{*})}, then derive \\frac{\\frac{\\partial}{\\partial f^{*}} \\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})}}{f^{*}} = - \\frac{\\cos{(C_{d} - f^{*})}}{f^{*}}, then obtain \\frac{\\frac{\\partial}{\\partial f^{*}} \\sin{(C_{d} - f^{*})}}{f^{*}} = - \\frac{\\cos{(C_{d} - f^{*})}}{f^{*}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})} = \\sin{(C_{d} - f^{*})} and \\frac{\\partial}{\\partial f^{*}} \\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})} = \\frac{\\partial}{\\partial f^{*}} \\sin{(C_{d} - f^{*})} and \\frac{\\frac{\\partial}{\\partial f^{*}} \\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})}}{f^{*}} = \\frac{\\frac{\\partial}{\\partial f^{*}} \\sin{(C_{d} - f^{*})}}{f^{*}} and \\frac{\\frac{\\partial}{\\partial f^{*}} \\operatorname{f_{\\mathbf{v}}}{(f^{*},C_{d})}}{f^{*}} = - \\frac{\\cos{(C_{d} - f^{*})}}{f^{*}} and \\frac{\\frac{\\partial}{\\partial f^{*}} \\sin{(C_{d} - f^{*})}}{f^{*}} = - \\frac{\\cos{(C_{d} - f^{*})}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('C_d', commutative=True)), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('f^*', commutative=True)"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))))"]]}, {"prompt": "Given H{(U,\\lambda)} = U + \\lambda, then derive \\int H{(U,\\lambda)} dU = \\frac{U^{2}}{2} + U \\lambda + \\phi_2, then obtain \\frac{U^{2}}{2} + U \\lambda + \\mathbf{J}_f = \\frac{U^{2}}{2} + U \\lambda + \\phi_2", "derivation": "H{(U,\\lambda)} = U + \\lambda and \\int H{(U,\\lambda)} dU = \\int (U + \\lambda) dU and \\int H{(U,\\lambda)} dU = \\frac{U^{2}}{2} + U \\lambda + \\phi_2 and \\int (U + \\lambda) dU = \\frac{U^{2}}{2} + U \\lambda + \\phi_2 and \\frac{U^{2}}{2} + U \\lambda + \\mathbf{J}_f = \\frac{U^{2}}{2} + U \\lambda + \\phi_2", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('H')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('H')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given c{(M,A_{x})} = \\int (A_{x} - M) dA_{x}, then obtain c{(M,A_{x})} \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} - (\\int (A_{x} - M) dA_{x})^{A_{x}} = \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} \\int (A_{x} - M) dA_{x} - (\\int (A_{x} - M) dA_{x})^{A_{x}}", "derivation": "c{(M,A_{x})} = \\int (A_{x} - M) dA_{x} and \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} = \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} - M) dA_{x} and c{(M,A_{x})} \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} - M) dA_{x} = (\\frac{\\partial}{\\partial A_{x}} \\int (A_{x} - M) dA_{x}) \\int (A_{x} - M) dA_{x} and c{(M,A_{x})} \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} = \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} \\int (A_{x} - M) dA_{x} and c{(M,A_{x})} \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} - (\\int (A_{x} - M) dA_{x})^{A_{x}} = \\frac{\\partial}{\\partial A_{x}} c{(M,A_{x})} \\int (A_{x} - M) dA_{x} - (\\int (A_{x} - M) dA_{x})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Mul(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Derivative(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Derivative(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Derivative(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Derivative(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True)))))"], [["minus", 4, "Pow(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Derivative(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))), Add(Mul(Derivative(Function('c')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True)))), Mul(Integer(-1), Pow(Integral(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain \\mathbf{A}{(\\hat{H})} + \\cos{(\\mathbf{A}{(\\hat{H})})} = \\cos{(\\hat{H})} + \\cos{(\\mathbf{A}{(\\hat{H})})}", "derivation": "\\mathbf{A}{(\\hat{H})} = \\cos{(\\hat{H})} and \\cos{(\\mathbf{A}{(\\hat{H})})} = \\cos{(\\cos{(\\hat{H})})} and \\mathbf{A}{(\\hat{H})} + \\cos{(\\cos{(\\hat{H})})} = \\cos{(\\hat{H})} + \\cos{(\\cos{(\\hat{H})})} and \\mathbf{A}{(\\hat{H})} + \\cos{(\\mathbf{A}{(\\hat{H})})} = \\cos{(\\hat{H})} + \\cos{(\\mathbf{A}{(\\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True))), cos(cos(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 1, "cos(cos(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), cos(cos(Symbol('\\\\hat{H}', commutative=True)))), Add(cos(Symbol('\\\\hat{H}', commutative=True)), cos(cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)))), Add(cos(Symbol('\\\\hat{H}', commutative=True)), cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\phi_2,Z)} = - Z + \\phi_2 and \\mathbf{s}{(Z,\\phi_2)} = 2 Z + (- Z + \\phi_2)^{2 Z} \\mathbf{P}{(\\phi_2,Z)}, then obtain (2 Z + (- Z + \\phi_2)^{Z} \\mathbf{P}{(\\phi_2,Z)} \\mathbf{P}^{Z}{(\\phi_2,Z)})^{Z} = \\mathbf{s}^{Z}{(Z,\\phi_2)}", "derivation": "\\mathbf{P}{(\\phi_2,Z)} = - Z + \\phi_2 and \\mathbf{P}^{Z}{(\\phi_2,Z)} = (- Z + \\phi_2)^{Z} and (- Z + \\phi_2)^{Z} \\mathbf{P}^{Z}{(\\phi_2,Z)} = (- Z + \\phi_2)^{2 Z} and (- Z + \\phi_2)^{Z} \\mathbf{P}{(\\phi_2,Z)} \\mathbf{P}^{Z}{(\\phi_2,Z)} = (- Z + \\phi_2)^{2 Z} \\mathbf{P}{(\\phi_2,Z)} and 2 Z + (- Z + \\phi_2)^{Z} \\mathbf{P}{(\\phi_2,Z)} \\mathbf{P}^{Z}{(\\phi_2,Z)} = 2 Z + (- Z + \\phi_2)^{2 Z} \\mathbf{P}{(\\phi_2,Z)} and \\mathbf{s}{(Z,\\phi_2)} = 2 Z + (- Z + \\phi_2)^{2 Z} \\mathbf{P}{(\\phi_2,Z)} and 2 Z + (- Z + \\phi_2)^{Z} \\mathbf{P}{(\\phi_2,Z)} \\mathbf{P}^{Z}{(\\phi_2,Z)} = \\mathbf{s}{(Z,\\phi_2)} and (2 Z + (- Z + \\phi_2)^{Z} \\mathbf{P}{(\\phi_2,Z)} \\mathbf{P}^{Z}{(\\phi_2,Z)})^{Z} = \\mathbf{s}^{Z}{(Z,\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 2, "Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["times", 3, "Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True))))"], [["add", 4, "Mul(Integer(2), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Function('\\\\mathbf{s}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["power", 7, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('Z', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain (\\frac{\\mathbf{B} \\frac{d}{d \\mathbf{B}} \\mathbf{S}{(\\mathbf{B})}}{\\mathbf{S}{(\\mathbf{B})}} + \\log{(\\mathbf{S}{(\\mathbf{B})})}) \\mathbf{S}^{\\mathbf{B}}{(\\mathbf{B})} = (\\mathbf{B} + \\log{(e^{\\mathbf{B}})}) (e^{\\mathbf{B}})^{\\mathbf{B}}", "derivation": "\\mathbf{S}{(\\mathbf{B})} = e^{\\mathbf{B}} and \\mathbf{S}^{\\mathbf{B}}{(\\mathbf{B})} = (e^{\\mathbf{B}})^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} \\mathbf{S}^{\\mathbf{B}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} (e^{\\mathbf{B}})^{\\mathbf{B}} and (\\frac{\\mathbf{B} \\frac{d}{d \\mathbf{B}} \\mathbf{S}{(\\mathbf{B})}}{\\mathbf{S}{(\\mathbf{B})}} + \\log{(\\mathbf{S}{(\\mathbf{B})})}) \\mathbf{S}^{\\mathbf{B}}{(\\mathbf{B})} = (\\mathbf{B} + \\log{(e^{\\mathbf{B}})}) (e^{\\mathbf{B}})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), log(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), log(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\chi{(r_{0})} = \\cos{(r_{0})}, then obtain - r_{0} + 3 \\chi^{r_{0}}{(r_{0})} + \\cos^{r_{0}}{(r_{0})} = - r_{0} + \\chi^{r_{0}}{(r_{0})} + 3 \\cos^{r_{0}}{(r_{0})}", "derivation": "\\chi{(r_{0})} = \\cos{(r_{0})} and \\chi^{r_{0}}{(r_{0})} = \\cos^{r_{0}}{(r_{0})} and 2 \\chi^{r_{0}}{(r_{0})} = \\chi^{r_{0}}{(r_{0})} + \\cos^{r_{0}}{(r_{0})} and - r_{0} + 2 \\chi^{r_{0}}{(r_{0})} = - r_{0} + \\chi^{r_{0}}{(r_{0})} + \\cos^{r_{0}}{(r_{0})} and - r_{0} + 3 \\chi^{r_{0}}{(r_{0})} + \\cos^{r_{0}}{(r_{0})} = - r_{0} + 2 \\chi^{r_{0}}{(r_{0})} + 2 \\cos^{r_{0}}{(r_{0})} and - r_{0} + 3 \\chi^{r_{0}}{(r_{0})} + \\cos^{r_{0}}{(r_{0})} = - r_{0} + \\chi^{r_{0}}{(r_{0})} + 3 \\cos^{r_{0}}{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["add", 2, "Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(2), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["add", 4, "Add(Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(3), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(2), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Mul(Integer(2), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(3), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(Function('\\\\chi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Mul(Integer(3), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = \\mathbb{I} - \\mathbf{E}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (\\mathbb{I} - \\mathbf{E}) = \\frac{d}{d \\mathbb{I}} 1", "derivation": "\\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = \\mathbb{I} - \\mathbf{E} and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = \\frac{\\partial}{\\partial \\mathbb{I}} (\\mathbb{I} - \\mathbf{E}) and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = 1 and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\operatorname{v_{2}}{(\\mathbb{I},\\mathbf{E})} = \\frac{d}{d \\mathbb{I}} 1 and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (\\mathbb{I} - \\mathbf{E}) = \\frac{d}{d \\mathbb{I}} 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(C)} = \\log{(C)}, then derive \\frac{d}{d C} \\int (\\hat{\\mathbf{r}}{(C)} - 1) dC = \\frac{\\partial}{\\partial C} (C \\log{(C)} - 2 C + \\varepsilon), then obtain \\frac{d}{d C} \\int (\\log{(C)} - 1) dC = \\frac{\\partial}{\\partial C} (C \\log{(C)} - 2 C + \\varepsilon)", "derivation": "\\hat{\\mathbf{r}}{(C)} = \\log{(C)} and \\hat{\\mathbf{r}}{(C)} - 1 = \\log{(C)} - 1 and \\int (\\hat{\\mathbf{r}}{(C)} - 1) dC = \\int (\\log{(C)} - 1) dC and \\frac{d}{d C} \\int (\\hat{\\mathbf{r}}{(C)} - 1) dC = \\frac{d}{d C} \\int (\\log{(C)} - 1) dC and \\frac{d}{d C} \\int (\\hat{\\mathbf{r}}{(C)} - 1) dC = \\frac{\\partial}{\\partial C} (C \\log{(C)} - 2 C + \\varepsilon) and \\frac{d}{d C} \\int (\\log{(C)} - 1) dC = \\frac{\\partial}{\\partial C} (C \\log{(C)} - 2 C + \\varepsilon)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C', commutative=True)), Integer(-1)), Add(log(Symbol('C', commutative=True)), Integer(-1)))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Integral(Add(log(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Add(log(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('C', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integral(Add(log(Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('C', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(H)} = \\log{(\\log{(H)})}, then derive \\frac{d}{d H} \\lambda{(H)} = \\frac{1}{H \\log{(H)}}, then obtain \\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)} \\frac{d}{d H} \\log{(\\log{(H)})} = \\frac{\\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)}}{H \\log{(H)}}", "derivation": "\\lambda{(H)} = \\log{(\\log{(H)})} and \\frac{d}{d H} \\lambda{(H)} = \\frac{d}{d H} \\log{(\\log{(H)})} and \\frac{d}{d H} \\lambda{(H)} = \\frac{1}{H \\log{(H)}} and \\frac{d}{d H} \\log{(\\log{(H)})} = \\frac{1}{H \\log{(H)}} and \\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)} = \\log{(\\log{(H)})} \\frac{d}{d H} \\log{(\\log{(H)})} and \\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)} = \\frac{\\log{(\\log{(H)})}}{H \\log{(H)}} and \\log{(\\log{(H)})} \\frac{d}{d H} \\log{(\\log{(H)})} = \\frac{\\log{(\\log{(H)})}}{H \\log{(H)}} and \\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)} \\frac{d}{d H} \\log{(\\log{(H)})} = \\frac{\\log{(\\log{(H)})} \\frac{d}{d H} \\lambda{(H)}}{H \\log{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))))"], [["times", 2, "log(log(Symbol('H', commutative=True)))"], "Equality(Mul(log(log(Symbol('H', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(log(log(Symbol('H', commutative=True))), Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(log(log(Symbol('H', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1)), log(log(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(log(log(Symbol('H', commutative=True))), Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1)), log(log(Symbol('H', commutative=True)))))"], [["times", 7, "Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(log(log(Symbol('H', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1)), log(log(Symbol('H', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(F_{g},V_{\\mathbf{B}})} = \\cos^{F_{g}}{(V_{\\mathbf{B}})} and \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})}, then obtain \\operatorname{v_{x}}^{F_{g}}{(V_{\\mathbf{B}})} - \\cos{(V_{\\mathbf{B}})} = - \\cos{(V_{\\mathbf{B}})} + \\cos^{F_{g}}{(V_{\\mathbf{B}})}", "derivation": "\\operatorname{c_{0}}{(F_{g},V_{\\mathbf{B}})} = \\cos^{F_{g}}{(V_{\\mathbf{B}})} and \\operatorname{c_{0}}{(F_{g},V_{\\mathbf{B}})} - \\cos{(V_{\\mathbf{B}})} = - \\cos{(V_{\\mathbf{B}})} + \\cos^{F_{g}}{(V_{\\mathbf{B}})} and \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\operatorname{c_{0}}{(F_{g},V_{\\mathbf{B}})} = \\operatorname{v_{x}}^{F_{g}}{(V_{\\mathbf{B}})} and \\operatorname{v_{x}}^{F_{g}}{(V_{\\mathbf{B}})} - \\cos{(V_{\\mathbf{B}})} = - \\cos{(V_{\\mathbf{B}})} + \\cos^{F_{g}}{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('F_g', commutative=True)))"], [["minus", 1, "cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('c_0')(Symbol('F_g', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Pow(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('F_g', commutative=True)), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given y{(B,\\sigma_p)} = - B + \\sigma_p, then derive \\frac{\\partial}{\\partial B} y{(B,\\sigma_p)} = -1, then obtain \\frac{\\partial^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}}{\\partial B^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}} (- B + \\sigma_p) = \\frac{\\partial}{\\partial B} (- B + \\sigma_p)", "derivation": "y{(B,\\sigma_p)} = - B + \\sigma_p and \\frac{\\partial}{\\partial B} y{(B,\\sigma_p)} = \\frac{\\partial}{\\partial B} (- B + \\sigma_p) and \\frac{\\partial}{\\partial B} y{(B,\\sigma_p)} = -1 and \\frac{\\partial}{\\partial B} (- B + \\sigma_p) = -1 and \\frac{\\partial^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}}{\\partial B^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}} y{(B,\\sigma_p)} = \\frac{\\partial}{\\partial B} (- B + \\sigma_p) and \\frac{\\partial^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}}{\\partial B^{- \\frac{\\partial}{\\partial B} (- B + \\sigma_p)}} (- B + \\sigma_p) = \\frac{\\partial}{\\partial B} (- B + \\sigma_p)", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('B', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('B', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('B', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('y')(Symbol('B', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(C,r)} = C + r, then obtain \\int (\\operatorname{v_{z}}{(C,r)} - \\frac{\\partial}{\\partial C} (C + r) + 1) dC = \\frac{C^{2}}{2} + C r + i", "derivation": "\\operatorname{v_{z}}{(C,r)} = C + r and \\frac{\\partial}{\\partial C} \\operatorname{v_{z}}{(C,r)} = \\frac{\\partial}{\\partial C} (C + r) and \\operatorname{v_{z}}{(C,r)} + 1 = C + r + 1 and \\operatorname{v_{z}}{(C,r)} - \\frac{\\partial}{\\partial C} \\operatorname{v_{z}}{(C,r)} + 1 = C + r - \\frac{\\partial}{\\partial C} \\operatorname{v_{z}}{(C,r)} + 1 and \\operatorname{v_{z}}{(C,r)} - \\frac{\\partial}{\\partial C} (C + r) + 1 = C + r - \\frac{\\partial}{\\partial C} (C + r) + 1 and \\int (\\operatorname{v_{z}}{(C,r)} - \\frac{\\partial}{\\partial C} (C + r) + 1) dC = \\int (C + r - \\frac{\\partial}{\\partial C} (C + r) + 1) dC and \\int (\\operatorname{v_{z}}{(C,r)} - \\frac{\\partial}{\\partial C} (C + r) + 1) dC = \\frac{C^{2}}{2} + C r + i", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Integer(1)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True), Integer(1)))"], [["minus", 3, "Derivative(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Add(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Derivative(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)))"], [["integrate", 5, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('C', commutative=True))), Integral(Add(Symbol('C', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Add(Function('v_z')(Symbol('C', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('C', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Symbol('C', commutative=True), Symbol('r', commutative=True)), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\mu{(v,\\theta_1)} = \\theta_1 + v, then obtain \\theta_1 \\int \\mu{(v,\\theta_1)} d\\theta_1 = \\theta_1 (\\dot{z} + \\frac{\\theta_1^{2}}{2} + \\theta_1 v)", "derivation": "\\mu{(v,\\theta_1)} = \\theta_1 + v and \\int \\mu{(v,\\theta_1)} d\\theta_1 = \\int (\\theta_1 + v) d\\theta_1 and \\theta_1 \\int \\mu{(v,\\theta_1)} d\\theta_1 = \\theta_1 \\int (\\theta_1 + v) d\\theta_1 and \\theta_1 \\int \\mu{(v,\\theta_1)} d\\theta_1 = \\theta_1 (\\dot{z} + \\frac{\\theta_1^{2}}{2} + \\theta_1 v)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Integral(Function('\\\\mu')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Integral(Function('\\\\mu')(Symbol('v', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2))), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(r_{0},l)} = r_{0} + \\cos{(l)} and Q{(\\mathbf{J}_f)} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f, then derive Q{(\\mathbf{J}_f)} - \\mathbf{J}{(r_{0},l)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f - \\mathbf{J}{(r_{0},l)}, then obtain - r_{0} + Q{(\\mathbf{J}_f)} - \\cos{(l)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f - r_{0} - \\cos{(l)}", "derivation": "\\mathbf{J}{(r_{0},l)} = r_{0} + \\cos{(l)} and Q{(\\mathbf{J}_f)} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and Q{(\\mathbf{J}_f)} - \\mathbf{J}{(r_{0},l)} = - \\mathbf{J}{(r_{0},l)} + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and Q{(\\mathbf{J}_f)} - \\mathbf{J}{(r_{0},l)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f - \\mathbf{J}{(r_{0},l)} and - r_{0} + Q{(\\mathbf{J}_f)} - \\cos{(l)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f - r_{0} - \\cos{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)), Add(Symbol('r_0', commutative=True), cos(Symbol('l', commutative=True))))"], ["get_premise", "Equality(Function('Q')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Function('Q')(Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True))), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('Q')(Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('r_0', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('Q')(Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\sigma_x,C_{d})} = \\int C_{d} \\sigma_x dC_{d}, then obtain (\\mathbf{r} + \\Psi{(\\sigma_x,C_{d})})^{\\sigma_x} = (\\frac{C_{d}^{2} \\sigma_x}{2} + \\hat{x})^{\\sigma_x}", "derivation": "\\Psi{(\\sigma_x,C_{d})} = \\int C_{d} \\sigma_x dC_{d} and \\frac{\\partial}{\\partial \\sigma_x} \\Psi{(\\sigma_x,C_{d})} = \\frac{\\partial}{\\partial \\sigma_x} \\int C_{d} \\sigma_x dC_{d} and \\int \\frac{\\partial}{\\partial \\sigma_x} \\Psi{(\\sigma_x,C_{d})} d\\sigma_x = \\int \\frac{\\partial}{\\partial \\sigma_x} \\int C_{d} \\sigma_x dC_{d} d\\sigma_x and (\\int \\frac{\\partial}{\\partial \\sigma_x} \\Psi{(\\sigma_x,C_{d})} d\\sigma_x)^{\\sigma_x} = (\\int \\frac{\\partial}{\\partial \\sigma_x} \\int C_{d} \\sigma_x dC_{d} d\\sigma_x)^{\\sigma_x} and (\\mathbf{r} + \\Psi{(\\sigma_x,C_{d})})^{\\sigma_x} = (\\frac{C_{d}^{2} \\sigma_x}{2} + \\hat{x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('C_d', commutative=True)), Integral(Mul(Symbol('C_d', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('C_d', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('C_d', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\Psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Derivative(Integral(Mul(Symbol('C_d', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\Psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('C_d', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(r)} = \\cos{(r)}, then obtain \\frac{d}{d r} (0^{r} + 1) \\frac{d}{d r} ((- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1) = \\frac{d}{d r} 2 \\frac{d}{d r} ((- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1)", "derivation": "\\operatorname{v_{x}}{(r)} = \\cos{(r)} and 0 = - \\operatorname{v_{x}}{(r)} + \\cos{(r)} and 0^{r} = (- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} and 0^{r} + 1 = (- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1 and (- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1 = 2 and \\frac{d}{d r} ((- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1) = \\frac{d}{d r} 2 and \\frac{d}{d r} (0^{r} + 1) = \\frac{d}{d r} 2 and \\frac{d}{d r} (0^{r} + 1) \\frac{d}{d r} ((- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1) = \\frac{d}{d r} 2 \\frac{d}{d r} ((- \\operatorname{v_{x}}{(r)} + \\cos{(r)})^{r} + 1)", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["minus", 1, "Function('v_x')(Symbol('r', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Integer(0), Symbol('r', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)), Integer(2))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Add(Pow(Integer(0), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["times", 7, "Derivative(Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Pow(Integer(0), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Derivative(Integer(2), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Function('v_x')(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Integer(1)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} = \\log{(\\dot{y})} and \\omega{(\\dot{y})} = \\log{(\\dot{y})}^{\\dot{y}}, then obtain e^{\\omega{(\\dot{y})}} + \\log{(\\dot{y})} = e^{\\log{(\\dot{y})}^{\\dot{y}}} + \\log{(\\dot{y})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\dot{y})} = \\log{(\\dot{y})} and \\operatorname{V_{\\mathbf{E}}}^{\\dot{y}}{(\\dot{y})} = \\log{(\\dot{y})}^{\\dot{y}} and e^{\\operatorname{V_{\\mathbf{E}}}^{\\dot{y}}{(\\dot{y})}} = e^{\\log{(\\dot{y})}^{\\dot{y}}} and e^{\\operatorname{V_{\\mathbf{E}}}^{\\dot{y}}{(\\dot{y})}} + \\log{(\\dot{y})} = e^{\\log{(\\dot{y})}^{\\dot{y}}} + \\log{(\\dot{y})} and \\omega{(\\dot{y})} = \\log{(\\dot{y})}^{\\dot{y}} and \\omega{(\\dot{y})} = \\operatorname{V_{\\mathbf{E}}}^{\\dot{y}}{(\\dot{y})} and e^{\\omega{(\\dot{y})}} + \\log{(\\dot{y})} = e^{\\log{(\\dot{y})}^{\\dot{y}}} + \\log{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), exp(Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["add", 3, "log(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(exp(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\dot{y}', commutative=True))), Add(exp(Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Function('\\\\omega')(Symbol('\\\\dot{y}', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(exp(Function('\\\\omega')(Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\dot{y}', commutative=True))), Add(exp(Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(i,\\mu_0)} = \\frac{i^{2}}{\\mu_0} and \\operatorname{v_{2}}{(i,\\mu_0)} = (\\frac{i^{2}}{\\mu_0})^{i}, then obtain \\int (- i + \\operatorname{v_{2}}{(i,\\mu_0)})^{2} d\\mu_0 = \\int (- i + (\\frac{i^{2}}{\\mu_0})^{i})^{2} d\\mu_0", "derivation": "\\mathbb{I}{(i,\\mu_0)} = \\frac{i^{2}}{\\mu_0} and \\mathbb{I}^{i}{(i,\\mu_0)} = (\\frac{i^{2}}{\\mu_0})^{i} and \\operatorname{v_{2}}{(i,\\mu_0)} = (\\frac{i^{2}}{\\mu_0})^{i} and \\operatorname{v_{2}}{(i,\\mu_0)} = \\mathbb{I}^{i}{(i,\\mu_0)} and - i + \\mathbb{I}^{i}{(i,\\mu_0)} = - i + (\\frac{i^{2}}{\\mu_0})^{i} and - i + \\operatorname{v_{2}}{(i,\\mu_0)} = - i + (\\frac{i^{2}}{\\mu_0})^{i} and (- i + \\operatorname{v_{2}}{(i,\\mu_0)})^{2} = (- i + (\\frac{i^{2}}{\\mu_0})^{i})^{2} and \\int (- i + \\operatorname{v_{2}}{(i,\\mu_0)})^{2} d\\mu_0 = \\int (- i + (\\frac{i^{2}}{\\mu_0})^{i})^{2} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('v_2')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 2, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True))))"], [["power", 6, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True))), Integer(2)))"], [["integrate", 7, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('i', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(2))), Symbol('i', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given L{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\cos{(f_{\\mathbf{p}})}, then derive L{(f_{\\mathbf{p}})} = - \\sin{(f_{\\mathbf{p}})}, then derive \\frac{d}{d f_{\\mathbf{p}}} L{(f_{\\mathbf{p}})} = - \\cos{(f_{\\mathbf{p}})}, then obtain \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} \\cos{(f_{\\mathbf{p}})} = - \\cos{(f_{\\mathbf{p}})}", "derivation": "L{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\cos{(f_{\\mathbf{p}})} and L{(f_{\\mathbf{p}})} = - \\sin{(f_{\\mathbf{p}})} and \\frac{d}{d f_{\\mathbf{p}}} L{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} - \\sin{(f_{\\mathbf{p}})} and \\frac{d}{d f_{\\mathbf{p}}} L{(f_{\\mathbf{p}})} = - \\cos{(f_{\\mathbf{p}})} and \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} \\cos{(f_{\\mathbf{p}})} = - \\cos{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain \\log{(\\Psi^{\\dagger}{(f^{\\prime})})} + \\frac{- f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})}}{f^{\\prime}} = \\log{(e^{f^{\\prime}})} + \\frac{- f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})}}{f^{\\prime}}", "derivation": "\\Psi^{\\dagger}{(f^{\\prime})} = e^{f^{\\prime}} and - f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})} = - f^{\\prime} + e^{f^{\\prime}} and \\frac{- f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})}}{f^{\\prime}} = \\frac{- f^{\\prime} + e^{f^{\\prime}}}{f^{\\prime}} and \\log{(\\Psi^{\\dagger}{(f^{\\prime})})} = \\log{(e^{f^{\\prime}})} and \\log{(\\Psi^{\\dagger}{(f^{\\prime})})} + \\frac{- f^{\\prime} + e^{f^{\\prime}}}{f^{\\prime}} = \\log{(e^{f^{\\prime}})} + \\frac{- f^{\\prime} + e^{f^{\\prime}}}{f^{\\prime}} and \\log{(\\Psi^{\\dagger}{(f^{\\prime})})} + \\frac{- f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})}}{f^{\\prime}} = \\log{(e^{f^{\\prime}})} + \\frac{- f^{\\prime} + \\Psi^{\\dagger}{(f^{\\prime})}}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))))"], [["log", 1], "Equality(log(Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))), log(exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 4, "Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Add(log(Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))))), Add(log(exp(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(log(Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))))), Add(log(exp(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(i,\\Omega)} = \\Omega - i, then obtain \\int e di = \\int e^{\\frac{\\Omega - i}{\\phi_{2}{(i,\\Omega)}}} di", "derivation": "\\phi_{2}{(i,\\Omega)} = \\Omega - i and - \\frac{\\phi_{2}{(i,\\Omega)}}{i} = - \\frac{\\Omega - i}{i} and 1 = \\frac{\\Omega - i}{\\phi_{2}{(i,\\Omega)}} and e = e^{\\frac{\\Omega - i}{\\phi_{2}{(i,\\Omega)}}} and \\int e di = \\int e^{\\frac{\\Omega - i}{\\phi_{2}{(i,\\Omega)}}} di", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], [["exp", 3], "Equality(E, exp(Mul(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)))))"], [["integrate", 4, "Symbol('i', commutative=True)"], "Equality(Integral(E, Tuple(Symbol('i', commutative=True))), Integral(exp(Mul(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given W{(P_{g})} = P_{g}, then obtain \\frac{\\partial}{\\partial P_{g}} (\\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int W{(P_{g})} dP_{g}) = \\frac{\\partial}{\\partial P_{g}} (\\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int P_{g} dP_{g})", "derivation": "W{(P_{g})} = P_{g} and \\int W{(P_{g})} dP_{g} = \\int P_{g} dP_{g} and \\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int W{(P_{g})} dP_{g} = \\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int P_{g} dP_{g} and \\frac{\\partial}{\\partial P_{g}} (\\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int W{(P_{g})} dP_{g}) = \\frac{\\partial}{\\partial P_{g}} (\\hat{\\mathbf{r}}{(i,P_{g})} \\sin{(\\frac{i}{P_{g}})} + \\int P_{g} dP_{g})", "srepr_derivation": [["renaming_premise", "Equality(Function('W')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('W')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('P_g', commutative=True)), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('i', commutative=True))))"], "Equality(Add(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('P_g', commutative=True)), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Integral(Function('W')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('P_g', commutative=True)), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Integral(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True)))))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Add(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('P_g', commutative=True)), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Integral(Function('W')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('P_g', commutative=True)), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('i', commutative=True)))), Integral(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(a^{\\dagger},Z)} = Z a^{\\dagger} and \\theta_{2}{(g_{\\varepsilon},f_{E})} = f_{E} + g_{\\varepsilon}, then obtain 2 f_{E} + 2 g_{\\varepsilon} + 1 = \\frac{Z a^{\\dagger}}{\\hat{H}{(a^{\\dagger},Z)}} + 2 f_{E} + 2 g_{\\varepsilon}", "derivation": "\\hat{H}{(a^{\\dagger},Z)} = Z a^{\\dagger} and 1 = \\frac{Z a^{\\dagger}}{\\hat{H}{(a^{\\dagger},Z)}} and \\theta_{2}{(g_{\\varepsilon},f_{E})} = f_{E} + g_{\\varepsilon} and f_{E} + g_{\\varepsilon} + \\theta_{2}{(g_{\\varepsilon},f_{E})} = 2 f_{E} + 2 g_{\\varepsilon} and f_{E} + g_{\\varepsilon} + \\theta_{2}{(g_{\\varepsilon},f_{E})} + 1 = \\frac{Z a^{\\dagger}}{\\hat{H}{(a^{\\dagger},Z)}} + f_{E} + g_{\\varepsilon} + \\theta_{2}{(g_{\\varepsilon},f_{E})} and 2 f_{E} + 2 g_{\\varepsilon} + 1 = \\frac{Z a^{\\dagger}}{\\hat{H}{(a^{\\dagger},Z)}} + 2 f_{E} + 2 g_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Symbol('Z', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 3, "Add(Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('f_E', commutative=True))), Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 2, "Add(Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Add(Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Add(Mul(Symbol('Z', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Symbol('f_E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(Mul(Symbol('Z', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given M{(\\nabla,T)} = e^{T \\nabla} and \\Omega{(\\nabla,T)} = T \\nabla, then obtain \\log{(\\frac{\\partial}{\\partial \\nabla} T M{(\\nabla,T)})} = \\log{(\\frac{\\partial}{\\partial \\nabla} T e^{\\Omega{(\\nabla,T)}})}", "derivation": "M{(\\nabla,T)} = e^{T \\nabla} and T M{(\\nabla,T)} = T e^{T \\nabla} and \\Omega{(\\nabla,T)} = T \\nabla and \\frac{\\partial}{\\partial \\nabla} T M{(\\nabla,T)} = \\frac{\\partial}{\\partial \\nabla} T e^{T \\nabla} and \\frac{\\partial}{\\partial \\nabla} T M{(\\nabla,T)} = \\frac{\\partial}{\\partial \\nabla} T e^{\\Omega{(\\nabla,T)}} and \\log{(\\frac{\\partial}{\\partial \\nabla} T M{(\\nabla,T)})} = \\log{(\\frac{\\partial}{\\partial \\nabla} T e^{\\Omega{(\\nabla,T)}})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Function('M')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True))), Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Symbol('T', commutative=True), Function('M')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Mul(Symbol('T', commutative=True), Function('M')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Mul(Symbol('T', commutative=True), exp(Function('\\\\Omega')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["log", 5], "Equality(log(Derivative(Mul(Symbol('T', commutative=True), Function('M')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), log(Derivative(Mul(Symbol('T', commutative=True), exp(Function('\\\\Omega')(Symbol('\\\\nabla', commutative=True), Symbol('T', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(x,c)} = x^{c} and \\rho_{f}{(x,c)} = x^{c}, then obtain 2 \\dot{\\mathbf{r}}{(x,c)} = \\dot{\\mathbf{r}}{(x,c)} + \\rho_{f}{(x,c)}", "derivation": "\\dot{\\mathbf{r}}{(x,c)} = x^{c} and \\rho_{f}{(x,c)} = x^{c} and 2 \\dot{\\mathbf{r}}{(x,c)} = x^{c} + \\dot{\\mathbf{r}}{(x,c)} and 2 \\dot{\\mathbf{r}}{(x,c)} = \\dot{\\mathbf{r}}{(x,c)} + \\rho_{f}{(x,c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)))"], [["add", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True))), Add(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Function('\\\\rho_f')(Symbol('x', commutative=True), Symbol('c', commutative=True))))"]]}, {"prompt": "Given k{(\\mathbf{H},\\phi_1)} = \\mathbf{H}^{\\phi_1}, then derive \\frac{\\partial}{\\partial \\phi_1} k{(\\mathbf{H},\\phi_1)} = \\mathbf{H}^{\\phi_1} \\log{(\\mathbf{H})}, then obtain k{(\\mathbf{H},\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} \\mathbf{H}^{\\phi_1} = \\mathbf{H}^{\\phi_1} k{(\\mathbf{H},\\phi_1)} \\log{(\\mathbf{H})}", "derivation": "k{(\\mathbf{H},\\phi_1)} = \\mathbf{H}^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} k{(\\mathbf{H},\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} \\mathbf{H}^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} k{(\\mathbf{H},\\phi_1)} = \\mathbf{H}^{\\phi_1} \\log{(\\mathbf{H})} and \\frac{\\partial}{\\partial \\phi_1} \\mathbf{H}^{\\phi_1} = \\mathbf{H}^{\\phi_1} \\log{(\\mathbf{H})} and k{(\\mathbf{H},\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} \\mathbf{H}^{\\phi_1} = \\mathbf{H}^{\\phi_1} k{(\\mathbf{H},\\phi_1)} \\log{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 4, "Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Derivative(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('k')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})} = - F_{g} - \\Psi_{nl} + \\mathbf{r}, then obtain - F_{g} + \\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})} = F_{g} (-1 - \\frac{\\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})}}{- F_{g} - \\Psi_{nl} + \\mathbf{r}}) - \\Psi_{nl} + \\mathbf{r}", "derivation": "\\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})} = - F_{g} - \\Psi_{nl} + \\mathbf{r} and \\frac{\\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})}}{- F_{g} - \\Psi_{nl} + \\mathbf{r}} = 1 and - F_{g} + \\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})} = - 2 F_{g} - \\Psi_{nl} + \\mathbf{r} and 1 + \\frac{\\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})}}{- F_{g} - \\Psi_{nl} + \\mathbf{r}} = 2 and - F_{g} + \\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})} = F_{g} (-1 - \\frac{\\operatorname{y^{\\prime}}{(F_{g},\\mathbf{r},\\Psi_{nl})}}{- F_{g} - \\Psi_{nl} + \\mathbf{r}}) - \\Psi_{nl} + \\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1))"], [["add", 1, "Mul(Integer(-1), Symbol('F_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(2))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), Add(Integer(-1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(g)} = \\sin{(g)}, then obtain \\hat{H}_l{(g)} = \\frac{\\sin^{2}{(g)}}{\\hat{H}_l{(g)}}", "derivation": "\\hat{H}_l{(g)} = \\sin{(g)} and \\hat{H}_l{(g)} \\sin{(g)} = \\sin^{2}{(g)} and \\sin{(g)} = \\frac{\\sin^{2}{(g)}}{\\hat{H}_l{(g)}} and \\hat{H}_l{(g)} = \\frac{\\sin^{2}{(g)}}{\\hat{H}_l{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["times", 1, "sin(Symbol('g', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True))), Pow(sin(Symbol('g', commutative=True)), Integer(2)))"], [["divide", 2, "Function('\\\\hat{H}_l')(Symbol('g', commutative=True))"], "Equality(sin(Symbol('g', commutative=True)), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\hat{H}_l')(Symbol('g', commutative=True)), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(V,\\hbar)} = V - \\hbar, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(V,\\hbar)} dV = \\frac{V^{2}}{2} - V \\hbar + V_{\\mathbf{E}}, then obtain \\int (V - \\hbar) dV = \\frac{V^{2}}{2} - V \\hbar + V_{\\mathbf{E}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(V,\\hbar)} = V - \\hbar and \\int \\operatorname{f_{\\mathbf{v}}}{(V,\\hbar)} dV = \\int (V - \\hbar) dV and \\int \\operatorname{f_{\\mathbf{v}}}{(V,\\hbar)} dV = \\frac{V^{2}}{2} - V \\hbar + V_{\\mathbf{E}} and \\int (V - \\hbar) dV = \\frac{V^{2}}{2} - V \\hbar + V_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('V', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('V', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('V', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given k{(W)} = \\cos{(\\log{(W)})}, then derive \\int k{(W)} dW = \\frac{W \\sin{(\\log{(W)})}}{2} + \\frac{W \\cos{(\\log{(W)})}}{2} + \\hat{x}, then obtain \\frac{W \\sin{(\\log{(W)})}}{2} + \\frac{W \\cos{(\\log{(W)})}}{2} + \\hat{x} = \\frac{W k{(W)}}{2} + \\frac{W \\sin{(\\log{(W)})}}{2} + \\hat{x}", "derivation": "k{(W)} = \\cos{(\\log{(W)})} and \\int k{(W)} dW = \\int \\cos{(\\log{(W)})} dW and \\int k{(W)} dW = \\frac{W \\sin{(\\log{(W)})}}{2} + \\frac{W \\cos{(\\log{(W)})}}{2} + \\hat{x} and \\int k{(W)} dW = \\frac{W k{(W)}}{2} + \\frac{W \\sin{(\\log{(W)})}}{2} + \\hat{x} and \\frac{W \\sin{(\\log{(W)})}}{2} + \\frac{W \\cos{(\\log{(W)})}}{2} + \\hat{x} = \\frac{W k{(W)}}{2} + \\frac{W \\sin{(\\log{(W)})}}{2} + \\hat{x}", "srepr_derivation": [["get_premise", "Equality(Function('k')(Symbol('W', commutative=True)), cos(log(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('k')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Symbol('W', commutative=True), sin(log(Symbol('W', commutative=True)))), Mul(Rational(1, 2), Symbol('W', commutative=True), cos(log(Symbol('W', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('k')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Symbol('W', commutative=True), Function('k')(Symbol('W', commutative=True))), Mul(Rational(1, 2), Symbol('W', commutative=True), sin(log(Symbol('W', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Rational(1, 2), Symbol('W', commutative=True), sin(log(Symbol('W', commutative=True)))), Mul(Rational(1, 2), Symbol('W', commutative=True), cos(log(Symbol('W', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Rational(1, 2), Symbol('W', commutative=True), Function('k')(Symbol('W', commutative=True))), Mul(Rational(1, 2), Symbol('W', commutative=True), sin(log(Symbol('W', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given q{(a^{\\dagger},W)} = e^{\\frac{W}{a^{\\dagger}}}, then obtain \\frac{q{(a^{\\dagger},W)}}{\\sin{(e^{\\frac{W}{a^{\\dagger}}})}} = \\frac{e^{\\frac{W}{a^{\\dagger}}}}{\\sin{(e^{\\frac{W}{a^{\\dagger}}})}}", "derivation": "q{(a^{\\dagger},W)} = e^{\\frac{W}{a^{\\dagger}}} and \\sin{(q{(a^{\\dagger},W)})} = \\sin{(e^{\\frac{W}{a^{\\dagger}}})} and \\frac{q{(a^{\\dagger},W)}}{\\sin{(q{(a^{\\dagger},W)})}} = \\frac{e^{\\frac{W}{a^{\\dagger}}}}{\\sin{(q{(a^{\\dagger},W)})}} and \\frac{q{(a^{\\dagger},W)}}{\\sin{(e^{\\frac{W}{a^{\\dagger}}})}} = \\frac{e^{\\frac{W}{a^{\\dagger}}}}{\\sin{(e^{\\frac{W}{a^{\\dagger}}})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)), exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))))"], [["sin", 1], "Equality(sin(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), sin(exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))))"], [["divide", 1, "sin(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)), Pow(sin(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Integer(-1))), Mul(exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))), Pow(sin(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)), Pow(sin(exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))), Integer(-1))), Mul(exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))), Pow(sin(exp(Mul(Symbol('W', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))), Integer(-1))))"]]}, {"prompt": "Given x{(\\varepsilon_0)} = \\log{(e^{\\varepsilon_0})}, then derive - e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} x{(\\varepsilon_0)} = 1 - e^{\\varepsilon_0}, then obtain - e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} \\log{(e^{\\varepsilon_0})} = 1 - e^{\\varepsilon_0}", "derivation": "x{(\\varepsilon_0)} = \\log{(e^{\\varepsilon_0})} and x{(\\varepsilon_0)} - e^{\\varepsilon_0} = - e^{\\varepsilon_0} + \\log{(e^{\\varepsilon_0})} and \\frac{d}{d \\varepsilon_0} (x{(\\varepsilon_0)} - e^{\\varepsilon_0}) = \\frac{d}{d \\varepsilon_0} (- e^{\\varepsilon_0} + \\log{(e^{\\varepsilon_0})}) and - e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} x{(\\varepsilon_0)} = 1 - e^{\\varepsilon_0} and - e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} \\log{(e^{\\varepsilon_0})} = 1 - e^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), log(exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))), log(exp(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))), log(exp(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))), Derivative(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))), Derivative(log(exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\omega)} = \\cos{(e^{\\omega})}, then obtain \\frac{\\int (\\hat{H}{(\\omega)} + 2 \\cos{(e^{\\omega})}) d\\omega}{3 \\hat{H}{(\\omega)}} = \\frac{\\int 3 \\cos{(e^{\\omega})} d\\omega}{3 \\hat{H}{(\\omega)}}", "derivation": "\\hat{H}{(\\omega)} = \\cos{(e^{\\omega})} and \\hat{H}{(\\omega)} + \\cos{(e^{\\omega})} = 2 \\cos{(e^{\\omega})} and \\hat{H}{(\\omega)} + 2 \\cos{(e^{\\omega})} = 3 \\cos{(e^{\\omega})} and \\int (\\hat{H}{(\\omega)} + 2 \\cos{(e^{\\omega})}) d\\omega = \\int 3 \\cos{(e^{\\omega})} d\\omega and \\frac{\\int (\\hat{H}{(\\omega)} + 2 \\cos{(e^{\\omega})}) d\\omega}{3 \\cos{(e^{\\omega})}} = \\frac{\\int 3 \\cos{(e^{\\omega})} d\\omega}{3 \\cos{(e^{\\omega})}} and \\frac{\\int (\\hat{H}{(\\omega)} + 2 \\cos{(e^{\\omega})}) d\\omega}{3 \\hat{H}{(\\omega)}} = \\frac{\\int 3 \\cos{(e^{\\omega})} d\\omega}{3 \\hat{H}{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), cos(exp(Symbol('\\\\omega', commutative=True))))"], [["add", 1, "cos(exp(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), cos(exp(Symbol('\\\\omega', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('\\\\omega', commutative=True)))))"], [["add", 2, "cos(exp(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\omega', commutative=True))))), Mul(Integer(3), cos(exp(Symbol('\\\\omega', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(3), cos(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 4, "Mul(Integer(3), cos(exp(Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Rational(1, 3), Pow(cos(exp(Symbol('\\\\omega', commutative=True))), Integer(-1)), Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Rational(1, 3), Pow(cos(exp(Symbol('\\\\omega', commutative=True))), Integer(-1)), Integral(Mul(Integer(3), cos(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Rational(1, 3), Pow(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), cos(exp(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Rational(1, 3), Pow(Function('\\\\hat{H}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Integral(Mul(Integer(3), cos(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(a,B)} = a + \\log{(B)} and \\mathbf{f}{(a,B)} = \\nabla{(a,B)} \\log{(B)}, then obtain \\mathbf{f}{(a,B)} = (a + \\log{(B)}) \\log{(B)}", "derivation": "\\nabla{(a,B)} = a + \\log{(B)} and \\nabla{(a,B)} \\log{(B)} = (a + \\log{(B)}) \\log{(B)} and \\mathbf{f}{(a,B)} = \\nabla{(a,B)} \\log{(B)} and \\mathbf{f}{(a,B)} = (a + \\log{(B)}) \\log{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('a', commutative=True), Symbol('B', commutative=True)), Add(Symbol('a', commutative=True), log(Symbol('B', commutative=True))))"], [["times", 1, "log(Symbol('B', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('a', commutative=True), Symbol('B', commutative=True)), log(Symbol('B', commutative=True))), Mul(Add(Symbol('a', commutative=True), log(Symbol('B', commutative=True))), log(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('a', commutative=True), Symbol('B', commutative=True)), Mul(Function('\\\\nabla')(Symbol('a', commutative=True), Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{f}')(Symbol('a', commutative=True), Symbol('B', commutative=True)), Mul(Add(Symbol('a', commutative=True), log(Symbol('B', commutative=True))), log(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(l)} = \\log{(l)}, then obtain 2 \\frac{d}{d l} \\dot{x}{(l)} - \\frac{2}{l} = 0", "derivation": "\\dot{x}{(l)} = \\log{(l)} and \\dot{x}{(l)} - \\log{(l)} = 0 and \\frac{d}{d l} (\\dot{x}{(l)} - \\log{(l)}) = \\frac{d}{d l} 0 and \\log{(l)} = - \\dot{x}{(l)} + 2 \\log{(l)} and \\frac{d}{d l} (2 \\dot{x}{(l)} - 2 \\log{(l)}) = \\frac{d}{d l} 0 and 2 \\frac{d}{d l} \\dot{x}{(l)} - \\frac{2}{l} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["minus", 1, "log(Symbol('l', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Symbol('l', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 1, "Add(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Mul(Integer(-1), log(Symbol('l', commutative=True))))"], "Equality(log(Symbol('l', commutative=True)), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('l', commutative=True))), Mul(Integer(2), log(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('l', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(2), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Pow(Symbol('l', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\dot{z}{(H)} = \\cos{(H)}, then derive 0^{H} = (\\sin^{2}{(H)} + \\sin{(H)} \\frac{d}{d H} \\dot{z}{(H)})^{H}, then obtain (- \\frac{d}{d H} \\dot{z}{(H)} \\frac{d}{d H} \\cos{(H)} + (\\frac{d}{d H} \\cos{(H)})^{2})^{H} = 1", "derivation": "\\dot{z}{(H)} = \\cos{(H)} and \\frac{d}{d H} \\dot{z}{(H)} = \\frac{d}{d H} \\cos{(H)} and \\frac{d}{d H} \\dot{z}{(H)} \\frac{d}{d H} \\cos{(H)} = (\\frac{d}{d H} \\cos{(H)})^{2} and 0 = - \\frac{d}{d H} \\dot{z}{(H)} \\frac{d}{d H} \\cos{(H)} + (\\frac{d}{d H} \\cos{(H)})^{2} and 0^{H} = (- \\frac{d}{d H} \\dot{z}{(H)} \\frac{d}{d H} \\cos{(H)} + (\\frac{d}{d H} \\cos{(H)})^{2})^{H} and 0^{H} = (\\sin^{2}{(H)} + \\sin{(H)} \\frac{d}{d H} \\dot{z}{(H)})^{H} and (\\sin^{2}{(H)} + \\sin{(H)} \\frac{d}{d H} \\dot{z}{(H)})^{H} = 1 and 0^{H} = 1 and (- \\frac{d}{d H} \\dot{z}{(H)} \\frac{d}{d H} \\cos{(H)} + (\\frac{d}{d H} \\cos{(H)})^{2})^{H} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["times", 2, "Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2)))"], [["minus", 3, "Mul(Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2))))"], [["power", 4, "Symbol('H', commutative=True)"], "Equality(Pow(Integer(0), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2))), Symbol('H', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Integer(0), Symbol('H', commutative=True)), Pow(Add(Pow(sin(Symbol('H', commutative=True)), Integer(2)), Mul(sin(Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Add(Pow(sin(Symbol('H', commutative=True)), Integer(2)), Mul(sin(Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Symbol('H', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Integer(0), Symbol('H', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2))), Symbol('H', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(I,m_{s})} = m_{s}^{I}, then derive \\operatorname{E_{\\lambda}}{(I,m_{s})} - \\frac{d}{d v_{y}} \\eta^{\\prime}{(v_{y})} = m_{s}^{I} - \\frac{d}{d v_{y}} \\eta^{\\prime}{(v_{y})}, then obtain - m_{s}^{I} + \\operatorname{E_{\\lambda}}{(I,m_{s})} = 0", "derivation": "\\operatorname{E_{\\lambda}}{(I,m_{s})} = m_{s}^{I} and v_{y} \\operatorname{E_{\\lambda}}{(I,m_{s})} = m_{s}^{I} v_{y} and v_{y} \\operatorname{E_{\\lambda}}{(I,m_{s})} - \\eta^{\\prime}{(v_{y})} = m_{s}^{I} v_{y} - \\eta^{\\prime}{(v_{y})} and \\frac{\\partial}{\\partial v_{y}} (v_{y} \\operatorname{E_{\\lambda}}{(I,m_{s})} - \\eta^{\\prime}{(v_{y})}) = \\frac{\\partial}{\\partial v_{y}} (m_{s}^{I} v_{y} - \\eta^{\\prime}{(v_{y})}) and \\operatorname{E_{\\lambda}}{(I,m_{s})} - \\frac{d}{d v_{y}} \\eta^{\\prime}{(v_{y})} = m_{s}^{I} - \\frac{d}{d v_{y}} \\eta^{\\prime}{(v_{y})} and - m_{s}^{I} + \\operatorname{E_{\\lambda}}{(I,m_{s})} = 0", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)))"], [["times", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True)))"], [["minus", 2, "Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Symbol('v_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)))), Add(Mul(Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('v_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)), Symbol('v_y', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))), Add(Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))))"], [["minus", 5, "Add(Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('I', commutative=True))), Function('E_{\\\\lambda}')(Symbol('I', commutative=True), Symbol('m_s', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\rho,\\hat{H})} = \\frac{\\hat{H}}{\\rho} and \\operatorname{v_{z}}{(\\rho,\\hat{H})} = (\\frac{\\hat{H} \\operatorname{A_{y}}{(\\rho,\\hat{H})}}{\\rho})^{\\rho}, then obtain \\operatorname{v_{z}}{(\\rho,\\hat{H})} = (\\frac{\\hat{H}^{2}}{\\rho^{2}})^{\\rho}", "derivation": "\\operatorname{A_{y}}{(\\rho,\\hat{H})} = \\frac{\\hat{H}}{\\rho} and \\frac{\\hat{H} \\operatorname{A_{y}}{(\\rho,\\hat{H})}}{\\rho} = \\frac{\\hat{H}^{2}}{\\rho^{2}} and \\operatorname{v_{z}}{(\\rho,\\hat{H})} = (\\frac{\\hat{H} \\operatorname{A_{y}}{(\\rho,\\hat{H})}}{\\rho})^{\\rho} and \\operatorname{v_{z}}{(\\rho,\\hat{H})} = (\\frac{\\hat{H}^{2}}{\\rho^{2}})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["times", 1, "Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('\\\\rho', commutative=True), Integer(-2))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_z')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Pow(Symbol('\\\\rho', commutative=True), Integer(-2))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)}, then derive \\operatorname{v_{y}}{(\\mu_0)} = \\frac{1}{\\mu_0}, then obtain (\\operatorname{v_{y}}{(\\mu_0)} - 1)^{\\mu_0} = (-1 + \\frac{1}{\\mu_0})^{\\mu_0}", "derivation": "\\operatorname{v_{y}}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and \\operatorname{v_{y}}{(\\mu_0)} - 1 = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} - 1 and \\operatorname{v_{y}}{(\\mu_0)} = \\frac{1}{\\mu_0} and \\frac{d}{d \\mu_0} \\log{(\\mu_0)} = \\frac{1}{\\mu_0} and \\operatorname{v_{y}}{(\\mu_0)} - 1 = -1 + \\frac{1}{\\mu_0} and (\\operatorname{v_{y}}{(\\mu_0)} - 1)^{\\mu_0} = (-1 + \\frac{1}{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('v_y')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Add(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 1], "Equality(Function('v_y')(Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('v_y')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Add(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["power", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Add(Function('v_y')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\psi,A_{2})} = A_{2}^{\\psi}, then obtain (\\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} \\operatorname{A_{x}}{(\\psi,A_{2})})^{\\psi} = (\\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} A_{2}^{\\psi})^{\\psi}", "derivation": "\\operatorname{A_{x}}{(\\psi,A_{2})} = A_{2}^{\\psi} and \\frac{\\partial}{\\partial \\psi} \\operatorname{A_{x}}{(\\psi,A_{2})} = \\frac{\\partial}{\\partial \\psi} A_{2}^{\\psi} and \\psi \\frac{\\partial}{\\partial \\psi} \\operatorname{A_{x}}{(\\psi,A_{2})} = \\psi \\frac{\\partial}{\\partial \\psi} A_{2}^{\\psi} and \\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} \\operatorname{A_{x}}{(\\psi,A_{2})} = \\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} A_{2}^{\\psi} and (\\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} \\operatorname{A_{x}}{(\\psi,A_{2})})^{\\psi} = (\\frac{\\partial}{\\partial \\psi} \\psi \\frac{\\partial}{\\partial \\psi} A_{2}^{\\psi})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Pow(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Derivative(Function('A_x')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi', commutative=True), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\psi', commutative=True), Derivative(Function('A_x')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi', commutative=True), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\psi', commutative=True), Derivative(Function('A_x')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\psi', commutative=True), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(J)} = \\log{(\\sin{(J)})} and \\operatorname{P_{e}}{(J)} = \\sin{(J)}, then obtain (\\varepsilon_{0}{(J)} + \\log{(\\sin{(J)})}) \\log{(\\sin{(J)})} = (\\varepsilon_{0}{(J)} + \\log{(\\operatorname{P_{e}}{(J)})}) \\log{(\\sin{(J)})}", "derivation": "\\varepsilon_{0}{(J)} = \\log{(\\sin{(J)})} and 2 \\varepsilon_{0}{(J)} = \\varepsilon_{0}{(J)} + \\log{(\\sin{(J)})} and \\operatorname{P_{e}}{(J)} = \\sin{(J)} and 2 \\varepsilon_{0}{(J)} = \\varepsilon_{0}{(J)} + \\log{(\\operatorname{P_{e}}{(J)})} and \\varepsilon_{0}{(J)} + \\log{(\\sin{(J)})} = \\varepsilon_{0}{(J)} + \\log{(\\operatorname{P_{e}}{(J)})} and (\\varepsilon_{0}{(J)} + \\log{(\\sin{(J)})}) \\log{(\\sin{(J)})} = (\\varepsilon_{0}{(J)} + \\log{(\\operatorname{P_{e}}{(J)})}) \\log{(\\sin{(J)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True))))"], [["add", 1, "Function('\\\\varepsilon_0')(Symbol('J', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('J', commutative=True))), Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('J', commutative=True))), Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(Function('P_e')(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(Function('P_e')(Symbol('J', commutative=True)))))"], [["times", 5, "log(sin(Symbol('J', commutative=True)))"], "Equality(Mul(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), log(sin(Symbol('J', commutative=True)))), Mul(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(Function('P_e')(Symbol('J', commutative=True)))), log(sin(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\hat{p}_0,A_{y})} = A_{y}^{\\hat{p}_0}, then obtain A_{y}^{\\hat{p}_0} - \\rho_{f}{(\\hat{p}_0,A_{y})} = 0", "derivation": "\\rho_{f}{(\\hat{p}_0,A_{y})} = A_{y}^{\\hat{p}_0} and - A_{y}^{\\hat{p}_0} + \\rho_{f}{(\\hat{p}_0,A_{y})} = 0 and - A_{y}^{\\hat{p}_0} + 2 \\rho_{f}{(\\hat{p}_0,A_{y})} = \\rho_{f}{(\\hat{p}_0,A_{y})} and - A_{y}^{\\hat{p}_0} + 2 \\rho_{f}{(\\hat{p}_0,A_{y})} = A_{y}^{\\hat{p}_0} and A_{y}^{\\hat{p}_0} - \\rho_{f}{(\\hat{p}_0,A_{y})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))), Integer(0))"], [["add", 1, "Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)))), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)))), Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Pow(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True)))), Integer(0))"]]}, {"prompt": "Given l{(\\mathbf{D})} = e^{\\sin{(\\mathbf{D})}}, then obtain (2 l{(\\mathbf{D})} + 2 e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} = (3 l{(\\mathbf{D})} + e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}}", "derivation": "l{(\\mathbf{D})} = e^{\\sin{(\\mathbf{D})}} and l{(\\mathbf{D})} + e^{\\sin{(\\mathbf{D})}} = 2 e^{\\sin{(\\mathbf{D})}} and 2 l{(\\mathbf{D})} + 2 e^{\\sin{(\\mathbf{D})}} = l{(\\mathbf{D})} + 3 e^{\\sin{(\\mathbf{D})}} and (2 l{(\\mathbf{D})} + 2 e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} = (l{(\\mathbf{D})} + 3 e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} and (3 l{(\\mathbf{D})} + e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} = (l{(\\mathbf{D})} + 3 e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} and (2 l{(\\mathbf{D})} + 2 e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}} = (3 l{(\\mathbf{D})} + e^{\\sin{(\\mathbf{D})}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 1, "exp(sin(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), exp(sin(Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Integer(2), exp(sin(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["add", 2, "Add(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('l')(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(3), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('l')(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(3), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Mul(Integer(3), Function('l')(Symbol('\\\\mathbf{D}', commutative=True))), exp(sin(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(3), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Add(Mul(Integer(2), Function('l')(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), exp(sin(Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Mul(Integer(3), Function('l')(Symbol('\\\\mathbf{D}', commutative=True))), exp(sin(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a^{\\dagger},G)} = \\log{(G a^{\\dagger})}, then obtain \\sin{(a^{\\dagger} \\operatorname{M_{E}}{(a^{\\dagger},G)} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)})} = \\sin{(a^{\\dagger} \\log{(G a^{\\dagger})} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)})}", "derivation": "\\operatorname{M_{E}}{(a^{\\dagger},G)} = \\log{(G a^{\\dagger})} and \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)} = \\log{(G a^{\\dagger})}^{G} and a^{\\dagger} \\operatorname{M_{E}}{(a^{\\dagger},G)} = a^{\\dagger} \\log{(G a^{\\dagger})} and a^{\\dagger} \\operatorname{M_{E}}{(a^{\\dagger},G)} - \\log{(G a^{\\dagger})}^{G} = a^{\\dagger} \\log{(G a^{\\dagger})} - \\log{(G a^{\\dagger})}^{G} and a^{\\dagger} \\operatorname{M_{E}}{(a^{\\dagger},G)} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)} = a^{\\dagger} \\log{(G a^{\\dagger})} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)} and \\sin{(a^{\\dagger} \\operatorname{M_{E}}{(a^{\\dagger},G)} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)})} = \\sin{(a^{\\dagger} \\log{(G a^{\\dagger})} - \\operatorname{M_{E}}^{G}{(a^{\\dagger},G)})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('G', commutative=True)))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["minus", 3, "Pow(log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('G', commutative=True))"], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('G', commutative=True)))), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Pow(log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"], [["sin", 5], "Equality(sin(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))))), sin(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('G', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))))))"]]}, {"prompt": "Given C{(\\Psi,n)} = - \\sin{(\\Psi - n)} and \\operatorname{f_{E}}{(\\Psi,n)} = \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int - \\sin{(\\Psi - n)} dn, then obtain \\operatorname{f_{E}}{(\\Psi,n)} = \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int C{(\\Psi,n)} dn", "derivation": "C{(\\Psi,n)} = - \\sin{(\\Psi - n)} and \\int C{(\\Psi,n)} dn = \\int - \\sin{(\\Psi - n)} dn and \\frac{\\partial}{\\partial \\Psi} \\int C{(\\Psi,n)} dn = \\frac{\\partial}{\\partial \\Psi} \\int - \\sin{(\\Psi - n)} dn and \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int C{(\\Psi,n)} dn = \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int - \\sin{(\\Psi - n)} dn and \\operatorname{f_{E}}{(\\Psi,n)} = \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int - \\sin{(\\Psi - n)} dn and \\operatorname{f_{E}}{(\\Psi,n)} = \\Psi + \\frac{\\partial}{\\partial \\Psi} \\int C{(\\Psi,n)} dn", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integral(Function('C')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Derivative(Integral(Function('C')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(Symbol('\\\\Psi', commutative=True), Derivative(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Derivative(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('f_E')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Derivative(Integral(Function('C')(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\sigma_x)} = \\cos{(\\sigma_x)}, then derive 0 = - \\sin{(\\sigma_x)} - \\frac{d}{d \\sigma_x} \\operatorname{n_{1}}{(\\sigma_x)}, then obtain 0 = - \\sin{(\\sigma_x)} - \\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)}", "derivation": "\\operatorname{n_{1}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and 0 = - \\operatorname{n_{1}}{(\\sigma_x)} + \\cos{(\\sigma_x)} and \\frac{d}{d \\sigma_x} 0 = \\frac{d}{d \\sigma_x} (- \\operatorname{n_{1}}{(\\sigma_x)} + \\cos{(\\sigma_x)}) and 0 = - \\sin{(\\sigma_x)} - \\frac{d}{d \\sigma_x} \\operatorname{n_{1}}{(\\sigma_x)} and 0 = - \\sin{(\\sigma_x)} - \\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Function('n_1')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Derivative(Function('n_1')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(x,h)} = h x and \\hat{x}_0{(x,h)} = (- h + \\operatorname{v_{x}}{(x,h)}) (h x - 1), then obtain \\sin{(h + \\hat{x}_0{(x,h)})} = \\sin{(h + (- h + \\operatorname{v_{x}}{(x,h)}) (\\operatorname{v_{x}}{(x,h)} - 1))}", "derivation": "\\operatorname{v_{x}}{(x,h)} = h x and \\operatorname{v_{x}}{(x,h)} - 1 = h x - 1 and \\hat{x}_0{(x,h)} = (- h + \\operatorname{v_{x}}{(x,h)}) (h x - 1) and h + \\hat{x}_0{(x,h)} = h + (- h + \\operatorname{v_{x}}{(x,h)}) (h x - 1) and h + \\hat{x}_0{(x,h)} = h + (- h + \\operatorname{v_{x}}{(x,h)}) (\\operatorname{v_{x}}{(x,h)} - 1) and \\sin{(h + \\hat{x}_0{(x,h)})} = \\sin{(h + (- h + \\operatorname{v_{x}}{(x,h)}) (\\operatorname{v_{x}}{(x,h)} - 1))}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Add(Mul(Symbol('h', commutative=True), Symbol('x', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('h', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Symbol('x', commutative=True)), Integer(-1))))"], [["minus", 3, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Symbol('x', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('h', commutative=True), Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True)), Integer(-1)))))"], [["sin", 5], "Equality(sin(Add(Symbol('h', commutative=True), Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('h', commutative=True)))), sin(Add(Symbol('h', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True))), Add(Function('v_x')(Symbol('x', commutative=True), Symbol('h', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{1})} = \\sin{(v_{1})} and \\operatorname{A_{y}}{(v_{1})} = v_{1} + \\sigma_{p}{(v_{1})} and \\Omega{(v_{1})} = \\sin{(v_{1})}, then obtain \\frac{\\operatorname{A_{y}}{(v_{1})} \\sigma_{p}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{(v_{1} + \\Omega{(v_{1})}) \\sigma_{p}{(v_{1})}}{\\sin{(v_{1})}}", "derivation": "\\sigma_{p}{(v_{1})} = \\sin{(v_{1})} and v_{1} + \\sigma_{p}{(v_{1})} = v_{1} + \\sin{(v_{1})} and \\operatorname{A_{y}}{(v_{1})} = v_{1} + \\sigma_{p}{(v_{1})} and \\Omega{(v_{1})} = \\sin{(v_{1})} and v_{1} + \\sigma_{p}{(v_{1})} = v_{1} + \\Omega{(v_{1})} and \\operatorname{A_{y}}{(v_{1})} = v_{1} + \\Omega{(v_{1})} and \\frac{\\operatorname{A_{y}}{(v_{1})} \\sigma_{p}{(v_{1})}}{\\sin{(v_{1})}} = \\frac{(v_{1} + \\Omega{(v_{1})}) \\sigma_{p}{(v_{1})}}{\\sin{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\sigma_p')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), sin(Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('v_1', commutative=True)), Add(Symbol('v_1', commutative=True), Function('\\\\sigma_p')(Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\sigma_p')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), Function('\\\\Omega')(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Function('A_y')(Symbol('v_1', commutative=True)), Add(Symbol('v_1', commutative=True), Function('\\\\Omega')(Symbol('v_1', commutative=True))))"], [["times", 6, "Mul(Function('\\\\sigma_p')(Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('A_y')(Symbol('v_1', commutative=True)), Function('\\\\sigma_p')(Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Add(Symbol('v_1', commutative=True), Function('\\\\Omega')(Symbol('v_1', commutative=True))), Function('\\\\sigma_p')(Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(z^{*},\\sigma_x)} = \\sigma_x z^{*} and i{(z^{*},\\sigma_x)} = - \\sigma_x z^{*} + \\hat{x}{(z^{*},\\sigma_x)}, then obtain \\frac{\\partial}{\\partial \\sigma_x} z^{*} i{(z^{*},\\sigma_x)} = \\frac{d}{d \\sigma_x} 0", "derivation": "\\hat{x}{(z^{*},\\sigma_x)} = \\sigma_x z^{*} and - \\sigma_x z^{*} + \\hat{x}{(z^{*},\\sigma_x)} = 0 and z^{*} (- \\sigma_x z^{*} + \\hat{x}{(z^{*},\\sigma_x)}) = 0 and \\frac{\\partial}{\\partial \\sigma_x} z^{*} (- \\sigma_x z^{*} + \\hat{x}{(z^{*},\\sigma_x)}) = \\frac{d}{d \\sigma_x} 0 and i{(z^{*},\\sigma_x)} = - \\sigma_x z^{*} + \\hat{x}{(z^{*},\\sigma_x)} and \\frac{\\partial}{\\partial \\sigma_x} z^{*} i{(z^{*},\\sigma_x)} = \\frac{d}{d \\sigma_x} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\hat{x}')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Integer(0))"], [["times", 2, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\hat{x}')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('z^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\hat{x}')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('i')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\hat{x}')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Symbol('z^*', commutative=True), Function('i')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(Z,\\mathbf{f})} = (e^{Z})^{\\mathbf{f}}, then obtain \\frac{\\partial}{\\partial \\mathbf{f}} (\\operatorname{P_{g}}{(Z,\\mathbf{f})} - (e^{Z})^{\\mathbf{f}}) = \\frac{d}{d \\mathbf{f}} 0", "derivation": "\\operatorname{P_{g}}{(Z,\\mathbf{f})} = (e^{Z})^{\\mathbf{f}} and \\mathbf{f} + \\operatorname{P_{g}}{(Z,\\mathbf{f})} = \\mathbf{f} + (e^{Z})^{\\mathbf{f}} and \\operatorname{P_{g}}{(Z,\\mathbf{f})} - (e^{Z})^{\\mathbf{f}} = 0 and \\frac{\\partial}{\\partial \\mathbf{f}} (\\operatorname{P_{g}}{(Z,\\mathbf{f})} - (e^{Z})^{\\mathbf{f}}) = \\frac{d}{d \\mathbf{f}} 0", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Symbol('Z', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('P_g')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(exp(Symbol('Z', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(exp(Symbol('Z', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Function('P_g')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('Z', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Add(Function('P_g')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('Z', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(y,W)} = \\cos{(W y)}, then derive - W \\sin{(W y)} + 2 W + \\frac{\\partial}{\\partial y} Q{(y,W)} = - 2 W \\sin{(W y)} + 2 W, then obtain \\frac{- W \\sin{(W y)} + 2 W + \\frac{\\partial}{\\partial y} \\cos{(W y)}}{Q{(y,W)}} = \\frac{- 2 W \\sin{(W y)} + 2 W}{Q{(y,W)}}", "derivation": "Q{(y,W)} = \\cos{(W y)} and W y + Q{(y,W)} = W y + \\cos{(W y)} and 2 W y + Q{(y,W)} + \\cos{(W y)} = 2 W y + 2 \\cos{(W y)} and \\frac{\\partial}{\\partial y} (2 W y + Q{(y,W)} + \\cos{(W y)}) = \\frac{\\partial}{\\partial y} (2 W y + 2 \\cos{(W y)}) and - W \\sin{(W y)} + 2 W + \\frac{\\partial}{\\partial y} Q{(y,W)} = - 2 W \\sin{(W y)} + 2 W and - W \\sin{(W y)} + 2 W + \\frac{\\partial}{\\partial y} \\cos{(W y)} = - 2 W \\sin{(W y)} + 2 W and \\frac{- W \\sin{(W y)} + 2 W + \\frac{\\partial}{\\partial y} \\cos{(W y)}}{Q{(y,W)}} = \\frac{- 2 W \\sin{(W y)} + 2 W}{Q{(y,W)}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))))"], [["add", 1, "Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)), Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))))"], [["add", 2, "Add(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('y', commutative=True)), Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('y', commutative=True)), Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('W', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True)), Derivative(Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True)), Derivative(cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True))))"], [["divide", 6, "Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True)), Derivative(cos(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), Pow(Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('W', commutative=True), sin(Mul(Symbol('W', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True))), Pow(Function('Q')(Symbol('y', commutative=True), Symbol('W', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given q{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})}, then obtain - \\frac{2 q{(\\Psi_{\\lambda})} - 2 \\log{(\\Psi_{\\lambda})}}{q{(\\Psi_{\\lambda})}} = - \\frac{- q{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}}{q{(\\Psi_{\\lambda})}}", "derivation": "q{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and q{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})} = 2 \\log{(\\Psi_{\\lambda})} and q{(\\Psi_{\\lambda})} + 2 \\log{(\\Psi_{\\lambda})} = 3 \\log{(\\Psi_{\\lambda})} and 2 q{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})} = 3 \\log{(\\Psi_{\\lambda})} and 2 q{(\\Psi_{\\lambda})} = 2 \\log{(\\Psi_{\\lambda})} and 3 q{(\\Psi_{\\lambda})} = 3 \\log{(\\Psi_{\\lambda})} and 2 q{(\\Psi_{\\lambda})} - 2 \\log{(\\Psi_{\\lambda})} = - q{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})} and - \\frac{2 q{(\\Psi_{\\lambda})} - 2 \\log{(\\Psi_{\\lambda})}}{q{(\\Psi_{\\lambda})}} = - \\frac{- q{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}}{q{(\\Psi_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["add", 2, "log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(3), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(3), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 4, "log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(3), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(3), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 6, "Add(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 7, "Mul(Integer(-1), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(2), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Pow(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given f{(v_{2},\\hat{H})} = e^{\\hat{H}^{v_{2}}}, then obtain (- \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + f^{v_{2}}{(v_{2},\\hat{H})}))^{\\hat{H}} = (- \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + (e^{\\hat{H}^{v_{2}}})^{v_{2}}))^{\\hat{H}}", "derivation": "f{(v_{2},\\hat{H})} = e^{\\hat{H}^{v_{2}}} and f^{v_{2}}{(v_{2},\\hat{H})} = (e^{\\hat{H}^{v_{2}}})^{v_{2}} and - \\hat{H}^{v_{2}} + f^{v_{2}}{(v_{2},\\hat{H})} = - \\hat{H}^{v_{2}} + (e^{\\hat{H}^{v_{2}}})^{v_{2}} and - \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + f^{v_{2}}{(v_{2},\\hat{H})}) = - \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + (e^{\\hat{H}^{v_{2}}})^{v_{2}}) and (- \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + f^{v_{2}}{(v_{2},\\hat{H})}))^{\\hat{H}} = (- \\hat{H}^{v_{2}} (- \\hat{H}^{v_{2}} + (e^{\\hat{H}^{v_{2}}})^{v_{2}}))^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('f')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('v_2', commutative=True)), Pow(exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"], [["minus", 2, "Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(Function('f')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(Function('f')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('v_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(Function('f')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('v_2', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Pow(exp(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\Omega)} = \\sin{(\\Omega)}, then derive \\frac{d}{d \\Omega} \\operatorname{F_{g}}{(\\Omega)} = \\cos{(\\Omega)}, then obtain \\frac{d^{2}}{d \\Omega^{2}} \\operatorname{F_{g}}{(\\Omega)} = \\frac{d}{d \\Omega} \\cos{(\\Omega)}", "derivation": "\\operatorname{F_{g}}{(\\Omega)} = \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\operatorname{F_{g}}{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\operatorname{F_{g}}{(\\Omega)} = \\cos{(\\Omega)} and \\cos{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\cos{(\\Omega)} = \\frac{d^{2}}{d \\Omega^{2}} \\sin{(\\Omega)} and \\frac{d^{2}}{d \\Omega^{2}} \\operatorname{F_{g}}{(\\Omega)} = \\frac{d^{2}}{d \\Omega^{2}} \\sin{(\\Omega)} and \\frac{d^{2}}{d \\Omega^{2}} \\operatorname{F_{g}}{(\\Omega)} = \\frac{d}{d \\Omega} \\cos{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), cos(Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('F_g')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('F_g')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(y,b)} = b y, then obtain y \\hat{p}{(y,b)} - \\hat{p}{(y,b)} + \\int b y dy = b y^{2} - \\hat{p}{(y,b)} + \\int b y dy", "derivation": "\\hat{p}{(y,b)} = b y and y \\hat{p}{(y,b)} = b y^{2} and \\int \\hat{p}{(y,b)} dy = \\int b y dy and y \\hat{p}{(y,b)} + \\int \\hat{p}{(y,b)} dy = b y^{2} + \\int \\hat{p}{(y,b)} dy and y \\hat{p}{(y,b)} + \\int b y dy = b y^{2} + \\int b y dy and y \\hat{p}{(y,b)} - \\hat{p}{(y,b)} + \\int b y dy = b y^{2} - \\hat{p}{(y,b)} + \\int b y dy", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), Pow(Symbol('y', commutative=True), Integer(2))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 2, "Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Add(Mul(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('y', commutative=True), Integer(2))), Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('y', commutative=True), Integer(2))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["minus", 5, "Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Symbol('y', commutative=True), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Mul(Symbol('b', commutative=True), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('y', commutative=True), Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\operatorname{f^{*}}{(\\Psi_{\\lambda})} = \\frac{2 \\operatorname{A_{y}}^{2}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}}, then obtain \\operatorname{f^{*}}{(\\Psi_{\\lambda})} = \\frac{2 \\operatorname{A_{y}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}}", "derivation": "\\operatorname{A_{y}}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\frac{\\operatorname{A_{y}}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} = \\frac{\\log{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} and \\frac{2 \\operatorname{A_{y}}^{2}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}} = \\frac{2 \\operatorname{A_{y}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}} and \\operatorname{f^{*}}{(\\Psi_{\\lambda})} = \\frac{2 \\operatorname{A_{y}}^{2}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}} and \\operatorname{f^{*}}{(\\Psi_{\\lambda})} = \\frac{2 \\operatorname{A_{y}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 2, "Mul(Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Pow(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))), Mul(Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Pow(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('f^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2)), Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given L{(\\mathbf{p})} = \\log{(\\log{(\\mathbf{p})})}, then derive \\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = \\log{(\\mathbf{p})} + \\frac{1}{\\mathbf{p} \\log{(\\mathbf{p})}}, then obtain (\\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})})^{\\mathbf{p}} = (\\log{(\\mathbf{p})} + \\frac{1}{\\mathbf{p} \\log{(\\mathbf{p})}})^{\\mathbf{p}}", "derivation": "L{(\\mathbf{p})} = \\log{(\\log{(\\mathbf{p})})} and \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\log{(\\log{(\\mathbf{p})})} and \\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = \\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} \\log{(\\log{(\\mathbf{p})})} and \\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = \\log{(\\mathbf{p})} + \\frac{1}{\\mathbf{p} \\log{(\\mathbf{p})}} and (\\log{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})})^{\\mathbf{p}} = (\\log{(\\mathbf{p})} + \\frac{1}{\\mathbf{p} \\log{(\\mathbf{p})}})^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), log(log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["add", 2, "log(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(log(log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)))))"], [["power", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)))), Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\mu_0)} = \\sin{(\\mu_0)}, then derive \\mu_0 \\int \\hat{p}_0{(\\mu_0)} d\\mu_0 = \\mu_0 (A_{x} - \\cos{(\\mu_0)}), then obtain \\frac{\\mu_0 \\int \\sin{(\\mu_0)} d\\mu_0}{A_{x} - \\cos{(\\mu_0)}} = \\mu_0", "derivation": "\\hat{p}_0{(\\mu_0)} = \\sin{(\\mu_0)} and \\int \\hat{p}_0{(\\mu_0)} d\\mu_0 = \\int \\sin{(\\mu_0)} d\\mu_0 and \\mu_0 \\int \\hat{p}_0{(\\mu_0)} d\\mu_0 = \\mu_0 \\int \\sin{(\\mu_0)} d\\mu_0 and \\mu_0 \\int \\hat{p}_0{(\\mu_0)} d\\mu_0 = \\mu_0 (A_{x} - \\cos{(\\mu_0)}) and \\frac{\\mu_0 \\int \\hat{p}_0{(\\mu_0)} d\\mu_0}{A_{x} - \\cos{(\\mu_0)}} = \\mu_0 and \\frac{\\mu_0 \\int \\sin{(\\mu_0)} d\\mu_0}{A_{x} - \\cos{(\\mu_0)}} = \\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["times", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True))))))"], [["divide", 4, "Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), Integer(-1)), Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))), Integer(-1)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"]]}, {"prompt": "Given \\mathbf{S}{(i)} = \\sin{(i)}, then obtain (i (i \\mathbf{S}{(i)} + i) \\mathbf{S}{(i)})^{i} = (i (i \\mathbf{S}{(i)} + i) \\sin{(i)})^{i}", "derivation": "\\mathbf{S}{(i)} = \\sin{(i)} and i \\mathbf{S}{(i)} = i \\sin{(i)} and i (i \\mathbf{S}{(i)} + i) \\mathbf{S}{(i)} = i (i \\mathbf{S}{(i)} + i) \\sin{(i)} and (i (i \\mathbf{S}{(i)} + i) \\mathbf{S}{(i)})^{i} = (i (i \\mathbf{S}{(i)} + i) \\sin{(i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], [["times", 2, "Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True))"], "Equality(Mul(Symbol('i', commutative=True), Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Symbol('i', commutative=True), Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{S}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(A_{1},f_{E})} = f_{E} e^{A_{1}}, then obtain - e^{A_{1}} + \\frac{\\operatorname{a^{\\dagger}}^{2}{(A_{1},f_{E})}}{A_{1}} = - e^{A_{1}} + \\frac{f_{E} \\operatorname{a^{\\dagger}}{(A_{1},f_{E})} e^{A_{1}}}{A_{1}}", "derivation": "\\operatorname{a^{\\dagger}}{(A_{1},f_{E})} = f_{E} e^{A_{1}} and \\frac{\\operatorname{a^{\\dagger}}{(A_{1},f_{E})}}{A_{1}} = \\frac{f_{E} e^{A_{1}}}{A_{1}} and \\frac{\\operatorname{a^{\\dagger}}^{2}{(A_{1},f_{E})}}{A_{1}} = \\frac{f_{E} \\operatorname{a^{\\dagger}}{(A_{1},f_{E})} e^{A_{1}}}{A_{1}} and - e^{A_{1}} + \\frac{\\operatorname{a^{\\dagger}}^{2}{(A_{1},f_{E})}}{A_{1}} = - e^{A_{1}} + \\frac{f_{E} \\operatorname{a^{\\dagger}}{(A_{1},f_{E})} e^{A_{1}}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), exp(Symbol('A_1', commutative=True))))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), exp(Symbol('A_1', commutative=True))))"], [["times", 2, "Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)), exp(Symbol('A_1', commutative=True))))"], [["minus", 3, "exp(Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Function('a^{\\\\dagger}')(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)), exp(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\Omega)} = \\cos{(\\Omega)}, then derive \\int \\delta{(\\Omega)} d\\Omega = h + \\sin{(\\Omega)}, then obtain \\Omega (\\theta_2 + \\sin{(\\Omega)}) = \\Omega (h + \\sin{(\\Omega)})", "derivation": "\\delta{(\\Omega)} = \\cos{(\\Omega)} and \\int \\delta{(\\Omega)} d\\Omega = \\int \\cos{(\\Omega)} d\\Omega and \\int \\delta{(\\Omega)} d\\Omega = h + \\sin{(\\Omega)} and \\int \\cos{(\\Omega)} d\\Omega = h + \\sin{(\\Omega)} and \\Omega \\int \\cos{(\\Omega)} d\\Omega = \\Omega (h + \\sin{(\\Omega)}) and \\Omega (\\theta_2 + \\sin{(\\Omega)}) = \\Omega (h + \\sin{(\\Omega)})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('h', commutative=True), sin(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('h', commutative=True), sin(Symbol('\\\\Omega', commutative=True))))"], [["times", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('h', commutative=True), sin(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('h', commutative=True), sin(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\operatorname{m_{s}}{(\\mathbf{J})} = \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})}, then obtain \\frac{d}{d \\mathbf{J}} \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J})}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} = \\sin^{\\mathbf{J}}{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\sin^{\\mathbf{J}}{(\\mathbf{J})} and \\operatorname{m_{s}}{(\\mathbf{J})} = \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} and \\operatorname{m_{s}}{(\\mathbf{J})} = \\sin^{\\mathbf{J}}{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(\\mathbf{E},t)} = \\mathbf{E} + t, then derive \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} + 1 = 2, then obtain \\frac{\\partial}{\\partial t} (\\mathbf{E} + t) + \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} = (\\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} + 1) \\frac{\\partial}{\\partial t} (\\mathbf{E} + t)", "derivation": "i{(\\mathbf{E},t)} = \\mathbf{E} + t and \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} = \\frac{\\partial}{\\partial t} (\\mathbf{E} + t) and \\frac{\\partial}{\\partial t} (\\mathbf{E} + t) + \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} = 2 \\frac{\\partial}{\\partial t} (\\mathbf{E} + t) and \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} + 1 = 2 and \\frac{\\partial}{\\partial t} (\\mathbf{E} + t) + \\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} = (\\frac{\\partial}{\\partial t} i{(\\mathbf{E},t)} + 1) \\frac{\\partial}{\\partial t} (\\mathbf{E} + t)", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Add(Derivative(Function('i')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(a)} = \\cos{(a)}, then derive \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)} = - \\sin{(a)}, then obtain - 2 (- \\sin{(a)} + \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)}) \\sin{(a)} = 4 \\sin^{2}{(a)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(a)} = \\cos{(a)} and \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)} = \\frac{d}{d a} \\cos{(a)} and \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)} = - \\sin{(a)} and - \\sin{(a)} + \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)} = - 2 \\sin{(a)} and - 2 (- \\sin{(a)} + \\frac{d}{d a} \\operatorname{f_{\\mathbf{p}}}{(a)}) \\sin{(a)} = 4 \\sin^{2}{(a)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('a', commutative=True))))"], [["add", 3, "Mul(Integer(-1), sin(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('a', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Integer(2), sin(Symbol('a', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), sin(Symbol('a', commutative=True))), Mul(Integer(4), Pow(sin(Symbol('a', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\hbar)} = \\cos{(\\hbar)} and \\operatorname{F_{x}}{(\\pi,\\sigma_p,\\mathbf{B})} = \\mathbf{B} + \\sigma_p^{\\pi}, then obtain \\operatorname{F_{x}}^{\\mathbf{B}}{(\\pi,\\sigma_p,\\mathbf{B})} \\Psi_{nl}^{- \\hbar}{(\\hbar)} = (\\mathbf{B} + \\sigma_p^{\\pi})^{\\mathbf{B}} \\Psi_{nl}^{- \\hbar}{(\\hbar)}", "derivation": "\\Psi_{nl}{(\\hbar)} = \\cos{(\\hbar)} and \\operatorname{F_{x}}{(\\pi,\\sigma_p,\\mathbf{B})} = \\mathbf{B} + \\sigma_p^{\\pi} and \\operatorname{F_{x}}^{\\mathbf{B}}{(\\pi,\\sigma_p,\\mathbf{B})} = (\\mathbf{B} + \\sigma_p^{\\pi})^{\\mathbf{B}} and \\operatorname{F_{x}}^{\\mathbf{B}}{(\\pi,\\sigma_p,\\mathbf{B})} \\cos^{- \\hbar}{(\\hbar)} = (\\mathbf{B} + \\sigma_p^{\\pi})^{\\mathbf{B}} \\cos^{- \\hbar}{(\\hbar)} and \\operatorname{F_{x}}^{\\mathbf{B}}{(\\pi,\\sigma_p,\\mathbf{B})} \\Psi_{nl}^{- \\hbar}{(\\hbar)} = (\\mathbf{B} + \\sigma_p^{\\pi})^{\\mathbf{B}} \\Psi_{nl}^{- \\hbar}{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], ["get_premise", "Equality(Function('F_x')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 3, "Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Function('F_x')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('F_x')(Symbol('\\\\pi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given U{(\\varepsilon,p)} = \\frac{\\varepsilon}{p}, then obtain \\frac{\\partial}{\\partial \\varepsilon} \\frac{U^{\\varepsilon}{(\\varepsilon,p)}}{U{(\\varepsilon,p)}} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{(\\frac{\\varepsilon}{p})^{\\varepsilon}}{U{(\\varepsilon,p)}}", "derivation": "U{(\\varepsilon,p)} = \\frac{\\varepsilon}{p} and U^{\\varepsilon}{(\\varepsilon,p)} = (\\frac{\\varepsilon}{p})^{\\varepsilon} and \\frac{U^{\\varepsilon}{(\\varepsilon,p)}}{U{(\\varepsilon,p)}} = \\frac{(\\frac{\\varepsilon}{p})^{\\varepsilon}}{U{(\\varepsilon,p)}} and \\frac{\\partial}{\\partial \\varepsilon} \\frac{U^{\\varepsilon}{(\\varepsilon,p)}}{U{(\\varepsilon,p)}} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{(\\frac{\\varepsilon}{p})^{\\varepsilon}}{U{(\\varepsilon,p)}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 2, "Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('U')(Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(\\Psi^{\\dagger},T)} = (e^{T})^{\\Psi^{\\dagger}}, then obtain - \\frac{T - f{(\\Psi^{\\dagger},T)} + \\frac{\\sin{(f{(\\Psi^{\\dagger},T)})}}{\\sin{((e^{T})^{\\Psi^{\\dagger}})}}}{T} = - \\frac{T - f{(\\Psi^{\\dagger},T)} + 1}{T}", "derivation": "f{(\\Psi^{\\dagger},T)} = (e^{T})^{\\Psi^{\\dagger}} and \\sin{(f{(\\Psi^{\\dagger},T)})} = \\sin{((e^{T})^{\\Psi^{\\dagger}})} and (- T + f{(\\Psi^{\\dagger},T)}) \\sin{(f{(\\Psi^{\\dagger},T)})} = (- T + f{(\\Psi^{\\dagger},T)}) \\sin{((e^{T})^{\\Psi^{\\dagger}})} and \\frac{\\sin{(f{(\\Psi^{\\dagger},T)})}}{\\sin{((e^{T})^{\\Psi^{\\dagger}})}} = 1 and T - f{(\\Psi^{\\dagger},T)} + \\frac{\\sin{(f{(\\Psi^{\\dagger},T)})}}{\\sin{((e^{T})^{\\Psi^{\\dagger}})}} = T - f{(\\Psi^{\\dagger},T)} + 1 and - \\frac{T - f{(\\Psi^{\\dagger},T)} + \\frac{\\sin{(f{(\\Psi^{\\dagger},T)})}}{\\sin{((e^{T})^{\\Psi^{\\dagger}})}}}{T} = - \\frac{T - f{(\\Psi^{\\dagger},T)} + 1}{T}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True)), Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), sin(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["divide", 3, "Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], "Equality(Mul(sin(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Pow(sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Mul(sin(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Pow(sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1)))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Integer(1)))"], [["divide", 5, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Mul(sin(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Pow(sin(Pow(exp(Symbol('T', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('T', commutative=True))), Integer(1))))"]]}, {"prompt": "Given k{(f_{E})} = \\cos{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} = 2 \\cos{(f_{E})}, then obtain k{(f_{E})} = \\frac{2 k^{2}{(f_{E})}}{\\operatorname{A_{1}}{(f_{E})}}", "derivation": "k{(f_{E})} = \\cos{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} = 2 \\cos{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} = 2 k{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} \\cos{(f_{E})} = 2 k{(f_{E})} \\cos{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} k{(f_{E})} = 2 k^{2}{(f_{E})} and k{(f_{E})} = \\frac{2 k^{2}{(f_{E})}}{\\operatorname{A_{1}}{(f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('f_E', commutative=True)), Mul(Integer(2), cos(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_1')(Symbol('f_E', commutative=True)), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True))))"], [["times", 3, "cos(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('A_1')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Mul(Integer(2), Function('k')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('A_1')(Symbol('f_E', commutative=True)), Function('k')(Symbol('f_E', commutative=True))), Mul(Integer(2), Pow(Function('k')(Symbol('f_E', commutative=True)), Integer(2))))"], [["divide", 5, "Function('A_1')(Symbol('f_E', commutative=True))"], "Equality(Function('k')(Symbol('f_E', commutative=True)), Mul(Integer(2), Pow(Function('A_1')(Symbol('f_E', commutative=True)), Integer(-1)), Pow(Function('k')(Symbol('f_E', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\varphi{(E_{x},m)} = E_{x} m, then derive \\frac{(\\phi + m) \\varphi{(E_{x},m)}}{E_{x} m} = \\phi + m, then obtain \\frac{(\\phi + m) \\varphi^{2}{(E_{x},m)}}{E_{x} m^{2}} = \\frac{(\\phi + m) \\varphi{(E_{x},m)}}{m}", "derivation": "\\varphi{(E_{x},m)} = E_{x} m and \\frac{\\varphi{(E_{x},m)}}{E_{x} m} = 1 and \\int \\frac{\\varphi{(E_{x},m)}}{E_{x} m} dm = \\int 1 dm and \\frac{\\varphi{(E_{x},m)} \\int \\frac{\\varphi{(E_{x},m)}}{E_{x} m} dm}{E_{x} m} = \\int \\frac{\\varphi{(E_{x},m)}}{E_{x} m} dm and \\frac{\\varphi{(E_{x},m)} \\int 1 dm}{E_{x} m} = \\int 1 dm and \\frac{(\\phi + m) \\varphi{(E_{x},m)}}{E_{x} m} = \\phi + m and \\frac{(\\phi + m) \\varphi{(E_{x},m)}}{E_{x} m^{2}} = \\frac{\\phi + m}{m} and \\frac{(\\phi + m) \\varphi^{2}{(E_{x},m)}}{E_{x} m^{2}} = \\frac{(\\phi + m) \\varphi{(E_{x},m)}}{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Mul(Symbol('E_x', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Integer(1), Tuple(Symbol('m', commutative=True))))"], [["times", 2, "Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True)), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True)), Integral(Integer(1), Tuple(Symbol('m', commutative=True)))), Integral(Integer(1), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True)))"], [["divide", 6, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-2)), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True))))"], [["times", 7, "Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-2)), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True)), Pow(Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True)), Integer(2))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('E_x', commutative=True), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(T)} = \\sin{(T)}, then derive \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT = T + k, then derive 2 T + \\mathbf{g} + k = T + k + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT, then obtain 2 T + \\mathbf{g} + k = 2 T + 2 k", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(T)} = \\sin{(T)} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} = 1 and \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT = \\int 1 dT and \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT = T + k and T + k + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT = 2 T + 2 k and T + k + \\int 1 dT = 2 T + 2 k and T + k + \\int 1 dT = T + k + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT and 2 T + \\mathbf{g} + k = T + k + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(T)}}{\\sin{(T)}} dT and 2 T + \\mathbf{g} + k = 2 T + 2 k", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["divide", 1, "sin(Symbol('T', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True))), Integral(Integer(1), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), Symbol('k', commutative=True)))"], [["add", 4, "Add(Symbol('T', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Symbol('k', commutative=True), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('T', commutative=True), Symbol('k', commutative=True), Integral(Integer(1), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('T', commutative=True), Symbol('k', commutative=True), Integral(Integer(1), Tuple(Symbol('T', commutative=True)))), Add(Symbol('T', commutative=True), Symbol('k', commutative=True), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Integer(2), Symbol('T', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('k', commutative=True)), Add(Symbol('T', commutative=True), Symbol('k', commutative=True), Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Add(Mul(Integer(2), Symbol('T', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('k', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(A)} = e^{\\cos{(A)}} and \\phi_{1}{(A)} = \\operatorname{C_{d}}{(A)} + e^{\\cos{(A)}}, then obtain \\log{(\\operatorname{C_{d}}{(A)} + e^{\\cos{(A)}})}^{A} = \\log{(\\phi_{1}{(A)})}^{A}", "derivation": "\\operatorname{C_{d}}{(A)} = e^{\\cos{(A)}} and \\phi_{1}{(A)} = \\operatorname{C_{d}}{(A)} + e^{\\cos{(A)}} and \\phi_{1}{(A)} = 2 \\operatorname{C_{d}}{(A)} and \\log{(\\phi_{1}{(A)})} = \\log{(2 \\operatorname{C_{d}}{(A)})} and \\log{(\\phi_{1}{(A)})}^{A} = \\log{(2 \\operatorname{C_{d}}{(A)})}^{A} and \\log{(\\operatorname{C_{d}}{(A)} + e^{\\cos{(A)}})}^{A} = \\log{(2 \\operatorname{C_{d}}{(A)})}^{A} and \\log{(\\operatorname{C_{d}}{(A)} + e^{\\cos{(A)}})}^{A} = \\log{(\\phi_{1}{(A)})}^{A}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('A', commutative=True)), Add(Function('C_d')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\phi_1')(Symbol('A', commutative=True)), Mul(Integer(2), Function('C_d')(Symbol('A', commutative=True))))"], [["log", 3], "Equality(log(Function('\\\\phi_1')(Symbol('A', commutative=True))), log(Mul(Integer(2), Function('C_d')(Symbol('A', commutative=True)))))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(log(Function('\\\\phi_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(log(Mul(Integer(2), Function('C_d')(Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(log(Add(Function('C_d')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True))))), Symbol('A', commutative=True)), Pow(log(Mul(Integer(2), Function('C_d')(Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(log(Add(Function('C_d')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True))))), Symbol('A', commutative=True)), Pow(log(Function('\\\\phi_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\sigma_x,L_{\\varepsilon})} = L_{\\varepsilon} + \\sigma_x, then obtain \\frac{\\partial}{\\partial L_{\\varepsilon}} (- L_{\\varepsilon} + \\operatorname{A_{x}}^{\\sigma_x}{(\\sigma_x,L_{\\varepsilon})}) = \\frac{\\partial}{\\partial L_{\\varepsilon}} (- L_{\\varepsilon} + (L_{\\varepsilon} + \\sigma_x)^{\\sigma_x})", "derivation": "\\operatorname{A_{x}}{(\\sigma_x,L_{\\varepsilon})} = L_{\\varepsilon} + \\sigma_x and \\operatorname{A_{x}}^{\\sigma_x}{(\\sigma_x,L_{\\varepsilon})} = (L_{\\varepsilon} + \\sigma_x)^{\\sigma_x} and - L_{\\varepsilon} + \\operatorname{A_{x}}^{\\sigma_x}{(\\sigma_x,L_{\\varepsilon})} = - L_{\\varepsilon} + (L_{\\varepsilon} + \\sigma_x)^{\\sigma_x} and \\frac{\\partial}{\\partial L_{\\varepsilon}} (- L_{\\varepsilon} + \\operatorname{A_{x}}^{\\sigma_x}{(\\sigma_x,L_{\\varepsilon})}) = \\frac{\\partial}{\\partial L_{\\varepsilon}} (- L_{\\varepsilon} + (L_{\\varepsilon} + \\sigma_x)^{\\sigma_x})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('A_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('A_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(Q)} = \\sin{(Q)} and \\operatorname{E_{x}}{(Q)} = \\frac{d}{d Q} T{(Q)} and U{(Q)} = \\sin{(Q)}, then obtain \\frac{(\\operatorname{E_{x}}{(Q)} - \\frac{d}{d Q} U{(Q)}) T{(Q)}}{\\operatorname{E_{x}}{(Q)} - \\cos{(Q)}} = 0", "derivation": "T{(Q)} = \\sin{(Q)} and \\operatorname{E_{x}}{(Q)} = \\frac{d}{d Q} T{(Q)} and \\operatorname{E_{x}}{(Q)} - \\frac{d}{d Q} \\sin{(Q)} = \\frac{d}{d Q} T{(Q)} - \\frac{d}{d Q} \\sin{(Q)} and \\operatorname{E_{x}}{(Q)} - \\frac{d}{d Q} \\sin{(Q)} = 0 and U{(Q)} = \\sin{(Q)} and \\operatorname{E_{x}}{(Q)} - \\frac{d}{d Q} U{(Q)} = 0 and \\frac{(\\operatorname{E_{x}}{(Q)} - \\frac{d}{d Q} U{(Q)}) T{(Q)}}{\\operatorname{E_{x}}{(Q)} - \\cos{(Q)}} = 0", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('Q', commutative=True)), Derivative(Function('T')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Add(Derivative(Function('T')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(0))"], ["renaming_premise", "Equality(Function('U')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('U')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Integer(0))"], [["divide", 6, "Mul(Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Pow(Function('T')(Symbol('Q', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Integer(-1)), Add(Function('E_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('U')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Function('T')(Symbol('Q', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(f_{E},\\sigma_x)} = \\sigma_x + \\sin{(f_{E})}, then derive \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{A_{2}}{(f_{E},\\sigma_x)} - 1 = 0, then obtain \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + \\sin{(f_{E})}) - 1 = 0", "derivation": "\\operatorname{A_{2}}{(f_{E},\\sigma_x)} = \\sigma_x + \\sin{(f_{E})} and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{A_{2}}{(f_{E},\\sigma_x)} = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + \\sin{(f_{E})}) and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{A_{2}}{(f_{E},\\sigma_x)} - 1 = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + \\sin{(f_{E})}) - 1 and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{A_{2}}{(f_{E},\\sigma_x)} - 1 = 0 and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + \\sin{(f_{E})}) - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('A_2')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('A_2')(Symbol('f_E', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{S}{(\\chi)} = \\cos{(\\chi)}, then obtain (\\chi \\mathbf{S}{(\\chi)} + \\chi) \\frac{d}{d \\chi} - \\chi \\mathbf{S}{(\\chi)} = (\\chi \\mathbf{S}{(\\chi)} + \\chi) \\frac{d}{d \\chi} - \\chi \\cos{(\\chi)}", "derivation": "\\mathbf{S}{(\\chi)} = \\cos{(\\chi)} and \\chi \\mathbf{S}{(\\chi)} = \\chi \\cos{(\\chi)} and - \\chi \\mathbf{S}{(\\chi)} = - \\chi \\cos{(\\chi)} and \\frac{d}{d \\chi} - \\chi \\mathbf{S}{(\\chi)} = \\frac{d}{d \\chi} - \\chi \\cos{(\\chi)} and (\\chi \\mathbf{S}{(\\chi)} + \\chi) \\frac{d}{d \\chi} - \\chi \\mathbf{S}{(\\chi)} = (\\chi \\mathbf{S}{(\\chi)} + \\chi) \\frac{d}{d \\chi} - \\chi \\cos{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 4, "Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbb{I},\\mathbf{S})} = \\cos{(\\mathbb{I} \\mathbf{S})} and E{(\\theta,\\hat{\\mathbf{r}},n_{2})} = \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}}, then obtain \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}} + \\frac{\\mathbf{p}{(\\mathbb{I},\\mathbf{S})}}{\\mathbb{I}} = \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}} + \\frac{\\cos{(\\mathbb{I} \\mathbf{S})}}{\\mathbb{I}}", "derivation": "\\mathbf{p}{(\\mathbb{I},\\mathbf{S})} = \\cos{(\\mathbb{I} \\mathbf{S})} and \\frac{\\mathbf{p}{(\\mathbb{I},\\mathbf{S})}}{\\mathbb{I}} = \\frac{\\cos{(\\mathbb{I} \\mathbf{S})}}{\\mathbb{I}} and E{(\\theta,\\hat{\\mathbf{r}},n_{2})} = \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}} and E{(\\theta,\\hat{\\mathbf{r}},n_{2})} + \\frac{\\mathbf{p}{(\\mathbb{I},\\mathbf{S})}}{\\mathbb{I}} = E{(\\theta,\\hat{\\mathbf{r}},n_{2})} + \\frac{\\cos{(\\mathbb{I} \\mathbf{S})}}{\\mathbb{I}} and \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}} + \\frac{\\mathbf{p}{(\\mathbb{I},\\mathbf{S})}}{\\mathbb{I}} = \\frac{\\hat{\\mathbf{r}} \\theta}{n_{2}} + \\frac{\\cos{(\\mathbb{I} \\mathbf{S})}}{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["add", 2, "Function('E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Function('E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Add(Function('E')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given S{(F_{g},\\hat{p})} = \\int (F_{g} + \\hat{p}) d\\hat{p}, then derive S{(F_{g},\\hat{p})} = F_{g} \\hat{p} + J + \\frac{\\hat{p}^{2}}{2}, then obtain \\frac{\\int (F_{g} + \\hat{p}) d\\hat{p}}{\\hat{p}^{2}} = \\frac{S{(F_{g},\\hat{p})}}{\\hat{p}^{2}}", "derivation": "S{(F_{g},\\hat{p})} = \\int (F_{g} + \\hat{p}) d\\hat{p} and S{(F_{g},\\hat{p})} = F_{g} \\hat{p} + J + \\frac{\\hat{p}^{2}}{2} and \\frac{S{(F_{g},\\hat{p})}}{\\hat{p}^{2}} = \\frac{F_{g} \\hat{p} + J + \\frac{\\hat{p}^{2}}{2}}{\\hat{p}^{2}} and \\frac{\\int (F_{g} + \\hat{p}) d\\hat{p}}{\\hat{p}^{2}} = \\frac{F_{g} \\hat{p} + J + \\frac{\\hat{p}^{2}}{2}}{\\hat{p}^{2}} and \\frac{\\int (F_{g} + \\hat{p}) d\\hat{p}}{\\hat{p}^{2}} = \\frac{S{(F_{g},\\hat{p})}}{\\hat{p}^{2}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('S')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)))))"], [["divide", 2, "Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Function('S')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Integral(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Function('S')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(s)} = \\sin{(s)}, then derive - u + \\cos{(s)} + \\int \\operatorname{v_{2}}{(s)} ds = 0, then obtain - u + \\cos{(s)} + \\int \\sin{(s)} ds = 0", "derivation": "\\operatorname{v_{2}}{(s)} = \\sin{(s)} and \\int \\operatorname{v_{2}}{(s)} ds = \\int \\sin{(s)} ds and \\int \\operatorname{v_{2}}{(s)} ds - \\int \\sin{(s)} ds = 0 and - u + \\cos{(s)} + \\int \\operatorname{v_{2}}{(s)} ds = 0 and - u + \\cos{(s)} + \\int \\sin{(s)} ds = 0", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["minus", 2, "Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Integral(Function('v_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), cos(Symbol('s', commutative=True)), Integral(Function('v_2')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), cos(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(V_{\\mathbf{E}})} = V_{\\mathbf{E}}, then obtain g{(V_{\\mathbf{E}})} \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{t_{2}}^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} = g{(V_{\\mathbf{E}})} \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}}^{V_{\\mathbf{E}}}", "derivation": "\\operatorname{t_{2}}{(V_{\\mathbf{E}})} = V_{\\mathbf{E}} and \\operatorname{t_{2}}^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} = V_{\\mathbf{E}}^{V_{\\mathbf{E}}} and \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{t_{2}}^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}}^{V_{\\mathbf{E}}} and g{(V_{\\mathbf{E}})} \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{t_{2}}^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} = g{(V_{\\mathbf{E}})} \\frac{d}{d V_{\\mathbf{E}}} V_{\\mathbf{E}}^{V_{\\mathbf{E}}}", "srepr_derivation": [["renaming_premise", "Equality(Function('t_2')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], [["power", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Pow(Function('t_2')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["times", 3, "Function('g')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Function('g')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Pow(Function('t_2')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(Function('g')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(C_{d})} = \\log{(C_{d})}, then obtain \\operatorname{C_{2}}{(C_{d})} = \\operatorname{C_{2}}{(C_{d})} - \\operatorname{C_{2}}^{C_{d}}{(C_{d})} + \\log{(C_{d})}^{C_{d}}", "derivation": "\\operatorname{C_{2}}{(C_{d})} = \\log{(C_{d})} and \\operatorname{C_{2}}^{C_{d}}{(C_{d})} = \\log{(C_{d})}^{C_{d}} and \\operatorname{C_{2}}{(C_{d})} + \\operatorname{C_{2}}^{C_{d}}{(C_{d})} = \\operatorname{C_{2}}{(C_{d})} + \\log{(C_{d})}^{C_{d}} and \\operatorname{C_{2}}{(C_{d})} = \\operatorname{C_{2}}{(C_{d})} - \\operatorname{C_{2}}^{C_{d}}{(C_{d})} + \\log{(C_{d})}^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], [["add", 2, "Function('C_2')(Symbol('C_d', commutative=True))"], "Equality(Add(Function('C_2')(Symbol('C_d', commutative=True)), Pow(Function('C_2')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Add(Function('C_2')(Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))))"], [["minus", 3, "Pow(Function('C_2')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Function('C_2')(Symbol('C_d', commutative=True)), Add(Function('C_2')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Pow(Function('C_2')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Pow(log(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{1},t,b)} = b + t - v_{1}, then derive v_{1} (\\frac{\\partial}{\\partial v_{1}} \\operatorname{f^{\\prime}}{(v_{1},t,b)} + 1) = 0, then obtain \\frac{\\partial}{\\partial b} v_{1} (\\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) + 1) = \\frac{d}{d b} 0", "derivation": "\\operatorname{f^{\\prime}}{(v_{1},t,b)} = b + t - v_{1} and \\frac{\\partial}{\\partial v_{1}} \\operatorname{f^{\\prime}}{(v_{1},t,b)} = \\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) and \\frac{\\partial}{\\partial v_{1}} \\operatorname{f^{\\prime}}{(v_{1},t,b)} + 1 = \\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) + 1 and v_{1} (\\frac{\\partial}{\\partial v_{1}} \\operatorname{f^{\\prime}}{(v_{1},t,b)} + 1) = v_{1} (\\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) + 1) and v_{1} (\\frac{\\partial}{\\partial v_{1}} \\operatorname{f^{\\prime}}{(v_{1},t,b)} + 1) = 0 and v_{1} (\\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) + 1) = 0 and \\frac{\\partial}{\\partial b} v_{1} (\\frac{\\partial}{\\partial v_{1}} (b + t - v_{1}) + 1) = \\frac{d}{d b} 0", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('t', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('t', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('t', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)))"], [["times", 3, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Add(Derivative(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('t', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('v_1', commutative=True), Add(Derivative(Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('v_1', commutative=True), Add(Derivative(Function('f^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('t', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('v_1', commutative=True), Add(Derivative(Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Integer(0))"], [["differentiate", 6, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Symbol('v_1', commutative=True), Add(Derivative(Add(Symbol('b', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(M_{E})} = \\log{(\\cos{(M_{E})})}, then derive \\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})} = - \\frac{\\sin{(M_{E})}}{\\cos{(M_{E})}}, then obtain ((\\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})})^{M_{E}})^{M_{E}} = ((- \\frac{\\sin{(M_{E})}}{\\cos{(M_{E})}})^{M_{E}})^{M_{E}}", "derivation": "\\operatorname{y^{\\prime}}{(M_{E})} = \\log{(\\cos{(M_{E})})} and \\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})} = \\frac{d}{d M_{E}} \\log{(\\cos{(M_{E})})} and \\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})} = - \\frac{\\sin{(M_{E})}}{\\cos{(M_{E})}} and (\\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})})^{M_{E}} = (- \\frac{\\sin{(M_{E})}}{\\cos{(M_{E})}})^{M_{E}} and ((\\frac{d}{d M_{E}} \\operatorname{y^{\\prime}}{(M_{E})})^{M_{E}})^{M_{E}} = ((- \\frac{\\sin{(M_{E})}}{\\cos{(M_{E})}})^{M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True)), log(cos(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(log(cos(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('M_E', commutative=True)), Pow(cos(Symbol('M_E', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('M_E', commutative=True)), Pow(cos(Symbol('M_E', commutative=True)), Integer(-1))), Symbol('M_E', commutative=True)))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Mul(Integer(-1), sin(Symbol('M_E', commutative=True)), Pow(cos(Symbol('M_E', commutative=True)), Integer(-1))), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(\\pi,C_{2})} = C_{2} \\pi, then obtain C_{2} \\pi \\iiint \\tilde{g}{(\\pi,C_{2})} d\\pi dC_{2} dC_{2} = C_{2} \\pi \\iiint C_{2} \\pi d\\pi dC_{2} dC_{2}", "derivation": "\\tilde{g}{(\\pi,C_{2})} = C_{2} \\pi and \\int \\tilde{g}{(\\pi,C_{2})} d\\pi = \\int C_{2} \\pi d\\pi and \\iint \\tilde{g}{(\\pi,C_{2})} d\\pi dC_{2} = \\iint C_{2} \\pi d\\pi dC_{2} and \\iiint \\tilde{g}{(\\pi,C_{2})} d\\pi dC_{2} dC_{2} = \\iiint C_{2} \\pi d\\pi dC_{2} dC_{2} and C_{2} \\pi \\iiint \\tilde{g}{(\\pi,C_{2})} d\\pi dC_{2} dC_{2} = C_{2} \\pi \\iiint C_{2} \\pi d\\pi dC_{2} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\pi', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\pi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\pi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\pi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["times", 4, "Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('\\\\pi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True), Integral(Mul(Symbol('C_2', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\phi{(a)} = e^{a}, then obtain \\frac{\\phi^{2}{(a)} e^{- a}}{a} = \\frac{\\phi{(a)}}{a}", "derivation": "\\phi{(a)} = e^{a} and \\frac{\\phi{(a)}}{a} = \\frac{e^{a}}{a} and \\frac{\\phi^{2}{(a)}}{a^{2}} = \\frac{\\phi{(a)} e^{a}}{a^{2}} and \\frac{\\phi^{2}{(a)} e^{- a}}{a} = \\frac{\\phi{(a)}}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["divide", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('a', commutative=True))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('a', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('a', commutative=True)))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Pow(Function('\\\\phi')(Symbol('a', commutative=True)), Integer(2))), Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Function('\\\\phi')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('a', commutative=True)))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('a', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(T)} = T, then derive \\frac{d}{d T} \\mathbf{M}{(T)} + 1 = 2, then obtain - T + \\mathbf{M}{(T)} = T (- \\frac{d}{d T} \\mathbf{M}{(T)} - 1) + (\\frac{d}{d T} \\mathbf{M}{(T)} + 1) \\mathbf{M}{(T)}", "derivation": "\\mathbf{M}{(T)} = T and 0 = T - \\mathbf{M}{(T)} and T - \\mathbf{M}{(T)} = 2 T - 2 \\mathbf{M}{(T)} and - T + \\mathbf{M}{(T)} = - 2 T + 2 \\mathbf{M}{(T)} and \\frac{d}{d T} \\mathbf{M}{(T)} = \\frac{d}{d T} T and \\frac{d}{d T} \\mathbf{M}{(T)} + 1 = \\frac{d}{d T} T + 1 and \\frac{d}{d T} \\mathbf{M}{(T)} + 1 = 2 and - T + \\mathbf{M}{(T)} = T (- \\frac{d}{d T} \\mathbf{M}{(T)} - 1) + (\\frac{d}{d T} \\mathbf{M}{(T)} + 1) \\mathbf{M}{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Symbol('T', commutative=True))"], [["minus", 1, "Function('\\\\mathbf{M}')(Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('T', commutative=True)))))"], [["add", 2, "Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('T', commutative=True))))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{M}')(Symbol('T', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbf{M}')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('T', commutative=True)))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Symbol('T', commutative=True), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["add", 5, 1], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('T', commutative=True), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 4, 7], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\mathbf{M}')(Symbol('T', commutative=True))), Add(Mul(Symbol('T', commutative=True), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1))), Mul(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Function('\\\\mathbf{M}')(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{J}_f)} = \\cos{(\\cos{(\\mathbf{J}_f)})} and \\operatorname{C_{2}}{(E,\\Psi_{nl})} = - E + e^{\\Psi_{nl}}, then obtain (E + \\operatorname{C_{2}}{(E,\\Psi_{nl})} - e^{\\Psi_{nl}}) \\int \\cos{(\\cos{(\\mathbf{J}_f)})} d\\mathbf{J}_f = 0", "derivation": "\\operatorname{P_{e}}{(\\mathbf{J}_f)} = \\cos{(\\cos{(\\mathbf{J}_f)})} and \\operatorname{C_{2}}{(E,\\Psi_{nl})} = - E + e^{\\Psi_{nl}} and E + \\operatorname{C_{2}}{(E,\\Psi_{nl})} - e^{\\Psi_{nl}} = 0 and (E + \\operatorname{C_{2}}{(E,\\Psi_{nl})} - e^{\\Psi_{nl}}) \\int \\operatorname{P_{e}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = 0 and (E + \\operatorname{C_{2}}{(E,\\Psi_{nl})} - e^{\\Psi_{nl}}) \\int \\cos{(\\cos{(\\mathbf{J}_f)})} d\\mathbf{J}_f = 0", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], ["get_premise", "Equality(Function('C_2')(Symbol('E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Function('C_2')(Symbol('E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(0))"], [["times", 3, "Integral(Function('P_e')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Mul(Add(Symbol('E', commutative=True), Function('C_2')(Symbol('E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(Function('P_e')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Symbol('E', commutative=True), Function('C_2')(Symbol('E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(cos(cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{H}_l{(v_{2},L)} = L - v_{2}, then obtain \\frac{v_{2} + 1}{L \\int \\frac{\\hat{H}_l{(v_{2},L)}}{L - v_{2}} dv_{2}} = \\frac{v_{2} + \\frac{L - v_{2}}{\\hat{H}_l{(v_{2},L)}}}{L \\int \\frac{\\hat{H}_l{(v_{2},L)}}{L - v_{2}} dv_{2}}", "derivation": "\\hat{H}_l{(v_{2},L)} = L - v_{2} and 1 = \\frac{L - v_{2}}{\\hat{H}_l{(v_{2},L)}} and v_{2} + 1 = v_{2} + \\frac{L - v_{2}}{\\hat{H}_l{(v_{2},L)}} and \\frac{v_{2} + 1}{L} = \\frac{v_{2} + \\frac{L - v_{2}}{\\hat{H}_l{(v_{2},L)}}}{L} and \\frac{v_{2} + 1}{L \\int \\frac{\\hat{H}_l{(v_{2},L)}}{L - v_{2}} dv_{2}} = \\frac{v_{2} + \\frac{L - v_{2}}{\\hat{H}_l{(v_{2},L)}}}{L \\int \\frac{\\hat{H}_l{(v_{2},L)}}{L - v_{2}} dv_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('v_2', commutative=True), Integer(1)), Add(Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))))"], [["divide", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Integer(1))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))))))"], [["divide", 4, "Integral(Mul(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Integer(1)), Pow(Integral(Mul(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))), Pow(Integral(Mul(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('v_2', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(n_{1})} = n_{1}, then derive \\frac{d}{d n_{1}} \\operatorname{E_{n}}{(n_{1})} = 1, then obtain \\nabla - \\frac{1}{n_{1}} = t_{1} - \\frac{1}{n_{1}}", "derivation": "\\operatorname{E_{n}}{(n_{1})} = n_{1} and \\frac{d}{d n_{1}} \\operatorname{E_{n}}{(n_{1})} = \\frac{d}{d n_{1}} n_{1} and \\frac{d}{d n_{1}} \\operatorname{E_{n}}{(n_{1})} = 1 and \\frac{\\frac{d}{d n_{1}} \\operatorname{E_{n}}{(n_{1})}}{n_{1}^{2}} = \\frac{1}{n_{1}^{2}} and \\frac{\\frac{d}{d n_{1}} n_{1}}{n_{1}^{2}} = \\frac{1}{n_{1}^{2}} and \\int \\frac{\\frac{d}{d n_{1}} n_{1}}{n_{1}^{2}} dn_{1} = \\int \\frac{1}{n_{1}^{2}} dn_{1} and \\nabla - \\frac{1}{n_{1}} = t_{1} - \\frac{1}{n_{1}}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Symbol('n_1', commutative=True), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Pow(Symbol('n_1', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Derivative(Function('E_n')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Pow(Symbol('n_1', commutative=True), Integer(-2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Derivative(Symbol('n_1', commutative=True), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Pow(Symbol('n_1', commutative=True), Integer(-2)))"], [["integrate", 5, "Symbol('n_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Derivative(Symbol('n_1', commutative=True), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Integer(-2)), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given z{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\hat{p}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} + z{(g^{\\prime}_{\\varepsilon})}, then obtain \\hat{p}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})}", "derivation": "z{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and g^{\\prime}_{\\varepsilon} + z{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})} and \\hat{p}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} + z{(g^{\\prime}_{\\varepsilon})} and \\hat{p}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{p}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\Psi^{\\dagger},v_{x})} = \\Psi^{\\dagger} v_{x}, then obtain - \\Psi^{\\dagger} + \\frac{\\int \\phi{(\\Psi^{\\dagger},v_{x})} dv_{x}}{\\int \\Psi^{\\dagger} v_{x} dv_{x}} = 1 - \\Psi^{\\dagger}", "derivation": "\\phi{(\\Psi^{\\dagger},v_{x})} = \\Psi^{\\dagger} v_{x} and \\int \\phi{(\\Psi^{\\dagger},v_{x})} dv_{x} = \\int \\Psi^{\\dagger} v_{x} dv_{x} and \\frac{\\int \\phi{(\\Psi^{\\dagger},v_{x})} dv_{x}}{\\int \\Psi^{\\dagger} v_{x} dv_{x}} = 1 and - \\Psi^{\\dagger} + \\frac{\\int \\phi{(\\Psi^{\\dagger},v_{x})} dv_{x}}{\\int \\Psi^{\\dagger} v_{x} dv_{x}} = 1 - \\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["divide", 2, "Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Integer(1))"], [["minus", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Pow(Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(-1)), Integral(Function('\\\\phi')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} = \\hat{p} \\sin{(v_{y})}, then obtain - \\frac{\\partial}{\\partial v_{y}} \\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} - \\frac{1}{v_{y}} = - \\frac{\\partial}{\\partial v_{y}} \\hat{p} \\sin{(v_{y})} - \\frac{1}{v_{y}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} = \\hat{p} \\sin{(v_{y})} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} = \\frac{\\partial}{\\partial v_{y}} \\hat{p} \\sin{(v_{y})} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} + \\frac{1}{v_{y}} = \\frac{\\partial}{\\partial v_{y}} \\hat{p} \\sin{(v_{y})} + \\frac{1}{v_{y}} and - \\frac{\\partial}{\\partial v_{y}} \\operatorname{V_{\\mathbf{E}}}{(v_{y},\\hat{p})} - \\frac{1}{v_{y}} = - \\frac{\\partial}{\\partial v_{y}} \\hat{p} \\sin{(v_{y})} - \\frac{1}{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('v_y', commutative=True))))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["add", 2, "Pow(Symbol('v_y', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Pow(Symbol('v_y', commutative=True), Integer(-1))), Add(Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Pow(Symbol('v_y', commutative=True), Integer(-1))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given I{(c)} = \\cos{(c)}, then obtain - z{(\\nabla)} + \\frac{d}{d c} (c + I^{c}{(c)}) = - z{(\\nabla)} + \\frac{d}{d c} (c + \\cos^{c}{(c)})", "derivation": "I{(c)} = \\cos{(c)} and I^{c}{(c)} = \\cos^{c}{(c)} and c + I^{c}{(c)} = c + \\cos^{c}{(c)} and \\frac{d}{d c} (c + I^{c}{(c)}) = \\frac{d}{d c} (c + \\cos^{c}{(c)}) and - z{(\\nabla)} + \\frac{d}{d c} (c + I^{c}{(c)}) = - z{(\\nabla)} + \\frac{d}{d c} (c + \\cos^{c}{(c)})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('I')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["add", 2, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Pow(Function('I')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Symbol('c', commutative=True), Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Symbol('c', commutative=True), Pow(Function('I')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Symbol('c', commutative=True), Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["minus", 4, "Function('z')(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('z')(Symbol('\\\\nabla', commutative=True))), Derivative(Add(Symbol('c', commutative=True), Pow(Function('I')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('z')(Symbol('\\\\nabla', commutative=True))), Derivative(Add(Symbol('c', commutative=True), Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given T{(m,\\mathbf{v})} = \\int \\frac{m}{\\mathbf{v}} d\\mathbf{v}, then derive \\frac{\\partial}{\\partial m} T{(m,\\mathbf{v})} + \\frac{1}{\\mathbf{v}} = \\frac{\\partial}{\\partial m} (\\int \\frac{m}{\\mathbf{v}} d\\mathbf{v} + \\frac{m}{\\mathbf{v}}), then obtain \\frac{\\partial}{\\partial m} T{(m,\\mathbf{v})} + \\frac{1}{\\mathbf{v}} = \\frac{\\partial}{\\partial m} (T{(m,\\mathbf{v})} + \\frac{m}{\\mathbf{v}})", "derivation": "T{(m,\\mathbf{v})} = \\int \\frac{m}{\\mathbf{v}} d\\mathbf{v} and T{(m,\\mathbf{v})} + \\frac{m}{\\mathbf{v}} = \\int \\frac{m}{\\mathbf{v}} d\\mathbf{v} + \\frac{m}{\\mathbf{v}} and \\frac{\\partial}{\\partial m} (T{(m,\\mathbf{v})} + \\frac{m}{\\mathbf{v}}) = \\frac{\\partial}{\\partial m} (\\int \\frac{m}{\\mathbf{v}} d\\mathbf{v} + \\frac{m}{\\mathbf{v}}) and \\frac{\\partial}{\\partial m} T{(m,\\mathbf{v})} + \\frac{1}{\\mathbf{v}} = \\frac{\\partial}{\\partial m} (\\int \\frac{m}{\\mathbf{v}} d\\mathbf{v} + \\frac{m}{\\mathbf{v}}) and \\frac{\\partial}{\\partial m} T{(m,\\mathbf{v})} + \\frac{1}{\\mathbf{v}} = \\frac{\\partial}{\\partial m} (T{(m,\\mathbf{v})} + \\frac{m}{\\mathbf{v}})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))"], "Equality(Add(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Add(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Derivative(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Derivative(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Derivative(Add(Function('T')(Symbol('m', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain \\operatorname{f^{*}}^{3}{(\\varepsilon)} \\cos{(\\varepsilon)} = \\cos^{4}{(\\varepsilon)}", "derivation": "\\operatorname{f^{*}}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\operatorname{f^{*}}{(\\varepsilon)} \\cos{(\\varepsilon)} = \\cos^{2}{(\\varepsilon)} and \\operatorname{f^{*}}^{2}{(\\varepsilon)} \\cos^{2}{(\\varepsilon)} = \\cos^{4}{(\\varepsilon)} and \\operatorname{f^{*}}^{3}{(\\varepsilon)} \\cos{(\\varepsilon)} = \\operatorname{f^{*}}^{2}{(\\varepsilon)} \\cos^{2}{(\\varepsilon)} and \\operatorname{f^{*}}^{3}{(\\varepsilon)} \\cos{(\\varepsilon)} = \\cos^{4}{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Integer(3)), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Integer(3)), cos(Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(4)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(r)} = \\log{(r)} and \\operatorname{x^{{\\}'}}{(r)} = \\operatorname{f^{\\prime}}{(r)} + \\log{(r)}, then obtain r + \\operatorname{f^{\\prime}}{(r)} + \\operatorname{x^{{\\}'}}{(r)} + \\log{(r)} = r + \\operatorname{f^{\\prime}}{(r)} + 3 \\log{(r)}", "derivation": "\\operatorname{f^{\\prime}}{(r)} = \\log{(r)} and \\operatorname{x^{{\\}'}}{(r)} = \\operatorname{f^{\\prime}}{(r)} + \\log{(r)} and \\operatorname{x^{{\\}'}}{(r)} = 2 \\log{(r)} and r + \\operatorname{f^{\\prime}}{(r)} + \\operatorname{x^{{\\}'}}{(r)} + \\log{(r)} = r + \\operatorname{f^{\\prime}}{(r)} + 3 \\log{(r)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('r', commutative=True)), Add(Function('f^{\\\\prime}')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('r', commutative=True)), Mul(Integer(2), log(Symbol('r', commutative=True))))"], [["add", 3, "Add(Symbol('r', commutative=True), Function('f^{\\\\prime}')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], "Equality(Add(Symbol('r', commutative=True), Function('f^{\\\\prime}')(Symbol('r', commutative=True)), Function('x^\\\\prime')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True))), Add(Symbol('r', commutative=True), Function('f^{\\\\prime}')(Symbol('r', commutative=True)), Mul(Integer(3), log(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{x}_0)} = \\cos{(\\sin{(\\hat{x}_0)})} and \\mathbf{A}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)}, then obtain 1 = \\frac{\\cos{(\\mathbf{A}{(\\hat{x}_0)})} + \\cos{(\\sin{(\\hat{x}_0)})}}{\\theta_{1}{(\\hat{x}_0)} + \\cos{(\\sin{(\\hat{x}_0)})}}", "derivation": "\\theta_{1}{(\\hat{x}_0)} = \\cos{(\\sin{(\\hat{x}_0)})} and \\mathbf{A}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\theta_{1}{(\\hat{x}_0)} + \\cos{(\\sin{(\\hat{x}_0)})} = 2 \\cos{(\\sin{(\\hat{x}_0)})} and \\theta_{1}{(\\hat{x}_0)} = \\cos{(\\mathbf{A}{(\\hat{x}_0)})} and \\cos{(\\mathbf{A}{(\\hat{x}_0)})} + \\cos{(\\sin{(\\hat{x}_0)})} = 2 \\cos{(\\sin{(\\hat{x}_0)})} and \\theta_{1}{(\\hat{x}_0)} + \\cos{(\\sin{(\\hat{x}_0)})} = \\cos{(\\mathbf{A}{(\\hat{x}_0)})} + \\cos{(\\sin{(\\hat{x}_0)})} and 1 = \\frac{\\cos{(\\mathbf{A}{(\\hat{x}_0)})} + \\cos{(\\sin{(\\hat{x}_0)})}}{\\theta_{1}{(\\hat{x}_0)} + \\cos{(\\sin{(\\hat{x}_0)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Add(cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["divide", 6, "Add(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Integer(-1)), Add(cos(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))), cos(sin(Symbol('\\\\hat{x}_0', commutative=True))))))"]]}, {"prompt": "Given a{(I,b)} = \\cos^{b}{(I)}, then obtain \\frac{\\partial}{\\partial I} (\\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{a{(I,b)}}{I}) = \\frac{\\partial}{\\partial I} (\\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{\\cos^{b}{(I)}}{I})", "derivation": "a{(I,b)} = \\cos^{b}{(I)} and \\int a{(I,b)} dI = \\int \\cos^{b}{(I)} dI and \\frac{a{(I,b)}}{I} = \\frac{\\cos^{b}{(I)}}{I} and \\cos^{b}{(I)} - \\int a{(I,b)} dI + \\frac{a{(I,b)}}{I} = \\cos^{b}{(I)} - \\int a{(I,b)} dI + \\frac{\\cos^{b}{(I)}}{I} and \\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{a{(I,b)}}{I} = \\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{\\cos^{b}{(I)}}{I} and \\frac{\\partial}{\\partial I} (\\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{a{(I,b)}}{I}) = \\frac{\\partial}{\\partial I} (\\cos^{b}{(I)} - \\int \\cos^{b}{(I)} dI + \\frac{\\cos^{b}{(I)}}{I})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True))), Integral(Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True))))"], "Equality(Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)))), Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)))), Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)))))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('a')(Symbol('I', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(cos(Symbol('I', commutative=True)), Symbol('b', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(k,v)} = \\cos^{k}{(v)}, then obtain - v + \\frac{v + s{(k,v)}}{\\int \\cos^{k}{(v)} dv} - s{(k,v)} = - v + \\frac{v + \\cos^{k}{(v)}}{\\int \\cos^{k}{(v)} dv} - s{(k,v)}", "derivation": "s{(k,v)} = \\cos^{k}{(v)} and v + s{(k,v)} = v + \\cos^{k}{(v)} and \\int s{(k,v)} dv = \\int \\cos^{k}{(v)} dv and \\frac{v + s{(k,v)}}{\\int s{(k,v)} dv} = \\frac{v + \\cos^{k}{(v)}}{\\int s{(k,v)} dv} and \\frac{v + s{(k,v)}}{\\int \\cos^{k}{(v)} dv} = \\frac{v + \\cos^{k}{(v)}}{\\int \\cos^{k}{(v)} dv} and - v + \\frac{v + s{(k,v)}}{\\int \\cos^{k}{(v)} dv} - s{(k,v)} = - v + \\frac{v + \\cos^{k}{(v)}}{\\int \\cos^{k}{(v)} dv} - s{(k,v)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 2, "Integral(Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Mul(Add(Symbol('v', commutative=True), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Pow(Integral(Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(Add(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True))), Pow(Integral(Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('v', commutative=True), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Pow(Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(Add(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True))), Pow(Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))))"], [["minus", 5, "Add(Symbol('v', commutative=True), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Add(Symbol('v', commutative=True), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Pow(Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(Integer(-1), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Add(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True))), Pow(Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(Integer(-1), Function('s')(Symbol('k', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given l{(a^{\\dagger},f^{*})} = \\frac{a^{\\dagger}}{f^{*}}, then obtain l{(a^{\\dagger},f^{*})} \\int \\frac{a^{\\dagger}}{f^{*}} da^{\\dagger} = \\frac{a^{\\dagger} \\int \\frac{a^{\\dagger}}{f^{*}} da^{\\dagger}}{f^{*}}", "derivation": "l{(a^{\\dagger},f^{*})} = \\frac{a^{\\dagger}}{f^{*}} and \\int l{(a^{\\dagger},f^{*})} da^{\\dagger} = \\int \\frac{a^{\\dagger}}{f^{*}} da^{\\dagger} and l{(a^{\\dagger},f^{*})} \\int l{(a^{\\dagger},f^{*})} da^{\\dagger} = \\frac{a^{\\dagger} \\int l{(a^{\\dagger},f^{*})} da^{\\dagger}}{f^{*}} and l{(a^{\\dagger},f^{*})} \\int \\frac{a^{\\dagger}}{f^{*}} da^{\\dagger} = \\frac{a^{\\dagger} \\int \\frac{a^{\\dagger}}{f^{*}} da^{\\dagger}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Integral(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Integral(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)), Integral(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^*', commutative=True)), Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)), Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\hat{p},t_{1},n)} = \\hat{p} t_{1} + n, then obtain \\hat{p} n + \\frac{- 2 n + 2 \\hat{x}{(\\hat{p},t_{1},n)}}{t_{1}} = \\hat{p} n + 2 \\hat{p}", "derivation": "\\hat{x}{(\\hat{p},t_{1},n)} = \\hat{p} t_{1} + n and - n + \\hat{x}{(\\hat{p},t_{1},n)} = \\hat{p} t_{1} and \\frac{- n + \\hat{x}{(\\hat{p},t_{1},n)}}{t_{1}} = \\hat{p} and \\frac{2 (- n + \\hat{x}{(\\hat{p},t_{1},n)})}{t_{1}} = \\hat{p} + \\frac{- n + \\hat{x}{(\\hat{p},t_{1},n)}}{t_{1}} and \\frac{- 2 n + 2 \\hat{x}{(\\hat{p},t_{1},n)}}{t_{1}} = 2 \\hat{p} and \\hat{p} n + \\frac{- 2 n + 2 \\hat{x}{(\\hat{p},t_{1},n)}}{t_{1}} = \\hat{p} n + 2 \\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True)), Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('n', commutative=True)))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)))"], [["divide", 2, "Symbol('t_1', commutative=True)"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True)))), Symbol('\\\\hat{p}', commutative=True))"], [["add", 3, "Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True))))), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)))"], [["add", 5, "Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True), Symbol('n', commutative=True)))))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given l{(i)} = \\frac{d}{d i} \\log{(i)}, then derive i l{(i)} = 1, then obtain i \\frac{d}{d i} \\log{(i)} + l{(i)} = l{(i)} + 1", "derivation": "l{(i)} = \\frac{d}{d i} \\log{(i)} and i l{(i)} = i \\frac{d}{d i} \\log{(i)} and i l{(i)} = 1 and i l{(i)} + l{(i)} = l{(i)} + 1 and i \\frac{d}{d i} \\log{(i)} + l{(i)} = l{(i)} + 1", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('i', commutative=True)), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('l')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('i', commutative=True), Function('l')(Symbol('i', commutative=True))), Integer(1))"], [["add", 3, "Function('l')(Symbol('i', commutative=True))"], "Equality(Add(Mul(Symbol('i', commutative=True), Function('l')(Symbol('i', commutative=True))), Function('l')(Symbol('i', commutative=True))), Add(Function('l')(Symbol('i', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('i', commutative=True), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Function('l')(Symbol('i', commutative=True))), Add(Function('l')(Symbol('i', commutative=True)), Integer(1)))"]]}, {"prompt": "Given Q{(\\mu)} = \\cos{(\\sin{(\\mu)})}, then obtain \\int (\\frac{d}{d \\mu} Q{(\\mu)})^{\\mu} d\\mu = \\int (\\frac{d}{d \\mu} \\cos{(\\sin{(\\mu)})})^{\\mu} d\\mu", "derivation": "Q{(\\mu)} = \\cos{(\\sin{(\\mu)})} and \\frac{d}{d \\mu} Q{(\\mu)} = \\frac{d}{d \\mu} \\cos{(\\sin{(\\mu)})} and (\\frac{d}{d \\mu} Q{(\\mu)})^{\\mu} = (\\frac{d}{d \\mu} \\cos{(\\sin{(\\mu)})})^{\\mu} and \\int (\\frac{d}{d \\mu} Q{(\\mu)})^{\\mu} d\\mu = \\int (\\frac{d}{d \\mu} \\cos{(\\sin{(\\mu)})})^{\\mu} d\\mu", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mu', commutative=True)), cos(sin(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Function('Q')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(cos(sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('Q')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(Derivative(cos(sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given A{(f^{*},b)} = (f^{*})^{b} and \\hat{H}{(S,F_{c})} = \\frac{F_{c}}{S}, then obtain \\hat{H}{(S,F_{c})} - \\frac{A^{f^{*}}{(f^{*},b)}}{f^{*}} = \\frac{F_{c}}{S} - \\frac{A^{f^{*}}{(f^{*},b)}}{f^{*}}", "derivation": "A{(f^{*},b)} = (f^{*})^{b} and A^{f^{*}}{(f^{*},b)} = ((f^{*})^{b})^{f^{*}} and \\frac{A^{f^{*}}{(f^{*},b)}}{f^{*}} = \\frac{((f^{*})^{b})^{f^{*}}}{f^{*}} and \\hat{H}{(S,F_{c})} = \\frac{F_{c}}{S} and \\hat{H}{(S,F_{c})} - \\frac{((f^{*})^{b})^{f^{*}}}{f^{*}} = \\frac{F_{c}}{S} - \\frac{((f^{*})^{b})^{f^{*}}}{f^{*}} and \\hat{H}{(S,F_{c})} - \\frac{A^{f^{*}}{(f^{*},b)}}{f^{*}} = \\frac{F_{c}}{S} - \\frac{A^{f^{*}}{(f^{*},b)}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('A')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))"], [["divide", 2, "Symbol('f^*', commutative=True)"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))))"], [["minus", 4, "Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))), Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Pow(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))), Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A')(Symbol('f^*', commutative=True), Symbol('b', commutative=True)), Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(T)} = \\sin{(T)}, then obtain (\\frac{1}{2})^{T} = (\\frac{\\int \\frac{\\sin{(T)}}{T} dT}{2 \\int \\frac{\\mu_{0}{(T)}}{T} dT})^{T}", "derivation": "\\mu_{0}{(T)} = \\sin{(T)} and \\frac{\\mu_{0}{(T)}}{T} = \\frac{\\sin{(T)}}{T} and \\int \\frac{\\mu_{0}{(T)}}{T} dT = \\int \\frac{\\sin{(T)}}{T} dT and \\frac{1}{2} = \\frac{\\int \\frac{\\sin{(T)}}{T} dT}{2 \\int \\frac{\\mu_{0}{(T)}}{T} dT} and (\\frac{1}{2})^{T} = (\\frac{\\int \\frac{\\sin{(T)}}{T} dT}{2 \\int \\frac{\\mu_{0}{(T)}}{T} dT})^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["power", 4, "Symbol('T', commutative=True)"], "Equality(Pow(Rational(1, 2), Symbol('T', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(Z,\\phi)} = Z - \\phi, then derive \\int \\hat{H}_{\\lambda}{(Z,\\phi)} d\\phi = Z \\phi - \\frac{\\phi^{2}}{2} + \\varphi^*, then obtain \\int (Z - \\phi) d\\phi = Z \\phi - \\frac{\\phi^{2}}{2} + \\varphi^*", "derivation": "\\hat{H}_{\\lambda}{(Z,\\phi)} = Z - \\phi and \\int \\hat{H}_{\\lambda}{(Z,\\phi)} d\\phi = \\int (Z - \\phi) d\\phi and \\int \\hat{H}_{\\lambda}{(Z,\\phi)} d\\phi = Z \\phi - \\frac{\\phi^{2}}{2} + \\varphi^* and \\int (Z - \\phi) d\\phi = Z \\phi - \\frac{\\phi^{2}}{2} + \\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given t{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain \\int 0 d\\mathbf{E} = \\int (\\int (t{(\\mathbf{E})} + e^{\\mathbf{E}}) d\\mathbf{E} - \\int 2 t{(\\mathbf{E})} d\\mathbf{E}) d\\mathbf{E}", "derivation": "t{(\\mathbf{E})} = e^{\\mathbf{E}} and 2 t{(\\mathbf{E})} = t{(\\mathbf{E})} + e^{\\mathbf{E}} and \\int 2 t{(\\mathbf{E})} d\\mathbf{E} = \\int (t{(\\mathbf{E})} + e^{\\mathbf{E}}) d\\mathbf{E} and 0 = \\int (t{(\\mathbf{E})} + e^{\\mathbf{E}}) d\\mathbf{E} - \\int 2 t{(\\mathbf{E})} d\\mathbf{E} and \\int 0 d\\mathbf{E} = \\int (\\int (t{(\\mathbf{E})} + e^{\\mathbf{E}}) d\\mathbf{E} - \\int 2 t{(\\mathbf{E})} d\\mathbf{E}) d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Function('t')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 3, "Integral(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Integral(Add(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(f_{E})} = \\sin{(\\sin{(f_{E})})}, then obtain \\frac{\\frac{d}{d f_{E}} (\\varepsilon{(f_{E})} - \\sin{(\\sin{(f_{E})})})}{- \\lambda{(g_{\\varepsilon})} + \\log{(\\cos{(g_{\\varepsilon})})}} = \\frac{\\frac{d}{d f_{E}} 0}{- \\lambda{(g_{\\varepsilon})} + \\log{(\\cos{(g_{\\varepsilon})})}}", "derivation": "\\varepsilon{(f_{E})} = \\sin{(\\sin{(f_{E})})} and \\varepsilon{(f_{E})} - \\sin{(\\sin{(f_{E})})} = 0 and \\frac{d}{d f_{E}} (\\varepsilon{(f_{E})} - \\sin{(\\sin{(f_{E})})}) = \\frac{d}{d f_{E}} 0 and \\frac{\\frac{d}{d f_{E}} (\\varepsilon{(f_{E})} - \\sin{(\\sin{(f_{E})})})}{- \\lambda{(g_{\\varepsilon})} + \\log{(\\cos{(g_{\\varepsilon})})}} = \\frac{\\frac{d}{d f_{E}} 0}{- \\lambda{(g_{\\varepsilon})} + \\log{(\\cos{(g_{\\varepsilon})})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('f_E', commutative=True)), sin(sin(Symbol('f_E', commutative=True))))"], [["minus", 1, "sin(sin(Symbol('f_E', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('f_E', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varepsilon')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('g_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('g_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(-1)), Derivative(Add(Function('\\\\varepsilon')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('g_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(v_{x})} = \\cos{(v_{x})} and q{(v_{x})} = - \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})}, then derive \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} = - \\sin{(v_{x})}, then obtain 0 = \\sin{(v_{x})} + \\frac{d}{d v_{x}} \\cos{(v_{x})}", "derivation": "\\operatorname{v_{y}}{(v_{x})} = \\cos{(v_{x})} and \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} = \\frac{d}{d v_{x}} \\cos{(v_{x})} and 0 = - \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} + \\frac{d}{d v_{x}} \\cos{(v_{x})} and q{(v_{x})} = - \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} and \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} = - \\sin{(v_{x})} and q{(v_{x})} = \\sin{(v_{x})} and \\sin{(v_{x})} = - \\frac{d}{d v_{x}} \\operatorname{v_{y}}{(v_{x})} and 0 = \\sin{(v_{x})} + \\frac{d}{d v_{x}} \\cos{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('v_x', commutative=True)), Mul(Integer(-1), Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('q')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(sin(Symbol('v_x', commutative=True)), Mul(Integer(-1), Derivative(Function('v_y')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Integer(0), Add(sin(Symbol('v_x', commutative=True)), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(A_{2},\\mu)} = A_{2} + \\mu, then obtain \\Psi^{\\dagger}{(A_{2},\\mu)} \\int \\frac{\\partial}{\\partial \\mu} \\Psi^{\\dagger}{(A_{2},\\mu)} dA_{2} = (A_{2} + \\mathbf{S}) \\Psi^{\\dagger}{(A_{2},\\mu)}", "derivation": "\\Psi^{\\dagger}{(A_{2},\\mu)} = A_{2} + \\mu and \\frac{\\partial}{\\partial \\mu} \\Psi^{\\dagger}{(A_{2},\\mu)} = \\frac{\\partial}{\\partial \\mu} (A_{2} + \\mu) and \\int \\frac{\\partial}{\\partial \\mu} \\Psi^{\\dagger}{(A_{2},\\mu)} dA_{2} = \\int \\frac{\\partial}{\\partial \\mu} (A_{2} + \\mu) dA_{2} and \\Psi^{\\dagger}{(A_{2},\\mu)} \\int \\frac{\\partial}{\\partial \\mu} \\Psi^{\\dagger}{(A_{2},\\mu)} dA_{2} = \\Psi^{\\dagger}{(A_{2},\\mu)} \\int \\frac{\\partial}{\\partial \\mu} (A_{2} + \\mu) dA_{2} and \\Psi^{\\dagger}{(A_{2},\\mu)} \\int \\frac{\\partial}{\\partial \\mu} \\Psi^{\\dagger}{(A_{2},\\mu)} dA_{2} = (A_{2} + \\mathbf{S}) \\Psi^{\\dagger}{(A_{2},\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))), Integral(Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))))"], [["times", 3, "Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True)))), Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True)))), Mul(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given G{(H,\\mathbf{p})} = H \\mathbf{p}, then derive - H \\mathbf{p} + \\frac{\\partial}{\\partial H} G{(H,\\mathbf{p})} = - H \\mathbf{p} + \\mathbf{p}, then obtain \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\frac{\\partial}{\\partial H} H \\mathbf{p}) = \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\mathbf{p})", "derivation": "G{(H,\\mathbf{p})} = H \\mathbf{p} and \\frac{\\partial}{\\partial H} G{(H,\\mathbf{p})} = \\frac{\\partial}{\\partial H} H \\mathbf{p} and - H \\mathbf{p} + \\frac{\\partial}{\\partial H} G{(H,\\mathbf{p})} = - H \\mathbf{p} + \\frac{\\partial}{\\partial H} H \\mathbf{p} and - H \\mathbf{p} + \\frac{\\partial}{\\partial H} G{(H,\\mathbf{p})} = - H \\mathbf{p} + \\mathbf{p} and \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\frac{\\partial}{\\partial H} G{(H,\\mathbf{p})}) = \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\mathbf{p}) and \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\frac{\\partial}{\\partial H} H \\mathbf{p}) = \\frac{\\partial}{\\partial H} (- H \\mathbf{p} + \\mathbf{p})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('G')(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('G')(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 4, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('G')(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(L)} = \\cos{(L)} and \\rho{(L)} = L + \\cos{(L)}, then derive \\int \\rho{(L)} dL = \\frac{L^{2}}{2} + v_{1} + \\sin{(L)}, then obtain \\iint (L + \\dot{z}{(L)}) dL dv_{1} = \\int (\\frac{L^{2}}{2} + v_{1} + \\sin{(L)}) dv_{1}", "derivation": "\\dot{z}{(L)} = \\cos{(L)} and L + \\dot{z}{(L)} = L + \\cos{(L)} and \\rho{(L)} = L + \\cos{(L)} and \\int \\rho{(L)} dL = \\int (L + \\cos{(L)}) dL and \\rho{(L)} = L + \\dot{z}{(L)} and \\int \\rho{(L)} dL = \\frac{L^{2}}{2} + v_{1} + \\sin{(L)} and \\iint \\rho{(L)} dL dv_{1} = \\int (\\frac{L^{2}}{2} + v_{1} + \\sin{(L)}) dv_{1} and \\iint (L + \\dot{z}{(L)}) dL dv_{1} = \\int (\\frac{L^{2}}{2} + v_{1} + \\sin{(L)}) dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["add", 1, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('\\\\dot{z}')(Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\rho')(Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Function('\\\\dot{z}')(Symbol('L', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('\\\\rho')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('v_1', commutative=True), sin(Symbol('L', commutative=True))))"], [["integrate", 6, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('v_1', commutative=True), sin(Symbol('L', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integral(Add(Symbol('L', commutative=True), Function('\\\\dot{z}')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Symbol('v_1', commutative=True), sin(Symbol('L', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} = \\mathbf{J}_M^{\\sigma_x} \\sigma_x, then obtain (\\int \\frac{\\int \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} d\\mathbf{J}_M}{\\int \\mathbf{J}_M^{\\sigma_x} \\sigma_x d\\mathbf{J}_M} d\\sigma_x)^{\\mathbf{J}_M} = (\\int 1 d\\sigma_x)^{\\mathbf{J}_M}", "derivation": "\\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} = \\mathbf{J}_M^{\\sigma_x} \\sigma_x and \\int \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} d\\mathbf{J}_M = \\int \\mathbf{J}_M^{\\sigma_x} \\sigma_x d\\mathbf{J}_M and \\frac{\\int \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} d\\mathbf{J}_M}{\\int \\mathbf{J}_M^{\\sigma_x} \\sigma_x d\\mathbf{J}_M} = 1 and \\int \\frac{\\int \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} d\\mathbf{J}_M}{\\int \\mathbf{J}_M^{\\sigma_x} \\sigma_x d\\mathbf{J}_M} d\\sigma_x = \\int 1 d\\sigma_x and (\\int \\frac{\\int \\bar{\\h}{(\\mathbf{J}_M,\\sigma_x)} d\\mathbf{J}_M}{\\int \\mathbf{J}_M^{\\sigma_x} \\sigma_x d\\mathbf{J}_M} d\\sigma_x)^{\\mathbf{J}_M} = (\\int 1 d\\sigma_x)^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 2, "Integral(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1)), Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(1))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1)), Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1)), Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(U)} = \\cos{(\\sin{(U)})}, then obtain ((\\mathbf{F}{(U)} - 1)^{U} + 1) (\\mathbf{F}{(U)} - \\cos{(\\sin{(U)})}) = ((\\cos{(\\sin{(U)})} - 1)^{U} + 1) (\\mathbf{F}{(U)} - \\cos{(\\sin{(U)})})", "derivation": "\\mathbf{F}{(U)} = \\cos{(\\sin{(U)})} and \\mathbf{F}{(U)} - 1 = \\cos{(\\sin{(U)})} - 1 and (\\mathbf{F}{(U)} - 1)^{U} = (\\cos{(\\sin{(U)})} - 1)^{U} and (\\mathbf{F}{(U)} - 1)^{U} + 1 = (\\cos{(\\sin{(U)})} - 1)^{U} + 1 and ((\\mathbf{F}{(U)} - 1)^{U} + 1) (\\mathbf{F}{(U)} - \\cos{(\\sin{(U)})}) = ((\\cos{(\\sin{(U)})} - 1)^{U} + 1) (\\mathbf{F}{(U)} - \\cos{(\\sin{(U)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), cos(sin(Symbol('U', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Integer(-1)), Add(cos(sin(Symbol('U', commutative=True))), Integer(-1)))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Add(cos(sin(Symbol('U', commutative=True))), Integer(-1)), Symbol('U', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Integer(1)), Add(Pow(Add(cos(sin(Symbol('U', commutative=True))), Integer(-1)), Symbol('U', commutative=True)), Integer(1)))"], [["times", 4, "Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('U', commutative=True)))))"], "Equality(Mul(Add(Pow(Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('U', commutative=True)))))), Mul(Add(Pow(Add(cos(sin(Symbol('U', commutative=True))), Integer(-1)), Symbol('U', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{F}')(Symbol('U', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('U', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(c_{0},\\mathbf{r})} = \\frac{\\cos{(c_{0})}}{\\mathbf{r}}, then obtain \\log{(\\frac{\\mathbf{r}}{\\cos{(c_{0})}})} = \\log{(\\frac{1}{\\operatorname{n_{1}}{(c_{0},\\mathbf{r})}})}", "derivation": "\\operatorname{n_{1}}{(c_{0},\\mathbf{r})} = \\frac{\\cos{(c_{0})}}{\\mathbf{r}} and 1 = \\frac{\\cos{(c_{0})}}{\\mathbf{r} \\operatorname{n_{1}}{(c_{0},\\mathbf{r})}} and \\frac{\\mathbf{r}}{\\cos{(c_{0})}} = \\frac{1}{\\operatorname{n_{1}}{(c_{0},\\mathbf{r})}} and \\log{(\\frac{\\mathbf{r}}{\\cos{(c_{0})}})} = \\log{(\\frac{1}{\\operatorname{n_{1}}{(c_{0},\\mathbf{r})}})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), cos(Symbol('c_0', commutative=True))))"], [["divide", 1, "Function('n_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Pow(Function('n_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Symbol('c_0', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), cos(Symbol('c_0', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(cos(Symbol('c_0', commutative=True)), Integer(-1))), Pow(Function('n_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))"], [["log", 3], "Equality(log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(cos(Symbol('c_0', commutative=True)), Integer(-1)))), log(Pow(Function('n_1')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given h{(m,\\dot{y})} = - \\dot{y} + m and \\mathbf{r}{(\\dot{y})} = - \\dot{y}, then derive \\frac{\\partial}{\\partial m} h{(m,\\dot{y})} = 1, then obtain \\frac{\\partial}{\\partial m} (m + \\mathbf{r}{(\\dot{y})}) = 1", "derivation": "h{(m,\\dot{y})} = - \\dot{y} + m and \\frac{\\partial}{\\partial m} h{(m,\\dot{y})} = \\frac{\\partial}{\\partial m} (- \\dot{y} + m) and \\frac{\\partial}{\\partial m} h{(m,\\dot{y})} = 1 and \\mathbf{r}{(\\dot{y})} = - \\dot{y} and \\frac{\\partial}{\\partial m} (- \\dot{y} + m) = 1 and \\frac{\\partial}{\\partial m} (m + \\mathbf{r}{(\\dot{y})}) = 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('m', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('m', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('m', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})}, then derive \\frac{d}{d \\varepsilon_0} \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = - \\sin{(\\sin{(\\varepsilon_0)})} \\cos{(\\varepsilon_0)}, then obtain \\frac{d}{d \\varepsilon_0} \\cos{(\\sin{(\\varepsilon_0)})} = - \\sin{(\\sin{(\\varepsilon_0)})} \\cos{(\\varepsilon_0)}", "derivation": "\\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\cos{(\\sin{(\\varepsilon_0)})} and \\frac{d}{d \\varepsilon_0} \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\cos{(\\sin{(\\varepsilon_0)})} and \\frac{d}{d \\varepsilon_0} \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = - \\sin{(\\sin{(\\varepsilon_0)})} \\cos{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\cos{(\\sin{(\\varepsilon_0)})} = - \\sin{(\\sin{(\\varepsilon_0)})} \\cos{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), cos(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\varepsilon_0', commutative=True))), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(sin(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\varepsilon_0', commutative=True))), cos(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\theta,a,M)} = M a - \\theta, then obtain \\int a (a (M a - \\theta))^{M} (a \\mathbf{M}{(\\theta,a,M)})^{M} (M a - \\theta) dM = \\int a (a (M a - \\theta))^{2 M} (M a - \\theta) dM", "derivation": "\\mathbf{M}{(\\theta,a,M)} = M a - \\theta and a \\mathbf{M}{(\\theta,a,M)} = a (M a - \\theta) and (a \\mathbf{M}{(\\theta,a,M)})^{M} = (a (M a - \\theta))^{M} and a (a \\mathbf{M}{(\\theta,a,M)})^{M} (M a - \\theta) = a (a (M a - \\theta))^{M} (M a - \\theta) and a (a (M a - \\theta))^{M} (a \\mathbf{M}{(\\theta,a,M)})^{M} (M a - \\theta) = a (a (M a - \\theta))^{2 M} (M a - \\theta) and \\int a (a (M a - \\theta))^{M} (a \\mathbf{M}{(\\theta,a,M)})^{M} (M a - \\theta) dM = \\int a (a (M a - \\theta))^{2 M} (M a - \\theta) dM", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True)), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Symbol('a', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Symbol('M', commutative=True)))"], [["times", 3, "Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], "Equality(Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Symbol('M', commutative=True)), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["times", 4, "Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Symbol('M', commutative=True))"], "Equality(Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(Symbol('a', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Integer(2), Symbol('M', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["integrate", 5, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(Symbol('a', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True), Symbol('a', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Pow(Mul(Symbol('a', commutative=True), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Integer(2), Symbol('M', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given G{(\\omega)} = \\omega and f{(\\omega)} = G{(\\omega)} + \\sin{(e^{\\omega})}, then obtain (\\int \\frac{d}{d \\omega} f{(\\omega)} d\\omega)^{\\omega} = (\\int \\frac{d}{d \\omega} (\\omega + \\sin{(e^{\\omega})}) d\\omega)^{\\omega}", "derivation": "G{(\\omega)} = \\omega and f{(\\omega)} = G{(\\omega)} + \\sin{(e^{\\omega})} and \\frac{d}{d \\omega} f{(\\omega)} = \\frac{d}{d \\omega} (G{(\\omega)} + \\sin{(e^{\\omega})}) and \\int \\frac{d}{d \\omega} f{(\\omega)} d\\omega = \\int \\frac{d}{d \\omega} (G{(\\omega)} + \\sin{(e^{\\omega})}) d\\omega and \\int \\frac{d}{d \\omega} f{(\\omega)} d\\omega = \\int \\frac{d}{d \\omega} (\\omega + \\sin{(e^{\\omega})}) d\\omega and (\\int \\frac{d}{d \\omega} f{(\\omega)} d\\omega)^{\\omega} = (\\int \\frac{d}{d \\omega} (\\omega + \\sin{(e^{\\omega})}) d\\omega)^{\\omega}", "srepr_derivation": [["renaming_premise", "Equality(Function('G')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\omega', commutative=True)), Add(Function('G')(Symbol('\\\\omega', commutative=True)), sin(exp(Symbol('\\\\omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Function('G')(Symbol('\\\\omega', commutative=True)), sin(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Derivative(Function('f')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(Add(Function('G')(Symbol('\\\\omega', commutative=True)), sin(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Function('f')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('f')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integral(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} = \\frac{\\hat{x}}{W}, then obtain \\frac{\\partial}{\\partial W} \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} dW}{W} + 1 = \\frac{\\partial}{\\partial W} \\frac{\\int \\frac{\\hat{x}}{W} dW}{W} + 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} = \\frac{\\hat{x}}{W} and \\int \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} dW = \\int \\frac{\\hat{x}}{W} dW and \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} dW}{W} = \\frac{\\int \\frac{\\hat{x}}{W} dW}{W} and \\frac{\\partial}{\\partial W} \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} dW}{W} = \\frac{\\partial}{\\partial W} \\frac{\\int \\frac{\\hat{x}}{W} dW}{W} and \\frac{\\partial}{\\partial W} \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},W)} dW}{W} + 1 = \\frac{\\partial}{\\partial W} \\frac{\\int \\frac{\\hat{x}}{W} dW}{W} + 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["divide", 2, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(M_{E},a)} = \\frac{a}{M_{E}} and \\operatorname{F_{g}}{(M_{E},a)} = M_{E} + (\\frac{a}{M_{E}})^{a}, then obtain e^{M_{E} + \\operatorname{v_{2}}^{a}{(M_{E},a)}} = e^{\\operatorname{F_{g}}{(M_{E},a)}}", "derivation": "\\operatorname{v_{2}}{(M_{E},a)} = \\frac{a}{M_{E}} and \\operatorname{v_{2}}^{a}{(M_{E},a)} = (\\frac{a}{M_{E}})^{a} and M_{E} + \\operatorname{v_{2}}^{a}{(M_{E},a)} = M_{E} + (\\frac{a}{M_{E}})^{a} and \\operatorname{F_{g}}{(M_{E},a)} = M_{E} + (\\frac{a}{M_{E}})^{a} and M_{E} + \\operatorname{v_{2}}^{a}{(M_{E},a)} = \\operatorname{F_{g}}{(M_{E},a)} and e^{M_{E} + \\operatorname{v_{2}}^{a}{(M_{E},a)}} = e^{\\operatorname{F_{g}}{(M_{E},a)}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('a', commutative=True)))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], [["add", 2, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Pow(Function('v_2')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Add(Symbol('M_E', commutative=True), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Add(Symbol('M_E', commutative=True), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('M_E', commutative=True), Pow(Function('v_2')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Function('F_g')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)))"], [["exp", 5], "Equality(exp(Add(Symbol('M_E', commutative=True), Pow(Function('v_2')(Symbol('M_E', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))), exp(Function('F_g')(Symbol('M_E', commutative=True), Symbol('a', commutative=True))))"]]}, {"prompt": "Given f{(M)} = \\sin{(M)}, then derive \\frac{d}{d M} \\int f{(M)} dM = \\frac{\\partial}{\\partial M} (\\mu - \\cos{(M)}), then obtain \\frac{\\partial}{\\partial M} (\\tilde{g}^* - \\cos{(M)}) = \\frac{\\partial}{\\partial M} (\\mu - \\cos{(M)})", "derivation": "f{(M)} = \\sin{(M)} and \\int f{(M)} dM = \\int \\sin{(M)} dM and \\frac{d}{d M} \\int f{(M)} dM = \\frac{d}{d M} \\int \\sin{(M)} dM and \\frac{d}{d M} \\int f{(M)} dM = \\frac{\\partial}{\\partial M} (\\mu - \\cos{(M)}) and \\frac{d}{d M} \\int \\sin{(M)} dM = \\frac{\\partial}{\\partial M} (\\mu - \\cos{(M)}) and \\frac{\\partial}{\\partial M} (\\tilde{g}^* - \\cos{(M)}) = \\frac{\\partial}{\\partial M} (\\mu - \\cos{(M)})", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(v)} = \\int e^{v} dv and \\operatorname{c_{0}}{(v)} = e^{v} and \\varphi^{*}{(r)} = \\log{(\\log{(r)})}, then obtain (\\mathbf{F}{(v)} - \\operatorname{c_{0}}{(v)}) \\varphi^{*}{(r)} = (\\mathbf{F}{(v)} - \\operatorname{c_{0}}{(v)}) \\log{(\\log{(r)})}", "derivation": "\\mathbf{F}{(v)} = \\int e^{v} dv and \\mathbf{F}{(v)} - e^{v} = - e^{v} + \\int e^{v} dv and \\operatorname{c_{0}}{(v)} = e^{v} and \\varphi^{*}{(r)} = \\log{(\\log{(r)})} and \\mathbf{F}{(v)} - \\operatorname{c_{0}}{(v)} = - \\operatorname{c_{0}}{(v)} + \\int e^{v} dv and (- \\operatorname{c_{0}}{(v)} + \\int e^{v} dv) \\varphi^{*}{(r)} = (- \\operatorname{c_{0}}{(v)} + \\int e^{v} dv) \\log{(\\log{(r)})} and (\\mathbf{F}{(v)} - \\operatorname{c_{0}}{(v)}) \\varphi^{*}{(r)} = (\\mathbf{F}{(v)} - \\operatorname{c_{0}}{(v)}) \\log{(\\log{(r)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v', commutative=True)), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["minus", 1, "exp(Symbol('v', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('r', commutative=True)), log(log(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('v', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Function('\\\\varphi^*')(Symbol('r', commutative=True))), Mul(Add(Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), log(log(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Add(Function('\\\\mathbf{F}')(Symbol('v', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True)))), Function('\\\\varphi^*')(Symbol('r', commutative=True))), Mul(Add(Function('\\\\mathbf{F}')(Symbol('v', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('v', commutative=True)))), log(log(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given s{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain (s{(\\mathbb{I})} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} = (e^{\\mathbb{I}} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}", "derivation": "s{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} s{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and s{(\\mathbb{I})} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} = e^{\\mathbb{I}} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and (s{(\\mathbb{I})} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} s{(\\mathbb{I})} = (e^{\\mathbb{I}} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} s{(\\mathbb{I})} and (s{(\\mathbb{I})} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} = (e^{\\mathbb{I}} + \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}) \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Add(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Derivative(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(Add(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Derivative(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Function('s')(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(Add(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(f^{*})} = \\cos{(f^{*})}, then obtain (\\int \\mathbf{P}{(f^{*})} df^{*})^{f^{*}} = (m_{s} + \\sin{(f^{*})})^{f^{*}}", "derivation": "\\mathbf{P}{(f^{*})} = \\cos{(f^{*})} and \\int \\mathbf{P}{(f^{*})} df^{*} = \\int \\cos{(f^{*})} df^{*} and (\\int \\mathbf{P}{(f^{*})} df^{*})^{f^{*}} = (\\int \\cos{(f^{*})} df^{*})^{f^{*}} and (\\int \\mathbf{P}{(f^{*})} df^{*})^{f^{*}} = (m_{s} + \\sin{(f^{*})})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{P}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{P}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), sin(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given x{(\\rho,\\theta_1)} = \\rho + \\theta_1, then obtain (\\theta_1 + \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)})^{\\theta_1} = (\\theta_1 + 1)^{\\theta_1}", "derivation": "x{(\\rho,\\theta_1)} = \\rho + \\theta_1 and \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\rho + \\theta_1) and - \\rho + \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)} = - \\rho + \\frac{\\partial}{\\partial \\theta_1} (\\rho + \\theta_1) and \\theta_1 + \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)} = \\theta_1 + \\frac{\\partial}{\\partial \\theta_1} (\\rho + \\theta_1) and (\\theta_1 + \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)})^{\\theta_1} = (\\theta_1 + \\frac{\\partial}{\\partial \\theta_1} (\\rho + \\theta_1))^{\\theta_1} and (\\theta_1 + \\frac{\\partial}{\\partial \\theta_1} x{(\\rho,\\theta_1)})^{\\theta_1} = (\\theta_1 + 1)^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Derivative(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["add", 3, "Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Derivative(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Symbol('\\\\theta_1', commutative=True), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta_1', commutative=True), Derivative(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Symbol('\\\\theta_1', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Symbol('\\\\theta_1', commutative=True), Derivative(Function('x')(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Integer(1)), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} = \\log{(\\ddot{x})}, then obtain \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} + \\int \\log{(\\ddot{x})} d\\ddot{x} = \\log{(\\ddot{x})} + \\int \\log{(\\ddot{x})} d\\ddot{x}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} = \\log{(\\ddot{x})} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} d\\ddot{x} = \\int \\log{(\\ddot{x})} d\\ddot{x} and \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} + \\int \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} d\\ddot{x} = \\log{(\\ddot{x})} + \\int \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} d\\ddot{x} and \\operatorname{V_{\\mathbf{B}}}{(\\ddot{x})} + \\int \\log{(\\ddot{x})} d\\ddot{x} = \\log{(\\ddot{x})} + \\int \\log{(\\ddot{x})} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 1, "Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(log(Symbol('\\\\ddot{x}', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\ddot{x}', commutative=True)), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(log(Symbol('\\\\ddot{x}', commutative=True)), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(A)} = \\cos{(e^{A})}, then obtain - \\frac{d}{d A} \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} + \\int \\operatorname{x^{{\\}'}}{(A)} dA = - \\frac{d}{d A} \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} + \\int \\cos{(e^{A})} dA", "derivation": "\\operatorname{x^{{\\}'}}{(A)} = \\cos{(e^{A})} and 1 = \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} and \\frac{d}{d A} 1 = \\frac{d}{d A} \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} and \\int \\operatorname{x^{{\\}'}}{(A)} dA = \\int \\cos{(e^{A})} dA and - \\frac{d}{d A} 1 + \\int \\operatorname{x^{{\\}'}}{(A)} dA = - \\frac{d}{d A} 1 + \\int \\cos{(e^{A})} dA and - \\frac{d}{d A} \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} + \\int \\operatorname{x^{{\\}'}}{(A)} dA = - \\frac{d}{d A} \\frac{\\cos{(e^{A})}}{\\operatorname{x^{{\\}'}}{(A)}} + \\int \\cos{(e^{A})} dA", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('A', commutative=True)), cos(exp(Symbol('A', commutative=True))))"], [["divide", 1, "Function('x^\\\\prime')(Symbol('A', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('A', commutative=True)), Integer(-1)), cos(exp(Symbol('A', commutative=True)))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('x^\\\\prime')(Symbol('A', commutative=True)), Integer(-1)), cos(exp(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["minus", 4, "Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1)))), Integral(Function('x^\\\\prime')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1)))), Integral(cos(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Function('x^\\\\prime')(Symbol('A', commutative=True)), Integer(-1)), cos(exp(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)))), Integral(Function('x^\\\\prime')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Derivative(Mul(Pow(Function('x^\\\\prime')(Symbol('A', commutative=True)), Integer(-1)), cos(exp(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)))), Integral(cos(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given u{(\\varepsilon_0,A_{x})} = A_{x} \\varepsilon_0, then obtain -1 = - A_{x} + \\frac{\\partial}{\\partial \\varepsilon_0} u{(\\varepsilon_0,A_{x})} - 1", "derivation": "u{(\\varepsilon_0,A_{x})} = A_{x} \\varepsilon_0 and 0 = A_{x} \\varepsilon_0 - u{(\\varepsilon_0,A_{x})} and \\frac{d}{d \\varepsilon_0} 0 = \\frac{\\partial}{\\partial \\varepsilon_0} (A_{x} \\varepsilon_0 - u{(\\varepsilon_0,A_{x})}) and - \\frac{d}{d \\varepsilon_0} 0 = - \\frac{\\partial}{\\partial \\varepsilon_0} (A_{x} \\varepsilon_0 - u{(\\varepsilon_0,A_{x})}) and - \\frac{d}{d \\varepsilon_0} 0 - 1 = - \\frac{\\partial}{\\partial \\varepsilon_0} (A_{x} \\varepsilon_0 - u{(\\varepsilon_0,A_{x})}) - 1 and -1 = - A_{x} + \\frac{\\partial}{\\partial \\varepsilon_0} u{(\\varepsilon_0,A_{x})} - 1", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Add(Mul(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Integer(-1), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Derivative(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given x{(\\psi^*)} = e^{\\psi^*}, then obtain - x{(\\psi^*)} e^{\\psi^*} + x{(\\psi^*)} = - x{(\\psi^*)} e^{\\psi^*} + e^{\\psi^*}", "derivation": "x{(\\psi^*)} = e^{\\psi^*} and x^{2}{(\\psi^*)} = x{(\\psi^*)} e^{\\psi^*} and - x^{2}{(\\psi^*)} + x{(\\psi^*)} = - x^{2}{(\\psi^*)} + e^{\\psi^*} and - x{(\\psi^*)} e^{\\psi^*} + x{(\\psi^*)} = - x{(\\psi^*)} e^{\\psi^*} + e^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Function('x')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Pow(Function('x')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Function('x')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 1, "Pow(Function('x')(Symbol('\\\\psi^*', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Function('x')(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\psi^*', commutative=True)), Integer(2))), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), Function('x')(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Function('x')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given S{(P_{g},\\mathbf{B})} = P_{g} + \\sin{(\\mathbf{B})}, then obtain \\int (2 P_{g} + S{(P_{g},\\mathbf{B})} + 2 \\sin{(\\mathbf{B})}) d\\mathbf{B} = \\int (P_{g} + 2 S{(P_{g},\\mathbf{B})} + \\sin{(\\mathbf{B})}) d\\mathbf{B}", "derivation": "S{(P_{g},\\mathbf{B})} = P_{g} + \\sin{(\\mathbf{B})} and 2 S{(P_{g},\\mathbf{B})} = P_{g} + S{(P_{g},\\mathbf{B})} + \\sin{(\\mathbf{B})} and 3 S{(P_{g},\\mathbf{B})} = P_{g} + 2 S{(P_{g},\\mathbf{B})} + \\sin{(\\mathbf{B})} and 3 S{(P_{g},\\mathbf{B})} = 2 P_{g} + S{(P_{g},\\mathbf{B})} + 2 \\sin{(\\mathbf{B})} and 2 P_{g} + S{(P_{g},\\mathbf{B})} + 2 \\sin{(\\mathbf{B})} = P_{g} + 2 S{(P_{g},\\mathbf{B})} + \\sin{(\\mathbf{B})} and \\int (2 P_{g} + S{(P_{g},\\mathbf{B})} + 2 \\sin{(\\mathbf{B})}) d\\mathbf{B} = \\int (P_{g} + 2 S{(P_{g},\\mathbf{B})} + \\sin{(\\mathbf{B})}) d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('P_g', commutative=True), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 1, "Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('P_g', commutative=True), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 2, "Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(3), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('P_g', commutative=True), Mul(Integer(2), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(2), Symbol('P_g', commutative=True)), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('P_g', commutative=True)), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Symbol('P_g', commutative=True), Mul(Integer(2), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('P_g', commutative=True)), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Mul(Integer(2), Function('S')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given L{(f_{E})} = \\cos{(f_{E})}, then derive - \\int L{(f_{E})} df_{E} = - \\rho - \\sin{(f_{E})}, then obtain - \\int \\cos{(f_{E})} df_{E} = - \\rho - \\sin{(f_{E})}", "derivation": "L{(f_{E})} = \\cos{(f_{E})} and \\int L{(f_{E})} df_{E} = \\int \\cos{(f_{E})} df_{E} and - \\int L{(f_{E})} df_{E} = - \\int \\cos{(f_{E})} df_{E} and - \\int L{(f_{E})} df_{E} = - \\rho - \\sin{(f_{E})} and - \\int \\cos{(f_{E})} df_{E} = - \\rho - \\sin{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('L')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('L')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Mul(Integer(-1), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('L')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(V)} = e^{V}, then derive - e^{V} = - \\frac{d}{d V} \\hat{\\mathbf{x}}{(V)}, then obtain \\hat{\\mathbf{x}}{(V)} \\frac{d}{d V} - \\frac{d}{d V} \\hat{\\mathbf{x}}{(V)} = \\hat{\\mathbf{x}}{(V)} \\frac{d}{d V} - \\hat{\\mathbf{x}}{(V)}", "derivation": "\\hat{\\mathbf{x}}{(V)} = e^{V} and 0 = - \\hat{\\mathbf{x}}{(V)} + e^{V} and - e^{V} = - \\hat{\\mathbf{x}}{(V)} and \\frac{d}{d V} - e^{V} = \\frac{d}{d V} - \\hat{\\mathbf{x}}{(V)} and - e^{V} = - \\frac{d}{d V} \\hat{\\mathbf{x}}{(V)} and \\frac{d}{d V} - \\frac{d}{d V} \\hat{\\mathbf{x}}{(V)} = \\frac{d}{d V} - \\hat{\\mathbf{x}}{(V)} and \\hat{\\mathbf{x}}{(V)} \\frac{d}{d V} - \\frac{d}{d V} \\hat{\\mathbf{x}}{(V)} = \\hat{\\mathbf{x}}{(V)} \\frac{d}{d V} - \\hat{\\mathbf{x}}{(V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["minus", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))), exp(Symbol('V', commutative=True))))"], [["minus", 2, "exp(Symbol('V', commutative=True))"], "Equality(Mul(Integer(-1), exp(Symbol('V', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), exp(Symbol('V', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Mul(Integer(-1), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["times", 6, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), Derivative(Mul(Integer(-1), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True)), Derivative(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(v_{z})} = e^{v_{z}}, then derive \\frac{d}{d v_{z}} \\rho_{b}{(v_{z})} = e^{v_{z}}, then obtain \\frac{d^{2}}{d v_{z}^{2}} e^{v_{z}} = \\frac{d}{d v_{z}} e^{v_{z}}", "derivation": "\\rho_{b}{(v_{z})} = e^{v_{z}} and \\frac{d}{d v_{z}} \\rho_{b}{(v_{z})} = \\frac{d}{d v_{z}} e^{v_{z}} and \\frac{d}{d v_{z}} \\rho_{b}{(v_{z})} = e^{v_{z}} and \\frac{d}{d v_{z}} e^{v_{z}} = e^{v_{z}} and \\rho_{b}{(v_{z})} = \\frac{d}{d v_{z}} e^{v_{z}} and \\frac{d^{2}}{d v_{z}^{2}} e^{v_{z}} = \\frac{d}{d v_{z}} e^{v_{z}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_b')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), exp(Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), exp(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\rho_b')(Symbol('v_z', commutative=True)), Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2))), Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then obtain 0 = (- g_{\\varepsilon} T{(g_{\\varepsilon})} + g_{\\varepsilon} \\log{(g_{\\varepsilon})}) (g_{\\varepsilon} \\log{(g_{\\varepsilon})} - \\log{(g_{\\varepsilon})})", "derivation": "T{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and g_{\\varepsilon} T{(g_{\\varepsilon})} = g_{\\varepsilon} \\log{(g_{\\varepsilon})} and 0 = - g_{\\varepsilon} T{(g_{\\varepsilon})} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} and g_{\\varepsilon} T{(g_{\\varepsilon})} - \\log{(g_{\\varepsilon})} = g_{\\varepsilon} \\log{(g_{\\varepsilon})} - \\log{(g_{\\varepsilon})} and 0 = (- g_{\\varepsilon} T{(g_{\\varepsilon})} + g_{\\varepsilon} \\log{(g_{\\varepsilon})}) (g_{\\varepsilon} T{(g_{\\varepsilon})} - \\log{(g_{\\varepsilon})}) and 0 = (- g_{\\varepsilon} T{(g_{\\varepsilon})} + g_{\\varepsilon} \\log{(g_{\\varepsilon})}) (g_{\\varepsilon} \\log{(g_{\\varepsilon})} - \\log{(g_{\\varepsilon})})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["minus", 2, "log(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["times", 3, "Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('T')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(\\varphi^*,s)} = \\log{((\\varphi^*)^{s})}, then obtain \\hat{p}^{s}{(\\varphi^*,s)} \\frac{\\partial}{\\partial s} \\hat{p}{(\\varphi^*,s)} = \\hat{p}^{s}{(\\varphi^*,s)} \\log{(\\varphi^*)}", "derivation": "\\hat{p}{(\\varphi^*,s)} = \\log{((\\varphi^*)^{s})} and \\frac{\\partial}{\\partial s} \\hat{p}{(\\varphi^*,s)} = \\frac{\\partial}{\\partial s} \\log{((\\varphi^*)^{s})} and \\hat{p}^{s}{(\\varphi^*,s)} \\frac{\\partial}{\\partial s} \\hat{p}{(\\varphi^*,s)} = \\hat{p}^{s}{(\\varphi^*,s)} \\frac{\\partial}{\\partial s} \\log{((\\varphi^*)^{s})} and \\hat{p}^{s}{(\\varphi^*,s)} \\frac{\\partial}{\\partial s} \\hat{p}{(\\varphi^*,s)} = \\hat{p}^{s}{(\\varphi^*,s)} \\log{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 2, "Pow(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Derivative(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{E})} = e^{\\cos{(\\mathbf{E})}}, then obtain \\operatorname{E_{\\lambda}}^{2}{(\\mathbf{E})} e^{\\cos{(\\mathbf{E})}} = \\operatorname{E_{\\lambda}}{(\\mathbf{E})} e^{2 \\cos{(\\mathbf{E})}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{E})} = e^{\\cos{(\\mathbf{E})}} and \\operatorname{E_{\\lambda}}{(\\mathbf{E})} e^{\\cos{(\\mathbf{E})}} = e^{2 \\cos{(\\mathbf{E})}} and \\operatorname{E_{\\lambda}}{(\\mathbf{E})} e^{2 \\cos{(\\mathbf{E})}} = e^{3 \\cos{(\\mathbf{E})}} and \\operatorname{E_{\\lambda}}^{2}{(\\mathbf{E})} e^{\\cos{(\\mathbf{E})}} = e^{3 \\cos{(\\mathbf{E})}} and \\operatorname{E_{\\lambda}}^{2}{(\\mathbf{E})} e^{\\cos{(\\mathbf{E})}} = \\operatorname{E_{\\lambda}}{(\\mathbf{E})} e^{2 \\cos{(\\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["times", 1, "exp(cos(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(cos(Symbol('\\\\mathbf{E}', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["times", 2, "exp(cos(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(2), cos(Symbol('\\\\mathbf{E}', commutative=True))))), exp(Mul(Integer(3), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), exp(cos(Symbol('\\\\mathbf{E}', commutative=True)))), exp(Mul(Integer(3), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), exp(cos(Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(2), cos(Symbol('\\\\mathbf{E}', commutative=True))))))"]]}, {"prompt": "Given L{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then derive (\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})^{\\mathbf{v}} = \\cos^{\\mathbf{v}}{(\\mathbf{v})}, then obtain \\sin{((\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})^{\\mathbf{v}})} = \\sin{(\\cos^{\\mathbf{v}}{(\\mathbf{v})})}", "derivation": "L{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\sin{(\\mathbf{v})} and (\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})^{\\mathbf{v}} = (\\frac{d}{d \\mathbf{v}} \\sin{(\\mathbf{v})})^{\\mathbf{v}} and (\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})^{\\mathbf{v}} = \\cos^{\\mathbf{v}}{(\\mathbf{v})} and \\sin{((\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})^{\\mathbf{v}})} = \\sin{(\\cos^{\\mathbf{v}}{(\\mathbf{v})})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True))), sin(Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given A{(g)} = \\sin{(g)} and \\operatorname{f^{\\prime}}{(g)} = e^{\\frac{A^{2}{(g)}}{\\int A^{2}{(g)} dg}}, then obtain \\operatorname{f^{\\prime}}{(g)} = e^{\\frac{A{(g)} \\sin{(g)}}{\\int A^{2}{(g)} dg}}", "derivation": "A{(g)} = \\sin{(g)} and A^{2}{(g)} = A{(g)} \\sin{(g)} and \\frac{A^{2}{(g)}}{\\int A^{2}{(g)} dg} = \\frac{A{(g)} \\sin{(g)}}{\\int A^{2}{(g)} dg} and e^{\\frac{A^{2}{(g)}}{\\int A^{2}{(g)} dg}} = e^{\\frac{A{(g)} \\sin{(g)}}{\\int A^{2}{(g)} dg}} and \\operatorname{f^{\\prime}}{(g)} = e^{\\frac{A^{2}{(g)}}{\\int A^{2}{(g)} dg}} and \\operatorname{f^{\\prime}}{(g)} = e^{\\frac{A{(g)} \\sin{(g)}}{\\int A^{2}{(g)} dg}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["times", 1, "Function('A')(Symbol('g', commutative=True))"], "Equality(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True))))"], [["divide", 2, "Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Function('A')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1))))"], [["exp", 3], "Equality(exp(Mul(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1)))), exp(Mul(Function('A')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1)))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('g', commutative=True)), exp(Mul(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('f^{\\\\prime}')(Symbol('g', commutative=True)), exp(Mul(Function('A')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)), Pow(Integral(Pow(Function('A')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given U{(\\lambda)} = e^{\\lambda}, then obtain U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2} + e^{\\lambda} (\\int U{(\\lambda)} d\\lambda) \\int e^{\\lambda} d\\lambda = 2 U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2}", "derivation": "U{(\\lambda)} = e^{\\lambda} and \\int U{(\\lambda)} d\\lambda = \\int e^{\\lambda} d\\lambda and U{(\\lambda)} \\int U{(\\lambda)} d\\lambda = e^{\\lambda} \\int U{(\\lambda)} d\\lambda and U{(\\lambda)} (\\int U{(\\lambda)} d\\lambda) \\int e^{\\lambda} d\\lambda = U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2} and U{(\\lambda)} (\\int U{(\\lambda)} d\\lambda) \\int e^{\\lambda} d\\lambda + U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2} = 2 U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2} and U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2} + e^{\\lambda} (\\int U{(\\lambda)} d\\lambda) \\int e^{\\lambda} d\\lambda = 2 U{(\\lambda)} (\\int e^{\\lambda} d\\lambda)^{2}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(exp(Symbol('\\\\lambda', commutative=True)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["times", 2, "Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], "Equality(Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2))))"], [["add", 4, "Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2)))"], "Equality(Add(Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2)))), Mul(Integer(2), Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2))), Mul(exp(Symbol('\\\\lambda', commutative=True)), Integral(Function('U')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Mul(Integer(2), Function('U')(Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(x,p)} = \\frac{p}{x} and m{(x,p)} = \\int \\operatorname{F_{x}}^{x}{(x,p)} dp, then obtain \\frac{\\partial}{\\partial p} \\int \\operatorname{F_{x}}^{x}{(x,p)} dp = \\frac{\\partial}{\\partial p} m{(x,p)}", "derivation": "\\operatorname{F_{x}}{(x,p)} = \\frac{p}{x} and \\operatorname{F_{x}}^{x}{(x,p)} = (\\frac{p}{x})^{x} and \\int \\operatorname{F_{x}}^{x}{(x,p)} dp = \\int (\\frac{p}{x})^{x} dp and \\frac{\\partial}{\\partial p} \\int \\operatorname{F_{x}}^{x}{(x,p)} dp = \\frac{\\partial}{\\partial p} \\int (\\frac{p}{x})^{x} dp and m{(x,p)} = \\int \\operatorname{F_{x}}^{x}{(x,p)} dp and m{(x,p)} = \\int (\\frac{p}{x})^{x} dp and \\frac{\\partial}{\\partial p} \\int \\operatorname{F_{x}}^{x}{(x,p)} dp = \\frac{\\partial}{\\partial p} m{(x,p)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Symbol('x', commutative=True)), Pow(Mul(Symbol('p', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True)))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Pow(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Mul(Symbol('p', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Symbol('p', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('m')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('m')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Mul(Symbol('p', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Derivative(Integral(Pow(Function('F_x')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Function('m')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(u)} = e^{e^{u}}, then derive (\\frac{u \\frac{d}{d u} \\operatorname{V_{\\mathbf{E}}}{(u)}}{\\operatorname{V_{\\mathbf{E}}}{(u)}} + \\log{(\\operatorname{V_{\\mathbf{E}}}{(u)})}) \\operatorname{V_{\\mathbf{E}}}^{u}{(u)} = (u e^{u} + \\log{(e^{e^{u}})}) (e^{e^{u}})^{u}, then obtain (\\frac{u \\frac{d}{d u} \\operatorname{V_{\\mathbf{E}}}{(u)}}{\\operatorname{V_{\\mathbf{E}}}{(u)}} + \\log{(\\operatorname{V_{\\mathbf{E}}}{(u)})}) (e^{e^{u}})^{u} = (u e^{u} + \\log{(e^{e^{u}})}) (e^{e^{u}})^{u}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(u)} = e^{e^{u}} and \\operatorname{V_{\\mathbf{E}}}^{u}{(u)} = (e^{e^{u}})^{u} and \\frac{d}{d u} \\operatorname{V_{\\mathbf{E}}}^{u}{(u)} = \\frac{d}{d u} (e^{e^{u}})^{u} and (\\frac{u \\frac{d}{d u} \\operatorname{V_{\\mathbf{E}}}{(u)}}{\\operatorname{V_{\\mathbf{E}}}{(u)}} + \\log{(\\operatorname{V_{\\mathbf{E}}}{(u)})}) \\operatorname{V_{\\mathbf{E}}}^{u}{(u)} = (u e^{u} + \\log{(e^{e^{u}})}) (e^{e^{u}})^{u} and (\\frac{u \\frac{d}{d u} \\operatorname{V_{\\mathbf{E}}}{(u)}}{\\operatorname{V_{\\mathbf{E}}}{(u)}} + \\log{(\\operatorname{V_{\\mathbf{E}}}{(u)})}) (e^{e^{u}})^{u} = (u e^{u} + \\log{(e^{e^{u}})}) (e^{e^{u}})^{u}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), exp(exp(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(exp(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(exp(exp(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('u', commutative=True), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Integer(-1)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), log(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)))), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Add(Mul(Symbol('u', commutative=True), exp(Symbol('u', commutative=True))), log(exp(exp(Symbol('u', commutative=True))))), Pow(exp(exp(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('u', commutative=True), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Integer(-1)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), log(Function('V_{\\\\mathbf{E}}')(Symbol('u', commutative=True)))), Pow(exp(exp(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Add(Mul(Symbol('u', commutative=True), exp(Symbol('u', commutative=True))), log(exp(exp(Symbol('u', commutative=True))))), Pow(exp(exp(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"]]}, {"prompt": "Given r{(\\mathbf{H})} = \\cos{(\\cos{(\\mathbf{H})})}, then obtain \\sin{(\\frac{d^{2}}{d \\mathbf{H}^{2}} r{(\\mathbf{H})})} = \\sin{(\\frac{d^{2}}{d \\mathbf{H}^{2}} \\cos{(\\cos{(\\mathbf{H})})})}", "derivation": "r{(\\mathbf{H})} = \\cos{(\\cos{(\\mathbf{H})})} and \\frac{d}{d \\mathbf{H}} r{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\cos{(\\cos{(\\mathbf{H})})} and \\frac{d^{2}}{d \\mathbf{H}^{2}} r{(\\mathbf{H})} = \\frac{d^{2}}{d \\mathbf{H}^{2}} \\cos{(\\cos{(\\mathbf{H})})} and \\sin{(\\frac{d^{2}}{d \\mathbf{H}^{2}} r{(\\mathbf{H})})} = \\sin{(\\frac{d^{2}}{d \\mathbf{H}^{2}} \\cos{(\\cos{(\\mathbf{H})})})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{H}', commutative=True)), cos(cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(cos(cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["sin", 3], "Equality(sin(Derivative(Function('r')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))), sin(Derivative(cos(cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given l{(\\mathbf{s},b)} = \\frac{b}{\\mathbf{s}}, then obtain \\frac{\\frac{\\partial}{\\partial b} \\int l{(\\mathbf{s},b)} d\\mathbf{s}}{\\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}} = \\frac{\\frac{\\partial}{\\partial b} \\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}}{\\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}}", "derivation": "l{(\\mathbf{s},b)} = \\frac{b}{\\mathbf{s}} and \\int l{(\\mathbf{s},b)} d\\mathbf{s} = \\int \\frac{b}{\\mathbf{s}} d\\mathbf{s} and \\frac{\\partial}{\\partial b} \\int l{(\\mathbf{s},b)} d\\mathbf{s} = \\frac{\\partial}{\\partial b} \\int \\frac{b}{\\mathbf{s}} d\\mathbf{s} and \\frac{\\frac{\\partial}{\\partial b} \\int l{(\\mathbf{s},b)} d\\mathbf{s}}{\\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}} = \\frac{\\frac{\\partial}{\\partial b} \\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}}{\\int \\frac{b}{\\mathbf{s}} d\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('b', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Function('l')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('l')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Mul(Derivative(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\phi_1,n_{2})} = n_{2}^{\\phi_1} and \\ddot{x}{(\\phi_1,n_{2})} = \\frac{\\mathbf{r}{(\\phi_1,n_{2})}}{n_{2}}, then obtain \\ddot{x}{(\\phi_1,n_{2})} = \\frac{n_{2}^{\\phi_1}}{n_{2}}", "derivation": "\\mathbf{r}{(\\phi_1,n_{2})} = n_{2}^{\\phi_1} and \\frac{\\mathbf{r}{(\\phi_1,n_{2})}}{n_{2}} = \\frac{n_{2}^{\\phi_1}}{n_{2}} and \\ddot{x}{(\\phi_1,n_{2})} = \\frac{\\mathbf{r}{(\\phi_1,n_{2})}}{n_{2}} and \\ddot{x}{(\\phi_1,n_{2})} = \\frac{n_{2}^{\\phi_1}}{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\Psi)} = \\frac{d}{d \\Psi} \\cos{(\\Psi)}, then derive \\hat{H}_l{(\\Psi)} = - \\sin{(\\Psi)}, then obtain \\log{(- \\frac{\\hat{H}_l{(\\Psi)}}{\\sin{(\\Psi)}})} = 0", "derivation": "\\hat{H}_l{(\\Psi)} = \\frac{d}{d \\Psi} \\cos{(\\Psi)} and \\hat{H}_l{(\\Psi)} = - \\sin{(\\Psi)} and \\frac{\\hat{H}_l{(\\Psi)}}{\\frac{d}{d \\Psi} \\cos{(\\Psi)}} = - \\frac{\\sin{(\\Psi)}}{\\frac{d}{d \\Psi} \\cos{(\\Psi)}} and - \\sin{(\\Psi)} = \\frac{d}{d \\Psi} \\cos{(\\Psi)} and - \\frac{\\hat{H}_l{(\\Psi)}}{\\sin{(\\Psi)}} = 1 and \\log{(- \\frac{\\hat{H}_l{(\\Psi)}}{\\sin{(\\Psi)}})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\Psi', commutative=True)), Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Derivative(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Integer(1))"], [["log", 5], "Equality(log(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\Omega{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime}, then derive \\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)} = \\varphi, then obtain \\int \\Omega{(f^{\\prime},\\varphi)} d\\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)} = \\int \\varphi f^{\\prime} d\\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)}", "derivation": "\\Omega{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime} and \\int \\Omega{(f^{\\prime},\\varphi)} d\\varphi = \\int \\varphi f^{\\prime} d\\varphi and \\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)} = \\frac{\\partial}{\\partial f^{\\prime}} \\varphi f^{\\prime} and \\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)} = \\varphi and \\int \\Omega{(f^{\\prime},\\varphi)} d\\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)} = \\int \\varphi f^{\\prime} d\\frac{\\partial}{\\partial f^{\\prime}} \\Omega{(f^{\\prime},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Derivative(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))), Integral(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Derivative(Function('\\\\Omega')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\phi{(f^{\\prime})} = \\log{(f^{\\prime})} and \\mathbf{H}{(f^{\\prime})} = \\log{(f^{\\prime})}, then derive \\int \\phi{(f^{\\prime})} df^{\\prime} = \\mathbb{I} + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}, then obtain \\iint \\mathbf{H}{(f^{\\prime})} df^{\\prime} df^{\\prime} = \\int (\\mathbb{I} + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) df^{\\prime}", "derivation": "\\phi{(f^{\\prime})} = \\log{(f^{\\prime})} and \\int \\phi{(f^{\\prime})} df^{\\prime} = \\int \\log{(f^{\\prime})} df^{\\prime} and \\mathbf{H}{(f^{\\prime})} = \\log{(f^{\\prime})} and \\mathbf{H}{(f^{\\prime})} = \\phi{(f^{\\prime})} and \\int \\phi{(f^{\\prime})} df^{\\prime} = \\mathbb{I} + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} and \\iint \\phi{(f^{\\prime})} df^{\\prime} df^{\\prime} = \\int (\\mathbb{I} + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) df^{\\prime} and \\iint \\mathbf{H}{(f^{\\prime})} df^{\\prime} df^{\\prime} = \\int (\\mathbb{I} + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{H}')(Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{J})} = \\int e^{\\mathbf{J}} d\\mathbf{J}, then obtain \\frac{d}{d \\mathbf{J}} (\\int \\operatorname{v_{2}}{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}} = \\frac{d}{d \\mathbf{J}} (\\iint e^{\\mathbf{J}} d\\mathbf{J} d\\mathbf{J})^{\\mathbf{J}}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{J})} = \\int e^{\\mathbf{J}} d\\mathbf{J} and \\int \\operatorname{v_{2}}{(\\mathbf{J})} d\\mathbf{J} = \\iint e^{\\mathbf{J}} d\\mathbf{J} d\\mathbf{J} and (\\int \\operatorname{v_{2}}{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}} = (\\iint e^{\\mathbf{J}} d\\mathbf{J} d\\mathbf{J})^{\\mathbf{J}} and \\frac{d}{d \\mathbf{J}} (\\int \\operatorname{v_{2}}{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}} = \\frac{d}{d \\mathbf{J}} (\\iint e^{\\mathbf{J}} d\\mathbf{J} d\\mathbf{J})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Integral(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(g,C_{d})} = C_{d} + g, then obtain 4 (C_{d} + g)^{2} \\Psi_{\\lambda}{(g,C_{d})} = 2 (C_{d} + g)^{2} (2 C_{d} + 2 g)", "derivation": "\\Psi_{\\lambda}{(g,C_{d})} = C_{d} + g and C_{d} + g + \\Psi_{\\lambda}{(g,C_{d})} = 2 C_{d} + 2 g and 2 \\Psi_{\\lambda}{(g,C_{d})} = 2 C_{d} + 2 g and 2 (C_{d} + g) \\Psi_{\\lambda}{(g,C_{d})} = (C_{d} + g) (2 C_{d} + 2 g) and 2 (C_{d} + g) (C_{d} + g + \\Psi_{\\lambda}{(g,C_{d})}) \\Psi_{\\lambda}{(g,C_{d})} = (C_{d} + g) (2 C_{d} + 2 g) (C_{d} + g + \\Psi_{\\lambda}{(g,C_{d})}) and 2 (C_{d} + g)^{2} = (C_{d} + g) (2 C_{d} + 2 g) and 2 (C_{d} + g) (2 C_{d} + 2 g) \\Psi_{\\lambda}{(g,C_{d})} = (C_{d} + g) (2 C_{d} + 2 g)^{2} and 4 (C_{d} + g)^{2} \\Psi_{\\lambda}{(g,C_{d})} = 2 (C_{d} + g)^{2} (2 C_{d} + 2 g)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)))"], [["add", 1, "Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True))))"], [["times", 3, "Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Integer(2), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Mul(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)))))"], [["times", 4, "Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Mul(Integer(2), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Mul(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True))), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Pow(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Integer(2))), Mul(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(2), Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True))), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Mul(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Integer(4), Pow(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Integer(2)), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('C_d', commutative=True), Symbol('g', commutative=True)), Integer(2)), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\theta_1)} = e^{\\theta_1}, then obtain \\varepsilon_{0}^{2}{(\\theta_1)} e^{2 \\theta_1} - \\varepsilon_{0}{(\\theta_1)} e^{\\theta_1} = - \\varepsilon_{0}{(\\theta_1)} e^{\\theta_1} + e^{4 \\theta_1}", "derivation": "\\varepsilon_{0}{(\\theta_1)} = e^{\\theta_1} and \\varepsilon_{0}{(\\theta_1)} e^{\\theta_1} = e^{2 \\theta_1} and \\varepsilon_{0}^{2}{(\\theta_1)} e^{2 \\theta_1} = e^{4 \\theta_1} and \\varepsilon_{0}^{2}{(\\theta_1)} e^{2 \\theta_1} - \\varepsilon_{0}{(\\theta_1)} e^{\\theta_1} = - \\varepsilon_{0}{(\\theta_1)} e^{\\theta_1} + e^{4 \\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))), exp(Mul(Integer(4), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), exp(Mul(Integer(4), Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = \\cos{(\\frac{\\varepsilon}{\\mathbf{g}})}, then obtain \\mathbf{g} \\frac{\\partial}{\\partial \\varepsilon} \\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = - \\sin{(\\frac{\\varepsilon}{\\mathbf{g}})}", "derivation": "\\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = \\cos{(\\frac{\\varepsilon}{\\mathbf{g}})} and \\mathbf{g} \\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = \\mathbf{g} \\cos{(\\frac{\\varepsilon}{\\mathbf{g}})} and \\frac{\\partial}{\\partial \\varepsilon} \\mathbf{g} \\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = \\frac{\\partial}{\\partial \\varepsilon} \\mathbf{g} \\cos{(\\frac{\\varepsilon}{\\mathbf{g}})} and \\mathbf{g} \\frac{\\partial}{\\partial \\varepsilon} \\varepsilon_{0}{(\\varepsilon,\\mathbf{g})} = - \\sin{(\\frac{\\varepsilon}{\\mathbf{g}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Symbol('\\\\mathbf{g}', commutative=True), cos(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), cos(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given i{(v_{z},A_{2})} = \\cos{(A_{2} + v_{z})}, then obtain \\frac{2 i{(v_{z},A_{2})} + 2 \\cos{(A_{2} + v_{z})}}{2 i{(v_{z},A_{2})}} = \\frac{i{(v_{z},A_{2})} + 3 \\cos{(A_{2} + v_{z})}}{2 i{(v_{z},A_{2})}}", "derivation": "i{(v_{z},A_{2})} = \\cos{(A_{2} + v_{z})} and i{(v_{z},A_{2})} + \\cos{(A_{2} + v_{z})} = 2 \\cos{(A_{2} + v_{z})} and 2 i{(v_{z},A_{2})} + 2 \\cos{(A_{2} + v_{z})} = i{(v_{z},A_{2})} + 3 \\cos{(A_{2} + v_{z})} and \\frac{2 i{(v_{z},A_{2})} + 2 \\cos{(A_{2} + v_{z})}}{2 i{(v_{z},A_{2})}} = \\frac{i{(v_{z},A_{2})} + 3 \\cos{(A_{2} + v_{z})}}{2 i{(v_{z},A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 1, "cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Add(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True)))))"], [["add", 2, "Add(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(2), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))), Add(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(3), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))))"], [["divide", 3, "Mul(Integer(2), Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Mul(Integer(2), Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(2), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))), Pow(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(3), cos(Add(Symbol('A_2', commutative=True), Symbol('v_z', commutative=True))))), Pow(Function('i')(Symbol('v_z', commutative=True), Symbol('A_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(E_{n})} = \\log{(\\cos{(E_{n})})}, then obtain \\frac{d}{d E_{n}} \\log{(\\cos{(E_{n})})} = \\frac{d}{d E_{n}} \\operatorname{n_{1}}{(E_{n})}", "derivation": "\\operatorname{n_{1}}{(E_{n})} = \\log{(\\cos{(E_{n})})} and \\operatorname{n_{1}}{(E_{n})} \\log{(\\cos{(E_{n})})} = \\log{(\\cos{(E_{n})})}^{2} and \\log{(\\cos{(E_{n})})} = \\frac{\\log{(\\cos{(E_{n})})}^{2}}{\\operatorname{n_{1}}{(E_{n})}} and \\frac{d}{d E_{n}} \\log{(\\cos{(E_{n})})} = \\frac{d}{d E_{n}} \\frac{\\log{(\\cos{(E_{n})})}^{2}}{\\operatorname{n_{1}}{(E_{n})}} and \\operatorname{n_{1}}{(E_{n})} = \\frac{\\log{(\\cos{(E_{n})})}^{2}}{\\operatorname{n_{1}}{(E_{n})}} and \\frac{d}{d E_{n}} \\log{(\\cos{(E_{n})})} = \\frac{d}{d E_{n}} \\operatorname{n_{1}}{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('E_n', commutative=True)), log(cos(Symbol('E_n', commutative=True))))"], [["times", 1, "log(cos(Symbol('E_n', commutative=True)))"], "Equality(Mul(Function('n_1')(Symbol('E_n', commutative=True)), log(cos(Symbol('E_n', commutative=True)))), Pow(log(cos(Symbol('E_n', commutative=True))), Integer(2)))"], [["divide", 2, "Function('n_1')(Symbol('E_n', commutative=True))"], "Equality(log(cos(Symbol('E_n', commutative=True))), Mul(Pow(Function('n_1')(Symbol('E_n', commutative=True)), Integer(-1)), Pow(log(cos(Symbol('E_n', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('E_n', commutative=True)"], "Equality(Derivative(log(cos(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('n_1')(Symbol('E_n', commutative=True)), Integer(-1)), Pow(log(cos(Symbol('E_n', commutative=True))), Integer(2))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('n_1')(Symbol('E_n', commutative=True)), Mul(Pow(Function('n_1')(Symbol('E_n', commutative=True)), Integer(-1)), Pow(log(cos(Symbol('E_n', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(log(cos(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Function('n_1')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(n_{1})} = \\cos{(n_{1})}, then obtain (\\mathbf{D}^{2}{(n_{1})} + \\mathbf{D}{(n_{1})} \\cos{(n_{1})})^{n_{1}} = (2 \\mathbf{D}{(n_{1})} \\cos{(n_{1})})^{n_{1}}", "derivation": "\\mathbf{D}{(n_{1})} = \\cos{(n_{1})} and \\mathbf{D}^{2}{(n_{1})} = \\mathbf{D}{(n_{1})} \\cos{(n_{1})} and \\mathbf{D}^{2}{(n_{1})} + \\mathbf{D}{(n_{1})} \\cos{(n_{1})} = 2 \\mathbf{D}{(n_{1})} \\cos{(n_{1})} and (\\mathbf{D}^{2}{(n_{1})} + \\mathbf{D}{(n_{1})} \\cos{(n_{1})})^{n_{1}} = (2 \\mathbf{D}{(n_{1})} \\cos{(n_{1})})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True))))"], [["add", 2, "Mul(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True))))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Pow(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(F_{g},\\hbar)} = F_{g} \\hbar, then derive F_{g} + \\mu_0 = \\int (\\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}})^{\\hbar} dF_{g}, then obtain F_{g} + \\mu_0 = \\int \\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}} dF_{g}", "derivation": "\\dot{y}{(F_{g},\\hbar)} = F_{g} \\hbar and 1 = \\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}} and 1 = (\\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}})^{\\hbar} and \\int 1 dF_{g} = \\int \\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}} dF_{g} and \\int 1 dF_{g} = \\int (\\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}})^{\\hbar} dF_{g} and F_{g} + \\mu_0 = \\int (\\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}})^{\\hbar} dF_{g} and F_{g} + \\mu_0 = \\int 1 dF_{g} and F_{g} + \\mu_0 = \\int \\frac{F_{g} \\hbar}{\\dot{y}{(F_{g},\\hbar)}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(1), Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Pow(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)} = \\chi f_{\\mathbf{p}} and E{(f_{\\mathbf{p}})} = f_{\\mathbf{p}}, then obtain \\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)} E{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} \\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)}", "derivation": "\\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)} = \\chi f_{\\mathbf{p}} and E{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} and \\chi f_{\\mathbf{p}} E{(f_{\\mathbf{p}})} = \\chi f_{\\mathbf{p}}^{2} and \\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)} E{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} \\operatorname{A_{y}}{(f_{\\mathbf{p}},\\chi)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["times", 2, "Mul(Symbol('\\\\chi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('E')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('A_y')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\chi', commutative=True)), Function('E')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('A_y')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given B{(\\psi,F_{c})} = - F_{c} + \\psi and t{(\\psi,F_{c})} = - F_{c} + \\psi, then obtain (- F_{c} (- F_{c} + \\psi))^{F_{c}} = (- F_{c} t{(\\psi,F_{c})})^{F_{c}}", "derivation": "B{(\\psi,F_{c})} = - F_{c} + \\psi and - F_{c} B{(\\psi,F_{c})} = - F_{c} (- F_{c} + \\psi) and t{(\\psi,F_{c})} = - F_{c} + \\psi and - F_{c} B{(\\psi,F_{c})} = - F_{c} t{(\\psi,F_{c})} and - F_{c} (- F_{c} + \\psi) = - F_{c} t{(\\psi,F_{c})} and (- F_{c} (- F_{c} + \\psi))^{F_{c}} = (- F_{c} t{(\\psi,F_{c})})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('F_c', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_c', commutative=True), Function('B')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('F_c', commutative=True), Function('B')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True), Function('t')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Symbol('F_c', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True), Function('t')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))))"], [["power", 5, "Symbol('F_c', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('F_c', commutative=True), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\psi', commutative=True))), Symbol('F_c', commutative=True)), Pow(Mul(Integer(-1), Symbol('F_c', commutative=True), Function('t')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} = e^{- \\mathbf{E} + t} and \\mathbf{S}{(t,\\mathbf{E})} = - \\mathbf{E} + t, then obtain \\mathbf{E} \\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} + t = \\mathbf{E} e^{\\mathbf{S}{(t,\\mathbf{E})}} + t", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} = e^{- \\mathbf{E} + t} and \\mathbf{E} \\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} = \\mathbf{E} e^{- \\mathbf{E} + t} and \\mathbf{E} \\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} + t = \\mathbf{E} e^{- \\mathbf{E} + t} + t and \\mathbf{S}{(t,\\mathbf{E})} = - \\mathbf{E} + t and \\mathbf{E} \\operatorname{f_{\\mathbf{v}}}{(t,\\mathbf{E})} + t = \\mathbf{E} e^{\\mathbf{S}{(t,\\mathbf{E})}} + t", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('t', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('t', commutative=True)))))"], [["add", 2, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('t', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('t', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Function('\\\\mathbf{S}')(Symbol('t', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given y{(E,\\mathbf{s})} = E \\mathbf{s}, then derive \\int (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} dE = A_{z}, then obtain \\frac{d}{d E} \\int 0^{E} dE = \\frac{\\partial}{\\partial E} \\int (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} dE", "derivation": "y{(E,\\mathbf{s})} = E \\mathbf{s} and - E \\mathbf{s} + y{(E,\\mathbf{s})} = 0 and (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} = 0^{E} and \\int (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} dE = \\int 0^{E} dE and \\int (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} dE = A_{z} and \\int 0^{E} dE = A_{z} and \\frac{d}{d E} \\int 0^{E} dE = \\frac{d}{d E} A_{z} and \\frac{d}{d E} \\int 0^{E} dE = \\frac{\\partial}{\\partial E} \\int (- E \\mathbf{s} + y{(E,\\mathbf{s})})^{E} dE", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Mul(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(0))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('E', commutative=True)), Pow(Integer(0), Symbol('E', commutative=True)))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Integer(0), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('A_z', commutative=True))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Pow(Integer(0), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('A_z', commutative=True))"], [["differentiate", 6, "Symbol('E', commutative=True)"], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Symbol('A_z', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('y')(Symbol('E', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a,\\pi)} = \\pi - a, then derive \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi = \\pi (\\hat{\\mathbf{r}} + \\frac{\\pi^{2}}{2} - \\pi a), then obtain - \\pi^{2} + \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi + \\operatorname{M_{E}}{(a,\\pi)} + 1 = - \\pi^{2} + \\pi (\\hat{\\mathbf{r}} + \\frac{\\pi^{2}}{2} - \\pi a) + \\operatorname{M_{E}}{(a,\\pi)} + 1", "derivation": "\\operatorname{M_{E}}{(a,\\pi)} = \\pi - a and \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi = \\int (\\pi - a) d\\pi and \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi = \\pi \\int (\\pi - a) d\\pi and \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi = \\pi (\\hat{\\mathbf{r}} + \\frac{\\pi^{2}}{2} - \\pi a) and - \\pi^{2} + \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi = - \\pi^{2} + \\pi (\\hat{\\mathbf{r}} + \\frac{\\pi^{2}}{2} - \\pi a) and - \\pi^{2} + \\pi \\int \\operatorname{M_{E}}{(a,\\pi)} d\\pi + \\operatorname{M_{E}}{(a,\\pi)} + 1 = - \\pi^{2} + \\pi (\\hat{\\mathbf{r}} + \\frac{\\pi^{2}}{2} - \\pi a) + \\operatorname{M_{E}}{(a,\\pi)} + 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["times", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Integral(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\pi', commutative=True), Integral(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Symbol('a', commutative=True)))))"], [["minus", 4, "Pow(Symbol('\\\\pi', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), Integral(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Symbol('a', commutative=True))))))"], [["add", 5, "Add(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), Integral(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Symbol('a', commutative=True)))), Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{J}{(S)} = \\cos{(e^{S})}, then derive (\\frac{d}{d S} \\mathbf{J}{(S)})^{S} = (- e^{S} \\sin{(e^{S})})^{S}, then obtain (- e^{S} \\sin{(e^{S})})^{S} = (\\frac{d}{d S} \\cos{(e^{S})})^{S}", "derivation": "\\mathbf{J}{(S)} = \\cos{(e^{S})} and \\frac{d}{d S} \\mathbf{J}{(S)} = \\frac{d}{d S} \\cos{(e^{S})} and (\\frac{d}{d S} \\mathbf{J}{(S)})^{S} = (\\frac{d}{d S} \\cos{(e^{S})})^{S} and (\\frac{d}{d S} \\mathbf{J}{(S)})^{S} = (- e^{S} \\sin{(e^{S})})^{S} and (- e^{S} \\sin{(e^{S})})^{S} = (\\frac{d}{d S} \\cos{(e^{S})})^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('S', commutative=True)), cos(exp(Symbol('S', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(cos(exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Mul(Integer(-1), exp(Symbol('S', commutative=True)), sin(exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Mul(Integer(-1), exp(Symbol('S', commutative=True)), sin(exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True)), Pow(Derivative(cos(exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\pi,\\chi)} = \\pi^{\\chi}, then derive \\pi \\frac{\\partial^{2}}{\\partial \\pi\\partial \\chi} \\Omega{(\\pi,\\chi)} + \\frac{\\partial}{\\partial \\chi} \\Omega{(\\pi,\\chi)} = \\pi^{\\chi} (\\chi \\log{(\\pi)} + \\log{(\\pi)} + 1), then obtain \\pi \\frac{\\partial^{2}}{\\partial \\pi\\partial \\chi} \\pi^{\\chi} + \\frac{\\partial}{\\partial \\chi} \\pi^{\\chi} = \\pi^{\\chi} (\\chi \\log{(\\pi)} + \\log{(\\pi)} + 1)", "derivation": "\\Omega{(\\pi,\\chi)} = \\pi^{\\chi} and \\pi \\Omega{(\\pi,\\chi)} = \\pi \\pi^{\\chi} and \\frac{\\partial}{\\partial \\pi} \\pi \\Omega{(\\pi,\\chi)} = \\frac{\\partial}{\\partial \\pi} \\pi \\pi^{\\chi} and \\frac{\\partial^{2}}{\\partial \\chi\\partial \\pi} \\pi \\Omega{(\\pi,\\chi)} = \\frac{\\partial^{2}}{\\partial \\chi\\partial \\pi} \\pi \\pi^{\\chi} and \\pi \\frac{\\partial^{2}}{\\partial \\pi\\partial \\chi} \\Omega{(\\pi,\\chi)} + \\frac{\\partial}{\\partial \\chi} \\Omega{(\\pi,\\chi)} = \\pi^{\\chi} (\\chi \\log{(\\pi)} + \\log{(\\pi)} + 1) and \\pi \\frac{\\partial^{2}}{\\partial \\pi\\partial \\chi} \\pi^{\\chi} + \\frac{\\partial}{\\partial \\chi} \\pi^{\\chi} = \\pi^{\\chi} (\\chi \\log{(\\pi)} + \\log{(\\pi)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Derivative(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Derivative(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\pi', commutative=True))), log(Symbol('\\\\pi', commutative=True)), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\pi', commutative=True))), log(Symbol('\\\\pi', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(H)} = e^{H}, then derive (\\int \\operatorname{f^{*}}{(H)} dH)^{H} = (v_{z} + e^{H})^{H}, then obtain (v_{z} + e^{H})^{- H} \\int (\\int e^{H} dH)^{H} dv_{z} = (v_{z} + e^{H})^{- H} \\int (v_{z} + e^{H})^{H} dv_{z}", "derivation": "\\operatorname{f^{*}}{(H)} = e^{H} and \\int \\operatorname{f^{*}}{(H)} dH = \\int e^{H} dH and (\\int \\operatorname{f^{*}}{(H)} dH)^{H} = (\\int e^{H} dH)^{H} and (\\int \\operatorname{f^{*}}{(H)} dH)^{H} = (v_{z} + e^{H})^{H} and (\\int e^{H} dH)^{H} = (v_{z} + e^{H})^{H} and \\int (\\int e^{H} dH)^{H} dv_{z} = \\int (v_{z} + e^{H})^{H} dv_{z} and (v_{z} + e^{H})^{- H} \\int (\\int e^{H} dH)^{H} dv_{z} = (v_{z} + e^{H})^{- H} \\int (v_{z} + e^{H})^{H} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Function('f^*')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('f^*')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["integrate", 5, "Symbol('v_z', commutative=True)"], "Equality(Integral(Pow(Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["divide", 6, "Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Integral(Pow(Integral(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Integral(Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon}, then obtain \\int \\mathbf{A}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} d\\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = \\int (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} d\\mathbf{A}{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} and \\mathbf{A}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} = (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} and \\int \\mathbf{A}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} dg^{\\prime}_{\\varepsilon} and \\int \\mathbf{A}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} d\\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = \\int (g^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} d\\mathbf{A}{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], [["power", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Pow(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Integral(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{p})} = \\sin{(\\log{(\\mathbf{p})})}, then obtain \\frac{d}{d \\mathbf{p}} \\mathbf{p} \\operatorname{m_{s}}^{\\mathbf{p}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\mathbf{p} \\sin^{\\mathbf{p}}{(\\log{(\\mathbf{p})})}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{p})} = \\sin{(\\log{(\\mathbf{p})})} and \\operatorname{m_{s}}^{\\mathbf{p}}{(\\mathbf{p})} = \\sin^{\\mathbf{p}}{(\\log{(\\mathbf{p})})} and \\mathbf{p} \\operatorname{m_{s}}^{\\mathbf{p}}{(\\mathbf{p})} = \\mathbf{p} \\sin^{\\mathbf{p}}{(\\log{(\\mathbf{p})})} and \\frac{d}{d \\mathbf{p}} \\mathbf{p} \\operatorname{m_{s}}^{\\mathbf{p}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\mathbf{p} \\sin^{\\mathbf{p}}{(\\log{(\\mathbf{p})})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{p}', commutative=True)), sin(log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(sin(log(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('m_s')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(sin(log(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('m_s')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(sin(log(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} = \\cos{(\\rho_f)}, then derive \\cos{(\\rho_f)} \\int \\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} d\\rho_f = (\\varphi + \\sin{(\\rho_f)}) \\cos{(\\rho_f)}, then obtain \\cos{(\\rho_f)} \\int \\cos{(\\rho_f)} d\\rho_f = (\\varphi + \\sin{(\\rho_f)}) \\cos{(\\rho_f)}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} = \\cos{(\\rho_f)} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} d\\rho_f = \\int \\cos{(\\rho_f)} d\\rho_f and \\cos{(\\rho_f)} \\int \\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} d\\rho_f = \\cos{(\\rho_f)} \\int \\cos{(\\rho_f)} d\\rho_f and \\cos{(\\rho_f)} \\int \\operatorname{V_{\\mathbf{B}}}{(\\rho_f)} d\\rho_f = (\\varphi + \\sin{(\\rho_f)}) \\cos{(\\rho_f)} and \\cos{(\\rho_f)} \\int \\cos{(\\rho_f)} d\\rho_f = (\\varphi + \\sin{(\\rho_f)}) \\cos{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(cos(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["times", 2, "cos(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\rho_f', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(cos(Symbol('\\\\rho_f', commutative=True)), Integral(cos(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(cos(Symbol('\\\\rho_f', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(cos(Symbol('\\\\rho_f', commutative=True)), Integral(cos(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\phi)} = \\log{(\\phi)}, then derive \\int \\hat{H}_{\\lambda}{(\\phi)} d\\phi = \\phi \\log{(\\phi)} - \\phi + z, then obtain \\iint \\hat{H}_{\\lambda}{(\\phi)} d\\phi d\\phi = \\int (\\phi \\log{(\\phi)} - \\phi + z) d\\phi", "derivation": "\\hat{H}_{\\lambda}{(\\phi)} = \\log{(\\phi)} and \\int \\hat{H}_{\\lambda}{(\\phi)} d\\phi = \\int \\log{(\\phi)} d\\phi and \\int \\hat{H}_{\\lambda}{(\\phi)} d\\phi = \\phi \\log{(\\phi)} - \\phi + z and \\iint \\hat{H}_{\\lambda}{(\\phi)} d\\phi d\\phi = \\int (\\phi \\log{(\\phi)} - \\phi + z) d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('z', commutative=True)))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(l)} = e^{e^{l}}, then derive - (\\mathbb{I} + \\hat{x}_0{(l)}) e^{- l} = - (\\sigma_p + e^{e^{l}}) e^{- l}, then obtain - (\\mathbb{I} + \\hat{x}_0{(l)}) e^{- l} = - (\\sigma_p + \\hat{x}_0{(l)}) e^{- l}", "derivation": "\\hat{x}_0{(l)} = e^{e^{l}} and \\frac{d}{d l} \\hat{x}_0{(l)} = \\frac{d}{d l} e^{e^{l}} and \\int \\frac{d}{d l} \\hat{x}_0{(l)} dl = \\int \\frac{d}{d l} e^{e^{l}} dl and e^{- l} \\int \\frac{d}{d l} \\hat{x}_0{(l)} dl = e^{- l} \\int \\frac{d}{d l} e^{e^{l}} dl and - e^{- l} \\int \\frac{d}{d l} \\hat{x}_0{(l)} dl = - e^{- l} \\int \\frac{d}{d l} e^{e^{l}} dl and - (\\mathbb{I} + \\hat{x}_0{(l)}) e^{- l} = - (\\sigma_p + e^{e^{l}}) e^{- l} and - (\\mathbb{I} + \\hat{x}_0{(l)}) e^{- l} = - (\\sigma_p + \\hat{x}_0{(l)}) e^{- l}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))))"], [["divide", 3, "exp(Symbol('l', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Integer(-1), Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\hat{x}_0')(Symbol('l', commutative=True))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Integer(-1), Add(Symbol('\\\\sigma_p', commutative=True), exp(exp(Symbol('l', commutative=True)))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(-1), Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\hat{x}_0')(Symbol('l', commutative=True))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Integer(-1), Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\hat{x}_0')(Symbol('l', commutative=True))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\rho_b)} = \\frac{d}{d \\rho_b} e^{\\rho_b}, then derive \\int \\varphi^{*}{(\\rho_b)} d\\rho_b = \\varphi + e^{\\rho_b}, then derive \\varphi^{*}{(\\rho_b)} = e^{\\rho_b}, then obtain \\int e^{\\rho_b} d\\rho_b = \\varphi + e^{\\rho_b}", "derivation": "\\varphi^{*}{(\\rho_b)} = \\frac{d}{d \\rho_b} e^{\\rho_b} and \\int \\varphi^{*}{(\\rho_b)} d\\rho_b = \\int \\frac{d}{d \\rho_b} e^{\\rho_b} d\\rho_b and \\int \\varphi^{*}{(\\rho_b)} d\\rho_b = \\varphi + e^{\\rho_b} and \\varphi^{*}{(\\rho_b)} = e^{\\rho_b} and \\int e^{\\rho_b} d\\rho_b = \\varphi + e^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\rho_b', commutative=True)), Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varphi^*')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{H})} = \\sin{(e^{\\mathbf{H}})}, then derive \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{S} + \\operatorname{Si}{(e^{\\mathbf{H}})}, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} (\\mathbf{S} + \\operatorname{Si}{(e^{\\mathbf{H}})})}{\\frac{d}{d \\mathbf{S}} \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H}}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{H})} = \\sin{(e^{\\mathbf{H}})} and \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H} = \\int \\sin{(e^{\\mathbf{H}})} d\\mathbf{H} and \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{S} + \\operatorname{Si}{(e^{\\mathbf{H}})} and \\frac{d}{d \\mathbf{S}} \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H} = \\frac{\\partial}{\\partial \\mathbf{S}} (\\mathbf{S} + \\operatorname{Si}{(e^{\\mathbf{H}})}) and 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{S}} (\\mathbf{S} + \\operatorname{Si}{(e^{\\mathbf{H}})})}{\\frac{d}{d \\mathbf{S}} \\int \\operatorname{n_{2}}{(\\mathbf{H})} d\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), sin(exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(sin(exp(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Si(exp(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Integral(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Si(exp(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Integral(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Si(exp(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Pow(Derivative(Integral(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{p},t,G)} = G + \\mathbf{p} + t and \\mu{(\\mathbf{p},t,G)} = \\varphi^{t}{(\\mathbf{p},t,G)}, then obtain \\mu{(\\mathbf{p},t,G)} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)} = \\varphi^{t}{(\\mathbf{p},t,G)} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)}", "derivation": "\\varphi{(\\mathbf{p},t,G)} = G + \\mathbf{p} + t and \\mu{(\\mathbf{p},t,G)} = \\varphi^{t}{(\\mathbf{p},t,G)} and \\mu{(\\mathbf{p},t,G)} = (G + \\mathbf{p} + t)^{t} and \\mu{(\\mathbf{p},t,G)} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)} = (G + \\mathbf{p} + t)^{t} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)} and \\mu{(\\mathbf{p},t,G)} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)} = \\varphi^{t}{(\\mathbf{p},t,G)} - \\frac{\\partial}{\\partial \\mathbf{p}} \\varphi{(\\mathbf{p},t,G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Pow(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["minus", 3, "Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))), Add(Pow(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('t', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(Z,\\psi^*)} = \\sin{(Z + \\psi^*)} and E{(Z,\\psi^*)} = \\frac{\\operatorname{E_{n}}{(Z,\\psi^*)}}{\\sin{(Z + \\psi^*)}}, then obtain \\psi^* E{(Z,\\psi^*)} = \\psi^*", "derivation": "\\operatorname{E_{n}}{(Z,\\psi^*)} = \\sin{(Z + \\psi^*)} and E{(Z,\\psi^*)} = \\frac{\\operatorname{E_{n}}{(Z,\\psi^*)}}{\\sin{(Z + \\psi^*)}} and E{(Z,\\psi^*)} = 1 and \\psi^* E{(Z,\\psi^*)} = \\psi^*", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), sin(Add(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1))"], [["times", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('E')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))"]]}, {"prompt": "Given \\psi^{*}{(a^{\\dagger},H)} = \\log{(H + a^{\\dagger})}, then derive \\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1 = -1 + \\frac{1}{H + a^{\\dagger}}, then obtain (\\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1)^{a^{\\dagger}} = (\\frac{\\partial}{\\partial H} \\log{(H + a^{\\dagger})} - 1)^{a^{\\dagger}}", "derivation": "\\psi^{*}{(a^{\\dagger},H)} = \\log{(H + a^{\\dagger})} and \\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} = \\frac{\\partial}{\\partial H} \\log{(H + a^{\\dagger})} and \\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1 = \\frac{\\partial}{\\partial H} \\log{(H + a^{\\dagger})} - 1 and \\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1 = -1 + \\frac{1}{H + a^{\\dagger}} and \\frac{\\partial}{\\partial H} \\log{(H + a^{\\dagger})} - 1 = -1 + \\frac{1}{H + a^{\\dagger}} and (\\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1)^{a^{\\dagger}} = (-1 + \\frac{1}{H + a^{\\dagger}})^{a^{\\dagger}} and (\\frac{\\partial}{\\partial H} \\psi^{*}{(a^{\\dagger},H)} - 1)^{a^{\\dagger}} = (\\frac{\\partial}{\\partial H} \\log{(H + a^{\\dagger})} - 1)^{a^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), log(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(log(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(log(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Add(Derivative(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Derivative(log(Add(Symbol('H', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given h{(\\sigma_x)} = \\log{(\\sigma_x)}, then derive \\frac{\\int h{(\\sigma_x)} d\\sigma_x}{A_{z} + \\sigma_x \\log{(\\sigma_x)} - \\sigma_x} = 1, then obtain \\frac{\\int h{(\\sigma_x)} d\\sigma_x}{A_{z} + \\sigma_x h{(\\sigma_x)} - \\sigma_x} = 1", "derivation": "h{(\\sigma_x)} = \\log{(\\sigma_x)} and \\int h{(\\sigma_x)} d\\sigma_x = \\int \\log{(\\sigma_x)} d\\sigma_x and \\frac{\\int h{(\\sigma_x)} d\\sigma_x}{\\int \\log{(\\sigma_x)} d\\sigma_x} = 1 and \\frac{\\int h{(\\sigma_x)} d\\sigma_x}{A_{z} + \\sigma_x \\log{(\\sigma_x)} - \\sigma_x} = 1 and \\frac{\\int h{(\\sigma_x)} d\\sigma_x}{A_{z} + \\sigma_x h{(\\sigma_x)} - \\sigma_x} = 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 2, "Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Integral(Function('h')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Pow(Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Integral(Function('h')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Symbol('\\\\sigma_x', commutative=True), Function('h')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Integral(Function('h')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{A}{(T)} = e^{T}, then derive 2 \\int \\mathbf{A}{(T)} dT = \\psi^* + e^{T} + \\int \\mathbf{A}{(T)} dT, then obtain (2 U + 2 e^{T})^{T} = (U + \\psi^* + \\mathbf{A}{(T)} + e^{T})^{T}", "derivation": "\\mathbf{A}{(T)} = e^{T} and \\int \\mathbf{A}{(T)} dT = \\int e^{T} dT and 2 \\int \\mathbf{A}{(T)} dT = \\int \\mathbf{A}{(T)} dT + \\int e^{T} dT and 2 \\int \\mathbf{A}{(T)} dT = \\psi^* + e^{T} + \\int \\mathbf{A}{(T)} dT and 2 \\int \\mathbf{A}{(T)} dT = \\psi^* + \\mathbf{A}{(T)} + \\int \\mathbf{A}{(T)} dT and (2 \\int \\mathbf{A}{(T)} dT)^{T} = (\\psi^* + \\mathbf{A}{(T)} + \\int \\mathbf{A}{(T)} dT)^{T} and (2 \\int e^{T} dT)^{T} = (\\psi^* + \\mathbf{A}{(T)} + \\int e^{T} dT)^{T} and (2 U + 2 e^{T})^{T} = (U + \\psi^* + \\mathbf{A}{(T)} + e^{T})^{T}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 2, "Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), exp(Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["power", 5, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Integer(2), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Mul(Integer(2), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(r_{0},y^{\\prime})} = r_{0} + y^{\\prime}, then obtain \\log{(\\int \\frac{\\mathbf{H}{(r_{0},y^{\\prime})}}{r_{0}} dy^{\\prime})} = \\log{(\\int \\frac{r_{0} + y^{\\prime}}{r_{0}} dy^{\\prime})}", "derivation": "\\mathbf{H}{(r_{0},y^{\\prime})} = r_{0} + y^{\\prime} and \\frac{\\mathbf{H}{(r_{0},y^{\\prime})}}{r_{0}} = \\frac{r_{0} + y^{\\prime}}{r_{0}} and \\int \\frac{\\mathbf{H}{(r_{0},y^{\\prime})}}{r_{0}} dy^{\\prime} = \\int \\frac{r_{0} + y^{\\prime}}{r_{0}} dy^{\\prime} and \\log{(\\int \\frac{\\mathbf{H}{(r_{0},y^{\\prime})}}{r_{0}} dy^{\\prime})} = \\log{(\\int \\frac{r_{0} + y^{\\prime}}{r_{0}} dy^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["log", 3], "Equality(log(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), log(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\eta)} = \\cos{(\\eta)}, then obtain \\frac{d}{d \\eta} (\\int \\operatorname{F_{x}}{(\\eta)} d\\eta - \\int \\cos{(\\eta)} d\\eta) = 0", "derivation": "\\operatorname{F_{x}}{(\\eta)} = \\cos{(\\eta)} and \\int \\operatorname{F_{x}}{(\\eta)} d\\eta = \\int \\cos{(\\eta)} d\\eta and \\int \\operatorname{F_{x}}{(\\eta)} d\\eta - \\int \\cos{(\\eta)} d\\eta = 0 and \\frac{d}{d \\eta} (\\int \\operatorname{F_{x}}{(\\eta)} d\\eta - \\int \\cos{(\\eta)} d\\eta) = \\frac{d}{d \\eta} 0 and \\frac{d}{d \\eta} (\\int \\operatorname{F_{x}}{(\\eta)} d\\eta - \\int \\cos{(\\eta)} d\\eta) = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["minus", 2, "Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Integral(Function('F_x')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Integral(Function('F_x')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Add(Integral(Function('F_x')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given S{(v)} = \\frac{d}{d v} e^{v}, then derive S{(v)} = e^{v}, then obtain \\sin{(\\frac{d}{d v} (S{(v)} + \\frac{d}{d v} e^{v}))} = \\sin{(\\frac{d}{d v} 2 \\frac{d}{d v} e^{v})}", "derivation": "S{(v)} = \\frac{d}{d v} e^{v} and S{(v)} = e^{v} and e^{v} = \\frac{d}{d v} e^{v} and S{(v)} + e^{v} = 2 e^{v} and S{(v)} + \\frac{d}{d v} e^{v} = 2 \\frac{d}{d v} e^{v} and \\frac{d}{d v} (S{(v)} + \\frac{d}{d v} e^{v}) = \\frac{d}{d v} 2 \\frac{d}{d v} e^{v} and \\sin{(\\frac{d}{d v} (S{(v)} + \\frac{d}{d v} e^{v}))} = \\sin{(\\frac{d}{d v} 2 \\frac{d}{d v} e^{v})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('S')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 2, "exp(Symbol('v', commutative=True))"], "Equality(Add(Function('S')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))), Mul(Integer(2), exp(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('S')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('S')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["sin", 6], "Equality(sin(Derivative(Add(Function('S')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1)))), sin(Derivative(Mul(Integer(2), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{2}{(J)} = \\cos{(J)}, then obtain e^{\\theta_{2}{(J)}} e^{\\cos{(J)}} \\frac{d}{d J} \\theta_{2}{(J)} = e^{\\theta_{2}{(J)}} e^{\\cos{(J)}} \\frac{d}{d J} \\cos{(J)}", "derivation": "\\theta_{2}{(J)} = \\cos{(J)} and \\frac{d}{d J} \\theta_{2}{(J)} = \\frac{d}{d J} \\cos{(J)} and e^{\\theta_{2}{(J)}} = e^{\\cos{(J)}} and e^{2 \\theta_{2}{(J)}} = e^{\\theta_{2}{(J)}} e^{\\cos{(J)}} and e^{2 \\theta_{2}{(J)}} \\frac{d}{d J} \\theta_{2}{(J)} = e^{2 \\theta_{2}{(J)}} \\frac{d}{d J} \\cos{(J)} and e^{\\theta_{2}{(J)}} e^{\\cos{(J)}} \\frac{d}{d J} \\theta_{2}{(J)} = e^{\\theta_{2}{(J)}} e^{\\cos{(J)}} \\frac{d}{d J} \\cos{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["exp", 1], "Equality(exp(Function('\\\\theta_2')(Symbol('J', commutative=True))), exp(cos(Symbol('J', commutative=True))))"], [["times", 3, "exp(Function('\\\\theta_2')(Symbol('J', commutative=True)))"], "Equality(exp(Mul(Integer(2), Function('\\\\theta_2')(Symbol('J', commutative=True)))), Mul(exp(Function('\\\\theta_2')(Symbol('J', commutative=True))), exp(cos(Symbol('J', commutative=True)))))"], [["times", 2, "exp(Mul(Integer(2), Function('\\\\theta_2')(Symbol('J', commutative=True))))"], "Equality(Mul(exp(Mul(Integer(2), Function('\\\\theta_2')(Symbol('J', commutative=True)))), Derivative(Function('\\\\theta_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(2), Function('\\\\theta_2')(Symbol('J', commutative=True)))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(exp(Function('\\\\theta_2')(Symbol('J', commutative=True))), exp(cos(Symbol('J', commutative=True))), Derivative(Function('\\\\theta_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(exp(Function('\\\\theta_2')(Symbol('J', commutative=True))), exp(cos(Symbol('J', commutative=True))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\delta,f)} = \\delta - f, then obtain - f (- f^{2} \\operatorname{n_{1}}{(\\delta,f)} - f (\\delta - f)) = - f (- f^{2} (\\delta - f) - f \\operatorname{n_{1}}{(\\delta,f)})", "derivation": "\\operatorname{n_{1}}{(\\delta,f)} = \\delta - f and - f \\operatorname{n_{1}}{(\\delta,f)} = - f (\\delta - f) and - f^{2} \\operatorname{n_{1}}{(\\delta,f)} = - f^{2} (\\delta - f) and - f^{2} \\operatorname{n_{1}}{(\\delta,f)} - f (\\delta - f) = - f^{2} (\\delta - f) - f (\\delta - f) and - f^{2} (\\delta - f) - f \\operatorname{n_{1}}{(\\delta,f)} = - f^{2} (\\delta - f) - f (\\delta - f) and - f^{2} \\operatorname{n_{1}}{(\\delta,f)} - f (\\delta - f) = - f^{2} (\\delta - f) - f \\operatorname{n_{1}}{(\\delta,f)} and - f (- f^{2} \\operatorname{n_{1}}{(\\delta,f)} - f (\\delta - f)) = - f (- f^{2} (\\delta - f) - f \\operatorname{n_{1}}{(\\delta,f)})", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('f', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('f', commutative=True), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["times", 2, "Symbol('f', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))))"], [["add", 2, "Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True)))))"], [["times", 6, "Mul(Integer(-1), Symbol('f', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('f', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))))), Mul(Integer(-1), Symbol('f', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(2)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(r_{0})} = e^{r_{0}}, then obtain 2 r_{0} \\operatorname{F_{H}}{(r_{0})} + \\operatorname{F_{H}}{(r_{0})} = 2 r_{0} \\operatorname{F_{H}}{(r_{0})} + e^{r_{0}}", "derivation": "\\operatorname{F_{H}}{(r_{0})} = e^{r_{0}} and r_{0} \\operatorname{F_{H}}{(r_{0})} = r_{0} e^{r_{0}} and r_{0} \\operatorname{F_{H}}{(r_{0})} + r_{0} e^{r_{0}} + \\operatorname{F_{H}}{(r_{0})} = r_{0} \\operatorname{F_{H}}{(r_{0})} + r_{0} e^{r_{0}} + e^{r_{0}} and 2 r_{0} \\operatorname{F_{H}}{(r_{0})} + \\operatorname{F_{H}}{(r_{0})} = 2 r_{0} \\operatorname{F_{H}}{(r_{0})} + e^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), exp(Symbol('r_0', commutative=True))))"], [["add", 1, "Add(Mul(Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), exp(Symbol('r_0', commutative=True))))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), exp(Symbol('r_0', commutative=True))), Function('F_H')(Symbol('r_0', commutative=True))), Add(Mul(Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), exp(Symbol('r_0', commutative=True))), exp(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), Function('F_H')(Symbol('r_0', commutative=True))), Add(Mul(Integer(2), Symbol('r_0', commutative=True), Function('F_H')(Symbol('r_0', commutative=True))), exp(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(F_{H})} = e^{F_{H}}, then obtain - F_{H} + \\mathbf{E}{(F_{H})} = - F_{H} - \\mathbf{E}{(F_{H})} + 2 e^{F_{H}}", "derivation": "\\mathbf{E}{(F_{H})} = e^{F_{H}} and - F_{H} + \\mathbf{E}{(F_{H})} = - F_{H} + e^{F_{H}} and - F_{H} = - F_{H} - \\mathbf{E}{(F_{H})} + e^{F_{H}} and - F_{H} + e^{F_{H}} = - F_{H} - \\mathbf{E}{(F_{H})} + 2 e^{F_{H}} and - F_{H} + \\mathbf{E}{(F_{H})} = - F_{H} - \\mathbf{E}{(F_{H})} + 2 e^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["minus", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))), exp(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))), Mul(Integer(2), exp(Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('F_H', commutative=True))), Mul(Integer(2), exp(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(h)} = \\sin{(h)} and k{(h)} = \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}{(h)}}{\\sin{(h)}}, then obtain - \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}^{2}{(h)}}{\\sin^{2}{(h)}} = \\operatorname{f_{E}}{(h)} - \\sin{(h)} - 1", "derivation": "\\operatorname{f_{E}}{(h)} = \\sin{(h)} and \\frac{\\operatorname{f_{E}}{(h)}}{\\sin{(h)}} = 1 and \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}{(h)}}{\\sin{(h)}} = - \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1 and k{(h)} = \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}{(h)}}{\\sin{(h)}} and k{(h)} = - \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1 and - k{(h)} = \\operatorname{f_{E}}{(h)} - \\sin{(h)} - 1 and k{(h)} = \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}^{2}{(h)}}{\\sin^{2}{(h)}} and - \\frac{(- \\operatorname{f_{E}}{(h)} + \\sin{(h)} + 1) \\operatorname{f_{E}}^{2}{(h)}}{\\sin^{2}{(h)}} = \\operatorname{f_{E}}{(h)} - \\sin{(h)} - 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["divide", 1, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Function('f_E')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)), Function('f_E')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('k')(Symbol('h', commutative=True)), Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)), Function('f_E')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('k')(Symbol('h', commutative=True)), Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('k')(Symbol('h', commutative=True))), Add(Function('f_E')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('k')(Symbol('h', commutative=True)), Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)), Pow(Function('f_E')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True)), Integer(1)), Pow(Function('f_E')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-2))), Add(Function('f_E')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given E{(C_{1},\\mathbf{H})} = C_{1} \\cos{(\\mathbf{H})}, then obtain - C_{1} E{(C_{1},\\mathbf{H})} \\sin{(\\mathbf{H})} = - C_{1}^{2} \\sin{(\\mathbf{H})} \\cos{(\\mathbf{H})}", "derivation": "E{(C_{1},\\mathbf{H})} = C_{1} \\cos{(\\mathbf{H})} and \\frac{\\partial}{\\partial \\mathbf{H}} E{(C_{1},\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} C_{1} \\cos{(\\mathbf{H})} and E{(C_{1},\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} E{(C_{1},\\mathbf{H})} = C_{1} \\cos{(\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} E{(C_{1},\\mathbf{H})} and E{(C_{1},\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} C_{1} \\cos{(\\mathbf{H})} = C_{1} \\cos{(\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} C_{1} \\cos{(\\mathbf{H})} and - C_{1} E{(C_{1},\\mathbf{H})} \\sin{(\\mathbf{H})} = - C_{1}^{2} \\sin{(\\mathbf{H})} \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Mul(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Mul(Symbol('C_1', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Symbol('C_1', commutative=True), Function('E')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(I)} = \\cos{(I)} and \\operatorname{P_{g}}{(I)} = \\int \\operatorname{A_{x}}{(I)} dI, then obtain \\operatorname{P_{g}}^{I}{(I)} = (\\nabla + \\sin{(I)})^{I}", "derivation": "\\operatorname{A_{x}}{(I)} = \\cos{(I)} and \\int \\operatorname{A_{x}}{(I)} dI = \\int \\cos{(I)} dI and \\operatorname{P_{g}}{(I)} = \\int \\operatorname{A_{x}}{(I)} dI and \\operatorname{P_{g}}{(I)} = \\int \\cos{(I)} dI and \\operatorname{P_{g}}^{I}{(I)} = (\\int \\cos{(I)} dI)^{I} and \\operatorname{P_{g}}^{I}{(I)} = (\\nabla + \\sin{(I)})^{I}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('I', commutative=True)), Integral(Function('A_x')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('P_g')(Symbol('I', commutative=True)), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Function('P_g')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Add(Symbol('\\\\nabla', commutative=True), sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\pi{(Z,\\tilde{g})} = Z^{\\tilde{g}}, then derive \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} = Z^{\\tilde{g}} \\log{(Z)}, then obtain 2 Z^{\\tilde{g}} \\log{(Z)} = Z^{\\tilde{g}} \\log{(Z)} + \\pi{(Z,\\tilde{g})} \\log{(Z)}", "derivation": "\\pi{(Z,\\tilde{g})} = Z^{\\tilde{g}} and \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} Z^{\\tilde{g}} and \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} = Z^{\\tilde{g}} \\log{(Z)} and 2 \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} = Z^{\\tilde{g}} \\log{(Z)} + \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} and 2 \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} = \\pi{(Z,\\tilde{g})} \\log{(Z)} + \\frac{\\partial}{\\partial \\tilde{g}} \\pi{(Z,\\tilde{g})} and 2 Z^{\\tilde{g}} \\log{(Z)} = Z^{\\tilde{g}} \\log{(Z)} + \\pi{(Z,\\tilde{g})} \\log{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True))))"], [["add", 3, "Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True))), Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(Mul(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True))), Derivative(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True))), Add(Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True))), Mul(Function('\\\\pi')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\Psi,\\mathbf{S})} = \\frac{\\Psi}{\\mathbf{S}}, then derive \\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} - 1 = Q + \\log{(\\mathbf{S})} - 1, then obtain \\log{(\\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} - 1)} = \\log{(Q + \\log{(\\mathbf{S})} - 1)}", "derivation": "\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})} = \\frac{\\Psi}{\\mathbf{S}} and \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} = \\frac{1}{\\mathbf{S}} and \\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} = \\int \\frac{1}{\\mathbf{S}} d\\mathbf{S} and \\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} - 1 = \\int \\frac{1}{\\mathbf{S}} d\\mathbf{S} - 1 and \\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} - 1 = Q + \\log{(\\mathbf{S})} - 1 and \\log{(\\int \\frac{\\operatorname{c_{0}}{(\\Psi,\\mathbf{S})}}{\\Psi} d\\mathbf{S} - 1)} = \\log{(Q + \\log{(\\mathbf{S})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Add(Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Add(Symbol('Q', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)))"], [["log", 5], "Equality(log(Add(Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('c_0')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))), log(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(U,\\rho,\\sigma_p)} = U \\rho^{\\sigma_p} and \\mathbf{g}{(U,\\rho,\\sigma_p)} = - \\mathbf{B}{(U,\\rho,\\sigma_p)}, then obtain U \\frac{\\partial}{\\partial \\rho} (\\sigma_p - \\mathbf{g}{(U,\\rho,\\sigma_p)}) = U \\frac{\\partial}{\\partial \\rho} (U \\rho^{\\sigma_p} + \\sigma_p)", "derivation": "\\mathbf{B}{(U,\\rho,\\sigma_p)} = U \\rho^{\\sigma_p} and \\sigma_p + \\mathbf{B}{(U,\\rho,\\sigma_p)} = U \\rho^{\\sigma_p} + \\sigma_p and \\mathbf{g}{(U,\\rho,\\sigma_p)} = - \\mathbf{B}{(U,\\rho,\\sigma_p)} and - \\mathbf{g}{(U,\\rho,\\sigma_p)} = \\mathbf{B}{(U,\\rho,\\sigma_p)} and \\sigma_p - \\mathbf{g}{(U,\\rho,\\sigma_p)} = U \\rho^{\\sigma_p} + \\sigma_p and \\frac{\\partial}{\\partial \\rho} (\\sigma_p - \\mathbf{g}{(U,\\rho,\\sigma_p)}) = \\frac{\\partial}{\\partial \\rho} (U \\rho^{\\sigma_p} + \\sigma_p) and U \\frac{\\partial}{\\partial \\rho} (\\sigma_p - \\mathbf{g}{(U,\\rho,\\sigma_p)}) = U \\frac{\\partial}{\\partial \\rho} (U \\rho^{\\sigma_p} + \\sigma_p)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{B}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('\\\\mathbf{B}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["times", 6, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Derivative(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('U', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Symbol('U', commutative=True), Derivative(Add(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(r,\\mathbf{g})} = \\cos{(\\mathbf{g}^{r})}, then obtain 2 \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\operatorname{A_{1}}{(r,\\mathbf{g})} = 2 \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\cos{(\\mathbf{g}^{r})}", "derivation": "\\operatorname{A_{1}}{(r,\\mathbf{g})} = \\cos{(\\mathbf{g}^{r})} and \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} = \\cos^{2}{(\\mathbf{g}^{r})} and \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\operatorname{A_{1}}{(r,\\mathbf{g})} + \\cos^{2}{(\\mathbf{g}^{r})} = \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\cos^{2}{(\\mathbf{g}^{r})} + \\cos{(\\mathbf{g}^{r})} and 2 \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\operatorname{A_{1}}{(r,\\mathbf{g})} = 2 \\operatorname{A_{1}}{(r,\\mathbf{g})} \\cos{(\\mathbf{g}^{r})} + \\cos{(\\mathbf{g}^{r})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True))))"], [["times", 1, "cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))"], "Equality(Mul(Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True))), Integer(2)))"], [["add", 1, "Add(Mul(Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True))), Integer(2)))"], "Equality(Add(Mul(Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True))), Integer(2))), Add(Mul(Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True))), Integer(2)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(2), Function('A_1')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{S},f^{\\prime})} = - \\mathbf{S} + f^{\\prime}, then obtain \\mathbf{S} - f^{\\prime} + \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{\\mathbf{f}{(\\mathbf{S},f^{\\prime})}}{f^{\\prime}})} = \\mathbf{S} - f^{\\prime} + \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{- \\mathbf{S} + f^{\\prime}}{f^{\\prime}})}", "derivation": "\\mathbf{f}{(\\mathbf{S},f^{\\prime})} = - \\mathbf{S} + f^{\\prime} and \\frac{\\mathbf{f}{(\\mathbf{S},f^{\\prime})}}{f^{\\prime}} = \\frac{- \\mathbf{S} + f^{\\prime}}{f^{\\prime}} and \\cos{(\\frac{\\mathbf{f}{(\\mathbf{S},f^{\\prime})}}{f^{\\prime}})} = \\cos{(\\frac{- \\mathbf{S} + f^{\\prime}}{f^{\\prime}})} and \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{\\mathbf{f}{(\\mathbf{S},f^{\\prime})}}{f^{\\prime}})} = \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{- \\mathbf{S} + f^{\\prime}}{f^{\\prime}})} and \\mathbf{S} - f^{\\prime} + \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{\\mathbf{f}{(\\mathbf{S},f^{\\prime})}}{f^{\\prime}})} = \\mathbf{S} - f^{\\prime} + \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(\\frac{- \\mathbf{S} + f^{\\prime}}{f^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(cos(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(i)} = \\sin{(i)}, then derive - \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\cos{(i)}} = - \\frac{\\sin{(i)}}{\\cos{(i)}}, then obtain \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\sin{(i)}} = 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(i)} = \\sin{(i)} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\frac{d}{d i} \\sin{(i)}} = \\frac{\\sin{(i)}}{\\frac{d}{d i} \\sin{(i)}} and - \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\frac{d}{d i} \\sin{(i)}} = - \\frac{\\sin{(i)}}{\\frac{d}{d i} \\sin{(i)}} and - \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\cos{(i)}} = - \\frac{\\sin{(i)}}{\\cos{(i)}} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(i)}}{\\sin{(i)}} = 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["divide", 1, "Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('i', commutative=True)), Pow(Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Symbol('i', commutative=True)), Pow(Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('i', commutative=True)), Pow(Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), sin(Symbol('i', commutative=True)), Pow(Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\delta{(\\nabla,B)} = B + \\nabla, then obtain \\cos{(B - \\delta{(\\nabla,B)})} = \\cos{(\\nabla)}", "derivation": "\\delta{(\\nabla,B)} = B + \\nabla and - B - \\nabla + \\delta{(\\nabla,B)} = 0 and - B + \\delta{(\\nabla,B)} = \\nabla and \\cos{(B - \\delta{(\\nabla,B)})} = \\cos{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Add(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('\\\\delta')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))), Integer(0))"], [["add", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\delta')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\nabla', commutative=True))"], [["cos", 3], "Equality(cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))))), cos(Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(F_{c})} = \\log{(F_{c})}, then obtain F_{c} \\tilde{g}^*{(F_{c})} + \\log{(F_{c} + \\int \\tilde{g}^*{(F_{c})} dF_{c})} = F_{c} \\log{(F_{c})} + \\log{(F_{c} + \\int \\tilde{g}^*{(F_{c})} dF_{c})}", "derivation": "\\tilde{g}^*{(F_{c})} = \\log{(F_{c})} and \\int \\tilde{g}^*{(F_{c})} dF_{c} = \\int \\log{(F_{c})} dF_{c} and F_{c} \\tilde{g}^*{(F_{c})} = F_{c} \\log{(F_{c})} and F_{c} + \\int \\tilde{g}^*{(F_{c})} dF_{c} = F_{c} + \\int \\log{(F_{c})} dF_{c} and F_{c} \\tilde{g}^*{(F_{c})} + \\log{(F_{c} + \\int \\log{(F_{c})} dF_{c})} = F_{c} \\log{(F_{c})} + \\log{(F_{c} + \\int \\log{(F_{c})} dF_{c})} and F_{c} \\tilde{g}^*{(F_{c})} + \\log{(F_{c} + \\int \\tilde{g}^*{(F_{c})} dF_{c})} = F_{c} \\log{(F_{c})} + \\log{(F_{c} + \\int \\tilde{g}^*{(F_{c})} dF_{c})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["times", 1, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))))"], [["add", 2, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Add(Symbol('F_c', commutative=True), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["add", 3, "log(Add(Symbol('F_c', commutative=True), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], "Equality(Add(Mul(Symbol('F_c', commutative=True), Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True))), log(Add(Symbol('F_c', commutative=True), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))), Add(Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))), log(Add(Symbol('F_c', commutative=True), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('F_c', commutative=True), Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True))), log(Add(Symbol('F_c', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))), Add(Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))), log(Add(Symbol('F_c', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)} = f_{\\mathbf{v}} + e^{\\Omega} and \\mathbb{I}{(f_{\\mathbf{v}},\\Omega)} = f_{\\mathbf{v}} + e^{\\Omega}, then obtain \\int e^{\\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)}} df_{\\mathbf{v}} = \\int e^{f_{\\mathbf{v}} + e^{\\Omega}} df_{\\mathbf{v}}", "derivation": "\\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)} = f_{\\mathbf{v}} + e^{\\Omega} and \\mathbb{I}{(f_{\\mathbf{v}},\\Omega)} = f_{\\mathbf{v}} + e^{\\Omega} and \\mathbb{I}{(f_{\\mathbf{v}},\\Omega)} = \\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)} and e^{\\mathbb{I}{(f_{\\mathbf{v}},\\Omega)}} = e^{f_{\\mathbf{v}} + e^{\\Omega}} and e^{\\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)}} = e^{f_{\\mathbf{v}} + e^{\\Omega}} and \\int e^{\\mathbf{J}_M{(f_{\\mathbf{v}},\\Omega)}} df_{\\mathbf{v}} = \\int e^{f_{\\mathbf{v}} + e^{\\Omega}} df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["exp", 2], "Equality(exp(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True))), exp(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True))), exp(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 5, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(exp(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(exp(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(t,n_{2})} = n_{2} + t, then obtain 4 \\rho_{f}^{2}{(t,n_{2})} = 2 (2 n_{2} + 2 t) \\rho_{f}{(t,n_{2})}", "derivation": "\\rho_{f}{(t,n_{2})} = n_{2} + t and n_{2} + t + \\rho_{f}{(t,n_{2})} = 2 n_{2} + 2 t and 2 \\rho_{f}{(t,n_{2})} = 2 n_{2} + 2 t and 4 \\rho_{f}^{2}{(t,n_{2})} = 2 (2 n_{2} + 2 t) \\rho_{f}{(t,n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('n_2', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Add(Symbol('n_2', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Symbol('n_2', commutative=True), Symbol('t', commutative=True), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Integer(2), Symbol('n_2', commutative=True)), Mul(Integer(2), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Integer(2), Symbol('n_2', commutative=True)), Mul(Integer(2), Symbol('t', commutative=True))))"], [["times", 3, "Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Integer(2), Symbol('n_2', commutative=True)), Mul(Integer(2), Symbol('t', commutative=True))), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given k{(g_{\\varepsilon},\\mu)} = \\frac{\\partial}{\\partial g_{\\varepsilon}} \\mu^{g_{\\varepsilon}}, then derive k{(g_{\\varepsilon},\\mu)} = \\mu^{g_{\\varepsilon}} \\log{(\\mu)}, then obtain - \\mu + \\frac{\\partial}{\\partial g_{\\varepsilon}} \\mu^{g_{\\varepsilon}} = - \\mu + \\mu^{g_{\\varepsilon}} \\log{(\\mu)}", "derivation": "k{(g_{\\varepsilon},\\mu)} = \\frac{\\partial}{\\partial g_{\\varepsilon}} \\mu^{g_{\\varepsilon}} and k{(g_{\\varepsilon},\\mu)} = \\mu^{g_{\\varepsilon}} \\log{(\\mu)} and \\frac{\\partial}{\\partial g_{\\varepsilon}} \\mu^{g_{\\varepsilon}} = \\mu^{g_{\\varepsilon}} \\log{(\\mu)} and - \\mu + \\frac{\\partial}{\\partial g_{\\varepsilon}} \\mu^{g_{\\varepsilon}} = - \\mu + \\mu^{g_{\\varepsilon}} \\log{(\\mu)}", "srepr_derivation": [["get_premise", "Equality(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["minus", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\tilde{g},h)} = \\sin{(\\frac{h}{\\tilde{g}})}, then obtain 1 = ((0^{\\tilde{g}})^{\\tilde{g}})^{\\tilde{g}}", "derivation": "\\mathbf{B}{(\\tilde{g},h)} = \\sin{(\\frac{h}{\\tilde{g}})} and \\mathbf{B}{(\\tilde{g},h)} - \\sin{(\\frac{h}{\\tilde{g}})} = 0 and (\\mathbf{B}{(\\tilde{g},h)} - \\sin{(\\frac{h}{\\tilde{g}})})^{\\tilde{g}} = 0^{\\tilde{g}} and ((\\mathbf{B}{(\\tilde{g},h)} - \\sin{(\\frac{h}{\\tilde{g}})})^{\\tilde{g}})^{\\tilde{g}} = (0^{\\tilde{g}})^{\\tilde{g}} and 1 = ((\\mathbf{B}{(\\tilde{g},h)} - \\sin{(\\frac{h}{\\tilde{g}})})^{\\tilde{g}})^{\\tilde{g}} and 1 = (0^{\\tilde{g}})^{\\tilde{g}} and 1 = ((0^{\\tilde{g}})^{\\tilde{g}})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))"], [["minus", 1, "sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{B}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Integer(0), Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\mathbf{B}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Function('\\\\mathbf{B}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('h', commutative=True))))), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 6, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integer(1), Pow(Pow(Pow(Integer(0), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\nabla{(b)} = b, then obtain 3 \\nabla{(b)} = 2 b + \\nabla{(b)}", "derivation": "\\nabla{(b)} = b and 2 \\nabla{(b)} = b + \\nabla{(b)} and 3 \\nabla{(b)} = b + 2 \\nabla{(b)} and 3 \\nabla{(b)} = 2 b + \\nabla{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('b', commutative=True)), Symbol('b', commutative=True))"], [["add", 1, "Function('\\\\nabla')(Symbol('b', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('b', commutative=True))), Add(Symbol('b', commutative=True), Function('\\\\nabla')(Symbol('b', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('\\\\nabla')(Symbol('b', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\nabla')(Symbol('b', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(2), Function('\\\\nabla')(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\nabla')(Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('b', commutative=True)), Function('\\\\nabla')(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(k,L)} = L - k and \\dot{x}{(\\mathbf{M},L)} = L \\mathbf{M}, then obtain \\frac{\\partial}{\\partial L} (L - k) (\\int \\dot{x}{(\\mathbf{M},L)} dL)^{L} = \\frac{\\partial}{\\partial L} (L - k) (\\int L \\mathbf{M} dL)^{L}", "derivation": "\\operatorname{P_{g}}{(k,L)} = L - k and \\dot{x}{(\\mathbf{M},L)} = L \\mathbf{M} and \\int \\dot{x}{(\\mathbf{M},L)} dL = \\int L \\mathbf{M} dL and (\\int \\dot{x}{(\\mathbf{M},L)} dL)^{L} = (\\int L \\mathbf{M} dL)^{L} and \\operatorname{P_{g}}{(k,L)} (\\int \\dot{x}{(\\mathbf{M},L)} dL)^{L} = \\operatorname{P_{g}}{(k,L)} (\\int L \\mathbf{M} dL)^{L} and (L - k) (\\int \\dot{x}{(\\mathbf{M},L)} dL)^{L} = (L - k) (\\int L \\mathbf{M} dL)^{L} and \\frac{\\partial}{\\partial L} (L - k) (\\int \\dot{x}{(\\mathbf{M},L)} dL)^{L} = \\frac{\\partial}{\\partial L} (L - k) (\\int L \\mathbf{M} dL)^{L}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('k', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["times", 4, "Function('P_g')(Symbol('k', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('k', commutative=True), Symbol('L', commutative=True)), Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Function('P_g')(Symbol('k', commutative=True), Symbol('L', commutative=True)), Pow(Integral(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Pow(Integral(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Pow(Integral(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(\\mathbb{I},\\mathbf{H})} = \\log{(\\mathbb{I})}^{\\mathbf{H}}, then obtain (\\int - \\mathbb{I} d\\mathbf{H})^{\\mathbf{H}} = (\\int (- \\mathbb{I} - \\pi{(\\mathbb{I},\\mathbf{H})} + \\log{(\\mathbb{I})}^{\\mathbf{H}}) d\\mathbf{H})^{\\mathbf{H}}", "derivation": "\\pi{(\\mathbb{I},\\mathbf{H})} = \\log{(\\mathbb{I})}^{\\mathbf{H}} and - \\mathbb{I} = - \\mathbb{I} - \\pi{(\\mathbb{I},\\mathbf{H})} + \\log{(\\mathbb{I})}^{\\mathbf{H}} and \\int - \\mathbb{I} d\\mathbf{H} = \\int (- \\mathbb{I} - \\pi{(\\mathbb{I},\\mathbf{H})} + \\log{(\\mathbb{I})}^{\\mathbf{H}}) d\\mathbf{H} and (\\int - \\mathbb{I} d\\mathbf{H})^{\\mathbf{H}} = (\\int (- \\mathbb{I} - \\pi{(\\mathbb{I},\\mathbf{H})} + \\log{(\\mathbb{I})}^{\\mathbf{H}}) d\\mathbf{H})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\pi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(Z,\\psi^*)} = \\sin{(\\frac{Z}{\\psi^*})}, then obtain \\frac{\\partial}{\\partial \\psi^*} \\frac{- \\psi^* + \\tilde{g}^{\\psi^*}{(Z,\\psi^*)}}{\\psi^*} = \\frac{\\partial}{\\partial \\psi^*} \\frac{- \\psi^* + \\sin^{\\psi^*}{(\\frac{Z}{\\psi^*})}}{\\psi^*}", "derivation": "\\tilde{g}{(Z,\\psi^*)} = \\sin{(\\frac{Z}{\\psi^*})} and \\tilde{g}^{\\psi^*}{(Z,\\psi^*)} = \\sin^{\\psi^*}{(\\frac{Z}{\\psi^*})} and - \\psi^* + \\tilde{g}^{\\psi^*}{(Z,\\psi^*)} = - \\psi^* + \\sin^{\\psi^*}{(\\frac{Z}{\\psi^*})} and \\frac{- \\psi^* + \\tilde{g}^{\\psi^*}{(Z,\\psi^*)}}{\\psi^*} = \\frac{- \\psi^* + \\sin^{\\psi^*}{(\\frac{Z}{\\psi^*})}}{\\psi^*} and \\frac{\\partial}{\\partial \\psi^*} \\frac{- \\psi^* + \\tilde{g}^{\\psi^*}{(Z,\\psi^*)}}{\\psi^*} = \\frac{\\partial}{\\partial \\psi^*} \\frac{- \\psi^* + \\sin^{\\psi^*}{(\\frac{Z}{\\psi^*})}}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), sin(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Symbol('\\\\psi^*', commutative=True))))"], [["times", 3, "Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('Z', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\psi^*,m_{s})} = \\sin{(\\psi^* - m_{s})}, then obtain (\\frac{\\mathbf{P}}{\\sigma_x} - \\psi + \\dot{x}{(\\psi^*,m_{s})} f{(c,f^{\\prime})})^{\\psi} = (\\frac{\\mathbf{P}}{\\sigma_x} - \\psi + f{(c,f^{\\prime})} \\sin{(\\psi^* - m_{s})})^{\\psi}", "derivation": "\\dot{x}{(\\psi^*,m_{s})} = \\sin{(\\psi^* - m_{s})} and \\dot{x}{(\\psi^*,m_{s})} f{(c,f^{\\prime})} = f{(c,f^{\\prime})} \\sin{(\\psi^* - m_{s})} and \\frac{\\mathbf{P}}{\\sigma_x} - \\psi + \\dot{x}{(\\psi^*,m_{s})} f{(c,f^{\\prime})} = \\frac{\\mathbf{P}}{\\sigma_x} - \\psi + f{(c,f^{\\prime})} \\sin{(\\psi^* - m_{s})} and (\\frac{\\mathbf{P}}{\\sigma_x} - \\psi + \\dot{x}{(\\psi^*,m_{s})} f{(c,f^{\\prime})})^{\\psi} = (\\frac{\\mathbf{P}}{\\sigma_x} - \\psi + f{(c,f^{\\prime})} \\sin{(\\psi^* - m_{s})})^{\\psi}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('m_s', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"], [["times", 1, "Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('m_s', commutative=True)), Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"], [["add", 2, "Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('m_s', commutative=True)), Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('m_s', commutative=True)), Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Function('f')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(i,z^{*})} = i + z^{*}, then obtain \\cos{(\\sin{(\\operatorname{a^{\\dagger}}^{z^{*}}{(i,z^{*})})})} = \\cos{(\\sin{((i + z^{*})^{z^{*}})})}", "derivation": "\\operatorname{a^{\\dagger}}{(i,z^{*})} = i + z^{*} and \\operatorname{a^{\\dagger}}^{z^{*}}{(i,z^{*})} = (i + z^{*})^{z^{*}} and \\sin{(\\operatorname{a^{\\dagger}}^{z^{*}}{(i,z^{*})})} = \\sin{((i + z^{*})^{z^{*}})} and \\cos{(\\sin{(\\operatorname{a^{\\dagger}}^{z^{*}}{(i,z^{*})})})} = \\cos{(\\sin{((i + z^{*})^{z^{*}})})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('i', commutative=True), Symbol('z^*', commutative=True)))"], [["power", 1, "Symbol('z^*', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('a^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))), sin(Pow(Add(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))))"], [["cos", 3], "Equality(cos(sin(Pow(Function('a^{\\\\dagger}')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))), cos(sin(Pow(Add(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given G{(\\Omega,f^{*},F_{N})} = f^{*} (F_{N} + \\Omega), then obtain F_{N} + \\Omega = F_{N} + \\Omega + 2 f^{*} (F_{N} + \\Omega) - 2 G{(\\Omega,f^{*},F_{N})}", "derivation": "G{(\\Omega,f^{*},F_{N})} = f^{*} (F_{N} + \\Omega) and F_{N} + \\Omega + G{(\\Omega,f^{*},F_{N})} = F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) and F_{N} + \\Omega = F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) - G{(\\Omega,f^{*},F_{N})} and F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) = F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) + f^{*} (F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) - G{(\\Omega,f^{*},F_{N})}) - G{(\\Omega,f^{*},F_{N})} and F_{N} + \\Omega = F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) + f^{*} (F_{N} + \\Omega + f^{*} (F_{N} + \\Omega) - G{(\\Omega,f^{*},F_{N})}) - 2 G{(\\Omega,f^{*},F_{N})} and F_{N} + \\Omega = F_{N} + \\Omega + 2 f^{*} (F_{N} + \\Omega) - 2 G{(\\Omega,f^{*},F_{N})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["minus", 2, "Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True))))), Mul(Integer(-1), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True))))), Mul(Integer(-1), Integer(2), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integer(2), Function('G')(Symbol('\\\\Omega', commutative=True), Symbol('f^*', commutative=True), Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hbar)} = e^{\\hbar}, then derive - \\operatorname{v_{2}}{(\\hbar)} + \\int \\operatorname{v_{2}}{(\\hbar)} d\\hbar = \\mathbf{A} - \\operatorname{v_{2}}{(\\hbar)} + e^{\\hbar}, then obtain - \\operatorname{v_{2}}{(\\hbar)} + \\int e^{\\hbar} d\\hbar = \\mathbf{A} - \\operatorname{v_{2}}{(\\hbar)} + e^{\\hbar}", "derivation": "\\operatorname{v_{2}}{(\\hbar)} = e^{\\hbar} and \\int \\operatorname{v_{2}}{(\\hbar)} d\\hbar = \\int e^{\\hbar} d\\hbar and - \\operatorname{v_{2}}{(\\hbar)} + \\int \\operatorname{v_{2}}{(\\hbar)} d\\hbar = - \\operatorname{v_{2}}{(\\hbar)} + \\int e^{\\hbar} d\\hbar and - \\operatorname{v_{2}}{(\\hbar)} + \\int \\operatorname{v_{2}}{(\\hbar)} d\\hbar = \\mathbf{A} - \\operatorname{v_{2}}{(\\hbar)} + e^{\\hbar} and - \\operatorname{v_{2}}{(\\hbar)} + \\int e^{\\hbar} d\\hbar = \\mathbf{A} - \\operatorname{v_{2}}{(\\hbar)} + e^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Function('v_2')(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), Integral(Function('v_2')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), Integral(Function('v_2')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), exp(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('v_2')(Symbol('\\\\hbar', commutative=True))), exp(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\theta_1)} = e^{\\theta_1} and \\hat{X}{(\\theta_1)} = \\frac{e^{\\theta_1}}{\\operatorname{n_{1}}{(\\theta_1)}}, then obtain 1 = \\hat{X}^{\\theta_1}{(\\theta_1)}", "derivation": "\\operatorname{n_{1}}{(\\theta_1)} = e^{\\theta_1} and 1 = \\frac{e^{\\theta_1}}{\\operatorname{n_{1}}{(\\theta_1)}} and \\hat{X}{(\\theta_1)} = \\frac{e^{\\theta_1}}{\\operatorname{n_{1}}{(\\theta_1)}} and 1 = \\hat{X}{(\\theta_1)} and 1 = \\hat{X}^{\\theta_1}{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["divide", 1, "Function('n_1')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('n_1')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Function('n_1')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integer(1), Pow(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(W)} = \\frac{d}{d W} \\cos{(W)}, then derive \\hat{x}_0^{W}{(W)} = (- \\sin{(W)})^{W}, then obtain (- \\sin{(W)})^{W} = (\\frac{d}{d W} \\cos{(W)})^{W}", "derivation": "\\hat{x}_0{(W)} = \\frac{d}{d W} \\cos{(W)} and \\hat{x}_0^{W}{(W)} = (\\frac{d}{d W} \\cos{(W)})^{W} and \\hat{x}_0^{W}{(W)} = (- \\sin{(W)})^{W} and (- \\sin{(W)})^{W} = (\\frac{d}{d W} \\cos{(W)})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('W', commutative=True)), Derivative(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Derivative(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Mul(Integer(-1), sin(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Derivative(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(M,n)} = \\log{(M - n)}, then obtain (\\frac{\\partial}{\\partial M} - n \\operatorname{c_{0}}{(M,n)})^{M} = (\\frac{\\partial}{\\partial M} - n \\log{(M - n)})^{M}", "derivation": "\\operatorname{c_{0}}{(M,n)} = \\log{(M - n)} and - n \\operatorname{c_{0}}{(M,n)} = - n \\log{(M - n)} and \\frac{\\partial}{\\partial M} - n \\operatorname{c_{0}}{(M,n)} = \\frac{\\partial}{\\partial M} - n \\log{(M - n)} and (\\frac{\\partial}{\\partial M} - n \\operatorname{c_{0}}{(M,n)})^{M} = (\\frac{\\partial}{\\partial M} - n \\log{(M - n)})^{M}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('M', commutative=True), Symbol('n', commutative=True)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["times", 1, "Mul(Integer(-1), Symbol('n', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('n', commutative=True), Function('c_0')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('n', commutative=True), Function('c_0')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('n', commutative=True), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(-1), Symbol('n', commutative=True), Function('c_0')(Symbol('M', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Integer(-1), Symbol('n', commutative=True), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\phi_2,\\varphi)} = \\frac{\\phi_2}{\\varphi} and i{(\\phi_2,\\varphi)} = \\frac{\\phi_2}{\\varphi}, then obtain 0 = \\frac{\\phi_2}{\\varphi} - \\hat{H}{(\\phi_2,\\varphi)}", "derivation": "\\hat{H}{(\\phi_2,\\varphi)} = \\frac{\\phi_2}{\\varphi} and i{(\\phi_2,\\varphi)} = \\frac{\\phi_2}{\\varphi} and i{(\\phi_2,\\varphi)} = \\hat{H}{(\\phi_2,\\varphi)} and 0 = \\frac{\\phi_2}{\\varphi} - i{(\\phi_2,\\varphi)} and 0 = \\frac{\\phi_2}{\\varphi} - \\hat{H}{(\\phi_2,\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["minus", 2, "Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(A_{2},\\theta_2)} = A_{2} + \\theta_2, then derive - \\int \\Psi_{nl}{(A_{2},\\theta_2)} dA_{2} = - \\frac{A_{2}^{2}}{2} - A_{2} \\theta_2 - i, then obtain -1 = \\frac{\\partial}{\\partial i} - \\int (A_{2} + \\theta_2) dA_{2}", "derivation": "\\Psi_{nl}{(A_{2},\\theta_2)} = A_{2} + \\theta_2 and \\int \\Psi_{nl}{(A_{2},\\theta_2)} dA_{2} = \\int (A_{2} + \\theta_2) dA_{2} and - \\int \\Psi_{nl}{(A_{2},\\theta_2)} dA_{2} = - \\int (A_{2} + \\theta_2) dA_{2} and - \\int \\Psi_{nl}{(A_{2},\\theta_2)} dA_{2} = - \\frac{A_{2}^{2}}{2} - A_{2} \\theta_2 - i and - \\frac{A_{2}^{2}}{2} - A_{2} \\theta_2 - i = - \\int (A_{2} + \\theta_2) dA_{2} and \\frac{\\partial}{\\partial i} (- \\frac{A_{2}^{2}}{2} - A_{2} \\theta_2 - i) = \\frac{\\partial}{\\partial i} - \\int (A_{2} + \\theta_2) dA_{2} and -1 = \\frac{\\partial}{\\partial i} - \\int (A_{2} + \\theta_2) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\Psi_{nl}')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\Psi_{nl}')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(-1), Derivative(Mul(Integer(-1), Integral(Add(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain (\\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}}) e^{\\mathbb{I}} + \\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}} = \\operatorname{A_{2}}{(\\mathbb{I})} + 2 e^{2 \\mathbb{I}} + e^{\\mathbb{I}}", "derivation": "\\operatorname{A_{2}}{(\\mathbb{I})} = e^{\\mathbb{I}} and \\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}} = 2 e^{\\mathbb{I}} and (\\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}}) e^{\\mathbb{I}} = 2 e^{2 \\mathbb{I}} and (\\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}}) e^{\\mathbb{I}} + \\operatorname{A_{2}}{(\\mathbb{I})} + e^{\\mathbb{I}} = \\operatorname{A_{2}}{(\\mathbb{I})} + 2 e^{2 \\mathbb{I}} + e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["add", 3, "Add(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Mul(Add(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), exp(Symbol('\\\\mathbb{I}', commutative=True))), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Add(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))), exp(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(A_{2})} = e^{A_{2}}, then obtain \\operatorname{a^{\\dagger}}{(A_{2})} \\int \\operatorname{a^{\\dagger}}{(A_{2})} dA_{2} = e^{A_{2}} \\int \\operatorname{a^{\\dagger}}{(A_{2})} dA_{2}", "derivation": "\\operatorname{a^{\\dagger}}{(A_{2})} = e^{A_{2}} and \\int \\operatorname{a^{\\dagger}}{(A_{2})} dA_{2} = \\int e^{A_{2}} dA_{2} and \\operatorname{a^{\\dagger}}{(A_{2})} \\int e^{A_{2}} dA_{2} = e^{A_{2}} \\int e^{A_{2}} dA_{2} and \\operatorname{a^{\\dagger}}{(A_{2})} \\int \\operatorname{a^{\\dagger}}{(A_{2})} dA_{2} = e^{A_{2}} \\int \\operatorname{a^{\\dagger}}{(A_{2})} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Mul(exp(Symbol('A_2', commutative=True)), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Mul(exp(Symbol('A_2', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\delta{(l)} = \\cos{(l)}, then obtain \\delta^{l}{(l)} \\cos^{2}{(l)} \\cos^{l}{(l)} + \\cos{(l)} \\cos^{l}{(l)} = \\cos^{2}{(l)} \\cos^{2 l}{(l)} + \\cos{(l)} \\cos^{l}{(l)}", "derivation": "\\delta{(l)} = \\cos{(l)} and \\delta^{l}{(l)} = \\cos^{l}{(l)} and \\delta^{l}{(l)} \\cos{(l)} = \\cos{(l)} \\cos^{l}{(l)} and \\delta^{l}{(l)} \\cos^{2}{(l)} \\cos^{l}{(l)} = \\cos^{2}{(l)} \\cos^{2 l}{(l)} and \\delta^{l}{(l)} \\cos^{2}{(l)} \\cos^{l}{(l)} + \\cos{(l)} \\cos^{l}{(l)} = \\cos^{2}{(l)} \\cos^{2 l}{(l)} + \\cos{(l)} \\cos^{l}{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["times", 2, "cos(Symbol('l', commutative=True))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["times", 3, "Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Mul(Pow(cos(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True)))))"], [["add", 4, "Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Add(Mul(Pow(cos(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True)))), Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hbar)} = \\log{(\\hbar)}, then derive \\int \\operatorname{F_{g}}{(\\hbar)} d\\hbar = \\hbar \\log{(\\hbar)} - \\hbar + t_{2}, then obtain \\cos{(\\int \\log{(\\hbar)} d\\hbar)} = \\cos{(\\hbar \\log{(\\hbar)} - \\hbar + t_{2})}", "derivation": "\\operatorname{F_{g}}{(\\hbar)} = \\log{(\\hbar)} and \\int \\operatorname{F_{g}}{(\\hbar)} d\\hbar = \\int \\log{(\\hbar)} d\\hbar and \\int \\operatorname{F_{g}}{(\\hbar)} d\\hbar = \\hbar \\log{(\\hbar)} - \\hbar + t_{2} and \\int \\log{(\\hbar)} d\\hbar = \\hbar \\log{(\\hbar)} - \\hbar + t_{2} and \\cos{(\\int \\log{(\\hbar)} d\\hbar)} = \\cos{(\\hbar \\log{(\\hbar)} - \\hbar + t_{2})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_g')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('t_2', commutative=True)))"], [["cos", 4], "Equality(cos(Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), cos(Add(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mu_0)} = \\log{(\\mu_0)}, then obtain \\int (\\int \\mathbf{J}_f{(\\mu_0)} d\\mu_0)^{\\mu_0} d\\mu_0 = \\int (\\int \\log{(\\mu_0)} d\\mu_0)^{\\mu_0} d\\mu_0", "derivation": "\\mathbf{J}_f{(\\mu_0)} = \\log{(\\mu_0)} and \\int \\mathbf{J}_f{(\\mu_0)} d\\mu_0 = \\int \\log{(\\mu_0)} d\\mu_0 and (\\int \\mathbf{J}_f{(\\mu_0)} d\\mu_0)^{\\mu_0} = (\\int \\log{(\\mu_0)} d\\mu_0)^{\\mu_0} and \\int (\\int \\mathbf{J}_f{(\\mu_0)} d\\mu_0)^{\\mu_0} d\\mu_0 = \\int (\\int \\log{(\\mu_0)} d\\mu_0)^{\\mu_0} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C_{d})} = \\sin{(e^{C_{d}})}, then obtain \\frac{\\int (- \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})})^{C_{d}} dC_{d}}{C_{d}} = \\frac{\\Psi}{C_{d}}", "derivation": "\\operatorname{F_{x}}{(C_{d})} = \\sin{(e^{C_{d}})} and \\operatorname{F_{x}}{(C_{d})} - \\sin{(e^{C_{d}})} = 0 and - \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})} = 0 and (- \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})})^{C_{d}} = 0^{C_{d}} and \\int (- \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})})^{C_{d}} dC_{d} = \\int 0^{C_{d}} dC_{d} and \\frac{\\int (- \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})})^{C_{d}} dC_{d}}{C_{d}} = \\frac{\\int 0^{C_{d}} dC_{d}}{C_{d}} and \\frac{\\int (- \\operatorname{F_{x}}{(C_{d})} + \\sin{(e^{C_{d}})})^{C_{d}} dC_{d}}{C_{d}} = \\frac{\\Psi}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C_d', commutative=True)), sin(exp(Symbol('C_d', commutative=True))))"], [["minus", 1, "sin(exp(Symbol('C_d', commutative=True)))"], "Equality(Add(Function('F_x')(Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('C_d', commutative=True))))), Integer(0))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('C_d', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Pow(Integer(0), Symbol('C_d', commutative=True)))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Pow(Integer(0), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 5, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Pow(Integer(0), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(x^\\prime)} = x^\\prime, then derive \\frac{d^{2}}{d (x^\\prime)^{2}} \\int \\hat{H}_{\\lambda}{(x^\\prime)} dx^\\prime = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (\\dot{y} + \\frac{(x^\\prime)^{2}}{2}), then obtain \\frac{d^{2}}{d (x^\\prime)^{2}} \\int x^\\prime dx^\\prime = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (\\dot{y} + \\frac{(x^\\prime)^{2}}{2})", "derivation": "\\hat{H}_{\\lambda}{(x^\\prime)} = x^\\prime and \\int \\hat{H}_{\\lambda}{(x^\\prime)} dx^\\prime = \\int x^\\prime dx^\\prime and \\frac{d}{d x^\\prime} \\int \\hat{H}_{\\lambda}{(x^\\prime)} dx^\\prime = \\frac{d}{d x^\\prime} \\int x^\\prime dx^\\prime and \\frac{d^{2}}{d (x^\\prime)^{2}} \\int \\hat{H}_{\\lambda}{(x^\\prime)} dx^\\prime = \\frac{d^{2}}{d (x^\\prime)^{2}} \\int x^\\prime dx^\\prime and \\frac{d^{2}}{d (x^\\prime)^{2}} \\int \\hat{H}_{\\lambda}{(x^\\prime)} dx^\\prime = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (\\dot{y} + \\frac{(x^\\prime)^{2}}{2}) and \\frac{d^{2}}{d (x^\\prime)^{2}} \\int x^\\prime dx^\\prime = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (\\dot{y} + \\frac{(x^\\prime)^{2}}{2})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{p},U)} = \\cos{(U - \\mathbf{p})}, then obtain U - \\mathbf{p} = (\\frac{\\cos{(U - \\mathbf{p})}}{\\dot{y}{(\\mathbf{p},U)}})^{U} (U - \\mathbf{p})", "derivation": "\\dot{y}{(\\mathbf{p},U)} = \\cos{(U - \\mathbf{p})} and 1 = \\frac{\\cos{(U - \\mathbf{p})}}{\\dot{y}{(\\mathbf{p},U)}} and 1 = (\\frac{\\cos{(U - \\mathbf{p})}}{\\dot{y}{(\\mathbf{p},U)}})^{U} and U - \\mathbf{p} = (\\frac{\\cos{(U - \\mathbf{p})}}{\\dot{y}{(\\mathbf{p},U)}})^{U} (U - \\mathbf{p})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('U', commutative=True)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('U', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('U', commutative=True)), Integer(-1)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('U', commutative=True)), Integer(-1)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))), Symbol('U', commutative=True)))"], [["times", 3, "Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('U', commutative=True)), Integer(-1)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)} = \\hat{H}_l \\theta_2, then derive (\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)})^{\\theta_2} = \\theta_2^{\\theta_2}, then obtain \\theta_2^{\\theta_2} = (\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\theta_2)^{\\theta_2}", "derivation": "\\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)} = \\hat{H}_l \\theta_2 and \\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)} = \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\theta_2 and (\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)})^{\\theta_2} = (\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\theta_2)^{\\theta_2} and (\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{E_{x}}{(\\hat{H}_l,\\theta_2)})^{\\theta_2} = \\theta_2^{\\theta_2} and \\theta_2^{\\theta_2} = (\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\theta_2)^{\\theta_2}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Derivative(Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(\\chi,g)} = \\frac{\\partial}{\\partial \\chi} g^{\\chi} and \\tilde{g}{(\\chi,g)} = \\frac{\\partial}{\\partial \\chi} g^{\\chi}, then obtain \\int \\mathbf{J}^{\\chi}{(\\chi,g)} dg = \\int \\tilde{g}^{\\chi}{(\\chi,g)} dg", "derivation": "\\mathbf{J}{(\\chi,g)} = \\frac{\\partial}{\\partial \\chi} g^{\\chi} and \\mathbf{J}^{\\chi}{(\\chi,g)} = (\\frac{\\partial}{\\partial \\chi} g^{\\chi})^{\\chi} and \\tilde{g}{(\\chi,g)} = \\frac{\\partial}{\\partial \\chi} g^{\\chi} and \\mathbf{J}^{\\chi}{(\\chi,g)} = \\tilde{g}^{\\chi}{(\\chi,g)} and \\int \\mathbf{J}^{\\chi}{(\\chi,g)} dg = \\int \\tilde{g}^{\\chi}{(\\chi,g)} dg", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Derivative(Pow(Symbol('g', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Derivative(Pow(Symbol('g', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Derivative(Pow(Symbol('g', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(U)} = \\log{(\\log{(U)})}, then derive \\int \\mathbf{p}{(U)} dU = S + U \\log{(\\log{(U)})} - \\operatorname{li}{(U)}, then obtain \\log{(U)} \\int \\mathbf{p}{(U)} dU = (S + U \\mathbf{p}{(U)} - \\operatorname{li}{(U)}) \\log{(U)}", "derivation": "\\mathbf{p}{(U)} = \\log{(\\log{(U)})} and \\int \\mathbf{p}{(U)} dU = \\int \\log{(\\log{(U)})} dU and \\int \\mathbf{p}{(U)} dU = S + U \\log{(\\log{(U)})} - \\operatorname{li}{(U)} and \\int \\log{(\\log{(U)})} dU = S + U \\log{(\\log{(U)})} - \\operatorname{li}{(U)} and \\int \\log{(\\log{(U)})} dU = S + U \\mathbf{p}{(U)} - \\operatorname{li}{(U)} and \\int \\mathbf{p}{(U)} dU = S + U \\mathbf{p}{(U)} - \\operatorname{li}{(U)} and \\log{(U)} \\int \\mathbf{p}{(U)} dU = (S + U \\mathbf{p}{(U)} - \\operatorname{li}{(U)}) \\log{(U)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), log(log(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('U', commutative=True), log(log(Symbol('U', commutative=True)))), Mul(Integer(-1), li(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('U', commutative=True), log(log(Symbol('U', commutative=True)))), Mul(Integer(-1), li(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('U', commutative=True), Function('\\\\mathbf{p}')(Symbol('U', commutative=True))), Mul(Integer(-1), li(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('U', commutative=True), Function('\\\\mathbf{p}')(Symbol('U', commutative=True))), Mul(Integer(-1), li(Symbol('U', commutative=True)))))"], [["times", 6, "log(Symbol('U', commutative=True))"], "Equality(Mul(log(Symbol('U', commutative=True)), Integral(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Add(Symbol('S', commutative=True), Mul(Symbol('U', commutative=True), Function('\\\\mathbf{p}')(Symbol('U', commutative=True))), Mul(Integer(-1), li(Symbol('U', commutative=True)))), log(Symbol('U', commutative=True))))"]]}, {"prompt": "Given s{(i)} = \\sin{(i)} and \\hat{H}{(i)} = \\frac{i^{2} s{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}}, then obtain \\hat{H}{(i)} \\frac{d}{d i} i^{2} \\sin{(i)} = \\frac{i^{2} \\sin{(i)} \\frac{d}{d i} i^{2} \\sin{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}}", "derivation": "s{(i)} = \\sin{(i)} and i s{(i)} = i \\sin{(i)} and i^{2} s{(i)} = i^{2} \\sin{(i)} and \\frac{d}{d i} i^{2} s{(i)} = \\frac{d}{d i} i^{2} \\sin{(i)} and \\hat{H}{(i)} = \\frac{i^{2} s{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}} and \\hat{H}{(i)} = \\frac{i^{2} \\sin{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}} and \\hat{H}{(i)} \\frac{d}{d i} i^{2} s{(i)} = \\frac{i^{2} \\sin{(i)} \\frac{d}{d i} i^{2} s{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}} and \\hat{H}{(i)} \\frac{d}{d i} i^{2} \\sin{(i)} = \\frac{i^{2} \\sin{(i)} \\frac{d}{d i} i^{2} \\sin{(i)}}{i^{2} \\sin{(i)} + i \\sin{(i)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('s')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], [["times", 2, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(2)), Function('s')(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), Function('s')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))), Integer(-1)), Function('s')(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\hat{H}')(Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))), Integer(-1)), sin(Symbol('i', commutative=True))))"], [["times", 6, "Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), Function('s')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('i', commutative=True)), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), Function('s')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))), Integer(-1)), sin(Symbol('i', commutative=True)), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), Function('s')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Function('\\\\hat{H}')(Symbol('i', commutative=True)), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Pow(Symbol('i', commutative=True), Integer(2)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))), Integer(-1)), sin(Symbol('i', commutative=True)), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(F_{x})} = e^{F_{x}}, then obtain F_{x} \\frac{d}{d F_{x}} \\frac{s{(F_{x})}}{F_{x}} = F_{x} \\frac{d}{d F_{x}} \\frac{e^{F_{x}}}{F_{x}}", "derivation": "s{(F_{x})} = e^{F_{x}} and \\frac{s{(F_{x})}}{F_{x}} = \\frac{e^{F_{x}}}{F_{x}} and \\frac{d}{d F_{x}} \\frac{s{(F_{x})}}{F_{x}} = \\frac{d}{d F_{x}} \\frac{e^{F_{x}}}{F_{x}} and F_{x} \\frac{d}{d F_{x}} \\frac{s{(F_{x})}}{F_{x}} = F_{x} \\frac{d}{d F_{x}} \\frac{e^{F_{x}}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["divide", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('s')(Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('s')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 3, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('s')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Symbol('F_x', commutative=True), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given c{(a^{\\dagger},\\lambda)} = \\frac{e^{\\lambda}}{a^{\\dagger}} and \\Psi_{\\lambda}{(\\lambda)} = \\lambda, then obtain (\\frac{d}{d \\lambda} \\Psi_{\\lambda}{(\\lambda)} + \\frac{1}{a^{\\dagger}}) c^{\\lambda}{(a^{\\dagger},\\lambda)} = (\\frac{e^{\\lambda}}{a^{\\dagger}})^{\\lambda} (\\frac{d}{d \\lambda} \\Psi_{\\lambda}{(\\lambda)} + \\frac{1}{a^{\\dagger}})", "derivation": "c{(a^{\\dagger},\\lambda)} = \\frac{e^{\\lambda}}{a^{\\dagger}} and \\Psi_{\\lambda}{(\\lambda)} = \\lambda and c^{\\lambda}{(a^{\\dagger},\\lambda)} = (\\frac{e^{\\lambda}}{a^{\\dagger}})^{\\lambda} and \\frac{d}{d \\lambda} \\Psi_{\\lambda}{(\\lambda)} = \\frac{d}{d \\lambda} \\lambda and (\\frac{d}{d \\lambda} \\lambda + \\frac{1}{a^{\\dagger}}) c^{\\lambda}{(a^{\\dagger},\\lambda)} = (\\frac{e^{\\lambda}}{a^{\\dagger}})^{\\lambda} (\\frac{d}{d \\lambda} \\lambda + \\frac{1}{a^{\\dagger}}) and (\\frac{d}{d \\lambda} \\Psi_{\\lambda}{(\\lambda)} + \\frac{1}{a^{\\dagger}}) c^{\\lambda}{(a^{\\dagger},\\lambda)} = (\\frac{e^{\\lambda}}{a^{\\dagger}})^{\\lambda} (\\frac{d}{d \\lambda} \\Psi_{\\lambda}{(\\lambda)} + \\frac{1}{a^{\\dagger}})", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), exp(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["divide", 3, "Pow(Add(Derivative(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Integer(-1))"], "Equality(Mul(Add(Derivative(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Pow(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Add(Derivative(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Pow(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Add(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given U{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)}, then derive \\hat{p}_0 \\frac{d}{d \\hat{p}_0} U{(\\hat{p}_0)} + U{(\\hat{p}_0)} = \\hat{p}_0 \\cos{(\\hat{p}_0)} + \\sin{(\\hat{p}_0)}, then obtain \\hat{p}_0 \\frac{d}{d \\hat{p}_0} U{(\\hat{p}_0)} + U{(\\hat{p}_0)} = \\hat{p}_0 \\cos{(\\hat{p}_0)} + U{(\\hat{p}_0)}", "derivation": "U{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\hat{p}_0 U{(\\hat{p}_0)} = \\hat{p}_0 \\sin{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} \\hat{p}_0 U{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\hat{p}_0 \\sin{(\\hat{p}_0)} and \\hat{p}_0 \\frac{d}{d \\hat{p}_0} U{(\\hat{p}_0)} + U{(\\hat{p}_0)} = \\hat{p}_0 \\cos{(\\hat{p}_0)} + \\sin{(\\hat{p}_0)} and \\hat{p}_0 \\frac{d}{d \\hat{p}_0} U{(\\hat{p}_0)} + U{(\\hat{p}_0)} = \\hat{p}_0 \\cos{(\\hat{p}_0)} + U{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('U')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('U')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('U')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Function('U')(Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True))), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('U')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Function('U')(Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True))), Function('U')(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\omega{(v_{z},v_{t})} = \\frac{v_{t}}{v_{z}} and \\operatorname{P_{g}}{(v_{z},v_{t})} = 2 v_{t} + (\\frac{v_{t}}{v_{z}})^{v_{z}} + \\omega^{v_{z}}{(v_{z},v_{t})}, then obtain 2 v_{t} + 2 (\\frac{v_{t}}{v_{z}})^{v_{z}} = 2 v_{t} + 2 \\omega^{v_{z}}{(v_{z},v_{t})}", "derivation": "\\omega{(v_{z},v_{t})} = \\frac{v_{t}}{v_{z}} and \\omega^{v_{z}}{(v_{z},v_{t})} = (\\frac{v_{t}}{v_{z}})^{v_{z}} and \\operatorname{P_{g}}{(v_{z},v_{t})} = 2 v_{t} + (\\frac{v_{t}}{v_{z}})^{v_{z}} + \\omega^{v_{z}}{(v_{z},v_{t})} and \\operatorname{P_{g}}{(v_{z},v_{t})} = 2 v_{t} + 2 (\\frac{v_{t}}{v_{z}})^{v_{z}} and \\operatorname{P_{g}}{(v_{z},v_{t})} = 2 v_{t} + 2 \\omega^{v_{z}}{(v_{z},v_{t})} and 2 v_{t} + 2 (\\frac{v_{t}}{v_{z}})^{v_{z}} = 2 v_{t} + 2 \\omega^{v_{z}}{(v_{z},v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_z', commutative=True)), Pow(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(2), Symbol('v_t', commutative=True)), Pow(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Symbol('v_z', commutative=True)), Pow(Function('\\\\omega')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('P_g')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Pow(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('P_g')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Pow(Function('\\\\omega')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Pow(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Symbol('v_z', commutative=True)))), Add(Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Pow(Function('\\\\omega')(Symbol('v_z', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})} = (S^{\\hat{x}})^{f_{E}} and \\operatorname{c_{0}}{(\\Psi_{\\lambda})} = \\log{(\\sin{(\\Psi_{\\lambda})})}, then obtain \\frac{- (S^{\\hat{x}})^{f_{E}} + \\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})}}{\\operatorname{c_{0}}{(\\Psi_{\\lambda})}} = 0", "derivation": "\\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})} = (S^{\\hat{x}})^{f_{E}} and - (S^{\\hat{x}})^{f_{E}} + \\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})} = 0 and \\operatorname{c_{0}}{(\\Psi_{\\lambda})} = \\log{(\\sin{(\\Psi_{\\lambda})})} and \\frac{- (S^{\\hat{x}})^{f_{E}} + \\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})}}{\\log{(\\sin{(\\Psi_{\\lambda})})}} = 0 and \\frac{- (S^{\\hat{x}})^{f_{E}} + \\dot{\\mathbf{r}}{(\\hat{x},S,f_{E})}}{\\operatorname{c_{0}}{(\\Psi_{\\lambda})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Pow(Pow(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f_E', commutative=True)))"], [["minus", 1, "Pow(Pow(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f_E', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Integer(0))"], ["get_premise", "Equality(Function('c_0')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 2, "log(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Pow(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f_E', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Pow(log(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Pow(Pow(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f_E', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Pow(Function('c_0')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given L{(v_{2})} = e^{v_{2}} and \\hat{\\mathbf{x}}{(\\theta)} = \\cos{(\\theta)}, then obtain (\\theta L^{v_{2}}{(v_{2})} \\hat{\\mathbf{x}}{(\\theta)})^{\\theta} = (\\theta L^{v_{2}}{(v_{2})} \\cos{(\\theta)})^{\\theta}", "derivation": "L{(v_{2})} = e^{v_{2}} and L^{v_{2}}{(v_{2})} = (e^{v_{2}})^{v_{2}} and \\hat{\\mathbf{x}}{(\\theta)} = \\cos{(\\theta)} and \\theta \\hat{\\mathbf{x}}{(\\theta)} = \\theta \\cos{(\\theta)} and \\theta \\hat{\\mathbf{x}}{(\\theta)} (e^{v_{2}})^{v_{2}} = \\theta (e^{v_{2}})^{v_{2}} \\cos{(\\theta)} and \\theta L^{v_{2}}{(v_{2})} \\hat{\\mathbf{x}}{(\\theta)} = \\theta L^{v_{2}}{(v_{2})} \\cos{(\\theta)} and (\\theta L^{v_{2}}{(v_{2})} \\hat{\\mathbf{x}}{(\\theta)})^{\\theta} = (\\theta L^{v_{2}}{(v_{2})} \\cos{(\\theta)})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('L')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(exp(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["times", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), cos(Symbol('\\\\theta', commutative=True))))"], [["times", 4, "Pow(exp(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(exp(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('L')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Function('L')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["power", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('L')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('L')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\dot{x},\\rho_b)} = \\dot{x} - \\rho_b, then obtain \\iint (- \\dot{x} + \\rho_b - \\operatorname{v_{x}}{(\\dot{x},\\rho_b)}) d\\dot{x} d\\rho_b = \\iint (- 2 \\dot{x} + 2 \\rho_b) d\\dot{x} d\\rho_b", "derivation": "\\operatorname{v_{x}}{(\\dot{x},\\rho_b)} = \\dot{x} - \\rho_b and - \\operatorname{v_{x}}{(\\dot{x},\\rho_b)} = - \\dot{x} + \\rho_b and - \\dot{x} + \\rho_b - \\operatorname{v_{x}}{(\\dot{x},\\rho_b)} = - 2 \\dot{x} + 2 \\rho_b and \\int (- \\dot{x} + \\rho_b - \\operatorname{v_{x}}{(\\dot{x},\\rho_b)}) d\\dot{x} = \\int (- 2 \\dot{x} + 2 \\rho_b) d\\dot{x} and \\iint (- \\dot{x} + \\rho_b - \\operatorname{v_{x}}{(\\dot{x},\\rho_b)}) d\\dot{x} d\\rho_b = \\iint (- 2 \\dot{x} + 2 \\rho_b) d\\dot{x} d\\rho_b", "srepr_derivation": [["get_premise", "Equality(Function('v_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Function('v_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Function('v_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Function('v_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(c_{0},G)} = G + c_{0}, then derive \\int \\mathbf{p}{(c_{0},G)} dG = \\frac{G^{2}}{2} + G c_{0} + n_{2}, then obtain \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{p}{(c_{0},G)} dG + 1 = \\frac{\\partial}{\\partial n_{2}} \\int (G + c_{0}) dG + 1", "derivation": "\\mathbf{p}{(c_{0},G)} = G + c_{0} and \\int \\mathbf{p}{(c_{0},G)} dG = \\int (G + c_{0}) dG and \\int \\mathbf{p}{(c_{0},G)} dG = \\frac{G^{2}}{2} + G c_{0} + n_{2} and \\frac{G^{2}}{2} + G c_{0} + n_{2} = \\int (G + c_{0}) dG and \\frac{\\partial}{\\partial n_{2}} (\\frac{G^{2}}{2} + G c_{0} + n_{2}) = \\frac{\\partial}{\\partial n_{2}} \\int (G + c_{0}) dG and \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{p}{(c_{0},G)} dG = \\frac{\\partial}{\\partial n_{2}} \\int (G + c_{0}) dG and \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{p}{(c_{0},G)} dG + 1 = \\frac{\\partial}{\\partial n_{2}} \\int (G + c_{0}) dG + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(2))), Mul(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(2))), Mul(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Symbol('n_2', commutative=True)), Integral(Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 4, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(2))), Mul(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["add", 6, 1], "Equality(Add(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Add(Symbol('G', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})}, then derive \\operatorname{F_{c}}{(z^{*})} = - \\sin{(z^{*})}, then obtain (\\frac{\\operatorname{F_{c}}{(z^{*})}}{\\cos{(z^{*})}})^{z^{*}} = (- \\frac{\\sin{(z^{*})}}{\\cos{(z^{*})}})^{z^{*}}", "derivation": "\\operatorname{F_{c}}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\operatorname{F_{c}}{(z^{*})} = - \\sin{(z^{*})} and \\frac{\\operatorname{F_{c}}{(z^{*})}}{\\cos{(z^{*})}} = \\frac{\\frac{d}{d z^{*}} \\cos{(z^{*})}}{\\cos{(z^{*})}} and - \\sin{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{\\operatorname{F_{c}}{(z^{*})}}{\\cos{(z^{*})}} = - \\frac{\\sin{(z^{*})}}{\\cos{(z^{*})}} and (\\frac{\\operatorname{F_{c}}{(z^{*})}}{\\cos{(z^{*})}})^{z^{*}} = (- \\frac{\\sin{(z^{*})}}{\\cos{(z^{*})}})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('z^*', commutative=True)), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_c')(Symbol('z^*', commutative=True)), Mul(Integer(-1), sin(Symbol('z^*', commutative=True))))"], [["divide", 1, "cos(Symbol('z^*', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))), Mul(Pow(cos(Symbol('z^*', commutative=True)), Integer(-1)), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('z^*', commutative=True))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('F_c')(Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('z^*', commutative=True)"], "Equality(Pow(Mul(Function('F_c')(Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))), Symbol('z^*', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then derive \\int \\mathbf{H}{(L_{\\varepsilon})} dL_{\\varepsilon} = Q + e^{L_{\\varepsilon}}, then obtain Q (Q + \\mathbf{H}{(L_{\\varepsilon})}) + e^{L_{\\varepsilon}} = Q (Q + e^{L_{\\varepsilon}}) + e^{L_{\\varepsilon}}", "derivation": "\\mathbf{H}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\int \\mathbf{H}{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int e^{L_{\\varepsilon}} dL_{\\varepsilon} and \\int \\mathbf{H}{(L_{\\varepsilon})} dL_{\\varepsilon} = Q + e^{L_{\\varepsilon}} and \\int \\mathbf{H}{(L_{\\varepsilon})} dL_{\\varepsilon} = Q + \\mathbf{H}{(L_{\\varepsilon})} and Q + \\mathbf{H}{(L_{\\varepsilon})} = Q + e^{L_{\\varepsilon}} and Q (Q + \\mathbf{H}{(L_{\\varepsilon})}) = Q (Q + e^{L_{\\varepsilon}}) and Q (Q + \\mathbf{H}{(L_{\\varepsilon})}) + e^{L_{\\varepsilon}} = Q (Q + e^{L_{\\varepsilon}}) + e^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('Q', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('Q', commutative=True), Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('Q', commutative=True), Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('Q', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 5, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Add(Symbol('Q', commutative=True), Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('Q', commutative=True), Add(Symbol('Q', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["add", 6, "exp(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Add(Symbol('Q', commutative=True), Function('\\\\mathbf{H}')(Symbol('L_{\\\\varepsilon}', commutative=True)))), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('Q', commutative=True), Add(Symbol('Q', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\chi{(m)} = \\cos{(m)}, then derive \\frac{d}{d m} \\chi{(m)} = - \\sin{(m)}, then obtain \\log{((\\frac{d}{d m} - \\sin{(m)})^{2})} = \\log{((\\frac{d^{2}}{d m^{2}} \\chi{(m)})^{2})}", "derivation": "\\chi{(m)} = \\cos{(m)} and \\frac{d}{d m} \\chi{(m)} = \\frac{d}{d m} \\cos{(m)} and \\frac{d}{d m} \\chi{(m)} = - \\sin{(m)} and \\frac{d}{d m} \\cos{(m)} = - \\sin{(m)} and \\frac{d^{2}}{d m^{2}} \\cos{(m)} = \\frac{d}{d m} - \\sin{(m)} and (\\frac{d^{2}}{d m^{2}} \\cos{(m)})^{2} = (\\frac{d}{d m} - \\sin{(m)})^{2} and (\\frac{d^{2}}{d m^{2}} \\cos{(m)})^{2} = (\\frac{d^{2}}{d m^{2}} \\chi{(m)})^{2} and (\\frac{d}{d m} - \\sin{(m)})^{2} = (\\frac{d^{2}}{d m^{2}} \\chi{(m)})^{2} and \\log{((\\frac{d}{d m} - \\sin{(m)})^{2})} = \\log{((\\frac{d^{2}}{d m^{2}} \\chi{(m)})^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('m', commutative=True))))"], [["differentiate", 4, "Symbol('m', commutative=True)"], "Equality(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 5, 2], "Equality(Pow(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Function('\\\\chi')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Derivative(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Function('\\\\chi')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Integer(2)))"], [["log", 8], "Equality(log(Pow(Derivative(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2))), log(Pow(Derivative(Function('\\\\chi')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Integer(2))))"]]}, {"prompt": "Given s{(x)} = \\log{(x)}, then obtain ((- s{(x)} + 2 \\log{(x)})^{x} + \\log{(x)})^{x} = (\\log{(x)} + \\log{(x)}^{x})^{x}", "derivation": "s{(x)} = \\log{(x)} and \\log{(x)} = - s{(x)} + 2 \\log{(x)} and s^{x}{(x)} = \\log{(x)}^{x} and s^{x}{(x)} = (- s{(x)} + 2 \\log{(x)})^{x} and (- s{(x)} + 2 \\log{(x)})^{x} = \\log{(x)}^{x} and (- s{(x)} + 2 \\log{(x)})^{x} + \\log{(x)} = \\log{(x)} + \\log{(x)}^{x} and ((- s{(x)} + 2 \\log{(x)})^{x} + \\log{(x)})^{x} = (\\log{(x)} + \\log{(x)}^{x})^{x}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["minus", 1, "Add(Function('s')(Symbol('x', commutative=True)), Mul(Integer(-1), log(Symbol('x', commutative=True))))"], "Equality(log(Symbol('x', commutative=True)), Add(Mul(Integer(-1), Function('s')(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True)))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('s')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('s')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('s')(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True)))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Mul(Integer(-1), Function('s')(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["minus", 5, "Mul(Integer(-1), log(Symbol('x', commutative=True)))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('s')(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Add(log(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["power", 6, "Symbol('x', commutative=True)"], "Equality(Pow(Add(Pow(Add(Mul(Integer(-1), Function('s')(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Add(log(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given U{(E,\\mathbf{g})} = E + \\mathbf{g} and \\rho{(E,\\mathbf{g})} = (\\frac{\\partial}{\\partial E} (- E - \\mathbf{g} + U{(E,\\mathbf{g})}))^{\\mathbf{g}} and \\mathbf{J}{(\\mathbf{g})} = (\\frac{d}{d E} 0)^{\\mathbf{g}}, then obtain \\rho^{E}{(E,\\mathbf{g})} = \\mathbf{J}^{E}{(\\mathbf{g})}", "derivation": "U{(E,\\mathbf{g})} = E + \\mathbf{g} and 0 = E + \\mathbf{g} - U{(E,\\mathbf{g})} and - E - \\mathbf{g} + U{(E,\\mathbf{g})} = 0 and \\frac{\\partial}{\\partial E} (- E - \\mathbf{g} + U{(E,\\mathbf{g})}) = \\frac{d}{d E} 0 and \\rho{(E,\\mathbf{g})} = (\\frac{\\partial}{\\partial E} (- E - \\mathbf{g} + U{(E,\\mathbf{g})}))^{\\mathbf{g}} and \\mathbf{J}{(\\mathbf{g})} = (\\frac{d}{d E} 0)^{\\mathbf{g}} and \\mathbf{J}{(\\mathbf{g})} = (\\frac{\\partial}{\\partial E} (- E - \\mathbf{g} + U{(E,\\mathbf{g})}))^{\\mathbf{g}} and \\rho^{E}{(E,\\mathbf{g})} = ((\\frac{\\partial}{\\partial E} (- E - \\mathbf{g} + U{(E,\\mathbf{g})}))^{\\mathbf{g}})^{E} and \\rho^{E}{(E,\\mathbf{g})} = \\mathbf{J}^{E}{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["minus", 2, "Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 5, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('E', commutative=True)), Pow(Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('U')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Pow(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('E', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{s})} = e^{\\mathbf{s}} and E{(\\mathbf{s})} = - \\frac{\\operatorname{c_{0}}{(\\mathbf{s})}}{1 - e^{\\mathbf{s}}}, then obtain E{(\\mathbf{s})} = - \\frac{\\operatorname{c_{0}}{(\\mathbf{s})}}{1 - \\operatorname{c_{0}}{(\\mathbf{s})}}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{s})} = e^{\\mathbf{s}} and - \\operatorname{c_{0}}{(\\mathbf{s})} = - e^{\\mathbf{s}} and 1 - \\operatorname{c_{0}}{(\\mathbf{s})} = 1 - e^{\\mathbf{s}} and E{(\\mathbf{s})} = - \\frac{\\operatorname{c_{0}}{(\\mathbf{s})}}{1 - e^{\\mathbf{s}}} and E{(\\mathbf{s})} = - \\frac{\\operatorname{c_{0}}{(\\mathbf{s})}}{1 - \\operatorname{c_{0}}{(\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True)))), Integer(-1)), Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('E')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Add(Integer(1), Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)))), Integer(-1)), Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})} = \\frac{\\sin{(\\mathbf{J}_M)}}{F_{H}} and \\mathbf{S}{(\\mathbf{J}_M,F_{H})} = (F_{H} \\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})})^{F_{H}}, then obtain \\mathbf{S}{(\\mathbf{J}_M,F_{H})} = \\sin^{F_{H}}{(\\mathbf{J}_M)}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})} = \\frac{\\sin{(\\mathbf{J}_M)}}{F_{H}} and F_{H} \\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})} = \\sin{(\\mathbf{J}_M)} and (F_{H} \\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})})^{F_{H}} = \\sin^{F_{H}}{(\\mathbf{J}_M)} and \\mathbf{S}{(\\mathbf{J}_M,F_{H})} = (F_{H} \\operatorname{C_{2}}{(\\mathbf{J}_M,F_{H})})^{F_{H}} and \\mathbf{S}{(\\mathbf{J}_M,F_{H})} = \\sin^{F_{H}}{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 1, "Pow(Symbol('F_H', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('F_H', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True))), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Mul(Symbol('F_H', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('F_H', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Pow(Mul(Symbol('F_H', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\chi{(G,\\mathbf{v},F_{N})} = \\frac{F_{N} + G}{\\mathbf{v}} and \\tilde{g}^*{(G,\\mathbf{v},F_{N})} = \\int \\frac{F_{N} + G}{\\mathbf{v}} dG, then obtain \\frac{\\tilde{g}^*{(G,\\mathbf{v},F_{N})}}{\\mathbf{v}} = \\frac{\\int \\chi{(G,\\mathbf{v},F_{N})} dG}{\\mathbf{v}}", "derivation": "\\chi{(G,\\mathbf{v},F_{N})} = \\frac{F_{N} + G}{\\mathbf{v}} and \\int \\chi{(G,\\mathbf{v},F_{N})} dG = \\int \\frac{F_{N} + G}{\\mathbf{v}} dG and \\tilde{g}^*{(G,\\mathbf{v},F_{N})} = \\int \\frac{F_{N} + G}{\\mathbf{v}} dG and \\tilde{g}^*{(G,\\mathbf{v},F_{N})} = \\int \\chi{(G,\\mathbf{v},F_{N})} dG and \\frac{\\tilde{g}^*{(G,\\mathbf{v},F_{N})}}{\\mathbf{v}} = \\frac{\\int \\chi{(G,\\mathbf{v},F_{N})} dG}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('G', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Integral(Function('\\\\chi')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["divide", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Integral(Function('\\\\chi')(Symbol('G', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given z{(H,y)} = \\cos{(H y)}, then obtain y - z{(H,y)} - 1 = y - \\cos{(H y)} - 1", "derivation": "z{(H,y)} = \\cos{(H y)} and - y + z{(H,y)} = - y + \\cos{(H y)} and - y + z{(H,y)} + 1 = - y + \\cos{(H y)} + 1 and y - z{(H,y)} - 1 = y - \\cos{(H y)} - 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('H', commutative=True), Symbol('y', commutative=True)), cos(Mul(Symbol('H', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), cos(Mul(Symbol('H', commutative=True), Symbol('y', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('z')(Symbol('H', commutative=True), Symbol('y', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), cos(Mul(Symbol('H', commutative=True), Symbol('y', commutative=True))), Integer(1)))"], [["times", 3, "Integer(-1)"], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('z')(Symbol('H', commutative=True), Symbol('y', commutative=True))), Integer(-1)), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Mul(Symbol('H', commutative=True), Symbol('y', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\rho_b,v_{t})} = \\sin{(\\rho_b - v_{t})} and T{(\\rho_b,v_{t})} = \\sin{(\\rho_b - v_{t})}, then obtain 1 = \\frac{- v_{t} + T{(\\rho_b,v_{t})}}{- v_{t} + \\operatorname{v_{z}}{(\\rho_b,v_{t})}}", "derivation": "\\operatorname{v_{z}}{(\\rho_b,v_{t})} = \\sin{(\\rho_b - v_{t})} and - v_{t} + \\operatorname{v_{z}}{(\\rho_b,v_{t})} = - v_{t} + \\sin{(\\rho_b - v_{t})} and T{(\\rho_b,v_{t})} = \\sin{(\\rho_b - v_{t})} and 1 = \\frac{- v_{t} + \\sin{(\\rho_b - v_{t})}}{- v_{t} + \\operatorname{v_{z}}{(\\rho_b,v_{t})}} and 1 = \\frac{- v_{t} + T{(\\rho_b,v_{t})}}{- v_{t} + \\operatorname{v_{z}}{(\\rho_b,v_{t})}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["add", 1, "Mul(Integer(-1), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_z')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), sin(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_z')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_z')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), sin(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('T')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v_z')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(q)} = \\sin{(q)}, then obtain 0 = (- \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq) \\operatorname{y^{\\prime}}{(q)} \\sin^{q}{(q)}", "derivation": "\\operatorname{y^{\\prime}}{(q)} = \\sin{(q)} and \\operatorname{y^{\\prime}}^{q}{(q)} = \\sin^{q}{(q)} and \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq = \\int \\sin^{q}{(q)} dq and 0 = - \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq and 0 = (- \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq) \\sin{(q)} and 0 = (- \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq) \\operatorname{y^{\\prime}}{(q)} and 0 = (- \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq) \\operatorname{y^{\\prime}}{(q)} \\operatorname{y^{\\prime}}^{q}{(q)} and 0 = (- \\int \\operatorname{y^{\\prime}}^{q}{(q)} dq + \\int \\sin^{q}{(q)} dq) \\operatorname{y^{\\prime}}{(q)} \\sin^{q}{(q)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["minus", 3, "Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["times", 4, "sin(Symbol('q', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), sin(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Function('y^{\\\\prime}')(Symbol('q', commutative=True))))"], [["times", 6, "Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integral(Pow(Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Function('y^{\\\\prime}')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\Psi{(a,\\omega)} = \\sin{(\\frac{a}{\\omega})}, then derive \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)})} = - \\sin{(\\frac{a \\Psi{(a,\\omega)} \\cos{(\\frac{a}{\\omega})}}{\\omega^{2}})}, then obtain \\frac{\\partial}{\\partial \\omega} \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)})} = \\frac{\\partial}{\\partial \\omega} - \\sin{(\\frac{a \\Psi{(a,\\omega)} \\cos{(\\frac{a}{\\omega})}}{\\omega^{2}})}", "derivation": "\\Psi{(a,\\omega)} = \\sin{(\\frac{a}{\\omega})} and \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\sin{(\\frac{a}{\\omega})} and \\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)} = \\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\sin{(\\frac{a}{\\omega})} and \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)})} = \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\sin{(\\frac{a}{\\omega})})} and \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)})} = - \\sin{(\\frac{a \\Psi{(a,\\omega)} \\cos{(\\frac{a}{\\omega})}}{\\omega^{2}})} and \\frac{\\partial}{\\partial \\omega} \\sin{(\\Psi{(a,\\omega)} \\frac{\\partial}{\\partial \\omega} \\Psi{(a,\\omega)})} = \\frac{\\partial}{\\partial \\omega} - \\sin{(\\frac{a \\Psi{(a,\\omega)} \\cos{(\\frac{a}{\\omega})}}{\\omega^{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(sin(Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), sin(Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(sin(Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Symbol('a', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))))"], [["differentiate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(sin(Mul(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Symbol('a', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(y)} = e^{y}, then obtain - \\frac{W{(y)} W^{y}{(y)}}{y I{(\\mathbf{r},\\mathbf{B})}} = - \\frac{W^{y}{(y)} e^{y}}{y I{(\\mathbf{r},\\mathbf{B})}}", "derivation": "W{(y)} = e^{y} and \\frac{W{(y)}}{y} = \\frac{e^{y}}{y} and - \\frac{W{(y)}}{y} = - \\frac{e^{y}}{y} and W^{y}{(y)} = (e^{y})^{y} and - \\frac{W{(y)} W^{y}{(y)}}{y} = - \\frac{W^{y}{(y)} e^{y}}{y} and - \\frac{W{(y)} (e^{y})^{y}}{y} = - \\frac{e^{y} (e^{y})^{y}}{y} and - \\frac{W{(y)} (e^{y})^{y}}{y I{(\\mathbf{r},\\mathbf{B})}} = - \\frac{e^{y} (e^{y})^{y}}{y I{(\\mathbf{r},\\mathbf{B})}} and - \\frac{W{(y)} W^{y}{(y)}}{y I{(\\mathbf{r},\\mathbf{B})}} = - \\frac{W^{y}{(y)} e^{y}}{y I{(\\mathbf{r},\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('W')(Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), exp(Symbol('y', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('W')(Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), exp(Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["times", 3, "Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('W')(Symbol('y', commutative=True)), Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), exp(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('W')(Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), exp(Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["divide", 6, "Function('I')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Function('W')(Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), exp(Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Function('W')(Symbol('y', commutative=True)), Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Pow(Function('W')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), exp(Symbol('y', commutative=True))))"]]}, {"prompt": "Given A{(\\mu)} = \\sin{(\\mu)}, then obtain \\mu (\\mu + 1) - \\sin^{2}{(\\mu)} = \\mu (\\mu - 1 + \\frac{2 \\sin{(\\mu)}}{A{(\\mu)}}) - \\sin^{2}{(\\mu)}", "derivation": "A{(\\mu)} = \\sin{(\\mu)} and A{(\\mu)} \\sin{(\\mu)} = \\sin^{2}{(\\mu)} and 1 = \\frac{\\sin{(\\mu)}}{A{(\\mu)}} and \\mu + 1 = \\mu + \\frac{\\sin{(\\mu)}}{A{(\\mu)}} and \\mu (\\mu + 1) = \\mu (\\mu + \\frac{\\sin{(\\mu)}}{A{(\\mu)}}) and \\mu (\\mu + \\frac{\\sin{(\\mu)}}{A{(\\mu)}}) = \\mu (\\mu - 1 + \\frac{2 \\sin{(\\mu)}}{A{(\\mu)}}) and \\mu (\\mu + 1) = \\mu (\\mu - 1 + \\frac{2 \\sin{(\\mu)}}{A{(\\mu)}}) and \\mu (\\mu + 1) - \\sin^{2}{(\\mu)} = \\mu (\\mu - 1 + \\frac{2 \\sin{(\\mu)}}{A{(\\mu)}}) - \\sin^{2}{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('A')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(2)))"], [["divide", 2, "Mul(Function('A')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))"], [["add", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Integer(1)), Add(Symbol('\\\\mu', commutative=True), Mul(Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True)))))"], [["times", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Mul(Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Mul(Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))), Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(-1), Mul(Integer(2), Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(-1), Mul(Integer(2), Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))))"], [["minus", 7, "Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(2))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(2)))), Add(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Integer(-1), Mul(Integer(2), Pow(Function('A')(Symbol('\\\\mu', commutative=True)), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\theta)} = \\theta and \\mathbf{F}{(\\theta)} = \\theta^{2}, then obtain \\int \\mathbf{F}{(\\theta)} d\\theta = \\int \\theta \\operatorname{t_{1}}{(\\theta)} d\\theta", "derivation": "\\operatorname{t_{1}}{(\\theta)} = \\theta and \\theta \\operatorname{t_{1}}{(\\theta)} = \\theta^{2} and \\mathbf{F}{(\\theta)} = \\theta^{2} and \\mathbf{F}{(\\theta)} = \\theta \\operatorname{t_{1}}{(\\theta)} and \\int \\mathbf{F}{(\\theta)} d\\theta = \\int \\theta \\operatorname{t_{1}}{(\\theta)} d\\theta", "srepr_derivation": [["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('t_1')(Symbol('\\\\theta', commutative=True))), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Function('t_1')(Symbol('\\\\theta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Symbol('\\\\theta', commutative=True), Function('t_1')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given s{(h)} = \\log{(h)}, then obtain h + s^{2}{(h)} + s{(h)} \\log{(h)} - \\frac{d}{d h} s^{2}{(h)} = h + 2 s{(h)} \\log{(h)} - \\frac{d}{d h} s^{2}{(h)}", "derivation": "s{(h)} = \\log{(h)} and s^{2}{(h)} = s{(h)} \\log{(h)} and s^{2}{(h)} + s{(h)} \\log{(h)} = 2 s{(h)} \\log{(h)} and h + s^{2}{(h)} + s{(h)} \\log{(h)} = h + 2 s{(h)} \\log{(h)} and \\frac{d}{d h} s^{2}{(h)} = \\frac{d}{d h} s{(h)} \\log{(h)} and h + s^{2}{(h)} + s{(h)} \\log{(h)} - \\frac{d}{d h} s{(h)} \\log{(h)} = h + 2 s{(h)} \\log{(h)} - \\frac{d}{d h} s{(h)} \\log{(h)} and h + s^{2}{(h)} + s{(h)} \\log{(h)} - \\frac{d}{d h} s^{2}{(h)} = h + 2 s{(h)} \\log{(h)} - \\frac{d}{d h} s^{2}{(h)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["times", 1, "Function('s')(Symbol('h', commutative=True))"], "Equality(Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))))"], [["add", 2, "Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], "Equality(Add(Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))), Mul(Integer(2), Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))))"], [["add", 3, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))), Add(Symbol('h', commutative=True), Mul(Integer(2), Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Symbol('h', commutative=True), Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Symbol('h', commutative=True), Mul(Integer(2), Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('h', commutative=True), Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Symbol('h', commutative=True), Mul(Integer(2), Function('s')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Pow(Function('s')(Symbol('h', commutative=True)), Integer(2)), Tuple(Symbol('h', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbb{I}{(m)} = \\sin{(m)}, then derive \\frac{d}{d m} \\mathbb{I}{(m)} = \\cos{(m)}, then obtain \\sin{(n_{1} + \\frac{d}{d m} \\mathbb{I}{(m)})} = \\sin{(n_{1} + \\cos{(m)})}", "derivation": "\\mathbb{I}{(m)} = \\sin{(m)} and \\frac{d}{d m} \\mathbb{I}{(m)} = \\frac{d}{d m} \\sin{(m)} and \\frac{d}{d m} \\mathbb{I}{(m)} = \\cos{(m)} and \\frac{d}{d m} \\sin{(m)} = \\cos{(m)} and n_{1} + \\frac{d}{d m} \\sin{(m)} = n_{1} + \\cos{(m)} and n_{1} + \\frac{d}{d m} \\mathbb{I}{(m)} = n_{1} + \\cos{(m)} and \\sin{(n_{1} + \\frac{d}{d m} \\mathbb{I}{(m)})} = \\sin{(n_{1} + \\cos{(m)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), cos(Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), cos(Symbol('m', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('n_1', commutative=True))"], "Equality(Add(Symbol('n_1', commutative=True), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Symbol('n_1', commutative=True), cos(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('n_1', commutative=True), Derivative(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Symbol('n_1', commutative=True), cos(Symbol('m', commutative=True))))"], [["sin", 6], "Equality(sin(Add(Symbol('n_1', commutative=True), Derivative(Function('\\\\mathbb{I}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))), sin(Add(Symbol('n_1', commutative=True), cos(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\rho)} = \\log{(\\rho)}, then obtain (\\frac{\\mathbf{p}{(\\rho)}}{(1 + \\frac{\\log{(\\rho)}}{\\rho}) \\log{(\\rho)}})^{\\rho} = (\\frac{1}{1 + \\frac{\\log{(\\rho)}}{\\rho}})^{\\rho}", "derivation": "\\mathbf{p}{(\\rho)} = \\log{(\\rho)} and \\frac{\\mathbf{p}{(\\rho)}}{\\rho} = \\frac{\\log{(\\rho)}}{\\rho} and \\frac{\\mathbf{p}{(\\rho)}}{\\log{(\\rho)}} = 1 and \\frac{\\mathbf{p}{(\\rho)}}{(1 + \\frac{\\mathbf{p}{(\\rho)}}{\\rho}) \\log{(\\rho)}} = \\frac{1}{1 + \\frac{\\mathbf{p}{(\\rho)}}{\\rho}} and (\\frac{\\mathbf{p}{(\\rho)}}{(1 + \\frac{\\mathbf{p}{(\\rho)}}{\\rho}) \\log{(\\rho)}})^{\\rho} = (\\frac{1}{1 + \\frac{\\mathbf{p}{(\\rho)}}{\\rho}})^{\\rho} and (\\frac{\\mathbf{p}{(\\rho)}}{(1 + \\frac{\\log{(\\rho)}}{\\rho}) \\log{(\\rho)}})^{\\rho} = (\\frac{1}{1 + \\frac{\\log{(\\rho)}}{\\rho}})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Symbol('\\\\rho', commutative=True))))"], [["divide", 1, "log(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 3, "Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True))))"], "Equality(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)))), Integer(-1)))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Symbol('\\\\rho', commutative=True)), Pow(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Symbol('\\\\rho', commutative=True)), Pow(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\dot{x},A_{y})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x}), then derive e^{\\lambda{(\\dot{x},A_{y})}} = e, then obtain 0 = e - e^{\\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x})}", "derivation": "\\lambda{(\\dot{x},A_{y})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x}) and e^{\\lambda{(\\dot{x},A_{y})}} = e^{\\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x})} and e^{\\lambda{(\\dot{x},A_{y})}} = e and e^{\\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x})} = e and - e^{\\lambda{(\\dot{x},A_{y})}} + e^{\\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x})} = e - e^{\\lambda{(\\dot{x},A_{y})}} and 0 = e - e^{\\frac{\\partial}{\\partial A_{y}} (A_{y} + \\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)), Derivative(Add(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["exp", 1], "Equality(exp(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))), exp(Derivative(Add(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(exp(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))), E)"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(Derivative(Add(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), E)"], [["minus", 4, "exp(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)))), exp(Derivative(Add(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))), Add(E, Mul(Integer(-1), exp(Function('\\\\lambda')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(E, Mul(Integer(-1), exp(Derivative(Add(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\theta_{2}{(v_{y},t_{2})} = - t_{2} + v_{y}, then derive \\frac{\\frac{\\partial}{\\partial v_{y}} \\theta_{2}{(v_{y},t_{2})}}{t_{2}} = \\frac{1}{t_{2}}, then obtain \\frac{\\frac{\\partial}{\\partial v_{y}} (- t_{2} + v_{y})}{t_{2}} = \\frac{1}{t_{2}}", "derivation": "\\theta_{2}{(v_{y},t_{2})} = - t_{2} + v_{y} and \\frac{\\theta_{2}{(v_{y},t_{2})}}{t_{2}} = \\frac{- t_{2} + v_{y}}{t_{2}} and \\frac{\\partial}{\\partial v_{y}} \\frac{\\theta_{2}{(v_{y},t_{2})}}{t_{2}} = \\frac{\\partial}{\\partial v_{y}} \\frac{- t_{2} + v_{y}}{t_{2}} and \\frac{\\frac{\\partial}{\\partial v_{y}} \\theta_{2}{(v_{y},t_{2})}}{t_{2}} = \\frac{1}{t_{2}} and \\frac{\\frac{\\partial}{\\partial v_{y}} (- t_{2} + v_{y})}{t_{2}} = \\frac{1}{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_y', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v_y', commutative=True)))"], [["divide", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('v_y', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('v_y', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_2')(Symbol('v_y', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Pow(Symbol('t_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Pow(Symbol('t_2', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{F}{(E,S)} = \\log{(E S)}, then derive \\log{(E S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} + \\frac{\\mathbf{F}{(E,S)}}{E} = 2 \\mathbf{F}{(E,S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)}, then obtain \\int (\\log{(E S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} + \\frac{\\mathbf{F}{(E,S)}}{E}) dE = \\int 2 \\mathbf{F}{(E,S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} dE", "derivation": "\\mathbf{F}{(E,S)} = \\log{(E S)} and \\mathbf{F}{(E,S)} \\log{(E S)} = \\log{(E S)}^{2} and \\mathbf{F}^{2}{(E,S)} = \\mathbf{F}{(E,S)} \\log{(E S)} and \\mathbf{F}^{2}{(E,S)} = \\log{(E S)}^{2} and \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} \\log{(E S)} = \\frac{\\partial}{\\partial E} \\log{(E S)}^{2} and \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} \\log{(E S)} = \\frac{\\partial}{\\partial E} \\mathbf{F}^{2}{(E,S)} and \\log{(E S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} + \\frac{\\mathbf{F}{(E,S)}}{E} = 2 \\mathbf{F}{(E,S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} and \\int (\\log{(E S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} + \\frac{\\mathbf{F}{(E,S)}}{E}) dE = \\int 2 \\mathbf{F}{(E,S)} \\frac{\\partial}{\\partial E} \\mathbf{F}{(E,S)} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))))"], [["times", 1, "log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True)))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))), Integer(2)))"], [["times", 1, "Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Integer(2)), Pow(log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))), Integer(2)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Mul(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Integer(2)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))), Derivative(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)))), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["integrate", 7, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Mul(log(Mul(Symbol('E', commutative=True), Symbol('S', commutative=True))), Derivative(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given J{(S,J,\\hat{H})} = J (S + \\hat{H}), then derive 0 = - S - \\hat{H} + \\frac{\\partial}{\\partial J} J{(S,J,\\hat{H})}, then obtain 0 = - S - \\hat{H} + \\frac{\\partial}{\\partial J} J (S + \\hat{H})", "derivation": "J{(S,J,\\hat{H})} = J (S + \\hat{H}) and 0 = J (S + \\hat{H}) - J{(S,J,\\hat{H})} and J (S + \\hat{H}) = 2 J (S + \\hat{H}) - J{(S,J,\\hat{H})} and \\frac{d}{d J} 0 = \\frac{\\partial}{\\partial J} (J (S + \\hat{H}) - J{(S,J,\\hat{H})}) and J{(S,J,\\hat{H})} = 2 J (S + \\hat{H}) - J{(S,J,\\hat{H})} and \\frac{d}{d J} 0 = \\frac{\\partial}{\\partial J} (- J (S + \\hat{H}) + J{(S,J,\\hat{H})}) and 0 = - S - \\hat{H} + \\frac{\\partial}{\\partial J} J{(S,J,\\hat{H})} and 0 = - S - \\hat{H} + \\frac{\\partial}{\\partial J} J (S + \\hat{H})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["add", 2, "Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(2), Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(2), Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Integer(0), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Function('J')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Mul(Symbol('J', commutative=True), Add(Symbol('S', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(x^\\prime)} = \\cos{(\\sin{(x^\\prime)})}, then obtain \\frac{d}{d x^\\prime} ((\\operatorname{f^{*}}^{x^\\prime}{(x^\\prime)})^{x^\\prime})^{x^\\prime} = \\frac{d}{d x^\\prime} ((\\cos^{x^\\prime}{(\\sin{(x^\\prime)})})^{x^\\prime})^{x^\\prime}", "derivation": "\\operatorname{f^{*}}{(x^\\prime)} = \\cos{(\\sin{(x^\\prime)})} and \\operatorname{f^{*}}^{x^\\prime}{(x^\\prime)} = \\cos^{x^\\prime}{(\\sin{(x^\\prime)})} and (\\operatorname{f^{*}}^{x^\\prime}{(x^\\prime)})^{x^\\prime} = (\\cos^{x^\\prime}{(\\sin{(x^\\prime)})})^{x^\\prime} and ((\\operatorname{f^{*}}^{x^\\prime}{(x^\\prime)})^{x^\\prime})^{x^\\prime} = ((\\cos^{x^\\prime}{(\\sin{(x^\\prime)})})^{x^\\prime})^{x^\\prime} and \\frac{d}{d x^\\prime} ((\\operatorname{f^{*}}^{x^\\prime}{(x^\\prime)})^{x^\\prime})^{x^\\prime} = \\frac{d}{d x^\\prime} ((\\cos^{x^\\prime}{(\\sin{(x^\\prime)})})^{x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('x^\\\\prime', commutative=True)), cos(sin(Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Pow(Function('f^*')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Pow(Pow(Function('f^*')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Pow(Pow(Function('f^*')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Pow(Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(a)} = \\sin{(a)}, then obtain \\frac{(\\frac{d}{d a} \\hat{H}{(a)})^{2}}{\\hat{H}{(a)} + \\sin{(a)}} = \\frac{\\frac{d}{d a} \\hat{H}{(a)} \\frac{d}{d a} \\sin{(a)}}{\\hat{H}{(a)} + \\sin{(a)}}", "derivation": "\\hat{H}{(a)} = \\sin{(a)} and \\frac{d}{d a} \\hat{H}{(a)} = \\frac{d}{d a} \\sin{(a)} and \\frac{\\frac{d}{d a} \\hat{H}{(a)}}{\\hat{H}{(a)} + \\sin{(a)}} = \\frac{\\frac{d}{d a} \\sin{(a)}}{\\hat{H}{(a)} + \\sin{(a)}} and \\frac{(\\frac{d}{d a} \\hat{H}{(a)})^{2}}{\\hat{H}{(a)} + \\sin{(a)}} = \\frac{\\frac{d}{d a} \\hat{H}{(a)} \\frac{d}{d a} \\sin{(a)}}{\\hat{H}{(a)} + \\sin{(a)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["divide", 2, "Add(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Add(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Integer(-1)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Function('\\\\hat{H}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Integer(-1)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Add(Function('\\\\hat{H}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(v_{z},t_{1})} = - t_{1} + \\cos{(v_{z})}, then obtain t_{1} + (t_{1} + r{(v_{z},t_{1})}) r{(v_{z},t_{1})} - \\cos{(v_{z})} = t_{1} + r{(v_{z},t_{1})} \\cos{(v_{z})} - \\cos{(v_{z})}", "derivation": "r{(v_{z},t_{1})} = - t_{1} + \\cos{(v_{z})} and t_{1} + r{(v_{z},t_{1})} = \\cos{(v_{z})} and (- t_{1} + \\cos{(v_{z})}) (t_{1} + r{(v_{z},t_{1})}) = (- t_{1} + \\cos{(v_{z})}) \\cos{(v_{z})} and (t_{1} + r{(v_{z},t_{1})}) r{(v_{z},t_{1})} = r{(v_{z},t_{1})} \\cos{(v_{z})} and t_{1} + (t_{1} + r{(v_{z},t_{1})}) r{(v_{z},t_{1})} - \\cos{(v_{z})} = t_{1} + r{(v_{z},t_{1})} \\cos{(v_{z})} - \\cos{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True)), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('t_1', commutative=True))"], "Equality(Add(Symbol('t_1', commutative=True), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True))), cos(Symbol('v_z', commutative=True)))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True))), Add(Symbol('t_1', commutative=True), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True))), cos(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Symbol('t_1', commutative=True), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True))), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True))), Mul(Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True)))"], "Equality(Add(Symbol('t_1', commutative=True), Mul(Add(Symbol('t_1', commutative=True), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True))), Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), cos(Symbol('v_z', commutative=True)))), Add(Symbol('t_1', commutative=True), Mul(Function('r')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True)), cos(Symbol('v_z', commutative=True))), Mul(Integer(-1), cos(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(H)} = \\frac{d}{d H} \\cos{(H)}, then derive \\bar{\\h}{(H)} = - \\sin{(H)}, then obtain \\int (\\sin{(H)} + \\frac{d}{d H} \\cos{(H)}) dH = \\int 0 dH", "derivation": "\\bar{\\h}{(H)} = \\frac{d}{d H} \\cos{(H)} and \\bar{\\h}{(H)} = - \\sin{(H)} and \\bar{\\h}{(H)} + \\sin{(H)} = 0 and \\int (\\bar{\\h}{(H)} + \\sin{(H)}) dH = \\int 0 dH and \\int (\\sin{(H)} + \\frac{d}{d H} \\cos{(H)}) dH = \\int 0 dH", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hbar')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), sin(Symbol('H', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Function('\\\\hbar')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Integer(0), Tuple(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(sin(Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True))), Integral(Integer(0), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},\\mathbf{M})} = \\sin^{A_{2}}{(\\mathbf{M})}, then obtain \\mathbf{S}{(A_{2},\\mathbf{M})} - 2 \\sin^{A_{2}}{(\\mathbf{M})} = - \\mathbf{S}{(A_{2},\\mathbf{M})}", "derivation": "\\mathbf{S}{(A_{2},\\mathbf{M})} = \\sin^{A_{2}}{(\\mathbf{M})} and \\mathbf{S}{(A_{2},\\mathbf{M})} + \\sin^{A_{2}}{(\\mathbf{M})} = 2 \\sin^{A_{2}}{(\\mathbf{M})} and - \\sin^{A_{2}}{(\\mathbf{M})} = - \\mathbf{S}{(A_{2},\\mathbf{M})} and - \\mathbf{S}{(A_{2},\\mathbf{M})} - \\sin^{A_{2}}{(\\mathbf{M})} = - 2 \\sin^{A_{2}}{(\\mathbf{M})} and - \\sin^{A_{2}}{(\\mathbf{M})} = \\mathbf{S}{(A_{2},\\mathbf{M})} - 2 \\sin^{A_{2}}{(\\mathbf{M})} and \\mathbf{S}{(A_{2},\\mathbf{M})} - 2 \\sin^{A_{2}}{(\\mathbf{M})} = - \\mathbf{S}{(A_{2},\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))))"], [["minus", 1, "Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True))), Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_2', commutative=True)))), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\theta_1,\\mathbf{f},\\mu)} = - \\mathbf{f} + \\mu + \\theta_1 and \\phi_{1}{(\\theta_1,\\mathbf{f},\\mu)} = - \\mathbf{f} + \\mu + \\theta_1, then obtain \\sin{(\\operatorname{F_{x}}{(\\theta_1,\\mathbf{f},\\mu)})} = \\sin{(\\phi_{1}{(\\theta_1,\\mathbf{f},\\mu)})}", "derivation": "\\operatorname{F_{x}}{(\\theta_1,\\mathbf{f},\\mu)} = - \\mathbf{f} + \\mu + \\theta_1 and \\sin{(\\operatorname{F_{x}}{(\\theta_1,\\mathbf{f},\\mu)})} = \\sin{(- \\mathbf{f} + \\mu + \\theta_1)} and \\phi_{1}{(\\theta_1,\\mathbf{f},\\mu)} = - \\mathbf{f} + \\mu + \\theta_1 and \\sin{(\\operatorname{F_{x}}{(\\theta_1,\\mathbf{f},\\mu)})} = \\sin{(\\phi_{1}{(\\theta_1,\\mathbf{f},\\mu)})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["sin", 1], "Equality(sin(Function('F_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu', commutative=True))), sin(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mu', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(sin(Function('F_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu', commutative=True))), sin(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given Q{(\\chi)} = \\chi, then derive A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} Q{(\\chi)} = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + 1, then obtain A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} 1 + \\frac{d}{d \\chi} \\chi = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} 1 + 1", "derivation": "Q{(\\chi)} = \\chi and \\frac{d}{d \\chi} Q{(\\chi)} = \\frac{d}{d \\chi} \\chi and - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} Q{(\\chi)} = - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} \\chi and A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} Q{(\\chi)} = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} \\chi and A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} Q{(\\chi)} = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + 1 and A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} \\chi = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + 1 and A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} 1 + \\frac{d}{d \\chi} \\chi = A_{y} - \\frac{\\chi}{Q{(\\chi)}} + \\frac{d}{d \\chi} 1 + 1", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Function('Q')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["add", 3, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Function('Q')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Function('Q')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Integer(1)))"], [["add", 6, "Derivative(Integer(1), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Integer(1), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Pow(Function('Q')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Derivative(Integer(1), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(A_{x},n)} = e^{A_{x} n}, then obtain (\\frac{\\operatorname{v_{1}}{(A_{x},n)}}{A_{x}^{2} n^{2}})^{A_{x}} = (\\frac{e^{A_{x} n}}{A_{x}^{2} n^{2}})^{A_{x}}", "derivation": "\\operatorname{v_{1}}{(A_{x},n)} = e^{A_{x} n} and \\frac{\\operatorname{v_{1}}{(A_{x},n)}}{A_{x} n} = \\frac{e^{A_{x} n}}{A_{x} n} and \\frac{\\operatorname{v_{1}}{(A_{x},n)}}{A_{x}^{2} n^{2}} = \\frac{e^{A_{x} n}}{A_{x}^{2} n^{2}} and (\\frac{\\operatorname{v_{1}}{(A_{x},n)}}{A_{x}^{2} n^{2}})^{A_{x}} = (\\frac{e^{A_{x} n}}{A_{x}^{2} n^{2}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('A_x', commutative=True), Symbol('n', commutative=True)), exp(Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True))))"], [["divide", 1, "Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Integer(-1)), Function('v_1')(Symbol('A_x', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Integer(-1)), exp(Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True)))))"], [["divide", 2, "Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-2)), Pow(Symbol('n', commutative=True), Integer(-2)), Function('v_1')(Symbol('A_x', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-2)), Pow(Symbol('n', commutative=True), Integer(-2)), exp(Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True)))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-2)), Pow(Symbol('n', commutative=True), Integer(-2)), Function('v_1')(Symbol('A_x', commutative=True), Symbol('n', commutative=True))), Symbol('A_x', commutative=True)), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-2)), Pow(Symbol('n', commutative=True), Integer(-2)), exp(Mul(Symbol('A_x', commutative=True), Symbol('n', commutative=True)))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{f},\\varphi)} = - \\varphi + \\cos{(\\mathbf{f})}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\operatorname{A_{1}}{(\\mathbf{f},\\varphi)}}{- \\varphi + \\cos{(\\mathbf{f})}} d\\mathbf{f} = \\int \\frac{d}{d \\mathbf{f}} 1 d\\mathbf{f}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{f},\\varphi)} = - \\varphi + \\cos{(\\mathbf{f})} and \\frac{\\operatorname{A_{1}}{(\\mathbf{f},\\varphi)}}{- \\varphi + \\cos{(\\mathbf{f})}} = 1 and \\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\operatorname{A_{1}}{(\\mathbf{f},\\varphi)}}{- \\varphi + \\cos{(\\mathbf{f})}} = \\frac{d}{d \\mathbf{f}} 1 and \\int \\frac{\\partial}{\\partial \\mathbf{f}} \\frac{\\operatorname{A_{1}}{(\\mathbf{f},\\varphi)}}{- \\varphi + \\cos{(\\mathbf{f})}} d\\mathbf{f} = \\int \\frac{d}{d \\mathbf{f}} 1 d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\phi_2)} = \\log{(\\phi_2)}, then obtain (\\mathbf{P}{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2)^{\\phi_2} = (\\log{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2)^{\\phi_2}", "derivation": "\\mathbf{P}{(\\phi_2)} = \\log{(\\phi_2)} and \\int \\mathbf{P}{(\\phi_2)} d\\phi_2 = \\int \\log{(\\phi_2)} d\\phi_2 and \\mathbf{P}{(\\phi_2)} + \\int \\log{(\\phi_2)} d\\phi_2 = \\log{(\\phi_2)} + \\int \\log{(\\phi_2)} d\\phi_2 and \\mathbf{P}{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2 = \\log{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2 and (\\mathbf{P}{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2)^{\\phi_2} = (\\log{(\\phi_2)} + \\int \\mathbf{P}{(\\phi_2)} d\\phi_2)^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "Integral(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Integral(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(log(Symbol('\\\\phi_2', commutative=True)), Integral(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(log(Symbol('\\\\phi_2', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(Add(log(Symbol('\\\\phi_2', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\delta,A_{y})} = \\frac{\\delta}{A_{y}}, then obtain - \\operatorname{F_{H}}{(L_{\\varepsilon})} + \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon} + 1 = (\\frac{\\delta}{A_{y}})^{A_{y}} \\mathbf{P}^{- A_{y}}{(\\delta,A_{y})} - \\operatorname{F_{H}}{(L_{\\varepsilon})} + \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon}", "derivation": "\\mathbf{P}{(\\delta,A_{y})} = \\frac{\\delta}{A_{y}} and \\mathbf{P}^{A_{y}}{(\\delta,A_{y})} = (\\frac{\\delta}{A_{y}})^{A_{y}} and 1 = (\\frac{\\delta}{A_{y}})^{A_{y}} \\mathbf{P}^{- A_{y}}{(\\delta,A_{y})} and 1 - \\operatorname{F_{H}}{(L_{\\varepsilon})} = (\\frac{\\delta}{A_{y}})^{A_{y}} \\mathbf{P}^{- A_{y}}{(\\delta,A_{y})} - \\operatorname{F_{H}}{(L_{\\varepsilon})} and - \\operatorname{F_{H}}{(L_{\\varepsilon})} + \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon} + 1 = (\\frac{\\delta}{A_{y}})^{A_{y}} \\mathbf{P}^{- A_{y}}{(\\delta,A_{y})} - \\operatorname{F_{H}}{(L_{\\varepsilon})} + \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Symbol('A_y', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Symbol('A_y', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))))"], [["minus", 3, "Function('F_H')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_H')(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Mul(Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Symbol('A_y', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))), Mul(Integer(-1), Function('F_H')(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["add", 4, "Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_H')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(1)), Add(Mul(Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Symbol('A_y', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))), Mul(Integer(-1), Function('F_H')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(i,g,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + i^{g}, then obtain \\frac{- \\hat{\\mathbf{x}} + \\phi_{1}{(i,g,\\hat{\\mathbf{x}})}}{\\phi_{1}{(i,g,\\hat{\\mathbf{x}})} + 1} = \\frac{- 2 \\hat{\\mathbf{x}} + i^{g}}{\\phi_{1}{(i,g,\\hat{\\mathbf{x}})} + 1}", "derivation": "\\phi_{1}{(i,g,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + i^{g} and - \\hat{\\mathbf{x}} + \\phi_{1}{(i,g,\\hat{\\mathbf{x}})} = - 2 \\hat{\\mathbf{x}} + i^{g} and \\phi_{1}{(i,g,\\hat{\\mathbf{x}})} + 1 = - \\hat{\\mathbf{x}} + i^{g} + 1 and \\frac{- \\hat{\\mathbf{x}} + \\phi_{1}{(i,g,\\hat{\\mathbf{x}})}}{- \\hat{\\mathbf{x}} + i^{g} + 1} = \\frac{- 2 \\hat{\\mathbf{x}} + i^{g}}{- \\hat{\\mathbf{x}} + i^{g} + 1} and \\frac{- \\hat{\\mathbf{x}} + \\phi_{1}{(i,g,\\hat{\\mathbf{x}})}}{\\phi_{1}{(i,g,\\hat{\\mathbf{x}})} + 1} = \\frac{- 2 \\hat{\\mathbf{x}} + i^{g}}{\\phi_{1}{(i,g,\\hat{\\mathbf{x}})} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True)), Integer(1)))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True)), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True)), Integer(1)), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True)), Integer(1)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Add(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(1)), Integer(-1))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('g', commutative=True))), Pow(Add(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(c_{0},Z)} = - c_{0} + \\cos{(Z)}, then obtain - \\frac{\\partial}{\\partial Z} (- c_{0} + \\mathbf{J}_M{(c_{0},Z)}) = - \\frac{\\partial}{\\partial Z} (- 2 c_{0} + \\cos{(Z)})", "derivation": "\\mathbf{J}_M{(c_{0},Z)} = - c_{0} + \\cos{(Z)} and - c_{0} + \\mathbf{J}_M{(c_{0},Z)} = - 2 c_{0} + \\cos{(Z)} and \\frac{\\partial}{\\partial Z} (- c_{0} + \\mathbf{J}_M{(c_{0},Z)}) = \\frac{\\partial}{\\partial Z} (- 2 c_{0} + \\cos{(Z)}) and - \\frac{\\partial}{\\partial Z} (- c_{0} + \\mathbf{J}_M{(c_{0},Z)}) = - \\frac{\\partial}{\\partial Z} (- 2 c_{0} + \\cos{(Z)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), cos(Symbol('Z', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), cos(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(\\eta)} = \\sin{(\\eta)}, then derive \\int \\chi{(\\eta)} d\\eta = y^{\\prime} - \\cos{(\\eta)}, then derive n_{2} - \\cos{(\\eta)} = y^{\\prime} - \\cos{(\\eta)}, then obtain 1 = \\frac{y^{\\prime} - \\cos{(\\eta)}}{n_{2} - \\cos{(\\eta)}}", "derivation": "\\chi{(\\eta)} = \\sin{(\\eta)} and \\int \\chi{(\\eta)} d\\eta = \\int \\sin{(\\eta)} d\\eta and \\int \\chi{(\\eta)} d\\eta = y^{\\prime} - \\cos{(\\eta)} and \\int \\sin{(\\eta)} d\\eta = y^{\\prime} - \\cos{(\\eta)} and n_{2} - \\cos{(\\eta)} = y^{\\prime} - \\cos{(\\eta)} and 1 = \\frac{y^{\\prime} - \\cos{(\\eta)}}{n_{2} - \\cos{(\\eta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(sin(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))), Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], [["divide", 5, "Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))), Integer(-1)), Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(C,\\varepsilon_0)} = \\sin{(C + \\varepsilon_0)} and \\Psi_{nl}{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain \\frac{\\Psi_{nl}{(\\mathbf{H})}}{1 + \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0}} = \\frac{\\cos{(\\mathbf{H})}}{1 + \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0}}", "derivation": "\\hat{H}{(C,\\varepsilon_0)} = \\sin{(C + \\varepsilon_0)} and \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0} = \\frac{\\sin{(C + \\varepsilon_0)}}{\\varepsilon_0} and 1 + \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0} = 1 + \\frac{\\sin{(C + \\varepsilon_0)}}{\\varepsilon_0} and \\Psi_{nl}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\frac{\\Psi_{nl}{(\\mathbf{H})}}{1 + \\frac{\\sin{(C + \\varepsilon_0)}}{\\varepsilon_0}} = \\frac{\\cos{(\\mathbf{H})}}{1 + \\frac{\\sin{(C + \\varepsilon_0)}}{\\varepsilon_0}} and \\frac{\\Psi_{nl}{(\\mathbf{H})}}{1 + \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0}} = \\frac{\\cos{(\\mathbf{H})}}{1 + \\frac{\\hat{H}{(C,\\varepsilon_0)}}{\\varepsilon_0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 4, "Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], "Equality(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))), Integer(-1)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Integer(-1)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(B)} = \\cos{(\\sin{(B)})}, then derive \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)} = - \\sin{(\\sin{(B)})} \\cos{(B)}, then obtain (\\cos{(B)} + \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)})^{B} = (- \\sin{(\\sin{(B)})} \\cos{(B)} + \\cos{(B)})^{B}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(B)} = \\cos{(\\sin{(B)})} and \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)} = \\frac{d}{d B} \\cos{(\\sin{(B)})} and \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)} = - \\sin{(\\sin{(B)})} \\cos{(B)} and \\cos{(B)} + \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)} = - \\sin{(\\sin{(B)})} \\cos{(B)} + \\cos{(B)} and \\cos{(B)} + \\frac{d}{d B} \\cos{(\\sin{(B)})} = - \\sin{(\\sin{(B)})} \\cos{(B)} + \\cos{(B)} and (\\cos{(B)} + \\frac{d}{d B} \\cos{(\\sin{(B)})})^{B} = (- \\sin{(\\sin{(B)})} \\cos{(B)} + \\cos{(B)})^{B} and (\\cos{(B)} + \\frac{d}{d B} \\operatorname{V_{\\mathbf{B}}}{(B)})^{B} = (- \\sin{(\\sin{(B)})} \\cos{(B)} + \\cos{(B)})^{B}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True)), cos(sin(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))))"], [["add", 3, "cos(Symbol('B', commutative=True))"], "Equality(Add(cos(Symbol('B', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(cos(Symbol('B', commutative=True)), Derivative(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(Add(cos(Symbol('B', commutative=True)), Derivative(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), sin(sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(cos(Symbol('B', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), sin(sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\Omega,F_{N})} = F_{N} - \\Omega and \\theta_{1}{(\\theta_2,\\mathbf{r})} = \\mathbf{r} \\theta_2 and f{(\\Omega,F_{N})} = - F_{N} + \\Omega + \\operatorname{A_{2}}{(\\Omega,F_{N})}, then obtain \\int \\theta_{1}{(\\theta_2,\\mathbf{r})} d\\theta_2 = f{(\\Omega,F_{N})} + \\int \\mathbf{r} \\theta_2 d\\theta_2", "derivation": "\\operatorname{A_{2}}{(\\Omega,F_{N})} = F_{N} - \\Omega and \\theta_{1}{(\\theta_2,\\mathbf{r})} = \\mathbf{r} \\theta_2 and \\int \\theta_{1}{(\\theta_2,\\mathbf{r})} d\\theta_2 = \\int \\mathbf{r} \\theta_2 d\\theta_2 and f{(\\Omega,F_{N})} = - F_{N} + \\Omega + \\operatorname{A_{2}}{(\\Omega,F_{N})} and f{(\\Omega,F_{N})} = 0 and f{(\\Omega,F_{N})} + \\int \\theta_{1}{(\\theta_2,\\mathbf{r})} d\\theta_2 = \\int \\theta_{1}{(\\theta_2,\\mathbf{r})} d\\theta_2 and f{(\\Omega,F_{N})} + \\int \\mathbf{r} \\theta_2 d\\theta_2 = \\int \\mathbf{r} \\theta_2 d\\theta_2 and \\int \\theta_{1}{(\\theta_2,\\mathbf{r})} d\\theta_2 = f{(\\Omega,F_{N})} + \\int \\mathbf{r} \\theta_2 d\\theta_2", "srepr_derivation": [["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('\\\\Omega', commutative=True), Function('A_2')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('f')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Integer(0))"], [["add", 5, "Integral(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Function('f')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Integral(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Function('f')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Function('f')(Symbol('\\\\Omega', commutative=True), Symbol('F_N', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(P_{g},G)} = G - P_{g}, then obtain \\int \\frac{\\mathbf{H}{(P_{g},G)}}{G^{2}} dG - \\frac{1}{G} = \\int \\frac{G - P_{g}}{G^{2}} dG - \\frac{1}{G}", "derivation": "\\mathbf{H}{(P_{g},G)} = G - P_{g} and - \\mathbf{H}{(P_{g},G)} = - G + P_{g} and - \\frac{\\mathbf{H}{(P_{g},G)}}{G} = \\frac{- G + P_{g}}{G} and - \\frac{G - P_{g}}{G} = \\frac{- G + P_{g}}{G} and - \\frac{\\mathbf{H}{(P_{g},G)}}{G} = - \\frac{G - P_{g}}{G} and \\frac{\\mathbf{H}{(P_{g},G)}}{G^{2}} = \\frac{G - P_{g}}{G^{2}} and \\int \\frac{\\mathbf{H}{(P_{g},G)}}{G^{2}} dG = \\int \\frac{G - P_{g}}{G^{2}} dG and \\int \\frac{\\mathbf{H}{(P_{g},G)}}{G^{2}} dG - \\frac{1}{G} = \\int \\frac{G - P_{g}}{G^{2}} dG - \\frac{1}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('P_g', commutative=True)))"], [["divide", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))))"], [["divide", 5, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))))"], [["integrate", 6, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["minus", 7, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Function('\\\\mathbf{H}')(Symbol('P_g', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))), Add(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))), Tuple(Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C_{1},L)} = \\frac{L}{C_{1}} and \\Psi{(C_{1},L)} = \\frac{L}{C_{1}}, then obtain L + \\operatorname{M_{E}}{(C_{1},L)} = L + \\Psi{(C_{1},L)}", "derivation": "\\operatorname{M_{E}}{(C_{1},L)} = \\frac{L}{C_{1}} and L + \\operatorname{M_{E}}{(C_{1},L)} = L + \\frac{L}{C_{1}} and \\Psi{(C_{1},L)} = \\frac{L}{C_{1}} and L + \\operatorname{M_{E}}{(C_{1},L)} = L + \\Psi{(C_{1},L)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C_1', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('L', commutative=True)))"], [["add", 1, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('C_1', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('L', commutative=True), Function('M_E')(Symbol('C_1', commutative=True), Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Function('\\\\Psi')(Symbol('C_1', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\tilde{g},n)} = \\tilde{g} - n and \\operatorname{z^{*}}{(\\tilde{g},n)} = \\tilde{g} - n, then obtain \\int 0 dn = \\int (\\tilde{g} - n - \\operatorname{f_{\\mathbf{p}}}{(\\tilde{g},n)}) dn", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\tilde{g},n)} = \\tilde{g} - n and \\operatorname{z^{*}}{(\\tilde{g},n)} = \\tilde{g} - n and \\operatorname{f_{\\mathbf{p}}}{(\\tilde{g},n)} = \\operatorname{z^{*}}{(\\tilde{g},n)} and 0 = \\tilde{g} - n - \\operatorname{z^{*}}{(\\tilde{g},n)} and \\int 0 dn = \\int (\\tilde{g} - n - \\operatorname{z^{*}}{(\\tilde{g},n)}) dn and \\int 0 dn = \\int (\\tilde{g} - n - \\operatorname{f_{\\mathbf{p}}}{(\\tilde{g},n)}) dn", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Function('z^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)))"], [["minus", 2, "Function('z^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Function('z^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Function('z^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Integer(0), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\Psi{(Q)} = \\sin{(Q)}, then obtain \\frac{d}{d Q} (\\int 0 dQ + \\iint (\\Psi{(Q)} - \\sin{(Q)}) dQ dQ) = \\frac{d}{d Q} (\\int 0 dQ + \\iint 0 dQ dQ)", "derivation": "\\Psi{(Q)} = \\sin{(Q)} and \\Psi{(Q)} - \\sin{(Q)} = 0 and \\int (\\Psi{(Q)} - \\sin{(Q)}) dQ = \\int 0 dQ and \\iint (\\Psi{(Q)} - \\sin{(Q)}) dQ dQ = \\iint 0 dQ dQ and \\int (\\Psi{(Q)} - \\sin{(Q)}) dQ + \\iint (\\Psi{(Q)} - \\sin{(Q)}) dQ dQ = \\int (\\Psi{(Q)} - \\sin{(Q)}) dQ + \\iint 0 dQ dQ and \\frac{d}{d Q} (\\int (\\Psi{(Q)} - \\sin{(Q)}) dQ + \\iint (\\Psi{(Q)} - \\sin{(Q)}) dQ dQ) = \\frac{d}{d Q} (\\int (\\Psi{(Q)} - \\sin{(Q)}) dQ + \\iint 0 dQ dQ) and \\frac{d}{d Q} (\\int 0 dQ + \\iint (\\Psi{(Q)} - \\sin{(Q)}) dQ dQ) = \\frac{d}{d Q} (\\int 0 dQ + \\iint 0 dQ dQ)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["minus", 1, "sin(Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["add", 4, "Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Add(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Integral(Add(Function('\\\\Psi')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(q)} = \\cos{(q)}, then derive \\frac{\\sin{(q)}}{\\cos^{2}{(q)}} = - \\frac{\\frac{d}{d q} \\operatorname{v_{t}}{(q)}}{\\operatorname{v_{t}}^{2}{(q)}}, then obtain \\frac{\\sin{(q)}}{\\cos^{2}{(q)}} = \\frac{\\sin{(q)}}{\\operatorname{v_{t}}{(q)} \\cos{(q)}}", "derivation": "\\operatorname{v_{t}}{(q)} = \\cos{(q)} and \\operatorname{v_{t}}{(q)} \\cos{(q)} = \\cos^{2}{(q)} and \\frac{1}{\\cos{(q)}} = \\frac{1}{\\operatorname{v_{t}}{(q)}} and \\frac{d}{d q} \\frac{1}{\\cos{(q)}} = \\frac{d}{d q} \\frac{1}{\\operatorname{v_{t}}{(q)}} and \\frac{\\sin{(q)}}{\\cos^{2}{(q)}} = - \\frac{\\frac{d}{d q} \\operatorname{v_{t}}{(q)}}{\\operatorname{v_{t}}^{2}{(q)}} and \\frac{\\sin{(q)}}{\\operatorname{v_{t}}{(q)} \\cos{(q)}} = - \\frac{\\frac{d}{d q} \\operatorname{v_{t}}{(q)}}{\\operatorname{v_{t}}^{2}{(q)}} and \\frac{\\sin{(q)}}{\\cos^{2}{(q)}} = \\frac{\\sin{(q)}}{\\operatorname{v_{t}}{(q)} \\cos{(q)}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["times", 1, "cos(Symbol('q', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Pow(cos(Symbol('q', commutative=True)), Integer(2)))"], [["divide", 1, "Mul(Function('v_t')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], "Equality(Pow(cos(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Pow(cos(Symbol('q', commutative=True)), Integer(-1)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-1)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(sin(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-2)), Derivative(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-1)), sin(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-2)), Derivative(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(sin(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-2))), Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Integer(-1)), sin(Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given M{(\\mathbf{D},\\varphi)} = \\mathbf{D} + \\varphi, then obtain - M^{\\mathbf{D}}{(\\mathbf{D},\\varphi)} + \\log{(M{(\\mathbf{D},\\varphi)})}^{\\mathbf{D}} = - M^{\\mathbf{D}}{(\\mathbf{D},\\varphi)} + \\log{(\\mathbf{D} + \\varphi)}^{\\mathbf{D}}", "derivation": "M{(\\mathbf{D},\\varphi)} = \\mathbf{D} + \\varphi and \\log{(M{(\\mathbf{D},\\varphi)})} = \\log{(\\mathbf{D} + \\varphi)} and \\log{(M{(\\mathbf{D},\\varphi)})}^{\\mathbf{D}} = \\log{(\\mathbf{D} + \\varphi)}^{\\mathbf{D}} and - M^{\\mathbf{D}}{(\\mathbf{D},\\varphi)} + \\log{(M{(\\mathbf{D},\\varphi)})}^{\\mathbf{D}} = - M^{\\mathbf{D}}{(\\mathbf{D},\\varphi)} + \\log{(\\mathbf{D} + \\varphi)}^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["log", 1], "Equality(log(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(log(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 3, "Pow(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Pow(log(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Pow(Function('M')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Pow(log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{r},W)} = \\frac{\\mathbf{r}}{W}, then obtain W \\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{p}{(\\mathbf{r},W)} = W (\\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W})^{2}", "derivation": "\\mathbf{p}{(\\mathbf{r},W)} = \\frac{\\mathbf{r}}{W} and \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{p}{(\\mathbf{r},W)} = \\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W} and W \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{p}{(\\mathbf{r},W)} = W \\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W} and W \\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{p}{(\\mathbf{r},W)} = W (\\frac{\\partial}{\\partial \\mathbf{r}} \\frac{\\mathbf{r}}{W})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Symbol('W', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('W', commutative=True), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(Symbol('W', commutative=True), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('W', commutative=True), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(Symbol('W', commutative=True), Pow(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\psi^{*}{(p)} = \\log{(p)}, then derive \\Omega + \\psi^{*}{(p)} = V_{\\mathbf{E}} + \\log{(p)}, then obtain \\Omega + \\psi^{*}{(p)} + \\frac{V_{\\mathbf{E}} + \\log{(p)}}{\\mathbf{r}} = \\Omega + \\log{(p)} + \\frac{V_{\\mathbf{E}} + \\log{(p)}}{\\mathbf{r}}", "derivation": "\\psi^{*}{(p)} = \\log{(p)} and \\frac{d}{d p} \\psi^{*}{(p)} = \\frac{d}{d p} \\log{(p)} and \\int \\frac{d}{d p} \\psi^{*}{(p)} dp = \\int \\frac{d}{d p} \\log{(p)} dp and \\Omega + \\psi^{*}{(p)} = V_{\\mathbf{E}} + \\log{(p)} and \\Omega + \\psi^{*}{(p)} = V_{\\mathbf{E}} + \\psi^{*}{(p)} and \\Omega + \\log{(p)} = V_{\\mathbf{E}} + \\log{(p)} and \\Omega + \\psi^{*}{(p)} = \\Omega + \\log{(p)} and \\Omega + \\psi^{*}{(p)} + \\frac{V_{\\mathbf{E}} + \\log{(p)}}{\\mathbf{r}} = \\Omega + \\log{(p)} + \\frac{V_{\\mathbf{E}} + \\log{(p)}}{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(Derivative(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\psi^*')(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\psi^*')(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\psi^*')(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\psi^*')(Symbol('p', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), log(Symbol('p', commutative=True))))"], [["add", 7, "Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Symbol('p', commutative=True))))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\psi^*')(Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Symbol('p', commutative=True))))), Add(Symbol('\\\\Omega', commutative=True), log(Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(J)} = \\cos{(J)}, then obtain J \\phi_{2}{(J)} + (- J + \\cos{(J)}) \\phi_{2}{(J)} = J \\phi_{2}{(J)} + (- J + \\cos{(J)}) \\cos{(J)}", "derivation": "\\phi_{2}{(J)} = \\cos{(J)} and - J + \\phi_{2}{(J)} = - J + \\cos{(J)} and J \\phi_{2}{(J)} = J \\cos{(J)} and (- J + \\phi_{2}{(J)}) \\phi_{2}{(J)} = (- J + \\phi_{2}{(J)}) \\cos{(J)} and J \\cos{(J)} + (- J + \\phi_{2}{(J)}) \\phi_{2}{(J)} = J \\cos{(J)} + (- J + \\phi_{2}{(J)}) \\cos{(J)} and J \\cos{(J)} + (- J + \\cos{(J)}) \\phi_{2}{(J)} = J \\cos{(J)} + (- J + \\cos{(J)}) \\cos{(J)} and J \\phi_{2}{(J)} + (- J + \\cos{(J)}) \\phi_{2}{(J)} = J \\phi_{2}{(J)} + (- J + \\cos{(J)}) \\cos{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\phi_2')(Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True))), Function('\\\\phi_2')(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True))))"], [["add", 4, "Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True)))"], "Equality(Add(Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True))), Function('\\\\phi_2')(Symbol('J', commutative=True)))), Add(Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Function('\\\\phi_2')(Symbol('J', commutative=True)))), Add(Mul(Symbol('J', commutative=True), cos(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Symbol('J', commutative=True), Function('\\\\phi_2')(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Function('\\\\phi_2')(Symbol('J', commutative=True)))), Add(Mul(Symbol('J', commutative=True), Function('\\\\phi_2')(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(a^{\\dagger},\\rho_f)} = - \\rho_f + e^{a^{\\dagger}}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\dot{y}{(a^{\\dagger},\\rho_f)} = e^{a^{\\dagger}}, then derive \\mathbf{f} + \\dot{y}{(a^{\\dagger},\\rho_f)} = f_{\\mathbf{p}} + e^{a^{\\dagger}}, then obtain \\mathbf{f} - \\rho_f + e^{a^{\\dagger}} = f_{\\mathbf{p}} + e^{a^{\\dagger}}", "derivation": "\\dot{y}{(a^{\\dagger},\\rho_f)} = - \\rho_f + e^{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\dot{y}{(a^{\\dagger},\\rho_f)} = \\frac{\\partial}{\\partial a^{\\dagger}} (- \\rho_f + e^{a^{\\dagger}}) and \\frac{\\partial}{\\partial a^{\\dagger}} \\dot{y}{(a^{\\dagger},\\rho_f)} = e^{a^{\\dagger}} and \\int \\frac{\\partial}{\\partial a^{\\dagger}} \\dot{y}{(a^{\\dagger},\\rho_f)} da^{\\dagger} = \\int e^{a^{\\dagger}} da^{\\dagger} and \\mathbf{f} + \\dot{y}{(a^{\\dagger},\\rho_f)} = f_{\\mathbf{p}} + e^{a^{\\dagger}} and \\mathbf{f} - \\rho_f + e^{a^{\\dagger}} = f_{\\mathbf{p}} + e^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{y}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\dot{y}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(i)} = e^{i}, then derive \\frac{d^{2}}{d i^{2}} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{\\partial^{2}}{\\partial i^{2}} (A_{1} + e^{i})^{i}, then obtain \\frac{d^{2}}{d i^{2}} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{\\partial^{2}}{\\partial i^{2}} (A_{1} + \\operatorname{L_{\\varepsilon}}{(i)})^{i}", "derivation": "\\operatorname{L_{\\varepsilon}}{(i)} = e^{i} and \\int \\operatorname{L_{\\varepsilon}}{(i)} di = \\int e^{i} di and (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = (\\int e^{i} di)^{i} and \\frac{d}{d i} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{d}{d i} (\\int e^{i} di)^{i} and \\frac{d^{2}}{d i^{2}} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{d^{2}}{d i^{2}} (\\int e^{i} di)^{i} and \\frac{d^{2}}{d i^{2}} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{\\partial^{2}}{\\partial i^{2}} (A_{1} + e^{i})^{i} and \\frac{d^{2}}{d i^{2}} (\\int \\operatorname{L_{\\varepsilon}}{(i)} di)^{i} = \\frac{\\partial^{2}}{\\partial i^{2}} (A_{1} + \\operatorname{L_{\\varepsilon}}{(i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Integral(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(Pow(Integral(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))))"], [["evaluate_integrals", 5], "Equality(Derivative(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(Pow(Add(Symbol('A_1', commutative=True), exp(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(Pow(Add(Symbol('A_1', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))))"]]}, {"prompt": "Given A{(\\Omega,W)} = \\frac{\\cos{(W)}}{\\Omega}, then obtain (\\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW)^{2 W} = ((2 \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW)^{W}) (\\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW)^{W}", "derivation": "A{(\\Omega,W)} = \\frac{\\cos{(W)}}{\\Omega} and A^{W}{(\\Omega,W)} = (\\frac{\\cos{(W)}}{\\Omega})^{W} and \\int A^{W}{(\\Omega,W)} dW = \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW and \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW = 2 \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW and (\\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW)^{W} = (2 \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW)^{W} and (\\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW)^{2 W} = ((2 \\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW)^{W}) (\\int (\\frac{\\cos{(W)}}{\\Omega})^{W} dW + \\int A^{W}{(\\Omega,W)} dW)^{W}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["add", 3, "Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(2), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Pow(Mul(Integer(2), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], [["times", 5, "Pow(Add(Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True))"], "Equality(Pow(Add(Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True))), Mul(Pow(Mul(Integer(2), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Pow(Add(Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Function('A')(Symbol('\\\\Omega', commutative=True), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\rho_f)} = \\log{(\\rho_f)} and J{(\\rho_f)} = \\mathbf{F}^{\\rho_f}{(\\rho_f)}, then obtain \\int \\log{(\\rho_f)}^{\\rho_f} d\\rho_f = \\int \\mathbf{F}^{\\rho_f}{(\\rho_f)} d\\rho_f", "derivation": "\\mathbf{F}{(\\rho_f)} = \\log{(\\rho_f)} and \\mathbf{F}^{\\rho_f}{(\\rho_f)} = \\log{(\\rho_f)}^{\\rho_f} and J{(\\rho_f)} = \\mathbf{F}^{\\rho_f}{(\\rho_f)} and \\int J{(\\rho_f)} d\\rho_f = \\int \\mathbf{F}^{\\rho_f}{(\\rho_f)} d\\rho_f and J{(\\rho_f)} = \\log{(\\rho_f)}^{\\rho_f} and \\int \\log{(\\rho_f)}^{\\rho_f} d\\rho_f = \\int \\mathbf{F}^{\\rho_f}{(\\rho_f)} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('J')(Symbol('\\\\rho_f', commutative=True)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Pow(log(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain (\\sin{(v)} + 1)^{\\rho_f} = (\\sin{(v)} + \\frac{\\sin{(\\rho_f)}}{\\mathbf{p}{(\\rho_f)}})^{\\rho_f}", "derivation": "\\mathbf{p}{(\\rho_f)} = \\sin{(\\rho_f)} and 1 = \\frac{\\sin{(\\rho_f)}}{\\mathbf{p}{(\\rho_f)}} and \\sin{(v)} + 1 = \\sin{(v)} + \\frac{\\sin{(\\rho_f)}}{\\mathbf{p}{(\\rho_f)}} and (\\sin{(v)} + 1)^{\\rho_f} = (\\sin{(v)} + \\frac{\\sin{(\\rho_f)}}{\\mathbf{p}{(\\rho_f)}})^{\\rho_f}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), sin(Symbol('v', commutative=True)))"], "Equality(Add(sin(Symbol('v', commutative=True)), Integer(1)), Add(sin(Symbol('v', commutative=True)), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True)))))"], [["power", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Add(sin(Symbol('v', commutative=True)), Integer(1)), Symbol('\\\\rho_f', commutative=True)), Pow(Add(sin(Symbol('v', commutative=True)), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(E_{x})} = \\cos{(E_{x})}, then obtain \\iint \\frac{\\Psi_{nl}{(E_{x})}}{E_{x}} dE_{x} dE_{x} = \\iint \\frac{\\cos{(E_{x})}}{E_{x}} dE_{x} dE_{x}", "derivation": "\\Psi_{nl}{(E_{x})} = \\cos{(E_{x})} and \\frac{\\Psi_{nl}{(E_{x})}}{E_{x}} = \\frac{\\cos{(E_{x})}}{E_{x}} and \\int \\frac{\\Psi_{nl}{(E_{x})}}{E_{x}} dE_{x} = \\int \\frac{\\cos{(E_{x})}}{E_{x}} dE_{x} and \\iint \\frac{\\Psi_{nl}{(E_{x})}}{E_{x}} dE_{x} dE_{x} = \\iint \\frac{\\cos{(E_{x})}}{E_{x}} dE_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(i,\\mathbf{J})} = i \\sin{(\\mathbf{J})}, then obtain - \\frac{\\sin{(i \\operatorname{n_{1}}{(i,\\mathbf{J})})}}{\\sin{(i^{2} \\sin{(\\mathbf{J})})}} = -1", "derivation": "\\operatorname{n_{1}}{(i,\\mathbf{J})} = i \\sin{(\\mathbf{J})} and i \\operatorname{n_{1}}{(i,\\mathbf{J})} = i^{2} \\sin{(\\mathbf{J})} and \\sin{(i \\operatorname{n_{1}}{(i,\\mathbf{J})})} = \\sin{(i^{2} \\sin{(\\mathbf{J})})} and \\frac{\\sin{(i \\operatorname{n_{1}}{(i,\\mathbf{J})})}}{\\sin{(i^{2} \\sin{(\\mathbf{J})})}} = 1 and - \\frac{\\sin{(i \\operatorname{n_{1}}{(i,\\mathbf{J})})}}{\\sin{(i^{2} \\sin{(\\mathbf{J})})}} = -1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('i', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), sin(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 3, "sin(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Mul(sin(Mul(Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Pow(sin(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1))), Integer(1))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(Mul(Symbol('i', commutative=True), Function('n_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Pow(sin(Mul(Pow(Symbol('i', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(-1))), Integer(-1))"]]}, {"prompt": "Given \\mathbf{F}{(v_{z})} = e^{e^{v_{z}}} and V{(v_{z})} = e^{v_{z}} e^{e^{v_{z}}}, then obtain v_{z} + V{(v_{z})} + e^{e^{v_{z}}} = v_{z} + \\mathbf{F}{(v_{z})} e^{v_{z}} + e^{e^{v_{z}}}", "derivation": "\\mathbf{F}{(v_{z})} = e^{e^{v_{z}}} and \\mathbf{F}{(v_{z})} e^{v_{z}} = e^{v_{z}} e^{e^{v_{z}}} and V{(v_{z})} = e^{v_{z}} e^{e^{v_{z}}} and v_{z} + V{(v_{z})} = v_{z} + e^{v_{z}} e^{e^{v_{z}}} and v_{z} + V{(v_{z})} + e^{e^{v_{z}}} = v_{z} + e^{v_{z}} e^{e^{v_{z}}} + e^{e^{v_{z}}} and v_{z} + V{(v_{z})} + e^{e^{v_{z}}} = v_{z} + \\mathbf{F}{(v_{z})} e^{v_{z}} + e^{e^{v_{z}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True))))"], [["times", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Mul(exp(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True)))))"], ["renaming_premise", "Equality(Function('V')(Symbol('v_z', commutative=True)), Mul(exp(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True)))))"], [["add", 3, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Function('V')(Symbol('v_z', commutative=True))), Add(Symbol('v_z', commutative=True), Mul(exp(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True))))))"], [["add", 4, "exp(exp(Symbol('v_z', commutative=True)))"], "Equality(Add(Symbol('v_z', commutative=True), Function('V')(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True)))), Add(Symbol('v_z', commutative=True), Mul(exp(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True)))), exp(exp(Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('v_z', commutative=True), Function('V')(Symbol('v_z', commutative=True)), exp(exp(Symbol('v_z', commutative=True)))), Add(Symbol('v_z', commutative=True), Mul(Function('\\\\mathbf{F}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), exp(exp(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{P})} = \\mathbf{P}, then derive \\int \\operatorname{A_{1}}{(\\mathbf{P})} d\\mathbf{P} = \\hat{H}_{\\lambda} + \\frac{\\mathbf{P}^{2}}{2}, then obtain \\frac{d}{d \\mathbf{P}} \\int \\operatorname{A_{1}}{(\\mathbf{P})} d\\mathbf{P} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\hat{H}_{\\lambda} + \\frac{\\mathbf{P}^{2}}{2})", "derivation": "\\operatorname{A_{1}}{(\\mathbf{P})} = \\mathbf{P} and \\int \\operatorname{A_{1}}{(\\mathbf{P})} d\\mathbf{P} = \\int \\mathbf{P} d\\mathbf{P} and \\int \\operatorname{A_{1}}{(\\mathbf{P})} d\\mathbf{P} = \\hat{H}_{\\lambda} + \\frac{\\mathbf{P}^{2}}{2} and \\frac{d}{d \\mathbf{P}} \\int \\operatorname{A_{1}}{(\\mathbf{P})} d\\mathbf{P} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\hat{H}_{\\lambda} + \\frac{\\mathbf{P}^{2}}{2})", "srepr_derivation": [["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_1')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Function('A_1')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{D},f)} = \\mathbf{D} + f and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} = \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})}, then obtain - \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} + \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})} = 0", "derivation": "\\operatorname{F_{x}}{(\\mathbf{D},f)} = \\mathbf{D} + f and \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})} = \\log{(\\mathbf{D} + f)} and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} = \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})} and - \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} + \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})} = - \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} + \\log{(\\mathbf{D} + f)} and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} = \\log{(\\mathbf{D} + f)} and - \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},f)} + \\log{(\\operatorname{F_{x}}{(\\mathbf{D},f)})} = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)))"], [["log", 1], "Equality(log(Function('F_x')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)), log(Function('F_x')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))))"], [["minus", 2, "Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))), log(Function('F_x')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True))), log(Function('F_x')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = e^{f^{*}}, then obtain (- \\operatorname{A_{y}}{(l,\\mathbf{B})} + \\operatorname{V_{\\mathbf{B}}}{(f^{*})} e^{f^{*}}) \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = (- \\operatorname{A_{y}}{(l,\\mathbf{B})} + e^{2 f^{*}}) \\operatorname{V_{\\mathbf{B}}}{(f^{*})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(f^{*})} = e^{f^{*}} and \\operatorname{V_{\\mathbf{B}}}{(f^{*})} e^{f^{*}} = e^{2 f^{*}} and - \\operatorname{A_{y}}{(l,\\mathbf{B})} + \\operatorname{V_{\\mathbf{B}}}{(f^{*})} e^{f^{*}} = - \\operatorname{A_{y}}{(l,\\mathbf{B})} + e^{2 f^{*}} and (- \\operatorname{A_{y}}{(l,\\mathbf{B})} + \\operatorname{V_{\\mathbf{B}}}{(f^{*})} e^{f^{*}}) \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = (- \\operatorname{A_{y}}{(l,\\mathbf{B})} + e^{2 f^{*}}) \\operatorname{V_{\\mathbf{B}}}{(f^{*})}", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["times", 1, "exp(Symbol('f^*', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True))), exp(Mul(Integer(2), Symbol('f^*', commutative=True))))"], [["minus", 2, "Function('A_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))), Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), exp(Mul(Integer(2), Symbol('f^*', commutative=True)))))"], [["times", 3, "Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))), Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True))), Mul(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), exp(Mul(Integer(2), Symbol('f^*', commutative=True)))), Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(q,f^{*})} = f^{*} + q, then derive \\int \\sigma_{p}{(q,f^{*})} dq = B + f^{*} q + \\frac{q^{2}}{2}, then obtain B + f^{*} q + \\frac{q^{2}}{2} = \\int f^{*} dq + \\int q dq", "derivation": "\\sigma_{p}{(q,f^{*})} = f^{*} + q and \\int \\sigma_{p}{(q,f^{*})} dq = \\int (f^{*} + q) dq and \\int \\sigma_{p}{(q,f^{*})} dq = B + f^{*} q + \\frac{q^{2}}{2} and \\int \\sigma_{p}{(q,f^{*})} dq = \\int f^{*} dq + \\int q dq and B + f^{*} q + \\frac{q^{2}}{2} = \\int f^{*} dq + \\int q dq", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('q', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('q', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_p')(Symbol('q', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('f^*', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))))"], [["expand", 2], "Equality(Integral(Function('\\\\sigma_p')(Symbol('q', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Integral(Symbol('f^*', commutative=True), Tuple(Symbol('q', commutative=True))), Integral(Symbol('q', commutative=True), Tuple(Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('B', commutative=True), Mul(Symbol('f^*', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Add(Integral(Symbol('f^*', commutative=True), Tuple(Symbol('q', commutative=True))), Integral(Symbol('q', commutative=True), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(H,v_{t})} = \\log{(H v_{t})}, then obtain \\int \\frac{\\sigma_{p}{(H,v_{t})}}{H} dH = \\eta^{\\prime} + \\frac{\\log{(H v_{t})}^{2}}{2}", "derivation": "\\sigma_{p}{(H,v_{t})} = \\log{(H v_{t})} and \\frac{\\sigma_{p}{(H,v_{t})}}{H} = \\frac{\\log{(H v_{t})}}{H} and \\int \\frac{\\sigma_{p}{(H,v_{t})}}{H} dH = \\int \\frac{\\log{(H v_{t})}}{H} dH and \\int \\frac{\\sigma_{p}{(H,v_{t})}}{H} dH = \\eta^{\\prime} + \\frac{\\log{(H v_{t})}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('v_t', commutative=True))))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), log(Mul(Symbol('H', commutative=True), Symbol('v_t', commutative=True)))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), log(Mul(Symbol('H', commutative=True), Symbol('v_t', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('H', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(log(Mul(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\theta_{2}{(h,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + h, then derive \\frac{\\partial}{\\partial h} \\theta_{2}{(h,f_{\\mathbf{v}})} = 1, then obtain \\frac{\\partial^{2}}{\\partial f_{\\mathbf{v}}^{2}} - \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) = \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} (-1)", "derivation": "\\theta_{2}{(h,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + h and \\frac{\\partial}{\\partial h} \\theta_{2}{(h,f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) and \\frac{\\partial}{\\partial h} \\theta_{2}{(h,f_{\\mathbf{v}})} = 1 and \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) = 1 and - \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) = -1 and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} - \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) = \\frac{d}{d f_{\\mathbf{v}}} (-1) and \\frac{\\partial^{2}}{\\partial f_{\\mathbf{v}}^{2}} - \\frac{\\partial}{\\partial h} (- f_{\\mathbf{v}} + h) = \\frac{d^{2}}{d f_{\\mathbf{v}}^{2}} (-1)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Integer(-1))"], [["differentiate", 5, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))), Derivative(Integer(-1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{H}{(A)} = \\sin{(A)}, then derive \\sin{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} = \\sin{(A)} \\cos{(A)}, then obtain \\mathbf{H}{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} + 1 = \\mathbf{H}{(A)} \\cos{(A)} + 1", "derivation": "\\mathbf{H}{(A)} = \\sin{(A)} and \\frac{d}{d A} \\mathbf{H}{(A)} = \\frac{d}{d A} \\sin{(A)} and \\sin{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} = \\sin{(A)} \\frac{d}{d A} \\sin{(A)} and \\sin{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} = \\sin{(A)} \\cos{(A)} and \\mathbf{H}{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} = \\mathbf{H}{(A)} \\cos{(A)} and \\mathbf{H}{(A)} \\frac{d}{d A} \\mathbf{H}{(A)} + 1 = \\mathbf{H}{(A)} \\cos{(A)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 2, "sin(Symbol('A', commutative=True))"], "Equality(Mul(sin(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(sin(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(sin(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))))"], [["add", 5, 1], "Equality(Add(Mul(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Function('\\\\mathbf{H}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\theta_{1}{(J)} = \\cos{(\\log{(J)})} and \\operatorname{r_{0}}{(J)} = \\frac{\\cos{(\\log{(J)})}}{J \\theta_{1}{(J)}}, then obtain \\frac{d}{d J} e^{\\frac{\\cos{(\\log{(J)})}}{J \\theta_{1}{(J)}}} = \\frac{d}{d J} e^{\\frac{1}{J}}", "derivation": "\\theta_{1}{(J)} = \\cos{(\\log{(J)})} and J \\theta_{1}{(J)} = J \\cos{(\\log{(J)})} and \\operatorname{r_{0}}{(J)} = \\frac{\\cos{(\\log{(J)})}}{J \\theta_{1}{(J)}} and \\operatorname{r_{0}}{(J)} = \\frac{1}{J} and e^{\\operatorname{r_{0}}{(J)}} = e^{\\frac{1}{J}} and \\frac{d}{d J} e^{\\operatorname{r_{0}}{(J)}} = \\frac{d}{d J} e^{\\frac{1}{J}} and \\frac{d}{d J} e^{\\frac{\\cos{(\\log{(J)})}}{J \\theta_{1}{(J)}}} = \\frac{d}{d J} e^{\\frac{1}{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('J', commutative=True)), cos(log(Symbol('J', commutative=True))))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\theta_1')(Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), cos(log(Symbol('J', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('J', commutative=True)), Integer(-1)), cos(log(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('r_0')(Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)))"], [["exp", 4], "Equality(exp(Function('r_0')(Symbol('J', commutative=True))), exp(Pow(Symbol('J', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(exp(Function('r_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('J', commutative=True)), Integer(-1)), cos(log(Symbol('J', commutative=True))))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\Omega)} = \\sin{(\\Omega)}, then obtain 2 (4 \\mathbf{D}{(\\Omega)} - \\sin{(\\Omega)}) \\mathbf{D}{(\\Omega)} = 6 \\mathbf{D}^{2}{(\\Omega)}", "derivation": "\\mathbf{D}{(\\Omega)} = \\sin{(\\Omega)} and \\mathbf{D}{(\\Omega)} - \\sin{(\\Omega)} = 0 and 2 \\mathbf{D}{(\\Omega)} - \\sin{(\\Omega)} = \\mathbf{D}{(\\Omega)} and 4 \\mathbf{D}{(\\Omega)} - \\sin{(\\Omega)} = 3 \\mathbf{D}{(\\Omega)} and 2 (4 \\mathbf{D}{(\\Omega)} - \\sin{(\\Omega)}) \\mathbf{D}{(\\Omega)} = 6 \\mathbf{D}^{2}{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Omega', commutative=True)))), Integer(0))"], [["add", 2, "Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Omega', commutative=True)))), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)))"], [["add", 3, "Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(4), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Omega', commutative=True)))), Mul(Integer(3), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))))"], [["times", 4, "Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Integer(2), Add(Mul(Integer(4), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Omega', commutative=True)))), Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True))), Mul(Integer(6), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\Omega', commutative=True)), Integer(2))))"]]}, {"prompt": "Given I{(\\mathbf{J}_P)} = \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P, then derive (\\mathbf{J}_P I{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\mathbf{J}_P (n_{2} + e^{\\mathbf{J}_P}))^{\\mathbf{J}_P}, then obtain (\\mathbf{J}_P \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P)^{\\mathbf{J}_P} = (\\mathbf{J}_P (n_{2} + e^{\\mathbf{J}_P}))^{\\mathbf{J}_P}", "derivation": "I{(\\mathbf{J}_P)} = \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P and \\mathbf{J}_P I{(\\mathbf{J}_P)} = \\mathbf{J}_P \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P and (\\mathbf{J}_P I{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\mathbf{J}_P \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P)^{\\mathbf{J}_P} and (\\mathbf{J}_P I{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\mathbf{J}_P (n_{2} + e^{\\mathbf{J}_P}))^{\\mathbf{J}_P} and (\\mathbf{J}_P \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P)^{\\mathbf{J}_P} = (\\mathbf{J}_P (n_{2} + e^{\\mathbf{J}_P}))^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('n_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('n_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(F_{x},\\hat{x}_0)} = \\frac{\\partial}{\\partial F_{x}} (F_{x} + \\hat{x}_0), then derive \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(F_{x},\\hat{x}_0)} = 0, then obtain - \\hat{x}_0 + \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial F_{x}} (F_{x} + \\hat{x}_0) = - \\hat{x}_0", "derivation": "\\mathbf{J}_P{(F_{x},\\hat{x}_0)} = \\frac{\\partial}{\\partial F_{x}} (F_{x} + \\hat{x}_0) and \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(F_{x},\\hat{x}_0)} = \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial F_{x}} (F_{x} + \\hat{x}_0) and \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(F_{x},\\hat{x}_0)} = 0 and - \\hat{x}_0 + \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(F_{x},\\hat{x}_0)} = - \\hat{x}_0 and - \\hat{x}_0 + \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial F_{x}} (F_{x} + \\hat{x}_0) = - \\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(0))"], [["minus", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\chi)} = \\sin{(\\log{(\\chi)})} and S{(\\chi)} = \\sin{(\\log{(\\chi)})}, then obtain \\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)} - \\sin{(\\log{(\\chi)})} = - \\sin{(\\log{(\\chi)})} + \\sin^{\\chi}{(\\log{(\\chi)})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\chi)} = \\sin{(\\log{(\\chi)})} and \\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)} = \\sin^{\\chi}{(\\log{(\\chi)})} and S{(\\chi)} = \\sin{(\\log{(\\chi)})} and - S{(\\chi)} + \\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)} = - S{(\\chi)} + \\sin^{\\chi}{(\\log{(\\chi)})} and \\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)} - \\sin{(\\log{(\\chi)})} = - \\sin{(\\log{(\\chi)})} + \\sin^{\\chi}{(\\log{(\\chi)})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), sin(log(Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(log(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\chi', commutative=True)), sin(log(Symbol('\\\\chi', commutative=True))))"], [["minus", 2, "Function('S')(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('\\\\chi', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\chi', commutative=True))), Pow(sin(log(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Add(Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True)))), Pow(sin(log(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\phi)} = \\sin{(\\phi)} and \\mathbf{M}{(\\phi)} = \\sin^{\\phi}{(\\phi)}, then obtain \\iint (\\mathbf{S}^{\\phi}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi d\\phi = \\iint (\\mathbf{M}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi d\\phi", "derivation": "\\mathbf{S}{(\\phi)} = \\sin{(\\phi)} and \\mathbf{S}^{\\phi}{(\\phi)} = \\sin^{\\phi}{(\\phi)} and \\mathbf{M}{(\\phi)} = \\sin^{\\phi}{(\\phi)} and \\mathbf{S}^{\\phi}{(\\phi)} = \\mathbf{M}{(\\phi)} and \\mathbf{S}^{\\phi}{(\\phi)} + \\frac{d}{d \\phi} 0 = \\mathbf{M}{(\\phi)} + \\frac{d}{d \\phi} 0 and \\int (\\mathbf{S}^{\\phi}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi = \\int (\\mathbf{M}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi and \\iint (\\mathbf{S}^{\\phi}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi d\\phi = \\iint (\\mathbf{M}{(\\phi)} + \\frac{d}{d \\phi} 0) d\\phi d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(sin(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\phi', commutative=True)), Pow(sin(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\phi', commutative=True)))"], [["add", 4, "Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{M}')(Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Function('\\\\mathbf{M}')(Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["integrate", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Function('\\\\mathbf{M}')(Symbol('\\\\phi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{H},x)} = \\frac{x}{\\mathbf{H}}, then derive \\frac{\\partial}{\\partial x} \\varepsilon{(\\mathbf{H},x)} = \\frac{1}{\\mathbf{H}}, then obtain \\mathbf{H} - (\\mathbf{H} + \\frac{1}{\\mathbf{H}})^{x} + \\frac{1}{\\mathbf{H}} = \\mathbf{H} - (\\mathbf{H} + \\frac{1}{\\mathbf{H}})^{x} + \\frac{\\partial}{\\partial x} \\frac{x}{\\mathbf{H}}", "derivation": "\\varepsilon{(\\mathbf{H},x)} = \\frac{x}{\\mathbf{H}} and \\frac{\\partial}{\\partial x} \\varepsilon{(\\mathbf{H},x)} = \\frac{\\partial}{\\partial x} \\frac{x}{\\mathbf{H}} and \\frac{\\partial}{\\partial x} \\varepsilon{(\\mathbf{H},x)} = \\frac{1}{\\mathbf{H}} and \\mathbf{H} + \\frac{\\partial}{\\partial x} \\varepsilon{(\\mathbf{H},x)} = \\mathbf{H} + \\frac{\\partial}{\\partial x} \\frac{x}{\\mathbf{H}} and \\mathbf{H} + \\frac{1}{\\mathbf{H}} = \\mathbf{H} + \\frac{\\partial}{\\partial x} \\frac{x}{\\mathbf{H}} and \\mathbf{H} - (\\mathbf{H} + \\frac{1}{\\mathbf{H}})^{x} + \\frac{1}{\\mathbf{H}} = \\mathbf{H} - (\\mathbf{H} + \\frac{1}{\\mathbf{H}})^{x} + \\frac{\\partial}{\\partial x} \\frac{x}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["add", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["minus", 5, "Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('x', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('x', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('x', commutative=True))), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(c)} = \\log{(c)}, then derive \\int \\mathbf{p}{(c)} dc = c \\log{(c)} - c + r, then obtain ((- c + \\log{(c)}) \\iint \\mathbf{p}{(c)} dc dc)^{c} = ((- c + \\log{(c)}) \\int (c \\mathbf{p}{(c)} - c + r) dc)^{c}", "derivation": "\\mathbf{p}{(c)} = \\log{(c)} and \\int \\mathbf{p}{(c)} dc = \\int \\log{(c)} dc and \\int \\mathbf{p}{(c)} dc = c \\log{(c)} - c + r and \\int \\mathbf{p}{(c)} dc = c \\mathbf{p}{(c)} - c + r and \\iint \\mathbf{p}{(c)} dc dc = \\int (c \\mathbf{p}{(c)} - c + r) dc and (- c + \\log{(c)}) \\iint \\mathbf{p}{(c)} dc dc = (- c + \\log{(c)}) \\int (c \\mathbf{p}{(c)} - c + r) dc and ((- c + \\log{(c)}) \\iint \\mathbf{p}{(c)} dc dc)^{c} = ((- c + \\log{(c)}) \\int (c \\mathbf{p}{(c)} - c + r) dc)^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), Function('\\\\mathbf{p}')(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('c', commutative=True), Function('\\\\mathbf{p}')(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('c', commutative=True), Function('\\\\mathbf{p}')(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["power", 6, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Integral(Function('\\\\mathbf{p}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('c', commutative=True), Function('\\\\mathbf{p}')(Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})}, then obtain \\frac{d}{d J_{\\varepsilon}} \\operatorname{t_{1}}{(J_{\\varepsilon})} + 1 = 1 + \\frac{1}{J_{\\varepsilon}}", "derivation": "\\operatorname{t_{1}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and J_{\\varepsilon} + \\operatorname{t_{1}}{(J_{\\varepsilon})} = J_{\\varepsilon} + \\log{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} (J_{\\varepsilon} + \\operatorname{t_{1}}{(J_{\\varepsilon})}) = \\frac{d}{d J_{\\varepsilon}} (J_{\\varepsilon} + \\log{(J_{\\varepsilon})}) and \\frac{d}{d J_{\\varepsilon}} \\operatorname{t_{1}}{(J_{\\varepsilon})} + 1 = 1 + \\frac{1}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(z)} = \\log{(z)}, then derive \\int z \\operatorname{z^{*}}{(z)} dz = f_{E} + \\frac{z^{2} \\log{(z)}}{2} - \\frac{z^{2}}{4}, then obtain - f_{E} - \\frac{z^{2} \\operatorname{z^{*}}{(z)}}{2} + \\frac{z^{2}}{4} + \\int z \\operatorname{z^{*}}{(z)} dz = 0", "derivation": "\\operatorname{z^{*}}{(z)} = \\log{(z)} and z \\operatorname{z^{*}}{(z)} = z \\log{(z)} and \\int z \\operatorname{z^{*}}{(z)} dz = \\int z \\log{(z)} dz and \\int z \\operatorname{z^{*}}{(z)} dz = f_{E} + \\frac{z^{2} \\log{(z)}}{2} - \\frac{z^{2}}{4} and \\int z \\operatorname{z^{*}}{(z)} dz = f_{E} + \\frac{z^{2} \\operatorname{z^{*}}{(z)}}{2} - \\frac{z^{2}}{4} and - f_{E} - \\frac{z^{2} \\operatorname{z^{*}}{(z)}}{2} + \\frac{z^{2}}{4} + \\int z \\operatorname{z^{*}}{(z)} dz = 0", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('z^*')(Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Symbol('z', commutative=True), Function('z^*')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('z', commutative=True), Function('z^*')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)), log(Symbol('z', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('z', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Symbol('z', commutative=True), Function('z^*')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)), Function('z^*')(Symbol('z', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('z', commutative=True), Integer(2)))))"], [["minus", 5, "Add(Symbol('f_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)), Function('z^*')(Symbol('z', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('z', commutative=True), Integer(2))))"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)), Function('z^*')(Symbol('z', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('z', commutative=True), Integer(2))), Integral(Mul(Symbol('z', commutative=True), Function('z^*')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(n_{1},B)} = n_{1} e^{B}, then obtain B - \\operatorname{A_{2}}^{2}{(n_{1},B)} - \\frac{\\operatorname{A_{2}}^{2}{(n_{1},B)}}{B} = B - \\operatorname{A_{2}}^{2}{(n_{1},B)} - \\frac{n_{1} \\operatorname{A_{2}}{(n_{1},B)} e^{B}}{B}", "derivation": "\\operatorname{A_{2}}{(n_{1},B)} = n_{1} e^{B} and \\operatorname{A_{2}}^{2}{(n_{1},B)} = n_{1} \\operatorname{A_{2}}{(n_{1},B)} e^{B} and - \\frac{\\operatorname{A_{2}}^{2}{(n_{1},B)}}{B} = - \\frac{n_{1} \\operatorname{A_{2}}{(n_{1},B)} e^{B}}{B} and B - \\operatorname{A_{2}}^{2}{(n_{1},B)} - \\frac{\\operatorname{A_{2}}^{2}{(n_{1},B)}}{B} = B - \\operatorname{A_{2}}^{2}{(n_{1},B)} - \\frac{n_{1} \\operatorname{A_{2}}{(n_{1},B)} e^{B}}{B}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('n_1', commutative=True), exp(Symbol('B', commutative=True))))"], [["times", 1, "Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True))"], "Equality(Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2)), Mul(Symbol('n_1', commutative=True), Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2)))"], "Equality(Add(Symbol('B', commutative=True), Mul(Integer(-1), Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2)))), Add(Symbol('B', commutative=True), Mul(Integer(-1), Pow(Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Function('A_2')(Symbol('n_1', commutative=True), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(h,\\hat{p}_0)} = \\frac{\\hat{p}_0}{h}, then derive \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{S}{(h,\\hat{p}_0)} = \\frac{1}{h}, then obtain - \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{S}{(h,\\hat{p}_0)} = - \\frac{1}{h}", "derivation": "\\mathbf{S}{(h,\\hat{p}_0)} = \\frac{\\hat{p}_0}{h} and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{S}{(h,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} \\frac{\\hat{p}_0}{h} and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{S}{(h,\\hat{p}_0)} = \\frac{1}{h} and - \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{S}{(h,\\hat{p}_0)} = - \\frac{1}{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Pow(Symbol('h', commutative=True), Integer(-1)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{S}')(Symbol('h', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain \\dot{y}{(y^{\\prime})} e^{- y^{\\prime}} = 1", "derivation": "\\dot{y}{(y^{\\prime})} = e^{y^{\\prime}} and \\dot{y}^{y^{\\prime}}{(y^{\\prime})} = (e^{y^{\\prime}})^{y^{\\prime}} and \\dot{y}{(y^{\\prime})} \\dot{y}^{y^{\\prime}}{(y^{\\prime})} = \\dot{y}^{y^{\\prime}}{(y^{\\prime})} e^{y^{\\prime}} and \\dot{y}{(y^{\\prime})} (e^{y^{\\prime}})^{y^{\\prime}} = e^{y^{\\prime}} (e^{y^{\\prime}})^{y^{\\prime}} and \\dot{y}{(y^{\\prime})} e^{- y^{\\prime}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(exp(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(exp(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(exp(Symbol('y^{\\\\prime}', commutative=True)), Pow(exp(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 4, "Mul(exp(Symbol('y^{\\\\prime}', commutative=True)), Pow(exp(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\chi)} = \\chi, then obtain (\\frac{\\int \\cos{(\\operatorname{F_{x}}{(\\chi)})} d\\chi}{\\cos{(\\chi)}})^{\\chi} = (\\frac{\\varphi + \\sin{(\\chi)}}{\\cos{(\\chi)}})^{\\chi}", "derivation": "\\operatorname{F_{x}}{(\\chi)} = \\chi and \\cos{(\\operatorname{F_{x}}{(\\chi)})} = \\cos{(\\chi)} and \\int \\cos{(\\operatorname{F_{x}}{(\\chi)})} d\\chi = \\int \\cos{(\\chi)} d\\chi and \\frac{\\int \\cos{(\\operatorname{F_{x}}{(\\chi)})} d\\chi}{\\cos{(\\chi)}} = \\frac{\\int \\cos{(\\chi)} d\\chi}{\\cos{(\\chi)}} and (\\frac{\\int \\cos{(\\operatorname{F_{x}}{(\\chi)})} d\\chi}{\\cos{(\\chi)}})^{\\chi} = (\\frac{\\int \\cos{(\\chi)} d\\chi}{\\cos{(\\chi)}})^{\\chi} and (\\frac{\\int \\cos{(\\operatorname{F_{x}}{(\\chi)})} d\\chi}{\\cos{(\\chi)}})^{\\chi} = (\\frac{\\varphi + \\sin{(\\chi)}}{\\cos{(\\chi)}})^{\\chi}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["cos", 1], "Equality(cos(Function('F_x')(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(cos(Function('F_x')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 3, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Function('F_x')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Function('F_x')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Mul(Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Function('F_x')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\chi', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given A{(A_{z},\\mathbf{M})} = \\cos{(\\frac{A_{z}}{\\mathbf{M}})} and \\mathbf{J}_P{(A_{z},\\mathbf{M})} = A^{A_{z}}{(A_{z},\\mathbf{M})}, then obtain \\mathbf{J}_P{(A_{z},\\mathbf{M})} - \\frac{1}{\\mathbf{M}} = \\cos^{A_{z}}{(\\frac{A_{z}}{\\mathbf{M}})} - \\frac{1}{\\mathbf{M}}", "derivation": "A{(A_{z},\\mathbf{M})} = \\cos{(\\frac{A_{z}}{\\mathbf{M}})} and A^{A_{z}}{(A_{z},\\mathbf{M})} = \\cos^{A_{z}}{(\\frac{A_{z}}{\\mathbf{M}})} and \\mathbf{J}_P{(A_{z},\\mathbf{M})} = A^{A_{z}}{(A_{z},\\mathbf{M})} and \\mathbf{J}_P{(A_{z},\\mathbf{M})} = \\cos^{A_{z}}{(\\frac{A_{z}}{\\mathbf{M}})} and \\mathbf{J}_P{(A_{z},\\mathbf{M})} - \\frac{1}{\\mathbf{M}} = \\cos^{A_{z}}{(\\frac{A_{z}}{\\mathbf{M}})} - \\frac{1}{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('A')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_z', commutative=True)), Pow(cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))), Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('A')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}_P')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))), Symbol('A_z', commutative=True)))"], [["minus", 4, "Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))), Add(Pow(cos(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))), Symbol('A_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{r}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain \\frac{\\mathbf{r}{(y^{\\prime})} + 3 e^{y^{\\prime}}}{y^{\\prime}} = \\frac{4 e^{y^{\\prime}}}{y^{\\prime}}", "derivation": "\\mathbf{r}{(y^{\\prime})} = e^{y^{\\prime}} and \\mathbf{r}{(y^{\\prime})} + e^{y^{\\prime}} = 2 e^{y^{\\prime}} and \\mathbf{r}{(y^{\\prime})} + 3 e^{y^{\\prime}} = 4 e^{y^{\\prime}} and \\frac{\\mathbf{r}{(y^{\\prime})} + 3 e^{y^{\\prime}}}{y^{\\prime}} = \\frac{4 e^{y^{\\prime}}}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(2), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 2, "Mul(Integer(2), exp(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(3), exp(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(4), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(3), exp(Symbol('y^{\\\\prime}', commutative=True))))), Mul(Integer(4), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(m,\\phi_1)} = \\phi_1^{m}, then derive \\frac{\\partial}{\\partial m} \\operatorname{y^{\\prime}}{(m,\\phi_1)} = \\phi_1^{m} \\log{(\\phi_1)}, then obtain \\phi_1^{m} \\log{(\\phi_1)} = \\frac{\\partial}{\\partial m} \\phi_1^{m}", "derivation": "\\operatorname{y^{\\prime}}{(m,\\phi_1)} = \\phi_1^{m} and \\frac{\\partial}{\\partial m} \\operatorname{y^{\\prime}}{(m,\\phi_1)} = \\frac{\\partial}{\\partial m} \\phi_1^{m} and \\frac{\\partial}{\\partial m} \\operatorname{y^{\\prime}}{(m,\\phi_1)} = \\phi_1^{m} \\log{(\\phi_1)} and \\phi_1^{m} \\log{(\\phi_1)} = \\frac{\\partial}{\\partial m} \\phi_1^{m}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('m', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('m', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('m', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))), Derivative(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} = \\frac{\\pi t_{1}}{v_{z}}, then obtain - \\frac{- \\pi + 2 \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}{\\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}} = - \\frac{\\frac{\\pi t_{1}}{v_{z}} - \\pi + \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}{\\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}", "derivation": "\\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} = \\frac{\\pi t_{1}}{v_{z}} and 2 \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} = \\frac{\\pi t_{1}}{v_{z}} + \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} and - \\pi + 2 \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} = \\frac{\\pi t_{1}}{v_{z}} - \\pi + \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)} and - \\frac{- \\pi + 2 \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}{\\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}} = - \\frac{\\frac{\\pi t_{1}}{v_{z}} - \\pi + \\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}{\\operatorname{F_{c}}{(v_{z},t_{1},\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["minus", 1, "Mul(Integer(-1), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)))), Pow(Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True))), Pow(Function('F_c')(Symbol('v_z', commutative=True), Symbol('t_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{x}{(y)} = \\cos{(y)}, then obtain \\frac{\\int \\frac{d}{d y} (\\sigma_{x}{(y)} - \\cos{(y)}) dy}{y} = \\frac{\\int 0 dy}{y}", "derivation": "\\sigma_{x}{(y)} = \\cos{(y)} and \\sigma_{x}{(y)} - \\cos{(y)} = 0 and \\frac{d}{d y} (\\sigma_{x}{(y)} - \\cos{(y)}) = \\frac{d}{d y} 0 and \\int \\frac{d}{d y} (\\sigma_{x}{(y)} - \\cos{(y)}) dy = \\int \\frac{d}{d y} 0 dy and \\frac{\\int \\frac{d}{d y} (\\sigma_{x}{(y)} - \\cos{(y)}) dy}{y} = \\frac{\\int \\frac{d}{d y} 0 dy}{y} and \\frac{\\int \\frac{d}{d y} (\\sigma_{x}{(y)} - \\cos{(y)}) dy}{y} = \\frac{\\int 0 dy}{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["minus", 1, "cos(Symbol('y', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Function('\\\\sigma_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\sigma_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["times", 4, "Pow(Symbol('y', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Derivative(Add(Function('\\\\sigma_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Derivative(Add(Function('\\\\sigma_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given s{(f^{\\prime},\\mu_0)} = \\frac{\\mu_0}{f^{\\prime}}, then derive \\frac{\\partial}{\\partial f^{\\prime}} s{(f^{\\prime},\\mu_0)} = - \\frac{\\mu_0}{(f^{\\prime})^{2}}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} \\frac{\\mu_0}{f^{\\prime}} = - \\frac{\\mu_0}{(f^{\\prime})^{2}}", "derivation": "s{(f^{\\prime},\\mu_0)} = \\frac{\\mu_0}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} s{(f^{\\prime},\\mu_0)} = \\frac{\\partial}{\\partial f^{\\prime}} \\frac{\\mu_0}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} s{(f^{\\prime},\\mu_0)} = - \\frac{\\mu_0}{(f^{\\prime})^{2}} and \\frac{\\partial}{\\partial f^{\\prime}} s{(f^{\\prime},\\mu_0)} = - \\frac{s{(f^{\\prime},\\mu_0)}}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} \\frac{\\mu_0}{f^{\\prime}} = - \\frac{\\mu_0}{(f^{\\prime})^{2}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('s')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('s')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mu)} = \\cos{(\\mu)} and \\operatorname{a^{\\dagger}}{(\\mu)} = 2 \\cos{(\\mu)}, then obtain 2 \\mathbf{A}{(\\mu)} = 2 \\cos{(\\mu)}", "derivation": "\\mathbf{A}{(\\mu)} = \\cos{(\\mu)} and \\mathbf{A}{(\\mu)} + \\cos{(\\mu)} = 2 \\cos{(\\mu)} and \\operatorname{a^{\\dagger}}{(\\mu)} = 2 \\cos{(\\mu)} and \\operatorname{a^{\\dagger}}{(\\mu)} = \\mathbf{A}{(\\mu)} + \\cos{(\\mu)} and \\operatorname{a^{\\dagger}}{(\\mu)} = 2 \\mathbf{A}{(\\mu)} and 2 \\mathbf{A}{(\\mu)} = \\mathbf{A}{(\\mu)} + \\cos{(\\mu)} and 2 \\mathbf{A}{(\\mu)} = 2 \\cos{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Add(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True))), Add(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\eta{(t_{2})} = \\cos{(e^{t_{2}})} and \\operatorname{A_{z}}{(t_{2})} = \\eta{(t_{2})} + e^{t_{2}}, then obtain \\frac{\\eta{(t_{2})}}{\\operatorname{A_{z}}{(t_{2})}} = \\frac{\\cos{(e^{t_{2}})}}{\\operatorname{A_{z}}{(t_{2})}}", "derivation": "\\eta{(t_{2})} = \\cos{(e^{t_{2}})} and \\eta{(t_{2})} + e^{t_{2}} = e^{t_{2}} + \\cos{(e^{t_{2}})} and \\frac{\\eta{(t_{2})}}{\\eta{(t_{2})} + e^{t_{2}}} = \\frac{\\cos{(e^{t_{2}})}}{\\eta{(t_{2})} + e^{t_{2}}} and \\frac{\\eta{(t_{2})}}{e^{t_{2}} + \\cos{(e^{t_{2}})}} = \\frac{\\cos{(e^{t_{2}})}}{e^{t_{2}} + \\cos{(e^{t_{2}})}} and \\operatorname{A_{z}}{(t_{2})} = \\eta{(t_{2})} + e^{t_{2}} and \\operatorname{A_{z}}{(t_{2})} = e^{t_{2}} + \\cos{(e^{t_{2}})} and \\frac{\\eta{(t_{2})}}{\\operatorname{A_{z}}{(t_{2})}} = \\frac{\\cos{(e^{t_{2}})}}{\\operatorname{A_{z}}{(t_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('t_2', commutative=True)), cos(exp(Symbol('t_2', commutative=True))))"], [["add", 1, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Add(exp(Symbol('t_2', commutative=True)), cos(exp(Symbol('t_2', commutative=True)))))"], [["divide", 1, "Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('t_2', commutative=True))), Mul(Pow(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(-1)), cos(exp(Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(exp(Symbol('t_2', commutative=True)), cos(exp(Symbol('t_2', commutative=True)))), Integer(-1)), Function('\\\\eta')(Symbol('t_2', commutative=True))), Mul(Pow(Add(exp(Symbol('t_2', commutative=True)), cos(exp(Symbol('t_2', commutative=True)))), Integer(-1)), cos(exp(Symbol('t_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('t_2', commutative=True)), Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('A_z')(Symbol('t_2', commutative=True)), Add(exp(Symbol('t_2', commutative=True)), cos(exp(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Function('A_z')(Symbol('t_2', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('t_2', commutative=True))), Mul(Pow(Function('A_z')(Symbol('t_2', commutative=True)), Integer(-1)), cos(exp(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(f)} = \\sin{(e^{f})} and z{(f)} = \\frac{d}{d f} \\int \\frac{\\mathbf{p}{(f)}}{\\sin{(e^{f})}} df, then obtain z{(f)} = \\frac{d}{d f} \\int 1 df", "derivation": "\\mathbf{p}{(f)} = \\sin{(e^{f})} and f \\mathbf{p}{(f)} = f \\sin{(e^{f})} and \\frac{\\mathbf{p}{(f)}}{\\sin{(e^{f})}} = 1 and \\int \\frac{\\mathbf{p}{(f)}}{\\sin{(e^{f})}} df = \\int 1 df and \\frac{d}{d f} \\int \\frac{\\mathbf{p}{(f)}}{\\sin{(e^{f})}} df = \\frac{d}{d f} \\int 1 df and z{(f)} = \\frac{d}{d f} \\int \\frac{\\mathbf{p}{(f)}}{\\sin{(e^{f})}} df and z{(f)} = \\frac{d}{d f} \\int 1 df", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), sin(exp(Symbol('f', commutative=True))))"], [["times", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Function('\\\\mathbf{p}')(Symbol('f', commutative=True))), Mul(Symbol('f', commutative=True), sin(exp(Symbol('f', commutative=True)))))"], [["divide", 2, "Mul(Symbol('f', commutative=True), sin(exp(Symbol('f', commutative=True))))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True))), Integral(Integer(1), Tuple(Symbol('f', commutative=True))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z')(Symbol('f', commutative=True)), Derivative(Integral(Mul(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), Pow(sin(exp(Symbol('f', commutative=True))), Integer(-1))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('z')(Symbol('f', commutative=True)), Derivative(Integral(Integer(1), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\frac{\\mathbf{E} \\mathbf{J}_M}{r}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\frac{\\mathbf{E}}{r}, then obtain r \\frac{\\partial}{\\partial \\mathbf{J}_M} l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\mathbf{E}", "derivation": "l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\frac{\\mathbf{E} \\mathbf{J}_M}{r} and \\frac{\\partial}{\\partial \\mathbf{J}_M} l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{E} \\mathbf{J}_M}{r} and \\frac{\\partial}{\\partial \\mathbf{J}_M} l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\frac{\\mathbf{E}}{r} and r \\frac{\\partial}{\\partial \\mathbf{J}_M} l{(r,\\mathbf{E},\\mathbf{J}_M)} = \\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('r', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["divide", 3, "Pow(Symbol('r', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('r', commutative=True), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Symbol('\\\\mathbf{E}', commutative=True))"]]}, {"prompt": "Given \\sigma_{x}{(\\omega)} = e^{\\omega}, then derive (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} \\int \\sigma_{x}{(\\omega)} d\\omega = (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} (J_{\\varepsilon} + e^{\\omega}), then obtain (\\Psi_{\\lambda} e^{\\omega})^{- \\omega} \\int \\sigma_{x}{(\\omega)} d\\omega = (\\Psi_{\\lambda} e^{\\omega})^{- \\omega} (J_{\\varepsilon} + e^{\\omega})", "derivation": "\\sigma_{x}{(\\omega)} = e^{\\omega} and \\Psi_{\\lambda} \\sigma_{x}{(\\omega)} = \\Psi_{\\lambda} e^{\\omega} and \\int \\sigma_{x}{(\\omega)} d\\omega = \\int e^{\\omega} d\\omega and (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} \\int \\sigma_{x}{(\\omega)} d\\omega = (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} \\int e^{\\omega} d\\omega and (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} \\int \\sigma_{x}{(\\omega)} d\\omega = (\\Psi_{\\lambda} \\sigma_{x}{(\\omega)})^{- \\omega} (J_{\\varepsilon} + e^{\\omega}) and (\\Psi_{\\lambda} e^{\\omega})^{- \\omega} \\int \\sigma_{x}{(\\omega)} d\\omega = (\\Psi_{\\lambda} e^{\\omega})^{- \\omega} (J_{\\varepsilon} + e^{\\omega})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 3, "Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('\\\\sigma_x')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\nabla,W)} = W + \\nabla, then obtain 2 e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} = e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} + e", "derivation": "\\mathbb{I}{(\\nabla,W)} = W + \\nabla and \\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)} = \\frac{\\partial}{\\partial \\nabla} (W + \\nabla) and (\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W} = (\\frac{\\partial}{\\partial \\nabla} (W + \\nabla))^{W} and e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} = e^{(\\frac{\\partial}{\\partial \\nabla} (W + \\nabla))^{W}} and 2 e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} = e^{(\\frac{\\partial}{\\partial \\nabla} (W + \\nabla))^{W}} + e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} and 2 e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} = e^{(\\frac{\\partial}{\\partial \\nabla} \\mathbb{I}{(\\nabla,W)})^{W}} + e", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True))), exp(Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True))))"], [["add", 4, "exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], "Equality(Mul(Integer(2), exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)))), Add(exp(Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True))), exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(2), exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True)))), Add(exp(Pow(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('W', commutative=True))), E))"]]}, {"prompt": "Given \\dot{z}{(k)} = \\log{(k)} and \\phi{(k)} = 0^{k}, then obtain \\phi{(k)} = 1", "derivation": "\\dot{z}{(k)} = \\log{(k)} and \\dot{z}^{k}{(k)} = \\log{(k)}^{k} and (\\dot{z}^{k}{(k)})^{k} = (\\log{(k)}^{k})^{k} and 0 = - (\\dot{z}^{k}{(k)})^{k} + (\\log{(k)}^{k})^{k} and 0^{k} = (- (\\dot{z}^{k}{(k)})^{k} + (\\log{(k)}^{k})^{k})^{k} and \\phi{(k)} = 0^{k} and \\phi{(k)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 3, "Pow(Pow(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Pow(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Integer(0), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Pow(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('k', commutative=True)), Pow(Integer(0), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\phi')(Symbol('k', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mu,s)} = \\frac{\\partial}{\\partial \\mu} (- \\mu + s), then derive \\operatorname{v_{y}}{(\\mu,s)} = -1, then obtain \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{- \\mu + s} = - \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{(- \\mu + s) \\frac{\\partial}{\\partial \\mu} (- \\mu + s)}", "derivation": "\\operatorname{v_{y}}{(\\mu,s)} = \\frac{\\partial}{\\partial \\mu} (- \\mu + s) and - \\operatorname{v_{y}}{(\\mu,s)} = - \\frac{\\partial}{\\partial \\mu} (- \\mu + s) and - \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{\\frac{\\partial}{\\partial \\mu} (- \\mu + s)} = -1 and \\operatorname{v_{y}}{(\\mu,s)} = -1 and \\operatorname{v_{y}}{(\\mu,s)} = - \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{\\frac{\\partial}{\\partial \\mu} (- \\mu + s)} and \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{- \\mu + s} = - \\frac{\\operatorname{v_{y}}{(\\mu,s)}}{(- \\mu + s) \\frac{\\partial}{\\partial \\mu} (- \\mu + s)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["divide", 2, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))), Integer(-1))"], [["evaluate_derivatives", 3], "Equality(Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Integer(-1)), Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Integer(-1)), Function('v_y')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(x)} = e^{x}, then obtain \\operatorname{A_{x}}^{2}{(x)} + \\operatorname{A_{x}}{(x)} \\int \\operatorname{A_{x}}{(x)} dx = \\operatorname{A_{x}}^{2}{(x)} + e^{x} \\int \\operatorname{A_{x}}{(x)} dx", "derivation": "\\operatorname{A_{x}}{(x)} = e^{x} and \\int \\operatorname{A_{x}}{(x)} dx = \\int e^{x} dx and \\operatorname{A_{x}}{(x)} \\int e^{x} dx = e^{x} \\int e^{x} dx and \\operatorname{A_{x}}^{2}{(x)} + \\operatorname{A_{x}}{(x)} \\int e^{x} dx = \\operatorname{A_{x}}^{2}{(x)} + e^{x} \\int e^{x} dx and \\operatorname{A_{x}}^{2}{(x)} + \\operatorname{A_{x}}{(x)} \\int \\operatorname{A_{x}}{(x)} dx = \\operatorname{A_{x}}^{2}{(x)} + e^{x} \\int \\operatorname{A_{x}}{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('x', commutative=True)), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(exp(Symbol('x', commutative=True)), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["add", 3, "Pow(Function('A_x')(Symbol('x', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('A_x')(Symbol('x', commutative=True)), Integer(2)), Mul(Function('A_x')(Symbol('x', commutative=True)), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(Pow(Function('A_x')(Symbol('x', commutative=True)), Integer(2)), Mul(exp(Symbol('x', commutative=True)), Integral(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('A_x')(Symbol('x', commutative=True)), Integer(2)), Mul(Function('A_x')(Symbol('x', commutative=True)), Integral(Function('A_x')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(Pow(Function('A_x')(Symbol('x', commutative=True)), Integer(2)), Mul(exp(Symbol('x', commutative=True)), Integral(Function('A_x')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given k{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\operatorname{A_{2}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\pi{(\\dot{\\mathbf{r}})} = \\frac{k{(\\dot{\\mathbf{r}})}}{\\dot{\\mathbf{r}}}, then obtain \\pi{(\\dot{\\mathbf{r}})} = \\frac{\\operatorname{A_{2}}{(\\dot{\\mathbf{r}})}}{\\dot{\\mathbf{r}}}", "derivation": "k{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\operatorname{A_{2}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\pi{(\\dot{\\mathbf{r}})} = \\frac{k{(\\dot{\\mathbf{r}})}}{\\dot{\\mathbf{r}}} and k{(\\dot{\\mathbf{r}})} = \\operatorname{A_{2}}{(\\dot{\\mathbf{r}})} and \\pi{(\\dot{\\mathbf{r}})} = \\frac{\\operatorname{A_{2}}{(\\dot{\\mathbf{r}})}}{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('k')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('A_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Function('\\\\pi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given B{(\\mathbb{I},f^{\\prime})} = \\frac{f^{\\prime}}{\\mathbb{I}}, then obtain (- \\mathbb{I} + B{(\\mathbb{I},f^{\\prime})} + \\frac{1}{\\mathbb{I}})^{f^{\\prime}} = (- \\mathbb{I} + \\frac{f^{\\prime}}{\\mathbb{I}} + \\frac{1}{\\mathbb{I}})^{f^{\\prime}}", "derivation": "B{(\\mathbb{I},f^{\\prime})} = \\frac{f^{\\prime}}{\\mathbb{I}} and B{(\\mathbb{I},f^{\\prime})} + \\frac{1}{\\mathbb{I}} = \\frac{f^{\\prime}}{\\mathbb{I}} + \\frac{1}{\\mathbb{I}} and - \\mathbb{I} + B{(\\mathbb{I},f^{\\prime})} + \\frac{1}{\\mathbb{I}} = - \\mathbb{I} + \\frac{f^{\\prime}}{\\mathbb{I}} + \\frac{1}{\\mathbb{I}} and (- \\mathbb{I} + B{(\\mathbb{I},f^{\\prime})} + \\frac{1}{\\mathbb{I}})^{f^{\\prime}} = (- \\mathbb{I} + \\frac{f^{\\prime}}{\\mathbb{I}} + \\frac{1}{\\mathbb{I}})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Add(Function('B')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["minus", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('B')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('B')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given z{(\\tilde{g}^*,g)} = \\frac{\\log{(\\tilde{g}^*)}}{g}, then obtain \\frac{\\log{(\\tilde{g}^*)}}{g} = \\tilde{g}^* \\cos{(\\frac{\\log{(\\tilde{g}^*)}}{g})} - \\tilde{g}^* \\cos{(z{(\\tilde{g}^*,g)})} + \\frac{\\log{(\\tilde{g}^*)}}{g}", "derivation": "z{(\\tilde{g}^*,g)} = \\frac{\\log{(\\tilde{g}^*)}}{g} and \\cos{(z{(\\tilde{g}^*,g)})} = \\cos{(\\frac{\\log{(\\tilde{g}^*)}}{g})} and \\tilde{g}^* \\cos{(z{(\\tilde{g}^*,g)})} = \\tilde{g}^* \\cos{(\\frac{\\log{(\\tilde{g}^*)}}{g})} and 0 = \\tilde{g}^* \\cos{(\\frac{\\log{(\\tilde{g}^*)}}{g})} - \\tilde{g}^* \\cos{(z{(\\tilde{g}^*,g)})} and \\frac{\\log{(\\tilde{g}^*)}}{g} = \\tilde{g}^* \\cos{(\\frac{\\log{(\\tilde{g}^*)}}{g})} - \\tilde{g}^* \\cos{(z{(\\tilde{g}^*,g)})} + \\frac{\\log{(\\tilde{g}^*)}}{g}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["cos", 1], "Equality(cos(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True))), cos(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["times", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), cos(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True)))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), cos(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True))))))"], [["minus", 3, "Mul(Symbol('\\\\tilde{g}^*', commutative=True), cos(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True))))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), cos(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True))))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), cos(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True))))))"], [["add", 4, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), cos(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True))))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), cos(Function('z')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('g', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(i,\\psi)} = \\psi i and \\operatorname{F_{c}}{(i)} = i^{9}, then obtain \\frac{i^{5} \\bar{\\h}{(i,\\psi)}}{\\psi^{3}} = \\frac{\\psi \\operatorname{F_{c}}{(i)}}{\\bar{\\h}^{3}{(i,\\psi)}}", "derivation": "\\bar{\\h}{(i,\\psi)} = \\psi i and \\psi i \\bar{\\h}{(i,\\psi)} = \\psi^{2} i^{2} and \\psi^{2} i^{2} \\bar{\\h}^{2}{(i,\\psi)} = \\psi^{4} i^{4} and \\psi^{4} i^{4} \\bar{\\h}^{4}{(i,\\psi)} = \\psi^{8} i^{8} and \\frac{i^{5} \\bar{\\h}{(i,\\psi)}}{\\psi^{3}} = \\frac{\\psi i^{9}}{\\bar{\\h}^{3}{(i,\\psi)}} and \\operatorname{F_{c}}{(i)} = i^{9} and \\frac{i^{5} \\bar{\\h}{(i,\\psi)}}{\\psi^{3}} = \\frac{\\psi \\operatorname{F_{c}}{(i)}}{\\bar{\\h}^{3}{(i,\\psi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True), Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4))))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4)), Pow(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(4))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(8)), Pow(Symbol('i', commutative=True), Integer(8))))"], [["times", 4, "Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-7)), Symbol('i', commutative=True), Pow(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-3)))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-3)), Pow(Symbol('i', commutative=True), Integer(5)), Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('i', commutative=True), Integer(9)), Pow(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-3))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Integer(9)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-3)), Pow(Symbol('i', commutative=True), Integer(5)), Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Function('F_c')(Symbol('i', commutative=True)), Pow(Function('\\\\hbar')(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-3))))"]]}, {"prompt": "Given c{(\\hat{x},\\mathbf{E})} = \\log{(\\hat{x})}^{\\mathbf{E}}, then obtain - (\\log{(\\hat{x})}^{\\mathbf{E}})^{\\hat{x}} + c{(\\hat{x},\\mathbf{E})} = - (\\log{(\\hat{x})}^{\\mathbf{E}})^{\\hat{x}} + \\log{(\\hat{x})}^{\\mathbf{E}}", "derivation": "c{(\\hat{x},\\mathbf{E})} = \\log{(\\hat{x})}^{\\mathbf{E}} and c^{\\hat{x}}{(\\hat{x},\\mathbf{E})} = (\\log{(\\hat{x})}^{\\mathbf{E}})^{\\hat{x}} and c{(\\hat{x},\\mathbf{E})} - c^{\\hat{x}}{(\\hat{x},\\mathbf{E})} = - c^{\\hat{x}}{(\\hat{x},\\mathbf{E})} + \\log{(\\hat{x})}^{\\mathbf{E}} and - (\\log{(\\hat{x})}^{\\mathbf{E}})^{\\hat{x}} + c{(\\hat{x},\\mathbf{E})} = - (\\log{(\\hat{x})}^{\\mathbf{E}})^{\\hat{x}} + \\log{(\\hat{x})}^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 1, "Pow(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Function('c')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Pow(log(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}}{t}, then obtain \\frac{C_{2}^{2}}{t} + \\frac{\\partial}{\\partial t} \\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}^{2}}{t} - \\frac{C_{2}}{t^{2}}", "derivation": "\\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}}{t} and C_{2} \\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}^{2}}{t} and \\frac{\\partial}{\\partial t} \\operatorname{v_{1}}{(t,C_{2})} = \\frac{\\partial}{\\partial t} \\frac{C_{2}}{t} and C_{2} \\operatorname{v_{1}}{(t,C_{2})} + \\frac{\\partial}{\\partial t} \\operatorname{v_{1}}{(t,C_{2})} = C_{2} \\operatorname{v_{1}}{(t,C_{2})} + \\frac{\\partial}{\\partial t} \\frac{C_{2}}{t} and \\frac{C_{2}^{2}}{t} + \\frac{\\partial}{\\partial t} \\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}^{2}}{t} + \\frac{\\partial}{\\partial t} \\frac{C_{2}}{t} and \\frac{C_{2}^{2}}{t} + \\frac{\\partial}{\\partial t} \\operatorname{v_{1}}{(t,C_{2})} = \\frac{C_{2}^{2}}{t} - \\frac{C_{2}}{t^{2}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('C_2', commutative=True), Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True))), Derivative(Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Symbol('C_2', commutative=True), Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True))), Derivative(Mul(Symbol('C_2', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-1))), Derivative(Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-1))), Derivative(Mul(Symbol('C_2', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-1))), Derivative(Function('v_1')(Symbol('t', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('t', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(f_{E})} = \\log{(f_{E})}, then obtain \\frac{- 3 \\operatorname{z^{*}}{(f_{E})} - 1}{\\operatorname{z^{*}}{(f_{E})} + 1} = \\frac{- \\operatorname{z^{*}}{(f_{E})} - 2 \\log{(f_{E})} - 1}{\\operatorname{z^{*}}{(f_{E})} + 1}", "derivation": "\\operatorname{z^{*}}{(f_{E})} = \\log{(f_{E})} and 2 \\operatorname{z^{*}}{(f_{E})} = \\operatorname{z^{*}}{(f_{E})} + \\log{(f_{E})} and 3 \\operatorname{z^{*}}{(f_{E})} + 1 = 2 \\operatorname{z^{*}}{(f_{E})} + \\log{(f_{E})} + 1 and - 3 \\operatorname{z^{*}}{(f_{E})} - 1 = - 2 \\operatorname{z^{*}}{(f_{E})} - \\log{(f_{E})} - 1 and - 3 \\operatorname{z^{*}}{(f_{E})} - 1 = - \\operatorname{z^{*}}{(f_{E})} - 2 \\log{(f_{E})} - 1 and \\frac{- 3 \\operatorname{z^{*}}{(f_{E})} - 1}{\\operatorname{z^{*}}{(f_{E})} + 1} = \\frac{- \\operatorname{z^{*}}{(f_{E})} - 2 \\log{(f_{E})} - 1}{\\operatorname{z^{*}}{(f_{E})} + 1}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Function('z^*')(Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(2), Function('z^*')(Symbol('f_E', commutative=True))), Add(Function('z^*')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))"], [["add", 2, "Add(Function('z^*')(Symbol('f_E', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(3), Function('z^*')(Symbol('f_E', commutative=True))), Integer(1)), Add(Mul(Integer(2), Function('z^*')(Symbol('f_E', commutative=True))), log(Symbol('f_E', commutative=True)), Integer(1)))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Integer(3), Function('z^*')(Symbol('f_E', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('z^*')(Symbol('f_E', commutative=True))), Mul(Integer(-1), log(Symbol('f_E', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Integer(3), Function('z^*')(Symbol('f_E', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('z^*')(Symbol('f_E', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('f_E', commutative=True))), Integer(-1)))"], [["divide", 5, "Add(Function('z^*')(Symbol('f_E', commutative=True)), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(3), Function('z^*')(Symbol('f_E', commutative=True))), Integer(-1)), Pow(Add(Function('z^*')(Symbol('f_E', commutative=True)), Integer(1)), Integer(-1))), Mul(Pow(Add(Function('z^*')(Symbol('f_E', commutative=True)), Integer(1)), Integer(-1)), Add(Mul(Integer(-1), Function('z^*')(Symbol('f_E', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('f_E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{p}{(E_{n})} = e^{e^{E_{n}}}, then obtain (\\iiint \\sigma_{p}{(E_{n})} dE_{n} dE_{n} dE_{n}) \\iiint e^{e^{E_{n}}} dE_{n} dE_{n} dE_{n} = (\\iiint e^{e^{E_{n}}} dE_{n} dE_{n} dE_{n})^{2}", "derivation": "\\sigma_{p}{(E_{n})} = e^{e^{E_{n}}} and \\int \\sigma_{p}{(E_{n})} dE_{n} = \\int e^{e^{E_{n}}} dE_{n} and \\iint \\sigma_{p}{(E_{n})} dE_{n} dE_{n} = \\iint e^{e^{E_{n}}} dE_{n} dE_{n} and \\iiint \\sigma_{p}{(E_{n})} dE_{n} dE_{n} dE_{n} = \\iiint e^{e^{E_{n}}} dE_{n} dE_{n} dE_{n} and (\\iiint \\sigma_{p}{(E_{n})} dE_{n} dE_{n} dE_{n}) \\iiint e^{e^{E_{n}}} dE_{n} dE_{n} dE_{n} = (\\iiint e^{e^{E_{n}}} dE_{n} dE_{n} dE_{n})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('E_n', commutative=True)), exp(exp(Symbol('E_n', commutative=True))))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["times", 4, "Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\sigma_p')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Pow(Integral(exp(exp(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\rho{(H,\\mathbf{s})} = \\frac{H}{\\mathbf{s}}, then obtain (\\mathbf{s}^{H})^{\\mathbf{s}} = ((\\frac{H}{\\rho{(H,\\mathbf{s})}})^{H})^{\\mathbf{s}}", "derivation": "\\rho{(H,\\mathbf{s})} = \\frac{H}{\\mathbf{s}} and \\mathbf{s} \\rho{(H,\\mathbf{s})} = H and \\mathbf{s} = \\frac{H}{\\rho{(H,\\mathbf{s})}} and \\mathbf{s}^{H} = (\\frac{H}{\\rho{(H,\\mathbf{s})}})^{H} and (\\mathbf{s}^{H})^{\\mathbf{s}} = ((\\frac{H}{\\rho{(H,\\mathbf{s})}})^{H})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('H', commutative=True))"], [["divide", 2, "Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Symbol('\\\\mathbf{s}', commutative=True), Mul(Symbol('H', commutative=True), Pow(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Pow(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('H', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Pow(Mul(Symbol('H', commutative=True), Pow(Function('\\\\rho')(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('H', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given u{(v_{x})} = \\cos{(v_{x})}, then obtain u^{12}{(v_{x})} = u^{9}{(v_{x})} \\cos^{3}{(v_{x})}", "derivation": "u{(v_{x})} = \\cos{(v_{x})} and u^{2}{(v_{x})} = u{(v_{x})} \\cos{(v_{x})} and u^{4}{(v_{x})} = u^{3}{(v_{x})} \\cos{(v_{x})} and u^{12}{(v_{x})} = u^{9}{(v_{x})} \\cos^{3}{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["times", 1, "Function('u')(Symbol('v_x', commutative=True))"], "Equality(Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Function('u')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True))))"], [["times", 2, "Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(2))"], "Equality(Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(4)), Mul(Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(3)), cos(Symbol('v_x', commutative=True))))"], [["power", 3, 3], "Equality(Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(12)), Mul(Pow(Function('u')(Symbol('v_x', commutative=True)), Integer(9)), Pow(cos(Symbol('v_x', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(S)} = \\log{(S)}, then obtain 0 = - \\frac{d}{d S} \\operatorname{a^{\\dagger}}{(S)} + \\frac{1}{S}", "derivation": "\\operatorname{a^{\\dagger}}{(S)} = \\log{(S)} and 0 = - \\operatorname{a^{\\dagger}}{(S)} + \\log{(S)} and \\frac{d}{d S} 0 = \\frac{d}{d S} (- \\operatorname{a^{\\dagger}}{(S)} + \\log{(S)}) and 0 = - \\frac{d}{d S} \\operatorname{a^{\\dagger}}{(S)} + \\frac{1}{S}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["minus", 1, "Function('a^{\\\\dagger}')(Symbol('S', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('a^{\\\\dagger}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Pow(Symbol('S', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = F_{N} - a^{\\dagger} and \\psi^{*}{(a^{\\dagger},F_{N})} = F_{N} - a^{\\dagger}, then obtain - a^{\\dagger} \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = - a^{\\dagger} \\psi^{*}{(a^{\\dagger},F_{N})}", "derivation": "\\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = F_{N} - a^{\\dagger} and a^{\\dagger} \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = a^{\\dagger} (F_{N} - a^{\\dagger}) and - a^{\\dagger} \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = - a^{\\dagger} (F_{N} - a^{\\dagger}) and \\psi^{*}{(a^{\\dagger},F_{N})} = F_{N} - a^{\\dagger} and \\psi^{*}{(a^{\\dagger},F_{N})} = \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} and a^{\\dagger} \\psi^{*}{(a^{\\dagger},F_{N})} = a^{\\dagger} (F_{N} - a^{\\dagger}) and - a^{\\dagger} \\operatorname{y^{\\prime}}{(a^{\\dagger},F_{N})} = - a^{\\dagger} \\psi^{*}{(a^{\\dagger},F_{N})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True)), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given k{(E,T)} = E + T and \\phi{(E,T)} = \\int (E + T) dT, then obtain \\frac{\\partial}{\\partial T} \\frac{\\phi{(E,T)}}{k{(E,T)}} = \\frac{\\partial}{\\partial T} \\frac{\\int k{(E,T)} dT}{k{(E,T)}}", "derivation": "k{(E,T)} = E + T and \\int k{(E,T)} dT = \\int (E + T) dT and \\phi{(E,T)} = \\int (E + T) dT and \\phi{(E,T)} = \\int k{(E,T)} dT and \\frac{\\phi{(E,T)}}{E + T} = \\frac{\\int k{(E,T)} dT}{E + T} and \\frac{\\partial}{\\partial T} \\frac{\\phi{(E,T)}}{E + T} = \\frac{\\partial}{\\partial T} \\frac{\\int k{(E,T)} dT}{E + T} and \\frac{\\partial}{\\partial T} \\frac{\\phi{(E,T)}}{k{(E,T)}} = \\frac{\\partial}{\\partial T} \\frac{\\int k{(E,T)} dT}{k{(E,T)}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Add(Symbol('E', commutative=True), Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integral(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integral(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["divide", 4, "Add(Symbol('E', commutative=True), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Integral(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["differentiate", 5, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Integral(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Pow(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Integral(Function('k')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(k)} = \\sin{(k)}, then obtain 2 k - \\sin{(k)} + \\int (- k + \\omega{(k)} + 2 \\sin{(k)}) dk = 2 k - \\sin{(k)} + \\int (- k + 2 \\omega{(k)} + \\sin{(k)}) dk", "derivation": "\\omega{(k)} = \\sin{(k)} and 2 \\omega{(k)} = \\omega{(k)} + \\sin{(k)} and - k + 3 \\omega{(k)} = - k + 2 \\omega{(k)} + \\sin{(k)} and - k + 3 \\omega{(k)} = - k + \\omega{(k)} + 2 \\sin{(k)} and \\int (- k + 3 \\omega{(k)}) dk = \\int (- k + \\omega{(k)} + 2 \\sin{(k)}) dk and \\int (- k + 3 \\omega{(k)}) dk = \\int (- k + 2 \\omega{(k)} + \\sin{(k)}) dk and \\omega{(k)} + \\sin{(k)} + \\int (- k + 3 \\omega{(k)}) dk = \\omega{(k)} + \\sin{(k)} + \\int (- k + 2 \\omega{(k)} + \\sin{(k)}) dk and 2 k - \\sin{(k)} + \\int (- k + 3 \\omega{(k)}) dk = 2 k - \\sin{(k)} + \\int (- k + 2 \\omega{(k)} + \\sin{(k)}) dk and 2 k - \\sin{(k)} + \\int (- k + \\omega{(k)} + 2 \\sin{(k)}) dk = 2 k - \\sin{(k)} + \\int (- k + 2 \\omega{(k)} + \\sin{(k)}) dk", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["add", 1, "Function('\\\\omega')(Symbol('k', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), Add(Function('\\\\omega')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\omega')(Symbol('k', commutative=True)), Mul(Integer(2), sin(Symbol('k', commutative=True)))))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\omega')(Symbol('k', commutative=True)), Mul(Integer(2), sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["add", 6, "Add(Function('\\\\omega')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)))), Add(Function('\\\\omega')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["minus", 7, "Add(Mul(Integer(-1), Integer(2), Symbol('k', commutative=True)), Function('\\\\omega')(Symbol('k', commutative=True)), Mul(Integer(2), sin(Symbol('k', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(-1), sin(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(3), Function('\\\\omega')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(-1), sin(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(-1), sin(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\omega')(Symbol('k', commutative=True)), Mul(Integer(2), sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(-1), sin(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('\\\\omega')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\varphi)} = \\log{(\\varphi)} and \\operatorname{f^{*}}{(\\varphi)} = \\log{(\\varphi)}^{\\varphi}, then obtain \\mathbf{P}{(\\varphi)} + \\operatorname{f^{*}}{(\\varphi)} = \\mathbf{P}{(\\varphi)} + \\log{(\\varphi)}^{\\varphi}", "derivation": "\\mathbf{P}{(\\varphi)} = \\log{(\\varphi)} and \\mathbf{P}^{\\varphi}{(\\varphi)} = \\log{(\\varphi)}^{\\varphi} and \\operatorname{f^{*}}{(\\varphi)} = \\log{(\\varphi)}^{\\varphi} and \\operatorname{f^{*}}{(\\varphi)} + \\log{(\\varphi)} = \\log{(\\varphi)} + \\log{(\\varphi)}^{\\varphi} and \\mathbf{P}{(\\varphi)} + \\operatorname{f^{*}}{(\\varphi)} = \\mathbf{P}{(\\varphi)} + \\mathbf{P}^{\\varphi}{(\\varphi)} and \\mathbf{P}{(\\varphi)} + \\operatorname{f^{*}}{(\\varphi)} = \\mathbf{P}{(\\varphi)} + \\log{(\\varphi)}^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["add", 3, "log(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('f^*')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Add(log(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Function('f^*')(Symbol('\\\\varphi', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Function('f^*')(Symbol('\\\\varphi', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given s{(b)} = \\log{(e^{b})}, then obtain b (s{(b)} + e^{b}) (s{(b)} + e^{b})^{b} = b (s{(b)} + e^{b}) (e^{b} + \\log{(e^{b})})^{b}", "derivation": "s{(b)} = \\log{(e^{b})} and s{(b)} + e^{b} = e^{b} + \\log{(e^{b})} and b (s{(b)} + e^{b}) = b (e^{b} + \\log{(e^{b})}) and (s{(b)} + e^{b})^{b} = (e^{b} + \\log{(e^{b})})^{b} and b (s{(b)} + e^{b})^{b} (e^{b} + \\log{(e^{b})}) = b (e^{b} + \\log{(e^{b})}) (e^{b} + \\log{(e^{b})})^{b} and b (s{(b)} + e^{b}) (s{(b)} + e^{b})^{b} = b (s{(b)} + e^{b}) (e^{b} + \\log{(e^{b})})^{b}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))"], [["add", 1, "exp(Symbol('b', commutative=True))"], "Equality(Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))))"], [["times", 2, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))), Mul(Symbol('b', commutative=True), Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))), Symbol('b', commutative=True)))"], [["times", 4, "Mul(Symbol('b', commutative=True), Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))))"], "Equality(Mul(Symbol('b', commutative=True), Pow(Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))), Mul(Symbol('b', commutative=True), Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))), Pow(Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('b', commutative=True), Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Pow(Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), Add(Function('s')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Pow(Add(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(F_{N})} = \\log{(e^{F_{N}})}, then obtain (- F_{N} + \\log{(e^{F_{N}})}) \\int (- F_{N} + \\Psi^{\\dagger}{(F_{N})})^{F_{N}} dF_{N} = (- F_{N} + \\log{(e^{F_{N}})}) \\int (- F_{N} + \\log{(e^{F_{N}})})^{F_{N}} dF_{N}", "derivation": "\\Psi^{\\dagger}{(F_{N})} = \\log{(e^{F_{N}})} and F_{N} + \\Psi^{\\dagger}{(F_{N})} = F_{N} + \\log{(e^{F_{N}})} and 2 F_{N} + \\Psi^{\\dagger}{(F_{N})} = 2 F_{N} + \\log{(e^{F_{N}})} and - F_{N} + \\Psi^{\\dagger}{(F_{N})} = - F_{N} + \\log{(e^{F_{N}})} and (- F_{N} + \\Psi^{\\dagger}{(F_{N})})^{F_{N}} = (- F_{N} + \\log{(e^{F_{N}})})^{F_{N}} and \\int (- F_{N} + \\Psi^{\\dagger}{(F_{N})})^{F_{N}} dF_{N} = \\int (- F_{N} + \\log{(e^{F_{N}})})^{F_{N}} dF_{N} and (- F_{N} + \\log{(e^{F_{N}})}) \\int (- F_{N} + \\Psi^{\\dagger}{(F_{N})})^{F_{N}} dF_{N} = (- F_{N} + \\log{(e^{F_{N}})}) \\int (- F_{N} + \\log{(e^{F_{N}})})^{F_{N}} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True))))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), log(exp(Symbol('F_N', commutative=True)))))"], [["add", 2, "Symbol('F_N', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Mul(Integer(3), Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))))"], [["power", 4, "Symbol('F_N', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)))"], [["integrate", 5, "Symbol('F_N', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["times", 6, "Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(exp(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = (\\mathbf{r} - f_{\\mathbf{p}})^{\\mu_0} and \\operatorname{c_{0}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = (\\mathbf{r} - f_{\\mathbf{p}})^{\\mu_0} \\operatorname{F_{c}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{c_{0}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (\\mathbf{r} - f_{\\mathbf{p}})^{2 \\mu_0}", "derivation": "\\operatorname{F_{c}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = (\\mathbf{r} - f_{\\mathbf{p}})^{\\mu_0} and \\operatorname{c_{0}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = (\\mathbf{r} - f_{\\mathbf{p}})^{\\mu_0} \\operatorname{F_{c}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{c_{0}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (\\mathbf{r} - f_{\\mathbf{p}})^{\\mu_0} \\operatorname{F_{c}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{c_{0}}{(\\mu_0,f_{\\mathbf{p}},\\mathbf{r})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (\\mathbf{r} - f_{\\mathbf{p}})^{2 \\mu_0}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Function('F_c')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Function('F_c')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('c_0')(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then derive g^{\\prime}_{\\varepsilon} + \\hat{x}_0{(\\hat{H}_l)} = \\psi^* + \\sin{(\\hat{H}_l)}, then obtain \\psi^* + \\sin{(\\hat{H}_l)} = \\psi^* + \\hat{x}_0{(\\hat{H}_l)}", "derivation": "\\hat{x}_0{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\hat{x}_0{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} and \\int \\frac{d}{d \\hat{H}_l} \\hat{x}_0{(\\hat{H}_l)} d\\hat{H}_l = \\int \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} d\\hat{H}_l and g^{\\prime}_{\\varepsilon} + \\hat{x}_0{(\\hat{H}_l)} = \\psi^* + \\sin{(\\hat{H}_l)} and g^{\\prime}_{\\varepsilon} + \\hat{x}_0{(\\hat{H}_l)} = \\psi^* + \\hat{x}_0{(\\hat{H}_l)} and \\psi^* + \\sin{(\\hat{H}_l)} = \\psi^* + \\hat{x}_0{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Derivative(sin(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given k{(L)} = \\log{(L)}, then derive \\int k{(L)} dL = L \\log{(L)} - L + \\theta_2, then obtain \\int k{(L)} dL = L k{(L)} - L + \\theta_2", "derivation": "k{(L)} = \\log{(L)} and \\int k{(L)} dL = \\int \\log{(L)} dL and \\int k{(L)} dL = L \\log{(L)} - L + \\theta_2 and \\int k{(L)} dL = L k{(L)} - L + \\theta_2", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Function('k')(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\tilde{g}{(\\mathbf{f})} = \\frac{\\cos{(\\mathbf{f})}}{\\mathbf{f}}, then obtain \\int (-1 + \\frac{\\operatorname{F_{g}}{(\\mathbf{f})}}{\\mathbf{f}}) d\\mathbf{f} = \\int (\\tilde{g}{(\\mathbf{f})} - 1) d\\mathbf{f}", "derivation": "\\operatorname{F_{g}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\frac{\\operatorname{F_{g}}{(\\mathbf{f})}}{\\mathbf{f}} = \\frac{\\cos{(\\mathbf{f})}}{\\mathbf{f}} and \\tilde{g}{(\\mathbf{f})} = \\frac{\\cos{(\\mathbf{f})}}{\\mathbf{f}} and \\frac{\\operatorname{F_{g}}{(\\mathbf{f})}}{\\mathbf{f}} = \\tilde{g}{(\\mathbf{f})} and -1 + \\frac{\\operatorname{F_{g}}{(\\mathbf{f})}}{\\mathbf{f}} = \\tilde{g}{(\\mathbf{f})} - 1 and \\int (-1 + \\frac{\\operatorname{F_{g}}{(\\mathbf{f})}}{\\mathbf{f}}) d\\mathbf{f} = \\int (\\tilde{g}{(\\mathbf{f})} - 1) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\mathbf{f}', commutative=True))), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given p{(\\phi_2,F_{N})} = F_{N} - \\phi_2, then obtain F_{N} + \\phi_2 + p^{\\phi_2}{(\\phi_2,F_{N})} = F_{N} + \\phi_2 + (F_{N} - \\phi_2)^{\\phi_2}", "derivation": "p{(\\phi_2,F_{N})} = F_{N} - \\phi_2 and p^{\\phi_2}{(\\phi_2,F_{N})} = (F_{N} - \\phi_2)^{\\phi_2} and \\phi_2 + p^{\\phi_2}{(\\phi_2,F_{N})} = \\phi_2 + (F_{N} - \\phi_2)^{\\phi_2} and F_{N} + \\phi_2 + p^{\\phi_2}{(\\phi_2,F_{N})} = F_{N} + \\phi_2 + (F_{N} - \\phi_2)^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Pow(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"], [["add", 3, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\phi_2', commutative=True), Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('F_N', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('\\\\phi_2', commutative=True), Pow(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{p},q,c)} = \\mathbf{p} (- c + q), then obtain - c + \\theta_{2}{(\\mathbf{p},q,c)} + \\theta_{2}^{c}{(\\mathbf{p},q,c)} = - c + (\\mathbf{p} (- c + q))^{c} + \\theta_{2}{(\\mathbf{p},q,c)}", "derivation": "\\theta_{2}{(\\mathbf{p},q,c)} = \\mathbf{p} (- c + q) and \\theta_{2}^{c}{(\\mathbf{p},q,c)} = (\\mathbf{p} (- c + q))^{c} and - c + \\theta_{2}^{c}{(\\mathbf{p},q,c)} = - c + (\\mathbf{p} (- c + q))^{c} and - c + \\theta_{2}{(\\mathbf{p},q,c)} + \\theta_{2}^{c}{(\\mathbf{p},q,c)} = - c + (\\mathbf{p} (- c + q))^{c} + \\theta_{2}{(\\mathbf{p},q,c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('q', commutative=True))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{p}', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('q', commutative=True))), Symbol('c', commutative=True)))"], [["minus", 2, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{p}', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('q', commutative=True))), Symbol('c', commutative=True))))"], [["add", 3, "Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{p}', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('q', commutative=True))), Symbol('c', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('q', commutative=True), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(v_{t})} = \\frac{d}{d v_{t}} \\log{(v_{t})} and \\pi{(v_{t})} = \\log{(v_{t})}, then derive \\operatorname{L_{\\varepsilon}}{(v_{t})} = \\frac{1}{v_{t}}, then obtain \\frac{d}{d v_{t}} (\\frac{1}{v_{t}})^{v_{t}} = \\frac{d}{d v_{t}} (\\frac{d}{d v_{t}} \\pi{(v_{t})})^{v_{t}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(v_{t})} = \\frac{d}{d v_{t}} \\log{(v_{t})} and \\operatorname{L_{\\varepsilon}}{(v_{t})} = \\frac{1}{v_{t}} and \\pi{(v_{t})} = \\log{(v_{t})} and \\operatorname{L_{\\varepsilon}}{(v_{t})} = \\frac{d}{d v_{t}} \\pi{(v_{t})} and \\operatorname{L_{\\varepsilon}}^{v_{t}}{(v_{t})} = (\\frac{d}{d v_{t}} \\pi{(v_{t})})^{v_{t}} and \\frac{d}{d v_{t}} \\operatorname{L_{\\varepsilon}}^{v_{t}}{(v_{t})} = \\frac{d}{d v_{t}} (\\frac{d}{d v_{t}} \\pi{(v_{t})})^{v_{t}} and \\frac{d}{d v_{t}} (\\frac{1}{v_{t}})^{v_{t}} = \\frac{d}{d v_{t}} (\\frac{d}{d v_{t}} \\pi{(v_{t})})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Derivative(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Derivative(Function('\\\\pi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["power", 4, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(Derivative(Function('\\\\pi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)))"], [["differentiate", 5, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Pow(Derivative(Function('\\\\pi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Pow(Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Pow(Derivative(Function('\\\\pi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(H,\\mathbf{B})} = H - \\mathbf{B}, then obtain \\dot{\\mathbf{r}}{(H,\\mathbf{B})} e^{\\frac{\\partial}{\\partial H} \\dot{\\mathbf{r}}{(H,\\mathbf{B})}} = e \\dot{\\mathbf{r}}{(H,\\mathbf{B})}", "derivation": "\\dot{\\mathbf{r}}{(H,\\mathbf{B})} = H - \\mathbf{B} and \\frac{\\partial}{\\partial H} \\dot{\\mathbf{r}}{(H,\\mathbf{B})} = \\frac{\\partial}{\\partial H} (H - \\mathbf{B}) and e^{\\frac{\\partial}{\\partial H} \\dot{\\mathbf{r}}{(H,\\mathbf{B})}} = e^{\\frac{\\partial}{\\partial H} (H - \\mathbf{B})} and \\dot{\\mathbf{r}}{(H,\\mathbf{B})} e^{\\frac{\\partial}{\\partial H} \\dot{\\mathbf{r}}{(H,\\mathbf{B})}} = \\dot{\\mathbf{r}}{(H,\\mathbf{B})} e^{\\frac{\\partial}{\\partial H} (H - \\mathbf{B})} and \\dot{\\mathbf{r}}{(H,\\mathbf{B})} e^{\\frac{\\partial}{\\partial H} \\dot{\\mathbf{r}}{(H,\\mathbf{B})}} = e \\dot{\\mathbf{r}}{(H,\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), exp(Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["times", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Mul(E, Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{J}_f,s)} = \\mathbf{J}_f^{s}, then obtain \\frac{\\partial}{\\partial s} (\\mathbf{J}_f^{s} + \\mathbb{I}{(\\mathbf{J}_f,s)})^{- s} \\mathbb{I}{(\\mathbf{J}_f,s)} = \\frac{\\partial}{\\partial s} \\mathbf{J}_f^{s} (\\mathbf{J}_f^{s} + \\mathbb{I}{(\\mathbf{J}_f,s)})^{- s}", "derivation": "\\mathbb{I}{(\\mathbf{J}_f,s)} = \\mathbf{J}_f^{s} and \\mathbf{J}_f^{s} + \\mathbb{I}{(\\mathbf{J}_f,s)} = 2 \\mathbf{J}_f^{s} and (2 \\mathbf{J}_f^{s})^{- s} \\mathbb{I}{(\\mathbf{J}_f,s)} = \\mathbf{J}_f^{s} (2 \\mathbf{J}_f^{s})^{- s} and \\frac{\\partial}{\\partial s} (2 \\mathbf{J}_f^{s})^{- s} \\mathbb{I}{(\\mathbf{J}_f,s)} = \\frac{\\partial}{\\partial s} \\mathbf{J}_f^{s} (2 \\mathbf{J}_f^{s})^{- s} and \\frac{\\partial}{\\partial s} (\\mathbf{J}_f^{s} + \\mathbb{I}{(\\mathbf{J}_f,s)})^{- s} \\mathbb{I}{(\\mathbf{J}_f,s)} = \\frac{\\partial}{\\partial s} \\mathbf{J}_f^{s} (\\mathbf{J}_f^{s} + \\mathbb{I}{(\\mathbf{J}_f,s)})^{- s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))))"], [["divide", 1, "Pow(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Pow(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Pow(Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Pow(Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(v_{2},z,W)} = - v_{2} + z^{W} and \\mathbf{p}{(v_{2})} = - v_{2}, then obtain (z + \\phi_{1}{(v_{2},z,W)})^{z} + (z + z^{W} + \\mathbf{p}{(v_{2})})^{z} = (z + \\phi_{1}{(v_{2},z,W)})^{z} + (- v_{2} + z + z^{W})^{z}", "derivation": "\\phi_{1}{(v_{2},z,W)} = - v_{2} + z^{W} and z + \\phi_{1}{(v_{2},z,W)} = - v_{2} + z + z^{W} and (z + \\phi_{1}{(v_{2},z,W)})^{z} = (- v_{2} + z + z^{W})^{z} and \\mathbf{p}{(v_{2})} = - v_{2} and z + \\phi_{1}{(v_{2},z,W)} = z + z^{W} + \\mathbf{p}{(v_{2})} and (z + z^{W} + \\mathbf{p}{(v_{2})})^{z} = (- v_{2} + z + z^{W})^{z} and (z + \\phi_{1}{(v_{2},z,W)})^{z} + (z + z^{W} + \\mathbf{p}{(v_{2})})^{z} = (z + \\phi_{1}{(v_{2},z,W)})^{z} + (- v_{2} + z + z^{W})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True))))"], [["add", 1, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Add(Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True)))"], [["add", 6, "Pow(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True))"], "Equality(Add(Pow(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True))), Symbol('z', commutative=True))), Add(Pow(Add(Symbol('z', commutative=True), Function('\\\\phi_1')(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('z', commutative=True), Pow(Symbol('z', commutative=True), Symbol('W', commutative=True))), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\phi_2)} = \\cos{(e^{\\phi_2})} and L{(\\phi_2)} = \\phi_2, then obtain \\frac{1}{\\phi_2^{2}} = \\frac{1}{\\phi_2 L{(\\phi_2)}}", "derivation": "\\operatorname{A_{y}}{(\\phi_2)} = \\cos{(e^{\\phi_2})} and L{(\\phi_2)} = \\phi_2 and \\operatorname{A_{y}}{(\\phi_2)} L{(\\phi_2)} = \\phi_2 \\operatorname{A_{y}}{(\\phi_2)} and \\frac{1}{\\phi_2} = \\frac{\\cos{(e^{\\phi_2})}}{\\phi_2 \\operatorname{A_{y}}{(\\phi_2)}} and \\frac{1}{\\phi_2} = \\frac{\\cos{(e^{\\phi_2})}}{\\operatorname{A_{y}}{(\\phi_2)} L{(\\phi_2)}} and \\frac{1}{\\phi_2} = \\frac{1}{L{(\\phi_2)}} and \\frac{1}{\\phi_2^{2}} = \\frac{1}{\\phi_2 L{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["times", 2, "Function('A_y')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('\\\\phi_2', commutative=True)), Function('L')(Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('\\\\phi_2', commutative=True), Function('A_y')(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\phi_2', commutative=True), Function('A_y')(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(exp(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Mul(Pow(Function('A_y')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(exp(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Function('L')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)))"], [["divide", 6, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Function('L')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\dot{y},W)} = W + \\dot{y}, then obtain (W + \\dot{y})^{12} \\mathbf{J}^{4}{(\\dot{y},W)} = (W + \\dot{y})^{16}", "derivation": "\\mathbf{J}{(\\dot{y},W)} = W + \\dot{y} and \\mathbf{J}^{2}{(\\dot{y},W)} = (W + \\dot{y}) \\mathbf{J}{(\\dot{y},W)} and \\mathbf{J}^{4}{(\\dot{y},W)} = (W + \\dot{y})^{2} \\mathbf{J}^{2}{(\\dot{y},W)} and (W + \\dot{y})^{2} \\mathbf{J}{(\\dot{y},W)} = (W + \\dot{y})^{3} and (W + \\dot{y})^{2} \\mathbf{J}^{2}{(\\dot{y},W)} = (W + \\dot{y})^{3} \\mathbf{J}{(\\dot{y},W)} and (W + \\dot{y})^{3} \\mathbf{J}{(\\dot{y},W)} = (W + \\dot{y})^{4} and (W + \\dot{y})^{12} \\mathbf{J}^{4}{(\\dot{y},W)} = (W + \\dot{y})^{16}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Integer(2)), Mul(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Integer(4)), Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Integer(2))))"], [["times", 1, "Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True))), Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(3)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(3)), Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(3)), Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True))), Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(4)))"], [["power", 6, 4], "Equality(Mul(Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(12)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('W', commutative=True)), Integer(4))), Pow(Add(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(16)))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{H},\\mathbf{r})} = \\hat{H} \\mathbf{r}, then obtain \\frac{\\int\\limits^{\\frac{\\theta_{1}{(\\hat{H},\\mathbf{r})}}{\\mathbf{r}}} \\theta_{1}{(\\hat{H},\\mathbf{r})} d\\hat{H}}{\\theta_{1}{(\\hat{H},\\mathbf{r})}} = \\frac{\\int\\limits^{\\frac{\\theta_{1}{(\\hat{H},\\mathbf{r})}}{\\mathbf{r}}} \\hat{H} \\mathbf{r} d\\hat{H}}{\\theta_{1}{(\\hat{H},\\mathbf{r})}}", "derivation": "\\theta_{1}{(\\hat{H},\\mathbf{r})} = \\hat{H} \\mathbf{r} and \\int \\theta_{1}{(\\hat{H},\\mathbf{r})} d\\hat{H} = \\int \\hat{H} \\mathbf{r} d\\hat{H} and \\frac{\\int \\theta_{1}{(\\hat{H},\\mathbf{r})} d\\hat{H}}{\\hat{H} \\mathbf{r}} = \\frac{\\int \\hat{H} \\mathbf{r} d\\hat{H}}{\\hat{H} \\mathbf{r}} and \\frac{\\theta_{1}{(\\hat{H},\\mathbf{r})}}{\\mathbf{r}} = \\hat{H} and \\frac{\\int\\limits^{\\frac{\\theta_{1}{(\\hat{H},\\mathbf{r})}}{\\mathbf{r}}} \\theta_{1}{(\\hat{H},\\mathbf{r})} d\\hat{H}}{\\theta_{1}{(\\hat{H},\\mathbf{r})}} = \\frac{\\int\\limits^{\\frac{\\theta_{1}{(\\hat{H},\\mathbf{r})}}{\\mathbf{r}}} \\hat{H} \\mathbf{r} d\\hat{H}}{\\theta_{1}{(\\hat{H},\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["divide", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Integral(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))))"]]}, {"prompt": "Given \\mu{(V_{\\mathbf{B}})} = \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}}, then derive \\mu^{2}{(V_{\\mathbf{B}})} = (\\mathbf{J}_M + e^{V_{\\mathbf{B}}}) \\mu{(V_{\\mathbf{B}})}, then obtain (\\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}})^{2} = (\\mathbf{J}_M + e^{V_{\\mathbf{B}}}) \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}}", "derivation": "\\mu{(V_{\\mathbf{B}})} = \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} and \\mu^{2}{(V_{\\mathbf{B}})} = \\mu{(V_{\\mathbf{B}})} \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} and \\mu^{2}{(V_{\\mathbf{B}})} = (\\mathbf{J}_M + e^{V_{\\mathbf{B}}}) \\mu{(V_{\\mathbf{B}})} and (\\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}})^{2} = (\\mathbf{J}_M + e^{V_{\\mathbf{B}}}) \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 1, "Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('\\\\mu')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\delta,\\omega)} = \\omega^{\\delta}, then obtain \\cos{(\\frac{\\frac{\\partial}{\\partial \\omega} \\varepsilon{(\\delta,\\omega)}}{\\omega})} = \\cos{(\\frac{\\delta \\omega^{\\delta}}{\\omega^{2}})}", "derivation": "\\varepsilon{(\\delta,\\omega)} = \\omega^{\\delta} and \\frac{\\partial}{\\partial \\omega} \\varepsilon{(\\delta,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\omega^{\\delta} and - \\frac{\\frac{\\partial}{\\partial \\omega} \\varepsilon{(\\delta,\\omega)}}{\\omega} = - \\frac{\\frac{\\partial}{\\partial \\omega} \\omega^{\\delta}}{\\omega} and \\cos{(\\frac{\\frac{\\partial}{\\partial \\omega} \\varepsilon{(\\delta,\\omega)}}{\\omega})} = \\cos{(\\frac{\\frac{\\partial}{\\partial \\omega} \\omega^{\\delta}}{\\omega})} and \\cos{(\\frac{\\frac{\\partial}{\\partial \\omega} \\varepsilon{(\\delta,\\omega)}}{\\omega})} = \\cos{(\\frac{\\delta \\omega^{\\delta}}{\\omega^{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), cos(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(I,\\mathbf{f})} = e^{\\frac{\\mathbf{f}}{I}} and \\Psi{(I,\\mathbf{f})} = \\frac{\\mathbf{f}}{I}, then obtain \\frac{e^{\\Psi{(I,\\mathbf{f})}}}{I} = \\frac{e^{\\frac{\\mathbf{f}}{I}}}{I}", "derivation": "\\eta^{\\prime}{(I,\\mathbf{f})} = e^{\\frac{\\mathbf{f}}{I}} and \\frac{\\eta^{\\prime}{(I,\\mathbf{f})}}{I} = \\frac{e^{\\frac{\\mathbf{f}}{I}}}{I} and \\Psi{(I,\\mathbf{f})} = \\frac{\\mathbf{f}}{I} and \\frac{\\eta^{\\prime}{(I,\\mathbf{f})}}{I} = \\frac{e^{\\Psi{(I,\\mathbf{f})}}}{I} and \\frac{e^{\\Psi{(I,\\mathbf{f})}}}{I} = \\frac{e^{\\frac{\\mathbf{f}}{I}}}{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Function('\\\\Psi')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Function('\\\\Psi')(Symbol('I', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\nabla,l)} = - \\nabla + l and l{(\\nabla)} = - \\nabla, then obtain l (- \\nabla + l) = l (l + l{(\\nabla)})", "derivation": "\\operatorname{t_{1}}{(\\nabla,l)} = - \\nabla + l and l{(\\nabla)} = - \\nabla and \\operatorname{t_{1}}{(\\nabla,l)} = l + l{(\\nabla)} and l \\operatorname{t_{1}}{(\\nabla,l)} = l (l + l{(\\nabla)}) and l (- \\nabla + l) = l (l + l{(\\nabla)})", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\nabla', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('l', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('t_1')(Symbol('\\\\nabla', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Function('l')(Symbol('\\\\nabla', commutative=True))))"], [["times", 3, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('t_1')(Symbol('\\\\nabla', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('l')(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Function('l')(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given A{(\\mathbf{M})} = \\sin{(e^{\\mathbf{M}})} and Q{(\\mathbf{M})} = \\sin{(e^{\\mathbf{M}})}, then derive v_{x} + \\operatorname{Si}{(e^{\\mathbf{M}})} = \\int Q{(\\mathbf{M})} d\\mathbf{M}, then obtain v_{x} + 2 \\operatorname{Si}{(e^{\\mathbf{M}})} = \\operatorname{Si}{(e^{\\mathbf{M}})} + \\int Q{(\\mathbf{M})} d\\mathbf{M}", "derivation": "A{(\\mathbf{M})} = \\sin{(e^{\\mathbf{M}})} and Q{(\\mathbf{M})} = \\sin{(e^{\\mathbf{M}})} and Q{(\\mathbf{M})} = A{(\\mathbf{M})} and \\int Q{(\\mathbf{M})} d\\mathbf{M} = \\int A{(\\mathbf{M})} d\\mathbf{M} and \\int \\sin{(e^{\\mathbf{M}})} d\\mathbf{M} = \\int A{(\\mathbf{M})} d\\mathbf{M} and \\int \\sin{(e^{\\mathbf{M}})} d\\mathbf{M} = \\int Q{(\\mathbf{M})} d\\mathbf{M} and v_{x} + \\operatorname{Si}{(e^{\\mathbf{M}})} = \\int Q{(\\mathbf{M})} d\\mathbf{M} and v_{x} + 2 \\operatorname{Si}{(e^{\\mathbf{M}})} = \\operatorname{Si}{(e^{\\mathbf{M}})} + \\int Q{(\\mathbf{M})} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), sin(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), sin(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Function('A')(Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(sin(exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('v_x', commutative=True), Si(exp(Symbol('\\\\mathbf{M}', commutative=True)))), Integral(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 7, "Si(exp(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Symbol('v_x', commutative=True), Mul(Integer(2), Si(exp(Symbol('\\\\mathbf{M}', commutative=True))))), Add(Si(exp(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\phi_1)} = e^{\\sin{(\\phi_1)}}, then derive \\hat{X} + \\phi_1 = \\int \\frac{e^{\\sin{(\\phi_1)}}}{\\varepsilon{(\\phi_1)}} d\\phi_1, then obtain (\\hat{X} + \\phi_1) \\int (\\hat{X} + \\phi_1) \\int 1 d\\phi_1 d\\phi_1 = (\\hat{X} + \\phi_1) \\int (\\hat{X} + \\phi_1)^{2} d\\phi_1", "derivation": "\\varepsilon{(\\phi_1)} = e^{\\sin{(\\phi_1)}} and 1 = \\frac{e^{\\sin{(\\phi_1)}}}{\\varepsilon{(\\phi_1)}} and \\int 1 d\\phi_1 = \\int \\frac{e^{\\sin{(\\phi_1)}}}{\\varepsilon{(\\phi_1)}} d\\phi_1 and \\hat{X} + \\phi_1 = \\int \\frac{e^{\\sin{(\\phi_1)}}}{\\varepsilon{(\\phi_1)}} d\\phi_1 and (\\hat{X} + \\phi_1) \\int 1 d\\phi_1 = (\\hat{X} + \\phi_1) \\int \\frac{e^{\\sin{(\\phi_1)}}}{\\varepsilon{(\\phi_1)}} d\\phi_1 and (\\hat{X} + \\phi_1) \\int 1 d\\phi_1 = (\\hat{X} + \\phi_1)^{2} and \\int (\\hat{X} + \\phi_1) \\int 1 d\\phi_1 d\\phi_1 = \\int (\\hat{X} + \\phi_1)^{2} d\\phi_1 and (\\hat{X} + \\phi_1) \\int (\\hat{X} + \\phi_1) \\int 1 d\\phi_1 d\\phi_1 = (\\hat{X} + \\phi_1) \\int (\\hat{X} + \\phi_1)^{2} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True)), exp(sin(Symbol('\\\\phi_1', commutative=True))))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\phi_1', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 3, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\phi_1', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True)))), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(2)))"], [["integrate", 6, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 7, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given z{(\\psi,G)} = \\frac{\\psi}{G}, then derive \\frac{\\partial}{\\partial G} z{(\\psi,G)} = - \\frac{\\psi}{G^{2}}, then obtain \\frac{\\partial}{\\partial G} z{(\\psi,G)} + \\frac{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G}}{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G} + 1} = \\frac{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G}}{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G} + 1} - \\frac{\\psi}{G^{2}}", "derivation": "z{(\\psi,G)} = \\frac{\\psi}{G} and \\frac{\\partial}{\\partial G} z{(\\psi,G)} = \\frac{\\partial}{\\partial G} \\frac{\\psi}{G} and \\frac{\\partial}{\\partial G} z{(\\psi,G)} = - \\frac{\\psi}{G^{2}} and \\frac{\\partial}{\\partial G} z{(\\psi,G)} + \\frac{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G}}{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G} + 1} = \\frac{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G}}{\\frac{\\partial}{\\partial G} \\frac{\\psi}{G} + 1} - \\frac{\\psi}{G^{2}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\psi', commutative=True)))"], [["add", 3, "Mul(Pow(Add(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], "Equality(Add(Derivative(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Pow(Add(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Add(Mul(Pow(Add(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H \\mathbf{J}, then derive \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H, then obtain \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} + \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H + \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})}", "derivation": "\\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H \\mathbf{J} and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} H \\mathbf{J} and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H and \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} + \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})} = H + \\operatorname{x^{{\\}'}}{(H,\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('H', commutative=True))"], [["add", 3, "Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Symbol('H', commutative=True), Function('x^\\\\prime')(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(U)} = \\sin{(U)} and \\Psi^{\\dagger}{(U)} = 2 \\operatorname{f_{\\mathbf{p}}}{(U)}, then obtain \\Psi^{\\dagger}^{2}{(U)} = 4 \\operatorname{f_{\\mathbf{p}}}^{2}{(U)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(U)} = \\sin{(U)} and \\Psi^{\\dagger}{(U)} = 2 \\operatorname{f_{\\mathbf{p}}}{(U)} and \\Psi^{\\dagger}{(U)} = 2 \\sin{(U)} and \\frac{\\Psi^{\\dagger}{(U)}}{\\operatorname{f_{\\mathbf{p}}}{(U)}} = \\frac{2 \\sin{(U)}}{\\operatorname{f_{\\mathbf{p}}}{(U)}} and \\frac{\\Psi^{\\dagger}^{2}{(U)}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(U)}} = \\frac{4 \\sin^{2}{(U)}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(U)}} and \\Psi^{\\dagger}^{2}{(U)} = 4 \\sin^{2}{(U)} and \\Psi^{\\dagger}^{2}{(U)} = 4 \\operatorname{f_{\\mathbf{p}}}^{2}{(U)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Mul(Integer(2), sin(Symbol('U', commutative=True))))"], [["divide", 3, "Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Integer(2)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(-2))), Mul(Integer(4), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(-2)), Pow(sin(Symbol('U', commutative=True)), Integer(2))))"], [["divide", 5, "Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(-2))"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(4), Pow(sin(Symbol('U', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(4), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{J}_M,\\mathbf{S})} = \\mathbf{J}_M - \\mathbf{S} and \\operatorname{F_{N}}{(\\mathbf{J}_M)} = 2 \\mathbf{J}_M, then derive \\frac{d}{d \\mathbf{J}_M} \\operatorname{F_{N}}{(\\mathbf{J}_M)} = 2, then obtain 2 \\mathbf{J}_M - \\mathbf{S} = \\mathbf{J}_M \\frac{d}{d \\mathbf{J}_M} 2 \\mathbf{J}_M - \\mathbf{S}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{J}_M,\\mathbf{S})} = \\mathbf{J}_M - \\mathbf{S} and \\mathbf{J}_M + \\operatorname{E_{x}}{(\\mathbf{J}_M,\\mathbf{S})} = 2 \\mathbf{J}_M - \\mathbf{S} and \\operatorname{F_{N}}{(\\mathbf{J}_M)} = 2 \\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\operatorname{F_{N}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} 2 \\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\operatorname{F_{N}}{(\\mathbf{J}_M)} = 2 and \\frac{d}{d \\mathbf{J}_M} 2 \\mathbf{J}_M = 2 and \\mathbf{J}_M + \\operatorname{E_{x}}{(\\mathbf{J}_M,\\mathbf{S})} = \\mathbf{J}_M \\frac{d}{d \\mathbf{J}_M} 2 \\mathbf{J}_M - \\mathbf{S} and 2 \\mathbf{J}_M - \\mathbf{S} = \\mathbf{J}_M \\frac{d}{d \\mathbf{J}_M} 2 \\mathbf{J}_M - \\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(2))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(2))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given h{(\\varphi^*,\\hat{x}_0)} = - \\hat{x}_0 + \\varphi^*, then obtain \\frac{2}{\\varphi^*} = \\frac{2 (- \\hat{x}_0 + \\varphi^*)}{\\varphi^* h{(\\varphi^*,\\hat{x}_0)}}", "derivation": "h{(\\varphi^*,\\hat{x}_0)} = - \\hat{x}_0 + \\varphi^* and 2 h{(\\varphi^*,\\hat{x}_0)} = - \\hat{x}_0 + \\varphi^* + h{(\\varphi^*,\\hat{x}_0)} and \\frac{2 h{(\\varphi^*,\\hat{x}_0)}}{\\varphi^*} = \\frac{- \\hat{x}_0 + \\varphi^* + h{(\\varphi^*,\\hat{x}_0)}}{\\varphi^*} and \\frac{2 (- \\hat{x}_0 + \\varphi^*)}{\\varphi^*} = \\frac{- 2 \\hat{x}_0 + 2 \\varphi^*}{\\varphi^*} and \\frac{2 h{(\\varphi^*,\\hat{x}_0)}}{\\varphi^*} = \\frac{- 2 \\hat{x}_0 + 2 \\varphi^*}{\\varphi^*} and \\frac{2 h{(\\varphi^*,\\hat{x}_0)}}{\\varphi^*} = \\frac{2 (- \\hat{x}_0 + \\varphi^*)}{\\varphi^*} and \\frac{2}{\\varphi^*} = \\frac{2 (- \\hat{x}_0 + \\varphi^*)}{\\varphi^* h{(\\varphi^*,\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 6, "Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('h')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\nabla)} = e^{\\nabla} and B{(\\nabla)} = 2 \\nabla, then obtain \\operatorname{m_{s}}{(\\nabla)} e^{\\nabla} = e^{B{(\\nabla)}}", "derivation": "\\operatorname{m_{s}}{(\\nabla)} = e^{\\nabla} and \\operatorname{m_{s}}{(\\nabla)} e^{\\nabla} = e^{2 \\nabla} and B{(\\nabla)} = 2 \\nabla and \\operatorname{m_{s}}{(\\nabla)} e^{\\nabla} = e^{B{(\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))), exp(Function('B')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(y)} = \\log{(y)}, then derive \\frac{d}{d y} \\hat{H}{(y)} = \\frac{1}{y}, then obtain \\int \\frac{\\frac{d}{d y} \\hat{H}{(y)}}{\\frac{d}{d y} \\frac{1}{y}} dy = \\int \\frac{1}{y \\frac{d}{d y} \\frac{1}{y}} dy", "derivation": "\\hat{H}{(y)} = \\log{(y)} and \\frac{d}{d y} \\hat{H}{(y)} = \\frac{d}{d y} \\log{(y)} and \\frac{d}{d y} \\hat{H}{(y)} = \\frac{1}{y} and \\frac{d^{2}}{d y^{2}} \\hat{H}{(y)} = \\frac{d}{d y} \\frac{1}{y} and \\frac{\\frac{d}{d y} \\hat{H}{(y)}}{\\frac{d^{2}}{d y^{2}} \\hat{H}{(y)}} = \\frac{1}{y \\frac{d^{2}}{d y^{2}} \\hat{H}{(y)}} and \\frac{\\frac{d}{d y} \\hat{H}{(y)}}{\\frac{d}{d y} \\frac{1}{y}} = \\frac{1}{y \\frac{d}{d y} \\frac{1}{y}} and \\int \\frac{\\frac{d}{d y} \\hat{H}{(y)}}{\\frac{d}{d y} \\frac{1}{y}} dy = \\int \\frac{1}{y \\frac{d}{d y} \\frac{1}{y}} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Pow(Symbol('y', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Derivative(Pow(Symbol('y', commutative=True), Integer(-1)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Integer(-1))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Derivative(Pow(Symbol('y', commutative=True), Integer(-1)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Derivative(Pow(Symbol('y', commutative=True), Integer(-1)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 6, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Pow(Derivative(Pow(Symbol('y', commutative=True), Integer(-1)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Derivative(Pow(Symbol('y', commutative=True), Integer(-1)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given L{(c)} = \\sin{(c)}, then obtain (- L{(c)} \\int L{(c)} dc)^{c} = (- \\sin{(c)} \\int L{(c)} dc)^{c}", "derivation": "L{(c)} = \\sin{(c)} and \\int L{(c)} dc = \\int \\sin{(c)} dc and - L{(c)} \\int \\sin{(c)} dc = - \\sin{(c)} \\int \\sin{(c)} dc and (- L{(c)} \\int \\sin{(c)} dc)^{c} = (- \\sin{(c)} \\int \\sin{(c)} dc)^{c} and (- L{(c)} \\int L{(c)} dc)^{c} = (- \\sin{(c)} \\int L{(c)} dc)^{c}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('L')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], "Equality(Mul(Integer(-1), Function('L')(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(Integer(-1), sin(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('L')(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Mul(Integer(-1), Function('L')(Symbol('c', commutative=True)), Integral(Function('L')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('c', commutative=True)), Integral(Function('L')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given J{(F_{c})} = \\cos{(F_{c})}, then obtain (J^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c})^{F_{c}} = (\\cos^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c})^{F_{c}}", "derivation": "J{(F_{c})} = \\cos{(F_{c})} and \\int J{(F_{c})} dF_{c} = \\int \\cos{(F_{c})} dF_{c} and J^{F_{c}}{(F_{c})} = \\cos^{F_{c}}{(F_{c})} and J^{F_{c}}{(F_{c})} - \\int J{(F_{c})} dF_{c} = \\cos^{F_{c}}{(F_{c})} - \\int J{(F_{c})} dF_{c} and J^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c} = \\cos^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c} and (J^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c})^{F_{c}} = (\\cos^{F_{c}}{(F_{c})} - \\int \\cos{(F_{c})} dF_{c})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('J')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('J')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["minus", 3, "Integral(Function('J')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))"], "Equality(Add(Pow(Function('J')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(Function('J')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))), Add(Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(Function('J')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('J')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))), Add(Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))))"], [["power", 5, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Pow(Function('J')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))), Symbol('F_c', commutative=True)), Pow(Add(Pow(cos(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(t_{2})} = \\sin{(t_{2})}, then obtain (f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\operatorname{v_{y}}{(t_{2})} \\sin{(t_{2})})^{t_{2}} = (f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\sin^{2}{(t_{2})})^{t_{2}}", "derivation": "\\operatorname{v_{y}}{(t_{2})} = \\sin{(t_{2})} and \\operatorname{v_{y}}{(t_{2})} \\sin{(t_{2})} = \\sin^{2}{(t_{2})} and \\frac{d}{d t_{2}} \\operatorname{v_{y}}{(t_{2})} \\sin{(t_{2})} = \\frac{d}{d t_{2}} \\sin^{2}{(t_{2})} and f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\operatorname{v_{y}}{(t_{2})} \\sin{(t_{2})} = f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\sin^{2}{(t_{2})} and (f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\operatorname{v_{y}}{(t_{2})} \\sin{(t_{2})})^{t_{2}} = (f^{*} (I + (f^{*})^{\\mathbf{H}}) + \\frac{d}{d t_{2}} \\sin^{2}{(t_{2})})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["times", 1, "sin(Symbol('t_2', commutative=True))"], "Equality(Mul(Function('v_y')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Pow(sin(Symbol('t_2', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Function('v_y')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('t_2', commutative=True)), Integer(2)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('f^*', commutative=True), Add(Symbol('I', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Add(Mul(Symbol('f^*', commutative=True), Add(Symbol('I', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Derivative(Mul(Function('v_y')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Mul(Symbol('f^*', commutative=True), Add(Symbol('I', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Derivative(Pow(sin(Symbol('t_2', commutative=True)), Integer(2)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('f^*', commutative=True), Add(Symbol('I', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Derivative(Mul(Function('v_y')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Symbol('t_2', commutative=True)), Pow(Add(Mul(Symbol('f^*', commutative=True), Add(Symbol('I', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Derivative(Pow(sin(Symbol('t_2', commutative=True)), Integer(2)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = - \\mathbf{r} + \\phi_2^{\\Psi_{nl}} and \\mathbf{A}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = \\mathbf{J}_f^{\\Psi_{nl}}{(\\phi_2,\\Psi_{nl},\\mathbf{r})}, then obtain \\mathbf{A}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = (- \\mathbf{r} + \\phi_2^{\\Psi_{nl}})^{\\Psi_{nl}}", "derivation": "\\mathbf{J}_f{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = - \\mathbf{r} + \\phi_2^{\\Psi_{nl}} and \\mathbf{J}_f^{\\Psi_{nl}}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = (- \\mathbf{r} + \\phi_2^{\\Psi_{nl}})^{\\Psi_{nl}} and \\mathbf{A}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = \\mathbf{J}_f^{\\Psi_{nl}}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} and \\mathbf{A}{(\\phi_2,\\Psi_{nl},\\mathbf{r})} = (- \\mathbf{r} + \\phi_2^{\\Psi_{nl}})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\hbar)} = e^{\\sin{(\\hbar)}} and a{(\\hbar)} = - \\hat{\\mathbf{r}}{(\\hbar)} + e^{\\sin{(\\hbar)}}, then obtain 0 = a{(\\hbar)} e^{- \\sin{(\\hbar)}}", "derivation": "\\hat{\\mathbf{r}}{(\\hbar)} = e^{\\sin{(\\hbar)}} and 0 = - \\hat{\\mathbf{r}}{(\\hbar)} + e^{\\sin{(\\hbar)}} and a{(\\hbar)} = - \\hat{\\mathbf{r}}{(\\hbar)} + e^{\\sin{(\\hbar)}} and 0 = a{(\\hbar)} and 0 = a{(\\hbar)} e^{- \\sin{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hbar', commutative=True)), exp(sin(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hbar', commutative=True))), exp(sin(Symbol('\\\\hbar', commutative=True)))))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hbar', commutative=True))), exp(sin(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Function('a')(Symbol('\\\\hbar', commutative=True)))"], [["divide", 4, "exp(sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Integer(0), Mul(Function('a')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v_{y},V,\\omega)} = V \\omega^{v_{y}} and \\phi_{2}{(v_{y},V,\\omega)} = V \\omega^{v_{y}} \\operatorname{f_{E}}{(v_{y},V,\\omega)}, then obtain \\omega^{- v_{y}} \\frac{\\partial}{\\partial \\omega} \\phi_{2}{(v_{y},V,\\omega)} = \\omega^{- v_{y}} \\frac{\\partial}{\\partial \\omega} V^{2} \\omega^{2 v_{y}}", "derivation": "\\operatorname{f_{E}}{(v_{y},V,\\omega)} = V \\omega^{v_{y}} and \\phi_{2}{(v_{y},V,\\omega)} = V \\omega^{v_{y}} \\operatorname{f_{E}}{(v_{y},V,\\omega)} and \\frac{\\partial}{\\partial \\omega} \\phi_{2}{(v_{y},V,\\omega)} = \\frac{\\partial}{\\partial \\omega} V \\omega^{v_{y}} \\operatorname{f_{E}}{(v_{y},V,\\omega)} and \\frac{\\partial}{\\partial \\omega} \\phi_{2}{(v_{y},V,\\omega)} = \\frac{\\partial}{\\partial \\omega} V^{2} \\omega^{2 v_{y}} and \\omega^{- v_{y}} \\frac{\\partial}{\\partial \\omega} \\phi_{2}{(v_{y},V,\\omega)} = \\omega^{- v_{y}} \\frac{\\partial}{\\partial \\omega} V^{2} \\omega^{2 v_{y}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('f_E')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('f_E')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(2)), Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(2), Symbol('v_y', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["divide", 4, "Pow(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True), Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(2)), Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(2), Symbol('v_y', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} = F_{c} + \\hat{p} + \\mathbf{J}_f, then obtain \\iint (- F_{c} - \\hat{p} - \\mathbf{J}_f + \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} + 1) dF_{c} d\\mathbf{J}_f = \\iint 1 dF_{c} d\\mathbf{J}_f", "derivation": "\\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} = F_{c} + \\hat{p} + \\mathbf{J}_f and - F_{c} - \\hat{p} - \\mathbf{J}_f + \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} = 0 and - F_{c} - \\hat{p} - \\mathbf{J}_f + \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} + 1 = 1 and \\int (- F_{c} - \\hat{p} - \\mathbf{J}_f + \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} + 1) dF_{c} = \\int 1 dF_{c} and \\iint (- F_{c} - \\hat{p} - \\mathbf{J}_f + \\chi{(\\mathbf{J}_f,\\hat{p},F_{c})} + 1) dF_{c} d\\mathbf{J}_f = \\iint 1 dF_{c} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 1, "Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Integer(1))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\chi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v_{z},p)} = \\sin{(p v_{z})}, then obtain \\sin{(p v_{z})} \\int (p v_{z} + \\operatorname{v_{t}}{(v_{z},p)}) dp = \\sin{(p v_{z})} \\int (p v_{z} + \\sin{(p v_{z})}) dp", "derivation": "\\operatorname{v_{t}}{(v_{z},p)} = \\sin{(p v_{z})} and p v_{z} + \\operatorname{v_{t}}{(v_{z},p)} = p v_{z} + \\sin{(p v_{z})} and \\int (p v_{z} + \\operatorname{v_{t}}{(v_{z},p)}) dp = \\int (p v_{z} + \\sin{(p v_{z})}) dp and \\sin{(p v_{z})} \\int (p v_{z} + \\operatorname{v_{t}}{(v_{z},p)}) dp = \\sin{(p v_{z})} \\int (p v_{z} + \\sin{(p v_{z})}) dp", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 1, "Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), Function('v_t')(Symbol('v_z', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)))))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), Function('v_t')(Symbol('v_z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('p', commutative=True))))"], [["times", 3, "sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Integral(Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), Function('v_t')(Symbol('v_z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Mul(sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Integral(Add(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)), sin(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then obtain - \\mathbf{g} - \\frac{\\mathbf{M}{(\\mathbf{g})} + \\cos{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} + \\frac{2 \\mathbf{M}{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} = - \\mathbf{g}", "derivation": "\\mathbf{M}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and 2 \\mathbf{M}{(\\mathbf{g})} = \\mathbf{M}{(\\mathbf{g})} + \\cos{(\\mathbf{g})} and \\frac{2 \\mathbf{M}{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} = \\frac{\\mathbf{M}{(\\mathbf{g})} + \\cos{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} and - \\mathbf{g} + \\frac{2 \\mathbf{M}{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} = - \\mathbf{g} + \\frac{\\mathbf{M}{(\\mathbf{g})} + \\cos{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} and - \\mathbf{g} - \\frac{\\mathbf{M}{(\\mathbf{g})} + \\cos{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} + \\frac{2 \\mathbf{M}{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} = - \\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 2, "cos(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"], [["minus", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))))"], [["minus", 4, "Mul(Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Add(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given t{(r)} = \\sin{(\\log{(r)})}, then obtain \\sin^{r}{(\\log{(r)})} + \\frac{d}{d r} (t^{r}{(r)} - \\sin{(\\log{(r)})}) = \\sin^{r}{(\\log{(r)})} + \\frac{d}{d r} (- \\sin{(\\log{(r)})} + \\sin^{r}{(\\log{(r)})})", "derivation": "t{(r)} = \\sin{(\\log{(r)})} and t^{r}{(r)} = \\sin^{r}{(\\log{(r)})} and t^{r}{(r)} - \\sin{(\\log{(r)})} = - \\sin{(\\log{(r)})} + \\sin^{r}{(\\log{(r)})} and \\frac{d}{d r} (t^{r}{(r)} - \\sin{(\\log{(r)})}) = \\frac{d}{d r} (- \\sin{(\\log{(r)})} + \\sin^{r}{(\\log{(r)})}) and \\sin^{r}{(\\log{(r)})} + \\frac{d}{d r} (t^{r}{(r)} - \\sin{(\\log{(r)})}) = \\sin^{r}{(\\log{(r)})} + \\frac{d}{d r} (- \\sin{(\\log{(r)})} + \\sin^{r}{(\\log{(r)})})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('r', commutative=True)), sin(log(Symbol('r', commutative=True))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('t')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["minus", 2, "sin(log(Symbol('r', commutative=True)))"], "Equality(Add(Pow(Function('t')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), sin(log(Symbol('r', commutative=True))))), Add(Mul(Integer(-1), sin(log(Symbol('r', commutative=True)))), Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Pow(Function('t')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), sin(log(Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(log(Symbol('r', commutative=True)))), Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["add", 4, "Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True))"], "Equality(Add(Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Derivative(Add(Pow(Function('t')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), sin(log(Symbol('r', commutative=True))))), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Derivative(Add(Mul(Integer(-1), sin(log(Symbol('r', commutative=True)))), Pow(sin(log(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(U)} = e^{U}, then obtain \\frac{V^{U}{(U)}}{(- U + 2 V{(U)}) (- U + e^{U})} = \\frac{(e^{U})^{U}}{(- U + 2 V{(U)}) (- U + e^{U})}", "derivation": "V{(U)} = e^{U} and - U + V{(U)} = - U + e^{U} and - U + 2 V{(U)} = - U + V{(U)} + e^{U} and V^{U}{(U)} = (e^{U})^{U} and - U + 2 V{(U)} = - U + 2 e^{U} and \\frac{V^{U}{(U)}}{(- U + e^{U}) (- U + V{(U)} + e^{U})} = \\frac{(e^{U})^{U}}{(- U + e^{U}) (- U + V{(U)} + e^{U})} and \\frac{V^{U}{(U)}}{(- U + e^{U}) (- U + 2 e^{U})} = \\frac{(e^{U})^{U}}{(- U + e^{U}) (- U + 2 e^{U})} and \\frac{V^{U}{(U)}}{(- U + 2 V{(U)}) (- U + e^{U})} = \\frac{(e^{U})^{U}}{(- U + 2 V{(U)}) (- U + e^{U})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))))"], [["add", 2, "Function('V')(Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('V')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('V')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('V')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('U', commutative=True)))))"], [["divide", 4, "Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Function('V')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('U', commutative=True)))), Integer(-1)), Pow(Function('V')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('U', commutative=True)))), Integer(-1)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('V')(Symbol('U', commutative=True)))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(Function('V')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('V')(Symbol('U', commutative=True)))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))), Integer(-1)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{p})} = \\log{(\\mathbf{p})}, then obtain \\frac{\\partial}{\\partial \\mathbf{r}} (- 2 \\Omega^{\\mathbf{r}} + t_{2} + \\Omega{(\\mathbf{p})}) = \\frac{\\partial}{\\partial \\mathbf{r}} (- 2 \\Omega^{\\mathbf{r}} + t_{2} + \\log{(\\mathbf{p})})", "derivation": "\\Omega{(\\mathbf{p})} = \\log{(\\mathbf{p})} and - \\Omega^{\\mathbf{r}} + t_{2} + \\Omega{(\\mathbf{p})} = - \\Omega^{\\mathbf{r}} + t_{2} + \\log{(\\mathbf{p})} and - 2 \\Omega^{\\mathbf{r}} + t_{2} + \\Omega{(\\mathbf{p})} = - 2 \\Omega^{\\mathbf{r}} + t_{2} + \\log{(\\mathbf{p})} and \\frac{\\partial}{\\partial \\mathbf{r}} (- 2 \\Omega^{\\mathbf{r}} + t_{2} + \\Omega{(\\mathbf{p})}) = \\frac{\\partial}{\\partial \\mathbf{r}} (- 2 \\Omega^{\\mathbf{r}} + t_{2} + \\log{(\\mathbf{p})})", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('t_2', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and \\varepsilon{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and E{(r_{0})} = r_{0}, then obtain \\frac{E{(r_{0})}}{I{(f_{\\mathbf{p}})}} = \\frac{r_{0}}{I{(f_{\\mathbf{p}})}}", "derivation": "I{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and \\varepsilon{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and E{(r_{0})} = r_{0} and I{(f_{\\mathbf{p}})} = \\varepsilon{(f_{\\mathbf{p}})} and \\frac{E{(r_{0})}}{\\log{(f_{\\mathbf{p}})}} = \\frac{r_{0}}{\\log{(f_{\\mathbf{p}})}} and \\frac{E{(r_{0})}}{\\varepsilon{(f_{\\mathbf{p}})}} = \\frac{r_{0}}{\\varepsilon{(f_{\\mathbf{p}})}} and \\frac{E{(r_{0})}}{I{(f_{\\mathbf{p}})}} = \\frac{r_{0}}{I{(f_{\\mathbf{p}})}}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\varepsilon')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["divide", 3, "log(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('E')(Symbol('r_0', commutative=True)), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Symbol('r_0', commutative=True), Pow(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('E')(Symbol('r_0', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Symbol('r_0', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Function('E')(Symbol('r_0', commutative=True)), Pow(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))), Mul(Symbol('r_0', commutative=True), Pow(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given b{(f_{\\mathbf{p}})} = e^{e^{f_{\\mathbf{p}}}}, then derive \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = z^{*} + \\operatorname{Ei}{(e^{f_{\\mathbf{p}}})}, then obtain z^{*} + \\operatorname{Ei}{(e^{f_{\\mathbf{p}}})} - \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = 0", "derivation": "b{(f_{\\mathbf{p}})} = e^{e^{f_{\\mathbf{p}}}} and \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int e^{e^{f_{\\mathbf{p}}}} df_{\\mathbf{p}} and \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} - \\int e^{e^{f_{\\mathbf{p}}}} df_{\\mathbf{p}} = 0 and \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = z^{*} + \\operatorname{Ei}{(e^{f_{\\mathbf{p}}})} and z^{*} + \\operatorname{Ei}{(e^{f_{\\mathbf{p}}})} - \\int e^{e^{f_{\\mathbf{p}}}} df_{\\mathbf{p}} = 0 and z^{*} + \\operatorname{Ei}{(e^{f_{\\mathbf{p}}})} - \\int b{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = 0", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(exp(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 2, "Integral(exp(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Add(Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Integral(exp(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('z^*', commutative=True), Ei(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('z^*', commutative=True), Ei(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Integral(exp(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('z^*', commutative=True), Ei(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given v{(\\tilde{g},f)} = \\sin{(\\tilde{g} - f)}, then derive \\int v{(\\tilde{g},f)} df = \\dot{\\mathbf{r}} + \\cos{(\\tilde{g} - f)}, then obtain (\\int v{(\\tilde{g},f)} df)^{f} = (\\int \\sin{(\\tilde{g} - f)} df)^{f}", "derivation": "v{(\\tilde{g},f)} = \\sin{(\\tilde{g} - f)} and \\int v{(\\tilde{g},f)} df = \\int \\sin{(\\tilde{g} - f)} df and \\int v{(\\tilde{g},f)} df = \\dot{\\mathbf{r}} + \\cos{(\\tilde{g} - f)} and \\int \\sin{(\\tilde{g} - f)} df = \\dot{\\mathbf{r}} + \\cos{(\\tilde{g} - f)} and (\\int v{(\\tilde{g},f)} df)^{f} = (\\dot{\\mathbf{r}} + \\cos{(\\tilde{g} - f)})^{f} and (\\int v{(\\tilde{g},f)} df)^{f} = (\\int \\sin{(\\tilde{g} - f)} df)^{f}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f', commutative=True)), sin(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(sin(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), cos(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), cos(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), cos(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(Function('v')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(sin(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{X},\\hat{\\mathbf{r}})} = \\hat{X} \\hat{\\mathbf{r}}, then derive \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{t_{2}}{(\\hat{X},\\hat{\\mathbf{r}})} = \\hat{X}, then obtain \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{X})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\cos{(\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{X} \\hat{\\mathbf{r}})}", "derivation": "\\operatorname{t_{2}}{(\\hat{X},\\hat{\\mathbf{r}})} = \\hat{X} \\hat{\\mathbf{r}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{t_{2}}{(\\hat{X},\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{X} \\hat{\\mathbf{r}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{t_{2}}{(\\hat{X},\\hat{\\mathbf{r}})} = \\hat{X} and \\hat{X} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{X} \\hat{\\mathbf{r}} and \\cos{(\\hat{X})} = \\cos{(\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{X} \\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{X})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\cos{(\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{X} \\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('\\\\hat{X}', commutative=True), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Symbol('\\\\hat{X}', commutative=True)), cos(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(cos(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(cos(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(E_{x},i)} = \\frac{\\partial}{\\partial E_{x}} (E_{x} + i), then obtain \\int \\frac{l{(E_{x},i)}}{E_{x}} dE_{x} = \\theta_1 + \\log{(E_{x})}", "derivation": "l{(E_{x},i)} = \\frac{\\partial}{\\partial E_{x}} (E_{x} + i) and \\frac{l{(E_{x},i)}}{E_{x}} = \\frac{\\frac{\\partial}{\\partial E_{x}} (E_{x} + i)}{E_{x}} and \\int \\frac{l{(E_{x},i)}}{E_{x}} dE_{x} = \\int \\frac{\\frac{\\partial}{\\partial E_{x}} (E_{x} + i)}{E_{x}} dE_{x} and \\int \\frac{l{(E_{x},i)}}{E_{x}} dE_{x} = \\theta_1 + \\log{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('E_x', commutative=True), Symbol('i', commutative=True)), Derivative(Add(Symbol('E_x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('l')(Symbol('E_x', commutative=True), Symbol('i', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('E_x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('l')(Symbol('E_x', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Add(Symbol('E_x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('l')(Symbol('E_x', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(T)} = \\log{(T)}, then derive \\int \\mathbf{J}_M{(T)} dT = T \\log{(T)} - T + t, then obtain \\frac{\\partial}{\\partial T} (T \\log{(T)} - T + v_{y}) = \\frac{d}{d T} \\int \\mathbf{J}_M{(T)} dT", "derivation": "\\mathbf{J}_M{(T)} = \\log{(T)} and \\int \\mathbf{J}_M{(T)} dT = \\int \\log{(T)} dT and \\int \\mathbf{J}_M{(T)} dT = T \\log{(T)} - T + t and \\frac{d}{d T} \\int \\mathbf{J}_M{(T)} dT = \\frac{\\partial}{\\partial T} (T \\log{(T)} - T + t) and \\frac{d}{d T} \\int \\log{(T)} dT = \\frac{\\partial}{\\partial T} (T \\log{(T)} - T + t) and \\frac{d}{d T} \\int \\log{(T)} dT = \\frac{d}{d T} \\int \\mathbf{J}_M{(T)} dT and \\frac{\\partial}{\\partial T} (T \\log{(T)} - T + v_{y}) = \\frac{d}{d T} \\int \\mathbf{J}_M{(T)} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbf{J}_M')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\phi_1,\\theta_2)} = \\phi_1 \\theta_2 and U{(\\phi_1,\\theta_2)} = \\phi_1 \\theta_2, then obtain \\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)} - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} = - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} + \\frac{\\partial}{\\partial \\theta_2} U{(\\phi_1,\\theta_2)}", "derivation": "B{(\\phi_1,\\theta_2)} = \\phi_1 \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\phi_1 \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)} - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} = \\frac{\\partial}{\\partial \\theta_2} \\phi_1 \\theta_2 - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} and U{(\\phi_1,\\theta_2)} = \\phi_1 \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)} - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} = - (\\frac{\\partial}{\\partial \\theta_2} B{(\\phi_1,\\theta_2)})^{\\theta_2} + \\frac{\\partial}{\\partial \\theta_2} U{(\\phi_1,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))), Add(Derivative(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))), Derivative(Function('U')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\pi)} = \\sin{(\\pi)}, then obtain (((\\frac{\\operatorname{C_{d}}^{2}{(\\pi)}}{\\sin{(\\pi)}})^{\\pi})^{\\pi})^{\\pi} = ((\\sin^{\\pi}{(\\pi)})^{\\pi})^{\\pi}", "derivation": "\\operatorname{C_{d}}{(\\pi)} = \\sin{(\\pi)} and \\operatorname{C_{d}}^{2}{(\\pi)} = \\operatorname{C_{d}}{(\\pi)} \\sin{(\\pi)} and \\frac{\\operatorname{C_{d}}^{2}{(\\pi)}}{\\sin{(\\pi)}} = \\operatorname{C_{d}}{(\\pi)} and \\operatorname{C_{d}}^{\\pi}{(\\pi)} = \\sin^{\\pi}{(\\pi)} and (\\operatorname{C_{d}}^{\\pi}{(\\pi)})^{\\pi} = (\\sin^{\\pi}{(\\pi)})^{\\pi} and ((\\frac{\\operatorname{C_{d}}^{2}{(\\pi)}}{\\sin{(\\pi)}})^{\\pi})^{\\pi} = (\\sin^{\\pi}{(\\pi)})^{\\pi} and (((\\frac{\\operatorname{C_{d}}^{2}{(\\pi)}}{\\sin{(\\pi)}})^{\\pi})^{\\pi})^{\\pi} = ((\\sin^{\\pi}{(\\pi)})^{\\pi})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Function('C_d')(Symbol('\\\\pi', commutative=True))"], "Equality(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Integer(2)), Mul(Function('C_d')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Function('C_d')(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Pow(Mul(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["power", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Pow(Function('C_d')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(c_{0},h)} = h^{c_{0}}, then derive \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = h^{c_{0}} \\log{(h)}, then obtain - h \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = - h \\mathbf{E}{(c_{0},h)} \\log{(h)}", "derivation": "\\mathbf{E}{(c_{0},h)} = h^{c_{0}} and \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = \\frac{\\partial}{\\partial c_{0}} h^{c_{0}} and \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = h^{c_{0}} \\log{(h)} and - h \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = - h h^{c_{0}} \\log{(h)} and - h \\frac{\\partial}{\\partial c_{0}} \\mathbf{E}{(c_{0},h)} = - h \\mathbf{E}{(c_{0},h)} \\log{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('h', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Pow(Symbol('h', commutative=True), Symbol('c_0', commutative=True)), log(Symbol('h', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('h', commutative=True), Derivative(Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('c_0', commutative=True)), log(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('h', commutative=True), Derivative(Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('h', commutative=True), Function('\\\\mathbf{E}')(Symbol('c_0', commutative=True), Symbol('h', commutative=True)), log(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)} = Q^{f_{E}} - q, then obtain \\frac{\\int Q^{f_{E}} dQ}{\\int 2 Q^{f_{E}} dQ + \\int - q dQ} + \\frac{\\int \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)} dQ}{\\int 2 Q^{f_{E}} dQ + \\int - q dQ} = 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)} = Q^{f_{E}} - q and Q^{f_{E}} + \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)} = 2 Q^{f_{E}} - q and \\int (Q^{f_{E}} + \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)}) dQ = \\int (2 Q^{f_{E}} - q) dQ and \\frac{\\int (Q^{f_{E}} + \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)}) dQ}{\\int (2 Q^{f_{E}} - q) dQ} = 1 and \\frac{\\int Q^{f_{E}} dQ}{\\int 2 Q^{f_{E}} dQ + \\int - q dQ} + \\frac{\\int \\operatorname{L_{\\varepsilon}}{(Q,f_{E},q)} dQ}{\\int 2 Q^{f_{E}} dQ + \\int - q dQ} = 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('f_E', commutative=True), Symbol('q', commutative=True)), Add(Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["add", 1, "Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Add(Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('f_E', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('f_E', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["divide", 3, "Integral(Add(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('Q', commutative=True)))"], "Equality(Mul(Integral(Add(Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('f_E', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Pow(Integral(Add(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integer(-1))), Integer(1))"], [["expand", 4], "Equality(Add(Mul(Pow(Add(Integral(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Integer(-1), Symbol('q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Integral(Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Add(Integral(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Integer(-1), Symbol('q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Integral(Function('L_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('f_E', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('Q', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\bar{\\h}{(C)} = \\log{(C)} and \\sigma_{p}{(C)} = \\log{(C)}, then obtain - (\\frac{d}{d C} (\\sigma_{p}^{C}{(C)})^{C})^{C} = - (\\frac{d}{d C} (\\log{(C)}^{C})^{C})^{C}", "derivation": "\\bar{\\h}{(C)} = \\log{(C)} and \\sigma_{p}{(C)} = \\log{(C)} and \\bar{\\h}^{C}{(C)} = \\log{(C)}^{C} and (\\bar{\\h}^{C}{(C)})^{C} = (\\log{(C)}^{C})^{C} and \\frac{d}{d C} (\\bar{\\h}^{C}{(C)})^{C} = \\frac{d}{d C} (\\log{(C)}^{C})^{C} and \\frac{d}{d C} (\\bar{\\h}^{C}{(C)})^{C} = \\frac{d}{d C} (\\sigma_{p}^{C}{(C)})^{C} and \\frac{d}{d C} (\\sigma_{p}^{C}{(C)})^{C} = \\frac{d}{d C} (\\log{(C)}^{C})^{C} and (\\frac{d}{d C} (\\sigma_{p}^{C}{(C)})^{C})^{C} = (\\frac{d}{d C} (\\log{(C)}^{C})^{C})^{C} and - (\\frac{d}{d C} (\\sigma_{p}^{C}{(C)})^{C})^{C} = - (\\frac{d}{d C} (\\log{(C)}^{C})^{C})^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hbar')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\hbar')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Pow(Pow(Function('\\\\hbar')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('\\\\sigma_p')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Pow(Pow(Function('\\\\sigma_p')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["power", 7, "Symbol('C', commutative=True)"], "Equality(Pow(Derivative(Pow(Pow(Function('\\\\sigma_p')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Pow(Derivative(Pow(Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)))"], [["divide", 8, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Derivative(Pow(Pow(Function('\\\\sigma_p')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))), Mul(Integer(-1), Pow(Derivative(Pow(Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)}, then derive \\mathbf{g}{(\\Omega)} - 1 = \\cos{(\\Omega)} - 1, then obtain \\frac{d}{d \\Omega} \\sin{(\\Omega)} - 1 = \\cos{(\\Omega)} - 1", "derivation": "\\mathbf{g}{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\mathbf{g}{(\\Omega)} - 1 = \\frac{d}{d \\Omega} \\sin{(\\Omega)} - 1 and \\mathbf{g}{(\\Omega)} - 1 = \\cos{(\\Omega)} - 1 and \\frac{d}{d \\Omega} \\sin{(\\Omega)} - 1 = \\cos{(\\Omega)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\Omega', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon{(f,l,\\mu)} = (\\frac{l}{\\mu})^{f}, then obtain \\frac{\\partial}{\\partial l} (2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + \\varepsilon{(f,l,\\mu)}) df) = \\frac{\\partial}{\\partial l} (2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + (\\frac{l}{\\mu})^{f}) df)", "derivation": "\\varepsilon{(f,l,\\mu)} = (\\frac{l}{\\mu})^{f} and - l + \\varepsilon{(f,l,\\mu)} = - l + (\\frac{l}{\\mu})^{f} and \\int (- l + \\varepsilon{(f,l,\\mu)}) df = \\int (- l + (\\frac{l}{\\mu})^{f}) df and 2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + \\varepsilon{(f,l,\\mu)}) df = 2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + (\\frac{l}{\\mu})^{f}) df and \\frac{\\partial}{\\partial l} (2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + \\varepsilon{(f,l,\\mu)}) df) = \\frac{\\partial}{\\partial l} (2 l - (\\frac{l}{\\mu})^{f} + \\int (- l + (\\frac{l}{\\mu})^{f}) df)", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('f', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True)))"], [["minus", 1, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\varepsilon')(Symbol('f', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))))"], [["integrate", 2, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\varepsilon')(Symbol('f', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\varepsilon')(Symbol('f', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(2), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\varepsilon')(Symbol('f', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('f', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(v_{y},f)} = e^{f + v_{y}}, then obtain e^{- 2 f - 2 v_{y}} \\int \\hat{H}{(v_{y},f)} df = e^{- 2 f - 2 v_{y}} \\int e^{f + v_{y}} df", "derivation": "\\hat{H}{(v_{y},f)} = e^{f + v_{y}} and \\int \\hat{H}{(v_{y},f)} df = \\int e^{f + v_{y}} df and e^{- f - v_{y}} \\int \\hat{H}{(v_{y},f)} df = e^{- f - v_{y}} \\int e^{f + v_{y}} df and e^{- 2 f - 2 v_{y}} \\int \\hat{H}{(v_{y},f)} df = e^{- 2 f - 2 v_{y}} \\int e^{f + v_{y}} df", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('f', commutative=True)), exp(Add(Symbol('f', commutative=True), Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(exp(Add(Symbol('f', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "exp(Add(Symbol('f', commutative=True), Symbol('v_y', commutative=True)))"], "Equality(Mul(exp(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)))), Integral(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(exp(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)))), Integral(exp(Add(Symbol('f', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('f', commutative=True)))))"], [["times", 3, "exp(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], "Equality(Mul(exp(Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)))), Integral(Function('\\\\hat{H}')(Symbol('v_y', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(exp(Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)))), Integral(exp(Add(Symbol('f', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\chi)} = \\log{(\\chi)}, then derive \\cos{(\\chi - \\frac{d}{d \\chi} \\hat{\\mathbf{r}}{(\\chi)} + \\frac{1}{\\chi})} = \\cos{(\\chi)}, then obtain \\cos{(\\chi - \\frac{d}{d \\chi} \\log{(\\chi)} + \\frac{1}{\\chi})} = \\cos{(\\chi)}", "derivation": "\\hat{\\mathbf{r}}{(\\chi)} = \\log{(\\chi)} and \\hat{\\mathbf{r}}{(\\chi)} - \\log{(\\chi)} = 0 and \\frac{d}{d \\chi} (\\hat{\\mathbf{r}}{(\\chi)} - \\log{(\\chi)}) = \\frac{d}{d \\chi} 0 and - \\chi + \\frac{d}{d \\chi} (\\hat{\\mathbf{r}}{(\\chi)} - \\log{(\\chi)}) = - \\chi + \\frac{d}{d \\chi} 0 and \\cos{(\\chi - \\frac{d}{d \\chi} (\\hat{\\mathbf{r}}{(\\chi)} - \\log{(\\chi)}))} = \\cos{(\\chi - \\frac{d}{d \\chi} 0)} and \\cos{(\\chi - \\frac{d}{d \\chi} \\hat{\\mathbf{r}}{(\\chi)} + \\frac{1}{\\chi})} = \\cos{(\\chi)} and \\cos{(\\chi - \\frac{d}{d \\chi} \\log{(\\chi)} + \\frac{1}{\\chi})} = \\cos{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["cos", 4], "Equality(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Derivative(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))))"], [["evaluate_derivatives", 5], "Equality(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), cos(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), cos(Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(S)} = \\cos{(S)}, then obtain \\frac{(S + \\operatorname{t_{2}}{(S)})^{2}}{(\\int \\cos{(S)} dS)^{2}} = \\frac{(S + \\operatorname{t_{2}}{(S)}) (S + \\cos{(S)})}{(\\int \\cos{(S)} dS)^{2}}", "derivation": "\\operatorname{t_{2}}{(S)} = \\cos{(S)} and S + \\operatorname{t_{2}}{(S)} = S + \\cos{(S)} and \\frac{S + \\operatorname{t_{2}}{(S)}}{\\int \\cos{(S)} dS} = \\frac{S + \\cos{(S)}}{\\int \\cos{(S)} dS} and \\frac{(S + \\operatorname{t_{2}}{(S)})^{2}}{(\\int \\cos{(S)} dS)^{2}} = \\frac{(S + \\operatorname{t_{2}}{(S)}) (S + \\cos{(S)})}{(\\int \\cos{(S)} dS)^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('t_2')(Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))))"], [["divide", 2, "Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Symbol('S', commutative=True), Function('t_2')(Symbol('S', commutative=True))), Pow(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))), Mul(Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Pow(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))))"], [["times", 3, "Mul(Add(Symbol('S', commutative=True), Function('t_2')(Symbol('S', commutative=True))), Pow(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('S', commutative=True), Function('t_2')(Symbol('S', commutative=True))), Integer(2)), Pow(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-2))), Mul(Add(Symbol('S', commutative=True), Function('t_2')(Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Pow(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(y^{\\prime})} = \\log{(\\log{(y^{\\prime})})} and \\operatorname{v_{x}}{(y^{\\prime})} = \\log{(y^{\\prime})}, then obtain \\frac{\\log{(\\operatorname{v_{x}}{(y^{\\prime})})}}{\\operatorname{t_{1}}{(y^{\\prime})}} = 1", "derivation": "\\operatorname{t_{1}}{(y^{\\prime})} = \\log{(\\log{(y^{\\prime})})} and \\frac{\\operatorname{t_{1}}{(y^{\\prime})}}{\\log{(\\log{(y^{\\prime})})}} = 1 and \\operatorname{v_{x}}{(y^{\\prime})} = \\log{(y^{\\prime})} and \\operatorname{t_{1}}{(y^{\\prime})} = \\log{(\\operatorname{v_{x}}{(y^{\\prime})})} and \\frac{\\log{(\\operatorname{v_{x}}{(y^{\\prime})})}}{\\log{(\\log{(y^{\\prime})})}} = 1 and \\frac{\\log{(\\operatorname{v_{x}}{(y^{\\prime})})}}{\\operatorname{t_{1}}{(y^{\\prime})}} = 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), log(log(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "log(log(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(log(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), log(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(log(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True))), Pow(log(log(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('t_1')(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), log(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\bar{\\h}{(u,\\psi)} = u^{\\psi}, then obtain \\frac{(- \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} \\bar{\\h}{(u,\\psi)}) \\bar{\\h}{(u,\\psi)}}{u} = \\frac{(- \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} u^{\\psi}) \\bar{\\h}{(u,\\psi)}}{u}", "derivation": "\\bar{\\h}{(u,\\psi)} = u^{\\psi} and \\frac{\\partial}{\\partial \\psi} \\bar{\\h}{(u,\\psi)} = \\frac{\\partial}{\\partial \\psi} u^{\\psi} and - \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} \\bar{\\h}{(u,\\psi)} = - \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} u^{\\psi} and \\frac{(- \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} \\bar{\\h}{(u,\\psi)}) \\bar{\\h}{(u,\\psi)}}{u} = \\frac{(- \\bar{\\h}{(u,\\psi)} + \\frac{\\partial}{\\partial \\psi} u^{\\psi}) \\bar{\\h}{(u,\\psi)}}{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Pow(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))), Derivative(Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))), Derivative(Pow(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["times", 3, "Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))), Derivative(Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))), Derivative(Pow(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Function('\\\\hbar')(Symbol('u', commutative=True), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(q)} = e^{\\cos{(q)}} and \\operatorname{C_{1}}{(q)} = \\cos{(q)}, then obtain (\\frac{e^{\\cos{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q} = (\\frac{e^{\\operatorname{C_{1}}{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q}", "derivation": "\\Psi_{nl}{(q)} = e^{\\cos{(q)}} and \\operatorname{C_{1}}{(q)} = \\cos{(q)} and \\Psi_{nl}{(q)} = e^{\\operatorname{C_{1}}{(q)}} and \\frac{\\Psi_{nl}{(q)}}{e^{\\cos{(q)}} - \\cos{(q)}} = \\frac{e^{\\operatorname{C_{1}}{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}} and (\\frac{\\Psi_{nl}{(q)}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q} = (\\frac{e^{\\operatorname{C_{1}}{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q} and \\frac{\\Psi_{nl}{(q)}}{e^{\\cos{(q)}} - \\cos{(q)}} = \\frac{e^{\\cos{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}} and (\\frac{e^{\\cos{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q} = (\\frac{e^{\\operatorname{C_{1}}{(q)}}}{e^{\\cos{(q)}} - \\cos{(q)}})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('q', commutative=True)), exp(cos(Symbol('q', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\Psi_{nl}')(Symbol('q', commutative=True)), exp(Function('C_1')(Symbol('q', commutative=True))))"], [["divide", 3, "Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True))))"], "Equality(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('q', commutative=True))), Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), exp(Function('C_1')(Symbol('q', commutative=True)))))"], [["power", 4, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), exp(Function('C_1')(Symbol('q', commutative=True)))), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('q', commutative=True))), Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), exp(cos(Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), exp(cos(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Pow(Mul(Pow(Add(exp(cos(Symbol('q', commutative=True))), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1)), exp(Function('C_1')(Symbol('q', commutative=True)))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(M_{E},\\theta)} = M_{E} + \\theta, then obtain ((- \\theta + \\dot{y}{(M_{E},\\theta)})^{\\theta})^{\\theta} = (M_{E}^{\\theta})^{\\theta}", "derivation": "\\dot{y}{(M_{E},\\theta)} = M_{E} + \\theta and - \\theta + \\dot{y}{(M_{E},\\theta)} = M_{E} and (- \\theta + \\dot{y}{(M_{E},\\theta)})^{\\theta} = M_{E}^{\\theta} and ((- \\theta + \\dot{y}{(M_{E},\\theta)})^{\\theta})^{\\theta} = (M_{E}^{\\theta})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\dot{y}')(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('M_E', commutative=True))"], [["power", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\dot{y}')(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\dot{y}')(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Pow(Symbol('M_E', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given l{(x^\\prime)} = e^{e^{x^\\prime}}, then derive (\\int l{(x^\\prime)} dx^\\prime)^{x^\\prime} = (\\dot{z} + \\operatorname{Ei}{(e^{x^\\prime})})^{x^\\prime}, then obtain (\\dot{z} + \\operatorname{Ei}{(e^{x^\\prime})})^{x^\\prime} = (\\int e^{e^{x^\\prime}} dx^\\prime)^{x^\\prime}", "derivation": "l{(x^\\prime)} = e^{e^{x^\\prime}} and \\int l{(x^\\prime)} dx^\\prime = \\int e^{e^{x^\\prime}} dx^\\prime and (\\int l{(x^\\prime)} dx^\\prime)^{x^\\prime} = (\\int e^{e^{x^\\prime}} dx^\\prime)^{x^\\prime} and (\\int l{(x^\\prime)} dx^\\prime)^{x^\\prime} = (\\dot{z} + \\operatorname{Ei}{(e^{x^\\prime})})^{x^\\prime} and (\\dot{z} + \\operatorname{Ei}{(e^{x^\\prime})})^{x^\\prime} = (\\int e^{e^{x^\\prime}} dx^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('l')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integral(Function('l')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(exp(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('l')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Ei(exp(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Ei(exp(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(exp(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(a,J_{\\varepsilon})} = J_{\\varepsilon} \\sin{(a)}, then obtain \\frac{2}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} = \\frac{\\frac{J_{\\varepsilon} \\sin{(a)}}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} + 1}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}}", "derivation": "\\operatorname{t_{2}}{(a,J_{\\varepsilon})} = J_{\\varepsilon} \\sin{(a)} and 1 = \\frac{J_{\\varepsilon} \\sin{(a)}}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} and 2 = \\frac{J_{\\varepsilon} \\sin{(a)}}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} + 1 and \\frac{2}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} = \\frac{\\frac{J_{\\varepsilon} \\sin{(a)}}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}} + 1}{\\operatorname{t_{2}}{(a,J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), sin(Symbol('a', commutative=True))))"], [["divide", 1, "Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), sin(Symbol('a', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), sin(Symbol('a', commutative=True))), Integer(1)))"], [["divide", 3, "Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), sin(Symbol('a', commutative=True))), Integer(1)), Pow(Function('t_2')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{r})} = \\cos{(\\log{(\\mathbf{r})})}, then obtain - \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})} = (- \\hat{p}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})}) \\hat{p}{(\\mathbf{r})} - \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})}", "derivation": "\\hat{p}{(\\mathbf{r})} = \\cos{(\\log{(\\mathbf{r})})} and \\hat{p}^{\\mathbf{r}}{(\\mathbf{r})} = \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})} and 0 = - \\hat{p}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})} and 0 = (- \\hat{p}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})}) \\hat{p}{(\\mathbf{r})} and - \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})} = (- \\hat{p}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})}) \\hat{p}{(\\mathbf{r})} - \\cos^{\\mathbf{r}}{(\\log{(\\mathbf{r})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 3, "Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 4, "Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Pow(cos(log(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)}, then derive \\hat{p}_0 + \\int \\operatorname{E_{n}}{(\\hat{p}_0)} d\\hat{p}_0 = \\hat{p}_0 + a - \\cos{(\\hat{p}_0)}, then obtain \\hat{p}_0 + \\int \\sin{(\\hat{p}_0)} d\\hat{p}_0 = \\hat{p}_0 + a - \\cos{(\\hat{p}_0)}", "derivation": "\\operatorname{E_{n}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\int \\operatorname{E_{n}}{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\sin{(\\hat{p}_0)} d\\hat{p}_0 and \\hat{p}_0 + \\int \\operatorname{E_{n}}{(\\hat{p}_0)} d\\hat{p}_0 = \\hat{p}_0 + \\int \\sin{(\\hat{p}_0)} d\\hat{p}_0 and \\hat{p}_0 + \\int \\operatorname{E_{n}}{(\\hat{p}_0)} d\\hat{p}_0 = \\hat{p}_0 + a - \\cos{(\\hat{p}_0)} and \\hat{p}_0 + \\int \\sin{(\\hat{p}_0)} d\\hat{p}_0 = \\hat{p}_0 + a - \\cos{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(sin(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Integral(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Integral(sin(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Integral(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Integral(sin(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(S)} = \\cos{(S)} and \\hat{X}{(S)} = 2 \\cos{(S)}, then obtain 4 \\cos^{2}{(S)} = 4 \\mathbf{H}^{2}{(S)}", "derivation": "\\mathbf{H}{(S)} = \\cos{(S)} and \\hat{X}{(S)} = 2 \\cos{(S)} and \\hat{X}^{2}{(S)} = 4 \\cos^{2}{(S)} and \\hat{X}^{2}{(S)} = 4 \\mathbf{H}^{2}{(S)} and 4 \\cos^{2}{(S)} = 4 \\mathbf{H}^{2}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Mul(Integer(2), cos(Symbol('S', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Integer(2)), Mul(Integer(4), Pow(cos(Symbol('S', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Integer(2)), Mul(Integer(4), Pow(Function('\\\\mathbf{H}')(Symbol('S', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(4), Pow(cos(Symbol('S', commutative=True)), Integer(2))), Mul(Integer(4), Pow(Function('\\\\mathbf{H}')(Symbol('S', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\varepsilon{(T)} = \\cos{(e^{T})}, then derive 1 + \\frac{\\int \\varepsilon{(T)} dT}{T} = 1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T}, then obtain ((1 + \\frac{\\int \\cos{(e^{T})} dT}{T})^{T})^{T} = ((1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T})^{T})^{T}", "derivation": "\\varepsilon{(T)} = \\cos{(e^{T})} and \\int \\varepsilon{(T)} dT = \\int \\cos{(e^{T})} dT and \\frac{\\int \\varepsilon{(T)} dT}{T} = \\frac{\\int \\cos{(e^{T})} dT}{T} and 1 + \\frac{\\int \\varepsilon{(T)} dT}{T} = 1 + \\frac{\\int \\cos{(e^{T})} dT}{T} and 1 + \\frac{\\int \\varepsilon{(T)} dT}{T} = 1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T} and (1 + \\frac{\\int \\varepsilon{(T)} dT}{T})^{T} = (1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T})^{T} and (1 + \\frac{\\int \\cos{(e^{T})} dT}{T})^{T} = (1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T})^{T} and ((1 + \\frac{\\int \\cos{(e^{T})} dT}{T})^{T})^{T} = ((1 + \\frac{r_{0} + \\operatorname{Ci}{(e^{T})}}{T})^{T})^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('T', commutative=True)), cos(exp(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["divide", 2, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Ci(exp(Symbol('T', commutative=True)))))))"], [["power", 5, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(Function('\\\\varepsilon')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Symbol('T', commutative=True)), Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Ci(exp(Symbol('T', commutative=True)))))), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))), Symbol('T', commutative=True)), Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Ci(exp(Symbol('T', commutative=True)))))), Symbol('T', commutative=True)))"], [["power", 7, "Symbol('T', commutative=True)"], "Equality(Pow(Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Add(Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Ci(exp(Symbol('T', commutative=True)))))), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\nabla{(F_{H})} = \\int \\cos{(F_{H})} dF_{H}, then derive (E_{x} + \\sin{(F_{H})}) \\nabla{(F_{H})} = (E_{x} + \\sin{(F_{H})})^{2}, then derive (E_{x} + \\sin{(F_{H})}) (\\pi + \\sin{(F_{H})}) = (E_{x} + \\sin{(F_{H})})^{2}, then obtain (E_{x} + \\sin{(F_{H})}) \\nabla{(F_{H})} = (E_{x} + \\sin{(F_{H})}) (\\pi + \\sin{(F_{H})})", "derivation": "\\nabla{(F_{H})} = \\int \\cos{(F_{H})} dF_{H} and \\nabla{(F_{H})} \\int \\cos{(F_{H})} dF_{H} = (\\int \\cos{(F_{H})} dF_{H})^{2} and (E_{x} + \\sin{(F_{H})}) \\nabla{(F_{H})} = (E_{x} + \\sin{(F_{H})})^{2} and (E_{x} + \\sin{(F_{H})}) \\int \\cos{(F_{H})} dF_{H} = (E_{x} + \\sin{(F_{H})})^{2} and (E_{x} + \\sin{(F_{H})}) (\\pi + \\sin{(F_{H})}) = (E_{x} + \\sin{(F_{H})})^{2} and (E_{x} + \\sin{(F_{H})}) \\nabla{(F_{H})} = (E_{x} + \\sin{(F_{H})}) (\\pi + \\sin{(F_{H})})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('F_H', commutative=True)), Integral(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["times", 1, "Integral(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Mul(Function('\\\\nabla')(Symbol('F_H', commutative=True)), Integral(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Pow(Integral(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Function('\\\\nabla')(Symbol('F_H', commutative=True))), Pow(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Integral(cos(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Pow(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Integer(2)))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('F_H', commutative=True)))), Pow(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Function('\\\\nabla')(Symbol('F_H', commutative=True))), Mul(Add(Symbol('E_x', commutative=True), sin(Symbol('F_H', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given u{(E_{n},v)} = E_{n} - v, then derive 0 = \\frac{\\partial}{\\partial v} (\\int (v + (E_{n} - v)^{v}) dE_{n} - \\int (v + u^{v}{(E_{n},v)}) dE_{n}), then obtain 0 = \\frac{d}{d v} 0", "derivation": "u{(E_{n},v)} = E_{n} - v and u^{v}{(E_{n},v)} = (E_{n} - v)^{v} and v + u^{v}{(E_{n},v)} = v + (E_{n} - v)^{v} and \\int (v + u^{v}{(E_{n},v)}) dE_{n} = \\int (v + (E_{n} - v)^{v}) dE_{n} and 0 = \\int (v + (E_{n} - v)^{v}) dE_{n} - \\int (v + u^{v}{(E_{n},v)}) dE_{n} and \\frac{d}{d v} 0 = \\frac{\\partial}{\\partial v} (\\int (v + (E_{n} - v)^{v}) dE_{n} - \\int (v + u^{v}{(E_{n},v)}) dE_{n}) and 0 = \\frac{\\partial}{\\partial v} (\\int (v + (E_{n} - v)^{v}) dE_{n} - \\int (v + u^{v}{(E_{n},v)}) dE_{n}) and 0 = \\frac{d}{d v} 0", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('v', commutative=True))"], "Equality(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Symbol('v', commutative=True), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], [["minus", 4, "Integral(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Symbol('v', commutative=True), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))))))"], [["differentiate", 5, "Symbol('v', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Integral(Add(Symbol('v', commutative=True), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Derivative(Add(Integral(Add(Symbol('v', commutative=True), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('v', commutative=True), Pow(Function('u')(Symbol('E_n', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('E_n', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integer(0), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain - 2 \\operatorname{v_{x}}{(\\mathbf{H})} + e^{\\mathbf{H}} = - 3 \\operatorname{v_{x}}{(\\mathbf{H})} + 2 e^{\\mathbf{H}}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{H})} = e^{\\mathbf{H}} and 0 = - \\operatorname{v_{x}}{(\\mathbf{H})} + e^{\\mathbf{H}} and - \\operatorname{v_{x}}{(\\mathbf{H})} = - 2 \\operatorname{v_{x}}{(\\mathbf{H})} + e^{\\mathbf{H}} and - 2 \\operatorname{v_{x}}{(\\mathbf{H})} + e^{\\mathbf{H}} = - 3 \\operatorname{v_{x}}{(\\mathbf{H})} + 2 e^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Integer(3), Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})}, then obtain - \\frac{\\int \\frac{d}{d J_{\\varepsilon}} \\mathbf{v}{(J_{\\varepsilon})} dJ_{\\varepsilon}}{J_{\\varepsilon}} = - \\frac{\\int \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} dJ_{\\varepsilon}}{J_{\\varepsilon}}", "derivation": "\\mathbf{v}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} \\mathbf{v}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and \\int \\frac{d}{d J_{\\varepsilon}} \\mathbf{v}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} dJ_{\\varepsilon} and - \\frac{\\int \\frac{d}{d J_{\\varepsilon}} \\mathbf{v}{(J_{\\varepsilon})} dJ_{\\varepsilon}}{J_{\\varepsilon}} = - \\frac{\\int \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} dJ_{\\varepsilon}}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{v}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(Derivative(Function('\\\\mathbf{v}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\chi{(x,l)} = x e^{l}, then obtain \\int (- x e^{l} + 2 \\chi{(x,l)}) dl = \\int \\chi{(x,l)} dl", "derivation": "\\chi{(x,l)} = x e^{l} and - x + \\chi{(x,l)} = x e^{l} - x and - x e^{l} + 2 \\chi{(x,l)} = \\chi{(x,l)} and \\int (- x e^{l} + 2 \\chi{(x,l)}) dl = \\int \\chi{(x,l)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('x', commutative=True), exp(Symbol('l', commutative=True))))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True))), Add(Mul(Symbol('x', commutative=True), exp(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('x', commutative=True), exp(Symbol('l', commutative=True))), Symbol('x', commutative=True), Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True), exp(Symbol('l', commutative=True))), Mul(Integer(2), Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)))), Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True), exp(Symbol('l', commutative=True))), Mul(Integer(2), Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Integral(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\hat{p},p)} = \\frac{p}{\\hat{p}}, then derive \\frac{\\partial^{2}}{\\partial p\\partial \\hat{p}} \\varepsilon_{0}{(\\hat{p},p)} = - \\frac{1}{\\hat{p}^{2}}, then obtain \\hat{p} \\frac{\\partial^{2}}{\\partial p\\partial \\hat{p}} \\varepsilon_{0}{(\\hat{p},p)} = - \\frac{1}{\\hat{p}}", "derivation": "\\varepsilon_{0}{(\\hat{p},p)} = \\frac{p}{\\hat{p}} and \\frac{\\partial}{\\partial p} \\varepsilon_{0}{(\\hat{p},p)} = \\frac{\\partial}{\\partial p} \\frac{p}{\\hat{p}} and \\frac{\\partial^{2}}{\\partial \\hat{p}\\partial p} \\varepsilon_{0}{(\\hat{p},p)} = \\frac{\\partial^{2}}{\\partial \\hat{p}\\partial p} \\frac{p}{\\hat{p}} and \\frac{\\partial^{2}}{\\partial p\\partial \\hat{p}} \\varepsilon_{0}{(\\hat{p},p)} = - \\frac{1}{\\hat{p}^{2}} and \\hat{p} \\frac{\\partial^{2}}{\\partial p\\partial \\hat{p}} \\varepsilon_{0}{(\\hat{p},p)} = - \\frac{1}{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2))))"], [["divide", 4, "Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{p}{(A_{1},c)} = - A_{1} + c, then obtain \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\iint \\sigma_{p}^{c}{(A_{1},c)} dA_{1} dA_{1}) = \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\iint (- A_{1} + c)^{c} dA_{1} dA_{1})", "derivation": "\\sigma_{p}{(A_{1},c)} = - A_{1} + c and \\sigma_{p}^{c}{(A_{1},c)} = (- A_{1} + c)^{c} and \\int \\sigma_{p}^{c}{(A_{1},c)} dA_{1} = \\int (- A_{1} + c)^{c} dA_{1} and \\iint \\sigma_{p}^{c}{(A_{1},c)} dA_{1} dA_{1} = \\iint (- A_{1} + c)^{c} dA_{1} dA_{1} and A_{1} + \\iint \\sigma_{p}^{c}{(A_{1},c)} dA_{1} dA_{1} = A_{1} + \\iint (- A_{1} + c)^{c} dA_{1} dA_{1} and \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\iint \\sigma_{p}^{c}{(A_{1},c)} dA_{1} dA_{1}) = \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\iint (- A_{1} + c)^{c} dA_{1} dA_{1})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Pow(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["integrate", 3, "Symbol('A_1', commutative=True)"], "Equality(Integral(Pow(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["add", 4, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Integral(Pow(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["differentiate", 5, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Symbol('A_1', commutative=True), Integral(Pow(Function('\\\\sigma_p')(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\omega)} = \\cos{(\\omega)} and T{(\\omega)} = \\omega, then obtain \\int \\operatorname{A_{2}}^{\\omega}{(\\omega)} dT{(\\omega)} = \\int \\cos^{\\omega}{(\\omega)} dT{(\\omega)}", "derivation": "\\operatorname{A_{2}}{(\\omega)} = \\cos{(\\omega)} and \\operatorname{A_{2}}^{\\omega}{(\\omega)} = \\cos^{\\omega}{(\\omega)} and T{(\\omega)} = \\omega and \\int \\operatorname{A_{2}}^{\\omega}{(\\omega)} d\\omega = \\int \\cos^{\\omega}{(\\omega)} d\\omega and \\int \\operatorname{A_{2}}^{\\omega}{(\\omega)} dT{(\\omega)} = \\int \\cos^{\\omega}{(\\omega)} dT{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Function('A_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Pow(Function('A_2')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Function('T')(Symbol('\\\\omega', commutative=True)))), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Function('T')(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given I{(a,\\mathbf{J}_f,\\sigma_x)} = \\frac{\\mathbf{J}_f}{\\sigma_x a}, then obtain - e^{\\Psi^{\\dagger}} - \\int \\frac{\\mathbf{J}_f}{\\sigma_x a} d\\sigma_x + \\int I{(a,\\mathbf{J}_f,\\sigma_x)} d\\sigma_x = - e^{\\Psi^{\\dagger}}", "derivation": "I{(a,\\mathbf{J}_f,\\sigma_x)} = \\frac{\\mathbf{J}_f}{\\sigma_x a} and \\int I{(a,\\mathbf{J}_f,\\sigma_x)} d\\sigma_x = \\int \\frac{\\mathbf{J}_f}{\\sigma_x a} d\\sigma_x and - \\int \\frac{\\mathbf{J}_f}{\\sigma_x a} d\\sigma_x + \\int I{(a,\\mathbf{J}_f,\\sigma_x)} d\\sigma_x = 0 and - e^{\\Psi^{\\dagger}} - \\int \\frac{\\mathbf{J}_f}{\\sigma_x a} d\\sigma_x + \\int I{(a,\\mathbf{J}_f,\\sigma_x)} d\\sigma_x = - e^{\\Psi^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Integral(Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Integer(0))"], [["minus", 3, "exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Integral(Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\varepsilon,\\pi)} = \\pi^{\\varepsilon}, then derive \\log{(\\frac{\\frac{\\partial}{\\partial \\varepsilon} \\hat{X}{(\\varepsilon,\\pi)}}{\\pi})} = \\log{(\\frac{\\pi^{\\varepsilon} \\log{(\\pi)}}{\\pi})}, then obtain \\log{(\\frac{\\frac{\\partial}{\\partial \\varepsilon} \\pi^{\\varepsilon}}{\\pi})} = \\log{(\\frac{\\pi^{\\varepsilon} \\log{(\\pi)}}{\\pi})}", "derivation": "\\hat{X}{(\\varepsilon,\\pi)} = \\pi^{\\varepsilon} and \\frac{\\hat{X}{(\\varepsilon,\\pi)}}{\\pi} = \\frac{\\pi^{\\varepsilon}}{\\pi} and \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\hat{X}{(\\varepsilon,\\pi)}}{\\pi} = \\frac{\\partial}{\\partial \\varepsilon} \\frac{\\pi^{\\varepsilon}}{\\pi} and \\log{(\\frac{\\partial}{\\partial \\varepsilon} \\frac{\\hat{X}{(\\varepsilon,\\pi)}}{\\pi})} = \\log{(\\frac{\\partial}{\\partial \\varepsilon} \\frac{\\pi^{\\varepsilon}}{\\pi})} and \\log{(\\frac{\\frac{\\partial}{\\partial \\varepsilon} \\hat{X}{(\\varepsilon,\\pi)}}{\\pi})} = \\log{(\\frac{\\pi^{\\varepsilon} \\log{(\\pi)}}{\\pi})} and \\log{(\\frac{\\frac{\\partial}{\\partial \\varepsilon} \\pi^{\\varepsilon}}{\\pi})} = \\log{(\\frac{\\pi^{\\varepsilon} \\log{(\\pi)}}{\\pi})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), log(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(log(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\varphi)} = \\log{(e^{\\varphi})}, then derive 0 = - \\sin{(\\frac{d}{d \\varphi} \\operatorname{f^{\\prime}}{(\\varphi)} - 1)}, then obtain 0 = - \\sin{(\\frac{d}{d \\varphi} \\log{(e^{\\varphi})} - 1)}", "derivation": "\\operatorname{f^{\\prime}}{(\\varphi)} = \\log{(e^{\\varphi})} and 0 = - \\operatorname{f^{\\prime}}{(\\varphi)} + \\log{(e^{\\varphi})} and \\frac{d}{d \\varphi} \\operatorname{f^{\\prime}}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(e^{\\varphi})} and \\frac{d}{d \\varphi} 0 = \\frac{d}{d \\varphi} (- \\operatorname{f^{\\prime}}{(\\varphi)} + \\log{(e^{\\varphi})}) and \\sin{(\\frac{d}{d \\varphi} 0)} = \\sin{(\\frac{d}{d \\varphi} (- \\operatorname{f^{\\prime}}{(\\varphi)} + \\log{(e^{\\varphi})}))} and 0 = - \\sin{(\\frac{d}{d \\varphi} \\operatorname{f^{\\prime}}{(\\varphi)} - 1)} and 0 = - \\sin{(\\frac{d}{d \\varphi} \\log{(e^{\\varphi})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True))))"], [["minus", 1, "Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))), log(exp(Symbol('\\\\varphi', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))), log(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["sin", 4], "Equality(sin(Derivative(Integer(0), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), sin(Derivative(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))), log(exp(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given S{(\\mathbf{P},\\mu)} = \\frac{\\mathbf{P}}{\\mu} and y{(\\varphi^*)} = \\log{(\\varphi^*)}, then derive - \\frac{\\mu \\int y{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} = - \\frac{\\mu (\\varphi^* \\log{(\\varphi^*)} - \\varphi^* + v_{z})}{\\mathbf{P}}, then obtain - \\frac{\\mu \\int \\log{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} = - \\frac{\\mu (\\varphi^* \\log{(\\varphi^*)} - \\varphi^* + v_{z})}{\\mathbf{P}}", "derivation": "S{(\\mathbf{P},\\mu)} = \\frac{\\mathbf{P}}{\\mu} and y{(\\varphi^*)} = \\log{(\\varphi^*)} and \\int y{(\\varphi^*)} d\\varphi^* = \\int \\log{(\\varphi^*)} d\\varphi^* and - \\frac{\\int y{(\\varphi^*)} d\\varphi^*}{S{(\\mathbf{P},\\mu)}} = - \\frac{\\int \\log{(\\varphi^*)} d\\varphi^*}{S{(\\mathbf{P},\\mu)}} and - \\frac{\\mu \\int y{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} = - \\frac{\\mu \\int \\log{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} and - \\frac{\\mu \\int y{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} = - \\frac{\\mu (\\varphi^* \\log{(\\varphi^*)} - \\varphi^* + v_{z})}{\\mathbf{P}} and - \\frac{\\mu \\int \\log{(\\varphi^*)} d\\varphi^*}{\\mathbf{P}} = - \\frac{\\mu (\\varphi^* \\log{(\\varphi^*)} - \\varphi^* + v_{z})}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('y')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('y')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Function('y')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Integral(Function('y')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Integral(Function('y')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Add(Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True), Add(Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(y,\\Psi_{\\lambda})} = \\Psi_{\\lambda} y, then obtain \\frac{\\partial}{\\partial y} (\\Psi_{\\lambda} + \\operatorname{n_{2}}^{\\Psi_{\\lambda}}{(y,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial y} (\\Psi_{\\lambda} + (\\Psi_{\\lambda} y)^{\\Psi_{\\lambda}})", "derivation": "\\operatorname{n_{2}}{(y,\\Psi_{\\lambda})} = \\Psi_{\\lambda} y and \\operatorname{n_{2}}^{\\Psi_{\\lambda}}{(y,\\Psi_{\\lambda})} = (\\Psi_{\\lambda} y)^{\\Psi_{\\lambda}} and \\Psi_{\\lambda} + \\operatorname{n_{2}}^{\\Psi_{\\lambda}}{(y,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + (\\Psi_{\\lambda} y)^{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial y} (\\Psi_{\\lambda} + \\operatorname{n_{2}}^{\\Psi_{\\lambda}}{(y,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial y} (\\Psi_{\\lambda} + (\\Psi_{\\lambda} y)^{\\Psi_{\\lambda}})", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('n_2')(Symbol('y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('n_2')(Symbol('y', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(C_{1})} = \\int e^{C_{1}} dC_{1}, then derive \\frac{\\hat{p}{(C_{1})}}{\\mathbf{v} + e^{C_{1}}} = 1, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} \\int \\frac{\\int e^{C_{1}} dC_{1}}{\\mathbf{v} + e^{C_{1}}} d\\mathbf{v} = \\frac{d}{d \\mathbf{v}} \\int 1 d\\mathbf{v}", "derivation": "\\hat{p}{(C_{1})} = \\int e^{C_{1}} dC_{1} and \\frac{\\hat{p}{(C_{1})}}{\\int e^{C_{1}} dC_{1}} = 1 and \\frac{\\hat{p}{(C_{1})}}{\\mathbf{v} + e^{C_{1}}} = 1 and \\frac{\\int e^{C_{1}} dC_{1}}{\\mathbf{v} + e^{C_{1}}} = 1 and \\int \\frac{\\int e^{C_{1}} dC_{1}}{\\mathbf{v} + e^{C_{1}}} d\\mathbf{v} = \\int 1 d\\mathbf{v} and \\frac{\\partial}{\\partial \\mathbf{v}} \\int \\frac{\\int e^{C_{1}} dC_{1}}{\\mathbf{v} + e^{C_{1}}} d\\mathbf{v} = \\frac{d}{d \\mathbf{v}} \\int 1 d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('C_1', commutative=True)), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["divide", 1, "Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('C_1', commutative=True)), Pow(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Integer(-1)), Function('\\\\hat{p}')(Symbol('C_1', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Integer(-1)), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Integer(1))"], [["integrate", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Integer(-1)), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('C_1', commutative=True))), Integer(-1)), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(t_{2})} = \\sin{(t_{2})}, then obtain t_{2} = t_{2} - g^{t_{2}}{(t_{2})} + \\sin^{t_{2}}{(t_{2})}", "derivation": "g{(t_{2})} = \\sin{(t_{2})} and g^{t_{2}}{(t_{2})} = \\sin^{t_{2}}{(t_{2})} and 0 = - g^{t_{2}}{(t_{2})} + \\sin^{t_{2}}{(t_{2})} and t_{2} = t_{2} - g^{t_{2}}{(t_{2})} + \\sin^{t_{2}}{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('g')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 2, "Pow(Function('g')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('g')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["add", 3, "Symbol('t_2', commutative=True)"], "Equality(Symbol('t_2', commutative=True), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Pow(Function('g')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(F_{g},F_{H})} = \\frac{F_{H}}{F_{g}}, then obtain \\int (\\frac{\\mathbf{v}{(F_{g},F_{H})}}{F_{g}})^{F_{H}} dF_{H} = \\int (\\frac{F_{H}}{F_{g}^{2}})^{F_{H}} dF_{H}", "derivation": "\\mathbf{v}{(F_{g},F_{H})} = \\frac{F_{H}}{F_{g}} and \\frac{\\mathbf{v}{(F_{g},F_{H})}}{F_{g}} = \\frac{F_{H}}{F_{g}^{2}} and (\\frac{\\mathbf{v}{(F_{g},F_{H})}}{F_{g}})^{F_{H}} = (\\frac{F_{H}}{F_{g}^{2}})^{F_{H}} and \\int (\\frac{\\mathbf{v}{(F_{g},F_{H})}}{F_{g}})^{F_{H}} dF_{H} = \\int (\\frac{F_{H}}{F_{g}^{2}})^{F_{H}} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_g', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('F_g', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('F_g', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(Symbol('F_g', commutative=True), Integer(-2))))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('F_g', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Mul(Symbol('F_H', commutative=True), Pow(Symbol('F_g', commutative=True), Integer(-2))), Symbol('F_H', commutative=True)))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('F_g', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Mul(Symbol('F_H', commutative=True), Pow(Symbol('F_g', commutative=True), Integer(-2))), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\eta)} = \\eta, then derive \\frac{d}{d \\eta} \\operatorname{n_{2}}{(\\eta)} = 1, then obtain \\frac{d}{d \\eta} \\eta = 1", "derivation": "\\operatorname{n_{2}}{(\\eta)} = \\eta and \\frac{d}{d \\eta} \\operatorname{n_{2}}{(\\eta)} = \\frac{d}{d \\eta} \\eta and \\frac{d}{d \\eta} \\operatorname{n_{2}}{(\\eta)} = 1 and \\frac{d}{d \\eta} \\eta = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('n_2')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(A_{y})} = e^{\\cos{(A_{y})}}, then obtain \\int 1 dA_{y} + \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} = \\int \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} dA_{y} + \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}}", "derivation": "\\operatorname{C_{1}}{(A_{y})} = e^{\\cos{(A_{y})}} and 1 = \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} and \\int 1 dA_{y} = \\int \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} dA_{y} and \\int 1 dA_{y} + \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} = \\int \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}} dA_{y} + \\frac{e^{\\cos{(A_{y})}}}{\\operatorname{C_{1}}{(A_{y})}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('A_y', commutative=True)), exp(cos(Symbol('A_y', commutative=True))))"], [["divide", 1, "Function('C_1')(Symbol('A_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True)))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_y', commutative=True))), Integral(Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))))"], [["add", 3, "Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True))))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('A_y', commutative=True))), Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True))))), Add(Integral(Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))), Mul(Pow(Function('C_1')(Symbol('A_y', commutative=True)), Integer(-1)), exp(cos(Symbol('A_y', commutative=True))))))"]]}, {"prompt": "Given \\varphi{(t_{1},y^{\\prime})} = t_{1} - y^{\\prime} and \\operatorname{a^{\\dagger}}{(t_{1},y^{\\prime})} = \\frac{\\partial}{\\partial t_{1}} \\varphi{(t_{1},y^{\\prime})}, then obtain \\operatorname{a^{\\dagger}}{(t_{1},y^{\\prime})} = \\frac{\\partial}{\\partial t_{1}} (t_{1} - y^{\\prime})", "derivation": "\\varphi{(t_{1},y^{\\prime})} = t_{1} - y^{\\prime} and \\frac{\\partial}{\\partial t_{1}} \\varphi{(t_{1},y^{\\prime})} = \\frac{\\partial}{\\partial t_{1}} (t_{1} - y^{\\prime}) and \\operatorname{a^{\\dagger}}{(t_{1},y^{\\prime})} = \\frac{\\partial}{\\partial t_{1}} \\varphi{(t_{1},y^{\\prime})} and \\operatorname{a^{\\dagger}}{(t_{1},y^{\\prime})} = \\frac{\\partial}{\\partial t_{1}} (t_{1} - y^{\\prime})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})}, then obtain \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} \\cos{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\mathbf{f} \\cos^{2}{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\mathbf{f} \\cos{(\\mathbf{f})} and \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}^{2}{(\\mathbf{f})} = \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} \\cos{(\\mathbf{f})} and \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}^{2}{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} \\cos{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} and \\mathbf{f} \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} \\cos{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})} = \\mathbf{f} \\cos^{2}{(\\mathbf{f})} - \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('\\\\mathbf{f}', commutative=True), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 2, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 3, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(cos(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(E,\\theta)} = \\theta^{E} and \\mathbf{H}{(A_{y})} = \\sin{(A_{y})}, then obtain A_{y} (\\mathbf{B} - \\operatorname{A_{y}}{(E,\\theta)} + \\mathbf{H}{(A_{y})}) = A_{y} (\\mathbf{B} - \\operatorname{A_{y}}{(E,\\theta)} + \\sin{(A_{y})})", "derivation": "\\operatorname{A_{y}}{(E,\\theta)} = \\theta^{E} and \\mathbf{H}{(A_{y})} = \\sin{(A_{y})} and - \\theta^{E} + \\mathbf{H}{(A_{y})} = - \\theta^{E} + \\sin{(A_{y})} and \\mathbf{B} - \\theta^{E} + \\mathbf{H}{(A_{y})} = \\mathbf{B} - \\theta^{E} + \\sin{(A_{y})} and A_{y} (\\mathbf{B} - \\theta^{E} + \\mathbf{H}{(A_{y})}) = A_{y} (\\mathbf{B} - \\theta^{E} + \\sin{(A_{y})}) and A_{y} (\\mathbf{B} - \\operatorname{A_{y}}{(E,\\theta)} + \\mathbf{H}{(A_{y})}) = A_{y} (\\mathbf{B} - \\operatorname{A_{y}}{(E,\\theta)} + \\sin{(A_{y})})", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["minus", 2, "Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), Function('\\\\mathbf{H}')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), sin(Symbol('A_y', commutative=True))))"], [["add", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), Function('\\\\mathbf{H}')(Symbol('A_y', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), sin(Symbol('A_y', commutative=True))))"], [["times", 4, "Symbol('A_y', commutative=True)"], "Equality(Mul(Symbol('A_y', commutative=True), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), Function('\\\\mathbf{H}')(Symbol('A_y', commutative=True)))), Mul(Symbol('A_y', commutative=True), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Symbol('E', commutative=True))), sin(Symbol('A_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('A_y', commutative=True), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Function('A_y')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True))), Function('\\\\mathbf{H}')(Symbol('A_y', commutative=True)))), Mul(Symbol('A_y', commutative=True), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Function('A_y')(Symbol('E', commutative=True), Symbol('\\\\theta', commutative=True))), sin(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given M{(x^\\prime)} = \\sin{(x^\\prime)} and \\operatorname{F_{N}}{(u,\\theta_2)} = \\theta_2 \\sin{(u)} and \\varphi^{*}{(n_{2})} = \\log{(e^{n_{2}})}, then obtain (\\theta_2 \\sin{(u)} + M^{x^\\prime}{(x^\\prime)}) \\varphi^{*}{(n_{2})} = (\\theta_2 \\sin{(u)} + \\sin^{x^\\prime}{(x^\\prime)}) \\varphi^{*}{(n_{2})}", "derivation": "M{(x^\\prime)} = \\sin{(x^\\prime)} and \\operatorname{F_{N}}{(u,\\theta_2)} = \\theta_2 \\sin{(u)} and M^{x^\\prime}{(x^\\prime)} = \\sin^{x^\\prime}{(x^\\prime)} and \\operatorname{F_{N}}{(u,\\theta_2)} + M^{x^\\prime}{(x^\\prime)} = \\operatorname{F_{N}}{(u,\\theta_2)} + \\sin^{x^\\prime}{(x^\\prime)} and \\varphi^{*}{(n_{2})} = \\log{(e^{n_{2}})} and \\theta_2 \\sin{(u)} + M^{x^\\prime}{(x^\\prime)} = \\theta_2 \\sin{(u)} + \\sin^{x^\\prime}{(x^\\prime)} and (\\theta_2 \\sin{(u)} + M^{x^\\prime}{(x^\\prime)}) \\log{(e^{n_{2}})} = (\\theta_2 \\sin{(u)} + \\sin^{x^\\prime}{(x^\\prime)}) \\log{(e^{n_{2}})} and (\\theta_2 \\sin{(u)} + M^{x^\\prime}{(x^\\prime)}) \\varphi^{*}{(n_{2})} = (\\theta_2 \\sin{(u)} + \\sin^{x^\\prime}{(x^\\prime)}) \\varphi^{*}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], ["get_premise", "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('M')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 3, "Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('M')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('n_2', commutative=True)), log(exp(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(Function('M')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["times", 6, "log(exp(Symbol('n_2', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(Function('M')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), log(exp(Symbol('n_2', commutative=True)))), Mul(Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), log(exp(Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(Function('M')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Function('\\\\varphi^*')(Symbol('n_2', commutative=True))), Mul(Add(Mul(Symbol('\\\\theta_2', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Function('\\\\varphi^*')(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain \\frac{d}{d \\mathbf{J}_P} (\\hat{p}^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = \\frac{d}{d \\mathbf{J}_P} ((e^{\\mathbf{J}_P})^{\\mathbf{J}_P})^{\\mathbf{J}_P}", "derivation": "\\hat{p}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\hat{p}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = (e^{\\mathbf{J}_P})^{\\mathbf{J}_P} and (\\hat{p}^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = ((e^{\\mathbf{J}_P})^{\\mathbf{J}_P})^{\\mathbf{J}_P} and \\frac{d}{d \\mathbf{J}_P} (\\hat{p}^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = \\frac{d}{d \\mathbf{J}_P} ((e^{\\mathbf{J}_P})^{\\mathbf{J}_P})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Pow(Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(\\dot{y})} = \\cos{(\\log{(\\dot{y})})}, then obtain \\int (x{(\\dot{y})} + 2 \\int x{(\\dot{y})} d\\dot{y}) d\\dot{y} = \\int (x{(\\dot{y})} + \\int x{(\\dot{y})} d\\dot{y} + \\int \\cos{(\\log{(\\dot{y})})} d\\dot{y}) d\\dot{y}", "derivation": "x{(\\dot{y})} = \\cos{(\\log{(\\dot{y})})} and \\int x{(\\dot{y})} d\\dot{y} = \\int \\cos{(\\log{(\\dot{y})})} d\\dot{y} and 2 \\int x{(\\dot{y})} d\\dot{y} = \\int x{(\\dot{y})} d\\dot{y} + \\int \\cos{(\\log{(\\dot{y})})} d\\dot{y} and x{(\\dot{y})} + 2 \\int x{(\\dot{y})} d\\dot{y} = x{(\\dot{y})} + \\int x{(\\dot{y})} d\\dot{y} + \\int \\cos{(\\log{(\\dot{y})})} d\\dot{y} and \\int (x{(\\dot{y})} + 2 \\int x{(\\dot{y})} d\\dot{y}) d\\dot{y} = \\int (x{(\\dot{y})} + \\int x{(\\dot{y})} d\\dot{y} + \\int \\cos{(\\log{(\\dot{y})})} d\\dot{y}) d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\dot{y}', commutative=True)), cos(log(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 2, "Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Add(Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 3, "Function('x')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))), Add(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Integral(Function('x')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(log(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(P_{g},\\mathbf{A})} = - \\mathbf{A} + e^{P_{g}}, then obtain - (\\hat{\\mathbf{r}}^{\\mathbf{A}}{(P_{g},\\mathbf{A})})^{P_{g}} + \\hat{\\mathbf{r}}{(P_{g},\\mathbf{A})} = - \\mathbf{A} - (\\hat{\\mathbf{r}}^{\\mathbf{A}}{(P_{g},\\mathbf{A})})^{P_{g}} + e^{P_{g}}", "derivation": "\\hat{\\mathbf{r}}{(P_{g},\\mathbf{A})} = - \\mathbf{A} + e^{P_{g}} and \\hat{\\mathbf{r}}^{\\mathbf{A}}{(P_{g},\\mathbf{A})} = (- \\mathbf{A} + e^{P_{g}})^{\\mathbf{A}} and - ((- \\mathbf{A} + e^{P_{g}})^{\\mathbf{A}})^{P_{g}} + \\hat{\\mathbf{r}}{(P_{g},\\mathbf{A})} = - \\mathbf{A} - ((- \\mathbf{A} + e^{P_{g}})^{\\mathbf{A}})^{P_{g}} + e^{P_{g}} and - (\\hat{\\mathbf{r}}^{\\mathbf{A}}{(P_{g},\\mathbf{A})})^{P_{g}} + \\hat{\\mathbf{r}}{(P_{g},\\mathbf{A})} = - \\mathbf{A} - (\\hat{\\mathbf{r}}^{\\mathbf{A}}{(P_{g},\\mathbf{A})})^{P_{g}} + e^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('P_g', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('P_g', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('P_g', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('P_g', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('P_g', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('P_g', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('P_g', commutative=True))), exp(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('P_g', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('P_g', commutative=True))), exp(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given i{(\\hat{p},A)} = A - \\hat{p}, then obtain A - \\hat{p} + i{(\\hat{p},A)} + i^{A}{(\\hat{p},A)} = (A - \\hat{p})^{A} + 2 i{(\\hat{p},A)}", "derivation": "i{(\\hat{p},A)} = A - \\hat{p} and i^{A}{(\\hat{p},A)} = (A - \\hat{p})^{A} and A - \\hat{p} + i^{A}{(\\hat{p},A)} = A - \\hat{p} + (A - \\hat{p})^{A} and A - \\hat{p} + (A - \\hat{p})^{A} + i{(\\hat{p},A)} = 2 A - 2 \\hat{p} + (A - \\hat{p})^{A} and A - \\hat{p} + i{(\\hat{p},A)} + i^{A}{(\\hat{p},A)} = 2 A - 2 \\hat{p} + (A - \\hat{p})^{A} and 2 i{(\\hat{p},A)} + i^{A}{(\\hat{p},A)} = 2 A - 2 \\hat{p} + i^{A}{(\\hat{p},A)} and (A - \\hat{p})^{A} + 2 i{(\\hat{p},A)} = 2 A - 2 \\hat{p} + (A - \\hat{p})^{A} and A - \\hat{p} + i{(\\hat{p},A)} + i^{A}{(\\hat{p},A)} = (A - \\hat{p})^{A} + 2 i{(\\hat{p},A)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True)))"], [["add", 2, "Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True))))"], [["add", 1, "Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True)))"], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True)), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(2), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))), Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(2), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Pow(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Add(Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(2), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\delta)} = \\sin{(\\delta)} and \\operatorname{M_{E}}{(\\delta)} = - \\operatorname{g_{\\varepsilon}}{(\\delta)}, then obtain 0 = \\operatorname{M_{E}}{(\\delta)} + \\sin{(\\delta)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\delta)} = \\sin{(\\delta)} and 0 = - \\operatorname{g_{\\varepsilon}}{(\\delta)} + \\sin{(\\delta)} and \\operatorname{M_{E}}{(\\delta)} = - \\operatorname{g_{\\varepsilon}}{(\\delta)} and 0 = \\operatorname{M_{E}}{(\\delta)} + \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True))), sin(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('M_E')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}}, then derive \\hat{H}_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F})} = (e^{\\mathbf{F}})^{\\mathbf{F}}, then obtain (e^{\\mathbf{F}})^{\\mathbf{F}} - (\\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}})^{\\mathbf{F}} = 0", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}} and \\hat{H}_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F})} = (\\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}})^{\\mathbf{F}} and \\hat{H}_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F})} = (e^{\\mathbf{F}})^{\\mathbf{F}} and \\hat{H}_{\\lambda}^{\\mathbf{F}}{(\\mathbf{F})} - (\\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}})^{\\mathbf{F}} = 0 and (e^{\\mathbf{F}})^{\\mathbf{F}} - (\\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}})^{\\mathbf{F}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 2, "Pow(Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mu_0)} = \\sin{(\\mu_0)}, then obtain \\frac{\\int (- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)} - 1) d\\mu_0}{- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)}} = \\frac{\\int (- \\mu_0 + \\sin{(\\mu_0)} - 1) d\\mu_0}{- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)}}", "derivation": "\\operatorname{F_{N}}{(\\mu_0)} = \\sin{(\\mu_0)} and - \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)} = - \\mu_0 + \\sin{(\\mu_0)} and - \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)} - 1 = - \\mu_0 + \\sin{(\\mu_0)} - 1 and \\int (- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)} - 1) d\\mu_0 = \\int (- \\mu_0 + \\sin{(\\mu_0)} - 1) d\\mu_0 and \\frac{\\int (- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)} - 1) d\\mu_0}{- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)}} = \\frac{\\int (- \\mu_0 + \\sin{(\\mu_0)} - 1) d\\mu_0}{- \\mu_0 + \\operatorname{F_{N}}{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('F_N')(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(B,l)} = l \\log{(B)} and \\varepsilon_{0}{(B,l)} = \\frac{\\hat{p}{(B,l)}}{\\log{(B)}}, then derive \\int (l + \\varepsilon_{0}{(B,l)}) dl = Q + \\frac{\\int l \\log{(B)} dl + \\int \\hat{p}{(B,l)} dl}{\\log{(B)}}, then obtain \\int (Q + \\frac{2 \\int l \\log{(B)} dl}{\\log{(B)}}) dl = \\iint (l + \\frac{\\hat{p}{(B,l)}}{\\log{(B)}}) dl dl", "derivation": "\\hat{p}{(B,l)} = l \\log{(B)} and \\varepsilon_{0}{(B,l)} = \\frac{\\hat{p}{(B,l)}}{\\log{(B)}} and l + \\varepsilon_{0}{(B,l)} = l + \\frac{\\hat{p}{(B,l)}}{\\log{(B)}} and \\int (l + \\varepsilon_{0}{(B,l)}) dl = \\int (l + \\frac{\\hat{p}{(B,l)}}{\\log{(B)}}) dl and \\int (l + \\varepsilon_{0}{(B,l)}) dl = Q + \\frac{\\int l \\log{(B)} dl + \\int \\hat{p}{(B,l)} dl}{\\log{(B)}} and \\int (l + \\varepsilon_{0}{(B,l)}) dl = Q + \\frac{2 \\int l \\log{(B)} dl}{\\log{(B)}} and Q + \\frac{2 \\int l \\log{(B)} dl}{\\log{(B)}} = \\int (l + \\frac{\\hat{p}{(B,l)}}{\\log{(B)}}) dl and \\int (Q + \\frac{2 \\int l \\log{(B)} dl}{\\log{(B)}}) dl = \\iint (l + \\frac{\\hat{p}{(B,l)}}{\\log{(B)}}) dl dl", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), log(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Mul(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('B', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('\\\\varepsilon_0')(Symbol('B', commutative=True), Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('B', commutative=True)), Integer(-1)))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Function('\\\\varepsilon_0')(Symbol('B', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('B', commutative=True)), Integer(-1)))), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Symbol('l', commutative=True), Function('\\\\varepsilon_0')(Symbol('B', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Add(Integral(Mul(Symbol('l', commutative=True), log(Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Pow(log(Symbol('B', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Symbol('l', commutative=True), Function('\\\\varepsilon_0')(Symbol('B', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Integer(2), Pow(log(Symbol('B', commutative=True)), Integer(-1)), Integral(Mul(Symbol('l', commutative=True), log(Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('Q', commutative=True), Mul(Integer(2), Pow(log(Symbol('B', commutative=True)), Integer(-1)), Integral(Mul(Symbol('l', commutative=True), log(Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True))))), Integral(Add(Symbol('l', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('B', commutative=True)), Integer(-1)))), Tuple(Symbol('l', commutative=True))))"], [["integrate", 7, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Mul(Integer(2), Pow(log(Symbol('B', commutative=True)), Integer(-1)), Integral(Mul(Symbol('l', commutative=True), log(Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('B', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('B', commutative=True)), Integer(-1)))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\varphi^*)} = e^{\\varphi^*} and \\operatorname{M_{E}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*}, then derive \\frac{d}{d \\varphi^*} \\theta_{2}{(\\varphi^*)} = e^{\\varphi^*}, then obtain \\operatorname{M_{E}}^{2}{(\\varphi^*)} = \\operatorname{M_{E}}{(\\varphi^*)} \\frac{d^{2}}{d (\\varphi^*)^{2}} \\theta_{2}{(\\varphi^*)}", "derivation": "\\theta_{2}{(\\varphi^*)} = e^{\\varphi^*} and \\frac{d}{d \\varphi^*} \\theta_{2}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*} and \\operatorname{M_{E}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*} and \\frac{d}{d \\varphi^*} \\theta_{2}{(\\varphi^*)} = e^{\\varphi^*} and \\frac{d^{2}}{d (\\varphi^*)^{2}} \\theta_{2}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*} and \\operatorname{M_{E}}{(\\varphi^*)} = \\frac{d^{2}}{d (\\varphi^*)^{2}} \\theta_{2}{(\\varphi^*)} and \\operatorname{M_{E}}^{2}{(\\varphi^*)} = \\operatorname{M_{E}}{(\\varphi^*)} \\frac{d^{2}}{d (\\varphi^*)^{2}} \\theta_{2}{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))))"], [["times", 6, "Function('M_E')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Mul(Function('M_E')(Symbol('\\\\varphi^*', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\pi{(\\theta)} = \\sin{(\\theta)}, then obtain \\int \\frac{\\sin^{12}{(\\theta)}}{\\pi^{11}{(\\theta)}} d\\theta = \\int \\pi{(\\theta)} d\\theta", "derivation": "\\pi{(\\theta)} = \\sin{(\\theta)} and 1 = \\frac{\\sin{(\\theta)}}{\\pi{(\\theta)}} and \\sin{(\\theta)} = \\frac{\\sin^{2}{(\\theta)}}{\\pi{(\\theta)}} and \\pi{(\\theta)} = \\frac{\\sin^{2}{(\\theta)}}{\\pi{(\\theta)}} and \\pi{(\\theta)} = \\frac{\\sin^{4}{(\\theta)}}{\\pi^{3}{(\\theta)}} and \\frac{\\sin^{4}{(\\theta)}}{\\pi^{3}{(\\theta)}} = \\frac{\\pi^{3}{(\\theta)}}{\\sin^{2}{(\\theta)}} and \\frac{\\sin^{6}{(\\theta)}}{\\pi^{5}{(\\theta)}} = \\pi{(\\theta)} and \\frac{\\sin^{12}{(\\theta)}}{\\pi^{11}{(\\theta)}} = \\pi{(\\theta)} and \\int \\frac{\\sin^{12}{(\\theta)}}{\\pi^{11}{(\\theta)}} d\\theta = \\int \\pi{(\\theta)} d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["divide", 1, "Function('\\\\pi')(Symbol('\\\\theta', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-1)), sin(Symbol('\\\\theta', commutative=True))))"], [["times", 2, "sin(Symbol('\\\\theta', commutative=True))"], "Equality(sin(Symbol('\\\\theta', commutative=True)), Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-3)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(4))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-3)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(4))), Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(3)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(-2))))"], [["times", 6, "Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-5)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(6))), Function('\\\\pi')(Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-11)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(12))), Function('\\\\pi')(Symbol('\\\\theta', commutative=True)))"], [["integrate", 8, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Integer(-11)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(12))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given c{(F_{c})} = F_{c}, then obtain (a^{\\dagger} + \\frac{c^{2}{(F_{c})}}{2})^{c{(F_{c})}} = (\\int F_{c} dc{(F_{c})})^{c{(F_{c})}}", "derivation": "c{(F_{c})} = F_{c} and \\int c{(F_{c})} dF_{c} = \\int F_{c} dF_{c} and (\\int c{(F_{c})} dF_{c})^{F_{c}} = (\\int F_{c} dF_{c})^{F_{c}} and (\\int c{(F_{c})} dc{(F_{c})})^{c{(F_{c})}} = (\\int F_{c} dc{(F_{c})})^{c{(F_{c})}} and (a^{\\dagger} + \\frac{c^{2}{(F_{c})}}{2})^{c{(F_{c})}} = (\\int F_{c} dc{(F_{c})})^{c{(F_{c})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('c')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True))))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integral(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Integral(Function('c')(Symbol('F_c', commutative=True)), Tuple(Function('c')(Symbol('F_c', commutative=True)))), Function('c')(Symbol('F_c', commutative=True))), Pow(Integral(Symbol('F_c', commutative=True), Tuple(Function('c')(Symbol('F_c', commutative=True)))), Function('c')(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Function('c')(Symbol('F_c', commutative=True)), Integer(2)))), Function('c')(Symbol('F_c', commutative=True))), Pow(Integral(Symbol('F_c', commutative=True), Tuple(Function('c')(Symbol('F_c', commutative=True)))), Function('c')(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{B})} = \\log{(\\cos{(\\mathbf{B})})}, then obtain - \\frac{0^{\\mathbf{B}}}{\\hat{\\mathbf{r}}^{2}{(\\mathbf{B})}} = - \\frac{(- \\hat{\\mathbf{r}}^{2}{(\\mathbf{B})} + \\hat{\\mathbf{r}}{(\\mathbf{B})} \\log{(\\cos{(\\mathbf{B})})})^{\\mathbf{B}}}{\\hat{\\mathbf{r}}^{2}{(\\mathbf{B})}}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{B})} = \\log{(\\cos{(\\mathbf{B})})} and \\hat{\\mathbf{r}}^{2}{(\\mathbf{B})} = \\hat{\\mathbf{r}}{(\\mathbf{B})} \\log{(\\cos{(\\mathbf{B})})} and 0 = - \\hat{\\mathbf{r}}^{2}{(\\mathbf{B})} + \\hat{\\mathbf{r}}{(\\mathbf{B})} \\log{(\\cos{(\\mathbf{B})})} and 0^{\\mathbf{B}} = (- \\hat{\\mathbf{r}}^{2}{(\\mathbf{B})} + \\hat{\\mathbf{r}}{(\\mathbf{B})} \\log{(\\cos{(\\mathbf{B})})})^{\\mathbf{B}} and - \\frac{0^{\\mathbf{B}}}{\\hat{\\mathbf{r}}^{2}{(\\mathbf{B})}} = - \\frac{(- \\hat{\\mathbf{r}}^{2}{(\\mathbf{B})} + \\hat{\\mathbf{r}}{(\\mathbf{B})} \\log{(\\cos{(\\mathbf{B})})})^{\\mathbf{B}}}{\\hat{\\mathbf{r}}^{2}{(\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(cos(Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(cos(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["minus", 2, "Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(cos(Symbol('\\\\mathbf{B}', commutative=True))))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(cos(Symbol('\\\\mathbf{B}', commutative=True))))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))), Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(cos(Symbol('\\\\mathbf{B}', commutative=True))))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\bar{\\h}{(A_{1},g)} = e^{A_{1} + g} and \\mathbf{v}{(A_{1},g)} = - \\bar{\\h}{(A_{1},g)}, then obtain - g \\bar{\\h}{(A_{1},g)} + \\mathbf{v}^{g}{(A_{1},g)} - \\sin{(g)} = - g \\bar{\\h}{(A_{1},g)} + (- \\bar{\\h}{(A_{1},g)})^{g} - \\sin{(g)}", "derivation": "\\bar{\\h}{(A_{1},g)} = e^{A_{1} + g} and \\mathbf{v}{(A_{1},g)} = - \\bar{\\h}{(A_{1},g)} and \\mathbf{v}^{g}{(A_{1},g)} = (- \\bar{\\h}{(A_{1},g)})^{g} and - g e^{A_{1} + g} + \\mathbf{v}^{g}{(A_{1},g)} = - g e^{A_{1} + g} + (- \\bar{\\h}{(A_{1},g)})^{g} and - g e^{A_{1} + g} + \\mathbf{v}^{g}{(A_{1},g)} - \\sin{(g)} = - g e^{A_{1} + g} + (- \\bar{\\h}{(A_{1},g)})^{g} - \\sin{(g)} and - g \\bar{\\h}{(A_{1},g)} + \\mathbf{v}^{g}{(A_{1},g)} - \\sin{(g)} = - g \\bar{\\h}{(A_{1},g)} + (- \\bar{\\h}{(A_{1},g)})^{g} - \\sin{(g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["minus", 3, "Mul(Symbol('g', commutative=True), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True)))), Pow(Function('\\\\mathbf{v}')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True)))), Pow(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True))))"], [["add", 4, "Mul(Integer(-1), sin(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True)))), Pow(Function('\\\\mathbf{v}')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Symbol('g', commutative=True), exp(Add(Symbol('A_1', commutative=True), Symbol('g', commutative=True)))), Pow(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Pow(Function('\\\\mathbf{v}')(Symbol('A_1', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Symbol('g', commutative=True), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Pow(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_1', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(Z)} = e^{Z}, then obtain \\frac{d}{d Z} e^{- Z} \\int Z \\hat{p}{(Z)} dZ = \\frac{d}{d Z} e^{- Z} \\int Z e^{Z} dZ", "derivation": "\\hat{p}{(Z)} = e^{Z} and Z \\hat{p}{(Z)} = Z e^{Z} and \\int Z \\hat{p}{(Z)} dZ = \\int Z e^{Z} dZ and e^{- Z} \\int Z \\hat{p}{(Z)} dZ = e^{- Z} \\int Z e^{Z} dZ and \\frac{d}{d Z} e^{- Z} \\int Z \\hat{p}{(Z)} dZ = \\frac{d}{d Z} e^{- Z} \\int Z e^{Z} dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["times", 1, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Function('\\\\hat{p}')(Symbol('Z', commutative=True))), Mul(Symbol('Z', commutative=True), exp(Symbol('Z', commutative=True))))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('\\\\hat{p}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["divide", 3, "exp(Symbol('Z', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Function('\\\\hat{p}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Function('\\\\hat{p}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then obtain \\int (- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}) d\\mathbf{g} = \\int \\frac{- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} d\\mathbf{g}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and 0 = - \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})} and 0 = \\frac{- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} and \\int 0 d\\mathbf{g} = \\int (- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}) d\\mathbf{g} and \\int 0 d\\mathbf{g} = \\int \\frac{- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} d\\mathbf{g} and \\int (- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}) d\\mathbf{g} = \\int \\frac{- \\operatorname{c_{0}}{(\\mathbf{g})} + \\sin{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True))), sin(Symbol('\\\\mathbf{g}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(z^{*},z)} = - z^{*} + \\log{(z)}, then derive \\frac{\\partial}{\\partial z^{*}} \\hat{p}{(z^{*},z)} = -1, then obtain (\\frac{\\partial}{\\partial z^{*}} \\hat{p}{(z^{*},z)})^{z^{*}} = (-1)^{z^{*}}", "derivation": "\\hat{p}{(z^{*},z)} = - z^{*} + \\log{(z)} and \\frac{\\partial}{\\partial z^{*}} \\hat{p}{(z^{*},z)} = \\frac{\\partial}{\\partial z^{*}} (- z^{*} + \\log{(z)}) and \\frac{\\partial}{\\partial z^{*}} \\hat{p}{(z^{*},z)} = -1 and (\\frac{\\partial}{\\partial z^{*}} \\hat{p}{(z^{*},z)})^{z^{*}} = (-1)^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('z^*', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('z^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('z', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('z^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(-1))"], [["power", 3, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{p}')(Symbol('z^*', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Integer(-1), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\dot{z})} = \\cos{(\\dot{z})}, then derive \\int (\\dot{z} + \\operatorname{C_{2}}{(\\dot{z})}) d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\mathbf{H} + \\sin{(\\dot{z})}, then obtain \\int (\\dot{z} + \\cos{(\\dot{z})}) d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\mathbf{H} + \\sin{(\\dot{z})}", "derivation": "\\operatorname{C_{2}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\dot{z} + \\operatorname{C_{2}}{(\\dot{z})} = \\dot{z} + \\cos{(\\dot{z})} and \\int (\\dot{z} + \\operatorname{C_{2}}{(\\dot{z})}) d\\dot{z} = \\int (\\dot{z} + \\cos{(\\dot{z})}) d\\dot{z} and \\int (\\dot{z} + \\operatorname{C_{2}}{(\\dot{z})}) d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\mathbf{H} + \\sin{(\\dot{z})} and \\int (\\dot{z} + \\cos{(\\dot{z})}) d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\mathbf{H} + \\sin{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('C_2')(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\dot{z}', commutative=True), Function('C_2')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\dot{z}', commutative=True), Function('C_2')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given L{(\\Omega,T)} = T - \\Omega, then obtain T - \\Omega + L{(\\Omega,T)} + 2 e^{L{(\\Omega,T)}} + \\int (T - \\Omega) dT = 2 T - 2 \\Omega + 2 e^{L{(\\Omega,T)}} + \\int (T - \\Omega) dT", "derivation": "L{(\\Omega,T)} = T - \\Omega and L{(\\Omega,T)} + e^{L{(\\Omega,T)}} = T - \\Omega + e^{L{(\\Omega,T)}} and T - \\Omega + L{(\\Omega,T)} + 2 e^{L{(\\Omega,T)}} = 2 T - 2 \\Omega + 2 e^{L{(\\Omega,T)}} and T - \\Omega + L{(\\Omega,T)} + 2 e^{L{(\\Omega,T)}} + \\int (T - \\Omega) dT = 2 T - 2 \\Omega + 2 e^{L{(\\Omega,T)}} + \\int (T - \\Omega) dT", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True))))"], "Equality(Add(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)))))"], [["add", 2, "Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True))))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)), Mul(Integer(2), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True))))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True))))))"], [["add", 3, "Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)), Mul(Integer(2), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)))), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), exp(Function('L')(Symbol('\\\\Omega', commutative=True), Symbol('T', commutative=True)))), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)}, then derive \\hat{x}{(\\varphi^*)} + \\sin{(\\varphi^*)} = 0, then obtain \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = 0", "derivation": "\\hat{x}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} and \\hat{x}{(\\varphi^*)} - \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = 0 and \\hat{x}{(\\varphi^*)} + \\sin{(\\varphi^*)} = 0 and \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True)), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(sin(Symbol('\\\\varphi^*', commutative=True)), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\cos{(\\Psi_{\\lambda} - \\mathbf{B})}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\sin{(\\Psi_{\\lambda} - \\mathbf{B})}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{B}^{2}} \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\sin{(\\Psi_{\\lambda} - \\mathbf{B})}", "derivation": "\\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\cos{(\\Psi_{\\lambda} - \\mathbf{B})} and \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} - 1 = \\cos{(\\Psi_{\\lambda} - \\mathbf{B})} - 1 and \\frac{\\partial}{\\partial \\mathbf{B}} (\\Psi{(\\Psi_{\\lambda},\\mathbf{B})} - 1) = \\frac{\\partial}{\\partial \\mathbf{B}} (\\cos{(\\Psi_{\\lambda} - \\mathbf{B})} - 1) and \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\sin{(\\Psi_{\\lambda} - \\mathbf{B})} and \\frac{\\partial^{2}}{\\partial \\mathbf{B}^{2}} \\Psi{(\\Psi_{\\lambda},\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\sin{(\\Psi_{\\lambda} - \\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\Psi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Add(cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(cos(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Derivative(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{r})} = \\log{(e^{\\mathbf{r}})}, then obtain \\int \\cos{(\\frac{d}{d \\mathbf{r}} \\dot{y}{(\\mathbf{r})})} d\\mathbf{r} = \\int \\cos{(\\frac{d}{d \\mathbf{r}} \\log{(e^{\\mathbf{r}})})} d\\mathbf{r}", "derivation": "\\dot{y}{(\\mathbf{r})} = \\log{(e^{\\mathbf{r}})} and \\frac{d}{d \\mathbf{r}} \\dot{y}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\log{(e^{\\mathbf{r}})} and \\cos{(\\frac{d}{d \\mathbf{r}} \\dot{y}{(\\mathbf{r})})} = \\cos{(\\frac{d}{d \\mathbf{r}} \\log{(e^{\\mathbf{r}})})} and \\int \\cos{(\\frac{d}{d \\mathbf{r}} \\dot{y}{(\\mathbf{r})})} d\\mathbf{r} = \\int \\cos{(\\frac{d}{d \\mathbf{r}} \\log{(e^{\\mathbf{r}})})} d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{r}', commutative=True)), log(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), cos(Derivative(log(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(cos(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(cos(Derivative(log(exp(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then derive \\frac{d}{d \\varepsilon_0} \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)}, then obtain \\int \\frac{d}{d \\varepsilon_0} \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\cos{(\\varepsilon_0)} d\\varepsilon_0", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\int \\frac{d}{d \\varepsilon_0} \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\cos{(\\varepsilon_0)} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(cos(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\nabla{(x^\\prime,l)} = - l + x^\\prime, then obtain \\frac{(l + \\nabla{(x^\\prime,l)})^{x^\\prime} + \\nabla{(x^\\prime,l)}}{l + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}} = \\frac{- l + x^\\prime + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}}{l + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}}", "derivation": "\\nabla{(x^\\prime,l)} = - l + x^\\prime and l + \\nabla{(x^\\prime,l)} = x^\\prime and (l + \\nabla{(x^\\prime,l)})^{x^\\prime} = (x^\\prime)^{x^\\prime} and (x^\\prime)^{x^\\prime} + \\nabla{(x^\\prime,l)} = - l + x^\\prime + (x^\\prime)^{x^\\prime} and (l + \\nabla{(x^\\prime,l)})^{x^\\prime} + \\nabla{(x^\\prime,l)} = - l + x^\\prime + (l + \\nabla{(x^\\prime,l)})^{x^\\prime} and \\frac{(l + \\nabla{(x^\\prime,l)})^{x^\\prime} + \\nabla{(x^\\prime,l)}}{l + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}} = \\frac{- l + x^\\prime + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}}{l + (l + \\nabla{(x^\\prime,l)})^{x^\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('x^\\\\prime', commutative=True), Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True))))"], [["divide", 5, "Add(Symbol('l', commutative=True), Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('l', commutative=True), Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Add(Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True)))), Mul(Pow(Add(Symbol('l', commutative=True), Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('x^\\\\prime', commutative=True), Pow(Add(Symbol('l', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True), Symbol('l', commutative=True))), Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given B{(C_{d})} = \\int \\cos{(C_{d})} dC_{d}, then obtain - W \\mathbf{A} - ((\\int \\cos{(C_{d})} dC_{d})^{C_{d}})^{C_{d}} + B{(C_{d})} = - W \\mathbf{A} - ((\\int \\cos{(C_{d})} dC_{d})^{C_{d}})^{C_{d}} + \\int \\cos{(C_{d})} dC_{d}", "derivation": "B{(C_{d})} = \\int \\cos{(C_{d})} dC_{d} and B^{C_{d}}{(C_{d})} = (\\int \\cos{(C_{d})} dC_{d})^{C_{d}} and (B^{C_{d}}{(C_{d})})^{C_{d}} = ((\\int \\cos{(C_{d})} dC_{d})^{C_{d}})^{C_{d}} and - W \\mathbf{A} - (B^{C_{d}}{(C_{d})})^{C_{d}} + B{(C_{d})} = - W \\mathbf{A} - (B^{C_{d}}{(C_{d})})^{C_{d}} + \\int \\cos{(C_{d})} dC_{d} and - W \\mathbf{A} - ((\\int \\cos{(C_{d})} dC_{d})^{C_{d}})^{C_{d}} + B{(C_{d})} = - W \\mathbf{A} - ((\\int \\cos{(C_{d})} dC_{d})^{C_{d}})^{C_{d}} + \\int \\cos{(C_{d})} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_d', commutative=True)), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('B')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Pow(Function('B')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Pow(Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], [["minus", 1, "Add(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Pow(Function('B')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('B')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Function('B')(Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('B')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Function('B')(Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Pow(Pow(Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(T)} = \\frac{d}{d T} \\cos{(T)}, then derive \\dot{y}{(T)} = - \\sin{(T)}, then obtain (\\frac{\\frac{d}{d T} \\cos{(T)}}{\\dot{y}{(T)}})^{T} + 1 = (\\frac{\\frac{d}{d T} \\cos{(T)}}{\\dot{y}{(T)}})^{T} - \\frac{\\sin{(T)}}{\\frac{d}{d T} \\cos{(T)}}", "derivation": "\\dot{y}{(T)} = \\frac{d}{d T} \\cos{(T)} and \\dot{y}{(T)} = - \\sin{(T)} and - \\sin{(T)} = \\frac{d}{d T} \\cos{(T)} and 1 = \\frac{\\frac{d}{d T} \\cos{(T)}}{\\dot{y}{(T)}} and 1 = - \\frac{\\sin{(T)}}{\\dot{y}{(T)}} and 1 = - \\frac{\\sin{(T)}}{\\frac{d}{d T} \\cos{(T)}} and (\\frac{\\frac{d}{d T} \\cos{(T)}}{\\dot{y}{(T)}})^{T} + 1 = (\\frac{\\frac{d}{d T} \\cos{(T)}}{\\dot{y}{(T)}})^{T} - \\frac{\\sin{(T)}}{\\frac{d}{d T} \\cos{(T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('T', commutative=True))), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('T', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Mul(Integer(-1), sin(Symbol('T', commutative=True)), Pow(Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))))"], [["add", 6, "Pow(Mul(Pow(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)), Integer(1)), Add(Pow(Mul(Pow(Function('\\\\dot{y}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)), Pow(Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given G{(S,A_{1},t_{2})} = A_{1} + S + t_{2}, then derive \\int G{(S,A_{1},t_{2})} dt_{2} = \\frac{t_{2}^{2}}{2} + t_{2} (A_{1} + S) + v_{x}, then obtain - \\frac{t_{2}^{2}}{2} + \\int (A_{1} + S + t_{2}) dt_{2} = t_{2} (A_{1} + S) + v_{x}", "derivation": "G{(S,A_{1},t_{2})} = A_{1} + S + t_{2} and \\int G{(S,A_{1},t_{2})} dt_{2} = \\int (A_{1} + S + t_{2}) dt_{2} and \\int G{(S,A_{1},t_{2})} dt_{2} = \\frac{t_{2}^{2}}{2} + t_{2} (A_{1} + S) + v_{x} and \\int (A_{1} + S + t_{2}) dt_{2} = \\frac{t_{2}^{2}}{2} + t_{2} (A_{1} + S) + v_{x} and - \\frac{t_{2}^{2}}{2} + \\int (A_{1} + S + t_{2}) dt_{2} = t_{2} (A_{1} + S) + v_{x}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('S', commutative=True), Symbol('A_1', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True), Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('G')(Symbol('S', commutative=True), Symbol('A_1', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('S', commutative=True), Symbol('A_1', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Mul(Symbol('t_2', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True))), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Mul(Symbol('t_2', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True))), Symbol('v_x', commutative=True)))"], [["minus", 4, "Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Integral(Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Mul(Symbol('t_2', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('S', commutative=True))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given g{(v_{t})} = \\cos{(v_{t})}, then obtain 1 - \\frac{\\int \\cos^{v_{t}}{(v_{t})} dv_{t}}{\\int g^{v_{t}}{(v_{t})} dv_{t}} = 0", "derivation": "g{(v_{t})} = \\cos{(v_{t})} and g^{v_{t}}{(v_{t})} = \\cos^{v_{t}}{(v_{t})} and \\int g^{v_{t}}{(v_{t})} dv_{t} = \\int \\cos^{v_{t}}{(v_{t})} dv_{t} and 1 = \\frac{\\int \\cos^{v_{t}}{(v_{t})} dv_{t}}{\\int g^{v_{t}}{(v_{t})} dv_{t}} and 1 - \\frac{\\int \\cos^{v_{t}}{(v_{t})} dv_{t}}{\\int g^{v_{t}}{(v_{t})} dv_{t}} = 0", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Pow(cos(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["divide", 3, "Integral(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Integral(Pow(cos(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["minus", 4, "Mul(Pow(Integral(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Integral(Pow(cos(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Integral(Pow(Function('g')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Integral(Pow(cos(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Integer(0))"]]}, {"prompt": "Given C{(l,\\mathbf{r})} = \\frac{\\partial}{\\partial l} (\\mathbf{r} + l), then obtain \\frac{\\partial}{\\partial l} \\int C{(l,\\mathbf{r})} d\\mathbf{r} + 1 = \\frac{\\partial}{\\partial l} (\\mathbf{r} + y^{\\prime}) + 1", "derivation": "C{(l,\\mathbf{r})} = \\frac{\\partial}{\\partial l} (\\mathbf{r} + l) and \\int C{(l,\\mathbf{r})} d\\mathbf{r} = \\int \\frac{\\partial}{\\partial l} (\\mathbf{r} + l) d\\mathbf{r} and \\frac{\\partial}{\\partial l} \\int C{(l,\\mathbf{r})} d\\mathbf{r} = \\frac{\\partial}{\\partial l} \\int \\frac{\\partial}{\\partial l} (\\mathbf{r} + l) d\\mathbf{r} and \\frac{\\partial}{\\partial l} \\int C{(l,\\mathbf{r})} d\\mathbf{r} + 1 = \\frac{\\partial}{\\partial l} \\int \\frac{\\partial}{\\partial l} (\\mathbf{r} + l) d\\mathbf{r} + 1 and \\frac{\\partial}{\\partial l} \\int C{(l,\\mathbf{r})} d\\mathbf{r} + 1 = \\frac{\\partial}{\\partial l} (\\mathbf{r} + y^{\\prime}) + 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Integral(Function('C')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('C')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_integrals", 4], "Equality(Add(Derivative(Integral(Function('C')(Symbol('l', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\nabla,y^{\\prime},A)} = - A + \\nabla y^{\\prime}, then derive \\int \\operatorname{C_{d}}{(\\nabla,y^{\\prime},A)} d\\nabla = - A \\nabla + F_{c} + \\frac{\\nabla^{2} y^{\\prime}}{2}, then obtain \\int (- A + \\nabla y^{\\prime}) d\\nabla = - A \\nabla + F_{c} + \\frac{\\nabla^{2} y^{\\prime}}{2}", "derivation": "\\operatorname{C_{d}}{(\\nabla,y^{\\prime},A)} = - A + \\nabla y^{\\prime} and \\int \\operatorname{C_{d}}{(\\nabla,y^{\\prime},A)} d\\nabla = \\int (- A + \\nabla y^{\\prime}) d\\nabla and \\int \\operatorname{C_{d}}{(\\nabla,y^{\\prime},A)} d\\nabla = - A \\nabla + F_{c} + \\frac{\\nabla^{2} y^{\\prime}}{2} and \\int (- A + \\nabla y^{\\prime}) d\\nabla = - A \\nabla + F_{c} + \\frac{\\nabla^{2} y^{\\prime}}{2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_d')(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2)), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2)), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(B)} = e^{B}, then obtain 0 = \\int (- \\mathbf{A}{(B)} + e^{B}) dB - \\int (- \\int 0 dB + \\int (- \\mathbf{A}{(B)} + e^{B}) dB) dB", "derivation": "\\mathbf{A}{(B)} = e^{B} and 0 = - \\mathbf{A}{(B)} + e^{B} and \\int 0 dB = \\int (- \\mathbf{A}{(B)} + e^{B}) dB and 0 = - \\int 0 dB + \\int (- \\mathbf{A}{(B)} + e^{B}) dB and \\int 0 dB = \\int (- \\int 0 dB + \\int (- \\mathbf{A}{(B)} + e^{B}) dB) dB and 0 = \\int (- \\mathbf{A}{(B)} + e^{B}) dB - \\int (- \\int 0 dB + \\int (- \\mathbf{A}{(B)} + e^{B}) dB) dB", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{A}')(Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["minus", 3, "Integral(Integer(0), Tuple(Symbol('B', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('B', commutative=True)))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))))"], [["integrate", 4, "Symbol('B', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('B', commutative=True)))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(0), Add(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('B', commutative=True)))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given g{(\\sigma_p)} = \\log{(\\sigma_p)}, then derive \\frac{d}{d \\sigma_p} g{(\\sigma_p)} = \\frac{1}{\\sigma_p}, then obtain (- \\log{(\\sigma_p)} + \\frac{1}{\\sigma_p}) (- \\log{(\\sigma_p)} + \\frac{d}{d \\sigma_p} g{(\\sigma_p)}) = (- \\log{(\\sigma_p)} + \\frac{1}{\\sigma_p})^{2}", "derivation": "g{(\\sigma_p)} = \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} g{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} g{(\\sigma_p)} = \\frac{1}{\\sigma_p} and - \\log{(\\sigma_p)} + \\frac{d}{d \\sigma_p} g{(\\sigma_p)} = - \\log{(\\sigma_p)} + \\frac{1}{\\sigma_p} and (- \\log{(\\sigma_p)} + \\frac{1}{\\sigma_p}) (- \\log{(\\sigma_p)} + \\frac{d}{d \\sigma_p} g{(\\sigma_p)}) = (- \\log{(\\sigma_p)} + \\frac{1}{\\sigma_p})^{2}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["minus", 3, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Derivative(Function('g')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], [["times", 4, "Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], "Equality(Mul(Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Derivative(Function('g')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))), Pow(Add(Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Integer(2)))"]]}, {"prompt": "Given p{(H,m_{s},\\mathbf{B})} = H^{m_{s}} \\mathbf{B} and T{(H,m_{s},\\mathbf{B})} = H^{m_{s}} \\mathbf{B}, then obtain \\mathbf{B} (- m_{s} + T{(H,m_{s},\\mathbf{B})}) = \\mathbf{B} (H^{m_{s}} \\mathbf{B} - m_{s})", "derivation": "p{(H,m_{s},\\mathbf{B})} = H^{m_{s}} \\mathbf{B} and T{(H,m_{s},\\mathbf{B})} = H^{m_{s}} \\mathbf{B} and - m_{s} + T{(H,m_{s},\\mathbf{B})} = H^{m_{s}} \\mathbf{B} - m_{s} and - m_{s} + T{(H,m_{s},\\mathbf{B})} = - m_{s} + p{(H,m_{s},\\mathbf{B})} and \\mathbf{B} (- m_{s} + T{(H,m_{s},\\mathbf{B})}) = \\mathbf{B} (- m_{s} + p{(H,m_{s},\\mathbf{B})}) and \\mathbf{B} (- m_{s} + T{(H,m_{s},\\mathbf{B})}) = \\mathbf{B} (H^{m_{s}} \\mathbf{B} - m_{s})", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('T')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Pow(Symbol('H', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('T')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('p')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('T')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('p')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('T')(Symbol('H', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Pow(Symbol('H', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(m)} = \\cos{(m)} and \\mathbf{S}{(m)} = \\cos{(m)}, then obtain (\\mathbf{S}^{m}{(m)})^{m} = (\\cos^{m}{(m)})^{m}", "derivation": "\\operatorname{F_{c}}{(m)} = \\cos{(m)} and \\mathbf{S}{(m)} = \\cos{(m)} and \\operatorname{F_{c}}{(m)} = \\mathbf{S}{(m)} and \\operatorname{F_{c}}^{m}{(m)} = \\cos^{m}{(m)} and \\mathbf{S}^{m}{(m)} = \\cos^{m}{(m)} and (\\mathbf{S}^{m}{(m)})^{m} = (\\cos^{m}{(m)})^{m}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_c')(Symbol('m', commutative=True)), Function('\\\\mathbf{S}')(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["power", 5, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{S}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and u{(\\mathbf{A})} = - \\operatorname{E_{x}}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}, then obtain \\frac{d}{d \\mathbf{A}} 0 = \\frac{d}{d \\mathbf{A}} - u{(\\mathbf{A})}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and 0 = - \\operatorname{E_{x}}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and u{(\\mathbf{A})} = - \\operatorname{E_{x}}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and 0 = u{(\\mathbf{A})} and 0 = - u{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} 0 = \\frac{d}{d \\mathbf{A}} - u{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Function('E_x')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Function('u')(Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Integer(0), Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(\\psi)} = \\psi and p{(\\psi)} = \\iint \\mu_{0}^{\\psi}{(\\psi)} d\\psi d\\psi, then obtain - c + p{(\\psi)} + 1 = - c + \\iint \\mu_{0}^{\\psi}{(\\psi)} d\\psi d\\psi + 1", "derivation": "\\mu_{0}{(\\psi)} = \\psi and \\mu_{0}^{\\psi}{(\\psi)} = \\psi^{\\psi} and \\int \\mu_{0}^{\\psi}{(\\psi)} d\\psi = \\int \\psi^{\\psi} d\\psi and \\iint \\mu_{0}^{\\psi}{(\\psi)} d\\psi d\\psi = \\iint \\psi^{\\psi} d\\psi d\\psi and p{(\\psi)} = \\iint \\mu_{0}^{\\psi}{(\\psi)} d\\psi d\\psi and p{(\\psi)} = \\iint \\psi^{\\psi} d\\psi d\\psi and - c + p{(\\psi)} = - c + \\iint \\psi^{\\psi} d\\psi d\\psi and - c + p{(\\psi)} + 1 = - c + \\iint \\psi^{\\psi} d\\psi d\\psi + 1 and - c + p{(\\psi)} + 1 = - c + \\iint \\mu_{0}^{\\psi}{(\\psi)} d\\psi d\\psi + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\psi', commutative=True)), Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('p')(Symbol('\\\\psi', commutative=True)), Integral(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["minus", 6, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["minus", 7, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 8, 4], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\dot{x}{(A_{1})} = e^{A_{1}}, then derive \\int \\dot{x}{(A_{1})} dA_{1} = F_{N} + e^{A_{1}}, then obtain Q + e^{A_{1}} = F_{N} + e^{A_{1}}", "derivation": "\\dot{x}{(A_{1})} = e^{A_{1}} and \\int \\dot{x}{(A_{1})} dA_{1} = \\int e^{A_{1}} dA_{1} and \\int \\dot{x}{(A_{1})} dA_{1} = F_{N} + e^{A_{1}} and \\int e^{A_{1}} dA_{1} = F_{N} + e^{A_{1}} and Q + e^{A_{1}} = F_{N} + e^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True)))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('Q', commutative=True), exp(Symbol('A_1', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given l{(\\chi)} = \\int \\cos{(\\chi)} d\\chi, then derive l{(\\chi)} = \\mathbf{s} + \\sin{(\\chi)}, then obtain \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}} d\\chi + 1 = \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}} d\\chi + \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}}", "derivation": "l{(\\chi)} = \\int \\cos{(\\chi)} d\\chi and 1 = \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} and l{(\\chi)} = \\mathbf{s} + \\sin{(\\chi)} and \\int 1 d\\chi = \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} d\\chi and \\int 1 d\\chi + 1 = \\int 1 d\\chi + \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} and \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} d\\chi + 1 = \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} d\\chi + \\frac{\\int \\cos{(\\chi)} d\\chi}{l{(\\chi)}} and \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}} d\\chi + 1 = \\int \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}} d\\chi + \\frac{\\int \\cos{(\\chi)} d\\chi}{\\mathbf{s} + \\sin{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\chi', commutative=True)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 1, "Function('l')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('l')(Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\chi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))), Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Integral(Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Mul(Pow(Function('l')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"]]}, {"prompt": "Given c{(\\psi,Q)} = Q \\psi and \\operatorname{x^{{\\}'}}{(\\psi,Q)} = (\\frac{\\partial}{\\partial Q} \\log{(c{(\\psi,Q)})}^{Q})^{Q}, then obtain \\operatorname{x^{{\\}'}}{(\\psi,Q)} = ((\\log{(\\log{(Q \\psi)})} + \\frac{1}{\\log{(Q \\psi)}}) \\log{(Q \\psi)}^{Q})^{Q}", "derivation": "c{(\\psi,Q)} = Q \\psi and \\log{(c{(\\psi,Q)})} = \\log{(Q \\psi)} and \\log{(c{(\\psi,Q)})}^{Q} = \\log{(Q \\psi)}^{Q} and \\frac{\\partial}{\\partial Q} \\log{(c{(\\psi,Q)})}^{Q} = \\frac{\\partial}{\\partial Q} \\log{(Q \\psi)}^{Q} and (\\frac{\\partial}{\\partial Q} \\log{(c{(\\psi,Q)})}^{Q})^{Q} = (\\frac{\\partial}{\\partial Q} \\log{(Q \\psi)}^{Q})^{Q} and \\operatorname{x^{{\\}'}}{(\\psi,Q)} = (\\frac{\\partial}{\\partial Q} \\log{(c{(\\psi,Q)})}^{Q})^{Q} and \\operatorname{x^{{\\}'}}{(\\psi,Q)} = (\\frac{\\partial}{\\partial Q} \\log{(Q \\psi)}^{Q})^{Q} and \\operatorname{x^{{\\}'}}{(\\psi,Q)} = ((\\log{(\\log{(Q \\psi)})} + \\frac{1}{\\log{(Q \\psi)}}) \\log{(Q \\psi)}^{Q})^{Q}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["log", 1], "Equality(log(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(log(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('Q', commutative=True)))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Pow(log(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Pow(log(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True)), Pow(Derivative(Pow(log(Function('c')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True)), Pow(Derivative(Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["evaluate_derivatives", 7], "Equality(Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True)), Pow(Mul(Add(log(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True)))), Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Integer(-1))), Pow(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(A_{z},\\phi_1)} = \\sin{(A_{z} + \\phi_1)} and \\operatorname{g_{\\varepsilon}}{(m,\\mathbf{H})} = \\mathbf{H} + m, then obtain - 2 (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1} + \\operatorname{g_{\\varepsilon}}{(m,\\mathbf{H})} = \\mathbf{H} + m - 2 (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1}", "derivation": "\\mathbf{S}{(A_{z},\\phi_1)} = \\sin{(A_{z} + \\phi_1)} and A_{z} \\mathbf{S}{(A_{z},\\phi_1)} = A_{z} \\sin{(A_{z} + \\phi_1)} and \\operatorname{g_{\\varepsilon}}{(m,\\mathbf{H})} = \\mathbf{H} + m and - (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1} - (A_{z} \\sin{(A_{z} + \\phi_1)})^{\\phi_1} + \\operatorname{g_{\\varepsilon}}{(m,\\mathbf{H})} = \\mathbf{H} + m - (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1} - (A_{z} \\sin{(A_{z} + \\phi_1)})^{\\phi_1} and - 2 (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1} + \\operatorname{g_{\\varepsilon}}{(m,\\mathbf{H})} = \\mathbf{H} + m - 2 (A_{z} \\mathbf{S}{(A_{z},\\phi_1)})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)), sin(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('A_z', commutative=True), sin(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], ["get_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('m', commutative=True)))"], [["minus", 3, "Add(Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), sin(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('A_z', commutative=True), sin(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('m', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('A_z', commutative=True), sin(Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('m', commutative=True), Mul(Integer(-1), Integer(2), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\dot{\\mathbf{r}},B)} = B - \\dot{\\mathbf{r}}, then obtain \\dot{\\mathbf{r}} + \\mathbf{H}{(\\dot{\\mathbf{r}},B)} = B", "derivation": "\\mathbf{H}{(\\dot{\\mathbf{r}},B)} = B - \\dot{\\mathbf{r}} and - \\mathbf{H}{(\\dot{\\mathbf{r}},B)} = - B + \\dot{\\mathbf{r}} and - \\dot{\\mathbf{r}} - \\mathbf{H}{(\\dot{\\mathbf{r}},B)} = - B and \\dot{\\mathbf{r}} + \\mathbf{H}{(\\dot{\\mathbf{r}},B)} = B", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["minus", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('B', commutative=True)))), Mul(Integer(-1), Symbol('B', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('B', commutative=True))), Symbol('B', commutative=True))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} = \\mathbf{A} - \\varphi and \\mathbf{E}{(\\mathbf{A},\\varphi)} = \\mathbf{A} - \\varphi - 1, then obtain \\mathbf{A} - 2 \\varphi - \\mathbf{E}{(\\mathbf{A},\\varphi)} - 2 = - \\varphi - \\mathbf{E}{(\\mathbf{A},\\varphi)} + \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} - 2", "derivation": "\\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} = \\mathbf{A} - \\varphi and \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} - 1 = \\mathbf{A} - \\varphi - 1 and \\mathbf{E}{(\\mathbf{A},\\varphi)} = \\mathbf{A} - \\varphi - 1 and \\mathbf{E}{(\\mathbf{A},\\varphi)} = \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} - 1 and - \\mathbf{A} + \\varphi + \\mathbf{E}{(\\mathbf{A},\\varphi)} + 1 = - \\mathbf{A} + \\varphi + \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} and 0 = - \\mathbf{A} + \\varphi + \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} and \\mathbf{A} - 2 \\varphi - \\mathbf{E}{(\\mathbf{A},\\varphi)} - 2 = - \\varphi - \\mathbf{E}{(\\mathbf{A},\\varphi)} + \\operatorname{v_{z}}{(\\mathbf{A},\\varphi)} - 2", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)))"], [["minus", 4, "Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2))"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-2)))"]]}, {"prompt": "Given \\ddot{x}{(I)} = \\int \\log{(I)} dI, then obtain - \\frac{\\ddot{x}{(I)}}{(\\ddot{x}{(I)} - \\log{(I)}) \\log{(I)}} = - \\frac{\\int \\log{(I)} dI}{(\\ddot{x}{(I)} - \\log{(I)}) \\log{(I)}}", "derivation": "\\ddot{x}{(I)} = \\int \\log{(I)} dI and \\ddot{x}{(I)} - \\log{(I)} = - \\log{(I)} + \\int \\log{(I)} dI and - \\frac{\\ddot{x}{(I)}}{\\log{(I)}} = - \\frac{\\int \\log{(I)} dI}{\\log{(I)}} and - \\frac{\\ddot{x}{(I)}}{(- \\log{(I)} + \\int \\log{(I)} dI) \\log{(I)}} = - \\frac{\\int \\log{(I)} dI}{(- \\log{(I)} + \\int \\log{(I)} dI) \\log{(I)}} and - \\frac{\\ddot{x}{(I)}}{(\\ddot{x}{(I)} - \\log{(I)}) \\log{(I)}} = - \\frac{\\int \\log{(I)} dI}{(\\ddot{x}{(I)} - \\log{(I)}) \\log{(I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["minus", 1, "log(Symbol('I', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Mul(Integer(-1), log(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["divide", 1, "Mul(Integer(-1), log(Symbol('I', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('I', commutative=True)), Integer(-1)), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), log(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), log(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integer(-1)), Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), log(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1)), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Add(Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Mul(Integer(-1), log(Symbol('I', commutative=True)))), Integer(-1)), Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Function('\\\\ddot{x}')(Symbol('I', commutative=True)), Mul(Integer(-1), log(Symbol('I', commutative=True)))), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1)), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given L{(\\theta,\\rho_b)} = \\rho_b - \\theta and t{(\\theta,\\rho_b)} = 2 \\rho_b - \\theta, then obtain \\rho_b + L{(\\theta,\\rho_b)} = t{(\\theta,\\rho_b)}", "derivation": "L{(\\theta,\\rho_b)} = \\rho_b - \\theta and \\rho_b + L{(\\theta,\\rho_b)} = 2 \\rho_b - \\theta and t{(\\theta,\\rho_b)} = 2 \\rho_b - \\theta and 1 = \\frac{2 \\rho_b - \\theta}{\\rho_b + L{(\\theta,\\rho_b)}} and 1 = \\frac{t{(\\theta,\\rho_b)}}{\\rho_b + L{(\\theta,\\rho_b)}} and \\rho_b + L{(\\theta,\\rho_b)} = t{(\\theta,\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Integer(-1)), Function('t')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["divide", 5, "Pow(Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Integer(-1))"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('L')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Function('t')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = - \\ddot{x} + \\eta + a, then derive \\frac{\\partial}{\\partial \\eta} \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = 1, then obtain \\frac{\\partial^{2}}{\\partial a\\partial \\eta} \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = \\frac{d}{d a} 1", "derivation": "\\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = - \\ddot{x} + \\eta + a and \\frac{\\partial}{\\partial \\eta} \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = \\frac{\\partial}{\\partial \\eta} (- \\ddot{x} + \\eta + a) and \\frac{\\partial}{\\partial \\eta} \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = 1 and \\frac{\\partial^{2}}{\\partial a\\partial \\eta} \\operatorname{n_{1}}{(\\eta,a,\\ddot{x})} = \\frac{d}{d a} 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\eta', commutative=True), Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(a,\\mathbf{D},v)} = (\\mathbf{D} + a)^{v} and \\Omega{(a,\\mathbf{D})} = \\mathbf{D} + a, then obtain - v + I{(a,\\mathbf{D},v)} - \\Omega{(a,\\mathbf{D})} = - v - \\Omega{(a,\\mathbf{D})} + \\Omega^{v}{(a,\\mathbf{D})}", "derivation": "I{(a,\\mathbf{D},v)} = (\\mathbf{D} + a)^{v} and - \\mathbf{D} - a + I{(a,\\mathbf{D},v)} = - \\mathbf{D} - a + (\\mathbf{D} + a)^{v} and - \\mathbf{D} - a - v + I{(a,\\mathbf{D},v)} = - \\mathbf{D} - a - v + (\\mathbf{D} + a)^{v} and \\Omega{(a,\\mathbf{D})} = \\mathbf{D} + a and - v + I{(a,\\mathbf{D},v)} - \\Omega{(a,\\mathbf{D})} = - v - \\Omega{(a,\\mathbf{D})} + \\Omega^{v}{(a,\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a', commutative=True)), Symbol('v', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a', commutative=True)), Symbol('v', commutative=True))))"], [["minus", 2, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a', commutative=True)), Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(I,g)} = g^{I}, then derive \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} = \\frac{I g^{I}}{g}, then obtain \\int \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} dI = \\int \\frac{I \\dot{z}{(I,g)}}{g} dI", "derivation": "\\dot{z}{(I,g)} = g^{I} and \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} = \\frac{\\partial}{\\partial g} g^{I} and \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} = \\frac{I g^{I}}{g} and \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} = \\frac{I \\dot{z}{(I,g)}}{g} and \\int \\frac{\\partial}{\\partial g} \\dot{z}{(I,g)} dI = \\int \\frac{I \\dot{z}{(I,g)}}{g} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Symbol('g', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Symbol('I', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Symbol('I', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True))))"], [["integrate", 4, "Symbol('I', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Symbol('I', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(H)} = e^{e^{H}}, then obtain 2 H \\dot{y}{(H)} e^{- H} e^{- e^{H}} = H (\\dot{y}{(H)} + e^{e^{H}}) e^{- H} e^{- e^{H}}", "derivation": "\\dot{y}{(H)} = e^{e^{H}} and 2 \\dot{y}{(H)} = \\dot{y}{(H)} + e^{e^{H}} and 2 H \\dot{y}{(H)} = H (\\dot{y}{(H)} + e^{e^{H}}) and 2 H \\dot{y}{(H)} e^{- e^{H}} = H (\\dot{y}{(H)} + e^{e^{H}}) e^{- e^{H}} and 2 H \\dot{y}{(H)} e^{- H} e^{- e^{H}} = H (\\dot{y}{(H)} + e^{e^{H}}) e^{- H} e^{- e^{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True))))"], [["add", 1, "Function('\\\\dot{y}')(Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('H', commutative=True))), Add(Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))))"], [["times", 2, "Symbol('H', commutative=True)"], "Equality(Mul(Integer(2), Symbol('H', commutative=True), Function('\\\\dot{y}')(Symbol('H', commutative=True))), Mul(Symbol('H', commutative=True), Add(Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True))))))"], [["divide", 3, "exp(exp(Symbol('H', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('H', commutative=True), Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('H', commutative=True))))), Mul(Symbol('H', commutative=True), Add(Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))), exp(Mul(Integer(-1), exp(Symbol('H', commutative=True))))))"], [["divide", 4, "exp(Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Symbol('H', commutative=True), Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(Mul(Integer(-1), Symbol('H', commutative=True))), exp(Mul(Integer(-1), exp(Symbol('H', commutative=True))))), Mul(Symbol('H', commutative=True), Add(Function('\\\\dot{y}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))), exp(Mul(Integer(-1), Symbol('H', commutative=True))), exp(Mul(Integer(-1), exp(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given n{(h)} = \\sin{(h)}, then obtain ((\\sin{(h)} + 1) \\sin{(h)} - \\frac{n{(h)}}{\\sin{(h)}})^{h} = ((\\sin{(h)} + 1) \\sin{(h)} - 1)^{h}", "derivation": "n{(h)} = \\sin{(h)} and n{(h)} + 1 = \\sin{(h)} + 1 and (n{(h)} + 1) \\sin{(h)} = (\\sin{(h)} + 1) \\sin{(h)} and - \\frac{n{(h)}}{\\sin{(h)}} = -1 and (\\sin{(h)} + 1) \\sin{(h)} - \\frac{n{(h)}}{\\sin{(h)}} = (\\sin{(h)} + 1) \\sin{(h)} - 1 and (n{(h)} + 1) \\sin{(h)} - \\frac{n{(h)}}{\\sin{(h)}} = (n{(h)} + 1) \\sin{(h)} - 1 and ((n{(h)} + 1) \\sin{(h)} - \\frac{n{(h)}}{\\sin{(h)}})^{h} = ((n{(h)} + 1) \\sin{(h)} - 1)^{h} and ((\\sin{(h)} + 1) \\sin{(h)} - \\frac{n{(h)}}{\\sin{(h)}})^{h} = ((\\sin{(h)} + 1) \\sin{(h)} - 1)^{h}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), Add(sin(Symbol('h', commutative=True)), Integer(1)))"], [["times", 2, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), sin(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Function('n')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Integer(-1))"], [["add", 4, "Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('n')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)))), Add(Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('n')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)))), Add(Mul(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Integer(-1)))"], [["power", 6, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Mul(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('n')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)))), Symbol('h', commutative=True)), Pow(Add(Mul(Add(Function('n')(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Integer(-1)), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Pow(Add(Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('n')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)))), Symbol('h', commutative=True)), Pow(Add(Mul(Add(sin(Symbol('h', commutative=True)), Integer(1)), sin(Symbol('h', commutative=True))), Integer(-1)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given z{(Z,\\dot{z})} = \\sin{(Z - \\dot{z})}, then obtain - Z (- Z + \\dot{z} + z{(Z,\\dot{z})}) + Z - \\dot{z} - \\sin{(Z - \\dot{z})} = - Z (- Z + \\dot{z} + \\sin{(Z - \\dot{z})}) + Z - \\dot{z} - \\sin{(Z - \\dot{z})}", "derivation": "z{(Z,\\dot{z})} = \\sin{(Z - \\dot{z})} and - Z + \\dot{z} + z{(Z,\\dot{z})} = - Z + \\dot{z} + \\sin{(Z - \\dot{z})} and - Z (- Z + \\dot{z} + z{(Z,\\dot{z})}) = - Z (- Z + \\dot{z} + \\sin{(Z - \\dot{z})}) and - Z (- Z + \\dot{z} + z{(Z,\\dot{z})}) + Z - \\dot{z} - \\sin{(Z - \\dot{z})} = - Z (- Z + \\dot{z} + \\sin{(Z - \\dot{z})}) + Z - \\dot{z} - \\sin{(Z - \\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('Z', commutative=True), Symbol('\\\\dot{z}', commutative=True)), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))"], [["minus", 1, "Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Function('z')(Symbol('Z', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))))"], [["times", 2, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('Z', commutative=True), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Function('z')(Symbol('Z', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(-1), Symbol('Z', commutative=True), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Function('z')(Symbol('Z', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))), Add(Mul(Integer(-1), Symbol('Z', commutative=True), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\dot{z}', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))), Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))))"]]}, {"prompt": "Given I{(Z)} = \\frac{d}{d Z} \\sin{(Z)}, then derive 2 I{(Z)} = I{(Z)} + \\cos{(Z)}, then obtain Z + 2 I{(Z)} = Z + I{(Z)} + \\cos{(Z)}", "derivation": "I{(Z)} = \\frac{d}{d Z} \\sin{(Z)} and 2 I{(Z)} = I{(Z)} + \\frac{d}{d Z} \\sin{(Z)} and 2 I{(Z)} = I{(Z)} + \\cos{(Z)} and Z + 2 I{(Z)} = Z + I{(Z)} + \\cos{(Z)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('Z', commutative=True)), Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 1, "Function('I')(Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Function('I')(Symbol('Z', commutative=True))), Add(Function('I')(Symbol('Z', commutative=True)), Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(2), Function('I')(Symbol('Z', commutative=True))), Add(Function('I')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))))"], [["add", 3, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Mul(Integer(2), Function('I')(Symbol('Z', commutative=True)))), Add(Symbol('Z', commutative=True), Function('I')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(A)} = \\frac{d}{d A} \\sin{(A)}, then derive \\operatorname{A_{1}}{(A)} = \\cos{(A)}, then obtain (- \\frac{d}{d A} \\sin{(A)})^{A} - 1 = (- \\operatorname{A_{1}}{(A)})^{A} - 1", "derivation": "\\operatorname{A_{1}}{(A)} = \\frac{d}{d A} \\sin{(A)} and \\operatorname{A_{1}}{(A)} = \\cos{(A)} and \\cos{(A)} = \\frac{d}{d A} \\sin{(A)} and - \\cos{(A)} = - \\frac{d}{d A} \\sin{(A)} and - \\cos{(A)} = - \\operatorname{A_{1}}{(A)} and (- \\cos{(A)})^{A} = (- \\operatorname{A_{1}}{(A)})^{A} and (- \\frac{d}{d A} \\sin{(A)})^{A} = (- \\operatorname{A_{1}}{(A)})^{A} and (- \\frac{d}{d A} \\sin{(A)})^{A} - 1 = (- \\operatorname{A_{1}}{(A)})^{A} - 1", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_1')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), cos(Symbol('A', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), cos(Symbol('A', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('A', commutative=True))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Integer(-1), cos(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Integer(-1), Function('A_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Mul(Integer(-1), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Symbol('A', commutative=True)), Pow(Mul(Integer(-1), Function('A_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["add", 7, "Integer(-1)"], "Equality(Add(Pow(Mul(Integer(-1), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Symbol('A', commutative=True)), Integer(-1)), Add(Pow(Mul(Integer(-1), Function('A_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given a{(\\mathbf{P},v_{z})} = \\mathbf{P} + \\log{(v_{z})}, then obtain 1 = \\frac{\\mathbf{P} + \\log{(v_{z})}}{a{(\\mathbf{P},v_{z})}}", "derivation": "a{(\\mathbf{P},v_{z})} = \\mathbf{P} + \\log{(v_{z})} and v_{z} a{(\\mathbf{P},v_{z})} = v_{z} (\\mathbf{P} + \\log{(v_{z})}) and v_{z}^{2} a^{2}{(\\mathbf{P},v_{z})} = v_{z}^{2} (\\mathbf{P} + \\log{(v_{z})}) a{(\\mathbf{P},v_{z})} and 1 = \\frac{\\mathbf{P} + \\log{(v_{z})}}{a{(\\mathbf{P},v_{z})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('v_z', commutative=True))))"], [["times", 1, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True))), Mul(Symbol('v_z', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('v_z', commutative=True)))))"], [["times", 2, "Mul(Symbol('v_z', commutative=True), Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(2)), Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True)), Integer(2))), Mul(Pow(Symbol('v_z', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('v_z', commutative=True))), Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('v_z', commutative=True), Integer(2)), Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True)), Integer(2)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('v_z', commutative=True))), Pow(Function('a')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\lambda)} = e^{\\lambda}, then derive (\\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda)^{\\lambda} = (\\rho + e^{\\lambda})^{\\lambda}, then obtain \\int (\\rho + e^{\\lambda})^{\\lambda} d\\rho = \\int (\\rho + \\operatorname{f^{*}}{(\\lambda)})^{\\lambda} d\\rho", "derivation": "\\operatorname{f^{*}}{(\\lambda)} = e^{\\lambda} and \\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda = \\int e^{\\lambda} d\\lambda and (\\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda)^{\\lambda} = (\\int e^{\\lambda} d\\lambda)^{\\lambda} and (\\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda)^{\\lambda} = (\\rho + e^{\\lambda})^{\\lambda} and (\\int \\operatorname{f^{*}}{(\\lambda)} d\\lambda)^{\\lambda} = (\\rho + \\operatorname{f^{*}}{(\\lambda)})^{\\lambda} and (\\rho + e^{\\lambda})^{\\lambda} = (\\rho + \\operatorname{f^{*}}{(\\lambda)})^{\\lambda} and \\int (\\rho + e^{\\lambda})^{\\lambda} d\\rho = \\int (\\rho + \\operatorname{f^{*}}{(\\lambda)})^{\\lambda} d\\rho", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Add(Symbol('\\\\rho', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Add(Symbol('\\\\rho', commutative=True), Function('f^*')(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Symbol('\\\\rho', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Add(Symbol('\\\\rho', commutative=True), Function('f^*')(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 6, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('\\\\rho', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Pow(Add(Symbol('\\\\rho', commutative=True), Function('f^*')(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(P_{e},q)} = \\frac{P_{e}}{q}, then derive \\frac{\\partial}{\\partial q} \\operatorname{n_{2}}{(P_{e},q)} = - \\frac{P_{e}}{q^{2}}, then obtain \\frac{\\partial}{\\partial q} \\operatorname{n_{2}}{(P_{e},q)} = - \\frac{\\operatorname{n_{2}}{(P_{e},q)}}{q}", "derivation": "\\operatorname{n_{2}}{(P_{e},q)} = \\frac{P_{e}}{q} and \\frac{\\partial}{\\partial q} \\operatorname{n_{2}}{(P_{e},q)} = \\frac{\\partial}{\\partial q} \\frac{P_{e}}{q} and \\frac{\\partial}{\\partial q} \\operatorname{n_{2}}{(P_{e},q)} = - \\frac{P_{e}}{q^{2}} and \\frac{\\partial}{\\partial q} \\operatorname{n_{2}}{(P_{e},q)} = - \\frac{\\operatorname{n_{2}}{(P_{e},q)}}{q}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('q', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('n_2')(Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('n_2')(Symbol('P_e', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\hat{H}_l)} = e^{\\hat{H}_l}, then derive \\frac{d}{d \\hat{H}_l} \\mathbf{E}{(\\hat{H}_l)} = e^{\\hat{H}_l}, then obtain \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} - 1 = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} - 1", "derivation": "\\mathbf{E}{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} \\mathbf{E}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} \\mathbf{E}{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} \\mathbf{E}{(\\hat{H}_l)} = \\mathbf{E}{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} \\mathbf{E}{(\\hat{H}_l)} - 1 = \\mathbf{E}{(\\hat{H}_l)} - 1 and \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} = e^{\\hat{H}_l} and \\mathbf{E}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} - 1 = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1)), Add(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 6], "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Integer(-1)), Add(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\Psi{(\\delta,\\dot{z},\\phi)} = \\frac{\\delta}{\\phi} + \\dot{z}, then obtain \\frac{2 \\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1 = \\frac{\\delta (\\frac{\\delta}{\\phi} + \\dot{z})}{\\phi} + \\frac{\\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1", "derivation": "\\Psi{(\\delta,\\dot{z},\\phi)} = \\frac{\\delta}{\\phi} + \\dot{z} and \\frac{\\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} = \\frac{\\delta (\\frac{\\delta}{\\phi} + \\dot{z})}{\\phi} and \\frac{\\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1 = \\frac{\\delta (\\frac{\\delta}{\\phi} + \\dot{z})}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1 and \\frac{2 \\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1 = \\frac{\\delta (\\frac{\\delta}{\\phi} + \\dot{z})}{\\phi} + \\frac{\\delta \\Psi{(\\delta,\\dot{z},\\phi)}}{\\phi} - \\frac{\\delta}{\\phi} - \\dot{z} - \\Psi{(\\delta,\\dot{z},\\phi)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 2, "Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\dot{z}', commutative=True), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(1))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)))"], [["add", 3, "Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\rho)} = \\log{(\\rho)}, then obtain \\frac{\\frac{2 \\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - 1 - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}} = \\frac{\\frac{\\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}}", "derivation": "\\operatorname{v_{t}}{(\\rho)} = \\log{(\\rho)} and \\frac{\\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} = 1 and \\frac{\\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - \\frac{1}{\\log{(\\rho)}} = 1 - \\frac{1}{\\log{(\\rho)}} and \\frac{\\frac{\\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}} = \\frac{1 - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}} and \\frac{\\frac{2 \\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - 1 - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}} = \\frac{\\frac{\\operatorname{v_{t}}{(\\rho)}}{\\log{(\\rho)}} - \\frac{1}{\\log{(\\rho)}}}{\\log{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('v_t')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))))"], [["divide", 3, "log(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Add(Mul(Function('v_t')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Add(Integer(1), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(2), Function('v_t')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Integer(-1), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Add(Mul(Function('v_t')(Symbol('\\\\rho', commutative=True)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(m_{s},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s}, then derive \\operatorname{P_{g}}^{m_{s}}{(m_{s},\\mu_0)} = m_{s}^{m_{s}}, then derive \\operatorname{P_{g}}{(m_{s},\\mu_0)} = m_{s}, then obtain \\operatorname{P_{g}}^{m_{s}}{(\\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s},\\mu_0)} = m_{s}^{m_{s}}", "derivation": "\\operatorname{P_{g}}{(m_{s},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s} and \\operatorname{P_{g}}^{m_{s}}{(m_{s},\\mu_0)} = (\\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s})^{m_{s}} and \\operatorname{P_{g}}^{m_{s}}{(m_{s},\\mu_0)} = m_{s}^{m_{s}} and (\\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s})^{m_{s}} = m_{s}^{m_{s}} and \\operatorname{P_{g}}{(m_{s},\\mu_0)} = m_{s} and m_{s} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s} and \\operatorname{P_{g}}{(\\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s} and \\operatorname{P_{g}}^{m_{s}}{(\\frac{\\partial}{\\partial \\mu_0} \\mu_0 m_{s},\\mu_0)} = m_{s}^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('m_s', commutative=True), Symbol('\\\\mu_0', commutative=True)), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('m_s', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('m_s', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('m_s', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('P_g')(Symbol('m_s', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('m_s', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('P_g')(Symbol('m_s', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('m_s', commutative=True))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Symbol('m_s', commutative=True), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Function('P_g')(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 7], "Equality(Pow(Function('P_g')(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given a{(x)} = - x, then obtain - (a{(x)} + \\int - x dx) \\int a{(x)} dx = - (a{(x)} + \\int - x dx) \\int - x dx", "derivation": "a{(x)} = - x and \\int a{(x)} dx = \\int - x dx and a{(x)} + \\int a{(x)} dx = a{(x)} + \\int - x dx and (a{(x)} + \\int a{(x)} dx) \\int a{(x)} dx = (a{(x)} + \\int a{(x)} dx) \\int - x dx and - (a{(x)} + \\int a{(x)} dx) \\int a{(x)} dx = - (a{(x)} + \\int a{(x)} dx) \\int - x dx and - (a{(x)} + \\int - x dx) \\int a{(x)} dx = - (a{(x)} + \\int - x dx) \\int - x dx", "srepr_derivation": [["renaming_premise", "Equality(Function('a')(Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["add", 2, "Function('a')(Symbol('x', commutative=True))"], "Equality(Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Function('a')(Symbol('x', commutative=True)), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["times", 2, "Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], "Equality(Mul(Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Function('a')(Symbol('x', commutative=True)), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Add(Function('a')(Symbol('x', commutative=True)), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Function('a')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Function('a')(Symbol('x', commutative=True)), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(Mul(Integer(-1), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(P_{g},Z)} = e^{P_{g} + Z}, then obtain P_{g} (\\varphi{(P_{g},Z)} + e^{P_{g} + Z}) + \\varphi{(P_{g},Z)} + e^{P_{g} + Z} = P_{g} (\\varphi{(P_{g},Z)} + e^{P_{g} + Z}) + 2 e^{P_{g} + Z}", "derivation": "\\varphi{(P_{g},Z)} = e^{P_{g} + Z} and \\varphi{(P_{g},Z)} + e^{P_{g} + Z} = 2 e^{P_{g} + Z} and P_{g} (\\varphi{(P_{g},Z)} + e^{P_{g} + Z}) = 2 P_{g} e^{P_{g} + Z} and 2 P_{g} e^{P_{g} + Z} + \\varphi{(P_{g},Z)} + e^{P_{g} + Z} = 2 P_{g} e^{P_{g} + Z} + 2 e^{P_{g} + Z} and P_{g} (\\varphi{(P_{g},Z)} + e^{P_{g} + Z}) + \\varphi{(P_{g},Z)} + e^{P_{g} + Z} = P_{g} (\\varphi{(P_{g},Z)} + e^{P_{g} + Z}) + 2 e^{P_{g} + Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))"], [["add", 1, "exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Add(Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))), Mul(Integer(2), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))))"], [["times", 2, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Add(Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))), Mul(Integer(2), Symbol('P_g', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))))"], [["add", 2, "Mul(Integer(2), Symbol('P_g', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('P_g', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))), Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))), Add(Mul(Integer(2), Symbol('P_g', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))), Mul(Integer(2), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Symbol('P_g', commutative=True), Add(Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))), Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)))), Add(Mul(Symbol('P_g', commutative=True), Add(Function('\\\\varphi')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))), Mul(Integer(2), exp(Add(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(t_{2})} = e^{t_{2}}, then derive \\int \\hat{\\mathbf{r}}{(t_{2})} dt_{2} = \\pi + e^{t_{2}}, then obtain t_{2} \\int \\hat{\\mathbf{r}}{(t_{2})} dt_{2} = t_{2} (\\pi + e^{t_{2}})", "derivation": "\\hat{\\mathbf{r}}{(t_{2})} = e^{t_{2}} and \\int \\hat{\\mathbf{r}}{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2} and \\int \\hat{\\mathbf{r}}{(t_{2})} dt_{2} = \\pi + e^{t_{2}} and t_{2} \\int \\hat{\\mathbf{r}}{(t_{2})} dt_{2} = t_{2} (\\pi + e^{t_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('\\\\pi', commutative=True), exp(Symbol('t_2', commutative=True))))"], [["times", 3, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Symbol('t_2', commutative=True), Add(Symbol('\\\\pi', commutative=True), exp(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(c,g)} = \\log{(\\frac{c}{g})}, then derive \\int \\varphi{(c,g)} dg - 1 = \\mathbf{J}_M + g \\log{(\\frac{c}{g})} + g - 1, then obtain \\mathbf{J}_M + g \\log{(\\frac{c}{g})} + g - 1 = \\int \\log{(\\frac{c}{g})} dg - 1", "derivation": "\\varphi{(c,g)} = \\log{(\\frac{c}{g})} and \\int \\varphi{(c,g)} dg = \\int \\log{(\\frac{c}{g})} dg and \\int \\varphi{(c,g)} dg - 1 = \\int \\log{(\\frac{c}{g})} dg - 1 and \\int \\varphi{(c,g)} dg - 1 = \\mathbf{J}_M + g \\log{(\\frac{c}{g})} + g - 1 and \\mathbf{J}_M + g \\log{(\\frac{c}{g})} + g - 1 = \\int \\log{(\\frac{c}{g})} dg - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('c', commutative=True), Symbol('g', commutative=True)), log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\varphi')(Symbol('c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)), Add(Integral(log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 3], "Equality(Add(Integral(Function('\\\\varphi')(Symbol('c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Symbol('g', commutative=True), log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('g', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Symbol('g', commutative=True), log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('g', commutative=True), Integer(-1)), Add(Integral(log(Mul(Symbol('c', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given m{(v_{1},\\dot{z})} = - \\dot{z} + v_{1}, then obtain \\iint 0 dv_{1} d\\dot{z} = \\iint (\\dot{z} (- \\dot{z} + v_{1}) - \\dot{z} m{(v_{1},\\dot{z})}) dv_{1} d\\dot{z}", "derivation": "m{(v_{1},\\dot{z})} = - \\dot{z} + v_{1} and \\dot{z} m{(v_{1},\\dot{z})} = \\dot{z} (- \\dot{z} + v_{1}) and \\dot{z} m{(v_{1},\\dot{z})} + \\dot{z} = \\dot{z} (- \\dot{z} + v_{1}) + \\dot{z} and \\dot{z} m{(v_{1},\\dot{z})} + \\dot{z} + v_{1} = \\dot{z} (- \\dot{z} + v_{1}) + \\dot{z} + v_{1} and 0 = \\dot{z} (- \\dot{z} + v_{1}) - \\dot{z} m{(v_{1},\\dot{z})} and \\int 0 dv_{1} = \\int (\\dot{z} (- \\dot{z} + v_{1}) - \\dot{z} m{(v_{1},\\dot{z})}) dv_{1} and \\iint 0 dv_{1} d\\dot{z} = \\iint (\\dot{z} (- \\dot{z} + v_{1}) - \\dot{z} m{(v_{1},\\dot{z})}) dv_{1} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["add", 3, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))), Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)))"], [["minus", 4, "Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], [["integrate", 5, "Symbol('v_1', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('v_1', commutative=True))))"], [["integrate", 6, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('m')(Symbol('v_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then derive \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + \\tilde{g}^*, then obtain \\frac{d}{d \\mathbf{B}} \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{B} \\operatorname{v_{1}}{(\\mathbf{B})} - \\mathbf{B} + \\tilde{g}^*)", "derivation": "\\operatorname{v_{1}}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + \\tilde{g}^* and \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\operatorname{v_{1}}{(\\mathbf{B})} - \\mathbf{B} + \\tilde{g}^* and \\frac{d}{d \\mathbf{B}} \\int \\operatorname{v_{1}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{B} \\operatorname{v_{1}}{(\\mathbf{B})} - \\mathbf{B} + \\tilde{g}^*)", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(\\psi^*,t_{2})} = \\frac{\\psi^*}{t_{2}}, then obtain (e^{(\\theta{(\\psi^*,t_{2})} + 1)^{\\psi^*}} - \\frac{1}{t_{2}})^{t_{2}} = (e^{(\\frac{\\psi^*}{t_{2}} + 1)^{\\psi^*}} - \\frac{1}{t_{2}})^{t_{2}}", "derivation": "\\theta{(\\psi^*,t_{2})} = \\frac{\\psi^*}{t_{2}} and \\theta{(\\psi^*,t_{2})} + 1 = \\frac{\\psi^*}{t_{2}} + 1 and (\\theta{(\\psi^*,t_{2})} + 1)^{\\psi^*} = (\\frac{\\psi^*}{t_{2}} + 1)^{\\psi^*} and e^{(\\theta{(\\psi^*,t_{2})} + 1)^{\\psi^*}} = e^{(\\frac{\\psi^*}{t_{2}} + 1)^{\\psi^*}} and e^{(\\theta{(\\psi^*,t_{2})} + 1)^{\\psi^*}} - \\frac{1}{t_{2}} = e^{(\\frac{\\psi^*}{t_{2}} + 1)^{\\psi^*}} - \\frac{1}{t_{2}} and (e^{(\\theta{(\\psi^*,t_{2})} + 1)^{\\psi^*}} - \\frac{1}{t_{2}})^{t_{2}} = (e^{(\\frac{\\psi^*}{t_{2}} + 1)^{\\psi^*}} - \\frac{1}{t_{2}})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(1)))"], [["power", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Integer(1)), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\psi^*', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Add(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Integer(1)), Symbol('\\\\psi^*', commutative=True))), exp(Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 4, "Pow(Symbol('t_2', commutative=True), Integer(-1))"], "Equality(Add(exp(Pow(Add(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Integer(1)), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Add(exp(Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)))))"], [["power", 5, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(exp(Pow(Add(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('t_2', commutative=True)), Integer(1)), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Symbol('t_2', commutative=True)), Pow(Add(exp(Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given c{(\\mathbf{H},\\mathbf{J}_f)} = \\frac{\\mathbf{H}}{\\mathbf{J}_f}, then obtain \\frac{\\mathbf{J}_f + \\theta_1 c{(\\mathbf{H},\\mathbf{J}_f)}}{\\frac{\\partial}{\\partial \\mathbf{J}_f} c{(\\mathbf{H},\\mathbf{J}_f)}} = \\frac{\\frac{\\mathbf{H} \\theta_1}{\\mathbf{J}_f} + \\mathbf{J}_f}{\\frac{\\partial}{\\partial \\mathbf{J}_f} c{(\\mathbf{H},\\mathbf{J}_f)}}", "derivation": "c{(\\mathbf{H},\\mathbf{J}_f)} = \\frac{\\mathbf{H}}{\\mathbf{J}_f} and \\theta_1 c{(\\mathbf{H},\\mathbf{J}_f)} = \\frac{\\mathbf{H} \\theta_1}{\\mathbf{J}_f} and \\mathbf{J}_f + \\theta_1 c{(\\mathbf{H},\\mathbf{J}_f)} = \\frac{\\mathbf{H} \\theta_1}{\\mathbf{J}_f} + \\mathbf{J}_f and \\frac{\\mathbf{J}_f + \\theta_1 c{(\\mathbf{H},\\mathbf{J}_f)}}{\\frac{\\partial}{\\partial \\mathbf{J}_f} c{(\\mathbf{H},\\mathbf{J}_f)}} = \\frac{\\frac{\\mathbf{H} \\theta_1}{\\mathbf{J}_f} + \\mathbf{J}_f}{\\frac{\\partial}{\\partial \\mathbf{J}_f} c{(\\mathbf{H},\\mathbf{J}_f)}}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 3, "Derivative(Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Pow(Derivative(Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Function('c')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(t_{1},\\hat{H}_l)} = (e^{t_{1}})^{\\hat{H}_l}, then derive \\frac{\\partial}{\\partial t_{1}} \\operatorname{P_{g}}{(t_{1},\\hat{H}_l)} = \\hat{H}_l (e^{t_{1}})^{\\hat{H}_l}, then obtain \\hat{H}_l (e^{t_{1}})^{\\hat{H}_l} = \\frac{\\partial}{\\partial t_{1}} (e^{t_{1}})^{\\hat{H}_l}", "derivation": "\\operatorname{P_{g}}{(t_{1},\\hat{H}_l)} = (e^{t_{1}})^{\\hat{H}_l} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{P_{g}}{(t_{1},\\hat{H}_l)} = \\frac{\\partial}{\\partial t_{1}} (e^{t_{1}})^{\\hat{H}_l} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{P_{g}}{(t_{1},\\hat{H}_l)} = \\hat{H}_l (e^{t_{1}})^{\\hat{H}_l} and \\hat{H}_l (e^{t_{1}})^{\\hat{H}_l} = \\frac{\\partial}{\\partial t_{1}} (e^{t_{1}})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(exp(Symbol('t_1', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('t_1', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(exp(Symbol('t_1', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(exp(Symbol('t_1', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Derivative(Pow(exp(Symbol('t_1', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(J,\\mu)} = e^{J - \\mu} and \\bar{\\h}{(J,\\mu)} = e^{\\cos{((J + e^{J - \\mu})^{\\mu})}}, then obtain \\bar{\\h}^{2}{(J,\\mu)} - \\bar{\\h}^{\\mu}{(J,\\mu)} = \\bar{\\h}{(J,\\mu)} e^{\\cos{((J + \\varphi^{*}{(J,\\mu)})^{\\mu})}} - \\bar{\\h}^{\\mu}{(J,\\mu)}", "derivation": "\\varphi^{*}{(J,\\mu)} = e^{J - \\mu} and J + \\varphi^{*}{(J,\\mu)} = J + e^{J - \\mu} and (J + \\varphi^{*}{(J,\\mu)})^{\\mu} = (J + e^{J - \\mu})^{\\mu} and \\cos{((J + \\varphi^{*}{(J,\\mu)})^{\\mu})} = \\cos{((J + e^{J - \\mu})^{\\mu})} and \\bar{\\h}{(J,\\mu)} = e^{\\cos{((J + e^{J - \\mu})^{\\mu})}} and \\bar{\\h}{(J,\\mu)} = e^{\\cos{((J + \\varphi^{*}{(J,\\mu)})^{\\mu})}} and \\bar{\\h}^{2}{(J,\\mu)} = \\bar{\\h}{(J,\\mu)} e^{\\cos{((J + \\varphi^{*}{(J,\\mu)})^{\\mu})}} and \\bar{\\h}^{2}{(J,\\mu)} - \\bar{\\h}^{\\mu}{(J,\\mu)} = \\bar{\\h}{(J,\\mu)} e^{\\cos{((J + \\varphi^{*}{(J,\\mu)})^{\\mu})}} - \\bar{\\h}^{\\mu}{(J,\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), exp(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('J', commutative=True), exp(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('J', commutative=True), exp(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), cos(Pow(Add(Symbol('J', commutative=True), exp(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), exp(cos(Pow(Add(Symbol('J', commutative=True), exp(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), exp(cos(Pow(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))))"], [["times", 6, "Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), exp(cos(Pow(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))))"], [["minus", 7, "Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Add(Mul(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), exp(cos(Pow(Add(Symbol('J', commutative=True), Function('\\\\varphi^*')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\dot{z})} = \\sin{(\\dot{z})}, then derive \\frac{d}{d \\dot{z}} \\psi{(\\dot{z})} = \\cos{(\\dot{z})}, then obtain 1 = (\\frac{\\frac{d}{d \\dot{z}} \\psi{(\\dot{z})}}{\\cos{(\\dot{z})}})^{\\dot{z}}", "derivation": "\\psi{(\\dot{z})} = \\sin{(\\dot{z})} and \\frac{d}{d \\dot{z}} \\psi{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\sin{(\\dot{z})} and 1 = \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\dot{z})}}{\\frac{d}{d \\dot{z}} \\psi{(\\dot{z})}} and \\frac{d}{d \\dot{z}} \\psi{(\\dot{z})} = \\cos{(\\dot{z})} and 1 = \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\dot{z})}}{\\cos{(\\dot{z})}} and 1 = \\frac{\\frac{d}{d \\dot{z}} \\psi{(\\dot{z})}}{\\cos{(\\dot{z})}} and 1 = (\\frac{\\frac{d}{d \\dot{z}} \\psi{(\\dot{z})}}{\\cos{(\\dot{z})}})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Mul(Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(1), Mul(Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(Function('\\\\psi')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{s})} = \\log{(\\cos{(\\mathbf{s})})}, then obtain \\frac{(\\frac{\\hat{H}{(\\mathbf{s})}}{\\log{(\\cos{(\\mathbf{s})})}})^{\\mathbf{s}}}{\\log{(\\cos{(\\mathbf{s})})}} = \\frac{1}{\\log{(\\cos{(\\mathbf{s})})}}", "derivation": "\\hat{H}{(\\mathbf{s})} = \\log{(\\cos{(\\mathbf{s})})} and 1 = \\frac{\\log{(\\cos{(\\mathbf{s})})}}{\\hat{H}{(\\mathbf{s})}} and \\frac{\\hat{H}{(\\mathbf{s})}}{\\log{(\\cos{(\\mathbf{s})})}} = 1 and (\\frac{\\hat{H}{(\\mathbf{s})}}{\\log{(\\cos{(\\mathbf{s})})}})^{\\mathbf{s}} = 1 and \\frac{(\\frac{\\hat{H}{(\\mathbf{s})}}{\\log{(\\cos{(\\mathbf{s})})}})^{\\mathbf{s}}}{\\log{(\\cos{(\\mathbf{s})})}} = \\frac{1}{\\log{(\\cos{(\\mathbf{s})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), log(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["divide", 2, "Mul(Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"], [["divide", 4, "log(cos(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Pow(log(cos(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(Q)} = \\cos{(Q)} and \\mathbb{I}{(Q)} = \\cos{(Q)}, then obtain \\frac{\\mathbb{I}{(Q)}}{- \\frac{- Q + \\operatorname{A_{2}}{(Q)}}{\\operatorname{A_{2}}{(Q)}} + \\mathbb{I}{(Q)}} = \\frac{\\operatorname{A_{2}}{(Q)}}{- \\frac{- Q + \\operatorname{A_{2}}{(Q)}}{\\operatorname{A_{2}}{(Q)}} + \\mathbb{I}{(Q)}}", "derivation": "\\operatorname{A_{2}}{(Q)} = \\cos{(Q)} and \\mathbb{I}{(Q)} = \\cos{(Q)} and \\mathbb{I}{(Q)} = \\operatorname{A_{2}}{(Q)} and \\frac{\\mathbb{I}{(Q)}}{- \\frac{- Q + \\cos{(Q)}}{\\cos{(Q)}} + \\mathbb{I}{(Q)}} = \\frac{\\operatorname{A_{2}}{(Q)}}{- \\frac{- Q + \\cos{(Q)}}{\\cos{(Q)}} + \\mathbb{I}{(Q)}} and \\frac{\\mathbb{I}{(Q)}}{- \\frac{- Q + \\operatorname{A_{2}}{(Q)}}{\\operatorname{A_{2}}{(Q)}} + \\mathbb{I}{(Q)}} = \\frac{\\operatorname{A_{2}}{(Q)}}{- \\frac{- Q + \\operatorname{A_{2}}{(Q)}}{\\operatorname{A_{2}}{(Q)}} + \\mathbb{I}{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('Q', commutative=True)), Function('A_2')(Symbol('Q', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Pow(cos(Symbol('Q', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Pow(cos(Symbol('Q', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Pow(cos(Symbol('Q', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Integer(-1)), Function('A_2')(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('Q', commutative=True))), Pow(Function('A_2')(Symbol('Q', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('Q', commutative=True))), Pow(Function('A_2')(Symbol('Q', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True))), Integer(-1)), Function('A_2')(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})} = \\frac{\\dot{y}}{v_{x}}, then obtain \\frac{\\partial}{\\partial \\dot{y}} \\int (- v_{x} + \\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})}) d\\dot{y} = \\frac{\\partial}{\\partial \\dot{y}} \\int (\\frac{\\dot{y}}{v_{x}} - v_{x}) d\\dot{y}", "derivation": "\\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})} = \\frac{\\dot{y}}{v_{x}} and - v_{x} + \\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})} = \\frac{\\dot{y}}{v_{x}} - v_{x} and \\int (- v_{x} + \\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})}) d\\dot{y} = \\int (\\frac{\\dot{y}}{v_{x}} - v_{x}) d\\dot{y} and \\frac{\\partial}{\\partial \\dot{y}} \\int (- v_{x} + \\operatorname{g_{\\varepsilon}}{(v_{x},\\dot{y})}) d\\dot{y} = \\frac{\\partial}{\\partial \\dot{y}} \\int (\\frac{\\dot{y}}{v_{x}} - v_{x}) d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(\\hat{H},\\hbar)} = \\hat{H} \\hbar and \\dot{z}{(\\hat{H},\\hbar)} = \\hat{H} \\hbar, then obtain (\\dot{z}{(\\hat{H},\\hbar)} - \\rho_{f}{(\\hat{H},\\hbar)})^{\\hbar} = 0^{\\hbar}", "derivation": "\\rho_{f}{(\\hat{H},\\hbar)} = \\hat{H} \\hbar and \\dot{z}{(\\hat{H},\\hbar)} = \\hat{H} \\hbar and \\dot{z}{(\\hat{H},\\hbar)} = \\rho_{f}{(\\hat{H},\\hbar)} and - \\hat{H} \\hbar + \\dot{z}{(\\hat{H},\\hbar)} = - \\hat{H} \\hbar + \\rho_{f}{(\\hat{H},\\hbar)} and (- \\hat{H} \\hbar + \\dot{z}{(\\hat{H},\\hbar)})^{\\hbar} = (- \\hat{H} \\hbar + \\rho_{f}{(\\hat{H},\\hbar)})^{\\hbar} and (\\dot{z}{(\\hat{H},\\hbar)} - \\rho_{f}{(\\hat{H},\\hbar)})^{\\hbar} = 0^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["minus", 3, "Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(x)} = \\int \\log{(x)} dx and \\pi{(x)} = \\int \\log{(x)} dx, then derive \\pi{(x)} \\log{(x)} = (\\sigma_p + x \\log{(x)} - x) \\log{(x)}, then obtain \\theta_{2}{(x)} \\log{(x)} = (\\sigma_p + x \\log{(x)} - x) \\log{(x)}", "derivation": "\\theta_{2}{(x)} = \\int \\log{(x)} dx and \\pi{(x)} = \\int \\log{(x)} dx and \\pi{(x)} \\log{(x)} = \\log{(x)} \\int \\log{(x)} dx and \\pi{(x)} \\log{(x)} = (\\sigma_p + x \\log{(x)} - x) \\log{(x)} and \\log{(x)} \\int \\log{(x)} dx = (\\sigma_p + x \\log{(x)} - x) \\log{(x)} and \\theta_{2}{(x)} \\log{(x)} = (\\sigma_p + x \\log{(x)} - x) \\log{(x)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_2')(Symbol('x', commutative=True)), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('x', commutative=True)), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["times", 2, "log(Symbol('x', commutative=True))"], "Equality(Mul(Function('\\\\pi')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Mul(log(Symbol('x', commutative=True)), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('\\\\pi')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), log(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(log(Symbol('x', commutative=True)), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), log(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\theta_2')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), log(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\theta_2)} = \\sin{(\\theta_2)}, then obtain e^{\\int (\\theta_2 + \\operatorname{n_{1}}{(\\theta_2)}) (\\theta_2 + \\sin{(\\theta_2)}) d\\theta_2} = e^{\\int (\\theta_2 + \\sin{(\\theta_2)})^{2} d\\theta_2}", "derivation": "\\operatorname{n_{1}}{(\\theta_2)} = \\sin{(\\theta_2)} and \\theta_2 + \\operatorname{n_{1}}{(\\theta_2)} = \\theta_2 + \\sin{(\\theta_2)} and (\\theta_2 + \\operatorname{n_{1}}{(\\theta_2)}) (\\theta_2 + \\sin{(\\theta_2)}) = (\\theta_2 + \\sin{(\\theta_2)})^{2} and \\int (\\theta_2 + \\operatorname{n_{1}}{(\\theta_2)}) (\\theta_2 + \\sin{(\\theta_2)}) d\\theta_2 = \\int (\\theta_2 + \\sin{(\\theta_2)})^{2} d\\theta_2 and e^{\\int (\\theta_2 + \\operatorname{n_{1}}{(\\theta_2)}) (\\theta_2 + \\sin{(\\theta_2)}) d\\theta_2} = e^{\\int (\\theta_2 + \\sin{(\\theta_2)})^{2} d\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('n_1')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('n_1')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True)))), Pow(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Integer(2)))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('n_1')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('n_1')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True)))), exp(Integral(Pow(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_2', commutative=True))), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given q{(\\mathbf{S})} = e^{\\mathbf{S}}, then derive \\int q{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{J}_M + e^{\\mathbf{S}}, then obtain \\mathbf{J}_M - e^{2 \\mathbf{S}} + e^{\\mathbf{S}} = \\mathbf{J}_M + q{(\\mathbf{S})} - e^{2 \\mathbf{S}}", "derivation": "q{(\\mathbf{S})} = e^{\\mathbf{S}} and \\int q{(\\mathbf{S})} d\\mathbf{S} = \\int e^{\\mathbf{S}} d\\mathbf{S} and \\int q{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{J}_M + e^{\\mathbf{S}} and \\int q{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{J}_M + q{(\\mathbf{S})} and - e^{2 \\mathbf{S}} + \\int q{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{J}_M + q{(\\mathbf{S})} - e^{2 \\mathbf{S}} and \\mathbf{J}_M - e^{2 \\mathbf{S}} + e^{\\mathbf{S}} = \\mathbf{J}_M + q{(\\mathbf{S})} - e^{2 \\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('q')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 4, "exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))), Integral(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)))), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given g{(\\mathbf{J}_M,h)} = - \\mathbf{J}_M + h, then obtain g{(\\mathbf{J}_M,h)} + \\frac{\\mathbf{J}_M - h}{h} = - \\mathbf{J}_M + h + \\frac{\\mathbf{J}_M - h}{h}", "derivation": "g{(\\mathbf{J}_M,h)} = - \\mathbf{J}_M + h and \\frac{g{(\\mathbf{J}_M,h)}}{h} = \\frac{- \\mathbf{J}_M + h}{h} and g{(\\mathbf{J}_M,h)} - \\frac{g{(\\mathbf{J}_M,h)}}{h} = - \\mathbf{J}_M + h - \\frac{g{(\\mathbf{J}_M,h)}}{h} and g{(\\mathbf{J}_M,h)} - \\frac{- \\mathbf{J}_M + h}{h} = - \\mathbf{J}_M + h - \\frac{- \\mathbf{J}_M + h}{h} and g{(\\mathbf{J}_M,h)} + \\frac{\\mathbf{J}_M - h}{h} = - \\mathbf{J}_M + h + \\frac{\\mathbf{J}_M - h}{h}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True)))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))"], "Equality(Add(Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('g')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('h', commutative=True), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\dot{z},h,G)} = \\frac{G}{\\dot{z}} - h, then obtain 1 + \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G} = \\dot{z} (\\frac{G}{\\dot{z}} - h - \\hat{H}_{\\lambda}{(\\dot{z},h,G)}) + 1 + \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G}", "derivation": "\\hat{H}_{\\lambda}{(\\dot{z},h,G)} = \\frac{G}{\\dot{z}} - h and 0 = \\frac{G}{\\dot{z}} - h - \\hat{H}_{\\lambda}{(\\dot{z},h,G)} and 0 = \\dot{z} (\\frac{G}{\\dot{z}} - h - \\hat{H}_{\\lambda}{(\\dot{z},h,G)}) and \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G} = \\dot{z} (\\frac{G}{\\dot{z}} - h - \\hat{H}_{\\lambda}{(\\dot{z},h,G)}) + \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G} and 1 + \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G} = \\dot{z} (\\frac{G}{\\dot{z}} - h - \\hat{H}_{\\lambda}{(\\dot{z},h,G)}) + 1 + \\frac{\\hat{H}_{\\lambda}^{G}{(\\dot{z},h,G)}}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["minus", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)))))"], [["divide", 2, "Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))"], "Equality(Integer(0), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True))))))"], [["add", 3, "Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True))))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True))))), Integer(1), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True), Symbol('h', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(C_{d},\\mu)} = - C_{d} + \\mu, then derive \\frac{\\partial}{\\partial \\mu} \\mathbf{F}{(C_{d},\\mu)} + 1 = 2, then obtain \\frac{d}{d C_{d}} 2 + \\frac{\\partial}{\\partial C_{d}} (\\frac{\\partial}{\\partial \\mu} (- C_{d} + \\mu) + 1) = 2 \\frac{d}{d C_{d}} 2", "derivation": "\\mathbf{F}{(C_{d},\\mu)} = - C_{d} + \\mu and - C_{d} + \\mu + \\mathbf{F}{(C_{d},\\mu)} = - 2 C_{d} + 2 \\mu and \\frac{\\partial}{\\partial \\mu} (- C_{d} + \\mu + \\mathbf{F}{(C_{d},\\mu)}) = \\frac{\\partial}{\\partial \\mu} (- 2 C_{d} + 2 \\mu) and \\frac{\\partial}{\\partial \\mu} \\mathbf{F}{(C_{d},\\mu)} + 1 = 2 and \\frac{\\partial}{\\partial \\mu} (- C_{d} + \\mu) + 1 = 2 and \\frac{\\partial}{\\partial C_{d}} (\\frac{\\partial}{\\partial \\mu} (- C_{d} + \\mu) + 1) = \\frac{d}{d C_{d}} 2 and \\frac{d}{d C_{d}} 2 + \\frac{\\partial}{\\partial C_{d}} (\\frac{\\partial}{\\partial \\mu} (- C_{d} + \\mu) + 1) = 2 \\frac{d}{d C_{d}} 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["differentiate", 5, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 6, "Derivative(Integer(2), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integer(2), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integer(2), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} e^{\\mathbf{g}}, then derive y{(\\mathbf{g})} = e^{\\mathbf{g}}, then obtain \\frac{d}{d \\mathbf{g}} e^{\\mathbf{g}} = \\frac{d^{2}}{d \\mathbf{g}^{2}} e^{\\mathbf{g}}", "derivation": "y{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} e^{\\mathbf{g}} and \\frac{d}{d \\mathbf{g}} y{(\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{g}^{2}} e^{\\mathbf{g}} and y{(\\mathbf{g})} = e^{\\mathbf{g}} and \\frac{d}{d \\mathbf{g}} e^{\\mathbf{g}} = \\frac{d^{2}}{d \\mathbf{g}^{2}} e^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{g}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 1], "Equality(Function('y')(Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(exp(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"]]}, {"prompt": "Given g{(\\varphi^*,\\theta,\\rho)} = - \\rho + \\frac{\\theta}{\\varphi^*}, then obtain \\varphi^* (- \\rho + \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*})) g{(\\varphi^*,\\theta,\\rho)} = \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}) (- \\rho + \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}))", "derivation": "g{(\\varphi^*,\\theta,\\rho)} = - \\rho + \\frac{\\theta}{\\varphi^*} and \\varphi^* g{(\\varphi^*,\\theta,\\rho)} = \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}) and - \\rho + \\varphi^* g{(\\varphi^*,\\theta,\\rho)} = - \\rho + \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}) and \\varphi^* (- \\rho + \\varphi^* g{(\\varphi^*,\\theta,\\rho)}) g{(\\varphi^*,\\theta,\\rho)} = \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}) (- \\rho + \\varphi^* g{(\\varphi^*,\\theta,\\rho)}) and \\varphi^* (- \\rho + \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*})) g{(\\varphi^*,\\theta,\\rho)} = \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}) (- \\rho + \\varphi^* (- \\rho + \\frac{\\theta}{\\varphi^*}))", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True)))), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))), Function('g')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))))))"]]}, {"prompt": "Given \\pi{(a)} = \\sin{(a)}, then derive \\dot{\\mathbf{r}} + \\pi{(a)} = \\mathbf{s} + \\sin{(a)}, then obtain (\\dot{\\mathbf{r}} + \\sin{(a)})^{\\mathbf{s}} = (\\mathbf{s} + \\pi{(a)})^{\\mathbf{s}}", "derivation": "\\pi{(a)} = \\sin{(a)} and \\frac{d}{d a} \\pi{(a)} = \\frac{d}{d a} \\sin{(a)} and \\int \\frac{d}{d a} \\pi{(a)} da = \\int \\frac{d}{d a} \\sin{(a)} da and \\dot{\\mathbf{r}} + \\pi{(a)} = \\mathbf{s} + \\sin{(a)} and \\dot{\\mathbf{r}} + \\sin{(a)} = \\mathbf{s} + \\sin{(a)} and \\dot{\\mathbf{r}} + \\pi{(a)} = \\mathbf{s} + \\pi{(a)} and \\mathbf{s} + \\sin{(a)} = \\mathbf{s} + \\pi{(a)} and (\\mathbf{s} + \\sin{(a)})^{\\mathbf{s}} = (\\mathbf{s} + \\pi{(a)})^{\\mathbf{s}} and (\\dot{\\mathbf{r}} + \\sin{(a)})^{\\mathbf{s}} = (\\mathbf{s} + \\pi{(a)})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\pi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Integral(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), sin(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))))"], [["power", 7, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('a', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), sin(Symbol('a', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\pi')(Symbol('a', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given x{(M_{E})} = e^{M_{E}}, then derive \\frac{d}{d M_{E}} x{(M_{E})} = e^{M_{E}}, then obtain x{(M_{E})} - \\frac{d}{d M_{E}} x{(M_{E})} = 0", "derivation": "x{(M_{E})} = e^{M_{E}} and x{(M_{E})} - e^{M_{E}} = 0 and \\frac{d}{d M_{E}} x{(M_{E})} = \\frac{d}{d M_{E}} e^{M_{E}} and \\frac{d}{d M_{E}} x{(M_{E})} = e^{M_{E}} and x{(M_{E})} - \\frac{d}{d M_{E}} x{(M_{E})} = 0", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["minus", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Function('x')(Symbol('M_E', commutative=True)), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Integer(0))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), exp(Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('x')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{P}{(b)} = \\log{(e^{b})}, then obtain 2 \\mathbf{P}{(b)} e^{b} + \\mathbf{P}{(b)} = \\mathbf{P}{(b)} e^{b} + e^{b} \\log{(e^{b})} + \\log{(e^{b})}", "derivation": "\\mathbf{P}{(b)} = \\log{(e^{b})} and \\mathbf{P}{(b)} e^{b} = e^{b} \\log{(e^{b})} and \\mathbf{P}{(b)} e^{b} + \\mathbf{P}{(b)} = \\mathbf{P}{(b)} e^{b} + \\log{(e^{b})} and 2 \\mathbf{P}{(b)} e^{b} + \\mathbf{P}{(b)} = \\mathbf{P}{(b)} e^{b} + \\mathbf{P}{(b)} + e^{b} \\log{(e^{b})} and 2 \\mathbf{P}{(b)} e^{b} + \\mathbf{P}{(b)} = \\mathbf{P}{(b)} e^{b} + e^{b} \\log{(e^{b})} + \\log{(e^{b})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))"], [["times", 1, "exp(Symbol('b', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Mul(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))))"], [["add", 1, "Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{P}')(Symbol('b', commutative=True))), Add(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), log(exp(Symbol('b', commutative=True)))))"], [["add", 2, "Add(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{P}')(Symbol('b', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{P}')(Symbol('b', commutative=True))), Add(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), Mul(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{P}')(Symbol('b', commutative=True))), Add(Mul(Function('\\\\mathbf{P}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Mul(exp(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True)))), log(exp(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} = P_{e}^{\\hat{H}_{\\lambda}}, then obtain ((\\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} + 1)^{P_{e}})^{\\hat{H}_{\\lambda}} = ((P_{e}^{\\hat{H}_{\\lambda}} + 1)^{P_{e}})^{\\hat{H}_{\\lambda}}", "derivation": "\\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} = P_{e}^{\\hat{H}_{\\lambda}} and \\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} + 1 = P_{e}^{\\hat{H}_{\\lambda}} + 1 and (\\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} + 1)^{P_{e}} = (P_{e}^{\\hat{H}_{\\lambda}} + 1)^{P_{e}} and ((\\operatorname{C_{2}}{(P_{e},\\hat{H}_{\\lambda})} + 1)^{P_{e}})^{\\hat{H}_{\\lambda}} = ((P_{e}^{\\hat{H}_{\\lambda}} + 1)^{P_{e}})^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('C_2')(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Add(Function('C_2')(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('P_e', commutative=True)), Pow(Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('P_e', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Pow(Add(Function('C_2')(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('P_e', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Pow(Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1)), Symbol('P_e', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\mathbf{J}_M)} = \\sin{(e^{\\mathbf{J}_M})}, then obtain \\iint \\varphi{(\\mathbf{J}_M)} \\sin{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M d\\mathbf{J}_M = \\iint \\sin^{2}{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M d\\mathbf{J}_M", "derivation": "\\varphi{(\\mathbf{J}_M)} = \\sin{(e^{\\mathbf{J}_M})} and \\varphi{(\\mathbf{J}_M)} \\sin{(e^{\\mathbf{J}_M})} = \\sin^{2}{(e^{\\mathbf{J}_M})} and \\int \\varphi{(\\mathbf{J}_M)} \\sin{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M = \\int \\sin^{2}{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M and \\iint \\varphi{(\\mathbf{J}_M)} \\sin{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M d\\mathbf{J}_M = \\iint \\sin^{2}{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 1, "sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then derive \\frac{d}{d \\mathbf{J}_M} \\mathbf{F}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain \\mathbf{F}{(\\mathbf{J}_M)} - \\cos{(\\mathbf{J}_M)} = \\mathbf{F}{(\\mathbf{J}_M)} - \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)}", "derivation": "\\mathbf{F}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\mathbf{F}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\mathbf{F}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\cos{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and - \\mathbf{F}{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)} = - \\mathbf{F}{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and \\mathbf{F}{(\\mathbf{J}_M)} - \\cos{(\\mathbf{J}_M)} = \\mathbf{F}{(\\mathbf{J}_M)} - \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["minus", 4, "Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["times", 5, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\nabla{(Z)} = e^{Z}, then obtain 1 = \\nabla^{- Z \\nabla^{- Z \\nabla^{- Z}{(Z)} (e^{Z})^{Z}}{(Z)} (e^{Z})^{Z}}{(Z)} (e^{Z})^{Z}", "derivation": "\\nabla{(Z)} = e^{Z} and \\nabla^{Z}{(Z)} = (e^{Z})^{Z} and 1 = \\nabla^{- Z}{(Z)} (e^{Z})^{Z} and - Z = - Z \\nabla^{- Z}{(Z)} (e^{Z})^{Z} and 1 = \\nabla^{- Z \\nabla^{- Z}{(Z)} (e^{Z})^{Z}}{(Z)} (e^{Z})^{Z} and 1 = \\nabla^{- Z \\nabla^{- Z \\nabla^{- Z}{(Z)} (e^{Z})^{Z}}{(Z)} (e^{Z})^{Z}}{(Z)} (e^{Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True), Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Mul(Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True), Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(1), Mul(Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True), Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True), Pow(Function('\\\\nabla')(Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))), Pow(exp(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given I{(\\theta_1,\\hat{p}_0,\\sigma_p)} = \\hat{p}_0 \\theta_1 + \\sigma_p, then obtain \\sin{(\\frac{\\partial}{\\partial \\sigma_p} (- \\hat{p}_0 \\theta_1 - \\sigma_p + I{(\\theta_1,\\hat{p}_0,\\sigma_p)}))} = \\sin{(\\frac{d}{d \\sigma_p} 0)}", "derivation": "I{(\\theta_1,\\hat{p}_0,\\sigma_p)} = \\hat{p}_0 \\theta_1 + \\sigma_p and - \\hat{p}_0 \\theta_1 - \\sigma_p + I{(\\theta_1,\\hat{p}_0,\\sigma_p)} = 0 and \\frac{\\partial}{\\partial \\sigma_p} (- \\hat{p}_0 \\theta_1 - \\sigma_p + I{(\\theta_1,\\hat{p}_0,\\sigma_p)}) = \\frac{d}{d \\sigma_p} 0 and \\sin{(\\frac{\\partial}{\\partial \\sigma_p} (- \\hat{p}_0 \\theta_1 - \\sigma_p + I{(\\theta_1,\\hat{p}_0,\\sigma_p)}))} = \\sin{(\\frac{d}{d \\sigma_p} 0)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), sin(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(f^{*})} = \\sin{(f^{*})}, then obtain 2^{2 f^{*}} \\sigma_{x}{(V_{\\mathbf{E}})} \\sin^{2 f^{*}}{(f^{*})} = (2 \\sin{(f^{*})})^{2 f^{*}} \\sigma_{x}{(V_{\\mathbf{E}})}", "derivation": "\\mathbf{g}{(f^{*})} = \\sin{(f^{*})} and 2 \\mathbf{g}{(f^{*})} = \\mathbf{g}{(f^{*})} + \\sin{(f^{*})} and (2 \\mathbf{g}{(f^{*})})^{f^{*}} = (\\mathbf{g}{(f^{*})} + \\sin{(f^{*})})^{f^{*}} and (2 \\mathbf{g}{(f^{*})})^{2 f^{*}} = (\\mathbf{g}{(f^{*})} + \\sin{(f^{*})})^{2 f^{*}} and 2^{2 f^{*}} \\mathbf{g}^{2 f^{*}}{(f^{*})} = (\\mathbf{g}{(f^{*})} + \\sin{(f^{*})})^{2 f^{*}} and 2^{2 f^{*}} \\sin^{2 f^{*}}{(f^{*})} = (2 \\sin{(f^{*})})^{2 f^{*}} and 2^{2 f^{*}} \\sigma_{x}{(V_{\\mathbf{E}})} \\sin^{2 f^{*}}{(f^{*})} = (2 \\sin{(f^{*})})^{2 f^{*}} \\sigma_{x}{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True))), Add(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True))), Mul(Integer(2), Symbol('f^*', commutative=True))), Pow(Add(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True))), Mul(Integer(2), Symbol('f^*', commutative=True))))"], [["expand", 4], "Equality(Mul(Pow(Integer(2), Mul(Integer(2), Symbol('f^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)))), Pow(Add(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True))), Mul(Integer(2), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Integer(2), Mul(Integer(2), Symbol('f^*', commutative=True))), Pow(sin(Symbol('f^*', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)))), Pow(Mul(Integer(2), sin(Symbol('f^*', commutative=True))), Mul(Integer(2), Symbol('f^*', commutative=True))))"], [["times", 6, "Function('\\\\sigma_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Pow(Integer(2), Mul(Integer(2), Symbol('f^*', commutative=True))), Function('\\\\sigma_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('f^*', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)))), Mul(Pow(Mul(Integer(2), sin(Symbol('f^*', commutative=True))), Mul(Integer(2), Symbol('f^*', commutative=True))), Function('\\\\sigma_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(c_{0})} = \\frac{d}{d c_{0}} \\log{(c_{0})}, then derive \\operatorname{t_{2}}{(c_{0})} = \\frac{1}{c_{0}}, then obtain \\operatorname{t_{2}}{(c_{0})} + \\frac{1}{c_{0}} = \\frac{d}{d c_{0}} \\log{(c_{0})} + \\frac{1}{c_{0}}", "derivation": "\\operatorname{t_{2}}{(c_{0})} = \\frac{d}{d c_{0}} \\log{(c_{0})} and 2 \\operatorname{t_{2}}{(c_{0})} = \\operatorname{t_{2}}{(c_{0})} + \\frac{d}{d c_{0}} \\log{(c_{0})} and \\operatorname{t_{2}}{(c_{0})} = \\frac{1}{c_{0}} and \\frac{2}{c_{0}} = \\frac{d}{d c_{0}} \\log{(c_{0})} + \\frac{1}{c_{0}} and \\frac{2}{c_{0}} = \\operatorname{t_{2}}{(c_{0})} + \\frac{1}{c_{0}} and \\operatorname{t_{2}}{(c_{0})} + \\frac{1}{c_{0}} = \\frac{d}{d c_{0}} \\log{(c_{0})} + \\frac{1}{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('c_0', commutative=True)), Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["add", 1, "Function('t_2')(Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(2), Function('t_2')(Symbol('c_0', commutative=True))), Add(Function('t_2')(Symbol('c_0', commutative=True)), Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('t_2')(Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Pow(Symbol('c_0', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Symbol('c_0', commutative=True), Integer(-1))), Add(Function('t_2')(Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('t_2')(Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Symbol('c_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given z{(S,f^{*})} = \\sin{(S + f^{*})} and \\operatorname{a^{\\dagger}}{(S,f^{*})} = f^{*} + \\sin{(S + f^{*})}, then obtain S + f^{*} + \\operatorname{a^{\\dagger}}{(S,f^{*})} = S + 2 f^{*} + z{(S,f^{*})}", "derivation": "z{(S,f^{*})} = \\sin{(S + f^{*})} and \\operatorname{a^{\\dagger}}{(S,f^{*})} = f^{*} + \\sin{(S + f^{*})} and S + f^{*} + \\operatorname{a^{\\dagger}}{(S,f^{*})} = S + 2 f^{*} + \\sin{(S + f^{*})} and S + f^{*} + \\operatorname{a^{\\dagger}}{(S,f^{*})} = S + 2 f^{*} + z{(S,f^{*})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)), sin(Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), sin(Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))))"], [["add", 2, "Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True)), sin(Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Symbol('S', commutative=True), Symbol('f^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('f^*', commutative=True)), Function('z')(Symbol('S', commutative=True), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given b{(G)} = \\cos{(G)}, then derive \\int b{(G)} dG = \\hat{H} + \\sin{(G)}, then obtain \\hat{H} + b{(G)} \\cos{(G)} + \\sin{(G)} = \\hat{H} + \\sin{(G)} + \\cos^{2}{(G)}", "derivation": "b{(G)} = \\cos{(G)} and b{(G)} \\cos{(G)} = \\cos^{2}{(G)} and \\int b{(G)} dG = \\int \\cos{(G)} dG and b{(G)} \\cos{(G)} + \\int \\cos{(G)} dG = \\cos^{2}{(G)} + \\int \\cos{(G)} dG and \\int b{(G)} dG = \\hat{H} + \\sin{(G)} and b{(G)} \\cos{(G)} + \\int b{(G)} dG = \\cos^{2}{(G)} + \\int b{(G)} dG and \\hat{H} + b{(G)} \\cos{(G)} + \\sin{(G)} = \\hat{H} + \\sin{(G)} + \\cos^{2}{(G)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["times", 1, "cos(Symbol('G', commutative=True))"], "Equality(Mul(Function('b')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Pow(cos(Symbol('G', commutative=True)), Integer(2)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('b')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Function('b')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Add(Pow(cos(Symbol('G', commutative=True)), Integer(2)), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('b')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), sin(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Function('b')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Integral(Function('b')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Add(Pow(cos(Symbol('G', commutative=True)), Integer(2)), Integral(Function('b')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Function('b')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), sin(Symbol('G', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), sin(Symbol('G', commutative=True)), Pow(cos(Symbol('G', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\varepsilon{(\\tilde{g}^*,m_{s},\\chi)} = \\chi + \\tilde{g}^* - m_{s}, then obtain (\\frac{\\varepsilon{(\\tilde{g}^*,m_{s},\\chi)}}{(\\chi + \\tilde{g}^* - m_{s})^{2}})^{\\tilde{g}^*} = (\\frac{1}{\\chi + \\tilde{g}^* - m_{s}})^{\\tilde{g}^*}", "derivation": "\\varepsilon{(\\tilde{g}^*,m_{s},\\chi)} = \\chi + \\tilde{g}^* - m_{s} and \\frac{\\varepsilon{(\\tilde{g}^*,m_{s},\\chi)}}{\\chi + \\tilde{g}^* - m_{s}} = 1 and \\frac{\\varepsilon{(\\tilde{g}^*,m_{s},\\chi)}}{(\\chi + \\tilde{g}^* - m_{s})^{2}} = \\frac{1}{\\chi + \\tilde{g}^* - m_{s}} and (\\frac{\\varepsilon{(\\tilde{g}^*,m_{s},\\chi)}}{(\\chi + \\tilde{g}^* - m_{s})^{2}})^{\\tilde{g}^*} = (\\frac{1}{\\chi + \\tilde{g}^* - m_{s}})^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(1))"], [["times", 2, "Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-2)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-1)))"], [["power", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-2)), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(p)} = \\sin{(p)}, then obtain p \\varepsilon_{0}^{2}{(p)} = p \\sin^{2}{(p)}", "derivation": "\\varepsilon_{0}{(p)} = \\sin{(p)} and p \\varepsilon_{0}{(p)} = p \\sin{(p)} and p \\varepsilon_{0}^{2}{(p)} = p \\varepsilon_{0}{(p)} \\sin{(p)} and p \\varepsilon_{0}{(p)} \\sin{(p)} = p \\sin^{2}{(p)} and p \\varepsilon_{0}^{2}{(p)} = p \\sin^{2}{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\varepsilon_0')(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), sin(Symbol('p', commutative=True))))"], [["times", 2, "Function('\\\\varepsilon_0')(Symbol('p', commutative=True))"], "Equality(Mul(Symbol('p', commutative=True), Pow(Function('\\\\varepsilon_0')(Symbol('p', commutative=True)), Integer(2))), Mul(Symbol('p', commutative=True), Function('\\\\varepsilon_0')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\varepsilon_0')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Pow(sin(Symbol('p', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('p', commutative=True), Pow(Function('\\\\varepsilon_0')(Symbol('p', commutative=True)), Integer(2))), Mul(Symbol('p', commutative=True), Pow(sin(Symbol('p', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\rho_{b}{(\\delta,S)} = \\delta + e^{S}, then obtain - \\delta + (- \\delta + \\rho_{b}{(\\delta,S)}) e^{S} + \\rho_{b}{(\\delta,S)} = (- \\delta + \\rho_{b}{(\\delta,S)}) e^{S} + e^{S}", "derivation": "\\rho_{b}{(\\delta,S)} = \\delta + e^{S} and - \\delta + \\rho_{b}{(\\delta,S)} = e^{S} and (- \\delta + \\rho_{b}{(\\delta,S)}) e^{S} = e^{2 S} and - \\delta + \\rho_{b}{(\\delta,S)} + e^{2 S} = e^{2 S} + e^{S} and - \\delta + (- \\delta + \\rho_{b}{(\\delta,S)}) e^{S} + \\rho_{b}{(\\delta,S)} = (- \\delta + \\rho_{b}{(\\delta,S)}) e^{S} + e^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), Add(Symbol('\\\\delta', commutative=True), exp(Symbol('S', commutative=True))))"], [["minus", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True))), exp(Symbol('S', commutative=True)))"], [["times", 2, "exp(Symbol('S', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), exp(Mul(Integer(2), Symbol('S', commutative=True))))"], [["add", 2, "exp(Mul(Integer(2), Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), exp(Mul(Integer(2), Symbol('S', commutative=True)))), Add(exp(Mul(Integer(2), Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))), exp(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(A_{z},C_{2})} = \\sin{(A_{z} C_{2})}, then derive \\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})} = C_{2} \\cos{(A_{z} C_{2})}, then obtain \\frac{\\log{(\\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})})}}{C_{2}} = \\frac{\\log{(C_{2} \\cos{(A_{z} C_{2})})}}{C_{2}}", "derivation": "\\operatorname{F_{g}}{(A_{z},C_{2})} = \\sin{(A_{z} C_{2})} and \\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})} = \\frac{\\partial}{\\partial A_{z}} \\sin{(A_{z} C_{2})} and \\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})} = C_{2} \\cos{(A_{z} C_{2})} and \\log{(\\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})})} = \\log{(C_{2} \\cos{(A_{z} C_{2})})} and \\frac{\\log{(\\frac{\\partial}{\\partial A_{z}} \\operatorname{F_{g}}{(A_{z},C_{2})})}}{C_{2}} = \\frac{\\log{(C_{2} \\cos{(A_{z} C_{2})})}}{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)), sin(Mul(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Symbol('C_2', commutative=True), cos(Mul(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)))))"], [["log", 3], "Equality(log(Derivative(Function('F_g')(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), log(Mul(Symbol('C_2', commutative=True), cos(Mul(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True))))))"], [["divide", 4, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), log(Derivative(Function('F_g')(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), log(Mul(Symbol('C_2', commutative=True), cos(Mul(Symbol('A_z', commutative=True), Symbol('C_2', commutative=True)))))))"]]}, {"prompt": "Given \\phi{(f^{*})} = e^{f^{*}}, then derive \\frac{d}{d f^{*}} \\phi{(f^{*})} = e^{f^{*}}, then obtain \\frac{d}{d f^{*}} \\phi{(f^{*})} = \\frac{d^{2}}{d (f^{*})^{2}} \\phi{(f^{*})}", "derivation": "\\phi{(f^{*})} = e^{f^{*}} and \\frac{d}{d f^{*}} \\phi{(f^{*})} = \\frac{d}{d f^{*}} e^{f^{*}} and \\frac{d}{d f^{*}} \\phi{(f^{*})} = e^{f^{*}} and \\frac{d}{d f^{*}} \\phi{(f^{*})} = \\frac{d^{2}}{d (f^{*})^{2}} \\phi{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), exp(Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\phi')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Function('\\\\phi')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given z{(B)} = \\frac{d}{d B} \\log{(B)}, then derive z{(B)} = \\frac{1}{B}, then derive \\int z{(B)} dB = E_{n} + \\log{(B)}, then obtain - C_{d} + \\int z{(B)} dB = - C_{d} + E_{n} + \\log{(B)}", "derivation": "z{(B)} = \\frac{d}{d B} \\log{(B)} and z{(B)} = \\frac{1}{B} and \\int z{(B)} dB = \\int \\frac{1}{B} dB and \\int \\frac{d}{d B} \\log{(B)} dB = \\int \\frac{1}{B} dB and \\int z{(B)} dB = \\int \\frac{d}{d B} \\log{(B)} dB and \\int z{(B)} dB = E_{n} + \\log{(B)} and - C_{d} + \\int z{(B)} dB = - C_{d} + E_{n} + \\log{(B)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('B', commutative=True)), Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('z')(Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Function('z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Symbol('B', commutative=True), Integer(-1)), Tuple(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Pow(Symbol('B', commutative=True), Integer(-1)), Tuple(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('E_n', commutative=True), log(Symbol('B', commutative=True))))"], [["minus", 6, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Integral(Function('z')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('E_n', commutative=True), log(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\mathbf{P},\\mathbf{g})} = \\mathbf{g}^{\\mathbf{P}}, then obtain \\int 4 \\pi^{2}{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g} = \\int 2 (\\mathbf{g}^{\\mathbf{P}} + \\pi{(\\mathbf{P},\\mathbf{g})}) \\pi{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g}", "derivation": "\\pi{(\\mathbf{P},\\mathbf{g})} = \\mathbf{g}^{\\mathbf{P}} and 2 \\pi{(\\mathbf{P},\\mathbf{g})} = \\mathbf{g}^{\\mathbf{P}} + \\pi{(\\mathbf{P},\\mathbf{g})} and 4 \\pi^{2}{(\\mathbf{P},\\mathbf{g})} = 2 (\\mathbf{g}^{\\mathbf{P}} + \\pi{(\\mathbf{P},\\mathbf{g})}) \\pi{(\\mathbf{P},\\mathbf{g})} and \\int 4 \\pi^{2}{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g} = \\int 2 (\\mathbf{g}^{\\mathbf{P}} + \\pi{(\\mathbf{P},\\mathbf{g})}) \\pi{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Mul(Integer(2), Add(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Integer(2), Add(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\Psi)} = \\cos{(\\log{(\\Psi)})} and \\mu_{0}{(\\Psi)} = \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})}, then obtain (\\hat{p} + \\frac{d}{d \\Psi} \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})})^{\\hat{p}} = (\\hat{p} + \\frac{d}{d \\Psi} 1)^{\\hat{p}}", "derivation": "\\operatorname{A_{1}}{(\\Psi)} = \\cos{(\\log{(\\Psi)})} and \\mu_{0}{(\\Psi)} = \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})} and \\mu_{0}{(\\Psi)} = 1 and \\frac{d}{d \\Psi} \\mu_{0}{(\\Psi)} = \\frac{d}{d \\Psi} 1 and \\frac{d}{d \\Psi} \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})} = \\frac{d}{d \\Psi} 1 and \\hat{p} + \\frac{d}{d \\Psi} \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})} = \\hat{p} + \\frac{d}{d \\Psi} 1 and (\\hat{p} + \\frac{d}{d \\Psi} \\operatorname{A_{1}}^{- \\Psi}{(\\Psi)} \\cos^{\\Psi}{(\\log{(\\Psi)})})^{\\hat{p}} = (\\hat{p} + \\frac{d}{d \\Psi} 1)^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\Psi', commutative=True)), cos(log(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(cos(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mu_0')(Symbol('\\\\Psi', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(cos(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["add", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Mul(Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(cos(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Mul(Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Pow(cos(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given s{(\\dot{x},y)} = - \\sin{(\\dot{x} - y)}, then derive - \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)} = \\cos{(\\dot{x} - y)}, then obtain (- \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)})^{y} = (- \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)})^{y}", "derivation": "s{(\\dot{x},y)} = - \\sin{(\\dot{x} - y)} and \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)} = \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)} and - \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)} = - \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)} and - \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)} = \\cos{(\\dot{x} - y)} and - \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)} = \\cos{(\\dot{x} - y)} and (- \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)})^{y} = \\cos^{y}{(\\dot{x} - y)} and (- \\frac{\\partial}{\\partial \\dot{x}} - \\sin{(\\dot{x} - y)})^{y} = (- \\frac{\\partial}{\\partial \\dot{x}} s{(\\dot{x},y)})^{y}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('s')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('s')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), cos(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), cos(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Symbol('y', commutative=True)), Pow(cos(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Mul(Integer(-1), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Symbol('y', commutative=True)), Pow(Mul(Integer(-1), Derivative(Function('s')(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(x^\\prime,a)} = \\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}, then obtain (\\frac{x^\\prime (x^\\prime)^{- a} \\mu_{0}{(x^\\prime,a)}}{a} + \\frac{x^\\prime (x^\\prime)^{- a}}{a})^{a} = (1 + \\frac{x^\\prime (x^\\prime)^{- a}}{a})^{a}", "derivation": "\\mu_{0}{(x^\\prime,a)} = \\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a} and \\frac{\\mu_{0}{(x^\\prime,a)}}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}} = 1 and \\frac{\\mu_{0}{(x^\\prime,a)}}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}} + \\frac{1}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}} = 1 + \\frac{1}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}} and (\\frac{\\mu_{0}{(x^\\prime,a)}}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}} + \\frac{1}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}})^{a} = (1 + \\frac{1}{\\frac{\\partial}{\\partial x^\\prime} (x^\\prime)^{a}})^{a} and (\\frac{x^\\prime (x^\\prime)^{- a} \\mu_{0}{(x^\\prime,a)}}{a} + \\frac{x^\\prime (x^\\prime)^{- a}}{a})^{a} = (1 + \\frac{x^\\prime (x^\\prime)^{- a}}{a})^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["add", 2, "Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Add(Integer(1), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Symbol('a', commutative=True)), Pow(Add(Integer(1), Pow(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Symbol('a', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))), Symbol('a', commutative=True)), Pow(Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{x})} = \\cos{(\\hat{x})}, then obtain \\operatorname{t_{2}}^{12}{(\\hat{x})} = \\operatorname{t_{2}}^{3}{(\\hat{x})} \\cos^{9}{(\\hat{x})}", "derivation": "\\operatorname{t_{2}}{(\\hat{x})} = \\cos{(\\hat{x})} and \\operatorname{t_{2}}^{2}{(\\hat{x})} = \\operatorname{t_{2}}{(\\hat{x})} \\cos{(\\hat{x})} and \\operatorname{t_{2}}^{4}{(\\hat{x})} = \\operatorname{t_{2}}^{2}{(\\hat{x})} \\cos^{2}{(\\hat{x})} and \\operatorname{t_{2}}^{2}{(\\hat{x})} \\cos^{2}{(\\hat{x})} = \\operatorname{t_{2}}{(\\hat{x})} \\cos^{3}{(\\hat{x})} and \\operatorname{t_{2}}^{4}{(\\hat{x})} = \\operatorname{t_{2}}{(\\hat{x})} \\cos^{3}{(\\hat{x})} and \\operatorname{t_{2}}^{12}{(\\hat{x})} = \\operatorname{t_{2}}^{3}{(\\hat{x})} \\cos^{9}{(\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Function('t_2')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Mul(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(4)), Mul(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(2))), Mul(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(4)), Mul(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(3))))"], [["power", 5, 3], "Equality(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(12)), Mul(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Integer(3)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(9))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{E},V_{\\mathbf{E}})} = \\mathbf{E} \\sin{(V_{\\mathbf{E}})} and \\operatorname{f_{\\mathbf{v}}}{(\\phi_2,t)} = \\phi_2^{t}, then obtain \\operatorname{f_{\\mathbf{v}}}{(\\phi_2,t)} - \\cos{(\\mathbf{E} \\sin{(V_{\\mathbf{E}})})} = \\phi_2^{t} - \\cos{(\\mathbf{E} \\sin{(V_{\\mathbf{E}})})}", "derivation": "\\tilde{g}{(\\mathbf{E},V_{\\mathbf{E}})} = \\mathbf{E} \\sin{(V_{\\mathbf{E}})} and \\cos{(\\tilde{g}{(\\mathbf{E},V_{\\mathbf{E}})})} = \\cos{(\\mathbf{E} \\sin{(V_{\\mathbf{E}})})} and \\operatorname{f_{\\mathbf{v}}}{(\\phi_2,t)} = \\phi_2^{t} and \\operatorname{f_{\\mathbf{v}}}{(\\phi_2,t)} - \\cos{(\\tilde{g}{(\\mathbf{E},V_{\\mathbf{E}})})} = \\phi_2^{t} - \\cos{(\\tilde{g}{(\\mathbf{E},V_{\\mathbf{E}})})} and \\operatorname{f_{\\mathbf{v}}}{(\\phi_2,t)} - \\cos{(\\mathbf{E} \\sin{(V_{\\mathbf{E}})})} = \\phi_2^{t} - \\cos{(\\mathbf{E} \\sin{(V_{\\mathbf{E}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], ["get_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)))"], [["minus", 3, "cos(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), cos(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))), Add(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), cos(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))), Add(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and \\operatorname{M_{E}}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}, then obtain - \\operatorname{M_{E}}{(\\mathbf{F})} - \\log{(\\mathbf{F})} = - \\log{(\\mathbf{F})} - \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and \\operatorname{M_{E}}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and \\operatorname{M_{E}}{(\\mathbf{F})} + \\operatorname{c_{0}}{(\\mathbf{F})} = \\operatorname{c_{0}}{(\\mathbf{F})} + \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and - \\operatorname{M_{E}}{(\\mathbf{F})} - \\operatorname{c_{0}}{(\\mathbf{F})} = - \\operatorname{c_{0}}{(\\mathbf{F})} - \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and - \\operatorname{M_{E}}{(\\mathbf{F})} - \\log{(\\mathbf{F})} = - \\log{(\\mathbf{F})} - \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["add", 2, "Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('\\\\mathbf{F}', commutative=True)), Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given a{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 + \\phi_1, then derive 2 = 1 + \\frac{1}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}}, then obtain \\frac{\\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + \\phi_1)}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}} + 1 = 1 + \\frac{1}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}}", "derivation": "a{(\\phi_1,\\hat{x}_0)} = \\hat{x}_0 + \\phi_1 and \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + \\phi_1) and 1 = \\frac{\\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + \\phi_1)}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}} and 2 = \\frac{\\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + \\phi_1)}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}} + 1 and 2 = 1 + \\frac{1}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}} and \\frac{\\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + \\phi_1)}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}} + 1 = 1 + \\frac{1}{\\frac{\\partial}{\\partial \\hat{x}_0} a{(\\phi_1,\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Pow(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))))"], [["add", 3, 1], "Equality(Integer(2), Add(Mul(Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Pow(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Integer(2), Add(Integer(1), Pow(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Pow(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))), Integer(1)), Add(Integer(1), Pow(Derivative(Function('a')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given y{(H,A)} = H^{A}, then derive - \\frac{H^{A} (H + y{(H,A)}) \\log{(H)}}{(H + H^{A})^{2}} + \\frac{\\frac{\\partial}{\\partial A} y{(H,A)}}{H + H^{A}} = 0, then obtain - \\frac{H^{A} \\log{(H)}}{H + H^{A}} + \\frac{\\frac{\\partial}{\\partial A} y{(H,A)}}{H + H^{A}} = 0", "derivation": "y{(H,A)} = H^{A} and H + y{(H,A)} = H + H^{A} and (H + y{(H,A)}) \\frac{\\partial}{\\partial H} (H + H^{A}) (H + y{(H,A)}) = (H + H^{A}) \\frac{\\partial}{\\partial H} (H + H^{A}) (H + y{(H,A)}) and \\frac{H + y{(H,A)}}{H + H^{A}} = 1 and \\frac{\\partial}{\\partial A} \\frac{H + y{(H,A)}}{H + H^{A}} = \\frac{d}{d A} 1 and - \\frac{H^{A} (H + y{(H,A)}) \\log{(H)}}{(H + H^{A})^{2}} + \\frac{\\frac{\\partial}{\\partial A} y{(H,A)}}{H + H^{A}} = 0 and - \\frac{H^{A} \\log{(H)}}{H + H^{A}} + \\frac{\\frac{\\partial}{\\partial A} y{(H,A)}}{H + H^{A}} = 0", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True))), Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))))"], [["times", 2, "Derivative(Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-1)), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Integer(1))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-1)), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True)), Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-2)), Add(Symbol('H', commutative=True), Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True))), log(Symbol('H', commutative=True))), Mul(Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-1)), Derivative(Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True)), Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-1)), log(Symbol('H', commutative=True))), Mul(Pow(Add(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('A', commutative=True))), Integer(-1)), Derivative(Function('y')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then obtain \\log{((\\frac{\\theta_{1}^{2}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}})} = \\log{((\\frac{\\theta_{1}{(\\mathbf{F})} \\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}})}", "derivation": "\\theta_{1}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\frac{\\theta_{1}^{2}{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\theta_{1}{(\\mathbf{F})} \\cos{(\\mathbf{F})}}{\\mathbf{F}} and (\\frac{\\theta_{1}^{2}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} = (\\frac{\\theta_{1}{(\\mathbf{F})} \\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} and \\log{((\\frac{\\theta_{1}^{2}{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}})} = \\log{((\\frac{\\theta_{1}{(\\mathbf{F})} \\cos{(\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["log", 3], "Equality(log(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True))), log(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)} = - \\varphi + \\log{(J_{\\varepsilon})}, then obtain \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial \\varphi} (\\varphi + \\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)}) = \\frac{d^{2}}{d J_{\\varepsilon}d \\varphi} \\log{(J_{\\varepsilon})}", "derivation": "\\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)} = - \\varphi + \\log{(J_{\\varepsilon})} and \\varphi + \\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)} = \\log{(J_{\\varepsilon})} and \\frac{\\partial}{\\partial \\varphi} (\\varphi + \\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)}) = \\frac{d}{d \\varphi} \\log{(J_{\\varepsilon})} and \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial \\varphi} (\\varphi + \\operatorname{F_{c}}{(J_{\\varepsilon},\\varphi)}) = \\frac{d^{2}}{d J_{\\varepsilon}d \\varphi} \\log{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('F_c')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varphi', commutative=True))), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varphi', commutative=True), Function('F_c')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varphi', commutative=True), Function('F_c')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain \\int (\\frac{d^{2}}{d \\mathbf{H}^{2}} (\\nabla{(\\mathbf{H})} + 1))^{2} d\\mathbf{H} = \\int (\\frac{d^{2}}{d \\mathbf{H}^{2}} (e^{\\mathbf{H}} + 1))^{2} d\\mathbf{H}", "derivation": "\\nabla{(\\mathbf{H})} = e^{\\mathbf{H}} and \\nabla{(\\mathbf{H})} + 1 = e^{\\mathbf{H}} + 1 and \\frac{d}{d \\mathbf{H}} (\\nabla{(\\mathbf{H})} + 1) = \\frac{d}{d \\mathbf{H}} (e^{\\mathbf{H}} + 1) and \\frac{d^{2}}{d \\mathbf{H}^{2}} (\\nabla{(\\mathbf{H})} + 1) = \\frac{d^{2}}{d \\mathbf{H}^{2}} (e^{\\mathbf{H}} + 1) and (\\frac{d^{2}}{d \\mathbf{H}^{2}} (\\nabla{(\\mathbf{H})} + 1))^{2} = (\\frac{d^{2}}{d \\mathbf{H}^{2}} (e^{\\mathbf{H}} + 1))^{2} and \\int (\\frac{d^{2}}{d \\mathbf{H}^{2}} (\\nabla{(\\mathbf{H})} + 1))^{2} d\\mathbf{H} = \\int (\\frac{d^{2}}{d \\mathbf{H}^{2}} (e^{\\mathbf{H}} + 1))^{2} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Add(exp(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(Add(exp(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["power", 4, 2], "Equality(Pow(Derivative(Add(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Add(exp(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(2)))"], [["integrate", 5, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Pow(Derivative(Add(Function('\\\\nabla')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(2)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(Derivative(Add(exp(Symbol('\\\\mathbf{H}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(2)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\varphi^*)} = \\log{(\\varphi^*)}, then derive - \\frac{\\int \\mathbf{M}{(\\varphi^*)} d\\varphi^*}{\\varphi^*} = - \\frac{\\lambda + \\varphi^* \\log{(\\varphi^*)} - \\varphi^*}{\\varphi^*}, then obtain - \\frac{\\int \\log{(\\varphi^*)} d\\varphi^*}{\\varphi^*} = \\frac{- \\lambda - \\varphi^* \\log{(\\varphi^*)} + \\varphi^*}{\\varphi^*}", "derivation": "\\mathbf{M}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\int \\mathbf{M}{(\\varphi^*)} d\\varphi^* = \\int \\log{(\\varphi^*)} d\\varphi^* and - \\frac{\\int \\mathbf{M}{(\\varphi^*)} d\\varphi^*}{\\varphi^*} = - \\frac{\\int \\log{(\\varphi^*)} d\\varphi^*}{\\varphi^*} and - \\frac{\\int \\mathbf{M}{(\\varphi^*)} d\\varphi^*}{\\varphi^*} = - \\frac{\\lambda + \\varphi^* \\log{(\\varphi^*)} - \\varphi^*}{\\varphi^*} and - \\frac{\\int \\log{(\\varphi^*)} d\\varphi^*}{\\varphi^*} = \\frac{- \\lambda - \\varphi^* \\log{(\\varphi^*)} + \\varphi^*}{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\hat{X})} = e^{\\hat{X}}, then derive \\int \\psi^{*}{(\\hat{X})} d\\hat{X} = \\sigma_x + e^{\\hat{X}}, then obtain \\frac{d}{d \\sigma_x} \\int \\psi^{*}{(\\hat{X})} d\\hat{X} = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + e^{\\hat{X}})", "derivation": "\\psi^{*}{(\\hat{X})} = e^{\\hat{X}} and \\int \\psi^{*}{(\\hat{X})} d\\hat{X} = \\int e^{\\hat{X}} d\\hat{X} and \\int \\psi^{*}{(\\hat{X})} d\\hat{X} = \\sigma_x + e^{\\hat{X}} and \\frac{d}{d \\sigma_x} \\int \\psi^{*}{(\\hat{X})} d\\hat{X} = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + e^{\\hat{X}})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(exp(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(m_{s},L_{\\varepsilon})} = L_{\\varepsilon}^{m_{s}}, then obtain \\frac{d}{d L_{\\varepsilon}} 0 = \\frac{\\partial}{\\partial L_{\\varepsilon}} ((L_{\\varepsilon}^{m_{s}})^{m_{s}} - \\operatorname{n_{1}}^{m_{s}}{(m_{s},L_{\\varepsilon})})", "derivation": "\\operatorname{n_{1}}{(m_{s},L_{\\varepsilon})} = L_{\\varepsilon}^{m_{s}} and \\operatorname{n_{1}}^{m_{s}}{(m_{s},L_{\\varepsilon})} = (L_{\\varepsilon}^{m_{s}})^{m_{s}} and (L_{\\varepsilon}^{m_{s}})^{m_{s}} + \\operatorname{n_{1}}^{m_{s}}{(m_{s},L_{\\varepsilon})} = 2 (L_{\\varepsilon}^{m_{s}})^{m_{s}} and 0 = (L_{\\varepsilon}^{m_{s}})^{m_{s}} - \\operatorname{n_{1}}^{m_{s}}{(m_{s},L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} 0 = \\frac{\\partial}{\\partial L_{\\varepsilon}} ((L_{\\varepsilon}^{m_{s}})^{m_{s}} - \\operatorname{n_{1}}^{m_{s}}{(m_{s},L_{\\varepsilon})})", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('m_s', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))))"], [["minus", 3, "Add(Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('m_s', commutative=True)))"], "Equality(Integer(0), Add(Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('m_s', commutative=True)))))"], [["differentiate", 4, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Pow(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Function('n_1')(Symbol('m_s', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('m_s', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(n_{1})} = \\sin{(e^{n_{1}})} and \\hat{\\mathbf{r}}{(n_{1})} = n_{1} \\hat{x}{(n_{1})}, then obtain \\frac{\\hat{\\mathbf{r}}{(n_{1})}}{x{(n_{1})}} = \\frac{n_{1} \\sin{(e^{n_{1}})}}{x{(n_{1})}}", "derivation": "\\hat{x}{(n_{1})} = \\sin{(e^{n_{1}})} and n_{1} \\hat{x}{(n_{1})} = n_{1} \\sin{(e^{n_{1}})} and \\hat{\\mathbf{r}}{(n_{1})} = n_{1} \\hat{x}{(n_{1})} and \\hat{\\mathbf{r}}{(n_{1})} = n_{1} \\sin{(e^{n_{1}})} and \\frac{\\hat{\\mathbf{r}}{(n_{1})}}{x{(n_{1})}} = \\frac{n_{1} \\sin{(e^{n_{1}})}}{x{(n_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('n_1', commutative=True)), sin(exp(Symbol('n_1', commutative=True))))"], [["times", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Function('\\\\hat{x}')(Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), sin(exp(Symbol('n_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n_1', commutative=True)), Mul(Symbol('n_1', commutative=True), Function('\\\\hat{x}')(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n_1', commutative=True)), Mul(Symbol('n_1', commutative=True), sin(exp(Symbol('n_1', commutative=True)))))"], [["divide", 4, "Function('x')(Symbol('n_1', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n_1', commutative=True)), Pow(Function('x')(Symbol('n_1', commutative=True)), Integer(-1))), Mul(Symbol('n_1', commutative=True), Pow(Function('x')(Symbol('n_1', commutative=True)), Integer(-1)), sin(exp(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(v_{x})} = \\int \\cos{(v_{x})} dv_{x}, then obtain \\sin{((v_{x} \\operatorname{g_{\\varepsilon}}{(v_{x})})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}})} = \\sin{((v_{x} \\int \\cos{(v_{x})} dv_{x})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and v_{x} \\operatorname{g_{\\varepsilon}}{(v_{x})} = v_{x} \\int \\cos{(v_{x})} dv_{x} and (v_{x} \\operatorname{g_{\\varepsilon}}{(v_{x})})^{v_{x}} = (v_{x} \\int \\cos{(v_{x})} dv_{x})^{v_{x}} and (v_{x} \\operatorname{g_{\\varepsilon}}{(v_{x})})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}} = (v_{x} \\int \\cos{(v_{x})} dv_{x})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}} and \\sin{((v_{x} \\operatorname{g_{\\varepsilon}}{(v_{x})})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}})} = \\sin{((v_{x} \\int \\cos{(v_{x})} dv_{x})^{v_{x}} + \\frac{1}{\\cos{(v_{x})}})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["times", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Symbol('v_x', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)))"], [["add", 3, "Pow(cos(Symbol('v_x', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Mul(Symbol('v_x', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Integer(-1))), Add(Pow(Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Integer(-1))))"], [["sin", 4], "Equality(sin(Add(Pow(Mul(Symbol('v_x', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Integer(-1)))), sin(Add(Pow(Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)), Pow(cos(Symbol('v_x', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbb{I}{(E_{n},f_{\\mathbf{v}},\\mathbf{r})} = E_{n} \\mathbf{r} - f_{\\mathbf{v}}, then derive \\int \\mathbb{I}{(E_{n},f_{\\mathbf{v}},\\mathbf{r})} df_{\\mathbf{v}} = E_{n} \\mathbf{r} f_{\\mathbf{v}} - \\frac{f_{\\mathbf{v}}^{2}}{2} + v_{x}, then obtain \\int (E_{n} \\mathbf{r} - f_{\\mathbf{v}}) df_{\\mathbf{v}} = E_{n} \\mathbf{r} f_{\\mathbf{v}} - \\frac{f_{\\mathbf{v}}^{2}}{2} + v_{x}", "derivation": "\\mathbb{I}{(E_{n},f_{\\mathbf{v}},\\mathbf{r})} = E_{n} \\mathbf{r} - f_{\\mathbf{v}} and \\int \\mathbb{I}{(E_{n},f_{\\mathbf{v}},\\mathbf{r})} df_{\\mathbf{v}} = \\int (E_{n} \\mathbf{r} - f_{\\mathbf{v}}) df_{\\mathbf{v}} and \\int \\mathbb{I}{(E_{n},f_{\\mathbf{v}},\\mathbf{r})} df_{\\mathbf{v}} = E_{n} \\mathbf{r} f_{\\mathbf{v}} - \\frac{f_{\\mathbf{v}}^{2}}{2} + v_{x} and \\int (E_{n} \\mathbf{r} - f_{\\mathbf{v}}) df_{\\mathbf{v}} = E_{n} \\mathbf{r} f_{\\mathbf{v}} - \\frac{f_{\\mathbf{v}}^{2}}{2} + v_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Add(Mul(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}, then derive \\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = f^{\\prime} - \\cos{(\\Psi^{\\dagger})}, then derive a^{\\dagger} - \\cos{(\\Psi^{\\dagger})} = f^{\\prime} - \\cos{(\\Psi^{\\dagger})}, then obtain \\sin{(\\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})} = \\sin{(a^{\\dagger} - \\cos{(\\Psi^{\\dagger})})}", "derivation": "\\operatorname{v_{z}}{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})} and \\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = f^{\\prime} - \\cos{(\\Psi^{\\dagger})} and \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = f^{\\prime} - \\cos{(\\Psi^{\\dagger})} and a^{\\dagger} - \\cos{(\\Psi^{\\dagger})} = f^{\\prime} - \\cos{(\\Psi^{\\dagger})} and \\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = a^{\\dagger} - \\cos{(\\Psi^{\\dagger})} and \\sin{(\\int \\operatorname{v_{z}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})} = \\sin{(a^{\\dagger} - \\cos{(\\Psi^{\\dagger})})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_z')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('v_z')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["sin", 6], "Equality(sin(Integral(Function('v_z')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), sin(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given \\delta{(\\mathbb{I},\\theta_1)} = - \\mathbb{I} + \\sin{(\\theta_1)}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\delta{(\\mathbb{I},\\theta_1)} = -1, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + \\sin{(\\theta_1)}) = -1", "derivation": "\\delta{(\\mathbb{I},\\theta_1)} = - \\mathbb{I} + \\sin{(\\theta_1)} and \\frac{\\partial}{\\partial \\mathbb{I}} \\delta{(\\mathbb{I},\\theta_1)} = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + \\sin{(\\theta_1)}) and \\frac{\\partial}{\\partial \\mathbb{I}} \\delta{(\\mathbb{I},\\theta_1)} = -1 and \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + \\sin{(\\theta_1)}) = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(g,l)} = - g + l, then obtain \\frac{\\partial}{\\partial g} (-1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(g,l)}}{- g + l}) = \\frac{d}{d g} 0", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(g,l)} = - g + l and \\frac{\\operatorname{V_{\\mathbf{E}}}{(g,l)}}{- g + l} = 1 and -1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(g,l)}}{- g + l} = 0 and \\frac{\\partial}{\\partial g} (-1 + \\frac{\\operatorname{V_{\\mathbf{E}}}{(g,l)}}{- g + l}) = \\frac{d}{d g} 0", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('l', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True), Symbol('l', commutative=True))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('l', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True), Symbol('l', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('l', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}{(C_{2})} = e^{C_{2}}, then obtain - \\frac{1}{2 C_{2} (\\mathbf{J}{(C_{2})} - 2 e^{C_{2}})^{2}} = - \\frac{e^{- 2 C_{2}}}{2 C_{2}}", "derivation": "\\mathbf{J}{(C_{2})} = e^{C_{2}} and \\mathbf{J}{(C_{2})} - e^{C_{2}} = 0 and \\mathbf{J}{(C_{2})} - 2 e^{C_{2}} = - e^{C_{2}} and \\frac{1}{(\\mathbf{J}{(C_{2})} - 2 e^{C_{2}})^{2}} = e^{- 2 C_{2}} and - \\frac{1}{2 C_{2} (\\mathbf{J}{(C_{2})} - 2 e^{C_{2}})^{2}} = - \\frac{e^{- 2 C_{2}}}{2 C_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["minus", 1, "exp(Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), exp(Symbol('C_2', commutative=True)))), Integer(0))"], [["add", 2, "Mul(Integer(-1), exp(Symbol('C_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('C_2', commutative=True)))), Mul(Integer(-1), exp(Symbol('C_2', commutative=True))))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Add(Function('\\\\mathbf{J}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('C_2', commutative=True)))), Integer(-2)), exp(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(-1)), Pow(Add(Function('\\\\mathbf{J}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('C_2', commutative=True)))), Integer(-2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Integer(2), Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given I{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain \\frac{\\cos{(\\mathbb{I})}}{2} + \\frac{d}{d \\mathbb{I}} \\cos{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\cos{(\\mathbb{I})} + \\frac{\\cos^{2}{(\\mathbb{I})}}{I{(\\mathbb{I})} + \\cos{(\\mathbb{I})}}", "derivation": "I{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and 2 I{(\\mathbb{I})} = I{(\\mathbb{I})} + \\cos{(\\mathbb{I})} and I{(\\mathbb{I})} \\cos{(\\mathbb{I})} = \\cos^{2}{(\\mathbb{I})} and \\frac{\\cos{(\\mathbb{I})}}{2} = \\frac{\\cos^{2}{(\\mathbb{I})}}{2 I{(\\mathbb{I})}} and \\frac{\\cos{(\\mathbb{I})}}{2} = \\frac{\\cos^{2}{(\\mathbb{I})}}{I{(\\mathbb{I})} + \\cos{(\\mathbb{I})}} and \\frac{\\cos{(\\mathbb{I})}}{2} + \\frac{d}{d \\mathbb{I}} \\cos{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\cos{(\\mathbb{I})} + \\frac{\\cos^{2}{(\\mathbb{I})}}{I{(\\mathbb{I})} + \\cos{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Function('I')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Integer(2), Function('I')(Symbol('\\\\mathbb{I}', commutative=True))), Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))"], [["divide", 3, "Mul(Integer(2), Function('I')(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Rational(1, 2), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Rational(1, 2), Pow(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Rational(1, 2), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2))))"], [["add", 5, "Derivative(cos(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Rational(1, 2), cos(Symbol('\\\\mathbb{I}', commutative=True))), Derivative(cos(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Add(Derivative(cos(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Pow(Add(Function('I')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\hat{H}{(v)} = \\log{(\\sin{(v)})} and q{(v)} = \\log{(\\sin{(v)})}, then obtain - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\hat{H}{(v)} = - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\log{(\\sin{(v)})}", "derivation": "\\hat{H}{(v)} = \\log{(\\sin{(v)})} and \\hat{H}{(v)} \\log{(\\sin{(v)})} = \\log{(\\sin{(v)})}^{2} and q{(v)} = \\log{(\\sin{(v)})} and q{(v)} - \\log{(\\sin{(v)})}^{2} = - \\log{(\\sin{(v)})}^{2} + \\log{(\\sin{(v)})} and - \\hat{H}{(v)} \\log{(\\sin{(v)})} + q{(v)} = - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\log{(\\sin{(v)})} and \\hat{H}^{2}{(v)} - \\hat{H}{(v)} \\log{(\\sin{(v)})} + q{(v)} = \\hat{H}^{2}{(v)} - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\log{(\\sin{(v)})} and q{(v)} = \\hat{H}{(v)} and - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\hat{H}{(v)} = - \\hat{H}{(v)} \\log{(\\sin{(v)})} + \\log{(\\sin{(v)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True))))"], [["times", 1, "log(sin(Symbol('v', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), Pow(log(sin(Symbol('v', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('q')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True))))"], [["minus", 3, "Pow(log(sin(Symbol('v', commutative=True))), Integer(2))"], "Equality(Add(Function('q')(Symbol('v', commutative=True)), Mul(Integer(-1), Pow(log(sin(Symbol('v', commutative=True))), Integer(2)))), Add(Mul(Integer(-1), Pow(log(sin(Symbol('v', commutative=True))), Integer(2))), log(sin(Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), Function('q')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), log(sin(Symbol('v', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Integer(2)))"], "Equality(Add(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), Function('q')(Symbol('v', commutative=True))), Add(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), log(sin(Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Function('q')(Symbol('v', commutative=True)), Function('\\\\hat{H}')(Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), Function('\\\\hat{H}')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v', commutative=True)), log(sin(Symbol('v', commutative=True)))), log(sin(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(Q)} = \\cos{(Q)} and \\operatorname{C_{d}}{(Q)} = \\cos{(Q)}, then obtain \\operatorname{C_{d}}{(Q)} + \\cos{(Q)} = 2 \\cos{(Q)}", "derivation": "\\lambda{(Q)} = \\cos{(Q)} and \\lambda{(Q)} + \\cos{(Q)} = 2 \\cos{(Q)} and \\operatorname{C_{d}}{(Q)} = \\cos{(Q)} and \\operatorname{C_{d}}{(Q)} = \\lambda{(Q)} and \\operatorname{C_{d}}{(Q)} + \\cos{(Q)} = 2 \\cos{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["add", 1, "cos(Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Mul(Integer(2), cos(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('C_d')(Symbol('Q', commutative=True)), Function('\\\\lambda')(Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('C_d')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Mul(Integer(2), cos(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\rho{(v_{1},J)} = J v_{1}, then derive L_{\\varepsilon} + \\rho{(v_{1},J)} = J v_{1} + P_{e}, then obtain \\frac{J v_{1} - J + L_{\\varepsilon}}{J v_{1}} = \\frac{J v_{1} - J + P_{e}}{J v_{1}}", "derivation": "\\rho{(v_{1},J)} = J v_{1} and \\frac{\\partial}{\\partial J} \\rho{(v_{1},J)} = \\frac{\\partial}{\\partial J} J v_{1} and \\int \\frac{\\partial}{\\partial J} \\rho{(v_{1},J)} dJ = \\int \\frac{\\partial}{\\partial J} J v_{1} dJ and L_{\\varepsilon} + \\rho{(v_{1},J)} = J v_{1} + P_{e} and - J + L_{\\varepsilon} + \\rho{(v_{1},J)} = J v_{1} - J + P_{e} and - J + L_{\\varepsilon} + \\rho{(v_{1},J)} = - J + P_{e} + \\rho{(v_{1},J)} and \\frac{- J + L_{\\varepsilon} + \\rho{(v_{1},J)}}{J v_{1}} = \\frac{- J + P_{e} + \\rho{(v_{1},J)}}{J v_{1}} and \\frac{J v_{1} - J + L_{\\varepsilon}}{J v_{1}} = \\frac{J v_{1} - J + P_{e}}{J v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Derivative(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True))), Add(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Symbol('P_e', commutative=True)))"], [["minus", 4, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True))), Add(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('P_e', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True))))"], [["divide", 6, "Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('P_e', commutative=True), Function('\\\\rho')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} = \\phi_2 + \\varphi^*, then derive \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2 = \\frac{\\phi_2^{2}}{2} + \\phi_2 \\varphi^* + \\varphi, then obtain \\frac{\\varphi^*}{\\cos{(\\int (\\phi_2 + \\varphi^*) d\\phi_2)}} = \\frac{\\varphi^* (\\frac{\\phi_2^{2}}{2} + \\phi_2 \\varphi^* + \\varphi)}{(\\cos{(\\int (\\phi_2 + \\varphi^*) d\\phi_2)}) \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2}", "derivation": "\\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} = \\phi_2 + \\varphi^* and \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2 = \\int (\\phi_2 + \\varphi^*) d\\phi_2 and \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2 = \\frac{\\phi_2^{2}}{2} + \\phi_2 \\varphi^* + \\varphi and \\varphi^* \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2 = \\varphi^* (\\frac{\\phi_2^{2}}{2} + \\phi_2 \\varphi^* + \\varphi) and \\frac{\\varphi^*}{\\cos{(\\int (\\phi_2 + \\varphi^*) d\\phi_2)}} = \\frac{\\varphi^* (\\frac{\\phi_2^{2}}{2} + \\phi_2 \\varphi^* + \\varphi)}{(\\cos{(\\int (\\phi_2 + \\varphi^*) d\\phi_2)}) \\int \\operatorname{v_{1}}{(\\phi_2,\\varphi^*)} d\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["times", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["divide", 4, "Mul(cos(Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integral(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(cos(Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integer(-1))), Mul(Symbol('\\\\varphi^*', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(cos(Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integer(-1)), Pow(Integral(Function('v_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\delta{(\\lambda)} = \\log{(\\sin{(\\lambda)})}, then derive \\frac{d}{d \\lambda} \\delta{(\\lambda)} = \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}}, then obtain \\frac{d^{2}}{d \\lambda^{2}} \\delta{(\\lambda)} = \\frac{\\sin{(\\lambda)} \\frac{d^{2}}{d \\lambda^{2}} \\delta{(\\lambda)} \\frac{d}{d \\lambda} \\log{(\\sin{(\\lambda)})}}{\\cos{(\\lambda)}}", "derivation": "\\delta{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and \\frac{d}{d \\lambda} \\delta{(\\lambda)} = \\frac{d}{d \\lambda} \\log{(\\sin{(\\lambda)})} and \\frac{d}{d \\lambda} \\delta{(\\lambda)} = \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}} and \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}} = \\frac{d}{d \\lambda} \\log{(\\sin{(\\lambda)})} and 1 = \\frac{\\sin{(\\lambda)} \\frac{d}{d \\lambda} \\log{(\\sin{(\\lambda)})}}{\\cos{(\\lambda)}} and \\frac{d^{2}}{d \\lambda^{2}} \\delta{(\\lambda)} = \\frac{\\sin{(\\lambda)} \\frac{d^{2}}{d \\lambda^{2}} \\delta{(\\lambda)} \\frac{d}{d \\lambda} \\log{(\\sin{(\\lambda)})}}{\\cos{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), log(sin(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True))), Derivative(log(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True)))"], "Equality(Integer(1), Mul(sin(Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Derivative(log(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["times", 5, "Derivative(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2)))"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(sin(Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Derivative(log(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\theta,r_{0})} = \\frac{r_{0}}{\\theta}, then obtain (\\frac{\\hat{p}_0{(\\theta,r_{0})} + \\frac{r_{0}}{\\theta}}{\\hat{p}_0{(\\theta,r_{0})}})^{r_{0}} = 2^{r_{0}}", "derivation": "\\hat{p}_0{(\\theta,r_{0})} = \\frac{r_{0}}{\\theta} and \\hat{p}_0{(\\theta,r_{0})} + \\frac{r_{0}}{\\theta} = \\frac{2 r_{0}}{\\theta} and \\theta \\hat{p}_0{(\\theta,r_{0})} = r_{0} and \\frac{\\hat{p}_0{(\\theta,r_{0})} + \\frac{r_{0}}{\\theta}}{\\hat{p}_0{(\\theta,r_{0})}} = \\frac{2 r_{0}}{\\theta \\hat{p}_0{(\\theta,r_{0})}} and \\frac{\\hat{p}_0{(\\theta,r_{0})} + \\frac{r_{0}}{\\theta}}{\\hat{p}_0{(\\theta,r_{0})}} = 2 and (\\frac{\\hat{p}_0{(\\theta,r_{0})} + \\frac{r_{0}}{\\theta}}{\\hat{p}_0{(\\theta,r_{0})}})^{r_{0}} = 2^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\theta', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))"], [["times", 2, "Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))), Integer(2))"], [["power", 5, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)), Pow(Integer(2), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mu)} = \\log{(e^{\\mu})} and \\mathbb{I}{(\\mu)} = e^{- \\mu} \\log{(e^{\\mu})}^{\\mu}, then obtain - (\\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu})^{\\mu} + \\mathbb{I}{(\\mu)} = - (\\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu})^{\\mu} + e^{- \\mu} \\log{(e^{\\mu})}^{\\mu}", "derivation": "\\eta^{\\prime}{(\\mu)} = \\log{(e^{\\mu})} and \\eta^{\\prime}^{\\mu}{(\\mu)} = \\log{(e^{\\mu})}^{\\mu} and \\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu} = e^{- \\mu} \\log{(e^{\\mu})}^{\\mu} and (\\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu})^{\\mu} = (e^{- \\mu} \\log{(e^{\\mu})}^{\\mu})^{\\mu} and \\mathbb{I}{(\\mu)} = e^{- \\mu} \\log{(e^{\\mu})}^{\\mu} and - (e^{- \\mu} \\log{(e^{\\mu})}^{\\mu})^{\\mu} + \\mathbb{I}{(\\mu)} = - (e^{- \\mu} \\log{(e^{\\mu})}^{\\mu})^{\\mu} + e^{- \\mu} \\log{(e^{\\mu})}^{\\mu} and - (\\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu})^{\\mu} + \\mathbb{I}{(\\mu)} = - (\\eta^{\\prime}^{\\mu}{(\\mu)} e^{- \\mu})^{\\mu} + e^{- \\mu} \\log{(e^{\\mu})}^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), log(exp(Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["divide", 2, "exp(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Pow(Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mu', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["minus", 5, "Pow(Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)} = M_{E} - \\mu_0 and \\operatorname{P_{e}}{(\\mu_0)} = \\mu_0, then obtain - \\sin{(\\mu_0 - \\frac{\\partial}{\\partial \\mu_0} \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)})} = - \\sin{(\\mu_0 - \\frac{\\partial}{\\partial \\mu_0} (M_{E} - \\mu_0))}", "derivation": "\\operatorname{y^{\\prime}}{(M_{E},\\mu_0)} = M_{E} - \\mu_0 and \\frac{\\partial}{\\partial \\mu_0} \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (M_{E} - \\mu_0) and \\operatorname{P_{e}}{(\\mu_0)} = \\mu_0 and - \\operatorname{P_{e}}{(\\mu_0)} + \\frac{\\partial}{\\partial \\mu_0} \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)} = - \\operatorname{P_{e}}{(\\mu_0)} + \\frac{\\partial}{\\partial \\mu_0} (M_{E} - \\mu_0) and - \\mu_0 + \\frac{\\partial}{\\partial \\mu_0} \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)} = - \\mu_0 + \\frac{\\partial}{\\partial \\mu_0} (M_{E} - \\mu_0) and - \\sin{(\\mu_0 - \\frac{\\partial}{\\partial \\mu_0} \\operatorname{y^{\\prime}}{(M_{E},\\mu_0)})} = - \\sin{(\\mu_0 - \\frac{\\partial}{\\partial \\mu_0} (M_{E} - \\mu_0))}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["minus", 2, "Function('P_e')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('P_e')(Symbol('\\\\mu_0', commutative=True))), Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('P_e')(Symbol('\\\\mu_0', commutative=True))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["sin", 5], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))), Mul(Integer(-1), sin(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} = \\frac{\\eta^{\\prime}}{x^\\prime}, then derive \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} = \\frac{1}{x^\\prime}, then obtain \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\eta^{\\prime}}{x^\\prime} = \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} + \\frac{1}{x^\\prime}", "derivation": "\\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} = \\frac{\\eta^{\\prime}}{x^\\prime} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\eta^{\\prime}}{x^\\prime} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} = \\frac{1}{x^\\prime} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\eta^{\\prime}}{x^\\prime} = \\frac{1}{x^\\prime} and \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\eta^{\\prime}}{x^\\prime} = \\operatorname{C_{d}}{(x^\\prime,\\eta^{\\prime})} + \\frac{1}{x^\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["add", 4, "Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Add(Function('C_d')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} = \\frac{\\cos{(b)}}{J_{\\varepsilon}} and \\operatorname{a^{\\dagger}}{(b)} = \\cos{(b)}, then obtain \\frac{\\partial}{\\partial b} \\int \\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} dJ_{\\varepsilon} = \\frac{\\partial}{\\partial b} \\int \\frac{\\operatorname{a^{\\dagger}}{(b)}}{J_{\\varepsilon}} dJ_{\\varepsilon}", "derivation": "\\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} = \\frac{\\cos{(b)}}{J_{\\varepsilon}} and \\operatorname{a^{\\dagger}}{(b)} = \\cos{(b)} and \\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} = \\frac{\\operatorname{a^{\\dagger}}{(b)}}{J_{\\varepsilon}} and \\int \\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\frac{\\operatorname{a^{\\dagger}}{(b)}}{J_{\\varepsilon}} dJ_{\\varepsilon} and \\frac{\\partial}{\\partial b} \\int \\operatorname{f^{\\prime}}{(b,J_{\\varepsilon})} dJ_{\\varepsilon} = \\frac{\\partial}{\\partial b} \\int \\frac{\\operatorname{a^{\\dagger}}{(b)}}{J_{\\varepsilon}} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), cos(Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('b', commutative=True))))"], [["integrate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('b', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('b', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(z^{*},W)} = - \\sin{(W - z^{*})} and \\operatorname{f_{\\mathbf{v}}}{(z^{*},W)} = W - z^{*} and \\Psi_{nl}{(z^{*},W)} = W + \\dot{z}{(z^{*},W)}, then obtain \\Psi_{nl}{(z^{*},W)} = W - \\sin{(\\operatorname{f_{\\mathbf{v}}}{(z^{*},W)})}", "derivation": "\\dot{z}{(z^{*},W)} = - \\sin{(W - z^{*})} and W + \\dot{z}{(z^{*},W)} = W - \\sin{(W - z^{*})} and \\operatorname{f_{\\mathbf{v}}}{(z^{*},W)} = W - z^{*} and W + \\dot{z}{(z^{*},W)} = W - \\sin{(\\operatorname{f_{\\mathbf{v}}}{(z^{*},W)})} and \\Psi_{nl}{(z^{*},W)} = W + \\dot{z}{(z^{*},W)} and \\Psi_{nl}{(z^{*},W)} = W - \\sin{(W - z^{*})} and W - \\sin{(W - z^{*})} = W - \\sin{(\\operatorname{f_{\\mathbf{v}}}{(z^{*},W)})} and \\Psi_{nl}{(z^{*},W)} = W - \\sin{(\\operatorname{f_{\\mathbf{v}}}{(z^{*},W)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["add", 1, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Function('\\\\dot{z}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))), Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), sin(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True), Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\rho{(f^{\\prime})} = \\sin{(f^{\\prime})}, then obtain - \\frac{2 \\frac{d}{d f^{\\prime}} \\rho{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} = \\frac{\\cos{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} - \\frac{3 \\sin{(f^{\\prime})} \\frac{d}{d f^{\\prime}} \\rho{(f^{\\prime})}}{\\rho^{4}{(f^{\\prime})}}", "derivation": "\\rho{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\frac{1}{\\rho{(f^{\\prime})}} = \\frac{\\sin{(f^{\\prime})}}{\\rho^{2}{(f^{\\prime})}} and \\frac{1}{\\rho^{2}{(f^{\\prime})}} = \\frac{\\sin{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} and \\frac{d}{d f^{\\prime}} \\frac{1}{\\rho^{2}{(f^{\\prime})}} = \\frac{d}{d f^{\\prime}} \\frac{\\sin{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} and - \\frac{2 \\frac{d}{d f^{\\prime}} \\rho{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} = \\frac{\\cos{(f^{\\prime})}}{\\rho^{3}{(f^{\\prime})}} - \\frac{3 \\sin{(f^{\\prime})} \\frac{d}{d f^{\\prime}} \\rho{(f^{\\prime})}}{\\rho^{4}{(f^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-2)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 2, "Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-2)), Mul(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-3)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-2)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-3)), sin(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Integer(2), Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-3)), Derivative(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Mul(Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-3)), cos(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Integer(3), Pow(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-4)), sin(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\rho')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{x}{(n_{1},y)} = e^{n_{1} y} and \\tilde{g}^*{(x^\\prime,\\mathbf{f})} = - \\mathbf{f} + e^{x^\\prime}, then obtain \\frac{\\sin{(\\dot{x}^{y}{(n_{1},y)})}}{- \\mathbf{f} + e^{x^\\prime}} = \\frac{\\sin{((e^{n_{1} y})^{y})}}{- \\mathbf{f} + e^{x^\\prime}}", "derivation": "\\dot{x}{(n_{1},y)} = e^{n_{1} y} and \\dot{x}^{y}{(n_{1},y)} = (e^{n_{1} y})^{y} and \\tilde{g}^*{(x^\\prime,\\mathbf{f})} = - \\mathbf{f} + e^{x^\\prime} and \\sin{(\\dot{x}^{y}{(n_{1},y)})} = \\sin{((e^{n_{1} y})^{y})} and \\frac{\\sin{(\\dot{x}^{y}{(n_{1},y)})}}{\\tilde{g}^*{(x^\\prime,\\mathbf{f})}} = \\frac{\\sin{((e^{n_{1} y})^{y})}}{\\tilde{g}^*{(x^\\prime,\\mathbf{f})}} and \\frac{\\sin{(\\dot{x}^{y}{(n_{1},y)})}}{- \\mathbf{f} + e^{x^\\prime}} = \\frac{\\sin{((e^{n_{1} y})^{y})}}{- \\mathbf{f} + e^{x^\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), exp(Mul(Symbol('n_1', commutative=True), Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Mul(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\dot{x}')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), sin(Pow(exp(Mul(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))))"], [["times", 4, "Pow(Function('\\\\tilde{g}^*')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), sin(Pow(Function('\\\\dot{x}')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), sin(Pow(exp(Mul(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), sin(Pow(Function('\\\\dot{x}')(Symbol('n_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), sin(Pow(exp(Mul(Symbol('n_1', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(P_{g})} = e^{P_{g}}, then obtain \\frac{(\\int 2 \\phi_{1}{(P_{g})} dP_{g})^{2}}{(\\phi_{1}{(P_{g})} + e^{P_{g}})^{2}} = \\frac{(\\int (\\phi_{1}{(P_{g})} + e^{P_{g}}) dP_{g})^{2}}{(\\phi_{1}{(P_{g})} + e^{P_{g}})^{2}}", "derivation": "\\phi_{1}{(P_{g})} = e^{P_{g}} and 2 \\phi_{1}{(P_{g})} = \\phi_{1}{(P_{g})} + e^{P_{g}} and \\int 2 \\phi_{1}{(P_{g})} dP_{g} = \\int (\\phi_{1}{(P_{g})} + e^{P_{g}}) dP_{g} and \\frac{\\int 2 \\phi_{1}{(P_{g})} dP_{g}}{\\phi_{1}{(P_{g})} + e^{P_{g}}} = \\frac{\\int (\\phi_{1}{(P_{g})} + e^{P_{g}}) dP_{g}}{\\phi_{1}{(P_{g})} + e^{P_{g}}} and \\frac{(\\int 2 \\phi_{1}{(P_{g})} dP_{g})^{2}}{(\\phi_{1}{(P_{g})} + e^{P_{g}})^{2}} = \\frac{(\\int (\\phi_{1}{(P_{g})} + e^{P_{g}}) dP_{g})^{2}}{(\\phi_{1}{(P_{g})} + e^{P_{g}})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["add", 1, "Function('\\\\phi_1')(Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('P_g', commutative=True))), Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\phi_1')(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["divide", 3, "Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Integer(-1)), Integral(Mul(Integer(2), Function('\\\\phi_1')(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)))), Mul(Pow(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Integer(-1)), Integral(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Pow(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Integer(-2)), Pow(Integral(Mul(Integer(2), Function('\\\\phi_1')(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(2))), Mul(Pow(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Integer(-2)), Pow(Integral(Add(Function('\\\\phi_1')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\varepsilon_{0}{(i)} = \\log{(\\sin{(i)})} and Z{(i)} = \\frac{- i + \\log{(\\sin{(i)})}}{\\log{(\\sin{(i)})}^{2}}, then obtain Z{(i)} = \\frac{- i + \\varepsilon_{0}{(i)}}{\\log{(\\sin{(i)})}^{2}}", "derivation": "\\varepsilon_{0}{(i)} = \\log{(\\sin{(i)})} and - i + \\varepsilon_{0}{(i)} = - i + \\log{(\\sin{(i)})} and \\frac{- i + \\varepsilon_{0}{(i)}}{\\log{(\\sin{(i)})}} = \\frac{- i + \\log{(\\sin{(i)})}}{\\log{(\\sin{(i)})}} and Z{(i)} = \\frac{- i + \\log{(\\sin{(i)})}}{\\log{(\\sin{(i)})}^{2}} and Z{(i)} = \\frac{- i + \\varepsilon_{0}{(i)}}{\\log{(\\sin{(i)})}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True))))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\varepsilon_0')(Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True)))))"], [["divide", 2, "log(sin(Symbol('i', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\varepsilon_0')(Symbol('i', commutative=True))), Pow(log(sin(Symbol('i', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True)))), Pow(log(sin(Symbol('i', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('i', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True)))), Pow(log(sin(Symbol('i', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('Z')(Symbol('i', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\varepsilon_0')(Symbol('i', commutative=True))), Pow(log(sin(Symbol('i', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain \\frac{e^{\\mathbf{P}} \\sin{(\\frac{\\operatorname{v_{z}}{(\\mathbf{P})}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})}}{\\mathbf{P}} = \\frac{e^{\\mathbf{P}} \\sin{(\\frac{e^{\\mathbf{P}}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})}}{\\mathbf{P}}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{\\operatorname{v_{z}}{(\\mathbf{P})}}{\\mathbf{P}} = \\frac{e^{\\mathbf{P}}}{\\mathbf{P}} and \\frac{\\operatorname{v_{z}}{(\\mathbf{P})}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}} = \\frac{e^{\\mathbf{P}}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}} and \\sin{(\\frac{\\operatorname{v_{z}}{(\\mathbf{P})}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})} = \\sin{(\\frac{e^{\\mathbf{P}}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})} and \\frac{e^{\\mathbf{P}} \\sin{(\\frac{\\operatorname{v_{z}}{(\\mathbf{P})}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})}}{\\mathbf{P}} = \\frac{e^{\\mathbf{P}} \\sin{(\\frac{e^{\\mathbf{P}}}{\\mathbf{P}} + \\frac{1}{\\mathbf{P}})}}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))))"], [["sin", 3], "Equality(sin(Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))), sin(Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))))"], [["times", 4, "Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True)), sin(Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True)), sin(Add(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\dot{z}{(\\eta,F_{x})} = F_{x} + \\sin{(\\eta)}, then obtain \\int (- H + \\dot{z}{(\\eta,F_{x})}) d\\eta = F_{x} \\eta - H \\eta + \\varepsilon_0 - \\cos{(\\eta)}", "derivation": "\\dot{z}{(\\eta,F_{x})} = F_{x} + \\sin{(\\eta)} and - H + \\dot{z}{(\\eta,F_{x})} = F_{x} - H + \\sin{(\\eta)} and \\int (- H + \\dot{z}{(\\eta,F_{x})}) d\\eta = \\int (F_{x} - H + \\sin{(\\eta)}) d\\eta and \\int (- H + \\dot{z}{(\\eta,F_{x})}) d\\eta = F_{x} \\eta - H \\eta + \\varepsilon_0 - \\cos{(\\eta)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), sin(Symbol('\\\\eta', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\eta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)}, then obtain - \\varphi{(\\tilde{g}^*)} \\sin{(\\tilde{g}^*)} + \\cos{(\\tilde{g}^*)} \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} = - 2 \\sin{(\\tilde{g}^*)} \\cos{(\\tilde{g}^*)}", "derivation": "\\varphi{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)} and \\varphi{(\\tilde{g}^*)} \\cos{(\\tilde{g}^*)} = \\cos^{2}{(\\tilde{g}^*)} and \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} \\cos{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\cos^{2}{(\\tilde{g}^*)} and - \\varphi{(\\tilde{g}^*)} \\sin{(\\tilde{g}^*)} + \\cos{(\\tilde{g}^*)} \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} = - 2 \\sin{(\\tilde{g}^*)} \\cos{(\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True))), Pow(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(cos(Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(r_{0})} = r_{0}, then obtain \\frac{d}{d r_{0}} 1 = \\frac{d}{d r_{0}} \\frac{r_{0} + \\operatorname{P_{g}}{(r_{0})}}{2 \\operatorname{P_{g}}{(r_{0})}}", "derivation": "\\operatorname{P_{g}}{(r_{0})} = r_{0} and 2 \\operatorname{P_{g}}{(r_{0})} = r_{0} + \\operatorname{P_{g}}{(r_{0})} and 1 = \\frac{r_{0} + \\operatorname{P_{g}}{(r_{0})}}{2 \\operatorname{P_{g}}{(r_{0})}} and \\frac{d}{d r_{0}} 1 = \\frac{d}{d r_{0}} \\frac{r_{0} + \\operatorname{P_{g}}{(r_{0})}}{2 \\operatorname{P_{g}}{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], [["add", 1, "Function('P_g')(Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), Function('P_g')(Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('P_g')(Symbol('r_0', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True))), Pow(Function('P_g')(Symbol('r_0', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Add(Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True))), Pow(Function('P_g')(Symbol('r_0', commutative=True)), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(A_{1})} = \\cos{(A_{1})} and \\theta_{2}{(A_{1})} = \\operatorname{J_{\\varepsilon}}{(A_{1})} - \\cos{(A_{1})}, then obtain (\\operatorname{J_{\\varepsilon}}{(A_{1})} - \\cos{(A_{1})})^{A_{1}} = 0^{A_{1}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(A_{1})} = \\cos{(A_{1})} and A_{1} + \\operatorname{J_{\\varepsilon}}{(A_{1})} = A_{1} + \\cos{(A_{1})} and \\operatorname{J_{\\varepsilon}}{(A_{1})} - \\cos{(A_{1})} = 0 and \\theta_{2}{(A_{1})} = \\operatorname{J_{\\varepsilon}}{(A_{1})} - \\cos{(A_{1})} and \\theta_{2}{(A_{1})} = 0 and \\theta_{2}^{A_{1}}{(A_{1})} = 0^{A_{1}} and (\\operatorname{J_{\\varepsilon}}{(A_{1})} - \\cos{(A_{1})})^{A_{1}} = 0^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))"], [["add", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), cos(Symbol('A_1', commutative=True))))"], [["minus", 2, "Add(Symbol('A_1', commutative=True), cos(Symbol('A_1', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('A_1', commutative=True)), Add(Function('J_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\theta_2')(Symbol('A_1', commutative=True)), Integer(0))"], [["power", 5, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Integer(0), Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Add(Function('J_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Pow(Integer(0), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(n)} = \\sin{(n)} and v{(g,\\mathbf{B})} = \\mathbf{B} + g, then obtain \\frac{\\int v^{\\mathbf{B}}{(g,\\mathbf{B})} d\\mathbf{B}}{\\log{(\\mathbf{J}_P{(n)})}} = \\frac{\\int (\\mathbf{B} + g)^{\\mathbf{B}} d\\mathbf{B}}{\\log{(\\mathbf{J}_P{(n)})}}", "derivation": "\\mathbf{J}_P{(n)} = \\sin{(n)} and v{(g,\\mathbf{B})} = \\mathbf{B} + g and v^{\\mathbf{B}}{(g,\\mathbf{B})} = (\\mathbf{B} + g)^{\\mathbf{B}} and \\int v^{\\mathbf{B}}{(g,\\mathbf{B})} d\\mathbf{B} = \\int (\\mathbf{B} + g)^{\\mathbf{B}} d\\mathbf{B} and \\frac{\\int v^{\\mathbf{B}}{(g,\\mathbf{B})} d\\mathbf{B}}{\\log{(\\sin{(n)})}} = \\frac{\\int (\\mathbf{B} + g)^{\\mathbf{B}} d\\mathbf{B}}{\\log{(\\sin{(n)})}} and \\frac{\\int v^{\\mathbf{B}}{(g,\\mathbf{B})} d\\mathbf{B}}{\\log{(\\mathbf{J}_P{(n)})}} = \\frac{\\int (\\mathbf{B} + g)^{\\mathbf{B}} d\\mathbf{B}}{\\log{(\\mathbf{J}_P{(n)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], ["get_premise", "Equality(Function('v')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('v')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Pow(Function('v')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 4, "log(sin(Symbol('n', commutative=True)))"], "Equality(Mul(Pow(log(sin(Symbol('n', commutative=True))), Integer(-1)), Integral(Pow(Function('v')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(log(sin(Symbol('n', commutative=True))), Integer(-1)), Integral(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(log(Function('\\\\mathbf{J}_P')(Symbol('n', commutative=True))), Integer(-1)), Integral(Pow(Function('v')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(log(Function('\\\\mathbf{J}_P')(Symbol('n', commutative=True))), Integer(-1)), Integral(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(M_{E},\\mathbf{S})} = \\log{(M_{E} \\mathbf{S})}, then obtain \\operatorname{A_{2}}^{3}{(M_{E},\\mathbf{S})} = \\operatorname{A_{2}}{(M_{E},\\mathbf{S})} \\log{(M_{E} \\mathbf{S})}^{2}", "derivation": "\\operatorname{A_{2}}{(M_{E},\\mathbf{S})} = \\log{(M_{E} \\mathbf{S})} and \\operatorname{A_{2}}^{2}{(M_{E},\\mathbf{S})} = \\operatorname{A_{2}}{(M_{E},\\mathbf{S})} \\log{(M_{E} \\mathbf{S})} and \\operatorname{A_{2}}^{3}{(M_{E},\\mathbf{S})} = \\operatorname{A_{2}}^{2}{(M_{E},\\mathbf{S})} \\log{(M_{E} \\mathbf{S})} and \\operatorname{A_{2}}^{3}{(M_{E},\\mathbf{S})} = \\operatorname{A_{2}}{(M_{E},\\mathbf{S})} \\log{(M_{E} \\mathbf{S})}^{2}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Pow(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Mul(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["times", 1, "Pow(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))"], "Equality(Pow(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(3)), Mul(Pow(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), log(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(3)), Mul(Function('A_2')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(log(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(h,C_{d})} = C_{d} + h and \\operatorname{A_{x}}{(h,C_{d})} = C_{d} + h, then obtain (\\operatorname{A_{x}}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})})^{2} = (\\operatorname{A_{x}}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})}) (\\hat{H}_{\\lambda}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})})", "derivation": "\\hat{H}_{\\lambda}{(h,C_{d})} = C_{d} + h and \\operatorname{A_{x}}{(h,C_{d})} = C_{d} + h and (C_{d} + h)^{C_{d}} + \\operatorname{A_{x}}{(h,C_{d})} = C_{d} + h + (C_{d} + h)^{C_{d}} and \\operatorname{A_{x}}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})} = \\hat{H}_{\\lambda}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})} and (\\operatorname{A_{x}}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})})^{2} = (\\operatorname{A_{x}}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})}) (\\hat{H}_{\\lambda}{(h,C_{d})} + \\hat{H}_{\\lambda}^{C_{d}}{(h,C_{d})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True)))"], [["add", 2, "Pow(Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Add(Pow(Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True)), Symbol('C_d', commutative=True)), Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True), Pow(Add(Symbol('C_d', commutative=True), Symbol('h', commutative=True)), Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))))"], [["times", 4, "Add(Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], "Equality(Pow(Add(Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Integer(2)), Mul(Add(Function('A_x')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})} = \\mathbf{g} \\tilde{g}, then derive P_{g} + \\mathbf{g} = \\int \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} d\\mathbf{g}, then obtain (P_{g} + \\mathbf{g})^{\\mathbf{g}} = (\\int \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} d\\mathbf{g})^{\\mathbf{g}}", "derivation": "\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})} = \\mathbf{g} \\tilde{g} and 1 = \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} and \\int 1 d\\mathbf{g} = \\int \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} d\\mathbf{g} and P_{g} + \\mathbf{g} = \\int \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} d\\mathbf{g} and P_{g} + \\mathbf{g} = \\int 1 d\\mathbf{g} and (P_{g} + \\mathbf{g})^{\\mathbf{g}} = (\\int 1 d\\mathbf{g})^{\\mathbf{g}} and (P_{g} + \\mathbf{g})^{\\mathbf{g}} = (\\int \\frac{\\mathbf{g} \\tilde{g}}{\\operatorname{A_{x}}{(\\tilde{g},\\mathbf{g})}} d\\mathbf{g})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Function('A_x')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{1})} = \\cos{(e^{A_{1}})}, then obtain e^{\\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}} \\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})} = e^{\\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}} \\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}", "derivation": "\\operatorname{A_{x}}{(A_{1})} = \\cos{(e^{A_{1}})} and \\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})} = \\frac{d}{d A_{1}} \\cos{(e^{A_{1}})} and e^{\\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})}} = e^{\\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}} and e^{\\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})}} \\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})} = e^{\\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})}} \\frac{d}{d A_{1}} \\cos{(e^{A_{1}})} and e^{\\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}} \\frac{d}{d A_{1}} \\operatorname{A_{x}}{(A_{1})} = e^{\\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}} \\frac{d}{d A_{1}} \\cos{(e^{A_{1}})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), exp(Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["times", 2, "exp(Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], "Equality(Mul(exp(Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(exp(Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(exp(Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Derivative(Function('A_x')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(exp(Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Derivative(cos(exp(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(t,\\lambda)} = - \\sin{(\\lambda - t)} and \\varphi^{*}{(t,\\lambda)} = \\lambda - t, then obtain (\\int - t \\sin{(\\varphi^{*}{(t,\\lambda)})} dt)^{\\lambda} = (\\int - t \\sin{(\\lambda - t)} dt)^{\\lambda}", "derivation": "\\operatorname{g_{\\varepsilon}}{(t,\\lambda)} = - \\sin{(\\lambda - t)} and t \\operatorname{g_{\\varepsilon}}{(t,\\lambda)} = - t \\sin{(\\lambda - t)} and \\int t \\operatorname{g_{\\varepsilon}}{(t,\\lambda)} dt = \\int - t \\sin{(\\lambda - t)} dt and (\\int t \\operatorname{g_{\\varepsilon}}{(t,\\lambda)} dt)^{\\lambda} = (\\int - t \\sin{(\\lambda - t)} dt)^{\\lambda} and \\varphi^{*}{(t,\\lambda)} = \\lambda - t and \\operatorname{g_{\\varepsilon}}{(t,\\lambda)} = - \\sin{(\\varphi^{*}{(t,\\lambda)})} and (\\int - t \\sin{(\\varphi^{*}{(t,\\lambda)})} dt)^{\\lambda} = (\\int - t \\sin{(\\lambda - t)} dt)^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))))"], [["times", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True), sin(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Symbol('t', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Symbol('t', commutative=True), sin(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Tuple(Symbol('t', commutative=True))))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('t', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('t', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('t', commutative=True), sin(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Tuple(Symbol('t', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('g_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Integral(Mul(Integer(-1), Symbol('t', commutative=True), sin(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('t', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('t', commutative=True), sin(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Tuple(Symbol('t', commutative=True))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(A_{z})} = e^{A_{z}} and \\bar{\\h}{(A_{z})} = e^{A_{z}}, then obtain \\frac{d}{d A_{z}} \\bar{\\h}{(A_{z})} = \\frac{d}{d A_{z}} e^{A_{z}}", "derivation": "\\operatorname{f^{\\prime}}{(A_{z})} = e^{A_{z}} and \\bar{\\h}{(A_{z})} = e^{A_{z}} and \\bar{\\h}{(A_{z})} = \\operatorname{f^{\\prime}}{(A_{z})} and \\frac{d}{d A_{z}} \\bar{\\h}{(A_{z})} = \\frac{d}{d A_{z}} \\operatorname{f^{\\prime}}{(A_{z})} and \\frac{d}{d A_{z}} e^{A_{z}} = \\frac{d}{d A_{z}} \\operatorname{f^{\\prime}}{(A_{z})} and \\frac{d}{d A_{z}} \\bar{\\h}{(A_{z})} = \\frac{d}{d A_{z}} e^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hbar')(Symbol('A_z', commutative=True)), Function('f^{\\\\prime}')(Symbol('A_z', commutative=True)))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Function('f^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Function('f^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('\\\\hbar')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(h)} = \\cos{(h)} and \\varphi^{*}{(\\hbar)} = \\int \\cos{(\\hbar)} d\\hbar, then derive \\varphi^{*}{(\\hbar)} + \\operatorname{x^{{\\}'}}{(h)} = W + \\operatorname{x^{{\\}'}}{(h)} + \\sin{(\\hbar)}, then obtain \\varphi^{*}{(\\hbar)} + \\operatorname{x^{{\\}'}}{(h)} - \\cos{(\\hbar)} = W + \\operatorname{x^{{\\}'}}{(h)} + \\sin{(\\hbar)} - \\cos{(\\hbar)}", "derivation": "\\operatorname{x^{{\\}'}}{(h)} = \\cos{(h)} and \\varphi^{*}{(\\hbar)} = \\int \\cos{(\\hbar)} d\\hbar and \\varphi^{*}{(\\hbar)} + \\cos{(h)} = \\cos{(h)} + \\int \\cos{(\\hbar)} d\\hbar and \\varphi^{*}{(\\hbar)} + \\operatorname{x^{{\\}'}}{(h)} = \\operatorname{x^{{\\}'}}{(h)} + \\int \\cos{(\\hbar)} d\\hbar and \\varphi^{*}{(\\hbar)} + \\operatorname{x^{{\\}'}}{(h)} = W + \\operatorname{x^{{\\}'}}{(h)} + \\sin{(\\hbar)} and \\varphi^{*}{(\\hbar)} + \\operatorname{x^{{\\}'}}{(h)} - \\cos{(\\hbar)} = W + \\operatorname{x^{{\\}'}}{(h)} + \\sin{(\\hbar)} - \\cos{(\\hbar)}", "srepr_derivation": [["get_premise", "Equality(Function('x^\\\\prime')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hbar', commutative=True)), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "cos(Symbol('h', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('h', commutative=True))), Add(cos(Symbol('h', commutative=True)), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\hbar', commutative=True)), Function('x^\\\\prime')(Symbol('h', commutative=True))), Add(Function('x^\\\\prime')(Symbol('h', commutative=True)), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\hbar', commutative=True)), Function('x^\\\\prime')(Symbol('h', commutative=True))), Add(Symbol('W', commutative=True), Function('x^\\\\prime')(Symbol('h', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))))"], [["minus", 5, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\hbar', commutative=True)), Function('x^\\\\prime')(Symbol('h', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('W', commutative=True), Function('x^\\\\prime')(Symbol('h', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\mu_0)} = \\cos{(\\sin{(\\mu_0)})}, then derive - \\frac{d}{d \\mu_0} \\chi{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})} \\cos{(\\mu_0)}, then obtain - \\frac{d}{d \\mu_0} \\cos{(\\sin{(\\mu_0)})} = \\sin{(\\sin{(\\mu_0)})} \\cos{(\\mu_0)}", "derivation": "\\chi{(\\mu_0)} = \\cos{(\\sin{(\\mu_0)})} and \\frac{d}{d \\mu_0} \\chi{(\\mu_0)} = \\frac{d}{d \\mu_0} \\cos{(\\sin{(\\mu_0)})} and - \\frac{d}{d \\mu_0} \\chi{(\\mu_0)} = - \\frac{d}{d \\mu_0} \\cos{(\\sin{(\\mu_0)})} and - \\frac{d}{d \\mu_0} \\chi{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})} \\cos{(\\mu_0)} and - \\frac{d}{d \\mu_0} \\cos{(\\sin{(\\mu_0)})} = \\sin{(\\sin{(\\mu_0)})} \\cos{(\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), cos(sin(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(sin(sin(Symbol('\\\\mu_0', commutative=True))), cos(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(sin(sin(Symbol('\\\\mu_0', commutative=True))), cos(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given c{(E_{n},v_{z})} = \\int (E_{n} - v_{z}) dE_{n}, then obtain \\frac{\\partial^{2}}{\\partial v_{z}\\partial E_{n}} \\int c{(E_{n},v_{z})} dE_{n} = \\frac{\\partial^{2}}{\\partial v_{z}\\partial E_{n}} \\iint (E_{n} - v_{z}) dE_{n} dE_{n}", "derivation": "c{(E_{n},v_{z})} = \\int (E_{n} - v_{z}) dE_{n} and \\int c{(E_{n},v_{z})} dE_{n} = \\iint (E_{n} - v_{z}) dE_{n} dE_{n} and \\frac{\\partial}{\\partial E_{n}} \\int c{(E_{n},v_{z})} dE_{n} = \\frac{\\partial}{\\partial E_{n}} \\iint (E_{n} - v_{z}) dE_{n} dE_{n} and \\frac{\\partial^{2}}{\\partial v_{z}\\partial E_{n}} \\int c{(E_{n},v_{z})} dE_{n} = \\frac{\\partial^{2}}{\\partial v_{z}\\partial E_{n}} \\iint (E_{n} - v_{z}) dE_{n} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('c')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('E_n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(f^{*})} = \\int \\cos{(f^{*})} df^{*}, then obtain 0^{f^{*}} = (- \\mathbf{f}{(f^{*})} + \\int \\cos{(f^{*})} df^{*})^{f^{*}}", "derivation": "\\mathbf{f}{(f^{*})} = \\int \\cos{(f^{*})} df^{*} and \\mathbf{f}{(f^{*})} + \\iint \\cos{(f^{*})} df^{*} df^{*} = \\int \\cos{(f^{*})} df^{*} + \\iint \\cos{(f^{*})} df^{*} df^{*} and 0 = - \\mathbf{f}{(f^{*})} + \\int \\cos{(f^{*})} df^{*} and 0^{f^{*}} = (- \\mathbf{f}{(f^{*})} + \\int \\cos{(f^{*})} df^{*})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["add", 1, "Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["minus", 2, "Add(Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["power", 3, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f^*', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\nabla,V_{\\mathbf{B}})} = \\nabla^{V_{\\mathbf{B}}}, then obtain V_{\\mathbf{B}} - \\nabla^{V_{\\mathbf{B}}} + \\operatorname{A_{2}}^{V_{\\mathbf{B}}}{(\\nabla,V_{\\mathbf{B}})} = V_{\\mathbf{B}} - \\nabla^{V_{\\mathbf{B}}} + (\\nabla^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "derivation": "\\operatorname{A_{2}}{(\\nabla,V_{\\mathbf{B}})} = \\nabla^{V_{\\mathbf{B}}} and \\operatorname{A_{2}}^{V_{\\mathbf{B}}}{(\\nabla,V_{\\mathbf{B}})} = (\\nabla^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} and V_{\\mathbf{B}} + \\operatorname{A_{2}}^{V_{\\mathbf{B}}}{(\\nabla,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + (\\nabla^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} and V_{\\mathbf{B}} - \\nabla^{V_{\\mathbf{B}}} + \\operatorname{A_{2}}^{V_{\\mathbf{B}}}{(\\nabla,V_{\\mathbf{B}})} = V_{\\mathbf{B}} - \\nabla^{V_{\\mathbf{B}}} + (\\nabla^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Function('A_2')(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 3, "Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Function('A_2')(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Pow(Symbol('\\\\nabla', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(F_{H})} = F_{H}, then derive (\\frac{d}{d F_{H}} \\rho_{f}{(F_{H})} + 1)^{F_{H}} = 2^{F_{H}}, then obtain (\\frac{d}{d \\rho_{f}{(F_{H})}} \\rho_{f}{(F_{H})} + 1)^{\\rho_{f}{(F_{H})}} = 2^{\\rho_{f}{(F_{H})}}", "derivation": "\\rho_{f}{(F_{H})} = F_{H} and \\frac{d}{d F_{H}} \\rho_{f}{(F_{H})} = \\frac{d}{d F_{H}} F_{H} and \\frac{d}{d F_{H}} \\rho_{f}{(F_{H})} + 1 = \\frac{d}{d F_{H}} F_{H} + 1 and (\\frac{d}{d F_{H}} \\rho_{f}{(F_{H})} + 1)^{F_{H}} = (\\frac{d}{d F_{H}} F_{H} + 1)^{F_{H}} and (\\frac{d}{d F_{H}} \\rho_{f}{(F_{H})} + 1)^{F_{H}} = 2^{F_{H}} and (\\frac{d}{d F_{H}} F_{H} + 1)^{F_{H}} = 2^{F_{H}} and (\\frac{d}{d \\rho_{f}{(F_{H})}} \\rho_{f}{(F_{H})} + 1)^{\\rho_{f}{(F_{H})}} = 2^{\\rho_{f}{(F_{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Symbol('F_H', commutative=True)), Pow(Add(Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Symbol('F_H', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Symbol('F_H', commutative=True)), Pow(Integer(2), Symbol('F_H', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)), Symbol('F_H', commutative=True)), Pow(Integer(2), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Derivative(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Tuple(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Integer(1))), Integer(1)), Function('\\\\rho_f')(Symbol('F_H', commutative=True))), Pow(Integer(2), Function('\\\\rho_f')(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\Psi{(s)} = \\cos{(s)}, then derive n_{1} + \\Psi{(s)} = f_{\\mathbf{v}} + \\cos{(s)}, then obtain f_{\\mathbf{v}} + \\Psi{(s)} = f_{\\mathbf{v}} + \\cos{(s)}", "derivation": "\\Psi{(s)} = \\cos{(s)} and \\frac{d}{d s} \\Psi{(s)} = \\frac{d}{d s} \\cos{(s)} and \\int \\frac{d}{d s} \\Psi{(s)} ds = \\int \\frac{d}{d s} \\cos{(s)} ds and n_{1} + \\Psi{(s)} = f_{\\mathbf{v}} + \\cos{(s)} and n_{1} + \\cos{(s)} = f_{\\mathbf{v}} + \\cos{(s)} and n_{1} + \\Psi{(s)} = f_{\\mathbf{v}} + \\Psi{(s)} and f_{\\mathbf{v}} + \\Psi{(s)} = f_{\\mathbf{v}} + \\cos{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))), Integral(Derivative(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('n_1', commutative=True), Function('\\\\Psi')(Symbol('s', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('n_1', commutative=True), cos(Symbol('s', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('n_1', commutative=True), Function('\\\\Psi')(Symbol('s', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\Psi')(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\Psi')(Symbol('s', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('s', commutative=True))))"]]}, {"prompt": "Given s{(\\varphi^*,\\Omega)} = \\sin{(\\Omega \\varphi^*)}, then derive \\frac{\\partial}{\\partial \\varphi^*} s{(\\varphi^*,\\Omega)} = \\Omega \\cos{(\\Omega \\varphi^*)}, then obtain (\\Omega \\cos{(\\Omega \\varphi^*)})^{\\varphi^*} = (\\frac{\\partial}{\\partial \\varphi^*} \\sin{(\\Omega \\varphi^*)})^{\\varphi^*}", "derivation": "s{(\\varphi^*,\\Omega)} = \\sin{(\\Omega \\varphi^*)} and \\frac{\\partial}{\\partial \\varphi^*} s{(\\varphi^*,\\Omega)} = \\frac{\\partial}{\\partial \\varphi^*} \\sin{(\\Omega \\varphi^*)} and s{(\\varphi^*,\\Omega)} + 1 = \\sin{(\\Omega \\varphi^*)} + 1 and \\frac{\\partial}{\\partial \\varphi^*} (s{(\\varphi^*,\\Omega)} + 1) = \\frac{\\partial}{\\partial \\varphi^*} (\\sin{(\\Omega \\varphi^*)} + 1) and \\frac{\\partial}{\\partial \\varphi^*} s{(\\varphi^*,\\Omega)} = \\Omega \\cos{(\\Omega \\varphi^*)} and \\Omega \\cos{(\\Omega \\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\sin{(\\Omega \\varphi^*)} and (\\Omega \\cos{(\\Omega \\varphi^*)})^{\\varphi^*} = (\\frac{\\partial}{\\partial \\varphi^*} \\sin{(\\Omega \\varphi^*)})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1)), Add(sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(1)))"], [["differentiate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Add(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Symbol('\\\\Omega', commutative=True), cos(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Symbol('\\\\Omega', commutative=True), cos(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Derivative(sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["power", 6, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\Omega', commutative=True), cos(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)), Pow(Derivative(sin(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\mathbb{I},x^\\prime)} = \\mathbb{I} x^\\prime, then obtain - \\mathbb{I} x^\\prime - \\mathbb{I} + 2 \\lambda{(\\mathbb{I},x^\\prime)} = - \\mathbb{I} + \\lambda{(\\mathbb{I},x^\\prime)}", "derivation": "\\lambda{(\\mathbb{I},x^\\prime)} = \\mathbb{I} x^\\prime and - \\mathbb{I} + \\lambda{(\\mathbb{I},x^\\prime)} = \\mathbb{I} x^\\prime - \\mathbb{I} and - \\mathbb{I} x^\\prime - \\mathbb{I} + \\lambda{(\\mathbb{I},x^\\prime)} = - \\mathbb{I} and - \\mathbb{I} x^\\prime - \\mathbb{I} + 2 \\lambda{(\\mathbb{I},x^\\prime)} = - \\mathbb{I} + \\lambda{(\\mathbb{I},x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given Z{(G,\\eta^{\\prime})} = \\eta^{\\prime} \\cos{(G)}, then derive \\int (G + Z{(G,\\eta^{\\prime})}) d\\eta^{\\prime} = G \\eta^{\\prime} + \\frac{(\\eta^{\\prime})^{2} \\cos{(G)}}{2} + x^\\prime, then obtain \\frac{\\partial}{\\partial \\eta^{\\prime}} (G \\eta^{\\prime} + \\frac{(\\eta^{\\prime})^{2} \\cos{(G)}}{2} + x^\\prime) = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int (G + \\eta^{\\prime} \\cos{(G)}) d\\eta^{\\prime}", "derivation": "Z{(G,\\eta^{\\prime})} = \\eta^{\\prime} \\cos{(G)} and G + Z{(G,\\eta^{\\prime})} = G + \\eta^{\\prime} \\cos{(G)} and \\int (G + Z{(G,\\eta^{\\prime})}) d\\eta^{\\prime} = \\int (G + \\eta^{\\prime} \\cos{(G)}) d\\eta^{\\prime} and \\int (G + Z{(G,\\eta^{\\prime})}) d\\eta^{\\prime} = G \\eta^{\\prime} + \\frac{(\\eta^{\\prime})^{2} \\cos{(G)}}{2} + x^\\prime and G \\eta^{\\prime} + \\frac{(\\eta^{\\prime})^{2} \\cos{(G)}}{2} + x^\\prime = \\int (G + \\eta^{\\prime} \\cos{(G)}) d\\eta^{\\prime} and \\frac{\\partial}{\\partial \\eta^{\\prime}} (G \\eta^{\\prime} + \\frac{(\\eta^{\\prime})^{2} \\cos{(G)}}{2} + x^\\prime) = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\int (G + \\eta^{\\prime} \\cos{(G)}) d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Symbol('G', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('G', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Symbol('G', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Add(Symbol('G', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2)), cos(Symbol('G', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2)), cos(Symbol('G', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Integral(Add(Symbol('G', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('G', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2)), cos(Symbol('G', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('G', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(c,n)} = \\frac{n}{c}, then obtain \\frac{\\partial}{\\partial n} \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\mu{(c,n)} + \\frac{1}{c}) dc = \\frac{\\partial}{\\partial n} \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\frac{n}{c} + \\frac{1}{c}) dc", "derivation": "\\mu{(c,n)} = \\frac{n}{c} and \\mu{(c,n)} + \\frac{1}{c} = \\frac{n}{c} + \\frac{1}{c} and \\int (\\mu{(c,n)} + \\frac{1}{c}) dc = \\int (\\frac{n}{c} + \\frac{1}{c}) dc and \\frac{\\partial}{\\partial n} \\int (\\mu{(c,n)} + \\frac{1}{c}) dc = \\frac{\\partial}{\\partial n} \\int (\\frac{n}{c} + \\frac{1}{c}) dc and \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\mu{(c,n)} + \\frac{1}{c}) dc = \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\frac{n}{c} + \\frac{1}{c}) dc and \\frac{\\partial}{\\partial n} \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\mu{(c,n)} + \\frac{1}{c}) dc = \\frac{\\partial}{\\partial n} \\mu{(c,n)} \\frac{\\partial}{\\partial n} \\int (\\frac{n}{c} + \\frac{1}{c}) dc", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["add", 1, "Pow(Symbol('c', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 4, "Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Derivative(Integral(Add(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Derivative(Integral(Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Derivative(Integral(Add(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mu')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Derivative(Integral(Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(S)} = \\log{(e^{S})}, then obtain \\int 0^{S} \\log{(e^{S})} dS = \\int 0 dS", "derivation": "\\operatorname{C_{2}}{(S)} = \\log{(e^{S})} and \\operatorname{C_{2}}{(S)} - \\log{(e^{S})} = 0 and (\\operatorname{C_{2}}{(S)} - \\log{(e^{S})})^{S} = 0^{S} and (\\operatorname{C_{2}}{(S)} - \\log{(e^{S})}) (\\operatorname{C_{2}}{(S)} - \\log{(e^{S})})^{S} = 0 and 0^{S} (\\operatorname{C_{2}}{(S)} - \\log{(e^{S})}) = 0 and (\\operatorname{C_{2}}{(S)} - \\log{(e^{S})})^{S} \\operatorname{C_{2}}{(S)} = 0 and 0^{S} \\log{(e^{S})} = 0 and \\int 0^{S} \\log{(e^{S})} dS = \\int 0 dS", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('S', commutative=True)), log(exp(Symbol('S', commutative=True))))"], [["minus", 1, "log(exp(Symbol('S', commutative=True)))"], "Equality(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Symbol('S', commutative=True)), Pow(Integer(0), Symbol('S', commutative=True)))"], [["times", 2, "Pow(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Symbol('S', commutative=True))"], "Equality(Mul(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Pow(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Symbol('S', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Integer(0), Symbol('S', commutative=True)), Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Function('C_2')(Symbol('S', commutative=True)), Mul(Integer(-1), log(exp(Symbol('S', commutative=True))))), Symbol('S', commutative=True)), Function('C_2')(Symbol('S', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Integer(0), Symbol('S', commutative=True)), log(exp(Symbol('S', commutative=True)))), Integer(0))"], [["integrate", 7, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Pow(Integer(0), Symbol('S', commutative=True)), log(exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(h)} = \\cos{(h)} and \\hat{x}{(h)} = \\cos{(h)} + (\\int \\cos{(h)} dh)^{h}, then obtain \\frac{\\mathbf{S}{(h)} + (\\int \\cos{(h)} dh)^{h}}{\\cos{(h)}} = \\frac{\\hat{x}{(h)}}{\\cos{(h)}}", "derivation": "\\mathbf{S}{(h)} = \\cos{(h)} and \\int \\mathbf{S}{(h)} dh = \\int \\cos{(h)} dh and (\\int \\mathbf{S}{(h)} dh)^{h} = (\\int \\cos{(h)} dh)^{h} and \\mathbf{S}{(h)} + (\\int \\mathbf{S}{(h)} dh)^{h} = \\cos{(h)} + (\\int \\mathbf{S}{(h)} dh)^{h} and \\frac{\\mathbf{S}{(h)} + (\\int \\mathbf{S}{(h)} dh)^{h}}{\\cos{(h)}} = \\frac{\\cos{(h)} + (\\int \\mathbf{S}{(h)} dh)^{h}}{\\cos{(h)}} and \\frac{\\mathbf{S}{(h)} + (\\int \\cos{(h)} dh)^{h}}{\\cos{(h)}} = \\frac{\\cos{(h)} + (\\int \\cos{(h)} dh)^{h}}{\\cos{(h)}} and \\hat{x}{(h)} = \\cos{(h)} + (\\int \\cos{(h)} dh)^{h} and \\frac{\\mathbf{S}{(h)} + (\\int \\cos{(h)} dh)^{h}}{\\cos{(h)}} = \\frac{\\hat{x}{(h)}}{\\cos{(h)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["add", 1, "Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Add(cos(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"], [["divide", 4, "cos(Symbol('h', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Add(cos(Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Add(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Mul(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(E_{n})} = e^{E_{n}} and c{(E_{n})} = \\frac{e^{E_{n}}}{\\mathbf{J}_f{(E_{n})}} and \\mathbf{g}{(E_{n})} = \\sin{(c{(E_{n})})}, then obtain \\iint \\mathbf{g}{(E_{n})} dE_{n} dE_{n} = \\iint \\sin{(1)} dE_{n} dE_{n}", "derivation": "\\mathbf{J}_f{(E_{n})} = e^{E_{n}} and 1 = \\frac{e^{E_{n}}}{\\mathbf{J}_f{(E_{n})}} and \\sin{(1)} = \\sin{(\\frac{e^{E_{n}}}{\\mathbf{J}_f{(E_{n})}})} and c{(E_{n})} = \\frac{e^{E_{n}}}{\\mathbf{J}_f{(E_{n})}} and \\sin{(1)} = \\sin{(c{(E_{n})})} and \\mathbf{g}{(E_{n})} = \\sin{(c{(E_{n})})} and \\mathbf{g}{(E_{n})} = \\sin{(1)} and \\int \\mathbf{g}{(E_{n})} dE_{n} = \\int \\sin{(1)} dE_{n} and \\iint \\mathbf{g}{(E_{n})} dE_{n} dE_{n} = \\iint \\sin{(1)} dE_{n} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('E_n', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_n', commutative=True)), Integer(-1)), exp(Symbol('E_n', commutative=True))))"], [["sin", 2], "Equality(sin(Integer(1)), sin(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_n', commutative=True)), Integer(-1)), exp(Symbol('E_n', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('E_n', commutative=True)), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_n', commutative=True)), Integer(-1)), exp(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Integer(1)), sin(Function('c')(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True)), sin(Function('c')(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True)), sin(Integer(1)))"], [["integrate", 7, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(sin(Integer(1)), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 8, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(sin(Integer(1)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = \\Psi_{nl} e^{\\mathbf{M}}, then derive - e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = 0, then obtain \\int (- e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})}) d\\mathbf{M} = \\int 0 d\\mathbf{M}", "derivation": "\\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = \\Psi_{nl} e^{\\mathbf{M}} and \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = \\frac{\\partial}{\\partial \\Psi_{nl}} \\Psi_{nl} e^{\\mathbf{M}} and - e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = - e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\Psi_{nl} e^{\\mathbf{M}} and - e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})} = 0 and \\int (- e^{\\mathbf{M}} + \\frac{\\partial}{\\partial \\Psi_{nl}} \\operatorname{x^{{\\}'}}{(\\Psi_{nl},\\mathbf{M})}) d\\mathbf{M} = \\int 0 d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(A_{x})} = \\cos{(\\log{(A_{x})})}, then obtain \\frac{d}{d A_{x}} \\rho_{f}^{A_{x}}{(A_{x})} \\cos{(\\log{(A_{x})})} = \\frac{d}{d A_{x}} \\cos{(\\log{(A_{x})})} \\cos^{A_{x}}{(\\log{(A_{x})})}", "derivation": "\\rho_{f}{(A_{x})} = \\cos{(\\log{(A_{x})})} and \\rho_{f}^{A_{x}}{(A_{x})} = \\cos^{A_{x}}{(\\log{(A_{x})})} and \\rho_{f}^{A_{x}}{(A_{x})} \\cos{(\\log{(A_{x})})} = \\cos{(\\log{(A_{x})})} \\cos^{A_{x}}{(\\log{(A_{x})})} and \\frac{d}{d A_{x}} \\rho_{f}^{A_{x}}{(A_{x})} \\cos{(\\log{(A_{x})})} = \\frac{d}{d A_{x}} \\cos{(\\log{(A_{x})})} \\cos^{A_{x}}{(\\log{(A_{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('A_x', commutative=True)), cos(log(Symbol('A_x', commutative=True))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(cos(log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["times", 2, "cos(log(Symbol('A_x', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\rho_f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), cos(log(Symbol('A_x', commutative=True)))), Mul(cos(log(Symbol('A_x', commutative=True))), Pow(cos(log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\rho_f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), cos(log(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(cos(log(Symbol('A_x', commutative=True))), Pow(cos(log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(n_{2})} = \\log{(n_{2})}, then obtain \\int \\frac{d}{d n_{2}} \\frac{(\\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})})^{2}}{n_{2}} dn_{2} = \\int \\frac{d}{d n_{2}} 0 dn_{2}", "derivation": "\\operatorname{M_{E}}{(n_{2})} = \\log{(n_{2})} and \\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})} = 0 and \\frac{\\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})}}{n_{2}} = 0 and \\frac{(\\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})})^{2}}{n_{2}} = 0 and \\frac{d}{d n_{2}} \\frac{(\\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})})^{2}}{n_{2}} = \\frac{d}{d n_{2}} 0 and \\int \\frac{d}{d n_{2}} \\frac{(\\operatorname{M_{E}}{(n_{2})} - \\log{(n_{2})})^{2}}{n_{2}} dn_{2} = \\int \\frac{d}{d n_{2}} 0 dn_{2}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["minus", 1, "log(Symbol('n_2', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True)))), Integer(0))"], [["divide", 2, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True))))), Integer(0))"], [["times", 3, "Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True))))"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True)))), Integer(2))), Integer(0))"], [["differentiate", 4, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True)))), Integer(2))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('n_2', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Add(Function('M_E')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True)))), Integer(2))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given z{(\\mu)} = \\int e^{\\mu} d\\mu, then obtain \\frac{z{(\\mu)} + \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}}}{\\int e^{\\mu} d\\mu} = \\frac{\\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} + \\frac{(\\int e^{\\mu} d\\mu)^{4}}{z^{3}{(\\mu)}}}{\\int e^{\\mu} d\\mu}", "derivation": "z{(\\mu)} = \\int e^{\\mu} d\\mu and 1 = \\frac{\\int e^{\\mu} d\\mu}{z{(\\mu)}} and \\int e^{\\mu} d\\mu = \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} and z{(\\mu)} = \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} and z{(\\mu)} = \\frac{(\\int e^{\\mu} d\\mu)^{4}}{z^{3}{(\\mu)}} and z{(\\mu)} + \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} = \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} + \\frac{(\\int e^{\\mu} d\\mu)^{4}}{z^{3}{(\\mu)}} and \\frac{z{(\\mu)} + \\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}}}{\\int e^{\\mu} d\\mu} = \\frac{\\frac{(\\int e^{\\mu} d\\mu)^{2}}{z{(\\mu)}} + \\frac{(\\int e^{\\mu} d\\mu)^{4}}{z^{3}{(\\mu)}}}{\\int e^{\\mu} d\\mu}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mu', commutative=True)), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["divide", 1, "Function('z')(Symbol('\\\\mu', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["times", 2, "Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('z')(Symbol('\\\\mu', commutative=True)), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('z')(Symbol('\\\\mu', commutative=True)), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-3)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(4))))"], [["add", 5, "Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2)))"], "Equality(Add(Function('z')(Symbol('\\\\mu', commutative=True)), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2)))), Add(Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2))), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-3)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(4)))))"], [["divide", 6, "Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Add(Function('z')(Symbol('\\\\mu', commutative=True)), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2)))), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(Add(Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2))), Mul(Pow(Function('z')(Symbol('\\\\mu', commutative=True)), Integer(-3)), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(4)))), Pow(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(a)} = \\log{(e^{a})}, then obtain \\frac{d}{d a} (\\operatorname{A_{2}}{(a)} e^{- a})^{a} = \\frac{d}{d a} (e^{- a} \\log{(e^{a})})^{a}", "derivation": "\\operatorname{A_{2}}{(a)} = \\log{(e^{a})} and \\operatorname{A_{2}}{(a)} e^{- a} = e^{- a} \\log{(e^{a})} and (\\operatorname{A_{2}}{(a)} e^{- a})^{a} = (e^{- a} \\log{(e^{a})})^{a} and \\frac{d}{d a} (\\operatorname{A_{2}}{(a)} e^{- a})^{a} = \\frac{d}{d a} (e^{- a} \\log{(e^{a})})^{a}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('a', commutative=True)), log(exp(Symbol('a', commutative=True))))"], [["divide", 1, "exp(Symbol('a', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('a', commutative=True))), log(exp(Symbol('a', commutative=True)))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Function('A_2')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Pow(Mul(exp(Mul(Integer(-1), Symbol('a', commutative=True))), log(exp(Symbol('a', commutative=True)))), Symbol('a', commutative=True)))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('A_2')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Mul(exp(Mul(Integer(-1), Symbol('a', commutative=True))), log(exp(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(H,A)} = A + \\log{(H)}, then derive \\int \\operatorname{F_{x}}{(H,A)} dH = H (A - 1) + H \\log{(H)} + U, then obtain \\frac{\\int (A + \\log{(H)}) dH}{H} = \\frac{H (A - 1) + H \\log{(H)} + U}{H}", "derivation": "\\operatorname{F_{x}}{(H,A)} = A + \\log{(H)} and \\int \\operatorname{F_{x}}{(H,A)} dH = \\int (A + \\log{(H)}) dH and \\int \\operatorname{F_{x}}{(H,A)} dH = H (A - 1) + H \\log{(H)} + U and \\int (A + \\log{(H)}) dH = H (A - 1) + H \\log{(H)} + U and \\frac{\\int (A + \\log{(H)}) dH}{H} = \\frac{H (A - 1) + H \\log{(H)} + U}{H}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), log(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('A', commutative=True), log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_x')(Symbol('H', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), Add(Symbol('A', commutative=True), Integer(-1))), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('A', commutative=True), log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), Add(Symbol('A', commutative=True), Integer(-1))), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Symbol('U', commutative=True)))"], [["divide", 4, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Integral(Add(Symbol('A', commutative=True), log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Symbol('H', commutative=True), Add(Symbol('A', commutative=True), Integer(-1))), Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\lambda)} = \\cos{(\\sin{(\\lambda)})}, then obtain (\\operatorname{v_{t}}^{\\lambda}{(\\lambda)} \\sin{(\\lambda)})^{\\lambda} = (\\sin{(\\lambda)} \\cos^{\\lambda}{(\\sin{(\\lambda)})})^{\\lambda}", "derivation": "\\operatorname{v_{t}}{(\\lambda)} = \\cos{(\\sin{(\\lambda)})} and \\operatorname{v_{t}}^{\\lambda}{(\\lambda)} = \\cos^{\\lambda}{(\\sin{(\\lambda)})} and \\operatorname{v_{t}}^{\\lambda}{(\\lambda)} \\sin{(\\lambda)} = \\sin{(\\lambda)} \\cos^{\\lambda}{(\\sin{(\\lambda)})} and (\\operatorname{v_{t}}^{\\lambda}{(\\lambda)} \\sin{(\\lambda)})^{\\lambda} = (\\sin{(\\lambda)} \\cos^{\\lambda}{(\\sin{(\\lambda)})})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\lambda', commutative=True)), cos(sin(Symbol('\\\\lambda', commutative=True))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(cos(sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["times", 2, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Function('v_t')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Mul(sin(Symbol('\\\\lambda', commutative=True)), Pow(cos(sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Mul(Pow(Function('v_t')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(sin(Symbol('\\\\lambda', commutative=True)), Pow(cos(sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\phi_2)} = e^{\\phi_2}, then derive \\int (1 - e^{\\phi_2}) d\\phi_2 = \\mathbf{J}_M - \\int - \\frac{e^{\\phi_2}}{\\theta_{1}{(\\phi_2)}} d\\phi_2 - \\int e^{\\phi_2} d\\phi_2, then obtain - (\\int (1 - e^{\\phi_2}) d\\phi_2) \\int e^{\\phi_2} d\\phi_2 = - (\\mathbf{J}_M - \\int (-1) d\\phi_2 - \\int e^{\\phi_2} d\\phi_2) \\int e^{\\phi_2} d\\phi_2", "derivation": "\\theta_{1}{(\\phi_2)} = e^{\\phi_2} and 1 = \\frac{e^{\\phi_2}}{\\theta_{1}{(\\phi_2)}} and 1 - e^{\\phi_2} = - e^{\\phi_2} + \\frac{e^{\\phi_2}}{\\theta_{1}{(\\phi_2)}} and \\int (1 - e^{\\phi_2}) d\\phi_2 = \\int (- e^{\\phi_2} + \\frac{e^{\\phi_2}}{\\theta_{1}{(\\phi_2)}}) d\\phi_2 and \\int (1 - e^{\\phi_2}) d\\phi_2 = \\mathbf{J}_M - \\int - \\frac{e^{\\phi_2}}{\\theta_{1}{(\\phi_2)}} d\\phi_2 - \\int e^{\\phi_2} d\\phi_2 and \\int (1 - e^{\\phi_2}) d\\phi_2 = \\mathbf{J}_M - \\int (-1) d\\phi_2 - \\int e^{\\phi_2} d\\phi_2 and - (\\int (1 - e^{\\phi_2}) d\\phi_2) \\int e^{\\phi_2} d\\phi_2 = - (\\mathbf{J}_M - \\int (-1) d\\phi_2 - \\int e^{\\phi_2} d\\phi_2) \\int e^{\\phi_2} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Integral(Mul(Integer(-1), Pow(Function('\\\\theta_1')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Integral(Integer(-1), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["times", 6, "Mul(Integer(-1), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], "Equality(Mul(Integer(-1), Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Integral(Integer(-1), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\Psi^{\\dagger},\\mathbf{v})} = \\Psi^{\\dagger} \\mathbf{v}, then obtain \\delta{(\\Psi^{\\dagger},\\mathbf{v})} - \\delta^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\mathbf{v})} = \\Psi^{\\dagger} \\mathbf{v} - \\delta^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\mathbf{v})}", "derivation": "\\delta{(\\Psi^{\\dagger},\\mathbf{v})} = \\Psi^{\\dagger} \\mathbf{v} and \\delta^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\mathbf{v})} = (\\Psi^{\\dagger} \\mathbf{v})^{\\Psi^{\\dagger}} and - (\\Psi^{\\dagger} \\mathbf{v})^{\\Psi^{\\dagger}} + \\delta{(\\Psi^{\\dagger},\\mathbf{v})} = \\Psi^{\\dagger} \\mathbf{v} - (\\Psi^{\\dagger} \\mathbf{v})^{\\Psi^{\\dagger}} and \\delta{(\\Psi^{\\dagger},\\mathbf{v})} - \\delta^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\mathbf{v})} = \\Psi^{\\dagger} \\mathbf{v} - \\delta^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(E)} = \\cos{(\\sin{(E)})}, then derive U + \\operatorname{v_{x}}{(E)} = P_{e} + \\cos{(\\sin{(E)})}, then obtain \\frac{\\partial}{\\partial U} (U + \\operatorname{v_{x}}{(E)}) = \\frac{\\partial}{\\partial U} (P_{e} + \\operatorname{v_{x}}{(E)})", "derivation": "\\operatorname{v_{x}}{(E)} = \\cos{(\\sin{(E)})} and \\frac{d}{d E} \\operatorname{v_{x}}{(E)} = \\frac{d}{d E} \\cos{(\\sin{(E)})} and \\int \\frac{d}{d E} \\operatorname{v_{x}}{(E)} dE = \\int \\frac{d}{d E} \\cos{(\\sin{(E)})} dE and U + \\operatorname{v_{x}}{(E)} = P_{e} + \\cos{(\\sin{(E)})} and U + \\operatorname{v_{x}}{(E)} = P_{e} + \\operatorname{v_{x}}{(E)} and \\frac{\\partial}{\\partial U} (U + \\operatorname{v_{x}}{(E)}) = \\frac{\\partial}{\\partial U} (P_{e} + \\operatorname{v_{x}}{(E)})", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('E', commutative=True)), cos(sin(Symbol('E', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Derivative(Function('v_x')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Derivative(cos(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('U', commutative=True), Function('v_x')(Symbol('E', commutative=True))), Add(Symbol('P_e', commutative=True), cos(sin(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('U', commutative=True), Function('v_x')(Symbol('E', commutative=True))), Add(Symbol('P_e', commutative=True), Function('v_x')(Symbol('E', commutative=True))))"], [["differentiate", 5, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Symbol('U', commutative=True), Function('v_x')(Symbol('E', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Function('v_x')(Symbol('E', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} = e^{J + \\hat{\\mathbf{x}}}, then derive (\\int \\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (v_{1} + e^{J + \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}}, then obtain (v_{1} + e^{J + \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} = (\\int e^{J + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}}", "derivation": "\\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} = e^{J + \\hat{\\mathbf{x}}} and \\int \\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int e^{J + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} and (\\int \\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (\\int e^{J + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} and (\\int \\mathbf{J}_M{(J,\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (v_{1} + e^{J + \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} and (v_{1} + e^{J + \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} = (\\int e^{J + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Integral(exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Symbol('v_1', commutative=True), exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('v_1', commutative=True), exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Integral(exp(Add(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(a,V_{\\mathbf{B}})} = \\log{(- V_{\\mathbf{B}} + a)}, then obtain V_{\\mathbf{B}} (\\int 0 dV_{\\mathbf{B}} - \\int (- \\varphi^{*}{(a,V_{\\mathbf{B}})} + \\log{(- V_{\\mathbf{B}} + a)}) dV_{\\mathbf{B}}) = 0", "derivation": "\\varphi^{*}{(a,V_{\\mathbf{B}})} = \\log{(- V_{\\mathbf{B}} + a)} and 0 = - \\varphi^{*}{(a,V_{\\mathbf{B}})} + \\log{(- V_{\\mathbf{B}} + a)} and \\int 0 dV_{\\mathbf{B}} = \\int (- \\varphi^{*}{(a,V_{\\mathbf{B}})} + \\log{(- V_{\\mathbf{B}} + a)}) dV_{\\mathbf{B}} and \\int 0 dV_{\\mathbf{B}} - \\int (- \\varphi^{*}{(a,V_{\\mathbf{B}})} + \\log{(- V_{\\mathbf{B}} + a)}) dV_{\\mathbf{B}} = 0 and V_{\\mathbf{B}} (\\int 0 dV_{\\mathbf{B}} - \\int (- \\varphi^{*}{(a,V_{\\mathbf{B}})} + \\log{(- V_{\\mathbf{B}} + a)}) dV_{\\mathbf{B}}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True))))"], [["minus", 1, "Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True)))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 3, "Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Integer(0))"], [["times", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Add(Integral(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('a', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given s{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive \\int s{(\\mathbf{D})} d\\mathbf{D} = \\Psi^{\\dagger} + e^{\\mathbf{D}}, then obtain \\iint s{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D} = \\int (\\Psi^{\\dagger} + s{(\\mathbf{D})}) d\\mathbf{D}", "derivation": "s{(\\mathbf{D})} = e^{\\mathbf{D}} and \\int s{(\\mathbf{D})} d\\mathbf{D} = \\int e^{\\mathbf{D}} d\\mathbf{D} and \\int s{(\\mathbf{D})} d\\mathbf{D} = \\Psi^{\\dagger} + e^{\\mathbf{D}} and \\int s{(\\mathbf{D})} d\\mathbf{D} = \\Psi^{\\dagger} + s{(\\mathbf{D})} and \\iint s{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D} = \\int (\\Psi^{\\dagger} + s{(\\mathbf{D})}) d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('s')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('s')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('s')(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('s')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given h{(\\dot{\\mathbf{r}},C_{1})} = C_{1}^{\\dot{\\mathbf{r}}}, then obtain \\frac{\\dot{\\mathbf{r}} h^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},C_{1})} \\frac{\\partial}{\\partial C_{1}} h{(\\dot{\\mathbf{r}},C_{1})}}{h{(\\dot{\\mathbf{r}},C_{1})}} = \\frac{\\dot{\\mathbf{r}}^{2} (C_{1}^{\\dot{\\mathbf{r}}})^{\\dot{\\mathbf{r}}}}{C_{1}}", "derivation": "h{(\\dot{\\mathbf{r}},C_{1})} = C_{1}^{\\dot{\\mathbf{r}}} and h^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},C_{1})} = (C_{1}^{\\dot{\\mathbf{r}}})^{\\dot{\\mathbf{r}}} and \\frac{\\partial}{\\partial C_{1}} h^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},C_{1})} = \\frac{\\partial}{\\partial C_{1}} (C_{1}^{\\dot{\\mathbf{r}}})^{\\dot{\\mathbf{r}}} and \\frac{\\dot{\\mathbf{r}} h^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},C_{1})} \\frac{\\partial}{\\partial C_{1}} h{(\\dot{\\mathbf{r}},C_{1})}}{h{(\\dot{\\mathbf{r}},C_{1})}} = \\frac{\\dot{\\mathbf{r}}^{2} (C_{1}^{\\dot{\\mathbf{r}}})^{\\dot{\\mathbf{r}}}}{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Pow(Symbol('C_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Pow(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('C_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), Pow(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(Function('h')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2)), Pow(Pow(Symbol('C_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\dot{x})} = \\int \\cos{(\\dot{x})} d\\dot{x}, then derive \\operatorname{M_{E}}{(\\dot{x})} = \\mu + \\sin{(\\dot{x})}, then derive \\operatorname{M_{E}}^{\\dot{x}}{(\\dot{x})} = (\\mathbf{E} + \\sin{(\\dot{x})})^{\\dot{x}}, then obtain (\\mu + \\sin{(\\dot{x})})^{\\dot{x}} = (\\mathbf{E} + \\sin{(\\dot{x})})^{\\dot{x}}", "derivation": "\\operatorname{M_{E}}{(\\dot{x})} = \\int \\cos{(\\dot{x})} d\\dot{x} and \\operatorname{M_{E}}{(\\dot{x})} = \\mu + \\sin{(\\dot{x})} and \\operatorname{M_{E}}^{\\dot{x}}{(\\dot{x})} = (\\int \\cos{(\\dot{x})} d\\dot{x})^{\\dot{x}} and \\operatorname{M_{E}}^{\\dot{x}}{(\\dot{x})} = (\\mathbf{E} + \\sin{(\\dot{x})})^{\\dot{x}} and (\\mu + \\sin{(\\dot{x})})^{\\dot{x}} = (\\int \\cos{(\\dot{x})} d\\dot{x})^{\\dot{x}} and (\\mu + \\sin{(\\dot{x})})^{\\dot{x}} = \\operatorname{M_{E}}^{\\dot{x}}{(\\dot{x})} and (\\mu + \\sin{(\\dot{x})})^{\\dot{x}} = (\\mathbf{E} + \\sin{(\\dot{x})})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\dot{x}', commutative=True)), Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('M_E')(Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Function('M_E')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('M_E')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(h,\\mathbf{E})} = \\mathbf{E} + h and \\mu_{0}{(h)} = - h, then obtain e^{\\iint (\\mathbf{v}{(h,\\mathbf{E})} + \\mu_{0}{(h)}) dh dh} = e^{\\iint \\mathbf{E} dh dh}", "derivation": "\\mathbf{v}{(h,\\mathbf{E})} = \\mathbf{E} + h and - h + \\mathbf{v}{(h,\\mathbf{E})} = \\mathbf{E} and \\mu_{0}{(h)} = - h and \\mathbf{v}{(h,\\mathbf{E})} + \\mu_{0}{(h)} = \\mathbf{E} and \\int (\\mathbf{v}{(h,\\mathbf{E})} + \\mu_{0}{(h)}) dh = \\int \\mathbf{E} dh and \\iint (\\mathbf{v}{(h,\\mathbf{E})} + \\mu_{0}{(h)}) dh dh = \\iint \\mathbf{E} dh dh and e^{\\iint (\\mathbf{v}{(h,\\mathbf{E})} + \\mu_{0}{(h)}) dh dh} = e^{\\iint \\mathbf{E} dh dh}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('h', commutative=True)))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('h', commutative=True))))"], [["integrate", 5, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["exp", 6], "Equality(exp(Integral(Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), exp(Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(r)} = r, then derive \\frac{d^{2}}{d r^{2}} \\operatorname{t_{2}}{(r)} = 0, then obtain \\frac{d^{3}}{d r^{3}} r = \\frac{d}{d r} 0", "derivation": "\\operatorname{t_{2}}{(r)} = r and \\frac{d}{d r} \\operatorname{t_{2}}{(r)} = \\frac{d}{d r} r and \\frac{d^{2}}{d r^{2}} \\operatorname{t_{2}}{(r)} = \\frac{d^{2}}{d r^{2}} r and \\frac{d^{2}}{d r^{2}} \\operatorname{t_{2}}{(r)} = 0 and \\frac{d^{2}}{d r^{2}} r = 0 and \\frac{d^{3}}{d r^{3}} r = \\frac{d}{d r} 0", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('r', commutative=True)), Symbol('r', commutative=True))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('t_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(2))), Integer(0))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(3))), Derivative(Integer(0), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(\\mathbf{D})} = \\cos{(\\mathbf{D})}, then obtain \\frac{d}{d \\mathbf{D}} U^{3 \\mathbf{D}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} U^{2 \\mathbf{D}}{(\\mathbf{D})} \\cos^{\\mathbf{D}}{(\\mathbf{D})}", "derivation": "U{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and U^{\\mathbf{D}}{(\\mathbf{D})} = \\cos^{\\mathbf{D}}{(\\mathbf{D})} and U^{2 \\mathbf{D}}{(\\mathbf{D})} = U^{\\mathbf{D}}{(\\mathbf{D})} \\cos^{\\mathbf{D}}{(\\mathbf{D})} and U^{3 \\mathbf{D}}{(\\mathbf{D})} = U^{2 \\mathbf{D}}{(\\mathbf{D})} \\cos^{\\mathbf{D}}{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} U^{3 \\mathbf{D}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} U^{2 \\mathbf{D}}{(\\mathbf{D})} \\cos^{\\mathbf{D}}{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["times", 2, "Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 3, "Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('U')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(I,\\hat{p}_0)} = \\log{(I \\hat{p}_0)}, then obtain - \\hat{p}_0 + \\int \\frac{\\varphi{(I,\\hat{p}_0)}}{\\log{(I \\hat{p}_0)}} dI = - \\hat{p}_0 + \\int 1 dI", "derivation": "\\varphi{(I,\\hat{p}_0)} = \\log{(I \\hat{p}_0)} and \\frac{\\varphi{(I,\\hat{p}_0)}}{\\log{(I \\hat{p}_0)}} = 1 and \\int \\frac{\\varphi{(I,\\hat{p}_0)}}{\\log{(I \\hat{p}_0)}} dI = \\int 1 dI and - \\hat{p}_0 + \\int \\frac{\\varphi{(I,\\hat{p}_0)}}{\\log{(I \\hat{p}_0)}} dI = - \\hat{p}_0 + \\int 1 dI", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["divide", 1, "log(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Tuple(Symbol('I', commutative=True))), Integral(Integer(1), Tuple(Symbol('I', commutative=True))))"], [["minus", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Integral(Mul(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Tuple(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Integral(Integer(1), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{E},\\mathbf{f})} = \\mathbf{E} \\mathbf{f}, then derive \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{A_{1}}{(\\mathbf{E},\\mathbf{f})} = \\mathbf{E}, then derive - V_{\\mathbf{B}} + v_{2} = 0, then obtain - V_{\\mathbf{B}} + \\mathbf{f} + v_{2} = \\mathbf{f}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{E},\\mathbf{f})} = \\mathbf{E} \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{A_{1}}{(\\mathbf{E},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{E} \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{A_{1}}{(\\mathbf{E},\\mathbf{f})} = \\mathbf{E} and \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{E} \\mathbf{f} = \\mathbf{E} and \\int \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{E} \\mathbf{f} d\\mathbf{E} = \\int \\mathbf{E} d\\mathbf{E} and - \\int \\mathbf{E} d\\mathbf{E} + \\int \\frac{\\partial}{\\partial \\mathbf{f}} \\mathbf{E} \\mathbf{f} d\\mathbf{E} = 0 and - V_{\\mathbf{B}} + v_{2} = 0 and - V_{\\mathbf{B}} + \\mathbf{f} + v_{2} = \\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\mathbf{E}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\mathbf{E}', commutative=True))"], [["integrate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 5, "Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Integral(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(0))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('v_2', commutative=True)), Integer(0))"], [["add", 7, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"]]}, {"prompt": "Given S{(a,n)} = \\cos{(a - n)}, then obtain (- 2 \\sin{(a - n)} + 2 \\frac{\\partial}{\\partial a} S{(a,n)}) (- n + S{(a,n)} + \\cos{(a - n)}) = - 4 (- n + 2 \\cos{(a - n)}) \\sin{(a - n)}", "derivation": "S{(a,n)} = \\cos{(a - n)} and - n + S{(a,n)} = - n + \\cos{(a - n)} and - n + S{(a,n)} + \\cos{(a - n)} = - n + 2 \\cos{(a - n)} and (- n + S{(a,n)} + \\cos{(a - n)})^{2} = (- n + 2 \\cos{(a - n)})^{2} and \\frac{\\partial}{\\partial a} (- n + S{(a,n)} + \\cos{(a - n)})^{2} = \\frac{\\partial}{\\partial a} (- n + 2 \\cos{(a - n)})^{2} and (- 2 \\sin{(a - n)} + 2 \\frac{\\partial}{\\partial a} S{(a,n)}) (- n + S{(a,n)} + \\cos{(a - n)}) = - 4 (- n + 2 \\cos{(a - n)}) \\sin{(a - n)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["add", 2, "cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), Integer(2)))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Mul(Integer(2), Derivative(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), Mul(Integer(-1), Integer(4), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(2), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))), sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"]]}, {"prompt": "Given G{(V_{\\mathbf{B}},\\dot{x})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\dot{x}) and \\operatorname{v_{y}}{(\\dot{x})} = \\dot{x}, then derive G{(V_{\\mathbf{B}},\\dot{x})} = 1, then obtain - \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\operatorname{v_{y}}{(\\dot{x})}) = -1", "derivation": "G{(V_{\\mathbf{B}},\\dot{x})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\dot{x}) and G{(V_{\\mathbf{B}},\\dot{x})} = 1 and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\dot{x}) = 1 and \\operatorname{v_{y}}{(\\dot{x})} = \\dot{x} and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\operatorname{v_{y}}{(\\dot{x})}) = 1 and - \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\operatorname{v_{y}}{(\\dot{x})}) = -1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_y')(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('v_y')(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Integer(-1))"]]}, {"prompt": "Given \\ddot{x}{(u,M_{E})} = M_{E}^{u}, then obtain - M_{E}^{- 4 u} = - \\frac{M_{E}^{- 2 u}}{\\ddot{x}^{2}{(u,M_{E})}}", "derivation": "\\ddot{x}{(u,M_{E})} = M_{E}^{u} and M_{E}^{u} \\ddot{x}{(u,M_{E})} = M_{E}^{2 u} and M_{E}^{u} = \\frac{M_{E}^{2 u}}{\\ddot{x}{(u,M_{E})}} and M_{E}^{2 u} = \\frac{M_{E}^{4 u}}{\\ddot{x}^{2}{(u,M_{E})}} and - M_{E}^{2 u} = - \\frac{M_{E}^{4 u}}{\\ddot{x}^{2}{(u,M_{E})}} and - M_{E}^{- 2 u} = - \\frac{1}{\\ddot{x}^{2}{(u,M_{E})}} and - M_{E}^{- 4 u} = - \\frac{M_{E}^{- 2 u}}{\\ddot{x}^{2}{(u,M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('u', commutative=True)))"], [["times", 1, "Pow(Symbol('M_E', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Symbol('u', commutative=True)), Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))), Pow(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))))"], [["divide", 2, "Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Pow(Symbol('M_E', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Mul(Integer(4), Symbol('u', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-2))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Mul(Integer(4), Symbol('u', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-2))))"], [["divide", 5, "Pow(Symbol('M_E', commutative=True), Mul(Integer(4), Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-2))))"], [["divide", 6, "Pow(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Integer(4), Symbol('u', commutative=True)))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} = a^{\\dagger} + \\sin{(\\varphi^*)}, then obtain \\int \\varphi^* (\\int \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} d\\varphi^*)^{a^{\\dagger}} d\\varphi^* = \\int \\varphi^* (\\int (a^{\\dagger} + \\sin{(\\varphi^*)}) d\\varphi^*)^{a^{\\dagger}} d\\varphi^*", "derivation": "\\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} = a^{\\dagger} + \\sin{(\\varphi^*)} and \\int \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} d\\varphi^* = \\int (a^{\\dagger} + \\sin{(\\varphi^*)}) d\\varphi^* and (\\int \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} d\\varphi^*)^{a^{\\dagger}} = (\\int (a^{\\dagger} + \\sin{(\\varphi^*)}) d\\varphi^*)^{a^{\\dagger}} and \\varphi^* (\\int \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} d\\varphi^*)^{a^{\\dagger}} = \\varphi^* (\\int (a^{\\dagger} + \\sin{(\\varphi^*)}) d\\varphi^*)^{a^{\\dagger}} and \\int \\varphi^* (\\int \\operatorname{f_{E}}{(a^{\\dagger},\\varphi^*)} d\\varphi^*)^{a^{\\dagger}} d\\varphi^* = \\int \\varphi^* (\\int (a^{\\dagger} + \\sin{(\\varphi^*)}) d\\varphi^*)^{a^{\\dagger}} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Integral(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Integral(Function('f_E')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\phi,\\hat{X})} = \\cos^{\\hat{X}}{(\\phi)} and \\hat{x}{(\\hat{X})} = \\cos{(\\log{(\\hat{X})})}, then obtain \\operatorname{P_{g}}^{\\phi}{(\\phi,\\hat{X})} \\frac{d}{d \\hat{X}} \\hat{x}{(\\hat{X})} = \\operatorname{P_{g}}^{\\phi}{(\\phi,\\hat{X})} \\frac{d}{d \\hat{X}} \\cos{(\\log{(\\hat{X})})}", "derivation": "\\operatorname{P_{g}}{(\\phi,\\hat{X})} = \\cos^{\\hat{X}}{(\\phi)} and \\hat{x}{(\\hat{X})} = \\cos{(\\log{(\\hat{X})})} and \\frac{d}{d \\hat{X}} \\hat{x}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(\\log{(\\hat{X})})} and (\\cos^{\\hat{X}}{(\\phi)})^{\\phi} \\frac{d}{d \\hat{X}} \\hat{x}{(\\hat{X})} = (\\cos^{\\hat{X}}{(\\phi)})^{\\phi} \\frac{d}{d \\hat{X}} \\cos{(\\log{(\\hat{X})})} and \\operatorname{P_{g}}^{\\phi}{(\\phi,\\hat{X})} \\frac{d}{d \\hat{X}} \\hat{x}{(\\hat{X})} = \\operatorname{P_{g}}^{\\phi}{(\\phi,\\hat{X})} \\frac{d}{d \\hat{X}} \\cos{(\\log{(\\hat{X})})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{X}', commutative=True)), cos(log(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["times", 3, "Pow(Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Pow(Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(cos(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('P_g')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Pow(Function('P_g')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(cos(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(f^{*},\\chi)} = e^{\\frac{\\chi}{f^{*}}} and \\operatorname{E_{n}}{(f^{*},\\chi)} = \\frac{\\chi}{f^{*}}, then obtain - \\chi + e^{\\frac{\\chi}{f^{*}}} = - \\chi + e^{\\operatorname{E_{n}}{(f^{*},\\chi)}}", "derivation": "\\operatorname{z^{*}}{(f^{*},\\chi)} = e^{\\frac{\\chi}{f^{*}}} and - \\chi + \\operatorname{z^{*}}{(f^{*},\\chi)} = - \\chi + e^{\\frac{\\chi}{f^{*}}} and \\operatorname{E_{n}}{(f^{*},\\chi)} = \\frac{\\chi}{f^{*}} and - \\chi + \\operatorname{z^{*}}{(f^{*},\\chi)} = - \\chi + e^{\\operatorname{E_{n}}{(f^{*},\\chi)}} and - \\chi + e^{\\frac{\\chi}{f^{*}}} = - \\chi + e^{\\operatorname{E_{n}}{(f^{*},\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('z^*')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('z^*')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Function('E_n')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Function('E_n')(Symbol('f^*', commutative=True), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given V{(f^{*})} = \\sin{(f^{*})}, then derive \\frac{d}{d f^{*}} V{(f^{*})} = \\cos{(f^{*})}, then obtain - f^{*} + \\frac{d}{d f^{*}} V{(f^{*})} = - f^{*} + \\cos{(f^{*})}", "derivation": "V{(f^{*})} = \\sin{(f^{*})} and \\frac{d}{d f^{*}} V{(f^{*})} = \\frac{d}{d f^{*}} \\sin{(f^{*})} and \\frac{d}{d f^{*}} V{(f^{*})} = \\cos{(f^{*})} and - f^{*} + \\frac{d}{d f^{*}} V{(f^{*})} = - f^{*} + \\cos{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), cos(Symbol('f^*', commutative=True)))"], [["minus", 3, "Symbol('f^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Derivative(Function('V')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given k{(\\phi_2,\\mathbb{I})} = \\mathbb{I} - \\phi_2, then obtain (\\log{(k{(\\phi_2,\\mathbb{I})})} - \\cos{(k{(\\phi_2,\\mathbb{I})})})^{\\mathbb{I}} = (\\log{(\\mathbb{I} - \\phi_2)} - \\cos{(k{(\\phi_2,\\mathbb{I})})})^{\\mathbb{I}}", "derivation": "k{(\\phi_2,\\mathbb{I})} = \\mathbb{I} - \\phi_2 and \\log{(k{(\\phi_2,\\mathbb{I})})} = \\log{(\\mathbb{I} - \\phi_2)} and \\cos{(k{(\\phi_2,\\mathbb{I})})} = \\cos{(\\mathbb{I} - \\phi_2)} and \\log{(k{(\\phi_2,\\mathbb{I})})} - \\cos{(\\mathbb{I} - \\phi_2)} = \\log{(\\mathbb{I} - \\phi_2)} - \\cos{(\\mathbb{I} - \\phi_2)} and \\log{(k{(\\phi_2,\\mathbb{I})})} - \\cos{(k{(\\phi_2,\\mathbb{I})})} = \\log{(\\mathbb{I} - \\phi_2)} - \\cos{(k{(\\phi_2,\\mathbb{I})})} and (\\log{(k{(\\phi_2,\\mathbb{I})})} - \\cos{(k{(\\phi_2,\\mathbb{I})})})^{\\mathbb{I}} = (\\log{(\\mathbb{I} - \\phi_2)} - \\cos{(k{(\\phi_2,\\mathbb{I})})})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["log", 1], "Equality(log(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), log(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"], [["cos", 1], "Equality(cos(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), cos(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 2, "cos(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], "Equality(Add(log(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))), Add(log(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(log(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), cos(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))), Add(log(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), cos(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))))"], [["power", 5, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Add(log(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), cos(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Add(log(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), cos(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given Z{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain \\frac{(Z{(\\mathbf{s})} - Z^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}}}{\\mathbf{s}} = \\frac{(- Z^{\\mathbf{s}}{(\\mathbf{s})} + e^{\\mathbf{s}})^{\\mathbf{s}}}{\\mathbf{s}}", "derivation": "Z{(\\mathbf{s})} = e^{\\mathbf{s}} and Z^{\\mathbf{s}}{(\\mathbf{s})} = (e^{\\mathbf{s}})^{\\mathbf{s}} and Z{(\\mathbf{s})} - (e^{\\mathbf{s}})^{\\mathbf{s}} = e^{\\mathbf{s}} - (e^{\\mathbf{s}})^{\\mathbf{s}} and Z{(\\mathbf{s})} - Z^{\\mathbf{s}}{(\\mathbf{s})} = - Z^{\\mathbf{s}}{(\\mathbf{s})} + e^{\\mathbf{s}} and (Z{(\\mathbf{s})} - Z^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}} = (- Z^{\\mathbf{s}}{(\\mathbf{s})} + e^{\\mathbf{s}})^{\\mathbf{s}} and \\frac{(Z{(\\mathbf{s})} - Z^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}}}{\\mathbf{s}} = \\frac{(- Z^{\\mathbf{s}}{(\\mathbf{s})} + e^{\\mathbf{s}})^{\\mathbf{s}}}{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Add(exp(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), exp(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 5, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), exp(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(I,\\mu)} = (e^{\\mu})^{I} and \\rho{(C_{1},c_{0})} = C_{1} c_{0}, then obtain \\mu \\rho^{c_{0}}{(C_{1},c_{0})} (e^{\\mu})^{I} = \\mu (C_{1} c_{0})^{c_{0}} (e^{\\mu})^{I}", "derivation": "\\ddot{x}{(I,\\mu)} = (e^{\\mu})^{I} and \\mu \\ddot{x}{(I,\\mu)} = \\mu (e^{\\mu})^{I} and \\rho{(C_{1},c_{0})} = C_{1} c_{0} and \\rho^{c_{0}}{(C_{1},c_{0})} = (C_{1} c_{0})^{c_{0}} and \\mu \\ddot{x}{(I,\\mu)} \\rho^{c_{0}}{(C_{1},c_{0})} = \\mu (C_{1} c_{0})^{c_{0}} \\ddot{x}{(I,\\mu)} and \\mu \\rho^{c_{0}}{(C_{1},c_{0})} (e^{\\mu})^{I} = \\mu (C_{1} c_{0})^{c_{0}} (e^{\\mu})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('I', commutative=True)))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\ddot{x}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('I', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Mul(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], [["times", 4, "Mul(Symbol('\\\\mu', commutative=True), Function('\\\\ddot{x}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\ddot{x}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\rho')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Mul(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Function('\\\\ddot{x}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('I', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Mul(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(H)} = e^{H} and \\mathbf{f}{(H)} = e^{H}, then derive \\frac{d^{2}}{d H^{2}} \\mathbf{f}{(H)} = e^{H}, then obtain e^{H} = \\frac{d^{2}}{d H^{2}} e^{H}", "derivation": "\\operatorname{A_{z}}{(H)} = e^{H} and \\frac{d}{d H} \\operatorname{A_{z}}{(H)} = \\frac{d}{d H} e^{H} and \\mathbf{f}{(H)} = e^{H} and \\operatorname{A_{z}}{(H)} = \\mathbf{f}{(H)} and \\frac{d}{d H} \\mathbf{f}{(H)} = \\frac{d}{d H} e^{H} and \\frac{d^{2}}{d H^{2}} \\mathbf{f}{(H)} = \\frac{d^{2}}{d H^{2}} e^{H} and \\frac{d^{2}}{d H^{2}} \\mathbf{f}{(H)} = e^{H} and e^{H} = \\frac{d^{2}}{d H^{2}} e^{H}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A_z')(Symbol('H', commutative=True)), Function('\\\\mathbf{f}')(Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), exp(Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(exp(Symbol('H', commutative=True)), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(f,Z,M_{E})} = \\frac{M_{E} Z}{f} and r{(f,Z,M_{E})} = \\frac{M_{E} Z}{f} and \\operatorname{y^{\\prime}}{(f,Z,M_{E})} = \\int \\operatorname{V_{\\mathbf{E}}}^{Z}{(f,Z,M_{E})} df and s{(f,Z,M_{E})} = \\int \\operatorname{V_{\\mathbf{E}}}^{Z}{(f,Z,M_{E})} df, then obtain \\int r^{Z}{(f,Z,M_{E})} df = \\operatorname{y^{\\prime}}{(f,Z,M_{E})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(f,Z,M_{E})} = \\frac{M_{E} Z}{f} and r{(f,Z,M_{E})} = \\frac{M_{E} Z}{f} and r{(f,Z,M_{E})} = \\operatorname{V_{\\mathbf{E}}}{(f,Z,M_{E})} and \\operatorname{y^{\\prime}}{(f,Z,M_{E})} = \\int \\operatorname{V_{\\mathbf{E}}}^{Z}{(f,Z,M_{E})} df and s{(f,Z,M_{E})} = \\int \\operatorname{V_{\\mathbf{E}}}^{Z}{(f,Z,M_{E})} df and s{(f,Z,M_{E})} = \\int r^{Z}{(f,Z,M_{E})} df and s{(f,Z,M_{E})} = \\operatorname{y^{\\prime}}{(f,Z,M_{E})} and \\int r^{Z}{(f,Z,M_{E})} df = \\operatorname{y^{\\prime}}{(f,Z,M_{E})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('r')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('Z', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('r')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Integral(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Integral(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('s')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Integral(Pow(Function('r')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('s')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integral(Pow(Function('r')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('f', commutative=True))), Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('Z', commutative=True), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given M{(I,A_{2})} = \\log{(A_{2} - I)}, then obtain \\frac{e^{- M{(I,A_{2})}}}{A_{2} - I} = \\frac{1}{(A_{2} - I)^{2}}", "derivation": "M{(I,A_{2})} = \\log{(A_{2} - I)} and - M{(I,A_{2})} = - \\log{(A_{2} - I)} and - M{(I,A_{2})} - \\log{(A_{2} - I)} = - 2 \\log{(A_{2} - I)} and \\frac{e^{- M{(I,A_{2})}}}{A_{2} - I} = \\frac{1}{(A_{2} - I)^{2}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('M')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["minus", 2, "log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('M')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))), Mul(Integer(-1), Integer(2), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["exp", 3], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Function('M')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))))), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\omega,a)} = \\omega - a, then derive - \\omega + \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = \\mathbf{p} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega, then derive \\mathbf{p} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega = \\mathbf{r} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega, then obtain - \\omega + \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = \\mathbf{r} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega", "derivation": "\\operatorname{F_{N}}{(\\omega,a)} = \\omega - a and \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = \\int (\\omega - a) d\\omega and - \\omega + \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = - \\omega + \\int (\\omega - a) d\\omega and - \\omega + \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = \\mathbf{p} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega and \\mathbf{p} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega = - \\omega + \\int (\\omega - a) d\\omega and \\mathbf{p} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega = \\mathbf{r} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega and - \\omega + \\int \\operatorname{F_{N}}{(\\omega,a)} d\\omega = \\mathbf{r} + \\frac{\\omega^{2}}{2} - \\omega a - \\omega", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(Function('F_N')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(Function('F_N')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(Function('F_N')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\Omega{(n_{2})} = \\cos{(n_{2})}, then obtain (\\frac{d}{d n_{2}} (n_{2} + \\Omega{(n_{2})}))^{n_{2}} = (\\frac{d}{d n_{2}} (n_{2} + \\cos{(n_{2})}))^{n_{2}}", "derivation": "\\Omega{(n_{2})} = \\cos{(n_{2})} and n_{2} + \\Omega{(n_{2})} = n_{2} + \\cos{(n_{2})} and \\frac{d}{d n_{2}} (n_{2} + \\Omega{(n_{2})}) = \\frac{d}{d n_{2}} (n_{2} + \\cos{(n_{2})}) and (\\frac{d}{d n_{2}} (n_{2} + \\Omega{(n_{2})}))^{n_{2}} = (\\frac{d}{d n_{2}} (n_{2} + \\cos{(n_{2})}))^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["add", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Symbol('n_2', commutative=True), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Add(Symbol('n_2', commutative=True), cos(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Add(Symbol('n_2', commutative=True), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Symbol('n_2', commutative=True), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('n_2', commutative=True), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Add(Symbol('n_2', commutative=True), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(x,\\rho_b)} = \\frac{\\rho_b}{x}, then obtain \\sin{(\\operatorname{P_{g}}^{\\rho_b}{(x,\\rho_b)})} - \\frac{1}{x} = \\sin{((\\frac{\\rho_b}{x})^{\\rho_b})} - \\frac{1}{x}", "derivation": "\\operatorname{P_{g}}{(x,\\rho_b)} = \\frac{\\rho_b}{x} and \\operatorname{P_{g}}^{\\rho_b}{(x,\\rho_b)} = (\\frac{\\rho_b}{x})^{\\rho_b} and \\sin{(\\operatorname{P_{g}}^{\\rho_b}{(x,\\rho_b)})} = \\sin{((\\frac{\\rho_b}{x})^{\\rho_b})} and \\sin{(\\operatorname{P_{g}}^{\\rho_b}{(x,\\rho_b)})} - \\frac{1}{x} = \\sin{((\\frac{\\rho_b}{x})^{\\rho_b})} - \\frac{1}{x}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), sin(Pow(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Pow(Symbol('x', commutative=True), Integer(-1))"], "Equality(Add(sin(Pow(Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)))), Add(sin(Pow(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\psi^{*}{(\\rho)} = \\sin{(\\sin{(\\rho)})}, then obtain e^{\\frac{d}{d \\rho} \\psi^{*}{(\\rho)}} = e^{\\cos{(\\rho)} \\cos{(\\sin{(\\rho)})}}", "derivation": "\\psi^{*}{(\\rho)} = \\sin{(\\sin{(\\rho)})} and \\frac{d}{d \\rho} \\psi^{*}{(\\rho)} = \\frac{d}{d \\rho} \\sin{(\\sin{(\\rho)})} and e^{\\frac{d}{d \\rho} \\psi^{*}{(\\rho)}} = e^{\\frac{d}{d \\rho} \\sin{(\\sin{(\\rho)})}} and e^{\\frac{d}{d \\rho} \\psi^{*}{(\\rho)}} = e^{\\cos{(\\rho)} \\cos{(\\sin{(\\rho)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\rho', commutative=True)), sin(sin(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\psi^*')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), exp(Derivative(sin(sin(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(exp(Derivative(Function('\\\\psi^*')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), exp(Mul(cos(Symbol('\\\\rho', commutative=True)), cos(sin(Symbol('\\\\rho', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(T)} = \\cos{(T)}, then obtain (\\frac{\\operatorname{g_{\\varepsilon}}^{3}{(T)}}{\\cos^{4}{(T)}})^{T} = (\\frac{1}{\\cos{(T)}})^{T}", "derivation": "\\operatorname{g_{\\varepsilon}}{(T)} = \\cos{(T)} and \\frac{\\operatorname{g_{\\varepsilon}}{(T)}}{\\cos{(T)}} = 1 and \\frac{\\operatorname{g_{\\varepsilon}}{(T)}}{\\cos^{2}{(T)}} = \\frac{1}{\\cos{(T)}} and (\\frac{\\operatorname{g_{\\varepsilon}}{(T)}}{\\cos^{2}{(T)}})^{T} = (\\frac{1}{\\cos{(T)}})^{T} and (\\frac{\\operatorname{g_{\\varepsilon}}^{3}{(T)}}{\\cos^{4}{(T)}})^{T} = (\\frac{\\operatorname{g_{\\varepsilon}}{(T)}}{\\cos^{2}{(T)}})^{T} and (\\frac{\\operatorname{g_{\\varepsilon}}^{3}{(T)}}{\\cos^{4}{(T)}})^{T} = (\\frac{1}{\\cos{(T)}})^{T}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["divide", 1, "cos(Symbol('T', commutative=True))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "cos(Symbol('T', commutative=True))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-2))), Pow(cos(Symbol('T', commutative=True)), Integer(-1)))"], [["power", 3, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-2))), Symbol('T', commutative=True)), Pow(Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Integer(3)), Pow(cos(Symbol('T', commutative=True)), Integer(-4))), Symbol('T', commutative=True)), Pow(Mul(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-2))), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Integer(3)), Pow(cos(Symbol('T', commutative=True)), Integer(-4))), Symbol('T', commutative=True)), Pow(Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(b)} = \\log{(b)}, then obtain (\\hat{x}_0^{2}{(b)} - \\hat{x}_0{(b)})^{2} \\hat{x}_0{(b)} \\log{(b)} = (\\hat{x}_0{(b)} \\log{(b)} - \\hat{x}_0{(b)})^{2} \\hat{x}_0{(b)} \\log{(b)}", "derivation": "\\hat{x}_0{(b)} = \\log{(b)} and \\hat{x}_0^{2}{(b)} = \\hat{x}_0{(b)} \\log{(b)} and \\hat{x}_0^{2}{(b)} - \\hat{x}_0{(b)} = \\hat{x}_0{(b)} \\log{(b)} - \\hat{x}_0{(b)} and (\\hat{x}_0^{2}{(b)} - \\hat{x}_0{(b)})^{2} = (\\hat{x}_0{(b)} \\log{(b)} - \\hat{x}_0{(b)})^{2} and (\\hat{x}_0^{2}{(b)} - \\hat{x}_0{(b)})^{2} \\hat{x}_0{(b)} \\log{(b)} = (\\hat{x}_0{(b)} \\log{(b)} - \\hat{x}_0{(b)})^{2} \\hat{x}_0{(b)} \\log{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["times", 1, "Function('\\\\hat{x}_0')(Symbol('b', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), Integer(2)), Mul(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))))"], [["minus", 2, "Function('\\\\hat{x}_0')(Symbol('b', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))), Add(Mul(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))), Integer(2)), Pow(Add(Mul(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))), Integer(2)))"], [["times", 4, "Mul(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Add(Pow(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))), Integer(2)), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))), Mul(Pow(Add(Mul(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)))), Integer(2)), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))))"]]}, {"prompt": "Given E{(\\mathbf{P})} = \\sin{(\\mathbf{P})} and \\operatorname{t_{2}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then derive \\frac{d}{d \\mathbf{P}} E{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then obtain \\frac{d}{d \\mathbf{P}} \\sin{(\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{P})}", "derivation": "E{(\\mathbf{P})} = \\sin{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} E{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\sin{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} E{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\operatorname{t_{2}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} E{(\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\sin{(\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('E')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given x{(P_{g},P_{e})} = P_{e} + P_{g}, then derive \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} = 1, then obtain \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} \\int (P_{e} + P_{g}) dP_{e} = \\int (P_{e} + P_{g}) dP_{e}", "derivation": "x{(P_{g},P_{e})} = P_{e} + P_{g} and \\int x{(P_{g},P_{e})} dP_{e} = \\int (P_{e} + P_{g}) dP_{e} and \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} = \\frac{\\partial}{\\partial P_{e}} (P_{e} + P_{g}) and \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} = 1 and \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} \\int x{(P_{g},P_{e})} dP_{e} = \\int x{(P_{g},P_{e})} dP_{e} and \\frac{\\partial}{\\partial P_{e}} x{(P_{g},P_{e})} \\int (P_{e} + P_{g}) dP_{e} = \\int (P_{e} + P_{g}) dP_{e}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('P_g', commutative=True)))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Integral(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))"], "Equality(Mul(Derivative(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integral(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Integral(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Derivative(Function('x')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integral(Add(Symbol('P_e', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Integral(Add(Symbol('P_e', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given I{(s,u)} = s^{u}, then derive \\frac{\\partial}{\\partial u} I{(s,u)} = s^{u} \\log{(s)}, then obtain - \\sin{((s^{u})^{u})} + \\frac{\\partial}{\\partial u} I{(s,u)} = I{(s,u)} \\log{(s)} - \\sin{((s^{u})^{u})}", "derivation": "I{(s,u)} = s^{u} and \\frac{\\partial}{\\partial u} I{(s,u)} = \\frac{\\partial}{\\partial u} s^{u} and \\frac{\\partial}{\\partial u} I{(s,u)} = s^{u} \\log{(s)} and \\frac{\\partial}{\\partial u} I{(s,u)} = I{(s,u)} \\log{(s)} and - \\sin{((s^{u})^{u})} + \\frac{\\partial}{\\partial u} I{(s,u)} = I{(s,u)} \\log{(s)} - \\sin{((s^{u})^{u})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)), log(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), log(Symbol('s', commutative=True))))"], [["minus", 4, "sin(Pow(Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))), Derivative(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Mul(Function('I')(Symbol('s', commutative=True), Symbol('u', commutative=True)), log(Symbol('s', commutative=True))), Mul(Integer(-1), sin(Pow(Pow(Symbol('s', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\dot{x}{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain \\frac{\\frac{d}{d \\tilde{g}} \\dot{x}{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\dot{x}{(\\tilde{g})}}{\\tilde{g}^{2}} = - \\frac{\\sin{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\cos{(\\tilde{g})}}{\\tilde{g}^{2}}", "derivation": "\\dot{x}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\frac{\\dot{x}{(\\tilde{g})}}{\\tilde{g}} = \\frac{\\cos{(\\tilde{g})}}{\\tilde{g}} and \\frac{d}{d \\tilde{g}} \\frac{\\dot{x}{(\\tilde{g})}}{\\tilde{g}} = \\frac{d}{d \\tilde{g}} \\frac{\\cos{(\\tilde{g})}}{\\tilde{g}} and \\frac{\\frac{d}{d \\tilde{g}} \\dot{x}{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\dot{x}{(\\tilde{g})}}{\\tilde{g}^{2}} = - \\frac{\\sin{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\cos{(\\tilde{g})}}{\\tilde{g}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Function('\\\\dot{x}')(Symbol('\\\\tilde{g}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), cos(Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(a,q)} = a q, then obtain \\int (a + \\log{(\\hat{\\mathbf{r}}{(a,q)})} + \\int a q dq) da = \\int (a + \\log{(a q)} + \\int a q dq) da", "derivation": "\\hat{\\mathbf{r}}{(a,q)} = a q and \\log{(\\hat{\\mathbf{r}}{(a,q)})} = \\log{(a q)} and \\log{(\\hat{\\mathbf{r}}{(a,q)})} + \\int a q dq = \\log{(a q)} + \\int a q dq and a + \\log{(\\hat{\\mathbf{r}}{(a,q)})} + \\int a q dq = a + \\log{(a q)} + \\int a q dq and \\int (a + \\log{(\\hat{\\mathbf{r}}{(a,q)})} + \\int a q dq) da = \\int (a + \\log{(a q)} + \\int a q dq) da", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('q', commutative=True))), log(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True))))"], [["add", 2, "Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(log(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["add", 3, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Symbol('a', commutative=True), log(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["integrate", 4, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('a', commutative=True), log(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('a', commutative=True), log(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{A},\\Omega)} = \\frac{\\mathbf{A}}{\\Omega}, then obtain \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\int 0 d\\Omega = \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\int (- \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\frac{\\mathbf{A}}{\\Omega}) d\\Omega", "derivation": "\\eta^{\\prime}{(\\mathbf{A},\\Omega)} = \\frac{\\mathbf{A}}{\\Omega} and 0 = - \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\frac{\\mathbf{A}}{\\Omega} and \\int 0 d\\Omega = \\int (- \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\frac{\\mathbf{A}}{\\Omega}) d\\Omega and \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\int 0 d\\Omega = \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\int (- \\eta^{\\prime}{(\\mathbf{A},\\Omega)} + \\frac{\\mathbf{A}}{\\Omega}) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\rho_{b}{(\\varphi^*)} = \\frac{\\sin{(\\varphi^*)}}{\\hat{H}{(\\varphi^*)}}, then obtain \\varphi^* = \\varphi^* \\rho_{b}{(\\varphi^*)}", "derivation": "\\hat{H}{(\\varphi^*)} = \\sin{(\\varphi^*)} and 1 = \\frac{\\sin{(\\varphi^*)}}{\\hat{H}{(\\varphi^*)}} and \\varphi^* = \\frac{\\varphi^* \\sin{(\\varphi^*)}}{\\hat{H}{(\\varphi^*)}} and \\rho_{b}{(\\varphi^*)} = \\frac{\\sin{(\\varphi^*)}}{\\hat{H}{(\\varphi^*)}} and \\varphi^* = \\varphi^* \\rho_{b}{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Symbol('\\\\varphi^*', commutative=True), Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), sin(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Function('\\\\hat{H}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Symbol('\\\\varphi^*', commutative=True), Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\rho_b')(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\mu{(F_{H})} = e^{F_{H}}, then obtain (2 \\mu{(F_{H})} - e^{F_{H}})^{F_{H}} = (e^{F_{H}})^{F_{H}}", "derivation": "\\mu{(F_{H})} = e^{F_{H}} and \\mu^{F_{H}}{(F_{H})} = (e^{F_{H}})^{F_{H}} and 2 \\mu{(F_{H})} = \\mu{(F_{H})} + e^{F_{H}} and 2 \\mu{(F_{H})} - e^{F_{H}} = \\mu{(F_{H})} and 2 \\mu{(F_{H})} - e^{F_{H}} = e^{F_{H}} and \\mu^{F_{H}}{(F_{H})} = (2 \\mu{(F_{H})} - e^{F_{H}})^{F_{H}} and (2 \\mu{(F_{H})} - e^{F_{H}})^{F_{H}} = (e^{F_{H}})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(exp(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["add", 1, "Function('\\\\mu')(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mu')(Symbol('F_H', commutative=True))), Add(Function('\\\\mu')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True))))"], [["minus", 3, "exp(Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mu')(Symbol('F_H', commutative=True))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True)))), Function('\\\\mu')(Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Add(Mul(Integer(2), Function('\\\\mu')(Symbol('F_H', commutative=True))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True)))), exp(Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('\\\\mu')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(2), Function('\\\\mu')(Symbol('F_H', commutative=True))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True)))), Symbol('F_H', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Pow(Add(Mul(Integer(2), Function('\\\\mu')(Symbol('F_H', commutative=True))), Mul(Integer(-1), exp(Symbol('F_H', commutative=True)))), Symbol('F_H', commutative=True)), Pow(exp(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(\\dot{x},Q)} = \\frac{\\dot{x}}{Q}, then obtain (Q \\mathbf{E}^{2}{(\\dot{x},Q)})^{\\dot{x}} = (\\dot{x} \\mathbf{E}{(\\dot{x},Q)})^{\\dot{x}}", "derivation": "\\mathbf{E}{(\\dot{x},Q)} = \\frac{\\dot{x}}{Q} and Q \\mathbf{E}{(\\dot{x},Q)} = \\dot{x} and Q \\mathbf{E}^{2}{(\\dot{x},Q)} = \\dot{x} \\mathbf{E}{(\\dot{x},Q)} and (Q \\mathbf{E}^{2}{(\\dot{x},Q)})^{\\dot{x}} = (\\dot{x} \\mathbf{E}{(\\dot{x},Q)})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\dot{x}', commutative=True))"], [["times", 2, "Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Symbol('Q', commutative=True), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Mul(Symbol('\\\\dot{x}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Symbol('Q', commutative=True), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\dot{x}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})}, then derive \\frac{d}{d z^{*}} \\mathbf{v}{(z^{*})} = - \\cos{(z^{*})}, then obtain - \\cos{(z^{*})} - \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} = 0", "derivation": "\\mathbf{v}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\mathbf{v}{(z^{*})} = \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\mathbf{v}{(z^{*})} - \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} = 0 and \\frac{d}{d z^{*}} \\mathbf{v}{(z^{*})} = - \\cos{(z^{*})} and - \\cos{(z^{*})} - \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True)), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))))"], [["minus", 2, "Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('z^*', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))))), Integer(0))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} = \\hat{H}_l - \\mathbf{p} + \\rho, then obtain - \\int \\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} d\\hat{H}_l = - A_{y} - \\frac{\\hat{H}_l^{2}}{2} - \\hat{H}_l (- \\mathbf{p} + \\rho)", "derivation": "\\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} = \\hat{H}_l - \\mathbf{p} + \\rho and \\int \\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} d\\hat{H}_l = \\int (\\hat{H}_l - \\mathbf{p} + \\rho) d\\hat{H}_l and - \\int \\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} d\\hat{H}_l = - \\int (\\hat{H}_l - \\mathbf{p} + \\rho) d\\hat{H}_l and - \\int \\dot{\\mathbf{r}}{(\\hat{H}_l,\\mathbf{p},\\rho)} d\\hat{H}_l = - A_{y} - \\frac{\\hat{H}_l^{2}}{2} - \\hat{H}_l (- \\mathbf{p} + \\rho)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})}, then obtain 1 = \\frac{F_{x} + \\Psi_{\\lambda} \\log{(\\Psi_{\\lambda})} - \\Psi_{\\lambda}}{\\int \\omega{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}}", "derivation": "\\omega{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\int \\omega{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int \\log{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} and 1 = \\frac{\\int \\log{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}}{\\int \\omega{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}} and 1 = \\frac{F_{x} + \\Psi_{\\lambda} \\log{(\\Psi_{\\lambda})} - \\Psi_{\\lambda}}{\\int \\omega{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Symbol('F_x', commutative=True), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Integral(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(p)} = \\log{(p)}, then derive \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = \\frac{\\partial}{\\partial p} (a + p), then obtain p + \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = p + \\frac{d}{d p} \\int 1 dp", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(p)} = \\log{(p)} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} = 1 and \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = \\int 1 dp and \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = \\frac{d}{d p} \\int 1 dp and \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = \\frac{\\partial}{\\partial p} (a + p) and p + \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = p + \\frac{\\partial}{\\partial p} (a + p) and \\frac{d}{d p} \\int 1 dp = \\frac{\\partial}{\\partial p} (a + p) and p + \\frac{d}{d p} \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(p)}}{\\log{(p)}} dp = p + \\frac{d}{d p} \\int 1 dp", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["divide", 1, "log(Symbol('p', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Integral(Integer(1), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 5, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Derivative(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Symbol('p', commutative=True), Derivative(Add(Symbol('a', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Symbol('p', commutative=True), Derivative(Integral(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Symbol('p', commutative=True), Derivative(Integral(Integer(1), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and l{(\\mathbf{D})} = ((\\operatorname{F_{N}}{(\\mathbf{D})} + \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}}, then obtain l{(\\mathbf{D})} = ((2 \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and \\operatorname{F_{N}}{(\\mathbf{D})} + \\cos{(\\mathbf{D})} = 2 \\cos{(\\mathbf{D})} and (\\operatorname{F_{N}}{(\\mathbf{D})} + \\cos{(\\mathbf{D})})^{\\mathbf{D}} = (2 \\cos{(\\mathbf{D})})^{\\mathbf{D}} and ((\\operatorname{F_{N}}{(\\mathbf{D})} + \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}} = ((2 \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}} and l{(\\mathbf{D})} = ((\\operatorname{F_{N}}{(\\mathbf{D})} + \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}} and l{(\\mathbf{D})} = ((2 \\cos{(\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Function('F_N')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Integer(2), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Pow(Add(Function('F_N')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Mul(Integer(2), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Add(Function('F_N')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('l')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Mul(Integer(2), cos(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(b)} = e^{b}, then obtain \\operatorname{f^{\\prime}}{(b)} + \\frac{d}{d b} (b + 1) = \\operatorname{f^{\\prime}}{(b)} + \\frac{d}{d b} (b + \\frac{e^{b}}{\\operatorname{f^{\\prime}}{(b)}})", "derivation": "\\operatorname{f^{\\prime}}{(b)} = e^{b} and 1 = \\frac{e^{b}}{\\operatorname{f^{\\prime}}{(b)}} and b + 1 = b + \\frac{e^{b}}{\\operatorname{f^{\\prime}}{(b)}} and \\frac{d}{d b} (b + 1) = \\frac{d}{d b} (b + \\frac{e^{b}}{\\operatorname{f^{\\prime}}{(b)}}) and \\operatorname{f^{\\prime}}{(b)} + \\frac{d}{d b} (b + 1) = \\operatorname{f^{\\prime}}{(b)} + \\frac{d}{d b} (b + \\frac{e^{b}}{\\operatorname{f^{\\prime}}{(b)}})", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["divide", 1, "Function('f^{\\\\prime}')(Symbol('b', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Integer(-1)), exp(Symbol('b', commutative=True))))"], [["add", 2, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Integer(1)), Add(Symbol('b', commutative=True), Mul(Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Integer(-1)), exp(Symbol('b', commutative=True)))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), Mul(Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Integer(-1)), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["add", 4, "Function('f^{\\\\prime}')(Symbol('b', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Derivative(Add(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Derivative(Add(Symbol('b', commutative=True), Mul(Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Integer(-1)), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A)} = \\int \\sin{(A)} dA, then derive (\\frac{\\operatorname{P_{e}}{(A)}}{A})^{A} = (\\frac{\\phi_1 - \\cos{(A)}}{A})^{A}, then obtain (\\frac{\\int \\sin{(A)} dA}{A})^{A} = (\\frac{\\phi_1 - \\cos{(A)}}{A})^{A}", "derivation": "\\operatorname{P_{e}}{(A)} = \\int \\sin{(A)} dA and \\frac{\\operatorname{P_{e}}{(A)}}{A} = \\frac{\\int \\sin{(A)} dA}{A} and (\\frac{\\operatorname{P_{e}}{(A)}}{A})^{A} = (\\frac{\\int \\sin{(A)} dA}{A})^{A} and (\\frac{\\operatorname{P_{e}}{(A)}}{A})^{A} = (\\frac{\\phi_1 - \\cos{(A)}}{A})^{A} and (\\frac{\\int \\sin{(A)} dA}{A})^{A} = (\\frac{\\phi_1 - \\cos{(A)}}{A})^{A}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A', commutative=True)), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["divide", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('P_e')(Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('P_e')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('P_e')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True))))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True))))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{E})} = \\sin{(\\mathbf{E})}, then derive (\\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} = (F_{g} - \\cos{(\\mathbf{E})})^{\\mathbf{E}}, then obtain \\frac{d}{d \\mathbf{E}} (\\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} = \\frac{\\partial}{\\partial \\mathbf{E}} (F_{g} - \\cos{(\\mathbf{E})})^{\\mathbf{E}}", "derivation": "\\hat{x}{(\\mathbf{E})} = \\sin{(\\mathbf{E})} and \\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E} = \\int \\sin{(\\mathbf{E})} d\\mathbf{E} and (\\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} = (\\int \\sin{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} and (\\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} = (F_{g} - \\cos{(\\mathbf{E})})^{\\mathbf{E}} and \\frac{d}{d \\mathbf{E}} (\\int \\hat{x}{(\\mathbf{E})} d\\mathbf{E})^{\\mathbf{E}} = \\frac{\\partial}{\\partial \\mathbf{E}} (F_{g} - \\cos{(\\mathbf{E})})^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(\\omega,\\delta)} = \\omega^{\\delta}, then derive a^{\\dagger} + 2 \\int \\omega d\\omega + 2 \\int a{(\\omega,\\delta)} d\\omega = \\int (2 \\omega + \\omega^{\\delta} + a{(\\omega,\\delta)}) d\\omega, then obtain a^{\\dagger} + 2 \\int \\omega d\\omega + 2 \\int \\omega^{\\delta} d\\omega = \\int (2 \\omega + 2 \\omega^{\\delta}) d\\omega", "derivation": "a{(\\omega,\\delta)} = \\omega^{\\delta} and \\omega + a{(\\omega,\\delta)} = \\omega + \\omega^{\\delta} and 2 \\omega + 2 a{(\\omega,\\delta)} = 2 \\omega + \\omega^{\\delta} + a{(\\omega,\\delta)} and \\int (2 \\omega + 2 a{(\\omega,\\delta)}) d\\omega = \\int (2 \\omega + \\omega^{\\delta} + a{(\\omega,\\delta)}) d\\omega and a^{\\dagger} + 2 \\int \\omega d\\omega + 2 \\int a{(\\omega,\\delta)} d\\omega = \\int (2 \\omega + \\omega^{\\delta} + a{(\\omega,\\delta)}) d\\omega and a^{\\dagger} + 2 \\int \\omega d\\omega + 2 \\int \\omega^{\\delta} d\\omega = \\int (2 \\omega + 2 \\omega^{\\delta}) d\\omega", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\omega', commutative=True), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), Add(Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))), Integral(Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Function('a')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Integer(2), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Integral(Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})} = \\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2}), then obtain \\frac{1}{t_{2}^{3} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})}} = \\frac{\\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2})}{t_{2}^{3} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(t_{2},f_{E})}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})} = \\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2}) and - \\frac{1}{t_{2}} = - \\frac{\\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2})}{t_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})}} and - \\frac{1}{t_{2}^{2}} = - \\frac{\\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2})}{t_{2}^{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})}} and \\frac{1}{t_{2}^{3} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2},f_{E})}} = \\frac{\\frac{\\partial}{\\partial t_{2}} (f_{E} - t_{2})}{t_{2}^{3} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(t_{2},f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Derivative(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["divide", 1, "Mul(Integer(-1), Symbol('t_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Derivative(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["divide", 2, "Symbol('t_2', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-2))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-2)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Derivative(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('t_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-3)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-3)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Integer(-2)), Derivative(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(W,\\mathbf{A})} = W \\mathbf{A}, then obtain 2 \\int L{(W,\\mathbf{A})} dW - 2 \\int L{(W,\\mathbf{A})} d\\mathbf{A} = \\int W \\mathbf{A} dW + \\int L{(W,\\mathbf{A})} dW - 2 \\int L{(W,\\mathbf{A})} d\\mathbf{A}", "derivation": "L{(W,\\mathbf{A})} = W \\mathbf{A} and \\int L{(W,\\mathbf{A})} dW = \\int W \\mathbf{A} dW and \\int L{(W,\\mathbf{A})} d\\mathbf{A} = \\int W \\mathbf{A} d\\mathbf{A} and \\int L{(W,\\mathbf{A})} dW - \\int L{(W,\\mathbf{A})} d\\mathbf{A} = \\int W \\mathbf{A} dW - \\int L{(W,\\mathbf{A})} d\\mathbf{A} and - \\int W \\mathbf{A} d\\mathbf{A} + 2 \\int L{(W,\\mathbf{A})} dW - \\int L{(W,\\mathbf{A})} d\\mathbf{A} = \\int W \\mathbf{A} dW - \\int W \\mathbf{A} d\\mathbf{A} + \\int L{(W,\\mathbf{A})} dW - \\int L{(W,\\mathbf{A})} d\\mathbf{A} and 2 \\int L{(W,\\mathbf{A})} dW - 2 \\int L{(W,\\mathbf{A})} d\\mathbf{A} = \\int W \\mathbf{A} dW + \\int L{(W,\\mathbf{A})} dW - 2 \\int L{(W,\\mathbf{A})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))), Add(Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["add", 4, "Add(Mul(Integer(-1), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Integer(2), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(-1), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))), Add(Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(2), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(-1), Integer(2), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))), Add(Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integer(2), Integral(Function('L')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))))"]]}, {"prompt": "Given G{(U,\\sigma_x)} = - U + e^{\\sigma_x}, then obtain G{(U,\\sigma_x)} \\frac{\\partial}{\\partial U} (- 2 U + 2 e^{\\sigma_x}) = (- U + e^{\\sigma_x}) \\frac{\\partial}{\\partial U} (- 2 U + 2 e^{\\sigma_x})", "derivation": "G{(U,\\sigma_x)} = - U + e^{\\sigma_x} and - U + G{(U,\\sigma_x)} + e^{\\sigma_x} = - 2 U + 2 e^{\\sigma_x} and G{(U,\\sigma_x)} \\frac{\\partial}{\\partial U} (- U + G{(U,\\sigma_x)} + e^{\\sigma_x}) = (- U + e^{\\sigma_x}) \\frac{\\partial}{\\partial U} (- U + G{(U,\\sigma_x)} + e^{\\sigma_x}) and G{(U,\\sigma_x)} \\frac{\\partial}{\\partial U} (- 2 U + 2 e^{\\sigma_x}) = (- U + e^{\\sigma_x}) \\frac{\\partial}{\\partial U} (- 2 U + 2 e^{\\sigma_x})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 1, "Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('G')(Symbol('U', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(E,s)} = E - s and \\rho_{f}{(E,s)} = 1 - l{(E,s)}, then derive - l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)} = 1 - l{(E,s)}, then obtain \\iint (1 - l{(E,s)}) dE ds = \\iint (- l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)}) dE ds", "derivation": "l{(E,s)} = E - s and \\frac{\\partial}{\\partial E} l{(E,s)} = \\frac{\\partial}{\\partial E} (E - s) and - l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)} = - l{(E,s)} + \\frac{\\partial}{\\partial E} (E - s) and - l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)} = 1 - l{(E,s)} and \\rho_{f}{(E,s)} = 1 - l{(E,s)} and \\rho_{f}{(E,s)} = - l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)} and \\int \\rho_{f}{(E,s)} dE = \\int (- l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)}) dE and \\iint \\rho_{f}{(E,s)} dE ds = \\iint (- l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)}) dE ds and \\iint (1 - l{(E,s)}) dE ds = \\iint (- l{(E,s)} + \\frac{\\partial}{\\partial E} l{(E,s)}) dE ds", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["add", 2, "Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Add(Integer(1), Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["integrate", 6, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('E', commutative=True))))"], [["integrate", 7, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('E', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('E', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Derivative(Function('l')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('E', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(J_{\\varepsilon},Q)} = J_{\\varepsilon} Q, then obtain 2 (J_{\\varepsilon} Q)^{Q} \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)} = - (- (J_{\\varepsilon} Q)^{Q} - \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)}) \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)}", "derivation": "\\operatorname{C_{d}}{(J_{\\varepsilon},Q)} = J_{\\varepsilon} Q and \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)} = (J_{\\varepsilon} Q)^{Q} and - (J_{\\varepsilon} Q)^{Q} = - \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)} and - 2 (J_{\\varepsilon} Q)^{Q} = - (J_{\\varepsilon} Q)^{Q} - \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)} and 2 (J_{\\varepsilon} Q)^{Q} \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)} = - (- (J_{\\varepsilon} Q)^{Q} - \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)}) \\operatorname{C_{d}}^{Q}{(J_{\\varepsilon},Q)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["minus", 2, "Add(Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))), Pow(Function('C_d')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\chi{(v_{x},z)} = \\frac{e^{v_{x}}}{z}, then obtain \\frac{\\partial}{\\partial z} \\log{(\\int \\chi{(v_{x},z)} e^{v_{x}} dv_{x})} = \\frac{\\partial}{\\partial z} \\log{(\\int \\frac{e^{2 v_{x}}}{z} dv_{x})}", "derivation": "\\chi{(v_{x},z)} = \\frac{e^{v_{x}}}{z} and \\chi{(v_{x},z)} e^{v_{x}} = \\frac{e^{2 v_{x}}}{z} and \\int \\chi{(v_{x},z)} e^{v_{x}} dv_{x} = \\int \\frac{e^{2 v_{x}}}{z} dv_{x} and \\log{(\\int \\chi{(v_{x},z)} e^{v_{x}} dv_{x})} = \\log{(\\int \\frac{e^{2 v_{x}}}{z} dv_{x})} and \\frac{\\partial}{\\partial z} \\log{(\\int \\chi{(v_{x},z)} e^{v_{x}} dv_{x})} = \\frac{\\partial}{\\partial z} \\log{(\\int \\frac{e^{2 v_{x}}}{z} dv_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Symbol('v_x', commutative=True))))"], [["times", 1, "exp(Symbol('v_x', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('z', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('z', commutative=True)), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["log", 3], "Equality(log(Integral(Mul(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('z', commutative=True)), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), log(Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(log(Integral(Mul(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('z', commutative=True)), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(log(Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(q)} = e^{q}, then obtain e^{q} \\frac{d}{d q} \\frac{(q + n^{q}{(q)}) e^{q}}{n{(q)}} = e^{q} \\frac{d}{d q} \\frac{(q + (e^{q})^{q}) e^{q}}{n{(q)}}", "derivation": "n{(q)} = e^{q} and n^{q}{(q)} = (e^{q})^{q} and q + n^{q}{(q)} = q + (e^{q})^{q} and \\frac{(q + n^{q}{(q)}) e^{q}}{n{(q)}} = \\frac{(q + (e^{q})^{q}) e^{q}}{n{(q)}} and \\frac{d}{d q} \\frac{(q + n^{q}{(q)}) e^{q}}{n{(q)}} = \\frac{d}{d q} \\frac{(q + (e^{q})^{q}) e^{q}}{n{(q)}} and e^{q} \\frac{d}{d q} \\frac{(q + n^{q}{(q)}) e^{q}}{n{(q)}} = e^{q} \\frac{d}{d q} \\frac{(q + (e^{q})^{q}) e^{q}}{n{(q)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('n')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(exp(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Add(Symbol('q', commutative=True), Pow(Function('n')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Add(Symbol('q', commutative=True), Pow(exp(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["divide", 3, "Mul(Function('n')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), Symbol('q', commutative=True))))"], "Equality(Mul(Add(Symbol('q', commutative=True), Pow(Function('n')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))), Mul(Add(Symbol('q', commutative=True), Pow(exp(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('q', commutative=True), Pow(Function('n')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('q', commutative=True), Pow(exp(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 5, "exp(Mul(Integer(-1), Symbol('q', commutative=True)))"], "Equality(Mul(exp(Symbol('q', commutative=True)), Derivative(Mul(Add(Symbol('q', commutative=True), Pow(Function('n')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(exp(Symbol('q', commutative=True)), Derivative(Mul(Add(Symbol('q', commutative=True), Pow(exp(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Pow(Function('n')(Symbol('q', commutative=True)), Integer(-1)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(F_{N},\\dot{z})} = \\cos{(F_{N} - \\dot{z})} and A{(C,p)} = C^{p}, then obtain - C^{p} + \\dot{z} + \\mathbf{J}{(F_{N},\\dot{z})} = - C^{p} + \\dot{z} + \\cos{(F_{N} - \\dot{z})}", "derivation": "\\mathbf{J}{(F_{N},\\dot{z})} = \\cos{(F_{N} - \\dot{z})} and A{(C,p)} = C^{p} and \\dot{z} - A{(C,p)} + \\mathbf{J}{(F_{N},\\dot{z})} = \\dot{z} - A{(C,p)} + \\cos{(F_{N} - \\dot{z})} and - C^{p} + \\dot{z} + \\mathbf{J}{(F_{N},\\dot{z})} = - C^{p} + \\dot{z} + \\cos{(F_{N} - \\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True)), cos(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))"], ["get_premise", "Equality(Function('A')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('C', commutative=True), Symbol('p', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('A')(Symbol('C', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Function('A')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Function('\\\\mathbf{J}')(Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Function('A')(Symbol('C', commutative=True), Symbol('p', commutative=True))), cos(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\dot{z}', commutative=True), Function('\\\\mathbf{J}')(Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\dot{z}', commutative=True), cos(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))))"]]}, {"prompt": "Given \\phi{(L)} = \\sin{(L)} and \\mathbf{E}{(L)} = \\phi{(L)} \\sin{(L)}, then obtain \\frac{\\mathbf{E}^{2}{(L)}}{\\phi{(L)} \\sin^{3}{(L)}} = \\frac{\\mathbf{E}{(L)}}{\\sin^{2}{(L)}}", "derivation": "\\phi{(L)} = \\sin{(L)} and \\phi{(L)} \\sin{(L)} = \\sin^{2}{(L)} and \\mathbf{E}{(L)} = \\phi{(L)} \\sin{(L)} and \\mathbf{E}{(L)} = \\sin^{2}{(L)} and \\frac{\\mathbf{E}{(L)}}{\\sin^{2}{(L)}} = 1 and \\frac{\\mathbf{E}{(L)}}{\\phi{(L)} \\sin{(L)}} = 1 and \\frac{\\mathbf{E}^{2}{(L)}}{\\phi{(L)} \\sin^{3}{(L)}} = \\frac{\\mathbf{E}{(L)}}{\\sin^{2}{(L)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["times", 1, "sin(Symbol('L', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Pow(sin(Symbol('L', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Mul(Function('\\\\phi')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(2)))"], [["divide", 4, "Pow(sin(Symbol('L', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-2))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Pow(Function('\\\\phi')(Symbol('L', commutative=True)), Integer(-1)), Pow(sin(Symbol('L', commutative=True)), Integer(-1))), Integer(1))"], [["times", 6, "Mul(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-2)))"], "Equality(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Integer(2)), Pow(Function('\\\\phi')(Symbol('L', commutative=True)), Integer(-1)), Pow(sin(Symbol('L', commutative=True)), Integer(-3))), Mul(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\rho_b)} = \\int \\cos{(\\rho_b)} d\\rho_b, then obtain - v_{z} + (\\operatorname{C_{2}}{(\\rho_b)} \\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)})^{\\rho_b} = - v_{z} + (\\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)} \\int \\cos{(\\rho_b)} d\\rho_b)^{\\rho_b}", "derivation": "\\operatorname{C_{2}}{(\\rho_b)} = \\int \\cos{(\\rho_b)} d\\rho_b and \\operatorname{C_{2}}{(\\rho_b)} \\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)} = \\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)} \\int \\cos{(\\rho_b)} d\\rho_b and (\\operatorname{C_{2}}{(\\rho_b)} \\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)})^{\\rho_b} = (\\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)} \\int \\cos{(\\rho_b)} d\\rho_b)^{\\rho_b} and - v_{z} + (\\operatorname{C_{2}}{(\\rho_b)} \\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)})^{\\rho_b} = - v_{z} + (\\operatorname{C_{2}}^{- \\rho_b}{(\\rho_b)} \\int \\cos{(\\rho_b)} d\\rho_b)^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["divide", 1, "Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Mul(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True)))"], [["minus", 3, "Symbol('v_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Mul(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Mul(Pow(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(h,U)} = U - h, then obtain (\\frac{\\partial}{\\partial h} \\int \\frac{- \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU}{U} dU)^{h} = (\\frac{d}{d h} \\int 0 dU)^{h}", "derivation": "\\theta_{2}{(h,U)} = U - h and \\int \\theta_{2}{(h,U)} dU = \\int (U - h) dU and - \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU = 0 and \\frac{- \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU}{U} = 0 and \\int \\frac{- \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU}{U} dU = \\int 0 dU and \\frac{\\partial}{\\partial h} \\int \\frac{- \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU}{U} dU = \\frac{d}{d h} \\int 0 dU and (\\frac{\\partial}{\\partial h} \\int \\frac{- \\int (U - h) dU + \\int \\theta_{2}{(h,U)} dU}{U} dU)^{h} = (\\frac{d}{d h} \\int 0 dU)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integer(0))"], [["divide", 3, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 5, "Symbol('h', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 6, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('U', commutative=True)))), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Derivative(Integral(Integer(0), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given s{(L_{\\varepsilon})} = \\sin{(\\cos{(L_{\\varepsilon})})} and \\rho_{b}{(L_{\\varepsilon})} = \\tilde{\\infty} \\sin{(\\cos{(L_{\\varepsilon})})}, then obtain \\tilde{\\infty} s{(L_{\\varepsilon})} \\sin{(\\cos{(L_{\\varepsilon})})} = 2 \\rho_{b}{(L_{\\varepsilon})} \\sin{(\\cos{(L_{\\varepsilon})})}", "derivation": "s{(L_{\\varepsilon})} = \\sin{(\\cos{(L_{\\varepsilon})})} and \\tilde{\\infty} s{(L_{\\varepsilon})} = \\tilde{\\infty} \\sin{(\\cos{(L_{\\varepsilon})})} and \\rho_{b}{(L_{\\varepsilon})} = \\tilde{\\infty} \\sin{(\\cos{(L_{\\varepsilon})})} and \\tilde{\\infty} s{(L_{\\varepsilon})} = \\rho_{b}{(L_{\\varepsilon})} and \\tilde{\\infty} s{(L_{\\varepsilon})} \\sin{(\\cos{(L_{\\varepsilon})})} = 2 \\rho_{b}{(L_{\\varepsilon})} \\sin{(\\cos{(L_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["divide", 1, 0], "Equality(Mul(zoo, Function('s')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(zoo, sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(zoo, sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(zoo, Function('s')(Symbol('L_{\\\\varepsilon}', commutative=True))), Function('\\\\rho_b')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["times", 4, "Mul(Integer(2), sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(zoo, Function('s')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), Function('\\\\rho_b')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given t{(H,F_{x})} = F_{x} H, then derive e^{t{(H,F_{x})}} \\frac{\\partial}{\\partial H} t{(H,F_{x})} = F_{x} e^{F_{x} H}, then obtain \\frac{\\partial}{\\partial F_{x}} \\cos{(e^{t{(H,F_{x})}} \\frac{\\partial}{\\partial H} t{(H,F_{x})})} = \\frac{\\partial}{\\partial F_{x}} \\cos{(F_{x} e^{F_{x} H})}", "derivation": "t{(H,F_{x})} = F_{x} H and e^{t{(H,F_{x})}} = e^{F_{x} H} and \\frac{\\partial}{\\partial H} e^{t{(H,F_{x})}} = \\frac{\\partial}{\\partial H} e^{F_{x} H} and e^{t{(H,F_{x})}} \\frac{\\partial}{\\partial H} t{(H,F_{x})} = F_{x} e^{F_{x} H} and \\cos{(e^{t{(H,F_{x})}} \\frac{\\partial}{\\partial H} t{(H,F_{x})})} = \\cos{(F_{x} e^{F_{x} H})} and \\frac{\\partial}{\\partial F_{x}} \\cos{(e^{t{(H,F_{x})}} \\frac{\\partial}{\\partial H} t{(H,F_{x})})} = \\frac{\\partial}{\\partial F_{x}} \\cos{(F_{x} e^{F_{x} H})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True)))"], [["exp", 1], "Equality(exp(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True))), exp(Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(exp(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True))), Derivative(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('F_x', commutative=True), exp(Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True)))))"], [["cos", 4], "Equality(cos(Mul(exp(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True))), Derivative(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), cos(Mul(Symbol('F_x', commutative=True), exp(Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True))))))"], [["differentiate", 5, "Symbol('F_x', commutative=True)"], "Equality(Derivative(cos(Mul(exp(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True))), Derivative(Function('t')(Symbol('H', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('F_x', commutative=True), exp(Mul(Symbol('F_x', commutative=True), Symbol('H', commutative=True))))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\varphi,\\dot{y})} = \\dot{y} + \\varphi, then obtain - ((\\dot{y} + \\varphi)^{\\dot{y}})^{\\varphi} + \\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})} = (\\dot{y} + \\varphi)^{\\dot{y}} - ((\\dot{y} + \\varphi)^{\\dot{y}})^{\\varphi}", "derivation": "\\dot{x}{(\\varphi,\\dot{y})} = \\dot{y} + \\varphi and \\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})} = (\\dot{y} + \\varphi)^{\\dot{y}} and (\\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})})^{\\varphi} = ((\\dot{y} + \\varphi)^{\\dot{y}})^{\\varphi} and - (\\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})})^{\\varphi} + \\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})} = (\\dot{y} + \\varphi)^{\\dot{y}} - (\\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})})^{\\varphi} and - ((\\dot{y} + \\varphi)^{\\dot{y}})^{\\varphi} + \\dot{x}^{\\dot{y}}{(\\varphi,\\dot{y})} = (\\dot{y} + \\varphi)^{\\dot{y}} - ((\\dot{y} + \\varphi)^{\\dot{y}})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["minus", 2, "Pow(Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Pow(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(y^{\\prime},\\varphi^*)} = \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* and s{(y^{\\prime},\\varphi^*)} = \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{F}{(y^{\\prime},\\varphi^*)}, then obtain s{(y^{\\prime},\\varphi^*)} - \\frac{\\partial}{\\partial y^{\\prime}} \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* = 0", "derivation": "\\mathbf{F}{(y^{\\prime},\\varphi^*)} = \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{F}{(y^{\\prime},\\varphi^*)} = \\frac{\\partial}{\\partial y^{\\prime}} \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{F}{(y^{\\prime},\\varphi^*)} - \\frac{\\partial}{\\partial y^{\\prime}} \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* = 0 and s{(y^{\\prime},\\varphi^*)} = \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{F}{(y^{\\prime},\\varphi^*)} and s{(y^{\\prime},\\varphi^*)} - \\frac{\\partial}{\\partial y^{\\prime}} \\int \\frac{y^{\\prime}}{\\varphi^*} d\\varphi^* = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Integral(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))), Integer(0))"], ["renaming_premise", "Equality(Function('s')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('s')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Derivative(Integral(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{D})} = \\cos{(\\sin{(\\mathbf{D})})} and Z{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then obtain \\cos{(\\sin{(\\mathbf{D})})} = \\cos{(Z{(\\mathbf{D})})}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{D})} = \\cos{(\\sin{(\\mathbf{D})})} and Z{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\operatorname{M_{E}}{(\\mathbf{D})} = \\cos{(Z{(\\mathbf{D})})} and \\cos{(\\sin{(\\mathbf{D})})} = \\cos{(Z{(\\mathbf{D})})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{D}', commutative=True)), cos(sin(Symbol('\\\\mathbf{D}', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('M_E')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Function('Z')(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(cos(sin(Symbol('\\\\mathbf{D}', commutative=True))), cos(Function('Z')(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} = \\frac{M_{E} - \\mathbf{B}}{I} and s{(I,\\mathbf{B},M_{E})} = \\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} - 1, then obtain \\frac{\\partial}{\\partial M_{E}} \\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} = \\frac{1}{I}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} = \\frac{M_{E} - \\mathbf{B}}{I} and s{(I,\\mathbf{B},M_{E})} = \\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} - 1 and s{(I,\\mathbf{B},M_{E})} = -1 + \\frac{M_{E} - \\mathbf{B}}{I} and \\frac{\\partial}{\\partial M_{E}} s{(I,\\mathbf{B},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (-1 + \\frac{M_{E} - \\mathbf{B}}{I}) and \\frac{\\partial}{\\partial M_{E}} (\\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} - 1) = \\frac{\\partial}{\\partial M_{E}} (-1 + \\frac{M_{E} - \\mathbf{B}}{I}) and \\frac{\\partial}{\\partial M_{E}} \\operatorname{f_{\\mathbf{p}}}{(I,\\mathbf{B},M_{E})} = \\frac{1}{I}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"], ["renaming_premise", "Equality(Function('s')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('s')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Add(Integer(-1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Pow(Symbol('I', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = (\\hbar + v_{1})^{v_{t}}, then derive \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = \\frac{v_{t} (\\hbar + v_{1})^{v_{t}} (v_{t} - 1)}{(\\hbar + v_{1})^{2}}, then obtain \\frac{\\partial^{2}}{\\partial v_{1}^{2}} (\\hbar + v_{1})^{v_{t}} = \\frac{v_{t} (\\hbar + v_{1})^{v_{t}} (v_{t} - 1)}{(\\hbar + v_{1})^{2}}", "derivation": "\\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = (\\hbar + v_{1})^{v_{t}} and \\frac{\\partial}{\\partial v_{1}} \\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = \\frac{\\partial}{\\partial v_{1}} (\\hbar + v_{1})^{v_{t}} and \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = \\frac{\\partial^{2}}{\\partial v_{1}^{2}} (\\hbar + v_{1})^{v_{t}} and \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\operatorname{A_{2}}{(v_{t},v_{1},\\hbar)} = \\frac{v_{t} (\\hbar + v_{1})^{v_{t}} (v_{t} - 1)}{(\\hbar + v_{1})^{2}} and \\frac{\\partial^{2}}{\\partial v_{1}^{2}} (\\hbar + v_{1})^{v_{t}} = \\frac{v_{t} (\\hbar + v_{1})^{v_{t}} (v_{t} - 1)}{(\\hbar + v_{1})^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('A_2')(Symbol('v_t', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('v_t', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('v_t', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Derivative(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_2')(Symbol('v_t', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Mul(Symbol('v_t', commutative=True), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Integer(-2)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)), Add(Symbol('v_t', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Mul(Symbol('v_t', commutative=True), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Integer(-2)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('v_1', commutative=True)), Symbol('v_t', commutative=True)), Add(Symbol('v_t', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\pi{(M,V_{\\mathbf{E}})} = - M + V_{\\mathbf{E}}, then derive \\frac{\\partial}{\\partial M} \\pi{(M,V_{\\mathbf{E}})} = -1, then obtain V_{\\mathbf{E}} \\frac{\\partial^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}}{\\partial M^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}} \\pi{(M,V_{\\mathbf{E}})} = V_{\\mathbf{E}} \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})", "derivation": "\\pi{(M,V_{\\mathbf{E}})} = - M + V_{\\mathbf{E}} and \\frac{\\partial}{\\partial M} \\pi{(M,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}}) and \\frac{\\partial}{\\partial M} \\pi{(M,V_{\\mathbf{E}})} = -1 and -1 = \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}}) and \\frac{\\partial^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}}{\\partial M^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}} \\pi{(M,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}}) and V_{\\mathbf{E}} \\frac{\\partial^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}}{\\partial M^{- \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})}} \\pi{(M,V_{\\mathbf{E}})} = V_{\\mathbf{E}} \\frac{\\partial}{\\partial M} (- M + V_{\\mathbf{E}})", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('M', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('M', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('M', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\pi')(Symbol('M', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 5, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Derivative(Function('\\\\pi')(Symbol('M', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(f_{E})} = \\cos{(\\log{(f_{E})})}, then obtain 4 \\operatorname{f_{\\mathbf{v}}}^{4}{(f_{E})} \\cos^{2}{(\\log{(f_{E})})} = (\\operatorname{f_{\\mathbf{v}}}{(f_{E})} + \\cos{(\\log{(f_{E})})})^{2} \\operatorname{f_{\\mathbf{v}}}^{2}{(f_{E})} \\cos^{2}{(\\log{(f_{E})})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(f_{E})} = \\cos{(\\log{(f_{E})})} and 2 \\operatorname{f_{\\mathbf{v}}}{(f_{E})} = \\operatorname{f_{\\mathbf{v}}}{(f_{E})} + \\cos{(\\log{(f_{E})})} and 2 \\operatorname{f_{\\mathbf{v}}}^{2}{(f_{E})} \\cos{(\\log{(f_{E})})} = (\\operatorname{f_{\\mathbf{v}}}{(f_{E})} + \\cos{(\\log{(f_{E})})}) \\operatorname{f_{\\mathbf{v}}}{(f_{E})} \\cos{(\\log{(f_{E})})} and 4 \\operatorname{f_{\\mathbf{v}}}^{4}{(f_{E})} \\cos^{2}{(\\log{(f_{E})})} = (\\operatorname{f_{\\mathbf{v}}}{(f_{E})} + \\cos{(\\log{(f_{E})})})^{2} \\operatorname{f_{\\mathbf{v}}}^{2}{(f_{E})} \\cos^{2}{(\\log{(f_{E})})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True))))"], [["add", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True))"], "Equality(Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True)))))"], [["times", 2, "Mul(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), Integer(2)), cos(log(Symbol('f_E', commutative=True)))), Mul(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True)))), Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), Integer(4)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(2))), Mul(Pow(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), cos(log(Symbol('f_E', commutative=True)))), Integer(2)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('f_E', commutative=True)), Integer(2)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mu_{0}{(\\delta,\\mathbf{M})} = \\cos{(\\delta^{\\mathbf{M}})} and \\operatorname{t_{2}}{(\\delta,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\mu_{0}{(\\delta,\\mathbf{M})}, then obtain \\operatorname{t_{2}}{(\\delta,\\mathbf{M})} = - \\delta^{\\mathbf{M}} \\log{(\\delta)} \\sin{(\\delta^{\\mathbf{M}})}", "derivation": "\\mu_{0}{(\\delta,\\mathbf{M})} = \\cos{(\\delta^{\\mathbf{M}})} and \\frac{\\partial}{\\partial \\mathbf{M}} \\mu_{0}{(\\delta,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\cos{(\\delta^{\\mathbf{M}})} and \\operatorname{t_{2}}{(\\delta,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\mu_{0}{(\\delta,\\mathbf{M})} and \\operatorname{t_{2}}{(\\delta,\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\cos{(\\delta^{\\mathbf{M}})} and \\operatorname{t_{2}}{(\\delta,\\mathbf{M})} = - \\delta^{\\mathbf{M}} \\log{(\\delta)} \\sin{(\\delta^{\\mathbf{M}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(cos(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\delta', commutative=True)), sin(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given I{(a,i)} = a i, then derive - a + \\frac{\\partial}{\\partial i} I{(a,i)} = 0, then obtain \\frac{\\partial}{\\partial i} (- a + \\frac{\\partial}{\\partial i} a i) = \\frac{d}{d i} 0", "derivation": "I{(a,i)} = a i and - a i + I{(a,i)} = 0 and \\frac{\\partial}{\\partial i} (- a i + I{(a,i)}) = \\frac{d}{d i} 0 and - a + \\frac{\\partial}{\\partial i} I{(a,i)} = 0 and \\frac{\\partial}{\\partial i} (- a + \\frac{\\partial}{\\partial i} I{(a,i)}) = \\frac{d}{d i} 0 and \\frac{\\partial}{\\partial i} (- a + \\frac{\\partial}{\\partial i} a i) = \\frac{d}{d i} 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Mul(Symbol('a', commutative=True), Symbol('i', commutative=True)))"], [["minus", 1, "Mul(Symbol('a', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True), Symbol('i', commutative=True)), Function('I')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True), Symbol('i', commutative=True)), Function('I')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Function('I')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Function('I')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Mul(Symbol('a', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(\\mathbf{r})} = \\mathbf{r}, then derive \\int y{(\\mathbf{r})} d\\mathbf{r} = \\frac{\\mathbf{r}^{2}}{2} + v_{1}, then derive \\frac{\\partial}{\\partial v_{1}} (\\mathbf{S} + \\frac{\\mathbf{r}^{2}}{2}) = \\frac{\\partial}{\\partial v_{1}} (\\frac{\\mathbf{r}^{2}}{2} + v_{1}), then derive \\frac{d}{d v_{1}} \\int \\mathbf{r} d\\mathbf{r} = 1, then obtain \\frac{\\partial}{\\partial v_{1}} (\\mathbf{S} + \\frac{\\mathbf{r}^{2}}{2}) = 1", "derivation": "y{(\\mathbf{r})} = \\mathbf{r} and \\int y{(\\mathbf{r})} d\\mathbf{r} = \\int \\mathbf{r} d\\mathbf{r} and \\int y{(\\mathbf{r})} d\\mathbf{r} = \\frac{\\mathbf{r}^{2}}{2} + v_{1} and \\int \\mathbf{r} d\\mathbf{r} = \\frac{\\mathbf{r}^{2}}{2} + v_{1} and \\frac{d}{d v_{1}} \\int \\mathbf{r} d\\mathbf{r} = \\frac{\\partial}{\\partial v_{1}} (\\frac{\\mathbf{r}^{2}}{2} + v_{1}) and \\frac{\\partial}{\\partial v_{1}} (\\mathbf{S} + \\frac{\\mathbf{r}^{2}}{2}) = \\frac{\\partial}{\\partial v_{1}} (\\frac{\\mathbf{r}^{2}}{2} + v_{1}) and \\frac{d}{d v_{1}} \\int \\mathbf{r} d\\mathbf{r} = 1 and \\frac{\\partial}{\\partial v_{1}} (\\frac{\\mathbf{r}^{2}}{2} + v_{1}) = 1 and \\frac{\\partial}{\\partial v_{1}} (\\mathbf{S} + \\frac{\\mathbf{r}^{2}}{2}) = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('y')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('y')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Symbol('v_1', commutative=True)))"], [["differentiate", 4, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given n{(b,u)} = b u, then derive \\frac{\\partial}{\\partial b} n{(b,u)} = u, then obtain \\frac{\\partial}{\\partial b} b u = u", "derivation": "n{(b,u)} = b u and \\frac{\\partial}{\\partial b} n{(b,u)} = \\frac{\\partial}{\\partial b} b u and \\frac{\\partial}{\\partial b} n{(b,u)} = u and \\frac{\\partial}{\\partial b} b u = u", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('b', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('b', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('b', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('b', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('u', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('b', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('u', commutative=True))"]]}, {"prompt": "Given g{(\\mathbf{J}_f,v_{2})} = e^{- \\mathbf{J}_f + v_{2}}, then obtain 2 g{(\\mathbf{J}_f,v_{2})} = g{(\\mathbf{J}_f,v_{2})} + e^{- \\mathbf{J}_f + v_{2}}", "derivation": "g{(\\mathbf{J}_f,v_{2})} = e^{- \\mathbf{J}_f + v_{2}} and g{(\\mathbf{J}_f,v_{2})} + e^{- \\mathbf{J}_f + v_{2}} = 2 e^{- \\mathbf{J}_f + v_{2}} and g{(\\mathbf{J}_f,v_{2})} - e^{- \\mathbf{J}_f + v_{2}} = 0 and 2 g{(\\mathbf{J}_f,v_{2})} = g{(\\mathbf{J}_f,v_{2})} + e^{- \\mathbf{J}_f + v_{2}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True))))"], [["add", 1, "exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True)))"], "Equality(Add(Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True)))), Mul(Integer(2), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True)))))"], [["minus", 2, "Mul(Integer(2), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True))))"], "Equality(Add(Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True))))), Integer(0))"], [["add", 3, "Add(Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True))))"], "Equality(Mul(Integer(2), Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True))), Add(Function('g')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given a{(z^{*},\\varphi^*)} = \\log{((\\varphi^*)^{z^{*}})}, then derive \\int a{(z^{*},\\varphi^*)} dz^{*} = n + \\frac{(z^{*})^{2} \\log{(\\varphi^*)}}{2}, then obtain \\frac{\\partial}{\\partial z^{*}} \\int \\log{((\\varphi^*)^{z^{*}})} dz^{*} = \\frac{\\partial}{\\partial z^{*}} (n + \\frac{(z^{*})^{2} \\log{(\\varphi^*)}}{2})", "derivation": "a{(z^{*},\\varphi^*)} = \\log{((\\varphi^*)^{z^{*}})} and \\int a{(z^{*},\\varphi^*)} dz^{*} = \\int \\log{((\\varphi^*)^{z^{*}})} dz^{*} and \\int a{(z^{*},\\varphi^*)} dz^{*} = n + \\frac{(z^{*})^{2} \\log{(\\varphi^*)}}{2} and \\frac{\\partial}{\\partial z^{*}} \\int a{(z^{*},\\varphi^*)} dz^{*} = \\frac{\\partial}{\\partial z^{*}} (n + \\frac{(z^{*})^{2} \\log{(\\varphi^*)}}{2}) and \\frac{\\partial}{\\partial z^{*}} \\int \\log{((\\varphi^*)^{z^{*}})} dz^{*} = \\frac{\\partial}{\\partial z^{*}} (n + \\frac{(z^{*})^{2} \\log{(\\varphi^*)}}{2})", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('z^*', commutative=True), Symbol('\\\\varphi^*', commutative=True)), log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('a')(Symbol('z^*', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a')(Symbol('z^*', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)), log(Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Integral(Function('a')(Symbol('z^*', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)), log(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)), log(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = e^{(\\eta^{\\prime})^{x^\\prime}}, then obtain e^{- (\\eta^{\\prime})^{x^\\prime}} \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = \\frac{(\\eta^{\\prime})^{x^\\prime} x^\\prime}{\\eta^{\\prime}}", "derivation": "\\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = e^{(\\eta^{\\prime})^{x^\\prime}} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} e^{(\\eta^{\\prime})^{x^\\prime}} and e^{- (\\eta^{\\prime})^{x^\\prime}} \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = e^{- (\\eta^{\\prime})^{x^\\prime}} \\frac{\\partial}{\\partial \\eta^{\\prime}} e^{(\\eta^{\\prime})^{x^\\prime}} and e^{- (\\eta^{\\prime})^{x^\\prime}} \\frac{\\partial}{\\partial \\eta^{\\prime}} \\operatorname{C_{2}}{(x^\\prime,\\eta^{\\prime})} = \\frac{(\\eta^{\\prime})^{x^\\prime} x^\\prime}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 2, "exp(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Derivative(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Derivative(exp(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Derivative(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(A_{x},\\mathbf{v})} = A_{x}^{\\mathbf{v}}, then obtain (A_{x}^{\\mathbf{v}})^{A_{x}} + \\int \\operatorname{F_{g}}{(A_{x},\\mathbf{v})} dA_{x} = (A_{x}^{\\mathbf{v}})^{A_{x}} + \\int A_{x}^{\\mathbf{v}} dA_{x}", "derivation": "\\operatorname{F_{g}}{(A_{x},\\mathbf{v})} = A_{x}^{\\mathbf{v}} and \\operatorname{F_{g}}^{A_{x}}{(A_{x},\\mathbf{v})} = (A_{x}^{\\mathbf{v}})^{A_{x}} and \\int \\operatorname{F_{g}}{(A_{x},\\mathbf{v})} dA_{x} = \\int A_{x}^{\\mathbf{v}} dA_{x} and \\operatorname{F_{g}}^{A_{x}}{(A_{x},\\mathbf{v})} + \\int \\operatorname{F_{g}}{(A_{x},\\mathbf{v})} dA_{x} = \\operatorname{F_{g}}^{A_{x}}{(A_{x},\\mathbf{v})} + \\int A_{x}^{\\mathbf{v}} dA_{x} and (A_{x}^{\\mathbf{v}})^{A_{x}} + \\int \\operatorname{F_{g}}{(A_{x},\\mathbf{v})} dA_{x} = (A_{x}^{\\mathbf{v}})^{A_{x}} + \\int A_{x}^{\\mathbf{v}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["add", 3, "Pow(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True))"], "Equality(Add(Pow(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)), Integral(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Pow(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)), Integral(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)), Integral(Function('F_g')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Pow(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('A_x', commutative=True)), Integral(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given b{(b,g^{\\prime}_{\\varepsilon})} = - b + \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain \\frac{b{(b,g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} - 2 \\cos{(g^{\\prime}_{\\varepsilon})} = \\frac{- b + 2 \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} - 2 \\cos{(g^{\\prime}_{\\varepsilon})}", "derivation": "b{(b,g^{\\prime}_{\\varepsilon})} = - b + \\cos{(g^{\\prime}_{\\varepsilon})} and b{(b,g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} = - b + 2 \\cos{(g^{\\prime}_{\\varepsilon})} and \\frac{b{(b,g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} = \\frac{- b + 2 \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} and \\frac{b{(b,g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} - 2 \\cos{(g^{\\prime}_{\\varepsilon})} = \\frac{- b + 2 \\cos{(g^{\\prime}_{\\varepsilon})}}{b{(b,g^{\\prime}_{\\varepsilon})}} - 2 \\cos{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["divide", 2, "Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Pow(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["minus", 3, "Mul(Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Add(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Pow(Function('b')(Symbol('b', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(\\theta)} = e^{\\theta} and h{(\\theta)} = (e^{\\theta})^{\\theta} and \\eta^{\\prime}{(\\theta)} = \\mathbf{f}^{\\theta}{(\\theta)}, then obtain \\eta^{\\prime}{(\\theta)} = h{(\\theta)}", "derivation": "\\mathbf{f}{(\\theta)} = e^{\\theta} and \\mathbf{f}^{\\theta}{(\\theta)} = (e^{\\theta})^{\\theta} and h{(\\theta)} = (e^{\\theta})^{\\theta} and h{(\\theta)} = \\mathbf{f}^{\\theta}{(\\theta)} and \\eta^{\\prime}{(\\theta)} = \\mathbf{f}^{\\theta}{(\\theta)} and \\eta^{\\prime}{(\\theta)} = h{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('h')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta', commutative=True)), Function('h')(Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(m)} = \\int \\cos{(m)} dm, then derive \\mu_{0}{(m)} = M + \\sin{(m)}, then derive E_{\\lambda} + \\sin{(m)} = M + \\sin{(m)}, then obtain (M + \\sin{(m)}) \\iint \\cos{(m)} dm dm + 1 = (M + \\sin{(m)}) \\int (M + \\sin{(m)}) dm + 1", "derivation": "\\mu_{0}{(m)} = \\int \\cos{(m)} dm and \\mu_{0}{(m)} = M + \\sin{(m)} and \\int \\cos{(m)} dm = M + \\sin{(m)} and \\iint \\cos{(m)} dm dm = \\int (M + \\sin{(m)}) dm and E_{\\lambda} + \\sin{(m)} = M + \\sin{(m)} and (E_{\\lambda} + \\sin{(m)}) \\iint \\cos{(m)} dm dm = (E_{\\lambda} + \\sin{(m)}) \\int (M + \\sin{(m)}) dm and (E_{\\lambda} + \\sin{(m)}) \\iint \\cos{(m)} dm dm + \\frac{\\mu_{0}{(m)}}{\\int \\cos{(m)} dm} = (E_{\\lambda} + \\sin{(m)}) \\int (M + \\sin{(m)}) dm + \\frac{\\mu_{0}{(m)}}{\\int \\cos{(m)} dm} and (E_{\\lambda} + \\sin{(m)}) \\iint \\cos{(m)} dm dm + 1 = (E_{\\lambda} + \\sin{(m)}) \\int (M + \\sin{(m)}) dm + 1 and (M + \\sin{(m)}) \\iint \\cos{(m)} dm dm + 1 = (M + \\sin{(m)}) \\int (M + \\sin{(m)}) dm + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('m', commutative=True)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mu_0')(Symbol('m', commutative=True)), Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))))"], [["times", 4, "Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True)))"], "Equality(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))))"], [["add", 6, "Mul(Function('\\\\mu_0')(Symbol('m', commutative=True)), Pow(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Function('\\\\mu_0')(Symbol('m', commutative=True)), Pow(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(-1)))), Add(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Function('\\\\mu_0')(Symbol('m', commutative=True)), Pow(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Integer(1)), Add(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('m', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Integer(1)))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Add(Mul(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Integer(1)), Add(Mul(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Integral(Add(Symbol('M', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\eta{(A)} = \\cos{(\\sin{(A)})} and \\varphi{(A)} = \\cos{(\\sin{(A)})}, then obtain \\varphi{(A)} + \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\eta{(A)} = 2 \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\cos{(\\sin{(A)})}", "derivation": "\\eta{(A)} = \\cos{(\\sin{(A)})} and \\frac{d}{d A} \\eta{(A)} = \\frac{d}{d A} \\cos{(\\sin{(A)})} and \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\eta{(A)} = \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\cos{(\\sin{(A)})} and \\varphi{(A)} = \\cos{(\\sin{(A)})} and \\varphi{(A)} + \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\cos{(\\sin{(A)})} = 2 \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\cos{(\\sin{(A)})} and \\varphi{(A)} + \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\eta{(A)} = 2 \\cos{(\\sin{(A)})} + \\frac{d}{d A} \\cos{(\\sin{(A)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('A', commutative=True)), cos(sin(Symbol('A', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["add", 2, "cos(sin(Symbol('A', commutative=True)))"], "Equality(Add(cos(sin(Symbol('A', commutative=True))), Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(cos(sin(Symbol('A', commutative=True))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('A', commutative=True)), cos(sin(Symbol('A', commutative=True))))"], [["add", 4, "Add(cos(sin(Symbol('A', commutative=True))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\varphi')(Symbol('A', commutative=True)), cos(sin(Symbol('A', commutative=True))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(2), cos(sin(Symbol('A', commutative=True)))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\varphi')(Symbol('A', commutative=True)), cos(sin(Symbol('A', commutative=True))), Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(2), cos(sin(Symbol('A', commutative=True)))), Derivative(cos(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(H)} = \\log{(H)}, then obtain \\operatorname{f^{*}}{(H)} \\log{(H)}^{H} = \\log{(H)} \\log{(H)}^{H}", "derivation": "\\operatorname{f^{*}}{(H)} = \\log{(H)} and \\operatorname{f^{*}}^{H}{(H)} = \\log{(H)}^{H} and \\operatorname{f^{*}}{(H)} \\operatorname{f^{*}}^{H}{(H)} = \\operatorname{f^{*}}^{H}{(H)} \\log{(H)} and \\operatorname{f^{*}}{(H)} \\log{(H)}^{H} = \\log{(H)} \\log{(H)}^{H}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["times", 1, "Pow(Function('f^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('H', commutative=True)), Pow(Function('f^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Mul(Pow(Function('f^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), log(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('f^*')(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Mul(log(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True))))"]]}, {"prompt": "Given h{(x,u)} = \\cos{(u + x)}, then derive \\int h{(x,u)} du = \\dot{y} + \\sin{(u + x)}, then obtain \\int \\cos{(u + x)} du = \\dot{y} + \\sin{(u + x)}", "derivation": "h{(x,u)} = \\cos{(u + x)} and \\int h{(x,u)} du = \\int \\cos{(u + x)} du and \\int h{(x,u)} du = \\dot{y} + \\sin{(u + x)} and \\int \\cos{(u + x)} du = \\dot{y} + \\sin{(u + x)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('x', commutative=True), Symbol('u', commutative=True)), cos(Add(Symbol('u', commutative=True), Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('h')(Symbol('x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(cos(Add(Symbol('u', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), sin(Add(Symbol('u', commutative=True), Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Add(Symbol('u', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), sin(Add(Symbol('u', commutative=True), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\hat{H})} = \\sin{(\\hat{H})}, then obtain \\cos{(\\hat{H} \\sin{(\\hat{H})})} = \\cos{(\\hat{H} \\sin{(\\hat{H})} + \\bar{\\h}{(\\hat{H})} - \\sin{(\\hat{H})})}", "derivation": "\\bar{\\h}{(\\hat{H})} = \\sin{(\\hat{H})} and - \\hat{H} + \\bar{\\h}{(\\hat{H})} = - \\hat{H} + \\sin{(\\hat{H})} and - \\hat{H} \\sin{(\\hat{H})} - \\hat{H} + \\bar{\\h}{(\\hat{H})} = - \\hat{H} \\sin{(\\hat{H})} - \\hat{H} + \\sin{(\\hat{H})} and - \\hat{H} \\sin{(\\hat{H})} = - \\hat{H} \\sin{(\\hat{H})} - \\bar{\\h}{(\\hat{H})} + \\sin{(\\hat{H})} and \\cos{(\\hat{H} \\sin{(\\hat{H})})} = \\cos{(\\hat{H} \\sin{(\\hat{H})} + \\bar{\\h}{(\\hat{H})} - \\sin{(\\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True))), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["cos", 4], "Equality(cos(Mul(Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)))), cos(Add(Mul(Symbol('\\\\hat{H}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Function('\\\\hbar')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(z)} = \\log{(e^{z})}, then derive \\int \\Psi_{\\lambda}{(z)} dz = f^{*} + \\frac{z^{2}}{2}, then obtain z (\\tilde{g} + \\frac{z^{2}}{2}) = z (f^{*} + \\frac{z^{2}}{2})", "derivation": "\\Psi_{\\lambda}{(z)} = \\log{(e^{z})} and \\int \\Psi_{\\lambda}{(z)} dz = \\int \\log{(e^{z})} dz and \\int \\Psi_{\\lambda}{(z)} dz = f^{*} + \\frac{z^{2}}{2} and \\int \\log{(e^{z})} dz = f^{*} + \\frac{z^{2}}{2} and z \\int \\log{(e^{z})} dz = z (f^{*} + \\frac{z^{2}}{2}) and z (\\tilde{g} + \\frac{z^{2}}{2}) = z (f^{*} + \\frac{z^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('z', commutative=True)), log(exp(Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(log(exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2)))))"], [["times", 4, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Integral(log(exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Mul(Symbol('z', commutative=True), Add(Symbol('f^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))))))"], [["evaluate_integrals", 5], "Equality(Mul(Symbol('z', commutative=True), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))))), Mul(Symbol('z', commutative=True), Add(Symbol('f^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(U,v_{z})} = U - v_{z}, then derive \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})} = 1, then obtain (1 - \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})})^{v_{z}} = 1", "derivation": "\\eta^{\\prime}{(U,v_{z})} = U - v_{z} and \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})} = \\frac{\\partial}{\\partial U} (U - v_{z}) and \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})} = 1 and - v_{z} + \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})} = 1 - v_{z} and 0 = 1 - \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})} and 0 = 1 - \\frac{\\partial}{\\partial U} (U - v_{z}) and 0^{v_{z}} = (1 - \\frac{\\partial}{\\partial U} (U - v_{z}))^{v_{z}} and 0^{v_{z}} = (1 - \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})})^{v_{z}} and (1 - \\frac{\\partial}{\\partial U} \\eta^{\\prime}{(U,v_{z})})^{v_{z}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Mul(Integer(-1), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["power", 6, "Symbol('v_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_z', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Pow(Integer(0), Symbol('v_z', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))), Symbol('v_z', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\pi,y)} = \\frac{\\partial}{\\partial \\pi} (\\pi + y), then derive \\operatorname{L_{\\varepsilon}}{(\\pi,y)} = 1, then obtain \\operatorname{L_{\\varepsilon}}^{\\pi}{(\\pi,y \\frac{\\partial}{\\partial \\pi} (\\pi + y))} = 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\pi,y)} = \\frac{\\partial}{\\partial \\pi} (\\pi + y) and \\operatorname{L_{\\varepsilon}}{(\\pi,y)} = 1 and \\operatorname{L_{\\varepsilon}}^{\\pi}{(\\pi,y)} = 1 and \\frac{\\partial}{\\partial \\pi} (\\pi + y) = 1 and y \\frac{\\partial}{\\partial \\pi} (\\pi + y) = y and \\operatorname{L_{\\varepsilon}}^{\\pi}{(\\pi,y \\frac{\\partial}{\\partial \\pi} (\\pi + y))} = 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Integer(1))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Symbol('y', commutative=True))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True), Mul(Symbol('y', commutative=True), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Symbol('\\\\pi', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\rho_b)} = \\cos{(\\rho_b)} and G{(\\pi,A_{1})} = A_{1} + \\pi, then obtain (1 + \\frac{\\operatorname{A_{1}}{(\\rho_b)}}{A_{1} + \\pi}) (A_{1} + \\pi - \\rho_b - G{(\\pi,A_{1})}) = (1 + \\frac{\\cos{(\\rho_b)}}{A_{1} + \\pi}) (A_{1} + \\pi - \\rho_b - G{(\\pi,A_{1})})", "derivation": "\\operatorname{A_{1}}{(\\rho_b)} = \\cos{(\\rho_b)} and G{(\\pi,A_{1})} = A_{1} + \\pi and \\frac{\\operatorname{A_{1}}{(\\rho_b)}}{G{(\\pi,A_{1})}} = \\frac{\\cos{(\\rho_b)}}{G{(\\pi,A_{1})}} and \\frac{\\operatorname{A_{1}}{(\\rho_b)}}{G{(\\pi,A_{1})}} + 1 = 1 + \\frac{\\cos{(\\rho_b)}}{G{(\\pi,A_{1})}} and 1 + \\frac{\\operatorname{A_{1}}{(\\rho_b)}}{A_{1} + \\pi} = 1 + \\frac{\\cos{(\\rho_b)}}{A_{1} + \\pi} and (1 + \\frac{\\operatorname{A_{1}}{(\\rho_b)}}{A_{1} + \\pi}) (A_{1} + \\pi - \\rho_b - G{(\\pi,A_{1})}) = (1 + \\frac{\\cos{(\\rho_b)}}{A_{1} + \\pi}) (A_{1} + \\pi - \\rho_b - G{(\\pi,A_{1})})", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)))"], ["get_premise", "Equality(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Function('A_1')(Symbol('\\\\rho_b', commutative=True)), Pow(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Mul(Pow(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Function('A_1')(Symbol('\\\\rho_b', commutative=True)), Pow(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Integer(1)), Add(Integer(1), Mul(Pow(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Function('A_1')(Symbol('\\\\rho_b', commutative=True)))), Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_b', commutative=True)))))"], [["times", 5, "Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True))))"], "Equality(Mul(Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Function('A_1')(Symbol('\\\\rho_b', commutative=True)))), Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True))))), Mul(Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_b', commutative=True)))), Add(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Function('G')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True))))))"]]}, {"prompt": "Given W{(\\mu_0)} = e^{\\mu_0} and \\dot{z}{(\\mu_0)} = \\mu_0, then obtain \\frac{\\frac{d}{d \\mu_0} 0}{- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\mu_0}}{\\mu_0}} = \\frac{\\frac{d}{d \\mu_0} (- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\dot{z}{(\\mu_0)}}}{\\mu_0})}{- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\mu_0}}{\\mu_0}}", "derivation": "W{(\\mu_0)} = e^{\\mu_0} and \\frac{W{(\\mu_0)}}{\\mu_0} = \\frac{e^{\\mu_0}}{\\mu_0} and 0 = - \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\mu_0}}{\\mu_0} and \\dot{z}{(\\mu_0)} = \\mu_0 and e^{\\dot{z}{(\\mu_0)}} = e^{\\mu_0} and 0 = - \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\dot{z}{(\\mu_0)}}}{\\mu_0} and \\frac{d}{d \\mu_0} 0 = \\frac{d}{d \\mu_0} (- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\dot{z}{(\\mu_0)}}}{\\mu_0}) and \\frac{\\frac{d}{d \\mu_0} 0}{- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\mu_0}}{\\mu_0}} = \\frac{\\frac{d}{d \\mu_0} (- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\dot{z}{(\\mu_0)}}}{\\mu_0})}{- \\frac{W{(\\mu_0)}}{\\mu_0} + \\frac{e^{\\mu_0}}{\\mu_0}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["exp", 4], "Equality(exp(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), exp(Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))))))"], [["differentiate", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 7, "Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True)))), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(t_{2},c)} = \\int (- c + t_{2}) dc, then obtain \\frac{\\partial}{\\partial c} c t_{2} \\operatorname{v_{t}}{(t_{2},c)} = \\frac{\\partial}{\\partial c} c t_{2} \\int (- c + t_{2}) dc", "derivation": "\\operatorname{v_{t}}{(t_{2},c)} = \\int (- c + t_{2}) dc and c \\operatorname{v_{t}}{(t_{2},c)} = c \\int (- c + t_{2}) dc and c t_{2} \\operatorname{v_{t}}{(t_{2},c)} = c t_{2} \\int (- c + t_{2}) dc and \\frac{\\partial}{\\partial c} c t_{2} \\operatorname{v_{t}}{(t_{2},c)} = \\frac{\\partial}{\\partial c} c t_{2} \\int (- c + t_{2}) dc", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('t_2', commutative=True), Symbol('c', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('v_t')(Symbol('t_2', commutative=True), Symbol('c', commutative=True))), Mul(Symbol('c', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["times", 2, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Symbol('t_2', commutative=True), Function('v_t')(Symbol('t_2', commutative=True), Symbol('c', commutative=True))), Mul(Symbol('c', commutative=True), Symbol('t_2', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Symbol('c', commutative=True), Symbol('t_2', commutative=True), Function('v_t')(Symbol('t_2', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Symbol('c', commutative=True), Symbol('t_2', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(g_{\\varepsilon},C_{d},z)} = C_{d} - g_{\\varepsilon} - z, then obtain - (\\int b B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} db)^{b} = - (\\int b (C_{d} - g_{\\varepsilon} - z)^{C_{d}} db)^{b}", "derivation": "B{(g_{\\varepsilon},C_{d},z)} = C_{d} - g_{\\varepsilon} - z and B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} = (C_{d} - g_{\\varepsilon} - z)^{C_{d}} and b B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} = b (C_{d} - g_{\\varepsilon} - z)^{C_{d}} and \\int b B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} db = \\int b (C_{d} - g_{\\varepsilon} - z)^{C_{d}} db and (\\int b B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} db)^{b} = (\\int b (C_{d} - g_{\\varepsilon} - z)^{C_{d}} db)^{b} and - (\\int b B^{C_{d}}{(g_{\\varepsilon},C_{d},z)} db)^{b} = - (\\int b (C_{d} - g_{\\varepsilon} - z)^{C_{d}} db)^{b}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('C_d', commutative=True)), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('C_d', commutative=True)))"], [["times", 2, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Pow(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('C_d', commutative=True))), Mul(Symbol('b', commutative=True), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('C_d', commutative=True))))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Symbol('b', commutative=True), Pow(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('b', commutative=True), Pow(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Mul(Symbol('b', commutative=True), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integral(Mul(Symbol('b', commutative=True), Pow(Function('B')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C_d', commutative=True), Symbol('z', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Symbol('b', commutative=True), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('C_d', commutative=True))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\varphi,y)} = \\frac{y}{\\varphi} and b{(\\varphi,y)} = \\operatorname{n_{2}}^{\\varphi}{(\\varphi,y)}, then obtain ((\\frac{y}{\\varphi})^{- \\varphi} b{(\\varphi,y)})^{y} \\operatorname{n_{2}}^{- \\varphi}{(\\varphi,y)} = \\operatorname{n_{2}}^{- \\varphi}{(\\varphi,y)}", "derivation": "\\operatorname{n_{2}}{(\\varphi,y)} = \\frac{y}{\\varphi} and \\operatorname{n_{2}}^{\\varphi}{(\\varphi,y)} = (\\frac{y}{\\varphi})^{\\varphi} and (\\frac{y}{\\varphi})^{- \\varphi} \\operatorname{n_{2}}^{\\varphi}{(\\varphi,y)} = 1 and b{(\\varphi,y)} = \\operatorname{n_{2}}^{\\varphi}{(\\varphi,y)} and (\\frac{y}{\\varphi})^{- \\varphi} b{(\\varphi,y)} = 1 and ((\\frac{y}{\\varphi})^{- \\varphi} b{(\\varphi,y)})^{y} = 1 and ((\\frac{y}{\\varphi})^{- \\varphi} b{(\\varphi,y)})^{y} \\operatorname{n_{2}}^{- \\varphi}{(\\varphi,y)} = \\operatorname{n_{2}}^{- \\varphi}{(\\varphi,y)}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["divide", 2, "Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Function('b')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True))), Integer(1))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Function('b')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(1))"], [["divide", 6, "Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Function('b')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Pow(Function('n_2')(Symbol('\\\\varphi', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon}, then derive \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\theta_{2}{(g^{\\prime}_{\\varepsilon})} - 1 = 0, then obtain (\\frac{d}{d g^{\\prime}_{\\varepsilon}} g^{\\prime}_{\\varepsilon} - 1) \\theta_{2}{(g^{\\prime}_{\\varepsilon})} = 0", "derivation": "\\theta_{2}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} and - g^{\\prime}_{\\varepsilon} + \\theta_{2}{(g^{\\prime}_{\\varepsilon})} = 0 and \\frac{d}{d g^{\\prime}_{\\varepsilon}} (- g^{\\prime}_{\\varepsilon} + \\theta_{2}{(g^{\\prime}_{\\varepsilon})}) = \\frac{d}{d g^{\\prime}_{\\varepsilon}} 0 and \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\theta_{2}{(g^{\\prime}_{\\varepsilon})} - 1 = 0 and \\frac{d}{d g^{\\prime}_{\\varepsilon}} g^{\\prime}_{\\varepsilon} - 1 = 0 and (\\frac{d}{d g^{\\prime}_{\\varepsilon}} g^{\\prime}_{\\varepsilon} - 1) \\theta_{2}{(g^{\\prime}_{\\varepsilon})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], [["minus", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["times", 5, "Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Derivative(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Function('\\\\theta_2')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = e^{v_{x}}, then derive \\frac{d}{d v_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = e^{v_{x}}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = \\frac{d}{d v_{x}} e^{v_{x}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = e^{v_{x}} and \\frac{d}{d v_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = \\frac{d}{d v_{x}} e^{v_{x}} and \\frac{d}{d v_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = e^{v_{x}} and \\frac{d}{d v_{x}} e^{v_{x}} = e^{v_{x}} and \\frac{d}{d v_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x})} = \\frac{d}{d v_{x}} e^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), exp(Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), exp(Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True)), Derivative(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\log{(\\mathbf{J}_P)}) \\operatorname{F_{H}}{(\\mathbf{J}_P)}, then obtain \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\operatorname{F_{H}}{(\\mathbf{J}_P)}) \\log{(\\mathbf{J}_P)}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and - \\mathbf{J}_P + \\operatorname{F_{H}}{(\\mathbf{J}_P)} = - \\mathbf{J}_P + \\log{(\\mathbf{J}_P)} and (- \\mathbf{J}_P + \\log{(\\mathbf{J}_P)}) \\operatorname{F_{H}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\log{(\\mathbf{J}_P)}) \\log{(\\mathbf{J}_P)} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\log{(\\mathbf{J}_P)}) \\operatorname{F_{H}}{(\\mathbf{J}_P)} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\log{(\\mathbf{J}_P)}) \\log{(\\mathbf{J}_P)} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_P)} = (- \\mathbf{J}_P + \\operatorname{F_{H}}{(\\mathbf{J}_P)}) \\log{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True))), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given x{(\\mu,\\eta^{\\prime})} = - \\eta^{\\prime} + \\mu, then obtain e^{\\sin{(\\frac{x{(\\mu,\\eta^{\\prime})}}{\\eta^{\\prime}})}} = e^{\\sin{(\\frac{- \\eta^{\\prime} + \\mu}{\\eta^{\\prime}})}}", "derivation": "x{(\\mu,\\eta^{\\prime})} = - \\eta^{\\prime} + \\mu and \\frac{x{(\\mu,\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{- \\eta^{\\prime} + \\mu}{\\eta^{\\prime}} and \\sin{(\\frac{x{(\\mu,\\eta^{\\prime})}}{\\eta^{\\prime}})} = \\sin{(\\frac{- \\eta^{\\prime} + \\mu}{\\eta^{\\prime}})} and e^{\\sin{(\\frac{x{(\\mu,\\eta^{\\prime})}}{\\eta^{\\prime}})}} = e^{\\sin{(\\frac{- \\eta^{\\prime} + \\mu}{\\eta^{\\prime}})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["exp", 3], "Equality(exp(sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))), exp(sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given L{(J,E_{\\lambda})} = E_{\\lambda} J and a{(E_{\\lambda})} = E_{\\lambda}, then derive - \\frac{E_{\\lambda} J^{2}}{2} + J a{(E_{\\lambda})} + v_{y} = - \\frac{E_{\\lambda} J^{2}}{2} + E_{\\lambda} J + g, then obtain E_{\\lambda} J - \\frac{J L{(J,E_{\\lambda})}}{2} + v_{y} = - \\frac{J L{(J,E_{\\lambda})}}{2} + g + L{(J,E_{\\lambda})}", "derivation": "L{(J,E_{\\lambda})} = E_{\\lambda} J and a{(E_{\\lambda})} = E_{\\lambda} and - E_{\\lambda} J + a{(E_{\\lambda})} = - E_{\\lambda} J + E_{\\lambda} and \\int (- E_{\\lambda} J + a{(E_{\\lambda})}) dJ = \\int (- E_{\\lambda} J + E_{\\lambda}) dJ and - \\frac{E_{\\lambda} J^{2}}{2} + J a{(E_{\\lambda})} + v_{y} = - \\frac{E_{\\lambda} J^{2}}{2} + E_{\\lambda} J + g and - \\frac{J L{(J,E_{\\lambda})}}{2} + J a{(E_{\\lambda})} + v_{y} = - \\frac{J L{(J,E_{\\lambda})}}{2} + g + L{(J,E_{\\lambda})} and E_{\\lambda} J - \\frac{J L{(J,E_{\\lambda})}}{2} + v_{y} = - \\frac{J L{(J,E_{\\lambda})}}{2} + g + L{(J,E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["minus", 2, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Function('a')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Function('a')(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Function('a')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('J', commutative=True), Function('a')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('g', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('g', commutative=True), Function('L')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\theta_1)} = \\sin{(\\sin{(\\theta_1)})}, then derive \\frac{d}{d \\theta_1} \\operatorname{M_{E}}{(\\theta_1)} = \\cos{(\\theta_1)} \\cos{(\\sin{(\\theta_1)})}, then obtain \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} = \\cos{(\\theta_1)} \\cos{(\\sin{(\\theta_1)})}", "derivation": "\\operatorname{M_{E}}{(\\theta_1)} = \\sin{(\\sin{(\\theta_1)})} and \\frac{d}{d \\theta_1} \\operatorname{M_{E}}{(\\theta_1)} = \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} and \\frac{d}{d \\theta_1} \\operatorname{M_{E}}{(\\theta_1)} = \\cos{(\\theta_1)} \\cos{(\\sin{(\\theta_1)})} and \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} = \\cos{(\\theta_1)} \\cos{(\\sin{(\\theta_1)})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), sin(sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\theta_1', commutative=True)), cos(sin(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\theta_1', commutative=True)), cos(sin(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given r{(\\lambda)} = \\sin{(\\lambda)}, then obtain ((\\frac{\\lambda + r^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda})^{\\lambda} = ((\\frac{\\lambda + \\sin^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda})^{\\lambda}", "derivation": "r{(\\lambda)} = \\sin{(\\lambda)} and r^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} and \\lambda + r^{\\lambda}{(\\lambda)} = \\lambda + \\sin^{\\lambda}{(\\lambda)} and \\frac{\\lambda + r^{\\lambda}{(\\lambda)}}{\\lambda} = \\frac{\\lambda + \\sin^{\\lambda}{(\\lambda)}}{\\lambda} and (\\frac{\\lambda + r^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda} = (\\frac{\\lambda + \\sin^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda} and ((\\frac{\\lambda + r^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda})^{\\lambda} = ((\\frac{\\lambda + \\sin^{\\lambda}{(\\lambda)}}{\\lambda})^{\\lambda})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('r')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["add", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Pow(Function('r')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["divide", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(Function('r')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(Function('r')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)))"], [["power", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(Function('r')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(q)} = \\log{(q)}, then obtain \\psi^{*}^{q}{(q)} \\cos{(\\psi^{*}{(q)})} = \\psi^{*}^{q}{(q)} \\cos{(\\log{(q)})}", "derivation": "\\psi^{*}{(q)} = \\log{(q)} and \\psi^{*}^{q}{(q)} = \\log{(q)}^{q} and \\cos{(\\psi^{*}{(q)})} = \\cos{(\\log{(q)})} and \\log{(q)}^{q} \\cos{(\\psi^{*}{(q)})} = \\log{(q)}^{q} \\cos{(\\log{(q)})} and \\psi^{*}^{q}{(q)} \\cos{(\\psi^{*}{(q)})} = \\psi^{*}^{q}{(q)} \\cos{(\\log{(q)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\psi^*')(Symbol('q', commutative=True))), cos(log(Symbol('q', commutative=True))))"], [["times", 3, "Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Function('\\\\psi^*')(Symbol('q', commutative=True)))), Mul(Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Function('\\\\psi^*')(Symbol('q', commutative=True)))), Mul(Pow(Function('\\\\psi^*')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(n_{2},\\Psi)} = \\log{(\\Psi n_{2})}, then derive \\int \\operatorname{E_{x}}{(n_{2},\\Psi)} dn_{2} = \\mathbb{I} + n_{2} \\log{(\\Psi n_{2})} - n_{2}, then obtain (\\mathbb{I} + n_{2} \\operatorname{E_{x}}{(n_{2},\\Psi)} - n_{2})^{n_{2}} = (\\int \\log{(\\Psi n_{2})} dn_{2})^{n_{2}}", "derivation": "\\operatorname{E_{x}}{(n_{2},\\Psi)} = \\log{(\\Psi n_{2})} and \\int \\operatorname{E_{x}}{(n_{2},\\Psi)} dn_{2} = \\int \\log{(\\Psi n_{2})} dn_{2} and (\\int \\operatorname{E_{x}}{(n_{2},\\Psi)} dn_{2})^{n_{2}} = (\\int \\log{(\\Psi n_{2})} dn_{2})^{n_{2}} and \\int \\operatorname{E_{x}}{(n_{2},\\Psi)} dn_{2} = \\mathbb{I} + n_{2} \\log{(\\Psi n_{2})} - n_{2} and \\int \\operatorname{E_{x}}{(n_{2},\\Psi)} dn_{2} = \\mathbb{I} + n_{2} \\operatorname{E_{x}}{(n_{2},\\Psi)} - n_{2} and (\\mathbb{I} + n_{2} \\operatorname{E_{x}}{(n_{2},\\Psi)} - n_{2})^{n_{2}} = (\\int \\log{(\\Psi n_{2})} dn_{2})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True)), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Integral(Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Integral(log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('n_2', commutative=True), log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('n_2', commutative=True), Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('n_2', commutative=True), Function('E_x')(Symbol('n_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Integral(log(Mul(Symbol('\\\\Psi', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} = \\Psi \\hat{H}_l - \\varepsilon_0, then derive - \\int \\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} d\\Psi = - \\frac{\\Psi^{2} \\hat{H}_l}{2} + \\Psi \\varepsilon_0 - t_{2}, then obtain - \\int (\\Psi \\hat{H}_l - \\varepsilon_0) d\\Psi = - \\frac{\\Psi^{2} \\hat{H}_l}{2} + \\Psi \\varepsilon_0 - t_{2}", "derivation": "\\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} = \\Psi \\hat{H}_l - \\varepsilon_0 and \\int \\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} d\\Psi = \\int (\\Psi \\hat{H}_l - \\varepsilon_0) d\\Psi and - \\int \\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} d\\Psi = - \\int (\\Psi \\hat{H}_l - \\varepsilon_0) d\\Psi and - \\int \\rho_{b}{(\\varepsilon_0,\\hat{H}_l,\\Psi)} d\\Psi = - \\frac{\\Psi^{2} \\hat{H}_l}{2} + \\Psi \\varepsilon_0 - t_{2} and - \\int (\\Psi \\hat{H}_l - \\varepsilon_0) d\\Psi = - \\frac{\\Psi^{2} \\hat{H}_l}{2} + \\Psi \\varepsilon_0 - t_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Integral(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then derive \\mathbf{F} \\frac{d}{d \\mathbf{F}} \\mathbf{r}{(\\mathbf{F})} + \\mathbf{r}{(\\mathbf{F})} = - \\mathbf{F} \\sin{(\\mathbf{F})} + \\cos{(\\mathbf{F})}, then obtain \\frac{\\mathbf{F} \\frac{d}{d \\mathbf{F}} \\mathbf{r}{(\\mathbf{F})} + \\mathbf{r}{(\\mathbf{F})}}{\\cos{(\\mathbf{F})}} = \\frac{- \\mathbf{F} \\sin{(\\mathbf{F})} + \\cos{(\\mathbf{F})}}{\\cos{(\\mathbf{F})}}", "derivation": "\\mathbf{r}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\mathbf{F} \\mathbf{r}{(\\mathbf{F})} = \\mathbf{F} \\cos{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\mathbf{r}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\cos{(\\mathbf{F})} and \\mathbf{F} \\frac{d}{d \\mathbf{F}} \\mathbf{r}{(\\mathbf{F})} + \\mathbf{r}{(\\mathbf{F})} = - \\mathbf{F} \\sin{(\\mathbf{F})} + \\cos{(\\mathbf{F})} and \\frac{\\mathbf{F} \\frac{d}{d \\mathbf{F}} \\mathbf{r}{(\\mathbf{F})} + \\mathbf{r}{(\\mathbf{F})}}{\\cos{(\\mathbf{F})}} = \\frac{- \\mathbf{F} \\sin{(\\mathbf{F})} + \\cos{(\\mathbf{F})}}{\\cos{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 4, "cos(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\mathbf{F}', commutative=True), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{F}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), cos(Symbol('\\\\mathbf{F}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_l{(v_{1})} = \\cos{(v_{1})}, then obtain v_{1}^{- v_{1}} (\\frac{v_{1} \\hat{H}_l{(v_{1})}}{\\cos{(v_{1})}})^{v_{1}} = 1", "derivation": "\\hat{H}_l{(v_{1})} = \\cos{(v_{1})} and \\frac{\\hat{H}_l{(v_{1})}}{\\cos{(v_{1})}} = 1 and \\frac{v_{1} \\hat{H}_l{(v_{1})}}{\\cos{(v_{1})}} = v_{1} and (\\frac{v_{1} \\hat{H}_l{(v_{1})}}{\\cos{(v_{1})}})^{v_{1}} = v_{1}^{v_{1}} and v_{1}^{- v_{1}} (\\frac{v_{1} \\hat{H}_l{(v_{1})}}{\\cos{(v_{1})}})^{v_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["divide", 1, "cos(Symbol('v_1', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('v_1', commutative=True))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Mul(Symbol('v_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Symbol('v_1', commutative=True)))"], [["divide", 4, "Pow(Symbol('v_1', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Pow(Mul(Symbol('v_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('v_1', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(P_{e})} = \\cos{(P_{e})}, then obtain \\frac{d}{d P_{e}} \\operatorname{v_{y}}{(P_{e})} \\int \\operatorname{v_{y}}^{2}{(P_{e})} dP_{e} = \\frac{d}{d P_{e}} \\operatorname{v_{y}}{(P_{e})} \\int \\operatorname{v_{y}}{(P_{e})} \\cos{(P_{e})} dP_{e}", "derivation": "\\operatorname{v_{y}}{(P_{e})} = \\cos{(P_{e})} and \\operatorname{v_{y}}^{2}{(P_{e})} = \\operatorname{v_{y}}{(P_{e})} \\cos{(P_{e})} and \\int \\operatorname{v_{y}}^{2}{(P_{e})} dP_{e} = \\int \\operatorname{v_{y}}{(P_{e})} \\cos{(P_{e})} dP_{e} and \\frac{d}{d P_{e}} \\operatorname{v_{y}}{(P_{e})} \\int \\operatorname{v_{y}}^{2}{(P_{e})} dP_{e} = \\frac{d}{d P_{e}} \\operatorname{v_{y}}{(P_{e})} \\int \\operatorname{v_{y}}{(P_{e})} \\cos{(P_{e})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["times", 1, "Function('v_y')(Symbol('P_e', commutative=True))"], "Equality(Pow(Function('v_y')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('v_y')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Pow(Function('v_y')(Symbol('P_e', commutative=True)), Integer(2)), Tuple(Symbol('P_e', commutative=True))), Integral(Mul(Function('v_y')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["times", 3, "Derivative(Function('v_y')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('v_y')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integral(Pow(Function('v_y')(Symbol('P_e', commutative=True)), Integer(2)), Tuple(Symbol('P_e', commutative=True)))), Mul(Derivative(Function('v_y')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integral(Mul(Function('v_y')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} = \\Psi_{nl}^{\\mathbf{g}}, then obtain (\\iint \\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} d\\Psi_{nl} d\\mathbf{g})^{\\Psi_{nl}} = (\\iint \\Psi_{nl}^{\\mathbf{g}} d\\Psi_{nl} d\\mathbf{g})^{\\Psi_{nl}}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} = \\Psi_{nl}^{\\mathbf{g}} and \\int \\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} d\\Psi_{nl} = \\int \\Psi_{nl}^{\\mathbf{g}} d\\Psi_{nl} and \\iint \\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} d\\Psi_{nl} d\\mathbf{g} = \\iint \\Psi_{nl}^{\\mathbf{g}} d\\Psi_{nl} d\\mathbf{g} and (\\iint \\operatorname{c_{0}}{(\\mathbf{g},\\Psi_{nl})} d\\Psi_{nl} d\\mathbf{g})^{\\Psi_{nl}} = (\\iint \\Psi_{nl}^{\\mathbf{g}} d\\Psi_{nl} d\\mathbf{g})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Integral(Function('c_0')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Integral(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\chi,\\rho)} = \\chi - \\rho, then derive \\frac{\\partial}{\\partial \\rho} \\varphi^{*}{(\\chi,\\rho)} = -1, then obtain \\frac{\\partial}{\\partial \\rho} (\\chi - \\rho) = -1", "derivation": "\\varphi^{*}{(\\chi,\\rho)} = \\chi - \\rho and \\varphi^{*}{(\\chi,\\rho)} - 1 = \\chi - \\rho - 1 and \\frac{\\partial}{\\partial \\rho} (\\varphi^{*}{(\\chi,\\rho)} - 1) = \\frac{\\partial}{\\partial \\rho} (\\chi - \\rho - 1) and \\frac{\\partial}{\\partial \\rho} \\varphi^{*}{(\\chi,\\rho)} = -1 and \\frac{\\partial}{\\partial \\rho} (\\chi - \\rho) = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\mathbf{s}{(B)} = \\sin{(\\log{(B)})} and \\operatorname{c_{0}}{(B)} = \\sin{(\\log{(B)})}, then obtain \\frac{d}{d B} \\sin{(\\log{(B)})} = \\frac{d}{d B} \\operatorname{c_{0}}{(B)}", "derivation": "\\mathbf{s}{(B)} = \\sin{(\\log{(B)})} and \\operatorname{c_{0}}{(B)} = \\sin{(\\log{(B)})} and \\frac{d}{d B} \\mathbf{s}{(B)} = \\frac{d}{d B} \\sin{(\\log{(B)})} and \\frac{d}{d B} \\mathbf{s}{(B)} = \\frac{d}{d B} \\operatorname{c_{0}}{(B)} and \\frac{d}{d B} \\sin{(\\log{(B)})} = \\frac{d}{d B} \\operatorname{c_{0}}{(B)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Function('c_0')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Function('c_0')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(c_{0})} = \\sin{(c_{0})}, then derive \\int f{(c_{0})} dc_{0} = \\hat{H}_{\\lambda} - \\cos{(c_{0})}, then obtain \\cos{(c_{0})} + \\int \\sin{(c_{0})} dc_{0} = \\hat{H}_{\\lambda}", "derivation": "f{(c_{0})} = \\sin{(c_{0})} and \\int f{(c_{0})} dc_{0} = \\int \\sin{(c_{0})} dc_{0} and \\int f{(c_{0})} dc_{0} = \\hat{H}_{\\lambda} - \\cos{(c_{0})} and \\cos{(c_{0})} + \\int f{(c_{0})} dc_{0} = \\hat{H}_{\\lambda} and \\cos{(c_{0})} + \\int \\sin{(c_{0})} dc_{0} = \\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('f')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))"], "Equality(Add(cos(Symbol('c_0', commutative=True)), Integral(Function('f')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('c_0', commutative=True)), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"]]}, {"prompt": "Given \\theta_{1}{(F_{N})} = \\log{(F_{N})} and \\operatorname{v_{2}}{(F_{N})} = F_{N} (F_{N} + \\log{(F_{N})}^{F_{N}}), then obtain F_{N} (F_{N} + \\theta_{1}^{F_{N}}{(F_{N})}) = \\operatorname{v_{2}}{(F_{N})}", "derivation": "\\theta_{1}{(F_{N})} = \\log{(F_{N})} and \\theta_{1}^{F_{N}}{(F_{N})} = \\log{(F_{N})}^{F_{N}} and F_{N} + \\theta_{1}^{F_{N}}{(F_{N})} = F_{N} + \\log{(F_{N})}^{F_{N}} and F_{N} (F_{N} + \\theta_{1}^{F_{N}}{(F_{N})}) = F_{N} (F_{N} + \\log{(F_{N})}^{F_{N}}) and \\operatorname{v_{2}}{(F_{N})} = F_{N} (F_{N} + \\log{(F_{N})}^{F_{N}}) and F_{N} (F_{N} + \\theta_{1}^{F_{N}}{(F_{N})}) = \\operatorname{v_{2}}{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["add", 2, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"], [["times", 3, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Function('v_2')(Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(T,r)} = T r and Z{(T,r)} = \\iint (T r)^{T} dr dT, then obtain T r + (\\iint (T r)^{T} dr dT)^{T} = T r + Z^{T}{(T,r)}", "derivation": "\\operatorname{E_{n}}{(T,r)} = T r and \\operatorname{E_{n}}^{T}{(T,r)} = (T r)^{T} and \\int \\operatorname{E_{n}}^{T}{(T,r)} dr = \\int (T r)^{T} dr and \\iint \\operatorname{E_{n}}^{T}{(T,r)} dr dT = \\iint (T r)^{T} dr dT and Z{(T,r)} = \\iint (T r)^{T} dr dT and \\iint \\operatorname{E_{n}}^{T}{(T,r)} dr dT = Z{(T,r)} and (\\iint \\operatorname{E_{n}}^{T}{(T,r)} dr dT)^{T} = Z^{T}{(T,r)} and \\operatorname{E_{n}}{(T,r)} + (\\iint \\operatorname{E_{n}}^{T}{(T,r)} dr dT)^{T} = \\operatorname{E_{n}}{(T,r)} + Z^{T}{(T,r)} and T r + (\\iint (T r)^{T} dr dT)^{T} = T r + Z^{T}{(T,r)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Pow(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Integral(Pow(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Function('Z')(Symbol('T', commutative=True), Symbol('r', commutative=True)))"], [["power", 6, "Symbol('T', commutative=True)"], "Equality(Pow(Integral(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True)), Pow(Function('Z')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)))"], [["add", 7, "Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Pow(Integral(Pow(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True))), Add(Function('E_n')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Pow(Function('Z')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Add(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Pow(Integral(Pow(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True))), Add(Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Pow(Function('Z')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Symbol('T', commutative=True))))"]]}, {"prompt": "Given g{(\\delta,W)} = W + \\delta, then derive \\int g{(\\delta,W)} dW = \\frac{W^{2}}{2} + W \\delta + a^{\\dagger}, then obtain \\frac{\\int g{(\\delta,W)} dW}{W} = \\frac{\\frac{W^{2}}{2} + W \\delta + a^{\\dagger}}{W}", "derivation": "g{(\\delta,W)} = W + \\delta and \\int g{(\\delta,W)} dW = \\int (W + \\delta) dW and \\int g{(\\delta,W)} dW = \\frac{W^{2}}{2} + W \\delta + a^{\\dagger} and \\frac{\\int g{(\\delta,W)} dW}{W} = \\frac{\\int (W + \\delta) dW}{W} and \\frac{W^{2}}{2} + W \\delta + a^{\\dagger} = \\int (W + \\delta) dW and \\frac{\\int g{(\\delta,W)} dW}{W} = \\frac{\\frac{W^{2}}{2} + W \\delta + a^{\\dagger}}{W}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 2, "Pow(Symbol('W', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(f_{\\mathbf{v}},c)} = \\frac{f_{\\mathbf{v}}}{c}, then derive \\frac{\\partial}{\\partial c} \\mathbf{f}{(f_{\\mathbf{v}},c)} = - \\frac{f_{\\mathbf{v}}}{c^{2}}, then obtain \\frac{\\partial}{\\partial c} \\mathbf{f}{(f_{\\mathbf{v}},c)} = - \\frac{\\mathbf{f}{(f_{\\mathbf{v}},c)}}{c}", "derivation": "\\mathbf{f}{(f_{\\mathbf{v}},c)} = \\frac{f_{\\mathbf{v}}}{c} and \\frac{\\partial}{\\partial c} \\mathbf{f}{(f_{\\mathbf{v}},c)} = \\frac{\\partial}{\\partial c} \\frac{f_{\\mathbf{v}}}{c} and \\frac{\\partial}{\\partial c} \\mathbf{f}{(f_{\\mathbf{v}},c)} = - \\frac{f_{\\mathbf{v}}}{c^{2}} and \\frac{\\partial}{\\partial c} \\mathbf{f}{(f_{\\mathbf{v}},c)} = - \\frac{\\mathbf{f}{(f_{\\mathbf{v}},c)}}{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-2)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},\\mathbf{f})} = A_{2} - \\mathbf{f}, then derive \\int \\mathbf{S}{(A_{2},\\mathbf{f})} dA_{2} = \\frac{A_{2}^{2}}{2} - A_{2} \\mathbf{f} + U, then obtain \\frac{A_{2}^{2}}{2} - A_{2} \\mathbf{f} + A_{2} + U - \\mathbf{f} = A_{2} - \\mathbf{f} + \\int (A_{2} - \\mathbf{f}) dA_{2}", "derivation": "\\mathbf{S}{(A_{2},\\mathbf{f})} = A_{2} - \\mathbf{f} and \\int \\mathbf{S}{(A_{2},\\mathbf{f})} dA_{2} = \\int (A_{2} - \\mathbf{f}) dA_{2} and \\int \\mathbf{S}{(A_{2},\\mathbf{f})} dA_{2} = \\frac{A_{2}^{2}}{2} - A_{2} \\mathbf{f} + U and A_{2} - \\mathbf{f} + \\int \\mathbf{S}{(A_{2},\\mathbf{f})} dA_{2} = A_{2} - \\mathbf{f} + \\int (A_{2} - \\mathbf{f}) dA_{2} and \\frac{A_{2}^{2}}{2} - A_{2} \\mathbf{f} + A_{2} + U - \\mathbf{f} = A_{2} - \\mathbf{f} + \\int (A_{2} - \\mathbf{f}) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('U', commutative=True)))"], [["add", 2, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('A_2', commutative=True), Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given Z{(L)} = e^{L}, then derive (\\int Z{(L)} dL)^{L} = (c + e^{L})^{L}, then obtain \\int (\\int e^{L} dL)^{L} dc = \\int (c + e^{L})^{L} dc", "derivation": "Z{(L)} = e^{L} and \\int Z{(L)} dL = \\int e^{L} dL and (\\int Z{(L)} dL)^{L} = (\\int e^{L} dL)^{L} and (\\int Z{(L)} dL)^{L} = (c + e^{L})^{L} and \\int (\\int Z{(L)} dL)^{L} dc = \\int (c + e^{L})^{L} dc and \\int (\\int e^{L} dL)^{L} dc = \\int (c + e^{L})^{L} dc", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Function('Z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('Z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Add(Symbol('c', commutative=True), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Pow(Integral(Function('Z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Add(Symbol('c', commutative=True), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Pow(Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Add(Symbol('c', commutative=True), exp(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(r)} = \\log{(r)}, then obtain \\frac{((- \\hat{p}_0{(r)} + \\log{(r)})^{r})^{r}}{\\Psi^{\\dagger}} = \\frac{1}{\\Psi^{\\dagger}}", "derivation": "\\hat{p}_0{(r)} = \\log{(r)} and 0 = - \\hat{p}_0{(r)} + \\log{(r)} and 0^{r} = (- \\hat{p}_0{(r)} + \\log{(r)})^{r} and (0^{r})^{r} = ((- \\hat{p}_0{(r)} + \\log{(r)})^{r})^{r} and ((- \\hat{p}_0{(r)} + \\log{(r)})^{r})^{r} = 1 and \\frac{((- \\hat{p}_0{(r)} + \\log{(r)})^{r})^{r}}{\\Psi^{\\dagger}} = \\frac{1}{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["minus", 1, "Function('\\\\hat{p}_0')(Symbol('r', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r', commutative=True))), log(Symbol('r', commutative=True))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r', commutative=True))), log(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r', commutative=True))), log(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r', commutative=True))), log(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Integer(1))"], [["divide", 5, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r', commutative=True))), log(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Symbol('r', commutative=True))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given k{(\\phi)} = \\sin{(\\log{(\\phi)})}, then derive \\int k{(\\phi)} d\\phi = S + \\frac{\\phi \\sin{(\\log{(\\phi)})}}{2} - \\frac{\\phi \\cos{(\\log{(\\phi)})}}{2}, then obtain 2 \\int k{(\\phi)} d\\phi = 2 S + \\phi k{(\\phi)} - \\phi \\cos{(\\log{(\\phi)})}", "derivation": "k{(\\phi)} = \\sin{(\\log{(\\phi)})} and \\int k{(\\phi)} d\\phi = \\int \\sin{(\\log{(\\phi)})} d\\phi and \\int k{(\\phi)} d\\phi = S + \\frac{\\phi \\sin{(\\log{(\\phi)})}}{2} - \\frac{\\phi \\cos{(\\log{(\\phi)})}}{2} and \\int k{(\\phi)} d\\phi = S + \\frac{\\phi k{(\\phi)}}{2} - \\frac{\\phi \\cos{(\\log{(\\phi)})}}{2} and 2 \\int k{(\\phi)} d\\phi = 2 S + \\phi k{(\\phi)} - \\phi \\cos{(\\log{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi', commutative=True)), sin(log(Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('k')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(sin(log(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('S', commutative=True), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True), sin(log(Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\phi', commutative=True), cos(log(Symbol('\\\\phi', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('k')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('S', commutative=True), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True), Function('k')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\phi', commutative=True), cos(log(Symbol('\\\\phi', commutative=True))))))"], [["divide", 4, "Rational(1, 2)"], "Equality(Mul(Integer(2), Integral(Function('k')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Function('k')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True), cos(log(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given h{(\\psi,f^{*})} = \\int (- \\psi + f^{*}) df^{*}, then obtain r + h{(\\psi,f^{*})} = - \\psi f^{*} + c_{0}", "derivation": "h{(\\psi,f^{*})} = \\int (- \\psi + f^{*}) df^{*} and \\frac{\\partial}{\\partial \\psi} h{(\\psi,f^{*})} = \\frac{\\partial}{\\partial \\psi} \\int (- \\psi + f^{*}) df^{*} and \\int \\frac{\\partial}{\\partial \\psi} h{(\\psi,f^{*})} d\\psi = \\int \\frac{\\partial}{\\partial \\psi} \\int (- \\psi + f^{*}) df^{*} d\\psi and r + h{(\\psi,f^{*})} = - \\psi f^{*} + c_{0}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Derivative(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('r', commutative=True), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(L)} = \\cos{(\\sin{(L)})}, then derive \\frac{d}{d L} \\mathbf{s}{(L)} = - \\sin{(\\sin{(L)})} \\cos{(L)}, then obtain \\frac{d^{2}}{d L^{2}} \\mathbf{s}{(L)} = \\sin{(L)} \\sin{(\\sin{(L)})} - \\cos^{2}{(L)} \\cos{(\\sin{(L)})}", "derivation": "\\mathbf{s}{(L)} = \\cos{(\\sin{(L)})} and \\frac{d}{d L} \\mathbf{s}{(L)} = \\frac{d}{d L} \\cos{(\\sin{(L)})} and \\frac{d}{d L} \\mathbf{s}{(L)} = - \\sin{(\\sin{(L)})} \\cos{(L)} and \\frac{d}{d L} \\cos{(\\sin{(L)})} = - \\sin{(\\sin{(L)})} \\cos{(L)} and \\frac{d^{2}}{d L^{2}} \\mathbf{s}{(L)} = \\frac{d}{d L} - \\sin{(\\sin{(L)})} \\cos{(L)} and \\frac{d^{2}}{d L^{2}} \\mathbf{s}{(L)} = \\frac{d^{2}}{d L^{2}} \\cos{(\\sin{(L)})} and \\frac{d^{2}}{d L^{2}} \\mathbf{s}{(L)} = \\sin{(L)} \\sin{(\\sin{(L)})} - \\cos^{2}{(L)} \\cos{(\\sin{(L)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), cos(sin(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Add(Mul(sin(Symbol('L', commutative=True)), sin(sin(Symbol('L', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('L', commutative=True)), Integer(2)), cos(sin(Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})} = \\Psi^{\\dagger} + g_{\\varepsilon}, then obtain \\frac{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})}}{g_{\\varepsilon}} = \\frac{1}{g_{\\varepsilon}}", "derivation": "\\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})} = \\Psi^{\\dagger} + g_{\\varepsilon} and \\frac{\\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})}}{g_{\\varepsilon}} = \\frac{\\Psi^{\\dagger} + g_{\\varepsilon}}{g_{\\varepsilon}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})}}{g_{\\varepsilon}} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger} + g_{\\varepsilon}}{g_{\\varepsilon}} and \\frac{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\mathbf{E}{(g_{\\varepsilon},\\Psi^{\\dagger})}}{g_{\\varepsilon}} = \\frac{1}{g_{\\varepsilon}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{E}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(S,y)} = \\log{(y^{S})}, then derive \\int \\Psi_{\\lambda}{(S,y)} dy = - S y + u + y \\log{(y^{S})}, then obtain (- S y + u + y \\log{(y^{S})})^{y} = (- S y + u + y \\Psi_{\\lambda}{(S,y)})^{y}", "derivation": "\\Psi_{\\lambda}{(S,y)} = \\log{(y^{S})} and \\int \\Psi_{\\lambda}{(S,y)} dy = \\int \\log{(y^{S})} dy and \\int \\Psi_{\\lambda}{(S,y)} dy = - S y + u + y \\log{(y^{S})} and \\int \\Psi_{\\lambda}{(S,y)} dy = - S y + u + y \\Psi_{\\lambda}{(S,y)} and - S y + u + y \\log{(y^{S})} = - S y + u + y \\Psi_{\\lambda}{(S,y)} and (- S y + u + y \\log{(y^{S})})^{y} = (- S y + u + y \\Psi_{\\lambda}{(S,y)})^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), log(Pow(Symbol('y', commutative=True), Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Pow(Symbol('y', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), log(Pow(Symbol('y', commutative=True), Symbol('S', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), log(Pow(Symbol('y', commutative=True), Symbol('S', commutative=True))))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)))))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), log(Pow(Symbol('y', commutative=True), Symbol('S', commutative=True))))), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True), Mul(Symbol('y', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(\\omega,u)} = \\omega - u, then derive \\int \\theta_{2}{(\\omega,u)} du = \\mathbf{v} + \\omega u - \\frac{u^{2}}{2}, then obtain \\omega (\\mathbf{v} + \\omega u - \\frac{u^{2}}{2}) = \\omega \\int \\theta_{2}{(\\omega,u)} du", "derivation": "\\theta_{2}{(\\omega,u)} = \\omega - u and \\int \\theta_{2}{(\\omega,u)} du = \\int (\\omega - u) du and \\int \\theta_{2}{(\\omega,u)} du = \\mathbf{v} + \\omega u - \\frac{u^{2}}{2} and \\int (\\omega - u) du = \\mathbf{v} + \\omega u - \\frac{u^{2}}{2} and \\omega \\int (\\omega - u) du = \\omega (\\mathbf{v} + \\omega u - \\frac{u^{2}}{2}) and \\omega \\int (\\omega - u) du = \\omega \\int \\theta_{2}{(\\omega,u)} du and \\omega (\\mathbf{v} + \\omega u - \\frac{u^{2}}{2}) = \\omega \\int \\theta_{2}{(\\omega,u)} du", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_2')(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["times", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('\\\\omega', commutative=True), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Symbol('\\\\omega', commutative=True), Integral(Function('\\\\theta_2')(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2))))), Mul(Symbol('\\\\omega', commutative=True), Integral(Function('\\\\theta_2')(Symbol('\\\\omega', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\mu{(\\hbar)} = e^{\\hbar}, then obtain (((\\mu{(\\hbar)} e^{- \\hbar})^{\\hbar})^{\\hbar})^{\\hbar} = 1", "derivation": "\\mu{(\\hbar)} = e^{\\hbar} and \\mu^{2}{(\\hbar)} = \\mu{(\\hbar)} e^{\\hbar} and \\mu{(\\hbar)} e^{- \\hbar} = 1 and (\\mu{(\\hbar)} e^{- \\hbar})^{\\hbar} = 1 and ((\\mu{(\\hbar)} e^{- \\hbar})^{\\hbar})^{\\hbar} = 1 and (((\\mu{(\\hbar)} e^{- \\hbar})^{\\hbar})^{\\hbar})^{\\hbar} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Function('\\\\mu')(Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))))"], [["divide", 2, "Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Integer(1))"], [["power", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Function('\\\\mu')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\varepsilon{(g,\\omega,v_{2})} = \\omega + g + v_{2}, then obtain \\sin{(\\frac{\\varepsilon{(g,\\omega,v_{2})}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}})} = \\sin{(\\frac{\\omega + g + v_{2}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}})}", "derivation": "\\varepsilon{(g,\\omega,v_{2})} = \\omega + g + v_{2} and \\omega \\varepsilon{(g,\\omega,v_{2})} = \\omega (\\omega + g + v_{2}) and \\frac{\\omega \\varepsilon{(g,\\omega,v_{2})}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}} = \\frac{\\omega (\\omega + g + v_{2})}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}} and \\frac{\\varepsilon{(g,\\omega,v_{2})}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}} = \\frac{\\omega + g + v_{2}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}} and \\sin{(\\frac{\\varepsilon{(g,\\omega,v_{2})}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}})} = \\sin{(\\frac{\\omega + g + v_{2}}{\\int (\\mathbf{M}^{z} + i) d\\mathbf{M}})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('v_2', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Symbol('v_2', commutative=True))))"], [["divide", 2, "Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))))"], [["divide", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))))"], [["sin", 4], "Equality(sin(Mul(Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1)))), sin(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('g', commutative=True), Symbol('v_2', commutative=True)), Pow(Integral(Add(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} = \\frac{\\mathbf{P}}{v_{y}} + \\rho_b, then obtain \\frac{(\\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P}) \\int \\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} d\\mathbf{P}}{v_{y}^{2}} = \\frac{(\\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P})^{2}}{v_{y}^{2}}", "derivation": "\\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} = \\frac{\\mathbf{P}}{v_{y}} + \\rho_b and \\int \\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} d\\mathbf{P} = \\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P} and \\frac{\\int \\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} d\\mathbf{P}}{v_{y}} = \\frac{\\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P}}{v_{y}} and \\frac{(\\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P}) \\int \\dot{y}{(\\rho_b,\\mathbf{P},v_{y})} d\\mathbf{P}}{v_{y}^{2}} = \\frac{(\\int (\\frac{\\mathbf{P}}{v_{y}} + \\rho_b) d\\mathbf{P})^{2}}{v_{y}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 2, "Symbol('v_y', commutative=True)"], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Integral(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Integer(-2)), Integral(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Pow(Symbol('v_y', commutative=True), Integer(-2)), Pow(Integral(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given E{(k,\\mathbf{A})} = \\mathbf{A}^{k} and \\rho_{f}{(k,\\mathbf{A})} = \\mathbf{A}^{k}, then obtain \\tilde{g} (\\mathbf{A}^{k})^{k} = \\tilde{g} \\rho_{f}^{k}{(k,\\mathbf{A})}", "derivation": "E{(k,\\mathbf{A})} = \\mathbf{A}^{k} and E^{k}{(k,\\mathbf{A})} = (\\mathbf{A}^{k})^{k} and \\rho_{f}{(k,\\mathbf{A})} = \\mathbf{A}^{k} and E{(k,\\mathbf{A})} = \\rho_{f}{(k,\\mathbf{A})} and E^{k}{(k,\\mathbf{A})} = \\rho_{f}^{k}{(k,\\mathbf{A})} and (\\mathbf{A}^{k})^{k} = \\rho_{f}^{k}{(k,\\mathbf{A})} and \\tilde{g} (\\mathbf{A}^{k})^{k} = \\tilde{g} \\rho_{f}^{k}{(k,\\mathbf{A})}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('E')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('E')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\rho_f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Function('E')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('k', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('k', commutative=True)))"], [["times", 6, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('\\\\rho_f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(E_{n},m,G)} = (E_{n} m)^{G}, then obtain - \\operatorname{f^{\\prime}}{(E_{n},m,G)} + \\int m \\operatorname{f^{\\prime}}{(E_{n},m,G)} dm = - \\operatorname{f^{\\prime}}{(E_{n},m,G)} + \\int m (E_{n} m)^{G} dm", "derivation": "\\operatorname{f^{\\prime}}{(E_{n},m,G)} = (E_{n} m)^{G} and m \\operatorname{f^{\\prime}}{(E_{n},m,G)} = m (E_{n} m)^{G} and \\int m \\operatorname{f^{\\prime}}{(E_{n},m,G)} dm = \\int m (E_{n} m)^{G} dm and - \\operatorname{f^{\\prime}}{(E_{n},m,G)} + \\int m \\operatorname{f^{\\prime}}{(E_{n},m,G)} dm = - \\operatorname{f^{\\prime}}{(E_{n},m,G)} + \\int m (E_{n} m)^{G} dm", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True)), Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('G', commutative=True)))"], [["times", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Symbol('m', commutative=True), Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["minus", 3, "Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True), Symbol('G', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Pow(Mul(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given g{(h,\\mu_0)} = h \\sin{(\\mu_0)}, then obtain h^{2} \\sin^{2}{(\\mu_0)} \\int g{(h,\\mu_0)} dh = h^{2} \\sin^{2}{(\\mu_0)} \\int h \\sin{(\\mu_0)} dh", "derivation": "g{(h,\\mu_0)} = h \\sin{(\\mu_0)} and h g{(h,\\mu_0)} \\sin{(\\mu_0)} = h^{2} \\sin^{2}{(\\mu_0)} and \\int g{(h,\\mu_0)} dh = \\int h \\sin{(\\mu_0)} dh and h g{(h,\\mu_0)} \\sin{(\\mu_0)} \\int g{(h,\\mu_0)} dh = h g{(h,\\mu_0)} \\sin{(\\mu_0)} \\int h \\sin{(\\mu_0)} dh and h^{2} \\sin^{2}{(\\mu_0)} \\int g{(h,\\mu_0)} dh = h^{2} \\sin^{2}{(\\mu_0)} \\int h \\sin{(\\mu_0)} dh", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('h', commutative=True), sin(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "Mul(Symbol('h', commutative=True), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Symbol('h', commutative=True), Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Mul(Symbol('h', commutative=True), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["times", 3, "Mul(Symbol('h', commutative=True), Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Symbol('h', commutative=True), Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)), Integral(Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Symbol('h', commutative=True), Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)), Integral(Mul(Symbol('h', commutative=True), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Integral(Function('g')(Symbol('h', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Integral(Mul(Symbol('h', commutative=True), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\varepsilon{(\\mathbf{H})} = \\mathbf{H} + \\frac{\\hat{H}_l{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}}, then obtain \\int \\varepsilon{(\\mathbf{H})} d\\mathbf{H} + \\frac{1}{\\sin{(\\mathbf{H})}} = \\int (\\mathbf{H} + 1) d\\mathbf{H} + \\frac{1}{\\sin{(\\mathbf{H})}}", "derivation": "\\hat{H}_l{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\frac{\\hat{H}_l{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}} = 1 and \\mathbf{H} + \\frac{\\hat{H}_l{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}} = \\mathbf{H} + 1 and \\varepsilon{(\\mathbf{H})} = \\mathbf{H} + \\frac{\\hat{H}_l{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}} and \\varepsilon{(\\mathbf{H})} = \\mathbf{H} + 1 and \\int \\varepsilon{(\\mathbf{H})} d\\mathbf{H} = \\int (\\mathbf{H} + 1) d\\mathbf{H} and \\int \\varepsilon{(\\mathbf{H})} d\\mathbf{H} + \\frac{1}{\\sin{(\\mathbf{H})}} = \\int (\\mathbf{H} + 1) d\\mathbf{H} + \\frac{1}{\\sin{(\\mathbf{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], [["integrate", 5, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 6, "Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))"], "Equality(Add(Integral(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Add(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(F_{N})} = \\cos{(F_{N})} and p{(F_{N})} = \\cos^{F_{N}}{(F_{N})}, then obtain \\operatorname{A_{1}}{(F_{N})} - p{(F_{N})} = - p{(F_{N})} + \\cos{(F_{N})}", "derivation": "\\operatorname{A_{1}}{(F_{N})} = \\cos{(F_{N})} and p{(F_{N})} = \\cos^{F_{N}}{(F_{N})} and \\operatorname{A_{1}}{(F_{N})} - \\operatorname{A_{1}}^{F_{N}}{(F_{N})} = - \\operatorname{A_{1}}^{F_{N}}{(F_{N})} + \\cos{(F_{N})} and p{(F_{N})} = \\operatorname{A_{1}}^{F_{N}}{(F_{N})} and \\operatorname{A_{1}}{(F_{N})} - p{(F_{N})} = - p{(F_{N})} + \\cos{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('F_N', commutative=True)), Pow(cos(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["minus", 1, "Pow(Function('A_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Function('A_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('A_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('p')(Symbol('F_N', commutative=True)), Pow(Function('A_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('A_1')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Function('p')(Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Function('p')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(F_{N},\\hat{p}_0)} = \\log{(F_{N}^{\\hat{p}_0})} and \\mathbf{v}{(F_{N},\\hat{p}_0)} = \\log{(\\operatorname{t_{2}}{(F_{N},\\hat{p}_0)})}, then obtain (\\int \\mathbf{v}^{F_{N}}{(F_{N},\\hat{p}_0)} d\\hat{p}_0)^{F_{N}} = (\\int \\log{(\\log{(F_{N}^{\\hat{p}_0})})}^{F_{N}} d\\hat{p}_0)^{F_{N}}", "derivation": "\\operatorname{t_{2}}{(F_{N},\\hat{p}_0)} = \\log{(F_{N}^{\\hat{p}_0})} and \\mathbf{v}{(F_{N},\\hat{p}_0)} = \\log{(\\operatorname{t_{2}}{(F_{N},\\hat{p}_0)})} and \\mathbf{v}{(F_{N},\\hat{p}_0)} = \\log{(\\log{(F_{N}^{\\hat{p}_0})})} and \\mathbf{v}^{F_{N}}{(F_{N},\\hat{p}_0)} = \\log{(\\log{(F_{N}^{\\hat{p}_0})})}^{F_{N}} and \\int \\mathbf{v}^{F_{N}}{(F_{N},\\hat{p}_0)} d\\hat{p}_0 = \\int \\log{(\\log{(F_{N}^{\\hat{p}_0})})}^{F_{N}} d\\hat{p}_0 and (\\int \\mathbf{v}^{F_{N}}{(F_{N},\\hat{p}_0)} d\\hat{p}_0)^{F_{N}} = (\\int \\log{(\\log{(F_{N}^{\\hat{p}_0})})}^{F_{N}} d\\hat{p}_0)^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Pow(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Function('t_2')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(log(Pow(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('F_N', commutative=True)), Pow(log(log(Pow(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('F_N', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Pow(log(log(Pow(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 5, "Symbol('F_N', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('F_N', commutative=True)), Pow(Integral(Pow(log(log(Pow(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given z{(J,\\mathbb{I})} = \\sin{(\\mathbb{I}^{J})}, then obtain z^{\\mathbb{I}}{(J,\\mathbb{I})} \\log{(z{(J,\\mathbb{I})})} \\sin{(\\mathbb{I}^{J})} = z^{\\mathbb{I}}{(J,\\mathbb{I})} \\log{(\\sin{(\\mathbb{I}^{J})})} \\sin{(\\mathbb{I}^{J})}", "derivation": "z{(J,\\mathbb{I})} = \\sin{(\\mathbb{I}^{J})} and \\log{(z{(J,\\mathbb{I})})} = \\log{(\\sin{(\\mathbb{I}^{J})})} and z^{\\mathbb{I}}{(J,\\mathbb{I})} = \\sin^{\\mathbb{I}}{(\\mathbb{I}^{J})} and z^{\\mathbb{I}}{(J,\\mathbb{I})} \\sin{(\\mathbb{I}^{J})} = \\sin{(\\mathbb{I}^{J})} \\sin^{\\mathbb{I}}{(\\mathbb{I}^{J})} and \\log{(z{(J,\\mathbb{I})})} \\sin{(\\mathbb{I}^{J})} \\sin^{\\mathbb{I}}{(\\mathbb{I}^{J})} = \\log{(\\sin{(\\mathbb{I}^{J})})} \\sin{(\\mathbb{I}^{J})} \\sin^{\\mathbb{I}}{(\\mathbb{I}^{J})} and z^{\\mathbb{I}}{(J,\\mathbb{I})} \\log{(z{(J,\\mathbb{I})})} \\sin{(\\mathbb{I}^{J})} = z^{\\mathbb{I}}{(J,\\mathbb{I})} \\log{(\\sin{(\\mathbb{I}^{J})})} \\sin{(\\mathbb{I}^{J})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))))"], [["log", 1], "Equality(log(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), log(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["times", 3, "sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))), Mul(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Pow(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 2, "Mul(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Pow(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(log(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Pow(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))), Mul(log(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Pow(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), log(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))), Mul(Pow(Function('z')(Symbol('J', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), log(sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))), sin(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(Q)} = e^{\\sin{(Q)}}, then obtain \\frac{\\mathbf{p} \\operatorname{F_{x}}^{4}{(Q)}}{\\eta} = \\frac{\\mathbf{p} \\operatorname{F_{x}}{(Q)} e^{3 \\sin{(Q)}}}{\\eta}", "derivation": "\\operatorname{F_{x}}{(Q)} = e^{\\sin{(Q)}} and \\operatorname{F_{x}}^{2}{(Q)} = \\operatorname{F_{x}}{(Q)} e^{\\sin{(Q)}} and \\operatorname{F_{x}}^{4}{(Q)} = \\operatorname{F_{x}}^{2}{(Q)} e^{2 \\sin{(Q)}} and \\operatorname{F_{x}}^{2}{(Q)} e^{2 \\sin{(Q)}} = \\operatorname{F_{x}}{(Q)} e^{3 \\sin{(Q)}} and \\operatorname{F_{x}}^{4}{(Q)} = \\operatorname{F_{x}}{(Q)} e^{3 \\sin{(Q)}} and \\frac{\\mathbf{p} \\operatorname{F_{x}}^{4}{(Q)}}{\\eta} = \\frac{\\mathbf{p} \\operatorname{F_{x}}{(Q)} e^{3 \\sin{(Q)}}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('Q', commutative=True)), exp(sin(Symbol('Q', commutative=True))))"], [["times", 1, "Function('F_x')(Symbol('Q', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(2)), Mul(Function('F_x')(Symbol('Q', commutative=True)), exp(sin(Symbol('Q', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(4)), Mul(Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(2)), exp(Mul(Integer(2), sin(Symbol('Q', commutative=True))))), Mul(Function('F_x')(Symbol('Q', commutative=True)), exp(Mul(Integer(3), sin(Symbol('Q', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(4)), Mul(Function('F_x')(Symbol('Q', commutative=True)), exp(Mul(Integer(3), sin(Symbol('Q', commutative=True))))))"], [["divide", 5, "Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('F_x')(Symbol('Q', commutative=True)), Integer(4))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Function('F_x')(Symbol('Q', commutative=True)), exp(Mul(Integer(3), sin(Symbol('Q', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} = \\sin{(\\dot{y} \\phi_1)}, then obtain (\\int 0 d\\phi_1 + \\frac{1}{2})^{\\phi_1} = (\\int (- \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)}) d\\phi_1 + \\frac{1}{2})^{\\phi_1}", "derivation": "\\operatorname{v_{z}}{(\\phi_1,\\dot{y})} = \\sin{(\\dot{y} \\phi_1)} and 0 = - \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)} and \\int 0 d\\phi_1 = \\int (- \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)}) d\\phi_1 and \\int 0 d\\phi_1 + 1 = \\int (- \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)}) d\\phi_1 + 1 and \\int 0 d\\phi_1 + \\frac{1}{2} = \\int (- \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)}) d\\phi_1 + \\frac{1}{2} and (\\int 0 d\\phi_1 + \\frac{1}{2})^{\\phi_1} = (\\int (- \\operatorname{v_{z}}{(\\phi_1,\\dot{y})} + \\sin{(\\dot{y} \\phi_1)}) d\\phi_1 + \\frac{1}{2})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Mul(Integer(-1), Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["add", 3, 1], "Equality(Add(Integral(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(1)), Add(Integral(Add(Mul(Integer(-1), Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(1)))"], [["add", 4, "Rational(-1, 2)"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True))), Rational(1, 2)), Add(Integral(Add(Mul(Integer(-1), Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Rational(1, 2)))"], [["power", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Integral(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True))), Rational(1, 2)), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Integral(Add(Mul(Integer(-1), Function('v_z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), sin(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Rational(1, 2)), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\omega{(A)} = \\sin{(A)} and \\operatorname{M_{E}}{(A)} = \\sin{(A)}, then obtain C_{d}^{\\mathbf{F}} + \\omega{(A)} + 1 = C_{d}^{\\mathbf{F}} + \\operatorname{M_{E}}{(A)} + 1", "derivation": "\\omega{(A)} = \\sin{(A)} and \\omega{(A)} + \\sin{(A)} = 2 \\sin{(A)} and \\operatorname{M_{E}}{(A)} = \\sin{(A)} and \\omega{(A)} = \\operatorname{M_{E}}{(A)} and C_{d}^{\\mathbf{F}} + \\omega{(A)} = C_{d}^{\\mathbf{F}} + \\operatorname{M_{E}}{(A)} and C_{d}^{\\mathbf{F}} + \\frac{\\omega{(A)} + \\sin{(A)}}{2 \\sin{(A)}} + \\omega{(A)} = C_{d}^{\\mathbf{F}} + \\frac{\\omega{(A)} + \\sin{(A)}}{2 \\sin{(A)}} + \\operatorname{M_{E}}{(A)} and C_{d}^{\\mathbf{F}} + \\omega{(A)} + 1 = C_{d}^{\\mathbf{F}} + \\operatorname{M_{E}}{(A)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["add", 1, "sin(Symbol('A', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Mul(Integer(2), sin(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\omega')(Symbol('A', commutative=True)), Function('M_E')(Symbol('A', commutative=True)))"], [["add", 4, "Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\omega')(Symbol('A', commutative=True))), Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('M_E')(Symbol('A', commutative=True))))"], [["add", 5, "Mul(Rational(1, 2), Add(Function('\\\\omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(-1)))"], "Equality(Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Rational(1, 2), Add(Function('\\\\omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(-1))), Function('\\\\omega')(Symbol('A', commutative=True))), Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Rational(1, 2), Add(Function('\\\\omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(-1))), Function('M_E')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\omega')(Symbol('A', commutative=True)), Integer(1)), Add(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('M_E')(Symbol('A', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\varphi{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J}, then derive \\varphi{(\\mathbf{J})} = \\hat{p}_0 + \\sin{(\\mathbf{J})}, then obtain \\frac{\\varphi{(\\mathbf{J})}}{2 \\hat{p}_0} = \\frac{\\dot{y} + \\sin{(\\mathbf{J})}}{2 \\hat{p}_0}", "derivation": "\\varphi{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and \\varphi{(\\mathbf{J})} = \\hat{p}_0 + \\sin{(\\mathbf{J})} and \\hat{p}_0 + \\sin{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and \\frac{\\hat{p}_0 + \\sin{(\\mathbf{J})}}{2 \\hat{p}_0} = \\frac{\\int \\cos{(\\mathbf{J})} d\\mathbf{J}}{2 \\hat{p}_0} and \\frac{\\varphi{(\\mathbf{J})}}{2 \\hat{p}_0} = \\frac{\\int \\cos{(\\mathbf{J})} d\\mathbf{J}}{2 \\hat{p}_0} and \\frac{\\varphi{(\\mathbf{J})}}{2 \\hat{p}_0} = \\frac{\\dot{y} + \\sin{(\\mathbf{J})}}{2 \\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(V)} = \\sin{(V)} and \\operatorname{f_{E}}{(V)} = \\sin{(V)}, then obtain \\frac{d}{d V} \\sin{(\\lambda{(V)} + \\sin{(V)})} = \\frac{d}{d V} \\sin{(\\lambda{(V)} + \\operatorname{f_{E}}{(V)})}", "derivation": "\\lambda{(V)} = \\sin{(V)} and 2 \\lambda{(V)} = \\lambda{(V)} + \\sin{(V)} and \\sin{(2 \\lambda{(V)})} = \\sin{(\\lambda{(V)} + \\sin{(V)})} and \\operatorname{f_{E}}{(V)} = \\sin{(V)} and \\sin{(2 \\lambda{(V)})} = \\sin{(\\lambda{(V)} + \\operatorname{f_{E}}{(V)})} and \\sin{(\\lambda{(V)} + \\sin{(V)})} = \\sin{(\\lambda{(V)} + \\operatorname{f_{E}}{(V)})} and \\frac{d}{d V} \\sin{(\\lambda{(V)} + \\sin{(V)})} = \\frac{d}{d V} \\sin{(\\lambda{(V)} + \\operatorname{f_{E}}{(V)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["add", 1, "Function('\\\\lambda')(Symbol('V', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\lambda')(Symbol('V', commutative=True))), Add(Function('\\\\lambda')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\lambda')(Symbol('V', commutative=True)))), sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Mul(Integer(2), Function('\\\\lambda')(Symbol('V', commutative=True)))), sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), Function('f_E')(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))), sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), Function('f_E')(Symbol('V', commutative=True)))))"], [["differentiate", 6, "Symbol('V', commutative=True)"], "Equality(Derivative(sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(sin(Add(Function('\\\\lambda')(Symbol('V', commutative=True)), Function('f_E')(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} = \\Psi_{nl} \\cos{(\\varphi)}, then obtain - 2 \\Psi_{nl} + \\int \\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} d\\varphi = - 2 \\Psi_{nl} + \\int \\Psi_{nl} \\cos{(\\varphi)} d\\varphi", "derivation": "\\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} = \\Psi_{nl} \\cos{(\\varphi)} and \\int \\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} d\\varphi = \\int \\Psi_{nl} \\cos{(\\varphi)} d\\varphi and - \\Psi_{nl} + \\int \\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} d\\varphi = - \\Psi_{nl} + \\int \\Psi_{nl} \\cos{(\\varphi)} d\\varphi and - 2 \\Psi_{nl} + \\int \\operatorname{P_{g}}{(\\varphi,\\Psi_{nl})} d\\varphi = - 2 \\Psi_{nl} + \\int \\Psi_{nl} \\cos{(\\varphi)} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["minus", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\delta,\\rho_b)} = e^{\\frac{\\rho_b}{\\delta}}, then derive \\int \\lambda{(\\delta,\\rho_b)} d\\delta = \\delta e^{\\frac{\\rho_b}{\\delta}} - \\rho_b \\operatorname{Ei}{(\\frac{\\rho_b}{\\delta})} + k, then obtain \\int \\lambda{(\\delta,\\rho_b)} d\\delta = \\delta \\lambda{(\\delta,\\rho_b)} - \\rho_b \\operatorname{Ei}{(\\frac{\\rho_b}{\\delta})} + k", "derivation": "\\lambda{(\\delta,\\rho_b)} = e^{\\frac{\\rho_b}{\\delta}} and \\int \\lambda{(\\delta,\\rho_b)} d\\delta = \\int e^{\\frac{\\rho_b}{\\delta}} d\\delta and \\int \\lambda{(\\delta,\\rho_b)} d\\delta = \\delta e^{\\frac{\\rho_b}{\\delta}} - \\rho_b \\operatorname{Ei}{(\\frac{\\rho_b}{\\delta})} + k and \\int \\lambda{(\\delta,\\rho_b)} d\\delta = \\delta \\lambda{(\\delta,\\rho_b)} - \\rho_b \\operatorname{Ei}{(\\frac{\\rho_b}{\\delta})} + k", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), exp(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Ei(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\lambda')(Symbol('\\\\delta', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Ei(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(\\ddot{x})} = \\sin{(\\ddot{x})}, then derive \\int \\mathbf{J}{(\\ddot{x})} d\\ddot{x} = \\mathbf{J}_P - \\cos{(\\ddot{x})}, then obtain (\\mathbf{J}_P - \\cos{(\\ddot{x})})^{\\ddot{x}} = (\\int \\sin{(\\ddot{x})} d\\ddot{x})^{\\ddot{x}}", "derivation": "\\mathbf{J}{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\int \\mathbf{J}{(\\ddot{x})} d\\ddot{x} = \\int \\sin{(\\ddot{x})} d\\ddot{x} and \\int \\mathbf{J}{(\\ddot{x})} d\\ddot{x} = \\mathbf{J}_P - \\cos{(\\ddot{x})} and (\\int \\mathbf{J}{(\\ddot{x})} d\\ddot{x})^{\\ddot{x}} = (\\int \\sin{(\\ddot{x})} d\\ddot{x})^{\\ddot{x}} and (\\mathbf{J}_P - \\cos{(\\ddot{x})})^{\\ddot{x}} = (\\int \\sin{(\\ddot{x})} d\\ddot{x})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(F_{c},H)} = F_{c} \\cos{(H)}, then obtain (\\int \\mathbb{I}{(F_{c},H)} \\cos{(H)} dF_{c})^{H} = (\\int F_{c} \\cos^{2}{(H)} dF_{c})^{H}", "derivation": "\\mathbb{I}{(F_{c},H)} = F_{c} \\cos{(H)} and \\mathbb{I}{(F_{c},H)} \\cos{(H)} = F_{c} \\cos^{2}{(H)} and \\int \\mathbb{I}{(F_{c},H)} \\cos{(H)} dF_{c} = \\int F_{c} \\cos^{2}{(H)} dF_{c} and (\\int \\mathbb{I}{(F_{c},H)} \\cos{(H)} dF_{c})^{H} = (\\int F_{c} \\cos^{2}{(H)} dF_{c})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('F_c', commutative=True), cos(Symbol('H', commutative=True))))"], [["times", 1, "cos(Symbol('H', commutative=True))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(cos(Symbol('H', commutative=True)), Integer(2))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Symbol('F_c', commutative=True), Pow(cos(Symbol('H', commutative=True)), Integer(2))), Tuple(Symbol('F_c', commutative=True))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Mul(Symbol('F_c', commutative=True), Pow(cos(Symbol('H', commutative=True)), Integer(2))), Tuple(Symbol('F_c', commutative=True))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given G{(\\phi_2,r)} = - \\phi_2 + \\log{(r)}, then obtain \\phi_2 - \\log{(r)} + \\log{(\\frac{\\int e^{G{(\\phi_2,r)}} d\\phi_2}{\\phi_2})} + \\frac{\\partial}{\\partial r} G{(\\phi_2,r)} = \\phi_2 - \\log{(r)} + \\log{(\\frac{\\int r e^{- \\phi_2} d\\phi_2}{\\phi_2})} + \\frac{\\partial}{\\partial r} G{(\\phi_2,r)}", "derivation": "G{(\\phi_2,r)} = - \\phi_2 + \\log{(r)} and e^{G{(\\phi_2,r)}} = r e^{- \\phi_2} and \\int e^{G{(\\phi_2,r)}} d\\phi_2 = \\int r e^{- \\phi_2} d\\phi_2 and \\frac{\\int e^{G{(\\phi_2,r)}} d\\phi_2}{\\phi_2} = \\frac{\\int r e^{- \\phi_2} d\\phi_2}{\\phi_2} and \\log{(\\frac{\\int e^{G{(\\phi_2,r)}} d\\phi_2}{\\phi_2})} = \\log{(\\frac{\\int r e^{- \\phi_2} d\\phi_2}{\\phi_2})} and \\phi_2 - \\log{(r)} + \\log{(\\frac{\\int e^{G{(\\phi_2,r)}} d\\phi_2}{\\phi_2})} + \\frac{\\partial}{\\partial r} G{(\\phi_2,r)} = \\phi_2 - \\log{(r)} + \\log{(\\frac{\\int r e^{- \\phi_2} d\\phi_2}{\\phi_2})} + \\frac{\\partial}{\\partial r} G{(\\phi_2,r)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), log(Symbol('r', commutative=True))))"], [["exp", 1], "Equality(exp(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(exp(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Symbol('r', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(exp(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('r', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["log", 4], "Equality(log(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(exp(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))), log(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('r', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), log(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), log(Symbol('r', commutative=True))), log(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(exp(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))), Derivative(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), log(Symbol('r', commutative=True))), log(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('r', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))))), Derivative(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(c)} = \\log{(e^{c})}, then obtain (\\int \\frac{d}{d c} (\\sigma_{p}{(c)} + e^{c})^{c} dc)^{c} = (\\int \\frac{d}{d c} (e^{c} + \\log{(e^{c})})^{c} dc)^{c}", "derivation": "\\sigma_{p}{(c)} = \\log{(e^{c})} and \\sigma_{p}{(c)} + e^{c} = e^{c} + \\log{(e^{c})} and (\\sigma_{p}{(c)} + e^{c})^{c} = (e^{c} + \\log{(e^{c})})^{c} and \\frac{d}{d c} (\\sigma_{p}{(c)} + e^{c})^{c} = \\frac{d}{d c} (e^{c} + \\log{(e^{c})})^{c} and \\int \\frac{d}{d c} (\\sigma_{p}{(c)} + e^{c})^{c} dc = \\int \\frac{d}{d c} (e^{c} + \\log{(e^{c})})^{c} dc and (\\int \\frac{d}{d c} (\\sigma_{p}{(c)} + e^{c})^{c} dc)^{c} = (\\int \\frac{d}{d c} (e^{c} + \\log{(e^{c})})^{c} dc)^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True))))"], [["add", 1, "exp(Symbol('c', commutative=True))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Add(exp(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Function('\\\\sigma_p')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Add(exp(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\sigma_p')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Add(exp(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Derivative(Pow(Add(Function('\\\\sigma_p')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Integral(Derivative(Pow(Add(exp(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))))"], [["power", 5, "Symbol('c', commutative=True)"], "Equality(Pow(Integral(Derivative(Pow(Add(Function('\\\\sigma_p')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Integral(Derivative(Pow(Add(exp(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\dot{\\mathbf{r}},\\varphi)} = \\dot{\\mathbf{r}} + \\varphi, then derive \\int \\varphi^{*}{(\\dot{\\mathbf{r}},\\varphi)} d\\varphi = \\dot{\\mathbf{r}} \\varphi + \\psi^* + \\frac{\\varphi^{2}}{2}, then obtain \\dot{\\mathbf{r}} \\varphi + \\psi^* + \\frac{\\varphi^{2}}{2} = \\int (\\dot{\\mathbf{r}} + \\varphi) d\\varphi", "derivation": "\\varphi^{*}{(\\dot{\\mathbf{r}},\\varphi)} = \\dot{\\mathbf{r}} + \\varphi and \\int \\varphi^{*}{(\\dot{\\mathbf{r}},\\varphi)} d\\varphi = \\int (\\dot{\\mathbf{r}} + \\varphi) d\\varphi and \\int \\varphi^{*}{(\\dot{\\mathbf{r}},\\varphi)} d\\varphi = \\dot{\\mathbf{r}} \\varphi + \\psi^* + \\frac{\\varphi^{2}}{2} and \\dot{\\mathbf{r}} \\varphi + \\psi^* + \\frac{\\varphi^{2}}{2} = \\int (\\dot{\\mathbf{r}} + \\varphi) d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\psi^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\psi^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given k{(U)} = \\sin{(\\cos{(U)})} and \\varepsilon_{0}{(f_{E})} = \\cos{(e^{f_{E}})}, then obtain \\varepsilon_{0}{(f_{E})} - \\iint \\sin{(\\cos{(U)})} dU dU = \\cos{(e^{f_{E}})} - \\iint \\sin{(\\cos{(U)})} dU dU", "derivation": "k{(U)} = \\sin{(\\cos{(U)})} and \\int k{(U)} dU = \\int \\sin{(\\cos{(U)})} dU and \\iint k{(U)} dU dU = \\iint \\sin{(\\cos{(U)})} dU dU and \\varepsilon_{0}{(f_{E})} = \\cos{(e^{f_{E}})} and \\varepsilon_{0}{(f_{E})} - \\iint k{(U)} dU dU = \\cos{(e^{f_{E}})} - \\iint k{(U)} dU dU and \\varepsilon_{0}{(f_{E})} - \\iint \\sin{(\\cos{(U)})} dU dU = \\cos{(e^{f_{E}})} - \\iint \\sin{(\\cos{(U)})} dU dU", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('U', commutative=True)), sin(cos(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), cos(exp(Symbol('f_E', commutative=True))))"], [["minus", 4, "Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Add(cos(exp(Symbol('f_E', commutative=True))), Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(sin(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Add(cos(exp(Symbol('f_E', commutative=True))), Mul(Integer(-1), Integral(sin(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given k{(H,L)} = \\frac{\\sin{(H)}}{L}, then obtain \\frac{2 L k{(H,L)}}{H} = L (\\frac{k{(H,L)}}{H} + \\frac{\\sin{(H)}}{H L})", "derivation": "k{(H,L)} = \\frac{\\sin{(H)}}{L} and \\frac{k{(H,L)}}{H} = \\frac{\\sin{(H)}}{H L} and \\frac{2 k{(H,L)}}{H} = \\frac{k{(H,L)}}{H} + \\frac{\\sin{(H)}}{H L} and \\frac{2 L k{(H,L)}}{H} = L (\\frac{k{(H,L)}}{H} + \\frac{\\sin{(H)}}{H L})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True))), Add(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True)))))"], [["times", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('L', commutative=True), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Add(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('k')(Symbol('H', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(I,c)} = I c, then obtain 0 = I c \\mathbf{J}_M{(I,c)} - \\mathbf{J}_M^{2}{(I,c)}", "derivation": "\\mathbf{J}_M{(I,c)} = I c and \\mathbf{J}_M^{2}{(I,c)} = I c \\mathbf{J}_M{(I,c)} and - I c \\mathbf{J}_M{(I,c)} + \\mathbf{J}_M^{2}{(I,c)} = 0 and 0 = I c \\mathbf{J}_M{(I,c)} - \\mathbf{J}_M^{2}{(I,c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('c', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Integer(2)), Mul(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True))))"], [["minus", 2, "Mul(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('c', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Pow(Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Integer(2))), Integer(0))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('c', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Pow(Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Integer(2)))"], "Equality(Integer(0), Add(Mul(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given T{(\\mathbf{P},Q)} = \\cos{(Q + \\mathbf{P})}, then obtain - \\mathbf{P} = - \\frac{\\mathbf{P} \\cos{(Q + \\mathbf{P})}}{T{(\\mathbf{P},Q)}}", "derivation": "T{(\\mathbf{P},Q)} = \\cos{(Q + \\mathbf{P})} and 1 = \\frac{\\cos{(Q + \\mathbf{P})}}{T{(\\mathbf{P},Q)}} and -1 = - \\frac{\\cos{(Q + \\mathbf{P})}}{T{(\\mathbf{P},Q)}} and - \\mathbf{P} = - \\frac{\\mathbf{P} \\cos{(Q + \\mathbf{P})}}{T{(\\mathbf{P},Q)}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Q', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 1, "Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["times", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True), Pow(Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given u{(\\hat{H}_l,v)} = \\hat{H}_l^{v} and k{(\\hat{H}_l,v)} = \\hat{H}_l^{v} + u{(\\hat{H}_l,v)}, then obtain k{(\\hat{H}_l,v)} = 2 \\hat{H}_l^{v}", "derivation": "u{(\\hat{H}_l,v)} = \\hat{H}_l^{v} and \\hat{H}_l^{v} + u{(\\hat{H}_l,v)} = 2 \\hat{H}_l^{v} and k{(\\hat{H}_l,v)} = \\hat{H}_l^{v} + u{(\\hat{H}_l,v)} and k{(\\hat{H}_l,v)} = 2 \\hat{H}_l^{v}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}}, then obtain \\frac{e^{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}}}{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}} = \\frac{e^{e^{\\dot{\\mathbf{r}}}}}{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}}", "derivation": "\\tilde{g}^*{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and e^{\\tilde{g}^*{(\\dot{\\mathbf{r}})}} = e^{e^{\\dot{\\mathbf{r}}}} and e^{\\tilde{g}^*{(\\dot{\\mathbf{r}})}} = e^{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}} and e^{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}} = e^{e^{\\dot{\\mathbf{r}}}} and \\frac{e^{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}}}{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}} = \\frac{e^{e^{\\dot{\\mathbf{r}}}}}{\\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(exp(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), exp(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["divide", 5, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(U,\\mu,A_{z})} = (U + \\mu)^{A_{z}}, then obtain \\operatorname{m_{s}}{(U,\\mu,A_{z})} + \\frac{(U + \\mu)^{A_{z}}}{U + \\mu} = (U + \\mu)^{A_{z}} + \\frac{(U + \\mu)^{A_{z}}}{U + \\mu}", "derivation": "\\operatorname{m_{s}}{(U,\\mu,A_{z})} = (U + \\mu)^{A_{z}} and \\frac{\\operatorname{m_{s}}{(U,\\mu,A_{z})}}{U + \\mu} = \\frac{(U + \\mu)^{A_{z}}}{U + \\mu} and \\operatorname{m_{s}}{(U,\\mu,A_{z})} + \\frac{\\operatorname{m_{s}}{(U,\\mu,A_{z})}}{U + \\mu} = (U + \\mu)^{A_{z}} + \\frac{\\operatorname{m_{s}}{(U,\\mu,A_{z})}}{U + \\mu} and \\operatorname{m_{s}}{(U,\\mu,A_{z})} + \\frac{(U + \\mu)^{A_{z}}}{U + \\mu} = (U + \\mu)^{A_{z}} + \\frac{(U + \\mu)^{A_{z}}}{U + \\mu}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True)))"], [["divide", 1, "Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True))))"], [["add", 1, "Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)))"], "Equality(Add(Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)))), Add(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('m_s')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('A_z', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True)))), Add(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(\\rho_b)} = \\log{(\\sin{(\\rho_b)})}, then obtain (\\log{(\\sin{(\\rho_b)})} + \\log{(\\sin{(\\rho_b)})}^{\\rho_b}) \\mu_{0}{(\\rho_b)} = (\\log{(\\sin{(\\rho_b)})} + \\log{(\\sin{(\\rho_b)})}^{\\rho_b}) \\log{(\\sin{(\\rho_b)})}", "derivation": "\\mu_{0}{(\\rho_b)} = \\log{(\\sin{(\\rho_b)})} and \\mu_{0}^{\\rho_b}{(\\rho_b)} = \\log{(\\sin{(\\rho_b)})}^{\\rho_b} and (\\mu_{0}^{\\rho_b}{(\\rho_b)} + \\log{(\\sin{(\\rho_b)})}) \\mu_{0}{(\\rho_b)} = (\\mu_{0}^{\\rho_b}{(\\rho_b)} + \\log{(\\sin{(\\rho_b)})}) \\log{(\\sin{(\\rho_b)})} and (\\log{(\\sin{(\\rho_b)})} + \\log{(\\sin{(\\rho_b)})}^{\\rho_b}) \\mu_{0}{(\\rho_b)} = (\\log{(\\sin{(\\rho_b)})} + \\log{(\\sin{(\\rho_b)})}^{\\rho_b}) \\log{(\\sin{(\\rho_b)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True)), log(sin(Symbol('\\\\rho_b', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(log(sin(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "Add(Pow(Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), log(sin(Symbol('\\\\rho_b', commutative=True))))"], "Equality(Mul(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), log(sin(Symbol('\\\\rho_b', commutative=True)))), Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True))), Mul(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), log(sin(Symbol('\\\\rho_b', commutative=True)))), log(sin(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(log(sin(Symbol('\\\\rho_b', commutative=True))), Pow(log(sin(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True))), Function('\\\\mu_0')(Symbol('\\\\rho_b', commutative=True))), Mul(Add(log(sin(Symbol('\\\\rho_b', commutative=True))), Pow(log(sin(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True))), log(sin(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\Psi)} = \\sin{(\\Psi)}, then obtain \\frac{d}{d \\Psi} \\operatorname{a^{\\dagger}}{(\\Psi)} \\int \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} d\\Psi = \\frac{d}{d \\Psi} \\operatorname{a^{\\dagger}}{(\\Psi)} \\int \\sin^{\\Psi}{(\\Psi)} d\\Psi", "derivation": "\\operatorname{a^{\\dagger}}{(\\Psi)} = \\sin{(\\Psi)} and \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} = \\sin^{\\Psi}{(\\Psi)} and \\int \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} d\\Psi = \\int \\sin^{\\Psi}{(\\Psi)} d\\Psi and \\sin{(\\Psi)} \\int \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} d\\Psi = \\sin{(\\Psi)} \\int \\sin^{\\Psi}{(\\Psi)} d\\Psi and \\frac{d}{d \\Psi} \\sin{(\\Psi)} \\int \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} d\\Psi = \\frac{d}{d \\Psi} \\sin{(\\Psi)} \\int \\sin^{\\Psi}{(\\Psi)} d\\Psi and \\frac{d}{d \\Psi} \\operatorname{a^{\\dagger}}{(\\Psi)} \\int \\operatorname{a^{\\dagger}}^{\\Psi}{(\\Psi)} d\\Psi = \\frac{d}{d \\Psi} \\operatorname{a^{\\dagger}}{(\\Psi)} \\int \\sin^{\\Psi}{(\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 3, "sin(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\Psi', commutative=True)), Integral(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(sin(Symbol('\\\\Psi', commutative=True)), Integral(Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(sin(Symbol('\\\\Psi', commutative=True)), Integral(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(sin(Symbol('\\\\Psi', commutative=True)), Integral(Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Integral(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\Psi', commutative=True)), Integral(Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(g,r)} = g + r, then derive (\\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr)^{r} = (\\frac{\\partial}{\\partial r} (\\Omega + g r + \\frac{r^{2}}{2}))^{r}, then obtain (\\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr)^{r} = (g + r)^{r}", "derivation": "\\psi^{*}{(g,r)} = g + r and \\int \\psi^{*}{(g,r)} dr = \\int (g + r) dr and \\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr = \\frac{\\partial}{\\partial r} \\int (g + r) dr and (\\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr)^{r} = (\\frac{\\partial}{\\partial r} \\int (g + r) dr)^{r} and (\\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr)^{r} = (\\frac{\\partial}{\\partial r} (\\Omega + g r + \\frac{r^{2}}{2}))^{r} and (\\frac{\\partial}{\\partial r} \\int \\psi^{*}{(g,r)} dr)^{r} = (g + r)^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Add(Symbol('g', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Integral(Add(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Derivative(Integral(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Add(Symbol('g', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\nabla{(C_{1},\\Omega)} = \\frac{\\Omega}{C_{1}}, then obtain - \\cos{(\\frac{\\Omega}{C_{1}})} - \\frac{\\nabla^{2}{(C_{1},\\Omega)}}{C_{1}} = - \\cos{(\\frac{\\Omega}{C_{1}})} - \\frac{\\Omega \\nabla{(C_{1},\\Omega)}}{C_{1}^{2}}", "derivation": "\\nabla{(C_{1},\\Omega)} = \\frac{\\Omega}{C_{1}} and - \\nabla{(C_{1},\\Omega)} = - \\frac{\\Omega}{C_{1}} and - \\nabla^{2}{(C_{1},\\Omega)} = - \\frac{\\Omega \\nabla{(C_{1},\\Omega)}}{C_{1}} and - \\frac{\\nabla^{2}{(C_{1},\\Omega)}}{C_{1}} = - \\frac{\\Omega \\nabla{(C_{1},\\Omega)}}{C_{1}^{2}} and - \\cos{(\\frac{\\Omega}{C_{1}})} - \\frac{\\nabla^{2}{(C_{1},\\Omega)}}{C_{1}} = - \\cos{(\\frac{\\Omega}{C_{1}})} - \\frac{\\Omega \\nabla{(C_{1},\\Omega)}}{C_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True), Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["times", 3, "Pow(Symbol('C_1', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-2)), Symbol('\\\\Omega', commutative=True), Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["minus", 4, "cos(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-2)), Symbol('\\\\Omega', commutative=True), Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(E,\\mathbf{D},\\Psi^{\\dagger})} = \\frac{E + \\mathbf{D}}{\\Psi^{\\dagger}}, then obtain - E - \\mathbf{D} - \\frac{E + \\mathbf{D}}{\\Psi^{\\dagger}} = - E - \\mathbf{D} + \\frac{- E - \\mathbf{D}}{\\Psi^{\\dagger}}", "derivation": "\\rho_{f}{(E,\\mathbf{D},\\Psi^{\\dagger})} = \\frac{E + \\mathbf{D}}{\\Psi^{\\dagger}} and - \\rho_{f}{(E,\\mathbf{D},\\Psi^{\\dagger})} = - \\frac{E + \\mathbf{D}}{\\Psi^{\\dagger}} and - \\rho_{f}{(E,\\mathbf{D},\\Psi^{\\dagger})} = \\frac{- E - \\mathbf{D}}{\\Psi^{\\dagger}} and - E - \\mathbf{D} - \\rho_{f}{(E,\\mathbf{D},\\Psi^{\\dagger})} = - E - \\mathbf{D} + \\frac{- E - \\mathbf{D}}{\\Psi^{\\dagger}} and - E - \\mathbf{D} - \\frac{E + \\mathbf{D}}{\\Psi^{\\dagger}} = - E - \\mathbf{D} + \\frac{- E - \\mathbf{D}}{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\hbar)} = \\cos{(\\hbar)}, then obtain \\log{(\\frac{\\partial}{\\partial v_{t}} (\\operatorname{r_{0}}{(\\hbar)} - \\sin{(v_{t})}))} \\cos^{\\hbar}{(\\hbar)} = \\log{(\\frac{\\partial}{\\partial v_{t}} (- \\sin{(v_{t})} + \\cos{(\\hbar)}))} \\cos^{\\hbar}{(\\hbar)}", "derivation": "\\operatorname{r_{0}}{(\\hbar)} = \\cos{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} - \\sin{(v_{t})} = - \\sin{(v_{t})} + \\cos{(\\hbar)} and \\frac{\\partial}{\\partial v_{t}} (\\operatorname{r_{0}}{(\\hbar)} - \\sin{(v_{t})}) = \\frac{\\partial}{\\partial v_{t}} (- \\sin{(v_{t})} + \\cos{(\\hbar)}) and \\log{(\\frac{\\partial}{\\partial v_{t}} (\\operatorname{r_{0}}{(\\hbar)} - \\sin{(v_{t})}))} = \\log{(\\frac{\\partial}{\\partial v_{t}} (- \\sin{(v_{t})} + \\cos{(\\hbar)}))} and \\log{(\\frac{\\partial}{\\partial v_{t}} (\\operatorname{r_{0}}{(\\hbar)} - \\sin{(v_{t})}))} \\cos^{\\hbar}{(\\hbar)} = \\log{(\\frac{\\partial}{\\partial v_{t}} (- \\sin{(v_{t})} + \\cos{(\\hbar)}))} \\cos^{\\hbar}{(\\hbar)}", "srepr_derivation": [["get_premise", "Equality(Function('r_0')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "sin(Symbol('v_t', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Add(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), log(Derivative(Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"], [["times", 4, "Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(log(Derivative(Add(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), Mul(log(Derivative(Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(t)} = \\int e^{t} dt, then derive \\frac{\\mathbb{I}{(t)}}{\\tilde{g}^* + e^{t}} = 1, then derive \\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} = 1, then obtain (\\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} + \\int e^{t} dt)^{C_{2}} + 1 = (\\int e^{t} dt + 1)^{C_{2}} + 1", "derivation": "\\mathbb{I}{(t)} = \\int e^{t} dt and \\mathbb{I}{(t)} e^{t} = e^{t} \\int e^{t} dt and \\frac{\\mathbb{I}{(t)}}{\\int e^{t} dt} = 1 and \\frac{\\mathbb{I}{(t)}}{\\tilde{g}^* + e^{t}} = 1 and \\frac{\\int e^{t} dt}{\\tilde{g}^* + e^{t}} = 1 and \\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} = 1 and \\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} + \\int e^{t} dt = \\int e^{t} dt + 1 and (\\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} + \\int e^{t} dt)^{C_{2}} = (\\int e^{t} dt + 1)^{C_{2}} and (\\frac{C_{2} + e^{t}}{\\tilde{g}^* + e^{t}} + \\int e^{t} dt)^{C_{2}} + 1 = (\\int e^{t} dt + 1)^{C_{2}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('t', commutative=True)), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 1, "exp(Symbol('t', commutative=True))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))), Mul(exp(Symbol('t', commutative=True)), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["divide", 2, "Mul(exp(Symbol('t', commutative=True)), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('t', commutative=True)), Pow(Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('t', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1)), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Integer(1))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('C_2', commutative=True), exp(Symbol('t', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1))), Integer(1))"], [["add", 6, "Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('C_2', commutative=True), exp(Symbol('t', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(1)))"], [["power", 7, "Symbol('C_2', commutative=True)"], "Equality(Pow(Add(Mul(Add(Symbol('C_2', commutative=True), exp(Symbol('t', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Symbol('C_2', commutative=True)), Pow(Add(Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(1)), Symbol('C_2', commutative=True)))"], [["minus", 8, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Add(Symbol('C_2', commutative=True), exp(Symbol('t', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('t', commutative=True))), Integer(-1))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Symbol('C_2', commutative=True)), Integer(1)), Add(Pow(Add(Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(1)), Symbol('C_2', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(v_{1})} = e^{v_{1}} and \\hat{H}{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})}, then derive 2 \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})} = e^{v_{1}} + \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})}, then obtain 2 \\hat{H}{(v_{1})} = \\hat{H}{(v_{1})} + \\operatorname{v_{x}}{(v_{1})}", "derivation": "\\operatorname{v_{x}}{(v_{1})} = e^{v_{1}} and 2 \\operatorname{v_{x}}{(v_{1})} = \\operatorname{v_{x}}{(v_{1})} + e^{v_{1}} and \\frac{d}{d v_{1}} 2 \\operatorname{v_{x}}{(v_{1})} = \\frac{d}{d v_{1}} (\\operatorname{v_{x}}{(v_{1})} + e^{v_{1}}) and 2 \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})} = e^{v_{1}} + \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})} and \\hat{H}{(v_{1})} = \\frac{d}{d v_{1}} \\operatorname{v_{x}}{(v_{1})} and 2 \\hat{H}{(v_{1})} = \\hat{H}{(v_{1})} + e^{v_{1}} and 2 \\hat{H}{(v_{1})} = \\hat{H}{(v_{1})} + \\operatorname{v_{x}}{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["add", 1, "Function('v_x')(Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(2), Function('v_x')(Symbol('v_1', commutative=True))), Add(Function('v_x')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('v_x')(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Function('v_x')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('v_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(exp(Symbol('v_1', commutative=True)), Derivative(Function('v_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('v_1', commutative=True)), Derivative(Function('v_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('v_1', commutative=True))), Add(Function('\\\\hat{H}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('v_1', commutative=True))), Add(Function('\\\\hat{H}')(Symbol('v_1', commutative=True)), Function('v_x')(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(f^{\\prime})} = \\cos{(f^{\\prime})} and g{(f^{\\prime})} = \\int \\cos{(f^{\\prime})} df^{\\prime}, then derive \\int \\mathbf{p}{(f^{\\prime})} df^{\\prime} = m_{s} + \\sin{(f^{\\prime})}, then obtain m_{s} + \\sin{(f^{\\prime})} = g{(f^{\\prime})}", "derivation": "\\mathbf{p}{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\int \\mathbf{p}{(f^{\\prime})} df^{\\prime} = \\int \\cos{(f^{\\prime})} df^{\\prime} and \\int \\mathbf{p}{(f^{\\prime})} df^{\\prime} = m_{s} + \\sin{(f^{\\prime})} and m_{s} + \\sin{(f^{\\prime})} = \\int \\cos{(f^{\\prime})} df^{\\prime} and g{(f^{\\prime})} = \\int \\cos{(f^{\\prime})} df^{\\prime} and m_{s} + \\sin{(f^{\\prime})} = g{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('m_s', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('m_s', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True))), Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('g')(Symbol('f^{\\\\prime}', commutative=True)), Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('m_s', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True))), Function('g')(Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(A_{2},p)} = - A_{2} + e^{p}, then derive \\frac{\\partial}{\\partial p} \\operatorname{n_{1}}{(A_{2},p)} = e^{p}, then obtain - A_{2} + \\mathbf{S} + e^{p} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) = - A_{2} + \\mathbf{g} + e^{p} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p})", "derivation": "\\operatorname{n_{1}}{(A_{2},p)} = - A_{2} + e^{p} and \\frac{\\partial}{\\partial p} \\operatorname{n_{1}}{(A_{2},p)} = \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) and \\frac{\\partial}{\\partial p} \\operatorname{n_{1}}{(A_{2},p)} = e^{p} and \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) = e^{p} and \\int \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) dp = \\int e^{p} dp and - A_{2} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) + \\int \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) dp = - A_{2} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) + \\int e^{p} dp and - A_{2} + \\mathbf{S} + e^{p} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p}) = - A_{2} + \\mathbf{g} + e^{p} + \\frac{\\partial}{\\partial p} (- A_{2} + e^{p})", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('A_2', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('A_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('A_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), exp(Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), exp(Symbol('p', commutative=True)))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('p', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('p', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\mathbb{I},\\mathbf{g})} = - \\mathbb{I} + \\mathbf{g}, then derive \\int S{(\\mathbb{I},\\mathbf{g})} d\\mathbb{I} = \\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g}, then obtain \\frac{\\partial}{\\partial \\eta} (\\hat{p}_0 - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g}) = \\frac{\\partial}{\\partial \\eta} (\\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g})", "derivation": "S{(\\mathbb{I},\\mathbf{g})} = - \\mathbb{I} + \\mathbf{g} and \\int S{(\\mathbb{I},\\mathbf{g})} d\\mathbb{I} = \\int (- \\mathbb{I} + \\mathbf{g}) d\\mathbb{I} and \\int S{(\\mathbb{I},\\mathbf{g})} d\\mathbb{I} = \\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g} and \\int (- \\mathbb{I} + \\mathbf{g}) d\\mathbb{I} = \\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g} and \\frac{\\partial}{\\partial \\eta} \\int (- \\mathbb{I} + \\mathbf{g}) d\\mathbb{I} = \\frac{\\partial}{\\partial \\eta} (\\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g}) and \\frac{\\partial}{\\partial \\eta} (\\hat{p}_0 - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g}) = \\frac{\\partial}{\\partial \\eta} (\\eta - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\mathbf{g})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('S')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(\\psi^*)} = \\sin{(\\psi^*)} and \\operatorname{g_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain \\sin{(\\psi^*)} = - g{(\\psi^*)} + 2 \\sin{(\\psi^*)}", "derivation": "g{(\\psi^*)} = \\sin{(\\psi^*)} and \\operatorname{g_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)} and - g{(\\psi^*)} + \\operatorname{g_{\\varepsilon}}{(\\psi^*)} + \\sin{(\\psi^*)} = - g{(\\psi^*)} + 2 \\sin{(\\psi^*)} and \\operatorname{g_{\\varepsilon}}{(\\psi^*)} = g{(\\psi^*)} and \\sin{(\\psi^*)} = - g{(\\psi^*)} + 2 \\sin{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "Add(Function('g')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('g')(Symbol('\\\\psi^*', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Function('g')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Function('g')(Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(sin(Symbol('\\\\psi^*', commutative=True)), Add(Mul(Integer(-1), Function('g')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(x)} = x, then obtain \\iint \\frac{\\frac{d}{d x} \\mathbf{S}{(x)}}{x} dx dx = \\iint \\frac{\\frac{d}{d x} x}{x} dx dx", "derivation": "\\mathbf{S}{(x)} = x and \\frac{d}{d x} \\mathbf{S}{(x)} = \\frac{d}{d x} x and \\frac{\\frac{d}{d x} \\mathbf{S}{(x)}}{x} = \\frac{\\frac{d}{d x} x}{x} and \\int \\frac{\\frac{d}{d x} \\mathbf{S}{(x)}}{x} dx = \\int \\frac{\\frac{d}{d x} x}{x} dx and \\iint \\frac{\\frac{d}{d x} \\mathbf{S}{(x)}}{x} dx dx = \\iint \\frac{\\frac{d}{d x} x}{x} dx dx", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Symbol('x', commutative=True))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True))))"], [["integrate", 4, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\varphi{(n_{2})} = \\sin{(n_{2})}, then obtain \\varphi^{n_{2}}{(n_{2})} - \\cos{(\\sin^{n_{2}}{(n_{2})})} = \\sin^{n_{2}}{(n_{2})} - \\cos{(\\sin^{n_{2}}{(n_{2})})}", "derivation": "\\varphi{(n_{2})} = \\sin{(n_{2})} and \\varphi^{n_{2}}{(n_{2})} = \\sin^{n_{2}}{(n_{2})} and \\cos{(\\varphi^{n_{2}}{(n_{2})})} = \\cos{(\\sin^{n_{2}}{(n_{2})})} and \\varphi^{n_{2}}{(n_{2})} - \\cos{(\\varphi^{n_{2}}{(n_{2})})} = \\sin^{n_{2}}{(n_{2})} - \\cos{(\\varphi^{n_{2}}{(n_{2})})} and \\varphi^{n_{2}}{(n_{2})} - \\cos{(\\sin^{n_{2}}{(n_{2})})} = \\sin^{n_{2}}{(n_{2})} - \\cos{(\\sin^{n_{2}}{(n_{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), cos(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"], [["minus", 2, "cos(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], "Equality(Add(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))), Add(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))), Add(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(z)} = e^{z}, then derive \\int \\operatorname{F_{H}}{(z)} dz = \\mathbf{F} + e^{z}, then obtain ((\\int \\operatorname{F_{H}}{(z)} dz)^{z})^{\\mathbf{F}} = ((\\mathbf{F} + \\operatorname{F_{H}}{(z)})^{z})^{\\mathbf{F}}", "derivation": "\\operatorname{F_{H}}{(z)} = e^{z} and \\int \\operatorname{F_{H}}{(z)} dz = \\int e^{z} dz and \\int \\operatorname{F_{H}}{(z)} dz = \\mathbf{F} + e^{z} and (\\int \\operatorname{F_{H}}{(z)} dz)^{z} = (\\int e^{z} dz)^{z} and \\int e^{z} dz = \\mathbf{F} + e^{z} and (\\int \\operatorname{F_{H}}{(z)} dz)^{z} = (\\mathbf{F} + e^{z})^{z} and (\\int \\operatorname{F_{H}}{(z)} dz)^{z} = (\\mathbf{F} + \\operatorname{F_{H}}{(z)})^{z} and ((\\int \\operatorname{F_{H}}{(z)} dz)^{z})^{\\mathbf{F}} = ((\\mathbf{F} + \\operatorname{F_{H}}{(z)})^{z})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('z', commutative=True))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('F_H')(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["power", 7, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Pow(Integral(Function('F_H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('F_H')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given z{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain - \\mathbf{M} + z{(\\mathbf{M})} + e^{\\mathbf{M}} = - \\mathbf{M} + 2 e^{\\mathbf{M}}", "derivation": "z{(\\mathbf{M})} = e^{\\mathbf{M}} and - \\mathbf{M} + z{(\\mathbf{M})} = - \\mathbf{M} + e^{\\mathbf{M}} and - \\mathbf{M} + 2 z{(\\mathbf{M})} = - \\mathbf{M} + z{(\\mathbf{M})} + e^{\\mathbf{M}} and - \\mathbf{M} + 2 z{(\\mathbf{M})} = - \\mathbf{M} + 2 e^{\\mathbf{M}} and - \\mathbf{M} + z{(\\mathbf{M})} + e^{\\mathbf{M}} = - \\mathbf{M} + 2 e^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given v{(\\phi_1,i)} = - \\phi_1 + \\sin{(i)}, then obtain 1 - \\frac{\\partial}{\\partial i} v{(\\phi_1,i)} = 1 - \\cos{(i)}", "derivation": "v{(\\phi_1,i)} = - \\phi_1 + \\sin{(i)} and - i + v{(\\phi_1,i)} = - \\phi_1 - i + \\sin{(i)} and i - v{(\\phi_1,i)} = \\phi_1 + i - \\sin{(i)} and \\frac{\\partial}{\\partial i} (i - v{(\\phi_1,i)}) = \\frac{\\partial}{\\partial i} (\\phi_1 + i - \\sin{(i)}) and 1 - \\frac{\\partial}{\\partial i} v{(\\phi_1,i)} = 1 - \\cos{(i)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), sin(Symbol('i', commutative=True))))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True)))), Add(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True), Mul(Integer(-1), sin(Symbol('i', commutative=True)))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), cos(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{p},F_{x})} = F_{x} \\hat{p}, then obtain - \\frac{\\int \\operatorname{A_{y}}{(\\hat{p},F_{x})} dF_{x}}{\\hat{p}} = - \\frac{\\int F_{x} \\hat{p} dF_{x}}{\\hat{p}}", "derivation": "\\operatorname{A_{y}}{(\\hat{p},F_{x})} = F_{x} \\hat{p} and \\int \\operatorname{A_{y}}{(\\hat{p},F_{x})} dF_{x} = \\int F_{x} \\hat{p} dF_{x} and \\frac{\\int \\operatorname{A_{y}}{(\\hat{p},F_{x})} dF_{x}}{\\hat{p}} = \\frac{\\int F_{x} \\hat{p} dF_{x}}{\\hat{p}} and - \\frac{\\int \\operatorname{A_{y}}{(\\hat{p},F_{x})} dF_{x}}{\\hat{p}} = - \\frac{\\int F_{x} \\hat{p} dF_{x}}{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["divide", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Integral(Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Integral(Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given s{(g,\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + g, then obtain (- 2 \\Psi^{\\dagger} + 2 g - s{(g,\\Psi^{\\dagger})}) (\\Psi^{\\dagger} - g + s{(g,\\Psi^{\\dagger})}) = (\\Psi^{\\dagger} - g + s{(g,\\Psi^{\\dagger})}) s{(g,\\Psi^{\\dagger})}", "derivation": "s{(g,\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + g and 0 = - \\Psi^{\\dagger} + g - s{(g,\\Psi^{\\dagger})} and - \\Psi^{\\dagger} = - 2 \\Psi^{\\dagger} + g - s{(g,\\Psi^{\\dagger})} and s{(g,\\Psi^{\\dagger})} = - 2 \\Psi^{\\dagger} + 2 g - s{(g,\\Psi^{\\dagger})} and (- \\Psi^{\\dagger} + g - s{(g,\\Psi^{\\dagger})}) s{(g,\\Psi^{\\dagger})} = (- 2 \\Psi^{\\dagger} + 2 g - s{(g,\\Psi^{\\dagger})}) (- \\Psi^{\\dagger} + g - s{(g,\\Psi^{\\dagger})}) and (- 2 \\Psi^{\\dagger} + 2 g - s{(g,\\Psi^{\\dagger})}) (\\Psi^{\\dagger} - g + s{(g,\\Psi^{\\dagger})}) = (\\Psi^{\\dagger} - g + s{(g,\\Psi^{\\dagger})}) s{(g,\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True)))"], [["minus", 1, "Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Mul(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('s')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\dot{z})} = e^{\\dot{z}}, then obtain \\frac{d}{d \\dot{z}} \\dot{z} \\operatorname{E_{x}}{(\\dot{z})} - 1 = \\frac{d}{d \\dot{z}} \\dot{z} e^{\\dot{z}} - 1", "derivation": "\\operatorname{E_{x}}{(\\dot{z})} = e^{\\dot{z}} and \\dot{z} \\operatorname{E_{x}}{(\\dot{z})} = \\dot{z} e^{\\dot{z}} and \\frac{d}{d \\dot{z}} \\dot{z} \\operatorname{E_{x}}{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\dot{z} e^{\\dot{z}} and \\frac{d}{d \\dot{z}} \\dot{z} \\operatorname{E_{x}}{(\\dot{z})} - 1 = \\frac{d}{d \\dot{z}} \\dot{z} e^{\\dot{z}} - 1", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('E_x')(Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Function('E_x')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Function('E_x')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{H}{(B)} = \\cos{(B)}, then derive B (B \\frac{d}{d B} \\mathbf{H}{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)} = B (- B \\sin{(B)} + \\cos{(B)}) \\mathbf{H}{(B)}, then obtain B (B \\frac{d}{d B} \\mathbf{H}{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)} = B (- B \\sin{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)}", "derivation": "\\mathbf{H}{(B)} = \\cos{(B)} and B \\mathbf{H}{(B)} = B \\cos{(B)} and \\frac{d}{d B} B \\mathbf{H}{(B)} = \\frac{d}{d B} B \\cos{(B)} and B \\mathbf{H}{(B)} \\frac{d}{d B} B \\mathbf{H}{(B)} = B \\mathbf{H}{(B)} \\frac{d}{d B} B \\cos{(B)} and B (B \\frac{d}{d B} \\mathbf{H}{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)} = B (- B \\sin{(B)} + \\cos{(B)}) \\mathbf{H}{(B)} and B (B \\frac{d}{d B} \\mathbf{H}{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)} = B (- B \\sin{(B)} + \\mathbf{H}{(B)}) \\mathbf{H}{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), cos(Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["times", 3, "Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True)))"], "Equality(Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('B', commutative=True), Add(Mul(Symbol('B', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True), sin(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('B', commutative=True), Add(Mul(Symbol('B', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True), sin(Symbol('B', commutative=True))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))), Function('\\\\mathbf{H}')(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(b,\\Psi)} = \\frac{\\log{(\\Psi)}}{b} and \\mu{(\\Psi)} = \\log{(\\Psi)}, then obtain \\int (0^{\\Psi})^{b} d\\Psi = \\int 1 d\\Psi", "derivation": "\\sigma_{x}{(b,\\Psi)} = \\frac{\\log{(\\Psi)}}{b} and \\mu{(\\Psi)} = \\log{(\\Psi)} and 0 = - \\sigma_{x}{(b,\\Psi)} + \\frac{\\log{(\\Psi)}}{b} and 0^{\\Psi} = (- \\sigma_{x}{(b,\\Psi)} + \\frac{\\log{(\\Psi)}}{b})^{\\Psi} and 0^{\\Psi} = (- \\sigma_{x}{(b,\\Psi)} + \\frac{\\mu{(\\Psi)}}{b})^{\\Psi} and (0^{\\Psi})^{b} = ((- \\sigma_{x}{(b,\\Psi)} + \\frac{\\mu{(\\Psi)}}{b})^{\\Psi})^{b} and ((- \\sigma_{x}{(b,\\Psi)} + \\frac{\\log{(\\Psi)}}{b})^{\\Psi})^{b} = 1 and (0^{\\Psi})^{b} = 1 and \\int (0^{\\Psi})^{b} d\\Psi = \\int 1 d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True)))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('b', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True)), Symbol('b', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Symbol('b', commutative=True)), Integer(1))"], [["integrate", 8, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Pow(Pow(Integer(0), Symbol('\\\\Psi', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(a,c)} = a c, then derive - \\frac{a (- \\frac{\\frac{\\partial}{\\partial a} \\sigma_{x}{(a,c)}}{a} + \\frac{\\sigma_{x}{(a,c)}}{a^{2}})}{\\sigma_{x}{(a,c)}} = 0, then obtain \\frac{\\partial}{\\partial c} - \\frac{a (- \\frac{\\frac{\\partial}{\\partial a} \\sigma_{x}{(a,c)}}{a} + \\frac{\\sigma_{x}{(a,c)}}{a^{2}})}{\\sigma_{x}{(a,c)}} = \\frac{d}{d c} 0", "derivation": "\\sigma_{x}{(a,c)} = a c and \\frac{\\sigma_{x}{(a,c)}}{a} = c and - \\frac{\\sigma_{x}{(a,c)}}{a} = - c and \\log{(- \\frac{\\sigma_{x}{(a,c)}}{a})} = \\log{(- c)} and \\frac{\\partial}{\\partial a} \\log{(- \\frac{\\sigma_{x}{(a,c)}}{a})} = \\frac{d}{d a} \\log{(- c)} and - \\frac{a (- \\frac{\\frac{\\partial}{\\partial a} \\sigma_{x}{(a,c)}}{a} + \\frac{\\sigma_{x}{(a,c)}}{a^{2}})}{\\sigma_{x}{(a,c)}} = 0 and \\frac{\\partial}{\\partial c} - \\frac{a (- \\frac{\\frac{\\partial}{\\partial a} \\sigma_{x}{(a,c)}}{a} + \\frac{\\sigma_{x}{(a,c)}}{a^{2}})}{\\sigma_{x}{(a,c)}} = \\frac{d}{d c} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('a', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)))"], [["log", 3], "Equality(log(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)))), log(Mul(Integer(-1), Symbol('c', commutative=True))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(log(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(log(Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Derivative(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)))), Pow(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Integer(0))"], [["differentiate", 6, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('a', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Derivative(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)))), Pow(Function('\\\\sigma_x')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(\\pi,A)} = A \\pi, then obtain - \\frac{(\\iint A \\pi d\\pi dA) \\iint W{(\\pi,A)} d\\pi dA}{\\pi} + \\frac{(\\iint W{(\\pi,A)} d\\pi dA)^{2}}{\\pi} = 0", "derivation": "W{(\\pi,A)} = A \\pi and \\int W{(\\pi,A)} d\\pi = \\int A \\pi d\\pi and \\iint W{(\\pi,A)} d\\pi dA = \\iint A \\pi d\\pi dA and (\\iint W{(\\pi,A)} d\\pi dA)^{2} = (\\iint A \\pi d\\pi dA) \\iint W{(\\pi,A)} d\\pi dA and \\frac{(\\iint W{(\\pi,A)} d\\pi dA)^{2}}{\\pi} = \\frac{(\\iint A \\pi d\\pi dA) \\iint W{(\\pi,A)} d\\pi dA}{\\pi} and - \\frac{(\\iint A \\pi d\\pi dA) \\iint W{(\\pi,A)} d\\pi dA}{\\pi} + \\frac{(\\iint W{(\\pi,A)} d\\pi dA)^{2}}{\\pi} = 0", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 3, "Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Pow(Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2)), Mul(Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["divide", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["minus", 5, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Integral(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} = \\log{(t_{2})}, then obtain \\frac{d}{d t_{2}} \\int \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - 2 \\log{(t_{2})}}{\\log{(t_{2})}} dt_{2} = \\frac{d}{d t_{2}} \\int (-1) dt_{2}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} = \\log{(t_{2})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - \\log{(t_{2})} = 0 and \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - 2 \\log{(t_{2})} = - \\log{(t_{2})} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - 2 \\log{(t_{2})}}{\\log{(t_{2})}} = -1 and \\int \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - 2 \\log{(t_{2})}}{\\log{(t_{2})}} dt_{2} = \\int (-1) dt_{2} and \\frac{d}{d t_{2}} \\int \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{2})} - 2 \\log{(t_{2})}}{\\log{(t_{2})}} dt_{2} = \\frac{d}{d t_{2}} \\int (-1) dt_{2}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True)))"], [["minus", 1, "log(Symbol('t_2', commutative=True))"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), log(Symbol('t_2', commutative=True)))), Integer(0))"], [["add", 2, "Mul(Integer(-1), log(Symbol('t_2', commutative=True)))"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('t_2', commutative=True)))), Mul(Integer(-1), log(Symbol('t_2', commutative=True))))"], [["divide", 3, "log(Symbol('t_2', commutative=True))"], "Equality(Mul(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('t_2', commutative=True)))), Pow(log(Symbol('t_2', commutative=True)), Integer(-1))), Integer(-1))"], [["integrate", 4, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('t_2', commutative=True)))), Pow(log(Symbol('t_2', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(-1), Tuple(Symbol('t_2', commutative=True))))"], [["differentiate", 5, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Integral(Mul(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('t_2', commutative=True)))), Pow(log(Symbol('t_2', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integral(Integer(-1), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\mathbf{B},\\mathbb{I})} = - \\mathbb{I} + \\mathbf{B}, then obtain \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\int - n{(\\mathbf{B},\\mathbb{I})} d\\mathbf{B} d\\mathbb{I} = \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\mathbb{I} - \\mathbf{B}) d\\mathbf{B} d\\mathbb{I}", "derivation": "n{(\\mathbf{B},\\mathbb{I})} = - \\mathbb{I} + \\mathbf{B} and - n{(\\mathbf{B},\\mathbb{I})} = \\mathbb{I} - \\mathbf{B} and \\int - n{(\\mathbf{B},\\mathbb{I})} d\\mathbf{B} = \\int (\\mathbb{I} - \\mathbf{B}) d\\mathbf{B} and \\frac{\\partial}{\\partial \\mathbb{I}} \\int - n{(\\mathbf{B},\\mathbb{I})} d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\mathbb{I} - \\mathbf{B}) d\\mathbf{B} and \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\int - n{(\\mathbf{B},\\mathbb{I})} d\\mathbf{B} d\\mathbb{I} = \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\mathbb{I} - \\mathbf{B}) d\\mathbf{B} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Derivative(Integral(Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Derivative(Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given L{(f)} = \\sin{(f)}, then obtain \\frac{d}{d f} \\sin^{f}{(L{(f)})} = \\frac{d}{d f} \\sin^{f}{(\\sin{(f)})}", "derivation": "L{(f)} = \\sin{(f)} and \\sin{(L{(f)})} = \\sin{(\\sin{(f)})} and \\sin^{f}{(L{(f)})} = \\sin^{f}{(\\sin{(f)})} and \\frac{d}{d f} \\sin^{f}{(L{(f)})} = \\frac{d}{d f} \\sin^{f}{(\\sin{(f)})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["sin", 1], "Equality(sin(Function('L')(Symbol('f', commutative=True))), sin(sin(Symbol('f', commutative=True))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(sin(Function('L')(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(sin(sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(sin(Function('L')(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(sin(sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(a)} = \\cos{(a)}, then derive 2 \\operatorname{C_{1}}{(a)} \\frac{d}{d a} \\operatorname{C_{1}}{(a)} + \\cos{(a)} = - \\operatorname{C_{1}}{(a)} \\sin{(a)} + \\cos{(a)} \\frac{d}{d a} \\operatorname{C_{1}}{(a)} + \\cos{(a)}, then obtain 2 \\cos{(a)} \\frac{d}{d a} \\cos{(a)} + \\cos{(a)} = - \\sin{(a)} \\cos{(a)} + \\cos{(a)} \\frac{d}{d a} \\cos{(a)} + \\cos{(a)}", "derivation": "\\operatorname{C_{1}}{(a)} = \\cos{(a)} and \\operatorname{C_{1}}^{2}{(a)} = \\operatorname{C_{1}}{(a)} \\cos{(a)} and \\frac{d}{d a} \\operatorname{C_{1}}^{2}{(a)} = \\frac{d}{d a} \\operatorname{C_{1}}{(a)} \\cos{(a)} and \\cos{(a)} + \\frac{d}{d a} \\operatorname{C_{1}}^{2}{(a)} = \\cos{(a)} + \\frac{d}{d a} \\operatorname{C_{1}}{(a)} \\cos{(a)} and 2 \\operatorname{C_{1}}{(a)} \\frac{d}{d a} \\operatorname{C_{1}}{(a)} + \\cos{(a)} = - \\operatorname{C_{1}}{(a)} \\sin{(a)} + \\cos{(a)} \\frac{d}{d a} \\operatorname{C_{1}}{(a)} + \\cos{(a)} and 2 \\cos{(a)} \\frac{d}{d a} \\cos{(a)} + \\cos{(a)} = - \\sin{(a)} \\cos{(a)} + \\cos{(a)} \\frac{d}{d a} \\cos{(a)} + \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["times", 1, "Function('C_1')(Symbol('a', commutative=True))"], "Equality(Pow(Function('C_1')(Symbol('a', commutative=True)), Integer(2)), Mul(Function('C_1')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Function('C_1')(Symbol('a', commutative=True)), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Function('C_1')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 3, "cos(Symbol('a', commutative=True))"], "Equality(Add(cos(Symbol('a', commutative=True)), Derivative(Pow(Function('C_1')(Symbol('a', commutative=True)), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(cos(Symbol('a', commutative=True)), Derivative(Mul(Function('C_1')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Function('C_1')(Symbol('a', commutative=True)), Derivative(Function('C_1')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Function('C_1')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Mul(cos(Symbol('a', commutative=True)), Derivative(Function('C_1')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(2), cos(Symbol('a', commutative=True)), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Symbol('a', commutative=True))), Add(Mul(Integer(-1), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Mul(cos(Symbol('a', commutative=True)), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), cos(Symbol('a', commutative=True))))"]]}, {"prompt": "Given Z{(T,p)} = \\frac{T}{p}, then obtain \\int (- \\frac{T}{p} + Z{(T,p)} \\int Z{(T,p)} dT) dp = \\int (- \\frac{T}{p} + Z{(T,p)} \\int \\frac{T}{p} dT) dp", "derivation": "Z{(T,p)} = \\frac{T}{p} and \\int Z{(T,p)} dT = \\int \\frac{T}{p} dT and \\frac{T \\int Z{(T,p)} dT}{p} = \\frac{T \\int \\frac{T}{p} dT}{p} and Z{(T,p)} \\int Z{(T,p)} dT = Z{(T,p)} \\int \\frac{T}{p} dT and - \\frac{T}{p} + Z{(T,p)} \\int Z{(T,p)} dT = - \\frac{T}{p} + Z{(T,p)} \\int \\frac{T}{p} dT and \\int (- \\frac{T}{p} + Z{(T,p)} \\int Z{(T,p)} dT) dp = \\int (- \\frac{T}{p} + Z{(T,p)} \\int \\frac{T}{p} dT) dp", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))))"], [["times", 2, "Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True)))))"], [["minus", 4, "Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))))))"], [["integrate", 5, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True))))), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Function('Z')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Symbol('T', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given C{(C_{1},B)} = - B + \\sin{(C_{1})}, then obtain A_{1} + C{(C_{1},B)} + q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} = t_{2} + q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} + \\sin{(C_{1})}", "derivation": "C{(C_{1},B)} = - B + \\sin{(C_{1})} and \\frac{\\partial}{\\partial C_{1}} C{(C_{1},B)} = \\frac{\\partial}{\\partial C_{1}} (- B + \\sin{(C_{1})}) and \\int \\frac{\\partial}{\\partial C_{1}} C{(C_{1},B)} dC_{1} = \\int \\frac{\\partial}{\\partial C_{1}} (- B + \\sin{(C_{1})}) dC_{1} and q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} + \\int \\frac{\\partial}{\\partial C_{1}} C{(C_{1},B)} dC_{1} = q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} + \\int \\frac{\\partial}{\\partial C_{1}} (- B + \\sin{(C_{1})}) dC_{1} and A_{1} + C{(C_{1},B)} + q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} = t_{2} + q{(g^{\\prime}_{\\varepsilon},\\mathbf{r})} + \\sin{(C_{1})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), sin(Symbol('C_1', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(Function('C')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))))"], [["add", 3, "Function('q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Function('q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Derivative(Function('C')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Add(Function('q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_1', commutative=True), Function('C')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), Function('q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('t_2', commutative=True), Function('q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{X})} = \\sin{(e^{\\hat{X}})}, then derive (\\frac{d}{d \\hat{X}} \\operatorname{P_{g}}{(\\hat{X})})^{\\hat{X}} = (e^{\\hat{X}} \\cos{(e^{\\hat{X}})})^{\\hat{X}}, then obtain (\\frac{d}{d \\hat{X}} \\sin{(e^{\\hat{X}})})^{\\hat{X}} = (e^{\\hat{X}} \\cos{(e^{\\hat{X}})})^{\\hat{X}}", "derivation": "\\operatorname{P_{g}}{(\\hat{X})} = \\sin{(e^{\\hat{X}})} and \\frac{d}{d \\hat{X}} \\operatorname{P_{g}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\sin{(e^{\\hat{X}})} and (\\frac{d}{d \\hat{X}} \\operatorname{P_{g}}{(\\hat{X})})^{\\hat{X}} = (\\frac{d}{d \\hat{X}} \\sin{(e^{\\hat{X}})})^{\\hat{X}} and (\\frac{d}{d \\hat{X}} \\operatorname{P_{g}}{(\\hat{X})})^{\\hat{X}} = (e^{\\hat{X}} \\cos{(e^{\\hat{X}})})^{\\hat{X}} and (\\frac{d}{d \\hat{X}} \\sin{(e^{\\hat{X}})})^{\\hat{X}} = (e^{\\hat{X}} \\cos{(e^{\\hat{X}})})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), sin(exp(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Derivative(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(sin(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(exp(Symbol('\\\\hat{X}', commutative=True)), cos(exp(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(sin(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(exp(Symbol('\\\\hat{X}', commutative=True)), cos(exp(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = \\Psi \\theta_2, then derive \\frac{\\partial^{3}}{\\partial \\theta_2^{2}\\partial \\Psi} \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = 0, then obtain c = \\int 0 d\\theta_2", "derivation": "\\operatorname{F_{H}}{(\\theta_2,\\Psi)} = \\Psi \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = \\frac{\\partial}{\\partial \\theta_2} \\Psi \\theta_2 and \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\theta_2} \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\theta_2} \\Psi \\theta_2 and \\frac{\\partial^{3}}{\\partial \\theta_2\\partial \\Psi\\partial \\theta_2} \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = \\frac{\\partial^{3}}{\\partial \\theta_2\\partial \\Psi\\partial \\theta_2} \\Psi \\theta_2 and \\frac{\\partial^{3}}{\\partial \\theta_2^{2}\\partial \\Psi} \\operatorname{F_{H}}{(\\theta_2,\\Psi)} = 0 and \\frac{\\partial^{3}}{\\partial \\theta_2^{2}\\partial \\Psi} \\Psi \\theta_2 = 0 and \\int \\frac{\\partial^{3}}{\\partial \\theta_2^{2}\\partial \\Psi} \\Psi \\theta_2 d\\theta_2 = \\int 0 d\\theta_2 and c = \\int 0 d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('F_H')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Integer(0))"], [["integrate", 6, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Symbol('c', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)} = \\phi_2 + \\sigma_p and B{(\\phi_2,\\sigma_p)} = (\\phi_2 + \\sigma_p + \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p}, then obtain \\frac{\\partial}{\\partial \\sigma_p} B{(\\phi_2,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (2 \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p}", "derivation": "\\operatorname{F_{g}}{(\\phi_2,\\sigma_p)} = \\phi_2 + \\sigma_p and 2 \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)} = \\phi_2 + \\sigma_p + \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)} and (2 \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p} = (\\phi_2 + \\sigma_p + \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p} and B{(\\phi_2,\\sigma_p)} = (\\phi_2 + \\sigma_p + \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} B{(\\phi_2,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (\\phi_2 + \\sigma_p + \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} B{(\\phi_2,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (2 \\operatorname{F_{g}}{(\\phi_2,\\sigma_p)})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('B')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(2), Function('F_g')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(V,\\hat{H})} = - V + \\hat{H}, then derive 2 \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 2 = \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 1, then obtain 2 \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} C{(V,\\hat{H})} = \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} C{(V,\\hat{H})}", "derivation": "C{(V,\\hat{H})} = - V + \\hat{H} and V - \\hat{H} + C{(V,\\hat{H})} = 0 and \\frac{\\partial}{\\partial \\hat{H}} (V - \\hat{H} + C{(V,\\hat{H})}) = \\frac{d}{d \\hat{H}} 0 and \\frac{\\partial}{\\partial \\hat{H}} (V - \\hat{H} + C{(V,\\hat{H})}) + \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 1 = \\frac{d}{d \\hat{H}} 0 + \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 1 and 2 \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 2 = \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 1 and \\frac{\\partial}{\\partial \\hat{H}} (2 \\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 2) = \\frac{\\partial}{\\partial \\hat{H}} (\\frac{\\partial}{\\partial \\hat{H}} C{(V,\\hat{H})} - 1) and 2 \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} C{(V,\\hat{H})} = \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} C{(V,\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["add", 3, "Add(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(-2)), Add(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(-2)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(2), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(v_{2},\\Psi_{nl})} = \\Psi_{nl} v_{2} and \\tilde{g}{(v_{2},\\Psi_{nl})} = \\Psi_{nl} v_{2}, then obtain v_{2} \\int \\tilde{g}{(v_{2},\\Psi_{nl})} d\\Psi_{nl} = v_{2} \\int \\Psi_{nl} v_{2} d\\Psi_{nl}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(v_{2},\\Psi_{nl})} = \\Psi_{nl} v_{2} and \\int \\operatorname{f_{\\mathbf{v}}}{(v_{2},\\Psi_{nl})} d\\Psi_{nl} = \\int \\Psi_{nl} v_{2} d\\Psi_{nl} and \\tilde{g}{(v_{2},\\Psi_{nl})} = \\Psi_{nl} v_{2} and \\tilde{g}{(v_{2},\\Psi_{nl})} = \\operatorname{f_{\\mathbf{v}}}{(v_{2},\\Psi_{nl})} and v_{2} \\int \\operatorname{f_{\\mathbf{v}}}{(v_{2},\\Psi_{nl})} d\\Psi_{nl} = v_{2} \\int \\Psi_{nl} v_{2} d\\Psi_{nl} and v_{2} \\int \\tilde{g}{(v_{2},\\Psi_{nl})} d\\Psi_{nl} = v_{2} \\int \\Psi_{nl} v_{2} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 2, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Symbol('v_2', commutative=True), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('v_2', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Symbol('v_2', commutative=True), Integral(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given H{(\\Omega)} = e^{\\Omega}, then derive \\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)} = e^{\\Omega} \\cos{(e^{\\Omega})}, then derive (\\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)})^{\\Omega} = (e^{\\Omega} \\cos{(e^{\\Omega})})^{\\Omega}, then obtain (\\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)})^{\\Omega} = (H{(\\Omega)} \\cos{(H{(\\Omega)})})^{\\Omega}", "derivation": "H{(\\Omega)} = e^{\\Omega} and \\sin{(H{(\\Omega)})} = \\sin{(e^{\\Omega})} and \\frac{d}{d \\Omega} \\sin{(H{(\\Omega)})} = \\frac{d}{d \\Omega} \\sin{(e^{\\Omega})} and \\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)} = e^{\\Omega} \\cos{(e^{\\Omega})} and \\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)} = H{(\\Omega)} \\cos{(H{(\\Omega)})} and (\\frac{d}{d \\Omega} \\sin{(H{(\\Omega)})})^{\\Omega} = (\\frac{d}{d \\Omega} \\sin{(e^{\\Omega})})^{\\Omega} and (\\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)})^{\\Omega} = (e^{\\Omega} \\cos{(e^{\\Omega})})^{\\Omega} and e^{\\Omega} \\cos{(e^{\\Omega})} = H{(\\Omega)} \\cos{(H{(\\Omega)})} and (\\cos{(H{(\\Omega)})} \\frac{d}{d \\Omega} H{(\\Omega)})^{\\Omega} = (H{(\\Omega)} \\cos{(H{(\\Omega)})})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["sin", 1], "Equality(sin(Function('H')(Symbol('\\\\Omega', commutative=True))), sin(exp(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(sin(Function('H')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('H')(Symbol('\\\\Omega', commutative=True))), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(cos(Function('H')(Symbol('\\\\Omega', commutative=True))), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), cos(Function('H')(Symbol('\\\\Omega', commutative=True)))))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Derivative(sin(Function('H')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Mul(cos(Function('H')(Symbol('\\\\Omega', commutative=True))), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Mul(exp(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(exp(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True)))), Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), cos(Function('H')(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Pow(Mul(cos(Function('H')(Symbol('\\\\Omega', commutative=True))), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), cos(Function('H')(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given S{(l,E_{x})} = l + e^{E_{x}}, then obtain E_{x} + S{(l,E_{x})} + \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + S{(l,E_{x})}) dE_{x} = E_{x} + S{(l,E_{x})} + \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + l + e^{E_{x}}) dE_{x}", "derivation": "S{(l,E_{x})} = l + e^{E_{x}} and E_{x} + S{(l,E_{x})} = E_{x} + l + e^{E_{x}} and \\int (E_{x} + S{(l,E_{x})}) dE_{x} = \\int (E_{x} + l + e^{E_{x}}) dE_{x} and \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + S{(l,E_{x})}) dE_{x} = \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + l + e^{E_{x}}) dE_{x} and E_{x} + S{(l,E_{x})} + \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + S{(l,E_{x})}) dE_{x} = E_{x} + S{(l,E_{x})} + \\log{(\\frac{\\varepsilon}{\\mu_0})} + \\int (E_{x} + l + e^{E_{x}}) dE_{x}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('l', commutative=True), exp(Symbol('E_x', commutative=True))))"], [["add", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('l', commutative=True), exp(Symbol('E_x', commutative=True))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Symbol('l', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["add", 3, "log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))), Add(log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Symbol('l', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))))"], [["add", 4, "Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))), Add(Symbol('E_x', commutative=True), Function('S')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Symbol('l', commutative=True), exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and \\theta{(\\mathbf{D})} = e^{e^{\\mathbf{D}}}, then obtain - \\theta{(\\mathbf{D})} + e^{e^{\\mathbf{D}}} = 0", "derivation": "\\mathbf{J}_M{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and \\theta{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and \\mathbf{J}_M{(\\mathbf{D})} - e^{e^{\\mathbf{D}}} = 0 and \\mathbf{J}_M{(\\mathbf{D})} - \\theta{(\\mathbf{D})} = 0 and - \\theta{(\\mathbf{D})} + e^{e^{\\mathbf{D}}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{D}', commutative=True)), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{D}', commutative=True)), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 1, "exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\mathbf{D}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\mathbf{D}', commutative=True))), exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}{(\\hat{x}_0,I,B)} = B + I + \\hat{x}_0, then obtain \\int (- B - I - \\hat{x}_0 + \\frac{\\tilde{g}{(\\hat{x}_0,I,B)}}{\\hat{x}_0}) dI = \\int (- B - I - \\hat{x}_0 + \\frac{B + I + \\hat{x}_0}{\\hat{x}_0}) dI", "derivation": "\\tilde{g}{(\\hat{x}_0,I,B)} = B + I + \\hat{x}_0 and \\frac{\\tilde{g}{(\\hat{x}_0,I,B)}}{\\hat{x}_0} = \\frac{B + I + \\hat{x}_0}{\\hat{x}_0} and - B - I - \\hat{x}_0 + \\frac{\\tilde{g}{(\\hat{x}_0,I,B)}}{\\hat{x}_0} = - B - I - \\hat{x}_0 + \\frac{B + I + \\hat{x}_0}{\\hat{x}_0} and \\int (- B - I - \\hat{x}_0 + \\frac{\\tilde{g}{(\\hat{x}_0,I,B)}}{\\hat{x}_0}) dI = \\int (- B - I - \\hat{x}_0 + \\frac{B + I + \\hat{x}_0}{\\hat{x}_0}) dI", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('I', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('I', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 2, "Add(Symbol('B', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('I', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('I', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(n_{2},C_{1})} = (e^{C_{1}})^{n_{2}}, then obtain - \\operatorname{F_{c}}{(n_{2},C_{1})} + \\cos{(n_{2} - \\operatorname{F_{c}}{(n_{2},C_{1})} + (e^{C_{1}})^{n_{2}})} = - \\operatorname{F_{c}}{(n_{2},C_{1})} + \\cos{(n_{2})}", "derivation": "\\operatorname{F_{c}}{(n_{2},C_{1})} = (e^{C_{1}})^{n_{2}} and \\operatorname{F_{c}}{(n_{2},C_{1})} - (e^{C_{1}})^{n_{2}} = 0 and - n_{2} + \\operatorname{F_{c}}{(n_{2},C_{1})} - (e^{C_{1}})^{n_{2}} = - n_{2} and \\cos{(n_{2} - \\operatorname{F_{c}}{(n_{2},C_{1})} + (e^{C_{1}})^{n_{2}})} = \\cos{(n_{2})} and - \\operatorname{F_{c}}{(n_{2},C_{1})} + \\cos{(n_{2} - \\operatorname{F_{c}}{(n_{2},C_{1})} + (e^{C_{1}})^{n_{2}})} = - \\operatorname{F_{c}}{(n_{2},C_{1})} + \\cos{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True)), Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True)))"], [["minus", 1, "Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True)))), Integer(0))"], [["minus", 2, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True)))"], [["cos", 3], "Equality(cos(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True))), Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True)))), cos(Symbol('n_2', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True))), cos(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True))), Pow(exp(Symbol('C_1', commutative=True)), Symbol('n_2', commutative=True))))), Add(Mul(Integer(-1), Function('F_c')(Symbol('n_2', commutative=True), Symbol('C_1', commutative=True))), cos(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\lambda{(A_{2},\\mathbf{v})} = \\frac{A_{2}}{\\mathbf{v}}, then obtain \\frac{A_{2}^{2}}{2} + F_{N} + \\lambda{(A_{2},\\mathbf{v})} = \\frac{A_{2}^{2}}{2} + \\frac{A_{2}}{\\mathbf{v}} + f_{E}", "derivation": "\\lambda{(A_{2},\\mathbf{v})} = \\frac{A_{2}}{\\mathbf{v}} and \\frac{\\partial}{\\partial A_{2}} \\lambda{(A_{2},\\mathbf{v})} = \\frac{\\partial}{\\partial A_{2}} \\frac{A_{2}}{\\mathbf{v}} and A_{2} + \\frac{\\partial}{\\partial A_{2}} \\lambda{(A_{2},\\mathbf{v})} = A_{2} + \\frac{\\partial}{\\partial A_{2}} \\frac{A_{2}}{\\mathbf{v}} and \\int (A_{2} + \\frac{\\partial}{\\partial A_{2}} \\lambda{(A_{2},\\mathbf{v})}) dA_{2} = \\int (A_{2} + \\frac{\\partial}{\\partial A_{2}} \\frac{A_{2}}{\\mathbf{v}}) dA_{2} and \\frac{A_{2}^{2}}{2} + F_{N} + \\lambda{(A_{2},\\mathbf{v})} = \\frac{A_{2}^{2}}{2} + \\frac{A_{2}}{\\mathbf{v}} + f_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["add", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Derivative(Function('\\\\lambda')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Symbol('A_2', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Symbol('A_2', commutative=True), Derivative(Function('\\\\lambda')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('F_N', commutative=True), Function('\\\\lambda')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given r{(J_{\\varepsilon},E_{x})} = E_{x} + J_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{\\partial}{\\partial J_{\\varepsilon}} r{(J_{\\varepsilon},E_{x})})^{J_{\\varepsilon}} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{\\partial}{\\partial J_{\\varepsilon}} (E_{x} + J_{\\varepsilon}))^{J_{\\varepsilon}}", "derivation": "r{(J_{\\varepsilon},E_{x})} = E_{x} + J_{\\varepsilon} and \\frac{\\partial}{\\partial J_{\\varepsilon}} r{(J_{\\varepsilon},E_{x})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (E_{x} + J_{\\varepsilon}) and (\\frac{\\partial}{\\partial J_{\\varepsilon}} r{(J_{\\varepsilon},E_{x})})^{J_{\\varepsilon}} = (\\frac{\\partial}{\\partial J_{\\varepsilon}} (E_{x} + J_{\\varepsilon}))^{J_{\\varepsilon}} and \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{\\partial}{\\partial J_{\\varepsilon}} r{(J_{\\varepsilon},E_{x})})^{J_{\\varepsilon}} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{\\partial}{\\partial J_{\\varepsilon}} (E_{x} + J_{\\varepsilon}))^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Derivative(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Add(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('E_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\varepsilon_0,\\varphi)} = \\varepsilon_0 \\varphi, then obtain (\\varepsilon_0 \\varphi)^{\\varepsilon_0} ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- 2 \\varphi} \\hat{H}_{\\lambda}^{\\varepsilon_0}{(\\varepsilon_0,\\varphi)} = (\\varepsilon_0 \\varphi)^{2 \\varepsilon_0} ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- 2 \\varphi}", "derivation": "\\hat{H}_{\\lambda}{(\\varepsilon_0,\\varphi)} = \\varepsilon_0 \\varphi and \\hat{H}_{\\lambda}^{\\varepsilon_0}{(\\varepsilon_0,\\varphi)} = (\\varepsilon_0 \\varphi)^{\\varepsilon_0} and ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- \\varphi} \\hat{H}_{\\lambda}^{\\varepsilon_0}{(\\varepsilon_0,\\varphi)} = (\\varepsilon_0 \\varphi)^{\\varepsilon_0} ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- \\varphi} and (\\varepsilon_0 \\varphi)^{\\varepsilon_0} ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- 2 \\varphi} \\hat{H}_{\\lambda}^{\\varepsilon_0}{(\\varepsilon_0,\\varphi)} = (\\varepsilon_0 \\varphi)^{2 \\varepsilon_0} ((\\varepsilon_0 \\varphi)^{\\varepsilon_0})^{- 2 \\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 2, "Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))))"], [["times", 3, "Mul(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], "Equality(Mul(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True))), Pow(Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given r{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and \\rho{(\\hat{x})} = \\cos{(\\hat{x})}, then derive r{(\\Psi_{nl})} \\cos{(\\hat{x})} = \\frac{\\cos{(\\hat{x})}}{\\Psi_{nl}}, then obtain \\rho{(\\hat{x})} \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} = \\frac{\\rho{(\\hat{x})}}{\\Psi_{nl}}", "derivation": "r{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and r{(\\Psi_{nl})} \\cos{(\\hat{x})} = \\cos{(\\hat{x})} \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and r{(\\Psi_{nl})} \\cos{(\\hat{x})} = \\frac{\\cos{(\\hat{x})}}{\\Psi_{nl}} and \\rho{(\\hat{x})} = \\cos{(\\hat{x})} and \\rho{(\\hat{x})} r{(\\Psi_{nl})} = \\frac{\\rho{(\\hat{x})}}{\\Psi_{nl}} and \\rho{(\\hat{x})} \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} = \\frac{\\rho{(\\hat{x})}}{\\Psi_{nl}}", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["times", 1, "cos(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('r')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))), Mul(cos(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('r')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\hat{x}', commutative=True)), Function('r')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\omega,\\dot{x})} = \\sin{(\\dot{x} - \\omega)}, then obtain \\frac{\\partial^{3}}{\\partial \\dot{x}^{3}} \\operatorname{C_{1}}{(\\omega,\\dot{x})} = - \\cos{(\\dot{x} - \\omega)}", "derivation": "\\operatorname{C_{1}}{(\\omega,\\dot{x})} = \\sin{(\\dot{x} - \\omega)} and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{1}}{(\\omega,\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} \\sin{(\\dot{x} - \\omega)} and \\frac{\\partial^{2}}{\\partial \\dot{x}^{2}} \\operatorname{C_{1}}{(\\omega,\\dot{x})} = \\frac{\\partial^{2}}{\\partial \\dot{x}^{2}} \\sin{(\\dot{x} - \\omega)} and \\frac{\\partial^{3}}{\\partial \\dot{x}^{3}} \\operatorname{C_{1}}{(\\omega,\\dot{x})} = \\frac{\\partial^{3}}{\\partial \\dot{x}^{3}} \\sin{(\\dot{x} - \\omega)} and \\frac{\\partial^{3}}{\\partial \\dot{x}^{3}} \\operatorname{C_{1}}{(\\omega,\\dot{x})} = - \\cos{(\\dot{x} - \\omega)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\dot{x}', commutative=True)), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))), Derivative(sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(3))), Derivative(sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(3))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(3))), Mul(Integer(-1), cos(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\psi^{*}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and h{(\\varepsilon_0)} = \\sin^{\\varepsilon_0}{(\\varepsilon_0)}, then obtain \\psi^{*}^{\\varepsilon_0}{(\\varepsilon_0)} \\sin{(\\varepsilon_0)} = h{(\\varepsilon_0)} \\sin{(\\varepsilon_0)}", "derivation": "\\psi^{*}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\psi^{*}^{\\varepsilon_0}{(\\varepsilon_0)} = \\sin^{\\varepsilon_0}{(\\varepsilon_0)} and h{(\\varepsilon_0)} = \\sin^{\\varepsilon_0}{(\\varepsilon_0)} and h{(\\varepsilon_0)} \\sin{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} \\sin^{\\varepsilon_0}{(\\varepsilon_0)} and \\psi^{*}^{\\varepsilon_0}{(\\varepsilon_0)} = h{(\\varepsilon_0)} and \\psi^{*}^{\\varepsilon_0}{(\\varepsilon_0)} \\sin{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} \\sin^{\\varepsilon_0}{(\\varepsilon_0)} and \\psi^{*}^{\\varepsilon_0}{(\\varepsilon_0)} \\sin{(\\varepsilon_0)} = h{(\\varepsilon_0)} \\sin{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 3, "sin(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Function('h')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Mul(sin(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\psi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Function('h')(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Mul(sin(Symbol('\\\\varepsilon_0', commutative=True)), Pow(sin(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Function('h')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(b,\\pi,E_{x})} = \\frac{b^{E_{x}}}{\\pi} and \\hat{H}_l{(A_{x},\\sigma_x)} = \\frac{\\partial}{\\partial A_{x}} \\sigma_x^{A_{x}}, then obtain \\hat{H}_l{(A_{x},\\sigma_x)} - \\operatorname{v_{t}}{(b,\\pi,E_{x})} = - \\operatorname{v_{t}}{(b,\\pi,E_{x})} + \\frac{\\partial}{\\partial A_{x}} \\sigma_x^{A_{x}}", "derivation": "\\operatorname{v_{t}}{(b,\\pi,E_{x})} = \\frac{b^{E_{x}}}{\\pi} and \\hat{H}_l{(A_{x},\\sigma_x)} = \\frac{\\partial}{\\partial A_{x}} \\sigma_x^{A_{x}} and \\hat{H}_l{(A_{x},\\sigma_x)} - \\frac{b^{E_{x}}}{\\pi} = \\frac{\\partial}{\\partial A_{x}} \\sigma_x^{A_{x}} - \\frac{b^{E_{x}}}{\\pi} and \\hat{H}_l{(A_{x},\\sigma_x)} - \\operatorname{v_{t}}{(b,\\pi,E_{x})} = - \\operatorname{v_{t}}{(b,\\pi,E_{x})} + \\frac{\\partial}{\\partial A_{x}} \\sigma_x^{A_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('v_t')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('E_x', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('A_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('A_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('E_x', commutative=True)))), Add(Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('E_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('A_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Function('v_t')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True))), Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(\\mathbf{J},\\hbar)} = \\hbar + \\mathbf{J} and \\operatorname{P_{e}}{(\\mathbf{J},\\hbar)} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} (\\hbar + \\mathbf{J}), then obtain \\sin{(\\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} a{(\\mathbf{J},\\hbar)})} = \\sin{(\\operatorname{P_{e}}{(\\mathbf{J},\\hbar)})}", "derivation": "a{(\\mathbf{J},\\hbar)} = \\hbar + \\mathbf{J} and \\frac{\\partial}{\\partial \\hbar} a{(\\mathbf{J},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (\\hbar + \\mathbf{J}) and \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} a{(\\mathbf{J},\\hbar)} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} (\\hbar + \\mathbf{J}) and \\operatorname{P_{e}}{(\\mathbf{J},\\hbar)} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} (\\hbar + \\mathbf{J}) and \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} a{(\\mathbf{J},\\hbar)} = \\operatorname{P_{e}}{(\\mathbf{J},\\hbar)} and \\sin{(\\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial \\hbar} a{(\\mathbf{J},\\hbar)})} = \\sin{(\\operatorname{P_{e}}{(\\mathbf{J},\\hbar)})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('a')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Function('P_e')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["sin", 5], "Equality(sin(Derivative(Function('a')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), sin(Function('P_e')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given M{(v)} = e^{v}, then derive M{(v)} - e^{v} = 0, then obtain (M{(v)} - \\frac{d}{d v} e^{v}) \\frac{d}{d v} (M{(v)} - e^{v}) = (M{(v)} - \\frac{d}{d v} e^{v}) \\frac{d}{d v} 0", "derivation": "M{(v)} = e^{v} and M{(v)} - \\frac{d}{d v} e^{v} = e^{v} - \\frac{d}{d v} e^{v} and M{(v)} - e^{v} = 0 and \\frac{d}{d v} (M{(v)} - e^{v}) = \\frac{d}{d v} 0 and (M{(v)} - \\frac{d}{d v} e^{v}) \\frac{d}{d v} (M{(v)} - e^{v}) = (M{(v)} - \\frac{d}{d v} e^{v}) \\frac{d}{d v} 0", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["minus", 1, "Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))), Add(exp(Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 4, "Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], "Equality(Mul(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))), Derivative(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Add(Function('M')(Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))), Derivative(Integer(0), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\varphi,E,M)} = E + \\frac{\\varphi}{M} and \\operatorname{t_{2}}{(E,\\varphi,M)} = (E + \\frac{\\varphi}{M}) \\int \\mathbf{J}{(\\varphi,E,M)} dE, then obtain \\operatorname{t_{2}}{(E,\\varphi,M)} = (E + \\frac{\\varphi}{M}) \\int (E + \\frac{\\varphi}{M}) dE", "derivation": "\\mathbf{J}{(\\varphi,E,M)} = E + \\frac{\\varphi}{M} and \\int \\mathbf{J}{(\\varphi,E,M)} dE = \\int (E + \\frac{\\varphi}{M}) dE and \\operatorname{t_{2}}{(E,\\varphi,M)} = (E + \\frac{\\varphi}{M}) \\int \\mathbf{J}{(\\varphi,E,M)} dE and \\operatorname{t_{2}}{(E,\\varphi,M)} = (E + \\frac{\\varphi}{M}) \\int (E + \\frac{\\varphi}{M}) dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\varphi', commutative=True), Symbol('E', commutative=True), Symbol('M', commutative=True)), Add(Symbol('E', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\varphi', commutative=True), Symbol('E', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Add(Symbol('E', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('E', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('M', commutative=True)), Mul(Add(Symbol('E', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\varphi', commutative=True), Symbol('E', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('t_2')(Symbol('E', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('M', commutative=True)), Mul(Add(Symbol('E', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('E', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\psi^*,B)} = B^{\\psi^*} and \\operatorname{f_{\\mathbf{v}}}{(\\psi^*,B)} = - B (- B + B^{\\psi^*}), then obtain \\operatorname{f_{\\mathbf{v}}}{(\\psi^*,B)} = - B (- B + \\eta{(\\psi^*,B)})", "derivation": "\\eta{(\\psi^*,B)} = B^{\\psi^*} and - B + \\eta{(\\psi^*,B)} = - B + B^{\\psi^*} and - B (- B + \\eta{(\\psi^*,B)}) = - B (- B + B^{\\psi^*}) and \\operatorname{f_{\\mathbf{v}}}{(\\psi^*,B)} = - B (- B + B^{\\psi^*}) and \\operatorname{f_{\\mathbf{v}}}{(\\psi^*,B)} = - B (- B + \\eta{(\\psi^*,B)})", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True)))), Mul(Integer(-1), Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(W,\\varphi)} = W \\sin{(\\varphi)}, then derive \\frac{\\partial}{\\partial \\varphi} \\operatorname{E_{\\lambda}}{(W,\\varphi)} = W \\cos{(\\varphi)}, then obtain W \\cos{(\\varphi)} = \\frac{\\partial}{\\partial \\varphi} W \\sin{(\\varphi)}", "derivation": "\\operatorname{E_{\\lambda}}{(W,\\varphi)} = W \\sin{(\\varphi)} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{E_{\\lambda}}{(W,\\varphi)} = \\frac{\\partial}{\\partial \\varphi} W \\sin{(\\varphi)} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{E_{\\lambda}}{(W,\\varphi)} = W \\cos{(\\varphi)} and W \\cos{(\\varphi)} = \\frac{\\partial}{\\partial \\varphi} W \\sin{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('W', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Symbol('W', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), cos(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('W', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Derivative(Mul(Symbol('W', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{A})} = \\mathbf{A}, then derive - \\frac{1}{\\mathbf{A}^{2}} = - \\frac{\\frac{d}{d \\mathbf{A}} \\varepsilon{(\\mathbf{A})}}{\\varepsilon^{2}{(\\mathbf{A})}}, then obtain - \\frac{1}{\\mathbf{A}^{2}} = - \\frac{\\frac{d}{d \\mathbf{A}} \\varepsilon{(\\mathbf{A})}}{\\mathbf{A} \\varepsilon{(\\mathbf{A})}}", "derivation": "\\varepsilon{(\\mathbf{A})} = \\mathbf{A} and \\varepsilon^{2}{(\\mathbf{A})} = \\mathbf{A} \\varepsilon{(\\mathbf{A})} and \\frac{1}{\\mathbf{A}} = \\frac{1}{\\varepsilon{(\\mathbf{A})}} and \\frac{d}{d \\mathbf{A}} \\frac{1}{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} \\frac{1}{\\varepsilon{(\\mathbf{A})}} and - \\frac{1}{\\mathbf{A}^{2}} = - \\frac{\\frac{d}{d \\mathbf{A}} \\varepsilon{(\\mathbf{A})}}{\\varepsilon^{2}{(\\mathbf{A})}} and - \\frac{1}{\\mathbf{A}^{2}} = - \\frac{\\frac{d}{d \\mathbf{A}} \\varepsilon{(\\mathbf{A})}}{\\mathbf{A} \\varepsilon{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], [["times", 1, "Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-2)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(C_{2})} = \\cos{(C_{2})}, then obtain (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})}) \\frac{d}{d C_{2}} 0 = (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})}) \\frac{d}{d C_{2}} (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})})", "derivation": "\\operatorname{F_{H}}{(C_{2})} = \\cos{(C_{2})} and 0 = - \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})} and \\frac{d}{d C_{2}} 0 = \\frac{d}{d C_{2}} (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})}) and (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})}) \\frac{d}{d C_{2}} 0 = (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})}) \\frac{d}{d C_{2}} (- \\operatorname{F_{H}}{(C_{2})} + \\cos{(C_{2})})", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["minus", 1, "Function('F_H')(Symbol('C_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["times", 3, "Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True))), Derivative(Integer(0), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True))), Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('C_2', commutative=True))), cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\omega,\\rho_b)} = \\omega - \\rho_b, then obtain - \\frac{\\omega + \\varepsilon_{0}{(\\omega,\\rho_b)}}{\\omega} = - \\frac{2 \\omega - \\rho_b}{\\omega}", "derivation": "\\varepsilon_{0}{(\\omega,\\rho_b)} = \\omega - \\rho_b and \\omega + \\varepsilon_{0}{(\\omega,\\rho_b)} = 2 \\omega - \\rho_b and \\frac{\\omega + \\varepsilon_{0}{(\\omega,\\rho_b)}}{\\omega} = \\frac{2 \\omega - \\rho_b}{\\omega} and - \\frac{\\omega + \\varepsilon_{0}{(\\omega,\\rho_b)}}{\\omega} = - \\frac{2 \\omega - \\rho_b}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["divide", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\delta)} = \\sin{(\\delta)}, then obtain \\frac{4 \\eta{(\\delta)}}{\\sin{(\\delta)}} = \\frac{2 (\\frac{\\eta{(\\delta)}}{\\sin{(\\delta)}} + 1) \\eta{(\\delta)}}{\\sin{(\\delta)}}", "derivation": "\\eta{(\\delta)} = \\sin{(\\delta)} and \\eta{(\\delta)} \\sin{(\\delta)} = \\sin^{2}{(\\delta)} and \\frac{\\eta{(\\delta)}}{\\sin{(\\delta)}} = 1 and \\frac{2 \\eta{(\\delta)}}{\\sin{(\\delta)}} = \\frac{\\eta{(\\delta)}}{\\sin{(\\delta)}} + 1 and \\frac{4 \\eta^{2}{(\\delta)}}{\\sin^{2}{(\\delta)}} = \\frac{2 (\\frac{\\eta{(\\delta)}}{\\sin{(\\delta)}} + 1) \\eta{(\\delta)}}{\\sin{(\\delta)}} and \\frac{4 \\eta{(\\delta)}}{\\sin{(\\delta)}} = \\frac{2 (\\frac{\\eta{(\\delta)}}{\\sin{(\\delta)}} + 1) \\eta{(\\delta)}}{\\sin{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(2)))"], [["divide", 1, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1))"], [["add", 3, "Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Add(Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1)))"], [["times", 4, "Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-2))), Mul(Integer(2), Add(Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(4), Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Integer(2), Add(Mul(Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\eta')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(C_{2})} = \\cos{(C_{2})}, then obtain - C_{2} + 2 \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} = - C_{2} + 2 \\operatorname{v_{y}}^{C_{2}}{(C_{2})} \\cos^{C_{2}}{(C_{2})}", "derivation": "\\operatorname{v_{y}}{(C_{2})} = \\cos{(C_{2})} and \\operatorname{v_{y}}^{C_{2}}{(C_{2})} = \\cos^{C_{2}}{(C_{2})} and \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} = \\operatorname{v_{y}}^{C_{2}}{(C_{2})} \\cos^{C_{2}}{(C_{2})} and - C_{2} + \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} = - C_{2} + \\operatorname{v_{y}}^{C_{2}}{(C_{2})} \\cos^{C_{2}}{(C_{2})} and - C_{2} + 2 \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} = - C_{2} + \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} + \\operatorname{v_{y}}^{C_{2}}{(C_{2})} \\cos^{C_{2}}{(C_{2})} and - C_{2} + 2 \\operatorname{v_{y}}^{2 C_{2}}{(C_{2})} = - C_{2} + 2 \\operatorname{v_{y}}^{C_{2}}{(C_{2})} \\cos^{C_{2}}{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["times", 2, "Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))), Mul(Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["minus", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))"], [["add", 4, "Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(2), Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))), Mul(Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(2), Pow(Function('v_y')(Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(2), Pow(Function('v_y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\tilde{g},\\varepsilon)} = \\tilde{g} \\varepsilon, then obtain ((- \\tilde{g} \\varepsilon + \\mathbf{J}{(\\tilde{g},\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = (0^{\\varepsilon})^{\\varepsilon}", "derivation": "\\mathbf{J}{(\\tilde{g},\\varepsilon)} = \\tilde{g} \\varepsilon and - \\tilde{g} \\varepsilon + \\mathbf{J}{(\\tilde{g},\\varepsilon)} = 0 and (- \\tilde{g} \\varepsilon + \\mathbf{J}{(\\tilde{g},\\varepsilon)})^{\\varepsilon} = 0^{\\varepsilon} and ((- \\tilde{g} \\varepsilon + \\mathbf{J}{(\\tilde{g},\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = (0^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given E{(b,\\tilde{g})} = \\int \\frac{b}{\\tilde{g}} db and Q{(b,\\tilde{g})} = - E{(b,\\tilde{g})}, then obtain ((Q{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db)^{b})^{b} = 1", "derivation": "E{(b,\\tilde{g})} = \\int \\frac{b}{\\tilde{g}} db and 0 = - E{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db and Q{(b,\\tilde{g})} = - E{(b,\\tilde{g})} and 0 = Q{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db and 0^{b} = (Q{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db)^{b} and 0^{b} = (E{(b,\\tilde{g})} + Q{(b,\\tilde{g})})^{b} and (Q{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db)^{b} = 1 and ((Q{(b,\\tilde{g})} + \\int \\frac{b}{\\tilde{g}} db)^{b})^{b} = 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 1, "Function('E')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Integer(0), Symbol('b', commutative=True)), Pow(Add(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integer(0), Symbol('b', commutative=True)), Pow(Add(Function('E')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Add(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Symbol('b', commutative=True)), Integer(1))"], [["power", 7, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Add(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Integer(1))"]]}, {"prompt": "Given I{(\\mu)} = \\sin{(\\mu)}, then derive \\frac{d}{d \\mu} I{(\\mu)} = \\cos{(\\mu)}, then obtain \\frac{d}{d \\mu} \\sin{(\\mu)} = \\cos{(\\mu)}", "derivation": "I{(\\mu)} = \\sin{(\\mu)} and \\frac{d}{d \\mu} I{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)} and \\frac{d}{d \\mu} I{(\\mu)} = \\cos{(\\mu)} and \\frac{d}{d \\mu} \\sin{(\\mu)} = \\cos{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), cos(Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), cos(Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} = \\frac{k}{\\Psi_{\\lambda}}, then obtain e^{\\int (\\int \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} dk)^{\\Psi_{\\lambda}} dk} = e^{\\int (\\int \\frac{k}{\\Psi_{\\lambda}} dk)^{\\Psi_{\\lambda}} dk}", "derivation": "\\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} = \\frac{k}{\\Psi_{\\lambda}} and \\int \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} dk = \\int \\frac{k}{\\Psi_{\\lambda}} dk and (\\int \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} dk)^{\\Psi_{\\lambda}} = (\\int \\frac{k}{\\Psi_{\\lambda}} dk)^{\\Psi_{\\lambda}} and \\int (\\int \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} dk)^{\\Psi_{\\lambda}} dk = \\int (\\int \\frac{k}{\\Psi_{\\lambda}} dk)^{\\Psi_{\\lambda}} dk and e^{\\int (\\int \\operatorname{t_{2}}{(k,\\Psi_{\\lambda})} dk)^{\\Psi_{\\lambda}} dk} = e^{\\int (\\int \\frac{k}{\\Psi_{\\lambda}} dk)^{\\Psi_{\\lambda}} dk}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('k', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('k', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["power", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integral(Function('t_2')(Symbol('k', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Integral(Function('t_2')(Symbol('k', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Pow(Integral(Function('t_2')(Symbol('k', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True)))), exp(Integral(Pow(Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}}, then obtain e^{\\hat{\\mathbf{x}}} = (- \\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} + (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}}) e^{\\hat{\\mathbf{x}}} + e^{\\hat{\\mathbf{x}}}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}} and 0 = - \\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} + (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}} and 0 = (- \\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} + (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}}) e^{\\hat{\\mathbf{x}}} and e^{\\hat{\\mathbf{x}}} = (- \\operatorname{F_{N}}{(\\mathbf{B},\\hat{\\mathbf{x}})} + (e^{\\hat{\\mathbf{x}}})^{\\mathbf{B}}) e^{\\hat{\\mathbf{x}}} + e^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 1, "Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["add", 3, "exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\dot{\\mathbf{r}},F_{x})} = \\sin{(F_{x} + \\dot{\\mathbf{r}})}, then obtain \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\nabla{(\\dot{\\mathbf{r}},F_{x})} = \\cos{(F_{x} + \\dot{\\mathbf{r}})}", "derivation": "\\nabla{(\\dot{\\mathbf{r}},F_{x})} = \\sin{(F_{x} + \\dot{\\mathbf{r}})} and F_{x} + \\nabla{(\\dot{\\mathbf{r}},F_{x})} = F_{x} + \\sin{(F_{x} + \\dot{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (F_{x} + \\nabla{(\\dot{\\mathbf{r}},F_{x})}) = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (F_{x} + \\sin{(F_{x} + \\dot{\\mathbf{r}})}) and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\nabla{(\\dot{\\mathbf{r}},F_{x})} = \\cos{(F_{x} + \\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Symbol('F_x', commutative=True), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), cos(Add(Symbol('F_x', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\eta{(g,J)} = J^{g}, then obtain \\int \\log{(\\int \\eta^{J}{(g,J)} dJ)} dg = \\int \\log{(\\int (J^{g})^{J} dJ)} dg", "derivation": "\\eta{(g,J)} = J^{g} and \\eta^{J}{(g,J)} = (J^{g})^{J} and \\int \\eta^{J}{(g,J)} dJ = \\int (J^{g})^{J} dJ and \\log{(\\int \\eta^{J}{(g,J)} dJ)} = \\log{(\\int (J^{g})^{J} dJ)} and \\int \\log{(\\int \\eta^{J}{(g,J)} dJ)} dg = \\int \\log{(\\int (J^{g})^{J} dJ)} dg", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('g', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Symbol('J', commutative=True), Symbol('g', commutative=True)), Symbol('J', commutative=True)))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Pow(Symbol('J', commutative=True), Symbol('g', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["log", 3], "Equality(log(Integral(Pow(Function('\\\\eta')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), log(Integral(Pow(Pow(Symbol('J', commutative=True), Symbol('g', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(log(Integral(Pow(Function('\\\\eta')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(log(Integral(Pow(Pow(Symbol('J', commutative=True), Symbol('g', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{S})} = \\log{(\\cos{(\\mathbf{S})})}, then derive \\frac{d}{d \\mathbf{S}} \\operatorname{v_{t}}{(\\mathbf{S})} = - \\frac{\\sin{(\\mathbf{S})}}{\\cos{(\\mathbf{S})}}, then obtain - \\frac{\\operatorname{v_{t}}{(\\mathbf{S})} \\cos{(\\mathbf{S})}}{\\sin{(\\mathbf{S})}} = - \\frac{\\log{(\\cos{(\\mathbf{S})})} \\cos{(\\mathbf{S})}}{\\sin{(\\mathbf{S})}}", "derivation": "\\operatorname{v_{t}}{(\\mathbf{S})} = \\log{(\\cos{(\\mathbf{S})})} and \\frac{d}{d \\mathbf{S}} \\operatorname{v_{t}}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\log{(\\cos{(\\mathbf{S})})} and \\frac{\\operatorname{v_{t}}{(\\mathbf{S})}}{\\frac{d}{d \\mathbf{S}} \\operatorname{v_{t}}{(\\mathbf{S})}} = \\frac{\\log{(\\cos{(\\mathbf{S})})}}{\\frac{d}{d \\mathbf{S}} \\operatorname{v_{t}}{(\\mathbf{S})}} and \\frac{d}{d \\mathbf{S}} \\operatorname{v_{t}}{(\\mathbf{S})} = - \\frac{\\sin{(\\mathbf{S})}}{\\cos{(\\mathbf{S})}} and - \\frac{\\operatorname{v_{t}}{(\\mathbf{S})} \\cos{(\\mathbf{S})}}{\\sin{(\\mathbf{S})}} = - \\frac{\\log{(\\cos{(\\mathbf{S})})} \\cos{(\\mathbf{S})}}{\\sin{(\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), log(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(log(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], "Equality(Mul(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(Derivative(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1))), Mul(log(cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(Derivative(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\varphi{(h,r)} = h r and Z{(h,r)} = \\frac{\\partial}{\\partial h} (h r - \\varphi{(h,r)}), then obtain - 2 h r + 2 \\frac{d^{2}}{d h^{2}} 0 = - 2 h r + \\frac{d^{2}}{d h^{2}} 0 + \\frac{\\partial^{2}}{\\partial h^{2}} (h r - \\varphi{(h,r)})", "derivation": "\\varphi{(h,r)} = h r and 0 = h r - \\varphi{(h,r)} and \\frac{d}{d h} 0 = \\frac{\\partial}{\\partial h} (h r - \\varphi{(h,r)}) and Z{(h,r)} = \\frac{\\partial}{\\partial h} (h r - \\varphi{(h,r)}) and \\frac{\\partial}{\\partial h} Z{(h,r)} = \\frac{\\partial^{2}}{\\partial h^{2}} (h r - \\varphi{(h,r)}) and \\frac{d}{d h} 0 = Z{(h,r)} and - h r + \\frac{\\partial}{\\partial h} Z{(h,r)} = - h r + \\frac{\\partial^{2}}{\\partial h^{2}} (h r - \\varphi{(h,r)}) and - 2 h r + 2 \\frac{\\partial}{\\partial h} Z{(h,r)} = - 2 h r + \\frac{\\partial^{2}}{\\partial h^{2}} (h r - \\varphi{(h,r)}) + \\frac{\\partial}{\\partial h} Z{(h,r)} and - 2 h r + 2 \\frac{d^{2}}{d h^{2}} 0 = - 2 h r + \\frac{d^{2}}{d h^{2}} 0 + \\frac{\\partial^{2}}{\\partial h^{2}} (h r - \\varphi{(h,r)})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)))"], [["minus", 1, "Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1))), Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)))"], [["minus", 5, "Mul(Symbol('h', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(2)))))"], [["add", 7, "Add(Mul(Integer(-1), Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Derivative(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Integer(2), Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(2))), Derivative(Function('Z')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(2))))), Add(Mul(Integer(-1), Integer(2), Symbol('h', commutative=True), Symbol('r', commutative=True)), Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('h', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{M}{(n,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + n, then derive \\frac{\\partial}{\\partial n} \\mathbf{M}{(n,\\Psi^{\\dagger})} - 1 = 0, then obtain \\frac{\\partial^{2}}{\\partial n^{2}} (- n + \\mathbf{M}{(n,\\Psi^{\\dagger})} + \\frac{\\partial}{\\partial n} \\mathbf{M}{(n,\\Psi^{\\dagger})} - 1) = \\frac{d^{2}}{d n^{2}} \\Psi^{\\dagger}", "derivation": "\\mathbf{M}{(n,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + n and - n + \\mathbf{M}{(n,\\Psi^{\\dagger})} = \\Psi^{\\dagger} and \\frac{\\partial}{\\partial n} (- n + \\mathbf{M}{(n,\\Psi^{\\dagger})}) = \\frac{d}{d n} \\Psi^{\\dagger} and \\frac{\\partial}{\\partial n} \\mathbf{M}{(n,\\Psi^{\\dagger})} - 1 = 0 and - n + \\frac{\\partial}{\\partial n} \\mathbf{M}{(n,\\Psi^{\\dagger})} - 1 = - n and \\frac{\\partial^{2}}{\\partial n^{2}} (- n + \\mathbf{M}{(n,\\Psi^{\\dagger})}) = \\frac{d^{2}}{d n^{2}} \\Psi^{\\dagger} and \\frac{\\partial^{2}}{\\partial n^{2}} (- n + \\mathbf{M}{(n,\\Psi^{\\dagger})} + \\frac{\\partial}{\\partial n} \\mathbf{M}{(n,\\Psi^{\\dagger})} - 1) = \\frac{d^{2}}{d n^{2}} \\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n', commutative=True)))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["minus", 4, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Mul(Integer(-1), Symbol('n', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Tuple(Symbol('n', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('n', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Tuple(Symbol('n', commutative=True), Integer(2))))"]]}, {"prompt": "Given Q{(v_{1},z^{*},A_{z})} = A_{z} + v_{1} - z^{*}, then obtain \\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})} (\\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})})^{z^{*}} = (\\frac{\\partial}{\\partial z^{*}} (A_{z} + v_{1} - z^{*}))^{z^{*}} \\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})}", "derivation": "Q{(v_{1},z^{*},A_{z})} = A_{z} + v_{1} - z^{*} and \\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})} = \\frac{\\partial}{\\partial z^{*}} (A_{z} + v_{1} - z^{*}) and (\\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})})^{z^{*}} = (\\frac{\\partial}{\\partial z^{*}} (A_{z} + v_{1} - z^{*}))^{z^{*}} and \\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})} (\\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})})^{z^{*}} = (\\frac{\\partial}{\\partial z^{*}} (A_{z} + v_{1} - z^{*}))^{z^{*}} \\frac{\\partial}{\\partial z^{*}} Q{(v_{1},z^{*},A_{z})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"], [["times", 3, "Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Pow(Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))), Mul(Pow(Derivative(Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Derivative(Function('Q')(Symbol('v_1', commutative=True), Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t_{1},m_{s})} = m_{s} - t_{1}, then derive \\frac{\\partial}{\\partial t_{1}} \\operatorname{F_{g}}{(t_{1},m_{s})} = -1, then derive A = \\int \\frac{d}{d m_{s}} (-1) dt_{1}, then obtain A = \\int 0 dt_{1}", "derivation": "\\operatorname{F_{g}}{(t_{1},m_{s})} = m_{s} - t_{1} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{F_{g}}{(t_{1},m_{s})} = \\frac{\\partial}{\\partial t_{1}} (m_{s} - t_{1}) and \\frac{\\partial}{\\partial t_{1}} \\operatorname{F_{g}}{(t_{1},m_{s})} = -1 and \\frac{\\partial^{2}}{\\partial m_{s}\\partial t_{1}} \\operatorname{F_{g}}{(t_{1},m_{s})} = \\frac{d}{d m_{s}} (-1) and \\frac{\\partial^{2}}{\\partial m_{s}\\partial t_{1}} (m_{s} - t_{1}) = \\frac{d}{d m_{s}} (-1) and \\int \\frac{\\partial^{2}}{\\partial m_{s}\\partial t_{1}} (m_{s} - t_{1}) dt_{1} = \\int \\frac{d}{d m_{s}} (-1) dt_{1} and A = \\int \\frac{d}{d m_{s}} (-1) dt_{1} and A = \\int 0 dt_{1}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t_1', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('t_1', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('t_1', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 3, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('t_1', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('t_1', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))), Integral(Derivative(Integer(-1), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Symbol('A', commutative=True), Integral(Derivative(Integer(-1), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_derivatives", 7], "Equality(Symbol('A', commutative=True), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{p},A_{1})} = A_{1}^{\\mathbf{p}} and \\sigma_{p}{(\\mathbf{p},A_{1})} = (A_{1}^{\\mathbf{p}})^{A_{1}}, then obtain \\hat{x} \\sigma_{p}{(\\mathbf{p},A_{1})} = \\hat{x} (A_{1}^{\\mathbf{p}})^{A_{1}}", "derivation": "\\mathbf{g}{(\\mathbf{p},A_{1})} = A_{1}^{\\mathbf{p}} and \\mathbf{g}^{A_{1}}{(\\mathbf{p},A_{1})} = (A_{1}^{\\mathbf{p}})^{A_{1}} and \\hat{x} \\mathbf{g}^{A_{1}}{(\\mathbf{p},A_{1})} = \\hat{x} (A_{1}^{\\mathbf{p}})^{A_{1}} and \\sigma_{p}{(\\mathbf{p},A_{1})} = (A_{1}^{\\mathbf{p}})^{A_{1}} and \\mathbf{g}^{A_{1}}{(\\mathbf{p},A_{1})} = \\sigma_{p}{(\\mathbf{p},A_{1})} and \\hat{x} \\sigma_{p}{(\\mathbf{p},A_{1})} = \\hat{x} (A_{1}^{\\mathbf{p}})^{A_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('A_1', commutative=True)))"], [["times", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Function('\\\\sigma_p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(A,\\varepsilon)} = A e^{\\varepsilon}, then obtain \\int \\frac{\\partial}{\\partial A} (\\varepsilon + \\int \\varepsilon \\hat{x}_0{(A,\\varepsilon)} dA) d\\varepsilon = \\int \\frac{\\partial}{\\partial A} (\\varepsilon + \\int A \\varepsilon e^{\\varepsilon} dA) d\\varepsilon", "derivation": "\\hat{x}_0{(A,\\varepsilon)} = A e^{\\varepsilon} and \\varepsilon \\hat{x}_0{(A,\\varepsilon)} = A \\varepsilon e^{\\varepsilon} and \\int \\varepsilon \\hat{x}_0{(A,\\varepsilon)} dA = \\int A \\varepsilon e^{\\varepsilon} dA and \\varepsilon + \\int \\varepsilon \\hat{x}_0{(A,\\varepsilon)} dA = \\varepsilon + \\int A \\varepsilon e^{\\varepsilon} dA and \\frac{\\partial}{\\partial A} (\\varepsilon + \\int \\varepsilon \\hat{x}_0{(A,\\varepsilon)} dA) = \\frac{\\partial}{\\partial A} (\\varepsilon + \\int A \\varepsilon e^{\\varepsilon} dA) and \\int \\frac{\\partial}{\\partial A} (\\varepsilon + \\int \\varepsilon \\hat{x}_0{(A,\\varepsilon)} dA) d\\varepsilon = \\int \\frac{\\partial}{\\partial A} (\\varepsilon + \\int A \\varepsilon e^{\\varepsilon} dA) d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('A', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["add", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))), Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\hat{x}_0')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True), exp(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given m{(\\theta_1,\\eta^{\\prime})} = \\eta^{\\prime} + \\theta_1, then derive \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} m{(\\theta_1,\\eta^{\\prime})}}{\\theta_1} = \\frac{1}{\\theta_1}, then obtain \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} + \\theta_1)}{\\theta_1} = \\frac{1}{\\theta_1}", "derivation": "m{(\\theta_1,\\eta^{\\prime})} = \\eta^{\\prime} + \\theta_1 and \\frac{\\partial}{\\partial \\eta^{\\prime}} m{(\\theta_1,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} + \\theta_1) and \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} m{(\\theta_1,\\eta^{\\prime})}}{\\theta_1} = \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} + \\theta_1)}{\\theta_1} and \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} m{(\\theta_1,\\eta^{\\prime})}}{\\theta_1} = \\frac{1}{\\theta_1} and \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} + \\theta_1)}{\\theta_1} = \\frac{1}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))"]]}, {"prompt": "Given n{(\\hat{H})} = \\log{(\\cos{(\\hat{H})})} and \\mathbf{M}{(\\hat{H})} = n{(\\hat{H})} + \\log{(\\cos{(\\hat{H})})}, then obtain 2 \\log{(\\cos{(\\hat{H})})} = n{(\\hat{H})} + \\log{(\\cos{(\\hat{H})})}", "derivation": "n{(\\hat{H})} = \\log{(\\cos{(\\hat{H})})} and \\mathbf{M}{(\\hat{H})} = n{(\\hat{H})} + \\log{(\\cos{(\\hat{H})})} and \\mathbf{M}{(\\hat{H})} = 2 \\log{(\\cos{(\\hat{H})})} and 2 \\log{(\\cos{(\\hat{H})})} = n{(\\hat{H})} + \\log{(\\cos{(\\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\hat{H}', commutative=True)), log(cos(Symbol('\\\\hat{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True)), Add(Function('n')(Symbol('\\\\hat{H}', commutative=True)), log(cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), log(cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), log(cos(Symbol('\\\\hat{H}', commutative=True)))), Add(Function('n')(Symbol('\\\\hat{H}', commutative=True)), log(cos(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given H{(f)} = \\log{(f)}, then derive - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + \\mathbf{J} = - \\frac{A_{x}^{2}}{2} - A_{x} \\log{(f)} + f^{\\prime}, then obtain - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + \\mathbf{J} = - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + f^{\\prime}", "derivation": "H{(f)} = \\log{(f)} and A_{x} + H{(f)} = A_{x} + \\log{(f)} and - A_{x} - H{(f)} = - A_{x} - \\log{(f)} and \\int (- A_{x} - H{(f)}) dA_{x} = \\int (- A_{x} - \\log{(f)}) dA_{x} and - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + \\mathbf{J} = - \\frac{A_{x}^{2}}{2} - A_{x} \\log{(f)} + f^{\\prime} and - \\frac{A_{x}^{2}}{2} - A_{x} \\log{(f)} + \\mathbf{J} = - \\frac{A_{x}^{2}}{2} - A_{x} \\log{(f)} + f^{\\prime} and - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + \\mathbf{J} = - \\frac{A_{x}^{2}}{2} - A_{x} H{(f)} + f^{\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["add", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Function('H')(Symbol('f', commutative=True))), Add(Symbol('A_x', commutative=True), log(Symbol('f', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), log(Symbol('f', commutative=True)))))"], [["integrate", 3, "Symbol('A_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('f', commutative=True)))), Tuple(Symbol('A_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), log(Symbol('f', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), Function('H')(Symbol('f', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), log(Symbol('f', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), log(Symbol('f', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), log(Symbol('f', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), Function('H')(Symbol('f', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_x', commutative=True), Function('H')(Symbol('f', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given V{(m)} = \\cos{(\\sin{(m)})} and Q{(m)} = V{(m)} - \\sin{(m)}, then obtain - \\cos{(m)} + \\frac{d}{d m} V{(m)} = - \\sin{(\\sin{(m)})} \\cos{(m)} - \\cos{(m)}", "derivation": "V{(m)} = \\cos{(\\sin{(m)})} and V{(m)} - \\sin{(m)} = - \\sin{(m)} + \\cos{(\\sin{(m)})} and Q{(m)} = V{(m)} - \\sin{(m)} and \\frac{d}{d m} Q{(m)} = \\frac{d}{d m} (V{(m)} - \\sin{(m)}) and \\frac{d}{d m} Q{(m)} = \\frac{d}{d m} (- \\sin{(m)} + \\cos{(\\sin{(m)})}) and \\frac{d}{d m} (V{(m)} - \\sin{(m)}) = \\frac{d}{d m} (- \\sin{(m)} + \\cos{(\\sin{(m)})}) and - \\cos{(m)} + \\frac{d}{d m} V{(m)} = - \\sin{(\\sin{(m)})} \\cos{(m)} - \\cos{(m)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True))))"], [["minus", 1, "sin(Symbol('m', commutative=True))"], "Equality(Add(Function('V')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), cos(sin(Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('m', commutative=True)), Add(Function('V')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True)))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Function('V')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('Q')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), cos(sin(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Function('V')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), cos(sin(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Integer(-1), cos(Symbol('m', commutative=True))), Derivative(Function('V')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(sin(Symbol('m', commutative=True))), cos(Symbol('m', commutative=True))), Mul(Integer(-1), cos(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given l{(A,\\dot{x})} = A \\dot{x}, then obtain (\\int (A \\dot{x} + l{(A,\\dot{x})}) d\\dot{x})^{2} = (\\int 2 A \\dot{x} d\\dot{x})^{2}", "derivation": "l{(A,\\dot{x})} = A \\dot{x} and A \\dot{x} + l{(A,\\dot{x})} = 2 A \\dot{x} and \\int (A \\dot{x} + l{(A,\\dot{x})}) d\\dot{x} = \\int 2 A \\dot{x} d\\dot{x} and (\\int (A \\dot{x} + l{(A,\\dot{x})}) d\\dot{x})^{2} = (\\int 2 A \\dot{x} d\\dot{x})^{2}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "Mul(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('l')(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('l')(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('l')(Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\hat{H}{(E,r_{0})} = \\cos{(E r_{0})}, then obtain E r_{0} \\hat{H}^{2}{(E,r_{0})} = E r_{0} \\hat{H}{(E,r_{0})} \\cos{(E r_{0})}", "derivation": "\\hat{H}{(E,r_{0})} = \\cos{(E r_{0})} and E r_{0} \\hat{H}{(E,r_{0})} = E r_{0} \\cos{(E r_{0})} and E r_{0} \\hat{H}{(E,r_{0})} \\cos{(E r_{0})} = E r_{0} \\cos^{2}{(E r_{0})} and E r_{0} \\hat{H}^{2}{(E,r_{0})} = E r_{0} \\hat{H}{(E,r_{0})} \\cos{(E r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True))))"], [["times", 1, "Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), Function('\\\\hat{H}')(Symbol('E', commutative=True), Symbol('r_0', commutative=True))), Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True)))))"], [["times", 2, "cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), Function('\\\\hat{H}')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True)))), Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), Pow(cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True), Function('\\\\hat{H}')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given h{(C_{d})} = \\cos{(e^{C_{d}})}, then obtain h{(C_{d})} + \\frac{d}{d C_{d}} h{(C_{d})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} - 1 = \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} h{(C_{d})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} - 1", "derivation": "h{(C_{d})} = \\cos{(e^{C_{d}})} and \\frac{d}{d C_{d}} h{(C_{d})} = \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} and h{(C_{d})} + \\frac{d}{d C_{d}} h{(C_{d})} = \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} h{(C_{d})} and h{(C_{d})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} = \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} and h{(C_{d})} + \\frac{d}{d C_{d}} h{(C_{d})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} - 1 = \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} h{(C_{d})} + \\frac{d}{d C_{d}} \\cos{(e^{C_{d}})} - 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Add(Function('h')(Symbol('C_d', commutative=True)), Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(cos(exp(Symbol('C_d', commutative=True))), Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('h')(Symbol('C_d', commutative=True)), Derivative(cos(exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(cos(exp(Symbol('C_d', commutative=True))), Derivative(cos(exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["add", 4, "Add(Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Function('h')(Symbol('C_d', commutative=True)), Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1)), Add(cos(exp(Symbol('C_d', commutative=True))), Derivative(Function('h')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(C)} = \\log{(C)}, then obtain \\frac{C^{2} \\operatorname{v_{2}}^{2}{(C)}}{\\log{(C)}^{2}} - 1 = \\frac{C^{2} \\operatorname{v_{2}}{(C)}}{\\log{(C)}} - 1", "derivation": "\\operatorname{v_{2}}{(C)} = \\log{(C)} and \\frac{\\operatorname{v_{2}}{(C)}}{\\log{(C)}} = 1 and \\frac{C \\operatorname{v_{2}}{(C)}}{\\log{(C)}} = C and \\frac{C^{2} \\operatorname{v_{2}}{(C)}}{\\log{(C)}} = C^{2} and \\frac{C^{2} \\operatorname{v_{2}}{(C)}}{\\log{(C)}} - 1 = C^{2} - 1 and \\frac{C^{2} \\operatorname{v_{2}}^{2}{(C)}}{\\log{(C)}^{2}} - 1 = \\frac{C^{2} \\operatorname{v_{2}}{(C)}}{\\log{(C)}} - 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["divide", 1, "log(Symbol('C', commutative=True))"], "Equality(Mul(Function('v_2')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('v_2')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Symbol('C', commutative=True))"], [["times", 3, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Function('v_2')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Pow(Symbol('C', commutative=True), Integer(2)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Function('v_2')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Integer(-1)), Add(Pow(Symbol('C', commutative=True), Integer(2)), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Pow(Function('v_2')(Symbol('C', commutative=True)), Integer(2)), Pow(log(Symbol('C', commutative=True)), Integer(-2))), Integer(-1)), Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Function('v_2')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Integer(-1)))"]]}, {"prompt": "Given \\varphi{(\\hat{x})} = \\sin{(\\hat{x})}, then obtain -1 = - 2 \\varphi{(\\hat{x})} + 2 \\sin{(\\hat{x})} - 1", "derivation": "\\varphi{(\\hat{x})} = \\sin{(\\hat{x})} and 0 = - \\varphi{(\\hat{x})} + \\sin{(\\hat{x})} and - \\varphi{(\\hat{x})} = - 2 \\varphi{(\\hat{x})} + \\sin{(\\hat{x})} and 0 = - 2 \\varphi{(\\hat{x})} + 2 \\sin{(\\hat{x})} and -1 = - 2 \\varphi{(\\hat{x})} + 2 \\sin{(\\hat{x})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 1, "Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 2, "Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{x}', commutative=True)))))"], [["minus", 4, 1], "Equality(Integer(-1), Add(Mul(Integer(-1), Integer(2), Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{x}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given l{(T)} = \\sin{(T)}, then derive \\int l{(T)} dT = \\rho - \\cos{(T)}, then obtain - \\tilde{g} = - T - y", "derivation": "l{(T)} = \\sin{(T)} and \\int l{(T)} dT = \\int \\sin{(T)} dT and \\int l{(T)} dT = \\rho - \\cos{(T)} and \\frac{d}{d \\rho} \\int l{(T)} dT = \\frac{\\partial}{\\partial \\rho} (\\rho - \\cos{(T)}) and \\int \\frac{d}{d \\rho} \\int l{(T)} dT dT = \\int \\frac{\\partial}{\\partial \\rho} (\\rho - \\cos{(T)}) dT and - \\int \\frac{d}{d \\rho} \\int l{(T)} dT dT = - \\int \\frac{\\partial}{\\partial \\rho} (\\rho - \\cos{(T)}) dT and - \\tilde{g} = - T - y", "srepr_derivation": [["get_premise", "Equality(Function('l')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Integral(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Derivative(Integral(Function('l')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Derivative(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\chi,A_{z})} = e^{A_{z}^{\\chi}}, then obtain 2 \\mathbf{f}{(\\chi,A_{z})} + 2 e^{A_{z}^{\\chi}} = \\mathbf{f}{(\\chi,A_{z})} + 3 e^{A_{z}^{\\chi}}", "derivation": "\\mathbf{f}{(\\chi,A_{z})} = e^{A_{z}^{\\chi}} and 2 \\mathbf{f}{(\\chi,A_{z})} = \\mathbf{f}{(\\chi,A_{z})} + e^{A_{z}^{\\chi}} and 2 \\mathbf{f}{(\\chi,A_{z})} + e^{A_{z}^{\\chi}} = \\mathbf{f}{(\\chi,A_{z})} + 2 e^{A_{z}^{\\chi}} and 2 \\mathbf{f}{(\\chi,A_{z})} + 2 e^{A_{z}^{\\chi}} = \\mathbf{f}{(\\chi,A_{z})} + 3 e^{A_{z}^{\\chi}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True)), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True))), Add(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True)), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["add", 2, "exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True))), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(2), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True))))))"], [["add", 3, "exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(2), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True))))), Add(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(3), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\chi', commutative=True))))))"]]}, {"prompt": "Given h{(t_{1},\\sigma_p)} = \\frac{t_{1}}{\\sigma_p} and y{(t_{1},\\sigma_p)} = \\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p, then obtain (\\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p)^{\\sigma_p} = y^{\\sigma_p}{(t_{1},\\sigma_p)}", "derivation": "h{(t_{1},\\sigma_p)} = \\frac{t_{1}}{\\sigma_p} and \\int h{(t_{1},\\sigma_p)} d\\sigma_p = \\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p and (\\int h{(t_{1},\\sigma_p)} d\\sigma_p)^{\\sigma_p} = (\\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p)^{\\sigma_p} and y{(t_{1},\\sigma_p)} = \\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p and (\\int h{(t_{1},\\sigma_p)} d\\sigma_p)^{\\sigma_p} = y^{\\sigma_p}{(t_{1},\\sigma_p)} and (\\int \\frac{t_{1}}{\\sigma_p} d\\sigma_p)^{\\sigma_p} = y^{\\sigma_p}{(t_{1},\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integral(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0, then derive 0 = \\nabla - \\operatorname{m_{s}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}, then obtain 0 = \\frac{\\nabla - \\operatorname{m_{s}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}}{- \\operatorname{m_{s}}{(\\varepsilon_0)} + \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0}", "derivation": "\\operatorname{m_{s}}{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and 0 = - \\operatorname{m_{s}}{(\\varepsilon_0)} + \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and 0 = \\nabla - \\operatorname{m_{s}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)} and 0 = \\frac{\\nabla - \\operatorname{m_{s}}{(\\varepsilon_0)} - \\cos{(\\varepsilon_0)}}{- \\operatorname{m_{s}}{(\\varepsilon_0)} + \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True)), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 1, "Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(0), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"]]}, {"prompt": "Given \\dot{x}{(\\hat{H}_{\\lambda})} = e^{e^{\\hat{H}_{\\lambda}}}, then obtain e^{\\frac{(\\dot{x}{(\\hat{H}_{\\lambda})} e^{- e^{\\hat{H}_{\\lambda}}})^{\\hat{H}_{\\lambda}}}{\\dot{x}{(\\hat{H}_{\\lambda})}}} = e^{\\frac{1}{\\dot{x}{(\\hat{H}_{\\lambda})}}}", "derivation": "\\dot{x}{(\\hat{H}_{\\lambda})} = e^{e^{\\hat{H}_{\\lambda}}} and \\dot{x}{(\\hat{H}_{\\lambda})} e^{- e^{\\hat{H}_{\\lambda}}} = 1 and (\\dot{x}{(\\hat{H}_{\\lambda})} e^{- e^{\\hat{H}_{\\lambda}}})^{\\hat{H}_{\\lambda}} = 1 and \\frac{(\\dot{x}{(\\hat{H}_{\\lambda})} e^{- e^{\\hat{H}_{\\lambda}}})^{\\hat{H}_{\\lambda}}}{\\dot{x}{(\\hat{H}_{\\lambda})}} = \\frac{1}{\\dot{x}{(\\hat{H}_{\\lambda})}} and e^{\\frac{(\\dot{x}{(\\hat{H}_{\\lambda})} e^{- e^{\\hat{H}_{\\lambda}}})^{\\hat{H}_{\\lambda}}}{\\dot{x}{(\\hat{H}_{\\lambda})}}} = e^{\\frac{1}{\\dot{x}{(\\hat{H}_{\\lambda})}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Integer(1))"], [["power", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))"], [["divide", 3, "Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Pow(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))"], [["exp", 4], "Equality(exp(Mul(Pow(Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)))), exp(Pow(Function('\\\\dot{x}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given H{(F_{g},L)} = \\frac{\\partial}{\\partial L} \\frac{\\log{(L)}}{F_{g}}, then derive H{(F_{g},L)} - \\log{(L)} + 1 = - \\log{(L)} + 1 + \\frac{1}{F_{g} L}, then obtain \\frac{H{(F_{g},L)} - \\log{(L)} + 1 + \\frac{1}{F_{g}}}{L} = \\frac{- \\log{(L)} + 1 + \\frac{1}{F_{g}} + \\frac{1}{F_{g} L}}{L}", "derivation": "H{(F_{g},L)} = \\frac{\\partial}{\\partial L} \\frac{\\log{(L)}}{F_{g}} and H{(F_{g},L)} - \\log{(L)} = - \\log{(L)} + \\frac{\\partial}{\\partial L} \\frac{\\log{(L)}}{F_{g}} and H{(F_{g},L)} - \\log{(L)} + 1 = - \\log{(L)} + \\frac{\\partial}{\\partial L} \\frac{\\log{(L)}}{F_{g}} + 1 and H{(F_{g},L)} - \\log{(L)} + 1 = - \\log{(L)} + 1 + \\frac{1}{F_{g} L} and H{(F_{g},L)} - \\log{(L)} + 1 + \\frac{1}{F_{g}} = - \\log{(L)} + 1 + \\frac{1}{F_{g}} + \\frac{1}{F_{g} L} and \\frac{H{(F_{g},L)} - \\log{(L)} + 1 + \\frac{1}{F_{g}}}{L} = \\frac{- \\log{(L)} + 1 + \\frac{1}{F_{g}} + \\frac{1}{F_{g} L}}{L}", "srepr_derivation": [["renaming_premise", "Equality(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 1, "log(Symbol('L', commutative=True))"], "Equality(Add(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('L', commutative=True))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1)), Add(Mul(Integer(-1), log(Symbol('L', commutative=True))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1)), Add(Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1)))))"], [["add", 4, "Pow(Symbol('F_g', commutative=True), Integer(-1))"], "Equality(Add(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1), Pow(Symbol('F_g', commutative=True), Integer(-1))), Add(Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1), Pow(Symbol('F_g', commutative=True), Integer(-1)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1)))))"], [["times", 5, "Pow(Symbol('L', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Function('H')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1), Pow(Symbol('F_g', commutative=True), Integer(-1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('L', commutative=True))), Integer(1), Pow(Symbol('F_g', commutative=True), Integer(-1)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('L', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\hat{x}_0{(P_{e})} = e^{P_{e}}, then derive \\frac{\\int \\hat{x}_0{(P_{e})} dP_{e}}{P_{e}} = \\frac{\\hat{p} + e^{P_{e}}}{P_{e}}, then obtain \\frac{\\int \\hat{x}_0{(P_{e})} dP_{e}}{P_{e}} = \\frac{\\hat{p} + \\hat{x}_0{(P_{e})}}{P_{e}}", "derivation": "\\hat{x}_0{(P_{e})} = e^{P_{e}} and \\int \\hat{x}_0{(P_{e})} dP_{e} = \\int e^{P_{e}} dP_{e} and \\frac{\\int \\hat{x}_0{(P_{e})} dP_{e}}{P_{e}} = \\frac{\\int e^{P_{e}} dP_{e}}{P_{e}} and \\frac{\\int \\hat{x}_0{(P_{e})} dP_{e}}{P_{e}} = \\frac{\\hat{p} + e^{P_{e}}}{P_{e}} and \\frac{\\int \\hat{x}_0{(P_{e})} dP_{e}}{P_{e}} = \\frac{\\hat{p} + \\hat{x}_0{(P_{e})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["divide", 2, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('P_e', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given B{(t)} = e^{t}, then derive \\int (t + \\sin{(B{(t)})}) dt = \\mathbf{J}_P + \\frac{t^{2}}{2} + \\operatorname{Si}{(e^{t})}, then obtain B{(t)} \\int (t + \\sin{(B{(t)})}) dt = (\\mathbf{J}_P + \\frac{t^{2}}{2} + \\operatorname{Si}{(e^{t})}) B{(t)}", "derivation": "B{(t)} = e^{t} and \\sin{(B{(t)})} = \\sin{(e^{t})} and t + \\sin{(B{(t)})} = t + \\sin{(e^{t})} and \\int (t + \\sin{(B{(t)})}) dt = \\int (t + \\sin{(e^{t})}) dt and \\int (t + \\sin{(B{(t)})}) dt = \\mathbf{J}_P + \\frac{t^{2}}{2} + \\operatorname{Si}{(e^{t})} and B{(t)} \\int (t + \\sin{(B{(t)})}) dt = (\\mathbf{J}_P + \\frac{t^{2}}{2} + \\operatorname{Si}{(e^{t})}) B{(t)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["sin", 1], "Equality(sin(Function('B')(Symbol('t', commutative=True))), sin(exp(Symbol('t', commutative=True))))"], [["add", 2, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), sin(Function('B')(Symbol('t', commutative=True)))), Add(Symbol('t', commutative=True), sin(exp(Symbol('t', commutative=True)))))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Symbol('t', commutative=True), sin(Function('B')(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('t', commutative=True), sin(exp(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Symbol('t', commutative=True), sin(Function('B')(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Si(exp(Symbol('t', commutative=True)))))"], [["times", 5, "Function('B')(Symbol('t', commutative=True))"], "Equality(Mul(Function('B')(Symbol('t', commutative=True)), Integral(Add(Symbol('t', commutative=True), sin(Function('B')(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Si(exp(Symbol('t', commutative=True)))), Function('B')(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\Psi)} = \\Psi, then obtain \\frac{\\partial}{\\partial t} \\frac{\\theta_{1}^{2}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}} = \\frac{\\partial}{\\partial t} \\frac{\\Psi \\theta_{1}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}}", "derivation": "\\theta_{1}{(\\Psi)} = \\Psi and \\theta_{1}^{2}{(\\Psi)} = \\Psi \\theta_{1}{(\\Psi)} and \\frac{\\theta_{1}^{2}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}} = \\frac{\\Psi \\theta_{1}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}} and \\frac{\\partial}{\\partial t} \\frac{\\theta_{1}^{2}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}} = \\frac{\\partial}{\\partial t} \\frac{\\Psi \\theta_{1}{(\\Psi)}}{\\hat{H}_{\\lambda}{(t,\\Psi)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], [["times", 1, "Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True))"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True)), Integer(2))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True)), Integer(2))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(\\mu,i,p)} = \\frac{i p}{\\mu}, then obtain - \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu} - \\frac{\\partial}{\\partial p} v{(\\mu,i,p)} = - 2 \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu}", "derivation": "v{(\\mu,i,p)} = \\frac{i p}{\\mu} and \\frac{\\partial}{\\partial p} v{(\\mu,i,p)} = \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu} and - \\frac{\\partial}{\\partial p} v{(\\mu,i,p)} = - \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu} and - \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu} - \\frac{\\partial}{\\partial p} v{(\\mu,i,p)} = - 2 \\frac{\\partial}{\\partial p} \\frac{i p}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('v')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["add", 3, "Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('v')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Mul(Integer(-1), Integer(2), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('i', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(B)} = e^{e^{B}} and \\mathbf{E}{(B)} = \\omega{(B)} e^{\\omega{(B)}} \\int e^{e^{B}} dB, then obtain \\mathbf{E}{(B)} = \\omega{(B)} e^{e^{e^{B}}} \\int \\omega{(B)} dB", "derivation": "\\omega{(B)} = e^{e^{B}} and \\int \\omega{(B)} dB = \\int e^{e^{B}} dB and e^{\\omega{(B)}} = e^{e^{e^{B}}} and \\omega{(B)} e^{\\omega{(B)}} = \\omega{(B)} e^{e^{e^{B}}} and \\omega{(B)} e^{\\omega{(B)}} \\int e^{e^{B}} dB = \\omega{(B)} e^{e^{e^{B}}} \\int e^{e^{B}} dB and \\mathbf{E}{(B)} = \\omega{(B)} e^{\\omega{(B)}} \\int e^{e^{B}} dB and \\omega{(B)} e^{\\omega{(B)}} \\int \\omega{(B)} dB = \\omega{(B)} e^{e^{e^{B}}} \\int \\omega{(B)} dB and \\mathbf{E}{(B)} = \\omega{(B)} e^{\\omega{(B)}} \\int \\omega{(B)} dB and \\mathbf{E}{(B)} = \\omega{(B)} e^{e^{e^{B}}} \\int \\omega{(B)} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('B', commutative=True)), exp(exp(Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(exp(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\omega')(Symbol('B', commutative=True))), exp(exp(exp(Symbol('B', commutative=True)))))"], [["times", 3, "Function('\\\\omega')(Symbol('B', commutative=True))"], "Equality(Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(Function('\\\\omega')(Symbol('B', commutative=True)))), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(exp(exp(Symbol('B', commutative=True))))))"], [["times", 4, "Integral(exp(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(Function('\\\\omega')(Symbol('B', commutative=True))), Integral(exp(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(exp(exp(Symbol('B', commutative=True)))), Integral(exp(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('B', commutative=True)), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(Function('\\\\omega')(Symbol('B', commutative=True))), Integral(exp(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(Function('\\\\omega')(Symbol('B', commutative=True))), Integral(Function('\\\\omega')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(exp(exp(Symbol('B', commutative=True)))), Integral(Function('\\\\omega')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Function('\\\\mathbf{E}')(Symbol('B', commutative=True)), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(Function('\\\\omega')(Symbol('B', commutative=True))), Integral(Function('\\\\omega')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Function('\\\\mathbf{E}')(Symbol('B', commutative=True)), Mul(Function('\\\\omega')(Symbol('B', commutative=True)), exp(exp(exp(Symbol('B', commutative=True)))), Integral(Function('\\\\omega')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{H},\\Psi)} = \\frac{\\Psi}{\\mathbf{H}}, then obtain e^{\\mathbf{H}} = (e^{\\frac{\\Psi}{\\mathbf{H} \\mathbf{E}{(\\mathbf{H},\\Psi)}}})^{\\mathbf{H}}", "derivation": "\\mathbf{E}{(\\mathbf{H},\\Psi)} = \\frac{\\Psi}{\\mathbf{H}} and 1 = \\frac{\\Psi}{\\mathbf{H} \\mathbf{E}{(\\mathbf{H},\\Psi)}} and e = e^{\\frac{\\Psi}{\\mathbf{H} \\mathbf{E}{(\\mathbf{H},\\Psi)}}} and e^{\\mathbf{H}} = (e^{\\frac{\\Psi}{\\mathbf{H} \\mathbf{E}{(\\mathbf{H},\\Psi)}}})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["divide", 1, "Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["exp", 2], "Equality(E, exp(Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)))))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(exp(Symbol('\\\\mathbf{H}', commutative=True)), Pow(exp(Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\psi^*,\\omega)} = \\psi^* \\cos{(\\omega)}, then derive \\frac{\\partial}{\\partial \\psi^*} \\hat{p}{(\\psi^*,\\omega)} = \\cos{(\\omega)}, then obtain \\frac{\\partial^{2}}{\\partial \\psi^*\\partial \\omega} \\hat{p}{(\\psi^*,\\omega)} = - \\sin{(\\omega)}", "derivation": "\\hat{p}{(\\psi^*,\\omega)} = \\psi^* \\cos{(\\omega)} and \\frac{\\partial}{\\partial \\psi^*} \\hat{p}{(\\psi^*,\\omega)} = \\frac{\\partial}{\\partial \\psi^*} \\psi^* \\cos{(\\omega)} and \\frac{\\partial}{\\partial \\psi^*} \\hat{p}{(\\psi^*,\\omega)} = \\cos{(\\omega)} and \\frac{\\partial^{2}}{\\partial \\omega\\partial \\psi^*} \\hat{p}{(\\psi^*,\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)} and \\frac{\\partial^{2}}{\\partial \\psi^*\\partial \\omega} \\hat{p}{(\\psi^*,\\omega)} = - \\sin{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), cos(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(Z)} = \\cos{(Z)}, then derive \\int \\frac{\\varepsilon{(Z)} - 1}{\\cos{(Z)} - 1} dZ = Z + \\hat{x}, then obtain \\int 1 dZ = Z + \\hat{x}", "derivation": "\\varepsilon{(Z)} = \\cos{(Z)} and \\varepsilon{(Z)} - 1 = \\cos{(Z)} - 1 and \\frac{\\varepsilon{(Z)} - 1}{\\cos{(Z)} - 1} = 1 and \\int \\frac{\\varepsilon{(Z)} - 1}{\\cos{(Z)} - 1} dZ = \\int 1 dZ and \\int \\frac{\\varepsilon{(Z)} - 1}{\\cos{(Z)} - 1} dZ = Z + \\hat{x} and \\int 1 dZ = Z + \\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1)), Add(cos(Symbol('Z', commutative=True)), Integer(-1)))"], [["divide", 2, "Add(cos(Symbol('Z', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('Z', commutative=True)), Integer(-1)), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('Z', commutative=True)), Integer(-1)), Integer(-1))), Tuple(Symbol('Z', commutative=True))), Integral(Integer(1), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Add(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('Z', commutative=True)), Integer(-1)), Integer(-1))), Tuple(Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Integer(1), Tuple(Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\omega,\\varphi)} = \\cos{(\\frac{\\omega}{\\varphi})} and \\mathbf{S}{(\\omega,\\varphi)} = - \\cos{(\\frac{\\omega}{\\varphi})}, then obtain \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\omega} (- \\hat{p}{(\\omega,\\varphi)})^{\\omega} = \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\omega} \\mathbf{S}^{\\omega}{(\\omega,\\varphi)}", "derivation": "\\hat{p}{(\\omega,\\varphi)} = \\cos{(\\frac{\\omega}{\\varphi})} and - \\hat{p}{(\\omega,\\varphi)} = - \\cos{(\\frac{\\omega}{\\varphi})} and \\mathbf{S}{(\\omega,\\varphi)} = - \\cos{(\\frac{\\omega}{\\varphi})} and (- \\hat{p}{(\\omega,\\varphi)})^{\\omega} = (- \\cos{(\\frac{\\omega}{\\varphi})})^{\\omega} and \\frac{\\partial}{\\partial \\omega} (- \\hat{p}{(\\omega,\\varphi)})^{\\omega} = \\frac{\\partial}{\\partial \\omega} (- \\cos{(\\frac{\\omega}{\\varphi})})^{\\omega} and \\frac{\\partial}{\\partial \\omega} (- \\hat{p}{(\\omega,\\varphi)})^{\\omega} = \\frac{\\partial}{\\partial \\omega} \\mathbf{S}^{\\omega}{(\\omega,\\varphi)} and \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\omega} (- \\hat{p}{(\\omega,\\varphi)})^{\\omega} = \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\omega} \\mathbf{S}^{\\omega}{(\\omega,\\varphi)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Integer(-1), cos(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), cos(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Pow(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(\\varepsilon_0,\\dot{y})} = \\varepsilon_0^{\\dot{y}}, then obtain \\varepsilon_0 - \\log{(\\hat{x}_0{(\\varepsilon_0,\\dot{y})})} = \\varepsilon_0 - \\log{(\\varepsilon_0^{\\dot{y}})}", "derivation": "\\hat{x}_0{(\\varepsilon_0,\\dot{y})} = \\varepsilon_0^{\\dot{y}} and \\log{(\\hat{x}_0{(\\varepsilon_0,\\dot{y})})} = \\log{(\\varepsilon_0^{\\dot{y}})} and - \\varepsilon_0 + \\log{(\\hat{x}_0{(\\varepsilon_0,\\dot{y})})} = - \\varepsilon_0 + \\log{(\\varepsilon_0^{\\dot{y}})} and \\varepsilon_0 - \\log{(\\hat{x}_0{(\\varepsilon_0,\\dot{y})})} = \\varepsilon_0 - \\log{(\\varepsilon_0^{\\dot{y}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), log(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), log(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), log(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = \\dot{z} - \\nabla, then obtain \\dot{z} \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} + \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = 2 \\dot{z} - \\nabla", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = \\dot{z} - \\nabla and \\dot{z} \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = \\dot{z} (\\dot{z} - \\nabla) and \\frac{\\partial}{\\partial \\dot{z}} \\dot{z} \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\dot{z} (\\dot{z} - \\nabla) and \\dot{z} \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} + \\operatorname{f_{\\mathbf{p}}}{(\\nabla,\\dot{z})} = 2 \\dot{z} - \\nabla", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(A_{2},t)} = A_{2} t, then obtain (A_{2} t)^{t} \\dot{y}{(A_{2},t)} + (A_{2} t)^{t} = A_{2} t (A_{2} t)^{t} + (A_{2} t)^{t}", "derivation": "\\dot{y}{(A_{2},t)} = A_{2} t and \\dot{y}^{t}{(A_{2},t)} = (A_{2} t)^{t} and \\dot{y}{(A_{2},t)} \\dot{y}^{t}{(A_{2},t)} = A_{2} t \\dot{y}^{t}{(A_{2},t)} and \\dot{y}{(A_{2},t)} \\dot{y}^{t}{(A_{2},t)} + \\dot{y}^{t}{(A_{2},t)} = A_{2} t \\dot{y}^{t}{(A_{2},t)} + \\dot{y}^{t}{(A_{2},t)} and (A_{2} t)^{t} \\dot{y}{(A_{2},t)} + (A_{2} t)^{t} = A_{2} t (A_{2} t)^{t} + (A_{2} t)^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["times", 1, "Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["add", 3, "Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))"], "Equality(Add(Mul(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Pow(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Function('\\\\dot{y}')(Symbol('A_2', commutative=True), Symbol('t', commutative=True))), Pow(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True), Pow(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(Mul(Symbol('A_2', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(t_{1},v_{y})} = t_{1} + \\sin{(v_{y})} and \\operatorname{n_{1}}{(t_{1},v_{y})} = (t_{1} + \\sin{(v_{y})})^{v_{y}}, then obtain \\int \\mathbf{J}^{v_{y}}{(t_{1},v_{y})} dt_{1} = \\int (t_{1} + \\sin{(v_{y})})^{v_{y}} dt_{1}", "derivation": "\\mathbf{J}{(t_{1},v_{y})} = t_{1} + \\sin{(v_{y})} and \\mathbf{J}^{v_{y}}{(t_{1},v_{y})} = (t_{1} + \\sin{(v_{y})})^{v_{y}} and \\operatorname{n_{1}}{(t_{1},v_{y})} = (t_{1} + \\sin{(v_{y})})^{v_{y}} and \\operatorname{n_{1}}{(t_{1},v_{y})} = \\mathbf{J}^{v_{y}}{(t_{1},v_{y})} and \\int \\operatorname{n_{1}}{(t_{1},v_{y})} dt_{1} = \\int (t_{1} + \\sin{(v_{y})})^{v_{y}} dt_{1} and \\int \\mathbf{J}^{v_{y}}{(t_{1},v_{y})} dt_{1} = \\int (t_{1} + \\sin{(v_{y})})^{v_{y}} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('t_1', commutative=True), sin(Symbol('v_y', commutative=True))))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('n_1')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"], [["integrate", 3, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Pow(Add(Symbol('t_1', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Pow(Add(Symbol('t_1', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given W{(F_{g})} = \\frac{d}{d F_{g}} \\cos{(F_{g})} and \\operatorname{z^{*}}{(F_{g})} = \\frac{d}{d F_{g}} \\cos{(F_{g})} and \\operatorname{g_{\\varepsilon}}{(F_{g})} = W{(F_{g})} - \\frac{d}{d F_{g}} \\cos{(F_{g})}, then derive W{(F_{g})} = - \\sin{(F_{g})}, then obtain \\operatorname{g_{\\varepsilon}}{(F_{g})} = \\operatorname{z^{*}}{(F_{g})} + \\sin{(F_{g})}", "derivation": "W{(F_{g})} = \\frac{d}{d F_{g}} \\cos{(F_{g})} and \\operatorname{z^{*}}{(F_{g})} = \\frac{d}{d F_{g}} \\cos{(F_{g})} and W{(F_{g})} = - \\sin{(F_{g})} and \\operatorname{g_{\\varepsilon}}{(F_{g})} = W{(F_{g})} - \\frac{d}{d F_{g}} \\cos{(F_{g})} and \\frac{d}{d F_{g}} \\cos{(F_{g})} = - \\sin{(F_{g})} and \\operatorname{z^{*}}{(F_{g})} = W{(F_{g})} and \\operatorname{g_{\\varepsilon}}{(F_{g})} = W{(F_{g})} + \\sin{(F_{g})} and \\operatorname{g_{\\varepsilon}}{(F_{g})} = \\operatorname{z^{*}}{(F_{g})} + \\sin{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('F_g', commutative=True)), Derivative(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('F_g', commutative=True)), Derivative(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('W')(Symbol('F_g', commutative=True)), Mul(Integer(-1), sin(Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_g', commutative=True)), Add(Function('W')(Symbol('F_g', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('z^*')(Symbol('F_g', commutative=True)), Function('W')(Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_g', commutative=True)), Add(Function('W')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_g', commutative=True)), Add(Function('z^*')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(T,A_{y})} = \\frac{A_{y}}{T}, then obtain \\frac{A_{y} \\operatorname{C_{1}}^{A_{y}}{(T,A_{y})} \\frac{\\partial}{\\partial T} \\operatorname{C_{1}}{(T,A_{y})}}{\\operatorname{C_{1}}{(T,A_{y})}} = - \\frac{A_{y} (\\frac{A_{y}}{T})^{A_{y}}}{T}", "derivation": "\\operatorname{C_{1}}{(T,A_{y})} = \\frac{A_{y}}{T} and \\operatorname{C_{1}}^{A_{y}}{(T,A_{y})} = (\\frac{A_{y}}{T})^{A_{y}} and \\frac{\\partial}{\\partial T} \\operatorname{C_{1}}^{A_{y}}{(T,A_{y})} = \\frac{\\partial}{\\partial T} (\\frac{A_{y}}{T})^{A_{y}} and \\frac{A_{y} \\operatorname{C_{1}}^{A_{y}}{(T,A_{y})} \\frac{\\partial}{\\partial T} \\operatorname{C_{1}}{(T,A_{y})}}{\\operatorname{C_{1}}{(T,A_{y})}} = - \\frac{A_{y} (\\frac{A_{y}}{T})^{A_{y}}}{T}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Mul(Symbol('A_y', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Symbol('A_y', commutative=True)))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Pow(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('A_y', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Symbol('A_y', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('A_y', commutative=True), Pow(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Derivative(Function('C_1')(Symbol('T', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_y', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Mul(Symbol('A_y', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\psi,z^{*})} = \\int (- \\psi + z^{*}) dz^{*}, then derive \\mathbf{H}{(\\psi,z^{*})} = - \\psi z^{*} + \\sigma_p + \\frac{(z^{*})^{2}}{2}, then obtain \\frac{\\mathbf{H}{(\\psi,z^{*})}}{(z^{*})^{2}} = \\frac{\\int (- \\psi + z^{*}) dz^{*}}{(z^{*})^{2}}", "derivation": "\\mathbf{H}{(\\psi,z^{*})} = \\int (- \\psi + z^{*}) dz^{*} and \\mathbf{H}{(\\psi,z^{*})} = - \\psi z^{*} + \\sigma_p + \\frac{(z^{*})^{2}}{2} and \\frac{\\mathbf{H}{(\\psi,z^{*})}}{(z^{*})^{2}} = \\frac{- \\psi z^{*} + \\sigma_p + \\frac{(z^{*})^{2}}{2}}{(z^{*})^{2}} and \\int (- \\psi + z^{*}) dz^{*} = - \\psi z^{*} + \\sigma_p + \\frac{(z^{*})^{2}}{2} and \\frac{\\mathbf{H}{(\\psi,z^{*})}}{(z^{*})^{2}} = \\frac{\\int (- \\psi + z^{*}) dz^{*}}{(z^{*})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"], [["divide", 2, "Pow(Symbol('z^*', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-2)), Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-2)), Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-2)), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given J{(V_{\\mathbf{B}},\\dot{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (V_{\\mathbf{B}} + \\dot{\\mathbf{r}}), then derive 1 = \\frac{1}{J{(V_{\\mathbf{B}},\\dot{\\mathbf{r}})}}, then obtain 1 = \\frac{1}{\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (V_{\\mathbf{B}} + \\dot{\\mathbf{r}})}", "derivation": "J{(V_{\\mathbf{B}},\\dot{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (V_{\\mathbf{B}} + \\dot{\\mathbf{r}}) and 1 = \\frac{\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (V_{\\mathbf{B}} + \\dot{\\mathbf{r}})}{J{(V_{\\mathbf{B}},\\dot{\\mathbf{r}})}} and 1 = \\frac{1}{J{(V_{\\mathbf{B}},\\dot{\\mathbf{r}})}} and 1 = \\frac{1}{\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (V_{\\mathbf{B}} + \\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["divide", 1, "Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(1), Pow(Function('J')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Pow(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given f{(y^{\\prime},\\Omega)} = \\log{(\\Omega - y^{\\prime})} and \\eta^{\\prime}{(y^{\\prime})} = y^{\\prime} and \\mathbf{B}{(y^{\\prime},\\Omega)} = \\log{(\\Omega - y^{\\prime})}, then obtain 0 = - f{(y^{\\prime},\\Omega)} + \\log{(\\Omega - y^{\\prime})}", "derivation": "f{(y^{\\prime},\\Omega)} = \\log{(\\Omega - y^{\\prime})} and \\eta^{\\prime}{(y^{\\prime})} = y^{\\prime} and \\mathbf{B}{(y^{\\prime},\\Omega)} = \\log{(\\Omega - y^{\\prime})} and \\eta^{\\prime}{(y^{\\prime})} + \\mathbf{B}{(y^{\\prime},\\Omega)} = \\eta^{\\prime}{(y^{\\prime})} + \\log{(\\Omega - y^{\\prime})} and y^{\\prime} + \\mathbf{B}{(y^{\\prime},\\Omega)} = y^{\\prime} + \\log{(\\Omega - y^{\\prime})} and y^{\\prime} + \\mathbf{B}{(y^{\\prime},\\Omega)} = y^{\\prime} + f{(y^{\\prime},\\Omega)} and y^{\\prime} + f{(y^{\\prime},\\Omega)} = y^{\\prime} + \\log{(\\Omega - y^{\\prime})} and 0 = - f{(y^{\\prime},\\Omega)} + \\log{(\\Omega - y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 3, "Function('\\\\eta^{\\\\prime}')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('y^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), Function('f')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('f')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["minus", 7, "Add(Symbol('y^{\\\\prime}', commutative=True), Function('f')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then derive \\int \\operatorname{f^{*}}{(\\mathbf{g})} d\\mathbf{g} = \\Psi_{\\lambda} - \\cos{(\\mathbf{g})}, then obtain (\\Psi_{\\lambda} - \\cos{(\\mathbf{g})}) (\\varepsilon - \\cos{(\\mathbf{g})}) = (\\varepsilon - \\cos{(\\mathbf{g})})^{2}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\int \\operatorname{f^{*}}{(\\mathbf{g})} d\\mathbf{g} = \\int \\sin{(\\mathbf{g})} d\\mathbf{g} and (\\int \\operatorname{f^{*}}{(\\mathbf{g})} d\\mathbf{g}) \\int \\sin{(\\mathbf{g})} d\\mathbf{g} = (\\int \\sin{(\\mathbf{g})} d\\mathbf{g})^{2} and \\int \\operatorname{f^{*}}{(\\mathbf{g})} d\\mathbf{g} = \\Psi_{\\lambda} - \\cos{(\\mathbf{g})} and (\\Psi_{\\lambda} - \\cos{(\\mathbf{g})}) \\int \\sin{(\\mathbf{g})} d\\mathbf{g} = (\\int \\sin{(\\mathbf{g})} d\\mathbf{g})^{2} and (\\Psi_{\\lambda} - \\cos{(\\mathbf{g})}) (\\varepsilon - \\cos{(\\mathbf{g})}) = (\\varepsilon - \\cos{(\\mathbf{g})})^{2}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Integral(Function('f^*')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Pow(Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{g}', commutative=True)))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Pow(Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{g}', commutative=True)))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{g}', commutative=True))))), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{g}', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{H}{(h,\\varepsilon_0)} = h^{\\varepsilon_0}, then obtain \\int \\varepsilon_0 (h + \\mathbf{H}{(h,\\varepsilon_0)})^{\\varepsilon_0} dh = \\int \\varepsilon_0 (h + h^{\\varepsilon_0})^{\\varepsilon_0} dh", "derivation": "\\mathbf{H}{(h,\\varepsilon_0)} = h^{\\varepsilon_0} and h + \\mathbf{H}{(h,\\varepsilon_0)} = h + h^{\\varepsilon_0} and (h + \\mathbf{H}{(h,\\varepsilon_0)})^{\\varepsilon_0} = (h + h^{\\varepsilon_0})^{\\varepsilon_0} and \\varepsilon_0 (h + \\mathbf{H}{(h,\\varepsilon_0)})^{\\varepsilon_0} = \\varepsilon_0 (h + h^{\\varepsilon_0})^{\\varepsilon_0} and \\int \\varepsilon_0 (h + \\mathbf{H}{(h,\\varepsilon_0)})^{\\varepsilon_0} dh = \\int \\varepsilon_0 (h + h^{\\varepsilon_0})^{\\varepsilon_0} dh", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Add(Symbol('h', commutative=True), Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Add(Symbol('h', commutative=True), Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Add(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Add(Symbol('h', commutative=True), Function('\\\\mathbf{H}')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Add(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(C_{1},\\pi)} = C_{1} + \\pi, then derive \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} = 1, then obtain \\pi + \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) + 1 = \\pi + 2", "derivation": "\\phi_{2}{(C_{1},\\pi)} = C_{1} + \\pi and \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} = \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) and \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} = 1 and \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) = 1 and \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) + 1 = 2 and \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} + 1 = 2 and \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) + 1 = \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} + 1 and \\pi + \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) + 1 = \\pi + \\frac{\\partial}{\\partial C_{1}} \\phi_{2}{(C_{1},\\pi)} + 1 and \\pi + \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\pi) + 1 = \\pi + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)))"], [["minus", 7, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\pi', commutative=True), Derivative(Function('\\\\phi_2')(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Add(Symbol('\\\\pi', commutative=True), Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\pi', commutative=True), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_f{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\operatorname{f_{\\mathbf{p}}}{(E,\\mathbf{p})} = \\mathbf{p}^{E}, then obtain \\mathbf{J}_f{(g_{\\varepsilon})} e^{\\mathbf{p}^{E}} = e^{\\mathbf{p}^{E}} \\log{(g_{\\varepsilon})}", "derivation": "\\mathbf{J}_f{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\operatorname{f_{\\mathbf{p}}}{(E,\\mathbf{p})} = \\mathbf{p}^{E} and e^{\\operatorname{f_{\\mathbf{p}}}{(E,\\mathbf{p})}} = e^{\\mathbf{p}^{E}} and \\mathbf{J}_f{(g_{\\varepsilon})} e^{\\operatorname{f_{\\mathbf{p}}}{(E,\\mathbf{p})}} = e^{\\operatorname{f_{\\mathbf{p}}}{(E,\\mathbf{p})}} \\log{(g_{\\varepsilon})} and \\mathbf{J}_f{(g_{\\varepsilon})} e^{\\mathbf{p}^{E}} = e^{\\mathbf{p}^{E}} \\log{(g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('E', commutative=True)))"], [["exp", 2], "Equality(exp(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), exp(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('E', commutative=True))))"], [["times", 1, "exp(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Mul(exp(Function('f_{\\\\mathbf{p}}')(Symbol('E', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('E', commutative=True)))), Mul(exp(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('E', commutative=True))), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given z{(\\hat{p})} = e^{\\hat{p}}, then derive \\frac{d}{d \\hat{p}} z{(\\hat{p})} = e^{\\hat{p}}, then obtain 2 = - \\frac{d}{d \\hat{p}} z{(\\hat{p})} + \\frac{d}{d \\hat{p}} e^{\\hat{p}} + 2", "derivation": "z{(\\hat{p})} = e^{\\hat{p}} and \\frac{d}{d \\hat{p}} z{(\\hat{p})} = \\frac{d}{d \\hat{p}} e^{\\hat{p}} and \\frac{d}{d \\hat{p}} z{(\\hat{p})} = e^{\\hat{p}} and \\frac{d}{d \\hat{p}} z{(\\hat{p})} + 1 = \\frac{d}{d \\hat{p}} e^{\\hat{p}} + 1 and \\frac{d}{d \\hat{p}} z{(\\hat{p})} = \\frac{d^{2}}{d \\hat{p}^{2}} z{(\\hat{p})} and \\frac{d}{d \\hat{p}} z{(\\hat{p})} - \\frac{d^{2}}{d \\hat{p}^{2}} z{(\\hat{p})} + 2 = - \\frac{d^{2}}{d \\hat{p}^{2}} z{(\\hat{p})} + \\frac{d}{d \\hat{p}} e^{\\hat{p}} + 2 and 2 = - \\frac{d}{d \\hat{p}} z{(\\hat{p})} + \\frac{d}{d \\hat{p}} e^{\\hat{p}} + 2", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["add", 2, 1], "Equality(Add(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))))"], [["minus", 4, "Add(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Integer(-1))"], "Equality(Add(Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Integer(2)), Add(Mul(Integer(-1), Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(2), Add(Mul(Integer(-1), Derivative(Function('z')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{p}{(x)} = \\log{(x)}, then obtain \\int \\frac{x \\mathbf{p}{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx = \\int \\frac{x \\log{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx", "derivation": "\\mathbf{p}{(x)} = \\log{(x)} and x \\mathbf{p}{(x)} = x \\log{(x)} and \\mathbf{p}{(x)} + \\log{(x)} = 2 \\log{(x)} and \\frac{x \\mathbf{p}{(x)}}{2 \\log{(x)}} = \\frac{x}{2} and \\frac{x \\mathbf{p}{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} = \\frac{x}{2} and \\frac{x \\log{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} = \\frac{x}{2} and \\int \\frac{x \\mathbf{p}{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx = \\int \\frac{x}{2} dx and \\int \\frac{x \\log{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx = \\int \\frac{x}{2} dx and \\int \\frac{x \\mathbf{p}{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx = \\int \\frac{x \\log{(x)}}{\\mathbf{p}{(x)} + \\log{(x)}} dx", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))))"], [["add", 1, "log(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Mul(Integer(2), log(Symbol('x', commutative=True))))"], [["divide", 2, "Mul(Integer(2), log(Symbol('x', commutative=True)))"], "Equality(Mul(Rational(1, 2), Symbol('x', commutative=True), Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('x', commutative=True))), Mul(Rational(1, 2), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), log(Symbol('x', commutative=True))), Mul(Rational(1, 2), Symbol('x', commutative=True)))"], [["integrate", 5, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Rational(1, 2), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["integrate", 6, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), log(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Rational(1, 2), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Integral(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), Pow(Add(Function('\\\\mathbf{p}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Integer(-1)), log(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given W{(\\phi_1,\\mathbf{g})} = - \\mathbf{g} + \\cos{(\\phi_1)}, then obtain W{(\\phi_1,\\mathbf{g})} \\frac{\\partial}{\\partial \\phi_1} (- \\mathbf{g} + \\cos{(\\phi_1)}) = (- \\mathbf{g} + \\cos{(\\phi_1)}) \\frac{\\partial}{\\partial \\phi_1} (- \\mathbf{g} + \\cos{(\\phi_1)})", "derivation": "W{(\\phi_1,\\mathbf{g})} = - \\mathbf{g} + \\cos{(\\phi_1)} and \\frac{\\partial}{\\partial \\phi_1} W{(\\phi_1,\\mathbf{g})} = \\frac{\\partial}{\\partial \\phi_1} (- \\mathbf{g} + \\cos{(\\phi_1)}) and W{(\\phi_1,\\mathbf{g})} \\frac{\\partial}{\\partial \\phi_1} W{(\\phi_1,\\mathbf{g})} = (- \\mathbf{g} + \\cos{(\\phi_1)}) \\frac{\\partial}{\\partial \\phi_1} W{(\\phi_1,\\mathbf{g})} and W{(\\phi_1,\\mathbf{g})} \\frac{\\partial}{\\partial \\phi_1} (- \\mathbf{g} + \\cos{(\\phi_1)}) = (- \\mathbf{g} + \\cos{(\\phi_1)}) \\frac{\\partial}{\\partial \\phi_1} (- \\mathbf{g} + \\cos{(\\phi_1)})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))"], "Equality(Mul(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Derivative(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('W')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(A_{2},\\mathbf{r})} = A_{2} \\mathbf{r} and a{(P_{g},i)} = e^{\\frac{P_{g}}{i}}, then obtain \\mathbf{r} (i{(A_{2},\\mathbf{r})} - \\sin{(a{(P_{g},i)})}) = \\mathbf{r} (A_{2} \\mathbf{r} - \\sin{(a{(P_{g},i)})})", "derivation": "i{(A_{2},\\mathbf{r})} = A_{2} \\mathbf{r} and a{(P_{g},i)} = e^{\\frac{P_{g}}{i}} and i{(A_{2},\\mathbf{r})} - \\sin{(e^{\\frac{P_{g}}{i}})} = A_{2} \\mathbf{r} - \\sin{(e^{\\frac{P_{g}}{i}})} and \\mathbf{r} (i{(A_{2},\\mathbf{r})} - \\sin{(e^{\\frac{P_{g}}{i}})}) = \\mathbf{r} (A_{2} \\mathbf{r} - \\sin{(e^{\\frac{P_{g}}{i}})}) and \\mathbf{r} (i{(A_{2},\\mathbf{r})} - \\sin{(a{(P_{g},i)})}) = \\mathbf{r} (A_{2} \\mathbf{r} - \\sin{(a{(P_{g},i)})})", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('P_g', commutative=True), Symbol('i', commutative=True)), exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))"], [["minus", 1, "sin(exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))"], "Equality(Add(Function('i')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))))), Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))))))"], [["times", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Function('i')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(exp(Mul(Symbol('P_g', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Function('i')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(Function('a')(Symbol('P_g', commutative=True), Symbol('i', commutative=True)))))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(Function('a')(Symbol('P_g', commutative=True), Symbol('i', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)}, then obtain \\cos{(\\mathbf{J}_f)} + \\int \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = t_{2} + \\sin{(\\mathbf{J}_f)} + \\cos{(\\mathbf{J}_f)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\cos{(\\mathbf{J}_f)} + \\int \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\cos{(\\mathbf{J}_f)} + \\int \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\cos{(\\mathbf{J}_f)} + \\int \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = t_{2} + \\sin{(\\mathbf{J}_f)} + \\cos{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(h)} = e^{\\sin{(h)}}, then derive \\frac{d}{d h} \\operatorname{v_{t}}{(h)} = e^{\\sin{(h)}} \\cos{(h)}, then obtain \\frac{d}{d h} (\\frac{d}{d h} e^{\\sin{(h)}})^{h} = \\frac{d}{d h} (e^{\\sin{(h)}} \\cos{(h)})^{h}", "derivation": "\\operatorname{v_{t}}{(h)} = e^{\\sin{(h)}} and \\frac{d}{d h} \\operatorname{v_{t}}{(h)} = \\frac{d}{d h} e^{\\sin{(h)}} and (\\frac{d}{d h} \\operatorname{v_{t}}{(h)})^{h} = (\\frac{d}{d h} e^{\\sin{(h)}})^{h} and \\frac{d}{d h} \\operatorname{v_{t}}{(h)} = e^{\\sin{(h)}} \\cos{(h)} and \\frac{d}{d h} e^{\\sin{(h)}} = e^{\\sin{(h)}} \\cos{(h)} and (\\frac{d}{d h} \\operatorname{v_{t}}{(h)})^{h} = (e^{\\sin{(h)}} \\cos{(h)})^{h} and (\\frac{d}{d h} e^{\\sin{(h)}})^{h} = (e^{\\sin{(h)}} \\cos{(h)})^{h} and \\frac{d}{d h} (\\frac{d}{d h} e^{\\sin{(h)}})^{h} = \\frac{d}{d h} (e^{\\sin{(h)}} \\cos{(h)})^{h}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('h', commutative=True)), exp(sin(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Function('v_t')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Derivative(exp(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(exp(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(exp(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Derivative(Function('v_t')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(exp(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Derivative(exp(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(exp(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["differentiate", 7, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Derivative(exp(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(Mul(exp(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{A})} = e^{\\cos{(\\mathbf{A})}}, then derive \\frac{d}{d \\mathbf{A}} \\sigma_{p}{(\\mathbf{A})} - 1 = - e^{\\cos{(\\mathbf{A})}} \\sin{(\\mathbf{A})} - 1, then obtain \\frac{d}{d \\mathbf{A}} e^{\\cos{(\\mathbf{A})}} - 1 = \\frac{d}{d \\mathbf{A}} \\sigma_{p}{(\\mathbf{A})} - 1", "derivation": "\\sigma_{p}{(\\mathbf{A})} = e^{\\cos{(\\mathbf{A})}} and - \\mathbf{A} + \\sigma_{p}{(\\mathbf{A})} = - \\mathbf{A} + e^{\\cos{(\\mathbf{A})}} and \\frac{d}{d \\mathbf{A}} (- \\mathbf{A} + \\sigma_{p}{(\\mathbf{A})}) = \\frac{d}{d \\mathbf{A}} (- \\mathbf{A} + e^{\\cos{(\\mathbf{A})}}) and \\frac{d}{d \\mathbf{A}} \\sigma_{p}{(\\mathbf{A})} - 1 = - e^{\\cos{(\\mathbf{A})}} \\sin{(\\mathbf{A})} - 1 and \\frac{d}{d \\mathbf{A}} e^{\\cos{(\\mathbf{A})}} - 1 = - e^{\\cos{(\\mathbf{A})}} \\sin{(\\mathbf{A})} - 1 and \\frac{d}{d \\mathbf{A}} e^{\\cos{(\\mathbf{A})}} - 1 = \\frac{d}{d \\mathbf{A}} \\sigma_{p}{(\\mathbf{A})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), exp(cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), exp(cos(Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{A}', commutative=True))), sin(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(exp(cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{A}', commutative=True))), sin(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(exp(cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given s{(F_{c},m)} = \\sin{(m^{F_{c}})} and h{(F_{c},m)} = \\log{((\\int s{(F_{c},m)} dF_{c})^{m})}, then obtain 1 = \\frac{\\log{((\\int \\sin{(m^{F_{c}})} dF_{c})^{m})}}{h{(F_{c},m)}}", "derivation": "s{(F_{c},m)} = \\sin{(m^{F_{c}})} and \\int s{(F_{c},m)} dF_{c} = \\int \\sin{(m^{F_{c}})} dF_{c} and h{(F_{c},m)} = \\log{((\\int s{(F_{c},m)} dF_{c})^{m})} and h{(F_{c},m)} = \\log{((\\int \\sin{(m^{F_{c}})} dF_{c})^{m})} and 1 = \\frac{\\log{((\\int \\sin{(m^{F_{c}})} dF_{c})^{m})}}{h{(F_{c},m)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), sin(Pow(Symbol('m', commutative=True), Symbol('F_c', commutative=True))))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('s')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(sin(Pow(Symbol('m', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), log(Pow(Integral(Function('s')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('h')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), log(Pow(Integral(sin(Pow(Symbol('m', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('m', commutative=True))))"], [["divide", 4, "Function('h')(Symbol('F_c', commutative=True), Symbol('m', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('h')(Symbol('F_c', commutative=True), Symbol('m', commutative=True)), Integer(-1)), log(Pow(Integral(sin(Pow(Symbol('m', commutative=True), Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\omega{(C_{d},\\Omega)} = \\Omega^{C_{d}} and \\operatorname{E_{x}}{(C_{d},\\Omega)} = \\Omega + 2 (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)}) \\omega{(C_{d},\\Omega)}, then obtain \\Omega^{C_{d}} \\operatorname{E_{x}}{(C_{d},\\Omega)} = \\Omega^{C_{d}} (\\Omega + (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)})^{2})", "derivation": "\\omega{(C_{d},\\Omega)} = \\Omega^{C_{d}} and 2 \\omega{(C_{d},\\Omega)} = \\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)} and 2 (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)}) \\omega{(C_{d},\\Omega)} = (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)})^{2} and \\Omega + 2 (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)}) \\omega{(C_{d},\\Omega)} = \\Omega + (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)})^{2} and \\operatorname{E_{x}}{(C_{d},\\Omega)} = \\Omega + 2 (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)}) \\omega{(C_{d},\\Omega)} and \\operatorname{E_{x}}{(C_{d},\\Omega)} = \\Omega + (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)})^{2} and \\Omega^{C_{d}} \\operatorname{E_{x}}{(C_{d},\\Omega)} = \\Omega^{C_{d}} (\\Omega + (\\Omega^{C_{d}} + \\omega{(C_{d},\\Omega)})^{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)))"], [["add", 1, "Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["times", 2, "Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Integer(2), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2)))"], [["add", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('E_x')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2))))"], [["times", 6, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('E_x')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Pow(Add(Pow(Symbol('\\\\Omega', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\omega')(Symbol('C_d', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain \\mathbf{s} + 2 \\hat{p}_0{(\\mathbf{s})} + \\cos{(\\mathbf{s})} = \\mathbf{s} + 3 \\cos{(\\mathbf{s})}", "derivation": "\\hat{p}_0{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and \\mathbf{s} + \\hat{p}_0{(\\mathbf{s})} = \\mathbf{s} + \\cos{(\\mathbf{s})} and \\mathbf{s} + \\hat{p}_0{(\\mathbf{s})} + \\cos{(\\mathbf{s})} = \\mathbf{s} + 2 \\cos{(\\mathbf{s})} and \\mathbf{s} + 2 \\hat{p}_0{(\\mathbf{s})} + \\cos{(\\mathbf{s})} = \\mathbf{s} + \\hat{p}_0{(\\mathbf{s})} + 2 \\cos{(\\mathbf{s})} and \\mathbf{s} + 2 \\hat{p}_0{(\\mathbf{s})} + \\cos{(\\mathbf{s})} = \\mathbf{s} + 3 \\cos{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["add", 3, "Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(3), cos(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\psi{(r_{0})} = r_{0}, then obtain (\\int \\psi{(r_{0})} d\\psi{(r_{0})})^{\\psi{(r_{0})}} = (\\int r_{0} d\\psi{(r_{0})})^{\\psi{(r_{0})}}", "derivation": "\\psi{(r_{0})} = r_{0} and \\int \\psi{(r_{0})} dr_{0} = \\int r_{0} dr_{0} and (\\int \\psi{(r_{0})} dr_{0})^{r_{0}} = (\\int r_{0} dr_{0})^{r_{0}} and (\\int \\psi{(r_{0})} d\\psi{(r_{0})})^{\\psi{(r_{0})}} = (\\int r_{0} d\\psi{(r_{0})})^{\\psi{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Symbol('r_0', commutative=True), Tuple(Symbol('r_0', commutative=True))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Integral(Function('\\\\psi')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Pow(Integral(Symbol('r_0', commutative=True), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Integral(Function('\\\\psi')(Symbol('r_0', commutative=True)), Tuple(Function('\\\\psi')(Symbol('r_0', commutative=True)))), Function('\\\\psi')(Symbol('r_0', commutative=True))), Pow(Integral(Symbol('r_0', commutative=True), Tuple(Function('\\\\psi')(Symbol('r_0', commutative=True)))), Function('\\\\psi')(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\phi)} = e^{\\phi}, then derive \\int \\Psi^{\\dagger}{(\\phi)} d\\phi = \\omega + e^{\\phi}, then obtain \\phi (- \\phi + \\Psi^{\\dagger}{(\\phi)}) + \\int e^{\\phi} d\\phi + 1 = \\omega + \\phi (- \\phi + \\Psi^{\\dagger}{(\\phi)}) + e^{\\phi} + 1", "derivation": "\\Psi^{\\dagger}{(\\phi)} = e^{\\phi} and - \\phi + \\Psi^{\\dagger}{(\\phi)} = - \\phi + e^{\\phi} and \\int \\Psi^{\\dagger}{(\\phi)} d\\phi = \\int e^{\\phi} d\\phi and \\int \\Psi^{\\dagger}{(\\phi)} d\\phi = \\omega + e^{\\phi} and \\int \\Psi^{\\dagger}{(\\phi)} d\\phi + 1 = \\omega + e^{\\phi} + 1 and \\phi (- \\phi + e^{\\phi}) + \\int \\Psi^{\\dagger}{(\\phi)} d\\phi + 1 = \\omega + \\phi (- \\phi + e^{\\phi}) + e^{\\phi} + 1 and \\phi (- \\phi + e^{\\phi}) + \\int e^{\\phi} d\\phi + 1 = \\omega + \\phi (- \\phi + e^{\\phi}) + e^{\\phi} + 1 and \\phi (- \\phi + \\Psi^{\\dagger}{(\\phi)}) + \\int e^{\\phi} d\\phi + 1 = \\omega + \\phi (- \\phi + \\Psi^{\\dagger}{(\\phi)}) + e^{\\phi} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\omega', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\omega', commutative=True), exp(Symbol('\\\\phi', commutative=True)), Integer(1)))"], [["minus", 5, "Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\theta_{1}{(U)} = \\cos{(\\cos{(U)})}, then obtain (\\theta_{1}^{U}{(U)} + \\sin{(\\theta_{1}{(U)})})^{U} = (\\theta_{1}^{U}{(U)} + \\sin{(\\cos{(\\cos{(U)})})})^{U}", "derivation": "\\theta_{1}{(U)} = \\cos{(\\cos{(U)})} and \\theta_{1}^{U}{(U)} = \\cos^{U}{(\\cos{(U)})} and \\sin{(\\theta_{1}{(U)})} = \\sin{(\\cos{(\\cos{(U)})})} and \\sin{(\\theta_{1}{(U)})} + \\cos^{U}{(\\cos{(U)})} = \\sin{(\\cos{(\\cos{(U)})})} + \\cos^{U}{(\\cos{(U)})} and \\theta_{1}^{U}{(U)} + \\sin{(\\theta_{1}{(U)})} = \\theta_{1}^{U}{(U)} + \\sin{(\\cos{(\\cos{(U)})})} and (\\theta_{1}^{U}{(U)} + \\sin{(\\theta_{1}{(U)})})^{U} = (\\theta_{1}^{U}{(U)} + \\sin{(\\cos{(\\cos{(U)})})})^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('U', commutative=True)), cos(cos(Symbol('U', commutative=True))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(cos(cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\theta_1')(Symbol('U', commutative=True))), sin(cos(cos(Symbol('U', commutative=True)))))"], [["add", 3, "Pow(cos(cos(Symbol('U', commutative=True))), Symbol('U', commutative=True))"], "Equality(Add(sin(Function('\\\\theta_1')(Symbol('U', commutative=True))), Pow(cos(cos(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Add(sin(cos(cos(Symbol('U', commutative=True)))), Pow(cos(cos(Symbol('U', commutative=True))), Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('\\\\theta_1')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), sin(Function('\\\\theta_1')(Symbol('U', commutative=True)))), Add(Pow(Function('\\\\theta_1')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), sin(cos(cos(Symbol('U', commutative=True))))))"], [["power", 5, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\theta_1')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), sin(Function('\\\\theta_1')(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Add(Pow(Function('\\\\theta_1')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), sin(cos(cos(Symbol('U', commutative=True))))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(T)} = e^{T}, then obtain T + \\operatorname{F_{H}}{(T)} - \\int T (T + e^{T}) dT = T + e^{T} - \\int T (T + e^{T}) dT", "derivation": "\\operatorname{F_{H}}{(T)} = e^{T} and T + \\operatorname{F_{H}}{(T)} = T + e^{T} and T (T + \\operatorname{F_{H}}{(T)}) = T (T + e^{T}) and \\int T (T + \\operatorname{F_{H}}{(T)}) dT = \\int T (T + e^{T}) dT and T + \\operatorname{F_{H}}{(T)} - \\int T (T + \\operatorname{F_{H}}{(T)}) dT = T + e^{T} - \\int T (T + \\operatorname{F_{H}}{(T)}) dT and T + \\operatorname{F_{H}}{(T)} - \\int T (T + e^{T}) dT = T + e^{T} - \\int T (T + e^{T}) dT", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True))))"], [["times", 2, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)))), Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('T', commutative=True), Function('F_H')(Symbol('T', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))))"]]}, {"prompt": "Given \\chi{(t_{1})} = \\log{(\\sin{(t_{1})})}, then derive \\frac{d}{d t_{1}} \\chi{(t_{1})} = \\frac{\\cos{(t_{1})}}{\\sin{(t_{1})}}, then obtain \\frac{\\sin{(t_{1})} \\frac{d}{d t_{1}} \\log{(\\sin{(t_{1})})}}{(C_{1} + 1) \\cos{(t_{1})}} = \\frac{1}{C_{1} + 1}", "derivation": "\\chi{(t_{1})} = \\log{(\\sin{(t_{1})})} and \\frac{d}{d t_{1}} \\chi{(t_{1})} = \\frac{d}{d t_{1}} \\log{(\\sin{(t_{1})})} and \\frac{d}{d t_{1}} \\chi{(t_{1})} = \\frac{\\cos{(t_{1})}}{\\sin{(t_{1})}} and \\frac{\\frac{d}{d t_{1}} \\chi{(t_{1})}}{C_{1} + 1} = \\frac{\\cos{(t_{1})}}{(C_{1} + 1) \\sin{(t_{1})}} and \\frac{\\sin{(t_{1})} \\frac{d}{d t_{1}} \\chi{(t_{1})}}{(C_{1} + 1) \\cos{(t_{1})}} = \\frac{1}{C_{1} + 1} and \\frac{\\sin{(t_{1})} \\frac{d}{d t_{1}} \\log{(\\sin{(t_{1})})}}{(C_{1} + 1) \\cos{(t_{1})}} = \\frac{1}{C_{1} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('t_1', commutative=True)), log(sin(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(log(sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('t_1', commutative=True)), Integer(-1)), cos(Symbol('t_1', commutative=True))))"], [["divide", 3, "Add(Symbol('C_1', commutative=True), Integer(1))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)), Pow(sin(Symbol('t_1', commutative=True)), Integer(-1)), cos(Symbol('t_1', commutative=True))))"], [["divide", 4, "Mul(Pow(sin(Symbol('t_1', commutative=True)), Integer(-1)), cos(Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)), sin(Symbol('t_1', commutative=True)), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)), sin(Symbol('t_1', commutative=True)), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1)), Derivative(log(sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Pow(Add(Symbol('C_1', commutative=True), Integer(1)), Integer(-1)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(M_{E})} = e^{M_{E}}, then obtain (\\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} - e^{M_{E}}) \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} = (- e^{M_{E}} + (e^{M_{E}})^{2 M_{E}}) \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}}", "derivation": "\\dot{\\mathbf{r}}{(M_{E})} = e^{M_{E}} and \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} = (e^{M_{E}})^{M_{E}} and \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} = (e^{M_{E}})^{2 M_{E}} and \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} - e^{M_{E}} = - e^{M_{E}} + (e^{M_{E}})^{2 M_{E}} and (\\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} - e^{M_{E}}) \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}} = (- e^{M_{E}} + (e^{M_{E}})^{2 M_{E}}) \\dot{\\mathbf{r}}^{M_{E}}{(M_{E})} (e^{M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["power", 1, "Symbol('M_E', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Pow(exp(Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('M_E', commutative=True))))"], [["minus", 3, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('M_E', commutative=True))), Pow(exp(Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('M_E', commutative=True)))))"], [["times", 4, "Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Mul(Integer(-1), exp(Symbol('M_E', commutative=True)))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Mul(Add(Mul(Integer(-1), exp(Symbol('M_E', commutative=True))), Pow(exp(Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('M_E', commutative=True)))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(exp(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\phi_2,\\Omega)} = \\Omega + \\phi_2, then obtain \\frac{\\partial}{\\partial \\Omega} (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi^{2}{(\\phi_2,\\Omega)} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} = 0", "derivation": "\\pi{(\\phi_2,\\Omega)} = \\Omega + \\phi_2 and (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} = (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) and (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} = (\\Omega + \\phi_2)^{2} (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) and \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\phi_2)^{2} (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) and \\frac{\\partial}{\\partial \\Omega} (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi^{2}{(\\phi_2,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} and \\frac{\\partial}{\\partial \\Omega} (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi^{2}{(\\phi_2,\\Omega)} - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\phi_2) (\\phi_2 + \\pi{(\\phi_2,\\Omega)}) \\pi{(\\phi_2,\\Omega)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["times", 1, "Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Integer(2)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Integer(2)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given c{(a^{\\dagger},q)} = a^{\\dagger} q, then obtain \\iint (- \\frac{\\partial}{\\partial q} \\int a^{\\dagger} q da^{\\dagger} + \\frac{\\partial}{\\partial q} \\int c{(a^{\\dagger},q)} da^{\\dagger}) dq da^{\\dagger} = \\iint 0 dq da^{\\dagger}", "derivation": "c{(a^{\\dagger},q)} = a^{\\dagger} q and \\int c{(a^{\\dagger},q)} da^{\\dagger} = \\int a^{\\dagger} q da^{\\dagger} and \\frac{\\partial}{\\partial q} \\int c{(a^{\\dagger},q)} da^{\\dagger} = \\frac{\\partial}{\\partial q} \\int a^{\\dagger} q da^{\\dagger} and - \\frac{\\partial}{\\partial q} \\int a^{\\dagger} q da^{\\dagger} + \\frac{\\partial}{\\partial q} \\int c{(a^{\\dagger},q)} da^{\\dagger} = 0 and \\int (- \\frac{\\partial}{\\partial q} \\int a^{\\dagger} q da^{\\dagger} + \\frac{\\partial}{\\partial q} \\int c{(a^{\\dagger},q)} da^{\\dagger}) dq = \\int 0 dq and \\iint (- \\frac{\\partial}{\\partial q} \\int a^{\\dagger} q da^{\\dagger} + \\frac{\\partial}{\\partial q} \\int c{(a^{\\dagger},q)} da^{\\dagger}) dq da^{\\dagger} = \\iint 0 dq da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Derivative(Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))), Integral(Integer(0), Tuple(Symbol('q', commutative=True))))"], [["integrate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Derivative(Integral(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Integer(0), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\hat{H}_l)} = \\hat{H}_l, then derive - \\frac{\\hbar^{2}}{2} + \\hbar (- \\hat{H}_l + \\operatorname{A_{z}}{(\\hat{H}_l)}) + \\theta_1 = \\int - \\hbar d\\hbar, then obtain - \\frac{\\hbar^{2}}{2} + \\theta_1 = \\int - \\hbar d\\hbar", "derivation": "\\operatorname{A_{z}}{(\\hat{H}_l)} = \\hat{H}_l and - \\hat{H}_l + \\operatorname{A_{z}}{(\\hat{H}_l)} = 0 and - \\hat{H}_l - \\hbar + \\operatorname{A_{z}}{(\\hat{H}_l)} = - \\hbar and \\int (- \\hat{H}_l - \\hbar + \\operatorname{A_{z}}{(\\hat{H}_l)}) d\\hbar = \\int - \\hbar d\\hbar and - \\frac{\\hbar^{2}}{2} + \\hbar (- \\hat{H}_l + \\operatorname{A_{z}}{(\\hat{H}_l)}) + \\theta_1 = \\int - \\hbar d\\hbar and - \\frac{\\hbar^{2}}{2} + \\theta_1 = \\int - \\hbar d\\hbar", "srepr_derivation": [["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], [["minus", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain \\tilde{\\infty} \\mathbf{M} \\operatorname{f^{\\prime}}{(\\mathbf{M})} = \\tilde{\\infty} \\mathbf{M} e^{\\mathbf{M}}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\operatorname{f^{\\prime}}^{\\mathbf{M}}{(\\mathbf{M})} = (e^{\\mathbf{M}})^{\\mathbf{M}} and \\frac{\\operatorname{f^{\\prime}}{(\\mathbf{M})}}{- \\operatorname{f^{\\prime}}^{\\mathbf{M}}{(\\mathbf{M})} + (e^{\\mathbf{M}})^{\\mathbf{M}}} = \\frac{e^{\\mathbf{M}}}{- \\operatorname{f^{\\prime}}^{\\mathbf{M}}{(\\mathbf{M})} + (e^{\\mathbf{M}})^{\\mathbf{M}}} and \\tilde{\\infty} \\operatorname{f^{\\prime}}{(\\mathbf{M})} = \\tilde{\\infty} e^{\\mathbf{M}} and \\tilde{\\infty} \\mathbf{M} \\operatorname{f^{\\prime}}{(\\mathbf{M})} = \\tilde{\\infty} \\mathbf{M} e^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1)), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1)), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(zoo, Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(zoo, exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(zoo, Symbol('\\\\mathbf{M}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(zoo, Symbol('\\\\mathbf{M}', commutative=True), exp(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(W)} = \\log{(\\log{(W)})} and \\operatorname{v_{x}}{(W)} = \\hat{H}_l{(W)} - \\log{(W)} and \\mathbf{J}_M{(W)} = - \\log{(W)}, then obtain - W \\hat{H}_l{(W)} \\operatorname{v_{x}}{(W)} + \\operatorname{v_{x}}{(W)} = - W \\hat{H}_l{(W)} \\operatorname{v_{x}}{(W)} + \\mathbf{J}_M{(W)} + \\log{(\\log{(W)})}", "derivation": "\\hat{H}_l{(W)} = \\log{(\\log{(W)})} and \\hat{H}_l{(W)} - \\log{(W)} = - \\log{(W)} + \\log{(\\log{(W)})} and \\operatorname{v_{x}}{(W)} = \\hat{H}_l{(W)} - \\log{(W)} and \\operatorname{v_{x}}{(W)} = - \\log{(W)} + \\log{(\\log{(W)})} and \\mathbf{J}_M{(W)} = - \\log{(W)} and \\operatorname{v_{x}}{(W)} = \\mathbf{J}_M{(W)} + \\log{(\\log{(W)})} and - W (- \\log{(W)} + \\log{(\\log{(W)})}) \\hat{H}_l{(W)} + \\operatorname{v_{x}}{(W)} = - W (- \\log{(W)} + \\log{(\\log{(W)})}) \\hat{H}_l{(W)} + \\mathbf{J}_M{(W)} + \\log{(\\log{(W)})} and - W \\hat{H}_l{(W)} \\operatorname{v_{x}}{(W)} + \\operatorname{v_{x}}{(W)} = - W \\hat{H}_l{(W)} \\operatorname{v_{x}}{(W)} + \\mathbf{J}_M{(W)} + \\log{(\\log{(W)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True))))"], [["minus", 1, "log(Symbol('W', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), log(log(Symbol('W', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('W', commutative=True)), Add(Function('\\\\hat{H}_l')(Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_x')(Symbol('W', commutative=True)), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), log(log(Symbol('W', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('v_x')(Symbol('W', commutative=True)), Add(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))))"], [["minus", 6, "Mul(Symbol('W', commutative=True), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), log(log(Symbol('W', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), log(log(Symbol('W', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('W', commutative=True))), Function('v_x')(Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Add(Mul(Integer(-1), log(Symbol('W', commutative=True))), log(log(Symbol('W', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('W', commutative=True))), Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\hat{H}_l')(Symbol('W', commutative=True)), Function('v_x')(Symbol('W', commutative=True))), Function('v_x')(Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\hat{H}_l')(Symbol('W', commutative=True)), Function('v_x')(Symbol('W', commutative=True))), Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{H},A_{z})} = \\hat{H}^{A_{z}}, then derive \\frac{\\partial}{\\partial A_{z}} \\mathbf{H}{(\\hat{H},A_{z})} = \\hat{H}^{A_{z}} \\log{(\\hat{H})}, then obtain \\frac{\\partial}{\\partial A_{z}} \\mathbf{H}{(\\hat{H},A_{z})} = \\mathbf{H}{(\\hat{H},A_{z})} \\log{(\\hat{H})}", "derivation": "\\mathbf{H}{(\\hat{H},A_{z})} = \\hat{H}^{A_{z}} and \\hat{H} + \\mathbf{H}{(\\hat{H},A_{z})} = \\hat{H} + \\hat{H}^{A_{z}} and \\hat{H} + \\mathbf{H}{(\\hat{H},A_{z})} + \\frac{1}{\\hat{H}} = \\hat{H} + \\hat{H}^{A_{z}} + \\frac{1}{\\hat{H}} and \\frac{\\partial}{\\partial A_{z}} (\\hat{H} + \\mathbf{H}{(\\hat{H},A_{z})} + \\frac{1}{\\hat{H}}) = \\frac{\\partial}{\\partial A_{z}} (\\hat{H} + \\hat{H}^{A_{z}} + \\frac{1}{\\hat{H}}) and \\frac{\\partial}{\\partial A_{z}} \\mathbf{H}{(\\hat{H},A_{z})} = \\hat{H}^{A_{z}} \\log{(\\hat{H})} and \\frac{\\partial}{\\partial A_{z}} \\mathbf{H}{(\\hat{H},A_{z})} = \\mathbf{H}{(\\hat{H},A_{z})} \\log{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Add(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_z', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given G{(\\rho)} = \\frac{d}{d \\rho} \\cos{(\\rho)}, then derive G^{\\rho}{(\\rho)} = (- \\sin{(\\rho)})^{\\rho}, then obtain - \\frac{d}{d \\rho} (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} = - \\frac{d}{d \\rho} G^{\\rho}{(\\rho)}", "derivation": "G{(\\rho)} = \\frac{d}{d \\rho} \\cos{(\\rho)} and G^{\\rho}{(\\rho)} = (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} and G^{\\rho}{(\\rho)} = (- \\sin{(\\rho)})^{\\rho} and (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} = (- \\sin{(\\rho)})^{\\rho} and \\frac{d}{d \\rho} (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} = \\frac{d}{d \\rho} (- \\sin{(\\rho)})^{\\rho} and \\frac{d}{d \\rho} (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} = \\frac{d}{d \\rho} G^{\\rho}{(\\rho)} and - \\frac{d}{d \\rho} (\\frac{d}{d \\rho} \\cos{(\\rho)})^{\\rho} = - \\frac{d}{d \\rho} G^{\\rho}{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\rho', commutative=True)), Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('G')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Pow(Function('G')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Pow(Function('G')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(i)} = \\log{(i)}, then obtain \\frac{\\frac{d}{d i} \\mathbf{M}^{i}{(i)}}{\\log{(i)}} = \\frac{\\frac{d}{d i} \\log{(i)}^{i}}{\\log{(i)}}", "derivation": "\\mathbf{M}{(i)} = \\log{(i)} and \\mathbf{M}^{i}{(i)} = \\log{(i)}^{i} and \\frac{d}{d i} \\mathbf{M}^{i}{(i)} = \\frac{d}{d i} \\log{(i)}^{i} and \\frac{\\frac{d}{d i} \\mathbf{M}^{i}{(i)}}{\\log{(i)}} = \\frac{\\frac{d}{d i} \\log{(i)}^{i}}{\\log{(i)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["divide", 3, "log(Symbol('i', commutative=True))"], "Equality(Mul(Pow(log(Symbol('i', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('i', commutative=True)), Integer(-1)), Derivative(Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given T{(\\lambda,y^{\\prime})} = \\lambda y^{\\prime}, then obtain \\lambda y^{\\prime} + 2 + (\\lambda y^{\\prime} + 1)^{- \\lambda} (T{(\\lambda,y^{\\prime})} + 1)^{\\lambda} = \\lambda y^{\\prime} + 3", "derivation": "T{(\\lambda,y^{\\prime})} = \\lambda y^{\\prime} and T{(\\lambda,y^{\\prime})} + 1 = \\lambda y^{\\prime} + 1 and (T{(\\lambda,y^{\\prime})} + 1)^{\\lambda} = (\\lambda y^{\\prime} + 1)^{\\lambda} and (\\lambda y^{\\prime} + 1)^{- \\lambda} (T{(\\lambda,y^{\\prime})} + 1)^{\\lambda} = 1 and \\lambda y^{\\prime} + 2 + (\\lambda y^{\\prime} + 1)^{- \\lambda} (T{(\\lambda,y^{\\prime})} + 1)^{\\lambda} = \\lambda y^{\\prime} + 3", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('T')(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Add(Function('T')(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True)), Pow(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True)))"], [["divide", 3, "Pow(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(Add(Function('T')(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True))), Integer(1))"], [["add", 4, "Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2))"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2), Mul(Pow(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Pow(Add(Function('T')(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('\\\\lambda', commutative=True)))), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(3)))"]]}, {"prompt": "Given V{(C_{2},p)} = C_{2} - p, then obtain \\frac{\\frac{\\frac{\\partial}{\\partial p} V{(C_{2},p)}}{C_{2} - p} + \\frac{V{(C_{2},p)}}{(C_{2} - p)^{2}}}{1 + \\frac{V{(C_{2},p)}}{C_{2} - p}} = 0", "derivation": "V{(C_{2},p)} = C_{2} - p and \\frac{V{(C_{2},p)}}{C_{2} - p} = 1 and 1 + \\frac{V{(C_{2},p)}}{C_{2} - p} = 2 and \\log{(1 + \\frac{V{(C_{2},p)}}{C_{2} - p})} = \\log{(2)} and \\frac{\\partial}{\\partial p} \\log{(1 + \\frac{V{(C_{2},p)}}{C_{2} - p})} = \\frac{d}{d p} \\log{(2)} and \\frac{\\frac{\\frac{\\partial}{\\partial p} V{(C_{2},p)}}{C_{2} - p} + \\frac{V{(C_{2},p)}}{(C_{2} - p)^{2}}}{1 + \\frac{V{(C_{2},p)}}{C_{2} - p}} = 0", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["divide", 1, "Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True))), Integer(1))"], [["add", 2, 1], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True)))), Integer(2))"], [["log", 3], "Equality(log(Add(Integer(1), Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True))))), log(Integer(2)))"], [["differentiate", 4, "Symbol('p', commutative=True)"], "Equality(Derivative(log(Add(Integer(1), Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(Integer(2)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Add(Integer(1), Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True)))), Integer(-1)), Add(Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-1)), Derivative(Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(-2)), Function('V')(Symbol('C_2', commutative=True), Symbol('p', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(M,G)} = \\frac{\\sin{(G)}}{M}, then obtain \\frac{\\partial}{\\partial G} M \\operatorname{v_{t}}^{G}{(M,G)} = \\frac{\\partial}{\\partial G} M (\\frac{\\sin{(G)}}{M})^{G}", "derivation": "\\operatorname{v_{t}}{(M,G)} = \\frac{\\sin{(G)}}{M} and \\operatorname{v_{t}}^{G}{(M,G)} = (\\frac{\\sin{(G)}}{M})^{G} and M \\operatorname{v_{t}}^{G}{(M,G)} = M (\\frac{\\sin{(G)}}{M})^{G} and \\frac{\\partial}{\\partial G} M \\operatorname{v_{t}}^{G}{(M,G)} = \\frac{\\partial}{\\partial G} M (\\frac{\\sin{(G)}}{M})^{G}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('M', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('G', commutative=True))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["times", 2, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Pow(Function('v_t')(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Symbol('M', commutative=True), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True))))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Symbol('M', commutative=True), Pow(Function('v_t')(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('M', commutative=True), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(J)} = \\log{(J)}, then obtain \\frac{d}{d J} (\\varepsilon_{0}{(J)} \\varepsilon_{0}^{J}{(J)})^{J} = \\frac{d}{d J} (\\varepsilon_{0}^{J}{(J)} \\log{(J)})^{J}", "derivation": "\\varepsilon_{0}{(J)} = \\log{(J)} and \\varepsilon_{0}^{J}{(J)} = \\log{(J)}^{J} and \\varepsilon_{0}{(J)} \\log{(J)}^{J} = \\log{(J)} \\log{(J)}^{J} and \\varepsilon_{0}{(J)} \\varepsilon_{0}^{J}{(J)} = \\varepsilon_{0}^{J}{(J)} \\log{(J)} and (\\varepsilon_{0}{(J)} \\varepsilon_{0}^{J}{(J)})^{J} = (\\varepsilon_{0}^{J}{(J)} \\log{(J)})^{J} and \\frac{d}{d J} (\\varepsilon_{0}{(J)} \\varepsilon_{0}^{J}{(J)})^{J} = \\frac{d}{d J} (\\varepsilon_{0}^{J}{(J)} \\log{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(log(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["times", 1, "Pow(log(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Pow(log(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(log(Symbol('J', commutative=True)), Pow(log(Symbol('J', commutative=True)), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), log(Symbol('J', commutative=True))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), log(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), log(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(T)} = \\int \\cos{(T)} dT, then obtain \\frac{\\cos{(T)} \\int \\hat{H}{(T)} dT}{T} = \\frac{\\cos{(T)} \\iint \\cos{(T)} dT dT}{T}", "derivation": "\\hat{H}{(T)} = \\int \\cos{(T)} dT and \\int \\hat{H}{(T)} dT = \\iint \\cos{(T)} dT dT and \\frac{\\int \\hat{H}{(T)} dT}{T} = \\frac{\\iint \\cos{(T)} dT dT}{T} and \\frac{\\cos{(T)} \\int \\hat{H}{(T)} dT}{T} = \\frac{\\cos{(T)} \\iint \\cos{(T)} dT dT}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('T', commutative=True)), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["divide", 2, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(Function('\\\\hat{H}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["times", 3, "cos(Symbol('T', commutative=True))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Symbol('T', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Symbol('T', commutative=True)), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\sigma_x)} = \\log{(\\sigma_x)}, then obtain 2 e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}} = e^{\\frac{1}{\\sigma_x}} + e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}}", "derivation": "\\mathbf{s}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} and e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}} = e^{\\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}} and 2 e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}} = e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}} + e^{\\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}} and 2 e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}} = e^{\\frac{1}{\\sigma_x}} + e^{\\frac{d}{d \\sigma_x} \\mathbf{s}{(\\sigma_x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), exp(Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["add", 3, "exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))), Add(exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), exp(Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))), Add(exp(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), exp(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)}, then derive (\\int \\operatorname{t_{2}}{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\rho_b - \\cos{(\\tilde{g}^*)})^{\\tilde{g}^*}, then obtain (\\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\rho_b - \\cos{(\\tilde{g}^*)})^{\\tilde{g}^*}", "derivation": "\\operatorname{t_{2}}{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)} and \\int \\operatorname{t_{2}}{(\\tilde{g}^*)} d\\tilde{g}^* = \\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^* and (\\int \\operatorname{t_{2}}{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} and (\\int \\operatorname{t_{2}}{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\rho_b - \\cos{(\\tilde{g}^*)})^{\\tilde{g}^*} and (\\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^*)^{\\tilde{g}^*} = (\\rho_b - \\cos{(\\tilde{g}^*)})^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["power", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\chi{(x,Q)} = x^{Q}, then derive Q + i = \\psi^* + \\int e^{\\frac{Q x^{Q}}{x}} e^{- \\frac{\\partial}{\\partial x} \\chi{(x,Q)}} dQ, then obtain Q + i = \\psi^* + \\int e^{\\frac{Q x^{Q}}{x}} e^{- \\frac{\\partial}{\\partial x} x^{Q}} dQ", "derivation": "\\chi{(x,Q)} = x^{Q} and \\frac{\\partial}{\\partial x} \\chi{(x,Q)} = \\frac{\\partial}{\\partial x} x^{Q} and 0 = \\frac{\\partial}{\\partial x} x^{Q} - \\frac{\\partial}{\\partial x} \\chi{(x,Q)} and 1 = e^{\\frac{\\partial}{\\partial x} x^{Q} - \\frac{\\partial}{\\partial x} \\chi{(x,Q)}} and \\int 1 dQ = \\int e^{\\frac{\\partial}{\\partial x} x^{Q} - \\frac{\\partial}{\\partial x} \\chi{(x,Q)}} dQ and Q + i = \\psi^* + \\int e^{\\frac{Q x^{Q}}{x}} e^{- \\frac{\\partial}{\\partial x} \\chi{(x,Q)}} dQ and Q + i = \\psi^* + \\int e^{\\frac{Q x^{Q}}{x}} e^{- \\frac{\\partial}{\\partial x} x^{Q}} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))"], [["exp", 3], "Equality(Integer(1), exp(Add(Derivative(Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('Q', commutative=True))), Integral(exp(Add(Derivative(Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Integral(Mul(exp(Mul(Symbol('Q', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)))), exp(Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Integral(Mul(exp(Mul(Symbol('Q', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)))), exp(Mul(Integer(-1), Derivative(Pow(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(C_{2},\\nabla)} = \\frac{\\nabla}{C_{2}} and z{(t,v_{z})} = t + v_{z}, then obtain \\cos{(\\hat{\\mathbf{x}}{(C_{2},\\nabla)} - z{(t,v_{z})})} = \\cos{(z{(t,v_{z})} - \\frac{\\nabla}{C_{2}})}", "derivation": "\\hat{\\mathbf{x}}{(C_{2},\\nabla)} = \\frac{\\nabla}{C_{2}} and z{(t,v_{z})} = t + v_{z} and - t - v_{z} + \\hat{\\mathbf{x}}{(C_{2},\\nabla)} = - t - v_{z} + \\frac{\\nabla}{C_{2}} and \\cos{(t + v_{z} - \\hat{\\mathbf{x}}{(C_{2},\\nabla)})} = \\cos{(t + v_{z} - \\frac{\\nabla}{C_{2}})} and \\cos{(\\hat{\\mathbf{x}}{(C_{2},\\nabla)} - z{(t,v_{z})})} = \\cos{(z{(t,v_{z})} - \\frac{\\nabla}{C_{2}})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('C_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))"], ["get_premise", "Equality(Function('z')(Symbol('t', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('t', commutative=True), Symbol('v_z', commutative=True)))"], [["minus", 1, "Add(Symbol('t', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('C_2', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True))))"], [["cos", 3], "Equality(cos(Add(Symbol('t', commutative=True), Symbol('v_z', commutative=True), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('C_2', commutative=True), Symbol('\\\\nabla', commutative=True))))), cos(Add(Symbol('t', commutative=True), Symbol('v_z', commutative=True), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(cos(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('C_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('t', commutative=True), Symbol('v_z', commutative=True))))), cos(Add(Function('z')(Symbol('t', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then derive \\int \\operatorname{F_{c}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\varphi^* + \\sin{(V_{\\mathbf{E}})}, then obtain \\int \\cos{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\varphi^* + \\sin{(V_{\\mathbf{E}})}", "derivation": "\\operatorname{F_{c}}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and \\int \\operatorname{F_{c}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\cos{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and \\int \\operatorname{F_{c}}{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\varphi^* + \\sin{(V_{\\mathbf{E}})} and \\int \\cos{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\varphi^* + \\sin{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_c')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mu)} = \\log{(e^{\\mu})} and \\operatorname{M_{E}}{(\\mu)} = - \\mathbf{J}_P{(\\mu)}, then obtain - \\log{(e^{\\mu})} + \\int \\mathbf{J}_P{(\\mu)} d\\mu = - \\log{(e^{\\mu})} + \\int \\log{(e^{\\mu})} d\\mu", "derivation": "\\mathbf{J}_P{(\\mu)} = \\log{(e^{\\mu})} and \\int \\mathbf{J}_P{(\\mu)} d\\mu = \\int \\log{(e^{\\mu})} d\\mu and - \\mathbf{J}_P{(\\mu)} + \\int \\mathbf{J}_P{(\\mu)} d\\mu = - \\mathbf{J}_P{(\\mu)} + \\int \\log{(e^{\\mu})} d\\mu and \\operatorname{M_{E}}{(\\mu)} = - \\mathbf{J}_P{(\\mu)} and \\operatorname{M_{E}}{(\\mu)} = - \\log{(e^{\\mu})} and - \\mathbf{J}_P{(\\mu)} = - \\log{(e^{\\mu})} and - \\log{(e^{\\mu})} + \\int \\mathbf{J}_P{(\\mu)} d\\mu = - \\log{(e^{\\mu})} + \\int \\log{(e^{\\mu})} d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True)), log(exp(Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(log(exp(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True))), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True))), Integral(log(exp(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('M_E')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), log(exp(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Add(Mul(Integer(-1), log(exp(Symbol('\\\\mu', commutative=True)))), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), log(exp(Symbol('\\\\mu', commutative=True)))), Integral(log(exp(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t)} = \\frac{d}{d t} \\cos{(t)}, then obtain \\frac{d}{d t} \\frac{\\operatorname{F_{g}}{(t)} + 1}{\\frac{d}{d t} (\\operatorname{F_{g}}{(t)} + \\cos{(t)} - \\frac{d}{d t} \\cos{(t)}) + 1} = \\frac{d}{d t} 1", "derivation": "\\operatorname{F_{g}}{(t)} = \\frac{d}{d t} \\cos{(t)} and \\operatorname{F_{g}}{(t)} + 1 = \\frac{d}{d t} \\cos{(t)} + 1 and \\operatorname{F_{g}}{(t)} - \\frac{d}{d t} \\cos{(t)} + 1 = 1 and \\frac{\\operatorname{F_{g}}{(t)} + 1}{\\frac{d}{d t} \\cos{(t)} + 1} = 1 and \\operatorname{F_{g}}{(t)} + \\cos{(t)} - \\frac{d}{d t} \\cos{(t)} + 1 = \\cos{(t)} + 1 and \\frac{d}{d t} \\frac{\\operatorname{F_{g}}{(t)} + 1}{\\frac{d}{d t} \\cos{(t)} + 1} = \\frac{d}{d t} 1 and \\operatorname{F_{g}}{(t)} + \\cos{(t)} - \\frac{d}{d t} \\cos{(t)} = \\cos{(t)} and \\frac{d}{d t} \\frac{\\operatorname{F_{g}}{(t)} + 1}{\\frac{d}{d t} (\\operatorname{F_{g}}{(t)} + \\cos{(t)} - \\frac{d}{d t} \\cos{(t)}) + 1} = \\frac{d}{d t} 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t', commutative=True)), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('F_g')(Symbol('t', commutative=True)), Integer(1)), Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)))"], [["minus", 2, "Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Add(Function('F_g')(Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(1)), Integer(1))"], [["divide", 2, "Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Function('F_g')(Symbol('t', commutative=True)), Integer(1)), Pow(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)), Integer(-1))), Integer(1))"], [["add", 3, "cos(Symbol('t', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(1)), Add(cos(Symbol('t', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Add(Function('F_g')(Symbol('t', commutative=True)), Integer(1)), Pow(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)), Integer(-1))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["minus", 5, 1], "Equality(Add(Function('F_g')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))), cos(Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Mul(Add(Function('F_g')(Symbol('t', commutative=True)), Integer(1)), Pow(Add(Derivative(Add(Function('F_g')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(1)), Integer(-1))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(\\mathbf{J}_M)} = \\mathbf{J}_M, then derive \\omega + \\frac{\\ddot{x}^{2}{(\\mathbf{J}_M)}}{2} = \\int \\mathbf{J}_M d\\ddot{x}{(\\mathbf{J}_M)}, then obtain \\int \\ddot{x}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\frac{\\mathbf{J}_M^{2}}{2} + \\omega", "derivation": "\\ddot{x}{(\\mathbf{J}_M)} = \\mathbf{J}_M and \\int \\ddot{x}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\int \\mathbf{J}_M d\\mathbf{J}_M and \\int \\ddot{x}{(\\mathbf{J}_M)} d\\ddot{x}{(\\mathbf{J}_M)} = \\int \\mathbf{J}_M d\\ddot{x}{(\\mathbf{J}_M)} and \\omega + \\frac{\\ddot{x}^{2}{(\\mathbf{J}_M)}}{2} = \\int \\mathbf{J}_M d\\ddot{x}{(\\mathbf{J}_M)} and \\int \\ddot{x}{(\\mathbf{J}_M)} d\\ddot{x}{(\\mathbf{J}_M)} = \\omega + \\frac{\\ddot{x}^{2}{(\\mathbf{J}_M)}}{2} and \\int \\mathbf{J}_M d\\mathbf{J}_M = \\frac{\\mathbf{J}_M^{2}}{2} + \\omega and \\int \\ddot{x}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\frac{\\mathbf{J}_M^{2}}{2} + \\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)))), Integral(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(A)} = e^{A}, then derive \\frac{d^{3}}{d A^{3}} \\sigma_{p}{(A)} = e^{A}, then obtain \\frac{d^{2}}{d A^{2}} \\sigma_{p}{(A)} = \\frac{d^{5}}{d A^{5}} \\sigma_{p}{(A)}", "derivation": "\\sigma_{p}{(A)} = e^{A} and \\frac{d}{d A} \\sigma_{p}{(A)} = \\frac{d}{d A} e^{A} and \\frac{d^{2}}{d A^{2}} \\sigma_{p}{(A)} = \\frac{d^{2}}{d A^{2}} e^{A} and \\frac{d^{3}}{d A^{3}} \\sigma_{p}{(A)} = \\frac{d^{3}}{d A^{3}} e^{A} and \\frac{d^{3}}{d A^{3}} \\sigma_{p}{(A)} = e^{A} and \\frac{d^{2}}{d A^{2}} \\sigma_{p}{(A)} = \\frac{d^{5}}{d A^{5}} \\sigma_{p}{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(3))), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(3))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(3))), exp(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(Function('\\\\sigma_p')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(5))))"]]}, {"prompt": "Given \\mathbf{g}{(I,L)} = L^{I} and b{(I,L)} = - L^{2 I} and \\mathbf{f}{(I,L)} = L^{2 I} - \\mathbf{g}{(I,L)} + \\int (\\mathbf{g}{(I,L)} + b{(I,L)}) dL, then obtain \\mathbf{f}{(I,L)} = L^{2 I} - L^{I} + \\int (L^{I} + b{(I,L)}) dL", "derivation": "\\mathbf{g}{(I,L)} = L^{I} and - L^{2 I} + \\mathbf{g}{(I,L)} = - L^{2 I} + L^{I} and b{(I,L)} = - L^{2 I} and \\mathbf{g}{(I,L)} + b{(I,L)} = L^{I} + b{(I,L)} and \\int (\\mathbf{g}{(I,L)} + b{(I,L)}) dL = \\int (L^{I} + b{(I,L)}) dL and L^{2 I} - \\mathbf{g}{(I,L)} + \\int (\\mathbf{g}{(I,L)} + b{(I,L)}) dL = L^{2 I} - \\mathbf{g}{(I,L)} + \\int (L^{I} + b{(I,L)}) dL and \\mathbf{f}{(I,L)} = L^{2 I} - \\mathbf{g}{(I,L)} + \\int (\\mathbf{g}{(I,L)} + b{(I,L)}) dL and \\mathbf{f}{(I,L)} = L^{2 I} - \\mathbf{g}{(I,L)} + \\int (L^{I} + b{(I,L)}) dL and \\mathbf{f}{(I,L)} = L^{2 I} - L^{I} + \\int (L^{I} + b{(I,L)}) dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)))"], [["minus", 1, "Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True)))), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True)))), Pow(Symbol('L', commutative=True), Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["minus", 5, "Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True)))), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)))"], "Equality(Add(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Integral(Add(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))), Add(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Integral(Add(Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Add(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Integral(Add(Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('\\\\mathbf{f}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Add(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Integral(Add(Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Function('\\\\mathbf{f}')(Symbol('I', commutative=True), Symbol('L', commutative=True)), Add(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('I', commutative=True))), Integral(Add(Pow(Symbol('L', commutative=True), Symbol('I', commutative=True)), Function('b')(Symbol('I', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(C,\\Psi^{\\dagger})} = C + \\sin{(\\Psi^{\\dagger})}, then derive \\int \\psi^{*}{(C,\\Psi^{\\dagger})} dC = \\frac{C^{2}}{2} + C \\sin{(\\Psi^{\\dagger})} + \\mathbf{S}, then obtain \\frac{\\partial}{\\partial C} \\int \\psi^{*}{(C,\\Psi^{\\dagger})} dC = \\frac{\\partial}{\\partial C} (\\frac{C^{2}}{2} + C \\sin{(\\Psi^{\\dagger})} + \\mathbf{S})", "derivation": "\\psi^{*}{(C,\\Psi^{\\dagger})} = C + \\sin{(\\Psi^{\\dagger})} and \\int \\psi^{*}{(C,\\Psi^{\\dagger})} dC = \\int (C + \\sin{(\\Psi^{\\dagger})}) dC and \\int \\psi^{*}{(C,\\Psi^{\\dagger})} dC = \\frac{C^{2}}{2} + C \\sin{(\\Psi^{\\dagger})} + \\mathbf{S} and \\frac{\\partial}{\\partial C} \\int \\psi^{*}{(C,\\Psi^{\\dagger})} dC = \\frac{\\partial}{\\partial C} (\\frac{C^{2}}{2} + C \\sin{(\\Psi^{\\dagger})} + \\mathbf{S})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('C', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Add(Symbol('C', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Symbol('C', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Symbol('C', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(U,c_{0})} = \\frac{U}{c_{0}}, then obtain \\eta{(U,c_{0})} - \\frac{\\partial}{\\partial U} \\eta{(U,c_{0})} = \\frac{U}{c_{0}} - \\frac{\\partial}{\\partial U} \\eta{(U,c_{0})}", "derivation": "\\eta{(U,c_{0})} = \\frac{U}{c_{0}} and \\frac{\\partial}{\\partial U} \\eta{(U,c_{0})} = \\frac{\\partial}{\\partial U} \\frac{U}{c_{0}} and \\eta{(U,c_{0})} - \\frac{\\partial}{\\partial U} \\frac{U}{c_{0}} = \\frac{U}{c_{0}} - \\frac{\\partial}{\\partial U} \\frac{U}{c_{0}} and \\eta{(U,c_{0})} - \\frac{\\partial}{\\partial U} \\eta{(U,c_{0})} = \\frac{U}{c_{0}} - \\frac{\\partial}{\\partial U} \\eta{(U,c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('U', commutative=True), Integer(1))))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Function('\\\\eta')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} = \\mathbf{F} + \\tilde{g}, then derive \\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F} = C_{2} + \\frac{\\mathbf{F}^{2}}{2} + \\mathbf{F} \\tilde{g}, then obtain \\frac{\\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F}}{2} = \\frac{C_{2}}{2} + \\frac{\\mathbf{F}^{2}}{4} + \\frac{\\mathbf{F} \\tilde{g}}{2}", "derivation": "\\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} = \\mathbf{F} + \\tilde{g} and \\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F} = \\int (\\mathbf{F} + \\tilde{g}) d\\mathbf{F} and \\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F} = C_{2} + \\frac{\\mathbf{F}^{2}}{2} + \\mathbf{F} \\tilde{g} and \\frac{\\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F}}{2} = \\frac{\\int (\\mathbf{F} + \\tilde{g}) d\\mathbf{F}}{2} and C_{2} + \\frac{\\mathbf{F}^{2}}{2} + \\mathbf{F} \\tilde{g} = \\int (\\mathbf{F} + \\tilde{g}) d\\mathbf{F} and \\frac{\\int \\operatorname{M_{E}}{(\\tilde{g},\\mathbf{F})} d\\mathbf{F}}{2} = \\frac{C_{2}}{2} + \\frac{\\mathbf{F}^{2}}{4} + \\frac{\\mathbf{F} \\tilde{g}}{2}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "Rational(1, 2)"], "Equality(Mul(Rational(1, 2), Integral(Function('M_E')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Rational(1, 2), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Rational(1, 2), Integral(Function('M_E')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('C_2', commutative=True)), Mul(Rational(1, 4), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Mul(Rational(1, 2), Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given S{(\\theta_2,\\dot{z})} = \\cos^{\\dot{z}}{(\\theta_2)}, then obtain \\frac{(\\int (S{(\\theta_2,\\dot{z})} - \\cos{(\\theta_2)}) d\\theta_2)^{\\dot{z}}}{\\theta_2} = \\frac{(\\int (- \\cos{(\\theta_2)} + \\cos^{\\dot{z}}{(\\theta_2)}) d\\theta_2)^{\\dot{z}}}{\\theta_2}", "derivation": "S{(\\theta_2,\\dot{z})} = \\cos^{\\dot{z}}{(\\theta_2)} and S{(\\theta_2,\\dot{z})} - \\cos{(\\theta_2)} = - \\cos{(\\theta_2)} + \\cos^{\\dot{z}}{(\\theta_2)} and \\int (S{(\\theta_2,\\dot{z})} - \\cos{(\\theta_2)}) d\\theta_2 = \\int (- \\cos{(\\theta_2)} + \\cos^{\\dot{z}}{(\\theta_2)}) d\\theta_2 and (\\int (S{(\\theta_2,\\dot{z})} - \\cos{(\\theta_2)}) d\\theta_2)^{\\dot{z}} = (\\int (- \\cos{(\\theta_2)} + \\cos^{\\dot{z}}{(\\theta_2)}) d\\theta_2)^{\\dot{z}} and \\frac{(\\int (S{(\\theta_2,\\dot{z})} - \\cos{(\\theta_2)}) d\\theta_2)^{\\dot{z}}}{\\theta_2} = \\frac{(\\int (- \\cos{(\\theta_2)} + \\cos^{\\dot{z}}{(\\theta_2)}) d\\theta_2)^{\\dot{z}}}{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('S')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Function('S')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Integral(Add(Function('S')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Integral(Add(Function('S')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Integral(Add(Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(a,\\mathbf{J})} = e^{\\mathbf{J} + a}, then obtain 1 = \\cos{((- \\hat{H}{(a,\\mathbf{J})} + e^{\\mathbf{J} + a}) \\hat{H}{(a,\\mathbf{J})})}", "derivation": "\\hat{H}{(a,\\mathbf{J})} = e^{\\mathbf{J} + a} and 0 = - \\hat{H}{(a,\\mathbf{J})} + e^{\\mathbf{J} + a} and 0 = - (- \\hat{H}{(a,\\mathbf{J})} + e^{\\mathbf{J} + a}) \\hat{H}{(a,\\mathbf{J})} and 1 = \\cos{((- \\hat{H}{(a,\\mathbf{J})} + e^{\\mathbf{J} + a}) \\hat{H}{(a,\\mathbf{J})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('a', commutative=True))))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('a', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('a', commutative=True)))), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["cos", 3], "Equality(Integer(1), cos(Mul(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('a', commutative=True)))), Function('\\\\hat{H}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\log{(\\rho_b^{A_{1}})}, then obtain \\hat{p}_0 \\frac{\\partial}{\\partial A_{1}} \\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\hat{p}_0 \\log{(\\rho_b)}", "derivation": "\\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\log{(\\rho_b^{A_{1}})} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\log{(\\rho_b^{A_{1}})} and \\hat{p}_0 \\frac{\\partial}{\\partial A_{1}} \\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\hat{p}_0 \\frac{\\partial}{\\partial A_{1}} \\log{(\\rho_b^{A_{1}})} and \\hat{p}_0 \\frac{\\partial}{\\partial A_{1}} \\operatorname{y^{\\prime}}{(\\rho_b,A_{1})} = \\hat{p}_0 \\log{(\\rho_b)}", "srepr_derivation": [["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{p}_0', commutative=True), Derivative(log(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{p}_0', commutative=True), log(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(F_{c},E_{\\lambda})} = \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c}), then obtain \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial F_{c}} (- E_{\\lambda} + \\sigma_{x}{(F_{c},E_{\\lambda})}) = \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial F_{c}} (- E_{\\lambda} + \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c}))", "derivation": "\\sigma_{x}{(F_{c},E_{\\lambda})} = \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c}) and - E_{\\lambda} + \\sigma_{x}{(F_{c},E_{\\lambda})} = - E_{\\lambda} + \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c}) and \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + \\sigma_{x}{(F_{c},E_{\\lambda})}) = \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c})) and \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial F_{c}} (- E_{\\lambda} + \\sigma_{x}{(F_{c},E_{\\lambda})}) = \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial F_{c}} (- E_{\\lambda} + \\frac{\\partial}{\\partial F_{c}} (- E_{\\lambda} + F_{c}))", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('F_c', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["add", 1, "Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('F_c', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('F_c', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('F_c', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Tuple(Symbol('F_c', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(c,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c) and \\operatorname{n_{1}}{(c)} = (-1)^{c}, then derive b{(c,\\dot{z})} = -1, then obtain \\frac{d}{d c} \\operatorname{n_{1}}{(c)} = \\frac{\\partial}{\\partial c} (\\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c))^{c}", "derivation": "b{(c,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c) and b{(c,\\dot{z})} = -1 and -1 = \\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c) and (-1)^{c} = (\\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c))^{c} and \\frac{d}{d c} (-1)^{c} = \\frac{\\partial}{\\partial c} (\\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c))^{c} and \\operatorname{n_{1}}{(c)} = (-1)^{c} and \\frac{d}{d c} \\operatorname{n_{1}}{(c)} = \\frac{\\partial}{\\partial c} (\\frac{\\partial}{\\partial \\dot{z}} (- \\dot{z} + c))^{c}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('c', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('b')(Symbol('c', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('c', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["differentiate", 4, "Symbol('c', commutative=True)"], "Equality(Derivative(Pow(Integer(-1), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('c', commutative=True)), Pow(Integer(-1), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Function('n_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\mathbf{E},l)} = \\mathbf{E} + l, then derive \\int I{(\\mathbf{E},l)} dl = \\mathbf{E} l + \\mathbf{g} + \\frac{l^{2}}{2}, then obtain \\mathbf{E} l + \\mathbf{g} + \\frac{l^{2}}{2} + l = l + \\int I{(\\mathbf{E},l)} dl", "derivation": "I{(\\mathbf{E},l)} = \\mathbf{E} + l and \\int I{(\\mathbf{E},l)} dl = \\int (\\mathbf{E} + l) dl and l + \\int I{(\\mathbf{E},l)} dl = l + \\int (\\mathbf{E} + l) dl and \\int I{(\\mathbf{E},l)} dl = \\mathbf{E} l + \\mathbf{g} + \\frac{l^{2}}{2} and \\mathbf{E} l + \\mathbf{g} + \\frac{l^{2}}{2} + l = l + \\int (\\mathbf{E} + l) dl and \\mathbf{E} l + \\mathbf{g} + \\frac{l^{2}}{2} + l = l + \\int I{(\\mathbf{E},l)} dl", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["add", 2, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Integral(Function('I')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Symbol('l', commutative=True), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Integral(Function('I')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\eta,\\varphi^*)} = \\log{(\\eta + \\varphi^*)}, then obtain e^{\\mathbf{J}_P{(\\eta,\\varphi^*)}} \\frac{\\partial}{\\partial \\eta} \\mathbf{J}_P{(\\eta,\\varphi^*)} = 1", "derivation": "\\mathbf{J}_P{(\\eta,\\varphi^*)} = \\log{(\\eta + \\varphi^*)} and e^{\\mathbf{J}_P{(\\eta,\\varphi^*)}} = \\eta + \\varphi^* and \\frac{\\partial}{\\partial \\eta} e^{\\mathbf{J}_P{(\\eta,\\varphi^*)}} = \\frac{\\partial}{\\partial \\eta} (\\eta + \\varphi^*) and e^{\\mathbf{J}_P{(\\eta,\\varphi^*)}} \\frac{\\partial}{\\partial \\eta} \\mathbf{J}_P{(\\eta,\\varphi^*)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(exp(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(t_{2})} = \\sin{(t_{2})}, then derive \\frac{d}{d t_{2}} \\operatorname{M_{E}}{(t_{2})} = \\cos{(t_{2})}, then obtain \\frac{\\frac{d}{d t_{2}} \\sin{(t_{2})}}{\\operatorname{M_{E}}{(t_{2})}} = \\frac{\\cos{(t_{2})}}{\\operatorname{M_{E}}{(t_{2})}}", "derivation": "\\operatorname{M_{E}}{(t_{2})} = \\sin{(t_{2})} and \\frac{d}{d t_{2}} \\operatorname{M_{E}}{(t_{2})} = \\frac{d}{d t_{2}} \\sin{(t_{2})} and \\frac{d}{d t_{2}} \\operatorname{M_{E}}{(t_{2})} = \\cos{(t_{2})} and \\frac{d}{d t_{2}} \\sin{(t_{2})} = \\cos{(t_{2})} and \\frac{\\frac{d}{d t_{2}} \\sin{(t_{2})}}{\\operatorname{M_{E}}{(t_{2})}} = \\frac{\\cos{(t_{2})}}{\\operatorname{M_{E}}{(t_{2})}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M_E')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), cos(Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), cos(Symbol('t_2', commutative=True)))"], [["divide", 4, "Function('M_E')(Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), Derivative(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Pow(Function('M_E')(Symbol('t_2', commutative=True)), Integer(-1)), cos(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(h,v_{t})} = \\cos{(\\frac{v_{t}}{h})}, then derive \\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} + 1 = 1 - \\frac{\\sin{(\\frac{v_{t}}{h})}}{h}, then obtain \\frac{\\partial}{\\partial v_{t}} (\\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} + 1) = \\frac{\\partial}{\\partial v_{t}} (1 - \\frac{\\sin{(\\frac{v_{t}}{h})}}{h})", "derivation": "\\operatorname{C_{1}}{(h,v_{t})} = \\cos{(\\frac{v_{t}}{h})} and \\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} = \\frac{\\partial}{\\partial v_{t}} \\cos{(\\frac{v_{t}}{h})} and \\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} + 1 = \\frac{\\partial}{\\partial v_{t}} \\cos{(\\frac{v_{t}}{h})} + 1 and \\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} + 1 = 1 - \\frac{\\sin{(\\frac{v_{t}}{h})}}{h} and \\frac{\\partial}{\\partial v_{t}} (\\frac{\\partial}{\\partial v_{t}} \\operatorname{C_{1}}{(h,v_{t})} + 1) = \\frac{\\partial}{\\partial v_{t}} (1 - \\frac{\\sin{(\\frac{v_{t}}{h})}}{h})", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('h', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('h', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('C_1')(Symbol('h', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('C_1')(Symbol('h', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))))))"], [["differentiate", 4, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('C_1')(Symbol('h', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('v_t', commutative=True))))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{S})} = e^{\\mathbf{S}} and \\delta{(E_{n},v_{1})} = E_{n} + \\cos{(v_{1})}, then obtain \\frac{\\partial}{\\partial E_{n}} \\delta{(E_{n},v_{1})} e^{- \\mathbf{S}} = \\frac{\\partial}{\\partial E_{n}} (E_{n} + \\cos{(v_{1})}) e^{- \\mathbf{S}}", "derivation": "\\mathbf{P}{(\\mathbf{S})} = e^{\\mathbf{S}} and \\delta{(E_{n},v_{1})} = E_{n} + \\cos{(v_{1})} and \\frac{\\delta{(E_{n},v_{1})}}{\\mathbf{P}{(\\mathbf{S})}} = \\frac{E_{n} + \\cos{(v_{1})}}{\\mathbf{P}{(\\mathbf{S})}} and \\delta{(E_{n},v_{1})} e^{- \\mathbf{S}} = (E_{n} + \\cos{(v_{1})}) e^{- \\mathbf{S}} and \\frac{\\partial}{\\partial E_{n}} \\delta{(E_{n},v_{1})} e^{- \\mathbf{S}} = \\frac{\\partial}{\\partial E_{n}} (E_{n} + \\cos{(v_{1})}) e^{- \\mathbf{S}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('E_n', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('E_n', commutative=True), cos(Symbol('v_1', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('E_n', commutative=True), Symbol('v_1', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Add(Symbol('E_n', commutative=True), cos(Symbol('v_1', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\delta')(Symbol('E_n', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Add(Symbol('E_n', commutative=True), cos(Symbol('v_1', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["differentiate", 4, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\delta')(Symbol('E_n', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('E_n', commutative=True), cos(Symbol('v_1', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(t_{1})} = \\cos{(t_{1})} and Z{(t_{1})} = \\frac{I{(t_{1})} - \\cos{(t_{1})}}{\\cos{(t_{1})}}, then obtain Z{(t_{1})} = 0", "derivation": "I{(t_{1})} = \\cos{(t_{1})} and I{(t_{1})} - \\cos{(t_{1})} = 0 and \\frac{I{(t_{1})} - \\cos{(t_{1})}}{\\cos{(t_{1})}} = 0 and Z{(t_{1})} = \\frac{I{(t_{1})} - \\cos{(t_{1})}}{\\cos{(t_{1})}} and Z{(t_{1})} = 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["minus", 1, "cos(Symbol('t_1', commutative=True))"], "Equality(Add(Function('I')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Integer(0))"], [["divide", 2, "cos(Symbol('t_1', commutative=True))"], "Equality(Mul(Add(Function('I')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1))), Integer(0))"], ["renaming_premise", "Equality(Function('Z')(Symbol('t_1', commutative=True)), Mul(Add(Function('I')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('Z')(Symbol('t_1', commutative=True)), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\hat{X},G)} = G \\log{(\\hat{X})}, then derive \\frac{\\partial^{2}}{\\partial G^{2}} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = 0, then derive \\frac{\\partial}{\\partial G} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = \\log{(\\hat{X})}, then obtain \\frac{\\partial^{2}}{\\partial G^{2}} G \\frac{\\partial}{\\partial G} G \\log{(\\hat{X})} = 0", "derivation": "\\operatorname{f^{\\prime}}{(\\hat{X},G)} = G \\log{(\\hat{X})} and \\frac{\\partial}{\\partial G} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = \\frac{\\partial}{\\partial G} G \\log{(\\hat{X})} and \\frac{\\partial^{2}}{\\partial G^{2}} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = \\frac{\\partial^{2}}{\\partial G^{2}} G \\log{(\\hat{X})} and \\frac{\\partial^{2}}{\\partial G^{2}} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = 0 and \\frac{\\partial}{\\partial G} \\operatorname{f^{\\prime}}{(\\hat{X},G)} = \\log{(\\hat{X})} and \\frac{\\partial}{\\partial G} G \\log{(\\hat{X})} = \\log{(\\hat{X})} and \\operatorname{f^{\\prime}}{(\\hat{X},G)} = G \\frac{\\partial}{\\partial G} G \\log{(\\hat{X})} and \\frac{\\partial^{2}}{\\partial G^{2}} G \\frac{\\partial}{\\partial G} G \\log{(\\hat{X})} = 0", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), log(Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), log(Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 6], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Derivative(Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Derivative(Mul(Symbol('G', commutative=True), Derivative(Mul(Symbol('G', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\theta{(v_{z})} = \\log{(v_{z})}, then obtain (\\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1) \\theta{(v_{z})} + \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 3 = (\\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1) \\theta{(v_{z})} + 4", "derivation": "\\theta{(v_{z})} = \\log{(v_{z})} and \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} = 1 and \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1 = 2 and \\theta{(v_{z})} + \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1 = \\theta{(v_{z})} + 2 and 2 \\theta{(v_{z})} + \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 3 = 2 \\theta{(v_{z})} + 4 and (\\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1) \\theta{(v_{z})} + \\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 3 = (\\frac{\\theta{(v_{z})}}{\\log{(v_{z})}} + 1) \\theta{(v_{z})} + 4", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["divide", 1, "log(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, 1], "Equality(Add(Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(1)), Integer(2))"], [["add", 3, "Function('\\\\theta')(Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('v_z', commutative=True)), Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(1)), Add(Function('\\\\theta')(Symbol('v_z', commutative=True)), Integer(2)))"], [["add", 4, "Add(Function('\\\\theta')(Symbol('v_z', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta')(Symbol('v_z', commutative=True))), Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(3)), Add(Mul(Integer(2), Function('\\\\theta')(Symbol('v_z', commutative=True))), Integer(4)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Add(Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\theta')(Symbol('v_z', commutative=True))), Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(3)), Add(Mul(Add(Mul(Function('\\\\theta')(Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\theta')(Symbol('v_z', commutative=True))), Integer(4)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(s)} = \\cos{(s)} and \\mathbf{A}{(s)} = \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})}, then obtain - \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})} = - \\mathbf{A}{(s)}", "derivation": "\\operatorname{F_{g}}{(s)} = \\cos{(s)} and \\operatorname{F_{g}}{(s)} + \\cos{(s)} = 2 \\cos{(s)} and \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})} = \\log{(2 \\cos{(s)})} and - \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})} = - \\log{(2 \\cos{(s)})} and \\mathbf{A}{(s)} = \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})} and \\mathbf{A}{(s)} = \\log{(2 \\cos{(s)})} and - \\log{(\\operatorname{F_{g}}{(s)} + \\cos{(s)})} = - \\mathbf{A}{(s)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["add", 1, "cos(Symbol('s', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))), Mul(Integer(2), cos(Symbol('s', commutative=True))))"], [["log", 2], "Equality(log(Add(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))), log(Mul(Integer(2), cos(Symbol('s', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), log(Add(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))))), Mul(Integer(-1), log(Mul(Integer(2), cos(Symbol('s', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), log(Add(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), log(Mul(Integer(2), cos(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Integer(-1), log(Add(Function('F_g')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))))), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('s', commutative=True))))"]]}, {"prompt": "Given a{(E,F_{H})} = \\log{(- E + F_{H})} and \\hat{H}_{\\lambda}{(E,F_{H})} = \\log{(- E + F_{H})}, then derive \\log{(\\frac{\\partial}{\\partial E} a{(E,F_{H})})} = \\log{(- \\frac{1}{- E + F_{H}})}, then obtain \\log{(\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,F_{H})})} = \\log{(- \\frac{1}{- E + F_{H}})}", "derivation": "a{(E,F_{H})} = \\log{(- E + F_{H})} and \\frac{\\partial}{\\partial E} a{(E,F_{H})} = \\frac{\\partial}{\\partial E} \\log{(- E + F_{H})} and \\log{(\\frac{\\partial}{\\partial E} a{(E,F_{H})})} = \\log{(\\frac{\\partial}{\\partial E} \\log{(- E + F_{H})})} and \\log{(\\frac{\\partial}{\\partial E} a{(E,F_{H})})} = \\log{(- \\frac{1}{- E + F_{H}})} and \\log{(\\frac{\\partial}{\\partial E} \\log{(- E + F_{H})})} = \\log{(- \\frac{1}{- E + F_{H}})} and \\hat{H}_{\\lambda}{(E,F_{H})} = \\log{(- E + F_{H})} and \\log{(\\frac{\\partial}{\\partial E} \\hat{H}_{\\lambda}{(E,F_{H})})} = \\log{(- \\frac{1}{- E + F_{H}})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('a')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Derivative(log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('a')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(Derivative(log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(log(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), log(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given I{(h)} = \\sin{(e^{h})} and \\mathbf{B}{(h)} = \\frac{d}{d h} I{(h)}, then derive \\mathbf{B}{(h)} = e^{h} \\cos{(e^{h})}, then obtain \\frac{d}{d h} I{(h)} = e^{h} \\cos{(e^{h})}", "derivation": "I{(h)} = \\sin{(e^{h})} and \\mathbf{B}{(h)} = \\frac{d}{d h} I{(h)} and \\mathbf{B}{(h)} = \\frac{d}{d h} \\sin{(e^{h})} and \\mathbf{B}{(h)} = e^{h} \\cos{(e^{h})} and \\frac{d}{d h} I{(h)} = e^{h} \\cos{(e^{h})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('h', commutative=True)), sin(exp(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('h', commutative=True)), Derivative(Function('I')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{B}')(Symbol('h', commutative=True)), Derivative(sin(exp(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\mathbf{B}')(Symbol('h', commutative=True)), Mul(exp(Symbol('h', commutative=True)), cos(exp(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('I')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(exp(Symbol('h', commutative=True)), cos(exp(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(F_{c},Z)} = Z + \\cos{(F_{c})}, then derive I + \\mathbf{J}_f{(F_{c},Z)} = a^{\\dagger} + \\cos{(F_{c})}, then obtain I + Z + \\cos{(F_{c})} = I + \\mathbf{J}_f{(F_{c},Z)}", "derivation": "\\mathbf{J}_f{(F_{c},Z)} = Z + \\cos{(F_{c})} and \\frac{\\partial}{\\partial F_{c}} \\mathbf{J}_f{(F_{c},Z)} = \\frac{\\partial}{\\partial F_{c}} (Z + \\cos{(F_{c})}) and \\int \\frac{\\partial}{\\partial F_{c}} \\mathbf{J}_f{(F_{c},Z)} dF_{c} = \\int \\frac{\\partial}{\\partial F_{c}} (Z + \\cos{(F_{c})}) dF_{c} and I + \\mathbf{J}_f{(F_{c},Z)} = a^{\\dagger} + \\cos{(F_{c})} and I + Z + \\cos{(F_{c})} = a^{\\dagger} + \\cos{(F_{c})} and I + Z + \\cos{(F_{c})} = I + \\mathbf{J}_f{(F_{c},Z)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), cos(Symbol('F_c', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), cos(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Tuple(Symbol('F_c', commutative=True))), Integral(Derivative(Add(Symbol('Z', commutative=True), cos(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('I', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('I', commutative=True), Symbol('Z', commutative=True), cos(Symbol('F_c', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('I', commutative=True), Symbol('Z', commutative=True), cos(Symbol('F_c', commutative=True))), Add(Symbol('I', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('F_c', commutative=True), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given E{(f^{*},u)} = f^{*} - u and \\operatorname{f^{\\prime}}{(f^{*},u)} = - f^{*} + u, then obtain - E{(f^{*},u)} = \\operatorname{f^{\\prime}}{(f^{*},u)}", "derivation": "E{(f^{*},u)} = f^{*} - u and - E{(f^{*},u)} = - f^{*} + u and \\operatorname{f^{\\prime}}{(f^{*},u)} = - f^{*} + u and - E{(f^{*},u)} = \\operatorname{f^{\\prime}}{(f^{*},u)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E')(Symbol('f^*', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Function('E')(Symbol('f^*', commutative=True), Symbol('u', commutative=True))), Function('f^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\mathbf{p} + \\tilde{g}, then obtain \\theta \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\theta", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\mathbf{p} + \\tilde{g} and \\theta \\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\theta (\\mathbf{p} + \\tilde{g}) and \\frac{\\partial}{\\partial \\tilde{g}} \\theta \\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} \\theta (\\mathbf{p} + \\tilde{g}) and \\theta \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{E_{\\lambda}}{(\\mathbf{p},\\tilde{g})} = \\theta", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\theta', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta', commutative=True), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\theta', commutative=True), Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Symbol('\\\\theta', commutative=True))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{X})} = \\hat{X}, then obtain \\hat{X} + 3 \\operatorname{F_{N}}{(\\hat{X})} = 3 \\hat{X} + \\operatorname{F_{N}}{(\\hat{X})}", "derivation": "\\operatorname{F_{N}}{(\\hat{X})} = \\hat{X} and 2 \\operatorname{F_{N}}{(\\hat{X})} = \\hat{X} + \\operatorname{F_{N}}{(\\hat{X})} and \\hat{X} + 3 \\operatorname{F_{N}}{(\\hat{X})} = 2 \\hat{X} + 2 \\operatorname{F_{N}}{(\\hat{X})} and \\hat{X} + 3 \\operatorname{F_{N}}{(\\hat{X})} = 3 \\hat{X} + \\operatorname{F_{N}}{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], [["add", 1, "Function('F_N')(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Integer(2), Function('F_N')(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Function('F_N')(Symbol('\\\\hat{X}', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\hat{X}', commutative=True), Function('F_N')(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(3), Function('F_N')(Symbol('\\\\hat{X}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(2), Function('F_N')(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(3), Function('F_N')(Symbol('\\\\hat{X}', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\hat{X}', commutative=True)), Function('F_N')(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given V{(Z)} = \\frac{d}{d Z} \\sin{(Z)}, then derive \\int V{(Z)} dZ = \\varphi^* + \\sin{(Z)}, then obtain 1 = \\frac{\\varphi^* + \\sin{(Z)}}{\\int V{(Z)} dZ}", "derivation": "V{(Z)} = \\frac{d}{d Z} \\sin{(Z)} and \\int V{(Z)} dZ = \\int \\frac{d}{d Z} \\sin{(Z)} dZ and \\int V{(Z)} dZ = \\varphi^* + \\sin{(Z)} and \\frac{\\int V{(Z)} dZ}{Z} = \\frac{\\varphi^* + \\sin{(Z)}}{Z} and 1 = \\frac{\\varphi^* + \\sin{(Z)}}{\\int V{(Z)} dZ}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('Z', commutative=True)), Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('Z', commutative=True))))"], [["divide", 3, "Symbol('Z', commutative=True)"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Integral(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('Z', commutative=True)))))"], [["divide", 4, "Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Integral(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('Z', commutative=True))), Pow(Integral(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\rho,p)} = \\rho \\log{(p)}, then obtain \\frac{\\partial}{\\partial \\rho} \\operatorname{f_{E}}{(\\rho,p)} = ((\\rho \\log{(p)})^{p} - \\operatorname{f_{E}}^{p}{(\\rho,p)} + 1) \\frac{\\partial}{\\partial \\rho} \\operatorname{f_{E}}{(\\rho,p)}", "derivation": "\\operatorname{f_{E}}{(\\rho,p)} = \\rho \\log{(p)} and \\frac{\\partial}{\\partial \\rho} \\operatorname{f_{E}}{(\\rho,p)} = \\frac{\\partial}{\\partial \\rho} \\rho \\log{(p)} and \\operatorname{f_{E}}^{p}{(\\rho,p)} = (\\rho \\log{(p)})^{p} and 0 = (\\rho \\log{(p)})^{p} - \\operatorname{f_{E}}^{p}{(\\rho,p)} and 1 = (\\rho \\log{(p)})^{p} - \\operatorname{f_{E}}^{p}{(\\rho,p)} + 1 and \\frac{\\partial}{\\partial \\rho} \\rho \\log{(p)} = ((\\rho \\log{(p)})^{p} - \\operatorname{f_{E}}^{p}{(\\rho,p)} + 1) \\frac{\\partial}{\\partial \\rho} \\rho \\log{(p)} and \\frac{\\partial}{\\partial \\rho} \\operatorname{f_{E}}{(\\rho,p)} = ((\\rho \\log{(p)})^{p} - \\operatorname{f_{E}}^{p}{(\\rho,p)} + 1) \\frac{\\partial}{\\partial \\rho} \\operatorname{f_{E}}{(\\rho,p)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["minus", 3, "Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)))))"], [["add", 4, 1], "Equality(Integer(1), Add(Pow(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(1)))"], [["times", 5, "Derivative(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))"], "Equality(Derivative(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Add(Pow(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(1)), Derivative(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Add(Pow(Mul(Symbol('\\\\rho', commutative=True), log(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(1)), Derivative(Function('f_E')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"]]}, {"prompt": "Given q{(\\delta,\\Omega)} = \\Omega + \\delta, then obtain \\delta (- \\delta - q{(\\delta,\\Omega)}) = \\delta (- \\Omega - 2 \\delta)", "derivation": "q{(\\delta,\\Omega)} = \\Omega + \\delta and \\delta + q{(\\delta,\\Omega)} = \\Omega + 2 \\delta and - \\delta - q{(\\delta,\\Omega)} = - \\Omega - 2 \\delta and \\delta (- \\delta - q{(\\delta,\\Omega)}) = \\delta (- \\Omega - 2 \\delta)", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True))))"], [["times", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Omega', commutative=True))))), Mul(Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\psi^*)} = \\log{(\\psi^*)} and \\Omega{(\\eta,\\mathbf{p})} = \\log{(\\eta^{\\mathbf{p}})}, then obtain (- \\mathbf{p} + \\nabla{(\\psi^*)}) (\\psi^* + \\log{(\\eta^{\\mathbf{p}})}) = (- \\mathbf{p} + \\log{(\\psi^*)}) (\\psi^* + \\log{(\\eta^{\\mathbf{p}})})", "derivation": "\\nabla{(\\psi^*)} = \\log{(\\psi^*)} and \\Omega{(\\eta,\\mathbf{p})} = \\log{(\\eta^{\\mathbf{p}})} and \\psi^* + \\Omega{(\\eta,\\mathbf{p})} = \\psi^* + \\log{(\\eta^{\\mathbf{p}})} and - \\mathbf{p} + \\nabla{(\\psi^*)} = - \\mathbf{p} + \\log{(\\psi^*)} and (- \\mathbf{p} + \\nabla{(\\psi^*)}) (\\psi^* + \\Omega{(\\eta,\\mathbf{p})}) = (- \\mathbf{p} + \\log{(\\psi^*)}) (\\psi^* + \\Omega{(\\eta,\\mathbf{p})}) and (- \\mathbf{p} + \\nabla{(\\psi^*)}) (\\psi^* + \\log{(\\eta^{\\mathbf{p}})}) = (- \\mathbf{p} + \\log{(\\psi^*)}) (\\psi^* + \\log{(\\eta^{\\mathbf{p}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], ["get_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))))"], [["times", 4, "Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\nabla')(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given t{(a)} = \\cos{(a)} and \\operatorname{P_{g}}{(a)} = t^{2}{(a)}, then obtain \\operatorname{P_{g}}{(a)} + \\frac{1}{a} = t{(a)} \\cos{(a)} + \\frac{1}{a}", "derivation": "t{(a)} = \\cos{(a)} and t^{2}{(a)} = t{(a)} \\cos{(a)} and \\operatorname{P_{g}}{(a)} = t^{2}{(a)} and \\operatorname{P_{g}}{(a)} = t{(a)} \\cos{(a)} and \\operatorname{P_{g}}{(a)} + \\frac{1}{a} = t{(a)} \\cos{(a)} + \\frac{1}{a}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["times", 1, "Function('t')(Symbol('a', commutative=True))"], "Equality(Pow(Function('t')(Symbol('a', commutative=True)), Integer(2)), Mul(Function('t')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('a', commutative=True)), Pow(Function('t')(Symbol('a', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('P_g')(Symbol('a', commutative=True)), Mul(Function('t')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))))"], [["add", 4, "Pow(Symbol('a', commutative=True), Integer(-1))"], "Equality(Add(Function('P_g')(Symbol('a', commutative=True)), Pow(Symbol('a', commutative=True), Integer(-1))), Add(Mul(Function('t')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Pow(Symbol('a', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\psi{(v_{y})} = e^{v_{y}}, then obtain \\int \\frac{\\frac{d}{d v_{y}} 2 \\psi{(v_{y})}}{\\frac{d}{d v_{y}} (\\psi{(v_{y})} + e^{v_{y}})} dv_{y} = \\int 1 dv_{y}", "derivation": "\\psi{(v_{y})} = e^{v_{y}} and 2 \\psi{(v_{y})} = \\psi{(v_{y})} + e^{v_{y}} and \\frac{d}{d v_{y}} 2 \\psi{(v_{y})} = \\frac{d}{d v_{y}} (\\psi{(v_{y})} + e^{v_{y}}) and \\frac{\\frac{d}{d v_{y}} 2 \\psi{(v_{y})}}{\\frac{d}{d v_{y}} (\\psi{(v_{y})} + e^{v_{y}})} = 1 and \\int \\frac{\\frac{d}{d v_{y}} 2 \\psi{(v_{y})}}{\\frac{d}{d v_{y}} (\\psi{(v_{y})} + e^{v_{y}})} dv_{y} = \\int 1 dv_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["add", 1, "Function('\\\\psi')(Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi')(Symbol('v_y', commutative=True))), Add(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\psi')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Add(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Integer(2), Function('\\\\psi')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Integer(1))"], [["integrate", 4, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Pow(Derivative(Add(Function('\\\\psi')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Integer(2), Function('\\\\psi')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True))), Integral(Integer(1), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given a{(F_{H})} = \\sin{(\\cos{(F_{H})})}, then obtain \\frac{\\log{(a^{F_{H}}{(F_{H})})}}{a^{F_{H}}{(F_{H})} + \\sin^{F_{H}}{(\\cos{(F_{H})})}} = \\frac{\\log{(\\sin^{F_{H}}{(\\cos{(F_{H})})})}}{a^{F_{H}}{(F_{H})} + \\sin^{F_{H}}{(\\cos{(F_{H})})}}", "derivation": "a{(F_{H})} = \\sin{(\\cos{(F_{H})})} and a^{F_{H}}{(F_{H})} = \\sin^{F_{H}}{(\\cos{(F_{H})})} and \\log{(a^{F_{H}}{(F_{H})})} = \\log{(\\sin^{F_{H}}{(\\cos{(F_{H})})})} and \\frac{\\log{(a^{F_{H}}{(F_{H})})}}{a^{F_{H}}{(F_{H})} + \\sin^{F_{H}}{(\\cos{(F_{H})})}} = \\frac{\\log{(\\sin^{F_{H}}{(\\cos{(F_{H})})})}}{a^{F_{H}}{(F_{H})} + \\sin^{F_{H}}{(\\cos{(F_{H})})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('F_H', commutative=True)), sin(cos(Symbol('F_H', commutative=True))))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), log(Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))))"], [["divide", 3, "Add(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], "Equality(Mul(Pow(Add(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))), Integer(-1)), log(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))), Mul(Pow(Add(Pow(Function('a')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))), Integer(-1)), log(Pow(sin(cos(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(k,t)} = \\frac{k}{t} and a{(k,m,t)} = \\frac{k}{m t}, then obtain (\\frac{k}{m t} + 1) \\sin{(a{(k,m,t)})} = (\\frac{k}{m t} + 1) \\sin{(\\frac{k}{m t})}", "derivation": "\\mu_{0}{(k,t)} = \\frac{k}{t} and \\frac{\\mu_{0}{(k,t)}}{m} = \\frac{k}{m t} and 1 + \\frac{\\mu_{0}{(k,t)}}{m} = \\frac{k}{m t} + 1 and a{(k,m,t)} = \\frac{k}{m t} and \\sin{(a{(k,m,t)})} = \\sin{(\\frac{k}{m t})} and (1 + \\frac{\\mu_{0}{(k,t)}}{m}) \\sin{(a{(k,m,t)})} = (1 + \\frac{\\mu_{0}{(k,t)}}{m}) \\sin{(\\frac{k}{m t})} and (\\frac{k}{m t} + 1) \\sin{(a{(k,m,t)})} = (\\frac{k}{m t} + 1) \\sin{(\\frac{k}{m t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('k', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True))), Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["add", 2, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('a')(Symbol('k', commutative=True), Symbol('m', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["sin", 4], "Equality(sin(Function('a')(Symbol('k', commutative=True), Symbol('m', commutative=True), Symbol('t', commutative=True))), sin(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["times", 5, "Add(Integer(1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True))))"], "Equality(Mul(Add(Integer(1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True)))), sin(Function('a')(Symbol('k', commutative=True), Symbol('m', commutative=True), Symbol('t', commutative=True)))), Mul(Add(Integer(1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('k', commutative=True), Symbol('t', commutative=True)))), sin(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))), Integer(1)), sin(Function('a')(Symbol('k', commutative=True), Symbol('m', commutative=True), Symbol('t', commutative=True)))), Mul(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))), Integer(1)), sin(Mul(Symbol('k', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{J}_P)} = \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P, then derive \\mathbf{F}{(\\mathbf{J}_P)} = C_{2} + e^{\\mathbf{J}_P}, then obtain C_{2} + \\iint e^{\\mathbf{J}_P} d\\mathbf{J}_P d\\mathbf{J}_P = C_{2} + \\int (C_{2} + e^{\\mathbf{J}_P}) d\\mathbf{J}_P", "derivation": "\\mathbf{F}{(\\mathbf{J}_P)} = \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P and \\mathbf{F}{(\\mathbf{J}_P)} = C_{2} + e^{\\mathbf{J}_P} and \\int e^{\\mathbf{J}_P} d\\mathbf{J}_P = C_{2} + e^{\\mathbf{J}_P} and \\iint e^{\\mathbf{J}_P} d\\mathbf{J}_P d\\mathbf{J}_P = \\int (C_{2} + e^{\\mathbf{J}_P}) d\\mathbf{J}_P and C_{2} + \\iint e^{\\mathbf{J}_P} d\\mathbf{J}_P d\\mathbf{J}_P = C_{2} + \\int (C_{2} + e^{\\mathbf{J}_P}) d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 4, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(n_{2},\\hat{H})} = \\hat{H} n_{2}, then obtain \\log{(- \\hat{H} n_{2} + \\int \\hat{p}{(n_{2},\\hat{H})} dn_{2})} = \\log{(- \\hat{H} n_{2} + \\int \\hat{H} n_{2} dn_{2})}", "derivation": "\\hat{p}{(n_{2},\\hat{H})} = \\hat{H} n_{2} and - \\hat{p}{(n_{2},\\hat{H})} = - \\hat{H} n_{2} and \\int \\hat{p}{(n_{2},\\hat{H})} dn_{2} = \\int \\hat{H} n_{2} dn_{2} and - \\hat{p}{(n_{2},\\hat{H})} + \\int \\hat{p}{(n_{2},\\hat{H})} dn_{2} = - \\hat{p}{(n_{2},\\hat{H})} + \\int \\hat{H} n_{2} dn_{2} and - \\hat{H} n_{2} + \\int \\hat{p}{(n_{2},\\hat{H})} dn_{2} = - \\hat{H} n_{2} + \\int \\hat{H} n_{2} dn_{2} and \\log{(- \\hat{H} n_{2} + \\int \\hat{p}{(n_{2},\\hat{H})} dn_{2})} = \\log{(- \\hat{H} n_{2} + \\int \\hat{H} n_{2} dn_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Integral(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\phi_{1}{(W)} = \\cos{(W)}, then derive \\int \\phi_{1}{(W)} dW = A_{x} + \\sin{(W)}, then obtain W + (\\int \\phi_{1}{(W)} dW)^{A_{x}} = W + (A_{x} + \\sin{(W)})^{A_{x}}", "derivation": "\\phi_{1}{(W)} = \\cos{(W)} and \\int \\phi_{1}{(W)} dW = \\int \\cos{(W)} dW and \\int \\phi_{1}{(W)} dW = A_{x} + \\sin{(W)} and (\\int \\phi_{1}{(W)} dW)^{A_{x}} = (A_{x} + \\sin{(W)})^{A_{x}} and W + (\\int \\phi_{1}{(W)} dW)^{A_{x}} = W + (A_{x} + \\sin{(W)})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('A_x', commutative=True), sin(Symbol('W', commutative=True))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('A_x', commutative=True)), Pow(Add(Symbol('A_x', commutative=True), sin(Symbol('W', commutative=True))), Symbol('A_x', commutative=True)))"], [["add", 4, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Pow(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('A_x', commutative=True))), Add(Symbol('W', commutative=True), Pow(Add(Symbol('A_x', commutative=True), sin(Symbol('W', commutative=True))), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} = \\log{(E_{\\lambda} - t)}, then derive \\frac{\\partial}{\\partial t} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} = - \\frac{1}{E_{\\lambda} - t}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} \\int - \\frac{1}{E_{\\lambda} - t} dt = \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} \\int \\frac{\\partial}{\\partial t} \\log{(E_{\\lambda} - t)} dt", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} = \\log{(E_{\\lambda} - t)} and \\frac{\\partial}{\\partial t} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} = \\frac{\\partial}{\\partial t} \\log{(E_{\\lambda} - t)} and \\frac{\\partial}{\\partial t} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} = - \\frac{1}{E_{\\lambda} - t} and - \\frac{1}{E_{\\lambda} - t} = \\frac{\\partial}{\\partial t} \\log{(E_{\\lambda} - t)} and \\int - \\frac{1}{E_{\\lambda} - t} dt = \\int \\frac{\\partial}{\\partial t} \\log{(E_{\\lambda} - t)} dt and \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} \\int - \\frac{1}{E_{\\lambda} - t} dt = \\operatorname{g^{\\prime}_{\\varepsilon}}{(t,E_{\\lambda})} \\int \\frac{\\partial}{\\partial t} \\log{(E_{\\lambda} - t)} dt", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), log(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('t', commutative=True))), Integral(Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["times", 5, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Mul(Integer(-1), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('t', commutative=True)))), Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given c{(f_{\\mathbf{p}},A)} = A^{f_{\\mathbf{p}}} and \\sigma_{x}{(f_{\\mathbf{p}},A)} = A^{f_{\\mathbf{p}}}, then obtain A + \\sigma_{x}{(f_{\\mathbf{p}},A)} = A + A^{f_{\\mathbf{p}}}", "derivation": "c{(f_{\\mathbf{p}},A)} = A^{f_{\\mathbf{p}}} and A + c{(f_{\\mathbf{p}},A)} = A + A^{f_{\\mathbf{p}}} and \\sigma_{x}{(f_{\\mathbf{p}},A)} = A^{f_{\\mathbf{p}}} and c{(f_{\\mathbf{p}},A)} = \\sigma_{x}{(f_{\\mathbf{p}},A)} and A + \\sigma_{x}{(f_{\\mathbf{p}},A)} = A + A^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True)), Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('A', commutative=True), Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(s)} = \\cos{(s)}, then obtain - \\cos{(s)} - \\frac{d}{d s} \\phi_{1}{(s)} + \\int \\cos{(\\iint \\phi_{1}{(s)} ds ds)} ds = - \\cos{(s)} - \\frac{d}{d s} \\phi_{1}{(s)} + \\int \\cos{(\\iint \\cos{(s)} ds ds)} ds", "derivation": "\\phi_{1}{(s)} = \\cos{(s)} and \\int \\phi_{1}{(s)} ds = \\int \\cos{(s)} ds and \\iint \\phi_{1}{(s)} ds ds = \\iint \\cos{(s)} ds ds and \\cos{(\\iint \\phi_{1}{(s)} ds ds)} = \\cos{(\\iint \\cos{(s)} ds ds)} and \\int \\cos{(\\iint \\phi_{1}{(s)} ds ds)} ds = \\int \\cos{(\\iint \\cos{(s)} ds ds)} ds and - \\cos{(s)} - \\frac{d}{d s} \\phi_{1}{(s)} + \\int \\cos{(\\iint \\phi_{1}{(s)} ds ds)} ds = - \\cos{(s)} - \\frac{d}{d s} \\phi_{1}{(s)} + \\int \\cos{(\\iint \\cos{(s)} ds ds)} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), cos(Integral(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(cos(Integral(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Integral(cos(Integral(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))))"], [["minus", 5, "Add(cos(Symbol('s', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integral(cos(Integral(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integral(cos(Integral(cos(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given J{(C_{d},\\hat{H}_l,\\mathbf{F})} = (\\hat{H}_l + \\mathbf{F})^{C_{d}}, then obtain \\frac{1}{J{(C_{d},\\hat{H}_l,\\mathbf{F})}} = \\frac{\\int (\\hat{H}_l + \\mathbf{F})^{C_{d}} d\\mathbf{F}}{J{(C_{d},\\hat{H}_l,\\mathbf{F})} \\int J{(C_{d},\\hat{H}_l,\\mathbf{F})} d\\mathbf{F}}", "derivation": "J{(C_{d},\\hat{H}_l,\\mathbf{F})} = (\\hat{H}_l + \\mathbf{F})^{C_{d}} and \\int J{(C_{d},\\hat{H}_l,\\mathbf{F})} d\\mathbf{F} = \\int (\\hat{H}_l + \\mathbf{F})^{C_{d}} d\\mathbf{F} and 1 = \\frac{\\int (\\hat{H}_l + \\mathbf{F})^{C_{d}} d\\mathbf{F}}{\\int J{(C_{d},\\hat{H}_l,\\mathbf{F})} d\\mathbf{F}} and \\frac{1}{J{(C_{d},\\hat{H}_l,\\mathbf{F})}} = \\frac{\\int (\\hat{H}_l + \\mathbf{F})^{C_{d}} d\\mathbf{F}}{J{(C_{d},\\hat{H}_l,\\mathbf{F})} \\int J{(C_{d},\\hat{H}_l,\\mathbf{F})} d\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 2, "Integral(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Integral(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"], [["divide", 3, "Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Pow(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Mul(Pow(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Integral(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Integral(Function('J')(Symbol('C_d', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(f^{*},\\rho)} = \\rho f^{*} and \\rho_{b}{(C_{2},\\mathbf{p})} = e^{C_{2}^{\\mathbf{p}}}, then obtain f^{*} + \\rho_{b}{(C_{2},\\mathbf{p})} = f^{*} + e^{C_{2}^{\\mathbf{p}}}", "derivation": "\\operatorname{A_{1}}{(f^{*},\\rho)} = \\rho f^{*} and f^{*} = \\rho f^{*} + f^{*} - \\operatorname{A_{1}}{(f^{*},\\rho)} and \\rho_{b}{(C_{2},\\mathbf{p})} = e^{C_{2}^{\\mathbf{p}}} and \\rho f^{*} + f^{*} - \\operatorname{A_{1}}{(f^{*},\\rho)} + \\rho_{b}{(C_{2},\\mathbf{p})} = \\rho f^{*} + f^{*} - \\operatorname{A_{1}}{(f^{*},\\rho)} + e^{C_{2}^{\\mathbf{p}}} and f^{*} + \\rho_{b}{(C_{2},\\mathbf{p})} = f^{*} + e^{C_{2}^{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Symbol('f^*', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Symbol('f^*', commutative=True), Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), exp(Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 3, "Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True))), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True), Mul(Integer(-1), Function('A_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho', commutative=True))), exp(Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('f^*', commutative=True), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('f^*', commutative=True), exp(Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given L{(b,U)} = U - b, then obtain U - 2 b + L{(b,U)} - \\frac{1}{\\sin^{3}{(b - 2 L{(b,U)})}} = U - 2 b + L{(b,U)} + \\frac{1}{\\sin^{3}{(U - 2 b + L{(b,U)})}}", "derivation": "L{(b,U)} = U - b and - b + L{(b,U)} = U - 2 b and U - 2 b + L{(b,U)} = 2 U - 3 b and - b + 2 L{(b,U)} = 2 U - 3 b and - \\sin{(b - 2 L{(b,U)})} = \\sin{(2 U - 3 b)} and - \\frac{1}{\\sin^{3}{(b - 2 L{(b,U)})}} = \\frac{1}{\\sin^{3}{(2 U - 3 b)}} and U - 2 b + L{(b,U)} - \\frac{1}{\\sin^{3}{(b - 2 L{(b,U)})}} = U - 2 b + L{(b,U)} + \\frac{1}{\\sin^{3}{(2 U - 3 b)}} and U - 2 b + L{(b,U)} - \\frac{1}{\\sin^{3}{(b - 2 L{(b,U)})}} = U - 2 b + L{(b,U)} + \\frac{1}{\\sin^{3}{(U - 2 b + L{(b,U)})}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True))), Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))))"], [["add", 1, "Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(2), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('b', commutative=True))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)))))), sin(Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('b', commutative=True)))))"], [["power", 5, "Integer(-3)"], "Equality(Mul(Integer(-1), Pow(sin(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True))))), Integer(-3))), Pow(sin(Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('b', commutative=True)))), Integer(-3)))"], [["add", 6, "Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(sin(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True))))), Integer(-3)))), Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Pow(sin(Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('b', commutative=True)))), Integer(-3))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(sin(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True))))), Integer(-3)))), Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Pow(sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True)), Function('L')(Symbol('b', commutative=True), Symbol('U', commutative=True)))), Integer(-3))))"]]}, {"prompt": "Given Z{(C_{d})} = \\cos{(C_{d})}, then obtain \\frac{Z{(C_{d})}}{\\cos^{2}{(C_{d})}} + \\cos{(C_{d})} = \\frac{Z{(C_{d})}}{\\cos^{2}{(C_{d})}} + \\frac{\\cos^{2}{(C_{d})}}{Z{(C_{d})}}", "derivation": "Z{(C_{d})} = \\cos{(C_{d})} and \\cos{(C_{d})} = \\frac{\\cos^{2}{(C_{d})}}{Z{(C_{d})}} and Z{(C_{d})} = \\frac{\\cos^{2}{(C_{d})}}{Z{(C_{d})}} and \\cos{(C_{d})} + \\frac{1}{Z{(C_{d})}} = \\frac{\\cos^{2}{(C_{d})}}{Z{(C_{d})}} + \\frac{1}{Z{(C_{d})}} and \\cos{(C_{d})} + \\frac{1}{Z{(C_{d})}} = Z{(C_{d})} + \\frac{1}{Z{(C_{d})}} and \\frac{Z{(C_{d})}}{\\cos^{2}{(C_{d})}} + \\cos{(C_{d})} = \\frac{Z{(C_{d})}}{\\cos^{2}{(C_{d})}} + \\frac{\\cos^{2}{(C_{d})}}{Z{(C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["times", 1, "Mul(Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1)), cos(Symbol('C_d', commutative=True)))"], "Equality(cos(Symbol('C_d', commutative=True)), Mul(Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1)), Pow(cos(Symbol('C_d', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('Z')(Symbol('C_d', commutative=True)), Mul(Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1)), Pow(cos(Symbol('C_d', commutative=True)), Integer(2))))"], [["add", 2, "Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1))"], "Equality(Add(cos(Symbol('C_d', commutative=True)), Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1)), Pow(cos(Symbol('C_d', commutative=True)), Integer(2))), Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(cos(Symbol('C_d', commutative=True)), Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1))), Add(Function('Z')(Symbol('C_d', commutative=True)), Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Function('Z')(Symbol('C_d', commutative=True)), Pow(cos(Symbol('C_d', commutative=True)), Integer(-2))), cos(Symbol('C_d', commutative=True))), Add(Mul(Function('Z')(Symbol('C_d', commutative=True)), Pow(cos(Symbol('C_d', commutative=True)), Integer(-2))), Mul(Pow(Function('Z')(Symbol('C_d', commutative=True)), Integer(-1)), Pow(cos(Symbol('C_d', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given i{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\iint (i{(\\mathbf{H})} - \\sin{(\\mathbf{H})}) d\\mathbf{H} d\\mathbf{H} = \\iint 0 d\\mathbf{H} d\\mathbf{H}", "derivation": "i{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and i{(\\mathbf{H})} - \\sin{(\\mathbf{H})} = 0 and \\int (i{(\\mathbf{H})} - \\sin{(\\mathbf{H})}) d\\mathbf{H} = \\int 0 d\\mathbf{H} and \\iint (i{(\\mathbf{H})} - \\sin{(\\mathbf{H})}) d\\mathbf{H} d\\mathbf{H} = \\iint 0 d\\mathbf{H} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{D},H)} = - H + \\mathbf{D}, then derive (- H + \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)})^{\\mathbf{D}} = (- H - 1)^{\\mathbf{D}}, then obtain \\sin{((- H - 1)^{\\mathbf{D}})} = \\sin{((- H + \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}))^{\\mathbf{D}})}", "derivation": "\\dot{z}{(\\mathbf{D},H)} = - H + \\mathbf{D} and \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)} = \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}) and - H + \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)} = - H + \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}) and (- H + \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)})^{\\mathbf{D}} = (- H + \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}))^{\\mathbf{D}} and \\sin{((- H + \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)})^{\\mathbf{D}})} = \\sin{((- H + \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}))^{\\mathbf{D}})} and (- H + \\frac{\\partial}{\\partial H} \\dot{z}{(\\mathbf{D},H)})^{\\mathbf{D}} = (- H - 1)^{\\mathbf{D}} and \\sin{((- H - 1)^{\\mathbf{D}})} = \\sin{((- H + \\frac{\\partial}{\\partial H} (- H + \\mathbf{D}))^{\\mathbf{D}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 2, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(sin(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(\\mu_0,\\nabla)} = \\log{(\\mu_0 + \\nabla)}, then obtain \\frac{d}{d \\nabla} \\int 1 d\\nabla = \\frac{\\partial}{\\partial \\nabla} \\int \\frac{\\log{(\\mu_0 + \\nabla)}}{\\phi_{1}{(\\mu_0,\\nabla)}} d\\nabla", "derivation": "\\phi_{1}{(\\mu_0,\\nabla)} = \\log{(\\mu_0 + \\nabla)} and 1 = \\frac{\\log{(\\mu_0 + \\nabla)}}{\\phi_{1}{(\\mu_0,\\nabla)}} and \\int 1 d\\nabla = \\int \\frac{\\log{(\\mu_0 + \\nabla)}}{\\phi_{1}{(\\mu_0,\\nabla)}} d\\nabla and \\frac{d}{d \\nabla} \\int 1 d\\nabla = \\frac{\\partial}{\\partial \\nabla} \\int \\frac{\\log{(\\mu_0 + \\nabla)}}{\\phi_{1}{(\\mu_0,\\nabla)}} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\omega)} = \\log{(\\omega)} and \\mu_{0}{(\\omega)} = \\rho_{b}{(\\omega)} \\log{(\\omega)} \\log{(\\omega)}^{- \\omega}, then obtain \\mu_{0}{(\\omega)} = \\log{(\\omega)}^{2} \\log{(\\omega)}^{- \\omega}", "derivation": "\\rho_{b}{(\\omega)} = \\log{(\\omega)} and \\rho_{b}{(\\omega)} \\log{(\\omega)}^{- \\omega} = \\log{(\\omega)} \\log{(\\omega)}^{- \\omega} and \\rho_{b}{(\\omega)} \\log{(\\omega)} \\log{(\\omega)}^{- 2 \\omega} = \\log{(\\omega)}^{2} \\log{(\\omega)}^{- 2 \\omega} and \\rho_{b}{(\\omega)} \\log{(\\omega)} \\log{(\\omega)}^{- \\omega} = \\log{(\\omega)}^{2} \\log{(\\omega)}^{- \\omega} and \\mu_{0}{(\\omega)} = \\rho_{b}{(\\omega)} \\log{(\\omega)} \\log{(\\omega)}^{- \\omega} and \\mu_{0}{(\\omega)} = \\log{(\\omega)}^{2} \\log{(\\omega)}^{- \\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Mul(log(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], [["times", 2, "Mul(log(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)))), Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)))))"], [["times", 3, "Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))), Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True)), Mul(Function('\\\\rho_b')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True)), Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\dot{x})} = \\cos{(\\dot{x})}, then obtain \\frac{d}{d \\dot{x}} (1 - \\frac{1}{\\cos{(\\dot{x})}}) = \\frac{d}{d \\dot{x}} (- \\frac{1}{\\cos{(\\dot{x})}} + \\frac{\\cos{(\\dot{x})}}{\\theta_{2}{(\\dot{x})}})", "derivation": "\\theta_{2}{(\\dot{x})} = \\cos{(\\dot{x})} and 1 = \\frac{\\cos{(\\dot{x})}}{\\theta_{2}{(\\dot{x})}} and 1 - \\frac{1}{\\cos{(\\dot{x})}} = - \\frac{1}{\\cos{(\\dot{x})}} + \\frac{\\cos{(\\dot{x})}}{\\theta_{2}{(\\dot{x})}} and \\frac{d}{d \\dot{x}} (1 - \\frac{1}{\\cos{(\\dot{x})}}) = \\frac{d}{d \\dot{x}} (- \\frac{1}{\\cos{(\\dot{x})}} + \\frac{\\cos{(\\dot{x})}}{\\theta_{2}{(\\dot{x})}})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 2, "Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{x}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(k)} = \\sin{(k)}, then derive - \\mathbf{p} + \\cos{(k)} + 2 \\int \\operatorname{E_{\\lambda}}{(k)} dk = \\int \\operatorname{E_{\\lambda}}{(k)} dk, then obtain \\int (- \\mathbf{p} + \\cos{(k)} + 2 \\int \\sin{(k)} dk) d\\mathbf{p} = \\iint \\sin{(k)} dk d\\mathbf{p}", "derivation": "\\operatorname{E_{\\lambda}}{(k)} = \\sin{(k)} and \\int \\operatorname{E_{\\lambda}}{(k)} dk = \\int \\sin{(k)} dk and \\int \\operatorname{E_{\\lambda}}{(k)} dk - \\int \\sin{(k)} dk = 0 and 2 \\int \\operatorname{E_{\\lambda}}{(k)} dk - \\int \\sin{(k)} dk = \\int \\operatorname{E_{\\lambda}}{(k)} dk and - \\mathbf{p} + \\cos{(k)} + 2 \\int \\operatorname{E_{\\lambda}}{(k)} dk = \\int \\operatorname{E_{\\lambda}}{(k)} dk and - \\mathbf{p} + \\cos{(k)} + 2 \\int \\operatorname{E_{\\lambda}}{(k)} dk = \\int \\sin{(k)} dk and - \\mathbf{p} + \\cos{(k)} + 2 \\int \\sin{(k)} dk = \\int \\sin{(k)} dk and \\int (- \\mathbf{p} + \\cos{(k)} + 2 \\int \\sin{(k)} dk) d\\mathbf{p} = \\iint \\sin{(k)} dk d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["minus", 2, "Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Integer(0))"], [["add", 3, "Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Mul(Integer(2), Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Integer(-1), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('k', commutative=True)), Mul(Integer(2), Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('k', commutative=True)), Mul(Integer(2), Integral(Function('E_{\\\\lambda}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('k', commutative=True)), Mul(Integer(2), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["integrate", 7, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('k', commutative=True)), Mul(Integer(2), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given L{(T)} = e^{T}, then obtain - T - e^{T} + \\int (T + L{(T)}) dT = - T - e^{T} + \\int (T + e^{T}) dT", "derivation": "L{(T)} = e^{T} and T + L{(T)} = T + e^{T} and \\int (T + L{(T)}) dT = \\int (T + e^{T}) dT and - T - e^{T} + \\int (T + L{(T)}) dT = - T - e^{T} + \\int (T + e^{T}) dT", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('L')(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Function('L')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["minus", 3, "Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), Function('L')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(t_{1},\\mathbf{J}_f)} = \\mathbf{J}_f t_{1}, then obtain \\mathbf{J}_f^{t_{1}} = (\\mathbf{J}_f t_{1} + \\mathbf{J}_f - \\mathbf{J}_M{(t_{1},\\mathbf{J}_f)})^{t_{1}}", "derivation": "\\mathbf{J}_M{(t_{1},\\mathbf{J}_f)} = \\mathbf{J}_f t_{1} and 0 = \\mathbf{J}_f t_{1} - \\mathbf{J}_M{(t_{1},\\mathbf{J}_f)} and \\mathbf{J}_f = \\mathbf{J}_f t_{1} + \\mathbf{J}_f - \\mathbf{J}_M{(t_{1},\\mathbf{J}_f)} and \\mathbf{J}_f^{t_{1}} = (\\mathbf{J}_f t_{1} + \\mathbf{J}_f - \\mathbf{J}_M{(t_{1},\\mathbf{J}_f)})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('t_1', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["add", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('t_1', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\mathbf{J}_M,F_{H})} = \\frac{\\mathbf{J}_M}{F_{H}}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} \\varphi{(\\mathbf{J}_M,F_{H})} = \\frac{1}{F_{H}}, then obtain \\frac{d}{d F_{H}} \\frac{1}{F_{H}} = \\frac{\\partial^{2}}{\\partial F_{H}\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{F_{H}}", "derivation": "\\varphi{(\\mathbf{J}_M,F_{H})} = \\frac{\\mathbf{J}_M}{F_{H}} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\varphi{(\\mathbf{J}_M,F_{H})} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{F_{H}} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\varphi{(\\mathbf{J}_M,F_{H})} = \\frac{1}{F_{H}} and \\frac{1}{F_{H}} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{F_{H}} and \\frac{d}{d F_{H}} \\frac{1}{F_{H}} = \\frac{\\partial^{2}}{\\partial F_{H}\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('F_H', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Pow(Symbol('F_H', commutative=True), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(\\dot{x},\\theta)} = \\cos^{\\theta}{(\\dot{x})}, then obtain (\\frac{\\partial}{\\partial \\theta} i{(\\dot{x},\\theta)} - 1)^{\\dot{x}} = (\\frac{\\partial}{\\partial \\theta} \\cos^{\\theta}{(\\dot{x})} - 1)^{\\dot{x}}", "derivation": "i{(\\dot{x},\\theta)} = \\cos^{\\theta}{(\\dot{x})} and \\frac{\\partial}{\\partial \\theta} i{(\\dot{x},\\theta)} = \\frac{\\partial}{\\partial \\theta} \\cos^{\\theta}{(\\dot{x})} and \\frac{\\partial}{\\partial \\theta} i{(\\dot{x},\\theta)} - 1 = \\frac{\\partial}{\\partial \\theta} \\cos^{\\theta}{(\\dot{x})} - 1 and (\\frac{\\partial}{\\partial \\theta} i{(\\dot{x},\\theta)} - 1)^{\\dot{x}} = (\\frac{\\partial}{\\partial \\theta} \\cos^{\\theta}{(\\dot{x})} - 1)^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('i')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1)))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Add(Derivative(Function('i')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Pow(Add(Derivative(Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given U{(l,\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda} l)}, then obtain - U{(l,\\hat{H}_{\\lambda})} - \\frac{U{(l,\\hat{H}_{\\lambda})}}{\\hat{H}_{\\lambda} l} = - U{(l,\\hat{H}_{\\lambda})} - \\frac{\\sin{(\\hat{H}_{\\lambda} l)}}{\\hat{H}_{\\lambda} l}", "derivation": "U{(l,\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda} l)} and \\frac{U{(l,\\hat{H}_{\\lambda})}}{\\hat{H}_{\\lambda} l} = \\frac{\\sin{(\\hat{H}_{\\lambda} l)}}{\\hat{H}_{\\lambda} l} and - \\frac{U{(l,\\hat{H}_{\\lambda})}}{\\hat{H}_{\\lambda} l} = - \\frac{\\sin{(\\hat{H}_{\\lambda} l)}}{\\hat{H}_{\\lambda} l} and - U{(l,\\hat{H}_{\\lambda})} - \\frac{U{(l,\\hat{H}_{\\lambda})}}{\\hat{H}_{\\lambda} l} = - U{(l,\\hat{H}_{\\lambda})} - \\frac{\\sin{(\\hat{H}_{\\lambda} l)}}{\\hat{H}_{\\lambda} l}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('l', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('l', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('l', commutative=True)))))"], [["minus", 3, "Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Function('U')(Symbol('l', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('l', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} = \\frac{\\log{(F_{N})}}{\\mathbf{S}}, then derive \\frac{\\partial}{\\partial \\mathbf{S}} \\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} - 1 = -1 - \\frac{\\log{(F_{N})}}{\\mathbf{S}^{2}}, then obtain -1 - \\frac{\\log{(F_{N})}}{\\mathbf{S}^{2}} = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{\\log{(F_{N})}}{\\mathbf{S}} - 1", "derivation": "\\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} = \\frac{\\log{(F_{N})}}{\\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} \\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{\\log{(F_{N})}}{\\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} \\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} - 1 = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{\\log{(F_{N})}}{\\mathbf{S}} - 1 and \\frac{\\partial}{\\partial \\mathbf{S}} \\dot{\\mathbf{r}}{(F_{N},\\mathbf{S})} - 1 = -1 - \\frac{\\log{(F_{N})}}{\\mathbf{S}^{2}} and -1 - \\frac{\\log{(F_{N})}}{\\mathbf{S}^{2}} = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{\\log{(F_{N})}}{\\mathbf{S}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('F_N', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-2)), log(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-2)), log(Symbol('F_N', commutative=True)))), Add(Derivative(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(n)} = \\cos{(n)}, then derive \\int \\operatorname{g_{\\varepsilon}}{(n)} dn = C_{2} + \\sin{(n)}, then derive m_{s} + \\sin{(n)} = C_{2} + \\sin{(n)}, then obtain m_{s} + \\sin{(n)} = \\int \\cos{(n)} dn", "derivation": "\\operatorname{g_{\\varepsilon}}{(n)} = \\cos{(n)} and \\int \\operatorname{g_{\\varepsilon}}{(n)} dn = \\int \\cos{(n)} dn and \\int \\operatorname{g_{\\varepsilon}}{(n)} dn = C_{2} + \\sin{(n)} and \\int \\cos{(n)} dn = C_{2} + \\sin{(n)} and m_{s} + \\sin{(n)} = C_{2} + \\sin{(n)} and m_{s} + \\sin{(n)} = \\int \\cos{(n)} dn", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('C_2', commutative=True), sin(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('C_2', commutative=True), sin(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('m_s', commutative=True), sin(Symbol('n', commutative=True))), Add(Symbol('C_2', commutative=True), sin(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('m_s', commutative=True), sin(Symbol('n', commutative=True))), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given G{(f^{*},m_{s})} = \\sin{(f^{*} m_{s})}, then obtain m_{s} (\\hat{H}_l + f^{*} (f^{*} + G{(f^{*},m_{s})})) G{(f^{*},m_{s})} = m_{s} (\\hat{H}_l + f^{*} (f^{*} + \\sin{(f^{*} m_{s})})) G{(f^{*},m_{s})}", "derivation": "G{(f^{*},m_{s})} = \\sin{(f^{*} m_{s})} and f^{*} + G{(f^{*},m_{s})} = f^{*} + \\sin{(f^{*} m_{s})} and f^{*} (f^{*} + G{(f^{*},m_{s})}) = f^{*} (f^{*} + \\sin{(f^{*} m_{s})}) and \\hat{H}_l + f^{*} (f^{*} + G{(f^{*},m_{s})}) = \\hat{H}_l + f^{*} (f^{*} + \\sin{(f^{*} m_{s})}) and m_{s} (\\hat{H}_l + f^{*} (f^{*} + G{(f^{*},m_{s})})) G{(f^{*},m_{s})} = m_{s} (\\hat{H}_l + f^{*} (f^{*} + \\sin{(f^{*} m_{s})})) G{(f^{*},m_{s})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)), sin(Mul(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))))"], [["add", 1, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))), Add(Symbol('f^*', commutative=True), sin(Mul(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)))))"], [["times", 2, "Symbol('f^*', commutative=True)"], "Equality(Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)))), Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), sin(Mul(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))))))"], [["add", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), sin(Mul(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)))))))"], [["times", 4, "Mul(Symbol('m_s', commutative=True), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Mul(Symbol('m_s', commutative=True), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))))), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Symbol('f^*', commutative=True), Add(Symbol('f^*', commutative=True), sin(Mul(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True)))))), Function('G')(Symbol('f^*', commutative=True), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = \\cos{(\\hat{H}_l \\mathbf{H})} and l{(\\hat{H}_l,\\mathbf{H})} = \\sin{(\\hat{H}_l \\mathbf{H})}, then derive \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = - \\hat{H}_l \\mathbf{H} \\sin{(\\hat{H}_l \\mathbf{H})}, then obtain \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = - \\hat{H}_l \\mathbf{H} l{(\\hat{H}_l,\\mathbf{H})}", "derivation": "\\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = \\cos{(\\hat{H}_l \\mathbf{H})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\hat{H}_l \\mathbf{H})} and \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\hat{H}_l \\mathbf{H})} and \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = - \\hat{H}_l \\mathbf{H} \\sin{(\\hat{H}_l \\mathbf{H})} and l{(\\hat{H}_l,\\mathbf{H})} = \\sin{(\\hat{H}_l \\mathbf{H})} and \\mathbf{H} \\frac{\\partial}{\\partial \\mathbf{H}} \\bar{\\h}{(\\hat{H}_l,\\mathbf{H})} = - \\hat{H}_l \\mathbf{H} l{(\\hat{H}_l,\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(cos(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('l')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(t_{2},\\psi^*)} = \\frac{t_{2}}{\\psi^*}, then derive \\frac{\\partial}{\\partial \\psi^*} \\mathbf{D}{(t_{2},\\psi^*)} = - \\frac{t_{2}}{(\\psi^*)^{2}}, then obtain \\frac{\\partial}{\\partial \\psi^*} \\frac{t_{2}}{\\psi^*} = - \\frac{\\mathbf{D}{(t_{2},\\psi^*)}}{\\psi^*}", "derivation": "\\mathbf{D}{(t_{2},\\psi^*)} = \\frac{t_{2}}{\\psi^*} and \\frac{\\partial}{\\partial \\psi^*} \\mathbf{D}{(t_{2},\\psi^*)} = \\frac{\\partial}{\\partial \\psi^*} \\frac{t_{2}}{\\psi^*} and \\frac{\\partial}{\\partial \\psi^*} \\mathbf{D}{(t_{2},\\psi^*)} = - \\frac{t_{2}}{(\\psi^*)^{2}} and \\frac{\\partial}{\\partial \\psi^*} \\mathbf{D}{(t_{2},\\psi^*)} = - \\frac{\\mathbf{D}{(t_{2},\\psi^*)}}{\\psi^*} and \\frac{\\partial}{\\partial \\psi^*} \\frac{t_{2}}{\\psi^*} = - \\frac{\\mathbf{D}{(t_{2},\\psi^*)}}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(F_{c},E_{x})} = e^{E_{x} F_{c}} and \\operatorname{A_{y}}{(\\chi,f_{E})} = \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}}, then obtain - E_{x} + \\hat{H}_{\\lambda}{(F_{c},E_{x})} \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}} = - E_{x} + e^{E_{x} F_{c}} \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}}", "derivation": "\\hat{H}_{\\lambda}{(F_{c},E_{x})} = e^{E_{x} F_{c}} and \\operatorname{A_{y}}{(\\chi,f_{E})} = \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}} and \\operatorname{A_{y}}{(\\chi,f_{E})} \\hat{H}_{\\lambda}{(F_{c},E_{x})} = \\operatorname{A_{y}}{(\\chi,f_{E})} e^{E_{x} F_{c}} and - E_{x} + \\operatorname{A_{y}}{(\\chi,f_{E})} \\hat{H}_{\\lambda}{(F_{c},E_{x})} = - E_{x} + \\operatorname{A_{y}}{(\\chi,f_{E})} e^{E_{x} F_{c}} and - E_{x} + \\hat{H}_{\\lambda}{(F_{c},E_{x})} \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}} = - E_{x} + e^{E_{x} F_{c}} \\frac{\\partial}{\\partial \\chi} \\chi^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_c', commutative=True), Symbol('E_x', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Symbol('F_c', commutative=True))))"], ["get_premise", "Equality(Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 1, "Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_c', commutative=True), Symbol('E_x', commutative=True))), Mul(Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Symbol('F_c', commutative=True)))))"], [["minus", 3, "Symbol('E_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_c', commutative=True), Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Function('A_y')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), exp(Mul(Symbol('E_x', commutative=True), Symbol('F_c', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_c', commutative=True), Symbol('E_x', commutative=True)), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(exp(Mul(Symbol('E_x', commutative=True), Symbol('F_c', commutative=True))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given x{(v)} = e^{e^{v}}, then obtain (v + \\frac{e^{e^{v}}}{x{(v)}})^{v} (v - 1 + \\frac{2 e^{e^{v}}}{x{(v)}})^{- v} = 1", "derivation": "x{(v)} = e^{e^{v}} and 1 = \\frac{e^{e^{v}}}{x{(v)}} and v + 1 = v + \\frac{e^{e^{v}}}{x{(v)}} and (v + 1)^{v} = (v + \\frac{e^{e^{v}}}{x{(v)}})^{v} and (v + 1)^{v} (v + \\frac{e^{e^{v}}}{x{(v)}})^{- v} = 1 and (v + \\frac{e^{e^{v}}}{x{(v)}})^{v} (v - 1 + \\frac{2 e^{e^{v}}}{x{(v)}})^{- v} = 1", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('v', commutative=True)), exp(exp(Symbol('v', commutative=True))))"], [["divide", 1, "Function('x')(Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True)))))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Integer(1)), Add(Symbol('v', commutative=True), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Symbol('v', commutative=True), Integer(1)), Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))), Symbol('v', commutative=True)))"], [["divide", 4, "Pow(Add(Symbol('v', commutative=True), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('v', commutative=True), Integer(1)), Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))), Mul(Integer(-1), Symbol('v', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Symbol('v', commutative=True), Mul(Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))), Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), Integer(-1), Mul(Integer(2), Pow(Function('x')(Symbol('v', commutative=True)), Integer(-1)), exp(exp(Symbol('v', commutative=True))))), Mul(Integer(-1), Symbol('v', commutative=True)))), Integer(1))"]]}, {"prompt": "Given t{(x,\\theta)} = \\theta x and U{(M_{E})} = e^{\\sin{(M_{E})}}, then obtain \\iint \\frac{x + U{(M_{E})}}{\\theta x (x + e^{\\sin{(M_{E})}})} d\\theta d\\theta = \\iint \\frac{1}{\\theta x} d\\theta d\\theta", "derivation": "t{(x,\\theta)} = \\theta x and U{(M_{E})} = e^{\\sin{(M_{E})}} and x + U{(M_{E})} = x + e^{\\sin{(M_{E})}} and \\frac{x + U{(M_{E})}}{t{(x,\\theta)}} = \\frac{x + e^{\\sin{(M_{E})}}}{t{(x,\\theta)}} and \\frac{x + U{(M_{E})}}{\\theta x} = \\frac{x + e^{\\sin{(M_{E})}}}{\\theta x} and \\frac{x + U{(M_{E})}}{\\theta x (x + e^{\\sin{(M_{E})}})} = \\frac{1}{\\theta x} and \\int \\frac{x + U{(M_{E})}}{\\theta x (x + e^{\\sin{(M_{E})}})} d\\theta = \\int \\frac{1}{\\theta x} d\\theta and \\iint \\frac{x + U{(M_{E})}}{\\theta x (x + e^{\\sin{(M_{E})}})} d\\theta d\\theta = \\iint \\frac{1}{\\theta x} d\\theta d\\theta", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('x', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('x', commutative=True)))"], ["get_premise", "Equality(Function('U')(Symbol('M_E', commutative=True)), exp(sin(Symbol('M_E', commutative=True))))"], [["add", 2, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True))), Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True)))))"], [["divide", 3, "Function('t')(Symbol('x', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True))), Pow(Function('t')(Symbol('x', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Mul(Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True)))), Pow(Function('t')(Symbol('x', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True))))))"], [["divide", 5, "Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True)))), Integer(-1))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["integrate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 7, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Function('U')(Symbol('M_E', commutative=True))), Pow(Add(Symbol('x', commutative=True), exp(sin(Symbol('M_E', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbb{I})} = \\sin{(\\cos{(\\mathbb{I})})} and \\theta{(\\mathbb{I})} = \\sin{(\\cos{(\\mathbb{I})})}, then obtain (-1 + \\frac{\\operatorname{t_{1}}{(\\mathbb{I})}}{\\mathbb{I}}) \\theta{(\\mathbb{I})} = (-1 + \\frac{\\operatorname{t_{1}}{(\\mathbb{I})}}{\\mathbb{I}}) \\sin{(\\cos{(\\mathbb{I})})}", "derivation": "\\operatorname{t_{1}}{(\\mathbb{I})} = \\sin{(\\cos{(\\mathbb{I})})} and \\frac{\\operatorname{t_{1}}{(\\mathbb{I})}}{\\mathbb{I}} = \\frac{\\sin{(\\cos{(\\mathbb{I})})}}{\\mathbb{I}} and \\theta{(\\mathbb{I})} = \\sin{(\\cos{(\\mathbb{I})})} and (-1 + \\frac{\\sin{(\\cos{(\\mathbb{I})})}}{\\mathbb{I}}) \\theta{(\\mathbb{I})} = (-1 + \\frac{\\sin{(\\cos{(\\mathbb{I})})}}{\\mathbb{I}}) \\sin{(\\cos{(\\mathbb{I})})} and (-1 + \\frac{\\operatorname{t_{1}}{(\\mathbb{I})}}{\\mathbb{I}}) \\theta{(\\mathbb{I})} = (-1 + \\frac{\\operatorname{t_{1}}{(\\mathbb{I})}}{\\mathbb{I}}) \\sin{(\\cos{(\\mathbb{I})})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbb{I}', commutative=True)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 3, "Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True)))))"], "Equality(Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True))))), Function('\\\\theta')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\mathbb{I}', commutative=True))))), sin(cos(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)))), Function('\\\\theta')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)))), sin(cos(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(z^{*})} = \\sin{(z^{*})}, then obtain 2 \\operatorname{C_{2}}^{2}{(z^{*})} + 2 \\operatorname{C_{2}}{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{C_{2}}^{2}{(z^{*})} = 2 \\operatorname{C_{2}}^{2}{(z^{*})} + 2 \\operatorname{C_{2}}{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{C_{2}}{(z^{*})} \\sin{(z^{*})}", "derivation": "\\operatorname{C_{2}}{(z^{*})} = \\sin{(z^{*})} and \\operatorname{C_{2}}^{2}{(z^{*})} = \\operatorname{C_{2}}{(z^{*})} \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{C_{2}}^{2}{(z^{*})} = \\frac{d}{d z^{*}} \\operatorname{C_{2}}{(z^{*})} \\sin{(z^{*})} and 2 \\operatorname{C_{2}}^{2}{(z^{*})} + 2 \\operatorname{C_{2}}{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{C_{2}}^{2}{(z^{*})} = 2 \\operatorname{C_{2}}^{2}{(z^{*})} + 2 \\operatorname{C_{2}}{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{C_{2}}{(z^{*})} \\sin{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["times", 1, "Function('C_2')(Symbol('z^*', commutative=True))"], "Equality(Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2)), Mul(Function('C_2')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Function('C_2')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["add", 3, "Add(Mul(Integer(2), Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2))), Mul(Integer(2), Function('C_2')(Symbol('z^*', commutative=True))))"], "Equality(Add(Mul(Integer(2), Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2))), Mul(Integer(2), Function('C_2')(Symbol('z^*', commutative=True))), Derivative(Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(Mul(Integer(2), Pow(Function('C_2')(Symbol('z^*', commutative=True)), Integer(2))), Mul(Integer(2), Function('C_2')(Symbol('z^*', commutative=True))), Derivative(Mul(Function('C_2')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(T)} = e^{T}, then obtain \\frac{d}{d T} (- \\hat{x}_0{(T)} e^{T} + \\int \\hat{x}_0{(T)} e^{T} dT) = \\frac{\\partial}{\\partial T} (\\dot{y} - \\hat{x}_0{(T)} e^{T} + \\frac{e^{2 T}}{2})", "derivation": "\\hat{x}_0{(T)} = e^{T} and \\hat{x}_0{(T)} e^{T} = e^{2 T} and \\int \\hat{x}_0{(T)} e^{T} dT = \\int e^{2 T} dT and - \\hat{x}_0{(T)} e^{T} + \\int \\hat{x}_0{(T)} e^{T} dT = - \\hat{x}_0{(T)} e^{T} + \\int e^{2 T} dT and \\frac{d}{d T} (- \\hat{x}_0{(T)} e^{T} + \\int \\hat{x}_0{(T)} e^{T} dT) = \\frac{d}{d T} (- \\hat{x}_0{(T)} e^{T} + \\int e^{2 T} dT) and \\frac{d}{d T} (- \\hat{x}_0{(T)} e^{T} + \\int \\hat{x}_0{(T)} e^{T} dT) = \\frac{\\partial}{\\partial T} (\\dot{y} - \\hat{x}_0{(T)} e^{T} + \\frac{e^{2 T}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["times", 1, "exp(Symbol('T', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), exp(Mul(Integer(2), Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integral(Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integral(Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integral(Mul(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('T', commutative=True))))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(\\eta)} = \\sin{(\\eta)} and m{(\\chi)} = \\chi, then derive \\int \\mathbb{I}{(\\eta)} d\\eta = \\chi - \\cos{(\\eta)}, then obtain \\int (- \\sin{(\\eta)} + \\int \\mathbb{I}{(\\eta)} d\\eta) d\\eta = \\int (m{(\\chi)} - \\sin{(\\eta)} - \\cos{(\\eta)}) d\\eta", "derivation": "\\mathbb{I}{(\\eta)} = \\sin{(\\eta)} and \\int \\mathbb{I}{(\\eta)} d\\eta = \\int \\sin{(\\eta)} d\\eta and \\int \\mathbb{I}{(\\eta)} d\\eta = \\chi - \\cos{(\\eta)} and m{(\\chi)} = \\chi and \\int \\mathbb{I}{(\\eta)} d\\eta = m{(\\chi)} - \\cos{(\\eta)} and - \\sin{(\\eta)} + \\int \\mathbb{I}{(\\eta)} d\\eta = m{(\\chi)} - \\sin{(\\eta)} - \\cos{(\\eta)} and \\int (- \\sin{(\\eta)} + \\int \\mathbb{I}{(\\eta)} d\\eta) d\\eta = \\int (m{(\\chi)} - \\sin{(\\eta)} - \\cos{(\\eta)}) d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(sin(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Function('m')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], [["minus", 5, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Function('m')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Function('m')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(S,\\dot{y})} = S e^{\\dot{y}} and \\lambda{(S,\\dot{y})} = (S e^{\\dot{y}})^{- S} \\operatorname{A_{2}}^{S}{(S,\\dot{y})}, then obtain \\lambda^{S}{(S,\\dot{y})} = 1", "derivation": "\\operatorname{A_{2}}{(S,\\dot{y})} = S e^{\\dot{y}} and \\operatorname{A_{2}}^{S}{(S,\\dot{y})} = (S e^{\\dot{y}})^{S} and \\lambda{(S,\\dot{y})} = (S e^{\\dot{y}})^{- S} \\operatorname{A_{2}}^{S}{(S,\\dot{y})} and \\lambda{(S,\\dot{y})} = 1 and \\lambda^{S}{(S,\\dot{y})} = 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('S', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('S', commutative=True)), Pow(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(Function('A_2')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\lambda')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('S', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('S', commutative=True)), Integer(1))"]]}, {"prompt": "Given Z{(G,F_{x})} = F_{x} - G, then obtain F_{x} + (G + Z{(G,F_{x})}) (\\int Z{(G,F_{x})} dG)^{F_{x}} = F_{x} + (G + Z{(G,F_{x})}) (\\int (F_{x} - G) dG)^{F_{x}}", "derivation": "Z{(G,F_{x})} = F_{x} - G and \\int Z{(G,F_{x})} dG = \\int (F_{x} - G) dG and (\\int Z{(G,F_{x})} dG)^{F_{x}} = (\\int (F_{x} - G) dG)^{F_{x}} and (G + Z{(G,F_{x})}) (\\int Z{(G,F_{x})} dG)^{F_{x}} = (G + Z{(G,F_{x})}) (\\int (F_{x} - G) dG)^{F_{x}} and F_{x} + (G + Z{(G,F_{x})}) (\\int Z{(G,F_{x})} dG)^{F_{x}} = F_{x} + (G + Z{(G,F_{x})}) (\\int (F_{x} - G) dG)^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Integral(Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True)), Pow(Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True)))"], [["times", 3, "Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True))), Pow(Integral(Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True))), Mul(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True))), Pow(Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True))))"], [["add", 4, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Mul(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True))), Pow(Integral(Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True)))), Add(Symbol('F_x', commutative=True), Mul(Add(Symbol('G', commutative=True), Function('Z')(Symbol('G', commutative=True), Symbol('F_x', commutative=True))), Pow(Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given S{(c_{0},\\varepsilon)} = - c_{0} + \\sin{(\\varepsilon)}, then obtain \\frac{\\frac{\\partial}{\\partial c_{0}} S^{\\varepsilon}{(c_{0},\\varepsilon)}}{\\int c_{0} S{(c_{0},\\varepsilon)} dc_{0}} = \\frac{\\frac{\\partial}{\\partial c_{0}} (- c_{0} + \\sin{(\\varepsilon)})^{\\varepsilon}}{\\int c_{0} S{(c_{0},\\varepsilon)} dc_{0}}", "derivation": "S{(c_{0},\\varepsilon)} = - c_{0} + \\sin{(\\varepsilon)} and S^{\\varepsilon}{(c_{0},\\varepsilon)} = (- c_{0} + \\sin{(\\varepsilon)})^{\\varepsilon} and \\frac{\\partial}{\\partial c_{0}} S^{\\varepsilon}{(c_{0},\\varepsilon)} = \\frac{\\partial}{\\partial c_{0}} (- c_{0} + \\sin{(\\varepsilon)})^{\\varepsilon} and \\frac{\\frac{\\partial}{\\partial c_{0}} S^{\\varepsilon}{(c_{0},\\varepsilon)}}{\\int c_{0} S{(c_{0},\\varepsilon)} dc_{0}} = \\frac{\\frac{\\partial}{\\partial c_{0}} (- c_{0} + \\sin{(\\varepsilon)})^{\\varepsilon}}{\\int c_{0} S{(c_{0},\\varepsilon)} dc_{0}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Pow(Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Mul(Symbol('c_0', commutative=True), Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('c_0', commutative=True)))"], "Equality(Mul(Derivative(Pow(Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Integral(Mul(Symbol('c_0', commutative=True), Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integer(-1))), Mul(Derivative(Pow(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Integral(Mul(Symbol('c_0', commutative=True), Function('S')(Symbol('c_0', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(s)} = \\cos{(s)} and \\mathbf{A}{(s)} = \\frac{\\cos{(s)}}{\\operatorname{P_{e}}{(s)}}, then obtain \\mathbf{A}^{s}{(s)} = 1", "derivation": "\\operatorname{P_{e}}{(s)} = \\cos{(s)} and \\mathbf{A}{(s)} = \\frac{\\cos{(s)}}{\\operatorname{P_{e}}{(s)}} and \\mathbf{A}^{s}{(s)} = (\\frac{\\cos{(s)}}{\\operatorname{P_{e}}{(s)}})^{s} and \\mathbf{A}^{s}{(s)} = 1", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), Mul(Pow(Function('P_e')(Symbol('s', commutative=True)), Integer(-1)), cos(Symbol('s', commutative=True))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Mul(Pow(Function('P_e')(Symbol('s', commutative=True)), Integer(-1)), cos(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(1))"]]}, {"prompt": "Given B{(\\phi_1,H)} = \\frac{\\phi_1}{H}, then derive \\frac{\\partial}{\\partial \\phi_1} B{(\\phi_1,H)} = \\frac{1}{H}, then obtain \\frac{1}{H} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{H}", "derivation": "B{(\\phi_1,H)} = \\frac{\\phi_1}{H} and \\frac{\\partial}{\\partial \\phi_1} B{(\\phi_1,H)} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{H} and \\frac{\\partial}{\\partial \\phi_1} B{(\\phi_1,H)} = \\frac{1}{H} and \\frac{1}{H} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{H}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('\\\\phi_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Pow(Symbol('H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('H', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(v_{1},z,G)} = G (- v_{1} + z), then obtain \\int (e^{\\frac{\\partial}{\\partial v_{1}} - v_{1} \\varphi^{*}{(v_{1},z,G)}})^{z} dG = \\int (e^{\\frac{\\partial}{\\partial v_{1}} - G v_{1} (- v_{1} + z)})^{z} dG", "derivation": "\\varphi^{*}{(v_{1},z,G)} = G (- v_{1} + z) and - v_{1} \\varphi^{*}{(v_{1},z,G)} = - G v_{1} (- v_{1} + z) and \\frac{\\partial}{\\partial v_{1}} - v_{1} \\varphi^{*}{(v_{1},z,G)} = \\frac{\\partial}{\\partial v_{1}} - G v_{1} (- v_{1} + z) and e^{\\frac{\\partial}{\\partial v_{1}} - v_{1} \\varphi^{*}{(v_{1},z,G)}} = e^{\\frac{\\partial}{\\partial v_{1}} - G v_{1} (- v_{1} + z)} and (e^{\\frac{\\partial}{\\partial v_{1}} - v_{1} \\varphi^{*}{(v_{1},z,G)}})^{z} = (e^{\\frac{\\partial}{\\partial v_{1}} - G v_{1} (- v_{1} + z)})^{z} and \\int (e^{\\frac{\\partial}{\\partial v_{1}} - v_{1} \\varphi^{*}{(v_{1},z,G)}})^{z} dG = \\int (e^{\\frac{\\partial}{\\partial v_{1}} - G v_{1} (- v_{1} + z)})^{z} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_1', commutative=True), Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('v_1', commutative=True), Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Mul(Integer(-1), Symbol('v_1', commutative=True), Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), exp(Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(exp(Derivative(Mul(Integer(-1), Symbol('v_1', commutative=True), Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('z', commutative=True)), Pow(exp(Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('z', commutative=True)))"], [["integrate", 5, "Symbol('G', commutative=True)"], "Equality(Integral(Pow(exp(Derivative(Mul(Integer(-1), Symbol('v_1', commutative=True), Function('\\\\varphi^*')(Symbol('v_1', commutative=True), Symbol('z', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('z', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Pow(exp(Derivative(Mul(Integer(-1), Symbol('G', commutative=True), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('z', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given W{(\\dot{\\mathbf{r}},A_{z})} = - \\dot{\\mathbf{r}} + \\cos{(A_{z})}, then obtain \\dot{\\mathbf{r}} + \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} W^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},A_{z})} = \\dot{\\mathbf{r}} + \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\cos{(A_{z})})^{\\dot{\\mathbf{r}}}", "derivation": "W{(\\dot{\\mathbf{r}},A_{z})} = - \\dot{\\mathbf{r}} + \\cos{(A_{z})} and W^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},A_{z})} = (- \\dot{\\mathbf{r}} + \\cos{(A_{z})})^{\\dot{\\mathbf{r}}} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} W^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},A_{z})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\cos{(A_{z})})^{\\dot{\\mathbf{r}}} and \\dot{\\mathbf{r}} + \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} W^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}},A_{z})} = \\dot{\\mathbf{r}} + \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\cos{(A_{z})})^{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('A_z', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('A_z', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('A_z', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Derivative(Pow(Function('W')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('A_z', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(J)} = \\log{(J)}, then obtain - \\frac{d}{d J} \\dot{\\mathbf{r}}{(J)} + \\iint \\dot{\\mathbf{r}}{(J)} dJ dJ = - \\frac{d}{d J} \\dot{\\mathbf{r}}{(J)} + \\iint \\log{(J)} dJ dJ", "derivation": "\\dot{\\mathbf{r}}{(J)} = \\log{(J)} and \\int \\dot{\\mathbf{r}}{(J)} dJ = \\int \\log{(J)} dJ and \\frac{d}{d J} \\dot{\\mathbf{r}}{(J)} = \\frac{d}{d J} \\log{(J)} and \\iint \\dot{\\mathbf{r}}{(J)} dJ dJ = \\iint \\log{(J)} dJ dJ and - \\frac{d}{d J} \\log{(J)} + \\iint \\dot{\\mathbf{r}}{(J)} dJ dJ = - \\frac{d}{d J} \\log{(J)} + \\iint \\log{(J)} dJ dJ and - \\frac{d}{d J} \\dot{\\mathbf{r}}{(J)} + \\iint \\dot{\\mathbf{r}}{(J)} dJ dJ = - \\frac{d}{d J} \\dot{\\mathbf{r}}{(J)} + \\iint \\log{(J)} dJ dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["minus", 4, "Derivative(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Derivative(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(log(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(A)} = e^{A}, then derive \\frac{d}{d A} \\operatorname{c_{0}}{(A)} = e^{A}, then obtain \\operatorname{c_{0}}^{A}{(A)} = (\\frac{d}{d A} \\operatorname{c_{0}}{(A)})^{A}", "derivation": "\\operatorname{c_{0}}{(A)} = e^{A} and \\operatorname{c_{0}}^{A}{(A)} = (e^{A})^{A} and \\frac{d}{d A} \\operatorname{c_{0}}{(A)} = \\frac{d}{d A} e^{A} and \\frac{d}{d A} \\operatorname{c_{0}}{(A)} = e^{A} and \\operatorname{c_{0}}^{A}{(A)} = (\\frac{d}{d A} \\operatorname{c_{0}}{(A)})^{A}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(exp(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c_0')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), exp(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('c_0')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Derivative(Function('c_0')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(r)} = \\sin{(\\log{(r)})} and \\operatorname{n_{1}}{(r)} = \\log{(r)}, then obtain \\operatorname{F_{g}}{(r)} - \\Psi_{nl}{(\\tilde{g}^*)} = - \\Psi_{nl}{(\\tilde{g}^*)} + \\sin{(\\operatorname{n_{1}}{(r)})}", "derivation": "\\operatorname{F_{g}}{(r)} = \\sin{(\\log{(r)})} and \\operatorname{n_{1}}{(r)} = \\log{(r)} and \\operatorname{F_{g}}{(r)} = \\sin{(\\operatorname{n_{1}}{(r)})} and \\operatorname{F_{g}}{(r)} - \\Psi_{nl}{(\\tilde{g}^*)} = - \\Psi_{nl}{(\\tilde{g}^*)} + \\sin{(\\operatorname{n_{1}}{(r)})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('r', commutative=True)), sin(log(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_g')(Symbol('r', commutative=True)), sin(Function('n_1')(Symbol('r', commutative=True))))"], [["minus", 3, "Function('\\\\Psi_{nl}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('r', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\tilde{g}^*', commutative=True))), sin(Function('n_1')(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then derive 2 \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})} = - \\sin{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})}, then obtain \\log{(2 \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})})} = \\log{(- \\sin{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})})}", "derivation": "\\varphi{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and 2 \\varphi{(a^{\\dagger})} = \\varphi{(a^{\\dagger})} + \\cos{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} 2 \\varphi{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} (\\varphi{(a^{\\dagger})} + \\cos{(a^{\\dagger})}) and 2 \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})} = - \\sin{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})} and \\log{(2 \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})})} = \\log{(- \\sin{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})})} and \\log{(2 \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})})} = \\log{(- \\sin{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 1, "Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["log", 4], "Equality(log(Mul(Integer(2), Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Integer(2), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = C_{1} \\varphi, then obtain C_{1} \\varphi + \\varphi + \\frac{\\partial}{\\partial C_{1}} \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = C_{1} \\varphi + 2 \\varphi", "derivation": "\\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = C_{1} \\varphi and \\frac{\\partial}{\\partial C_{1}} \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = \\frac{\\partial}{\\partial C_{1}} C_{1} \\varphi and \\varphi + \\frac{\\partial}{\\partial C_{1}} \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = \\varphi + \\frac{\\partial}{\\partial C_{1}} C_{1} \\varphi and C_{1} \\varphi + \\varphi + \\frac{\\partial}{\\partial C_{1}} \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = C_{1} \\varphi + \\varphi + \\frac{\\partial}{\\partial C_{1}} C_{1} \\varphi and C_{1} \\varphi + \\varphi + \\frac{\\partial}{\\partial C_{1}} \\operatorname{y^{\\prime}}{(C_{1},\\varphi)} = C_{1} \\varphi + 2 \\varphi", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Symbol('\\\\varphi', commutative=True), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["add", 3, "Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True), Derivative(Function('y^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given m{(\\sigma_p,f_{E})} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p f_{E}, then obtain m{(\\sigma_p,f_{E})} - \\frac{\\partial}{\\partial f_{E}} m{(\\sigma_p,f_{E})} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p f_{E} - \\frac{\\partial}{\\partial f_{E}} m{(\\sigma_p,f_{E})}", "derivation": "m{(\\sigma_p,f_{E})} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p f_{E} and \\frac{\\partial}{\\partial f_{E}} m{(\\sigma_p,f_{E})} = \\frac{\\partial^{2}}{\\partial f_{E}\\partial \\sigma_p} \\sigma_p f_{E} and m{(\\sigma_p,f_{E})} - \\frac{\\partial^{2}}{\\partial f_{E}\\partial \\sigma_p} \\sigma_p f_{E} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p f_{E} - \\frac{\\partial^{2}}{\\partial f_{E}\\partial \\sigma_p} \\sigma_p f_{E} and m{(\\sigma_p,f_{E})} - \\frac{\\partial}{\\partial f_{E}} m{(\\sigma_p,f_{E})} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p f_{E} - \\frac{\\partial}{\\partial f_{E}} m{(\\sigma_p,f_{E})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1)))"], "Equality(Add(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))), Add(Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))), Add(Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('m')(Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))))"]]}, {"prompt": "Given v{(J)} = \\frac{d}{d J} \\sin{(J)}, then derive v{(J)} = \\cos{(J)}, then derive \\frac{d}{d J} v{(J)} = - \\sin{(J)}, then obtain \\frac{\\frac{d}{d J} v{(J)}}{\\frac{d^{2}}{d J^{2}} v{(J)}} = - \\frac{\\sin{(J)}}{\\frac{d^{2}}{d J^{2}} v{(J)}}", "derivation": "v{(J)} = \\frac{d}{d J} \\sin{(J)} and v{(J)} = \\cos{(J)} and \\frac{d}{d J} v{(J)} = \\frac{d}{d J} \\cos{(J)} and \\frac{d^{2}}{d J^{2}} v{(J)} = \\frac{d^{2}}{d J^{2}} \\cos{(J)} and \\frac{d}{d J} v{(J)} = - \\sin{(J)} and \\frac{\\frac{d}{d J} v{(J)}}{\\frac{d^{2}}{d J^{2}} \\cos{(J)}} = - \\frac{\\sin{(J)}}{\\frac{d^{2}}{d J^{2}} \\cos{(J)}} and \\frac{\\frac{d}{d J} v{(J)}}{\\frac{d^{2}}{d J^{2}} v{(J)}} = - \\frac{\\sin{(J)}}{\\frac{d^{2}}{d J^{2}} v{(J)}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J', commutative=True)), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('J', commutative=True))))"], [["divide", 5, "Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(-1))), Mul(Integer(-1), sin(Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(-1))), Mul(Integer(-1), sin(Symbol('J', commutative=True)), Pow(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given M{(y^{\\prime})} = \\log{(\\log{(y^{\\prime})})}, then derive \\int M{(y^{\\prime})} dy^{\\prime} = \\mathbf{F} + y^{\\prime} \\log{(\\log{(y^{\\prime})})} - \\operatorname{li}{(y^{\\prime})}, then obtain \\mathbf{F} + y^{\\prime} \\log{(\\log{(y^{\\prime})})} - \\log{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})} = \\mathbf{F} + y^{\\prime} M{(y^{\\prime})} - \\log{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})}", "derivation": "M{(y^{\\prime})} = \\log{(\\log{(y^{\\prime})})} and \\int M{(y^{\\prime})} dy^{\\prime} = \\int \\log{(\\log{(y^{\\prime})})} dy^{\\prime} and \\int M{(y^{\\prime})} dy^{\\prime} = \\mathbf{F} + y^{\\prime} \\log{(\\log{(y^{\\prime})})} - \\operatorname{li}{(y^{\\prime})} and \\int M{(y^{\\prime})} dy^{\\prime} = \\mathbf{F} + y^{\\prime} M{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})} and - \\log{(y^{\\prime})} + \\int M{(y^{\\prime})} dy^{\\prime} = \\mathbf{F} + y^{\\prime} M{(y^{\\prime})} - \\log{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})} and \\mathbf{F} + y^{\\prime} \\log{(\\log{(y^{\\prime})})} - \\log{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})} = \\mathbf{F} + y^{\\prime} M{(y^{\\prime})} - \\log{(y^{\\prime})} - \\operatorname{li}{(y^{\\prime})}", "srepr_derivation": [["get_premise", "Equality(Function('M')(Symbol('y^{\\\\prime}', commutative=True)), log(log(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(log(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), li(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('M')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), Function('M')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), li(Symbol('y^{\\\\prime}', commutative=True)))))"], [["minus", 4, "log(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('y^{\\\\prime}', commutative=True))), Integral(Function('M')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), Function('M')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), li(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(log(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), li(Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), Function('M')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), li(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(g_{\\varepsilon})} = \\cos{(\\cos{(g_{\\varepsilon})})}, then obtain -1 = - \\frac{\\eta^{\\prime}{(g_{\\varepsilon})}}{- \\eta^{\\prime}{(g_{\\varepsilon})} + 2 \\cos{(\\cos{(g_{\\varepsilon})})}}", "derivation": "\\eta^{\\prime}{(g_{\\varepsilon})} = \\cos{(\\cos{(g_{\\varepsilon})})} and \\cos{(\\cos{(g_{\\varepsilon})})} = - \\eta^{\\prime}{(g_{\\varepsilon})} + 2 \\cos{(\\cos{(g_{\\varepsilon})})} and \\eta^{\\prime}{(g_{\\varepsilon})} = - \\eta^{\\prime}{(g_{\\varepsilon})} + 2 \\cos{(\\cos{(g_{\\varepsilon})})} and -1 = - \\frac{- \\eta^{\\prime}{(g_{\\varepsilon})} + 2 \\cos{(\\cos{(g_{\\varepsilon})})}}{\\eta^{\\prime}{(g_{\\varepsilon})}} and -1 = - \\frac{\\eta^{\\prime}{(g_{\\varepsilon})}}{- \\eta^{\\prime}{(g_{\\varepsilon})} + 2 \\cos{(\\cos{(g_{\\varepsilon})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], "Equality(cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["divide", 3, "Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(-1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(U,A_{x})} = A_{x} U, then derive \\frac{\\partial}{\\partial U} \\operatorname{c_{0}}{(U,A_{x})} = A_{x}, then obtain 0 = A_{x} - \\frac{\\partial}{\\partial U} U \\frac{\\partial}{\\partial U} A_{x} U", "derivation": "\\operatorname{c_{0}}{(U,A_{x})} = A_{x} U and \\frac{\\partial}{\\partial U} \\operatorname{c_{0}}{(U,A_{x})} = \\frac{\\partial}{\\partial U} A_{x} U and U \\frac{\\partial}{\\partial U} \\operatorname{c_{0}}{(U,A_{x})} = U \\frac{\\partial}{\\partial U} A_{x} U and \\frac{\\partial}{\\partial U} \\operatorname{c_{0}}{(U,A_{x})} = A_{x} and A_{x} + \\frac{\\partial}{\\partial U} \\operatorname{c_{0}}{(U,A_{x})} = 2 A_{x} and A_{x} + \\frac{\\partial}{\\partial U} A_{x} U = 2 A_{x} and 0 = A_{x} - \\frac{\\partial}{\\partial U} A_{x} U and A_{x} U = U \\frac{\\partial}{\\partial U} A_{x} U and 0 = A_{x} - \\frac{\\partial}{\\partial U} U \\frac{\\partial}{\\partial U} A_{x} U", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('U', commutative=True), Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('U', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["times", 2, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Derivative(Function('c_0')(Symbol('U', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c_0')(Symbol('U', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('A_x', commutative=True))"], [["add", 4, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Derivative(Function('c_0')(Symbol('U', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('A_x', commutative=True)))"], [["minus", 6, "Add(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Integer(0), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True), Integer(1))))))"]]}, {"prompt": "Given T{(\\mathbf{s})} = \\int \\cos{(\\mathbf{s})} d\\mathbf{s}, then derive T{(\\mathbf{s})} = y + \\sin{(\\mathbf{s})}, then derive \\Omega + \\sin{(\\mathbf{s})} = y + \\sin{(\\mathbf{s})}, then obtain \\int \\cos{(\\mathbf{s})} d\\mathbf{s} = \\Omega + \\sin{(\\mathbf{s})}", "derivation": "T{(\\mathbf{s})} = \\int \\cos{(\\mathbf{s})} d\\mathbf{s} and T{(\\mathbf{s})} = y + \\sin{(\\mathbf{s})} and \\int \\cos{(\\mathbf{s})} d\\mathbf{s} = y + \\sin{(\\mathbf{s})} and \\Omega + \\sin{(\\mathbf{s})} = y + \\sin{(\\mathbf{s})} and \\int \\cos{(\\mathbf{s})} d\\mathbf{s} = \\Omega + \\sin{(\\mathbf{s})}", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('\\\\mathbf{s}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('T')(Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('y', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('y', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('y', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}, then obtain \\int e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{H}{(\\hat{\\mathbf{r}})} + \\log{(n)})} d\\hat{\\mathbf{r}} = \\int e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(n)})} d\\hat{\\mathbf{r}}", "derivation": "\\hat{H}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} and \\hat{H}{(\\hat{\\mathbf{r}})} + \\log{(n)} = \\hat{\\mathbf{r}} + \\log{(n)} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{H}{(\\hat{\\mathbf{r}})} + \\log{(n)}) = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(n)}) and e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{H}{(\\hat{\\mathbf{r}})} + \\log{(n)})} = e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(n)})} and \\int e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{H}{(\\hat{\\mathbf{r}})} + \\log{(n)})} d\\hat{\\mathbf{r}} = \\int e^{\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(n)})} d\\hat{\\mathbf{r}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], [["add", 1, "log(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('n', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Add(Function('\\\\hat{H}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), exp(Derivative(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(exp(Derivative(Add(Function('\\\\hat{H}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(exp(Derivative(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{2})} = \\sin{(A_{2})} and \\operatorname{A_{y}}{(A_{2})} = (\\operatorname{P_{g}}{(A_{2})} + \\sin{(A_{2})}) \\operatorname{P_{g}}{(A_{2})}, then obtain \\frac{d}{d A_{2}} \\operatorname{A_{y}}{(A_{2})} = \\frac{d}{d A_{2}} 2 \\operatorname{P_{g}}^{2}{(A_{2})}", "derivation": "\\operatorname{P_{g}}{(A_{2})} = \\sin{(A_{2})} and \\operatorname{P_{g}}{(A_{2})} + \\sin{(A_{2})} = 2 \\sin{(A_{2})} and \\operatorname{A_{y}}{(A_{2})} = (\\operatorname{P_{g}}{(A_{2})} + \\sin{(A_{2})}) \\operatorname{P_{g}}{(A_{2})} and \\operatorname{A_{y}}{(A_{2})} = 2 \\operatorname{P_{g}}{(A_{2})} \\sin{(A_{2})} and \\operatorname{A_{y}}{(A_{2})} = 2 \\operatorname{P_{g}}^{2}{(A_{2})} and \\frac{d}{d A_{2}} \\operatorname{A_{y}}{(A_{2})} = \\frac{d}{d A_{2}} 2 \\operatorname{P_{g}}^{2}{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["add", 1, "sin(Symbol('A_2', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Mul(Integer(2), sin(Symbol('A_2', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('A_2', commutative=True)), Mul(Add(Function('P_g')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Function('P_g')(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('A_y')(Symbol('A_2', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('A_y')(Symbol('A_2', commutative=True)), Mul(Integer(2), Pow(Function('P_g')(Symbol('A_2', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('P_g')(Symbol('A_2', commutative=True)), Integer(2))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(U,i,\\varphi)} = i (- U + \\varphi), then derive - i + \\frac{\\partial}{\\partial U} C{(U,i,\\varphi)} = - 2 i, then obtain \\frac{\\partial}{\\partial U} (- i + \\frac{\\partial}{\\partial U} i (- U + \\varphi)) = \\frac{d}{d U} - 2 i", "derivation": "C{(U,i,\\varphi)} = i (- U + \\varphi) and i (- U + \\varphi) + C{(U,i,\\varphi)} = 2 i (- U + \\varphi) and \\frac{\\partial}{\\partial U} (i (- U + \\varphi) + C{(U,i,\\varphi)}) = \\frac{\\partial}{\\partial U} 2 i (- U + \\varphi) and - i + \\frac{\\partial}{\\partial U} C{(U,i,\\varphi)} = - 2 i and - i + \\frac{\\partial}{\\partial U} i (- U + \\varphi) = - 2 i and - i + \\frac{\\partial}{\\partial U} i (- U + \\varphi) = - i + \\frac{\\partial}{\\partial U} C{(U,i,\\varphi)} and \\frac{\\partial}{\\partial U} (- i + \\frac{\\partial}{\\partial U} i (- U + \\varphi)) = \\frac{\\partial}{\\partial U} (- i + \\frac{\\partial}{\\partial U} C{(U,i,\\varphi)}) and \\frac{\\partial}{\\partial U} (- i + \\frac{\\partial}{\\partial U} i (- U + \\varphi)) = \\frac{d}{d U} - 2 i", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('C')(Symbol('U', commutative=True), Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), Symbol('i', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v_{z})} = \\sin{(v_{z})}, then obtain \\operatorname{f_{E}}^{- v_{z}}{(v_{z})} \\sin^{3 v_{z}}{(v_{z})} = \\sin^{2 v_{z}}{(v_{z})}", "derivation": "\\operatorname{f_{E}}{(v_{z})} = \\sin{(v_{z})} and \\operatorname{f_{E}}^{v_{z}}{(v_{z})} = \\sin^{v_{z}}{(v_{z})} and \\operatorname{f_{E}}^{v_{z}}{(v_{z})} \\sin^{v_{z}}{(v_{z})} = \\sin^{2 v_{z}}{(v_{z})} and \\operatorname{f_{E}}^{2 v_{z}}{(v_{z})} \\sin^{2 v_{z}}{(v_{z})} = \\sin^{4 v_{z}}{(v_{z})} and \\operatorname{f_{E}}^{v_{z}}{(v_{z})} \\sin^{v_{z}}{(v_{z})} = \\operatorname{f_{E}}^{- v_{z}}{(v_{z})} \\sin^{3 v_{z}}{(v_{z})} and \\operatorname{f_{E}}^{- v_{z}}{(v_{z})} \\sin^{3 v_{z}}{(v_{z})} = \\sin^{2 v_{z}}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(sin(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["times", 2, "Pow(sin(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(sin(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(2), Symbol('v_z', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Mul(Integer(2), Symbol('v_z', commutative=True))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(2), Symbol('v_z', commutative=True)))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(4), Symbol('v_z', commutative=True))))"], [["divide", 4, "Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(sin(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], "Equality(Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(sin(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(3), Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Function('f_E')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(3), Symbol('v_z', commutative=True)))), Pow(sin(Symbol('v_z', commutative=True)), Mul(Integer(2), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\pi,\\rho)} = \\frac{\\rho}{\\pi}, then derive \\hat{H}_{\\lambda} + 2 \\mathbf{D}{(\\pi,\\rho)} = \\mathbf{A} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi}, then obtain \\hat{H}_{\\lambda} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi} = \\mathbf{A} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi}", "derivation": "\\mathbf{D}{(\\pi,\\rho)} = \\frac{\\rho}{\\pi} and 2 \\mathbf{D}{(\\pi,\\rho)} = \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi} and \\frac{\\partial}{\\partial \\rho} 2 \\mathbf{D}{(\\pi,\\rho)} = \\frac{\\partial}{\\partial \\rho} (\\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi}) and \\int \\frac{\\partial}{\\partial \\rho} 2 \\mathbf{D}{(\\pi,\\rho)} d\\rho = \\int \\frac{\\partial}{\\partial \\rho} (\\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi}) d\\rho and \\hat{H}_{\\lambda} + 2 \\mathbf{D}{(\\pi,\\rho)} = \\mathbf{A} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi} and \\hat{H}_{\\lambda} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi} = \\mathbf{A} + \\mathbf{D}{(\\pi,\\rho)} + \\frac{\\rho}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given A{(S,v_{y})} = S + \\sin{(v_{y})}, then derive \\int A{(S,v_{y})} dS = \\frac{S^{2}}{2} + S \\sin{(v_{y})} + h, then obtain - \\cos{(L)} + \\int (S + \\sin{(v_{y})}) dS = \\frac{S^{2}}{2} + S \\sin{(v_{y})} + h - \\cos{(L)}", "derivation": "A{(S,v_{y})} = S + \\sin{(v_{y})} and \\int A{(S,v_{y})} dS = \\int (S + \\sin{(v_{y})}) dS and \\int A{(S,v_{y})} dS = \\frac{S^{2}}{2} + S \\sin{(v_{y})} + h and - \\cos{(L)} + \\int A{(S,v_{y})} dS = \\frac{S^{2}}{2} + S \\sin{(v_{y})} + h - \\cos{(L)} and - \\cos{(L)} + \\int (S + \\sin{(v_{y})}) dS = \\frac{S^{2}}{2} + S \\sin{(v_{y})} + h - \\cos{(L)}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('S', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('A')(Symbol('S', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A')(Symbol('S', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('h', commutative=True)))"], [["minus", 3, "cos(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('L', commutative=True))), Integral(Function('A')(Symbol('S', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('L', commutative=True))), Integral(Add(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), sin(Symbol('v_y', commutative=True))), Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{P},m,\\varphi^*)} = \\frac{\\varphi^* m}{\\mathbf{P}} and A{(\\mathbf{P},m,\\varphi^*)} = \\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}}, then obtain \\frac{\\partial}{\\partial \\varphi^*} \\log{(A{(\\mathbf{P},m,\\varphi^*)})} = \\frac{\\partial}{\\partial \\varphi^*} \\log{(\\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}})}", "derivation": "\\mathbf{v}{(\\mathbf{P},m,\\varphi^*)} = \\frac{\\varphi^* m}{\\mathbf{P}} and \\mathbf{P} + \\mathbf{v}{(\\mathbf{P},m,\\varphi^*)} = \\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}} and \\log{(\\mathbf{P} + \\mathbf{v}{(\\mathbf{P},m,\\varphi^*)})} = \\log{(\\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}})} and A{(\\mathbf{P},m,\\varphi^*)} = \\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}} and \\log{(\\mathbf{P} + \\mathbf{v}{(\\mathbf{P},m,\\varphi^*)})} = \\log{(A{(\\mathbf{P},m,\\varphi^*)})} and \\log{(A{(\\mathbf{P},m,\\varphi^*)})} = \\log{(\\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}})} and \\frac{\\partial}{\\partial \\varphi^*} \\log{(A{(\\mathbf{P},m,\\varphi^*)})} = \\frac{\\partial}{\\partial \\varphi^*} \\log{(\\mathbf{P} + \\frac{\\varphi^* m}{\\mathbf{P}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True))))"], [["log", 2], "Equality(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), log(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), log(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True))), log(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(log(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})}, then obtain \\eta^{\\prime} \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} + \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\eta^{\\prime} \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} + \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})}", "derivation": "\\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} and \\eta^{\\prime} \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\eta^{\\prime} \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} and \\eta^{\\prime} \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} + \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\eta^{\\prime} \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} + \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} and \\eta^{\\prime} \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} + \\operatorname{v_{2}}{(\\hat{\\mathbf{x}},\\eta^{\\prime})} = \\eta^{\\prime} \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})} + \\cos{(\\eta^{\\prime} + \\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["add", 1, "Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Function('v_2')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{f})} = e^{\\mathbf{f}}, then obtain -1 + \\frac{e^{2 \\mathbf{f}}}{\\operatorname{v_{1}}{(\\mathbf{f})}} = e^{\\mathbf{f}} - 1", "derivation": "\\operatorname{v_{1}}{(\\mathbf{f})} = e^{\\mathbf{f}} and e^{\\mathbf{f}} = \\frac{e^{2 \\mathbf{f}}}{\\operatorname{v_{1}}{(\\mathbf{f})}} and \\operatorname{v_{1}}{(\\mathbf{f})} - 1 = e^{\\mathbf{f}} - 1 and \\operatorname{v_{1}}{(\\mathbf{f})} = \\frac{e^{2 \\mathbf{f}}}{\\operatorname{v_{1}}{(\\mathbf{f})}} and -1 + \\frac{e^{2 \\mathbf{f}}}{\\operatorname{v_{1}}{(\\mathbf{f})}} = e^{\\mathbf{f}} - 1", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 1, "Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(exp(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Integer(-1), Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True))))), Add(exp(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(n,F_{x})} = \\frac{\\partial}{\\partial F_{x}} n^{F_{x}}, then derive \\operatorname{f_{E}}{(n,F_{x})} = n^{F_{x}} \\log{(n)}, then obtain \\frac{n^{2 F_{x}} e^{2 \\operatorname{f_{E}}{(n,F_{x})}}}{n} = \\frac{n^{2 F_{x}} e^{n^{F_{x}} \\log{(n)}} e^{\\operatorname{f_{E}}{(n,F_{x})}}}{n}", "derivation": "\\operatorname{f_{E}}{(n,F_{x})} = \\frac{\\partial}{\\partial F_{x}} n^{F_{x}} and \\operatorname{f_{E}}{(n,F_{x})} = n^{F_{x}} \\log{(n)} and e^{\\operatorname{f_{E}}{(n,F_{x})}} = e^{n^{F_{x}} \\log{(n)}} and n^{F_{x}} e^{\\operatorname{f_{E}}{(n,F_{x})}} = n^{F_{x}} e^{n^{F_{x}} \\log{(n)}} and \\frac{n^{F_{x}} e^{\\operatorname{f_{E}}{(n,F_{x})}}}{n} = \\frac{n^{F_{x}} e^{n^{F_{x}} \\log{(n)}}}{n} and \\frac{n^{2 F_{x}} e^{2 \\operatorname{f_{E}}{(n,F_{x})}}}{n} = \\frac{n^{2 F_{x}} e^{n^{F_{x}} \\log{(n)}} e^{\\operatorname{f_{E}}{(n,F_{x})}}}{n}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('n', commutative=True))))"], [["exp", 2], "Equality(exp(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True))), exp(Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('n', commutative=True)))))"], [["times", 3, "Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), exp(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), exp(Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('n', commutative=True))))))"], [["divide", 4, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), exp(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), exp(Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('n', commutative=True))))))"], [["times", 5, "Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), exp(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True))))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Mul(Integer(2), Symbol('F_x', commutative=True))), exp(Mul(Integer(2), Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True))))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Mul(Integer(2), Symbol('F_x', commutative=True))), exp(Mul(Pow(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('n', commutative=True)))), exp(Function('f_E')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(h,\\hbar)} = \\cos{(\\frac{\\hbar}{h})} and \\mathbf{v}{(h,\\hbar)} = \\cos{(\\frac{\\hbar}{h})}, then obtain (h^{2})^{\\hbar} = (\\frac{h^{2} \\cos{(\\frac{\\hbar}{h})}}{\\mathbf{v}{(h,\\hbar)}})^{\\hbar}", "derivation": "\\hat{x}_0{(h,\\hbar)} = \\cos{(\\frac{\\hbar}{h})} and h \\hat{x}_0{(h,\\hbar)} = h \\cos{(\\frac{\\hbar}{h})} and h^{2} \\hat{x}_0^{2}{(h,\\hbar)} = h^{2} \\hat{x}_0{(h,\\hbar)} \\cos{(\\frac{\\hbar}{h})} and \\mathbf{v}{(h,\\hbar)} = \\cos{(\\frac{\\hbar}{h})} and \\hat{x}_0{(h,\\hbar)} = \\mathbf{v}{(h,\\hbar)} and h^{2} = \\frac{h^{2} \\cos{(\\frac{\\hbar}{h})}}{\\hat{x}_0{(h,\\hbar)}} and h^{2} = \\frac{h^{2} \\cos{(\\frac{\\hbar}{h})}}{\\mathbf{v}{(h,\\hbar)}} and (h^{2})^{\\hbar} = (\\frac{h^{2} \\cos{(\\frac{\\hbar}{h})}}{\\mathbf{v}{(h,\\hbar)}})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('h', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))))"], [["times", 2, "Mul(Symbol('h', commutative=True), Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 3, "Pow(Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))"], "Equality(Pow(Symbol('h', commutative=True), Integer(2)), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Function('\\\\hat{x}_0')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Symbol('h', commutative=True), Integer(2)), Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))))"], [["power", 7, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Symbol('h', commutative=True), Integer(2)), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Pow(Symbol('h', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(\\mu)} = \\log{(\\mu)}, then obtain \\mu \\rho_{f}^{2}{(\\mu)} \\log{(\\mu)}^{2} = \\mu \\rho_{f}{(\\mu)} \\log{(\\mu)}^{3}", "derivation": "\\rho_{f}{(\\mu)} = \\log{(\\mu)} and \\mu \\rho_{f}{(\\mu)} = \\mu \\log{(\\mu)} and \\mu \\rho_{f}{(\\mu)} \\log{(\\mu)} = \\mu \\log{(\\mu)}^{2} and \\mu \\rho_{f}^{2}{(\\mu)} = \\mu \\rho_{f}{(\\mu)} \\log{(\\mu)} and \\mu \\rho_{f}^{3}{(\\mu)} \\log{(\\mu)} = \\mu \\rho_{f}^{2}{(\\mu)} \\log{(\\mu)}^{2} and \\mu \\rho_{f}^{2}{(\\mu)} \\log{(\\mu)}^{2} = \\mu \\rho_{f}{(\\mu)} \\log{(\\mu)}^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), log(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mu', commutative=True), log(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["times", 3, "Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Integer(2))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Integer(3)), log(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(3))))"]]}, {"prompt": "Given k{(T,\\dot{x})} = e^{T \\dot{x}}, then obtain (T k{(T,\\dot{x})} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T e^{T \\dot{x}} dT = (T e^{T \\dot{x}} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T e^{T \\dot{x}} dT", "derivation": "k{(T,\\dot{x})} = e^{T \\dot{x}} and T k{(T,\\dot{x})} = T e^{T \\dot{x}} and \\int T k{(T,\\dot{x})} dT = \\int T e^{T \\dot{x}} dT and T k{(T,\\dot{x})} - \\int T e^{T \\dot{x}} dT = T e^{T \\dot{x}} - \\int T e^{T \\dot{x}} dT and (T k{(T,\\dot{x})} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T k{(T,\\dot{x})} dT = (T e^{T \\dot{x}} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T k{(T,\\dot{x})} dT and (T k{(T,\\dot{x})} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T e^{T \\dot{x}} dT = (T e^{T \\dot{x}} - \\int T e^{T \\dot{x}} dT) \\frac{\\partial}{\\partial T} \\int T e^{T \\dot{x}} dT", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["times", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Add(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))))"], [["times", 4, "Derivative(Integral(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Derivative(Integral(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Derivative(Integral(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Mul(Symbol('T', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Derivative(Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))))), Derivative(Integral(Mul(Symbol('T', commutative=True), exp(Mul(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(G)} = \\log{(e^{G})} and \\hat{\\mathbf{r}}{(G)} = \\log{(e^{G})}, then derive \\frac{d}{d G} \\mathbf{M}{(G)} = 1, then obtain e^{- G} \\frac{d}{d G} \\hat{\\mathbf{r}}{(G)} = e^{- G}", "derivation": "\\mathbf{M}{(G)} = \\log{(e^{G})} and \\frac{d}{d G} \\mathbf{M}{(G)} = \\frac{d}{d G} \\log{(e^{G})} and \\frac{d}{d G} \\mathbf{M}{(G)} = 1 and e^{- G} \\frac{d}{d G} \\mathbf{M}{(G)} = e^{- G} and \\hat{\\mathbf{r}}{(G)} = \\log{(e^{G})} and \\hat{\\mathbf{r}}{(G)} = \\mathbf{M}{(G)} and e^{- G} \\frac{d}{d G} \\hat{\\mathbf{r}}{(G)} = e^{- G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(log(exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "exp(Symbol('G', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('G', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True)), log(exp(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True)), Function('\\\\mathbf{M}')(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(exp(Mul(Integer(-1), Symbol('G', commutative=True))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\chi{(g^{\\prime}_{\\varepsilon},Z)} = Z + g^{\\prime}_{\\varepsilon}, then obtain \\int 1 dg^{\\prime}_{\\varepsilon} = \\int (Z + g^{\\prime}_{\\varepsilon} - \\chi{(g^{\\prime}_{\\varepsilon},Z)} + 1) dg^{\\prime}_{\\varepsilon}", "derivation": "\\chi{(g^{\\prime}_{\\varepsilon},Z)} = Z + g^{\\prime}_{\\varepsilon} and 0 = Z + g^{\\prime}_{\\varepsilon} - \\chi{(g^{\\prime}_{\\varepsilon},Z)} and 1 = Z + g^{\\prime}_{\\varepsilon} - \\chi{(g^{\\prime}_{\\varepsilon},Z)} + 1 and \\int 1 dg^{\\prime}_{\\varepsilon} = \\int (Z + g^{\\prime}_{\\varepsilon} - \\chi{(g^{\\prime}_{\\varepsilon},Z)} + 1) dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Integer(0), Add(Symbol('Z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(1), Add(Symbol('Z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Integer(1)))"], [["integrate", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Integer(1)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given b{(\\sigma_p)} = e^{e^{\\sigma_p}} and \\theta{(\\sigma_p)} = e^{e^{\\sigma_p}}, then obtain \\frac{\\theta{(\\sigma_p)}}{\\sigma_p} = \\frac{e^{e^{\\sigma_p}}}{\\sigma_p}", "derivation": "b{(\\sigma_p)} = e^{e^{\\sigma_p}} and \\frac{b{(\\sigma_p)}}{\\sigma_p} = \\frac{e^{e^{\\sigma_p}}}{\\sigma_p} and \\theta{(\\sigma_p)} = e^{e^{\\sigma_p}} and \\theta{(\\sigma_p)} = b{(\\sigma_p)} and \\frac{\\theta{(\\sigma_p)}}{\\sigma_p} = \\frac{e^{e^{\\sigma_p}}}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\sigma_p', commutative=True)), exp(exp(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('b')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\sigma_p', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), exp(exp(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Function('b')(Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(H)} = \\sin{(\\cos{(H)})}, then obtain \\mathbf{B}{(H)} - \\int \\sin{(\\cos{(H)})} dH = \\sin{(\\cos{(H)})} - \\int \\sin{(\\cos{(H)})} dH", "derivation": "\\mathbf{B}{(H)} = \\sin{(\\cos{(H)})} and \\int \\mathbf{B}{(H)} dH = \\int \\sin{(\\cos{(H)})} dH and \\mathbf{B}{(H)} - \\int \\mathbf{B}{(H)} dH = \\sin{(\\cos{(H)})} - \\int \\mathbf{B}{(H)} dH and \\mathbf{B}{(H)} - \\int \\sin{(\\cos{(H)})} dH = \\sin{(\\cos{(H)})} - \\int \\sin{(\\cos{(H)})} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), sin(cos(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["minus", 1, "Integral(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(sin(cos(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('H', commutative=True)), Mul(Integer(-1), Integral(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))), Add(sin(cos(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(S,t)} = S - t, then derive - \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(S,t)} = 1, then obtain t - \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(S,t)} = t + 1", "derivation": "\\operatorname{F_{g}}{(S,t)} = S - t and - \\operatorname{F_{g}}{(S,t)} = - S + t and \\frac{\\partial}{\\partial t} - \\operatorname{F_{g}}{(S,t)} = \\frac{\\partial}{\\partial t} (- S + t) and - \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(S,t)} = 1 and t - \\frac{\\partial}{\\partial t} \\operatorname{F_{g}}{(S,t)} = t + 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('S', commutative=True), Symbol('t', commutative=True)), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_g')(Symbol('S', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('F_g')(Symbol('S', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('F_g')(Symbol('S', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(1))"], [["add", 4, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(-1), Derivative(Function('F_g')(Symbol('S', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))), Add(Symbol('t', commutative=True), Integer(1)))"]]}, {"prompt": "Given c{(C_{2},q)} = C_{2} q, then derive 0 = q - \\frac{\\partial}{\\partial C_{2}} c{(C_{2},q)}, then obtain 1 = \\cos{(q - \\frac{\\partial}{\\partial C_{2}} C_{2} q)}", "derivation": "c{(C_{2},q)} = C_{2} q and 0 = C_{2} q - c{(C_{2},q)} and \\frac{d}{d C_{2}} 0 = \\frac{\\partial}{\\partial C_{2}} (C_{2} q - c{(C_{2},q)}) and 0 = q - \\frac{\\partial}{\\partial C_{2}} c{(C_{2},q)} and 0 = q - \\frac{\\partial}{\\partial C_{2}} C_{2} q and 1 = \\cos{(q - \\frac{\\partial}{\\partial C_{2}} C_{2} q)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Function('c')(Symbol('C_2', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('C_2', commutative=True), Symbol('q', commutative=True)))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('C_2', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Symbol('q', commutative=True), Mul(Integer(-1), Derivative(Function('c')(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Symbol('q', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))))"], [["cos", 5], "Equality(Integer(1), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\rho_{b}{(E,\\mu_0)} = E \\mu_0 and \\eta^{\\prime}{(\\mu_0)} = \\mu_0, then obtain (\\frac{(- E \\mu_0 + \\rho_{b}{(E,\\mu_0)})^{E}}{\\mu_0})^{\\mu_0} = (\\frac{0^{E}}{\\mu_0})^{\\mu_0}", "derivation": "\\rho_{b}{(E,\\mu_0)} = E \\mu_0 and \\eta^{\\prime}{(\\mu_0)} = \\mu_0 and - E \\mu_0 + \\rho_{b}{(E,\\mu_0)} = 0 and (- E \\mu_0 + \\rho_{b}{(E,\\mu_0)})^{E} = 0^{E} and \\frac{(- E \\mu_0 + \\rho_{b}{(E,\\mu_0)})^{E}}{\\eta^{\\prime}{(\\mu_0)}} = \\frac{0^{E}}{\\eta^{\\prime}{(\\mu_0)}} and (\\frac{(- E \\mu_0 + \\rho_{b}{(E,\\mu_0)})^{E}}{\\eta^{\\prime}{(\\mu_0)}})^{\\mu_0} = (\\frac{0^{E}}{\\eta^{\\prime}{(\\mu_0)}})^{\\mu_0} and (\\frac{(- E \\mu_0 + \\rho_{b}{(E,\\mu_0)})^{E}}{\\mu_0})^{\\mu_0} = (\\frac{0^{E}}{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["minus", 1, "Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(0))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('E', commutative=True)), Pow(Integer(0), Symbol('E', commutative=True)))"], [["divide", 4, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('E', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Mul(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('E', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('E', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} = \\cos{(\\hbar n)}, then obtain (\\operatorname{V_{\\mathbf{B}}}^{\\hbar}{(n,\\hbar)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn)^{\\hbar} = (\\cos^{\\hbar}{(\\hbar n)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn)^{\\hbar}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} = \\cos{(\\hbar n)} and \\operatorname{V_{\\mathbf{B}}}^{\\hbar}{(n,\\hbar)} = \\cos^{\\hbar}{(\\hbar n)} and \\operatorname{V_{\\mathbf{B}}}^{\\hbar}{(n,\\hbar)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn = \\cos^{\\hbar}{(\\hbar n)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn and (\\operatorname{V_{\\mathbf{B}}}^{\\hbar}{(n,\\hbar)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn)^{\\hbar} = (\\cos^{\\hbar}{(\\hbar n)} - \\int \\operatorname{V_{\\mathbf{B}}}{(n,\\hbar)} dn)^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(cos(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["minus", 2, "Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n', commutative=True))))), Add(Pow(cos(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n', commutative=True))))))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n', commutative=True))))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Pow(cos(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n', commutative=True))))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(t_{1},\\mathbf{F})} = \\mathbf{F} t_{1}, then obtain \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{P}{(t_{1},\\mathbf{F})} \\mathbf{P}^{t_{1}}{(t_{1},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} t_{1})^{t_{1}} \\mathbf{P}{(t_{1},\\mathbf{F})}", "derivation": "\\mathbf{P}{(t_{1},\\mathbf{F})} = \\mathbf{F} t_{1} and \\mathbf{P}^{t_{1}}{(t_{1},\\mathbf{F})} = (\\mathbf{F} t_{1})^{t_{1}} and \\mathbf{P}{(t_{1},\\mathbf{F})} \\mathbf{P}^{t_{1}}{(t_{1},\\mathbf{F})} = (\\mathbf{F} t_{1})^{t_{1}} \\mathbf{P}{(t_{1},\\mathbf{F})} and \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{P}{(t_{1},\\mathbf{F})} \\mathbf{P}^{t_{1}}{(t_{1},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} t_{1})^{t_{1}} \\mathbf{P}{(t_{1},\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('t_1', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('t_1', commutative=True))), Mul(Pow(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Function('\\\\mathbf{P}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(J,v_{z})} = J v_{z}, then obtain v_{z} + \\frac{(J i{(J,v_{z})})^{J} i{(J,v_{z})}}{v_{z}} = v_{z} + \\frac{(J^{2} v_{z})^{J} i{(J,v_{z})}}{v_{z}}", "derivation": "i{(J,v_{z})} = J v_{z} and J i{(J,v_{z})} = J^{2} v_{z} and (J i{(J,v_{z})})^{J} = (J^{2} v_{z})^{J} and J (J i{(J,v_{z})})^{J} i{(J,v_{z})} = J (J^{2} v_{z})^{J} i{(J,v_{z})} and \\frac{(J i{(J,v_{z})})^{J} i{(J,v_{z})}}{v_{z}} = \\frac{(J^{2} v_{z})^{J} i{(J,v_{z})}}{v_{z}} and v_{z} + \\frac{(J i{(J,v_{z})})^{J} i{(J,v_{z})}}{v_{z}} = v_{z} + \\frac{(J^{2} v_{z})^{J} i{(J,v_{z})}}{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v_z', commutative=True)))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Symbol('J', commutative=True)), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v_z', commutative=True)), Symbol('J', commutative=True)))"], [["times", 3, "Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(Symbol('J', commutative=True), Pow(Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Mul(Symbol('J', commutative=True), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v_z', commutative=True)), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 4, "Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v_z', commutative=True)), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 5, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Mul(Symbol('J', commutative=True), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))), Add(Symbol('v_z', commutative=True), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v_z', commutative=True)), Symbol('J', commutative=True)), Function('i')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given p{(\\hbar)} = \\cos{(\\sin{(\\hbar)})}, then obtain 1 = \\frac{\\frac{d}{d \\hbar} \\cos^{\\hbar}{(\\sin{(\\hbar)})}}{\\frac{d}{d \\hbar} p^{\\hbar}{(\\hbar)}}", "derivation": "p{(\\hbar)} = \\cos{(\\sin{(\\hbar)})} and p^{\\hbar}{(\\hbar)} = \\cos^{\\hbar}{(\\sin{(\\hbar)})} and \\frac{d}{d \\hbar} p^{\\hbar}{(\\hbar)} = \\frac{d}{d \\hbar} \\cos^{\\hbar}{(\\sin{(\\hbar)})} and 1 = \\frac{\\frac{d}{d \\hbar} \\cos^{\\hbar}{(\\sin{(\\hbar)})}}{\\frac{d}{d \\hbar} p^{\\hbar}{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\hbar', commutative=True)), cos(sin(Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('p')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(cos(sin(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Pow(Function('p')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Pow(cos(sin(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Pow(Function('p')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Pow(Function('p')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Derivative(Pow(cos(sin(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(S)} = \\log{(\\log{(S)})}, then derive \\int \\dot{y}{(S)} dS = S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)}, then derive \\frac{\\partial}{\\partial f} (F_{x} + S \\log{(\\log{(S)})} - \\operatorname{li}{(S)}) = \\frac{\\partial}{\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)}), then obtain \\frac{\\partial^{2}}{\\partial S\\partial f} (F_{x} + S \\log{(\\log{(S)})} - \\operatorname{li}{(S)}) = \\frac{\\partial^{2}}{\\partial S\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)})", "derivation": "\\dot{y}{(S)} = \\log{(\\log{(S)})} and \\int \\dot{y}{(S)} dS = \\int \\log{(\\log{(S)})} dS and \\int \\dot{y}{(S)} dS = S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)} and \\frac{d}{d f} \\int \\dot{y}{(S)} dS = \\frac{\\partial}{\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)}) and \\frac{d}{d f} \\int \\log{(\\log{(S)})} dS = \\frac{\\partial}{\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)}) and \\frac{\\partial}{\\partial f} (F_{x} + S \\log{(\\log{(S)})} - \\operatorname{li}{(S)}) = \\frac{\\partial}{\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)}) and \\frac{\\partial^{2}}{\\partial S\\partial f} (F_{x} + S \\log{(\\log{(S)})} - \\operatorname{li}{(S)}) = \\frac{\\partial^{2}}{\\partial S\\partial f} (S \\log{(\\log{(S)})} + f - \\operatorname{li}{(S)})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('S', commutative=True)), log(log(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('S', commutative=True)))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\dot{y}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(log(log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), log(log(Symbol('S', commutative=True)))), Symbol('f', commutative=True), Mul(Integer(-1), li(Symbol('S', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(n_{2},\\dot{z})} = - \\dot{z} + n_{2}, then obtain \\iint 2 \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} dn_{2} dn_{2} = \\iint ((- \\dot{z} + n_{2})^{n_{2}} + \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})}) dn_{2} dn_{2}", "derivation": "\\sigma_{p}{(n_{2},\\dot{z})} = - \\dot{z} + n_{2} and \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} = (- \\dot{z} + n_{2})^{n_{2}} and 2 \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} = (- \\dot{z} + n_{2})^{n_{2}} + \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} and \\int 2 \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} dn_{2} = \\int ((- \\dot{z} + n_{2})^{n_{2}} + \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})}) dn_{2} and \\iint 2 \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})} dn_{2} dn_{2} = \\iint ((- \\dot{z} + n_{2})^{n_{2}} + \\sigma_{p}^{n_{2}}{(n_{2},\\dot{z})}) dn_{2} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["add", 2, "Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["integrate", 4, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\chi{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then derive \\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon} = \\hat{p} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}, then obtain (\\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon})^{g_{\\varepsilon}} = (\\hat{p} + g_{\\varepsilon} \\chi{(g_{\\varepsilon})} - g_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "\\chi{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon} and \\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon} = \\hat{p} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon} and \\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon} = \\hat{p} + g_{\\varepsilon} \\chi{(g_{\\varepsilon})} - g_{\\varepsilon} and (\\int \\chi{(g_{\\varepsilon})} dg_{\\varepsilon})^{g_{\\varepsilon}} = (\\hat{p} + g_{\\varepsilon} \\chi{(g_{\\varepsilon})} - g_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\chi')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\pi{(k,f)} = f k, then derive 0 = f - \\frac{\\partial}{\\partial k} \\pi{(k,f)}, then obtain \\frac{\\frac{\\partial}{\\partial k} \\pi{(k,f)}}{\\frac{\\partial}{\\partial k} f k} = - \\frac{f - \\frac{\\partial}{\\partial k} f k - \\frac{\\partial}{\\partial k} \\pi{(k,f)}}{\\frac{\\partial}{\\partial k} f k}", "derivation": "\\pi{(k,f)} = f k and 0 = f k - \\pi{(k,f)} and \\frac{d}{d k} 0 = \\frac{\\partial}{\\partial k} (f k - \\pi{(k,f)}) and 0 = f - \\frac{\\partial}{\\partial k} \\pi{(k,f)} and 0 = f - \\frac{\\partial}{\\partial k} f k and - \\frac{\\partial}{\\partial k} \\pi{(k,f)} = f - \\frac{\\partial}{\\partial k} f k - \\frac{\\partial}{\\partial k} \\pi{(k,f)} and \\frac{\\frac{\\partial}{\\partial k} \\pi{(k,f)}}{\\frac{\\partial}{\\partial k} f k} = - \\frac{f - \\frac{\\partial}{\\partial k} f k - \\frac{\\partial}{\\partial k} \\pi{(k,f)}}{\\frac{\\partial}{\\partial k} f k}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)))"], [["minus", 1, "Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Symbol('f', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Symbol('f', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["add", 5, "Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Symbol('f', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["divide", 6, "Mul(Integer(-1), Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Symbol('f', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('k', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Pow(Derivative(Mul(Symbol('f', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\nabla{(r_{0},\\Psi_{\\lambda})} = \\frac{\\sin{(\\Psi_{\\lambda})}}{r_{0}}, then obtain \\int (r_{0} \\nabla{(r_{0},\\Psi_{\\lambda})} + \\nabla{(r_{0},\\Psi_{\\lambda})}) d\\Psi_{\\lambda} = \\int (\\nabla{(r_{0},\\Psi_{\\lambda})} + \\sin{(\\Psi_{\\lambda})}) d\\Psi_{\\lambda}", "derivation": "\\nabla{(r_{0},\\Psi_{\\lambda})} = \\frac{\\sin{(\\Psi_{\\lambda})}}{r_{0}} and r_{0} \\nabla{(r_{0},\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})} and r_{0} \\nabla{(r_{0},\\Psi_{\\lambda})} + \\nabla{(r_{0},\\Psi_{\\lambda})} = \\nabla{(r_{0},\\Psi_{\\lambda})} + \\sin{(\\Psi_{\\lambda})} and \\int (r_{0} \\nabla{(r_{0},\\Psi_{\\lambda})} + \\nabla{(r_{0},\\Psi_{\\lambda})}) d\\Psi_{\\lambda} = \\int (\\nabla{(r_{0},\\Psi_{\\lambda})} + \\sin{(\\Psi_{\\lambda})}) d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 1, "Pow(Symbol('r_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 2, "Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Add(Function('\\\\nabla')(Symbol('r_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(g_{\\varepsilon},A_{z})} = A_{z} g_{\\varepsilon}, then obtain (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} = 2 (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} - \\operatorname{J_{\\varepsilon}}^{g_{\\varepsilon}}{(g_{\\varepsilon},A_{z})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(g_{\\varepsilon},A_{z})} = A_{z} g_{\\varepsilon} and \\operatorname{J_{\\varepsilon}}^{g_{\\varepsilon}}{(g_{\\varepsilon},A_{z})} = (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} and (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} + \\operatorname{J_{\\varepsilon}}^{g_{\\varepsilon}}{(g_{\\varepsilon},A_{z})} = 2 (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} and (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} = 2 (A_{z} g_{\\varepsilon})^{g_{\\varepsilon}} - \\operatorname{J_{\\varepsilon}}^{g_{\\varepsilon}}{(g_{\\varepsilon},A_{z})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('J_{\\\\varepsilon}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Pow(Function('J_{\\\\varepsilon}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(2), Pow(Mul(Symbol('A_z', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})} = A_{z} - T + \\mathbf{A}, then obtain (- 2 T + \\int \\frac{\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})}}{A_{z} - T + \\mathbf{A}} dA_{z})^{A_{z}} = (- 2 T + \\int 1 dA_{z})^{A_{z}}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})} = A_{z} - T + \\mathbf{A} and \\frac{\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})}}{A_{z} - T + \\mathbf{A}} = 1 and \\int \\frac{\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})}}{A_{z} - T + \\mathbf{A}} dA_{z} = \\int 1 dA_{z} and - 2 T + \\int \\frac{\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})}}{A_{z} - T + \\mathbf{A}} dA_{z} = - 2 T + \\int 1 dA_{z} and (- 2 T + \\int \\frac{\\operatorname{C_{d}}{(\\mathbf{A},T,A_{z})}}{A_{z} - T + \\mathbf{A}} dA_{z})^{A_{z}} = (- 2 T + \\int 1 dA_{z})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('T', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 1, "Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('C_d')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('T', commutative=True), Symbol('A_z', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('C_d')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('T', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Integer(2), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True)), Integral(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('C_d')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('T', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True)), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True)))))"], [["power", 4, "Symbol('A_z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True)), Integral(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('C_d')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('T', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True)), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = \\mathbf{J} - \\mathbf{f}, then derive 1 - \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = 0, then obtain \\mathbf{f} (1 - \\frac{\\partial}{\\partial \\mathbf{J}} (\\mathbf{J} - \\mathbf{f})) = 0", "derivation": "\\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = \\mathbf{J} - \\mathbf{f} and - \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = - \\mathbf{J} + \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{J}} - \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{J}} (- \\mathbf{J} + \\mathbf{f}) and \\frac{\\partial}{\\partial \\mathbf{J}} - \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} + 1 = \\frac{\\partial}{\\partial \\mathbf{J}} (- \\mathbf{J} + \\mathbf{f}) + 1 and 1 - \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{A_{y}}{(\\mathbf{J},\\mathbf{f})} = 0 and 1 - \\frac{\\partial}{\\partial \\mathbf{J}} (\\mathbf{J} - \\mathbf{f}) = 0 and \\mathbf{f} (1 - \\frac{\\partial}{\\partial \\mathbf{J}} (\\mathbf{J} - \\mathbf{f})) = 0", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('A_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('A_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Mul(Integer(-1), Function('A_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Function('A_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))), Integer(0))"], [["times", 6, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given g{(Z)} = e^{Z}, then derive \\int g{(Z)} dZ = v_{2} + e^{Z}, then obtain (v_{2} + g{(Z)} + e^{Z})^{v_{2}} = (v_{2} + 2 g{(Z)})^{v_{2}}", "derivation": "g{(Z)} = e^{Z} and \\int g{(Z)} dZ = \\int e^{Z} dZ and \\int g{(Z)} dZ = v_{2} + e^{Z} and \\int g{(Z)} dZ = v_{2} + g{(Z)} and \\int e^{Z} dZ = v_{2} + e^{Z} and g{(Z)} + \\int e^{Z} dZ = v_{2} + g{(Z)} + e^{Z} and g{(Z)} + \\int e^{Z} dZ = v_{2} + 2 g{(Z)} and (g{(Z)} + \\int e^{Z} dZ)^{v_{2}} = (v_{2} + 2 g{(Z)})^{v_{2}} and (v_{2} + g{(Z)} + e^{Z})^{v_{2}} = (v_{2} + 2 g{(Z)})^{v_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('g')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('g')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('v_2', commutative=True), exp(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('g')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('v_2', commutative=True), Function('g')(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('v_2', commutative=True), exp(Symbol('Z', commutative=True))))"], [["add", 5, "Function('g')(Symbol('Z', commutative=True))"], "Equality(Add(Function('g')(Symbol('Z', commutative=True)), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Symbol('v_2', commutative=True), Function('g')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Function('g')(Symbol('Z', commutative=True)), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Symbol('v_2', commutative=True), Mul(Integer(2), Function('g')(Symbol('Z', commutative=True)))))"], [["power", 7, "Symbol('v_2', commutative=True)"], "Equality(Pow(Add(Function('g')(Symbol('Z', commutative=True)), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Symbol('v_2', commutative=True)), Pow(Add(Symbol('v_2', commutative=True), Mul(Integer(2), Function('g')(Symbol('Z', commutative=True)))), Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Pow(Add(Symbol('v_2', commutative=True), Function('g')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))), Symbol('v_2', commutative=True)), Pow(Add(Symbol('v_2', commutative=True), Mul(Integer(2), Function('g')(Symbol('Z', commutative=True)))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given E{(S)} = e^{S}, then derive E^{2}{(S)} + \\int E{(S)} dS = f_{\\mathbf{v}} + E^{2}{(S)} + e^{S}, then obtain (\\hat{X} + E^{2}{(S)} + e^{S}) \\int E{(S)} dS = (f_{\\mathbf{v}} + E^{2}{(S)} + E{(S)}) \\int E{(S)} dS", "derivation": "E{(S)} = e^{S} and \\int E{(S)} dS = \\int e^{S} dS and E^{2}{(S)} + \\int E{(S)} dS = E^{2}{(S)} + \\int e^{S} dS and E^{2}{(S)} + \\int E{(S)} dS = f_{\\mathbf{v}} + E^{2}{(S)} + e^{S} and E^{2}{(S)} + \\int E{(S)} dS = f_{\\mathbf{v}} + E^{2}{(S)} + E{(S)} and E^{2}{(S)} + \\int e^{S} dS = f_{\\mathbf{v}} + E^{2}{(S)} + E{(S)} and (E^{2}{(S)} + \\int e^{S} dS) \\int E{(S)} dS = (f_{\\mathbf{v}} + E^{2}{(S)} + E{(S)}) \\int E{(S)} dS and (\\hat{X} + E^{2}{(S)} + e^{S}) \\int E{(S)} dS = (f_{\\mathbf{v}} + E^{2}{(S)} + E{(S)}) \\int E{(S)} dS", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["add", 2, "Pow(Function('E')(Symbol('S', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), exp(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Function('E')(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Function('E')(Symbol('S', commutative=True))))"], [["times", 6, "Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Function('E')(Symbol('S', commutative=True))), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), exp(Symbol('S', commutative=True))), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('S', commutative=True)), Integer(2)), Function('E')(Symbol('S', commutative=True))), Integral(Function('E')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)} = - A_{2} - \\sigma_x + q, then obtain 0 = \\frac{- A_{2} - \\sigma_x + q - \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)}}{2 (- A_{2} - \\sigma_x + q) \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)}}", "derivation": "\\Psi_{\\lambda}{(A_{2},\\sigma_x,q)} = - A_{2} - \\sigma_x + q and 0 = - A_{2} - \\sigma_x + q - \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)} and 0 = \\frac{- A_{2} - \\sigma_x + q - \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)}}{- A_{2} - \\sigma_x + q} and 0 = \\frac{- A_{2} - \\sigma_x + q - \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)}}{2 (- A_{2} - \\sigma_x + q) \\Psi_{\\lambda}{(A_{2},\\sigma_x,q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))))))"], [["divide", 3, "Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)))"], "Equality(Integer(0), Mul(Rational(1, 2), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('q', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)))), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(c)} = \\frac{d}{d c} \\sin{(c)}, then derive 2 \\Psi^{\\dagger}{(c)} = \\Psi^{\\dagger}{(c)} + \\cos{(c)}, then obtain \\frac{\\Psi^{\\dagger}{(c)}}{\\Psi^{\\dagger}{(c)} + \\cos{(c)}} = \\frac{\\frac{d}{d c} \\sin{(c)}}{\\Psi^{\\dagger}{(c)} + \\cos{(c)}}", "derivation": "\\Psi^{\\dagger}{(c)} = \\frac{d}{d c} \\sin{(c)} and 2 \\Psi^{\\dagger}{(c)} = \\Psi^{\\dagger}{(c)} + \\frac{d}{d c} \\sin{(c)} and \\frac{\\Psi^{\\dagger}{(c)}}{\\Psi^{\\dagger}{(c)} + \\frac{d}{d c} \\sin{(c)}} = \\frac{\\frac{d}{d c} \\sin{(c)}}{\\Psi^{\\dagger}{(c)} + \\frac{d}{d c} \\sin{(c)}} and 2 \\Psi^{\\dagger}{(c)} = \\Psi^{\\dagger}{(c)} + \\cos{(c)} and \\Psi^{\\dagger}{(c)} + \\cos{(c)} = \\Psi^{\\dagger}{(c)} + \\frac{d}{d c} \\sin{(c)} and \\frac{\\Psi^{\\dagger}{(c)}}{\\Psi^{\\dagger}{(c)} + \\cos{(c)}} = \\frac{\\frac{d}{d c} \\sin{(c)}}{\\Psi^{\\dagger}{(c)} + \\cos{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["add", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["divide", 1, "Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True))), Mul(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Integer(-1)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True))), Mul(Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integer(-1)), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given p{(\\phi_2,g)} = \\phi_2 g, then obtain \\phi_2 p^{2}{(\\phi_2,g)} + g = \\phi_2^{3} g^{2} + g", "derivation": "p{(\\phi_2,g)} = \\phi_2 g and \\phi_2 p{(\\phi_2,g)} = \\phi_2^{2} g and \\phi_2 p^{2}{(\\phi_2,g)} = \\phi_2^{2} g p{(\\phi_2,g)} and \\phi_2 p^{2}{(\\phi_2,g)} + g = \\phi_2^{2} g p{(\\phi_2,g)} + g and \\phi_2^{2} g p{(\\phi_2,g)} = \\phi_2^{3} g^{2} and \\phi_2 p^{2}{(\\phi_2,g)} + g = \\phi_2^{3} g^{2} + g", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))"], [["times", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Symbol('g', commutative=True)))"], [["times", 2, "Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Symbol('g', commutative=True), Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))"], [["add", 3, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Integer(2))), Symbol('g', commutative=True)), Add(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Symbol('g', commutative=True), Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Symbol('g', commutative=True), Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(3)), Pow(Symbol('g', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('p')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Integer(2))), Symbol('g', commutative=True)), Add(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(3)), Pow(Symbol('g', commutative=True), Integer(2))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\theta{(g_{\\varepsilon},\\hbar)} = \\hbar g_{\\varepsilon}, then obtain \\hbar g_{\\varepsilon} + g_{\\varepsilon} + \\theta{(g_{\\varepsilon},\\hbar)} = 2 \\hbar g_{\\varepsilon} + g_{\\varepsilon}", "derivation": "\\theta{(g_{\\varepsilon},\\hbar)} = \\hbar g_{\\varepsilon} and \\frac{\\theta{(g_{\\varepsilon},\\hbar)}}{\\hbar} = g_{\\varepsilon} and \\hbar g_{\\varepsilon} + \\theta{(g_{\\varepsilon},\\hbar)} = 2 \\hbar g_{\\varepsilon} and \\hbar g_{\\varepsilon} + \\theta{(g_{\\varepsilon},\\hbar)} + \\frac{\\theta{(g_{\\varepsilon},\\hbar)}}{\\hbar} = 2 \\hbar g_{\\varepsilon} + \\frac{\\theta{(g_{\\varepsilon},\\hbar)}}{\\hbar} and \\hbar g_{\\varepsilon} + g_{\\varepsilon} + \\theta{(g_{\\varepsilon},\\hbar)} = 2 \\hbar g_{\\varepsilon} + g_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["add", 1, "Mul(Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 3, "Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\theta')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(t_{1},z)} = t_{1} + z, then derive \\frac{\\partial}{\\partial t_{1}} \\operatorname{J_{\\varepsilon}}{(t_{1},z)} = 1, then obtain \\cos{(\\frac{\\partial}{\\partial t_{1}} (t_{1} + z))} - \\frac{\\frac{\\partial}{\\partial t_{1}} (t_{1} + z)}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}} = \\cos{(\\frac{\\partial}{\\partial t_{1}} (t_{1} + z))} - \\frac{1}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(t_{1},z)} = t_{1} + z and \\frac{\\partial}{\\partial t_{1}} \\operatorname{J_{\\varepsilon}}{(t_{1},z)} = \\frac{\\partial}{\\partial t_{1}} (t_{1} + z) and \\frac{\\partial}{\\partial t_{1}} \\operatorname{J_{\\varepsilon}}{(t_{1},z)} = 1 and \\frac{\\partial}{\\partial t_{1}} (t_{1} + z) = 1 and - \\frac{\\frac{\\partial}{\\partial t_{1}} (t_{1} + z)}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}} = - \\frac{1}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}} and \\cos{(\\frac{\\partial}{\\partial t_{1}} (t_{1} + z))} - \\frac{\\frac{\\partial}{\\partial t_{1}} (t_{1} + z)}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}} = \\cos{(\\frac{\\partial}{\\partial t_{1}} (t_{1} + z))} - \\frac{1}{\\operatorname{J_{\\varepsilon}}{(t_{1},z)}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Integer(-1))))"], [["add", 5, "cos(Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], "Equality(Add(cos(Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(cos(Derivative(Add(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{v}{(a,E)} = e^{\\frac{a}{E}}, then obtain (\\mathbf{v}{(a,E)} + \\frac{1}{E})^{2 E} = (\\mathbf{v}{(a,E)} + \\frac{1}{E})^{E} (e^{\\frac{a}{E}} + \\frac{1}{E})^{E}", "derivation": "\\mathbf{v}{(a,E)} = e^{\\frac{a}{E}} and \\mathbf{v}{(a,E)} + \\frac{1}{E} = e^{\\frac{a}{E}} + \\frac{1}{E} and (\\mathbf{v}{(a,E)} + \\frac{1}{E})^{E} = (e^{\\frac{a}{E}} + \\frac{1}{E})^{E} and (\\mathbf{v}{(a,E)} + \\frac{1}{E})^{2 E} = (\\mathbf{v}{(a,E)} + \\frac{1}{E})^{E} (e^{\\frac{a}{E}} + \\frac{1}{E})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), exp(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["add", 1, "Pow(Symbol('E', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1))), Add(exp(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Pow(Symbol('E', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1))), Symbol('E', commutative=True)), Pow(Add(exp(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Pow(Symbol('E', commutative=True), Integer(-1))), Symbol('E', commutative=True)))"], [["times", 3, "Pow(Add(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1))), Symbol('E', commutative=True))"], "Equality(Pow(Add(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('E', commutative=True))), Mul(Pow(Add(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1))), Symbol('E', commutative=True)), Pow(Add(exp(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Pow(Symbol('E', commutative=True), Integer(-1))), Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(C)} = \\sin{(C)}, then obtain 1 = \\frac{\\cos{(C)}}{\\frac{d}{d C} \\operatorname{f_{\\mathbf{v}}}{(C)}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(C)} = \\sin{(C)} and \\frac{d}{d C} \\operatorname{f_{\\mathbf{v}}}{(C)} = \\frac{d}{d C} \\sin{(C)} and 1 = \\frac{\\frac{d}{d C} \\sin{(C)}}{\\frac{d}{d C} \\operatorname{f_{\\mathbf{v}}}{(C)}} and 1 = \\frac{\\cos{(C)}}{\\frac{d}{d C} \\operatorname{f_{\\mathbf{v}}}{(C)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(cos(Symbol('C', commutative=True)), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{F}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)}, then derive \\mathbf{F}{(\\eta)} = - \\sin{(\\eta)}, then obtain - (\\mathbf{F}{(\\eta)} - \\sin{(\\eta)}) \\sin{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} = (\\mathbf{F}{(\\eta)} - \\sin{(\\eta)}) \\sin^{2}{(\\eta)}", "derivation": "\\mathbf{F}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)} and \\mathbf{F}{(\\eta)} = - \\sin{(\\eta)} and 2 \\mathbf{F}{(\\eta)} = \\mathbf{F}{(\\eta)} - \\sin{(\\eta)} and \\frac{d}{d \\eta} \\cos{(\\eta)} = - \\sin{(\\eta)} and - \\sin{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} = \\sin^{2}{(\\eta)} and - 2 \\mathbf{F}{(\\eta)} \\sin{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} = 2 \\mathbf{F}{(\\eta)} \\sin^{2}{(\\eta)} and - (\\mathbf{F}{(\\eta)} - \\sin{(\\eta)}) \\sin{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} = (\\mathbf{F}{(\\eta)} - \\sin{(\\eta)}) \\sin^{2}{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))"], [["times", 4, "Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)))"], [["times", 5, "Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(-1), Add(Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), sin(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Add(Function('\\\\mathbf{F}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2))))"]]}, {"prompt": "Given A{(\\dot{y})} = e^{\\dot{y}}, then derive \\frac{d}{d \\dot{y}} A{(\\dot{y})} = e^{\\dot{y}}, then obtain \\frac{d}{d \\dot{y}} e^{\\dot{y}} = e^{\\dot{y}}", "derivation": "A{(\\dot{y})} = e^{\\dot{y}} and \\frac{d}{d \\dot{y}} A{(\\dot{y})} = \\frac{d}{d \\dot{y}} e^{\\dot{y}} and \\frac{d}{d \\dot{y}} A{(\\dot{y})} = e^{\\dot{y}} and \\frac{d}{d \\dot{y}} e^{\\dot{y}} = e^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), exp(Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\omega)} = \\cos{(\\omega)}, then obtain \\operatorname{n_{2}}^{- \\omega}{(\\omega)} \\int \\operatorname{n_{2}}{(\\omega)} d\\omega = (\\Omega + \\sin{(\\omega)}) \\operatorname{n_{2}}^{- \\omega}{(\\omega)}", "derivation": "\\operatorname{n_{2}}{(\\omega)} = \\cos{(\\omega)} and \\int \\operatorname{n_{2}}{(\\omega)} d\\omega = \\int \\cos{(\\omega)} d\\omega and \\cos^{- \\omega}{(\\omega)} \\int \\operatorname{n_{2}}{(\\omega)} d\\omega = \\cos^{- \\omega}{(\\omega)} \\int \\cos{(\\omega)} d\\omega and \\operatorname{n_{2}}^{- \\omega}{(\\omega)} \\int \\operatorname{n_{2}}{(\\omega)} d\\omega = \\operatorname{n_{2}}^{- \\omega}{(\\omega)} \\int \\cos{(\\omega)} d\\omega and \\operatorname{n_{2}}^{- \\omega}{(\\omega)} \\int \\operatorname{n_{2}}{(\\omega)} d\\omega = (\\Omega + \\sin{(\\omega)}) \\operatorname{n_{2}}^{- \\omega}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 2, "Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('n_2')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('n_2')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('n_2')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Function('n_2')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Function('n_2')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(Function('n_2')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Pow(Function('n_2')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given B{(g,H)} = \\sin{(g^{H})} and \\hat{X}{(U,\\tilde{g}^*)} = - \\tilde{g}^* + \\hat{H}{(U,\\tilde{g}^*)}, then obtain H (B^{H}{(g,H)})^{g} + \\frac{\\tilde{g}^* \\hat{X}{(U,\\tilde{g}^*)}}{U} = H (\\sin^{H}{(g^{H})})^{g} + \\frac{\\tilde{g}^* \\hat{X}{(U,\\tilde{g}^*)}}{U}", "derivation": "B{(g,H)} = \\sin{(g^{H})} and B^{H}{(g,H)} = \\sin^{H}{(g^{H})} and (B^{H}{(g,H)})^{g} = (\\sin^{H}{(g^{H})})^{g} and H (B^{H}{(g,H)})^{g} = H (\\sin^{H}{(g^{H})})^{g} and \\hat{X}{(U,\\tilde{g}^*)} = - \\tilde{g}^* + \\hat{H}{(U,\\tilde{g}^*)} and H (B^{H}{(g,H)})^{g} + \\frac{\\tilde{g}^* (- \\tilde{g}^* + \\hat{H}{(U,\\tilde{g}^*)})}{U} = H (\\sin^{H}{(g^{H})})^{g} + \\frac{\\tilde{g}^* (- \\tilde{g}^* + \\hat{H}{(U,\\tilde{g}^*)})}{U} and H (B^{H}{(g,H)})^{g} + \\frac{\\tilde{g}^* \\hat{X}{(U,\\tilde{g}^*)}}{U} = H (\\sin^{H}{(g^{H})})^{g} + \\frac{\\tilde{g}^* \\hat{X}{(U,\\tilde{g}^*)}}{U}", "srepr_derivation": [["get_premise", "Equality(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Pow(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Symbol('g', commutative=True)))"], [["times", 3, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Pow(Pow(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('g', commutative=True))), Mul(Symbol('H', commutative=True), Pow(Pow(sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 4, "Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Add(Mul(Symbol('H', commutative=True), Pow(Pow(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))), Add(Mul(Symbol('H', commutative=True), Pow(Pow(sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Symbol('H', commutative=True), Pow(Pow(Function('B')(Symbol('g', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Symbol('H', commutative=True), Pow(Pow(sin(Pow(Symbol('g', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain \\mathbf{S}{(\\mathbb{I})} \\cos^{2}{(\\mathbb{I})} = \\cos^{3}{(\\mathbb{I})}", "derivation": "\\mathbf{S}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\mathbf{S}{(\\mathbb{I})} \\cos{(\\mathbb{I})} = \\cos^{2}{(\\mathbb{I})} and \\mathbf{S}^{2}{(\\mathbb{I})} \\cos{(\\mathbb{I})} = \\mathbf{S}{(\\mathbb{I})} \\cos^{2}{(\\mathbb{I})} and \\mathbf{S}{(\\mathbb{I})} \\cos^{2}{(\\mathbb{I})} = \\cos^{3}{(\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))"], [["times", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda})} and \\varepsilon{(\\hat{H}_{\\lambda})} = \\sin{(\\operatorname{z^{*}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})}, then obtain \\varepsilon^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\sin^{\\hat{H}_{\\lambda}}{(\\sin^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})}", "derivation": "\\operatorname{z^{*}}{(\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda})} and \\operatorname{z^{*}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\sin^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} and \\sin{(\\operatorname{z^{*}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})} = \\sin{(\\sin^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})} and \\varepsilon{(\\hat{H}_{\\lambda})} = \\sin{(\\operatorname{z^{*}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})} and \\varepsilon{(\\hat{H}_{\\lambda})} = \\sin{(\\sin^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})} and \\varepsilon^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\sin^{\\hat{H}_{\\lambda}}{(\\sin^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), sin(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Pow(Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varepsilon')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["power", 5, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(sin(Pow(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\tilde{g}^*)} = \\cos{(\\sin{(\\tilde{g}^*)})}, then obtain - \\int \\operatorname{E_{n}}{(\\tilde{g}^*)} d\\tilde{g}^* = - \\operatorname{E_{n}}^{\\tilde{g}^*}{(\\tilde{g}^*)} + \\cos^{\\tilde{g}^*}{(\\sin{(\\tilde{g}^*)})} - \\int \\operatorname{E_{n}}{(\\tilde{g}^*)} d\\tilde{g}^*", "derivation": "\\operatorname{E_{n}}{(\\tilde{g}^*)} = \\cos{(\\sin{(\\tilde{g}^*)})} and \\operatorname{E_{n}}^{\\tilde{g}^*}{(\\tilde{g}^*)} = \\cos^{\\tilde{g}^*}{(\\sin{(\\tilde{g}^*)})} and 0 = - \\operatorname{E_{n}}^{\\tilde{g}^*}{(\\tilde{g}^*)} + \\cos^{\\tilde{g}^*}{(\\sin{(\\tilde{g}^*)})} and - \\int \\operatorname{E_{n}}{(\\tilde{g}^*)} d\\tilde{g}^* = - \\operatorname{E_{n}}^{\\tilde{g}^*}{(\\tilde{g}^*)} + \\cos^{\\tilde{g}^*}{(\\sin{(\\tilde{g}^*)})} - \\int \\operatorname{E_{n}}{(\\tilde{g}^*)} d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["power", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 2, "Pow(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 3, "Integral(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{M})} = \\log{(\\mathbf{M})}, then derive \\operatorname{v_{2}}{(E_{n})} \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = (\\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + z) \\operatorname{v_{2}}{(E_{n})}, then obtain \\operatorname{v_{2}}{(E_{n})} \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = (\\mathbf{M} \\hat{H}{(\\mathbf{M})} - \\mathbf{M} + z) \\operatorname{v_{2}}{(E_{n})}", "derivation": "\\hat{H}{(\\mathbf{M})} = \\log{(\\mathbf{M})} and \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = \\int \\log{(\\mathbf{M})} d\\mathbf{M} and \\operatorname{v_{2}}{(E_{n})} \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = \\operatorname{v_{2}}{(E_{n})} \\int \\log{(\\mathbf{M})} d\\mathbf{M} and \\operatorname{v_{2}}{(E_{n})} \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = (\\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + z) \\operatorname{v_{2}}{(E_{n})} and \\operatorname{v_{2}}{(E_{n})} \\int \\hat{H}{(\\mathbf{M})} d\\mathbf{M} = (\\mathbf{M} \\hat{H}{(\\mathbf{M})} - \\mathbf{M} + z) \\operatorname{v_{2}}{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(log(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 2, "Function('v_2')(Symbol('E_n', commutative=True))"], "Equality(Mul(Function('v_2')(Symbol('E_n', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Function('v_2')(Symbol('E_n', commutative=True)), Integral(log(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('v_2')(Symbol('E_n', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('z', commutative=True)), Function('v_2')(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('v_2')(Symbol('E_n', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('z', commutative=True)), Function('v_2')(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} = - \\phi + \\varepsilon_0 - r, then obtain - \\phi \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} + \\frac{\\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)}}{- \\phi + \\varepsilon_0 - r} = - \\phi \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} + 1", "derivation": "\\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} = - \\phi + \\varepsilon_0 - r and \\frac{\\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)}}{- \\phi + \\varepsilon_0 - r} = 1 and \\phi \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} = \\phi (- \\phi + \\varepsilon_0 - r) and - \\phi (- \\phi + \\varepsilon_0 - r) + \\frac{\\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)}}{- \\phi + \\varepsilon_0 - r} = - \\phi (- \\phi + \\varepsilon_0 - r) + 1 and - \\phi \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} + \\frac{\\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)}}{- \\phi + \\varepsilon_0 - r} = - \\phi \\operatorname{a^{\\dagger}}{(\\varepsilon_0,\\phi,r)} + 1", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True))), Integer(1))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], [["minus", 2, "Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\hat{p}{(\\chi)} = e^{\\chi}, then derive - \\chi + \\frac{d}{d \\chi} \\hat{p}{(\\chi)} = - \\chi + e^{\\chi}, then obtain - \\chi + \\frac{d}{d \\chi} e^{\\chi} = - \\chi + e^{\\chi}", "derivation": "\\hat{p}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\hat{p}{(\\chi)} = \\frac{d}{d \\chi} e^{\\chi} and - \\chi + \\frac{d}{d \\chi} \\hat{p}{(\\chi)} = - \\chi + \\frac{d}{d \\chi} e^{\\chi} and - \\chi + \\frac{d}{d \\chi} \\hat{p}{(\\chi)} = - \\chi + e^{\\chi} and - \\chi + \\frac{d}{d \\chi} \\hat{p}{(\\chi)} = - \\chi + \\hat{p}{(\\chi)} and - \\chi + \\frac{d}{d \\chi} e^{\\chi} = - \\chi + e^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\Psi{(I)} = \\log{(\\cos{(I)})}, then derive \\cos{(\\Psi{(I)})} \\frac{d}{d I} \\Psi{(I)} = - \\frac{\\sin{(I)} \\cos{(\\log{(\\cos{(I)})})}}{\\cos{(I)}}, then obtain - \\frac{\\sin{(I)} \\cos{(\\log{(\\cos{(I)})})}}{\\cos{(I)}} = - \\frac{\\sin{(I)} \\cos{(\\Psi{(I)})}}{\\cos{(I)}}", "derivation": "\\Psi{(I)} = \\log{(\\cos{(I)})} and \\sin{(\\Psi{(I)})} = \\sin{(\\log{(\\cos{(I)})})} and \\frac{d}{d I} \\sin{(\\Psi{(I)})} = \\frac{d}{d I} \\sin{(\\log{(\\cos{(I)})})} and \\cos{(\\Psi{(I)})} \\frac{d}{d I} \\Psi{(I)} = - \\frac{\\sin{(I)} \\cos{(\\log{(\\cos{(I)})})}}{\\cos{(I)}} and \\cos{(\\Psi{(I)})} \\frac{d}{d I} \\Psi{(I)} = - \\frac{\\sin{(I)} \\cos{(\\Psi{(I)})}}{\\cos{(I)}} and - \\frac{\\sin{(I)} \\cos{(\\log{(\\cos{(I)})})}}{\\cos{(I)}} = - \\frac{\\sin{(I)} \\cos{(\\Psi{(I)})}}{\\cos{(I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('I', commutative=True)), log(cos(Symbol('I', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\Psi')(Symbol('I', commutative=True))), sin(log(cos(Symbol('I', commutative=True)))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(sin(Function('\\\\Psi')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(log(cos(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('\\\\Psi')(Symbol('I', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), cos(log(cos(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(cos(Function('\\\\Psi')(Symbol('I', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), cos(Function('\\\\Psi')(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), sin(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), cos(log(cos(Symbol('I', commutative=True))))), Mul(Integer(-1), sin(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1)), cos(Function('\\\\Psi')(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(Z)} = \\sin{(Z)}, then derive \\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ = \\frac{\\partial}{\\partial Z} (v - \\cos{(Z)}), then obtain \\cos{(\\frac{d}{d Z} \\int \\sin{(Z)} dZ)} = \\cos{(\\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ)}", "derivation": "\\phi_{2}{(Z)} = \\sin{(Z)} and \\int \\phi_{2}{(Z)} dZ = \\int \\sin{(Z)} dZ and \\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ = \\frac{d}{d Z} \\int \\sin{(Z)} dZ and \\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ = \\frac{\\partial}{\\partial Z} (v - \\cos{(Z)}) and \\cos{(\\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ)} = \\cos{(\\frac{\\partial}{\\partial Z} (v - \\cos{(Z)}))} and \\cos{(\\frac{d}{d Z} \\int \\sin{(Z)} dZ)} = \\cos{(\\frac{\\partial}{\\partial Z} (v - \\cos{(Z)}))} and \\cos{(\\frac{d}{d Z} \\int \\sin{(Z)} dZ)} = \\cos{(\\frac{d}{d Z} \\int \\phi_{2}{(Z)} dZ)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Integral(Function('\\\\phi_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), cos(Derivative(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(cos(Derivative(Integral(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), cos(Derivative(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(cos(Derivative(Integral(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), cos(Derivative(Integral(Function('\\\\phi_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(v_{x},t)} = \\int (t - v_{x}) dt, then derive y{(v_{x},t)} = \\mathbf{B} + \\frac{t^{2}}{2} - t v_{x}, then obtain \\frac{2 (\\mathbf{B} + \\frac{t^{2}}{2} - t v_{x})}{t^{2}} = \\frac{2 \\int (t - v_{x}) dt}{t^{2}}", "derivation": "y{(v_{x},t)} = \\int (t - v_{x}) dt and y{(v_{x},t)} = \\mathbf{B} + \\frac{t^{2}}{2} - t v_{x} and \\frac{2 y{(v_{x},t)}}{t^{2}} = \\frac{2 \\int (t - v_{x}) dt}{t^{2}} and \\frac{2 (\\mathbf{B} + \\frac{t^{2}}{2} - t v_{x})}{t^{2}} = \\frac{2 \\int (t - v_{x}) dt}{t^{2}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_x', commutative=True), Symbol('t', commutative=True)), Integral(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('y')(Symbol('v_x', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v_x', commutative=True))))"], [["divide", 1, "Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('t', commutative=True), Integer(-2)), Function('y')(Symbol('v_x', commutative=True), Symbol('t', commutative=True))), Mul(Integer(2), Pow(Symbol('t', commutative=True), Integer(-2)), Integral(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Pow(Symbol('t', commutative=True), Integer(-2)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v_x', commutative=True)))), Mul(Integer(2), Pow(Symbol('t', commutative=True), Integer(-2)), Integral(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(c)} = \\log{(c)} and \\Omega{(c)} = - c \\log{(c)} + (c \\rho_{b}{(c)})^{c}, then obtain \\int (- c \\log{(c)} + (c \\rho_{b}{(c)})^{c}) dc = \\int (- c \\log{(c)} + (c \\log{(c)})^{c}) dc", "derivation": "\\rho_{b}{(c)} = \\log{(c)} and c \\rho_{b}{(c)} = c \\log{(c)} and (c \\rho_{b}{(c)})^{c} = (c \\log{(c)})^{c} and - c \\log{(c)} + (c \\rho_{b}{(c)})^{c} = - c \\log{(c)} + (c \\log{(c)})^{c} and \\Omega{(c)} = - c \\log{(c)} + (c \\rho_{b}{(c)})^{c} and \\Omega{(c)} = - c \\log{(c)} + (c \\log{(c)})^{c} and \\int \\Omega{(c)} dc = \\int (- c \\log{(c)} + (c \\log{(c)})^{c}) dc and \\int (- c \\log{(c)} + (c \\rho_{b}{(c)})^{c}) dc = \\int (- c \\log{(c)} + (c \\log{(c)})^{c}) dc", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('\\\\rho_b')(Symbol('c', commutative=True))), Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Symbol('c', commutative=True), Function('\\\\rho_b')(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["minus", 3, "Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), Function('\\\\rho_b')(Symbol('c', commutative=True))), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), Function('\\\\rho_b')(Symbol('c', commutative=True))), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\Omega')(Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Symbol('c', commutative=True))))"], [["integrate", 6, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), Function('\\\\rho_b')(Symbol('c', commutative=True))), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Pow(Mul(Symbol('c', commutative=True), log(Symbol('c', commutative=True))), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(C_{2},F_{N})} = \\log{(C_{2} F_{N})}, then obtain - C_{2} \\operatorname{F_{H}}{(C_{2},F_{N})} = - C_{2} \\log{(C_{2} F_{N})} + \\operatorname{F_{H}}{(C_{2},F_{N})} - \\log{(C_{2} F_{N})}", "derivation": "\\operatorname{F_{H}}{(C_{2},F_{N})} = \\log{(C_{2} F_{N})} and C_{2} \\operatorname{F_{H}}{(C_{2},F_{N})} = C_{2} \\log{(C_{2} F_{N})} and \\operatorname{F_{H}}{(C_{2},F_{N})} - \\log{(C_{2} F_{N})} = 0 and - C_{2} \\log{(C_{2} F_{N})} + \\operatorname{F_{H}}{(C_{2},F_{N})} - \\log{(C_{2} F_{N})} = - C_{2} \\log{(C_{2} F_{N})} and - C_{2} \\operatorname{F_{H}}{(C_{2},F_{N})} = - C_{2} \\log{(C_{2} F_{N})} and - C_{2} \\operatorname{F_{H}}{(C_{2},F_{N})} = - C_{2} \\log{(C_{2} F_{N})} + \\operatorname{F_{H}}{(C_{2},F_{N})} - \\log{(C_{2} F_{N})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))), Mul(Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))))"], [["minus", 1, "log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Add(Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))))), Integer(0))"], [["minus", 3, "Mul(Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))), Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))))), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('C_2', commutative=True), Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Symbol('C_2', commutative=True), Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)))), Function('F_H')(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_2', commutative=True), Symbol('F_N', commutative=True))))))"]]}, {"prompt": "Given \\pi{(\\delta)} = \\sin{(\\delta)} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_{\\lambda})} = e^{\\hat{H}_{\\lambda}}, then obtain \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\delta} \\sin^{\\delta}{(\\delta)} = e^{\\hat{H}_{\\lambda}} - \\frac{d}{d \\delta} \\sin^{\\delta}{(\\delta)}", "derivation": "\\pi{(\\delta)} = \\sin{(\\delta)} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_{\\lambda})} = e^{\\hat{H}_{\\lambda}} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\delta} \\pi^{\\delta}{(\\delta)} = e^{\\hat{H}_{\\lambda}} - \\frac{d}{d \\delta} \\pi^{\\delta}{(\\delta)} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_{\\lambda})} - \\frac{d}{d \\delta} \\sin^{\\delta}{(\\delta)} = e^{\\hat{H}_{\\lambda}} - \\frac{d}{d \\delta} \\sin^{\\delta}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["minus", 2, "Derivative(Pow(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Derivative(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Derivative(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{y}{(n_{2},\\dot{z})} = \\sin{(\\dot{z}^{n_{2}})} and M{(n_{2},\\dot{z})} = \\sin{(\\dot{z}^{n_{2}})}, then obtain \\frac{\\partial}{\\partial n_{2}} M{(n_{2},\\dot{z})} = \\frac{\\partial}{\\partial n_{2}} \\dot{y}{(n_{2},\\dot{z})}", "derivation": "\\dot{y}{(n_{2},\\dot{z})} = \\sin{(\\dot{z}^{n_{2}})} and \\frac{\\partial}{\\partial n_{2}} \\dot{y}{(n_{2},\\dot{z})} = \\frac{\\partial}{\\partial n_{2}} \\sin{(\\dot{z}^{n_{2}})} and M{(n_{2},\\dot{z})} = \\sin{(\\dot{z}^{n_{2}})} and M{(n_{2},\\dot{z})} = \\dot{y}{(n_{2},\\dot{z})} and \\frac{\\partial}{\\partial n_{2}} M{(n_{2},\\dot{z})} = \\frac{\\partial}{\\partial n_{2}} \\sin{(\\dot{z}^{n_{2}})} and \\frac{\\partial}{\\partial n_{2}} M{(n_{2},\\dot{z})} = \\frac{\\partial}{\\partial n_{2}} \\dot{y}{(n_{2},\\dot{z})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), sin(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), sin(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('M')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\dot{y}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('M')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Function('M')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Function('\\\\dot{y}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\hat{X})} = \\sin{(\\hat{X})}, then derive \\int \\mathbf{S}{(\\hat{X})} d\\hat{X} = E_{\\lambda} - \\cos{(\\hat{X})}, then obtain \\int \\sin{(\\hat{X})} d\\hat{X} = E_{\\lambda} - \\cos{(\\hat{X})}", "derivation": "\\mathbf{S}{(\\hat{X})} = \\sin{(\\hat{X})} and \\int \\mathbf{S}{(\\hat{X})} d\\hat{X} = \\int \\sin{(\\hat{X})} d\\hat{X} and \\int \\mathbf{S}{(\\hat{X})} d\\hat{X} = E_{\\lambda} - \\cos{(\\hat{X})} and \\int \\sin{(\\hat{X})} d\\hat{X} = E_{\\lambda} - \\cos{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(sin(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given A{(J,\\rho_f)} = J \\rho_f, then obtain \\log{(A{(J,\\rho_f)} \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} + \\int A{(J,\\rho_f)} dJ = \\log{(J \\rho_f \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} + \\int A{(J,\\rho_f)} dJ", "derivation": "A{(J,\\rho_f)} = J \\rho_f and A{(J,\\rho_f)} \\frac{\\partial}{\\partial J} A{(J,\\rho_f)} = J \\rho_f \\frac{\\partial}{\\partial J} A{(J,\\rho_f)} and \\log{(A{(J,\\rho_f)} \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} = \\log{(J \\rho_f \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} and \\log{(A{(J,\\rho_f)} \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} + \\int A{(J,\\rho_f)} dJ = \\log{(J \\rho_f \\frac{\\partial}{\\partial J} A{(J,\\rho_f)})} + \\int A{(J,\\rho_f)} dJ", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["log", 2], "Equality(log(Mul(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), log(Mul(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))))"], [["add", 3, "Integral(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(log(Mul(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Integral(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(log(Mul(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True), Derivative(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Integral(Function('A')(Symbol('J', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} = \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})}, then obtain 2 \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} + \\frac{1}{f_{E}} = 2 \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} + \\frac{1}{f_{E}}", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} = \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} and \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} + \\frac{1}{f_{E}} = \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} + \\frac{1}{f_{E}} and \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} + \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} + \\frac{1}{f_{E}} = 2 \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} + \\frac{1}{f_{E}} and 2 \\operatorname{y^{\\prime}}{(\\hat{\\mathbf{r}},f_{E})} + \\frac{1}{f_{E}} = 2 \\sin{(\\frac{\\hat{\\mathbf{r}}}{f_{E}})} + \\frac{1}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f_E', commutative=True)), sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)))))"], [["add", 1, "Pow(Symbol('f_E', commutative=True), Integer(-1))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f_E', commutative=True)), Pow(Symbol('f_E', commutative=True), Integer(-1))), Add(sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)))), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["add", 1, "Add(sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)))), Pow(Symbol('f_E', commutative=True), Integer(-1)))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f_E', commutative=True)), sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)))), Pow(Symbol('f_E', commutative=True), Integer(-1))), Add(Mul(Integer(2), sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('f_E', commutative=True))), Pow(Symbol('f_E', commutative=True), Integer(-1))), Add(Mul(Integer(2), sin(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))), Pow(Symbol('f_E', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\psi^*,\\mathbf{v},W)} = \\frac{\\mathbf{v}}{W \\psi^*} and \\ddot{x}{(\\mathbf{v})} = \\mathbf{v}, then obtain \\frac{\\partial}{\\partial \\psi^*} (\\frac{\\mathbf{v}}{W \\psi^*})^{\\psi^*} \\ddot{x}{(\\mathbf{v})} = \\frac{\\partial}{\\partial \\psi^*} \\mathbf{v} (\\frac{\\mathbf{v}}{W \\psi^*})^{\\psi^*}", "derivation": "\\operatorname{v_{1}}{(\\psi^*,\\mathbf{v},W)} = \\frac{\\mathbf{v}}{W \\psi^*} and \\ddot{x}{(\\mathbf{v})} = \\mathbf{v} and \\ddot{x}{(\\mathbf{v})} \\operatorname{v_{1}}^{\\psi^*}{(\\psi^*,\\mathbf{v},W)} = \\mathbf{v} \\operatorname{v_{1}}^{\\psi^*}{(\\psi^*,\\mathbf{v},W)} and \\frac{\\partial}{\\partial \\psi^*} \\ddot{x}{(\\mathbf{v})} \\operatorname{v_{1}}^{\\psi^*}{(\\psi^*,\\mathbf{v},W)} = \\frac{\\partial}{\\partial \\psi^*} \\mathbf{v} \\operatorname{v_{1}}^{\\psi^*}{(\\psi^*,\\mathbf{v},W)} and \\frac{\\partial}{\\partial \\psi^*} (\\frac{\\mathbf{v}}{W \\psi^*})^{\\psi^*} \\ddot{x}{(\\mathbf{v})} = \\frac{\\partial}{\\partial \\psi^*} \\mathbf{v} (\\frac{\\mathbf{v}}{W \\psi^*})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))"], [["times", 2, "Pow(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('v_1')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Symbol('\\\\psi^*', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(r)} = \\cos{(r)}, then obtain - r + \\int \\tilde{g}^*^{r}{(r)} dr = - r - \\int \\tilde{g}^*^{r}{(r)} dr + 2 \\int \\cos^{r}{(r)} dr", "derivation": "\\tilde{g}^*{(r)} = \\cos{(r)} and \\tilde{g}^*^{r}{(r)} = \\cos^{r}{(r)} and \\int \\tilde{g}^*^{r}{(r)} dr = \\int \\cos^{r}{(r)} dr and - r + \\int \\tilde{g}^*^{r}{(r)} dr = - r + \\int \\cos^{r}{(r)} dr and - r = - r - \\int \\tilde{g}^*^{r}{(r)} dr + \\int \\cos^{r}{(r)} dr and - r + \\int \\cos^{r}{(r)} dr = - r - \\int \\tilde{g}^*^{r}{(r)} dr + 2 \\int \\cos^{r}{(r)} dr and - r + \\int \\tilde{g}^*^{r}{(r)} dr = - r - \\int \\tilde{g}^*^{r}{(r)} dr + 2 \\int \\cos^{r}{(r)} dr", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["minus", 3, "Symbol('r', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["minus", 4, "Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Integer(2), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Integer(2), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{J})} = \\cos{(e^{\\mathbf{J}})}, then obtain 1 - 2 \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos{(e^{\\mathbf{J}})} = - 2 \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos{(e^{\\mathbf{J}})} + \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos^{\\mathbf{J}}{(e^{\\mathbf{J}})}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{J})} = \\cos{(e^{\\mathbf{J}})} and \\operatorname{F_{x}}^{\\mathbf{J}}{(\\mathbf{J})} = \\cos^{\\mathbf{J}}{(e^{\\mathbf{J}})} and 1 = \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos^{\\mathbf{J}}{(e^{\\mathbf{J}})} and 1 - 2 \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos{(e^{\\mathbf{J}})} = - 2 \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos{(e^{\\mathbf{J}})} + \\operatorname{F_{x}}^{- \\mathbf{J}}{(\\mathbf{J})} \\cos^{\\mathbf{J}}{(e^{\\mathbf{J}})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), cos(exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 2, "Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(cos(exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), cos(exp(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integer(2), Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), cos(exp(Symbol('\\\\mathbf{J}', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), cos(exp(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(cos(exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(S,\\rho)} = S^{\\rho} and \\operatorname{P_{g}}{(S,\\rho)} = \\frac{1}{\\operatorname{M_{E}}{(S,\\rho)}}, then obtain \\iint \\frac{1}{\\operatorname{M_{E}}{(S,\\rho)}} dS dS = \\iint S^{- \\rho} dS dS", "derivation": "\\operatorname{M_{E}}{(S,\\rho)} = S^{\\rho} and \\operatorname{P_{g}}{(S,\\rho)} = \\frac{1}{\\operatorname{M_{E}}{(S,\\rho)}} and \\operatorname{P_{g}}{(S,\\rho)} = S^{- \\rho} and \\int \\operatorname{P_{g}}{(S,\\rho)} dS = \\int S^{- \\rho} dS and \\int \\frac{1}{\\operatorname{M_{E}}{(S,\\rho)}} dS = \\int S^{- \\rho} dS and \\iint \\frac{1}{\\operatorname{M_{E}}{(S,\\rho)}} dS dS = \\iint S^{- \\rho} dS dS", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Function('M_E')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_g')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Pow(Function('M_E')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Tuple(Symbol('S', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["integrate", 5, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Function('M_E')(Symbol('S', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(S,t_{1})} = \\log{(S + t_{1})} and \\operatorname{F_{H}}{(S,t_{1})} = S + t_{1}, then obtain ((S + \\log{(\\operatorname{F_{H}}{(S,t_{1})})})^{t_{1}})^{S} = ((S + \\log{(S + t_{1})})^{t_{1}})^{S}", "derivation": "\\bar{\\h}{(S,t_{1})} = \\log{(S + t_{1})} and S + \\bar{\\h}{(S,t_{1})} = S + \\log{(S + t_{1})} and (S + \\bar{\\h}{(S,t_{1})})^{t_{1}} = (S + \\log{(S + t_{1})})^{t_{1}} and ((S + \\bar{\\h}{(S,t_{1})})^{t_{1}})^{S} = ((S + \\log{(S + t_{1})})^{t_{1}})^{S} and \\operatorname{F_{H}}{(S,t_{1})} = S + t_{1} and S + \\bar{\\h}{(S,t_{1})} = S + \\log{(\\operatorname{F_{H}}{(S,t_{1})})} and ((S + \\log{(\\operatorname{F_{H}}{(S,t_{1})})})^{t_{1}})^{S} = ((S + \\log{(S + t_{1})})^{t_{1}})^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('t_1', commutative=True)), log(Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True))))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('S', commutative=True), log(Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))))"], [["power", 2, "Symbol('t_1', commutative=True)"], "Equality(Pow(Add(Symbol('S', commutative=True), Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Pow(Add(Symbol('S', commutative=True), log(Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('S', commutative=True), Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Add(Symbol('S', commutative=True), log(Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('S', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Symbol('S', commutative=True), Function('\\\\hbar')(Symbol('S', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('S', commutative=True), log(Function('F_H')(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Pow(Add(Symbol('S', commutative=True), log(Function('F_H')(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Add(Symbol('S', commutative=True), log(Add(Symbol('S', commutative=True), Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Symbol('S', commutative=True)))"]]}, {"prompt": "Given m{(F_{c},\\delta)} = F_{c} \\delta and z{(F_{c},\\delta)} = F_{c} \\delta, then obtain z^{2}{(F_{c},\\delta)} = F_{c} \\delta z{(F_{c},\\delta)}", "derivation": "m{(F_{c},\\delta)} = F_{c} \\delta and m^{2}{(F_{c},\\delta)} = F_{c} \\delta m{(F_{c},\\delta)} and z{(F_{c},\\delta)} = F_{c} \\delta and z{(F_{c},\\delta)} = m{(F_{c},\\delta)} and z^{2}{(F_{c},\\delta)} = F_{c} \\delta z{(F_{c},\\delta)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Function('m')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('m')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True), Function('m')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('z')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('z')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True), Function('z')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(r)} = e^{r} and s{(r)} = e^{r}, then obtain \\int \\operatorname{c_{0}}^{r}{(r)} dr = \\int s^{r}{(r)} dr", "derivation": "\\operatorname{c_{0}}{(r)} = e^{r} and \\operatorname{c_{0}}^{r}{(r)} = (e^{r})^{r} and s{(r)} = e^{r} and \\operatorname{c_{0}}^{r}{(r)} = s^{r}{(r)} and \\int \\operatorname{c_{0}}^{r}{(r)} dr = \\int s^{r}{(r)} dr", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('s')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('c_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Function('c_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\lambda,i)} = i^{\\lambda}, then derive (\\lambda + \\int i^{\\lambda} di) (- \\frac{\\lambda i^{\\lambda}}{i} + \\frac{\\partial}{\\partial i} \\operatorname{f^{*}}{(\\lambda,i)}) = 0, then obtain (\\lambda + \\int i^{\\lambda} di) (- \\frac{\\lambda i^{\\lambda}}{i} + \\frac{\\partial}{\\partial i} i^{\\lambda}) = 0", "derivation": "\\operatorname{f^{*}}{(\\lambda,i)} = i^{\\lambda} and - i^{\\lambda} + \\operatorname{f^{*}}{(\\lambda,i)} = 0 and \\frac{\\partial}{\\partial i} (- i^{\\lambda} + \\operatorname{f^{*}}{(\\lambda,i)}) = \\frac{d}{d i} 0 and (\\lambda + \\int i^{\\lambda} di) \\frac{\\partial}{\\partial i} (- i^{\\lambda} + \\operatorname{f^{*}}{(\\lambda,i)}) = (\\lambda + \\int i^{\\lambda} di) \\frac{d}{d i} 0 and (\\lambda + \\int i^{\\lambda} di) (- \\frac{\\lambda i^{\\lambda}}{i} + \\frac{\\partial}{\\partial i} \\operatorname{f^{*}}{(\\lambda,i)}) = 0 and (\\lambda + \\int i^{\\lambda} di) (- \\frac{\\lambda i^{\\lambda}}{i} + \\frac{\\partial}{\\partial i} i^{\\lambda}) = 0", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["minus", 1, "Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('i', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["times", 3, "Add(Symbol('\\\\lambda', commutative=True), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True)))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Add(Symbol('\\\\lambda', commutative=True), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))), Derivative(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True))), Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{E}{(c,y^{\\prime})} = - c + y^{\\prime}, then obtain (\\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + \\frac{\\mathbf{E}{(c,y^{\\prime})}}{- c + y^{\\prime}}))^{y^{\\prime}} = (\\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + 1))^{y^{\\prime}}", "derivation": "\\mathbf{E}{(c,y^{\\prime})} = - c + y^{\\prime} and (- c + y^{\\prime}) \\mathbf{E}{(c,y^{\\prime})} = (- c + y^{\\prime})^{2} and \\frac{\\mathbf{E}{(c,y^{\\prime})}}{- c + y^{\\prime}} = 1 and c - y^{\\prime} + \\frac{\\mathbf{E}{(c,y^{\\prime})}}{- c + y^{\\prime}} = c - y^{\\prime} + 1 and \\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + \\frac{\\mathbf{E}{(c,y^{\\prime})}}{- c + y^{\\prime}}) = \\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + 1) and (\\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + \\frac{\\mathbf{E}{(c,y^{\\prime})}}{- c + y^{\\prime}}))^{y^{\\prime}} = (\\frac{\\partial}{\\partial y^{\\prime}} (c - y^{\\prime} + 1))^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)))"], [["divide", 2, "Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(1))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('c', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\rho,A_{1})} = A_{1} \\rho, then obtain (\\operatorname{v_{x}}^{A_{1}}{(\\rho,A_{1})})^{\\rho} + 1 = ((A_{1} \\rho)^{A_{1}})^{\\rho} + 1", "derivation": "\\operatorname{v_{x}}{(\\rho,A_{1})} = A_{1} \\rho and \\operatorname{v_{x}}^{A_{1}}{(\\rho,A_{1})} = (A_{1} \\rho)^{A_{1}} and (\\operatorname{v_{x}}^{A_{1}}{(\\rho,A_{1})})^{\\rho} = ((A_{1} \\rho)^{A_{1}})^{\\rho} and (\\operatorname{v_{x}}^{A_{1}}{(\\rho,A_{1})})^{\\rho} + 1 = ((A_{1} \\rho)^{A_{1}})^{\\rho} + 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Mul(Symbol('A_1', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('A_1', commutative=True)))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Function('v_x')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Mul(Symbol('A_1', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('A_1', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["add", 3, 1], "Equality(Add(Pow(Pow(Function('v_x')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(1)), Add(Pow(Pow(Mul(Symbol('A_1', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('A_1', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\sigma_{x}{(t_{2})} = e^{t_{2}}, then obtain - t_{2} + \\sigma_{x}{(t_{2})} - e^{t_{2}} + \\log{(t_{2})} = - t_{2} + \\sigma_{x}{(t_{2})} - e^{t_{2}} + \\log{(t_{2} - \\sigma_{x}{(t_{2})} + e^{t_{2}})}", "derivation": "\\sigma_{x}{(t_{2})} = e^{t_{2}} and t_{2} + \\sigma_{x}{(t_{2})} = t_{2} + e^{t_{2}} and t_{2} = t_{2} - \\sigma_{x}{(t_{2})} + e^{t_{2}} and \\log{(t_{2})} = \\log{(t_{2} - \\sigma_{x}{(t_{2})} + e^{t_{2}})} and - t_{2} + \\sigma_{x}{(t_{2})} - e^{t_{2}} + \\log{(t_{2})} = - t_{2} + \\sigma_{x}{(t_{2})} - e^{t_{2}} + \\log{(t_{2} - \\sigma_{x}{(t_{2})} + e^{t_{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Function('\\\\sigma_x')(Symbol('t_2', commutative=True))), Add(Symbol('t_2', commutative=True), exp(Symbol('t_2', commutative=True))))"], [["minus", 2, "Function('\\\\sigma_x')(Symbol('t_2', commutative=True))"], "Equality(Symbol('t_2', commutative=True), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))))"], [["log", 3], "Equality(log(Symbol('t_2', commutative=True)), log(Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True)))))"], [["minus", 4, "Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('\\\\sigma_x')(Symbol('t_2', commutative=True)), Mul(Integer(-1), exp(Symbol('t_2', commutative=True))), log(Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('\\\\sigma_x')(Symbol('t_2', commutative=True)), Mul(Integer(-1), exp(Symbol('t_2', commutative=True))), log(Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(n)} = e^{n} and S{(n)} = \\mathbf{r}{(n)} + e^{n}, then obtain \\frac{3 n e^{n}}{\\mathbf{r}{(n)} + 2 e^{n}} = \\frac{n (S{(n)} + e^{n})}{\\mathbf{r}{(n)} + 2 e^{n}}", "derivation": "\\mathbf{r}{(n)} = e^{n} and \\mathbf{r}{(n)} + e^{n} = 2 e^{n} and S{(n)} = \\mathbf{r}{(n)} + e^{n} and S{(n)} = 2 \\mathbf{r}{(n)} and S{(n)} = 2 e^{n} and S{(n)} + e^{n} = 2 \\mathbf{r}{(n)} + e^{n} and 3 e^{n} = 2 \\mathbf{r}{(n)} + e^{n} and 3 e^{n} = S{(n)} + e^{n} and \\frac{3 n e^{n}}{\\mathbf{r}{(n)} + 2 e^{n}} = \\frac{n (S{(n)} + e^{n})}{\\mathbf{r}{(n)} + 2 e^{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["add", 1, "exp(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Mul(Integer(2), exp(Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('n', commutative=True)), Add(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('S')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('S')(Symbol('n', commutative=True)), Mul(Integer(2), exp(Symbol('n', commutative=True))))"], [["add", 4, "exp(Symbol('n', commutative=True))"], "Equality(Add(Function('S')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Integer(3), exp(Symbol('n', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('n', commutative=True))), exp(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(3), exp(Symbol('n', commutative=True))), Add(Function('S')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))))"], [["divide", 8, "Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), Mul(Integer(2), exp(Symbol('n', commutative=True)))))"], "Equality(Mul(Integer(3), Symbol('n', commutative=True), Pow(Add(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), Mul(Integer(2), exp(Symbol('n', commutative=True)))), Integer(-1)), exp(Symbol('n', commutative=True))), Mul(Symbol('n', commutative=True), Add(Function('S')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True))), Pow(Add(Function('\\\\mathbf{r}')(Symbol('n', commutative=True)), Mul(Integer(2), exp(Symbol('n', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}_0{(W,u)} = W - u, then obtain \\log{(- W + \\int - \\hat{p}_0^{2}{(W,u)} du)} = \\log{(- W + \\int - (W - u) \\hat{p}_0{(W,u)} du)}", "derivation": "\\hat{p}_0{(W,u)} = W - u and - \\hat{p}_0^{2}{(W,u)} = - (W - u) \\hat{p}_0{(W,u)} and \\int - \\hat{p}_0^{2}{(W,u)} du = \\int - (W - u) \\hat{p}_0{(W,u)} du and - W + \\int - \\hat{p}_0^{2}{(W,u)} du = - W + \\int - (W - u) \\hat{p}_0{(W,u)} du and \\log{(- W + \\int - \\hat{p}_0^{2}{(W,u)} du)} = \\log{(- W + \\int - (W - u) \\hat{p}_0{(W,u)} du)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)), Integer(2))), Mul(Integer(-1), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(-1), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["minus", 3, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Mul(Integer(-1), Pow(Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Mul(Integer(-1), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["log", 4], "Equality(log(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Mul(Integer(-1), Pow(Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True))))), log(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Mul(Integer(-1), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Function('\\\\hat{p}_0')(Symbol('W', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))))"]]}, {"prompt": "Given G{(\\mu_0,\\mathbf{r})} = \\mu_0^{\\mathbf{r}}, then obtain 0 = \\frac{\\int \\mu_0^{\\mathbf{r}} d\\mathbf{r} - \\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r}}{\\log{(\\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r})}}", "derivation": "G{(\\mu_0,\\mathbf{r})} = \\mu_0^{\\mathbf{r}} and \\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r} = \\int \\mu_0^{\\mathbf{r}} d\\mathbf{r} and 0 = \\int \\mu_0^{\\mathbf{r}} d\\mathbf{r} - \\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r} and \\log{(\\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r})} = \\log{(\\int \\mu_0^{\\mathbf{r}} d\\mathbf{r})} and 0 = \\frac{\\int \\mu_0^{\\mathbf{r}} d\\mathbf{r} - \\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r}}{\\log{(\\int \\mu_0^{\\mathbf{r}} d\\mathbf{r})}} and 0 = \\frac{\\int \\mu_0^{\\mathbf{r}} d\\mathbf{r} - \\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r}}{\\log{(\\int G{(\\mu_0,\\mathbf{r})} d\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 2, "Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Integer(0), Add(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))))"], [["log", 2], "Equality(log(Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), log(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["divide", 3, "log(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], "Equality(Integer(0), Mul(Add(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))), Pow(log(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(0), Mul(Add(Integral(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))), Pow(log(Integral(Function('G')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given z{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain z^{2}{(\\mathbf{s})} + 2 e^{\\mathbf{s}} = z{(\\mathbf{s})} e^{\\mathbf{s}} + 2 e^{\\mathbf{s}}", "derivation": "z{(\\mathbf{s})} = e^{\\mathbf{s}} and z^{2}{(\\mathbf{s})} = z{(\\mathbf{s})} e^{\\mathbf{s}} and z^{2}{(\\mathbf{s})} + e^{\\mathbf{s}} = z{(\\mathbf{s})} e^{\\mathbf{s}} + e^{\\mathbf{s}} and z^{2}{(\\mathbf{s})} + 2 e^{\\mathbf{s}} = z{(\\mathbf{s})} e^{\\mathbf{s}} + 2 e^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 1, "Function('z')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Pow(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Pow(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Pow(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Mul(Integer(2), exp(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Function('z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hat{x}_0,v)} = \\hat{x}_0 \\cos{(v)}, then obtain \\int 1 d\\hat{x}_0 + 1 = \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} + \\int 1 d\\hat{x}_0", "derivation": "\\mathbf{J}_f{(\\hat{x}_0,v)} = \\hat{x}_0 \\cos{(v)} and 1 = \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} and \\int 1 d\\hat{x}_0 = \\int \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} d\\hat{x}_0 and \\int \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} d\\hat{x}_0 + 1 = \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} + \\int \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} d\\hat{x}_0 and \\int 1 d\\hat{x}_0 + 1 = \\frac{\\hat{x}_0 \\cos{(v)}}{\\mathbf{J}_f{(\\hat{x}_0,v)}} + \\int 1 d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), cos(Symbol('v', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["add", 2, "Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integral(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{p},\\theta)} = \\theta^{\\mathbf{p}}, then derive 0 = \\frac{\\mathbf{p} \\theta^{\\mathbf{p}}}{\\theta} - \\frac{\\partial}{\\partial \\theta} \\operatorname{v_{2}}{(\\mathbf{p},\\theta)}, then obtain 0 = \\frac{\\mathbf{p} \\theta^{\\mathbf{p}}}{\\theta} - \\frac{\\partial}{\\partial \\theta} \\theta^{\\mathbf{p}}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{p},\\theta)} = \\theta^{\\mathbf{p}} and \\frac{\\partial}{\\partial \\theta} \\operatorname{v_{2}}{(\\mathbf{p},\\theta)} = \\frac{\\partial}{\\partial \\theta} \\theta^{\\mathbf{p}} and 0 = \\frac{\\partial}{\\partial \\theta} \\theta^{\\mathbf{p}} - \\frac{\\partial}{\\partial \\theta} \\operatorname{v_{2}}{(\\mathbf{p},\\theta)} and 0 = \\frac{\\mathbf{p} \\theta^{\\mathbf{p}}}{\\theta} - \\frac{\\partial}{\\partial \\theta} \\operatorname{v_{2}}{(\\mathbf{p},\\theta)} and 0 = \\frac{\\mathbf{p} \\theta^{\\mathbf{p}}}{\\theta} - \\frac{\\partial}{\\partial \\theta} \\theta^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('v_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('v_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Derivative(Function('v_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{x}{(E_{n},\\mathbf{p})} = \\frac{E_{n}}{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})} = \\frac{1}{\\mathbf{p}}, then obtain - \\frac{\\mathbf{p} \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})}}{\\dot{x}{(E_{n},\\mathbf{p})}} = - \\frac{1}{\\dot{x}{(E_{n},\\mathbf{p})}}", "derivation": "\\dot{x}{(E_{n},\\mathbf{p})} = \\frac{E_{n}}{\\mathbf{p}} and \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})} = \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\mathbf{p}} and \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})} = \\frac{1}{\\mathbf{p}} and \\mathbf{p} \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})} = 1 and - \\frac{\\mathbf{p} \\frac{\\partial}{\\partial E_{n}} \\dot{x}{(E_{n},\\mathbf{p})}}{\\dot{x}{(E_{n},\\mathbf{p})}} = - \\frac{1}{\\dot{x}{(E_{n},\\mathbf{p})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["divide", 3, "Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Derivative(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Integer(1))"], [["divide", 4, "Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{A},F_{x})} = \\sin{(F_{x} + \\mathbf{A})}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{S}{(\\mathbf{A},F_{x})} = \\cos{(F_{x} + \\mathbf{A})}, then obtain (\\sin{(F_{x} + \\mathbf{A})} + \\frac{\\partial}{\\partial \\mathbf{A}} \\sin{(F_{x} + \\mathbf{A})})^{\\mathbf{A}} = (\\sin{(F_{x} + \\mathbf{A})} + \\cos{(F_{x} + \\mathbf{A})})^{\\mathbf{A}}", "derivation": "\\mathbf{S}{(\\mathbf{A},F_{x})} = \\sin{(F_{x} + \\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{S}{(\\mathbf{A},F_{x})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\sin{(F_{x} + \\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{S}{(\\mathbf{A},F_{x})} = \\cos{(F_{x} + \\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\sin{(F_{x} + \\mathbf{A})} = \\cos{(F_{x} + \\mathbf{A})} and \\sin{(F_{x} + \\mathbf{A})} + \\frac{\\partial}{\\partial \\mathbf{A}} \\sin{(F_{x} + \\mathbf{A})} = \\sin{(F_{x} + \\mathbf{A})} + \\cos{(F_{x} + \\mathbf{A})} and (\\sin{(F_{x} + \\mathbf{A})} + \\frac{\\partial}{\\partial \\mathbf{A}} \\sin{(F_{x} + \\mathbf{A})})^{\\mathbf{A}} = (\\sin{(F_{x} + \\mathbf{A})} + \\cos{(F_{x} + \\mathbf{A})})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_x', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), cos(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), cos(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 4, "sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Derivative(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Add(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), cos(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Add(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Derivative(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), cos(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given J{(\\pi,\\Psi_{\\lambda})} = - \\pi + e^{\\Psi_{\\lambda}}, then obtain \\pi (- \\pi + e^{\\Psi_{\\lambda}}) (\\pi + J{(\\pi,\\Psi_{\\lambda})}) = \\pi (- \\pi + e^{\\Psi_{\\lambda}}) e^{\\Psi_{\\lambda}}", "derivation": "J{(\\pi,\\Psi_{\\lambda})} = - \\pi + e^{\\Psi_{\\lambda}} and \\pi + J{(\\pi,\\Psi_{\\lambda})} - e^{\\Psi_{\\lambda}} = 0 and \\pi + J{(\\pi,\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\pi (\\pi + J{(\\pi,\\Psi_{\\lambda})}) = \\pi e^{\\Psi_{\\lambda}} and \\pi (- \\pi + e^{\\Psi_{\\lambda}}) (\\pi + J{(\\pi,\\Psi_{\\lambda})}) = \\pi (- \\pi + e^{\\Psi_{\\lambda}}) e^{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('J')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(0))"], [["add", 2, "exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('J')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\pi', commutative=True), Function('J')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Function('J')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + k, then obtain \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} \\int (k + \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})}) dk = \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} \\int (- \\hat{\\mathbf{x}} + 2 k) dk", "derivation": "\\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + k and k + \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + 2 k and \\int (k + \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})}) dk = \\int (- \\hat{\\mathbf{x}} + 2 k) dk and (- \\hat{\\mathbf{x}} + k) \\int (k + \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})}) dk = (- \\hat{\\mathbf{x}} + k) \\int (- \\hat{\\mathbf{x}} + 2 k) dk and \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} \\int (k + \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})}) dk = \\operatorname{m_{s}}{(k,\\hat{\\mathbf{x}})} \\int (- \\hat{\\mathbf{x}} + 2 k) dk", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Symbol('k', commutative=True), Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('k', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('k', commutative=True)), Integral(Add(Symbol('k', commutative=True), Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integral(Add(Symbol('k', commutative=True), Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Function('m_s')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(M,\\mathbf{J})} = M + \\mathbf{J}, then obtain F_{c} + M + \\mathbf{J} + \\operatorname{P_{g}}^{M}{(M,\\mathbf{J})} = F_{c} + M + \\mathbf{J} + (M + \\mathbf{J})^{M}", "derivation": "\\operatorname{P_{g}}{(M,\\mathbf{J})} = M + \\mathbf{J} and \\operatorname{P_{g}}^{M}{(M,\\mathbf{J})} = (M + \\mathbf{J})^{M} and F_{c} + \\operatorname{P_{g}}{(M,\\mathbf{J})} = F_{c} + M + \\mathbf{J} and F_{c} + \\operatorname{P_{g}}{(M,\\mathbf{J})} + \\operatorname{P_{g}}^{M}{(M,\\mathbf{J})} = F_{c} + (M + \\mathbf{J})^{M} + \\operatorname{P_{g}}{(M,\\mathbf{J})} and F_{c} + M + \\mathbf{J} + \\operatorname{P_{g}}^{M}{(M,\\mathbf{J})} = F_{c} + M + \\mathbf{J} + (M + \\mathbf{J})^{M}", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True)), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True)))"], [["add", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('F_c', commutative=True), Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 2, "Add(Symbol('F_c', commutative=True), Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Symbol('F_c', commutative=True), Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True))), Add(Symbol('F_c', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True)), Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('F_c', commutative=True), Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('P_g')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True))), Add(Symbol('F_c', commutative=True), Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('M', commutative=True))))"]]}, {"prompt": "Given n{(f^{*},\\mathbf{s})} = (f^{*})^{\\mathbf{s}} and \\mathbb{I}{(f^{*},\\mathbf{s})} = \\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})}, then obtain e^{\\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})}} = e^{\\mathbb{I}{(f^{*},\\mathbf{s})}}", "derivation": "n{(f^{*},\\mathbf{s})} = (f^{*})^{\\mathbf{s}} and \\sin{(n{(f^{*},\\mathbf{s})})} = \\sin{((f^{*})^{\\mathbf{s}})} and \\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})} = \\sin{(\\sin{((f^{*})^{\\mathbf{s}})})} and e^{\\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})}} = e^{\\sin{(\\sin{((f^{*})^{\\mathbf{s}})})}} and \\mathbb{I}{(f^{*},\\mathbf{s})} = \\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})} and \\mathbb{I}{(f^{*},\\mathbf{s})} = \\sin{(\\sin{((f^{*})^{\\mathbf{s}})})} and e^{\\sin{(\\sin{(n{(f^{*},\\mathbf{s})})})}} = e^{\\mathbb{I}{(f^{*},\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), sin(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["sin", 2], "Equality(sin(sin(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), sin(sin(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["exp", 3], "Equality(exp(sin(sin(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))), exp(sin(sin(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), sin(sin(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), sin(sin(Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(exp(sin(sin(Function('n')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))), exp(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{J}_M,v_{2},\\dot{y})} = (\\mathbf{J}_M + v_{2})^{\\dot{y}}, then obtain \\int (\\operatorname{F_{N}}^{\\dot{y}}{(\\mathbf{J}_M,v_{2},\\dot{y})})^{\\dot{y}} d\\mathbf{J}_M = \\int (((\\mathbf{J}_M + v_{2})^{\\dot{y}})^{\\dot{y}})^{\\dot{y}} d\\mathbf{J}_M", "derivation": "\\operatorname{F_{N}}{(\\mathbf{J}_M,v_{2},\\dot{y})} = (\\mathbf{J}_M + v_{2})^{\\dot{y}} and \\operatorname{F_{N}}^{\\dot{y}}{(\\mathbf{J}_M,v_{2},\\dot{y})} = ((\\mathbf{J}_M + v_{2})^{\\dot{y}})^{\\dot{y}} and (\\operatorname{F_{N}}^{\\dot{y}}{(\\mathbf{J}_M,v_{2},\\dot{y})})^{\\dot{y}} = (((\\mathbf{J}_M + v_{2})^{\\dot{y}})^{\\dot{y}})^{\\dot{y}} and \\int (\\operatorname{F_{N}}^{\\dot{y}}{(\\mathbf{J}_M,v_{2},\\dot{y})})^{\\dot{y}} d\\mathbf{J}_M = \\int (((\\mathbf{J}_M + v_{2})^{\\dot{y}})^{\\dot{y}})^{\\dot{y}} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Pow(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Pow(Pow(Function('F_N')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(Pow(Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given A{(S,Q)} = Q + S and \\rho_{b}{(S,Q)} = Q + S, then obtain (A{(S,Q)} + \\frac{\\partial}{\\partial S} \\rho_{b}{(S,Q)}) \\cos{(Q + S)} = (A{(S,Q)} + \\frac{\\partial}{\\partial S} A{(S,Q)}) \\cos{(Q + S)}", "derivation": "A{(S,Q)} = Q + S and \\rho_{b}{(S,Q)} = Q + S and \\frac{\\partial}{\\partial S} \\rho_{b}{(S,Q)} = \\frac{\\partial}{\\partial S} (Q + S) and \\frac{\\partial}{\\partial S} \\rho_{b}{(S,Q)} = \\frac{\\partial}{\\partial S} A{(S,Q)} and \\frac{\\partial}{\\partial S} (Q + S) = \\frac{\\partial}{\\partial S} A{(S,Q)} and A{(S,Q)} + \\frac{\\partial}{\\partial S} (Q + S) = A{(S,Q)} + \\frac{\\partial}{\\partial S} A{(S,Q)} and A{(S,Q)} + \\frac{\\partial}{\\partial S} \\rho_{b}{(S,Q)} = A{(S,Q)} + \\frac{\\partial}{\\partial S} A{(S,Q)} and (A{(S,Q)} + \\frac{\\partial}{\\partial S} \\rho_{b}{(S,Q)}) \\cos{(Q + S)} = (A{(S,Q)} + \\frac{\\partial}{\\partial S} A{(S,Q)}) \\cos{(Q + S)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\rho_b')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 5, "Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["times", 7, "cos(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)))"], "Equality(Mul(Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('\\\\rho_b')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), cos(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)))), Mul(Add(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('A')(Symbol('S', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), cos(Add(Symbol('Q', commutative=True), Symbol('S', commutative=True)))))"]]}, {"prompt": "Given W{(f^{*},F_{H})} = \\cos{((f^{*})^{F_{H}})}, then obtain (\\int W{(f^{*},F_{H})} df^{*})^{f^{*}} + \\frac{\\phi_2^{2}}{\\hat{x}} = (\\int \\cos{((f^{*})^{F_{H}})} df^{*})^{f^{*}} + \\frac{\\phi_2^{2}}{\\hat{x}}", "derivation": "W{(f^{*},F_{H})} = \\cos{((f^{*})^{F_{H}})} and \\int W{(f^{*},F_{H})} df^{*} = \\int \\cos{((f^{*})^{F_{H}})} df^{*} and (\\int W{(f^{*},F_{H})} df^{*})^{f^{*}} = (\\int \\cos{((f^{*})^{F_{H}})} df^{*})^{f^{*}} and (\\int W{(f^{*},F_{H})} df^{*})^{f^{*}} + \\frac{\\phi_2^{2}}{\\hat{x}} = (\\int \\cos{((f^{*})^{F_{H}})} df^{*})^{f^{*}} + \\frac{\\phi_2^{2}}{\\hat{x}}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True)), cos(Pow(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True))))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('W')(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(cos(Pow(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Integral(cos(Pow(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["add", 3, "Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))"], "Equality(Add(Pow(Integral(Function('W')(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))), Add(Pow(Integral(cos(Pow(Symbol('f^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given b{(\\mathbf{f})} = e^{\\mathbf{f}}, then obtain e^{- b^{\\mathbf{f}}{(\\mathbf{f})}} = e^{- \\frac{e^{\\mathbf{f}} (e^{\\mathbf{f}})^{\\mathbf{f}}}{b{(\\mathbf{f})}}}", "derivation": "b{(\\mathbf{f})} = e^{\\mathbf{f}} and b^{\\mathbf{f}}{(\\mathbf{f})} = (e^{\\mathbf{f}})^{\\mathbf{f}} and 1 = \\frac{e^{\\mathbf{f}}}{b{(\\mathbf{f})}} and (e^{\\mathbf{f}})^{\\mathbf{f}} = \\frac{e^{\\mathbf{f}} (e^{\\mathbf{f}})^{\\mathbf{f}}}{b{(\\mathbf{f})}} and - b^{\\mathbf{f}}{(\\mathbf{f})} = - (e^{\\mathbf{f}})^{\\mathbf{f}} and e^{- b^{\\mathbf{f}}{(\\mathbf{f})}} = e^{- (e^{\\mathbf{f}})^{\\mathbf{f}}} and e^{- b^{\\mathbf{f}}{(\\mathbf{f})}} = e^{- \\frac{e^{\\mathbf{f}} (e^{\\mathbf{f}})^{\\mathbf{f}}}{b{(\\mathbf{f})}}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 1, "Function('b')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 3, "Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), exp(Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(exp(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), exp(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\psi,f^{*})} = e^{\\psi + f^{*}}, then obtain 2 \\psi + 2 f^{*} + (\\hat{x}^{\\psi}{(\\psi,f^{*})})^{\\psi} + 2 e^{\\psi + f^{*}} = 2 \\psi + 2 f^{*} + ((e^{\\psi + f^{*}})^{\\psi})^{\\psi} + 2 e^{\\psi + f^{*}}", "derivation": "\\hat{x}{(\\psi,f^{*})} = e^{\\psi + f^{*}} and \\hat{x}^{\\psi}{(\\psi,f^{*})} = (e^{\\psi + f^{*}})^{\\psi} and (\\hat{x}^{\\psi}{(\\psi,f^{*})})^{\\psi} = ((e^{\\psi + f^{*}})^{\\psi})^{\\psi} and 2 \\psi + 2 f^{*} + (\\hat{x}^{\\psi}{(\\psi,f^{*})})^{\\psi} + 2 e^{\\psi + f^{*}} = 2 \\psi + 2 f^{*} + ((e^{\\psi + f^{*}})^{\\psi})^{\\psi} + 2 e^{\\psi + f^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["add", 3, "Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)), Mul(Integer(2), exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)), Pow(Pow(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))))), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Symbol('f^*', commutative=True)), Pow(Pow(exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), exp(Add(Symbol('\\\\psi', commutative=True), Symbol('f^*', commutative=True))))))"]]}, {"prompt": "Given \\theta{(f^{*},\\eta)} = \\frac{\\eta}{f^{*}}, then derive (\\frac{\\partial}{\\partial \\eta} \\theta{(f^{*},\\eta)})^{f^{*}} = (\\frac{1}{f^{*}})^{f^{*}}, then obtain (\\frac{1}{f^{*}})^{f^{*}} = (\\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{f^{*}})^{f^{*}}", "derivation": "\\theta{(f^{*},\\eta)} = \\frac{\\eta}{f^{*}} and \\frac{\\partial}{\\partial \\eta} \\theta{(f^{*},\\eta)} = \\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{f^{*}} and (\\frac{\\partial}{\\partial \\eta} \\theta{(f^{*},\\eta)})^{f^{*}} = (\\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{f^{*}})^{f^{*}} and (\\frac{\\partial}{\\partial \\eta} \\theta{(f^{*},\\eta)})^{f^{*}} = (\\frac{1}{f^{*}})^{f^{*}} and (\\frac{1}{f^{*}})^{f^{*}} = (\\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{f^{*}})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\theta')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\theta')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(P_{e})} = \\int \\sin{(P_{e})} dP_{e}, then derive \\operatorname{f^{\\prime}}{(P_{e})} = \\mathbf{E} - \\cos{(P_{e})}, then obtain 2 \\mathbf{E} - 2 \\cos{(P_{e})} + \\int \\sin{(P_{e})} dP_{e} = \\mathbf{E} + \\operatorname{f^{\\prime}}{(P_{e})} - \\cos{(P_{e})} + \\int \\sin{(P_{e})} dP_{e}", "derivation": "\\operatorname{f^{\\prime}}{(P_{e})} = \\int \\sin{(P_{e})} dP_{e} and \\operatorname{f^{\\prime}}{(P_{e})} + \\int \\sin{(P_{e})} dP_{e} = 2 \\int \\sin{(P_{e})} dP_{e} and 2 \\operatorname{f^{\\prime}}{(P_{e})} + \\int \\sin{(P_{e})} dP_{e} = \\operatorname{f^{\\prime}}{(P_{e})} + 2 \\int \\sin{(P_{e})} dP_{e} and \\operatorname{f^{\\prime}}{(P_{e})} = \\mathbf{E} - \\cos{(P_{e})} and 2 \\mathbf{E} - 2 \\cos{(P_{e})} + \\int \\sin{(P_{e})} dP_{e} = \\mathbf{E} - \\cos{(P_{e})} + 2 \\int \\sin{(P_{e})} dP_{e} and 2 \\mathbf{E} - 2 \\cos{(P_{e})} + \\int \\sin{(P_{e})} dP_{e} = \\mathbf{E} + \\operatorname{f^{\\prime}}{(P_{e})} - \\cos{(P_{e})} + \\int \\sin{(P_{e})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('P_e', commutative=True)), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["add", 1, "Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('P_e', commutative=True)), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))))"], [["add", 2, "Function('f^{\\\\prime}')(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('P_e', commutative=True))), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Function('f^{\\\\prime}')(Symbol('P_e', commutative=True)), Mul(Integer(2), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))))"], [["evaluate_integrals", 1], "Equality(Function('f^{\\\\prime}')(Symbol('P_e', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('P_e', commutative=True))), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), cos(Symbol('P_e', commutative=True))), Mul(Integer(2), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('P_e', commutative=True))), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('f^{\\\\prime}')(Symbol('P_e', commutative=True)), Mul(Integer(-1), cos(Symbol('P_e', commutative=True))), Integral(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\chi,p)} = \\log{(\\chi p)}, then derive \\sin{(\\frac{\\partial^{2}}{\\partial p\\partial \\chi} \\hat{x}{(\\chi,p)})} = 0, then obtain \\frac{\\partial}{\\partial p} \\sin{(\\frac{\\partial^{2}}{\\partial p\\partial \\chi} \\log{(\\chi p)})} = \\frac{d}{d p} 0", "derivation": "\\hat{x}{(\\chi,p)} = \\log{(\\chi p)} and \\frac{\\partial}{\\partial p} \\hat{x}{(\\chi,p)} = \\frac{\\partial}{\\partial p} \\log{(\\chi p)} and \\frac{\\partial^{2}}{\\partial \\chi\\partial p} \\hat{x}{(\\chi,p)} = \\frac{\\partial^{2}}{\\partial \\chi\\partial p} \\log{(\\chi p)} and \\sin{(\\frac{\\partial^{2}}{\\partial \\chi\\partial p} \\hat{x}{(\\chi,p)})} = \\sin{(\\frac{\\partial^{2}}{\\partial \\chi\\partial p} \\log{(\\chi p)})} and \\sin{(\\frac{\\partial^{2}}{\\partial p\\partial \\chi} \\hat{x}{(\\chi,p)})} = 0 and \\frac{\\partial}{\\partial p} \\sin{(\\frac{\\partial^{2}}{\\partial p\\partial \\chi} \\hat{x}{(\\chi,p)})} = \\frac{d}{d p} 0 and \\frac{\\partial}{\\partial p} \\sin{(\\frac{\\partial^{2}}{\\partial p\\partial \\chi} \\log{(\\chi p)})} = \\frac{d}{d p} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), log(Mul(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), sin(Derivative(log(Mul(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(sin(Derivative(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('p', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(sin(Derivative(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('p', commutative=True), Integer(1)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(sin(Derivative(log(Mul(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('p', commutative=True), Integer(1)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\pi)} = \\cos{(\\pi)}, then derive \\int (\\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi) d\\pi = \\frac{\\pi^{2}}{2} + \\pi \\sin{(\\pi)} + n_{1} + \\cos{(\\pi)}, then obtain \\int (\\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi) d\\pi = \\frac{\\pi^{2}}{2} + \\pi \\sin{(\\pi)} + n_{1} + \\hat{\\mathbf{x}}{(\\pi)}", "derivation": "\\hat{\\mathbf{x}}{(\\pi)} = \\cos{(\\pi)} and \\pi \\hat{\\mathbf{x}}{(\\pi)} = \\pi \\cos{(\\pi)} and \\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi = \\pi \\cos{(\\pi)} + \\pi and \\int (\\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi) d\\pi = \\int (\\pi \\cos{(\\pi)} + \\pi) d\\pi and \\int (\\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi) d\\pi = \\frac{\\pi^{2}}{2} + \\pi \\sin{(\\pi)} + n_{1} + \\cos{(\\pi)} and \\int (\\pi \\hat{\\mathbf{x}}{(\\pi)} + \\pi) d\\pi = \\frac{\\pi^{2}}{2} + \\pi \\sin{(\\pi)} + n_{1} + \\hat{\\mathbf{x}}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Add(Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Symbol('n_1', commutative=True), cos(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Symbol('n_1', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\Psi)} = \\sin{(\\Psi)}, then obtain - \\Psi + 2 \\Omega{(\\Psi)} = - \\Psi + 2 \\sin{(\\Psi)}", "derivation": "\\Omega{(\\Psi)} = \\sin{(\\Psi)} and - \\Psi + \\Omega{(\\Psi)} = - \\Psi + \\sin{(\\Psi)} and - \\Psi + 2 \\Omega{(\\Psi)} = - \\Psi + \\Omega{(\\Psi)} + \\sin{(\\Psi)} and - \\Psi + 2 \\Omega{(\\Psi)} = - \\Psi + 2 \\sin{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(a,p)} = \\frac{p}{a}, then obtain a - v_{z}^{\\mathbf{r}} + ((\\int (a + \\mathbf{F}{(a,p)}) da)^{a})^{p} + \\frac{p}{a} = a - v_{z}^{\\mathbf{r}} + ((\\int (a + \\frac{p}{a}) da)^{a})^{p} + \\frac{p}{a}", "derivation": "\\mathbf{F}{(a,p)} = \\frac{p}{a} and a + \\mathbf{F}{(a,p)} = a + \\frac{p}{a} and \\int (a + \\mathbf{F}{(a,p)}) da = \\int (a + \\frac{p}{a}) da and (\\int (a + \\mathbf{F}{(a,p)}) da)^{a} = (\\int (a + \\frac{p}{a}) da)^{a} and ((\\int (a + \\mathbf{F}{(a,p)}) da)^{a})^{p} = ((\\int (a + \\frac{p}{a}) da)^{a})^{p} and a + ((\\int (a + \\mathbf{F}{(a,p)}) da)^{a})^{p} + \\frac{p}{a} = a + ((\\int (a + \\frac{p}{a}) da)^{a})^{p} + \\frac{p}{a} and a - v_{z}^{\\mathbf{r}} + ((\\int (a + \\mathbf{F}{(a,p)}) da)^{a})^{p} + \\frac{p}{a} = a - v_{z}^{\\mathbf{r}} + ((\\int (a + \\frac{p}{a}) da)^{a})^{p} + \\frac{p}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Pow(Integral(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(Integral(Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)))"], [["add", 5, "Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], "Equality(Add(Symbol('a', commutative=True), Pow(Pow(Integral(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Add(Symbol('a', commutative=True), Pow(Pow(Integral(Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["minus", 6, "Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Pow(Integral(Add(Symbol('a', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Pow(Integral(Add(Symbol('a', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('p', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\nabla{(c,t)} = t \\sin{(c)}, then derive \\frac{\\partial}{\\partial t} \\nabla{(c,t)} = \\sin{(c)}, then obtain \\frac{\\int \\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)} dt}{\\sin^{2}{(c)}} = \\frac{\\int \\sin^{2}{(c)} dt}{\\sin^{2}{(c)}}", "derivation": "\\nabla{(c,t)} = t \\sin{(c)} and \\frac{\\partial}{\\partial t} \\nabla{(c,t)} = \\frac{\\partial}{\\partial t} t \\sin{(c)} and \\frac{\\partial}{\\partial t} \\nabla{(c,t)} = \\sin{(c)} and \\frac{\\partial}{\\partial t} t \\sin{(c)} = \\sin{(c)} and \\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)} = \\sin^{2}{(c)} and \\int \\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)} dt = \\int \\sin^{2}{(c)} dt and \\frac{\\int \\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)} dt}{\\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)}} = \\frac{\\int \\sin^{2}{(c)} dt}{\\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)}} and \\frac{\\int \\sin{(c)} \\frac{\\partial}{\\partial t} t \\sin{(c)} dt}{\\sin^{2}{(c)}} = \\frac{\\int \\sin^{2}{(c)} dt}{\\sin^{2}{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('c', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('c', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('c', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), sin(Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), sin(Symbol('c', commutative=True)))"], [["times", 4, "sin(Symbol('c', commutative=True))"], "Equality(Mul(sin(Symbol('c', commutative=True)), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Pow(sin(Symbol('c', commutative=True)), Integer(2)))"], [["integrate", 5, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(sin(Symbol('c', commutative=True)), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True))), Integral(Pow(sin(Symbol('c', commutative=True)), Integer(2)), Tuple(Symbol('t', commutative=True))))"], [["divide", 6, "Mul(sin(Symbol('c', commutative=True)), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], "Equality(Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Pow(Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(sin(Symbol('c', commutative=True)), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True)))), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Pow(Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Integral(Pow(sin(Symbol('c', commutative=True)), Integer(2)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-2)), Integral(Mul(sin(Symbol('c', commutative=True)), Derivative(Mul(Symbol('t', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True)))), Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-2)), Integral(Pow(sin(Symbol('c', commutative=True)), Integer(2)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\nabla)} = \\cos{(\\nabla)}, then derive C_{1} + \\phi_{1}{(\\nabla)} = \\theta_2 + \\cos{(\\nabla)}, then obtain \\frac{C_{1} + \\cos{(\\nabla)}}{\\bar{\\h}{(\\nabla,\\theta_2)}} = \\frac{\\theta_2 + \\phi_{1}{(\\nabla)}}{\\bar{\\h}{(\\nabla,\\theta_2)}}", "derivation": "\\phi_{1}{(\\nabla)} = \\cos{(\\nabla)} and \\frac{d}{d \\nabla} \\phi_{1}{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)} and \\int \\frac{d}{d \\nabla} \\phi_{1}{(\\nabla)} d\\nabla = \\int \\frac{d}{d \\nabla} \\cos{(\\nabla)} d\\nabla and C_{1} + \\phi_{1}{(\\nabla)} = \\theta_2 + \\cos{(\\nabla)} and C_{1} + \\phi_{1}{(\\nabla)} = \\theta_2 + \\phi_{1}{(\\nabla)} and \\theta_2 + \\cos{(\\nabla)} = \\theta_2 + \\phi_{1}{(\\nabla)} and C_{1} + \\cos{(\\nabla)} = \\theta_2 + \\cos{(\\nabla)} and C_{1} + \\cos{(\\nabla)} = \\theta_2 + \\phi_{1}{(\\nabla)} and \\frac{C_{1} + \\cos{(\\nabla)}}{\\bar{\\h}{(\\nabla,\\theta_2)}} = \\frac{\\theta_2 + \\phi_{1}{(\\nabla)}}{\\bar{\\h}{(\\nabla,\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('C_1', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Symbol('C_1', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))))"], [["divide", 8, "Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\theta,f_{E})} = \\theta^{f_{E}} and c{(\\theta,f_{E})} = \\mathbf{g}^{\\theta}{(\\theta,f_{E})}, then obtain \\frac{\\partial}{\\partial f_{E}} c{(\\theta,f_{E})} = \\frac{\\partial}{\\partial f_{E}} (\\theta^{f_{E}})^{\\theta}", "derivation": "\\mathbf{g}{(\\theta,f_{E})} = \\theta^{f_{E}} and \\mathbf{g}^{\\theta}{(\\theta,f_{E})} = (\\theta^{f_{E}})^{\\theta} and c{(\\theta,f_{E})} = \\mathbf{g}^{\\theta}{(\\theta,f_{E})} and c{(\\theta,f_{E})} = (\\theta^{f_{E}})^{\\theta} and \\frac{\\partial}{\\partial f_{E}} c{(\\theta,f_{E})} = \\frac{\\partial}{\\partial f_{E}} (\\theta^{f_{E}})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Pow(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('c')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Pow(Pow(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 4, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(f_{E},F_{x})} = F_{x}^{f_{E}}, then obtain \\sin{(\\iint E{(f_{E},F_{x})} dF_{x} dF_{x} - 1)} = \\sin{(\\iint F_{x}^{f_{E}} dF_{x} dF_{x} - 1)}", "derivation": "E{(f_{E},F_{x})} = F_{x}^{f_{E}} and \\int E{(f_{E},F_{x})} dF_{x} = \\int F_{x}^{f_{E}} dF_{x} and \\iint E{(f_{E},F_{x})} dF_{x} dF_{x} = \\iint F_{x}^{f_{E}} dF_{x} dF_{x} and \\iint E{(f_{E},F_{x})} dF_{x} dF_{x} - 1 = \\iint F_{x}^{f_{E}} dF_{x} dF_{x} - 1 and \\sin{(\\iint E{(f_{E},F_{x})} dF_{x} dF_{x} - 1)} = \\sin{(\\iint F_{x}^{f_{E}} dF_{x} dF_{x} - 1)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f_E', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('F_x', commutative=True), Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('E')(Symbol('f_E', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Pow(Symbol('F_x', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["integrate", 2, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('E')(Symbol('f_E', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Pow(Symbol('F_x', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Function('E')(Symbol('f_E', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1)), Add(Integral(Pow(Symbol('F_x', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1)))"], [["sin", 4], "Equality(sin(Add(Integral(Function('E')(Symbol('f_E', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1))), sin(Add(Integral(Pow(Symbol('F_x', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(\\psi)} = \\int \\cos{(\\psi)} d\\psi, then derive \\Omega{(\\psi)} = m + \\sin{(\\psi)}, then obtain \\frac{\\partial}{\\partial \\psi} (- \\psi + m + 3 \\sin{(\\psi)}) = \\frac{d}{d \\psi} (- \\psi + 2 \\sin{(\\psi)} + \\int \\cos{(\\psi)} d\\psi)", "derivation": "\\Omega{(\\psi)} = \\int \\cos{(\\psi)} d\\psi and \\Omega{(\\psi)} = m + \\sin{(\\psi)} and - \\psi + \\Omega{(\\psi)} = - \\psi + \\int \\cos{(\\psi)} d\\psi and - \\psi + \\Omega{(\\psi)} + \\sin{(\\psi)} = - \\psi + \\sin{(\\psi)} + \\int \\cos{(\\psi)} d\\psi and - \\psi + \\Omega{(\\psi)} + 2 \\sin{(\\psi)} = - \\psi + 2 \\sin{(\\psi)} + \\int \\cos{(\\psi)} d\\psi and \\frac{d}{d \\psi} (- \\psi + \\Omega{(\\psi)} + 2 \\sin{(\\psi)}) = \\frac{d}{d \\psi} (- \\psi + 2 \\sin{(\\psi)} + \\int \\cos{(\\psi)} d\\psi) and \\frac{\\partial}{\\partial \\psi} (- \\psi + m + 3 \\sin{(\\psi)}) = \\frac{d}{d \\psi} (- \\psi + 2 \\sin{(\\psi)} + \\int \\cos{(\\psi)} d\\psi)", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\psi', commutative=True)), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\Omega')(Symbol('\\\\psi', commutative=True)), Add(Symbol('m', commutative=True), sin(Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["add", 3, "sin(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["add", 4, "sin(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\psi', commutative=True))), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Omega')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\psi', commutative=True))), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True), Mul(Integer(3), sin(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\psi', commutative=True))), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(n_{1})} = \\log{(n_{1})}, then obtain \\int \\frac{\\int I{(n_{1})} dn_{1}}{I{(n_{1})}} dn_{1} = \\int \\frac{\\int \\log{(n_{1})} dn_{1}}{I{(n_{1})}} dn_{1}", "derivation": "I{(n_{1})} = \\log{(n_{1})} and \\int I{(n_{1})} dn_{1} = \\int \\log{(n_{1})} dn_{1} and \\frac{\\int I{(n_{1})} dn_{1}}{I{(n_{1})}} = \\frac{\\int \\log{(n_{1})} dn_{1}}{I{(n_{1})}} and \\int \\frac{\\int I{(n_{1})} dn_{1}}{I{(n_{1})}} dn_{1} = \\int \\frac{\\int \\log{(n_{1})} dn_{1}}{I{(n_{1})}} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('I')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["divide", 2, "Function('I')(Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Function('I')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(Function('I')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(Pow(Function('I')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["integrate", 3, "Symbol('n_1', commutative=True)"], "Equality(Integral(Mul(Pow(Function('I')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(Function('I')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Pow(Function('I')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(log(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(Z)} = \\cos{(Z)}, then obtain \\frac{d}{d Z} (\\int Z (\\hat{H}_l{(Z)} + \\cos{(Z)}) dZ)^{Z} = \\frac{d}{d Z} (\\int 2 Z \\cos{(Z)} dZ)^{Z}", "derivation": "\\hat{H}_l{(Z)} = \\cos{(Z)} and \\hat{H}_l{(Z)} + \\cos{(Z)} = 2 \\cos{(Z)} and Z (\\hat{H}_l{(Z)} + \\cos{(Z)}) = 2 Z \\cos{(Z)} and \\int Z (\\hat{H}_l{(Z)} + \\cos{(Z)}) dZ = \\int 2 Z \\cos{(Z)} dZ and (\\int Z (\\hat{H}_l{(Z)} + \\cos{(Z)}) dZ)^{Z} = (\\int 2 Z \\cos{(Z)} dZ)^{Z} and \\frac{d}{d Z} (\\int Z (\\hat{H}_l{(Z)} + \\cos{(Z)}) dZ)^{Z} = \\frac{d}{d Z} (\\int 2 Z \\cos{(Z)} dZ)^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["add", 1, "cos(Symbol('Z', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))), Mul(Integer(2), cos(Symbol('Z', commutative=True))))"], [["times", 2, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))), Mul(Integer(2), Symbol('Z', commutative=True), cos(Symbol('Z', commutative=True))))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Integer(2), Symbol('Z', commutative=True), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('Z', commutative=True), Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Integral(Mul(Integer(2), Symbol('Z', commutative=True), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Integral(Mul(Symbol('Z', commutative=True), Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Integer(2), Symbol('Z', commutative=True), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(A_{y})} = \\cos{(A_{y})} and \\mathbf{M}{(A_{y})} = (\\cos^{A_{y}}{(A_{y})})^{A_{y}}, then obtain ((r^{A_{y}}{(A_{y})})^{A_{y}})^{A_{y}} = \\mathbf{M}^{A_{y}}{(A_{y})}", "derivation": "r{(A_{y})} = \\cos{(A_{y})} and r^{A_{y}}{(A_{y})} = \\cos^{A_{y}}{(A_{y})} and (r^{A_{y}}{(A_{y})})^{A_{y}} = (\\cos^{A_{y}}{(A_{y})})^{A_{y}} and \\mathbf{M}{(A_{y})} = (\\cos^{A_{y}}{(A_{y})})^{A_{y}} and (r^{A_{y}}{(A_{y})})^{A_{y}} = \\mathbf{M}{(A_{y})} and ((r^{A_{y}}{(A_{y})})^{A_{y}})^{A_{y}} = \\mathbf{M}^{A_{y}}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('r')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Pow(Function('r')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Pow(cos(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('A_y', commutative=True)), Pow(Pow(cos(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('r')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Function('\\\\mathbf{M}')(Symbol('A_y', commutative=True)))"], [["power", 5, "Symbol('A_y', commutative=True)"], "Equality(Pow(Pow(Pow(Function('r')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Function('\\\\mathbf{M}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(W,P_{e})} = \\frac{\\sin{(W)}}{P_{e}}, then derive \\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} + 1 = 1 + \\frac{\\cos{(W)}}{P_{e}}, then obtain \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} \\frac{\\sin{(W)}}{P_{e}} - \\frac{\\cos{(W)}}{P_{e}}) = \\frac{d}{d W} 0", "derivation": "\\theta_{1}{(W,P_{e})} = \\frac{\\sin{(W)}}{P_{e}} and \\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} = \\frac{\\partial}{\\partial W} \\frac{\\sin{(W)}}{P_{e}} and \\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} + 1 = \\frac{\\partial}{\\partial W} \\frac{\\sin{(W)}}{P_{e}} + 1 and \\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} + 1 = 1 + \\frac{\\cos{(W)}}{P_{e}} and \\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} - \\frac{\\cos{(W)}}{P_{e}} = 0 and \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} \\theta_{1}{(W,P_{e})} - \\frac{\\cos{(W)}}{P_{e}}) = \\frac{d}{d W} 0 and \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} \\frac{\\sin{(W)}}{P_{e}} - \\frac{\\cos{(W)}}{P_{e}}) = \\frac{d}{d W} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), sin(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))))"], [["minus", 4, "Add(Integer(1), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True))))"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))), Integer(0))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\theta_1')(Symbol('W', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Derivative(Add(Derivative(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} e^{g_{\\varepsilon}} and l{(g_{\\varepsilon})} = g{(g_{\\varepsilon})} - 1, then obtain - g_{\\varepsilon} + l{(g_{\\varepsilon})} = - g_{\\varepsilon} + e^{g_{\\varepsilon}} - 1", "derivation": "g{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} e^{g_{\\varepsilon}} and g{(g_{\\varepsilon})} - 1 = \\frac{d}{d g_{\\varepsilon}} e^{g_{\\varepsilon}} - 1 and l{(g_{\\varepsilon})} = g{(g_{\\varepsilon})} - 1 and - g_{\\varepsilon} + l{(g_{\\varepsilon})} = - g_{\\varepsilon} + g{(g_{\\varepsilon})} - 1 and - g_{\\varepsilon} + l{(g_{\\varepsilon})} = - g_{\\varepsilon} + \\frac{d}{d g_{\\varepsilon}} e^{g_{\\varepsilon}} - 1 and - g_{\\varepsilon} + l{(g_{\\varepsilon})} = - g_{\\varepsilon} + e^{g_{\\varepsilon}} - 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Derivative(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["minus", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\theta_{2}{(f^{*},A_{1})} = \\frac{\\cos{(A_{1})}}{f^{*}}, then obtain \\cos{(f^{*} + \\theta_{2}{(f^{*},A_{1})})} = \\cos{(f^{*} + \\frac{\\cos{(A_{1})}}{f^{*}})}", "derivation": "\\theta_{2}{(f^{*},A_{1})} = \\frac{\\cos{(A_{1})}}{f^{*}} and - \\theta_{2}{(f^{*},A_{1})} = - \\frac{\\cos{(A_{1})}}{f^{*}} and - f^{*} - \\theta_{2}{(f^{*},A_{1})} = - f^{*} - \\frac{\\cos{(A_{1})}}{f^{*}} and \\cos{(f^{*} + \\theta_{2}{(f^{*},A_{1})})} = \\cos{(f^{*} + \\frac{\\cos{(A_{1})}}{f^{*}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('f^*', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('f^*', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))))"], [["minus", 2, "Symbol('f^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('f^*', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Symbol('f^*', commutative=True), Function('\\\\theta_2')(Symbol('f^*', commutative=True), Symbol('A_1', commutative=True)))), cos(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} = z^{m_{s}} and m{(z,m_{s})} = z^{m_{s}}, then obtain \\frac{(\\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} + 1) \\frac{d}{d z} 0}{m{(z,m_{s})}} = \\frac{(m{(z,m_{s})} + 1) \\frac{d}{d z} 0}{m{(z,m_{s})}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} = z^{m_{s}} and \\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} + 1 = z^{m_{s}} + 1 and z^{- m_{s}} (\\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} + 1) = z^{- m_{s}} (z^{m_{s}} + 1) and z^{- m_{s}} (\\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} + 1) \\frac{d}{d z} 0 = z^{- m_{s}} (z^{m_{s}} + 1) \\frac{d}{d z} 0 and m{(z,m_{s})} = z^{m_{s}} and \\frac{(\\operatorname{V_{\\mathbf{E}}}{(z,m_{s})} + 1) \\frac{d}{d z} 0}{m{(z,m_{s})}} = \\frac{(m{(z,m_{s})} + 1) \\frac{d}{d z} 0}{m{(z,m_{s})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Add(Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)))"], [["divide", 2, "Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1))), Mul(Pow(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Add(Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1))))"], [["times", 3, "Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Add(Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('m_s', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Add(Function('V_{\\\\mathbf{E}}')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Pow(Function('m')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Add(Function('m')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Pow(Function('m')(Symbol('z', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu{(A_{2})} = e^{A_{2}} and \\operatorname{f_{E}}{(A_{2},\\eta^{\\prime})} = \\eta^{\\prime} + \\mu{(A_{2})}, then derive \\int \\mu{(A_{2})} dA_{2} = \\eta^{\\prime} + e^{A_{2}}, then obtain \\int \\mu{(A_{2})} dA_{2} = \\operatorname{f_{E}}{(A_{2},\\eta^{\\prime})}", "derivation": "\\mu{(A_{2})} = e^{A_{2}} and \\int \\mu{(A_{2})} dA_{2} = \\int e^{A_{2}} dA_{2} and \\int \\mu{(A_{2})} dA_{2} = \\eta^{\\prime} + e^{A_{2}} and \\int \\mu{(A_{2})} dA_{2} = \\eta^{\\prime} + \\mu{(A_{2})} and \\operatorname{f_{E}}{(A_{2},\\eta^{\\prime})} = \\eta^{\\prime} + \\mu{(A_{2})} and \\int \\mu{(A_{2})} dA_{2} = \\operatorname{f_{E}}{(A_{2},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mu')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('A_2', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('\\\\mu')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Function('f_E')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(\\dot{x})} = \\int e^{\\dot{x}} d\\dot{x}, then derive - a + \\hat{X}{(\\dot{x})} - e^{\\dot{x}} = 0, then obtain - \\frac{- 2 a - z + \\hat{X}{(\\dot{x})} - 2 e^{\\dot{x}} + \\int e^{\\dot{x}} d\\dot{x}}{a} = - \\frac{- a - z + \\hat{X}{(\\dot{x})} - e^{\\dot{x}}}{a}", "derivation": "\\hat{X}{(\\dot{x})} = \\int e^{\\dot{x}} d\\dot{x} and \\hat{X}{(\\dot{x})} - \\int e^{\\dot{x}} d\\dot{x} = 0 and - a + \\hat{X}{(\\dot{x})} - e^{\\dot{x}} = 0 and - a - z + \\hat{X}{(\\dot{x})} - e^{\\dot{x}} = - z and - a - z - e^{\\dot{x}} + \\int e^{\\dot{x}} d\\dot{x} = - z and - 2 a - z + \\hat{X}{(\\dot{x})} - 2 e^{\\dot{x}} + \\int e^{\\dot{x}} d\\dot{x} = - a - z + \\hat{X}{(\\dot{x})} - e^{\\dot{x}} and - \\frac{- 2 a - z + \\hat{X}{(\\dot{x})} - 2 e^{\\dot{x}} + \\int e^{\\dot{x}} d\\dot{x}}{a} = - \\frac{- a - z + \\hat{X}{(\\dot{x})} - e^{\\dot{x}}}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 1, "Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{x}', commutative=True)))), Integer(0))"], [["minus", 3, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{x}', commutative=True))), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\dot{x}', commutative=True))), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{x}', commutative=True)))))"], [["divide", 6, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\dot{x}', commutative=True))), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{x}', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\Psi)} = \\sin{(\\Psi)} and x{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain (x{(\\dot{z})} + \\frac{d}{d \\Psi} \\hat{\\mathbf{r}}{(\\Psi)})^{\\Psi} = (\\sin{(\\dot{z})} + \\frac{d}{d \\Psi} \\hat{\\mathbf{r}}{(\\Psi)})^{\\Psi}", "derivation": "\\hat{\\mathbf{r}}{(\\Psi)} = \\sin{(\\Psi)} and \\frac{d}{d \\Psi} \\hat{\\mathbf{r}}{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)} and x{(\\dot{z})} = \\sin{(\\dot{z})} and x{(\\dot{z})} + \\frac{d}{d \\Psi} \\sin{(\\Psi)} = \\sin{(\\dot{z})} + \\frac{d}{d \\Psi} \\sin{(\\Psi)} and (x{(\\dot{z})} + \\frac{d}{d \\Psi} \\sin{(\\Psi)})^{\\Psi} = (\\sin{(\\dot{z})} + \\frac{d}{d \\Psi} \\sin{(\\Psi)})^{\\Psi} and (x{(\\dot{z})} + \\frac{d}{d \\Psi} \\hat{\\mathbf{r}}{(\\Psi)})^{\\Psi} = (\\sin{(\\dot{z})} + \\frac{d}{d \\Psi} \\hat{\\mathbf{r}}{(\\Psi)})^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('x')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 3, "Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Add(Function('x')(Symbol('\\\\dot{z}', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\dot{z}', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Add(Function('x')(Symbol('\\\\dot{z}', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\Psi', commutative=True)), Pow(Add(sin(Symbol('\\\\dot{z}', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\Psi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Add(Function('x')(Symbol('\\\\dot{z}', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\Psi', commutative=True)), Pow(Add(sin(Symbol('\\\\dot{z}', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given q{(n_{2},\\mathbf{J}_M,a^{\\dagger})} = - \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}}, then obtain \\mathbf{J}_M + \\frac{1}{(- \\mathbf{J}_M + q{(n_{2},\\mathbf{J}_M,a^{\\dagger})})^{2}} - \\frac{n_{2}}{a^{\\dagger}} = \\mathbf{J}_M + \\frac{1}{(- 2 \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}})^{2}} - \\frac{n_{2}}{a^{\\dagger}}", "derivation": "q{(n_{2},\\mathbf{J}_M,a^{\\dagger})} = - \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}} and - \\mathbf{J}_M + q{(n_{2},\\mathbf{J}_M,a^{\\dagger})} = - 2 \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}} and \\frac{1}{(- \\mathbf{J}_M + q{(n_{2},\\mathbf{J}_M,a^{\\dagger})})^{2}} = \\frac{1}{(- 2 \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}})^{2}} and \\mathbf{J}_M + \\frac{1}{(- \\mathbf{J}_M + q{(n_{2},\\mathbf{J}_M,a^{\\dagger})})^{2}} - \\frac{n_{2}}{a^{\\dagger}} = \\mathbf{J}_M + \\frac{1}{(- 2 \\mathbf{J}_M + \\frac{n_{2}}{a^{\\dagger}})^{2}} - \\frac{n_{2}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('q')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('q')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Integer(-2)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('q')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integer(-2)), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Integer(-2)), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\mathbf{S},\\hat{p}_0)} = \\frac{\\hat{p}_0}{\\mathbf{S}}, then obtain \\hat{p}_0 \\mu^{\\mathbf{S}}{(\\mathbf{S},\\hat{p}_0)} = \\hat{p}_0 (\\frac{\\hat{p}_0}{\\mathbf{S}})^{\\mathbf{S}}", "derivation": "\\mu{(\\mathbf{S},\\hat{p}_0)} = \\frac{\\hat{p}_0}{\\mathbf{S}} and \\mu^{\\mathbf{S}}{(\\mathbf{S},\\hat{p}_0)} = (\\frac{\\hat{p}_0}{\\mathbf{S}})^{\\mathbf{S}} and \\mathbf{S} \\mu^{\\mathbf{S}}{(\\mathbf{S},\\hat{p}_0)} = \\mathbf{S} (\\frac{\\hat{p}_0}{\\mathbf{S}})^{\\mathbf{S}} and \\hat{p}_0 \\mu^{\\mathbf{S}}{(\\mathbf{S},\\hat{p}_0)} = \\hat{p}_0 (\\frac{\\hat{p}_0}{\\mathbf{S}})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["divide", 2, "Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 3, "Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given r{(C_{1})} = \\cos{(\\sin{(C_{1})})}, then obtain e^{- (e^{\\int r{(C_{1})} dC_{1}})^{C_{1}}} = e^{- (e^{\\int \\cos{(\\sin{(C_{1})})} dC_{1}})^{C_{1}}}", "derivation": "r{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\int r{(C_{1})} dC_{1} = \\int \\cos{(\\sin{(C_{1})})} dC_{1} and e^{\\int r{(C_{1})} dC_{1}} = e^{\\int \\cos{(\\sin{(C_{1})})} dC_{1}} and (e^{\\int r{(C_{1})} dC_{1}})^{C_{1}} = (e^{\\int \\cos{(\\sin{(C_{1})})} dC_{1}})^{C_{1}} and - (e^{\\int r{(C_{1})} dC_{1}})^{C_{1}} = - (e^{\\int \\cos{(\\sin{(C_{1})})} dC_{1}})^{C_{1}} and e^{- (e^{\\int r{(C_{1})} dC_{1}})^{C_{1}}} = e^{- (e^{\\int \\cos{(\\sin{(C_{1})})} dC_{1}})^{C_{1}}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C_1', commutative=True)), cos(sin(Symbol('C_1', commutative=True))))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('r')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(cos(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('r')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), exp(Integral(cos(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(exp(Integral(Function('r')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)), Pow(exp(Integral(cos(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(exp(Integral(Function('r')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))), Mul(Integer(-1), Pow(exp(Integral(cos(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(Integer(-1), Pow(exp(Integral(Function('r')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))), exp(Mul(Integer(-1), Pow(exp(Integral(cos(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(\\hat{p},y)} = - \\hat{p} + y, then obtain \\frac{\\partial}{\\partial \\hat{p}} (- \\hat{p} + 2 \\rho_{b}{(\\hat{p},y)}) (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)}) = \\frac{\\partial}{\\partial \\hat{p}} (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)})^{2}", "derivation": "\\rho_{b}{(\\hat{p},y)} = - \\hat{p} + y and - \\hat{p} + \\rho_{b}{(\\hat{p},y)} = - 2 \\hat{p} + y and - \\hat{p} + 2 \\rho_{b}{(\\hat{p},y)} = - 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)} and (- \\hat{p} + 2 \\rho_{b}{(\\hat{p},y)}) (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)}) = (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)})^{2} and \\frac{\\partial}{\\partial \\hat{p}} (- \\hat{p} + 2 \\rho_{b}{(\\hat{p},y)}) (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)}) = \\frac{\\partial}{\\partial \\hat{p}} (- 2 \\hat{p} + y + \\rho_{b}{(\\hat{p},y)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True)))"], [["add", 2, "Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True))), Integer(2)))"], [["differentiate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho_b')(Symbol('\\\\hat{p}', commutative=True), Symbol('y', commutative=True))), Integer(2)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(v_{2})} = e^{v_{2}}, then derive \\frac{d}{d v_{2}} \\hat{p}{(v_{2})} = e^{v_{2}}, then obtain \\int \\frac{d}{d v_{2}} \\hat{p}{(v_{2})} dv_{2} = \\int e^{v_{2}} dv_{2}", "derivation": "\\hat{p}{(v_{2})} = e^{v_{2}} and \\frac{d}{d v_{2}} \\hat{p}{(v_{2})} = \\frac{d}{d v_{2}} e^{v_{2}} and \\frac{d}{d v_{2}} \\hat{p}{(v_{2})} = e^{v_{2}} and \\int \\frac{d}{d v_{2}} \\hat{p}{(v_{2})} dv_{2} = \\int e^{v_{2}} dv_{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), exp(Symbol('v_2', commutative=True)))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given H{(\\delta)} = \\log{(\\cos{(\\delta)})}, then obtain \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta + H{(\\delta)}}}{g^{\\prime}_{\\varepsilon} \\operatorname{f^{\\prime}}{(C_{1})}} = \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta} \\cos{(\\delta)}}{g^{\\prime}_{\\varepsilon} \\operatorname{f^{\\prime}}{(C_{1})}}", "derivation": "H{(\\delta)} = \\log{(\\cos{(\\delta)})} and \\delta + H{(\\delta)} = \\delta + \\log{(\\cos{(\\delta)})} and e^{\\delta + H{(\\delta)}} = e^{\\delta} \\cos{(\\delta)} and \\frac{d}{d \\delta} e^{\\delta + H{(\\delta)}} = \\frac{d}{d \\delta} e^{\\delta} \\cos{(\\delta)} and \\delta \\frac{d}{d \\delta} e^{\\delta + H{(\\delta)}} = \\delta \\frac{d}{d \\delta} e^{\\delta} \\cos{(\\delta)} and \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta + H{(\\delta)}}}{g^{\\prime}_{\\varepsilon}} = \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta} \\cos{(\\delta)}}{g^{\\prime}_{\\varepsilon}} and \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta + H{(\\delta)}}}{g^{\\prime}_{\\varepsilon} \\operatorname{f^{\\prime}}{(C_{1})}} = \\frac{\\delta \\frac{d}{d \\delta} e^{\\delta} \\cos{(\\delta)}}{g^{\\prime}_{\\varepsilon} \\operatorname{f^{\\prime}}{(C_{1})}}", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('\\\\delta', commutative=True)), log(cos(Symbol('\\\\delta', commutative=True))))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), log(cos(Symbol('\\\\delta', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True)))), Mul(exp(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["times", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Symbol('\\\\delta', commutative=True), Derivative(Mul(exp(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["divide", 5, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Mul(exp(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["divide", 6, "Function('f^{\\\\prime}')(Symbol('C_1', commutative=True))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('C_1', commutative=True)), Integer(-1)), Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Function('H')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('C_1', commutative=True)), Integer(-1)), Derivative(Mul(exp(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(v)} = \\cos{(v)}, then obtain \\frac{d}{d v} (- v + \\operatorname{F_{N}}{(v)}) \\cos{(v)} = \\frac{d}{d v} (- v + \\cos{(v)}) \\cos{(v)}", "derivation": "\\operatorname{F_{N}}{(v)} = \\cos{(v)} and - v + \\operatorname{F_{N}}{(v)} = - v + \\cos{(v)} and (- v + \\operatorname{F_{N}}{(v)}) \\cos{(v)} = (- v + \\cos{(v)}) \\cos{(v)} and \\frac{d}{d v} (- v + \\operatorname{F_{N}}{(v)}) \\cos{(v)} = \\frac{d}{d v} (- v + \\cos{(v)}) \\cos{(v)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('F_N')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))))"], [["times", 2, "cos(Symbol('v', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('F_N')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('F_N')(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{E})} = \\cos{(\\log{(\\mathbf{E})})}, then derive 0 = - \\frac{d}{d \\mathbf{E}} \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{\\sin{(\\log{(\\mathbf{E})})}}{\\mathbf{E}}, then obtain \\operatorname{C_{d}}{(\\mathbf{E})} = \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{d}{d \\mathbf{E}} \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{\\sin{(\\log{(\\mathbf{E})})}}{\\mathbf{E}}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{E})} = \\cos{(\\log{(\\mathbf{E})})} and 0 = - \\operatorname{C_{d}}{(\\mathbf{E})} + \\cos{(\\log{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} 0 = \\frac{d}{d \\mathbf{E}} (- \\operatorname{C_{d}}{(\\mathbf{E})} + \\cos{(\\log{(\\mathbf{E})})}) and 0 = - \\frac{d}{d \\mathbf{E}} \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{\\sin{(\\log{(\\mathbf{E})})}}{\\mathbf{E}} and \\operatorname{C_{d}}{(\\mathbf{E})} = \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{d}{d \\mathbf{E}} \\operatorname{C_{d}}{(\\mathbf{E})} - \\frac{\\sin{(\\log{(\\mathbf{E})})}}{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True)), cos(log(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 1, "Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True))), cos(log(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True))), cos(log(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["add", 4, "Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True)), Add(Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Derivative(Function('C_d')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\mathbf{E}', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(A,B)} = A^{B}, then obtain - \\theta_{1}{(A,B)} + \\int (- A^{B} + \\theta_{1}{(A,B)}) dB = - \\theta_{1}{(A,B)} + \\int 0 dB", "derivation": "\\theta_{1}{(A,B)} = A^{B} and - A^{B} + \\theta_{1}{(A,B)} = 0 and \\int (- A^{B} + \\theta_{1}{(A,B)}) dB = \\int 0 dB and - A^{B} + \\int (- A^{B} + \\theta_{1}{(A,B)}) dB = - A^{B} + \\int 0 dB and - \\theta_{1}{(A,B)} + \\int (- A^{B} + \\theta_{1}{(A,B)}) dB = - \\theta_{1}{(A,B)} + \\int 0 dB", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True)))"], [["minus", 1, "Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Integer(0), Tuple(Symbol('B', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Integral(Integer(0), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('B', commutative=True))), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('A', commutative=True), Symbol('B', commutative=True))), Integral(Integer(0), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})}, then derive r_{0} \\mathbf{F}{(r_{0})} = - r_{0} \\sin{(r_{0})}, then obtain r_{0} \\mathbf{F}{(r_{0})} - (r_{0} \\mathbf{F}{(r_{0})})^{r_{0}} = - r_{0} \\sin{(r_{0})} - (r_{0} \\mathbf{F}{(r_{0})})^{r_{0}}", "derivation": "\\mathbf{F}{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(r_{0})} and r_{0} \\mathbf{F}{(r_{0})} = r_{0} \\frac{d}{d r_{0}} \\cos{(r_{0})} and r_{0} \\mathbf{F}{(r_{0})} = - r_{0} \\sin{(r_{0})} and (r_{0} \\mathbf{F}{(r_{0})})^{r_{0}} = (- r_{0} \\sin{(r_{0})})^{r_{0}} and r_{0} \\mathbf{F}{(r_{0})} - (- r_{0} \\sin{(r_{0})})^{r_{0}} = - r_{0} \\sin{(r_{0})} - (- r_{0} \\sin{(r_{0})})^{r_{0}} and r_{0} \\mathbf{F}{(r_{0})} - (r_{0} \\mathbf{F}{(r_{0})})^{r_{0}} = - r_{0} \\sin{(r_{0})} - (r_{0} \\mathbf{F}{(r_{0})})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True)), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), Derivative(cos(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Pow(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["minus", 3, "Pow(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True), sin(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})} = e^{\\frac{\\mathbf{r}}{\\eta}}, then obtain \\mathbf{r} + \\frac{\\eta + \\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}} = \\mathbf{r} + \\frac{\\eta + e^{\\frac{\\mathbf{r}}{\\eta}}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})} = e^{\\frac{\\mathbf{r}}{\\eta}} and \\eta + \\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})} = \\eta + e^{\\frac{\\mathbf{r}}{\\eta}} and \\frac{\\eta + \\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}} = \\frac{\\eta + e^{\\frac{\\mathbf{r}}{\\eta}}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}} and \\mathbf{r} + \\frac{\\eta + \\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}} = \\mathbf{r} + \\frac{\\eta + e^{\\frac{\\mathbf{r}}{\\eta}}}{\\operatorname{f_{\\mathbf{v}}}{(\\eta,\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\eta', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["divide", 2, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\eta', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\eta', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"], [["add", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Add(Symbol('\\\\eta', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Add(Symbol('\\\\eta', commutative=True), exp(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\varphi^{*}{(A_{y},\\hat{p}_0)} = e^{\\hat{p}_0^{A_{y}}} and W{(A_{y},\\hat{p}_0)} = e^{\\hat{p}_0^{A_{y}}}, then obtain \\iint W{(A_{y},\\hat{p}_0)} dA_{y} dA_{y} = \\iint e^{\\hat{p}_0^{A_{y}}} dA_{y} dA_{y}", "derivation": "\\varphi^{*}{(A_{y},\\hat{p}_0)} = e^{\\hat{p}_0^{A_{y}}} and \\int \\varphi^{*}{(A_{y},\\hat{p}_0)} dA_{y} = \\int e^{\\hat{p}_0^{A_{y}}} dA_{y} and \\iint \\varphi^{*}{(A_{y},\\hat{p}_0)} dA_{y} dA_{y} = \\iint e^{\\hat{p}_0^{A_{y}}} dA_{y} dA_{y} and W{(A_{y},\\hat{p}_0)} = e^{\\hat{p}_0^{A_{y}}} and W{(A_{y},\\hat{p}_0)} = \\varphi^{*}{(A_{y},\\hat{p}_0)} and \\iint W{(A_{y},\\hat{p}_0)} dA_{y} dA_{y} = \\iint e^{\\hat{p}_0^{A_{y}}} dA_{y} dA_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], ["renaming_premise", "Equality(Function('W')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('W')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\varphi^*')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('W')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(E,C_{2})} = C_{2} E and \\rho_{f}{(E,C_{2})} = C_{2} E, then obtain - \\dot{y}{(E,C_{2})} - \\rho_{f}{(E,C_{2})} + \\rho_{f}^{C_{2}}{(E,C_{2})} = (C_{2} E)^{C_{2}} - \\dot{y}{(E,C_{2})} - \\rho_{f}{(E,C_{2})}", "derivation": "\\dot{y}{(E,C_{2})} = C_{2} E and \\dot{y}^{C_{2}}{(E,C_{2})} = (C_{2} E)^{C_{2}} and \\rho_{f}{(E,C_{2})} = C_{2} E and \\dot{y}{(E,C_{2})} = \\rho_{f}{(E,C_{2})} and \\rho_{f}^{C_{2}}{(E,C_{2})} = (C_{2} E)^{C_{2}} and - \\rho_{f}{(E,C_{2})} + \\rho_{f}^{C_{2}}{(E,C_{2})} = (C_{2} E)^{C_{2}} - \\rho_{f}{(E,C_{2})} and - \\dot{y}{(E,C_{2})} - \\rho_{f}{(E,C_{2})} + \\rho_{f}^{C_{2}}{(E,C_{2})} = (C_{2} E)^{C_{2}} - \\dot{y}{(E,C_{2})} - \\rho_{f}{(E,C_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)), Symbol('C_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)), Symbol('C_2', commutative=True)))"], [["minus", 5, "Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))), Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Pow(Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)), Symbol('C_2', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)))))"], [["minus", 6, "Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))), Pow(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Pow(Mul(Symbol('C_2', commutative=True), Symbol('E', commutative=True)), Symbol('C_2', commutative=True)), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(E)} = e^{E}, then obtain e^{E} + \\frac{d}{d E} - (\\hat{H}_l{(E)} e^{- E})^{E} \\hat{H}_l{(E)} e^{E} = e^{E} + \\frac{d}{d E} - \\hat{H}_l{(E)} e^{E}", "derivation": "\\hat{H}_l{(E)} = e^{E} and \\hat{H}_l{(E)} e^{- E} = 1 and (\\hat{H}_l{(E)} e^{- E})^{E} = 1 and - (\\hat{H}_l{(E)} e^{- E})^{E} \\hat{H}_l{(E)} e^{E} = - \\hat{H}_l{(E)} e^{E} and \\frac{d}{d E} - (\\hat{H}_l{(E)} e^{- E})^{E} \\hat{H}_l{(E)} e^{E} = \\frac{d}{d E} - \\hat{H}_l{(E)} e^{E} and e^{E} + \\frac{d}{d E} - (\\hat{H}_l{(E)} e^{- E})^{E} \\hat{H}_l{(E)} e^{E} = e^{E} + \\frac{d}{d E} - \\hat{H}_l{(E)} e^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["divide", 1, "exp(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), Symbol('E', commutative=True)))), Integer(1))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Integer(1))"], [["times", 3, "Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Mul(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Mul(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["add", 5, "exp(Symbol('E', commutative=True))"], "Equality(Add(exp(Symbol('E', commutative=True)), Derivative(Mul(Integer(-1), Pow(Mul(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(exp(Symbol('E', commutative=True)), Derivative(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(m_{s})} = \\sin{(m_{s})}, then obtain \\frac{I^{2}{(m_{s})} \\sin^{2}{(m_{s})} + I{(m_{s})} \\sin^{3}{(m_{s})}}{I{(m_{s})} \\sin^{3}{(m_{s})}} = 2", "derivation": "I{(m_{s})} = \\sin{(m_{s})} and I^{2}{(m_{s})} = I{(m_{s})} \\sin{(m_{s})} and I^{4}{(m_{s})} = I^{2}{(m_{s})} \\sin^{2}{(m_{s})} and I^{2}{(m_{s})} \\sin^{2}{(m_{s})} = I{(m_{s})} \\sin^{3}{(m_{s})} and I^{4}{(m_{s})} = I{(m_{s})} \\sin^{3}{(m_{s})} and I^{4}{(m_{s})} + I{(m_{s})} \\sin^{3}{(m_{s})} = 2 I{(m_{s})} \\sin^{3}{(m_{s})} and \\frac{I^{4}{(m_{s})} + I{(m_{s})} \\sin^{3}{(m_{s})}}{I{(m_{s})} \\sin^{3}{(m_{s})}} = 2 and \\frac{I^{2}{(m_{s})} \\sin^{2}{(m_{s})} + I{(m_{s})} \\sin^{3}{(m_{s})}}{I{(m_{s})} \\sin^{3}{(m_{s})}} = 2", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["times", 1, "Function('I')(Symbol('m_s', commutative=True))"], "Equality(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(4)), Mul(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2))), Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(4)), Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3))))"], [["add", 5, "Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3)))"], "Equality(Add(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(4)), Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3)))), Mul(Integer(2), Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3))))"], [["divide", 6, "Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3)))"], "Equality(Mul(Add(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(4)), Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3)))), Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(-1)), Pow(sin(Symbol('m_s', commutative=True)), Integer(-3))), Integer(2))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Add(Mul(Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2))), Mul(Function('I')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Integer(3)))), Pow(Function('I')(Symbol('m_s', commutative=True)), Integer(-1)), Pow(sin(Symbol('m_s', commutative=True)), Integer(-3))), Integer(2))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(V,H)} = - H + e^{V}, then obtain \\operatorname{f_{E}}{(V,H)} + 1 = (0^{H})^{V} + \\operatorname{f_{E}}{(V,H)}", "derivation": "\\operatorname{f_{E}}{(V,H)} = - H + e^{V} and H + \\operatorname{f_{E}}{(V,H)} - e^{V} = 0 and (H + \\operatorname{f_{E}}{(V,H)} - e^{V})^{H} = 0^{H} and ((H + \\operatorname{f_{E}}{(V,H)} - e^{V})^{H})^{V} = (0^{H})^{V} and 1 = ((H + \\operatorname{f_{E}}{(V,H)} - e^{V})^{H})^{V} and 1 = (0^{H})^{V} and - H + e^{V} + 1 = - H + (0^{H})^{V} + e^{V} and \\operatorname{f_{E}}{(V,H)} + 1 = (0^{H})^{V} + \\operatorname{f_{E}}{(V,H)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('V', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('V', commutative=True)))"], "Equality(Add(Symbol('H', commutative=True), Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Symbol('H', commutative=True), Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Symbol('H', commutative=True)), Pow(Integer(0), Symbol('H', commutative=True)))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('H', commutative=True), Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Symbol('H', commutative=True)), Symbol('V', commutative=True)), Pow(Pow(Integer(0), Symbol('H', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Symbol('H', commutative=True), Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), exp(Symbol('V', commutative=True)))), Symbol('H', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('H', commutative=True)), Symbol('V', commutative=True)))"], [["add", 6, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('V', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), exp(Symbol('V', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Pow(Integer(0), Symbol('H', commutative=True)), Symbol('V', commutative=True)), exp(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True)), Integer(1)), Add(Pow(Pow(Integer(0), Symbol('H', commutative=True)), Symbol('V', commutative=True)), Function('f_E')(Symbol('V', commutative=True), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} = - \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P, then obtain \\mathbf{D} \\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} - \\mathbf{E} = \\mathbf{D} (- \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P) - \\mathbf{E}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} = - \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P and - \\mathbf{D} \\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} = - \\mathbf{D} (- \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P) and - \\mathbf{D} \\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} + \\mathbf{E} = - \\mathbf{D} (- \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P) + \\mathbf{E} and \\mathbf{D} \\operatorname{v_{z}}{(\\mathbf{E},\\mathbf{D},\\mathbf{J}_P)} - \\mathbf{E} = \\mathbf{D} (- \\mathbf{D} + \\mathbf{E} \\mathbf{J}_P) - \\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Function('v_z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["add", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Function('v_z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('v_z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{H},m)} = \\hat{H} + m, then obtain (\\int \\operatorname{t_{2}}{(\\hat{H},m)} dm)^{m} = (\\int \\hat{H} dm + \\int m dm)^{m}", "derivation": "\\operatorname{t_{2}}{(\\hat{H},m)} = \\hat{H} + m and \\int \\operatorname{t_{2}}{(\\hat{H},m)} dm = \\int (\\hat{H} + m) dm and (\\int \\operatorname{t_{2}}{(\\hat{H},m)} dm)^{m} = (\\int (\\hat{H} + m) dm)^{m} and \\int \\operatorname{t_{2}}{(\\hat{H},m)} dm = \\int \\hat{H} dm + \\int m dm and (\\int \\hat{H} dm + \\int m dm)^{m} = (\\int (\\hat{H} + m) dm)^{m} and (\\int \\operatorname{t_{2}}{(\\hat{H},m)} dm)^{m} = (\\int \\hat{H} dm + \\int m dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["expand", 2], "Equality(Integral(Function('t_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Integral(Symbol('\\\\hat{H}', commutative=True), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Integral(Symbol('\\\\hat{H}', commutative=True), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Add(Integral(Symbol('\\\\hat{H}', commutative=True), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\chi{(v)} = e^{v}, then obtain v \\frac{d}{d v} \\int \\chi{(v)} dv = v \\frac{d}{d v} \\int e^{v} dv", "derivation": "\\chi{(v)} = e^{v} and \\int \\chi{(v)} dv = \\int e^{v} dv and \\frac{d}{d v} \\int \\chi{(v)} dv = \\frac{d}{d v} \\int e^{v} dv and v \\frac{d}{d v} \\int \\chi{(v)} dv = v \\frac{d}{d v} \\int e^{v} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\chi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 3, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Derivative(Integral(Function('\\\\chi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Symbol('v', commutative=True), Derivative(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(f^{*},m)} = \\frac{\\partial}{\\partial f^{*}} (f^{*} + m), then derive \\int b^{m}{(f^{*},m)} dm = W + m, then obtain \\int (\\frac{\\partial}{\\partial f^{*}} (f^{*} + m))^{m} dm - 1 = W + m - 1", "derivation": "b{(f^{*},m)} = \\frac{\\partial}{\\partial f^{*}} (f^{*} + m) and b^{m}{(f^{*},m)} = (\\frac{\\partial}{\\partial f^{*}} (f^{*} + m))^{m} and \\int b^{m}{(f^{*},m)} dm = \\int (\\frac{\\partial}{\\partial f^{*}} (f^{*} + m))^{m} dm and \\int b^{m}{(f^{*},m)} dm = W + m and \\int b^{m}{(f^{*},m)} dm - 1 = W + m - 1 and \\int (\\frac{\\partial}{\\partial f^{*}} (f^{*} + m))^{m} dm - 1 = W + m - 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Derivative(Add(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('b')(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Derivative(Add(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('m', commutative=True)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Function('b')(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Derivative(Add(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Pow(Function('b')(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('W', commutative=True), Symbol('m', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Integral(Pow(Function('b')(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('m', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integral(Pow(Derivative(Add(Symbol('f^*', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('m', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\omega{(p)} = \\cos{(p)}, then obtain \\log{(\\log{(\\omega{(p)})})} - 1 = \\log{(\\log{(\\cos{(p)})})} - 1", "derivation": "\\omega{(p)} = \\cos{(p)} and \\log{(\\omega{(p)})} = \\log{(\\cos{(p)})} and \\log{(\\log{(\\omega{(p)})})} = \\log{(\\log{(\\cos{(p)})})} and \\log{(\\log{(\\omega{(p)})})} - 1 = \\log{(\\log{(\\cos{(p)})})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\omega')(Symbol('p', commutative=True))), log(cos(Symbol('p', commutative=True))))"], [["log", 2], "Equality(log(log(Function('\\\\omega')(Symbol('p', commutative=True)))), log(log(cos(Symbol('p', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(log(log(Function('\\\\omega')(Symbol('p', commutative=True)))), Integer(-1)), Add(log(log(cos(Symbol('p', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given W{(J_{\\varepsilon})} = \\log{(e^{J_{\\varepsilon}})}, then obtain W^{3}{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})} = W^{2}{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})}^{2}", "derivation": "W{(J_{\\varepsilon})} = \\log{(e^{J_{\\varepsilon}})} and W{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})} = \\log{(e^{J_{\\varepsilon}})}^{2} and W^{2}{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})}^{2} = \\log{(e^{J_{\\varepsilon}})}^{4} and W^{3}{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})} = W^{2}{(J_{\\varepsilon})} \\log{(e^{J_{\\varepsilon}})}^{2}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "log(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(2))), Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(3)), log(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Pow(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\phi{(b,\\tilde{g}^*)} = (e^{b})^{\\tilde{g}^*}, then obtain \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{e^{\\phi^{2}{(b,\\tilde{g}^*)}}}{\\phi{(b,\\tilde{g}^*)}} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{e^{\\phi{(b,\\tilde{g}^*)} (e^{b})^{\\tilde{g}^*}}}{\\phi{(b,\\tilde{g}^*)}}", "derivation": "\\phi{(b,\\tilde{g}^*)} = (e^{b})^{\\tilde{g}^*} and \\phi^{2}{(b,\\tilde{g}^*)} = \\phi{(b,\\tilde{g}^*)} (e^{b})^{\\tilde{g}^*} and e^{\\phi^{2}{(b,\\tilde{g}^*)}} = e^{\\phi{(b,\\tilde{g}^*)} (e^{b})^{\\tilde{g}^*}} and \\frac{e^{\\phi^{2}{(b,\\tilde{g}^*)}}}{\\phi{(b,\\tilde{g}^*)}} = \\frac{e^{\\phi{(b,\\tilde{g}^*)} (e^{b})^{\\tilde{g}^*}}}{\\phi{(b,\\tilde{g}^*)}} and \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{e^{\\phi^{2}{(b,\\tilde{g}^*)}}}{\\phi{(b,\\tilde{g}^*)}} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{e^{\\phi{(b,\\tilde{g}^*)} (e^{b})^{\\tilde{g}^*}}}{\\phi{(b,\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))), exp(Mul(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["divide", 3, "Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))), Mul(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Mul(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Mul(Function('\\\\phi')(Symbol('b', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(s,Z)} = s + \\sin{(Z)}, then obtain - \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int (s + \\sin{(Z)}) ds) + \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int k{(s,Z)} ds) = 0", "derivation": "k{(s,Z)} = s + \\sin{(Z)} and \\int k{(s,Z)} ds = \\int (s + \\sin{(Z)}) ds and s + \\sin{(Z)} + \\int k{(s,Z)} ds = s + \\sin{(Z)} + \\int (s + \\sin{(Z)}) ds and k{(s,Z)} + \\int k{(s,Z)} ds = k{(s,Z)} + \\int (s + \\sin{(Z)}) ds and \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int k{(s,Z)} ds) = \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int (s + \\sin{(Z)}) ds) and 2 \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int k{(s,Z)} ds) = \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int (s + \\sin{(Z)}) ds) + \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int k{(s,Z)} ds) and - \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int (s + \\sin{(Z)}) ds) + \\frac{\\partial}{\\partial s} (k{(s,Z)} + \\int k{(s,Z)} ds) = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["add", 2, "Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True)))"], "Equality(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["add", 5, "Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["minus", 6, "Add(Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('s', commutative=True), sin(Symbol('Z', commutative=True))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))), Derivative(Add(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Function('k')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given Z{(\\hat{H})} = e^{\\hat{H}}, then obtain \\cos{(\\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{Z{(\\hat{H})}}{\\hat{H}})} = \\cos{(\\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{e^{\\hat{H}}}{\\hat{H}})}", "derivation": "Z{(\\hat{H})} = e^{\\hat{H}} and \\frac{Z{(\\hat{H})}}{\\hat{H}} = \\frac{e^{\\hat{H}}}{\\hat{H}} and \\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{Z{(\\hat{H})}}{\\hat{H}} = \\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{e^{\\hat{H}}}{\\hat{H}} and \\cos{(\\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{Z{(\\hat{H})}}{\\hat{H}})} = \\cos{(\\operatorname{A_{1}}{(x^\\prime,\\mathbf{M})} + \\frac{e^{\\hat{H}}}{\\hat{H}})}", "srepr_derivation": [["get_premise", "Equality(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('Z')(Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 2, "Function('A_1')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('Z')(Symbol('\\\\hat{H}', commutative=True)))), Add(Function('A_1')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{H}', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Function('A_1')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('Z')(Symbol('\\\\hat{H}', commutative=True))))), cos(Add(Function('A_1')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given s{(c,\\mu)} = \\cos{(\\mu^{c})} and \\operatorname{C_{2}}{(\\mu)} = \\log{(\\mu)}, then derive \\frac{\\partial}{\\partial c} s{(c,\\mu)} = - \\mu^{c} \\log{(\\mu)} \\sin{(\\mu^{c})}, then obtain e^{(\\frac{\\partial}{\\partial c} \\cos{(\\mu^{c})})^{c}} = e^{(- \\mu^{c} \\operatorname{C_{2}}{(\\mu)} \\sin{(\\mu^{c})})^{c}}", "derivation": "s{(c,\\mu)} = \\cos{(\\mu^{c})} and \\frac{\\partial}{\\partial c} s{(c,\\mu)} = \\frac{\\partial}{\\partial c} \\cos{(\\mu^{c})} and \\frac{\\partial}{\\partial c} s{(c,\\mu)} = - \\mu^{c} \\log{(\\mu)} \\sin{(\\mu^{c})} and (\\frac{\\partial}{\\partial c} s{(c,\\mu)})^{c} = (- \\mu^{c} \\log{(\\mu)} \\sin{(\\mu^{c})})^{c} and (\\frac{\\partial}{\\partial c} \\cos{(\\mu^{c})})^{c} = (- \\mu^{c} \\log{(\\mu)} \\sin{(\\mu^{c})})^{c} and \\operatorname{C_{2}}{(\\mu)} = \\log{(\\mu)} and e^{(\\frac{\\partial}{\\partial c} \\cos{(\\mu^{c})})^{c}} = e^{(- \\mu^{c} \\log{(\\mu)} \\sin{(\\mu^{c})})^{c}} and e^{(\\frac{\\partial}{\\partial c} \\cos{(\\mu^{c})})^{c}} = e^{(- \\mu^{c} \\operatorname{C_{2}}{(\\mu)} \\sin{(\\mu^{c})})^{c}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('c', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('c', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('c', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)), log(Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Function('s')(Symbol('c', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)), log(Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)), log(Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Derivative(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True))), exp(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)), log(Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(exp(Pow(Derivative(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True))), exp(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)), Function('C_2')(Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(f_{E})} = \\cos{(f_{E})} and q{(f_{E})} = \\mathbf{H}{(f_{E})} \\cos{(f_{E})}, then obtain 0 = \\frac{- \\mathbf{H}{(f_{E})} \\cos{(f_{E})} + \\cos^{2}{(f_{E})}}{\\mathbf{H}{(f_{E})} \\cos{(f_{E})}}", "derivation": "\\mathbf{H}{(f_{E})} = \\cos{(f_{E})} and \\mathbf{H}{(f_{E})} \\cos{(f_{E})} = \\cos^{2}{(f_{E})} and q{(f_{E})} = \\mathbf{H}{(f_{E})} \\cos{(f_{E})} and 0 = \\mathbf{H}{(f_{E})} \\cos{(f_{E})} - q{(f_{E})} and 0 = \\frac{\\mathbf{H}{(f_{E})} \\cos{(f_{E})} - q{(f_{E})}}{q{(f_{E})}} and 0 = \\frac{- q{(f_{E})} + \\cos^{2}{(f_{E})}}{q{(f_{E})}} and 0 = \\frac{- \\mathbf{H}{(f_{E})} \\cos{(f_{E})} + \\cos^{2}{(f_{E})}}{\\mathbf{H}{(f_{E})} \\cos{(f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["times", 1, "cos(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Pow(cos(Symbol('f_E', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('q')(Symbol('f_E', commutative=True)), Mul(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))))"], [["minus", 3, "Function('q')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Mul(Integer(-1), Function('q')(Symbol('f_E', commutative=True)))))"], [["divide", 4, "Function('q')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Mul(Integer(-1), Function('q')(Symbol('f_E', commutative=True)))), Pow(Function('q')(Symbol('f_E', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('q')(Symbol('f_E', commutative=True))), Pow(cos(Symbol('f_E', commutative=True)), Integer(2))), Pow(Function('q')(Symbol('f_E', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Pow(cos(Symbol('f_E', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), Integer(-1)), Pow(cos(Symbol('f_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\theta_1)} = e^{\\cos{(\\theta_1)}}, then obtain \\frac{d}{d \\theta_1} (- \\theta_1 + \\operatorname{y^{\\prime}}{(\\theta_1)})^{\\theta_1} = \\frac{d}{d \\theta_1} (- \\theta_1 + e^{\\cos{(\\theta_1)}})^{\\theta_1}", "derivation": "\\operatorname{y^{\\prime}}{(\\theta_1)} = e^{\\cos{(\\theta_1)}} and - \\theta_1 + \\operatorname{y^{\\prime}}{(\\theta_1)} = - \\theta_1 + e^{\\cos{(\\theta_1)}} and (- \\theta_1 + \\operatorname{y^{\\prime}}{(\\theta_1)})^{\\theta_1} = (- \\theta_1 + e^{\\cos{(\\theta_1)}})^{\\theta_1} and \\frac{d}{d \\theta_1} (- \\theta_1 + \\operatorname{y^{\\prime}}{(\\theta_1)})^{\\theta_1} = \\frac{d}{d \\theta_1} (- \\theta_1 + e^{\\cos{(\\theta_1)}})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True)), exp(cos(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(cos(Symbol('\\\\theta_1', commutative=True)))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(cos(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(cos(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(T,\\Psi^{\\dagger})} = T + \\sin{(\\Psi^{\\dagger})} and \\mathbf{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger}, then obtain \\Psi^{\\dagger} + \\mathbf{M}{(T,\\Psi^{\\dagger})} - 1 = T + \\Psi^{\\dagger} + \\sin{(\\Psi^{\\dagger})} - 1", "derivation": "\\mathbf{M}{(T,\\Psi^{\\dagger})} = T + \\sin{(\\Psi^{\\dagger})} and \\mathbf{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} and \\mathbf{M}{(T,\\Psi^{\\dagger})} + \\mathbf{p}{(\\Psi^{\\dagger})} = T + \\mathbf{p}{(\\Psi^{\\dagger})} + \\sin{(\\Psi^{\\dagger})} and \\Psi^{\\dagger} + \\mathbf{M}{(T,\\Psi^{\\dagger})} = T + \\Psi^{\\dagger} + \\sin{(\\Psi^{\\dagger})} and \\Psi^{\\dagger} + \\mathbf{M}{(T,\\Psi^{\\dagger})} - 1 = T + \\Psi^{\\dagger} + \\sin{(\\Psi^{\\dagger})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('T', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('T', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\mathbf{M}')(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\mathbf{M}')(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} = F_{g} + \\mathbf{H}, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} \\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} \\log{(F_{g} + \\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} (F_{g} + \\mathbf{H}) \\log{(F_{g} + \\mathbf{H})}", "derivation": "\\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} = F_{g} + \\mathbf{H} and \\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} \\log{(F_{g} + \\mathbf{H})} = (F_{g} + \\mathbf{H}) \\log{(F_{g} + \\mathbf{H})} and \\mathbf{H} \\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} \\log{(F_{g} + \\mathbf{H})} = \\mathbf{H} (F_{g} + \\mathbf{H}) \\log{(F_{g} + \\mathbf{H})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} \\dot{\\mathbf{r}}{(F_{g},\\mathbf{H})} \\log{(F_{g} + \\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} (F_{g} + \\mathbf{H}) \\log{(F_{g} + \\mathbf{H})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(T,J_{\\varepsilon})} = \\cos{(J_{\\varepsilon}^{T})}, then obtain \\sin{((- 2 J_{\\varepsilon}^{T} + y{(T,J_{\\varepsilon})})^{T})} = \\sin{((- 2 J_{\\varepsilon}^{T} + \\cos{(J_{\\varepsilon}^{T})})^{T})}", "derivation": "y{(T,J_{\\varepsilon})} = \\cos{(J_{\\varepsilon}^{T})} and - J_{\\varepsilon}^{T} + y{(T,J_{\\varepsilon})} = - J_{\\varepsilon}^{T} + \\cos{(J_{\\varepsilon}^{T})} and - 2 J_{\\varepsilon}^{T} + y{(T,J_{\\varepsilon})} = - 2 J_{\\varepsilon}^{T} + \\cos{(J_{\\varepsilon}^{T})} and (- 2 J_{\\varepsilon}^{T} + y{(T,J_{\\varepsilon})})^{T} = (- 2 J_{\\varepsilon}^{T} + \\cos{(J_{\\varepsilon}^{T})})^{T} and \\sin{((- 2 J_{\\varepsilon}^{T} + y{(T,J_{\\varepsilon})})^{T})} = \\sin{((- 2 J_{\\varepsilon}^{T} + \\cos{(J_{\\varepsilon}^{T})})^{T})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('T', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))))"], [["minus", 1, "Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Function('y')(Symbol('T', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), cos(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)))))"], [["minus", 2, "Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Function('y')(Symbol('T', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), cos(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)))))"], [["power", 3, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Function('y')(Symbol('T', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), cos(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), Function('y')(Symbol('T', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('T', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True))), cos(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}}, then obtain 2 \\mathbf{J} + 2 \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} = 2 \\mathbf{J} + \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}})^{2}", "derivation": "\\operatorname{v_{t}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} and \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} = (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}})^{2} and \\mathbf{J} + \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} = \\mathbf{J} + (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}})^{2} and 2 \\mathbf{J} + 2 \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} = 2 \\mathbf{J} + \\operatorname{v_{t}}{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}})^{2}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))"], "Equality(Mul(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(2)))"], [["add", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))), Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(2))))"], [["add", 3, "Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given A{(z^{*})} = \\log{(z^{*})} and \\hat{X}{(z^{*})} = \\log{(z^{*})}, then obtain (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} = (\\frac{d}{d z^{*}} \\hat{X}{(z^{*})})^{z^{*}}", "derivation": "A{(z^{*})} = \\log{(z^{*})} and \\frac{d}{d z^{*}} A{(z^{*})} = \\frac{d}{d z^{*}} \\log{(z^{*})} and \\hat{X}{(z^{*})} = \\log{(z^{*})} and \\frac{d}{d z^{*}} A{(z^{*})} = \\frac{d}{d z^{*}} \\hat{X}{(z^{*})} and \\frac{d}{d z^{*}} \\log{(z^{*})} = \\frac{d}{d z^{*}} \\hat{X}{(z^{*})} and (\\frac{d}{d z^{*}} A{(z^{*})})^{z^{*}} = (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} and (\\frac{d}{d z^{*}} A{(z^{*})})^{z^{*}} = (\\frac{d}{d z^{*}} \\hat{X}{(z^{*})})^{z^{*}} and (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} = (\\frac{d}{d z^{*}} \\hat{X}{(z^{*})})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('A')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Function('\\\\hat{X}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Function('\\\\hat{X}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Function('A')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Derivative(Function('A')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(Function('\\\\hat{X}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(Function('\\\\hat{X}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given v{(i)} = i \\cos{(i)} and y{(i)} = - i \\sin{(i)}, then derive \\frac{d}{d i} v{(i)} = - i \\sin{(i)} + \\cos{(i)}, then obtain (\\frac{d}{d i} v{(i)})^{i} = (y{(i)} + \\cos{(i)})^{i}", "derivation": "v{(i)} = i \\cos{(i)} and \\frac{d}{d i} v{(i)} = \\frac{d}{d i} i \\cos{(i)} and \\frac{d}{d i} v{(i)} = - i \\sin{(i)} + \\cos{(i)} and \\frac{d}{d i} i \\cos{(i)} = - i \\sin{(i)} + \\cos{(i)} and (\\frac{d}{d i} v{(i)})^{i} = (\\frac{d}{d i} i \\cos{(i)})^{i} and y{(i)} = - i \\sin{(i)} and \\frac{d}{d i} i \\cos{(i)} = y{(i)} + \\cos{(i)} and (\\frac{d}{d i} v{(i)})^{i} = (y{(i)} + \\cos{(i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('i', commutative=True)), Mul(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Add(Mul(Integer(-1), Symbol('i', commutative=True), sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Add(Mul(Integer(-1), Symbol('i', commutative=True), sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Function('v')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Add(Function('y')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Pow(Derivative(Function('v')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Add(Function('y')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\hbar)} = \\cos{(\\hbar)}, then obtain (\\frac{d}{d \\hbar} (\\operatorname{t_{1}}{(\\hbar)} + \\cos{(\\hbar)}) + \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)})^{\\hbar} = (2 \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)})^{\\hbar}", "derivation": "\\operatorname{t_{1}}{(\\hbar)} = \\cos{(\\hbar)} and \\operatorname{t_{1}}{(\\hbar)} + \\cos{(\\hbar)} = 2 \\cos{(\\hbar)} and \\frac{d}{d \\hbar} (\\operatorname{t_{1}}{(\\hbar)} + \\cos{(\\hbar)}) = \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)} and \\frac{d}{d \\hbar} (\\operatorname{t_{1}}{(\\hbar)} + \\cos{(\\hbar)}) + \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)} = 2 \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)} and (\\frac{d}{d \\hbar} (\\operatorname{t_{1}}{(\\hbar)} + \\cos{(\\hbar)}) + \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)})^{\\hbar} = (2 \\frac{d}{d \\hbar} 2 \\cos{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('t_1')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('t_1')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Function('t_1')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Derivative(Add(Function('t_1')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Integer(2), Derivative(Mul(Integer(2), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} = \\sin{(\\hat{H}_l \\varphi^*)}, then obtain \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\int \\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} d\\hat{H}_l) = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\int \\sin{(\\hat{H}_l \\varphi^*)} d\\hat{H}_l)", "derivation": "\\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} = \\sin{(\\hat{H}_l \\varphi^*)} and \\int \\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} d\\hat{H}_l = \\int \\sin{(\\hat{H}_l \\varphi^*)} d\\hat{H}_l and \\hat{H}_l + \\int \\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} d\\hat{H}_l = \\hat{H}_l + \\int \\sin{(\\hat{H}_l \\varphi^*)} d\\hat{H}_l and \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\int \\operatorname{n_{2}}{(\\hat{H}_l,\\varphi^*)} d\\hat{H}_l) = \\frac{\\partial}{\\partial \\hat{H}_l} (\\hat{H}_l + \\int \\sin{(\\hat{H}_l \\varphi^*)} d\\hat{H}_l)", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True)), sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Integral(Function('n_2')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Symbol('\\\\hat{H}_l', commutative=True), Integral(sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Integral(Function('n_2')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Integral(sin(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(a,u)} = - a + \\cos{(u)}, then derive \\int Z{(a,u)} da = - \\frac{a^{2}}{2} + a \\cos{(u)} + m, then obtain - u + \\iint (- a + \\cos{(u)}) da dm = - u + \\int (- \\frac{a^{2}}{2} + a \\cos{(u)} + m) dm", "derivation": "Z{(a,u)} = - a + \\cos{(u)} and \\int Z{(a,u)} da = \\int (- a + \\cos{(u)}) da and \\int Z{(a,u)} da = - \\frac{a^{2}}{2} + a \\cos{(u)} + m and a + Z{(a,u)} = \\cos{(u)} and \\int Z{(a,u)} da = - \\frac{a^{2}}{2} + a (a + Z{(a,u)}) + m and \\iint Z{(a,u)} da dm = \\int (- \\frac{a^{2}}{2} + a (a + Z{(a,u)}) + m) dm and - u + \\iint Z{(a,u)} da dm = - u + \\int (- \\frac{a^{2}}{2} + a (a + Z{(a,u)}) + m) dm and - u + \\iint (- a + \\cos{(u)}) da dm = - u + \\int (- \\frac{a^{2}}{2} + a \\cos{(u)} + m) dm", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('u', commutative=True))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Symbol('a', commutative=True), cos(Symbol('u', commutative=True))), Symbol('m', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Add(Symbol('a', commutative=True), Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True))), cos(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Symbol('a', commutative=True), Add(Symbol('a', commutative=True), Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)))), Symbol('m', commutative=True)))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Symbol('a', commutative=True), Add(Symbol('a', commutative=True), Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["minus", 6, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Symbol('a', commutative=True), Add(Symbol('a', commutative=True), Function('Z')(Symbol('a', commutative=True), Symbol('u', commutative=True)))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Symbol('a', commutative=True), cos(Symbol('u', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(r)} = e^{r}, then obtain - \\dot{x}{(r)} + 2 e^{r} + \\int (\\dot{x}{(r)} + e^{r}) dr = - \\dot{x}{(r)} + 2 e^{r} + \\int 2 e^{r} dr", "derivation": "\\dot{x}{(r)} = e^{r} and \\dot{x}{(r)} + e^{r} = 2 e^{r} and \\int (\\dot{x}{(r)} + e^{r}) dr = \\int 2 e^{r} dr and - e^{r} + \\int (\\dot{x}{(r)} + e^{r}) dr = - e^{r} + \\int 2 e^{r} dr and - \\dot{x}{(r)} + \\int (\\dot{x}{(r)} + e^{r}) dr = - \\dot{x}{(r)} + \\int 2 e^{r} dr and - \\dot{x}{(r)} + 2 e^{r} + \\int (\\dot{x}{(r)} + e^{r}) dr = - \\dot{x}{(r)} + 2 e^{r} + \\int 2 e^{r} dr", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["add", 1, "exp(Symbol('r', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), Mul(Integer(2), exp(Symbol('r', commutative=True))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["minus", 3, "exp(Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('r', commutative=True))), Integral(Add(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('r', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('r', commutative=True))), Integral(Add(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('r', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"], [["add", 5, "Mul(Integer(2), exp(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('r', commutative=True))), Mul(Integer(2), exp(Symbol('r', commutative=True))), Integral(Add(Function('\\\\dot{x}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('r', commutative=True))), Mul(Integer(2), exp(Symbol('r', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{v},v_{x})} = e^{\\mathbf{v} v_{x}}, then obtain \\frac{\\partial}{\\partial v_{x}} \\int (\\mathbf{v} v_{x} + \\operatorname{M_{E}}{(\\mathbf{v},v_{x})}) dv_{x} = \\frac{\\partial}{\\partial v_{x}} \\int (\\mathbf{v} v_{x} + e^{\\mathbf{v} v_{x}}) dv_{x}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{v},v_{x})} = e^{\\mathbf{v} v_{x}} and \\mathbf{v} v_{x} + \\operatorname{M_{E}}{(\\mathbf{v},v_{x})} = \\mathbf{v} v_{x} + e^{\\mathbf{v} v_{x}} and \\int (\\mathbf{v} v_{x} + \\operatorname{M_{E}}{(\\mathbf{v},v_{x})}) dv_{x} = \\int (\\mathbf{v} v_{x} + e^{\\mathbf{v} v_{x}}) dv_{x} and \\frac{\\partial}{\\partial v_{x}} \\int (\\mathbf{v} v_{x} + \\operatorname{M_{E}}{(\\mathbf{v},v_{x})}) dv_{x} = \\frac{\\partial}{\\partial v_{x}} \\int (\\mathbf{v} v_{x} + e^{\\mathbf{v} v_{x}}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), exp(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True))), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), exp(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), exp(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)), exp(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(\\chi,Q)} = Q \\chi, then obtain - \\frac{Q^{2} \\chi}{2} - \\nabla + \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ = - \\frac{Q^{2} \\chi}{2} - \\nabla + \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ", "derivation": "\\Omega{(\\chi,Q)} = Q \\chi and \\int \\Omega{(\\chi,Q)} dQ = \\int Q \\chi dQ and \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ = \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ and \\int \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ dQ = \\int \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ dQ and \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ - \\int \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ dQ = \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ - \\int \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ dQ and \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ - \\int \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ dQ = \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ - \\int \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ dQ and - \\frac{Q^{2} \\chi}{2} - \\nabla + \\frac{\\partial}{\\partial Q} \\int \\Omega{(\\chi,Q)} dQ = - \\frac{Q^{2} \\chi}{2} - \\nabla + \\frac{\\partial}{\\partial Q} \\int Q \\chi dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))"], [["minus", 3, "Integral(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))), Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))), Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(\\hat{X},F_{x})} = \\hat{X} + \\log{(F_{x})}, then obtain 2 F_{x} (\\hat{X} + \\log{(F_{x})}) = F_{x} (2 \\hat{X} + 2 \\log{(F_{x})})", "derivation": "V{(\\hat{X},F_{x})} = \\hat{X} + \\log{(F_{x})} and 2 V{(\\hat{X},F_{x})} = \\hat{X} + V{(\\hat{X},F_{x})} + \\log{(F_{x})} and 2 F_{x} V{(\\hat{X},F_{x})} = F_{x} (\\hat{X} + V{(\\hat{X},F_{x})} + \\log{(F_{x})}) and 2 F_{x} (\\hat{X} + \\log{(F_{x})}) = F_{x} (2 \\hat{X} + 2 \\log{(F_{x})})", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), log(Symbol('F_x', commutative=True))))"], [["add", 1, "Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True))))"], [["times", 2, "Symbol('F_x', commutative=True)"], "Equality(Mul(Integer(2), Symbol('F_x', commutative=True), Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), Add(Symbol('\\\\hat{X}', commutative=True), Function('V')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Symbol('F_x', commutative=True), Add(Symbol('\\\\hat{X}', commutative=True), log(Symbol('F_x', commutative=True)))), Mul(Symbol('F_x', commutative=True), Add(Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(2), log(Symbol('F_x', commutative=True))))))"]]}, {"prompt": "Given C{(L)} = \\sin{(L)}, then derive - C^{L}{(L)} + \\frac{d}{d L} C{(L)} = - C^{L}{(L)} + \\cos{(L)}, then obtain \\int (- C^{L}{(L)} + \\frac{d}{d L} C{(L)}) dL = \\int (- C^{L}{(L)} + \\cos{(L)}) dL", "derivation": "C{(L)} = \\sin{(L)} and \\frac{d}{d L} C{(L)} = \\frac{d}{d L} \\sin{(L)} and - C^{L}{(L)} + \\frac{d}{d L} C{(L)} = - C^{L}{(L)} + \\frac{d}{d L} \\sin{(L)} and - C^{L}{(L)} + \\frac{d}{d L} C{(L)} = - C^{L}{(L)} + \\cos{(L)} and \\int (- C^{L}{(L)} + \\frac{d}{d L} C{(L)}) dL = \\int (- C^{L}{(L)} + \\cos{(L)}) dL", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Derivative(Function('C')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Derivative(Function('C')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Derivative(Function('C')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Function('C')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then obtain (- 2 \\mathbf{F} + \\operatorname{C_{2}}{(\\mathbf{F})}) \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = (- 2 \\mathbf{F} + \\cos{(\\mathbf{F})}) \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and - \\mathbf{F} + \\operatorname{C_{2}}{(\\mathbf{F})} = - \\mathbf{F} + \\cos{(\\mathbf{F})} and - 2 \\mathbf{F} + \\operatorname{C_{2}}{(\\mathbf{F})} = - 2 \\mathbf{F} + \\cos{(\\mathbf{F})} and \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and (- 2 \\mathbf{F} + \\operatorname{C_{2}}{(\\mathbf{F})}) \\int \\cos{(\\mathbf{F})} d\\mathbf{F} = (- 2 \\mathbf{F} + \\cos{(\\mathbf{F})}) \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and (- 2 \\mathbf{F} + \\operatorname{C_{2}}{(\\mathbf{F})}) \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = (- 2 \\mathbf{F} + \\cos{(\\mathbf{F})}) \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 3, "Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given U{(\\mathbf{r})} = e^{\\mathbf{r}} and \\mathbf{M}{(\\mathbf{r})} = e^{\\mathbf{r}}, then obtain 0 = - \\mathbf{M}^{2}{(\\mathbf{r})} + \\mathbf{M}{(\\mathbf{r})} e^{\\mathbf{r}}", "derivation": "U{(\\mathbf{r})} = e^{\\mathbf{r}} and U^{2}{(\\mathbf{r})} = U{(\\mathbf{r})} e^{\\mathbf{r}} and \\mathbf{M}{(\\mathbf{r})} = e^{\\mathbf{r}} and 0 = - U^{2}{(\\mathbf{r})} + U{(\\mathbf{r})} e^{\\mathbf{r}} and U{(\\mathbf{r})} = \\mathbf{M}{(\\mathbf{r})} and 0 = - \\mathbf{M}^{2}{(\\mathbf{r})} + \\mathbf{M}{(\\mathbf{r})} e^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "Function('U')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Mul(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 2, "Pow(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))), Mul(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('U')(Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))), Mul(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(h,\\varepsilon_0)} = h^{\\varepsilon_0}, then obtain \\int \\frac{\\partial}{\\partial h} - h^{\\varepsilon_0} d\\varepsilon_0 = \\int \\frac{\\partial}{\\partial h} - \\operatorname{M_{E}}{(h,\\varepsilon_0)} d\\varepsilon_0", "derivation": "\\operatorname{M_{E}}{(h,\\varepsilon_0)} = h^{\\varepsilon_0} and - h^{\\varepsilon_0} + \\operatorname{M_{E}}{(h,\\varepsilon_0)} = 0 and - h^{\\varepsilon_0} = - \\operatorname{M_{E}}{(h,\\varepsilon_0)} and \\frac{\\partial}{\\partial h} - h^{\\varepsilon_0} = \\frac{\\partial}{\\partial h} - \\operatorname{M_{E}}{(h,\\varepsilon_0)} and \\int \\frac{\\partial}{\\partial h} - h^{\\varepsilon_0} d\\varepsilon_0 = \\int \\frac{\\partial}{\\partial h} - \\operatorname{M_{E}}{(h,\\varepsilon_0)} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Integer(0))"], [["minus", 2, "Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Derivative(Mul(Integer(-1), Function('M_E')(Symbol('h', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\theta)} = \\sin{(\\theta)}, then obtain \\operatorname{A_{z}}^{3}{(\\theta)} \\sin^{2}{(\\theta)} = \\operatorname{A_{z}}{(\\theta)} \\sin^{4}{(\\theta)}", "derivation": "\\operatorname{A_{z}}{(\\theta)} = \\sin{(\\theta)} and \\operatorname{A_{z}}^{2}{(\\theta)} = \\operatorname{A_{z}}{(\\theta)} \\sin{(\\theta)} and \\operatorname{A_{z}}^{3}{(\\theta)} = \\operatorname{A_{z}}^{2}{(\\theta)} \\sin{(\\theta)} and \\operatorname{A_{z}}^{3}{(\\theta)} \\sin^{2}{(\\theta)} = \\operatorname{A_{z}}^{2}{(\\theta)} \\sin^{3}{(\\theta)} and \\operatorname{A_{z}}^{3}{(\\theta)} \\sin^{2}{(\\theta)} = \\operatorname{A_{z}}{(\\theta)} \\sin^{4}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Function('A_z')(Symbol('\\\\theta', commutative=True))"], "Equality(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Function('A_z')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True))))"], [["times", 2, "Function('A_z')(Symbol('\\\\theta', commutative=True))"], "Equality(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(3)), Mul(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(2)), sin(Symbol('\\\\theta', commutative=True))))"], [["times", 3, "Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(3)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2))), Mul(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('A_z')(Symbol('\\\\theta', commutative=True)), Integer(3)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(2))), Mul(Function('A_z')(Symbol('\\\\theta', commutative=True)), Pow(sin(Symbol('\\\\theta', commutative=True)), Integer(4))))"]]}, {"prompt": "Given \\ddot{x}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\operatorname{y^{\\prime}}{(\\sigma_p)} = - \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}, then derive \\sigma_p \\operatorname{y^{\\prime}}{(\\sigma_p)} = -1, then obtain \\frac{d}{d \\sigma_p} \\sigma_p \\operatorname{y^{\\prime}}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} (-1)", "derivation": "\\ddot{x}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\operatorname{y^{\\prime}}{(\\sigma_p)} = - \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\operatorname{y^{\\prime}}{(\\sigma_p)} = - \\ddot{x}{(\\sigma_p)} and \\frac{\\operatorname{y^{\\prime}}{(\\sigma_p)}}{\\ddot{x}{(\\sigma_p)}} = -1 and \\frac{\\operatorname{y^{\\prime}}{(\\sigma_p)}}{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}} = -1 and \\sigma_p \\operatorname{y^{\\prime}}{(\\sigma_p)} = -1 and \\frac{d}{d \\sigma_p} \\sigma_p \\operatorname{y^{\\prime}}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} (-1)", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 3, "Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))), Integer(-1))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))"], [["differentiate", 6, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(E_{x})} = \\log{(E_{x})}, then obtain 0 = \\frac{(- \\delta{(E_{x})} + \\log{(E_{x})})^{2} \\delta{(E_{x})}}{E_{x} \\log{(E_{x})}}", "derivation": "\\delta{(E_{x})} = \\log{(E_{x})} and 0 = - \\delta{(E_{x})} + \\log{(E_{x})} and 0 = \\frac{- \\delta{(E_{x})} + \\log{(E_{x})}}{\\log{(E_{x})}} and 0 = \\frac{(- \\delta{(E_{x})} + \\log{(E_{x})}) \\delta{(E_{x})}}{E_{x} \\log{(E_{x})}} and 0 = \\frac{(- \\delta{(E_{x})} + \\log{(E_{x})})^{2} \\delta{(E_{x})}}{E_{x} \\log{(E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["minus", 1, "Function('\\\\delta')(Symbol('E_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_x', commutative=True))), log(Symbol('E_x', commutative=True))))"], [["divide", 2, "log(Symbol('E_x', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_x', commutative=True))), log(Symbol('E_x', commutative=True))), Pow(log(Symbol('E_x', commutative=True)), Integer(-1))))"], [["times", 3, "Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_x', commutative=True))), log(Symbol('E_x', commutative=True))), Function('\\\\delta')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1))))"], [["times", 4, "Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_x', commutative=True))), log(Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_x', commutative=True))), log(Symbol('E_x', commutative=True))), Integer(2)), Function('\\\\delta')(Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given Q{(g,E_{\\lambda})} = \\sin{(E_{\\lambda} g)} and k{(g,E_{\\lambda})} = 2 \\frac{\\partial}{\\partial E_{\\lambda}} Q{(g,E_{\\lambda})}, then obtain k^{E_{\\lambda}}{(g,E_{\\lambda})} = (2 \\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda} g)})^{E_{\\lambda}}", "derivation": "Q{(g,E_{\\lambda})} = \\sin{(E_{\\lambda} g)} and k{(g,E_{\\lambda})} = 2 \\frac{\\partial}{\\partial E_{\\lambda}} Q{(g,E_{\\lambda})} and k{(g,E_{\\lambda})} = 2 \\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda} g)} and k^{E_{\\lambda}}{(g,E_{\\lambda})} = (2 \\frac{\\partial}{\\partial E_{\\lambda}} Q{(g,E_{\\lambda})})^{E_{\\lambda}} and 2 \\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda} g)} = 2 \\frac{\\partial}{\\partial E_{\\lambda}} Q{(g,E_{\\lambda})} and k^{E_{\\lambda}}{(g,E_{\\lambda})} = (2 \\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda} g)})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('k')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('k')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(2), Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('k')(Symbol('g', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(2), Derivative(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(I,v_{x},A)} = \\frac{A + I}{v_{x}}, then derive \\frac{I \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)}}{v_{x}} = \\frac{I}{v_{x}^{2}}, then obtain \\int \\frac{I \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)}}{v_{x}} dA = \\int \\frac{I}{v_{x}^{2}} dA", "derivation": "\\mathbf{M}{(I,v_{x},A)} = \\frac{A + I}{v_{x}} and \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)} = \\frac{\\partial}{\\partial I} \\frac{A + I}{v_{x}} and I \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)} = I \\frac{\\partial}{\\partial I} \\frac{A + I}{v_{x}} and I \\frac{\\partial}{\\partial I} \\frac{A + I}{v_{x}} \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)} = I (\\frac{\\partial}{\\partial I} \\frac{A + I}{v_{x}})^{2} and \\frac{I \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)}}{v_{x}} = \\frac{I}{v_{x}^{2}} and \\int \\frac{I \\frac{\\partial}{\\partial I} \\mathbf{M}{(I,v_{x},A)}}{v_{x}} dA = \\int \\frac{I}{v_{x}^{2}} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 2, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Derivative(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Symbol('I', commutative=True), Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('I', commutative=True), Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Symbol('I', commutative=True), Pow(Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('I', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Symbol('I', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-2))))"], [["integrate", 5, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Symbol('I', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('v_x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True))), Integral(Mul(Symbol('I', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-2))), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(E_{x})} = e^{E_{x}}, then obtain \\operatorname{r_{0}}{(E_{x})} + \\iint (\\operatorname{r_{0}}{(E_{x})} - e^{E_{x}})^{E_{x}} dE_{x} dE_{x} = \\operatorname{r_{0}}{(E_{x})} + \\iint 0^{E_{x}} dE_{x} dE_{x}", "derivation": "\\operatorname{r_{0}}{(E_{x})} = e^{E_{x}} and \\operatorname{r_{0}}{(E_{x})} - e^{E_{x}} = 0 and (\\operatorname{r_{0}}{(E_{x})} - e^{E_{x}})^{E_{x}} = 0^{E_{x}} and \\int (\\operatorname{r_{0}}{(E_{x})} - e^{E_{x}})^{E_{x}} dE_{x} = \\int 0^{E_{x}} dE_{x} and \\iint (\\operatorname{r_{0}}{(E_{x})} - e^{E_{x}})^{E_{x}} dE_{x} dE_{x} = \\iint 0^{E_{x}} dE_{x} dE_{x} and \\operatorname{r_{0}}{(E_{x})} + \\iint (\\operatorname{r_{0}}{(E_{x})} - e^{E_{x}})^{E_{x}} dE_{x} dE_{x} = \\operatorname{r_{0}}{(E_{x})} + \\iint 0^{E_{x}} dE_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], [["minus", 1, "exp(Symbol('E_x', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Add(Function('r_0')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Pow(Integer(0), Symbol('E_x', commutative=True)))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Pow(Add(Function('r_0')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(Integer(0), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 4, "Symbol('E_x', commutative=True)"], "Equality(Integral(Pow(Add(Function('r_0')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(Integer(0), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["add", 5, "Function('r_0')(Symbol('E_x', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('E_x', commutative=True)), Integral(Pow(Add(Function('r_0')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(Function('r_0')(Symbol('E_x', commutative=True)), Integral(Pow(Integer(0), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\eta)} = e^{e^{\\eta}}, then obtain 2 \\eta \\frac{d}{d \\eta} \\operatorname{A_{1}}{(\\eta)} = \\eta (e^{\\eta} e^{e^{\\eta}} + \\frac{d}{d \\eta} \\operatorname{A_{1}}{(\\eta)})", "derivation": "\\operatorname{A_{1}}{(\\eta)} = e^{e^{\\eta}} and 2 \\operatorname{A_{1}}{(\\eta)} = \\operatorname{A_{1}}{(\\eta)} + e^{e^{\\eta}} and \\frac{d}{d \\eta} 2 \\operatorname{A_{1}}{(\\eta)} = \\frac{d}{d \\eta} (\\operatorname{A_{1}}{(\\eta)} + e^{e^{\\eta}}) and \\eta \\frac{d}{d \\eta} 2 \\operatorname{A_{1}}{(\\eta)} = \\eta \\frac{d}{d \\eta} (\\operatorname{A_{1}}{(\\eta)} + e^{e^{\\eta}}) and 2 \\eta \\frac{d}{d \\eta} \\operatorname{A_{1}}{(\\eta)} = \\eta (e^{\\eta} e^{e^{\\eta}} + \\frac{d}{d \\eta} \\operatorname{A_{1}}{(\\eta)})", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Function('A_1')(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('A_1')(Symbol('\\\\eta', commutative=True))), Add(Function('A_1')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('A_1')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Function('A_1')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Derivative(Mul(Integer(2), Function('A_1')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Symbol('\\\\eta', commutative=True), Derivative(Add(Function('A_1')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Symbol('\\\\eta', commutative=True), Derivative(Function('A_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Symbol('\\\\eta', commutative=True), Add(Mul(exp(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True)))), Derivative(Function('A_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\tilde{g}{(U,r_{0})} = U r_{0}, then obtain \\frac{\\partial}{\\partial U} \\frac{(\\frac{\\partial}{\\partial r_{0}} \\tilde{g}{(U,r_{0})})^{r_{0}}}{r_{0}} = \\frac{\\partial}{\\partial U} \\frac{(\\frac{\\partial}{\\partial r_{0}} U r_{0})^{r_{0}}}{r_{0}}", "derivation": "\\tilde{g}{(U,r_{0})} = U r_{0} and \\frac{\\partial}{\\partial r_{0}} \\tilde{g}{(U,r_{0})} = \\frac{\\partial}{\\partial r_{0}} U r_{0} and (\\frac{\\partial}{\\partial r_{0}} \\tilde{g}{(U,r_{0})})^{r_{0}} = (\\frac{\\partial}{\\partial r_{0}} U r_{0})^{r_{0}} and \\frac{(\\frac{\\partial}{\\partial r_{0}} \\tilde{g}{(U,r_{0})})^{r_{0}}}{r_{0}} = \\frac{(\\frac{\\partial}{\\partial r_{0}} U r_{0})^{r_{0}}}{r_{0}} and \\frac{\\partial}{\\partial U} \\frac{(\\frac{\\partial}{\\partial r_{0}} \\tilde{g}{(U,r_{0})})^{r_{0}}}{r_{0}} = \\frac{\\partial}{\\partial U} \\frac{(\\frac{\\partial}{\\partial r_{0}} U r_{0})^{r_{0}}}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\tilde{g}')(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)), Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True)))"], [["divide", 3, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True))))"], [["differentiate", 4, "Symbol('U', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Derivative(Mul(Symbol('U', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Symbol('r_0', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(u)} = \\sin{(u)}, then obtain \\cos{(\\int \\operatorname{F_{N}}^{u}{(u)} \\log{(\\operatorname{F_{N}}{(u)})} du)} = \\cos{(\\int \\log{(\\operatorname{F_{N}}{(u)})} \\sin^{u}{(u)} du)}", "derivation": "\\operatorname{F_{N}}{(u)} = \\sin{(u)} and \\log{(\\operatorname{F_{N}}{(u)})} = \\log{(\\sin{(u)})} and \\operatorname{F_{N}}^{u}{(u)} = \\sin^{u}{(u)} and \\operatorname{F_{N}}^{u}{(u)} \\log{(\\sin{(u)})} = \\log{(\\sin{(u)})} \\sin^{u}{(u)} and \\operatorname{F_{N}}^{u}{(u)} \\log{(\\operatorname{F_{N}}{(u)})} = \\log{(\\operatorname{F_{N}}{(u)})} \\sin^{u}{(u)} and \\int \\operatorname{F_{N}}^{u}{(u)} \\log{(\\operatorname{F_{N}}{(u)})} du = \\int \\log{(\\operatorname{F_{N}}{(u)})} \\sin^{u}{(u)} du and \\cos{(\\int \\operatorname{F_{N}}^{u}{(u)} \\log{(\\operatorname{F_{N}}{(u)})} du)} = \\cos{(\\int \\log{(\\operatorname{F_{N}}{(u)})} \\sin^{u}{(u)} du)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["log", 1], "Equality(log(Function('F_N')(Symbol('u', commutative=True))), log(sin(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["times", 3, "log(sin(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Function('F_N')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), log(sin(Symbol('u', commutative=True)))), Mul(log(sin(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('F_N')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), log(Function('F_N')(Symbol('u', commutative=True)))), Mul(log(Function('F_N')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True))))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Pow(Function('F_N')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), log(Function('F_N')(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Mul(log(Function('F_N')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["cos", 6], "Equality(cos(Integral(Mul(Pow(Function('F_N')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), log(Function('F_N')(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), cos(Integral(Mul(log(Function('F_N')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\omega{(x)} = e^{e^{x}}, then obtain ((- x + \\int \\omega{(x)} dx) e^{e^{x}} - (- x + \\int e^{e^{x}} dx) \\omega{(x)})^{x} = (- (- x + \\int e^{e^{x}} dx) \\omega{(x)} + (- x + \\int e^{e^{x}} dx) e^{e^{x}})^{x}", "derivation": "\\omega{(x)} = e^{e^{x}} and \\int \\omega{(x)} dx = \\int e^{e^{x}} dx and - x + \\int \\omega{(x)} dx = - x + \\int e^{e^{x}} dx and (- x + \\int \\omega{(x)} dx) e^{e^{x}} = (- x + \\int e^{e^{x}} dx) e^{e^{x}} and (- x + \\int \\omega{(x)} dx) e^{e^{x}} - (- x + \\int e^{e^{x}} dx) \\omega{(x)} = - (- x + \\int e^{e^{x}} dx) \\omega{(x)} + (- x + \\int e^{e^{x}} dx) e^{e^{x}} and ((- x + \\int \\omega{(x)} dx) e^{e^{x}} - (- x + \\int e^{e^{x}} dx) \\omega{(x)})^{x} = (- (- x + \\int e^{e^{x}} dx) \\omega{(x)} + (- x + \\int e^{e^{x}} dx) e^{e^{x}})^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('x', commutative=True)), exp(exp(Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["minus", 2, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Function('\\\\omega')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))))"], [["times", 3, "exp(exp(Symbol('x', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Function('\\\\omega')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True)))))"], [["minus", 4, "Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Function('\\\\omega')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True))))))"], [["power", 5, "Symbol('x', commutative=True)"], "Equality(Pow(Add(Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Function('\\\\omega')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), exp(exp(Symbol('x', commutative=True))))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\delta{(\\mathbf{S})} = e^{\\mathbf{S}}, then derive \\int \\delta{(\\mathbf{S})} d\\mathbf{S} = \\theta_1 + e^{\\mathbf{S}}, then derive f + e^{\\mathbf{S}} = \\theta_1 + e^{\\mathbf{S}}, then obtain f + e^{\\mathbf{S}} = \\theta_1 + \\delta{(\\mathbf{S})}", "derivation": "\\delta{(\\mathbf{S})} = e^{\\mathbf{S}} and \\int \\delta{(\\mathbf{S})} d\\mathbf{S} = \\int e^{\\mathbf{S}} d\\mathbf{S} and \\int \\delta{(\\mathbf{S})} d\\mathbf{S} = \\theta_1 + e^{\\mathbf{S}} and \\int \\delta{(\\mathbf{S})} d\\mathbf{S} = \\theta_1 + \\delta{(\\mathbf{S})} and \\int e^{\\mathbf{S}} d\\mathbf{S} = \\theta_1 + e^{\\mathbf{S}} and \\theta_1 + e^{\\mathbf{S}} = \\theta_1 + \\delta{(\\mathbf{S})} and f + e^{\\mathbf{S}} = \\theta_1 + e^{\\mathbf{S}} and f + e^{\\mathbf{S}} = \\theta_1 + \\delta{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(k)} = e^{e^{k}}, then obtain (2 \\operatorname{F_{N}}{(k)})^{k} - \\operatorname{F_{N}}{(k)} = (\\operatorname{F_{N}}{(k)} + e^{e^{k}})^{k} - \\operatorname{F_{N}}{(k)}", "derivation": "\\operatorname{F_{N}}{(k)} = e^{e^{k}} and 2 \\operatorname{F_{N}}{(k)} = \\operatorname{F_{N}}{(k)} + e^{e^{k}} and (2 \\operatorname{F_{N}}{(k)})^{k} = (\\operatorname{F_{N}}{(k)} + e^{e^{k}})^{k} and (2 \\operatorname{F_{N}}{(k)})^{k} - \\operatorname{F_{N}}{(k)} = (\\operatorname{F_{N}}{(k)} + e^{e^{k}})^{k} - \\operatorname{F_{N}}{(k)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True))))"], [["add", 1, "Function('F_N')(Symbol('k', commutative=True))"], "Equality(Mul(Integer(2), Function('F_N')(Symbol('k', commutative=True))), Add(Function('F_N')(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('F_N')(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Function('F_N')(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["minus", 3, "Function('F_N')(Symbol('k', commutative=True))"], "Equality(Add(Pow(Mul(Integer(2), Function('F_N')(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Mul(Integer(-1), Function('F_N')(Symbol('k', commutative=True)))), Add(Pow(Add(Function('F_N')(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Mul(Integer(-1), Function('F_N')(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(F_{c},\\theta)} = e^{- F_{c} + \\theta} and \\mathbf{E}{(a)} = \\sin{(a)}, then obtain \\int (\\mathbf{E}{(a)} + \\int \\operatorname{v_{2}}{(F_{c},\\theta)} d\\theta) d\\theta = \\int (\\mathbf{E}{(a)} + \\int e^{- F_{c} + \\theta} d\\theta) d\\theta", "derivation": "\\operatorname{v_{2}}{(F_{c},\\theta)} = e^{- F_{c} + \\theta} and \\mathbf{E}{(a)} = \\sin{(a)} and \\int \\operatorname{v_{2}}{(F_{c},\\theta)} d\\theta = \\int e^{- F_{c} + \\theta} d\\theta and \\sin{(a)} + \\int \\operatorname{v_{2}}{(F_{c},\\theta)} d\\theta = \\sin{(a)} + \\int e^{- F_{c} + \\theta} d\\theta and \\int (\\sin{(a)} + \\int \\operatorname{v_{2}}{(F_{c},\\theta)} d\\theta) d\\theta = \\int (\\sin{(a)} + \\int e^{- F_{c} + \\theta} d\\theta) d\\theta and \\int (\\mathbf{E}{(a)} + \\int \\operatorname{v_{2}}{(F_{c},\\theta)} d\\theta) d\\theta = \\int (\\mathbf{E}{(a)} + \\int e^{- F_{c} + \\theta} d\\theta) d\\theta", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('F_c', commutative=True), Symbol('\\\\theta', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\theta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('F_c', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["add", 3, "sin(Symbol('a', commutative=True))"], "Equality(Add(sin(Symbol('a', commutative=True)), Integral(Function('v_2')(Symbol('F_c', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(sin(Symbol('a', commutative=True)), Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Add(sin(Symbol('a', commutative=True)), Integral(Function('v_2')(Symbol('F_c', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(sin(Symbol('a', commutative=True)), Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Add(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), Integral(Function('v_2')(Symbol('F_c', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), Integral(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\Psi_{\\lambda})} = \\int e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} and m{(\\Psi_{\\lambda})} = \\Psi_{\\lambda}, then obtain \\int \\operatorname{C_{d}}{(\\Psi_{\\lambda})} dm{(\\Psi_{\\lambda})} = \\iint e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} dm{(\\Psi_{\\lambda})}", "derivation": "\\operatorname{C_{d}}{(\\Psi_{\\lambda})} = \\int e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} and \\int \\operatorname{C_{d}}{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\iint e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} d\\Psi_{\\lambda} and m{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} and \\int \\operatorname{C_{d}}{(\\Psi_{\\lambda})} dm{(\\Psi_{\\lambda})} = \\iint e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} dm{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integral(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Function('m')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integral(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Function('m')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(L)} = \\log{(\\cos{(L)})}, then derive \\frac{d}{d L} \\mathbf{P}{(L)} = - \\frac{\\sin{(L)}}{\\cos{(L)}}, then derive l + \\mathbf{P}{(L)} = \\int - \\frac{\\sin{(L)}}{\\cos{(L)}} dL, then obtain \\int (l + \\mathbf{P}{(L)}) dL = \\iint \\frac{d}{d L} \\mathbf{P}{(L)} dL dL", "derivation": "\\mathbf{P}{(L)} = \\log{(\\cos{(L)})} and \\frac{d}{d L} \\mathbf{P}{(L)} = \\frac{d}{d L} \\log{(\\cos{(L)})} and \\frac{d}{d L} \\mathbf{P}{(L)} = - \\frac{\\sin{(L)}}{\\cos{(L)}} and \\int \\frac{d}{d L} \\mathbf{P}{(L)} dL = \\int - \\frac{\\sin{(L)}}{\\cos{(L)}} dL and l + \\mathbf{P}{(L)} = \\int - \\frac{\\sin{(L)}}{\\cos{(L)}} dL and l + \\mathbf{P}{(L)} = \\int \\frac{d}{d L} \\mathbf{P}{(L)} dL and \\int (l + \\mathbf{P}{(L)}) dL = \\iint \\frac{d}{d L} \\mathbf{P}{(L)} dL dL", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('l', commutative=True), Function('\\\\mathbf{P}')(Symbol('L', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('l', commutative=True), Function('\\\\mathbf{P}')(Symbol('L', commutative=True))), Integral(Derivative(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"], [["integrate", 6, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Function('\\\\mathbf{P}')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(A_{x},\\Psi)} = \\frac{\\sin{(A_{x})}}{\\Psi}, then obtain \\sin{(\\int (- \\Psi + \\mathbb{I}{(A_{x},\\Psi)}) d\\Psi)} = \\sin{(\\int (- \\Psi + \\frac{\\sin{(A_{x})}}{\\Psi}) d\\Psi)}", "derivation": "\\mathbb{I}{(A_{x},\\Psi)} = \\frac{\\sin{(A_{x})}}{\\Psi} and - \\Psi + \\mathbb{I}{(A_{x},\\Psi)} = - \\Psi + \\frac{\\sin{(A_{x})}}{\\Psi} and \\int (- \\Psi + \\mathbb{I}{(A_{x},\\Psi)}) d\\Psi = \\int (- \\Psi + \\frac{\\sin{(A_{x})}}{\\Psi}) d\\Psi and \\sin{(\\int (- \\Psi + \\mathbb{I}{(A_{x},\\Psi)}) d\\Psi)} = \\sin{(\\int (- \\Psi + \\frac{\\sin{(A_{x})}}{\\Psi}) d\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('A_x', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('A_x', commutative=True))))"], [["minus", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbb{I}')(Symbol('A_x', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('A_x', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbb{I}')(Symbol('A_x', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('A_x', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbb{I}')(Symbol('A_x', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), sin(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('A_x', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(v_{x},F_{H})} = F_{H} v_{x}, then obtain 0 = \\frac{\\partial}{\\partial v_{x}} (\\int F_{H} v_{x} dF_{H} - \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H})^{F_{H}}", "derivation": "\\mathbf{A}{(v_{x},F_{H})} = F_{H} v_{x} and \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H} = \\int F_{H} v_{x} dF_{H} and 0 = \\int F_{H} v_{x} dF_{H} - \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H} and 0^{F_{H}} = (\\int F_{H} v_{x} dF_{H} - \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H})^{F_{H}} and \\frac{d}{d v_{x}} 0^{F_{H}} = \\frac{\\partial}{\\partial v_{x}} (\\int F_{H} v_{x} dF_{H} - \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H})^{F_{H}} and 0 = \\frac{\\partial}{\\partial v_{x}} (\\int F_{H} v_{x} dF_{H} - \\int \\mathbf{A}{(v_{x},F_{H})} dF_{H})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Integer(0), Symbol('F_H', commutative=True)), Pow(Add(Integral(Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Symbol('F_H', commutative=True)))"], [["differentiate", 4, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('F_H', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Pow(Add(Integral(Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Symbol('F_H', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Derivative(Pow(Add(Integral(Mul(Symbol('F_H', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Symbol('F_H', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(a^{\\dagger})} = e^{a^{\\dagger}}, then derive \\hat{\\mathbf{r}} = \\int (- \\hat{X}{(a^{\\dagger})} + e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger}, then obtain \\int 0^{a^{\\dagger}} da^{\\dagger} = \\hat{\\mathbf{r}}", "derivation": "\\hat{X}{(a^{\\dagger})} = e^{a^{\\dagger}} and 0 = - \\hat{X}{(a^{\\dagger})} + e^{a^{\\dagger}} and 0^{a^{\\dagger}} = (- \\hat{X}{(a^{\\dagger})} + e^{a^{\\dagger}})^{a^{\\dagger}} and \\int 0^{a^{\\dagger}} da^{\\dagger} = \\int (- \\hat{X}{(a^{\\dagger})} + e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger} and \\hat{\\mathbf{r}} = \\int (- \\hat{X}{(a^{\\dagger})} + e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger} and \\int 0^{a^{\\dagger}} da^{\\dagger} = \\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then obtain \\int \\frac{\\operatorname{A_{2}}^{2 \\hat{H}_l}{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l = \\int \\frac{\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} \\log{(\\hat{H}_l)}^{\\hat{H}_l}}{\\hat{H}_l} d\\hat{H}_l", "derivation": "\\operatorname{A_{2}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}^{\\hat{H}_l} and \\operatorname{A_{2}}^{2 \\hat{H}_l}{(\\hat{H}_l)} = \\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} \\log{(\\hat{H}_l)}^{\\hat{H}_l} and \\frac{\\operatorname{A_{2}}^{2 \\hat{H}_l}{(\\hat{H}_l)}}{\\hat{H}_l} = \\frac{\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} \\log{(\\hat{H}_l)}^{\\hat{H}_l}}{\\hat{H}_l} and \\int \\frac{\\operatorname{A_{2}}^{2 \\hat{H}_l}{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l = \\int \\frac{\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} \\log{(\\hat{H}_l)}^{\\hat{H}_l}}{\\hat{H}_l} d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 2, "Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(y)} = \\cos{(\\cos{(y)})}, then obtain - \\cos{(\\cos{(y)})} = \\frac{- \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} + 1}{\\cos{(y)}} - \\cos{(\\cos{(y)})}", "derivation": "\\hat{H}{(y)} = \\cos{(\\cos{(y)})} and \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} = 1 and \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} - \\cos{(\\cos{(y)})} = 1 - \\cos{(\\cos{(y)})} and 0 = - \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} + 1 and 0 = \\frac{- \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} + 1}{\\cos{(y)}} and - \\cos{(\\cos{(y)})} = \\frac{- \\frac{\\hat{H}{(y)}}{\\cos{(\\cos{(y)})}} + 1}{\\cos{(y)}} - \\cos{(\\cos{(y)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))"], [["divide", 1, "cos(cos(Symbol('y', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "cos(cos(Symbol('y', commutative=True)))"], "Equality(Add(Mul(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(cos(Symbol('y', commutative=True))))), Add(Integer(1), Mul(Integer(-1), cos(cos(Symbol('y', commutative=True))))))"], [["minus", 3, "Add(Mul(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(cos(Symbol('y', commutative=True)))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Integer(1)))"], [["divide", 4, "cos(Symbol('y', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Integer(1)), Pow(cos(Symbol('y', commutative=True)), Integer(-1))))"], [["minus", 5, "cos(cos(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), cos(cos(Symbol('y', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('y', commutative=True)), Pow(cos(cos(Symbol('y', commutative=True))), Integer(-1))), Integer(1)), Pow(cos(Symbol('y', commutative=True)), Integer(-1))), Mul(Integer(-1), cos(cos(Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{F})} = \\log{(e^{\\mathbf{F}})} and \\operatorname{C_{2}}{(\\mathbf{F})} = - \\tilde{g}{(\\mathbf{F})} + \\log{(e^{\\mathbf{F}})}, then obtain \\frac{\\operatorname{C_{2}}{(\\mathbf{F})} \\tilde{g}^{- \\mathbf{F}}{(\\mathbf{F})}}{(- \\tilde{g}{(\\mathbf{F})} + \\log{(e^{\\mathbf{F}})}) \\frac{d}{d \\hat{X}} \\Psi_{\\lambda}{(\\hat{X})} \\frac{d}{d \\hat{X}} \\log{(e^{\\hat{X}})}} = 0", "derivation": "\\tilde{g}{(\\mathbf{F})} = \\log{(e^{\\mathbf{F}})} and \\operatorname{C_{2}}{(\\mathbf{F})} = - \\tilde{g}{(\\mathbf{F})} + \\log{(e^{\\mathbf{F}})} and \\operatorname{C_{2}}{(\\mathbf{F})} = 0 and \\operatorname{C_{2}}{(\\mathbf{F})} \\tilde{g}^{- \\mathbf{F}}{(\\mathbf{F})} = 0 and \\frac{\\operatorname{C_{2}}{(\\mathbf{F})} \\tilde{g}^{- \\mathbf{F}}{(\\mathbf{F})}}{- \\tilde{g}{(\\mathbf{F})} + \\log{(e^{\\mathbf{F}})}} = 0 and \\frac{\\operatorname{C_{2}}{(\\mathbf{F})} \\tilde{g}^{- \\mathbf{F}}{(\\mathbf{F})}}{(- \\tilde{g}{(\\mathbf{F})} + \\log{(e^{\\mathbf{F}})}) \\frac{d}{d \\hat{X}} \\Psi_{\\lambda}{(\\hat{X})} \\frac{d}{d \\hat{X}} \\log{(e^{\\hat{X}})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True)), log(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True))), log(exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(0))"], [["divide", 3, "Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(0))"], [["divide", 4, "Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True))), log(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True))), log(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(0))"], [["divide", 5, "Derivative(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{X}', commutative=True)), Derivative(log(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True))), log(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Derivative(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{X}', commutative=True)), Derivative(log(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\Omega)} = \\log{(\\Omega)}, then obtain \\int 4 e^{2 \\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} d\\Omega = \\int 2 (\\Omega + e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}}) e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} d\\Omega", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\Omega)} = \\log{(\\Omega)} and e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} = \\Omega and 2 e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} = \\Omega + e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} and 4 e^{2 \\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} = 2 (\\Omega + e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}}) e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} and \\int 4 e^{2 \\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} d\\Omega = \\int 2 (\\Omega + e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}}) e^{\\operatorname{V_{\\mathbf{E}}}{(\\Omega)}} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["exp", 1], "Equality(exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], [["add", 2, "exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Integer(2), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))))"], [["times", 3, "Mul(Integer(2), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(Integer(4), exp(Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True))))), Mul(Integer(2), Add(Symbol('\\\\Omega', commutative=True), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Integer(4), exp(Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(2), Add(Symbol('\\\\Omega', commutative=True), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))), exp(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(r,\\mathbf{S})} = r + \\cos{(\\mathbf{S})}, then derive C + \\mathbf{g}{(r,\\mathbf{S})} = \\ddot{x} + r, then obtain \\frac{C + r + \\cos{(\\mathbf{S})}}{\\ddot{x} + r + \\frac{r \\sin^{Z}{(Z)}}{\\sin{(Z)}} + \\sin^{Z}{(Z)}} = \\frac{C + \\mathbf{g}{(r,\\mathbf{S})}}{\\ddot{x} + r + \\frac{r \\sin^{Z}{(Z)}}{\\sin{(Z)}} + \\sin^{Z}{(Z)}}", "derivation": "\\mathbf{g}{(r,\\mathbf{S})} = r + \\cos{(\\mathbf{S})} and \\frac{\\partial}{\\partial r} \\mathbf{g}{(r,\\mathbf{S})} = \\frac{\\partial}{\\partial r} (r + \\cos{(\\mathbf{S})}) and \\int \\frac{\\partial}{\\partial r} \\mathbf{g}{(r,\\mathbf{S})} dr = \\int \\frac{\\partial}{\\partial r} (r + \\cos{(\\mathbf{S})}) dr and C + \\mathbf{g}{(r,\\mathbf{S})} = \\ddot{x} + r and C + r + \\cos{(\\mathbf{S})} = \\ddot{x} + r and C + r + \\cos{(\\mathbf{S})} = C + \\mathbf{g}{(r,\\mathbf{S})} and \\frac{C + r + \\cos{(\\mathbf{S})}}{\\ddot{x} + r + \\frac{r \\sin^{Z}{(Z)}}{\\sin{(Z)}} + \\sin^{Z}{(Z)}} = \\frac{C + \\mathbf{g}{(r,\\mathbf{S})}}{\\ddot{x} + r + \\frac{r \\sin^{Z}{(Z)}}{\\sin{(Z)}} + \\sin^{Z}{(Z)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Integral(Derivative(Add(Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('C', commutative=True), Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('C', commutative=True), Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 6, "Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r', commutative=True), Mul(Symbol('r', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], "Equality(Mul(Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r', commutative=True), Mul(Symbol('r', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Integer(-1))), Mul(Add(Symbol('C', commutative=True), Function('\\\\mathbf{g}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('r', commutative=True), Mul(Symbol('r', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given s{(\\mathbf{A},A_{1})} = \\mathbf{A} + e^{A_{1}}, then obtain (\\mathbf{A}^{2} s{(\\mathbf{A},A_{1})})^{A_{1}} = (\\mathbf{A}^{2} (\\mathbf{A} + e^{A_{1}}))^{A_{1}}", "derivation": "s{(\\mathbf{A},A_{1})} = \\mathbf{A} + e^{A_{1}} and \\mathbf{A} s{(\\mathbf{A},A_{1})} = \\mathbf{A} (\\mathbf{A} + e^{A_{1}}) and \\mathbf{A}^{2} s{(\\mathbf{A},A_{1})} = \\mathbf{A}^{2} (\\mathbf{A} + e^{A_{1}}) and (\\mathbf{A}^{2} s{(\\mathbf{A},A_{1})})^{A_{1}} = (\\mathbf{A}^{2} (\\mathbf{A} + e^{A_{1}}))^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_1', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_1', commutative=True)))))"], [["times", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Function('s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_1', commutative=True)))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Function('s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\psi^*,\\mathbf{P})} = \\cos{(\\frac{\\psi^*}{\\mathbf{P}})}, then obtain 2 \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} + \\frac{1}{\\mathbf{P}} = \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} + \\cos^{\\psi^*}{(\\frac{\\psi^*}{\\mathbf{P}})} + \\frac{1}{\\mathbf{P}}", "derivation": "\\operatorname{f^{*}}{(\\psi^*,\\mathbf{P})} = \\cos{(\\frac{\\psi^*}{\\mathbf{P}})} and \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} = \\cos^{\\psi^*}{(\\frac{\\psi^*}{\\mathbf{P}})} and \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} + \\frac{1}{\\mathbf{P}} = \\cos^{\\psi^*}{(\\frac{\\psi^*}{\\mathbf{P}})} + \\frac{1}{\\mathbf{P}} and 2 \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} + \\frac{1}{\\mathbf{P}} = \\operatorname{f^{*}}^{\\psi^*}{(\\psi^*,\\mathbf{P})} + \\cos^{\\psi^*}{(\\frac{\\psi^*}{\\mathbf{P}})} + \\frac{1}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), cos(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))), Add(Pow(cos(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))))"], [["add", 3, "Pow(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))), Add(Pow(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given s{(\\phi,W)} = \\cos{(\\phi^{W})} and \\hat{H}_{\\lambda}{(\\phi)} = 2 \\phi, then obtain s^{2 \\phi}{(\\phi,W)} + \\cos{(\\phi^{W})} = \\cos{(\\phi^{W})} + \\cos^{2 \\phi}{(\\phi^{W})}", "derivation": "s{(\\phi,W)} = \\cos{(\\phi^{W})} and s^{\\phi}{(\\phi,W)} = \\cos^{\\phi}{(\\phi^{W})} and s^{2 \\phi}{(\\phi,W)} = s^{\\phi}{(\\phi,W)} \\cos^{\\phi}{(\\phi^{W})} and \\hat{H}_{\\lambda}{(\\phi)} = 2 \\phi and s^{\\hat{H}_{\\lambda}{(\\phi)}}{(\\phi,W)} = s^{\\phi}{(\\phi,W)} \\cos^{\\phi}{(\\phi^{W})} and s^{\\hat{H}_{\\lambda}{(\\phi)}}{(\\phi,W)} = \\cos^{2 \\phi}{(\\phi^{W})} and s^{\\hat{H}_{\\lambda}{(\\phi)}}{(\\phi,W)} = s^{2 \\phi}{(\\phi,W)} and s^{2 \\phi}{(\\phi,W)} = \\cos^{2 \\phi}{(\\phi^{W})} and s^{2 \\phi}{(\\phi,W)} + \\cos{(\\phi^{W})} = \\cos{(\\phi^{W})} + \\cos^{2 \\phi}{(\\phi^{W})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["times", 2, "Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Mul(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True))), Mul(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True))), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi', commutative=True))), Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["add", 8, "cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Pow(Function('s')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)))), Add(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Pow(cos(Pow(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})} = V_{\\mathbf{B}} + z^{*}, then obtain \\frac{1}{2} = \\frac{V_{\\mathbf{B}} + z^{*}}{V_{\\mathbf{B}} + z^{*} + \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})}}", "derivation": "\\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})} = V_{\\mathbf{B}} + z^{*} and 2 \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})} = V_{\\mathbf{B}} + z^{*} + \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})} and \\frac{1}{2} = \\frac{V_{\\mathbf{B}} + z^{*}}{2 \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})}} and \\frac{1}{2} = \\frac{V_{\\mathbf{B}} + z^{*}}{V_{\\mathbf{B}} + z^{*} + \\mathbf{J}_f{(V_{\\mathbf{B}},z^{*})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True)), Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z^*', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{M})} = e^{\\mathbf{M}}, then derive \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain \\mathbf{M} + 1 = \\mathbf{M} + \\frac{\\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})}}{\\frac{d^{2}}{d \\mathbf{M}^{2}} \\bar{\\h}{(\\mathbf{M})}}", "derivation": "\\bar{\\h}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} = e^{\\mathbf{M}} and \\frac{d^{2}}{d \\mathbf{M}^{2}} \\bar{\\h}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})} and 1 = \\frac{\\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})}}{\\frac{d^{2}}{d \\mathbf{M}^{2}} \\bar{\\h}{(\\mathbf{M})}} and \\mathbf{M} + 1 = \\mathbf{M} + \\frac{\\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})}}{\\frac{d^{2}}{d \\mathbf{M}^{2}} \\bar{\\h}{(\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))), Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))"], "Equality(Integer(1), Mul(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))), Integer(-1))))"], [["add", 6, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))), Integer(-1)))))"]]}, {"prompt": "Given \\pi{(v_{x})} = e^{v_{x}}, then obtain \\int 2 v_{x} \\pi{(v_{x})} dv_{x} = \\int v_{x} (\\pi{(v_{x})} + e^{v_{x}}) dv_{x}", "derivation": "\\pi{(v_{x})} = e^{v_{x}} and 2 \\pi{(v_{x})} = \\pi{(v_{x})} + e^{v_{x}} and 2 v_{x} \\pi{(v_{x})} = v_{x} (\\pi{(v_{x})} + e^{v_{x}}) and \\int 2 v_{x} \\pi{(v_{x})} dv_{x} = \\int v_{x} (\\pi{(v_{x})} + e^{v_{x}}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["add", 1, "Function('\\\\pi')(Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('v_x', commutative=True))), Add(Function('\\\\pi')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["times", 2, "Symbol('v_x', commutative=True)"], "Equality(Mul(Integer(2), Symbol('v_x', commutative=True), Function('\\\\pi')(Symbol('v_x', commutative=True))), Mul(Symbol('v_x', commutative=True), Add(Function('\\\\pi')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))))"], [["integrate", 3, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Integer(2), Symbol('v_x', commutative=True), Function('\\\\pi')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('v_x', commutative=True), Add(Function('\\\\pi')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\theta,\\rho)} = \\int \\rho \\theta d\\rho, then obtain \\int \\frac{\\partial}{\\partial \\theta} \\frac{\\operatorname{A_{x}}{(\\theta,\\rho)}}{\\rho \\theta} d\\rho = \\int \\frac{\\partial}{\\partial \\theta} \\frac{\\int \\rho \\theta d\\rho}{\\rho \\theta} d\\rho", "derivation": "\\operatorname{A_{x}}{(\\theta,\\rho)} = \\int \\rho \\theta d\\rho and \\frac{\\operatorname{A_{x}}{(\\theta,\\rho)}}{\\rho \\theta} = \\frac{\\int \\rho \\theta d\\rho}{\\rho \\theta} and \\frac{\\partial}{\\partial \\theta} \\frac{\\operatorname{A_{x}}{(\\theta,\\rho)}}{\\rho \\theta} = \\frac{\\partial}{\\partial \\theta} \\frac{\\int \\rho \\theta d\\rho}{\\rho \\theta} and \\int \\frac{\\partial}{\\partial \\theta} \\frac{\\operatorname{A_{x}}{(\\theta,\\rho)}}{\\rho \\theta} d\\rho = \\int \\frac{\\partial}{\\partial \\theta} \\frac{\\int \\rho \\theta d\\rho}{\\rho \\theta} d\\rho", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given J{(n,H)} = \\int (H - n) dn and \\operatorname{m_{s}}{(n)} = - n, then obtain (H - n) ((H - n) \\int (H - n) dn + \\operatorname{m_{s}}{(n)}) = (H - n) (- n + (H - n) \\int (H - n) dn)", "derivation": "J{(n,H)} = \\int (H - n) dn and (H - n) J{(n,H)} = (H - n) \\int (H - n) dn and \\operatorname{m_{s}}{(n)} = - n and (H - n) J{(n,H)} + \\operatorname{m_{s}}{(n)} = - n + (H - n) J{(n,H)} and (H - n) ((H - n) J{(n,H)} + \\operatorname{m_{s}}{(n)}) = (H - n) (- n + (H - n) J{(n,H)}) and (H - n) ((H - n) \\int (H - n) dn + \\operatorname{m_{s}}{(n)}) = (H - n) (- n + (H - n) \\int (H - n) dn)", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["times", 1, "Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True))), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True)))"], [["add", 3, "Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True))), Function('m_s')(Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True)))))"], [["times", 4, "Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True))), Function('m_s')(Symbol('n', commutative=True)))), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('J')(Symbol('n', commutative=True), Symbol('H', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), Function('m_s')(Symbol('n', commutative=True)))), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\rho)} = \\cos{(\\rho)}, then derive \\frac{d}{d \\rho} \\mathbf{F}{(\\rho)} = - \\sin{(\\rho)}, then obtain \\cos{(\\frac{d}{d \\rho} \\cos{(\\rho)})} \\frac{d}{d \\rho} \\mathbf{F}{(\\rho)} = - \\sin{(\\rho)} \\cos{(\\frac{d}{d \\rho} \\cos{(\\rho)})}", "derivation": "\\mathbf{F}{(\\rho)} = \\cos{(\\rho)} and \\frac{d}{d \\rho} \\mathbf{F}{(\\rho)} = \\frac{d}{d \\rho} \\cos{(\\rho)} and \\frac{d}{d \\rho} \\mathbf{F}{(\\rho)} = - \\sin{(\\rho)} and \\cos{(\\frac{d}{d \\rho} \\cos{(\\rho)})} \\frac{d}{d \\rho} \\mathbf{F}{(\\rho)} = - \\sin{(\\rho)} \\cos{(\\frac{d}{d \\rho} \\cos{(\\rho)})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))))"], [["times", 3, "cos(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], "Equality(Mul(cos(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True)), cos(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{F}{(i)} = \\sin{(i)}, then obtain \\frac{\\mathbf{F}{(i)} \\sin^{2}{(i)}}{i^{2}} = \\frac{\\sin^{3}{(i)}}{i^{2}}", "derivation": "\\mathbf{F}{(i)} = \\sin{(i)} and \\mathbf{F}{(i)} \\sin{(i)} = \\sin^{2}{(i)} and \\frac{\\mathbf{F}{(i)} \\sin{(i)}}{i} = \\frac{\\sin^{2}{(i)}}{i} and \\frac{\\mathbf{F}{(i)} \\sin{(i)}}{i^{2}} = \\frac{\\sin^{2}{(i)}}{i^{2}} and \\frac{\\mathbf{F}{(i)} \\sin^{2}{(i)}}{i^{2}} = \\frac{\\sin^{3}{(i)}}{i^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["times", 1, "sin(Symbol('i', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Pow(sin(Symbol('i', commutative=True)), Integer(2)))"], [["divide", 2, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(sin(Symbol('i', commutative=True)), Integer(2))))"], [["times", 3, "Pow(Symbol('i', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Pow(sin(Symbol('i', commutative=True)), Integer(2))))"], [["times", 4, "sin(Symbol('i', commutative=True))"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Function('\\\\mathbf{F}')(Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Integer(2))), Mul(Pow(Symbol('i', commutative=True), Integer(-2)), Pow(sin(Symbol('i', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(V,I)} = e^{I V}, then obtain (\\operatorname{v_{z}}{(V,I)} - \\operatorname{v_{z}}^{V}{(V,I)})^{I} = (- \\operatorname{v_{z}}^{V}{(V,I)} + e^{I V})^{I}", "derivation": "\\operatorname{v_{z}}{(V,I)} = e^{I V} and \\operatorname{v_{z}}^{V}{(V,I)} = (e^{I V})^{V} and \\operatorname{v_{z}}{(V,I)} - (e^{I V})^{V} = e^{I V} - (e^{I V})^{V} and \\operatorname{v_{z}}{(V,I)} - \\operatorname{v_{z}}^{V}{(V,I)} = - \\operatorname{v_{z}}^{V}{(V,I)} + e^{I V} and (\\operatorname{v_{z}}{(V,I)} - \\operatorname{v_{z}}^{V}{(V,I)})^{I} = (- \\operatorname{v_{z}}^{V}{(V,I)} + e^{I V})^{I}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Symbol('V', commutative=True)), Pow(exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["minus", 1, "Pow(exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))), Symbol('V', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))), Symbol('V', commutative=True)))), Add(exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))), Mul(Integer(-1), Pow(exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True))), Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Symbol('V', commutative=True))), exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True)))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Symbol('V', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('v_z')(Symbol('V', commutative=True), Symbol('I', commutative=True)), Symbol('V', commutative=True))), exp(Mul(Symbol('I', commutative=True), Symbol('V', commutative=True)))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given H{(r,s)} = \\cos{(\\frac{s}{r})}, then obtain r + s + H{(r,s)} - \\cos{(\\frac{s}{r})} = r + s", "derivation": "H{(r,s)} = \\cos{(\\frac{s}{r})} and r + H{(r,s)} = r + \\cos{(\\frac{s}{r})} and r + s + H{(r,s)} = r + s + \\cos{(\\frac{s}{r})} and r + s + H{(r,s)} - \\cos{(\\frac{s}{r})} = r + s", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('r', commutative=True), Symbol('s', commutative=True)), cos(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["add", 1, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Function('H')(Symbol('r', commutative=True), Symbol('s', commutative=True))), Add(Symbol('r', commutative=True), cos(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('s', commutative=True)))))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Symbol('s', commutative=True), Function('H')(Symbol('r', commutative=True), Symbol('s', commutative=True))), Add(Symbol('r', commutative=True), Symbol('s', commutative=True), cos(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('s', commutative=True)))))"], [["minus", 3, "cos(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], "Equality(Add(Symbol('r', commutative=True), Symbol('s', commutative=True), Function('H')(Symbol('r', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('s', commutative=True))))), Add(Symbol('r', commutative=True), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\hat{x}_0)} = e^{\\hat{x}_0}, then obtain 0 = \\frac{(- \\mathbf{F}{(\\hat{x}_0)} + e^{\\hat{x}_0}) e^{\\hat{x}_0}}{\\mathbf{F}{(\\hat{x}_0)}}", "derivation": "\\mathbf{F}{(\\hat{x}_0)} = e^{\\hat{x}_0} and 0 = - \\mathbf{F}{(\\hat{x}_0)} + e^{\\hat{x}_0} and 0 = \\frac{- \\mathbf{F}{(\\hat{x}_0)} + e^{\\hat{x}_0}}{\\mathbf{F}{(\\hat{x}_0)}} and 0 = \\frac{(- \\mathbf{F}{(\\hat{x}_0)} + e^{\\hat{x}_0}) e^{\\hat{x}_0}}{\\mathbf{F}{(\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True))), exp(Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True))), exp(Symbol('\\\\hat{x}_0', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))))"], [["times", 3, "exp(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True))), exp(Symbol('\\\\hat{x}_0', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), exp(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(l,G)} = l^{G} and \\operatorname{C_{1}}{(l,G)} = - \\int \\frac{d}{d l} \\int 0 dl dG + \\frac{\\int 0 dl}{\\mathbf{r}{(l,G)}}, then obtain \\operatorname{C_{1}}{(l,G)} = - \\int \\frac{\\partial}{\\partial l} \\int (- l^{G} + \\mathbf{r}{(l,G)}) dl dG + \\frac{\\int (- l^{G} + \\mathbf{r}{(l,G)}) dl}{\\mathbf{r}{(l,G)}}", "derivation": "\\mathbf{r}{(l,G)} = l^{G} and - l^{G} + \\mathbf{r}{(l,G)} = 0 and \\int (- l^{G} + \\mathbf{r}{(l,G)}) dl = \\int 0 dl and \\frac{\\int (- l^{G} + \\mathbf{r}{(l,G)}) dl}{\\mathbf{r}{(l,G)}} = \\frac{\\int 0 dl}{\\mathbf{r}{(l,G)}} and \\operatorname{C_{1}}{(l,G)} = - \\int \\frac{d}{d l} \\int 0 dl dG + \\frac{\\int 0 dl}{\\mathbf{r}{(l,G)}} and \\operatorname{C_{1}}{(l,G)} = - \\int \\frac{d}{d l} \\int 0 dl dG + \\frac{\\int (- l^{G} + \\mathbf{r}{(l,G)}) dl}{\\mathbf{r}{(l,G)}} and \\operatorname{C_{1}}{(l,G)} = - \\int \\frac{\\partial}{\\partial l} \\int (- l^{G} + \\mathbf{r}{(l,G)}) dl dG + \\frac{\\int (- l^{G} + \\mathbf{r}{(l,G)}) dl}{\\mathbf{r}{(l,G)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True)))"], [["minus", 1, "Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Integer(0), Tuple(Symbol('l', commutative=True))))"], [["divide", 3, "Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Integral(Derivative(Integral(Integer(0), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('l', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('C_1')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Integral(Derivative(Integral(Integer(0), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('l', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Function('C_1')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Integral(Derivative(Integral(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('G', commutative=True))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('l', commutative=True))))))"]]}, {"prompt": "Given k{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M}, then derive \\frac{d}{d \\mathbf{J}_M} k{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M}, then obtain 1 = e^{- \\mathbf{J}_M} \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M}", "derivation": "k{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} k{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} k{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M} and e^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M} and 1 = e^{- \\mathbf{J}_M} \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["divide", 4, "exp(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(1), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(J_{\\varepsilon},Z)} = - Z + \\cos{(J_{\\varepsilon})}, then obtain \\int (4 Z + 4 \\mathbf{M}{(J_{\\varepsilon},Z)} - 4 \\cos{(J_{\\varepsilon})}) dZ = \\int 0 dZ", "derivation": "\\mathbf{M}{(J_{\\varepsilon},Z)} = - Z + \\cos{(J_{\\varepsilon})} and Z + \\mathbf{M}{(J_{\\varepsilon},Z)} - \\cos{(J_{\\varepsilon})} = 0 and Z + \\mathbf{M}{(J_{\\varepsilon},Z)} = \\cos{(J_{\\varepsilon})} and \\cos{(J_{\\varepsilon})} = - Z - \\mathbf{M}{(J_{\\varepsilon},Z)} + 2 \\cos{(J_{\\varepsilon})} and 2 Z + 2 \\mathbf{M}{(J_{\\varepsilon},Z)} - 2 \\cos{(J_{\\varepsilon})} = 0 and 4 Z + 4 \\mathbf{M}{(J_{\\varepsilon},Z)} - 4 \\cos{(J_{\\varepsilon})} = 0 and \\int (4 Z + 4 \\mathbf{M}{(J_{\\varepsilon},Z)} - 4 \\cos{(J_{\\varepsilon})}) dZ = \\int 0 dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["add", 2, "cos(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["minus", 3, "Add(Symbol('Z', commutative=True), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], "Equality(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(2), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(4), Symbol('Z', commutative=True)), Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["integrate", 6, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(4), Symbol('Z', commutative=True)), Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('Z', commutative=True))), Integral(Integer(0), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\delta{(n_{1})} = \\sin{(n_{1})}, then obtain - \\delta{(n_{1})} \\sin^{3}{(n_{1})} + \\delta^{n_{1}}{(n_{1})} = - \\delta{(n_{1})} \\sin^{3}{(n_{1})} + \\sin^{n_{1}}{(n_{1})}", "derivation": "\\delta{(n_{1})} = \\sin{(n_{1})} and \\delta{(n_{1})} \\sin{(n_{1})} = \\sin^{2}{(n_{1})} and \\delta^{n_{1}}{(n_{1})} = \\sin^{n_{1}}{(n_{1})} and \\delta{(n_{1})} \\sin^{3}{(n_{1})} = \\sin^{4}{(n_{1})} and \\delta^{n_{1}}{(n_{1})} - \\sin^{4}{(n_{1})} = - \\sin^{4}{(n_{1})} + \\sin^{n_{1}}{(n_{1})} and - \\delta{(n_{1})} \\sin^{3}{(n_{1})} + \\delta^{n_{1}}{(n_{1})} = - \\delta{(n_{1})} \\sin^{3}{(n_{1})} + \\sin^{n_{1}}{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('n_1', commutative=True)), sin(Symbol('n_1', commutative=True)))"], [["times", 1, "sin(Symbol('n_1', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('n_1', commutative=True)), sin(Symbol('n_1', commutative=True))), Pow(sin(Symbol('n_1', commutative=True)), Integer(2)))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(sin(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["times", 2, "Pow(sin(Symbol('n_1', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\delta')(Symbol('n_1', commutative=True)), Pow(sin(Symbol('n_1', commutative=True)), Integer(3))), Pow(sin(Symbol('n_1', commutative=True)), Integer(4)))"], [["minus", 3, "Pow(sin(Symbol('n_1', commutative=True)), Integer(4))"], "Equality(Add(Pow(Function('\\\\delta')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('n_1', commutative=True)), Integer(4)))), Add(Mul(Integer(-1), Pow(sin(Symbol('n_1', commutative=True)), Integer(4))), Pow(sin(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('n_1', commutative=True)), Pow(sin(Symbol('n_1', commutative=True)), Integer(3))), Pow(Function('\\\\delta')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('n_1', commutative=True)), Pow(sin(Symbol('n_1', commutative=True)), Integer(3))), Pow(sin(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(Q,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} Q)}, then obtain \\frac{\\varphi^{*}{(Q,L_{\\varepsilon})}}{\\frac{\\partial}{\\partial Q} \\sin{(L_{\\varepsilon} Q)}} = \\frac{\\sin{(L_{\\varepsilon} Q)}}{\\frac{\\partial}{\\partial Q} \\sin{(L_{\\varepsilon} Q)}}", "derivation": "\\varphi^{*}{(Q,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} Q)} and \\frac{\\partial}{\\partial Q} \\varphi^{*}{(Q,L_{\\varepsilon})} = \\frac{\\partial}{\\partial Q} \\sin{(L_{\\varepsilon} Q)} and \\frac{\\varphi^{*}{(Q,L_{\\varepsilon})}}{\\frac{\\partial}{\\partial Q} \\varphi^{*}{(Q,L_{\\varepsilon})}} = \\frac{\\sin{(L_{\\varepsilon} Q)}}{\\frac{\\partial}{\\partial Q} \\varphi^{*}{(Q,L_{\\varepsilon})}} and \\frac{\\varphi^{*}{(Q,L_{\\varepsilon})}}{\\frac{\\partial}{\\partial Q} \\sin{(L_{\\varepsilon} Q)}} = \\frac{\\sin{(L_{\\varepsilon} Q)}}{\\frac{\\partial}{\\partial Q} \\sin{(L_{\\varepsilon} Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Pow(Derivative(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\varphi^*')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Pow(Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\hat{X}{(J,z^{*})} = (z^{*})^{J} and \\pi{(J,z^{*})} = (\\hat{X}^{J}{(J,z^{*})})^{z^{*}}, then obtain \\frac{\\partial}{\\partial z^{*}} \\hat{X}^{J}{(J,z^{*})} \\pi{(J,z^{*})} = \\frac{\\partial}{\\partial z^{*}} ((z^{*})^{J})^{J} \\pi{(J,z^{*})}", "derivation": "\\hat{X}{(J,z^{*})} = (z^{*})^{J} and \\hat{X}^{J}{(J,z^{*})} = ((z^{*})^{J})^{J} and (\\hat{X}^{J}{(J,z^{*})})^{z^{*}} \\hat{X}^{J}{(J,z^{*})} = ((z^{*})^{J})^{J} (\\hat{X}^{J}{(J,z^{*})})^{z^{*}} and \\pi{(J,z^{*})} = (\\hat{X}^{J}{(J,z^{*})})^{z^{*}} and \\hat{X}^{J}{(J,z^{*})} \\pi{(J,z^{*})} = ((z^{*})^{J})^{J} \\pi{(J,z^{*})} and \\frac{\\partial}{\\partial z^{*}} \\hat{X}^{J}{(J,z^{*})} \\pi{(J,z^{*})} = \\frac{\\partial}{\\partial z^{*}} ((z^{*})^{J})^{J} \\pi{(J,z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('z^*', commutative=True), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Symbol('z^*', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["times", 2, "Pow(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Symbol('z^*', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True))), Mul(Pow(Pow(Symbol('z^*', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Symbol('z^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Pow(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Pow(Symbol('z^*', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('z^*', commutative=True))))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True), Symbol('z^*', commutative=True)), Symbol('J', commutative=True)), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Pow(Symbol('z^*', commutative=True), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\mathbf{v})} = \\cos{(\\mathbf{v})}, then obtain \\frac{d^{2}}{d \\mathbf{v}^{2}} \\psi{(\\mathbf{v})} + 1 = 1 - \\cos{(\\mathbf{v})}", "derivation": "\\psi{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\frac{d}{d \\mathbf{v}} \\psi{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} and \\frac{d^{2}}{d \\mathbf{v}^{2}} \\psi{(\\mathbf{v})} = \\frac{d^{2}}{d \\mathbf{v}^{2}} \\cos{(\\mathbf{v})} and \\frac{d^{2}}{d \\mathbf{v}^{2}} \\psi{(\\mathbf{v})} + 1 = \\frac{d^{2}}{d \\mathbf{v}^{2}} \\cos{(\\mathbf{v})} + 1 and \\frac{d^{2}}{d \\mathbf{v}^{2}} \\psi{(\\mathbf{v})} + 1 = 1 - \\cos{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Integer(1)), Add(Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Integer(1)), Add(Integer(1), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then obtain \\int \\frac{d^{2}}{d \\varepsilon_0^{2}} \\operatorname{A_{x}}{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\frac{d^{2}}{d \\varepsilon_0^{2}} \\sin{(\\varepsilon_0)} d\\varepsilon_0", "derivation": "\\operatorname{A_{x}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\operatorname{A_{x}}{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\sin{(\\varepsilon_0)} and \\frac{d^{2}}{d \\varepsilon_0^{2}} \\operatorname{A_{x}}{(\\varepsilon_0)} = \\frac{d^{2}}{d \\varepsilon_0^{2}} \\sin{(\\varepsilon_0)} and \\int \\frac{d^{2}}{d \\varepsilon_0^{2}} \\operatorname{A_{x}}{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\frac{d^{2}}{d \\varepsilon_0^{2}} \\sin{(\\varepsilon_0)} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Derivative(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Derivative(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given h{(V,n_{2},J_{\\varepsilon})} = \\frac{J_{\\varepsilon} n_{2}}{V}, then obtain - h{(V,n_{2},J_{\\varepsilon})} + \\int (h{(V,n_{2},J_{\\varepsilon})} - 1) dn_{2} = - h{(V,n_{2},J_{\\varepsilon})} + \\int (\\frac{J_{\\varepsilon} n_{2}}{V} - 1) dn_{2}", "derivation": "h{(V,n_{2},J_{\\varepsilon})} = \\frac{J_{\\varepsilon} n_{2}}{V} and h{(V,n_{2},J_{\\varepsilon})} - 1 = \\frac{J_{\\varepsilon} n_{2}}{V} - 1 and \\int (h{(V,n_{2},J_{\\varepsilon})} - 1) dn_{2} = \\int (\\frac{J_{\\varepsilon} n_{2}}{V} - 1) dn_{2} and - n_{2} + \\int (h{(V,n_{2},J_{\\varepsilon})} - 1) dn_{2} = - n_{2} + \\int (\\frac{J_{\\varepsilon} n_{2}}{V} - 1) dn_{2} and - h{(V,n_{2},J_{\\varepsilon})} + \\int (h{(V,n_{2},J_{\\varepsilon})} - 1) dn_{2} = - h{(V,n_{2},J_{\\varepsilon})} + \\int (\\frac{J_{\\varepsilon} n_{2}}{V} - 1) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Integer(-1)))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Add(Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Integral(Add(Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Integral(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Function('h')(Symbol('V', commutative=True), Symbol('n_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Integer(-1)), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\mu{(A,E_{x})} = \\sin{(A E_{x})}, then obtain \\int \\mu{(A,E_{x})} dE_{x} + (\\int \\mu{(A,E_{x})} dE_{x})^{E_{x}} = \\int \\mu{(A,E_{x})} dE_{x} + (\\int \\sin{(A E_{x})} dE_{x})^{E_{x}}", "derivation": "\\mu{(A,E_{x})} = \\sin{(A E_{x})} and \\int \\mu{(A,E_{x})} dE_{x} = \\int \\sin{(A E_{x})} dE_{x} and (\\int \\mu{(A,E_{x})} dE_{x})^{E_{x}} = (\\int \\sin{(A E_{x})} dE_{x})^{E_{x}} and \\int \\mu{(A,E_{x})} dE_{x} + (\\int \\mu{(A,E_{x})} dE_{x})^{E_{x}} = \\int \\mu{(A,E_{x})} dE_{x} + (\\int \\sin{(A E_{x})} dE_{x})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), sin(Mul(Symbol('A', commutative=True), Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(sin(Mul(Symbol('A', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Pow(Integral(sin(Mul(Symbol('A', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["add", 3, "Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))), Add(Integral(Function('\\\\mu')(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Pow(Integral(sin(Mul(Symbol('A', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given a{(T,y^{\\prime})} = e^{\\frac{T}{y^{\\prime}}}, then derive \\eta + a{(T,y^{\\prime})} = \\varphi^* + e^{\\frac{T}{y^{\\prime}}}, then obtain \\eta + e^{\\frac{T}{y^{\\prime}}} = \\eta + a{(T,y^{\\prime})}", "derivation": "a{(T,y^{\\prime})} = e^{\\frac{T}{y^{\\prime}}} and \\frac{\\partial}{\\partial T} a{(T,y^{\\prime})} = \\frac{\\partial}{\\partial T} e^{\\frac{T}{y^{\\prime}}} and \\int \\frac{\\partial}{\\partial T} a{(T,y^{\\prime})} dT = \\int \\frac{\\partial}{\\partial T} e^{\\frac{T}{y^{\\prime}}} dT and \\eta + a{(T,y^{\\prime})} = \\varphi^* + e^{\\frac{T}{y^{\\prime}}} and \\eta + e^{\\frac{T}{y^{\\prime}}} = \\varphi^* + e^{\\frac{T}{y^{\\prime}}} and \\eta + e^{\\frac{T}{y^{\\prime}}} = \\eta + a{(T,y^{\\prime})}", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('T', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('T', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Function('a')(Symbol('T', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('a')(Symbol('T', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\eta', commutative=True), exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))), Add(Symbol('\\\\varphi^*', commutative=True), exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\eta', commutative=True), exp(Mul(Symbol('T', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))), Add(Symbol('\\\\eta', commutative=True), Function('a')(Symbol('T', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\psi^*)} = e^{\\psi^*}, then derive \\frac{d^{2}}{d (\\psi^*)^{2}} \\hat{p}_0{(\\psi^*)} = e^{\\psi^*}, then obtain e^{\\psi^*} = \\frac{d^{2}}{d (\\psi^*)^{2}} e^{\\psi^*}", "derivation": "\\hat{p}_0{(\\psi^*)} = e^{\\psi^*} and \\frac{d}{d \\psi^*} \\hat{p}_0{(\\psi^*)} = \\frac{d}{d \\psi^*} e^{\\psi^*} and \\frac{d^{2}}{d (\\psi^*)^{2}} \\hat{p}_0{(\\psi^*)} = \\frac{d^{2}}{d (\\psi^*)^{2}} e^{\\psi^*} and \\frac{d^{2}}{d (\\psi^*)^{2}} \\hat{p}_0{(\\psi^*)} = e^{\\psi^*} and e^{\\psi^*} = \\frac{d^{2}}{d (\\psi^*)^{2}} e^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), exp(Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Symbol('\\\\psi^*', commutative=True)), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{P},A_{z})} = \\cos{(A_{z} + \\mathbf{P})}, then obtain \\mathbf{v} + \\theta_{1}{(\\mathbf{P},A_{z})} = \\Psi_{nl} + \\cos{(A_{z} + \\mathbf{P})}", "derivation": "\\theta_{1}{(\\mathbf{P},A_{z})} = \\cos{(A_{z} + \\mathbf{P})} and \\frac{\\partial}{\\partial \\mathbf{P}} \\theta_{1}{(\\mathbf{P},A_{z})} = \\frac{\\partial}{\\partial \\mathbf{P}} \\cos{(A_{z} + \\mathbf{P})} and \\int \\frac{\\partial}{\\partial \\mathbf{P}} \\theta_{1}{(\\mathbf{P},A_{z})} d\\mathbf{P} = \\int \\frac{\\partial}{\\partial \\mathbf{P}} \\cos{(A_{z} + \\mathbf{P})} d\\mathbf{P} and \\mathbf{v} + \\theta_{1}{(\\mathbf{P},A_{z})} = \\Psi_{nl} + \\cos{(A_{z} + \\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_z', commutative=True)), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\theta_1')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(x^\\prime)} = x^\\prime, then obtain (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)} - \\frac{1}{\\hat{H}{(x^\\prime)}})^{x^\\prime} = (\\frac{d}{d x^\\prime} x^\\prime - \\frac{1}{\\hat{H}{(x^\\prime)}})^{x^\\prime}", "derivation": "\\hat{H}{(x^\\prime)} = x^\\prime and \\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)} = \\frac{d}{d x^\\prime} x^\\prime and \\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)} - \\frac{1}{\\hat{H}{(x^\\prime)}} = \\frac{d}{d x^\\prime} x^\\prime - \\frac{1}{\\hat{H}{(x^\\prime)}} and (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)} - \\frac{1}{\\hat{H}{(x^\\prime)}})^{x^\\prime} = (\\frac{d}{d x^\\prime} x^\\prime - \\frac{1}{\\hat{H}{(x^\\prime)}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))), Add(Derivative(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Derivative(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{\\mathbf{x}},f^{\\prime})} = \\hat{\\mathbf{x}} + \\log{(f^{\\prime})} and \\theta_{2}{(f^{\\prime})} = \\log{(f^{\\prime})}, then obtain \\sin{((\\hat{\\mathbf{x}} + \\log{(f^{\\prime})})^{\\hat{\\mathbf{x}}})} = \\sin{((\\hat{\\mathbf{x}} + \\theta_{2}{(f^{\\prime})})^{\\hat{\\mathbf{x}}})}", "derivation": "\\operatorname{A_{y}}{(\\hat{\\mathbf{x}},f^{\\prime})} = \\hat{\\mathbf{x}} + \\log{(f^{\\prime})} and \\theta_{2}{(f^{\\prime})} = \\log{(f^{\\prime})} and \\operatorname{A_{y}}{(\\hat{\\mathbf{x}},f^{\\prime})} = \\hat{\\mathbf{x}} + \\theta_{2}{(f^{\\prime})} and \\hat{\\mathbf{x}} + \\log{(f^{\\prime})} = \\hat{\\mathbf{x}} + \\theta_{2}{(f^{\\prime})} and (\\hat{\\mathbf{x}} + \\log{(f^{\\prime})})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} + \\theta_{2}{(f^{\\prime})})^{\\hat{\\mathbf{x}}} and \\sin{((\\hat{\\mathbf{x}} + \\log{(f^{\\prime})})^{\\hat{\\mathbf{x}}})} = \\sin{((\\hat{\\mathbf{x}} + \\theta_{2}{(f^{\\prime})})^{\\hat{\\mathbf{x}}})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["sin", 5], "Equality(sin(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), sin(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given H{(P_{e},G)} = \\frac{P_{e}}{G} and \\mathbf{A}{(P_{e},G)} = \\frac{P_{e}}{G}, then obtain 1 = (\\frac{G \\mathbf{A}{(P_{e},G)}}{P_{e}})^{G}", "derivation": "H{(P_{e},G)} = \\frac{P_{e}}{G} and \\mathbf{A}{(P_{e},G)} = \\frac{P_{e}}{G} and \\mathbf{A}{(P_{e},G)} = H{(P_{e},G)} and 1 = \\frac{H{(P_{e},G)}}{\\mathbf{A}{(P_{e},G)}} and 1 = \\frac{G H{(P_{e},G)}}{P_{e}} and 1 = \\frac{G \\mathbf{A}{(P_{e},G)}}{P_{e}} and 1 = (\\frac{G \\mathbf{A}{(P_{e},G)}}{P_{e}})^{G}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('P_e', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Function('H')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)))"], [["divide", 3, "Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(1), Mul(Function('H')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Symbol('G', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('H')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Mul(Symbol('G', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))))"], [["power", 6, "Symbol('G', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('P_e', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\operatorname{M_{E}}{(a^{\\dagger})} e^{a^{\\dagger}}, then obtain \\operatorname{x^{{\\}'}}{(a^{\\dagger})} + 1 = \\operatorname{M_{E}}^{2}{(a^{\\dagger})} + 1", "derivation": "\\operatorname{M_{E}}{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\operatorname{M_{E}}{(a^{\\dagger})} e^{a^{\\dagger}} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = e^{2 a^{\\dagger}} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} + \\frac{e^{a^{\\dagger}}}{\\operatorname{M_{E}}{(a^{\\dagger})}} = e^{2 a^{\\dagger}} + \\frac{e^{a^{\\dagger}}}{\\operatorname{M_{E}}{(a^{\\dagger})}} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} + 1 = \\operatorname{M_{E}}^{2}{(a^{\\dagger})} + 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 3, "Mul(Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1)), Add(Pow(Function('M_E')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(I,u)} = \\sin{(I + u)}, then obtain \\iint \\frac{1}{I} dI dI = \\iint \\frac{\\sin{(\\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u)}}{I \\operatorname{F_{g}}{(I,u)}} dI dI", "derivation": "\\operatorname{F_{g}}{(I,u)} = \\sin{(I + u)} and I \\operatorname{F_{g}}{(I,u)} = I \\sin{(I + u)} and I = \\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} and I + u = \\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u and \\operatorname{F_{g}}{(I,u)} = \\sin{(\\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u)} and \\frac{1}{I} = \\frac{\\sin{(\\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u)}}{I \\operatorname{F_{g}}{(I,u)}} and \\int \\frac{1}{I} dI = \\int \\frac{\\sin{(\\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u)}}{I \\operatorname{F_{g}}{(I,u)}} dI and \\iint \\frac{1}{I} dI dI = \\iint \\frac{\\sin{(\\frac{I \\sin{(I + u)}}{\\operatorname{F_{g}}{(I,u)}} + u)}}{I \\operatorname{F_{g}}{(I,u)}} dI dI", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True))))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('I', commutative=True), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))))"], [["divide", 2, "Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True))"], "Equality(Symbol('I', commutative=True), Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))))"], [["add", 3, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)), Add(Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), sin(Add(Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True))))"], [["divide", 5, "Mul(Symbol('I', commutative=True), Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)))"], "Equality(Pow(Symbol('I', commutative=True), Integer(-1)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))))"], [["integrate", 6, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))), Tuple(Symbol('I', commutative=True))))"], [["integrate", 7, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Mul(Symbol('I', commutative=True), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('u', commutative=True)), Integer(-1)), sin(Add(Symbol('I', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\phi_1)} = e^{\\phi_1} and H{(\\phi_1)} = - \\operatorname{v_{1}}{(\\phi_1)} e^{\\phi_1} + \\operatorname{v_{1}}{(\\phi_1)}, then obtain (\\frac{d}{d \\phi_1} H^{4}{(\\phi_1)})^{\\phi_1} = (\\frac{d}{d \\phi_1} (- \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)})^{4})^{\\phi_1}", "derivation": "\\operatorname{v_{1}}{(\\phi_1)} = e^{\\phi_1} and H{(\\phi_1)} = - \\operatorname{v_{1}}{(\\phi_1)} e^{\\phi_1} + \\operatorname{v_{1}}{(\\phi_1)} and H{(\\phi_1)} = - \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)} and H^{2}{(\\phi_1)} = (- \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)})^{2} and H^{4}{(\\phi_1)} = (- \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)})^{4} and \\frac{d}{d \\phi_1} H^{4}{(\\phi_1)} = \\frac{d}{d \\phi_1} (- \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)})^{4} and (\\frac{d}{d \\phi_1} H^{4}{(\\phi_1)})^{\\phi_1} = (\\frac{d}{d \\phi_1} (- \\operatorname{v_{1}}^{2}{(\\phi_1)} + \\operatorname{v_{1}}{(\\phi_1)})^{4})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Function('v_1')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('H')(Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('H')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Pow(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Function('H')(Symbol('\\\\phi_1', commutative=True)), Integer(4)), Pow(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))), Integer(4)))"], [["differentiate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Function('H')(Symbol('\\\\phi_1', commutative=True)), Integer(4)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))), Integer(4)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["power", 6, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('H')(Symbol('\\\\phi_1', commutative=True)), Integer(4)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Pow(Add(Mul(Integer(-1), Pow(Function('v_1')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Function('v_1')(Symbol('\\\\phi_1', commutative=True))), Integer(4)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(b,U)} = U b, then derive - U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)} = - U b + b, then obtain U b + \\int (- U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)}) db = U b + \\int (- U b + b) db", "derivation": "\\mathbf{M}{(b,U)} = U b and \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)} = \\frac{\\partial}{\\partial U} U b and - U \\frac{\\partial}{\\partial U} U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)} = - U \\frac{\\partial}{\\partial U} U b + \\frac{\\partial}{\\partial U} U b and - U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)} = - U b + b and \\int (- U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)}) db = \\int (- U b + b) db and U \\frac{\\partial}{\\partial U} U b + \\int (- U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)}) db = U \\frac{\\partial}{\\partial U} U b + \\int (- U b + b) db and - U b + \\mathbf{M}{(b,U)} + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)} = - U b + b + \\mathbf{M}{(b,U)} and \\frac{\\partial}{\\partial U} U b = b and U b + \\int (- U b + \\frac{\\partial}{\\partial U} \\mathbf{M}{(b,U)}) db = U b + \\int (- U b + b) db", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], "Equality(Add(Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True)))), Add(Mul(Symbol('U', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["add", 4, "Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True), Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Derivative(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('b', commutative=True))"], [["substitute_LHS_for_RHS", 6, 8], "Equality(Add(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True)))), Add(Mul(Symbol('U', commutative=True), Symbol('b', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{z},\\varepsilon)} = A_{z} + \\varepsilon, then derive \\int \\operatorname{F_{x}}{(A_{z},\\varepsilon)} dA_{z} = \\frac{A_{z}^{2}}{2} + A_{z} \\varepsilon + g_{\\varepsilon}, then obtain \\frac{\\int (A_{z} + \\varepsilon) dA_{z}}{A_{z}} = \\frac{\\frac{A_{z}^{2}}{2} + A_{z} \\varepsilon + g_{\\varepsilon}}{A_{z}}", "derivation": "\\operatorname{F_{x}}{(A_{z},\\varepsilon)} = A_{z} + \\varepsilon and \\int \\operatorname{F_{x}}{(A_{z},\\varepsilon)} dA_{z} = \\int (A_{z} + \\varepsilon) dA_{z} and \\int \\operatorname{F_{x}}{(A_{z},\\varepsilon)} dA_{z} = \\frac{A_{z}^{2}}{2} + A_{z} \\varepsilon + g_{\\varepsilon} and \\frac{\\int \\operatorname{F_{x}}{(A_{z},\\varepsilon)} dA_{z}}{A_{z}} = \\frac{\\frac{A_{z}^{2}}{2} + A_{z} \\varepsilon + g_{\\varepsilon}}{A_{z}} and \\frac{\\int (A_{z} + \\varepsilon) dA_{z}}{A_{z}} = \\frac{\\frac{A_{z}^{2}}{2} + A_{z} \\varepsilon + g_{\\varepsilon}}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 3, "Symbol('A_z', commutative=True)"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Function('F_x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Mul(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(s)} = e^{s} and x{(s)} = s + e^{s}, then obtain s = - \\operatorname{a^{\\dagger}}{(s)} + x{(s)}", "derivation": "\\operatorname{a^{\\dagger}}{(s)} = e^{s} and s + \\operatorname{a^{\\dagger}}{(s)} = s + e^{s} and x{(s)} = s + e^{s} and s + \\operatorname{a^{\\dagger}}{(s)} = x{(s)} and s = - \\operatorname{a^{\\dagger}}{(s)} + x{(s)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["add", 1, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('a^{\\\\dagger}')(Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), exp(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('s', commutative=True)), Add(Symbol('s', commutative=True), exp(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('s', commutative=True), Function('a^{\\\\dagger}')(Symbol('s', commutative=True))), Function('x')(Symbol('s', commutative=True)))"], [["minus", 4, "Function('a^{\\\\dagger}')(Symbol('s', commutative=True))"], "Equality(Symbol('s', commutative=True), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('s', commutative=True))), Function('x')(Symbol('s', commutative=True))))"]]}, {"prompt": "Given l{(H,B)} = \\frac{\\partial}{\\partial B} B H, then obtain - F_{N} + \\frac{\\int l{(H,B)} dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H} = - F_{N} + \\frac{\\int \\frac{\\partial}{\\partial B} B H dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H}", "derivation": "l{(H,B)} = \\frac{\\partial}{\\partial B} B H and \\int l{(H,B)} dB = \\int \\frac{\\partial}{\\partial B} B H dB and \\frac{\\partial}{\\partial B} l{(H,B)} = \\frac{\\partial^{2}}{\\partial B^{2}} B H and \\frac{\\int l{(H,B)} dB}{\\frac{\\partial}{\\partial B} l{(H,B)}} = \\frac{\\int \\frac{\\partial}{\\partial B} B H dB}{\\frac{\\partial}{\\partial B} l{(H,B)}} and \\frac{\\int l{(H,B)} dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H} = \\frac{\\int \\frac{\\partial}{\\partial B} B H dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H} and - F_{N} + \\frac{\\int l{(H,B)} dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H} = - F_{N} + \\frac{\\int \\frac{\\partial}{\\partial B} B H dB}{\\frac{\\partial^{2}}{\\partial B^{2}} B H}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["divide", 2, "Derivative(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1)), Integral(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Derivative(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Integer(-1)), Integral(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Integer(-1)), Integral(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True)))))"], [["minus", 5, "Symbol('F_N', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Integer(-1)), Integral(Function('l')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Integer(-1)), Integral(Derivative(Mul(Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} = e^{v v_{x}}, then obtain \\frac{\\partial}{\\partial v_{x}} (v - v_{x} + e^{v v_{x}}) = \\frac{\\partial}{\\partial v_{x}} (v - v_{x} - \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} + 2 e^{v v_{x}})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} = e^{v v_{x}} and - v_{x} + \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} = - v_{x} + e^{v v_{x}} and - v_{x} = - v_{x} - \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} + e^{v v_{x}} and - v_{x} + e^{v v_{x}} = - v_{x} - \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} + 2 e^{v v_{x}} and v - v_{x} + e^{v v_{x}} = v - v_{x} - \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} + 2 e^{v v_{x}} and \\frac{\\partial}{\\partial v_{x}} (v - v_{x} + e^{v v_{x}}) = \\frac{\\partial}{\\partial v_{x}} (v - v_{x} - \\operatorname{f_{\\mathbf{p}}}{(v_{x},v)} + 2 e^{v v_{x}})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True)), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True))))"], [["minus", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True)))))"], [["minus", 2, "Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_x', commutative=True)), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True))))))"], [["add", 4, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True)))), Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True))))))"], [["differentiate", 5, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), exp(Mul(Symbol('v', commutative=True), Symbol('v_x', commutative=True))))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(P_{g},n)} = \\frac{n}{P_{g}}, then obtain \\frac{\\sin{(\\eta{(P_{g},n)})} + \\frac{n}{P_{g}^{2}}}{\\sin{(\\eta{(P_{g},n)})}} = \\frac{\\sin{(\\frac{n}{P_{g}})} + \\frac{n}{P_{g}^{2}}}{\\sin{(\\eta{(P_{g},n)})}}", "derivation": "\\eta{(P_{g},n)} = \\frac{n}{P_{g}} and \\frac{\\eta{(P_{g},n)}}{P_{g}} = \\frac{n}{P_{g}^{2}} and \\sin{(\\eta{(P_{g},n)})} = \\sin{(\\frac{n}{P_{g}})} and \\sin{(\\eta{(P_{g},n)})} + \\frac{\\eta{(P_{g},n)}}{P_{g}} = \\sin{(\\frac{n}{P_{g}})} + \\frac{\\eta{(P_{g},n)}}{P_{g}} and \\sin{(\\eta{(P_{g},n)})} + \\frac{n}{P_{g}^{2}} = \\sin{(\\frac{n}{P_{g}})} + \\frac{n}{P_{g}^{2}} and \\frac{\\sin{(\\eta{(P_{g},n)})} + \\frac{n}{P_{g}^{2}}}{\\sin{(\\eta{(P_{g},n)})}} = \\frac{\\sin{(\\frac{n}{P_{g}})} + \\frac{n}{P_{g}^{2}}}{\\sin{(\\eta{(P_{g},n)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["divide", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-2)), Symbol('n', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))), Add(sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-2)), Symbol('n', commutative=True))), Add(sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-2)), Symbol('n', commutative=True))))"], [["divide", 5, "sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Add(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-2)), Symbol('n', commutative=True))), Pow(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Integer(-1))), Mul(Add(sin(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-2)), Symbol('n', commutative=True))), Pow(sin(Function('\\\\eta')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(E_{n},\\hat{H},\\tilde{g})} = E_{n} + \\tilde{g}^{\\hat{H}}, then obtain - E_{n} + \\frac{\\int k{(E_{n},\\hat{H},\\tilde{g})} d\\hat{H}}{E_{n}} = - E_{n} + \\frac{\\int (E_{n} + \\tilde{g}^{\\hat{H}}) d\\hat{H}}{E_{n}}", "derivation": "k{(E_{n},\\hat{H},\\tilde{g})} = E_{n} + \\tilde{g}^{\\hat{H}} and \\int k{(E_{n},\\hat{H},\\tilde{g})} d\\hat{H} = \\int (E_{n} + \\tilde{g}^{\\hat{H}}) d\\hat{H} and \\frac{\\int k{(E_{n},\\hat{H},\\tilde{g})} d\\hat{H}}{E_{n}} = \\frac{\\int (E_{n} + \\tilde{g}^{\\hat{H}}) d\\hat{H}}{E_{n}} and - E_{n} + \\frac{\\int k{(E_{n},\\hat{H},\\tilde{g})} d\\hat{H}}{E_{n}} = - E_{n} + \\frac{\\int (E_{n} + \\tilde{g}^{\\hat{H}}) d\\hat{H}}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 2, "Symbol('E_n', commutative=True)"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Integral(Function('k')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Integral(Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 3, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Integral(Function('k')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Integral(Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given M{(\\mathbf{f},\\sigma_p)} = - \\mathbf{f} + e^{\\sigma_p}, then derive \\int M{(\\mathbf{f},\\sigma_p)} d\\mathbf{f} = \\hat{x} - \\frac{\\mathbf{f}^{2}}{2} + \\mathbf{f} e^{\\sigma_p}, then obtain \\int (- \\mathbf{f} + e^{\\sigma_p}) d\\mathbf{f} = \\hat{x} - \\frac{\\mathbf{f}^{2}}{2} + \\mathbf{f} e^{\\sigma_p}", "derivation": "M{(\\mathbf{f},\\sigma_p)} = - \\mathbf{f} + e^{\\sigma_p} and \\int M{(\\mathbf{f},\\sigma_p)} d\\mathbf{f} = \\int (- \\mathbf{f} + e^{\\sigma_p}) d\\mathbf{f} and \\int M{(\\mathbf{f},\\sigma_p)} d\\mathbf{f} = \\hat{x} - \\frac{\\mathbf{f}^{2}}{2} + \\mathbf{f} e^{\\sigma_p} and \\int (- \\mathbf{f} + e^{\\sigma_p}) d\\mathbf{f} = \\hat{x} - \\frac{\\mathbf{f}^{2}}{2} + \\mathbf{f} e^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain - \\frac{\\operatorname{F_{c}}{(a^{\\dagger})} - \\log{(a^{\\dagger})}}{\\log{(a^{\\dagger})}} + \\operatorname{F_{c}}^{2}{(a^{\\dagger})} + \\log{(a^{\\dagger})} = \\operatorname{F_{c}}^{2}{(a^{\\dagger})} + \\log{(a^{\\dagger})}", "derivation": "\\operatorname{F_{c}}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and \\operatorname{F_{c}}{(a^{\\dagger})} - \\log{(a^{\\dagger})} = 0 and \\frac{\\operatorname{F_{c}}{(a^{\\dagger})} - \\log{(a^{\\dagger})}}{\\log{(a^{\\dagger})}} = 0 and - \\frac{\\operatorname{F_{c}}{(a^{\\dagger})} - \\log{(a^{\\dagger})}}{\\log{(a^{\\dagger})}} = 0 and - \\frac{\\operatorname{F_{c}}{(a^{\\dagger})} - \\log{(a^{\\dagger})}}{\\log{(a^{\\dagger})}} + \\operatorname{F_{c}}^{2}{(a^{\\dagger})} + \\log{(a^{\\dagger})} = \\operatorname{F_{c}}^{2}{(a^{\\dagger})} + \\log{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 1, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(0))"], [["divide", 2, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(0))"], [["add", 4, "Add(Pow(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Pow(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), log(Symbol('a^{\\\\dagger}', commutative=True))), Add(Pow(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), log(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\omega{(f^{*},\\dot{\\mathbf{r}})} = (f^{*})^{\\dot{\\mathbf{r}}}, then obtain \\frac{\\omega{(f^{*},\\dot{\\mathbf{r}})}}{\\int (f^{*})^{\\dot{\\mathbf{r}}} df^{*}} = \\frac{(f^{*})^{\\dot{\\mathbf{r}}}}{\\int (f^{*})^{\\dot{\\mathbf{r}}} df^{*}}", "derivation": "\\omega{(f^{*},\\dot{\\mathbf{r}})} = (f^{*})^{\\dot{\\mathbf{r}}} and \\int \\omega{(f^{*},\\dot{\\mathbf{r}})} df^{*} = \\int (f^{*})^{\\dot{\\mathbf{r}}} df^{*} and \\frac{\\omega{(f^{*},\\dot{\\mathbf{r}})}}{\\int \\omega{(f^{*},\\dot{\\mathbf{r}})} df^{*}} = \\frac{(f^{*})^{\\dot{\\mathbf{r}}}}{\\int \\omega{(f^{*},\\dot{\\mathbf{r}})} df^{*}} and \\frac{\\omega{(f^{*},\\dot{\\mathbf{r}})}}{\\int (f^{*})^{\\dot{\\mathbf{r}}} df^{*}} = \\frac{(f^{*})^{\\dot{\\mathbf{r}}}}{\\int (f^{*})^{\\dot{\\mathbf{r}}} df^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integer(-1))), Mul(Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\omega')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integer(-1))), Mul(Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(Pow(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(i)} = e^{i} and n{(i)} = 3 \\varphi{(i)} + e^{i}, then obtain \\frac{1}{\\varphi{(i)}} = \\frac{3 \\varphi{(i)} + e^{i}}{\\varphi{(i)} n{(i)}}", "derivation": "\\varphi{(i)} = e^{i} and 2 \\varphi{(i)} = \\varphi{(i)} + e^{i} and 3 \\varphi{(i)} = 2 \\varphi{(i)} + e^{i} and 4 \\varphi{(i)} = 3 \\varphi{(i)} + e^{i} and n{(i)} = 3 \\varphi{(i)} + e^{i} and n{(i)} = 4 \\varphi{(i)} and \\frac{1}{\\varphi{(i)}} = \\frac{3 \\varphi{(i)} + e^{i}}{4 \\varphi^{2}{(i)}} and \\frac{1}{\\varphi{(i)}} = \\frac{3 \\varphi{(i)} + e^{i}}{\\varphi{(i)} n{(i)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["add", 1, "Function('\\\\varphi')(Symbol('i', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi')(Symbol('i', commutative=True))), Add(Function('\\\\varphi')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True))))"], [["add", 2, "Function('\\\\varphi')(Symbol('i', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\varphi')(Symbol('i', commutative=True))), Add(Mul(Integer(2), Function('\\\\varphi')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))))"], [["add", 3, "Function('\\\\varphi')(Symbol('i', commutative=True))"], "Equality(Mul(Integer(4), Function('\\\\varphi')(Symbol('i', commutative=True))), Add(Mul(Integer(3), Function('\\\\varphi')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('i', commutative=True)), Add(Mul(Integer(3), Function('\\\\varphi')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('n')(Symbol('i', commutative=True)), Mul(Integer(4), Function('\\\\varphi')(Symbol('i', commutative=True))))"], [["divide", 4, "Mul(Integer(4), Pow(Function('\\\\varphi')(Symbol('i', commutative=True)), Integer(2)))"], "Equality(Pow(Function('\\\\varphi')(Symbol('i', commutative=True)), Integer(-1)), Mul(Rational(1, 4), Add(Mul(Integer(3), Function('\\\\varphi')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))), Pow(Function('\\\\varphi')(Symbol('i', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Function('\\\\varphi')(Symbol('i', commutative=True)), Integer(-1)), Mul(Add(Mul(Integer(3), Function('\\\\varphi')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))), Pow(Function('\\\\varphi')(Symbol('i', commutative=True)), Integer(-1)), Pow(Function('n')(Symbol('i', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\lambda{(z^{*})} = \\cos{(z^{*})}, then derive \\frac{d}{d z^{*}} \\lambda{(z^{*})} = - \\sin{(z^{*})}, then obtain \\frac{d}{d z^{*}} \\lambda{(z^{*})} - 1 = - \\sin{(z^{*})} - 1", "derivation": "\\lambda{(z^{*})} = \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\lambda{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\lambda{(z^{*})} = - \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\lambda{(z^{*})} - 1 = - \\sin{(z^{*})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('z^*', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(t_{1})} = t_{1}, then derive \\int \\operatorname{C_{2}}{(t_{1})} dt_{1} = p + \\frac{t_{1}^{2}}{2}, then obtain \\mathbf{r} + \\frac{\\operatorname{C_{2}}^{2}{(t_{1})}}{2} = p + \\frac{\\operatorname{C_{2}}^{2}{(t_{1})}}{2}", "derivation": "\\operatorname{C_{2}}{(t_{1})} = t_{1} and \\int \\operatorname{C_{2}}{(t_{1})} dt_{1} = \\int t_{1} dt_{1} and \\int \\operatorname{C_{2}}{(t_{1})} dt_{1} = p + \\frac{t_{1}^{2}}{2} and \\int \\operatorname{C_{2}}{(t_{1})} d\\operatorname{C_{2}}{(t_{1})} = p + \\frac{\\operatorname{C_{2}}^{2}{(t_{1})}}{2} and \\mathbf{r} + \\frac{\\operatorname{C_{2}}^{2}{(t_{1})}}{2} = p + \\frac{\\operatorname{C_{2}}^{2}{(t_{1})}}{2}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('p', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('C_2')(Symbol('t_1', commutative=True)), Tuple(Function('C_2')(Symbol('t_1', commutative=True)))), Add(Symbol('p', commutative=True), Mul(Rational(1, 2), Pow(Function('C_2')(Symbol('t_1', commutative=True)), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Rational(1, 2), Pow(Function('C_2')(Symbol('t_1', commutative=True)), Integer(2)))), Add(Symbol('p', commutative=True), Mul(Rational(1, 2), Pow(Function('C_2')(Symbol('t_1', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given v{(n_{2},H,v_{z})} = \\frac{H}{v_{z}} - n_{2}, then obtain \\frac{H}{v_{z}} + 2 n_{2} = \\frac{H}{v_{z}} + \\frac{n_{2} (\\frac{H}{v_{z}} - n_{2})}{v{(n_{2},H,v_{z})}} + n_{2}", "derivation": "v{(n_{2},H,v_{z})} = \\frac{H}{v_{z}} - n_{2} and 1 = \\frac{\\frac{H}{v_{z}} - n_{2}}{v{(n_{2},H,v_{z})}} and n_{2} = \\frac{n_{2} (\\frac{H}{v_{z}} - n_{2})}{v{(n_{2},H,v_{z})}} and 2 n_{2} = \\frac{n_{2} (\\frac{H}{v_{z}} - n_{2})}{v{(n_{2},H,v_{z})}} + n_{2} and \\frac{H}{v_{z}} + 2 n_{2} = \\frac{H}{v_{z}} + \\frac{n_{2} (\\frac{H}{v_{z}} - n_{2})}{v{(n_{2},H,v_{z})}} + n_{2}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["divide", 1, "Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Pow(Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["times", 2, "Symbol('n_2', commutative=True)"], "Equality(Symbol('n_2', commutative=True), Mul(Symbol('n_2', commutative=True), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Pow(Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["add", 3, "Symbol('n_2', commutative=True)"], "Equality(Mul(Integer(2), Symbol('n_2', commutative=True)), Add(Mul(Symbol('n_2', commutative=True), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Pow(Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Symbol('n_2', commutative=True)))"], [["add", 4, "Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('n_2', commutative=True))), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Symbol('n_2', commutative=True), Add(Mul(Symbol('H', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Pow(Function('v')(Symbol('n_2', commutative=True), Symbol('H', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\hbar)} = \\cos{(\\hbar)}, then obtain \\cos^{2}{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\mathbf{A}{(\\hbar)} d\\hbar = \\cos^{2}{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\cos{(\\hbar)} d\\hbar", "derivation": "\\mathbf{A}{(\\hbar)} = \\cos{(\\hbar)} and \\mathbf{A}{(\\hbar)} \\cos{(\\hbar)} = \\cos^{2}{(\\hbar)} and \\int \\mathbf{A}{(\\hbar)} d\\hbar = \\int \\cos{(\\hbar)} d\\hbar and \\frac{d}{d \\hbar} \\int \\mathbf{A}{(\\hbar)} d\\hbar = \\frac{d}{d \\hbar} \\int \\cos{(\\hbar)} d\\hbar and \\mathbf{A}{(\\hbar)} \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\mathbf{A}{(\\hbar)} d\\hbar = \\mathbf{A}{(\\hbar)} \\cos{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\cos{(\\hbar)} d\\hbar and \\cos^{2}{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\mathbf{A}{(\\hbar)} d\\hbar = \\cos^{2}{(\\hbar)} + \\frac{d}{d \\hbar} \\int \\cos{(\\hbar)} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["add", 4, "Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Derivative(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Derivative(Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2)), Derivative(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2)), Derivative(Integral(cos(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(\\phi,\\delta)} = \\log{(\\frac{\\delta}{\\phi})}, then derive \\frac{\\partial}{\\partial \\delta} \\chi{(\\phi,\\delta)} = \\frac{1}{\\delta}, then obtain \\log{(\\frac{\\frac{\\partial}{\\partial \\delta} \\log{(\\frac{\\delta}{\\phi})}}{\\phi})} = \\log{(\\frac{1}{\\delta \\phi})}", "derivation": "\\chi{(\\phi,\\delta)} = \\log{(\\frac{\\delta}{\\phi})} and \\frac{\\partial}{\\partial \\delta} \\chi{(\\phi,\\delta)} = \\frac{\\partial}{\\partial \\delta} \\log{(\\frac{\\delta}{\\phi})} and \\frac{\\partial}{\\partial \\delta} \\chi{(\\phi,\\delta)} = \\frac{1}{\\delta} and \\frac{\\frac{\\partial}{\\partial \\delta} \\chi{(\\phi,\\delta)}}{\\phi} = \\frac{1}{\\delta \\phi} and \\log{(\\frac{\\frac{\\partial}{\\partial \\delta} \\chi{(\\phi,\\delta)}}{\\phi})} = \\log{(\\frac{1}{\\delta \\phi})} and \\log{(\\frac{\\frac{\\partial}{\\partial \\delta} \\log{(\\frac{\\delta}{\\phi})}}{\\phi})} = \\log{(\\frac{1}{\\delta \\phi})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), log(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["times", 3, "Pow(Symbol('\\\\phi', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], [["log", 4], "Equality(log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Derivative(log(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given W{(\\theta_1,l,\\theta)} = \\frac{\\theta_1}{\\theta l} and M{(\\theta_1)} = \\theta_1 and \\dot{z}{(\\theta_1,l,\\theta)} = \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}}, then obtain \\frac{\\theta l M{(\\theta_1)}}{\\theta_1} + \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}} = \\theta l + \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}}", "derivation": "W{(\\theta_1,l,\\theta)} = \\frac{\\theta_1}{\\theta l} and M{(\\theta_1)} = \\theta_1 and \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}} = \\frac{\\theta_1}{W{(\\theta_1,l,\\theta)}} and \\dot{z}{(\\theta_1,l,\\theta)} = \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}} and \\frac{\\theta l M{(\\theta_1)}}{\\theta_1} = \\theta l and \\frac{\\theta l M{(\\theta_1)}}{\\theta_1} + \\dot{z}{(\\theta_1,l,\\theta)} = \\theta l + \\dot{z}{(\\theta_1,l,\\theta)} and \\frac{\\theta l M{(\\theta_1)}}{\\theta_1} + \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}} = \\theta l + \\frac{M{(\\theta_1)}}{W{(\\theta_1,l,\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], [["divide", 2, "Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('M')(Symbol('\\\\theta_1', commutative=True)), Pow(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Function('M')(Symbol('\\\\theta_1', commutative=True)), Pow(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('l', commutative=True), Function('M')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Symbol('l', commutative=True)))"], [["add", 5, "Function('\\\\dot{z}')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('l', commutative=True), Function('M')(Symbol('\\\\theta_1', commutative=True))), Function('\\\\dot{z}')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('l', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('l', commutative=True), Function('M')(Symbol('\\\\theta_1', commutative=True))), Mul(Function('M')(Symbol('\\\\theta_1', commutative=True)), Pow(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('l', commutative=True)), Mul(Function('M')(Symbol('\\\\theta_1', commutative=True)), Pow(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(G)} = e^{e^{G}} and \\hat{p}_0{(G)} = e^{G}, then obtain G e^{2 G} e^{\\hat{p}_0{(G)}} = G e^{2 G} e^{e^{G}}", "derivation": "\\operatorname{P_{g}}{(G)} = e^{e^{G}} and \\hat{p}_0{(G)} = e^{G} and \\operatorname{P_{g}}{(G)} = e^{\\hat{p}_0{(G)}} and e^{\\hat{p}_0{(G)}} = e^{e^{G}} and G e^{2 G} e^{\\hat{p}_0{(G)}} = G e^{2 G} e^{e^{G}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('G', commutative=True)), exp(exp(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('P_g')(Symbol('G', commutative=True)), exp(Function('\\\\hat{p}_0')(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(exp(Function('\\\\hat{p}_0')(Symbol('G', commutative=True))), exp(exp(Symbol('G', commutative=True))))"], [["times", 4, "Mul(Symbol('G', commutative=True), exp(Mul(Integer(2), Symbol('G', commutative=True))))"], "Equality(Mul(Symbol('G', commutative=True), exp(Mul(Integer(2), Symbol('G', commutative=True))), exp(Function('\\\\hat{p}_0')(Symbol('G', commutative=True)))), Mul(Symbol('G', commutative=True), exp(Mul(Integer(2), Symbol('G', commutative=True))), exp(exp(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(s)} = \\sin{(s)}, then obtain (\\int (\\eta^{\\prime}{(s)} + \\iint \\eta^{\\prime}{(s)} ds ds) ds)^{s} = (\\int (\\eta^{\\prime}{(s)} + \\iint \\sin{(s)} ds ds) ds)^{s}", "derivation": "\\eta^{\\prime}{(s)} = \\sin{(s)} and \\int \\eta^{\\prime}{(s)} ds = \\int \\sin{(s)} ds and \\iint \\eta^{\\prime}{(s)} ds ds = \\iint \\sin{(s)} ds ds and \\sin{(s)} + \\iint \\eta^{\\prime}{(s)} ds ds = \\sin{(s)} + \\iint \\sin{(s)} ds ds and \\eta^{\\prime}{(s)} + \\iint \\eta^{\\prime}{(s)} ds ds = \\eta^{\\prime}{(s)} + \\iint \\sin{(s)} ds ds and \\int (\\eta^{\\prime}{(s)} + \\iint \\eta^{\\prime}{(s)} ds ds) ds = \\int (\\eta^{\\prime}{(s)} + \\iint \\sin{(s)} ds ds) ds and (\\int (\\eta^{\\prime}{(s)} + \\iint \\eta^{\\prime}{(s)} ds ds) ds)^{s} = (\\int (\\eta^{\\prime}{(s)} + \\iint \\sin{(s)} ds ds) ds)^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["add", 3, "sin(Symbol('s', commutative=True))"], "Equality(Add(sin(Symbol('s', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(sin(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["integrate", 5, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))))"], [["power", 6, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"]]}, {"prompt": "Given Q{(\\mathbf{M})} = \\cos{(\\mathbf{M})}, then derive \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})} = - \\sin{(\\mathbf{M})}, then obtain \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})} \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} = - \\sin{(\\mathbf{M})} \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})}", "derivation": "Q{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})} = - \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} = - \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})} \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} = - \\sin{(\\mathbf{M})} \\frac{d}{d \\mathbf{M}} Q{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 4, "Derivative(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Function('Q')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(\\eta^{\\prime})} = \\eta^{\\prime} and \\hat{x}_0{(\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p}{(\\eta^{\\prime})}, then obtain 4 \\hat{p}^{2}{(\\eta^{\\prime})} = \\hat{x}_0^{2}{(\\eta^{\\prime})}", "derivation": "\\hat{p}{(\\eta^{\\prime})} = \\eta^{\\prime} and 2 \\hat{p}{(\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p}{(\\eta^{\\prime})} and \\hat{x}_0{(\\eta^{\\prime})} = \\eta^{\\prime} + \\hat{p}{(\\eta^{\\prime})} and 2 \\hat{p}{(\\eta^{\\prime})} = \\hat{x}_0{(\\eta^{\\prime})} and 4 \\hat{p}^{2}{(\\eta^{\\prime})} = \\hat{x}_0^{2}{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{p}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{P}{(f^{\\prime})} = \\cos{(f^{\\prime})}, then derive e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}} = e^{f + \\sin{(f^{\\prime})}}, then obtain - u = - u + e^{f + \\sin{(f^{\\prime})}} - e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}}", "derivation": "\\mathbf{P}{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\int \\mathbf{P}{(f^{\\prime})} df^{\\prime} = \\int \\cos{(f^{\\prime})} df^{\\prime} and e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}} = e^{\\int \\cos{(f^{\\prime})} df^{\\prime}} and e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}} = e^{f + \\sin{(f^{\\prime})}} and - u + e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}} = - u + e^{f + \\sin{(f^{\\prime})}} and - u = - u + e^{f + \\sin{(f^{\\prime})}} - e^{\\int \\mathbf{P}{(f^{\\prime})} df^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), exp(Integral(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), exp(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), exp(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), exp(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True))))))"], [["minus", 5, "exp(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), exp(Add(Symbol('f', commutative=True), sin(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), exp(Integral(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))))"]]}, {"prompt": "Given a{(M)} = \\cos{(M)}, then obtain a{(M)} + \\cos{(M)} - \\frac{d}{d M} (a{(M)} + \\cos{(M)}) = 2 \\cos{(M)} - \\frac{d}{d M} (a{(M)} + \\cos{(M)})", "derivation": "a{(M)} = \\cos{(M)} and a{(M)} + \\cos{(M)} = 2 \\cos{(M)} and \\frac{d}{d M} (a{(M)} + \\cos{(M)}) = \\frac{d}{d M} 2 \\cos{(M)} and a{(M)} + \\cos{(M)} - \\frac{d}{d M} 2 \\cos{(M)} = 2 \\cos{(M)} - \\frac{d}{d M} 2 \\cos{(M)} and a{(M)} + \\cos{(M)} - \\frac{d}{d M} (a{(M)} + \\cos{(M)}) = 2 \\cos{(M)} - \\frac{d}{d M} (a{(M)} + \\cos{(M)})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["add", 1, "cos(Symbol('M', commutative=True))"], "Equality(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(2), cos(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)), Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))), Add(Mul(Integer(2), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)), Mul(Integer(-1), Derivative(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))), Add(Mul(Integer(2), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Derivative(Add(Function('a')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given C{(\\chi,\\mathbf{H},\\Omega)} = (- \\chi + \\mathbf{H})^{\\Omega} and z{(\\chi)} = - \\chi, then obtain \\frac{\\frac{\\partial}{\\partial \\Omega} (- \\chi + \\mathbf{H})^{\\Omega}}{\\mathbf{H} + z{(\\chi)}} = \\frac{\\frac{\\partial}{\\partial \\Omega} (\\mathbf{H} + z{(\\chi)})^{\\Omega}}{\\mathbf{H} + z{(\\chi)}}", "derivation": "C{(\\chi,\\mathbf{H},\\Omega)} = (- \\chi + \\mathbf{H})^{\\Omega} and \\frac{\\partial}{\\partial \\Omega} C{(\\chi,\\mathbf{H},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (- \\chi + \\mathbf{H})^{\\Omega} and z{(\\chi)} = - \\chi and \\frac{\\frac{\\partial}{\\partial \\Omega} C{(\\chi,\\mathbf{H},\\Omega)}}{- \\chi + \\mathbf{H}} = \\frac{\\frac{\\partial}{\\partial \\Omega} (- \\chi + \\mathbf{H})^{\\Omega}}{- \\chi + \\mathbf{H}} and \\frac{\\frac{\\partial}{\\partial \\Omega} C{(\\chi,\\mathbf{H},\\Omega)}}{\\mathbf{H} + z{(\\chi)}} = \\frac{\\frac{\\partial}{\\partial \\Omega} (\\mathbf{H} + z{(\\chi)})^{\\Omega}}{\\mathbf{H} + z{(\\chi)}} and \\frac{\\frac{\\partial}{\\partial \\Omega} (- \\chi + \\mathbf{H})^{\\Omega}}{\\mathbf{H} + z{(\\chi)}} = \\frac{\\frac{\\partial}{\\partial \\Omega} (\\mathbf{H} + z{(\\chi)})^{\\Omega}}{\\mathbf{H} + z{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Derivative(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Integer(-1)), Derivative(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('z')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(\\mathbb{I},F_{H})} = \\mathbb{I} \\log{(F_{H})} and \\Psi{(F_{H})} = \\log{(F_{H})}, then obtain (\\frac{\\partial}{\\partial \\mathbb{I}} b{(\\mathbb{I},F_{H})})^{\\mathbb{I}} = (\\frac{\\partial}{\\partial \\mathbb{I}} \\mathbb{I} \\Psi{(F_{H})})^{\\mathbb{I}}", "derivation": "b{(\\mathbb{I},F_{H})} = \\mathbb{I} \\log{(F_{H})} and \\Psi{(F_{H})} = \\log{(F_{H})} and \\frac{\\partial}{\\partial \\mathbb{I}} b{(\\mathbb{I},F_{H})} = \\frac{\\partial}{\\partial \\mathbb{I}} \\mathbb{I} \\log{(F_{H})} and \\frac{\\partial}{\\partial \\mathbb{I}} b{(\\mathbb{I},F_{H})} = \\frac{\\partial}{\\partial \\mathbb{I}} \\mathbb{I} \\Psi{(F_{H})} and (\\frac{\\partial}{\\partial \\mathbb{I}} b{(\\mathbb{I},F_{H})})^{\\mathbb{I}} = (\\frac{\\partial}{\\partial \\mathbb{I}} \\mathbb{I} \\Psi{(F_{H})})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\Psi')(Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Derivative(Function('b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\Psi')(Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\rho)} = e^{\\rho} and \\operatorname{z^{*}}{(\\rho)} = 3 \\mathbf{s}{(\\rho)}, then obtain \\operatorname{z^{*}}{(\\rho)} + e^{\\rho} = 3 \\mathbf{s}{(\\rho)} + e^{\\rho}", "derivation": "\\mathbf{s}{(\\rho)} = e^{\\rho} and 2 \\mathbf{s}{(\\rho)} = \\mathbf{s}{(\\rho)} + e^{\\rho} and 3 \\mathbf{s}{(\\rho)} + e^{\\rho} = 2 \\mathbf{s}{(\\rho)} + 2 e^{\\rho} and \\operatorname{z^{*}}{(\\rho)} = 3 \\mathbf{s}{(\\rho)} and 3 \\mathbf{s}{(\\rho)} + e^{\\rho} = \\mathbf{s}{(\\rho)} + 3 e^{\\rho} and \\operatorname{z^{*}}{(\\rho)} + e^{\\rho} = \\mathbf{s}{(\\rho)} + 3 e^{\\rho} and \\operatorname{z^{*}}{(\\rho)} + e^{\\rho} = 3 \\mathbf{s}{(\\rho)} + e^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"], [["add", 2, "Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\rho', commutative=True)))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\rho', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('z^*')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('z^*')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(3), Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(c,F_{x})} = \\sin{(F_{x}^{c})}, then obtain - F_{x} + 3 \\bar{\\h}{(c,F_{x})} - \\sin{(F_{x}^{c})} - \\sin{((F_{x} - \\bar{\\h}{(c,F_{x})} + \\sin{(F_{x}^{c})})^{c})} = - F_{x} + \\bar{\\h}{(c,F_{x})}", "derivation": "\\bar{\\h}{(c,F_{x})} = \\sin{(F_{x}^{c})} and - F_{x} + \\bar{\\h}{(c,F_{x})} = - F_{x} + \\sin{(F_{x}^{c})} and - F_{x} + \\bar{\\h}{(c,F_{x})} - \\sin{(F_{x}^{c})} = - F_{x} and - F_{x} + 2 \\bar{\\h}{(c,F_{x})} - \\sin{(F_{x}^{c})} = - F_{x} + \\bar{\\h}{(c,F_{x})} and - F_{x} + 2 \\bar{\\h}{(c,F_{x})} - \\sin{(F_{x}^{c})} = - F_{x} + \\sin{(F_{x}^{c})} and - F_{x} + 3 \\bar{\\h}{(c,F_{x})} - \\sin{(F_{x}^{c})} - \\sin{((F_{x} - \\bar{\\h}{(c,F_{x})} + \\sin{(F_{x}^{c})})^{c})} = - F_{x} + \\bar{\\h}{(c,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True)), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True))))"], [["minus", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True)))))"], [["minus", 2, "sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True))))), Mul(Integer(-1), Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(2), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(2), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(3), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True)))), Mul(Integer(-1), sin(Pow(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))), sin(Pow(Symbol('F_x', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hbar')(Symbol('c', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(f,c_{0})} = - c_{0} + f, then obtain c_{0} \\bar{\\h}{(f,c_{0})} + \\sin{(c_{0} (- c_{0} + f))} = c_{0} (- c_{0} + f) + \\sin{(c_{0} (- c_{0} + f))}", "derivation": "\\bar{\\h}{(f,c_{0})} = - c_{0} + f and c_{0} \\bar{\\h}{(f,c_{0})} = c_{0} (- c_{0} + f) and \\sin{(c_{0} \\bar{\\h}{(f,c_{0})})} = \\sin{(c_{0} (- c_{0} + f))} and c_{0} \\bar{\\h}{(f,c_{0})} + \\sin{(c_{0} \\bar{\\h}{(f,c_{0})})} = c_{0} (- c_{0} + f) + \\sin{(c_{0} \\bar{\\h}{(f,c_{0})})} and c_{0} \\bar{\\h}{(f,c_{0})} + \\sin{(c_{0} (- c_{0} + f))} = c_{0} (- c_{0} + f) + \\sin{(c_{0} (- c_{0} + f))}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True)))"], [["times", 1, "Symbol('c_0', commutative=True)"], "Equality(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))), Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True)))), sin(Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True)))))"], [["add", 2, "sin(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))))"], "Equality(Add(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))), sin(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))))), Add(Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True))), sin(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('c_0', commutative=True), Function('\\\\hbar')(Symbol('f', commutative=True), Symbol('c_0', commutative=True))), sin(Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True))))), Add(Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True))), sin(Mul(Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} = \\frac{n}{f^{*}}, then obtain \\frac{\\partial^{2}}{\\partial n^{2}} (\\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} + 1) = \\frac{\\partial^{2}}{\\partial n^{2}} (1 + \\frac{n}{f^{*}})", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} = \\frac{n}{f^{*}} and \\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} + 1 = 1 + \\frac{n}{f^{*}} and \\frac{\\partial}{\\partial n} (\\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} + 1) = \\frac{\\partial}{\\partial n} (1 + \\frac{n}{f^{*}}) and \\frac{\\partial^{2}}{\\partial n^{2}} (\\operatorname{f_{\\mathbf{v}}}{(f^{*},n)} + 1) = \\frac{\\partial^{2}}{\\partial n^{2}} (1 + \\frac{n}{f^{*}})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f^*', commutative=True), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(Add(Integer(1), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(2))))"]]}, {"prompt": "Given y{(x^\\prime,\\mathbf{J}_P)} = e^{\\mathbf{J}_P + x^\\prime} and J{(\\mathbf{J}_P)} = \\mathbf{J}_P, then derive \\int y{(x^\\prime,\\mathbf{J}_P)} d\\mathbf{J}_P = V_{\\mathbf{B}} + e^{\\mathbf{J}_P + x^\\prime}, then obtain \\int e^{\\mathbf{J}_P + x^\\prime} dJ{(\\mathbf{J}_P)} = V_{\\mathbf{B}} + e^{x^\\prime + J{(\\mathbf{J}_P)}}", "derivation": "y{(x^\\prime,\\mathbf{J}_P)} = e^{\\mathbf{J}_P + x^\\prime} and \\int y{(x^\\prime,\\mathbf{J}_P)} d\\mathbf{J}_P = \\int e^{\\mathbf{J}_P + x^\\prime} d\\mathbf{J}_P and \\int y{(x^\\prime,\\mathbf{J}_P)} d\\mathbf{J}_P = V_{\\mathbf{B}} + e^{\\mathbf{J}_P + x^\\prime} and J{(\\mathbf{J}_P)} = \\mathbf{J}_P and \\int e^{\\mathbf{J}_P + x^\\prime} d\\mathbf{J}_P = V_{\\mathbf{B}} + e^{\\mathbf{J}_P + x^\\prime} and \\int e^{\\mathbf{J}_P + x^\\prime} dJ{(\\mathbf{J}_P)} = V_{\\mathbf{B}} + e^{x^\\prime + J{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(exp(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Function('J')(Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Symbol('x^\\\\prime', commutative=True), Function('J')(Symbol('\\\\mathbf{J}_P', commutative=True))))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{H},\\Omega)} = \\Omega + \\hat{H}, then obtain \\sin{(e^{- \\hat{H}})} + \\frac{1}{2} = \\sin{(e^{- \\hat{H} + \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} - \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}}})} + \\frac{1}{2}", "derivation": "\\phi_{1}{(\\hat{H},\\Omega)} = \\Omega + \\hat{H} and \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}} = \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} and - \\hat{H} = - \\hat{H} + \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} - \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}} and e^{- \\hat{H}} = e^{- \\hat{H} + \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} - \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}}} and \\sin{(e^{- \\hat{H}})} = \\sin{(e^{- \\hat{H} + \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} - \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}}})} and \\sin{(e^{- \\hat{H}})} + \\frac{1}{2} = \\sin{(e^{- \\hat{H} + \\frac{\\Omega + \\hat{H}}{2 \\hat{H}} - \\frac{\\phi_{1}{(\\hat{H},\\Omega)}}{2 \\hat{H}}})} + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\hat{H}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["sin", 4], "Equality(sin(exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), sin(exp(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True)))))))"], [["minus", 5, "Rational(-1, 2)"], "Equality(Add(sin(exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Rational(1, 2)), Add(sin(exp(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Rational(1, 2)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(f,\\theta)} = \\theta^{f} and \\operatorname{t_{2}}{(f,\\theta)} = \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} \\log{(\\theta)} + \\theta^{f} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)}, then derive 2 \\operatorname{P_{e}}{(f,\\theta)} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} = \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} \\log{(\\theta)} + \\theta^{f} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)}, then obtain 2 \\operatorname{P_{e}}{(f,\\theta)} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} = \\operatorname{t_{2}}{(f,\\theta)}", "derivation": "\\operatorname{P_{e}}{(f,\\theta)} = \\theta^{f} and \\operatorname{P_{e}}^{2}{(f,\\theta)} = \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} and \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}^{2}{(f,\\theta)} = \\frac{\\partial}{\\partial f} \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} and 2 \\operatorname{P_{e}}{(f,\\theta)} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} = \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} \\log{(\\theta)} + \\theta^{f} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} and \\operatorname{t_{2}}{(f,\\theta)} = \\theta^{f} \\operatorname{P_{e}}{(f,\\theta)} \\log{(\\theta)} + \\theta^{f} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} and 2 \\operatorname{P_{e}}{(f,\\theta)} \\frac{\\partial}{\\partial f} \\operatorname{P_{e}}{(f,\\theta)} = \\operatorname{t_{2}}{(f,\\theta)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Derivative(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Symbol('f', commutative=True)), Derivative(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Function('P_e')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(V)} = \\log{(V)}, then derive \\frac{d}{d V} \\mathbf{M}{(V)} = \\frac{1}{V}, then obtain \\log{(V)}^{- V} \\frac{d}{d V} \\log{(V)} = \\frac{\\log{(V)}^{- V}}{V}", "derivation": "\\mathbf{M}{(V)} = \\log{(V)} and \\frac{d}{d V} \\mathbf{M}{(V)} = \\frac{d}{d V} \\log{(V)} and \\frac{d}{d V} \\mathbf{M}{(V)} = \\frac{1}{V} and \\log{(V)}^{- V} \\frac{d}{d V} \\mathbf{M}{(V)} = \\frac{\\log{(V)}^{- V}}{V} and \\log{(V)}^{- V} \\frac{d}{d V} \\log{(V)} = \\frac{\\log{(V)}^{- V}}{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Pow(Symbol('V', commutative=True), Integer(-1)))"], [["divide", 3, "Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], "Equality(Mul(Pow(log(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(log(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(log(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))), Derivative(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(log(Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))))"]]}, {"prompt": "Given u{(\\sigma_p)} = \\sigma_p, then obtain u^{\\sigma_p}{(\\sigma_p)} + \\frac{1}{\\sigma_p} - \\frac{\\operatorname{M_{E}}{(y,\\sigma_p)}}{\\sigma_p y} = \\sigma_p^{\\sigma_p} + \\frac{1}{\\sigma_p} - \\frac{\\operatorname{M_{E}}{(y,\\sigma_p)}}{\\sigma_p y}", "derivation": "u{(\\sigma_p)} = \\sigma_p and u^{\\sigma_p}{(\\sigma_p)} = \\sigma_p^{\\sigma_p} and u^{\\sigma_p}{(\\sigma_p)} + \\frac{1}{\\sigma_p} = \\sigma_p^{\\sigma_p} + \\frac{1}{\\sigma_p} and u^{\\sigma_p}{(\\sigma_p)} + \\frac{1}{\\sigma_p} - \\frac{\\operatorname{M_{E}}{(y,\\sigma_p)}}{\\sigma_p y} = \\sigma_p^{\\sigma_p} + \\frac{1}{\\sigma_p} - \\frac{\\operatorname{M_{E}}{(y,\\sigma_p)}}{\\sigma_p y}", "srepr_derivation": [["renaming_premise", "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Pow(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\phi,F_{x})} = \\sin{(\\phi^{F_{x}})}, then obtain F_{x} (\\hat{\\mathbf{r}}{(\\phi,F_{x})} + \\int \\hat{\\mathbf{r}}{(\\phi,F_{x})} d\\phi) = F_{x} (\\hat{\\mathbf{r}}{(\\phi,F_{x})} + \\int \\sin{(\\phi^{F_{x}})} d\\phi)", "derivation": "\\hat{\\mathbf{r}}{(\\phi,F_{x})} = \\sin{(\\phi^{F_{x}})} and \\int \\hat{\\mathbf{r}}{(\\phi,F_{x})} d\\phi = \\int \\sin{(\\phi^{F_{x}})} d\\phi and \\sin{(\\phi^{F_{x}})} + \\int \\hat{\\mathbf{r}}{(\\phi,F_{x})} d\\phi = \\sin{(\\phi^{F_{x}})} + \\int \\sin{(\\phi^{F_{x}})} d\\phi and F_{x} (\\sin{(\\phi^{F_{x}})} + \\int \\hat{\\mathbf{r}}{(\\phi,F_{x})} d\\phi) = F_{x} (\\sin{(\\phi^{F_{x}})} + \\int \\sin{(\\phi^{F_{x}})} d\\phi) and F_{x} (\\hat{\\mathbf{r}}{(\\phi,F_{x})} + \\int \\hat{\\mathbf{r}}{(\\phi,F_{x})} d\\phi) = F_{x} (\\hat{\\mathbf{r}}{(\\phi,F_{x})} + \\int \\sin{(\\phi^{F_{x}})} d\\phi)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 2, "sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Add(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Integral(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["times", 3, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Add(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))), Mul(Symbol('F_x', commutative=True), Add(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Integral(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('F_x', commutative=True), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))), Mul(Symbol('F_x', commutative=True), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True)), Integral(sin(Pow(Symbol('\\\\phi', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = Z + \\mathbf{H}, then obtain \\frac{\\partial}{\\partial Z} (Z + 2 \\mathbf{H} + \\hat{\\mathbf{x}}{(\\mathbf{H},Z)}) = \\frac{\\partial}{\\partial Z} (2 Z + 3 \\mathbf{H})", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = Z + \\mathbf{H} and \\mathbf{H} + \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = Z + 2 \\mathbf{H} and Z + 2 \\mathbf{H} + \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = 2 Z + 3 \\mathbf{H} and \\mathbf{H} + 2 \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = 2 Z + 3 \\mathbf{H} and \\frac{\\partial}{\\partial Z} (\\mathbf{H} + 2 \\hat{\\mathbf{x}}{(\\mathbf{H},Z)}) = \\frac{\\partial}{\\partial Z} (2 Z + 3 \\mathbf{H}) and Z + 2 \\mathbf{H} + \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} = \\mathbf{H} + 2 \\hat{\\mathbf{x}}{(\\mathbf{H},Z)} and \\frac{\\partial}{\\partial Z} (Z + 2 \\mathbf{H} + \\hat{\\mathbf{x}}{(\\mathbf{H},Z)}) = \\frac{\\partial}{\\partial Z} (2 Z + 3 \\mathbf{H})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "Add(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)))), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\rho)} = \\rho and \\hat{p}_0{(\\rho)} = 2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}), then obtain \\frac{d}{d \\rho} ((\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} - 1) = \\frac{d}{d \\rho} (2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}) - 1)", "derivation": "\\operatorname{A_{y}}{(\\rho)} = \\rho and \\rho + \\operatorname{A_{y}}{(\\rho)} = 2 \\rho and (\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} = 2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}) and \\hat{p}_0{(\\rho)} = 2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}) and (\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} = \\hat{p}_0{(\\rho)} and (\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} - 1 = \\hat{p}_0{(\\rho)} - 1 and (\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} - 1 = 2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}) - 1 and \\frac{d}{d \\rho} ((\\rho + \\operatorname{A_{y}}{(\\rho)})^{2} - 1) = \\frac{d}{d \\rho} (2 \\rho (\\rho + \\operatorname{A_{y}}{(\\rho)}) - 1)", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))"], [["add", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Mul(Integer(2), Symbol('\\\\rho', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Integer(2)), Mul(Integer(2), Symbol('\\\\rho', commutative=True), Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True), Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Integer(2)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))"], [["minus", 5, 1], "Equality(Add(Pow(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Integer(2)), Integer(-1)), Add(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Integer(2)), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\rho', commutative=True), Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True)))), Integer(-1)))"], [["differentiate", 7, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True))), Integer(2)), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\rho', commutative=True), Add(Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\chi)} = \\log{(\\chi)}, then obtain \\frac{d}{d \\chi} \\int \\operatorname{A_{y}}{(\\chi)} d\\chi - 1 = \\frac{\\partial}{\\partial \\chi} (\\chi \\log{(\\chi)} - \\chi + \\delta) - 1", "derivation": "\\operatorname{A_{y}}{(\\chi)} = \\log{(\\chi)} and \\int \\operatorname{A_{y}}{(\\chi)} d\\chi = \\int \\log{(\\chi)} d\\chi and \\frac{d}{d \\chi} \\int \\operatorname{A_{y}}{(\\chi)} d\\chi = \\frac{d}{d \\chi} \\int \\log{(\\chi)} d\\chi and \\frac{d}{d \\chi} \\int \\operatorname{A_{y}}{(\\chi)} d\\chi - 1 = \\frac{d}{d \\chi} \\int \\log{(\\chi)} d\\chi - 1 and \\frac{d}{d \\chi} \\int \\operatorname{A_{y}}{(\\chi)} d\\chi - 1 = \\frac{\\partial}{\\partial \\chi} (\\chi \\log{(\\chi)} - \\chi + \\delta) - 1", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Function('A_y')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('A_y')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Derivative(Integral(Function('A_y')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(u)} = \\sin{(\\cos{(u)})} and m{(u)} = \\sin{(\\cos{(u)})}, then obtain \\frac{\\frac{d}{d u} 1}{\\frac{d}{d u} 1 + \\int 1 du} = \\frac{\\frac{d}{d u} \\frac{\\sin{(\\cos{(u)})}}{m{(u)}}}{\\frac{d}{d u} 1 + \\int 1 du}", "derivation": "\\operatorname{A_{y}}{(u)} = \\sin{(\\cos{(u)})} and 1 = \\frac{\\sin{(\\cos{(u)})}}{\\operatorname{A_{y}}{(u)}} and \\frac{d}{d u} 1 = \\frac{d}{d u} \\frac{\\sin{(\\cos{(u)})}}{\\operatorname{A_{y}}{(u)}} and m{(u)} = \\sin{(\\cos{(u)})} and \\operatorname{A_{y}}{(u)} = m{(u)} and \\frac{d}{d u} 1 = \\frac{d}{d u} \\frac{\\sin{(\\cos{(u)})}}{m{(u)}} and \\frac{\\frac{d}{d u} 1}{\\frac{d}{d u} 1 + \\int 1 du} = \\frac{\\frac{d}{d u} \\frac{\\sin{(\\cos{(u)})}}{m{(u)}}}{\\frac{d}{d u} 1 + \\int 1 du}", "srepr_derivation": [["get_premise", "Equality(Function('A_y')(Symbol('u', commutative=True)), sin(cos(Symbol('u', commutative=True))))"], [["divide", 1, "Function('A_y')(Symbol('u', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('A_y')(Symbol('u', commutative=True)), Integer(-1)), sin(cos(Symbol('u', commutative=True)))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('A_y')(Symbol('u', commutative=True)), Integer(-1)), sin(cos(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('m')(Symbol('u', commutative=True)), sin(cos(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('A_y')(Symbol('u', commutative=True)), Function('m')(Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('m')(Symbol('u', commutative=True)), Integer(-1)), sin(cos(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 6, "Add(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"], "Equality(Mul(Pow(Add(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('u', commutative=True)))), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Pow(Add(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('u', commutative=True)))), Integer(-1)), Derivative(Mul(Pow(Function('m')(Symbol('u', commutative=True)), Integer(-1)), sin(cos(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(x^\\prime,\\hbar)} = (e^{\\hbar})^{x^\\prime}, then obtain \\eta{(x^\\prime,\\hbar)} - \\frac{\\int (e^{\\hbar})^{x^\\prime} dx^\\prime}{\\hbar} = (e^{\\hbar})^{x^\\prime} - \\frac{\\int (e^{\\hbar})^{x^\\prime} dx^\\prime}{\\hbar}", "derivation": "\\eta{(x^\\prime,\\hbar)} = (e^{\\hbar})^{x^\\prime} and \\int \\eta{(x^\\prime,\\hbar)} dx^\\prime = \\int (e^{\\hbar})^{x^\\prime} dx^\\prime and \\frac{\\int \\eta{(x^\\prime,\\hbar)} dx^\\prime}{\\hbar} = \\frac{\\int (e^{\\hbar})^{x^\\prime} dx^\\prime}{\\hbar} and \\eta{(x^\\prime,\\hbar)} - \\frac{\\int \\eta{(x^\\prime,\\hbar)} dx^\\prime}{\\hbar} = (e^{\\hbar})^{x^\\prime} - \\frac{\\int \\eta{(x^\\prime,\\hbar)} dx^\\prime}{\\hbar} and \\eta{(x^\\prime,\\hbar)} - \\frac{\\int (e^{\\hbar})^{x^\\prime} dx^\\prime}{\\hbar} = (e^{\\hbar})^{x^\\prime} - \\frac{\\int (e^{\\hbar})^{x^\\prime} dx^\\prime}{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], "Equality(Add(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Add(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Add(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(exp(Symbol('\\\\hbar', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\varphi,Q)} = Q \\log{(\\varphi)} and \\operatorname{C_{d}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then obtain (Q \\log{(\\varphi)})^{Q} + \\operatorname{C_{d}}{(V_{\\mathbf{B}})} = (Q \\log{(\\varphi)})^{Q} + \\log{(V_{\\mathbf{B}})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\varphi,Q)} = Q \\log{(\\varphi)} and \\operatorname{V_{\\mathbf{B}}}^{Q}{(\\varphi,Q)} = (Q \\log{(\\varphi)})^{Q} and \\operatorname{C_{d}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\operatorname{C_{d}}{(V_{\\mathbf{B}})} + \\operatorname{V_{\\mathbf{B}}}^{Q}{(\\varphi,Q)} = \\operatorname{V_{\\mathbf{B}}}^{Q}{(\\varphi,Q)} + \\log{(V_{\\mathbf{B}})} and (Q \\log{(\\varphi)})^{Q} + \\operatorname{C_{d}}{(V_{\\mathbf{B}})} = (Q \\log{(\\varphi)})^{Q} + \\log{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\varphi', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\varphi', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Mul(Symbol('Q', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Symbol('Q', commutative=True)))"], ["get_premise", "Equality(Function('C_d')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 3, "Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\varphi', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Add(Function('C_d')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\varphi', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\varphi', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Mul(Symbol('Q', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Symbol('Q', commutative=True)), Function('C_d')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Pow(Mul(Symbol('Q', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Symbol('Q', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\Psi{(b,c)} = b + c, then derive \\int \\Psi{(b,c)} db = \\lambda + \\frac{b^{2}}{2} + b c, then obtain \\frac{\\partial}{\\partial \\lambda} \\frac{(\\lambda + \\frac{b^{2}}{2} + b c)^{b}}{b^{2}} = \\frac{\\partial}{\\partial \\lambda} \\frac{(\\int (b + c) db)^{b}}{b^{2}}", "derivation": "\\Psi{(b,c)} = b + c and \\int \\Psi{(b,c)} db = \\int (b + c) db and \\int \\Psi{(b,c)} db = \\lambda + \\frac{b^{2}}{2} + b c and (\\int \\Psi{(b,c)} db)^{b} = (\\int (b + c) db)^{b} and (\\lambda + \\frac{b^{2}}{2} + b c)^{b} = (\\int (b + c) db)^{b} and \\frac{(\\lambda + \\frac{b^{2}}{2} + b c)^{b}}{b^{2}} = \\frac{(\\int (b + c) db)^{b}}{b^{2}} and \\frac{\\partial}{\\partial \\lambda} \\frac{(\\lambda + \\frac{b^{2}}{2} + b c)^{b}}{b^{2}} = \\frac{\\partial}{\\partial \\lambda} \\frac{(\\int (b + c) db)^{b}}{b^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Add(Symbol('b', commutative=True), Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Add(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('c', commutative=True))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Add(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('c', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Add(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["divide", 5, "Pow(Symbol('b', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('c', commutative=True))), Symbol('b', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Pow(Integral(Add(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), Symbol('c', commutative=True))), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Pow(Integral(Add(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(E,M)} = E M and \\lambda{(E,M)} = E M + E, then obtain \\mathbf{H}^{E}{(E,M)} = (\\frac{\\lambda{(E,M)} \\mathbf{H}{(E,M)}}{E + \\mathbf{H}{(E,M)}})^{E}", "derivation": "\\mathbf{H}{(E,M)} = E M and E + \\mathbf{H}{(E,M)} = E M + E and E M (E + \\mathbf{H}{(E,M)}) = E M (E M + E) and \\lambda{(E,M)} = E M + E and E M (E + \\mathbf{H}{(E,M)}) = E M \\lambda{(E,M)} and (E + \\mathbf{H}{(E,M)}) \\mathbf{H}{(E,M)} = \\lambda{(E,M)} \\mathbf{H}{(E,M)} and \\mathbf{H}{(E,M)} = \\frac{\\lambda{(E,M)} \\mathbf{H}{(E,M)}}{E + \\mathbf{H}{(E,M)}} and \\mathbf{H}^{E}{(E,M)} = (\\frac{\\lambda{(E,M)} \\mathbf{H}{(E,M)}}{E + \\mathbf{H}{(E,M)}})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('M', commutative=True)))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Add(Mul(Symbol('E', commutative=True), Symbol('M', commutative=True)), Symbol('E', commutative=True)))"], [["times", 2, "Mul(Symbol('E', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('M', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)))), Mul(Symbol('E', commutative=True), Symbol('M', commutative=True), Add(Mul(Symbol('E', commutative=True), Symbol('M', commutative=True)), Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Add(Mul(Symbol('E', commutative=True), Symbol('M', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('E', commutative=True), Symbol('M', commutative=True), Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)))), Mul(Symbol('E', commutative=True), Symbol('M', commutative=True), Function('\\\\lambda')(Symbol('E', commutative=True), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Mul(Function('\\\\lambda')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))))"], [["divide", 6, "Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)))"], "Equality(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Integer(-1)), Function('\\\\lambda')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))))"], [["power", 7, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Pow(Add(Symbol('E', commutative=True), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Integer(-1)), Function('\\\\lambda')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('M', commutative=True))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given s{(\\chi,n)} = \\frac{\\partial}{\\partial \\chi} \\chi n, then derive s{(\\chi,n)} = n, then derive \\int s{(\\chi,n)} dn = L_{\\varepsilon} + \\frac{n^{2}}{2}, then obtain (\\int s{(\\chi,n)} ds{(\\chi,n)})^{\\chi} = (L_{\\varepsilon} + \\frac{s^{2}{(\\chi,n)}}{2})^{\\chi}", "derivation": "s{(\\chi,n)} = \\frac{\\partial}{\\partial \\chi} \\chi n and s{(\\chi,n)} = n and \\int s{(\\chi,n)} dn = \\int n dn and \\int s{(\\chi,n)} dn = L_{\\varepsilon} + \\frac{n^{2}}{2} and \\int s{(\\chi,n)} ds{(\\chi,n)} = L_{\\varepsilon} + \\frac{s^{2}{(\\chi,n)}}{2} and (\\int s{(\\chi,n)} ds{(\\chi,n)})^{\\chi} = (L_{\\varepsilon} + \\frac{s^{2}{(\\chi,n)}}{2})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Tuple(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Integer(2)))))"], [["power", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Tuple(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Function('s')(Symbol('\\\\chi', commutative=True), Symbol('n', commutative=True)), Integer(2)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(V_{\\mathbf{B}},A_{z})} = (e^{V_{\\mathbf{B}}})^{A_{z}}, then obtain ((\\mathbf{F}{(V_{\\mathbf{B}},A_{z})} + e^{V_{\\mathbf{B}}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})})^{V_{\\mathbf{B}}} = ((e^{V_{\\mathbf{B}}} + (e^{V_{\\mathbf{B}}})^{A_{z}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})})^{V_{\\mathbf{B}}}", "derivation": "\\mathbf{F}{(V_{\\mathbf{B}},A_{z})} = (e^{V_{\\mathbf{B}}})^{A_{z}} and \\mathbf{F}{(V_{\\mathbf{B}},A_{z})} + e^{V_{\\mathbf{B}}} = e^{V_{\\mathbf{B}}} + (e^{V_{\\mathbf{B}}})^{A_{z}} and (\\mathbf{F}{(V_{\\mathbf{B}},A_{z})} + e^{V_{\\mathbf{B}}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})} = (e^{V_{\\mathbf{B}}} + (e^{V_{\\mathbf{B}}})^{A_{z}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})} and ((\\mathbf{F}{(V_{\\mathbf{B}},A_{z})} + e^{V_{\\mathbf{B}}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})})^{V_{\\mathbf{B}}} = ((e^{V_{\\mathbf{B}}} + (e^{V_{\\mathbf{B}}})^{A_{z}}) \\mathbf{F}{(V_{\\mathbf{B}},A_{z})})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('A_z', commutative=True)))"], [["add", 1, "exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True)), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('A_z', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True)), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True))), Mul(Add(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('A_z', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True))))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True)), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Mul(Add(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('A_z', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('A_z', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} = \\eta + n_{1}^{C_{d}}, then obtain \\int 4 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} dC_{d} = \\int (\\eta + n_{1}^{C_{d}} + 3 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)}) dC_{d}", "derivation": "\\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} = \\eta + n_{1}^{C_{d}} and 2 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} = \\eta + n_{1}^{C_{d}} + \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} and 4 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} = \\eta + n_{1}^{C_{d}} + 3 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} and \\int 4 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)} dC_{d} = \\int (\\eta + n_{1}^{C_{d}} + 3 \\Psi^{\\dagger}{(n_{1},C_{d},\\eta)}) dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True))))"], [["add", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(3), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Integer(4), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Symbol('\\\\eta', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(3), Function('\\\\Psi^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given I{(\\mathbf{F})} = e^{\\mathbf{F}} and n{(\\hat{x}_0,\\mathbf{F})} = \\hat{x}_0 + \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})}, then derive \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})} = e^{\\mathbf{F}}, then derive \\int I{(\\mathbf{F})} d\\mathbf{F} = \\hat{x}_0 + e^{\\mathbf{F}}, then obtain n{(\\hat{x}_0,\\mathbf{F})} = \\int I{(\\mathbf{F})} d\\mathbf{F}", "derivation": "I{(\\mathbf{F})} = e^{\\mathbf{F}} and \\int I{(\\mathbf{F})} d\\mathbf{F} = \\int e^{\\mathbf{F}} d\\mathbf{F} and \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} e^{\\mathbf{F}} and \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})} = e^{\\mathbf{F}} and \\int I{(\\mathbf{F})} d\\mathbf{F} = \\hat{x}_0 + e^{\\mathbf{F}} and \\int I{(\\mathbf{F})} d\\mathbf{F} = \\hat{x}_0 + \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})} and n{(\\hat{x}_0,\\mathbf{F})} = \\hat{x}_0 + \\frac{d}{d \\mathbf{F}} I{(\\mathbf{F})} and n{(\\hat{x}_0,\\mathbf{F})} = \\int I{(\\mathbf{F})} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Derivative(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Derivative(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Function('n')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integral(Function('I')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given U{(C_{1})} = e^{C_{1}}, then obtain \\frac{d}{d C_{1}} (\\frac{- C_{1} + U{(C_{1})} - 1}{- C_{1} + e^{C_{1}} - 1} + \\frac{1}{- C_{1} + e^{C_{1}} - 1}) = \\frac{d}{d C_{1}} (1 + \\frac{1}{- C_{1} + e^{C_{1}} - 1})", "derivation": "U{(C_{1})} = e^{C_{1}} and U{(C_{1})} - 1 = e^{C_{1}} - 1 and - C_{1} + U{(C_{1})} - 1 = - C_{1} + e^{C_{1}} - 1 and \\frac{- C_{1} + U{(C_{1})} - 1}{- C_{1} + e^{C_{1}} - 1} = 1 and \\frac{- C_{1} + U{(C_{1})} - 1}{- C_{1} + e^{C_{1}} - 1} + \\frac{1}{- C_{1} + e^{C_{1}} - 1} = 1 + \\frac{1}{- C_{1} + e^{C_{1}} - 1} and \\frac{d}{d C_{1}} (\\frac{- C_{1} + U{(C_{1})} - 1}{- C_{1} + e^{C_{1}} - 1} + \\frac{1}{- C_{1} + e^{C_{1}} - 1}) = \\frac{d}{d C_{1}} (1 + \\frac{1}{- C_{1} + e^{C_{1}} - 1})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('U')(Symbol('C_1', commutative=True)), Integer(-1)), Add(exp(Symbol('C_1', commutative=True)), Integer(-1)))"], [["minus", 2, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('U')(Symbol('C_1', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('U')(Symbol('C_1', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Integer(1))"], [["add", 4, "Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('U')(Symbol('C_1', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Add(Integer(1), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))))"], [["differentiate", 5, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('U')(Symbol('C_1', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Integer(1), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)), Integer(-1)), Integer(-1))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(\\hat{x})} = \\log{(\\hat{x})}, then derive \\int s{(\\hat{x})} d\\hat{x} = \\hat{x} \\log{(\\hat{x})} - \\hat{x} + \\mathbf{A}, then obtain \\log{(\\hat{x})}^{2} \\int s{(\\hat{x})} d\\hat{x} = (\\hat{x} s{(\\hat{x})} - \\hat{x} + \\mathbf{A}) \\log{(\\hat{x})}^{2}", "derivation": "s{(\\hat{x})} = \\log{(\\hat{x})} and \\int s{(\\hat{x})} d\\hat{x} = \\int \\log{(\\hat{x})} d\\hat{x} and \\int s{(\\hat{x})} d\\hat{x} = \\hat{x} \\log{(\\hat{x})} - \\hat{x} + \\mathbf{A} and \\int s{(\\hat{x})} d\\hat{x} = \\hat{x} s{(\\hat{x})} - \\hat{x} + \\mathbf{A} and \\int \\log{(\\hat{x})} d\\hat{x} = \\hat{x} s{(\\hat{x})} - \\hat{x} + \\mathbf{A} and \\log{(\\hat{x})}^{2} \\int \\log{(\\hat{x})} d\\hat{x} = (\\hat{x} s{(\\hat{x})} - \\hat{x} + \\mathbf{A}) \\log{(\\hat{x})}^{2} and \\log{(\\hat{x})}^{2} \\int s{(\\hat{x})} d\\hat{x} = (\\hat{x} s{(\\hat{x})} - \\hat{x} + \\mathbf{A}) \\log{(\\hat{x})}^{2}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Function('s')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Function('s')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 5, "Pow(log(Symbol('\\\\hat{x}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(log(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Function('s')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(log(Symbol('\\\\hat{x}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(log(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Function('s')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(log(Symbol('\\\\hat{x}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(W,v_{1})} = W + v_{1} and \\operatorname{t_{2}}{(\\phi_1)} = \\log{(\\phi_1)}, then obtain - W + \\operatorname{t_{2}}{(\\phi_1)} = - W + \\log{(\\phi_1)}", "derivation": "\\operatorname{v_{y}}{(W,v_{1})} = W + v_{1} and \\operatorname{t_{2}}{(\\phi_1)} = \\log{(\\phi_1)} and v_{1} + \\operatorname{t_{2}}{(\\phi_1)} = v_{1} + \\log{(\\phi_1)} and v_{1} + \\operatorname{t_{2}}{(\\phi_1)} - \\operatorname{v_{y}}{(W,v_{1})} = v_{1} - \\operatorname{v_{y}}{(W,v_{1})} + \\log{(\\phi_1)} and - W + \\operatorname{t_{2}}{(\\phi_1)} = - W + \\log{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))"], ["get_premise", "Equality(Function('t_2')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["add", 2, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('t_2')(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('v_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 3, "Function('v_y')(Symbol('W', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Symbol('v_1', commutative=True), Function('t_2')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))), Add(Symbol('v_1', commutative=True), Mul(Integer(-1), Function('v_y')(Symbol('W', commutative=True), Symbol('v_1', commutative=True))), log(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('t_2')(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(n)} = \\log{(\\cos{(n)})} and \\mathbf{P}{(n)} = \\log{(\\cos{(n)})}, then obtain \\frac{\\operatorname{C_{1}}{(n)}}{\\cos{(n)}} = \\frac{\\mathbf{P}{(n)}}{\\cos{(n)}}", "derivation": "\\operatorname{C_{1}}{(n)} = \\log{(\\cos{(n)})} and \\mathbf{P}{(n)} = \\log{(\\cos{(n)})} and \\frac{\\operatorname{C_{1}}{(n)}}{\\cos{(n)}} = \\frac{\\log{(\\cos{(n)})}}{\\cos{(n)}} and \\frac{\\operatorname{C_{1}}{(n)}}{\\cos{(n)}} = \\frac{\\mathbf{P}{(n)}}{\\cos{(n)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('n', commutative=True)), log(cos(Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('n', commutative=True)), log(cos(Symbol('n', commutative=True))))"], [["divide", 1, "cos(Symbol('n', commutative=True))"], "Equality(Mul(Function('C_1')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Mul(log(cos(Symbol('n', commutative=True))), Pow(cos(Symbol('n', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('C_1')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Mul(Function('\\\\mathbf{P}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\theta{(E_{x})} = \\int \\cos{(E_{x})} dE_{x}, then derive - E_{x} + \\theta{(E_{x})} = B - E_{x} + \\sin{(E_{x})}, then obtain \\frac{\\partial}{\\partial B} (B - E_{x} + \\sin{(E_{x})}) = \\frac{\\partial}{\\partial B} (- E_{x} + J + \\sin{(E_{x})})", "derivation": "\\theta{(E_{x})} = \\int \\cos{(E_{x})} dE_{x} and - E_{x} + \\theta{(E_{x})} = - E_{x} + \\int \\cos{(E_{x})} dE_{x} and - E_{x} + \\theta{(E_{x})} = B - E_{x} + \\sin{(E_{x})} and B - E_{x} + \\sin{(E_{x})} = - E_{x} + \\int \\cos{(E_{x})} dE_{x} and \\frac{\\partial}{\\partial B} (B - E_{x} + \\sin{(E_{x})}) = \\frac{d}{d B} (- E_{x} + \\int \\cos{(E_{x})} dE_{x}) and \\frac{\\partial}{\\partial B} (B - E_{x} + \\sin{(E_{x})}) = \\frac{\\partial}{\\partial B} (- E_{x} + J + \\sin{(E_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["minus", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\theta')(Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\theta')(Symbol('E_x', commutative=True))), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["differentiate", 4, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\hbar)} = e^{e^{\\hbar}} and \\mathbf{J}_P{(\\hbar)} = - e^{\\hbar}, then obtain \\hbar \\dot{x}{(\\hbar)} e^{\\mathbf{J}_P{(\\hbar)}} = \\hbar", "derivation": "\\dot{x}{(\\hbar)} = e^{e^{\\hbar}} and \\dot{x}{(\\hbar)} e^{- e^{\\hbar}} = 1 and \\hbar \\dot{x}{(\\hbar)} e^{- e^{\\hbar}} = \\hbar and \\mathbf{J}_P{(\\hbar)} = - e^{\\hbar} and \\hbar \\dot{x}{(\\hbar)} e^{\\mathbf{J}_P{(\\hbar)}} = \\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hbar', commutative=True)), exp(exp(Symbol('\\\\hbar', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hbar', commutative=True))))), Integer(1))"], [["divide", 2, "Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\hbar', commutative=True))))), Symbol('\\\\hbar', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hbar', commutative=True)), exp(Function('\\\\mathbf{J}_P')(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True))"]]}, {"prompt": "Given W{(t_{2},\\sigma_x)} = \\int \\frac{t_{2}}{\\sigma_x} dt_{2}, then obtain 0 = \\frac{\\partial}{\\partial t_{2}} (- \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} + 1)^{\\sigma_x}", "derivation": "W{(t_{2},\\sigma_x)} = \\int \\frac{t_{2}}{\\sigma_x} dt_{2} and \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} = 1 and 0 = - \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} + 1 and 0^{\\sigma_x} = (- \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} + 1)^{\\sigma_x} and \\frac{d}{d t_{2}} 0^{\\sigma_x} = \\frac{\\partial}{\\partial t_{2}} (- \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} + 1)^{\\sigma_x} and 0 = \\frac{\\partial}{\\partial t_{2}} (- \\frac{W{(t_{2},\\sigma_x)}}{\\int \\frac{t_{2}}{\\sigma_x} dt_{2}} + 1)^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["divide", 1, "Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1)))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Derivative(Pow(Add(Mul(Integer(-1), Function('W')(Symbol('t_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(t_{1},F_{H})} = F_{H} t_{1} and y{(t_{1},F_{H})} = \\int F_{H} t_{1} dt_{1}, then obtain y{(t_{1},F_{H})} = \\int \\hat{\\mathbf{r}}{(t_{1},F_{H})} dt_{1}", "derivation": "\\hat{\\mathbf{r}}{(t_{1},F_{H})} = F_{H} t_{1} and \\int \\hat{\\mathbf{r}}{(t_{1},F_{H})} dt_{1} = \\int F_{H} t_{1} dt_{1} and y{(t_{1},F_{H})} = \\int F_{H} t_{1} dt_{1} and y{(t_{1},F_{H})} = \\int \\hat{\\mathbf{r}}{(t_{1},F_{H})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Symbol('F_H', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Integral(Mul(Symbol('F_H', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('y')(Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given Q{(F_{c},J_{\\varepsilon})} = - F_{c} + J_{\\varepsilon} and \\eta{(F_{c},J_{\\varepsilon})} = (- F_{c} + J_{\\varepsilon}) Q{(F_{c},J_{\\varepsilon})}, then obtain (- F_{c} + J_{\\varepsilon}) Q{(F_{c},J_{\\varepsilon})} \\eta{(F_{c},J_{\\varepsilon})} = (- F_{c} + J_{\\varepsilon})^{3} Q{(F_{c},J_{\\varepsilon})}", "derivation": "Q{(F_{c},J_{\\varepsilon})} = - F_{c} + J_{\\varepsilon} and \\eta{(F_{c},J_{\\varepsilon})} = (- F_{c} + J_{\\varepsilon}) Q{(F_{c},J_{\\varepsilon})} and \\eta{(F_{c},J_{\\varepsilon})} = (- F_{c} + J_{\\varepsilon})^{2} and (- F_{c} + J_{\\varepsilon}) Q{(F_{c},J_{\\varepsilon})} \\eta{(F_{c},J_{\\varepsilon})} = (- F_{c} + J_{\\varepsilon})^{3} Q{(F_{c},J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('Q')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["times", 3, "Mul(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('Q')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('Q')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('\\\\eta')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(3)), Function('Q')(Symbol('F_c', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(q)} = \\sin{(q)}, then obtain \\Psi + \\frac{\\frac{d}{d q} (q + \\operatorname{g_{\\varepsilon}}{(q)})}{\\frac{d}{d q} (q + \\sin{(q)})} = \\Psi + 1", "derivation": "\\operatorname{g_{\\varepsilon}}{(q)} = \\sin{(q)} and q + \\operatorname{g_{\\varepsilon}}{(q)} = q + \\sin{(q)} and \\frac{d}{d q} (q + \\operatorname{g_{\\varepsilon}}{(q)}) = \\frac{d}{d q} (q + \\sin{(q)}) and \\frac{\\frac{d}{d q} (q + \\operatorname{g_{\\varepsilon}}{(q)})}{\\frac{d}{d q} (q + \\sin{(q)})} = 1 and \\Psi + \\frac{\\frac{d}{d q} (q + \\operatorname{g_{\\varepsilon}}{(q)})}{\\frac{d}{d q} (q + \\sin{(q)})} = \\Psi + 1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Add(Symbol('q', commutative=True), sin(Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Symbol('q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Add(Symbol('q', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('q', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["add", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Derivative(Add(Symbol('q', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('q', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1)))), Add(Symbol('\\\\Psi', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\phi_1,M_{E})} = \\sin{(M_{E} + \\phi_1)}, then obtain 2 \\operatorname{t_{1}}{(\\phi_1,M_{E})} \\sin{(M_{E} + \\phi_1)} = 2 \\sin^{2}{(M_{E} + \\phi_1)}", "derivation": "\\operatorname{t_{1}}{(\\phi_1,M_{E})} = \\sin{(M_{E} + \\phi_1)} and \\operatorname{t_{1}}{(\\phi_1,M_{E})} + \\sin{(M_{E} + \\phi_1)} = 2 \\sin{(M_{E} + \\phi_1)} and (\\operatorname{t_{1}}{(\\phi_1,M_{E})} + \\sin{(M_{E} + \\phi_1)}) \\operatorname{t_{1}}{(\\phi_1,M_{E})} = (\\operatorname{t_{1}}{(\\phi_1,M_{E})} + \\sin{(M_{E} + \\phi_1)}) \\sin{(M_{E} + \\phi_1)} and 2 \\operatorname{t_{1}}{(\\phi_1,M_{E})} \\sin{(M_{E} + \\phi_1)} = 2 \\sin^{2}{(M_{E} + \\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["add", 1, "sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["times", 1, "Add(Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], "Equality(Mul(Add(Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True))), Mul(Add(Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('t_1')(Symbol('\\\\phi_1', commutative=True), Symbol('M_E', commutative=True)), sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Pow(sin(Add(Symbol('M_E', commutative=True), Symbol('\\\\phi_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{r}{(\\sigma_p)} = e^{\\sigma_p} and \\dot{y}{(l,\\mathbf{H})} = \\cos{(\\mathbf{H} l)}, then obtain \\dot{y}{(l,\\mathbf{H})} - e^{\\sigma_p} = - e^{\\sigma_p} + \\cos{(\\mathbf{H} l)}", "derivation": "\\mathbf{r}{(\\sigma_p)} = e^{\\sigma_p} and \\dot{y}{(l,\\mathbf{H})} = \\cos{(\\mathbf{H} l)} and \\dot{y}{(l,\\mathbf{H})} - \\mathbf{r}{(\\sigma_p)} = - \\mathbf{r}{(\\sigma_p)} + \\cos{(\\mathbf{H} l)} and \\dot{y}{(l,\\mathbf{H})} - e^{\\sigma_p} = - e^{\\sigma_p} + \\cos{(\\mathbf{H} l)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('l', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{r}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\sigma_p', commutative=True))), cos(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\dot{y}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True))), cos(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given s{(a^{\\dagger})} = \\cos{(e^{a^{\\dagger}})} and \\operatorname{E_{n}}{(a^{\\dagger})} = e^{a^{\\dagger}}, then derive \\int s{(a^{\\dagger})} da^{\\dagger} = c + \\operatorname{Ci}{(e^{a^{\\dagger}})}, then obtain \\iint \\cos{(\\operatorname{E_{n}}{(a^{\\dagger})})} da^{\\dagger} da^{\\dagger} = \\int (c + \\operatorname{Ci}{(e^{a^{\\dagger}})}) da^{\\dagger}", "derivation": "s{(a^{\\dagger})} = \\cos{(e^{a^{\\dagger}})} and \\int s{(a^{\\dagger})} da^{\\dagger} = \\int \\cos{(e^{a^{\\dagger}})} da^{\\dagger} and \\int s{(a^{\\dagger})} da^{\\dagger} = c + \\operatorname{Ci}{(e^{a^{\\dagger}})} and \\operatorname{E_{n}}{(a^{\\dagger})} = e^{a^{\\dagger}} and s{(a^{\\dagger})} = \\cos{(\\operatorname{E_{n}}{(a^{\\dagger})})} and \\int s{(a^{\\dagger})} da^{\\dagger} = \\int \\cos{(\\operatorname{E_{n}}{(a^{\\dagger})})} da^{\\dagger} and \\int \\cos{(\\operatorname{E_{n}}{(a^{\\dagger})})} da^{\\dagger} = c + \\operatorname{Ci}{(e^{a^{\\dagger}})} and \\iint \\cos{(\\operatorname{E_{n}}{(a^{\\dagger})})} da^{\\dagger} da^{\\dagger} = \\int (c + \\operatorname{Ci}{(e^{a^{\\dagger}})}) da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('a^{\\\\dagger}', commutative=True)), cos(exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(cos(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('c', commutative=True), Ci(exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('s')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Function('E_n')(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(cos(Function('E_n')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integral(cos(Function('E_n')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('c', commutative=True), Ci(exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 7, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(cos(Function('E_n')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('c', commutative=True), Ci(exp(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} = V + s, then derive \\frac{\\partial^{2}}{\\partial s^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} = 0, then obtain \\sin{(\\frac{\\partial^{2}}{\\partial s^{2}} (V + s))} = 0", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} = V + s and \\frac{\\partial}{\\partial s} \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} = \\frac{\\partial}{\\partial s} (V + s) and \\frac{\\partial}{\\partial s} \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} - 1 = \\frac{\\partial}{\\partial s} (V + s) - 1 and \\frac{\\partial}{\\partial s} (\\frac{\\partial}{\\partial s} \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} - 1) = \\frac{\\partial}{\\partial s} (\\frac{\\partial}{\\partial s} (V + s) - 1) and \\frac{\\partial^{2}}{\\partial s^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,s)} = 0 and \\frac{\\partial^{2}}{\\partial s^{2}} (V + s) = 0 and \\sin{(\\frac{\\partial^{2}}{\\partial s^{2}} (V + s))} = 0", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(0))"], [["sin", 6], "Equality(sin(Derivative(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2)))), Integer(0))"]]}, {"prompt": "Given \\hat{x}{(v_{t},Z)} = Z - v_{t}, then obtain v_{t}^{4} (Z - v_{t})^{2} \\hat{x}^{4}{(v_{t},Z)} = v_{t}^{4} (Z - v_{t})^{4} \\hat{x}^{2}{(v_{t},Z)}", "derivation": "\\hat{x}{(v_{t},Z)} = Z - v_{t} and v_{t} \\hat{x}{(v_{t},Z)} = v_{t} (Z - v_{t}) and - v_{t} \\hat{x}{(v_{t},Z)} = - v_{t} (Z - v_{t}) and - v_{t} \\hat{x}^{2}{(v_{t},Z)} = - v_{t} (Z - v_{t}) \\hat{x}{(v_{t},Z)} and v_{t}^{2} (Z - v_{t}) \\hat{x}^{2}{(v_{t},Z)} = v_{t}^{2} (Z - v_{t})^{2} \\hat{x}{(v_{t},Z)} and v_{t}^{4} (Z - v_{t})^{2} \\hat{x}^{4}{(v_{t},Z)} = v_{t}^{4} (Z - v_{t})^{4} \\hat{x}^{2}{(v_{t},Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["divide", 1, "Pow(Symbol('v_t', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v_t', commutative=True), Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('v_t', commutative=True), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('v_t', commutative=True), Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["times", 3, "Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_t', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('v_t', commutative=True), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('v_t', commutative=True), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(2)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Integer(2))), Mul(Pow(Symbol('v_t', commutative=True), Integer(2)), Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Integer(2)), Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))))"], [["power", 5, 2], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(4)), Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Integer(2)), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Integer(4))), Mul(Pow(Symbol('v_t', commutative=True), Integer(4)), Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Integer(4)), Pow(Function('\\\\hat{x}')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Integer(2))))"]]}, {"prompt": "Given q{(P_{e},F_{x})} = F_{x} P_{e}, then derive \\frac{\\partial}{\\partial F_{x}} q{(P_{e},F_{x})} = P_{e}, then obtain q{(\\frac{\\partial}{\\partial F_{x}} F_{x} P_{e},F_{x})} = F_{x} \\frac{\\partial}{\\partial F_{x}} F_{x} P_{e}", "derivation": "q{(P_{e},F_{x})} = F_{x} P_{e} and \\frac{\\partial}{\\partial F_{x}} q{(P_{e},F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x} P_{e} and \\frac{\\partial}{\\partial F_{x}} q{(P_{e},F_{x})} = P_{e} and \\frac{\\partial}{\\partial F_{x}} F_{x} P_{e} = P_{e} and q{(\\frac{\\partial}{\\partial F_{x}} F_{x} P_{e},F_{x})} = F_{x} \\frac{\\partial}{\\partial F_{x}} F_{x} P_{e}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('P_e', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('P_e', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('P_e', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('P_e', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('P_e', commutative=True))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('q')(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain \\frac{d^{2}}{d \\theta_1^{2}} (2 \\phi_{1}{(\\theta_1)} - 2 \\sin{(\\theta_1)}) = \\frac{d^{2}}{d \\theta_1^{2}} 0", "derivation": "\\phi_{1}{(\\theta_1)} = \\sin{(\\theta_1)} and \\phi_{1}{(\\theta_1)} - \\sin{(\\theta_1)} = 0 and 2 \\phi_{1}{(\\theta_1)} - 2 \\sin{(\\theta_1)} = \\phi_{1}{(\\theta_1)} - \\sin{(\\theta_1)} and 2 \\phi_{1}{(\\theta_1)} - 2 \\sin{(\\theta_1)} = 0 and \\frac{d}{d \\theta_1} (2 \\phi_{1}{(\\theta_1)} - 2 \\sin{(\\theta_1)}) = \\frac{d}{d \\theta_1} 0 and \\frac{d}{d \\theta_1} (\\phi_{1}{(\\theta_1)} - \\sin{(\\theta_1)}) = \\frac{d}{d \\theta_1} 0 and \\frac{d^{2}}{d \\theta_1^{2}} (\\phi_{1}{(\\theta_1)} - \\sin{(\\theta_1)}) = \\frac{d^{2}}{d \\theta_1^{2}} 0 and \\frac{d^{2}}{d \\theta_1^{2}} (2 \\phi_{1}{(\\theta_1)} - 2 \\sin{(\\theta_1)}) = \\frac{d^{2}}{d \\theta_1^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Integer(0))"], [["add", 2, "Add(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given G{(\\pi)} = \\log{(\\pi)}, then obtain \\cos{((\\pi G^{2}{(\\pi)} + G{(\\pi)}) \\int (\\pi + G^{2}{(\\pi)}) d\\pi)} = \\cos{((\\pi G^{2}{(\\pi)} + G{(\\pi)}) \\int (\\pi + G{(\\pi)} \\log{(\\pi)}) d\\pi)}", "derivation": "G{(\\pi)} = \\log{(\\pi)} and G^{2}{(\\pi)} = G{(\\pi)} \\log{(\\pi)} and \\pi + G^{2}{(\\pi)} = \\pi + G{(\\pi)} \\log{(\\pi)} and \\int (\\pi + G^{2}{(\\pi)}) d\\pi = \\int (\\pi + G{(\\pi)} \\log{(\\pi)}) d\\pi and (\\pi G^{2}{(\\pi)} + \\log{(\\pi)}) \\int (\\pi + G^{2}{(\\pi)}) d\\pi = (\\pi G^{2}{(\\pi)} + \\log{(\\pi)}) \\int (\\pi + G{(\\pi)} \\log{(\\pi)}) d\\pi and \\cos{((\\pi G^{2}{(\\pi)} + \\log{(\\pi)}) \\int (\\pi + G^{2}{(\\pi)}) d\\pi)} = \\cos{((\\pi G^{2}{(\\pi)} + \\log{(\\pi)}) \\int (\\pi + G{(\\pi)} \\log{(\\pi)}) d\\pi)} and \\cos{((\\pi G^{2}{(\\pi)} + G{(\\pi)}) \\int (\\pi + G^{2}{(\\pi)}) d\\pi)} = \\cos{((\\pi G^{2}{(\\pi)} + G{(\\pi)}) \\int (\\pi + G{(\\pi)} \\log{(\\pi)}) d\\pi)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Function('G')(Symbol('\\\\pi', commutative=True))"], "Equality(Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2)), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Add(Symbol('\\\\pi', commutative=True), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["times", 4, "Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), log(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), log(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), log(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["cos", 5], "Equality(cos(Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), log(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\pi', commutative=True))))), cos(Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), log(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(cos(Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Function('G')(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\pi', commutative=True))))), cos(Mul(Add(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('G')(Symbol('\\\\pi', commutative=True)), Integer(2))), Function('G')(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Function('G')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given r{(F_{c})} = \\int e^{F_{c}} dF_{c}, then derive r{(F_{c})} = r + e^{F_{c}}, then obtain - r{(F_{c})} + \\int r{(F_{c})} dF_{c} = - r{(F_{c})} + \\int (r + e^{F_{c}}) dF_{c}", "derivation": "r{(F_{c})} = \\int e^{F_{c}} dF_{c} and r{(F_{c})} = r + e^{F_{c}} and \\int r{(F_{c})} dF_{c} = \\int (r + e^{F_{c}}) dF_{c} and - r{(F_{c})} + \\int r{(F_{c})} dF_{c} = - r{(F_{c})} + \\int (r + e^{F_{c}}) dF_{c}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('F_c', commutative=True)), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('r')(Symbol('F_c', commutative=True)), Add(Symbol('r', commutative=True), exp(Symbol('F_c', commutative=True))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('r')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], [["minus", 3, "Function('r')(Symbol('F_c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('r')(Symbol('F_c', commutative=True))), Integral(Function('r')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Add(Mul(Integer(-1), Function('r')(Symbol('F_c', commutative=True))), Integral(Add(Symbol('r', commutative=True), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\theta{(S,\\mathbf{s})} = S + \\mathbf{s}, then obtain \\int \\theta{(S,\\mathbf{s})} d\\mathbf{s} + \\frac{S + \\mathbf{s}}{S} = \\int (S + \\mathbf{s}) d\\mathbf{s} + \\frac{S + \\mathbf{s}}{S}", "derivation": "\\theta{(S,\\mathbf{s})} = S + \\mathbf{s} and \\frac{\\theta{(S,\\mathbf{s})}}{S} = \\frac{S + \\mathbf{s}}{S} and \\int \\theta{(S,\\mathbf{s})} d\\mathbf{s} = \\int (S + \\mathbf{s}) d\\mathbf{s} and \\int \\theta{(S,\\mathbf{s})} d\\mathbf{s} + \\frac{\\theta{(S,\\mathbf{s})}}{S} = \\int (S + \\mathbf{s}) d\\mathbf{s} + \\frac{\\theta{(S,\\mathbf{s})}}{S} and \\int \\theta{(S,\\mathbf{s})} d\\mathbf{s} + \\frac{S + \\mathbf{s}}{S} = \\int (S + \\mathbf{s}) d\\mathbf{s} + \\frac{S + \\mathbf{s}}{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Integral(Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Add(Integral(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integral(Function('\\\\theta')(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Add(Integral(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{H})} = \\sin{(\\hat{H})}, then obtain ((\\frac{\\sin{(\\operatorname{F_{g}}{(\\hat{H})})}}{\\hat{H}})^{\\hat{H}} (\\frac{\\sin{(\\sin{(\\hat{H})})}}{\\hat{H}})^{- \\hat{H}})^{\\hat{H}} = 1", "derivation": "\\operatorname{F_{g}}{(\\hat{H})} = \\sin{(\\hat{H})} and \\sin{(\\operatorname{F_{g}}{(\\hat{H})})} = \\sin{(\\sin{(\\hat{H})})} and \\frac{\\sin{(\\operatorname{F_{g}}{(\\hat{H})})}}{\\hat{H}} = \\frac{\\sin{(\\sin{(\\hat{H})})}}{\\hat{H}} and (\\frac{\\sin{(\\operatorname{F_{g}}{(\\hat{H})})}}{\\hat{H}})^{\\hat{H}} = (\\frac{\\sin{(\\sin{(\\hat{H})})}}{\\hat{H}})^{\\hat{H}} and (\\frac{\\sin{(\\operatorname{F_{g}}{(\\hat{H})})}}{\\hat{H}})^{\\hat{H}} (\\frac{\\sin{(\\sin{(\\hat{H})})}}{\\hat{H}})^{- \\hat{H}} = 1 and ((\\frac{\\sin{(\\operatorname{F_{g}}{(\\hat{H})})}}{\\hat{H}})^{\\hat{H}} (\\frac{\\sin{(\\sin{(\\hat{H})})}}{\\hat{H}})^{- \\hat{H}})^{\\hat{H}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('F_g')(Symbol('\\\\hat{H}', commutative=True))), sin(sin(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\hat{H}', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 4, "Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Integer(1))"], [["power", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} = \\mathbf{J}_f + \\psi, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} - 1 = 0, then derive \\frac{\\partial^{2}}{\\partial \\psi\\partial \\mathbf{J}_f} \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} = 0, then obtain \\frac{\\partial^{2}}{\\partial \\psi\\partial \\mathbf{J}_f} (\\mathbf{J}_f + \\psi) = 0", "derivation": "\\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} = \\mathbf{J}_f + \\psi and - \\mathbf{J}_f - \\psi + \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} = 0 and \\frac{\\partial}{\\partial \\mathbf{J}_f} (- \\mathbf{J}_f - \\psi + \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)}) = \\frac{d}{d \\mathbf{J}_f} 0 and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} - 1 = 0 and \\frac{\\partial}{\\partial \\psi} (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} - 1) = \\frac{d}{d \\psi} 0 and \\frac{\\partial^{2}}{\\partial \\psi\\partial \\mathbf{J}_f} \\hat{\\mathbf{x}}{(\\psi,\\mathbf{J}_f)} = 0 and \\frac{\\partial^{2}}{\\partial \\psi\\partial \\mathbf{J}_f} (\\mathbf{J}_f + \\psi) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["differentiate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given W{(y^{\\prime})} = \\sin{(y^{\\prime})} and \\mathbf{J}{(\\psi^*,\\mathbf{r})} = \\mathbf{r} - \\psi^*, then obtain \\mathbf{J}{(\\psi^*,\\mathbf{r})} + e^{\\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})}} = \\mathbf{r} - \\psi^* + e^{\\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})}}", "derivation": "W{(y^{\\prime})} = \\sin{(y^{\\prime})} and W^{y^{\\prime}}{(y^{\\prime})} = \\sin^{y^{\\prime}}{(y^{\\prime})} and \\sin{(W^{y^{\\prime}}{(y^{\\prime})})} = \\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})} and e^{\\sin{(W^{y^{\\prime}}{(y^{\\prime})})}} = e^{\\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})}} and \\mathbf{J}{(\\psi^*,\\mathbf{r})} = \\mathbf{r} - \\psi^* and \\mathbf{J}{(\\psi^*,\\mathbf{r})} + e^{\\sin{(W^{y^{\\prime}}{(y^{\\prime})})}} = \\mathbf{r} - \\psi^* + e^{\\sin{(W^{y^{\\prime}}{(y^{\\prime})})}} and \\mathbf{J}{(\\psi^*,\\mathbf{r})} + e^{\\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})}} = \\mathbf{r} - \\psi^* + e^{\\sin{(\\sin^{y^{\\prime}}{(y^{\\prime})})}}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), sin(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], [["exp", 3], "Equality(exp(sin(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))), exp(sin(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["add", 5, "exp(sin(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), exp(sin(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(sin(Pow(Function('W')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), exp(sin(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(sin(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\hat{p},A_{y})} = A_{y} + \\sin{(\\hat{p})} and t{(\\psi,\\rho_f)} = \\psi \\rho_f, then obtain - (\\psi \\rho_f)^{\\psi} + \\hat{\\mathbf{x}}{(\\hat{p},A_{y})} = A_{y} - (\\psi \\rho_f)^{\\psi} + \\sin{(\\hat{p})}", "derivation": "\\hat{\\mathbf{x}}{(\\hat{p},A_{y})} = A_{y} + \\sin{(\\hat{p})} and t{(\\psi,\\rho_f)} = \\psi \\rho_f and t^{\\psi}{(\\psi,\\rho_f)} = (\\psi \\rho_f)^{\\psi} and \\hat{\\mathbf{x}}{(\\hat{p},A_{y})} - t^{\\psi}{(\\psi,\\rho_f)} = A_{y} - t^{\\psi}{(\\psi,\\rho_f)} + \\sin{(\\hat{p})} and - (\\psi \\rho_f)^{\\psi} + \\hat{\\mathbf{x}}{(\\hat{p},A_{y})} = A_{y} - (\\psi \\rho_f)^{\\psi} + \\sin{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}', commutative=True))))"], ["get_premise", "Equality(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Pow(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True)))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True))), sin(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\psi', commutative=True))), sin(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given h{(A)} = \\cos{(A)} and \\mathbf{E}{(A)} = \\cos{(A)}, then obtain A^{2} \\mathbf{E}{(A)} = \\frac{A^{2} \\mathbf{E}^{2}{(A)}}{h{(A)}}", "derivation": "h{(A)} = \\cos{(A)} and A h{(A)} = A \\cos{(A)} and A^{2} h{(A)} \\cos{(A)} = A^{2} \\cos^{2}{(A)} and \\mathbf{E}{(A)} = \\cos{(A)} and A^{2} \\cos{(A)} = \\frac{A^{2} \\cos^{2}{(A)}}{h{(A)}} and A^{2} \\mathbf{E}{(A)} = \\frac{A^{2} \\mathbf{E}^{2}{(A)}}{h{(A)}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["times", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('h')(Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True))))"], [["times", 2, "Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True)))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Function('h')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(cos(Symbol('A', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["divide", 3, "Function('h')(Symbol('A', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(2)), cos(Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Function('h')(Symbol('A', commutative=True)), Integer(-1)), Pow(cos(Symbol('A', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Function('\\\\mathbf{E}')(Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{E}')(Symbol('A', commutative=True)), Integer(2)), Pow(Function('h')(Symbol('A', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\mathbf{J}_f{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then obtain 1 = \\frac{\\mathbf{J}_f{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}}", "derivation": "\\hat{x}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and 1 = \\frac{\\cos{(\\mathbf{g})}}{\\hat{x}{(\\mathbf{g})}} and \\mathbf{J}_f{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and 1 = \\frac{\\mathbf{J}_f{(\\mathbf{g})}}{\\hat{x}{(\\mathbf{g})}} and 1 = \\frac{\\mathbf{J}_f{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 1, "Function('\\\\hat{x}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(\\theta_2,v_{y})} = - \\theta_2 + e^{v_{y}}, then obtain - \\mathbf{B}{(\\theta_2,v_{y})} + \\iint \\mathbf{B}{(\\theta_2,v_{y})} dv_{y} d\\theta_2 = - \\mathbf{B}{(\\theta_2,v_{y})} + \\iint (- \\theta_2 + e^{v_{y}}) dv_{y} d\\theta_2", "derivation": "\\mathbf{B}{(\\theta_2,v_{y})} = - \\theta_2 + e^{v_{y}} and \\int \\mathbf{B}{(\\theta_2,v_{y})} dv_{y} = \\int (- \\theta_2 + e^{v_{y}}) dv_{y} and \\iint \\mathbf{B}{(\\theta_2,v_{y})} dv_{y} d\\theta_2 = \\iint (- \\theta_2 + e^{v_{y}}) dv_{y} d\\theta_2 and - \\mathbf{B}{(\\theta_2,v_{y})} + \\iint \\mathbf{B}{(\\theta_2,v_{y})} dv_{y} d\\theta_2 = - \\mathbf{B}{(\\theta_2,v_{y})} + \\iint (- \\theta_2 + e^{v_{y}}) dv_{y} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then obtain 1 + \\frac{\\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}} = \\frac{2 \\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\operatorname{P_{g}}{(\\mathbf{J}_P)} + \\log{(\\mathbf{J}_P)} = 2 \\log{(\\mathbf{J}_P)} and \\frac{\\operatorname{P_{g}}{(\\mathbf{J}_P)} + \\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}} = \\frac{2 \\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}} and 1 + \\frac{\\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}} = \\frac{2 \\log{(\\mathbf{J}_P)}}{\\operatorname{P_{g}}{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "log(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 2, "Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Add(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["expand", 3], "Equality(Add(Integer(1), Mul(Pow(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(2), Pow(Function('P_g')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(f^{*},v_{t})} = f^{*} - v_{t}, then obtain \\frac{\\partial}{\\partial v_{t}} \\operatorname{x^{{\\}'}}{(f^{*},v_{t})} - 1 = -2", "derivation": "\\operatorname{x^{{\\}'}}{(f^{*},v_{t})} = f^{*} - v_{t} and - v_{t} + \\operatorname{x^{{\\}'}}{(f^{*},v_{t})} = f^{*} - 2 v_{t} and \\frac{\\partial}{\\partial v_{t}} (- v_{t} + \\operatorname{x^{{\\}'}}{(f^{*},v_{t})}) = \\frac{\\partial}{\\partial v_{t}} (f^{*} - 2 v_{t}) and \\frac{\\partial}{\\partial v_{t}} \\operatorname{x^{{\\}'}}{(f^{*},v_{t})} - 1 = -2", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('f^*', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('x^\\\\prime')(Symbol('f^*', commutative=True), Symbol('v_t', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True))))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('x^\\\\prime')(Symbol('f^*', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('x^\\\\prime')(Symbol('f^*', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"]]}, {"prompt": "Given B{(\\rho)} = \\sin{(\\cos{(\\rho)})}, then derive \\log{(\\frac{d^{2}}{d \\rho^{2}} B{(\\rho)})} = \\log{(- \\sin^{2}{(\\rho)} \\sin{(\\cos{(\\rho)})} - \\cos{(\\rho)} \\cos{(\\cos{(\\rho)})})}, then obtain \\log{(\\frac{d^{2}}{d \\rho^{2}} B{(\\rho)})} = \\log{(- B{(\\rho)} \\sin^{2}{(\\rho)} - \\cos{(\\rho)} \\cos{(\\cos{(\\rho)})})}", "derivation": "B{(\\rho)} = \\sin{(\\cos{(\\rho)})} and \\frac{d}{d \\rho} B{(\\rho)} = \\frac{d}{d \\rho} \\sin{(\\cos{(\\rho)})} and \\frac{d^{2}}{d \\rho^{2}} B{(\\rho)} = \\frac{d^{2}}{d \\rho^{2}} \\sin{(\\cos{(\\rho)})} and \\log{(\\frac{d^{2}}{d \\rho^{2}} B{(\\rho)})} = \\log{(\\frac{d^{2}}{d \\rho^{2}} \\sin{(\\cos{(\\rho)})})} and \\log{(\\frac{d^{2}}{d \\rho^{2}} B{(\\rho)})} = \\log{(- \\sin^{2}{(\\rho)} \\sin{(\\cos{(\\rho)})} - \\cos{(\\rho)} \\cos{(\\cos{(\\rho)})})} and \\log{(\\frac{d^{2}}{d \\rho^{2}} B{(\\rho)})} = \\log{(- B{(\\rho)} \\sin^{2}{(\\rho)} - \\cos{(\\rho)} \\cos{(\\cos{(\\rho)})})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\rho', commutative=True)), sin(cos(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))), Derivative(sin(cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(2))))"], [["log", 3], "Equality(log(Derivative(Function('B')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2)))), log(Derivative(sin(cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(log(Derivative(Function('B')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2)))), log(Mul(Integer(-1), Add(Mul(Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(2)), sin(cos(Symbol('\\\\rho', commutative=True)))), Mul(cos(Symbol('\\\\rho', commutative=True)), cos(cos(Symbol('\\\\rho', commutative=True))))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(log(Derivative(Function('B')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(2)))), log(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\rho', commutative=True)), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)), cos(cos(Symbol('\\\\rho', commutative=True)))))))"]]}, {"prompt": "Given W{(\\dot{y})} = e^{\\dot{y}}, then obtain \\int (W{(\\dot{y})} e^{\\dot{y}} - e^{\\dot{y}}) d\\dot{y} = f^{\\prime} + \\frac{e^{2 \\dot{y}}}{2} - e^{\\dot{y}}", "derivation": "W{(\\dot{y})} = e^{\\dot{y}} and W{(\\dot{y})} e^{\\dot{y}} = e^{2 \\dot{y}} and W{(\\dot{y})} e^{\\dot{y}} - e^{\\dot{y}} = e^{2 \\dot{y}} - e^{\\dot{y}} and \\int (W{(\\dot{y})} e^{\\dot{y}} - e^{\\dot{y}}) d\\dot{y} = \\int (e^{2 \\dot{y}} - e^{\\dot{y}}) d\\dot{y} and \\int (W{(\\dot{y})} e^{\\dot{y}} - e^{\\dot{y}}) d\\dot{y} = f^{\\prime} + \\frac{e^{2 \\dot{y}}}{2} - e^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Function('W')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Mul(Function('W')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))), Add(exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Mul(Function('W')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Function('W')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given i{(c_{0})} = \\frac{d}{d c_{0}} \\cos{(c_{0})}, then derive i{(c_{0})} = - \\sin{(c_{0})}, then obtain c_{0} + i{(c_{0})} \\cos{(c_{0})} = c_{0} + \\cos{(c_{0})} \\frac{d}{d c_{0}} \\cos{(c_{0})}", "derivation": "i{(c_{0})} = \\frac{d}{d c_{0}} \\cos{(c_{0})} and i{(c_{0})} = - \\sin{(c_{0})} and - \\sin{(c_{0})} = \\frac{d}{d c_{0}} \\cos{(c_{0})} and - \\sin{(c_{0})} \\cos{(c_{0})} = \\cos{(c_{0})} \\frac{d}{d c_{0}} \\cos{(c_{0})} and i{(c_{0})} \\cos{(c_{0})} = \\cos{(c_{0})} \\frac{d}{d c_{0}} \\cos{(c_{0})} and c_{0} + i{(c_{0})} \\cos{(c_{0})} = c_{0} + \\cos{(c_{0})} \\frac{d}{d c_{0}} \\cos{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('c_0', commutative=True)), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('i')(Symbol('c_0', commutative=True)), Mul(Integer(-1), sin(Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('c_0', commutative=True))), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["times", 3, "cos(Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True))), Mul(cos(Symbol('c_0', commutative=True)), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('i')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True))), Mul(cos(Symbol('c_0', commutative=True)), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["add", 5, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Mul(Function('i')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))), Add(Symbol('c_0', commutative=True), Mul(cos(Symbol('c_0', commutative=True)), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))))"]]}, {"prompt": "Given H{(i,F_{N})} = - F_{N} + i, then derive \\frac{\\partial}{\\partial i} H{(i,F_{N})} = 1, then derive i + \\frac{\\partial}{\\partial i} H{(i,F_{N})} = i + 1, then obtain - i (i + \\frac{\\partial}{\\partial i} H{(i,F_{N})}) = - i (i + 1)", "derivation": "H{(i,F_{N})} = - F_{N} + i and \\frac{\\partial}{\\partial i} H{(i,F_{N})} = \\frac{\\partial}{\\partial i} (- F_{N} + i) and \\frac{\\partial}{\\partial i} H{(i,F_{N})} = 1 and \\frac{\\frac{\\partial}{\\partial i} H{(i,F_{N})}}{\\frac{\\partial}{\\partial i} (- F_{N} + i)} = \\frac{1}{\\frac{\\partial}{\\partial i} (- F_{N} + i)} and i + \\frac{\\frac{\\partial}{\\partial i} H{(i,F_{N})}}{\\frac{\\partial}{\\partial i} (- F_{N} + i)} = i + \\frac{1}{\\frac{\\partial}{\\partial i} (- F_{N} + i)} and i + \\frac{\\partial}{\\partial i} H{(i,F_{N})} = i + 1 and i + \\frac{\\partial}{\\partial i} (- F_{N} + i) = i + 1 and - i (i + \\frac{\\partial}{\\partial i} (- F_{N} + i)) = - i (i + 1) and - i (i + \\frac{\\partial}{\\partial i} H{(i,F_{N})}) = - i (i + 1)", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)))"], [["add", 4, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Add(Symbol('i', commutative=True), Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Add(Symbol('i', commutative=True), Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('i', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Integer(1)))"], [["times", 7, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('i', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('i', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('i', commutative=True), Derivative(Function('H')(Symbol('i', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(F_{c},\\sigma_x)} = \\frac{\\sigma_x}{F_{c}}, then obtain \\int (F_{c} + A{(F_{c},\\sigma_x)} + 1) d\\sigma_x - 1 = \\int (F_{c} + 1 + \\frac{\\sigma_x}{F_{c}}) d\\sigma_x - 1", "derivation": "A{(F_{c},\\sigma_x)} = \\frac{\\sigma_x}{F_{c}} and F_{c} + A{(F_{c},\\sigma_x)} = F_{c} + \\frac{\\sigma_x}{F_{c}} and F_{c} + A{(F_{c},\\sigma_x)} + 1 = F_{c} + 1 + \\frac{\\sigma_x}{F_{c}} and \\int (F_{c} + A{(F_{c},\\sigma_x)} + 1) d\\sigma_x = \\int (F_{c} + 1 + \\frac{\\sigma_x}{F_{c}}) d\\sigma_x and \\int (F_{c} + A{(F_{c},\\sigma_x)} + 1) d\\sigma_x - 1 = \\int (F_{c} + 1 + \\frac{\\sigma_x}{F_{c}}) d\\sigma_x - 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)))"], [["add", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Function('A')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Symbol('F_c', commutative=True), Function('A')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)), Add(Symbol('F_c', commutative=True), Integer(1), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Symbol('F_c', commutative=True), Function('A')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Integer(1), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Integral(Add(Symbol('F_c', commutative=True), Function('A')(Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Integral(Add(Symbol('F_c', commutative=True), Integer(1), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\sigma_{x}{(f^{*},z^{*})} = \\cos{(f^{*} + z^{*})} and \\mathbf{J}_P{(f^{*},z^{*})} = \\frac{\\partial}{\\partial f^{*}} \\cos^{f^{*}}{(f^{*} + z^{*})}, then obtain (\\frac{\\partial}{\\partial f^{*}} \\sigma_{x}^{f^{*}}{(f^{*},z^{*})})^{f^{*}} = \\mathbf{J}_P^{f^{*}}{(f^{*},z^{*})}", "derivation": "\\sigma_{x}{(f^{*},z^{*})} = \\cos{(f^{*} + z^{*})} and \\sigma_{x}^{f^{*}}{(f^{*},z^{*})} = \\cos^{f^{*}}{(f^{*} + z^{*})} and \\frac{\\partial}{\\partial f^{*}} \\sigma_{x}^{f^{*}}{(f^{*},z^{*})} = \\frac{\\partial}{\\partial f^{*}} \\cos^{f^{*}}{(f^{*} + z^{*})} and \\mathbf{J}_P{(f^{*},z^{*})} = \\frac{\\partial}{\\partial f^{*}} \\cos^{f^{*}}{(f^{*} + z^{*})} and \\frac{\\partial}{\\partial f^{*}} \\sigma_{x}^{f^{*}}{(f^{*},z^{*})} = \\mathbf{J}_P{(f^{*},z^{*})} and (\\frac{\\partial}{\\partial f^{*}} \\sigma_{x}^{f^{*}}{(f^{*},z^{*})})^{f^{*}} = \\mathbf{J}_P^{f^{*}}{(f^{*},z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), cos(Add(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(cos(Add(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\sigma_x')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Pow(cos(Add(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Derivative(Pow(cos(Add(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('\\\\sigma_x')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Function('\\\\mathbf{J}_P')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)))"], [["power", 5, "Symbol('f^*', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\sigma_x')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('f^*', commutative=True), Symbol('z^*', commutative=True)), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\Psi{(h,\\eta^{\\prime})} = \\cos{(\\eta^{\\prime} - h)}, then obtain \\frac{\\Psi{(h,\\eta^{\\prime})} + \\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}}{\\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}} = \\frac{\\Psi{(h,\\eta^{\\prime})} + \\int \\cos{(\\eta^{\\prime} - h)} d\\eta^{\\prime}}{\\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}}", "derivation": "\\Psi{(h,\\eta^{\\prime})} = \\cos{(\\eta^{\\prime} - h)} and \\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime} = \\int \\cos{(\\eta^{\\prime} - h)} d\\eta^{\\prime} and \\Psi{(h,\\eta^{\\prime})} + \\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime} = \\Psi{(h,\\eta^{\\prime})} + \\int \\cos{(\\eta^{\\prime} - h)} d\\eta^{\\prime} and \\frac{\\Psi{(h,\\eta^{\\prime})} + \\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}}{\\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}} = \\frac{\\Psi{(h,\\eta^{\\prime})} + \\int \\cos{(\\eta^{\\prime} - h)} d\\eta^{\\prime}}{\\int \\Psi{(h,\\eta^{\\prime})} d\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 2, "Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Pow(Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Add(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Pow(Integral(Function('\\\\Psi')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given Z{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\operatorname{F_{N}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain \\log{(\\operatorname{F_{N}}{(\\mathbf{p})} + \\cos{(\\mathbf{p})})} = \\log{(2 \\operatorname{F_{N}}{(\\mathbf{p})})}", "derivation": "Z{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\operatorname{F_{N}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and Z{(\\mathbf{p})} = \\operatorname{F_{N}}{(\\mathbf{p})} and \\operatorname{F_{N}}{(\\mathbf{p})} + Z{(\\mathbf{p})} = 2 \\operatorname{F_{N}}{(\\mathbf{p})} and \\log{(\\operatorname{F_{N}}{(\\mathbf{p})} + Z{(\\mathbf{p})})} = \\log{(2 \\operatorname{F_{N}}{(\\mathbf{p})})} and \\log{(\\operatorname{F_{N}}{(\\mathbf{p})} + \\cos{(\\mathbf{p})})} = \\log{(2 \\operatorname{F_{N}}{(\\mathbf{p})})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 3, "Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)), Function('Z')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True))))"], [["log", 4], "Equality(log(Add(Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)), Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)))), log(Mul(Integer(2), Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Add(Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))), log(Mul(Integer(2), Function('F_N')(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given g{(t)} = e^{t}, then obtain \\frac{g{(t)}}{t \\frac{d}{d t} g^{t}{(t)}} = \\frac{e^{t}}{t \\frac{d}{d t} g^{t}{(t)}}", "derivation": "g{(t)} = e^{t} and \\frac{g{(t)}}{t} = \\frac{e^{t}}{t} and g^{t}{(t)} = (e^{t})^{t} and \\frac{d}{d t} g^{t}{(t)} = \\frac{d}{d t} (e^{t})^{t} and \\frac{g{(t)}}{t \\frac{d}{d t} (e^{t})^{t}} = \\frac{e^{t}}{t \\frac{d}{d t} (e^{t})^{t}} and \\frac{g{(t)}}{t \\frac{d}{d t} g^{t}{(t)}} = \\frac{e^{t}}{t \\frac{d}{d t} g^{t}{(t)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["divide", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('g')(Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(Symbol('t', commutative=True))))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('g')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Function('g')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('g')(Symbol('t', commutative=True)), Pow(Derivative(Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(Symbol('t', commutative=True)), Pow(Derivative(Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('g')(Symbol('t', commutative=True)), Pow(Derivative(Pow(Function('g')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(Symbol('t', commutative=True)), Pow(Derivative(Pow(Function('g')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(F_{x})} = \\log{(F_{x})}, then obtain 6 \\operatorname{J_{\\varepsilon}}{(F_{x})} - 4 \\log{(F_{x})} = 2 \\operatorname{J_{\\varepsilon}}{(F_{x})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(F_{x})} = \\log{(F_{x})} and 0 = - \\operatorname{J_{\\varepsilon}}{(F_{x})} + \\log{(F_{x})} and \\operatorname{J_{\\varepsilon}}{(F_{x})} - \\log{(F_{x})} = 0 and 2 \\operatorname{J_{\\varepsilon}}{(F_{x})} - \\log{(F_{x})} = \\operatorname{J_{\\varepsilon}}{(F_{x})} and 2 \\operatorname{J_{\\varepsilon}}{(F_{x})} - \\log{(F_{x})} = \\log{(F_{x})} and 4 \\operatorname{J_{\\varepsilon}}{(F_{x})} - 3 \\log{(F_{x})} = \\log{(F_{x})} and 6 \\operatorname{J_{\\varepsilon}}{(F_{x})} - 4 \\log{(F_{x})} = 2 \\operatorname{J_{\\varepsilon}}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], [["minus", 1, "Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), log(Symbol('F_x', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), log(Symbol('F_x', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True)), Mul(Integer(-1), log(Symbol('F_x', commutative=True)))), Integer(0))"], [["add", 3, "Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), Mul(Integer(-1), log(Symbol('F_x', commutative=True)))), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Add(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), Mul(Integer(-1), log(Symbol('F_x', commutative=True)))), log(Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(4), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), Mul(Integer(-1), Integer(3), log(Symbol('F_x', commutative=True)))), log(Symbol('F_x', commutative=True)))"], [["add", 6, "Add(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), Mul(Integer(-1), log(Symbol('F_x', commutative=True))))"], "Equality(Add(Mul(Integer(6), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))), Mul(Integer(-1), Integer(4), log(Symbol('F_x', commutative=True)))), Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{v},y)} = - \\mathbf{v} + y, then obtain (1 - \\frac{y}{\\mathbf{v}})^{y} = (- \\frac{- \\mathbf{v} + y}{\\mathbf{v}})^{y}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{v},y)} = - \\mathbf{v} + y and - \\frac{\\operatorname{A_{z}}{(\\mathbf{v},y)}}{\\mathbf{v}} = - \\frac{- \\mathbf{v} + y}{\\mathbf{v}} and (- \\frac{\\operatorname{A_{z}}{(\\mathbf{v},y)}}{\\mathbf{v}})^{y} = (- \\frac{- \\mathbf{v} + y}{\\mathbf{v}})^{y} and - \\frac{\\operatorname{A_{z}}{(\\mathbf{v},y)}}{\\mathbf{v}} = 1 - \\frac{y}{\\mathbf{v}} and (1 - \\frac{y}{\\mathbf{v}})^{y} = (\\frac{\\mathbf{v} - y}{\\mathbf{v}})^{y} and (- \\frac{- \\mathbf{v} + y}{\\mathbf{v}})^{y} = (\\frac{\\mathbf{v} - y}{\\mathbf{v}})^{y} and (1 - \\frac{y}{\\mathbf{v}})^{y} = (- \\frac{- \\mathbf{v} + y}{\\mathbf{v}})^{y}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y', commutative=True))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["expand", 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(h,\\mathbf{r})} = \\mathbf{r} h, then derive \\frac{\\partial}{\\partial \\mathbf{r}} \\operatorname{C_{2}}{(h,\\mathbf{r})} = h, then obtain \\operatorname{C_{2}}{(\\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h,\\mathbf{r})} = \\mathbf{r} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h", "derivation": "\\operatorname{C_{2}}{(h,\\mathbf{r})} = \\mathbf{r} h and \\frac{\\partial}{\\partial \\mathbf{r}} \\operatorname{C_{2}}{(h,\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h and \\frac{\\partial}{\\partial \\mathbf{r}} \\operatorname{C_{2}}{(h,\\mathbf{r})} = h and h = \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h and \\operatorname{C_{2}}{(\\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h,\\mathbf{r})} = \\mathbf{r} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} h", "srepr_derivation": [["get_premise", "Equality(Function('C_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Symbol('h', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('h', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('C_2')(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(C)} = \\sin{(C)} and \\phi_{2}{(C)} = 2 \\sin{(C)}, then obtain \\log{(\\frac{d}{d C} \\frac{\\phi_{2}{(C)}}{C})} = \\log{(\\frac{d}{d C} \\frac{2 \\sin{(C)}}{C})}", "derivation": "u{(C)} = \\sin{(C)} and u{(C)} + \\sin{(C)} = 2 \\sin{(C)} and \\frac{u{(C)} + \\sin{(C)}}{C} = \\frac{2 \\sin{(C)}}{C} and \\phi_{2}{(C)} = 2 \\sin{(C)} and \\frac{u{(C)} + \\sin{(C)}}{C} = \\frac{\\phi_{2}{(C)}}{C} and \\frac{\\phi_{2}{(C)}}{C} = \\frac{2 \\sin{(C)}}{C} and \\frac{d}{d C} \\frac{\\phi_{2}{(C)}}{C} = \\frac{d}{d C} \\frac{2 \\sin{(C)}}{C} and \\log{(\\frac{d}{d C} \\frac{\\phi_{2}{(C)}}{C})} = \\log{(\\frac{d}{d C} \\frac{2 \\sin{(C)}}{C})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["add", 1, "sin(Symbol('C', commutative=True))"], "Equality(Add(Function('u')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Integer(2), sin(Symbol('C', commutative=True))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Function('u')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('C', commutative=True)), Mul(Integer(2), sin(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Function('u')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('C', commutative=True))), Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True))))"], [["differentiate", 6, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["log", 7], "Equality(log(Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), log(Derivative(Mul(Integer(2), Pow(Symbol('C', commutative=True), Integer(-1)), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(t_{2})} = \\cos{(t_{2})}, then obtain - t_{2} \\int \\tilde{g}^*{(t_{2})} dt_{2} + \\tilde{g}^*{(t_{2})} \\int \\tilde{g}^*{(t_{2})} dt_{2} = - t_{2} \\int \\tilde{g}^*{(t_{2})} dt_{2} + \\tilde{g}^*{(t_{2})} \\int \\cos{(t_{2})} dt_{2}", "derivation": "\\tilde{g}^*{(t_{2})} = \\cos{(t_{2})} and \\int \\tilde{g}^*{(t_{2})} dt_{2} = \\int \\cos{(t_{2})} dt_{2} and \\cos{(t_{2})} \\int \\tilde{g}^*{(t_{2})} dt_{2} = \\cos{(t_{2})} \\int \\cos{(t_{2})} dt_{2} and \\tilde{g}^*{(t_{2})} \\int \\tilde{g}^*{(t_{2})} dt_{2} = \\tilde{g}^*{(t_{2})} \\int \\cos{(t_{2})} dt_{2} and - t_{2} \\int \\tilde{g}^*{(t_{2})} dt_{2} + \\tilde{g}^*{(t_{2})} \\int \\tilde{g}^*{(t_{2})} dt_{2} = - t_{2} \\int \\tilde{g}^*{(t_{2})} dt_{2} + \\tilde{g}^*{(t_{2})} \\int \\cos{(t_{2})} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["times", 2, "cos(Symbol('t_2', commutative=True))"], "Equality(Mul(cos(Symbol('t_2', commutative=True)), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(cos(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["minus", 4, "Mul(Symbol('t_2', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given p{(U,x)} = U x, then derive (\\int (- x + p{(U,x)}) dU)^{x} = (\\frac{U^{2} x}{2} - U x + r)^{x}, then obtain (\\int (U x - x) dU)^{2 x} = (\\frac{U p{(U,x)}}{2} + r - p{(U,x)})^{2 x}", "derivation": "p{(U,x)} = U x and - x + p{(U,x)} = U x - x and \\int (- x + p{(U,x)}) dU = \\int (U x - x) dU and (\\int (- x + p{(U,x)}) dU)^{x} = (\\int (U x - x) dU)^{x} and (\\int (- x + p{(U,x)}) dU)^{x} = (\\frac{U^{2} x}{2} - U x + r)^{x} and (\\int (U x - x) dU)^{x} = (\\frac{U^{2} x}{2} - U x + r)^{x} and (\\int (U x - x) dU)^{2 x} = (\\frac{U^{2} x}{2} - U x + r)^{2 x} and (\\int (U x - x) dU)^{2 x} = (\\frac{U p{(U,x)}}{2} + r - p{(U,x)})^{2 x}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True))), Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('x', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('x', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2)), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('U', commutative=True), Symbol('x', commutative=True)), Symbol('r', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('x', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2)), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('U', commutative=True), Symbol('x', commutative=True)), Symbol('r', commutative=True)), Symbol('x', commutative=True)))"], [["power", 6, 2], "Equality(Pow(Integral(Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True))), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2)), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('U', commutative=True), Symbol('x', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Pow(Integral(Add(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('U', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True))), Pow(Add(Mul(Rational(1, 2), Symbol('U', commutative=True), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True))), Symbol('r', commutative=True), Mul(Integer(-1), Function('p')(Symbol('U', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(2), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)} = - \\mathbb{I} + e^{\\Omega} and \\operatorname{E_{\\lambda}}{(\\theta_1)} = \\theta_1, then obtain \\operatorname{E_{\\lambda}}^{\\theta_1}{(\\theta_1)} - e^{\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)}} = \\theta_1^{\\theta_1} - e^{\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)} = - \\mathbb{I} + e^{\\Omega} and e^{\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)}} = e^{- \\mathbb{I} + e^{\\Omega}} and \\operatorname{E_{\\lambda}}{(\\theta_1)} = \\theta_1 and \\operatorname{E_{\\lambda}}^{\\theta_1}{(\\theta_1)} = \\theta_1^{\\theta_1} and \\operatorname{E_{\\lambda}}^{\\theta_1}{(\\theta_1)} - e^{- \\mathbb{I} + e^{\\Omega}} = \\theta_1^{\\theta_1} - e^{- \\mathbb{I} + e^{\\Omega}} and \\operatorname{E_{\\lambda}}^{\\theta_1}{(\\theta_1)} - e^{\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)}} = \\theta_1^{\\theta_1} - e^{\\operatorname{x^{{\\}'}}{(\\mathbb{I},\\Omega)}}", "srepr_derivation": [["get_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\Omega', commutative=True))))"], [["exp", 1], "Equality(exp(Function('x^\\\\prime')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 4, "exp(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\Omega', commutative=True))))"], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Function('x^\\\\prime')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Function('x^\\\\prime')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(E,\\hat{H}_l)} = \\frac{\\hat{H}_l}{E}, then derive \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} = - \\frac{\\hat{H}_l}{E^{2}}, then obtain 0 = - \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} - \\frac{\\hat{H}_l}{E^{2}}", "derivation": "\\operatorname{A_{z}}{(E,\\hat{H}_l)} = \\frac{\\hat{H}_l}{E} and \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} = \\frac{\\partial}{\\partial E} \\frac{\\hat{H}_l}{E} and 0 = \\frac{\\partial}{\\partial E} \\frac{\\hat{H}_l}{E} - \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} and \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} = - \\frac{\\hat{H}_l}{E^{2}} and \\frac{\\partial}{\\partial E} \\frac{\\hat{H}_l}{E} = - \\frac{\\hat{H}_l}{E^{2}} and 0 = - \\frac{\\partial}{\\partial E} \\operatorname{A_{z}}{(E,\\hat{H}_l)} - \\frac{\\hat{H}_l}{E^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-2)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-2)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('A_z')(Symbol('E', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-2)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given t{(p)} = \\sin{(p)}, then obtain - 0^{p} - \\sin{(p)} = - 0^{p} - t{(p)}", "derivation": "t{(p)} = \\sin{(p)} and t{(p)} + \\sin{(p)} = 2 \\sin{(p)} and t{(p)} - \\sin{(p)} = 0 and (t{(p)} - \\sin{(p)})^{p} = 0^{p} and - \\sin{(p)} = - t{(p)} and - (t{(p)} - \\sin{(p)})^{p} - \\sin{(p)} = - (t{(p)} - \\sin{(p)})^{p} - t{(p)} and - 0^{p} - \\sin{(p)} = - 0^{p} - t{(p)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["add", 1, "sin(Symbol('p', commutative=True))"], "Equality(Add(Function('t')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Integer(2), sin(Symbol('p', commutative=True))))"], [["minus", 2, "Mul(Integer(2), sin(Symbol('p', commutative=True)))"], "Equality(Add(Function('t')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Function('t')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True)), Pow(Integer(0), Symbol('p', commutative=True)))"], [["minus", 3, "Function('t')(Symbol('p', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('p', commutative=True))), Mul(Integer(-1), Function('t')(Symbol('p', commutative=True))))"], [["minus", 5, "Pow(Add(Function('t')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('t')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True))), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Function('t')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True))), Mul(Integer(-1), Function('t')(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('p', commutative=True))), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('p', commutative=True))), Mul(Integer(-1), Function('t')(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\delta{(c)} = \\int \\sin{(c)} dc, then derive \\frac{\\delta{(c)}}{\\operatorname{P_{g}}{(c)}} = \\frac{V_{\\mathbf{B}} - \\cos{(c)}}{\\operatorname{P_{g}}{(c)}}, then obtain \\int \\frac{\\delta{(c)} \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} dV_{\\mathbf{B}} = \\int \\frac{(V_{\\mathbf{B}} - \\cos{(c)}) \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} dV_{\\mathbf{B}}", "derivation": "\\delta{(c)} = \\int \\sin{(c)} dc and \\frac{\\delta{(c)}}{\\operatorname{P_{g}}{(c)}} = \\frac{\\int \\sin{(c)} dc}{\\operatorname{P_{g}}{(c)}} and \\frac{\\delta{(c)}}{\\operatorname{P_{g}}{(c)}} = \\frac{V_{\\mathbf{B}} - \\cos{(c)}}{\\operatorname{P_{g}}{(c)}} and \\frac{\\delta{(c)} \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} = \\frac{(V_{\\mathbf{B}} - \\cos{(c)}) \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} and \\int \\frac{\\delta{(c)} \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} dV_{\\mathbf{B}} = \\int \\frac{(V_{\\mathbf{B}} - \\cos{(c)}) \\sin{(c)}}{\\operatorname{P_{g}}{(c)}} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('c', commutative=True)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["divide", 1, "Function('P_g')(Symbol('c', commutative=True))"], "Equality(Mul(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\delta')(Symbol('c', commutative=True))), Mul(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), Integral(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\delta')(Symbol('c', commutative=True))), Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1))))"], [["times", 3, "sin(Symbol('c', commutative=True))"], "Equality(Mul(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\delta')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))))"], [["integrate", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\delta')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), cos(Symbol('c', commutative=True)))), Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(g,\\sigma_x)} = g^{\\sigma_x} and \\operatorname{g_{\\varepsilon}}{(g,\\sigma_x)} = (g g^{\\sigma_x})^{\\sigma_x}, then obtain \\operatorname{g_{\\varepsilon}}^{g}{(g,\\sigma_x)} = ((g \\operatorname{F_{H}}{(g,\\sigma_x)})^{\\sigma_x})^{g}", "derivation": "\\operatorname{F_{H}}{(g,\\sigma_x)} = g^{\\sigma_x} and g \\operatorname{F_{H}}{(g,\\sigma_x)} = g g^{\\sigma_x} and (g \\operatorname{F_{H}}{(g,\\sigma_x)})^{\\sigma_x} = (g g^{\\sigma_x})^{\\sigma_x} and \\operatorname{g_{\\varepsilon}}{(g,\\sigma_x)} = (g g^{\\sigma_x})^{\\sigma_x} and \\operatorname{g_{\\varepsilon}}{(g,\\sigma_x)} = (g \\operatorname{F_{H}}{(g,\\sigma_x)})^{\\sigma_x} and \\operatorname{g_{\\varepsilon}}^{g}{(g,\\sigma_x)} = ((g \\operatorname{F_{H}}{(g,\\sigma_x)})^{\\sigma_x})^{g}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('g', commutative=True), Pow(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Mul(Symbol('g', commutative=True), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('g', commutative=True), Pow(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('g', commutative=True), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Mul(Symbol('g', commutative=True), Function('F_H')(Symbol('g', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(\\hat{x})} = \\cos{(e^{\\hat{x}})}, then obtain \\hat{X}{(\\hat{x})} \\hat{X}^{\\hat{x}}{(\\hat{x})} \\cos{(e^{\\hat{x}})} = \\hat{X}{(\\hat{x})} \\cos{(e^{\\hat{x}})} \\cos^{\\hat{x}}{(e^{\\hat{x}})}", "derivation": "\\hat{X}{(\\hat{x})} = \\cos{(e^{\\hat{x}})} and \\hat{X}^{\\hat{x}}{(\\hat{x})} = \\cos^{\\hat{x}}{(e^{\\hat{x}})} and \\hat{X}{(\\hat{x})} \\hat{X}^{\\hat{x}}{(\\hat{x})} = \\hat{X}{(\\hat{x})} \\cos^{\\hat{x}}{(e^{\\hat{x}})} and \\hat{X}{(\\hat{x})} \\hat{X}^{\\hat{x}}{(\\hat{x})} \\cos{(e^{\\hat{x}})} = \\hat{X}{(\\hat{x})} \\cos{(e^{\\hat{x}})} \\cos^{\\hat{x}}{(e^{\\hat{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), cos(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(cos(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 2, "Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Mul(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))))"], [["times", 3, "cos(exp(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), cos(exp(Symbol('\\\\hat{x}', commutative=True)))), Mul(Function('\\\\hat{X}')(Symbol('\\\\hat{x}', commutative=True)), cos(exp(Symbol('\\\\hat{x}', commutative=True))), Pow(cos(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(f_{\\mathbf{p}},M)} = M f_{\\mathbf{p}}, then obtain (- (\\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}})^{f_{\\mathbf{p}}} \\cos{(1)} + \\cos{(1)})^{f_{\\mathbf{p}}} = 0^{f_{\\mathbf{p}}}", "derivation": "\\mathbf{g}{(f_{\\mathbf{p}},M)} = M f_{\\mathbf{p}} and 1 = \\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}} and 1 = (\\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}})^{f_{\\mathbf{p}}} and \\cos{(1)} = (\\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}})^{f_{\\mathbf{p}}} \\cos{(1)} and - (\\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}})^{f_{\\mathbf{p}}} \\cos{(1)} + \\cos{(1)} = 0 and (- (\\frac{M f_{\\mathbf{p}}}{\\mathbf{g}{(f_{\\mathbf{p}},M)}})^{f_{\\mathbf{p}}} \\cos{(1)} + \\cos{(1)})^{f_{\\mathbf{p}}} = 0^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(1), Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 3, "cos(Integer(1))"], "Equality(cos(Integer(1)), Mul(Pow(Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Integer(1))))"], [["minus", 4, "Mul(Pow(Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Integer(1)))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Integer(1))), cos(Integer(1))), Integer(0))"], [["power", 5, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Integer(1))), cos(Integer(1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Integer(0), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\Psi,\\hbar)} = \\Psi \\hbar and \\dot{y}{(\\Psi,\\hbar)} = \\Psi \\hbar - \\Psi, then obtain \\int \\frac{\\Psi \\hbar - \\Psi}{\\Psi} d\\Psi = \\int \\frac{\\dot{y}{(\\Psi,\\hbar)}}{\\Psi} d\\Psi", "derivation": "\\operatorname{v_{z}}{(\\Psi,\\hbar)} = \\Psi \\hbar and - \\Psi + \\operatorname{v_{z}}{(\\Psi,\\hbar)} = \\Psi \\hbar - \\Psi and \\frac{- \\Psi + \\operatorname{v_{z}}{(\\Psi,\\hbar)}}{\\Psi} = \\frac{\\Psi \\hbar - \\Psi}{\\Psi} and \\dot{y}{(\\Psi,\\hbar)} = \\Psi \\hbar - \\Psi and \\frac{- \\Psi + \\operatorname{v_{z}}{(\\Psi,\\hbar)}}{\\Psi} = \\frac{\\dot{y}{(\\Psi,\\hbar)}}{\\Psi} and \\frac{\\Psi \\hbar - \\Psi}{\\Psi} = \\frac{\\dot{y}{(\\Psi,\\hbar)}}{\\Psi} and \\int \\frac{\\Psi \\hbar - \\Psi}{\\Psi} d\\Psi = \\int \\frac{\\dot{y}{(\\Psi,\\hbar)}}{\\Psi} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('v_z')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('v_z')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('v_z')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 6, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given T{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} + T{(\\theta_1)} = 0^{\\theta_1} (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} + T{(\\theta_1)}", "derivation": "T{(\\theta_1)} = \\sin{(\\theta_1)} and T{(\\theta_1)} - \\sin{(\\theta_1)} = 0 and (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} = 0^{\\theta_1} and (T{(\\theta_1)} - \\sin{(\\theta_1)})^{2 \\theta_1} = 0^{\\theta_1} (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} and (T{(\\theta_1)} - \\sin{(\\theta_1)})^{2 \\theta_1} + T{(\\theta_1)} = 0^{\\theta_1} (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} + T{(\\theta_1)} and (T{(\\theta_1)} - \\sin{(\\theta_1)})^{2 \\theta_1} = (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} and (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} + T{(\\theta_1)} = 0^{\\theta_1} (T{(\\theta_1)} - \\sin{(\\theta_1)})^{\\theta_1} + T{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Pow(Integer(0), Symbol('\\\\theta_1', commutative=True)))"], [["times", 3, "Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True))"], "Equality(Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True))))"], [["add", 4, "Function('T')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))), Function('T')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Pow(Integer(0), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True))), Function('T')(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))), Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Function('T')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Pow(Integer(0), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Function('T')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True))), Function('T')(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given t{(\\delta)} = \\sin{(\\delta)}, then obtain 1 = (\\frac{t^{\\delta}{(\\delta)} + \\sin{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}})^{\\delta}", "derivation": "t{(\\delta)} = \\sin{(\\delta)} and t^{\\delta}{(\\delta)} = \\sin^{\\delta}{(\\delta)} and t{(\\delta)} + t^{\\delta}{(\\delta)} = t^{\\delta}{(\\delta)} + \\sin{(\\delta)} and t{(\\delta)} + \\sin^{\\delta}{(\\delta)} = \\sin{(\\delta)} + \\sin^{\\delta}{(\\delta)} and \\frac{t{(\\delta)} + \\sin^{\\delta}{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}} = \\frac{\\sin{(\\delta)} + \\sin^{\\delta}{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}} and (\\frac{t{(\\delta)} + \\sin^{\\delta}{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}})^{\\delta} = (\\frac{\\sin{(\\delta)} + \\sin^{\\delta}{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}})^{\\delta} and 1 = (\\frac{t^{\\delta}{(\\delta)} + \\sin{(\\delta)}}{t{(\\delta)} + t^{\\delta}{(\\delta)}})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["divide", 4, "Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Mul(Pow(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))))"], [["power", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Pow(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('t')(Symbol('\\\\delta', commutative=True)), Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Pow(Function('t')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(n_{2},\\mathbf{J}_M)} = \\frac{n_{2}}{\\mathbf{J}_M}, then obtain \\frac{\\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 + \\frac{n_{2}}{\\mathbf{J}_M}}{n_{2}} = \\frac{1 + \\frac{2 n_{2}}{\\mathbf{J}_M}}{n_{2}}", "derivation": "\\mathbf{S}{(n_{2},\\mathbf{J}_M)} = \\frac{n_{2}}{\\mathbf{J}_M} and \\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 = 1 + \\frac{n_{2}}{\\mathbf{J}_M} and 2 \\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 = \\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 + \\frac{n_{2}}{\\mathbf{J}_M} and 2 \\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 = 1 + \\frac{2 n_{2}}{\\mathbf{J}_M} and \\frac{2 \\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1}{n_{2}} = \\frac{1 + \\frac{2 n_{2}}{\\mathbf{J}_M}}{n_{2}} and \\frac{\\mathbf{S}{(n_{2},\\mathbf{J}_M)} + 1 + \\frac{n_{2}}{\\mathbf{J}_M}}{n_{2}} = \\frac{1 + \\frac{2 n_{2}}{\\mathbf{J}_M}}{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1)), Add(Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["divide", 4, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(2), Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Integer(1), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{S}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Integer(1), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\psi,\\mathbf{f})} = \\mathbf{f} \\psi and y{(\\psi,\\mathbf{f})} = \\int \\mathbf{H}{(\\psi,\\mathbf{f})} d\\mathbf{f}, then obtain \\frac{\\partial}{\\partial \\mathbf{f}} y{(\\psi,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\mathbf{f} \\psi d\\mathbf{f}", "derivation": "\\mathbf{H}{(\\psi,\\mathbf{f})} = \\mathbf{f} \\psi and \\int \\mathbf{H}{(\\psi,\\mathbf{f})} d\\mathbf{f} = \\int \\mathbf{f} \\psi d\\mathbf{f} and y{(\\psi,\\mathbf{f})} = \\int \\mathbf{H}{(\\psi,\\mathbf{f})} d\\mathbf{f} and y{(\\psi,\\mathbf{f})} = \\int \\mathbf{f} \\psi d\\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} y{(\\psi,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\mathbf{f} \\psi d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(g,\\mathbf{B})} = \\sin{(\\mathbf{B}^{g})}, then obtain (\\frac{\\partial}{\\partial g} (\\mathbf{B} + \\operatorname{A_{2}}^{\\mathbf{B}}{(g,\\mathbf{B})}))^{\\mathbf{B}} = (\\frac{\\partial}{\\partial g} (\\mathbf{B} + \\sin^{\\mathbf{B}}{(\\mathbf{B}^{g})}))^{\\mathbf{B}}", "derivation": "\\operatorname{A_{2}}{(g,\\mathbf{B})} = \\sin{(\\mathbf{B}^{g})} and \\operatorname{A_{2}}^{\\mathbf{B}}{(g,\\mathbf{B})} = \\sin^{\\mathbf{B}}{(\\mathbf{B}^{g})} and \\mathbf{B} + \\operatorname{A_{2}}^{\\mathbf{B}}{(g,\\mathbf{B})} = \\mathbf{B} + \\sin^{\\mathbf{B}}{(\\mathbf{B}^{g})} and \\frac{\\partial}{\\partial g} (\\mathbf{B} + \\operatorname{A_{2}}^{\\mathbf{B}}{(g,\\mathbf{B})}) = \\frac{\\partial}{\\partial g} (\\mathbf{B} + \\sin^{\\mathbf{B}}{(\\mathbf{B}^{g})}) and (\\frac{\\partial}{\\partial g} (\\mathbf{B} + \\operatorname{A_{2}}^{\\mathbf{B}}{(g,\\mathbf{B})}))^{\\mathbf{B}} = (\\frac{\\partial}{\\partial g} (\\mathbf{B} + \\sin^{\\mathbf{B}}{(\\mathbf{B}^{g})}))^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), sin(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(sin(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(sin(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(sin(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbf{B}', commutative=True), Pow(sin(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},i)} = - i + e^{\\Psi_{\\lambda}}, then derive \\int \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},i)} di = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k, then derive \\pi - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + 2 k, then obtain \\pi - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k = k + \\int (- i + e^{\\Psi_{\\lambda}}) di", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},i)} = - i + e^{\\Psi_{\\lambda}} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},i)} di = \\int (- i + e^{\\Psi_{\\lambda}}) di and \\int \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},i)} di = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k and \\int (- i + e^{\\Psi_{\\lambda}}) di = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k and k + \\int (- i + e^{\\Psi_{\\lambda}}) di = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + 2 k and \\pi - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k = - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + 2 k and \\pi - \\frac{i^{2}}{2} + i e^{\\Psi_{\\lambda}} + k = k + \\int (- i + e^{\\Psi_{\\lambda}}) di", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('i', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('k', commutative=True)))"], [["add", 4, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), Symbol('k', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('k', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('k', commutative=True)), Add(Symbol('k', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(n)} = e^{n}, then obtain (- \\phi_{1}{(n)} + \\frac{e^{n}}{\\phi_{1}{(n)}}) \\log{(\\phi_{1}{(n)})} = (- \\phi_{1}{(n)} + \\frac{e^{n}}{\\phi_{1}{(n)}}) \\log{(e^{n})}", "derivation": "\\phi_{1}{(n)} = e^{n} and \\log{(\\phi_{1}{(n)})} = \\log{(e^{n})} and 1 = \\frac{e^{n}}{\\phi_{1}{(n)}} and 1 - \\phi_{1}{(n)} = - \\phi_{1}{(n)} + \\frac{e^{n}}{\\phi_{1}{(n)}} and (1 - \\phi_{1}{(n)}) \\log{(\\phi_{1}{(n)})} = (1 - \\phi_{1}{(n)}) \\log{(e^{n})} and (- \\phi_{1}{(n)} + \\frac{e^{n}}{\\phi_{1}{(n)}}) \\log{(\\phi_{1}{(n)})} = (- \\phi_{1}{(n)} + \\frac{e^{n}}{\\phi_{1}{(n)}}) \\log{(e^{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\phi_1')(Symbol('n', commutative=True))), log(exp(Symbol('n', commutative=True))))"], [["divide", 1, "Function('\\\\phi_1')(Symbol('n', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\phi_1')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))))"], [["minus", 3, "Function('\\\\phi_1')(Symbol('n', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True))), Mul(Pow(Function('\\\\phi_1')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))"], [["times", 2, "Add(Integer(1), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True))))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True)))), log(Function('\\\\phi_1')(Symbol('n', commutative=True)))), Mul(Add(Integer(1), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True)))), log(exp(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True))), Mul(Pow(Function('\\\\phi_1')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), log(Function('\\\\phi_1')(Symbol('n', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n', commutative=True))), Mul(Pow(Function('\\\\phi_1')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), log(exp(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(A,\\theta_2)} = A \\theta_2, then obtain \\frac{\\int \\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2} dA}{\\hat{H}_{\\lambda} + \\frac{\\operatorname{A_{z}}^{2}{(A,\\theta_2)}}{2 \\theta_2^{2}}} = \\frac{\\frac{A^{2}}{2} + y^{\\prime}}{\\hat{H}_{\\lambda} + \\frac{\\operatorname{A_{z}}^{2}{(A,\\theta_2)}}{2 \\theta_2^{2}}}", "derivation": "\\operatorname{A_{z}}{(A,\\theta_2)} = A \\theta_2 and \\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2} = A and \\int \\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2} dA = \\int A dA and \\frac{\\int \\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2} dA}{\\int\\limits^{\\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2}} A dA} = \\frac{\\int A dA}{\\int\\limits^{\\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2}} A dA} and \\frac{\\int \\frac{\\operatorname{A_{z}}{(A,\\theta_2)}}{\\theta_2} dA}{\\hat{H}_{\\lambda} + \\frac{\\operatorname{A_{z}}^{2}{(A,\\theta_2)}}{2 \\theta_2^{2}}} = \\frac{\\frac{A^{2}}{2} + y^{\\prime}}{\\hat{H}_{\\lambda} + \\frac{\\operatorname{A_{z}}^{2}{(A,\\theta_2)}}{2 \\theta_2^{2}}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('A', commutative=True))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True))))"], [["divide", 3, "Integral(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], "Equality(Mul(Pow(Integral(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('A', commutative=True)))), Mul(Integral(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True))), Pow(Integral(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-2)), Pow(Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)))), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('A', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-2)), Pow(Function('A_z')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(n_{2},\\hbar)} = \\hbar n_{2}, then derive \\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} - 1 = n_{2} - 1, then obtain \\frac{C_{2} (\\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} - 1) \\operatorname{m_{s}}{(\\delta,i)}}{E_{x}} = \\frac{C_{2} (n_{2} - 1) \\operatorname{m_{s}}{(\\delta,i)}}{E_{x}}", "derivation": "\\mathbf{B}{(n_{2},\\hbar)} = \\hbar n_{2} and \\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\hbar n_{2} and \\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} - 1 = \\frac{\\partial}{\\partial \\hbar} \\hbar n_{2} - 1 and \\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} - 1 = n_{2} - 1 and \\frac{C_{2} (\\frac{\\partial}{\\partial \\hbar} \\mathbf{B}{(n_{2},\\hbar)} - 1) \\operatorname{m_{s}}{(\\delta,i)}}{E_{x}} = \\frac{C_{2} (n_{2} - 1) \\operatorname{m_{s}}{(\\delta,i)}}{E_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('n_2', commutative=True), Integer(-1)))"], [["times", 4, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\delta', commutative=True), Symbol('i', commutative=True)))"], "Equality(Mul(Symbol('C_2', commutative=True), Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1)), Function('m_s')(Symbol('\\\\delta', commutative=True), Symbol('i', commutative=True))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('n_2', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\delta', commutative=True), Symbol('i', commutative=True))))"]]}, {"prompt": "Given i{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\mathbf{J} u and A{(c)} = \\log{(c)}, then derive i{(\\mathbf{J},u)} = \\mathbf{J}, then derive - 2 \\mathbf{J} - u + A{(c)} = - 2 \\mathbf{J} - u + \\log{(c)}, then obtain (- u + A{(c)} - 2 i{(\\mathbf{J},u)}) (- u - 2 i{(\\mathbf{J},u)} + \\log{(c)}) = (- u - 2 i{(\\mathbf{J},u)} + \\log{(c)})^{2}", "derivation": "i{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\mathbf{J} u and i{(\\mathbf{J},u)} = \\mathbf{J} and A{(c)} = \\log{(c)} and - \\mathbf{J} - u + A{(c)} - \\frac{\\partial}{\\partial u} \\mathbf{J} u = - \\mathbf{J} - u + \\log{(c)} - \\frac{\\partial}{\\partial u} \\mathbf{J} u and - 2 \\mathbf{J} - u + A{(c)} = - 2 \\mathbf{J} - u + \\log{(c)} and (- 2 \\mathbf{J} - u + A{(c)}) (- 2 \\mathbf{J} - u + \\log{(c)}) = (- 2 \\mathbf{J} - u + \\log{(c)})^{2} and (- u + A{(c)} - 2 i{(\\mathbf{J},u)}) (- u - 2 i{(\\mathbf{J},u)} + \\log{(c)}) = (- u - 2 i{(\\mathbf{J},u)} + \\log{(c)})^{2}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], ["get_premise", "Equality(Function('A')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["minus", 3, "Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A')(Symbol('c', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('c', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A')(Symbol('c', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('c', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('c', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A')(Symbol('c', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('c', commutative=True)))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('c', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('A')(Symbol('c', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))), log(Symbol('c', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))), log(Symbol('c', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{s}{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)}, then derive \\mathbf{s}{(\\omega)} = - \\sin{(\\omega)}, then obtain - \\sin{(\\omega)} + \\frac{d}{d \\omega} \\mathbf{s}{(\\omega)} + \\frac{d}{d \\omega} \\cos{(\\omega)} = - 2 \\sin{(\\omega)} + \\frac{d}{d \\omega} \\mathbf{s}{(\\omega)}", "derivation": "\\mathbf{s}{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)} and \\mathbf{s}{(\\omega)} = - \\sin{(\\omega)} and \\frac{d}{d \\omega} \\cos{(\\omega)} = - \\sin{(\\omega)} and - \\sin{(\\omega)} + \\frac{d}{d \\omega} \\mathbf{s}{(\\omega)} + \\frac{d}{d \\omega} \\cos{(\\omega)} = - 2 \\sin{(\\omega)} + \\frac{d}{d \\omega} \\mathbf{s}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True)), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), sin(Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\pi,t)} = \\pi + t, then obtain t ((\\pi + t) \\bar{\\h}^{2}{(\\pi,t)})^{t} = t ((\\pi + t)^{3})^{t}", "derivation": "\\bar{\\h}{(\\pi,t)} = \\pi + t and (\\pi + t) \\bar{\\h}{(\\pi,t)} = (\\pi + t)^{2} and (\\pi + t) \\bar{\\h}^{2}{(\\pi,t)} = (\\pi + t)^{2} \\bar{\\h}{(\\pi,t)} and ((\\pi + t) \\bar{\\h}^{2}{(\\pi,t)})^{t} = ((\\pi + t)^{2} \\bar{\\h}{(\\pi,t)})^{t} and t ((\\pi + t) \\bar{\\h}^{2}{(\\pi,t)})^{t} = t ((\\pi + t)^{2} \\bar{\\h}{(\\pi,t)})^{t} and t ((\\pi + t)^{2} \\bar{\\h}{(\\pi,t)})^{t} = t ((\\pi + t)^{3})^{t} and t ((\\pi + t) \\bar{\\h}^{2}{(\\pi,t)})^{t} = t ((\\pi + t)^{3})^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))), Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2)))"], [["times", 1, "Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2))), Symbol('t', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["times", 4, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2))), Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), Pow(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('t', commutative=True), Pow(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), Pow(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(3)), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Symbol('t', commutative=True), Pow(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(2))), Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), Pow(Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Integer(3)), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\phi_1,v)} = \\phi_1 + v and \\hat{X}{(\\phi_1,v)} = - \\operatorname{a^{\\dagger}}{(\\phi_1,v)}, then obtain - \\operatorname{a^{\\dagger}}{(\\phi_1,v)} = - \\phi_1 - v", "derivation": "\\operatorname{a^{\\dagger}}{(\\phi_1,v)} = \\phi_1 + v and \\hat{X}{(\\phi_1,v)} = - \\operatorname{a^{\\dagger}}{(\\phi_1,v)} and \\hat{X}{(\\phi_1,v)} = - \\phi_1 - v and - \\operatorname{a^{\\dagger}}{(\\phi_1,v)} = - \\phi_1 - v", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{X}')(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\phi_1', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"]]}, {"prompt": "Given Z{(\\mathbf{J}_M)} = \\cos{(\\sin{(\\mathbf{J}_M)})}, then derive \\frac{d}{d \\mathbf{J}_M} Z{(\\mathbf{J}_M)} = - \\sin{(\\sin{(\\mathbf{J}_M)})} \\cos{(\\mathbf{J}_M)}, then obtain \\cos{(\\mathbf{J}_M)} \\frac{d}{d \\mathbf{J}_M} \\cos{(\\sin{(\\mathbf{J}_M)})} = - \\sin{(\\sin{(\\mathbf{J}_M)})} \\cos^{2}{(\\mathbf{J}_M)}", "derivation": "Z{(\\mathbf{J}_M)} = \\cos{(\\sin{(\\mathbf{J}_M)})} and \\frac{d}{d \\mathbf{J}_M} Z{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\cos{(\\sin{(\\mathbf{J}_M)})} and \\frac{d}{d \\mathbf{J}_M} Z{(\\mathbf{J}_M)} = - \\sin{(\\sin{(\\mathbf{J}_M)})} \\cos{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\cos{(\\sin{(\\mathbf{J}_M)})} = - \\sin{(\\sin{(\\mathbf{J}_M)})} \\cos{(\\mathbf{J}_M)} and \\cos{(\\mathbf{J}_M)} \\frac{d}{d \\mathbf{J}_M} \\cos{(\\sin{(\\mathbf{J}_M)})} = - \\sin{(\\sin{(\\mathbf{J}_M)})} \\cos^{2}{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 4, "cos(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Mul(Integer(-1), sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\chi{(z^{*},H)} = H z^{*}, then obtain \\frac{- H^{3} + H z^{*} \\chi^{2}{(z^{*},H)}}{H^{3} (z^{*})^{3}} = \\frac{- H^{3} + H^{2} (z^{*})^{2} \\chi{(z^{*},H)}}{H^{3} (z^{*})^{3}}", "derivation": "\\chi{(z^{*},H)} = H z^{*} and H \\chi{(z^{*},H)} = H^{2} z^{*} and H^{2} z^{*} \\chi{(z^{*},H)} = H^{3} (z^{*})^{2} and H^{2} (z^{*})^{2} \\chi{(z^{*},H)} = H^{3} (z^{*})^{3} and H z^{*} \\chi^{2}{(z^{*},H)} = H^{2} (z^{*})^{2} \\chi{(z^{*},H)} and - H^{3} + H z^{*} \\chi^{2}{(z^{*},H)} = - H^{3} + H^{2} (z^{*})^{2} \\chi{(z^{*},H)} and \\frac{- H^{3} + H z^{*} \\chi^{2}{(z^{*},H)}}{H^{3} (z^{*})^{3}} = \\frac{- H^{3} + H^{2} (z^{*})^{2} \\chi{(z^{*},H)}}{H^{3} (z^{*})^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('z^*', commutative=True)))"], [["times", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(2)), Symbol('z^*', commutative=True)))"], [["times", 2, "Mul(Symbol('H', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(2)), Symbol('z^*', commutative=True), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(3)), Pow(Symbol('z^*', commutative=True), Integer(2))))"], [["times", 3, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(3)), Pow(Symbol('z^*', commutative=True), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('H', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Integer(2))), Mul(Pow(Symbol('H', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))))"], [["minus", 5, "Pow(Symbol('H', commutative=True), Integer(3))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(3))), Mul(Symbol('H', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(3))), Mul(Pow(Symbol('H', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)))))"], [["divide", 6, "Mul(Pow(Symbol('H', commutative=True), Integer(3)), Pow(Symbol('z^*', commutative=True), Integer(3)))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-3)), Pow(Symbol('z^*', commutative=True), Integer(-3)), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(3))), Mul(Symbol('H', commutative=True), Symbol('z^*', commutative=True), Pow(Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Integer(2))))), Mul(Pow(Symbol('H', commutative=True), Integer(-3)), Pow(Symbol('z^*', commutative=True), Integer(-3)), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(3))), Mul(Pow(Symbol('H', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2)), Function('\\\\chi')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))))))"]]}, {"prompt": "Given A{(h,\\varphi)} = \\int \\frac{\\varphi}{h} dh and \\tilde{g}{(h,\\varphi)} = \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh, then obtain \\int \\frac{\\partial}{\\partial h} A{(h,\\varphi)} d\\varphi = \\int \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh d\\varphi", "derivation": "A{(h,\\varphi)} = \\int \\frac{\\varphi}{h} dh and \\frac{\\partial}{\\partial h} A{(h,\\varphi)} = \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh and \\tilde{g}{(h,\\varphi)} = \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh and \\tilde{g}{(h,\\varphi)} = \\frac{\\partial}{\\partial h} A{(h,\\varphi)} and \\int \\tilde{g}{(h,\\varphi)} d\\varphi = \\int \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh d\\varphi and \\int \\frac{\\partial}{\\partial h} A{(h,\\varphi)} d\\varphi = \\int \\frac{\\partial}{\\partial h} \\int \\frac{\\varphi}{h} dh d\\varphi", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Integral(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Function('A')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Derivative(Function('A')(Symbol('h', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi_1,\\Psi^{\\dagger})} = (\\Psi^{\\dagger} + \\phi_1)^{\\Psi^{\\dagger}} and t{(\\phi_1,\\Psi^{\\dagger})} = \\int (\\Psi^{\\dagger} + \\phi_1)^{\\Psi^{\\dagger}} d\\phi_1, then obtain t{(\\phi_1,\\Psi^{\\dagger})} = \\int \\dot{\\mathbf{r}}{(\\phi_1,\\Psi^{\\dagger})} d\\phi_1", "derivation": "\\dot{\\mathbf{r}}{(\\phi_1,\\Psi^{\\dagger})} = (\\Psi^{\\dagger} + \\phi_1)^{\\Psi^{\\dagger}} and \\int \\dot{\\mathbf{r}}{(\\phi_1,\\Psi^{\\dagger})} d\\phi_1 = \\int (\\Psi^{\\dagger} + \\phi_1)^{\\Psi^{\\dagger}} d\\phi_1 and t{(\\phi_1,\\Psi^{\\dagger})} = \\int (\\Psi^{\\dagger} + \\phi_1)^{\\Psi^{\\dagger}} d\\phi_1 and t{(\\phi_1,\\Psi^{\\dagger})} = \\int \\dot{\\mathbf{r}}{(\\phi_1,\\Psi^{\\dagger})} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given H{(\\hat{H})} = \\sin{(\\hat{H})}, then derive \\int H{(\\hat{H})} d\\hat{H} = b - \\cos{(\\hat{H})}, then obtain \\frac{- \\int H{(\\hat{H})} d\\hat{H} + \\int \\sin{(\\hat{H})} d\\hat{H}}{\\int H{(\\hat{H})} d\\hat{H}} = 0", "derivation": "H{(\\hat{H})} = \\sin{(\\hat{H})} and \\int H{(\\hat{H})} d\\hat{H} = \\int \\sin{(\\hat{H})} d\\hat{H} and \\int H{(\\hat{H})} d\\hat{H} = b - \\cos{(\\hat{H})} and \\int \\sin{(\\hat{H})} d\\hat{H} = b - \\cos{(\\hat{H})} and - \\int H{(\\hat{H})} d\\hat{H} + \\int \\sin{(\\hat{H})} d\\hat{H} = b - \\cos{(\\hat{H})} - \\int H{(\\hat{H})} d\\hat{H} and \\frac{- \\int H{(\\hat{H})} d\\hat{H} + \\int \\sin{(\\hat{H})} d\\hat{H}}{b - \\cos{(\\hat{H})}} = \\frac{b - \\cos{(\\hat{H})} - \\int H{(\\hat{H})} d\\hat{H}}{b - \\cos{(\\hat{H})}} and \\frac{- \\int H{(\\hat{H})} d\\hat{H} + \\int \\sin{(\\hat{H})} d\\hat{H}}{\\int H{(\\hat{H})} d\\hat{H}} = 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 4, "Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"], [["divide", 5, "Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Mul(Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Pow(Integral(Function('H')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given n{(f_{E})} = \\log{(f_{E})}, then obtain \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(n{(f_{E})} \\log{(f_{E})}^{- f_{E}})} = \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(\\log{(f_{E})} \\log{(f_{E})}^{- f_{E}})}", "derivation": "n{(f_{E})} = \\log{(f_{E})} and n{(f_{E})} \\log{(f_{E})}^{- f_{E}} = \\log{(f_{E})} \\log{(f_{E})}^{- f_{E}} and \\log{(n{(f_{E})} \\log{(f_{E})}^{- f_{E}})} = \\log{(\\log{(f_{E})} \\log{(f_{E})}^{- f_{E}})} and \\frac{d}{d f_{E}} \\log{(n{(f_{E})} \\log{(f_{E})}^{- f_{E}})} = \\frac{d}{d f_{E}} \\log{(\\log{(f_{E})} \\log{(f_{E})}^{- f_{E}})} and \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(n{(f_{E})} \\log{(f_{E})}^{- f_{E}})} = \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(\\log{(f_{E})} \\log{(f_{E})}^{- f_{E}})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["divide", 1, "Pow(log(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Mul(Function('n')(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True)))), Mul(log(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Function('n')(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))), log(Mul(log(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(log(Mul(Function('n')(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(log(Mul(log(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["times", 4, "log(Symbol('f_E', commutative=True))"], "Equality(Mul(log(Symbol('f_E', commutative=True)), Derivative(log(Mul(Function('n')(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(log(Symbol('f_E', commutative=True)), Derivative(log(Mul(log(Symbol('f_E', commutative=True)), Pow(log(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(h)} = \\sin{(e^{h})} and Q{(S)} = e^{S}, then obtain (Q{(S)} - i{(h)} e^{h}) e^{- h} = (- i{(h)} e^{h} + e^{S}) e^{- h}", "derivation": "i{(h)} = \\sin{(e^{h})} and Q{(S)} = e^{S} and Q{(S)} - e^{h} \\sin{(e^{h})} = e^{S} - e^{h} \\sin{(e^{h})} and Q{(S)} - i{(h)} e^{h} = - i{(h)} e^{h} + e^{S} and (Q{(S)} - i{(h)} e^{h}) e^{- h} = (- i{(h)} e^{h} + e^{S}) e^{- h}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('h', commutative=True)), sin(exp(Symbol('h', commutative=True))))"], ["get_premise", "Equality(Function('Q')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["minus", 2, "Mul(exp(Symbol('h', commutative=True)), sin(exp(Symbol('h', commutative=True))))"], "Equality(Add(Function('Q')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)), sin(exp(Symbol('h', commutative=True))))), Add(exp(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)), sin(exp(Symbol('h', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('Q')(Symbol('S', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Function('i')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), exp(Symbol('S', commutative=True))))"], [["divide", 4, "exp(Symbol('h', commutative=True))"], "Equality(Mul(Add(Function('Q')(Symbol('S', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))), exp(Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('i')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), exp(Symbol('S', commutative=True))), exp(Mul(Integer(-1), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}}, then derive - e^{f_{\\mathbf{p}}} + \\int \\mathbb{I}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\chi, then obtain - \\mathbb{I}{(f_{\\mathbf{p}})} + \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} = \\chi", "derivation": "\\mathbb{I}{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and \\int \\mathbb{I}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} and - e^{f_{\\mathbf{p}}} + \\int \\mathbb{I}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = - e^{f_{\\mathbf{p}}} + \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} and - e^{f_{\\mathbf{p}}} + \\int \\mathbb{I}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\chi and - \\mathbb{I}{(f_{\\mathbf{p}})} + \\int \\mathbb{I}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\chi and - \\mathbb{I}{(f_{\\mathbf{p}})} + \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} = \\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 2, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('\\\\chi', commutative=True))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('\\\\chi', commutative=True))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('\\\\chi', commutative=True))"]]}, {"prompt": "Given V{(\\delta)} = \\sin{(\\delta)}, then obtain V^{2}{(\\delta)} V^{\\delta}{(\\delta)} = V{(\\delta)} V^{\\delta}{(\\delta)} \\sin{(\\delta)}", "derivation": "V{(\\delta)} = \\sin{(\\delta)} and V^{\\delta}{(\\delta)} = \\sin^{\\delta}{(\\delta)} and V{(\\delta)} \\sin^{\\delta}{(\\delta)} = \\sin{(\\delta)} \\sin^{\\delta}{(\\delta)} and V{(\\delta)} \\sin{(\\delta)} \\sin^{\\delta}{(\\delta)} = \\sin^{2}{(\\delta)} \\sin^{\\delta}{(\\delta)} and V^{2}{(\\delta)} \\sin^{\\delta}{(\\delta)} = V{(\\delta)} \\sin{(\\delta)} \\sin^{\\delta}{(\\delta)} and V^{2}{(\\delta)} V^{\\delta}{(\\delta)} = V{(\\delta)} V^{\\delta}{(\\delta)} \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('V')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["times", 1, "Mul(sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Function('V')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('V')(Symbol('\\\\delta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Function('V')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('V')(Symbol('\\\\delta', commutative=True)), Integer(2)), Pow(Function('V')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Function('V')(Symbol('\\\\delta', commutative=True)), Pow(Function('V')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(F_{N})} = \\cos{(\\sin{(F_{N})})}, then obtain 1 - F_{N} = - F_{N} + \\frac{2 \\cos{(\\sin{(F_{N})})}}{\\phi_{1}{(F_{N})} + \\cos{(\\sin{(F_{N})})}}", "derivation": "\\phi_{1}{(F_{N})} = \\cos{(\\sin{(F_{N})})} and \\phi_{1}{(F_{N})} + \\cos{(\\sin{(F_{N})})} = 2 \\cos{(\\sin{(F_{N})})} and 1 = \\frac{2 \\cos{(\\sin{(F_{N})})}}{\\phi_{1}{(F_{N})} + \\cos{(\\sin{(F_{N})})}} and 1 - F_{N} = - F_{N} + \\frac{2 \\cos{(\\sin{(F_{N})})}}{\\phi_{1}{(F_{N})} + \\cos{(\\sin{(F_{N})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_N', commutative=True)), cos(sin(Symbol('F_N', commutative=True))))"], [["add", 1, "cos(sin(Symbol('F_N', commutative=True)))"], "Equality(Add(Function('\\\\phi_1')(Symbol('F_N', commutative=True)), cos(sin(Symbol('F_N', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('F_N', commutative=True)))))"], [["divide", 2, "Add(Function('\\\\phi_1')(Symbol('F_N', commutative=True)), cos(sin(Symbol('F_N', commutative=True))))"], "Equality(Integer(1), Mul(Integer(2), Pow(Add(Function('\\\\phi_1')(Symbol('F_N', commutative=True)), cos(sin(Symbol('F_N', commutative=True)))), Integer(-1)), cos(sin(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Symbol('F_N', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(2), Pow(Add(Function('\\\\phi_1')(Symbol('F_N', commutative=True)), cos(sin(Symbol('F_N', commutative=True)))), Integer(-1)), cos(sin(Symbol('F_N', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}}, then obtain \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} 1}{\\tilde{g}{(\\dot{\\mathbf{r}})}} = \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} \\frac{e^{\\dot{\\mathbf{r}}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}}", "derivation": "\\tilde{g}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and 1 = \\frac{e^{\\dot{\\mathbf{r}}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}} and \\frac{d}{d \\dot{\\mathbf{r}}} 1 = \\frac{d}{d \\dot{\\mathbf{r}}} \\frac{e^{\\dot{\\mathbf{r}}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}} and \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} 1}{\\tilde{g}{(\\dot{\\mathbf{r}})}} = \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} \\frac{e^{\\dot{\\mathbf{r}}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}}}{\\tilde{g}{(\\dot{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["times", 3, "Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} = \\Psi_{nl} y + m_{s}, then obtain \\frac{\\int \\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} d\\Psi_{nl}}{\\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})}} = \\frac{\\int (\\Psi_{nl} y + m_{s}) d\\Psi_{nl}}{\\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})}}", "derivation": "\\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} = \\Psi_{nl} y + m_{s} and \\int \\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} d\\Psi_{nl} = \\int (\\Psi_{nl} y + m_{s}) d\\Psi_{nl} and \\frac{\\int \\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} d\\Psi_{nl}}{\\Psi_{nl} y + m_{s}} = \\frac{\\int (\\Psi_{nl} y + m_{s}) d\\Psi_{nl}}{\\Psi_{nl} y + m_{s}} and \\frac{\\int \\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})} d\\Psi_{nl}}{\\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})}} = \\frac{\\int (\\Psi_{nl} y + m_{s}) d\\Psi_{nl}}{\\operatorname{A_{z}}{(y,m_{s},\\Psi_{nl})}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 2, "Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)), Integer(-1)), Integral(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Pow(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Integral(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Pow(Function('A_z')(Symbol('y', commutative=True), Symbol('m_s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('y', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\chi,\\varphi)} = \\frac{\\chi}{\\varphi}, then obtain \\int \\tilde{g} (\\frac{\\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} + \\operatorname{P_{g}}{(\\chi,\\varphi)} - 1) d\\chi = \\int \\tilde{g} (\\frac{2 \\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} - 1) d\\chi", "derivation": "\\operatorname{P_{g}}{(\\chi,\\varphi)} = \\frac{\\chi}{\\varphi} and \\operatorname{P_{g}}{(\\chi,\\varphi)} - 1 = \\frac{\\chi}{\\varphi} - 1 and \\frac{\\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} + \\operatorname{P_{g}}{(\\chi,\\varphi)} - 1 = \\frac{2 \\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} - 1 and \\tilde{g} (\\frac{\\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} + \\operatorname{P_{g}}{(\\chi,\\varphi)} - 1) = \\tilde{g} (\\frac{2 \\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} - 1) and \\int \\tilde{g} (\\frac{\\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} + \\operatorname{P_{g}}{(\\chi,\\varphi)} - 1) d\\chi = \\int \\tilde{g} (\\frac{2 \\chi}{\\varphi} - \\operatorname{F_{N}}{(\\chi,\\varphi)} - 1) d\\chi", "srepr_derivation": [["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_g')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Integer(-1)))"], [["add", 2, "Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('P_g')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1)))"], [["times", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('P_g')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))), Mul(Symbol('\\\\tilde{g}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\tilde{g}', commutative=True), Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('P_g')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\tilde{g}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(a)} = e^{a}, then obtain (- a - \\mu_{0}{(a)}) e^{- a} + \\int ((a + \\mu_{0}{(a)}) e^{- a})^{a} da = (- a - \\mu_{0}{(a)}) e^{- a} + \\int ((a + e^{a}) e^{- a})^{a} da", "derivation": "\\mu_{0}{(a)} = e^{a} and a + \\mu_{0}{(a)} = a + e^{a} and (a + \\mu_{0}{(a)}) e^{- a} = (a + e^{a}) e^{- a} and ((a + \\mu_{0}{(a)}) e^{- a})^{a} = ((a + e^{a}) e^{- a})^{a} and \\int ((a + \\mu_{0}{(a)}) e^{- a})^{a} da = \\int ((a + e^{a}) e^{- a})^{a} da and - (a + \\mu_{0}{(a)}) e^{- a} + \\int ((a + \\mu_{0}{(a)}) e^{- a})^{a} da = - (a + \\mu_{0}{(a)}) e^{- a} + \\int ((a + e^{a}) e^{- a})^{a} da and (- a - \\mu_{0}{(a)}) e^{- a} + \\int ((a + \\mu_{0}{(a)}) e^{- a})^{a} da = (- a - \\mu_{0}{(a)}) e^{- a} + \\int ((a + e^{a}) e^{- a})^{a} da", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))))"], [["divide", 2, "exp(Symbol('a', commutative=True))"], "Equality(Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Mul(Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Pow(Mul(Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)))"], [["integrate", 4, "Symbol('a', commutative=True)"], "Equality(Integral(Pow(Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["minus", 5, "Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True)))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), Function('\\\\mu_0')(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True)))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integral(Pow(Mul(Add(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\psi)} = \\cos{(\\psi)}, then derive \\frac{d}{d \\psi} \\hat{x}{(\\psi)} = - \\sin{(\\psi)}, then obtain v^{2}{(\\theta,\\hat{X})} (\\frac{d}{d \\psi} \\hat{x}{(\\psi)})^{\\psi} = v^{2}{(\\theta,\\hat{X})} (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi}", "derivation": "\\hat{x}{(\\psi)} = \\cos{(\\psi)} and \\frac{d}{d \\psi} \\hat{x}{(\\psi)} = \\frac{d}{d \\psi} \\cos{(\\psi)} and \\frac{d}{d \\psi} \\hat{x}{(\\psi)} = - \\sin{(\\psi)} and \\frac{d}{d \\psi} \\cos{(\\psi)} = - \\sin{(\\psi)} and (\\frac{d}{d \\psi} \\hat{x}{(\\psi)})^{\\psi} = (- \\sin{(\\psi)})^{\\psi} and (\\frac{d}{d \\psi} \\hat{x}{(\\psi)})^{\\psi} = (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi} and v^{2}{(\\theta,\\hat{X})} (\\frac{d}{d \\psi} \\hat{x}{(\\psi)})^{\\psi} = v^{2}{(\\theta,\\hat{X})} (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)))"], [["times", 6, "Pow(Function('v')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('v')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Pow(Derivative(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True))), Mul(Pow(Function('v')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Pow(Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{c})} = \\log{(F_{c})}, then derive \\frac{d}{d F_{c}} \\operatorname{x^{{\\}'}}{(F_{c})} = \\frac{1}{F_{c}}, then obtain \\frac{1}{F_{c} \\frac{d}{d F_{c}} \\log{(F_{c})}} = 1", "derivation": "\\operatorname{x^{{\\}'}}{(F_{c})} = \\log{(F_{c})} and \\frac{d}{d F_{c}} \\operatorname{x^{{\\}'}}{(F_{c})} = \\frac{d}{d F_{c}} \\log{(F_{c})} and \\frac{\\frac{d}{d F_{c}} \\operatorname{x^{{\\}'}}{(F_{c})}}{\\frac{d}{d F_{c}} \\log{(F_{c})}} = 1 and \\frac{d}{d F_{c}} \\operatorname{x^{{\\}'}}{(F_{c})} = \\frac{1}{F_{c}} and \\frac{1}{F_{c} \\frac{d}{d F_{c}} \\log{(F_{c})}} = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('x^\\\\prime')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Pow(Symbol('F_c', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Derivative(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\hat{H}{(\\phi_1,B)} = \\log{(B \\phi_1)}, then obtain 1 - \\hat{H}{(\\phi_1,B)} = - \\hat{H}{(\\phi_1,B)} + \\frac{\\int \\log{(B \\phi_1)}^{B} d\\phi_1}{\\int \\hat{H}^{B}{(\\phi_1,B)} d\\phi_1}", "derivation": "\\hat{H}{(\\phi_1,B)} = \\log{(B \\phi_1)} and \\hat{H}^{B}{(\\phi_1,B)} = \\log{(B \\phi_1)}^{B} and \\int \\hat{H}^{B}{(\\phi_1,B)} d\\phi_1 = \\int \\log{(B \\phi_1)}^{B} d\\phi_1 and 1 = \\frac{\\int \\log{(B \\phi_1)}^{B} d\\phi_1}{\\int \\hat{H}^{B}{(\\phi_1,B)} d\\phi_1} and 1 - \\hat{H}{(\\phi_1,B)} = - \\hat{H}{(\\phi_1,B)} + \\frac{\\int \\log{(B \\phi_1)}^{B} d\\phi_1}{\\int \\hat{H}^{B}{(\\phi_1,B)} d\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), log(Mul(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(log(Mul(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('B', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(log(Mul(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["divide", 3, "Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Integral(Pow(log(Mul(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["minus", 4, "Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Integral(Pow(Function('\\\\hat{H}')(Symbol('\\\\phi_1', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Integral(Pow(log(Mul(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(u)} = e^{u} and \\Psi{(u)} = (e^{u})^{u}, then obtain \\Psi{(u)} = \\mathbf{A}^{u}{(u)}", "derivation": "\\mathbf{A}{(u)} = e^{u} and \\mathbf{A}^{u}{(u)} = (e^{u})^{u} and \\Psi{(u)} = (e^{u})^{u} and \\Psi{(u)} = \\mathbf{A}^{u}{(u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\Psi')(Symbol('u', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(f_{E})} = e^{f_{E}} and \\operatorname{t_{1}}{(f_{E})} = (e^{f_{E}})^{f_{E}}, then obtain \\operatorname{t_{1}}^{f_{E}}{(f_{E})} = ((e^{f_{E}})^{f_{E}})^{f_{E}}", "derivation": "\\operatorname{n_{2}}{(f_{E})} = e^{f_{E}} and \\operatorname{n_{2}}^{f_{E}}{(f_{E})} = (e^{f_{E}})^{f_{E}} and (\\operatorname{n_{2}}^{f_{E}}{(f_{E})})^{f_{E}} = ((e^{f_{E}})^{f_{E}})^{f_{E}} and \\operatorname{t_{1}}{(f_{E})} = (e^{f_{E}})^{f_{E}} and \\operatorname{t_{1}}{(f_{E})} = \\operatorname{n_{2}}^{f_{E}}{(f_{E})} and \\operatorname{t_{1}}^{f_{E}}{(f_{E})} = ((e^{f_{E}})^{f_{E}})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Pow(Function('n_2')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('f_E', commutative=True)), Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('t_1')(Symbol('f_E', commutative=True)), Pow(Function('n_2')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('t_1')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(\\theta_1,T)} = \\frac{\\theta_1}{T}, then obtain (T (\\int (T + 1) d\\theta_1)^{2})^{\\theta_1} = (T (\\int (T + 1) d\\theta_1) \\int (T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T}) d\\theta_1)^{\\theta_1}", "derivation": "\\rho_{b}{(\\theta_1,T)} = \\frac{\\theta_1}{T} and T + \\rho_{b}{(\\theta_1,T)} = T + \\frac{\\theta_1}{T} and T + 1 = T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T} and \\int (T + 1) d\\theta_1 = \\int (T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T}) d\\theta_1 and T \\int (T + 1) d\\theta_1 = T \\int (T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T}) d\\theta_1 and T (\\int (T + 1) d\\theta_1)^{2} = T (\\int (T + 1) d\\theta_1) \\int (T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T}) d\\theta_1 and (T (\\int (T + 1) d\\theta_1)^{2})^{\\theta_1} = (T (\\int (T + 1) d\\theta_1) \\int (T - \\rho_{b}{(\\theta_1,T)} + 1 + \\frac{\\theta_1}{T}) d\\theta_1)^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 2, "Add(Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))"], "Equality(Add(Symbol('T', commutative=True), Integer(1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 4, "Pow(Symbol('T', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('T', commutative=True), Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('T', commutative=True), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["times", 5, "Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Symbol('T', commutative=True), Pow(Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))), Mul(Symbol('T', commutative=True), Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["power", 6, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Mul(Symbol('T', commutative=True), Pow(Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Integral(Add(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True))), Integer(1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\chi{(E_{x})} = \\cos{(E_{x})}, then obtain ((\\int \\chi{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} + \\cos{(E_{x})} = ((x + \\sin{(E_{x})})^{E_{x}})^{E_{x}} + \\cos{(E_{x})}", "derivation": "\\chi{(E_{x})} = \\cos{(E_{x})} and \\int \\chi{(E_{x})} dE_{x} = \\int \\cos{(E_{x})} dE_{x} and (\\int \\chi{(E_{x})} dE_{x})^{E_{x}} = (\\int \\cos{(E_{x})} dE_{x})^{E_{x}} and ((\\int \\chi{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} = ((\\int \\cos{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} and ((\\int \\chi{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} + \\cos{(E_{x})} = ((\\int \\cos{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} + \\cos{(E_{x})} and ((\\int \\chi{(E_{x})} dE_{x})^{E_{x}})^{E_{x}} + \\cos{(E_{x})} = ((x + \\sin{(E_{x})})^{E_{x}})^{E_{x}} + \\cos{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\chi')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Pow(Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["power", 3, "Symbol('E_x', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\chi')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["add", 4, "cos(Symbol('E_x', commutative=True))"], "Equality(Add(Pow(Pow(Integral(Function('\\\\chi')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), Add(Pow(Pow(Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Pow(Pow(Integral(Function('\\\\chi')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), Add(Pow(Pow(Add(Symbol('x', commutative=True), sin(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(c_{0})} = \\cos{(c_{0})}, then derive \\frac{d}{d c_{0}} \\varepsilon{(c_{0})} = - \\sin{(c_{0})}, then obtain \\log{(\\frac{d}{d c_{0}} \\varepsilon{(c_{0})})} = \\log{(- \\sin{(c_{0})})}", "derivation": "\\varepsilon{(c_{0})} = \\cos{(c_{0})} and \\frac{d}{d c_{0}} \\varepsilon{(c_{0})} = \\frac{d}{d c_{0}} \\cos{(c_{0})} and \\frac{d}{d c_{0}} \\varepsilon{(c_{0})} = - \\sin{(c_{0})} and \\log{(\\frac{d}{d c_{0}} \\varepsilon{(c_{0})})} = \\log{(- \\sin{(c_{0})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('c_0', commutative=True))))"], [["log", 3], "Equality(log(Derivative(Function('\\\\varepsilon')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), log(Mul(Integer(-1), sin(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given c{(V,i)} = V - i, then derive \\frac{\\partial}{\\partial i} c{(V,i)} = -1, then obtain \\iint (\\int \\frac{\\partial}{\\partial i} c{(V,i)} di)^{i} di di = \\iint (\\int (-1) di)^{i} di di", "derivation": "c{(V,i)} = V - i and - V + c{(V,i)} = - i and \\frac{\\partial}{\\partial i} (- V + c{(V,i)}) = \\frac{d}{d i} - i and \\frac{\\partial}{\\partial i} c{(V,i)} = -1 and \\int \\frac{\\partial}{\\partial i} c{(V,i)} di = \\int (-1) di and (\\int \\frac{\\partial}{\\partial i} c{(V,i)} di)^{i} = (\\int (-1) di)^{i} and \\int (\\int \\frac{\\partial}{\\partial i} c{(V,i)} di)^{i} di = \\int (\\int (-1) di)^{i} di and \\iint (\\int \\frac{\\partial}{\\partial i} c{(V,i)} di)^{i} di di = \\iint (\\int (-1) di)^{i} di di", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["minus", 1, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))"], [["integrate", 4, "Symbol('i', commutative=True)"], "Equality(Integral(Derivative(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Integer(-1), Tuple(Symbol('i', commutative=True))))"], [["power", 5, "Symbol('i', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(Integer(-1), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["integrate", 6, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Integral(Derivative(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Integral(Integer(-1), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["integrate", 7, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Integral(Derivative(Function('c')(Symbol('V', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Integral(Integer(-1), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(m)} = \\sin{(m)} and \\operatorname{n_{2}}{(m)} = \\frac{d}{d m} \\sin{(m)}, then derive \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = \\cos{(m)}, then derive \\operatorname{n_{2}}{(m)} = \\cos{(m)}, then obtain \\operatorname{n_{2}}{(m)} + \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = 2 \\operatorname{n_{2}}{(m)}", "derivation": "\\operatorname{f^{\\prime}}{(m)} = \\sin{(m)} and \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = \\frac{d}{d m} \\sin{(m)} and \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = \\cos{(m)} and \\cos{(m)} + \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = 2 \\cos{(m)} and \\operatorname{n_{2}}{(m)} = \\frac{d}{d m} \\sin{(m)} and \\operatorname{n_{2}}{(m)} = \\cos{(m)} and \\operatorname{n_{2}}{(m)} + \\frac{d}{d m} \\operatorname{f^{\\prime}}{(m)} = 2 \\operatorname{n_{2}}{(m)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), cos(Symbol('m', commutative=True)))"], [["add", 3, "cos(Symbol('m', commutative=True))"], "Equality(Add(cos(Symbol('m', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('m', commutative=True)), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('n_2')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Function('n_2')(Symbol('m', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Integer(2), Function('n_2')(Symbol('m', commutative=True))))"]]}, {"prompt": "Given n{(h,c)} = c h and \\mu_{0}{(h,c)} = c + n{(h,c)}, then obtain (c h \\frac{\\partial}{\\partial h} n{(h,c)} + c (c + n{(h,c)})) (- c h + c + n{(h,c)}) = c (c h \\frac{\\partial}{\\partial h} n{(h,c)} + c (c + n{(h,c)}))", "derivation": "n{(h,c)} = c h and c + n{(h,c)} = c h + c and \\mu_{0}{(h,c)} = c + n{(h,c)} and - c h + \\mu_{0}{(h,c)} = - c h + c + n{(h,c)} and - c h + \\mu_{0}{(h,c)} = c and - c h + c + n{(h,c)} = c and (- c h + c + n{(h,c)}) \\frac{\\partial}{\\partial h} c h (c + n{(h,c)}) = c \\frac{\\partial}{\\partial h} c h (c + n{(h,c)}) and (c h \\frac{\\partial}{\\partial h} n{(h,c)} + c (c + n{(h,c)})) (- c h + c + n{(h,c)}) = c (c h \\frac{\\partial}{\\partial h} n{(h,c)} + c (c + n{(h,c)}))", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True)), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))))"], [["minus", 3, "Mul(Symbol('c', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))"], [["times", 6, "Derivative(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))), Derivative(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('c', commutative=True), Derivative(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 7], "Equality(Mul(Add(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True), Derivative(Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('c', commutative=True), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True))))), Add(Mul(Integer(-1), Symbol('c', commutative=True), Symbol('h', commutative=True)), Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)))), Mul(Symbol('c', commutative=True), Add(Mul(Symbol('c', commutative=True), Symbol('h', commutative=True), Derivative(Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('c', commutative=True), Add(Symbol('c', commutative=True), Function('n')(Symbol('h', commutative=True), Symbol('c', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)} = \\frac{\\hbar \\tilde{g}}{J}, then obtain \\iint \\frac{\\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)}}{\\hbar} d\\tilde{g} d\\hbar = \\iint \\frac{\\tilde{g}}{J} d\\tilde{g} d\\hbar", "derivation": "\\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)} = \\frac{\\hbar \\tilde{g}}{J} and \\frac{\\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)}}{\\hbar} = \\frac{\\tilde{g}}{J} and \\int \\frac{\\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)}}{\\hbar} d\\tilde{g} = \\int \\frac{\\tilde{g}}{J} d\\tilde{g} and \\iint \\frac{\\operatorname{n_{2}}{(J,\\tilde{g},\\hbar)}}{\\hbar} d\\tilde{g} d\\hbar = \\iint \\frac{\\tilde{g}}{J} d\\tilde{g} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('n_2')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(b)} = \\cos{(e^{b})}, then obtain b + 1 + \\frac{\\operatorname{M_{E}}{(b)}}{b} = b + 1 + \\frac{\\cos{(e^{b})}}{b}", "derivation": "\\operatorname{M_{E}}{(b)} = \\cos{(e^{b})} and \\frac{\\operatorname{M_{E}}{(b)}}{b} = \\frac{\\cos{(e^{b})}}{b} and b + \\frac{\\operatorname{M_{E}}{(b)}}{b} = b + \\frac{\\cos{(e^{b})}}{b} and b + 1 + \\frac{\\operatorname{M_{E}}{(b)}}{b} = b + 1 + \\frac{\\cos{(e^{b})}}{b}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('b', commutative=True)), cos(exp(Symbol('b', commutative=True))))"], [["divide", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('M_E')(Symbol('b', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(exp(Symbol('b', commutative=True)))))"], [["add", 2, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('M_E')(Symbol('b', commutative=True)))), Add(Symbol('b', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(exp(Symbol('b', commutative=True))))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Symbol('b', commutative=True), Integer(1), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('M_E')(Symbol('b', commutative=True)))), Add(Symbol('b', commutative=True), Integer(1), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(exp(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given c{(E_{\\lambda})} = \\log{(\\sin{(E_{\\lambda})})}, then obtain - c{(E_{\\lambda})} = - c{(E_{\\lambda})} - \\frac{d}{d E_{\\lambda}} c{(E_{\\lambda})} + \\frac{\\cos{(E_{\\lambda})}}{\\sin{(E_{\\lambda})}}", "derivation": "c{(E_{\\lambda})} = \\log{(\\sin{(E_{\\lambda})})} and 0 = - c{(E_{\\lambda})} + \\log{(\\sin{(E_{\\lambda})})} and \\frac{d}{d E_{\\lambda}} 0 = \\frac{d}{d E_{\\lambda}} (- c{(E_{\\lambda})} + \\log{(\\sin{(E_{\\lambda})})}) and - c{(E_{\\lambda})} + \\frac{d}{d E_{\\lambda}} 0 = - c{(E_{\\lambda})} + \\frac{d}{d E_{\\lambda}} (- c{(E_{\\lambda})} + \\log{(\\sin{(E_{\\lambda})})}) and - c{(E_{\\lambda})} = - c{(E_{\\lambda})} - \\frac{d}{d E_{\\lambda}} c{(E_{\\lambda})} + \\frac{\\cos{(E_{\\lambda})}}{\\sin{(E_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E_{\\\\lambda}', commutative=True)), log(sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Function('c')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), log(sin(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), log(sin(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), Derivative(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), Derivative(Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), log(sin(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Function('c')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Derivative(Function('c')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Q)} = e^{Q}, then derive \\int (2 Q + \\operatorname{y^{\\prime}}{(Q)} e^{Q}) dQ = Q^{2} + v_{2} + \\frac{e^{2 Q}}{2}, then obtain Q^{2} + v_{2} + \\frac{e^{2 Q}}{2} = \\int (2 Q + e^{2 Q}) dQ", "derivation": "\\operatorname{y^{\\prime}}{(Q)} = e^{Q} and \\operatorname{y^{\\prime}}{(Q)} e^{Q} = e^{2 Q} and 2 Q + \\operatorname{y^{\\prime}}{(Q)} e^{Q} = 2 Q + e^{2 Q} and \\int (2 Q + \\operatorname{y^{\\prime}}{(Q)} e^{Q}) dQ = \\int (2 Q + e^{2 Q}) dQ and \\int (2 Q + \\operatorname{y^{\\prime}}{(Q)} e^{Q}) dQ = Q^{2} + v_{2} + \\frac{e^{2 Q}}{2} and \\int (2 Q + \\operatorname{y^{\\prime}}{(Q)} e^{Q}) dQ = Q^{2} + v_{2} + \\frac{\\operatorname{y^{\\prime}}^{2}{(Q)}}{2} and Q^{2} + v_{2} + \\frac{e^{2 Q}}{2} = Q^{2} + v_{2} + \\frac{\\operatorname{y^{\\prime}}^{2}{(Q)}}{2} and \\int (2 Q + e^{2 Q}) dQ = Q^{2} + v_{2} + \\frac{\\operatorname{y^{\\prime}}^{2}{(Q)}}{2} and Q^{2} + v_{2} + \\frac{e^{2 Q}}{2} = \\int (2 Q + e^{2 Q}) dQ", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["times", 1, "exp(Symbol('Q', commutative=True))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), exp(Mul(Integer(2), Symbol('Q', commutative=True))))"], [["add", 2, "Mul(Integer(2), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))), Add(Mul(Integer(2), Symbol('Q', commutative=True)), exp(Mul(Integer(2), Symbol('Q', commutative=True)))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), exp(Mul(Integer(2), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('Q', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), Mul(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), Pow(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('Q', commutative=True))))), Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), Pow(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), exp(Mul(Integer(2), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), Pow(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Add(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('v_2', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('Q', commutative=True))))), Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True)), exp(Mul(Integer(2), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given r{(Z,\\Omega)} = \\cos{(Z + \\Omega)}, then obtain (\\int (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) (r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)}) d\\Omega)^{2} = (\\int - (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) \\cos{(Z + \\Omega)} d\\Omega)^{2}", "derivation": "r{(Z,\\Omega)} = \\cos{(Z + \\Omega)} and r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)} = - \\cos{(Z + \\Omega)} and (Z + \\Omega) (r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)}) = - (Z + \\Omega) \\cos{(Z + \\Omega)} and (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) (r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)}) = - (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) \\cos{(Z + \\Omega)} and \\int (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) (r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)}) d\\Omega = \\int - (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) \\cos{(Z + \\Omega)} d\\Omega and (\\int (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) (r{(Z,\\Omega)} - 2 \\cos{(Z + \\Omega)}) d\\Omega)^{2} = (\\int - (2 \\cos{(Z + \\Omega)})^{- \\Omega} (Z + \\Omega) \\cos{(Z + \\Omega)} d\\Omega)^{2}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["minus", 1, "Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True))))"], "Equality(Add(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True))))), Mul(Integer(-1), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["times", 2, "Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Mul(Integer(-1), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["divide", 3, "Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Mul(Integer(-1), Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(-1), Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 5, 2], "Equality(Pow(Integral(Mul(Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(-1), Pow(Mul(Integer(2), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('Z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)))"]]}, {"prompt": "Given J{(T)} = \\cos{(T)}, then derive \\int J{(T)} dT = \\rho + \\sin{(T)}, then obtain \\frac{d}{d \\rho} 1 = \\frac{d}{d \\rho} \\frac{\\int \\cos{(T)} dT}{\\int J{(T)} dT}", "derivation": "J{(T)} = \\cos{(T)} and \\int J{(T)} dT = \\int \\cos{(T)} dT and \\int J{(T)} dT = \\rho + \\sin{(T)} and \\frac{\\int J{(T)} dT}{\\int \\cos{(T)} dT} = \\frac{\\rho + \\sin{(T)}}{\\int \\cos{(T)} dT} and \\frac{d}{d \\rho} \\frac{\\int J{(T)} dT}{\\int \\cos{(T)} dT} = \\frac{\\partial}{\\partial \\rho} \\frac{\\rho + \\sin{(T)}}{\\int \\cos{(T)} dT} and \\frac{d}{d \\rho} 1 = \\frac{\\partial}{\\partial \\rho} \\frac{\\rho + \\sin{(T)}}{\\int J{(T)} dT} and \\rho + \\sin{(T)} = \\int \\cos{(T)} dT and \\frac{d}{d \\rho} 1 = \\frac{d}{d \\rho} \\frac{\\int \\cos{(T)} dT}{\\int J{(T)} dT}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\rho', commutative=True), sin(Symbol('T', commutative=True))))"], [["divide", 3, "Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Pow(Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('T', commutative=True))), Pow(Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Pow(Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('T', commutative=True))), Pow(Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('T', commutative=True))), Pow(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('T', commutative=True))), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Pow(Integral(Function('J')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(-1)), Integral(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(h,\\theta)} = \\theta h and \\omega{(h,\\theta)} = \\theta h, then obtain 0 = \\frac{\\operatorname{A_{y}}^{2}{(h,\\theta)} - \\operatorname{A_{y}}{(h,\\theta)} \\omega{(h,\\theta)}}{h \\sin{(\\theta^{2} h^{2})}}", "derivation": "\\operatorname{A_{y}}{(h,\\theta)} = \\theta h and \\omega{(h,\\theta)} = \\theta h and \\operatorname{A_{y}}{(h,\\theta)} \\omega{(h,\\theta)} = \\theta h \\operatorname{A_{y}}{(h,\\theta)} and \\theta h \\omega{(h,\\theta)} = \\theta^{2} h^{2} and \\theta h \\omega{(h,\\theta)} + h^{2} = \\theta^{2} h^{2} + h^{2} and 0 = \\theta^{2} h^{2} - \\theta h \\omega{(h,\\theta)} and 0 = \\operatorname{A_{y}}^{2}{(h,\\theta)} - \\operatorname{A_{y}}{(h,\\theta)} \\omega{(h,\\theta)} and 0 = \\frac{\\operatorname{A_{y}}^{2}{(h,\\theta)} - \\operatorname{A_{y}}{(h,\\theta)} \\omega{(h,\\theta)}}{h} and 0 = \\frac{\\operatorname{A_{y}}^{2}{(h,\\theta)} - \\operatorname{A_{y}}{(h,\\theta)} \\omega{(h,\\theta)}}{h \\sin{(\\theta^{2} h^{2})}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True)))"], [["times", 2, "Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True), Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(2))))"], [["add", 4, "Pow(Symbol('h', commutative=True), Integer(2))"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Symbol('h', commutative=True), Integer(2))), Add(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(2))), Pow(Symbol('h', commutative=True), Integer(2))))"], [["minus", 5, "Add(Mul(Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Symbol('h', commutative=True), Integer(2)))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Symbol('h', commutative=True), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Add(Pow(Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["divide", 7, "Symbol('h', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Pow(Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True))))))"], [["divide", 8, "sin(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(2))))"], "Equality(Integer(0), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Pow(Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('A_y')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\omega')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)))), Pow(sin(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(2)))), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(g)} = \\cos{(g)} and \\omega{(g,\\mathbf{B})} = - \\mathbf{B} + \\frac{d}{d g} \\cos{(g)}, then derive \\frac{d}{d g} \\hat{x}{(g)} = - \\sin{(g)}, then obtain \\omega{(g,\\mathbf{B})} = - \\mathbf{B} - \\sin{(g)}", "derivation": "\\hat{x}{(g)} = \\cos{(g)} and \\frac{d}{d g} \\hat{x}{(g)} = \\frac{d}{d g} \\cos{(g)} and \\frac{d}{d g} \\hat{x}{(g)} = - \\sin{(g)} and \\frac{d}{d g} \\cos{(g)} = - \\sin{(g)} and - \\mathbf{B} + \\frac{d}{d g} \\cos{(g)} = - \\mathbf{B} - \\sin{(g)} and \\omega{(g,\\mathbf{B})} = - \\mathbf{B} + \\frac{d}{d g} \\cos{(g)} and \\omega{(g,\\mathbf{B})} = - \\mathbf{B} - \\sin{(g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('g', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then obtain 2 (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} - \\sin{(a^{\\dagger})} = (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} - \\sin{(a^{\\dagger})} + 1", "derivation": "\\operatorname{t_{2}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}} = 1 and (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} = 1 and (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} - \\sin{(a^{\\dagger})} = 1 - \\sin{(a^{\\dagger})} and 2 (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} - \\sin{(a^{\\dagger})} = (\\frac{\\operatorname{t_{2}}{(a^{\\dagger})}}{\\sin{(a^{\\dagger})}})^{a^{\\dagger}} - \\sin{(a^{\\dagger})} + 1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "sin(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True)), Integer(1))"], [["minus", 3, "sin(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Pow(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Integer(1), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["add", 4, "Pow(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Pow(Mul(Function('t_2')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} = (M^{G})^{f_{\\mathbf{v}}}, then obtain (((\\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} + 1)^{M})^{f_{\\mathbf{v}}})^{G} = ((((M^{G})^{f_{\\mathbf{v}}} + 1)^{M})^{f_{\\mathbf{v}}})^{G}", "derivation": "\\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} = (M^{G})^{f_{\\mathbf{v}}} and \\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} + 1 = (M^{G})^{f_{\\mathbf{v}}} + 1 and (\\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} + 1)^{M} = ((M^{G})^{f_{\\mathbf{v}}} + 1)^{M} and ((\\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} + 1)^{M})^{f_{\\mathbf{v}}} = (((M^{G})^{f_{\\mathbf{v}}} + 1)^{M})^{f_{\\mathbf{v}}} and (((\\operatorname{f^{\\prime}}{(M,f_{\\mathbf{v}},G)} + 1)^{M})^{f_{\\mathbf{v}}})^{G} = ((((M^{G})^{f_{\\mathbf{v}}} + 1)^{M})^{f_{\\mathbf{v}}})^{G}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('G', commutative=True)), Pow(Pow(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('G', commutative=True)), Integer(1)), Add(Pow(Pow(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Add(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('G', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Pow(Add(Pow(Pow(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Symbol('M', commutative=True)))"], [["power", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Pow(Add(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('G', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Pow(Add(Pow(Pow(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["power", 4, "Symbol('G', commutative=True)"], "Equality(Pow(Pow(Pow(Add(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('G', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('G', commutative=True)), Pow(Pow(Pow(Add(Pow(Pow(Symbol('M', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\omega{(v,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v}, then obtain \\cos{(\\omega{(v,\\Psi^{\\dagger})} - 1)} + \\cos{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1)} = 2 \\cos{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1)}", "derivation": "\\omega{(v,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} and \\omega{(v,\\Psi^{\\dagger})} - 1 = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1 and \\cos{(\\omega{(v,\\Psi^{\\dagger})} - 1)} = \\cos{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1)} and \\cos{(\\omega{(v,\\Psi^{\\dagger})} - 1)} + \\cos{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1)} = 2 \\cos{(\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\frac{\\Psi^{\\dagger}}{v} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)))"], [["cos", 2], "Equality(cos(Add(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))), cos(Add(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1))))"], [["add", 3, "cos(Add(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(cos(Add(Function('\\\\omega')(Symbol('v', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))), cos(Add(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)))), Mul(Integer(2), cos(Add(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given B{(\\hat{x}_0,\\mathbf{v})} = \\hat{x}_0 - \\mathbf{v} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\mathbf{v})} = 2 \\hat{x}_0 - \\mathbf{v}, then obtain \\cos{(3 \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})})} = \\cos{(4 \\hat{x}_0 - \\mathbf{v})}", "derivation": "B{(\\hat{x}_0,\\mathbf{v})} = \\hat{x}_0 - \\mathbf{v} and - \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})} = - \\mathbf{v} and \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})} = 2 \\hat{x}_0 - \\mathbf{v} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\mathbf{v})} = 2 \\hat{x}_0 - \\mathbf{v} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\mathbf{v})} = \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})} and 2 \\hat{x}_0 + \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\mathbf{v})} = 4 \\hat{x}_0 - \\mathbf{v} and 3 \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})} = 4 \\hat{x}_0 - \\mathbf{v} and \\cos{(3 \\hat{x}_0 + B{(\\hat{x}_0,\\mathbf{v})})} = \\cos{(4 \\hat{x}_0 - \\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(4), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(3), Symbol('\\\\hat{x}_0', commutative=True)), Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(4), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["cos", 7], "Equality(cos(Add(Mul(Integer(3), Symbol('\\\\hat{x}_0', commutative=True)), Function('B')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), cos(Add(Mul(Integer(4), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} = \\frac{\\mathbf{J}}{F_{x} r_{0}}, then obtain \\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} + 1 - \\frac{\\mathbf{J}}{F_{x} r_{0}} = 1", "derivation": "\\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} = \\frac{\\mathbf{J}}{F_{x} r_{0}} and - r_{0} + \\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} = - r_{0} + \\frac{\\mathbf{J}}{F_{x} r_{0}} and \\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} - \\frac{\\mathbf{J}}{F_{x} r_{0}} = 0 and \\operatorname{v_{y}}{(r_{0},\\mathbf{J},F_{x})} + 1 - \\frac{\\mathbf{J}}{F_{x} r_{0}} = 1", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('v_y')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], "Equality(Add(Function('v_y')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))), Integer(0))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('v_y')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))), Integer(1))"]]}, {"prompt": "Given a{(x^\\prime)} = \\sin{(x^\\prime)} and \\mathbf{H}{(x^\\prime)} = - x^\\prime and \\mu_{0}{(x^\\prime)} = - (a{(x^\\prime)} + \\sin{(x^\\prime)})^{2} + \\sin{(\\mathbf{H}{(x^\\prime)})}, then obtain \\mu_{0}{(x^\\prime)} = - 4 \\sin^{2}{(x^\\prime)} - \\sin{(x^\\prime)}", "derivation": "a{(x^\\prime)} = \\sin{(x^\\prime)} and a{(x^\\prime)} + \\sin{(x^\\prime)} = 2 \\sin{(x^\\prime)} and \\mathbf{H}{(x^\\prime)} = - x^\\prime and \\sin{(\\mathbf{H}{(x^\\prime)})} = - \\sin{(x^\\prime)} and - (a{(x^\\prime)} + \\sin{(x^\\prime)})^{2} + \\sin{(\\mathbf{H}{(x^\\prime)})} = - (a{(x^\\prime)} + \\sin{(x^\\prime)})^{2} - \\sin{(x^\\prime)} and - 4 \\sin^{2}{(x^\\prime)} + \\sin{(\\mathbf{H}{(x^\\prime)})} = - 4 \\sin^{2}{(x^\\prime)} - \\sin{(x^\\prime)} and \\mu_{0}{(x^\\prime)} = - (a{(x^\\prime)} + \\sin{(x^\\prime)})^{2} + \\sin{(\\mathbf{H}{(x^\\prime)})} and \\mu_{0}{(x^\\prime)} = - 4 \\sin^{2}{(x^\\prime)} + \\sin{(\\mathbf{H}{(x^\\prime)})} and \\mu_{0}{(x^\\prime)} = - 4 \\sin^{2}{(x^\\prime)} - \\sin{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "sin(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), sin(Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))"], [["sin", 3], "Equality(sin(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 4, "Pow(Add(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Integer(2))), sin(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Integer(2))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Integer(4), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(2))), sin(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Integer(4), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Pow(Add(Function('a')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Integer(2))), sin(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Integer(4), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(2))), sin(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Integer(4), Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given r{(a^{\\dagger},\\mathbf{F})} = - \\mathbf{F} + a^{\\dagger}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{\\partial}{\\partial a^{\\dagger}} r{(a^{\\dagger},\\mathbf{F})} d\\mathbf{F} = \\frac{\\partial}{\\partial a^{\\dagger}} (\\hat{H}_l + \\mathbf{F})", "derivation": "r{(a^{\\dagger},\\mathbf{F})} = - \\mathbf{F} + a^{\\dagger} and \\frac{\\partial}{\\partial a^{\\dagger}} r{(a^{\\dagger},\\mathbf{F})} = \\frac{\\partial}{\\partial a^{\\dagger}} (- \\mathbf{F} + a^{\\dagger}) and \\int \\frac{\\partial}{\\partial a^{\\dagger}} r{(a^{\\dagger},\\mathbf{F})} d\\mathbf{F} = \\int \\frac{\\partial}{\\partial a^{\\dagger}} (- \\mathbf{F} + a^{\\dagger}) d\\mathbf{F} and \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{\\partial}{\\partial a^{\\dagger}} r{(a^{\\dagger},\\mathbf{F})} d\\mathbf{F} = \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{\\partial}{\\partial a^{\\dagger}} (- \\mathbf{F} + a^{\\dagger}) d\\mathbf{F} and \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\frac{\\partial}{\\partial a^{\\dagger}} r{(a^{\\dagger},\\mathbf{F})} d\\mathbf{F} = \\frac{\\partial}{\\partial a^{\\dagger}} (\\hat{H}_l + \\mathbf{F})", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Derivative(Function('r')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('r')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Derivative(Function('r')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(J)} = \\frac{d}{d J} \\cos{(J)}, then derive \\mu{(J)} = - \\sin{(J)}, then obtain 1 = \\frac{- J - \\sin{(J)}}{- J + \\mu{(J)}}", "derivation": "\\mu{(J)} = \\frac{d}{d J} \\cos{(J)} and \\mu{(J)} = - \\sin{(J)} and - J + \\mu{(J)} = - J - \\sin{(J)} and - J + \\frac{d}{d J} \\cos{(J)} = - J - \\sin{(J)} and \\frac{- J + \\frac{d}{d J} \\cos{(J)}}{- J + \\mu{(J)}} = \\frac{- J - \\sin{(J)}}{- J + \\mu{(J)}} and 1 = \\frac{- J - \\sin{(J)}}{- J + \\mu{(J)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('J', commutative=True)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu')(Symbol('J', commutative=True)), Mul(Integer(-1), sin(Symbol('J', commutative=True))))"], [["minus", 2, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), sin(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), sin(Symbol('J', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), sin(Symbol('J', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), sin(Symbol('J', commutative=True))))))"]]}, {"prompt": "Given Q{(z,y)} = \\frac{z}{y} and \\varphi{(z,y)} = \\frac{z}{y}, then obtain \\frac{(y Q{(z,y)})^{y} \\varphi{(z,y)}}{z} = \\frac{(y Q{(z,y)})^{y}}{y}", "derivation": "Q{(z,y)} = \\frac{z}{y} and y Q{(z,y)} = z and \\varphi{(z,y)} = \\frac{z}{y} and \\frac{\\varphi{(z,y)}}{y Q{(z,y)}} = \\frac{z}{y^{2} Q{(z,y)}} and \\frac{\\varphi{(z,y)}}{z} = \\frac{1}{y} and \\frac{(y Q{(z,y)})^{y} \\varphi{(z,y)}}{z} = \\frac{(y Q{(z,y)})^{y}}{y}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["divide", 1, "Pow(Symbol('y', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('y', commutative=True), Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Symbol('z', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('z', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["divide", 3, "Mul(Symbol('y', commutative=True), Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\varphi')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-2)), Symbol('z', commutative=True), Pow(Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Pow(Symbol('y', commutative=True), Integer(-1)))"], [["times", 5, "Pow(Mul(Symbol('y', commutative=True), Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Mul(Symbol('y', commutative=True), Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Function('\\\\varphi')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Mul(Symbol('y', commutative=True), Function('Q')(Symbol('z', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})} = \\frac{\\mathbf{H}}{\\Psi_{nl}}, then obtain (- \\Psi_{nl} + \\log{(- \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})})})^{\\Psi_{nl}} = (- \\Psi_{nl} + \\log{(- \\frac{\\mathbf{H}}{\\Psi_{nl}})})^{\\Psi_{nl}}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})} = \\frac{\\mathbf{H}}{\\Psi_{nl}} and - \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})} = - \\frac{\\mathbf{H}}{\\Psi_{nl}} and \\log{(- \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})})} = \\log{(- \\frac{\\mathbf{H}}{\\Psi_{nl}})} and - \\Psi_{nl} + \\log{(- \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})})} = - \\Psi_{nl} + \\log{(- \\frac{\\mathbf{H}}{\\Psi_{nl}})} and (- \\Psi_{nl} + \\log{(- \\operatorname{E_{x}}{(\\mathbf{H},\\Psi_{nl})})})^{\\Psi_{nl}} = (- \\Psi_{nl} + \\log{(- \\frac{\\mathbf{H}}{\\Psi_{nl}})})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["log", 2], "Equality(log(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), log(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["power", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(U)} = \\log{(U)} and \\mathbf{A}{(\\psi^*,U)} = \\psi^* \\log{(U)}, then obtain \\log{(U)} + \\int \\mathbf{A}{(\\psi^*,U)} dU = \\log{(U)} + \\int \\psi^* \\log{(U)} dU", "derivation": "\\operatorname{a^{\\dagger}}{(U)} = \\log{(U)} and \\mathbf{A}{(\\psi^*,U)} = \\psi^* \\log{(U)} and \\mathbf{A}{(\\psi^*,U)} = \\psi^* \\operatorname{a^{\\dagger}}{(U)} and \\int \\mathbf{A}{(\\psi^*,U)} dU = \\int \\psi^* \\operatorname{a^{\\dagger}}{(U)} dU and \\log{(U)} + \\int \\mathbf{A}{(\\psi^*,U)} dU = \\log{(U)} + \\int \\psi^* \\operatorname{a^{\\dagger}}{(U)} dU and \\operatorname{a^{\\dagger}}{(U)} + \\int \\mathbf{A}{(\\psi^*,U)} dU = \\operatorname{a^{\\dagger}}{(U)} + \\int \\psi^* \\operatorname{a^{\\dagger}}{(U)} dU and \\log{(U)} + \\int \\mathbf{A}{(\\psi^*,U)} dU = \\log{(U)} + \\int \\psi^* \\log{(U)} dU", "srepr_derivation": [["get_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('\\\\psi^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["add", 4, "log(Symbol('U', commutative=True))"], "Equality(Add(log(Symbol('U', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(log(Symbol('U', commutative=True)), Integral(Mul(Symbol('\\\\psi^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('U', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Function('a^{\\\\dagger}')(Symbol('U', commutative=True)), Integral(Mul(Symbol('\\\\psi^*', commutative=True), Function('a^{\\\\dagger}')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(log(Symbol('U', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(log(Symbol('U', commutative=True)), Integral(Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},v_{y})} = e^{v_{y}^{A_{2}}} and \\rho_{b}{(A_{2},v_{y})} = e^{v_{y}^{A_{2}}}, then obtain \\frac{d}{d v_{y}} 1 = \\frac{\\partial}{\\partial v_{y}} (\\frac{\\rho_{b}{(A_{2},v_{y})}}{\\mathbf{S}{(A_{2},v_{y})}})^{v_{y}}", "derivation": "\\mathbf{S}{(A_{2},v_{y})} = e^{v_{y}^{A_{2}}} and 1 = \\frac{e^{v_{y}^{A_{2}}}}{\\mathbf{S}{(A_{2},v_{y})}} and 1 = (\\frac{e^{v_{y}^{A_{2}}}}{\\mathbf{S}{(A_{2},v_{y})}})^{v_{y}} and \\rho_{b}{(A_{2},v_{y})} = e^{v_{y}^{A_{2}}} and 1 = (\\frac{\\rho_{b}{(A_{2},v_{y})}}{\\mathbf{S}{(A_{2},v_{y})}})^{v_{y}} and \\frac{d}{d v_{y}} 1 = \\frac{\\partial}{\\partial v_{y}} (\\frac{\\rho_{b}{(A_{2},v_{y})}}{\\mathbf{S}{(A_{2},v_{y})}})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), exp(Pow(Symbol('v_y', commutative=True), Symbol('A_2', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), exp(Pow(Symbol('v_y', commutative=True), Symbol('A_2', commutative=True)))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), exp(Pow(Symbol('v_y', commutative=True), Symbol('A_2', commutative=True)))), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), exp(Pow(Symbol('v_y', commutative=True), Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('\\\\rho_b')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["differentiate", 5, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('\\\\rho_b')(Symbol('A_2', commutative=True), Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(n_{1},z^{*})} = - n_{1} + z^{*}, then obtain \\int (- n_{1} + z^{*} - \\mu{(n_{1},z^{*})})^{n_{1}} dz^{*} = \\int 1 dz^{*}", "derivation": "\\mu{(n_{1},z^{*})} = - n_{1} + z^{*} and 0 = - n_{1} + z^{*} - \\mu{(n_{1},z^{*})} and 0^{n_{1}} = (- n_{1} + z^{*} - \\mu{(n_{1},z^{*})})^{n_{1}} and \\int 0^{n_{1}} dz^{*} = \\int (- n_{1} + z^{*} - \\mu{(n_{1},z^{*})})^{n_{1}} dz^{*} and \\int (- n_{1} + z^{*} - \\mu{(n_{1},z^{*})})^{n_{1}} dz^{*} = \\int 1 dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('z^*', commutative=True)))"], [["minus", 1, "Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)))), Symbol('n_1', commutative=True)))"], [["integrate", 3, "Symbol('z^*', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('n_1', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)))), Symbol('n_1', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('n_1', commutative=True), Symbol('z^*', commutative=True)))), Symbol('n_1', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Integer(1), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain (((\\varphi{(\\mathbf{S})} e^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}} = (((e^{2 \\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}}", "derivation": "\\varphi{(\\mathbf{S})} = e^{\\mathbf{S}} and \\varphi{(\\mathbf{S})} e^{\\mathbf{S}} = e^{2 \\mathbf{S}} and (\\varphi{(\\mathbf{S})} e^{\\mathbf{S}})^{\\mathbf{S}} = (e^{2 \\mathbf{S}})^{\\mathbf{S}} and ((\\varphi{(\\mathbf{S})} e^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}} = ((e^{2 \\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}} and (((\\varphi{(\\mathbf{S})} e^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}} = (((e^{2 \\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given M{(C_{d})} = \\cos{(e^{C_{d}})}, then obtain M{(C_{d})} e^{C_{d}} + e^{C_{d}} \\frac{d}{d C_{d}} M{(C_{d})} - \\cos{(e^{C_{d}})} = - e^{2 C_{d}} \\sin{(e^{C_{d}})} + e^{C_{d}} \\cos{(e^{C_{d}})} - \\cos{(e^{C_{d}})}", "derivation": "M{(C_{d})} = \\cos{(e^{C_{d}})} and M{(C_{d})} e^{C_{d}} = e^{C_{d}} \\cos{(e^{C_{d}})} and \\frac{d}{d C_{d}} M{(C_{d})} e^{C_{d}} = \\frac{d}{d C_{d}} e^{C_{d}} \\cos{(e^{C_{d}})} and - \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} M{(C_{d})} e^{C_{d}} = - \\cos{(e^{C_{d}})} + \\frac{d}{d C_{d}} e^{C_{d}} \\cos{(e^{C_{d}})} and M{(C_{d})} e^{C_{d}} + e^{C_{d}} \\frac{d}{d C_{d}} M{(C_{d})} - \\cos{(e^{C_{d}})} = - e^{2 C_{d}} \\sin{(e^{C_{d}})} + e^{C_{d}} \\cos{(e^{C_{d}})} - \\cos{(e^{C_{d}})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True))))"], [["times", 1, "exp(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('M')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True))), Mul(exp(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True)))))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Function('M')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 3, "cos(exp(Symbol('C_d', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(exp(Symbol('C_d', commutative=True)))), Derivative(Mul(Function('M')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(exp(Symbol('C_d', commutative=True)))), Derivative(Mul(exp(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Function('M')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True))), Mul(exp(Symbol('C_d', commutative=True)), Derivative(Function('M')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), cos(exp(Symbol('C_d', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('C_d', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))), Mul(exp(Symbol('C_d', commutative=True)), cos(exp(Symbol('C_d', commutative=True)))), Mul(Integer(-1), cos(exp(Symbol('C_d', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(V_{\\mathbf{B}},y)} = - V_{\\mathbf{B}} + y, then obtain e^{\\frac{\\partial}{\\partial y} (y + \\mathbf{F}{(V_{\\mathbf{B}},y)})} = e^{\\frac{\\partial}{\\partial y} (- V_{\\mathbf{B}} + 2 y)}", "derivation": "\\mathbf{F}{(V_{\\mathbf{B}},y)} = - V_{\\mathbf{B}} + y and y + \\mathbf{F}{(V_{\\mathbf{B}},y)} = - V_{\\mathbf{B}} + 2 y and \\frac{\\partial}{\\partial y} (y + \\mathbf{F}{(V_{\\mathbf{B}},y)}) = \\frac{\\partial}{\\partial y} (- V_{\\mathbf{B}} + 2 y) and e^{\\frac{\\partial}{\\partial y} (y + \\mathbf{F}{(V_{\\mathbf{B}},y)})} = e^{\\frac{\\partial}{\\partial y} (- V_{\\mathbf{B}} + 2 y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Symbol('y', commutative=True), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Add(Symbol('y', commutative=True), Function('\\\\mathbf{F}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), exp(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(b,c)} = - c + \\sin{(b)}, then derive \\frac{\\partial}{\\partial c} r{(b,c)} = -1, then obtain \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int \\frac{\\frac{\\partial}{\\partial c} r{(b,c)}}{\\sin{(b)}} dc}{\\sin{(b)}} - \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int - \\frac{1}{\\sin{(b)}} dc}{\\sin{(b)}} = 0", "derivation": "r{(b,c)} = - c + \\sin{(b)} and \\frac{\\partial}{\\partial c} r{(b,c)} = \\frac{\\partial}{\\partial c} (- c + \\sin{(b)}) and \\frac{\\partial}{\\partial c} r{(b,c)} = -1 and \\frac{\\frac{\\partial}{\\partial c} r{(b,c)}}{\\sin{(b)}} = - \\frac{1}{\\sin{(b)}} and \\int \\frac{\\frac{\\partial}{\\partial c} r{(b,c)}}{\\sin{(b)}} dc = \\int - \\frac{1}{\\sin{(b)}} dc and \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int \\frac{\\frac{\\partial}{\\partial c} r{(b,c)}}{\\sin{(b)}} dc}{\\sin{(b)}} = \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int - \\frac{1}{\\sin{(b)}} dc}{\\sin{(b)}} and \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int \\frac{\\frac{\\partial}{\\partial c} r{(b,c)}}{\\sin{(b)}} dc}{\\sin{(b)}} - \\frac{\\frac{\\partial}{\\partial c} r{(b,c)} \\int - \\frac{1}{\\sin{(b)}} dc}{\\sin{(b)}} = 0", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), sin(Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), sin(Symbol('b', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(-1))"], [["divide", 3, "sin(Symbol('b', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Tuple(Symbol('c', commutative=True))), Integral(Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True))))"], [["times", 5, "Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], "Equality(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integral(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Tuple(Symbol('c', commutative=True)))), Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True)))))"], [["minus", 6, "Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True))))"], "Equality(Add(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integral(Mul(Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Tuple(Symbol('c', commutative=True)))), Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Pow(sin(Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('c', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(H)} = e^{H} and \\varphi^{*}{(H)} = \\frac{d}{d H} (H + \\operatorname{V_{\\mathbf{B}}}{(H)}), then obtain \\varphi^{*}{(H)} = \\frac{d}{d H} (H + e^{H})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(H)} = e^{H} and H + \\operatorname{V_{\\mathbf{B}}}{(H)} = H + e^{H} and \\frac{d}{d H} (H + \\operatorname{V_{\\mathbf{B}}}{(H)}) = \\frac{d}{d H} (H + e^{H}) and \\varphi^{*}{(H)} = \\frac{d}{d H} (H + \\operatorname{V_{\\mathbf{B}}}{(H)}) and \\varphi^{*}{(H)} = \\frac{d}{d H} (H + e^{H})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), exp(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Symbol('H', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('H', commutative=True)), Derivative(Add(Symbol('H', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varphi^*')(Symbol('H', commutative=True)), Derivative(Add(Symbol('H', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(m_{s})} = \\log{(m_{s})}, then obtain \\frac{2 \\dot{x}{(m_{s})}}{\\int (\\dot{x}{(m_{s})} + \\log{(m_{s})}) dm_{s}} = \\frac{\\dot{x}{(m_{s})} + \\log{(m_{s})}}{\\int (\\dot{x}{(m_{s})} + \\log{(m_{s})}) dm_{s}}", "derivation": "\\dot{x}{(m_{s})} = \\log{(m_{s})} and 2 \\dot{x}{(m_{s})} = \\dot{x}{(m_{s})} + \\log{(m_{s})} and \\int 2 \\dot{x}{(m_{s})} dm_{s} = \\int (\\dot{x}{(m_{s})} + \\log{(m_{s})}) dm_{s} and \\frac{2 \\dot{x}{(m_{s})}}{\\int 2 \\dot{x}{(m_{s})} dm_{s}} = \\frac{\\dot{x}{(m_{s})} + \\log{(m_{s})}}{\\int 2 \\dot{x}{(m_{s})} dm_{s}} and \\frac{2 \\dot{x}{(m_{s})}}{\\int (\\dot{x}{(m_{s})} + \\log{(m_{s})}) dm_{s}} = \\frac{\\dot{x}{(m_{s})} + \\log{(m_{s})}}{\\int (\\dot{x}{(m_{s})} + \\log{(m_{s})}) dm_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], [["add", 1, "Function('\\\\dot{x}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True))), Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"], [["divide", 2, "Integral(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Pow(Integral(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integer(-1))), Mul(Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Pow(Integral(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Pow(Integral(Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integer(-1))), Mul(Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Pow(Integral(Add(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given l{(r_{0},\\mathbf{p},\\theta)} = - \\mathbf{p} + \\theta + r_{0}, then obtain (\\theta + r_{0}) (- \\mathbf{p} + \\theta + r_{0}) l{(r_{0},\\mathbf{p},\\theta)} = (\\theta + r_{0}) (- \\mathbf{p} + \\theta + r_{0})^{2}", "derivation": "l{(r_{0},\\mathbf{p},\\theta)} = - \\mathbf{p} + \\theta + r_{0} and (- \\mathbf{p} + \\theta + r_{0}) l{(r_{0},\\mathbf{p},\\theta)} = (- \\mathbf{p} + \\theta + r_{0})^{2} and \\mathbf{p} + l{(r_{0},\\mathbf{p},\\theta)} = \\theta + r_{0} and (\\mathbf{p} + l{(r_{0},\\mathbf{p},\\theta)}) (- \\mathbf{p} + \\theta + r_{0}) l{(r_{0},\\mathbf{p},\\theta)} = (\\mathbf{p} + l{(r_{0},\\mathbf{p},\\theta)}) (- \\mathbf{p} + \\theta + r_{0})^{2} and (\\theta + r_{0}) (- \\mathbf{p} + \\theta + r_{0}) l{(r_{0},\\mathbf{p},\\theta)} = (\\theta + r_{0}) (- \\mathbf{p} + \\theta + r_{0})^{2}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(2)))"], [["add", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\mathbf{p}', commutative=True), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Function('l')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)), Integer(2))))"]]}, {"prompt": "Given L{(\\mathbf{J})} = \\mathbf{J}, then derive f + 2 \\int L{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} L{(\\mathbf{J})} d\\mathbf{J} = \\int \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} d\\mathbf{J}, then obtain \\sin{(f + 2 \\int L{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} L{(\\mathbf{J})} d\\mathbf{J})} = \\sin{(\\int \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} d\\mathbf{J})}", "derivation": "L{(\\mathbf{J})} = \\mathbf{J} and L^{2}{(\\mathbf{J})} = \\mathbf{J} L{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} L^{2}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} and \\int \\frac{d}{d \\mathbf{J}} L^{2}{(\\mathbf{J})} d\\mathbf{J} = \\int \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} d\\mathbf{J} and f + 2 \\int L{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} L{(\\mathbf{J})} d\\mathbf{J} = \\int \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} d\\mathbf{J} and \\sin{(f + 2 \\int L{(\\mathbf{J})} \\frac{d}{d \\mathbf{J}} L{(\\mathbf{J})} d\\mathbf{J})} = \\sin{(\\int \\frac{d}{d \\mathbf{J}} \\mathbf{J} L{(\\mathbf{J})} d\\mathbf{J})}", "srepr_derivation": [["renaming_premise", "Equality(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], [["times", 1, "Function('L')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Pow(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('L')(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Pow(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('L')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('L')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('f', commutative=True), Mul(Integer(2), Integral(Mul(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))), Integral(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('L')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["sin", 5], "Equality(sin(Add(Symbol('f', commutative=True), Mul(Integer(2), Integral(Mul(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))), sin(Integral(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('L')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J} + \\operatorname{F_{g}}{(\\mathbf{J})} + e^{\\mathbf{J}})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J} + 2 e^{\\mathbf{J}})}", "derivation": "\\operatorname{F_{g}}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\operatorname{F_{g}}{(\\mathbf{J})} + e^{\\mathbf{J}} = 2 e^{\\mathbf{J}} and \\mathbf{J} + \\operatorname{F_{g}}{(\\mathbf{J})} + e^{\\mathbf{J}} = \\mathbf{J} + 2 e^{\\mathbf{J}} and \\log{(\\mathbf{J} + \\operatorname{F_{g}}{(\\mathbf{J})} + e^{\\mathbf{J}})} = \\log{(\\mathbf{J} + 2 e^{\\mathbf{J}})} and \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J} + \\operatorname{F_{g}}{(\\mathbf{J})} + e^{\\mathbf{J}})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J} + 2 e^{\\mathbf{J}})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('F_g')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["log", 3], "Equality(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('F_g')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('F_g')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(M,v_{z})} = M v_{z}, then obtain \\Psi_{nl}{(M,v_{z})} - \\sin{(\\Psi_{nl}{(M,v_{z})})} = M v_{z} - \\sin{(\\Psi_{nl}{(M,v_{z})})}", "derivation": "\\Psi_{nl}{(M,v_{z})} = M v_{z} and \\sin{(\\Psi_{nl}{(M,v_{z})})} = \\sin{(M v_{z})} and \\Psi_{nl}{(M,v_{z})} - \\sin{(M v_{z})} = M v_{z} - \\sin{(M v_{z})} and \\Psi_{nl}{(M,v_{z})} - \\sin{(\\Psi_{nl}{(M,v_{z})})} = M v_{z} - \\sin{(\\Psi_{nl}{(M,v_{z})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True))), sin(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True))))"], [["minus", 1, "sin(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True))))), Add(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), sin(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True))))), Add(Mul(Symbol('M', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), sin(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\rho{(\\eta)} = \\log{(\\eta)}, then obtain \\frac{\\rho{(\\eta)}}{\\frac{d}{d \\eta} \\rho{(\\eta)}} = \\frac{\\log{(\\eta)}}{\\frac{d}{d \\eta} \\rho{(\\eta)}}", "derivation": "\\rho{(\\eta)} = \\log{(\\eta)} and \\frac{d}{d \\eta} \\rho{(\\eta)} = \\frac{d}{d \\eta} \\log{(\\eta)} and \\frac{\\rho{(\\eta)}}{\\frac{d}{d \\eta} \\log{(\\eta)}} = \\frac{\\log{(\\eta)}}{\\frac{d}{d \\eta} \\log{(\\eta)}} and \\frac{\\rho{(\\eta)}}{\\frac{d}{d \\eta} \\rho{(\\eta)}} = \\frac{\\log{(\\eta)}}{\\frac{d}{d \\eta} \\rho{(\\eta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Pow(Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('\\\\eta', commutative=True)), Pow(Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Pow(Derivative(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('\\\\eta', commutative=True)), Pow(Derivative(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{s}{(r_{0})} = \\log{(r_{0})}, then derive \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} = \\frac{1}{r_{0}}, then obtain \\int \\frac{d}{d r_{0}} \\log{(r_{0})} dr_{0} = \\int \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} dr_{0}", "derivation": "\\mathbf{s}{(r_{0})} = \\log{(r_{0})} and \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} = \\frac{d}{d r_{0}} \\log{(r_{0})} and \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} = \\frac{1}{r_{0}} and \\int \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} dr_{0} = \\int \\frac{1}{r_{0}} dr_{0} and \\int \\frac{d}{d r_{0}} \\log{(r_{0})} dr_{0} = \\int \\frac{1}{r_{0}} dr_{0} and \\int \\frac{d}{d r_{0}} \\log{(r_{0})} dr_{0} = \\int \\frac{d}{d r_{0}} \\mathbf{s}{(r_{0})} dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Pow(Symbol('r_0', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))), Integral(Pow(Symbol('r_0', commutative=True), Integer(-1)), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))), Integral(Pow(Symbol('r_0', commutative=True), Integer(-1)), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))), Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(J)} = \\sin{(J)}, then obtain \\hat{\\mathbf{x}}{(J)} - \\sin^{J}{(J)} = \\sin{(J)} - \\sin^{J}{(J)}", "derivation": "\\hat{\\mathbf{x}}{(J)} = \\sin{(J)} and \\hat{\\mathbf{x}}^{J}{(J)} = \\sin^{J}{(J)} and \\hat{\\mathbf{x}}{(J)} - \\hat{\\mathbf{x}}^{J}{(J)} = - \\hat{\\mathbf{x}}^{J}{(J)} + \\sin{(J)} and \\hat{\\mathbf{x}}{(J)} - \\sin^{J}{(J)} = \\sin{(J)} - \\sin^{J}{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), sin(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('J', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)))), Add(sin(Symbol('J', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('J', commutative=True)), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})} = \\frac{\\mathbf{A}}{\\mathbf{g}} and \\operatorname{v_{t}}{(\\mathbf{g},\\mathbf{A})} = \\frac{\\mathbf{A}}{\\mathbf{g}}, then obtain (\\frac{\\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}})^{\\mathbf{g}} = (\\frac{\\operatorname{v_{t}}{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}})^{\\mathbf{g}}", "derivation": "\\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})} = \\frac{\\mathbf{A}}{\\mathbf{g}} and \\frac{\\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}} = \\frac{\\mathbf{A}}{\\mathbf{g}^{2}} and (\\frac{\\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}})^{\\mathbf{g}} = (\\frac{\\mathbf{A}}{\\mathbf{g}^{2}})^{\\mathbf{g}} and \\operatorname{v_{t}}{(\\mathbf{g},\\mathbf{A})} = \\frac{\\mathbf{A}}{\\mathbf{g}} and (\\frac{\\mathbf{J}_f{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}})^{\\mathbf{g}} = (\\frac{\\operatorname{v_{t}}{(\\mathbf{g},\\mathbf{A})}}{\\mathbf{g}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2))))"], [["power", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2))), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('v_t')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\ddot{x})} = \\int \\cos{(\\ddot{x})} d\\ddot{x} and v{(E_{x})} = E_{x}, then derive \\varphi{(\\ddot{x})} = E_{x} + \\sin{(\\ddot{x})}, then derive A_{y} + E_{x} + 2 \\sin{(\\ddot{x})} = E_{x} + v{(E_{x})} + 2 \\sin{(\\ddot{x})}, then obtain \\sin{(A_{y} + E_{x} + 2 \\sin{(\\ddot{x})})} = \\sin{(E_{x} + v{(E_{x})} + 2 \\sin{(\\ddot{x})})}", "derivation": "\\varphi{(\\ddot{x})} = \\int \\cos{(\\ddot{x})} d\\ddot{x} and \\varphi{(\\ddot{x})} = E_{x} + \\sin{(\\ddot{x})} and v{(E_{x})} = E_{x} and \\varphi{(\\ddot{x})} = v{(E_{x})} + \\sin{(\\ddot{x})} and \\int \\cos{(\\ddot{x})} d\\ddot{x} = v{(E_{x})} + \\sin{(\\ddot{x})} and E_{x} + \\sin{(\\ddot{x})} + \\int \\cos{(\\ddot{x})} d\\ddot{x} = E_{x} + v{(E_{x})} + 2 \\sin{(\\ddot{x})} and A_{y} + E_{x} + 2 \\sin{(\\ddot{x})} = E_{x} + v{(E_{x})} + 2 \\sin{(\\ddot{x})} and \\sin{(A_{y} + E_{x} + 2 \\sin{(\\ddot{x})})} = \\sin{(E_{x} + v{(E_{x})} + 2 \\sin{(\\ddot{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\ddot{x}', commutative=True)), Integral(cos(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\varphi')(Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('E_x', commutative=True), sin(Symbol('\\\\ddot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varphi')(Symbol('\\\\ddot{x}', commutative=True)), Add(Function('v')(Symbol('E_x', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(cos(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Function('v')(Symbol('E_x', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 5, "Add(Symbol('E_x', commutative=True), sin(Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Symbol('E_x', commutative=True), sin(Symbol('\\\\ddot{x}', commutative=True)), Integral(cos(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(Symbol('E_x', commutative=True), Function('v')(Symbol('E_x', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\ddot{x}', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True), Mul(Integer(2), sin(Symbol('\\\\ddot{x}', commutative=True)))), Add(Symbol('E_x', commutative=True), Function('v')(Symbol('E_x', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\ddot{x}', commutative=True)))))"], [["sin", 7], "Equality(sin(Add(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True), Mul(Integer(2), sin(Symbol('\\\\ddot{x}', commutative=True))))), sin(Add(Symbol('E_x', commutative=True), Function('v')(Symbol('E_x', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\ddot{x}', commutative=True))))))"]]}, {"prompt": "Given b{(u)} = \\log{(u)}, then obtain \\frac{d}{d u} 2 b{(u)} (\\int b{(u)} du)^{u} = \\frac{d}{d u} (b{(u)} + \\log{(u)}) (\\int b{(u)} du)^{u}", "derivation": "b{(u)} = \\log{(u)} and 2 b{(u)} = b{(u)} + \\log{(u)} and \\int b{(u)} du = \\int \\log{(u)} du and (\\int b{(u)} du)^{u} = (\\int \\log{(u)} du)^{u} and 2 b{(u)} (\\int \\log{(u)} du)^{u} = (b{(u)} + \\log{(u)}) (\\int \\log{(u)} du)^{u} and 2 b{(u)} (\\int b{(u)} du)^{u} = (b{(u)} + \\log{(u)}) (\\int b{(u)} du)^{u} and \\frac{d}{d u} 2 b{(u)} (\\int b{(u)} du)^{u} = \\frac{d}{d u} (b{(u)} + \\log{(u)}) (\\int b{(u)} du)^{u}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["add", 1, "Function('b')(Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Function('b')(Symbol('u', commutative=True))), Add(Function('b')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["times", 2, "Pow(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Function('b')(Symbol('u', commutative=True)), Pow(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Add(Function('b')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Function('b')(Symbol('u', commutative=True)), Pow(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Add(Function('b')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('b')(Symbol('u', commutative=True)), Pow(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Add(Function('b')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\omega)} = \\omega, then derive \\int n{(\\omega)} d\\omega = \\frac{\\omega^{2}}{2} + \\theta_2, then obtain \\log{(\\int \\omega d\\omega)} = \\log{(\\frac{\\omega^{2}}{2} + \\theta_2)}", "derivation": "n{(\\omega)} = \\omega and \\int n{(\\omega)} d\\omega = \\int \\omega d\\omega and \\int n{(\\omega)} d\\omega = \\frac{\\omega^{2}}{2} + \\theta_2 and \\int \\omega d\\omega = \\frac{\\omega^{2}}{2} + \\theta_2 and \\log{(\\int \\omega d\\omega)} = \\log{(\\frac{\\omega^{2}}{2} + \\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True)))"], [["log", 4], "Equality(log(Integral(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True)))), log(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\Omega,H,\\Psi_{nl})} = \\frac{H + \\Omega}{\\Psi_{nl}} and L{(H,\\Omega)} = H + \\Omega, then obtain \\log{((\\frac{L{(H,\\Omega)}}{\\Psi_{nl}})^{\\Omega})} = \\log{((\\frac{H + \\Omega}{\\Psi_{nl}})^{\\Omega})}", "derivation": "\\operatorname{A_{x}}{(\\Omega,H,\\Psi_{nl})} = \\frac{H + \\Omega}{\\Psi_{nl}} and \\operatorname{A_{x}}^{\\Omega}{(\\Omega,H,\\Psi_{nl})} = (\\frac{H + \\Omega}{\\Psi_{nl}})^{\\Omega} and L{(H,\\Omega)} = H + \\Omega and \\log{(\\operatorname{A_{x}}^{\\Omega}{(\\Omega,H,\\Psi_{nl})})} = \\log{((\\frac{H + \\Omega}{\\Psi_{nl}})^{\\Omega})} and \\operatorname{A_{x}}{(\\Omega,H,\\Psi_{nl})} = \\frac{L{(H,\\Omega)}}{\\Psi_{nl}} and \\log{((\\frac{L{(H,\\Omega)}}{\\Psi_{nl}})^{\\Omega})} = \\log{((\\frac{H + \\Omega}{\\Psi_{nl}})^{\\Omega})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Omega', commutative=True))), log(Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A_x')(Symbol('\\\\Omega', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('L')(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(log(Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('L')(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))), log(Pow(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x,\\mu)} = \\mu x and \\operatorname{t_{1}}{(x,\\mu)} = (\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x}, then obtain \\mu ((\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x})^{x} = \\mu \\operatorname{t_{1}}^{x}{(x,\\mu)}", "derivation": "\\operatorname{A_{y}}{(x,\\mu)} = \\mu x and \\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x} = \\mu and (\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x} = \\mu^{x} and \\operatorname{t_{1}}{(x,\\mu)} = (\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x} and \\operatorname{t_{1}}{(x,\\mu)} = \\mu^{x} and ((\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x})^{x} = (\\mu^{x})^{x} and \\mu ((\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x})^{x} = \\mu (\\mu^{x})^{x} and \\mu ((\\frac{\\operatorname{A_{y}}{(x,\\mu)}}{x})^{x})^{x} = \\mu \\operatorname{t_{1}}^{x}{(x,\\mu)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('x', commutative=True)))"], [["divide", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('x', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('t_1')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('x', commutative=True)))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 6, "Pow(Symbol('\\\\mu', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Pow(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Pow(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('A_y')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Function('t_1')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('x', commutative=True))))"]]}, {"prompt": "Given I{(a)} = e^{a} and \\mathbf{J}_M{(a)} = a e^{a}, then obtain a I{(a)} + a = a + \\mathbf{J}_M{(a)}", "derivation": "I{(a)} = e^{a} and a I{(a)} = a e^{a} and a I{(a)} + a = a e^{a} + a and \\mathbf{J}_M{(a)} = a e^{a} and a I{(a)} + a = a + \\mathbf{J}_M{(a)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('I')(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))))"], [["add", 2, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Symbol('a', commutative=True), Function('I')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Add(Mul(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), exp(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('a', commutative=True), Function('I')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Add(Symbol('a', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\varphi^*)} = \\log{(\\varphi^*)}, then obtain - \\frac{d}{d \\varphi^*} \\int \\log{(\\varphi^*)} d\\varphi^* + \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* = - \\frac{d}{d \\varphi^*} \\int \\log{(\\varphi^*)} d\\varphi^* + \\int \\log{(\\varphi^*)} d\\varphi^*", "derivation": "\\operatorname{A_{z}}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* = \\int \\log{(\\varphi^*)} d\\varphi^* and \\frac{d}{d \\varphi^*} \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* = \\frac{d}{d \\varphi^*} \\int \\log{(\\varphi^*)} d\\varphi^* and - \\frac{d}{d \\varphi^*} \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* + \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* = - \\frac{d}{d \\varphi^*} \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* + \\int \\log{(\\varphi^*)} d\\varphi^* and - \\frac{d}{d \\varphi^*} \\int \\log{(\\varphi^*)} d\\varphi^* + \\int \\operatorname{A_{z}}{(\\varphi^*)} d\\varphi^* = - \\frac{d}{d \\varphi^*} \\int \\log{(\\varphi^*)} d\\varphi^* + \\int \\log{(\\varphi^*)} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Derivative(Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integral(Function('A_z')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Derivative(Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integral(log(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\psi^*,v)} = \\psi^* v, then obtain v = v + (\\psi^* v)^{\\psi^*} - \\tilde{g}^{\\psi^*}{(\\psi^*,v)}", "derivation": "\\tilde{g}{(\\psi^*,v)} = \\psi^* v and \\tilde{g}^{\\psi^*}{(\\psi^*,v)} = (\\psi^* v)^{\\psi^*} and v + \\tilde{g}^{\\psi^*}{(\\psi^*,v)} = v + (\\psi^* v)^{\\psi^*} and v = v + (\\psi^* v)^{\\psi^*} - \\tilde{g}^{\\psi^*}{(\\psi^*,v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Add(Symbol('v', commutative=True), Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Symbol('v', commutative=True), Add(Symbol('v', commutative=True), Pow(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})}, then obtain \\nabla^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})}^{\\Psi^{\\dagger}}", "derivation": "\\nabla{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} and \\nabla^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})}^{\\Psi^{\\dagger}} and \\nabla^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})}^{\\Psi^{\\dagger}} = \\log{(\\Psi^{\\dagger})}^{2 \\Psi^{\\dagger}} and \\nabla^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} \\log{(\\Psi^{\\dagger})}^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["times", 2, "Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["times", 3, "Mul(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given z{(\\psi,m_{s})} = \\frac{\\psi}{m_{s}}, then obtain \\iiint z{(\\psi,m_{s})} dm_{s} dm_{s} d\\psi = \\iiint \\frac{\\psi}{m_{s}} dm_{s} dm_{s} d\\psi", "derivation": "z{(\\psi,m_{s})} = \\frac{\\psi}{m_{s}} and \\int z{(\\psi,m_{s})} dm_{s} = \\int \\frac{\\psi}{m_{s}} dm_{s} and \\iint z{(\\psi,m_{s})} dm_{s} dm_{s} = \\iint \\frac{\\psi}{m_{s}} dm_{s} dm_{s} and \\iiint z{(\\psi,m_{s})} dm_{s} dm_{s} d\\psi = \\iiint \\frac{\\psi}{m_{s}} dm_{s} dm_{s} d\\psi", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True))))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\psi', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\hat{H}_l)} = e^{\\hat{H}_l}, then derive \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} \\mathbf{M}{(\\hat{H}_l)} = \\psi^{*}{(v_{1},A_{z})} e^{\\hat{H}_l}, then obtain \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} = \\psi^{*}{(v_{1},A_{z})} e^{\\hat{H}_l}", "derivation": "\\mathbf{M}{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} \\mathbf{M}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} \\mathbf{M}{(\\hat{H}_l)} = \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} \\mathbf{M}{(\\hat{H}_l)} = \\psi^{*}{(v_{1},A_{z})} e^{\\hat{H}_l} and \\psi^{*}{(v_{1},A_{z})} \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} = \\psi^{*}{(v_{1},A_{z})} e^{\\hat{H}_l}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Function('\\\\psi^*')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\hat{x})} = e^{\\hat{x}}, then obtain (\\hat{H}_{\\lambda}{(\\hat{x})} - e^{2 \\hat{x}})^{2} - e^{2 \\hat{x}} + e^{\\hat{x}} = (- e^{2 \\hat{x}} + e^{\\hat{x}})^{2} - e^{2 \\hat{x}} + e^{\\hat{x}}", "derivation": "\\hat{H}_{\\lambda}{(\\hat{x})} = e^{\\hat{x}} and \\hat{H}_{\\lambda}{(\\hat{x})} e^{\\hat{x}} = e^{2 \\hat{x}} and - \\hat{H}_{\\lambda}{(\\hat{x})} e^{\\hat{x}} + \\hat{H}_{\\lambda}{(\\hat{x})} = - \\hat{H}_{\\lambda}{(\\hat{x})} e^{\\hat{x}} + e^{\\hat{x}} and \\hat{H}_{\\lambda}{(\\hat{x})} - e^{2 \\hat{x}} = - e^{2 \\hat{x}} + e^{\\hat{x}} and (\\hat{H}_{\\lambda}{(\\hat{x})} - e^{2 \\hat{x}})^{2} = (- e^{2 \\hat{x}} + e^{\\hat{x}})^{2} and (\\hat{H}_{\\lambda}{(\\hat{x})} - e^{2 \\hat{x}})^{2} - e^{2 \\hat{x}} + e^{\\hat{x}} = (- e^{2 \\hat{x}} + e^{\\hat{x}})^{2} - e^{2 \\hat{x}} + e^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True))), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))), Integer(2)), Pow(Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True))), Integer(2)))"], [["add", 5, "Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Pow(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))), Integer(2)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True))), Integer(2)), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)))), exp(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(C_{1},\\rho_b)} = \\frac{\\rho_b}{C_{1}} and \\sigma_{x}{(G)} = \\sin{(G)}, then obtain - 2 \\Psi_{nl}{(C_{1},\\rho_b)} \\sigma_{x}^{G}{(G)} = - 2 \\Psi_{nl}{(C_{1},\\rho_b)} \\sin^{G}{(G)}", "derivation": "\\Psi_{nl}{(C_{1},\\rho_b)} = \\frac{\\rho_b}{C_{1}} and 2 \\Psi_{nl}{(C_{1},\\rho_b)} = \\Psi_{nl}{(C_{1},\\rho_b)} + \\frac{\\rho_b}{C_{1}} and \\sigma_{x}{(G)} = \\sin{(G)} and \\sigma_{x}^{G}{(G)} = \\sin^{G}{(G)} and - 2 \\Psi_{nl}{(C_{1},\\rho_b)} = - \\Psi_{nl}{(C_{1},\\rho_b)} - \\frac{\\rho_b}{C_{1}} and (- \\Psi_{nl}{(C_{1},\\rho_b)} - \\frac{\\rho_b}{C_{1}}) \\sigma_{x}^{G}{(G)} = (- \\Psi_{nl}{(C_{1},\\rho_b)} - \\frac{\\rho_b}{C_{1}}) \\sin^{G}{(G)} and - 2 \\Psi_{nl}{(C_{1},\\rho_b)} \\sigma_{x}^{G}{(G)} = - 2 \\Psi_{nl}{(C_{1},\\rho_b)} \\sin^{G}{(G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))))"], ["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))), Pow(sin(Symbol('G', commutative=True)), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('C_1', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(z^{*},H)} = \\frac{z^{*}}{H} and \\sigma_{x}{(z^{*},H)} = \\frac{z^{*}}{H}, then obtain \\frac{z^{*} \\operatorname{J_{\\varepsilon}}{(z^{*},H)}}{H} = \\frac{(z^{*})^{2}}{H^{2}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(z^{*},H)} = \\frac{z^{*}}{H} and \\sigma_{x}{(z^{*},H)} = \\frac{z^{*}}{H} and \\sigma_{x}{(z^{*},H)} = \\operatorname{J_{\\varepsilon}}{(z^{*},H)} and \\frac{z^{*} \\sigma_{x}{(z^{*},H)}}{H} = \\frac{(z^{*})^{2}}{H^{2}} and \\frac{z^{*} \\operatorname{J_{\\varepsilon}}{(z^{*},H)}}{H} = \\frac{(z^{*})^{2}}{H^{2}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\sigma_x')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Symbol('z^*', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Symbol('z^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\lambda{(\\chi)} = e^{\\chi} and \\operatorname{P_{g}}{(\\chi)} = \\chi e^{\\chi}, then obtain \\frac{d}{d \\chi} \\int \\frac{d}{d \\chi} \\int \\chi e^{\\chi} d\\chi d\\chi = \\frac{d}{d \\chi} \\int \\frac{d}{d \\chi} \\int \\chi \\lambda{(\\chi)} d\\chi d\\chi", "derivation": "\\lambda{(\\chi)} = e^{\\chi} and \\operatorname{P_{g}}{(\\chi)} = \\chi e^{\\chi} and \\operatorname{P_{g}}{(\\chi)} = \\chi \\lambda{(\\chi)} and \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\int \\chi \\lambda{(\\chi)} d\\chi and \\frac{d}{d \\chi} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi = \\frac{d}{d \\chi} \\int \\chi \\lambda{(\\chi)} d\\chi and \\int \\frac{d}{d \\chi} \\int \\operatorname{P_{g}}{(\\chi)} d\\chi d\\chi = \\int \\frac{d}{d \\chi} \\int \\chi \\lambda{(\\chi)} d\\chi d\\chi and \\int \\frac{d}{d \\chi} \\int \\chi e^{\\chi} d\\chi d\\chi = \\int \\frac{d}{d \\chi} \\int \\chi \\lambda{(\\chi)} d\\chi d\\chi and \\frac{d}{d \\chi} \\int \\frac{d}{d \\chi} \\int \\chi e^{\\chi} d\\chi d\\chi = \\frac{d}{d \\chi} \\int \\frac{d}{d \\chi} \\int \\chi \\lambda{(\\chi)} d\\chi d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('P_g')(Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('P_g')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Integral(Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(Derivative(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\lambda')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{v},A,\\dot{z})} = A + \\dot{z} - \\mathbf{v}, then obtain A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})} + \\int (A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})}) dA = A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})} + \\int (2 A + \\dot{z} - \\mathbf{v}) dA", "derivation": "\\rho_{b}{(\\mathbf{v},A,\\dot{z})} = A + \\dot{z} - \\mathbf{v} and A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})} = 2 A + \\dot{z} - \\mathbf{v} and \\int (A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})}) dA = \\int (2 A + \\dot{z} - \\mathbf{v}) dA and 2 A + \\dot{z} - \\mathbf{v} + \\int (A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})}) dA = 2 A + \\dot{z} - \\mathbf{v} + \\int (2 A + \\dot{z} - \\mathbf{v}) dA and A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})} + \\int (A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})}) dA = A + \\rho_{b}{(\\mathbf{v},A,\\dot{z})} + \\int (2 A + \\dot{z} - \\mathbf{v}) dA", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 1, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["add", 3, "Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('A', commutative=True)))), Add(Symbol('A', commutative=True), Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('A', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given W{(I,t)} = I t, then obtain - I t - W{(I,t)} \\int W{(I,t)} dI - W{(I,t)} = - I t - W{(I,t)} \\int I t dI - W{(I,t)}", "derivation": "W{(I,t)} = I t and 2 W{(I,t)} = I t + W{(I,t)} and \\int W{(I,t)} dI = \\int I t dI and - W{(I,t)} \\int W{(I,t)} dI = - W{(I,t)} \\int I t dI and - W{(I,t)} \\int W{(I,t)} dI - 2 W{(I,t)} = - W{(I,t)} \\int I t dI - 2 W{(I,t)} and - I t - W{(I,t)} \\int W{(I,t)} dI - W{(I,t)} = - I t - W{(I,t)} \\int I t dI - W{(I,t)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))"], "Equality(Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["minus", 4, "Mul(Integer(2), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Integer(2), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Integer(2), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('I', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Function('W')(Symbol('I', commutative=True), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\psi,L)} = \\frac{\\psi}{L} and \\operatorname{V_{\\mathbf{E}}}{(L)} = \\frac{1}{L}, then obtain \\log{(\\int \\frac{1}{L \\operatorname{V_{\\mathbf{E}}}{(L)}} dL)} = \\log{(\\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL)}", "derivation": "\\bar{\\h}{(\\psi,L)} = \\frac{\\psi}{L} and 1 = \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} and \\int 1 dL = \\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL and \\log{(\\int 1 dL)} = \\log{(\\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL)} and \\operatorname{V_{\\mathbf{E}}}{(L)} = \\frac{1}{L} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(L)}}{\\log{(\\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL)}} = \\frac{1}{L \\log{(\\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL)}} and 1 = \\frac{1}{L \\operatorname{V_{\\mathbf{E}}}{(L)}} and \\int 1 dL = \\int \\frac{1}{L \\operatorname{V_{\\mathbf{E}}}{(L)}} dL and \\log{(\\int \\frac{1}{L \\operatorname{V_{\\mathbf{E}}}{(L)}} dL)} = \\log{(\\int \\frac{\\psi}{L \\bar{\\h}{(\\psi,L)}} dL)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('L', commutative=True))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True))))"], [["log", 3], "Equality(log(Integral(Integer(1), Tuple(Symbol('L', commutative=True)))), log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Integer(-1)))"], [["divide", 5, "log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True))))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Pow(log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))), Integer(-1))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))), Integer(-1))))"], [["divide", 6, "Mul(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Pow(log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Integer(-1))))"], [["integrate", 7, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('L', commutative=True))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 8], "Equality(log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))), log(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\psi', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given S{(\\Psi)} = e^{\\sin{(\\Psi)}}, then obtain e^{\\sin{(\\Psi)}} + \\int (\\frac{S{(\\Psi)}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi = e^{\\sin{(\\Psi)}} + \\int (\\frac{e^{\\sin{(\\Psi)}}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi", "derivation": "S{(\\Psi)} = e^{\\sin{(\\Psi)}} and \\frac{S{(\\Psi)}}{\\sin{(\\Psi)}} = \\frac{e^{\\sin{(\\Psi)}}}{\\sin{(\\Psi)}} and (\\frac{S{(\\Psi)}}{\\sin{(\\Psi)}})^{\\Psi} = (\\frac{e^{\\sin{(\\Psi)}}}{\\sin{(\\Psi)}})^{\\Psi} and \\int (\\frac{S{(\\Psi)}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi = \\int (\\frac{e^{\\sin{(\\Psi)}}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi and e^{\\sin{(\\Psi)}} + \\int (\\frac{S{(\\Psi)}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi = e^{\\sin{(\\Psi)}} + \\int (\\frac{e^{\\sin{(\\Psi)}}}{\\sin{(\\Psi)}})^{\\Psi} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Psi', commutative=True)), exp(sin(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Function('S')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(exp(sin(Symbol('\\\\Psi', commutative=True))), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Mul(Function('S')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)), Pow(Mul(exp(sin(Symbol('\\\\Psi', commutative=True))), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Pow(Mul(Function('S')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Mul(exp(sin(Symbol('\\\\Psi', commutative=True))), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["add", 4, "exp(sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(exp(sin(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Mul(Function('S')(Symbol('\\\\Psi', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(exp(sin(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Mul(exp(sin(Symbol('\\\\Psi', commutative=True))), Pow(sin(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(a,Q)} = Q + a, then derive 0 = 1 - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)}, then obtain - Q - a + \\hat{X}{(a,Q)} + 1 = - Q - a + \\hat{X}{(a,Q)} + (e^{- Q - a + \\hat{X}{(a,Q)} - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)} + 1})^{a}", "derivation": "\\hat{X}{(a,Q)} = Q + a and 0 = Q + a - \\hat{X}{(a,Q)} and \\frac{d}{d Q} 0 = \\frac{\\partial}{\\partial Q} (Q + a - \\hat{X}{(a,Q)}) and 0 = 1 - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)} and \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)} = Q + a - \\hat{X}{(a,Q)} + \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)} and 1 = e^{1 - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)}} and 1 = (e^{1 - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)}})^{a} and - Q - a + \\hat{X}{(a,Q)} + 1 = - Q - a + \\hat{X}{(a,Q)} + (e^{1 - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)}})^{a} and - Q - a + \\hat{X}{(a,Q)} + 1 = - Q - a + \\hat{X}{(a,Q)} + (e^{- Q - a + \\hat{X}{(a,Q)} - \\frac{\\partial}{\\partial Q} \\hat{X}{(a,Q)} + 1})^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('a', commutative=True)))"], [["minus", 1, "Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Symbol('Q', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["add", 2, "Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Add(Symbol('Q', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True))), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["exp", 4], "Equality(Integer(1), exp(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Integer(1), Pow(exp(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))), Symbol('a', commutative=True)))"], [["minus", 7, "Add(Symbol('Q', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Pow(exp(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('a', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(1))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\delta{(b,f^{*})} = \\int b f^{*} df^{*}, then obtain 2 b + 2 \\delta{(b,f^{*})} = 2 b + 2 \\int b f^{*} df^{*}", "derivation": "\\delta{(b,f^{*})} = \\int b f^{*} df^{*} and b + \\delta{(b,f^{*})} = b + \\int b f^{*} df^{*} and 2 b + 2 \\delta{(b,f^{*})} = 2 b + \\delta{(b,f^{*})} + \\int b f^{*} df^{*} and 2 b + 2 \\delta{(b,f^{*})} + \\int b f^{*} df^{*} = 2 b + \\delta{(b,f^{*})} + 2 \\int b f^{*} df^{*} and 2 b + \\delta{(b,f^{*})} + \\int b f^{*} df^{*} = 2 b + 2 \\int b f^{*} df^{*} and 2 b + 2 \\delta{(b,f^{*})} = 2 b + 2 \\int b f^{*} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True))), Add(Symbol('b', commutative=True), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["add", 2, "Add(Symbol('b', commutative=True), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('b', commutative=True)), Mul(Integer(2), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)))), Add(Mul(Integer(2), Symbol('b', commutative=True)), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["add", 1, "Add(Mul(Integer(2), Symbol('b', commutative=True)), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('b', commutative=True)), Mul(Integer(2), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Mul(Integer(2), Symbol('b', commutative=True)), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))))"], [["minus", 4, "Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('b', commutative=True)), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Mul(Integer(2), Symbol('b', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(2), Symbol('b', commutative=True)), Mul(Integer(2), Function('\\\\delta')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)))), Add(Mul(Integer(2), Symbol('b', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))))"]]}, {"prompt": "Given L{(k)} = \\sin{(e^{k})}, then obtain (2 L{(k)} + \\sin{(e^{k})}) L{(k)} + \\sin{(e^{k})} = (L{(k)} + 2 \\sin{(e^{k})}) L{(k)} + \\sin{(e^{k})}", "derivation": "L{(k)} = \\sin{(e^{k})} and L{(k)} + \\sin{(e^{k})} = 2 \\sin{(e^{k})} and L{(k)} + 2 \\sin{(e^{k})} = 3 \\sin{(e^{k})} and 2 L{(k)} + \\sin{(e^{k})} = 3 \\sin{(e^{k})} and 2 L{(k)} + \\sin{(e^{k})} = L{(k)} + 2 \\sin{(e^{k})} and (2 L{(k)} + \\sin{(e^{k})}) L{(k)} = (L{(k)} + 2 \\sin{(e^{k})}) L{(k)} and (2 L{(k)} + \\sin{(e^{k})}) L{(k)} + \\sin{(e^{k})} = (L{(k)} + 2 \\sin{(e^{k})}) L{(k)} + \\sin{(e^{k})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('k', commutative=True)), sin(exp(Symbol('k', commutative=True))))"], [["add", 1, "sin(exp(Symbol('k', commutative=True)))"], "Equality(Add(Function('L')(Symbol('k', commutative=True)), sin(exp(Symbol('k', commutative=True)))), Mul(Integer(2), sin(exp(Symbol('k', commutative=True)))))"], [["add", 1, "Mul(Integer(2), sin(exp(Symbol('k', commutative=True))))"], "Equality(Add(Function('L')(Symbol('k', commutative=True)), Mul(Integer(2), sin(exp(Symbol('k', commutative=True))))), Mul(Integer(3), sin(exp(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))), Mul(Integer(3), sin(exp(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))), Add(Function('L')(Symbol('k', commutative=True)), Mul(Integer(2), sin(exp(Symbol('k', commutative=True))))))"], [["times", 5, "Function('L')(Symbol('k', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))), Function('L')(Symbol('k', commutative=True))), Mul(Add(Function('L')(Symbol('k', commutative=True)), Mul(Integer(2), sin(exp(Symbol('k', commutative=True))))), Function('L')(Symbol('k', commutative=True))))"], [["minus", 6, "Mul(Integer(-1), sin(exp(Symbol('k', commutative=True))))"], "Equality(Add(Mul(Add(Mul(Integer(2), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))), Add(Mul(Add(Function('L')(Symbol('k', commutative=True)), Mul(Integer(2), sin(exp(Symbol('k', commutative=True))))), Function('L')(Symbol('k', commutative=True))), sin(exp(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(U)} = \\cos{(U)} and \\Psi_{nl}{(U,\\mathbf{E})} = \\mathbf{E} + \\sin{(U)}, then derive \\int \\lambda{(U)} dU = \\mathbf{E} + \\sin{(U)}, then obtain \\Psi_{nl}{(U,\\mathbf{E})} = \\int \\cos{(U)} dU", "derivation": "\\lambda{(U)} = \\cos{(U)} and \\int \\lambda{(U)} dU = \\int \\cos{(U)} dU and \\int \\lambda{(U)} dU = \\mathbf{E} + \\sin{(U)} and \\mathbf{E} + \\sin{(U)} = \\int \\cos{(U)} dU and \\Psi_{nl}{(U,\\mathbf{E})} = \\mathbf{E} + \\sin{(U)} and \\Psi_{nl}{(U,\\mathbf{E})} = \\int \\cos{(U)} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), sin(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\Psi_{nl}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given H{(t,v_{t})} = t + \\log{(v_{t})}, then obtain H{(t,v_{t})} - \\int \\log{(v_{t})} dv_{t} = t + \\log{(v_{t})} - \\int \\log{(v_{t})} dv_{t}", "derivation": "H{(t,v_{t})} = t + \\log{(v_{t})} and - t + H{(t,v_{t})} = \\log{(v_{t})} and \\int (- t + H{(t,v_{t})}) dv_{t} = \\int \\log{(v_{t})} dv_{t} and H{(t,v_{t})} - \\int (- t + H{(t,v_{t})}) dv_{t} = t + \\log{(v_{t})} - \\int (- t + H{(t,v_{t})}) dv_{t} and H{(t,v_{t})} - \\int \\log{(v_{t})} dv_{t} = t + \\log{(v_{t})} - \\int \\log{(v_{t})} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('t', commutative=True), log(Symbol('v_t', commutative=True))))"], [["minus", 1, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True))), log(Symbol('v_t', commutative=True)))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["minus", 1, "Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Add(Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))), Add(Symbol('t', commutative=True), log(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('H')(Symbol('t', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Add(Symbol('t', commutative=True), log(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\pi)} = \\log{(e^{\\pi})}, then obtain \\frac{d}{d \\pi} (\\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int \\operatorname{J_{\\varepsilon}}{(\\pi)} e^{- \\pi} d\\pi) = \\frac{d}{d \\pi} (\\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int e^{- \\pi} \\log{(e^{\\pi})} d\\pi)", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\pi)} = \\log{(e^{\\pi})} and \\operatorname{J_{\\varepsilon}}{(\\pi)} e^{- \\pi} = e^{- \\pi} \\log{(e^{\\pi})} and \\int \\operatorname{J_{\\varepsilon}}{(\\pi)} e^{- \\pi} d\\pi = \\int e^{- \\pi} \\log{(e^{\\pi})} d\\pi and \\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int \\operatorname{J_{\\varepsilon}}{(\\pi)} e^{- \\pi} d\\pi = \\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int e^{- \\pi} \\log{(e^{\\pi})} d\\pi and \\frac{d}{d \\pi} (\\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int \\operatorname{J_{\\varepsilon}}{(\\pi)} e^{- \\pi} d\\pi) = \\frac{d}{d \\pi} (\\operatorname{J_{\\varepsilon}}{(\\pi)} + \\int e^{- \\pi} \\log{(e^{\\pi})} d\\pi)", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(exp(Symbol('\\\\pi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 3, "Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(Mul(exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Integral(Mul(exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(C)} = \\cos{(C)}, then derive \\frac{d}{d C} \\dot{x}{(C)} = - \\sin{(C)}, then obtain \\sin{(C)} + \\frac{d}{d C} \\dot{x}{(C)} = 0", "derivation": "\\dot{x}{(C)} = \\cos{(C)} and \\frac{d}{d C} \\dot{x}{(C)} = \\frac{d}{d C} \\cos{(C)} and \\frac{d}{d C} \\dot{x}{(C)} = - \\sin{(C)} and \\sin{(C)} + \\frac{d}{d C} \\dot{x}{(C)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), sin(Symbol('C', commutative=True)))"], "Equality(Add(sin(Symbol('C', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\hat{H}{(Z,p)} = \\int Z^{p} dp, then obtain ((- p + \\hat{H}{(Z,p)} \\int Z^{p} dp)^{p}) \\hat{H}{(Z,p)} \\int Z^{p} dp = ((- p + (\\int Z^{p} dp)^{2})^{p}) \\hat{H}{(Z,p)} \\int Z^{p} dp", "derivation": "\\hat{H}{(Z,p)} = \\int Z^{p} dp and \\hat{H}{(Z,p)} \\int Z^{p} dp = (\\int Z^{p} dp)^{2} and - p + \\hat{H}{(Z,p)} \\int Z^{p} dp = - p + (\\int Z^{p} dp)^{2} and (- p + \\hat{H}{(Z,p)} \\int Z^{p} dp)^{p} = (- p + (\\int Z^{p} dp)^{2})^{p} and ((- p + \\hat{H}{(Z,p)} \\int Z^{p} dp)^{p}) \\hat{H}{(Z,p)} \\int Z^{p} dp = ((- p + (\\int Z^{p} dp)^{2})^{p}) \\hat{H}{(Z,p)} \\int Z^{p} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["times", 1, "Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Pow(Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(2)))"], [["minus", 2, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(2))))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(2))), Symbol('p', commutative=True)))"], [["times", 4, "Mul(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Symbol('p', commutative=True)), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(2))), Symbol('p', commutative=True)), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\eta)} = e^{\\eta} and \\mu_{0}{(\\eta)} = \\operatorname{E_{x}}{(\\eta)} e^{\\eta}, then obtain \\operatorname{E_{x}}^{2}{(\\eta)} = e^{2 \\eta}", "derivation": "\\operatorname{E_{x}}{(\\eta)} = e^{\\eta} and \\operatorname{E_{x}}^{2}{(\\eta)} = \\operatorname{E_{x}}{(\\eta)} e^{\\eta} and \\mu_{0}{(\\eta)} = \\operatorname{E_{x}}{(\\eta)} e^{\\eta} and \\mu_{0}{(\\eta)} = e^{2 \\eta} and \\operatorname{E_{x}}^{2}{(\\eta)} = \\mu_{0}{(\\eta)} and \\operatorname{E_{x}}^{2}{(\\eta)} = e^{2 \\eta}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('\\\\eta', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('\\\\eta', commutative=True)), Integer(2)), Mul(Function('E_x')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\eta', commutative=True)), Mul(Function('E_x')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mu_0')(Symbol('\\\\eta', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('E_x')(Symbol('\\\\eta', commutative=True)), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('E_x')(Symbol('\\\\eta', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(F_{g},y^{\\prime})} = F_{g} + y^{\\prime}, then obtain \\frac{\\partial}{\\partial F_{g}} (F_{g} (- y^{\\prime} + \\operatorname{v_{z}}{(F_{g},y^{\\prime})}) + y^{\\prime}) = \\frac{\\partial}{\\partial F_{g}} (F_{g}^{2} + y^{\\prime})", "derivation": "\\operatorname{v_{z}}{(F_{g},y^{\\prime})} = F_{g} + y^{\\prime} and - y^{\\prime} + \\operatorname{v_{z}}{(F_{g},y^{\\prime})} = F_{g} and F_{g} (- y^{\\prime} + \\operatorname{v_{z}}{(F_{g},y^{\\prime})}) = F_{g}^{2} and F_{g} (- y^{\\prime} + \\operatorname{v_{z}}{(F_{g},y^{\\prime})}) + y^{\\prime} = F_{g}^{2} + y^{\\prime} and \\frac{\\partial}{\\partial F_{g}} (F_{g} (- y^{\\prime} + \\operatorname{v_{z}}{(F_{g},y^{\\prime})}) + y^{\\prime}) = \\frac{\\partial}{\\partial F_{g}} (F_{g}^{2} + y^{\\prime})", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('v_z')(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('F_g', commutative=True))"], [["times", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('v_z')(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Pow(Symbol('F_g', commutative=True), Integer(2)))"], [["add", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('v_z')(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Add(Pow(Symbol('F_g', commutative=True), Integer(2)), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 4, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('v_z')(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('F_g', commutative=True), Integer(2)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(A_{z})} = \\cos{(A_{z})}, then obtain - \\sin{(A_{z})} \\frac{d^{2}}{d A_{z}^{2}} \\mathbb{I}{(A_{z})} - \\cos{(A_{z})} \\frac{d}{d A_{z}} \\mathbb{I}{(A_{z})} = 2 \\sin{(A_{z})} \\cos{(A_{z})}", "derivation": "\\mathbb{I}{(A_{z})} = \\cos{(A_{z})} and \\frac{d}{d A_{z}} \\mathbb{I}{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})} and \\frac{d}{d A_{z}} \\mathbb{I}{(A_{z})} \\frac{d}{d A_{z}} \\cos{(A_{z})} = (\\frac{d}{d A_{z}} \\cos{(A_{z})})^{2} and \\frac{d}{d A_{z}} \\frac{d}{d A_{z}} \\mathbb{I}{(A_{z})} \\frac{d}{d A_{z}} \\cos{(A_{z})} = \\frac{d}{d A_{z}} (\\frac{d}{d A_{z}} \\cos{(A_{z})})^{2} and - \\sin{(A_{z})} \\frac{d^{2}}{d A_{z}^{2}} \\mathbb{I}{(A_{z})} - \\cos{(A_{z})} \\frac{d}{d A_{z}} \\mathbb{I}{(A_{z})} = 2 \\sin{(A_{z})} \\cos{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["times", 2, "Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('A_z', commutative=True)), Derivative(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2)))), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)), Derivative(Function('\\\\mathbb{I}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))), Mul(Integer(2), sin(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given Z{(\\omega,c,\\eta)} = \\eta \\omega c, then obtain (\\frac{\\eta \\omega c}{Z{(\\omega,c,\\eta)}} + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)} = (\\frac{2 \\eta \\omega c}{Z{(\\omega,c,\\eta)}} - 1 + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)}", "derivation": "Z{(\\omega,c,\\eta)} = \\eta \\omega c and 1 = \\frac{\\eta \\omega c}{Z{(\\omega,c,\\eta)}} and 1 + \\frac{1}{Z{(\\omega,c,\\eta)}} = \\frac{\\eta \\omega c}{Z{(\\omega,c,\\eta)}} + \\frac{1}{Z{(\\omega,c,\\eta)}} and (1 + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)} = (\\frac{\\eta \\omega c}{Z{(\\omega,c,\\eta)}} + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)} and (\\frac{\\eta \\omega c}{Z{(\\omega,c,\\eta)}} + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)} = (\\frac{2 \\eta \\omega c}{Z{(\\omega,c,\\eta)}} - 1 + \\frac{1}{Z{(\\omega,c,\\eta)}}) Z{(\\omega,c,\\eta)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"], [["add", 2, "Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"], [["times", 3, "Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Add(Integer(1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('\\\\eta', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(-1), Pow(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(c_{0},v)} = \\log{(c_{0} + v)} and y{(v)} = - v, then obtain \\frac{y{(v)}}{- v + \\bar{\\h}{(c_{0},v)}} = - \\frac{v - \\bar{\\h}{(c_{0},v)} + \\log{(c_{0} + v)}}{- v + \\bar{\\h}{(c_{0},v)} - \\log{(c_{0} + v)} + \\log{(c_{0} + v - \\bar{\\h}{(c_{0},v)} + \\log{(c_{0} + v)})}}", "derivation": "\\bar{\\h}{(c_{0},v)} = \\log{(c_{0} + v)} and - v + \\bar{\\h}{(c_{0},v)} = - v + \\log{(c_{0} + v)} and - v + \\bar{\\h}{(c_{0},v)} - \\log{(c_{0} + v)} = - v and y{(v)} = - v and \\frac{y{(v)}}{- v + \\log{(c_{0} + v)}} = - \\frac{v}{- v + \\log{(c_{0} + v)}} and \\frac{y{(v)}}{- v + \\bar{\\h}{(c_{0},v)}} = - \\frac{v - \\bar{\\h}{(c_{0},v)} + \\log{(c_{0} + v)}}{- v + \\bar{\\h}{(c_{0},v)} - \\log{(c_{0} + v)} + \\log{(c_{0} + v - \\bar{\\h}{(c_{0},v)} + \\log{(c_{0} + v)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True))))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))))"], [["minus", 2, "log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True))))), Mul(Integer(-1), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('v', commutative=True)), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))), Integer(-1)), Function('y')(Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True))), Integer(-1)), Function('y')(Symbol('v', commutative=True))), Mul(Integer(-1), Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True))), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\hbar')(Symbol('c_0', commutative=True), Symbol('v', commutative=True))), log(Add(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))))), Integer(-1))))"]]}, {"prompt": "Given \\rho{(\\hat{x}_0)} = \\hat{x}_0, then obtain 0 = (\\hat{x}_0 - \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)})^{\\mathbf{J}_P} - (- \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)} + \\rho{(\\hat{x}_0)})^{\\mathbf{J}_P}", "derivation": "\\rho{(\\hat{x}_0)} = \\hat{x}_0 and - \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)} + \\rho{(\\hat{x}_0)} = \\hat{x}_0 - \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)} and (- \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)} + \\rho{(\\hat{x}_0)})^{\\mathbf{J}_P} = (\\hat{x}_0 - \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)})^{\\mathbf{J}_P} and 0 = (\\hat{x}_0 - \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)})^{\\mathbf{J}_P} - (- \\hat{H}_l{(\\hat{x}_0,\\mathbf{J}_P,p)} + \\rho{(\\hat{x}_0)})^{\\mathbf{J}_P}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], [["minus", 1, "Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Function('\\\\rho')(Symbol('\\\\hat{x}_0', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Function('\\\\rho')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Function('\\\\rho')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Function('\\\\rho')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\phi_2,m)} = m + \\cos{(\\phi_2)}, then obtain (\\int ((A_{x} + \\mathbf{J}_M) \\rho_{f}{(\\phi_2,m)})^{\\mathbf{J}_M} dA_{x})^{\\phi_2} = (\\int ((A_{x} + \\mathbf{J}_M) (m + \\cos{(\\phi_2)}))^{\\mathbf{J}_M} dA_{x})^{\\phi_2}", "derivation": "\\rho_{f}{(\\phi_2,m)} = m + \\cos{(\\phi_2)} and (A_{x} + \\mathbf{J}_M) \\rho_{f}{(\\phi_2,m)} = (A_{x} + \\mathbf{J}_M) (m + \\cos{(\\phi_2)}) and ((A_{x} + \\mathbf{J}_M) \\rho_{f}{(\\phi_2,m)})^{\\mathbf{J}_M} = ((A_{x} + \\mathbf{J}_M) (m + \\cos{(\\phi_2)}))^{\\mathbf{J}_M} and \\int ((A_{x} + \\mathbf{J}_M) \\rho_{f}{(\\phi_2,m)})^{\\mathbf{J}_M} dA_{x} = \\int ((A_{x} + \\mathbf{J}_M) (m + \\cos{(\\phi_2)}))^{\\mathbf{J}_M} dA_{x} and (\\int ((A_{x} + \\mathbf{J}_M) \\rho_{f}{(\\phi_2,m)})^{\\mathbf{J}_M} dA_{x})^{\\phi_2} = (\\int ((A_{x} + \\mathbf{J}_M) (m + \\cos{(\\phi_2)}))^{\\mathbf{J}_M} dA_{x})^{\\phi_2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True), Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True), Symbol('m', commutative=True))), Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('\\\\phi_2', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True), Symbol('m', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["integrate", 3, "Symbol('A_x', commutative=True)"], "Equality(Integral(Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True), Symbol('m', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Integral(Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True), Symbol('m', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Integral(Pow(Mul(Add(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given L{(E)} = \\sin{(E)}, then derive \\int L{(E)} dE = \\mu - \\cos{(E)}, then obtain (\\frac{d}{d E} (\\int L{(E)} dE)^{E}) \\int (L{(E)} + \\sin{(E)}) dE = \\frac{\\partial}{\\partial E} (\\mu - \\cos{(E)})^{E} \\int (L{(E)} + \\sin{(E)}) dE", "derivation": "L{(E)} = \\sin{(E)} and \\int L{(E)} dE = \\int \\sin{(E)} dE and \\int L{(E)} dE = \\mu - \\cos{(E)} and (\\int L{(E)} dE)^{E} = (\\int \\sin{(E)} dE)^{E} and \\mu - \\cos{(E)} = \\int \\sin{(E)} dE and \\frac{d}{d E} (\\int L{(E)} dE)^{E} = \\frac{d}{d E} (\\int \\sin{(E)} dE)^{E} and (\\frac{d}{d E} (\\int L{(E)} dE)^{E}) \\int (L{(E)} + \\sin{(E)}) dE = (\\frac{d}{d E} (\\int \\sin{(E)} dE)^{E}) \\int (L{(E)} + \\sin{(E)}) dE and (\\frac{d}{d E} (\\int L{(E)} dE)^{E}) \\int (L{(E)} + \\sin{(E)}) dE = \\frac{\\partial}{\\partial E} (\\mu - \\cos{(E)})^{E} \\int (L{(E)} + \\sin{(E)}) dE", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(sin(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('E', commutative=True)))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(sin(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('E', commutative=True)))), Integral(sin(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["times", 6, "Integral(Add(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))"], "Equality(Mul(Derivative(Pow(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integral(Add(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Mul(Derivative(Pow(Integral(sin(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integral(Add(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Derivative(Pow(Integral(Function('L')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integral(Add(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Mul(Derivative(Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integral(Add(Function('L')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given x{(J,v)} = J - v, then derive \\frac{\\partial}{\\partial v} \\int J x{(J,v)} dv = \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + \\mathbf{s}), then derive \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + x) = \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + \\mathbf{s}), then obtain J^{2} - J v = \\frac{\\partial}{\\partial v} \\int J (J - v) dv", "derivation": "x{(J,v)} = J - v and J x{(J,v)} = J (J - v) and \\int J x{(J,v)} dv = \\int J (J - v) dv and \\frac{\\partial}{\\partial v} \\int J x{(J,v)} dv = \\frac{\\partial}{\\partial v} \\int J (J - v) dv and \\frac{\\partial}{\\partial v} \\int J x{(J,v)} dv = \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + \\mathbf{s}) and \\frac{\\partial}{\\partial v} \\int J (J - v) dv = \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + \\mathbf{s}) and \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + x) = \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + \\mathbf{s}) and \\frac{\\partial}{\\partial v} (J^{2} v - \\frac{J v^{2}}{2} + x) = \\frac{\\partial}{\\partial v} \\int J (J - v) dv and J^{2} - J v = \\frac{\\partial}{\\partial v} \\int J (J - v) dv", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('J', commutative=True), Symbol('v', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('x')(Symbol('J', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Symbol('J', commutative=True), Function('x')(Symbol('J', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('J', commutative=True), Function('x')(Symbol('J', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Symbol('J', commutative=True), Function('x')(Symbol('J', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('x', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('J', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('x', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 8], "Equality(Add(Pow(Symbol('J', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('v', commutative=True))), Derivative(Integral(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(C_{d},\\eta)} = - C_{d} + \\eta and \\operatorname{A_{z}}{(C_{d})} = - C_{d}, then obtain 2 \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d} = - \\frac{C_{d}^{2}}{2} + C_{d} \\eta + E_{\\lambda} + \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d}", "derivation": "Z{(C_{d},\\eta)} = - C_{d} + \\eta and \\int Z{(C_{d},\\eta)} dC_{d} = \\int (- C_{d} + \\eta) dC_{d} and \\operatorname{A_{z}}{(C_{d})} = - C_{d} and 2 \\int Z{(C_{d},\\eta)} dC_{d} = \\int (- C_{d} + \\eta) dC_{d} + \\int Z{(C_{d},\\eta)} dC_{d} and Z{(C_{d},\\eta)} = \\eta + \\operatorname{A_{z}}{(C_{d})} and 2 \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d} = \\int (- C_{d} + \\eta) dC_{d} + \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d} and 2 \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d} = - \\frac{C_{d}^{2}}{2} + C_{d} \\eta + E_{\\lambda} + \\int (\\eta + \\operatorname{A_{z}}{(C_{d})}) dC_{d}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True)))"], [["add", 2, "Integral(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('Z')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Function('A_z')(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Integral(Add(Symbol('\\\\eta', commutative=True), Function('A_z')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Symbol('\\\\eta', commutative=True), Function('A_z')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Integer(2), Integral(Add(Symbol('\\\\eta', commutative=True), Function('A_z')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Mul(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True), Integral(Add(Symbol('\\\\eta', commutative=True), Function('A_z')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{v},W,\\mathbf{J}_P)} = (W \\mathbf{J}_P)^{\\mathbf{v}} and E{(m,\\sigma_p)} = \\log{(\\sigma_p m)}, then obtain \\operatorname{t_{2}}^{- 2 \\mathbf{J}_P}{(\\mathbf{v},W,\\mathbf{J}_P)} \\frac{\\partial}{\\partial m} E{(m,\\sigma_p)} = \\operatorname{t_{2}}^{- 2 \\mathbf{J}_P}{(\\mathbf{v},W,\\mathbf{J}_P)} \\frac{\\partial}{\\partial m} \\log{(\\sigma_p m)}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{v},W,\\mathbf{J}_P)} = (W \\mathbf{J}_P)^{\\mathbf{v}} and E{(m,\\sigma_p)} = \\log{(\\sigma_p m)} and \\frac{\\partial}{\\partial m} E{(m,\\sigma_p)} = \\frac{\\partial}{\\partial m} \\log{(\\sigma_p m)} and ((W \\mathbf{J}_P)^{\\mathbf{v}})^{- 2 \\mathbf{J}_P} \\frac{\\partial}{\\partial m} E{(m,\\sigma_p)} = ((W \\mathbf{J}_P)^{\\mathbf{v}})^{- 2 \\mathbf{J}_P} \\frac{\\partial}{\\partial m} \\log{(\\sigma_p m)} and \\operatorname{t_{2}}^{- 2 \\mathbf{J}_P}{(\\mathbf{v},W,\\mathbf{J}_P)} \\frac{\\partial}{\\partial m} E{(m,\\sigma_p)} = \\operatorname{t_{2}}^{- 2 \\mathbf{J}_P}{(\\mathbf{v},W,\\mathbf{J}_P)} \\frac{\\partial}{\\partial m} \\log{(\\sigma_p m)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], ["get_premise", "Equality(Function('E')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["divide", 3, "Pow(Pow(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Pow(Pow(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(Function('E')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Pow(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(log(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(Function('E')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Function('t_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(log(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain - \\sigma_p \\cos{(\\sigma_p)} + \\operatorname{C_{d}}{(\\sigma_p)} = - \\sigma_p \\cos{(\\sigma_p)} + \\cos{(\\sigma_p)}", "derivation": "\\operatorname{C_{d}}{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\sigma_p \\operatorname{C_{d}}{(\\sigma_p)} = \\sigma_p \\cos{(\\sigma_p)} and - \\sigma_p \\operatorname{C_{d}}{(\\sigma_p)} + \\operatorname{C_{d}}{(\\sigma_p)} = - \\sigma_p \\operatorname{C_{d}}{(\\sigma_p)} + \\cos{(\\sigma_p)} and - \\sigma_p \\cos{(\\sigma_p)} + \\operatorname{C_{d}}{(\\sigma_p)} = - \\sigma_p \\cos{(\\sigma_p)} + \\cos{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\sigma_p', commutative=True), Function('C_d')(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(n_{2},\\psi^*)} = \\int \\frac{n_{2}}{\\psi^*} dn_{2}, then obtain 0 = \\frac{\\partial}{\\partial \\psi^*} \\frac{\\int \\frac{n_{2}}{\\psi^*} dn_{2}}{\\operatorname{v_{2}}{(n_{2},\\psi^*)}}", "derivation": "\\operatorname{v_{2}}{(n_{2},\\psi^*)} = \\int \\frac{n_{2}}{\\psi^*} dn_{2} and \\frac{n_{2} \\operatorname{v_{2}}{(n_{2},\\psi^*)}}{\\psi^*} = \\frac{n_{2} \\int \\frac{n_{2}}{\\psi^*} dn_{2}}{\\psi^*} and \\frac{\\operatorname{v_{2}}{(n_{2},\\psi^*)}}{\\int \\frac{n_{2}}{\\psi^*} dn_{2}} = 1 and 1 = \\frac{\\int \\frac{n_{2}}{\\psi^*} dn_{2}}{\\operatorname{v_{2}}{(n_{2},\\psi^*)}} and \\frac{d}{d \\psi^*} 1 = \\frac{\\partial}{\\partial \\psi^*} \\frac{\\int \\frac{n_{2}}{\\psi^*} dn_{2}}{\\operatorname{v_{2}}{(n_{2},\\psi^*)}} and 0 = \\frac{\\partial}{\\partial \\psi^*} \\frac{\\int \\frac{n_{2}}{\\psi^*} dn_{2}}{\\operatorname{v_{2}}{(n_{2},\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], "Equality(Mul(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 3, "Mul(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Derivative(Mul(Pow(Function('v_2')(Symbol('n_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain (\\frac{d}{d \\tilde{g}} \\operatorname{m_{s}}{(\\tilde{g})})^{\\tilde{g}} = (- \\sin{(\\tilde{g})})^{\\tilde{g}}", "derivation": "\\operatorname{m_{s}}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\operatorname{m_{s}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\cos{(\\tilde{g})} and (\\frac{d}{d \\tilde{g}} \\operatorname{m_{s}}{(\\tilde{g})})^{\\tilde{g}} = (\\frac{d}{d \\tilde{g}} \\cos{(\\tilde{g})})^{\\tilde{g}} and (\\frac{d}{d \\tilde{g}} \\operatorname{m_{s}}{(\\tilde{g})})^{\\tilde{g}} = (- \\sin{(\\tilde{g})})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Derivative(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(U)} = \\sin{(U)}, then obtain 0 = - \\frac{\\mathbf{g}{(U)}}{U} + \\frac{\\sin{(U)}}{U}", "derivation": "\\mathbf{g}{(U)} = \\sin{(U)} and \\frac{\\mathbf{g}{(U)}}{U} = \\frac{\\sin{(U)}}{U} and \\mathbf{g}{(U)} + \\frac{\\mathbf{g}{(U)}}{U} = \\mathbf{g}{(U)} + \\frac{\\sin{(U)}}{U} and 0 = - \\frac{\\mathbf{g}{(U)}}{U} + \\frac{\\sin{(U)}}{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["divide", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('U', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{g}')(Symbol('U', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('U', commutative=True)))), Add(Function('\\\\mathbf{g}')(Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), sin(Symbol('U', commutative=True)))))"], [["minus", 3, "Add(Function('\\\\mathbf{g}')(Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('U', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('U', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), sin(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given C{(v_{z})} = \\log{(v_{z})} and \\operatorname{E_{x}}{(v_{z})} = - \\log{(v_{z})}, then obtain \\int\\limits^{e^{C{(v_{z})}}} \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\int\\limits^{e^{C{(v_{z})}}} - \\log{(v_{z})} dv_{z}", "derivation": "C{(v_{z})} = \\log{(v_{z})} and e^{C{(v_{z})}} = v_{z} and \\operatorname{E_{x}}{(v_{z})} = - \\log{(v_{z})} and \\int \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\int - \\log{(v_{z})} dv_{z} and \\int\\limits^{e^{C{(v_{z})}}} \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\int\\limits^{e^{C{(v_{z})}}} - \\log{(v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["exp", 1], "Equality(exp(Function('C')(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(Symbol('v_z', commutative=True))))"], [["integrate", 3, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Mul(Integer(-1), log(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('E_x')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), exp(Function('C')(Symbol('v_z', commutative=True))))), Integral(Mul(Integer(-1), log(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), exp(Function('C')(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(g,n_{1})} = \\frac{\\cos{(g)}}{n_{1}}, then obtain \\mathbf{P}{(g,n_{1})} + \\mathbf{P}^{g}{(g,n_{1})} - \\frac{\\cos{(g)}}{n_{1}} = (\\frac{\\cos{(g)}}{n_{1}})^{g}", "derivation": "\\mathbf{P}{(g,n_{1})} = \\frac{\\cos{(g)}}{n_{1}} and \\mathbf{P}{(g,n_{1})} - \\frac{\\cos{(g)}}{n_{1}} = 0 and \\mathbf{P}^{g}{(g,n_{1})} = (\\frac{\\cos{(g)}}{n_{1}})^{g} and \\mathbf{P}{(g,n_{1})} + \\mathbf{P}^{g}{(g,n_{1})} - \\frac{\\cos{(g)}}{n_{1}} = \\mathbf{P}^{g}{(g,n_{1})} and \\mathbf{P}{(g,n_{1})} + \\mathbf{P}^{g}{(g,n_{1})} - \\frac{\\cos{(g)}}{n_{1}} = (\\frac{\\cos{(g)}}{n_{1}})^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Integer(0))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["add", 2, "Pow(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Symbol('g', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Pow(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))), Pow(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\mu,\\hat{x},\\mathbf{g})} = \\hat{x}^{\\mu} \\mathbf{g}, then obtain ((\\mathbf{P}^{\\hat{x}}{(\\mu,\\hat{x},\\mathbf{g})})^{\\mu})^{\\hat{x}} = (((\\hat{x}^{\\mu} \\mathbf{g})^{\\hat{x}})^{\\mu})^{\\hat{x}}", "derivation": "\\mathbf{P}{(\\mu,\\hat{x},\\mathbf{g})} = \\hat{x}^{\\mu} \\mathbf{g} and \\mathbf{P}^{\\hat{x}}{(\\mu,\\hat{x},\\mathbf{g})} = (\\hat{x}^{\\mu} \\mathbf{g})^{\\hat{x}} and (\\mathbf{P}^{\\hat{x}}{(\\mu,\\hat{x},\\mathbf{g})})^{\\mu} = ((\\hat{x}^{\\mu} \\mathbf{g})^{\\hat{x}})^{\\mu} and ((\\mathbf{P}^{\\hat{x}}{(\\mu,\\hat{x},\\mathbf{g})})^{\\mu})^{\\hat{x}} = (((\\hat{x}^{\\mu} \\mathbf{g})^{\\hat{x}})^{\\mu})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Pow(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\hat{H},\\phi)} = \\hat{H} e^{\\phi}, then derive e^{\\phi} + \\frac{\\partial}{\\partial \\hat{H}} \\mathbf{B}{(\\hat{H},\\phi)} = 2 e^{\\phi}, then obtain e^{\\phi} + \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} e^{\\phi} = 2 e^{\\phi}", "derivation": "\\mathbf{B}{(\\hat{H},\\phi)} = \\hat{H} e^{\\phi} and \\hat{H} e^{\\phi} + \\mathbf{B}{(\\hat{H},\\phi)} = 2 \\hat{H} e^{\\phi} and \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} e^{\\phi} + \\mathbf{B}{(\\hat{H},\\phi)}) = \\frac{\\partial}{\\partial \\hat{H}} 2 \\hat{H} e^{\\phi} and e^{\\phi} + \\frac{\\partial}{\\partial \\hat{H}} \\mathbf{B}{(\\hat{H},\\phi)} = 2 e^{\\phi} and e^{\\phi} + \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} e^{\\phi} = 2 e^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(exp(Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\nabla{(r,\\rho)} = \\cos{(\\frac{\\rho}{r})}, then obtain \\cos{(\\frac{\\rho}{r})} \\iint \\nabla{(r,\\rho)} d\\rho dr = \\cos{(\\frac{\\rho}{r})} \\iint \\cos{(\\frac{\\rho}{r})} d\\rho dr", "derivation": "\\nabla{(r,\\rho)} = \\cos{(\\frac{\\rho}{r})} and \\int \\nabla{(r,\\rho)} d\\rho = \\int \\cos{(\\frac{\\rho}{r})} d\\rho and \\iint \\nabla{(r,\\rho)} d\\rho dr = \\iint \\cos{(\\frac{\\rho}{r})} d\\rho dr and \\cos{(\\frac{\\rho}{r})} \\iint \\nabla{(r,\\rho)} d\\rho dr = \\cos{(\\frac{\\rho}{r})} \\iint \\cos{(\\frac{\\rho}{r})} d\\rho dr", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["times", 3, "cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], "Equality(Mul(cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Integral(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Integral(cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain \\frac{\\frac{d}{d \\hat{H}} \\operatorname{n_{1}}^{2}{(\\hat{H})}}{\\frac{d}{d \\hat{H}} \\operatorname{n_{1}}{(\\hat{H})} \\cos{(\\hat{H})}} = 1", "derivation": "\\operatorname{n_{1}}{(\\hat{H})} = \\cos{(\\hat{H})} and \\operatorname{n_{1}}^{2}{(\\hat{H})} = \\operatorname{n_{1}}{(\\hat{H})} \\cos{(\\hat{H})} and \\frac{d}{d \\hat{H}} \\operatorname{n_{1}}^{2}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\operatorname{n_{1}}{(\\hat{H})} \\cos{(\\hat{H})} and \\frac{\\frac{d}{d \\hat{H}} \\operatorname{n_{1}}^{2}{(\\hat{H})}}{\\frac{d}{d \\hat{H}} \\operatorname{n_{1}}{(\\hat{H})} \\cos{(\\hat{H})}} = 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["times", 1, "Function('n_1')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Pow(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Pow(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Mul(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Mul(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(-1)), Derivative(Pow(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(z)} = \\log{(z)} and \\mathbf{H}{(z)} = \\operatorname{v_{2}}{(z)} + \\log{(z)}, then obtain 4 \\log{(z)}^{2} = 2 (\\operatorname{v_{2}}{(z)} + \\log{(z)}) \\log{(z)}", "derivation": "\\operatorname{v_{2}}{(z)} = \\log{(z)} and 2 \\operatorname{v_{2}}{(z)} = \\operatorname{v_{2}}{(z)} + \\log{(z)} and \\mathbf{H}{(z)} = \\operatorname{v_{2}}{(z)} + \\log{(z)} and 4 \\operatorname{v_{2}}^{2}{(z)} = 2 (\\operatorname{v_{2}}{(z)} + \\log{(z)}) \\operatorname{v_{2}}{(z)} and 4 \\operatorname{v_{2}}^{2}{(z)} = 2 \\mathbf{H}{(z)} \\operatorname{v_{2}}{(z)} and 4 \\log{(z)}^{2} = 2 \\mathbf{H}{(z)} \\log{(z)} and 4 \\log{(z)}^{2} = 2 (\\operatorname{v_{2}}{(z)} + \\log{(z)}) \\log{(z)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["add", 1, "Function('v_2')(Symbol('z', commutative=True))"], "Equality(Mul(Integer(2), Function('v_2')(Symbol('z', commutative=True))), Add(Function('v_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Add(Function('v_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('v_2')(Symbol('z', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('v_2')(Symbol('z', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('v_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))), Function('v_2')(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(4), Pow(Function('v_2')(Symbol('z', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Function('v_2')(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(4), Pow(log(Symbol('z', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(4), Pow(log(Symbol('z', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('v_2')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))), log(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(b,F_{H})} = \\int (F_{H} + b) dF_{H}, then obtain 0 = \\frac{F_{H} \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H}}{\\int - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H} db}", "derivation": "\\operatorname{E_{x}}{(b,F_{H})} = \\int (F_{H} + b) dF_{H} and \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} = \\iint (F_{H} + b) dF_{H} dF_{H} and F_{H} \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} = F_{H} \\iint (F_{H} + b) dF_{H} dF_{H} and - F_{H} \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} = - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H} and 0 = F_{H} \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H} and 0 = \\frac{F_{H} \\int \\operatorname{E_{x}}{(b,F_{H})} dF_{H} - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H}}{\\int - F_{H} \\iint (F_{H} + b) dF_{H} dF_{H} db}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["times", 2, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["add", 4, "Mul(Symbol('F_H', commutative=True), Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], "Equality(Integer(0), Add(Mul(Symbol('F_H', commutative=True), Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))))"], [["divide", 5, "Integral(Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Tuple(Symbol('b', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Symbol('F_H', commutative=True), Integral(Function('E_x')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Pow(Integral(Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given f{(\\lambda,\\Psi)} = \\Psi \\lambda, then obtain \\Psi \\lambda + \\int f{(\\lambda,\\Psi)} d\\Psi = 2 \\Psi \\lambda - f{(\\lambda,\\Psi)} + \\int f{(\\lambda,\\Psi)} d\\Psi", "derivation": "f{(\\lambda,\\Psi)} = \\Psi \\lambda and \\int f{(\\lambda,\\Psi)} d\\Psi = \\int \\Psi \\lambda d\\Psi and f{(\\lambda,\\Psi)} + \\int \\Psi \\lambda d\\Psi = \\Psi \\lambda + \\int \\Psi \\lambda d\\Psi and 0 = \\Psi \\lambda - f{(\\lambda,\\Psi)} and \\Psi \\lambda + \\int \\Psi \\lambda d\\Psi = 2 \\Psi \\lambda - f{(\\lambda,\\Psi)} + \\int \\Psi \\lambda d\\Psi and \\Psi \\lambda + \\int f{(\\lambda,\\Psi)} d\\Psi = 2 \\Psi \\lambda - f{(\\lambda,\\Psi)} + \\int f{(\\lambda,\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["add", 1, "Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["minus", 3, "Add(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["add", 4, "Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Function('f')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\delta,s)} = \\delta s, then obtain \\frac{\\partial}{\\partial s} (- \\delta s + \\int \\mathbf{r}{(\\delta,s)} d\\delta) = \\frac{\\partial}{\\partial s} (- \\delta s + \\int \\delta s d\\delta)", "derivation": "\\mathbf{r}{(\\delta,s)} = \\delta s and \\int \\mathbf{r}{(\\delta,s)} d\\delta = \\int \\delta s d\\delta and - \\delta s + \\int \\mathbf{r}{(\\delta,s)} d\\delta = - \\delta s + \\int \\delta s d\\delta and \\frac{\\partial}{\\partial s} (- \\delta s + \\int \\mathbf{r}{(\\delta,s)} d\\delta) = \\frac{\\partial}{\\partial s} (- \\delta s + \\int \\delta s d\\delta)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(F_{N},c_{0})} = \\cos{(F_{N} + c_{0})}, then obtain - F_{N} - c_{0} + \\frac{\\partial}{\\partial c_{0}} \\lambda{(F_{N},c_{0})} = - F_{N} - c_{0} - \\sin{(F_{N} + c_{0})}", "derivation": "\\lambda{(F_{N},c_{0})} = \\cos{(F_{N} + c_{0})} and \\frac{\\partial}{\\partial c_{0}} \\lambda{(F_{N},c_{0})} = \\frac{\\partial}{\\partial c_{0}} \\cos{(F_{N} + c_{0})} and - F_{N} - c_{0} + \\frac{\\partial}{\\partial c_{0}} \\lambda{(F_{N},c_{0})} = - F_{N} - c_{0} + \\frac{\\partial}{\\partial c_{0}} \\cos{(F_{N} + c_{0})} and - F_{N} - c_{0} + \\frac{\\partial}{\\partial c_{0}} \\lambda{(F_{N},c_{0})} = - F_{N} - c_{0} - \\sin{(F_{N} + c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True)), cos(Add(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True))))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["minus", 2, "Add(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Derivative(cos(Add(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('F_N', commutative=True), Symbol('c_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} = \\frac{f_{\\mathbf{v}}}{v_{1}}, then obtain \\iint (-1) df_{\\mathbf{v}} dv_{1} = \\iint (\\frac{f_{\\mathbf{v}}}{v_{1}} - \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} - 1) df_{\\mathbf{v}} dv_{1}", "derivation": "\\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} = \\frac{f_{\\mathbf{v}}}{v_{1}} and \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} - 1 = \\frac{f_{\\mathbf{v}}}{v_{1}} - 1 and -1 = \\frac{f_{\\mathbf{v}}}{v_{1}} - \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} - 1 and \\int (-1) df_{\\mathbf{v}} = \\int (\\frac{f_{\\mathbf{v}}}{v_{1}} - \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} - 1) df_{\\mathbf{v}} and \\iint (-1) df_{\\mathbf{v}} dv_{1} = \\iint (\\frac{f_{\\mathbf{v}}}{v_{1}} - \\operatorname{A_{y}}{(f_{\\mathbf{v}},v_{1})} - 1) df_{\\mathbf{v}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Integer(-1)))"], [["minus", 2, "Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(-1), Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Mul(Integer(-1), Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True))), Integer(-1)))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Mul(Integer(-1), Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True))), Integer(-1)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["integrate", 4, "Symbol('v_1', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Mul(Integer(-1), Function('A_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_1', commutative=True))), Integer(-1)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(F_{g},\\mathbf{g})} = \\sin{(F_{g} + \\mathbf{g})}, then derive \\cos{(F_{g} + \\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\ddot{x}{(F_{g},\\mathbf{g})} = 2 \\cos{(F_{g} + \\mathbf{g})}, then obtain \\cos{(F_{g} + \\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\sin{(F_{g} + \\mathbf{g})} = 2 \\cos{(F_{g} + \\mathbf{g})}", "derivation": "\\ddot{x}{(F_{g},\\mathbf{g})} = \\sin{(F_{g} + \\mathbf{g})} and \\ddot{x}{(F_{g},\\mathbf{g})} + \\sin{(F_{g} + \\mathbf{g})} = 2 \\sin{(F_{g} + \\mathbf{g})} and \\frac{\\partial}{\\partial \\mathbf{g}} (\\ddot{x}{(F_{g},\\mathbf{g})} + \\sin{(F_{g} + \\mathbf{g})}) = \\frac{\\partial}{\\partial \\mathbf{g}} 2 \\sin{(F_{g} + \\mathbf{g})} and \\cos{(F_{g} + \\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\ddot{x}{(F_{g},\\mathbf{g})} = 2 \\cos{(F_{g} + \\mathbf{g})} and \\cos{(F_{g} + \\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\sin{(F_{g} + \\mathbf{g})} = 2 \\cos{(F_{g} + \\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 1, "sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Integer(2), sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\ddot{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Derivative(sin(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(B)} = e^{B}, then derive \\theta_1 + \\operatorname{F_{g}}{(B)} = \\mu + e^{B}, then obtain \\theta_1 + e^{B} = \\theta_1 + \\operatorname{F_{g}}{(B)}", "derivation": "\\operatorname{F_{g}}{(B)} = e^{B} and \\frac{d}{d B} \\operatorname{F_{g}}{(B)} = \\frac{d}{d B} e^{B} and \\int \\frac{d}{d B} \\operatorname{F_{g}}{(B)} dB = \\int \\frac{d}{d B} e^{B} dB and \\theta_1 + \\operatorname{F_{g}}{(B)} = \\mu + e^{B} and \\theta_1 + e^{B} = \\mu + e^{B} and \\theta_1 + e^{B} = \\theta_1 + \\operatorname{F_{g}}{(B)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Function('F_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('F_g')(Symbol('B', commutative=True))), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('B', commutative=True))), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('B', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Function('F_g')(Symbol('B', commutative=True))))"]]}, {"prompt": "Given W{(J)} = J, then obtain W{(J)} - \\int \\frac{J}{W{(J)}} dJ = J - \\int \\frac{J}{W{(J)}} dJ", "derivation": "W{(J)} = J and 1 = \\frac{J}{W{(J)}} and \\int 1 dJ = \\int \\frac{J}{W{(J)}} dJ and W{(J)} - \\int 1 dJ = J - \\int 1 dJ and W{(J)} - \\int \\frac{J}{W{(J)}} dJ = J - \\int \\frac{J}{W{(J)}} dJ", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], [["divide", 1, "Function('W')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Symbol('J', commutative=True), Pow(Function('W')(Symbol('J', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Pow(Function('W')(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))))"], [["minus", 1, "Integral(Integer(1), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Function('W')(Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('J', commutative=True))))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('J', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('W')(Symbol('J', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('J', commutative=True), Pow(Function('W')(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('J', commutative=True), Pow(Function('W')(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{B}{(v)} = \\cos{(v)}, then derive \\frac{d}{d v} \\mathbf{B}{(v)} = - \\sin{(v)}, then obtain - \\frac{\\frac{d}{d v} \\cos{(v)}}{\\int \\frac{\\mathbf{B}{(v)}}{v} dv} = \\frac{\\sin{(v)}}{\\int \\frac{\\mathbf{B}{(v)}}{v} dv}", "derivation": "\\mathbf{B}{(v)} = \\cos{(v)} and \\frac{d}{d v} \\mathbf{B}{(v)} = \\frac{d}{d v} \\cos{(v)} and \\frac{d}{d v} \\mathbf{B}{(v)} = - \\sin{(v)} and \\frac{d}{d v} \\cos{(v)} = - \\sin{(v)} and - \\frac{d}{d v} \\cos{(v)} = \\sin{(v)} and - \\frac{\\frac{d}{d v} \\cos{(v)}}{\\int \\frac{\\mathbf{B}{(v)}}{v} dv} = \\frac{\\sin{(v)}}{\\int \\frac{\\mathbf{B}{(v)}}{v} dv}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), sin(Symbol('v', commutative=True)))"], [["divide", 5, "Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))"], "Equality(Mul(Integer(-1), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(sin(Symbol('v', commutative=True)), Pow(Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given m{(M,F_{N},s)} = - F_{N} - M + s, then derive \\int m{(M,F_{N},s)} dM = - \\frac{M^{2}}{2} + M (- F_{N} + s) + U, then obtain F_{N} (- F_{N} + s + 2 \\int m{(M,F_{N},s)} dM)^{2} = F_{N} (- F_{N} - \\frac{M^{2}}{2} + M (- F_{N} + s) + U + s + \\int m{(M,F_{N},s)} dM)^{2}", "derivation": "m{(M,F_{N},s)} = - F_{N} - M + s and \\int m{(M,F_{N},s)} dM = \\int (- F_{N} - M + s) dM and \\int m{(M,F_{N},s)} dM = - \\frac{M^{2}}{2} + M (- F_{N} + s) + U and - F_{N} + s + 2 \\int m{(M,F_{N},s)} dM = - F_{N} - \\frac{M^{2}}{2} + M (- F_{N} + s) + U + s + \\int m{(M,F_{N},s)} dM and (- F_{N} + s + 2 \\int m{(M,F_{N},s)} dM)^{2} = (- F_{N} - \\frac{M^{2}}{2} + M (- F_{N} + s) + U + s + \\int m{(M,F_{N},s)} dM)^{2} and F_{N} (- F_{N} + s + 2 \\int m{(M,F_{N},s)} dM)^{2} = F_{N} (- F_{N} - \\frac{M^{2}}{2} + M (- F_{N} + s) + U + s + \\int m{(M,F_{N},s)} dM)^{2}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True))), Symbol('U', commutative=True)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True), Mul(Integer(2), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True))), Symbol('U', commutative=True), Symbol('s', commutative=True), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True), Mul(Integer(2), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True))), Symbol('U', commutative=True), Symbol('s', commutative=True), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True)))), Integer(2)))"], [["times", 5, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True), Mul(Integer(2), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True))))), Integer(2))), Mul(Symbol('F_N', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('s', commutative=True))), Symbol('U', commutative=True), Symbol('s', commutative=True), Integral(Function('m')(Symbol('M', commutative=True), Symbol('F_N', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('M', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(V,T)} = T V, then obtain 2 T + V \\operatorname{y^{\\prime}}{(V,T)} = T V^{2} + 2 T", "derivation": "\\operatorname{y^{\\prime}}{(V,T)} = T V and V \\operatorname{y^{\\prime}}{(V,T)} = T V^{2} and \\frac{\\operatorname{y^{\\prime}}{(V,T)}}{V} = T and T + \\frac{\\operatorname{y^{\\prime}}{(V,T)}}{V} = 2 T and T + V \\operatorname{y^{\\prime}}{(V,T)} + \\frac{\\operatorname{y^{\\prime}}{(V,T)}}{V} = T V^{2} + T + \\frac{\\operatorname{y^{\\prime}}{(V,T)}}{V} and 2 T + V \\operatorname{y^{\\prime}}{(V,T)} = T V^{2} + 2 T", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True))), Mul(Symbol('T', commutative=True), Pow(Symbol('V', commutative=True), Integer(2))))"], [["divide", 2, "Pow(Symbol('V', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True))), Symbol('T', commutative=True))"], [["add", 3, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(2), Symbol('T', commutative=True)))"], [["add", 2, "Add(Symbol('T', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True))))"], "Equality(Add(Symbol('T', commutative=True), Mul(Symbol('V', commutative=True), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True)))), Add(Mul(Symbol('T', commutative=True), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('T', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Symbol('V', commutative=True), Function('y^{\\\\prime}')(Symbol('V', commutative=True), Symbol('T', commutative=True)))), Add(Mul(Symbol('T', commutative=True), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Integer(2), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(F_{c})} = \\int \\sin{(F_{c})} dF_{c}, then derive \\operatorname{g_{\\varepsilon}}{(F_{c})} = v_{z} - \\cos{(F_{c})}, then obtain \\iint \\sin{(F_{c})} dF_{c} dF_{c} = \\int (v_{z} - \\cos{(F_{c})}) dF_{c}", "derivation": "\\operatorname{g_{\\varepsilon}}{(F_{c})} = \\int \\sin{(F_{c})} dF_{c} and \\operatorname{g_{\\varepsilon}}{(F_{c})} = v_{z} - \\cos{(F_{c})} and \\int \\sin{(F_{c})} dF_{c} = v_{z} - \\cos{(F_{c})} and \\iint \\sin{(F_{c})} dF_{c} dF_{c} = \\int (v_{z} - \\cos{(F_{c})}) dF_{c}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A,m)} = \\cos{(A + m)}, then obtain \\frac{\\partial}{\\partial m} \\int (m + \\operatorname{P_{e}}{(A,m)}) dA = \\frac{\\partial}{\\partial m} \\int (m + \\cos{(A + m)}) dA", "derivation": "\\operatorname{P_{e}}{(A,m)} = \\cos{(A + m)} and m + \\operatorname{P_{e}}{(A,m)} = m + \\cos{(A + m)} and \\int (m + \\operatorname{P_{e}}{(A,m)}) dA = \\int (m + \\cos{(A + m)}) dA and \\frac{\\partial}{\\partial m} \\int (m + \\operatorname{P_{e}}{(A,m)}) dA = \\frac{\\partial}{\\partial m} \\int (m + \\cos{(A + m)}) dA", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('m', commutative=True))))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('m', commutative=True))), Add(Symbol('m', commutative=True), cos(Add(Symbol('A', commutative=True), Symbol('m', commutative=True)))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Symbol('m', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('m', commutative=True), cos(Add(Symbol('A', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('A', commutative=True))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('m', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('m', commutative=True), cos(Add(Symbol('A', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(n)} = \\cos{(n)} and \\mathbf{v}{(n)} = \\operatorname{V_{\\mathbf{B}}}^{n}{(n)}, then obtain - \\int \\mathbf{v}{(n)} dn = - \\int \\cos^{n}{(n)} dn", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(n)} = \\cos{(n)} and \\operatorname{V_{\\mathbf{B}}}^{n}{(n)} = \\cos^{n}{(n)} and \\mathbf{v}{(n)} = \\operatorname{V_{\\mathbf{B}}}^{n}{(n)} and \\mathbf{v}{(n)} = \\cos^{n}{(n)} and \\int \\mathbf{v}{(n)} dn = \\int \\cos^{n}{(n)} dn and - \\int \\mathbf{v}{(n)} dn = - \\int \\cos^{n}{(n)} dn", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Integer(-1), Integral(Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\hat{H})} = e^{\\hat{H}} and k{(\\hat{H})} = e^{\\hat{H}}, then obtain e^{2 \\hat{H}} = k{(\\hat{H})} e^{\\hat{H}}", "derivation": "\\mathbf{E}{(\\hat{H})} = e^{\\hat{H}} and \\mathbf{E}^{2}{(\\hat{H})} = \\mathbf{E}{(\\hat{H})} e^{\\hat{H}} and k{(\\hat{H})} = e^{\\hat{H}} and \\mathbf{E}^{2}{(\\hat{H})} = \\mathbf{E}{(\\hat{H})} k{(\\hat{H})} and e^{2 \\hat{H}} = k{(\\hat{H})} e^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\hat{H}', commutative=True)), Function('k')(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True))), Mul(Function('k')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(r_{0},\\phi,G)} = G \\phi r_{0} and T{(g^{\\prime}_{\\varepsilon})} = e^{g^{\\prime}_{\\varepsilon}}, then obtain - G^{2} \\phi r_{0} + T{(g^{\\prime}_{\\varepsilon})} = - G^{2} \\phi r_{0} + e^{g^{\\prime}_{\\varepsilon}}", "derivation": "\\operatorname{M_{E}}{(r_{0},\\phi,G)} = G \\phi r_{0} and T{(g^{\\prime}_{\\varepsilon})} = e^{g^{\\prime}_{\\varepsilon}} and - G \\operatorname{M_{E}}{(r_{0},\\phi,G)} + T{(g^{\\prime}_{\\varepsilon})} = - G \\operatorname{M_{E}}{(r_{0},\\phi,G)} + e^{g^{\\prime}_{\\varepsilon}} and - G^{2} \\phi r_{0} + T{(g^{\\prime}_{\\varepsilon})} = - G^{2} \\phi r_{0} + e^{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('r_0', commutative=True)))"], ["get_premise", "Equality(Function('T')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 2, "Mul(Symbol('G', commutative=True), Function('M_E')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True), Function('M_E')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True))), Function('T')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True), Function('M_E')(Symbol('r_0', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True))), exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True), Symbol('r_0', commutative=True)), Function('T')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True), Symbol('r_0', commutative=True)), exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given T{(M_{E})} = \\cos{(\\sin{(M_{E})})} and \\operatorname{F_{N}}{(M_{E})} = \\frac{d}{d M_{E}} T{(M_{E})}, then obtain \\operatorname{F_{N}}{(M_{E})} = \\frac{d}{d M_{E}} \\cos{(\\sin{(M_{E})})}", "derivation": "T{(M_{E})} = \\cos{(\\sin{(M_{E})})} and \\frac{d}{d M_{E}} T{(M_{E})} = \\frac{d}{d M_{E}} \\cos{(\\sin{(M_{E})})} and \\operatorname{F_{N}}{(M_{E})} = \\frac{d}{d M_{E}} T{(M_{E})} and \\operatorname{F_{N}}{(M_{E})} = \\frac{d}{d M_{E}} \\cos{(\\sin{(M_{E})})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('M_E', commutative=True)), cos(sin(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('M_E', commutative=True)), Derivative(Function('T')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_N')(Symbol('M_E', commutative=True)), Derivative(cos(sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\delta)} = \\delta and \\operatorname{n_{2}}{(\\delta)} = (\\frac{d}{d \\delta} \\delta)^{\\delta}, then obtain \\delta + \\operatorname{n_{2}}{(\\delta)} = \\delta + (\\frac{d}{d \\delta} \\delta)^{\\delta}", "derivation": "\\operatorname{F_{H}}{(\\delta)} = \\delta and \\frac{d}{d \\delta} \\operatorname{F_{H}}{(\\delta)} = \\frac{d}{d \\delta} \\delta and (\\frac{d}{d \\delta} \\operatorname{F_{H}}{(\\delta)})^{\\delta} = (\\frac{d}{d \\delta} \\delta)^{\\delta} and \\operatorname{n_{2}}{(\\delta)} = (\\frac{d}{d \\delta} \\delta)^{\\delta} and \\operatorname{n_{2}}{(\\delta)} = (\\frac{d}{d \\delta} \\operatorname{F_{H}}{(\\delta)})^{\\delta} and \\delta + \\operatorname{n_{2}}{(\\delta)} = \\delta + (\\frac{d}{d \\delta} \\operatorname{F_{H}}{(\\delta)})^{\\delta} and \\delta + \\operatorname{n_{2}}{(\\delta)} = \\delta + (\\frac{d}{d \\delta} \\delta)^{\\delta}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('\\\\delta', commutative=True)), Pow(Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('n_2')(Symbol('\\\\delta', commutative=True)), Pow(Derivative(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["add", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('n_2')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(Derivative(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('n_2')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given B{(I,\\mathbf{D})} = \\frac{I}{\\mathbf{D}}, then derive \\mathbf{J}_P + B{(I,\\mathbf{D})} = \\frac{I}{\\mathbf{D}} + p, then obtain (\\mathbf{J}_P + B{(I,\\mathbf{D})})^{I} = (\\frac{I}{\\mathbf{D}} + p)^{I}", "derivation": "B{(I,\\mathbf{D})} = \\frac{I}{\\mathbf{D}} and \\frac{\\partial}{\\partial \\mathbf{D}} B{(I,\\mathbf{D})} = \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{I}{\\mathbf{D}} and \\int \\frac{\\partial}{\\partial \\mathbf{D}} B{(I,\\mathbf{D})} d\\mathbf{D} = \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\frac{I}{\\mathbf{D}} d\\mathbf{D} and \\mathbf{J}_P + B{(I,\\mathbf{D})} = \\frac{I}{\\mathbf{D}} + p and (\\mathbf{J}_P + B{(I,\\mathbf{D})})^{I} = (\\frac{I}{\\mathbf{D}} + p)^{I}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('B')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Symbol('p', commutative=True)))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('B')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Symbol('p', commutative=True)), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\chi{(C_{d})} = \\sin{(C_{d})} and \\operatorname{P_{e}}{(\\eta,\\mathbf{H})} = \\eta^{\\mathbf{H}}, then obtain \\sin{(\\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\chi{(C_{d})} + 1)} = \\sin{(\\eta^{\\mathbf{H}} \\chi{(C_{d})} + 1)}", "derivation": "\\chi{(C_{d})} = \\sin{(C_{d})} and \\operatorname{P_{e}}{(\\eta,\\mathbf{H})} = \\eta^{\\mathbf{H}} and \\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\chi{(C_{d})} = \\eta^{\\mathbf{H}} \\chi{(C_{d})} and \\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\chi{(C_{d})} + 1 = \\eta^{\\mathbf{H}} \\chi{(C_{d})} + 1 and \\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\sin{(C_{d})} + 1 = \\eta^{\\mathbf{H}} \\sin{(C_{d})} + 1 and \\sin{(\\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\sin{(C_{d})} + 1)} = \\sin{(\\eta^{\\mathbf{H}} \\sin{(C_{d})} + 1)} and \\sin{(\\operatorname{P_{e}}{(\\eta,\\mathbf{H})} \\chi{(C_{d})} + 1)} = \\sin{(\\eta^{\\mathbf{H}} \\chi{(C_{d})} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], ["get_premise", "Equality(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 2, "Function('\\\\chi')(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))), Integer(1)), Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('C_d', commutative=True))), Integer(1)), Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('C_d', commutative=True))), Integer(1)))"], [["sin", 5], "Equality(sin(Add(Mul(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('C_d', commutative=True))), Integer(1))), sin(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('C_d', commutative=True))), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(sin(Add(Mul(Function('P_e')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))), Integer(1))), sin(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\chi')(Symbol('C_d', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(L)} = \\cos{(\\sin{(L)})}, then derive \\frac{d}{d L} \\mathbf{J}_f{(L)} = - \\sin{(\\sin{(L)})} \\cos{(L)}, then obtain - \\sin{(L)} + \\frac{d}{d L} \\mathbf{J}_f{(L)} = - \\sin{(L)} - \\sin{(\\sin{(L)})} \\cos{(L)}", "derivation": "\\mathbf{J}_f{(L)} = \\cos{(\\sin{(L)})} and \\frac{d}{d L} \\mathbf{J}_f{(L)} = \\frac{d}{d L} \\cos{(\\sin{(L)})} and \\frac{d}{d L} \\mathbf{J}_f{(L)} = - \\sin{(\\sin{(L)})} \\cos{(L)} and \\frac{d}{d L} \\cos{(\\sin{(L)})} = - \\sin{(\\sin{(L)})} \\cos{(L)} and - \\sin{(L)} + \\frac{d}{d L} \\cos{(\\sin{(L)})} = - \\sin{(L)} - \\sin{(\\sin{(L)})} \\cos{(L)} and - \\sin{(L)} + \\frac{d}{d L} \\mathbf{J}_f{(L)} = - \\sin{(L)} - \\sin{(\\sin{(L)})} \\cos{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('L', commutative=True)), cos(sin(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))))"], [["minus", 4, "sin(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Derivative(cos(sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('L', commutative=True))), Mul(Integer(-1), sin(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(C_{2},\\hat{x})} = C_{2} - \\hat{x}, then derive \\int \\mathbf{f}{(C_{2},\\hat{x})} dC_{2} = \\frac{C_{2}^{2}}{2} - C_{2} \\hat{x} + V_{\\mathbf{B}}, then obtain \\frac{C_{2}^{2}}{2} - C_{2} \\hat{x} + V_{\\mathbf{B}} = \\int (C_{2} - \\hat{x}) dC_{2}", "derivation": "\\mathbf{f}{(C_{2},\\hat{x})} = C_{2} - \\hat{x} and \\int \\mathbf{f}{(C_{2},\\hat{x})} dC_{2} = \\int (C_{2} - \\hat{x}) dC_{2} and \\int \\mathbf{f}{(C_{2},\\hat{x})} dC_{2} = \\frac{C_{2}^{2}}{2} - C_{2} \\hat{x} + V_{\\mathbf{B}} and \\frac{C_{2}^{2}}{2} - C_{2} \\hat{x} + V_{\\mathbf{B}} = \\int (C_{2} - \\hat{x}) dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(E_{n},C_{1})} = \\frac{C_{1}}{E_{n}}, then derive \\frac{\\partial}{\\partial E_{n}} \\theta_{2}{(E_{n},C_{1})} = - \\frac{C_{1}}{E_{n}^{2}}, then obtain \\frac{C_{1}}{E_{n}} - \\frac{C_{1}}{E_{n}^{2}} = \\frac{C_{1}}{E_{n}} + \\frac{\\partial}{\\partial E_{n}} \\frac{C_{1}}{E_{n}}", "derivation": "\\theta_{2}{(E_{n},C_{1})} = \\frac{C_{1}}{E_{n}} and \\frac{\\partial}{\\partial E_{n}} \\theta_{2}{(E_{n},C_{1})} = \\frac{\\partial}{\\partial E_{n}} \\frac{C_{1}}{E_{n}} and \\frac{\\partial}{\\partial E_{n}} \\theta_{2}{(E_{n},C_{1})} = - \\frac{C_{1}}{E_{n}^{2}} and \\frac{C_{1}}{E_{n}} + \\frac{\\partial}{\\partial E_{n}} \\theta_{2}{(E_{n},C_{1})} = \\frac{C_{1}}{E_{n}} + \\frac{\\partial}{\\partial E_{n}} \\frac{C_{1}}{E_{n}} and \\frac{C_{1}}{E_{n}} - \\frac{C_{1}}{E_{n}^{2}} = \\frac{C_{1}}{E_{n}} + \\frac{\\partial}{\\partial E_{n}} \\frac{C_{1}}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('C_1', commutative=True)), Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-2))))"], [["add", 2, "Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Derivative(Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Derivative(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-2)))), Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Derivative(Mul(Symbol('C_1', commutative=True), Pow(Symbol('E_n', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi{(x,A_{1})} = A_{1} - x, then obtain (A_{1} - x)^{x} \\Psi^{x}{(x,A_{1})} = (A_{1} - x)^{x (1 + \\frac{\\Psi{(x,A_{1})}}{A_{1} - x})}", "derivation": "\\Psi{(x,A_{1})} = A_{1} - x and \\frac{\\Psi{(x,A_{1})}}{A_{1} - x} = 1 and \\Psi^{x}{(x,A_{1})} = (A_{1} - x)^{x} and 1 + \\frac{\\Psi{(x,A_{1})}}{A_{1} - x} = 2 and (A_{1} - x)^{x} \\Psi^{x}{(x,A_{1})} = (A_{1} - x)^{2 x} and (A_{1} - x)^{x} \\Psi^{x}{(x,A_{1})} = (A_{1} - x)^{x (1 + \\frac{\\Psi{(x,A_{1})}}{A_{1} - x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["divide", 1, "Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True))), Integer(1))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)), Symbol('x', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["add", 2, 1], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)))), Integer(2))"], [["times", 3, "Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)), Symbol('x', commutative=True))), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)), Symbol('x', commutative=True))), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), Add(Integer(1), Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('A_1', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{s}{(k)} = \\frac{d}{d k} \\sin{(k)}, then derive \\mathbf{s}{(k)} = \\cos{(k)}, then derive \\mathbf{s}{(k)} \\sin{(k)} - 1 = \\sin{(k)} \\cos{(k)} - 1, then obtain \\frac{d}{d k} (\\mathbf{s}{(k)} \\sin{(k)} - 1) = \\frac{d}{d k} (\\sin{(k)} \\frac{d}{d k} \\sin{(k)} - 1)", "derivation": "\\mathbf{s}{(k)} = \\frac{d}{d k} \\sin{(k)} and \\mathbf{s}{(k)} = \\cos{(k)} and \\mathbf{s}{(k)} \\sin{(k)} = \\sin{(k)} \\frac{d}{d k} \\sin{(k)} and \\sin{(k)} \\cos{(k)} = \\sin{(k)} \\frac{d}{d k} \\sin{(k)} and \\mathbf{s}{(k)} \\sin{(k)} - 1 = \\sin{(k)} \\frac{d}{d k} \\sin{(k)} - 1 and \\mathbf{s}{(k)} \\sin{(k)} - 1 = \\sin{(k)} \\cos{(k)} - 1 and \\frac{d}{d k} (\\mathbf{s}{(k)} \\sin{(k)} - 1) = \\frac{d}{d k} (\\sin{(k)} \\cos{(k)} - 1) and \\frac{d}{d k} (\\mathbf{s}{(k)} \\sin{(k)} - 1) = \\frac{d}{d k} (\\sin{(k)} \\frac{d}{d k} \\sin{(k)} - 1)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["times", 1, "sin(Symbol('k', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), Mul(sin(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(sin(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Mul(sin(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["minus", 3, 1], "Equality(Add(Mul(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), Integer(-1)), Add(Mul(sin(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), Integer(-1)), Add(Mul(sin(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Integer(-1)))"], [["differentiate", 6, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Mul(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(sin(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Derivative(Add(Mul(Function('\\\\mathbf{s}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(sin(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(k)} = \\cos{(k)} and \\hat{p}_0{(k)} = \\frac{1}{\\cos{(k)}}, then obtain \\frac{1}{\\cos{(k)}} = \\frac{1}{\\operatorname{v_{2}}{(k)}}", "derivation": "\\operatorname{v_{2}}{(k)} = \\cos{(k)} and \\hat{p}_0{(k)} = \\frac{1}{\\cos{(k)}} and \\hat{p}_0{(k)} = \\frac{1}{\\operatorname{v_{2}}{(k)}} and \\frac{1}{\\cos{(k)}} = \\frac{1}{\\operatorname{v_{2}}{(k)}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{p}_0')(Symbol('k', commutative=True)), Pow(Function('v_2')(Symbol('k', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(cos(Symbol('k', commutative=True)), Integer(-1)), Pow(Function('v_2')(Symbol('k', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\theta{(A_{x},F_{x})} = A_{x} - F_{x}, then obtain - F_{x} + (A_{x} - 3 F_{x}) (- 2 F_{x} + \\theta{(A_{x},F_{x})}) + \\theta{(A_{x},F_{x})} = - F_{x} + (A_{x} - 3 F_{x})^{2} + \\theta{(A_{x},F_{x})}", "derivation": "\\theta{(A_{x},F_{x})} = A_{x} - F_{x} and - F_{x} + \\theta{(A_{x},F_{x})} = A_{x} - 2 F_{x} and - 2 F_{x} + \\theta{(A_{x},F_{x})} = A_{x} - 3 F_{x} and (A_{x} - 3 F_{x}) (- 2 F_{x} + \\theta{(A_{x},F_{x})}) = (A_{x} - 3 F_{x})^{2} and - F_{x} + (A_{x} - 3 F_{x}) (- 2 F_{x} + \\theta{(A_{x},F_{x})}) + \\theta{(A_{x},F_{x})} = - F_{x} + (A_{x} - 3 F_{x})^{2} + \\theta{(A_{x},F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('F_x', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True))))"], [["minus", 2, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True))))"], [["times", 3, "Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True)))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True)))), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True))), Integer(2)))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True)))), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('F_x', commutative=True))), Integer(2)), Function('\\\\theta')(Symbol('A_x', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given J{(g_{\\varepsilon},z^{*})} = g_{\\varepsilon} - z^{*}, then derive (\\int J{(g_{\\varepsilon},z^{*})} dz^{*})^{z^{*}} = (\\mathbf{H} + g_{\\varepsilon} z^{*} - \\frac{(z^{*})^{2}}{2})^{z^{*}}, then obtain (\\int (g_{\\varepsilon} - z^{*}) dz^{*})^{z^{*}} = (\\mathbf{H} + g_{\\varepsilon} z^{*} - \\frac{(z^{*})^{2}}{2})^{z^{*}}", "derivation": "J{(g_{\\varepsilon},z^{*})} = g_{\\varepsilon} - z^{*} and \\int J{(g_{\\varepsilon},z^{*})} dz^{*} = \\int (g_{\\varepsilon} - z^{*}) dz^{*} and (\\int J{(g_{\\varepsilon},z^{*})} dz^{*})^{z^{*}} = (\\int (g_{\\varepsilon} - z^{*}) dz^{*})^{z^{*}} and (\\int J{(g_{\\varepsilon},z^{*})} dz^{*})^{z^{*}} = (\\mathbf{H} + g_{\\varepsilon} z^{*} - \\frac{(z^{*})^{2}}{2})^{z^{*}} and (\\int (g_{\\varepsilon} - z^{*}) dz^{*})^{z^{*}} = (\\mathbf{H} + g_{\\varepsilon} z^{*} - \\frac{(z^{*})^{2}}{2})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Integral(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Pow(Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\mathbf{D})} = \\log{(e^{\\mathbf{D}})}, then obtain \\sigma_{x}{(\\mathbf{D})} e^{\\mathbf{D}} = e^{\\mathbf{D}} \\log{(e^{\\mathbf{D}})}", "derivation": "\\sigma_{x}{(\\mathbf{D})} = \\log{(e^{\\mathbf{D}})} and e^{\\sigma_{x}{(\\mathbf{D})}} = e^{\\mathbf{D}} and \\sigma_{x}{(\\mathbf{D})} e^{\\sigma_{x}{(\\mathbf{D})}} = e^{\\sigma_{x}{(\\mathbf{D})}} \\log{(e^{\\mathbf{D}})} and \\sigma_{x}{(\\mathbf{D})} e^{\\mathbf{D}} = e^{\\mathbf{D}} \\log{(e^{\\mathbf{D}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), log(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["times", 1, "exp(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)))), Mul(exp(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True))), log(exp(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True))), Mul(exp(Symbol('\\\\mathbf{D}', commutative=True)), log(exp(Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given h{(x)} = e^{x}, then obtain 3 h{(x)} + e^{x} = h{(x)} + 3 e^{x}", "derivation": "h{(x)} = e^{x} and h{(x)} + e^{x} = 2 e^{x} and 2 h{(x)} + 2 e^{x} = h{(x)} + 3 e^{x} and 3 h{(x)} + e^{x} = h{(x)} + 3 e^{x}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["add", 1, "exp(Symbol('x', commutative=True))"], "Equality(Add(Function('h')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True))), Mul(Integer(2), exp(Symbol('x', commutative=True))))"], [["add", 2, "Add(Function('h')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('h')(Symbol('x', commutative=True))), Mul(Integer(2), exp(Symbol('x', commutative=True)))), Add(Function('h')(Symbol('x', commutative=True)), Mul(Integer(3), exp(Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('h')(Symbol('x', commutative=True))), exp(Symbol('x', commutative=True))), Add(Function('h')(Symbol('x', commutative=True)), Mul(Integer(3), exp(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} = \\cos{(\\frac{a^{\\dagger}}{s})}, then obtain (- \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)})})^{s} = (- \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\cos{(\\frac{a^{\\dagger}}{s})})})^{s}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} = \\cos{(\\frac{a^{\\dagger}}{s})} and \\log{(\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)})} = \\log{(\\cos{(\\frac{a^{\\dagger}}{s})})} and - \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)})} = - \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\cos{(\\frac{a^{\\dagger}}{s})})} and (- \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)})})^{s} = (- \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},s)} + \\log{(\\cos{(\\frac{a^{\\dagger}}{s})})})^{s}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["log", 1], "Equality(log(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), log(cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], [["minus", 2, "Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), log(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), log(cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), log(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), log(cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))), Symbol('s', commutative=True)))"]]}, {"prompt": "Given B{(C_{1})} = \\cos{(\\log{(C_{1})})}, then obtain \\frac{d}{d C_{1}} (B{(C_{1})} - \\cos{(\\log{(C_{1})})}) = \\frac{d}{d C_{1}} 0", "derivation": "B{(C_{1})} = \\cos{(\\log{(C_{1})})} and - \\log{(C_{1})} = - B{(C_{1})} - \\log{(C_{1})} + \\cos{(\\log{(C_{1})})} and B{(C_{1})} - \\cos{(\\log{(C_{1})})} = 0 and \\frac{d}{d C_{1}} (B{(C_{1})} - \\cos{(\\log{(C_{1})})}) = \\frac{d}{d C_{1}} 0", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_1', commutative=True)), cos(log(Symbol('C_1', commutative=True))))"], [["minus", 1, "Add(Function('B')(Symbol('C_1', commutative=True)), log(Symbol('C_1', commutative=True)))"], "Equality(Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True))), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), cos(log(Symbol('C_1', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True))), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), cos(log(Symbol('C_1', commutative=True))))"], "Equality(Add(Function('B')(Symbol('C_1', commutative=True)), Mul(Integer(-1), cos(log(Symbol('C_1', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Function('B')(Symbol('C_1', commutative=True)), Mul(Integer(-1), cos(log(Symbol('C_1', commutative=True))))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(m,F_{H})} = \\sin{(\\frac{m}{F_{H}})} and k{(m,F_{H})} = \\operatorname{V_{\\mathbf{B}}}^{m}{(m,F_{H})}, then obtain \\frac{\\partial}{\\partial F_{H}} k^{m}{(m,F_{H})} = \\frac{\\partial}{\\partial F_{H}} (\\sin^{m}{(\\frac{m}{F_{H}})})^{m}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(m,F_{H})} = \\sin{(\\frac{m}{F_{H}})} and \\operatorname{V_{\\mathbf{B}}}^{m}{(m,F_{H})} = \\sin^{m}{(\\frac{m}{F_{H}})} and (\\operatorname{V_{\\mathbf{B}}}^{m}{(m,F_{H})})^{m} = (\\sin^{m}{(\\frac{m}{F_{H}})})^{m} and \\frac{\\partial}{\\partial F_{H}} (\\operatorname{V_{\\mathbf{B}}}^{m}{(m,F_{H})})^{m} = \\frac{\\partial}{\\partial F_{H}} (\\sin^{m}{(\\frac{m}{F_{H}})})^{m} and k{(m,F_{H})} = \\operatorname{V_{\\mathbf{B}}}^{m}{(m,F_{H})} and \\frac{\\partial}{\\partial F_{H}} k^{m}{(m,F_{H})} = \\frac{\\partial}{\\partial F_{H}} (\\sin^{m}{(\\frac{m}{F_{H}})})^{m}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('k')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Pow(Function('k')(Symbol('m', commutative=True), Symbol('F_H', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(C_{1},u)} = \\log{(\\frac{u}{C_{1}})}, then obtain \\int (2 \\theta_{2}{(C_{1},u)} - 2 \\log{(\\frac{u}{C_{1}})}) dC_{1} = \\int 0 dC_{1}", "derivation": "\\theta_{2}{(C_{1},u)} = \\log{(\\frac{u}{C_{1}})} and - u + \\theta_{2}{(C_{1},u)} = - u + \\log{(\\frac{u}{C_{1}})} and \\theta_{2}{(C_{1},u)} - \\log{(\\frac{u}{C_{1}})} = 0 and \\int (\\theta_{2}{(C_{1},u)} - \\log{(\\frac{u}{C_{1}})}) dC_{1} = \\int 0 dC_{1} and 2 \\theta_{2}{(C_{1},u)} - \\log{(\\frac{u}{C_{1}})} = \\theta_{2}{(C_{1},u)} and \\int (2 \\theta_{2}{(C_{1},u)} - 2 \\log{(\\frac{u}{C_{1}})}) dC_{1} = \\int 0 dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], "Equality(Add(Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))), Integer(0))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Add(Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))), Tuple(Symbol('C_1', commutative=True))), Integral(Integer(0), Tuple(Symbol('C_1', commutative=True))))"], [["add", 3, "Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))), Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('C_1', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Integer(2), log(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))), Tuple(Symbol('C_1', commutative=True))), Integral(Integer(0), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given v{(L)} = \\log{(L)}, then derive \\frac{d}{d L} v{(L)} - 1 = -1 + \\frac{1}{L}, then obtain \\frac{d}{d L} (\\frac{d}{d L} v{(L)} - 1) = \\frac{d}{d L} (-1 + \\frac{1}{L})", "derivation": "v{(L)} = \\log{(L)} and - L + v{(L)} = - L + \\log{(L)} and \\frac{d}{d L} (- L + v{(L)}) = \\frac{d}{d L} (- L + \\log{(L)}) and \\frac{d}{d L} v{(L)} - 1 = -1 + \\frac{1}{L} and \\frac{d}{d L} (\\frac{d}{d L} v{(L)} - 1) = \\frac{d}{d L} (-1 + \\frac{1}{L})", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('v')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('v')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('v')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('v')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(M)} = e^{M}, then obtain - 2 \\rho{(M)} e^{M} + \\int \\frac{d}{d M} \\int (\\rho{(M)} + e^{M}) dM dM = \\hat{p} - 2 \\rho{(M)} e^{M} + 2 e^{M}", "derivation": "\\rho{(M)} = e^{M} and \\rho{(M)} + e^{M} = 2 e^{M} and \\int (\\rho{(M)} + e^{M}) dM = \\int 2 e^{M} dM and \\frac{d}{d M} \\int (\\rho{(M)} + e^{M}) dM = \\frac{d}{d M} \\int 2 e^{M} dM and \\int \\frac{d}{d M} \\int (\\rho{(M)} + e^{M}) dM dM = \\int \\frac{d}{d M} \\int 2 e^{M} dM dM and - 2 \\rho{(M)} e^{M} + \\int \\frac{d}{d M} \\int (\\rho{(M)} + e^{M}) dM dM = - 2 \\rho{(M)} e^{M} + \\int \\frac{d}{d M} \\int 2 e^{M} dM dM and - 2 \\rho{(M)} e^{M} + \\int \\frac{d}{d M} \\int (\\rho{(M)} + e^{M}) dM dM = \\hat{p} - 2 \\rho{(M)} e^{M} + 2 e^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["add", 1, "exp(Symbol('M', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Mul(Integer(2), exp(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 3, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Integral(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(Integral(Mul(Integer(2), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["minus", 5, "Mul(Integer(2), Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Integral(Derivative(Integral(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Integral(Derivative(Integral(Mul(Integer(2), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Integral(Derivative(Integral(Add(Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\rho')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True))), Mul(Integer(2), exp(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(m)} = \\log{(m)}, then derive (2 \\frac{d}{d m} \\varphi^{*}{(m)} + 1)^{2} = (\\frac{d}{d m} \\varphi^{*}{(m)} + 1 + \\frac{1}{m})^{2}, then obtain (2 \\frac{d}{d m} \\log{(m)} + 1)^{2} = (\\frac{d}{d m} \\log{(m)} + 1 + \\frac{1}{m})^{2}", "derivation": "\\varphi^{*}{(m)} = \\log{(m)} and 2 \\varphi^{*}{(m)} = \\varphi^{*}{(m)} + \\log{(m)} and m + 2 \\varphi^{*}{(m)} = m + \\varphi^{*}{(m)} + \\log{(m)} and \\frac{d}{d m} (m + 2 \\varphi^{*}{(m)}) = \\frac{d}{d m} (m + \\varphi^{*}{(m)} + \\log{(m)}) and (\\frac{d}{d m} (m + 2 \\varphi^{*}{(m)}))^{2} = (\\frac{d}{d m} (m + \\varphi^{*}{(m)} + \\log{(m)}))^{2} and (2 \\frac{d}{d m} \\varphi^{*}{(m)} + 1)^{2} = (\\frac{d}{d m} \\varphi^{*}{(m)} + 1 + \\frac{1}{m})^{2} and (2 \\frac{d}{d m} \\log{(m)} + 1)^{2} = (\\frac{d}{d m} \\log{(m)} + 1 + \\frac{1}{m})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["add", 1, "Function('\\\\varphi^*')(Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('m', commutative=True))), Add(Function('\\\\varphi^*')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))))"], [["add", 2, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('m', commutative=True)))), Add(Symbol('m', commutative=True), Function('\\\\varphi^*')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), Function('\\\\varphi^*')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(Add(Symbol('m', commutative=True), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Symbol('m', commutative=True), Function('\\\\varphi^*')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Mul(Integer(2), Derivative(Function('\\\\varphi^*')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Integer(1)), Integer(2)), Pow(Add(Derivative(Function('\\\\varphi^*')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(2)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Add(Mul(Integer(2), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Integer(1)), Integer(2)), Pow(Add(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{s})} = e^{\\cos{(\\mathbf{s})}}, then obtain - \\mathbf{s} + 2 \\operatorname{z^{*}}{(\\mathbf{s})} = - \\mathbf{s} + 2 e^{\\cos{(\\mathbf{s})}}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{s})} = e^{\\cos{(\\mathbf{s})}} and - \\mathbf{s} + \\operatorname{z^{*}}{(\\mathbf{s})} = - \\mathbf{s} + e^{\\cos{(\\mathbf{s})}} and - \\mathbf{s} + 2 \\operatorname{z^{*}}{(\\mathbf{s})} = - \\mathbf{s} + \\operatorname{z^{*}}{(\\mathbf{s})} + e^{\\cos{(\\mathbf{s})}} and - \\mathbf{s} + 2 \\operatorname{z^{*}}{(\\mathbf{s})} = - \\mathbf{s} + 2 e^{\\cos{(\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True)), exp(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), exp(cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True)), exp(cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), Function('z^*')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), exp(cos(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(m,p)} = p + \\log{(m)}, then derive \\frac{m (\\operatorname{P_{g}}{(m,p)} - \\log{(m)})^{m} \\frac{\\partial}{\\partial p} \\operatorname{P_{g}}{(m,p)}}{\\operatorname{P_{g}}{(m,p)} - \\log{(m)}} = \\frac{m p^{m}}{p}, then obtain \\frac{m p^{m} \\frac{\\partial}{\\partial p} (p + \\log{(m)})}{p} = \\frac{m p^{m}}{p}", "derivation": "\\operatorname{P_{g}}{(m,p)} = p + \\log{(m)} and \\operatorname{P_{g}}{(m,p)} - \\log{(m)} = p and (\\operatorname{P_{g}}{(m,p)} - \\log{(m)})^{m} = p^{m} and \\frac{\\partial}{\\partial p} (\\operatorname{P_{g}}{(m,p)} - \\log{(m)})^{m} = \\frac{\\partial}{\\partial p} p^{m} and \\frac{m (\\operatorname{P_{g}}{(m,p)} - \\log{(m)})^{m} \\frac{\\partial}{\\partial p} \\operatorname{P_{g}}{(m,p)}}{\\operatorname{P_{g}}{(m,p)} - \\log{(m)}} = \\frac{m p^{m}}{p} and \\frac{m p^{m} \\frac{\\partial}{\\partial p} \\operatorname{P_{g}}{(m,p)}}{p} = \\frac{m p^{m}}{p} and \\frac{m p^{m} \\frac{\\partial}{\\partial p} (p + \\log{(m)})}{p} = \\frac{m p^{m}}{p}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Add(Symbol('p', commutative=True), log(Symbol('m', commutative=True))))"], [["minus", 1, "log(Symbol('m', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Symbol('p', commutative=True))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Add(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Symbol('p', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('m', commutative=True), Pow(Add(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Integer(-1)), Pow(Add(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), log(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Derivative(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True)), Derivative(Function('P_g')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True)), Derivative(Add(Symbol('p', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(E,\\rho)} = E - \\rho, then derive \\rho \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\rho)} - \\mathbf{D}{(E,\\rho)} = \\rho - \\mathbf{D}{(E,\\rho)}, then obtain e^{\\rho \\frac{\\partial}{\\partial E} (E - \\rho) - \\mathbf{D}{(E,\\rho)}} = e^{\\rho - \\mathbf{D}{(E,\\rho)}}", "derivation": "\\mathbf{D}{(E,\\rho)} = E - \\rho and \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\rho)} = \\frac{\\partial}{\\partial E} (E - \\rho) and \\rho \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\rho)} = \\rho \\frac{\\partial}{\\partial E} (E - \\rho) and \\rho \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\rho)} - \\mathbf{D}{(E,\\rho)} = \\rho \\frac{\\partial}{\\partial E} (E - \\rho) - \\mathbf{D}{(E,\\rho)} and \\rho \\frac{\\partial}{\\partial E} \\mathbf{D}{(E,\\rho)} - \\mathbf{D}{(E,\\rho)} = \\rho - \\mathbf{D}{(E,\\rho)} and \\rho \\frac{\\partial}{\\partial E} (E - \\rho) - \\mathbf{D}{(E,\\rho)} = \\rho - \\mathbf{D}{(E,\\rho)} and e^{\\rho \\frac{\\partial}{\\partial E} (E - \\rho) - \\mathbf{D}{(E,\\rho)}} = e^{\\rho - \\mathbf{D}{(E,\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho', commutative=True), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["minus", 3, "Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Mul(Symbol('\\\\rho', commutative=True), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["exp", 6], "Equality(exp(Add(Mul(Symbol('\\\\rho', commutative=True), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))))), exp(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))))))"]]}, {"prompt": "Given n{(t_{2},\\varphi)} = \\varphi^{t_{2}}, then derive \\frac{\\partial}{\\partial \\varphi} n{(t_{2},\\varphi)} = \\frac{\\varphi^{t_{2}} t_{2}}{\\varphi}, then obtain - \\sin{(\\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} - \\frac{\\varphi^{t_{2}} t_{2}}{\\varphi})} = 0", "derivation": "n{(t_{2},\\varphi)} = \\varphi^{t_{2}} and \\frac{\\partial}{\\partial \\varphi} n{(t_{2},\\varphi)} = \\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} and - \\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} + \\frac{\\partial}{\\partial \\varphi} n{(t_{2},\\varphi)} = 0 and \\frac{\\partial}{\\partial \\varphi} n{(t_{2},\\varphi)} = \\frac{\\varphi^{t_{2}} t_{2}}{\\varphi} and \\frac{\\partial}{\\partial \\varphi} n{(t_{2},\\varphi)} = \\frac{t_{2} n{(t_{2},\\varphi)}}{\\varphi} and - \\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} + \\frac{t_{2} n{(t_{2},\\varphi)}}{\\varphi} = 0 and - \\sin{(\\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} - \\frac{t_{2} n{(t_{2},\\varphi)}}{\\varphi})} = 0 and - \\sin{(\\frac{\\partial}{\\partial \\varphi} \\varphi^{t_{2}} - \\frac{\\varphi^{t_{2}} t_{2}}{\\varphi})} = 0", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Derivative(Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)))), Integer(0))"], [["sin", 6], "Equality(Mul(Integer(-1), sin(Add(Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Function('n')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)))))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Integer(-1), sin(Add(Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\ddot{x}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})}, then obtain \\int \\frac{\\ddot{x}{(J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon})}} dJ_{\\varepsilon} - 1 = \\int 1 dJ_{\\varepsilon} - 1", "derivation": "\\ddot{x}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\frac{\\ddot{x}{(J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon})}} = 1 and \\int \\frac{\\ddot{x}{(J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon})}} dJ_{\\varepsilon} = \\int 1 dJ_{\\varepsilon} and \\int \\frac{\\ddot{x}{(J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon})}} dJ_{\\varepsilon} - 1 = \\int 1 dJ_{\\varepsilon} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "sin(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\ddot{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Mul(Function('\\\\ddot{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{H}{(A,m)} = \\sin{(A - m)} and f{(\\mathbf{p},v_{1})} = \\frac{\\log{(v_{1})}}{\\mathbf{p}}, then obtain \\frac{\\mathbf{p}^{2} \\mathbf{H}{(A,m)}}{\\log{(v_{1})}} = \\frac{\\mathbf{p}^{2} \\sin{(A - m)}}{\\log{(v_{1})}}", "derivation": "\\mathbf{H}{(A,m)} = \\sin{(A - m)} and f{(\\mathbf{p},v_{1})} = \\frac{\\log{(v_{1})}}{\\mathbf{p}} and \\mathbf{p} \\mathbf{H}{(A,m)} = \\mathbf{p} \\sin{(A - m)} and \\frac{\\mathbf{p} \\mathbf{H}{(A,m)}}{f{(\\mathbf{p},v_{1})}} = \\frac{\\mathbf{p} \\sin{(A - m)}}{f{(\\mathbf{p},v_{1})}} and \\frac{\\mathbf{p}^{2} \\mathbf{H}{(A,m)}}{\\log{(v_{1})}} = \\frac{\\mathbf{p}^{2} \\sin{(A - m)}}{\\log{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A', commutative=True), Symbol('m', commutative=True)), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))))"], ["get_premise", "Equality(Function('f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), log(Symbol('v_1', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{H}')(Symbol('A', commutative=True), Symbol('m', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["divide", 3, "Function('f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{H}')(Symbol('A', commutative=True), Symbol('m', commutative=True)), Pow(Function('f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_1', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)), Function('\\\\mathbf{H}')(Symbol('A', commutative=True), Symbol('m', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)), Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\phi{(\\phi_1,y)} = \\frac{\\phi_1}{y}, then obtain (((\\frac{\\phi^{y}{(\\phi_1,y)}}{y})^{\\phi_1})^{\\phi_1})^{y} = (((\\frac{(\\frac{\\phi_1}{y})^{y}}{y})^{\\phi_1})^{\\phi_1})^{y}", "derivation": "\\phi{(\\phi_1,y)} = \\frac{\\phi_1}{y} and \\phi^{y}{(\\phi_1,y)} = (\\frac{\\phi_1}{y})^{y} and \\frac{\\phi^{y}{(\\phi_1,y)}}{y} = \\frac{(\\frac{\\phi_1}{y})^{y}}{y} and (\\frac{\\phi^{y}{(\\phi_1,y)}}{y})^{\\phi_1} = (\\frac{(\\frac{\\phi_1}{y})^{y}}{y})^{\\phi_1} and ((\\frac{\\phi^{y}{(\\phi_1,y)}}{y})^{\\phi_1})^{\\phi_1} = ((\\frac{(\\frac{\\phi_1}{y})^{y}}{y})^{\\phi_1})^{\\phi_1} and (((\\frac{\\phi^{y}{(\\phi_1,y)}}{y})^{\\phi_1})^{\\phi_1})^{y} = (((\\frac{(\\frac{\\phi_1}{y})^{y}}{y})^{\\phi_1})^{\\phi_1})^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True)))"], [["divide", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('y', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(g)} = \\sin{(e^{g})}, then obtain \\frac{d}{d g} \\int (- e^{g} + \\log{(\\hat{x}_0{(g)})}) dg = \\frac{d}{d g} \\int (- e^{g} + \\log{(\\sin{(e^{g})})}) dg", "derivation": "\\hat{x}_0{(g)} = \\sin{(e^{g})} and \\log{(\\hat{x}_0{(g)})} = \\log{(\\sin{(e^{g})})} and - e^{g} + \\log{(\\hat{x}_0{(g)})} = - e^{g} + \\log{(\\sin{(e^{g})})} and \\int (- e^{g} + \\log{(\\hat{x}_0{(g)})}) dg = \\int (- e^{g} + \\log{(\\sin{(e^{g})})}) dg and \\frac{d}{d g} \\int (- e^{g} + \\log{(\\hat{x}_0{(g)})}) dg = \\frac{d}{d g} \\int (- e^{g} + \\log{(\\sin{(e^{g})})}) dg", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\hat{x}_0')(Symbol('g', commutative=True))), log(sin(exp(Symbol('g', commutative=True)))))"], [["minus", 2, "exp(Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(sin(exp(Symbol('g', commutative=True))))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(sin(exp(Symbol('g', commutative=True))))), Tuple(Symbol('g', commutative=True))))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(Function('\\\\hat{x}_0')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), log(sin(exp(Symbol('g', commutative=True))))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\phi_2,\\delta)} = \\log{(\\delta^{\\phi_2})}, then derive \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\mathbf{J}_P{(\\phi_2,\\delta)} = \\frac{1}{\\delta}, then obtain 1 = \\frac{1}{\\delta \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\log{(\\delta^{\\phi_2})}}", "derivation": "\\mathbf{J}_P{(\\phi_2,\\delta)} = \\log{(\\delta^{\\phi_2})} and \\frac{\\partial}{\\partial \\delta} \\mathbf{J}_P{(\\phi_2,\\delta)} = \\frac{\\partial}{\\partial \\delta} \\log{(\\delta^{\\phi_2})} and \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\mathbf{J}_P{(\\phi_2,\\delta)} = \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\log{(\\delta^{\\phi_2})} and \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\mathbf{J}_P{(\\phi_2,\\delta)} = \\frac{1}{\\delta} and \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\log{(\\delta^{\\phi_2})} = \\frac{1}{\\delta} and 1 = \\frac{1}{\\delta \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\delta} \\log{(\\delta^{\\phi_2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["divide", 5, "Derivative(log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Derivative(log(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given V{(U)} = \\log{(U)} and \\operatorname{z^{*}}{(U)} = - V{(U)}, then obtain \\frac{- V{(U)} + \\operatorname{z^{*}}{(U)} + \\log{(U)}}{- V{(U)} + \\log{(U)}} = - \\frac{V{(U)}}{- V{(U)} + \\log{(U)}}", "derivation": "V{(U)} = \\log{(U)} and \\operatorname{z^{*}}{(U)} = - V{(U)} and - V{(U)} + \\operatorname{z^{*}}{(U)} + \\log{(U)} = - 2 V{(U)} + \\log{(U)} and \\frac{- V{(U)} + \\operatorname{z^{*}}{(U)} + \\log{(U)}}{- V{(U)} + \\log{(U)}} = \\frac{- 2 V{(U)} + \\log{(U)}}{- V{(U)} + \\log{(U)}} and \\operatorname{z^{*}}{(U)} = - \\log{(U)} and - V{(U)} = - 2 V{(U)} + \\log{(U)} and \\frac{- V{(U)} + \\operatorname{z^{*}}{(U)} + \\log{(U)}}{- V{(U)} + \\log{(U)}} = - \\frac{V{(U)}}{- V{(U)} + \\log{(U)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('U', commutative=True)), Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), Function('z^*')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), Function('z^*')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))), Pow(Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('z^*')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), Function('z^*')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('V')(Symbol('U', commutative=True))), log(Symbol('U', commutative=True))), Integer(-1)), Function('V')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\nabla,A_{y})} = e^{A_{y} - \\nabla}, then derive \\int \\mathbf{D}{(\\nabla,A_{y})} d\\nabla = A_{2} - e^{A_{y} - \\nabla}, then obtain A_{2} - e^{A_{y} - \\nabla} = \\int e^{A_{y} - \\nabla} d\\nabla", "derivation": "\\mathbf{D}{(\\nabla,A_{y})} = e^{A_{y} - \\nabla} and \\int \\mathbf{D}{(\\nabla,A_{y})} d\\nabla = \\int e^{A_{y} - \\nabla} d\\nabla and \\int \\mathbf{D}{(\\nabla,A_{y})} d\\nabla = A_{2} - e^{A_{y} - \\nabla} and A_{2} - e^{A_{y} - \\nabla} = \\int e^{A_{y} - \\nabla} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('A_y', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), exp(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), exp(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))), Integral(exp(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\omega{(q)} = e^{q}, then derive \\frac{d}{d q} \\omega{(q)} = e^{q}, then obtain \\int 0 dq = \\int (- e^{q} + \\frac{d}{d q} e^{q}) dq", "derivation": "\\omega{(q)} = e^{q} and - q + \\omega{(q)} = - q + e^{q} and 0 = - \\omega{(q)} + e^{q} and \\frac{d}{d q} \\omega{(q)} = \\frac{d}{d q} e^{q} and \\frac{d}{d q} \\omega{(q)} = e^{q} and \\frac{d}{d q} e^{q} = e^{q} and 0 = - \\omega{(q)} + \\frac{d}{d q} e^{q} and 0 = - e^{q} + \\frac{d}{d q} e^{q} and \\int 0 dq = \\int (- e^{q} + \\frac{d}{d q} e^{q}) dq", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), exp(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), exp(Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('q', commutative=True))), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Symbol('q', commutative=True))), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["integrate", 8, "Symbol('q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('q', commutative=True))), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given v{(l)} = \\cos{(l)}, then derive 1 = \\frac{L_{\\varepsilon} + \\log{(\\cos{(l)})}}{P_{e} + \\log{(v{(l)})}}, then obtain 1 = (\\frac{L_{\\varepsilon} + \\log{(v{(l)})}}{P_{e} + \\log{(v{(l)})}})^{l}", "derivation": "v{(l)} = \\cos{(l)} and \\log{(v{(l)})} = \\log{(\\cos{(l)})} and \\frac{d}{d l} \\log{(v{(l)})} = \\frac{d}{d l} \\log{(\\cos{(l)})} and \\int \\frac{d}{d l} \\log{(v{(l)})} dl = \\int \\frac{d}{d l} \\log{(\\cos{(l)})} dl and 1 = \\frac{\\int \\frac{d}{d l} \\log{(\\cos{(l)})} dl}{\\int \\frac{d}{d l} \\log{(v{(l)})} dl} and 1 = \\frac{L_{\\varepsilon} + \\log{(\\cos{(l)})}}{P_{e} + \\log{(v{(l)})}} and 1 = (\\frac{L_{\\varepsilon} + \\log{(\\cos{(l)})}}{P_{e} + \\log{(v{(l)})}})^{l} and 1 = (\\frac{L_{\\varepsilon} + \\log{(v{(l)})}}{P_{e} + \\log{(v{(l)})}})^{l}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["log", 1], "Equality(log(Function('v')(Symbol('l', commutative=True))), log(cos(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(log(Function('v')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(log(Function('v')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(log(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))))"], [["divide", 4, "Integral(Derivative(log(Function('v')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Derivative(log(Function('v')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integer(-1)), Integral(Derivative(log(cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Integer(1), Mul(Add(Symbol('L_{\\\\varepsilon}', commutative=True), log(cos(Symbol('l', commutative=True)))), Pow(Add(Symbol('P_e', commutative=True), log(Function('v')(Symbol('l', commutative=True)))), Integer(-1))))"], [["power", 6, "Symbol('l', commutative=True)"], "Equality(Integer(1), Pow(Mul(Add(Symbol('L_{\\\\varepsilon}', commutative=True), log(cos(Symbol('l', commutative=True)))), Pow(Add(Symbol('P_e', commutative=True), log(Function('v')(Symbol('l', commutative=True)))), Integer(-1))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Integer(1), Pow(Mul(Add(Symbol('L_{\\\\varepsilon}', commutative=True), log(Function('v')(Symbol('l', commutative=True)))), Pow(Add(Symbol('P_e', commutative=True), log(Function('v')(Symbol('l', commutative=True)))), Integer(-1))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given Q{(I,\\Psi^{\\dagger})} = I \\Psi^{\\dagger}, then derive \\int (- (\\Psi^{\\dagger})^{2} + Q{(I,\\Psi^{\\dagger})}) d\\Psi^{\\dagger} = \\frac{I (\\Psi^{\\dagger})^{2}}{2} + V_{\\mathbf{B}} - \\frac{(\\Psi^{\\dagger})^{3}}{3}, then obtain \\int (I \\Psi^{\\dagger} - (\\Psi^{\\dagger})^{2}) d\\Psi^{\\dagger} = \\frac{I (\\Psi^{\\dagger})^{2}}{2} + V_{\\mathbf{B}} - \\frac{(\\Psi^{\\dagger})^{3}}{3}", "derivation": "Q{(I,\\Psi^{\\dagger})} = I \\Psi^{\\dagger} and - (\\Psi^{\\dagger})^{2} + Q{(I,\\Psi^{\\dagger})} = I \\Psi^{\\dagger} - (\\Psi^{\\dagger})^{2} and \\int (- (\\Psi^{\\dagger})^{2} + Q{(I,\\Psi^{\\dagger})}) d\\Psi^{\\dagger} = \\int (I \\Psi^{\\dagger} - (\\Psi^{\\dagger})^{2}) d\\Psi^{\\dagger} and \\int (- (\\Psi^{\\dagger})^{2} + Q{(I,\\Psi^{\\dagger})}) d\\Psi^{\\dagger} = \\frac{I (\\Psi^{\\dagger})^{2}}{2} + V_{\\mathbf{B}} - \\frac{(\\Psi^{\\dagger})^{3}}{3} and \\int (I \\Psi^{\\dagger} - (\\Psi^{\\dagger})^{2}) d\\Psi^{\\dagger} = \\frac{I (\\Psi^{\\dagger})^{2}}{2} + V_{\\mathbf{B}} - \\frac{(\\Psi^{\\dagger})^{3}}{3}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Function('Q')(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2)))))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Function('Q')(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Function('Q')(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Rational(1, 2), Symbol('I', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Rational(1, 3), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(3)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Rational(1, 2), Symbol('I', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Rational(1, 3), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(3)))))"]]}, {"prompt": "Given b{(\\hat{\\mathbf{x}},q,v_{z})} = - \\hat{\\mathbf{x}} + q v_{z}, then obtain \\cos{(q - \\frac{- \\hat{\\mathbf{x}} + q v_{z}}{b{(\\hat{\\mathbf{x}},q,v_{z})}})} = \\cos{(q - \\frac{2 (- \\hat{\\mathbf{x}} + q v_{z})}{b{(\\hat{\\mathbf{x}},q,v_{z})}} + 1)}", "derivation": "b{(\\hat{\\mathbf{x}},q,v_{z})} = - \\hat{\\mathbf{x}} + q v_{z} and -1 = - \\frac{- \\hat{\\mathbf{x}} + q v_{z}}{b{(\\hat{\\mathbf{x}},q,v_{z})}} and q - 1 = q - \\frac{- \\hat{\\mathbf{x}} + q v_{z}}{b{(\\hat{\\mathbf{x}},q,v_{z})}} and \\cos{(q - 1)} = \\cos{(q - \\frac{- \\hat{\\mathbf{x}} + q v_{z}}{b{(\\hat{\\mathbf{x}},q,v_{z})}})} and \\cos{(q - \\frac{- \\hat{\\mathbf{x}} + q v_{z}}{b{(\\hat{\\mathbf{x}},q,v_{z})}})} = \\cos{(q - \\frac{2 (- \\hat{\\mathbf{x}} + q v_{z})}{b{(\\hat{\\mathbf{x}},q,v_{z})}} + 1)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)))))"], [["cos", 3], "Equality(cos(Add(Symbol('q', commutative=True), Integer(-1))), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))), cos(Add(Symbol('q', commutative=True), Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('b')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(r,A_{1})} = \\log{(r^{A_{1}})} and S{(r,A_{1})} = \\log{(r^{A_{1}})}, then obtain r^{- A_{1}} \\log{(\\frac{r^{A_{1}} S{(r,A_{1})}}{\\log{(r^{A_{1}})}})} = r^{- A_{1}} \\log{(r^{A_{1}})}", "derivation": "\\mathbf{J}_P{(r,A_{1})} = \\log{(r^{A_{1}})} and r^{- A_{1}} \\mathbf{J}_P{(r,A_{1})} = r^{- A_{1}} \\log{(r^{A_{1}})} and S{(r,A_{1})} = \\log{(r^{A_{1}})} and \\frac{r^{A_{1}} S{(r,A_{1})}}{\\log{(r^{A_{1}})}} = r^{A_{1}} and \\mathbf{J}_P{(r,A_{1})} = \\log{(\\frac{r^{A_{1}} S{(r,A_{1})}}{\\log{(r^{A_{1}})}})} and r^{- A_{1}} \\log{(\\frac{r^{A_{1}} S{(r,A_{1})}}{\\log{(r^{A_{1}})}})} = r^{- A_{1}} \\log{(r^{A_{1}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))))"], [["divide", 1, "Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('S')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Pow(log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Integer(-1))), Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{J}_P')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), log(Mul(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Pow(log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Pow(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), log(Mul(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Function('S')(Symbol('r', commutative=True), Symbol('A_1', commutative=True)), Pow(log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True))), Integer(-1))))), Mul(Pow(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), log(Pow(Symbol('r', commutative=True), Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given T{(\\phi_2,E_{\\lambda})} = E_{\\lambda} + \\phi_2, then obtain E_{\\lambda} + \\int T^{3}{(\\phi_2,E_{\\lambda})} d\\phi_2 = E_{\\lambda} + \\int (E_{\\lambda} + \\phi_2) T^{2}{(\\phi_2,E_{\\lambda})} d\\phi_2", "derivation": "T{(\\phi_2,E_{\\lambda})} = E_{\\lambda} + \\phi_2 and T^{2}{(\\phi_2,E_{\\lambda})} = (E_{\\lambda} + \\phi_2) T{(\\phi_2,E_{\\lambda})} and (E_{\\lambda} + \\phi_2) T^{2}{(\\phi_2,E_{\\lambda})} = (E_{\\lambda} + \\phi_2)^{2} T{(\\phi_2,E_{\\lambda})} and T^{3}{(\\phi_2,E_{\\lambda})} = (E_{\\lambda} + \\phi_2) T^{2}{(\\phi_2,E_{\\lambda})} and \\int T^{3}{(\\phi_2,E_{\\lambda})} d\\phi_2 = \\int (E_{\\lambda} + \\phi_2) T^{2}{(\\phi_2,E_{\\lambda})} d\\phi_2 and E_{\\lambda} + \\int T^{3}{(\\phi_2,E_{\\lambda})} d\\phi_2 = E_{\\lambda} + \\int (E_{\\lambda} + \\phi_2) T^{2}{(\\phi_2,E_{\\lambda})} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 1, "Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Integer(2)), Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(3)), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(3)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["add", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Integral(Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(3)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Integral(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Function('T')(Symbol('\\\\phi_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given V{(M,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{M})}, then derive \\frac{\\partial}{\\partial \\mathbf{f}} V{(M,\\mathbf{f})} = - \\frac{\\sin{(\\frac{\\mathbf{f}}{M})}}{M}, then obtain - \\sin{(\\frac{\\mathbf{f}}{M})} = M \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{M})}", "derivation": "V{(M,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{M})} and \\frac{\\partial}{\\partial \\mathbf{f}} V{(M,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{M})} and \\frac{\\partial}{\\partial \\mathbf{f}} V{(M,\\mathbf{f})} = - \\frac{\\sin{(\\frac{\\mathbf{f}}{M})}}{M} and - \\frac{\\sin{(\\frac{\\mathbf{f}}{M})}}{M} = \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{M})} and - \\sin{(\\frac{\\mathbf{f}}{M})} = M \\frac{\\partial}{\\partial \\mathbf{f}} \\cos{(\\frac{\\mathbf{f}}{M})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), Derivative(cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["divide", 4, "Pow(Symbol('M', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Symbol('M', commutative=True), Derivative(cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(n_{1})} = \\log{(\\cos{(n_{1})})}, then obtain - (- 2 n_{1} + W{(n_{1})}) \\cos{(n_{1})} = - (- 2 n_{1} + \\log{(\\cos{(n_{1})})}) \\cos{(n_{1})}", "derivation": "W{(n_{1})} = \\log{(\\cos{(n_{1})})} and - n_{1} + W{(n_{1})} = - n_{1} + \\log{(\\cos{(n_{1})})} and - 2 n_{1} + W{(n_{1})} = - 2 n_{1} + \\log{(\\cos{(n_{1})})} and (- 2 n_{1} + W{(n_{1})}) \\cos{(n_{1})} = (- 2 n_{1} + \\log{(\\cos{(n_{1})})}) \\cos{(n_{1})} and - (- 2 n_{1} + W{(n_{1})}) \\cos{(n_{1})} = - (- 2 n_{1} + \\log{(\\cos{(n_{1})})}) \\cos{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('n_1', commutative=True)), log(cos(Symbol('n_1', commutative=True))))"], [["minus", 1, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('W')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), log(cos(Symbol('n_1', commutative=True)))))"], [["minus", 2, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), Function('W')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), log(cos(Symbol('n_1', commutative=True)))))"], [["times", 3, "cos(Symbol('n_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), Function('W')(Symbol('n_1', commutative=True))), cos(Symbol('n_1', commutative=True))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), log(cos(Symbol('n_1', commutative=True)))), cos(Symbol('n_1', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), Function('W')(Symbol('n_1', commutative=True))), cos(Symbol('n_1', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), log(cos(Symbol('n_1', commutative=True)))), cos(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given l{(\\mathbf{p},\\Omega)} = \\frac{\\Omega}{\\mathbf{p}}, then obtain \\int (\\iint l{(\\mathbf{p},\\Omega)} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} d\\mathbf{p} = \\int (\\iint \\frac{\\Omega}{\\mathbf{p}} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} d\\mathbf{p}", "derivation": "l{(\\mathbf{p},\\Omega)} = \\frac{\\Omega}{\\mathbf{p}} and \\int l{(\\mathbf{p},\\Omega)} d\\mathbf{p} = \\int \\frac{\\Omega}{\\mathbf{p}} d\\mathbf{p} and \\iint l{(\\mathbf{p},\\Omega)} d\\mathbf{p} d\\Omega = \\iint \\frac{\\Omega}{\\mathbf{p}} d\\mathbf{p} d\\Omega and (\\iint l{(\\mathbf{p},\\Omega)} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} = (\\iint \\frac{\\Omega}{\\mathbf{p}} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} and \\int (\\iint l{(\\mathbf{p},\\Omega)} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} d\\mathbf{p} = \\int (\\iint \\frac{\\Omega}{\\mathbf{p}} d\\mathbf{p} d\\Omega)^{\\mathbf{p}} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Integral(Function('l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Pow(Integral(Function('l')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(C_{2})} = \\sin{(e^{C_{2}})}, then obtain \\frac{d}{d C_{2}} (\\eta^{\\prime}{(C_{2})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}}) = \\frac{d}{d C_{2}} (\\sin{(e^{C_{2}})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}})", "derivation": "\\eta^{\\prime}{(C_{2})} = \\sin{(e^{C_{2}})} and \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}} = \\frac{\\sin{(e^{C_{2}})}}{C_{2}} and \\eta^{\\prime}{(C_{2})} + \\frac{\\sin{(e^{C_{2}})}}{C_{2}} = \\sin{(e^{C_{2}})} + \\frac{\\sin{(e^{C_{2}})}}{C_{2}} and \\eta^{\\prime}{(C_{2})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}} = \\sin{(e^{C_{2}})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}} and \\frac{d}{d C_{2}} (\\eta^{\\prime}{(C_{2})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}}) = \\frac{d}{d C_{2}} (\\sin{(e^{C_{2}})} + \\frac{\\eta^{\\prime}{(C_{2})}}{C_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)), sin(exp(Symbol('C_2', commutative=True))))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(exp(Symbol('C_2', commutative=True)))))"], [["add", 1, "Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(exp(Symbol('C_2', commutative=True))))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(exp(Symbol('C_2', commutative=True))))), Add(sin(exp(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(exp(Symbol('C_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)))), Add(sin(exp(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)))))"], [["differentiate", 4, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(sin(exp(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} = \\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})}, then obtain g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + e^{g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + 1} + 1 = g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + e^{\\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})} + 1} + 1", "derivation": "g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} = \\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})} and g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + 1 = \\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})} + 1 and e^{g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + 1} = e^{\\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})} + 1} and g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + e^{g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + 1} + 1 = g{(\\varepsilon_0,g^{\\prime}_{\\varepsilon})} + e^{\\cos{(\\frac{\\varepsilon_0}{g^{\\prime}_{\\varepsilon}})} + 1} + 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(cos(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))), Integer(1)))"], [["exp", 2], "Equality(exp(Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))), exp(Add(cos(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))), Integer(1))))"], [["add", 3, "Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))"], "Equality(Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), exp(Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))), Integer(1)), Add(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), exp(Add(cos(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))), Integer(1))), Integer(1)))"]]}, {"prompt": "Given W{(\\varphi^*,\\Psi)} = \\Psi \\varphi^*, then derive \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} = 0, then obtain \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} + \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} W{(\\varphi^*,\\Psi)} = \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} W{(\\varphi^*,\\Psi)}", "derivation": "W{(\\varphi^*,\\Psi)} = \\Psi \\varphi^* and \\frac{\\partial}{\\partial \\varphi^*} W{(\\varphi^*,\\Psi)} = \\frac{\\partial}{\\partial \\varphi^*} \\Psi \\varphi^* and \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} = \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} \\Psi \\varphi^* and \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} = 0 and \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} + \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} \\Psi \\varphi^* = \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} \\Psi \\varphi^* and \\frac{\\partial^{2}}{\\partial (\\varphi^*)^{2}} W{(\\varphi^*,\\Psi)} + \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} W{(\\varphi^*,\\Psi)} = \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\varphi^*} W{(\\varphi^*,\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Integer(0))"], [["add", 4, "Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Derivative(Function('W')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(a,v_{t})} = a^{v_{t}} and \\hat{\\mathbf{r}}{(a,v_{t})} = - \\mathbf{S}{(a,v_{t})}, then obtain 0 = (a^{v_{t}} - \\mathbf{S}{(a,v_{t})}) (a^{v_{t}} + \\hat{\\mathbf{r}}{(a,v_{t})} - \\mathbf{S}{(a,v_{t})})", "derivation": "\\mathbf{S}{(a,v_{t})} = a^{v_{t}} and 0 = a^{v_{t}} - \\mathbf{S}{(a,v_{t})} and \\hat{\\mathbf{r}}{(a,v_{t})} = - \\mathbf{S}{(a,v_{t})} and a^{v_{t}} + \\hat{\\mathbf{r}}{(a,v_{t})} - \\mathbf{S}{(a,v_{t})} = a^{v_{t}} - 2 \\mathbf{S}{(a,v_{t})} and 0 = (a^{v_{t}} - 2 \\mathbf{S}{(a,v_{t})}) (a^{v_{t}} - \\mathbf{S}{(a,v_{t})}) and 0 = (a^{v_{t}} - \\mathbf{S}{(a,v_{t})}) (a^{v_{t}} + \\hat{\\mathbf{r}}{(a,v_{t})} - \\mathbf{S}{(a,v_{t})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))))"], [["add", 3, "Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))))"], "Equality(Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))), Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))))"], [["times", 2, "Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))))"], "Equality(Integer(0), Mul(Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))), Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(0), Mul(Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)))), Add(Pow(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(C_{2})} = \\log{(C_{2})}, then obtain \\int \\frac{d}{d C_{2}} (- C_{2} + \\operatorname{f^{\\prime}}{(C_{2})}) dC_{2} = \\int \\frac{d}{d C_{2}} (- C_{2} + \\log{(C_{2})}) dC_{2}", "derivation": "\\operatorname{f^{\\prime}}{(C_{2})} = \\log{(C_{2})} and - C_{2} + \\operatorname{f^{\\prime}}{(C_{2})} = - C_{2} + \\log{(C_{2})} and \\frac{d}{d C_{2}} (- C_{2} + \\operatorname{f^{\\prime}}{(C_{2})}) = \\frac{d}{d C_{2}} (- C_{2} + \\log{(C_{2})}) and \\int \\frac{d}{d C_{2}} (- C_{2} + \\operatorname{f^{\\prime}}{(C_{2})}) dC_{2} = \\int \\frac{d}{d C_{2}} (- C_{2} + \\log{(C_{2})}) dC_{2}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["minus", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\Psi{(g_{\\varepsilon})} = \\cos{(e^{g_{\\varepsilon}})} and \\mu_{0}{(g_{\\varepsilon})} = - \\frac{d}{d g_{\\varepsilon}} \\cos{(e^{g_{\\varepsilon}})}, then obtain \\mu_{0}{(g_{\\varepsilon})} = - \\frac{d}{d g_{\\varepsilon}} \\Psi{(g_{\\varepsilon})}", "derivation": "\\Psi{(g_{\\varepsilon})} = \\cos{(e^{g_{\\varepsilon}})} and \\frac{d}{d g_{\\varepsilon}} \\Psi{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} \\cos{(e^{g_{\\varepsilon}})} and \\mu_{0}{(g_{\\varepsilon})} = - \\frac{d}{d g_{\\varepsilon}} \\cos{(e^{g_{\\varepsilon}})} and \\mu_{0}{(g_{\\varepsilon})} = - \\frac{d}{d g_{\\varepsilon}} \\Psi{(g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(cos(exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\Psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(L)} = L, then derive \\frac{d}{d L} \\operatorname{v_{z}}{(L)} = 1, then obtain \\frac{\\frac{d}{d \\operatorname{v_{z}}{(L)}} \\operatorname{v_{z}}{(L)}}{\\operatorname{v_{z}}{(L)} + 2} = \\frac{1}{\\operatorname{v_{z}}{(L)} + 2}", "derivation": "\\operatorname{v_{z}}{(L)} = L and \\frac{d}{d L} \\operatorname{v_{z}}{(L)} = \\frac{d}{d L} L and \\frac{d}{d L} \\operatorname{v_{z}}{(L)} = 1 and \\frac{d}{d L} L = 1 and \\frac{\\frac{d}{d L} L}{L + 2} = \\frac{1}{L + 2} and \\frac{\\frac{d}{d \\operatorname{v_{z}}{(L)}} \\operatorname{v_{z}}{(L)}}{\\operatorname{v_{z}}{(L)} + 2} = \\frac{1}{\\operatorname{v_{z}}{(L)} + 2}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Add(Symbol('L', commutative=True), Integer(2))"], "Equality(Mul(Pow(Add(Symbol('L', commutative=True), Integer(2)), Integer(-1)), Derivative(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True), Integer(1)))), Pow(Add(Symbol('L', commutative=True), Integer(2)), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Add(Function('v_z')(Symbol('L', commutative=True)), Integer(2)), Integer(-1)), Derivative(Function('v_z')(Symbol('L', commutative=True)), Tuple(Function('v_z')(Symbol('L', commutative=True)), Integer(1)))), Pow(Add(Function('v_z')(Symbol('L', commutative=True)), Integer(2)), Integer(-1)))"]]}, {"prompt": "Given \\rho{(C_{d})} = \\cos{(C_{d})}, then derive \\frac{d}{d C_{d}} \\rho{(C_{d})} = - \\sin{(C_{d})}, then obtain \\frac{d^{2}}{d C_{d}^{2}} \\rho{(C_{d})} = - \\cos{(C_{d})}", "derivation": "\\rho{(C_{d})} = \\cos{(C_{d})} and \\frac{d}{d C_{d}} \\rho{(C_{d})} = \\frac{d}{d C_{d}} \\cos{(C_{d})} and \\frac{d}{d C_{d}} \\rho{(C_{d})} = - \\sin{(C_{d})} and \\frac{d^{2}}{d C_{d}^{2}} \\rho{(C_{d})} = \\frac{d}{d C_{d}} - \\sin{(C_{d})} and \\frac{d^{2}}{d C_{d}^{2}} \\rho{(C_{d})} = - \\cos{(C_{d})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_d', commutative=True))))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\hat{p})} = \\cos{(\\hat{p})}, then derive \\hat{p} + \\sigma_p = \\int \\frac{\\cos{(\\hat{p})}}{\\mathbf{s}{(\\hat{p})}} d\\hat{p}, then obtain \\iint 1 d\\hat{p} d\\hat{p} = \\int (\\hat{p} + \\sigma_p) d\\hat{p}", "derivation": "\\mathbf{s}{(\\hat{p})} = \\cos{(\\hat{p})} and 1 = \\frac{\\cos{(\\hat{p})}}{\\mathbf{s}{(\\hat{p})}} and \\int 1 d\\hat{p} = \\int \\frac{\\cos{(\\hat{p})}}{\\mathbf{s}{(\\hat{p})}} d\\hat{p} and \\hat{p} + \\sigma_p = \\int \\frac{\\cos{(\\hat{p})}}{\\mathbf{s}{(\\hat{p})}} d\\hat{p} and \\int 1 d\\hat{p} = \\hat{p} + \\sigma_p and \\iint 1 d\\hat{p} d\\hat{p} = \\int (\\hat{p} + \\sigma_p) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{s}')(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given u{(W)} = \\cos{(W)}, then derive \\int u{(W)} dW = \\hbar + \\sin{(W)}, then obtain W + \\int u{(W)} dW = W + \\hbar + \\sin{(W)}", "derivation": "u{(W)} = \\cos{(W)} and \\int u{(W)} dW = \\int \\cos{(W)} dW and W + \\int u{(W)} dW = W + \\int \\cos{(W)} dW and \\int u{(W)} dW = \\hbar + \\sin{(W)} and \\hbar + \\sin{(W)} = \\int \\cos{(W)} dW and W + \\int u{(W)} dW = W + \\hbar + \\sin{(W)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["add", 2, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), sin(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('\\\\hbar', commutative=True), sin(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Symbol('W', commutative=True), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True), sin(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\hat{H},\\Psi,u)} = (\\hat{H} - u)^{\\Psi}, then obtain (\\hat{H} - u)^{2 \\Psi} = \\frac{(\\hat{H} - u)^{4 \\Psi}}{\\Psi_{\\lambda}^{2}{(\\hat{H},\\Psi,u)}}", "derivation": "\\Psi_{\\lambda}{(\\hat{H},\\Psi,u)} = (\\hat{H} - u)^{\\Psi} and (\\hat{H} - u)^{\\Psi} \\Psi_{\\lambda}{(\\hat{H},\\Psi,u)} = (\\hat{H} - u)^{2 \\Psi} and (\\hat{H} - u)^{\\Psi} = \\frac{(\\hat{H} - u)^{2 \\Psi}}{\\Psi_{\\lambda}{(\\hat{H},\\Psi,u)}} and (\\hat{H} - u)^{2 \\Psi} = \\frac{(\\hat{H} - u)^{4 \\Psi}}{\\Psi_{\\lambda}^{2}{(\\hat{H},\\Psi,u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('u', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["times", 1, "Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('\\\\Psi', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('u', commutative=True))), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('u', commutative=True))"], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi', commutative=True))), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('u', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Mul(Integer(4), Symbol('\\\\Psi', commutative=True))), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('u', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given i{(J,C)} = \\frac{J}{C} and \\mathbf{J}_P{(H)} = \\sin{(\\sin{(H)})}, then obtain (\\frac{J}{C})^{J} + \\mathbf{J}_P^{H}{(H)} - 1 = (\\frac{J}{C})^{J} + \\sin^{H}{(\\sin{(H)})} - 1", "derivation": "i{(J,C)} = \\frac{J}{C} and i^{J}{(J,C)} = (\\frac{J}{C})^{J} and i^{J}{(J,C)} - 1 = (\\frac{J}{C})^{J} - 1 and \\mathbf{J}_P{(H)} = \\sin{(\\sin{(H)})} and \\mathbf{J}_P^{H}{(H)} = \\sin^{H}{(\\sin{(H)})} and \\mathbf{J}_P^{H}{(H)} + i^{J}{(J,C)} - 1 = i^{J}{(J,C)} + \\sin^{H}{(\\sin{(H)})} - 1 and (\\frac{J}{C})^{J} + \\mathbf{J}_P^{H}{(H)} - 1 = (\\frac{J}{C})^{J} + \\sin^{H}{(\\sin{(H)})} - 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Symbol('J', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Symbol('J', commutative=True)), Integer(-1)), Add(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Integer(-1)))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["power", 4, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(sin(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["add", 5, "Add(Pow(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Symbol('J', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Function('\\\\mathbf{J}_P')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Symbol('J', commutative=True)), Integer(-1)), Add(Pow(Function('i')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Symbol('J', commutative=True)), Pow(sin(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Add(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(sin(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given u{(T,m,z^{*})} = m^{T} + z^{*}, then obtain 4 u^{2}{(T,m,z^{*})} = (m^{T} + z^{*} + u{(T,m,z^{*})})^{2}", "derivation": "u{(T,m,z^{*})} = m^{T} + z^{*} and m^{T} + z^{*} + u{(T,m,z^{*})} = 2 m^{T} + 2 z^{*} and 2 u{(T,m,z^{*})} = 2 m^{T} + 2 z^{*} and 2 u{(T,m,z^{*})} = m^{T} + z^{*} + u{(T,m,z^{*})} and 4 u^{2}{(T,m,z^{*})} = (m^{T} + z^{*} + u{(T,m,z^{*})})^{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True)), Add(Pow(Symbol('m', commutative=True), Symbol('T', commutative=True)), Symbol('z^*', commutative=True)))"], [["add", 1, "Add(Pow(Symbol('m', commutative=True), Symbol('T', commutative=True)), Symbol('z^*', commutative=True))"], "Equality(Add(Pow(Symbol('m', commutative=True), Symbol('T', commutative=True)), Symbol('z^*', commutative=True), Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True))), Add(Pow(Symbol('m', commutative=True), Symbol('T', commutative=True)), Symbol('z^*', commutative=True), Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Pow(Add(Pow(Symbol('m', commutative=True), Symbol('T', commutative=True)), Symbol('z^*', commutative=True), Function('u')(Symbol('T', commutative=True), Symbol('m', commutative=True), Symbol('z^*', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\omega{(\\hat{\\mathbf{x}},\\mathbf{H})} = \\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}, then obtain - \\frac{2 \\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} + 2 = 0", "derivation": "\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})} = \\mathbf{H} \\sin{(\\hat{\\mathbf{x}})} and 1 = \\frac{\\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} and - \\frac{\\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} + 1 = 0 and \\frac{\\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} + 1 = \\frac{2 \\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} and - \\frac{2 \\mathbf{H} \\sin{(\\hat{\\mathbf{x}})}}{\\omega{(\\hat{\\mathbf{x}},\\mathbf{H})}} + 2 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["divide", 1, "Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(1)), Integer(0))"], [["add", 2, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(1)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(2)), Integer(0))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(J_{\\varepsilon},L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{J_{\\varepsilon}}, then obtain - L_{\\varepsilon} + \\operatorname{y^{\\prime}}^{L_{\\varepsilon}}{(J_{\\varepsilon},L_{\\varepsilon})} + \\frac{1}{J_{\\varepsilon}} = - L_{\\varepsilon} + (\\frac{L_{\\varepsilon}}{J_{\\varepsilon}})^{L_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}}", "derivation": "\\operatorname{y^{\\prime}}{(J_{\\varepsilon},L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{J_{\\varepsilon}} and \\operatorname{y^{\\prime}}^{L_{\\varepsilon}}{(J_{\\varepsilon},L_{\\varepsilon})} = (\\frac{L_{\\varepsilon}}{J_{\\varepsilon}})^{L_{\\varepsilon}} and \\operatorname{y^{\\prime}}^{L_{\\varepsilon}}{(J_{\\varepsilon},L_{\\varepsilon})} + \\frac{1}{J_{\\varepsilon}} = (\\frac{L_{\\varepsilon}}{J_{\\varepsilon}})^{L_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}} and - L_{\\varepsilon} + \\operatorname{y^{\\prime}}^{L_{\\varepsilon}}{(J_{\\varepsilon},L_{\\varepsilon})} + \\frac{1}{J_{\\varepsilon}} = - L_{\\varepsilon} + (\\frac{L_{\\varepsilon}}{J_{\\varepsilon}})^{L_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Add(Pow(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["minus", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(S)} = \\log{(\\log{(S)})} and W{(S)} = \\log{(\\log{(S)})}, then obtain 1 = ((W^{S}{(S)})^{S})^{S} ((\\Psi_{nl}^{S}{(S)})^{S})^{- S}", "derivation": "\\Psi_{nl}{(S)} = \\log{(\\log{(S)})} and \\Psi_{nl}^{S}{(S)} = \\log{(\\log{(S)})}^{S} and W{(S)} = \\log{(\\log{(S)})} and \\Psi_{nl}^{S}{(S)} = W^{S}{(S)} and (\\Psi_{nl}^{S}{(S)})^{S} = (W^{S}{(S)})^{S} and ((\\Psi_{nl}^{S}{(S)})^{S})^{S} = ((W^{S}{(S)})^{S})^{S} and 1 = ((W^{S}{(S)})^{S})^{S} ((\\Psi_{nl}^{S}{(S)})^{S})^{- S}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), log(log(Symbol('S', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(log(log(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('W')(Symbol('S', commutative=True)), log(log(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Function('W')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Function('W')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["power", 5, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Pow(Function('W')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["divide", 6, "Pow(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(Pow(Function('W')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)))))"]]}, {"prompt": "Given E{(f_{\\mathbf{v}},Q)} = \\frac{f_{\\mathbf{v}}}{Q}, then obtain E{(f_{\\mathbf{v}},Q)} - \\frac{1}{Q} = - \\frac{1}{Q} + \\frac{f_{\\mathbf{v}}^{2}}{Q^{2} E{(f_{\\mathbf{v}},Q)}}", "derivation": "E{(f_{\\mathbf{v}},Q)} = \\frac{f_{\\mathbf{v}}}{Q} and 1 = \\frac{f_{\\mathbf{v}}}{Q E{(f_{\\mathbf{v}},Q)}} and \\frac{f_{\\mathbf{v}}}{Q} = \\frac{f_{\\mathbf{v}}^{2}}{Q^{2} E{(f_{\\mathbf{v}},Q)}} and E{(f_{\\mathbf{v}},Q)} = \\frac{f_{\\mathbf{v}}^{2}}{Q^{2} E{(f_{\\mathbf{v}},Q)}} and E{(f_{\\mathbf{v}},Q)} - \\frac{1}{Q} = - \\frac{1}{Q} + \\frac{f_{\\mathbf{v}}^{2}}{Q^{2} E{(f_{\\mathbf{v}},Q)}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 1, "Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["times", 2, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2)), Pow(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2)), Pow(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["minus", 4, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Add(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-2)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(2)), Pow(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} = e^{\\Psi_{nl} - \\mu_0}, then derive \\int \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} d\\Psi_{nl} = l + e^{\\Psi_{nl} - \\mu_0}, then obtain \\int \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} d\\Psi_{nl} = l + \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} = e^{\\Psi_{nl} - \\mu_0} and \\int \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} d\\Psi_{nl} = \\int e^{\\Psi_{nl} - \\mu_0} d\\Psi_{nl} and \\int \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} d\\Psi_{nl} = l + e^{\\Psi_{nl} - \\mu_0} and \\int \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})} d\\Psi_{nl} = l + \\operatorname{g_{\\varepsilon}}{(\\mu_0,\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('l', commutative=True), exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('l', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\psi^*)} = e^{\\psi^*}, then derive \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)} = e^{\\psi^*}, then obtain (\\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)})^{2} = \\operatorname{A_{z}}{(\\psi^*)} \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)}", "derivation": "\\operatorname{A_{z}}{(\\psi^*)} = e^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)} = \\frac{d}{d \\psi^*} e^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)} = e^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)} = \\operatorname{A_{z}}{(\\psi^*)} and \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)} \\frac{d}{d \\psi^*} e^{\\psi^*} = \\operatorname{A_{z}}{(\\psi^*)} \\frac{d}{d \\psi^*} e^{\\psi^*} and (\\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)})^{2} = \\operatorname{A_{z}}{(\\psi^*)} \\frac{d}{d \\psi^*} \\operatorname{A_{z}}{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), exp(Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Function('A_z')(Symbol('\\\\psi^*', commutative=True)))"], [["times", 4, "Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(2)), Mul(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Derivative(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(r_{0})} = \\log{(r_{0})}, then obtain - \\hat{p}_0{(r_{0})} \\hat{p}_0^{r_{0}}{(r_{0})} + \\hat{p}_0^{r_{0}}{(r_{0})} = - \\hat{p}_0{(r_{0})} \\hat{p}_0^{r_{0}}{(r_{0})} + \\log{(r_{0})}^{r_{0}}", "derivation": "\\hat{p}_0{(r_{0})} = \\log{(r_{0})} and \\hat{p}_0^{r_{0}}{(r_{0})} = \\log{(r_{0})}^{r_{0}} and \\hat{p}_0{(r_{0})} \\hat{p}_0^{r_{0}}{(r_{0})} = \\hat{p}_0{(r_{0})} \\log{(r_{0})}^{r_{0}} and - \\hat{p}_0{(r_{0})} \\log{(r_{0})}^{r_{0}} + \\hat{p}_0^{r_{0}}{(r_{0})} = - \\hat{p}_0{(r_{0})} \\log{(r_{0})}^{r_{0}} + \\log{(r_{0})}^{r_{0}} and - \\hat{p}_0{(r_{0})} \\hat{p}_0^{r_{0}}{(r_{0})} + \\hat{p}_0^{r_{0}}{(r_{0})} = - \\hat{p}_0{(r_{0})} \\hat{p}_0^{r_{0}}{(r_{0})} + \\log{(r_{0})}^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["times", 2, "Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Mul(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["minus", 2, "Mul(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given n{(C)} = \\log{(C)}, then obtain T + 3 n{(C)} + \\log{(C)} = T + n{(C)} + 3 \\log{(C)}", "derivation": "n{(C)} = \\log{(C)} and 2 n{(C)} = n{(C)} + \\log{(C)} and 3 n{(C)} + \\log{(C)} = 2 n{(C)} + 2 \\log{(C)} and 3 n{(C)} + \\log{(C)} = n{(C)} + 3 \\log{(C)} and T + 3 n{(C)} + \\log{(C)} = T + n{(C)} + 3 \\log{(C)}", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["add", 1, "Function('n')(Symbol('C', commutative=True))"], "Equality(Mul(Integer(2), Function('n')(Symbol('C', commutative=True))), Add(Function('n')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))))"], [["add", 2, "Add(Function('n')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('n')(Symbol('C', commutative=True))), log(Symbol('C', commutative=True))), Add(Mul(Integer(2), Function('n')(Symbol('C', commutative=True))), Mul(Integer(2), log(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('n')(Symbol('C', commutative=True))), log(Symbol('C', commutative=True))), Add(Function('n')(Symbol('C', commutative=True)), Mul(Integer(3), log(Symbol('C', commutative=True)))))"], [["add", 4, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(3), Function('n')(Symbol('C', commutative=True))), log(Symbol('C', commutative=True))), Add(Symbol('T', commutative=True), Function('n')(Symbol('C', commutative=True)), Mul(Integer(3), log(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(v_{2})} = \\sin{(v_{2})}, then obtain \\frac{\\log{(\\mathbb{I}{(v_{2})})} + \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\sin{(v_{2})})})}} = \\frac{2 \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\sin{(v_{2})})})}}", "derivation": "\\mathbb{I}{(v_{2})} = \\sin{(v_{2})} and \\log{(\\mathbb{I}{(v_{2})})} = \\log{(\\sin{(v_{2})})} and \\log{(\\mathbb{I}{(v_{2})})} + \\log{(\\sin{(v_{2})})} = 2 \\log{(\\sin{(v_{2})})} and \\sin{(\\log{(\\mathbb{I}{(v_{2})})})} = \\sin{(\\log{(\\sin{(v_{2})})})} and \\frac{\\log{(\\mathbb{I}{(v_{2})})} + \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\mathbb{I}{(v_{2})})})}} = \\frac{2 \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\mathbb{I}{(v_{2})})})}} and \\frac{\\log{(\\mathbb{I}{(v_{2})})} + \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\sin{(v_{2})})})}} = \\frac{2 \\log{(\\sin{(v_{2})})}}{\\sin{(\\log{(\\sin{(v_{2})})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True))), log(sin(Symbol('v_2', commutative=True))))"], [["add", 2, "log(sin(Symbol('v_2', commutative=True)))"], "Equality(Add(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True))), log(sin(Symbol('v_2', commutative=True)))), Mul(Integer(2), log(sin(Symbol('v_2', commutative=True)))))"], [["sin", 2], "Equality(sin(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True)))), sin(log(sin(Symbol('v_2', commutative=True)))))"], [["divide", 3, "sin(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True))))"], "Equality(Mul(Add(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True))), log(sin(Symbol('v_2', commutative=True)))), Pow(sin(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True)))), Integer(-1))), Mul(Integer(2), log(sin(Symbol('v_2', commutative=True))), Pow(sin(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(log(Function('\\\\mathbb{I}')(Symbol('v_2', commutative=True))), log(sin(Symbol('v_2', commutative=True)))), Pow(sin(log(sin(Symbol('v_2', commutative=True)))), Integer(-1))), Mul(Integer(2), log(sin(Symbol('v_2', commutative=True))), Pow(sin(log(sin(Symbol('v_2', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given Z{(h,A_{z})} = - A_{z} + h, then derive \\int Z{(h,A_{z})} dA_{z} = - \\frac{A_{z}^{2}}{2} + A_{z} h + \\mathbf{F}, then obtain - \\frac{A_{z}^{2}}{2} + \\iint Z{(h,A_{z})} dA_{z} dA_{z} = - \\frac{A_{z}^{2}}{2} + \\int (A_{z} h + \\mathbf{F} - \\frac{(h - Z{(h,A_{z})})^{2}}{2}) dA_{z}", "derivation": "Z{(h,A_{z})} = - A_{z} + h and - h + Z{(h,A_{z})} = - A_{z} and \\int Z{(h,A_{z})} dA_{z} = \\int (- A_{z} + h) dA_{z} and \\int Z{(h,A_{z})} dA_{z} = - \\frac{A_{z}^{2}}{2} + A_{z} h + \\mathbf{F} and \\int Z{(h,A_{z})} dA_{z} = A_{z} h + \\mathbf{F} - \\frac{(h - Z{(h,A_{z})})^{2}}{2} and \\iint Z{(h,A_{z})} dA_{z} dA_{z} = \\int (A_{z} h + \\mathbf{F} - \\frac{(h - Z{(h,A_{z})})^{2}}{2}) dA_{z} and - \\frac{A_{z}^{2}}{2} + \\iint Z{(h,A_{z})} dA_{z} dA_{z} = - \\frac{A_{z}^{2}}{2} + \\int (A_{z} h + \\mathbf{F} - \\frac{(h - Z{(h,A_{z})})^{2}}{2}) dA_{z}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True)))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Mul(Symbol('A_z', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)))), Integer(2)))))"], [["integrate", 5, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Mul(Symbol('A_z', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)))), Integer(2)))), Tuple(Symbol('A_z', commutative=True))))"], [["add", 6, "Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Integral(Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Integral(Add(Mul(Symbol('A_z', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('Z')(Symbol('h', commutative=True), Symbol('A_z', commutative=True)))), Integer(2)))), Tuple(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(E_{n})} = e^{E_{n}}, then derive \\frac{d}{d E_{n}} \\operatorname{P_{e}}{(E_{n})} = e^{E_{n}}, then obtain 2 \\frac{d}{d E_{n}} e^{E_{n}} = e^{E_{n}} + \\frac{d}{d E_{n}} e^{E_{n}}", "derivation": "\\operatorname{P_{e}}{(E_{n})} = e^{E_{n}} and \\frac{d}{d E_{n}} \\operatorname{P_{e}}{(E_{n})} = \\frac{d}{d E_{n}} e^{E_{n}} and \\frac{d}{d E_{n}} \\operatorname{P_{e}}{(E_{n})} = e^{E_{n}} and 2 \\frac{d}{d E_{n}} \\operatorname{P_{e}}{(E_{n})} = e^{E_{n}} + \\frac{d}{d E_{n}} \\operatorname{P_{e}}{(E_{n})} and 2 \\frac{d}{d E_{n}} e^{E_{n}} = e^{E_{n}} + \\frac{d}{d E_{n}} e^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), exp(Symbol('E_n', commutative=True)))"], [["add", 3, "Derivative(Function('P_e')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('P_e')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Add(exp(Symbol('E_n', commutative=True)), Derivative(Function('P_e')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Add(exp(Symbol('E_n', commutative=True)), Derivative(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(f_{\\mathbf{v}},r)} = f_{\\mathbf{v}} + r, then obtain 2 \\hat{H}_l^{2 f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} = (f_{\\mathbf{v}} + r)^{f_{\\mathbf{v}}} \\hat{H}_l^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} + \\hat{H}_l^{2 f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)}", "derivation": "\\hat{H}_l{(f_{\\mathbf{v}},r)} = f_{\\mathbf{v}} + r and \\hat{H}_l^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} = (f_{\\mathbf{v}} + r)^{f_{\\mathbf{v}}} and \\hat{H}_l^{2 f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} = (f_{\\mathbf{v}} + r)^{f_{\\mathbf{v}}} \\hat{H}_l^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} and 2 \\hat{H}_l^{2 f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} = (f_{\\mathbf{v}} + r)^{f_{\\mathbf{v}}} \\hat{H}_l^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)} + \\hat{H}_l^{2 f_{\\mathbf{v}}}{(f_{\\mathbf{v}},r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 2, "Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 3, "Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given B{(g)} = \\log{(g)} and \\dot{\\mathbf{r}}{(g)} = \\log{(g)}^{2}, then obtain \\frac{((B^{2}{(g)})^{g})^{g}}{g} = \\frac{((B{(g)} \\log{(g)})^{g})^{g}}{g}", "derivation": "B{(g)} = \\log{(g)} and B{(g)} \\log{(g)} = \\log{(g)}^{2} and \\dot{\\mathbf{r}}{(g)} = \\log{(g)}^{2} and \\dot{\\mathbf{r}}{(g)} = B{(g)} \\log{(g)} and \\dot{\\mathbf{r}}{(g)} = B^{2}{(g)} and \\dot{\\mathbf{r}}^{g}{(g)} = (B{(g)} \\log{(g)})^{g} and (\\dot{\\mathbf{r}}^{g}{(g)})^{g} = ((B{(g)} \\log{(g)})^{g})^{g} and \\frac{(\\dot{\\mathbf{r}}^{g}{(g)})^{g}}{g} = \\frac{((B{(g)} \\log{(g)})^{g})^{g}}{g} and \\frac{((B^{2}{(g)})^{g})^{g}}{g} = \\frac{((B{(g)} \\log{(g)})^{g})^{g}}{g}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["times", 1, "log(Symbol('g', commutative=True))"], "Equality(Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Pow(log(Symbol('g', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Pow(Function('B')(Symbol('g', commutative=True)), Integer(2)))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["power", 6, "Symbol('g', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["divide", 7, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Pow(Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Pow(Pow(Function('B')(Symbol('g', commutative=True)), Integer(2)), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Pow(Mul(Function('B')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\pi,v_{1})} = v_{1} + e^{\\pi} and \\mathbf{p}{(\\pi)} = e^{\\pi}, then obtain ((\\frac{\\hat{H}_{\\lambda}{(\\pi,v_{1})}}{v_{1}})^{v_{1}})^{\\pi} = ((\\frac{v_{1} + \\mathbf{p}{(\\pi)}}{v_{1}})^{v_{1}})^{\\pi}", "derivation": "\\hat{H}_{\\lambda}{(\\pi,v_{1})} = v_{1} + e^{\\pi} and \\frac{\\hat{H}_{\\lambda}{(\\pi,v_{1})}}{v_{1}} = \\frac{v_{1} + e^{\\pi}}{v_{1}} and (\\frac{\\hat{H}_{\\lambda}{(\\pi,v_{1})}}{v_{1}})^{v_{1}} = (\\frac{v_{1} + e^{\\pi}}{v_{1}})^{v_{1}} and ((\\frac{\\hat{H}_{\\lambda}{(\\pi,v_{1})}}{v_{1}})^{v_{1}})^{\\pi} = ((\\frac{v_{1} + e^{\\pi}}{v_{1}})^{v_{1}})^{\\pi} and \\mathbf{p}{(\\pi)} = e^{\\pi} and ((\\frac{\\hat{H}_{\\lambda}{(\\pi,v_{1})}}{v_{1}})^{v_{1}})^{\\pi} = ((\\frac{v_{1} + \\mathbf{p}{(\\pi)}}{v_{1}})^{v_{1}})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\pi', commutative=True)))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\pi', commutative=True)))), Symbol('v_1', commutative=True)))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\pi', commutative=True)))), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\pi', commutative=True)))), Symbol('v_1', commutative=True)), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\chi{(A)} = \\frac{1}{A}, then obtain (\\log{(\\chi{(A)})} - \\int (\\frac{1}{A})^{A} dA)^{A} = (\\log{(\\frac{1}{A})} - \\int (\\frac{1}{A})^{A} dA)^{A}", "derivation": "\\chi{(A)} = \\frac{1}{A} and \\log{(\\chi{(A)})} = \\log{(\\frac{1}{A})} and \\chi^{A}{(A)} = (\\frac{1}{A})^{A} and \\int \\chi^{A}{(A)} dA = \\int (\\frac{1}{A})^{A} dA and \\log{(\\chi{(A)})} - \\int \\chi^{A}{(A)} dA = \\log{(\\frac{1}{A})} - \\int \\chi^{A}{(A)} dA and (\\log{(\\chi{(A)})} - \\int \\chi^{A}{(A)} dA)^{A} = (\\log{(\\frac{1}{A})} - \\int \\chi^{A}{(A)} dA)^{A} and (\\log{(\\chi{(A)})} - \\int (\\frac{1}{A})^{A} dA)^{A} = (\\log{(\\frac{1}{A})} - \\int (\\frac{1}{A})^{A} dA)^{A}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\chi')(Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Integer(-1)))"], [["log", 1], "Equality(log(Function('\\\\chi')(Symbol('A', commutative=True))), log(Pow(Symbol('A', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A', commutative=True)))"], [["integrate", 3, "Symbol('A', commutative=True)"], "Equality(Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["minus", 2, "Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(log(Function('\\\\chi')(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Add(log(Pow(Symbol('A', commutative=True), Integer(-1))), Mul(Integer(-1), Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Add(log(Function('\\\\chi')(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Symbol('A', commutative=True)), Pow(Add(log(Pow(Symbol('A', commutative=True), Integer(-1))), Mul(Integer(-1), Integral(Pow(Function('\\\\chi')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Add(log(Function('\\\\chi')(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Symbol('A', commutative=True)), Pow(Add(log(Pow(Symbol('A', commutative=True), Integer(-1))), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(A_{2},v_{x})} = A_{2} - v_{x} and \\operatorname{v_{y}}{(A_{2},v_{x})} = \\int (v_{x} + \\operatorname{C_{2}}{(A_{2},v_{x})}) dv_{x}, then obtain \\operatorname{v_{y}}{(A_{2},v_{x})} = \\int A_{2} dv_{x}", "derivation": "\\operatorname{C_{2}}{(A_{2},v_{x})} = A_{2} - v_{x} and v_{x} + \\operatorname{C_{2}}{(A_{2},v_{x})} = A_{2} and \\int (v_{x} + \\operatorname{C_{2}}{(A_{2},v_{x})}) dv_{x} = \\int A_{2} dv_{x} and \\operatorname{v_{y}}{(A_{2},v_{x})} = \\int (v_{x} + \\operatorname{C_{2}}{(A_{2},v_{x})}) dv_{x} and \\operatorname{v_{y}}{(A_{2},v_{x})} = \\int A_{2} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["add", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Function('C_2')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True))), Symbol('A_2', commutative=True))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Symbol('v_x', commutative=True), Function('C_2')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Symbol('A_2', commutative=True), Tuple(Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True)), Integral(Add(Symbol('v_x', commutative=True), Function('C_2')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('v_y')(Symbol('A_2', commutative=True), Symbol('v_x', commutative=True)), Integral(Symbol('A_2', commutative=True), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given b{(\\psi)} = \\log{(\\psi)}, then obtain - 2 \\psi + 3 b{(\\psi)} + 2 \\log{(\\psi)} = - 2 \\psi + 2 b{(\\psi)} + 3 \\log{(\\psi)}", "derivation": "b{(\\psi)} = \\log{(\\psi)} and 2 b{(\\psi)} = b{(\\psi)} + \\log{(\\psi)} and - \\psi + 2 b{(\\psi)} = - \\psi + b{(\\psi)} + \\log{(\\psi)} and - 2 \\psi + 3 b{(\\psi)} + \\log{(\\psi)} = - 2 \\psi + 2 b{(\\psi)} + 2 \\log{(\\psi)} and - 2 \\psi + 3 b{(\\psi)} + 2 \\log{(\\psi)} = - 2 \\psi + 2 b{(\\psi)} + 3 \\log{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Function('b')(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Integer(2), Function('b')(Symbol('\\\\psi', commutative=True))), Add(Function('b')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"], [["minus", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Function('b')(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(3), Function('b')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Function('b')(Symbol('\\\\psi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))))"], [["add", 4, "log(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(3), Function('b')(Symbol('\\\\psi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Function('b')(Symbol('\\\\psi', commutative=True))), Mul(Integer(3), log(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given y{(l,n)} = \\cos{(l + n)} and \\dot{\\mathbf{r}}{(l,n)} = l + \\cos{(l + n)}, then obtain \\frac{\\dot{\\mathbf{r}}{(l,n)}}{y{(l,n)}} = \\frac{l + y{(l,n)}}{y{(l,n)}}", "derivation": "y{(l,n)} = \\cos{(l + n)} and \\dot{\\mathbf{r}}{(l,n)} = l + \\cos{(l + n)} and \\frac{\\dot{\\mathbf{r}}{(l,n)}}{\\cos{(l + n)}} = \\frac{l + \\cos{(l + n)}}{\\cos{(l + n)}} and \\frac{\\dot{\\mathbf{r}}{(l,n)}}{y{(l,n)}} = \\frac{l + y{(l,n)}}{y{(l,n)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('y')(Symbol('l', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('n', commutative=True)), Add(Symbol('l', commutative=True), cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True)))))"], [["divide", 2, "cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('n', commutative=True)), Pow(cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True))), Integer(-1))), Mul(Add(Symbol('l', commutative=True), cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True)))), Pow(cos(Add(Symbol('l', commutative=True), Symbol('n', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('n', commutative=True)), Pow(Function('y')(Symbol('l', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Add(Symbol('l', commutative=True), Function('y')(Symbol('l', commutative=True), Symbol('n', commutative=True))), Pow(Function('y')(Symbol('l', commutative=True), Symbol('n', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given l{(f^{\\prime},t)} = \\sin{(f^{\\prime} t)}, then obtain (\\frac{d}{d t} 0)^{t} = (\\frac{\\partial}{\\partial t} (- l{(f^{\\prime},t)} + \\sin{(f^{\\prime} t)}))^{t}", "derivation": "l{(f^{\\prime},t)} = \\sin{(f^{\\prime} t)} and 0 = - l{(f^{\\prime},t)} + \\sin{(f^{\\prime} t)} and \\frac{d}{d t} 0 = \\frac{\\partial}{\\partial t} (- l{(f^{\\prime},t)} + \\sin{(f^{\\prime} t)}) and (\\frac{d}{d t} 0)^{t} = (\\frac{\\partial}{\\partial t} (- l{(f^{\\prime},t)} + \\sin{(f^{\\prime} t)}))^{t}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True))))"], [["minus", 1, "Function('l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True))), sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True)))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True))), sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Function('l')(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True))), sin(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\omega{(g,A)} = A + g, then obtain \\log{(g + \\omega^{A}{(g,A)})} = \\log{(g + (A + g)^{A})}", "derivation": "\\omega{(g,A)} = A + g and \\omega^{A}{(g,A)} = (A + g)^{A} and g + \\omega^{A}{(g,A)} = g + (A + g)^{A} and \\log{(g + \\omega^{A}{(g,A)})} = \\log{(g + (A + g)^{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('g', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Add(Symbol('A', commutative=True), Symbol('g', commutative=True)), Symbol('A', commutative=True)))"], [["add", 2, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Pow(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Add(Symbol('g', commutative=True), Pow(Add(Symbol('A', commutative=True), Symbol('g', commutative=True)), Symbol('A', commutative=True))))"], [["log", 3], "Equality(log(Add(Symbol('g', commutative=True), Pow(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)))), log(Add(Symbol('g', commutative=True), Pow(Add(Symbol('A', commutative=True), Symbol('g', commutative=True)), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(H)} = \\log{(H)}, then obtain \\int \\frac{d}{d H} \\int \\operatorname{t_{2}}{(H)} dH dH = H \\log{(H)} - H + \\hat{H}_{\\lambda}", "derivation": "\\operatorname{t_{2}}{(H)} = \\log{(H)} and \\int \\operatorname{t_{2}}{(H)} dH = \\int \\log{(H)} dH and \\frac{d}{d H} \\int \\operatorname{t_{2}}{(H)} dH = \\frac{d}{d H} \\int \\log{(H)} dH and \\int \\frac{d}{d H} \\int \\operatorname{t_{2}}{(H)} dH dH = \\int \\frac{d}{d H} \\int \\log{(H)} dH dH and \\int \\frac{d}{d H} \\int \\operatorname{t_{2}}{(H)} dH dH = H \\log{(H)} - H + \\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Integral(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Integral(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given M{(\\mu)} = e^{\\mu}, then obtain (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{- \\mu} - 2)} - 2)} - 2)^{\\mu} = (M{(\\mu)} e^{- \\mu} - 2)^{\\mu}", "derivation": "M{(\\mu)} = e^{\\mu} and M{(\\mu)} e^{- \\mu} = 1 and M{(\\mu)} e^{- \\mu} - 1 = 0 and M{(\\mu)} e^{- \\mu} - 2 = -1 and (M{(\\mu)} e^{- \\mu} - 2)^{\\mu} = (-1)^{\\mu} and (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{- \\mu} - 2)} - 2)^{\\mu} = (M{(\\mu)} e^{- \\mu} - 2)^{\\mu} and (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{- \\mu} - 2)} - 2)^{\\mu} = (-1)^{\\mu} and (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{\\mu (M{(\\mu)} e^{- \\mu} - 2)} - 2)} - 2)^{\\mu} = (M{(\\mu)} e^{- \\mu} - 2)^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-1)), Integer(0))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2)), Integer(-1))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2)), Symbol('\\\\mu', commutative=True)), Pow(Integer(-1), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2))))), Integer(-2)), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2)), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2))))), Integer(-2)), Symbol('\\\\mu', commutative=True)), Pow(Integer(-1), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2))))), Integer(-2))))), Integer(-2)), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Function('M')(Symbol('\\\\mu', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Integer(-2)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{z})} = \\log{(\\log{(v_{z})})}, then obtain (\\frac{d}{d v_{z}} \\operatorname{y^{\\prime}}{(v_{z})})^{v_{z}} = (\\frac{1}{v_{z} \\log{(v_{z})}})^{v_{z}}", "derivation": "\\operatorname{y^{\\prime}}{(v_{z})} = \\log{(\\log{(v_{z})})} and \\frac{d}{d v_{z}} \\operatorname{y^{\\prime}}{(v_{z})} = \\frac{d}{d v_{z}} \\log{(\\log{(v_{z})})} and (\\frac{d}{d v_{z}} \\operatorname{y^{\\prime}}{(v_{z})})^{v_{z}} = (\\frac{d}{d v_{z}} \\log{(\\log{(v_{z})})})^{v_{z}} and (\\frac{d}{d v_{z}} \\operatorname{y^{\\prime}}{(v_{z})})^{v_{z}} = (\\frac{1}{v_{z} \\log{(v_{z})}})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_z', commutative=True)), log(log(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(log(log(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Derivative(Function('y^{\\\\prime}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('v_z', commutative=True)), Pow(Derivative(log(log(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('v_z', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('y^{\\\\prime}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('v_z', commutative=True)), Pow(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(log(Symbol('v_z', commutative=True)), Integer(-1))), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given U{(r_{0},f^{*})} = f^{*} + r_{0} and \\hat{x}{(r_{0})} = r_{0}, then obtain r_{0} (- r_{0} + U{(r_{0},f^{*})}) = f^{*} r_{0}", "derivation": "U{(r_{0},f^{*})} = f^{*} + r_{0} and - r_{0} + U{(r_{0},f^{*})} = f^{*} and \\hat{x}{(r_{0})} = r_{0} and (- r_{0} + U{(r_{0},f^{*})}) \\hat{x}{(r_{0})} = f^{*} \\hat{x}{(r_{0})} and r_{0} (- r_{0} + U{(r_{0},f^{*})}) = f^{*} r_{0}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('r_0', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Symbol('r_0', commutative=True)))"], [["minus", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('U')(Symbol('r_0', commutative=True), Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], [["times", 2, "Function('\\\\hat{x}')(Symbol('r_0', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('U')(Symbol('r_0', commutative=True), Symbol('f^*', commutative=True))), Function('\\\\hat{x}')(Symbol('r_0', commutative=True))), Mul(Symbol('f^*', commutative=True), Function('\\\\hat{x}')(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('r_0', commutative=True), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('U')(Symbol('r_0', commutative=True), Symbol('f^*', commutative=True)))), Mul(Symbol('f^*', commutative=True), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given c{(\\lambda,J)} = \\cos{(J + \\lambda)}, then derive \\int \\frac{c{(\\lambda,J)}}{\\cos{(J + \\lambda)}} dJ = J + m, then obtain \\frac{d}{d J} \\int 1 dJ = 1", "derivation": "c{(\\lambda,J)} = \\cos{(J + \\lambda)} and \\frac{c{(\\lambda,J)}}{\\cos{(J + \\lambda)}} = 1 and \\int \\frac{c{(\\lambda,J)}}{\\cos{(J + \\lambda)}} dJ = \\int 1 dJ and \\int \\frac{c{(\\lambda,J)}}{\\cos{(J + \\lambda)}} dJ = J + m and \\int 1 dJ = J + m and \\frac{d}{d J} \\int 1 dJ = \\frac{\\partial}{\\partial J} (J + m) and \\frac{d}{d J} \\int 1 dJ = 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\lambda', commutative=True), Symbol('J', commutative=True)), cos(Add(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "cos(Add(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('c')(Symbol('\\\\lambda', commutative=True), Symbol('J', commutative=True)), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Function('c')(Symbol('\\\\lambda', commutative=True), Symbol('J', commutative=True)), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Integral(Integer(1), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('c')(Symbol('\\\\lambda', commutative=True), Symbol('J', commutative=True)), Pow(cos(Add(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given H{(v_{z},f_{E})} = f_{E} + v_{z}, then derive \\int H{(v_{z},f_{E})} df_{E} = \\Omega + \\frac{f_{E}^{2}}{2} + f_{E} v_{z}, then obtain \\frac{\\int H{(v_{z},f_{E})} df_{E}}{\\mathbf{f}} = \\frac{\\Omega + \\frac{f_{E}^{2}}{2} + f_{E} v_{z}}{\\mathbf{f}}", "derivation": "H{(v_{z},f_{E})} = f_{E} + v_{z} and \\int H{(v_{z},f_{E})} df_{E} = \\int (f_{E} + v_{z}) df_{E} and \\int H{(v_{z},f_{E})} df_{E} = \\Omega + \\frac{f_{E}^{2}}{2} + f_{E} v_{z} and \\frac{\\int H{(v_{z},f_{E})} df_{E}}{\\mathbf{f}} = \\frac{\\int (f_{E} + v_{z}) df_{E}}{\\mathbf{f}} and \\int (f_{E} + v_{z}) df_{E} = \\Omega + \\frac{f_{E}^{2}}{2} + f_{E} v_{z} and \\frac{\\int H{(v_{z},f_{E})} df_{E}}{\\mathbf{f}} = \\frac{\\Omega + \\frac{f_{E}^{2}}{2} + f_{E} v_{z}}{\\mathbf{f}}", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('H')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('H')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Mul(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Integral(Function('H')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Integral(Add(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Mul(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Integral(Function('H')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Mul(Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(r)} = \\sin{(e^{r})} and g{(r)} = (\\int 0 dr) \\int (- \\operatorname{c_{0}}{(r)} + \\sin{(e^{r})}) dr, then obtain \\frac{e^{r} (\\int 0 dr)^{2}}{\\operatorname{c_{0}}{(r)} - e^{r}} = \\frac{g{(r)} e^{r}}{\\operatorname{c_{0}}{(r)} - e^{r}}", "derivation": "\\operatorname{c_{0}}{(r)} = \\sin{(e^{r})} and 0 = - \\operatorname{c_{0}}{(r)} + \\sin{(e^{r})} and \\int 0 dr = \\int (- \\operatorname{c_{0}}{(r)} + \\sin{(e^{r})}) dr and (\\int 0 dr)^{2} = (\\int 0 dr) \\int (- \\operatorname{c_{0}}{(r)} + \\sin{(e^{r})}) dr and g{(r)} = (\\int 0 dr) \\int (- \\operatorname{c_{0}}{(r)} + \\sin{(e^{r})}) dr and (\\int 0 dr)^{2} = g{(r)} and e^{r} (\\int 0 dr)^{2} = g{(r)} e^{r} and \\frac{e^{r} (\\int 0 dr)^{2}}{\\operatorname{c_{0}}{(r)} - e^{r}} = \\frac{g{(r)} e^{r}}{\\operatorname{c_{0}}{(r)} - e^{r}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('r', commutative=True)), sin(exp(Symbol('r', commutative=True))))"], [["minus", 1, "Function('c_0')(Symbol('r', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))))"], [["times", 3, "Integral(Integer(0), Tuple(Symbol('r', commutative=True)))"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integer(2)), Mul(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('g')(Symbol('r', commutative=True)), Mul(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integer(2)), Function('g')(Symbol('r', commutative=True)))"], [["times", 6, "exp(Symbol('r', commutative=True))"], "Equality(Mul(exp(Symbol('r', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integer(2))), Mul(Function('g')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))))"], [["divide", 7, "Add(Function('c_0')(Symbol('r', commutative=True)), Mul(Integer(-1), exp(Symbol('r', commutative=True))))"], "Equality(Mul(Pow(Add(Function('c_0')(Symbol('r', commutative=True)), Mul(Integer(-1), exp(Symbol('r', commutative=True)))), Integer(-1)), exp(Symbol('r', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('r', commutative=True))), Integer(2))), Mul(Pow(Add(Function('c_0')(Symbol('r', commutative=True)), Mul(Integer(-1), exp(Symbol('r', commutative=True)))), Integer(-1)), Function('g')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\nabla{(a^{\\dagger},z)} = a^{\\dagger} \\cos{(z)}, then obtain - a^{\\dagger} \\cos{(z)} = - \\nabla{(a^{\\dagger},z)}", "derivation": "\\nabla{(a^{\\dagger},z)} = a^{\\dagger} \\cos{(z)} and a^{\\dagger} \\cos{(z)} + \\nabla{(a^{\\dagger},z)} = 2 a^{\\dagger} \\cos{(z)} and a^{\\dagger} \\cos{(z)} = 2 a^{\\dagger} \\cos{(z)} - \\nabla{(a^{\\dagger},z)} and - a^{\\dagger} \\cos{(z)} = - \\nabla{(a^{\\dagger},z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))))"], [["add", 1, "Mul(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))), Function('\\\\nabla')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('z', commutative=True))), Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))))"], [["minus", 2, "Function('\\\\nabla')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))), Add(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('z', commutative=True))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given s{(\\phi_1,F_{c})} = F_{c} \\phi_1 and B{(\\dot{z})} = \\log{(\\dot{z})}, then obtain \\frac{\\partial}{\\partial \\dot{z}} \\frac{B{(\\dot{z})}}{s{(\\phi_1,F_{c})}} = \\frac{\\partial}{\\partial \\dot{z}} \\frac{\\log{(\\dot{z})}}{s{(\\phi_1,F_{c})}}", "derivation": "s{(\\phi_1,F_{c})} = F_{c} \\phi_1 and B{(\\dot{z})} = \\log{(\\dot{z})} and \\frac{B{(\\dot{z})}}{F_{c} \\phi_1} = \\frac{\\log{(\\dot{z})}}{F_{c} \\phi_1} and \\frac{B{(\\dot{z})}}{s{(\\phi_1,F_{c})}} = \\frac{\\log{(\\dot{z})}}{s{(\\phi_1,F_{c})}} and \\frac{\\partial}{\\partial \\dot{z}} \\frac{B{(\\dot{z})}}{s{(\\phi_1,F_{c})}} = \\frac{\\partial}{\\partial \\dot{z}} \\frac{\\log{(\\dot{z})}}{s{(\\phi_1,F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\phi_1', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], ["get_premise", "Equality(Function('B')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Mul(Symbol('F_c', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('B')(Symbol('\\\\dot{z}', commutative=True)), Pow(Function('s')(Symbol('\\\\phi_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1))), Mul(Pow(Function('s')(Symbol('\\\\phi_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1)), log(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Function('B')(Symbol('\\\\dot{z}', commutative=True)), Pow(Function('s')(Symbol('\\\\phi_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('s')(Symbol('\\\\phi_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1)), log(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})}, then obtain 2 W{(\\mathbf{s})} - 1 + \\frac{\\log{(e^{\\mathbf{s}})}}{\\mathbf{s}} = W{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})} - 1 + \\frac{\\log{(e^{\\mathbf{s}})}}{\\mathbf{s}}", "derivation": "W{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})} and 2 W{(\\mathbf{s})} = W{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})} and \\frac{W{(\\mathbf{s})}}{\\mathbf{s}} = \\frac{\\log{(e^{\\mathbf{s}})}}{\\mathbf{s}} and 2 W{(\\mathbf{s})} - 1 + \\frac{W{(\\mathbf{s})}}{\\mathbf{s}} = W{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})} - 1 + \\frac{W{(\\mathbf{s})}}{\\mathbf{s}} and 2 W{(\\mathbf{s})} - 1 + \\frac{\\log{(e^{\\mathbf{s}})}}{\\mathbf{s}} = W{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})} - 1 + \\frac{\\log{(e^{\\mathbf{s}})}}{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 1, "Function('W')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(2), Function('W')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('W')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["divide", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["add", 2, "Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{s}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('W')(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Function('W')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('W')(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))))), Add(Function('W')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given \\dot{z}{(P_{e},c_{0})} = \\cos{(P_{e} + c_{0})} and \\hat{p}{(P_{e})} = P_{e}, then derive \\int (\\hat{p}{(P_{e})} - 1) dP_{e} = \\frac{P_{e}^{2}}{2} - P_{e} + \\delta, then obtain \\int (P_{e} - 1) dP_{e} = \\frac{P_{e}^{2}}{2} - P_{e} + \\delta", "derivation": "\\dot{z}{(P_{e},c_{0})} = \\cos{(P_{e} + c_{0})} and \\hat{p}{(P_{e})} = P_{e} and \\hat{p}{(P_{e})} - \\frac{\\cos{(P_{e} + c_{0})}}{\\dot{z}{(P_{e},c_{0})}} = P_{e} - \\frac{\\cos{(P_{e} + c_{0})}}{\\dot{z}{(P_{e},c_{0})}} and \\hat{p}{(P_{e})} - 1 = P_{e} - 1 and \\int (\\hat{p}{(P_{e})} - 1) dP_{e} = \\int (P_{e} - 1) dP_{e} and \\int (\\hat{p}{(P_{e})} - 1) dP_{e} = \\frac{P_{e}^{2}}{2} - P_{e} + \\delta and \\int (P_{e} - 1) dP_{e} = \\frac{P_{e}^{2}}{2} - P_{e} + \\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), cos(Add(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], [["minus", 2, "Mul(Pow(Function('\\\\dot{z}')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), cos(Add(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), cos(Add(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), cos(Add(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), Add(Symbol('P_e', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Integer(-1)), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), Tuple(Symbol('P_e', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Integral(Add(Symbol('P_e', commutative=True), Integer(-1)), Tuple(Symbol('P_e', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(z^{*})} = e^{e^{z^{*}}}, then obtain \\operatorname{f_{\\mathbf{v}}}{(z^{*})} + \\frac{e^{z^{*}} - e^{e^{z^{*}}}}{z^{*}} = e^{e^{z^{*}}} + \\frac{e^{z^{*}} - e^{e^{z^{*}}}}{z^{*}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(z^{*})} = e^{e^{z^{*}}} and \\operatorname{f_{\\mathbf{v}}}{(z^{*})} - e^{z^{*}} = - e^{z^{*}} + e^{e^{z^{*}}} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(z^{*})} - e^{z^{*}}}{z^{*}} = \\frac{- e^{z^{*}} + e^{e^{z^{*}}}}{z^{*}} and \\operatorname{f_{\\mathbf{v}}}{(z^{*})} - \\frac{- e^{z^{*}} + e^{e^{z^{*}}}}{z^{*}} = e^{e^{z^{*}}} - \\frac{- e^{z^{*}} + e^{e^{z^{*}}}}{z^{*}} and \\operatorname{f_{\\mathbf{v}}}{(z^{*})} + \\frac{e^{z^{*}} - e^{e^{z^{*}}}}{z^{*}} = e^{e^{z^{*}}} + \\frac{e^{z^{*}} - e^{e^{z^{*}}}}{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True)), exp(exp(Symbol('z^*', commutative=True))))"], [["minus", 1, "exp(Symbol('z^*', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), exp(Symbol('z^*', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), exp(exp(Symbol('z^*', commutative=True)))))"], [["divide", 2, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), exp(Symbol('z^*', commutative=True))))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), exp(exp(Symbol('z^*', commutative=True))))))"], [["minus", 1, "Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), exp(exp(Symbol('z^*', commutative=True)))))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), exp(exp(Symbol('z^*', commutative=True)))))), Add(exp(exp(Symbol('z^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('z^*', commutative=True))), exp(exp(Symbol('z^*', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('z^*', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('z^*', commutative=True))))))), Add(exp(exp(Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('z^*', commutative=True))))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)} = Q h - \\hat{H}_l and \\operatorname{f_{\\mathbf{v}}}{(h,\\hat{H}_l,Q)} = \\frac{\\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)}}{Q h}, then obtain \\cos{(\\frac{\\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)}}{Q h})} = \\cos{(\\frac{Q h - \\hat{H}_l}{Q h})}", "derivation": "\\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)} = Q h - \\hat{H}_l and \\operatorname{f_{\\mathbf{v}}}{(h,\\hat{H}_l,Q)} = \\frac{\\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)}}{Q h} and \\operatorname{f_{\\mathbf{v}}}{(h,\\hat{H}_l,Q)} = \\frac{Q h - \\hat{H}_l}{Q h} and \\cos{(\\operatorname{f_{\\mathbf{v}}}{(h,\\hat{H}_l,Q)})} = \\cos{(\\frac{Q h - \\hat{H}_l}{Q h})} and \\cos{(\\frac{\\hat{\\mathbf{x}}{(h,\\hat{H}_l,Q)}}{Q h})} = \\cos{(\\frac{Q h - \\hat{H}_l}{Q h})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Symbol('Q', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Symbol('Q', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["cos", 3], "Equality(cos(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))), cos(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Symbol('Q', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(cos(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('h', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True)))), cos(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Symbol('Q', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))))"]]}, {"prompt": "Given p{(F_{x})} = e^{\\cos{(F_{x})}} and \\sigma_{p}{(m_{s})} = \\sin{(m_{s})}, then obtain - m_{s} + \\sigma_{p}^{m_{s}}{(m_{s})} e^{- 2 \\cos{(F_{x})}} = - m_{s} + e^{- 2 \\cos{(F_{x})}} \\sin^{m_{s}}{(m_{s})}", "derivation": "p{(F_{x})} = e^{\\cos{(F_{x})}} and \\sigma_{p}{(m_{s})} = \\sin{(m_{s})} and \\sigma_{p}^{m_{s}}{(m_{s})} = \\sin^{m_{s}}{(m_{s})} and \\frac{\\sigma_{p}^{m_{s}}{(m_{s})}}{p^{2}{(F_{x})}} = \\frac{\\sin^{m_{s}}{(m_{s})}}{p^{2}{(F_{x})}} and - m_{s} + \\frac{\\sigma_{p}^{m_{s}}{(m_{s})}}{p^{2}{(F_{x})}} = - m_{s} + \\frac{\\sin^{m_{s}}{(m_{s})}}{p^{2}{(F_{x})}} and - m_{s} + \\sigma_{p}^{m_{s}}{(m_{s})} e^{- 2 \\cos{(F_{x})}} = - m_{s} + e^{- 2 \\cos{(F_{x})}} \\sin^{m_{s}}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True))))"], ["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["divide", 3, "Pow(Function('p')(Symbol('F_x', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Function('p')(Symbol('F_x', commutative=True)), Integer(-2))), Mul(Pow(Function('p')(Symbol('F_x', commutative=True)), Integer(-2)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))))"], [["minus", 4, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Pow(Function('\\\\sigma_p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Function('p')(Symbol('F_x', commutative=True)), Integer(-2)))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Pow(Function('p')(Symbol('F_x', commutative=True)), Integer(-2)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Pow(Function('\\\\sigma_p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('F_x', commutative=True)))))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(exp(Mul(Integer(-1), Integer(2), cos(Symbol('F_x', commutative=True)))), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(t_{1},l)} = l + t_{1}, then derive \\int \\hat{x}{(t_{1},l)} dl = \\hat{X} + \\frac{l^{2}}{2} + l t_{1}, then obtain \\theta_2 + \\frac{l^{2}}{2} + l t_{1} = \\hat{X} + \\frac{l^{2}}{2} + l t_{1}", "derivation": "\\hat{x}{(t_{1},l)} = l + t_{1} and \\int \\hat{x}{(t_{1},l)} dl = \\int (l + t_{1}) dl and \\int \\hat{x}{(t_{1},l)} dl = \\hat{X} + \\frac{l^{2}}{2} + l t_{1} and \\int (l + t_{1}) dl = \\hat{X} + \\frac{l^{2}}{2} + l t_{1} and \\theta_2 + \\frac{l^{2}}{2} + l t_{1} = \\hat{X} + \\frac{l^{2}}{2} + l t_{1}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('t_1', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('t_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('t_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('l', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\dot{z})} = \\cos{(\\dot{z})}, then obtain \\int \\frac{\\frac{- \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} + 1}{\\cos{(\\dot{z})}} + \\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} d\\dot{z} = \\int 1 d\\dot{z}", "derivation": "\\mu{(\\dot{z})} = \\cos{(\\dot{z})} and \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} = 1 and \\int \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} d\\dot{z} = \\int 1 d\\dot{z} and 0 = - \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} + 1 and 0 = \\frac{- \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} + 1}{\\cos{(\\dot{z})}} and \\mu{(\\dot{z})} = \\frac{- \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} + 1}{\\cos{(\\dot{z})}} + \\mu{(\\dot{z})} and \\int \\frac{\\frac{- \\frac{\\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} + 1}{\\cos{(\\dot{z})}} + \\mu{(\\dot{z})}}{\\cos{(\\dot{z})}} d\\dot{z} = \\int 1 d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 2, "Mul(Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1)))"], [["times", 4, "Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["add", 5, "Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integral(Mul(Add(Mul(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Function('\\\\mu')(Symbol('\\\\dot{z}', commutative=True))), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\tilde{g})} = e^{\\tilde{g}} and b{(\\tilde{g})} = - \\operatorname{r_{0}}{(\\tilde{g})} + e^{\\tilde{g}}, then derive 0 = \\frac{d}{d \\tilde{g}} b{(\\tilde{g})}, then obtain 0^{\\tilde{g}} = (\\frac{d}{d \\tilde{g}} b{(\\tilde{g})})^{\\tilde{g}}", "derivation": "\\operatorname{r_{0}}{(\\tilde{g})} = e^{\\tilde{g}} and \\tilde{g} + \\operatorname{r_{0}}{(\\tilde{g})} = \\tilde{g} + e^{\\tilde{g}} and 0 = - \\operatorname{r_{0}}{(\\tilde{g})} + e^{\\tilde{g}} and b{(\\tilde{g})} = - \\operatorname{r_{0}}{(\\tilde{g})} + e^{\\tilde{g}} and \\frac{d}{d \\tilde{g}} 0 = \\frac{d}{d \\tilde{g}} (- \\operatorname{r_{0}}{(\\tilde{g})} + e^{\\tilde{g}}) and \\frac{d}{d \\tilde{g}} 0 = \\frac{d}{d \\tilde{g}} b{(\\tilde{g})} and 0 = \\frac{d}{d \\tilde{g}} b{(\\tilde{g})} and 0^{\\tilde{g}} = (\\frac{d}{d \\tilde{g}} b{(\\tilde{g})})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Function('r_0')(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\tilde{g}', commutative=True), Function('r_0')(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('r_0')(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Integer(-1), Function('r_0')(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('r_0')(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Function('b')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Derivative(Function('b')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["power", 7, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\tilde{g}', commutative=True)), Pow(Derivative(Function('b')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given v{(\\omega,\\Psi)} = \\frac{\\omega}{\\Psi}, then obtain \\tilde{\\infty} (2 v{(\\omega,\\Psi)} - \\frac{\\omega}{\\Psi}) = \\frac{\\tilde{\\infty} \\omega}{\\Psi}", "derivation": "v{(\\omega,\\Psi)} = \\frac{\\omega}{\\Psi} and v{(\\omega,\\Psi)} - \\frac{\\omega}{\\Psi} = 0 and 2 v{(\\omega,\\Psi)} - \\frac{\\omega}{\\Psi} = v{(\\omega,\\Psi)} and 2 v{(\\omega,\\Psi)} - \\frac{\\omega}{\\Psi} = \\frac{\\omega}{\\Psi} and \\tilde{\\infty} (2 v{(\\omega,\\Psi)} - \\frac{\\omega}{\\Psi}) = \\frac{\\tilde{\\infty} \\omega}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Integer(0))"], [["add", 2, "Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"], [["divide", 4, 0], "Equality(Mul(zoo, Add(Mul(Integer(2), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))), Mul(zoo, Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{M},\\rho)} = \\frac{\\mathbf{M}}{\\rho}, then obtain (- \\rho_{b}{(\\mathbf{M},\\rho)} - 1)^{\\rho} = (- \\frac{\\mathbf{M}}{\\rho} - 1)^{\\rho}", "derivation": "\\rho_{b}{(\\mathbf{M},\\rho)} = \\frac{\\mathbf{M}}{\\rho} and \\rho_{b}{(\\mathbf{M},\\rho)} + 1 = \\frac{\\mathbf{M}}{\\rho} + 1 and - \\rho_{b}{(\\mathbf{M},\\rho)} - 1 = - \\frac{\\mathbf{M}}{\\rho} - 1 and (- \\rho_{b}{(\\mathbf{M},\\rho)} - 1)^{\\rho} = (- \\frac{\\mathbf{M}}{\\rho} - 1)^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Integer(1)))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Integer(-1)))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1)), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Integer(-1)), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi,Z)} = \\cos{(\\frac{Z}{\\phi})} and \\tilde{g}{(\\phi,Z)} = (\\cos{(\\frac{Z}{\\phi})} + \\int \\cos{(\\frac{Z}{\\phi})} d\\phi)^{\\phi}, then obtain \\tilde{g}{(\\phi,Z)} = (\\cos{(\\frac{Z}{\\phi})} + \\int \\operatorname{t_{2}}{(\\phi,Z)} d\\phi)^{\\phi}", "derivation": "\\operatorname{t_{2}}{(\\phi,Z)} = \\cos{(\\frac{Z}{\\phi})} and \\int \\operatorname{t_{2}}{(\\phi,Z)} d\\phi = \\int \\cos{(\\frac{Z}{\\phi})} d\\phi and \\cos{(\\frac{Z}{\\phi})} + \\int \\operatorname{t_{2}}{(\\phi,Z)} d\\phi = \\cos{(\\frac{Z}{\\phi})} + \\int \\cos{(\\frac{Z}{\\phi})} d\\phi and (\\cos{(\\frac{Z}{\\phi})} + \\int \\operatorname{t_{2}}{(\\phi,Z)} d\\phi)^{\\phi} = (\\cos{(\\frac{Z}{\\phi})} + \\int \\cos{(\\frac{Z}{\\phi})} d\\phi)^{\\phi} and \\tilde{g}{(\\phi,Z)} = (\\cos{(\\frac{Z}{\\phi})} + \\int \\cos{(\\frac{Z}{\\phi})} d\\phi)^{\\phi} and \\tilde{g}{(\\phi,Z)} = (\\cos{(\\frac{Z}{\\phi})} + \\int \\operatorname{t_{2}}{(\\phi,Z)} d\\phi)^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 2, "cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], "Equality(Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["power", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Pow(Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Pow(Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\tilde{g}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Pow(Add(cos(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))), Integral(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\delta)} = \\sin{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} = \\delta \\hat{p}_0{(\\delta)}, then obtain \\operatorname{z^{*}}{(\\delta)} - \\sin{(\\delta)} = \\delta \\hat{p}_0{(\\delta)} - \\sin{(\\delta)}", "derivation": "\\hat{p}_0{(\\delta)} = \\sin{(\\delta)} and \\delta \\hat{p}_0{(\\delta)} = \\delta \\sin{(\\delta)} and \\delta \\hat{p}_0{(\\delta)} - \\sin{(\\delta)} = \\delta \\sin{(\\delta)} - \\sin{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} = \\delta \\hat{p}_0{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} - \\sin{(\\delta)} = \\delta \\sin{(\\delta)} - \\sin{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} - \\sin{(\\delta)} = \\delta \\hat{p}_0{(\\delta)} - \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), sin(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('z^*')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('z^*')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mu,y^{\\prime})} = y^{\\prime} + e^{\\mu} and \\operatorname{E_{x}}{(\\mu,y^{\\prime})} = - \\mu + \\mathbf{E}{(\\mu,y^{\\prime})}, then obtain \\operatorname{E_{x}}{(\\mu,y^{\\prime})} = - \\mu + y^{\\prime} + e^{\\mu}", "derivation": "\\mathbf{E}{(\\mu,y^{\\prime})} = y^{\\prime} + e^{\\mu} and - \\mu + \\mathbf{E}{(\\mu,y^{\\prime})} = - \\mu + y^{\\prime} + e^{\\mu} and \\operatorname{E_{x}}{(\\mu,y^{\\prime})} = - \\mu + \\mathbf{E}{(\\mu,y^{\\prime})} and \\operatorname{E_{x}}{(\\mu,y^{\\prime})} = - \\mu + y^{\\prime} + e^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f_{E})} = \\log{(f_{E})}, then obtain \\operatorname{n_{1}}^{f_{E}}{(f_{E})} + \\int \\operatorname{n_{1}}{(f_{E})} df_{E} = f_{E} \\log{(f_{E})} - f_{E} + v + \\operatorname{n_{1}}^{f_{E}}{(f_{E})}", "derivation": "\\operatorname{n_{1}}{(f_{E})} = \\log{(f_{E})} and \\int \\operatorname{n_{1}}{(f_{E})} df_{E} = \\int \\log{(f_{E})} df_{E} and \\log{(f_{E})}^{f_{E}} + \\int \\operatorname{n_{1}}{(f_{E})} df_{E} = \\log{(f_{E})}^{f_{E}} + \\int \\log{(f_{E})} df_{E} and \\operatorname{n_{1}}^{f_{E}}{(f_{E})} + \\int \\operatorname{n_{1}}{(f_{E})} df_{E} = \\operatorname{n_{1}}^{f_{E}}{(f_{E})} + \\int \\log{(f_{E})} df_{E} and \\operatorname{n_{1}}^{f_{E}}{(f_{E})} + \\int \\operatorname{n_{1}}{(f_{E})} df_{E} = f_{E} \\log{(f_{E})} - f_{E} + v + \\operatorname{n_{1}}^{f_{E}}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["add", 2, "Pow(log(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Add(Pow(log(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('n_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Add(Pow(log(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integral(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Pow(Function('n_1')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('n_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Add(Pow(Function('n_1')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integral(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Pow(Function('n_1')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('n_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Add(Mul(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('f_E', commutative=True)), Symbol('v', commutative=True), Pow(Function('n_1')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(S)} = \\log{(S)}, then obtain \\iint (- S \\log{(S)} + \\frac{d}{d S} S \\mathbf{J}_M{(S)}) dS dS = \\iint (- S \\log{(S)} + \\frac{d}{d S} S \\log{(S)}) dS dS", "derivation": "\\mathbf{J}_M{(S)} = \\log{(S)} and S \\mathbf{J}_M{(S)} = S \\log{(S)} and \\frac{d}{d S} S \\mathbf{J}_M{(S)} = \\frac{d}{d S} S \\log{(S)} and - S \\log{(S)} + \\frac{d}{d S} S \\mathbf{J}_M{(S)} = - S \\log{(S)} + \\frac{d}{d S} S \\log{(S)} and \\int (- S \\log{(S)} + \\frac{d}{d S} S \\mathbf{J}_M{(S)}) dS = \\int (- S \\log{(S)} + \\frac{d}{d S} S \\log{(S)}) dS and \\iint (- S \\log{(S)} + \\frac{d}{d S} S \\mathbf{J}_M{(S)}) dS dS = \\iint (- S \\log{(S)} + \\frac{d}{d S} S \\log{(S)}) dS dS", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Symbol('S', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True))))"], [["integrate", 5, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(Z)} = \\log{(Z)}, then derive \\int \\mathbf{D}{(Z)} dZ = Z \\log{(Z)} - Z + v_{z}, then obtain v_{z} + \\int \\mathbf{D}{(Z)} dZ = Z \\mathbf{D}{(Z)} - Z + 2 v_{z}", "derivation": "\\mathbf{D}{(Z)} = \\log{(Z)} and \\int \\mathbf{D}{(Z)} dZ = \\int \\log{(Z)} dZ and \\int \\mathbf{D}{(Z)} dZ = Z \\log{(Z)} - Z + v_{z} and \\int \\log{(Z)} dZ = Z \\log{(Z)} - Z + v_{z} and \\int \\log{(Z)} dZ = Z \\mathbf{D}{(Z)} - Z + v_{z} and Z \\log{(Z)} - Z + v_{z} = Z \\mathbf{D}{(Z)} - Z + v_{z} and \\int \\mathbf{D}{(Z)} dZ = Z \\mathbf{D}{(Z)} - Z + v_{z} and v_{z} + \\int \\mathbf{D}{(Z)} dZ = Z \\mathbf{D}{(Z)} - Z + 2 v_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Function('\\\\mathbf{D}')(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)), Add(Mul(Symbol('Z', commutative=True), Function('\\\\mathbf{D}')(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Function('\\\\mathbf{D}')(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('v_z', commutative=True)))"], [["add", 7, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Integral(Function('\\\\mathbf{D}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Symbol('Z', commutative=True), Function('\\\\mathbf{D}')(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given z{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})}, then obtain \\frac{d}{d \\ddot{x}} \\int (z{(\\ddot{x})} - \\sin{(\\ddot{x})}) d\\ddot{x} = \\frac{d}{d \\ddot{x}} \\int (\\log{(\\sin{(\\ddot{x})})} - \\sin{(\\ddot{x})}) d\\ddot{x}", "derivation": "z{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})} and z{(\\ddot{x})} - \\sin{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})} - \\sin{(\\ddot{x})} and \\int (z{(\\ddot{x})} - \\sin{(\\ddot{x})}) d\\ddot{x} = \\int (\\log{(\\sin{(\\ddot{x})})} - \\sin{(\\ddot{x})}) d\\ddot{x} and \\frac{d}{d \\ddot{x}} \\int (z{(\\ddot{x})} - \\sin{(\\ddot{x})}) d\\ddot{x} = \\frac{d}{d \\ddot{x}} \\int (\\log{(\\sin{(\\ddot{x})})} - \\sin{(\\ddot{x})}) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\ddot{x}', commutative=True)), log(sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Add(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Add(Function('z')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Integral(Add(Function('z')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Integral(Add(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\varepsilon)} = \\log{(\\varepsilon)}, then derive \\frac{d}{d \\varepsilon} V{(\\varepsilon)} = \\frac{1}{\\varepsilon}, then obtain A_{y} + V{(\\varepsilon)} = a + \\log{(\\varepsilon)}", "derivation": "V{(\\varepsilon)} = \\log{(\\varepsilon)} and \\frac{d}{d \\varepsilon} V{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)} and \\frac{d}{d \\varepsilon} V{(\\varepsilon)} = \\frac{1}{\\varepsilon} and \\int \\frac{d}{d \\varepsilon} V{(\\varepsilon)} d\\varepsilon = \\int \\frac{1}{\\varepsilon} d\\varepsilon and A_{y} + V{(\\varepsilon)} = a + \\log{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Function('V')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_y', commutative=True), Function('V')(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('a', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(F_{N})} = \\frac{d}{d F_{N}} \\sin{(F_{N})}, then derive \\frac{\\mathbf{J}_f{(F_{N})}}{F_{N}} = \\frac{\\cos{(F_{N})}}{F_{N}}, then obtain \\frac{d}{d F_{N}} \\sin{(F_{N})} + \\frac{\\frac{d}{d F_{N}} \\sin{(F_{N})}}{F_{N}} = \\frac{d}{d F_{N}} \\sin{(F_{N})} + \\frac{\\cos{(F_{N})}}{F_{N}}", "derivation": "\\mathbf{J}_f{(F_{N})} = \\frac{d}{d F_{N}} \\sin{(F_{N})} and \\frac{\\mathbf{J}_f{(F_{N})}}{F_{N}} = \\frac{\\frac{d}{d F_{N}} \\sin{(F_{N})}}{F_{N}} and \\frac{\\mathbf{J}_f{(F_{N})}}{F_{N}} = \\frac{\\cos{(F_{N})}}{F_{N}} and \\frac{\\frac{d}{d F_{N}} \\sin{(F_{N})}}{F_{N}} = \\frac{\\cos{(F_{N})}}{F_{N}} and \\frac{d}{d F_{N}} \\sin{(F_{N})} + \\frac{\\frac{d}{d F_{N}} \\sin{(F_{N})}}{F_{N}} = \\frac{d}{d F_{N}} \\sin{(F_{N})} + \\frac{\\cos{(F_{N})}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))))"], [["add", 4, "Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Add(Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))), Add(Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given f{(\\delta,\\mathbf{H})} = - \\delta + \\mathbf{H}, then obtain \\frac{\\partial}{\\partial \\delta} 2 f{(\\delta,\\mathbf{H})} = \\frac{\\partial}{\\partial \\delta} (- 2 \\delta + 2 \\mathbf{H})", "derivation": "f{(\\delta,\\mathbf{H})} = - \\delta + \\mathbf{H} and - \\delta + \\mathbf{H} + f{(\\delta,\\mathbf{H})} = - 2 \\delta + 2 \\mathbf{H} and 2 f{(\\delta,\\mathbf{H})} = - 2 \\delta + 2 \\mathbf{H} and \\frac{\\partial}{\\partial \\delta} 2 f{(\\delta,\\mathbf{H})} = \\frac{\\partial}{\\partial \\delta} (- 2 \\delta + 2 \\mathbf{H})", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Function('f')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('f')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('f')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(\\eta,\\hat{p}_0)} = \\eta + \\log{(\\hat{p}_0)}, then obtain \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} \\mathbf{P}{(\\eta,\\hat{p}_0)} + 1 = \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} (\\eta + \\log{(\\hat{p}_0)}) + 1", "derivation": "\\mathbf{P}{(\\eta,\\hat{p}_0)} = \\eta + \\log{(\\hat{p}_0)} and \\frac{\\partial}{\\partial \\eta} \\mathbf{P}{(\\eta,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\eta} (\\eta + \\log{(\\hat{p}_0)}) and \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} \\mathbf{P}{(\\eta,\\hat{p}_0)} = \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} (\\eta + \\log{(\\hat{p}_0)}) and \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} \\mathbf{P}{(\\eta,\\hat{p}_0)} + 1 = \\log{(\\hat{p}_0)} + \\frac{\\partial}{\\partial \\eta} (\\eta + \\log{(\\hat{p}_0)}) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["add", 2, "log(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(log(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(log(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Add(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(log(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1)), Add(log(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Add(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(l)} = \\cos{(l)} and \\Psi{(l)} = \\cos{(l)}, then obtain 0 = (- \\Psi{(l)} + \\cos{(l)})^{2}", "derivation": "\\operatorname{C_{d}}{(l)} = \\cos{(l)} and 0 = - \\operatorname{C_{d}}{(l)} + \\cos{(l)} and 0 = (- \\operatorname{C_{d}}{(l)} + \\cos{(l)})^{2} and \\Psi{(l)} = \\cos{(l)} and \\operatorname{C_{d}}{(l)} = \\Psi{(l)} and 0 = (- \\Psi{(l)} + \\cos{(l)})^{2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["minus", 1, "Function('C_d')(Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Function('C_d')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True)))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('C_d')(Symbol('l', commutative=True)), Function('\\\\Psi')(Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(U)} = \\sin{(U)}, then obtain \\int P_{e} dU = \\int (P_{e} - 2 \\operatorname{t_{1}}{(U)} + 2 \\sin{(U)}) dU", "derivation": "\\operatorname{t_{1}}{(U)} = \\sin{(U)} and 0 = - \\operatorname{t_{1}}{(U)} + \\sin{(U)} and P_{e} = P_{e} - \\operatorname{t_{1}}{(U)} + \\sin{(U)} and \\int P_{e} dU = \\int (P_{e} - \\operatorname{t_{1}}{(U)} + \\sin{(U)}) dU and \\int (P_{e} - \\operatorname{t_{1}}{(U)} + \\sin{(U)}) dU = \\int (P_{e} - 2 \\operatorname{t_{1}}{(U)} + 2 \\sin{(U)}) dU and \\int P_{e} dU = \\int (P_{e} - 2 \\operatorname{t_{1}}{(U)} + 2 \\sin{(U)}) dU", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["minus", 1, "Function('t_1')(Symbol('U', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t_1')(Symbol('U', commutative=True))), sin(Symbol('U', commutative=True))))"], [["add", 2, "Symbol('P_e', commutative=True)"], "Equality(Symbol('P_e', commutative=True), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('U', commutative=True))), sin(Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Symbol('P_e', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('U', commutative=True))), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('U', commutative=True))), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('U', commutative=True))), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Symbol('P_e', commutative=True), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('U', commutative=True))), Mul(Integer(2), sin(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given u{(\\hat{p},E_{\\lambda})} = E_{\\lambda} \\hat{p}, then obtain \\frac{\\partial}{\\partial \\hat{p}} u^{E_{\\lambda}}{(\\hat{p},E_{\\lambda})} \\int u{(\\hat{p},E_{\\lambda})} dE_{\\lambda} = \\frac{\\partial}{\\partial \\hat{p}} u^{E_{\\lambda}}{(\\hat{p},E_{\\lambda})} \\int E_{\\lambda} \\hat{p} dE_{\\lambda}", "derivation": "u{(\\hat{p},E_{\\lambda})} = E_{\\lambda} \\hat{p} and \\int u{(\\hat{p},E_{\\lambda})} dE_{\\lambda} = \\int E_{\\lambda} \\hat{p} dE_{\\lambda} and u^{E_{\\lambda}}{(\\hat{p},E_{\\lambda})} = (E_{\\lambda} \\hat{p})^{E_{\\lambda}} and \\frac{\\partial}{\\partial \\hat{p}} (E_{\\lambda} \\hat{p})^{E_{\\lambda}} \\int u{(\\hat{p},E_{\\lambda})} dE_{\\lambda} = \\frac{\\partial}{\\partial \\hat{p}} (E_{\\lambda} \\hat{p})^{E_{\\lambda}} \\int E_{\\lambda} \\hat{p} dE_{\\lambda} and \\frac{\\partial}{\\partial \\hat{p}} u^{E_{\\lambda}}{(\\hat{p},E_{\\lambda})} \\int u{(\\hat{p},E_{\\lambda})} dE_{\\lambda} = \\frac{\\partial}{\\partial \\hat{p}} u^{E_{\\lambda}}{(\\hat{p},E_{\\lambda})} \\int E_{\\lambda} \\hat{p} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 2, "Derivative(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Derivative(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Derivative(Pow(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Derivative(Pow(Function('u')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given L{(E,\\mu_0)} = \\log{(E \\mu_0)} and \\delta{(E,\\mu_0)} = E \\mu_0 L{(E,\\mu_0)} + L^{\\mu_0}{(E,\\mu_0)}, then obtain \\delta{(E,\\mu_0)} \\log{(E \\mu_0)}^{\\mu_0} = (E \\mu_0 L{(E,\\mu_0)} + \\log{(E \\mu_0)}^{\\mu_0}) \\log{(E \\mu_0)}^{\\mu_0}", "derivation": "L{(E,\\mu_0)} = \\log{(E \\mu_0)} and E \\mu_0 L{(E,\\mu_0)} = E \\mu_0 \\log{(E \\mu_0)} and L^{\\mu_0}{(E,\\mu_0)} = \\log{(E \\mu_0)}^{\\mu_0} and E \\mu_0 \\log{(E \\mu_0)} + L^{\\mu_0}{(E,\\mu_0)} = E \\mu_0 \\log{(E \\mu_0)} + \\log{(E \\mu_0)}^{\\mu_0} and E \\mu_0 L{(E,\\mu_0)} + L^{\\mu_0}{(E,\\mu_0)} = E \\mu_0 L{(E,\\mu_0)} + \\log{(E \\mu_0)}^{\\mu_0} and \\delta{(E,\\mu_0)} = E \\mu_0 L{(E,\\mu_0)} + L^{\\mu_0}{(E,\\mu_0)} and (E \\mu_0 L{(E,\\mu_0)} + L^{\\mu_0}{(E,\\mu_0)}) \\log{(E \\mu_0)}^{\\mu_0} = (E \\mu_0 L{(E,\\mu_0)} + \\log{(E \\mu_0)}^{\\mu_0}) \\log{(E \\mu_0)}^{\\mu_0} and \\delta{(E,\\mu_0)} \\log{(E \\mu_0)}^{\\mu_0} = (E \\mu_0 L{(E,\\mu_0)} + \\log{(E \\mu_0)}^{\\mu_0}) \\log{(E \\mu_0)}^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["add", 3, "Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Pow(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["times", 5, "Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Mul(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Mul(Add(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Function('L')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\Psi^{\\dagger},G)} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}}, then derive \\mathbf{A}{(\\Psi^{\\dagger},G)} = G^{\\Psi^{\\dagger}} \\log{(G)}, then obtain (- \\mathbf{A}{(\\Psi^{\\dagger},G)} + \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}}) \\mathbf{A}{(\\Psi^{\\dagger},G)} = (G^{\\Psi^{\\dagger}} \\log{(G)} - \\mathbf{A}{(\\Psi^{\\dagger},G)}) \\mathbf{A}{(\\Psi^{\\dagger},G)}", "derivation": "\\mathbf{A}{(\\Psi^{\\dagger},G)} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}} and \\mathbf{A}{(\\Psi^{\\dagger},G)} = G^{\\Psi^{\\dagger}} \\log{(G)} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}} = G^{\\Psi^{\\dagger}} \\log{(G)} and - \\mathbf{A}{(\\Psi^{\\dagger},G)} + \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}} = G^{\\Psi^{\\dagger}} \\log{(G)} - \\mathbf{A}{(\\Psi^{\\dagger},G)} and (- \\mathbf{A}{(\\Psi^{\\dagger},G)} + \\frac{\\partial}{\\partial \\Psi^{\\dagger}} G^{\\Psi^{\\dagger}}) \\mathbf{A}{(\\Psi^{\\dagger},G)} = (G^{\\Psi^{\\dagger}} \\log{(G)} - \\mathbf{A}{(\\Psi^{\\dagger},G)}) \\mathbf{A}{(\\Psi^{\\dagger},G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Derivative(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('G', commutative=True))))"], [["minus", 3, "Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Derivative(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('G', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)))))"], [["times", 4, "Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Derivative(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))), Mul(Add(Mul(Pow(Symbol('G', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('G', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True)))), Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('G', commutative=True))))"]]}, {"prompt": "Given W{(\\mathbf{v})} = \\sin{(\\log{(\\mathbf{v})})} and B{(\\mathbf{v})} = W{(\\mathbf{v})} - \\log{(\\mathbf{v})} - \\sin{(\\log{(\\mathbf{v})})}, then obtain B{(\\mathbf{v})} = - \\log{(\\mathbf{v})}", "derivation": "W{(\\mathbf{v})} = \\sin{(\\log{(\\mathbf{v})})} and W{(\\mathbf{v})} - \\log{(\\mathbf{v})} = - \\log{(\\mathbf{v})} + \\sin{(\\log{(\\mathbf{v})})} and W{(\\mathbf{v})} - \\log{(\\mathbf{v})} - \\sin{(\\log{(\\mathbf{v})})} = - \\log{(\\mathbf{v})} and B{(\\mathbf{v})} = W{(\\mathbf{v})} - \\log{(\\mathbf{v})} - \\sin{(\\log{(\\mathbf{v})})} and B{(\\mathbf{v})} = - \\log{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{v}', commutative=True)), sin(log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Function('W')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True))), sin(log(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["minus", 2, "sin(log(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Function('W')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\mathbf{v}', commutative=True))))), Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\mathbf{v}', commutative=True)), Add(Function('W')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\mathbf{v}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('B')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\delta{(A_{2},\\dot{y})} = \\log{(A_{2} + \\dot{y})}, then obtain 2 = 0^{\\dot{y}} + 1", "derivation": "\\delta{(A_{2},\\dot{y})} = \\log{(A_{2} + \\dot{y})} and A_{2} + \\dot{y} + \\delta{(A_{2},\\dot{y})} = A_{2} + \\dot{y} + \\log{(A_{2} + \\dot{y})} and \\delta{(A_{2},\\dot{y})} - \\log{(A_{2} + \\dot{y})} = 0 and (\\delta{(A_{2},\\dot{y})} - \\log{(A_{2} + \\dot{y})})^{\\dot{y}} = 0^{\\dot{y}} and 2 (\\delta{(A_{2},\\dot{y})} - \\log{(A_{2} + \\dot{y})})^{\\dot{y}} = 0^{\\dot{y}} + (\\delta{(A_{2},\\dot{y})} - \\log{(A_{2} + \\dot{y})})^{\\dot{y}} and 2 = (\\delta{(A_{2},\\dot{y})} - \\log{(A_{2} + \\dot{y})})^{\\dot{y}} + 1 and 2 = 0^{\\dot{y}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["add", 1, "Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True), Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["minus", 2, "Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Integer(0))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 4, "Pow(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(2), Add(Pow(Add(Function('\\\\delta')(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('A_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integer(2), Add(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} = \\log{(\\hat{H} + \\hat{p})} and S{(\\hat{p},\\hat{H})} = \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})}, then obtain \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} + \\hat{p} \\log{(\\hat{H} + \\hat{p})} = \\hat{p} \\log{(\\hat{H} + \\hat{p})} + S{(\\hat{p},\\hat{H})}", "derivation": "\\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} = \\log{(\\hat{H} + \\hat{p})} and \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} = \\hat{p} \\log{(\\hat{H} + \\hat{p})} and \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} + \\hat{p} \\log{(\\hat{H} + \\hat{p})} = 2 \\hat{p} \\log{(\\hat{H} + \\hat{p})} and S{(\\hat{p},\\hat{H})} = \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} and \\hat{p} \\log{(\\hat{H} + \\hat{p})} + S{(\\hat{p},\\hat{H})} = 2 \\hat{p} \\log{(\\hat{H} + \\hat{p})} and \\hat{p} \\hat{\\mathbf{x}}{(\\hat{p},\\hat{H})} + \\hat{p} \\log{(\\hat{H} + \\hat{p})} = \\hat{p} \\log{(\\hat{H} + \\hat{p})} + S{(\\hat{p},\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["add", 2, "Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given m{(\\rho)} = \\log{(\\log{(\\rho)})}, then derive \\frac{d}{d \\rho} m{(\\rho)} = \\frac{1}{\\rho \\log{(\\rho)}}, then obtain (- \\frac{d}{d \\rho} m{(\\rho)} + \\frac{d}{d \\rho} \\log{(\\log{(\\rho)})}) m{(\\rho)} = 0", "derivation": "m{(\\rho)} = \\log{(\\log{(\\rho)})} and \\frac{d}{d \\rho} m{(\\rho)} = \\frac{d}{d \\rho} \\log{(\\log{(\\rho)})} and \\frac{d}{d \\rho} m{(\\rho)} = \\frac{1}{\\rho \\log{(\\rho)}} and \\frac{d}{d \\rho} m{(\\rho)} - \\frac{1}{\\rho \\log{(\\rho)}} = 0 and (\\frac{d}{d \\rho} m{(\\rho)} - \\frac{1}{\\rho \\log{(\\rho)}}) m{(\\rho)} = 0 and (\\frac{d}{d \\rho} \\log{(\\log{(\\rho)})} - \\frac{1}{\\rho \\log{(\\rho)}}) m{(\\rho)} = 0 and (- \\frac{d}{d \\rho} m{(\\rho)} + \\frac{d}{d \\rho} \\log{(\\log{(\\rho)})}) m{(\\rho)} = 0", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\rho', commutative=True)), log(log(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))"], "Equality(Add(Derivative(Function('m')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Integer(0))"], [["times", 4, "Function('m')(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Add(Derivative(Function('m')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Function('m')(Symbol('\\\\rho', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Derivative(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1)))), Function('m')(Symbol('\\\\rho', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Mul(Integer(-1), Derivative(Function('m')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Derivative(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Function('m')(Symbol('\\\\rho', commutative=True))), Integer(0))"]]}, {"prompt": "Given E{(J,F_{c},u)} = F_{c} J u, then obtain \\int (- 3 F_{c} J u + 4 E{(J,F_{c},u)}) dJ = \\int (- F_{c} J u + 2 E{(J,F_{c},u)}) dJ", "derivation": "E{(J,F_{c},u)} = F_{c} J u and - F_{c} J u + E{(J,F_{c},u)} = 0 and - F_{c} J u + 2 E{(J,F_{c},u)} = E{(J,F_{c},u)} and \\int (- F_{c} J u + 2 E{(J,F_{c},u)}) dJ = \\int E{(J,F_{c},u)} dJ and \\int (- 3 F_{c} J u + 4 E{(J,F_{c},u)}) dJ = \\int (- F_{c} J u + 2 E{(J,F_{c},u)}) dJ", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Mul(Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True))), Integer(0))"], [["add", 2, "Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('J', commutative=True))), Integral(Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Add(Mul(Integer(-1), Integer(3), Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)), Mul(Integer(4), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('J', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Function('E')(Symbol('J', commutative=True), Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\hat{\\mathbf{x}},\\theta_2)} = \\theta_2 + \\log{(\\hat{\\mathbf{x}})} and \\hat{p}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then obtain \\frac{\\partial}{\\partial \\theta_2} \\mathbf{M}^{\\theta_2}{(\\hat{\\mathbf{x}},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + \\log{(\\hat{\\mathbf{x}})})^{\\theta_2}", "derivation": "\\mathbf{M}{(\\hat{\\mathbf{x}},\\theta_2)} = \\theta_2 + \\log{(\\hat{\\mathbf{x}})} and \\mathbf{M}^{\\theta_2}{(\\hat{\\mathbf{x}},\\theta_2)} = (\\theta_2 + \\log{(\\hat{\\mathbf{x}})})^{\\theta_2} and \\hat{p}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and \\mathbf{M}^{\\theta_2}{(\\hat{\\mathbf{x}},\\theta_2)} = (\\theta_2 + \\hat{p}{(\\hat{\\mathbf{x}})})^{\\theta_2} and \\frac{\\partial}{\\partial \\theta_2} \\mathbf{M}^{\\theta_2}{(\\hat{\\mathbf{x}},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + \\hat{p}{(\\hat{\\mathbf{x}})})^{\\theta_2} and \\frac{\\partial}{\\partial \\theta_2} \\mathbf{M}^{\\theta_2}{(\\hat{\\mathbf{x}},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 + \\log{(\\hat{\\mathbf{x}})})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(Q,\\Omega)} = \\Omega^{Q} and \\mathbf{v}{(Q,\\Omega)} = \\sin{(\\mu{(Q,\\Omega)})}, then obtain \\mu^{\\Omega}{(Q,\\Omega)} + \\frac{\\partial}{\\partial Q} \\mathbf{v}{(Q,\\Omega)} = \\mu^{\\Omega}{(Q,\\Omega)} + \\frac{\\partial}{\\partial Q} \\sin{(\\Omega^{Q})}", "derivation": "\\mu{(Q,\\Omega)} = \\Omega^{Q} and \\sin{(\\mu{(Q,\\Omega)})} = \\sin{(\\Omega^{Q})} and \\frac{\\partial}{\\partial Q} \\sin{(\\mu{(Q,\\Omega)})} = \\frac{\\partial}{\\partial Q} \\sin{(\\Omega^{Q})} and \\mathbf{v}{(Q,\\Omega)} = \\sin{(\\mu{(Q,\\Omega)})} and \\frac{\\partial}{\\partial Q} \\mathbf{v}{(Q,\\Omega)} = \\frac{\\partial}{\\partial Q} \\sin{(\\Omega^{Q})} and \\mu^{\\Omega}{(Q,\\Omega)} + \\frac{\\partial}{\\partial Q} \\mathbf{v}{(Q,\\Omega)} = \\mu^{\\Omega}{(Q,\\Omega)} + \\frac{\\partial}{\\partial Q} \\sin{(\\Omega^{Q})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('Q', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True))), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(sin(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 5, "Pow(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Pow(Function('\\\\mu')(Symbol('Q', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Derivative(sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(a,E_{x})} = E_{x} - a, then derive - a \\frac{\\partial^{2}}{\\partial a^{2}} z{(a,E_{x})} - 2 \\frac{\\partial}{\\partial a} z{(a,E_{x})} = 2, then obtain \\frac{\\partial}{\\partial a} \\int (- a \\frac{\\partial^{2}}{\\partial a^{2}} z{(a,E_{x})} - 2 \\frac{\\partial}{\\partial a} z{(a,E_{x})}) dE_{x} = \\frac{d}{d a} \\int 2 dE_{x}", "derivation": "z{(a,E_{x})} = E_{x} - a and - a z{(a,E_{x})} = - a (E_{x} - a) and \\frac{\\partial}{\\partial a} - a z{(a,E_{x})} = \\frac{\\partial}{\\partial a} - a (E_{x} - a) and \\frac{\\partial^{2}}{\\partial a^{2}} - a z{(a,E_{x})} = \\frac{\\partial^{2}}{\\partial a^{2}} - a (E_{x} - a) and - a \\frac{\\partial^{2}}{\\partial a^{2}} z{(a,E_{x})} - 2 \\frac{\\partial}{\\partial a} z{(a,E_{x})} = 2 and \\int (- a \\frac{\\partial^{2}}{\\partial a^{2}} z{(a,E_{x})} - 2 \\frac{\\partial}{\\partial a} z{(a,E_{x})}) dE_{x} = \\int 2 dE_{x} and \\frac{\\partial}{\\partial a} \\int (- a \\frac{\\partial^{2}}{\\partial a^{2}} z{(a,E_{x})} - 2 \\frac{\\partial}{\\partial a} z{(a,E_{x})}) dE_{x} = \\frac{d}{d a} \\int 2 dE_{x}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('a', commutative=True), Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('a', commutative=True), Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Mul(Integer(-1), Integer(2), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Integer(2))"], [["integrate", 5, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Mul(Integer(-1), Integer(2), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Tuple(Symbol('E_x', commutative=True))), Integral(Integer(2), Tuple(Symbol('E_x', commutative=True))))"], [["differentiate", 6, "Symbol('a', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Mul(Integer(-1), Integer(2), Derivative(Function('z')(Symbol('a', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integral(Integer(2), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(\\varepsilon)} = \\sin{(\\varepsilon)}, then obtain \\varepsilon \\frac{d}{d \\varepsilon} \\nabla{(\\varepsilon)} + \\nabla{(\\varepsilon)} = \\varepsilon \\cos{(\\varepsilon)} + \\sin{(\\varepsilon)}", "derivation": "\\nabla{(\\varepsilon)} = \\sin{(\\varepsilon)} and \\varepsilon \\nabla{(\\varepsilon)} = \\varepsilon \\sin{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\varepsilon \\nabla{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\varepsilon \\sin{(\\varepsilon)} and \\varepsilon \\frac{d}{d \\varepsilon} \\nabla{(\\varepsilon)} + \\nabla{(\\varepsilon)} = \\varepsilon \\cos{(\\varepsilon)} + \\sin{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(J,r)} = J^{r}, then obtain 1 - \\frac{1}{J} = 1 - \\frac{J^{r}}{J \\mathbf{F}{(J,r)}}", "derivation": "\\mathbf{F}{(J,r)} = J^{r} and 1 = \\frac{J^{r}}{\\mathbf{F}{(J,r)}} and - \\frac{1}{J} = - \\frac{J^{r}}{J \\mathbf{F}{(J,r)}} and 1 - \\frac{1}{J} = 1 - \\frac{J^{r}}{J \\mathbf{F}{(J,r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('r', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{F}')(Symbol('J', commutative=True), Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('J', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('J', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('J', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('J', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}}, then derive \\int \\operatorname{E_{x}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\frac{V_{\\mathbf{B}}^{2}}{2} + \\rho_b, then obtain \\Psi_{\\lambda} \\int \\operatorname{E_{x}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\Psi_{\\lambda} (\\frac{V_{\\mathbf{B}}^{2}}{2} + \\rho_b)", "derivation": "\\operatorname{E_{x}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} and \\int \\operatorname{E_{x}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\int V_{\\mathbf{B}} dV_{\\mathbf{B}} and \\int \\operatorname{E_{x}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\frac{V_{\\mathbf{B}}^{2}}{2} + \\rho_b and \\Psi_{\\lambda} \\int \\operatorname{E_{x}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\Psi_{\\lambda} (\\frac{V_{\\mathbf{B}}^{2}}{2} + \\rho_b)", "srepr_derivation": [["renaming_premise", "Equality(Function('E_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Symbol('V_{\\\\mathbf{B}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(2))), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 3, "Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('E_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(2))), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given V{(C_{d},\\varepsilon)} = \\varepsilon \\log{(C_{d})} and \\lambda{(C_{d},\\varepsilon)} = V^{C_{d}}{(C_{d},\\varepsilon)}, then obtain \\frac{V^{C_{d}}{(C_{d},\\varepsilon)}}{\\varepsilon^{2} \\log{(C_{d})}} = \\frac{\\lambda{(C_{d},\\varepsilon)}}{\\varepsilon^{2} \\log{(C_{d})}}", "derivation": "V{(C_{d},\\varepsilon)} = \\varepsilon \\log{(C_{d})} and V^{C_{d}}{(C_{d},\\varepsilon)} = (\\varepsilon \\log{(C_{d})})^{C_{d}} and \\frac{V^{C_{d}}{(C_{d},\\varepsilon)}}{\\varepsilon^{2} \\log{(C_{d})}} = \\frac{(\\varepsilon \\log{(C_{d})})^{C_{d}}}{\\varepsilon^{2} \\log{(C_{d})}} and \\lambda{(C_{d},\\varepsilon)} = V^{C_{d}}{(C_{d},\\varepsilon)} and \\lambda{(C_{d},\\varepsilon)} = (\\varepsilon \\log{(C_{d})})^{C_{d}} and \\frac{V^{C_{d}}{(C_{d},\\varepsilon)}}{\\varepsilon^{2} \\log{(C_{d})}} = \\frac{\\lambda{(C_{d},\\varepsilon)}}{\\varepsilon^{2} \\log{(C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('C_d', commutative=True))))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('V')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('C_d', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["divide", 2, "Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), log(Symbol('C_d', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-2)), Pow(Function('V')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-2)), Pow(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('V')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('C_d', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\lambda')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-2)), Pow(Function('V')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('C_d', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-2)), Function('\\\\lambda')(Symbol('C_d', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi{(c)} = \\log{(e^{c})}, then derive (\\frac{d}{d c} \\phi{(c)})^{c} + 1 = 2, then obtain ((\\frac{d}{d c} \\log{(e^{c})})^{c} + 1) \\phi{(c)} - \\phi^{c}{(c)} = \\phi{(c)} - \\phi^{c}{(c)} + \\log{(e^{c})}", "derivation": "\\phi{(c)} = \\log{(e^{c})} and 2 \\phi{(c)} = \\phi{(c)} + \\log{(e^{c})} and 2 \\phi{(c)} - \\phi^{c}{(c)} = \\phi{(c)} - \\phi^{c}{(c)} + \\log{(e^{c})} and \\frac{d}{d c} \\phi{(c)} = \\frac{d}{d c} \\log{(e^{c})} and (\\frac{d}{d c} \\phi{(c)})^{c} = (\\frac{d}{d c} \\log{(e^{c})})^{c} and (\\frac{d}{d c} \\phi{(c)})^{c} + 1 = (\\frac{d}{d c} \\log{(e^{c})})^{c} + 1 and (\\frac{d}{d c} \\phi{(c)})^{c} + 1 = 2 and (\\frac{d}{d c} \\log{(e^{c})})^{c} + 1 = 2 and ((\\frac{d}{d c} \\log{(e^{c})})^{c} + 1) \\phi{(c)} - \\phi^{c}{(c)} = \\phi{(c)} - \\phi^{c}{(c)} + \\log{(e^{c})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True))))"], [["add", 1, "Function('\\\\phi')(Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('c', commutative=True))), Add(Function('\\\\phi')(Symbol('c', commutative=True)), log(exp(Symbol('c', commutative=True)))))"], [["minus", 2, "Pow(Function('\\\\phi')(Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\phi')(Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Add(Function('\\\\phi')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), log(exp(Symbol('c', commutative=True)))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(log(exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 4, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\phi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Derivative(log(exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Pow(Derivative(Function('\\\\phi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Integer(1)), Add(Pow(Derivative(log(exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Integer(1)))"], [["evaluate_derivatives", 6], "Equality(Add(Pow(Derivative(Function('\\\\phi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Pow(Derivative(log(exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 3, 8], "Equality(Add(Mul(Add(Pow(Derivative(log(exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Integer(1)), Function('\\\\phi')(Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Add(Function('\\\\phi')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), log(exp(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\delta)} = \\cos{(\\delta)}, then derive \\frac{d}{d \\delta} \\hat{x}_0{(\\delta)} = - \\sin{(\\delta)}, then obtain - \\hat{x}_0{(\\delta)} + \\frac{d}{d \\delta} \\hat{x}_0{(\\delta)} = - \\hat{x}_0{(\\delta)} - \\sin{(\\delta)}", "derivation": "\\hat{x}_0{(\\delta)} = \\cos{(\\delta)} and \\frac{d}{d \\delta} \\hat{x}_0{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and \\frac{d}{d \\delta} \\hat{x}_0{(\\delta)} = - \\sin{(\\delta)} and - \\hat{x}_0{(\\delta)} + \\frac{d}{d \\delta} \\hat{x}_0{(\\delta)} = - \\hat{x}_0{(\\delta)} - \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True))))"], [["minus", 3, "Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True))), Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(a)} = \\cos{(a)}, then obtain (\\frac{d}{d a} 1)^{2} = \\frac{d}{d a} 1 \\frac{d}{d a} (- \\dot{\\mathbf{r}}{(a)} + \\cos{(a)} + 1)", "derivation": "\\dot{\\mathbf{r}}{(a)} = \\cos{(a)} and \\dot{\\mathbf{r}}{(a)} - \\cos{(a)} = 0 and \\dot{\\mathbf{r}}{(a)} - \\cos{(a)} + 1 = 1 and 1 = - \\dot{\\mathbf{r}}{(a)} + \\cos{(a)} + 1 and \\frac{d}{d a} 1 = \\frac{d}{d a} (- \\dot{\\mathbf{r}}{(a)} + \\cos{(a)} + 1) and (\\frac{d}{d a} 1)^{2} = \\frac{d}{d a} 1 \\frac{d}{d a} (- \\dot{\\mathbf{r}}{(a)} + \\cos{(a)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["minus", 1, "cos(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Mul(Integer(-1), cos(Symbol('a', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Mul(Integer(-1), cos(Symbol('a', commutative=True))), Integer(1)), Integer(1))"], [["minus", 3, "Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Mul(Integer(-1), cos(Symbol('a', commutative=True))))"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True))), cos(Symbol('a', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True))), cos(Symbol('a', commutative=True)), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 5, "Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True))), cos(Symbol('a', commutative=True)), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu_{0}{(r)} = e^{r}, then derive \\frac{d}{d r} \\mu_{0}{(r)} = e^{r}, then obtain (\\frac{d^{2}}{d r^{2}} e^{r})^{2 r} = (\\frac{d}{d r} e^{r})^{2 r}", "derivation": "\\mu_{0}{(r)} = e^{r} and \\frac{d}{d r} \\mu_{0}{(r)} = \\frac{d}{d r} e^{r} and \\frac{d}{d r} \\mu_{0}{(r)} = e^{r} and \\frac{d}{d r} e^{r} = e^{r} and \\frac{d^{2}}{d r^{2}} \\mu_{0}{(r)} = \\frac{d}{d r} \\mu_{0}{(r)} and \\frac{d^{2}}{d r^{2}} e^{r} = \\frac{d}{d r} e^{r} and (\\frac{d^{2}}{d r^{2}} e^{r})^{r} = (\\frac{d}{d r} e^{r})^{r} and (\\frac{d^{2}}{d r^{2}} e^{r})^{2 r} = (\\frac{d}{d r} e^{r})^{2 r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), exp(Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), exp(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('\\\\mu_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(Function('\\\\mu_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 6, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Symbol('r', commutative=True)), Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["power", 7, 2], "Equality(Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Mul(Integer(2), Symbol('r', commutative=True))), Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Integer(2), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{\\mathbf{g}}{W}, then derive \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{1}{W}, then obtain (\\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W})^{2} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W}}{W}", "derivation": "\\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{\\mathbf{g}}{W} and \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W} and \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{1}{W} and \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W} \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{A_{x}}{(W,\\mathbf{g})} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W}}{W} and (\\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W})^{2} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\frac{\\mathbf{g}}{W}}{W}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Pow(Symbol('W', commutative=True), Integer(-1)))"], [["times", 3, "Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Function('A_x')(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\delta,v_{t})} = \\cos{(\\delta v_{t})} and \\operatorname{n_{1}}{(\\delta,v_{t})} = \\delta v_{t} + \\dot{\\mathbf{r}}{(\\delta,v_{t})} + \\cos{(\\delta v_{t})}, then obtain \\operatorname{n_{1}}{(\\delta,v_{t})} = \\delta v_{t} + 2 \\cos{(\\delta v_{t})}", "derivation": "\\dot{\\mathbf{r}}{(\\delta,v_{t})} = \\cos{(\\delta v_{t})} and \\delta v_{t} + \\dot{\\mathbf{r}}{(\\delta,v_{t})} = \\delta v_{t} + \\cos{(\\delta v_{t})} and \\delta v_{t} + \\dot{\\mathbf{r}}{(\\delta,v_{t})} + \\cos{(\\delta v_{t})} = \\delta v_{t} + 2 \\cos{(\\delta v_{t})} and \\operatorname{n_{1}}{(\\delta,v_{t})} = \\delta v_{t} + \\dot{\\mathbf{r}}{(\\delta,v_{t})} + \\cos{(\\delta v_{t})} and \\operatorname{n_{1}}{(\\delta,v_{t})} = \\delta v_{t} + 2 \\cos{(\\delta v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)))))"], [["add", 2, "cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('n_1')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(E,\\mathbf{J},f)} = \\frac{E}{f} + \\mathbf{J} and \\operatorname{P_{g}}{(E,\\mathbf{J},f)} = \\int (\\frac{E}{f} + \\mathbf{J}) df, then obtain \\iint \\tilde{g}^*{(E,\\mathbf{J},f)} df df = \\iint (\\frac{E}{f} + \\mathbf{J}) df df", "derivation": "\\tilde{g}^*{(E,\\mathbf{J},f)} = \\frac{E}{f} + \\mathbf{J} and \\int \\tilde{g}^*{(E,\\mathbf{J},f)} df = \\int (\\frac{E}{f} + \\mathbf{J}) df and \\operatorname{P_{g}}{(E,\\mathbf{J},f)} = \\int (\\frac{E}{f} + \\mathbf{J}) df and \\int \\operatorname{P_{g}}{(E,\\mathbf{J},f)} df = \\iint (\\frac{E}{f} + \\mathbf{J}) df df and \\int \\tilde{g}^*{(E,\\mathbf{J},f)} df = \\operatorname{P_{g}}{(E,\\mathbf{J},f)} and \\iint \\tilde{g}^*{(E,\\mathbf{J},f)} df df = \\iint (\\frac{E}{f} + \\mathbf{J}) df df", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Add(Mul(Symbol('E', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Integral(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Function('P_g')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\mathbf{p},Q)} = \\frac{\\cos{(Q)}}{\\mathbf{p}}, then derive \\hat{p} + \\phi{(\\mathbf{p},Q)} = E_{n} + \\frac{\\cos{(Q)}}{\\mathbf{p}}, then obtain \\int (\\hat{p} + \\phi{(\\mathbf{p},Q)}) d\\mathbf{p} = \\int (E_{n} + \\phi{(\\mathbf{p},Q)}) d\\mathbf{p}", "derivation": "\\phi{(\\mathbf{p},Q)} = \\frac{\\cos{(Q)}}{\\mathbf{p}} and \\frac{\\partial}{\\partial Q} \\phi{(\\mathbf{p},Q)} = \\frac{\\partial}{\\partial Q} \\frac{\\cos{(Q)}}{\\mathbf{p}} and \\int \\frac{\\partial}{\\partial Q} \\phi{(\\mathbf{p},Q)} dQ = \\int \\frac{\\partial}{\\partial Q} \\frac{\\cos{(Q)}}{\\mathbf{p}} dQ and \\hat{p} + \\phi{(\\mathbf{p},Q)} = E_{n} + \\frac{\\cos{(Q)}}{\\mathbf{p}} and \\hat{p} + \\phi{(\\mathbf{p},Q)} = E_{n} + \\phi{(\\mathbf{p},Q)} and \\int (\\hat{p} + \\phi{(\\mathbf{p},Q)}) d\\mathbf{p} = \\int (E_{n} + \\phi{(\\mathbf{p},Q)}) d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), cos(Symbol('Q', commutative=True))))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), cos(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('E_n', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Function('\\\\phi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\theta_2,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\theta_2} and E{(\\theta_2,\\mathbf{J}_P)} = \\lambda{(\\theta_2,\\mathbf{J}_P)} - 1, then obtain E{(\\theta_2,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\theta_2} - 1", "derivation": "\\lambda{(\\theta_2,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\theta_2} and \\lambda{(\\theta_2,\\mathbf{J}_P)} - 1 = \\frac{\\mathbf{J}_P}{\\theta_2} - 1 and E{(\\theta_2,\\mathbf{J}_P)} = \\lambda{(\\theta_2,\\mathbf{J}_P)} - 1 and E{(\\theta_2,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\theta_2} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Function('\\\\lambda')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Integer(-1)))"]]}, {"prompt": "Given \\phi_{2}{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain \\frac{d}{d \\tilde{g}} \\frac{2 \\phi_{2}^{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} = \\frac{d}{d \\tilde{g}} (\\frac{\\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} + 1) \\phi_{2}{(\\tilde{g})}", "derivation": "\\phi_{2}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\frac{\\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} = 1 and \\frac{2 \\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} = \\frac{\\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} + 1 and \\frac{2 \\phi_{2}^{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} = (\\frac{\\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} + 1) \\phi_{2}{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\frac{2 \\phi_{2}^{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} = \\frac{d}{d \\tilde{g}} (\\frac{\\phi_{2}{(\\tilde{g})}}{\\cos{(\\tilde{g})}} + 1) \\phi_{2}{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Mul(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Add(Mul(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Integer(1)))"], [["times", 3, "Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Mul(Add(Mul(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Integer(1)), Function('\\\\phi_2')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\lambda,\\delta)} = - \\delta + \\lambda, then obtain (\\delta + \\psi{(\\lambda,\\delta)} - 1) \\int (\\delta + \\psi{(\\lambda,\\delta)} - 1) d\\delta = (\\delta + \\psi{(\\lambda,\\delta)} - 1) \\int (\\lambda - 1) d\\delta", "derivation": "\\psi{(\\lambda,\\delta)} = - \\delta + \\lambda and \\delta + \\psi{(\\lambda,\\delta)} = \\lambda and \\delta + \\psi{(\\lambda,\\delta)} - 1 = \\lambda - 1 and \\int (\\delta + \\psi{(\\lambda,\\delta)} - 1) d\\delta = \\int (\\lambda - 1) d\\delta and (\\delta + \\psi{(\\lambda,\\delta)} - 1) \\int (\\delta + \\psi{(\\lambda,\\delta)} - 1) d\\delta = (\\delta + \\psi{(\\lambda,\\delta)} - 1) \\int (\\lambda - 1) d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('\\\\lambda', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 4, "Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Symbol('\\\\delta', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\lambda', commutative=True), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\mu{(M_{E})} = \\log{(M_{E})}, then derive \\int \\mu{(M_{E})} dM_{E} = M_{E} \\log{(M_{E})} - M_{E} + s, then obtain \\iint \\mu{(M_{E})} dM_{E} dM_{E} = \\int (M_{E} \\mu{(M_{E})} - M_{E} + s) dM_{E}", "derivation": "\\mu{(M_{E})} = \\log{(M_{E})} and \\int \\mu{(M_{E})} dM_{E} = \\int \\log{(M_{E})} dM_{E} and \\int \\mu{(M_{E})} dM_{E} = M_{E} \\log{(M_{E})} - M_{E} + s and \\int \\mu{(M_{E})} dM_{E} = M_{E} \\mu{(M_{E})} - M_{E} + s and \\iint \\mu{(M_{E})} dM_{E} dM_{E} = \\int (M_{E} \\mu{(M_{E})} - M_{E} + s) dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mu')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mu')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mu')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} = Z \\mathbf{P}, then obtain - (\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} dZ)^{\\mathbf{P}} = - (\\int Z \\mathbf{P} dZ)^{\\mathbf{P}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} = Z \\mathbf{P} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} dZ = \\int Z \\mathbf{P} dZ and (\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} dZ)^{\\mathbf{P}} = (\\int Z \\mathbf{P} dZ)^{\\mathbf{P}} and - (\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{P},Z)} dZ)^{\\mathbf{P}} = - (\\int Z \\mathbf{P} dZ)^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Integral(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\Omega)} = \\sin{(\\Omega)}, then derive \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} + \\int \\operatorname{c_{0}}{(\\Omega)} d\\Omega = \\theta_1 + \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} - \\cos{(\\Omega)}, then obtain \\sin^{\\Omega}{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega = \\theta_1 + \\sin^{\\Omega}{(\\Omega)} - \\cos{(\\Omega)}", "derivation": "\\operatorname{c_{0}}{(\\Omega)} = \\sin{(\\Omega)} and \\int \\operatorname{c_{0}}{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} = \\sin^{\\Omega}{(\\Omega)} and \\sin^{\\Omega}{(\\Omega)} + \\int \\operatorname{c_{0}}{(\\Omega)} d\\Omega = \\sin^{\\Omega}{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega and \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} + \\int \\operatorname{c_{0}}{(\\Omega)} d\\Omega = \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega and \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} + \\int \\operatorname{c_{0}}{(\\Omega)} d\\Omega = \\theta_1 + \\operatorname{c_{0}}^{\\Omega}{(\\Omega)} - \\cos{(\\Omega)} and \\sin^{\\Omega}{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega = \\theta_1 + \\sin^{\\Omega}{(\\Omega)} - \\cos{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["add", 2, "Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Pow(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Pow(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Pow(Function('c_0')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(P_{e},u)} = (e^{P_{e}})^{u}, then obtain \\operatorname{v_{t}}{(P_{e},u)} e^{- P_{e}} \\log{((e^{P_{e}})^{u})} = e^{- P_{e}} (e^{P_{e}})^{u} \\log{((e^{P_{e}})^{u})}", "derivation": "\\operatorname{v_{t}}{(P_{e},u)} = (e^{P_{e}})^{u} and \\log{(\\operatorname{v_{t}}{(P_{e},u)})} = \\log{((e^{P_{e}})^{u})} and \\operatorname{v_{t}}{(P_{e},u)} e^{- P_{e}} = e^{- P_{e}} (e^{P_{e}})^{u} and \\operatorname{v_{t}}{(P_{e},u)} e^{- P_{e}} \\log{(\\operatorname{v_{t}}{(P_{e},u)})} = e^{- P_{e}} (e^{P_{e}})^{u} \\log{(\\operatorname{v_{t}}{(P_{e},u)})} and \\operatorname{v_{t}}{(P_{e},u)} e^{- P_{e}} \\log{((e^{P_{e}})^{u})} = e^{- P_{e}} (e^{P_{e}})^{u} \\log{((e^{P_{e}})^{u})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True)))"], [["log", 1], "Equality(log(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True))), log(Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True))))"], [["divide", 1, "exp(Symbol('P_e', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True))))"], [["times", 3, "log(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), log(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True)), log(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('v_t')(Symbol('P_e', commutative=True), Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), log(Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True)), log(Pow(exp(Symbol('P_e', commutative=True)), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{J}_f)} = e^{e^{\\mathbf{J}_f}}, then obtain \\lambda{(\\mathbf{J}_f)} (e^{e^{\\mathbf{J}_f}})^{\\mathbf{J}_f} = e^{e^{\\mathbf{J}_f}} (e^{e^{\\mathbf{J}_f}})^{\\mathbf{J}_f}", "derivation": "\\lambda{(\\mathbf{J}_f)} = e^{e^{\\mathbf{J}_f}} and \\lambda^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = (e^{e^{\\mathbf{J}_f}})^{\\mathbf{J}_f} and \\lambda{(\\mathbf{J}_f)} \\lambda^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = \\lambda^{\\mathbf{J}_f}{(\\mathbf{J}_f)} e^{e^{\\mathbf{J}_f}} and \\lambda{(\\mathbf{J}_f)} (e^{e^{\\mathbf{J}_f}})^{\\mathbf{J}_f} = e^{e^{\\mathbf{J}_f}} (e^{e^{\\mathbf{J}_f}})^{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(exp(exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given T{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain \\cos{(\\int T{(\\sigma_p)} d\\sigma_p)} = \\cos{(\\eta^{\\prime} + \\sigma_p \\log{(\\sigma_p)} - \\sigma_p)}", "derivation": "T{(\\sigma_p)} = \\log{(\\sigma_p)} and \\int T{(\\sigma_p)} d\\sigma_p = \\int \\log{(\\sigma_p)} d\\sigma_p and \\cos{(\\int T{(\\sigma_p)} d\\sigma_p)} = \\cos{(\\int \\log{(\\sigma_p)} d\\sigma_p)} and \\cos{(\\int T{(\\sigma_p)} d\\sigma_p)} = \\cos{(\\eta^{\\prime} + \\sigma_p \\log{(\\sigma_p)} - \\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('T')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('T')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), cos(Integral(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(cos(Integral(Function('T')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), cos(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\sigma_p', commutative=True), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(L)} = \\cos{(L)} and \\operatorname{V_{\\mathbf{E}}}{(L)} = \\mathbf{v}^{2}{(L)}, then obtain \\operatorname{V_{\\mathbf{E}}}^{2}{(L)} = \\operatorname{V_{\\mathbf{E}}}{(L)} \\cos^{2}{(L)}", "derivation": "\\mathbf{v}{(L)} = \\cos{(L)} and \\mathbf{v}^{2}{(L)} = \\mathbf{v}{(L)} \\cos{(L)} and \\mathbf{v}^{4}{(L)} = \\mathbf{v}^{2}{(L)} \\cos^{2}{(L)} and \\operatorname{V_{\\mathbf{E}}}{(L)} = \\mathbf{v}^{2}{(L)} and \\operatorname{V_{\\mathbf{E}}}^{2}{(L)} = \\operatorname{V_{\\mathbf{E}}}{(L)} \\cos^{2}{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{v}')(Symbol('L', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), Integer(2)), Pow(cos(Symbol('L', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('L', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Integer(2)), Mul(Function('V_{\\\\mathbf{E}}')(Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then derive \\int \\mathbf{v}{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon}, then obtain \\hat{p} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} = \\mathbf{r} \\mathbf{v}{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon}", "derivation": "\\mathbf{v}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\int \\mathbf{v}{(\\mathbf{r})} d\\mathbf{r} = \\int \\log{(\\mathbf{r})} d\\mathbf{r} and \\int \\mathbf{v}{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon} and \\int \\log{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon} and \\int \\log{(\\mathbf{r})} d\\mathbf{r} = \\mathbf{r} \\mathbf{v}{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon} and \\hat{p} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r} = \\mathbf{r} \\mathbf{v}{(\\mathbf{r})} - \\mathbf{r} + g_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\theta{(\\mathbf{J}_P,m_{s})} = m_{s}^{\\mathbf{J}_P}, then derive m_{s}^{\\mathbf{J}_P} \\log{(m_{s})} + \\frac{\\partial}{\\partial \\mathbf{J}_P} \\theta{(\\mathbf{J}_P,m_{s})} = 2 m_{s}^{\\mathbf{J}_P} \\log{(m_{s})}, then obtain m_{s}^{\\mathbf{J}_P} \\log{(m_{s})} + \\frac{\\partial}{\\partial \\mathbf{J}_P} m_{s}^{\\mathbf{J}_P} = 2 m_{s}^{\\mathbf{J}_P} \\log{(m_{s})}", "derivation": "\\theta{(\\mathbf{J}_P,m_{s})} = m_{s}^{\\mathbf{J}_P} and m_{s}^{\\mathbf{J}_P} + \\theta{(\\mathbf{J}_P,m_{s})} = 2 m_{s}^{\\mathbf{J}_P} and \\frac{\\partial}{\\partial \\mathbf{J}_P} (m_{s}^{\\mathbf{J}_P} + \\theta{(\\mathbf{J}_P,m_{s})}) = \\frac{\\partial}{\\partial \\mathbf{J}_P} 2 m_{s}^{\\mathbf{J}_P} and m_{s}^{\\mathbf{J}_P} \\log{(m_{s})} + \\frac{\\partial}{\\partial \\mathbf{J}_P} \\theta{(\\mathbf{J}_P,m_{s})} = 2 m_{s}^{\\mathbf{J}_P} \\log{(m_{s})} and m_{s}^{\\mathbf{J}_P} \\log{(m_{s})} + \\frac{\\partial}{\\partial \\mathbf{J}_P} m_{s}^{\\mathbf{J}_P} = 2 m_{s}^{\\mathbf{J}_P} \\log{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(2), Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('m_s', commutative=True))), Derivative(Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('m_s', commutative=True))), Derivative(Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given x{(s)} = \\sin{(s)}, then obtain \\frac{- \\cos{(s)} + \\int \\sin^{2}{(s)} ds}{(x{(s)} + \\sin{(s)})^{2}} = \\frac{- \\cos{(s)} + \\int \\sin^{\\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}}}{(s)} ds}{(x{(s)} + \\sin{(s)})^{2}}", "derivation": "x{(s)} = \\sin{(s)} and x{(s)} + \\sin{(s)} = 2 \\sin{(s)} and x{(s)} \\sin{(s)} = \\sin^{2}{(s)} and \\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}} = 2 and \\int x{(s)} \\sin{(s)} ds = \\int \\sin^{2}{(s)} ds and \\int x{(s)} \\sin{(s)} ds = \\int \\sin^{\\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}}}{(s)} ds and \\int \\sin^{2}{(s)} ds = \\int \\sin^{\\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}}}{(s)} ds and - \\cos{(s)} + \\int \\sin^{2}{(s)} ds = - \\cos{(s)} + \\int \\sin^{\\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}}}{(s)} ds and \\frac{- \\cos{(s)} + \\int \\sin^{2}{(s)} ds}{(x{(s)} + \\sin{(s)})^{2}} = \\frac{- \\cos{(s)} + \\int \\sin^{\\frac{x{(s)} + \\sin{(s)}}{\\sin{(s)}}}{(s)} ds}{(x{(s)} + \\sin{(s)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["add", 1, "sin(Symbol('s', commutative=True))"], "Equality(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Mul(Integer(2), sin(Symbol('s', commutative=True))))"], [["times", 1, "sin(Symbol('s', commutative=True))"], "Equality(Mul(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(2)))"], [["divide", 2, "sin(Symbol('s', commutative=True))"], "Equality(Mul(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1))), Integer(2))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Integer(2)), Tuple(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Mul(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Mul(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1)))), Tuple(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Pow(sin(Symbol('s', commutative=True)), Integer(2)), Tuple(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Mul(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1)))), Tuple(Symbol('s', commutative=True))))"], [["minus", 7, "cos(Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Integer(2)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Mul(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1)))), Tuple(Symbol('s', commutative=True)))))"], [["divide", 8, "Pow(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Integer(2))"], "Equality(Mul(Pow(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Integer(2)), Tuple(Symbol('s', commutative=True))))), Mul(Pow(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Mul(Add(Function('x')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(sin(Symbol('s', commutative=True)), Integer(-1)))), Tuple(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given S{(t_{2},\\mathbf{P})} = e^{\\frac{t_{2}}{\\mathbf{P}}}, then derive \\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} + 1 = 1 + \\frac{e^{\\frac{t_{2}}{\\mathbf{P}}}}{\\mathbf{P}}, then obtain (\\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} + 1) e^{\\frac{t_{2}}{\\mathbf{P}}} = (1 + \\frac{e^{\\frac{t_{2}}{\\mathbf{P}}}}{\\mathbf{P}}) e^{\\frac{t_{2}}{\\mathbf{P}}}", "derivation": "S{(t_{2},\\mathbf{P})} = e^{\\frac{t_{2}}{\\mathbf{P}}} and \\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} = \\frac{\\partial}{\\partial t_{2}} e^{\\frac{t_{2}}{\\mathbf{P}}} and \\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} + 1 = \\frac{\\partial}{\\partial t_{2}} e^{\\frac{t_{2}}{\\mathbf{P}}} + 1 and \\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} + 1 = 1 + \\frac{e^{\\frac{t_{2}}{\\mathbf{P}}}}{\\mathbf{P}} and (\\frac{\\partial}{\\partial t_{2}} S{(t_{2},\\mathbf{P})} + 1) e^{\\frac{t_{2}}{\\mathbf{P}}} = (1 + \\frac{e^{\\frac{t_{2}}{\\mathbf{P}}}}{\\mathbf{P}}) e^{\\frac{t_{2}}{\\mathbf{P}}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))))"], [["times", 4, "exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], "Equality(Mul(Add(Derivative(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1)), exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))), Mul(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))), exp(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} = e^{\\mathbf{F}}, then derive 1 = \\frac{n + e^{\\mathbf{F}}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}}, then obtain 1 - \\frac{\\int e^{\\mathbf{F}} d\\mathbf{F}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}} = \\frac{n + e^{\\mathbf{F}}}{\\int e^{\\mathbf{F}} d\\mathbf{F}} - \\frac{\\int e^{\\mathbf{F}} d\\mathbf{F}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} = e^{\\mathbf{F}} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F} = \\int e^{\\mathbf{F}} d\\mathbf{F} and 1 = \\frac{\\int e^{\\mathbf{F}} d\\mathbf{F}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}} and 1 = \\frac{n + e^{\\mathbf{F}}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}} and 1 = \\frac{n + e^{\\mathbf{F}}}{\\int e^{\\mathbf{F}} d\\mathbf{F}} and 1 - \\frac{\\int e^{\\mathbf{F}} d\\mathbf{F}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}} = \\frac{n + e^{\\mathbf{F}}}{\\int e^{\\mathbf{F}} d\\mathbf{F}} - \\frac{\\int e^{\\mathbf{F}} d\\mathbf{F}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{F})} d\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 2, "Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Symbol('n', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Add(Symbol('n', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"], [["minus", 5, "Mul(Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))), Add(Mul(Add(Symbol('n', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(h,a^{\\dagger})} = \\log{(a^{\\dagger} h)} and \\hat{X}{(h,a^{\\dagger})} = a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})}, then obtain a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)} = \\hat{X}{(h,a^{\\dagger})} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)}", "derivation": "\\operatorname{v_{1}}{(h,a^{\\dagger})} = \\log{(a^{\\dagger} h)} and a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})} = a^{\\dagger} + \\log{(a^{\\dagger} h)} and a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)} = a^{\\dagger} + \\log{(a^{\\dagger} h)} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)} and \\hat{X}{(h,a^{\\dagger})} = a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})} and \\hat{X}{(h,a^{\\dagger})} = a^{\\dagger} + \\log{(a^{\\dagger} h)} and a^{\\dagger} + \\operatorname{v_{1}}{(h,a^{\\dagger})} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)} = \\hat{X}{(h,a^{\\dagger})} + \\frac{\\partial}{\\partial a^{\\dagger}} \\log{(a^{\\dagger} h)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True)))))"], [["add", 2, "Derivative(log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Derivative(log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(log(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi{(C)} = e^{C}, then obtain (\\int e^{\\frac{C (\\psi{(C)} - e^{C} + 1)}{\\psi{(C)} - e^{C}}} dC)^{C} = (\\int e^{\\frac{C}{\\psi{(C)} - e^{C}}} dC)^{C}", "derivation": "\\psi{(C)} = e^{C} and \\psi{(C)} - e^{C} = 0 and \\psi{(C)} - e^{C} + 1 = 1 and \\frac{\\psi{(C)} - e^{C} + 1}{\\psi{(C)} - e^{C}} = \\frac{1}{\\psi{(C)} - e^{C}} and \\frac{C (\\psi{(C)} - e^{C} + 1)}{\\psi{(C)} - e^{C}} = \\frac{C}{\\psi{(C)} - e^{C}} and e^{\\frac{C (\\psi{(C)} - e^{C} + 1)}{\\psi{(C)} - e^{C}}} = e^{\\frac{C}{\\psi{(C)} - e^{C}}} and \\int e^{\\frac{C (\\psi{(C)} - e^{C} + 1)}{\\psi{(C)} - e^{C}}} dC = \\int e^{\\frac{C}{\\psi{(C)} - e^{C}}} dC and (\\int e^{\\frac{C (\\psi{(C)} - e^{C} + 1)}{\\psi{(C)} - e^{C}}} dC)^{C} = (\\int e^{\\frac{C}{\\psi{(C)} - e^{C}}} dC)^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["minus", 1, "exp(Symbol('C', commutative=True))"], "Equality(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1)), Integer(1))"], [["divide", 3, "Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)), Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1))), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)))"], [["times", 4, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)), Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1))), Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1))))"], [["exp", 5], "Equality(exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)), Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1)))), exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)))))"], [["integrate", 6, "Symbol('C', commutative=True)"], "Equality(Integral(exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)), Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1)))), Tuple(Symbol('C', commutative=True))), Integral(exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)))), Tuple(Symbol('C', commutative=True))))"], [["power", 7, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)), Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True))), Integer(1)))), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(exp(Mul(Symbol('C', commutative=True), Pow(Add(Function('\\\\psi')(Symbol('C', commutative=True)), Mul(Integer(-1), exp(Symbol('C', commutative=True)))), Integer(-1)))), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given H{(L)} = \\sin{(L)}, then derive (- \\frac{H{(L)} \\cos{(L)}}{\\sin^{2}{(L)}} + \\frac{\\frac{d}{d L} H{(L)}}{\\sin{(L)}}) \\cos{(\\frac{H{(L)}}{\\sin{(L)}})} = 0, then obtain (- \\frac{\\cos{(L)}}{\\sin{(L)}} + \\frac{\\frac{d}{d L} \\sin{(L)}}{\\sin{(L)}}) \\cos{(1)} = 0", "derivation": "H{(L)} = \\sin{(L)} and \\frac{H{(L)}}{\\sin{(L)}} = 1 and \\sin{(\\frac{H{(L)}}{\\sin{(L)}})} = \\sin{(1)} and \\frac{d}{d L} \\sin{(\\frac{H{(L)}}{\\sin{(L)}})} = \\frac{d}{d L} \\sin{(1)} and (- \\frac{H{(L)} \\cos{(L)}}{\\sin^{2}{(L)}} + \\frac{\\frac{d}{d L} H{(L)}}{\\sin{(L)}}) \\cos{(\\frac{H{(L)}}{\\sin{(L)}})} = 0 and (- \\frac{\\cos{(L)}}{\\sin{(L)}} + \\frac{\\frac{d}{d L} \\sin{(L)}}{\\sin{(L)}}) \\cos{(1)} = 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["divide", 1, "sin(Symbol('L', commutative=True))"], "Equality(Mul(Function('H')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-1))), Integer(1))"], [["sin", 2], "Equality(sin(Mul(Function('H')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-1)))), sin(Integer(1)))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(sin(Mul(Function('H')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-1)))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Mul(Integer(-1), Function('H')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-2)), cos(Symbol('L', commutative=True))), Mul(Pow(sin(Symbol('L', commutative=True)), Integer(-1)), Derivative(Function('H')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), cos(Mul(Function('H')(Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Integer(-1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Mul(Integer(-1), Pow(sin(Symbol('L', commutative=True)), Integer(-1)), cos(Symbol('L', commutative=True))), Mul(Pow(sin(Symbol('L', commutative=True)), Integer(-1)), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), cos(Integer(1))), Integer(0))"]]}, {"prompt": "Given c{(\\Psi)} = \\log{(e^{\\Psi})}, then obtain \\frac{d}{d \\Psi} (c^{3}{(\\Psi)} \\log{(e^{\\Psi})} - c^{3}{(\\Psi)}) = \\frac{d}{d \\Psi} (- c^{3}{(\\Psi)} + c^{2}{(\\Psi)} \\log{(e^{\\Psi})}^{2})", "derivation": "c{(\\Psi)} = \\log{(e^{\\Psi})} and c{(\\Psi)} \\log{(e^{\\Psi})} = \\log{(e^{\\Psi})}^{2} and c^{2}{(\\Psi)} \\log{(e^{\\Psi})}^{2} = \\log{(e^{\\Psi})}^{4} and c^{3}{(\\Psi)} \\log{(e^{\\Psi})} = c^{2}{(\\Psi)} \\log{(e^{\\Psi})}^{2} and c^{3}{(\\Psi)} \\log{(e^{\\Psi})} - c^{3}{(\\Psi)} = - c^{3}{(\\Psi)} + c^{2}{(\\Psi)} \\log{(e^{\\Psi})}^{2} and \\frac{d}{d \\Psi} (c^{3}{(\\Psi)} \\log{(e^{\\Psi})} - c^{3}{(\\Psi)}) = \\frac{d}{d \\Psi} (- c^{3}{(\\Psi)} + c^{2}{(\\Psi)} \\log{(e^{\\Psi})}^{2})", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Psi', commutative=True)), log(exp(Symbol('\\\\Psi', commutative=True))))"], [["times", 1, "log(exp(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Function('c')(Symbol('\\\\Psi', commutative=True)), log(exp(Symbol('\\\\Psi', commutative=True)))), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(2))), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3)), log(exp(Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(2))))"], [["minus", 4, "Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3))"], "Equality(Add(Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3)), log(exp(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3)))), Add(Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3))), Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(2)))))"], [["differentiate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Add(Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3)), log(exp(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(3))), Mul(Pow(Function('c')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\Psi', commutative=True))), Integer(2)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\eta)} = \\log{(\\eta)}, then obtain \\frac{\\frac{d}{d \\eta} b{(\\eta)}}{b{(\\eta)} \\frac{d^{2}}{d \\eta^{2}} b{(\\eta)}} = \\frac{\\frac{d}{d \\eta} \\log{(\\eta)}}{b{(\\eta)} \\frac{d^{2}}{d \\eta^{2}} b{(\\eta)}}", "derivation": "b{(\\eta)} = \\log{(\\eta)} and \\frac{d}{d \\eta} b{(\\eta)} = \\frac{d}{d \\eta} \\log{(\\eta)} and \\frac{\\frac{d}{d \\eta} b{(\\eta)}}{b{(\\eta)}} = \\frac{\\frac{d}{d \\eta} \\log{(\\eta)}}{b{(\\eta)}} and \\frac{\\frac{d}{d \\eta} b{(\\eta)}}{b{(\\eta)} \\frac{d^{2}}{d \\eta^{2}} b{(\\eta)}} = \\frac{\\frac{d}{d \\eta} \\log{(\\eta)}}{b{(\\eta)} \\frac{d^{2}}{d \\eta^{2}} b{(\\eta)}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 2, "Function('b')(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Function('b')(Symbol('\\\\eta', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Pow(Function('b')(Symbol('\\\\eta', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["divide", 3, "Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2)))"], "Equality(Mul(Pow(Function('b')(Symbol('\\\\eta', commutative=True)), Integer(-1)), Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Pow(Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(-1))), Mul(Pow(Function('b')(Symbol('\\\\eta', commutative=True)), Integer(-1)), Pow(Derivative(Function('b')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(2))), Integer(-1)), Derivative(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(\\sigma_x,m)} = \\sigma_x - m, then obtain (\\int \\nabla{(\\sigma_x,m)} d\\sigma_x - \\frac{1}{\\sigma_x})^{\\sigma_x} = (\\int (\\sigma_x - m) d\\sigma_x - \\frac{1}{\\sigma_x})^{\\sigma_x}", "derivation": "\\nabla{(\\sigma_x,m)} = \\sigma_x - m and \\int \\nabla{(\\sigma_x,m)} d\\sigma_x = \\int (\\sigma_x - m) d\\sigma_x and \\int \\nabla{(\\sigma_x,m)} d\\sigma_x - \\frac{1}{\\sigma_x} = \\int (\\sigma_x - m) d\\sigma_x - \\frac{1}{\\sigma_x} and (\\int \\nabla{(\\sigma_x,m)} d\\sigma_x - \\frac{1}{\\sigma_x})^{\\sigma_x} = (\\int (\\sigma_x - m) d\\sigma_x - \\frac{1}{\\sigma_x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Add(Integral(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Add(Integral(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Integral(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given T{(P_{g})} = e^{P_{g}}, then derive - P_{g} + \\int T{(P_{g})} dP_{g} = - P_{g} + \\hat{p} + e^{P_{g}}, then obtain - P_{g} + \\int e^{P_{g}} dP_{g} = - P_{g} + \\hat{p} + e^{P_{g}}", "derivation": "T{(P_{g})} = e^{P_{g}} and \\int T{(P_{g})} dP_{g} = \\int e^{P_{g}} dP_{g} and - P_{g} + \\int T{(P_{g})} dP_{g} = - P_{g} + \\int e^{P_{g}} dP_{g} and - P_{g} + \\int T{(P_{g})} dP_{g} = - P_{g} + \\hat{p} + e^{P_{g}} and - P_{g} + \\int e^{P_{g}} dP_{g} = - P_{g} + \\hat{p} + e^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\hat{p}', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('\\\\hat{p}', commutative=True), exp(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(f,P_{e})} = P_{e} - f and \\operatorname{r_{0}}{(P_{e})} = P_{e} and \\mathbf{s}{(f)} = - \\frac{1}{f}, then derive \\frac{\\partial}{\\partial P_{e}} \\mathbf{J}_P{(f,P_{e})} = 1, then obtain \\mathbf{s}{(f)} \\frac{\\partial}{\\partial \\operatorname{r_{0}}{(P_{e})}} (\\operatorname{r_{0}}{(P_{e})} + \\frac{1}{\\mathbf{s}{(f)}}) = \\mathbf{s}{(f)}", "derivation": "\\mathbf{J}_P{(f,P_{e})} = P_{e} - f and \\frac{\\partial}{\\partial P_{e}} \\mathbf{J}_P{(f,P_{e})} = \\frac{\\partial}{\\partial P_{e}} (P_{e} - f) and \\frac{\\partial}{\\partial P_{e}} \\mathbf{J}_P{(f,P_{e})} = 1 and \\frac{\\partial}{\\partial P_{e}} (P_{e} - f) = 1 and \\operatorname{r_{0}}{(P_{e})} = P_{e} and \\frac{\\partial}{\\partial \\operatorname{r_{0}}{(P_{e})}} (- f + \\operatorname{r_{0}}{(P_{e})}) = 1 and - \\frac{\\frac{\\partial}{\\partial \\operatorname{r_{0}}{(P_{e})}} (- f + \\operatorname{r_{0}}{(P_{e})})}{f} = - \\frac{1}{f} and \\mathbf{s}{(f)} = - \\frac{1}{f} and \\mathbf{s}{(f)} \\frac{\\partial}{\\partial \\operatorname{r_{0}}{(P_{e})}} (\\operatorname{r_{0}}{(P_{e})} + \\frac{1}{\\mathbf{s}{(f)}}) = \\mathbf{s}{(f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('r_0')(Symbol('P_e', commutative=True))), Tuple(Function('r_0')(Symbol('P_e', commutative=True)), Integer(1))), Integer(1))"], [["divide", 6, "Mul(Integer(-1), Symbol('f', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('r_0')(Symbol('P_e', commutative=True))), Tuple(Function('r_0')(Symbol('P_e', commutative=True)), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('f', commutative=True)), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('f', commutative=True)), Derivative(Add(Function('r_0')(Symbol('P_e', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('f', commutative=True)), Integer(-1))), Tuple(Function('r_0')(Symbol('P_e', commutative=True)), Integer(1)))), Function('\\\\mathbf{s}')(Symbol('f', commutative=True)))"]]}, {"prompt": "Given r{(C_{1})} = \\cos{(\\log{(C_{1})})} and z{(C_{1})} = \\log{(C_{1})}, then obtain \\cos{(\\log{(C_{1})})} - \\frac{d}{d C_{1}} \\cos{(\\log{(C_{1})})} = \\cos{(z{(C_{1})})} - \\frac{d}{d C_{1}} \\cos{(\\log{(C_{1})})}", "derivation": "r{(C_{1})} = \\cos{(\\log{(C_{1})})} and z{(C_{1})} = \\log{(C_{1})} and r{(C_{1})} = \\cos{(z{(C_{1})})} and \\cos{(\\log{(C_{1})})} = \\cos{(z{(C_{1})})} and \\cos{(\\log{(C_{1})})} - \\frac{d}{d C_{1}} \\cos{(\\log{(C_{1})})} = \\cos{(z{(C_{1})})} - \\frac{d}{d C_{1}} \\cos{(\\log{(C_{1})})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C_1', commutative=True)), cos(log(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('C_1', commutative=True)), log(Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('r')(Symbol('C_1', commutative=True)), cos(Function('z')(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(cos(log(Symbol('C_1', commutative=True))), cos(Function('z')(Symbol('C_1', commutative=True))))"], [["minus", 4, "Derivative(cos(log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Add(cos(log(Symbol('C_1', commutative=True))), Mul(Integer(-1), Derivative(cos(log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))), Add(cos(Function('z')(Symbol('C_1', commutative=True))), Mul(Integer(-1), Derivative(cos(log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(k)} = \\cos{(k)} and \\phi{(k)} = \\int (\\operatorname{F_{g}}{(k)} \\cos{(k)})^{k} dk, then obtain \\phi{(k)} = \\int (\\cos^{2}{(k)})^{k} dk", "derivation": "\\operatorname{F_{g}}{(k)} = \\cos{(k)} and \\operatorname{F_{g}}{(k)} \\cos{(k)} = \\cos^{2}{(k)} and (\\operatorname{F_{g}}{(k)} \\cos{(k)})^{k} = (\\cos^{2}{(k)})^{k} and \\int (\\operatorname{F_{g}}{(k)} \\cos{(k)})^{k} dk = \\int (\\cos^{2}{(k)})^{k} dk and \\phi{(k)} = \\int (\\operatorname{F_{g}}{(k)} \\cos{(k)})^{k} dk and \\phi{(k)} = \\int (\\cos^{2}{(k)})^{k} dk", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["times", 1, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Pow(cos(Symbol('k', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Mul(Function('F_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Symbol('k', commutative=True)))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Mul(Function('F_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('k', commutative=True)), Integral(Pow(Mul(Function('F_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\phi')(Symbol('k', commutative=True)), Integral(Pow(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given U{(v_{2},m_{s})} = m_{s} - v_{2}, then derive \\frac{\\partial}{\\partial m_{s}} U{(v_{2},m_{s})} = 1, then obtain - (- v_{2} + U{(v_{2},m_{s})}) \\frac{\\partial}{\\partial m_{s}} (- v_{2} + U{(v_{2},m_{s})}) + \\frac{\\partial}{\\partial m_{s}} (m_{s} - v_{2}) = - (- v_{2} + U{(v_{2},m_{s})}) \\frac{\\partial}{\\partial m_{s}} (- v_{2} + U{(v_{2},m_{s})}) + 1", "derivation": "U{(v_{2},m_{s})} = m_{s} - v_{2} and - v_{2} + U{(v_{2},m_{s})} = m_{s} - 2 v_{2} and \\frac{\\partial}{\\partial m_{s}} (- v_{2} + U{(v_{2},m_{s})}) = \\frac{\\partial}{\\partial m_{s}} (m_{s} - 2 v_{2}) and \\frac{\\partial}{\\partial m_{s}} U{(v_{2},m_{s})} = 1 and \\frac{\\partial}{\\partial m_{s}} (m_{s} - v_{2}) = 1 and - (- v_{2} + U{(v_{2},m_{s})}) \\frac{\\partial}{\\partial m_{s}} (- v_{2} + U{(v_{2},m_{s})}) + \\frac{\\partial}{\\partial m_{s}} (m_{s} - v_{2}) = - (- v_{2} + U{(v_{2},m_{s})}) \\frac{\\partial}{\\partial m_{s}} (- v_{2} + U{(v_{2},m_{s})}) + 1", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1))"], [["minus", 5, "Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Derivative(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{f}{(\\chi)} = e^{\\chi}, then derive \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} = e^{\\chi}, then obtain \\chi \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} = \\chi \\frac{d}{d \\chi} e^{\\chi}", "derivation": "\\mathbf{f}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} = \\frac{d}{d \\chi} e^{\\chi} and \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} = e^{\\chi} and e^{\\chi} = \\frac{d}{d \\chi} e^{\\chi} and \\mathbf{f}{(\\chi)} = \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} and \\chi e^{\\chi} = \\chi \\frac{d}{d \\chi} e^{\\chi} and \\chi \\mathbf{f}{(\\chi)} = \\chi \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} and \\chi \\mathbf{f}{(\\chi)} = \\chi \\frac{d}{d \\chi} e^{\\chi} and \\chi \\frac{d}{d \\chi} \\mathbf{f}{(\\chi)} = \\chi \\frac{d}{d \\chi} e^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), exp(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\chi', commutative=True)), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Mul(Symbol('\\\\chi', commutative=True), Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Symbol('\\\\chi', commutative=True), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(a,x)} = \\log{(a)}^{x}, then obtain \\frac{\\partial}{\\partial x} \\varepsilon_{0}{(a,x)} = \\log{(a)}^{x} \\log{(\\log{(a)})}", "derivation": "\\varepsilon_{0}{(a,x)} = \\log{(a)}^{x} and \\varepsilon_{0}{(a,x)} + \\log{(a)} = \\log{(a)} + \\log{(a)}^{x} and \\frac{\\partial}{\\partial x} (\\varepsilon_{0}{(a,x)} + \\log{(a)}) = \\frac{\\partial}{\\partial x} (\\log{(a)} + \\log{(a)}^{x}) and \\frac{\\partial}{\\partial x} \\varepsilon_{0}{(a,x)} = \\log{(a)}^{x} \\log{(\\log{(a)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('x', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('x', commutative=True)))"], [["add", 1, "log(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('x', commutative=True)), log(Symbol('a', commutative=True))), Add(log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('x', commutative=True)), log(Symbol('a', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(log(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('a', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Pow(log(Symbol('a', commutative=True)), Symbol('x', commutative=True)), log(log(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\Psi)} = \\log{(e^{\\Psi})}, then derive e^{\\int \\operatorname{A_{z}}{(\\Psi)} d\\Psi} = e^{\\frac{\\Psi^{2}}{2} + z^{*}}, then obtain e^{\\int \\log{(e^{\\Psi})} d\\Psi} = e^{\\frac{\\Psi^{2}}{2} + z^{*}}", "derivation": "\\operatorname{A_{z}}{(\\Psi)} = \\log{(e^{\\Psi})} and \\int \\operatorname{A_{z}}{(\\Psi)} d\\Psi = \\int \\log{(e^{\\Psi})} d\\Psi and e^{\\int \\operatorname{A_{z}}{(\\Psi)} d\\Psi} = e^{\\int \\log{(e^{\\Psi})} d\\Psi} and e^{\\int \\operatorname{A_{z}}{(\\Psi)} d\\Psi} = e^{\\frac{\\Psi^{2}}{2} + z^{*}} and e^{\\int \\log{(e^{\\Psi})} d\\Psi} = e^{\\frac{\\Psi^{2}}{2} + z^{*}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\Psi', commutative=True)), log(exp(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(log(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('A_z')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), exp(Integral(log(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('A_z')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Integral(log(exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbb{I},r)} = \\mathbb{I} r and \\eta{(\\mathbb{I},r)} = \\mathbb{I} r, then obtain 0 = \\int \\mathbb{I} r d\\mathbb{I} - \\int \\eta{(\\mathbb{I},r)} d\\mathbb{I}", "derivation": "\\mathbf{J}{(\\mathbb{I},r)} = \\mathbb{I} r and \\int \\mathbf{J}{(\\mathbb{I},r)} d\\mathbb{I} = \\int \\mathbb{I} r d\\mathbb{I} and 0 = \\int \\mathbb{I} r d\\mathbb{I} - \\int \\mathbf{J}{(\\mathbb{I},r)} d\\mathbb{I} and \\eta{(\\mathbb{I},r)} = \\mathbb{I} r and \\eta{(\\mathbb{I},r)} = \\mathbf{J}{(\\mathbb{I},r)} and 0 = \\int \\mathbb{I} r d\\mathbb{I} - \\int \\eta{(\\mathbb{I},r)} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\eta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(0), Add(Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\eta')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(f^{*},E_{\\lambda})} = E_{\\lambda} f^{*} and g{(E_{\\lambda})} = E_{\\lambda}, then obtain - E_{\\lambda} + g{(E_{\\lambda})} + \\int E_{\\lambda} f^{*} dE_{\\lambda} = \\int E_{\\lambda} f^{*} dE_{\\lambda}", "derivation": "\\Psi_{nl}{(f^{*},E_{\\lambda})} = E_{\\lambda} f^{*} and g{(E_{\\lambda})} = E_{\\lambda} and - E_{\\lambda} + g{(E_{\\lambda})} = 0 and - E_{\\lambda} + g{(E_{\\lambda})} + \\int \\Psi_{nl}{(f^{*},E_{\\lambda})} dE_{\\lambda} = \\int \\Psi_{nl}{(f^{*},E_{\\lambda})} dE_{\\lambda} and - E_{\\lambda} + g{(E_{\\lambda})} + \\int E_{\\lambda} f^{*} dE_{\\lambda} = \\int E_{\\lambda} f^{*} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["minus", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('g')(Symbol('E_{\\\\lambda}', commutative=True))), Integer(0))"], [["add", 3, "Integral(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('g')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Integral(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('g')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\eta,H)} = H + \\eta and \\hat{\\mathbf{r}}{(\\eta,H)} = \\frac{H + \\eta}{\\int (H + \\eta) d\\eta}, then obtain \\eta \\hat{\\mathbf{r}}{(\\eta,H)} = \\frac{\\eta \\varepsilon{(\\eta,H)}}{\\int (H + \\eta) d\\eta}", "derivation": "\\varepsilon{(\\eta,H)} = H + \\eta and \\int \\varepsilon{(\\eta,H)} d\\eta = \\int (H + \\eta) d\\eta and \\frac{\\varepsilon{(\\eta,H)}}{\\int \\varepsilon{(\\eta,H)} d\\eta} = \\frac{H + \\eta}{\\int \\varepsilon{(\\eta,H)} d\\eta} and \\frac{\\varepsilon{(\\eta,H)}}{\\int (H + \\eta) d\\eta} = \\frac{H + \\eta}{\\int (H + \\eta) d\\eta} and \\hat{\\mathbf{r}}{(\\eta,H)} = \\frac{H + \\eta}{\\int (H + \\eta) d\\eta} and \\hat{\\mathbf{r}}{(\\eta,H)} = \\frac{\\varepsilon{(\\eta,H)}}{\\int (H + \\eta) d\\eta} and \\eta \\hat{\\mathbf{r}}{(\\eta,H)} = \\frac{\\eta \\varepsilon{(\\eta,H)}}{\\int (H + \\eta) d\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Pow(Integral(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Integral(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Mul(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Mul(Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["times", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True))), Mul(Symbol('\\\\eta', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\eta', commutative=True), Symbol('H', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given l{(i)} = e^{\\sin{(i)}}, then derive \\frac{d}{d i} l{(i)} = e^{\\sin{(i)}} \\cos{(i)}, then obtain \\frac{d}{d i} l{(i)} = l{(i)} \\cos{(i)}", "derivation": "l{(i)} = e^{\\sin{(i)}} and \\frac{d}{d i} l{(i)} = \\frac{d}{d i} e^{\\sin{(i)}} and \\frac{d}{d i} l{(i)} = e^{\\sin{(i)}} \\cos{(i)} and \\frac{d}{d i} l{(i)} = l{(i)} \\cos{(i)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('l')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(exp(sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('l')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Function('l')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\chi,\\theta)} = \\chi - \\theta, then derive \\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)} = -1, then obtain \\frac{\\int \\frac{\\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)}}{\\chi} d\\chi}{\\sin{(H)}} = \\frac{\\int - \\frac{1}{\\chi} d\\chi}{\\sin{(H)}}", "derivation": "\\operatorname{F_{H}}{(\\chi,\\theta)} = \\chi - \\theta and \\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\chi - \\theta) and \\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)} = -1 and \\frac{\\partial}{\\partial \\theta} (\\chi - \\theta) = -1 and \\frac{\\frac{\\partial}{\\partial \\theta} (\\chi - \\theta)}{\\chi} = - \\frac{1}{\\chi} and \\frac{\\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)}}{\\chi} = - \\frac{1}{\\chi} and \\int \\frac{\\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)}}{\\chi} d\\chi = \\int - \\frac{1}{\\chi} d\\chi and \\frac{\\int \\frac{\\frac{\\partial}{\\partial \\theta} \\operatorname{F_{H}}{(\\chi,\\theta)}}{\\chi} d\\chi}{\\sin{(H)}} = \\frac{\\int - \\frac{1}{\\chi} d\\chi}{\\sin{(H)}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(-1))"], [["divide", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["integrate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 7, "sin(Symbol('H', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('H', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Function('F_H')(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(sin(Symbol('H', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x,H)} = e^{H - x} and \\rho_{b}{(x,H)} = H - x, then obtain e^{2 \\rho_{b}{(x,H)}} = e^{2 H - 2 x}", "derivation": "\\operatorname{A_{y}}{(x,H)} = e^{H - x} and \\operatorname{A_{y}}{(x,H)} e^{H - x} = e^{2 H - 2 x} and \\operatorname{A_{y}}^{2}{(x,H)} = e^{2 H - 2 x} and \\rho_{b}{(x,H)} = H - x and \\operatorname{A_{y}}{(x,H)} = e^{\\rho_{b}{(x,H)}} and e^{2 \\rho_{b}{(x,H)}} = e^{2 H - 2 x}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["times", 1, "exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], "Equality(Mul(Function('A_y')(Symbol('x', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))), exp(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Pow(Function('A_y')(Symbol('x', commutative=True), Symbol('H', commutative=True)), Integer(2)), exp(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('A_y')(Symbol('x', commutative=True), Symbol('H', commutative=True)), exp(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(exp(Mul(Integer(2), Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('H', commutative=True)))), exp(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{H})} = \\int \\cos{(\\mathbf{H})} d\\mathbf{H}, then derive \\mathbf{H} + \\operatorname{t_{2}}{(\\mathbf{H})} = \\mathbf{H} + \\mathbf{p} + \\sin{(\\mathbf{H})}, then obtain \\mathbf{H} + \\int \\cos{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{H} + \\mathbf{p} + \\sin{(\\mathbf{H})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{H})} = \\int \\cos{(\\mathbf{H})} d\\mathbf{H} and \\mathbf{H} + \\operatorname{t_{2}}{(\\mathbf{H})} = \\mathbf{H} + \\int \\cos{(\\mathbf{H})} d\\mathbf{H} and \\mathbf{H} + \\operatorname{t_{2}}{(\\mathbf{H})} = \\mathbf{H} + \\mathbf{p} + \\sin{(\\mathbf{H})} and \\mathbf{H} + \\int \\cos{(\\mathbf{H})} d\\mathbf{H} = \\mathbf{H} + \\mathbf{p} + \\sin{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{H}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{J}_M)} = \\mathbf{J}_M, then derive \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_f{(\\mathbf{J}_M)} = 1, then obtain - \\frac{\\mathbf{J}_M}{\\ddot{x}{(\\mathbf{J}_M)}} + \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_M = - \\frac{\\mathbf{J}_M}{\\ddot{x}{(\\mathbf{J}_M)}} + 1", "derivation": "\\mathbf{J}_f{(\\mathbf{J}_M)} = \\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_f{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_f{(\\mathbf{J}_M)} = 1 and \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_f{(\\mathbf{J}_M)} - \\frac{\\mathbf{J}_f{(\\mathbf{J}_M)}}{\\ddot{x}{(\\mathbf{J}_M)}} = 1 - \\frac{\\mathbf{J}_f{(\\mathbf{J}_M)}}{\\ddot{x}{(\\mathbf{J}_M)}} and - \\frac{\\mathbf{J}_M}{\\ddot{x}{(\\mathbf{J}_M)}} + \\frac{d}{d \\mathbf{J}_M} \\mathbf{J}_M = - \\frac{\\mathbf{J}_M}{\\ddot{x}{(\\mathbf{J}_M)}} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))), Derivative(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\rho_{b}{(\\theta_1)} = \\cos{(\\sin{(\\theta_1)})} and \\hat{\\mathbf{r}}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain \\hat{\\mathbf{r}}{(\\theta_1)} + \\cos{(\\sin{(\\theta_1)})} = \\sin{(\\theta_1)} + \\cos{(\\sin{(\\theta_1)})}", "derivation": "\\rho_{b}{(\\theta_1)} = \\cos{(\\sin{(\\theta_1)})} and \\hat{\\mathbf{r}}{(\\theta_1)} = \\sin{(\\theta_1)} and \\rho_{b}{(\\theta_1)} = \\cos{(\\hat{\\mathbf{r}}{(\\theta_1)})} and \\hat{\\mathbf{r}}{(\\theta_1)} + \\cos{(\\hat{\\mathbf{r}}{(\\theta_1)})} = \\sin{(\\theta_1)} + \\cos{(\\hat{\\mathbf{r}}{(\\theta_1)})} and \\hat{\\mathbf{r}}{(\\theta_1)} + \\rho_{b}{(\\theta_1)} = \\rho_{b}{(\\theta_1)} + \\sin{(\\theta_1)} and \\hat{\\mathbf{r}}{(\\theta_1)} + \\cos{(\\sin{(\\theta_1)})} = \\sin{(\\theta_1)} + \\cos{(\\sin{(\\theta_1)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True)), cos(sin(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True)), cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True))))"], [["add", 2, "cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)), cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)))), Add(sin(Symbol('\\\\theta_1', commutative=True)), cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True))), Add(Function('\\\\rho_b')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True)), cos(sin(Symbol('\\\\theta_1', commutative=True)))), Add(sin(Symbol('\\\\theta_1', commutative=True)), cos(sin(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(T)} = \\log{(T)}, then obtain \\int \\frac{\\int \\hat{x}_0{(T)} dT}{\\hat{x}_0{(T)}} dT = \\int \\frac{\\int \\log{(T)} dT}{\\hat{x}_0{(T)}} dT", "derivation": "\\hat{x}_0{(T)} = \\log{(T)} and \\int \\hat{x}_0{(T)} dT = \\int \\log{(T)} dT and \\frac{\\int \\hat{x}_0{(T)} dT}{\\log{(T)}} = \\frac{\\int \\log{(T)} dT}{\\log{(T)}} and \\frac{\\int \\hat{x}_0{(T)} dT}{\\hat{x}_0{(T)}} = \\frac{\\int \\log{(T)} dT}{\\hat{x}_0{(T)}} and \\int \\frac{\\int \\hat{x}_0{(T)} dT}{\\hat{x}_0{(T)}} dT = \\int \\frac{\\int \\log{(T)} dT}{\\hat{x}_0{(T)}} dT", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["divide", 2, "log(Symbol('T', commutative=True))"], "Equality(Mul(Pow(log(Symbol('T', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(log(Symbol('T', commutative=True)), Integer(-1)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Integer(-1)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('T', commutative=True)), Integer(-1)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\delta)} = e^{\\delta}, then derive \\int \\mathbf{B}{(\\delta)} d\\delta = \\hat{\\mathbf{r}} + e^{\\delta}, then obtain \\frac{\\partial}{\\partial \\delta} (\\hat{\\mathbf{r}} + e^{\\delta})^{2} = \\frac{\\partial}{\\partial \\delta} (\\hat{\\mathbf{r}} + e^{\\delta}) \\int e^{\\delta} d\\delta", "derivation": "\\mathbf{B}{(\\delta)} = e^{\\delta} and \\int \\mathbf{B}{(\\delta)} d\\delta = \\int e^{\\delta} d\\delta and (\\int \\mathbf{B}{(\\delta)} d\\delta)^{2} = (\\int \\mathbf{B}{(\\delta)} d\\delta) \\int e^{\\delta} d\\delta and \\frac{d}{d \\delta} (\\int \\mathbf{B}{(\\delta)} d\\delta)^{2} = \\frac{d}{d \\delta} (\\int \\mathbf{B}{(\\delta)} d\\delta) \\int e^{\\delta} d\\delta and \\int \\mathbf{B}{(\\delta)} d\\delta = \\hat{\\mathbf{r}} + e^{\\delta} and \\frac{\\partial}{\\partial \\delta} (\\hat{\\mathbf{r}} + e^{\\delta})^{2} = \\frac{\\partial}{\\partial \\delta} (\\hat{\\mathbf{r}} + e^{\\delta}) \\int e^{\\delta} d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 2, "Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(2)), Mul(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(2)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Integer(2)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\delta', commutative=True))), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\pi,E_{x})} = E_{x}^{\\pi}, then obtain \\operatorname{r_{0}}^{2}{(\\pi,E_{x})} = E_{x}^{2 \\pi}", "derivation": "\\operatorname{r_{0}}{(\\pi,E_{x})} = E_{x}^{\\pi} and \\operatorname{r_{0}}^{2}{(\\pi,E_{x})} = E_{x}^{\\pi} \\operatorname{r_{0}}{(\\pi,E_{x})} and 1 = \\frac{E_{x}^{\\pi}}{\\operatorname{r_{0}}{(\\pi,E_{x})}} and E_{x}^{\\pi} \\operatorname{r_{0}}{(\\pi,E_{x})} = E_{x}^{2 \\pi} and \\operatorname{r_{0}}^{2}{(\\pi,E_{x})} = E_{x}^{2 \\pi}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Pow(Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)), Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True))))"], [["divide", 2, "Pow(Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))))"], [["times", 3, "Mul(Pow(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)), Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)), Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True))), Pow(Symbol('E_x', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('r_0')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Pow(Symbol('E_x', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given y{(J,r)} = \\int (J + r) dJ, then obtain \\frac{J^{2}}{2} + J r - 2 J + g - 2 r + y{(J,r)} - 1 = J^{2} + 2 J r - 2 J + 2 g - 2 r - 1", "derivation": "y{(J,r)} = \\int (J + r) dJ and - J - r + y{(J,r)} = - J - r + \\int (J + r) dJ and - J - r + y{(J,r)} - 1 = - J - r + \\int (J + r) dJ - 1 and - 2 J - 2 r + y{(J,r)} + \\int (J + r) dJ - 1 = - 2 J - 2 r + 2 \\int (J + r) dJ - 1 and \\frac{J^{2}}{2} + J r - 2 J + g - 2 r + y{(J,r)} - 1 = J^{2} + 2 J r - 2 J + 2 g - 2 r - 1", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["minus", 1, "Add(Symbol('J', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Function('y')(Symbol('J', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Function('y')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Function('y')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), Integral(Add(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True)))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Function('y')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Add(Pow(Symbol('J', commutative=True), Integer(2)), Mul(Integer(2), Symbol('J', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{X},F_{x})} = \\hat{X}^{F_{x}}, then obtain (F_{x} + \\mathbf{H}{(\\hat{X},F_{x})}) \\mathbf{H}^{\\hat{X}}{(\\hat{X},F_{x})} = (F_{x} + \\mathbf{H}{(\\hat{X},F_{x})}) (\\hat{X}^{F_{x}})^{\\hat{X}}", "derivation": "\\mathbf{H}{(\\hat{X},F_{x})} = \\hat{X}^{F_{x}} and F_{x} + \\mathbf{H}{(\\hat{X},F_{x})} = F_{x} + \\hat{X}^{F_{x}} and \\mathbf{H}^{\\hat{X}}{(\\hat{X},F_{x})} = (\\hat{X}^{F_{x}})^{\\hat{X}} and (F_{x} + \\hat{X}^{F_{x}}) \\mathbf{H}^{\\hat{X}}{(\\hat{X},F_{x})} = (F_{x} + \\hat{X}^{F_{x}}) (\\hat{X}^{F_{x}})^{\\hat{X}} and (F_{x} + \\mathbf{H}{(\\hat{X},F_{x})}) \\mathbf{H}^{\\hat{X}}{(\\hat{X},F_{x})} = (F_{x} + \\mathbf{H}{(\\hat{X},F_{x})}) (\\hat{X}^{F_{x}})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)))"], [["add", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 3, "Add(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Add(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Add(Symbol('F_x', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Symbol('F_x', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Add(Symbol('F_x', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True))), Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given c{(\\delta)} = e^{e^{\\delta}} and \\mathbf{P}{(\\mathbf{A})} = \\log{(e^{\\mathbf{A}})}, then obtain 1 = \\frac{\\log{(e^{\\mathbf{A}})}}{\\mathbf{P}{(\\mathbf{A})}}", "derivation": "c{(\\delta)} = e^{e^{\\delta}} and \\mathbf{P}{(\\mathbf{A})} = \\log{(e^{\\mathbf{A}})} and \\mathbf{P}{(\\mathbf{A})} - c{(\\delta)} + e^{e^{\\delta}} = - c{(\\delta)} + e^{e^{\\delta}} + \\log{(e^{\\mathbf{A}})} and 1 = \\frac{- c{(\\delta)} + e^{e^{\\delta}} + \\log{(e^{\\mathbf{A}})}}{\\mathbf{P}{(\\mathbf{A})} - c{(\\delta)} + e^{e^{\\delta}}} and - \\frac{- c{(\\delta)} + e^{e^{\\delta}}}{c{(\\delta)}} + 1 = - \\frac{- c{(\\delta)} + e^{e^{\\delta}}}{c{(\\delta)}} + \\frac{- c{(\\delta)} + e^{e^{\\delta}} + \\log{(e^{\\mathbf{A}})}}{\\mathbf{P}{(\\mathbf{A})} - c{(\\delta)} + e^{e^{\\delta}}} and 1 = \\frac{\\log{(e^{\\mathbf{A}})}}{\\mathbf{P}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\delta', commutative=True)), exp(exp(Symbol('\\\\delta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), log(exp(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True))), log(exp(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["divide", 3, "Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True))), log(exp(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["minus", 4, "Mul(Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Pow(Function('c')(Symbol('\\\\delta', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Pow(Function('c')(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Pow(Function('c')(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Pow(Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\delta', commutative=True))), exp(exp(Symbol('\\\\delta', commutative=True))), log(exp(Symbol('\\\\mathbf{A}', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), log(exp(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given v{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}}, then obtain 0 = \\frac{- \\cos{(v{(\\dot{\\mathbf{r}})})} + \\cos{(e^{\\dot{\\mathbf{r}}})}}{\\int e^{\\dot{\\mathbf{r}}} d\\dot{\\mathbf{r}}}", "derivation": "v{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\cos{(v{(\\dot{\\mathbf{r}})})} = \\cos{(e^{\\dot{\\mathbf{r}}})} and 0 = - \\cos{(v{(\\dot{\\mathbf{r}})})} + \\cos{(e^{\\dot{\\mathbf{r}}})} and \\int v{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int e^{\\dot{\\mathbf{r}}} d\\dot{\\mathbf{r}} and 0 = \\frac{- \\cos{(v{(\\dot{\\mathbf{r}})})} + \\cos{(e^{\\dot{\\mathbf{r}}})}}{\\int v{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}} and 0 = \\frac{- \\cos{(v{(\\dot{\\mathbf{r}})})} + \\cos{(e^{\\dot{\\mathbf{r}}})}}{\\int e^{\\dot{\\mathbf{r}}} d\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["minus", 2, "cos(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), cos(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["divide", 3, "Integral(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), cos(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Pow(Integral(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), cos(Function('v')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Pow(Integral(exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\Psi_{nl})} = \\sin{(\\Psi_{nl})}, then obtain \\Psi_{nl} \\mathbf{S}^{\\Psi_{nl}}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} = \\Psi_{nl} \\sin{(\\Psi_{nl})} \\sin^{\\Psi_{nl}}{(\\Psi_{nl})}", "derivation": "\\mathbf{S}{(\\Psi_{nl})} = \\sin{(\\Psi_{nl})} and \\mathbf{S}^{\\Psi_{nl}}{(\\Psi_{nl})} = \\sin^{\\Psi_{nl}}{(\\Psi_{nl})} and \\Psi_{nl} \\mathbf{S}^{\\Psi_{nl}}{(\\Psi_{nl})} = \\Psi_{nl} \\sin^{\\Psi_{nl}}{(\\Psi_{nl})} and \\Psi_{nl} \\mathbf{S}^{\\Psi_{nl}}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} = \\Psi_{nl} \\sin{(\\Psi_{nl})} \\sin^{\\Psi_{nl}}{(\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["times", 3, "sin(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given h{(n)} = \\cos{(n)}, then obtain \\frac{n + h{(n)}}{n + \\cos{(n)}} + \\frac{d}{d n} 1 = \\frac{d}{d n} 1 + 1", "derivation": "h{(n)} = \\cos{(n)} and n + h{(n)} = n + \\cos{(n)} and \\frac{n + h{(n)}}{n + \\cos{(n)}} = 1 and \\frac{d}{d n} \\frac{n + h{(n)}}{n + \\cos{(n)}} = \\frac{d}{d n} 1 and \\frac{n + h{(n)}}{n + \\cos{(n)}} + \\frac{d}{d n} \\frac{n + h{(n)}}{n + \\cos{(n)}} = \\frac{d}{d n} \\frac{n + h{(n)}}{n + \\cos{(n)}} + 1 and \\frac{n + h{(n)}}{n + \\cos{(n)}} + \\frac{d}{d n} 1 = \\frac{d}{d n} 1 + 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))))"], [["divide", 2, "Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True)))"], "Equality(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Derivative(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Derivative(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Add(Symbol('n', commutative=True), Function('h')(Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), cos(Symbol('n', commutative=True))), Integer(-1))), Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{J}_f,r)} = - r + \\sin{(\\mathbf{J}_f)}, then obtain 0 = - \\frac{(- r + \\sin{(\\mathbf{J}_f)}) \\frac{\\partial}{\\partial r} \\mu_{0}{(\\mathbf{J}_f,r)}}{\\mu_{0}^{2}{(\\mathbf{J}_f,r)}} - \\frac{1}{\\mu_{0}{(\\mathbf{J}_f,r)}}", "derivation": "\\mu_{0}{(\\mathbf{J}_f,r)} = - r + \\sin{(\\mathbf{J}_f)} and 1 = \\frac{- r + \\sin{(\\mathbf{J}_f)}}{\\mu_{0}{(\\mathbf{J}_f,r)}} and \\frac{d}{d r} 1 = \\frac{\\partial}{\\partial r} \\frac{- r + \\sin{(\\mathbf{J}_f)}}{\\mu_{0}{(\\mathbf{J}_f,r)}} and 0 = - \\frac{(- r + \\sin{(\\mathbf{J}_f)}) \\frac{\\partial}{\\partial r} \\mu_{0}{(\\mathbf{J}_f,r)}}{\\mu_{0}^{2}{(\\mathbf{J}_f,r)}} - \\frac{1}{\\mu_{0}{(\\mathbf{J}_f,r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["divide", 1, "Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('r', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Integer(-2)), Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(F_{N},\\eta)} = F_{N} + \\eta, then obtain \\frac{e^{- F_{N} - \\eta + \\Psi^{\\dagger}{(F_{N},\\eta)}}}{F_{N} + \\eta} = \\frac{1}{F_{N} + \\eta}", "derivation": "\\Psi^{\\dagger}{(F_{N},\\eta)} = F_{N} + \\eta and - F_{N} - \\eta + \\Psi^{\\dagger}{(F_{N},\\eta)} = 0 and e^{- F_{N} - \\eta + \\Psi^{\\dagger}{(F_{N},\\eta)}} = 1 and \\frac{e^{- F_{N} - \\eta + \\Psi^{\\dagger}{(F_{N},\\eta)}}}{F_{N} + \\eta} = \\frac{1}{F_{N} + \\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "Add(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True)))), Integer(1))"], [["divide", 3, "Add(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True))))), Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{P}{(n_{2},t)} = - n_{2} + \\sin{(t)}, then derive \\int \\mathbf{P}{(n_{2},t)} dn_{2} = \\rho_b - \\frac{n_{2}^{2}}{2} + n_{2} \\sin{(t)}, then obtain \\int \\mathbf{P}{(n_{2},t)} dn_{2} = \\rho_b - \\frac{n_{2}^{2}}{2} + n_{2} (n_{2} + \\mathbf{P}{(n_{2},t)})", "derivation": "\\mathbf{P}{(n_{2},t)} = - n_{2} + \\sin{(t)} and n_{2} + \\mathbf{P}{(n_{2},t)} = \\sin{(t)} and \\int \\mathbf{P}{(n_{2},t)} dn_{2} = \\int (- n_{2} + \\sin{(t)}) dn_{2} and \\int \\mathbf{P}{(n_{2},t)} dn_{2} = \\rho_b - \\frac{n_{2}^{2}}{2} + n_{2} \\sin{(t)} and \\int \\mathbf{P}{(n_{2},t)} dn_{2} = \\rho_b - \\frac{n_{2}^{2}}{2} + n_{2} (n_{2} + \\mathbf{P}{(n_{2},t)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('t', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Add(Symbol('n_2', commutative=True), Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True))), sin(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Mul(Symbol('n_2', commutative=True), sin(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Mul(Symbol('n_2', commutative=True), Add(Symbol('n_2', commutative=True), Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True), Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\mu{(i)} = \\log{(i)} and n{(i)} = - \\mu^{i}{(i)}, then obtain 0^{i} - \\mu^{i}{(i)} = (\\mu^{i}{(i)} + n{(i)})^{i} - \\mu^{i}{(i)}", "derivation": "\\mu{(i)} = \\log{(i)} and \\mu^{i}{(i)} = \\log{(i)}^{i} and 0 = - \\mu^{i}{(i)} + \\log{(i)}^{i} and 0^{i} = (- \\mu^{i}{(i)} + \\log{(i)}^{i})^{i} and n{(i)} = - \\mu^{i}{(i)} and 0^{i} = (n{(i)} + \\log{(i)}^{i})^{i} and 0^{i} - \\log{(i)}^{i} = (n{(i)} + \\log{(i)}^{i})^{i} - \\log{(i)}^{i} and 0^{i} - \\mu^{i}{(i)} = (\\mu^{i}{(i)} + n{(i)})^{i} - \\mu^{i}{(i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Integer(0), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integer(0), Symbol('i', commutative=True)), Pow(Add(Function('n')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["minus", 6, "Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)))), Add(Pow(Add(Function('n')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('i', commutative=True)), Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Pow(Integer(0), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True)))), Add(Pow(Add(Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Function('n')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('i', commutative=True)), Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(g,v_{1})} = e^{\\frac{v_{1}}{g}}, then obtain \\frac{\\partial^{2}}{\\partial g\\partial v_{1}} - \\operatorname{P_{e}}{(g,v_{1})} = \\frac{\\partial^{2}}{\\partial g\\partial v_{1}} (- 2 \\operatorname{P_{e}}{(g,v_{1})} + e^{\\frac{v_{1}}{g}})", "derivation": "\\operatorname{P_{e}}{(g,v_{1})} = e^{\\frac{v_{1}}{g}} and 0 = - \\operatorname{P_{e}}{(g,v_{1})} + e^{\\frac{v_{1}}{g}} and - \\operatorname{P_{e}}{(g,v_{1})} = - 2 \\operatorname{P_{e}}{(g,v_{1})} + e^{\\frac{v_{1}}{g}} and \\frac{\\partial}{\\partial v_{1}} - \\operatorname{P_{e}}{(g,v_{1})} = \\frac{\\partial}{\\partial v_{1}} (- 2 \\operatorname{P_{e}}{(g,v_{1})} + e^{\\frac{v_{1}}{g}}) and \\frac{\\partial^{2}}{\\partial g\\partial v_{1}} - \\operatorname{P_{e}}{(g,v_{1})} = \\frac{\\partial^{2}}{\\partial g\\partial v_{1}} (- 2 \\operatorname{P_{e}}{(g,v_{1})} + e^{\\frac{v_{1}}{g}})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["minus", 1, "Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), exp(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), exp(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), exp(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('P_e')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), exp(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(F_{x})} = \\cos{(F_{x})}, then obtain \\frac{d}{d F_{x}} (\\operatorname{C_{2}}^{F_{x}}{(F_{x})} - \\cos{(F_{x})}) = \\frac{d}{d F_{x}} (- \\cos{(F_{x})} + \\cos^{F_{x}}{(F_{x})})", "derivation": "\\operatorname{C_{2}}{(F_{x})} = \\cos{(F_{x})} and \\operatorname{C_{2}}^{F_{x}}{(F_{x})} = \\cos^{F_{x}}{(F_{x})} and \\operatorname{C_{2}}^{F_{x}}{(F_{x})} - \\cos{(F_{x})} = - \\cos{(F_{x})} + \\cos^{F_{x}}{(F_{x})} and \\frac{d}{d F_{x}} (\\operatorname{C_{2}}^{F_{x}}{(F_{x})} - \\cos{(F_{x})}) = \\frac{d}{d F_{x}} (- \\cos{(F_{x})} + \\cos^{F_{x}}{(F_{x})})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(cos(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["minus", 2, "cos(Symbol('F_x', commutative=True))"], "Equality(Add(Pow(Function('C_2')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Pow(cos(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Add(Pow(Function('C_2')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Pow(cos(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain ((e^{\\mathbf{B}})^{\\mathbf{B}})^{\\mathbf{B}} \\int \\log{(\\dot{x}^{\\mathbf{B}}{(\\mathbf{B})})} d\\mathbf{B} = ((e^{\\mathbf{B}})^{\\mathbf{B}})^{\\mathbf{B}} \\int \\log{((e^{\\mathbf{B}})^{\\mathbf{B}})} d\\mathbf{B}", "derivation": "\\dot{x}{(\\mathbf{B})} = e^{\\mathbf{B}} and \\dot{x}^{\\mathbf{B}}{(\\mathbf{B})} = (e^{\\mathbf{B}})^{\\mathbf{B}} and \\log{(\\dot{x}^{\\mathbf{B}}{(\\mathbf{B})})} = \\log{((e^{\\mathbf{B}})^{\\mathbf{B}})} and \\int \\log{(\\dot{x}^{\\mathbf{B}}{(\\mathbf{B})})} d\\mathbf{B} = \\int \\log{((e^{\\mathbf{B}})^{\\mathbf{B}})} d\\mathbf{B} and ((e^{\\mathbf{B}})^{\\mathbf{B}})^{\\mathbf{B}} \\int \\log{(\\dot{x}^{\\mathbf{B}}{(\\mathbf{B})})} d\\mathbf{B} = ((e^{\\mathbf{B}})^{\\mathbf{B}})^{\\mathbf{B}} \\int \\log{((e^{\\mathbf{B}})^{\\mathbf{B}})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), log(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(log(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 4, "Pow(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Pow(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{S},s)} = - \\mathbf{S} + s, then obtain (- 2 \\mathbf{S} + 2 s) \\int 4 \\tilde{g}^{2}{(\\mathbf{S},s)} ds = (- 2 \\mathbf{S} + 2 s) \\int (- 2 \\mathbf{S} + 2 s)^{2} ds", "derivation": "\\tilde{g}{(\\mathbf{S},s)} = - \\mathbf{S} + s and - \\mathbf{S} + s + \\tilde{g}{(\\mathbf{S},s)} = - 2 \\mathbf{S} + 2 s and (- \\mathbf{S} + s + \\tilde{g}{(\\mathbf{S},s)})^{2} = (- 2 \\mathbf{S} + 2 s)^{2} and 4 \\tilde{g}^{2}{(\\mathbf{S},s)} = (- 2 \\mathbf{S} + 2 s)^{2} and \\int 4 \\tilde{g}^{2}{(\\mathbf{S},s)} ds = \\int (- 2 \\mathbf{S} + 2 s)^{2} ds and (- 2 \\mathbf{S} + 2 s) \\int 4 \\tilde{g}^{2}{(\\mathbf{S},s)} ds = (- 2 \\mathbf{S} + 2 s) \\int (- 2 \\mathbf{S} + 2 s)^{2} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('s', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('s', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('s', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(4), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integer(2)))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Tuple(Symbol('s', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integer(2)), Tuple(Symbol('s', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integral(Mul(Integer(4), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Tuple(Symbol('s', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Integer(2)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given V{(n_{2},\\theta_2)} = \\theta_2 n_{2}, then obtain V{(n_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} V{(n_{2},\\theta_2)} = \\theta_2 n_{2} \\frac{\\partial}{\\partial \\theta_2} V{(n_{2},\\theta_2)}", "derivation": "V{(n_{2},\\theta_2)} = \\theta_2 n_{2} and - n_{2} + V{(n_{2},\\theta_2)} = \\theta_2 n_{2} - n_{2} and \\frac{\\partial}{\\partial \\theta_2} (- n_{2} + V{(n_{2},\\theta_2)}) = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 n_{2} - n_{2}) and V{(n_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 n_{2} - n_{2}) = \\theta_2 n_{2} \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 n_{2} - n_{2}) and V{(n_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} (- n_{2} + V{(n_{2},\\theta_2)}) = \\theta_2 n_{2} \\frac{\\partial}{\\partial \\theta_2} (- n_{2} + V{(n_{2},\\theta_2)}) and V{(n_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} V{(n_{2},\\theta_2)} = \\theta_2 n_{2} \\frac{\\partial}{\\partial \\theta_2} V{(n_{2},\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Mul(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True), Derivative(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('n_2', commutative=True), Derivative(Function('V')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(\\mathbf{s},\\ddot{x})} = \\ddot{x} - \\mathbf{s}, then obtain I{(\\mathbf{s},\\ddot{x})} - \\frac{1}{\\ddot{x} - 2 \\mathbf{s}} = \\ddot{x} - \\mathbf{s} - \\frac{1}{\\ddot{x} - 2 \\mathbf{s}}", "derivation": "I{(\\mathbf{s},\\ddot{x})} = \\ddot{x} - \\mathbf{s} and - \\mathbf{s} + I{(\\mathbf{s},\\ddot{x})} = \\ddot{x} - 2 \\mathbf{s} and I{(\\mathbf{s},\\ddot{x})} - \\frac{1}{- \\mathbf{s} + I{(\\mathbf{s},\\ddot{x})}} = \\ddot{x} - \\mathbf{s} - \\frac{1}{- \\mathbf{s} + I{(\\mathbf{s},\\ddot{x})}} and I{(\\mathbf{s},\\ddot{x})} - \\frac{1}{\\ddot{x} - 2 \\mathbf{s}} = \\ddot{x} - \\mathbf{s} - \\frac{1}{\\ddot{x} - 2 \\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))"], "Equality(Add(Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1)))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('I')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} = \\mathbf{H} + x^\\prime, then obtain (\\mathbf{H} + x^\\prime + 3 \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})})^{x^\\prime} = (3 \\mathbf{H} + 3 x^\\prime + \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})})^{x^\\prime}", "derivation": "\\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} = \\mathbf{H} + x^\\prime and \\mathbf{H} + x^\\prime + \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} = 2 \\mathbf{H} + 2 x^\\prime and 2 \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} = 2 \\mathbf{H} + 2 x^\\prime and \\mathbf{H} + x^\\prime + 3 \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} = 3 \\mathbf{H} + 3 x^\\prime + \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})} and (\\mathbf{H} + x^\\prime + 3 \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})})^{x^\\prime} = (3 \\mathbf{H} + 3 x^\\prime + \\hat{\\mathbf{x}}{(x^\\prime,\\mathbf{H})})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True), Mul(Integer(3), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(3), Symbol('x^\\\\prime', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True), Mul(Integer(3), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(3), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(3), Symbol('x^\\\\prime', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\nabla{(b,\\psi^*)} = \\int (\\psi^* + b) d\\psi^*, then derive \\frac{\\nabla{(b,\\psi^*)}}{b} = \\frac{C_{d} + \\frac{(\\psi^*)^{2}}{2} + \\psi^* b}{b}, then obtain \\frac{\\int (\\psi^* + b) d\\psi^*}{b} = \\frac{C_{d} + \\frac{(\\psi^*)^{2}}{2} + \\psi^* b}{b}", "derivation": "\\nabla{(b,\\psi^*)} = \\int (\\psi^* + b) d\\psi^* and \\frac{\\nabla{(b,\\psi^*)}}{b} = \\frac{\\int (\\psi^* + b) d\\psi^*}{b} and \\frac{\\nabla{(b,\\psi^*)}}{b} = \\frac{C_{d} + \\frac{(\\psi^*)^{2}}{2} + \\psi^* b}{b} and \\frac{\\int (\\psi^* + b) d\\psi^*}{b} = \\frac{C_{d} + \\frac{(\\psi^*)^{2}}{2} + \\psi^* b}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('b', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('b', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('b', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\psi^*', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given Q{(V)} = \\cos{(V)} and v{(V)} = Q{(V)} - 1 + \\frac{Q{(V)}}{V}, then obtain \\cos{(V)} - 1 + \\frac{\\cos{(V)}}{V} = Q{(V)} - 1 + \\frac{\\cos{(V)}}{V}", "derivation": "Q{(V)} = \\cos{(V)} and \\frac{Q{(V)}}{V} = \\frac{\\cos{(V)}}{V} and Q{(V)} + \\frac{Q{(V)}}{V} = Q{(V)} + \\frac{\\cos{(V)}}{V} and Q{(V)} - 1 + \\frac{Q{(V)}}{V} = Q{(V)} - 1 + \\frac{\\cos{(V)}}{V} and v{(V)} = Q{(V)} - 1 + \\frac{Q{(V)}}{V} and v{(V)} = \\cos{(V)} - 1 + \\frac{\\cos{(V)}}{V} and v{(V)} = Q{(V)} - 1 + \\frac{\\cos{(V)}}{V} and \\cos{(V)} - 1 + \\frac{\\cos{(V)}}{V} = Q{(V)} - 1 + \\frac{\\cos{(V)}}{V}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('Q')(Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True))))"], [["add", 2, "Function('Q')(Symbol('V', commutative=True))"], "Equality(Add(Function('Q')(Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('Q')(Symbol('V', commutative=True)))), Add(Function('Q')(Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('Q')(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('Q')(Symbol('V', commutative=True)))), Add(Function('Q')(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))))"], ["renaming_premise", "Equality(Function('v')(Symbol('V', commutative=True)), Add(Function('Q')(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('Q')(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('v')(Symbol('V', commutative=True)), Add(cos(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('v')(Symbol('V', commutative=True)), Add(Function('Q')(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(cos(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))), Add(Function('Q')(Symbol('V', commutative=True)), Integer(-1), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given g{(\\Psi_{\\lambda},A,E_{x})} = A^{E_{x}} \\Psi_{\\lambda} and \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})} = A A^{E_{x}} \\Psi_{\\lambda}, then obtain A^{- E_{x}} (- C + \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})}) = A^{- E_{x}} (A g{(\\Psi_{\\lambda},A,E_{x})} - C)", "derivation": "g{(\\Psi_{\\lambda},A,E_{x})} = A^{E_{x}} \\Psi_{\\lambda} and A g{(\\Psi_{\\lambda},A,E_{x})} = A A^{E_{x}} \\Psi_{\\lambda} and \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})} = A A^{E_{x}} \\Psi_{\\lambda} and - C + \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})} = A A^{E_{x}} \\Psi_{\\lambda} - C and A^{- E_{x}} (- C + \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})}) = A^{- E_{x}} (A A^{E_{x}} \\Psi_{\\lambda} - C) and A^{- E_{x}} (- C + \\operatorname{c_{0}}{(A,\\Psi_{\\lambda},E_{x})}) = A^{- E_{x}} (A g{(\\Psi_{\\lambda},A,E_{x})} - C)", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('g')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('A', commutative=True), Symbol('E_x', commutative=True))), Mul(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('A', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["minus", 3, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('c_0')(Symbol('A', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True))))"], [["divide", 4, "Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('c_0')(Symbol('A', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('E_x', commutative=True)))), Mul(Pow(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('c_0')(Symbol('A', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('E_x', commutative=True)))), Mul(Pow(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Add(Mul(Symbol('A', commutative=True), Function('g')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('A', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\Omega)} = \\cos{(e^{\\Omega})}, then obtain \\iint \\frac{\\dot{y}{(\\Omega)}}{\\cos{(e^{\\Omega})}} d\\Omega d\\Omega = \\iint 1 d\\Omega d\\Omega", "derivation": "\\dot{y}{(\\Omega)} = \\cos{(e^{\\Omega})} and \\frac{\\dot{y}{(\\Omega)}}{\\cos{(e^{\\Omega})}} = 1 and \\int \\frac{\\dot{y}{(\\Omega)}}{\\cos{(e^{\\Omega})}} d\\Omega = \\int 1 d\\Omega and \\iint \\frac{\\dot{y}{(\\Omega)}}{\\cos{(e^{\\Omega})}} d\\Omega d\\Omega = \\iint 1 d\\Omega d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "cos(exp(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\Omega', commutative=True)), Pow(cos(exp(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{y}')(Symbol('\\\\Omega', commutative=True)), Pow(cos(exp(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{y}')(Symbol('\\\\Omega', commutative=True)), Pow(cos(exp(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given I{(\\theta_1,\\mathbf{J}_P)} = \\mathbf{J}_P + \\theta_1, then obtain \\mathbf{J}_P + \\theta_1 + 2 I{(\\theta_1,\\mathbf{J}_P)} = 2 \\mathbf{J}_P + 2 \\theta_1 + I{(\\theta_1,\\mathbf{J}_P)}", "derivation": "I{(\\theta_1,\\mathbf{J}_P)} = \\mathbf{J}_P + \\theta_1 and 2 I{(\\theta_1,\\mathbf{J}_P)} = \\mathbf{J}_P + \\theta_1 + I{(\\theta_1,\\mathbf{J}_P)} and 3 I{(\\theta_1,\\mathbf{J}_P)} = \\mathbf{J}_P + \\theta_1 + 2 I{(\\theta_1,\\mathbf{J}_P)} and 3 I{(\\theta_1,\\mathbf{J}_P)} = 2 \\mathbf{J}_P + 2 \\theta_1 + I{(\\theta_1,\\mathbf{J}_P)} and \\mathbf{J}_P + \\theta_1 + 2 I{(\\theta_1,\\mathbf{J}_P)} = 2 \\mathbf{J}_P + 2 \\theta_1 + I{(\\theta_1,\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Integer(2), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_1', commutative=True), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 2, "Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Integer(3), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_1', commutative=True), Mul(Integer(2), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_1', commutative=True), Mul(Integer(2), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given V{(m)} = \\cos{(m)}, then obtain (4 V^{2}{(m)} (\\frac{d}{d m} V{(m)})^{2})^{m} = ((- V{(m)} \\sin{(m)} + \\cos{(m)} \\frac{d}{d m} V{(m)})^{2})^{m}", "derivation": "V{(m)} = \\cos{(m)} and V^{2}{(m)} = V{(m)} \\cos{(m)} and \\frac{d}{d m} V^{2}{(m)} = \\frac{d}{d m} V{(m)} \\cos{(m)} and (\\frac{d}{d m} V^{2}{(m)})^{2} = (\\frac{d}{d m} V{(m)} \\cos{(m)})^{2} and ((\\frac{d}{d m} V^{2}{(m)})^{2})^{m} = ((\\frac{d}{d m} V{(m)} \\cos{(m)})^{2})^{m} and (4 V^{2}{(m)} (\\frac{d}{d m} V{(m)})^{2})^{m} = ((- V{(m)} \\sin{(m)} + \\cos{(m)} \\frac{d}{d m} V{(m)})^{2})^{m}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["times", 1, "Function('V')(Symbol('m', commutative=True))"], "Equality(Pow(Function('V')(Symbol('m', commutative=True)), Integer(2)), Mul(Function('V')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Pow(Function('V')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Function('V')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Pow(Function('V')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Function('V')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Derivative(Pow(Function('V')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Symbol('m', commutative=True)), Pow(Pow(Derivative(Mul(Function('V')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Symbol('m', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Mul(Integer(4), Pow(Function('V')(Symbol('m', commutative=True)), Integer(2)), Pow(Derivative(Function('V')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2))), Symbol('m', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('V')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(cos(Symbol('m', commutative=True)), Derivative(Function('V')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))), Integer(2)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given A{(\\omega)} = \\sin{(\\omega)} and g{(x)} = e^{x}, then obtain g{(x)} - \\sin^{2}{(\\omega)} = e^{x} - \\sin^{2}{(\\omega)}", "derivation": "A{(\\omega)} = \\sin{(\\omega)} and A^{2}{(\\omega)} = A{(\\omega)} \\sin{(\\omega)} and A{(\\omega)} \\sin{(\\omega)} = \\sin^{2}{(\\omega)} and A^{2}{(\\omega)} = \\sin^{2}{(\\omega)} and g{(x)} = e^{x} and - A^{2}{(\\omega)} + g{(x)} = - A^{2}{(\\omega)} + e^{x} and g{(x)} - \\sin^{2}{(\\omega)} = e^{x} - \\sin^{2}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Function('A')(Symbol('\\\\omega', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\omega', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('A')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('A')(Symbol('\\\\omega', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2)))"], ["get_premise", "Equality(Function('g')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["minus", 5, "Pow(Function('A')(Symbol('\\\\omega', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('A')(Symbol('\\\\omega', commutative=True)), Integer(2))), Function('g')(Symbol('x', commutative=True))), Add(Mul(Integer(-1), Pow(Function('A')(Symbol('\\\\omega', commutative=True)), Integer(2))), exp(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('g')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2)))), Add(exp(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\Psi{(F_{x})} = \\sin{(F_{x})} and \\operatorname{v_{y}}{(F_{x})} = (\\frac{d}{d F_{x}} \\int \\Psi{(F_{x})} dF_{x})^{F_{x}}, then obtain \\operatorname{v_{y}}{(F_{x})} = (\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x})^{F_{x}}", "derivation": "\\Psi{(F_{x})} = \\sin{(F_{x})} and \\int \\Psi{(F_{x})} dF_{x} = \\int \\sin{(F_{x})} dF_{x} and \\frac{d}{d F_{x}} \\int \\Psi{(F_{x})} dF_{x} = \\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x} and (\\frac{d}{d F_{x}} \\int \\Psi{(F_{x})} dF_{x})^{F_{x}} = (\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x})^{F_{x}} and \\operatorname{v_{y}}{(F_{x})} = (\\frac{d}{d F_{x}} \\int \\Psi{(F_{x})} dF_{x})^{F_{x}} and \\operatorname{v_{y}}{(F_{x})} = (\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Psi')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\Psi')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('F_x', commutative=True)), Pow(Derivative(Integral(Function('\\\\Psi')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('v_y')(Symbol('F_x', commutative=True)), Pow(Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given T{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}, then derive \\int T{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f + f_{E}, then obtain \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f + f_{E} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f", "derivation": "T{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and \\int T{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\int T{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f + f_{E} and \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f + f_{E} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('f_E', commutative=True)), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(s)} = \\sin{(s)} and W{(s)} = \\sin^{s}{(s)} + \\frac{1}{\\operatorname{y^{\\prime}}{(s)}}, then obtain \\sin^{s}{(s)} + \\frac{1}{\\operatorname{y^{\\prime}}{(s)}} = \\sin^{s}{(s)} + \\frac{1}{\\sin{(s)}}", "derivation": "\\operatorname{y^{\\prime}}{(s)} = \\sin{(s)} and W{(s)} = \\sin^{s}{(s)} + \\frac{1}{\\operatorname{y^{\\prime}}{(s)}} and W{(s)} = \\sin^{s}{(s)} + \\frac{1}{\\sin{(s)}} and \\sin^{s}{(s)} + \\frac{1}{\\operatorname{y^{\\prime}}{(s)}} = \\sin^{s}{(s)} + \\frac{1}{\\sin{(s)}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('W')(Symbol('s', commutative=True)), Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('s', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('W')(Symbol('s', commutative=True)), Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('s', commutative=True)), Integer(-1))), Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\pi{(h,\\mathbf{J}_M)} = e^{\\mathbf{J}_M h}, then obtain (- \\mathbf{J}_M h - \\pi{(h,\\mathbf{J}_M)} e^{- 2 \\mathbf{J}_M h})^{\\mathbf{J}_M} = (- \\mathbf{J}_M h - e^{- \\mathbf{J}_M h})^{\\mathbf{J}_M}", "derivation": "\\pi{(h,\\mathbf{J}_M)} = e^{\\mathbf{J}_M h} and \\pi{(h,\\mathbf{J}_M)} e^{- \\mathbf{J}_M h} = 1 and - \\pi{(h,\\mathbf{J}_M)} e^{- \\mathbf{J}_M h} = -1 and - \\pi{(h,\\mathbf{J}_M)} e^{- 2 \\mathbf{J}_M h} = - e^{- \\mathbf{J}_M h} and - \\mathbf{J}_M h - \\pi{(h,\\mathbf{J}_M)} e^{- 2 \\mathbf{J}_M h} = - \\mathbf{J}_M h - e^{- \\mathbf{J}_M h} and (- \\mathbf{J}_M h - \\pi{(h,\\mathbf{J}_M)} e^{- 2 \\mathbf{J}_M h})^{\\mathbf{J}_M} = (- \\mathbf{J}_M h - e^{- \\mathbf{J}_M h})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))))"], [["divide", 1, "exp(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))), Integer(-1))"], [["divide", 3, "exp(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))))))"], [["power", 5, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('h', commutative=True))))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given n{(n_{2},P_{g},\\mathbf{s})} = P_{g} + \\mathbf{s} - n_{2}, then obtain \\frac{\\partial^{2}}{\\partial P_{g}\\partial n_{2}} \\int n{(n_{2},P_{g},\\mathbf{s})} dP_{g} = \\frac{\\partial^{2}}{\\partial P_{g}\\partial n_{2}} \\int (P_{g} + \\mathbf{s} - n_{2}) dP_{g}", "derivation": "n{(n_{2},P_{g},\\mathbf{s})} = P_{g} + \\mathbf{s} - n_{2} and \\int n{(n_{2},P_{g},\\mathbf{s})} dP_{g} = \\int (P_{g} + \\mathbf{s} - n_{2}) dP_{g} and \\frac{\\partial}{\\partial n_{2}} \\int n{(n_{2},P_{g},\\mathbf{s})} dP_{g} = \\frac{\\partial}{\\partial n_{2}} \\int (P_{g} + \\mathbf{s} - n_{2}) dP_{g} and \\frac{\\partial^{2}}{\\partial P_{g}\\partial n_{2}} \\int n{(n_{2},P_{g},\\mathbf{s})} dP_{g} = \\frac{\\partial^{2}}{\\partial P_{g}\\partial n_{2}} \\int (P_{g} + \\mathbf{s} - n_{2}) dP_{g}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('n_2', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('n')(Symbol('n_2', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(Function('n')(Symbol('n_2', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Integral(Function('n')(Symbol('n_2', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(n)} = \\frac{d}{d n} \\log{(n)}, then derive \\operatorname{A_{z}}{(n)} = \\frac{1}{n}, then obtain \\frac{\\frac{d}{d n} \\operatorname{A_{z}}^{n}{(n)}}{\\frac{d}{d n} \\log{(n)}} = \\frac{\\frac{d}{d n} (\\frac{1}{n})^{n}}{\\frac{d}{d n} \\log{(n)}}", "derivation": "\\operatorname{A_{z}}{(n)} = \\frac{d}{d n} \\log{(n)} and \\operatorname{A_{z}}{(n)} = \\frac{1}{n} and \\operatorname{A_{z}}^{n}{(n)} = (\\frac{1}{n})^{n} and \\frac{d}{d n} \\operatorname{A_{z}}^{n}{(n)} = \\frac{d}{d n} (\\frac{1}{n})^{n} and \\frac{\\frac{d}{d n} \\operatorname{A_{z}}^{n}{(n)}}{\\frac{d}{d n} \\log{(n)}} = \\frac{\\frac{d}{d n} (\\frac{1}{n})^{n}}{\\frac{d}{d n} \\log{(n)}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('n', commutative=True)), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_z')(Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Function('A_z')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Function('A_z')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))), Mul(Derivative(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})} = \\mathbf{p} m_{s} and \\hat{p}{(\\mathbf{p},m_{s})} = - m_{s} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})}, then obtain 2 \\hat{p}^{\\mathbf{p}}{(\\mathbf{p},m_{s})} = (\\mathbf{p} m_{s} - m_{s})^{\\mathbf{p}} + \\hat{p}^{\\mathbf{p}}{(\\mathbf{p},m_{s})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})} = \\mathbf{p} m_{s} and - m_{s} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})} = \\mathbf{p} m_{s} - m_{s} and \\hat{p}{(\\mathbf{p},m_{s})} = - m_{s} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})} and (- m_{s} + \\operatorname{L_{\\varepsilon}}{(\\mathbf{p},m_{s})})^{\\mathbf{p}} = (\\mathbf{p} m_{s} - m_{s})^{\\mathbf{p}} and \\hat{p}^{\\mathbf{p}}{(\\mathbf{p},m_{s})} = (\\mathbf{p} m_{s} - m_{s})^{\\mathbf{p}} and 2 \\hat{p}^{\\mathbf{p}}{(\\mathbf{p},m_{s})} = (\\mathbf{p} m_{s} - m_{s})^{\\mathbf{p}} + \\hat{p}^{\\mathbf{p}}{(\\mathbf{p},m_{s})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)))"], [["minus", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 5, "Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Add(Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} = \\log{(- \\hat{p}_0 + \\rho)}, then derive \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} = - \\frac{1}{- \\hat{p}_0 + \\rho}, then obtain \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} d\\rho = \\int - \\frac{1}{- \\hat{p}_0 + \\rho} d\\rho", "derivation": "\\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} = \\log{(- \\hat{p}_0 + \\rho)} and \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} = \\frac{\\partial}{\\partial \\hat{p}_0} \\log{(- \\hat{p}_0 + \\rho)} and \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} = - \\frac{1}{- \\hat{p}_0 + \\rho} and \\int \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{v_{z}}{(\\hat{p}_0,\\rho)} d\\rho = \\int - \\frac{1}{- \\hat{p}_0 + \\rho} d\\rho", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\rho', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Function('v_z')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given G{(\\hat{p},c,q)} = \\hat{p} c q, then obtain (\\hat{p} c q)^{\\hat{p}} + G^{\\hat{p}}{(\\hat{p},c,q)} - \\int \\hat{p} c q dq - \\int G{(\\hat{p},c,q)} dq = 2 (\\hat{p} c q)^{\\hat{p}} - \\int \\hat{p} c q dq - \\int G{(\\hat{p},c,q)} dq", "derivation": "G{(\\hat{p},c,q)} = \\hat{p} c q and \\int G{(\\hat{p},c,q)} dq = \\int \\hat{p} c q dq and G^{\\hat{p}}{(\\hat{p},c,q)} = (\\hat{p} c q)^{\\hat{p}} and G^{\\hat{p}}{(\\hat{p},c,q)} - \\int \\hat{p} c q dq = (\\hat{p} c q)^{\\hat{p}} - \\int \\hat{p} c q dq and G^{\\hat{p}}{(\\hat{p},c,q)} - \\int G{(\\hat{p},c,q)} dq = (\\hat{p} c q)^{\\hat{p}} - \\int G{(\\hat{p},c,q)} dq and (\\hat{p} c q)^{\\hat{p}} + G^{\\hat{p}}{(\\hat{p},c,q)} - \\int \\hat{p} c q dq - \\int G{(\\hat{p},c,q)} dq = 2 (\\hat{p} c q)^{\\hat{p}} - \\int \\hat{p} c q dq - \\int G{(\\hat{p},c,q)} dq", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 3, "Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Pow(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Add(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Add(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))))"], [["add", 5, "Add(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], "Equality(Add(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Add(Mul(Integer(2), Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\mathbf{P}{(\\varphi^*)} = \\frac{1}{\\log{(\\varphi^*)}}, then obtain \\frac{d}{d \\varphi^*} \\mathbf{P}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\frac{1}{\\mathbf{r}{(\\varphi^*)}}", "derivation": "\\mathbf{r}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\mathbf{P}{(\\varphi^*)} = \\frac{1}{\\log{(\\varphi^*)}} and \\frac{d}{d \\varphi^*} \\mathbf{P}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\frac{1}{\\log{(\\varphi^*)}} and \\frac{d}{d \\varphi^*} \\mathbf{P}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\frac{1}{\\mathbf{r}{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(V)} = \\sin{(V)}, then derive \\frac{V^{2}}{2} + V + v_{y} = \\int (V + \\frac{\\sin{(V)}}{\\Psi{(V)}}) dV, then obtain \\frac{V^{2}}{2} + V + v_{y} = \\frac{V^{2}}{2} + V + \\mathbf{p}", "derivation": "\\Psi{(V)} = \\sin{(V)} and 1 = \\frac{\\sin{(V)}}{\\Psi{(V)}} and V + 1 = V + \\frac{\\sin{(V)}}{\\Psi{(V)}} and \\int (V + 1) dV = \\int (V + \\frac{\\sin{(V)}}{\\Psi{(V)}}) dV and \\frac{V^{2}}{2} + V + v_{y} = \\int (V + \\frac{\\sin{(V)}}{\\Psi{(V)}}) dV and \\frac{V^{2}}{2} + V + v_{y} = \\int (V + 1) dV and \\frac{V^{2}}{2} + V + v_{y} = \\frac{V^{2}}{2} + V + \\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["divide", 1, "Function('\\\\Psi')(Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True))))"], [["add", 2, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Integer(1)), Add(Symbol('V', commutative=True), Mul(Pow(Function('\\\\Psi')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True)))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True))), Integral(Add(Symbol('V', commutative=True), Mul(Pow(Function('\\\\Psi')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integral(Add(Symbol('V', commutative=True), Mul(Pow(Function('\\\\Psi')(Symbol('V', commutative=True)), Integer(-1)), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integral(Add(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\phi,C)} = C^{\\phi} and h{(\\phi,C)} = C^{\\phi}, then obtain (\\int h{(\\phi,C)} d\\phi)^{C} = (\\int C^{\\phi} d\\phi)^{C}", "derivation": "\\operatorname{F_{g}}{(\\phi,C)} = C^{\\phi} and \\int \\operatorname{F_{g}}{(\\phi,C)} d\\phi = \\int C^{\\phi} d\\phi and (\\int \\operatorname{F_{g}}{(\\phi,C)} d\\phi)^{C} = (\\int C^{\\phi} d\\phi)^{C} and h{(\\phi,C)} = C^{\\phi} and \\operatorname{F_{g}}{(\\phi,C)} = h{(\\phi,C)} and (\\int h{(\\phi,C)} d\\phi)^{C} = (\\int C^{\\phi} d\\phi)^{C}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Function('F_g')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('F_g')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Function('h')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Integral(Function('h')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(F_{N})} = \\log{(\\sin{(F_{N})})} and B{(F_{N})} = F_{N}, then obtain \\int 0 dB{(F_{N})} = \\int (-1 + \\frac{\\log{(\\sin{(F_{N})})}}{\\mathbf{E}{(F_{N})}}) dB{(F_{N})}", "derivation": "\\mathbf{E}{(F_{N})} = \\log{(\\sin{(F_{N})})} and 1 = \\frac{\\log{(\\sin{(F_{N})})}}{\\mathbf{E}{(F_{N})}} and 0 = -1 + \\frac{\\log{(\\sin{(F_{N})})}}{\\mathbf{E}{(F_{N})}} and \\int 0 dF_{N} = \\int (-1 + \\frac{\\log{(\\sin{(F_{N})})}}{\\mathbf{E}{(F_{N})}}) dF_{N} and B{(F_{N})} = F_{N} and \\int 0 dB{(F_{N})} = \\int (-1 + \\frac{\\log{(\\sin{(F_{N})})}}{\\mathbf{E}{(F_{N})}}) dB{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True)), log(sin(Symbol('F_N', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True)), Integer(-1)), log(sin(Symbol('F_N', commutative=True)))))"], [["minus", 2, 1], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True)), Integer(-1)), log(sin(Symbol('F_N', commutative=True))))))"], [["integrate", 3, "Symbol('F_N', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True)), Integer(-1)), log(sin(Symbol('F_N', commutative=True))))), Tuple(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Integer(0), Tuple(Function('B')(Symbol('F_N', commutative=True)))), Integral(Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True)), Integer(-1)), log(sin(Symbol('F_N', commutative=True))))), Tuple(Function('B')(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(r_{0},r)} = r_{0} \\sin{(r)} and \\mathbf{s}{(r)} = \\sin{(r)}, then obtain \\frac{\\partial}{\\partial r_{0}} \\mathbf{S}{(r_{0},r)} = \\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(r)}", "derivation": "\\mathbf{S}{(r_{0},r)} = r_{0} \\sin{(r)} and \\mathbf{s}{(r)} = \\sin{(r)} and r_{0} \\mathbf{s}{(r)} = r_{0} \\sin{(r)} and \\mathbf{S}{(r_{0},r)} = r_{0} \\mathbf{s}{(r)} and \\frac{\\partial}{\\partial r_{0}} r_{0} \\mathbf{s}{(r)} = \\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(r)} and \\frac{\\partial}{\\partial r_{0}} \\mathbf{S}{(r_{0},r)} = \\frac{\\partial}{\\partial r_{0}} r_{0} \\sin{(r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('r_0', commutative=True), sin(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["times", 2, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('r', commutative=True))), Mul(Symbol('r_0', commutative=True), sin(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{S}')(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('r', commutative=True))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('r_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('r', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), sin(Symbol('r', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('r_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), sin(Symbol('r', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{1})} = \\sin{(v_{1})}, then obtain 1 - v_{1} = - v_{1} + \\frac{\\mathbf{E}{(v_{1})} + \\sin{(v_{1})}}{2 \\mathbf{E}{(v_{1})}}", "derivation": "\\mathbf{E}{(v_{1})} = \\sin{(v_{1})} and 2 \\mathbf{E}{(v_{1})} = \\mathbf{E}{(v_{1})} + \\sin{(v_{1})} and 1 = \\frac{\\mathbf{E}{(v_{1})} + \\sin{(v_{1})}}{2 \\mathbf{E}{(v_{1})}} and 1 - v_{1} = - v_{1} + \\frac{\\mathbf{E}{(v_{1})} + \\sin{(v_{1})}}{2 \\mathbf{E}{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True))), Add(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), Integer(-1))))"], [["minus", 3, "Symbol('v_1', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Rational(1, 2), Add(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('v_1', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(A_{z},\\sigma_p)} = e^{\\frac{\\sigma_p}{A_{z}}}, then obtain e^{- \\operatorname{F_{g}}{(A_{z},\\sigma_p)} - e^{\\frac{\\sigma_p}{A_{z}}}} = e^{- 2 e^{\\frac{\\sigma_p}{A_{z}}}}", "derivation": "\\operatorname{F_{g}}{(A_{z},\\sigma_p)} = e^{\\frac{\\sigma_p}{A_{z}}} and \\operatorname{F_{g}}{(A_{z},\\sigma_p)} + e^{\\frac{\\sigma_p}{A_{z}}} = 2 e^{\\frac{\\sigma_p}{A_{z}}} and - \\operatorname{F_{g}}{(A_{z},\\sigma_p)} - e^{\\frac{\\sigma_p}{A_{z}}} = - 2 e^{\\frac{\\sigma_p}{A_{z}}} and e^{- \\operatorname{F_{g}}{(A_{z},\\sigma_p)} - e^{\\frac{\\sigma_p}{A_{z}}}} = e^{- 2 e^{\\frac{\\sigma_p}{A_{z}}}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 1, "exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Function('F_g')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True)), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(2), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('F_g')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))))), Mul(Integer(-1), Integer(2), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Mul(Integer(-1), Function('F_g')(Symbol('A_z', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))))), exp(Mul(Integer(-1), Integer(2), exp(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)}, then obtain - \\frac{d}{d \\mathbf{J}_f} \\operatorname{z^{*}}{(\\mathbf{J}_f)} + \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\operatorname{z^{*}}{(\\mathbf{J}_f)} = - \\frac{d}{d \\mathbf{J}_f} \\operatorname{z^{*}}{(\\mathbf{J}_f)} + \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\sin{(\\mathbf{J}_f)}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)} and \\frac{d}{d \\mathbf{J}_f} \\operatorname{z^{*}}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)} and \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\operatorname{z^{*}}{(\\mathbf{J}_f)} = \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\sin{(\\mathbf{J}_f)} and - \\frac{d}{d \\mathbf{J}_f} \\operatorname{z^{*}}{(\\mathbf{J}_f)} + \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\operatorname{z^{*}}{(\\mathbf{J}_f)} = - \\frac{d}{d \\mathbf{J}_f} \\operatorname{z^{*}}{(\\mathbf{J}_f)} + \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\sin{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))))"], [["minus", 3, "Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Derivative(Function('z^*')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\tilde{g}{(c_{0})} = c_{0} \\cos{(c_{0})} and \\operatorname{a^{\\dagger}}{(c_{0})} = \\cos{(c_{0})}, then obtain 1 = \\frac{c_{0} \\operatorname{a^{\\dagger}}{(c_{0})}}{\\tilde{g}{(c_{0})}}", "derivation": "\\tilde{g}{(c_{0})} = c_{0} \\cos{(c_{0})} and 1 = \\frac{c_{0} \\cos{(c_{0})}}{\\tilde{g}{(c_{0})}} and \\operatorname{a^{\\dagger}}{(c_{0})} = \\cos{(c_{0})} and 1 = \\frac{c_{0} \\operatorname{a^{\\dagger}}{(c_{0})}}{\\tilde{g}{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('c_0', commutative=True)), Mul(Symbol('c_0', commutative=True), cos(Symbol('c_0', commutative=True))))"], [["divide", 1, "Function('\\\\tilde{g}')(Symbol('c_0', commutative=True))"], "Equality(Integer(1), Mul(Symbol('c_0', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('c_0', commutative=True)), Integer(-1)), cos(Symbol('c_0', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Symbol('c_0', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('c_0', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(M,A_{x})} = \\log{(\\frac{A_{x}}{M})}, then derive M + \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})} = M - \\frac{1}{M}, then obtain (M + \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})})^{A_{x}} = (M - \\frac{1}{M})^{A_{x}}", "derivation": "\\operatorname{P_{g}}{(M,A_{x})} = \\log{(\\frac{A_{x}}{M})} and \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})} = \\frac{\\partial}{\\partial M} \\log{(\\frac{A_{x}}{M})} and M + \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})} = M + \\frac{\\partial}{\\partial M} \\log{(\\frac{A_{x}}{M})} and M + \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})} = M - \\frac{1}{M} and (M + \\frac{\\partial}{\\partial M} \\operatorname{P_{g}}{(M,A_{x})})^{A_{x}} = (M - \\frac{1}{M})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), log(Mul(Symbol('A_x', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('A_x', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["add", 2, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Derivative(Function('P_g')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(Symbol('M', commutative=True), Derivative(log(Mul(Symbol('A_x', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('M', commutative=True), Derivative(Function('P_g')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Symbol('M', commutative=True), Derivative(Function('P_g')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Symbol('A_x', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(V)} = \\cos{(V)}, then derive \\frac{\\frac{d}{d V} \\operatorname{v_{t}}{(V)}}{V} = - \\frac{\\sin{(V)}}{V}, then obtain \\frac{\\frac{d}{d V} \\cos{(V)}}{V} = - \\frac{\\sin{(V)}}{V}", "derivation": "\\operatorname{v_{t}}{(V)} = \\cos{(V)} and \\frac{d}{d V} \\operatorname{v_{t}}{(V)} = \\frac{d}{d V} \\cos{(V)} and \\frac{\\frac{d}{d V} \\operatorname{v_{t}}{(V)}}{V} = \\frac{\\frac{d}{d V} \\cos{(V)}}{V} and \\frac{\\frac{d}{d V} \\operatorname{v_{t}}{(V)}}{V} = - \\frac{\\sin{(V)}}{V} and \\frac{\\frac{d}{d V} \\cos{(V)}}{V} = - \\frac{\\sin{(V)}}{V}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Derivative(Function('v_t')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Derivative(Function('v_t')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('V', commutative=True))))"]]}, {"prompt": "Given J{(c,\\hat{p}_0)} = \\hat{p}_0 + c and Z{(c,\\hat{p}_0)} = \\hat{p}_0 + c, then derive c \\frac{\\partial}{\\partial \\hat{p}_0} Z{(c,\\hat{p}_0)} = c, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} c \\frac{\\partial}{\\partial \\hat{p}_0} Z{(c,\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} c", "derivation": "J{(c,\\hat{p}_0)} = \\hat{p}_0 + c and c J{(c,\\hat{p}_0)} = c (\\hat{p}_0 + c) and Z{(c,\\hat{p}_0)} = \\hat{p}_0 + c and J{(c,\\hat{p}_0)} = Z{(c,\\hat{p}_0)} and c Z{(c,\\hat{p}_0)} = c (\\hat{p}_0 + c) and \\frac{\\partial}{\\partial \\hat{p}_0} c Z{(c,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} c (\\hat{p}_0 + c) and c \\frac{\\partial}{\\partial \\hat{p}_0} Z{(c,\\hat{p}_0)} = c and \\frac{\\partial}{\\partial \\hat{p}_0} c \\frac{\\partial}{\\partial \\hat{p}_0} Z{(c,\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} c", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('c', commutative=True)))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('J')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('c', commutative=True), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('J')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('c', commutative=True), Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('c', commutative=True), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('c', commutative=True), Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Symbol('c', commutative=True), Derivative(Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Symbol('c', commutative=True))"], [["differentiate", 7, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('c', commutative=True), Derivative(Function('Z')(Symbol('c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Symbol('c', commutative=True), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(L)} = \\log{(L)} and T{(L)} = \\log{(L)}, then obtain \\frac{d}{d L} T{(L)} = \\frac{d}{d L} \\operatorname{f_{\\mathbf{p}}}{(L)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(L)} = \\log{(L)} and T{(L)} = \\log{(L)} and \\frac{d}{d L} T{(L)} = \\frac{d}{d L} \\log{(L)} and \\frac{d}{d L} T{(L)} = \\frac{d}{d L} \\operatorname{f_{\\mathbf{p}}}{(L)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('T')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(\\theta,v_{t})} = \\frac{v_{t}}{\\theta}, then obtain - \\theta \\ddot{x}^{\\theta}{(\\theta,v_{t})} + \\ddot{x}^{\\theta}{(\\theta,v_{t})} = - \\theta \\ddot{x}^{\\theta}{(\\theta,v_{t})} + (\\frac{v_{t}}{\\theta})^{\\theta}", "derivation": "\\ddot{x}{(\\theta,v_{t})} = \\frac{v_{t}}{\\theta} and \\ddot{x}^{\\theta}{(\\theta,v_{t})} = (\\frac{v_{t}}{\\theta})^{\\theta} and \\theta \\ddot{x}^{\\theta}{(\\theta,v_{t})} = \\theta (\\frac{v_{t}}{\\theta})^{\\theta} and - \\theta (\\frac{v_{t}}{\\theta})^{\\theta} + \\ddot{x}^{\\theta}{(\\theta,v_{t})} = - \\theta (\\frac{v_{t}}{\\theta})^{\\theta} + (\\frac{v_{t}}{\\theta})^{\\theta} and - \\theta \\ddot{x}^{\\theta}{(\\theta,v_{t})} + \\ddot{x}^{\\theta}{(\\theta,v_{t})} = - \\theta \\ddot{x}^{\\theta}{(\\theta,v_{t})} + (\\frac{v_{t}}{\\theta})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["times", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('\\\\theta', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('v_t', commutative=True)), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(i)} = \\cos{(\\sin{(i)})}, then derive \\frac{d}{d i} \\mathbf{J}_P{(i)} = - \\sin{(\\sin{(i)})} \\cos{(i)}, then obtain (\\frac{d}{d i} \\mathbf{J}_P{(i)})^{i} - \\frac{d}{d i} \\cos{(\\sin{(i)})} = (- \\sin{(\\sin{(i)})} \\cos{(i)})^{i} - \\frac{d}{d i} \\cos{(\\sin{(i)})}", "derivation": "\\mathbf{J}_P{(i)} = \\cos{(\\sin{(i)})} and \\frac{d}{d i} \\mathbf{J}_P{(i)} = \\frac{d}{d i} \\cos{(\\sin{(i)})} and \\frac{d}{d i} \\mathbf{J}_P{(i)} = - \\sin{(\\sin{(i)})} \\cos{(i)} and (\\frac{d}{d i} \\mathbf{J}_P{(i)})^{i} = (- \\sin{(\\sin{(i)})} \\cos{(i)})^{i} and (\\frac{d}{d i} \\mathbf{J}_P{(i)})^{i} - \\frac{d}{d i} \\cos{(\\sin{(i)})} = (- \\sin{(\\sin{(i)})} \\cos{(i)})^{i} - \\frac{d}{d i} \\cos{(\\sin{(i)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(sin(Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Mul(Integer(-1), sin(sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["minus", 4, "Derivative(cos(sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Add(Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Mul(Integer(-1), Derivative(cos(sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))), Add(Pow(Mul(Integer(-1), sin(sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Mul(Integer(-1), Derivative(cos(sin(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(B)} = e^{B}, then derive (\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B} = (e^{B})^{B}, then obtain \\frac{(\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B}}{B} = \\frac{(\\frac{d}{d B} e^{B})^{B}}{B}", "derivation": "\\operatorname{P_{g}}{(B)} = e^{B} and \\frac{d}{d B} \\operatorname{P_{g}}{(B)} = \\frac{d}{d B} e^{B} and \\operatorname{P_{g}}^{B}{(B)} = (e^{B})^{B} and (\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B} = (\\frac{d}{d B} e^{B})^{B} and (\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B} = (e^{B})^{B} and (\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B} = \\operatorname{P_{g}}^{B}{(B)} and \\frac{(\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B}}{B} = \\frac{(e^{B})^{B}}{B} and (\\frac{d}{d B} e^{B})^{B} = (e^{B})^{B} and \\frac{(\\frac{d}{d B} \\operatorname{P_{g}}{(B)})^{B}}{B} = \\frac{(\\frac{d}{d B} e^{B})^{B}}{B}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(Function('P_g')(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["divide", 5, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(r_{0})} = e^{r_{0}} and i{(r_{0})} = r_{0}, then obtain r_{0} + 2 e^{r_{0}} = 3 r_{0} - 2 i{(r_{0})} + 2 e^{r_{0}}", "derivation": "\\hat{\\mathbf{r}}{(r_{0})} = e^{r_{0}} and i{(r_{0})} = r_{0} and 2 \\hat{\\mathbf{r}}{(r_{0})} + i{(r_{0})} = r_{0} + 2 \\hat{\\mathbf{r}}{(r_{0})} and i{(r_{0})} + 2 e^{r_{0}} = r_{0} + 2 e^{r_{0}} and 2 e^{r_{0}} = r_{0} - i{(r_{0})} + 2 e^{r_{0}} and r_{0} + 2 e^{r_{0}} = 2 r_{0} - i{(r_{0})} + 2 e^{r_{0}} and 2 r_{0} - i{(r_{0})} + 2 e^{r_{0}} = 3 r_{0} - 2 i{(r_{0})} + 2 e^{r_{0}} and r_{0} + 2 e^{r_{0}} = 3 r_{0} - 2 i{(r_{0})} + 2 e^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], [["add", 2, "Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r_0', commutative=True))), Function('i')(Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('i')(Symbol('r_0', commutative=True)), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))), Add(Symbol('r_0', commutative=True), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))))"], [["minus", 4, "Function('i')(Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), exp(Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('i')(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('r_0', commutative=True), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))), Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))), Add(Mul(Integer(3), Symbol('r_0', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Symbol('r_0', commutative=True), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))), Add(Mul(Integer(3), Symbol('r_0', commutative=True)), Mul(Integer(-1), Integer(2), Function('i')(Symbol('r_0', commutative=True))), Mul(Integer(2), exp(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given J{(\\psi^*,u)} = \\psi^* u, then obtain - 2 \\psi^* + 2 \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} = - \\psi^* + \\frac{\\partial}{\\partial u} J{(\\psi^*,u)}", "derivation": "J{(\\psi^*,u)} = \\psi^* u and \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} = \\frac{\\partial}{\\partial u} \\psi^* u and - \\psi^* + \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} = - \\psi^* + \\frac{\\partial}{\\partial u} \\psi^* u and - 2 \\psi^* + 2 \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} = - 2 \\psi^* + \\frac{\\partial}{\\partial u} \\psi^* u + \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} and - 2 \\psi^* + 2 \\frac{\\partial}{\\partial u} J{(\\psi^*,u)} = - \\psi^* + \\frac{\\partial}{\\partial u} J{(\\psi^*,u)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(2), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(2), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(n_{1},q)} = \\cos{(\\frac{q}{n_{1}})}, then obtain 0 = \\frac{n_{1} \\chi{(n_{1},q)}}{s \\cos{(\\frac{q}{n_{1}})}} - \\frac{n_{1}}{s}", "derivation": "\\chi{(n_{1},q)} = \\cos{(\\frac{q}{n_{1}})} and - \\chi{(n_{1},q)} = - \\cos{(\\frac{q}{n_{1}})} and - n_{1} \\chi{(n_{1},q)} = - n_{1} \\cos{(\\frac{q}{n_{1}})} and - \\frac{n_{1} \\chi{(n_{1},q)}}{\\cos{(\\frac{q}{n_{1}})}} = - n_{1} and - \\frac{n_{1} \\chi{(n_{1},q)}}{s \\cos{(\\frac{q}{n_{1}})}} = - \\frac{n_{1}}{s} and - n_{1} - \\frac{n_{1} \\chi{(n_{1},q)}}{s \\cos{(\\frac{q}{n_{1}})}} = - n_{1} - \\frac{n_{1}}{s} and 0 = \\frac{n_{1} \\chi{(n_{1},q)}}{s \\cos{(\\frac{q}{n_{1}})}} - \\frac{n_{1}}{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True)))))"], [["divide", 2, "Pow(Symbol('n_1', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Symbol('n_1', commutative=True), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('n_1', commutative=True), cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True)))))"], [["divide", 3, "cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('n_1', commutative=True), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), Pow(cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('n_1', commutative=True)))"], [["divide", 4, "Symbol('s', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), Pow(cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["minus", 5, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), Pow(cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), Pow(cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(-1))))"], "Equality(Integer(0), Add(Mul(Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('q', commutative=True)), Pow(cos(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('n_1', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{J}_P)} = \\mathbf{J}_P, then derive \\varepsilon_0 + f^{\\prime} + \\frac{\\mathbf{J}^{2}{(\\mathbf{J}_P)}}{2} = f^{\\prime} + \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)}, then obtain \\varepsilon_0 + f^{\\prime} + z + \\frac{\\mathbf{J}^{2}{(\\mathbf{J}_P)}}{2} = f^{\\prime} + z + \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)}", "derivation": "\\mathbf{J}{(\\mathbf{J}_P)} = \\mathbf{J}_P and \\int \\mathbf{J}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\mathbf{J}_P d\\mathbf{J}_P and \\int \\mathbf{J}{(\\mathbf{J}_P)} d\\mathbf{J}{(\\mathbf{J}_P)} = \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)} and f^{\\prime} + \\int \\mathbf{J}{(\\mathbf{J}_P)} d\\mathbf{J}{(\\mathbf{J}_P)} = f^{\\prime} + \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)} and \\varepsilon_0 + f^{\\prime} + \\frac{\\mathbf{J}^{2}{(\\mathbf{J}_P)}}{2} = f^{\\prime} + \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)} and \\varepsilon_0 + f^{\\prime} + z + \\frac{\\mathbf{J}^{2}{(\\mathbf{J}_P)}}{2} = f^{\\prime} + z + \\int \\mathbf{J}_P d\\mathbf{J}{(\\mathbf{J}_P)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)))), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["add", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))))), Add(Symbol('f^{\\\\prime}', commutative=True), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))), Add(Symbol('f^{\\\\prime}', commutative=True), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["minus", 5, "Mul(Integer(-1), Symbol('z', commutative=True))"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('z', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))), Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('z', commutative=True), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{J}_P', commutative=True))))))"]]}, {"prompt": "Given \\theta{(l)} = e^{l}, then derive \\cos{(\\theta{(l)})} \\frac{d}{d l} \\theta{(l)} = e^{l} \\cos{(e^{l})}, then obtain \\cos{(\\theta{(l)})} \\frac{d}{d l} \\theta{(l)} = \\theta{(l)} \\cos{(\\theta{(l)})}", "derivation": "\\theta{(l)} = e^{l} and \\sin{(\\theta{(l)})} = \\sin{(e^{l})} and \\frac{d}{d l} \\sin{(\\theta{(l)})} = \\frac{d}{d l} \\sin{(e^{l})} and \\cos{(\\theta{(l)})} \\frac{d}{d l} \\theta{(l)} = e^{l} \\cos{(e^{l})} and \\cos{(e^{l})} \\frac{d}{d l} e^{l} = e^{l} \\cos{(e^{l})} and \\cos{(\\theta{(l)})} \\frac{d}{d l} \\theta{(l)} = \\theta{(l)} \\cos{(\\theta{(l)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\theta')(Symbol('l', commutative=True))), sin(exp(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(sin(Function('\\\\theta')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('\\\\theta')(Symbol('l', commutative=True))), Derivative(Function('\\\\theta')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(exp(Symbol('l', commutative=True)), cos(exp(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(cos(exp(Symbol('l', commutative=True))), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(exp(Symbol('l', commutative=True)), cos(exp(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(cos(Function('\\\\theta')(Symbol('l', commutative=True))), Derivative(Function('\\\\theta')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Function('\\\\theta')(Symbol('l', commutative=True)), cos(Function('\\\\theta')(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given M{(\\omega)} = \\log{(\\omega)}, then derive \\frac{d}{d \\omega} M{(\\omega)} = \\frac{1}{\\omega}, then obtain \\frac{1}{\\omega \\log{(\\omega)}} = \\frac{\\frac{d}{d \\omega} \\log{(\\omega)}}{\\log{(\\omega)}}", "derivation": "M{(\\omega)} = \\log{(\\omega)} and \\frac{d}{d \\omega} M{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\omega)} and \\frac{d}{d \\omega} M{(\\omega)} = \\frac{1}{\\omega} and \\frac{\\frac{d}{d \\omega} M{(\\omega)}}{\\log{(\\omega)}} = \\frac{\\frac{d}{d \\omega} \\log{(\\omega)}}{\\log{(\\omega)}} and \\frac{1}{\\omega \\log{(\\omega)}} = \\frac{\\frac{d}{d \\omega} \\log{(\\omega)}}{\\log{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], [["divide", 2, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(\\dot{x})} = e^{\\sin{(\\dot{x})}}, then derive \\frac{d}{d \\dot{x}} u{(\\dot{x})} = e^{\\sin{(\\dot{x})}} \\cos{(\\dot{x})}, then obtain (e^{\\sin{(\\dot{x})}} \\cos{(\\dot{x})})^{\\dot{x}} = (\\frac{d}{d \\dot{x}} e^{\\sin{(\\dot{x})}})^{\\dot{x}}", "derivation": "u{(\\dot{x})} = e^{\\sin{(\\dot{x})}} and \\frac{d}{d \\dot{x}} u{(\\dot{x})} = \\frac{d}{d \\dot{x}} e^{\\sin{(\\dot{x})}} and \\frac{d}{d \\dot{x}} u{(\\dot{x})} = e^{\\sin{(\\dot{x})}} \\cos{(\\dot{x})} and (\\frac{d}{d \\dot{x}} u{(\\dot{x})})^{\\dot{x}} = (\\frac{d}{d \\dot{x}} e^{\\sin{(\\dot{x})}})^{\\dot{x}} and (e^{\\sin{(\\dot{x})}} \\cos{(\\dot{x})})^{\\dot{x}} = (\\frac{d}{d \\dot{x}} e^{\\sin{(\\dot{x})}})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Mul(exp(sin(Symbol('\\\\dot{x}', commutative=True))), cos(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Derivative(Function('u')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)), Pow(Derivative(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(exp(sin(Symbol('\\\\dot{x}', commutative=True))), cos(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Derivative(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given r{(\\nabla)} = \\nabla, then obtain \\frac{\\frac{d}{d \\nabla} e^{r{(\\nabla)}}}{\\nabla} = \\frac{\\frac{d}{d \\nabla} e^{\\nabla}}{\\nabla}", "derivation": "r{(\\nabla)} = \\nabla and e^{r{(\\nabla)}} = e^{\\nabla} and \\frac{d}{d \\nabla} e^{r{(\\nabla)}} = \\frac{d}{d \\nabla} e^{\\nabla} and \\frac{\\frac{d}{d \\nabla} e^{r{(\\nabla)}}}{\\nabla} = \\frac{\\frac{d}{d \\nabla} e^{\\nabla}}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["exp", 1], "Equality(exp(Function('r')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(exp(Function('r')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(exp(Function('r')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(L)} = \\sin{(L)} and h{(\\pi,r)} = \\pi r, then obtain \\frac{\\partial}{\\partial L} (\\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)} + h{(\\pi,r)}) = \\frac{\\partial}{\\partial L} (2 \\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)})", "derivation": "\\operatorname{F_{c}}{(L)} = \\sin{(L)} and \\operatorname{F_{c}}^{2}{(L)} = \\operatorname{F_{c}}{(L)} \\sin{(L)} and h{(\\pi,r)} = \\pi r and - \\operatorname{F_{c}}{(L)} \\sin{(L)} + h{(\\pi,r)} = \\pi r - \\operatorname{F_{c}}{(L)} \\sin{(L)} and - \\operatorname{F_{c}}^{2}{(L)} + h{(\\pi,r)} = \\pi r - \\operatorname{F_{c}}^{2}{(L)} and \\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)} + h{(\\pi,r)} = 2 \\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)} and \\frac{\\partial}{\\partial L} (\\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)} + h{(\\pi,r)}) = \\frac{\\partial}{\\partial L} (2 \\pi r - \\operatorname{F_{c}}^{2}{(L)} - \\operatorname{F_{c}}{(L)} \\sin{(L)})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["times", 1, "Function('F_c')(Symbol('L', commutative=True))"], "Equality(Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2)), Mul(Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], ["get_premise", "Equality(Function('h')(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)))"], [["minus", 3, "Mul(Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2))), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2)))))"], [["add", 5, "Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2))), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2))), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2))), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('L', commutative=True)), Integer(2))), Mul(Integer(-1), Function('F_c')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p} = \\varphi, then obtain 0 = \\rho \\varphi - \\rho \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p}", "derivation": "z{(\\mathbf{p})} = e^{\\mathbf{p}} and z{(\\mathbf{p})} - e^{\\mathbf{p}} = 0 and (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} = 0^{\\mathbf{p}} and \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p} = \\int 0^{\\mathbf{p}} d\\mathbf{p} and \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p} = \\varphi and \\rho \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p} = \\rho \\varphi and 0 = \\rho \\varphi - \\rho \\int (z{(\\mathbf{p})} - e^{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\varphi', commutative=True))"], [["times", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Integral(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["minus", 6, "Mul(Symbol('\\\\rho', commutative=True), Integral(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Integral(Pow(Add(Function('z')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(P_{g},f_{E})} = P_{g} + f_{E} and \\operatorname{v_{t}}{(P_{g},f_{E})} = \\frac{\\theta_{1}{(P_{g},f_{E})}}{P_{g} + f_{E}}, then obtain - \\operatorname{v_{t}}{(P_{g},f_{E})} + \\iint \\operatorname{v_{t}}{(P_{g},f_{E})} dP_{g} dP_{g} = - \\operatorname{v_{t}}{(P_{g},f_{E})} + \\iint 1 dP_{g} dP_{g}", "derivation": "\\theta_{1}{(P_{g},f_{E})} = P_{g} + f_{E} and \\operatorname{v_{t}}{(P_{g},f_{E})} = \\frac{\\theta_{1}{(P_{g},f_{E})}}{P_{g} + f_{E}} and \\int \\operatorname{v_{t}}{(P_{g},f_{E})} dP_{g} = \\int \\frac{\\theta_{1}{(P_{g},f_{E})}}{P_{g} + f_{E}} dP_{g} and \\int \\operatorname{v_{t}}{(P_{g},f_{E})} dP_{g} = \\int 1 dP_{g} and \\iint \\operatorname{v_{t}}{(P_{g},f_{E})} dP_{g} dP_{g} = \\iint 1 dP_{g} dP_{g} and - \\operatorname{v_{t}}{(P_{g},f_{E})} + \\iint \\operatorname{v_{t}}{(P_{g},f_{E})} dP_{g} dP_{g} = - \\operatorname{v_{t}}{(P_{g},f_{E})} + \\iint 1 dP_{g} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Add(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Mul(Pow(Add(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_g', commutative=True))))"], [["integrate", 4, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 5, "Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True))), Integral(Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Function('v_t')(Symbol('P_g', commutative=True), Symbol('f_E', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(h)} = \\cos{(e^{h})} and x{(h)} = \\cos{(e^{h})}, then obtain 1 = (\\operatorname{V_{\\mathbf{E}}}{(h)} \\operatorname{V_{\\mathbf{E}}}^{- h}{(h)} \\cos^{h}{(e^{h})})^{- h} x^{h}{(h)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(h)} = \\cos{(e^{h})} and \\operatorname{V_{\\mathbf{E}}}^{h}{(h)} = \\cos^{h}{(e^{h})} and 1 = \\operatorname{V_{\\mathbf{E}}}^{- h}{(h)} \\cos^{h}{(e^{h})} and \\operatorname{V_{\\mathbf{E}}}{(h)} = \\operatorname{V_{\\mathbf{E}}}{(h)} \\operatorname{V_{\\mathbf{E}}}^{- h}{(h)} \\cos^{h}{(e^{h})} and x{(h)} = \\cos{(e^{h})} and 1 = \\operatorname{V_{\\mathbf{E}}}^{- h}{(h)} x^{h}{(h)} and 1 = (\\operatorname{V_{\\mathbf{E}}}{(h)} \\operatorname{V_{\\mathbf{E}}}^{- h}{(h)} \\cos^{h}{(e^{h})})^{- h} x^{h}{(h)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), cos(exp(Symbol('h', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(exp(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["divide", 2, "Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(cos(exp(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"], [["times", 3, "Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True))"], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Mul(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(cos(exp(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('h', commutative=True)), cos(exp(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(1), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(Function('x')(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integer(1), Mul(Pow(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(cos(exp(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(Function('x')(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given b{(A_{x})} = e^{A_{x}}, then derive \\frac{d}{d A_{x}} b{(A_{x})} = e^{A_{x}}, then derive \\frac{d^{2}}{d A_{x}^{2}} b{(A_{x})} = e^{A_{x}}, then obtain b{(A_{x})} + \\frac{d^{3}}{d A_{x}^{3}} b{(A_{x})} = b{(A_{x})} + \\frac{d}{d A_{x}} e^{A_{x}}", "derivation": "b{(A_{x})} = e^{A_{x}} and \\frac{d}{d A_{x}} b{(A_{x})} = \\frac{d}{d A_{x}} e^{A_{x}} and b{(A_{x})} + \\frac{d}{d A_{x}} b{(A_{x})} = b{(A_{x})} + \\frac{d}{d A_{x}} e^{A_{x}} and \\frac{d}{d A_{x}} b{(A_{x})} = e^{A_{x}} and \\frac{d^{2}}{d A_{x}^{2}} b{(A_{x})} = \\frac{d}{d A_{x}} e^{A_{x}} and \\frac{d^{2}}{d A_{x}^{2}} b{(A_{x})} = e^{A_{x}} and \\frac{d}{d A_{x}} b{(A_{x})} = \\frac{d^{3}}{d A_{x}^{3}} b{(A_{x})} and b{(A_{x})} + \\frac{d^{3}}{d A_{x}^{3}} b{(A_{x})} = b{(A_{x})} + \\frac{d}{d A_{x}} e^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(exp(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["add", 2, "Function('b')(Symbol('A_x', commutative=True))"], "Equality(Add(Function('b')(Symbol('A_x', commutative=True)), Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Function('b')(Symbol('A_x', commutative=True)), Derivative(exp(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), exp(Symbol('A_x', commutative=True)))"], [["differentiate", 4, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(2))), Derivative(exp(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(2))), exp(Symbol('A_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(3))))"], [["substitute_LHS_for_RHS", 3, 7], "Equality(Add(Function('b')(Symbol('A_x', commutative=True)), Derivative(Function('b')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(3)))), Add(Function('b')(Symbol('A_x', commutative=True)), Derivative(exp(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\psi^*)} = \\log{(\\psi^*)}, then derive \\frac{d}{d \\psi^*} \\operatorname{f_{E}}{(\\psi^*)} = \\frac{1}{\\psi^*}, then obtain \\frac{\\frac{d}{d \\psi^*} \\operatorname{f_{E}}{(\\psi^*)}}{\\tilde{g}{(\\rho_f)}} = \\frac{1}{\\psi^* \\tilde{g}{(\\rho_f)}}", "derivation": "\\operatorname{f_{E}}{(\\psi^*)} = \\log{(\\psi^*)} and \\frac{d}{d \\psi^*} \\operatorname{f_{E}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and \\frac{d}{d \\psi^*} \\operatorname{f_{E}}{(\\psi^*)} = \\frac{1}{\\psi^*} and \\frac{\\frac{d}{d \\psi^*} \\operatorname{f_{E}}{(\\psi^*)}}{\\tilde{g}{(\\rho_f)}} = \\frac{1}{\\psi^* \\tilde{g}{(\\rho_f)}}", "srepr_derivation": [["get_premise", "Equality(Function('f_E')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))"], [["divide", 3, "Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(Function('f_E')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given f{(\\mathbf{J}_f,\\theta_2)} = \\theta_2 + \\cos{(\\mathbf{J}_f)} and \\dot{\\mathbf{r}}{(\\mathbf{J}_f,\\theta_2)} = \\frac{\\mathbf{J}_f (\\theta_2 + \\cos{(\\mathbf{J}_f)})}{\\cos{(\\mathbf{J}_f)}}, then obtain \\int \\dot{\\mathbf{r}}{(\\mathbf{J}_f,\\theta_2)} d\\theta_2 = \\int \\frac{\\mathbf{J}_f f{(\\mathbf{J}_f,\\theta_2)}}{\\cos{(\\mathbf{J}_f)}} d\\theta_2", "derivation": "f{(\\mathbf{J}_f,\\theta_2)} = \\theta_2 + \\cos{(\\mathbf{J}_f)} and \\mathbf{J}_f f{(\\mathbf{J}_f,\\theta_2)} = \\mathbf{J}_f (\\theta_2 + \\cos{(\\mathbf{J}_f)}) and \\dot{\\mathbf{r}}{(\\mathbf{J}_f,\\theta_2)} = \\frac{\\mathbf{J}_f (\\theta_2 + \\cos{(\\mathbf{J}_f)})}{\\cos{(\\mathbf{J}_f)}} and \\dot{\\mathbf{r}}{(\\mathbf{J}_f,\\theta_2)} = \\frac{\\mathbf{J}_f f{(\\mathbf{J}_f,\\theta_2)}}{\\cos{(\\mathbf{J}_f)}} and \\int \\dot{\\mathbf{r}}{(\\mathbf{J}_f,\\theta_2)} d\\theta_2 = \\int \\frac{\\mathbf{J}_f f{(\\mathbf{J}_f,\\theta_2)}}{\\cos{(\\mathbf{J}_f)}} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('f')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('f')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('f')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\varepsilon_0,x)} = \\cos{(\\varepsilon_0 + x)}, then obtain \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\Psi^{x}{(\\varepsilon_0,x)} = \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\cos^{x}{(\\varepsilon_0 + x)}", "derivation": "\\Psi{(\\varepsilon_0,x)} = \\cos{(\\varepsilon_0 + x)} and \\Psi^{x}{(\\varepsilon_0,x)} = \\cos^{x}{(\\varepsilon_0 + x)} and \\frac{\\partial}{\\partial \\varepsilon_0} \\Psi^{x}{(\\varepsilon_0,x)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\cos^{x}{(\\varepsilon_0 + x)} and \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\Psi^{x}{(\\varepsilon_0,x)} = \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\cos^{x}{(\\varepsilon_0 + x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True)), cos(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(cos(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(cos(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Derivative(Pow(cos(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))))"]]}, {"prompt": "Given n{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and E{(\\lambda)} = \\frac{d}{d \\lambda} - \\log{(\\sin{(\\lambda)})}, then derive E{(\\lambda)} = - \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}}, then obtain \\frac{d}{d \\lambda} - n{(\\lambda)} = - \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}}", "derivation": "n{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and - n{(\\lambda)} = - \\log{(\\sin{(\\lambda)})} and \\frac{d}{d \\lambda} - n{(\\lambda)} = \\frac{d}{d \\lambda} - \\log{(\\sin{(\\lambda)})} and E{(\\lambda)} = \\frac{d}{d \\lambda} - \\log{(\\sin{(\\lambda)})} and E{(\\lambda)} = - \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}} and \\frac{d}{d \\lambda} - \\log{(\\sin{(\\lambda)})} = - \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}} and \\frac{d}{d \\lambda} - n{(\\lambda)} = - \\frac{\\cos{(\\lambda)}}{\\sin{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\lambda', commutative=True)), log(sin(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('n')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), log(sin(Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('n')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(sin(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\lambda', commutative=True)), Derivative(Mul(Integer(-1), log(sin(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('E')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(-1), log(sin(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Derivative(Mul(Integer(-1), Function('n')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)), cos(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given z{(F_{g},f)} = F_{g} + f, then obtain \\frac{\\frac{\\partial}{\\partial F_{g}} (- \\mathbf{F}{(F_{g})} + z{(F_{g},f)})}{\\operatorname{v_{x}}{(F_{g},f)}} = \\frac{\\frac{\\partial}{\\partial F_{g}} (F_{g} + f - \\mathbf{F}{(F_{g})})}{\\operatorname{v_{x}}{(F_{g},f)}}", "derivation": "z{(F_{g},f)} = F_{g} + f and - \\mathbf{F}{(F_{g})} + z{(F_{g},f)} = F_{g} + f - \\mathbf{F}{(F_{g})} and \\frac{\\partial}{\\partial F_{g}} (- \\mathbf{F}{(F_{g})} + z{(F_{g},f)}) = \\frac{\\partial}{\\partial F_{g}} (F_{g} + f - \\mathbf{F}{(F_{g})}) and \\frac{\\frac{\\partial}{\\partial F_{g}} (- \\mathbf{F}{(F_{g})} + z{(F_{g},f)})}{\\operatorname{v_{x}}{(F_{g},f)}} = \\frac{\\frac{\\partial}{\\partial F_{g}} (F_{g} + f - \\mathbf{F}{(F_{g})})}{\\operatorname{v_{x}}{(F_{g},f)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('z')(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('f', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True))), Function('z')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Add(Symbol('F_g', commutative=True), Symbol('f', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True)))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True))), Function('z')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Symbol('f', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["divide", 3, "Function('v_x')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Function('v_x')(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True))), Function('z')(Symbol('F_g', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Function('v_x')(Symbol('F_g', commutative=True), Symbol('f', commutative=True)), Integer(-1)), Derivative(Add(Symbol('F_g', commutative=True), Symbol('f', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(C_{2},F_{x})} = \\frac{C_{2}}{F_{x}}, then obtain \\frac{(\\frac{C_{2} \\hat{\\mathbf{r}}{(C_{2},F_{x})}}{F_{x}})^{2 F_{x}}}{F_{x}} = \\frac{(\\frac{C_{2}^{2}}{F_{x}^{2}})^{2 F_{x}}}{F_{x}}", "derivation": "\\hat{\\mathbf{r}}{(C_{2},F_{x})} = \\frac{C_{2}}{F_{x}} and \\frac{C_{2} \\hat{\\mathbf{r}}{(C_{2},F_{x})}}{F_{x}} = \\frac{C_{2}^{2}}{F_{x}^{2}} and (\\frac{C_{2} \\hat{\\mathbf{r}}{(C_{2},F_{x})}}{F_{x}})^{F_{x}} = (\\frac{C_{2}^{2}}{F_{x}^{2}})^{F_{x}} and (\\frac{C_{2} \\hat{\\mathbf{r}}{(C_{2},F_{x})}}{F_{x}})^{2 F_{x}} = (\\frac{C_{2}^{2}}{F_{x}^{2}})^{2 F_{x}} and \\frac{(\\frac{C_{2} \\hat{\\mathbf{r}}{(C_{2},F_{x})}}{F_{x}})^{2 F_{x}}}{F_{x}} = \\frac{(\\frac{C_{2}^{2}}{F_{x}^{2}})^{2 F_{x}}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1))))"], [["times", 1, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('F_x', commutative=True), Integer(-2))))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('F_x', commutative=True), Integer(-2))), Symbol('F_x', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('F_x', commutative=True))), Pow(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('F_x', commutative=True), Integer(-2))), Mul(Integer(2), Symbol('F_x', commutative=True))))"], [["divide", 4, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('F_x', commutative=True)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Pow(Symbol('F_x', commutative=True), Integer(-2))), Mul(Integer(2), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given G{(v_{z})} = \\cos{(v_{z})} and \\operatorname{v_{2}}{(v_{z})} = \\frac{\\int \\cos{(v_{z})} dv_{z}}{\\int G{(v_{z})} dv_{z}}, then obtain 1 = (\\operatorname{v_{2}}^{v_{z}}{(v_{z})})^{v_{z}}", "derivation": "G{(v_{z})} = \\cos{(v_{z})} and \\int G{(v_{z})} dv_{z} = \\int \\cos{(v_{z})} dv_{z} and (\\int G{(v_{z})} dv_{z})^{2} = (\\int G{(v_{z})} dv_{z}) \\int \\cos{(v_{z})} dv_{z} and 1 = \\frac{\\int \\cos{(v_{z})} dv_{z}}{\\int G{(v_{z})} dv_{z}} and 1 = (\\frac{\\int \\cos{(v_{z})} dv_{z}}{\\int G{(v_{z})} dv_{z}})^{v_{z}} and \\operatorname{v_{2}}{(v_{z})} = \\frac{\\int \\cos{(v_{z})} dv_{z}}{\\int G{(v_{z})} dv_{z}} and 1 = \\operatorname{v_{2}}^{v_{z}}{(v_{z})} and 1 = (\\operatorname{v_{2}}^{v_{z}}{(v_{z})})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["times", 2, "Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Pow(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(2)), Mul(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["divide", 3, "Pow(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(2))"], "Equality(Integer(1), Mul(Pow(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["power", 4, "Symbol('v_z', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('v_z', commutative=True)), Mul(Pow(Integral(Function('G')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integer(1), Pow(Function('v_2')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["power", 7, "Symbol('v_z', commutative=True)"], "Equality(Integer(1), Pow(Pow(Function('v_2')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{t},E_{x})} = E_{x} v_{t}, then obtain - E_{x} v_{t} + \\frac{- E_{x} v_{t} + \\operatorname{t_{2}}{(v_{t},E_{x})}}{E_{x} v_{t}} = - E_{x} v_{t}", "derivation": "\\operatorname{t_{2}}{(v_{t},E_{x})} = E_{x} v_{t} and - E_{x} v_{t} + \\operatorname{t_{2}}{(v_{t},E_{x})} = 0 and \\frac{- E_{x} v_{t} + \\operatorname{t_{2}}{(v_{t},E_{x})}}{E_{x} v_{t}} = 0 and - E_{x} v_{t} + \\frac{- E_{x} v_{t} + \\operatorname{t_{2}}{(v_{t},E_{x})}}{E_{x} v_{t}} = - E_{x} v_{t}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_t', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Mul(Symbol('E_x', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)), Function('t_2')(Symbol('v_t', commutative=True), Symbol('E_x', commutative=True))), Integer(0))"], [["divide", 2, "Mul(Symbol('E_x', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)), Function('t_2')(Symbol('v_t', commutative=True), Symbol('E_x', commutative=True)))), Integer(0))"], [["add", 3, "Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)), Function('t_2')(Symbol('v_t', commutative=True), Symbol('E_x', commutative=True))))), Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(v_{z})} = \\log{(\\cos{(v_{z})})}, then obtain \\int (\\operatorname{E_{n}}{(v_{z})} + \\operatorname{E_{n}}^{v_{z}}{(v_{z})}) dv_{z} - 1 = \\int (\\operatorname{E_{n}}{(v_{z})} + \\log{(\\cos{(v_{z})})}^{v_{z}}) dv_{z} - 1", "derivation": "\\operatorname{E_{n}}{(v_{z})} = \\log{(\\cos{(v_{z})})} and \\operatorname{E_{n}}^{v_{z}}{(v_{z})} = \\log{(\\cos{(v_{z})})}^{v_{z}} and \\operatorname{E_{n}}{(v_{z})} + \\operatorname{E_{n}}^{v_{z}}{(v_{z})} = \\operatorname{E_{n}}{(v_{z})} + \\log{(\\cos{(v_{z})})}^{v_{z}} and \\int (\\operatorname{E_{n}}{(v_{z})} + \\operatorname{E_{n}}^{v_{z}}{(v_{z})}) dv_{z} = \\int (\\operatorname{E_{n}}{(v_{z})} + \\log{(\\cos{(v_{z})})}^{v_{z}}) dv_{z} and \\int (\\operatorname{E_{n}}{(v_{z})} + \\operatorname{E_{n}}^{v_{z}}{(v_{z})}) dv_{z} - 1 = \\int (\\operatorname{E_{n}}{(v_{z})} + \\log{(\\cos{(v_{z})})}^{v_{z}}) dv_{z} - 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('v_z', commutative=True)), log(cos(Symbol('v_z', commutative=True))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(log(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 2, "Function('E_n')(Symbol('v_z', commutative=True))"], "Equality(Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(Function('E_n')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(log(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))))"], [["integrate", 3, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(Function('E_n')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(log(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integral(Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(Function('E_n')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integer(-1)), Add(Integral(Add(Function('E_n')(Symbol('v_z', commutative=True)), Pow(log(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_f{(h)} = e^{h}, then derive - \\mathbf{J}_f{(h)} + \\frac{d}{d h} \\mathbf{J}_f{(h)} = - \\mathbf{J}_f{(h)} + e^{h}, then obtain - e^{h} + \\frac{d}{d h} e^{h} = 0", "derivation": "\\mathbf{J}_f{(h)} = e^{h} and \\frac{d}{d h} \\mathbf{J}_f{(h)} = \\frac{d}{d h} e^{h} and - \\mathbf{J}_f{(h)} + \\frac{d}{d h} \\mathbf{J}_f{(h)} = - \\mathbf{J}_f{(h)} + \\frac{d}{d h} e^{h} and - \\mathbf{J}_f{(h)} + \\frac{d}{d h} \\mathbf{J}_f{(h)} = - \\mathbf{J}_f{(h)} + e^{h} and - e^{h} + \\frac{d}{d h} e^{h} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True))), exp(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('h', commutative=True))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given T{(J,\\nabla,\\hat{p}_0)} = - J + \\hat{p}_0 \\nabla, then obtain \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} + \\frac{\\partial}{\\partial J} T^{\\hat{p}_0}{(J,\\nabla,\\hat{p}_0)} + 1 = 2 \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} + 1", "derivation": "T{(J,\\nabla,\\hat{p}_0)} = - J + \\hat{p}_0 \\nabla and T^{\\hat{p}_0}{(J,\\nabla,\\hat{p}_0)} = (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} and \\frac{\\partial}{\\partial J} T^{\\hat{p}_0}{(J,\\nabla,\\hat{p}_0)} = \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} and \\frac{\\partial}{\\partial J} T^{\\hat{p}_0}{(J,\\nabla,\\hat{p}_0)} + 1 = \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} + 1 and \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} + \\frac{\\partial}{\\partial J} T^{\\hat{p}_0}{(J,\\nabla,\\hat{p}_0)} + 1 = 2 \\frac{\\partial}{\\partial J} (- J + \\hat{p}_0 \\nabla)^{\\hat{p}_0} + 1", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('J', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('T')(Symbol('J', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Function('T')(Symbol('J', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Pow(Function('T')(Symbol('J', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(1)))"], [["add", 4, "Derivative(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Function('T')(Symbol('J', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(2), Derivative(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\Omega{(\\omega)} = e^{\\omega}, then derive 2 e^{\\int \\Omega{(\\omega)} d\\omega} = e^{\\hat{x} + e^{\\omega}} + e^{\\int \\Omega{(\\omega)} d\\omega}, then obtain \\frac{\\partial}{\\partial \\omega} (e^{\\hat{x} + e^{\\omega}} + e^{\\int \\Omega{(\\omega)} d\\omega}) = \\frac{d}{d \\omega} (e^{\\int \\Omega{(\\omega)} d\\omega} + e^{\\int e^{\\omega} d\\omega})", "derivation": "\\Omega{(\\omega)} = e^{\\omega} and \\int \\Omega{(\\omega)} d\\omega = \\int e^{\\omega} d\\omega and e^{\\int \\Omega{(\\omega)} d\\omega} = e^{\\int e^{\\omega} d\\omega} and 2 e^{\\int \\Omega{(\\omega)} d\\omega} = e^{\\int \\Omega{(\\omega)} d\\omega} + e^{\\int e^{\\omega} d\\omega} and \\frac{d}{d \\omega} 2 e^{\\int \\Omega{(\\omega)} d\\omega} = \\frac{d}{d \\omega} (e^{\\int \\Omega{(\\omega)} d\\omega} + e^{\\int e^{\\omega} d\\omega}) and 2 e^{\\int \\Omega{(\\omega)} d\\omega} = e^{\\hat{x} + e^{\\omega}} + e^{\\int \\Omega{(\\omega)} d\\omega} and \\frac{\\partial}{\\partial \\omega} (e^{\\hat{x} + e^{\\omega}} + e^{\\int \\Omega{(\\omega)} d\\omega}) = \\frac{d}{d \\omega} (e^{\\int \\Omega{(\\omega)} d\\omega} + e^{\\int e^{\\omega} d\\omega})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), exp(Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["add", 3, "exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Integer(2), exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), exp(Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Mul(Integer(2), exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), exp(Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(exp(Add(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\omega', commutative=True)))), exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Add(exp(Add(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\omega', commutative=True)))), exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(exp(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), exp(Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(J)} = \\cos{(J)} and k{(J)} = \\int \\cos^{J}{(J)} dJ, then obtain \\frac{d}{d J} k^{J}{(J)} = \\frac{d}{d J} (\\int \\Psi^{\\dagger}^{J}{(J)} dJ)^{J}", "derivation": "\\Psi^{\\dagger}{(J)} = \\cos{(J)} and \\Psi^{\\dagger}^{J}{(J)} = \\cos^{J}{(J)} and \\int \\Psi^{\\dagger}^{J}{(J)} dJ = \\int \\cos^{J}{(J)} dJ and k{(J)} = \\int \\cos^{J}{(J)} dJ and k{(J)} = \\int \\Psi^{\\dagger}^{J}{(J)} dJ and k^{J}{(J)} = (\\int \\Psi^{\\dagger}^{J}{(J)} dJ)^{J} and \\frac{d}{d J} k^{J}{(J)} = \\frac{d}{d J} (\\int \\Psi^{\\dagger}^{J}{(J)} dJ)^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(cos(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(cos(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('J', commutative=True)), Integral(Pow(cos(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('k')(Symbol('J', commutative=True)), Integral(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(Function('k')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Integral(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["differentiate", 6, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Function('k')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(\\mathbf{J}_M)} = \\sin{(e^{\\mathbf{J}_M})}, then obtain A_{z} + \\frac{J{(\\mathbf{J}_M)}}{\\sin{(e^{\\mathbf{J}_M})}} + \\operatorname{Si}{(e^{\\mathbf{J}_M})} = A_{z} + \\operatorname{Si}{(e^{\\mathbf{J}_M})} + 1", "derivation": "J{(\\mathbf{J}_M)} = \\sin{(e^{\\mathbf{J}_M})} and \\frac{J{(\\mathbf{J}_M)}}{\\sin{(e^{\\mathbf{J}_M})}} = 1 and \\frac{J{(\\mathbf{J}_M)}}{\\sin{(e^{\\mathbf{J}_M})}} + \\int \\sin{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M = \\int \\sin{(e^{\\mathbf{J}_M})} d\\mathbf{J}_M + 1 and A_{z} + \\frac{J{(\\mathbf{J}_M)}}{\\sin{(e^{\\mathbf{J}_M})}} + \\operatorname{Si}{(e^{\\mathbf{J}_M})} = A_{z} + \\operatorname{Si}{(e^{\\mathbf{J}_M})} + 1", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 1, "sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Integral(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Integral(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Integral(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1)))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_z', commutative=True), Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Si(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Symbol('A_z', commutative=True), Si(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(1)))"]]}, {"prompt": "Given a{(v_{t})} = v_{t} and \\ddot{x}{(v_{t})} = v_{t}, then obtain - v_{t}^{2} + v_{t} s{(\\mathbf{H})} = - v_{t}^{2} + \\frac{v_{t}^{2} s{(\\mathbf{H})}}{\\ddot{x}{(v_{t})}}", "derivation": "a{(v_{t})} = v_{t} and \\ddot{x}{(v_{t})} = v_{t} and \\ddot{x}{(v_{t})} a{(v_{t})} = v_{t} a{(v_{t})} and a{(v_{t})} s{(\\mathbf{H})} = \\frac{v_{t} a{(v_{t})} s{(\\mathbf{H})}}{\\ddot{x}{(v_{t})}} and - v_{t} a{(v_{t})} + a{(v_{t})} s{(\\mathbf{H})} = - v_{t} a{(v_{t})} + \\frac{v_{t} a{(v_{t})} s{(\\mathbf{H})}}{\\ddot{x}{(v_{t})}} and - v_{t}^{2} + v_{t} s{(\\mathbf{H})} = - v_{t}^{2} + \\frac{v_{t}^{2} s{(\\mathbf{H})}}{\\ddot{x}{(v_{t})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('a')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], [["times", 2, "Function('a')(Symbol('v_t', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Function('a')(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), Function('a')(Symbol('v_t', commutative=True))))"], [["divide", 3, "Mul(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('a')(Symbol('v_t', commutative=True)), Function('s')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('v_t', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Integer(-1)), Function('a')(Symbol('v_t', commutative=True)), Function('s')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 4, "Mul(Symbol('v_t', commutative=True), Function('a')(Symbol('v_t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True), Function('a')(Symbol('v_t', commutative=True))), Mul(Function('a')(Symbol('v_t', commutative=True)), Function('s')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True), Function('a')(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Integer(-1)), Function('a')(Symbol('v_t', commutative=True)), Function('s')(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(2))), Mul(Symbol('v_t', commutative=True), Function('s')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(2))), Mul(Pow(Symbol('v_t', commutative=True), Integer(2)), Pow(Function('\\\\ddot{x}')(Symbol('v_t', commutative=True)), Integer(-1)), Function('s')(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(A_{x},M_{E})} = A_{x} + M_{E}, then derive \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})} = 1, then obtain (\\log{(v_{z})} + \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})})^{M_{E}} = (\\log{(v_{z})} + 1)^{M_{E}}", "derivation": "\\hat{x}_0{(A_{x},M_{E})} = A_{x} + M_{E} and \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (A_{x} + M_{E}) and \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})} = 1 and \\frac{\\partial}{\\partial M_{E}} (A_{x} + M_{E}) = 1 and \\log{(v_{z})} + \\frac{\\partial}{\\partial M_{E}} (A_{x} + M_{E}) = \\log{(v_{z})} + 1 and \\log{(v_{z})} + \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})} = \\log{(v_{z})} + 1 and (\\log{(v_{z})} + \\frac{\\partial}{\\partial M_{E}} \\hat{x}_0{(A_{x},M_{E})})^{M_{E}} = (\\log{(v_{z})} + 1)^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "log(Symbol('v_z', commutative=True))"], "Equality(Add(log(Symbol('v_z', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(log(Symbol('v_z', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(log(Symbol('v_z', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(log(Symbol('v_z', commutative=True)), Integer(1)))"], [["power", 6, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(log(Symbol('v_z', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Symbol('M_E', commutative=True)), Pow(Add(log(Symbol('v_z', commutative=True)), Integer(1)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\mathbf{J}_f{(f_{\\mathbf{v}})} = \\int \\operatorname{E_{n}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}}, then obtain \\mathbf{J}_f^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} = (\\int \\log{(f_{\\mathbf{v}})} df_{\\mathbf{v}})^{f_{\\mathbf{v}}}", "derivation": "\\operatorname{E_{n}}{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\int \\operatorname{E_{n}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int \\log{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and \\mathbf{J}_f{(f_{\\mathbf{v}})} = \\int \\operatorname{E_{n}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and \\mathbf{J}_f{(f_{\\mathbf{v}})} = \\int \\log{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and \\mathbf{J}_f^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} = (\\int \\log{(f_{\\mathbf{v}})} df_{\\mathbf{v}})^{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{J}_f')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Integral(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(m,Q)} = Q + m, then derive \\frac{e^{- \\frac{\\partial}{\\partial m} \\mathbf{J}_f{(m,Q)}}}{Q + m} = \\frac{1}{e (Q + m)}, then obtain \\frac{e^{- \\frac{\\partial}{\\partial m} (Q + m)}}{Q + m} = \\frac{1}{e (Q + m)}", "derivation": "\\mathbf{J}_f{(m,Q)} = Q + m and - \\mathbf{J}_f{(m,Q)} = - Q - m and \\frac{\\partial}{\\partial m} - \\mathbf{J}_f{(m,Q)} = \\frac{\\partial}{\\partial m} (- Q - m) and e^{\\frac{\\partial}{\\partial m} - \\mathbf{J}_f{(m,Q)}} = e^{\\frac{\\partial}{\\partial m} (- Q - m)} and \\frac{e^{\\frac{\\partial}{\\partial m} - \\mathbf{J}_f{(m,Q)}}}{Q + m} = \\frac{e^{\\frac{\\partial}{\\partial m} (- Q - m)}}{Q + m} and \\frac{e^{- \\frac{\\partial}{\\partial m} \\mathbf{J}_f{(m,Q)}}}{Q + m} = \\frac{1}{e (Q + m)} and \\frac{e^{- \\frac{\\partial}{\\partial m} (Q + m)}}{Q + m} = \\frac{1}{e (Q + m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), exp(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["divide", 4, "Add(Symbol('Q', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Derivative(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))), Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))), Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Derivative(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))), Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Integer(-1)), exp(Integer(-1))))"]]}, {"prompt": "Given i{(\\chi)} = \\cos{(\\chi)} and \\phi{(W,f_{E})} = \\frac{\\partial}{\\partial W} (- W + f_{E}), then derive \\phi{(W,f_{E})} = -1, then obtain \\frac{\\phi{(W,f_{E})}}{\\cos{(\\chi)}} = - \\frac{1}{\\cos{(\\chi)}}", "derivation": "i{(\\chi)} = \\cos{(\\chi)} and \\phi{(W,f_{E})} = \\frac{\\partial}{\\partial W} (- W + f_{E}) and \\phi{(W,f_{E})} = -1 and \\frac{\\phi{(W,f_{E})}}{i{(\\chi)}} = - \\frac{1}{i{(\\chi)}} and \\frac{\\phi{(W,f_{E})}}{\\cos{(\\chi)}} = - \\frac{1}{\\cos{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], ["get_premise", "Equality(Function('\\\\phi')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Function('\\\\phi')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))"], [["divide", 3, "Function('i')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('i')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('i')(Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('\\\\phi')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)), Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(cos(Symbol('\\\\chi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{F})} = e^{e^{\\mathbf{F}}}, then obtain \\int \\operatorname{x^{{\\}'}}^{2}{(\\mathbf{F})} e^{\\mathbf{F}} d\\mathbf{F} = \\int \\operatorname{x^{{\\}'}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{e^{\\mathbf{F}}} d\\mathbf{F}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{F})} = e^{e^{\\mathbf{F}}} and \\operatorname{x^{{\\}'}}{(\\mathbf{F})} e^{\\mathbf{F}} = e^{\\mathbf{F}} e^{e^{\\mathbf{F}}} and \\operatorname{x^{{\\}'}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{e^{\\mathbf{F}}} = e^{\\mathbf{F}} e^{2 e^{\\mathbf{F}}} and \\operatorname{x^{{\\}'}}^{2}{(\\mathbf{F})} e^{\\mathbf{F}} = \\operatorname{x^{{\\}'}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{e^{\\mathbf{F}}} and \\int \\operatorname{x^{{\\}'}}^{2}{(\\mathbf{F})} e^{\\mathbf{F}} d\\mathbf{F} = \\int \\operatorname{x^{{\\}'}}{(\\mathbf{F})} e^{\\mathbf{F}} e^{e^{\\mathbf{F}}} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["times", 2, "exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(2), exp(Symbol('\\\\mathbf{F}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Function('x^\\\\prime')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)), exp(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\operatorname{A_{1}}{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}}, then obtain \\operatorname{A_{1}}{(\\mathbf{H})} \\log{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} = \\operatorname{A_{1}}{(\\mathbf{H})} \\log{(\\mathbf{H})} + \\frac{1}{\\mathbf{H}}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{H})} = \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\operatorname{A_{1}}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\operatorname{A_{1}}{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\operatorname{A_{1}}{(\\mathbf{H})} \\log{(\\mathbf{H})} + \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} = \\operatorname{A_{1}}{(\\mathbf{H})} \\log{(\\mathbf{H})} + \\frac{1}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["add", 4, "Mul(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Mul(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given x{(P_{g})} = \\sin{(P_{g})} and \\rho_{f}{(P_{g})} = \\frac{x{(P_{g})}}{\\sin{(P_{g})}} and t{(P_{g})} = \\sin{(P_{g})}, then obtain \\frac{1}{t{(P_{g})}} = \\frac{1}{x{(P_{g})}}", "derivation": "x{(P_{g})} = \\sin{(P_{g})} and \\rho_{f}{(P_{g})} = \\frac{x{(P_{g})}}{\\sin{(P_{g})}} and \\frac{\\rho_{f}{(P_{g})}}{\\sin{(P_{g})}} = \\frac{x{(P_{g})}}{\\sin^{2}{(P_{g})}} and t{(P_{g})} = \\sin{(P_{g})} and \\frac{\\rho_{f}{(P_{g})}}{x{(P_{g})}} = \\frac{1}{x{(P_{g})}} and \\frac{1}{\\sin{(P_{g})}} = \\frac{1}{x{(P_{g})}} and \\frac{1}{t{(P_{g})}} = \\frac{1}{x{(P_{g})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('P_g', commutative=True)), Mul(Function('x')(Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))))"], [["divide", 2, "sin(Symbol('P_g', commutative=True))"], "Equality(Mul(Function('\\\\rho_f')(Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-1))), Mul(Function('x')(Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Integer(-2))))"], ["renaming_premise", "Equality(Function('t')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\rho_f')(Symbol('P_g', commutative=True)), Pow(Function('x')(Symbol('P_g', commutative=True)), Integer(-1))), Pow(Function('x')(Symbol('P_g', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(sin(Symbol('P_g', commutative=True)), Integer(-1)), Pow(Function('x')(Symbol('P_g', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Function('t')(Symbol('P_g', commutative=True)), Integer(-1)), Pow(Function('x')(Symbol('P_g', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given M{(\\hat{p},\\mathbf{D})} = \\cos{(\\mathbf{D}^{\\hat{p}})} and n{(\\hat{p},\\mathbf{D})} = - \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1, then obtain n^{\\hat{p}}{(\\hat{p},\\mathbf{D})} = (- \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1)^{\\hat{p}}", "derivation": "M{(\\hat{p},\\mathbf{D})} = \\cos{(\\mathbf{D}^{\\hat{p}})} and - \\mathbf{D} + M{(\\hat{p},\\mathbf{D})} = - \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} and - \\mathbf{D} + M{(\\hat{p},\\mathbf{D})} + 1 = - \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1 and (- \\mathbf{D} + M{(\\hat{p},\\mathbf{D})} + 1)^{\\hat{p}} = (- \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1)^{\\hat{p}} and n{(\\hat{p},\\mathbf{D})} = - \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1 and n{(\\hat{p},\\mathbf{D})} = - \\mathbf{D} + M{(\\hat{p},\\mathbf{D})} + 1 and n^{\\hat{p}}{(\\hat{p},\\mathbf{D})} = (- \\mathbf{D} + \\cos{(\\mathbf{D}^{\\hat{p}})} + 1)^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('M')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('M')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(1)))"], [["power", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('M')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1)), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(1)), Symbol('\\\\hat{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('M')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Pow(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Integer(1)), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given Q{(B)} = \\log{(B)} and z{(B)} = \\frac{d}{d B} \\log{(B)}, then obtain - Q{(B)} + \\frac{d}{d B} Q{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} = - Q{(B)} + z{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H}", "derivation": "Q{(B)} = \\log{(B)} and \\frac{d}{d B} Q{(B)} = \\frac{d}{d B} \\log{(B)} and \\frac{d}{d B} Q{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} = \\frac{d}{d B} \\log{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} and - Q{(B)} + \\frac{d}{d B} Q{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} = - Q{(B)} + \\frac{d}{d B} \\log{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} and z{(B)} = \\frac{d}{d B} \\log{(B)} and - Q{(B)} + \\frac{d}{d B} Q{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H} = - Q{(B)} + z{(B)} - \\int \\hat{x}{(\\hat{H},\\rho_f,\\chi)} d\\hat{H}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["minus", 2, "Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Derivative(Function('Q')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"], [["minus", 3, "Function('Q')(Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('B', commutative=True))), Derivative(Function('Q')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(Mul(Integer(-1), Function('Q')(Symbol('B', commutative=True))), Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"], ["renaming_premise", "Equality(Function('z')(Symbol('B', commutative=True)), Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('B', commutative=True))), Derivative(Function('Q')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(Mul(Integer(-1), Function('Q')(Symbol('B', commutative=True))), Function('z')(Symbol('B', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\psi^*)} = e^{\\psi^*}, then derive \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)} = e^{\\psi^*}, then obtain (- \\frac{- \\psi^* + \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)}}{\\psi^*})^{\\psi^*} = (- \\frac{- \\psi^* + e^{\\psi^*}}{\\psi^*})^{\\psi^*}", "derivation": "\\operatorname{m_{s}}{(\\psi^*)} = e^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)} = \\frac{d}{d \\psi^*} e^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)} = e^{\\psi^*} and - \\psi^* + \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)} = - \\psi^* + e^{\\psi^*} and - \\frac{- \\psi^* + \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)}}{\\psi^*} = - \\frac{- \\psi^* + e^{\\psi^*}}{\\psi^*} and (- \\frac{- \\psi^* + \\frac{d}{d \\psi^*} \\operatorname{m_{s}}{(\\psi^*)}}{\\psi^*})^{\\psi^*} = (- \\frac{- \\psi^* + e^{\\psi^*}}{\\psi^*})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), exp(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))))"], [["power", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\Omega{(h,r)} = h^{r} and \\operatorname{v_{1}}{(h,r)} = h^{r}, then derive h \\frac{\\partial}{\\partial r} \\Omega{(h,r)} = h h^{r} \\log{(h)}, then obtain h \\frac{\\partial}{\\partial r} \\Omega{(h,r)} = h \\operatorname{v_{1}}{(h,r)} \\log{(h)}", "derivation": "\\Omega{(h,r)} = h^{r} and h \\Omega{(h,r)} = h h^{r} and \\frac{\\partial}{\\partial r} h \\Omega{(h,r)} = \\frac{\\partial}{\\partial r} h h^{r} and \\operatorname{v_{1}}{(h,r)} = h^{r} and h \\frac{\\partial}{\\partial r} \\Omega{(h,r)} = h h^{r} \\log{(h)} and h \\frac{\\partial}{\\partial r} \\Omega{(h,r)} = h \\operatorname{v_{1}}{(h,r)} \\log{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('r', commutative=True)))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Symbol('h', commutative=True), Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('h', commutative=True), Derivative(Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Pow(Symbol('h', commutative=True), Symbol('r', commutative=True)), log(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('h', commutative=True), Derivative(Function('\\\\Omega')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Function('v_1')(Symbol('h', commutative=True), Symbol('r', commutative=True)), log(Symbol('h', commutative=True))))"]]}, {"prompt": "Given H{(x,F_{N})} = \\cos{(F_{N} + x)}, then obtain \\frac{\\partial}{\\partial x} (F_{N} + x + 2 H{(x,F_{N})} - \\cos{(F_{N} + x)} - \\cos{(F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)})}) = \\frac{\\partial}{\\partial x} (F_{N} + x)", "derivation": "H{(x,F_{N})} = \\cos{(F_{N} + x)} and H{(x,F_{N})} - \\cos{(F_{N} + x)} = 0 and F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)} = F_{N} + x and \\frac{\\partial}{\\partial x} (F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)}) = \\frac{\\partial}{\\partial x} (F_{N} + x) and \\frac{\\partial}{\\partial x} (F_{N} + x + 2 H{(x,F_{N})} - \\cos{(F_{N} + x)} - \\cos{(F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)})}) = \\frac{\\partial}{\\partial x} (F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)}) and \\frac{\\partial}{\\partial x} (F_{N} + x + 2 H{(x,F_{N})} - \\cos{(F_{N} + x)} - \\cos{(F_{N} + x + H{(x,F_{N})} - \\cos{(F_{N} + x)})}) = \\frac{\\partial}{\\partial x} (F_{N} + x)", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))))"], [["minus", 1, "cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))"], "Equality(Add(Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))))), Integer(0))"], [["add", 2, "Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))))), Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Mul(Integer(2), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))))))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True))))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Mul(Integer(2), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True), Function('H')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))))))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta}, then derive \\hat{H}{(\\eta)} = e^{\\eta}, then obtain \\hat{H}{(\\eta)} - 1 = \\frac{d}{d \\eta} \\hat{H}{(\\eta)} - 1", "derivation": "\\hat{H}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\hat{H}{(\\eta)} = e^{\\eta} and \\hat{H}{(\\eta)} = \\frac{d}{d \\eta} \\hat{H}{(\\eta)} and \\hat{H}{(\\eta)} - 1 = \\frac{d}{d \\eta} \\hat{H}{(\\eta)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), Integer(-1)), Add(Derivative(Function('\\\\hat{H}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\omega)} = e^{\\omega}, then derive \\operatorname{A_{z}}{(\\omega)} e^{\\omega} = e^{2 \\omega}, then derive \\operatorname{A_{z}}{(\\omega)} e^{\\omega} - e^{2 \\omega} = 0, then obtain (\\operatorname{A_{z}}{(\\omega)} e^{\\omega} - e^{2 \\omega})^{\\omega} = 0^{\\omega}", "derivation": "\\operatorname{A_{z}}{(\\omega)} = e^{\\omega} and \\operatorname{A_{z}}{(\\omega)} \\frac{d}{d \\omega} e^{\\omega} = e^{\\omega} \\frac{d}{d \\omega} e^{\\omega} and \\operatorname{A_{z}}{(\\omega)} e^{\\omega} = e^{2 \\omega} and \\operatorname{A_{z}}{(\\omega)} e^{\\omega} - \\operatorname{A_{z}}{(\\omega)} \\frac{d}{d \\omega} e^{\\omega} = - \\operatorname{A_{z}}{(\\omega)} \\frac{d}{d \\omega} e^{\\omega} + e^{2 \\omega} and \\operatorname{A_{z}}{(\\omega)} e^{\\omega} - e^{\\omega} \\frac{d}{d \\omega} e^{\\omega} = e^{2 \\omega} - e^{\\omega} \\frac{d}{d \\omega} e^{\\omega} and \\operatorname{A_{z}}{(\\omega)} e^{\\omega} - e^{2 \\omega} = 0 and (\\operatorname{A_{z}}{(\\omega)} e^{\\omega} - e^{2 \\omega})^{\\omega} = 0^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], "Equality(Add(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Add(exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\omega', commutative=True)), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True))))), Integer(0))"], [["power", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Mul(Function('A_z')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True))))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given A{(\\sigma_p)} = e^{\\sigma_p}, then obtain 0^{\\sigma_p} = (- \\frac{\\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} - A{(\\sigma_p)} + e^{\\sigma_p}}{- \\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} + A{(\\sigma_p)}})^{\\sigma_p}", "derivation": "A{(\\sigma_p)} = e^{\\sigma_p} and 0 = - A{(\\sigma_p)} + e^{\\sigma_p} and 0 = - \\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} and A{(\\sigma_p)} = - \\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} + A{(\\sigma_p)} and 0 = - \\frac{\\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} - A{(\\sigma_p)} + e^{\\sigma_p}}{- \\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} + A{(\\sigma_p)}} and 0^{\\sigma_p} = (- \\frac{\\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} - A{(\\sigma_p)} + e^{\\sigma_p}}{- \\frac{- A{(\\sigma_p)} + e^{\\sigma_p}}{A{(\\sigma_p)}} + A{(\\sigma_p)}})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Function('A')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"], [["minus", 3, "Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Function('A')(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Function('A')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Add(Mul(Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Function('A')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Add(Mul(Add(Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))), Pow(Function('A')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('A')(Symbol('\\\\sigma_p', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given n{(\\varphi,v_{t})} = - \\varphi + \\cos{(v_{t})}, then obtain 3 \\varphi + n{(\\varphi,v_{t})} + 1 = 2 \\varphi + \\cos{(v_{t})} + 1", "derivation": "n{(\\varphi,v_{t})} = - \\varphi + \\cos{(v_{t})} and - \\varphi + n{(\\varphi,v_{t})} = - 2 \\varphi + \\cos{(v_{t})} and 3 \\varphi + n{(\\varphi,v_{t})} = 2 \\varphi + \\cos{(v_{t})} and 3 \\varphi + n{(\\varphi,v_{t})} + 1 = 2 \\varphi + \\cos{(v_{t})} + 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\varphi', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), cos(Symbol('v_t', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('n')(Symbol('\\\\varphi', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\varphi', commutative=True)), cos(Symbol('v_t', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Integer(4), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True)), Function('n')(Symbol('\\\\varphi', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), cos(Symbol('v_t', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True)), Function('n')(Symbol('\\\\varphi', commutative=True), Symbol('v_t', commutative=True)), Integer(1)), Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), cos(Symbol('v_t', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{E}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})}, then derive \\frac{d}{d v_{2}} \\mathbf{E}{(v_{2})} = - \\frac{1}{v_{2}^{2}}, then obtain \\frac{d}{d v_{2}} \\log{(v_{2})} - \\frac{1}{v_{2}^{2}} = \\frac{d}{d v_{2}} \\log{(v_{2})} + \\frac{d^{2}}{d v_{2}^{2}} \\log{(v_{2})}", "derivation": "\\mathbf{E}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})} and \\frac{d}{d v_{2}} \\mathbf{E}{(v_{2})} = \\frac{d^{2}}{d v_{2}^{2}} \\log{(v_{2})} and \\frac{d}{d v_{2}} \\mathbf{E}{(v_{2})} + \\frac{d}{d v_{2}} \\log{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})} + \\frac{d^{2}}{d v_{2}^{2}} \\log{(v_{2})} and \\frac{d}{d v_{2}} \\mathbf{E}{(v_{2})} = - \\frac{1}{v_{2}^{2}} and \\frac{d}{d v_{2}} \\log{(v_{2})} - \\frac{1}{v_{2}^{2}} = \\frac{d}{d v_{2}} \\log{(v_{2})} + \\frac{d^{2}}{d v_{2}^{2}} \\log{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(2))))"], [["add", 2, "Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-2)))), Add(Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\phi_{1}{(n_{2})} = e^{n_{2}}, then derive (- \\frac{d}{d n_{2}} \\phi_{1}{(n_{2})})^{n_{2}} = (- e^{n_{2}})^{n_{2}}, then obtain (- \\frac{d}{d n_{2}} \\phi_{1}{(n_{2})})^{n_{2}} = (- \\phi_{1}{(n_{2})})^{n_{2}}", "derivation": "\\phi_{1}{(n_{2})} = e^{n_{2}} and - \\phi_{1}{(n_{2})} = - e^{n_{2}} and \\frac{d}{d n_{2}} - \\phi_{1}{(n_{2})} = \\frac{d}{d n_{2}} - e^{n_{2}} and (\\frac{d}{d n_{2}} - \\phi_{1}{(n_{2})})^{n_{2}} = (\\frac{d}{d n_{2}} - e^{n_{2}})^{n_{2}} and (- \\frac{d}{d n_{2}} \\phi_{1}{(n_{2})})^{n_{2}} = (- e^{n_{2}})^{n_{2}} and (- \\frac{d}{d n_{2}} \\phi_{1}{(n_{2})})^{n_{2}} = (- \\phi_{1}{(n_{2})})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n_2', commutative=True))), Mul(Integer(-1), exp(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Mul(Integer(-1), exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True)), Pow(Mul(Integer(-1), exp(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{x}_0,f^{*})} = e^{\\frac{\\hat{x}_0}{f^{*}}}, then obtain (\\frac{\\partial}{\\partial \\hat{x}_0} \\operatorname{P_{g}}^{f^{*}}{(\\hat{x}_0,f^{*})})^{f^{*}} = (\\frac{\\partial}{\\partial \\hat{x}_0} (e^{\\frac{\\hat{x}_0}{f^{*}}})^{f^{*}})^{f^{*}}", "derivation": "\\operatorname{P_{g}}{(\\hat{x}_0,f^{*})} = e^{\\frac{\\hat{x}_0}{f^{*}}} and \\operatorname{P_{g}}^{f^{*}}{(\\hat{x}_0,f^{*})} = (e^{\\frac{\\hat{x}_0}{f^{*}}})^{f^{*}} and \\frac{\\partial}{\\partial \\hat{x}_0} \\operatorname{P_{g}}^{f^{*}}{(\\hat{x}_0,f^{*})} = \\frac{\\partial}{\\partial \\hat{x}_0} (e^{\\frac{\\hat{x}_0}{f^{*}}})^{f^{*}} and (\\frac{\\partial}{\\partial \\hat{x}_0} \\operatorname{P_{g}}^{f^{*}}{(\\hat{x}_0,f^{*})})^{f^{*}} = (\\frac{\\partial}{\\partial \\hat{x}_0} (e^{\\frac{\\hat{x}_0}{f^{*}}})^{f^{*}})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)))), Symbol('f^*', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Pow(Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)))), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('f^*', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Derivative(Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1)))), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(f^{*})} = \\sin{(f^{*})}, then derive \\cos{(\\int \\frac{\\operatorname{v_{2}}{(f^{*})}}{f^{*}} df^{*})} = \\cos{(\\theta_2 + \\operatorname{Si}{(f^{*})})}, then obtain \\cos{(\\theta_2 + \\operatorname{Si}{(f^{*})})} = \\cos{(\\int \\frac{\\sin{(f^{*})}}{f^{*}} df^{*})}", "derivation": "\\operatorname{v_{2}}{(f^{*})} = \\sin{(f^{*})} and \\frac{\\operatorname{v_{2}}{(f^{*})}}{f^{*}} = \\frac{\\sin{(f^{*})}}{f^{*}} and \\int \\frac{\\operatorname{v_{2}}{(f^{*})}}{f^{*}} df^{*} = \\int \\frac{\\sin{(f^{*})}}{f^{*}} df^{*} and \\cos{(\\int \\frac{\\operatorname{v_{2}}{(f^{*})}}{f^{*}} df^{*})} = \\cos{(\\int \\frac{\\sin{(f^{*})}}{f^{*}} df^{*})} and \\cos{(\\int \\frac{\\operatorname{v_{2}}{(f^{*})}}{f^{*}} df^{*})} = \\cos{(\\theta_2 + \\operatorname{Si}{(f^{*})})} and \\cos{(\\theta_2 + \\operatorname{Si}{(f^{*})})} = \\cos{(\\int \\frac{\\sin{(f^{*})}}{f^{*}} df^{*})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["divide", 1, "Symbol('f^*', commutative=True)"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('v_2')(Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('v_2')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('v_2')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))), cos(Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(cos(Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('v_2')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))), cos(Add(Symbol('\\\\theta_2', commutative=True), Si(Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(cos(Add(Symbol('\\\\theta_2', commutative=True), Si(Symbol('f^*', commutative=True)))), cos(Integral(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(f^{*})} = \\cos{(f^{*})}, then obtain f^{*} \\frac{d}{d f^{*}} (f^{*} + \\operatorname{V_{\\mathbf{E}}}{(f^{*})}) (\\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{E}}}{(f^{*})})^{f^{*}} = f^{*} \\frac{d}{d f^{*}} (f^{*} + \\cos{(f^{*})}) (\\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{E}}}{(f^{*})})^{f^{*}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(f^{*})} = \\cos{(f^{*})} and f^{*} + \\operatorname{V_{\\mathbf{E}}}{(f^{*})} = f^{*} + \\cos{(f^{*})} and \\frac{d}{d f^{*}} (f^{*} + \\operatorname{V_{\\mathbf{E}}}{(f^{*})}) = \\frac{d}{d f^{*}} (f^{*} + \\cos{(f^{*})}) and f^{*} \\frac{d}{d f^{*}} (f^{*} + \\operatorname{V_{\\mathbf{E}}}{(f^{*})}) = f^{*} \\frac{d}{d f^{*}} (f^{*} + \\cos{(f^{*})}) and f^{*} \\frac{d}{d f^{*}} (f^{*} + \\operatorname{V_{\\mathbf{E}}}{(f^{*})}) (\\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{E}}}{(f^{*})})^{f^{*}} = f^{*} \\frac{d}{d f^{*}} (f^{*} + \\cos{(f^{*})}) (\\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{E}}}{(f^{*})})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["add", 1, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True))), Add(Symbol('f^*', commutative=True), cos(Symbol('f^*', commutative=True))))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Add(Symbol('f^*', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["times", 3, "Symbol('f^*', commutative=True)"], "Equality(Mul(Symbol('f^*', commutative=True), Derivative(Add(Symbol('f^*', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Mul(Symbol('f^*', commutative=True), Derivative(Add(Symbol('f^*', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["times", 4, "Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('f^*', commutative=True))"], "Equality(Mul(Symbol('f^*', commutative=True), Derivative(Add(Symbol('f^*', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('f^*', commutative=True))), Mul(Symbol('f^*', commutative=True), Derivative(Add(Symbol('f^*', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\chi{(V,A)} = A + V and \\theta{(\\theta_2,\\rho)} = \\frac{\\theta_2}{\\rho}, then obtain \\rho (A + V) \\theta{(\\theta_2,\\rho)} = \\theta_2 (A + V)", "derivation": "\\chi{(V,A)} = A + V and \\theta{(\\theta_2,\\rho)} = \\frac{\\theta_2}{\\rho} and \\chi{(V,A)} \\theta{(\\theta_2,\\rho)} = \\frac{\\theta_2 \\chi{(V,A)}}{\\rho} and \\rho \\chi{(V,A)} \\theta{(\\theta_2,\\rho)} = \\theta_2 \\chi{(V,A)} and \\rho (A + V) \\theta{(\\theta_2,\\rho)} = \\theta_2 (A + V)", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('V', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 2, "Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True))))"], [["divide", 3, "Pow(Symbol('\\\\rho', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Function('\\\\chi')(Symbol('V', commutative=True), Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\rho', commutative=True), Add(Symbol('A', commutative=True), Symbol('V', commutative=True)), Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('A', commutative=True), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\mathbb{I}{(g_{\\varepsilon})} = g_{\\varepsilon}, then obtain g_{\\varepsilon} \\mathbb{I}{(g_{\\varepsilon})} \\operatorname{v_{z}}{(g_{\\varepsilon})} = g_{\\varepsilon} \\mathbb{I}{(g_{\\varepsilon})} \\log{(g_{\\varepsilon})}", "derivation": "\\operatorname{v_{z}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\mathbb{I}{(g_{\\varepsilon})} = g_{\\varepsilon} and \\mathbb{I}^{2}{(g_{\\varepsilon})} = g_{\\varepsilon} \\mathbb{I}{(g_{\\varepsilon})} and \\mathbb{I}^{2}{(g_{\\varepsilon})} \\operatorname{v_{z}}{(g_{\\varepsilon})} = \\mathbb{I}^{2}{(g_{\\varepsilon})} \\log{(g_{\\varepsilon})} and g_{\\varepsilon} \\mathbb{I}{(g_{\\varepsilon})} \\operatorname{v_{z}}{(g_{\\varepsilon})} = g_{\\varepsilon} \\mathbb{I}{(g_{\\varepsilon})} \\log{(g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["times", 2, "Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Pow(Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Function('v_z')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Function('v_z')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbb{I}')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbf{J})} = \\log{(\\mathbf{J})}, then obtain \\int \\frac{\\int \\mathbf{J}_P{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} d\\mathbf{J} = \\int \\frac{\\int \\log{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} d\\mathbf{J}", "derivation": "\\mathbf{J}_P{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\int \\mathbf{J}_P{(\\mathbf{J})} d\\mathbf{J} = \\int \\log{(\\mathbf{J})} d\\mathbf{J} and \\frac{\\int \\mathbf{J}_P{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} = \\frac{\\int \\log{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} and \\int \\frac{\\int \\mathbf{J}_P{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} d\\mathbf{J} = \\int \\frac{\\int \\log{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}_P{(\\mathbf{J})}} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{g})} = e^{\\mathbf{g}}, then derive - e^{\\mathbf{g}} = - \\frac{d}{d \\mathbf{g}} \\operatorname{F_{H}}{(\\mathbf{g})}, then obtain - \\operatorname{F_{H}}{(\\mathbf{g})} = - \\frac{d}{d \\mathbf{g}} \\operatorname{F_{H}}{(\\mathbf{g})}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{g})} = e^{\\mathbf{g}} and 0 = - \\operatorname{F_{H}}{(\\mathbf{g})} + e^{\\mathbf{g}} and \\frac{d}{d \\mathbf{g}} 0 = \\frac{d}{d \\mathbf{g}} (- \\operatorname{F_{H}}{(\\mathbf{g})} + e^{\\mathbf{g}}) and - e^{\\mathbf{g}} + \\frac{d}{d \\mathbf{g}} 0 = - e^{\\mathbf{g}} + \\frac{d}{d \\mathbf{g}} (- \\operatorname{F_{H}}{(\\mathbf{g})} + e^{\\mathbf{g}}) and - e^{\\mathbf{g}} = - \\frac{d}{d \\mathbf{g}} \\operatorname{F_{H}}{(\\mathbf{g})} and - \\operatorname{F_{H}}{(\\mathbf{g})} = - \\frac{d}{d \\mathbf{g}} \\operatorname{F_{H}}{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True))), exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True))), exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["minus", 3, "exp(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))), Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True))), exp(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(Function('F_H')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},Q)} = \\cos{(Q + \\dot{\\mathbf{r}})} and \\mathbf{M}{(\\psi^*)} = e^{\\psi^*}, then obtain \\frac{\\mathbf{M}{(\\psi^*)} \\cos{(Q + \\dot{\\mathbf{r}})}}{\\psi^*} = \\frac{e^{\\psi^*} \\cos{(Q + \\dot{\\mathbf{r}})}}{\\psi^*}", "derivation": "\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},Q)} = \\cos{(Q + \\dot{\\mathbf{r}})} and \\mathbf{M}{(\\psi^*)} = e^{\\psi^*} and \\frac{\\mathbf{M}{(\\psi^*)}}{\\psi^*} = \\frac{e^{\\psi^*}}{\\psi^*} and \\frac{\\mathbf{M}{(\\psi^*)} \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},Q)}}{\\psi^*} = \\frac{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},Q)} e^{\\psi^*}}{\\psi^*} and \\frac{\\mathbf{M}{(\\psi^*)} \\cos{(Q + \\dot{\\mathbf{r}})}}{\\psi^*} = \\frac{e^{\\psi^*} \\cos{(Q + \\dot{\\mathbf{r}})}}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["divide", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["times", 3, "Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\psi^*', commutative=True)), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\psi^*', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(W,H)} = H W, then derive \\int \\frac{\\partial}{\\partial W} \\Psi_{\\lambda}{(W,H)} dH = A_{1} + \\frac{H^{2}}{2}, then obtain \\frac{\\iint \\frac{\\partial}{\\partial W} H W dH dH}{\\Psi_{\\lambda}{(W,H)}} = \\frac{\\int (A_{1} + \\frac{H^{2}}{2}) dH}{\\Psi_{\\lambda}{(W,H)}}", "derivation": "\\Psi_{\\lambda}{(W,H)} = H W and \\frac{\\partial}{\\partial W} \\Psi_{\\lambda}{(W,H)} = \\frac{\\partial}{\\partial W} H W and \\int \\frac{\\partial}{\\partial W} \\Psi_{\\lambda}{(W,H)} dH = \\int \\frac{\\partial}{\\partial W} H W dH and \\int \\frac{\\partial}{\\partial W} \\Psi_{\\lambda}{(W,H)} dH = A_{1} + \\frac{H^{2}}{2} and \\iint \\frac{\\partial}{\\partial W} \\Psi_{\\lambda}{(W,H)} dH dH = \\int (A_{1} + \\frac{H^{2}}{2}) dH and \\iint \\frac{\\partial}{\\partial W} H W dH dH = \\int (A_{1} + \\frac{H^{2}}{2}) dH and \\frac{\\iint \\frac{\\partial}{\\partial W} H W dH dH}{\\Psi_{\\lambda}{(W,H)}} = \\frac{\\int (A_{1} + \\frac{H^{2}}{2}) dH}{\\Psi_{\\lambda}{(W,H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Mul(Symbol('H', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)))))"], [["integrate", 4, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)))), Tuple(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(Mul(Symbol('H', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)))), Tuple(Symbol('H', commutative=True))))"], [["divide", 6, "Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Integral(Derivative(Mul(Symbol('H', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Integral(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)))), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given h{(\\mathbf{s},z^{*})} = 2 \\mathbf{s} z^{*}, then derive \\frac{\\partial}{\\partial \\mathbf{s}} h{(\\mathbf{s},z^{*})} = 2 z^{*}, then obtain \\frac{\\frac{\\partial}{\\partial \\mathbf{s}} h{(\\mathbf{s},z^{*})}}{\\dot{\\mathbf{r}}{(\\mathbf{s},z^{*})}} = \\frac{2 z^{*}}{\\dot{\\mathbf{r}}{(\\mathbf{s},z^{*})}}", "derivation": "h{(\\mathbf{s},z^{*})} = 2 \\mathbf{s} z^{*} and \\frac{\\partial}{\\partial \\mathbf{s}} h{(\\mathbf{s},z^{*})} = \\frac{\\partial}{\\partial \\mathbf{s}} 2 \\mathbf{s} z^{*} and \\frac{\\partial}{\\partial \\mathbf{s}} h{(\\mathbf{s},z^{*})} = 2 z^{*} and \\frac{\\frac{\\partial}{\\partial \\mathbf{s}} h{(\\mathbf{s},z^{*})}}{\\dot{\\mathbf{r}}{(\\mathbf{s},z^{*})}} = \\frac{2 z^{*}}{\\dot{\\mathbf{r}}{(\\mathbf{s},z^{*})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Mul(Integer(2), Symbol('z^*', commutative=True)))"], [["divide", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Integer(-1)), Derivative(Function('h')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('z^*', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(a,\\hat{p}_0)} = a \\cos{(\\hat{p}_0)}, then obtain \\sin{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + \\Omega{(a,\\hat{p}_0)}))} = \\sin{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + a \\cos{(\\hat{p}_0)}))}", "derivation": "\\Omega{(a,\\hat{p}_0)} = a \\cos{(\\hat{p}_0)} and \\hat{p}_0 + \\Omega{(a,\\hat{p}_0)} = \\hat{p}_0 + a \\cos{(\\hat{p}_0)} and \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + \\Omega{(a,\\hat{p}_0)}) = \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + a \\cos{(\\hat{p}_0)}) and \\sin{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + \\Omega{(a,\\hat{p}_0)}))} = \\sin{(\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 + a \\cos{(\\hat{p}_0)}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), sin(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{p}_0', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(h,H)} = H h, then obtain \\frac{H + W{(h,H)}}{H h + H} = 1", "derivation": "W{(h,H)} = H h and H + W{(h,H)} = H h + H and \\frac{H + W{(h,H)}}{W{(h,H)}} = \\frac{H h + H}{W{(h,H)}} and \\frac{H + W{(h,H)}}{H h + H} = 1", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), Symbol('h', commutative=True)), Symbol('H', commutative=True)))"], [["divide", 2, "Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Add(Symbol('H', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True))), Pow(Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('H', commutative=True), Symbol('h', commutative=True)), Symbol('H', commutative=True)), Pow(Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Add(Mul(Symbol('H', commutative=True), Symbol('h', commutative=True)), Symbol('H', commutative=True)), Pow(Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('H', commutative=True))), Pow(Add(Mul(Symbol('H', commutative=True), Symbol('h', commutative=True)), Symbol('H', commutative=True)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\hat{p}{(H)} = \\log{(H)}, then obtain \\frac{\\hat{p}{(H)} \\log{(H)}}{(\\int \\log{(H)}^{H} dH)^{2}} = \\frac{\\log{(H)}^{2}}{(\\int \\log{(H)}^{H} dH)^{2}}", "derivation": "\\hat{p}{(H)} = \\log{(H)} and \\hat{p}^{H}{(H)} = \\log{(H)}^{H} and \\int \\hat{p}^{H}{(H)} dH = \\int \\log{(H)}^{H} dH and \\frac{\\hat{p}{(H)}}{\\int \\hat{p}^{H}{(H)} dH} = \\frac{\\log{(H)}}{\\int \\hat{p}^{H}{(H)} dH} and \\frac{\\hat{p}{(H)} \\log{(H)}}{(\\int \\hat{p}^{H}{(H)} dH)^{2}} = \\frac{\\log{(H)}^{2}}{(\\int \\hat{p}^{H}{(H)} dH)^{2}} and \\frac{\\hat{p}{(H)} \\log{(H)}}{(\\int \\log{(H)}^{H} dH)^{2}} = \\frac{\\log{(H)}^{2}}{(\\int \\log{(H)}^{H} dH)^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["divide", 1, "Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(log(Symbol('H', commutative=True)), Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["times", 4, "Mul(log(Symbol('H', commutative=True)), Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)), Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-2))), Mul(Pow(log(Symbol('H', commutative=True)), Integer(2)), Pow(Integral(Pow(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Function('\\\\hat{p}')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)), Pow(Integral(Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-2))), Mul(Pow(log(Symbol('H', commutative=True)), Integer(2)), Pow(Integral(Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{x})} = e^{e^{v_{x}}}, then obtain (\\mathbf{J}_P^{2 v_{x}}{(v_{x})})^{v_{x}} = (\\mathbf{J}_P^{v_{x}}{(v_{x})} (e^{e^{v_{x}}})^{v_{x}})^{v_{x}}", "derivation": "\\mathbf{J}_P{(v_{x})} = e^{e^{v_{x}}} and \\mathbf{J}_P^{v_{x}}{(v_{x})} = (e^{e^{v_{x}}})^{v_{x}} and \\mathbf{J}_P^{2 v_{x}}{(v_{x})} = \\mathbf{J}_P^{v_{x}}{(v_{x})} (e^{e^{v_{x}}})^{v_{x}} and (\\mathbf{J}_P^{2 v_{x}}{(v_{x})})^{v_{x}} = (\\mathbf{J}_P^{v_{x}}{(v_{x})} (e^{e^{v_{x}}})^{v_{x}})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), exp(exp(Symbol('v_x', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(exp(exp(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["times", 2, "Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(exp(exp(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(exp(exp(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\mu)} = e^{\\mu}, then obtain \\log{(\\int \\psi^{*}{(\\mu)} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}})} = \\log{(\\int e^{\\mu} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}})}", "derivation": "\\psi^{*}{(\\mu)} = e^{\\mu} and \\int \\psi^{*}{(\\mu)} d\\mu = \\int e^{\\mu} d\\mu and \\int \\psi^{*}{(\\mu)} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}} = \\int e^{\\mu} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}} and \\log{(\\int \\psi^{*}{(\\mu)} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}})} = \\log{(\\int e^{\\mu} d\\mu - \\frac{e^{\\mu}}{\\psi^{*}^{2}{(\\mu)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Integer(-2)), exp(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Integral(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Integer(-2)), exp(Symbol('\\\\mu', commutative=True)))), Add(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Integer(-2)), exp(Symbol('\\\\mu', commutative=True)))))"], [["log", 3], "Equality(log(Add(Integral(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Integer(-2)), exp(Symbol('\\\\mu', commutative=True))))), log(Add(Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('\\\\mu', commutative=True)), Integer(-2)), exp(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(a^{\\dagger},l)} = (a^{\\dagger})^{l}, then obtain (- l + \\frac{\\dot{\\mathbf{r}}{(a^{\\dagger},l)}}{a^{\\dagger}})^{a^{\\dagger}} = (- l + \\frac{(a^{\\dagger})^{l}}{a^{\\dagger}})^{a^{\\dagger}}", "derivation": "\\dot{\\mathbf{r}}{(a^{\\dagger},l)} = (a^{\\dagger})^{l} and \\frac{\\dot{\\mathbf{r}}{(a^{\\dagger},l)}}{a^{\\dagger}} = \\frac{(a^{\\dagger})^{l}}{a^{\\dagger}} and - l + \\frac{\\dot{\\mathbf{r}}{(a^{\\dagger},l)}}{a^{\\dagger}} = - l + \\frac{(a^{\\dagger})^{l}}{a^{\\dagger}} and (- l + \\frac{\\dot{\\mathbf{r}}{(a^{\\dagger},l)}}{a^{\\dagger}})^{a^{\\dagger}} = (- l + \\frac{(a^{\\dagger})^{l}}{a^{\\dagger}})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)))"], [["divide", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True))))"], [["minus", 2, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)))))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('l', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{M},\\dot{y})} = \\dot{y}^{\\mathbf{M}}, then derive \\mathbf{M} + \\mathbf{S} = \\int \\frac{- \\dot{y} + \\dot{y}^{\\mathbf{M}}}{- \\dot{y} + \\mathbf{D}{(\\mathbf{M},\\dot{y})}} d\\mathbf{M}, then obtain \\int (\\mathbf{M} + \\mathbf{S}) d\\mathbf{M} = \\iint 1 d\\mathbf{M} d\\mathbf{M}", "derivation": "\\mathbf{D}{(\\mathbf{M},\\dot{y})} = \\dot{y}^{\\mathbf{M}} and - \\dot{y} + \\mathbf{D}{(\\mathbf{M},\\dot{y})} = - \\dot{y} + \\dot{y}^{\\mathbf{M}} and 1 = \\frac{- \\dot{y} + \\dot{y}^{\\mathbf{M}}}{- \\dot{y} + \\mathbf{D}{(\\mathbf{M},\\dot{y})}} and \\int 1 d\\mathbf{M} = \\int \\frac{- \\dot{y} + \\dot{y}^{\\mathbf{M}}}{- \\dot{y} + \\mathbf{D}{(\\mathbf{M},\\dot{y})}} d\\mathbf{M} and \\mathbf{M} + \\mathbf{S} = \\int \\frac{- \\dot{y} + \\dot{y}^{\\mathbf{M}}}{- \\dot{y} + \\mathbf{D}{(\\mathbf{M},\\dot{y})}} d\\mathbf{M} and \\mathbf{M} + \\mathbf{S} = \\int 1 d\\mathbf{M} and \\int (\\mathbf{M} + \\mathbf{S}) d\\mathbf{M} = \\iint 1 d\\mathbf{M} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["integrate", 6, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\hat{p},F_{c})} = \\frac{e^{F_{c}}}{\\hat{p}}, then obtain \\frac{\\operatorname{t_{1}}^{2}{(\\hat{p},F_{c})}}{\\hat{p}} = \\frac{\\operatorname{t_{1}}{(\\hat{p},F_{c})} e^{F_{c}}}{\\hat{p}^{2}}", "derivation": "\\operatorname{t_{1}}{(\\hat{p},F_{c})} = \\frac{e^{F_{c}}}{\\hat{p}} and \\frac{\\operatorname{t_{1}}{(\\hat{p},F_{c})}}{\\hat{p}} = \\frac{e^{F_{c}}}{\\hat{p}^{2}} and \\frac{\\operatorname{t_{1}}{(\\hat{p},F_{c})} e^{F_{c}}}{\\hat{p}^{2}} = \\frac{e^{2 F_{c}}}{\\hat{p}^{3}} and \\frac{\\operatorname{t_{1}}^{2}{(\\hat{p},F_{c})}}{\\hat{p}} = \\frac{\\operatorname{t_{1}}{(\\hat{p},F_{c})} e^{F_{c}}}{\\hat{p}^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), exp(Symbol('F_c', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Function('t_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-3)), exp(Mul(Integer(2), Symbol('F_c', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Function('t_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Function('t_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\lambda)} = \\int \\log{(\\lambda)} d\\lambda, then derive \\mathbf{H}{(\\lambda)} = \\hat{H}_{\\lambda} + \\lambda \\log{(\\lambda)} - \\lambda, then obtain (\\hat{H}_{\\lambda} + \\lambda \\log{(\\lambda)} - \\lambda) \\log{(\\lambda)} = \\log{(\\lambda)} \\int \\log{(\\lambda)} d\\lambda", "derivation": "\\mathbf{H}{(\\lambda)} = \\int \\log{(\\lambda)} d\\lambda and \\mathbf{H}{(\\lambda)} \\log{(\\lambda)} = \\log{(\\lambda)} \\int \\log{(\\lambda)} d\\lambda and \\mathbf{H}{(\\lambda)} = \\hat{H}_{\\lambda} + \\lambda \\log{(\\lambda)} - \\lambda and (\\hat{H}_{\\lambda} + \\lambda \\log{(\\lambda)} - \\lambda) \\log{(\\lambda)} = \\log{(\\lambda)} \\int \\log{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Mul(log(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), log(Symbol('\\\\lambda', commutative=True))), Mul(log(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(r)} = \\log{(e^{r})}, then obtain \\sin{(\\frac{\\int \\hat{p}{(r)} dr}{r})} = \\sin{(\\frac{\\int \\log{(e^{r})} dr}{r})}", "derivation": "\\hat{p}{(r)} = \\log{(e^{r})} and \\int \\hat{p}{(r)} dr = \\int \\log{(e^{r})} dr and \\frac{\\int \\hat{p}{(r)} dr}{r} = \\frac{\\int \\log{(e^{r})} dr}{r} and \\sin{(\\frac{\\int \\hat{p}{(r)} dr}{r})} = \\sin{(\\frac{\\int \\log{(e^{r})} dr}{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('r', commutative=True)), log(exp(Symbol('r', commutative=True))))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(log(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["divide", 2, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Integral(log(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Integral(Function('\\\\hat{p}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))), sin(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Integral(log(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given u{(f^{*},H)} = - H + f^{*}, then derive \\frac{\\partial}{\\partial H} u{(f^{*},H)} = -1, then obtain \\frac{\\partial}{\\partial f^{*}} - u{(f^{*},H)} = \\frac{\\partial}{\\partial f^{*}} (H - f^{*})", "derivation": "u{(f^{*},H)} = - H + f^{*} and \\frac{\\partial}{\\partial H} u{(f^{*},H)} = \\frac{\\partial}{\\partial H} (- H + f^{*}) and \\frac{u{(f^{*},H)}}{\\frac{\\partial}{\\partial H} u{(f^{*},H)}} = \\frac{- H + f^{*}}{\\frac{\\partial}{\\partial H} u{(f^{*},H)}} and \\frac{\\partial}{\\partial H} u{(f^{*},H)} = -1 and \\frac{\\partial}{\\partial f^{*}} \\frac{u{(f^{*},H)}}{\\frac{\\partial}{\\partial H} u{(f^{*},H)}} = \\frac{\\partial}{\\partial f^{*}} \\frac{- H + f^{*}}{\\frac{\\partial}{\\partial H} u{(f^{*},H)}} and \\frac{\\partial}{\\partial f^{*}} - u{(f^{*},H)} = \\frac{\\partial}{\\partial f^{*}} (H - f^{*})", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Pow(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f^*', commutative=True)), Pow(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Mul(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Pow(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f^*', commutative=True)), Pow(Derivative(Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(-1), Function('u')(Symbol('f^*', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(I)} = \\int e^{I} dI, then obtain 2 \\cos{(r_{0})} + \\cos{(\\int \\operatorname{A_{2}}{(I)} dI)} + \\cos{(\\iint e^{I} dI dI)} = 2 \\cos{(r_{0})} + 2 \\cos{(\\iint e^{I} dI dI)}", "derivation": "\\operatorname{A_{2}}{(I)} = \\int e^{I} dI and \\int \\operatorname{A_{2}}{(I)} dI = \\iint e^{I} dI dI and \\cos{(\\int \\operatorname{A_{2}}{(I)} dI)} = \\cos{(\\iint e^{I} dI dI)} and \\cos{(r_{0})} + \\cos{(\\int \\operatorname{A_{2}}{(I)} dI)} = \\cos{(r_{0})} + \\cos{(\\iint e^{I} dI dI)} and 2 \\cos{(r_{0})} + \\cos{(\\int \\operatorname{A_{2}}{(I)} dI)} + \\cos{(\\iint e^{I} dI dI)} = 2 \\cos{(r_{0})} + 2 \\cos{(\\iint e^{I} dI dI)}", "srepr_derivation": [["get_premise", "Equality(Function('A_2')(Symbol('I', commutative=True)), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('A_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), cos(Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["add", 3, "cos(Symbol('r_0', commutative=True))"], "Equality(Add(cos(Symbol('r_0', commutative=True)), cos(Integral(Function('A_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Add(cos(Symbol('r_0', commutative=True)), cos(Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))))"], [["add", 4, "Add(cos(Symbol('r_0', commutative=True)), cos(Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], "Equality(Add(Mul(Integer(2), cos(Symbol('r_0', commutative=True))), cos(Integral(Function('A_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), cos(Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Add(Mul(Integer(2), cos(Symbol('r_0', commutative=True))), Mul(Integer(2), cos(Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{g}{(r_{0})} = \\sin{(\\sin{(r_{0})})}, then derive W + r_{0} = \\int \\frac{\\sin{(\\sin{(r_{0})})}}{\\mathbf{g}{(r_{0})}} dr_{0}, then obtain \\int 1 dr_{0} = W + r_{0}", "derivation": "\\mathbf{g}{(r_{0})} = \\sin{(\\sin{(r_{0})})} and 1 = \\frac{\\sin{(\\sin{(r_{0})})}}{\\mathbf{g}{(r_{0})}} and \\int 1 dr_{0} = \\int \\frac{\\sin{(\\sin{(r_{0})})}}{\\mathbf{g}{(r_{0})}} dr_{0} and W + r_{0} = \\int \\frac{\\sin{(\\sin{(r_{0})})}}{\\mathbf{g}{(r_{0})}} dr_{0} and \\int 1 dr_{0} = W + r_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('r_0', commutative=True)), sin(sin(Symbol('r_0', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{g}')(Symbol('r_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(sin(Symbol('r_0', commutative=True)))))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(sin(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('W', commutative=True), Symbol('r_0', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(sin(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('r_0', commutative=True))), Add(Symbol('W', commutative=True), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\phi_2,g)} = \\cos{(\\phi_2 g)}, then obtain - g + 2 \\operatorname{F_{N}}{(\\phi_2,g)} - \\cos{(\\phi_2 g)} - 1 = - g + \\cos{(\\phi_2 g)} - 1", "derivation": "\\operatorname{F_{N}}{(\\phi_2,g)} = \\cos{(\\phi_2 g)} and - g + \\operatorname{F_{N}}{(\\phi_2,g)} = - g + \\cos{(\\phi_2 g)} and - g + \\operatorname{F_{N}}{(\\phi_2,g)} - \\cos{(\\phi_2 g)} = - g and - g + 2 \\operatorname{F_{N}}{(\\phi_2,g)} - \\cos{(\\phi_2 g)} = - g + \\operatorname{F_{N}}{(\\phi_2,g)} and - g + 2 \\operatorname{F_{N}}{(\\phi_2,g)} - \\cos{(\\phi_2 g)} = - g + \\cos{(\\phi_2 g)} and - g + 2 \\operatorname{F_{N}}{(\\phi_2,g)} - \\cos{(\\phi_2 g)} - 1 = - g + \\cos{(\\phi_2 g)} - 1", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))))"], [["minus", 2, "cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(2), Function('F_N')(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), cos(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('g', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given v{(\\mathbf{E})} = e^{\\mathbf{E}}, then derive \\mathbf{E} \\frac{d}{d \\mathbf{E}} v{(\\mathbf{E})} + v{(\\mathbf{E})} = \\mathbf{E} e^{\\mathbf{E}} + e^{\\mathbf{E}}, then obtain \\mathbf{E} e^{\\mathbf{E}} + m_{s} = \\mathbf{E} e^{\\mathbf{E}} + a", "derivation": "v{(\\mathbf{E})} = e^{\\mathbf{E}} and \\mathbf{E} v{(\\mathbf{E})} = \\mathbf{E} e^{\\mathbf{E}} and \\frac{d}{d \\mathbf{E}} \\mathbf{E} v{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\mathbf{E} e^{\\mathbf{E}} and \\mathbf{E} \\frac{d}{d \\mathbf{E}} v{(\\mathbf{E})} + v{(\\mathbf{E})} = \\mathbf{E} e^{\\mathbf{E}} + e^{\\mathbf{E}} and \\mathbf{E} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} + e^{\\mathbf{E}} = \\mathbf{E} e^{\\mathbf{E}} + e^{\\mathbf{E}} and \\int (\\mathbf{E} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} + e^{\\mathbf{E}}) d\\mathbf{E} = \\int (\\mathbf{E} e^{\\mathbf{E}} + e^{\\mathbf{E}}) d\\mathbf{E} and \\mathbf{E} e^{\\mathbf{E}} + m_{s} = \\mathbf{E} e^{\\mathbf{E}} + a", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('v')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('v')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Derivative(Function('v')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Function('v')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('m_s', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(u,E_{x})} = E_{x} - u, then obtain \\int 0^{E_{x}} du = \\int 1 du", "derivation": "\\operatorname{C_{2}}{(u,E_{x})} = E_{x} - u and 0 = E_{x} - u - \\operatorname{C_{2}}{(u,E_{x})} and 0^{E_{x}} = (E_{x} - u - \\operatorname{C_{2}}{(u,E_{x})})^{E_{x}} and \\int 0^{E_{x}} du = \\int (E_{x} - u - \\operatorname{C_{2}}{(u,E_{x})})^{E_{x}} du and \\int (E_{x} - u - \\operatorname{C_{2}}{(u,E_{x})})^{E_{x}} du = \\int 1 du and \\int 0^{E_{x}} du = \\int 1 du", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["minus", 1, "Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Integer(0), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True)))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E_x', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('E_x', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('u', commutative=True), Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Pow(Integer(0), Symbol('E_x', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given Z{(V)} = e^{\\sin{(V)}}, then obtain Z^{V}{(V)} + (\\int Z{(V)} Z^{V}{(V)} dV)^{V} = Z^{V}{(V)} + (\\int Z{(V)} (e^{\\sin{(V)}})^{V} dV)^{V}", "derivation": "Z{(V)} = e^{\\sin{(V)}} and Z^{V}{(V)} = (e^{\\sin{(V)}})^{V} and Z{(V)} Z^{V}{(V)} = Z{(V)} (e^{\\sin{(V)}})^{V} and \\int Z{(V)} Z^{V}{(V)} dV = \\int Z{(V)} (e^{\\sin{(V)}})^{V} dV and (\\int Z{(V)} Z^{V}{(V)} dV)^{V} = (\\int Z{(V)} (e^{\\sin{(V)}})^{V} dV)^{V} and (e^{\\sin{(V)}})^{V} + (\\int Z{(V)} Z^{V}{(V)} dV)^{V} = (e^{\\sin{(V)}})^{V} + (\\int Z{(V)} (e^{\\sin{(V)}})^{V} dV)^{V} and Z^{V}{(V)} + (\\int Z{(V)} Z^{V}{(V)} dV)^{V} = Z^{V}{(V)} + (\\int Z{(V)} (e^{\\sin{(V)}})^{V} dV)^{V}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('V', commutative=True)), exp(sin(Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["times", 2, "Function('Z')(Symbol('V', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Mul(Function('Z')(Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["add", 5, "Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))"], "Equality(Add(Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Add(Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Add(Pow(Function('Z')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Integral(Mul(Function('Z')(Symbol('V', commutative=True)), Pow(exp(sin(Symbol('V', commutative=True))), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(n,\\varepsilon)} = - \\sin{(\\varepsilon - n)} and x{(n,\\varepsilon)} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon}, then obtain x{(n,\\varepsilon)} + \\frac{1}{\\varepsilon} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon} + \\frac{1}{\\varepsilon}", "derivation": "\\dot{z}{(n,\\varepsilon)} = - \\sin{(\\varepsilon - n)} and \\frac{\\dot{z}{(n,\\varepsilon)}}{\\varepsilon} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon} and \\frac{\\dot{z}{(n,\\varepsilon)}}{\\varepsilon} + \\frac{1}{\\varepsilon} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon} + \\frac{1}{\\varepsilon} and x{(n,\\varepsilon)} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon} and \\frac{\\dot{z}{(n,\\varepsilon)}}{\\varepsilon} = x{(n,\\varepsilon)} and x{(n,\\varepsilon)} + \\frac{1}{\\varepsilon} = - \\frac{\\sin{(\\varepsilon - n)}}{\\varepsilon} + \\frac{1}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["divide", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["add", 2, "Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('x')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('x')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Function('x')(Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(M_{E})} = \\int \\sin{(M_{E})} dM_{E} and \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} = \\mathbf{E} + \\rho, then obtain M_{E}^{3} \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} \\mathbf{J}_P{(M_{E})} (\\int \\sin{(M_{E})} dM_{E})^{2} = M_{E}^{3} \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} (\\int \\sin{(M_{E})} dM_{E})^{3}", "derivation": "\\mathbf{J}_P{(M_{E})} = \\int \\sin{(M_{E})} dM_{E} and M_{E} \\mathbf{J}_P{(M_{E})} = M_{E} \\int \\sin{(M_{E})} dM_{E} and \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} = \\mathbf{E} + \\rho and M_{E}^{3} (\\mathbf{E} + \\rho) \\mathbf{J}_P{(M_{E})} (\\int \\sin{(M_{E})} dM_{E})^{2} = M_{E}^{3} (\\mathbf{E} + \\rho) (\\int \\sin{(M_{E})} dM_{E})^{3} and M_{E}^{3} \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} \\mathbf{J}_P{(M_{E})} (\\int \\sin{(M_{E})} dM_{E})^{2} = M_{E}^{3} \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\rho)} (\\int \\sin{(M_{E})} dM_{E})^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('M_E', commutative=True)), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], ["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(2)))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(3)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('M_E', commutative=True)), Pow(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(2))), Mul(Pow(Symbol('M_E', commutative=True), Integer(3)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(3)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('M_E', commutative=True)), Pow(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(2))), Mul(Pow(Symbol('M_E', commutative=True), Integer(3)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(3))))"]]}, {"prompt": "Given H{(\\Omega)} = \\sin{(\\Omega)}, then obtain H{(\\Omega)} \\frac{d}{d \\Omega} H{(\\Omega)} = H{(\\Omega)} \\cos{(\\Omega)}", "derivation": "H{(\\Omega)} = \\sin{(\\Omega)} and \\frac{d}{d \\Omega} H{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and H{(\\Omega)} \\frac{d}{d \\Omega} H{(\\Omega)} = H{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} and H{(\\Omega)} \\frac{d}{d \\Omega} H{(\\Omega)} = H{(\\Omega)} \\cos{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["times", 2, "Function('H')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('H')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Function('H')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given B{(\\phi,f,\\varphi)} = \\varphi^{\\phi} f, then derive \\varphi \\frac{\\partial}{\\partial \\phi} B{(\\phi,f,\\varphi)} - 1 = \\varphi \\varphi^{\\phi} f \\log{(\\varphi)} - 1, then obtain \\varphi \\frac{\\partial}{\\partial \\phi} \\varphi^{\\phi} f - \\log{(\\varphi)} - 1 = \\varphi \\varphi^{\\phi} f \\log{(\\varphi)} - \\log{(\\varphi)} - 1", "derivation": "B{(\\phi,f,\\varphi)} = \\varphi^{\\phi} f and \\varphi B{(\\phi,f,\\varphi)} = \\varphi \\varphi^{\\phi} f and - \\phi + \\varphi B{(\\phi,f,\\varphi)} = - \\phi + \\varphi \\varphi^{\\phi} f and \\frac{\\partial}{\\partial \\phi} (- \\phi + \\varphi B{(\\phi,f,\\varphi)}) = \\frac{\\partial}{\\partial \\phi} (- \\phi + \\varphi \\varphi^{\\phi} f) and \\varphi \\frac{\\partial}{\\partial \\phi} B{(\\phi,f,\\varphi)} - 1 = \\varphi \\varphi^{\\phi} f \\log{(\\varphi)} - 1 and \\varphi \\frac{\\partial}{\\partial \\phi} \\varphi^{\\phi} f - 1 = \\varphi \\varphi^{\\phi} f \\log{(\\varphi)} - 1 and \\varphi \\frac{\\partial}{\\partial \\phi} \\varphi^{\\phi} f - \\log{(\\varphi)} - 1 = \\varphi \\varphi^{\\phi} f \\log{(\\varphi)} - \\log{(\\varphi)} - 1", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\phi', commutative=True), Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('B')(Symbol('\\\\phi', commutative=True), Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True)))"], [["minus", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Function('B')(Symbol('\\\\phi', commutative=True), Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Function('B')(Symbol('\\\\phi', commutative=True), Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Function('B')(Symbol('\\\\phi', commutative=True), Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)))"], [["minus", 6, "log(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('f', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon{(\\Psi)} = e^{\\Psi}, then obtain \\Psi \\int \\varepsilon{(\\Psi)} d\\Psi + \\phi = \\Psi (\\hat{x} + e^{\\Psi}) + \\phi", "derivation": "\\varepsilon{(\\Psi)} = e^{\\Psi} and \\int \\varepsilon{(\\Psi)} d\\Psi = \\int e^{\\Psi} d\\Psi and \\Psi \\int \\varepsilon{(\\Psi)} d\\Psi = \\Psi \\int e^{\\Psi} d\\Psi and \\Psi \\int \\varepsilon{(\\Psi)} d\\Psi + \\phi = \\Psi \\int e^{\\Psi} d\\Psi + \\phi and \\Psi \\int \\varepsilon{(\\Psi)} d\\Psi + \\phi = \\Psi (\\hat{x} + e^{\\Psi}) + \\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Integral(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\Psi', commutative=True), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["add", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\Psi', commutative=True), Integral(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('\\\\Psi', commutative=True), Integral(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(C)} = \\log{(C)}, then obtain C \\int \\frac{\\Psi_{nl}{(C)}}{C} dC = C \\int \\frac{\\log{(C)}}{C} dC", "derivation": "\\Psi_{nl}{(C)} = \\log{(C)} and \\frac{\\Psi_{nl}{(C)}}{C} = \\frac{\\log{(C)}}{C} and \\int \\frac{\\Psi_{nl}{(C)}}{C} dC = \\int \\frac{\\log{(C)}}{C} dC and C \\int \\frac{\\Psi_{nl}{(C)}}{C} dC = C \\int \\frac{\\log{(C)}}{C} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["divide", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), log(Symbol('C', commutative=True))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["divide", 3, "Pow(Symbol('C', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('C', commutative=True), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\Psi)} = \\sin{(\\Psi)}, then derive \\int \\mathbf{r}{(\\Psi)} d\\Psi = c_{0} - \\cos{(\\Psi)}, then obtain c_{0} - \\mathbf{r}{(\\Psi)} - \\cos{(\\Psi)} = - \\mathbf{r}{(\\Psi)} + \\int \\sin{(\\Psi)} d\\Psi", "derivation": "\\mathbf{r}{(\\Psi)} = \\sin{(\\Psi)} and \\int \\mathbf{r}{(\\Psi)} d\\Psi = \\int \\sin{(\\Psi)} d\\Psi and - \\mathbf{r}{(\\Psi)} + \\int \\mathbf{r}{(\\Psi)} d\\Psi = - \\mathbf{r}{(\\Psi)} + \\int \\sin{(\\Psi)} d\\Psi and \\int \\mathbf{r}{(\\Psi)} d\\Psi = c_{0} - \\cos{(\\Psi)} and c_{0} - \\mathbf{r}{(\\Psi)} - \\cos{(\\Psi)} = - \\mathbf{r}{(\\Psi)} + \\int \\sin{(\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True))), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\dot{z})} = \\sin{(\\sin{(\\dot{z})})}, then obtain \\sin{(\\dot{z})} \\int (- \\dot{z} + \\operatorname{J_{\\varepsilon}}{(\\dot{z})}) d\\dot{z} = \\sin{(\\dot{z})} \\int (- \\dot{z} + \\sin{(\\sin{(\\dot{z})})}) d\\dot{z}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\dot{z})} = \\sin{(\\sin{(\\dot{z})})} and - \\dot{z} + \\operatorname{J_{\\varepsilon}}{(\\dot{z})} = - \\dot{z} + \\sin{(\\sin{(\\dot{z})})} and \\int (- \\dot{z} + \\operatorname{J_{\\varepsilon}}{(\\dot{z})}) d\\dot{z} = \\int (- \\dot{z} + \\sin{(\\sin{(\\dot{z})})}) d\\dot{z} and \\sin{(\\dot{z})} \\int (- \\dot{z} + \\operatorname{J_{\\varepsilon}}{(\\dot{z})}) d\\dot{z} = \\sin{(\\dot{z})} \\int (- \\dot{z} + \\sin{(\\sin{(\\dot{z})})}) d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\dot{z}', commutative=True)), sin(sin(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(sin(Symbol('\\\\dot{z}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(sin(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 3, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(sin(Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), sin(sin(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given b{(n_{1},\\hat{x},\\sigma_p)} = \\frac{\\hat{x} n_{1}}{\\sigma_p}, then derive \\frac{\\partial}{\\partial \\sigma_p} b{(n_{1},\\hat{x},\\sigma_p)} = - \\frac{\\hat{x} n_{1}}{\\sigma_p^{2}}, then obtain \\frac{\\partial}{\\partial \\sigma_p} b{(n_{1},\\hat{x},\\sigma_p)} = - \\frac{b{(n_{1},\\hat{x},\\sigma_p)}}{\\sigma_p}", "derivation": "b{(n_{1},\\hat{x},\\sigma_p)} = \\frac{\\hat{x} n_{1}}{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} b{(n_{1},\\hat{x},\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\hat{x} n_{1}}{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} b{(n_{1},\\hat{x},\\sigma_p)} = - \\frac{\\hat{x} n_{1}}{\\sigma_p^{2}} and \\frac{\\partial}{\\partial \\sigma_p} b{(n_{1},\\hat{x},\\sigma_p)} = - \\frac{b{(n_{1},\\hat{x},\\sigma_p)}}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('n_1', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('n_1', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('n_1', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-2)), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('b')(Symbol('n_1', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('b')(Symbol('n_1', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\phi_2,\\hat{H})} = - \\hat{H} + \\phi_2, then obtain \\frac{(\\frac{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}{- \\hat{H} + \\phi_2})^{\\phi_2} + 1}{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}} = \\frac{2}{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}", "derivation": "\\operatorname{r_{0}}{(\\phi_2,\\hat{H})} = - \\hat{H} + \\phi_2 and \\frac{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}{- \\hat{H} + \\phi_2} = 1 and (\\frac{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}{- \\hat{H} + \\phi_2})^{\\phi_2} = 1 and (\\frac{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}{- \\hat{H} + \\phi_2})^{\\phi_2} + 1 = 2 and \\frac{(\\frac{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}{- \\hat{H} + \\phi_2})^{\\phi_2} + 1}{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}} = \\frac{2}{\\operatorname{r_{0}}{(\\phi_2,\\hat{H})}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Integer(1))"], [["add", 3, 1], "Equality(Add(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Integer(1)), Integer(2))"], [["divide", 4, "Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Add(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Integer(1)), Pow(Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given f{(P_{g})} = \\sin{(P_{g})}, then obtain e^{(P_{g} + f^{P_{g}}{(P_{g})})^{P_{g}}} = e^{(P_{g} + \\sin^{P_{g}}{(P_{g})})^{P_{g}}}", "derivation": "f{(P_{g})} = \\sin{(P_{g})} and f^{P_{g}}{(P_{g})} = \\sin^{P_{g}}{(P_{g})} and P_{g} + f^{P_{g}}{(P_{g})} = P_{g} + \\sin^{P_{g}}{(P_{g})} and (P_{g} + f^{P_{g}}{(P_{g})})^{P_{g}} = (P_{g} + \\sin^{P_{g}}{(P_{g})})^{P_{g}} and e^{(P_{g} + f^{P_{g}}{(P_{g})})^{P_{g}}} = e^{(P_{g} + \\sin^{P_{g}}{(P_{g})})^{P_{g}}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["power", 1, "Symbol('P_g', commutative=True)"], "Equality(Pow(Function('f')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(sin(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('P_g', commutative=True))"], "Equality(Add(Symbol('P_g', commutative=True), Pow(Function('f')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Add(Symbol('P_g', commutative=True), Pow(sin(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Add(Symbol('P_g', commutative=True), Pow(Function('f')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Add(Symbol('P_g', commutative=True), Pow(sin(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Add(Symbol('P_g', commutative=True), Pow(Function('f')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True))), exp(Pow(Add(Symbol('P_g', commutative=True), Pow(sin(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\pi)} = \\log{(e^{\\pi})}, then obtain \\frac{d}{d \\pi} (- \\frac{\\operatorname{a^{\\dagger}}{(\\pi)}}{\\log{(e^{\\pi})}} + \\log{(e^{\\pi})}) = \\frac{d}{d \\pi} (\\log{(e^{\\pi})} - 1)", "derivation": "\\operatorname{a^{\\dagger}}{(\\pi)} = \\log{(e^{\\pi})} and \\frac{\\operatorname{a^{\\dagger}}{(\\pi)}}{\\log{(e^{\\pi})}} = 1 and - \\frac{\\operatorname{a^{\\dagger}}{(\\pi)}}{\\log{(e^{\\pi})}} = -1 and - \\frac{\\operatorname{a^{\\dagger}}{(\\pi)}}{\\log{(e^{\\pi})}} + \\log{(e^{\\pi})} = \\log{(e^{\\pi})} - 1 and \\frac{d}{d \\pi} (- \\frac{\\operatorname{a^{\\dagger}}{(\\pi)}}{\\log{(e^{\\pi})}} + \\log{(e^{\\pi})}) = \\frac{d}{d \\pi} (\\log{(e^{\\pi})} - 1)", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "log(exp(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1))), Integer(-1))"], [["add", 3, "log(exp(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1))), log(exp(Symbol('\\\\pi', commutative=True)))), Add(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1))), log(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(log(exp(Symbol('\\\\pi', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(M)} = \\cos{(M)}, then obtain \\frac{\\frac{d}{d M} (\\omega{(M)} + \\cos{(M)})}{(\\omega{(M)} + \\cos{(M)})^{2}} = \\frac{\\frac{d}{d M} 2 \\cos{(M)}}{(\\omega{(M)} + \\cos{(M)})^{2}}", "derivation": "\\omega{(M)} = \\cos{(M)} and \\omega{(M)} + \\cos{(M)} = 2 \\cos{(M)} and (\\omega{(M)} + \\cos{(M)})^{2} = 4 \\cos^{2}{(M)} and \\frac{d}{d M} (\\omega{(M)} + \\cos{(M)}) = \\frac{d}{d M} 2 \\cos{(M)} and \\frac{\\frac{d}{d M} (\\omega{(M)} + \\cos{(M)})}{4 \\cos^{2}{(M)}} = \\frac{\\frac{d}{d M} 2 \\cos{(M)}}{4 \\cos^{2}{(M)}} and \\frac{\\frac{d}{d M} (\\omega{(M)} + \\cos{(M)})}{(\\omega{(M)} + \\cos{(M)})^{2}} = \\frac{\\frac{d}{d M} 2 \\cos{(M)}}{(\\omega{(M)} + \\cos{(M)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["add", 1, "cos(Symbol('M', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(2), cos(Symbol('M', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Integer(2)), Mul(Integer(4), Pow(cos(Symbol('M', commutative=True)), Integer(2))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Integer(4), Pow(cos(Symbol('M', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 4), Pow(cos(Symbol('M', commutative=True)), Integer(-2)), Derivative(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Rational(1, 4), Pow(cos(Symbol('M', commutative=True)), Integer(-2)), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Integer(-2)), Derivative(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Add(Function('\\\\omega')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Integer(-2)), Derivative(Mul(Integer(2), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(\\mu)} = \\cos{(\\cos{(\\mu)})} and h{(Z)} = \\cos{(Z)}, then obtain \\frac{h{(Z)}}{\\cos{(\\cos{(\\mu)})}} + \\cos{(\\mu)} \\cos{(\\cos{(\\mu)})} = \\frac{\\cos{(Z)}}{\\cos{(\\cos{(\\mu)})}} + \\cos{(\\mu)} \\cos{(\\cos{(\\mu)})}", "derivation": "C{(\\mu)} = \\cos{(\\cos{(\\mu)})} and C{(\\mu)} \\cos{(\\mu)} = \\cos{(\\mu)} \\cos{(\\cos{(\\mu)})} and h{(Z)} = \\cos{(Z)} and \\frac{h{(Z)}}{\\cos{(\\cos{(\\mu)})}} = \\frac{\\cos{(Z)}}{\\cos{(\\cos{(\\mu)})}} and C{(\\mu)} \\cos{(\\mu)} + \\frac{h{(Z)}}{\\cos{(\\cos{(\\mu)})}} = C{(\\mu)} \\cos{(\\mu)} + \\frac{\\cos{(Z)}}{\\cos{(\\cos{(\\mu)})}} and \\frac{h{(Z)}}{\\cos{(\\cos{(\\mu)})}} + \\cos{(\\mu)} \\cos{(\\cos{(\\mu)})} = \\frac{\\cos{(Z)}}{\\cos{(\\cos{(\\mu)})}} + \\cos{(\\mu)} \\cos{(\\cos{(\\mu)})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('C')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(cos(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True)))))"], ["get_premise", "Equality(Function('h')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["divide", 3, "cos(cos(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(cos(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1))))"], [["add", 4, "Mul(Function('C')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Function('C')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(Function('h')(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1)))), Add(Mul(Function('C')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(cos(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Function('h')(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(cos(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True))))), Add(Mul(cos(Symbol('Z', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(cos(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{E}{(F_{x})} = \\sin{(\\log{(F_{x})})}, then derive \\int \\frac{\\mathbf{E}{(F_{x})} \\log{(F_{x})}}{\\sin{(\\log{(F_{x})})}} dF_{x} = F_{x} \\log{(F_{x})} - F_{x} + M_{E}, then obtain \\int \\log{(F_{x})} dF_{x} = F_{x} \\log{(F_{x})} - F_{x} + M_{E}", "derivation": "\\mathbf{E}{(F_{x})} = \\sin{(\\log{(F_{x})})} and \\frac{\\mathbf{E}{(F_{x})}}{\\log{(F_{x})}} = \\frac{\\sin{(\\log{(F_{x})})}}{\\log{(F_{x})}} and \\frac{\\mathbf{E}{(F_{x})} \\log{(F_{x})}}{\\sin{(\\log{(F_{x})})}} = \\log{(F_{x})} and \\int \\frac{\\mathbf{E}{(F_{x})} \\log{(F_{x})}}{\\sin{(\\log{(F_{x})})}} dF_{x} = \\int \\log{(F_{x})} dF_{x} and \\int \\frac{\\mathbf{E}{(F_{x})} \\log{(F_{x})}}{\\sin{(\\log{(F_{x})})}} dF_{x} = F_{x} \\log{(F_{x})} - F_{x} + M_{E} and \\int \\log{(F_{x})} dF_{x} = F_{x} \\log{(F_{x})} - F_{x} + M_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_x', commutative=True)), sin(log(Symbol('F_x', commutative=True))))"], [["divide", 1, "log(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('F_x', commutative=True)), Integer(-1)), sin(log(Symbol('F_x', commutative=True)))))"], [["divide", 2, "Mul(Pow(log(Symbol('F_x', commutative=True)), Integer(-2)), sin(log(Symbol('F_x', commutative=True))))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)), Pow(sin(log(Symbol('F_x', commutative=True))), Integer(-1))), log(Symbol('F_x', commutative=True)))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{E}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)), Pow(sin(log(Symbol('F_x', commutative=True))), Integer(-1))), Tuple(Symbol('F_x', commutative=True))), Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Function('\\\\mathbf{E}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)), Pow(sin(log(Symbol('F_x', commutative=True))), Integer(-1))), Tuple(Symbol('F_x', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('M_E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{E},H)} = - H + \\log{(\\mathbf{E})} and \\operatorname{c_{0}}{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} \\operatorname{v_{t}}{(\\mathbf{E},H)} = \\frac{\\partial}{\\partial \\mathbf{E}} (- H + \\operatorname{c_{0}}{(\\mathbf{E})})", "derivation": "\\operatorname{v_{t}}{(\\mathbf{E},H)} = - H + \\log{(\\mathbf{E})} and \\operatorname{c_{0}}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and \\frac{\\partial}{\\partial \\mathbf{E}} \\operatorname{v_{t}}{(\\mathbf{E},H)} = \\frac{\\partial}{\\partial \\mathbf{E}} (- H + \\log{(\\mathbf{E})}) and \\frac{\\partial}{\\partial \\mathbf{E}} \\operatorname{v_{t}}{(\\mathbf{E},H)} = \\frac{\\partial}{\\partial \\mathbf{E}} (- H + \\operatorname{c_{0}}{(\\mathbf{E})})", "srepr_derivation": [["get_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('c_0')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(\\phi_1)} = \\sin{(\\phi_1)}, then obtain (2 l{(\\phi_1)} - 2 \\sin{(\\phi_1)} + \\frac{1}{\\sin{(\\phi_1)}}) \\sin{(\\phi_1)} = 1", "derivation": "l{(\\phi_1)} = \\sin{(\\phi_1)} and l{(\\phi_1)} - \\sin{(\\phi_1)} = 0 and l{(\\phi_1)} - \\sin{(\\phi_1)} + \\frac{1}{\\sin{(\\phi_1)}} = \\frac{1}{\\sin{(\\phi_1)}} and (l{(\\phi_1)} - \\sin{(\\phi_1)} + \\frac{1}{\\sin{(\\phi_1)}}) \\sin{(\\phi_1)} = 1 and (2 l{(\\phi_1)} - 2 \\sin{(\\phi_1)} + \\frac{1}{\\sin{(\\phi_1)}}) \\sin{(\\phi_1)} = 1", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('l')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"], [["add", 2, "Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))"], "Equality(Add(Function('l')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_1', commutative=True))), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1)))"], [["times", 3, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Add(Function('l')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_1', commutative=True))), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), sin(Symbol('\\\\phi_1', commutative=True))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(2), Function('l')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\phi_1', commutative=True))), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(-1))), sin(Symbol('\\\\phi_1', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\mathbf{g}{(n)} = \\cos{(\\sin{(n)})}, then obtain (\\sin{(\\sin{(n)})} + \\frac{d}{d n} \\mathbf{g}{(n)})^{n} = (- \\sin{(\\sin{(n)})} \\cos{(n)} + \\sin{(\\sin{(n)})})^{n}", "derivation": "\\mathbf{g}{(n)} = \\cos{(\\sin{(n)})} and \\frac{d}{d n} \\mathbf{g}{(n)} = \\frac{d}{d n} \\cos{(\\sin{(n)})} and \\sin{(\\sin{(n)})} + \\frac{d}{d n} \\mathbf{g}{(n)} = \\sin{(\\sin{(n)})} + \\frac{d}{d n} \\cos{(\\sin{(n)})} and (\\sin{(\\sin{(n)})} + \\frac{d}{d n} \\mathbf{g}{(n)})^{n} = (\\sin{(\\sin{(n)})} + \\frac{d}{d n} \\cos{(\\sin{(n)})})^{n} and (\\sin{(\\sin{(n)})} + \\frac{d}{d n} \\mathbf{g}{(n)})^{n} = (- \\sin{(\\sin{(n)})} \\cos{(n)} + \\sin{(\\sin{(n)})})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('n', commutative=True)), cos(sin(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 2, "sin(sin(Symbol('n', commutative=True)))"], "Equality(Add(sin(sin(Symbol('n', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(sin(sin(Symbol('n', commutative=True))), Derivative(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Add(sin(sin(Symbol('n', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Symbol('n', commutative=True)), Pow(Add(sin(sin(Symbol('n', commutative=True))), Derivative(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(sin(sin(Symbol('n', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), sin(sin(Symbol('n', commutative=True))), cos(Symbol('n', commutative=True))), sin(sin(Symbol('n', commutative=True)))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(b,\\hbar)} = - \\hbar + b, then obtain \\int \\frac{- 2 \\hbar + 2 b}{- \\hbar + b} d\\hbar = \\int \\frac{- \\hbar + b + \\mathbb{I}{(b,\\hbar)}}{\\mathbb{I}{(b,\\hbar)}} d\\hbar", "derivation": "\\mathbb{I}{(b,\\hbar)} = - \\hbar + b and 2 \\mathbb{I}{(b,\\hbar)} = - \\hbar + b + \\mathbb{I}{(b,\\hbar)} and 2 = \\frac{- \\hbar + b + \\mathbb{I}{(b,\\hbar)}}{\\mathbb{I}{(b,\\hbar)}} and \\int 2 d\\hbar = \\int \\frac{- \\hbar + b + \\mathbb{I}{(b,\\hbar)}}{\\mathbb{I}{(b,\\hbar)}} d\\hbar and 2 = \\frac{- 2 \\hbar + 2 b}{- \\hbar + b} and \\int \\frac{- 2 \\hbar + 2 b}{- \\hbar + b} d\\hbar = \\int \\frac{- \\hbar + b + \\mathbb{I}{(b,\\hbar)}}{\\mathbb{I}{(b,\\hbar)}} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True)))"], [["add", 1, "Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True), Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["divide", 2, "Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(2), Mul(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True), Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Integer(2), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True), Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(2), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('b', commutative=True), Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(S,\\ddot{x})} = S^{\\ddot{x}}, then derive \\ddot{x} \\frac{\\partial}{\\partial \\ddot{x}} \\mathbf{M}{(S,\\ddot{x})} + \\mathbf{M}{(S,\\ddot{x})} = S^{\\ddot{x}} \\ddot{x} \\log{(S)} + S^{\\ddot{x}}, then obtain S^{\\ddot{x}} + \\ddot{x} \\frac{\\partial}{\\partial \\ddot{x}} S^{\\ddot{x}} = S^{\\ddot{x}} \\ddot{x} \\log{(S)} + S^{\\ddot{x}}", "derivation": "\\mathbf{M}{(S,\\ddot{x})} = S^{\\ddot{x}} and \\ddot{x} \\mathbf{M}{(S,\\ddot{x})} = S^{\\ddot{x}} \\ddot{x} and \\frac{\\partial}{\\partial \\ddot{x}} \\ddot{x} \\mathbf{M}{(S,\\ddot{x})} = \\frac{\\partial}{\\partial \\ddot{x}} S^{\\ddot{x}} \\ddot{x} and \\ddot{x} \\frac{\\partial}{\\partial \\ddot{x}} \\mathbf{M}{(S,\\ddot{x})} + \\mathbf{M}{(S,\\ddot{x})} = S^{\\ddot{x}} \\ddot{x} \\log{(S)} + S^{\\ddot{x}} and S^{\\ddot{x}} + \\ddot{x} \\frac{\\partial}{\\partial \\ddot{x}} S^{\\ddot{x}} = S^{\\ddot{x}} \\ddot{x} \\log{(S)} + S^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\mathbf{M}')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\mathbf{M}')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), Derivative(Function('\\\\mathbf{M}')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Function('\\\\mathbf{M}')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), log(Symbol('S', commutative=True))), Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Derivative(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), log(Symbol('S', commutative=True))), Pow(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given m{(f_{E})} = e^{\\cos{(f_{E})}}, then obtain - \\frac{- m{(f_{E})} + \\frac{d}{d f_{E}} m{(f_{E})}}{m{(f_{E})}} = - \\frac{- m{(f_{E})} - e^{\\cos{(f_{E})}} \\sin{(f_{E})}}{m{(f_{E})}}", "derivation": "m{(f_{E})} = e^{\\cos{(f_{E})}} and \\frac{d}{d f_{E}} m{(f_{E})} = \\frac{d}{d f_{E}} e^{\\cos{(f_{E})}} and - m{(f_{E})} + \\frac{d}{d f_{E}} m{(f_{E})} = - m{(f_{E})} + \\frac{d}{d f_{E}} e^{\\cos{(f_{E})}} and - \\frac{- m{(f_{E})} + \\frac{d}{d f_{E}} m{(f_{E})}}{m{(f_{E})}} = - \\frac{- m{(f_{E})} + \\frac{d}{d f_{E}} e^{\\cos{(f_{E})}}}{m{(f_{E})}} and - \\frac{- m{(f_{E})} + \\frac{d}{d f_{E}} m{(f_{E})}}{m{(f_{E})}} = - \\frac{- m{(f_{E})} - e^{\\cos{(f_{E})}} \\sin{(f_{E})}}{m{(f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('f_E', commutative=True)), exp(cos(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["minus", 2, "Function('m')(Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Derivative(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Derivative(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(Function('m')(Symbol('f_E', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(Function('m')(Symbol('f_E', commutative=True)), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Derivative(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(Function('m')(Symbol('f_E', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('f_E', commutative=True))), sin(Symbol('f_E', commutative=True)))), Pow(Function('m')(Symbol('f_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}}, then derive \\varepsilon_{0}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then obtain e^{f_{\\mathbf{v}}} \\frac{d}{d f_{\\mathbf{v}}} \\varepsilon_{0}{(f_{\\mathbf{v}})} \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} = e^{f_{\\mathbf{v}}} (\\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}})^{2}", "derivation": "\\varepsilon_{0}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} and \\varepsilon_{0}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\frac{d}{d f_{\\mathbf{v}}} \\varepsilon_{0}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} and e^{f_{\\mathbf{v}}} \\frac{d}{d f_{\\mathbf{v}}} \\varepsilon_{0}{(f_{\\mathbf{v}})} \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} = e^{f_{\\mathbf{v}}} (\\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varepsilon_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["times", 3, "Mul(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], "Equality(Mul(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Function('\\\\varepsilon_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), Mul(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given g{(S,\\pi)} = \\cos{(S - \\pi)}, then obtain ((\\int g{(S,\\pi)} d\\pi)^{\\pi})^{S} = ((\\int \\cos{(S - \\pi)} d\\pi)^{\\pi})^{S}", "derivation": "g{(S,\\pi)} = \\cos{(S - \\pi)} and \\int g{(S,\\pi)} d\\pi = \\int \\cos{(S - \\pi)} d\\pi and (\\int g{(S,\\pi)} d\\pi)^{\\pi} = (\\int \\cos{(S - \\pi)} d\\pi)^{\\pi} and ((\\int g{(S,\\pi)} d\\pi)^{\\pi})^{S} = ((\\int \\cos{(S - \\pi)} d\\pi)^{\\pi})^{S}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('g')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Function('g')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Integral(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Integral(Function('g')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Integral(cos(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Symbol('S', commutative=True)))"]]}, {"prompt": "Given y{(F_{g})} = - F_{g} + \\log{(F_{g})}, then obtain (y^{F_{g}}{(F_{g})})^{F_{g}} + y{(F_{g})} = ((- F_{g} + \\log{(F_{g})})^{F_{g}})^{F_{g}} + y{(F_{g})}", "derivation": "y{(F_{g})} = - F_{g} + \\log{(F_{g})} and y^{F_{g}}{(F_{g})} = (- F_{g} + \\log{(F_{g})})^{F_{g}} and (y^{F_{g}}{(F_{g})})^{F_{g}} = ((- F_{g} + \\log{(F_{g})})^{F_{g}})^{F_{g}} and (y^{F_{g}}{(F_{g})})^{F_{g}} + y{(F_{g})} = ((- F_{g} + \\log{(F_{g})})^{F_{g}})^{F_{g}} + y{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('F_g', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('y')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Pow(Function('y')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["add", 3, "Function('y')(Symbol('F_g', commutative=True))"], "Equality(Add(Pow(Pow(Function('y')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('y')(Symbol('F_g', commutative=True))), Add(Pow(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('y')(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(I,\\mu)} = \\cos{(\\mu^{I})}, then obtain - \\mathbf{M}{(I,\\mu)} \\cos{(\\mu^{I})} + \\int \\frac{\\mathbf{M}{(I,\\mu)}}{\\cos{(\\mu^{I})}} d\\mu - \\frac{1}{\\mu} = - \\mathbf{M}{(I,\\mu)} \\cos{(\\mu^{I})} + \\int 1 d\\mu - \\frac{1}{\\mu}", "derivation": "\\mathbf{M}{(I,\\mu)} = \\cos{(\\mu^{I})} and \\mathbf{M}{(I,\\mu)} \\cos{(\\mu^{I})} = \\cos^{2}{(\\mu^{I})} and \\frac{\\mathbf{M}{(I,\\mu)}}{\\cos{(\\mu^{I})}} = 1 and \\int \\frac{\\mathbf{M}{(I,\\mu)}}{\\cos{(\\mu^{I})}} d\\mu = \\int 1 d\\mu and - \\cos^{2}{(\\mu^{I})} + \\int \\frac{\\mathbf{M}{(I,\\mu)}}{\\cos{(\\mu^{I})}} d\\mu - \\frac{1}{\\mu} = - \\cos^{2}{(\\mu^{I})} + \\int 1 d\\mu - \\frac{1}{\\mu} and - \\mathbf{M}{(I,\\mu)} \\cos{(\\mu^{I})} + \\int \\frac{\\mathbf{M}{(I,\\mu)}}{\\cos{(\\mu^{I})}} d\\mu - \\frac{1}{\\mu} = - \\mathbf{M}{(I,\\mu)} \\cos{(\\mu^{I})} + \\int 1 d\\mu - \\frac{1}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))))"], [["times", 1, "cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True)))), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(2)))"], [["divide", 2, "Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 4, "Add(Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(2)), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(2))), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(2))), Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True)))), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu', commutative=True), Symbol('I', commutative=True)))), Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\sigma_p,H)} = \\sin{(\\sigma_p^{H})}, then obtain \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\int (- H + \\mathbf{S}{(\\sigma_p,H)}) d\\sigma_p}{\\int (- H + \\sin{(\\sigma_p^{H})}) d\\sigma_p} = \\frac{d}{d \\sigma_p} 1", "derivation": "\\mathbf{S}{(\\sigma_p,H)} = \\sin{(\\sigma_p^{H})} and - H + \\mathbf{S}{(\\sigma_p,H)} = - H + \\sin{(\\sigma_p^{H})} and \\int (- H + \\mathbf{S}{(\\sigma_p,H)}) d\\sigma_p = \\int (- H + \\sin{(\\sigma_p^{H})}) d\\sigma_p and \\frac{\\int (- H + \\mathbf{S}{(\\sigma_p,H)}) d\\sigma_p}{\\int (- H + \\sin{(\\sigma_p^{H})}) d\\sigma_p} = 1 and \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\int (- H + \\mathbf{S}{(\\sigma_p,H)}) d\\sigma_p}{\\int (- H + \\sin{(\\sigma_p^{H})}) d\\sigma_p} = \\frac{d}{d \\sigma_p} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 3, "Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(f,n)} = f n, then derive \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} = f, then obtain \\sigma_{p}{(f,n)} \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} - e^{\\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)}} = f \\sigma_{p}{(f,n)} - e^{\\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)}}", "derivation": "\\sigma_{p}{(f,n)} = f n and \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} = \\frac{\\partial}{\\partial n} f n and \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} = f and \\sigma_{p}{(f,n)} \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} = f \\sigma_{p}{(f,n)} and \\sigma_{p}{(f,n)} \\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)} - e^{\\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)}} = f \\sigma_{p}{(f,n)} - e^{\\frac{\\partial}{\\partial n} \\sigma_{p}{(f,n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('f', commutative=True))"], [["times", 3, "Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Symbol('f', commutative=True), Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True))))"], [["minus", 4, "exp(Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))), Add(Mul(Symbol('f', commutative=True), Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), exp(Derivative(Function('\\\\sigma_p')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given L{(\\sigma_p)} = \\sin{(\\cos{(\\sigma_p)})} and \\mathbf{M}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} L{(\\sigma_p)}, then derive \\frac{d}{d \\sigma_p} L{(\\sigma_p)} = - \\sin{(\\sigma_p)} \\cos{(\\cos{(\\sigma_p)})}, then obtain \\frac{\\mathbf{M}{(\\sigma_p)}}{\\sigma_p + L{(\\sigma_p)}} = - \\frac{\\sin{(\\sigma_p)} \\cos{(\\cos{(\\sigma_p)})}}{\\sigma_p + L{(\\sigma_p)}}", "derivation": "L{(\\sigma_p)} = \\sin{(\\cos{(\\sigma_p)})} and \\frac{d}{d \\sigma_p} L{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\sin{(\\cos{(\\sigma_p)})} and \\mathbf{M}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} L{(\\sigma_p)} and \\frac{d}{d \\sigma_p} L{(\\sigma_p)} = - \\sin{(\\sigma_p)} \\cos{(\\cos{(\\sigma_p)})} and \\frac{\\frac{d}{d \\sigma_p} L{(\\sigma_p)}}{\\sigma_p + L{(\\sigma_p)}} = - \\frac{\\sin{(\\sigma_p)} \\cos{(\\cos{(\\sigma_p)})}}{\\sigma_p + L{(\\sigma_p)}} and \\frac{\\mathbf{M}{(\\sigma_p)}}{\\sigma_p + L{(\\sigma_p)}} = - \\frac{\\sin{(\\sigma_p)} \\cos{(\\cos{(\\sigma_p)})}}{\\sigma_p + L{(\\sigma_p)}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\sigma_p', commutative=True)), sin(cos(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["divide", 4, "Add(Symbol('\\\\sigma_p', commutative=True), Function('L')(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\sigma_p', commutative=True), Function('L')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Derivative(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Function('L')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Symbol('\\\\sigma_p', commutative=True), Function('L')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Function('L')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain 2 \\mathbf{g}{(\\mathbf{p})} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} = 2 e^{\\mathbf{p}} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}}", "derivation": "\\mathbf{g}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\mathbf{g}{(\\mathbf{p})} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} = e^{\\mathbf{p}} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} and \\mathbf{g}{(\\mathbf{p})} + e^{\\mathbf{p}} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} = 2 e^{\\mathbf{p}} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} and 2 \\mathbf{g}{(\\mathbf{p})} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}} = 2 e^{\\mathbf{p}} + \\frac{e^{\\mathbf{p}}}{\\mathbf{g}{(\\mathbf{p})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Add(exp(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["add", 2, "exp(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(2), exp(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(2), exp(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given H{(\\phi_2,f^{\\prime},l)} = \\phi_2 (f^{\\prime})^{l}, then obtain (f^{\\prime})^{- l} l H{(\\phi_2,f^{\\prime},l)} = (f^{\\prime})^{- l} (l - \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} + \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime}) H{(\\phi_2,f^{\\prime},l)}", "derivation": "H{(\\phi_2,f^{\\prime},l)} = \\phi_2 (f^{\\prime})^{l} and \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime} = \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} and - \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime} = - \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} and 0 = - \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} + \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime} and l = l - \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} + \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime} and (f^{\\prime})^{- l} l H{(\\phi_2,f^{\\prime},l)} = (f^{\\prime})^{- l} (l - \\int \\phi_2 (f^{\\prime})^{l} df^{\\prime} + \\int H{(\\phi_2,f^{\\prime},l)} df^{\\prime}) H{(\\phi_2,f^{\\prime},l)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["add", 3, "Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["add", 4, "Symbol('l', commutative=True)"], "Equality(Symbol('l', commutative=True), Add(Symbol('l', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["times", 5, "Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('l', commutative=True), Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Integral(Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Function('H')(Symbol('\\\\phi_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hat{x})} = \\log{(\\hat{x})} and \\bar{\\h}{(V_{\\mathbf{B}},C_{2})} = C_{2} + V_{\\mathbf{B}}, then obtain \\operatorname{F_{c}}{(\\hat{x})} \\bar{\\h}^{2}{(V_{\\mathbf{B}},C_{2})} = (C_{2} + V_{\\mathbf{B}}) \\operatorname{F_{c}}{(\\hat{x})} \\bar{\\h}{(V_{\\mathbf{B}},C_{2})}", "derivation": "\\operatorname{F_{c}}{(\\hat{x})} = \\log{(\\hat{x})} and \\bar{\\h}{(V_{\\mathbf{B}},C_{2})} = C_{2} + V_{\\mathbf{B}} and \\bar{\\h}^{2}{(V_{\\mathbf{B}},C_{2})} = (C_{2} + V_{\\mathbf{B}}) \\bar{\\h}{(V_{\\mathbf{B}},C_{2})} and \\bar{\\h}^{2}{(V_{\\mathbf{B}},C_{2})} \\log{(\\hat{x})} = (C_{2} + V_{\\mathbf{B}}) \\bar{\\h}{(V_{\\mathbf{B}},C_{2})} \\log{(\\hat{x})} and \\operatorname{F_{c}}{(\\hat{x})} \\bar{\\h}^{2}{(V_{\\mathbf{B}},C_{2})} = (C_{2} + V_{\\mathbf{B}}) \\operatorname{F_{c}}{(\\hat{x})} \\bar{\\h}{(V_{\\mathbf{B}},C_{2})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 2, "Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True)), Integer(2)), Mul(Add(Symbol('C_2', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True))))"], [["times", 3, "log(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True)), Integer(2)), log(Symbol('\\\\hat{x}', commutative=True))), Mul(Add(Symbol('C_2', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('F_c')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True)), Integer(2))), Mul(Add(Symbol('C_2', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('F_c')(Symbol('\\\\hat{x}', commutative=True)), Function('\\\\hbar')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(I,\\chi)} = I \\chi, then obtain \\frac{\\partial}{\\partial \\chi} (I \\chi (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} - I \\chi (I \\chi)^{- \\chi}) = \\frac{\\partial}{\\partial \\chi} (I^{2} \\chi^{2} (I \\chi)^{- \\chi} - I \\chi (I \\chi)^{- \\chi})", "derivation": "\\theta_{1}{(I,\\chi)} = I \\chi and (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} = I \\chi (I \\chi)^{- \\chi} and (I \\chi)^{- \\chi} \\theta_{1}^{2}{(I,\\chi)} = I \\chi (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} and I \\chi (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} = I^{2} \\chi^{2} (I \\chi)^{- \\chi} and I \\chi (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} - I \\chi (I \\chi)^{- \\chi} = I^{2} \\chi^{2} (I \\chi)^{- \\chi} - I \\chi (I \\chi)^{- \\chi} and \\frac{\\partial}{\\partial \\chi} (I \\chi (I \\chi)^{- \\chi} \\theta_{1}{(I,\\chi)} - I \\chi (I \\chi)^{- \\chi}) = \\frac{\\partial}{\\partial \\chi} (I^{2} \\chi^{2} (I \\chi)^{- \\chi} - I \\chi (I \\chi)^{- \\chi})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"], [["times", 1, "Mul(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"], [["minus", 4, "Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], "Equality(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))), Add(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))))"], [["differentiate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\theta_1')(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\theta_2)} = \\sin{(\\theta_2)}, then obtain \\int (\\frac{\\operatorname{A_{x}}{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2}) d\\theta_2 = \\hat{H}_{\\lambda} + \\log{(\\theta_2)} + \\operatorname{Si}{(\\theta_2)}", "derivation": "\\operatorname{A_{x}}{(\\theta_2)} = \\sin{(\\theta_2)} and \\frac{\\operatorname{A_{x}}{(\\theta_2)}}{\\theta_2} = \\frac{\\sin{(\\theta_2)}}{\\theta_2} and \\frac{\\operatorname{A_{x}}{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2} = \\frac{\\sin{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2} and \\int (\\frac{\\operatorname{A_{x}}{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2}) d\\theta_2 = \\int (\\frac{\\sin{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2}) d\\theta_2 and \\int (\\frac{\\operatorname{A_{x}}{(\\theta_2)}}{\\theta_2} + \\frac{1}{\\theta_2}) d\\theta_2 = \\hat{H}_{\\lambda} + \\log{(\\theta_2)} + \\operatorname{Si}{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta_2', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta_2', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\theta_2', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), log(Symbol('\\\\theta_2', commutative=True)), Si(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)} = \\dot{\\mathbf{r}}^{q}, then obtain \\frac{\\partial}{\\partial q} (q + \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)})^{\\dot{\\mathbf{r}}} + 1 = \\frac{\\partial}{\\partial q} (\\dot{\\mathbf{r}}^{q} + q)^{\\dot{\\mathbf{r}}} + 1", "derivation": "\\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)} = \\dot{\\mathbf{r}}^{q} and q + \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)} = \\dot{\\mathbf{r}}^{q} + q and (q + \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)})^{\\dot{\\mathbf{r}}} = (\\dot{\\mathbf{r}}^{q} + q)^{\\dot{\\mathbf{r}}} and \\frac{\\partial}{\\partial q} (q + \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)})^{\\dot{\\mathbf{r}}} = \\frac{\\partial}{\\partial q} (\\dot{\\mathbf{r}}^{q} + q)^{\\dot{\\mathbf{r}}} and \\frac{\\partial}{\\partial q} (q + \\Psi_{\\lambda}{(\\dot{\\mathbf{r}},q)})^{\\dot{\\mathbf{r}}} + 1 = \\frac{\\partial}{\\partial q} (\\dot{\\mathbf{r}}^{q} + q)^{\\dot{\\mathbf{r}}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True))), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Add(Symbol('q', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('q', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(Pow(Add(Symbol('q', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\lambda,t_{1})} = \\lambda t_{1}, then obtain (t_{1} + \\iint (\\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})}) dt_{1} dt_{1})^{t_{1}} = (t_{1} + \\iint (\\lambda t_{1} + \\lambda) dt_{1} dt_{1})^{t_{1}}", "derivation": "\\operatorname{v_{y}}{(\\lambda,t_{1})} = \\lambda t_{1} and \\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})} = \\lambda t_{1} + \\lambda and \\int (\\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})}) dt_{1} = \\int (\\lambda t_{1} + \\lambda) dt_{1} and \\iint (\\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})}) dt_{1} dt_{1} = \\iint (\\lambda t_{1} + \\lambda) dt_{1} dt_{1} and t_{1} + \\iint (\\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})}) dt_{1} dt_{1} = t_{1} + \\iint (\\lambda t_{1} + \\lambda) dt_{1} dt_{1} and (t_{1} + \\iint (\\lambda + \\operatorname{v_{y}}{(\\lambda,t_{1})}) dt_{1} dt_{1})^{t_{1}} = (t_{1} + \\iint (\\lambda t_{1} + \\lambda) dt_{1} dt_{1})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True))), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\lambda', commutative=True), Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["integrate", 3, "Symbol('t_1', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\lambda', commutative=True), Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["add", 4, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Add(Symbol('t_1', commutative=True), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"], [["power", 5, "Symbol('t_1', commutative=True)"], "Equality(Pow(Add(Symbol('t_1', commutative=True), Integral(Add(Symbol('\\\\lambda', commutative=True), Function('v_y')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(P_{e})} = \\sin{(P_{e})}, then derive \\frac{d}{d P_{e}} \\operatorname{P_{g}}{(P_{e})} = \\cos{(P_{e})}, then obtain \\iint \\cos{(P_{e})} dP_{e} dP_{e} = \\iint \\frac{d}{d P_{e}} \\sin{(P_{e})} dP_{e} dP_{e}", "derivation": "\\operatorname{P_{g}}{(P_{e})} = \\sin{(P_{e})} and \\frac{d}{d P_{e}} \\operatorname{P_{g}}{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(P_{e})} and \\frac{d}{d P_{e}} \\operatorname{P_{g}}{(P_{e})} = \\cos{(P_{e})} and \\cos{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(P_{e})} and \\int \\cos{(P_{e})} dP_{e} = \\int \\frac{d}{d P_{e}} \\sin{(P_{e})} dP_{e} and \\iint \\cos{(P_{e})} dP_{e} dP_{e} = \\iint \\frac{d}{d P_{e}} \\sin{(P_{e})} dP_{e} dP_{e}", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), cos(Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('P_e', commutative=True)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('P_e', commutative=True))))"], [["integrate", 5, "Symbol('P_e', commutative=True)"], "Equality(Integral(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\pi)} = \\sin{(\\pi)} and \\varepsilon{(\\pi)} = \\cos{(\\pi)}, then derive -1 = \\cos{(\\pi)} - \\frac{d}{d \\pi} \\mathbf{J}_M{(\\pi)} - 1, then derive -1 = \\varepsilon{(\\pi)} - \\cos{(\\pi)} - 1, then obtain \\int (-1) d\\pi = \\int (\\varepsilon{(\\pi)} - \\cos{(\\pi)} - 1) d\\pi", "derivation": "\\mathbf{J}_M{(\\pi)} = \\sin{(\\pi)} and - \\pi = - \\pi - \\mathbf{J}_M{(\\pi)} + \\sin{(\\pi)} and \\frac{d}{d \\pi} - \\pi = \\frac{d}{d \\pi} (- \\pi - \\mathbf{J}_M{(\\pi)} + \\sin{(\\pi)}) and -1 = \\cos{(\\pi)} - \\frac{d}{d \\pi} \\mathbf{J}_M{(\\pi)} - 1 and \\varepsilon{(\\pi)} = \\cos{(\\pi)} and -1 = \\varepsilon{(\\pi)} - \\frac{d}{d \\pi} \\mathbf{J}_M{(\\pi)} - 1 and -1 = \\varepsilon{(\\pi)} - \\frac{d}{d \\pi} \\sin{(\\pi)} - 1 and -1 = \\varepsilon{(\\pi)} - \\cos{(\\pi)} - 1 and \\int (-1) d\\pi = \\int (\\varepsilon{(\\pi)} - \\cos{(\\pi)} - 1) d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True))), sin(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True))), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(-1), Add(cos(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(-1), Add(Function('\\\\varepsilon')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(-1), Add(Function('\\\\varepsilon')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Integer(-1)))"], [["evaluate_derivatives", 7], "Equality(Integer(-1), Add(Function('\\\\varepsilon')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))), Integer(-1)))"], [["integrate", 8, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Function('\\\\varepsilon')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(v_{t})} = e^{v_{t}}, then obtain (v_{t} + e^{v_{t}}) \\frac{d}{d v_{t}} \\operatorname{z^{*}}{(v_{t})} = (v_{t} + e^{v_{t}}) \\frac{d}{d v_{t}} e^{v_{t}}", "derivation": "\\operatorname{z^{*}}{(v_{t})} = e^{v_{t}} and v_{t} + \\operatorname{z^{*}}{(v_{t})} = v_{t} + e^{v_{t}} and \\frac{d}{d v_{t}} \\operatorname{z^{*}}{(v_{t})} = \\frac{d}{d v_{t}} e^{v_{t}} and (v_{t} + \\operatorname{z^{*}}{(v_{t})}) \\frac{d}{d v_{t}} \\operatorname{z^{*}}{(v_{t})} = (v_{t} + \\operatorname{z^{*}}{(v_{t})}) \\frac{d}{d v_{t}} e^{v_{t}} and (v_{t} + e^{v_{t}}) \\frac{d}{d v_{t}} \\operatorname{z^{*}}{(v_{t})} = (v_{t} + e^{v_{t}}) \\frac{d}{d v_{t}} e^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["add", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Symbol('v_t', commutative=True), Function('z^*')(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["times", 3, "Add(Symbol('v_t', commutative=True), Function('z^*')(Symbol('v_t', commutative=True)))"], "Equality(Mul(Add(Symbol('v_t', commutative=True), Function('z^*')(Symbol('v_t', commutative=True))), Derivative(Function('z^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Mul(Add(Symbol('v_t', commutative=True), Function('z^*')(Symbol('v_t', commutative=True))), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Derivative(Function('z^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Mul(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} = \\frac{\\cos{(\\mathbf{E})}}{\\eta}, then obtain 2 \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} = \\frac{2 \\cos{(\\mathbf{E})}}{\\eta}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{E},\\eta)} = \\frac{\\cos{(\\mathbf{E})}}{\\eta} and 2 \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} = \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} + \\frac{\\cos{(\\mathbf{E})}}{\\eta} and \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} + \\frac{\\cos{(\\mathbf{E})}}{\\eta} = \\frac{2 \\cos{(\\mathbf{E})}}{\\eta} and 2 \\operatorname{F_{c}}{(\\mathbf{E},\\eta)} = \\frac{2 \\cos{(\\mathbf{E})}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 1, "Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(J_{\\varepsilon})} = \\cos{(e^{J_{\\varepsilon}})} and s{(\\phi_2,E)} = \\sin^{E}{(\\phi_2)}, then obtain a{(p)} \\sin{(\\int s{(\\phi_2,E)} d\\phi_2)} + \\operatorname{f^{\\prime}}{(J_{\\varepsilon})} = a{(p)} \\sin{(\\int \\sin^{E}{(\\phi_2)} d\\phi_2)} + \\operatorname{f^{\\prime}}{(J_{\\varepsilon})}", "derivation": "\\operatorname{f^{\\prime}}{(J_{\\varepsilon})} = \\cos{(e^{J_{\\varepsilon}})} and s{(\\phi_2,E)} = \\sin^{E}{(\\phi_2)} and \\int s{(\\phi_2,E)} d\\phi_2 = \\int \\sin^{E}{(\\phi_2)} d\\phi_2 and \\sin{(\\int s{(\\phi_2,E)} d\\phi_2)} = \\sin{(\\int \\sin^{E}{(\\phi_2)} d\\phi_2)} and a{(p)} \\sin{(\\int s{(\\phi_2,E)} d\\phi_2)} = a{(p)} \\sin{(\\int \\sin^{E}{(\\phi_2)} d\\phi_2)} and a{(p)} \\sin{(\\int s{(\\phi_2,E)} d\\phi_2)} + \\cos{(e^{J_{\\varepsilon}})} = a{(p)} \\sin{(\\int \\sin^{E}{(\\phi_2)} d\\phi_2)} + \\cos{(e^{J_{\\varepsilon}})} and a{(p)} \\sin{(\\int s{(\\phi_2,E)} d\\phi_2)} + \\operatorname{f^{\\prime}}{(J_{\\varepsilon})} = a{(p)} \\sin{(\\int \\sin^{E}{(\\phi_2)} d\\phi_2)} + \\operatorname{f^{\\prime}}{(J_{\\varepsilon})}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["get_premise", "Equality(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), sin(Integral(Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["times", 4, "Function('a')(Symbol('p', commutative=True))"], "Equality(Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["add", 5, "cos(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), cos(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), cos(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Function('f^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Function('a')(Symbol('p', commutative=True)), sin(Integral(Pow(sin(Symbol('\\\\phi_2', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Function('f^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given i{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then derive \\int i{(\\mathbf{D})} d\\mathbf{D} = \\rho_f - \\cos{(\\mathbf{D})}, then obtain \\int (\\rho_f - \\cos{(\\mathbf{D})})^{\\mathbf{D}} d\\mathbf{D} = \\int (\\int \\sin{(\\mathbf{D})} d\\mathbf{D})^{\\mathbf{D}} d\\mathbf{D}", "derivation": "i{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\int i{(\\mathbf{D})} d\\mathbf{D} = \\int \\sin{(\\mathbf{D})} d\\mathbf{D} and (\\int i{(\\mathbf{D})} d\\mathbf{D})^{\\mathbf{D}} = (\\int \\sin{(\\mathbf{D})} d\\mathbf{D})^{\\mathbf{D}} and \\int i{(\\mathbf{D})} d\\mathbf{D} = \\rho_f - \\cos{(\\mathbf{D})} and (\\rho_f - \\cos{(\\mathbf{D})})^{\\mathbf{D}} = (\\int \\sin{(\\mathbf{D})} d\\mathbf{D})^{\\mathbf{D}} and \\int (\\rho_f - \\cos{(\\mathbf{D})})^{\\mathbf{D}} d\\mathbf{D} = \\int (\\int \\sin{(\\mathbf{D})} d\\mathbf{D})^{\\mathbf{D}} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('i')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Integral(Function('i')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('i')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Pow(Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given q{(\\mathbf{s})} = \\sin{(\\mathbf{s})}, then obtain 27 q^{3}{(\\mathbf{s})} = (2 q{(\\mathbf{s})} + \\sin{(\\mathbf{s})})^{3}", "derivation": "q{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and 2 q{(\\mathbf{s})} = q{(\\mathbf{s})} + \\sin{(\\mathbf{s})} and 3 q{(\\mathbf{s})} = 2 q{(\\mathbf{s})} + \\sin{(\\mathbf{s})} and 3 q{(\\mathbf{s})} = q{(\\mathbf{s})} + 2 \\sin{(\\mathbf{s})} and 27 q^{3}{(\\mathbf{s})} = (q{(\\mathbf{s})} + 2 \\sin{(\\mathbf{s})})^{3} and q{(\\mathbf{s})} + 2 \\sin{(\\mathbf{s})} = 2 q{(\\mathbf{s})} + \\sin{(\\mathbf{s})} and 27 q^{3}{(\\mathbf{s})} = (2 q{(\\mathbf{s})} + \\sin{(\\mathbf{s})})^{3}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 1, "Function('q')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(2), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 2, "Function('q')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(3), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(2), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["power", 4, 3], "Equality(Mul(Integer(27), Pow(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(3))), Pow(Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{s}', commutative=True)))), Integer(3)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(2), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(27), Pow(Function('q')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(3))), Pow(Add(Mul(Integer(2), Function('q')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))), Integer(3)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(F_{c},y)} = F_{c}^{y}, then derive \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},y)} = \\frac{F_{c}^{y} y}{F_{c}}, then obtain \\frac{\\partial}{\\partial F_{c}} F_{c}^{y} + \\frac{F_{c}^{y} y}{F_{c}} = \\frac{2 F_{c}^{y} y}{F_{c}}", "derivation": "\\operatorname{t_{1}}{(F_{c},y)} = F_{c}^{y} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},y)} = \\frac{\\partial}{\\partial F_{c}} F_{c}^{y} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},y)} = \\frac{F_{c}^{y} y}{F_{c}} and \\frac{\\partial}{\\partial F_{c}} F_{c}^{y} = \\frac{F_{c}^{y} y}{F_{c}} and \\frac{\\partial}{\\partial F_{c}} F_{c}^{y} + \\frac{F_{c}^{y} y}{F_{c}} = \\frac{2 F_{c}^{y} y}{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["add", 4, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Derivative(Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(2), Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\rho_b,n)} = \\rho_b + n, then derive 2 \\frac{\\partial}{\\partial n} \\operatorname{t_{2}}{(\\rho_b,n)} = \\frac{\\partial}{\\partial n} \\operatorname{t_{2}}{(\\rho_b,n)} + 1, then obtain \\cos{(2 \\frac{\\partial}{\\partial n} (\\rho_b + n))} = \\cos{(\\frac{\\partial}{\\partial n} (\\rho_b + n) + 1)}", "derivation": "\\operatorname{t_{2}}{(\\rho_b,n)} = \\rho_b + n and 2 \\operatorname{t_{2}}{(\\rho_b,n)} = \\rho_b + n + \\operatorname{t_{2}}{(\\rho_b,n)} and \\frac{\\partial}{\\partial n} \\operatorname{t_{2}}{(\\rho_b,n)} = \\frac{\\partial}{\\partial n} (\\rho_b + n) and \\frac{\\partial}{\\partial n} 2 \\operatorname{t_{2}}{(\\rho_b,n)} = \\frac{\\partial}{\\partial n} (\\rho_b + n + \\operatorname{t_{2}}{(\\rho_b,n)}) and 2 \\frac{\\partial}{\\partial n} \\operatorname{t_{2}}{(\\rho_b,n)} = \\frac{\\partial}{\\partial n} \\operatorname{t_{2}}{(\\rho_b,n)} + 1 and 2 \\frac{\\partial}{\\partial n} (\\rho_b + n) = \\frac{\\partial}{\\partial n} (\\rho_b + n) + 1 and \\cos{(2 \\frac{\\partial}{\\partial n} (\\rho_b + n))} = \\cos{(\\frac{\\partial}{\\partial n} (\\rho_b + n) + 1)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True), Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Derivative(Function('t_2')(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)))"], [["cos", 6], "Equality(cos(Mul(Integer(2), Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), cos(Add(Derivative(Add(Symbol('\\\\rho_b', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(f^{\\prime})} = \\sin{(f^{\\prime})}, then obtain \\operatorname{y^{\\prime}}^{f^{\\prime}}{(f^{\\prime})} = (- \\operatorname{y^{\\prime}}{(f^{\\prime})} + 2 \\sin{(f^{\\prime})})^{f^{\\prime}}", "derivation": "\\operatorname{y^{\\prime}}{(f^{\\prime})} = \\sin{(f^{\\prime})} and 0 = - \\operatorname{y^{\\prime}}{(f^{\\prime})} + \\sin{(f^{\\prime})} and \\sin{(f^{\\prime})} = - \\operatorname{y^{\\prime}}{(f^{\\prime})} + 2 \\sin{(f^{\\prime})} and \\operatorname{y^{\\prime}}^{f^{\\prime}}{(f^{\\prime})} = \\sin^{f^{\\prime}}{(f^{\\prime})} and \\operatorname{y^{\\prime}}^{f^{\\prime}}{(f^{\\prime})} = (- \\operatorname{y^{\\prime}}{(f^{\\prime})} + 2 \\sin{(f^{\\prime})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True))), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "sin(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(sin(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), sin(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(M,Q)} = \\frac{Q}{M}, then obtain \\sin^{M}{(\\int \\frac{\\partial}{\\partial Q} \\int - \\operatorname{C_{1}}{(M,Q)} dM dQ)} = \\sin^{M}{(\\int \\frac{\\partial}{\\partial Q} \\int - \\frac{Q}{M} dM dQ)}", "derivation": "\\operatorname{C_{1}}{(M,Q)} = \\frac{Q}{M} and - \\operatorname{C_{1}}{(M,Q)} = - \\frac{Q}{M} and \\int - \\operatorname{C_{1}}{(M,Q)} dM = \\int - \\frac{Q}{M} dM and \\frac{\\partial}{\\partial Q} \\int - \\operatorname{C_{1}}{(M,Q)} dM = \\frac{\\partial}{\\partial Q} \\int - \\frac{Q}{M} dM and \\int \\frac{\\partial}{\\partial Q} \\int - \\operatorname{C_{1}}{(M,Q)} dM dQ = \\int \\frac{\\partial}{\\partial Q} \\int - \\frac{Q}{M} dM dQ and \\sin{(\\int \\frac{\\partial}{\\partial Q} \\int - \\operatorname{C_{1}}{(M,Q)} dM dQ)} = \\sin{(\\int \\frac{\\partial}{\\partial Q} \\int - \\frac{Q}{M} dM dQ)} and \\sin^{M}{(\\int \\frac{\\partial}{\\partial Q} \\int - \\operatorname{C_{1}}{(M,Q)} dM dQ)} = \\sin^{M}{(\\int \\frac{\\partial}{\\partial Q} \\int - \\frac{Q}{M} dM dQ)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Integral(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Integral(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))"], [["sin", 5], "Equality(sin(Integral(Derivative(Integral(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), sin(Integral(Derivative(Integral(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))))"], [["power", 6, "Symbol('M', commutative=True)"], "Equality(Pow(sin(Integral(Derivative(Integral(Mul(Integer(-1), Function('C_1')(Symbol('M', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), Symbol('M', commutative=True)), Pow(sin(Integral(Derivative(Integral(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{E},n)} = \\mathbf{E} n, then obtain (\\mathbf{E} + 2 \\mathbf{H}{(\\mathbf{E},n)}) (\\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)}) = (\\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)})^{2}", "derivation": "\\mathbf{H}{(\\mathbf{E},n)} = \\mathbf{E} n and \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)} = \\mathbf{E} n + \\mathbf{E} and \\mathbf{E} + 2 \\mathbf{H}{(\\mathbf{E},n)} = \\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)} and \\mathbf{E} + 2 \\mathbf{H}{(\\mathbf{E},n)} = 2 \\mathbf{E} n + \\mathbf{E} and 2 \\mathbf{E} n + \\mathbf{E} = \\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)} and (\\mathbf{E} + 2 \\mathbf{H}{(\\mathbf{E},n)}) (2 \\mathbf{E} n + \\mathbf{E}) = (2 \\mathbf{E} n + \\mathbf{E})^{2} and (\\mathbf{E} + 2 \\mathbf{H}{(\\mathbf{E},n)}) (\\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)}) = (\\mathbf{E} n + \\mathbf{E} + \\mathbf{H}{(\\mathbf{E},n)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 2, "Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True))))"], [["times", 4, "Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)))), Pow(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('n', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\nabla{(k,\\mathbf{v})} = k \\cos{(\\mathbf{v})} and \\hat{\\mathbf{r}}{(k,\\mathbf{v})} = k \\cos{(\\mathbf{v})}, then obtain \\sin{(\\nabla{(k,\\mathbf{v})})} = \\sin{(\\hat{\\mathbf{r}}{(k,\\mathbf{v})})}", "derivation": "\\nabla{(k,\\mathbf{v})} = k \\cos{(\\mathbf{v})} and \\sin{(\\nabla{(k,\\mathbf{v})})} = \\sin{(k \\cos{(\\mathbf{v})})} and \\hat{\\mathbf{r}}{(k,\\mathbf{v})} = k \\cos{(\\mathbf{v})} and \\sin{(\\nabla{(k,\\mathbf{v})})} = \\sin{(\\hat{\\mathbf{r}}{(k,\\mathbf{v})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('k', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), sin(Mul(Symbol('k', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('k', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(sin(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), sin(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(E)} = e^{E} and \\varphi^{*}{(E)} = \\frac{e^{E}}{E}, then obtain \\frac{2 e^{E}}{E} = \\frac{\\hat{X}{(E)}}{E} + \\frac{e^{E}}{E}", "derivation": "\\hat{X}{(E)} = e^{E} and \\frac{\\hat{X}{(E)}}{E} = \\frac{e^{E}}{E} and \\varphi^{*}{(E)} = \\frac{e^{E}}{E} and \\varphi^{*}{(E)} = \\frac{\\hat{X}{(E)}}{E} and 2 \\varphi^{*}{(E)} = \\varphi^{*}{(E)} + \\frac{\\hat{X}{(E)}}{E} and \\frac{2 e^{E}}{E} = \\frac{\\hat{X}{(E)}}{E} + \\frac{e^{E}}{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["divide", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), exp(Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), exp(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varphi^*')(Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E', commutative=True))))"], [["add", 4, "Function('\\\\varphi^*')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('E', commutative=True))), Add(Function('\\\\varphi^*')(Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), exp(Symbol('E', commutative=True))), Add(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), exp(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given h{(i,s)} = i + s, then obtain \\iint (i + s + h{(i,s)}) di di = \\iint (2 i + 2 s) di di", "derivation": "h{(i,s)} = i + s and i + s + h{(i,s)} = 2 i + 2 s and \\int (i + s + h{(i,s)}) di = \\int (2 i + 2 s) di and \\iint (i + s + h{(i,s)}) di di = \\iint (2 i + 2 s) di di", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Add(Symbol('i', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Add(Symbol('i', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Symbol('i', commutative=True), Symbol('s', commutative=True), Function('h')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Add(Symbol('i', commutative=True), Symbol('s', commutative=True), Function('h')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Add(Symbol('i', commutative=True), Symbol('s', commutative=True), Function('h')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('s', commutative=True))), Tuple(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given H{(W)} = e^{W} and \\mathbf{M}{(W)} = H{(W)} e^{- W}, then obtain \\mathbf{M}{(W)} = 1", "derivation": "H{(W)} = e^{W} and H{(W)} e^{- W} = 1 and \\mathbf{M}{(W)} = H{(W)} e^{- W} and \\mathbf{M}{(W)} = 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["divide", 1, "exp(Symbol('W', commutative=True))"], "Equality(Mul(Function('H')(Symbol('W', commutative=True)), exp(Mul(Integer(-1), Symbol('W', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Mul(Function('H')(Symbol('W', commutative=True)), exp(Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{F}{(C_{1},W)} = C_{1} W and \\dot{x}{(C_{1},W)} = W \\mathbf{F}{(C_{1},W)}, then obtain - (- \\dot{x}{(C_{1},W)} + \\frac{\\mathbf{F}{(C_{1},W)}}{W}) \\dot{x}{(C_{1},W)} = - (C_{1} - \\dot{x}{(C_{1},W)}) \\dot{x}{(C_{1},W)}", "derivation": "\\mathbf{F}{(C_{1},W)} = C_{1} W and W \\mathbf{F}{(C_{1},W)} = C_{1} W^{2} and \\frac{\\mathbf{F}{(C_{1},W)}}{W} = C_{1} and - W \\mathbf{F}{(C_{1},W)} + \\frac{\\mathbf{F}{(C_{1},W)}}{W} = C_{1} - W \\mathbf{F}{(C_{1},W)} and \\dot{x}{(C_{1},W)} = W \\mathbf{F}{(C_{1},W)} and - \\dot{x}{(C_{1},W)} + \\frac{\\mathbf{F}{(C_{1},W)}}{W} = C_{1} - \\dot{x}{(C_{1},W)} and - (- \\dot{x}{(C_{1},W)} + \\frac{\\mathbf{F}{(C_{1},W)}}{W}) \\dot{x}{(C_{1},W)} = - (C_{1} - \\dot{x}{(C_{1},W)}) \\dot{x}{(C_{1},W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('C_1', commutative=True), Pow(Symbol('W', commutative=True), Integer(2))))"], [["divide", 2, "Pow(Symbol('W', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Symbol('C_1', commutative=True))"], [["minus", 3, "Mul(Symbol('W', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))))"], [["times", 6, "Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True)))), Function('\\\\dot{x}')(Symbol('C_1', commutative=True), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(z)} = \\cos{(z)}, then obtain \\frac{d}{d z} \\frac{\\operatorname{a^{\\dagger}}{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}} = \\frac{d}{d z} \\frac{\\cos{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}}", "derivation": "\\operatorname{a^{\\dagger}}{(z)} = \\cos{(z)} and \\operatorname{a^{\\dagger}}^{2}{(z)} = \\operatorname{a^{\\dagger}}{(z)} \\cos{(z)} and \\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)} = \\operatorname{a^{\\dagger}}{(z)} \\cos{(z)} - \\cos{(z)} and \\frac{\\operatorname{a^{\\dagger}}{(z)}}{\\operatorname{a^{\\dagger}}{(z)} \\cos{(z)} - \\cos{(z)}} = \\frac{\\cos{(z)}}{\\operatorname{a^{\\dagger}}{(z)} \\cos{(z)} - \\cos{(z)}} and \\frac{\\operatorname{a^{\\dagger}}{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}} = \\frac{\\cos{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}} and \\frac{d}{d z} \\frac{\\operatorname{a^{\\dagger}}{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}} = \\frac{d}{d z} \\frac{\\cos{(z)}}{\\operatorname{a^{\\dagger}}^{2}{(z)} - \\cos{(z)}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["times", 1, "Function('a^{\\\\dagger}')(Symbol('z', commutative=True))"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))))"], [["minus", 2, "cos(Symbol('z', commutative=True))"], "Equality(Add(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Add(Mul(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["divide", 1, "Add(Mul(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Mul(Integer(-1), cos(Symbol('z', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True))), Mul(Pow(Add(Mul(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), cos(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True))), Mul(Pow(Add(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), cos(Symbol('z', commutative=True))))"], [["differentiate", 5, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(-1)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\hat{H})} = e^{\\hat{H}}, then obtain \\frac{d}{d \\hat{H}} \\frac{\\hat{H} e^{2 \\hat{H}}}{\\operatorname{v_{z}}{(\\hat{H})}} = \\frac{d}{d \\hat{H}} \\hat{H} e^{\\hat{H}}", "derivation": "\\operatorname{v_{z}}{(\\hat{H})} = e^{\\hat{H}} and \\hat{H} \\operatorname{v_{z}}{(\\hat{H})} = \\hat{H} e^{\\hat{H}} and e^{\\hat{H}} = \\frac{e^{2 \\hat{H}}}{\\operatorname{v_{z}}{(\\hat{H})}} and \\hat{H} \\operatorname{v_{z}}{(\\hat{H})} = \\frac{\\hat{H} e^{2 \\hat{H}}}{\\operatorname{v_{z}}{(\\hat{H})}} and \\frac{\\hat{H} e^{2 \\hat{H}}}{\\operatorname{v_{z}}{(\\hat{H})}} = \\hat{H} e^{\\hat{H}} and \\frac{d}{d \\hat{H}} \\frac{\\hat{H} e^{2 \\hat{H}}}{\\operatorname{v_{z}}{(\\hat{H})}} = \\frac{d}{d \\hat{H}} \\hat{H} e^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('v_z')(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 1, "Mul(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))))"], "Equality(exp(Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('v_z')(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Function('v_z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(U,\\chi)} = U \\chi, then obtain 0 = \\chi (2 U \\chi - 2 Z{(U,\\chi)})", "derivation": "Z{(U,\\chi)} = U \\chi and 0 = U \\chi - Z{(U,\\chi)} and 0 = \\chi (U \\chi - Z{(U,\\chi)}) and U \\chi = 2 U \\chi - Z{(U,\\chi)} and 0 = \\chi (2 U \\chi - 2 Z{(U,\\chi)})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["times", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\chi', commutative=True), Add(Mul(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True))))))"], [["add", 2, "Mul(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(2), Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Symbol('\\\\chi', commutative=True), Add(Mul(Integer(2), Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Integer(2), Function('Z')(Symbol('U', commutative=True), Symbol('\\\\chi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\phi_1,l)} = \\phi_1 l, then obtain \\frac{\\frac{\\partial}{\\partial l} \\frac{\\operatorname{v_{y}}{(\\phi_1,l)}}{\\phi_1 l}}{\\phi_1} = \\frac{\\frac{d}{d l} 1}{\\phi_1}", "derivation": "\\operatorname{v_{y}}{(\\phi_1,l)} = \\phi_1 l and \\frac{\\operatorname{v_{y}}{(\\phi_1,l)}}{\\phi_1 l} = 1 and \\frac{\\partial}{\\partial l} \\frac{\\operatorname{v_{y}}{(\\phi_1,l)}}{\\phi_1 l} = \\frac{d}{d l} 1 and \\frac{\\frac{\\partial}{\\partial l} \\frac{\\operatorname{v_{y}}{(\\phi_1,l)}}{\\phi_1 l}}{\\phi_1} = \\frac{\\frac{d}{d l} 1}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(h,\\rho)} = \\log{(\\rho h)}, then obtain (\\frac{\\log{(\\rho h)}}{h})^{h} + \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{\\rho} = (\\frac{\\log{(\\rho h)}}{h})^{h} + \\frac{\\log{(\\rho h)}}{\\rho}", "derivation": "\\hat{H}_{\\lambda}{(h,\\rho)} = \\log{(\\rho h)} and \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{\\rho h} = \\frac{\\log{(\\rho h)}}{\\rho h} and \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{h} = \\frac{\\log{(\\rho h)}}{h} and \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{\\rho} = \\frac{\\log{(\\rho h)}}{\\rho} and (\\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{h})^{h} + \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{\\rho} = (\\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{h})^{h} + \\frac{\\log{(\\rho h)}}{\\rho} and (\\frac{\\log{(\\rho h)}}{h})^{h} + \\frac{\\hat{H}_{\\lambda}{(h,\\rho)}}{\\rho} = (\\frac{\\log{(\\rho h)}}{h})^{h} + \\frac{\\log{(\\rho h)}}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True)))))"], [["times", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True)))))"], [["divide", 2, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True)))))"], [["add", 4, "Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('h', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\rho', commutative=True), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\Psi{(A_{2},\\hbar)} = \\sin{(A_{2} \\hbar)}, then derive - \\frac{A_{2} \\hbar \\Psi{(A_{2},\\hbar)} \\cos{(A_{2} \\hbar)}}{\\sin^{2}{(A_{2} \\hbar)}} + \\frac{A_{2} \\frac{\\partial}{\\partial A_{2}} \\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} + \\frac{\\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} = 1, then obtain - \\frac{A_{2} \\hbar \\cos{(A_{2} \\hbar)}}{\\sin{(A_{2} \\hbar)}} + \\frac{A_{2} \\frac{\\partial}{\\partial A_{2}} \\sin{(A_{2} \\hbar)}}{\\sin{(A_{2} \\hbar)}} + 1 = 1", "derivation": "\\Psi{(A_{2},\\hbar)} = \\sin{(A_{2} \\hbar)} and \\frac{A_{2} \\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} = A_{2} and \\frac{\\partial}{\\partial A_{2}} \\frac{A_{2} \\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} = \\frac{d}{d A_{2}} A_{2} and - \\frac{A_{2} \\hbar \\Psi{(A_{2},\\hbar)} \\cos{(A_{2} \\hbar)}}{\\sin^{2}{(A_{2} \\hbar)}} + \\frac{A_{2} \\frac{\\partial}{\\partial A_{2}} \\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} + \\frac{\\Psi{(A_{2},\\hbar)}}{\\sin{(A_{2} \\hbar)}} = 1 and - \\frac{A_{2} \\hbar \\cos{(A_{2} \\hbar)}}{\\sin{(A_{2} \\hbar)}} + \\frac{A_{2} \\frac{\\partial}{\\partial A_{2}} \\sin{(A_{2} \\hbar)}}{\\sin{(A_{2} \\hbar)}} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))))"], "Equality(Mul(Symbol('A_2', commutative=True), Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1))), Symbol('A_2', commutative=True))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('A_2', commutative=True), Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Symbol('A_2', commutative=True), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-2)), cos(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('A_2', commutative=True), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Derivative(Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Function('\\\\Psi')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), cos(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('A_2', commutative=True), Pow(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Derivative(sin(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(v_{t})} = \\log{(v_{t})}, then obtain 1 = (((\\int \\operatorname{g_{\\varepsilon}}{(v_{t})} dv_{t})^{v_{t}})^{- v_{t}}) ((\\int \\log{(v_{t})} dv_{t})^{v_{t}})^{v_{t}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(v_{t})} = \\log{(v_{t})} and \\int \\operatorname{g_{\\varepsilon}}{(v_{t})} dv_{t} = \\int \\log{(v_{t})} dv_{t} and (\\int \\operatorname{g_{\\varepsilon}}{(v_{t})} dv_{t})^{v_{t}} = (\\int \\log{(v_{t})} dv_{t})^{v_{t}} and ((\\int \\operatorname{g_{\\varepsilon}}{(v_{t})} dv_{t})^{v_{t}})^{v_{t}} = ((\\int \\log{(v_{t})} dv_{t})^{v_{t}})^{v_{t}} and 1 = (((\\int \\operatorname{g_{\\varepsilon}}{(v_{t})} dv_{t})^{v_{t}})^{- v_{t}}) ((\\int \\log{(v_{t})} dv_{t})^{v_{t}})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(Pow(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["divide", 4, "Pow(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Pow(Pow(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\chi{(Q)} = \\log{(Q)}, then derive \\log{(Q)} \\frac{d}{d Q} \\chi{(Q)} + \\frac{\\chi{(Q)}}{Q} = \\frac{2 \\log{(Q)}}{Q}, then obtain - \\frac{\\log{(Q)} \\frac{d}{d Q} \\chi{(Q)} + \\frac{\\chi{(Q)}}{Q}}{\\log{(Q)}^{2}} = - \\frac{2}{Q \\log{(Q)}}", "derivation": "\\chi{(Q)} = \\log{(Q)} and \\chi{(Q)} \\log{(Q)} = \\log{(Q)}^{2} and \\frac{d}{d Q} \\chi{(Q)} \\log{(Q)} = \\frac{d}{d Q} \\log{(Q)}^{2} and \\log{(Q)} \\frac{d}{d Q} \\chi{(Q)} + \\frac{\\chi{(Q)}}{Q} = \\frac{2 \\log{(Q)}}{Q} and \\chi{(Q)} - \\log{(Q)}^{2} - \\log{(Q)} = - \\log{(Q)}^{2} and \\frac{\\log{(Q)} \\frac{d}{d Q} \\chi{(Q)} + \\frac{\\chi{(Q)}}{Q}}{\\chi{(Q)} - \\log{(Q)}^{2} - \\log{(Q)}} = \\frac{2 \\log{(Q)}}{Q (\\chi{(Q)} - \\log{(Q)}^{2} - \\log{(Q)})} and - \\frac{\\log{(Q)} \\frac{d}{d Q} \\chi{(Q)} + \\frac{\\chi{(Q)}}{Q}}{\\log{(Q)}^{2}} = - \\frac{2}{Q \\log{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["times", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Pow(log(Symbol('Q', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\chi')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(log(Symbol('Q', commutative=True)), Derivative(Function('\\\\chi')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('Q', commutative=True)))), Mul(Integer(2), Pow(Symbol('Q', commutative=True), Integer(-1)), log(Symbol('Q', commutative=True))))"], [["minus", 1, "Add(Pow(log(Symbol('Q', commutative=True)), Integer(2)), log(Symbol('Q', commutative=True)))"], "Equality(Add(Function('\\\\chi')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Q', commutative=True)), Integer(2))), Mul(Integer(-1), log(Symbol('Q', commutative=True)))), Mul(Integer(-1), Pow(log(Symbol('Q', commutative=True)), Integer(2))))"], [["divide", 4, "Add(Function('\\\\chi')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Q', commutative=True)), Integer(2))), Mul(Integer(-1), log(Symbol('Q', commutative=True))))"], "Equality(Mul(Add(Mul(log(Symbol('Q', commutative=True)), Derivative(Function('\\\\chi')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('Q', commutative=True)))), Pow(Add(Function('\\\\chi')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Q', commutative=True)), Integer(2))), Mul(Integer(-1), log(Symbol('Q', commutative=True)))), Integer(-1))), Mul(Integer(2), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Add(Function('\\\\chi')(Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('Q', commutative=True)), Integer(2))), Mul(Integer(-1), log(Symbol('Q', commutative=True)))), Integer(-1)), log(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Integer(-1), Add(Mul(log(Symbol('Q', commutative=True)), Derivative(Function('\\\\chi')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('Q', commutative=True)))), Pow(log(Symbol('Q', commutative=True)), Integer(-2))), Mul(Integer(-1), Integer(2), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(I,F_{H})} = F_{H} + I, then derive \\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1 = 0, then obtain - F_{H} (t - \\frac{2 (\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1)}{I}) = - F_{H} (t - \\frac{\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1}{I})", "derivation": "\\operatorname{E_{n}}{(I,F_{H})} = F_{H} + I and - F_{H} - I + \\operatorname{E_{n}}{(I,F_{H})} = 0 and \\frac{\\partial}{\\partial I} (- F_{H} - I + \\operatorname{E_{n}}{(I,F_{H})}) = \\frac{d}{d I} 0 and \\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1 = 0 and - \\frac{\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1}{I} = 0 and t - \\frac{\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1}{I} = t and - F_{H} (t - \\frac{\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1}{I}) = - F_{H} t and - F_{H} (t - \\frac{2 (\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1)}{I}) = - F_{H} (t - \\frac{\\frac{\\partial}{\\partial I} \\operatorname{E_{n}}{(I,F_{H})} - 1}{I})", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)))"], [["minus", 1, "Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["divide", 4, "Mul(Integer(-1), Symbol('I', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["add", 5, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)))), Symbol('t', commutative=True))"], [["times", 6, "Mul(Integer(-1), Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))))), Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))))), Mul(Integer(-1), Symbol('F_H', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))))))"]]}, {"prompt": "Given \\rho_{b}{(A_{y})} = e^{\\sin{(A_{y})}}, then obtain (\\sin{(A_{y})} + \\int \\rho_{b}{(A_{y})} dA_{y}) \\int \\rho_{b}{(A_{y})} dA_{y} = (\\sin{(A_{y})} + \\int \\rho_{b}{(A_{y})} dA_{y}) \\int e^{\\sin{(A_{y})}} dA_{y}", "derivation": "\\rho_{b}{(A_{y})} = e^{\\sin{(A_{y})}} and \\int \\rho_{b}{(A_{y})} dA_{y} = \\int e^{\\sin{(A_{y})}} dA_{y} and \\sin{(A_{y})} + \\int \\rho_{b}{(A_{y})} dA_{y} = \\sin{(A_{y})} + \\int e^{\\sin{(A_{y})}} dA_{y} and (\\sin{(A_{y})} + \\int e^{\\sin{(A_{y})}} dA_{y}) \\int \\rho_{b}{(A_{y})} dA_{y} = (\\sin{(A_{y})} + \\int e^{\\sin{(A_{y})}} dA_{y}) \\int e^{\\sin{(A_{y})}} dA_{y} and (\\sin{(A_{y})} + \\int \\rho_{b}{(A_{y})} dA_{y}) \\int \\rho_{b}{(A_{y})} dA_{y} = (\\sin{(A_{y})} + \\int \\rho_{b}{(A_{y})} dA_{y}) \\int e^{\\sin{(A_{y})}} dA_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), exp(sin(Symbol('A_y', commutative=True))))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], [["add", 2, "sin(Symbol('A_y', commutative=True))"], "Equality(Add(sin(Symbol('A_y', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(sin(Symbol('A_y', commutative=True)), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))))"], [["times", 2, "Add(sin(Symbol('A_y', commutative=True)), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], "Equality(Mul(Add(sin(Symbol('A_y', commutative=True)), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))), Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Add(sin(Symbol('A_y', commutative=True)), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(sin(Symbol('A_y', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Add(sin(Symbol('A_y', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Integral(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(C_{1},\\Psi)} = C_{1} - \\Psi, then obtain \\frac{\\partial}{\\partial \\Psi} (C_{1} - \\Psi + \\operatorname{P_{g}}{(C_{1},\\Psi)}) = \\frac{\\partial}{\\partial \\Psi} 2 \\operatorname{P_{g}}{(C_{1},\\Psi)}", "derivation": "\\operatorname{P_{g}}{(C_{1},\\Psi)} = C_{1} - \\Psi and C_{1} - \\Psi + \\operatorname{P_{g}}{(C_{1},\\Psi)} = 2 C_{1} - 2 \\Psi and \\frac{\\partial}{\\partial \\Psi} (C_{1} - \\Psi + \\operatorname{P_{g}}{(C_{1},\\Psi)}) = \\frac{\\partial}{\\partial \\Psi} (2 C_{1} - 2 \\Psi) and \\frac{\\partial}{\\partial \\Psi} 2 \\operatorname{P_{g}}{(C_{1},\\Psi)} = \\frac{\\partial}{\\partial \\Psi} (2 C_{1} - 2 \\Psi) and \\frac{\\partial}{\\partial \\Psi} (C_{1} - \\Psi + \\operatorname{P_{g}}{(C_{1},\\Psi)}) = \\frac{\\partial}{\\partial \\Psi} 2 \\operatorname{P_{g}}{(C_{1},\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["add", 1, "Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Mul(Integer(2), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(x)} = \\sin{(x)}, then obtain \\frac{d}{d x} \\frac{I^{x}{(x)}}{x + I^{x}{(x)}} = \\frac{d}{d x} \\frac{\\sin^{x}{(x)}}{x + I^{x}{(x)}}", "derivation": "I{(x)} = \\sin{(x)} and I^{x}{(x)} = \\sin^{x}{(x)} and x + I^{x}{(x)} = x + \\sin^{x}{(x)} and \\frac{I^{x}{(x)}}{x + \\sin^{x}{(x)}} = \\frac{\\sin^{x}{(x)}}{x + \\sin^{x}{(x)}} and \\frac{I^{x}{(x)}}{x + I^{x}{(x)}} = \\frac{\\sin^{x}{(x)}}{x + I^{x}{(x)}} and \\frac{d}{d x} \\frac{I^{x}{(x)}}{x + I^{x}{(x)}} = \\frac{d}{d x} \\frac{\\sin^{x}{(x)}}{x + I^{x}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["add", 2, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Add(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["divide", 2, "Add(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Add(Symbol('x', commutative=True), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('x', commutative=True), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Add(Symbol('x', commutative=True), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('x', commutative=True), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('x', commutative=True), Pow(Function('I')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(A,r,E_{\\lambda})} = (r^{E_{\\lambda}})^{A}, then obtain \\int (\\hat{H}{(A,r,E_{\\lambda})} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda}) dE_{\\lambda} = \\int ((r^{E_{\\lambda}})^{A} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda}) dE_{\\lambda}", "derivation": "\\hat{H}{(A,r,E_{\\lambda})} = (r^{E_{\\lambda}})^{A} and \\int \\hat{H}{(A,r,E_{\\lambda})} dE_{\\lambda} = \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda} and \\hat{H}{(A,r,E_{\\lambda})} - \\int \\hat{H}{(A,r,E_{\\lambda})} dE_{\\lambda} = (r^{E_{\\lambda}})^{A} - \\int \\hat{H}{(A,r,E_{\\lambda})} dE_{\\lambda} and \\hat{H}{(A,r,E_{\\lambda})} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda} = (r^{E_{\\lambda}})^{A} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda} and \\int (\\hat{H}{(A,r,E_{\\lambda})} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda}) dE_{\\lambda} = \\int ((r^{E_{\\lambda}})^{A} - \\int (r^{E_{\\lambda}})^{A} dE_{\\lambda}) dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Integral(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))), Add(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))), Add(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))))"], [["integrate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{H}')(Symbol('A', commutative=True), Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(E_{\\lambda},A_{1})} = \\frac{E_{\\lambda}}{A_{1}}, then derive \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(E_{\\lambda},A_{1})} = - \\frac{E_{\\lambda}}{A_{1}^{2}}, then obtain - \\frac{E_{\\lambda} \\mathbf{B}{(E_{\\lambda},A_{1})} \\frac{\\partial}{\\partial A_{1}} \\frac{E_{\\lambda}}{A_{1}}}{A_{1}} = \\frac{E_{\\lambda}^{2} \\mathbf{B}{(E_{\\lambda},A_{1})}}{A_{1}^{3}}", "derivation": "\\mathbf{B}{(E_{\\lambda},A_{1})} = \\frac{E_{\\lambda}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(E_{\\lambda},A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\frac{E_{\\lambda}}{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(E_{\\lambda},A_{1})} = - \\frac{E_{\\lambda}}{A_{1}^{2}} and \\frac{\\partial}{\\partial A_{1}} \\frac{E_{\\lambda}}{A_{1}} = - \\frac{E_{\\lambda}}{A_{1}^{2}} and E_{\\lambda} \\frac{\\partial}{\\partial A_{1}} \\frac{E_{\\lambda}}{A_{1}} = - \\frac{E_{\\lambda}^{2}}{A_{1}^{2}} and - \\frac{E_{\\lambda} \\mathbf{B}{(E_{\\lambda},A_{1})} \\frac{\\partial}{\\partial A_{1}} \\frac{E_{\\lambda}}{A_{1}}}{A_{1}} = \\frac{E_{\\lambda}^{2} \\mathbf{B}{(E_{\\lambda},A_{1})}}{A_{1}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))))"], [["times", 5, "Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True)), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-3)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(2)), Function('\\\\mathbf{B}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\omega)} = e^{\\sin{(\\omega)}} and \\operatorname{E_{n}}{(\\omega)} = \\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega, then obtain \\operatorname{E_{n}}^{2}{(\\omega)} = (\\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega)^{2}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\omega)} = e^{\\sin{(\\omega)}} and \\operatorname{J_{\\varepsilon}}{(\\omega)} e^{\\sin{(\\omega)}} = e^{2 \\sin{(\\omega)}} and \\int \\operatorname{J_{\\varepsilon}}{(\\omega)} e^{\\sin{(\\omega)}} d\\omega = \\int e^{2 \\sin{(\\omega)}} d\\omega and \\iint \\operatorname{J_{\\varepsilon}}{(\\omega)} e^{\\sin{(\\omega)}} d\\omega d\\omega = \\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega and (\\iint \\operatorname{J_{\\varepsilon}}{(\\omega)} e^{\\sin{(\\omega)}} d\\omega d\\omega)^{2} = (\\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega)^{2} and \\operatorname{E_{n}}{(\\omega)} = \\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega and \\iint \\operatorname{J_{\\varepsilon}}{(\\omega)} e^{\\sin{(\\omega)}} d\\omega d\\omega = \\operatorname{E_{n}}{(\\omega)} and \\operatorname{E_{n}}^{2}{(\\omega)} = (\\iint e^{2 \\sin{(\\omega)}} d\\omega d\\omega)^{2}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "exp(sin(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)), Pow(Integral(exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\omega', commutative=True)), Integral(exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integral(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Function('E_n')(Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Pow(Function('E_n')(Symbol('\\\\omega', commutative=True)), Integer(2)), Pow(Integral(exp(Mul(Integer(2), sin(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} = (e^{F_{H}})^{\\hat{H}_{\\lambda}}, then obtain F_{H} (F_{H} \\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} + (e^{F_{H}})^{\\hat{H}_{\\lambda}}) (e^{F_{H}})^{\\hat{H}_{\\lambda}} = F_{H} (F_{H} (e^{F_{H}})^{\\hat{H}_{\\lambda}} + (e^{F_{H}})^{\\hat{H}_{\\lambda}}) (e^{F_{H}})^{\\hat{H}_{\\lambda}}", "derivation": "\\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} = (e^{F_{H}})^{\\hat{H}_{\\lambda}} and F_{H} \\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} = F_{H} (e^{F_{H}})^{\\hat{H}_{\\lambda}} and F_{H} \\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} + (e^{F_{H}})^{\\hat{H}_{\\lambda}} = F_{H} (e^{F_{H}})^{\\hat{H}_{\\lambda}} + (e^{F_{H}})^{\\hat{H}_{\\lambda}} and F_{H} (F_{H} \\operatorname{z^{*}}{(\\hat{H}_{\\lambda},F_{H})} + (e^{F_{H}})^{\\hat{H}_{\\lambda}}) (e^{F_{H}})^{\\hat{H}_{\\lambda}} = F_{H} (F_{H} (e^{F_{H}})^{\\hat{H}_{\\lambda}} + (e^{F_{H}})^{\\hat{H}_{\\lambda}}) (e^{F_{H}})^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('F_H', commutative=True)), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["times", 1, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["add", 2, "Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Symbol('F_H', commutative=True), Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('F_H', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('F_H', commutative=True), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["times", 3, "Mul(Symbol('F_H', commutative=True), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Symbol('F_H', commutative=True), Add(Mul(Symbol('F_H', commutative=True), Function('z^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('F_H', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Symbol('F_H', commutative=True), Add(Mul(Symbol('F_H', commutative=True), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(exp(Symbol('F_H', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(v_{y})} = \\sin{(v_{y})}, then obtain (\\frac{d}{d v_{y}} \\operatorname{V_{\\mathbf{B}}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2)^{v_{y}} = (2 \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2)^{v_{y}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(v_{y})} = \\sin{(v_{y})} and \\frac{d}{d v_{y}} \\operatorname{V_{\\mathbf{B}}}{(v_{y})} = \\frac{d}{d v_{y}} \\sin{(v_{y})} and \\frac{d}{d v_{y}} \\operatorname{V_{\\mathbf{B}}}{(v_{y})} - 1 = \\frac{d}{d v_{y}} \\sin{(v_{y})} - 1 and \\frac{d}{d v_{y}} \\operatorname{V_{\\mathbf{B}}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2 = 2 \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2 and (\\frac{d}{d v_{y}} \\operatorname{V_{\\mathbf{B}}}{(v_{y})} + \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2)^{v_{y}} = (2 \\frac{d}{d v_{y}} \\sin{(v_{y})} - 2)^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)))"], [["add", 3, "Add(Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-2)), Add(Mul(Integer(2), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Integer(-2)))"], [["power", 4, "Symbol('v_y', commutative=True)"], "Equality(Pow(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-2)), Symbol('v_y', commutative=True)), Pow(Add(Mul(Integer(2), Derivative(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Integer(-2)), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given Z{(u)} = \\sin{(u)}, then obtain 0 = \\frac{(- Z{(u)} + \\sin{(u)})^{2}}{\\sin{(u)}}", "derivation": "Z{(u)} = \\sin{(u)} and Z{(u)} + \\sin{(u)} = 2 \\sin{(u)} and 0 = - Z{(u)} + \\sin{(u)} and 0 = (- Z{(u)} + \\sin{(u)})^{2} and 0 = \\frac{(- Z{(u)} + \\sin{(u)})^{2}}{\\sin{(u)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["add", 1, "sin(Symbol('u', commutative=True))"], "Equality(Add(Function('Z')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))), Mul(Integer(2), sin(Symbol('u', commutative=True))))"], [["minus", 2, "Add(Function('Z')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('Z')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Function('Z')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True)))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('Z')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True))), Integer(2)))"], [["divide", 4, "sin(Symbol('u', commutative=True))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Function('Z')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True))), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)} = - Q + \\sigma_p, then obtain - Q + \\sigma_p + (\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)})^{\\sigma_p} = - Q + \\sigma_p + 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)} = - Q + \\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (- Q + \\sigma_p) and (\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)})^{\\sigma_p} = (\\frac{\\partial}{\\partial \\sigma_p} (- Q + \\sigma_p))^{\\sigma_p} and - Q + \\sigma_p + (\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)})^{\\sigma_p} = - Q + \\sigma_p + (\\frac{\\partial}{\\partial \\sigma_p} (- Q + \\sigma_p))^{\\sigma_p} and - Q + \\sigma_p + (\\frac{\\partial}{\\partial \\sigma_p} \\operatorname{f_{\\mathbf{v}}}{(Q,\\sigma_p)})^{\\sigma_p} = - Q + \\sigma_p + 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(f^{\\prime})} = \\sin{(f^{\\prime})}, then obtain \\operatorname{C_{1}}{(f^{\\prime})} = \\frac{\\operatorname{C_{1}}{(f^{\\prime})}}{2} + \\frac{\\sin{(f^{\\prime})}}{2}", "derivation": "\\operatorname{C_{1}}{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\operatorname{C_{1}}{(f^{\\prime})} + \\sin{(f^{\\prime})} = 2 \\sin{(f^{\\prime})} and \\frac{\\operatorname{C_{1}}{(f^{\\prime})}}{2 \\sin{(f^{\\prime})}} = \\frac{1}{2} and \\frac{\\operatorname{C_{1}}{(f^{\\prime})}}{\\operatorname{C_{1}}{(f^{\\prime})} + \\sin{(f^{\\prime})}} = \\frac{1}{2} and \\operatorname{C_{1}}{(f^{\\prime})} = \\frac{\\operatorname{C_{1}}{(f^{\\prime})}}{2} + \\frac{\\sin{(f^{\\prime})}}{2}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "sin(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 1, "Mul(Integer(2), sin(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Rational(1, 2))"], [["times", 4, "Add(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Rational(1, 2), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Rational(1, 2), sin(Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given W{(v,v_{t})} = v v_{t}, then obtain \\frac{\\partial^{3}}{\\partial v_{t}^{2}\\partial v} \\int W{(v,v_{t})} dv_{t} = \\frac{\\partial^{3}}{\\partial v_{t}^{2}\\partial v} \\int v v_{t} dv_{t}", "derivation": "W{(v,v_{t})} = v v_{t} and \\int W{(v,v_{t})} dv_{t} = \\int v v_{t} dv_{t} and \\frac{\\partial}{\\partial v} \\int W{(v,v_{t})} dv_{t} = \\frac{\\partial}{\\partial v} \\int v v_{t} dv_{t} and \\frac{\\partial^{2}}{\\partial v_{t}\\partial v} \\int W{(v,v_{t})} dv_{t} = \\frac{\\partial^{2}}{\\partial v_{t}\\partial v} \\int v v_{t} dv_{t} and \\frac{\\partial^{3}}{\\partial v_{t}^{2}\\partial v} \\int W{(v,v_{t})} dv_{t} = \\frac{\\partial^{3}}{\\partial v_{t}^{2}\\partial v} \\int v v_{t} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('v', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('W')(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Function('W')(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Function('W')(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Function('W')(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(2))), Derivative(Integral(Mul(Symbol('v', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(2))))"]]}, {"prompt": "Given x{(\\nabla,B)} = \\frac{\\partial}{\\partial B} B \\nabla, then derive x{(\\nabla,B)} = \\nabla, then obtain x{(\\frac{\\partial}{\\partial B} B \\nabla,B)} = \\frac{\\partial}{\\partial B} B \\nabla", "derivation": "x{(\\nabla,B)} = \\frac{\\partial}{\\partial B} B \\nabla and x{(\\nabla,B)} = \\nabla and \\nabla = \\frac{\\partial}{\\partial B} B \\nabla and x{(\\frac{\\partial}{\\partial B} B \\nabla,B)} = \\frac{\\partial}{\\partial B} B \\nabla", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('x')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Symbol('\\\\nabla', commutative=True), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Function('x')(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(T,\\theta)} = \\cos^{T}{(\\theta)} and \\hat{\\mathbf{x}}{(\\theta)} = \\cos{(\\theta)}, then obtain (\\cos^{T}{(\\theta)})^{T} = (\\hat{\\mathbf{x}}^{T}{(\\theta)})^{T}", "derivation": "\\operatorname{E_{x}}{(T,\\theta)} = \\cos^{T}{(\\theta)} and \\operatorname{E_{x}}^{T}{(T,\\theta)} = (\\cos^{T}{(\\theta)})^{T} and \\hat{\\mathbf{x}}{(\\theta)} = \\cos{(\\theta)} and \\operatorname{E_{x}}^{T}{(T,\\theta)} = (\\hat{\\mathbf{x}}^{T}{(\\theta)})^{T} and (\\cos^{T}{(\\theta)})^{T} = (\\hat{\\mathbf{x}}^{T}{(\\theta)})^{T}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('E_x')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"]]}, {"prompt": "Given a{(c,A_{z})} = \\frac{A_{z}}{c} and \\operatorname{v_{2}}{(c,A_{z})} = (\\frac{A_{z}}{c})^{A_{z}}, then obtain 0^{A_{z}} = (- a^{A_{z}}{(c,A_{z})} + \\operatorname{v_{2}}{(c,A_{z})})^{A_{z}}", "derivation": "a{(c,A_{z})} = \\frac{A_{z}}{c} and a^{A_{z}}{(c,A_{z})} = (\\frac{A_{z}}{c})^{A_{z}} and 0 = (\\frac{A_{z}}{c})^{A_{z}} - a^{A_{z}}{(c,A_{z})} and \\operatorname{v_{2}}{(c,A_{z})} = (\\frac{A_{z}}{c})^{A_{z}} and 0 = - a^{A_{z}}{(c,A_{z})} + \\operatorname{v_{2}}{(c,A_{z})} and 0^{A_{z}} = (- a^{A_{z}}{(c,A_{z})} + \\operatorname{v_{2}}{(c,A_{z})})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('A_z', commutative=True)))"], [["minus", 2, "Pow(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Symbol('A_z', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('A_z', commutative=True)), Mul(Integer(-1), Pow(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True))), Function('v_2')(Symbol('c', commutative=True), Symbol('A_z', commutative=True))))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('a')(Symbol('c', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True))), Function('v_2')(Symbol('c', commutative=True), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain - \\operatorname{t_{2}}{(\\mathbf{E})} e^{- \\mathbf{E}} + e^{- \\mathbf{E}} \\frac{d}{d \\mathbf{E}} \\operatorname{t_{2}}{(\\mathbf{E})} = 0", "derivation": "\\operatorname{t_{2}}{(\\mathbf{E})} = e^{\\mathbf{E}} and \\operatorname{t_{2}}{(\\mathbf{E})} e^{- \\mathbf{E}} = 1 and \\frac{d}{d \\mathbf{E}} \\operatorname{t_{2}}{(\\mathbf{E})} e^{- \\mathbf{E}} = \\frac{d}{d \\mathbf{E}} 1 and - \\operatorname{t_{2}}{(\\mathbf{E})} e^{- \\mathbf{E}} + e^{- \\mathbf{E}} \\frac{d}{d \\mathbf{E}} \\operatorname{t_{2}}{(\\mathbf{E})} = 0", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Function('t_2')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Function('t_2')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Derivative(Function('t_2')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given u{(\\Psi)} = e^{\\Psi}, then derive \\frac{d}{d \\Psi} u{(\\Psi)} = e^{\\Psi}, then obtain u{(\\Psi)} + \\frac{d}{d \\Psi} u{(\\Psi)} - \\frac{d}{d \\Psi} e^{\\Psi} = u{(\\Psi)} - \\frac{d}{d \\Psi} e^{\\Psi} + \\frac{d^{2}}{d \\Psi^{2}} e^{\\Psi}", "derivation": "u{(\\Psi)} = e^{\\Psi} and \\frac{d}{d \\Psi} u{(\\Psi)} = \\frac{d}{d \\Psi} e^{\\Psi} and \\frac{d}{d \\Psi} u{(\\Psi)} = e^{\\Psi} and e^{\\Psi} = \\frac{d}{d \\Psi} e^{\\Psi} and \\frac{d}{d \\Psi} u{(\\Psi)} = \\frac{d^{2}}{d \\Psi^{2}} e^{\\Psi} and u{(\\Psi)} + \\frac{d}{d \\Psi} u{(\\Psi)} = u{(\\Psi)} + \\frac{d^{2}}{d \\Psi^{2}} e^{\\Psi} and u{(\\Psi)} + \\frac{d}{d \\Psi} u{(\\Psi)} - \\frac{d}{d \\Psi} e^{\\Psi} = u{(\\Psi)} - \\frac{d}{d \\Psi} e^{\\Psi} + \\frac{d^{2}}{d \\Psi^{2}} e^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), exp(Symbol('\\\\Psi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\Psi', commutative=True)), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('u')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2))))"], [["add", 5, "Function('u')(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\Psi', commutative=True)), Derivative(Function('u')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(Function('u')(Symbol('\\\\Psi', commutative=True)), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2)))))"], [["minus", 6, "Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Add(Function('u')(Symbol('\\\\Psi', commutative=True)), Derivative(Function('u')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Add(Function('u')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2)))))"]]}, {"prompt": "Given B{(F_{g})} = \\frac{d}{d F_{g}} e^{F_{g}} and \\operatorname{F_{H}}{(F_{g})} = e^{F_{g}}, then derive B^{F_{g}}{(F_{g})} = (e^{F_{g}})^{F_{g}}, then obtain \\operatorname{F_{H}}{(F_{g})} - (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}} = e^{F_{g}} - (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}}", "derivation": "B{(F_{g})} = \\frac{d}{d F_{g}} e^{F_{g}} and B^{F_{g}}{(F_{g})} = (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}} and B^{F_{g}}{(F_{g})} = (e^{F_{g}})^{F_{g}} and \\operatorname{F_{H}}{(F_{g})} = e^{F_{g}} and (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}} = (e^{F_{g}})^{F_{g}} and \\operatorname{F_{H}}{(F_{g})} - (e^{F_{g}})^{F_{g}} = e^{F_{g}} - (e^{F_{g}})^{F_{g}} and \\operatorname{F_{H}}{(F_{g})} - (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}} = e^{F_{g}} - (\\frac{d}{d F_{g}} e^{F_{g}})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('F_g', commutative=True)), Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('B')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('B')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["minus", 4, "Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('F_g', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))), Add(exp(Symbol('F_g', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('F_H')(Symbol('F_g', commutative=True)), Mul(Integer(-1), Pow(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)))), Add(exp(Symbol('F_g', commutative=True)), Mul(Integer(-1), Pow(Derivative(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(S)} = \\cos{(\\cos{(S)})}, then obtain \\frac{d^{2}}{d S^{2}} \\theta_{1}{(S)} = - \\sin^{2}{(S)} \\cos{(\\cos{(S)})} + \\sin{(\\cos{(S)})} \\cos{(S)}", "derivation": "\\theta_{1}{(S)} = \\cos{(\\cos{(S)})} and \\frac{d}{d S} \\theta_{1}{(S)} = \\frac{d}{d S} \\cos{(\\cos{(S)})} and \\frac{d^{2}}{d S^{2}} \\theta_{1}{(S)} = \\frac{d^{2}}{d S^{2}} \\cos{(\\cos{(S)})} and \\frac{d^{2}}{d S^{2}} \\theta_{1}{(S)} = - \\sin^{2}{(S)} \\cos{(\\cos{(S)})} + \\sin{(\\cos{(S)})} \\cos{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('S', commutative=True)), cos(cos(Symbol('S', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(cos(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\theta_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Add(Mul(Integer(-1), Pow(sin(Symbol('S', commutative=True)), Integer(2)), cos(cos(Symbol('S', commutative=True)))), Mul(sin(cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\theta_1,\\sigma_x)} = \\sigma_x + \\sin{(\\theta_1)}, then derive - \\theta_1 + \\int \\phi{(\\theta_1,\\sigma_x)} d\\sigma_x = J + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1, then obtain \\mathbf{F} + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1 = J + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1", "derivation": "\\phi{(\\theta_1,\\sigma_x)} = \\sigma_x + \\sin{(\\theta_1)} and \\int \\phi{(\\theta_1,\\sigma_x)} d\\sigma_x = \\int (\\sigma_x + \\sin{(\\theta_1)}) d\\sigma_x and - \\theta_1 + \\int \\phi{(\\theta_1,\\sigma_x)} d\\sigma_x = - \\theta_1 + \\int (\\sigma_x + \\sin{(\\theta_1)}) d\\sigma_x and - \\theta_1 + \\int \\phi{(\\theta_1,\\sigma_x)} d\\sigma_x = J + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1 and - \\theta_1 + \\int (\\sigma_x + \\sin{(\\theta_1)}) d\\sigma_x = J + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1 and \\mathbf{F} + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1 = J + \\frac{\\sigma_x^{2}}{2} + \\sigma_x \\sin{(\\theta_1)} - \\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\phi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integral(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\phi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Integral(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given M{(t,q)} = q + \\log{(t)}, then obtain - \\frac{2 M{(t,q)}}{t} = - \\frac{q + \\log{(t)}}{t} - \\frac{M{(t,q)}}{t}", "derivation": "M{(t,q)} = q + \\log{(t)} and \\frac{M{(t,q)}}{t} = \\frac{q + \\log{(t)}}{t} and \\frac{2 M{(t,q)}}{t} = \\frac{q + \\log{(t)}}{t} + \\frac{M{(t,q)}}{t} and - \\frac{2 M{(t,q)}}{t} = - \\frac{q + \\log{(t)}}{t} - \\frac{M{(t,q)}}{t}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Add(Symbol('q', commutative=True), log(Symbol('t', commutative=True))))"], [["divide", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Symbol('t', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Add(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Symbol('t', commutative=True)))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integer(2), Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Symbol('t', commutative=True)))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('M')(Symbol('t', commutative=True), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(C)} = \\cos{(C)}, then obtain \\frac{- \\mathbf{F}{(C)} + \\frac{2 \\mathbf{F}{(C)}}{\\cos{(C)}} - 1}{C} = \\frac{- \\mathbf{F}{(C)} + \\frac{\\mathbf{F}{(C)}}{\\cos{(C)}}}{C}", "derivation": "\\mathbf{F}{(C)} = \\cos{(C)} and \\frac{\\mathbf{F}{(C)}}{\\cos{(C)}} = 1 and - \\mathbf{F}{(C)} + \\frac{\\mathbf{F}{(C)}}{\\cos{(C)}} = 1 - \\mathbf{F}{(C)} and \\frac{- \\mathbf{F}{(C)} + \\frac{\\mathbf{F}{(C)}}{\\cos{(C)}}}{C} = \\frac{1 - \\mathbf{F}{(C)}}{C} and \\frac{- \\mathbf{F}{(C)} + \\frac{2 \\mathbf{F}{(C)}}{\\cos{(C)}} - 1}{C} = \\frac{- \\mathbf{F}{(C)} + \\frac{\\mathbf{F}{(C)}}{\\cos{(C)}}}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["divide", 1, "cos(Symbol('C', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), Pow(cos(Symbol('C', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Function('\\\\mathbf{F}')(Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True))), Mul(Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), Pow(cos(Symbol('C', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True)))))"], [["divide", 3, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True))), Mul(Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), Pow(cos(Symbol('C', commutative=True)), Integer(-1))))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), Pow(cos(Symbol('C', commutative=True)), Integer(-1))), Integer(-1))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('C', commutative=True))), Mul(Function('\\\\mathbf{F}')(Symbol('C', commutative=True)), Pow(cos(Symbol('C', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(l)} = \\cos{(l)}, then obtain \\hat{\\mathbf{x}}{(l)} \\cos{(l)} - \\sin{(\\int \\hat{\\mathbf{x}}{(l)} dl)} = - \\sin{(\\int \\hat{\\mathbf{x}}{(l)} dl)} + \\cos^{2}{(l)}", "derivation": "\\hat{\\mathbf{x}}{(l)} = \\cos{(l)} and \\hat{\\mathbf{x}}{(l)} \\cos{(l)} = \\cos^{2}{(l)} and \\int \\hat{\\mathbf{x}}{(l)} dl = \\int \\cos{(l)} dl and \\sin{(\\int \\hat{\\mathbf{x}}{(l)} dl)} = \\sin{(\\int \\cos{(l)} dl)} and \\hat{\\mathbf{x}}{(l)} \\cos{(l)} - \\sin{(\\int \\cos{(l)} dl)} = - \\sin{(\\int \\cos{(l)} dl)} + \\cos^{2}{(l)} and \\hat{\\mathbf{x}}{(l)} \\cos{(l)} - \\sin{(\\int \\hat{\\mathbf{x}}{(l)} dl)} = - \\sin{(\\int \\hat{\\mathbf{x}}{(l)} dl)} + \\cos^{2}{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["times", 1, "cos(Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(2)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), sin(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["minus", 2, "sin(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], "Equality(Add(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Mul(Integer(-1), sin(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))), Add(Mul(Integer(-1), sin(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Pow(cos(Symbol('l', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Mul(Integer(-1), sin(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))), Add(Mul(Integer(-1), sin(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Pow(cos(Symbol('l', commutative=True)), Integer(2))))"]]}, {"prompt": "Given T{(A_{x})} = \\sin{(A_{x})}, then derive \\int T{(A_{x})} dA_{x} = \\mathbf{g} - \\cos{(A_{x})}, then obtain \\dot{\\mathbf{r}} - T{(A_{x})} - \\cos{(A_{x})} = \\mathbf{g} - T{(A_{x})} - \\cos{(A_{x})}", "derivation": "T{(A_{x})} = \\sin{(A_{x})} and \\int T{(A_{x})} dA_{x} = \\int \\sin{(A_{x})} dA_{x} and \\int T{(A_{x})} dA_{x} = \\mathbf{g} - \\cos{(A_{x})} and - T{(A_{x})} + \\int T{(A_{x})} dA_{x} = \\mathbf{g} - T{(A_{x})} - \\cos{(A_{x})} and - \\sin{(A_{x})} + \\int \\sin{(A_{x})} dA_{x} = \\mathbf{g} - \\sin{(A_{x})} - \\cos{(A_{x})} and - T{(A_{x})} + \\int \\sin{(A_{x})} dA_{x} = \\mathbf{g} - T{(A_{x})} - \\cos{(A_{x})} and \\dot{\\mathbf{r}} - T{(A_{x})} - \\cos{(A_{x})} = \\mathbf{g} - T{(A_{x})} - \\cos{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('A_x', commutative=True)), sin(Symbol('A_x', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('T')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('T')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))))"], [["minus", 3, "Function('T')(Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Integral(Function('T')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('A_x', commutative=True))), Integral(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('A_x', commutative=True))), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Integral(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Function('T')(Symbol('A_x', commutative=True))), Mul(Integer(-1), cos(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})}, then obtain (2 \\hat{H}{(f_{\\mathbf{v}})} + \\sin{(f_{\\mathbf{v}})})^{2} = 9 \\sin^{2}{(f_{\\mathbf{v}})}", "derivation": "\\hat{H}{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})} and \\hat{H}{(f_{\\mathbf{v}})} + \\sin{(f_{\\mathbf{v}})} = 2 \\sin{(f_{\\mathbf{v}})} and \\hat{H}{(f_{\\mathbf{v}})} + 2 \\sin{(f_{\\mathbf{v}})} = 3 \\sin{(f_{\\mathbf{v}})} and 2 \\hat{H}{(f_{\\mathbf{v}})} + \\sin{(f_{\\mathbf{v}})} = 3 \\sin{(f_{\\mathbf{v}})} and (2 \\hat{H}{(f_{\\mathbf{v}})} + \\sin{(f_{\\mathbf{v}})})^{2} = 9 \\sin^{2}{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 1, "sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(2), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 1, "Mul(Integer(2), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(2), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(Integer(3), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(3), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(2)), Mul(Integer(9), Pow(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given g{(F_{g})} = e^{F_{g}} and \\mathbf{f}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain (g{(F_{g})} - e^{F_{g}}) \\mathbf{f}{(y^{\\prime})} + \\mathbf{f}{(y^{\\prime})} + g{(F_{g})} - e^{F_{g}} = \\mathbf{f}{(y^{\\prime})} + g{(F_{g})} - e^{F_{g}}", "derivation": "g{(F_{g})} = e^{F_{g}} and g{(F_{g})} - e^{F_{g}} = 0 and \\mathbf{f}{(y^{\\prime})} = e^{y^{\\prime}} and (g{(F_{g})} - e^{F_{g}}) \\mathbf{f}{(y^{\\prime})} = 0 and (g{(F_{g})} - e^{F_{g}}) \\mathbf{f}{(y^{\\prime})} + e^{y^{\\prime}} = e^{y^{\\prime}} and (g{(F_{g})} - e^{F_{g}}) \\mathbf{f}{(y^{\\prime})} + \\mathbf{f}{(y^{\\prime})} = \\mathbf{f}{(y^{\\prime})} and (g{(F_{g})} - e^{F_{g}}) \\mathbf{f}{(y^{\\prime})} + \\mathbf{f}{(y^{\\prime})} + g{(F_{g})} - e^{F_{g}} = \\mathbf{f}{(y^{\\prime})} + g{(F_{g})} - e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["minus", 1, "exp(Symbol('F_g', commutative=True))"], "Equality(Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))), Integer(0))"], [["add", 4, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))), exp(Symbol('y^{\\\\prime}', commutative=True))), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 6, "Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True))))"], "Equality(Add(Mul(Add(Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Add(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Function('g')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I}, then obtain 2 \\dot{x}^{I} + \\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I} + 2 \\rho_{b}{(I,\\dot{x})}", "derivation": "\\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I} and 2 \\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I} + \\rho_{b}{(I,\\dot{x})} and 3 \\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I} + 2 \\rho_{b}{(I,\\dot{x})} and 3 \\rho_{b}{(I,\\dot{x})} = 2 \\dot{x}^{I} + \\rho_{b}{(I,\\dot{x})} and 2 \\dot{x}^{I} + \\rho_{b}{(I,\\dot{x})} = \\dot{x}^{I} + 2 \\rho_{b}{(I,\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True)))"], [["add", 1, "Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True)), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["add", 2, "Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True))), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True))), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('I', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(u)} = \\sin{(u)}, then derive \\frac{d}{d u} \\mu_{0}{(u)} = \\cos{(u)}, then obtain 1 - \\psi = - \\frac{\\psi \\frac{d}{d u} \\mu_{0}{(u)}}{\\cos{(u)}} + 1", "derivation": "\\mu_{0}{(u)} = \\sin{(u)} and \\frac{d}{d u} \\mu_{0}{(u)} = \\frac{d}{d u} \\sin{(u)} and \\frac{d}{d u} \\mu_{0}{(u)} = \\cos{(u)} and \\cos{(u)} = \\frac{d}{d u} \\sin{(u)} and \\psi \\cos{(u)} = \\psi \\frac{d}{d u} \\sin{(u)} and \\psi \\cos{(u)} = \\psi \\frac{d}{d u} \\mu_{0}{(u)} and - \\psi = - \\frac{\\psi \\frac{d}{d u} \\mu_{0}{(u)}}{\\cos{(u)}} and 1 - \\psi = - \\frac{\\psi \\frac{d}{d u} \\mu_{0}{(u)}}{\\cos{(u)}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu_0')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), cos(Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('u', commutative=True)), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('u', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('u', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["divide", 6, "Mul(Integer(-1), cos(Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Pow(cos(Symbol('u', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu_0')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["minus", 7, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Pow(cos(Symbol('u', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu_0')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given f{(\\dot{z},y)} = \\sin{(\\dot{z} + y)}, then obtain (f{(\\dot{z},y)} - \\sin{(\\dot{z} + y)}) \\frac{\\partial}{\\partial \\dot{z}} f{(\\dot{z},y)} + (- \\cos{(\\dot{z} + y)} + \\frac{\\partial}{\\partial \\dot{z}} f{(\\dot{z},y)}) f{(\\dot{z},y)} = 0", "derivation": "f{(\\dot{z},y)} = \\sin{(\\dot{z} + y)} and f{(\\dot{z},y)} - \\sin{(\\dot{z} + y)} = 0 and (f{(\\dot{z},y)} - \\sin{(\\dot{z} + y)}) f{(\\dot{z},y)} = 0 and \\frac{\\partial}{\\partial \\dot{z}} (f{(\\dot{z},y)} - \\sin{(\\dot{z} + y)}) f{(\\dot{z},y)} = \\frac{d}{d \\dot{z}} 0 and (f{(\\dot{z},y)} - \\sin{(\\dot{z} + y)}) \\frac{\\partial}{\\partial \\dot{z}} f{(\\dot{z},y)} + (- \\cos{(\\dot{z} + y)} + \\frac{\\partial}{\\partial \\dot{z}} f{(\\dot{z},y)}) f{(\\dot{z},y)} = 0", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))))), Integer(0))"], [["times", 2, "Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Add(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))))), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Add(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))))), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True))))), Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), cos(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)))), Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('y', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\sigma_{x}{(\\theta_1,v_{y})} = \\log{(v_{y}^{\\theta_1})}, then derive \\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} = \\frac{\\theta_1}{v_{y}}, then obtain \\int (\\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} - 1) dv_{y} = \\int (\\frac{\\theta_1}{v_{y}} - 1) dv_{y}", "derivation": "\\sigma_{x}{(\\theta_1,v_{y})} = \\log{(v_{y}^{\\theta_1})} and \\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} = \\frac{\\partial}{\\partial v_{y}} \\log{(v_{y}^{\\theta_1})} and \\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} = \\frac{\\theta_1}{v_{y}} and \\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} - 1 = \\frac{\\theta_1}{v_{y}} - 1 and \\int (\\frac{\\partial}{\\partial v_{y}} \\sigma_{x}{(\\theta_1,v_{y})} - 1) dv_{y} = \\int (\\frac{\\theta_1}{v_{y}} - 1) dv_{y}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)), log(Pow(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('v_y', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Integer(-1)))"], [["integrate", 4, "Symbol('v_y', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given b{(J,\\hat{\\mathbf{x}})} = (e^{J})^{\\hat{\\mathbf{x}}}, then obtain - J - (- J + (e^{J})^{\\hat{\\mathbf{x}}}) e^{J} + b{(J,\\hat{\\mathbf{x}})} = - J - (- J + (e^{J})^{\\hat{\\mathbf{x}}}) e^{J} + (e^{J})^{\\hat{\\mathbf{x}}}", "derivation": "b{(J,\\hat{\\mathbf{x}})} = (e^{J})^{\\hat{\\mathbf{x}}} and - J + b{(J,\\hat{\\mathbf{x}})} = - J + (e^{J})^{\\hat{\\mathbf{x}}} and (- J + b{(J,\\hat{\\mathbf{x}})}) e^{J} = (- J + (e^{J})^{\\hat{\\mathbf{x}}}) e^{J} and - J - (- J + b{(J,\\hat{\\mathbf{x}})}) e^{J} + b{(J,\\hat{\\mathbf{x}})} = - J - (- J + b{(J,\\hat{\\mathbf{x}})}) e^{J} + (e^{J})^{\\hat{\\mathbf{x}}} and - J - (- J + (e^{J})^{\\hat{\\mathbf{x}}}) e^{J} + b{(J,\\hat{\\mathbf{x}})} = - J - (- J + (e^{J})^{\\hat{\\mathbf{x}}}) e^{J} + (e^{J})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 2, "exp(Symbol('J', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))))"], [["minus", 2, "Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))), Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given U{(\\omega,L_{\\varepsilon})} = e^{L_{\\varepsilon} + \\omega}, then obtain (\\int U^{L_{\\varepsilon}}{(\\omega,L_{\\varepsilon})} d\\omega)^{\\omega} = (\\int (e^{L_{\\varepsilon} + \\omega})^{L_{\\varepsilon}} d\\omega)^{\\omega}", "derivation": "U{(\\omega,L_{\\varepsilon})} = e^{L_{\\varepsilon} + \\omega} and U^{L_{\\varepsilon}}{(\\omega,L_{\\varepsilon})} = (e^{L_{\\varepsilon} + \\omega})^{L_{\\varepsilon}} and \\int U^{L_{\\varepsilon}}{(\\omega,L_{\\varepsilon})} d\\omega = \\int (e^{L_{\\varepsilon} + \\omega})^{L_{\\varepsilon}} d\\omega and (\\int U^{L_{\\varepsilon}}{(\\omega,L_{\\varepsilon})} d\\omega)^{\\omega} = (\\int (e^{L_{\\varepsilon} + \\omega})^{L_{\\varepsilon}} d\\omega)^{\\omega}", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('\\\\omega', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\omega', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(exp(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Function('U')(Symbol('\\\\omega', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(exp(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integral(Pow(Function('U')(Symbol('\\\\omega', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integral(Pow(exp(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\mathbf{v},L)} = L + \\mathbf{v} and \\mathbf{p}{(\\mathbf{v},L)} = (L + \\mathbf{v})^{2}, then obtain ((L + \\mathbf{v}) \\varepsilon_{0}{(\\mathbf{v},L)})^{L} = (\\varepsilon_{0}^{2}{(\\mathbf{v},L)})^{L}", "derivation": "\\varepsilon_{0}{(\\mathbf{v},L)} = L + \\mathbf{v} and (L + \\mathbf{v}) \\varepsilon_{0}{(\\mathbf{v},L)} = (L + \\mathbf{v})^{2} and ((L + \\mathbf{v}) \\varepsilon_{0}{(\\mathbf{v},L)})^{L} = ((L + \\mathbf{v})^{2})^{L} and \\mathbf{p}{(\\mathbf{v},L)} = (L + \\mathbf{v})^{2} and \\mathbf{p}{(\\mathbf{v},L)} = \\varepsilon_{0}^{2}{(\\mathbf{v},L)} and \\varepsilon_{0}^{2}{(\\mathbf{v},L)} = (L + \\mathbf{v})^{2} and ((L + \\mathbf{v}) \\varepsilon_{0}{(\\mathbf{v},L)})^{L} = (\\varepsilon_{0}^{2}{(\\mathbf{v},L)})^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 1, "Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True))), Pow(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Pow(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Pow(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Integer(2)), Pow(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Mul(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('L', commutative=True)), Integer(2)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given i{(a)} = e^{a}, then obtain (\\sigma_p + e^{a}) i{(a)} = (\\sigma_p + e^{a}) e^{a}", "derivation": "i{(a)} = e^{a} and \\sigma_p + i{(a)} = \\sigma_p + e^{a} and (\\sigma_p + i{(a)}) i{(a)} = (\\sigma_p + i{(a)}) e^{a} and (\\sigma_p + e^{a}) i{(a)} = (\\sigma_p + e^{a}) e^{a}", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('i')(Symbol('a', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('a', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\sigma_p', commutative=True), Function('i')(Symbol('a', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('i')(Symbol('a', commutative=True))), Function('i')(Symbol('a', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('i')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('a', commutative=True))), Function('i')(Symbol('a', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(s,\\hat{X})} = s + \\cos{(\\hat{X})} and \\theta_{1}{(s,\\hat{X})} = s (s + \\cos{(\\hat{X})})^{\\hat{X}}, then obtain (s + \\cos{(\\hat{X})}) \\frac{\\partial}{\\partial s} \\theta_{1}{(s,\\hat{X})} = (s + \\cos{(\\hat{X})}) \\frac{\\partial}{\\partial s} s (s + \\cos{(\\hat{X})})^{\\hat{X}}", "derivation": "\\dot{\\mathbf{r}}{(s,\\hat{X})} = s + \\cos{(\\hat{X})} and \\theta_{1}{(s,\\hat{X})} = s (s + \\cos{(\\hat{X})})^{\\hat{X}} and \\frac{\\partial}{\\partial s} \\theta_{1}{(s,\\hat{X})} = \\frac{\\partial}{\\partial s} s (s + \\cos{(\\hat{X})})^{\\hat{X}} and \\dot{\\mathbf{r}}{(s,\\hat{X})} \\frac{\\partial}{\\partial s} \\theta_{1}{(s,\\hat{X})} = \\dot{\\mathbf{r}}{(s,\\hat{X})} \\frac{\\partial}{\\partial s} s (s + \\cos{(\\hat{X})})^{\\hat{X}} and (s + \\cos{(\\hat{X})}) \\frac{\\partial}{\\partial s} \\theta_{1}{(s,\\hat{X})} = (s + \\cos{(\\hat{X})}) \\frac{\\partial}{\\partial s} s (s + \\cos{(\\hat{X})})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Derivative(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Derivative(Function('\\\\theta_1')(Symbol('s', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Derivative(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), cos(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(V_{\\mathbf{B}},\\mu_0)} = \\mu_0 \\log{(V_{\\mathbf{B}})}, then derive \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\chi{(V_{\\mathbf{B}},\\mu_0)} = \\mu_0 + \\frac{\\mu_0}{V_{\\mathbf{B}}}, then obtain \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mu_0 \\log{(V_{\\mathbf{B}})} = \\mu_0 + \\frac{\\mu_0}{V_{\\mathbf{B}}}", "derivation": "\\chi{(V_{\\mathbf{B}},\\mu_0)} = \\mu_0 \\log{(V_{\\mathbf{B}})} and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\chi{(V_{\\mathbf{B}},\\mu_0)} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mu_0 \\log{(V_{\\mathbf{B}})} and \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\chi{(V_{\\mathbf{B}},\\mu_0)} = \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mu_0 \\log{(V_{\\mathbf{B}})} and \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\chi{(V_{\\mathbf{B}},\\mu_0)} = \\mu_0 + \\frac{\\mu_0}{V_{\\mathbf{B}}} and \\mu_0 + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mu_0 \\log{(V_{\\mathbf{B}})} = \\mu_0 + \\frac{\\mu_0}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('\\\\chi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Add(Symbol('\\\\mu_0', commutative=True), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('\\\\chi')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{d},\\eta)} = C_{d} - \\eta, then derive \\int (C_{d} + \\Psi^{\\dagger}{(C_{d},\\eta)}) d\\eta = A_{z} + 2 C_{d} \\eta - \\frac{\\eta^{2}}{2}, then obtain \\eta (A_{z} + 2 C_{d} \\eta - \\frac{\\eta^{2}}{2}) = \\eta \\int (2 C_{d} - \\eta) d\\eta", "derivation": "\\Psi^{\\dagger}{(C_{d},\\eta)} = C_{d} - \\eta and C_{d} + \\Psi^{\\dagger}{(C_{d},\\eta)} = 2 C_{d} - \\eta and \\int (C_{d} + \\Psi^{\\dagger}{(C_{d},\\eta)}) d\\eta = \\int (2 C_{d} - \\eta) d\\eta and \\eta \\int (C_{d} + \\Psi^{\\dagger}{(C_{d},\\eta)}) d\\eta = \\eta \\int (2 C_{d} - \\eta) d\\eta and \\int (C_{d} + \\Psi^{\\dagger}{(C_{d},\\eta)}) d\\eta = A_{z} + 2 C_{d} \\eta - \\frac{\\eta^{2}}{2} and \\eta (A_{z} + 2 C_{d} \\eta - \\frac{\\eta^{2}}{2}) = \\eta \\int (2 C_{d} - \\eta) d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Symbol('C_d', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["times", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Integral(Add(Symbol('C_d', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Symbol('\\\\eta', commutative=True), Integral(Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('C_d', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('A_z', commutative=True), Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Symbol('\\\\eta', commutative=True), Add(Symbol('A_z', commutative=True), Mul(Integer(2), Symbol('C_d', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))))), Mul(Symbol('\\\\eta', commutative=True), Integral(Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\tilde{g})} = e^{\\tilde{g}}, then derive \\int \\nabla{(\\tilde{g})} d\\tilde{g} = p + e^{\\tilde{g}}, then derive \\hat{\\mathbf{r}} + e^{\\tilde{g}} = p + e^{\\tilde{g}}, then obtain (\\hat{\\mathbf{r}} + e^{\\tilde{g}})^{\\tilde{g}} = (p + e^{\\tilde{g}})^{\\tilde{g}}", "derivation": "\\nabla{(\\tilde{g})} = e^{\\tilde{g}} and \\int \\nabla{(\\tilde{g})} d\\tilde{g} = \\int e^{\\tilde{g}} d\\tilde{g} and \\int \\nabla{(\\tilde{g})} d\\tilde{g} = p + e^{\\tilde{g}} and \\int e^{\\tilde{g}} d\\tilde{g} = p + e^{\\tilde{g}} and \\hat{\\mathbf{r}} + e^{\\tilde{g}} = p + e^{\\tilde{g}} and (\\hat{\\mathbf{r}} + e^{\\tilde{g}})^{\\tilde{g}} = (p + e^{\\tilde{g}})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('p', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('p', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('p', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 5, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Add(Symbol('p', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(I)} = \\sin{(I)} and \\operatorname{n_{2}}{(I)} = \\log{(\\sin{(I)})} and \\mu{(I)} = \\operatorname{n_{2}}{(I)} + \\int \\log{(\\sin{(I)})} dI, then obtain \\operatorname{n_{2}}{(I)} + \\int \\log{(\\mathbf{f}{(I)})} dI = \\log{(\\sin{(I)})} + \\int \\log{(\\mathbf{f}{(I)})} dI", "derivation": "\\mathbf{f}{(I)} = \\sin{(I)} and \\log{(\\mathbf{f}{(I)})} = \\log{(\\sin{(I)})} and \\operatorname{n_{2}}{(I)} = \\log{(\\sin{(I)})} and \\int \\log{(\\mathbf{f}{(I)})} dI = \\int \\log{(\\sin{(I)})} dI and \\mu{(I)} = \\operatorname{n_{2}}{(I)} + \\int \\log{(\\sin{(I)})} dI and \\mu{(I)} = \\operatorname{n_{2}}{(I)} + \\int \\log{(\\mathbf{f}{(I)})} dI and \\mu{(I)} = \\log{(\\sin{(I)})} + \\int \\log{(\\mathbf{f}{(I)})} dI and \\operatorname{n_{2}}{(I)} + \\int \\log{(\\mathbf{f}{(I)})} dI = \\log{(\\sin{(I)})} + \\int \\log{(\\mathbf{f}{(I)})} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), log(sin(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('I', commutative=True)), log(sin(Symbol('I', commutative=True))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(log(sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('I', commutative=True)), Add(Function('n_2')(Symbol('I', commutative=True)), Integral(log(sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mu')(Symbol('I', commutative=True)), Add(Function('n_2')(Symbol('I', commutative=True)), Integral(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Function('\\\\mu')(Symbol('I', commutative=True)), Add(log(sin(Symbol('I', commutative=True))), Integral(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Function('n_2')(Symbol('I', commutative=True)), Integral(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Add(log(sin(Symbol('I', commutative=True))), Integral(log(Function('\\\\mathbf{f}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\operatorname{f^{*}}{(\\sigma_p)} = \\int 2 \\cos{(\\sigma_p)} d\\sigma_p, then obtain \\operatorname{f^{*}}{(\\sigma_p)} = \\int (\\varphi{(\\sigma_p)} + \\cos{(\\sigma_p)}) d\\sigma_p", "derivation": "\\varphi{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\varphi{(\\sigma_p)} + \\cos{(\\sigma_p)} = 2 \\cos{(\\sigma_p)} and \\int (\\varphi{(\\sigma_p)} + \\cos{(\\sigma_p)}) d\\sigma_p = \\int 2 \\cos{(\\sigma_p)} d\\sigma_p and \\operatorname{f^{*}}{(\\sigma_p)} = \\int 2 \\cos{(\\sigma_p)} d\\sigma_p and \\operatorname{f^{*}}{(\\sigma_p)} = \\int (\\varphi{(\\sigma_p)} + \\cos{(\\sigma_p)}) d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\sigma_p', commutative=True)), Integral(Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('f^*')(Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Function('\\\\varphi')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(F_{N})} = \\log{(F_{N})}, then obtain 2 F_{N} + \\theta_{1}^{F_{N}}{(F_{N})} + \\log{(F_{N})}^{F_{N}} = 2 F_{N} + 2 \\log{(F_{N})}^{F_{N}}", "derivation": "\\theta_{1}{(F_{N})} = \\log{(F_{N})} and \\theta_{1}^{F_{N}}{(F_{N})} = \\log{(F_{N})}^{F_{N}} and F_{N} + \\theta_{1}^{F_{N}}{(F_{N})} = F_{N} + \\log{(F_{N})}^{F_{N}} and 2 F_{N} + \\theta_{1}^{F_{N}}{(F_{N})} + \\log{(F_{N})}^{F_{N}} = 2 F_{N} + 2 \\log{(F_{N})}^{F_{N}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], [["add", 2, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))))"], [["add", 3, "Add(Symbol('F_N', commutative=True), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Pow(log(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given J{(u)} = \\sin{(u)}, then obtain J^{u}{(u)} - (\\int J^{u}{(u)} du)^{u} = \\sin^{u}{(u)} - (\\int J^{u}{(u)} du)^{u}", "derivation": "J{(u)} = \\sin{(u)} and J^{u}{(u)} = \\sin^{u}{(u)} and \\int J^{u}{(u)} du = \\int \\sin^{u}{(u)} du and J^{u}{(u)} - (\\int \\sin^{u}{(u)} du)^{u} = \\sin^{u}{(u)} - (\\int \\sin^{u}{(u)} du)^{u} and J^{u}{(u)} - (\\int J^{u}{(u)} du)^{u} = \\sin^{u}{(u)} - (\\int J^{u}{(u)} du)^{u}", "srepr_derivation": [["get_premise", "Equality(Function('J')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["minus", 2, "Pow(Integral(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))"], "Equality(Add(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))), Add(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))), Add(Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Function('J')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then obtain (\\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\operatorname{P_{g}}{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (\\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\sin{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and \\mathbf{S} \\operatorname{P_{g}}{(\\mathbf{S})} = \\mathbf{S} \\sin{(\\mathbf{S})} and \\int \\mathbf{S} \\operatorname{P_{g}}{(\\mathbf{S})} d\\mathbf{S} = \\int \\mathbf{S} \\sin{(\\mathbf{S})} d\\mathbf{S} and \\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\operatorname{P_{g}}{(\\mathbf{S})} d\\mathbf{S} = \\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\sin{(\\mathbf{S})} d\\mathbf{S} and (\\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\operatorname{P_{g}}{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (\\frac{d}{d \\mathbf{S}} \\int \\mathbf{S} \\sin{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('P_g')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('P_g')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('P_g')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Derivative(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('P_g')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(P_{g},\\delta)} = P_{g} + \\delta, then derive e^{\\mathbf{J}{(P_{g},\\delta)}} \\frac{\\partial}{\\partial \\delta} \\mathbf{J}{(P_{g},\\delta)} = e^{P_{g} + \\delta}, then obtain \\Psi^{\\dagger} e^{P_{g} + \\delta} \\frac{\\partial}{\\partial \\delta} (P_{g} + \\delta) = \\Psi^{\\dagger} e^{P_{g} + \\delta}", "derivation": "\\mathbf{J}{(P_{g},\\delta)} = P_{g} + \\delta and e^{\\mathbf{J}{(P_{g},\\delta)}} = e^{P_{g} + \\delta} and \\frac{\\partial}{\\partial \\delta} e^{\\mathbf{J}{(P_{g},\\delta)}} = \\frac{\\partial}{\\partial \\delta} e^{P_{g} + \\delta} and \\frac{\\partial}{\\partial \\delta} \\mathbf{J}{(P_{g},\\delta)} = \\frac{\\partial}{\\partial \\delta} (P_{g} + \\delta) and e^{\\mathbf{J}{(P_{g},\\delta)}} \\frac{\\partial}{\\partial \\delta} \\mathbf{J}{(P_{g},\\delta)} = e^{P_{g} + \\delta} and e^{\\mathbf{J}{(P_{g},\\delta)}} \\frac{\\partial}{\\partial \\delta} (P_{g} + \\delta) = e^{P_{g} + \\delta} and \\Psi^{\\dagger} e^{\\mathbf{J}{(P_{g},\\delta)}} \\frac{\\partial}{\\partial \\delta} (P_{g} + \\delta) = \\Psi^{\\dagger} e^{P_{g} + \\delta} and \\Psi^{\\dagger} e^{P_{g} + \\delta} \\frac{\\partial}{\\partial \\delta} (P_{g} + \\delta) = \\Psi^{\\dagger} e^{P_{g} + \\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(exp(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Derivative(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(exp(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["times", 6, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Add(Symbol('P_g', commutative=True), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)} = \\hat{\\mathbf{r}} + x, then obtain (\\iint \\cos{(\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)})} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}})^{x} = (\\iint \\cos{(\\hat{\\mathbf{r}} + x)} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}})^{x}", "derivation": "\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)} = \\hat{\\mathbf{r}} + x and \\cos{(\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)})} = \\cos{(\\hat{\\mathbf{r}} + x)} and \\int \\cos{(\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)})} d\\hat{\\mathbf{r}} = \\int \\cos{(\\hat{\\mathbf{r}} + x)} d\\hat{\\mathbf{r}} and \\iint \\cos{(\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)})} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}} = \\iint \\cos{(\\hat{\\mathbf{r}} + x)} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}} and (\\iint \\cos{(\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},x)})} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}})^{x} = (\\iint \\cos{(\\hat{\\mathbf{r}} + x)} d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}})^{x}", "srepr_derivation": [["get_premise", "Equality(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True)))"], [["cos", 1], "Equality(cos(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(cos(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(cos(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 4, "Symbol('x', commutative=True)"], "Equality(Pow(Integral(cos(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(F_{x})} = e^{F_{x}} and \\mathbf{s}{(F_{x})} = e^{F_{x}}, then obtain \\frac{\\hat{X}{(F_{x})}}{\\mathbf{s}{(F_{x})}} - e^{F_{x}} = 1 - e^{F_{x}}", "derivation": "\\hat{X}{(F_{x})} = e^{F_{x}} and \\mathbf{s}{(F_{x})} = e^{F_{x}} and \\hat{X}{(F_{x})} = \\mathbf{s}{(F_{x})} and \\frac{\\hat{X}{(F_{x})}}{\\mathbf{s}{(F_{x})}} = 1 and \\frac{\\hat{X}{(F_{x})}}{\\mathbf{s}{(F_{x})}} - e^{F_{x}} = 1 - e^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{X}')(Symbol('F_x', commutative=True)), Function('\\\\mathbf{s}')(Symbol('F_x', commutative=True)))"], [["divide", 3, "Function('\\\\mathbf{s}')(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('F_x', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('F_x', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 4, "exp(Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Function('\\\\hat{X}')(Symbol('F_x', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('F_x', commutative=True)), Integer(-1))), Mul(Integer(-1), exp(Symbol('F_x', commutative=True)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given B{(f^{\\prime})} = \\log{(f^{\\prime})}, then obtain \\int \\sin{(1)} df^{\\prime} = \\int \\sin{(\\frac{\\log{(f^{\\prime})}}{B{(f^{\\prime})}})} df^{\\prime}", "derivation": "B{(f^{\\prime})} = \\log{(f^{\\prime})} and 1 = \\frac{\\log{(f^{\\prime})}}{B{(f^{\\prime})}} and \\sin{(1)} = \\sin{(\\frac{\\log{(f^{\\prime})}}{B{(f^{\\prime})}})} and \\int \\sin{(1)} df^{\\prime} = \\int \\sin{(\\frac{\\log{(f^{\\prime})}}{B{(f^{\\prime})}})} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Function('B')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('B')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["sin", 2], "Equality(sin(Integer(1)), sin(Mul(Pow(Function('B')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), log(Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(sin(Integer(1)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(sin(Mul(Pow(Function('B')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), log(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given m{(S)} = \\cos{(S)}, then derive \\int m{(S)} dS = \\mathbf{S} + \\sin{(S)}, then derive (r_{0} + \\sin{(S)}) \\int m{(S)} dS - \\cos{(S)} = - \\cos{(S)} + (\\int m{(S)} dS)^{2}, then obtain \\frac{(r_{0} + \\sin{(S)}) \\int m{(S)} dS - \\cos{(S)}}{\\int m{(S)} dS} = \\frac{- \\cos{(S)} + (\\int m{(S)} dS)^{2}}{\\int m{(S)} dS}", "derivation": "m{(S)} = \\cos{(S)} and \\int m{(S)} dS = \\int \\cos{(S)} dS and \\int m{(S)} dS = \\mathbf{S} + \\sin{(S)} and \\int \\cos{(S)} dS = \\mathbf{S} + \\sin{(S)} and (\\int m{(S)} dS) \\int \\cos{(S)} dS = (\\mathbf{S} + \\sin{(S)}) \\int m{(S)} dS and (\\int m{(S)} dS) \\int \\cos{(S)} dS = (\\int m{(S)} dS)^{2} and - \\cos{(S)} + (\\int m{(S)} dS) \\int \\cos{(S)} dS = - \\cos{(S)} + (\\int m{(S)} dS)^{2} and (r_{0} + \\sin{(S)}) \\int m{(S)} dS - \\cos{(S)} = - \\cos{(S)} + (\\int m{(S)} dS)^{2} and \\frac{(r_{0} + \\sin{(S)}) \\int m{(S)} dS - \\cos{(S)}}{\\int m{(S)} dS} = \\frac{- \\cos{(S)} + (\\int m{(S)} dS)^{2}}{\\int m{(S)} dS}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('S', commutative=True))))"], [["times", 4, "Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('S', commutative=True))), Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(2)))"], [["minus", 6, "cos(Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('S', commutative=True))), Mul(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))), Add(Mul(Integer(-1), cos(Symbol('S', commutative=True))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(2))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Add(Symbol('r_0', commutative=True), sin(Symbol('S', commutative=True))), Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('S', commutative=True))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(2))))"], [["divide", 8, "Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Mul(Add(Symbol('r_0', commutative=True), sin(Symbol('S', commutative=True))), Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), cos(Symbol('S', commutative=True))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(2))), Pow(Integral(Function('m')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\delta{(\\phi_2)} = \\cos{(\\log{(\\phi_2)})}, then derive \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} = - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2}, then obtain \\frac{\\sin{(\\log{(\\phi_2)})} + \\frac{d}{d \\phi_2} \\cos{(\\log{(\\phi_2)})}}{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)}} = \\frac{\\sin{(\\log{(\\phi_2)})} - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2}}{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)}}", "derivation": "\\delta{(\\phi_2)} = \\cos{(\\log{(\\phi_2)})} and \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} = \\frac{d}{d \\phi_2} \\cos{(\\log{(\\phi_2)})} and \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} = - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2} and \\frac{d}{d \\phi_2} \\cos{(\\log{(\\phi_2)})} = - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2} and \\sin{(\\log{(\\phi_2)})} + \\frac{d}{d \\phi_2} \\cos{(\\log{(\\phi_2)})} = \\sin{(\\log{(\\phi_2)})} - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2} and \\frac{\\sin{(\\log{(\\phi_2)})} + \\frac{d}{d \\phi_2} \\cos{(\\log{(\\phi_2)})}}{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)}} = \\frac{\\sin{(\\log{(\\phi_2)})} - \\frac{\\sin{(\\log{(\\phi_2)})}}{\\phi_2}}{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), cos(log(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(log(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(log(Symbol('\\\\phi_2', commutative=True)))))"], [["add", 4, "sin(log(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(sin(log(Symbol('\\\\phi_2', commutative=True))), Derivative(cos(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(sin(log(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(log(Symbol('\\\\phi_2', commutative=True))))))"], [["divide", 5, "Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Mul(Add(sin(log(Symbol('\\\\phi_2', commutative=True))), Derivative(cos(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(sin(log(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(log(Symbol('\\\\phi_2', commutative=True))))), Pow(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{M}{(M,f_{E})} = \\log{(M f_{E})} and \\operatorname{A_{y}}{(M,f_{E})} = M + \\mathbf{M}{(M,f_{E})}, then obtain \\frac{\\partial}{\\partial M} \\operatorname{A_{y}}{(M,f_{E})} = \\frac{\\partial}{\\partial M} (M + \\log{(M f_{E})})", "derivation": "\\mathbf{M}{(M,f_{E})} = \\log{(M f_{E})} and M + \\mathbf{M}{(M,f_{E})} = M + \\log{(M f_{E})} and \\operatorname{A_{y}}{(M,f_{E})} = M + \\mathbf{M}{(M,f_{E})} and \\operatorname{A_{y}}{(M,f_{E})} = M + \\log{(M f_{E})} and \\frac{\\partial}{\\partial M} \\operatorname{A_{y}}{(M,f_{E})} = \\frac{\\partial}{\\partial M} (M + \\log{(M f_{E})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), log(Mul(Symbol('M', commutative=True), Symbol('f_E', commutative=True))))"], [["add", 1, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('f_E', commutative=True))), Add(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('f_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('M', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('A_y')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('f_E', commutative=True)))))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mu,\\varphi,\\eta^{\\prime})} = (\\eta^{\\prime} - \\varphi)^{\\mu} and G{(\\varphi)} = \\varphi, then obtain \\int \\frac{\\mathbf{A}^{\\varphi}{(\\mu,\\varphi,\\eta^{\\prime})}}{\\varphi} d\\mu = \\int \\frac{((\\eta^{\\prime} - \\varphi)^{\\mu})^{\\varphi}}{\\varphi} d\\mu", "derivation": "\\mathbf{A}{(\\mu,\\varphi,\\eta^{\\prime})} = (\\eta^{\\prime} - \\varphi)^{\\mu} and \\mathbf{A}^{\\varphi}{(\\mu,\\varphi,\\eta^{\\prime})} = ((\\eta^{\\prime} - \\varphi)^{\\mu})^{\\varphi} and G{(\\varphi)} = \\varphi and \\frac{\\mathbf{A}^{\\varphi}{(\\mu,\\varphi,\\eta^{\\prime})}}{G{(\\varphi)}} = \\frac{((\\eta^{\\prime} - \\varphi)^{\\mu})^{\\varphi}}{G{(\\varphi)}} and \\int \\frac{\\mathbf{A}^{\\varphi}{(\\mu,\\varphi,\\eta^{\\prime})}}{G{(\\varphi)}} d\\mu = \\int \\frac{((\\eta^{\\prime} - \\varphi)^{\\mu})^{\\varphi}}{G{(\\varphi)}} d\\mu and \\int \\frac{\\mathbf{A}^{\\varphi}{(\\mu,\\varphi,\\eta^{\\prime})}}{\\varphi} d\\mu = \\int \\frac{((\\eta^{\\prime} - \\varphi)^{\\mu})^{\\varphi}}{\\varphi} d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], [["divide", 2, "Function('G')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Function('G')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Function('G')(Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Pow(Function('G')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Pow(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Function('G')(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\omega,f_{E})} = \\omega f_{E}, then obtain f_{E}^{2} + \\cos{(\\omega^{2} f_{E}^{5} \\phi^{2}{(\\omega,f_{E})})} = f_{E}^{2} + \\cos{(\\omega^{3} f_{E}^{6} \\phi{(\\omega,f_{E})})}", "derivation": "\\phi{(\\omega,f_{E})} = \\omega f_{E} and f_{E} \\phi{(\\omega,f_{E})} = \\omega f_{E}^{2} and \\omega f_{E}^{3} \\phi{(\\omega,f_{E})} = \\omega^{2} f_{E}^{4} and \\omega f_{E}^{3} \\phi^{2}{(\\omega,f_{E})} = \\omega^{2} f_{E}^{4} \\phi{(\\omega,f_{E})} and \\omega^{2} f_{E}^{4} \\phi{(\\omega,f_{E})} = \\omega^{3} f_{E}^{5} and \\omega^{2} f_{E}^{5} \\phi^{2}{(\\omega,f_{E})} = \\omega^{3} f_{E}^{6} \\phi{(\\omega,f_{E})} and \\cos{(\\omega^{2} f_{E}^{5} \\phi^{2}{(\\omega,f_{E})})} = \\cos{(\\omega^{3} f_{E}^{6} \\phi{(\\omega,f_{E})})} and f_{E}^{2} + \\cos{(\\omega^{2} f_{E}^{5} \\phi^{2}{(\\omega,f_{E})})} = f_{E}^{2} + \\cos{(\\omega^{3} f_{E}^{6} \\phi{(\\omega,f_{E})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)))"], [["times", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(2))))"], [["times", 2, "Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(2)))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(3)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(4))))"], [["times", 1, "Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(3)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(3)), Pow(Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(4)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(4)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(3)), Pow(Symbol('f_E', commutative=True), Integer(5))))"], [["times", 5, "Mul(Symbol('f_E', commutative=True), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(5)), Pow(Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(3)), Pow(Symbol('f_E', commutative=True), Integer(6)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))))"], [["cos", 6], "Equality(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(5)), Pow(Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)), Integer(2)))), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(3)), Pow(Symbol('f_E', commutative=True), Integer(6)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)))))"], [["add", 7, "Pow(Symbol('f_E', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('f_E', commutative=True), Integer(2)), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Pow(Symbol('f_E', commutative=True), Integer(5)), Pow(Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True)), Integer(2))))), Add(Pow(Symbol('f_E', commutative=True), Integer(2)), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(3)), Pow(Symbol('f_E', commutative=True), Integer(6)), Function('\\\\phi')(Symbol('\\\\omega', commutative=True), Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given n{(q,B)} = B + q, then derive q \\int n{(q,B)} dB = q (\\frac{B^{2}}{2} + B q + \\mathbf{J}), then obtain q \\int (B + q) dB = q (\\frac{B^{2}}{2} + B q + \\mathbf{J})", "derivation": "n{(q,B)} = B + q and \\int n{(q,B)} dB = \\int (B + q) dB and q \\int n{(q,B)} dB = q \\int (B + q) dB and q \\int n{(q,B)} dB = q (\\frac{B^{2}}{2} + B q + \\mathbf{J}) and q \\int (B + q) dB = q (\\frac{B^{2}}{2} + B q + \\mathbf{J})", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('n')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Symbol('B', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["times", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Integral(Function('n')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Symbol('q', commutative=True), Integral(Add(Symbol('B', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('q', commutative=True), Integral(Function('n')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Symbol('q', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Symbol('B', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('q', commutative=True), Integral(Add(Symbol('B', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Symbol('q', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Symbol('B', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(I,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{I}, then derive \\frac{\\partial}{\\partial I} \\operatorname{n_{1}}{(I,\\mathbf{J}_M)} = - \\frac{\\mathbf{J}_M}{I^{2}}, then obtain (\\frac{\\mathbf{J}_M}{I})^{I} + \\frac{\\partial}{\\partial I} \\frac{\\mathbf{J}_M}{I} = (\\frac{\\mathbf{J}_M}{I})^{I} - \\frac{\\mathbf{J}_M}{I^{2}}", "derivation": "\\operatorname{n_{1}}{(I,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{I} and \\frac{\\partial}{\\partial I} \\operatorname{n_{1}}{(I,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial I} \\frac{\\mathbf{J}_M}{I} and \\frac{\\partial}{\\partial I} \\operatorname{n_{1}}{(I,\\mathbf{J}_M)} = - \\frac{\\mathbf{J}_M}{I^{2}} and \\frac{\\partial}{\\partial I} \\frac{\\mathbf{J}_M}{I} = - \\frac{\\mathbf{J}_M}{I^{2}} and (\\frac{\\mathbf{J}_M}{I})^{I} + \\frac{\\partial}{\\partial I} \\frac{\\mathbf{J}_M}{I} = (\\frac{\\mathbf{J}_M}{I})^{I} - \\frac{\\mathbf{J}_M}{I^{2}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 4, "Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('I', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('I', commutative=True)), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given y{(F_{H})} = \\int \\sin{(F_{H})} dF_{H} and \\operatorname{f_{E}}{(F_{H})} = \\frac{\\int y{(F_{H})} dF_{H}}{F_{H}}, then obtain \\operatorname{f_{E}}{(F_{H})} - y{(F_{H})} = - y{(F_{H})} + \\frac{\\iint \\sin{(F_{H})} dF_{H} dF_{H}}{F_{H}}", "derivation": "y{(F_{H})} = \\int \\sin{(F_{H})} dF_{H} and \\int y{(F_{H})} dF_{H} = \\iint \\sin{(F_{H})} dF_{H} dF_{H} and \\operatorname{f_{E}}{(F_{H})} = \\frac{\\int y{(F_{H})} dF_{H}}{F_{H}} and \\operatorname{f_{E}}{(F_{H})} = \\frac{\\iint \\sin{(F_{H})} dF_{H} dF_{H}}{F_{H}} and \\operatorname{f_{E}}{(F_{H})} - y{(F_{H})} = - y{(F_{H})} + \\frac{\\iint \\sin{(F_{H})} dF_{H} dF_{H}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('F_H', commutative=True)), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('y')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Integral(Function('y')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('f_E')(Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["minus", 4, "Function('y')(Symbol('F_H', commutative=True))"], "Equality(Add(Function('f_E')(Symbol('F_H', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('F_H', commutative=True)))), Add(Mul(Integer(-1), Function('y')(Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(F_{x})} = \\log{(\\sin{(F_{x})})}, then obtain (\\iint \\operatorname{v_{t}}{(F_{x})} dF_{x} dF_{x})^{F_{x}} = (\\iint \\log{(\\sin{(F_{x})})} dF_{x} dF_{x})^{F_{x}}", "derivation": "\\operatorname{v_{t}}{(F_{x})} = \\log{(\\sin{(F_{x})})} and \\int \\operatorname{v_{t}}{(F_{x})} dF_{x} = \\int \\log{(\\sin{(F_{x})})} dF_{x} and \\iint \\operatorname{v_{t}}{(F_{x})} dF_{x} dF_{x} = \\iint \\log{(\\sin{(F_{x})})} dF_{x} dF_{x} and (\\iint \\operatorname{v_{t}}{(F_{x})} dF_{x} dF_{x})^{F_{x}} = (\\iint \\log{(\\sin{(F_{x})})} dF_{x} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('F_x', commutative=True)), log(sin(Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(log(sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["integrate", 2, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(log(sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Integral(Function('v_t')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Integral(log(sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(\\hat{x}_0,W)} = e^{\\hat{x}_0^{W}}, then obtain (\\hat{x}_0 + (\\mathbf{D}{(\\hat{x}_0,W)} - e^{\\hat{x}_0^{W}})^{W})^{W} = (0^{W} + \\hat{x}_0)^{W}", "derivation": "\\mathbf{D}{(\\hat{x}_0,W)} = e^{\\hat{x}_0^{W}} and \\mathbf{D}{(\\hat{x}_0,W)} - e^{\\hat{x}_0^{W}} = 0 and (\\mathbf{D}{(\\hat{x}_0,W)} - e^{\\hat{x}_0^{W}})^{W} = 0^{W} and \\hat{x}_0 + (\\mathbf{D}{(\\hat{x}_0,W)} - e^{\\hat{x}_0^{W}})^{W} = 0^{W} + \\hat{x}_0 and (\\hat{x}_0 + (\\mathbf{D}{(\\hat{x}_0,W)} - e^{\\hat{x}_0^{W}})^{W})^{W} = (0^{W} + \\hat{x}_0)^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)), exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True))))"], [["minus", 1, "exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True))))), Symbol('W', commutative=True)), Pow(Integer(0), Symbol('W', commutative=True)))"], [["add", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Pow(Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True))))), Symbol('W', commutative=True))), Add(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Pow(Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('W', commutative=True))))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('W', commutative=True)))"]]}, {"prompt": "Given y{(v_{1})} = e^{e^{v_{1}}}, then obtain 0 = (- y{(v_{1})} e^{- v_{1}} + e^{- v_{1}} e^{e^{v_{1}}}) e^{- e^{v_{1}}}", "derivation": "y{(v_{1})} = e^{e^{v_{1}}} and y{(v_{1})} e^{- v_{1}} = e^{- v_{1}} e^{e^{v_{1}}} and 0 = - y{(v_{1})} e^{- v_{1}} + e^{- v_{1}} e^{e^{v_{1}}} and 0 = (- y{(v_{1})} e^{- v_{1}} + e^{- v_{1}} e^{e^{v_{1}}}) e^{- e^{v_{1}}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True))))"], [["divide", 1, "exp(Symbol('v_1', commutative=True))"], "Equality(Mul(Function('y')(Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_1', commutative=True))), exp(exp(Symbol('v_1', commutative=True)))))"], [["minus", 2, "Mul(Function('y')(Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y')(Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_1', commutative=True))), exp(exp(Symbol('v_1', commutative=True))))))"], [["divide", 3, "exp(exp(Symbol('v_1', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('y')(Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_1', commutative=True))), exp(exp(Symbol('v_1', commutative=True))))), exp(Mul(Integer(-1), exp(Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(W,v)} = - W + v, then derive \\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} - 1 = -2, then obtain (\\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} - 1) \\frac{\\partial}{\\partial W} (- W + v) = - 2 \\frac{\\partial}{\\partial W} (- W + v)", "derivation": "\\operatorname{C_{d}}{(W,v)} = - W + v and \\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} = \\frac{\\partial}{\\partial W} (- W + v) and \\frac{\\partial}{\\partial W} (- W + v) + \\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} = 2 \\frac{\\partial}{\\partial W} (- W + v) and \\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} - 1 = -2 and (\\frac{\\partial}{\\partial W} \\operatorname{C_{d}}{(W,v)} - 1) \\frac{\\partial}{\\partial W} (- W + v) = - 2 \\frac{\\partial}{\\partial W} (- W + v)", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Function('C_d')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('C_d')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"], [["times", 4, "Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('C_d')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given B{(\\mathbf{B},\\pi)} = \\frac{\\mathbf{B}}{\\pi}, then obtain - (\\int B{(\\mathbf{B},\\pi)} d\\mathbf{B})^{\\mathbf{B}} + \\frac{B{(\\mathbf{B},\\pi)}}{\\mathbf{B}} = - (\\int B{(\\mathbf{B},\\pi)} d\\mathbf{B})^{\\mathbf{B}} + \\frac{1}{\\pi}", "derivation": "B{(\\mathbf{B},\\pi)} = \\frac{\\mathbf{B}}{\\pi} and \\frac{B{(\\mathbf{B},\\pi)}}{\\mathbf{B}} = \\frac{1}{\\pi} and \\int B{(\\mathbf{B},\\pi)} d\\mathbf{B} = \\int \\frac{\\mathbf{B}}{\\pi} d\\mathbf{B} and (\\int B{(\\mathbf{B},\\pi)} d\\mathbf{B})^{\\mathbf{B}} = (\\int \\frac{\\mathbf{B}}{\\pi} d\\mathbf{B})^{\\mathbf{B}} and - (\\int \\frac{\\mathbf{B}}{\\pi} d\\mathbf{B})^{\\mathbf{B}} + \\frac{B{(\\mathbf{B},\\pi)}}{\\mathbf{B}} = - (\\int \\frac{\\mathbf{B}}{\\pi} d\\mathbf{B})^{\\mathbf{B}} + \\frac{1}{\\pi} and - (\\int B{(\\mathbf{B},\\pi)} d\\mathbf{B})^{\\mathbf{B}} + \\frac{B{(\\mathbf{B},\\pi)}}{\\mathbf{B}} = - (\\int B{(\\mathbf{B},\\pi)} d\\mathbf{B})^{\\mathbf{B}} + \\frac{1}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integral(Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 2, "Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given z{(f_{\\mathbf{v}},c)} = c - f_{\\mathbf{v}}, then obtain (- c - z{(f_{\\mathbf{v}},c)}) (c - f_{\\mathbf{v}}) = (- 2 c + f_{\\mathbf{v}}) (c - f_{\\mathbf{v}})", "derivation": "z{(f_{\\mathbf{v}},c)} = c - f_{\\mathbf{v}} and - z{(f_{\\mathbf{v}},c)} = - c + f_{\\mathbf{v}} and - c - z{(f_{\\mathbf{v}},c)} = - 2 c + f_{\\mathbf{v}} and (- c - z{(f_{\\mathbf{v}},c)}) (c - f_{\\mathbf{v}}) = (- 2 c + f_{\\mathbf{v}}) (c - f_{\\mathbf{v}})", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 3, "Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('c', commutative=True)))), Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{S},\\hat{x})} = \\cos{(\\hat{x} + \\mathbf{S})} and \\sigma_{p}{(C_{d})} = \\sin{(e^{C_{d}})}, then obtain \\sigma_{p}{(C_{d})} - \\cos{(\\hat{x} + \\mathbf{S})} = \\sin{(e^{C_{d}})} - \\cos{(\\hat{x} + \\mathbf{S})}", "derivation": "\\mathbf{J}{(\\mathbf{S},\\hat{x})} = \\cos{(\\hat{x} + \\mathbf{S})} and \\sigma_{p}{(C_{d})} = \\sin{(e^{C_{d}})} and - \\mathbf{J}{(\\mathbf{S},\\hat{x})} + \\sigma_{p}{(C_{d})} = - \\mathbf{J}{(\\mathbf{S},\\hat{x})} + \\sin{(e^{C_{d}})} and \\sigma_{p}{(C_{d})} - \\cos{(\\hat{x} + \\mathbf{S})} = \\sin{(e^{C_{d}})} - \\cos{(\\hat{x} + \\mathbf{S})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), cos(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('C_d', commutative=True)), sin(exp(Symbol('C_d', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Function('\\\\sigma_p')(Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), sin(exp(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\sigma_p')(Symbol('C_d', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))), Add(sin(exp(Symbol('C_d', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(I)} = \\cos{(I)}, then obtain \\dot{\\mathbf{r}}^{3}{(I)} \\cos{(I)} = \\cos^{4}{(I)}", "derivation": "\\dot{\\mathbf{r}}{(I)} = \\cos{(I)} and \\dot{\\mathbf{r}}{(I)} \\cos{(I)} = \\cos^{2}{(I)} and \\dot{\\mathbf{r}}^{2}{(I)} \\cos^{2}{(I)} = \\cos^{4}{(I)} and \\dot{\\mathbf{r}}^{3}{(I)} \\cos{(I)} = \\dot{\\mathbf{r}}^{2}{(I)} \\cos^{2}{(I)} and \\dot{\\mathbf{r}}^{3}{(I)} \\cos{(I)} = \\cos^{4}{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["times", 1, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True))), Pow(cos(Symbol('I', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Integer(2)), Pow(cos(Symbol('I', commutative=True)), Integer(2))), Pow(cos(Symbol('I', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Integer(3)), cos(Symbol('I', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Integer(2)), Pow(cos(Symbol('I', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Integer(3)), cos(Symbol('I', commutative=True))), Pow(cos(Symbol('I', commutative=True)), Integer(4)))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)} and E{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)}, then derive \\frac{d}{d \\hat{x}_0} \\mathbf{r}{(\\hat{x}_0)} = - \\sin{(\\hat{x}_0)}, then obtain \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} = - E{(\\hat{x}_0)}", "derivation": "\\mathbf{r}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\mathbf{r}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\mathbf{r}{(\\hat{x}_0)} = - \\sin{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} = - \\sin{(\\hat{x}_0)} and E{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} = - E{(\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{x}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Integer(-1), Function('E')(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\nabla,v)} = v^{\\nabla}, then obtain \\int (\\nabla \\mu{(\\nabla,v)})^{v} dv = \\int (\\nabla v^{\\nabla})^{v} dv", "derivation": "\\mu{(\\nabla,v)} = v^{\\nabla} and \\nabla \\mu{(\\nabla,v)} = \\nabla v^{\\nabla} and (\\nabla \\mu{(\\nabla,v)})^{v} = (\\nabla v^{\\nabla})^{v} and \\int (\\nabla \\mu{(\\nabla,v)})^{v} dv = \\int (\\nabla v^{\\nabla})^{v} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('v', commutative=True)))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\mathbf{S},q)} = \\mathbf{S} - q, then derive \\int \\frac{\\rho{(\\mathbf{S},q)} - 1}{q} dq = F_{g} - q - (1 - \\mathbf{S}) \\log{(q)}, then obtain \\frac{\\partial}{\\partial F_{g}} \\int \\frac{\\mathbf{S} - q - 1}{q} dq = 1", "derivation": "\\rho{(\\mathbf{S},q)} = \\mathbf{S} - q and \\rho{(\\mathbf{S},q)} - 1 = \\mathbf{S} - q - 1 and \\frac{\\rho{(\\mathbf{S},q)} - 1}{q} = \\frac{\\mathbf{S} - q - 1}{q} and \\int \\frac{\\rho{(\\mathbf{S},q)} - 1}{q} dq = \\int \\frac{\\mathbf{S} - q - 1}{q} dq and \\int \\frac{\\rho{(\\mathbf{S},q)} - 1}{q} dq = F_{g} - q - (1 - \\mathbf{S}) \\log{(q)} and \\int \\frac{\\mathbf{S} - q - 1}{q} dq = F_{g} - q - (1 - \\mathbf{S}) \\log{(q)} and \\frac{\\partial}{\\partial F_{g}} \\int \\frac{\\mathbf{S} - q - 1}{q} dq = \\frac{\\partial}{\\partial F_{g}} (F_{g} - q - (1 - \\mathbf{S}) \\log{(q)}) and \\frac{\\partial}{\\partial F_{g}} \\int \\frac{\\mathbf{S} - q - 1}{q} dq = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1)))"], [["divide", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), log(Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), log(Symbol('q', commutative=True)))))"], [["differentiate", 6, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), log(Symbol('q', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given u{(S,W)} = S + W, then derive \\int u{(S,W)} dW = S W + \\frac{W^{2}}{2} + g, then obtain W \\int (S + W) dW = W (S W + \\frac{W^{2}}{2} + g)", "derivation": "u{(S,W)} = S + W and \\int u{(S,W)} dW = \\int (S + W) dW and \\int u{(S,W)} dW = S W + \\frac{W^{2}}{2} + g and \\int (S + W) dW = S W + \\frac{W^{2}}{2} + g and W \\int (S + W) dW = W (S W + \\frac{W^{2}}{2} + g)", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('S', commutative=True), Symbol('W', commutative=True)), Add(Symbol('S', commutative=True), Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('u')(Symbol('S', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('S', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('W', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('S', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('W', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('g', commutative=True)))"], [["times", 4, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Integral(Add(Symbol('S', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('W', commutative=True), Add(Mul(Symbol('S', commutative=True), Symbol('W', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{v})} = \\int \\sin{(\\mathbf{v})} d\\mathbf{v}, then obtain \\frac{d}{d \\mathbf{v}} \\hat{\\mathbf{x}}^{3}{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\hat{\\mathbf{x}}^{2}{(\\mathbf{v})} \\int \\sin{(\\mathbf{v})} d\\mathbf{v}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{v})} = \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\hat{\\mathbf{x}}^{2}{(\\mathbf{v})} = \\hat{\\mathbf{x}}{(\\mathbf{v})} \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\hat{\\mathbf{x}}^{3}{(\\mathbf{v})} = \\hat{\\mathbf{x}}^{2}{(\\mathbf{v})} \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\frac{d}{d \\mathbf{v}} \\hat{\\mathbf{x}}^{3}{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\hat{\\mathbf{x}}^{2}{(\\mathbf{v})} \\int \\sin{(\\mathbf{v})} d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["times", 2, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(3)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(\\mathbf{A})} = \\sin{(\\sin{(\\mathbf{A})})}, then obtain \\frac{(- \\mathbf{A} + \\sin{(\\sin{(\\mathbf{A})})}) r{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} = \\frac{(- \\mathbf{A} + \\sin{(\\sin{(\\mathbf{A})})}) \\sin{(\\sin{(\\mathbf{A})})}}{\\sin{(\\mathbf{A})}}", "derivation": "r{(\\mathbf{A})} = \\sin{(\\sin{(\\mathbf{A})})} and - \\mathbf{A} + r{(\\mathbf{A})} = - \\mathbf{A} + \\sin{(\\sin{(\\mathbf{A})})} and \\frac{r{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} = \\frac{\\sin{(\\sin{(\\mathbf{A})})}}{\\sin{(\\mathbf{A})}} and \\frac{(- \\mathbf{A} + r{(\\mathbf{A})}) r{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} = \\frac{(- \\mathbf{A} + r{(\\mathbf{A})}) \\sin{(\\sin{(\\mathbf{A})})}}{\\sin{(\\mathbf{A})}} and \\frac{(- \\mathbf{A} + \\sin{(\\sin{(\\mathbf{A})})}) r{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} = \\frac{(- \\mathbf{A} + \\sin{(\\sin{(\\mathbf{A})})}) \\sin{(\\sin{(\\mathbf{A})})}}{\\sin{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{A}', commutative=True)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('r')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["divide", 1, "sin(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('r')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('r')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('r')(Symbol('\\\\mathbf{A}', commutative=True))), Function('r')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('r')(Symbol('\\\\mathbf{A}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))), Function('r')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given B{(M_{E})} = \\log{(\\sin{(M_{E})})} and \\operatorname{x^{{\\}'}}{(M_{E})} = \\log{(\\sin{(M_{E})})}, then obtain \\int (M_{E} + \\operatorname{x^{{\\}'}}{(M_{E})})^{M_{E}} dM_{E} = \\int (M_{E} + B{(M_{E})})^{M_{E}} dM_{E}", "derivation": "B{(M_{E})} = \\log{(\\sin{(M_{E})})} and M_{E} + B{(M_{E})} = M_{E} + \\log{(\\sin{(M_{E})})} and (M_{E} + B{(M_{E})})^{M_{E}} = (M_{E} + \\log{(\\sin{(M_{E})})})^{M_{E}} and \\operatorname{x^{{\\}'}}{(M_{E})} = \\log{(\\sin{(M_{E})})} and M_{E} + B{(M_{E})} = M_{E} + \\operatorname{x^{{\\}'}}{(M_{E})} and (M_{E} + \\operatorname{x^{{\\}'}}{(M_{E})})^{M_{E}} = (M_{E} + \\log{(\\sin{(M_{E})})})^{M_{E}} and (M_{E} + \\operatorname{x^{{\\}'}}{(M_{E})})^{M_{E}} = (M_{E} + B{(M_{E})})^{M_{E}} and \\int (M_{E} + \\operatorname{x^{{\\}'}}{(M_{E})})^{M_{E}} dM_{E} = \\int (M_{E} + B{(M_{E})})^{M_{E}} dM_{E}", "srepr_derivation": [["get_premise", "Equality(Function('B')(Symbol('M_E', commutative=True)), log(sin(Symbol('M_E', commutative=True))))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('B')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), log(sin(Symbol('M_E', commutative=True)))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Symbol('M_E', commutative=True), Function('B')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), log(sin(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('M_E', commutative=True)), log(sin(Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('M_E', commutative=True), Function('B')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Function('x^\\\\prime')(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Symbol('M_E', commutative=True), Function('x^\\\\prime')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), log(sin(Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Symbol('M_E', commutative=True), Function('x^\\\\prime')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), Function('B')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"], [["integrate", 7, "Symbol('M_E', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('M_E', commutative=True), Function('x^\\\\prime')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Pow(Add(Symbol('M_E', commutative=True), Function('B')(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\omega,\\pi)} = \\omega^{\\pi}, then obtain \\mu^{- \\pi}{(\\omega,\\pi)} (\\int \\mu{(\\omega,\\pi)} d\\omega)^{\\pi} = \\mu^{- \\pi}{(\\omega,\\pi)} (\\int \\omega^{\\pi} d\\omega)^{\\pi}", "derivation": "\\mu{(\\omega,\\pi)} = \\omega^{\\pi} and \\int \\mu{(\\omega,\\pi)} d\\omega = \\int \\omega^{\\pi} d\\omega and (\\int \\mu{(\\omega,\\pi)} d\\omega)^{\\pi} = (\\int \\omega^{\\pi} d\\omega)^{\\pi} and \\mu^{- \\pi}{(\\omega,\\pi)} (\\int \\mu{(\\omega,\\pi)} d\\omega)^{\\pi} = \\mu^{- \\pi}{(\\omega,\\pi)} (\\int \\omega^{\\pi} d\\omega)^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["divide", 3, "Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Pow(Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(f^{*},\\theta_1,\\eta)} = \\eta + \\theta_1 + f^{*}, then derive \\int \\operatorname{v_{z}}{(f^{*},\\theta_1,\\eta)} df^{*} = \\mathbf{J}_f + \\frac{(f^{*})^{2}}{2} + f^{*} (\\eta + \\theta_1), then obtain \\frac{\\partial}{\\partial \\theta_1} \\int (\\eta + \\theta_1 + f^{*}) df^{*} = \\frac{\\partial}{\\partial \\theta_1} (\\mathbf{J}_f + \\frac{(f^{*})^{2}}{2} + f^{*} (\\eta + \\theta_1))", "derivation": "\\operatorname{v_{z}}{(f^{*},\\theta_1,\\eta)} = \\eta + \\theta_1 + f^{*} and \\int \\operatorname{v_{z}}{(f^{*},\\theta_1,\\eta)} df^{*} = \\int (\\eta + \\theta_1 + f^{*}) df^{*} and \\int \\operatorname{v_{z}}{(f^{*},\\theta_1,\\eta)} df^{*} = \\mathbf{J}_f + \\frac{(f^{*})^{2}}{2} + f^{*} (\\eta + \\theta_1) and \\int (\\eta + \\theta_1 + f^{*}) df^{*} = \\mathbf{J}_f + \\frac{(f^{*})^{2}}{2} + f^{*} (\\eta + \\theta_1) and \\frac{\\partial}{\\partial \\theta_1} \\int (\\eta + \\theta_1 + f^{*}) df^{*} = \\frac{\\partial}{\\partial \\theta_1} (\\mathbf{J}_f + \\frac{(f^{*})^{2}}{2} + f^{*} (\\eta + \\theta_1))", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('f^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('f^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_z')(Symbol('f^*', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Mul(Symbol('f^*', commutative=True), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Mul(Symbol('f^*', commutative=True), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Mul(Symbol('f^*', commutative=True), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(S)} = e^{\\cos{(S)}}, then derive (\\frac{S \\frac{d}{d S} \\hat{X}{(S)}}{\\hat{X}{(S)}} + \\log{(\\hat{X}{(S)})}) \\hat{X}^{S}{(S)} = (- S \\sin{(S)} + \\log{(e^{\\cos{(S)}})}) (e^{\\cos{(S)}})^{S}, then obtain (\\frac{S \\frac{d}{d S} \\hat{X}{(S)}}{\\hat{X}{(S)}} + \\log{(\\hat{X}{(S)})}) (e^{\\cos{(S)}})^{S} = (- S \\sin{(S)} + \\log{(e^{\\cos{(S)}})}) (e^{\\cos{(S)}})^{S}", "derivation": "\\hat{X}{(S)} = e^{\\cos{(S)}} and \\hat{X}^{S}{(S)} = (e^{\\cos{(S)}})^{S} and \\frac{d}{d S} \\hat{X}^{S}{(S)} = \\frac{d}{d S} (e^{\\cos{(S)}})^{S} and (\\frac{S \\frac{d}{d S} \\hat{X}{(S)}}{\\hat{X}{(S)}} + \\log{(\\hat{X}{(S)})}) \\hat{X}^{S}{(S)} = (- S \\sin{(S)} + \\log{(e^{\\cos{(S)}})}) (e^{\\cos{(S)}})^{S} and (\\frac{S \\frac{d}{d S} \\hat{X}{(S)}}{\\hat{X}{(S)}} + \\log{(\\hat{X}{(S)})}) (e^{\\cos{(S)}})^{S} = (- S \\sin{(S)} + \\log{(e^{\\cos{(S)}})}) (e^{\\cos{(S)}})^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('S', commutative=True)), exp(cos(Symbol('S', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(exp(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(exp(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('S', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), log(Function('\\\\hat{X}')(Symbol('S', commutative=True)))), Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), log(exp(cos(Symbol('S', commutative=True))))), Pow(exp(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('S', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), log(Function('\\\\hat{X}')(Symbol('S', commutative=True)))), Pow(exp(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), log(exp(cos(Symbol('S', commutative=True))))), Pow(exp(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\rho)} = \\int e^{\\rho} d\\rho, then derive \\hat{x}_0{(\\rho)} = c_{0} + e^{\\rho}, then derive c_{0} + e^{\\rho} = r_{0} + e^{\\rho}, then obtain \\int \\frac{1}{\\theta_1} d\\theta_1 = \\int \\frac{c_{0} + e^{\\rho}}{\\theta_1 \\hat{x}_0{(\\rho)}} d\\theta_1", "derivation": "\\hat{x}_0{(\\rho)} = \\int e^{\\rho} d\\rho and \\hat{x}_0{(\\rho)} = c_{0} + e^{\\rho} and c_{0} + e^{\\rho} = \\int e^{\\rho} d\\rho and c_{0} + e^{\\rho} = r_{0} + e^{\\rho} and \\hat{x}_0{(\\rho)} = r_{0} + e^{\\rho} and 1 = \\frac{r_{0} + e^{\\rho}}{\\hat{x}_0{(\\rho)}} and 1 = \\frac{c_{0} + e^{\\rho}}{\\hat{x}_0{(\\rho)}} and \\frac{1}{\\theta_1} = \\frac{c_{0} + e^{\\rho}}{\\theta_1 \\hat{x}_0{(\\rho)}} and \\int \\frac{1}{\\theta_1} d\\theta_1 = \\int \\frac{c_{0} + e^{\\rho}}{\\theta_1 \\hat{x}_0{(\\rho)}} d\\theta_1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Add(Symbol('r_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Add(Symbol('r_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["divide", 5, "Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('r_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integer(1), Mul(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["divide", 7, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["integrate", 8, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(a^{\\dagger})} = \\cos{(\\sin{(a^{\\dagger})})}, then obtain - \\frac{d}{d a^{\\dagger}} (\\int \\rho_{b}{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = - \\frac{d}{d a^{\\dagger}} (\\int \\cos{(\\sin{(a^{\\dagger})})} da^{\\dagger})^{a^{\\dagger}}", "derivation": "\\rho_{b}{(a^{\\dagger})} = \\cos{(\\sin{(a^{\\dagger})})} and \\int \\rho_{b}{(a^{\\dagger})} da^{\\dagger} = \\int \\cos{(\\sin{(a^{\\dagger})})} da^{\\dagger} and (\\int \\rho_{b}{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = (\\int \\cos{(\\sin{(a^{\\dagger})})} da^{\\dagger})^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} (\\int \\rho_{b}{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = \\frac{d}{d a^{\\dagger}} (\\int \\cos{(\\sin{(a^{\\dagger})})} da^{\\dagger})^{a^{\\dagger}} and - \\frac{d}{d a^{\\dagger}} (\\int \\rho_{b}{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = - \\frac{d}{d a^{\\dagger}} (\\int \\cos{(\\sin{(a^{\\dagger})})} da^{\\dagger})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('a^{\\\\dagger}', commutative=True)), cos(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho_b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\rho_b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(Integral(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Pow(Integral(Function('\\\\rho_b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Pow(Integral(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(L)} = e^{L}, then derive \\int \\mathbf{P}{(L)} dL = t + e^{L}, then obtain 2 e^{L} \\int e^{L} dL = 2 (t + e^{L}) e^{L}", "derivation": "\\mathbf{P}{(L)} = e^{L} and \\int \\mathbf{P}{(L)} dL = \\int e^{L} dL and \\int \\mathbf{P}{(L)} dL = t + e^{L} and \\int e^{L} dL = t + e^{L} and 2 e^{L} \\int e^{L} dL = 2 (t + e^{L}) e^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('L', commutative=True))))"], [["times", 4, "Mul(Integer(2), exp(Symbol('L', commutative=True)))"], "Equality(Mul(Integer(2), exp(Symbol('L', commutative=True)), Integral(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Mul(Integer(2), Add(Symbol('t', commutative=True), exp(Symbol('L', commutative=True))), exp(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(J,\\hat{H}_l)} = \\int (J + \\hat{H}_l) dJ, then derive \\mu_{0}{(J,\\hat{H}_l)} = \\frac{J^{2}}{2} + J \\hat{H}_l + n, then obtain \\frac{J^{2}}{2} + n = - J \\hat{H}_l + \\int (J + \\hat{H}_l) dJ", "derivation": "\\mu_{0}{(J,\\hat{H}_l)} = \\int (J + \\hat{H}_l) dJ and \\mu_{0}{(J,\\hat{H}_l)} = \\frac{J^{2}}{2} + J \\hat{H}_l + n and - J \\hat{H}_l + \\mu_{0}{(J,\\hat{H}_l)} = - J \\hat{H}_l + \\int (J + \\hat{H}_l) dJ and \\frac{J^{2}}{2} + n = - J \\hat{H}_l + \\int (J + \\hat{H}_l) dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mu_0')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('n', commutative=True)))"], [["minus", 1, "Mul(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mu_0')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(F_{g},\\varphi)} = F_{g} - \\varphi, then derive (\\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)})^{\\varphi} = 1, then obtain \\int (\\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)})^{\\varphi} dF_{g} = \\int 1 dF_{g}", "derivation": "\\mathbf{E}{(F_{g},\\varphi)} = F_{g} - \\varphi and \\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)} = \\frac{\\partial}{\\partial F_{g}} (F_{g} - \\varphi) and (\\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)})^{\\varphi} = (\\frac{\\partial}{\\partial F_{g}} (F_{g} - \\varphi))^{\\varphi} and (\\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)})^{\\varphi} = 1 and \\int (\\frac{\\partial}{\\partial F_{g}} \\mathbf{E}{(F_{g},\\varphi)})^{\\varphi} dF_{g} = \\int 1 dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('F_g', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(W)} = \\cos{(W)} and \\hat{H}{(W)} = 2 \\operatorname{L_{\\varepsilon}}{(W)}, then obtain \\frac{d}{d W} (\\operatorname{L_{\\varepsilon}}{(W)} + \\cos{(W)}) = \\frac{d}{d W} 2 \\cos{(W)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(W)} = \\cos{(W)} and 2 \\operatorname{L_{\\varepsilon}}{(W)} = \\operatorname{L_{\\varepsilon}}{(W)} + \\cos{(W)} and \\frac{d}{d W} 2 \\operatorname{L_{\\varepsilon}}{(W)} = \\frac{d}{d W} (\\operatorname{L_{\\varepsilon}}{(W)} + \\cos{(W)}) and \\hat{H}{(W)} = 2 \\operatorname{L_{\\varepsilon}}{(W)} and \\frac{d}{d W} \\hat{H}{(W)} = \\frac{d}{d W} (\\operatorname{L_{\\varepsilon}}{(W)} + \\cos{(W)}) and \\frac{d}{d W} \\hat{H}{(W)} = \\frac{d}{d W} 2 \\cos{(W)} and \\frac{d}{d W} (\\operatorname{L_{\\varepsilon}}{(W)} + \\cos{(W)}) = \\frac{d}{d W} 2 \\cos{(W)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["add", 1, "Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True))"], "Equality(Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True))), Add(Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Add(Function('L_{\\\\varepsilon}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(t_{2})} = t_{2}, then derive \\nabla + \\frac{T^{2}{(t_{2})}}{2} = \\int t_{2} dT{(t_{2})}, then obtain \\mathbf{g} + \\nabla + \\frac{t_{2}^{2}}{2} = \\mathbf{g} + \\int t_{2} dt_{2}", "derivation": "T{(t_{2})} = t_{2} and \\int T{(t_{2})} dt_{2} = \\int t_{2} dt_{2} and \\int T{(t_{2})} dT{(t_{2})} = \\int t_{2} dT{(t_{2})} and \\nabla + \\frac{T^{2}{(t_{2})}}{2} = \\int t_{2} dT{(t_{2})} and \\mathbf{g} + \\nabla + \\frac{T^{2}{(t_{2})}}{2} = \\mathbf{g} + \\int t_{2} dT{(t_{2})} and \\mathbf{g} + \\nabla + \\frac{t_{2}^{2}}{2} = \\mathbf{g} + \\int t_{2} dt_{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('T')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('T')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('T')(Symbol('t_2', commutative=True)), Tuple(Function('T')(Symbol('t_2', commutative=True)))), Integral(Symbol('t_2', commutative=True), Tuple(Function('T')(Symbol('t_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\nabla', commutative=True), Mul(Rational(1, 2), Pow(Function('T')(Symbol('t_2', commutative=True)), Integer(2)))), Integral(Symbol('t_2', commutative=True), Tuple(Function('T')(Symbol('t_2', commutative=True)))))"], [["add", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Rational(1, 2), Pow(Function('T')(Symbol('t_2', commutative=True)), Integer(2)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Integral(Symbol('t_2', commutative=True), Tuple(Function('T')(Symbol('t_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(g)} = \\log{(g)}, then obtain \\log{(- g + \\frac{d}{d g} (\\int \\mathbb{I}{(g)} dg) \\int \\log{(g)} dg)} = \\log{(- g + \\frac{d}{d g} (\\int \\log{(g)} dg)^{2})}", "derivation": "\\mathbb{I}{(g)} = \\log{(g)} and \\int \\mathbb{I}{(g)} dg = \\int \\log{(g)} dg and (\\int \\mathbb{I}{(g)} dg) \\int \\log{(g)} dg = (\\int \\log{(g)} dg)^{2} and \\frac{d}{d g} (\\int \\mathbb{I}{(g)} dg) \\int \\log{(g)} dg = \\frac{d}{d g} (\\int \\log{(g)} dg)^{2} and - g + \\frac{d}{d g} (\\int \\mathbb{I}{(g)} dg) \\int \\log{(g)} dg = - g + \\frac{d}{d g} (\\int \\log{(g)} dg)^{2} and \\log{(- g + \\frac{d}{d g} (\\int \\mathbb{I}{(g)} dg) \\int \\log{(g)} dg)} = \\log{(- g + \\frac{d}{d g} (\\int \\log{(g)} dg)^{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["times", 2, "Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Mul(Integral(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Pow(Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Mul(Integral(Function('\\\\mathbb{I}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Pow(Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given t{(c,H)} = H^{c} and \\operatorname{F_{N}}{(c,H)} = H^{c}, then obtain \\frac{\\frac{\\partial}{\\partial c} t{(c,H)}}{c} - \\frac{t{(c,H)}}{c^{2}} = \\frac{\\frac{\\partial}{\\partial c} \\operatorname{F_{N}}{(c,H)}}{c} - \\frac{\\operatorname{F_{N}}{(c,H)}}{c^{2}}", "derivation": "t{(c,H)} = H^{c} and \\frac{t{(c,H)}}{c} = \\frac{H^{c}}{c} and \\frac{\\partial}{\\partial c} \\frac{t{(c,H)}}{c} = \\frac{\\partial}{\\partial c} \\frac{H^{c}}{c} and \\operatorname{F_{N}}{(c,H)} = H^{c} and \\frac{\\partial}{\\partial c} \\frac{t{(c,H)}}{c} = \\frac{\\partial}{\\partial c} \\frac{\\operatorname{F_{N}}{(c,H)}}{c} and \\frac{\\frac{\\partial}{\\partial c} t{(c,H)}}{c} - \\frac{t{(c,H)}}{c^{2}} = \\frac{\\frac{\\partial}{\\partial c} \\operatorname{F_{N}}{(c,H)}}{c} - \\frac{\\operatorname{F_{N}}{(c,H)}}{c^{2}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('F_N')(Symbol('c', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-2)), Function('t')(Symbol('c', commutative=True), Symbol('H', commutative=True)))), Add(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-2)), Function('F_N')(Symbol('c', commutative=True), Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\theta)} = e^{\\sin{(\\theta)}}, then derive - \\theta + \\frac{d}{d \\theta} \\operatorname{t_{1}}{(\\theta)} = - \\theta + e^{\\sin{(\\theta)}} \\cos{(\\theta)}, then obtain - \\theta + 2 \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} = - \\theta + e^{\\sin{(\\theta)}} \\cos{(\\theta)} + \\frac{d}{d \\theta} e^{\\sin{(\\theta)}}", "derivation": "\\operatorname{t_{1}}{(\\theta)} = e^{\\sin{(\\theta)}} and \\frac{d}{d \\theta} \\operatorname{t_{1}}{(\\theta)} = \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} and - \\theta + \\frac{d}{d \\theta} \\operatorname{t_{1}}{(\\theta)} = - \\theta + \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} and - \\theta + \\frac{d}{d \\theta} \\operatorname{t_{1}}{(\\theta)} = - \\theta + e^{\\sin{(\\theta)}} \\cos{(\\theta)} and - \\theta + \\frac{d}{d \\theta} \\operatorname{t_{1}}{(\\theta)} + \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} = - \\theta + e^{\\sin{(\\theta)}} \\cos{(\\theta)} + \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} and - \\theta + 2 \\frac{d}{d \\theta} e^{\\sin{(\\theta)}} = - \\theta + e^{\\sin{(\\theta)}} \\cos{(\\theta)} + \\frac{d}{d \\theta} e^{\\sin{(\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\theta', commutative=True)), exp(sin(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Function('t_1')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Function('t_1')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(exp(sin(Symbol('\\\\theta', commutative=True))), cos(Symbol('\\\\theta', commutative=True)))))"], [["add", 4, "Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Function('t_1')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(exp(sin(Symbol('\\\\theta', commutative=True))), cos(Symbol('\\\\theta', commutative=True))), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(exp(sin(Symbol('\\\\theta', commutative=True))), cos(Symbol('\\\\theta', commutative=True))), Derivative(exp(sin(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\varepsilon)} = e^{e^{\\varepsilon}} and \\theta_{2}{(\\varepsilon)} = e^{e^{\\varepsilon}}, then obtain S{(\\varepsilon)} e^{3 e^{\\varepsilon}} = e^{4 e^{\\varepsilon}}", "derivation": "S{(\\varepsilon)} = e^{e^{\\varepsilon}} and \\theta_{2}{(\\varepsilon)} = e^{e^{\\varepsilon}} and S{(\\varepsilon)} \\theta_{2}{(\\varepsilon)} = \\theta_{2}{(\\varepsilon)} e^{e^{\\varepsilon}} and S{(\\varepsilon)} \\theta_{2}^{2}{(\\varepsilon)} e^{e^{\\varepsilon}} = \\theta_{2}^{2}{(\\varepsilon)} e^{2 e^{\\varepsilon}} and S{(\\varepsilon)} e^{3 e^{\\varepsilon}} = e^{4 e^{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\varepsilon', commutative=True)), exp(exp(Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True)), exp(exp(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Function('S')(Symbol('\\\\varepsilon', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True)), exp(exp(Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 3, "Mul(Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True)), exp(exp(Symbol('\\\\varepsilon', commutative=True))))"], "Equality(Mul(Function('S')(Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), exp(exp(Symbol('\\\\varepsilon', commutative=True)))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('\\\\varepsilon', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('S')(Symbol('\\\\varepsilon', commutative=True)), exp(Mul(Integer(3), exp(Symbol('\\\\varepsilon', commutative=True))))), exp(Mul(Integer(4), exp(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given G{(a^{\\dagger},\\mu_0,x^\\prime)} = \\mu_0 (a^{\\dagger} + x^\\prime), then obtain 1 = \\frac{\\mu_0 (a^{\\dagger} + x^\\prime) + 2 x^\\prime + G{(a^{\\dagger},\\mu_0,x^\\prime)}}{2 x^\\prime + 2 G{(a^{\\dagger},\\mu_0,x^\\prime)}}", "derivation": "G{(a^{\\dagger},\\mu_0,x^\\prime)} = \\mu_0 (a^{\\dagger} + x^\\prime) and x^\\prime + G{(a^{\\dagger},\\mu_0,x^\\prime)} = \\mu_0 (a^{\\dagger} + x^\\prime) + x^\\prime and 2 x^\\prime + 2 G{(a^{\\dagger},\\mu_0,x^\\prime)} = \\mu_0 (a^{\\dagger} + x^\\prime) + 2 x^\\prime + G{(a^{\\dagger},\\mu_0,x^\\prime)} and 1 = \\frac{\\mu_0 (a^{\\dagger} + x^\\prime) + 2 x^\\prime + G{(a^{\\dagger},\\mu_0,x^\\prime)}}{2 x^\\prime + 2 G{(a^{\\dagger},\\mu_0,x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 2, "Add(Symbol('x^\\\\prime', commutative=True), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Integer(-1)), Add(Mul(Symbol('\\\\mu_0', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True)), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\rho,\\Psi_{nl})} = \\Psi_{nl}^{\\rho}, then obtain \\frac{\\partial}{\\partial \\rho} \\Psi_{nl}^{\\rho} \\mathbf{P}^{3}{(\\rho,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\rho} \\Psi_{nl}^{2 \\rho} \\mathbf{P}^{2}{(\\rho,\\Psi_{nl})}", "derivation": "\\mathbf{P}{(\\rho,\\Psi_{nl})} = \\Psi_{nl}^{\\rho} and \\Psi_{nl}^{\\rho} \\mathbf{P}{(\\rho,\\Psi_{nl})} = \\Psi_{nl}^{2 \\rho} and \\Psi_{nl}^{2 \\rho} \\mathbf{P}^{2}{(\\rho,\\Psi_{nl})} = \\Psi_{nl}^{4 \\rho} and \\Psi_{nl}^{\\rho} \\mathbf{P}^{3}{(\\rho,\\Psi_{nl})} = \\Psi_{nl}^{2 \\rho} \\mathbf{P}^{2}{(\\rho,\\Psi_{nl})} and \\frac{\\partial}{\\partial \\rho} \\Psi_{nl}^{\\rho} \\mathbf{P}^{3}{(\\rho,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\rho} \\Psi_{nl}^{2 \\rho} \\mathbf{P}^{2}{(\\rho,\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\rho', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), Symbol('\\\\rho', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(4), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(3))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2))))"], [["differentiate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(3))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(S)} = \\log{(S)}, then derive \\frac{d}{d S} \\theta{(S)} + \\frac{1}{S} = \\frac{2}{S}, then obtain \\int (\\frac{d}{d S} \\theta{(S)} + \\frac{1}{S}) dS = \\int \\frac{2}{S} dS", "derivation": "\\theta{(S)} = \\log{(S)} and \\theta{(S)} + \\log{(S)} = 2 \\log{(S)} and \\frac{d}{d S} (\\theta{(S)} + \\log{(S)}) = \\frac{d}{d S} 2 \\log{(S)} and \\frac{d}{d S} \\theta{(S)} + \\frac{1}{S} = \\frac{2}{S} and \\int (\\frac{d}{d S} \\theta{(S)} + \\frac{1}{S}) dS = \\int \\frac{2}{S} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["add", 1, "log(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Mul(Integer(2), log(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('S', commutative=True), Integer(-1))))"], [["integrate", 4, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\theta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('S', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('S', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(F_{N},\\mathbf{F})} = e^{F_{N}^{\\mathbf{F}}}, then obtain 2 \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} = 2 \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}}", "derivation": "\\mathbf{v}{(F_{N},\\mathbf{F})} = e^{F_{N}^{\\mathbf{F}}} and \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} = \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}} and 2 \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} = \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} + \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}} and \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} + \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}} = 2 \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}} and 2 \\frac{\\partial}{\\partial F_{N}} \\mathbf{v}{(F_{N},\\mathbf{F})} = 2 \\frac{\\partial}{\\partial F_{N}} e^{F_{N}^{\\mathbf{F}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["add", 2, "Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{v}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} = \\frac{v_{y}}{\\mu} and \\lambda{(x)} = \\cos{(x)}, then obtain \\int 0 dx = \\int \\frac{- \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} + \\frac{v_{y}}{\\mu}}{\\cos{(x)}} dx", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} = \\frac{v_{y}}{\\mu} and 0 = - \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} + \\frac{v_{y}}{\\mu} and \\lambda{(x)} = \\cos{(x)} and 0 = \\frac{- \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} + \\frac{v_{y}}{\\mu}}{\\lambda{(x)}} and \\int 0 dx = \\int \\frac{- \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} + \\frac{v_{y}}{\\mu}}{\\lambda{(x)}} dx and \\int 0 dx = \\int \\frac{- \\operatorname{V_{\\mathbf{E}}}{(\\mu,v_{y})} + \\frac{v_{y}}{\\mu}}{\\cos{(x)}} dx", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], [["minus", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))))"], ["get_premise", "Equality(Function('\\\\lambda')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["divide", 2, "Function('\\\\lambda')(Symbol('x', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Pow(Function('\\\\lambda')(Symbol('x', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('x', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('x', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Pow(Function('\\\\lambda')(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Integer(0), Tuple(Symbol('x', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mu', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\hat{x})} = \\sin{(\\hat{x})}, then obtain \\frac{d}{d \\hat{x}} (\\int \\operatorname{A_{x}}^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} = \\frac{d}{d \\hat{x}} (\\int \\sin^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}}", "derivation": "\\operatorname{A_{x}}{(\\hat{x})} = \\sin{(\\hat{x})} and \\operatorname{A_{x}}^{\\hat{x}}{(\\hat{x})} = \\sin^{\\hat{x}}{(\\hat{x})} and \\int \\operatorname{A_{x}}^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\int \\sin^{\\hat{x}}{(\\hat{x})} d\\hat{x} and (\\int \\operatorname{A_{x}}^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} = (\\int \\sin^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} and \\frac{d}{d \\hat{x}} (\\int \\operatorname{A_{x}}^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} = \\frac{d}{d \\hat{x}} (\\int \\sin^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Pow(Function('A_x')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('A_x')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Pow(Integral(Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Pow(Integral(Pow(Function('A_x')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(C_{2},\\rho_b)} = e^{\\rho_b^{C_{2}}}, then obtain \\frac{\\rho_b}{C_{2}} = \\frac{\\rho_b e^{\\rho_b^{C_{2}}}}{C_{2} W{(C_{2},\\rho_b)}}", "derivation": "W{(C_{2},\\rho_b)} = e^{\\rho_b^{C_{2}}} and 1 = \\frac{e^{\\rho_b^{C_{2}}}}{W{(C_{2},\\rho_b)}} and \\rho_b = \\frac{\\rho_b e^{\\rho_b^{C_{2}}}}{W{(C_{2},\\rho_b)}} and \\frac{\\rho_b}{C_{2}} = \\frac{\\rho_b e^{\\rho_b^{C_{2}}}}{C_{2} W{(C_{2},\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_2', commutative=True))))"], [["divide", 1, "Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_2', commutative=True)))))"], [["times", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Symbol('\\\\rho_b', commutative=True), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_2', commutative=True)))))"], [["divide", 3, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True), Pow(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(v_{t},y)} = - y + \\log{(v_{t})} and \\psi{(v_{t},y)} = y + \\frac{\\partial}{\\partial y} (\\mathbf{J}{(v_{t},y)} - \\log{(v_{t})}), then derive y + \\frac{\\partial}{\\partial y} \\mathbf{J}{(v_{t},y)} = y - 1, then derive \\psi{(v_{t},y)} = y + \\frac{\\partial}{\\partial y} \\mathbf{J}{(v_{t},y)}, then obtain \\psi{(v_{t},y)} = y - 1", "derivation": "\\mathbf{J}{(v_{t},y)} = - y + \\log{(v_{t})} and \\mathbf{J}{(v_{t},y)} - \\log{(v_{t})} = - y and \\frac{\\partial}{\\partial y} (\\mathbf{J}{(v_{t},y)} - \\log{(v_{t})}) = \\frac{d}{d y} - y and y + \\frac{\\partial}{\\partial y} (\\mathbf{J}{(v_{t},y)} - \\log{(v_{t})}) = y + \\frac{d}{d y} - y and \\psi{(v_{t},y)} = y + \\frac{\\partial}{\\partial y} (\\mathbf{J}{(v_{t},y)} - \\log{(v_{t})}) and y + \\frac{\\partial}{\\partial y} \\mathbf{J}{(v_{t},y)} = y - 1 and \\psi{(v_{t},y)} = y + \\frac{\\partial}{\\partial y} \\mathbf{J}{(v_{t},y)} and \\psi{(v_{t},y)} = y - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Symbol('v_t', commutative=True))))"], [["minus", 1, "log(Symbol('v_t', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), Symbol('y', commutative=True)))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('v_t', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Integer(-1), Symbol('y', commutative=True))"], "Equality(Add(Symbol('y', commutative=True), Derivative(Add(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('v_t', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Symbol('y', commutative=True), Derivative(Mul(Integer(-1), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Derivative(Add(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('v_t', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('y', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Symbol('y', commutative=True), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\psi')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Function('\\\\psi')(Symbol('v_t', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\hat{p}{(A_{z})} = \\sin{(\\log{(A_{z})})}, then obtain \\int (\\frac{d}{d A_{z}} \\hat{p}^{A_{z}}{(A_{z})} - 1) dA_{z} = \\int (\\frac{d}{d A_{z}} \\sin^{A_{z}}{(\\log{(A_{z})})} - 1) dA_{z}", "derivation": "\\hat{p}{(A_{z})} = \\sin{(\\log{(A_{z})})} and \\hat{p}^{A_{z}}{(A_{z})} = \\sin^{A_{z}}{(\\log{(A_{z})})} and \\frac{d}{d A_{z}} \\hat{p}^{A_{z}}{(A_{z})} = \\frac{d}{d A_{z}} \\sin^{A_{z}}{(\\log{(A_{z})})} and \\frac{d}{d A_{z}} \\hat{p}^{A_{z}}{(A_{z})} - 1 = \\frac{d}{d A_{z}} \\sin^{A_{z}}{(\\log{(A_{z})})} - 1 and \\int (\\frac{d}{d A_{z}} \\hat{p}^{A_{z}}{(A_{z})} - 1) dA_{z} = \\int (\\frac{d}{d A_{z}} \\sin^{A_{z}}{(\\log{(A_{z})})} - 1) dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('A_z', commutative=True)), sin(log(Symbol('A_z', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(sin(log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["differentiate", 2, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{p}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Pow(sin(log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Pow(Function('\\\\hat{p}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(sin(log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 4, "Symbol('A_z', commutative=True)"], "Equality(Integral(Add(Derivative(Pow(Function('\\\\hat{p}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Derivative(Pow(sin(log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(m,Q)} = Q^{m}, then obtain Q^{m} (- Q^{m} + \\operatorname{E_{n}}{(m,Q)}) = 0", "derivation": "\\operatorname{E_{n}}{(m,Q)} = Q^{m} and - Q^{m} + \\operatorname{E_{n}}{(m,Q)} = 0 and Q^{m} (- Q^{m} + \\operatorname{E_{n}}{(m,Q)}) \\operatorname{E_{n}}{(m,Q)} = 0 and Q^{m} (- Q^{m} + \\operatorname{E_{n}}{(m,Q)}) = 0", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True))), Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Integer(0))"], [["times", 2, "Mul(Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True))), Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True))), Integer(0))"], [["divide", 3, "Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Symbol('m', commutative=True))), Function('E_n')(Symbol('m', commutative=True), Symbol('Q', commutative=True)))), Integer(0))"]]}, {"prompt": "Given u{(J)} = \\sin{(J)}, then derive \\frac{\\frac{d}{d J} u{(J)}}{- \\cos{(J)} + \\frac{d}{d J} u{(J)}} = \\frac{\\cos{(J)}}{- \\cos{(J)} + \\frac{d}{d J} u{(J)}}, then obtain \\int \\frac{\\frac{d}{d J} \\sin{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} dJ = \\int \\frac{\\cos{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} dJ", "derivation": "u{(J)} = \\sin{(J)} and \\frac{d}{d J} u{(J)} = \\frac{d}{d J} \\sin{(J)} and \\frac{\\frac{d}{d J} u{(J)}}{\\frac{d}{d J} u{(J)} - \\frac{d}{d J} \\sin{(J)}} = \\frac{\\frac{d}{d J} \\sin{(J)}}{\\frac{d}{d J} u{(J)} - \\frac{d}{d J} \\sin{(J)}} and \\frac{\\frac{d}{d J} u{(J)}}{- \\cos{(J)} + \\frac{d}{d J} u{(J)}} = \\frac{\\cos{(J)}}{- \\cos{(J)} + \\frac{d}{d J} u{(J)}} and \\frac{\\frac{d}{d J} \\sin{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} = \\frac{\\cos{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} and \\int \\frac{\\frac{d}{d J} \\sin{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} dJ = \\int \\frac{\\cos{(J)}}{- \\cos{(J)} + \\frac{d}{d J} \\sin{(J)}} dJ", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["divide", 2, "Add(Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], "Equality(Mul(Pow(Add(Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Integer(-1)), Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Pow(Add(Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Integer(-1)), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(Function('u')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), cos(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), cos(Symbol('J', commutative=True))))"], [["integrate", 5, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(-1)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\phi,\\mathbf{P})} = \\cos^{\\phi}{(\\mathbf{P})}, then obtain ((- \\phi + \\lambda{(\\phi,\\mathbf{P})})^{\\phi})^{\\mathbf{P}} = ((- \\phi + \\cos^{\\phi}{(\\mathbf{P})})^{\\phi})^{\\mathbf{P}}", "derivation": "\\lambda{(\\phi,\\mathbf{P})} = \\cos^{\\phi}{(\\mathbf{P})} and - \\phi + \\lambda{(\\phi,\\mathbf{P})} = - \\phi + \\cos^{\\phi}{(\\mathbf{P})} and (- \\phi + \\lambda{(\\phi,\\mathbf{P})})^{\\phi} = (- \\phi + \\cos^{\\phi}{(\\mathbf{P})})^{\\phi} and ((- \\phi + \\lambda{(\\phi,\\mathbf{P})})^{\\phi})^{\\mathbf{P}} = ((- \\phi + \\cos^{\\phi}{(\\mathbf{P})})^{\\phi})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\lambda')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\lambda')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\lambda')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\phi', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(b,t)} = b + e^{t}, then derive \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} = 1, then obtain (2 e^{t} + 2 \\frac{\\partial}{\\partial b} (b + e^{t}))^{t} = (2 e^{t} + \\frac{\\partial}{\\partial b} (b + e^{t}) + 1)^{t}", "derivation": "\\mathbf{J}_f{(b,t)} = b + e^{t} and \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} = \\frac{\\partial}{\\partial b} (b + e^{t}) and \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} = 1 and e^{t} + \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} = e^{t} + 1 and 2 e^{t} + 2 \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} = 2 e^{t} + \\frac{\\partial}{\\partial b} \\mathbf{J}_f{(b,t)} + 1 and 2 e^{t} + 2 \\frac{\\partial}{\\partial b} (b + e^{t}) = 2 e^{t} + \\frac{\\partial}{\\partial b} (b + e^{t}) + 1 and (2 e^{t} + 2 \\frac{\\partial}{\\partial b} (b + e^{t}))^{t} = (2 e^{t} + \\frac{\\partial}{\\partial b} (b + e^{t}) + 1)^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "exp(Symbol('t', commutative=True))"], "Equality(Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(exp(Symbol('t', commutative=True)), Integer(1)))"], [["add", 4, "Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Mul(Integer(2), Derivative(Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))), Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Derivative(Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"], [["power", 6, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Mul(Integer(2), Derivative(Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))), Symbol('t', commutative=True)), Pow(Add(Mul(Integer(2), exp(Symbol('t', commutative=True))), Derivative(Add(Symbol('b', commutative=True), exp(Symbol('t', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)), Symbol('t', commutative=True)))"]]}, {"prompt": "Given Q{(\\hbar)} = \\sin{(\\hbar)}, then obtain \\frac{d}{d \\hbar} (\\hbar + Q{(\\hbar)} Q^{- \\hbar}{(\\hbar)}) = \\frac{d}{d \\hbar} (\\hbar + Q^{- \\hbar}{(\\hbar)} \\sin{(\\hbar)})", "derivation": "Q{(\\hbar)} = \\sin{(\\hbar)} and Q{(\\hbar)} Q^{- \\hbar}{(\\hbar)} = Q^{- \\hbar}{(\\hbar)} \\sin{(\\hbar)} and \\hbar + Q{(\\hbar)} Q^{- \\hbar}{(\\hbar)} = \\hbar + Q^{- \\hbar}{(\\hbar)} \\sin{(\\hbar)} and \\frac{d}{d \\hbar} (\\hbar + Q{(\\hbar)} Q^{- \\hbar}{(\\hbar)}) = \\frac{d}{d \\hbar} (\\hbar + Q^{- \\hbar}{(\\hbar)} \\sin{(\\hbar)})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('\\\\hbar', commutative=True)), Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Function('Q')(Symbol('\\\\hbar', commutative=True)), Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))), Add(Symbol('\\\\hbar', commutative=True), Mul(Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Function('Q')(Symbol('\\\\hbar', commutative=True)), Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Pow(Function('Q')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\lambda)} = \\cos{(\\lambda)}, then derive \\frac{d}{d \\lambda} h{(\\lambda)} = - \\sin{(\\lambda)}, then obtain ((- \\sin{(\\lambda)})^{\\lambda})^{\\lambda} = ((\\frac{d}{d \\lambda} h{(\\lambda)})^{\\lambda})^{\\lambda}", "derivation": "h{(\\lambda)} = \\cos{(\\lambda)} and \\frac{d}{d \\lambda} h{(\\lambda)} = \\frac{d}{d \\lambda} \\cos{(\\lambda)} and \\frac{d}{d \\lambda} h{(\\lambda)} = - \\sin{(\\lambda)} and \\frac{d}{d \\lambda} \\cos{(\\lambda)} = - \\sin{(\\lambda)} and (\\frac{d}{d \\lambda} \\cos{(\\lambda)})^{\\lambda} = (- \\sin{(\\lambda)})^{\\lambda} and (\\frac{d}{d \\lambda} \\cos{(\\lambda)})^{\\lambda} = (\\frac{d}{d \\lambda} h{(\\lambda)})^{\\lambda} and ((\\frac{d}{d \\lambda} \\cos{(\\lambda)})^{\\lambda})^{\\lambda} = ((\\frac{d}{d \\lambda} h{(\\lambda)})^{\\lambda})^{\\lambda} and ((- \\sin{(\\lambda)})^{\\lambda})^{\\lambda} = ((\\frac{d}{d \\lambda} h{(\\lambda)})^{\\lambda})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(Function('h')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)))"], [["power", 6, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Pow(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Derivative(Function('h')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["evaluate_derivatives", 7], "Equality(Pow(Pow(Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Derivative(Function('h')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given a{(\\chi,q)} = \\chi + q, then obtain - 2 q + (- q + a{(\\chi,q)})^{\\chi} + a{(\\chi,q)} = \\chi - q + (- q + a{(\\chi,q)})^{\\chi}", "derivation": "a{(\\chi,q)} = \\chi + q and - q + a{(\\chi,q)} = \\chi and - 2 q + a{(\\chi,q)} = \\chi - q and - 2 q + (- q + a{(\\chi,q)})^{\\chi} + a{(\\chi,q)} = \\chi - q + (- q + a{(\\chi,q)})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\chi', commutative=True))"], [["minus", 2, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\chi', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('a')(Symbol('\\\\chi', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{x},\\delta)} = - v_{x} + \\sin{(\\delta)}, then obtain \\int (- v_{x} + \\int \\operatorname{y^{\\prime}}{(v_{x},\\delta)} d\\delta) d\\delta = \\int (- v_{x} + \\int (- v_{x} + \\sin{(\\delta)}) d\\delta) d\\delta", "derivation": "\\operatorname{y^{\\prime}}{(v_{x},\\delta)} = - v_{x} + \\sin{(\\delta)} and \\int \\operatorname{y^{\\prime}}{(v_{x},\\delta)} d\\delta = \\int (- v_{x} + \\sin{(\\delta)}) d\\delta and - v_{x} + \\int \\operatorname{y^{\\prime}}{(v_{x},\\delta)} d\\delta = - v_{x} + \\int (- v_{x} + \\sin{(\\delta)}) d\\delta and \\int (- v_{x} + \\int \\operatorname{y^{\\prime}}{(v_{x},\\delta)} d\\delta) d\\delta = \\int (- v_{x} + \\int (- v_{x} + \\sin{(\\delta)}) d\\delta) d\\delta", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Function('y^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Function('y^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\dot{z})} = \\log{(\\dot{z})}, then obtain \\frac{d}{d \\dot{z}} (- \\dot{z} + \\bar{\\h}{(\\dot{z})})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (- \\dot{z} + \\log{(\\dot{z})})^{\\dot{z}}", "derivation": "\\bar{\\h}{(\\dot{z})} = \\log{(\\dot{z})} and - \\dot{z} + \\bar{\\h}{(\\dot{z})} = - \\dot{z} + \\log{(\\dot{z})} and (- \\dot{z} + \\bar{\\h}{(\\dot{z})})^{\\dot{z}} = (- \\dot{z} + \\log{(\\dot{z})})^{\\dot{z}} and \\frac{d}{d \\dot{z}} (- \\dot{z} + \\bar{\\h}{(\\dot{z})})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (- \\dot{z} + \\log{(\\dot{z})})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\hbar')(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\dot{x})} = \\int \\cos{(\\dot{x})} d\\dot{x}, then derive \\frac{d}{d \\dot{x}} \\operatorname{v_{1}}{(\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (\\lambda + \\sin{(\\dot{x})}), then obtain \\frac{\\frac{d}{d \\dot{x}} \\operatorname{v_{1}}{(\\dot{x})}}{\\lambda + \\sin{(\\dot{x})}} = \\frac{\\frac{\\partial}{\\partial \\dot{x}} (\\lambda + \\sin{(\\dot{x})})}{\\lambda + \\sin{(\\dot{x})}}", "derivation": "\\operatorname{v_{1}}{(\\dot{x})} = \\int \\cos{(\\dot{x})} d\\dot{x} and \\frac{d}{d \\dot{x}} \\operatorname{v_{1}}{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\int \\cos{(\\dot{x})} d\\dot{x} and \\frac{d}{d \\dot{x}} \\operatorname{v_{1}}{(\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (\\lambda + \\sin{(\\dot{x})}) and \\frac{\\frac{d}{d \\dot{x}} \\operatorname{v_{1}}{(\\dot{x})}}{\\lambda + \\sin{(\\dot{x})}} = \\frac{\\frac{\\partial}{\\partial \\dot{x}} (\\lambda + \\sin{(\\dot{x})})}{\\lambda + \\sin{(\\dot{x})}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\dot{x}', commutative=True)), Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('v_1')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["divide", 3, "Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), Derivative(Function('v_1')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), Derivative(Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given k{(\\rho)} = \\cos{(\\rho)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\rho)} = k^{\\rho}{(\\rho)}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\rho}{(\\rho)} = (\\cos^{\\rho}{(\\rho)})^{\\rho}", "derivation": "k{(\\rho)} = \\cos{(\\rho)} and k^{\\rho}{(\\rho)} = \\cos^{\\rho}{(\\rho)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\rho)} = k^{\\rho}{(\\rho)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\rho)} = \\cos^{\\rho}{(\\rho)} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\rho}{(\\rho)} = (\\cos^{\\rho}{(\\rho)})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\rho', commutative=True)), Pow(Function('k')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given u{(h)} = \\cos{(h)}, then derive \\int u{(h)} dh = \\varepsilon + \\sin{(h)}, then obtain \\frac{\\int \\cos{(h)} dh}{\\varepsilon} = \\frac{\\varepsilon + \\sin{(h)}}{\\varepsilon}", "derivation": "u{(h)} = \\cos{(h)} and \\int u{(h)} dh = \\int \\cos{(h)} dh and \\int u{(h)} dh = \\varepsilon + \\sin{(h)} and \\int \\cos{(h)} dh = \\varepsilon + \\sin{(h)} and \\frac{\\int \\cos{(h)} dh}{\\varepsilon} = \\frac{\\varepsilon + \\sin{(h)}}{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('u')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('h', commutative=True))))"], [["divide", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\psi)} = \\psi, then obtain \\frac{\\psi}{\\tilde{g}^* \\varepsilon{(\\psi)}} + 1 = \\frac{\\psi^{3}}{\\tilde{g}^* \\varepsilon^{3}{(\\psi)}} + 1", "derivation": "\\varepsilon{(\\psi)} = \\psi and \\frac{\\varepsilon{(\\psi)}}{\\tilde{g}^*} = \\frac{\\psi}{\\tilde{g}^*} and \\frac{1}{\\tilde{g}^*} = \\frac{\\psi}{\\tilde{g}^* \\varepsilon{(\\psi)}} and 1 + \\frac{1}{\\tilde{g}^*} = \\frac{\\psi}{\\tilde{g}^* \\varepsilon{(\\psi)}} + 1 and \\frac{\\psi}{\\tilde{g}^* \\varepsilon{(\\psi)}} + 1 = \\frac{\\psi^{2}}{\\tilde{g}^* \\varepsilon^{2}{(\\psi)}} + 1 and 1 + \\frac{1}{\\tilde{g}^*} = \\frac{\\psi^{2}}{\\tilde{g}^* \\varepsilon^{2}{(\\psi)}} + 1 and \\frac{\\psi}{\\tilde{g}^* \\varepsilon{(\\psi)}} + 1 = \\frac{\\psi^{3}}{\\tilde{g}^* \\varepsilon^{3}{(\\psi)}} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["divide", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))))"], [["divide", 2, "Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True))"], "Equality(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Add(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-2))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Integer(1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-2))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(3)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('\\\\psi', commutative=True)), Integer(-3))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} = g^{\\prime}_{\\varepsilon} (A_{1} + y) and \\operatorname{v_{y}}{(y,A_{1},g^{\\prime}_{\\varepsilon})} = \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} - \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})}, then derive \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} = g^{\\prime}_{\\varepsilon}, then obtain \\operatorname{v_{y}}{(y,A_{1},g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})}", "derivation": "\\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} = g^{\\prime}_{\\varepsilon} (A_{1} + y) and \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} = \\frac{\\partial}{\\partial A_{1}} g^{\\prime}_{\\varepsilon} (A_{1} + y) and \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} = g^{\\prime}_{\\varepsilon} and \\operatorname{v_{y}}{(y,A_{1},g^{\\prime}_{\\varepsilon})} = \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} - \\frac{\\partial}{\\partial A_{1}} \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})} and \\operatorname{v_{y}}{(y,A_{1},g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\mathbf{B}{(g^{\\prime}_{\\varepsilon},y,A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('y', commutative=True), Symbol('A_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('v_y')(Symbol('y', commutative=True), Symbol('A_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given y{(r)} = e^{r} and \\mathbf{F}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain - \\cos{(e^{r})} + \\int \\mathbf{F}{(\\mathbf{P})} d\\mathbf{P} = - \\cos{(e^{r})} + \\int \\log{(\\mathbf{P})} d\\mathbf{P}", "derivation": "y{(r)} = e^{r} and \\cos{(y{(r)})} = \\cos{(e^{r})} and \\mathbf{F}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\int \\mathbf{F}{(\\mathbf{P})} d\\mathbf{P} = \\int \\log{(\\mathbf{P})} d\\mathbf{P} and - \\cos{(y{(r)})} + \\int \\mathbf{F}{(\\mathbf{P})} d\\mathbf{P} = - \\cos{(y{(r)})} + \\int \\log{(\\mathbf{P})} d\\mathbf{P} and - \\cos{(e^{r})} + \\int \\mathbf{F}{(\\mathbf{P})} d\\mathbf{P} = - \\cos{(e^{r})} + \\int \\log{(\\mathbf{P})} d\\mathbf{P}", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["cos", 1], "Equality(cos(Function('y')(Symbol('r', commutative=True))), cos(exp(Symbol('r', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 4, "cos(Function('y')(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Function('y')(Symbol('r', commutative=True)))), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Integer(-1), cos(Function('y')(Symbol('r', commutative=True)))), Integral(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), cos(exp(Symbol('r', commutative=True)))), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Integer(-1), cos(exp(Symbol('r', commutative=True)))), Integral(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(t_{2},n)} = n + t_{2}, then derive 0 = - \\sin{(\\frac{\\partial}{\\partial t_{2}} \\mathbf{v}{(t_{2},n)})} + \\sin{(1)}, then obtain \\sin{(\\frac{\\partial}{\\partial t_{2}} (n + t_{2}))} = \\sin{(1)}", "derivation": "\\mathbf{v}{(t_{2},n)} = n + t_{2} and \\frac{\\partial}{\\partial t_{2}} \\mathbf{v}{(t_{2},n)} = \\frac{\\partial}{\\partial t_{2}} (n + t_{2}) and \\sin{(\\frac{\\partial}{\\partial t_{2}} \\mathbf{v}{(t_{2},n)})} = \\sin{(\\frac{\\partial}{\\partial t_{2}} (n + t_{2}))} and 0 = \\sin{(\\frac{\\partial}{\\partial t_{2}} (n + t_{2}))} - \\sin{(\\frac{\\partial}{\\partial t_{2}} \\mathbf{v}{(t_{2},n)})} and 0 = - \\sin{(\\frac{\\partial}{\\partial t_{2}} \\mathbf{v}{(t_{2},n)})} + \\sin{(1)} and 0 = - \\sin{(\\frac{\\partial}{\\partial t_{2}} (n + t_{2}))} + \\sin{(1)} and \\sin{(\\frac{\\partial}{\\partial t_{2}} (n + t_{2}))} = \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), sin(Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["minus", 3, "sin(Derivative(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(sin(Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Derivative(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Derivative(Function('\\\\mathbf{v}')(Symbol('t_2', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), sin(Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), sin(Integer(1))))"], [["add", 6, "sin(Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], "Equality(sin(Derivative(Add(Symbol('n', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), sin(Integer(1)))"]]}, {"prompt": "Given G{(J,B)} = \\frac{\\partial}{\\partial J} (B + J), then derive \\int G{(J,B)} dB = B + M, then derive B + v_{1} = B + M, then obtain (\\int \\frac{\\partial}{\\partial J} (B + J) dB)^{B} = (B + v_{1})^{B}", "derivation": "G{(J,B)} = \\frac{\\partial}{\\partial J} (B + J) and \\int G{(J,B)} dB = \\int \\frac{\\partial}{\\partial J} (B + J) dB and \\int G{(J,B)} dB = B + M and \\int \\frac{\\partial}{\\partial J} (B + J) dB = B + M and B + v_{1} = B + M and \\int \\frac{\\partial}{\\partial J} (B + J) dB = B + v_{1} and (\\int \\frac{\\partial}{\\partial J} (B + J) dB)^{B} = (B + v_{1})^{B}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('G')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Add(Symbol('B', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('J', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(Add(Symbol('B', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), Symbol('M', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('B', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('B', commutative=True), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Add(Symbol('B', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), Symbol('v_1', commutative=True)))"], [["power", 6, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Derivative(Add(Symbol('B', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Symbol('B', commutative=True), Symbol('v_1', commutative=True)), Symbol('B', commutative=True)))"]]}, {"prompt": "Given L{(\\mathbf{P},\\tilde{g})} = \\frac{\\cos{(\\mathbf{P})}}{\\tilde{g}}, then derive \\frac{\\partial}{\\partial \\mathbf{P}} L{(\\mathbf{P},\\tilde{g})} = - \\frac{\\sin{(\\mathbf{P})}}{\\tilde{g}}, then obtain \\sin{(\\mathbf{P})} + \\frac{\\partial}{\\partial \\mathbf{P}} L{(\\mathbf{P},\\tilde{g})} = \\sin{(\\mathbf{P})} - \\frac{\\sin{(\\mathbf{P})}}{\\tilde{g}}", "derivation": "L{(\\mathbf{P},\\tilde{g})} = \\frac{\\cos{(\\mathbf{P})}}{\\tilde{g}} and \\frac{\\partial}{\\partial \\mathbf{P}} L{(\\mathbf{P},\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\cos{(\\mathbf{P})}}{\\tilde{g}} and \\frac{\\partial}{\\partial \\mathbf{P}} L{(\\mathbf{P},\\tilde{g})} = - \\frac{\\sin{(\\mathbf{P})}}{\\tilde{g}} and \\sin{(\\mathbf{P})} + \\frac{\\partial}{\\partial \\mathbf{P}} L{(\\mathbf{P},\\tilde{g})} = \\sin{(\\mathbf{P})} - \\frac{\\sin{(\\mathbf{P})}}{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 3, "sin(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(sin(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('L')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(A_{y})} = \\log{(A_{y})}, then derive \\int \\varepsilon_{0}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + \\phi_2, then obtain A_{y} \\varepsilon_{0}{(A_{y})} - A_{y} + \\phi_2 = \\int \\log{(A_{y})} dA_{y}", "derivation": "\\varepsilon_{0}{(A_{y})} = \\log{(A_{y})} and \\int \\varepsilon_{0}{(A_{y})} dA_{y} = \\int \\log{(A_{y})} dA_{y} and \\int \\varepsilon_{0}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + \\phi_2 and \\int \\varepsilon_{0}{(A_{y})} dA_{y} = A_{y} \\varepsilon_{0}{(A_{y})} - A_{y} + \\phi_2 and A_{y} \\varepsilon_{0}{(A_{y})} - A_{y} + \\phi_2 = \\int \\log{(A_{y})} dA_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), log(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('A_y', commutative=True), Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + \\mathbb{I}, then obtain \\int 1 dA_{x} = \\int \\frac{- A_{x} \\ddot{x} - \\Psi^{\\dagger} + \\mathbb{I}}{- A_{x} \\ddot{x} + \\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})}} dA_{x}", "derivation": "\\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + \\mathbb{I} and - A_{x} \\ddot{x} + \\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})} = - A_{x} \\ddot{x} - \\Psi^{\\dagger} + \\mathbb{I} and 1 = \\frac{- A_{x} \\ddot{x} - \\Psi^{\\dagger} + \\mathbb{I}}{- A_{x} \\ddot{x} + \\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})}} and \\int 1 dA_{x} = \\int \\frac{- A_{x} \\ddot{x} - \\Psi^{\\dagger} + \\mathbb{I}}{- A_{x} \\ddot{x} + \\operatorname{z^{*}}{(\\mathbb{I},\\Psi^{\\dagger})}} dA_{x}", "srepr_derivation": [["get_premise", "Equality(Function('z^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Mul(Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Function('z^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Function('z^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Function('z^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 3, "Symbol('A_x', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Function('z^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given l{(\\nabla,\\tilde{g})} = \\int (- \\nabla + \\tilde{g}) d\\nabla, then derive l{(\\nabla,\\tilde{g})} = \\mathbf{s} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g}, then derive \\nabla = \\frac{\\partial}{\\partial \\tilde{g}} \\int (- \\nabla + \\tilde{g}) d\\nabla, then obtain \\nabla = \\frac{\\partial}{\\partial \\tilde{g}} (\\mathbf{A} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g})", "derivation": "l{(\\nabla,\\tilde{g})} = \\int (- \\nabla + \\tilde{g}) d\\nabla and l{(\\nabla,\\tilde{g})} = \\mathbf{s} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g} and \\mathbf{s} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g} = \\int (- \\nabla + \\tilde{g}) d\\nabla and \\frac{\\partial}{\\partial \\tilde{g}} (\\mathbf{s} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g}) = \\frac{\\partial}{\\partial \\tilde{g}} \\int (- \\nabla + \\tilde{g}) d\\nabla and \\nabla = \\frac{\\partial}{\\partial \\tilde{g}} \\int (- \\nabla + \\tilde{g}) d\\nabla and \\nabla = \\frac{\\partial}{\\partial \\tilde{g}} (\\mathbf{A} - \\frac{\\nabla^{2}}{2} + \\nabla \\tilde{g})", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('l')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Symbol('\\\\nabla', commutative=True), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Symbol('\\\\nabla', commutative=True), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(Z)} = e^{Z}, then derive \\frac{d}{d Z} \\operatorname{E_{n}}{(Z)} = e^{Z}, then obtain \\int \\frac{d}{d Z} e^{Z} dZ = \\int \\frac{d^{2}}{d Z^{2}} e^{Z} dZ", "derivation": "\\operatorname{E_{n}}{(Z)} = e^{Z} and \\frac{d}{d Z} \\operatorname{E_{n}}{(Z)} = \\frac{d}{d Z} e^{Z} and \\frac{d}{d Z} \\operatorname{E_{n}}{(Z)} = e^{Z} and \\frac{d}{d Z} \\operatorname{E_{n}}{(Z)} = \\frac{d^{2}}{d Z^{2}} \\operatorname{E_{n}}{(Z)} and \\frac{d}{d Z} e^{Z} = \\frac{d^{2}}{d Z^{2}} e^{Z} and \\int \\frac{d}{d Z} e^{Z} dZ = \\int \\frac{d^{2}}{d Z^{2}} e^{Z} dZ", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), exp(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('E_n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('E_n')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(\\hat{H})} = \\hat{H} and \\delta{(\\hat{H})} = \\frac{1}{\\dot{y}{(\\hat{H})}}, then obtain \\frac{1}{\\hat{H} (- \\hat{H} + a + n_{2})} = \\frac{1}{(- \\hat{H} + a + n_{2}) \\dot{y}{(\\hat{H})}}", "derivation": "\\dot{y}{(\\hat{H})} = \\hat{H} and \\delta{(\\hat{H})} = \\frac{1}{\\dot{y}{(\\hat{H})}} and \\delta{(\\hat{H})} = \\frac{1}{\\hat{H}} and \\frac{1}{\\hat{H}} = \\frac{1}{\\dot{y}{(\\hat{H})}} and \\frac{1}{\\hat{H} (- \\hat{H} + a + n_{2})} = \\frac{1}{(- \\hat{H} + a + n_{2}) \\dot{y}{(\\hat{H})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('a', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('a', commutative=True), Symbol('n_2', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('a', commutative=True), Symbol('n_2', commutative=True)), Integer(-1)), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{B},v)} = - \\mathbf{B} + v, then derive \\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B} = B - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} v, then obtain 2 e^{\\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B}} = e^{B - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} v} + e^{\\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B}}", "derivation": "\\hat{x}{(\\mathbf{B},v)} = - \\mathbf{B} + v and \\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B} = \\int (- \\mathbf{B} + v) d\\mathbf{B} and \\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B} = B - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} v and e^{\\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B}} = e^{B - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} v} and 2 e^{\\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B}} = e^{B - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} v} + e^{\\int \\hat{x}{(\\mathbf{B},v)} d\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('B', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), exp(Add(Symbol('B', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)))))"], [["add", 4, "exp(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Mul(Integer(2), exp(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Add(exp(Add(Symbol('B', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)))), exp(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} = J + (f^{*})^{\\mathbf{p}}, then obtain - \\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p} + \\frac{\\int \\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} d\\mathbf{p}}{J} = - \\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p} + \\frac{\\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p}}{J}", "derivation": "\\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} = J + (f^{*})^{\\mathbf{p}} and \\int \\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} d\\mathbf{p} = \\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p} and \\frac{\\int \\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} d\\mathbf{p}}{J} = \\frac{\\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p}}{J} and - \\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p} + \\frac{\\int \\operatorname{r_{0}}{(f^{*},J,\\mathbf{p})} d\\mathbf{p}}{J} = - \\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p} + \\frac{\\int (J + (f^{*})^{\\mathbf{p}}) d\\mathbf{p}}{J}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('f^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('f^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["divide", 2, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Function('r_0')(Symbol('f^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 3, "Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Function('r_0')(Symbol('f^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))), Add(Mul(Integer(-1), Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Integral(Add(Symbol('J', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(m,l)} = l m, then obtain \\frac{\\partial}{\\partial l} (\\frac{\\partial}{\\partial l} \\operatorname{E_{n}}{(m,l)} - 1) = \\frac{\\partial}{\\partial l} (\\frac{\\partial}{\\partial l} l m - 1)", "derivation": "\\operatorname{E_{n}}{(m,l)} = l m and \\frac{\\partial}{\\partial l} \\operatorname{E_{n}}{(m,l)} = \\frac{\\partial}{\\partial l} l m and \\frac{\\partial}{\\partial l} \\operatorname{E_{n}}{(m,l)} - 1 = \\frac{\\partial}{\\partial l} l m - 1 and \\frac{\\partial}{\\partial l} (\\frac{\\partial}{\\partial l} \\operatorname{E_{n}}{(m,l)} - 1) = \\frac{\\partial}{\\partial l} (\\frac{\\partial}{\\partial l} l m - 1)", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('m', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Symbol('l', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('E_n')(Symbol('m', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Derivative(Mul(Symbol('l', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive \\int \\operatorname{E_{n}}{(\\mathbf{D})} d\\mathbf{D} = f + e^{\\mathbf{D}}, then obtain f + e^{\\mathbf{D}} = C_{2} + e^{\\mathbf{D}}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{D})} = e^{\\mathbf{D}} and \\int \\operatorname{E_{n}}{(\\mathbf{D})} d\\mathbf{D} = \\int e^{\\mathbf{D}} d\\mathbf{D} and \\int \\operatorname{E_{n}}{(\\mathbf{D})} d\\mathbf{D} = f + e^{\\mathbf{D}} and f + e^{\\mathbf{D}} = \\int e^{\\mathbf{D}} d\\mathbf{D} and f + e^{\\mathbf{D}} = C_{2} + e^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\Psi{(a,\\ddot{x})} = - \\ddot{x} + a, then obtain (\\ddot{x} + \\Psi{(a,\\ddot{x})}) \\Psi{(a,\\ddot{x})} = a \\Psi{(a,\\ddot{x})}", "derivation": "\\Psi{(a,\\ddot{x})} = - \\ddot{x} + a and \\ddot{x} + \\Psi{(a,\\ddot{x})} = a and (- \\ddot{x} + a) (\\ddot{x} + \\Psi{(a,\\ddot{x})}) = a (- \\ddot{x} + a) and (\\ddot{x} + \\Psi{(a,\\ddot{x})}) \\Psi{(a,\\ddot{x})} = a \\Psi{(a,\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('a', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('a', commutative=True))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('a', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('a', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Mul(Symbol('a', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('a', commutative=True), Function('\\\\Psi')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given U{(\\psi,G)} = - G + \\psi, then obtain G \\psi (\\frac{\\partial}{\\partial \\psi} U{(\\psi,G)} - 1) + G (G - \\psi + U{(\\psi,G)}) + (- G + \\psi)^{G} = (- G + \\psi)^{G}", "derivation": "U{(\\psi,G)} = - G + \\psi and G - \\psi + U{(\\psi,G)} = 0 and G (G - \\psi + U{(\\psi,G)}) = 0 and G \\psi (G - \\psi + U{(\\psi,G)}) = 0 and \\frac{\\partial}{\\partial \\psi} G \\psi (G - \\psi + U{(\\psi,G)}) = \\frac{d}{d \\psi} 0 and (- G + \\psi)^{G} + \\frac{\\partial}{\\partial \\psi} G \\psi (G - \\psi + U{(\\psi,G)}) = (- G + \\psi)^{G} + \\frac{d}{d \\psi} 0 and G \\psi (\\frac{\\partial}{\\partial \\psi} U{(\\psi,G)} - 1) + G (G - \\psi + U{(\\psi,G)}) + (- G + \\psi)^{G} = (- G + \\psi)^{G}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True))), Integer(0))"], [["times", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)))), Integer(0))"], [["times", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Symbol('\\\\psi', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\psi', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["add", 5, "Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('G', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('G', commutative=True)), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\psi', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('G', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Symbol('G', commutative=True), Symbol('\\\\psi', commutative=True), Add(Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('U')(Symbol('\\\\psi', commutative=True), Symbol('G', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('G', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given i{(J,G)} = \\cos{(J^{G})}, then obtain F_{N} + G = \\int \\frac{\\cos{(J^{G})}}{i{(J,G)}} dG", "derivation": "i{(J,G)} = \\cos{(J^{G})} and 1 = \\frac{\\cos{(J^{G})}}{i{(J,G)}} and \\int 1 dG = \\int \\frac{\\cos{(J^{G})}}{i{(J,G)}} dG and F_{N} + G = \\int \\frac{\\cos{(J^{G})}}{i{(J,G)}} dG", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('J', commutative=True), Symbol('G', commutative=True)), cos(Pow(Symbol('J', commutative=True), Symbol('G', commutative=True))))"], [["divide", 1, "Function('i')(Symbol('J', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('i')(Symbol('J', commutative=True), Symbol('G', commutative=True)), Integer(-1)), cos(Pow(Symbol('J', commutative=True), Symbol('G', commutative=True)))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Function('i')(Symbol('J', commutative=True), Symbol('G', commutative=True)), Integer(-1)), cos(Pow(Symbol('J', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Integral(Mul(Pow(Function('i')(Symbol('J', commutative=True), Symbol('G', commutative=True)), Integer(-1)), cos(Pow(Symbol('J', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t_{1},F_{N})} = \\sin{(t_{1}^{F_{N}})}, then derive - t_{1}^{F_{N}} \\log{(t_{1})} \\cos{(t_{1}^{F_{N}})} + \\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{g}}{(t_{1},F_{N})} = 0, then obtain - \\sin{(t_{1}^{F_{N}} \\log{(t_{1})} \\cos{(t_{1}^{F_{N}})} - \\frac{\\partial}{\\partial F_{N}} \\sin{(t_{1}^{F_{N}})})} = 0", "derivation": "\\operatorname{F_{g}}{(t_{1},F_{N})} = \\sin{(t_{1}^{F_{N}})} and \\operatorname{F_{g}}{(t_{1},F_{N})} - \\sin{(t_{1}^{F_{N}})} = 0 and \\frac{\\partial}{\\partial F_{N}} (\\operatorname{F_{g}}{(t_{1},F_{N})} - \\sin{(t_{1}^{F_{N}})}) = \\frac{d}{d F_{N}} 0 and - t_{1}^{F_{N}} \\log{(t_{1})} \\cos{(t_{1}^{F_{N}})} + \\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{g}}{(t_{1},F_{N})} = 0 and - t_{1}^{F_{N}} \\log{(t_{1})} \\cos{(t_{1}^{F_{N}})} + \\frac{\\partial}{\\partial F_{N}} \\sin{(t_{1}^{F_{N}})} = 0 and - \\sin{(t_{1}^{F_{N}} \\log{(t_{1})} \\cos{(t_{1}^{F_{N}})} - \\frac{\\partial}{\\partial F_{N}} \\sin{(t_{1}^{F_{N}})})} = 0", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True))))"], [["minus", 1, "sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Add(Function('F_g')(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Function('F_g')(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), log(Symbol('t_1', commutative=True)), cos(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)))), Derivative(Function('F_g')(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), log(Symbol('t_1', commutative=True)), cos(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)))), Derivative(sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Integer(0))"], [["sin", 5], "Equality(Mul(Integer(-1), sin(Add(Mul(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)), log(Symbol('t_1', commutative=True)), cos(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True)))), Mul(Integer(-1), Derivative(sin(Pow(Symbol('t_1', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))))), Integer(0))"]]}, {"prompt": "Given c{(B)} = \\log{(B)} and l{(t_{1})} = \\cos{(t_{1})}, then obtain \\int (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) l{(t_{1})} dt_{1} = \\int (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) \\cos{(t_{1})} dt_{1}", "derivation": "c{(B)} = \\log{(B)} and c{(B)} - \\log{(B)} = 0 and l{(t_{1})} = \\cos{(t_{1})} and l{(t_{1})} \\cos{(\\hat{p})} = \\cos{(\\hat{p})} \\cos{(t_{1})} and c{(B)} - \\log{(B)} + \\cos{(\\hat{p})} = \\cos{(\\hat{p})} and (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) l{(t_{1})} = (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) \\cos{(t_{1})} and \\int (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) l{(t_{1})} dt_{1} = \\int (c{(B)} - \\log{(B)} + \\cos{(\\hat{p})}) \\cos{(t_{1})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], [["minus", 1, "log(Symbol('B', commutative=True))"], "Equality(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('l')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["times", 3, "cos(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Function('l')(Symbol('t_1', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(cos(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('t_1', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), Function('l')(Symbol('t_1', commutative=True))), Mul(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('t_1', commutative=True))))"], [["integrate", 6, "Symbol('t_1', commutative=True)"], "Equality(Integral(Mul(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), Function('l')(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Add(Function('c')(Symbol('B', commutative=True)), Mul(Integer(-1), log(Symbol('B', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\phi_1,\\dot{y})} = \\dot{y} + \\phi_1, then derive \\int - \\frac{\\operatorname{A_{2}}{(\\phi_1,\\dot{y})}}{\\dot{y} + \\phi_1 - 2 \\operatorname{A_{2}}{(\\phi_1,\\dot{y})}} d\\phi_1 = A_{z} + \\phi_1, then obtain \\int 1 d\\phi_1 = A_{z} + \\phi_1", "derivation": "\\operatorname{A_{2}}{(\\phi_1,\\dot{y})} = \\dot{y} + \\phi_1 and 0 = \\dot{y} + \\phi_1 - \\operatorname{A_{2}}{(\\phi_1,\\dot{y})} and - \\operatorname{A_{2}}{(\\phi_1,\\dot{y})} = \\dot{y} + \\phi_1 - 2 \\operatorname{A_{2}}{(\\phi_1,\\dot{y})} and - \\frac{\\operatorname{A_{2}}{(\\phi_1,\\dot{y})}}{\\dot{y} + \\phi_1 - 2 \\operatorname{A_{2}}{(\\phi_1,\\dot{y})}} = 1 and \\int - \\frac{\\operatorname{A_{2}}{(\\phi_1,\\dot{y})}}{\\dot{y} + \\phi_1 - 2 \\operatorname{A_{2}}{(\\phi_1,\\dot{y})}} d\\phi_1 = \\int 1 d\\phi_1 and \\int - \\frac{\\operatorname{A_{2}}{(\\phi_1,\\dot{y})}}{\\dot{y} + \\phi_1 - 2 \\operatorname{A_{2}}{(\\phi_1,\\dot{y})}} d\\phi_1 = A_{z} + \\phi_1 and \\int 1 d\\phi_1 = A_{z} + \\phi_1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(1))"], [["integrate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('A_z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given u{(\\hat{p})} = e^{\\hat{p}} and B{(\\hat{p})} = e^{e^{\\int e^{\\hat{p}} d\\hat{p}}}, then obtain \\int e^{e^{\\int e^{\\hat{p}} d\\hat{p}}} d\\hat{p} = \\int e^{e^{\\int u{(\\hat{p})} d\\hat{p}}} d\\hat{p}", "derivation": "u{(\\hat{p})} = e^{\\hat{p}} and \\int u{(\\hat{p})} d\\hat{p} = \\int e^{\\hat{p}} d\\hat{p} and e^{\\int u{(\\hat{p})} d\\hat{p}} = e^{\\int e^{\\hat{p}} d\\hat{p}} and e^{e^{\\int u{(\\hat{p})} d\\hat{p}}} = e^{e^{\\int e^{\\hat{p}} d\\hat{p}}} and B{(\\hat{p})} = e^{e^{\\int e^{\\hat{p}} d\\hat{p}}} and B{(\\hat{p})} = e^{e^{\\int u{(\\hat{p})} d\\hat{p}}} and \\int B{(\\hat{p})} d\\hat{p} = \\int e^{e^{\\int u{(\\hat{p})} d\\hat{p}}} d\\hat{p} and \\int e^{e^{\\int e^{\\hat{p}} d\\hat{p}}} d\\hat{p} = \\int e^{e^{\\int u{(\\hat{p})} d\\hat{p}}} d\\hat{p}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))), exp(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"], [["exp", 3], "Equality(exp(exp(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), exp(exp(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('B')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))))"], [["integrate", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(exp(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integral(exp(exp(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(exp(Integral(Function('u')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given q{(\\theta_2)} = \\sin{(\\cos{(\\theta_2)})} and W{(\\theta_2)} = \\int q{(\\theta_2)} d\\theta_2, then obtain W^{\\theta_2}{(\\theta_2)} = (\\int q{(\\theta_2)} d\\theta_2)^{\\theta_2}", "derivation": "q{(\\theta_2)} = \\sin{(\\cos{(\\theta_2)})} and \\int q{(\\theta_2)} d\\theta_2 = \\int \\sin{(\\cos{(\\theta_2)})} d\\theta_2 and W{(\\theta_2)} = \\int q{(\\theta_2)} d\\theta_2 and W{(\\theta_2)} = \\int \\sin{(\\cos{(\\theta_2)})} d\\theta_2 and W^{\\theta_2}{(\\theta_2)} = (\\int \\sin{(\\cos{(\\theta_2)})} d\\theta_2)^{\\theta_2} and W^{\\theta_2}{(\\theta_2)} = (\\int q{(\\theta_2)} d\\theta_2)^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\theta_2', commutative=True)), sin(cos(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(sin(cos(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('W')(Symbol('\\\\theta_2', commutative=True)), Integral(Function('q')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('W')(Symbol('\\\\theta_2', commutative=True)), Integral(sin(cos(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(sin(cos(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('W')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(Function('q')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} = \\log{(\\mathbf{M})} and \\operatorname{a^{\\dagger}}{(\\mathbf{M})} = 2 \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})}, then obtain \\operatorname{a^{\\dagger}}{(\\mathbf{M})} \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})} = (\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} + \\log{(\\mathbf{M})}) \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} = \\log{(\\mathbf{M})} and 2 \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} = \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} + \\log{(\\mathbf{M})} and \\operatorname{a^{\\dagger}}{(\\mathbf{M})} = 2 \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} and \\operatorname{a^{\\dagger}}{(\\mathbf{M})} \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})} = 2 \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})} and \\operatorname{a^{\\dagger}}{(\\mathbf{M})} \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})} = (\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{M})} + \\log{(\\mathbf{M})}) \\operatorname{f_{\\mathbf{v}}}^{- \\mathbf{M}}{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 3, "Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(2), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given H{(\\dot{\\mathbf{r}})} = \\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}, then derive H^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} = (x - \\cos{(\\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}}, then obtain (x - \\cos{(\\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}} = (\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})^{\\dot{\\mathbf{r}}}", "derivation": "H{(\\dot{\\mathbf{r}})} = \\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and H^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} = (\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})^{\\dot{\\mathbf{r}}} and H^{\\dot{\\mathbf{r}}}{(\\dot{\\mathbf{r}})} = (x - \\cos{(\\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}} and (x - \\cos{(\\dot{\\mathbf{r}})})^{\\dot{\\mathbf{r}}} = (\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})^{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('H')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{A})} = e^{\\mathbf{A}}, then derive \\frac{d}{d \\mathbf{A}} \\mathbf{s}{(\\mathbf{A})} + 1 = e^{\\mathbf{A}} + 1, then obtain \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + 1 = e^{\\mathbf{A}} + 1", "derivation": "\\mathbf{s}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\mathbf{A} + \\mathbf{s}{(\\mathbf{A})} = \\mathbf{A} + e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} (\\mathbf{A} + \\mathbf{s}{(\\mathbf{A})}) = \\frac{d}{d \\mathbf{A}} (\\mathbf{A} + e^{\\mathbf{A}}) and \\frac{d}{d \\mathbf{A}} \\mathbf{s}{(\\mathbf{A})} + 1 = e^{\\mathbf{A}} + 1 and \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + 1 = e^{\\mathbf{A}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\dot{x}{(M,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} (- M + \\mathbf{J}), then derive \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{x}{(M,\\mathbf{J})} = 0, then obtain \\frac{\\frac{\\partial}{\\partial \\mathbf{J}} \\dot{x}{(M,\\mathbf{J})}}{\\frac{\\partial^{2}}{\\partial \\mathbf{J}^{2}} (- M + \\mathbf{J})} = 0", "derivation": "\\dot{x}{(M,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} (- M + \\mathbf{J}) and \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{x}{(M,\\mathbf{J})} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}^{2}} (- M + \\mathbf{J}) and \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{x}{(M,\\mathbf{J})} = 0 and \\frac{\\frac{\\partial}{\\partial \\mathbf{J}} \\dot{x}{(M,\\mathbf{J})}}{\\frac{\\partial^{2}}{\\partial \\mathbf{J}^{2}} (- M + \\mathbf{J})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(0))"], [["times", 3, "Pow(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))), Integer(-1))"], "Equality(Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given Z{(\\mu_0)} = \\cos{(e^{\\mu_0})} and \\mathbf{A}{(\\mu_0)} = \\cos{(e^{\\mu_0})}, then obtain \\frac{d}{d \\mu_0} 1 + e^{- 2 \\mu_0} = \\frac{d}{d \\mu_0} Z^{- \\mu_0}{(\\mu_0)} \\mathbf{A}^{\\mu_0}{(\\mu_0)} + e^{- 2 \\mu_0}", "derivation": "Z{(\\mu_0)} = \\cos{(e^{\\mu_0})} and Z^{\\mu_0}{(\\mu_0)} = \\cos^{\\mu_0}{(e^{\\mu_0})} and Z^{2 \\mu_0}{(\\mu_0)} = Z^{\\mu_0}{(\\mu_0)} \\cos^{\\mu_0}{(e^{\\mu_0})} and Z^{2 \\mu_0}{(\\mu_0)} e^{\\mu_0} = Z^{\\mu_0}{(\\mu_0)} e^{\\mu_0} \\cos^{\\mu_0}{(e^{\\mu_0})} and 1 = Z^{- \\mu_0}{(\\mu_0)} \\cos^{\\mu_0}{(e^{\\mu_0})} and \\mathbf{A}{(\\mu_0)} = \\cos{(e^{\\mu_0})} and 1 = Z^{- \\mu_0}{(\\mu_0)} \\mathbf{A}^{\\mu_0}{(\\mu_0)} and \\frac{d}{d \\mu_0} 1 = \\frac{d}{d \\mu_0} Z^{- \\mu_0}{(\\mu_0)} \\mathbf{A}^{\\mu_0}{(\\mu_0)} and \\frac{d}{d \\mu_0} 1 + e^{- 2 \\mu_0} = \\frac{d}{d \\mu_0} Z^{- \\mu_0}{(\\mu_0)} \\mathbf{A}^{\\mu_0}{(\\mu_0)} + e^{- 2 \\mu_0}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mu_0', commutative=True)), cos(exp(Symbol('\\\\mu_0', commutative=True))))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(cos(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["times", 2, "Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], "Equality(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(cos(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], [["times", 3, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)), Pow(cos(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], [["divide", 4, "Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Pow(cos(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True)), cos(exp(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["add", 8, "exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)))), Add(Derivative(Mul(Pow(Function('Z')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\psi^*)} = \\log{(\\psi^*)}, then obtain (\\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\dot{z}{(\\psi^*)}}{\\psi^*})^{2} = (\\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\log{(\\psi^*)}}{\\psi^*})^{2}", "derivation": "\\dot{z}{(\\psi^*)} = \\log{(\\psi^*)} and \\frac{\\dot{z}{(\\psi^*)}}{\\psi^*} = \\frac{\\log{(\\psi^*)}}{\\psi^*} and \\frac{d}{d \\psi^*} \\frac{\\dot{z}{(\\psi^*)}}{\\psi^*} = \\frac{d}{d \\psi^*} \\frac{\\log{(\\psi^*)}}{\\psi^*} and \\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\dot{z}{(\\psi^*)}}{\\psi^*} = \\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\log{(\\psi^*)}}{\\psi^*} and (\\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\dot{z}{(\\psi^*)}}{\\psi^*})^{2} = (\\frac{d^{2}}{d (\\psi^*)^{2}} \\frac{\\log{(\\psi^*)}}{\\psi^*})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), log(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), log(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), log(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))))"], [["power", 4, 2], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), log(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = \\frac{\\mathbf{v}}{a}, then obtain - 2 \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = - 2 \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{a}", "derivation": "\\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = \\frac{\\mathbf{v}}{a} and \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{a} and - \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = - \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{a} and - 2 \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{y^{\\prime}}{(a,\\mathbf{v})} = - 2 \\mathbf{v} + \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{a}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\theta_1,\\delta)} = \\theta_1 + \\sin{(\\delta)}, then derive (\\theta_1 + \\sin{(\\delta)}) \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = - (\\theta_1 + \\sin{(\\delta)}) \\sin{(\\delta)}, then obtain \\dot{x}{(\\theta_1,\\delta)} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = - \\dot{x}{(\\theta_1,\\delta)} \\sin{(\\delta)}", "derivation": "\\dot{x}{(\\theta_1,\\delta)} = \\theta_1 + \\sin{(\\delta)} and \\frac{\\partial}{\\partial \\delta} \\dot{x}{(\\theta_1,\\delta)} = \\frac{\\partial}{\\partial \\delta} (\\theta_1 + \\sin{(\\delta)}) and \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = \\frac{\\partial^{2}}{\\partial \\delta^{2}} (\\theta_1 + \\sin{(\\delta)}) and (\\theta_1 + \\sin{(\\delta)}) \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = (\\theta_1 + \\sin{(\\delta)}) \\frac{\\partial^{2}}{\\partial \\delta^{2}} (\\theta_1 + \\sin{(\\delta)}) and (\\theta_1 + \\sin{(\\delta)}) \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = - (\\theta_1 + \\sin{(\\delta)}) \\sin{(\\delta)} and \\dot{x}{(\\theta_1,\\delta)} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\dot{x}{(\\theta_1,\\delta)} = - \\dot{x}{(\\theta_1,\\delta)} \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"], [["times", 3, "Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Derivative(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Derivative(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), Derivative(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))), Mul(Integer(-1), Add(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\delta', commutative=True))), sin(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\omega,\\mathbf{p})} = \\frac{\\partial}{\\partial \\omega} (\\mathbf{p} + \\omega), then derive \\operatorname{A_{1}}^{\\omega}{(\\omega,\\mathbf{p})} = 1, then obtain \\frac{\\partial}{\\partial \\omega} \\operatorname{A_{1}}^{\\omega}{(\\omega,\\mathbf{p})} = \\frac{d}{d \\omega} 1", "derivation": "\\operatorname{A_{1}}{(\\omega,\\mathbf{p})} = \\frac{\\partial}{\\partial \\omega} (\\mathbf{p} + \\omega) and \\operatorname{A_{1}}^{\\omega}{(\\omega,\\mathbf{p})} = (\\frac{\\partial}{\\partial \\omega} (\\mathbf{p} + \\omega))^{\\omega} and \\operatorname{A_{1}}^{\\omega}{(\\omega,\\mathbf{p})} = 1 and \\frac{\\partial}{\\partial \\omega} \\operatorname{A_{1}}^{\\omega}{(\\omega,\\mathbf{p})} = \\frac{d}{d \\omega} 1", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('A_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\omega', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Pow(Function('A_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\mathbb{I},\\Psi_{nl})} = e^{\\Psi_{nl} - \\mathbb{I}}, then obtain \\int (H{(\\mathbb{I},\\Psi_{nl})} + 1)^{\\Psi_{nl}} d\\Psi_{nl} = \\int (e^{\\Psi_{nl} - \\mathbb{I}} + 1)^{\\Psi_{nl}} d\\Psi_{nl}", "derivation": "H{(\\mathbb{I},\\Psi_{nl})} = e^{\\Psi_{nl} - \\mathbb{I}} and H{(\\mathbb{I},\\Psi_{nl})} + 1 = e^{\\Psi_{nl} - \\mathbb{I}} + 1 and (H{(\\mathbb{I},\\Psi_{nl})} + 1)^{\\Psi_{nl}} = (e^{\\Psi_{nl} - \\mathbb{I}} + 1)^{\\Psi_{nl}} and \\int (H{(\\mathbb{I},\\Psi_{nl})} + 1)^{\\Psi_{nl}} d\\Psi_{nl} = \\int (e^{\\Psi_{nl} - \\mathbb{I}} + 1)^{\\Psi_{nl}} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)), Add(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))), Integer(1)))"], [["power", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))), Integer(1)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Pow(Add(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Add(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))), Integer(1)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given u{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain \\int (- \\mathbf{p} u{(\\mathbf{p})} - \\mathbf{p}) d\\mathbf{p} = \\int (- \\mathbf{p} e^{\\mathbf{p}} - \\mathbf{p}) d\\mathbf{p}", "derivation": "u{(\\mathbf{p})} = e^{\\mathbf{p}} and - \\mathbf{p} u{(\\mathbf{p})} = - \\mathbf{p} e^{\\mathbf{p}} and - \\mathbf{p} u{(\\mathbf{p})} - \\mathbf{p} = - \\mathbf{p} e^{\\mathbf{p}} - \\mathbf{p} and \\int (- \\mathbf{p} u{(\\mathbf{p})} - \\mathbf{p}) d\\mathbf{p} = \\int (- \\mathbf{p} e^{\\mathbf{p}} - \\mathbf{p}) d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Function('u')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Function('u')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Function('u')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)}, then derive 1 = \\frac{\\cos{(\\Psi)}}{\\operatorname{L_{\\varepsilon}}{(\\Psi)}}, then obtain \\frac{d}{d \\Psi} 1 = \\frac{d}{d \\Psi} \\frac{\\cos{(\\Psi)}}{\\operatorname{L_{\\varepsilon}}{(\\Psi)}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)} and 1 = \\frac{\\frac{d}{d \\Psi} \\sin{(\\Psi)}}{\\operatorname{L_{\\varepsilon}}{(\\Psi)}} and 1 = \\frac{\\cos{(\\Psi)}}{\\operatorname{L_{\\varepsilon}}{(\\Psi)}} and \\frac{d}{d \\Psi} 1 = \\frac{d}{d \\Psi} \\frac{\\cos{(\\Psi)}}{\\operatorname{L_{\\varepsilon}}{(\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["divide", 1, "Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(1), Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(C_{1},\\mathbf{s})} = C_{1} + \\mathbf{s}, then obtain \\frac{\\partial}{\\partial C_{1}} ((C_{1} + \\mathbf{s}) (- \\mathbf{s} + \\varepsilon{(C_{1},\\mathbf{s})}))^{C_{1}} = \\frac{\\partial}{\\partial C_{1}} (C_{1} (C_{1} + \\mathbf{s}))^{C_{1}}", "derivation": "\\varepsilon{(C_{1},\\mathbf{s})} = C_{1} + \\mathbf{s} and - \\mathbf{s} + \\varepsilon{(C_{1},\\mathbf{s})} = C_{1} and (C_{1} + \\mathbf{s}) (- \\mathbf{s} + \\varepsilon{(C_{1},\\mathbf{s})}) = C_{1} (C_{1} + \\mathbf{s}) and ((C_{1} + \\mathbf{s}) (- \\mathbf{s} + \\varepsilon{(C_{1},\\mathbf{s})}))^{C_{1}} = (C_{1} (C_{1} + \\mathbf{s}))^{C_{1}} and \\frac{\\partial}{\\partial C_{1}} ((C_{1} + \\mathbf{s}) (- \\mathbf{s} + \\varepsilon{(C_{1},\\mathbf{s})}))^{C_{1}} = \\frac{\\partial}{\\partial C_{1}} (C_{1} (C_{1} + \\mathbf{s}))^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('C_1', commutative=True))"], [["times", 2, "Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Symbol('C_1', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('C_1', commutative=True)), Pow(Mul(Symbol('C_1', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('C_1', commutative=True)))"], [["differentiate", 4, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Pow(Mul(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('C_1', commutative=True), Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(x^\\prime)} = \\cos{(\\cos{(x^\\prime)})} and \\rho_{f}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain x^\\prime + \\cos{(\\rho_{f}{(x^\\prime)})} = x^\\prime + \\cos{(\\cos{(x^\\prime)})}", "derivation": "\\mathbf{s}{(x^\\prime)} = \\cos{(\\cos{(x^\\prime)})} and x^\\prime + \\mathbf{s}{(x^\\prime)} = x^\\prime + \\cos{(\\cos{(x^\\prime)})} and \\rho_{f}{(x^\\prime)} = \\cos{(x^\\prime)} and \\mathbf{s}{(x^\\prime)} = \\cos{(\\rho_{f}{(x^\\prime)})} and x^\\prime + \\cos{(\\rho_{f}{(x^\\prime)})} = x^\\prime + \\cos{(\\cos{(x^\\prime)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), cos(cos(Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), cos(cos(Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), cos(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('x^\\\\prime', commutative=True), cos(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), cos(cos(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(h)} = \\frac{d}{d h} e^{h}, then obtain \\frac{d}{d h} \\mathbf{S}{(h)} e^{- h} - e^{- h} = \\frac{d}{d h} e^{- h} \\frac{d}{d h} e^{h} - e^{- h}", "derivation": "\\mathbf{S}{(h)} = \\frac{d}{d h} e^{h} and \\mathbf{S}{(h)} e^{- h} = e^{- h} \\frac{d}{d h} e^{h} and \\frac{d}{d h} \\mathbf{S}{(h)} e^{- h} = \\frac{d}{d h} e^{- h} \\frac{d}{d h} e^{h} and \\frac{d}{d h} \\mathbf{S}{(h)} e^{- h} - e^{- h} = \\frac{d}{d h} e^{- h} \\frac{d}{d h} e^{h} - e^{- h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 1, "exp(Symbol('h', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('h', commutative=True))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Symbol('h', commutative=True))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 3, "exp(Mul(Integer(-1), Symbol('h', commutative=True)))"], "Equality(Add(Derivative(Mul(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('h', commutative=True))))), Add(Derivative(Mul(exp(Mul(Integer(-1), Symbol('h', commutative=True))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\dot{x})} = \\log{(\\dot{x})} and M{(\\dot{x})} = - \\Psi^{\\dagger}{(\\dot{x})}, then obtain (- \\frac{M{(\\dot{x})}}{\\Psi^{\\dagger}{(\\dot{x})}})^{\\dot{x}} + \\hat{p}_0{(I,p)} + 1 = \\hat{p}_0{(I,p)} + 2", "derivation": "\\Psi^{\\dagger}{(\\dot{x})} = \\log{(\\dot{x})} and M{(\\dot{x})} = - \\Psi^{\\dagger}{(\\dot{x})} and - \\frac{M{(\\dot{x})}}{\\Psi^{\\dagger}{(\\dot{x})}} = 1 and - \\frac{M{(\\dot{x})}}{\\log{(\\dot{x})}} = 1 and (- \\frac{M{(\\dot{x})}}{\\log{(\\dot{x})}})^{\\dot{x}} = 1 and (- \\frac{M{(\\dot{x})}}{\\log{(\\dot{x})}})^{\\dot{x}} + 1 = 2 and (- \\frac{M{(\\dot{x})}}{\\Psi^{\\dagger}{(\\dot{x})}})^{\\dot{x}} + 1 = 2 and (- \\frac{M{(\\dot{x})}}{\\Psi^{\\dagger}{(\\dot{x})}})^{\\dot{x}} + \\hat{p}_0{(I,p)} + 1 = \\hat{p}_0{(I,p)} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), Integer(1))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Pow(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Pow(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), Integer(1)), Integer(2))"], [["add", 7, "Function('\\\\hat{p}_0')(Symbol('I', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Pow(Mul(Integer(-1), Function('M')(Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Integer(1)), Add(Function('\\\\hat{p}_0')(Symbol('I', commutative=True), Symbol('p', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} = \\frac{\\dot{z}}{A \\hbar}, then obtain (- \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} + \\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar})^{A} = (- \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar} + \\frac{\\dot{z}}{A \\hbar})^{A}", "derivation": "\\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} = \\frac{\\dot{z}}{A \\hbar} and \\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar} = - \\frac{1}{\\hbar} + \\frac{\\dot{z}}{A \\hbar} and - \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} + \\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar} = - \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar} + \\frac{\\dot{z}}{A \\hbar} and (- \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} + \\Psi^{\\dagger}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar})^{A} = (- \\Psi^{\\dagger}^{2}{(A,\\dot{z},\\hbar)} - \\frac{1}{\\hbar} + \\frac{\\dot{z}}{A \\hbar})^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))"], [["minus", 2, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\sigma_x,n,q)} = \\sigma_x n q, then obtain \\int (- n q + \\int \\operatorname{E_{x}}{(\\sigma_x,n,q)} d\\sigma_x) dn = \\int (- n q + \\int \\sigma_x n q d\\sigma_x) dn", "derivation": "\\operatorname{E_{x}}{(\\sigma_x,n,q)} = \\sigma_x n q and \\frac{\\operatorname{E_{x}}{(\\sigma_x,n,q)}}{\\sigma_x} = n q and \\int \\operatorname{E_{x}}{(\\sigma_x,n,q)} d\\sigma_x = \\int \\sigma_x n q d\\sigma_x and \\int \\operatorname{E_{x}}{(\\sigma_x,n,q)} d\\sigma_x - \\frac{\\operatorname{E_{x}}{(\\sigma_x,n,q)}}{\\sigma_x} = \\int \\sigma_x n q d\\sigma_x - \\frac{\\operatorname{E_{x}}{(\\sigma_x,n,q)}}{\\sigma_x} and - n q + \\int \\operatorname{E_{x}}{(\\sigma_x,n,q)} d\\sigma_x = - n q + \\int \\sigma_x n q d\\sigma_x and \\int (- n q + \\int \\operatorname{E_{x}}{(\\sigma_x,n,q)} d\\sigma_x) dn = \\int (- n q + \\int \\sigma_x n q d\\sigma_x) dn", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Mul(Symbol('n', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)))"], "Equality(Add(Integral(Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)))), Add(Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True), Symbol('q', commutative=True)), Integral(Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('n', commutative=True), Symbol('q', commutative=True)), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["integrate", 5, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True), Symbol('q', commutative=True)), Integral(Function('E_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n', commutative=True), Symbol('q', commutative=True)), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(a,v_{t})} = \\sin{(a - v_{t})}, then derive v_{t} \\frac{\\partial}{\\partial a} \\phi_{2}{(a,v_{t})} = v_{t} \\cos{(a - v_{t})}, then obtain v_{t} \\frac{\\partial}{\\partial a} \\sin{(a - v_{t})} = v_{t} \\cos{(a - v_{t})}", "derivation": "\\phi_{2}{(a,v_{t})} = \\sin{(a - v_{t})} and v_{t} \\phi_{2}{(a,v_{t})} = v_{t} \\sin{(a - v_{t})} and \\frac{\\partial}{\\partial a} v_{t} \\phi_{2}{(a,v_{t})} = \\frac{\\partial}{\\partial a} v_{t} \\sin{(a - v_{t})} and v_{t} \\frac{\\partial}{\\partial a} \\phi_{2}{(a,v_{t})} = v_{t} \\cos{(a - v_{t})} and v_{t} \\frac{\\partial}{\\partial a} \\sin{(a - v_{t})} = v_{t} \\cos{(a - v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["times", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('\\\\phi_2')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Symbol('v_t', commutative=True), Function('\\\\phi_2')(Symbol('a', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_t', commutative=True), sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('v_t', commutative=True), Derivative(Function('\\\\phi_2')(Symbol('a', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Symbol('v_t', commutative=True), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('v_t', commutative=True), Derivative(sin(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Symbol('v_t', commutative=True), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}_0{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)}, then obtain (2 \\hat{x}_0{(\\hat{H}_l)})^{\\hat{H}_l} - 2 \\hat{x}_0{(\\hat{H}_l)} = (\\hat{x}_0{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - 2 \\hat{x}_0{(\\hat{H}_l)}", "derivation": "\\hat{x}_0{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and 2 \\hat{x}_0{(\\hat{H}_l)} = \\hat{x}_0{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} and (2 \\hat{x}_0{(\\hat{H}_l)})^{\\hat{H}_l} = (\\hat{x}_0{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} and (2 \\hat{x}_0{(\\hat{H}_l)})^{\\hat{H}_l} - 2 \\hat{x}_0{(\\hat{H}_l)} = (\\hat{x}_0{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - 2 \\hat{x}_0{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Pow(Add(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(n_{2})} = e^{n_{2}}, then derive \\frac{d}{d n_{2}} \\operatorname{v_{y}}{(n_{2})} = e^{n_{2}}, then obtain \\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}} = \\frac{d}{d n_{2}} e^{n_{2}}", "derivation": "\\operatorname{v_{y}}{(n_{2})} = e^{n_{2}} and \\frac{d}{d n_{2}} \\operatorname{v_{y}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\frac{d}{d n_{2}} \\operatorname{v_{y}}{(n_{2})} = e^{n_{2}} and \\operatorname{v_{y}}{(n_{2})} = \\frac{d}{d n_{2}} \\operatorname{v_{y}}{(n_{2})} and \\operatorname{v_{y}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\frac{d^{2}}{d n_{2}^{2}} \\operatorname{v_{y}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\frac{d^{2}}{d n_{2}^{2}} \\operatorname{v_{y}}{(n_{2})} = \\operatorname{v_{y}}{(n_{2})} and \\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}} = \\frac{d}{d n_{2}} e^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), exp(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_y')(Symbol('n_2', commutative=True)), Derivative(Function('v_y')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('v_y')(Symbol('n_2', commutative=True)), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(2))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(2))), Function('v_y')(Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(3))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\varphi)} = \\cos{(\\varphi)}, then obtain \\int 0 d\\varphi = \\int (- 2 \\mathbf{S}{(\\varphi)} + 2 \\cos{(\\varphi)}) d\\varphi", "derivation": "\\mathbf{S}{(\\varphi)} = \\cos{(\\varphi)} and 0 = - \\mathbf{S}{(\\varphi)} + \\cos{(\\varphi)} and - \\mathbf{S}{(\\varphi)} = - 2 \\mathbf{S}{(\\varphi)} + \\cos{(\\varphi)} and 0 = - 2 \\mathbf{S}{(\\varphi)} + 2 \\cos{(\\varphi)} and \\int 0 d\\varphi = \\int (- 2 \\mathbf{S}{(\\varphi)} + 2 \\cos{(\\varphi)}) d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\varphi', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{S}')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} \\sin{(x)}, then obtain \\frac{(\\hat{H}_{\\lambda} \\sin{(x)})^{x} \\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})}}{x} = \\frac{\\hat{H}_{\\lambda} (\\hat{H}_{\\lambda} \\sin{(x)})^{x} \\sin{(x)}}{x}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} \\sin{(x)} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})}}{x} = \\frac{\\hat{H}_{\\lambda} \\sin{(x)}}{x} and \\operatorname{V_{\\mathbf{E}}}^{x}{(x,\\hat{H}_{\\lambda})} = (\\hat{H}_{\\lambda} \\sin{(x)})^{x} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})} \\operatorname{V_{\\mathbf{E}}}^{x}{(x,\\hat{H}_{\\lambda})}}{x} = \\frac{\\hat{H}_{\\lambda} \\operatorname{V_{\\mathbf{E}}}^{x}{(x,\\hat{H}_{\\lambda})} \\sin{(x)}}{x} and \\frac{(\\hat{H}_{\\lambda} \\sin{(x)})^{x} \\operatorname{V_{\\mathbf{E}}}{(x,\\hat{H}_{\\lambda})}}{x} = \\frac{\\hat{H}_{\\lambda} (\\hat{H}_{\\lambda} \\sin{(x)})^{x} \\sin{(x)}}{x}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('x', commutative=True))))"], [["divide", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), sin(Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('x', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["times", 2, "Pow(Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('x', commutative=True))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(f,v_{1})} = \\frac{v_{1}}{f}, then obtain (\\frac{f \\operatorname{t_{2}}{(f,v_{1})}}{v_{1}} + \\frac{1}{v_{1}})^{f} = (1 + \\frac{1}{v_{1}})^{f}", "derivation": "\\operatorname{t_{2}}{(f,v_{1})} = \\frac{v_{1}}{f} and \\frac{f \\operatorname{t_{2}}{(f,v_{1})}}{v_{1}} = 1 and \\frac{f \\operatorname{t_{2}}{(f,v_{1})}}{v_{1}} + \\frac{1}{v_{1}} = 1 + \\frac{1}{v_{1}} and (\\frac{f \\operatorname{t_{2}}{(f,v_{1})}}{v_{1}} + \\frac{1}{v_{1}})^{f} = (1 + \\frac{1}{v_{1}})^{f}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('f', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))"], "Equality(Mul(Symbol('f', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('t_2')(Symbol('f', commutative=True), Symbol('v_1', commutative=True))), Integer(1))"], [["add", 2, "Pow(Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('f', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('t_2')(Symbol('f', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Add(Integer(1), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('f', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('t_2')(Symbol('f', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('f', commutative=True)), Pow(Add(Integer(1), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\Omega{(c)} = \\sin{(c)} and p{(c)} = \\frac{\\Omega{(c)} + \\sin{(c)}}{2 \\sin{(c)}}, then obtain p^{c}{(c)} = 1", "derivation": "\\Omega{(c)} = \\sin{(c)} and \\Omega{(c)} + \\sin{(c)} = 2 \\sin{(c)} and \\frac{\\Omega{(c)} + \\sin{(c)}}{2 \\sin{(c)}} = 1 and (\\frac{\\Omega{(c)} + \\sin{(c)}}{2 \\sin{(c)}})^{c} = 1 and p{(c)} = \\frac{\\Omega{(c)} + \\sin{(c)}}{2 \\sin{(c)}} and p^{c}{(c)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["add", 1, "sin(Symbol('c', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Mul(Integer(2), sin(Symbol('c', commutative=True))))"], [["divide", 2, "Mul(Integer(2), sin(Symbol('c', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('\\\\Omega')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Add(Function('\\\\Omega')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))), Symbol('c', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('p')(Symbol('c', commutative=True)), Mul(Rational(1, 2), Add(Function('\\\\Omega')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('p')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(v_{1})} = \\cos{(v_{1})}, then obtain \\frac{\\frac{d}{d v_{1}} \\operatorname{m_{s}}{(v_{1})}}{v_{1}} - \\frac{\\operatorname{m_{s}}{(v_{1})}}{v_{1}^{2}} = - \\frac{\\sin{(v_{1})}}{v_{1}} - \\frac{\\cos{(v_{1})}}{v_{1}^{2}}", "derivation": "\\operatorname{m_{s}}{(v_{1})} = \\cos{(v_{1})} and \\frac{\\operatorname{m_{s}}{(v_{1})}}{v_{1}} = \\frac{\\cos{(v_{1})}}{v_{1}} and \\frac{d}{d v_{1}} \\frac{\\operatorname{m_{s}}{(v_{1})}}{v_{1}} = \\frac{d}{d v_{1}} \\frac{\\cos{(v_{1})}}{v_{1}} and \\frac{\\frac{d}{d v_{1}} \\operatorname{m_{s}}{(v_{1})}}{v_{1}} - \\frac{\\operatorname{m_{s}}{(v_{1})}}{v_{1}^{2}} = - \\frac{\\sin{(v_{1})}}{v_{1}} - \\frac{\\cos{(v_{1})}}{v_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('m_s')(Symbol('v_1', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('m_s')(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Function('m_s')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-2)), Function('m_s')(Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), sin(Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-2)), cos(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\chi,M_{E})} = M_{E} \\chi, then obtain M_{E} \\chi \\int (M_{E} \\chi + M_{E} \\mathbf{p}{(\\chi,M_{E})}) dM_{E} = M_{E} \\chi (\\frac{M_{E}^{3} \\chi}{3} + \\frac{M_{E}^{2} \\chi}{2} + \\phi)", "derivation": "\\mathbf{p}{(\\chi,M_{E})} = M_{E} \\chi and M_{E} \\mathbf{p}{(\\chi,M_{E})} = M_{E}^{2} \\chi and M_{E} \\chi + M_{E} \\mathbf{p}{(\\chi,M_{E})} = M_{E}^{2} \\chi + M_{E} \\chi and \\int (M_{E} \\chi + M_{E} \\mathbf{p}{(\\chi,M_{E})}) dM_{E} = \\int (M_{E}^{2} \\chi + M_{E} \\chi) dM_{E} and M_{E} \\chi \\int (M_{E} \\chi + M_{E} \\mathbf{p}{(\\chi,M_{E})}) dM_{E} = M_{E} \\chi \\int (M_{E}^{2} \\chi + M_{E} \\chi) dM_{E} and M_{E} \\chi \\int (M_{E} \\chi + M_{E} \\mathbf{p}{(\\chi,M_{E})}) dM_{E} = M_{E} \\chi (\\frac{M_{E}^{3} \\chi}{3} + \\frac{M_{E}^{2} \\chi}{2} + \\phi)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)))"], [["add", 2, "Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True)))), Add(Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["times", 4, "Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True), Integral(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True), Integral(Add(Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('M_E', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True), Integral(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('M_E', commutative=True), Symbol('\\\\chi', commutative=True), Add(Mul(Rational(1, 3), Pow(Symbol('M_E', commutative=True), Integer(3)), Symbol('\\\\chi', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(u,M_{E})} = M_{E} - u, then obtain (\\operatorname{A_{y}}{(u,M_{E})} - 1) (M_{E} - u - 1)^{3} \\operatorname{A_{y}}^{2}{(u,M_{E})} = (M_{E} - u - 1)^{4} \\operatorname{A_{y}}^{2}{(u,M_{E})}", "derivation": "\\operatorname{A_{y}}{(u,M_{E})} = M_{E} - u and \\operatorname{A_{y}}{(u,M_{E})} - 1 = M_{E} - u - 1 and (\\operatorname{A_{y}}{(u,M_{E})} - 1) \\operatorname{A_{y}}{(u,M_{E})} = (M_{E} - u - 1) \\operatorname{A_{y}}{(u,M_{E})} and (\\operatorname{A_{y}}{(u,M_{E})} - 1) (M_{E} - u - 1) \\operatorname{A_{y}}{(u,M_{E})} = (M_{E} - u - 1)^{2} \\operatorname{A_{y}}{(u,M_{E})} and (\\operatorname{A_{y}}{(u,M_{E})} - 1) (M_{E} - u - 1)^{3} \\operatorname{A_{y}}^{2}{(u,M_{E})} = (M_{E} - u - 1)^{4} \\operatorname{A_{y}}^{2}{(u,M_{E})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)))"], [["times", 2, "Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Add(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))), Mul(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))))"], [["times", 3, "Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))), Mul(Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Integer(2)), Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True))))"], [["times", 4, "Mul(Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Integer(2)), Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Integer(3)), Pow(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Integer(-1)), Integer(4)), Pow(Function('A_y')(Symbol('u', commutative=True), Symbol('M_E', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\ddot{x}{(\\Psi,\\theta)} = \\cos{(\\theta^{\\Psi})} and \\varepsilon{(\\Psi,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})}, then obtain \\frac{\\frac{\\partial}{\\partial \\theta} \\ddot{x}{(\\Psi,\\theta)}}{\\cos{(\\theta^{\\Psi})}} = \\frac{\\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})}}{\\cos{(\\theta^{\\Psi})}}", "derivation": "\\ddot{x}{(\\Psi,\\theta)} = \\cos{(\\theta^{\\Psi})} and \\frac{\\partial}{\\partial \\theta} \\ddot{x}{(\\Psi,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})} and \\varepsilon{(\\Psi,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})} and \\frac{\\varepsilon{(\\Psi,\\theta)}}{\\cos{(\\theta^{\\Psi})}} = \\frac{\\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})}}{\\cos{(\\theta^{\\Psi})}} and \\varepsilon{(\\Psi,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\ddot{x}{(\\Psi,\\theta)} and \\frac{\\frac{\\partial}{\\partial \\theta} \\ddot{x}{(\\Psi,\\theta)}}{\\cos{(\\theta^{\\Psi})}} = \\frac{\\frac{\\partial}{\\partial \\theta} \\cos{(\\theta^{\\Psi})}}{\\cos{(\\theta^{\\Psi})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["divide", 3, "cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(Pow(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Pow(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\theta', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(A_{y},m_{s})} = \\frac{\\cos{(m_{s})}}{A_{y}}, then derive \\frac{\\partial}{\\partial A_{y}} I{(A_{y},m_{s})} = - \\frac{\\cos{(m_{s})}}{A_{y}^{2}}, then obtain \\frac{\\partial}{\\partial A_{y}} I{(A_{y},m_{s})} = - \\frac{I{(A_{y},m_{s})}}{A_{y}}", "derivation": "I{(A_{y},m_{s})} = \\frac{\\cos{(m_{s})}}{A_{y}} and \\frac{\\partial}{\\partial A_{y}} I{(A_{y},m_{s})} = \\frac{\\partial}{\\partial A_{y}} \\frac{\\cos{(m_{s})}}{A_{y}} and \\frac{\\partial}{\\partial A_{y}} I{(A_{y},m_{s})} = - \\frac{\\cos{(m_{s})}}{A_{y}^{2}} and \\frac{\\partial}{\\partial A_{y}} I{(A_{y},m_{s})} = - \\frac{I{(A_{y},m_{s})}}{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('A_y', commutative=True), Symbol('m_s', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), cos(Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('A_y', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), cos(Symbol('m_s', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('A_y', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-2)), cos(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('I')(Symbol('A_y', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-1)), Function('I')(Symbol('A_y', commutative=True), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(n_{1},\\mathbf{p})} = \\mathbf{p} n_{1}, then obtain \\frac{n_{1} \\bar{\\h}{(n_{1},\\mathbf{p})} + n_{1}}{\\mathbf{p}^{2} n_{1}^{2} + \\bar{\\h}{(n_{1},\\mathbf{p})}} = \\frac{\\mathbf{p} n_{1}^{2} + n_{1}}{\\mathbf{p}^{2} n_{1}^{2} + \\bar{\\h}{(n_{1},\\mathbf{p})}}", "derivation": "\\bar{\\h}{(n_{1},\\mathbf{p})} = \\mathbf{p} n_{1} and n_{1} \\bar{\\h}{(n_{1},\\mathbf{p})} = \\mathbf{p} n_{1}^{2} and n_{1} \\bar{\\h}{(n_{1},\\mathbf{p})} + n_{1} = \\mathbf{p} n_{1}^{2} + n_{1} and \\frac{n_{1} \\bar{\\h}{(n_{1},\\mathbf{p})} + n_{1}}{\\mathbf{p}^{2} n_{1}^{2} + \\bar{\\h}{(n_{1},\\mathbf{p})}} = \\frac{\\mathbf{p} n_{1}^{2} + n_{1}}{\\mathbf{p}^{2} n_{1}^{2} + \\bar{\\h}{(n_{1},\\mathbf{p})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_1', commutative=True)))"], [["times", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(2))))"], [["add", 2, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Symbol('n_1', commutative=True), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Symbol('n_1', commutative=True)), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(2))), Symbol('n_1', commutative=True)))"], [["divide", 3, "Add(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integer(-1)), Add(Mul(Symbol('n_1', commutative=True), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Symbol('n_1', commutative=True))), Mul(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(2))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Function('\\\\hbar')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(y,u)} = \\frac{\\log{(u)}}{y} and \\operatorname{v_{x}}{(y,u)} = (\\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)})^{y}, then obtain \\sin{(\\operatorname{v_{x}}^{u}{(y,u)} - 1)} = \\sin{(((\\frac{\\partial}{\\partial u} \\frac{\\log{(u)}}{y})^{y})^{u} - 1)}", "derivation": "\\mathbf{g}{(y,u)} = \\frac{\\log{(u)}}{y} and \\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)} = \\frac{\\partial}{\\partial u} \\frac{\\log{(u)}}{y} and \\operatorname{v_{x}}{(y,u)} = (\\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)})^{y} and \\operatorname{v_{x}}^{u}{(y,u)} = ((\\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)})^{y})^{u} and \\operatorname{v_{x}}^{u}{(y,u)} - 1 = ((\\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)})^{y})^{u} - 1 and \\sin{(\\operatorname{v_{x}}^{u}{(y,u)} - 1)} = \\sin{(((\\frac{\\partial}{\\partial u} \\mathbf{g}{(y,u)})^{y})^{u} - 1)} and \\sin{(\\operatorname{v_{x}}^{u}{(y,u)} - 1)} = \\sin{(((\\frac{\\partial}{\\partial u} \\frac{\\log{(u)}}{y})^{y})^{u} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('y', commutative=True)))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('u', commutative=True)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Pow(Function('v_x')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Add(Pow(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Integer(-1)))"], [["sin", 5], "Equality(sin(Add(Pow(Function('v_x')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(-1))), sin(Add(Pow(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(sin(Add(Pow(Function('v_x')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(-1))), sin(Add(Pow(Pow(Derivative(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given t{(\\theta_2)} = \\log{(\\theta_2)}^{\\theta_2}, then obtain \\frac{\\log{(t^{\\theta_2}{(\\theta_2)})}}{t{(\\theta_2)}} = \\frac{\\log{((\\log{(\\theta_2)}^{\\theta_2})^{\\theta_2})}}{t{(\\theta_2)}}", "derivation": "t{(\\theta_2)} = \\log{(\\theta_2)}^{\\theta_2} and t^{\\theta_2}{(\\theta_2)} = (\\log{(\\theta_2)}^{\\theta_2})^{\\theta_2} and \\log{(t^{\\theta_2}{(\\theta_2)})} = \\log{((\\log{(\\theta_2)}^{\\theta_2})^{\\theta_2})} and \\frac{\\log{(t^{\\theta_2}{(\\theta_2)})}}{t{(\\theta_2)}} = \\frac{\\log{((\\log{(\\theta_2)}^{\\theta_2})^{\\theta_2})}}{t{(\\theta_2)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), Pow(log(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(log(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), log(Pow(Pow(log(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 3, "Function('t')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), log(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), log(Pow(Pow(log(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(z^{*},v)} = \\sin{(v z^{*})}, then obtain z^{*} + \\int v z^{*} \\mathbf{g}{(z^{*},v)} dz^{*} = z^{*} + \\int v z^{*} \\sin{(v z^{*})} dz^{*}", "derivation": "\\mathbf{g}{(z^{*},v)} = \\sin{(v z^{*})} and v z^{*} \\mathbf{g}{(z^{*},v)} = v z^{*} \\sin{(v z^{*})} and \\int v z^{*} \\mathbf{g}{(z^{*},v)} dz^{*} = \\int v z^{*} \\sin{(v z^{*})} dz^{*} and z^{*} + \\int v z^{*} \\mathbf{g}{(z^{*},v)} dz^{*} = z^{*} + \\int v z^{*} \\sin{(v z^{*})} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True), Symbol('v', commutative=True)), sin(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True))))"], [["times", 1, "Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), sin(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True)))))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), sin(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True))))"], [["add", 3, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Integral(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('z^*', commutative=True)))), Add(Symbol('z^*', commutative=True), Integral(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True), sin(Mul(Symbol('v', commutative=True), Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(f,C_{d})} = \\cos{(C_{d} f)} and \\mathbf{M}{(C_{d},f)} = \\int \\dot{y}{(f,C_{d})} df, then obtain - \\frac{\\int \\dot{y}{(f,C_{d})} df}{\\cos{(C_{d} f)}} = - \\frac{\\int \\cos{(C_{d} f)} df}{\\cos{(C_{d} f)}}", "derivation": "\\dot{y}{(f,C_{d})} = \\cos{(C_{d} f)} and \\int \\dot{y}{(f,C_{d})} df = \\int \\cos{(C_{d} f)} df and \\mathbf{M}{(C_{d},f)} = \\int \\dot{y}{(f,C_{d})} df and \\mathbf{M}{(C_{d},f)} = \\int \\cos{(C_{d} f)} df and - \\frac{\\mathbf{M}{(C_{d},f)}}{\\cos{(C_{d} f)}} = - \\frac{\\int \\cos{(C_{d} f)} df}{\\cos{(C_{d} f)}} and - \\frac{\\int \\dot{y}{(f,C_{d})} df}{\\cos{(C_{d} f)}} = - \\frac{\\int \\cos{(C_{d} f)} df}{\\cos{(C_{d} f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('f', commutative=True), Symbol('C_d', commutative=True)), cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('f', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('C_d', commutative=True), Symbol('f', commutative=True)), Integral(Function('\\\\dot{y}')(Symbol('f', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('C_d', commutative=True), Symbol('f', commutative=True)), Integral(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('C_d', commutative=True), Symbol('f', commutative=True)), Pow(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Integral(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Integral(Function('\\\\dot{y}')(Symbol('f', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Integer(-1), Pow(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Integral(cos(Mul(Symbol('C_d', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given b{(\\psi^*)} = \\psi^*, then derive \\frac{d}{d \\psi^*} b{(\\psi^*)} = 1, then derive \\frac{\\partial}{\\partial \\psi^*} (A_{x} + b{(\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (\\psi^* + \\rho_f), then obtain (\\frac{\\partial}{\\partial \\psi^*} (A_{x} + b{(\\psi^*)}))^{\\rho_f} = (\\frac{\\partial}{\\partial \\psi^*} (\\psi^* + \\rho_f))^{\\rho_f}", "derivation": "b{(\\psi^*)} = \\psi^* and \\frac{d}{d \\psi^*} b{(\\psi^*)} = \\frac{d}{d \\psi^*} \\psi^* and \\frac{d}{d \\psi^*} b{(\\psi^*)} = 1 and \\int \\frac{d}{d \\psi^*} b{(\\psi^*)} d\\psi^* = \\int 1 d\\psi^* and \\frac{d}{d \\psi^*} \\int \\frac{d}{d \\psi^*} b{(\\psi^*)} d\\psi^* = \\frac{d}{d \\psi^*} \\int 1 d\\psi^* and \\frac{\\partial}{\\partial \\psi^*} (A_{x} + b{(\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (\\psi^* + \\rho_f) and (\\frac{\\partial}{\\partial \\psi^*} (A_{x} + b{(\\psi^*)}))^{\\rho_f} = (\\frac{\\partial}{\\partial \\psi^*} (\\psi^* + \\rho_f))^{\\rho_f}", "srepr_derivation": [["renaming_premise", "Equality(Function('b')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Derivative(Function('b')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('b')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Function('b')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["power", 6, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('A_x', commutative=True), Function('b')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} = \\hat{x}_0 + v_{1}, then obtain - \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} + \\int \\operatorname{m_{s}}^{2}{(\\hat{x}_0,v_{1})} d\\hat{x}_0 = - \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} + \\int (\\hat{x}_0 + v_{1}) \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} d\\hat{x}_0", "derivation": "\\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} = \\hat{x}_0 + v_{1} and \\operatorname{m_{s}}^{2}{(\\hat{x}_0,v_{1})} = (\\hat{x}_0 + v_{1}) \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} and \\int \\operatorname{m_{s}}^{2}{(\\hat{x}_0,v_{1})} d\\hat{x}_0 = \\int (\\hat{x}_0 + v_{1}) \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} d\\hat{x}_0 and - \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} + \\int \\operatorname{m_{s}}^{2}{(\\hat{x}_0,v_{1})} d\\hat{x}_0 = - \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} + \\int (\\hat{x}_0 + v_{1}) \\operatorname{m_{s}}{(\\hat{x}_0,v_{1})} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)))"], [["times", 1, "Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Pow(Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Pow(Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 3, "Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))), Integral(Pow(Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True)), Function('m_s')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(U,\\mathbf{D})} = U - \\mathbf{D}, then obtain \\int - U \\sigma_{x}{(U,\\mathbf{D})} d\\mathbf{D} = \\int - U (U - \\mathbf{D}) d\\mathbf{D}", "derivation": "\\sigma_{x}{(U,\\mathbf{D})} = U - \\mathbf{D} and U \\sigma_{x}{(U,\\mathbf{D})} = U (U - \\mathbf{D}) and - U \\sigma_{x}{(U,\\mathbf{D})} = - U (U - \\mathbf{D}) and \\int - U \\sigma_{x}{(U,\\mathbf{D})} d\\mathbf{D} = \\int - U (U - \\mathbf{D}) d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('\\\\sigma_x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('U', commutative=True), Function('\\\\sigma_x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('U', commutative=True), Function('\\\\sigma_x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Integer(-1), Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then obtain (\\frac{d^{2}}{d \\mathbf{g}^{2}} \\chi{(\\mathbf{g})})^{2} = (\\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin{(\\mathbf{g})})^{2}", "derivation": "\\chi{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} \\chi{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\sin{(\\mathbf{g})} and \\frac{d^{2}}{d \\mathbf{g}^{2}} \\chi{(\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin{(\\mathbf{g})} and (\\frac{d^{2}}{d \\mathbf{g}^{2}} \\chi{(\\mathbf{g})})^{2} = (\\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin{(\\mathbf{g})})^{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(z)} = e^{z}, then derive - z + \\int \\operatorname{m_{s}}{(z)} dz = \\hat{p} - z + e^{z}, then derive z (\\hat{p} - z + e^{z}) = z (f^{*} - z + e^{z}), then obtain z^{2} (- z + \\int e^{z} dz)^{2} = z^{2} (- z + \\int e^{z} dz) (f^{*} - z + e^{z})", "derivation": "\\operatorname{m_{s}}{(z)} = e^{z} and \\int \\operatorname{m_{s}}{(z)} dz = \\int e^{z} dz and - z + \\int \\operatorname{m_{s}}{(z)} dz = - z + \\int e^{z} dz and - z + \\int \\operatorname{m_{s}}{(z)} dz = \\hat{p} - z + e^{z} and \\hat{p} - z + e^{z} = - z + \\int e^{z} dz and z (\\hat{p} - z + e^{z}) = z (- z + \\int e^{z} dz) and z (\\hat{p} - z + e^{z}) = z (f^{*} - z + e^{z}) and z^{2} (- z + \\int e^{z} dz) (\\hat{p} - z + e^{z}) = z^{2} (- z + \\int e^{z} dz) (f^{*} - z + e^{z}) and z^{2} (- z + \\int e^{z} dz)^{2} = z^{2} (- z + \\int e^{z} dz) (f^{*} - z + e^{z})", "srepr_derivation": [["get_premise", "Equality(Function('m_s')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["minus", 2, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Function('m_s')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Function('m_s')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["times", 5, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Mul(Symbol('z', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))), Mul(Symbol('z', commutative=True), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))))"], [["times", 7, "Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integer(2))), Mul(Pow(Symbol('z', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(F_{H})} = F_{H}, then derive \\int \\operatorname{z^{*}}{(F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + \\mathbf{p}, then derive (\\frac{F_{H}^{2}}{2} + M_{E})^{2} = (\\frac{F_{H}^{2}}{2} + \\mathbf{p})^{2}, then obtain (\\frac{F_{H}^{2}}{2} + M_{E})^{2} = (\\int \\operatorname{z^{*}}{(F_{H})} dF_{H})^{2}", "derivation": "\\operatorname{z^{*}}{(F_{H})} = F_{H} and \\int \\operatorname{z^{*}}{(F_{H})} dF_{H} = \\int F_{H} dF_{H} and \\int \\operatorname{z^{*}}{(F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + \\mathbf{p} and \\int F_{H} dF_{H} = \\frac{F_{H}^{2}}{2} + \\mathbf{p} and (\\int F_{H} dF_{H})^{2} = (\\frac{F_{H}^{2}}{2} + \\mathbf{p})^{2} and (\\frac{F_{H}^{2}}{2} + M_{E})^{2} = (\\frac{F_{H}^{2}}{2} + \\mathbf{p})^{2} and (\\frac{F_{H}^{2}}{2} + M_{E})^{2} = (\\int \\operatorname{z^{*}}{(F_{H})} dF_{H})^{2}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True))), Integer(2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('M_E', commutative=True)), Integer(2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('M_E', commutative=True)), Integer(2)), Pow(Integral(Function('z^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\omega{(\\phi_1,c_{0})} = - \\phi_1 + c_{0}, then obtain \\phi_1 (- \\phi_1 + c_{0}) = - \\phi_1 (\\phi_1 - c_{0})", "derivation": "\\omega{(\\phi_1,c_{0})} = - \\phi_1 + c_{0} and - \\omega{(\\phi_1,c_{0})} = \\phi_1 - c_{0} and \\phi_1 \\omega{(\\phi_1,c_{0})} = - \\phi_1 (\\phi_1 - c_{0}) and \\phi_1 (- \\phi_1 + c_{0}) = - \\phi_1 (\\phi_1 - c_{0})", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('c_0', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\omega')(Symbol('\\\\phi_1', commutative=True), Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given x{(A_{z},\\varepsilon)} = e^{A_{z}^{\\varepsilon}} and h{(A_{z})} = 0^{A_{z}}, then obtain \\int (- x{(A_{z},\\varepsilon)} + e^{A_{z}^{\\varepsilon}})^{A_{z}} dA_{z} = \\int 1 dA_{z}", "derivation": "x{(A_{z},\\varepsilon)} = e^{A_{z}^{\\varepsilon}} and 0 = - x{(A_{z},\\varepsilon)} + e^{A_{z}^{\\varepsilon}} and 0^{A_{z}} = (- x{(A_{z},\\varepsilon)} + e^{A_{z}^{\\varepsilon}})^{A_{z}} and h{(A_{z})} = 0^{A_{z}} and h{(A_{z})} = 1 and \\int h{(A_{z})} dA_{z} = \\int 1 dA_{z} and h{(A_{z})} = (- x{(A_{z},\\varepsilon)} + e^{A_{z}^{\\varepsilon}})^{A_{z}} and \\int (- x{(A_{z},\\varepsilon)} + e^{A_{z}^{\\varepsilon}})^{A_{z}} dA_{z} = \\int 1 dA_{z}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 1, "Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('A_z', commutative=True)), Pow(Integer(0), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('h')(Symbol('A_z', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('h')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('h')(Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Symbol('A_z', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('x')(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), exp(Pow(Symbol('A_z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v_{1},\\mathbf{F})} = \\mathbf{F} v_{1}, then derive 0 = - \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\operatorname{M_{E}}{(v_{1},\\mathbf{F})}, then obtain 0 = - \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\mathbf{F} v_{1}", "derivation": "\\operatorname{M_{E}}{(v_{1},\\mathbf{F})} = \\mathbf{F} v_{1} and \\frac{\\partial}{\\partial v_{1}} \\operatorname{M_{E}}{(v_{1},\\mathbf{F})} = \\frac{\\partial}{\\partial v_{1}} \\mathbf{F} v_{1} and 0 = \\frac{\\partial}{\\partial v_{1}} \\mathbf{F} v_{1} - \\frac{\\partial}{\\partial v_{1}} \\operatorname{M_{E}}{(v_{1},\\mathbf{F})} and \\frac{d}{d v_{1}} 0 = \\frac{\\partial}{\\partial v_{1}} (\\frac{\\partial}{\\partial v_{1}} \\mathbf{F} v_{1} - \\frac{\\partial}{\\partial v_{1}} \\operatorname{M_{E}}{(v_{1},\\mathbf{F})}) and 0 = - \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\operatorname{M_{E}}{(v_{1},\\mathbf{F})} and 0 = - \\frac{\\partial^{2}}{\\partial v_{1}^{2}} \\mathbf{F} v_{1}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Integer(-1), Derivative(Function('M_E')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\delta{(a,v_{x})} = v_{x} \\sin{(a)}, then obtain a + v_{x} \\cos{(a)} - 3 \\delta{(a,v_{x})} = a - v_{x} \\sin{(a)} + v_{x} \\cos{(a)} - 2 \\delta{(a,v_{x})}", "derivation": "\\delta{(a,v_{x})} = v_{x} \\sin{(a)} and - a + \\delta{(a,v_{x})} = - a + v_{x} \\sin{(a)} and a - \\delta{(a,v_{x})} = a - v_{x} \\sin{(a)} and a - 2 \\delta{(a,v_{x})} = a - v_{x} \\sin{(a)} - \\delta{(a,v_{x})} and a - 3 \\delta{(a,v_{x})} = a - v_{x} \\sin{(a)} - 2 \\delta{(a,v_{x})} and a - 3 \\delta{(a,v_{x})} + \\frac{\\partial}{\\partial a} v_{x} \\sin{(a)} = a - v_{x} \\sin{(a)} - 2 \\delta{(a,v_{x})} + \\frac{\\partial}{\\partial a} v_{x} \\sin{(a)} and a + v_{x} \\cos{(a)} - 3 \\delta{(a,v_{x})} = a - v_{x} \\sin{(a)} + v_{x} \\cos{(a)} - 2 \\delta{(a,v_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Mul(Integer(-1), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))))"], [["minus", 4, "Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(3), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))))"], [["add", 5, "Derivative(Mul(Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(3), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True))), Derivative(Mul(Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True))), Derivative(Mul(Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(Symbol('a', commutative=True), Mul(Symbol('v_x', commutative=True), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Integer(3), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Symbol('a', commutative=True))), Mul(Symbol('v_x', commutative=True), cos(Symbol('a', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('a', commutative=True), Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given u{(A_{x})} = A_{x} and \\dot{\\mathbf{r}}{(A_{x},\\rho_f)} = - (- A_{x}^{\\rho_f} + u{(A_{x})})^{A_{x}}, then obtain \\int \\dot{\\mathbf{r}}^{A_{x}}{(A_{x},\\rho_f)} d\\rho_f = \\int (- (A_{x} - A_{x}^{\\rho_f})^{A_{x}})^{A_{x}} d\\rho_f", "derivation": "u{(A_{x})} = A_{x} and \\dot{\\mathbf{r}}{(A_{x},\\rho_f)} = - (- A_{x}^{\\rho_f} + u{(A_{x})})^{A_{x}} and \\dot{\\mathbf{r}}^{A_{x}}{(A_{x},\\rho_f)} = (- (- A_{x}^{\\rho_f} + u{(A_{x})})^{A_{x}})^{A_{x}} and \\int \\dot{\\mathbf{r}}^{A_{x}}{(A_{x},\\rho_f)} d\\rho_f = \\int (- (- A_{x}^{\\rho_f} + u{(A_{x})})^{A_{x}})^{A_{x}} d\\rho_f and \\int \\dot{\\mathbf{r}}^{A_{x}}{(A_{x},\\rho_f)} d\\rho_f = \\int (- (A_{x} - A_{x}^{\\rho_f})^{A_{x}})^{A_{x}} d\\rho_f", "srepr_derivation": [["renaming_premise", "Equality(Function('u')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True))), Function('u')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('A_x', commutative=True)), Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True))), Function('u')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True))), Function('u')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\Psi{(f_{\\mathbf{p}},m)} = f_{\\mathbf{p}} + m and \\varphi{(f_{\\mathbf{p}},m)} = f_{\\mathbf{p}} + m, then obtain (f_{\\mathbf{p}} + m)^{m} + \\Psi{(f_{\\mathbf{p}},m)} \\varphi{(f_{\\mathbf{p}},m)} = (f_{\\mathbf{p}} + m)^{m} + \\varphi^{2}{(f_{\\mathbf{p}},m)}", "derivation": "\\Psi{(f_{\\mathbf{p}},m)} = f_{\\mathbf{p}} + m and (f_{\\mathbf{p}} + m) \\Psi{(f_{\\mathbf{p}},m)} = (f_{\\mathbf{p}} + m)^{2} and \\varphi{(f_{\\mathbf{p}},m)} = f_{\\mathbf{p}} + m and \\Psi{(f_{\\mathbf{p}},m)} \\varphi{(f_{\\mathbf{p}},m)} = \\varphi^{2}{(f_{\\mathbf{p}},m)} and (f_{\\mathbf{p}} + m)^{m} + \\Psi{(f_{\\mathbf{p}},m)} \\varphi{(f_{\\mathbf{p}},m)} = (f_{\\mathbf{p}} + m)^{m} + \\varphi^{2}{(f_{\\mathbf{p}},m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)))"], [["times", 1, "Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Function('\\\\Psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True))), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\Psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True))), Pow(Function('\\\\varphi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Integer(2)))"], [["add", 4, "Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))"], "Equality(Add(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Mul(Function('\\\\Psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)))), Add(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Function('\\\\varphi')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)}, then derive \\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = C_{d} + \\sin{(\\mathbf{J}_f)}, then obtain (\\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f)^{\\mathbf{J}_f} = (C_{d} + \\sin{(\\mathbf{J}_f)})^{\\mathbf{J}_f}", "derivation": "\\operatorname{c_{0}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f = C_{d} + \\sin{(\\mathbf{J}_f)} and (\\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f)^{\\mathbf{J}_f} = (\\int \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f)^{\\mathbf{J}_f} and \\int \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f = C_{d} + \\sin{(\\mathbf{J}_f)} and (\\int \\operatorname{c_{0}}{(\\mathbf{J}_f)} d\\mathbf{J}_f)^{\\mathbf{J}_f} = (C_{d} + \\sin{(\\mathbf{J}_f)})^{\\mathbf{J}_f}", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('C_d', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Integral(Function('c_0')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('C_d', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Integral(Function('c_0')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Add(Symbol('C_d', commutative=True), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given b{(\\theta_1)} = e^{\\theta_1}, then obtain \\theta_1 - b{(\\theta_1)} = \\theta_1 + b{(\\theta_1)} - 2 e^{\\theta_1}", "derivation": "b{(\\theta_1)} = e^{\\theta_1} and - \\theta_1 + b{(\\theta_1)} = - \\theta_1 + e^{\\theta_1} and \\theta_1 - b{(\\theta_1)} = \\theta_1 - e^{\\theta_1} and 2 \\theta_1 - 2 b{(\\theta_1)} = 2 \\theta_1 - b{(\\theta_1)} - e^{\\theta_1} and 3 \\theta_1 - 2 b{(\\theta_1)} - e^{\\theta_1} = 3 \\theta_1 - b{(\\theta_1)} - 2 e^{\\theta_1} and \\theta_1 - e^{\\theta_1} = \\theta_1 + b{(\\theta_1)} - 2 e^{\\theta_1} and \\theta_1 - b{(\\theta_1)} = \\theta_1 + b{(\\theta_1)} - 2 e^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('b')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Function('b')(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('b')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), Function('b')(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('b')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 4, "Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), Function('b')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('b')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 5, "Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), Function('b')(Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Function('b')(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given x{(k,v_{z})} = k + \\log{(v_{z})}, then derive \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})} = \\frac{1}{v_{z}}, then obtain \\cos{(\\log{(m_{s})} + \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})})} = \\cos{(\\log{(m_{s})} + \\frac{1}{v_{z}})}", "derivation": "x{(k,v_{z})} = k + \\log{(v_{z})} and \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})} = \\frac{\\partial}{\\partial v_{z}} (k + \\log{(v_{z})}) and \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})} = \\frac{1}{v_{z}} and \\log{(m_{s})} + \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})} = \\log{(m_{s})} + \\frac{1}{v_{z}} and \\cos{(\\log{(m_{s})} + \\frac{\\partial}{\\partial v_{z}} x{(k,v_{z})})} = \\cos{(\\log{(m_{s})} + \\frac{1}{v_{z}})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('k', commutative=True), log(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), log(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Pow(Symbol('v_z', commutative=True), Integer(-1)))"], [["add", 3, "log(Symbol('m_s', commutative=True))"], "Equality(Add(log(Symbol('m_s', commutative=True)), Derivative(Function('x')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(log(Symbol('m_s', commutative=True)), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["cos", 4], "Equality(cos(Add(log(Symbol('m_s', commutative=True)), Derivative(Function('x')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))), cos(Add(log(Symbol('m_s', commutative=True)), Pow(Symbol('v_z', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(c,m_{s},L)} = L m_{s} - c and \\bar{\\h}{(k,C_{d})} = C_{d} k, then obtain C_{d} k + \\eta^{\\prime}{(c,m_{s},L)} = C_{d} k + L m_{s} - c", "derivation": "\\eta^{\\prime}{(c,m_{s},L)} = L m_{s} - c and \\bar{\\h}{(k,C_{d})} = C_{d} k and \\eta^{\\prime}{(c,m_{s},L)} + \\bar{\\h}{(k,C_{d})} = L m_{s} - c + \\bar{\\h}{(k,C_{d})} and C_{d} k + \\eta^{\\prime}{(c,m_{s},L)} = C_{d} k + L m_{s} - c", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True), Symbol('m_s', commutative=True), Symbol('L', commutative=True)), Add(Mul(Symbol('L', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('k', commutative=True)))"], [["add", 1, "Function('\\\\hbar')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True), Symbol('m_s', commutative=True), Symbol('L', commutative=True)), Function('\\\\hbar')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hbar')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Symbol('C_d', commutative=True), Symbol('k', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('c', commutative=True), Symbol('m_s', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\theta{(a,l)} = \\frac{\\partial}{\\partial l} l^{a}, then derive \\theta{(a,l)} = \\frac{a l^{a}}{l}, then obtain l^{a} + \\frac{\\partial}{\\partial l} \\frac{a l^{a}}{l} = l^{a} + \\frac{\\partial}{\\partial l} \\theta{(a,l)}", "derivation": "\\theta{(a,l)} = \\frac{\\partial}{\\partial l} l^{a} and \\frac{\\partial}{\\partial l} \\theta{(a,l)} = \\frac{\\partial^{2}}{\\partial l^{2}} l^{a} and \\theta{(a,l)} = \\frac{a l^{a}}{l} and \\frac{\\partial}{\\partial l} \\frac{a l^{a}}{l} = \\frac{\\partial^{2}}{\\partial l^{2}} l^{a} and \\frac{\\partial}{\\partial l} \\frac{a l^{a}}{l} = \\frac{\\partial}{\\partial l} \\theta{(a,l)} and l^{a} + \\frac{\\partial}{\\partial l} \\frac{a l^{a}}{l} = l^{a} + \\frac{\\partial}{\\partial l} \\theta{(a,l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Derivative(Pow(Symbol('l', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Symbol('l', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Symbol('l', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 5, "Pow(Symbol('l', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Pow(Symbol('l', commutative=True), Symbol('a', commutative=True)), Derivative(Mul(Symbol('a', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Pow(Symbol('l', commutative=True), Symbol('a', commutative=True)), Derivative(Function('\\\\theta')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(v_{1})} = \\sin{(v_{1})}, then obtain \\int (\\dot{y}^{2}{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})}) dv_{1} = \\int (\\dot{y}{(v_{1})} \\sin{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})}) dv_{1}", "derivation": "\\dot{y}{(v_{1})} = \\sin{(v_{1})} and \\dot{y}^{2}{(v_{1})} = \\dot{y}{(v_{1})} \\sin{(v_{1})} and \\dot{y}^{2}{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})} = \\dot{y}{(v_{1})} \\sin{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})} and \\int (\\dot{y}^{2}{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})}) dv_{1} = \\int (\\dot{y}{(v_{1})} \\sin{(v_{1})} - \\frac{d}{d v_{1}} \\dot{y}{(v_{1})}) dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('v_1', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Integer(2)), Mul(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))))"], [["minus", 2, "Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))), Add(Mul(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{J},n)} = \\frac{\\mathbf{J}}{n}, then derive \\int \\frac{\\frac{\\partial}{\\partial n} \\hat{H}_l{(\\mathbf{J},n)}}{n} dn = \\frac{\\mathbf{J}}{2 n^{2}} + \\phi_2, then obtain \\frac{\\mathbf{J}}{2 n^{2}} + \\nabla = \\frac{\\mathbf{J}}{2 n^{2}} + \\phi_2", "derivation": "\\hat{H}_l{(\\mathbf{J},n)} = \\frac{\\mathbf{J}}{n} and \\frac{\\partial}{\\partial n} \\hat{H}_l{(\\mathbf{J},n)} = \\frac{\\partial}{\\partial n} \\frac{\\mathbf{J}}{n} and \\frac{\\frac{\\partial}{\\partial n} \\hat{H}_l{(\\mathbf{J},n)}}{n} = \\frac{\\frac{\\partial}{\\partial n} \\frac{\\mathbf{J}}{n}}{n} and \\int \\frac{\\frac{\\partial}{\\partial n} \\hat{H}_l{(\\mathbf{J},n)}}{n} dn = \\int \\frac{\\frac{\\partial}{\\partial n} \\frac{\\mathbf{J}}{n}}{n} dn and \\int \\frac{\\frac{\\partial}{\\partial n} \\hat{H}_l{(\\mathbf{J},n)}}{n} dn = \\frac{\\mathbf{J}}{2 n^{2}} + \\phi_2 and \\int \\frac{\\frac{\\partial}{\\partial n} \\frac{\\mathbf{J}}{n}}{n} dn = \\frac{\\mathbf{J}}{2 n^{2}} + \\phi_2 and \\frac{\\mathbf{J}}{2 n^{2}} + \\nabla = \\frac{\\mathbf{J}}{2 n^{2}} + \\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 2, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2))), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2))), Symbol('\\\\phi_2', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2))), Symbol('\\\\nabla', commutative=True)), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(m_{s})} = \\cos{(m_{s})}, then obtain \\int - \\frac{\\Psi_{\\lambda}^{2}{(m_{s})}}{\\cos{(m_{s})}} dm_{s} = \\int - \\Psi_{\\lambda}{(m_{s})} dm_{s}", "derivation": "\\Psi_{\\lambda}{(m_{s})} = \\cos{(m_{s})} and \\frac{\\Psi_{\\lambda}{(m_{s})}}{\\cos{(m_{s})}} = 1 and \\frac{\\Psi_{\\lambda}^{2}{(m_{s})}}{\\cos{(m_{s})}} = \\Psi_{\\lambda}{(m_{s})} and - \\frac{\\Psi_{\\lambda}^{2}{(m_{s})}}{\\cos{(m_{s})}} = - \\Psi_{\\lambda}{(m_{s})} and \\int - \\frac{\\Psi_{\\lambda}^{2}{(m_{s})}}{\\cos{(m_{s})}} dm_{s} = \\int - \\Psi_{\\lambda}{(m_{s})} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), cos(Symbol('m_s', commutative=True)))"], [["divide", 1, "cos(Symbol('m_s', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1))), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))))"], [["integrate", 4, "Symbol('m_s', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1))), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given l{(\\mu_0)} = e^{\\mu_0}, then derive \\int l{(\\mu_0)} d\\mu_0 = F_{c} + e^{\\mu_0}, then obtain \\int e^{\\mu_0} d\\mu_0 = F_{c} + e^{\\mu_0}", "derivation": "l{(\\mu_0)} = e^{\\mu_0} and \\int l{(\\mu_0)} d\\mu_0 = \\int e^{\\mu_0} d\\mu_0 and \\int l{(\\mu_0)} d\\mu_0 = F_{c} + e^{\\mu_0} and \\int e^{\\mu_0} d\\mu_0 = F_{c} + e^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('l')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('F_c', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('F_c', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given m{(h,a)} = a + h, then obtain \\cos{(\\int m{(h,a)} dh + \\frac{\\int (a + h) dh}{h})} = \\cos{(\\int (a + h) dh + \\frac{\\int (a + h) dh}{h})}", "derivation": "m{(h,a)} = a + h and \\int m{(h,a)} dh = \\int (a + h) dh and \\frac{\\int m{(h,a)} dh}{h} = \\frac{\\int (a + h) dh}{h} and \\int m{(h,a)} dh + \\frac{\\int m{(h,a)} dh}{h} = \\int (a + h) dh + \\frac{\\int m{(h,a)} dh}{h} and \\cos{(\\int m{(h,a)} dh + \\frac{\\int m{(h,a)} dh}{h})} = \\cos{(\\int (a + h) dh + \\frac{\\int m{(h,a)} dh}{h})} and \\cos{(\\int m{(h,a)} dh + \\frac{\\int (a + h) dh}{h})} = \\cos{(\\int (a + h) dh + \\frac{\\int (a + h) dh}{h})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Add(Symbol('a', commutative=True), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["divide", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))))"], "Equality(Add(Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))))), Add(Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))))))"], [["cos", 4], "Equality(cos(Add(Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True)))))), cos(Add(Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(cos(Add(Integral(Function('m')(Symbol('h', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))), cos(Add(Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Symbol('a', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))))"]]}, {"prompt": "Given \\dot{z}{(m,\\mathbf{A})} = \\mathbf{A} + m, then derive \\frac{\\partial}{\\partial m} \\dot{z}{(m,\\mathbf{A})} = 1, then obtain \\mathbf{A} \\frac{\\partial}{\\partial m} \\dot{z}{(m,\\mathbf{A})} = \\mathbf{A}", "derivation": "\\dot{z}{(m,\\mathbf{A})} = \\mathbf{A} + m and \\frac{\\partial}{\\partial m} \\dot{z}{(m,\\mathbf{A})} = \\frac{\\partial}{\\partial m} (\\mathbf{A} + m) and \\frac{\\partial}{\\partial m} \\dot{z}{(m,\\mathbf{A})} = 1 and \\mathbf{A} \\frac{\\partial}{\\partial m} \\dot{z}{(m,\\mathbf{A})} = \\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('\\\\mathbf{A}', commutative=True))"]]}, {"prompt": "Given \\varphi^{*}{(t)} = \\sin{(\\log{(t)})} and \\operatorname{L_{\\varepsilon}}{(t)} = 2 \\varphi^{*}{(t)}, then obtain 2 t + \\operatorname{L_{\\varepsilon}}{(t)} = 2 t + 2 \\varphi^{*}{(t)}", "derivation": "\\varphi^{*}{(t)} = \\sin{(\\log{(t)})} and t + \\varphi^{*}{(t)} = t + \\sin{(\\log{(t)})} and 2 t + 2 \\varphi^{*}{(t)} = 2 t + \\varphi^{*}{(t)} + \\sin{(\\log{(t)})} and 2 t + \\varphi^{*}{(t)} = 2 t + \\sin{(\\log{(t)})} and 2 t + 2 \\varphi^{*}{(t)} = 2 t + 2 \\sin{(\\log{(t)})} and \\operatorname{L_{\\varepsilon}}{(t)} = 2 \\varphi^{*}{(t)} and 2 t + \\operatorname{L_{\\varepsilon}}{(t)} = 2 t + 2 \\sin{(\\log{(t)})} and 2 t + \\operatorname{L_{\\varepsilon}}{(t)} = 2 t + 2 \\varphi^{*}{(t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True)), sin(log(Symbol('t', commutative=True))))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\varphi^*')(Symbol('t', commutative=True))), Add(Symbol('t', commutative=True), sin(log(Symbol('t', commutative=True)))))"], [["add", 2, "Add(Symbol('t', commutative=True), Function('\\\\varphi^*')(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Function('\\\\varphi^*')(Symbol('t', commutative=True)), sin(log(Symbol('t', commutative=True)))))"], [["add", 2, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('t', commutative=True)), Function('\\\\varphi^*')(Symbol('t', commutative=True))), Add(Mul(Integer(2), Symbol('t', commutative=True)), sin(log(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), sin(log(Symbol('t', commutative=True))))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(2), Symbol('t', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), sin(log(Symbol('t', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Mul(Integer(2), Symbol('t', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given s{(\\Psi_{\\lambda},Q)} = \\frac{Q}{\\Psi_{\\lambda}}, then derive Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} s{(\\Psi_{\\lambda},Q)} = - \\frac{Q^{2}}{\\Psi_{\\lambda}^{2}}, then obtain - \\frac{Q^{2}}{\\Psi_{\\lambda}^{2}} = Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{Q}{\\Psi_{\\lambda}}", "derivation": "s{(\\Psi_{\\lambda},Q)} = \\frac{Q}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} s{(\\Psi_{\\lambda},Q)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{Q}{\\Psi_{\\lambda}} and Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} s{(\\Psi_{\\lambda},Q)} = Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{Q}{\\Psi_{\\lambda}} and Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} s{(\\Psi_{\\lambda},Q)} = - \\frac{Q^{2}}{\\Psi_{\\lambda}^{2}} and - \\frac{Q^{2}}{\\Psi_{\\lambda}^{2}} = Q \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{Q}{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Function('s')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Symbol('Q', commutative=True), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Function('s')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2))), Mul(Symbol('Q', commutative=True), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(J)} = e^{J}, then obtain 1 - \\int \\frac{e^{J}}{y{(J)}} dJ = - \\int \\frac{e^{J}}{y{(J)}} dJ + \\frac{e^{J}}{y{(J)}}", "derivation": "y{(J)} = e^{J} and 1 = \\frac{e^{J}}{y{(J)}} and \\int 1 dJ = \\int \\frac{e^{J}}{y{(J)}} dJ and 1 - \\int 1 dJ = - \\int 1 dJ + \\frac{e^{J}}{y{(J)}} and 1 - \\int \\frac{e^{J}}{y{(J)}} dJ = - \\int \\frac{e^{J}}{y{(J)}} dJ + \\frac{e^{J}}{y{(J)}}", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["divide", 1, "Function('y')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["minus", 2, "Integral(Integer(1), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('J', commutative=True))))), Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))), Add(Mul(Integer(-1), Integral(Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Function('y')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given S{(\\dot{z})} = e^{\\dot{z}}, then derive \\int S{(\\dot{z})} d\\dot{z} = \\rho_b + e^{\\dot{z}}, then obtain \\int e^{\\dot{z}} d\\dot{z} = \\rho_b + e^{\\dot{z}}", "derivation": "S{(\\dot{z})} = e^{\\dot{z}} and \\int S{(\\dot{z})} d\\dot{z} = \\int e^{\\dot{z}} d\\dot{z} and \\int S{(\\dot{z})} d\\dot{z} = \\rho_b + e^{\\dot{z}} and \\int e^{\\dot{z}} d\\dot{z} = \\rho_b + e^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(exp(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('S')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\theta_1,v_{2})} = \\theta_1 v_{2} and \\chi{(\\theta_1,v_{2})} = \\operatorname{C_{1}}{(\\theta_1,v_{2})} - 1, then obtain t_{1} (\\theta_1 v_{2} - 1) = t_{1} \\chi{(\\theta_1,v_{2})}", "derivation": "\\operatorname{C_{1}}{(\\theta_1,v_{2})} = \\theta_1 v_{2} and \\operatorname{C_{1}}{(\\theta_1,v_{2})} - 1 = \\theta_1 v_{2} - 1 and t_{1} (\\operatorname{C_{1}}{(\\theta_1,v_{2})} - 1) = t_{1} (\\theta_1 v_{2} - 1) and \\chi{(\\theta_1,v_{2})} = \\operatorname{C_{1}}{(\\theta_1,v_{2})} - 1 and \\chi{(\\theta_1,v_{2})} = \\theta_1 v_{2} - 1 and t_{1} (\\operatorname{C_{1}}{(\\theta_1,v_{2})} - 1) = t_{1} \\chi{(\\theta_1,v_{2})} and t_{1} (\\theta_1 v_{2} - 1) = t_{1} \\chi{(\\theta_1,v_{2})}", "srepr_derivation": [["get_premise", "Equality(Function('C_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))"], [["times", 2, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Mul(Symbol('t_1', commutative=True), Add(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Symbol('t_1', commutative=True), Add(Function('C_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Mul(Symbol('t_1', commutative=True), Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('t_1', commutative=True), Add(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Mul(Symbol('t_1', commutative=True), Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(a^{\\dagger},r,z)} = a^{\\dagger} + r + z, then derive - a^{\\dagger} + \\int \\operatorname{C_{1}}{(a^{\\dagger},r,z)} dz = L - a^{\\dagger} + \\frac{z^{2}}{2} + z (a^{\\dagger} + r), then obtain - a^{\\dagger} + \\int (a^{\\dagger} + r + z) dz = L - a^{\\dagger} + \\frac{z^{2}}{2} + z (a^{\\dagger} + r)", "derivation": "\\operatorname{C_{1}}{(a^{\\dagger},r,z)} = a^{\\dagger} + r + z and \\int \\operatorname{C_{1}}{(a^{\\dagger},r,z)} dz = \\int (a^{\\dagger} + r + z) dz and - a^{\\dagger} + \\int \\operatorname{C_{1}}{(a^{\\dagger},r,z)} dz = - a^{\\dagger} + \\int (a^{\\dagger} + r + z) dz and - a^{\\dagger} + \\int \\operatorname{C_{1}}{(a^{\\dagger},r,z)} dz = L - a^{\\dagger} + \\frac{z^{2}}{2} + z (a^{\\dagger} + r) and - a^{\\dagger} + \\int (a^{\\dagger} + r + z) dz = L - a^{\\dagger} + \\frac{z^{2}}{2} + z (a^{\\dagger} + r)", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["minus", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('C_1')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('C_1')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))), Mul(Symbol('z', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))), Mul(Symbol('z', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given Q{(\\Omega)} = \\sin{(e^{\\Omega})}, then derive (\\frac{d}{d \\Omega} 0)^{\\Omega} = (\\frac{\\partial}{\\partial \\Omega} (\\hat{H}_l + \\operatorname{Si}{(e^{\\Omega})} - \\int Q{(\\Omega)} d\\Omega))^{\\Omega}, then derive 0^{\\Omega} = (\\frac{\\partial}{\\partial \\Omega} (\\hat{H}_l + \\operatorname{Si}{(e^{\\Omega})} - \\int Q{(\\Omega)} d\\Omega))^{\\Omega}, then obtain (0^{\\Omega})^{\\Omega} = ((\\frac{d}{d \\Omega} 0)^{\\Omega})^{\\Omega}", "derivation": "Q{(\\Omega)} = \\sin{(e^{\\Omega})} and \\int Q{(\\Omega)} d\\Omega = \\int \\sin{(e^{\\Omega})} d\\Omega and 0 = - \\int Q{(\\Omega)} d\\Omega + \\int \\sin{(e^{\\Omega})} d\\Omega and \\frac{d}{d \\Omega} 0 = \\frac{d}{d \\Omega} (- \\int Q{(\\Omega)} d\\Omega + \\int \\sin{(e^{\\Omega})} d\\Omega) and (\\frac{d}{d \\Omega} 0)^{\\Omega} = (\\frac{d}{d \\Omega} (- \\int Q{(\\Omega)} d\\Omega + \\int \\sin{(e^{\\Omega})} d\\Omega))^{\\Omega} and (\\frac{d}{d \\Omega} 0)^{\\Omega} = (\\frac{\\partial}{\\partial \\Omega} (\\hat{H}_l + \\operatorname{Si}{(e^{\\Omega})} - \\int Q{(\\Omega)} d\\Omega))^{\\Omega} and 0^{\\Omega} = (\\frac{\\partial}{\\partial \\Omega} (\\hat{H}_l + \\operatorname{Si}{(e^{\\Omega})} - \\int Q{(\\Omega)} d\\Omega))^{\\Omega} and 0^{\\Omega} = (\\frac{d}{d \\Omega} 0)^{\\Omega} and (0^{\\Omega})^{\\Omega} = ((\\frac{d}{d \\Omega} 0)^{\\Omega})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\Omega', commutative=True)), sin(exp(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["minus", 2, "Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integral(sin(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Si(exp(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Integer(0), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Si(exp(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Integer(0), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["power", 8, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{2},\\mathbf{f})} = \\mathbf{f} + v_{2}, then obtain \\int 2 (\\mathbf{f} + v_{2})^{2} d\\mathbf{f} = \\int (\\mathbf{f} + v_{2}) (2 \\mathbf{f} + 2 v_{2}) d\\mathbf{f}", "derivation": "\\operatorname{x^{{\\}'}}{(v_{2},\\mathbf{f})} = \\mathbf{f} + v_{2} and \\mathbf{f} + v_{2} + \\operatorname{x^{{\\}'}}{(v_{2},\\mathbf{f})} = 2 \\mathbf{f} + 2 v_{2} and 2 \\operatorname{x^{{\\}'}}{(v_{2},\\mathbf{f})} = 2 \\mathbf{f} + 2 v_{2} and 2 (\\mathbf{f} + v_{2}) \\operatorname{x^{{\\}'}}{(v_{2},\\mathbf{f})} = (\\mathbf{f} + v_{2}) (2 \\mathbf{f} + 2 v_{2}) and 2 (\\mathbf{f} + v_{2})^{2} = (\\mathbf{f} + v_{2}) (2 \\mathbf{f} + 2 v_{2}) and \\int 2 (\\mathbf{f} + v_{2})^{2} d\\mathbf{f} = \\int (\\mathbf{f} + v_{2}) (2 \\mathbf{f} + 2 v_{2}) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True), Function('x^\\\\prime')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('x^\\\\prime')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["times", 3, "Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(2), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Function('x^\\\\prime')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and v{(\\hat{H}_l)} = \\sin^{\\hat{H}_l}{(\\hat{H}_l)}, then obtain \\frac{v{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}} = \\frac{\\sin^{\\hat{H}_l}{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}}", "derivation": "\\operatorname{C_{d}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\operatorname{C_{d}}^{\\hat{H}_l}{(\\hat{H}_l)} = \\sin^{\\hat{H}_l}{(\\hat{H}_l)} and v{(\\hat{H}_l)} = \\sin^{\\hat{H}_l}{(\\hat{H}_l)} and v{(\\hat{H}_l)} = \\operatorname{C_{d}}^{\\hat{H}_l}{(\\hat{H}_l)} and \\frac{v{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}} = \\frac{\\operatorname{C_{d}}^{\\hat{H}_l}{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}} and \\frac{v{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}} = \\frac{\\sin^{\\hat{H}_l}{(\\hat{H}_l)}}{\\operatorname{C_{d}}{(\\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('v')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["divide", 4, "Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Function('v')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Function('v')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Function('C_d')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\delta,f_{E})} = f_{E} \\log{(\\delta)}, then obtain f_{E} (\\log{(\\delta)}^{2} - 1 + \\frac{2 \\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}}) = f_{E} (\\log{(\\delta)}^{2} + \\frac{\\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}})", "derivation": "\\sigma_{x}{(\\delta,f_{E})} = f_{E} \\log{(\\delta)} and \\sigma_{x}{(\\delta,f_{E})} \\log{(\\delta)} = f_{E} \\log{(\\delta)}^{2} and \\frac{\\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}} = 1 and \\log{(\\delta)}^{2} + \\frac{\\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}} = \\log{(\\delta)}^{2} + 1 and f_{E} (\\log{(\\delta)}^{2} + \\frac{\\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}}) = f_{E} (\\log{(\\delta)}^{2} + 1) and f_{E} (\\log{(\\delta)}^{2} - 1 + \\frac{2 \\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}}) = f_{E} (\\log{(\\delta)}^{2} + \\frac{\\sigma_{x}{(\\delta,f_{E})}}{f_{E} \\log{(\\delta)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), log(Symbol('\\\\delta', commutative=True))))"], [["times", 1, "log(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2))))"], [["divide", 2, "Mul(Symbol('f_E', commutative=True), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))), Integer(1))"], [["add", 3, "Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2))"], "Equality(Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)))), Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Integer(1)))"], [["divide", 4, "Pow(Symbol('f_E', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('f_E', commutative=True), Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))))), Mul(Symbol('f_E', commutative=True), Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('f_E', commutative=True), Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Integer(-1), Mul(Integer(2), Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))))), Mul(Symbol('f_E', commutative=True), Add(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\delta', commutative=True), Symbol('f_E', commutative=True)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given U{(W,\\eta^{\\prime})} = \\int (W + \\eta^{\\prime}) dW, then derive U{(W,\\eta^{\\prime})} = \\frac{W^{2}}{2} + W \\eta^{\\prime} + \\hat{X}, then obtain - W^{2} + 2 \\int (W + \\eta^{\\prime}) dW = - \\frac{W^{2}}{2} + W \\eta^{\\prime} + \\hat{X} + \\int (W + \\eta^{\\prime}) dW", "derivation": "U{(W,\\eta^{\\prime})} = \\int (W + \\eta^{\\prime}) dW and U{(W,\\eta^{\\prime})} = \\frac{W^{2}}{2} + W \\eta^{\\prime} + \\hat{X} and \\int (W + \\eta^{\\prime}) dW = \\frac{W^{2}}{2} + W \\eta^{\\prime} + \\hat{X} and - W^{2} + 2 \\int (W + \\eta^{\\prime}) dW = - \\frac{W^{2}}{2} + W \\eta^{\\prime} + \\hat{X} + \\int (W + \\eta^{\\prime}) dW", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('U')(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 3, "Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(2))), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Integer(2), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\hat{X}', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given G{(\\mathbf{B},\\mathbf{S})} = \\log{(\\mathbf{B} \\mathbf{S})} and l{(\\mathbf{B},\\mathbf{S})} = \\iint \\log{(\\mathbf{B} \\mathbf{S})} d\\mathbf{S} d\\mathbf{B}, then obtain \\iint G{(\\mathbf{B},\\mathbf{S})} d\\mathbf{S} d\\mathbf{B} = l{(\\mathbf{B},\\mathbf{S})}", "derivation": "G{(\\mathbf{B},\\mathbf{S})} = \\log{(\\mathbf{B} \\mathbf{S})} and \\int G{(\\mathbf{B},\\mathbf{S})} d\\mathbf{S} = \\int \\log{(\\mathbf{B} \\mathbf{S})} d\\mathbf{S} and \\iint G{(\\mathbf{B},\\mathbf{S})} d\\mathbf{S} d\\mathbf{B} = \\iint \\log{(\\mathbf{B} \\mathbf{S})} d\\mathbf{S} d\\mathbf{B} and l{(\\mathbf{B},\\mathbf{S})} = \\iint \\log{(\\mathbf{B} \\mathbf{S})} d\\mathbf{S} d\\mathbf{B} and \\iint G{(\\mathbf{B},\\mathbf{S})} d\\mathbf{S} d\\mathbf{B} = l{(\\mathbf{B},\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Function('l')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\eta{(G,L)} = G + L, then derive \\frac{\\partial}{\\partial G} \\eta{(G,L)} = 1, then obtain \\int \\frac{\\partial}{\\partial G} (G + L) dL = \\int 1 dL", "derivation": "\\eta{(G,L)} = G + L and \\frac{\\partial}{\\partial G} \\eta{(G,L)} = \\frac{\\partial}{\\partial G} (G + L) and \\frac{\\partial}{\\partial G} \\eta{(G,L)} = 1 and \\frac{\\partial}{\\partial G} (G + L) = 1 and \\int \\frac{\\partial}{\\partial G} (G + L) dL = \\int 1 dL", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('L', commutative=True)), Add(Symbol('G', commutative=True), Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('G', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('G', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('G', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('G', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\mathbf{g}{(x^\\prime,\\varphi^*)} = (x^\\prime)^{2} \\cos{(\\varphi^*)}, then obtain (x^\\prime)^{4} \\operatorname{J_{\\varepsilon}}^{2}{(\\varphi^*)} = (x^\\prime)^{4} \\cos^{2}{(\\varphi^*)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\varphi^*)} = \\cos{(\\varphi^*)} and (x^\\prime)^{2} \\operatorname{J_{\\varepsilon}}{(\\varphi^*)} = (x^\\prime)^{2} \\cos{(\\varphi^*)} and \\mathbf{g}{(x^\\prime,\\varphi^*)} = (x^\\prime)^{2} \\cos{(\\varphi^*)} and \\mathbf{g}{(x^\\prime,\\varphi^*)} = (x^\\prime)^{2} \\operatorname{J_{\\varepsilon}}{(\\varphi^*)} and \\mathbf{g}^{2}{(x^\\prime,\\varphi^*)} = (x^\\prime)^{4} \\cos^{2}{(\\varphi^*)} and (x^\\prime)^{4} \\operatorname{J_{\\varepsilon}}^{2}{(\\varphi^*)} = (x^\\prime)^{4} \\cos^{2}{(\\varphi^*)}", "srepr_derivation": [["get_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), cos(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(4)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(4)), Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(4)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}_f{(A_{y},\\rho)} = \\log{(A_{y} - \\rho)}, then obtain \\rho \\int - \\frac{\\mathbf{J}_f{(A_{y},\\rho)}}{\\rho} d\\rho + 1 = \\rho \\int - \\frac{\\log{(A_{y} - \\rho)}}{\\rho} d\\rho + 1", "derivation": "\\mathbf{J}_f{(A_{y},\\rho)} = \\log{(A_{y} - \\rho)} and - \\frac{\\mathbf{J}_f{(A_{y},\\rho)}}{\\rho} = - \\frac{\\log{(A_{y} - \\rho)}}{\\rho} and \\int - \\frac{\\mathbf{J}_f{(A_{y},\\rho)}}{\\rho} d\\rho = \\int - \\frac{\\log{(A_{y} - \\rho)}}{\\rho} d\\rho and \\rho \\int - \\frac{\\mathbf{J}_f{(A_{y},\\rho)}}{\\rho} d\\rho = \\rho \\int - \\frac{\\log{(A_{y} - \\rho)}}{\\rho} d\\rho and \\rho \\int - \\frac{\\mathbf{J}_f{(A_{y},\\rho)}}{\\rho} d\\rho + 1 = \\rho \\int - \\frac{\\log{(A_{y} - \\rho)}}{\\rho} d\\rho + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_y', commutative=True), Symbol('\\\\rho', commutative=True)), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('A_y', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))))"], [["integrate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('A_y', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["times", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('A_y', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True)))), Mul(Symbol('\\\\rho', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Tuple(Symbol('\\\\rho', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('A_y', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1)), Add(Mul(Symbol('\\\\rho', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), log(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))), Tuple(Symbol('\\\\rho', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given p{(W)} = e^{W}, then obtain \\int ((p{(W)} - e^{W})^{W} - \\frac{\\iint (p{(W)} - e^{W}) dW dW}{p{(W)}}) dW = \\int ((p{(W)} - e^{W})^{W} - \\frac{\\iint 0 dW dW}{p{(W)}}) dW", "derivation": "p{(W)} = e^{W} and p{(W)} - e^{W} = 0 and \\int (p{(W)} - e^{W}) dW = \\int 0 dW and \\iint (p{(W)} - e^{W}) dW dW = \\iint 0 dW dW and - e^{- W} \\iint (p{(W)} - e^{W}) dW dW = - e^{- W} \\iint 0 dW dW and - \\frac{\\iint (p{(W)} - e^{W}) dW dW}{p{(W)}} = - \\frac{\\iint 0 dW dW}{p{(W)}} and (p{(W)} - e^{W})^{W} - \\frac{\\iint (p{(W)} - e^{W}) dW dW}{p{(W)}} = (p{(W)} - e^{W})^{W} - \\frac{\\iint 0 dW dW}{p{(W)}} and \\int ((p{(W)} - e^{W})^{W} - \\frac{\\iint (p{(W)} - e^{W}) dW dW}{p{(W)}}) dW = \\int ((p{(W)} - e^{W})^{W} - \\frac{\\iint 0 dW dW}{p{(W)}}) dW", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["minus", 1, "exp(Symbol('W', commutative=True))"], "Equality(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Integer(0), Tuple(Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Integer(0), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), exp(Symbol('W', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('W', commutative=True))), Integral(Integer(0), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["add", 6, "Pow(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Symbol('W', commutative=True))"], "Equality(Add(Pow(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Pow(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], [["integrate", 7, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Pow(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Tuple(Symbol('W', commutative=True))), Integral(Add(Pow(Add(Function('p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Function('p')(Symbol('W', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\Psi_{nl})} = \\cos{(e^{\\Psi_{nl}})}, then obtain \\int (\\Psi_{nl} + \\mathbf{E}{(\\Psi_{nl})} + \\cos{(e^{\\Psi_{nl}})}) d\\Psi_{nl} = \\int (\\Psi_{nl} + 2 \\cos{(e^{\\Psi_{nl}})}) d\\Psi_{nl}", "derivation": "\\mathbf{E}{(\\Psi_{nl})} = \\cos{(e^{\\Psi_{nl}})} and \\mathbf{E}{(\\Psi_{nl})} + \\cos{(e^{\\Psi_{nl}})} = 2 \\cos{(e^{\\Psi_{nl}})} and \\Psi_{nl} + \\mathbf{E}{(\\Psi_{nl})} + \\cos{(e^{\\Psi_{nl}})} = \\Psi_{nl} + 2 \\cos{(e^{\\Psi_{nl}})} and \\int (\\Psi_{nl} + \\mathbf{E}{(\\Psi_{nl})} + \\cos{(e^{\\Psi_{nl}})}) d\\Psi_{nl} = \\int (\\Psi_{nl} + 2 \\cos{(e^{\\Psi_{nl}})}) d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 1, "cos(exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["add", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(2), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given E{(z)} = e^{z}, then derive \\frac{d}{d z} E{(z)} = e^{z}, then derive \\int E{(z)} dz = k + e^{z}, then obtain \\int e^{z} dz = k + \\frac{d}{d z} E{(z)}", "derivation": "E{(z)} = e^{z} and \\frac{d}{d z} E{(z)} = \\frac{d}{d z} e^{z} and \\int E{(z)} dz = \\int e^{z} dz and \\frac{d}{d z} E{(z)} = e^{z} and \\int E{(z)} dz = k + e^{z} and \\int e^{z} dz = k + e^{z} and \\int e^{z} dz = k + \\frac{d}{d z} E{(z)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), exp(Symbol('z', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integral(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('k', commutative=True), exp(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('k', commutative=True), exp(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('k', commutative=True), Derivative(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(F_{H})} = \\log{(F_{H})} and r{(F_{H})} = \\log{(F_{H})}, then obtain \\frac{\\hat{H}_{\\lambda}^{2}{(F_{H})}}{\\log{(F_{H})}} + r{(F_{H})} = 2 r{(F_{H})}", "derivation": "\\hat{H}_{\\lambda}{(F_{H})} = \\log{(F_{H})} and \\hat{H}_{\\lambda}{(F_{H})} + \\log{(F_{H})} = 2 \\log{(F_{H})} and \\hat{H}_{\\lambda}^{2}{(F_{H})} = \\hat{H}_{\\lambda}{(F_{H})} \\log{(F_{H})} and \\frac{\\hat{H}_{\\lambda}^{2}{(F_{H})}}{\\log{(F_{H})}} = \\hat{H}_{\\lambda}{(F_{H})} and r{(F_{H})} = \\log{(F_{H})} and \\hat{H}_{\\lambda}{(F_{H})} + r{(F_{H})} = 2 r{(F_{H})} and \\frac{\\hat{H}_{\\lambda}^{2}{(F_{H})}}{\\log{(F_{H})}} + r{(F_{H})} = 2 r{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["add", 1, "log(Symbol('F_H', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True))), Mul(Integer(2), log(Symbol('F_H', commutative=True))))"], [["times", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), Integer(2)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True))))"], [["divide", 3, "log(Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), Integer(2)), Pow(log(Symbol('F_H', commutative=True)), Integer(-1))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)))"], ["renaming_premise", "Equality(Function('r')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), Function('r')(Symbol('F_H', commutative=True))), Mul(Integer(2), Function('r')(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_H', commutative=True)), Integer(2)), Pow(log(Symbol('F_H', commutative=True)), Integer(-1))), Function('r')(Symbol('F_H', commutative=True))), Mul(Integer(2), Function('r')(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(Q)} = \\cos{(Q)} and p{(Q)} = 2 \\operatorname{F_{H}}{(Q)}, then obtain p{(Q)} + \\cos^{2}{(Q)} = \\cos^{2}{(Q)} + 2 \\cos{(Q)}", "derivation": "\\operatorname{F_{H}}{(Q)} = \\cos{(Q)} and p{(Q)} = 2 \\operatorname{F_{H}}{(Q)} and \\operatorname{F_{H}}^{2}{(Q)} + p{(Q)} = \\operatorname{F_{H}}^{2}{(Q)} + 2 \\operatorname{F_{H}}{(Q)} and p{(Q)} + \\cos^{2}{(Q)} = \\cos^{2}{(Q)} + 2 \\cos{(Q)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('Q', commutative=True)), Mul(Integer(2), Function('F_H')(Symbol('Q', commutative=True))))"], [["add", 2, "Pow(Function('F_H')(Symbol('Q', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('F_H')(Symbol('Q', commutative=True)), Integer(2)), Function('p')(Symbol('Q', commutative=True))), Add(Pow(Function('F_H')(Symbol('Q', commutative=True)), Integer(2)), Mul(Integer(2), Function('F_H')(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('p')(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Integer(2))), Add(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Mul(Integer(2), cos(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(y)} = \\cos{(y)} and \\operatorname{E_{x}}{(y)} = \\operatorname{y^{\\prime}}{(y)} + \\cos{(y)}, then obtain \\operatorname{E_{x}}{(y)} - \\operatorname{y^{\\prime}}{(y)} = \\operatorname{y^{\\prime}}{(y)}", "derivation": "\\operatorname{y^{\\prime}}{(y)} = \\cos{(y)} and \\operatorname{E_{x}}{(y)} = \\operatorname{y^{\\prime}}{(y)} + \\cos{(y)} and \\operatorname{E_{x}}{(y)} - \\cos{(y)} = \\operatorname{y^{\\prime}}{(y)} and \\operatorname{E_{x}}{(y)} - \\cos{(y)} = \\cos{(y)} and \\operatorname{E_{x}}{(y)} - \\operatorname{y^{\\prime}}{(y)} = \\operatorname{y^{\\prime}}{(y)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('y', commutative=True)), Add(Function('y^{\\\\prime}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))))"], [["minus", 2, "cos(Symbol('y', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Function('y^{\\\\prime}')(Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('E_x')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), cos(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('E_x')(Symbol('y', commutative=True)), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('y', commutative=True)))), Function('y^{\\\\prime}')(Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\lambda{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then obtain 0 = \\frac{- \\mathbf{B} \\ddot{x}{(\\mathbf{B})} + \\mathbf{B} \\log{(\\mathbf{B})}}{\\log{(\\mathbf{B})}}", "derivation": "\\ddot{x}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\mathbf{B} \\ddot{x}{(\\mathbf{B})} = \\mathbf{B} \\log{(\\mathbf{B})} and \\lambda{(\\mathbf{B})} = \\log{(\\mathbf{B})} and 0 = - \\mathbf{B} \\ddot{x}{(\\mathbf{B})} + \\mathbf{B} \\log{(\\mathbf{B})} and 0 = \\frac{- \\mathbf{B} \\ddot{x}{(\\mathbf{B})} + \\mathbf{B} \\log{(\\mathbf{B})}}{\\lambda{(\\mathbf{B})}} and 0 = \\frac{- \\mathbf{B} \\ddot{x}{(\\mathbf{B})} + \\mathbf{B} \\log{(\\mathbf{B})}}{\\log{(\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["divide", 4, "Function('\\\\lambda')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True)))), Pow(log(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(m)} = m, then derive \\int \\operatorname{J_{\\varepsilon}}{(m)} dm = \\mathbf{P} + \\frac{m^{2}}{2}, then obtain 0 = \\mathbf{P} m (\\mathbf{P} + \\frac{m^{2}}{2} - \\int \\operatorname{J_{\\varepsilon}}{(m)} dm)", "derivation": "\\operatorname{J_{\\varepsilon}}{(m)} = m and \\int \\operatorname{J_{\\varepsilon}}{(m)} dm = \\int m dm and \\int \\operatorname{J_{\\varepsilon}}{(m)} dm = \\mathbf{P} + \\frac{m^{2}}{2} and 0 = \\mathbf{P} + \\frac{m^{2}}{2} - \\int \\operatorname{J_{\\varepsilon}}{(m)} dm and 0 = m (\\mathbf{P} + \\frac{m^{2}}{2} - \\int \\operatorname{J_{\\varepsilon}}{(m)} dm) and 0 = \\mathbf{P} m (\\mathbf{P} + \\frac{m^{2}}{2} - \\int \\operatorname{J_{\\varepsilon}}{(m)} dm)", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2)))))"], [["minus", 3, "Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2))), Mul(Integer(-1), Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))))"], [["times", 4, "Symbol('m', commutative=True)"], "Equality(Integer(0), Mul(Symbol('m', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2))), Mul(Integer(-1), Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))))"], [["times", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('m', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2))), Mul(Integer(-1), Integral(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(A_{x})} = \\frac{d}{d A_{x}} \\sin{(A_{x})} and \\dot{x}{(A_{x})} = \\sin{(A_{x})}, then obtain (\\operatorname{V_{\\mathbf{B}}}^{A_{x}}{(A_{x})})^{A_{x}} = ((\\frac{d}{d A_{x}} \\dot{x}{(A_{x})})^{A_{x}})^{A_{x}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(A_{x})} = \\frac{d}{d A_{x}} \\sin{(A_{x})} and \\operatorname{V_{\\mathbf{B}}}^{A_{x}}{(A_{x})} = (\\frac{d}{d A_{x}} \\sin{(A_{x})})^{A_{x}} and \\dot{x}{(A_{x})} = \\sin{(A_{x})} and (\\operatorname{V_{\\mathbf{B}}}^{A_{x}}{(A_{x})})^{A_{x}} = ((\\frac{d}{d A_{x}} \\sin{(A_{x})})^{A_{x}})^{A_{x}} and (\\operatorname{V_{\\mathbf{B}}}^{A_{x}}{(A_{x})})^{A_{x}} = ((\\frac{d}{d A_{x}} \\dot{x}{(A_{x})})^{A_{x}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True)), Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('A_x', commutative=True)), sin(Symbol('A_x', commutative=True)))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Derivative(sin(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Derivative(Function('\\\\dot{x}')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given W{(h,\\mathbf{P})} = \\cos{(\\mathbf{P} + h)}, then obtain h + W{(h,\\mathbf{P})} - \\frac{\\partial}{\\partial h} (W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)}) = h + \\cos{(\\mathbf{P} + h)} - \\frac{\\partial}{\\partial h} (W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)})", "derivation": "W{(h,\\mathbf{P})} = \\cos{(\\mathbf{P} + h)} and h + W{(h,\\mathbf{P})} = h + \\cos{(\\mathbf{P} + h)} and W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)} = 0 and \\frac{\\partial}{\\partial h} (W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)}) = \\frac{d}{d h} 0 and h + W{(h,\\mathbf{P})} - \\frac{d}{d h} 0 = h + \\cos{(\\mathbf{P} + h)} - \\frac{d}{d h} 0 and h + W{(h,\\mathbf{P})} - \\frac{\\partial}{\\partial h} (W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)}) = h + \\cos{(\\mathbf{P} + h)} - \\frac{\\partial}{\\partial h} (W{(h,\\mathbf{P})} - \\cos{(\\mathbf{P} + h)})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('h', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))))"], [["minus", 1, "cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))"], "Equality(Add(Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Symbol('h', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Symbol('h', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('h', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('h', commutative=True), Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Derivative(Add(Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Symbol('h', commutative=True), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Mul(Integer(-1), Derivative(Add(Function('W')(Symbol('h', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True), Integer(1))))))"]]}, {"prompt": "Given I{(\\phi_2,C_{d},r)} = - C_{d} + \\phi_2 + r, then obtain \\frac{((I^{\\phi_2}{(\\phi_2,C_{d},r)})^{C_{d}})^{C_{d}} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\phi_2 + r)^{\\phi_2}}{C_{d}} = \\frac{(((- C_{d} + \\phi_2 + r)^{\\phi_2})^{C_{d}})^{C_{d}} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\phi_2 + r)^{\\phi_2}}{C_{d}}", "derivation": "I{(\\phi_2,C_{d},r)} = - C_{d} + \\phi_2 + r and I^{\\phi_2}{(\\phi_2,C_{d},r)} = (- C_{d} + \\phi_2 + r)^{\\phi_2} and (I^{\\phi_2}{(\\phi_2,C_{d},r)})^{C_{d}} = ((- C_{d} + \\phi_2 + r)^{\\phi_2})^{C_{d}} and ((I^{\\phi_2}{(\\phi_2,C_{d},r)})^{C_{d}})^{C_{d}} = (((- C_{d} + \\phi_2 + r)^{\\phi_2})^{C_{d}})^{C_{d}} and \\frac{((I^{\\phi_2}{(\\phi_2,C_{d},r)})^{C_{d}})^{C_{d}} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\phi_2 + r)^{\\phi_2}}{C_{d}} = \\frac{(((- C_{d} + \\phi_2 + r)^{\\phi_2})^{C_{d}})^{C_{d}} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\phi_2 + r)^{\\phi_2}}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Pow(Function('I')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)))"], [["power", 3, "Symbol('C_d', commutative=True)"], "Equality(Pow(Pow(Pow(Function('I')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Pow(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], [["times", 4, "Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Pow(Pow(Pow(Function('I')(Symbol('\\\\phi_2', commutative=True), Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Pow(Pow(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\phi_2', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(\\psi)} = \\log{(\\psi)}, then obtain \\frac{d}{d \\psi} \\psi (- \\psi + I{(\\psi)}) = \\frac{d}{d \\psi} \\psi (- \\psi + \\log{(\\psi)})", "derivation": "I{(\\psi)} = \\log{(\\psi)} and - \\psi + I{(\\psi)} = - \\psi + \\log{(\\psi)} and \\psi (- \\psi + I{(\\psi)}) = \\psi (- \\psi + \\log{(\\psi)}) and \\frac{d}{d \\psi} \\psi (- \\psi + I{(\\psi)}) = \\frac{d}{d \\psi} \\psi (- \\psi + \\log{(\\psi)})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('I')(Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('I')(Symbol('\\\\psi', commutative=True)))), Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('I')(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(\\varepsilon_0,\\pi)} = \\pi - \\varepsilon_0, then obtain \\frac{\\varepsilon_0 (\\pi - \\varepsilon_0) (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi}) x{(\\varepsilon_0,\\pi)}}{\\pi} = \\frac{\\varepsilon_0 (\\pi - \\varepsilon_0)^{2} (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi})}{\\pi}", "derivation": "x{(\\varepsilon_0,\\pi)} = \\pi - \\varepsilon_0 and \\varepsilon_0 x{(\\varepsilon_0,\\pi)} = \\varepsilon_0 (\\pi - \\varepsilon_0) and \\varepsilon_0 (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi}) x{(\\varepsilon_0,\\pi)} = \\varepsilon_0 (\\pi - \\varepsilon_0) (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi}) and \\frac{\\varepsilon_0 (\\pi - \\varepsilon_0) (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi}) x{(\\varepsilon_0,\\pi)}}{\\pi} = \\frac{\\varepsilon_0 (\\pi - \\varepsilon_0)^{2} (- \\varepsilon_0 + \\frac{\\pi - \\varepsilon_0}{\\pi})}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["times", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))), Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))))"], [["times", 3, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))), Function('x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} = F_{x} \\eta^{\\prime}, then obtain \\frac{\\frac{\\partial}{\\partial F_{x}} \\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} - 1}{\\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})}} = \\frac{\\frac{\\partial}{\\partial F_{x}} F_{x} \\eta^{\\prime} - 1}{\\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})}}", "derivation": "\\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} = F_{x} \\eta^{\\prime} and \\frac{\\partial}{\\partial F_{x}} \\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x} \\eta^{\\prime} and \\frac{\\partial}{\\partial F_{x}} \\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} - 1 = \\frac{\\partial}{\\partial F_{x}} F_{x} \\eta^{\\prime} - 1 and \\frac{\\frac{\\partial}{\\partial F_{x}} \\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})} - 1}{\\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})}} = \\frac{\\frac{\\partial}{\\partial F_{x}} F_{x} \\eta^{\\prime} - 1}{\\operatorname{E_{n}}{(\\eta^{\\prime},F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 3, "Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Add(Derivative(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Mul(Add(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(C,\\phi,\\varphi)} = (\\phi \\varphi)^{C}, then derive \\frac{\\partial}{\\partial C} \\Psi_{nl}{(C,\\phi,\\varphi)} = (\\phi \\varphi)^{C} \\log{(\\phi \\varphi)}, then obtain (\\phi \\varphi)^{C} \\log{(\\phi \\varphi)} = \\Psi_{nl}{(C,\\phi,\\varphi)} \\log{(\\phi \\varphi)}", "derivation": "\\Psi_{nl}{(C,\\phi,\\varphi)} = (\\phi \\varphi)^{C} and \\frac{\\partial}{\\partial C} \\Psi_{nl}{(C,\\phi,\\varphi)} = \\frac{\\partial}{\\partial C} (\\phi \\varphi)^{C} and \\frac{\\partial}{\\partial C} \\Psi_{nl}{(C,\\phi,\\varphi)} = (\\phi \\varphi)^{C} \\log{(\\phi \\varphi)} and \\frac{\\partial}{\\partial C} \\Psi_{nl}{(C,\\phi,\\varphi)} = \\Psi_{nl}{(C,\\phi,\\varphi)} \\log{(\\phi \\varphi)} and \\frac{\\partial}{\\partial C} (\\phi \\varphi)^{C} = \\Psi_{nl}{(C,\\phi,\\varphi)} \\log{(\\phi \\varphi)} and (\\phi \\varphi)^{C} \\log{(\\phi \\varphi)} = \\Psi_{nl}{(C,\\phi,\\varphi)} \\log{(\\phi \\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('C', commutative=True)), log(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('C', commutative=True)), log(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given y{(f^{\\prime})} = \\log{(f^{\\prime})} and \\operatorname{F_{g}}{(f^{\\prime})} = - f^{\\prime} + y{(f^{\\prime})}, then obtain \\int (- f^{\\prime} + y{(f^{\\prime})})^{f^{\\prime}} df^{\\prime} = \\int \\operatorname{F_{g}}^{f^{\\prime}}{(f^{\\prime})} df^{\\prime}", "derivation": "y{(f^{\\prime})} = \\log{(f^{\\prime})} and - f^{\\prime} + y{(f^{\\prime})} = - f^{\\prime} + \\log{(f^{\\prime})} and (- f^{\\prime} + y{(f^{\\prime})})^{f^{\\prime}} = (- f^{\\prime} + \\log{(f^{\\prime})})^{f^{\\prime}} and \\operatorname{F_{g}}{(f^{\\prime})} = - f^{\\prime} + y{(f^{\\prime})} and \\operatorname{F_{g}}{(f^{\\prime})} = - f^{\\prime} + \\log{(f^{\\prime})} and (- f^{\\prime} + y{(f^{\\prime})})^{f^{\\prime}} = \\operatorname{F_{g}}^{f^{\\prime}}{(f^{\\prime})} and \\int (- f^{\\prime} + y{(f^{\\prime})})^{f^{\\prime}} df^{\\prime} = \\int \\operatorname{F_{g}}^{f^{\\prime}}{(f^{\\prime})} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('y')(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('y')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('y')(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('y')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 6, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('y')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Pow(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(m)} = \\sin{(m)} and \\operatorname{r_{0}}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)}, then obtain \\cos{(\\frac{\\operatorname{r_{0}}{(\\hat{x}_0)}}{\\sin{(m)}} + \\sin{(m)})} = \\cos{(\\frac{\\log{(\\hat{x}_0)}}{\\sin{(m)}} + \\sin{(m)})}", "derivation": "\\varepsilon{(m)} = \\sin{(m)} and \\operatorname{r_{0}}{(\\hat{x}_0)} = \\log{(\\hat{x}_0)} and \\frac{\\operatorname{r_{0}}{(\\hat{x}_0)}}{\\sin{(m)}} = \\frac{\\log{(\\hat{x}_0)}}{\\sin{(m)}} and \\varepsilon{(m)} + \\frac{\\operatorname{r_{0}}{(\\hat{x}_0)}}{\\sin{(m)}} = \\varepsilon{(m)} + \\frac{\\log{(\\hat{x}_0)}}{\\sin{(m)}} and \\cos{(\\varepsilon{(m)} + \\frac{\\operatorname{r_{0}}{(\\hat{x}_0)}}{\\sin{(m)}})} = \\cos{(\\varepsilon{(m)} + \\frac{\\log{(\\hat{x}_0)}}{\\sin{(m)}})} and \\cos{(\\frac{\\operatorname{r_{0}}{(\\hat{x}_0)}}{\\sin{(m)}} + \\sin{(m)})} = \\cos{(\\frac{\\log{(\\hat{x}_0)}}{\\sin{(m)}} + \\sin{(m)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], ["get_premise", "Equality(Function('r_0')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 2, "sin(Symbol('m', commutative=True))"], "Equality(Mul(Function('r_0')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))), Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))))"], [["add", 3, "Function('\\\\varepsilon')(Symbol('m', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('m', commutative=True)), Mul(Function('r_0')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1)))), Add(Function('\\\\varepsilon')(Symbol('m', commutative=True)), Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1)))))"], [["cos", 4], "Equality(cos(Add(Function('\\\\varepsilon')(Symbol('m', commutative=True)), Mul(Function('r_0')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))))), cos(Add(Function('\\\\varepsilon')(Symbol('m', commutative=True)), Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(cos(Add(Mul(Function('r_0')(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))), sin(Symbol('m', commutative=True)))), cos(Add(Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(-1))), sin(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given G{(C_{2},A_{z})} = - A_{z} + C_{2}, then obtain \\frac{- C_{2} + G{(C_{2},A_{z})}}{C_{2}} = - \\frac{C_{2} - G{(C_{2},A_{z})}}{C_{2}}", "derivation": "G{(C_{2},A_{z})} = - A_{z} + C_{2} and - C_{2} + G{(C_{2},A_{z})} = - A_{z} and \\frac{- C_{2} + G{(C_{2},A_{z})}}{C_{2}} = - \\frac{A_{z}}{C_{2}} and \\frac{- C_{2} + G{(C_{2},A_{z})}}{C_{2}} = - \\frac{C_{2} - G{(C_{2},A_{z})}}{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('C_2', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('C_2', commutative=True)))"], [["minus", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('G')(Symbol('C_2', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True)))"], [["divide", 2, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('G')(Symbol('C_2', commutative=True), Symbol('A_z', commutative=True)))), Mul(Integer(-1), Symbol('A_z', commutative=True), Pow(Symbol('C_2', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('G')(Symbol('C_2', commutative=True), Symbol('A_z', commutative=True)))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Function('G')(Symbol('C_2', commutative=True), Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given \\Psi{(g,u)} = g + u and \\operatorname{v_{z}}{(g,u)} = \\int \\Psi{(g,u)} du, then obtain \\frac{\\Psi{(g,u)}}{\\operatorname{v_{z}}{(g,u)}} = \\frac{g + u}{\\operatorname{v_{z}}{(g,u)}}", "derivation": "\\Psi{(g,u)} = g + u and \\int \\Psi{(g,u)} du = \\int (g + u) du and \\frac{\\Psi{(g,u)}}{\\int (g + u) du} = \\frac{g + u}{\\int (g + u) du} and \\frac{\\Psi{(g,u)}}{\\int \\Psi{(g,u)} du} = \\frac{g + u}{\\int \\Psi{(g,u)} du} and \\operatorname{v_{z}}{(g,u)} = \\int \\Psi{(g,u)} du and \\frac{\\Psi{(g,u)}}{\\operatorname{v_{z}}{(g,u)}} = \\frac{g + u}{\\operatorname{v_{z}}{(g,u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Add(Symbol('g', commutative=True), Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["divide", 1, "Integral(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Integral(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))), Mul(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Integral(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Integral(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))), Mul(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Integral(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Integral(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Function('\\\\Psi')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Function('v_z')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Mul(Add(Symbol('g', commutative=True), Symbol('u', commutative=True)), Pow(Function('v_z')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}}, then obtain \\eta^{\\prime} (2 \\operatorname{P_{g}}{(\\eta^{\\prime})} + e^{\\eta^{\\prime}}) = 3 \\eta^{\\prime} e^{\\eta^{\\prime}}", "derivation": "\\operatorname{P_{g}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}} and \\operatorname{P_{g}}{(\\eta^{\\prime})} + e^{\\eta^{\\prime}} = 2 e^{\\eta^{\\prime}} and \\operatorname{P_{g}}{(\\eta^{\\prime})} + 2 e^{\\eta^{\\prime}} = 3 e^{\\eta^{\\prime}} and \\eta^{\\prime} (\\operatorname{P_{g}}{(\\eta^{\\prime})} + 2 e^{\\eta^{\\prime}}) = 3 \\eta^{\\prime} e^{\\eta^{\\prime}} and 2 \\operatorname{P_{g}}{(\\eta^{\\prime})} + e^{\\eta^{\\prime}} = 3 e^{\\eta^{\\prime}} and 2 \\operatorname{P_{g}}{(\\eta^{\\prime})} + e^{\\eta^{\\prime}} = \\operatorname{P_{g}}{(\\eta^{\\prime})} + 2 e^{\\eta^{\\prime}} and \\eta^{\\prime} (2 \\operatorname{P_{g}}{(\\eta^{\\prime})} + e^{\\eta^{\\prime}}) = 3 \\eta^{\\prime} e^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 1, "Mul(Integer(2), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(3), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))), Mul(Integer(3), Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(3), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(2), Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Integer(2), Function('P_g')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(3), Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given m{(C_{d},G)} = \\frac{G}{C_{d}}, then obtain 2 G + 2 m{(C_{d},G)} = 2 G + \\frac{2 G}{C_{d}}", "derivation": "m{(C_{d},G)} = \\frac{G}{C_{d}} and m{(C_{d},G)} + \\frac{G}{C_{d}} = \\frac{2 G}{C_{d}} and G + m{(C_{d},G)} = G + \\frac{G}{C_{d}} and 2 G + 2 m{(C_{d},G)} = 2 G + m{(C_{d},G)} + \\frac{G}{C_{d}} and 2 G + 2 m{(C_{d},G)} = 2 G + \\frac{2 G}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True))"], "Equality(Add(Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True))), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True)))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True))))"], [["add", 3, "Add(Symbol('G', commutative=True), Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Function('m')(Symbol('C_d', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{p},U)} = U \\hat{p}, then derive \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{z^{*}}{(\\hat{p},U)} = U, then obtain \\int \\operatorname{z^{*}}{(\\hat{p},\\frac{\\partial}{\\partial \\hat{p}} U \\hat{p})} d\\hat{p} = \\int \\hat{p} \\frac{\\partial}{\\partial \\hat{p}} U \\hat{p} d\\hat{p}", "derivation": "\\operatorname{z^{*}}{(\\hat{p},U)} = U \\hat{p} and \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{z^{*}}{(\\hat{p},U)} = \\frac{\\partial}{\\partial \\hat{p}} U \\hat{p} and \\int \\operatorname{z^{*}}{(\\hat{p},U)} d\\hat{p} = \\int U \\hat{p} d\\hat{p} and \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{z^{*}}{(\\hat{p},U)} = U and \\frac{\\partial}{\\partial \\hat{p}} U \\hat{p} = U and \\int \\operatorname{z^{*}}{(\\hat{p},\\frac{\\partial}{\\partial \\hat{p}} U \\hat{p})} d\\hat{p} = \\int \\hat{p} \\frac{\\partial}{\\partial \\hat{p}} U \\hat{p} d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('U', commutative=True))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('U', commutative=True))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given p{(f^{\\prime})} = \\sin{(f^{\\prime})}, then derive \\int p{(f^{\\prime})} df^{\\prime} = g_{\\varepsilon} - \\cos{(f^{\\prime})}, then obtain \\int \\sin{(f^{\\prime})} df^{\\prime} + 1 = \\int p{(f^{\\prime})} df^{\\prime} + 1", "derivation": "p{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\int p{(f^{\\prime})} df^{\\prime} = \\int \\sin{(f^{\\prime})} df^{\\prime} and \\int p{(f^{\\prime})} df^{\\prime} = g_{\\varepsilon} - \\cos{(f^{\\prime})} and \\int \\sin{(f^{\\prime})} df^{\\prime} = g_{\\varepsilon} - \\cos{(f^{\\prime})} and \\int \\sin{(f^{\\prime})} df^{\\prime} + 1 = g_{\\varepsilon} - \\cos{(f^{\\prime})} + 1 and \\int \\sin{(f^{\\prime})} df^{\\prime} + 1 = \\int p{(f^{\\prime})} df^{\\prime} + 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)), Add(Integral(Function('p')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given E{(f_{E})} = \\log{(f_{E})}, then obtain \\frac{d}{d f_{E}} (E{(f_{E})} + \\cos{(\\sin{(E{(f_{E})})})}) = \\frac{d}{d f_{E}} (E{(f_{E})} + \\cos{(\\sin{(\\log{(f_{E})})})})", "derivation": "E{(f_{E})} = \\log{(f_{E})} and \\sin{(E{(f_{E})})} = \\sin{(\\log{(f_{E})})} and \\cos{(\\sin{(E{(f_{E})})})} = \\cos{(\\sin{(\\log{(f_{E})})})} and E{(f_{E})} + \\cos{(\\sin{(E{(f_{E})})})} = E{(f_{E})} + \\cos{(\\sin{(\\log{(f_{E})})})} and \\frac{d}{d f_{E}} (E{(f_{E})} + \\cos{(\\sin{(E{(f_{E})})})}) = \\frac{d}{d f_{E}} (E{(f_{E})} + \\cos{(\\sin{(\\log{(f_{E})})})})", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["sin", 1], "Equality(sin(Function('E')(Symbol('f_E', commutative=True))), sin(log(Symbol('f_E', commutative=True))))"], [["cos", 2], "Equality(cos(sin(Function('E')(Symbol('f_E', commutative=True)))), cos(sin(log(Symbol('f_E', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Function('E')(Symbol('f_E', commutative=True)))"], "Equality(Add(Function('E')(Symbol('f_E', commutative=True)), cos(sin(Function('E')(Symbol('f_E', commutative=True))))), Add(Function('E')(Symbol('f_E', commutative=True)), cos(sin(log(Symbol('f_E', commutative=True))))))"], [["differentiate", 4, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Function('E')(Symbol('f_E', commutative=True)), cos(sin(Function('E')(Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Function('E')(Symbol('f_E', commutative=True)), cos(sin(log(Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(m,i)} = m^{i}, then derive \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{i m^{i}}{m}, then obtain \\frac{i \\operatorname{V_{\\mathbf{E}}}{(m,i)}}{m} + \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{i \\operatorname{V_{\\mathbf{E}}}{(m,i)}}{m} + m^{i}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(m,i)} = m^{i} and \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{\\partial}{\\partial m} m^{i} and \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{i m^{i}}{m} and \\operatorname{V_{\\mathbf{E}}}{(m,i)} + \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} = m^{i} + \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} and \\frac{\\partial}{\\partial m} \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{i \\operatorname{V_{\\mathbf{E}}}{(m,i)}}{m} and \\frac{i \\operatorname{V_{\\mathbf{E}}}{(m,i)}}{m} + \\operatorname{V_{\\mathbf{E}}}{(m,i)} = \\frac{i \\operatorname{V_{\\mathbf{E}}}{(m,i)}}{m} + m^{i}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Symbol('i', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Symbol('i', commutative=True))))"], [["add", 1, "Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Pow(Symbol('m', commutative=True), Symbol('i', commutative=True)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Symbol('i', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Symbol('i', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True))), Add(Mul(Symbol('i', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('m', commutative=True), Symbol('i', commutative=True))), Pow(Symbol('m', commutative=True), Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\dot{z})} = \\cos{(\\dot{z})}, then derive \\int \\mathbf{f}{(\\dot{z})} d\\dot{z} = \\dot{y} + \\sin{(\\dot{z})}, then obtain (\\int \\cos{(\\dot{z})} d\\dot{z})^{\\dot{z}} = (\\dot{y} + \\sin{(\\dot{z})})^{\\dot{z}}", "derivation": "\\mathbf{f}{(\\dot{z})} = \\cos{(\\dot{z})} and \\int \\mathbf{f}{(\\dot{z})} d\\dot{z} = \\int \\cos{(\\dot{z})} d\\dot{z} and \\int \\mathbf{f}{(\\dot{z})} d\\dot{z} = \\dot{y} + \\sin{(\\dot{z})} and \\int \\cos{(\\dot{z})} d\\dot{z} = \\dot{y} + \\sin{(\\dot{z})} and (\\int \\cos{(\\dot{z})} d\\dot{z})^{\\dot{z}} = (\\dot{y} + \\sin{(\\dot{z})})^{\\dot{z}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbb{I},p)} = \\mathbb{I} + p, then obtain \\cos{(\\mathbb{I} (\\mathbb{I} + p) - \\operatorname{F_{g}}{(\\mathbb{I},p)})} = \\cos{(- \\mathbb{I} (\\mathbb{I} + p) + \\mathbb{I} + p)}", "derivation": "\\operatorname{F_{g}}{(\\mathbb{I},p)} = \\mathbb{I} + p and \\mathbb{I} \\operatorname{F_{g}}{(\\mathbb{I},p)} = \\mathbb{I} (\\mathbb{I} + p) and - \\mathbb{I} \\operatorname{F_{g}}{(\\mathbb{I},p)} + \\operatorname{F_{g}}{(\\mathbb{I},p)} = - \\mathbb{I} \\operatorname{F_{g}}{(\\mathbb{I},p)} + \\mathbb{I} + p and - \\mathbb{I} (\\mathbb{I} + p) + \\operatorname{F_{g}}{(\\mathbb{I},p)} = - \\mathbb{I} (\\mathbb{I} + p) + \\mathbb{I} + p and \\cos{(\\mathbb{I} (\\mathbb{I} + p) - \\operatorname{F_{g}}{(\\mathbb{I},p)})} = \\cos{(- \\mathbb{I} (\\mathbb{I} + p) + \\mathbb{I} + p)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)))"], [["cos", 4], "Equality(cos(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))))), cos(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given u{(\\varepsilon_0,A_{1})} = \\cos{(\\frac{A_{1}}{\\varepsilon_0})}, then obtain 0 = - \\frac{\\partial}{\\partial A_{1}} u{(\\varepsilon_0,A_{1})} + \\frac{\\partial}{\\partial A_{1}} \\cos{(\\frac{A_{1}}{\\varepsilon_0})}", "derivation": "u{(\\varepsilon_0,A_{1})} = \\cos{(\\frac{A_{1}}{\\varepsilon_0})} and \\frac{\\partial}{\\partial A_{1}} u{(\\varepsilon_0,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\cos{(\\frac{A_{1}}{\\varepsilon_0})} and u{(\\varepsilon_0,A_{1})} + \\frac{\\partial}{\\partial A_{1}} u{(\\varepsilon_0,A_{1})} = u{(\\varepsilon_0,A_{1})} + \\frac{\\partial}{\\partial A_{1}} \\cos{(\\frac{A_{1}}{\\varepsilon_0})} and 0 = - \\frac{\\partial}{\\partial A_{1}} u{(\\varepsilon_0,A_{1})} + \\frac{\\partial}{\\partial A_{1}} \\cos{(\\frac{A_{1}}{\\varepsilon_0})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), cos(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["add", 2, "Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Derivative(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Derivative(cos(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["minus", 3, "Add(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Derivative(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('u')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\nabla,C)} = C - \\nabla and \\mathbb{I}{(\\nabla,C)} = \\frac{\\partial}{\\partial \\nabla} \\frac{C - \\nabla}{\\nabla}, then obtain \\mathbb{I}{(\\nabla,C)} = \\frac{\\partial}{\\partial \\nabla} \\frac{\\varepsilon_{0}{(\\nabla,C)}}{\\nabla}", "derivation": "\\varepsilon_{0}{(\\nabla,C)} = C - \\nabla and \\frac{\\varepsilon_{0}{(\\nabla,C)}}{\\nabla} = \\frac{C - \\nabla}{\\nabla} and \\frac{\\partial}{\\partial \\nabla} \\frac{\\varepsilon_{0}{(\\nabla,C)}}{\\nabla} = \\frac{\\partial}{\\partial \\nabla} \\frac{C - \\nabla}{\\nabla} and \\mathbb{I}{(\\nabla,C)} = \\frac{\\partial}{\\partial \\nabla} \\frac{C - \\nabla}{\\nabla} and \\mathbb{I}{(\\nabla,C)} = \\frac{\\partial}{\\partial \\nabla} \\frac{\\varepsilon_{0}{(\\nabla,C)}}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(l)} = e^{l} and \\lambda{(l)} = e^{l}, then derive \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} = \\tilde{\\infty} e^{l}, then obtain \\int \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} \\lambda{(l)} dl = \\int \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} e^{l} dl", "derivation": "\\operatorname{L_{\\varepsilon}}{(l)} = e^{l} and \\lambda{(l)} = e^{l} and \\operatorname{L_{\\varepsilon}}{(l)} = \\lambda{(l)} and \\frac{\\operatorname{L_{\\varepsilon}}{(l)}}{\\frac{d}{d l} 0} = \\frac{e^{l}}{\\frac{d}{d l} 0} and \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} = \\tilde{\\infty} e^{l} and \\tilde{\\infty} \\lambda{(l)} = \\tilde{\\infty} e^{l} and \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} \\lambda{(l)} = \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} e^{l} and \\int \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} \\lambda{(l)} dl = \\int \\tilde{\\infty} \\operatorname{L_{\\varepsilon}}{(l)} e^{l} dl", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), Function('\\\\lambda')(Symbol('l', commutative=True)))"], [["divide", 1, "Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1))), Mul(exp(Symbol('l', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(zoo, Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True))), Mul(zoo, exp(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(zoo, Function('\\\\lambda')(Symbol('l', commutative=True))), Mul(zoo, exp(Symbol('l', commutative=True))))"], [["times", 6, "Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True))"], "Equality(Mul(zoo, Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), Function('\\\\lambda')(Symbol('l', commutative=True))), Mul(zoo, Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True))))"], [["integrate", 7, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(zoo, Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), Function('\\\\lambda')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Mul(zoo, Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given p{(\\pi,n)} = e^{\\pi n}, then derive \\frac{\\partial}{\\partial n} p{(\\pi,n)} = \\pi e^{\\pi n}, then obtain \\frac{\\partial}{\\partial n} p{(\\pi,n)} = \\pi p{(\\pi,n)}", "derivation": "p{(\\pi,n)} = e^{\\pi n} and \\frac{\\partial}{\\partial n} p{(\\pi,n)} = \\frac{\\partial}{\\partial n} e^{\\pi n} and \\frac{\\partial}{\\partial n} p{(\\pi,n)} = \\pi e^{\\pi n} and \\frac{\\partial}{\\partial n} p{(\\pi,n)} = \\pi p{(\\pi,n)}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Symbol('\\\\pi', commutative=True), exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('p')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Symbol('\\\\pi', commutative=True), Function('p')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given Q{(g,E,\\psi)} = g + \\frac{\\psi}{E}, then obtain e^{\\frac{\\partial}{\\partial g} Q{(g,E,\\psi)}} + \\frac{\\partial}{\\partial g} Q{(g,E,\\psi)} = e^{\\frac{\\partial}{\\partial g} Q{(g,E,\\psi)}} + \\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E})", "derivation": "Q{(g,E,\\psi)} = g + \\frac{\\psi}{E} and \\frac{\\partial}{\\partial g} Q{(g,E,\\psi)} = \\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E}) and e^{\\frac{\\partial}{\\partial g} Q{(g,E,\\psi)}} = e^{\\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E})} and e^{\\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E})} + \\frac{\\partial}{\\partial g} Q{(g,E,\\psi)} = e^{\\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E})} + \\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E}) and e^{\\frac{\\partial}{\\partial g} Q{(g,E,\\psi)}} + \\frac{\\partial}{\\partial g} Q{(g,E,\\psi)} = e^{\\frac{\\partial}{\\partial g} Q{(g,E,\\psi)}} + \\frac{\\partial}{\\partial g} (g + \\frac{\\psi}{E})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), exp(Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["add", 2, "exp(Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], "Equality(Add(exp(Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(exp(Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(exp(Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(exp(Derivative(Function('Q')(Symbol('g', commutative=True), Symbol('E', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Add(Symbol('g', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\dot{x},\\rho)} = - \\dot{x} + \\rho, then obtain \\frac{\\int \\operatorname{n_{2}}{(\\dot{x},\\rho)} d\\dot{x}}{\\operatorname{n_{2}}{(\\dot{x},\\rho)}} = \\frac{\\int (- \\dot{x} + \\rho) d\\dot{x}}{\\operatorname{n_{2}}{(\\dot{x},\\rho)}}", "derivation": "\\operatorname{n_{2}}{(\\dot{x},\\rho)} = - \\dot{x} + \\rho and \\int \\operatorname{n_{2}}{(\\dot{x},\\rho)} d\\dot{x} = \\int (- \\dot{x} + \\rho) d\\dot{x} and \\frac{\\int \\operatorname{n_{2}}{(\\dot{x},\\rho)} d\\dot{x}}{- \\dot{x} + \\rho} = \\frac{\\int (- \\dot{x} + \\rho) d\\dot{x}}{- \\dot{x} + \\rho} and \\frac{\\int \\operatorname{n_{2}}{(\\dot{x},\\rho)} d\\dot{x}}{\\operatorname{n_{2}}{(\\dot{x},\\rho)}} = \\frac{\\int (- \\dot{x} + \\rho) d\\dot{x}}{\\operatorname{n_{2}}{(\\dot{x},\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integral(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integral(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Pow(Function('n_2')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mu_0)} = \\log{(\\mu_0)}, then derive \\int \\hat{\\mathbf{r}}{(\\mu_0)} d\\mu_0 = \\mu_0 \\log{(\\mu_0)} - \\mu_0 + y^{\\prime}, then obtain \\log{(\\mu_0 \\hat{\\mathbf{r}}{(\\mu_0)} - \\mu_0 + y^{\\prime})} = \\log{(\\mu_0 \\log{(\\mu_0)} - \\mu_0 + \\phi_1)}", "derivation": "\\hat{\\mathbf{r}}{(\\mu_0)} = \\log{(\\mu_0)} and \\int \\hat{\\mathbf{r}}{(\\mu_0)} d\\mu_0 = \\int \\log{(\\mu_0)} d\\mu_0 and \\int \\hat{\\mathbf{r}}{(\\mu_0)} d\\mu_0 = \\mu_0 \\log{(\\mu_0)} - \\mu_0 + y^{\\prime} and \\int \\hat{\\mathbf{r}}{(\\mu_0)} d\\mu_0 = \\mu_0 \\hat{\\mathbf{r}}{(\\mu_0)} - \\mu_0 + y^{\\prime} and \\mu_0 \\hat{\\mathbf{r}}{(\\mu_0)} - \\mu_0 + y^{\\prime} = \\int \\log{(\\mu_0)} d\\mu_0 and \\log{(\\mu_0 \\hat{\\mathbf{r}}{(\\mu_0)} - \\mu_0 + y^{\\prime})} = \\log{(\\int \\log{(\\mu_0)} d\\mu_0)} and \\log{(\\mu_0 \\hat{\\mathbf{r}}{(\\mu_0)} - \\mu_0 + y^{\\prime})} = \\log{(\\mu_0 \\log{(\\mu_0)} - \\mu_0 + \\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["log", 5], "Equality(log(Add(Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), log(Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(log(Add(Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), log(Add(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)} = \\frac{A_{2} \\sigma_p}{q}, then obtain q (q + \\frac{\\partial}{\\partial q} \\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)}) = q (q + \\frac{\\partial}{\\partial q} \\frac{A_{2} \\sigma_p}{q})", "derivation": "\\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)} = \\frac{A_{2} \\sigma_p}{q} and \\frac{\\partial}{\\partial q} \\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)} = \\frac{\\partial}{\\partial q} \\frac{A_{2} \\sigma_p}{q} and q + \\frac{\\partial}{\\partial q} \\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)} = q + \\frac{\\partial}{\\partial q} \\frac{A_{2} \\sigma_p}{q} and q (q + \\frac{\\partial}{\\partial q} \\operatorname{J_{\\varepsilon}}{(A_{2},\\sigma_p,q)}) = q (q + \\frac{\\partial}{\\partial q} \\frac{A_{2} \\sigma_p}{q})", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 2, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Symbol('q', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["divide", 3, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('q', commutative=True), Add(Symbol('q', commutative=True), Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Mul(Symbol('q', commutative=True), Add(Symbol('q', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\lambda)} = \\lambda and z{(\\mathbf{H})} = \\mathbf{H}, then obtain \\frac{(\\frac{\\lambda}{\\mathbf{H}} + \\mathbf{H}) \\operatorname{F_{N}}{(\\lambda)}}{z{(\\mathbf{H})}} = \\frac{\\lambda (\\frac{\\lambda}{\\mathbf{H}} + \\mathbf{H})}{z{(\\mathbf{H})}}", "derivation": "\\operatorname{F_{N}}{(\\lambda)} = \\lambda and z{(\\mathbf{H})} = \\mathbf{H} and \\frac{\\operatorname{F_{N}}{(\\lambda)}}{\\mathbf{H}} = \\frac{\\lambda}{\\mathbf{H}} and \\frac{\\operatorname{F_{N}}{(\\lambda)}}{z{(\\mathbf{H})}} = \\frac{\\lambda}{z{(\\mathbf{H})}} and \\frac{(\\frac{\\lambda}{\\mathbf{H}} + \\mathbf{H}) \\operatorname{F_{N}}{(\\lambda)}}{z{(\\mathbf{H})}} = \\frac{\\lambda (\\frac{\\lambda}{\\mathbf{H}} + \\mathbf{H})}{z{(\\mathbf{H})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["divide", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('F_N')(Symbol('\\\\lambda', commutative=True)), Pow(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"], [["times", 4, "Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)), Function('F_N')(Symbol('\\\\lambda', commutative=True)), Pow(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(A_{2},\\mu_0)} = \\mu_0^{A_{2}} and \\operatorname{M_{E}}{(A_{2},\\mu_0)} = - \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} \\log{(\\mu_0)}, then derive \\int \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} d\\mu_0 = F_{N} + \\mu_0, then obtain - F_{N} - \\mu_0 + \\operatorname{M_{E}}{(A_{2},\\mu_0)} = - F_{N} - \\mu_0 - \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} \\log{(\\mu_0)}", "derivation": "\\mathbf{p}{(A_{2},\\mu_0)} = \\mu_0^{A_{2}} and \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} = 1 and \\int \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} d\\mu_0 = \\int 1 d\\mu_0 and \\int \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} d\\mu_0 = F_{N} + \\mu_0 and \\int 1 d\\mu_0 = F_{N} + \\mu_0 and \\operatorname{M_{E}}{(A_{2},\\mu_0)} = - \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} \\log{(\\mu_0)} and \\operatorname{M_{E}}{(A_{2},\\mu_0)} - \\int 1 d\\mu_0 = - \\int 1 d\\mu_0 - \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} \\log{(\\mu_0)} and - F_{N} - \\mu_0 + \\operatorname{M_{E}}{(A_{2},\\mu_0)} = - F_{N} - \\mu_0 - \\mu_0^{- A_{2}} \\mathbf{p}{(A_{2},\\mu_0)} \\log{(\\mu_0)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_2', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 6, "Integral(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Function('M_E')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True))))), Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('M_E')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('A_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(p)} = \\cos{(p)}, then derive \\frac{d}{d p} \\mathbf{M}{(p)} = - \\sin{(p)}, then obtain \\log{((- p + \\frac{d}{d p} \\mathbf{M}{(p)})^{p})} = \\log{((- p - \\sin{(p)})^{p})}", "derivation": "\\mathbf{M}{(p)} = \\cos{(p)} and \\frac{d}{d p} \\mathbf{M}{(p)} = \\frac{d}{d p} \\cos{(p)} and \\frac{d}{d p} \\mathbf{M}{(p)} = - \\sin{(p)} and \\frac{d}{d p} \\cos{(p)} = - \\sin{(p)} and - p + \\frac{d}{d p} \\cos{(p)} = - p - \\sin{(p)} and - p + \\frac{d}{d p} \\mathbf{M}{(p)} = - p - \\sin{(p)} and (- p + \\frac{d}{d p} \\mathbf{M}{(p)})^{p} = (- p - \\sin{(p)})^{p} and \\log{((- p + \\frac{d}{d p} \\mathbf{M}{(p)})^{p})} = \\log{((- p - \\sin{(p)})^{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('p', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))))"], [["power", 6, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True)))"], [["log", 7], "Equality(log(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Symbol('p', commutative=True))), log(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(E,c_{0})} = E - c_{0}, then derive V_{\\mathbf{E}} + \\phi_{2}{(E,c_{0})} = E + y, then obtain E + V_{\\mathbf{E}} - c_{0} = E + y", "derivation": "\\phi_{2}{(E,c_{0})} = E - c_{0} and \\frac{\\partial}{\\partial E} \\phi_{2}{(E,c_{0})} = \\frac{\\partial}{\\partial E} (E - c_{0}) and \\int \\frac{\\partial}{\\partial E} \\phi_{2}{(E,c_{0})} dE = \\int \\frac{\\partial}{\\partial E} (E - c_{0}) dE and V_{\\mathbf{E}} + \\phi_{2}{(E,c_{0})} = E + y and E + V_{\\mathbf{E}} - c_{0} = E + y", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Derivative(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('E', commutative=True), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('E', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Add(Symbol('E', commutative=True), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(r_{0},\\dot{x})} = r_{0}^{\\dot{x}}, then obtain (r_{0}^{\\dot{x}} + \\operatorname{C_{1}}{(r_{0},\\dot{x})}) \\operatorname{C_{1}}{(r_{0},\\dot{x})} = r_{0}^{\\dot{x}} (r_{0}^{\\dot{x}} + \\operatorname{C_{1}}{(r_{0},\\dot{x})})", "derivation": "\\operatorname{C_{1}}{(r_{0},\\dot{x})} = r_{0}^{\\dot{x}} and r_{0}^{\\dot{x}} + \\operatorname{C_{1}}{(r_{0},\\dot{x})} = 2 r_{0}^{\\dot{x}} and 2 r_{0}^{\\dot{x}} \\operatorname{C_{1}}{(r_{0},\\dot{x})} = 2 r_{0}^{2 \\dot{x}} and (r_{0}^{\\dot{x}} + \\operatorname{C_{1}}{(r_{0},\\dot{x})}) \\operatorname{C_{1}}{(r_{0},\\dot{x})} = r_{0}^{\\dot{x}} (r_{0}^{\\dot{x}} + \\operatorname{C_{1}}{(r_{0},\\dot{x})})", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["times", 1, "Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('C_1')(Symbol('r_0', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} = (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A}, then obtain \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} - \\int (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} d\\mathbf{S} = (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} - \\int (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} d\\mathbf{S}", "derivation": "\\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} = (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} and \\int \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} d\\mathbf{S} = \\int (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} d\\mathbf{S} and \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} - \\int \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} d\\mathbf{S} = (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} - \\int \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} d\\mathbf{S} and \\operatorname{C_{2}}{(\\hat{\\mathbf{r}},A,\\mathbf{S})} - \\int (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} d\\mathbf{S} = (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} - \\int (\\hat{\\mathbf{r}}^{\\mathbf{S}})^{A} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 1, "Integral(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integral(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))), Add(Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('C_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))), Add(Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{J},\\mathbf{g})} = - \\sin{(\\mathbf{J} - \\mathbf{g})}, then obtain \\int \\mathbf{J} \\mathbf{S}^{\\mathbf{g}}{(\\mathbf{J},\\mathbf{g})} d\\mathbf{g} = \\int \\mathbf{J} (- \\sin{(\\mathbf{J} - \\mathbf{g})})^{\\mathbf{g}} d\\mathbf{g}", "derivation": "\\mathbf{S}{(\\mathbf{J},\\mathbf{g})} = - \\sin{(\\mathbf{J} - \\mathbf{g})} and \\mathbf{S}^{\\mathbf{g}}{(\\mathbf{J},\\mathbf{g})} = (- \\sin{(\\mathbf{J} - \\mathbf{g})})^{\\mathbf{g}} and \\mathbf{J} \\mathbf{S}^{\\mathbf{g}}{(\\mathbf{J},\\mathbf{g})} = \\mathbf{J} (- \\sin{(\\mathbf{J} - \\mathbf{g})})^{\\mathbf{g}} and \\int \\mathbf{J} \\mathbf{S}^{\\mathbf{g}}{(\\mathbf{J},\\mathbf{g})} d\\mathbf{g} = \\int \\mathbf{J} (- \\sin{(\\mathbf{J} - \\mathbf{g})})^{\\mathbf{g}} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))), Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given y{(I,W)} = - W + e^{I}, then obtain 1 + \\frac{y^{2}{(I,W)}}{I} = 1 + \\frac{(- W + e^{I})^{2}}{I}", "derivation": "y{(I,W)} = - W + e^{I} and \\frac{y{(I,W)}}{I} = \\frac{- W + e^{I}}{I} and \\frac{y^{2}{(I,W)}}{I} = \\frac{(- W + e^{I}) y{(I,W)}}{I} and \\frac{(- W + e^{I}) y{(I,W)}}{I} = \\frac{(- W + e^{I})^{2}}{I} and \\frac{y^{2}{(I,W)}}{I} = \\frac{(- W + e^{I})^{2}}{I} and 1 + \\frac{y^{2}{(I,W)}}{I} = 1 + \\frac{(- W + e^{I})^{2}}{I}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True)))))"], [["times", 2, "Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True)), Integer(2))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))), Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))), Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True)), Integer(2))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2))))"], [["add", 5, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('y')(Symbol('I', commutative=True), Symbol('W', commutative=True)), Integer(2)))), Add(Integer(1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given J{(\\psi^*)} = e^{\\psi^*} and l{(\\psi^*)} = \\psi^*, then obtain \\operatorname{E_{x}}{(\\psi^*)} + l{(\\psi^*)} = \\operatorname{E_{x}}{(\\psi^*)} + l{(\\psi^*)} - \\int J{(\\psi^*)} dl{(\\psi^*)} + \\int e^{\\psi^*} dl{(\\psi^*)}", "derivation": "J{(\\psi^*)} = e^{\\psi^*} and \\int J{(\\psi^*)} d\\psi^* = \\int e^{\\psi^*} d\\psi^* and \\psi^* + \\int J{(\\psi^*)} d\\psi^* = \\psi^* + \\int e^{\\psi^*} d\\psi^* and \\psi^* = \\psi^* - \\int J{(\\psi^*)} d\\psi^* + \\int e^{\\psi^*} d\\psi^* and l{(\\psi^*)} = \\psi^* and l{(\\psi^*)} = l{(\\psi^*)} - \\int J{(\\psi^*)} dl{(\\psi^*)} + \\int e^{\\psi^*} dl{(\\psi^*)} and \\operatorname{E_{x}}{(\\psi^*)} + l{(\\psi^*)} = \\operatorname{E_{x}}{(\\psi^*)} + l{(\\psi^*)} - \\int J{(\\psi^*)} dl{(\\psi^*)} + \\int e^{\\psi^*} dl{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["minus", 3, "Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Symbol('\\\\psi^*', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('l')(Symbol('\\\\psi^*', commutative=True)), Add(Function('l')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Function('l')(Symbol('\\\\psi^*', commutative=True))))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Function('l')(Symbol('\\\\psi^*', commutative=True))))))"], [["add", 6, "Function('E_x')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('\\\\psi^*', commutative=True)), Function('l')(Symbol('\\\\psi^*', commutative=True))), Add(Function('E_x')(Symbol('\\\\psi^*', commutative=True)), Function('l')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Function('l')(Symbol('\\\\psi^*', commutative=True))))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Function('l')(Symbol('\\\\psi^*', commutative=True))))))"]]}, {"prompt": "Given L{(\\mathbf{p})} = e^{\\mathbf{p}} and \\mathbf{F}{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain e^{\\mathbf{p}} = \\frac{d}{d \\mathbf{p}} \\mathbf{F}{(\\mathbf{p})}", "derivation": "L{(\\mathbf{p})} = e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} L{(\\mathbf{p})} = e^{\\mathbf{p}} and \\mathbf{F}{(\\mathbf{p})} = e^{\\mathbf{p}} and e^{\\mathbf{p}} = \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} and L{(\\mathbf{p})} = \\mathbf{F}{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\mathbf{F}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} and e^{\\mathbf{p}} = \\frac{d}{d \\mathbf{p}} \\mathbf{F}{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(exp(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(V,E_{x})} = E_{x}^{V}, then obtain \\int \\frac{\\partial}{\\partial V} \\operatorname{f^{*}}^{E_{x}}{(V,E_{x})} dV = \\int \\frac{\\partial}{\\partial V} (E_{x}^{V})^{E_{x}} dV", "derivation": "\\operatorname{f^{*}}{(V,E_{x})} = E_{x}^{V} and \\operatorname{f^{*}}^{E_{x}}{(V,E_{x})} = (E_{x}^{V})^{E_{x}} and \\frac{\\partial}{\\partial V} \\operatorname{f^{*}}^{E_{x}}{(V,E_{x})} = \\frac{\\partial}{\\partial V} (E_{x}^{V})^{E_{x}} and \\int \\frac{\\partial}{\\partial V} \\operatorname{f^{*}}^{E_{x}}{(V,E_{x})} dV = \\int \\frac{\\partial}{\\partial V} (E_{x}^{V})^{E_{x}} dV", "srepr_derivation": [["get_premise", "Equality(Function('f^*')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('V', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Symbol('E_x', commutative=True)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Function('f^*')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('f^*')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Pow(Pow(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Symbol('E_x', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(A_{y})} = \\int e^{A_{y}} dA_{y}, then obtain - ((E_{\\lambda} + e^{A_{y}})^{A_{y}})^{A_{y}} + (\\mathbf{E}^{A_{y}}{(A_{y})})^{A_{y}} = 0", "derivation": "\\mathbf{E}{(A_{y})} = \\int e^{A_{y}} dA_{y} and \\mathbf{E}^{A_{y}}{(A_{y})} = (\\int e^{A_{y}} dA_{y})^{A_{y}} and (\\mathbf{E}^{A_{y}}{(A_{y})})^{A_{y}} = ((\\int e^{A_{y}} dA_{y})^{A_{y}})^{A_{y}} and (\\mathbf{E}^{A_{y}}{(A_{y})})^{A_{y}} - ((\\int e^{A_{y}} dA_{y})^{A_{y}})^{A_{y}} = 0 and - ((E_{\\lambda} + e^{A_{y}})^{A_{y}})^{A_{y}} + (\\mathbf{E}^{A_{y}}{(A_{y})})^{A_{y}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A_y', commutative=True)), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Pow(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["minus", 3, "Pow(Pow(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Pow(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))), Integer(0))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Pow(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Pow(Pow(Function('\\\\mathbf{E}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbb{I},l)} = \\log{(\\mathbb{I})}^{l} and \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} = \\log{(\\mathbb{I})}^{l}, then obtain \\frac{\\partial}{\\partial l} \\log{(\\mathbb{I})}^{2 l} = \\frac{\\partial}{\\partial l} \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} \\log{(\\mathbb{I})}^{l}", "derivation": "\\mathbf{P}{(\\mathbb{I},l)} = \\log{(\\mathbb{I})}^{l} and \\mathbf{P}^{2}{(\\mathbb{I},l)} = \\mathbf{P}{(\\mathbb{I},l)} \\log{(\\mathbb{I})}^{l} and \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} = \\log{(\\mathbb{I})}^{l} and \\mathbf{P}^{2}{(\\mathbb{I},l)} = \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} \\mathbf{P}{(\\mathbb{I},l)} and \\log{(\\mathbb{I})}^{2 l} = \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} \\log{(\\mathbb{I})}^{l} and \\frac{\\partial}{\\partial l} \\log{(\\mathbb{I})}^{2 l} = \\frac{\\partial}{\\partial l} \\operatorname{J_{\\varepsilon}}{(\\mathbb{I},l)} \\log{(\\mathbb{I})}^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('l', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Integer(2)), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('l', commutative=True))))"], [["differentiate", 5, "Symbol('l', commutative=True)"], "Equality(Derivative(Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(a^{\\dagger},\\sigma_p)} = \\sigma_p + a^{\\dagger}, then obtain \\int 0 d\\sigma_p = - \\sigma_p - a^{\\dagger} + Q{(a^{\\dagger},\\sigma_p)} + \\int (\\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)}) d\\sigma_p", "derivation": "Q{(a^{\\dagger},\\sigma_p)} = \\sigma_p + a^{\\dagger} and - \\sigma_p - a^{\\dagger} + Q{(a^{\\dagger},\\sigma_p)} = 0 and 0 = \\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)} and \\int 0 d\\sigma_p = \\int (\\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)}) d\\sigma_p and - \\sigma_p - a^{\\dagger} + Q{(a^{\\dagger},\\sigma_p)} + \\int 0 d\\sigma_p = \\int 0 d\\sigma_p and - \\sigma_p - a^{\\dagger} + Q{(a^{\\dagger},\\sigma_p)} + \\int (\\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)}) d\\sigma_p = \\int (\\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)}) d\\sigma_p and \\int 0 d\\sigma_p = - \\sigma_p - a^{\\dagger} + Q{(a^{\\dagger},\\sigma_p)} + \\int (\\sigma_p + a^{\\dagger} - Q{(a^{\\dagger},\\sigma_p)}) d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(0))"], [["minus", 1, "Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Integral(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Integral(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mu_0,p)} = \\frac{p}{\\mu_0}, then obtain \\iint \\operatorname{n_{1}}{(\\mu_0,p)} dp dp - 1 = \\iint \\frac{p}{\\mu_0} dp dp - 1", "derivation": "\\operatorname{n_{1}}{(\\mu_0,p)} = \\frac{p}{\\mu_0} and \\int \\operatorname{n_{1}}{(\\mu_0,p)} dp = \\int \\frac{p}{\\mu_0} dp and \\iint \\operatorname{n_{1}}{(\\mu_0,p)} dp dp = \\iint \\frac{p}{\\mu_0} dp dp and \\iint \\operatorname{n_{1}}{(\\mu_0,p)} dp dp - 1 = \\iint \\frac{p}{\\mu_0} dp dp - 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mu_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\mu_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\mu_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Function('n_1')(Symbol('\\\\mu_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)), Add(Integral(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given m{(\\phi_1,\\theta)} = \\frac{\\phi_1}{\\theta} and \\ddot{x}{(\\phi_1,\\theta)} = \\frac{\\phi_1}{\\theta}, then obtain \\frac{\\partial}{\\partial \\phi_1} e^{\\ddot{x}{(\\phi_1,\\theta)}} = \\frac{\\partial}{\\partial \\phi_1} e^{\\frac{\\phi_1}{\\theta}}", "derivation": "m{(\\phi_1,\\theta)} = \\frac{\\phi_1}{\\theta} and \\ddot{x}{(\\phi_1,\\theta)} = \\frac{\\phi_1}{\\theta} and e^{m{(\\phi_1,\\theta)}} = e^{\\frac{\\phi_1}{\\theta}} and \\frac{\\partial}{\\partial \\phi_1} e^{m{(\\phi_1,\\theta)}} = \\frac{\\partial}{\\partial \\phi_1} e^{\\frac{\\phi_1}{\\theta}} and m{(\\phi_1,\\theta)} = \\ddot{x}{(\\phi_1,\\theta)} and \\frac{\\partial}{\\partial \\phi_1} e^{\\ddot{x}{(\\phi_1,\\theta)}} = \\frac{\\partial}{\\partial \\phi_1} e^{\\frac{\\phi_1}{\\theta}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["exp", 1], "Equality(exp(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True))), exp(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))))"], [["differentiate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(exp(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(exp(Function('\\\\ddot{x}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(f_{E},A_{1})} = \\cos{(A_{1} f_{E})}, then obtain (- A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} l{(f_{E},A_{1})})^{f_{E}} = (- A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} \\cos{(A_{1} f_{E})})^{f_{E}}", "derivation": "l{(f_{E},A_{1})} = \\cos{(A_{1} f_{E})} and \\frac{\\partial}{\\partial A_{1}} l{(f_{E},A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\cos{(A_{1} f_{E})} and \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} l{(f_{E},A_{1})} = \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} \\cos{(A_{1} f_{E})} and - A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} l{(f_{E},A_{1})} = - A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} \\cos{(A_{1} f_{E})} and (- A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} l{(f_{E},A_{1})})^{f_{E}} = (- A_{1} + \\cos{(A_{1} f_{E})} \\frac{\\partial}{\\partial A_{1}} \\cos{(A_{1} f_{E})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('f_E', commutative=True), Symbol('A_1', commutative=True)), cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('f_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["times", 2, "cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(Function('l')(Symbol('f_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["minus", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(Function('l')(Symbol('f_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(Function('l')(Symbol('f_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))), Symbol('f_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Derivative(cos(Mul(Symbol('A_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(m,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m, then derive 1 = \\frac{m}{\\operatorname{t_{2}}{(m,\\varepsilon)}}, then derive \\operatorname{t_{2}}{(m,\\varepsilon)} = m, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m}{\\operatorname{t_{2}}{(\\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m,\\varepsilon)}}", "derivation": "\\operatorname{t_{2}}{(m,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m and 1 = \\frac{\\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m}{\\operatorname{t_{2}}{(m,\\varepsilon)}} and 1 = \\frac{m}{\\operatorname{t_{2}}{(m,\\varepsilon)}} and \\operatorname{t_{2}}{(m,\\varepsilon)} = m and m = \\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m and 1 = \\frac{\\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m}{\\operatorname{t_{2}}{(\\frac{\\partial}{\\partial \\varepsilon} \\varepsilon m,\\varepsilon)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('m', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["divide", 1, "Function('t_2')(Symbol('m', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t_2')(Symbol('m', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(1), Mul(Symbol('m', commutative=True), Pow(Function('t_2')(Symbol('m', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["evaluate_derivatives", 1], "Equality(Function('t_2')(Symbol('m', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('m', commutative=True))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Symbol('m', commutative=True), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(1), Mul(Pow(Function('t_2')(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(\\eta,\\mathbf{S})} = - \\eta + \\mathbf{S}, then obtain \\frac{(b{(\\eta,\\mathbf{S})} + 1)^{\\eta}}{(- \\eta + \\mathbf{S}) b{(\\eta,\\mathbf{S})}} = \\frac{(- \\eta + \\mathbf{S} + 1)^{\\eta}}{(- \\eta + \\mathbf{S}) b{(\\eta,\\mathbf{S})}}", "derivation": "b{(\\eta,\\mathbf{S})} = - \\eta + \\mathbf{S} and b{(\\eta,\\mathbf{S})} + 1 = - \\eta + \\mathbf{S} + 1 and (b{(\\eta,\\mathbf{S})} + 1)^{\\eta} = (- \\eta + \\mathbf{S} + 1)^{\\eta} and \\frac{(b{(\\eta,\\mathbf{S})} + 1)^{\\eta}}{b{(\\eta,\\mathbf{S})}} = \\frac{(- \\eta + \\mathbf{S} + 1)^{\\eta}}{b{(\\eta,\\mathbf{S})}} and \\frac{(b{(\\eta,\\mathbf{S})} + 1)^{\\eta}}{(- \\eta + \\mathbf{S}) b{(\\eta,\\mathbf{S})}} = \\frac{(- \\eta + \\mathbf{S} + 1)^{\\eta}}{(- \\eta + \\mathbf{S}) b{(\\eta,\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Add(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Symbol('\\\\eta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Integer(1)), Symbol('\\\\eta', commutative=True)))"], [["divide", 3, "Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Add(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Symbol('\\\\eta', commutative=True)), Pow(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Integer(1)), Symbol('\\\\eta', commutative=True)), Pow(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Pow(Add(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Symbol('\\\\eta', commutative=True)), Pow(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Integer(1)), Symbol('\\\\eta', commutative=True)), Pow(Function('b')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain \\frac{d}{d L_{\\varepsilon}} \\int 0 dL_{\\varepsilon} = \\frac{d}{d L_{\\varepsilon}} \\int (- \\mathbf{B}{(L_{\\varepsilon})} + e^{L_{\\varepsilon}}) dL_{\\varepsilon}", "derivation": "\\mathbf{B}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and 0 = - \\mathbf{B}{(L_{\\varepsilon})} + e^{L_{\\varepsilon}} and \\int 0 dL_{\\varepsilon} = \\int (- \\mathbf{B}{(L_{\\varepsilon})} + e^{L_{\\varepsilon}}) dL_{\\varepsilon} and \\frac{d}{d L_{\\varepsilon}} \\int 0 dL_{\\varepsilon} = \\frac{d}{d L_{\\varepsilon}} \\int (- \\mathbf{B}{(L_{\\varepsilon})} + e^{L_{\\varepsilon}}) dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('L_{\\\\varepsilon}', commutative=True))), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('L_{\\\\varepsilon}', commutative=True))), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('L_{\\\\varepsilon}', commutative=True))), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\theta_2)} = \\log{(\\theta_2)}, then obtain - \\log{(\\theta_2)} = - \\Psi_{nl}{(\\theta_2)}", "derivation": "\\Psi_{nl}{(\\theta_2)} = \\log{(\\theta_2)} and \\theta_2 + \\Psi_{nl}{(\\theta_2)} = \\theta_2 + \\log{(\\theta_2)} and 0 = - \\Psi_{nl}{(\\theta_2)} + \\log{(\\theta_2)} and - \\log{(\\theta_2)} = - \\Psi_{nl}{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True))), log(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "log(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(-1), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\varepsilon)} = \\cos{(\\log{(\\varepsilon)})}, then obtain - \\frac{\\mathbf{S}{(\\varepsilon)}}{\\cos{(\\log{(\\varepsilon)})} \\int \\cos{(\\log{(\\varepsilon)})} d\\varepsilon} = - \\frac{1}{\\int \\cos{(\\log{(\\varepsilon)})} d\\varepsilon}", "derivation": "\\mathbf{S}{(\\varepsilon)} = \\cos{(\\log{(\\varepsilon)})} and \\frac{\\mathbf{S}{(\\varepsilon)}}{\\cos{(\\log{(\\varepsilon)})}} = 1 and - \\frac{\\mathbf{S}{(\\varepsilon)}}{\\cos{(\\log{(\\varepsilon)})}} = -1 and \\int \\mathbf{S}{(\\varepsilon)} d\\varepsilon = \\int \\cos{(\\log{(\\varepsilon)})} d\\varepsilon and - \\frac{\\mathbf{S}{(\\varepsilon)}}{\\cos{(\\log{(\\varepsilon)})} \\int \\mathbf{S}{(\\varepsilon)} d\\varepsilon} = - \\frac{1}{\\int \\mathbf{S}{(\\varepsilon)} d\\varepsilon} and - \\frac{\\mathbf{S}{(\\varepsilon)}}{\\cos{(\\log{(\\varepsilon)})} \\int \\cos{(\\log{(\\varepsilon)})} d\\varepsilon} = - \\frac{1}{\\int \\cos{(\\log{(\\varepsilon)})} d\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), cos(log(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "cos(log(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(log(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(log(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))), Integer(-1))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 3, "Integral(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(log(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(log(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Pow(Integral(cos(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(cos(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\pi{(l,r)} = \\frac{l}{r}, then derive (\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r} = (- \\frac{l}{r^{2}})^{r}, then obtain ((\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r})^{l} = ((- \\frac{\\pi{(l,r)}}{r})^{r})^{l}", "derivation": "\\pi{(l,r)} = \\frac{l}{r} and \\frac{\\partial}{\\partial r} \\pi{(l,r)} = \\frac{\\partial}{\\partial r} \\frac{l}{r} and (\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r} = (\\frac{\\partial}{\\partial r} \\frac{l}{r})^{r} and ((\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r})^{l} = ((\\frac{\\partial}{\\partial r} \\frac{l}{r})^{r})^{l} and (\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r} = (- \\frac{l}{r^{2}})^{r} and (\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r} = (- \\frac{\\pi{(l,r)}}{r})^{r} and (- \\frac{\\pi{(l,r)}}{r})^{r} = (\\frac{\\partial}{\\partial r} \\frac{l}{r})^{r} and ((\\frac{\\partial}{\\partial r} \\pi{(l,r)})^{r})^{l} = ((- \\frac{\\pi{(l,r)}}{r})^{r})^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Symbol('l', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Mul(Integer(-1), Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-2))), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Derivative(Mul(Symbol('l', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 7], "Equality(Pow(Pow(Derivative(Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\mathbf{v}{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda}, then obtain \\frac{d}{d E_{\\lambda}} \\mathbf{v}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\int \\mu_{0}{(E_{\\lambda})} dE_{\\lambda}", "derivation": "\\mu_{0}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\int \\mu_{0}{(E_{\\lambda})} dE_{\\lambda} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and \\mathbf{v}{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and \\mathbf{v}{(E_{\\lambda})} = \\int \\mu_{0}{(E_{\\lambda})} dE_{\\lambda} and \\frac{d}{d E_{\\lambda}} \\mathbf{v}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\int \\mu_{0}{(E_{\\lambda})} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["differentiate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mu_0')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(c)} = \\cos{(c)}, then derive (\\frac{d}{d c} \\psi{(c)})^{c} = (- \\sin{(c)})^{c}, then obtain \\int (- \\sin{(c)})^{c} dc = \\int (\\frac{d}{d c} \\cos{(c)})^{c} dc", "derivation": "\\psi{(c)} = \\cos{(c)} and \\frac{d}{d c} \\psi{(c)} = \\frac{d}{d c} \\cos{(c)} and (\\frac{d}{d c} \\psi{(c)})^{c} = (\\frac{d}{d c} \\cos{(c)})^{c} and (\\frac{d}{d c} \\psi{(c)})^{c} = (- \\sin{(c)})^{c} and (- \\sin{(c)})^{c} = (\\frac{d}{d c} \\cos{(c)})^{c} and \\int (- \\sin{(c)})^{c} dc = \\int (\\frac{d}{d c} \\cos{(c)})^{c} dc", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\psi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\psi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Mul(Integer(-1), sin(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["integrate", 5, "Symbol('c', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), sin(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(E)} = e^{\\sin{(E)}}, then obtain (\\varepsilon_{0}^{E}{(E)} e^{- \\sin{(E)}} - e^{- \\sin{(E)}} (e^{\\sin{(E)}})^{E})^{E} = 0^{E}", "derivation": "\\varepsilon_{0}{(E)} = e^{\\sin{(E)}} and \\varepsilon_{0}^{E}{(E)} = (e^{\\sin{(E)}})^{E} and \\varepsilon_{0}^{E}{(E)} e^{- \\sin{(E)}} = e^{- \\sin{(E)}} (e^{\\sin{(E)}})^{E} and \\varepsilon_{0}^{E}{(E)} e^{- \\sin{(E)}} - \\sin{(E)} = - \\sin{(E)} + e^{- \\sin{(E)}} (e^{\\sin{(E)}})^{E} and \\varepsilon_{0}^{E}{(E)} e^{- \\sin{(E)}} - e^{- \\sin{(E)}} (e^{\\sin{(E)}})^{E} = 0 and (\\varepsilon_{0}^{E}{(E)} e^{- \\sin{(E)}} - e^{- \\sin{(E)}} (e^{\\sin{(E)}})^{E})^{E} = 0^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), exp(sin(Symbol('E', commutative=True))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["divide", 2, "exp(sin(Symbol('E', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True))))), Mul(exp(Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["minus", 3, "sin(Symbol('E', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True))))), Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('E', commutative=True))), Mul(exp(Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), sin(Symbol('E', commutative=True))), Mul(exp(Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], "Equality(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True))))), Mul(Integer(-1), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)))), Integer(0))"], [["power", 5, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True))))), Mul(Integer(-1), exp(Mul(Integer(-1), sin(Symbol('E', commutative=True)))), Pow(exp(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Pow(Integer(0), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and \\mathbf{P}{(\\mathbf{D})} = \\cos{(\\mathbf{D})}, then obtain 2 \\mathbf{D} + \\mathbf{M}{(\\mathbf{D})} + \\cos{(\\mathbf{D})} = 2 \\mathbf{D} + 2 \\cos{(\\mathbf{D})}", "derivation": "\\mathbf{M}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and \\mathbf{D} + \\mathbf{M}{(\\mathbf{D})} = \\mathbf{D} + \\cos{(\\mathbf{D})} and \\mathbf{P}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and 2 \\mathbf{D} + \\mathbf{M}{(\\mathbf{D})} = 2 \\mathbf{D} + \\cos{(\\mathbf{D})} and 2 \\mathbf{D} + \\mathbf{M}{(\\mathbf{D})} + \\mathbf{P}{(\\mathbf{D})} = 2 \\mathbf{D} + \\mathbf{P}{(\\mathbf{D})} + \\cos{(\\mathbf{D})} and 2 \\mathbf{D} + \\mathbf{M}{(\\mathbf{D})} + \\cos{(\\mathbf{D})} = 2 \\mathbf{D} + 2 \\cos{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{J},\\delta)} = \\delta \\mathbf{J}, then obtain \\mathbf{J} \\int \\delta \\operatorname{t_{1}}{(\\mathbf{J},\\delta)} d\\mathbf{J} = \\mathbf{J} \\int \\delta^{2} \\mathbf{J} d\\mathbf{J}", "derivation": "\\operatorname{t_{1}}{(\\mathbf{J},\\delta)} = \\delta \\mathbf{J} and \\delta \\operatorname{t_{1}}{(\\mathbf{J},\\delta)} = \\delta^{2} \\mathbf{J} and \\int \\delta \\operatorname{t_{1}}{(\\mathbf{J},\\delta)} d\\mathbf{J} = \\int \\delta^{2} \\mathbf{J} d\\mathbf{J} and \\mathbf{J} \\int \\delta \\operatorname{t_{1}}{(\\mathbf{J},\\delta)} d\\mathbf{J} = \\mathbf{J} \\int \\delta^{2} \\mathbf{J} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\delta', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Integral(Mul(Symbol('\\\\delta', commutative=True), Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(x^\\prime,\\mathbf{D})} = \\sin{(\\mathbf{D} x^\\prime)}, then obtain \\int \\frac{\\int x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} d\\mathbf{D} = \\int \\frac{\\int x^\\prime \\sin{(\\mathbf{D} x^\\prime)} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} d\\mathbf{D}", "derivation": "\\mathbf{g}{(x^\\prime,\\mathbf{D})} = \\sin{(\\mathbf{D} x^\\prime)} and x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})} = x^\\prime \\sin{(\\mathbf{D} x^\\prime)} and \\int x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})} dx^\\prime = \\int x^\\prime \\sin{(\\mathbf{D} x^\\prime)} dx^\\prime and \\frac{\\int x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} = \\frac{\\int x^\\prime \\sin{(\\mathbf{D} x^\\prime)} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} and \\int \\frac{\\int x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} d\\mathbf{D} = \\int \\frac{\\int x^\\prime \\sin{(\\mathbf{D} x^\\prime)} dx^\\prime}{x^\\prime \\mathbf{g}{(x^\\prime,\\mathbf{D})}} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), sin(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('x^\\\\prime', commutative=True), sin(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 3, "Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), sin(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), sin(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})} = - \\mathbf{H} + g_{\\varepsilon}, then obtain \\frac{(- g_{\\varepsilon} + \\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})})^{\\mathbf{H}}}{\\frac{d}{d \\mathbf{H}} V{(\\mathbf{H})}} = \\frac{(- \\mathbf{H})^{\\mathbf{H}}}{\\frac{d}{d \\mathbf{H}} V{(\\mathbf{H})}}", "derivation": "\\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})} = - \\mathbf{H} + g_{\\varepsilon} and - g_{\\varepsilon} + \\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})} = - \\mathbf{H} and (- g_{\\varepsilon} + \\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})})^{\\mathbf{H}} = (- \\mathbf{H})^{\\mathbf{H}} and \\frac{(- g_{\\varepsilon} + \\mathbf{F}{(g_{\\varepsilon},\\mathbf{H})})^{\\mathbf{H}}}{\\frac{d}{d \\mathbf{H}} V{(\\mathbf{H})}} = \\frac{(- \\mathbf{H})^{\\mathbf{H}}}{\\frac{d}{d \\mathbf{H}} V{(\\mathbf{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 3, "Derivative(Function('V')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(Function('V')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(Function('V')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(G)} = \\sin{(G)}, then derive \\frac{d}{d G} \\operatorname{v_{x}}{(G)} = \\cos{(G)}, then derive \\rho + \\sin{(G)} = n_{1} + \\sin{(G)}, then obtain 2 \\rho + 2 \\sin{(G)} - \\int \\frac{d}{d G} \\sin{(G)} dG = \\rho + n_{1} + 2 \\sin{(G)} - \\int \\frac{d}{d G} \\sin{(G)} dG", "derivation": "\\operatorname{v_{x}}{(G)} = \\sin{(G)} and \\frac{d}{d G} \\operatorname{v_{x}}{(G)} = \\frac{d}{d G} \\sin{(G)} and \\frac{d}{d G} \\operatorname{v_{x}}{(G)} = \\cos{(G)} and \\frac{d}{d G} \\sin{(G)} = \\cos{(G)} and \\int \\frac{d}{d G} \\sin{(G)} dG = \\int \\cos{(G)} dG and \\rho + \\sin{(G)} = n_{1} + \\sin{(G)} and 2 \\rho + 2 \\sin{(G)} = \\rho + n_{1} + 2 \\sin{(G)} and 2 \\rho + 2 \\sin{(G)} - \\int \\frac{d}{d G} \\sin{(G)} dG = \\rho + n_{1} + 2 \\sin{(G)} - \\int \\frac{d}{d G} \\sin{(G)} dG", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), cos(Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), cos(Symbol('G', commutative=True)))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integral(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('G', commutative=True))), Add(Symbol('n_1', commutative=True), sin(Symbol('G', commutative=True))))"], [["add", 6, "Add(Symbol('\\\\rho', commutative=True), sin(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Mul(Integer(2), sin(Symbol('G', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(2), sin(Symbol('G', commutative=True)))))"], [["minus", 7, "Integral(Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Mul(Integer(2), sin(Symbol('G', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))), Add(Symbol('\\\\rho', commutative=True), Symbol('n_1', commutative=True), Mul(Integer(2), sin(Symbol('G', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(m,z)} = \\frac{m}{z}, then derive - z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)} = - z + \\frac{1}{z}, then obtain \\frac{\\partial}{\\partial m} (- z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)}) = \\frac{d}{d m} (- z + \\frac{1}{z})", "derivation": "\\hat{x}{(m,z)} = \\frac{m}{z} and \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)} = \\frac{\\partial}{\\partial m} \\frac{m}{z} and - z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)} = - z + \\frac{\\partial}{\\partial m} \\frac{m}{z} and - z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)} = - z + \\frac{1}{z} and \\frac{\\partial}{\\partial m} (- z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)}) = \\frac{\\partial}{\\partial m} (- z + \\frac{\\partial}{\\partial m} \\frac{m}{z}) and - z + \\frac{\\partial}{\\partial m} \\frac{m}{z} = - z + \\frac{1}{z} and \\frac{\\partial}{\\partial m} (- z + \\frac{\\partial}{\\partial m} \\hat{x}{(m,z)}) = \\frac{d}{d m} (- z + \\frac{1}{z})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('m', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Symbol('m', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Mul(Symbol('m', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Mul(Symbol('m', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Mul(Symbol('m', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('m', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then obtain \\int \\frac{2 \\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} d\\mathbf{g} = \\int (\\frac{\\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} + 1) d\\mathbf{g}", "derivation": "\\operatorname{A_{x}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\frac{\\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} = 1 and \\frac{2 \\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} = \\frac{\\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} + 1 and \\int \\frac{2 \\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} d\\mathbf{g} = \\int (\\frac{\\operatorname{A_{x}}{(\\mathbf{g})}}{\\sin{(\\mathbf{g})}} + 1) d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Mul(Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Add(Mul(Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Integer(1)))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Mul(Function('A_x')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given C{(A_{2},\\nabla)} = A_{2} \\nabla, then derive \\frac{\\partial}{\\partial A_{2}} C{(A_{2},\\nabla)} = \\nabla, then obtain \\int \\frac{\\partial}{\\partial A_{2}} A_{2} \\nabla d\\nabla = \\int \\nabla d\\nabla", "derivation": "C{(A_{2},\\nabla)} = A_{2} \\nabla and \\frac{\\partial}{\\partial A_{2}} C{(A_{2},\\nabla)} = \\frac{\\partial}{\\partial A_{2}} A_{2} \\nabla and \\frac{\\partial}{\\partial A_{2}} C{(A_{2},\\nabla)} = \\nabla and \\frac{\\partial}{\\partial A_{2}} A_{2} \\nabla = \\nabla and \\int \\frac{\\partial}{\\partial A_{2}} A_{2} \\nabla d\\nabla = \\int \\nabla d\\nabla", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Symbol('\\\\nabla', commutative=True), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given m{(f_{E})} = e^{f_{E}}, then derive \\cos{(\\int m{(f_{E})} df_{E})} = \\cos{(\\varepsilon_0 + e^{f_{E}})}, then obtain - m{(f_{E})} + \\cos{(\\int m{(f_{E})} df_{E})} = - m{(f_{E})} + \\cos{(\\varepsilon_0 + m{(f_{E})})}", "derivation": "m{(f_{E})} = e^{f_{E}} and \\int m{(f_{E})} df_{E} = \\int e^{f_{E}} df_{E} and \\cos{(\\int m{(f_{E})} df_{E})} = \\cos{(\\int e^{f_{E}} df_{E})} and \\cos{(\\int m{(f_{E})} df_{E})} = \\cos{(\\varepsilon_0 + e^{f_{E}})} and - m{(f_{E})} + \\cos{(\\int m{(f_{E})} df_{E})} = - m{(f_{E})} + \\cos{(\\varepsilon_0 + e^{f_{E}})} and - m{(f_{E})} + \\cos{(\\int m{(f_{E})} df_{E})} = - m{(f_{E})} + \\cos{(\\varepsilon_0 + m{(f_{E})})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(exp(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), cos(Integral(exp(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(cos(Integral(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), cos(Add(Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('f_E', commutative=True)))))"], [["minus", 4, "Function('m')(Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), cos(Integral(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), cos(Add(Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('f_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), cos(Integral(Function('m')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))), Add(Mul(Integer(-1), Function('m')(Symbol('f_E', commutative=True))), cos(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('m')(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{x}_0,p)} = \\hat{x}_0 p, then obtain \\hat{x}_0^{2} p^{2} = \\hat{x}_0 p \\theta_{1}{(\\hat{x}_0,p)}", "derivation": "\\theta_{1}{(\\hat{x}_0,p)} = \\hat{x}_0 p and \\theta_{1}^{2}{(\\hat{x}_0,p)} = \\hat{x}_0 p \\theta_{1}{(\\hat{x}_0,p)} and p \\theta_{1}{(\\hat{x}_0,p)} = \\hat{x}_0 p^{2} and \\theta_{1}^{2}{(\\hat{x}_0,p)} = \\hat{x}_0^{2} p^{2} and \\hat{x}_0^{2} p^{2} = \\hat{x}_0 p \\theta_{1}{(\\hat{x}_0,p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Integer(2)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True), Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True))))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('p', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True), Function('\\\\theta_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given g{(E_{\\lambda},n)} = n^{E_{\\lambda}} and h{(E_{\\lambda},n)} = - n^{E_{\\lambda}} (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g{(E_{\\lambda},n)}, then obtain h{(E_{\\lambda},n)} = - (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g^{2}{(E_{\\lambda},n)}", "derivation": "g{(E_{\\lambda},n)} = n^{E_{\\lambda}} and - g^{2}{(E_{\\lambda},n)} = - n^{E_{\\lambda}} g{(E_{\\lambda},n)} and - (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g^{2}{(E_{\\lambda},n)} = - n^{E_{\\lambda}} (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g{(E_{\\lambda},n)} and h{(E_{\\lambda},n)} = - n^{E_{\\lambda}} (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g{(E_{\\lambda},n)} and h{(E_{\\lambda},n)} = - (n^{E_{\\lambda}} - g{(E_{\\lambda},n)}) g^{2}{(E_{\\lambda},n)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True))))"], [["times", 2, "Add(Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)))), Pow(Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)))), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)))), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('h')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Add(Pow(Symbol('n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)))), Pow(Function('g')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}_f{(J)} = \\cos{(J)}, then obtain \\mathbf{J}_f^{3}{(J)} = \\mathbf{J}_f^{2}{(J)} \\cos{(J)}", "derivation": "\\mathbf{J}_f{(J)} = \\cos{(J)} and \\mathbf{J}_f^{2}{(J)} = \\mathbf{J}_f{(J)} \\cos{(J)} and \\mathbf{J}_f^{2}{(J)} \\cos{(J)} = \\mathbf{J}_f{(J)} \\cos^{2}{(J)} and \\mathbf{J}_f^{3}{(J)} = \\mathbf{J}_f^{2}{(J)} \\cos{(J)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))))"], [["times", 1, "Mul(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), Integer(2)), cos(Symbol('J', commutative=True))), Mul(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), Pow(cos(Symbol('J', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True)), Integer(2)), cos(Symbol('J', commutative=True))))"]]}, {"prompt": "Given a{(l)} = \\cos{(l)}, then derive \\int a{(l)} dl = f^{*} + \\sin{(l)}, then obtain (\\cos{(\\iint a{(l)} dl dl)}) \\int a{(l)} dl = (\\cos{(\\iint \\cos{(l)} dl dl)}) \\int a{(l)} dl", "derivation": "a{(l)} = \\cos{(l)} and \\int a{(l)} dl = \\int \\cos{(l)} dl and \\int a{(l)} dl = f^{*} + \\sin{(l)} and \\iint a{(l)} dl dl = \\int (f^{*} + \\sin{(l)}) dl and \\iint \\cos{(l)} dl dl = \\int (f^{*} + \\sin{(l)}) dl and \\iint a{(l)} dl dl = \\iint \\cos{(l)} dl dl and \\cos{(\\iint a{(l)} dl dl)} = \\cos{(\\iint \\cos{(l)} dl dl)} and (\\cos{(\\iint a{(l)} dl dl)}) \\int a{(l)} dl = (\\cos{(\\iint \\cos{(l)} dl dl)}) \\int a{(l)} dl", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('f^*', commutative=True), sin(Symbol('l', commutative=True))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["cos", 6], "Equality(cos(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), cos(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["times", 7, "Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(cos(Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(cos(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\varphi,L)} = - L + \\varphi and \\operatorname{v_{y}}{(\\varphi,L)} = \\frac{L - \\varphi}{L}, then obtain - \\frac{\\frac{\\partial}{\\partial L} \\operatorname{F_{N}}{(\\varphi,L)}}{L} + \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L^{2}} = \\frac{\\partial}{\\partial L} \\operatorname{v_{y}}{(\\varphi,L)}", "derivation": "\\operatorname{F_{N}}{(\\varphi,L)} = - L + \\varphi and - \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L} = - \\frac{- L + \\varphi}{L} and \\frac{\\partial}{\\partial L} - \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L} = \\frac{\\partial}{\\partial L} - \\frac{- L + \\varphi}{L} and \\frac{\\partial}{\\partial L} - \\frac{- L + \\varphi}{L} = \\frac{\\partial}{\\partial L} \\frac{L - \\varphi}{L} and \\operatorname{v_{y}}{(\\varphi,L)} = \\frac{L - \\varphi}{L} and \\frac{\\partial}{\\partial L} - \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L} = \\frac{\\partial}{\\partial L} \\frac{L - \\varphi}{L} and \\frac{\\partial}{\\partial L} - \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L} = \\frac{\\partial}{\\partial L} \\operatorname{v_{y}}{(\\varphi,L)} and - \\frac{\\frac{\\partial}{\\partial L} \\operatorname{F_{N}}{(\\varphi,L)}}{L} + \\frac{\\operatorname{F_{N}}{(\\varphi,L)}}{L^{2}} = \\frac{\\partial}{\\partial L} \\operatorname{v_{y}}{(\\varphi,L)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('L', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Function('v_y')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-2)), Function('F_N')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)))), Derivative(Function('v_y')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given p{(\\eta)} = e^{\\eta}, then derive 1 = \\frac{e^{\\eta}}{\\frac{d}{d \\eta} p{(\\eta)}}, then obtain \\int 1 d\\eta = \\int \\frac{p{(\\eta)}}{\\frac{d}{d \\eta} p{(\\eta)}} d\\eta", "derivation": "p{(\\eta)} = e^{\\eta} and \\frac{d}{d \\eta} p{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and 1 = \\frac{\\frac{d}{d \\eta} e^{\\eta}}{\\frac{d}{d \\eta} p{(\\eta)}} and 1 = \\frac{e^{\\eta}}{\\frac{d}{d \\eta} p{(\\eta)}} and 1 = \\frac{e^{\\eta}}{\\frac{d}{d \\eta} e^{\\eta}} and 1 = \\frac{p{(\\eta)}}{\\frac{d}{d \\eta} p{(\\eta)}} and \\int 1 d\\eta = \\int \\frac{p{(\\eta)}}{\\frac{d}{d \\eta} p{(\\eta)}} d\\eta", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(exp(Symbol('\\\\eta', commutative=True)), Pow(Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(exp(Symbol('\\\\eta', commutative=True)), Pow(Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(1), Mul(Function('p')(Symbol('\\\\eta', commutative=True)), Pow(Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Mul(Function('p')(Symbol('\\\\eta', commutative=True)), Pow(Derivative(Function('p')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(v_{t},F_{x})} = - F_{x} + v_{t}, then obtain v_{1} + \\frac{\\partial}{\\partial v_{t}} - \\frac{\\mathbf{M}{(v_{t},F_{x})}}{F_{x}} = v_{1} + \\frac{\\partial}{\\partial v_{t}} \\frac{F_{x} - v_{t}}{F_{x}}", "derivation": "\\mathbf{M}{(v_{t},F_{x})} = - F_{x} + v_{t} and - \\mathbf{M}{(v_{t},F_{x})} = F_{x} - v_{t} and - \\frac{\\mathbf{M}{(v_{t},F_{x})}}{F_{x}} = \\frac{F_{x} - v_{t}}{F_{x}} and \\frac{\\partial}{\\partial v_{t}} - \\frac{\\mathbf{M}{(v_{t},F_{x})}}{F_{x}} = \\frac{\\partial}{\\partial v_{t}} \\frac{F_{x} - v_{t}}{F_{x}} and v_{1} + \\frac{\\partial}{\\partial v_{t}} - \\frac{\\mathbf{M}{(v_{t},F_{x})}}{F_{x}} = v_{1} + \\frac{\\partial}{\\partial v_{t}} \\frac{F_{x} - v_{t}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v_t', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('v_t', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v_t', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["divide", 2, "Symbol('F_x', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('v_t', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('v_t', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["add", 4, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Derivative(Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('v_t', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Add(Symbol('v_1', commutative=True), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\varepsilon)} = e^{\\cos{(\\varepsilon)}} and \\chi{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then obtain \\chi{(\\mathbb{I})} + \\cos{(\\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}})} = \\sin{(\\mathbb{I})} + \\cos{(\\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}})}", "derivation": "\\operatorname{v_{1}}{(\\varepsilon)} = e^{\\cos{(\\varepsilon)}} and 1 = \\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}} and \\cos{(1)} = \\cos{(\\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}})} and \\chi{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\chi{(\\mathbb{I})} + \\cos{(1)} = \\sin{(\\mathbb{I})} + \\cos{(1)} and \\chi{(\\mathbb{I})} + \\cos{(\\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}})} = \\sin{(\\mathbb{I})} + \\cos{(\\frac{e^{\\cos{(\\varepsilon)}}}{\\operatorname{v_{1}}{(\\varepsilon)}})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\varepsilon', commutative=True)), exp(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Function('v_1')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_1')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["cos", 2], "Equality(cos(Integer(1)), cos(Mul(Pow(Function('v_1')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\varepsilon', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 4, "cos(Integer(1))"], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Integer(1))), Add(sin(Symbol('\\\\mathbb{I}', commutative=True)), cos(Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Mul(Pow(Function('v_1')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\varepsilon', commutative=True)))))), Add(sin(Symbol('\\\\mathbb{I}', commutative=True)), cos(Mul(Pow(Function('v_1')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\varepsilon', commutative=True)))))))"]]}, {"prompt": "Given \\mu{(c)} = \\int \\log{(c)} dc, then obtain 4 \\mu^{2}{(c)} = 4 \\mu{(c)} \\int \\log{(c)} dc", "derivation": "\\mu{(c)} = \\int \\log{(c)} dc and 2 \\mu{(c)} = \\mu{(c)} + \\int \\log{(c)} dc and 4 \\mu^{2}{(c)} = 2 (\\mu{(c)} + \\int \\log{(c)} dc) \\mu{(c)} and (\\mu{(c)} + \\int \\log{(c)} dc) \\mu{(c)} = (\\mu{(c)} + \\int \\log{(c)} dc) \\int \\log{(c)} dc and 4 \\mu^{2}{(c)} = 2 (\\mu{(c)} + \\int \\log{(c)} dc) \\int \\log{(c)} dc and 4 \\mu^{2}{(c)} = 4 \\mu{(c)} \\int \\log{(c)} dc", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 1, "Function('\\\\mu')(Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mu')(Symbol('c', commutative=True))), Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["times", 2, "Mul(Integer(2), Function('\\\\mu')(Symbol('c', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Function('\\\\mu')(Symbol('c', commutative=True))))"], [["times", 1, "Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Function('\\\\mu')(Symbol('c', commutative=True))), Mul(Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(4), Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Integer(2))), Mul(Integer(4), Function('\\\\mu')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(s,F_{H})} = F_{H} s, then obtain \\varphi^* + s = \\int e^{F_{H} s - \\operatorname{f^{\\prime}}{(s,F_{H})}} ds", "derivation": "\\operatorname{f^{\\prime}}{(s,F_{H})} = F_{H} s and 0 = F_{H} s - \\operatorname{f^{\\prime}}{(s,F_{H})} and 1 = e^{F_{H} s - \\operatorname{f^{\\prime}}{(s,F_{H})}} and \\int 1 ds = \\int e^{F_{H} s - \\operatorname{f^{\\prime}}{(s,F_{H})}} ds and \\varphi^* + s = \\int e^{F_{H} s - \\operatorname{f^{\\prime}}{(s,F_{H})}} ds", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('F_H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True)))))"], [["exp", 2], "Equality(Integer(1), exp(Add(Mul(Symbol('F_H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True))))))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Integral(exp(Add(Mul(Symbol('F_H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True))))), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Integral(exp(Add(Mul(Symbol('F_H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('s', commutative=True), Symbol('F_H', commutative=True))))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\pi{(H,B)} = B^{H}, then obtain - B^{H} + (B^{H})^{H} \\pi^{H}{(H,B)} + (\\pi^{H}{(H,B)})^{H} + \\pi{(H,B)} = (B^{H})^{H} \\pi^{H}{(H,B)} + ((B^{H})^{H})^{H}", "derivation": "\\pi{(H,B)} = B^{H} and - B^{H} + \\pi{(H,B)} = 0 and \\pi^{H}{(H,B)} = (B^{H})^{H} and (\\pi^{H}{(H,B)})^{H} = ((B^{H})^{H})^{H} and - B^{H} + (\\pi^{H}{(H,B)})^{H} + \\pi{(H,B)} = (\\pi^{H}{(H,B)})^{H} and - B^{H} + (\\pi^{H}{(H,B)})^{H} + \\pi{(H,B)} = ((B^{H})^{H})^{H} and - B^{H} + (B^{H})^{H} \\pi^{H}{(H,B)} + (\\pi^{H}{(H,B)})^{H} + \\pi{(H,B)} = (B^{H})^{H} \\pi^{H}{(H,B)} + ((B^{H})^{H})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)))"], [["minus", 1, "Pow(Symbol('B', commutative=True), Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('H', commutative=True))), Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Integer(0))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["add", 2, "Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('H', commutative=True))), Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('H', commutative=True))), Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Pow(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["add", 6, "Mul(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True))), Pow(Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Add(Mul(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\pi')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Symbol('H', commutative=True))), Pow(Pow(Pow(Symbol('B', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\varphi^*,\\dot{y})} = \\dot{y} + \\varphi^*, then obtain \\log{(\\varphi^* \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{x}}{(\\varphi^*,\\dot{y})})} = \\log{(\\varphi^* \\frac{\\partial}{\\partial \\dot{y}} (\\dot{y} + \\varphi^*))}", "derivation": "\\operatorname{v_{x}}{(\\varphi^*,\\dot{y})} = \\dot{y} + \\varphi^* and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{x}}{(\\varphi^*,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} (\\dot{y} + \\varphi^*) and \\varphi^* \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{x}}{(\\varphi^*,\\dot{y})} = \\varphi^* \\frac{\\partial}{\\partial \\dot{y}} (\\dot{y} + \\varphi^*) and \\log{(\\varphi^* \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{x}}{(\\varphi^*,\\dot{y})})} = \\log{(\\varphi^* \\frac{\\partial}{\\partial \\dot{y}} (\\dot{y} + \\varphi^*))}", "srepr_derivation": [["get_premise", "Equality(Function('v_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Derivative(Function('v_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Mul(Symbol('\\\\varphi^*', commutative=True), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))))"], [["log", 3], "Equality(log(Mul(Symbol('\\\\varphi^*', commutative=True), Derivative(Function('v_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))), log(Mul(Symbol('\\\\varphi^*', commutative=True), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given T{(\\mu)} = \\log{(\\log{(\\mu)})}, then obtain \\sin{(\\frac{d}{d \\mu} 2 T{(\\mu)})} = \\sin{(\\frac{d}{d \\mu} (T{(\\mu)} + \\log{(\\log{(\\mu)})}))}", "derivation": "T{(\\mu)} = \\log{(\\log{(\\mu)})} and 2 T{(\\mu)} = T{(\\mu)} + \\log{(\\log{(\\mu)})} and \\frac{d}{d \\mu} 2 T{(\\mu)} = \\frac{d}{d \\mu} (T{(\\mu)} + \\log{(\\log{(\\mu)})}) and \\sin{(\\frac{d}{d \\mu} 2 T{(\\mu)})} = \\sin{(\\frac{d}{d \\mu} (T{(\\mu)} + \\log{(\\log{(\\mu)})}))}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mu', commutative=True)), log(log(Symbol('\\\\mu', commutative=True))))"], [["add", 1, "Function('T')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(2), Function('T')(Symbol('\\\\mu', commutative=True))), Add(Function('T')(Symbol('\\\\mu', commutative=True)), log(log(Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('T')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Function('T')(Symbol('\\\\mu', commutative=True)), log(log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Mul(Integer(2), Function('T')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), sin(Derivative(Add(Function('T')(Symbol('\\\\mu', commutative=True)), log(log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain (\\frac{d}{d \\rho_f} \\iint l{(\\rho_f)} d\\rho_f d\\rho_f)^{\\rho_f} = (\\frac{d}{d \\rho_f} \\iint \\sin{(\\rho_f)} d\\rho_f d\\rho_f)^{\\rho_f}", "derivation": "l{(\\rho_f)} = \\sin{(\\rho_f)} and \\int l{(\\rho_f)} d\\rho_f = \\int \\sin{(\\rho_f)} d\\rho_f and \\iint l{(\\rho_f)} d\\rho_f d\\rho_f = \\iint \\sin{(\\rho_f)} d\\rho_f d\\rho_f and \\frac{d}{d \\rho_f} \\iint l{(\\rho_f)} d\\rho_f d\\rho_f = \\frac{d}{d \\rho_f} \\iint \\sin{(\\rho_f)} d\\rho_f d\\rho_f and (\\frac{d}{d \\rho_f} \\iint l{(\\rho_f)} d\\rho_f d\\rho_f)^{\\rho_f} = (\\frac{d}{d \\rho_f} \\iint \\sin{(\\rho_f)} d\\rho_f d\\rho_f)^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Integral(Function('l')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('l')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(\\eta)} = \\sin{(\\eta)}, then obtain ((\\eta + \\hat{x}_0{(\\eta)}) (\\eta + \\sin{(\\eta)}) \\hat{x}_0{(\\eta)})^{\\eta} = ((\\eta + \\sin{(\\eta)})^{2} \\hat{x}_0{(\\eta)})^{\\eta}", "derivation": "\\hat{x}_0{(\\eta)} = \\sin{(\\eta)} and \\eta + \\hat{x}_0{(\\eta)} = \\eta + \\sin{(\\eta)} and (\\eta + \\hat{x}_0{(\\eta)}) \\hat{x}_0{(\\eta)} = (\\eta + \\sin{(\\eta)}) \\hat{x}_0{(\\eta)} and (\\eta + \\hat{x}_0{(\\eta)}) (\\eta + \\sin{(\\eta)}) \\hat{x}_0{(\\eta)} = (\\eta + \\sin{(\\eta)})^{2} \\hat{x}_0{(\\eta)} and ((\\eta + \\hat{x}_0{(\\eta)}) (\\eta + \\sin{(\\eta)}) \\hat{x}_0{(\\eta)})^{\\eta} = ((\\eta + \\sin{(\\eta)})^{2} \\hat{x}_0{(\\eta)})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))))"], [["times", 2, "Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\eta', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Mul(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))))"], [["times", 3, "Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\eta', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Integer(2)), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))))"], [["power", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\eta', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Integer(2)), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{B},v_{2})} = e^{\\frac{v_{2}}{\\mathbf{B}}}, then obtain (\\frac{\\partial}{\\partial \\mathbf{B}} \\log{(\\operatorname{C_{2}}{(\\mathbf{B},v_{2})})})^{v_{2}} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\log{(e^{\\frac{v_{2}}{\\mathbf{B}}})})^{v_{2}}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{B},v_{2})} = e^{\\frac{v_{2}}{\\mathbf{B}}} and \\log{(\\operatorname{C_{2}}{(\\mathbf{B},v_{2})})} = \\log{(e^{\\frac{v_{2}}{\\mathbf{B}}})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\log{(\\operatorname{C_{2}}{(\\mathbf{B},v_{2})})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\log{(e^{\\frac{v_{2}}{\\mathbf{B}}})} and (\\frac{\\partial}{\\partial \\mathbf{B}} \\log{(\\operatorname{C_{2}}{(\\mathbf{B},v_{2})})})^{v_{2}} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\log{(e^{\\frac{v_{2}}{\\mathbf{B}}})})^{v_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_2', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))))"], [["log", 1], "Equality(log(Function('C_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_2', commutative=True))), log(exp(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(log(Function('C_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(log(exp(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('v_2', commutative=True)"], "Equality(Pow(Derivative(log(Function('C_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('v_2', commutative=True)), Pow(Derivative(log(exp(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(s,\\rho)} = \\rho^{s} and z{(s,\\rho)} = \\int \\rho^{s} ds, then obtain z{(s,\\rho)} = \\int \\operatorname{x^{{\\}'}}{(s,\\rho)} ds", "derivation": "\\operatorname{x^{{\\}'}}{(s,\\rho)} = \\rho^{s} and \\int \\operatorname{x^{{\\}'}}{(s,\\rho)} ds = \\int \\rho^{s} ds and z{(s,\\rho)} = \\int \\rho^{s} ds and z{(s,\\rho)} = \\int \\operatorname{x^{{\\}'}}{(s,\\rho)} ds", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Symbol('\\\\rho', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Pow(Symbol('\\\\rho', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('z')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('x^\\\\prime')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(g)} = \\log{(\\cos{(g)})}, then obtain (- g + \\log{(\\cos{(g)})}) \\frac{d}{d g} - \\frac{\\hat{x}{(g)} + \\log{(\\cos{(g)})}}{\\log{(\\cos{(g)})}} = (- g + \\log{(\\cos{(g)})}) \\frac{d}{d g} (-2)", "derivation": "\\hat{x}{(g)} = \\log{(\\cos{(g)})} and \\hat{x}{(g)} + \\log{(\\cos{(g)})} = 2 \\log{(\\cos{(g)})} and - \\frac{\\hat{x}{(g)} + \\log{(\\cos{(g)})}}{\\log{(\\cos{(g)})}} = -2 and \\frac{d}{d g} - \\frac{\\hat{x}{(g)} + \\log{(\\cos{(g)})}}{\\log{(\\cos{(g)})}} = \\frac{d}{d g} (-2) and (- g + \\log{(\\cos{(g)})}) \\frac{d}{d g} - \\frac{\\hat{x}{(g)} + \\log{(\\cos{(g)})}}{\\log{(\\cos{(g)})}} = (- g + \\log{(\\cos{(g)})}) \\frac{d}{d g} (-2)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True))))"], [["add", 1, "log(cos(Symbol('g', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Mul(Integer(2), log(cos(Symbol('g', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), log(cos(Symbol('g', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\hat{x}')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Pow(log(cos(Symbol('g', commutative=True))), Integer(-1))), Integer(-2))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Function('\\\\hat{x}')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Pow(log(cos(Symbol('g', commutative=True))), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integer(-2), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Derivative(Mul(Integer(-1), Add(Function('\\\\hat{x}')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Pow(log(cos(Symbol('g', commutative=True))), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True)))), Derivative(Integer(-2), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hbar)} = e^{\\cos{(\\hbar)}} and \\operatorname{C_{1}}{(\\hbar)} = \\int \\frac{\\int 0 d\\hbar}{\\cos{(\\hbar)}} d\\hbar, then obtain \\operatorname{C_{1}}{(\\hbar)} = \\int \\frac{\\int (\\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}}) d\\hbar}{\\cos{(\\hbar)}} d\\hbar", "derivation": "\\operatorname{A_{y}}{(\\hbar)} = e^{\\cos{(\\hbar)}} and \\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}} = 0 and \\int (\\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}}) d\\hbar = \\int 0 d\\hbar and \\frac{\\int (\\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}}) d\\hbar}{\\cos{(\\hbar)}} = \\frac{\\int 0 d\\hbar}{\\cos{(\\hbar)}} and \\int \\frac{\\int (\\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}}) d\\hbar}{\\cos{(\\hbar)}} d\\hbar = \\int \\frac{\\int 0 d\\hbar}{\\cos{(\\hbar)}} d\\hbar and \\operatorname{C_{1}}{(\\hbar)} = \\int \\frac{\\int 0 d\\hbar}{\\cos{(\\hbar)}} d\\hbar and \\operatorname{C_{1}}{(\\hbar)} = \\int \\frac{\\int (\\operatorname{A_{y}}{(\\hbar)} - e^{\\cos{(\\hbar)}}) d\\hbar}{\\cos{(\\hbar)}} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hbar', commutative=True)), exp(cos(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "exp(cos(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('A_y')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\hbar', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Function('A_y')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\hbar', commutative=True))))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["divide", 3, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Add(Function('A_y')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\hbar', commutative=True))))), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Add(Function('A_y')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\hbar', commutative=True))))), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\hbar', commutative=True)), Integral(Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('C_1')(Symbol('\\\\hbar', commutative=True)), Integral(Mul(Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Add(Function('A_y')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\hbar', commutative=True))))), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{J})} = \\sin{(\\mathbf{J})}, then derive \\int \\hat{x}_0{(\\mathbf{J})} d\\mathbf{J} = \\dot{z} - \\cos{(\\mathbf{J})}, then obtain \\dot{z} - \\cos{(\\mathbf{J})} = \\mathbf{M} - \\cos{(\\mathbf{J})}", "derivation": "\\hat{x}_0{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\int \\hat{x}_0{(\\mathbf{J})} d\\mathbf{J} = \\int \\sin{(\\mathbf{J})} d\\mathbf{J} and \\int \\hat{x}_0{(\\mathbf{J})} d\\mathbf{J} = \\dot{z} - \\cos{(\\mathbf{J})} and \\dot{z} - \\cos{(\\mathbf{J})} = \\int \\sin{(\\mathbf{J})} d\\mathbf{J} and \\dot{z} - \\cos{(\\mathbf{J})} = \\mathbf{M} - \\cos{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Integral(sin(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)} = V_{\\mathbf{B}} + \\frac{\\mu}{r}, then obtain (- \\frac{1}{r})^{r} = (\\frac{- V_{\\mathbf{B}} - \\frac{\\mu}{r}}{r \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}})^{r}", "derivation": "\\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)} = V_{\\mathbf{B}} + \\frac{\\mu}{r} and 1 = \\frac{V_{\\mathbf{B}} + \\frac{\\mu}{r}}{\\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}} and \\frac{1}{r} = \\frac{V_{\\mathbf{B}} + \\frac{\\mu}{r}}{r \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}} and - \\frac{1}{r} = - \\frac{V_{\\mathbf{B}} + \\frac{\\mu}{r}}{r \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}} and - \\frac{1}{r} = \\frac{- V_{\\mathbf{B}} - \\frac{\\mu}{r}}{r \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}} and (- \\frac{1}{r})^{r} = (\\frac{- V_{\\mathbf{B}} - \\frac{\\mu}{r}}{r \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\mu,r)}})^{r}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))))"], [["divide", 1, "Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Pow(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["divide", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Symbol('r', commutative=True), Integer(-1)), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Pow(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Pow(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Pow(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('r', commutative=True)), Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Pow(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Integer(-1))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\omega{(r_{0})} = \\cos{(\\cos{(r_{0})})}, then obtain - \\omega{(r_{0})} + \\sin{(\\omega^{r_{0}}{(r_{0})})} = - \\omega{(r_{0})} + \\sin{(\\cos^{r_{0}}{(\\cos{(r_{0})})})}", "derivation": "\\omega{(r_{0})} = \\cos{(\\cos{(r_{0})})} and \\omega^{r_{0}}{(r_{0})} = \\cos^{r_{0}}{(\\cos{(r_{0})})} and \\sin{(\\omega^{r_{0}}{(r_{0})})} = \\sin{(\\cos^{r_{0}}{(\\cos{(r_{0})})})} and - \\omega{(r_{0})} + \\sin{(\\omega^{r_{0}}{(r_{0})})} = - \\omega{(r_{0})} + \\sin{(\\cos^{r_{0}}{(\\cos{(r_{0})})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('r_0', commutative=True)), cos(cos(Symbol('r_0', commutative=True))))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\omega')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), sin(Pow(cos(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))))"], [["minus", 3, "Function('\\\\omega')(Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('r_0', commutative=True))), sin(Pow(Function('\\\\omega')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('r_0', commutative=True))), sin(Pow(cos(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}, then derive \\rho{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f}, then obtain (- \\mathbf{J}_f + \\rho{(\\mathbf{J}_f)}) \\rho{(\\mathbf{J}_f)} = (- \\mathbf{J}_f + e^{\\mathbf{J}_f}) \\rho{(\\mathbf{J}_f)}", "derivation": "\\rho{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f} and - \\mathbf{J}_f + \\rho{(\\mathbf{J}_f)} = - \\mathbf{J}_f + \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f} and (- \\mathbf{J}_f + \\rho{(\\mathbf{J}_f)}) \\rho{(\\mathbf{J}_f)} = (- \\mathbf{J}_f + \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}) \\rho{(\\mathbf{J}_f)} and \\rho{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and e^{\\mathbf{J}_f} = \\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f} and (- \\mathbf{J}_f + \\rho{(\\mathbf{J}_f)}) \\rho{(\\mathbf{J}_f)} = (- \\mathbf{J}_f + e^{\\mathbf{J}_f}) \\rho{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["times", 2, "Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\rho')(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given L{(\\delta)} = \\cos{(\\delta)}, then obtain 1 = \\frac{- \\delta + \\frac{d}{d \\delta} \\cos{(\\delta)}}{- \\delta + \\frac{d}{d \\delta} L{(\\delta)}}", "derivation": "L{(\\delta)} = \\cos{(\\delta)} and \\frac{d}{d \\delta} L{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and - \\delta + \\frac{d}{d \\delta} L{(\\delta)} = - \\delta + \\frac{d}{d \\delta} \\cos{(\\delta)} and 1 = \\frac{- \\delta + \\frac{d}{d \\delta} \\cos{(\\delta)}}{- \\delta + \\frac{d}{d \\delta} L{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Function('L')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Function('L')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Function('L')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given J{(\\hat{x},f_{E})} = \\sin{(\\hat{x} - f_{E})}, then obtain J{(\\hat{x},f_{E})} \\int \\hat{x} J{(\\hat{x},f_{E})} df_{E} = J{(\\hat{x},f_{E})} \\int \\hat{x} \\sin{(\\hat{x} - f_{E})} df_{E}", "derivation": "J{(\\hat{x},f_{E})} = \\sin{(\\hat{x} - f_{E})} and \\hat{x} J{(\\hat{x},f_{E})} = \\hat{x} \\sin{(\\hat{x} - f_{E})} and \\int \\hat{x} J{(\\hat{x},f_{E})} df_{E} = \\int \\hat{x} \\sin{(\\hat{x} - f_{E})} df_{E} and J{(\\hat{x},f_{E})} \\int \\hat{x} J{(\\hat{x},f_{E})} df_{E} = J{(\\hat{x},f_{E})} \\int \\hat{x} \\sin{(\\hat{x} - f_{E})} df_{E}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True)), sin(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)))))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), sin(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), sin(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True))))"], [["times", 3, "Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True)))), Mul(Function('J')(Symbol('\\\\hat{x}', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), sin(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))), Tuple(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(M,U)} = - M + e^{U}, then obtain - \\frac{\\frac{\\partial}{\\partial M} \\varepsilon^{M}{(M,U)}}{M} = - \\frac{\\frac{\\partial}{\\partial M} (- M + e^{U})^{M}}{M}", "derivation": "\\varepsilon{(M,U)} = - M + e^{U} and \\varepsilon^{M}{(M,U)} = (- M + e^{U})^{M} and \\frac{\\partial}{\\partial M} \\varepsilon^{M}{(M,U)} = \\frac{\\partial}{\\partial M} (- M + e^{U})^{M} and - \\frac{\\frac{\\partial}{\\partial M} \\varepsilon^{M}{(M,U)}}{M} = - \\frac{\\frac{\\partial}{\\partial M} (- M + e^{U})^{M}}{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('M', commutative=True), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), exp(Symbol('U', commutative=True))))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('M', commutative=True), Symbol('U', commutative=True)), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), exp(Symbol('U', commutative=True))), Symbol('M', commutative=True)))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\varepsilon')(Symbol('M', commutative=True), Symbol('U', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), exp(Symbol('U', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), Symbol('M', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(Pow(Function('\\\\varepsilon')(Symbol('M', commutative=True), Symbol('U', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), exp(Symbol('U', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = \\frac{V_{\\mathbf{B}}}{\\phi}, then obtain V_{\\mathbf{B}} \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} - 2 \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = V_{\\mathbf{B}} \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} - \\frac{V_{\\mathbf{B}}}{\\phi} - \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)}", "derivation": "\\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = \\frac{V_{\\mathbf{B}}}{\\phi} and - \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = - \\frac{V_{\\mathbf{B}}}{\\phi} and - 2 \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = - \\frac{V_{\\mathbf{B}}}{\\phi} - \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} and V_{\\mathbf{B}} \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} - 2 \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} = V_{\\mathbf{B}} \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)} - \\frac{V_{\\mathbf{B}}}{\\phi} - \\operatorname{t_{1}}{(V_{\\mathbf{B}},\\phi)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], [["add", 2, "Mul(Integer(-1), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(g,E_{n})} = E_{n} \\sin{(g)} and \\pi{(g,E_{n})} = E_{n} \\sin{(g)}, then obtain \\cos{(\\sin{(\\hat{H}_{\\lambda}{(g,E_{n})})})} - \\frac{1}{\\omega} = \\cos{(\\sin{(\\pi{(g,E_{n})})})} - \\frac{1}{\\omega}", "derivation": "\\hat{H}_{\\lambda}{(g,E_{n})} = E_{n} \\sin{(g)} and \\pi{(g,E_{n})} = E_{n} \\sin{(g)} and \\hat{H}_{\\lambda}{(g,E_{n})} = \\pi{(g,E_{n})} and \\sin{(\\hat{H}_{\\lambda}{(g,E_{n})})} = \\sin{(\\pi{(g,E_{n})})} and \\sin{(E_{n} \\sin{(g)})} = \\sin{(\\pi{(g,E_{n})})} and \\cos{(\\sin{(E_{n} \\sin{(g)})})} = \\cos{(\\sin{(\\pi{(g,E_{n})})})} and \\cos{(\\sin{(E_{n} \\sin{(g)})})} - \\frac{1}{\\omega} = \\cos{(\\sin{(\\pi{(g,E_{n})})})} - \\frac{1}{\\omega} and \\cos{(\\sin{(\\hat{H}_{\\lambda}{(g,E_{n})})})} - \\frac{1}{\\omega} = \\cos{(\\sin{(\\pi{(g,E_{n})})})} - \\frac{1}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), sin(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), sin(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)))"], [["sin", 3], "Equality(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('E_n', commutative=True))), sin(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(sin(Mul(Symbol('E_n', commutative=True), sin(Symbol('g', commutative=True)))), sin(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True))))"], [["cos", 5], "Equality(cos(sin(Mul(Symbol('E_n', commutative=True), sin(Symbol('g', commutative=True))))), cos(sin(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)))))"], [["minus", 6, "Pow(Symbol('\\\\omega', commutative=True), Integer(-1))"], "Equality(Add(cos(sin(Mul(Symbol('E_n', commutative=True), sin(Symbol('g', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Add(cos(sin(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(cos(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Add(cos(sin(Function('\\\\pi')(Symbol('g', commutative=True), Symbol('E_n', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given S{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain \\int \\tilde{\\infty}^{\\mathbf{J}} ((S{(\\mathbf{J})} - e^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} d\\mathbf{J} = \\int \\tilde{\\infty}^{\\mathbf{J}} (0^{\\mathbf{J}})^{\\mathbf{J}} d\\mathbf{J}", "derivation": "S{(\\mathbf{J})} = e^{\\mathbf{J}} and S{(\\mathbf{J})} - e^{\\mathbf{J}} = 0 and (S{(\\mathbf{J})} - e^{\\mathbf{J}})^{\\mathbf{J}} = 0^{\\mathbf{J}} and ((S{(\\mathbf{J})} - e^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} = (0^{\\mathbf{J}})^{\\mathbf{J}} and \\tilde{\\infty}^{\\mathbf{J}} ((S{(\\mathbf{J})} - e^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} = \\tilde{\\infty}^{\\mathbf{J}} (0^{\\mathbf{J}})^{\\mathbf{J}} and \\int \\tilde{\\infty}^{\\mathbf{J}} ((S{(\\mathbf{J})} - e^{\\mathbf{J}})^{\\mathbf{J}})^{\\mathbf{J}} d\\mathbf{J} = \\int \\tilde{\\infty}^{\\mathbf{J}} (0^{\\mathbf{J}})^{\\mathbf{J}} d\\mathbf{J}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Add(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Pow(Add(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 4, "Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Add(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(zoo, Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Mul(Pow(zoo, Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Add(Function('S')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Pow(zoo, Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given n{(\\mathbf{D},i)} = \\frac{\\mathbf{D}}{i} and \\operatorname{r_{0}}{(i)} = \\frac{1}{i}, then obtain 2 \\operatorname{r_{0}}{(i)} - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} = \\operatorname{r_{0}}{(i)} - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} + \\frac{1}{i}", "derivation": "n{(\\mathbf{D},i)} = \\frac{\\mathbf{D}}{i} and i n{(\\mathbf{D},i)} = \\mathbf{D} and \\operatorname{r_{0}}{(i)} = \\frac{1}{i} and \\operatorname{r_{0}}{(i)} - \\int n{(\\mathbf{D},i)} d\\mathbf{D} = - \\int n{(\\mathbf{D},i)} d\\mathbf{D} + \\frac{1}{i} and \\operatorname{r_{0}}{(i)} - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} = - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} + \\frac{1}{i} and 2 \\operatorname{r_{0}}{(i)} - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} = \\operatorname{r_{0}}{(i)} - \\int\\limits^{i n{(\\mathbf{D},i)}} n{(\\mathbf{D},i)} d\\mathbf{D} + \\frac{1}{i}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))"], [["minus", 3, "Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('r_0')(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('r_0')(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('i', commutative=True), Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True))))))), Add(Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('i', commutative=True), Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)))))), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["add", 5, "Function('r_0')(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('r_0')(Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('i', commutative=True), Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True))))))), Add(Function('r_0')(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('i', commutative=True), Function('n')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)))))), Pow(Symbol('i', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(b,L_{\\varepsilon})} = \\frac{b}{L_{\\varepsilon}}, then obtain \\frac{d}{d L_{\\varepsilon}} L_{\\varepsilon} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} - \\mathbf{D}{(b,L_{\\varepsilon})} + \\frac{b}{L_{\\varepsilon}})", "derivation": "\\mathbf{D}{(b,L_{\\varepsilon})} = \\frac{b}{L_{\\varepsilon}} and L_{\\varepsilon} + \\mathbf{D}{(b,L_{\\varepsilon})} = L_{\\varepsilon} + \\frac{b}{L_{\\varepsilon}} and L_{\\varepsilon} = L_{\\varepsilon} - \\mathbf{D}{(b,L_{\\varepsilon})} + \\frac{b}{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} L_{\\varepsilon} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} - \\mathbf{D}{(b,L_{\\varepsilon})} + \\frac{b}{L_{\\varepsilon}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))"], [["add", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{D}')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{D}')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True))))"], [["differentiate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Symbol('L_{\\\\varepsilon}', commutative=True), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(S,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{S}, then derive T{(S,\\mathbf{J}_M)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} T{(S,\\mathbf{J}_M)} - \\frac{1}{S} = T{(S,\\mathbf{J}_M)}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{S} + \\frac{\\mathbf{J}_M}{S} - \\frac{1}{S} = \\frac{\\mathbf{J}_M}{S}", "derivation": "T{(S,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{S} and T{(S,\\mathbf{J}_M)} - \\frac{\\mathbf{J}_M}{S} = 0 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (T{(S,\\mathbf{J}_M)} - \\frac{\\mathbf{J}_M}{S}) = \\frac{d}{d \\mathbf{J}_M} 0 and T{(S,\\mathbf{J}_M)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} (T{(S,\\mathbf{J}_M)} - \\frac{\\mathbf{J}_M}{S}) = T{(S,\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} 0 and T{(S,\\mathbf{J}_M)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} T{(S,\\mathbf{J}_M)} - \\frac{1}{S} = T{(S,\\mathbf{J}_M)} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{S} + \\frac{\\mathbf{J}_M}{S} - \\frac{1}{S} = \\frac{\\mathbf{J}_M}{S}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["add", 3, "Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)))), Function('T')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(I,J)} = I + J, then obtain - \\mathbf{J}_M \\mathbf{p} + \\frac{\\partial}{\\partial J} (I I^{J} + I^{J} (- J + \\mathbf{D}{(I,J)})) = - \\mathbf{J}_M \\mathbf{p} + \\frac{\\partial}{\\partial J} 2 I I^{J}", "derivation": "\\mathbf{D}{(I,J)} = I + J and - J + \\mathbf{D}{(I,J)} = I and (- J + \\mathbf{D}{(I,J)})^{J} = I^{J} and (- J + \\mathbf{D}{(I,J)}) (- J + \\mathbf{D}{(I,J)})^{J} = I (- J + \\mathbf{D}{(I,J)})^{J} and I^{J} (- J + \\mathbf{D}{(I,J)}) = I I^{J} and I I^{J} + I^{J} (- J + \\mathbf{D}{(I,J)}) = 2 I I^{J} and \\frac{\\partial}{\\partial J} (I I^{J} + I^{J} (- J + \\mathbf{D}{(I,J)})) = \\frac{\\partial}{\\partial J} 2 I I^{J} and - \\mathbf{J}_M \\mathbf{p} + \\frac{\\partial}{\\partial J} (I I^{J} + I^{J} (- J + \\mathbf{D}{(I,J)})) = - \\mathbf{J}_M \\mathbf{p} + \\frac{\\partial}{\\partial J} 2 I I^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True)), Add(Symbol('I', commutative=True), Symbol('J', commutative=True)))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Symbol('I', commutative=True))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)))"], [["times", 2, "Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Symbol('I', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True)))), Mul(Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))))"], [["add", 5, "Mul(Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)))"], "Equality(Add(Mul(Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))))), Mul(Integer(2), Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))))"], [["differentiate", 6, "Symbol('J', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 7, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Add(Mul(Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('J', commutative=True))))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Mul(Integer(2), Symbol('I', commutative=True), Pow(Symbol('I', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{x})} = \\int e^{\\hat{x}} d\\hat{x}, then obtain (\\hat{x} + \\phi_{1}{(\\hat{x})} - e^{\\hat{x}})^{\\hat{x}} = (S + \\hat{x})^{\\hat{x}}", "derivation": "\\phi_{1}{(\\hat{x})} = \\int e^{\\hat{x}} d\\hat{x} and \\hat{x} + \\phi_{1}{(\\hat{x})} = \\hat{x} + \\int e^{\\hat{x}} d\\hat{x} and \\hat{x} + \\phi_{1}{(\\hat{x})} - e^{\\hat{x}} = \\hat{x} - e^{\\hat{x}} + \\int e^{\\hat{x}} d\\hat{x} and (\\hat{x} + \\phi_{1}{(\\hat{x})} - e^{\\hat{x}})^{\\hat{x}} = (\\hat{x} - e^{\\hat{x}} + \\int e^{\\hat{x}} d\\hat{x})^{\\hat{x}} and (\\hat{x} + \\phi_{1}{(\\hat{x})} - e^{\\hat{x}})^{\\hat{x}} = (S + \\hat{x})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True)), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["minus", 2, "exp(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then derive \\int \\mathbf{B}{(\\mathbf{E})} d\\mathbf{E} = I + \\mathbf{E} \\log{(\\mathbf{E})} - \\mathbf{E}, then obtain \\iint \\mathbf{B}{(\\mathbf{E})} d\\mathbf{E} d\\mathbf{E} = \\int (I + \\mathbf{E} \\log{(\\mathbf{E})} - \\mathbf{E}) d\\mathbf{E}", "derivation": "\\mathbf{B}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and \\int \\mathbf{B}{(\\mathbf{E})} d\\mathbf{E} = \\int \\log{(\\mathbf{E})} d\\mathbf{E} and \\int \\mathbf{B}{(\\mathbf{E})} d\\mathbf{E} = I + \\mathbf{E} \\log{(\\mathbf{E})} - \\mathbf{E} and \\iint \\mathbf{B}{(\\mathbf{E})} d\\mathbf{E} d\\mathbf{E} = \\int (I + \\mathbf{E} \\log{(\\mathbf{E})} - \\mathbf{E}) d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(log(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('I', commutative=True), Mul(Symbol('\\\\mathbf{E}', commutative=True), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('I', commutative=True), Mul(Symbol('\\\\mathbf{E}', commutative=True), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\Omega)} = \\sin{(\\Omega)} and \\phi_{2}{(\\Omega)} = \\int \\dot{x}{(\\Omega)} d\\Omega, then obtain \\cos{(\\frac{d}{d \\Omega} \\phi_{2}{(\\Omega)})} = \\cos{(\\frac{\\partial}{\\partial \\Omega} (I - \\cos{(\\Omega)}))}", "derivation": "\\dot{x}{(\\Omega)} = \\sin{(\\Omega)} and \\int \\dot{x}{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\frac{d}{d \\Omega} \\int \\dot{x}{(\\Omega)} d\\Omega = \\frac{d}{d \\Omega} \\int \\sin{(\\Omega)} d\\Omega and \\phi_{2}{(\\Omega)} = \\int \\dot{x}{(\\Omega)} d\\Omega and \\phi_{2}{(\\Omega)} = \\int \\sin{(\\Omega)} d\\Omega and \\frac{d}{d \\Omega} \\int \\dot{x}{(\\Omega)} d\\Omega = \\frac{d}{d \\Omega} \\phi_{2}{(\\Omega)} and \\cos{(\\frac{d}{d \\Omega} \\int \\dot{x}{(\\Omega)} d\\Omega)} = \\cos{(\\frac{d}{d \\Omega} \\int \\sin{(\\Omega)} d\\Omega)} and \\cos{(\\frac{d}{d \\Omega} \\phi_{2}{(\\Omega)})} = \\cos{(\\frac{d}{d \\Omega} \\int \\sin{(\\Omega)} d\\Omega)} and \\cos{(\\frac{d}{d \\Omega} \\phi_{2}{(\\Omega)})} = \\cos{(\\frac{\\partial}{\\partial \\Omega} (I - \\cos{(\\Omega)}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\Omega', commutative=True)), Integral(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\phi_2')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Integral(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Function('\\\\phi_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Integral(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), cos(Derivative(Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(cos(Derivative(Function('\\\\phi_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), cos(Derivative(Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_integrals", 8], "Equality(cos(Derivative(Function('\\\\phi_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), cos(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{z}{(\\delta,Z)} = - Z + \\delta, then obtain ((Z - \\delta + \\dot{z}{(\\delta,Z)}) \\dot{z}{(\\delta,Z)})^{\\delta} = 0^{\\delta}", "derivation": "\\dot{z}{(\\delta,Z)} = - Z + \\delta and Z - \\delta + \\dot{z}{(\\delta,Z)} = 0 and (Z - \\delta + \\dot{z}{(\\delta,Z)}) \\dot{z}{(\\delta,Z)} = 0 and ((Z - \\delta + \\dot{z}{(\\delta,Z)}) \\dot{z}{(\\delta,Z)})^{\\delta} = 0^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))), Integer(0))"], [["times", 2, "Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))), Integer(0))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integer(0), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{D})} = \\log{(\\mathbf{D})} and \\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} = \\tilde{g}^* - g_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} - \\log{(\\mathbf{D})}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\tilde{g}^* - g_{\\varepsilon} - \\log{(\\mathbf{D})})", "derivation": "\\theta_{1}{(\\mathbf{D})} = \\log{(\\mathbf{D})} and \\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} = \\tilde{g}^* - g_{\\varepsilon} and - \\theta_{1}{(\\mathbf{D})} + \\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} = \\tilde{g}^* - g_{\\varepsilon} - \\theta_{1}{(\\mathbf{D})} and \\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} - \\log{(\\mathbf{D})} = \\tilde{g}^* - g_{\\varepsilon} - \\log{(\\mathbf{D})} and \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\operatorname{v_{y}}{(g_{\\varepsilon},\\tilde{g}^*)} - \\log{(\\mathbf{D})}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\tilde{g}^* - g_{\\varepsilon} - \\log{(\\mathbf{D})})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True)))"], ["get_premise", "Equality(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True))), Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["differentiate", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger}, then obtain 1 = \\Psi^{\\dagger} - \\mathbf{p}{(\\Psi^{\\dagger})} + 1", "derivation": "\\mathbf{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} and 2 \\mathbf{p}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\mathbf{p}{(\\Psi^{\\dagger})} and 0 = \\Psi^{\\dagger} - \\mathbf{p}{(\\Psi^{\\dagger})} and 1 = \\Psi^{\\dagger} - \\mathbf{p}{(\\Psi^{\\dagger})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(1), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\omega{(C_{1})} = e^{C_{1}}, then obtain \\int \\frac{\\int \\frac{d}{d C_{1}} \\omega{(C_{1})} dC_{1}}{\\omega{(C_{1})}} dC_{1} = \\int \\frac{\\int \\frac{d}{d C_{1}} e^{C_{1}} dC_{1}}{\\omega{(C_{1})}} dC_{1}", "derivation": "\\omega{(C_{1})} = e^{C_{1}} and \\frac{d}{d C_{1}} \\omega{(C_{1})} = \\frac{d}{d C_{1}} e^{C_{1}} and \\int \\frac{d}{d C_{1}} \\omega{(C_{1})} dC_{1} = \\int \\frac{d}{d C_{1}} e^{C_{1}} dC_{1} and e^{- C_{1}} \\int \\frac{d}{d C_{1}} \\omega{(C_{1})} dC_{1} = e^{- C_{1}} \\int \\frac{d}{d C_{1}} e^{C_{1}} dC_{1} and \\frac{\\int \\frac{d}{d C_{1}} \\omega{(C_{1})} dC_{1}}{\\omega{(C_{1})}} = \\frac{\\int \\frac{d}{d C_{1}} e^{C_{1}} dC_{1}}{\\omega{(C_{1})}} and \\int \\frac{\\int \\frac{d}{d C_{1}} \\omega{(C_{1})} dC_{1}}{\\omega{(C_{1})}} dC_{1} = \\int \\frac{\\int \\frac{d}{d C_{1}} e^{C_{1}} dC_{1}}{\\omega{(C_{1})}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))))"], [["divide", 3, "exp(Symbol('C_1', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('C_1', commutative=True))), Integral(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('C_1', commutative=True))), Integral(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\omega')(Symbol('C_1', commutative=True)), Integer(-1)), Integral(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Mul(Pow(Function('\\\\omega')(Symbol('C_1', commutative=True)), Integer(-1)), Integral(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\omega')(Symbol('C_1', commutative=True)), Integer(-1)), Integral(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Pow(Function('\\\\omega')(Symbol('C_1', commutative=True)), Integer(-1)), Integral(Derivative(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{f_{\\mathbf{p}} + r_{0}}{f^{\\prime}}, then derive \\frac{\\partial}{\\partial r_{0}} \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{1}{f^{\\prime}}, then obtain \\frac{\\partial^{2}}{\\partial r_{0}^{2}} \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{d}{d r_{0}} \\frac{1}{f^{\\prime}}", "derivation": "\\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{f_{\\mathbf{p}} + r_{0}}{f^{\\prime}} and \\frac{\\partial}{\\partial r_{0}} \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{\\partial}{\\partial r_{0}} \\frac{f_{\\mathbf{p}} + r_{0}}{f^{\\prime}} and \\frac{\\partial}{\\partial r_{0}} \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{1}{f^{\\prime}} and \\frac{\\partial^{2}}{\\partial r_{0}^{2}} \\operatorname{F_{N}}{(f_{\\mathbf{p}},r_{0},f^{\\prime})} = \\frac{d}{d r_{0}} \\frac{1}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(2))), Derivative(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\cos{(v_{1})}, then obtain \\int \\frac{d^{2}}{d v_{1}^{2}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})} dv_{1} = \\int \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} dv_{1}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\cos{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and \\frac{d^{2}}{d v_{1}^{2}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})} = \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} and \\int \\frac{d^{2}}{d v_{1}^{2}} \\operatorname{V_{\\mathbf{E}}}{(v_{1})} dv_{1} = \\int \\frac{d^{2}}{d v_{1}^{2}} \\cos{(v_{1})} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Tuple(Symbol('v_1', commutative=True))), Integral(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(2))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given s{(\\theta_2,l)} = \\frac{\\log{(\\theta_2)}}{l} and a{(\\theta_2)} = \\log{(\\theta_2)}, then obtain (s{(\\theta_2,l)} + 1)^{\\theta_2} = (1 + \\frac{a{(\\theta_2)}}{l})^{\\theta_2}", "derivation": "s{(\\theta_2,l)} = \\frac{\\log{(\\theta_2)}}{l} and a{(\\theta_2)} = \\log{(\\theta_2)} and s{(\\theta_2,l)} = \\frac{a{(\\theta_2)}}{l} and \\frac{a{(\\theta_2)}}{l} = \\frac{\\log{(\\theta_2)}}{l} and 1 + \\frac{a{(\\theta_2)}}{l} = 1 + \\frac{\\log{(\\theta_2)}}{l} and s{(\\theta_2,l)} + 1 = 1 + \\frac{\\log{(\\theta_2)}}{l} and s{(\\theta_2,l)} + 1 = 1 + \\frac{a{(\\theta_2)}}{l} and (s{(\\theta_2,l)} + 1)^{\\theta_2} = (1 + \\frac{a{(\\theta_2)}}{l})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('s')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('a')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('a')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True))))"], [["add", 4, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('a')(Symbol('\\\\theta_2', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('s')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('s')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('a')(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 7, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Add(Function('s')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Integer(1)), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('a')(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(x,\\mathbf{S})} = \\int \\mathbf{S}^{x} d\\mathbf{S} and \\hat{x}_0{(x,\\mathbf{S})} = (\\int \\mathbf{S}^{x} d\\mathbf{S})^{\\mathbf{S}}, then obtain \\operatorname{F_{c}}^{\\mathbf{S}}{(x,\\mathbf{S})} = \\hat{x}_0{(x,\\mathbf{S})}", "derivation": "\\operatorname{F_{c}}{(x,\\mathbf{S})} = \\int \\mathbf{S}^{x} d\\mathbf{S} and \\operatorname{F_{c}}^{\\mathbf{S}}{(x,\\mathbf{S})} = (\\int \\mathbf{S}^{x} d\\mathbf{S})^{\\mathbf{S}} and \\hat{x}_0{(x,\\mathbf{S})} = (\\int \\mathbf{S}^{x} d\\mathbf{S})^{\\mathbf{S}} and \\operatorname{F_{c}}^{\\mathbf{S}}{(x,\\mathbf{S})} = \\hat{x}_0{(x,\\mathbf{S})}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('F_c')(Symbol('x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(P_{e})} = \\sin{(\\log{(P_{e})})}, then obtain \\operatorname{n_{1}}{(P_{e})} + \\int \\operatorname{n_{1}}{(P_{e})} dP_{e} = \\frac{P_{e} \\sin{(\\log{(P_{e})})}}{2} - \\frac{P_{e} \\cos{(\\log{(P_{e})})}}{2} + \\hbar + \\operatorname{n_{1}}{(P_{e})}", "derivation": "\\operatorname{n_{1}}{(P_{e})} = \\sin{(\\log{(P_{e})})} and \\int \\operatorname{n_{1}}{(P_{e})} dP_{e} = \\int \\sin{(\\log{(P_{e})})} dP_{e} and \\operatorname{n_{1}}{(P_{e})} + \\int \\operatorname{n_{1}}{(P_{e})} dP_{e} = \\operatorname{n_{1}}{(P_{e})} + \\int \\sin{(\\log{(P_{e})})} dP_{e} and \\operatorname{n_{1}}{(P_{e})} + \\int \\operatorname{n_{1}}{(P_{e})} dP_{e} = \\frac{P_{e} \\sin{(\\log{(P_{e})})}}{2} - \\frac{P_{e} \\cos{(\\log{(P_{e})})}}{2} + \\hbar + \\operatorname{n_{1}}{(P_{e})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('P_e', commutative=True)), sin(log(Symbol('P_e', commutative=True))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(sin(log(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["add", 2, "Function('n_1')(Symbol('P_e', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('P_e', commutative=True)), Integral(Function('n_1')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Function('n_1')(Symbol('P_e', commutative=True)), Integral(sin(log(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('n_1')(Symbol('P_e', commutative=True)), Integral(Function('n_1')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('P_e', commutative=True), sin(log(Symbol('P_e', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('P_e', commutative=True), cos(log(Symbol('P_e', commutative=True)))), Symbol('\\\\hbar', commutative=True), Function('n_1')(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(V)} = \\cos{(\\cos{(V)})}, then obtain \\operatorname{E_{n}}{(V)} + \\cos{(V)} + \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1 = \\cos{(V)} + 2 \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1", "derivation": "\\operatorname{E_{n}}{(V)} = \\cos{(\\cos{(V)})} and \\frac{d}{d V} \\operatorname{E_{n}}{(V)} = \\frac{d}{d V} \\cos{(\\cos{(V)})} and \\operatorname{E_{n}}{(V)} + 1 = \\cos{(\\cos{(V)})} + 1 and \\operatorname{E_{n}}{(V)} - \\frac{d}{d V} \\cos{(\\cos{(V)})} + 1 = \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\cos{(\\cos{(V)})} + 1 and \\operatorname{E_{n}}{(V)} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1 = \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1 and \\operatorname{E_{n}}{(V)} + \\cos{(V)} + \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1 = \\cos{(V)} + 2 \\cos{(\\cos{(V)})} - \\frac{d}{d V} \\operatorname{E_{n}}{(V)} + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('E_n')(Symbol('V', commutative=True)), Integer(1)), Add(cos(cos(Symbol('V', commutative=True))), Integer(1)))"], [["minus", 3, "Derivative(cos(cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))"], "Equality(Add(Function('E_n')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)), Add(cos(cos(Symbol('V', commutative=True))), Mul(Integer(-1), Derivative(cos(cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('E_n')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(Function('E_n')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)), Add(cos(cos(Symbol('V', commutative=True))), Mul(Integer(-1), Derivative(Function('E_n')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)))"], [["add", 5, "Add(cos(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))))"], "Equality(Add(Function('E_n')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)), cos(cos(Symbol('V', commutative=True))), Mul(Integer(-1), Derivative(Function('E_n')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)), Add(cos(Symbol('V', commutative=True)), Mul(Integer(2), cos(cos(Symbol('V', commutative=True)))), Mul(Integer(-1), Derivative(Function('E_n')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega} and \\mathbf{S}{(\\mathbf{H},\\Omega)} = \\int (\\frac{\\mathbf{H}}{\\Omega})^{\\Omega} d\\mathbf{H}, then obtain \\mathbf{S}{(\\mathbf{H},\\Omega)} = \\int \\phi_{1}^{\\Omega}{(\\mathbf{H},\\Omega)} d\\mathbf{H}", "derivation": "\\phi_{1}{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega} and \\phi_{1}^{\\Omega}{(\\mathbf{H},\\Omega)} = (\\frac{\\mathbf{H}}{\\Omega})^{\\Omega} and \\int \\phi_{1}^{\\Omega}{(\\mathbf{H},\\Omega)} d\\mathbf{H} = \\int (\\frac{\\mathbf{H}}{\\Omega})^{\\Omega} d\\mathbf{H} and \\mathbf{S}{(\\mathbf{H},\\Omega)} = \\int (\\frac{\\mathbf{H}}{\\Omega})^{\\Omega} d\\mathbf{H} and \\mathbf{S}{(\\mathbf{H},\\Omega)} = \\int \\phi_{1}^{\\Omega}{(\\mathbf{H},\\Omega)} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{A})} = \\sin{(\\log{(\\mathbf{A})})} and \\operatorname{E_{x}}{(f)} = \\cos{(f)} and \\operatorname{F_{g}}{(f)} = \\cos{(f)}, then obtain \\operatorname{F_{g}}{(f)} - \\sin^{2}{(\\log{(\\mathbf{A})})} = - \\sin^{2}{(\\log{(\\mathbf{A})})} + \\cos{(f)}", "derivation": "\\dot{x}{(\\mathbf{A})} = \\sin{(\\log{(\\mathbf{A})})} and \\operatorname{E_{x}}{(f)} = \\cos{(f)} and \\operatorname{E_{x}}{(f)} - \\dot{x}{(\\mathbf{A})} \\sin{(\\log{(\\mathbf{A})})} = - \\dot{x}{(\\mathbf{A})} \\sin{(\\log{(\\mathbf{A})})} + \\cos{(f)} and \\operatorname{E_{x}}{(f)} - \\dot{x}^{2}{(\\mathbf{A})} = - \\dot{x}^{2}{(\\mathbf{A})} + \\cos{(f)} and \\operatorname{E_{x}}{(f)} - \\sin^{2}{(\\log{(\\mathbf{A})})} = - \\sin^{2}{(\\log{(\\mathbf{A})})} + \\cos{(f)} and \\operatorname{F_{g}}{(f)} = \\cos{(f)} and \\operatorname{E_{x}}{(f)} = \\operatorname{F_{g}}{(f)} and \\operatorname{F_{g}}{(f)} - \\sin^{2}{(\\log{(\\mathbf{A})})} = - \\sin^{2}{(\\log{(\\mathbf{A})})} + \\cos{(f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(log(Symbol('\\\\mathbf{A}', commutative=True))))"], ["get_premise", "Equality(Function('E_x')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["minus", 2, "Mul(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(log(Symbol('\\\\mathbf{A}', commutative=True))))"], "Equality(Add(Function('E_x')(Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(log(Symbol('\\\\mathbf{A}', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(log(Symbol('\\\\mathbf{A}', commutative=True)))), cos(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('E_x')(Symbol('f', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), cos(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('E_x')(Symbol('f', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))), Add(Mul(Integer(-1), Pow(sin(log(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2))), cos(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Function('E_x')(Symbol('f', commutative=True)), Function('F_g')(Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Function('F_g')(Symbol('f', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))), Add(Mul(Integer(-1), Pow(sin(log(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2))), cos(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(E,m)} = E - m, then obtain (\\operatorname{E_{\\lambda}}^{m}{(E,m)})^{E} + 1 = ((E - m)^{m})^{E} + 1", "derivation": "\\operatorname{E_{\\lambda}}{(E,m)} = E - m and \\operatorname{E_{\\lambda}}^{m}{(E,m)} = (E - m)^{m} and (\\operatorname{E_{\\lambda}}^{m}{(E,m)})^{E} = ((E - m)^{m})^{E} and (\\operatorname{E_{\\lambda}}^{m}{(E,m)})^{E} + 1 = ((E - m)^{m})^{E} + 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('m', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Pow(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('E', commutative=True)), Pow(Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('E', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Pow(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('E', commutative=True)), Integer(1)), Add(Pow(Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('E', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\theta_{1}{(z^{*})} = e^{z^{*}} and \\operatorname{r_{0}}{(z^{*})} = \\frac{1}{z^{*}}, then obtain \\frac{d}{d z^{*}} (\\operatorname{r_{0}}{(z^{*})} e^{- z^{*}})^{z^{*}} = \\frac{d}{d z^{*}} (\\frac{e^{- z^{*}}}{z^{*}})^{z^{*}}", "derivation": "\\theta_{1}{(z^{*})} = e^{z^{*}} and \\operatorname{r_{0}}{(z^{*})} = \\frac{1}{z^{*}} and \\frac{\\operatorname{r_{0}}{(z^{*})}}{\\theta_{1}{(z^{*})}} = \\frac{1}{z^{*} \\theta_{1}{(z^{*})}} and (\\frac{\\operatorname{r_{0}}{(z^{*})}}{\\theta_{1}{(z^{*})}})^{z^{*}} = (\\frac{1}{z^{*} \\theta_{1}{(z^{*})}})^{z^{*}} and (\\operatorname{r_{0}}{(z^{*})} e^{- z^{*}})^{z^{*}} = (\\frac{e^{- z^{*}}}{z^{*}})^{z^{*}} and \\frac{d}{d z^{*}} (\\operatorname{r_{0}}{(z^{*})} e^{- z^{*}})^{z^{*}} = \\frac{d}{d z^{*}} (\\frac{e^{- z^{*}}}{z^{*}})^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('z^*', commutative=True)), exp(Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('z^*', commutative=True)), Pow(Symbol('z^*', commutative=True), Integer(-1)))"], [["divide", 2, "Function('\\\\theta_1')(Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('z^*', commutative=True)), Integer(-1)), Function('r_0')(Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('z^*', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('z^*', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\theta_1')(Symbol('z^*', commutative=True)), Integer(-1)), Function('r_0')(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Pow(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('z^*', commutative=True)), Integer(-1))), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(Function('r_0')(Symbol('z^*', commutative=True)), exp(Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Pow(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('r_0')(Symbol('z^*', commutative=True)), exp(Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('z^*', commutative=True)))), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(t_{1},\\theta)} = \\cos{(\\theta + t_{1})}, then derive \\int f{(t_{1},\\theta)} dt_{1} = u + \\sin{(\\theta + t_{1})}, then obtain \\int \\cos{(\\theta + t_{1})} dt_{1} = u + \\sin{(\\theta + t_{1})}", "derivation": "f{(t_{1},\\theta)} = \\cos{(\\theta + t_{1})} and \\int f{(t_{1},\\theta)} dt_{1} = \\int \\cos{(\\theta + t_{1})} dt_{1} and \\int f{(t_{1},\\theta)} dt_{1} = u + \\sin{(\\theta + t_{1})} and \\int \\cos{(\\theta + t_{1})} dt_{1} = u + \\sin{(\\theta + t_{1})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(cos(Add(Symbol('\\\\theta', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('u', commutative=True), sin(Add(Symbol('\\\\theta', commutative=True), Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Add(Symbol('\\\\theta', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('u', commutative=True), sin(Add(Symbol('\\\\theta', commutative=True), Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(W)} = \\log{(e^{W})} and \\phi_{2}{(W)} = \\int \\dot{z}{(W)} dW, then derive \\int \\dot{z}{(W)} dW - 1 = \\frac{W^{2}}{2} + v_{t} - 1, then obtain \\frac{\\partial}{\\partial v_{2}} \\frac{\\phi_{2}{(W)} - 1}{v_{2}} = \\frac{\\partial}{\\partial v_{2}} \\frac{\\frac{W^{2}}{2} + v_{t} - 1}{v_{2}}", "derivation": "\\dot{z}{(W)} = \\log{(e^{W})} and \\phi_{2}{(W)} = \\int \\dot{z}{(W)} dW and \\phi_{2}{(W)} = \\int \\log{(e^{W})} dW and \\phi_{2}{(W)} - \\frac{\\int \\dot{z}{(W)} dW}{\\phi_{2}{(W)}} = \\int \\log{(e^{W})} dW - \\frac{\\int \\dot{z}{(W)} dW}{\\phi_{2}{(W)}} and \\int \\dot{z}{(W)} dW - 1 = \\int \\log{(e^{W})} dW - 1 and \\int \\dot{z}{(W)} dW - 1 = \\frac{W^{2}}{2} + v_{t} - 1 and \\phi_{2}{(W)} - 1 = \\frac{W^{2}}{2} + v_{t} - 1 and \\frac{\\phi_{2}{(W)} - 1}{v_{2}} = \\frac{\\frac{W^{2}}{2} + v_{t} - 1}{v_{2}} and \\frac{\\partial}{\\partial v_{2}} \\frac{\\phi_{2}{(W)} - 1}{v_{2}} = \\frac{\\partial}{\\partial v_{2}} \\frac{\\frac{W^{2}}{2} + v_{t} - 1}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('W', commutative=True)), log(exp(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integral(log(exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["minus", 3, "Mul(Pow(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1)), Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], "Equality(Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1)), Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Integral(log(exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1)), Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1)), Add(Integral(log(exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 5], "Equality(Add(Integral(Function('\\\\dot{z}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('v_t', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('v_t', commutative=True), Integer(-1)))"], [["divide", 7, "Symbol('v_2', commutative=True)"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('v_t', commutative=True), Integer(-1))))"], [["differentiate", 8, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(\\lambda,n_{1},c)} = \\lambda c + n_{1}, then obtain \\int (\\iint \\chi{(\\lambda,n_{1},c)} dn_{1} d\\lambda + 1) d\\lambda = \\int (\\iint (\\lambda c + n_{1}) dn_{1} d\\lambda + 1) d\\lambda", "derivation": "\\chi{(\\lambda,n_{1},c)} = \\lambda c + n_{1} and \\int \\chi{(\\lambda,n_{1},c)} dn_{1} = \\int (\\lambda c + n_{1}) dn_{1} and \\iint \\chi{(\\lambda,n_{1},c)} dn_{1} d\\lambda = \\iint (\\lambda c + n_{1}) dn_{1} d\\lambda and \\iint \\chi{(\\lambda,n_{1},c)} dn_{1} d\\lambda + 1 = \\iint (\\lambda c + n_{1}) dn_{1} d\\lambda + 1 and \\int (\\iint \\chi{(\\lambda,n_{1},c)} dn_{1} d\\lambda + 1) d\\lambda = \\int (\\iint (\\lambda c + n_{1}) dn_{1} d\\lambda + 1) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\lambda', commutative=True), Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('c', commutative=True)), Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\lambda', commutative=True), Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('c', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\lambda', commutative=True), Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('c', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["add", 3, 1], "Equality(Add(Integral(Function('\\\\chi')(Symbol('\\\\lambda', commutative=True), Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(1)), Add(Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('c', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(1)))"], [["integrate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Integral(Function('\\\\chi')(Symbol('\\\\lambda', commutative=True), Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(1)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('c', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(1)), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})} = e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}}, then obtain - \\frac{\\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + \\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})})}{\\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})}} = - \\frac{\\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}})}{\\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})}}", "derivation": "\\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})} = e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}} and - \\hat{\\mathbf{r}} + \\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})} = - \\hat{\\mathbf{r}} + e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}} and - \\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + \\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})}) = - \\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}}) and - \\frac{\\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + \\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})})}{\\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})}} = - \\frac{\\hat{\\mathbf{r}} (- \\hat{\\mathbf{r}} + e^{\\frac{L_{\\varepsilon}}{\\mathbf{r}}})}{\\mathbf{E}{(\\mathbf{r},L_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)))))))"], [["divide", 3, "Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\rho_f,\\mathbf{f})} = - \\sin{(\\mathbf{f} - \\rho_f)}, then obtain - \\varepsilon_{0}^{2}{(\\rho_f,\\mathbf{f})} = - \\sin^{2}{(\\mathbf{f} - \\rho_f)}", "derivation": "\\varepsilon_{0}{(\\rho_f,\\mathbf{f})} = - \\sin{(\\mathbf{f} - \\rho_f)} and - \\varepsilon_{0}{(\\rho_f,\\mathbf{f})} = \\sin{(\\mathbf{f} - \\rho_f)} and - \\varepsilon_{0}^{2}{(\\rho_f,\\mathbf{f})} = \\varepsilon_{0}{(\\rho_f,\\mathbf{f})} \\sin{(\\mathbf{f} - \\rho_f)} and - \\sin^{2}{(\\mathbf{f} - \\rho_f)} = \\varepsilon_{0}{(\\rho_f,\\mathbf{f})} \\sin{(\\mathbf{f} - \\rho_f)} and - \\varepsilon_{0}^{2}{(\\rho_f,\\mathbf{f})} = - \\sin^{2}{(\\mathbf{f} - \\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))"], [["times", 2, "Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(2))), Mul(Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(P_{e})} = \\sin{(P_{e})}, then obtain 2 \\operatorname{t_{1}}{(P_{e})} + \\frac{- \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})}}{P_{e}} - \\frac{1}{P_{e}} = \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})} + \\frac{- \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})}}{P_{e}} - \\frac{1}{P_{e}}", "derivation": "\\operatorname{t_{1}}{(P_{e})} = \\sin{(P_{e})} and \\operatorname{t_{1}}{(P_{e})} - \\frac{1}{P_{e}} = \\sin{(P_{e})} - \\frac{1}{P_{e}} and 2 \\operatorname{t_{1}}{(P_{e})} - \\frac{1}{P_{e}} = \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})} - \\frac{1}{P_{e}} and 2 \\operatorname{t_{1}}{(P_{e})} + \\frac{- \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})}}{P_{e}} - \\frac{1}{P_{e}} = \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})} + \\frac{- \\operatorname{t_{1}}{(P_{e})} + \\sin{(P_{e})}}{P_{e}} - \\frac{1}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["minus", 1, "Pow(Symbol('P_e', commutative=True), Integer(-1))"], "Equality(Add(Function('t_1')(Symbol('P_e', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(sin(Symbol('P_e', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], [["add", 2, "Function('t_1')(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('t_1')(Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(Function('t_1')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], [["add", 3, "Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('t_1')(Symbol('P_e', commutative=True))), sin(Symbol('P_e', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('t_1')(Symbol('P_e', commutative=True))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('t_1')(Symbol('P_e', commutative=True))), sin(Symbol('P_e', commutative=True)))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(Function('t_1')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('t_1')(Symbol('P_e', commutative=True))), sin(Symbol('P_e', commutative=True)))), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given H{(z)} = \\sin{(z)}, then derive 0 = \\cos{(z)} - \\frac{d}{d z} H{(z)}, then obtain \\int \\sin^{z}{(z)} dz = \\int (\\sin^{z}{(z)} + \\cos{(z)} - \\frac{d}{d z} \\sin{(z)}) dz", "derivation": "H{(z)} = \\sin{(z)} and H^{z}{(z)} = \\sin^{z}{(z)} and 0 = - H{(z)} + \\sin{(z)} and \\frac{d}{d z} 0 = \\frac{d}{d z} (- H{(z)} + \\sin{(z)}) and 0 = \\cos{(z)} - \\frac{d}{d z} H{(z)} and \\sin^{z}{(z)} = \\sin^{z}{(z)} + \\cos{(z)} - \\frac{d}{d z} H{(z)} and H^{z}{(z)} = H^{z}{(z)} + \\cos{(z)} - \\frac{d}{d z} H{(z)} and \\sin^{z}{(z)} = \\sin^{z}{(z)} + \\cos{(z)} - \\frac{d}{d z} \\sin{(z)} and \\int \\sin^{z}{(z)} dz = \\int (\\sin^{z}{(z)} + \\cos{(z)} - \\frac{d}{d z} \\sin{(z)}) dz", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('H')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["minus", 1, "Function('H')(Symbol('z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('H')(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('H')(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(cos(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["add", 5, "Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True))"], "Equality(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Add(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Function('H')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Add(Pow(Function('H')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('H')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Add(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["integrate", 8, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Pow(sin(Symbol('z', commutative=True)), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given z{(G,y)} = y + \\log{(G)} and \\operatorname{E_{n}}{(y^{\\prime},\\hbar)} = - \\hbar + \\log{(y^{\\prime})}, then obtain \\operatorname{E_{n}}{(y^{\\prime},\\hbar)} = - \\hbar - y (y + \\log{(G)}) + y z{(G,y)} + \\log{(y^{\\prime})}", "derivation": "z{(G,y)} = y + \\log{(G)} and y z{(G,y)} = y (y + \\log{(G)}) and \\operatorname{E_{n}}{(y^{\\prime},\\hbar)} = - \\hbar + \\log{(y^{\\prime})} and - y (y + \\log{(G)}) + y z{(G,y)} = 0 and - \\hbar - y (y + \\log{(G)}) + y z{(G,y)} = - \\hbar and \\operatorname{E_{n}}{(y^{\\prime},\\hbar)} = - \\hbar - y (y + \\log{(G)}) + y z{(G,y)} + \\log{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True))))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('z')(Symbol('G', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True)))))"], ["get_premise", "Equality(Function('E_n')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 2, "Mul(Symbol('y', commutative=True), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True)))), Mul(Symbol('y', commutative=True), Function('z')(Symbol('G', commutative=True), Symbol('y', commutative=True)))), Integer(0))"], [["minus", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True)))), Mul(Symbol('y', commutative=True), Function('z')(Symbol('G', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('E_n')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('y', commutative=True), log(Symbol('G', commutative=True)))), Mul(Symbol('y', commutative=True), Function('z')(Symbol('G', commutative=True), Symbol('y', commutative=True))), log(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given S{(x,z)} = x + z, then obtain \\frac{(x + z)^{2}}{x^{2}} = \\frac{(1 + \\frac{z}{x}) (x + z)}{x}", "derivation": "S{(x,z)} = x + z and \\frac{S{(x,z)}}{x} = \\frac{x + z}{x} and \\frac{S{(x,z)}}{x} = 1 + \\frac{z}{x} and \\frac{x + z}{x} = 1 + \\frac{z}{x} and \\frac{(x + z)^{2}}{x^{2}} = \\frac{(1 + \\frac{z}{x}) (x + z)}{x}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Add(Symbol('x', commutative=True), Symbol('z', commutative=True)))"], [["divide", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('S')(Symbol('x', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Symbol('z', commutative=True))))"], [["expand", 2], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('S')(Symbol('x', commutative=True), Symbol('z', commutative=True))), Add(Integer(1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Symbol('z', commutative=True))), Add(Integer(1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-2)), Pow(Add(Symbol('x', commutative=True), Symbol('z', commutative=True)), Integer(2))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Integer(1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Add(Symbol('x', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mu{(A_{2})} = \\sin{(A_{2})} and \\varepsilon_{0}{(v_{2})} = \\cos{(v_{2})}, then obtain e^{- \\mu{(A_{2})} \\sin^{A_{2}}{(A_{2})} + \\varepsilon_{0}{(v_{2})}} = e^{- \\mu{(A_{2})} \\sin^{A_{2}}{(A_{2})} + \\cos{(v_{2})}}", "derivation": "\\mu{(A_{2})} = \\sin{(A_{2})} and \\mu^{A_{2}}{(A_{2})} = \\sin^{A_{2}}{(A_{2})} and \\varepsilon_{0}{(v_{2})} = \\cos{(v_{2})} and - \\mu{(A_{2})} \\mu^{A_{2}}{(A_{2})} + \\varepsilon_{0}{(v_{2})} = - \\mu{(A_{2})} \\mu^{A_{2}}{(A_{2})} + \\cos{(v_{2})} and e^{- \\mu{(A_{2})} \\mu^{A_{2}}{(A_{2})} + \\varepsilon_{0}{(v_{2})}} = e^{- \\mu{(A_{2})} \\mu^{A_{2}}{(A_{2})} + \\cos{(v_{2})}} and e^{- \\mu{(A_{2})} \\sin^{A_{2}}{(A_{2})} + \\varepsilon_{0}{(v_{2})}} = e^{- \\mu{(A_{2})} \\sin^{A_{2}}{(A_{2})} + \\cos{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["power", 1, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["minus", 3, "Mul(Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), Function('\\\\varepsilon_0')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), cos(Symbol('v_2', commutative=True))))"], [["exp", 4], "Equality(exp(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), Function('\\\\varepsilon_0')(Symbol('v_2', commutative=True)))), exp(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(Function('\\\\mu')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), cos(Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(exp(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), Function('\\\\varepsilon_0')(Symbol('v_2', commutative=True)))), exp(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('A_2', commutative=True)), Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), cos(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\hat{x})} = \\sin{(\\hat{x})}, then derive \\int (\\hat{x} + \\theta{(\\hat{x})}) d\\hat{x} = U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})}, then obtain (U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})})^{U} = (\\int (\\hat{x} + \\theta{(\\hat{x})}) d\\hat{x})^{U}", "derivation": "\\theta{(\\hat{x})} = \\sin{(\\hat{x})} and \\hat{x} + \\theta{(\\hat{x})} = \\hat{x} + \\sin{(\\hat{x})} and \\int (\\hat{x} + \\theta{(\\hat{x})}) d\\hat{x} = \\int (\\hat{x} + \\sin{(\\hat{x})}) d\\hat{x} and \\int (\\hat{x} + \\theta{(\\hat{x})}) d\\hat{x} = U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})} and U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})} = \\int (\\hat{x} + \\sin{(\\hat{x})}) d\\hat{x} and (U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})})^{U} = (\\int (\\hat{x} + \\sin{(\\hat{x})}) d\\hat{x})^{U} and (U + \\frac{\\hat{x}^{2}}{2} - \\cos{(\\hat{x})})^{U} = (\\int (\\hat{x} + \\theta{(\\hat{x})}) d\\hat{x})^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('U', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('U', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 5, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Symbol('U', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Symbol('U', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Symbol('U', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Symbol('U', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(k,y^{\\prime})} = k e^{y^{\\prime}}, then obtain \\operatorname{A_{y}}{(k,y^{\\prime})} = k ((k e^{2 y^{\\prime}})^{y^{\\prime}} - (\\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}})^{y^{\\prime}} + e^{y^{\\prime}})", "derivation": "\\operatorname{A_{y}}{(k,y^{\\prime})} = k e^{y^{\\prime}} and \\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}} = k e^{2 y^{\\prime}} and (\\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}})^{y^{\\prime}} = (k e^{2 y^{\\prime}})^{y^{\\prime}} and 0 = (k e^{2 y^{\\prime}})^{y^{\\prime}} - (\\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}})^{y^{\\prime}} and e^{y^{\\prime}} = (k e^{2 y^{\\prime}})^{y^{\\prime}} - (\\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}})^{y^{\\prime}} + e^{y^{\\prime}} and \\operatorname{A_{y}}{(k,y^{\\prime})} = k ((k e^{2 y^{\\prime}})^{y^{\\prime}} - (\\operatorname{A_{y}}{(k,y^{\\prime})} e^{y^{\\prime}})^{y^{\\prime}} + e^{y^{\\prime}})", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('k', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 1, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Mul(Symbol('k', commutative=True), exp(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('k', commutative=True), exp(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 3, "Pow(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Symbol('k', commutative=True), exp(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 4, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(exp(Symbol('y^{\\\\prime}', commutative=True)), Add(Pow(Mul(Symbol('k', commutative=True), exp(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('k', commutative=True), Add(Pow(Mul(Symbol('k', commutative=True), exp(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Mul(Function('A_y')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))), exp(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(t_{2})} = \\sin{(t_{2})}, then obtain \\frac{\\varphi^{*}{(t_{2})}}{\\int \\varphi^{*}{(t_{2})} dt_{2}} - \\frac{1}{\\int \\varphi^{*}{(t_{2})} dt_{2}} = \\frac{\\sin{(t_{2})}}{\\int \\varphi^{*}{(t_{2})} dt_{2}} - \\frac{1}{\\int \\varphi^{*}{(t_{2})} dt_{2}}", "derivation": "\\varphi^{*}{(t_{2})} = \\sin{(t_{2})} and \\int \\varphi^{*}{(t_{2})} dt_{2} = \\int \\sin{(t_{2})} dt_{2} and \\frac{\\varphi^{*}{(t_{2})}}{\\int \\sin{(t_{2})} dt_{2}} = \\frac{\\sin{(t_{2})}}{\\int \\sin{(t_{2})} dt_{2}} and \\frac{\\varphi^{*}{(t_{2})}}{\\int \\sin{(t_{2})} dt_{2}} - \\frac{1}{\\int \\sin{(t_{2})} dt_{2}} = \\frac{\\sin{(t_{2})}}{\\int \\sin{(t_{2})} dt_{2}} - \\frac{1}{\\int \\sin{(t_{2})} dt_{2}} and \\frac{\\varphi^{*}{(t_{2})}}{\\int \\varphi^{*}{(t_{2})} dt_{2}} - \\frac{1}{\\int \\varphi^{*}{(t_{2})} dt_{2}} = \\frac{\\sin{(t_{2})}}{\\int \\varphi^{*}{(t_{2})} dt_{2}} - \\frac{1}{\\int \\varphi^{*}{(t_{2})} dt_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["divide", 1, "Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Mul(sin(Symbol('t_2', commutative=True)), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))))"], [["minus", 3, "Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1)))), Add(Mul(sin(Symbol('t_2', commutative=True)), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(sin(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1)))), Add(Mul(sin(Symbol('t_2', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Function('\\\\varphi^*')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\Omega{(F_{g},l)} = F_{g} + l, then derive \\int (- F_{g} + \\Omega{(F_{g},l)}) dl = \\delta + \\frac{l^{2}}{2}, then obtain \\delta + \\frac{l^{2}}{2} = \\int l dl", "derivation": "\\Omega{(F_{g},l)} = F_{g} + l and - F_{g} + \\Omega{(F_{g},l)} = l and \\int (- F_{g} + \\Omega{(F_{g},l)}) dl = \\int l dl and \\int (- F_{g} + \\Omega{(F_{g},l)}) dl = \\delta + \\frac{l^{2}}{2} and \\delta + \\frac{l^{2}}{2} = \\int l dl", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('F_g', commutative=True), Symbol('l', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Omega')(Symbol('F_g', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Omega')(Symbol('F_g', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Symbol('l', commutative=True), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Omega')(Symbol('F_g', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2)))), Integral(Symbol('l', commutative=True), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\Psi_{nl})} = \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl}, then obtain 0 = \\theta_{1}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} - \\sin{(\\Psi_{nl})} \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl}", "derivation": "\\theta_{1}{(\\Psi_{nl})} = \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl} and \\theta_{1}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} = \\sin{(\\Psi_{nl})} \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl} and 0 = - \\theta_{1}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} + \\sin{(\\Psi_{nl})} \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl} and 0 = \\theta_{1}{(\\Psi_{nl})} \\sin{(\\Psi_{nl})} - \\sin{(\\Psi_{nl})} \\int \\sin{(\\Psi_{nl})} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["minus", 2, "Mul(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["divide", 3, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(x)} = \\sin{(x)}, then obtain 3 \\operatorname{v_{y}}{(x)} - \\sin{(x)} = \\operatorname{v_{y}}{(x)} + \\sin{(x)}", "derivation": "\\operatorname{v_{y}}{(x)} = \\sin{(x)} and 2 \\operatorname{v_{y}}{(x)} = \\operatorname{v_{y}}{(x)} + \\sin{(x)} and 3 \\operatorname{v_{y}}{(x)} - \\sin{(x)} = 2 \\operatorname{v_{y}}{(x)} and 3 \\operatorname{v_{y}}{(x)} - \\sin{(x)} = \\operatorname{v_{y}}{(x)} + \\sin{(x)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["add", 1, "Function('v_y')(Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('v_y')(Symbol('x', commutative=True))), Add(Function('v_y')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"], [["add", 1, "Add(Mul(Integer(2), Function('v_y')(Symbol('x', commutative=True))), Mul(Integer(-1), sin(Symbol('x', commutative=True))))"], "Equality(Add(Mul(Integer(3), Function('v_y')(Symbol('x', commutative=True))), Mul(Integer(-1), sin(Symbol('x', commutative=True)))), Mul(Integer(2), Function('v_y')(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('v_y')(Symbol('x', commutative=True))), Mul(Integer(-1), sin(Symbol('x', commutative=True)))), Add(Function('v_y')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(I,v_{y})} = e^{I v_{y}} and \\mathbf{H}{(I,v_{y})} = \\frac{\\partial}{\\partial I} e^{I v_{y}}, then obtain \\iint e^{\\mathbf{H}{(I,v_{y})}} dI dv_{y} = \\iint e^{\\frac{\\partial}{\\partial I} \\hat{\\mathbf{r}}{(I,v_{y})}} dI dv_{y}", "derivation": "\\hat{\\mathbf{r}}{(I,v_{y})} = e^{I v_{y}} and \\frac{\\partial}{\\partial I} \\hat{\\mathbf{r}}{(I,v_{y})} = \\frac{\\partial}{\\partial I} e^{I v_{y}} and \\mathbf{H}{(I,v_{y})} = \\frac{\\partial}{\\partial I} e^{I v_{y}} and e^{\\mathbf{H}{(I,v_{y})}} = e^{\\frac{\\partial}{\\partial I} e^{I v_{y}}} and e^{\\mathbf{H}{(I,v_{y})}} = e^{\\frac{\\partial}{\\partial I} \\hat{\\mathbf{r}}{(I,v_{y})}} and \\int e^{\\mathbf{H}{(I,v_{y})}} dI = \\int e^{\\frac{\\partial}{\\partial I} \\hat{\\mathbf{r}}{(I,v_{y})}} dI and \\iint e^{\\mathbf{H}{(I,v_{y})}} dI dv_{y} = \\iint e^{\\frac{\\partial}{\\partial I} \\hat{\\mathbf{r}}{(I,v_{y})}} dI dv_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), exp(Mul(Symbol('I', commutative=True), Symbol('v_y', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), Derivative(exp(Mul(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), exp(Derivative(exp(Mul(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(exp(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), exp(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('I', commutative=True)"], "Equality(Integral(exp(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(exp(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True))))"], [["integrate", 6, "Symbol('v_y', commutative=True)"], "Equality(Integral(exp(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(E,\\delta)} = \\sin{(E \\delta)}, then obtain \\delta \\rho_{b}{(E,\\delta)} \\sin{(E \\delta)} - \\sin^{2}{(E \\delta)} = \\delta \\sin^{2}{(E \\delta)} - \\sin^{2}{(E \\delta)}", "derivation": "\\rho_{b}{(E,\\delta)} = \\sin{(E \\delta)} and \\rho_{b}{(E,\\delta)} \\sin{(E \\delta)} = \\sin^{2}{(E \\delta)} and \\delta \\rho_{b}{(E,\\delta)} \\sin{(E \\delta)} = \\delta \\sin^{2}{(E \\delta)} and \\delta \\rho_{b}{(E,\\delta)} \\sin{(E \\delta)} - \\sin^{2}{(E \\delta)} = \\delta \\sin^{2}{(E \\delta)} - \\sin^{2}{(E \\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["times", 1, "sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)))), Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2)))"], [["times", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)))), Mul(Symbol('\\\\delta', commutative=True), Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2))))"], [["minus", 3, "Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)))), Mul(Integer(-1), Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2)))), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(sin(Mul(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(f^{\\prime})} = \\sin{(f^{\\prime})}, then derive \\int \\operatorname{v_{t}}{(f^{\\prime})} df^{\\prime} = \\mathbf{B} - \\cos{(f^{\\prime})}, then obtain \\int \\sin{(f^{\\prime})} df^{\\prime} = \\mathbf{B} - \\cos{(f^{\\prime})}", "derivation": "\\operatorname{v_{t}}{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\int \\operatorname{v_{t}}{(f^{\\prime})} df^{\\prime} = \\int \\sin{(f^{\\prime})} df^{\\prime} and \\int \\operatorname{v_{t}}{(f^{\\prime})} df^{\\prime} = \\mathbf{B} - \\cos{(f^{\\prime})} and \\int \\sin{(f^{\\prime})} df^{\\prime} = \\mathbf{B} - \\cos{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given A{(b,F_{H})} = \\frac{F_{H}}{b} and a{(b,F_{H})} = (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}}, then obtain \\iint (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} db dF_{H} = \\iint a{(b,F_{H})} db dF_{H}", "derivation": "A{(b,F_{H})} = \\frac{F_{H}}{b} and \\frac{b A{(b,F_{H})}}{F_{H}} = 1 and (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} = 1 and a{(b,F_{H})} = (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} and a{(b,F_{H})} = 1 and \\int a{(b,F_{H})} db = \\int 1 db and \\int (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} db = \\int 1 db and \\int (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} db = \\int a{(b,F_{H})} db and \\iint (\\frac{b A{(b,F_{H})}}{F_{H}})^{F_{H}} db dF_{H} = \\iint a{(b,F_{H})} db dF_{H}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('F_H', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Integer(1))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('a')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('a')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('b', commutative=True)"], "Equality(Integral(Function('a')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Integer(1), Tuple(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Integer(1), Tuple(Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integral(Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Function('a')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["integrate", 8, "Symbol('F_H', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('A')(Symbol('b', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Function('a')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(V,\\omega)} = \\frac{V}{\\omega}, then obtain \\frac{- V + \\int \\omega \\operatorname{g_{\\varepsilon}}{(V,\\omega)} d\\omega}{\\operatorname{g_{\\varepsilon}}{(V,\\omega)}} = \\frac{- V + \\int V d\\omega}{\\operatorname{g_{\\varepsilon}}{(V,\\omega)}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(V,\\omega)} = \\frac{V}{\\omega} and \\omega \\operatorname{g_{\\varepsilon}}{(V,\\omega)} = V and \\int \\omega \\operatorname{g_{\\varepsilon}}{(V,\\omega)} d\\omega = \\int V d\\omega and - V + \\int \\omega \\operatorname{g_{\\varepsilon}}{(V,\\omega)} d\\omega = - V + \\int V d\\omega and \\frac{- V + \\int \\omega \\operatorname{g_{\\varepsilon}}{(V,\\omega)} d\\omega}{\\operatorname{g_{\\varepsilon}}{(V,\\omega)}} = \\frac{- V + \\int V d\\omega}{\\operatorname{g_{\\varepsilon}}{(V,\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('V', commutative=True))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Symbol('V', commutative=True), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Symbol('V', commutative=True), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["divide", 4, "Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Pow(Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Symbol('V', commutative=True), Tuple(Symbol('\\\\omega', commutative=True)))), Pow(Function('g_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{B},f^{*})} = \\log{(\\mathbf{B} f^{*})}, then derive 0 = - \\frac{\\partial}{\\partial f^{*}} \\mu_{0}{(\\mathbf{B},f^{*})} + \\frac{1}{f^{*}}, then derive \\int 0 df^{*} = \\dot{\\mathbf{r}}, then obtain \\int (- \\frac{\\partial}{\\partial f^{*}} \\log{(\\mathbf{B} f^{*})} + \\frac{1}{f^{*}}) df^{*} = \\dot{\\mathbf{r}}", "derivation": "\\mu_{0}{(\\mathbf{B},f^{*})} = \\log{(\\mathbf{B} f^{*})} and 0 = - \\mu_{0}{(\\mathbf{B},f^{*})} + \\log{(\\mathbf{B} f^{*})} and \\frac{d}{d f^{*}} 0 = \\frac{\\partial}{\\partial f^{*}} (- \\mu_{0}{(\\mathbf{B},f^{*})} + \\log{(\\mathbf{B} f^{*})}) and 0 = - \\frac{\\partial}{\\partial f^{*}} \\mu_{0}{(\\mathbf{B},f^{*})} + \\frac{1}{f^{*}} and 0 = - \\frac{\\partial}{\\partial f^{*}} \\log{(\\mathbf{B} f^{*})} + \\frac{1}{f^{*}} and \\int 0 df^{*} = \\int (- \\frac{\\partial}{\\partial f^{*}} \\log{(\\mathbf{B} f^{*})} + \\frac{1}{f^{*}}) df^{*} and \\int 0 df^{*} = \\dot{\\mathbf{r}} and \\int (- \\frac{\\partial}{\\partial f^{*}} \\log{(\\mathbf{B} f^{*})} + \\frac{1}{f^{*}}) df^{*} = \\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True)), log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))), log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True)))))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))), log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('f^*', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integral(Add(Mul(Integer(-1), Derivative(log(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Pow(Symbol('f^*', commutative=True), Integer(-1))), Tuple(Symbol('f^*', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"]]}, {"prompt": "Given q{(c)} = \\sin{(\\sin{(c)})}, then obtain q{(c)} \\sin{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} q{(c)} = - (\\sin{(c)} \\cos{(\\sin{(c)})} + \\sin{(\\sin{(c)})} \\cos^{2}{(c)}) q{(c)} \\sin{(\\sin{(c)})}", "derivation": "q{(c)} = \\sin{(\\sin{(c)})} and q{(c)} \\sin{(\\sin{(c)})} = \\sin^{2}{(\\sin{(c)})} and \\frac{d}{d c} q{(c)} = \\frac{d}{d c} \\sin{(\\sin{(c)})} and \\frac{d^{2}}{d c^{2}} q{(c)} = \\frac{d^{2}}{d c^{2}} \\sin{(\\sin{(c)})} and \\sin^{2}{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} q{(c)} = \\sin^{2}{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} \\sin{(\\sin{(c)})} and q{(c)} \\sin{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} q{(c)} = q{(c)} \\sin{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} \\sin{(\\sin{(c)})} and q{(c)} \\sin{(\\sin{(c)})} \\frac{d^{2}}{d c^{2}} q{(c)} = - (\\sin{(c)} \\cos{(\\sin{(c)})} + \\sin{(\\sin{(c)})} \\cos^{2}{(c)}) q{(c)} \\sin{(\\sin{(c)})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))))"], [["times", 1, "sin(sin(Symbol('c', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True)))), Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2))))"], [["times", 4, "Pow(sin(sin(Symbol('c', commutative=True))), Integer(2))"], "Equality(Mul(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(Function('q')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Mul(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))), Derivative(Function('q')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Mul(Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))), Derivative(Function('q')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2)))), Mul(Integer(-1), Add(Mul(sin(Symbol('c', commutative=True)), cos(sin(Symbol('c', commutative=True)))), Mul(sin(sin(Symbol('c', commutative=True))), Pow(cos(Symbol('c', commutative=True)), Integer(2)))), Function('q')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{H}_l,Z)} = Z + \\hat{H}_l, then obtain (2 \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)})^{\\hat{H}_l} = ((Z + \\hat{H}_l)^{Z} + \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)})^{\\hat{H}_l}", "derivation": "\\operatorname{z^{*}}{(\\hat{H}_l,Z)} = Z + \\hat{H}_l and \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)} = (Z + \\hat{H}_l)^{Z} and 2 \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)} = (Z + \\hat{H}_l)^{Z} + \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)} and (2 \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)})^{\\hat{H}_l} = ((Z + \\hat{H}_l)^{Z} + \\operatorname{z^{*}}^{Z}{(\\hat{H}_l,Z)})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('Z', commutative=True)))"], [["add", 2, "Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Add(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given a{(\\Psi_{nl})} = \\cos{(e^{\\Psi_{nl}})} and \\theta_{1}{(\\Psi_{nl})} = \\int (a{(\\Psi_{nl})} - 1) d\\Psi_{nl}, then obtain \\theta_{1}{(\\Psi_{nl})} = \\int (\\cos{(e^{\\Psi_{nl}})} - 1) d\\Psi_{nl}", "derivation": "a{(\\Psi_{nl})} = \\cos{(e^{\\Psi_{nl}})} and a{(\\Psi_{nl})} - 1 = \\cos{(e^{\\Psi_{nl}})} - 1 and \\int (a{(\\Psi_{nl})} - 1) d\\Psi_{nl} = \\int (\\cos{(e^{\\Psi_{nl}})} - 1) d\\Psi_{nl} and \\theta_{1}{(\\Psi_{nl})} = \\int (a{(\\Psi_{nl})} - 1) d\\Psi_{nl} and \\theta_{1}{(\\Psi_{nl})} = \\int (\\cos{(e^{\\Psi_{nl}})} - 1) d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 1, 1], "Equality(Add(Function('a')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Add(cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Add(Function('a')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Add(Function('a')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\theta_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Add(cos(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\Psi{(z^{*})} = \\cos{(z^{*})} and J{(\\mu)} = \\mu, then derive \\frac{d}{d \\mu} J{(\\mu)} = 1, then obtain \\frac{\\frac{d}{d \\mu} \\mu}{2 \\cos{(z^{*})}} = \\frac{1}{2 \\cos{(z^{*})}}", "derivation": "\\Psi{(z^{*})} = \\cos{(z^{*})} and \\Psi{(z^{*})} + \\cos{(z^{*})} = 2 \\cos{(z^{*})} and J{(\\mu)} = \\mu and \\frac{d}{d \\mu} J{(\\mu)} = \\frac{d}{d \\mu} \\mu and \\frac{d}{d \\mu} J{(\\mu)} = 1 and \\frac{d}{d \\mu} \\mu = 1 and \\frac{\\frac{d}{d \\mu} \\mu}{\\Psi{(z^{*})} + \\cos{(z^{*})}} = \\frac{1}{\\Psi{(z^{*})} + \\cos{(z^{*})}} and \\frac{\\frac{d}{d \\mu} \\mu}{2 \\cos{(z^{*})}} = \\frac{1}{2 \\cos{(z^{*})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["add", 1, "cos(Symbol('z^*', commutative=True))"], "Equality(Add(Function('\\\\Psi')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True))), Mul(Integer(2), cos(Symbol('z^*', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('J')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1))"], [["divide", 6, "Add(Function('\\\\Psi')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\Psi')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True))), Integer(-1)), Derivative(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Pow(Add(Function('\\\\Psi')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1)), Derivative(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('z^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given x{(t_{1})} = \\log{(t_{1})}, then obtain 2 x^{t_{1}}{(t_{1})} \\log{(t_{1})} = (x{(t_{1})} + \\log{(t_{1})}) x^{t_{1}}{(t_{1})}", "derivation": "x{(t_{1})} = \\log{(t_{1})} and 2 x{(t_{1})} = x{(t_{1})} + \\log{(t_{1})} and x{(t_{1})} x^{t_{1}}{(t_{1})} = x^{t_{1}}{(t_{1})} \\log{(t_{1})} and 2 x{(t_{1})} x^{t_{1}}{(t_{1})} = (x{(t_{1})} + \\log{(t_{1})}) x^{t_{1}}{(t_{1})} and 2 x^{t_{1}}{(t_{1})} \\log{(t_{1})} = (x{(t_{1})} + \\log{(t_{1})}) x^{t_{1}}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["add", 1, "Function('x')(Symbol('t_1', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('t_1', commutative=True))), Add(Function('x')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True))))"], [["times", 1, "Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], "Equality(Mul(Function('x')(Symbol('t_1', commutative=True)), Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))), Mul(Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True))))"], [["times", 2, "Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('t_1', commutative=True)), Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))), Mul(Add(Function('x')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True))), Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True))), Mul(Add(Function('x')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True))), Pow(Function('x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain \\operatorname{A_{y}}{(\\hat{H})} = - \\operatorname{A_{y}}^{2}{(\\hat{H})} + \\operatorname{A_{y}}{(\\hat{H})} \\cos{(\\hat{H})} + \\operatorname{A_{y}}{(\\hat{H})}", "derivation": "\\operatorname{A_{y}}{(\\hat{H})} = \\cos{(\\hat{H})} and \\operatorname{A_{y}}^{2}{(\\hat{H})} = \\operatorname{A_{y}}{(\\hat{H})} \\cos{(\\hat{H})} and 0 = - \\operatorname{A_{y}}^{2}{(\\hat{H})} + \\operatorname{A_{y}}{(\\hat{H})} \\cos{(\\hat{H})} and \\operatorname{A_{y}}{(\\hat{H})} = - \\operatorname{A_{y}}^{2}{(\\hat{H})} + \\operatorname{A_{y}}{(\\hat{H})} \\cos{(\\hat{H})} + \\operatorname{A_{y}}{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["times", 1, "Function('A_y')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Pow(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 2, "Pow(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Mul(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["add", 3, "Function('A_y')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), Integer(2))), Mul(Function('A_y')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))), Function('A_y')(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given V{(\\Omega)} = \\Omega, then obtain \\frac{2 \\Omega^{2} V{(\\Omega)} \\frac{d}{d \\Omega} V{(\\Omega)} + 2 \\Omega V^{2}{(\\Omega)}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}} = \\frac{4 \\Omega^{3}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}}", "derivation": "V{(\\Omega)} = \\Omega and \\Omega V{(\\Omega)} = \\Omega^{2} and \\Omega^{2} V^{2}{(\\Omega)} = \\Omega^{4} and \\frac{d}{d \\Omega} \\Omega^{2} V^{2}{(\\Omega)} = \\frac{d}{d \\Omega} \\Omega^{4} and \\frac{\\frac{d}{d \\Omega} \\Omega^{2} V^{2}{(\\Omega)}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}} = \\frac{\\frac{d}{d \\Omega} \\Omega^{4}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}} and \\frac{2 \\Omega^{2} V{(\\Omega)} \\frac{d}{d \\Omega} V{(\\Omega)} + 2 \\Omega V^{2}{(\\Omega)}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}} = \\frac{4 \\Omega^{3}}{\\int (- E{(P_{g})} + \\sin{(P_{g})}) dP_{g}}", "srepr_derivation": [["renaming_premise", "Equality(Function('V')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["times", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('V')(Symbol('\\\\Omega', commutative=True))), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Pow(Function('V')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Pow(Symbol('\\\\Omega', commutative=True), Integer(4)))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Pow(Function('V')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\Omega', commutative=True), Integer(4)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["divide", 4, "Integral(Add(Mul(Integer(-1), Function('E')(Symbol('P_g', commutative=True))), sin(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)))"], "Equality(Mul(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Pow(Function('V')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Pow(Integral(Add(Mul(Integer(-1), Function('E')(Symbol('P_g', commutative=True))), sin(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(-1))), Mul(Derivative(Pow(Symbol('\\\\Omega', commutative=True), Integer(4)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Pow(Integral(Add(Mul(Integer(-1), Function('E')(Symbol('P_g', commutative=True))), sin(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Mul(Integer(2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Function('V')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('V')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Function('V')(Symbol('\\\\Omega', commutative=True)), Integer(2)))), Pow(Integral(Add(Mul(Integer(-1), Function('E')(Symbol('P_g', commutative=True))), sin(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(-1))), Mul(Integer(4), Pow(Symbol('\\\\Omega', commutative=True), Integer(3)), Pow(Integral(Add(Mul(Integer(-1), Function('E')(Symbol('P_g', commutative=True))), sin(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(\\psi,\\mathbf{J}_f,C)} = \\mathbf{J}_f^{C} + \\psi and h{(\\mathbf{J}_f,C)} = \\mathbf{J}_f^{C}, then obtain \\frac{\\ddot{x}{(\\psi,\\mathbf{J}_f,C)}}{\\mathbf{J}_f} = \\frac{\\mathbf{J}_f^{C} + \\psi}{\\mathbf{J}_f}", "derivation": "\\ddot{x}{(\\psi,\\mathbf{J}_f,C)} = \\mathbf{J}_f^{C} + \\psi and h{(\\mathbf{J}_f,C)} = \\mathbf{J}_f^{C} and \\ddot{x}{(\\psi,\\mathbf{J}_f,C)} = \\psi + h{(\\mathbf{J}_f,C)} and \\frac{\\ddot{x}{(\\psi,\\mathbf{J}_f,C)}}{\\mathbf{J}_f} = \\frac{\\psi + h{(\\mathbf{J}_f,C)}}{\\mathbf{J}_f} and \\frac{\\mathbf{J}_f^{C} + \\psi}{\\mathbf{J}_f} = \\frac{\\psi + h{(\\mathbf{J}_f,C)}}{\\mathbf{J}_f} and \\frac{\\ddot{x}{(\\psi,\\mathbf{J}_f,C)}}{\\mathbf{J}_f} = \\frac{\\mathbf{J}_f^{C} + \\psi}{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\ddot{x}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Function('h')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True))))"], [["divide", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Function('h')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Function('h')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(M,Z)} = Z^{M}, then obtain \\frac{- M + Z^{M} + \\hat{\\mathbf{r}}{(M,Z)}}{Z} = \\frac{- M + 2 Z^{M}}{Z}", "derivation": "\\hat{\\mathbf{r}}{(M,Z)} = Z^{M} and Z^{M} + \\hat{\\mathbf{r}}{(M,Z)} = 2 Z^{M} and - M + Z^{M} + \\hat{\\mathbf{r}}{(M,Z)} = - M + 2 Z^{M} and \\frac{- M + Z^{M} + \\hat{\\mathbf{r}}{(M,Z)}}{Z} = \\frac{- M + 2 Z^{M}}{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True)))"], [["add", 1, "Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True))"], "Equality(Add(Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('Z', commutative=True))), Mul(Integer(2), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True))))"], [["minus", 2, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True)))))"], [["divide", 3, "Symbol('Z', commutative=True)"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('Z', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Pow(Symbol('Z', commutative=True), Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(G,\\varepsilon)} = \\cos{(\\frac{\\varepsilon}{G})}, then obtain \\int G \\log{(G \\operatorname{f^{*}}{(G,\\varepsilon)})} \\cos{(\\frac{\\varepsilon}{G})} dG = \\int G \\log{(G \\cos{(\\frac{\\varepsilon}{G})})} \\cos{(\\frac{\\varepsilon}{G})} dG", "derivation": "\\operatorname{f^{*}}{(G,\\varepsilon)} = \\cos{(\\frac{\\varepsilon}{G})} and G \\operatorname{f^{*}}{(G,\\varepsilon)} = G \\cos{(\\frac{\\varepsilon}{G})} and \\log{(G \\operatorname{f^{*}}{(G,\\varepsilon)})} = \\log{(G \\cos{(\\frac{\\varepsilon}{G})})} and G \\log{(G \\operatorname{f^{*}}{(G,\\varepsilon)})} \\cos{(\\frac{\\varepsilon}{G})} = G \\log{(G \\cos{(\\frac{\\varepsilon}{G})})} \\cos{(\\frac{\\varepsilon}{G})} and \\int G \\log{(G \\operatorname{f^{*}}{(G,\\varepsilon)})} \\cos{(\\frac{\\varepsilon}{G})} dG = \\int G \\log{(G \\cos{(\\frac{\\varepsilon}{G})})} \\cos{(\\frac{\\varepsilon}{G})} dG", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Symbol('G', commutative=True), Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), log(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))))"], [["times", 3, "Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], "Equality(Mul(Symbol('G', commutative=True), log(Mul(Symbol('G', commutative=True), Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Mul(Symbol('G', commutative=True), log(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Symbol('G', commutative=True), log(Mul(Symbol('G', commutative=True), Function('f^*')(Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Symbol('G', commutative=True), log(Mul(Symbol('G', commutative=True), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))), cos(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given s{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then derive 0 = \\mathbf{B} + e^{\\Psi_{nl}} - \\int s{(\\Psi_{nl})} d\\Psi_{nl}, then obtain 0 = \\mathbf{B} + e^{\\Psi_{nl}} - \\int e^{\\Psi_{nl}} d\\Psi_{nl}", "derivation": "s{(\\Psi_{nl})} = e^{\\Psi_{nl}} and \\int s{(\\Psi_{nl})} d\\Psi_{nl} = \\int e^{\\Psi_{nl}} d\\Psi_{nl} and 0 = - \\int s{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\Psi_{nl}} d\\Psi_{nl} and 0 = \\mathbf{B} + e^{\\Psi_{nl}} - \\int s{(\\Psi_{nl})} d\\Psi_{nl} and 0 = \\mathbf{B} + e^{\\Psi_{nl}} - \\int e^{\\Psi_{nl}} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 2, "Integral(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Integral(Function('s')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Symbol('\\\\mathbf{B}', commutative=True), exp(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"]]}, {"prompt": "Given t{(\\mathbf{r})} = \\int \\log{(\\mathbf{r})} d\\mathbf{r}, then derive \\frac{t{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = \\frac{\\Psi_{\\lambda} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r}}{\\log{(\\mathbf{r})}}, then obtain \\frac{\\Psi_{\\lambda}}{\\log{(\\mathbf{r})}} + \\mathbf{r} - \\frac{\\mathbf{r}}{\\log{(\\mathbf{r})}} = \\frac{\\Psi_{\\lambda} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r}}{\\log{(\\mathbf{r})}}", "derivation": "t{(\\mathbf{r})} = \\int \\log{(\\mathbf{r})} d\\mathbf{r} and \\frac{t{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = \\frac{\\int \\log{(\\mathbf{r})} d\\mathbf{r}}{\\log{(\\mathbf{r})}} and \\frac{t{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = \\frac{\\Psi_{\\lambda} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r}}{\\log{(\\mathbf{r})}} and \\frac{t{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = \\frac{\\Psi_{\\lambda}}{\\log{(\\mathbf{r})}} + \\mathbf{r} - \\frac{\\mathbf{r}}{\\log{(\\mathbf{r})}} and \\frac{\\Psi_{\\lambda}}{\\log{(\\mathbf{r})}} + \\mathbf{r} - \\frac{\\mathbf{r}}{\\log{(\\mathbf{r})}} = \\frac{\\Psi_{\\lambda} + \\mathbf{r} \\log{(\\mathbf{r})} - \\mathbf{r}}{\\log{(\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{r}', commutative=True)), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 1, "log(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('t')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"], [["expand", 3], "Equality(Mul(Function('t')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)))), Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True))), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(B)} = e^{\\sin{(B)}} and \\operatorname{F_{x}}{(B)} = B + \\hat{p}{(B)}, then obtain \\frac{d}{d B} \\operatorname{F_{x}}{(B)} = \\frac{d}{d B} (B + e^{\\sin{(B)}})", "derivation": "\\hat{p}{(B)} = e^{\\sin{(B)}} and B + \\hat{p}{(B)} = B + e^{\\sin{(B)}} and \\operatorname{F_{x}}{(B)} = B + \\hat{p}{(B)} and \\operatorname{F_{x}}{(B)} = B + e^{\\sin{(B)}} and \\frac{d}{d B} \\operatorname{F_{x}}{(B)} = \\frac{d}{d B} (B + e^{\\sin{(B)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('B', commutative=True)), exp(sin(Symbol('B', commutative=True))))"], [["add", 1, "Symbol('B', commutative=True)"], "Equality(Add(Symbol('B', commutative=True), Function('\\\\hat{p}')(Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), exp(sin(Symbol('B', commutative=True)))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Function('\\\\hat{p}')(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_x')(Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), exp(sin(Symbol('B', commutative=True)))))"], [["differentiate", 4, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), exp(sin(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\theta_2)} = e^{\\theta_2} and \\bar{\\h}{(v_{x})} = \\cos{(v_{x})}, then obtain \\frac{\\bar{\\h}{(v_{x})}}{\\int (\\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2}) d\\theta_2} = \\frac{\\cos{(v_{x})}}{\\int (\\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2}) d\\theta_2}", "derivation": "\\Psi^{\\dagger}{(\\theta_2)} = e^{\\theta_2} and \\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2} = 0 and \\bar{\\h}{(v_{x})} = \\cos{(v_{x})} and \\int (\\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2}) d\\theta_2 = \\int 0 d\\theta_2 and \\frac{\\bar{\\h}{(v_{x})}}{\\int 0 d\\theta_2} = \\frac{\\cos{(v_{x})}}{\\int 0 d\\theta_2} and \\frac{\\bar{\\h}{(v_{x})}}{\\int (\\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2}) d\\theta_2} = \\frac{\\cos{(v_{x})}}{\\int (\\Psi^{\\dagger}{(\\theta_2)} - e^{\\theta_2}) d\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 3, "Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))), Mul(cos(Symbol('v_x', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('\\\\hbar')(Symbol('v_x', commutative=True)), Pow(Integral(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))), Mul(cos(Symbol('v_x', commutative=True)), Pow(Integral(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\psi{(V,m_{s})} = m_{s} + e^{V} and \\dot{y}{(V,m_{s})} = \\frac{m_{s} + e^{V}}{V}, then obtain \\iint \\frac{\\psi{(V,m_{s})}}{V} dV dm_{s} = \\iint \\dot{y}{(V,m_{s})} dV dm_{s}", "derivation": "\\psi{(V,m_{s})} = m_{s} + e^{V} and \\frac{\\psi{(V,m_{s})}}{V} = \\frac{m_{s} + e^{V}}{V} and \\dot{y}{(V,m_{s})} = \\frac{m_{s} + e^{V}}{V} and \\frac{\\psi{(V,m_{s})}}{V} = \\dot{y}{(V,m_{s})} and \\int \\frac{\\psi{(V,m_{s})}}{V} dV = \\int \\dot{y}{(V,m_{s})} dV and \\iint \\frac{\\psi{(V,m_{s})}}{V} dV dm_{s} = \\iint \\dot{y}{(V,m_{s})} dV dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), exp(Symbol('V', commutative=True))))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('V', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('m_s', commutative=True), exp(Symbol('V', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('m_s', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('m_s', commutative=True), exp(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('V', commutative=True), Symbol('m_s', commutative=True))), Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('m_s', commutative=True)))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('V', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["integrate", 5, "Symbol('m_s', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('V', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given y{(p)} = \\log{(p)}, then obtain e^{(p + \\log{(p)}^{p}) (- p + y^{p}{(p)} - \\log{(p)}^{p})} = e^{- p (p + \\log{(p)}^{p})}", "derivation": "y{(p)} = \\log{(p)} and y^{p}{(p)} = \\log{(p)}^{p} and p + y^{p}{(p)} = p + \\log{(p)}^{p} and - p + y^{p}{(p)} - \\log{(p)}^{p} = - p and (p + y^{p}{(p)}) (- p + y^{p}{(p)} - \\log{(p)}^{p}) = - p (p + y^{p}{(p)}) and (p + \\log{(p)}^{p}) (- p + y^{p}{(p)} - \\log{(p)}^{p}) = - p (p + \\log{(p)}^{p}) and e^{(p + \\log{(p)}^{p}) (- p + y^{p}{(p)} - \\log{(p)}^{p})} = e^{- p (p + \\log{(p)}^{p})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["add", 2, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["minus", 2, "Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))), Mul(Integer(-1), Symbol('p', commutative=True)))"], [["times", 4, "Add(Symbol('p', commutative=True), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], "Equality(Mul(Add(Symbol('p', commutative=True), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))))), Mul(Integer(-1), Symbol('p', commutative=True), Add(Symbol('p', commutative=True), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))))), Mul(Integer(-1), Symbol('p', commutative=True), Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))))"], [["exp", 6], "Equality(exp(Mul(Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))))), exp(Mul(Integer(-1), Symbol('p', commutative=True), Add(Symbol('p', commutative=True), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(i,E_{\\lambda})} = \\frac{E_{\\lambda}}{i}, then derive \\frac{\\partial}{\\partial E_{\\lambda}} \\theta_{1}{(i,E_{\\lambda})} - \\frac{1}{i} = 0, then obtain \\frac{e^{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i} - \\frac{1}{i}}}{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i}} = \\frac{1}{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i}}", "derivation": "\\theta_{1}{(i,E_{\\lambda})} = \\frac{E_{\\lambda}}{i} and - \\frac{E_{\\lambda}}{i} + \\theta_{1}{(i,E_{\\lambda})} = 0 and \\frac{\\partial}{\\partial E_{\\lambda}} (- \\frac{E_{\\lambda}}{i} + \\theta_{1}{(i,E_{\\lambda})}) = \\frac{d}{d E_{\\lambda}} 0 and \\frac{\\partial}{\\partial E_{\\lambda}} \\theta_{1}{(i,E_{\\lambda})} - \\frac{1}{i} = 0 and \\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i} - \\frac{1}{i} = 0 and e^{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i} - \\frac{1}{i}} = 1 and \\frac{e^{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i} - \\frac{1}{i}}}{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i}} = \\frac{1}{\\frac{\\partial}{\\partial E_{\\lambda}} \\frac{E_{\\lambda}}{i}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('i', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["minus", 1, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\theta_1')(Symbol('i', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\theta_1')(Symbol('i', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('i', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(0))"], [["exp", 5], "Equality(exp(Add(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1))))), Integer(1))"], [["divide", 6, "Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Mul(exp(Add(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1))))), Pow(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon_{0}{(J)} = \\cos{(J)}, then obtain u - \\int \\frac{\\varepsilon_{0}{(J)}}{\\sin{(J)}} dJ - \\int \\frac{\\cos{(J)}}{\\sin{(J)}} dJ = \\int \\frac{2 \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} dJ", "derivation": "\\varepsilon_{0}{(J)} = \\cos{(J)} and \\varepsilon_{0}{(J)} + \\cos{(J)} = 2 \\cos{(J)} and \\frac{\\varepsilon_{0}{(J)} + \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} = \\frac{2 \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} and \\int \\frac{\\varepsilon_{0}{(J)} + \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} dJ = \\int \\frac{2 \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} dJ and u - \\int \\frac{\\varepsilon_{0}{(J)}}{\\sin{(J)}} dJ - \\int \\frac{\\cos{(J)}}{\\sin{(J)}} dJ = \\int \\frac{2 \\cos{(J)}}{\\frac{d}{d J} \\cos{(J)}} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["add", 1, "cos(Symbol('J', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Mul(Integer(2), cos(Symbol('J', commutative=True))))"], [["divide", 2, "Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(2), cos(Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True))), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('u', commutative=True), Mul(Integer(-1), Integral(Mul(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True)))), Mul(Integer(-1), Integral(Mul(Pow(sin(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))), Integral(Mul(Integer(2), cos(Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(E,\\mathbb{I})} = \\int \\frac{E}{\\mathbb{I}} d\\mathbb{I}, then obtain \\int \\frac{\\mathbf{J}_M{(E,\\mathbb{I})} - \\frac{1}{\\mathbb{I}}}{\\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} - \\frac{1}{\\mathbb{I}}} dE = \\int 1 dE", "derivation": "\\mathbf{J}_M{(E,\\mathbb{I})} = \\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} and \\mathbf{J}_M{(E,\\mathbb{I})} - \\frac{1}{\\mathbb{I}} = \\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} - \\frac{1}{\\mathbb{I}} and (\\mathbf{J}_M{(E,\\mathbb{I})} - \\frac{1}{\\mathbb{I}}) \\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} = (\\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} - \\frac{1}{\\mathbb{I}}) \\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} and \\frac{\\mathbf{J}_M{(E,\\mathbb{I})} - \\frac{1}{\\mathbb{I}}}{\\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} - \\frac{1}{\\mathbb{I}}} = 1 and \\int \\frac{\\mathbf{J}_M{(E,\\mathbb{I})} - \\frac{1}{\\mathbb{I}}}{\\int \\frac{E}{\\mathbb{I}} d\\mathbb{I} - \\frac{1}{\\mathbb{I}}} dE = \\int 1 dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], [["times", 2, "Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Add(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["divide", 3, "Mul(Add(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Pow(Add(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integer(-1))), Integer(1))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Pow(Add(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('E', commutative=True))), Integral(Integer(1), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\nabla)} = \\int e^{\\nabla} d\\nabla, then obtain \\hat{H} - e^{- \\nabla} = \\int \\frac{e^{- \\nabla} \\int e^{\\nabla} d\\nabla}{\\operatorname{A_{z}}{(\\nabla)}} d\\nabla", "derivation": "\\operatorname{A_{z}}{(\\nabla)} = \\int e^{\\nabla} d\\nabla and 1 = \\frac{\\int e^{\\nabla} d\\nabla}{\\operatorname{A_{z}}{(\\nabla)}} and e^{- \\nabla} = \\frac{e^{- \\nabla} \\int e^{\\nabla} d\\nabla}{\\operatorname{A_{z}}{(\\nabla)}} and \\int e^{- \\nabla} d\\nabla = \\int \\frac{e^{- \\nabla} \\int e^{\\nabla} d\\nabla}{\\operatorname{A_{z}}{(\\nabla)}} d\\nabla and \\hat{H} - e^{- \\nabla} = \\int \\frac{e^{- \\nabla} \\int e^{\\nabla} d\\nabla}{\\operatorname{A_{z}}{(\\nabla)}} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\nabla', commutative=True)), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Function('A_z')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('A_z')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 2, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Function('A_z')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Pow(Function('A_z')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Integral(Mul(Pow(Function('A_z')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\varepsilon_0,L)} = - L + \\varepsilon_0, then obtain \\log{(\\int \\operatorname{E_{x}}{(\\varepsilon_0,L)} dL)} = \\log{(\\int (- L + \\varepsilon_0) dL)}", "derivation": "\\operatorname{E_{x}}{(\\varepsilon_0,L)} = - L + \\varepsilon_0 and \\int \\operatorname{E_{x}}{(\\varepsilon_0,L)} dL = \\int (- L + \\varepsilon_0) dL and \\int \\operatorname{E_{x}}{(\\varepsilon_0,L)} dL = \\int - L dL + \\int \\varepsilon_0 dL and \\log{(\\int \\operatorname{E_{x}}{(\\varepsilon_0,L)} dL)} = \\log{(\\int - L dL + \\int \\varepsilon_0 dL)} and \\int (- L + \\varepsilon_0) dL = \\int - L dL + \\int \\varepsilon_0 dL and \\log{(\\int \\operatorname{E_{x}}{(\\varepsilon_0,L)} dL)} = \\log{(\\int (- L + \\varepsilon_0) dL)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Integral(Mul(Integer(-1), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('L', commutative=True)))))"], [["log", 3], "Equality(log(Integral(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), log(Add(Integral(Mul(Integer(-1), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Integral(Mul(Integer(-1), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Integral(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), log(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(F_{x})} = \\log{(\\cos{(F_{x})})} and \\operatorname{a^{\\dagger}}{(F_{x})} = \\log{(\\cos{(F_{x})})}, then obtain \\operatorname{a^{\\dagger}}{(F_{x})} \\cos{(F_{x})} = \\log{(\\cos{(F_{x})})} \\cos{(F_{x})}", "derivation": "\\operatorname{v_{2}}{(F_{x})} = \\log{(\\cos{(F_{x})})} and \\operatorname{a^{\\dagger}}{(F_{x})} = \\log{(\\cos{(F_{x})})} and \\operatorname{a^{\\dagger}}{(F_{x})} = \\operatorname{v_{2}}{(F_{x})} and \\operatorname{a^{\\dagger}}{(F_{x})} \\cos{(F_{x})} = \\operatorname{v_{2}}{(F_{x})} \\cos{(F_{x})} and \\log{(\\cos{(F_{x})})} \\cos{(F_{x})} = \\operatorname{v_{2}}{(F_{x})} \\cos{(F_{x})} and \\operatorname{a^{\\dagger}}{(F_{x})} \\cos{(F_{x})} = \\log{(\\cos{(F_{x})})} \\cos{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('F_x', commutative=True)), log(cos(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), log(cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), Function('v_2')(Symbol('F_x', commutative=True)))"], [["times", 3, "cos(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Mul(Function('v_2')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(log(cos(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))), Mul(Function('v_2')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Mul(log(cos(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(t)} = e^{\\sin{(t)}}, then obtain t \\sigma_{x}{(t)} = t e^{\\sigma_{x}{(t)} e^{- \\sin{(t)}} \\sin{(t)}}", "derivation": "\\sigma_{x}{(t)} = e^{\\sin{(t)}} and t \\sigma_{x}{(t)} = t e^{\\sin{(t)}} and t \\sigma_{x}{(t)} \\sin{(t)} = t e^{\\sin{(t)}} \\sin{(t)} and \\sigma_{x}{(t)} e^{- \\sin{(t)}} \\sin{(t)} = \\sin{(t)} and t \\sigma_{x}{(t)} = t e^{\\sigma_{x}{(t)} e^{- \\sin{(t)}} \\sin{(t)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('t', commutative=True)), exp(sin(Symbol('t', commutative=True))))"], [["times", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\sigma_x')(Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), exp(sin(Symbol('t', commutative=True)))))"], [["times", 2, "sin(Symbol('t', commutative=True))"], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\sigma_x')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), exp(sin(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))))"], [["divide", 3, "Mul(Symbol('t', commutative=True), exp(sin(Symbol('t', commutative=True))))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('t', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True)))), sin(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\sigma_x')(Symbol('t', commutative=True))), Mul(Symbol('t', commutative=True), exp(Mul(Function('\\\\sigma_x')(Symbol('t', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('t', commutative=True)))), sin(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(U,\\hat{X})} = (e^{\\hat{X}})^{U}, then obtain 0 = (- ((e^{\\hat{X}})^{U})^{\\hat{X}} + \\mu_{0}^{\\hat{X}}{(U,\\hat{X})}) ((e^{\\hat{X}})^{U})^{\\hat{X}}", "derivation": "\\mu_{0}{(U,\\hat{X})} = (e^{\\hat{X}})^{U} and \\mu_{0}^{\\hat{X}}{(U,\\hat{X})} = ((e^{\\hat{X}})^{U})^{\\hat{X}} and - \\mu_{0}^{\\hat{X}}{(U,\\hat{X})} = - ((e^{\\hat{X}})^{U})^{\\hat{X}} and 0 = - ((e^{\\hat{X}})^{U})^{\\hat{X}} + \\mu_{0}^{\\hat{X}}{(U,\\hat{X})} and 0 = (- ((e^{\\hat{X}})^{U})^{\\hat{X}} + \\mu_{0}^{\\hat{X}}{(U,\\hat{X})}) ((e^{\\hat{X}})^{U})^{\\hat{X}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(-1), Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["times", 4, "Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Pow(Pow(exp(Symbol('\\\\hat{X}', commutative=True)), Symbol('U', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain 1 = \\frac{e^{4 \\mathbf{B}}}{\\operatorname{F_{x}}^{4}{(\\mathbf{B})}}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{B})} = e^{\\mathbf{B}} and 1 = \\frac{e^{\\mathbf{B}}}{\\operatorname{F_{x}}{(\\mathbf{B})}} and \\frac{e^{\\mathbf{B}}}{\\operatorname{F_{x}}{(\\mathbf{B})}} = \\frac{e^{2 \\mathbf{B}}}{\\operatorname{F_{x}}^{2}{(\\mathbf{B})}} and \\frac{e^{2 \\mathbf{B}}}{\\operatorname{F_{x}}^{2}{(\\mathbf{B})}} = \\frac{e^{4 \\mathbf{B}}}{\\operatorname{F_{x}}^{4}{(\\mathbf{B})}} and 1 = \\frac{e^{2 \\mathbf{B}}}{\\operatorname{F_{x}}^{2}{(\\mathbf{B})}} and 1 = \\frac{e^{4 \\mathbf{B}}}{\\operatorname{F_{x}}^{4}{(\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 1, "Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 2, "Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-4)), exp(Mul(Integer(4), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-4)), exp(Mul(Integer(4), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\mu{(k,q)} = \\log{(- k + q)} and I{(k,q)} = \\int \\log{(- k + q)} dk, then obtain - \\frac{\\int \\mu{(k,q)} dk}{k} = - \\frac{I{(k,q)}}{k}", "derivation": "\\mu{(k,q)} = \\log{(- k + q)} and \\int \\mu{(k,q)} dk = \\int \\log{(- k + q)} dk and I{(k,q)} = \\int \\log{(- k + q)} dk and \\int \\mu{(k,q)} dk = I{(k,q)} and - \\frac{\\int \\mu{(k,q)} dk}{k} = - \\frac{I{(k,q)}}{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('q', commutative=True)), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Integral(log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('k', commutative=True))), Function('I')(Symbol('k', commutative=True), Symbol('q', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Function('I')(Symbol('k', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(r,\\hat{p}_0)} = \\frac{\\partial}{\\partial r} (\\hat{p}_0 + r), then derive \\int \\theta_{2}{(r,\\hat{p}_0)} dr = \\mathbf{p} + r, then obtain (\\mathbf{S} + r)^{\\mathbf{p}} = (\\mathbf{p} + r)^{\\mathbf{p}}", "derivation": "\\theta_{2}{(r,\\hat{p}_0)} = \\frac{\\partial}{\\partial r} (\\hat{p}_0 + r) and \\int \\theta_{2}{(r,\\hat{p}_0)} dr = \\int \\frac{\\partial}{\\partial r} (\\hat{p}_0 + r) dr and \\int \\theta_{2}{(r,\\hat{p}_0)} dr = \\mathbf{p} + r and \\int \\frac{\\partial}{\\partial r} (\\hat{p}_0 + r) dr = \\mathbf{p} + r and (\\int \\frac{\\partial}{\\partial r} (\\hat{p}_0 + r) dr)^{\\mathbf{p}} = (\\mathbf{p} + r)^{\\mathbf{p}} and (\\mathbf{S} + r)^{\\mathbf{p}} = (\\mathbf{p} + r)^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_2')(Symbol('r', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Integral(Derivative(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given U{(E_{\\lambda},F_{x})} = \\cos{(E_{\\lambda} + F_{x})}, then derive \\int U{(E_{\\lambda},F_{x})} dF_{x} = V_{\\mathbf{E}} + \\sin{(E_{\\lambda} + F_{x})}, then obtain \\int \\cos{(E_{\\lambda} + F_{x})} dF_{x} = V_{\\mathbf{E}} + \\sin{(E_{\\lambda} + F_{x})}", "derivation": "U{(E_{\\lambda},F_{x})} = \\cos{(E_{\\lambda} + F_{x})} and \\int U{(E_{\\lambda},F_{x})} dF_{x} = \\int \\cos{(E_{\\lambda} + F_{x})} dF_{x} and \\int U{(E_{\\lambda},F_{x})} dF_{x} = V_{\\mathbf{E}} + \\sin{(E_{\\lambda} + F_{x})} and \\int \\cos{(E_{\\lambda} + F_{x})} dF_{x} = V_{\\mathbf{E}} + \\sin{(E_{\\lambda} + F_{x})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), cos(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('U')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(cos(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('U')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\varepsilon)} = \\sin{(\\varepsilon)}, then obtain \\int 0 d\\varepsilon - 1 = \\cos{(\\operatorname{c_{0}}{(\\varepsilon)} - \\sin{(\\varepsilon)})} \\int 0 d\\varepsilon - 1", "derivation": "\\operatorname{c_{0}}{(\\varepsilon)} = \\sin{(\\varepsilon)} and 0 = - \\operatorname{c_{0}}{(\\varepsilon)} + \\sin{(\\varepsilon)} and \\int 0 d\\varepsilon = \\int (- \\operatorname{c_{0}}{(\\varepsilon)} + \\sin{(\\varepsilon)}) d\\varepsilon and 1 = \\cos{(\\operatorname{c_{0}}{(\\varepsilon)} - \\sin{(\\varepsilon)})} and \\int (- \\operatorname{c_{0}}{(\\varepsilon)} + \\sin{(\\varepsilon)}) d\\varepsilon = \\cos{(\\operatorname{c_{0}}{(\\varepsilon)} - \\sin{(\\varepsilon)})} \\int (- \\operatorname{c_{0}}{(\\varepsilon)} + \\sin{(\\varepsilon)}) d\\varepsilon and \\int 0 d\\varepsilon = \\cos{(\\operatorname{c_{0}}{(\\varepsilon)} - \\sin{(\\varepsilon)})} \\int 0 d\\varepsilon and \\int 0 d\\varepsilon - 1 = \\cos{(\\operatorname{c_{0}}{(\\varepsilon)} - \\sin{(\\varepsilon)})} \\int 0 d\\varepsilon - 1", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "Function('c_0')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["cos", 2], "Equality(Integer(1), cos(Add(Function('c_0')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))))"], [["times", 4, "Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Mul(cos(Add(Function('c_0')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))), Integral(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\varepsilon', commutative=True))), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))), Mul(cos(Add(Function('c_0')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Add(Mul(cos(Add(Function('c_0')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} + y^{\\prime} and \\operatorname{L_{\\varepsilon}}{(\\mathbf{v},y^{\\prime})} = (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}}, then obtain - \\mathbf{v} \\operatorname{L_{\\varepsilon}}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}}", "derivation": "\\hat{x}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} + y^{\\prime} and \\hat{x}^{y^{\\prime}}{(\\mathbf{v},y^{\\prime})} = (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}} and - \\mathbf{v} \\hat{x}^{y^{\\prime}}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}} and \\operatorname{L_{\\varepsilon}}{(\\mathbf{v},y^{\\prime})} = (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}} and - \\mathbf{v} \\hat{x}^{y^{\\prime}}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} \\operatorname{L_{\\varepsilon}}{(\\mathbf{v},y^{\\prime})} and - \\mathbf{v} \\operatorname{L_{\\varepsilon}}{(\\mathbf{v},y^{\\prime})} = - \\mathbf{v} (- \\mathbf{v} + y^{\\prime})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(A)} = \\log{(A)} and Z{(A)} = \\log{(A)}, then obtain \\log{(A)} + \\frac{d}{d A} \\operatorname{m_{s}}{(A)} = \\log{(A)} + \\frac{d}{d A} Z{(A)}", "derivation": "\\operatorname{m_{s}}{(A)} = \\log{(A)} and \\frac{d}{d A} \\operatorname{m_{s}}{(A)} = \\frac{d}{d A} \\log{(A)} and Z{(A)} = \\log{(A)} and \\frac{d}{d A} \\operatorname{m_{s}}{(A)} = \\frac{d}{d A} Z{(A)} and \\log{(A)} + \\frac{d}{d A} \\operatorname{m_{s}}{(A)} = \\log{(A)} + \\frac{d}{d A} Z{(A)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('m_s')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Function('Z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["add", 4, "log(Symbol('A', commutative=True))"], "Equality(Add(log(Symbol('A', commutative=True)), Derivative(Function('m_s')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(log(Symbol('A', commutative=True)), Derivative(Function('Z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\rho_b)} = \\int \\sin{(\\rho_b)} d\\rho_b, then derive \\operatorname{n_{2}}{(\\rho_b)} = \\rho - \\cos{(\\rho_b)}, then obtain \\frac{d}{d \\rho_b} \\operatorname{n_{2}}{(\\rho_b)} \\cos{(\\rho_b)} = \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} \\int \\sin{(\\rho_b)} d\\rho_b", "derivation": "\\operatorname{n_{2}}{(\\rho_b)} = \\int \\sin{(\\rho_b)} d\\rho_b and \\operatorname{n_{2}}{(\\rho_b)} = \\rho - \\cos{(\\rho_b)} and \\operatorname{n_{2}}{(\\rho_b)} \\cos{(\\rho_b)} = (\\rho - \\cos{(\\rho_b)}) \\cos{(\\rho_b)} and \\cos{(\\rho_b)} \\int \\sin{(\\rho_b)} d\\rho_b = (\\rho - \\cos{(\\rho_b)}) \\cos{(\\rho_b)} and \\operatorname{n_{2}}{(\\rho_b)} \\cos{(\\rho_b)} = \\cos{(\\rho_b)} \\int \\sin{(\\rho_b)} d\\rho_b and \\frac{d}{d \\rho_b} \\operatorname{n_{2}}{(\\rho_b)} \\cos{(\\rho_b)} = \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} \\int \\sin{(\\rho_b)} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\rho_b', commutative=True)), Integral(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n_2')(Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_b', commutative=True)))))"], [["times", 2, "cos(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('n_2')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Mul(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_b', commutative=True)))), cos(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(cos(Symbol('\\\\rho_b', commutative=True)), Integral(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_b', commutative=True)))), cos(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('n_2')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Mul(cos(Symbol('\\\\rho_b', commutative=True)), Integral(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Function('n_2')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('\\\\rho_b', commutative=True)), Integral(sin(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = - C_{2} - \\phi + b, then obtain - b + \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = - b - \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)}", "derivation": "\\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = - C_{2} - \\phi + b and \\frac{\\partial}{\\partial \\phi} \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = \\frac{\\partial}{\\partial \\phi} (- C_{2} - \\phi + b) and \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} (- C_{2} - \\phi + b) and - b + \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = - b + \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} (- C_{2} - \\phi + b) and - b + \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} \\frac{\\partial}{\\partial \\phi} \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)} = - b - \\operatorname{f^{\\prime}}{(b,C_{2},\\phi)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["times", 2, "Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["minus", 3, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(E_{\\lambda},B)} = \\frac{B}{E_{\\lambda}}, then derive \\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},B)} = - \\frac{B}{E_{\\lambda}^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},B)}}{E_{\\lambda}^{2}} = - \\frac{B}{E_{\\lambda}^{4}}", "derivation": "\\hat{p}_0{(E_{\\lambda},B)} = \\frac{B}{E_{\\lambda}} and \\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},B)} = \\frac{\\partial}{\\partial E_{\\lambda}} \\frac{B}{E_{\\lambda}} and \\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},B)} = - \\frac{B}{E_{\\lambda}^{2}} and \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},B)}}{E_{\\lambda}^{2}} = - \\frac{B}{E_{\\lambda}^{4}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-2))))"], [["times", 3, "Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-2)), Derivative(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-4))))"]]}, {"prompt": "Given f{(A_{x})} = \\log{(A_{x})}, then obtain f^{A_{x}}{(A_{x})} - \\log{(A_{x})} - 2 \\log{(A_{x})}^{A_{x}} = - \\log{(A_{x})} - \\log{(A_{x})}^{A_{x}}", "derivation": "f{(A_{x})} = \\log{(A_{x})} and f^{A_{x}}{(A_{x})} = \\log{(A_{x})}^{A_{x}} and f^{A_{x}}{(A_{x})} + \\log{(A_{x})} = \\log{(A_{x})} + \\log{(A_{x})}^{A_{x}} and f^{A_{x}}{(A_{x})} - \\log{(A_{x})}^{A_{x}} = 0 and f^{A_{x}}{(A_{x})} - \\log{(A_{x})} - 2 \\log{(A_{x})}^{A_{x}} = - \\log{(A_{x})} - \\log{(A_{x})}^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["add", 2, "log(Symbol('A_x', commutative=True))"], "Equality(Add(Pow(Function('f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True))), Add(log(Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))))"], [["minus", 3, "Add(log(Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], "Equality(Add(Pow(Function('f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))), Integer(0))"], [["minus", 4, "Add(log(Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], "Equality(Add(Pow(Function('f')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Integer(-1), Integer(2), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\dot{\\mathbf{r}} - \\tilde{g}^*, then obtain (- 2 \\tilde{g}^* + 2 \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)})^{2} = (\\dot{\\mathbf{r}} - 3 \\tilde{g}^* + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)})^{2}", "derivation": "\\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\dot{\\mathbf{r}} - \\tilde{g}^* and - \\tilde{g}^* + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\dot{\\mathbf{r}} - 2 \\tilde{g}^* and - 2 \\tilde{g}^* + 2 \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\dot{\\mathbf{r}} - 3 \\tilde{g}^* + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} and (- 2 \\tilde{g}^* + 2 \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)})^{2} = (\\dot{\\mathbf{r}} - 3 \\tilde{g}^* + \\operatorname{A_{y}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)})^{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Integer(3), Symbol('\\\\tilde{g}^*', commutative=True)), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(2)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Integer(3), Symbol('\\\\tilde{g}^*', commutative=True)), Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{g}{(A_{y})} = \\log{(A_{y})}, then derive \\int \\mathbf{g}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + \\eta^{\\prime}, then obtain \\int \\mathbf{g}{(A_{y})} dA_{y} + 1 = A_{y} \\mathbf{g}{(A_{y})} - A_{y} + \\eta^{\\prime} + 1", "derivation": "\\mathbf{g}{(A_{y})} = \\log{(A_{y})} and \\int \\mathbf{g}{(A_{y})} dA_{y} = \\int \\log{(A_{y})} dA_{y} and \\int \\mathbf{g}{(A_{y})} dA_{y} = A_{y} \\log{(A_{y})} - A_{y} + \\eta^{\\prime} and \\int \\mathbf{g}{(A_{y})} dA_{y} = A_{y} \\mathbf{g}{(A_{y})} - A_{y} + \\eta^{\\prime} and \\int \\mathbf{g}{(A_{y})} dA_{y} + 1 = A_{y} \\mathbf{g}{(A_{y})} - A_{y} + \\eta^{\\prime} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), log(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integer(1)), Add(Mul(Symbol('A_y', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"]]}, {"prompt": "Given W{(C,\\Psi_{nl})} = \\int C^{\\Psi_{nl}} d\\Psi_{nl} and L{(C,\\Psi_{nl})} = - (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})})^{2}, then obtain L{(C,\\Psi_{nl})} = - (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})}) (- C^{\\Psi_{nl}} + \\int C^{\\Psi_{nl}} d\\Psi_{nl})", "derivation": "W{(C,\\Psi_{nl})} = \\int C^{\\Psi_{nl}} d\\Psi_{nl} and - C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})} = - C^{\\Psi_{nl}} + \\int C^{\\Psi_{nl}} d\\Psi_{nl} and (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})})^{2} = (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})}) (- C^{\\Psi_{nl}} + \\int C^{\\Psi_{nl}} d\\Psi_{nl}) and L{(C,\\Psi_{nl})} = - (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})})^{2} and L{(C,\\Psi_{nl})} = - (- C^{\\Psi_{nl}} + W{(C,\\Psi_{nl})}) (- C^{\\Psi_{nl}} + \\int C^{\\Psi_{nl}} d\\Psi_{nl})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 1, "Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"], ["renaming_premise", "Equality(Function('L')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('L')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Function('W')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Pow(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(V,\\hat{p})} = V \\hat{p} and \\hat{H}_{\\lambda}{(V,\\hat{p})} = \\int \\operatorname{m_{s}}^{V}{(V,\\hat{p})} dV, then obtain \\hat{H}_{\\lambda}{(V,\\hat{p})} \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} = \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} \\int (V \\hat{p})^{V} dV", "derivation": "\\operatorname{m_{s}}{(V,\\hat{p})} = V \\hat{p} and \\operatorname{m_{s}}^{V}{(V,\\hat{p})} = (V \\hat{p})^{V} and \\int \\operatorname{m_{s}}^{V}{(V,\\hat{p})} dV = \\int (V \\hat{p})^{V} dV and \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} \\int \\operatorname{m_{s}}^{V}{(V,\\hat{p})} dV = \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} \\int (V \\hat{p})^{V} dV and \\hat{H}_{\\lambda}{(V,\\hat{p})} = \\int \\operatorname{m_{s}}^{V}{(V,\\hat{p})} dV and \\hat{H}_{\\lambda}{(V,\\hat{p})} \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} = \\operatorname{m_{s}}^{- V}{(V,\\hat{p})} \\int (V \\hat{p})^{V} dV", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Pow(Mul(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(Mul(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["divide", 3, "Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))), Integral(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))), Integral(Pow(Mul(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integral(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), Mul(Pow(Function('m_s')(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))), Integral(Pow(Mul(Symbol('V', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{H})} = e^{\\cos{(\\mathbf{H})}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} = e^{\\cos{(\\mathbf{H})}} \\cos{(\\mathbf{H})}, then obtain \\int \\operatorname{v_{x}}{(\\mathbf{H})} \\cos{(\\mathbf{H})} d\\mathbf{H} = \\int e^{\\cos{(\\mathbf{H})}} \\cos{(\\mathbf{H})} d\\mathbf{H}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{H})} = e^{\\cos{(\\mathbf{H})}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} = e^{\\cos{(\\mathbf{H})}} \\cos{(\\mathbf{H})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} d\\mathbf{H} = \\int e^{\\cos{(\\mathbf{H})}} \\cos{(\\mathbf{H})} d\\mathbf{H} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{H})} = \\operatorname{v_{x}}{(\\mathbf{H})} \\cos{(\\mathbf{H})} and \\int \\operatorname{v_{x}}{(\\mathbf{H})} \\cos{(\\mathbf{H})} d\\mathbf{H} = \\int e^{\\cos{(\\mathbf{H})}} \\cos{(\\mathbf{H})} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), exp(cos(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(exp(cos(Symbol('\\\\mathbf{H}', commutative=True))), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(exp(cos(Symbol('\\\\mathbf{H}', commutative=True))), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integral(Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(exp(cos(Symbol('\\\\mathbf{H}', commutative=True))), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given f{(F_{x})} = \\sin{(F_{x})}, then derive \\int f{(F_{x})} dF_{x} = \\dot{x} - \\cos{(F_{x})}, then obtain \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} - \\cos{(F_{x})}) = \\frac{d}{d \\dot{x}} \\int \\sin{(F_{x})} dF_{x}", "derivation": "f{(F_{x})} = \\sin{(F_{x})} and \\int f{(F_{x})} dF_{x} = \\int \\sin{(F_{x})} dF_{x} and \\int f{(F_{x})} dF_{x} = \\dot{x} - \\cos{(F_{x})} and \\dot{x} - \\cos{(F_{x})} = \\int \\sin{(F_{x})} dF_{x} and \\frac{d}{d \\dot{x}} \\int f{(F_{x})} dF_{x} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} - \\cos{(F_{x})}) and \\frac{d}{d \\dot{x}} \\int f{(F_{x})} dF_{x} = \\frac{d}{d \\dot{x}} \\int \\sin{(F_{x})} dF_{x} and \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} - \\cos{(F_{x})}) = \\frac{d}{d \\dot{x}} \\int \\sin{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('f')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Integral(Function('f')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integral(Function('f')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(i,\\dot{y})} = \\sin{(\\dot{y} + i)}, then obtain \\frac{\\partial}{\\partial i} \\log{(\\dot{y} + \\int n^{i}{(i,\\dot{y})} di)} = \\frac{\\partial}{\\partial i} \\log{(\\dot{y} + \\int \\sin^{i}{(\\dot{y} + i)} di)}", "derivation": "n{(i,\\dot{y})} = \\sin{(\\dot{y} + i)} and n^{i}{(i,\\dot{y})} = \\sin^{i}{(\\dot{y} + i)} and \\int n^{i}{(i,\\dot{y})} di = \\int \\sin^{i}{(\\dot{y} + i)} di and \\dot{y} + \\int n^{i}{(i,\\dot{y})} di = \\dot{y} + \\int \\sin^{i}{(\\dot{y} + i)} di and \\log{(\\dot{y} + \\int n^{i}{(i,\\dot{y})} di)} = \\log{(\\dot{y} + \\int \\sin^{i}{(\\dot{y} + i)} di)} and \\frac{\\partial}{\\partial i} \\log{(\\dot{y} + \\int n^{i}{(i,\\dot{y})} di)} = \\frac{\\partial}{\\partial i} \\log{(\\dot{y} + \\int \\sin^{i}{(\\dot{y} + i)} di)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["add", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["log", 4], "Equality(log(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), log(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(log(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(Function('n')(Symbol('i', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Pow(sin(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(\\phi_1,\\lambda)} = \\frac{\\phi_1}{\\lambda}, then obtain \\frac{\\lambda^{2} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} + \\frac{1}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} = \\frac{\\lambda^{4} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\phi_1^{2}} + \\frac{1}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}}", "derivation": "\\sigma_{x}{(\\phi_1,\\lambda)} = \\frac{\\phi_1}{\\lambda} and \\frac{\\sigma_{x}{(\\phi_1,\\lambda)}}{\\lambda} = \\frac{\\phi_1}{\\lambda^{2}} and \\frac{\\lambda^{2}}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} = \\frac{\\lambda^{4}}{\\phi_1^{2}} and \\frac{\\lambda^{2} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} = \\frac{\\lambda^{4} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\phi_1^{2}} and \\frac{\\lambda^{2} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} + \\frac{1}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}} = \\frac{\\lambda^{4} (\\frac{\\phi_1}{\\lambda})^{- \\phi_1}}{\\phi_1^{2}} + \\frac{1}{\\sigma_{x}^{2}{(\\phi_1,\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\phi_1', commutative=True)))"], [["power", 2, "Integer(-2)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(2)), Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(4)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2))))"], [["divide", 3, "Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(2)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(4)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))))"], [["add", 4, "Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))"], "Equality(Add(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(2)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))), Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))), Add(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(4)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Pow(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given U{(\\varepsilon_0,u)} = \\frac{u}{\\varepsilon_0} and H{(\\varepsilon_0,u)} = u + \\frac{u}{\\varepsilon_0}, then obtain (u + U{(\\varepsilon_0,u)})^{u} - U{(\\varepsilon_0,u)} = H^{u}{(\\varepsilon_0,u)} - U{(\\varepsilon_0,u)}", "derivation": "U{(\\varepsilon_0,u)} = \\frac{u}{\\varepsilon_0} and u + U{(\\varepsilon_0,u)} = u + \\frac{u}{\\varepsilon_0} and H{(\\varepsilon_0,u)} = u + \\frac{u}{\\varepsilon_0} and (u + U{(\\varepsilon_0,u)})^{u} = (u + \\frac{u}{\\varepsilon_0})^{u} and (u + U{(\\varepsilon_0,u)})^{u} - U{(\\varepsilon_0,u)} = (u + \\frac{u}{\\varepsilon_0})^{u} - U{(\\varepsilon_0,u)} and (u + U{(\\varepsilon_0,u)})^{u} - U{(\\varepsilon_0,u)} = H^{u}{(\\varepsilon_0,u)} - U{(\\varepsilon_0,u)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Add(Symbol('u', commutative=True), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["minus", 4, "Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Pow(Add(Symbol('u', commutative=True), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)))), Add(Pow(Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Symbol('u', commutative=True), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)))), Add(Pow(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given c{(v_{t})} = \\log{(e^{v_{t}})}, then obtain \\frac{d}{d v_{t}} v_{t} = \\frac{d}{d v_{t}} \\frac{v_{t} \\log{(e^{v_{t}})}}{c{(v_{t})}}", "derivation": "c{(v_{t})} = \\log{(e^{v_{t}})} and v_{t} c{(v_{t})} = v_{t} \\log{(e^{v_{t}})} and v_{t} = \\frac{v_{t} \\log{(e^{v_{t}})}}{c{(v_{t})}} and \\frac{d}{d v_{t}} v_{t} = \\frac{d}{d v_{t}} \\frac{v_{t} \\log{(e^{v_{t}})}}{c{(v_{t})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('v_t', commutative=True)), log(exp(Symbol('v_t', commutative=True))))"], [["times", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('c')(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), log(exp(Symbol('v_t', commutative=True)))))"], [["divide", 2, "Function('c')(Symbol('v_t', commutative=True))"], "Equality(Symbol('v_t', commutative=True), Mul(Symbol('v_t', commutative=True), Pow(Function('c')(Symbol('v_t', commutative=True)), Integer(-1)), log(exp(Symbol('v_t', commutative=True)))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Symbol('v_t', commutative=True), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_t', commutative=True), Pow(Function('c')(Symbol('v_t', commutative=True)), Integer(-1)), log(exp(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(t_{1})} = \\sin{(\\sin{(t_{1})})} and c{(t_{1})} = \\sin{(\\sin{(t_{1})})}, then obtain c^{t_{1}}{(t_{1})} = \\sin^{t_{1}}{(\\sin{(t_{1})})}", "derivation": "\\varepsilon{(t_{1})} = \\sin{(\\sin{(t_{1})})} and \\varepsilon^{t_{1}}{(t_{1})} = \\sin^{t_{1}}{(\\sin{(t_{1})})} and c{(t_{1})} = \\sin{(\\sin{(t_{1})})} and \\varepsilon^{t_{1}}{(t_{1})} = c^{t_{1}}{(t_{1})} and c^{t_{1}}{(t_{1})} = \\sin^{t_{1}}{(\\sin{(t_{1})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), sin(sin(Symbol('t_1', commutative=True))))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(sin(sin(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('t_1', commutative=True)), sin(sin(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Function('c')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('c')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(sin(sin(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} = A_{x}^{\\mathbb{I}} + \\mathbf{H}, then obtain 0 = 1 - \\frac{\\int \\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} dA_{x}}{\\int (A_{x}^{\\mathbb{I}} + \\mathbf{H}) dA_{x}}", "derivation": "\\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} = A_{x}^{\\mathbb{I}} + \\mathbf{H} and \\int \\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} dA_{x} = \\int (A_{x}^{\\mathbb{I}} + \\mathbf{H}) dA_{x} and \\frac{\\int \\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} dA_{x}}{\\int (A_{x}^{\\mathbb{I}} + \\mathbf{H}) dA_{x}} = 1 and 0 = 1 - \\frac{\\int \\operatorname{M_{E}}{(A_{x},\\mathbb{I},\\mathbf{H})} dA_{x}}{\\int (A_{x}^{\\mathbb{I}} + \\mathbf{H}) dA_{x}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["divide", 2, "Integral(Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integer(-1)), Integral(Function('M_E')(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Integer(1))"], [["minus", 3, "Mul(Pow(Integral(Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integer(-1)), Integral(Function('M_E')(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Integral(Add(Pow(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integer(-1)), Integral(Function('M_E')(Symbol('A_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(T,F_{x})} = \\frac{\\partial}{\\partial T} F_{x} T and \\hat{p}{(T,F_{x})} = (\\frac{\\partial}{\\partial T} F_{x} T)^{T}, then obtain \\hat{p}{(T,F_{x})} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x} = (\\frac{\\partial}{\\partial T} F_{x} T)^{T} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x}", "derivation": "\\operatorname{C_{2}}{(T,F_{x})} = \\frac{\\partial}{\\partial T} F_{x} T and \\operatorname{C_{2}}^{T}{(T,F_{x})} = (\\frac{\\partial}{\\partial T} F_{x} T)^{T} and \\hat{p}{(T,F_{x})} = (\\frac{\\partial}{\\partial T} F_{x} T)^{T} and \\hat{p}{(T,F_{x})} = \\operatorname{C_{2}}^{T}{(T,F_{x})} and \\hat{p}{(T,F_{x})} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x} = \\operatorname{C_{2}}^{T}{(T,F_{x})} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x} and \\hat{p}{(T,F_{x})} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x} = (\\frac{\\partial}{\\partial T} F_{x} T)^{T} + \\int (\\frac{\\partial}{\\partial T} F_{x} T)^{T} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Symbol('T', commutative=True)), Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Pow(Function('C_2')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Symbol('T', commutative=True)))"], [["add", 4, "Integral(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Integral(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Pow(Function('C_2')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Symbol('T', commutative=True)), Integral(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Integral(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Add(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Integral(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(\\pi)} = e^{\\pi}, then derive \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = e^{\\pi}, then obtain \\int (- c{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\hat{x}{(\\pi)} + \\frac{d}{d \\pi} e^{\\pi}) d\\pi = \\int (- c{(\\pi)} + 2 \\frac{d}{d \\pi} e^{\\pi}) d\\pi", "derivation": "\\hat{x}{(\\pi)} = e^{\\pi} and \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} and \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = e^{\\pi} and \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = \\hat{x}{(\\pi)} and \\frac{d^{2}}{d \\pi^{2}} \\hat{x}{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} and \\frac{d^{2}}{d \\pi^{2}} \\hat{x}{(\\pi)} + \\frac{d}{d \\pi} e^{\\pi} = 2 \\frac{d}{d \\pi} e^{\\pi} and - c{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\hat{x}{(\\pi)} + \\frac{d}{d \\pi} e^{\\pi} = - c{(\\pi)} + 2 \\frac{d}{d \\pi} e^{\\pi} and \\int (- c{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\hat{x}{(\\pi)} + \\frac{d}{d \\pi} e^{\\pi}) d\\pi = \\int (- c{(\\pi)} + 2 \\frac{d}{d \\pi} e^{\\pi}) d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), exp(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 5, "Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["minus", 6, "Function('c')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('c')(Symbol('\\\\pi', commutative=True))), Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["integrate", 7, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('c')(Symbol('\\\\pi', commutative=True))), Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c')(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given T{(S,z)} = S + z, then derive \\int T{(S,z)} dS = J + \\frac{S^{2}}{2} + S z, then obtain (J + \\frac{S^{2}}{2} + S z) \\iint T{(S,z)} dS dS = (J + \\frac{S^{2}}{2} + S z) \\iint (S + z) dS dS", "derivation": "T{(S,z)} = S + z and \\int T{(S,z)} dS = \\int (S + z) dS and \\iint T{(S,z)} dS dS = \\iint (S + z) dS dS and (\\int (S + z) dS) \\iint T{(S,z)} dS dS = (\\int (S + z) dS) \\iint (S + z) dS dS and \\int T{(S,z)} dS = J + \\frac{S^{2}}{2} + S z and (\\int T{(S,z)} dS) \\iint T{(S,z)} dS dS = (\\int T{(S,z)} dS) \\iint (S + z) dS dS and (J + \\frac{S^{2}}{2} + S z) \\iint T{(S,z)} dS dS = (J + \\frac{S^{2}}{2} + S z) \\iint (S + z) dS dS", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Add(Symbol('S', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["times", 3, "Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integral(Function('T')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(F_{x})} = \\cos{(F_{x})} and \\phi{(F_{x})} = \\cos{(F_{x})}, then obtain \\Psi_{\\lambda}{(F_{x})} = (- \\Psi_{\\lambda}{(F_{x})} + \\phi{(F_{x})}) (- \\phi{(F_{x})} + \\int (\\Psi_{\\lambda}{(F_{x})} - \\phi{(F_{x})}) dF_{x}) + \\Psi_{\\lambda}{(F_{x})}", "derivation": "\\Psi_{\\lambda}{(F_{x})} = \\cos{(F_{x})} and \\phi{(F_{x})} = \\cos{(F_{x})} and 0 = - \\phi{(F_{x})} + \\cos{(F_{x})} and \\phi{(F_{x})} = \\Psi_{\\lambda}{(F_{x})} and 0 = - \\Psi_{\\lambda}{(F_{x})} + \\cos{(F_{x})} and 0 = - \\Psi_{\\lambda}{(F_{x})} + \\phi{(F_{x})} and 0 = (- \\Psi_{\\lambda}{(F_{x})} + \\phi{(F_{x})}) (- \\phi{(F_{x})} + \\int (\\Psi_{\\lambda}{(F_{x})} - \\phi{(F_{x})}) dF_{x}) and \\Psi_{\\lambda}{(F_{x})} = (- \\Psi_{\\lambda}{(F_{x})} + \\phi{(F_{x})}) (- \\phi{(F_{x})} + \\int (\\Psi_{\\lambda}{(F_{x})} - \\phi{(F_{x})}) dF_{x}) + \\Psi_{\\lambda}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["minus", 2, "Function('\\\\phi')(Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\phi')(Symbol('F_x', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))), Function('\\\\phi')(Symbol('F_x', commutative=True))))"], [["times", 6, "Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True))), Integral(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))), Function('\\\\phi')(Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True))), Integral(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))))"], [["add", 7, "Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))"], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))), Function('\\\\phi')(Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True))), Integral(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))), Function('\\\\Psi_{\\\\lambda}')(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\rho_f)} = e^{\\rho_f}, then derive \\int (- \\rho_f + \\operatorname{C_{d}}{(\\rho_f)}) d\\rho_f = \\rho - \\frac{\\rho_f^{2}}{2} + e^{\\rho_f}, then obtain \\int (- \\rho_f + e^{\\rho_f}) d\\rho_f = \\rho - \\frac{\\rho_f^{2}}{2} + \\operatorname{C_{d}}{(\\rho_f)}", "derivation": "\\operatorname{C_{d}}{(\\rho_f)} = e^{\\rho_f} and - \\rho_f + \\operatorname{C_{d}}{(\\rho_f)} = - \\rho_f + e^{\\rho_f} and \\int (- \\rho_f + \\operatorname{C_{d}}{(\\rho_f)}) d\\rho_f = \\int (- \\rho_f + e^{\\rho_f}) d\\rho_f and \\int (- \\rho_f + \\operatorname{C_{d}}{(\\rho_f)}) d\\rho_f = \\rho - \\frac{\\rho_f^{2}}{2} + e^{\\rho_f} and \\int (- \\rho_f + e^{\\rho_f}) d\\rho_f = \\rho - \\frac{\\rho_f^{2}}{2} + e^{\\rho_f} and \\int (- \\rho_f + e^{\\rho_f}) d\\rho_f = \\rho - \\frac{\\rho_f^{2}}{2} + \\operatorname{C_{d}}{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('C_d')(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('C_d')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('C_d')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))), exp(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))), exp(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2))), Function('C_d')(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given A{(k,V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}} - k)}, then derive \\int \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} dV_{\\mathbf{E}} = V_{\\mathbf{E}} + \\mathbf{B}, then obtain \\frac{\\int \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} dV_{\\mathbf{E}}}{V_{\\mathbf{E}}} = \\frac{V_{\\mathbf{E}} + \\mathbf{B}}{V_{\\mathbf{E}}}", "derivation": "A{(k,V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}} - k)} and \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} = 1 and \\int \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} dV_{\\mathbf{E}} = \\int 1 dV_{\\mathbf{E}} and \\int \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} dV_{\\mathbf{E}} = V_{\\mathbf{E}} + \\mathbf{B} and \\frac{\\int \\frac{A{(k,V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}} - k)}} dV_{\\mathbf{E}}}{V_{\\mathbf{E}}} = \\frac{V_{\\mathbf{E}} + \\mathbf{B}}{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["divide", 1, "sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], "Equality(Mul(Function('A')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Mul(Function('A')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('A')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 4, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Integral(Mul(Function('A')(Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\lambda{(n_{2},\\hbar)} = \\hbar n_{2}, then obtain - (\\hbar n_{2})^{\\hbar} + \\lambda{(n_{2},\\hbar)} = \\hbar n_{2} - (\\hbar n_{2})^{\\hbar}", "derivation": "\\lambda{(n_{2},\\hbar)} = \\hbar n_{2} and \\lambda^{\\hbar}{(n_{2},\\hbar)} = (\\hbar n_{2})^{\\hbar} and \\lambda{(n_{2},\\hbar)} - \\lambda^{\\hbar}{(n_{2},\\hbar)} = \\hbar n_{2} - \\lambda^{\\hbar}{(n_{2},\\hbar)} and - (\\hbar n_{2})^{\\hbar} + \\lambda{(n_{2},\\hbar)} = \\hbar n_{2} - (\\hbar n_{2})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))), Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\hbar', commutative=True))), Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(J,W)} = J^{W} and Q{(J,W)} = J^{W} \\log{(J)} + \\log{(J)}, then derive \\frac{\\partial}{\\partial W} \\bar{\\h}{(J,W)} = J^{W} \\log{(J)}, then derive \\frac{\\partial}{\\partial W} Q{(J,W)} = J^{W} \\log{(J)}^{2}, then obtain \\int \\frac{\\partial}{\\partial W} Q{(J,W)} dJ = \\int J^{W} \\log{(J)}^{2} dJ", "derivation": "\\bar{\\h}{(J,W)} = J^{W} and \\frac{\\partial}{\\partial W} \\bar{\\h}{(J,W)} = \\frac{\\partial}{\\partial W} J^{W} and \\frac{\\partial}{\\partial W} \\bar{\\h}{(J,W)} = J^{W} \\log{(J)} and \\frac{\\partial}{\\partial W} J^{W} = J^{W} \\log{(J)} and Q{(J,W)} = J^{W} \\log{(J)} + \\log{(J)} and \\frac{\\partial}{\\partial W} Q{(J,W)} = \\frac{\\partial}{\\partial W} (J^{W} \\log{(J)} + \\log{(J)}) and \\frac{\\partial}{\\partial W} Q{(J,W)} = \\frac{\\partial}{\\partial W} (\\log{(J)} + \\frac{\\partial}{\\partial W} J^{W}) and \\frac{\\partial}{\\partial W} Q{(J,W)} = J^{W} \\log{(J)}^{2} and \\int \\frac{\\partial}{\\partial W} Q{(J,W)} dJ = \\int J^{W} \\log{(J)}^{2} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), log(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), log(Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Add(Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), log(Symbol('J', commutative=True))), log(Symbol('J', commutative=True))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), log(Symbol('J', commutative=True))), log(Symbol('J', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Function('Q')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(log(Symbol('J', commutative=True)), Derivative(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('Q')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('J', commutative=True)), Integer(2))))"], [["integrate", 8, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('Q')(Symbol('J', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('J', commutative=True)), Integer(2))), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(v_{1},A_{1})} = - A_{1} + v_{1}, then obtain 6 A_{1} - 6 v_{1} + 6 \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 = 1", "derivation": "\\hat{H}_{\\lambda}{(v_{1},A_{1})} = - A_{1} + v_{1} and A_{1} - v_{1} + \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 = 1 and 2 A_{1} - 2 v_{1} + 2 \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 = A_{1} - v_{1} + \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 and 2 A_{1} - 2 v_{1} + 2 \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 = 1 and 2 A_{1} - 2 v_{1} + 3 \\hat{H}_{\\lambda}{(v_{1},A_{1})} = \\hat{H}_{\\lambda}{(v_{1},A_{1})} and 6 A_{1} - 6 v_{1} + 6 \\hat{H}_{\\lambda}{(v_{1},A_{1})} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)), Integer(1)), Integer(1))"], [["add", 2, "Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Integer(1)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Integer(1)), Integer(1))"], [["add", 4, "Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)), Mul(Integer(3), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(6), Symbol('A_1', commutative=True)), Mul(Integer(-1), Integer(6), Symbol('v_1', commutative=True)), Mul(Integer(6), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\varphi{(t_{1},\\mu_0)} = \\mu_0 + t_{1}, then derive \\int \\varphi{(t_{1},\\mu_0)} d\\mu_0 = \\ddot{x} + \\frac{\\mu_0^{2}}{2} + \\mu_0 t_{1}, then obtain \\int (\\mu_0 + t_{1}) d\\mu_0 = \\ddot{x} + \\frac{\\mu_0^{2}}{2} + \\mu_0 t_{1}", "derivation": "\\varphi{(t_{1},\\mu_0)} = \\mu_0 + t_{1} and \\int \\varphi{(t_{1},\\mu_0)} d\\mu_0 = \\int (\\mu_0 + t_{1}) d\\mu_0 and \\int \\varphi{(t_{1},\\mu_0)} d\\mu_0 = \\ddot{x} + \\frac{\\mu_0^{2}}{2} + \\mu_0 t_{1} and \\int (\\mu_0 + t_{1}) d\\mu_0 = \\ddot{x} + \\frac{\\mu_0^{2}}{2} + \\mu_0 t_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Symbol('\\\\mu_0', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\mu_0', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(a^{\\dagger},A_{y})} = A_{y}^{a^{\\dagger}} and \\hat{H}{(a^{\\dagger})} = a^{\\dagger}, then obtain \\frac{\\partial}{\\partial \\hat{H}{(a^{\\dagger})}} \\int \\operatorname{F_{c}}{(a^{\\dagger},A_{y})} d\\hat{H}{(a^{\\dagger})} = \\frac{\\partial}{\\partial \\hat{H}{(a^{\\dagger})}} \\int A_{y}^{a^{\\dagger}} d\\hat{H}{(a^{\\dagger})}", "derivation": "\\operatorname{F_{c}}{(a^{\\dagger},A_{y})} = A_{y}^{a^{\\dagger}} and \\hat{H}{(a^{\\dagger})} = a^{\\dagger} and \\int \\operatorname{F_{c}}{(a^{\\dagger},A_{y})} da^{\\dagger} = \\int A_{y}^{a^{\\dagger}} da^{\\dagger} and \\frac{\\partial}{\\partial a^{\\dagger}} \\int \\operatorname{F_{c}}{(a^{\\dagger},A_{y})} da^{\\dagger} = \\frac{\\partial}{\\partial a^{\\dagger}} \\int A_{y}^{a^{\\dagger}} da^{\\dagger} and \\frac{\\partial}{\\partial \\hat{H}{(a^{\\dagger})}} \\int \\operatorname{F_{c}}{(a^{\\dagger},A_{y})} d\\hat{H}{(a^{\\dagger})} = \\frac{\\partial}{\\partial \\hat{H}{(a^{\\dagger})}} \\int A_{y}^{a^{\\dagger}} d\\hat{H}{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('A_y', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Symbol('A_y', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('A_y', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Integral(Function('F_c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1))), Derivative(Integral(Pow(Symbol('A_y', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Function('\\\\hat{H}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(n_{2},C)} = C + \\cos{(n_{2})} and \\nabla{(n_{2},C)} = - \\frac{- C + \\operatorname{A_{x}}{(n_{2},C)} - \\cos{(n_{2})}}{C}, then obtain \\frac{\\nabla{(n_{2},C)} + \\int \\nabla{(n_{2},C)} dn_{2}}{\\int \\nabla{(n_{2},C)} dn_{2}} = 1", "derivation": "\\operatorname{A_{x}}{(n_{2},C)} = C + \\cos{(n_{2})} and - C + \\operatorname{A_{x}}{(n_{2},C)} - \\cos{(n_{2})} = 0 and - \\frac{- C + \\operatorname{A_{x}}{(n_{2},C)} - \\cos{(n_{2})}}{C} = 0 and \\nabla{(n_{2},C)} = - \\frac{- C + \\operatorname{A_{x}}{(n_{2},C)} - \\cos{(n_{2})}}{C} and \\nabla{(n_{2},C)} = 0 and \\nabla{(n_{2},C)} + \\int \\nabla{(n_{2},C)} dn_{2} = \\int \\nabla{(n_{2},C)} dn_{2} and \\frac{\\nabla{(n_{2},C)} + \\int \\nabla{(n_{2},C)} dn_{2}}{\\int \\nabla{(n_{2},C)} dn_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), cos(Symbol('n_2', commutative=True))))"], [["minus", 1, "Add(Symbol('C', commutative=True), cos(Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('A_x')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), Symbol('C', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('A_x')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('A_x')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Integer(0))"], [["add", 5, "Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Add(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["divide", 6, "Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Add(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Pow(Integral(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{J},s)} = \\frac{\\mathbf{J}}{s} and \\hat{x}_0{(\\hat{p},l)} = \\frac{\\hat{p}}{l}, then obtain - l + \\frac{s \\hat{x}_0{(\\hat{p},l)}}{\\frac{\\mathbf{J}}{s} + s} = \\frac{\\hat{p} s}{l (\\frac{\\mathbf{J}}{s} + s)} - l", "derivation": "\\mathbf{F}{(\\mathbf{J},s)} = \\frac{\\mathbf{J}}{s} and \\hat{x}_0{(\\hat{p},l)} = \\frac{\\hat{p}}{l} and \\frac{\\hat{x}_0{(\\hat{p},l)}}{s + \\mathbf{F}{(\\mathbf{J},s)}} = \\frac{\\hat{p}}{l (s + \\mathbf{F}{(\\mathbf{J},s)})} and \\frac{s \\hat{x}_0{(\\hat{p},l)}}{s + \\mathbf{F}{(\\mathbf{J},s)}} = \\frac{\\hat{p} s}{l (s + \\mathbf{F}{(\\mathbf{J},s)})} and - l + \\frac{s \\hat{x}_0{(\\hat{p},l)}}{s + \\mathbf{F}{(\\mathbf{J},s)}} = \\frac{\\hat{p} s}{l (s + \\mathbf{F}{(\\mathbf{J},s)})} - l and - l + \\frac{s \\hat{x}_0{(\\hat{p},l)}}{\\frac{\\mathbf{J}}{s} + s} = \\frac{\\hat{p} s}{l (\\frac{\\mathbf{J}}{s} + s)} - l", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["divide", 2, "Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1))))"], [["divide", 3, "Pow(Symbol('s', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1))))"], [["minus", 4, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Symbol('s', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('s', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Symbol('s', commutative=True), Pow(Add(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('s', commutative=True), Pow(Add(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Symbol('s', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('l', commutative=True))))"]]}, {"prompt": "Given A{(H)} = \\sin{(H)}, then derive A{(H)} + \\int A{(H)} dH = L + A{(H)} - \\cos{(H)}, then derive \\int A{(H)} dH = \\dot{y} - \\cos{(H)}, then obtain (L + A{(H)} - \\cos{(H)}) \\int A{(H)} dH = (\\dot{y} - \\cos{(H)}) (L + A{(H)} - \\cos{(H)})", "derivation": "A{(H)} = \\sin{(H)} and \\int A{(H)} dH = \\int \\sin{(H)} dH and A{(H)} + \\int A{(H)} dH = A{(H)} + \\int \\sin{(H)} dH and A{(H)} + \\int A{(H)} dH = L + A{(H)} - \\cos{(H)} and \\int A{(H)} dH = \\dot{y} - \\cos{(H)} and (A{(H)} + \\int A{(H)} dH) \\int A{(H)} dH = (\\dot{y} - \\cos{(H)}) (A{(H)} + \\int A{(H)} dH) and (L + A{(H)} - \\cos{(H)}) \\int A{(H)} dH = (\\dot{y} - \\cos{(H)}) (L + A{(H)} - \\cos{(H)})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["add", 2, "Function('A')(Symbol('H', commutative=True))"], "Equality(Add(Function('A')(Symbol('H', commutative=True)), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Function('A')(Symbol('H', commutative=True)), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('A')(Symbol('H', commutative=True)), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Symbol('L', commutative=True), Function('A')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"], [["times", 5, "Add(Function('A')(Symbol('H', commutative=True)), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], "Equality(Mul(Add(Function('A')(Symbol('H', commutative=True)), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Add(Function('A')(Symbol('H', commutative=True)), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Symbol('L', commutative=True), Function('A')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Integral(Function('A')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Add(Symbol('L', commutative=True), Function('A')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given M{(C_{1},q)} = C_{1}^{q} and h{(C_{1},q)} = \\frac{M{(C_{1},q)}}{C_{1}}, then obtain \\iint \\frac{C_{1}^{q} h{(C_{1},q)}}{C_{1}} dC_{1} dq = \\iint \\frac{C_{1}^{2 q}}{C_{1}^{2}} dC_{1} dq", "derivation": "M{(C_{1},q)} = C_{1}^{q} and \\frac{M{(C_{1},q)}}{C_{1}} = \\frac{C_{1}^{q}}{C_{1}} and \\frac{C_{1}^{q} M{(C_{1},q)}}{C_{1}^{2}} = \\frac{C_{1}^{2 q}}{C_{1}^{2}} and h{(C_{1},q)} = \\frac{M{(C_{1},q)}}{C_{1}} and \\frac{C_{1}^{q} h{(C_{1},q)}}{C_{1}} = \\frac{C_{1}^{2 q}}{C_{1}^{2}} and \\int \\frac{C_{1}^{q} h{(C_{1},q)}}{C_{1}} dC_{1} = \\int \\frac{C_{1}^{2 q}}{C_{1}^{2}} dC_{1} and \\iint \\frac{C_{1}^{q} h{(C_{1},q)}}{C_{1}} dC_{1} dq = \\iint \\frac{C_{1}^{2 q}}{C_{1}^{2}} dC_{1} dq", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))"], [["divide", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('M')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Function('M')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_1', commutative=True), Mul(Integer(2), Symbol('q', commutative=True)))))"], ["renaming_premise", "Equality(Function('h')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Function('M')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Function('h')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_1', commutative=True), Mul(Integer(2), Symbol('q', commutative=True)))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Function('h')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_1', commutative=True), Mul(Integer(2), Symbol('q', commutative=True)))), Tuple(Symbol('C_1', commutative=True))))"], [["integrate", 6, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Function('h')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-2)), Pow(Symbol('C_1', commutative=True), Mul(Integer(2), Symbol('q', commutative=True)))), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\phi_2,p)} = - \\sin{(\\phi_2 - p)}, then derive - \\cos{(\\phi_2 - p)} + \\frac{\\partial}{\\partial p} \\hat{p}_0{(\\phi_2,p)} = 0, then obtain - \\cos{(\\phi_2 - p)} + \\frac{\\partial}{\\partial p} - \\sin{(\\phi_2 - p)} = 0", "derivation": "\\hat{p}_0{(\\phi_2,p)} = - \\sin{(\\phi_2 - p)} and \\phi_2 + \\hat{p}_0{(\\phi_2,p)} = \\phi_2 - \\sin{(\\phi_2 - p)} and \\phi_2 + \\hat{p}_0{(\\phi_2,p)} + \\sin{(\\phi_2 - p)} = \\phi_2 and \\frac{\\partial}{\\partial p} (\\phi_2 + \\hat{p}_0{(\\phi_2,p)} + \\sin{(\\phi_2 - p)}) = \\frac{d}{d p} \\phi_2 and - \\cos{(\\phi_2 - p)} + \\frac{\\partial}{\\partial p} \\hat{p}_0{(\\phi_2,p)} = 0 and - \\cos{(\\phi_2 - p)} + \\frac{\\partial}{\\partial p} - \\sin{(\\phi_2 - p)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\phi_2', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))))"], [["add", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\phi_2', commutative=True), Symbol('p', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))))))"], [["add", 2, "sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\phi_2', commutative=True), Symbol('p', commutative=True)), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Symbol('\\\\phi_2', commutative=True))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\phi_2', commutative=True), Symbol('p', commutative=True)), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given z{(l)} = \\cos{(l)}, then derive \\int z{(l)} dl = \\mathbf{H} + \\sin{(l)}, then derive (\\mathbf{H} + \\sin{(l)})^{l} = (\\mathbf{g} + \\sin{(l)})^{l}, then obtain \\cos{(l)} (\\int z{(l)} dl)^{l} = (\\mathbf{H} + \\sin{(l)})^{l} \\cos{(l)}", "derivation": "z{(l)} = \\cos{(l)} and \\int z{(l)} dl = \\int \\cos{(l)} dl and \\int z{(l)} dl = \\mathbf{H} + \\sin{(l)} and \\mathbf{H} + \\sin{(l)} = \\int \\cos{(l)} dl and (\\mathbf{H} + \\sin{(l)})^{l} = (\\int \\cos{(l)} dl)^{l} and (\\mathbf{H} + \\sin{(l)})^{l} = (\\mathbf{g} + \\sin{(l)})^{l} and (\\int z{(l)} dl)^{l} = (\\mathbf{g} + \\sin{(l)})^{l} and \\cos{(l)} (\\int z{(l)} dl)^{l} = (\\mathbf{g} + \\sin{(l)})^{l} \\cos{(l)} and \\cos{(l)} (\\int z{(l)} dl)^{l} = (\\mathbf{H} + \\sin{(l)})^{l} \\cos{(l)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('z')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Integral(Function('z')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["times", 7, "cos(Symbol('l', commutative=True))"], "Equality(Mul(cos(Symbol('l', commutative=True)), Pow(Integral(Function('z')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Mul(cos(Symbol('l', commutative=True)), Pow(Integral(Function('z')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('l', commutative=True))), Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then derive \\int \\Psi_{\\lambda}{(a^{\\dagger})} da^{\\dagger} = q - \\cos{(a^{\\dagger})}, then obtain q - 2 \\cos{(a^{\\dagger})} = - \\cos{(a^{\\dagger})} + \\int \\sin{(a^{\\dagger})} da^{\\dagger}", "derivation": "\\Psi_{\\lambda}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\int \\Psi_{\\lambda}{(a^{\\dagger})} da^{\\dagger} = \\int \\sin{(a^{\\dagger})} da^{\\dagger} and \\int \\Psi_{\\lambda}{(a^{\\dagger})} da^{\\dagger} = q - \\cos{(a^{\\dagger})} and - \\cos{(a^{\\dagger})} + \\int \\Psi_{\\lambda}{(a^{\\dagger})} da^{\\dagger} = - \\cos{(a^{\\dagger})} + \\int \\sin{(a^{\\dagger})} da^{\\dagger} and q - 2 \\cos{(a^{\\dagger})} = - \\cos{(a^{\\dagger})} + \\int \\sin{(a^{\\dagger})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["minus", 2, "cos(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('q', commutative=True), Mul(Integer(-1), Integer(2), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f,T)} = T + f, then obtain (\\frac{\\partial}{\\partial f} (\\frac{\\operatorname{n_{1}}{(f,T)}}{f})^{T})^{T} = (\\frac{\\partial}{\\partial f} (\\frac{T + f}{f})^{T})^{T}", "derivation": "\\operatorname{n_{1}}{(f,T)} = T + f and \\frac{\\operatorname{n_{1}}{(f,T)}}{f} = \\frac{T + f}{f} and (\\frac{\\operatorname{n_{1}}{(f,T)}}{f})^{T} = (\\frac{T + f}{f})^{T} and \\frac{\\partial}{\\partial f} (\\frac{\\operatorname{n_{1}}{(f,T)}}{f})^{T} = \\frac{\\partial}{\\partial f} (\\frac{T + f}{f})^{T} and (\\frac{\\partial}{\\partial f} (\\frac{\\operatorname{n_{1}}{(f,T)}}{f})^{T})^{T} = (\\frac{\\partial}{\\partial f} (\\frac{T + f}{f})^{T})^{T}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('n_1')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True))))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('n_1')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Symbol('T', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True))), Symbol('T', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('n_1')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 4, "Symbol('T', commutative=True)"], "Equality(Pow(Derivative(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('n_1')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('T', commutative=True)), Pow(Derivative(Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given H{(i,\\varphi^*)} = \\varphi^* + \\log{(i)} and \\mathbf{J}_P{(i,\\varphi^*)} = H^{i}{(i,\\varphi^*)}, then obtain H^{i}{(i,\\varphi^*)} \\mathbf{J}_P{(i,\\varphi^*)} = H^{2 i}{(i,\\varphi^*)}", "derivation": "H{(i,\\varphi^*)} = \\varphi^* + \\log{(i)} and H^{i}{(i,\\varphi^*)} = (\\varphi^* + \\log{(i)})^{i} and \\mathbf{J}_P{(i,\\varphi^*)} = H^{i}{(i,\\varphi^*)} and (\\varphi^* + \\log{(i)})^{i} \\mathbf{J}_P{(i,\\varphi^*)} = (\\varphi^* + \\log{(i)})^{i} H^{i}{(i,\\varphi^*)} and H^{i}{(i,\\varphi^*)} \\mathbf{J}_P{(i,\\varphi^*)} = H^{2 i}{(i,\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), log(Symbol('i', commutative=True))))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('i', commutative=True)), Pow(Add(Symbol('\\\\varphi^*', commutative=True), log(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('i', commutative=True)))"], [["times", 3, "Pow(Add(Symbol('\\\\varphi^*', commutative=True), log(Symbol('i', commutative=True))), Symbol('i', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\varphi^*', commutative=True), log(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Add(Symbol('\\\\varphi^*', commutative=True), log(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('i', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Pow(Function('H')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(z^{*},H)} = \\cos{(H + z^{*})}, then obtain (\\int (\\int \\ddot{x}{(z^{*},H)} dH - \\int \\cos{(H + z^{*})} dH) \\int \\cos{(H + z^{*})} dH dz^{*})^{H} = (\\int 0 dz^{*})^{H}", "derivation": "\\ddot{x}{(z^{*},H)} = \\cos{(H + z^{*})} and \\int \\ddot{x}{(z^{*},H)} dH = \\int \\cos{(H + z^{*})} dH and \\int \\ddot{x}{(z^{*},H)} dH - \\int \\cos{(H + z^{*})} dH = 0 and (\\int \\ddot{x}{(z^{*},H)} dH - \\int \\cos{(H + z^{*})} dH) \\int \\cos{(H + z^{*})} dH = 0 and \\int (\\int \\ddot{x}{(z^{*},H)} dH - \\int \\cos{(H + z^{*})} dH) \\int \\cos{(H + z^{*})} dH dz^{*} = \\int 0 dz^{*} and (\\int (\\int \\ddot{x}{(z^{*},H)} dH - \\int \\cos{(H + z^{*})} dH) \\int \\cos{(H + z^{*})} dH dz^{*})^{H} = (\\int 0 dz^{*})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["minus", 2, "Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Integral(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True))))), Integer(0))"], [["times", 3, "Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Add(Integral(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True))))), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True)))), Integer(0))"], [["integrate", 4, "Symbol('z^*', commutative=True)"], "Equality(Integral(Mul(Add(Integral(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True))))), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('z^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('z^*', commutative=True))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Mul(Add(Integral(Function('\\\\ddot{x}')(Symbol('z^*', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True))))), Integral(cos(Add(Symbol('H', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('z^*', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('z^*', commutative=True))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\mathbf{v} \\log{(I)}, then derive \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\frac{\\mathbf{v}}{I}, then obtain \\frac{\\partial}{\\partial I} \\mathbf{v} \\log{(I)} + \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\frac{\\partial}{\\partial I} \\mathbf{v} \\log{(I)} + \\frac{\\mathbf{v}}{I}", "derivation": "\\operatorname{P_{e}}{(I,\\mathbf{v})} = \\mathbf{v} \\log{(I)} and \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\frac{\\partial}{\\partial I} \\mathbf{v} \\log{(I)} and \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\frac{\\mathbf{v}}{I} and \\frac{\\partial}{\\partial I} \\mathbf{v} \\log{(I)} + \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\frac{\\partial}{\\partial I} \\mathbf{v} \\log{(I)} + \\frac{\\mathbf{v}}{I}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 3, "Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given u{(s,E_{x})} = \\sin{(\\frac{E_{x}}{s})}, then obtain \\iint (u{(s,E_{x})} - 1) dE_{x} dE_{x} = \\iint (\\sin{(\\frac{E_{x}}{s})} - 1) dE_{x} dE_{x}", "derivation": "u{(s,E_{x})} = \\sin{(\\frac{E_{x}}{s})} and u{(s,E_{x})} - 1 = \\sin{(\\frac{E_{x}}{s})} - 1 and \\int (u{(s,E_{x})} - 1) dE_{x} = \\int (\\sin{(\\frac{E_{x}}{s})} - 1) dE_{x} and \\iint (u{(s,E_{x})} - 1) dE_{x} dE_{x} = \\iint (\\sin{(\\frac{E_{x}}{s})} - 1) dE_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('s', commutative=True), Symbol('E_x', commutative=True)), sin(Mul(Symbol('E_x', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('u')(Symbol('s', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Add(sin(Mul(Symbol('E_x', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(-1)))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('s', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(sin(Mul(Symbol('E_x', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(-1)), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('s', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(sin(Mul(Symbol('E_x', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(-1)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(x^\\prime)} = \\log{(x^\\prime)}, then obtain - \\hat{X}^{2}{(x^\\prime)} \\log{(x^\\prime)}^{2} = - \\hat{X}{(x^\\prime)} \\log{(x^\\prime)}^{3}", "derivation": "\\hat{X}{(x^\\prime)} = \\log{(x^\\prime)} and \\hat{X}^{2}{(x^\\prime)} = \\hat{X}{(x^\\prime)} \\log{(x^\\prime)} and \\hat{X}^{4}{(x^\\prime)} = \\hat{X}^{2}{(x^\\prime)} \\log{(x^\\prime)}^{2} and - \\hat{X}^{4}{(x^\\prime)} = - \\hat{X}^{2}{(x^\\prime)} \\log{(x^\\prime)}^{2} and - \\hat{X}^{2}{(x^\\prime)} \\log{(x^\\prime)}^{2} = - \\hat{X}{(x^\\prime)} \\log{(x^\\prime)}^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(2)), Mul(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(2)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(2))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(4))), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(2)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Integer(2)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('x^\\\\prime', commutative=True)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(m,p)} = m p and W{(\\omega,y^{\\prime})} = \\omega y^{\\prime}, then obtain \\sin{(\\frac{\\log{(W{(\\omega,y^{\\prime})})}}{\\int m p dp})} = \\sin{(\\frac{\\log{(\\omega y^{\\prime})}}{\\int m p dp})}", "derivation": "\\operatorname{t_{2}}{(m,p)} = m p and \\int \\operatorname{t_{2}}{(m,p)} dp = \\int m p dp and W{(\\omega,y^{\\prime})} = \\omega y^{\\prime} and \\log{(W{(\\omega,y^{\\prime})})} = \\log{(\\omega y^{\\prime})} and \\frac{\\log{(W{(\\omega,y^{\\prime})})}}{\\int m p dp} = \\frac{\\log{(\\omega y^{\\prime})}}{\\int m p dp} and \\frac{\\log{(W{(\\omega,y^{\\prime})})}}{\\int \\operatorname{t_{2}}{(m,p)} dp} = \\frac{\\log{(\\omega y^{\\prime})}}{\\int \\operatorname{t_{2}}{(m,p)} dp} and \\sin{(\\frac{\\log{(W{(\\omega,y^{\\prime})})}}{\\int \\operatorname{t_{2}}{(m,p)} dp})} = \\sin{(\\frac{\\log{(\\omega y^{\\prime})}}{\\int \\operatorname{t_{2}}{(m,p)} dp})} and \\sin{(\\frac{\\log{(W{(\\omega,y^{\\prime})})}}{\\int m p dp})} = \\sin{(\\frac{\\log{(\\omega y^{\\prime})}}{\\int m p dp})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], ["get_premise", "Equality(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["log", 3], "Equality(log(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), log(Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 4, "Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Mul(log(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))), Mul(log(Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(log(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))), Mul(log(Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))))"], [["sin", 6], "Equality(sin(Mul(log(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)))), sin(Mul(log(Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Function('t_2')(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(sin(Mul(log(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)))), sin(Mul(log(Mul(Symbol('\\\\omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Integral(Mul(Symbol('m', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\hat{p}{(\\chi)} = e^{\\chi} and \\operatorname{a^{\\dagger}}{(E_{x})} = \\int e^{E_{x}} dE_{x}, then obtain (- (\\chi e^{\\chi})^{\\chi} + \\operatorname{a^{\\dagger}}{(E_{x})})^{\\chi} = (- (\\chi e^{\\chi})^{\\chi} + \\int e^{E_{x}} dE_{x})^{\\chi}", "derivation": "\\hat{p}{(\\chi)} = e^{\\chi} and \\chi \\hat{p}{(\\chi)} = \\chi e^{\\chi} and (\\chi \\hat{p}{(\\chi)})^{\\chi} = (\\chi e^{\\chi})^{\\chi} and \\operatorname{a^{\\dagger}}{(E_{x})} = \\int e^{E_{x}} dE_{x} and - (\\chi \\hat{p}{(\\chi)})^{\\chi} + \\operatorname{a^{\\dagger}}{(E_{x})} = - (\\chi \\hat{p}{(\\chi)})^{\\chi} + \\int e^{E_{x}} dE_{x} and - (\\chi e^{\\chi})^{\\chi} + \\operatorname{a^{\\dagger}}{(E_{x})} = - (\\chi e^{\\chi})^{\\chi} + \\int e^{E_{x}} dE_{x} and (- (\\chi e^{\\chi})^{\\chi} + \\operatorname{a^{\\dagger}}{(E_{x})})^{\\chi} = (- (\\chi e^{\\chi})^{\\chi} + \\int e^{E_{x}} dE_{x})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], ["get_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('E_x', commutative=True)), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["minus", 4, "Pow(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Function('a^{\\\\dagger}')(Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Function('a^{\\\\dagger}')(Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["power", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Function('a^{\\\\dagger}')(Symbol('E_x', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(v_{t},\\sigma_p)} = \\sigma_p - v_{t}, then derive \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\frac{\\sigma_p^{2}}{2} - \\sigma_p v_{t} + n_{2}, then obtain \\frac{\\partial}{\\partial v_{t}} \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\frac{\\partial}{\\partial v_{t}} (\\frac{\\sigma_p^{2}}{2} - \\sigma_p v_{t} + n_{2})", "derivation": "\\mathbf{B}{(v_{t},\\sigma_p)} = \\sigma_p - v_{t} and \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\int (\\sigma_p - v_{t}) d\\sigma_p and \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\frac{\\sigma_p^{2}}{2} - \\sigma_p v_{t} + n_{2} and \\int (\\sigma_p - v_{t}) d\\sigma_p = \\frac{\\sigma_p^{2}}{2} - \\sigma_p v_{t} + n_{2} and \\frac{\\partial}{\\partial v_{t}} \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\frac{\\partial}{\\partial v_{t}} \\int (\\sigma_p - v_{t}) d\\sigma_p and \\frac{\\partial}{\\partial v_{t}} \\int \\mathbf{B}{(v_{t},\\sigma_p)} d\\sigma_p = \\frac{\\partial}{\\partial v_{t}} (\\frac{\\sigma_p^{2}}{2} - \\sigma_p v_{t} + n_{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('v_t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('v_t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('v_t', commutative=True)), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('v_t', commutative=True)), Symbol('n_2', commutative=True)))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('v_t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integral(Function('\\\\mathbf{B}')(Symbol('v_t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('v_t', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(u)} = \\sin{(u)}, then obtain \\frac{d}{d u} (\\operatorname{t_{1}}^{u}{(u)} + \\operatorname{t_{1}}^{u}{(u)} \\sin^{- u}{(u)}) = \\frac{d}{d u} (\\operatorname{t_{1}}^{u}{(u)} + 1)", "derivation": "\\operatorname{t_{1}}{(u)} = \\sin{(u)} and \\operatorname{t_{1}}^{u}{(u)} = \\sin^{u}{(u)} and \\operatorname{t_{1}}^{u}{(u)} \\sin^{- u}{(u)} = 1 and \\operatorname{t_{1}}^{u}{(u)} + \\operatorname{t_{1}}^{u}{(u)} \\sin^{- u}{(u)} = \\operatorname{t_{1}}^{u}{(u)} + 1 and \\frac{d}{d u} (\\operatorname{t_{1}}^{u}{(u)} + \\operatorname{t_{1}}^{u}{(u)} \\sin^{- u}{(u)}) = \\frac{d}{d u} (\\operatorname{t_{1}}^{u}{(u)} + 1)", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["divide", 2, "Pow(sin(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(1))"], [["add", 3, "Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Add(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))), Add(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Pow(Function('t_1')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(M,\\mathbf{B},i)} = \\frac{\\mathbf{B} + i}{M} and I{(V_{\\mathbf{B}},J)} = \\cos{(J V_{\\mathbf{B}})}, then obtain (\\cos{(J V_{\\mathbf{B}})} - \\frac{\\mathbf{B} + i}{M})^{i} = (\\cos{(J V_{\\mathbf{B}})} + \\frac{- \\mathbf{B} - i}{M})^{i}", "derivation": "\\rho_{b}{(M,\\mathbf{B},i)} = \\frac{\\mathbf{B} + i}{M} and - \\rho_{b}{(M,\\mathbf{B},i)} = - \\frac{\\mathbf{B} + i}{M} and - \\rho_{b}{(M,\\mathbf{B},i)} = \\frac{- \\mathbf{B} - i}{M} and - \\frac{\\mathbf{B} + i}{M} = \\frac{- \\mathbf{B} - i}{M} and I{(V_{\\mathbf{B}},J)} = \\cos{(J V_{\\mathbf{B}})} and I{(V_{\\mathbf{B}},J)} - \\frac{\\mathbf{B} + i}{M} = I{(V_{\\mathbf{B}},J)} + \\frac{- \\mathbf{B} - i}{M} and (I{(V_{\\mathbf{B}},J)} - \\frac{\\mathbf{B} + i}{M})^{i} = (I{(V_{\\mathbf{B}},J)} + \\frac{- \\mathbf{B} - i}{M})^{i} and (\\cos{(J V_{\\mathbf{B}})} - \\frac{\\mathbf{B} + i}{M})^{i} = (\\cos{(J V_{\\mathbf{B}})} + \\frac{- \\mathbf{B} - i}{M})^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))))"], ["get_premise", "Equality(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True)), cos(Mul(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 4, "Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True))"], "Equality(Add(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True)))), Add(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))))"], [["power", 6, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Add(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Add(cos(Mul(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Add(cos(Mul(Symbol('J', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given Z{(\\rho_f)} = \\int e^{\\rho_f} d\\rho_f, then derive Z{(\\rho_f)} = \\mathbf{M} + e^{\\rho_f}, then derive - V_{\\mathbf{B}} + \\mathbf{M} = 0, then obtain - I - V_{\\mathbf{B}} + \\mathbf{M} - e^{\\rho_f} = - I - e^{\\rho_f}", "derivation": "Z{(\\rho_f)} = \\int e^{\\rho_f} d\\rho_f and Z{(\\rho_f)} = \\mathbf{M} + e^{\\rho_f} and \\mathbf{M} + e^{\\rho_f} = \\int e^{\\rho_f} d\\rho_f and \\mathbf{M} + e^{\\rho_f} - \\int e^{\\rho_f} d\\rho_f = 0 and - V_{\\mathbf{B}} + \\mathbf{M} = 0 and - V_{\\mathbf{B}} + \\mathbf{M} - \\int e^{\\rho_f} d\\rho_f = - \\int e^{\\rho_f} d\\rho_f and - I - V_{\\mathbf{B}} + \\mathbf{M} - e^{\\rho_f} = - I - e^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('Z')(Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), exp(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), exp(Symbol('\\\\rho_f', commutative=True))), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 3, "Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), exp(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Integer(0))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Integer(0))"], [["add", 5, "Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))), Mul(Integer(-1), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\rho_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\Psi,\\Psi^{\\dagger})} = (\\Psi^{\\dagger})^{\\Psi}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta{(\\Psi,\\Psi^{\\dagger})} = \\frac{\\Psi (\\Psi^{\\dagger})^{\\Psi}}{\\Psi^{\\dagger}}, then obtain \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger})^{\\Psi} = \\frac{\\Psi \\theta{(\\Psi,\\Psi^{\\dagger})}}{\\Psi^{\\dagger}}", "derivation": "\\theta{(\\Psi,\\Psi^{\\dagger})} = (\\Psi^{\\dagger})^{\\Psi} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta{(\\Psi,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger})^{\\Psi} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta{(\\Psi,\\Psi^{\\dagger})} = \\frac{\\Psi (\\Psi^{\\dagger})^{\\Psi}}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\theta{(\\Psi,\\Psi^{\\dagger})} = \\frac{\\Psi \\theta{(\\Psi,\\Psi^{\\dagger})}}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger})^{\\Psi} = \\frac{\\Psi \\theta{(\\Psi,\\Psi^{\\dagger})}}{\\Psi^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given C{(F_{N})} = e^{F_{N}}, then obtain \\frac{\\int \\sin{(C{(F_{N})})} dF_{N}}{\\sin{(e^{F_{N}})}} = \\frac{\\int \\sin{(e^{F_{N}})} dF_{N}}{\\sin{(e^{F_{N}})}}", "derivation": "C{(F_{N})} = e^{F_{N}} and \\sin{(C{(F_{N})})} = \\sin{(e^{F_{N}})} and \\int \\sin{(C{(F_{N})})} dF_{N} = \\int \\sin{(e^{F_{N}})} dF_{N} and \\frac{\\int \\sin{(C{(F_{N})})} dF_{N}}{\\sin{(e^{F_{N}})}} = \\frac{\\int \\sin{(e^{F_{N}})} dF_{N}}{\\sin{(e^{F_{N}})}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["sin", 1], "Equality(sin(Function('C')(Symbol('F_N', commutative=True))), sin(exp(Symbol('F_N', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(sin(Function('C')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(sin(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["divide", 3, "sin(exp(Symbol('F_N', commutative=True)))"], "Equality(Mul(Pow(sin(exp(Symbol('F_N', commutative=True))), Integer(-1)), Integral(sin(Function('C')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Mul(Pow(sin(exp(Symbol('F_N', commutative=True))), Integer(-1)), Integral(sin(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\delta{(A_{z})} = e^{\\cos{(A_{z})}}, then obtain A_{z} \\delta{(A_{z})} + \\delta{(A_{z})} = A_{z} e^{\\cos{(A_{z})}} + e^{\\cos{(A_{z})}}", "derivation": "\\delta{(A_{z})} = e^{\\cos{(A_{z})}} and A_{z} \\delta{(A_{z})} = A_{z} e^{\\cos{(A_{z})}} and A_{z} \\delta{(A_{z})} + \\delta{(A_{z})} = A_{z} \\delta{(A_{z})} + e^{\\cos{(A_{z})}} and A_{z} \\delta{(A_{z})} + \\delta{(A_{z})} = A_{z} e^{\\cos{(A_{z})}} + \\delta{(A_{z})} and A_{z} e^{\\cos{(A_{z})}} + \\delta{(A_{z})} = A_{z} e^{\\cos{(A_{z})}} + e^{\\cos{(A_{z})}} and A_{z} \\delta{(A_{z})} + \\delta{(A_{z})} = A_{z} e^{\\cos{(A_{z})}} + e^{\\cos{(A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('A_z', commutative=True)), exp(cos(Symbol('A_z', commutative=True))))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), exp(cos(Symbol('A_z', commutative=True)))))"], [["add", 1, "Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True)))"], "Equality(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True))), Function('\\\\delta')(Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True))), exp(cos(Symbol('A_z', commutative=True)))))"], [["add", 2, "Function('\\\\delta')(Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True))), Function('\\\\delta')(Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), exp(cos(Symbol('A_z', commutative=True)))), Function('\\\\delta')(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Symbol('A_z', commutative=True), exp(cos(Symbol('A_z', commutative=True)))), Function('\\\\delta')(Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), exp(cos(Symbol('A_z', commutative=True)))), exp(cos(Symbol('A_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\delta')(Symbol('A_z', commutative=True))), Function('\\\\delta')(Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), exp(cos(Symbol('A_z', commutative=True)))), exp(cos(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{X})} = e^{e^{\\hat{X}}} and \\theta_{2}{(\\hat{X})} = 2 e^{\\hat{X}}, then obtain \\mathbf{r}{(\\hat{X})} e^{- \\theta_{2}{(\\hat{X})}} e^{e^{\\hat{X}}} = 1", "derivation": "\\mathbf{r}{(\\hat{X})} = e^{e^{\\hat{X}}} and \\mathbf{r}{(\\hat{X})} e^{e^{\\hat{X}}} = e^{2 e^{\\hat{X}}} and \\theta_{2}{(\\hat{X})} = 2 e^{\\hat{X}} and \\mathbf{r}{(\\hat{X})} e^{e^{\\hat{X}}} = e^{\\theta_{2}{(\\hat{X})}} and \\mathbf{r}{(\\hat{X})} e^{- \\theta_{2}{(\\hat{X})}} e^{e^{\\hat{X}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 1, "exp(exp(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('\\\\hat{X}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), exp(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 4, "exp(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\hat{X}', commutative=True)), exp(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)))), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given E{(A_{x})} = A_{x} and \\operatorname{a^{\\dagger}}{(A_{x})} = \\frac{A_{x}}{E{(A_{x})}}, then obtain A_{x} \\operatorname{a^{\\dagger}}{(A_{x})} - E{(A_{x})} = A_{x} - E{(A_{x})}", "derivation": "E{(A_{x})} = A_{x} and \\operatorname{a^{\\dagger}}{(A_{x})} = \\frac{A_{x}}{E{(A_{x})}} and E{(A_{x})} \\operatorname{a^{\\dagger}}{(A_{x})} = A_{x} and A_{x} \\operatorname{a^{\\dagger}}{(A_{x})} = A_{x} and A_{x} \\operatorname{a^{\\dagger}}{(A_{x})} - E{(A_{x})} = A_{x} - E{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('A_x', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Function('E')(Symbol('A_x', commutative=True)), Integer(-1))))"], [["times", 2, "Function('E')(Symbol('A_x', commutative=True))"], "Equality(Mul(Function('E')(Symbol('A_x', commutative=True)), Function('a^{\\\\dagger}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A_x', commutative=True), Function('a^{\\\\dagger}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))"], [["minus", 4, "Function('E')(Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Symbol('A_x', commutative=True), Function('a^{\\\\dagger}')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Function('E')(Symbol('A_x', commutative=True)))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('E')(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\rho_b)} = \\cos{(\\rho_b)}, then derive \\int \\mathbf{J}_f{(\\rho_b)} d\\rho_b = \\phi_2 + \\sin{(\\rho_b)}, then derive \\varepsilon + \\sin{(\\rho_b)} = \\phi_2 + \\sin{(\\rho_b)}, then obtain (\\phi_2 + \\sin{(\\rho_b)}) (\\varepsilon + \\sin{(\\rho_b)}) = (\\phi_2 + \\sin{(\\rho_b)})^{2}", "derivation": "\\mathbf{J}_f{(\\rho_b)} = \\cos{(\\rho_b)} and \\int \\mathbf{J}_f{(\\rho_b)} d\\rho_b = \\int \\cos{(\\rho_b)} d\\rho_b and \\int \\mathbf{J}_f{(\\rho_b)} d\\rho_b = \\phi_2 + \\sin{(\\rho_b)} and \\int \\cos{(\\rho_b)} d\\rho_b = \\phi_2 + \\sin{(\\rho_b)} and \\varepsilon + \\sin{(\\rho_b)} = \\phi_2 + \\sin{(\\rho_b)} and (\\phi_2 + \\sin{(\\rho_b)}) (\\varepsilon + \\sin{(\\rho_b)}) = (\\phi_2 + \\sin{(\\rho_b)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))))"], [["times", 5, "Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\rho_b', commutative=True)))), Pow(Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_P{(E,T)} = E T, then obtain (\\frac{E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)})^{T} = (\\frac{2 E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)} - 1)^{T}", "derivation": "\\mathbf{J}_P{(E,T)} = E T and 1 = \\frac{E T}{\\mathbf{J}_P{(E,T)}} and 1 - \\mathbf{J}_P^{2}{(E,T)} = \\frac{E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)} and (1 - \\mathbf{J}_P^{2}{(E,T)})^{T} = (\\frac{E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)})^{T} and (\\frac{E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)})^{T} = (\\frac{2 E T}{\\mathbf{J}_P{(E,T)}} - \\mathbf{J}_P^{2}{(E,T)} - 1)^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('T', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True))"], "Equality(Integer(1), Mul(Symbol('E', commutative=True), Symbol('T', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))))"], [["minus", 2, "Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Add(Mul(Symbol('E', commutative=True), Symbol('T', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2)))))"], [["power", 3, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Symbol('T', commutative=True)), Pow(Add(Mul(Symbol('E', commutative=True), Symbol('T', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Symbol('E', commutative=True), Symbol('T', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2)))), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('E', commutative=True), Symbol('T', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('E', commutative=True), Symbol('T', commutative=True)), Integer(2))), Integer(-1)), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(h)} = \\sin{(h)}, then obtain \\frac{d}{d h} \\mathbf{S}^{3}{(h)} \\sin{(h)} = \\frac{d}{d h} \\mathbf{S}^{2}{(h)} \\sin^{2}{(h)}", "derivation": "\\mathbf{S}{(h)} = \\sin{(h)} and \\mathbf{S}{(h)} \\sin{(h)} = \\sin^{2}{(h)} and \\mathbf{S}^{2}{(h)} \\sin^{2}{(h)} = \\sin^{4}{(h)} and \\mathbf{S}^{3}{(h)} \\sin{(h)} = \\mathbf{S}^{2}{(h)} \\sin^{2}{(h)} and \\mathbf{S}^{3}{(h)} \\sin{(h)} = \\sin^{4}{(h)} and \\frac{d}{d h} \\mathbf{S}^{3}{(h)} \\sin{(h)} = \\frac{d}{d h} \\sin^{4}{(h)} and \\frac{d}{d h} \\mathbf{S}^{3}{(h)} \\sin{(h)} = \\frac{d}{d h} \\mathbf{S}^{2}{(h)} \\sin^{2}{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["times", 1, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Pow(sin(Symbol('h', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(3)), sin(Symbol('h', commutative=True))), Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(3)), sin(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(4)))"], [["differentiate", 5, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(3)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('h', commutative=True)), Integer(4)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Derivative(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(3)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(E_{\\lambda},\\hbar)} = E_{\\lambda}^{\\hbar} and \\eta{(t_{2})} = \\sin{(t_{2})}, then obtain E_{\\lambda}^{- \\hbar} (\\eta^{t_{2}}{(t_{2})} + \\sigma_{x}{(E_{\\lambda},\\hbar)}) = E_{\\lambda}^{- \\hbar} (\\sigma_{x}{(E_{\\lambda},\\hbar)} + \\sin^{t_{2}}{(t_{2})})", "derivation": "\\sigma_{x}{(E_{\\lambda},\\hbar)} = E_{\\lambda}^{\\hbar} and \\eta{(t_{2})} = \\sin{(t_{2})} and \\eta^{t_{2}}{(t_{2})} = \\sin^{t_{2}}{(t_{2})} and E_{\\lambda}^{\\hbar} + \\eta^{t_{2}}{(t_{2})} = E_{\\lambda}^{\\hbar} + \\sin^{t_{2}}{(t_{2})} and \\eta^{t_{2}}{(t_{2})} + \\sigma_{x}{(E_{\\lambda},\\hbar)} = \\sigma_{x}{(E_{\\lambda},\\hbar)} + \\sin^{t_{2}}{(t_{2})} and E_{\\lambda}^{- \\hbar} (\\eta^{t_{2}}{(t_{2})} + \\sigma_{x}{(E_{\\lambda},\\hbar)}) = E_{\\lambda}^{- \\hbar} (\\sigma_{x}{(E_{\\lambda},\\hbar)} + \\sin^{t_{2}}{(t_{2})})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Function('\\\\sigma_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Function('\\\\sigma_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["divide", 5, "Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Add(Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Function('\\\\sigma_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Add(Function('\\\\sigma_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given b{(\\omega)} = \\log{(e^{\\omega})} and \\hat{X}{(\\omega)} = b{(\\omega)} + \\frac{d}{d \\omega} \\log{(e^{\\omega})}, then obtain (\\frac{d}{d \\omega} \\hat{X}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} (b{(\\omega)} + \\frac{d}{d \\omega} b{(\\omega)}))^{\\omega}", "derivation": "b{(\\omega)} = \\log{(e^{\\omega})} and \\frac{d}{d \\omega} b{(\\omega)} = \\frac{d}{d \\omega} \\log{(e^{\\omega})} and b{(\\omega)} + \\frac{d}{d \\omega} b{(\\omega)} = b{(\\omega)} + \\frac{d}{d \\omega} \\log{(e^{\\omega})} and \\hat{X}{(\\omega)} = b{(\\omega)} + \\frac{d}{d \\omega} \\log{(e^{\\omega})} and \\frac{d}{d \\omega} \\hat{X}{(\\omega)} = \\frac{d}{d \\omega} (b{(\\omega)} + \\frac{d}{d \\omega} \\log{(e^{\\omega})}) and (\\frac{d}{d \\omega} \\hat{X}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} (b{(\\omega)} + \\frac{d}{d \\omega} \\log{(e^{\\omega})}))^{\\omega} and (\\frac{d}{d \\omega} \\hat{X}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} (b{(\\omega)} + \\frac{d}{d \\omega} b{(\\omega)}))^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\omega', commutative=True)), log(exp(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 2, "Function('b')(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(Function('b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(log(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True)), Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(log(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(log(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(log(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Function('b')(Symbol('\\\\omega', commutative=True)), Derivative(Function('b')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\delta{(t_{1},\\mathbf{p})} = \\frac{t_{1}}{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial t_{1}} \\delta{(t_{1},\\mathbf{p})} = \\frac{1}{\\mathbf{p}}, then obtain - \\mathbf{p} + \\frac{\\partial}{\\partial t_{1}} \\frac{t_{1}}{\\mathbf{p}} = - \\mathbf{p} + \\frac{1}{\\mathbf{p}}", "derivation": "\\delta{(t_{1},\\mathbf{p})} = \\frac{t_{1}}{\\mathbf{p}} and \\frac{\\partial}{\\partial t_{1}} \\delta{(t_{1},\\mathbf{p})} = \\frac{\\partial}{\\partial t_{1}} \\frac{t_{1}}{\\mathbf{p}} and \\frac{\\partial}{\\partial t_{1}} \\delta{(t_{1},\\mathbf{p})} = \\frac{1}{\\mathbf{p}} and \\frac{\\partial}{\\partial t_{1}} \\frac{t_{1}}{\\mathbf{p}} = \\frac{1}{\\mathbf{p}} and - \\mathbf{p} + \\frac{\\partial}{\\partial t_{1}} \\frac{t_{1}}{\\mathbf{p}} = - \\mathbf{p} + \\frac{1}{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["minus", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{x}{(n_{1})} = \\cos{(n_{1})}, then obtain \\sigma_{x}^{2}{(n_{1})} \\sigma_{x}^{n_{1}}{(n_{1})} = \\sigma_{x}^{2}{(n_{1})} \\cos^{n_{1}}{(n_{1})}", "derivation": "\\sigma_{x}{(n_{1})} = \\cos{(n_{1})} and \\sigma_{x}^{n_{1}}{(n_{1})} = \\cos^{n_{1}}{(n_{1})} and \\sigma_{x}^{2}{(n_{1})} = \\sigma_{x}{(n_{1})} \\cos{(n_{1})} and \\sigma_{x}{(n_{1})} \\sigma_{x}^{n_{1}}{(n_{1})} \\cos{(n_{1})} = \\sigma_{x}{(n_{1})} \\cos{(n_{1})} \\cos^{n_{1}}{(n_{1})} and \\sigma_{x}^{2}{(n_{1})} \\sigma_{x}^{n_{1}}{(n_{1})} = \\sigma_{x}^{2}{(n_{1})} \\cos^{n_{1}}{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["times", 1, "Function('\\\\sigma_x')(Symbol('n_1', commutative=True))"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Integer(2)), Mul(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True))))"], [["times", 2, "Mul(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True))), Mul(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Integer(2)), Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Mul(Pow(Function('\\\\sigma_x')(Symbol('n_1', commutative=True)), Integer(2)), Pow(cos(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\varphi{(U)} = \\sin{(U)}, then derive \\frac{d}{d U} \\varphi{(U)} = \\cos{(U)}, then obtain 1 = \\frac{- U + \\cos{(U)} \\int \\sin{(U)} dU}{- U + \\frac{d}{d U} \\sin{(U)} \\int \\sin{(U)} dU}", "derivation": "\\varphi{(U)} = \\sin{(U)} and \\frac{d}{d U} \\varphi{(U)} = \\frac{d}{d U} \\sin{(U)} and \\frac{d}{d U} \\varphi{(U)} = \\cos{(U)} and \\frac{d}{d U} \\varphi{(U)} \\int \\sin{(U)} dU = \\cos{(U)} \\int \\sin{(U)} dU and \\frac{d}{d U} \\sin{(U)} \\int \\sin{(U)} dU = \\cos{(U)} \\int \\sin{(U)} dU and - U + \\frac{d}{d U} \\sin{(U)} \\int \\sin{(U)} dU = - U + \\cos{(U)} \\int \\sin{(U)} dU and \\frac{- U + \\frac{d}{d U} \\sin{(U)} \\int \\sin{(U)} dU}{\\frac{d}{d U} \\int \\varphi{(U)} dU} = \\frac{- U + \\cos{(U)} \\int \\sin{(U)} dU}{\\frac{d}{d U} \\int \\varphi{(U)} dU} and 1 = \\frac{- U + \\cos{(U)} \\int \\sin{(U)} dU}{- U + \\frac{d}{d U} \\sin{(U)} \\int \\sin{(U)} dU}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), cos(Symbol('U', commutative=True)))"], [["times", 3, "Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Derivative(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["minus", 5, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["divide", 6, "Derivative(Integral(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Pow(Derivative(Integral(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Pow(Derivative(Integral(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 7, "Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Pow(Derivative(Integral(Function('\\\\varphi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(F_{N})} = e^{e^{F_{N}}}, then obtain (e^{F_{N}} e^{e^{F_{N}}} \\frac{d}{d F_{N}} \\tilde{g}^*{(F_{N})})^{F_{N}} = (e^{2 F_{N}} e^{2 e^{F_{N}}})^{F_{N}}", "derivation": "\\tilde{g}^*{(F_{N})} = e^{e^{F_{N}}} and \\frac{d}{d F_{N}} \\tilde{g}^*{(F_{N})} = \\frac{d}{d F_{N}} e^{e^{F_{N}}} and \\frac{d}{d F_{N}} \\tilde{g}^*{(F_{N})} \\frac{d}{d F_{N}} e^{e^{F_{N}}} = (\\frac{d}{d F_{N}} e^{e^{F_{N}}})^{2} and (\\frac{d}{d F_{N}} \\tilde{g}^*{(F_{N})} \\frac{d}{d F_{N}} e^{e^{F_{N}}})^{F_{N}} = ((\\frac{d}{d F_{N}} e^{e^{F_{N}}})^{2})^{F_{N}} and (e^{F_{N}} e^{e^{F_{N}}} \\frac{d}{d F_{N}} \\tilde{g}^*{(F_{N})})^{F_{N}} = (e^{2 F_{N}} e^{2 e^{F_{N}}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), exp(exp(Symbol('F_N', commutative=True))))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["times", 2, "Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(2)))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Mul(Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Symbol('F_N', commutative=True)), Pow(Pow(Derivative(exp(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(2)), Symbol('F_N', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(exp(Symbol('F_N', commutative=True)), exp(exp(Symbol('F_N', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Symbol('F_N', commutative=True)), Pow(Mul(exp(Mul(Integer(2), Symbol('F_N', commutative=True))), exp(Mul(Integer(2), exp(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given i{(\\hat{p}_0,L)} = \\cos{(L - \\hat{p}_0)} and \\mathbf{J}_P{(\\hat{p}_0,L)} = L - \\hat{p}_0, then obtain - \\cos{(\\hat{\\mathbf{x}}^{G})} + \\int G i{(\\hat{p}_0,L)} d\\hat{p}_0 = - \\cos{(\\hat{\\mathbf{x}}^{G})} + \\int G \\cos{(\\mathbf{J}_P{(\\hat{p}_0,L)})} d\\hat{p}_0", "derivation": "i{(\\hat{p}_0,L)} = \\cos{(L - \\hat{p}_0)} and \\mathbf{J}_P{(\\hat{p}_0,L)} = L - \\hat{p}_0 and i{(\\hat{p}_0,L)} = \\cos{(\\mathbf{J}_P{(\\hat{p}_0,L)})} and G i{(\\hat{p}_0,L)} = G \\cos{(\\mathbf{J}_P{(\\hat{p}_0,L)})} and \\int G i{(\\hat{p}_0,L)} d\\hat{p}_0 = \\int G \\cos{(\\mathbf{J}_P{(\\hat{p}_0,L)})} d\\hat{p}_0 and - \\cos{(\\hat{\\mathbf{x}}^{G})} + \\int G i{(\\hat{p}_0,L)} d\\hat{p}_0 = - \\cos{(\\hat{\\mathbf{x}}^{G})} + \\int G \\cos{(\\mathbf{J}_P{(\\hat{p}_0,L)})} d\\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('i')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)), cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True))))"], [["times", 3, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('i')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('G', commutative=True), cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Mul(Symbol('G', commutative=True), Function('i')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Mul(Symbol('G', commutative=True), cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 5, "cos(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('G', commutative=True)))), Integral(Mul(Symbol('G', commutative=True), Function('i')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), cos(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('G', commutative=True)))), Integral(Mul(Symbol('G', commutative=True), cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('L', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given A{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then derive e^{\\int A{(\\varepsilon_0)} d\\varepsilon_0} = e^{\\delta - \\cos{(\\varepsilon_0)}}, then obtain \\varepsilon_0 e^{\\mu - \\cos{(\\varepsilon_0)}} = \\varepsilon_0 e^{\\delta - \\cos{(\\varepsilon_0)}}", "derivation": "A{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\int A{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and e^{\\int A{(\\varepsilon_0)} d\\varepsilon_0} = e^{\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0} and e^{\\int A{(\\varepsilon_0)} d\\varepsilon_0} = e^{\\delta - \\cos{(\\varepsilon_0)}} and e^{\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0} = e^{\\delta - \\cos{(\\varepsilon_0)}} and \\varepsilon_0 e^{\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0} = \\varepsilon_0 e^{\\delta - \\cos{(\\varepsilon_0)}} and \\varepsilon_0 e^{\\mu - \\cos{(\\varepsilon_0)}} = \\varepsilon_0 e^{\\delta - \\cos{(\\varepsilon_0)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('A')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), exp(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('A')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"], [["divide", 5, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), exp(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))), Mul(Symbol('\\\\varepsilon_0', commutative=True), exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))))"], [["evaluate_integrals", 6], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), exp(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))), Mul(Symbol('\\\\varepsilon_0', commutative=True), exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))))"]]}, {"prompt": "Given \\varphi{(v_{x})} = \\log{(\\cos{(v_{x})})} and \\varepsilon_{0}{(v_{x})} = \\cos{(v_{x})}, then obtain \\int (v_{x} - \\log{(\\varepsilon_{0}{(v_{x})})} + \\log{(\\cos{(v_{x})})}) dv_{x} = \\int v_{x} dv_{x}", "derivation": "\\varphi{(v_{x})} = \\log{(\\cos{(v_{x})})} and \\varphi{(v_{x})} + \\log{(\\cos{(v_{x})})} = 2 \\log{(\\cos{(v_{x})})} and \\varphi{(v_{x})} - \\log{(\\cos{(v_{x})})} = 0 and \\varepsilon_{0}{(v_{x})} = \\cos{(v_{x})} and \\varphi{(v_{x})} - \\log{(\\varepsilon_{0}{(v_{x})})} = 0 and - \\log{(\\varepsilon_{0}{(v_{x})})} + \\log{(\\cos{(v_{x})})} = 0 and v_{x} - \\log{(\\varepsilon_{0}{(v_{x})})} + \\log{(\\cos{(v_{x})})} = v_{x} and \\int (v_{x} - \\log{(\\varepsilon_{0}{(v_{x})})} + \\log{(\\cos{(v_{x})})}) dv_{x} = \\int v_{x} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True))))"], [["add", 1, "log(cos(Symbol('v_x', commutative=True)))"], "Equality(Add(Function('\\\\varphi')(Symbol('v_x', commutative=True)), log(cos(Symbol('v_x', commutative=True)))), Mul(Integer(2), log(cos(Symbol('v_x', commutative=True)))))"], [["minus", 2, "Mul(Integer(2), log(cos(Symbol('v_x', commutative=True))))"], "Equality(Add(Function('\\\\varphi')(Symbol('v_x', commutative=True)), Mul(Integer(-1), log(cos(Symbol('v_x', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\varphi')(Symbol('v_x', commutative=True)), Mul(Integer(-1), log(Function('\\\\varepsilon_0')(Symbol('v_x', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), log(Function('\\\\varepsilon_0')(Symbol('v_x', commutative=True)))), log(cos(Symbol('v_x', commutative=True)))), Integer(0))"], [["add", 6, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), log(Function('\\\\varepsilon_0')(Symbol('v_x', commutative=True)))), log(cos(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True))"], [["integrate", 7, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), log(Function('\\\\varepsilon_0')(Symbol('v_x', commutative=True)))), log(cos(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))), Integral(Symbol('v_x', commutative=True), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given M{(\\theta_2,\\mathbf{S})} = \\mathbf{S} + \\theta_2, then derive 0 = 1 - \\frac{\\partial}{\\partial \\theta_2} M{(\\theta_2,\\mathbf{S})}, then obtain 1 = e^{1 - \\frac{\\partial}{\\partial \\theta_2} (\\mathbf{S} + \\theta_2)}", "derivation": "M{(\\theta_2,\\mathbf{S})} = \\mathbf{S} + \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} M{(\\theta_2,\\mathbf{S})} = \\frac{\\partial}{\\partial \\theta_2} (\\mathbf{S} + \\theta_2) and 0 = \\frac{\\partial}{\\partial \\theta_2} (\\mathbf{S} + \\theta_2) - \\frac{\\partial}{\\partial \\theta_2} M{(\\theta_2,\\mathbf{S})} and 0 = 1 - \\frac{\\partial}{\\partial \\theta_2} M{(\\theta_2,\\mathbf{S})} and 1 = e^{1 - \\frac{\\partial}{\\partial \\theta_2} M{(\\theta_2,\\mathbf{S})}} and 1 = e^{1 - \\frac{\\partial}{\\partial \\theta_2} (\\mathbf{S} + \\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))))"], [["exp", 4], "Equality(Integer(1), exp(Add(Integer(1), Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), exp(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(H)} = \\cos{(H)} and \\Omega{(H)} = \\cos^{2}{(H)} \\cos^{H}{(H)}, then obtain \\frac{\\Omega{(H)} \\operatorname{a^{\\dagger}}^{- H}{(H)}}{\\cos^{2}{(H)}} = \\frac{\\operatorname{a^{\\dagger}}^{2}{(H)}}{\\cos^{2}{(H)}}", "derivation": "\\operatorname{a^{\\dagger}}{(H)} = \\cos{(H)} and \\Omega{(H)} = \\cos^{2}{(H)} \\cos^{H}{(H)} and \\Omega{(H)} = \\operatorname{a^{\\dagger}}^{2}{(H)} \\operatorname{a^{\\dagger}}^{H}{(H)} and \\frac{\\Omega{(H)} \\operatorname{a^{\\dagger}}^{- H}{(H)}}{\\cos^{2}{(H)}} = \\frac{\\operatorname{a^{\\dagger}}^{2}{(H)}}{\\cos^{2}{(H)}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('H', commutative=True)), Mul(Pow(cos(Symbol('H', commutative=True)), Integer(2)), Pow(cos(Symbol('H', commutative=True)), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Omega')(Symbol('H', commutative=True)), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), Integer(2)), Pow(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), Symbol('H', commutative=True))))"], [["divide", 3, "Mul(Pow(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(cos(Symbol('H', commutative=True)), Integer(2)))"], "Equality(Mul(Function('\\\\Omega')(Symbol('H', commutative=True)), Pow(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Pow(cos(Symbol('H', commutative=True)), Integer(-2))), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('H', commutative=True)), Integer(2)), Pow(cos(Symbol('H', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\rho_{f}{(n)} = \\frac{d}{d n} \\sin{(n)} and \\operatorname{C_{d}}{(n)} = \\frac{d}{d n} \\sin{(n)}, then obtain \\frac{d}{d n} \\operatorname{C_{d}}{(n)} = \\frac{d^{2}}{d n^{2}} \\sin{(n)}", "derivation": "\\rho_{f}{(n)} = \\frac{d}{d n} \\sin{(n)} and \\operatorname{C_{d}}{(n)} = \\frac{d}{d n} \\sin{(n)} and \\operatorname{C_{d}}{(n)} = \\rho_{f}{(n)} and \\frac{d}{d n} \\operatorname{C_{d}}{(n)} = \\frac{d}{d n} \\rho_{f}{(n)} and \\frac{d}{d n} \\operatorname{C_{d}}{(n)} = \\frac{d^{2}}{d n^{2}} \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('n', commutative=True)), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('n', commutative=True)), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('C_d')(Symbol('n', commutative=True)), Function('\\\\rho_f')(Symbol('n', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Function('\\\\rho_f')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Function('C_d')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} = \\frac{p + v_{2}}{a^{\\dagger}}, then obtain \\frac{\\partial}{\\partial v_{2}} (p + v_{2}) \\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} + 1 = \\frac{\\partial}{\\partial v_{2}} \\frac{(p + v_{2})^{2}}{a^{\\dagger}} + 1", "derivation": "\\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} = \\frac{p + v_{2}}{a^{\\dagger}} and (p + v_{2}) \\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} = \\frac{(p + v_{2})^{2}}{a^{\\dagger}} and \\frac{\\partial}{\\partial v_{2}} (p + v_{2}) \\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} = \\frac{\\partial}{\\partial v_{2}} \\frac{(p + v_{2})^{2}}{a^{\\dagger}} and \\frac{\\partial}{\\partial v_{2}} (p + v_{2}) \\operatorname{F_{H}}{(a^{\\dagger},v_{2},p)} + 1 = \\frac{\\partial}{\\partial v_{2}} \\frac{(p + v_{2})^{2}}{a^{\\dagger}} + 1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_2', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True))))"], [["times", 1, "Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Function('F_H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_2', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Integer(2))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Function('F_H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_2', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Mul(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Function('F_H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_2', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Add(Symbol('p', commutative=True), Symbol('v_2', commutative=True)), Integer(2))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{H},f^{*})} = e^{(f^{*})^{\\hat{H}}}, then obtain \\frac{\\partial}{\\partial \\hat{H}} (\\int \\operatorname{F_{g}}{(\\hat{H},f^{*})} d\\hat{H} - 1) = \\frac{\\partial}{\\partial \\hat{H}} (\\int e^{(f^{*})^{\\hat{H}}} d\\hat{H} - 1)", "derivation": "\\operatorname{F_{g}}{(\\hat{H},f^{*})} = e^{(f^{*})^{\\hat{H}}} and \\int \\operatorname{F_{g}}{(\\hat{H},f^{*})} d\\hat{H} = \\int e^{(f^{*})^{\\hat{H}}} d\\hat{H} and \\int \\operatorname{F_{g}}{(\\hat{H},f^{*})} d\\hat{H} - 1 = \\int e^{(f^{*})^{\\hat{H}}} d\\hat{H} - 1 and \\frac{\\partial}{\\partial \\hat{H}} (\\int \\operatorname{F_{g}}{(\\hat{H},f^{*})} d\\hat{H} - 1) = \\frac{\\partial}{\\partial \\hat{H}} (\\int e^{(f^{*})^{\\hat{H}}} d\\hat{H} - 1)", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('f^*', commutative=True)), exp(Pow(Symbol('f^*', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(exp(Pow(Symbol('f^*', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integral(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)), Add(Integral(exp(Pow(Symbol('f^*', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Integral(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Integral(exp(Pow(Symbol('f^*', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(l,E_{x})} = \\frac{\\partial}{\\partial l} \\frac{l}{E_{x}}, then obtain E_{x} (- y{(l,E_{x})} + \\frac{\\partial}{\\partial l} y{(l,E_{x})}) = - E_{x} y{(l,E_{x})}", "derivation": "y{(l,E_{x})} = \\frac{\\partial}{\\partial l} \\frac{l}{E_{x}} and \\frac{\\partial}{\\partial l} y{(l,E_{x})} = \\frac{\\partial^{2}}{\\partial l^{2}} \\frac{l}{E_{x}} and - y{(l,E_{x})} + \\frac{\\partial}{\\partial l} y{(l,E_{x})} = - y{(l,E_{x})} + \\frac{\\partial^{2}}{\\partial l^{2}} \\frac{l}{E_{x}} and E_{x} (- y{(l,E_{x})} + \\frac{\\partial}{\\partial l} y{(l,E_{x})}) = E_{x} (- y{(l,E_{x})} + \\frac{\\partial^{2}}{\\partial l^{2}} \\frac{l}{E_{x}}) and E_{x} (- y{(l,E_{x})} + \\frac{\\partial}{\\partial l} y{(l,E_{x})}) = - E_{x} y{(l,E_{x})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))))"], [["minus", 2, "Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Derivative(Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2)))))"], [["times", 3, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Add(Mul(Integer(-1), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Derivative(Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Mul(Symbol('E_x', commutative=True), Add(Mul(Integer(-1), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('E_x', commutative=True), Add(Mul(Integer(-1), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))), Derivative(Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('E_x', commutative=True), Function('y')(Symbol('l', commutative=True), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given h{(f^{*},\\pi)} = \\frac{\\pi}{f^{*}}, then obtain \\frac{\\partial}{\\partial \\pi} (f^{*})^{2} (- \\pi + \\frac{h{(f^{*},\\pi)}}{f^{*}}) = \\frac{\\partial}{\\partial \\pi} (f^{*})^{2} (- \\pi + \\frac{\\pi}{(f^{*})^{2}})", "derivation": "h{(f^{*},\\pi)} = \\frac{\\pi}{f^{*}} and \\frac{h{(f^{*},\\pi)}}{f^{*}} = \\frac{\\pi}{(f^{*})^{2}} and - \\pi + \\frac{h{(f^{*},\\pi)}}{f^{*}} = - \\pi + \\frac{\\pi}{(f^{*})^{2}} and (f^{*})^{2} (- \\pi + \\frac{h{(f^{*},\\pi)}}{f^{*}}) = (f^{*})^{2} (- \\pi + \\frac{\\pi}{(f^{*})^{2}}) and \\frac{\\partial}{\\partial \\pi} (f^{*})^{2} (- \\pi + \\frac{h{(f^{*},\\pi)}}{f^{*}}) = \\frac{\\partial}{\\partial \\pi} (f^{*})^{2} (- \\pi + \\frac{\\pi}{(f^{*})^{2}})", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('f^*', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('f^*', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('h')(Symbol('f^*', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-2))))"], [["minus", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('h')(Symbol('f^*', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-2)))))"], [["divide", 3, "Pow(Symbol('f^*', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('h')(Symbol('f^*', commutative=True), Symbol('\\\\pi', commutative=True))))), Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-2))))))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('h')(Symbol('f^*', commutative=True), Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f^*', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-2))))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(T)} = \\sin{(\\sin{(T)})} and \\Omega{(T)} = \\sin{(T)}, then obtain ((\\int \\sin{(\\Omega{(T)})} dT) \\int \\sin{(\\sin{(T)})} dT)^{T} = ((\\int \\operatorname{V_{\\mathbf{B}}}{(T)} dT) \\int \\sin{(\\sin{(T)})} dT)^{T}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(T)} = \\sin{(\\sin{(T)})} and \\int \\operatorname{V_{\\mathbf{B}}}{(T)} dT = \\int \\sin{(\\sin{(T)})} dT and (\\int \\operatorname{V_{\\mathbf{B}}}{(T)} dT) \\int \\sin{(\\sin{(T)})} dT = (\\int \\sin{(\\sin{(T)})} dT)^{2} and \\Omega{(T)} = \\sin{(T)} and \\operatorname{V_{\\mathbf{B}}}{(T)} = \\sin{(\\Omega{(T)})} and (\\int \\sin{(\\Omega{(T)})} dT) \\int \\sin{(\\sin{(T)})} dT = (\\int \\sin{(\\sin{(T)})} dT)^{2} and (\\int \\sin{(\\Omega{(T)})} dT) \\int \\sin{(\\sin{(T)})} dT = (\\int \\operatorname{V_{\\mathbf{B}}}{(T)} dT) \\int \\sin{(\\sin{(T)})} dT and ((\\int \\sin{(\\Omega{(T)})} dT) \\int \\sin{(\\sin{(T)})} dT)^{T} = ((\\int \\operatorname{V_{\\mathbf{B}}}{(T)} dT) \\int \\sin{(\\sin{(T)})} dT)^{T}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), sin(sin(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["times", 2, "Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Pow(Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), sin(Function('\\\\Omega')(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integral(sin(Function('\\\\Omega')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Pow(Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integral(sin(Function('\\\\Omega')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Mul(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["power", 7, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Integral(sin(Function('\\\\Omega')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Mul(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(V,n_{2})} = - V + \\log{(n_{2})} and \\mathbf{g}{(V,n_{2})} = - \\frac{- n_{2} + \\operatorname{V_{\\mathbf{E}}}{(V,n_{2})}}{V}, then obtain \\mathbf{g}{(V,n_{2})} = - \\frac{- V - n_{2} + \\log{(n_{2})}}{V}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(V,n_{2})} = - V + \\log{(n_{2})} and - n_{2} + \\operatorname{V_{\\mathbf{E}}}{(V,n_{2})} = - V - n_{2} + \\log{(n_{2})} and - \\frac{- n_{2} + \\operatorname{V_{\\mathbf{E}}}{(V,n_{2})}}{V} = - \\frac{- V - n_{2} + \\log{(n_{2})}}{V} and \\mathbf{g}{(V,n_{2})} = - \\frac{- n_{2} + \\operatorname{V_{\\mathbf{E}}}{(V,n_{2})}}{V} and \\mathbf{g}{(V,n_{2})} = - \\frac{- V - n_{2} + \\log{(n_{2})}}{V}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('n_2', commutative=True))))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('V', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('V', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(h,v)} = h \\cos{(v)}, then derive v \\cos{(v)} \\frac{\\partial}{\\partial h} \\operatorname{a^{\\dagger}}{(h,v)} = v \\cos^{2}{(v)}, then obtain \\cos{(v \\cos{(v)} \\frac{\\partial}{\\partial h} \\operatorname{a^{\\dagger}}{(h,v)})} = \\cos{(v \\cos^{2}{(v)})}", "derivation": "\\operatorname{a^{\\dagger}}{(h,v)} = h \\cos{(v)} and \\operatorname{a^{\\dagger}}{(h,v)} \\cos{(v)} = h \\cos^{2}{(v)} and \\operatorname{a^{\\dagger}}{(h,v)} \\cos{(v)} + \\cos{(v)} = h \\cos^{2}{(v)} + \\cos{(v)} and v (\\operatorname{a^{\\dagger}}{(h,v)} \\cos{(v)} + \\cos{(v)}) = v (h \\cos^{2}{(v)} + \\cos{(v)}) and \\frac{\\partial}{\\partial h} v (\\operatorname{a^{\\dagger}}{(h,v)} \\cos{(v)} + \\cos{(v)}) = \\frac{\\partial}{\\partial h} v (h \\cos^{2}{(v)} + \\cos{(v)}) and v \\cos{(v)} \\frac{\\partial}{\\partial h} \\operatorname{a^{\\dagger}}{(h,v)} = v \\cos^{2}{(v)} and \\cos{(v \\cos{(v)} \\frac{\\partial}{\\partial h} \\operatorname{a^{\\dagger}}{(h,v)})} = \\cos{(v \\cos^{2}{(v)})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('h', commutative=True), cos(Symbol('v', commutative=True))))"], [["times", 1, "cos(Symbol('v', commutative=True))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), Mul(Symbol('h', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2))))"], [["add", 2, "cos(Symbol('v', commutative=True))"], "Equality(Add(Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True))), Add(Mul(Symbol('h', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2))), cos(Symbol('v', commutative=True))))"], [["times", 3, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Add(Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True)))), Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2))), cos(Symbol('v', commutative=True)))))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Symbol('v', commutative=True), Add(Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), cos(Symbol('v', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('h', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2))), cos(Symbol('v', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('v', commutative=True), cos(Symbol('v', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2))))"], [["cos", 6], "Equality(cos(Mul(Symbol('v', commutative=True), cos(Symbol('v', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))), cos(Mul(Symbol('v', commutative=True), Pow(cos(Symbol('v', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\rho)} = \\int e^{\\rho} d\\rho, then derive A_{y} + \\mathbf{M}{(\\rho)} + e^{\\rho} = 2 A_{y} + 2 e^{\\rho}, then obtain A_{y} + e^{\\rho} + \\int e^{\\rho} d\\rho = A_{y} + \\mathbf{M}{(\\rho)} + e^{\\rho}", "derivation": "\\mathbf{M}{(\\rho)} = \\int e^{\\rho} d\\rho and \\mathbf{M}{(\\rho)} + \\int e^{\\rho} d\\rho = 2 \\int e^{\\rho} d\\rho and A_{y} + \\mathbf{M}{(\\rho)} + e^{\\rho} = 2 A_{y} + 2 e^{\\rho} and A_{y} + e^{\\rho} + \\int e^{\\rho} d\\rho = 2 A_{y} + 2 e^{\\rho} and A_{y} + e^{\\rho} + \\int e^{\\rho} d\\rho = A_{y} + \\mathbf{M}{(\\rho)} + e^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["add", 1, "Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('A_y', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('A_y', commutative=True), exp(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('A_y', commutative=True), exp(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Add(Symbol('A_y', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given L{(S,\\delta)} = \\log{(S \\delta)}, then obtain 0 = \\frac{- S L{(S,\\delta)} + S \\log{(S \\delta - 2 L{(S,\\delta)} + \\log{(S \\delta)} + \\log{(S \\delta - L{(S,\\delta)} + \\log{(S \\delta)})})}}{\\int S \\delta (- L{(S,\\delta)} + \\log{(S \\delta)}) dS}", "derivation": "L{(S,\\delta)} = \\log{(S \\delta)} and 0 = - L{(S,\\delta)} + \\log{(S \\delta)} and 0 = S (- L{(S,\\delta)} + \\log{(S \\delta)}) and S \\delta = S \\delta - L{(S,\\delta)} + \\log{(S \\delta)} and 0 = S (- L{(S,\\delta)} + \\log{(S \\delta - L{(S,\\delta)} + \\log{(S \\delta)})}) and 0 = S (- L{(S,\\delta)} + \\log{(S \\delta - 2 L{(S,\\delta)} + \\log{(S \\delta)} + \\log{(S \\delta - L{(S,\\delta)} + \\log{(S \\delta)})})}) and 0 = - S L{(S,\\delta)} + S \\log{(S \\delta - 2 L{(S,\\delta)} + \\log{(S \\delta)} + \\log{(S \\delta - L{(S,\\delta)} + \\log{(S \\delta)})})} and 0 = \\frac{- S L{(S,\\delta)} + S \\log{(S \\delta - 2 L{(S,\\delta)} + \\log{(S \\delta)} + \\log{(S \\delta - L{(S,\\delta)} + \\log{(S \\delta)})})}}{\\int S \\delta (- L{(S,\\delta)} + \\log{(S \\delta)}) dS}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["times", 2, "Symbol('S', commutative=True)"], "Equality(Integer(0), Mul(Symbol('S', commutative=True), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))))"], [["add", 2, "Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Symbol('S', commutative=True), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Mul(Symbol('S', commutative=True), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))))))))"], [["expand", 6], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('S', commutative=True), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Symbol('S', commutative=True), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))))))))"], [["divide", 7, "Integral(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('S', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Symbol('S', commutative=True), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))))))), Pow(Integral(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Add(Mul(Integer(-1), Function('L')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))), log(Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('S', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(f^{*},E_{x})} = \\log{(E_{x} - f^{*})} and \\mathbb{I}{(f^{*},E_{x})} = E_{x} - f^{*}, then obtain \\log{(\\mathbb{I}{(f^{*},E_{x})})}^{f^{*}} = \\log{(E_{x} - f^{*})}^{f^{*}}", "derivation": "\\operatorname{t_{2}}{(f^{*},E_{x})} = \\log{(E_{x} - f^{*})} and \\operatorname{t_{2}}^{f^{*}}{(f^{*},E_{x})} = \\log{(E_{x} - f^{*})}^{f^{*}} and \\mathbb{I}{(f^{*},E_{x})} = E_{x} - f^{*} and \\operatorname{t_{2}}^{f^{*}}{(f^{*},E_{x})} = \\log{(\\mathbb{I}{(f^{*},E_{x})})}^{f^{*}} and \\log{(\\mathbb{I}{(f^{*},E_{x})})}^{f^{*}} = \\log{(E_{x} - f^{*})}^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Symbol('f^*', commutative=True)), Pow(log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('t_2')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Symbol('f^*', commutative=True)), Pow(log(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True))), Symbol('f^*', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(log(Function('\\\\mathbb{I}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True))), Symbol('f^*', commutative=True)), Pow(log(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}, then derive \\int \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\frac{\\mathbf{J}_f^{2} \\log{(\\mathbf{J}_f)}}{2} - \\frac{\\mathbf{J}_f^{2}}{4} + k, then obtain \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} \\int \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f (\\frac{\\mathbf{J}_f^{2} \\log{(\\mathbf{J}_f)}}{2} - \\frac{\\mathbf{J}_f^{2}}{4} + k) \\mathbf{S}{(\\mathbf{J}_f)}", "derivation": "\\mathbf{S}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} = \\mathbf{J}_f \\log{(\\mathbf{J}_f)} and \\int \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int \\mathbf{J}_f \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\int \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\frac{\\mathbf{J}_f^{2} \\log{(\\mathbf{J}_f)}}{2} - \\frac{\\mathbf{J}_f^{2}}{4} + k and \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} \\int \\mathbf{J}_f \\mathbf{S}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f (\\frac{\\mathbf{J}_f^{2} \\log{(\\mathbf{J}_f)}}{2} - \\frac{\\mathbf{J}_f^{2}}{4} + k) \\mathbf{S}{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Symbol('k', commutative=True)))"], [["times", 4, "Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Symbol('k', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given c{(\\theta_2,C)} = C \\theta_2, then obtain 2 C + 2 \\int c{(\\theta_2,C)} dC = 2 C + \\int C \\theta_2 dC + \\int c{(\\theta_2,C)} dC", "derivation": "c{(\\theta_2,C)} = C \\theta_2 and \\int c{(\\theta_2,C)} dC = \\int C \\theta_2 dC and C + \\int c{(\\theta_2,C)} dC = C + \\int C \\theta_2 dC and 2 C + 2 \\int c{(\\theta_2,C)} dC = 2 C + \\int C \\theta_2 dC + \\int c{(\\theta_2,C)} dC", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["add", 2, "Symbol('C', commutative=True)"], "Equality(Add(Symbol('C', commutative=True), Integral(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Symbol('C', commutative=True), Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["add", 3, "Add(Symbol('C', commutative=True), Integral(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('C', commutative=True)), Mul(Integer(2), Integral(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))), Add(Mul(Integer(2), Symbol('C', commutative=True)), Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Function('c')(Symbol('\\\\theta_2', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(A_{z})} = \\log{(A_{z})}, then obtain (\\iint \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} dA_{z} dA_{z} - \\iint \\log{(A_{z})}^{A_{z}} dA_{z} dA_{z})^{A_{z}} = 0^{A_{z}}", "derivation": "\\dot{\\mathbf{r}}{(A_{z})} = \\log{(A_{z})} and \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} = \\log{(A_{z})}^{A_{z}} and \\int \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} dA_{z} = \\int \\log{(A_{z})}^{A_{z}} dA_{z} and \\iint \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} dA_{z} dA_{z} = \\iint \\log{(A_{z})}^{A_{z}} dA_{z} dA_{z} and \\iint \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} dA_{z} dA_{z} - \\iint \\log{(A_{z})}^{A_{z}} dA_{z} dA_{z} = 0 and (\\iint \\dot{\\mathbf{r}}^{A_{z}}{(A_{z})} dA_{z} dA_{z} - \\iint \\log{(A_{z})}^{A_{z}} dA_{z} dA_{z})^{A_{z}} = 0^{A_{z}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), log(Symbol('A_z', commutative=True)))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["minus", 4, "Integral(Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))"], "Equality(Add(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))), Integer(0))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Add(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))), Symbol('A_z', commutative=True)), Pow(Integer(0), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given A{(F_{g})} = \\log{(F_{g})}, then obtain \\frac{d}{d F_{g}} (\\frac{A^{4}{(F_{g})}}{\\log{(F_{g})}^{2}} + A{(F_{g})}) = \\frac{d}{d F_{g}} (A{(F_{g})} \\log{(F_{g})} + A{(F_{g})})", "derivation": "A{(F_{g})} = \\log{(F_{g})} and A^{2}{(F_{g})} = A{(F_{g})} \\log{(F_{g})} and A^{2}{(F_{g})} + A{(F_{g})} = A{(F_{g})} \\log{(F_{g})} + A{(F_{g})} and \\frac{d}{d F_{g}} (A^{2}{(F_{g})} + A{(F_{g})}) = \\frac{d}{d F_{g}} (A{(F_{g})} \\log{(F_{g})} + A{(F_{g})}) and \\frac{A^{2}{(F_{g})}}{\\log{(F_{g})}} = A{(F_{g})} and \\frac{A^{4}{(F_{g})}}{\\log{(F_{g})}^{2}} = A^{2}{(F_{g})} and \\frac{d}{d F_{g}} (\\frac{A^{4}{(F_{g})}}{\\log{(F_{g})}^{2}} + A{(F_{g})}) = \\frac{d}{d F_{g}} (A{(F_{g})} \\log{(F_{g})} + A{(F_{g})})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))"], [["times", 1, "Function('A')(Symbol('F_g', commutative=True))"], "Equality(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))))"], [["add", 2, "Function('A')(Symbol('F_g', commutative=True))"], "Equality(Add(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(2)), Function('A')(Symbol('F_g', commutative=True))), Add(Mul(Function('A')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Function('A')(Symbol('F_g', commutative=True))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(2)), Function('A')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Mul(Function('A')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Function('A')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(2)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Function('A')(Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(4)), Pow(log(Symbol('F_g', commutative=True)), Integer(-2))), Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Derivative(Add(Mul(Pow(Function('A')(Symbol('F_g', commutative=True)), Integer(4)), Pow(log(Symbol('F_g', commutative=True)), Integer(-2))), Function('A')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Mul(Function('A')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Function('A')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\phi_1,\\mu_0)} = e^{\\mu_0^{\\phi_1}}, then derive 0^{\\phi_1} = (- \\frac{\\partial}{\\partial \\mu_0} t{(\\phi_1,\\mu_0)} + \\frac{\\mu_0^{\\phi_1} \\phi_1 e^{\\mu_0^{\\phi_1}}}{\\mu_0})^{\\phi_1}, then obtain t{(\\phi_1,\\mu_0)} = e", "derivation": "t{(\\phi_1,\\mu_0)} = e^{\\mu_0^{\\phi_1}} and 0 = - t{(\\phi_1,\\mu_0)} + e^{\\mu_0^{\\phi_1}} and \\frac{d}{d \\mu_0} 0 = \\frac{\\partial}{\\partial \\mu_0} (- t{(\\phi_1,\\mu_0)} + e^{\\mu_0^{\\phi_1}}) and (\\frac{d}{d \\mu_0} 0)^{\\phi_1} = (\\frac{\\partial}{\\partial \\mu_0} (- t{(\\phi_1,\\mu_0)} + e^{\\mu_0^{\\phi_1}}))^{\\phi_1} and 0^{\\phi_1} = (- \\frac{\\partial}{\\partial \\mu_0} t{(\\phi_1,\\mu_0)} + \\frac{\\mu_0^{\\phi_1} \\phi_1 e^{\\mu_0^{\\phi_1}}}{\\mu_0})^{\\phi_1} and t{(\\phi_1,\\mu_0)} = e", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), exp(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), exp(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), exp(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), exp(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Integer(0), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True), exp(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi_1', commutative=True))))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Function('t')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), E)"]]}, {"prompt": "Given \\hat{H}{(W,l,\\rho_b)} = (l^{W})^{\\rho_b}, then derive \\frac{\\partial}{\\partial W} \\hat{H}{(W,l,\\rho_b)} = \\rho_b (l^{W})^{\\rho_b} \\log{(l)}, then obtain \\sin{(\\rho_b (l^{W})^{\\rho_b} \\log{(l)})} = \\sin{(\\frac{\\partial}{\\partial W} (l^{W})^{\\rho_b})}", "derivation": "\\hat{H}{(W,l,\\rho_b)} = (l^{W})^{\\rho_b} and \\frac{\\partial}{\\partial W} \\hat{H}{(W,l,\\rho_b)} = \\frac{\\partial}{\\partial W} (l^{W})^{\\rho_b} and \\sin{(\\frac{\\partial}{\\partial W} \\hat{H}{(W,l,\\rho_b)})} = \\sin{(\\frac{\\partial}{\\partial W} (l^{W})^{\\rho_b})} and \\frac{\\partial}{\\partial W} \\hat{H}{(W,l,\\rho_b)} = \\rho_b (l^{W})^{\\rho_b} \\log{(l)} and \\sin{(\\rho_b (l^{W})^{\\rho_b} \\log{(l)})} = \\sin{(\\frac{\\partial}{\\partial W} (l^{W})^{\\rho_b})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('l', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('l', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('l', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), sin(Derivative(Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('l', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)), log(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(sin(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)), log(Symbol('l', commutative=True)))), sin(Derivative(Pow(Pow(Symbol('l', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(r)} = e^{r}, then obtain - e^{r} + \\frac{d^{2}}{d r^{2}} \\operatorname{v_{1}}{(r)} = - e^{r} + \\frac{d^{2}}{d r^{2}} e^{r}", "derivation": "\\operatorname{v_{1}}{(r)} = e^{r} and \\frac{d}{d r} \\operatorname{v_{1}}{(r)} = \\frac{d}{d r} e^{r} and \\frac{d^{2}}{d r^{2}} \\operatorname{v_{1}}{(r)} = \\frac{d^{2}}{d r^{2}} e^{r} and - e^{r} + \\frac{d^{2}}{d r^{2}} \\operatorname{v_{1}}{(r)} = - e^{r} + \\frac{d^{2}}{d r^{2}} e^{r}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["minus", 3, "exp(Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('r', commutative=True))), Derivative(Function('v_1')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))), Add(Mul(Integer(-1), exp(Symbol('r', commutative=True))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)} = - V_{\\mathbf{B}} + \\eta + \\mathbb{I}, then derive \\int (V_{\\mathbf{B}} + \\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)}) d\\eta = \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + g, then obtain \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + \\mathbf{p} = \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + g", "derivation": "\\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)} = - V_{\\mathbf{B}} + \\eta + \\mathbb{I} and V_{\\mathbf{B}} + \\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)} = \\eta + \\mathbb{I} and \\int (V_{\\mathbf{B}} + \\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)}) d\\eta = \\int (\\eta + \\mathbb{I}) d\\eta and \\int (V_{\\mathbf{B}} + \\hat{p}_0{(V_{\\mathbf{B}},\\mathbb{I},\\eta)}) d\\eta = \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + g and \\int (\\eta + \\mathbb{I}) d\\eta = \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + g and \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + \\mathbf{p} = \\frac{\\eta^{2}}{2} + \\eta \\mathbb{I} + g", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\hat{p}_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\hat{p}_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\hat{p}_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('g', commutative=True)))"]]}, {"prompt": "Given g{(f_{\\mathbf{v}},v_{x})} = v_{x} + \\sin{(f_{\\mathbf{v}})}, then obtain \\frac{\\partial}{\\partial v_{x}} (g^{v_{x}}{(f_{\\mathbf{v}},v_{x})})^{v_{x}} = \\frac{\\partial}{\\partial v_{x}} ((v_{x} + \\sin{(f_{\\mathbf{v}})})^{v_{x}})^{v_{x}}", "derivation": "g{(f_{\\mathbf{v}},v_{x})} = v_{x} + \\sin{(f_{\\mathbf{v}})} and g^{v_{x}}{(f_{\\mathbf{v}},v_{x})} = (v_{x} + \\sin{(f_{\\mathbf{v}})})^{v_{x}} and (g^{v_{x}}{(f_{\\mathbf{v}},v_{x})})^{v_{x}} = ((v_{x} + \\sin{(f_{\\mathbf{v}})})^{v_{x}})^{v_{x}} and \\frac{\\partial}{\\partial v_{x}} (g^{v_{x}}{(f_{\\mathbf{v}},v_{x})})^{v_{x}} = \\frac{\\partial}{\\partial v_{x}} ((v_{x} + \\sin{(f_{\\mathbf{v}})})^{v_{x}})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True)), Add(Symbol('v_x', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Add(Symbol('v_x', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('v_x', commutative=True)))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Pow(Function('g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Pow(Add(Symbol('v_x', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Symbol('v_x', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(a^{\\dagger},\\Omega)} = \\Omega^{a^{\\dagger}}, then obtain (\\frac{\\log{(\\int I{(a^{\\dagger},\\Omega)} da^{\\dagger})}}{\\Omega})^{a^{\\dagger}} = (\\frac{\\log{(\\int \\Omega^{a^{\\dagger}} da^{\\dagger})}}{\\Omega})^{a^{\\dagger}}", "derivation": "I{(a^{\\dagger},\\Omega)} = \\Omega^{a^{\\dagger}} and \\int I{(a^{\\dagger},\\Omega)} da^{\\dagger} = \\int \\Omega^{a^{\\dagger}} da^{\\dagger} and \\log{(\\int I{(a^{\\dagger},\\Omega)} da^{\\dagger})} = \\log{(\\int \\Omega^{a^{\\dagger}} da^{\\dagger})} and \\frac{\\log{(\\int I{(a^{\\dagger},\\Omega)} da^{\\dagger})}}{\\Omega} = \\frac{\\log{(\\int \\Omega^{a^{\\dagger}} da^{\\dagger})}}{\\Omega} and (\\frac{\\log{(\\int I{(a^{\\dagger},\\Omega)} da^{\\dagger})}}{\\Omega})^{a^{\\dagger}} = (\\frac{\\log{(\\int \\Omega^{a^{\\dagger}} da^{\\dagger})}}{\\Omega})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('I')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('I')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), log(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["divide", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Integral(Function('I')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))))"], [["power", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Integral(Function('I')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(C_{2},c_{0})} = \\cos{(C_{2} - c_{0})}, then obtain \\sin{(\\frac{\\partial}{\\partial C_{2}} \\operatorname{f_{E}}{(C_{2},c_{0})} + 1)} = - \\sin{(\\sin{(C_{2} - c_{0})} - 1)}", "derivation": "\\operatorname{f_{E}}{(C_{2},c_{0})} = \\cos{(C_{2} - c_{0})} and C_{2} + \\operatorname{f_{E}}{(C_{2},c_{0})} = C_{2} + \\cos{(C_{2} - c_{0})} and \\frac{\\partial}{\\partial C_{2}} (C_{2} + \\operatorname{f_{E}}{(C_{2},c_{0})}) = \\frac{\\partial}{\\partial C_{2}} (C_{2} + \\cos{(C_{2} - c_{0})}) and \\sin{(\\frac{\\partial}{\\partial C_{2}} (C_{2} + \\operatorname{f_{E}}{(C_{2},c_{0})}))} = \\sin{(\\frac{\\partial}{\\partial C_{2}} (C_{2} + \\cos{(C_{2} - c_{0})}))} and \\sin{(\\frac{\\partial}{\\partial C_{2}} \\operatorname{f_{E}}{(C_{2},c_{0})} + 1)} = - \\sin{(\\sin{(C_{2} - c_{0})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('C_2', commutative=True), Symbol('c_0', commutative=True)), cos(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))"], [["add", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Function('f_E')(Symbol('C_2', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('C_2', commutative=True), cos(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Symbol('C_2', commutative=True), Function('f_E')(Symbol('C_2', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Symbol('C_2', commutative=True), cos(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Add(Symbol('C_2', commutative=True), Function('f_E')(Symbol('C_2', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))), sin(Derivative(Add(Symbol('C_2', commutative=True), cos(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(sin(Add(Derivative(Function('f_E')(Symbol('C_2', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1))), Mul(Integer(-1), sin(Add(sin(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))), Integer(-1)))))"]]}, {"prompt": "Given \\varphi^{*}{(G,L_{\\varepsilon})} = G - L_{\\varepsilon}, then obtain L_{\\varepsilon} + \\int (G - L_{\\varepsilon}) dG + \\int \\varphi^{*}{(G,L_{\\varepsilon})} dG = L_{\\varepsilon} + 2 \\int (G - L_{\\varepsilon}) dG", "derivation": "\\varphi^{*}{(G,L_{\\varepsilon})} = G - L_{\\varepsilon} and \\int \\varphi^{*}{(G,L_{\\varepsilon})} dG = \\int (G - L_{\\varepsilon}) dG and \\int (G - L_{\\varepsilon}) dG + \\int \\varphi^{*}{(G,L_{\\varepsilon})} dG = 2 \\int (G - L_{\\varepsilon}) dG and L_{\\varepsilon} + \\int (G - L_{\\varepsilon}) dG + \\int \\varphi^{*}{(G,L_{\\varepsilon})} dG = L_{\\varepsilon} + 2 \\int (G - L_{\\varepsilon}) dG", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('G', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('G', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["add", 2, "Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Function('\\\\varphi^*')(Symbol('G', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Integer(2), Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Function('\\\\varphi^*')(Symbol('G', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True)))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('G', commutative=True))))))"]]}, {"prompt": "Given Z{(\\mathbf{P},\\dot{z})} = e^{\\dot{z} + \\mathbf{P}}, then obtain (\\frac{d}{d \\mathbf{P}} 1)^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\mathbf{P}} Z{(\\mathbf{P},\\dot{z})} e^{- \\dot{z} - \\mathbf{P}})^{\\mathbf{P}}", "derivation": "Z{(\\mathbf{P},\\dot{z})} = e^{\\dot{z} + \\mathbf{P}} and 1 = \\frac{e^{\\dot{z} + \\mathbf{P}}}{Z{(\\mathbf{P},\\dot{z})}} and \\frac{d}{d \\mathbf{P}} 1 = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{e^{\\dot{z} + \\mathbf{P}}}{Z{(\\mathbf{P},\\dot{z})}} and \\frac{d}{d \\mathbf{P}} 1 = \\frac{\\partial}{\\partial \\mathbf{P}} e^{- \\dot{z} - \\mathbf{P}} e^{\\dot{z} + \\mathbf{P}} and (\\frac{d}{d \\mathbf{P}} 1)^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\mathbf{P}} e^{- \\dot{z} - \\mathbf{P}} e^{\\dot{z} + \\mathbf{P}})^{\\mathbf{P}} and (\\frac{d}{d \\mathbf{P}} 1)^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\mathbf{P}} Z{(\\mathbf{P},\\dot{z})} e^{- \\dot{z} - \\mathbf{P}})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), exp(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 1, "Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))), exp(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))), exp(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Mul(Function('Z')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(B)} = \\frac{d}{d B} e^{B}, then derive \\operatorname{F_{H}}{(B)} = e^{B}, then derive \\operatorname{F_{H}}^{B}{(B)} + 1 = (e^{B})^{B} + 1, then obtain ((\\frac{d}{d B} \\operatorname{F_{H}}{(B)})^{B} + 1) \\frac{d}{d B} e^{B} = ((e^{B})^{B} + 1) \\frac{d}{d B} e^{B}", "derivation": "\\operatorname{F_{H}}{(B)} = \\frac{d}{d B} e^{B} and \\operatorname{F_{H}}{(B)} = e^{B} and \\operatorname{F_{H}}^{B}{(B)} = (\\frac{d}{d B} e^{B})^{B} and \\operatorname{F_{H}}^{B}{(B)} = (\\frac{d}{d B} \\operatorname{F_{H}}{(B)})^{B} and \\operatorname{F_{H}}^{B}{(B)} + 1 = (\\frac{d}{d B} e^{B})^{B} + 1 and \\operatorname{F_{H}}^{B}{(B)} + 1 = (e^{B})^{B} + 1 and (\\operatorname{F_{H}}^{B}{(B)} + 1) \\frac{d}{d B} e^{B} = ((e^{B})^{B} + 1) \\frac{d}{d B} e^{B} and ((\\frac{d}{d B} \\operatorname{F_{H}}{(B)})^{B} + 1) \\frac{d}{d B} e^{B} = ((e^{B})^{B} + 1) \\frac{d}{d B} e^{B}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('B', commutative=True)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_H')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('F_H')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(Function('F_H')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["add", 3, 1], "Equality(Add(Pow(Function('F_H')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)), Add(Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Integer(1)))"], [["evaluate_derivatives", 5], "Equality(Add(Pow(Function('F_H')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)), Add(Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)))"], [["divide", 6, "Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))"], "Equality(Mul(Add(Pow(Function('F_H')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Add(Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Add(Pow(Derivative(Function('F_H')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)), Integer(1)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Add(Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(\\mathbf{M})} = \\mathbf{M}, then obtain - \\frac{(- A_{1} + \\eta{(\\mathbf{M})}) (A_{1} - \\mathbf{M})}{\\mathbf{M}} = - \\frac{(- A_{1} + \\mathbf{M}) (A_{1} - \\mathbf{M})}{\\mathbf{M}}", "derivation": "\\eta{(\\mathbf{M})} = \\mathbf{M} and - A_{1} + \\eta{(\\mathbf{M})} = - A_{1} + \\mathbf{M} and (- A_{1} + \\eta{(\\mathbf{M})}) (A_{1} - \\mathbf{M}) = (- A_{1} + \\mathbf{M}) (A_{1} - \\mathbf{M}) and - \\frac{(- A_{1} + \\eta{(\\mathbf{M})}) (A_{1} - \\mathbf{M})}{\\mathbf{M}} = - \\frac{(- A_{1} + \\mathbf{M}) (A_{1} - \\mathbf{M})}{\\mathbf{M}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], [["minus", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 2, "Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(v_{1})} = e^{v_{1}}, then obtain \\frac{\\hat{x}_0{(v_{1})} \\int (v_{1} + \\hat{x}_0{(v_{1})} - e^{v_{1}}) dv_{1}}{\\int v_{1} dv_{1}} = \\hat{x}_0{(v_{1})}", "derivation": "\\hat{x}_0{(v_{1})} = e^{v_{1}} and v_{1} + \\hat{x}_0{(v_{1})} = v_{1} + e^{v_{1}} and v_{1} + \\hat{x}_0{(v_{1})} - e^{v_{1}} = v_{1} and \\int (v_{1} + \\hat{x}_0{(v_{1})} - e^{v_{1}}) dv_{1} = \\int v_{1} dv_{1} and \\hat{x}_0{(v_{1})} \\int (v_{1} + \\hat{x}_0{(v_{1})} - e^{v_{1}}) dv_{1} = \\hat{x}_0{(v_{1})} \\int v_{1} dv_{1} and \\frac{\\hat{x}_0{(v_{1})} \\int (v_{1} + \\hat{x}_0{(v_{1})} - e^{v_{1}}) dv_{1}}{\\int v_{1} dv_{1}} = \\hat{x}_0{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True))))"], [["minus", 2, "exp(Symbol('v_1', commutative=True))"], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Add(Symbol('v_1', commutative=True), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True))), Integral(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True))))"], [["times", 4, "Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Integral(Add(Symbol('v_1', commutative=True), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True)))), Mul(Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Integral(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True)))))"], [["divide", 5, "Integral(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Pow(Integral(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True))), Integer(-1)), Integral(Add(Symbol('v_1', commutative=True), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)), Mul(Integer(-1), exp(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True)))), Function('\\\\hat{x}_0')(Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\chi,v_{1})} = \\sin{(\\chi + v_{1})}, then derive \\int \\frac{\\hat{x}{(\\chi,v_{1})}}{\\sin{(\\chi + v_{1})}} dv_{1} = f_{E} + v_{1}, then obtain \\int 1 dv_{1} = f_{E} + v_{1}", "derivation": "\\hat{x}{(\\chi,v_{1})} = \\sin{(\\chi + v_{1})} and \\frac{\\hat{x}{(\\chi,v_{1})}}{\\sin{(\\chi + v_{1})}} = 1 and \\int \\frac{\\hat{x}{(\\chi,v_{1})}}{\\sin{(\\chi + v_{1})}} dv_{1} = \\int 1 dv_{1} and \\int \\frac{\\hat{x}{(\\chi,v_{1})}}{\\sin{(\\chi + v_{1})}} dv_{1} = f_{E} + v_{1} and \\int 1 dv_{1} = f_{E} + v_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), sin(Add(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 1, "sin(Add(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Pow(sin(Add(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('v_1', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Pow(sin(Add(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Integer(-1))), Tuple(Symbol('v_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\hat{x}')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Pow(sin(Add(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Integer(-1))), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('f_E', commutative=True), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('f_E', commutative=True), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given m{(n_{2})} = \\sin{(\\sin{(n_{2})})}, then obtain \\frac{d}{d n_{2}} - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} \\int - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} dn_{2} = \\frac{d}{d n_{2}} (-1) \\int - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} dn_{2}", "derivation": "m{(n_{2})} = \\sin{(\\sin{(n_{2})})} and - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} = -1 and \\frac{d}{d n_{2}} - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} = \\frac{d}{d n_{2}} (-1) and \\frac{d}{d n_{2}} - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} \\int - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} dn_{2} = \\frac{d}{d n_{2}} (-1) \\int - \\frac{m{(n_{2})}}{\\sin{(\\sin{(n_{2})})}} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('n_2', commutative=True)), sin(sin(Symbol('n_2', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), sin(sin(Symbol('n_2', commutative=True))))"], "Equality(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Integer(-1))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["times", 3, "Integral(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Derivative(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True)))), Mul(Derivative(Integer(-1), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Function('m')(Symbol('n_2', commutative=True)), Pow(sin(sin(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}}, then obtain \\int \\lambda{(\\Psi_{\\lambda})} e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\hat{p}_0 + \\frac{e^{2 \\Psi_{\\lambda}}}{2}", "derivation": "\\lambda{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\lambda{(\\Psi_{\\lambda})} e^{\\Psi_{\\lambda}} = e^{2 \\Psi_{\\lambda}} and \\int \\lambda{(\\Psi_{\\lambda})} e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\int e^{2 \\Psi_{\\lambda}} d\\Psi_{\\lambda} and \\int \\lambda{(\\Psi_{\\lambda})} e^{\\Psi_{\\lambda}} d\\Psi_{\\lambda} = \\hat{p}_0 + \\frac{e^{2 \\Psi_{\\lambda}}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\lambda')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\lambda')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}{(i,\\mathbf{M})} = \\mathbf{M} + i, then obtain i \\frac{\\partial}{\\partial \\mathbf{M}} \\int i \\tilde{g}^{2}{(i,\\mathbf{M})} di = i \\frac{\\partial}{\\partial \\mathbf{M}} \\int i (\\mathbf{M} + i)^{2} di", "derivation": "\\tilde{g}{(i,\\mathbf{M})} = \\mathbf{M} + i and i \\tilde{g}{(i,\\mathbf{M})} = i (\\mathbf{M} + i) and i \\tilde{g}^{2}{(i,\\mathbf{M})} = i (\\mathbf{M} + i) \\tilde{g}{(i,\\mathbf{M})} and i (\\mathbf{M} + i) \\tilde{g}{(i,\\mathbf{M})} = i (\\mathbf{M} + i)^{2} and i \\tilde{g}^{2}{(i,\\mathbf{M})} = i (\\mathbf{M} + i)^{2} and \\int i \\tilde{g}^{2}{(i,\\mathbf{M})} di = \\int i (\\mathbf{M} + i)^{2} di and \\frac{\\partial}{\\partial \\mathbf{M}} \\int i \\tilde{g}^{2}{(i,\\mathbf{M})} di = \\frac{\\partial}{\\partial \\mathbf{M}} \\int i (\\mathbf{M} + i)^{2} di and i \\frac{\\partial}{\\partial \\mathbf{M}} \\int i \\tilde{g}^{2}{(i,\\mathbf{M})} di = i \\frac{\\partial}{\\partial \\mathbf{M}} \\int i (\\mathbf{M} + i)^{2} di", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('i', commutative=True), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True))))"], [["times", 2, "Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Symbol('i', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Mul(Symbol('i', commutative=True), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('i', commutative=True), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('i', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('i', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Mul(Symbol('i', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Integer(2))))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Symbol('i', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))), Integral(Mul(Symbol('i', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('i', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('i', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["times", 7, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Derivative(Integral(Mul(Symbol('i', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Mul(Symbol('i', commutative=True), Derivative(Integral(Mul(Symbol('i', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('i', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})}, then derive \\phi_{1}{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then obtain \\phi_{1}{(\\mathbf{S})} - \\cos^{\\mathbf{S}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} - \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "derivation": "\\phi_{1}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})} and \\phi_{1}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\phi_{1}{(\\mathbf{S})} - (\\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})})^{\\mathbf{S}} = \\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})} - (\\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})})^{\\mathbf{S}} and \\frac{d}{d \\mathbf{S}} \\sin{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\phi_{1}{(\\mathbf{S})} - \\cos^{\\mathbf{S}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} - \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "Pow(Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))), Add(Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Add(cos(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(E)} = \\sin{(\\sin{(E)})}, then derive (\\frac{d}{d E} \\Psi_{nl}{(E)})^{E} = (\\cos{(E)} \\cos{(\\sin{(E)})})^{E}, then obtain \\frac{d}{d E} (\\frac{d}{d E} \\Psi_{nl}{(E)})^{E} = \\frac{d}{d E} (\\cos{(E)} \\cos{(\\sin{(E)})})^{E}", "derivation": "\\Psi_{nl}{(E)} = \\sin{(\\sin{(E)})} and \\frac{d}{d E} \\Psi_{nl}{(E)} = \\frac{d}{d E} \\sin{(\\sin{(E)})} and (\\frac{d}{d E} \\Psi_{nl}{(E)})^{E} = (\\frac{d}{d E} \\sin{(\\sin{(E)})})^{E} and (\\frac{d}{d E} \\Psi_{nl}{(E)})^{E} = (\\cos{(E)} \\cos{(\\sin{(E)})})^{E} and \\frac{d}{d E} (\\frac{d}{d E} \\Psi_{nl}{(E)})^{E} = \\frac{d}{d E} (\\cos{(E)} \\cos{(\\sin{(E)})})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), sin(sin(Symbol('E', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(sin(sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Mul(cos(Symbol('E', commutative=True)), cos(sin(Symbol('E', commutative=True)))), Symbol('E', commutative=True)))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Mul(cos(Symbol('E', commutative=True)), cos(sin(Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{p})} = \\log{(\\mathbf{p})}, then obtain (\\mathbf{A}{(\\mathbf{p})} + \\log{(\\mathbf{p})})^{2} \\log{(\\mathbf{p})}^{2} = 4 \\log{(\\mathbf{p})}^{4}", "derivation": "\\mathbf{A}{(\\mathbf{p})} = \\log{(\\mathbf{p})} and \\mathbf{A}{(\\mathbf{p})} + \\log{(\\mathbf{p})} = 2 \\log{(\\mathbf{p})} and (\\mathbf{A}{(\\mathbf{p})} + \\log{(\\mathbf{p})})^{2} = 4 \\log{(\\mathbf{p})}^{2} and (\\mathbf{A}{(\\mathbf{p})} + \\log{(\\mathbf{p})})^{2} \\log{(\\mathbf{p})}^{2} = 4 \\log{(\\mathbf{p})}^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Integer(2)), Mul(Integer(4), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))))"], [["times", 3, "Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))), Integer(2)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), Mul(Integer(4), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Integer(4))))"]]}, {"prompt": "Given m{(E_{x})} = \\sin{(E_{x})}, then obtain \\dot{\\mathbf{r}} + m{(E_{x})} = g^{\\prime}_{\\varepsilon} + \\sin{(E_{x})}", "derivation": "m{(E_{x})} = \\sin{(E_{x})} and \\frac{d}{d E_{x}} m{(E_{x})} = \\frac{d}{d E_{x}} \\sin{(E_{x})} and \\int \\frac{d}{d E_{x}} m{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} \\sin{(E_{x})} dE_{x} and \\dot{\\mathbf{r}} + m{(E_{x})} = g^{\\prime}_{\\varepsilon} + \\sin{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Function('m')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('m')(Symbol('E_x', commutative=True))), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), sin(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(M,Z)} = \\frac{\\cos{(Z)}}{M}, then obtain (\\frac{e^{Z + \\phi_{2}{(M,Z)}}}{M})^{M} = (\\frac{e^{Z + \\frac{\\cos{(Z)}}{M}}}{M})^{M}", "derivation": "\\phi_{2}{(M,Z)} = \\frac{\\cos{(Z)}}{M} and Z + \\phi_{2}{(M,Z)} = Z + \\frac{\\cos{(Z)}}{M} and e^{Z + \\phi_{2}{(M,Z)}} = e^{Z + \\frac{\\cos{(Z)}}{M}} and \\frac{e^{Z + \\phi_{2}{(M,Z)}}}{M} = \\frac{e^{Z + \\frac{\\cos{(Z)}}{M}}}{M} and (\\frac{e^{Z + \\phi_{2}{(M,Z)}}}{M})^{M} = (\\frac{e^{Z + \\frac{\\cos{(Z)}}{M}}}{M})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('Z', commutative=True))))"], [["add", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('Z', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Symbol('Z', commutative=True), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('Z', commutative=True)))), exp(Add(Symbol('Z', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('Z', commutative=True))))))"], [["divide", 3, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Add(Symbol('Z', commutative=True), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('Z', commutative=True))))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Add(Symbol('Z', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('Z', commutative=True)))))))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Add(Symbol('Z', commutative=True), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('Z', commutative=True))))), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Add(Symbol('Z', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('Z', commutative=True)))))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given Q{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)}, then derive 1 = \\frac{\\cos{(\\mu)}}{Q{(\\mu)}}, then obtain (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{\\mu} Q^{2}{(\\mu)} = (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{2 \\mu} Q^{2}{(\\mu)}", "derivation": "Q{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)} and 1 = \\frac{\\frac{d}{d \\mu} \\sin{(\\mu)}}{Q{(\\mu)}} and 1 = \\frac{\\cos{(\\mu)}}{Q{(\\mu)}} and 1 = \\frac{\\cos{(\\mu)}}{\\frac{d}{d \\mu} \\sin{(\\mu)}} and 1 = (\\frac{\\cos{(\\mu)}}{\\frac{d}{d \\mu} \\sin{(\\mu)}})^{\\mu} and 1 = (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{\\mu} and \\frac{d}{d \\mu} \\sin{(\\mu)} = (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{\\mu} \\frac{d}{d \\mu} \\sin{(\\mu)} and Q{(\\mu)} = (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{\\mu} Q{(\\mu)} and (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{\\mu} Q^{2}{(\\mu)} = (\\frac{\\cos{(\\mu)}}{Q{(\\mu)}})^{2 \\mu} Q^{2}{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mu', commutative=True)), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["divide", 1, "Function('Q')(Symbol('\\\\mu', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(1), Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Mul(cos(Symbol('\\\\mu', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integer(1), Pow(Mul(cos(Symbol('\\\\mu', commutative=True)), Pow(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1))), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(1), Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["times", 6, "Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Function('Q')(Symbol('\\\\mu', commutative=True)), Mul(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Function('Q')(Symbol('\\\\mu', commutative=True))))"], [["times", 8, "Mul(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Function('Q')(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Symbol('\\\\mu', commutative=True))), Pow(Function('Q')(Symbol('\\\\mu', commutative=True)), Integer(2))))"]]}, {"prompt": "Given z{(m_{s},\\Omega)} = \\Omega m_{s}, then derive \\int (z{(m_{s},\\Omega)} - 1) d\\Omega = I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega, then obtain \\sin{(I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega)} = \\sin{(\\frac{\\Omega^{2} m_{s}}{2} - \\Omega + \\theta_1)}", "derivation": "z{(m_{s},\\Omega)} = \\Omega m_{s} and z{(m_{s},\\Omega)} - 1 = \\Omega m_{s} - 1 and \\int (z{(m_{s},\\Omega)} - 1) d\\Omega = \\int (\\Omega m_{s} - 1) d\\Omega and \\int (z{(m_{s},\\Omega)} - 1) d\\Omega = I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega and I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega = \\int (\\Omega m_{s} - 1) d\\Omega and \\sin{(I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega)} = \\sin{(\\int (\\Omega m_{s} - 1) d\\Omega)} and \\sin{(I + \\frac{\\Omega^{2} m_{s}}{2} - \\Omega)} = \\sin{(\\frac{\\Omega^{2} m_{s}}{2} - \\Omega + \\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('z')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Function('z')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('z')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["sin", 5], "Equality(sin(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), sin(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(sin(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), sin(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given Z{(n)} = \\cos{(\\sin{(n)})} and \\operatorname{V_{\\mathbf{E}}}{(n)} = \\frac{d}{d n} Z{(n)}, then derive \\frac{d}{d n} Z{(n)} = - \\sin{(\\sin{(n)})} \\cos{(n)}, then obtain (\\frac{d}{d n} Z{(n)})^{n} = \\operatorname{V_{\\mathbf{E}}}^{n}{(n)}", "derivation": "Z{(n)} = \\cos{(\\sin{(n)})} and \\frac{d}{d n} Z{(n)} = \\frac{d}{d n} \\cos{(\\sin{(n)})} and \\frac{d}{d n} Z{(n)} = - \\sin{(\\sin{(n)})} \\cos{(n)} and \\operatorname{V_{\\mathbf{E}}}{(n)} = \\frac{d}{d n} Z{(n)} and (\\frac{d}{d n} Z{(n)})^{n} = (- \\sin{(\\sin{(n)})} \\cos{(n)})^{n} and \\operatorname{V_{\\mathbf{E}}}{(n)} = - \\sin{(\\sin{(n)})} \\cos{(n)} and (\\frac{d}{d n} Z{(n)})^{n} = \\operatorname{V_{\\mathbf{E}}}^{n}{(n)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('n', commutative=True)), cos(sin(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('n', commutative=True))), cos(Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), Derivative(Function('Z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('Z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Mul(Integer(-1), sin(sin(Symbol('n', commutative=True))), cos(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('n', commutative=True))), cos(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Derivative(Function('Z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given b{(v)} = \\log{(\\log{(v)})} and \\operatorname{v_{2}}{(v)} = \\log{(\\log{(v)})}, then obtain \\frac{d}{d v} \\operatorname{v_{2}}{(v)} = \\frac{d}{d v} \\log{(\\log{(v)})}", "derivation": "b{(v)} = \\log{(\\log{(v)})} and \\frac{d}{d v} b{(v)} = \\frac{d}{d v} \\log{(\\log{(v)})} and \\operatorname{v_{2}}{(v)} = \\log{(\\log{(v)})} and \\operatorname{v_{2}}{(v)} = b{(v)} and \\frac{d}{d v} \\operatorname{v_{2}}{(v)} = \\frac{d}{d v} \\log{(\\log{(v)})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('v', commutative=True)), log(log(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('v', commutative=True)), log(log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('v_2')(Symbol('v', commutative=True)), Function('b')(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('v_2')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\hbar)} = \\sin{(\\hbar)}, then obtain \\hbar + 2 t{(\\hbar)} + \\sin{(\\hbar)} = \\hbar + 3 \\sin{(\\hbar)}", "derivation": "t{(\\hbar)} = \\sin{(\\hbar)} and t{(\\hbar)} + \\sin{(\\hbar)} = 2 \\sin{(\\hbar)} and t{(\\hbar)} + 2 \\sin{(\\hbar)} = 3 \\sin{(\\hbar)} and 2 t{(\\hbar)} + \\sin{(\\hbar)} = 3 \\sin{(\\hbar)} and \\hbar + 2 t{(\\hbar)} + \\sin{(\\hbar)} = \\hbar + 3 \\sin{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('t')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "sin(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('t')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(3), sin(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('t')(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Mul(Integer(3), sin(Symbol('\\\\hbar', commutative=True))))"], [["add", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(2), Function('t')(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(3), sin(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\rho,H)} = \\cos^{H}{(\\rho)} and \\lambda{(\\rho)} = \\rho, then obtain \\frac{\\lambda{(\\rho)} - 1}{\\cos{(\\rho)} + \\cos^{H}{(\\rho)}} = \\frac{\\rho - 1}{\\cos{(\\rho)} + \\cos^{H}{(\\rho)}}", "derivation": "\\mathbf{J}_P{(\\rho,H)} = \\cos^{H}{(\\rho)} and \\mathbf{J}_P{(\\rho,H)} + \\cos{(\\rho)} = \\cos{(\\rho)} + \\cos^{H}{(\\rho)} and \\lambda{(\\rho)} = \\rho and \\lambda{(\\rho)} - \\frac{\\cos^{H}{(\\rho)}}{\\mathbf{J}_P{(\\rho,H)}} = \\rho - \\frac{\\cos^{H}{(\\rho)}}{\\mathbf{J}_P{(\\rho,H)}} and \\lambda{(\\rho)} - 1 = \\rho - 1 and \\frac{\\lambda{(\\rho)} - 1}{\\mathbf{J}_P{(\\rho,H)} + \\cos{(\\rho)}} = \\frac{\\rho - 1}{\\mathbf{J}_P{(\\rho,H)} + \\cos{(\\rho)}} and \\frac{\\lambda{(\\rho)} - 1}{\\cos{(\\rho)} + \\cos^{H}{(\\rho)}} = \\frac{\\rho - 1}{\\cos{(\\rho)} + \\cos^{H}{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), cos(Symbol('\\\\rho', commutative=True))), Add(cos(Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))"], [["minus", 3, "Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Symbol('\\\\rho', commutative=True), Integer(-1)))"], [["divide", 5, "Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Add(Function('\\\\lambda')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Pow(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), cos(Symbol('\\\\rho', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), cos(Symbol('\\\\rho', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Add(Function('\\\\lambda')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Add(cos(Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('H', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{1}{(\\omega)} = \\sin{(\\omega)}, then obtain \\iint \\frac{d^{2}}{d \\omega^{2}} (e^{\\theta_{1}{(\\omega)}})^{\\omega} d\\omega d\\omega = \\iint \\frac{d^{2}}{d \\omega^{2}} (e^{\\sin{(\\omega)}})^{\\omega} d\\omega d\\omega", "derivation": "\\theta_{1}{(\\omega)} = \\sin{(\\omega)} and e^{\\theta_{1}{(\\omega)}} = e^{\\sin{(\\omega)}} and (e^{\\theta_{1}{(\\omega)}})^{\\omega} = (e^{\\sin{(\\omega)}})^{\\omega} and \\frac{d}{d \\omega} (e^{\\theta_{1}{(\\omega)}})^{\\omega} = \\frac{d}{d \\omega} (e^{\\sin{(\\omega)}})^{\\omega} and \\frac{d^{2}}{d \\omega^{2}} (e^{\\theta_{1}{(\\omega)}})^{\\omega} = \\frac{d^{2}}{d \\omega^{2}} (e^{\\sin{(\\omega)}})^{\\omega} and \\int \\frac{d^{2}}{d \\omega^{2}} (e^{\\theta_{1}{(\\omega)}})^{\\omega} d\\omega = \\int \\frac{d^{2}}{d \\omega^{2}} (e^{\\sin{(\\omega)}})^{\\omega} d\\omega and \\iint \\frac{d^{2}}{d \\omega^{2}} (e^{\\theta_{1}{(\\omega)}})^{\\omega} d\\omega d\\omega = \\iint \\frac{d^{2}}{d \\omega^{2}} (e^{\\sin{(\\omega)}})^{\\omega} d\\omega d\\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), exp(sin(Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Pow(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Pow(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Derivative(Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["integrate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Derivative(Pow(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Derivative(Pow(exp(Function('\\\\theta_1')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(Pow(exp(sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(f,E_{x})} = \\frac{\\partial}{\\partial f} E_{x} f, then derive \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{E_{x} f} = \\frac{1}{f}, then derive \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{E_{x}} = 1, then obtain \\int \\frac{\\frac{\\partial}{\\partial f} E_{x} f}{E_{x}} dE_{x} = \\int 1 dE_{x}", "derivation": "\\operatorname{L_{\\varepsilon}}{(f,E_{x})} = \\frac{\\partial}{\\partial f} E_{x} f and \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{E_{x} f} = \\frac{\\frac{\\partial}{\\partial f} E_{x} f}{E_{x} f} and \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{E_{x} f} = \\frac{1}{f} and \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{\\frac{\\partial}{\\partial f} E_{x} f} = \\frac{E_{x}}{\\frac{\\partial}{\\partial f} E_{x} f} and \\frac{\\operatorname{L_{\\varepsilon}}{(f,E_{x})}}{E_{x}} = 1 and \\frac{\\frac{\\partial}{\\partial f} E_{x} f}{E_{x}} = 1 and \\int \\frac{\\frac{\\partial}{\\partial f} E_{x} f}{E_{x}} dE_{x} = \\int 1 dE_{x}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('E_x', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["divide", 1, "Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('E_x', commutative=True))), Pow(Symbol('f', commutative=True), Integer(-1)))"], [["divide", 3, "Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('E_x', commutative=True)), Pow(Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('E_x', commutative=True), Pow(Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('E_x', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Integer(1))"], [["integrate", 6, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Tuple(Symbol('E_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given C{(\\dot{z})} = \\cos{(\\dot{z})}, then obtain C{(\\dot{z})} \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})} = \\cos{(\\dot{z})} \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})}", "derivation": "C{(\\dot{z})} = \\cos{(\\dot{z})} and \\frac{d}{d \\dot{z}} C{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})} and C{(\\dot{z})} \\frac{d}{d \\dot{z}} C{(\\dot{z})} = \\cos{(\\dot{z})} \\frac{d}{d \\dot{z}} C{(\\dot{z})} and C{(\\dot{z})} \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})} = \\cos{(\\dot{z})} \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Mul(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Derivative(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\dot{z}', commutative=True)), Derivative(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('C')(Symbol('\\\\dot{z}', commutative=True)), Derivative(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\dot{z}', commutative=True)), Derivative(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(c_{0})} = \\log{(c_{0})}, then obtain c_{0} \\operatorname{t_{2}}{(c_{0})} + c_{0} \\log{(c_{0})} - \\operatorname{t_{2}}{(c_{0})} = 2 c_{0} \\log{(c_{0})} - \\operatorname{t_{2}}{(c_{0})}", "derivation": "\\operatorname{t_{2}}{(c_{0})} = \\log{(c_{0})} and c_{0} \\operatorname{t_{2}}{(c_{0})} = c_{0} \\log{(c_{0})} and c_{0} \\operatorname{t_{2}}{(c_{0})} + c_{0} \\log{(c_{0})} = 2 c_{0} \\log{(c_{0})} and c_{0} \\operatorname{t_{2}}{(c_{0})} + c_{0} \\log{(c_{0})} - \\operatorname{t_{2}}{(c_{0})} = 2 c_{0} \\log{(c_{0})} - \\operatorname{t_{2}}{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["times", 1, "Symbol('c_0', commutative=True)"], "Equality(Mul(Symbol('c_0', commutative=True), Function('t_2')(Symbol('c_0', commutative=True))), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))))"], [["add", 2, "Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Symbol('c_0', commutative=True), Function('t_2')(Symbol('c_0', commutative=True))), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True)))), Mul(Integer(2), Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))))"], [["minus", 3, "Function('t_2')(Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Symbol('c_0', commutative=True), Function('t_2')(Symbol('c_0', commutative=True))), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Function('t_2')(Symbol('c_0', commutative=True)))), Add(Mul(Integer(2), Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Function('t_2')(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(F_{g},A_{1})} = A_{1} F_{g}, then derive e^{\\frac{\\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{2}}{(F_{g},A_{1})}}{A_{1}}} = e, then obtain e^{\\frac{\\frac{\\partial}{\\partial F_{g}} A_{1} F_{g}}{A_{1}}} = e", "derivation": "\\operatorname{v_{2}}{(F_{g},A_{1})} = A_{1} F_{g} and \\frac{\\operatorname{v_{2}}{(F_{g},A_{1})}}{A_{1}} = F_{g} and \\frac{\\partial}{\\partial F_{g}} \\frac{\\operatorname{v_{2}}{(F_{g},A_{1})}}{A_{1}} = \\frac{d}{d F_{g}} F_{g} and e^{\\frac{\\partial}{\\partial F_{g}} \\frac{\\operatorname{v_{2}}{(F_{g},A_{1})}}{A_{1}}} = e^{\\frac{d}{d F_{g}} F_{g}} and e^{\\frac{\\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{2}}{(F_{g},A_{1})}}{A_{1}}} = e and e^{\\frac{\\frac{\\partial}{\\partial F_{g}} A_{1} F_{g}}{A_{1}}} = e", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('F_g', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True)))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('v_2')(Symbol('F_g', commutative=True), Symbol('A_1', commutative=True))), Symbol('F_g', commutative=True))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('v_2')(Symbol('F_g', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('v_2')(Symbol('F_g', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), exp(Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(exp(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(Function('v_2')(Symbol('F_g', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))), E)"], [["substitute_LHS_for_RHS", 5, 1], "Equality(exp(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))), E)"]]}, {"prompt": "Given \\varphi^{*}{(u)} = \\log{(u)} and c{(u)} = \\log{(u)}, then obtain c{(u)} - \\log{(u)}^{2} = - \\log{(u)}^{2} + \\log{(u)}", "derivation": "\\varphi^{*}{(u)} = \\log{(u)} and \\varphi^{*}{(u)} \\log{(u)} = \\log{(u)}^{2} and c{(u)} = \\log{(u)} and - \\varphi^{*}{(u)} \\log{(u)} + c{(u)} = - \\varphi^{*}{(u)} \\log{(u)} + \\log{(u)} and c{(u)} - \\log{(u)}^{2} = - \\log{(u)}^{2} + \\log{(u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["times", 1, "log(Symbol('u', commutative=True))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(log(Symbol('u', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('c')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["minus", 3, "Mul(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Function('c')(Symbol('u', commutative=True))), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), log(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('c')(Symbol('u', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2))), log(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\omega,\\psi^*)} = \\frac{e^{\\psi^*}}{\\omega} and \\operatorname{P_{e}}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} \\hat{X}{(\\omega,\\psi^*)}, then obtain \\frac{\\partial}{\\partial \\omega} \\operatorname{P_{e}}^{\\omega}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} (\\frac{\\partial}{\\partial \\omega} \\frac{e^{\\psi^*}}{\\omega})^{\\omega}", "derivation": "\\hat{X}{(\\omega,\\psi^*)} = \\frac{e^{\\psi^*}}{\\omega} and \\frac{\\partial}{\\partial \\omega} \\hat{X}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} \\frac{e^{\\psi^*}}{\\omega} and \\operatorname{P_{e}}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} \\hat{X}{(\\omega,\\psi^*)} and \\operatorname{P_{e}}^{\\omega}{(\\omega,\\psi^*)} = (\\frac{\\partial}{\\partial \\omega} \\hat{X}{(\\omega,\\psi^*)})^{\\omega} and \\frac{\\partial}{\\partial \\omega} \\operatorname{P_{e}}^{\\omega}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} (\\frac{\\partial}{\\partial \\omega} \\hat{X}{(\\omega,\\psi^*)})^{\\omega} and \\frac{\\partial}{\\partial \\omega} \\operatorname{P_{e}}^{\\omega}{(\\omega,\\psi^*)} = \\frac{\\partial}{\\partial \\omega} (\\frac{\\partial}{\\partial \\omega} \\frac{e^{\\psi^*}}{\\omega})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\theta_1,u)} = \\theta_1 + u, then obtain \\operatorname{t_{1}}{(\\theta_1,u)} \\int (\\theta_1 + u) d\\theta_1 = (\\theta_1 + u) \\int (\\theta_1 + u) d\\theta_1", "derivation": "\\operatorname{t_{1}}{(\\theta_1,u)} = \\theta_1 + u and \\int \\operatorname{t_{1}}{(\\theta_1,u)} d\\theta_1 = \\int (\\theta_1 + u) d\\theta_1 and \\operatorname{t_{1}}{(\\theta_1,u)} \\int \\operatorname{t_{1}}{(\\theta_1,u)} d\\theta_1 = (\\theta_1 + u) \\int \\operatorname{t_{1}}{(\\theta_1,u)} d\\theta_1 and \\operatorname{t_{1}}{(\\theta_1,u)} \\int (\\theta_1 + u) d\\theta_1 = (\\theta_1 + u) \\int (\\theta_1 + u) d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(x)} = (e^{x})^{x} and \\bar{\\h}{(x)} = (e^{x})^{x}, then obtain (\\frac{d}{d x} \\operatorname{L_{\\varepsilon}}^{x}{(x)})^{x} = (\\frac{d}{d x} ((e^{x})^{x})^{x})^{x}", "derivation": "\\operatorname{L_{\\varepsilon}}{(x)} = (e^{x})^{x} and \\bar{\\h}{(x)} = (e^{x})^{x} and \\bar{\\h}{(x)} = \\operatorname{L_{\\varepsilon}}{(x)} and \\bar{\\h}^{x}{(x)} = ((e^{x})^{x})^{x} and \\frac{d}{d x} \\bar{\\h}^{x}{(x)} = \\frac{d}{d x} ((e^{x})^{x})^{x} and (\\frac{d}{d x} \\bar{\\h}^{x}{(x)})^{x} = (\\frac{d}{d x} ((e^{x})^{x})^{x})^{x} and (\\frac{d}{d x} \\operatorname{L_{\\varepsilon}}^{x}{(x)})^{x} = (\\frac{d}{d x} ((e^{x})^{x})^{x})^{x}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('x', commutative=True)), Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('x', commutative=True)), Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hbar')(Symbol('x', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('x', commutative=True)))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 4, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hbar')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["power", 5, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\hbar')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Pow(Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Derivative(Pow(Function('L_{\\\\varepsilon}')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Pow(Pow(exp(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(v_{2})} = \\sin{(v_{2})}, then obtain \\frac{\\operatorname{F_{g}}{(v_{2})}}{\\cos{(v_{2})}} = \\frac{\\sin{(v_{2})}}{\\cos{(v_{2})}}", "derivation": "\\operatorname{F_{g}}{(v_{2})} = \\sin{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{F_{g}}{(v_{2})} = \\frac{d}{d v_{2}} \\sin{(v_{2})} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{\\frac{d}{d v_{2}} \\operatorname{F_{g}}{(v_{2})}} = \\frac{\\sin{(v_{2})}}{\\frac{d}{d v_{2}} \\operatorname{F_{g}}{(v_{2})}} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{\\frac{d}{d v_{2}} \\sin{(v_{2})}} = \\frac{\\sin{(v_{2})}}{\\frac{d}{d v_{2}} \\sin{(v_{2})}} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{\\cos{(v_{2})}} = \\frac{\\sin{(v_{2})}}{\\cos{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Mul(Function('F_g')(Symbol('v_2', commutative=True)), Pow(Derivative(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Symbol('v_2', commutative=True)), Pow(Derivative(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('F_g')(Symbol('v_2', commutative=True)), Pow(Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Symbol('v_2', commutative=True)), Pow(Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Function('F_g')(Symbol('v_2', commutative=True)), Pow(cos(Symbol('v_2', commutative=True)), Integer(-1))), Mul(sin(Symbol('v_2', commutative=True)), Pow(cos(Symbol('v_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(k)} = \\sin{(k)}, then derive (\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k} = \\cos^{k}{(k)}, then obtain \\frac{\\partial}{\\partial C_{2}} \\frac{(\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k}}{\\operatorname{m_{s}}{(E,C_{2})}} = \\frac{\\partial}{\\partial C_{2}} \\frac{\\cos^{k}{(k)}}{\\operatorname{m_{s}}{(E,C_{2})}}", "derivation": "\\operatorname{A_{x}}{(k)} = \\sin{(k)} and \\frac{d}{d k} \\operatorname{A_{x}}{(k)} = \\frac{d}{d k} \\sin{(k)} and (\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k} = (\\frac{d}{d k} \\sin{(k)})^{k} and (\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k} = \\cos^{k}{(k)} and \\frac{(\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k}}{\\operatorname{m_{s}}{(E,C_{2})}} = \\frac{\\cos^{k}{(k)}}{\\operatorname{m_{s}}{(E,C_{2})}} and \\frac{\\partial}{\\partial C_{2}} \\frac{(\\frac{d}{d k} \\operatorname{A_{x}}{(k)})^{k}}{\\operatorname{m_{s}}{(E,C_{2})}} = \\frac{\\partial}{\\partial C_{2}} \\frac{\\cos^{k}{(k)}}{\\operatorname{m_{s}}{(E,C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Function('A_x')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('A_x')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["divide", 4, "Function('m_s')(Symbol('E', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Function('m_s')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Pow(Derivative(Function('A_x')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True))), Mul(Pow(Function('m_s')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))))"], [["differentiate", 5, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('m_s')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Pow(Derivative(Function('A_x')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('m_s')(Symbol('E', commutative=True), Symbol('C_2', commutative=True)), Integer(-1)), Pow(cos(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\theta_1,n)} = \\theta_1 - n, then obtain (\\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{r}}{(\\theta_1,n)}}{n})^{\\theta_1} = (\\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1 - n)}{n})^{\\theta_1}", "derivation": "\\hat{\\mathbf{r}}{(\\theta_1,n)} = \\theta_1 - n and \\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{r}}{(\\theta_1,n)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1 - n) and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{r}}{(\\theta_1,n)}}{n} = \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1 - n)}{n} and \\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{r}}{(\\theta_1,n)}}{n} = \\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1 - n)}{n} and (\\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{r}}{(\\theta_1,n)}}{n})^{\\theta_1} = (\\theta_1 + \\frac{\\frac{\\partial}{\\partial \\theta_1} (\\theta_1 - n)}{n})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["add", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given b{(\\mathbf{B},\\mathbf{J}_f)} = \\mathbf{B} - \\mathbf{J}_f, then obtain \\iint (- \\mathbf{J}_f + b{(\\mathbf{B},\\mathbf{J}_f)}) d\\mathbf{B} d\\mathbf{J}_f = \\iint (\\mathbf{B} - 2 \\mathbf{J}_f) d\\mathbf{B} d\\mathbf{J}_f", "derivation": "b{(\\mathbf{B},\\mathbf{J}_f)} = \\mathbf{B} - \\mathbf{J}_f and - \\mathbf{J}_f + b{(\\mathbf{B},\\mathbf{J}_f)} = \\mathbf{B} - 2 \\mathbf{J}_f and \\int (- \\mathbf{J}_f + b{(\\mathbf{B},\\mathbf{J}_f)}) d\\mathbf{B} = \\int (\\mathbf{B} - 2 \\mathbf{J}_f) d\\mathbf{B} and \\iint (- \\mathbf{J}_f + b{(\\mathbf{B},\\mathbf{J}_f)}) d\\mathbf{B} d\\mathbf{J}_f = \\iint (\\mathbf{B} - 2 \\mathbf{J}_f) d\\mathbf{B} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('b')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('b')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('b')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given C{(g,\\phi)} = \\phi g, then obtain \\frac{\\partial^{2}}{\\partial g\\partial \\phi} C{(g,\\phi)} + 1 = 2", "derivation": "C{(g,\\phi)} = \\phi g and - g + C{(g,\\phi)} = \\phi g - g and \\phi g - 2 g + C{(g,\\phi)} = 2 \\phi g - 2 g and \\frac{\\partial}{\\partial g} (\\phi g - 2 g + C{(g,\\phi)}) = \\frac{\\partial}{\\partial g} (2 \\phi g - 2 g) and \\frac{\\partial^{2}}{\\partial \\phi\\partial g} (\\phi g - 2 g + C{(g,\\phi)}) = \\frac{\\partial^{2}}{\\partial \\phi\\partial g} (2 \\phi g - 2 g) and \\frac{\\partial^{2}}{\\partial g\\partial \\phi} C{(g,\\phi)} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["add", 2, "Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(1)), Integer(2))"]]}, {"prompt": "Given T{(\\mathbf{H})} = \\cos{(\\log{(\\mathbf{H})})}, then obtain \\log{(\\frac{T{(\\mathbf{H})} + \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}})} = \\log{(\\frac{2 \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}})}", "derivation": "T{(\\mathbf{H})} = \\cos{(\\log{(\\mathbf{H})})} and T{(\\mathbf{H})} + \\cos{(\\log{(\\mathbf{H})})} = 2 \\cos{(\\log{(\\mathbf{H})})} and \\frac{T{(\\mathbf{H})} + \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}} = \\frac{2 \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}} and \\log{(\\frac{T{(\\mathbf{H})} + \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}})} = \\log{(\\frac{2 \\cos{(\\log{(\\mathbf{H})})}}{\\int 2 \\cos{(\\log{(\\mathbf{H})})} d\\mathbf{H}})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{H}', commutative=True)), cos(log(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "cos(log(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Function('T')(Symbol('\\\\mathbf{H}', commutative=True)), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["divide", 2, "Integral(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Add(Function('T')(Symbol('\\\\mathbf{H}', commutative=True)), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Pow(Integral(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))), Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Integral(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1))))"], [["log", 3], "Equality(log(Mul(Add(Function('T')(Symbol('\\\\mathbf{H}', commutative=True)), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Pow(Integral(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1)))), log(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Integral(Mul(Integer(2), cos(log(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} = \\log{(y^{\\prime})}, then derive \\int \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = x + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime}, then obtain \\theta_1 + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime} = x + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime}", "derivation": "\\operatorname{J_{\\varepsilon}}{(y^{\\prime})} = \\log{(y^{\\prime})} and \\int \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = \\int \\log{(y^{\\prime})} dy^{\\prime} and \\int \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = x + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime} and \\int \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = x + y^{\\prime} \\operatorname{J_{\\varepsilon}}{(y^{\\prime})} - y^{\\prime} and \\int \\log{(y^{\\prime})} dy^{\\prime} = x + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime} and \\theta_1 + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime} = x + y^{\\prime} \\log{(y^{\\prime})} - y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('x', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('x', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('x', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('x', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\rho_f)} = e^{\\rho_f}, then derive \\frac{d^{2}}{d \\rho_f^{2}} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (Q + e^{\\rho_f}), then obtain \\frac{d^{2}}{d \\rho_f^{2}} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (Q + \\theta_{2}{(\\rho_f)})", "derivation": "\\theta_{2}{(\\rho_f)} = e^{\\rho_f} and \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\int e^{\\rho_f} d\\rho_f and \\frac{d}{d \\rho_f} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{d}{d \\rho_f} \\int e^{\\rho_f} d\\rho_f and \\frac{d^{2}}{d \\rho_f^{2}} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{d^{2}}{d \\rho_f^{2}} \\int e^{\\rho_f} d\\rho_f and \\frac{d^{2}}{d \\rho_f^{2}} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (Q + e^{\\rho_f}) and \\frac{d^{2}}{d \\rho_f^{2}} \\int \\theta_{2}{(\\rho_f)} d\\rho_f = \\frac{\\partial^{2}}{\\partial \\rho_f^{2}} (Q + \\theta_{2}{(\\rho_f)})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Derivative(Integral(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Derivative(Add(Symbol('Q', commutative=True), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Integral(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Derivative(Add(Symbol('Q', commutative=True), Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))))"]]}, {"prompt": "Given f{(\\pi)} = \\log{(\\pi)}, then obtain 2 \\pi + f{(\\pi)} + \\log{(\\pi)} = 2 \\pi + 2 (\\pi + f{(\\pi)})^{\\pi} - 2 (\\pi + \\log{(\\pi)})^{\\pi} + 2 \\log{(\\pi)}", "derivation": "f{(\\pi)} = \\log{(\\pi)} and \\pi + f{(\\pi)} = \\pi + \\log{(\\pi)} and (\\pi + f{(\\pi)})^{\\pi} = (\\pi + \\log{(\\pi)})^{\\pi} and \\pi + (\\pi + f{(\\pi)})^{\\pi} + f{(\\pi)} = \\pi + (\\pi + \\log{(\\pi)})^{\\pi} + f{(\\pi)} and 2 \\pi + f{(\\pi)} + \\log{(\\pi)} = 2 \\pi + 2 \\log{(\\pi)} and \\pi + (\\pi + f{(\\pi)})^{\\pi} + f{(\\pi)} + 2 \\log{(\\pi)} = \\pi + (\\pi + \\log{(\\pi)})^{\\pi} + f{(\\pi)} + 2 \\log{(\\pi)} and (\\pi + f{(\\pi)})^{\\pi} - (\\pi + \\log{(\\pi)})^{\\pi} + 2 \\log{(\\pi)} = 2 \\log{(\\pi)} and 2 \\pi + f{(\\pi)} + \\log{(\\pi)} = 2 \\pi + (\\pi + f{(\\pi)})^{\\pi} - (\\pi + \\log{(\\pi)})^{\\pi} + 2 \\log{(\\pi)} and 2 \\pi + f{(\\pi)} + \\log{(\\pi)} = 2 \\pi + 2 (\\pi + f{(\\pi)})^{\\pi} - 2 (\\pi + \\log{(\\pi)})^{\\pi} + 2 \\log{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["add", 3, "Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"], [["add", 4, "Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"], [["minus", 6, "Add(Symbol('\\\\pi', commutative=True), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Function('f')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('\\\\pi', commutative=True), Function('f')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(V)} = \\sin{(V)}, then derive \\frac{\\frac{d}{d V} \\operatorname{F_{H}}{(V)}}{\\operatorname{F_{H}}{(V)}} = \\frac{\\cos{(V)}}{\\sin{(V)}}, then obtain \\frac{\\frac{d}{d V} \\sin{(V)}}{\\sin{(V)}} = \\frac{\\cos{(V)}}{\\sin{(V)}}", "derivation": "\\operatorname{F_{H}}{(V)} = \\sin{(V)} and \\log{(\\operatorname{F_{H}}{(V)})} = \\log{(\\sin{(V)})} and \\frac{d}{d V} \\log{(\\operatorname{F_{H}}{(V)})} = \\frac{d}{d V} \\log{(\\sin{(V)})} and \\frac{\\frac{d}{d V} \\operatorname{F_{H}}{(V)}}{\\operatorname{F_{H}}{(V)}} = \\frac{\\cos{(V)}}{\\sin{(V)}} and \\frac{\\frac{d}{d V} \\sin{(V)}}{\\sin{(V)}} = \\frac{\\cos{(V)}}{\\sin{(V)}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["log", 1], "Equality(log(Function('F_H')(Symbol('V', commutative=True))), log(sin(Symbol('V', commutative=True))))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(log(Function('F_H')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(log(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('F_H')(Symbol('V', commutative=True)), Integer(-1)), Derivative(Function('F_H')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))))"]]}, {"prompt": "Given L{(\\mathbf{v})} = \\cos{(\\mathbf{v})}, then derive \\log{(\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})} = \\log{(- \\sin{(\\mathbf{v})})}, then obtain \\log{(\\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})})} = \\log{(- \\sin{(\\mathbf{v})})}", "derivation": "L{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} and \\log{(\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})} = \\log{(\\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})})} and \\log{(\\frac{d}{d \\mathbf{v}} L{(\\mathbf{v})})} = \\log{(- \\sin{(\\mathbf{v})})} and \\log{(\\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})})} = \\log{(- \\sin{(\\mathbf{v})})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), log(Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('L')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), log(Mul(Integer(-1), sin(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), log(Mul(Integer(-1), sin(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\omega,a)} = \\omega - a, then obtain (- a (\\omega - a)^{a} \\operatorname{v_{t}}{(\\omega,a)})^{\\omega} = (- a (\\omega - a) (\\omega - a)^{a})^{\\omega}", "derivation": "\\operatorname{v_{t}}{(\\omega,a)} = \\omega - a and - a \\operatorname{v_{t}}{(\\omega,a)} = - a (\\omega - a) and \\operatorname{v_{t}}^{a}{(\\omega,a)} = (\\omega - a)^{a} and - a \\operatorname{v_{t}}{(\\omega,a)} \\operatorname{v_{t}}^{a}{(\\omega,a)} = - a (\\omega - a) \\operatorname{v_{t}}^{a}{(\\omega,a)} and - a (\\omega - a)^{a} \\operatorname{v_{t}}{(\\omega,a)} = - a (\\omega - a) (\\omega - a)^{a} and (- a (\\omega - a)^{a} \\operatorname{v_{t}}{(\\omega,a)})^{\\omega} = (- a (\\omega - a) (\\omega - a)^{a})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["times", 2, "Pow(Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Pow(Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Pow(Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Symbol('a', commutative=True)), Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Symbol('a', commutative=True))))"], [["power", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Symbol('a', commutative=True)), Function('v_t')(Symbol('\\\\omega', commutative=True), Symbol('a', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Integer(-1), Symbol('a', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Symbol('a', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(m,i,v)} = (i + m)^{v}, then obtain \\frac{\\partial}{\\partial i} (- m + 2 (i + m)^{v} + \\mathbf{v}{(m,i,v)}) = \\frac{\\partial}{\\partial i} (- m + (i + m)^{v} + 2 \\mathbf{v}{(m,i,v)})", "derivation": "\\mathbf{v}{(m,i,v)} = (i + m)^{v} and 2 \\mathbf{v}{(m,i,v)} = (i + m)^{v} + \\mathbf{v}{(m,i,v)} and - m + \\mathbf{v}{(m,i,v)} = - m + (i + m)^{v} and - m + (i + m)^{v} + 2 \\mathbf{v}{(m,i,v)} = - m + 2 (i + m)^{v} + \\mathbf{v}{(m,i,v)} and - m + 3 \\mathbf{v}{(m,i,v)} = - m + 2 (i + m)^{v} + \\mathbf{v}{(m,i,v)} and - m + 3 \\mathbf{v}{(m,i,v)} = - m + (i + m)^{v} + 2 \\mathbf{v}{(m,i,v)} and \\frac{\\partial}{\\partial i} (- m + 3 \\mathbf{v}{(m,i,v)}) = \\frac{\\partial}{\\partial i} (- m + (i + m)^{v} + 2 \\mathbf{v}{(m,i,v)}) and \\frac{\\partial}{\\partial i} (- m + 2 (i + m)^{v} + \\mathbf{v}{(m,i,v)}) = \\frac{\\partial}{\\partial i} (- m + (i + m)^{v} + 2 \\mathbf{v}{(m,i,v)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))), Add(Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True))), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True))), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))))"], [["differentiate", 6, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True))), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Symbol('i', commutative=True), Symbol('m', commutative=True)), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('m', commutative=True), Symbol('i', commutative=True), Symbol('v', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(s,H)} = H^{s}, then obtain \\frac{\\partial}{\\partial s} \\int H^{s} ds = \\frac{\\partial}{\\partial s} \\int (H (H^{s} - \\operatorname{t_{2}}{(s,H)}) + H^{s}) ds", "derivation": "\\operatorname{t_{2}}{(s,H)} = H^{s} and 0 = H^{s} - \\operatorname{t_{2}}{(s,H)} and 0 = H (H^{s} - \\operatorname{t_{2}}{(s,H)}) and H^{s} = H (H^{s} - \\operatorname{t_{2}}{(s,H)}) + H^{s} and \\int H^{s} ds = \\int (H (H^{s} - \\operatorname{t_{2}}{(s,H)}) + H^{s}) ds and \\frac{\\partial}{\\partial s} \\int H^{s} ds = \\frac{\\partial}{\\partial s} \\int (H (H^{s} - \\operatorname{t_{2}}{(s,H)}) + H^{s}) ds", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True)))))"], [["times", 2, "Symbol('H', commutative=True)"], "Equality(Integer(0), Mul(Symbol('H', commutative=True), Add(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True))))))"], [["add", 3, "Pow(Symbol('H', commutative=True), Symbol('s', commutative=True))"], "Equality(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Add(Mul(Symbol('H', commutative=True), Add(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True))))), Pow(Symbol('H', commutative=True), Symbol('s', commutative=True))))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Symbol('H', commutative=True), Add(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True))))), Pow(Symbol('H', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 5, "Symbol('s', commutative=True)"], "Equality(Derivative(Integral(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('H', commutative=True), Add(Pow(Symbol('H', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('s', commutative=True), Symbol('H', commutative=True))))), Pow(Symbol('H', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(t,v)} = \\frac{\\partial}{\\partial t} \\frac{v}{t}, then derive \\mu_{0}{(t,v)} = - \\frac{v}{t^{2}}, then obtain \\mu + \\int \\frac{\\partial}{\\partial t} \\frac{v}{t} dt = \\mu + \\int - \\frac{v}{t^{2}} dt", "derivation": "\\mu_{0}{(t,v)} = \\frac{\\partial}{\\partial t} \\frac{v}{t} and \\mu_{0}{(t,v)} = - \\frac{v}{t^{2}} and \\int \\mu_{0}{(t,v)} dt = \\int \\frac{\\partial}{\\partial t} \\frac{v}{t} dt and \\int - \\frac{v}{t^{2}} dt = \\int \\frac{\\partial}{\\partial t} \\frac{v}{t} dt and \\int \\mu_{0}{(t,v)} dt = \\int - \\frac{v}{t^{2}} dt and \\mu + \\int \\mu_{0}{(t,v)} dt = \\mu + \\int - \\frac{v}{t^{2}} dt and \\mu + \\int \\frac{\\partial}{\\partial t} \\frac{v}{t} dt = \\mu + \\int - \\frac{v}{t^{2}} dt", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["add", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Integral(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('\\\\mu', commutative=True), Integral(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Integral(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('v', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(k,t_{1},q)} = (k - t_{1})^{q} and \\sigma_{x}{(k,t_{1},q)} = (k - t_{1})^{- q}, then obtain \\int - (k - t_{1})^{q} \\mathbf{v}{(k,t_{1},q)} \\sigma_{x}^{2}{(k,t_{1},q)} dq = \\int - (k - t_{1})^{q} \\sigma_{x}{(k,t_{1},q)} dq", "derivation": "\\mathbf{v}{(k,t_{1},q)} = (k - t_{1})^{q} and - \\mathbf{v}{(k,t_{1},q)} = - (k - t_{1})^{q} and - (k - t_{1})^{- q} \\mathbf{v}{(k,t_{1},q)} = -1 and \\sigma_{x}{(k,t_{1},q)} = (k - t_{1})^{- q} and - \\mathbf{v}{(k,t_{1},q)} \\sigma_{x}{(k,t_{1},q)} = -1 and \\int - \\mathbf{v}{(k,t_{1},q)} \\sigma_{x}{(k,t_{1},q)} dq = \\int (-1) dq and - (k - t_{1})^{q} \\sigma_{x}{(k,t_{1},q)} = -1 and \\int - (k - t_{1})^{q} \\mathbf{v}{(k,t_{1},q)} \\sigma_{x}^{2}{(k,t_{1},q)} dq = \\int - (k - t_{1})^{q} \\sigma_{x}{(k,t_{1},q)} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Integer(-1))"], [["integrate", 5, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Integer(-1), Tuple(Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True)), Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Integer(-1))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True)), Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Integer(2))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), Pow(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('q', commutative=True)), Function('\\\\sigma_x')(Symbol('k', commutative=True), Symbol('t_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(A,\\varepsilon)} = \\cos{(A + \\varepsilon)}, then derive \\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)} = - \\sin{(A + \\varepsilon)}, then obtain ((\\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)})^{\\varepsilon})^{A} = ((- \\sin{(A + \\varepsilon)})^{\\varepsilon})^{A}", "derivation": "\\mathbf{v}{(A,\\varepsilon)} = \\cos{(A + \\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} \\cos{(A + \\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)} = - \\sin{(A + \\varepsilon)} and (\\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)})^{\\varepsilon} = (- \\sin{(A + \\varepsilon)})^{\\varepsilon} and ((\\frac{\\partial}{\\partial \\varepsilon} \\mathbf{v}{(A,\\varepsilon)})^{\\varepsilon})^{A} = ((- \\sin{(A + \\varepsilon)})^{\\varepsilon})^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{v}')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('\\\\mathbf{v}')(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Symbol('A', commutative=True)), Pow(Pow(Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True)), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{2},A)} = C_{2}^{A}, then obtain C_{2}^{A} + \\frac{\\partial}{\\partial C_{2}} \\operatorname{E_{n}}{(C_{2},A)} = \\frac{A C_{2}^{A}}{C_{2}} + C_{2}^{A}", "derivation": "\\operatorname{E_{n}}{(C_{2},A)} = C_{2}^{A} and \\frac{\\partial}{\\partial C_{2}} \\operatorname{E_{n}}{(C_{2},A)} = \\frac{\\partial}{\\partial C_{2}} C_{2}^{A} and C_{2}^{A} + \\frac{\\partial}{\\partial C_{2}} \\operatorname{E_{n}}{(C_{2},A)} = C_{2}^{A} + \\frac{\\partial}{\\partial C_{2}} C_{2}^{A} and C_{2}^{A} + \\frac{\\partial}{\\partial C_{2}} \\operatorname{E_{n}}{(C_{2},A)} = \\frac{A C_{2}^{A}}{C_{2}} + C_{2}^{A}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["add", 2, "Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True))"], "Equality(Add(Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Derivative(Function('E_n')(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Derivative(Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Derivative(Function('E_n')(Symbol('C_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('C_2', commutative=True), Integer(-1)), Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True))), Pow(Symbol('C_2', commutative=True), Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(I,\\delta,\\tilde{g})} = I \\delta \\tilde{g} and s{(I,\\delta,\\tilde{g})} = - \\tilde{g} + \\operatorname{P_{e}}{(I,\\delta,\\tilde{g})}, then obtain s{(I,\\delta,\\tilde{g})} = I \\delta \\tilde{g} - \\tilde{g}", "derivation": "\\operatorname{P_{e}}{(I,\\delta,\\tilde{g})} = I \\delta \\tilde{g} and - \\tilde{g} + \\operatorname{P_{e}}{(I,\\delta,\\tilde{g})} = I \\delta \\tilde{g} - \\tilde{g} and s{(I,\\delta,\\tilde{g})} = - \\tilde{g} + \\operatorname{P_{e}}{(I,\\delta,\\tilde{g})} and s{(I,\\delta,\\tilde{g})} = I \\delta \\tilde{g} - \\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('s')(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\mu_0,\\ddot{x})} = \\ddot{x} + \\mu_0 and \\varepsilon_{0}{(\\mu_0,\\ddot{x})} = \\int (\\ddot{x} + \\mu_0) d\\mu_0 + \\int \\pi{(\\mu_0,\\ddot{x})} d\\mu_0, then obtain \\varepsilon_{0}{(\\mu_0,\\ddot{x})} = 2 \\ddot{x} \\mu_0 + \\mu_0^{2} + 2 \\theta_1", "derivation": "\\pi{(\\mu_0,\\ddot{x})} = \\ddot{x} + \\mu_0 and \\int \\pi{(\\mu_0,\\ddot{x})} d\\mu_0 = \\int (\\ddot{x} + \\mu_0) d\\mu_0 and \\int (\\ddot{x} + \\mu_0) d\\mu_0 + \\int \\pi{(\\mu_0,\\ddot{x})} d\\mu_0 = 2 \\int (\\ddot{x} + \\mu_0) d\\mu_0 and \\varepsilon_{0}{(\\mu_0,\\ddot{x})} = \\int (\\ddot{x} + \\mu_0) d\\mu_0 + \\int \\pi{(\\mu_0,\\ddot{x})} d\\mu_0 and \\varepsilon_{0}{(\\mu_0,\\ddot{x})} = 2 \\int (\\ddot{x} + \\mu_0) d\\mu_0 and \\varepsilon_{0}{(\\mu_0,\\ddot{x})} = 2 \\ddot{x} \\mu_0 + \\mu_0^{2} + 2 \\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Integer(2), Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(m)} = e^{m}, then obtain \\frac{d}{d m} (0^{m})^{m} = \\frac{d}{d m} ((\\dot{x}{(m)} - e^{m})^{m})^{m}", "derivation": "\\dot{x}{(m)} = e^{m} and \\dot{x}{(m)} - e^{m} = 0 and (\\dot{x}{(m)} - e^{m})^{m} = 0^{m} and ((\\dot{x}{(m)} - e^{m})^{m})^{m} = (0^{m})^{m} and 1 = ((\\dot{x}{(m)} - e^{m})^{m})^{m} and \\frac{d}{d m} 1 = \\frac{d}{d m} ((\\dot{x}{(m)} - e^{m})^{m})^{m} and \\frac{d}{d m} 1 = \\frac{d}{d m} (0^{m})^{m} and \\frac{d}{d m} (0^{m})^{m} = \\frac{d}{d m} ((\\dot{x}{(m)} - e^{m})^{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["minus", 1, "exp(Symbol('m', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Integer(0), Symbol('m', commutative=True)))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["differentiate", 5, "Symbol('m', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Derivative(Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\omega)} = \\cos{(e^{\\omega})}, then derive - H - \\operatorname{Ci}{(e^{\\omega})} + \\int \\operatorname{f^{*}}{(\\omega)} d\\omega = 0, then obtain (- H - \\operatorname{Ci}{(e^{\\omega})} + \\int \\cos{(e^{\\omega})} d\\omega)^{\\omega} = 0^{\\omega}", "derivation": "\\operatorname{f^{*}}{(\\omega)} = \\cos{(e^{\\omega})} and \\int \\operatorname{f^{*}}{(\\omega)} d\\omega = \\int \\cos{(e^{\\omega})} d\\omega and \\int \\operatorname{f^{*}}{(\\omega)} d\\omega - \\int \\cos{(e^{\\omega})} d\\omega = 0 and - H - \\operatorname{Ci}{(e^{\\omega})} + \\int \\operatorname{f^{*}}{(\\omega)} d\\omega = 0 and (- H - \\operatorname{Ci}{(e^{\\omega})} + \\int \\operatorname{f^{*}}{(\\omega)} d\\omega)^{\\omega} = 0^{\\omega} and (- H - \\operatorname{Ci}{(e^{\\omega})} + \\int \\cos{(e^{\\omega})} d\\omega)^{\\omega} = 0^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\omega', commutative=True)), cos(exp(Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(cos(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Integral(cos(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Integral(cos(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Ci(exp(Symbol('\\\\omega', commutative=True)))), Integral(Function('f^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Ci(exp(Symbol('\\\\omega', commutative=True)))), Integral(Function('f^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Ci(exp(Symbol('\\\\omega', commutative=True)))), Integral(cos(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\pi)} = e^{\\pi}, then obtain \\operatorname{C_{2}}{(\\pi)} \\operatorname{C_{2}}^{\\pi}{(\\pi)} = \\operatorname{C_{2}}^{\\pi}{(\\pi)} e^{\\pi}", "derivation": "\\operatorname{C_{2}}{(\\pi)} = e^{\\pi} and \\operatorname{C_{2}}^{\\pi}{(\\pi)} = (e^{\\pi})^{\\pi} and \\operatorname{C_{2}}{(\\pi)} (e^{\\pi})^{\\pi} = e^{\\pi} (e^{\\pi})^{\\pi} and \\operatorname{C_{2}}{(\\pi)} \\operatorname{C_{2}}^{\\pi}{(\\pi)} = \\operatorname{C_{2}}^{\\pi}{(\\pi)} e^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('\\\\pi', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(exp(Symbol('\\\\pi', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('C_2')(Symbol('\\\\pi', commutative=True)), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})} = \\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})}, then obtain \\frac{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})}}{\\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})}} = \\frac{\\cos{(\\frac{\\Psi^{\\dagger}}{\\lambda})}}{\\lambda \\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})} = \\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})} and \\log{(\\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})})} = \\log{(\\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\log{(\\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\log{(\\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})})} and \\frac{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})}}{\\operatorname{f_{\\mathbf{p}}}{(\\lambda,\\Psi^{\\dagger})}} = \\frac{\\cos{(\\frac{\\Psi^{\\dagger}}{\\lambda})}}{\\lambda \\sin{(\\frac{\\Psi^{\\dagger}}{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"], [["log", 1], "Equality(log(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), log(sin(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))))"], [["differentiate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(log(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(log(sin(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Pow(sin(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Integer(-1)), cos(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given x{(U,v,\\ddot{x})} = U v - \\ddot{x}, then obtain (U - v + x{(U,v,\\ddot{x})})^{v} = (U v + U - \\ddot{x} - v)^{v}", "derivation": "x{(U,v,\\ddot{x})} = U v - \\ddot{x} and U + x{(U,v,\\ddot{x})} = U v + U - \\ddot{x} and U - v + x{(U,v,\\ddot{x})} = U v + U - \\ddot{x} - v and (U - v + x{(U,v,\\ddot{x})})^{v} = (U v + U - \\ddot{x} - v)^{v}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('U', commutative=True), Symbol('v', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('x')(Symbol('U', commutative=True), Symbol('v', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Function('x')(Symbol('U', commutative=True), Symbol('v', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Function('x')(Symbol('U', commutative=True), Symbol('v', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Symbol('U', commutative=True), Symbol('v', commutative=True)), Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(M_{E})} = \\sin{(M_{E})}, then obtain \\frac{\\sin{(M_{E})} (\\frac{d}{d M_{E}} 1)^{2}}{\\operatorname{m_{s}}{(M_{E})}} = \\frac{\\sin{(M_{E})} \\frac{d}{d M_{E}} 1 \\frac{d}{d M_{E}} (\\frac{\\sin{(M_{E})}}{\\operatorname{m_{s}}{(M_{E})}})^{M_{E}}}{\\operatorname{m_{s}}{(M_{E})}}", "derivation": "\\operatorname{m_{s}}{(M_{E})} = \\sin{(M_{E})} and 1 = \\frac{\\sin{(M_{E})}}{\\operatorname{m_{s}}{(M_{E})}} and 1 = (\\frac{\\sin{(M_{E})}}{\\operatorname{m_{s}}{(M_{E})}})^{M_{E}} and \\frac{d}{d M_{E}} 1 = \\frac{d}{d M_{E}} (\\frac{\\sin{(M_{E})}}{\\operatorname{m_{s}}{(M_{E})}})^{M_{E}} and \\frac{\\sin{(M_{E})} (\\frac{d}{d M_{E}} 1)^{2}}{\\operatorname{m_{s}}{(M_{E})}} = \\frac{\\sin{(M_{E})} \\frac{d}{d M_{E}} 1 \\frac{d}{d M_{E}} (\\frac{\\sin{(M_{E})}}{\\operatorname{m_{s}}{(M_{E})}})^{M_{E}}}{\\operatorname{m_{s}}{(M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["divide", 1, "Function('m_s')(Symbol('M_E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["times", 4, "Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True)), Derivative(Integer(1), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True)), Derivative(Integer(1), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('m_s')(Symbol('M_E', commutative=True)), Integer(-1)), sin(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(A_{z})} = \\log{(A_{z})}, then obtain (A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\hat{\\mathbf{x}}{(A_{z})})^{A_{z}})^{A_{z}} = (A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\log{(A_{z})})^{A_{z}})^{A_{z}}", "derivation": "\\hat{\\mathbf{x}}{(A_{z})} = \\log{(A_{z})} and A_{z} \\hat{\\mathbf{x}}{(A_{z})} = A_{z} \\log{(A_{z})} and (A_{z} \\hat{\\mathbf{x}}{(A_{z})})^{A_{z}} = (A_{z} \\log{(A_{z})})^{A_{z}} and A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\hat{\\mathbf{x}}{(A_{z})})^{A_{z}} = A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\log{(A_{z})})^{A_{z}} and (A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\hat{\\mathbf{x}}{(A_{z})})^{A_{z}})^{A_{z}} = (A_{z} \\hat{\\mathbf{x}}{(A_{z})} + (A_{z} \\log{(A_{z})})^{A_{z}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True)), log(Symbol('A_z', commutative=True)))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["add", 3, "Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True)))"], "Equality(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Pow(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))))"], [["power", 4, "Symbol('A_z', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Pow(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_z', commutative=True))), Pow(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\mathbf{P})} = e^{\\mathbf{P}}, then derive \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain (\\int (\\mu{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})}) d\\mathbf{P})^{2} = (\\int 2 \\mu{(\\mathbf{P})} d\\mathbf{P})^{2}", "derivation": "\\mu{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})} = e^{\\mathbf{P}} and e^{\\mathbf{P}} + \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})} = 2 e^{\\mathbf{P}} and \\mu{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})} = 2 \\mu{(\\mathbf{P})} and \\int (\\mu{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})}) d\\mathbf{P} = \\int 2 \\mu{(\\mathbf{P})} d\\mathbf{P} and (\\int (\\mu{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\mu{(\\mathbf{P})}) d\\mathbf{P})^{2} = (\\int 2 \\mu{(\\mathbf{P})} d\\mathbf{P})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 3, "exp(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(exp(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Mul(Integer(2), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 6, 2], "Equality(Pow(Integral(Add(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), Function('\\\\mu')(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given I{(x)} = \\sin{(x)}, then derive \\frac{d^{2}}{d x^{2}} \\int I{(x)} dx = \\frac{\\partial^{2}}{\\partial x^{2}} (q - \\cos{(x)}), then obtain \\frac{d^{3}}{d qd x^{2}} \\int \\sin{(x)} dx = 0", "derivation": "I{(x)} = \\sin{(x)} and \\int I{(x)} dx = \\int \\sin{(x)} dx and \\frac{d}{d x} \\int I{(x)} dx = \\frac{d}{d x} \\int \\sin{(x)} dx and \\frac{d^{2}}{d x^{2}} \\int I{(x)} dx = \\frac{d^{2}}{d x^{2}} \\int \\sin{(x)} dx and \\frac{d^{2}}{d x^{2}} \\int I{(x)} dx = \\frac{\\partial^{2}}{\\partial x^{2}} (q - \\cos{(x)}) and \\frac{d^{2}}{d x^{2}} \\int \\sin{(x)} dx = \\frac{\\partial^{2}}{\\partial x^{2}} (q - \\cos{(x)}) and \\frac{d^{3}}{d qd x^{2}} \\int \\sin{(x)} dx = \\frac{\\partial^{3}}{\\partial q\\partial x^{2}} (q - \\cos{(x)}) and \\frac{d^{3}}{d qd x^{2}} \\int \\sin{(x)} dx = 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('I')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(Function('I')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(Function('I')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('I')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(2))))"], [["differentiate", 6, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(2)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(2)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(v_{y})} = e^{v_{y}}, then obtain \\frac{\\partial}{\\partial v_{y}} (v_{y} + \\sigma_{p}{(E_{n})}) \\int \\operatorname{P_{g}}{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} (v_{y} + \\sigma_{p}{(E_{n})}) \\int e^{v_{y}} dv_{y}", "derivation": "\\operatorname{P_{g}}{(v_{y})} = e^{v_{y}} and \\int \\operatorname{P_{g}}{(v_{y})} dv_{y} = \\int e^{v_{y}} dv_{y} and (v_{y} + \\sigma_{p}{(E_{n})}) \\int \\operatorname{P_{g}}{(v_{y})} dv_{y} = (v_{y} + \\sigma_{p}{(E_{n})}) \\int e^{v_{y}} dv_{y} and \\frac{\\partial}{\\partial v_{y}} (v_{y} + \\sigma_{p}{(E_{n})}) \\int \\operatorname{P_{g}}{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} (v_{y} + \\sigma_{p}{(E_{n})}) \\int e^{v_{y}} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["times", 2, "Add(Symbol('v_y', commutative=True), Function('\\\\sigma_p')(Symbol('E_n', commutative=True)))"], "Equality(Mul(Add(Symbol('v_y', commutative=True), Function('\\\\sigma_p')(Symbol('E_n', commutative=True))), Integral(Function('P_g')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Mul(Add(Symbol('v_y', commutative=True), Function('\\\\sigma_p')(Symbol('E_n', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('v_y', commutative=True), Function('\\\\sigma_p')(Symbol('E_n', commutative=True))), Integral(Function('P_g')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('v_y', commutative=True), Function('\\\\sigma_p')(Symbol('E_n', commutative=True))), Integral(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\chi)} = \\cos{(\\chi)}, then derive - \\cos{(\\chi)} = - \\sin{(\\chi)} - \\cos{(\\chi)} - \\frac{d}{d \\chi} \\mathbf{g}{(\\chi)}, then obtain - \\mathbf{g}{(\\chi)} = - \\mathbf{g}{(\\chi)} - \\sin{(\\chi)} - \\frac{d}{d \\chi} \\mathbf{g}{(\\chi)}", "derivation": "\\mathbf{g}{(\\chi)} = \\cos{(\\chi)} and 0 = - \\mathbf{g}{(\\chi)} + \\cos{(\\chi)} and \\frac{d}{d \\chi} 0 = \\frac{d}{d \\chi} (- \\mathbf{g}{(\\chi)} + \\cos{(\\chi)}) and - \\cos{(\\chi)} + \\frac{d}{d \\chi} 0 = - \\cos{(\\chi)} + \\frac{d}{d \\chi} (- \\mathbf{g}{(\\chi)} + \\cos{(\\chi)}) and - \\cos{(\\chi)} = - \\sin{(\\chi)} - \\cos{(\\chi)} - \\frac{d}{d \\chi} \\mathbf{g}{(\\chi)} and - \\mathbf{g}{(\\chi)} = - \\mathbf{g}{(\\chi)} - \\sin{(\\chi)} - \\frac{d}{d \\chi} \\mathbf{g}{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 3, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))), Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(v_{2})} = \\log{(v_{2})}, then derive \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{1}{v_{2}}, then obtain \\frac{1}{v_{2}} = \\frac{d}{d v_{2}} \\log{(v_{2})}", "derivation": "\\operatorname{A_{x}}{(v_{2})} = \\log{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{1}{v_{2}} and \\frac{1}{v_{2}} = \\frac{d}{d v_{2}} \\log{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Pow(Symbol('v_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('v_2', commutative=True), Integer(-1)), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(E_{x})} = \\sin{(\\sin{(E_{x})})} and \\operatorname{v_{x}}{(E_{x})} = \\frac{\\mathbf{P}{(E_{x})}}{\\sin{(\\sin{(E_{x})})}}, then derive \\int \\operatorname{v_{x}}{(E_{x})} dE_{x} = E_{x} + m_{s}, then obtain E_{x} + m_{s} = \\int 1 dE_{x}", "derivation": "\\mathbf{P}{(E_{x})} = \\sin{(\\sin{(E_{x})})} and \\frac{\\mathbf{P}{(E_{x})}}{\\sin{(\\sin{(E_{x})})}} = 1 and \\int \\frac{\\mathbf{P}{(E_{x})}}{\\sin{(\\sin{(E_{x})})}} dE_{x} = \\int 1 dE_{x} and \\operatorname{v_{x}}{(E_{x})} = \\frac{\\mathbf{P}{(E_{x})}}{\\sin{(\\sin{(E_{x})})}} and \\operatorname{v_{x}}{(E_{x})} = 1 and \\int \\operatorname{v_{x}}{(E_{x})} dE_{x} = \\int 1 dE_{x} and \\int \\operatorname{v_{x}}{(E_{x})} dE_{x} = E_{x} + m_{s} and \\int \\frac{\\mathbf{P}{(E_{x})}}{\\sin{(\\sin{(E_{x})})}} dE_{x} = E_{x} + m_{s} and E_{x} + m_{s} = \\int 1 dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), sin(sin(Symbol('E_x', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('E_x', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Pow(sin(sin(Symbol('E_x', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Pow(sin(sin(Symbol('E_x', commutative=True))), Integer(-1))), Tuple(Symbol('E_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('E_x', commutative=True)), Mul(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Pow(sin(sin(Symbol('E_x', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('v_x')(Symbol('E_x', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Function('v_x')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Integral(Mul(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Pow(sin(sin(Symbol('E_x', commutative=True))), Integer(-1))), Tuple(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 8], "Equality(Add(Symbol('E_x', commutative=True), Symbol('m_s', commutative=True)), Integral(Integer(1), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given I{(f_{\\mathbf{p}},L)} = \\frac{\\partial}{\\partial L} (L + f_{\\mathbf{p}}), then derive I{(f_{\\mathbf{p}},L)} = 1, then derive \\int I{(f_{\\mathbf{p}},L)} dL = L + \\sigma_p, then obtain (\\int I{(f_{\\mathbf{p}},L)} dL)^{\\sigma_p} = (L + \\sigma_p)^{\\sigma_p}", "derivation": "I{(f_{\\mathbf{p}},L)} = \\frac{\\partial}{\\partial L} (L + f_{\\mathbf{p}}) and I{(f_{\\mathbf{p}},L)} = 1 and \\frac{\\partial}{\\partial L} (L + f_{\\mathbf{p}}) = 1 and \\int \\frac{\\partial}{\\partial L} (L + f_{\\mathbf{p}}) dL = \\int 1 dL and \\int I{(f_{\\mathbf{p}},L)} dL = \\int \\frac{\\partial}{\\partial L} (L + f_{\\mathbf{p}}) dL and \\int I{(f_{\\mathbf{p}},L)} dL = \\int 1 dL and \\int I{(f_{\\mathbf{p}},L)} dL = L + \\sigma_p and (\\int I{(f_{\\mathbf{p}},L)} dL)^{\\sigma_p} = (L + \\sigma_p)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Derivative(Add(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True))))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Add(Symbol('L', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["power", 7, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integral(Function('I')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f^{*},l,\\eta)} = (f^{*} + l)^{\\eta} and \\sigma_{p}{(f^{*},l,\\eta)} = \\int \\frac{(f^{*} + l)^{\\eta}}{l} dl, then obtain \\int \\frac{\\operatorname{F_{H}}{(f^{*},l,\\eta)}}{l} dl = \\sigma_{p}{(f^{*},l,\\eta)}", "derivation": "\\operatorname{F_{H}}{(f^{*},l,\\eta)} = (f^{*} + l)^{\\eta} and \\frac{\\operatorname{F_{H}}{(f^{*},l,\\eta)}}{l} = \\frac{(f^{*} + l)^{\\eta}}{l} and \\int \\frac{\\operatorname{F_{H}}{(f^{*},l,\\eta)}}{l} dl = \\int \\frac{(f^{*} + l)^{\\eta}}{l} dl and \\sigma_{p}{(f^{*},l,\\eta)} = \\int \\frac{(f^{*} + l)^{\\eta}}{l} dl and \\int \\frac{\\operatorname{F_{H}}{(f^{*},l,\\eta)}}{l} dl = \\sigma_{p}{(f^{*},l,\\eta)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('F_H')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\eta', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('F_H')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('F_H')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('l', commutative=True))), Function('\\\\sigma_p')(Symbol('f^*', commutative=True), Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\psi^*)} = \\psi^*, then derive M_{E} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} \\log{(\\operatorname{A_{y}}{(\\psi^*)})} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} = \\psi^* \\log{(\\psi^*)} - \\psi^* + i, then obtain M_{E} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} \\log{(\\psi^*)} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} = \\psi^* \\log{(\\psi^*)} - \\psi^* + i", "derivation": "\\operatorname{A_{y}}{(\\psi^*)} = \\psi^* and \\log{(\\operatorname{A_{y}}{(\\psi^*)})} = \\log{(\\psi^*)} and \\int \\log{(\\operatorname{A_{y}}{(\\psi^*)})} d\\psi^* = \\int \\log{(\\psi^*)} d\\psi^* and M_{E} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} \\log{(\\operatorname{A_{y}}{(\\psi^*)})} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} = \\psi^* \\log{(\\psi^*)} - \\psi^* + i and M_{E} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} \\log{(\\psi^*)} + \\tilde{\\infty} \\operatorname{A_{y}}{(\\psi^*)} = \\psi^* \\log{(\\psi^*)} - \\psi^* + i", "srepr_derivation": [["renaming_premise", "Equality(Function('A_y')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["log", 1], "Equality(log(Function('A_y')(Symbol('\\\\psi^*', commutative=True))), log(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(log(Function('A_y')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('M_E', commutative=True), Mul(zoo, Function('A_y')(Symbol('\\\\psi^*', commutative=True)), log(Function('A_y')(Symbol('\\\\psi^*', commutative=True)))), Mul(zoo, Function('A_y')(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('M_E', commutative=True), Mul(zoo, Function('A_y')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Mul(zoo, Function('A_y')(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}} k)}, then obtain \\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} \\cos^{2}{(\\dot{\\mathbf{r}} k)} = \\cos^{3}{(\\dot{\\mathbf{r}} k)}", "derivation": "\\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}} k)} and \\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} \\cos{(\\dot{\\mathbf{r}} k)} = \\cos^{2}{(\\dot{\\mathbf{r}} k)} and \\operatorname{A_{z}}^{2}{(k,\\dot{\\mathbf{r}})} \\cos{(\\dot{\\mathbf{r}} k)} = \\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} \\cos^{2}{(\\dot{\\mathbf{r}} k)} and \\operatorname{A_{z}}{(k,\\dot{\\mathbf{r}})} \\cos^{2}{(\\dot{\\mathbf{r}} k)} = \\cos^{3}{(\\dot{\\mathbf{r}} k)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True))))"], [["times", 1, "cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True)))"], "Equality(Mul(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True)))), Pow(cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True))), Integer(2)))"], [["times", 2, "Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(2)), cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True)))), Mul(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True))), Integer(2))), Pow(cos(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('k', commutative=True))), Integer(3)))"]]}, {"prompt": "Given \\hat{X}{(f)} = \\sin{(\\cos{(f)})}, then obtain \\frac{d}{d f} \\log{(\\hat{X}{(f)})} + \\frac{d}{d f} \\log{(\\sin{(\\cos{(f)})})} = 2 \\frac{d}{d f} \\log{(\\sin{(\\cos{(f)})})}", "derivation": "\\hat{X}{(f)} = \\sin{(\\cos{(f)})} and \\log{(\\hat{X}{(f)})} = \\log{(\\sin{(\\cos{(f)})})} and \\frac{d}{d f} \\log{(\\hat{X}{(f)})} = \\frac{d}{d f} \\log{(\\sin{(\\cos{(f)})})} and \\frac{d}{d f} \\log{(\\hat{X}{(f)})} + \\frac{d}{d f} \\log{(\\sin{(\\cos{(f)})})} = 2 \\frac{d}{d f} \\log{(\\sin{(\\cos{(f)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('f', commutative=True)), sin(cos(Symbol('f', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\hat{X}')(Symbol('f', commutative=True))), log(sin(cos(Symbol('f', commutative=True)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(log(Function('\\\\hat{X}')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(sin(cos(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["add", 3, "Derivative(log(sin(cos(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Derivative(log(Function('\\\\hat{X}')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(sin(cos(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(log(sin(cos(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(n_{1})} = e^{n_{1}}, then obtain - (- n_{1} + \\operatorname{P_{e}}{(n_{1})})^{n_{1}} + \\operatorname{P_{e}}{(n_{1})} = - (- n_{1} + \\operatorname{P_{e}}{(n_{1})})^{n_{1}} + e^{n_{1}}", "derivation": "\\operatorname{P_{e}}{(n_{1})} = e^{n_{1}} and - n_{1} + \\operatorname{P_{e}}{(n_{1})} = - n_{1} + e^{n_{1}} and (- n_{1} + \\operatorname{P_{e}}{(n_{1})})^{n_{1}} = (- n_{1} + e^{n_{1}})^{n_{1}} and - (- n_{1} + e^{n_{1}})^{n_{1}} + \\operatorname{P_{e}}{(n_{1})} = - (- n_{1} + e^{n_{1}})^{n_{1}} + e^{n_{1}} and - (- n_{1} + \\operatorname{P_{e}}{(n_{1})})^{n_{1}} + \\operatorname{P_{e}}{(n_{1})} = - (- n_{1} + \\operatorname{P_{e}}{(n_{1})})^{n_{1}} + e^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["minus", 1, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('P_e')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('P_e')(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))), Function('P_e')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))), exp(Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('P_e')(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))), Function('P_e')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('P_e')(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))), exp(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given H{(\\varepsilon_0,r_{0},v)} = \\varepsilon_0 + v^{r_{0}}, then obtain - r_{0} + \\int (\\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} + \\int H{(\\varepsilon_0,r_{0},v)} dr_{0}) dv = - r_{0} + \\int 2 \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} dv", "derivation": "H{(\\varepsilon_0,r_{0},v)} = \\varepsilon_0 + v^{r_{0}} and \\int H{(\\varepsilon_0,r_{0},v)} dr_{0} = \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} and \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} + \\int H{(\\varepsilon_0,r_{0},v)} dr_{0} = 2 \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} and \\int (\\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} + \\int H{(\\varepsilon_0,r_{0},v)} dr_{0}) dv = \\int 2 \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} dv and - r_{0} + \\int (\\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} + \\int H{(\\varepsilon_0,r_{0},v)} dr_{0}) dv = - r_{0} + \\int 2 \\int (\\varepsilon_0 + v^{r_{0}}) dr_{0} dv", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["add", 2, "Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Mul(Integer(2), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Integer(2), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["minus", 4, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Integral(Add(Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Integral(Mul(Integer(2), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(Symbol('v', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given u{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then obtain u{(\\mathbf{g})} \\int \\frac{u{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} d\\mathbf{g} = u{(\\mathbf{g})} \\int 1 d\\mathbf{g}", "derivation": "u{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\frac{u{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} = 1 and \\int \\frac{u{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} d\\mathbf{g} = \\int 1 d\\mathbf{g} and \\cos{(\\mathbf{g})} \\int \\frac{u{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} d\\mathbf{g} = \\cos{(\\mathbf{g})} \\int 1 d\\mathbf{g} and u{(\\mathbf{g})} \\int \\frac{u{(\\mathbf{g})}}{\\cos{(\\mathbf{g})}} d\\mathbf{g} = u{(\\mathbf{g})} \\int 1 d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 3, "Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))"], "Equality(Mul(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integral(Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Mul(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Integral(Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Function('u')(Symbol('\\\\mathbf{g}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} = \\hat{X} + \\mathbf{f} and \\rho_{f}{(\\mathbf{f},\\hat{X})} = \\hat{\\mathbf{r}}^{2}{(\\mathbf{f},\\hat{X})}, then obtain (\\hat{X} + \\mathbf{f}) \\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} = (\\hat{X} + \\mathbf{f})^{2}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} = \\hat{X} + \\mathbf{f} and \\hat{\\mathbf{r}}^{2}{(\\mathbf{f},\\hat{X})} = (\\hat{X} + \\mathbf{f}) \\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} and \\rho_{f}{(\\mathbf{f},\\hat{X})} = \\hat{\\mathbf{r}}^{2}{(\\mathbf{f},\\hat{X})} and \\rho_{f}{(\\mathbf{f},\\hat{X})} = (\\hat{X} + \\mathbf{f}) \\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} and \\rho_{f}{(\\mathbf{f},\\hat{X})} = (\\hat{X} + \\mathbf{f})^{2} and (\\hat{X} + \\mathbf{f}) \\hat{\\mathbf{r}}{(\\mathbf{f},\\hat{X})} = (\\hat{X} + \\mathbf{f})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given a{(v_{2})} = \\sin{(v_{2})}, then obtain \\frac{2 \\sin{(v_{2})}}{a{(v_{2})} + \\sin{(v_{2})}} = \\frac{2 \\sin^{2}{(v_{2})}}{(a{(v_{2})} + \\sin{(v_{2})}) a{(v_{2})}}", "derivation": "a{(v_{2})} = \\sin{(v_{2})} and 2 a{(v_{2})} = a{(v_{2})} + \\sin{(v_{2})} and \\frac{1}{2} = \\frac{\\sin{(v_{2})}}{2 a{(v_{2})}} and \\frac{1}{2} = \\frac{\\sin{(v_{2})}}{a{(v_{2})} + \\sin{(v_{2})}} and \\frac{\\sin{(v_{2})}}{a{(v_{2})} + \\sin{(v_{2})}} = \\frac{\\sin^{2}{(v_{2})}}{(a{(v_{2})} + \\sin{(v_{2})}) a{(v_{2})}} and \\frac{2 \\sin{(v_{2})}}{a{(v_{2})} + \\sin{(v_{2})}} = \\frac{2 \\sin^{2}{(v_{2})}}{(a{(v_{2})} + \\sin{(v_{2})}) a{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["add", 1, "Function('a')(Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(2), Function('a')(Symbol('v_2', commutative=True))), Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('a')(Symbol('v_2', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('a')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Pow(Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(-1)), sin(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(-1)), sin(Symbol('v_2', commutative=True))), Mul(Pow(Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(-1)), Pow(Function('a')(Symbol('v_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Integer(2))))"], [["divide", 5, "Rational(1, 2)"], "Equality(Mul(Integer(2), Pow(Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(-1)), sin(Symbol('v_2', commutative=True))), Mul(Integer(2), Pow(Add(Function('a')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(-1)), Pow(Function('a')(Symbol('v_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mu_{0}{(f,\\varphi)} = e^{\\frac{f}{\\varphi}} and \\Omega{(f,\\varphi)} = - \\mu_{0}{(f,\\varphi)}, then obtain \\frac{d}{d f} 0 = \\frac{\\partial}{\\partial f} \\frac{(\\Omega{(f,\\varphi)} + e^{\\frac{f}{\\varphi}})^{2}}{\\varphi}", "derivation": "\\mu_{0}{(f,\\varphi)} = e^{\\frac{f}{\\varphi}} and 0 = - \\mu_{0}{(f,\\varphi)} + e^{\\frac{f}{\\varphi}} and 0 = (- \\mu_{0}{(f,\\varphi)} + e^{\\frac{f}{\\varphi}})^{2} and \\Omega{(f,\\varphi)} = - \\mu_{0}{(f,\\varphi)} and 0 = \\frac{(- \\mu_{0}{(f,\\varphi)} + e^{\\frac{f}{\\varphi}})^{2}}{\\varphi} and 0 = \\frac{(\\Omega{(f,\\varphi)} + e^{\\frac{f}{\\varphi}})^{2}}{\\varphi} and \\frac{d}{d f} 0 = \\frac{\\partial}{\\partial f} \\frac{(\\Omega{(f,\\varphi)} + e^{\\frac{f}{\\varphi}})^{2}}{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["divide", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True))), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(0), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Add(Function('\\\\Omega')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Integer(2))))"], [["differentiate", 6, "Symbol('f', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Add(Function('\\\\Omega')(Symbol('f', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('f', commutative=True)))), Integer(2))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(b,v_{x})} = \\int b^{v_{x}} db, then obtain \\frac{\\partial}{\\partial b} (- v_{x} + \\int \\operatorname{n_{1}}{(b,v_{x})} dv_{x}) = \\frac{\\partial}{\\partial b} (- v_{x} + \\iint b^{v_{x}} db dv_{x})", "derivation": "\\operatorname{n_{1}}{(b,v_{x})} = \\int b^{v_{x}} db and \\int \\operatorname{n_{1}}{(b,v_{x})} dv_{x} = \\iint b^{v_{x}} db dv_{x} and - v_{x} + \\int \\operatorname{n_{1}}{(b,v_{x})} dv_{x} = - v_{x} + \\iint b^{v_{x}} db dv_{x} and \\frac{\\partial}{\\partial b} (- v_{x} + \\int \\operatorname{n_{1}}{(b,v_{x})} dv_{x}) = \\frac{\\partial}{\\partial b} (- v_{x} + \\iint b^{v_{x}} db dv_{x})", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Integral(Pow(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["minus", 2, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Function('n_1')(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Pow(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Function('n_1')(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Integral(Pow(Symbol('b', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(E)} = E and \\Omega{(E)} = \\frac{d}{d E} B{(E)}, then obtain E (\\Omega^{E}{(E)} (\\frac{d}{d E} E)^{- E} + (\\frac{d}{d E} E)^{E}) = E ((\\frac{d}{d E} E)^{E} + 1)", "derivation": "B{(E)} = E and \\frac{d}{d E} B{(E)} = \\frac{d}{d E} E and (\\frac{d}{d E} B{(E)})^{E} = (\\frac{d}{d E} E)^{E} and \\frac{d}{d E} E (\\frac{d}{d E} B{(E)})^{E} = \\frac{d}{d E} E (\\frac{d}{d E} E)^{E} and (\\frac{d}{d E} E)^{- E} (\\frac{d}{d E} B{(E)})^{E} = 1 and (\\frac{d}{d E} E)^{E} + (\\frac{d}{d E} E)^{- E} (\\frac{d}{d E} B{(E)})^{E} = (\\frac{d}{d E} E)^{E} + 1 and E ((\\frac{d}{d E} E)^{E} + (\\frac{d}{d E} E)^{- E} (\\frac{d}{d E} B{(E)})^{E}) = E ((\\frac{d}{d E} E)^{E} + 1) and \\Omega{(E)} = \\frac{d}{d E} B{(E)} and E (\\Omega^{E}{(E)} (\\frac{d}{d E} E)^{- E} + (\\frac{d}{d E} E)^{E}) = E ((\\frac{d}{d E} E)^{E} + 1)", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('E', commutative=True)), Symbol('E', commutative=True))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["times", 3, "Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Mul(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["divide", 4, "Mul(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], "Equality(Mul(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Integer(1))"], [["add", 5, "Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))"], "Equality(Add(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Mul(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))), Add(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1)))"], [["times", 6, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Add(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Mul(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))), Mul(Symbol('E', commutative=True), Add(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('E', commutative=True)), Derivative(Function('B')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Mul(Symbol('E', commutative=True), Add(Mul(Pow(Function('\\\\Omega')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('E', commutative=True)))), Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))), Mul(Symbol('E', commutative=True), Add(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(n_{2})} = \\int \\sin{(n_{2})} dn_{2}, then derive (U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})} = (U - \\cos{(n_{2})})^{2}, then obtain - \\operatorname{E_{\\lambda}}{(n_{2})} \\sin{((U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})})} = - (2 U - 2 \\cos{(n_{2})}) \\sin{((U - \\cos{(n_{2})})^{2})}", "derivation": "\\operatorname{E_{\\lambda}}{(n_{2})} = \\int \\sin{(n_{2})} dn_{2} and \\operatorname{E_{\\lambda}}{(n_{2})} \\int \\sin{(n_{2})} dn_{2} = (\\int \\sin{(n_{2})} dn_{2})^{2} and (U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})} = (U - \\cos{(n_{2})})^{2} and \\cos{((U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})})} = \\cos{((U - \\cos{(n_{2})})^{2})} and \\frac{\\partial}{\\partial U} \\cos{((U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})})} = \\frac{\\partial}{\\partial U} \\cos{((U - \\cos{(n_{2})})^{2})} and - \\operatorname{E_{\\lambda}}{(n_{2})} \\sin{((U - \\cos{(n_{2})}) \\operatorname{E_{\\lambda}}{(n_{2})})} = - (2 U - 2 \\cos{(n_{2})}) \\sin{((U - \\cos{(n_{2})})^{2})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["times", 1, "Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Pow(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True))), Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)))"], [["cos", 3], "Equality(cos(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)))), cos(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2))))"], [["differentiate", 4, "Symbol('U', commutative=True)"], "Equality(Derivative(cos(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)), sin(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True))))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('n_2', commutative=True)))), sin(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given t{(f,y^{\\prime})} = (y^{\\prime})^{f} and n{(f,y^{\\prime})} = ((y^{\\prime})^{f})^{y^{\\prime}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + (((y^{\\prime})^{f})^{y^{\\prime}})^{f}) = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + n^{f}{(f,y^{\\prime})})", "derivation": "t{(f,y^{\\prime})} = (y^{\\prime})^{f} and t^{y^{\\prime}}{(f,y^{\\prime})} = ((y^{\\prime})^{f})^{y^{\\prime}} and n{(f,y^{\\prime})} = ((y^{\\prime})^{f})^{y^{\\prime}} and t^{y^{\\prime}}{(f,y^{\\prime})} = n{(f,y^{\\prime})} and (t^{y^{\\prime}}{(f,y^{\\prime})})^{f} = n^{f}{(f,y^{\\prime})} and (((y^{\\prime})^{f})^{y^{\\prime}})^{f} = n^{f}{(f,y^{\\prime})} and \\mathbf{J}_P + (((y^{\\prime})^{f})^{y^{\\prime}})^{f} = \\mathbf{J}_P + n^{f}{(f,y^{\\prime})} and \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + (((y^{\\prime})^{f})^{y^{\\prime}})^{f}) = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + n^{f}{(f,y^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('t')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('t')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Function('t')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Pow(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True)))"], [["add", 6, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Pow(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Pow(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Function('n')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(\\lambda,v_{1})} = \\cos{(\\lambda - v_{1})} and \\mu{(x)} = \\sin{(x)}, then obtain \\frac{\\partial}{\\partial \\lambda} \\frac{\\hat{X}{(\\lambda,v_{1})}}{\\mu{(x)} \\cos{(\\lambda - v_{1})}} = \\frac{d}{d \\lambda} \\frac{1}{\\mu{(x)}}", "derivation": "\\hat{X}{(\\lambda,v_{1})} = \\cos{(\\lambda - v_{1})} and \\frac{\\hat{X}{(\\lambda,v_{1})}}{\\cos{(\\lambda - v_{1})}} = 1 and \\mu{(x)} = \\sin{(x)} and \\frac{\\hat{X}{(\\lambda,v_{1})}}{\\sin{(x)} \\cos{(\\lambda - v_{1})}} = \\frac{1}{\\sin{(x)}} and \\frac{\\partial}{\\partial \\lambda} \\frac{\\hat{X}{(\\lambda,v_{1})}}{\\sin{(x)} \\cos{(\\lambda - v_{1})}} = \\frac{d}{d \\lambda} \\frac{1}{\\sin{(x)}} and \\frac{\\partial}{\\partial \\lambda} \\frac{\\hat{X}{(\\lambda,v_{1})}}{\\mu{(x)} \\cos{(\\lambda - v_{1})}} = \\frac{d}{d \\lambda} \\frac{1}{\\mu{(x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"], [["divide", 1, "cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Integer(-1))), Integer(1))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["divide", 2, "sin(Symbol('x', commutative=True))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Integer(-1)), Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Integer(-1))), Pow(sin(Symbol('x', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{X}')(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Integer(-1)), Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('x', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Mul(Function('\\\\hat{X}')(Symbol('\\\\lambda', commutative=True), Symbol('v_1', commutative=True)), Pow(Function('\\\\mu')(Symbol('x', commutative=True)), Integer(-1)), Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mu')(Symbol('x', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then obtain 4 - \\sin{(\\mathbf{S})} = (0^{\\mathbf{S}} + 1)^{2} - \\sin{(\\mathbf{S})}", "derivation": "h{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and h{(\\mathbf{S})} - \\sin{(\\mathbf{S})} = 0 and (h{(\\mathbf{S})} - \\sin{(\\mathbf{S})})^{\\mathbf{S}} = 0^{\\mathbf{S}} and (h{(\\mathbf{S})} - \\sin{(\\mathbf{S})})^{\\mathbf{S}} + 1 = 0^{\\mathbf{S}} + 1 and 2 = (h{(\\mathbf{S})} - \\sin{(\\mathbf{S})})^{\\mathbf{S}} + 1 and 4 = ((h{(\\mathbf{S})} - \\sin{(\\mathbf{S})})^{\\mathbf{S}} + 1)^{2} and 4 = (0^{\\mathbf{S}} + 1)^{2} and 4 - \\sin{(\\mathbf{S})} = (0^{\\mathbf{S}} + 1)^{2} - \\sin{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Add(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Add(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(2), Add(Pow(Add(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)))"], [["power", 5, 2], "Equality(Integer(4), Pow(Add(Pow(Add(Function('h')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(4), Pow(Add(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Integer(2)))"], [["add", 7, "Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Integer(4), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Pow(Add(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Integer(1)), Integer(2)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mu,v)} = \\frac{\\log{(\\mu)}}{v}, then obtain \\frac{v \\operatorname{v_{x}}{(\\mu,v)}}{\\log{(\\mu)}} + \\operatorname{v_{x}}{(\\mu,v)} = \\operatorname{v_{x}}{(\\mu,v)} + 1", "derivation": "\\operatorname{v_{x}}{(\\mu,v)} = \\frac{\\log{(\\mu)}}{v} and \\frac{v \\operatorname{v_{x}}{(\\mu,v)}}{\\log{(\\mu)}} = 1 and \\frac{v \\operatorname{v_{x}}{(\\mu,v)}}{\\log{(\\mu)}} + \\frac{\\log{(\\mu)}}{v} = 1 + \\frac{\\log{(\\mu)}}{v} and \\frac{v \\operatorname{v_{x}}{(\\mu,v)}}{\\log{(\\mu)}} + \\operatorname{v_{x}}{(\\mu,v)} = \\operatorname{v_{x}}{(\\mu,v)} + 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Symbol('v', commutative=True), Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Symbol('v', commutative=True), Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Symbol('v', commutative=True), Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))), Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Add(Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(A_{x})} = e^{A_{x}}, then obtain \\int (\\frac{d}{d A_{x}} (A_{x} - \\operatorname{n_{2}}{(A_{x})}))^{A_{x}} dA_{x} = \\int (\\frac{d}{d A_{x}} (A_{x} - e^{A_{x}}))^{A_{x}} dA_{x}", "derivation": "\\operatorname{n_{2}}{(A_{x})} = e^{A_{x}} and - A_{x} + \\operatorname{n_{2}}{(A_{x})} = - A_{x} + e^{A_{x}} and A_{x} - \\operatorname{n_{2}}{(A_{x})} = A_{x} - e^{A_{x}} and \\frac{d}{d A_{x}} (A_{x} - \\operatorname{n_{2}}{(A_{x})}) = \\frac{d}{d A_{x}} (A_{x} - e^{A_{x}}) and (\\frac{d}{d A_{x}} (A_{x} - \\operatorname{n_{2}}{(A_{x})}))^{A_{x}} = (\\frac{d}{d A_{x}} (A_{x} - e^{A_{x}}))^{A_{x}} and \\int (\\frac{d}{d A_{x}} (A_{x} - \\operatorname{n_{2}}{(A_{x})}))^{A_{x}} dA_{x} = \\int (\\frac{d}{d A_{x}} (A_{x} - e^{A_{x}}))^{A_{x}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["minus", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('n_2')(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True)))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["power", 4, "Symbol('A_x', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Pow(Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)))"], [["integrate", 5, "Symbol('A_x', commutative=True)"], "Equality(Integral(Pow(Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Derivative(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given M{(g)} = e^{g}, then obtain (\\int M^{2}{(g)} dg)^{g} = (\\int M{(g)} e^{g} dg)^{g}", "derivation": "M{(g)} = e^{g} and M^{2}{(g)} = M{(g)} e^{g} and \\int M^{2}{(g)} dg = \\int M{(g)} e^{g} dg and (\\int M^{2}{(g)} dg)^{g} = (\\int M{(g)} e^{g} dg)^{g}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["times", 1, "Function('M')(Symbol('g', commutative=True))"], "Equality(Pow(Function('M')(Symbol('g', commutative=True)), Integer(2)), Mul(Function('M')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Pow(Function('M')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Integral(Mul(Function('M')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["power", 3, "Symbol('g', commutative=True)"], "Equality(Pow(Integral(Pow(Function('M')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Integral(Mul(Function('M')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given A{(t_{2},z^{*})} = (z^{*})^{t_{2}}, then obtain \\int (- (z^{*})^{t_{2}} + A{(t_{2},z^{*})}) \\hat{H}_l^{- \\Psi^{\\dagger}}{(\\Psi^{\\dagger})} dz^{*} = \\int 0 dz^{*}", "derivation": "A{(t_{2},z^{*})} = (z^{*})^{t_{2}} and z^{*} + A{(t_{2},z^{*})} = z^{*} + (z^{*})^{t_{2}} and - (z^{*})^{t_{2}} + A{(t_{2},z^{*})} = 0 and (- (z^{*})^{t_{2}} + A{(t_{2},z^{*})}) \\hat{H}_l^{- \\Psi^{\\dagger}}{(\\Psi^{\\dagger})} = 0 and \\int (- (z^{*})^{t_{2}} + A{(t_{2},z^{*})}) \\hat{H}_l^{- \\Psi^{\\dagger}}{(\\Psi^{\\dagger})} dz^{*} = \\int 0 dz^{*}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Function('A')(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True))), Add(Symbol('z^*', commutative=True), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True))))"], [["minus", 2, "Add(Symbol('z^*', commutative=True), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True))), Function('A')(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True))), Integer(0))"], [["divide", 3, "Pow(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True))), Function('A')(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Integer(0))"], [["integrate", 4, "Symbol('z^*', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Symbol('t_2', commutative=True))), Function('A')(Symbol('t_2', commutative=True), Symbol('z^*', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Tuple(Symbol('z^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(h)} = \\sin{(h)} and \\mathbf{p}{(h)} = \\frac{\\sin{(\\frac{\\sin{(h)}}{h})}}{\\sin{(\\frac{\\operatorname{x^{{\\}'}}{(h)}}{h})}}, then obtain \\mathbf{p}^{h}{(h)} = 1", "derivation": "\\operatorname{x^{{\\}'}}{(h)} = \\sin{(h)} and \\frac{\\operatorname{x^{{\\}'}}{(h)}}{h} = \\frac{\\sin{(h)}}{h} and \\sin{(\\frac{\\operatorname{x^{{\\}'}}{(h)}}{h})} = \\sin{(\\frac{\\sin{(h)}}{h})} and \\mathbf{p}{(h)} = \\frac{\\sin{(\\frac{\\sin{(h)}}{h})}}{\\sin{(\\frac{\\operatorname{x^{{\\}'}}{(h)}}{h})}} and \\mathbf{p}^{h}{(h)} = (\\frac{\\sin{(\\frac{\\sin{(h)}}{h})}}{\\sin{(\\frac{\\operatorname{x^{{\\}'}}{(h)}}{h})}})^{h} and \\mathbf{p}^{h}{(h)} = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('h', commutative=True)))), sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), Mul(Pow(sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('h', commutative=True)))), Integer(-1)), sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Pow(sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('h', commutative=True)))), Integer(-1)), sin(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(G)} = \\cos{(G)}, then derive - \\sin{(G)} + \\frac{d}{d G} \\operatorname{V_{\\mathbf{E}}}{(G)} = - 2 \\sin{(G)}, then obtain - \\sin{(G)} + \\frac{d}{d G} \\cos{(G)} = - \\sin{(G)} + \\frac{d}{d G} \\operatorname{V_{\\mathbf{E}}}{(G)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(G)} = \\cos{(G)} and \\operatorname{V_{\\mathbf{E}}}{(G)} + \\cos{(G)} = 2 \\cos{(G)} and \\frac{d}{d G} (\\operatorname{V_{\\mathbf{E}}}{(G)} + \\cos{(G)}) = \\frac{d}{d G} 2 \\cos{(G)} and \\frac{d}{d G} \\operatorname{V_{\\mathbf{E}}}{(G)} = \\frac{d}{d G} \\cos{(G)} and - \\sin{(G)} + \\frac{d}{d G} \\operatorname{V_{\\mathbf{E}}}{(G)} = - 2 \\sin{(G)} and - \\sin{(G)} + \\frac{d}{d G} \\cos{(G)} = - 2 \\sin{(G)} and - \\sin{(G)} + \\frac{d}{d G} \\cos{(G)} = - \\sin{(G)} + \\frac{d}{d G} \\operatorname{V_{\\mathbf{E}}}{(G)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["add", 1, "cos(Symbol('G', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Mul(Integer(2), cos(Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(z)} = \\sin{(z)}, then obtain (\\tilde{g}^*{(z)} \\frac{d}{d z} \\tilde{g}^*{(z)})^{z} = (\\tilde{g}^*{(z)} \\cos{(z)})^{z}", "derivation": "\\tilde{g}^*{(z)} = \\sin{(z)} and \\frac{d}{d z} \\tilde{g}^*{(z)} = \\frac{d}{d z} \\sin{(z)} and \\tilde{g}^*{(z)} \\frac{d}{d z} \\tilde{g}^*{(z)} = \\tilde{g}^*{(z)} \\frac{d}{d z} \\sin{(z)} and (\\tilde{g}^*{(z)} \\frac{d}{d z} \\tilde{g}^*{(z)})^{z} = (\\tilde{g}^*{(z)} \\frac{d}{d z} \\sin{(z)})^{z} and (\\tilde{g}^*{(z)} \\frac{d}{d z} \\tilde{g}^*{(z)})^{z} = (\\tilde{g}^*{(z)} \\cos{(z)})^{z}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\tilde{g}^*')(Symbol('z', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Derivative(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Derivative(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Symbol('z', commutative=True)), Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Derivative(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Symbol('z', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Derivative(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Symbol('z', commutative=True)), Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\rho,A_{1})} = \\rho^{A_{1}}, then obtain A_{1} + \\frac{\\partial}{\\partial A_{1}} \\operatorname{F_{g}}{(\\rho,A_{1})} = A_{1} + \\rho^{A_{1}} \\log{(\\rho)}", "derivation": "\\operatorname{F_{g}}{(\\rho,A_{1})} = \\rho^{A_{1}} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{F_{g}}{(\\rho,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\rho^{A_{1}} and A_{1} + \\frac{\\partial}{\\partial A_{1}} \\operatorname{F_{g}}{(\\rho,A_{1})} = A_{1} + \\frac{\\partial}{\\partial A_{1}} \\rho^{A_{1}} and A_{1} + \\frac{\\partial}{\\partial A_{1}} \\operatorname{F_{g}}{(\\rho,A_{1})} = A_{1} + \\rho^{A_{1}} \\log{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["add", 2, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Derivative(Function('F_g')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Symbol('A_1', commutative=True), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('A_1', commutative=True), Derivative(Function('F_g')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Symbol('A_1', commutative=True), Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), log(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(a)} = \\sin{(a)}, then derive \\frac{d}{d a} \\operatorname{F_{H}}{(a)} = \\cos{(a)}, then obtain \\frac{d}{d a} \\sin{(a)} = \\cos{(a)}", "derivation": "\\operatorname{F_{H}}{(a)} = \\sin{(a)} and \\frac{d}{d a} \\operatorname{F_{H}}{(a)} = \\frac{d}{d a} \\sin{(a)} and \\frac{d}{d a} \\operatorname{F_{H}}{(a)} = \\cos{(a)} and \\frac{d}{d a} \\sin{(a)} = \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), cos(Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), cos(Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} = T + \\hbar - \\varepsilon_0, then obtain \\hbar \\frac{\\partial}{\\partial \\hbar} \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} + 1 = \\hbar + 1", "derivation": "\\mathbf{E}{(\\varepsilon_0,T,\\hbar)} = T + \\hbar - \\varepsilon_0 and \\frac{\\partial}{\\partial \\hbar} \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (T + \\hbar - \\varepsilon_0) and \\hbar \\frac{\\partial}{\\partial \\hbar} \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} = \\hbar \\frac{\\partial}{\\partial \\hbar} (T + \\hbar - \\varepsilon_0) and \\hbar \\frac{\\partial}{\\partial \\hbar} \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} + \\frac{\\partial}{\\partial \\hbar} (T + \\hbar - \\varepsilon_0) = \\hbar \\frac{\\partial}{\\partial \\hbar} (T + \\hbar - \\varepsilon_0) + \\frac{\\partial}{\\partial \\hbar} (T + \\hbar - \\varepsilon_0) and \\hbar \\frac{\\partial}{\\partial \\hbar} \\mathbf{E}{(\\varepsilon_0,T,\\hbar)} + 1 = \\hbar + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Symbol('\\\\hbar', commutative=True), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Integer(1)), Add(Symbol('\\\\hbar', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\psi{(x,M_{E})} = M_{E} + x, then derive \\int \\psi{(x,M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l, then obtain \\int (\\int (M_{E} + x) dM_{E} + \\frac{1}{2}) dx - \\frac{1}{2} = \\int (\\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l + \\frac{1}{2}) dx - \\frac{1}{2}", "derivation": "\\psi{(x,M_{E})} = M_{E} + x and \\int \\psi{(x,M_{E})} dM_{E} = \\int (M_{E} + x) dM_{E} and \\int \\psi{(x,M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l and \\int (M_{E} + x) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l and \\int (M_{E} + x) dM_{E} + \\frac{1}{2} = \\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l + \\frac{1}{2} and \\int (\\int (M_{E} + x) dM_{E} + \\frac{1}{2}) dx = \\int (\\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l + \\frac{1}{2}) dx and \\int (\\int (M_{E} + x) dM_{E} + \\frac{1}{2}) dx - \\frac{1}{2} = \\int (\\frac{M_{E}^{2}}{2} + M_{E} x + \\hat{H}_l + \\frac{1}{2}) dx - \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('x', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('x', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 4, "Rational(1, 2)"], "Equality(Add(Integral(Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Rational(1, 2)), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Rational(1, 2)))"], [["integrate", 5, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Integral(Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Rational(1, 2)), Tuple(Symbol('x', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Rational(1, 2)), Tuple(Symbol('x', commutative=True))))"], [["minus", 6, "Rational(1, 2)"], "Equality(Add(Integral(Add(Integral(Add(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Rational(1, 2)), Tuple(Symbol('x', commutative=True))), Rational(-1, 2)), Add(Integral(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Rational(1, 2)), Tuple(Symbol('x', commutative=True))), Rational(-1, 2)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} = \\log{(- V_{\\mathbf{B}} + a)}, then obtain (\\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\operatorname{M_{E}}^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},a)})^{V_{\\mathbf{B}}} = (\\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\log{(- V_{\\mathbf{B}} + a)}^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "derivation": "\\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} = \\log{(- V_{\\mathbf{B}} + a)} and \\operatorname{M_{E}}^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},a)} = \\log{(- V_{\\mathbf{B}} + a)}^{V_{\\mathbf{B}}} and \\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\operatorname{M_{E}}^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},a)} = \\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\log{(- V_{\\mathbf{B}} + a)}^{V_{\\mathbf{B}}} and (\\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\operatorname{M_{E}}^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},a)})^{V_{\\mathbf{B}}} = (\\operatorname{M_{E}}{(V_{\\mathbf{B}},a)} + \\log{(- V_{\\mathbf{B}} + a)}^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 2, "Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Pow(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Add(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Pow(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Add(Function('M_E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('a', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(z)} = \\sin{(\\cos{(z)})} and \\pi{(h,A_{2})} = A_{2} + \\sin{(h)}, then obtain - \\frac{\\operatorname{f_{\\mathbf{p}}}{(z)}}{\\pi{(h,A_{2})} \\sin{(h)} \\sin{(\\cos{(z)})}} = - \\frac{1}{\\pi{(h,A_{2})} \\sin{(h)}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(z)} = \\sin{(\\cos{(z)})} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(z)}}{\\sin{(\\cos{(z)})}} = 1 and \\pi{(h,A_{2})} = A_{2} + \\sin{(h)} and - \\frac{\\operatorname{f_{\\mathbf{p}}}{(z)}}{\\sin{(h)} \\sin{(\\cos{(z)})}} = - \\frac{1}{\\sin{(h)}} and - \\frac{\\operatorname{f_{\\mathbf{p}}}{(z)}}{(A_{2} + \\sin{(h)}) \\sin{(h)} \\sin{(\\cos{(z)})}} = - \\frac{1}{(A_{2} + \\sin{(h)}) \\sin{(h)}} and - \\frac{\\operatorname{f_{\\mathbf{p}}}{(z)}}{\\pi{(h,A_{2})} \\sin{(h)} \\sin{(\\cos{(z)})}} = - \\frac{1}{\\pi{(h,A_{2})} \\sin{(h)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('z', commutative=True)), sin(cos(Symbol('z', commutative=True))))"], [["divide", 1, "sin(cos(Symbol('z', commutative=True)))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('z', commutative=True)), Pow(sin(cos(Symbol('z', commutative=True))), Integer(-1))), Integer(1))"], ["get_premise", "Equality(Function('\\\\pi')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), sin(Symbol('h', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), sin(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('z', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('h', commutative=True)), Integer(-1))))"], [["divide", 4, "Add(Symbol('A_2', commutative=True), sin(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('A_2', commutative=True), sin(Symbol('h', commutative=True))), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('z', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Add(Symbol('A_2', commutative=True), sin(Symbol('h', commutative=True))), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('z', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(cos(Symbol('z', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho{(\\Psi,\\tilde{g})} = \\frac{\\tilde{g}}{\\Psi}, then obtain - \\tilde{g} + (\\frac{\\partial}{\\partial \\tilde{g}} \\rho{(\\Psi,\\tilde{g})})^{\\Psi} = - \\tilde{g} + (\\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\Psi})^{\\Psi}", "derivation": "\\rho{(\\Psi,\\tilde{g})} = \\frac{\\tilde{g}}{\\Psi} and \\frac{\\partial}{\\partial \\tilde{g}} \\rho{(\\Psi,\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\Psi} and (\\frac{\\partial}{\\partial \\tilde{g}} \\rho{(\\Psi,\\tilde{g})})^{\\Psi} = (\\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\Psi})^{\\Psi} and - \\tilde{g} + (\\frac{\\partial}{\\partial \\tilde{g}} \\rho{(\\Psi,\\tilde{g})})^{\\Psi} = - \\tilde{g} + (\\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\Psi})^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)))"], [["minus", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Derivative(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(r)} = \\sin{(r)}, then derive 0 = \\frac{\\cos{(r)}}{\\operatorname{P_{e}}{(r)}} - \\frac{\\sin{(r)} \\frac{d}{d r} \\operatorname{P_{e}}{(r)}}{\\operatorname{P_{e}}^{2}{(r)}}, then obtain - \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}} = \\frac{\\cos{(r)}}{\\sin{(r)}} - \\frac{\\frac{d}{d r} \\sin{(r)}}{\\sin{(r)}} - \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}}", "derivation": "\\operatorname{P_{e}}{(r)} = \\sin{(r)} and 1 = \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}} and \\frac{d}{d r} 1 = \\frac{d}{d r} \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}} and 0 = \\frac{\\cos{(r)}}{\\operatorname{P_{e}}{(r)}} - \\frac{\\sin{(r)} \\frac{d}{d r} \\operatorname{P_{e}}{(r)}}{\\operatorname{P_{e}}^{2}{(r)}} and 0 = \\frac{\\cos{(r)}}{\\sin{(r)}} - \\frac{\\frac{d}{d r} \\sin{(r)}}{\\sin{(r)}} and - \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}} = \\frac{\\cos{(r)}}{\\sin{(r)}} - \\frac{\\frac{d}{d r} \\sin{(r)}}{\\sin{(r)}} - \\frac{\\sin{(r)}}{\\operatorname{P_{e}}{(r)}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["divide", 1, "Function('P_e')(Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), cos(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-2)), sin(Symbol('r', commutative=True)), Derivative(Function('P_e')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Pow(sin(Symbol('r', commutative=True)), Integer(-1)), cos(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('r', commutative=True)), Integer(-1)), Derivative(sin(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["minus", 5, "Mul(Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))), Add(Mul(Pow(sin(Symbol('r', commutative=True)), Integer(-1)), cos(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('r', commutative=True)), Integer(-1)), Derivative(sin(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('P_e')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given n{(f^{\\prime})} = \\cos{(f^{\\prime})}, then obtain - \\frac{\\frac{d}{d f^{\\prime}} n{(f^{\\prime})}}{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}} = -1", "derivation": "n{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} n{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} and \\frac{\\frac{d}{d f^{\\prime}} n{(f^{\\prime})}}{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}} = 1 and - \\frac{\\frac{d}{d f^{\\prime}} n{(f^{\\prime})}}{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}} = -1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('n')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('n')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\hbar)} = \\sin{(\\hbar)}, then obtain 0 = \\frac{- \\operatorname{P_{e}}^{2}{(\\hbar)} + \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)}}{\\operatorname{P_{e}}^{2}{(\\hbar)} - \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}}", "derivation": "\\operatorname{P_{e}}{(\\hbar)} = \\sin{(\\hbar)} and \\operatorname{P_{e}}^{2}{(\\hbar)} = \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)} and 0 = - \\operatorname{P_{e}}^{2}{(\\hbar)} + \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)} and 0 = \\frac{- \\operatorname{P_{e}}^{2}{(\\hbar)} + \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)}}{\\operatorname{P_{e}}^{2}{(\\hbar)} - \\operatorname{P_{e}}{(\\hbar)} \\sin{(\\hbar)} + \\sin{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))))"], [["divide", 3, "Add(Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))), Pow(Add(Pow(Function('P_e')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Integer(-1), Function('P_e')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(a^{\\dagger},y^{\\prime})} = a^{\\dagger} - y^{\\prime}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\varphi{(a^{\\dagger},y^{\\prime})} = 1, then obtain y^{\\prime} \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} - y^{\\prime}) = y^{\\prime}", "derivation": "\\varphi{(a^{\\dagger},y^{\\prime})} = a^{\\dagger} - y^{\\prime} and \\frac{\\partial}{\\partial a^{\\dagger}} \\varphi{(a^{\\dagger},y^{\\prime})} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} - y^{\\prime}) and \\frac{\\partial}{\\partial a^{\\dagger}} \\varphi{(a^{\\dagger},y^{\\prime})} = 1 and \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} - y^{\\prime}) = 1 and y^{\\prime} \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} - y^{\\prime}) = y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Symbol('y^{\\\\prime}', commutative=True))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(c_{0},s)} = \\sin^{c_{0}}{(s)} and \\mathbf{A}{(s)} = \\frac{d}{d c_{0}} \\int 0 ds, then obtain \\mathbf{A}{(s)} = \\frac{\\partial}{\\partial c_{0}} \\int (\\operatorname{E_{x}}{(c_{0},s)} - \\sin^{c_{0}}{(s)}) ds", "derivation": "\\operatorname{E_{x}}{(c_{0},s)} = \\sin^{c_{0}}{(s)} and \\operatorname{E_{x}}{(c_{0},s)} - \\sin^{c_{0}}{(s)} = 0 and \\int (\\operatorname{E_{x}}{(c_{0},s)} - \\sin^{c_{0}}{(s)}) ds = \\int 0 ds and \\frac{\\partial}{\\partial c_{0}} \\int (\\operatorname{E_{x}}{(c_{0},s)} - \\sin^{c_{0}}{(s)}) ds = \\frac{d}{d c_{0}} \\int 0 ds and \\mathbf{A}{(s)} = \\frac{d}{d c_{0}} \\int 0 ds and \\mathbf{A}{(s)} = \\frac{\\partial}{\\partial c_{0}} \\int (\\operatorname{E_{x}}{(c_{0},s)} - \\sin^{c_{0}}{(s)}) ds", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True)))"], [["minus", 1, "Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True)))), Tuple(Symbol('s', commutative=True))), Integral(Integer(0), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Integral(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True)))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), Derivative(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mathbf{A}')(Symbol('s', commutative=True)), Derivative(Integral(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('s', commutative=True)), Symbol('c_0', commutative=True)))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{x})} = e^{v_{x}} and C{(v_{x})} = e^{v_{x}}, then obtain 0 = v_{x} (C{(v_{x})} - \\operatorname{F_{c}}{(v_{x})})^{2}", "derivation": "\\operatorname{F_{c}}{(v_{x})} = e^{v_{x}} and 0 = - \\operatorname{F_{c}}{(v_{x})} + e^{v_{x}} and C{(v_{x})} = e^{v_{x}} and 0 = C{(v_{x})} - \\operatorname{F_{c}}{(v_{x})} and 0 = (C{(v_{x})} - \\operatorname{F_{c}}{(v_{x})})^{2} and 0 = v_{x} (C{(v_{x})} - \\operatorname{F_{c}}{(v_{x})})^{2}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["minus", 1, "Function('F_c')(Symbol('v_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('C')(Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True)))))"], [["times", 4, "Add(Function('C')(Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True))))"], "Equality(Integer(0), Pow(Add(Function('C')(Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True)))), Integer(2)))"], [["times", 5, "Symbol('v_x', commutative=True)"], "Equality(Integer(0), Mul(Symbol('v_x', commutative=True), Pow(Add(Function('C')(Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\dot{x},v_{z})} = \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}}, then obtain \\frac{\\partial^{2}}{\\partial \\dot{x}^{2}} \\operatorname{f^{*}}{(\\dot{x},v_{z})} - \\int \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} d\\dot{x} = \\frac{\\partial^{3}}{\\partial \\dot{x}^{2}\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} - \\int \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} d\\dot{x}", "derivation": "\\operatorname{f^{*}}{(\\dot{x},v_{z})} = \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{f^{*}}{(\\dot{x},v_{z})} = \\frac{\\partial^{2}}{\\partial \\dot{x}\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} and \\frac{\\partial^{2}}{\\partial \\dot{x}^{2}} \\operatorname{f^{*}}{(\\dot{x},v_{z})} = \\frac{\\partial^{3}}{\\partial \\dot{x}^{2}\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} and \\frac{\\partial^{2}}{\\partial \\dot{x}^{2}} \\operatorname{f^{*}}{(\\dot{x},v_{z})} - \\int \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} d\\dot{x} = \\frac{\\partial^{3}}{\\partial \\dot{x}^{2}\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} - \\int \\frac{\\partial}{\\partial v_{z}} \\frac{v_{z}}{\\dot{x}} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))))"], [["minus", 3, "Integral(Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(Derivative(Function('f^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))), Mul(Integer(-1), Integral(Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{x}', commutative=True))))), Add(Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(2))), Mul(Integer(-1), Integral(Derivative(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{x}', commutative=True))))))"]]}, {"prompt": "Given z{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and i{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then obtain \\hat{\\mathbf{x}} + i{(\\hat{\\mathbf{x}})} - \\sin{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}", "derivation": "z{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\hat{\\mathbf{x}} + z{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + \\sin{(\\hat{\\mathbf{x}})} and 2 \\hat{\\mathbf{x}} + z{(\\hat{\\mathbf{x}})} = 2 \\hat{\\mathbf{x}} + \\sin{(\\hat{\\mathbf{x}})} and \\hat{\\mathbf{x}} + z{(\\hat{\\mathbf{x}})} - \\sin{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} and i{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and z{(\\hat{\\mathbf{x}})} = i{(\\hat{\\mathbf{x}})} and \\hat{\\mathbf{x}} + i{(\\hat{\\mathbf{x}})} - \\sin{(\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('z')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('z')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('z')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('z')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('i')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('i')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"]]}, {"prompt": "Given \\chi{(F_{H})} = e^{F_{H}}, then derive \\frac{d}{d F_{H}} \\chi{(F_{H})} = e^{F_{H}}, then obtain \\frac{d}{d F_{H}} \\chi{(F_{H})} = \\chi{(F_{H})}", "derivation": "\\chi{(F_{H})} = e^{F_{H}} and \\frac{d}{d F_{H}} \\chi{(F_{H})} = \\frac{d}{d F_{H}} e^{F_{H}} and \\frac{d}{d F_{H}} \\chi{(F_{H})} = e^{F_{H}} and \\frac{d}{d F_{H}} e^{F_{H}} = e^{F_{H}} and \\frac{d}{d F_{H}} \\chi{(F_{H})} = \\chi{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), exp(Symbol('F_H', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), exp(Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\chi')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Function('\\\\chi')(Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} = \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})}, then obtain \\frac{\\int 2 \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} da^{\\dagger}}{\\int (\\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} + \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})}) da^{\\dagger}} = 1", "derivation": "\\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} = \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})} and 2 \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} = \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} + \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})} and \\int 2 \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} da^{\\dagger} = \\int (\\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} + \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})}) da^{\\dagger} and \\frac{\\int 2 \\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} da^{\\dagger}}{\\int (\\operatorname{F_{N}}{(a^{\\dagger},\\mathbf{J}_P)} + \\sin{(\\frac{a^{\\dagger}}{\\mathbf{J}_P})}) da^{\\dagger}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Integer(2), Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Add(Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 3, "Integral(Add(Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1)), Integral(Mul(Integer(2), Function('F_N')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(y,\\ddot{x})} = \\log{(\\ddot{x})}^{y}, then obtain \\log{(\\ddot{x})}^{y} \\int \\operatorname{v_{t}}{(y,\\ddot{x})} d\\ddot{x} = (G + (- \\log{(\\ddot{x})})^{- y} \\log{(\\ddot{x})}^{y} \\Gamma(y + 1, - \\log{(\\ddot{x})})) \\log{(\\ddot{x})}^{y}", "derivation": "\\operatorname{v_{t}}{(y,\\ddot{x})} = \\log{(\\ddot{x})}^{y} and \\int \\operatorname{v_{t}}{(y,\\ddot{x})} d\\ddot{x} = \\int \\log{(\\ddot{x})}^{y} d\\ddot{x} and \\log{(\\ddot{x})}^{y} \\int \\operatorname{v_{t}}{(y,\\ddot{x})} d\\ddot{x} = \\log{(\\ddot{x})}^{y} \\int \\log{(\\ddot{x})}^{y} d\\ddot{x} and \\log{(\\ddot{x})}^{y} \\int \\operatorname{v_{t}}{(y,\\ddot{x})} d\\ddot{x} = (G + (- \\log{(\\ddot{x})})^{- y} \\log{(\\ddot{x})}^{y} \\Gamma(y + 1, - \\log{(\\ddot{x})})) \\log{(\\ddot{x})}^{y}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('y', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('y', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["times", 2, "Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True))"], "Equality(Mul(Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), Integral(Function('v_t')(Symbol('y', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Mul(Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), Integral(Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), Integral(Function('v_t')(Symbol('y', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Mul(Add(Symbol('G', commutative=True), Mul(Pow(Mul(Integer(-1), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))), Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True)), uppergamma(Add(Symbol('y', commutative=True), Integer(1)), Mul(Integer(-1), log(Symbol('\\\\ddot{x}', commutative=True)))))), Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given Q{(\\varepsilon,\\dot{y})} = \\sin{(\\dot{y} - \\varepsilon)}, then obtain \\frac{\\partial}{\\partial \\varepsilon} (\\frac{\\partial}{\\partial \\varepsilon} Q{(\\varepsilon,\\dot{y})} - \\frac{\\partial}{\\partial \\varepsilon} \\sin{(\\dot{y} - \\varepsilon)}) = \\frac{d}{d \\varepsilon} 0", "derivation": "Q{(\\varepsilon,\\dot{y})} = \\sin{(\\dot{y} - \\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} Q{(\\varepsilon,\\dot{y})} = \\frac{\\partial}{\\partial \\varepsilon} \\sin{(\\dot{y} - \\varepsilon)} and \\frac{\\partial}{\\partial \\varepsilon} Q{(\\varepsilon,\\dot{y})} - \\frac{\\partial}{\\partial \\varepsilon} \\sin{(\\dot{y} - \\varepsilon)} = 0 and \\frac{\\partial}{\\partial \\varepsilon} (\\frac{\\partial}{\\partial \\varepsilon} Q{(\\varepsilon,\\dot{y})} - \\frac{\\partial}{\\partial \\varepsilon} \\sin{(\\dot{y} - \\varepsilon)}) = \\frac{d}{d \\varepsilon} 0", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(sin(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('Q')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('Q')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)}, then derive - \\eta + H{(\\eta)} = - \\eta - \\sin{(\\eta)}, then obtain \\frac{- \\frac{\\eta^{2}}{2} + g^{\\prime}_{\\varepsilon} + \\cos{(\\eta)}}{\\eta} = \\frac{\\int (- \\eta - \\sin{(\\eta)}) d\\eta}{\\eta}", "derivation": "H{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)} and - \\eta + H{(\\eta)} = - \\eta + \\frac{d}{d \\eta} \\cos{(\\eta)} and - \\eta + H{(\\eta)} = - \\eta - \\sin{(\\eta)} and - \\eta + \\frac{d}{d \\eta} \\cos{(\\eta)} = - \\eta - \\sin{(\\eta)} and \\int (- \\eta + \\frac{d}{d \\eta} \\cos{(\\eta)}) d\\eta = \\int (- \\eta - \\sin{(\\eta)}) d\\eta and \\frac{\\int (- \\eta + \\frac{d}{d \\eta} \\cos{(\\eta)}) d\\eta}{\\eta} = \\frac{\\int (- \\eta - \\sin{(\\eta)}) d\\eta}{\\eta} and \\frac{- \\frac{\\eta^{2}}{2} + g^{\\prime}_{\\varepsilon} + \\cos{(\\eta)}}{\\eta} = \\frac{\\int (- \\eta - \\sin{(\\eta)}) d\\eta}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('H')(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('H')(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["divide", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), cos(Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(A)} = \\log{(A)}, then obtain (\\operatorname{r_{0}}{(A)} + 2 \\log{(A)} - \\int 2 \\log{(A)} dA) \\int 2 \\log{(A)} dA = (3 \\log{(A)} - \\int 2 \\log{(A)} dA) \\int 2 \\log{(A)} dA", "derivation": "\\operatorname{r_{0}}{(A)} = \\log{(A)} and \\operatorname{r_{0}}{(A)} + \\log{(A)} = 2 \\log{(A)} and \\int (\\operatorname{r_{0}}{(A)} + \\log{(A)}) dA = \\int 2 \\log{(A)} dA and \\operatorname{r_{0}}{(A)} + 2 \\log{(A)} = 3 \\log{(A)} and \\operatorname{r_{0}}{(A)} + 2 \\log{(A)} - \\int (\\operatorname{r_{0}}{(A)} + \\log{(A)}) dA = 3 \\log{(A)} - \\int (\\operatorname{r_{0}}{(A)} + \\log{(A)}) dA and \\operatorname{r_{0}}{(A)} + 2 \\log{(A)} - \\int 2 \\log{(A)} dA = 3 \\log{(A)} - \\int 2 \\log{(A)} dA and (\\operatorname{r_{0}}{(A)} + 2 \\log{(A)} - \\int 2 \\log{(A)} dA) \\int 2 \\log{(A)} dA = (3 \\log{(A)} - \\int 2 \\log{(A)} dA) \\int 2 \\log{(A)} dA", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["add", 1, "log(Symbol('A', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True))), Mul(Integer(2), log(Symbol('A', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["add", 2, "log(Symbol('A', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('A', commutative=True)), Mul(Integer(2), log(Symbol('A', commutative=True)))), Mul(Integer(3), log(Symbol('A', commutative=True))))"], [["minus", 4, "Integral(Add(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Function('r_0')(Symbol('A', commutative=True)), Mul(Integer(2), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Add(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))), Add(Mul(Integer(3), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Add(Function('r_0')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('r_0')(Symbol('A', commutative=True)), Mul(Integer(2), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))), Add(Mul(Integer(3), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))))"], [["times", 6, "Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))"], "Equality(Mul(Add(Function('r_0')(Symbol('A', commutative=True)), Mul(Integer(2), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), Mul(Add(Mul(Integer(3), log(Symbol('A', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))), Integral(Mul(Integer(2), log(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(l)} = \\log{(e^{l})} and \\operatorname{A_{x}}{(l)} = - \\log{(e^{l})}, then obtain - \\frac{\\operatorname{n_{1}}{(l)}}{\\log{(e^{l})}} = -1", "derivation": "\\operatorname{n_{1}}{(l)} = \\log{(e^{l})} and \\operatorname{A_{x}}{(l)} = - \\log{(e^{l})} and \\operatorname{A_{x}}{(l)} = - \\operatorname{n_{1}}{(l)} and \\frac{\\operatorname{A_{x}}{(l)}}{\\log{(e^{l})}} = -1 and - \\frac{\\operatorname{n_{1}}{(l)}}{\\log{(e^{l})}} = -1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('l', commutative=True)), log(exp(Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('l', commutative=True)), Mul(Integer(-1), log(exp(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_x')(Symbol('l', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('l', commutative=True))))"], [["divide", 2, "log(exp(Symbol('l', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('l', commutative=True)), Pow(log(exp(Symbol('l', commutative=True))), Integer(-1))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Function('n_1')(Symbol('l', commutative=True)), Pow(log(exp(Symbol('l', commutative=True))), Integer(-1))), Integer(-1))"]]}, {"prompt": "Given \\mathbf{s}{(\\varphi)} = e^{\\varphi}, then derive \\frac{d}{d \\varphi} \\mathbf{s}{(\\varphi)} = e^{\\varphi}, then obtain \\frac{d^{2}}{d \\varphi^{2}} e^{\\varphi} = \\frac{d}{d \\varphi} e^{\\varphi}", "derivation": "\\mathbf{s}{(\\varphi)} = e^{\\varphi} and \\frac{d}{d \\varphi} \\mathbf{s}{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and \\frac{d}{d \\varphi} \\mathbf{s}{(\\varphi)} = e^{\\varphi} and \\frac{d^{2}}{d \\varphi^{2}} \\mathbf{s}{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and \\frac{d^{2}}{d \\varphi^{2}} \\mathbf{s}{(\\varphi)} = \\frac{d}{d \\varphi} \\mathbf{s}{(\\varphi)} and \\frac{d^{2}}{d \\varphi^{2}} e^{\\varphi} = \\frac{d}{d \\varphi} e^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), exp(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(2))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)}, then derive \\frac{d}{d \\mathbf{J}_P} \\operatorname{A_{1}}{(\\mathbf{J}_P)} = - \\sin{(\\mathbf{J}_P)}, then obtain - \\sin{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{A_{1}}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\operatorname{A_{1}}{(\\mathbf{J}_P)} = - \\sin{(\\mathbf{J}_P)} and - \\sin{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\cos{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{J})} = \\cos{(\\sin{(\\mathbf{J})})}, then obtain \\hat{H}_{\\lambda}{(\\mathbf{J})} + \\int \\cos{(\\sin{(\\mathbf{J})})} d\\mathbf{J} = \\cos{(\\sin{(\\mathbf{J})})} + \\int \\cos{(\\sin{(\\mathbf{J})})} d\\mathbf{J}", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{J})} = \\cos{(\\sin{(\\mathbf{J})})} and \\int \\hat{H}_{\\lambda}{(\\mathbf{J})} d\\mathbf{J} = \\int \\cos{(\\sin{(\\mathbf{J})})} d\\mathbf{J} and \\hat{H}_{\\lambda}{(\\mathbf{J})} + \\int \\hat{H}_{\\lambda}{(\\mathbf{J})} d\\mathbf{J} = \\cos{(\\sin{(\\mathbf{J})})} + \\int \\hat{H}_{\\lambda}{(\\mathbf{J})} d\\mathbf{J} and \\hat{H}_{\\lambda}{(\\mathbf{J})} + \\int \\cos{(\\sin{(\\mathbf{J})})} d\\mathbf{J} = \\cos{(\\sin{(\\mathbf{J})})} + \\int \\cos{(\\sin{(\\mathbf{J})})} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Add(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Add(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = (e^{\\hat{\\mathbf{r}}})^{\\Psi_{nl}}, then derive \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = \\Psi_{nl} (e^{\\hat{\\mathbf{r}}})^{\\Psi_{nl}}, then obtain \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = \\Psi_{nl} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})}", "derivation": "\\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = (e^{\\hat{\\mathbf{r}}})^{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (e^{\\hat{\\mathbf{r}}})^{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = \\Psi_{nl} (e^{\\hat{\\mathbf{r}}})^{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})} = \\Psi_{nl} \\mathbf{r}{(\\hat{\\mathbf{r}},\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given I{(t_{2},P_{e},J_{\\varepsilon})} = P_{e} (J_{\\varepsilon} + t_{2}), then obtain \\log{(P_{e} + \\frac{\\partial}{\\partial t_{2}} I{(t_{2},P_{e},J_{\\varepsilon})})} = \\log{(P_{e} + \\frac{\\partial}{\\partial t_{2}} P_{e} (J_{\\varepsilon} + t_{2}))}", "derivation": "I{(t_{2},P_{e},J_{\\varepsilon})} = P_{e} (J_{\\varepsilon} + t_{2}) and \\frac{\\partial}{\\partial t_{2}} I{(t_{2},P_{e},J_{\\varepsilon})} = \\frac{\\partial}{\\partial t_{2}} P_{e} (J_{\\varepsilon} + t_{2}) and P_{e} + \\frac{\\partial}{\\partial t_{2}} I{(t_{2},P_{e},J_{\\varepsilon})} = P_{e} + \\frac{\\partial}{\\partial t_{2}} P_{e} (J_{\\varepsilon} + t_{2}) and \\log{(P_{e} + \\frac{\\partial}{\\partial t_{2}} I{(t_{2},P_{e},J_{\\varepsilon})})} = \\log{(P_{e} + \\frac{\\partial}{\\partial t_{2}} P_{e} (J_{\\varepsilon} + t_{2}))}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('P_e', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["add", 2, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Symbol('P_e', commutative=True), Derivative(Mul(Symbol('P_e', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["log", 3], "Equality(log(Add(Symbol('P_e', commutative=True), Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))), log(Add(Symbol('P_e', commutative=True), Derivative(Mul(Symbol('P_e', commutative=True), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\sigma_{x}{(\\varphi)} = \\log{(\\sin{(\\varphi)})}, then obtain e^{\\sin{(2 \\sigma_{x}{(\\varphi)})}} \\sin{(\\varphi)} = e^{\\sin{(\\sigma_{x}{(\\varphi)} + \\log{(\\sin{(\\varphi)})})}} \\sin{(\\varphi)}", "derivation": "\\sigma_{x}{(\\varphi)} = \\log{(\\sin{(\\varphi)})} and 2 \\sigma_{x}{(\\varphi)} = \\sigma_{x}{(\\varphi)} + \\log{(\\sin{(\\varphi)})} and \\sin{(2 \\sigma_{x}{(\\varphi)})} = \\sin{(\\sigma_{x}{(\\varphi)} + \\log{(\\sin{(\\varphi)})})} and e^{\\sin{(2 \\sigma_{x}{(\\varphi)})}} = e^{\\sin{(\\sigma_{x}{(\\varphi)} + \\log{(\\sin{(\\varphi)})})}} and e^{\\sin{(2 \\sigma_{x}{(\\varphi)})}} \\sin{(\\varphi)} = e^{\\sin{(\\sigma_{x}{(\\varphi)} + \\log{(\\sin{(\\varphi)})})}} \\sin{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True))), Add(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)))), sin(Add(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True))))))"], [["exp", 3], "Equality(exp(sin(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True))))), exp(sin(Add(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True)))))))"], [["times", 4, "sin(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(exp(sin(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True))))), sin(Symbol('\\\\varphi', commutative=True))), Mul(exp(sin(Add(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True)))))), sin(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given S{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)}, then derive S{(\\mathbf{J}_M)} - 1 = \\cos{(\\mathbf{J}_M)} - 1, then obtain \\cos{(\\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} - 1)} = \\cos{(\\cos{(\\mathbf{J}_M)} - 1)}", "derivation": "S{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and S{(\\mathbf{J}_M)} - 1 = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} - 1 and S{(\\mathbf{J}_M)} - 1 = \\cos{(\\mathbf{J}_M)} - 1 and \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} - 1 = \\cos{(\\mathbf{J}_M)} - 1 and \\cos{(\\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} - 1)} = \\cos{(\\cos{(\\mathbf{J}_M)} - 1)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('S')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('S')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))"], [["cos", 4], "Equality(cos(Add(Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))), cos(Add(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(a,i)} = i^{a}, then obtain (\\int \\frac{\\partial}{\\partial i} \\dot{y}^{i}{(a,i)} da)^{a} = (\\int \\frac{\\partial}{\\partial i} (i^{a})^{i} da)^{a}", "derivation": "\\dot{y}{(a,i)} = i^{a} and \\dot{y}^{i}{(a,i)} = (i^{a})^{i} and \\frac{\\partial}{\\partial i} \\dot{y}^{i}{(a,i)} = \\frac{\\partial}{\\partial i} (i^{a})^{i} and \\int \\frac{\\partial}{\\partial i} \\dot{y}^{i}{(a,i)} da = \\int \\frac{\\partial}{\\partial i} (i^{a})^{i} da and (\\int \\frac{\\partial}{\\partial i} \\dot{y}^{i}{(a,i)} da)^{a} = (\\int \\frac{\\partial}{\\partial i} (i^{a})^{i} da)^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('a', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(Pow(Symbol('i', commutative=True), Symbol('a', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\dot{y}')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('a', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\dot{y}')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Integral(Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('a', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(Derivative(Pow(Function('\\\\dot{y}')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(Derivative(Pow(Pow(Symbol('i', commutative=True), Symbol('a', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(L)} = \\cos{(L)} and f{(L)} = - \\cos{(L)}, then obtain \\frac{d}{d L} - \\mathbf{s}{(L)} \\cos{(L)} = \\frac{d}{d L} f{(L)} \\cos{(L)}", "derivation": "\\mathbf{s}{(L)} = \\cos{(L)} and f{(L)} = - \\cos{(L)} and f{(L)} \\cos{(L)} = - \\cos^{2}{(L)} and \\frac{d}{d L} f{(L)} \\cos{(L)} = \\frac{d}{d L} - \\cos^{2}{(L)} and f{(L)} = - \\mathbf{s}{(L)} and \\frac{d}{d L} - \\mathbf{s}{(L)} \\cos{(L)} = \\frac{d}{d L} - \\cos^{2}{(L)} and \\frac{d}{d L} - \\mathbf{s}{(L)} \\cos{(L)} = \\frac{d}{d L} f{(L)} \\cos{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('L', commutative=True))))"], [["times", 2, "cos(Symbol('L', commutative=True))"], "Equality(Mul(Function('f')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('L', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Function('f')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(cos(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f')(Symbol('L', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(cos(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Function('f')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(z)} = \\sin{(z)}, then obtain - 5 \\operatorname{r_{0}}{(z)} + 6 \\sin{(z)} = - 8 \\operatorname{r_{0}}{(z)} + 9 \\sin{(z)}", "derivation": "\\operatorname{r_{0}}{(z)} = \\sin{(z)} and \\sin{(z)} = - \\operatorname{r_{0}}{(z)} + 2 \\sin{(z)} and \\sin{(z)} + 1 = - \\operatorname{r_{0}}{(z)} + 2 \\sin{(z)} + 1 and 0 = - \\operatorname{r_{0}}{(z)} + \\sin{(z)} and - \\operatorname{r_{0}}{(z)} + 2 \\sin{(z)} = - 2 \\operatorname{r_{0}}{(z)} + 3 \\sin{(z)} and \\sin{(z)} = - 2 \\operatorname{r_{0}}{(z)} + 3 \\sin{(z)} and - 5 \\operatorname{r_{0}}{(z)} + 6 \\sin{(z)} = - 8 \\operatorname{r_{0}}{(z)} + 9 \\sin{(z)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["minus", 1, "Add(Function('r_0')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))"], "Equality(sin(Symbol('z', commutative=True)), Add(Mul(Integer(-1), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(2), sin(Symbol('z', commutative=True)))))"], [["add", 2, 1], "Equality(Add(sin(Symbol('z', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(2), sin(Symbol('z', commutative=True))), Integer(1)))"], [["minus", 3, "Add(sin(Symbol('z', commutative=True)), Integer(1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('r_0')(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(2), sin(Symbol('z', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(2), sin(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(3), sin(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(sin(Symbol('z', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(3), sin(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Integer(5), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(6), sin(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Integer(8), Function('r_0')(Symbol('z', commutative=True))), Mul(Integer(9), sin(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given y{(\\hat{\\mathbf{x}},Q)} = \\hat{\\mathbf{x}} \\log{(Q)}, then obtain \\frac{(\\hat{\\mathbf{x}} \\log{(Q)})^{- Q} (- (\\hat{\\mathbf{x}} \\log{(Q)})^{Q} + y^{Q}{(\\hat{\\mathbf{x}},Q)})}{\\hat{\\mathbf{x}}} = 0", "derivation": "y{(\\hat{\\mathbf{x}},Q)} = \\hat{\\mathbf{x}} \\log{(Q)} and y^{Q}{(\\hat{\\mathbf{x}},Q)} = (\\hat{\\mathbf{x}} \\log{(Q)})^{Q} and - (\\hat{\\mathbf{x}} \\log{(Q)})^{Q} + y^{Q}{(\\hat{\\mathbf{x}},Q)} = 0 and (\\hat{\\mathbf{x}} \\log{(Q)})^{- Q} (- (\\hat{\\mathbf{x}} \\log{(Q)})^{Q} + y^{Q}{(\\hat{\\mathbf{x}},Q)}) = 0 and \\frac{(\\hat{\\mathbf{x}} \\log{(Q)})^{- Q} (- (\\hat{\\mathbf{x}} \\log{(Q)})^{Q} + y^{Q}{(\\hat{\\mathbf{x}},Q)})}{\\hat{\\mathbf{x}}} = 0", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["minus", 2, "Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Pow(Function('y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Integer(0))"], [["divide", 3, "Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Pow(Function('y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))), Integer(0))"], [["divide", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Pow(Function('y')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\theta{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(\\hat{x})}, then derive (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})^{2} = \\frac{2 (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})}{\\hat{x}}, then obtain (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})^{2} - \\theta{(\\hat{x})} = - \\theta{(\\hat{x})} + \\frac{2 (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})}{\\hat{x}}", "derivation": "\\theta{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} and \\theta{(\\hat{x})} + \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} = 2 \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} and (\\theta{(\\hat{x})} + \\frac{d}{d \\hat{x}} \\log{(\\hat{x})})^{2} = 2 (\\theta{(\\hat{x})} + \\frac{d}{d \\hat{x}} \\log{(\\hat{x})}) \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} and (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})^{2} = \\frac{2 (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})}{\\hat{x}} and (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})^{2} - \\theta{(\\hat{x})} = - \\theta{(\\hat{x})} + \\frac{2 (\\theta{(\\hat{x})} + \\frac{1}{\\hat{x}})}{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["times", 2, "Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], "Equality(Pow(Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(2), Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))), Integer(2)), Mul(Integer(2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))))"], [["minus", 4, "Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Pow(Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))), Integer(2)), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('\\\\hat{x}', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(s,v_{1})} = \\frac{v_{1}}{s}, then obtain \\operatorname{v_{x}}{(s,v_{1})} = - \\operatorname{v_{x}}{(s,v_{1})} + \\frac{2 v_{1}}{s}", "derivation": "\\operatorname{v_{x}}{(s,v_{1})} = \\frac{v_{1}}{s} and s + \\operatorname{v_{x}}{(s,v_{1})} = s + \\frac{v_{1}}{s} and \\frac{v_{1}}{s} = - \\operatorname{v_{x}}{(s,v_{1})} + \\frac{2 v_{1}}{s} and \\operatorname{v_{x}}{(s,v_{1})} = - \\operatorname{v_{x}}{(s,v_{1})} + \\frac{2 v_{1}}{s}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Add(Symbol('s', commutative=True), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(2), Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Function('v_x')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(2), Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\lambda,C_{d},L)} = \\lambda + \\frac{L}{C_{d}}, then derive \\frac{\\partial}{\\partial L} \\operatorname{A_{x}}{(\\lambda,C_{d},L)} = \\frac{1}{C_{d}}, then obtain 0^{\\lambda} = (\\frac{\\partial}{\\partial L} (\\lambda + \\frac{L}{C_{d}}) - \\frac{1}{C_{d}})^{\\lambda}", "derivation": "\\operatorname{A_{x}}{(\\lambda,C_{d},L)} = \\lambda + \\frac{L}{C_{d}} and \\frac{\\partial}{\\partial L} \\operatorname{A_{x}}{(\\lambda,C_{d},L)} = \\frac{\\partial}{\\partial L} (\\lambda + \\frac{L}{C_{d}}) and \\frac{\\partial}{\\partial L} \\operatorname{A_{x}}{(\\lambda,C_{d},L)} = \\frac{1}{C_{d}} and \\frac{\\partial}{\\partial L} \\operatorname{A_{x}}{(\\lambda,C_{d},L)} - \\frac{1}{C_{d}} = \\frac{\\partial}{\\partial L} (\\lambda + \\frac{L}{C_{d}}) - \\frac{1}{C_{d}} and (\\frac{\\partial}{\\partial L} \\operatorname{A_{x}}{(\\lambda,C_{d},L)} - \\frac{1}{C_{d}})^{\\lambda} = (\\frac{\\partial}{\\partial L} (\\lambda + \\frac{L}{C_{d}}) - \\frac{1}{C_{d}})^{\\lambda} and 0^{\\lambda} = (\\frac{\\partial}{\\partial L} (\\lambda + \\frac{L}{C_{d}}) - \\frac{1}{C_{d}})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\lambda', commutative=True), Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\lambda', commutative=True), Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('\\\\lambda', commutative=True), Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Pow(Symbol('C_d', commutative=True), Integer(-1)))"], [["minus", 2, "Pow(Symbol('C_d', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('A_x')(Symbol('\\\\lambda', commutative=True), Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Add(Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Add(Derivative(Function('A_x')(Symbol('\\\\lambda', commutative=True), Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Symbol('\\\\lambda', commutative=True)), Pow(Add(Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Integer(0), Symbol('\\\\lambda', commutative=True)), Pow(Add(Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(E,\\Psi_{\\lambda})} = \\log{(E \\Psi_{\\lambda})}, then obtain \\frac{\\partial}{\\partial E} (1 + \\frac{\\operatorname{F_{g}}{(E,\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}) = \\frac{\\partial}{\\partial E} (1 + \\frac{\\log{(E \\Psi_{\\lambda})}}{\\Psi_{\\lambda}})", "derivation": "\\operatorname{F_{g}}{(E,\\Psi_{\\lambda})} = \\log{(E \\Psi_{\\lambda})} and \\frac{\\operatorname{F_{g}}{(E,\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} = \\frac{\\log{(E \\Psi_{\\lambda})}}{\\Psi_{\\lambda}} and 1 + \\frac{\\operatorname{F_{g}}{(E,\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} = 1 + \\frac{\\log{(E \\Psi_{\\lambda})}}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial E} (1 + \\frac{\\operatorname{F_{g}}{(E,\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}) = \\frac{\\partial}{\\partial E} (1 + \\frac{\\log{(E \\Psi_{\\lambda})}}{\\Psi_{\\lambda}})", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('F_g')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('F_g')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))))"], [["differentiate", 3, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('F_g')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(E)} = \\cos{(E)}, then obtain \\frac{- E \\mathbb{I}{(E)} + \\mathbb{I}{(E)}}{\\mathbb{I}{(E)}} = \\frac{- E \\mathbb{I}{(E)} + \\cos{(E)}}{\\mathbb{I}{(E)}}", "derivation": "\\mathbb{I}{(E)} = \\cos{(E)} and E \\mathbb{I}{(E)} = E \\cos{(E)} and - E \\cos{(E)} + \\mathbb{I}{(E)} = - E \\cos{(E)} + \\cos{(E)} and - E \\mathbb{I}{(E)} + \\mathbb{I}{(E)} = - E \\mathbb{I}{(E)} + \\cos{(E)} and \\frac{- E \\mathbb{I}{(E)} + \\mathbb{I}{(E)}}{\\mathbb{I}{(E)}} = \\frac{- E \\mathbb{I}{(E)} + \\cos{(E)}}{\\mathbb{I}{(E)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["times", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))))"], [["minus", 1, "Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))))"], [["divide", 4, "Function('\\\\mathbb{I}')(Symbol('E', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('E', commutative=True), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('E', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('E', commutative=True), Function('\\\\mathbb{I}')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given I{(E,G)} = e^{E^{G}} and \\operatorname{C_{2}}{(t_{1},T)} = T t_{1}, then obtain - (\\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}})^{t_{1}} + e^{E^{G}} + 1 = e^{E^{G}}", "derivation": "I{(E,G)} = e^{E^{G}} and \\operatorname{C_{2}}{(t_{1},T)} = T t_{1} and 1 = \\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}} and 1 = (\\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}})^{t_{1}} and 1 - (\\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}})^{t_{1}} = 0 and - (\\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}})^{t_{1}} + I{(E,G)} + 1 = I{(E,G)} and - (\\frac{T t_{1}}{\\operatorname{C_{2}}{(t_{1},T)}})^{t_{1}} + e^{E^{G}} + 1 = e^{E^{G}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('E', commutative=True), Symbol('G', commutative=True)), exp(Pow(Symbol('E', commutative=True), Symbol('G', commutative=True))))"], ["get_premise", "Equality(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True)))"], [["divide", 2, "Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True))"], "Equality(Integer(1), Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Symbol('t_1', commutative=True)))"], [["minus", 4, "Pow(Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Symbol('t_1', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Symbol('t_1', commutative=True)))), Integer(0))"], [["add", 5, "Function('I')(Symbol('E', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Symbol('t_1', commutative=True))), Function('I')(Symbol('E', commutative=True), Symbol('G', commutative=True)), Integer(1)), Function('I')(Symbol('E', commutative=True), Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('t_1', commutative=True), Pow(Function('C_2')(Symbol('t_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Symbol('t_1', commutative=True))), exp(Pow(Symbol('E', commutative=True), Symbol('G', commutative=True))), Integer(1)), exp(Pow(Symbol('E', commutative=True), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\psi{(\\theta,C)} = C + \\theta and \\mathbf{p}{(\\theta,C)} = C \\theta, then obtain \\frac{\\partial}{\\partial C} C \\theta \\int \\psi{(\\theta,C)} dC = \\frac{\\partial}{\\partial C} C \\theta \\int (C + \\theta) dC", "derivation": "\\psi{(\\theta,C)} = C + \\theta and \\int \\psi{(\\theta,C)} dC = \\int (C + \\theta) dC and C \\theta \\int \\psi{(\\theta,C)} dC = C \\theta \\int (C + \\theta) dC and \\mathbf{p}{(\\theta,C)} = C \\theta and \\mathbf{p}{(\\theta,C)} \\int \\psi{(\\theta,C)} dC = \\mathbf{p}{(\\theta,C)} \\int (C + \\theta) dC and \\frac{\\partial}{\\partial C} \\mathbf{p}{(\\theta,C)} \\int \\psi{(\\theta,C)} dC = \\frac{\\partial}{\\partial C} \\mathbf{p}{(\\theta,C)} \\int (C + \\theta) dC and \\frac{\\partial}{\\partial C} C \\theta \\int \\psi{(\\theta,C)} dC = \\frac{\\partial}{\\partial C} C \\theta \\int (C + \\theta) dC", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["times", 2, "Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True), Integral(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True), Integral(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Integral(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Function('\\\\mathbf{p}')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Integral(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["differentiate", 5, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Integral(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Integral(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True), Integral(Function('\\\\psi')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True), Integral(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(\\phi_1)} = \\sin{(\\phi_1)} and B{(\\phi_1)} = y{(\\phi_1)} + \\frac{d}{d \\phi_1} 0, then obtain B{(\\phi_1)} = y{(\\phi_1)} + \\frac{d}{d \\phi_1} (- y{(\\phi_1)} + \\sin{(\\phi_1)})", "derivation": "y{(\\phi_1)} = \\sin{(\\phi_1)} and 0 = - y{(\\phi_1)} + \\sin{(\\phi_1)} and \\frac{d}{d \\phi_1} 0 = \\frac{d}{d \\phi_1} (- y{(\\phi_1)} + \\sin{(\\phi_1)}) and y{(\\phi_1)} + \\frac{d}{d \\phi_1} 0 = y{(\\phi_1)} + \\frac{d}{d \\phi_1} (- y{(\\phi_1)} + \\sin{(\\phi_1)}) and B{(\\phi_1)} = y{(\\phi_1)} + \\frac{d}{d \\phi_1} 0 and B{(\\phi_1)} = y{(\\phi_1)} + \\frac{d}{d \\phi_1} (- y{(\\phi_1)} + \\sin{(\\phi_1)})", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "Function('y')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y')(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["add", 3, "Function('y')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Add(Function('y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\phi_1', commutative=True)), Add(Function('y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('B')(Symbol('\\\\phi_1', commutative=True)), Add(Function('y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\mathbf{M}{(\\hat{p}_0)} = \\int (\\hat{p}_0 + \\sin{(\\hat{p}_0)}) d\\hat{p}_0, then obtain \\int (\\hat{p}_0 + \\operatorname{F_{x}}{(\\hat{p}_0)}) d\\hat{p}_0 = \\mathbf{M}{(\\hat{p}_0)}", "derivation": "\\operatorname{F_{x}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\hat{p}_0 + \\operatorname{F_{x}}{(\\hat{p}_0)} = \\hat{p}_0 + \\sin{(\\hat{p}_0)} and \\int (\\hat{p}_0 + \\operatorname{F_{x}}{(\\hat{p}_0)}) d\\hat{p}_0 = \\int (\\hat{p}_0 + \\sin{(\\hat{p}_0)}) d\\hat{p}_0 and \\mathbf{M}{(\\hat{p}_0)} = \\int (\\hat{p}_0 + \\sin{(\\hat{p}_0)}) d\\hat{p}_0 and \\int (\\hat{p}_0 + \\operatorname{F_{x}}{(\\hat{p}_0)}) d\\hat{p}_0 = \\mathbf{M}{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('F_x')(Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('F_x')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{p}_0', commutative=True)), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('F_x')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Function('\\\\mathbf{M}')(Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given p{(A_{y})} = \\int \\cos{(A_{y})} dA_{y}, then obtain \\frac{d}{d A_{y}} p^{A_{y}}{(A_{y})} = \\frac{\\partial}{\\partial A_{y}} (\\chi + \\sin{(A_{y})})^{A_{y}}", "derivation": "p{(A_{y})} = \\int \\cos{(A_{y})} dA_{y} and p^{A_{y}}{(A_{y})} = (\\int \\cos{(A_{y})} dA_{y})^{A_{y}} and \\frac{d}{d A_{y}} p^{A_{y}}{(A_{y})} = \\frac{d}{d A_{y}} (\\int \\cos{(A_{y})} dA_{y})^{A_{y}} and \\frac{d}{d A_{y}} p^{A_{y}}{(A_{y})} = \\frac{\\partial}{\\partial A_{y}} (\\chi + \\sin{(A_{y})})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('A_y', commutative=True)), Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('p')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Pow(Function('p')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Pow(Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Pow(Function('p')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\chi', commutative=True), sin(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)} = - \\mu_0 + k^{\\varphi}, then obtain 4 \\hat{\\mathbf{x}}^{2}{(k,\\mu_0,\\varphi)} = 2 (- 2 \\mu_0 + 2 k^{\\varphi}) \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)}", "derivation": "\\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)} = - \\mu_0 + k^{\\varphi} and 2 \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)} = - \\mu_0 + k^{\\varphi} + \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)} and 4 \\hat{\\mathbf{x}}^{2}{(k,\\mu_0,\\varphi)} = 2 (- \\mu_0 + k^{\\varphi} + \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)}) \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)} and 4 (- \\mu_0 + k^{\\varphi})^{2} = 2 (- 2 \\mu_0 + 2 k^{\\varphi}) (- \\mu_0 + k^{\\varphi}) and 4 \\hat{\\mathbf{x}}^{2}{(k,\\mu_0,\\varphi)} = 2 (- 2 \\mu_0 + 2 k^{\\varphi}) \\hat{\\mathbf{x}}{(k,\\mu_0,\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(2))), Mul(Integer(2), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(2))), Mul(Integer(2), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Pow(Symbol('k', commutative=True), Symbol('\\\\varphi', commutative=True)))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('k', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(E_{n},\\omega)} = \\cos{(E_{n} + \\omega)}, then obtain \\int (\\cos{(\\hat{x}_0{(E_{n},\\omega)} \\cos{(E_{n} + \\omega)})} + \\cos{(E_{n} + \\omega)}) dE_{n} = \\int (\\cos{(E_{n} + \\omega)} + \\cos{(\\cos^{2}{(E_{n} + \\omega)})}) dE_{n}", "derivation": "\\hat{x}_0{(E_{n},\\omega)} = \\cos{(E_{n} + \\omega)} and \\hat{x}_0{(E_{n},\\omega)} \\cos{(E_{n} + \\omega)} = \\cos^{2}{(E_{n} + \\omega)} and \\cos{(\\hat{x}_0{(E_{n},\\omega)} \\cos{(E_{n} + \\omega)})} = \\cos{(\\cos^{2}{(E_{n} + \\omega)})} and \\cos{(\\hat{x}_0{(E_{n},\\omega)} \\cos{(E_{n} + \\omega)})} + \\cos{(E_{n} + \\omega)} = \\cos{(E_{n} + \\omega)} + \\cos{(\\cos^{2}{(E_{n} + \\omega)})} and \\int (\\cos{(\\hat{x}_0{(E_{n},\\omega)} \\cos{(E_{n} + \\omega)})} + \\cos{(E_{n} + \\omega)}) dE_{n} = \\int (\\cos{(E_{n} + \\omega)} + \\cos{(\\cos^{2}{(E_{n} + \\omega)})}) dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["times", 1, "cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))), Pow(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2)))"], [["cos", 2], "Equality(cos(Mul(Function('\\\\hat{x}_0')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))))), cos(Pow(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2))))"], [["add", 3, "cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(cos(Mul(Function('\\\\hat{x}_0')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))))), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), cos(Pow(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2)))))"], [["integrate", 4, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(cos(Mul(Function('\\\\hat{x}_0')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))))), cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), cos(Pow(cos(Add(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(2)))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{p},x^\\prime)} = \\frac{\\hat{p}}{x^\\prime}, then obtain 0 = \\iint \\frac{\\hat{p}}{x^\\prime} d\\hat{p} dx^\\prime - \\iint \\operatorname{A_{y}}{(\\hat{p},x^\\prime)} d\\hat{p} dx^\\prime", "derivation": "\\operatorname{A_{y}}{(\\hat{p},x^\\prime)} = \\frac{\\hat{p}}{x^\\prime} and \\int \\operatorname{A_{y}}{(\\hat{p},x^\\prime)} d\\hat{p} = \\int \\frac{\\hat{p}}{x^\\prime} d\\hat{p} and \\iint \\operatorname{A_{y}}{(\\hat{p},x^\\prime)} d\\hat{p} dx^\\prime = \\iint \\frac{\\hat{p}}{x^\\prime} d\\hat{p} dx^\\prime and 0 = \\iint \\frac{\\hat{p}}{x^\\prime} d\\hat{p} dx^\\prime - \\iint \\operatorname{A_{y}}{(\\hat{p},x^\\prime)} d\\hat{p} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 3, "Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(x,x^\\prime)} = x x^\\prime, then derive \\int \\frac{\\partial}{\\partial x^\\prime} \\int \\theta_{1}{(x,x^\\prime)} dx^\\prime dx = \\ddot{x} + \\frac{x^{2} x^\\prime}{2}, then obtain \\int \\frac{\\partial}{\\partial x^\\prime} \\int x x^\\prime dx^\\prime dx = \\ddot{x} + \\frac{x^{2} x^\\prime}{2}", "derivation": "\\theta_{1}{(x,x^\\prime)} = x x^\\prime and \\int \\theta_{1}{(x,x^\\prime)} dx^\\prime = \\int x x^\\prime dx^\\prime and \\frac{\\partial}{\\partial x^\\prime} \\int \\theta_{1}{(x,x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} \\int x x^\\prime dx^\\prime and \\int \\frac{\\partial}{\\partial x^\\prime} \\int \\theta_{1}{(x,x^\\prime)} dx^\\prime dx = \\int \\frac{\\partial}{\\partial x^\\prime} \\int x x^\\prime dx^\\prime dx and \\int \\frac{\\partial}{\\partial x^\\prime} \\int \\theta_{1}{(x,x^\\prime)} dx^\\prime dx = \\ddot{x} + \\frac{x^{2} x^\\prime}{2} and \\int \\frac{\\partial}{\\partial x^\\prime} \\int x x^\\prime dx^\\prime dx = \\ddot{x} + \\frac{x^{2} x^\\prime}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\theta_1')(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\theta_1')(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Integral(Function('\\\\theta_1')(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Derivative(Integral(Mul(Symbol('x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(c_{0},\\mathbf{F})} = \\cos{(\\mathbf{F} - c_{0})}, then obtain \\iint - c_{0} \\log{(\\sigma_{x}{(c_{0},\\mathbf{F})})} d\\mathbf{F} d\\mathbf{F} = \\iint - c_{0} \\log{(\\cos{(\\mathbf{F} - c_{0})})} d\\mathbf{F} d\\mathbf{F}", "derivation": "\\sigma_{x}{(c_{0},\\mathbf{F})} = \\cos{(\\mathbf{F} - c_{0})} and \\log{(\\sigma_{x}{(c_{0},\\mathbf{F})})} = \\log{(\\cos{(\\mathbf{F} - c_{0})})} and - c_{0} \\log{(\\sigma_{x}{(c_{0},\\mathbf{F})})} = - c_{0} \\log{(\\cos{(\\mathbf{F} - c_{0})})} and \\int - c_{0} \\log{(\\sigma_{x}{(c_{0},\\mathbf{F})})} d\\mathbf{F} = \\int - c_{0} \\log{(\\cos{(\\mathbf{F} - c_{0})})} d\\mathbf{F} and \\iint - c_{0} \\log{(\\sigma_{x}{(c_{0},\\mathbf{F})})} d\\mathbf{F} d\\mathbf{F} = \\iint - c_{0} \\log{(\\cos{(\\mathbf{F} - c_{0})})} d\\mathbf{F} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))"], [["log", 1], "Equality(log(Function('\\\\sigma_x')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), log(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))))"], [["times", 2, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('c_0', commutative=True), log(Function('\\\\sigma_x')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(-1), Symbol('c_0', commutative=True), log(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), log(Function('\\\\sigma_x')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), log(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), log(Function('\\\\sigma_x')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Integer(-1), Symbol('c_0', commutative=True), log(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\lambda)} = \\sin{(\\lambda)}, then obtain \\cos{(\\hat{x}^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\hat{x}{(\\lambda)} = \\cos{(\\sin^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\hat{x}{(\\lambda)}", "derivation": "\\hat{x}{(\\lambda)} = \\sin{(\\lambda)} and \\frac{d}{d \\lambda} \\hat{x}{(\\lambda)} = \\frac{d}{d \\lambda} \\sin{(\\lambda)} and \\hat{x}^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} and \\cos{(\\hat{x}^{\\lambda}{(\\lambda)})} = \\cos{(\\sin^{\\lambda}{(\\lambda)})} and \\cos{(\\hat{x}^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\sin{(\\lambda)} = \\cos{(\\sin^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\sin{(\\lambda)} and \\cos{(\\hat{x}^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\hat{x}{(\\lambda)} = \\cos{(\\sin^{\\lambda}{(\\lambda)})} + \\frac{d}{d \\lambda} \\hat{x}{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), cos(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["add", 4, "Derivative(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(cos(Pow(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Derivative(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(cos(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Derivative(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(cos(Pow(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Derivative(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(cos(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Derivative(Function('\\\\hat{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} = \\cos{(\\frac{\\hat{p}_0}{r_{0}})}, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} (r_{0} \\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} + r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})}) = \\frac{\\partial}{\\partial \\hat{p}_0} 2 r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} = \\cos{(\\frac{\\hat{p}_0}{r_{0}})} and r_{0} \\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} = r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})} and r_{0} \\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} + r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})} = 2 r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})} and \\frac{\\partial}{\\partial \\hat{p}_0} (r_{0} \\operatorname{f_{\\mathbf{p}}}{(r_{0},\\hat{p}_0)} + r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})}) = \\frac{\\partial}{\\partial \\hat{p}_0} 2 r_{0} \\cos{(\\frac{\\hat{p}_0}{r_{0}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))"], [["divide", 1, "Pow(Symbol('r_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('r_0', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))))"], [["add", 2, "Mul(Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))"], "Equality(Add(Mul(Symbol('r_0', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))), Mul(Integer(2), Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))))"], [["differentiate", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('r_0', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('r_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('r_0', commutative=True), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{z},A_{1})} = A_{1} + e^{v_{z}}, then derive \\frac{\\partial}{\\partial A_{1}} \\mathbf{J}_P{(v_{z},A_{1})} = 1, then obtain (\\frac{\\partial}{\\partial A_{1}} \\mathbf{J}_P{(v_{z},A_{1})})^{v_{z}} = 1", "derivation": "\\mathbf{J}_P{(v_{z},A_{1})} = A_{1} + e^{v_{z}} and \\frac{\\partial}{\\partial A_{1}} \\mathbf{J}_P{(v_{z},A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} + e^{v_{z}}) and \\frac{\\partial}{\\partial A_{1}} \\mathbf{J}_P{(v_{z},A_{1})} = 1 and (\\frac{\\partial}{\\partial A_{1}} \\mathbf{J}_P{(v_{z},A_{1})})^{v_{z}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_z', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), exp(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('v_z', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('v_z', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('v_z', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('v_z', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Symbol('v_z', commutative=True)), Integer(1))"]]}, {"prompt": "Given b{(\\hbar)} = \\log{(\\log{(\\hbar)})}, then obtain b{(\\hbar)} + b^{\\hbar}{(\\hbar)} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar = b^{\\hbar}{(\\hbar)} + \\log{(\\log{(\\hbar)})} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar", "derivation": "b{(\\hbar)} = \\log{(\\log{(\\hbar)})} and b^{\\hbar}{(\\hbar)} = \\log{(\\log{(\\hbar)})}^{\\hbar} and b{(\\hbar)} + \\log{(\\log{(\\hbar)})}^{\\hbar} = \\log{(\\log{(\\hbar)})} + \\log{(\\log{(\\hbar)})}^{\\hbar} and b{(\\hbar)} + \\log{(\\log{(\\hbar)})}^{\\hbar} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar = \\log{(\\log{(\\hbar)})} + \\log{(\\log{(\\hbar)})}^{\\hbar} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar and b{(\\hbar)} + b^{\\hbar}{(\\hbar)} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar = b^{\\hbar}{(\\hbar)} + \\log{(\\log{(\\hbar)})} - \\int \\log{(\\log{(\\hbar)})}^{\\hbar} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\hbar', commutative=True)), log(log(Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('b')(Symbol('\\\\hbar', commutative=True)), Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Add(log(log(Symbol('\\\\hbar', commutative=True))), Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))))"], [["minus", 3, "Integral(Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('b')(Symbol('\\\\hbar', commutative=True)), Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Add(log(log(Symbol('\\\\hbar', commutative=True))), Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('b')(Symbol('\\\\hbar', commutative=True)), Pow(Function('b')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Add(Pow(Function('b')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), log(log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integral(Pow(log(log(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\chi)} = e^{\\chi}, then derive \\int \\frac{d}{d \\chi} 0 d\\chi = l + \\operatorname{A_{y}}{(\\chi)} - e^{\\chi}, then obtain \\int 0 d\\chi = l", "derivation": "\\operatorname{A_{y}}{(\\chi)} = e^{\\chi} and 0 = - \\operatorname{A_{y}}{(\\chi)} + e^{\\chi} and 0 = \\operatorname{A_{y}}{(\\chi)} - e^{\\chi} and \\frac{d}{d \\chi} 0 = \\frac{d}{d \\chi} (\\operatorname{A_{y}}{(\\chi)} - e^{\\chi}) and \\int \\frac{d}{d \\chi} 0 d\\chi = \\int \\frac{d}{d \\chi} (\\operatorname{A_{y}}{(\\chi)} - e^{\\chi}) d\\chi and \\int \\frac{d}{d \\chi} 0 d\\chi = l + \\operatorname{A_{y}}{(\\chi)} - e^{\\chi} and \\int \\frac{d}{d \\chi} 0 d\\chi = l and \\int 0 d\\chi = l", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Function('A_y')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_y')(Symbol('\\\\chi', commutative=True))), exp(Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('A_y')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Function('A_y')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Add(Function('A_y')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('l', commutative=True), Function('A_y')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('l', commutative=True))"], [["evaluate_derivatives", 7], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('l', commutative=True))"]]}, {"prompt": "Given W{(G)} = \\cos{(e^{G})}, then obtain (\\frac{d}{d G} \\int W{(G)} dG)^{G} = (\\frac{\\partial}{\\partial G} (\\mathbb{I} + \\operatorname{Ci}{(e^{G})}))^{G}", "derivation": "W{(G)} = \\cos{(e^{G})} and \\int W{(G)} dG = \\int \\cos{(e^{G})} dG and \\frac{d}{d G} \\int W{(G)} dG = \\frac{d}{d G} \\int \\cos{(e^{G})} dG and (\\frac{d}{d G} \\int W{(G)} dG)^{G} = (\\frac{d}{d G} \\int \\cos{(e^{G})} dG)^{G} and (\\frac{d}{d G} \\int W{(G)} dG)^{G} = (\\frac{\\partial}{\\partial G} (\\mathbb{I} + \\operatorname{Ci}{(e^{G})}))^{G}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('W')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(cos(exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Integral(Function('W')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integral(cos(exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('W')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Pow(Derivative(Integral(cos(exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('W')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Ci(exp(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} = \\psi^* + \\frac{r_{0}}{c_{0}} and \\operatorname{A_{z}}{(t_{1})} = \\cos{(\\sin{(t_{1})})}, then obtain - r_{0} \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} + \\operatorname{A_{z}}{(t_{1})} - 1 = - r_{0} \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} + \\cos{(\\sin{(t_{1})})} - 1", "derivation": "\\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} = \\psi^* + \\frac{r_{0}}{c_{0}} and r_{0} \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} = r_{0} (\\psi^* + \\frac{r_{0}}{c_{0}}) and \\operatorname{A_{z}}{(t_{1})} = \\cos{(\\sin{(t_{1})})} and - r_{0} (\\psi^* + \\frac{r_{0}}{c_{0}}) + \\operatorname{A_{z}}{(t_{1})} - 1 = - r_{0} (\\psi^* + \\frac{r_{0}}{c_{0}}) + \\cos{(\\sin{(t_{1})})} - 1 and - r_{0} \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} + \\operatorname{A_{z}}{(t_{1})} - 1 = - r_{0} \\operatorname{A_{2}}{(\\psi^*,r_{0},c_{0})} + \\cos{(\\sin{(t_{1})})} - 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\psi^*', commutative=True), Symbol('r_0', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('A_2')(Symbol('\\\\psi^*', commutative=True), Symbol('r_0', commutative=True), Symbol('c_0', commutative=True))), Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))))"], ["get_premise", "Equality(Function('A_z')(Symbol('t_1', commutative=True)), cos(sin(Symbol('t_1', commutative=True))))"], [["minus", 3, "Add(Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))), Function('A_z')(Symbol('t_1', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('r_0', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))), cos(sin(Symbol('t_1', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True), Function('A_2')(Symbol('\\\\psi^*', commutative=True), Symbol('r_0', commutative=True), Symbol('c_0', commutative=True))), Function('A_z')(Symbol('t_1', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('r_0', commutative=True), Function('A_2')(Symbol('\\\\psi^*', commutative=True), Symbol('r_0', commutative=True), Symbol('c_0', commutative=True))), cos(sin(Symbol('t_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\varphi^{*}{(\\chi)} = \\sin{(\\chi)}, then derive \\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi = t_{1} + \\operatorname{Si}{(\\chi)}, then obtain - \\frac{d}{d \\chi} \\int \\frac{\\sin{(\\chi)}}{\\chi} d\\chi + (\\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi)^{\\chi} = (t_{1} + \\operatorname{Si}{(\\chi)})^{\\chi} - \\frac{d}{d \\chi} \\int \\frac{\\sin{(\\chi)}}{\\chi} d\\chi", "derivation": "\\varphi^{*}{(\\chi)} = \\sin{(\\chi)} and \\frac{\\varphi^{*}{(\\chi)}}{\\chi} = \\frac{\\sin{(\\chi)}}{\\chi} and \\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi = \\int \\frac{\\sin{(\\chi)}}{\\chi} d\\chi and \\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi = t_{1} + \\operatorname{Si}{(\\chi)} and (\\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi)^{\\chi} = (t_{1} + \\operatorname{Si}{(\\chi)})^{\\chi} and - \\frac{d}{d \\chi} \\int \\frac{\\sin{(\\chi)}}{\\chi} d\\chi + (\\int \\frac{\\varphi^{*}{(\\chi)}}{\\chi} d\\chi)^{\\chi} = (t_{1} + \\operatorname{Si}{(\\chi)})^{\\chi} - \\frac{d}{d \\chi} \\int \\frac{\\sin{(\\chi)}}{\\chi} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('t_1', commutative=True), Si(Symbol('\\\\chi', commutative=True))))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), Si(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["minus", 5, "Derivative(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Add(Pow(Add(Symbol('t_1', commutative=True), Si(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Derivative(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{H}{(n_{1},\\tilde{g}^*)} = \\tilde{g}^* - n_{1}, then obtain (- \\frac{- 2 \\tilde{g}^* + 2 n_{1} + 2 \\hat{H}{(n_{1},\\tilde{g}^*)}}{\\tilde{g}^*})^{\\tilde{g}^*} = 0^{\\tilde{g}^*}", "derivation": "\\hat{H}{(n_{1},\\tilde{g}^*)} = \\tilde{g}^* - n_{1} and - \\tilde{g}^* + n_{1} + \\hat{H}{(n_{1},\\tilde{g}^*)} = 0 and - 2 \\tilde{g}^* + n_{1} + \\hat{H}{(n_{1},\\tilde{g}^*)} = - \\tilde{g}^* and - 2 \\tilde{g}^* + 2 n_{1} + 2 \\hat{H}{(n_{1},\\tilde{g}^*)} = 0 and - \\frac{- 2 \\tilde{g}^* + 2 n_{1} + 2 \\hat{H}{(n_{1},\\tilde{g}^*)}}{\\tilde{g}^*} = 0 and (- \\frac{- 2 \\tilde{g}^* + 2 n_{1} + 2 \\hat{H}{(n_{1},\\tilde{g}^*)}}{\\tilde{g}^*})^{\\tilde{g}^*} = 0^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('n_1', commutative=True), Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('n_1', commutative=True), Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))), Integer(0))"], [["power", 5, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('n_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then derive \\frac{d}{d \\mathbf{P}} \\operatorname{v_{x}}{(\\mathbf{P})} + 1 + \\frac{1}{\\mathbf{P}} = 1 + \\frac{2}{\\mathbf{P}}, then obtain \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} + 1 + \\frac{1}{\\mathbf{P}} = 1 + \\frac{2}{\\mathbf{P}}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\operatorname{v_{x}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\operatorname{v_{x}}{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} = 2 \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\operatorname{v_{x}}{(\\mathbf{P})} + \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} + 1 = 2 \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} + 1 and \\frac{d}{d \\mathbf{P}} \\operatorname{v_{x}}{(\\mathbf{P})} + 1 + \\frac{1}{\\mathbf{P}} = 1 + \\frac{2}{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} + 1 + \\frac{1}{\\mathbf{P}} = 1 + \\frac{2}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["add", 3, 1], "Equality(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(2), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))), Add(Integer(1), Mul(Integer(2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))), Add(Integer(1), Mul(Integer(2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given A{(\\varepsilon)} = \\cos{(e^{\\varepsilon})} and h{(\\varepsilon)} = \\cos{(e^{\\varepsilon})}, then obtain \\frac{d}{d \\varepsilon} \\cos^{\\varepsilon}{(e^{\\varepsilon})} = \\frac{d}{d \\varepsilon} A^{\\varepsilon}{(\\varepsilon)}", "derivation": "A{(\\varepsilon)} = \\cos{(e^{\\varepsilon})} and h{(\\varepsilon)} = \\cos{(e^{\\varepsilon})} and h{(\\varepsilon)} = A{(\\varepsilon)} and h^{\\varepsilon}{(\\varepsilon)} = A^{\\varepsilon}{(\\varepsilon)} and \\cos^{\\varepsilon}{(e^{\\varepsilon})} = A^{\\varepsilon}{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\cos^{\\varepsilon}{(e^{\\varepsilon})} = \\frac{d}{d \\varepsilon} A^{\\varepsilon}{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\varepsilon', commutative=True)), cos(exp(Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\varepsilon', commutative=True)), cos(exp(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('h')(Symbol('\\\\varepsilon', commutative=True)), Function('A')(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('A')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(cos(exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('A')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Pow(cos(exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Pow(Function('A')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(E_{n})} = \\log{(E_{n})}, then obtain \\frac{d^{2}}{d E_{n}^{2}} (c^{2}{(E_{n})})^{E_{n}} = \\frac{d^{2}}{d E_{n}^{2}} (c{(E_{n})} \\log{(E_{n})})^{E_{n}}", "derivation": "c{(E_{n})} = \\log{(E_{n})} and c^{2}{(E_{n})} = c{(E_{n})} \\log{(E_{n})} and (c^{2}{(E_{n})})^{E_{n}} = (c{(E_{n})} \\log{(E_{n})})^{E_{n}} and \\frac{d}{d E_{n}} (c^{2}{(E_{n})})^{E_{n}} = \\frac{d}{d E_{n}} (c{(E_{n})} \\log{(E_{n})})^{E_{n}} and \\frac{d^{2}}{d E_{n}^{2}} (c^{2}{(E_{n})})^{E_{n}} = \\frac{d^{2}}{d E_{n}^{2}} (c{(E_{n})} \\log{(E_{n})})^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], [["times", 1, "Function('c')(Symbol('E_n', commutative=True))"], "Equality(Pow(Function('c')(Symbol('E_n', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))))"], [["power", 2, "Symbol('E_n', commutative=True)"], "Equality(Pow(Pow(Function('c')(Symbol('E_n', commutative=True)), Integer(2)), Symbol('E_n', commutative=True)), Pow(Mul(Function('c')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)))"], [["differentiate", 3, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('c')(Symbol('E_n', commutative=True)), Integer(2)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Pow(Mul(Function('c')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('c')(Symbol('E_n', commutative=True)), Integer(2)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2))), Derivative(Pow(Mul(Function('c')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(E_{\\lambda},\\ddot{x})} = E_{\\lambda} + \\ddot{x}, then obtain \\int \\ddot{x} \\operatorname{v_{x}}^{E_{\\lambda}}{(E_{\\lambda},\\ddot{x})} d\\ddot{x} = \\int \\ddot{x} (E_{\\lambda} + \\ddot{x})^{E_{\\lambda}} d\\ddot{x}", "derivation": "\\operatorname{v_{x}}{(E_{\\lambda},\\ddot{x})} = E_{\\lambda} + \\ddot{x} and \\operatorname{v_{x}}^{E_{\\lambda}}{(E_{\\lambda},\\ddot{x})} = (E_{\\lambda} + \\ddot{x})^{E_{\\lambda}} and \\ddot{x} \\operatorname{v_{x}}^{E_{\\lambda}}{(E_{\\lambda},\\ddot{x})} = \\ddot{x} (E_{\\lambda} + \\ddot{x})^{E_{\\lambda}} and \\int \\ddot{x} \\operatorname{v_{x}}^{E_{\\lambda}}{(E_{\\lambda},\\ddot{x})} d\\ddot{x} = \\int \\ddot{x} (E_{\\lambda} + \\ddot{x})^{E_{\\lambda}} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Function('v_x')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}}, then obtain \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} \\phi_{2}{(\\dot{\\mathbf{r}})} = \\phi_{2}^{2}{(\\dot{\\mathbf{r}})}", "derivation": "\\phi_{2}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} = e^{\\dot{\\mathbf{r}}} and \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} = \\phi_{2}{(\\dot{\\mathbf{r}})} and \\dot{\\mathbf{r}} \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\phi_{2}{(\\dot{\\mathbf{r}})} and \\operatorname{F_{c}}{(\\dot{\\mathbf{r}})} \\phi_{2}{(\\dot{\\mathbf{r}})} = \\phi_{2}^{2}{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["divide", 3, "Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('F_c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Function('F_c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(Z)} = \\cos{(\\log{(Z)})}, then obtain (\\frac{\\operatorname{A_{x}}{(Z)}}{\\int \\operatorname{A_{x}}{(Z)} dZ})^{Z} = (\\frac{\\cos{(\\log{(Z)})}}{\\int \\operatorname{A_{x}}{(Z)} dZ})^{Z}", "derivation": "\\operatorname{A_{x}}{(Z)} = \\cos{(\\log{(Z)})} and \\int \\operatorname{A_{x}}{(Z)} dZ = \\int \\cos{(\\log{(Z)})} dZ and \\frac{\\operatorname{A_{x}}{(Z)}}{\\int \\cos{(\\log{(Z)})} dZ} = \\frac{\\cos{(\\log{(Z)})}}{\\int \\cos{(\\log{(Z)})} dZ} and \\frac{\\operatorname{A_{x}}{(Z)}}{\\int \\operatorname{A_{x}}{(Z)} dZ} = \\frac{\\cos{(\\log{(Z)})}}{\\int \\operatorname{A_{x}}{(Z)} dZ} and (\\frac{\\operatorname{A_{x}}{(Z)}}{\\int \\operatorname{A_{x}}{(Z)} dZ})^{Z} = (\\frac{\\cos{(\\log{(Z)})}}{\\int \\operatorname{A_{x}}{(Z)} dZ})^{Z}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True))))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["divide", 1, "Integral(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('Z', commutative=True)), Pow(Integral(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Mul(cos(log(Symbol('Z', commutative=True))), Pow(Integral(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('A_x')(Symbol('Z', commutative=True)), Pow(Integral(Function('A_x')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Mul(cos(log(Symbol('Z', commutative=True))), Pow(Integral(Function('A_x')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Function('A_x')(Symbol('Z', commutative=True)), Pow(Integral(Function('A_x')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)), Pow(Mul(cos(log(Symbol('Z', commutative=True))), Pow(Integral(Function('A_x')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{\\mathbf{x}})} = \\int \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}}, then derive \\log{(\\operatorname{a^{\\dagger}}{(\\hat{\\mathbf{x}})})} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})}, then obtain \\log{(\\dot{\\mathbf{r}} + \\sin{(\\hat{\\mathbf{x}})})}^{\\rho} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})}^{\\rho}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{\\mathbf{x}})} = \\int \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} and \\log{(\\operatorname{a^{\\dagger}}{(\\hat{\\mathbf{x}})})} = \\log{(\\int \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})} and \\log{(\\operatorname{a^{\\dagger}}{(\\hat{\\mathbf{x}})})} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})} and \\log{(\\int \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})} and \\log{(\\int \\cos{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})}^{\\rho} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})}^{\\rho} and \\log{(\\dot{\\mathbf{r}} + \\sin{(\\hat{\\mathbf{x}})})}^{\\rho} = \\log{(\\rho + \\sin{(\\hat{\\mathbf{x}})})}^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integral(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["log", 1], "Equality(log(Function('a^{\\\\dagger}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Integral(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(log(Function('a^{\\\\dagger}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(log(Integral(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), log(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(log(Integral(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(log(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\rho', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(log(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(log(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\psi^*)} = \\log{(\\psi^*)}, then obtain (\\frac{d}{d \\psi^*} \\rho{(\\psi^*)})^{2} = \\frac{\\frac{d}{d \\psi^*} \\rho{(\\psi^*)}}{\\psi^*}", "derivation": "\\rho{(\\psi^*)} = \\log{(\\psi^*)} and \\frac{d}{d \\psi^*} \\rho{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and (\\frac{d}{d \\psi^*} \\rho{(\\psi^*)})^{2} = \\frac{d}{d \\psi^*} \\rho{(\\psi^*)} \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and (\\frac{d}{d \\psi^*} \\rho{(\\psi^*)})^{2} = \\frac{\\frac{d}{d \\psi^*} \\rho{(\\psi^*)}}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Derivative(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\varepsilon_0,t)} = \\log{(- \\varepsilon_0 + t)} and k{(\\varepsilon_0,t)} = \\frac{\\partial}{\\partial t} \\log{(- \\varepsilon_0 + t)}, then obtain \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\varepsilon_0,t)} = k{(\\varepsilon_0,t)}", "derivation": "\\operatorname{z^{*}}{(\\varepsilon_0,t)} = \\log{(- \\varepsilon_0 + t)} and \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\varepsilon_0,t)} = \\frac{\\partial}{\\partial t} \\log{(- \\varepsilon_0 + t)} and k{(\\varepsilon_0,t)} = \\frac{\\partial}{\\partial t} \\log{(- \\varepsilon_0 + t)} and \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\varepsilon_0,t)} = k{(\\varepsilon_0,t)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t', commutative=True)), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('z^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Function('k')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(t_{2})} = \\log{(t_{2})}, then derive \\frac{d}{d t_{2}} \\int \\mathbf{p}{(t_{2})} dt_{2} = \\frac{\\partial}{\\partial t_{2}} (t_{2} \\log{(t_{2})} - t_{2} + z), then obtain \\frac{\\frac{d}{d t_{2}} \\int \\mathbf{p}{(t_{2})} dt_{2}}{\\Psi + t_{2} \\log{(t_{2})} - t_{2}} = \\frac{\\frac{\\partial}{\\partial t_{2}} (t_{2} \\log{(t_{2})} - t_{2} + z)}{\\Psi + t_{2} \\log{(t_{2})} - t_{2}}", "derivation": "\\mathbf{p}{(t_{2})} = \\log{(t_{2})} and \\int \\mathbf{p}{(t_{2})} dt_{2} = \\int \\log{(t_{2})} dt_{2} and \\frac{d}{d t_{2}} \\int \\mathbf{p}{(t_{2})} dt_{2} = \\frac{d}{d t_{2}} \\int \\log{(t_{2})} dt_{2} and \\frac{d}{d t_{2}} \\int \\mathbf{p}{(t_{2})} dt_{2} = \\frac{\\partial}{\\partial t_{2}} (t_{2} \\log{(t_{2})} - t_{2} + z) and \\frac{\\frac{d}{d t_{2}} \\int \\mathbf{p}{(t_{2})} dt_{2}}{\\Psi + t_{2} \\log{(t_{2})} - t_{2}} = \\frac{\\frac{\\partial}{\\partial t_{2}} (t_{2} \\log{(t_{2})} - t_{2} + z)}{\\Psi + t_{2} \\log{(t_{2})} - t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('t_2', commutative=True), log(Symbol('t_2', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["divide", 4, "Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('t_2', commutative=True), log(Symbol('t_2', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('t_2', commutative=True), log(Symbol('t_2', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True))), Integer(-1)), Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('t_2', commutative=True), log(Symbol('t_2', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True))), Integer(-1)), Derivative(Add(Mul(Symbol('t_2', commutative=True), log(Symbol('t_2', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\rho)} = \\log{(\\sin{(\\rho)})} and \\nabla{(\\rho)} = \\frac{\\hat{x}_0{(\\rho)}}{\\sin{(\\rho)}}, then obtain \\nabla{(\\rho)} - \\log{(\\sin{(\\rho)})} = - \\log{(\\sin{(\\rho)})} + \\frac{\\log{(\\sin{(\\rho)})}}{\\sin{(\\rho)}}", "derivation": "\\hat{x}_0{(\\rho)} = \\log{(\\sin{(\\rho)})} and \\frac{\\hat{x}_0{(\\rho)}}{\\sin{(\\rho)}} = \\frac{\\log{(\\sin{(\\rho)})}}{\\sin{(\\rho)}} and \\nabla{(\\rho)} = \\frac{\\hat{x}_0{(\\rho)}}{\\sin{(\\rho)}} and \\nabla{(\\rho)} = \\frac{\\log{(\\sin{(\\rho)})}}{\\sin{(\\rho)}} and \\nabla{(\\rho)} - \\log{(\\sin{(\\rho)})} = - \\log{(\\sin{(\\rho)})} + \\frac{\\log{(\\sin{(\\rho)})}}{\\sin{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), log(sin(Symbol('\\\\rho', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(log(sin(Symbol('\\\\rho', commutative=True))), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\rho', commutative=True)), Mul(Function('\\\\hat{x}_0')(Symbol('\\\\rho', commutative=True)), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\nabla')(Symbol('\\\\rho', commutative=True)), Mul(log(sin(Symbol('\\\\rho', commutative=True))), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["minus", 4, "log(sin(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), log(sin(Symbol('\\\\rho', commutative=True))))), Add(Mul(Integer(-1), log(sin(Symbol('\\\\rho', commutative=True)))), Mul(log(sin(Symbol('\\\\rho', commutative=True))), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(F_{g},\\hbar)} = F_{g} - \\hbar, then derive \\frac{\\partial}{\\partial \\hbar} \\operatorname{P_{e}}{(F_{g},\\hbar)} = -1, then obtain \\operatorname{P_{e}}{(F_{g},\\hbar)} \\frac{\\partial}{\\partial \\hbar} (F_{g} - \\hbar) = - \\operatorname{P_{e}}{(F_{g},\\hbar)}", "derivation": "\\operatorname{P_{e}}{(F_{g},\\hbar)} = F_{g} - \\hbar and \\operatorname{P_{e}}{(F_{g},\\hbar)} - 1 = F_{g} - \\hbar - 1 and \\frac{\\partial}{\\partial \\hbar} (\\operatorname{P_{e}}{(F_{g},\\hbar)} - 1) = \\frac{\\partial}{\\partial \\hbar} (F_{g} - \\hbar - 1) and \\frac{\\partial}{\\partial \\hbar} \\operatorname{P_{e}}{(F_{g},\\hbar)} = -1 and \\frac{\\partial}{\\partial \\hbar} (F_{g} - \\hbar) = -1 and \\operatorname{P_{e}}{(F_{g},\\hbar)} \\frac{\\partial}{\\partial \\hbar} (F_{g} - \\hbar) = - \\operatorname{P_{e}}{(F_{g},\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1))"], [["times", 5, "Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Integer(-1), Function('P_e')(Symbol('F_g', commutative=True), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\psi{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and J{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} - 1, then obtain (e^{g_{\\varepsilon}} - 1)^{g_{\\varepsilon}} = (\\psi{(g_{\\varepsilon})} - 1)^{g_{\\varepsilon}}", "derivation": "\\psi{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and \\psi{(g_{\\varepsilon})} - 1 = e^{g_{\\varepsilon}} - 1 and J{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} - 1 and J{(g_{\\varepsilon})} = \\psi{(g_{\\varepsilon})} - 1 and J^{g_{\\varepsilon}}{(g_{\\varepsilon})} = (\\psi{(g_{\\varepsilon})} - 1)^{g_{\\varepsilon}} and (e^{g_{\\varepsilon}} - 1)^{g_{\\varepsilon}} = (\\psi{(g_{\\varepsilon})} - 1)^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["power", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('J')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Function('\\\\psi')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})} = (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda}, then obtain \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})} = \\lambda (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda} \\log{(\\psi)}", "derivation": "\\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})} = (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda} and - \\lambda + \\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})} = - \\lambda + (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda} and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} (- \\lambda + \\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})}) = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} (- \\lambda + (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda}) and \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\tilde{g}^*{(\\psi,\\lambda,g^{\\prime}_{\\varepsilon})} = \\lambda (\\psi^{g^{\\prime}_{\\varepsilon}})^{\\lambda} \\log{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["minus", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(a)} = \\log{(a)} and \\operatorname{v_{y}}{(\\phi_1)} = \\cos{(e^{\\phi_1})}, then obtain \\operatorname{v_{y}}{(\\phi_1)} \\frac{d}{d a} (- \\mu_{0}{(a)} + \\log{(a)}) + \\log{(a)} = \\log{(a)} + \\cos{(e^{\\phi_1})} \\frac{d}{d a} (- \\mu_{0}{(a)} + \\log{(a)})", "derivation": "\\mu_{0}{(a)} = \\log{(a)} and 0 = - \\mu_{0}{(a)} + \\log{(a)} and \\frac{d}{d a} 0 = \\frac{d}{d a} (- \\mu_{0}{(a)} + \\log{(a)}) and \\operatorname{v_{y}}{(\\phi_1)} = \\cos{(e^{\\phi_1})} and \\operatorname{v_{y}}{(\\phi_1)} \\frac{d}{d a} 0 = \\cos{(e^{\\phi_1})} \\frac{d}{d a} 0 and \\operatorname{v_{y}}{(\\phi_1)} \\frac{d}{d a} 0 + \\log{(a)} = \\log{(a)} + \\cos{(e^{\\phi_1})} \\frac{d}{d a} 0 and \\operatorname{v_{y}}{(\\phi_1)} \\frac{d}{d a} (- \\mu_{0}{(a)} + \\log{(a)}) + \\log{(a)} = \\log{(a)} + \\cos{(e^{\\phi_1})} \\frac{d}{d a} (- \\mu_{0}{(a)} + \\log{(a)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('a', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), cos(exp(Symbol('\\\\phi_1', commutative=True))))"], [["times", 4, "Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(cos(exp(Symbol('\\\\phi_1', commutative=True))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["minus", 5, "Mul(Integer(-1), log(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))), log(Symbol('a', commutative=True))), Add(log(Symbol('a', commutative=True)), Mul(cos(exp(Symbol('\\\\phi_1', commutative=True))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), log(Symbol('a', commutative=True))), Add(log(Symbol('a', commutative=True)), Mul(cos(exp(Symbol('\\\\phi_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(a)} = \\sin{(a)} and \\lambda{(a)} = a - \\sin{(a)}, then obtain \\psi^* \\cos{(a - \\operatorname{c_{0}}{(a)})} = \\psi^* \\cos{(\\lambda{(a)})}", "derivation": "\\operatorname{c_{0}}{(a)} = \\sin{(a)} and - a + \\operatorname{c_{0}}{(a)} = - a + \\sin{(a)} and \\cos{(a - \\operatorname{c_{0}}{(a)})} = \\cos{(a - \\sin{(a)})} and \\psi^* \\cos{(a - \\operatorname{c_{0}}{(a)})} = \\psi^* \\cos{(a - \\sin{(a)})} and \\lambda{(a)} = a - \\sin{(a)} and \\psi^* \\cos{(a - \\operatorname{c_{0}}{(a)})} = \\psi^* \\cos{(\\lambda{(a)})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('c_0')(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True))))), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True))))))"], [["times", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True)))))), Mul(Symbol('\\\\psi^*', commutative=True), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('a', commutative=True)), Add(Symbol('a', commutative=True), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), cos(Add(Symbol('a', commutative=True), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True)))))), Mul(Symbol('\\\\psi^*', commutative=True), cos(Function('\\\\lambda')(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(v)} = \\int \\cos{(v)} dv, then obtain (((\\int \\nabla{(v)} dv)^{v}) \\iint \\cos{(v)} dv dv)^{v} = ((\\iint \\cos{(v)} dv dv) (\\iint \\cos{(v)} dv dv)^{v})^{v}", "derivation": "\\nabla{(v)} = \\int \\cos{(v)} dv and \\int \\nabla{(v)} dv = \\iint \\cos{(v)} dv dv and (\\int \\nabla{(v)} dv)^{v} = (\\iint \\cos{(v)} dv dv)^{v} and ((\\int \\nabla{(v)} dv)^{v}) \\iint \\cos{(v)} dv dv = (\\iint \\cos{(v)} dv dv) (\\iint \\cos{(v)} dv dv)^{v} and (((\\int \\nabla{(v)} dv)^{v}) \\iint \\cos{(v)} dv dv)^{v} = ((\\iint \\cos{(v)} dv dv) (\\iint \\cos{(v)} dv dv)^{v})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Integral(Function('\\\\nabla')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["times", 3, "Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('\\\\nabla')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Pow(Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Pow(Integral(Function('\\\\nabla')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(Mul(Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Pow(Integral(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(g)} = e^{g}, then derive \\int (\\operatorname{F_{x}}{(g)} + \\int e^{g} dg) dg = M_{E} + 2 e^{g}, then obtain \\frac{\\int (e^{g} + \\int e^{g} dg) dg}{\\hat{X}} = \\frac{M_{E} + 2 \\operatorname{F_{x}}{(g)}}{\\hat{X}}", "derivation": "\\operatorname{F_{x}}{(g)} = e^{g} and \\int \\operatorname{F_{x}}{(g)} dg = \\int e^{g} dg and \\operatorname{F_{x}}{(g)} + \\int \\operatorname{F_{x}}{(g)} dg = e^{g} + \\int \\operatorname{F_{x}}{(g)} dg and \\operatorname{F_{x}}{(g)} + \\int e^{g} dg = e^{g} + \\int e^{g} dg and \\int (\\operatorname{F_{x}}{(g)} + \\int e^{g} dg) dg = \\int (e^{g} + \\int e^{g} dg) dg and \\int (\\operatorname{F_{x}}{(g)} + \\int e^{g} dg) dg = M_{E} + 2 e^{g} and \\int (e^{g} + \\int e^{g} dg) dg = M_{E} + 2 e^{g} and \\frac{\\int (e^{g} + \\int e^{g} dg) dg}{\\hat{X}} = \\frac{M_{E} + 2 e^{g}}{\\hat{X}} and \\frac{\\int (e^{g} + \\int e^{g} dg) dg}{\\hat{X}} = \\frac{M_{E} + 2 \\operatorname{F_{x}}{(g)}}{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["add", 1, "Integral(Function('F_x')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Function('F_x')(Symbol('g', commutative=True)), Integral(Function('F_x')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(exp(Symbol('g', commutative=True)), Integral(Function('F_x')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('F_x')(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Function('F_x')(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Function('F_x')(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Integer(2), exp(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Integer(2), exp(Symbol('g', commutative=True)))))"], [["divide", 7, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(2), exp(Symbol('g', commutative=True))))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(2), Function('F_x')(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(J_{\\varepsilon},M_{E})} = J_{\\varepsilon} + e^{M_{E}}, then derive \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{F}{(J_{\\varepsilon},M_{E})} = 1, then obtain \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{F}{(J_{\\varepsilon},M_{E})} - 1 = 0", "derivation": "\\mathbf{F}{(J_{\\varepsilon},M_{E})} = J_{\\varepsilon} + e^{M_{E}} and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{F}{(J_{\\varepsilon},M_{E})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} + e^{M_{E}}) and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{F}{(J_{\\varepsilon},M_{E})} = 1 and \\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} + e^{M_{E}}) = 1 and \\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} + e^{M_{E}}) - 1 = 0 and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{F}{(J_{\\varepsilon},M_{E})} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, 1], "Equality(Add(Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\theta{(\\delta,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}), then derive \\theta{(\\delta,y^{\\prime})} = -1, then obtain \\frac{\\frac{d}{d \\delta} (-1)^{\\delta}}{\\frac{\\partial}{\\partial \\delta} (\\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}))^{\\delta}} = 1", "derivation": "\\theta{(\\delta,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}) and \\theta{(\\delta,y^{\\prime})} = -1 and -1 = \\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}) and (-1)^{\\delta} = (\\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}))^{\\delta} and \\frac{d}{d \\delta} (-1)^{\\delta} = \\frac{\\partial}{\\partial \\delta} (\\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}))^{\\delta} and \\frac{\\frac{d}{d \\delta} (-1)^{\\delta}}{\\frac{\\partial}{\\partial \\delta} (\\frac{\\partial}{\\partial y^{\\prime}} (\\delta - y^{\\prime}))^{\\delta}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\delta', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta')(Symbol('\\\\delta', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Integer(-1), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Pow(Integer(-1), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Integer(-1), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Pow(Derivative(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain \\frac{\\mathbf{A} + \\dot{\\mathbf{r}}{(\\mathbf{A})} + 1}{\\int \\dot{\\mathbf{r}}{(\\mathbf{A})} d\\mathbf{A}} = \\frac{\\mathbf{A} + \\cos{(\\mathbf{A})} + 1}{\\int \\dot{\\mathbf{r}}{(\\mathbf{A})} d\\mathbf{A}}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\dot{\\mathbf{r}}{(\\mathbf{A})} + 1 = \\cos{(\\mathbf{A})} + 1 and \\mathbf{A} + \\dot{\\mathbf{r}}{(\\mathbf{A})} + 1 = \\mathbf{A} + \\cos{(\\mathbf{A})} + 1 and \\int \\dot{\\mathbf{r}}{(\\mathbf{A})} d\\mathbf{A} = \\int \\cos{(\\mathbf{A})} d\\mathbf{A} and \\frac{\\mathbf{A} + \\dot{\\mathbf{r}}{(\\mathbf{A})} + 1}{\\int \\cos{(\\mathbf{A})} d\\mathbf{A}} = \\frac{\\mathbf{A} + \\cos{(\\mathbf{A})} + 1}{\\int \\cos{(\\mathbf{A})} d\\mathbf{A}} and \\frac{\\mathbf{A} + \\dot{\\mathbf{r}}{(\\mathbf{A})} + 1}{\\int \\dot{\\mathbf{r}}{(\\mathbf{A})} d\\mathbf{A}} = \\frac{\\mathbf{A} + \\cos{(\\mathbf{A})} + 1}{\\int \\dot{\\mathbf{r}}{(\\mathbf{A})} d\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Add(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)))"], [["add", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 3, "Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Pow(Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Pow(Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Pow(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Pow(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(l)} = \\cos{(l)}, then derive \\frac{d}{d l} \\operatorname{F_{N}}{(l)} = - \\sin{(l)}, then obtain ((- \\frac{d}{d l} \\operatorname{F_{N}}{(l)} + \\frac{d}{d l} \\cos{(l)}) c{(l)})^{l} = ((- \\sin{(l)} - \\frac{d}{d l} \\operatorname{F_{N}}{(l)}) c{(l)})^{l}", "derivation": "\\operatorname{F_{N}}{(l)} = \\cos{(l)} and \\frac{d}{d l} \\operatorname{F_{N}}{(l)} = \\frac{d}{d l} \\cos{(l)} and \\frac{d}{d l} \\operatorname{F_{N}}{(l)} = - \\sin{(l)} and \\frac{d}{d l} \\cos{(l)} = - \\sin{(l)} and - \\frac{d}{d l} \\operatorname{F_{N}}{(l)} + \\frac{d}{d l} \\cos{(l)} = - \\sin{(l)} - \\frac{d}{d l} \\operatorname{F_{N}}{(l)} and (- \\frac{d}{d l} \\operatorname{F_{N}}{(l)} + \\frac{d}{d l} \\cos{(l)}) c{(l)} = (- \\sin{(l)} - \\frac{d}{d l} \\operatorname{F_{N}}{(l)}) c{(l)} and ((- \\frac{d}{d l} \\operatorname{F_{N}}{(l)} + \\frac{d}{d l} \\cos{(l)}) c{(l)})^{l} = ((- \\sin{(l)} - \\frac{d}{d l} \\operatorname{F_{N}}{(l)}) c{(l)})^{l}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('l', commutative=True))))"], [["minus", 4, "Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["times", 5, "Function('c')(Symbol('l', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('c')(Symbol('l', commutative=True))), Mul(Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Function('c')(Symbol('l', commutative=True))))"], [["power", 6, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Derivative(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Function('c')(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('F_N')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Function('c')(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(f_{E})} = f_{E}, then obtain \\int \\dot{x}^{- f_{E}}{(f_{E})} \\int 0 df_{E} df_{E} = \\int \\dot{x}^{- f_{E}}{(f_{E})} \\int (f_{E} - \\dot{x}{(f_{E})}) df_{E} df_{E}", "derivation": "\\dot{x}{(f_{E})} = f_{E} and 0 = f_{E} - \\dot{x}{(f_{E})} and \\int 0 df_{E} = \\int (f_{E} - \\dot{x}{(f_{E})}) df_{E} and \\dot{x}^{- f_{E}}{(f_{E})} \\int 0 df_{E} = \\dot{x}^{- f_{E}}{(f_{E})} \\int (f_{E} - \\dot{x}{(f_{E})}) df_{E} and \\int \\dot{x}^{- f_{E}}{(f_{E})} \\int 0 df_{E} df_{E} = \\int \\dot{x}^{- f_{E}}{(f_{E})} \\int (f_{E} - \\dot{x}{(f_{E})}) df_{E} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], [["minus", 1, "Function('\\\\dot{x}')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f_E', commutative=True)))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))))"], [["divide", 3, "Pow(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('f_E', commutative=True)))), Mul(Pow(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))), Integral(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True)))))"], [["integrate", 4, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Pow(Function('\\\\dot{x}')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))), Integral(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)} = \\hat{\\mathbf{x}}^{x^\\prime}, then derive (\\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)})^{x^\\prime} = (\\hat{\\mathbf{x}}^{x^\\prime} \\log{(\\hat{\\mathbf{x}})})^{x^\\prime}, then obtain V_{\\mathbf{B}} + (\\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)})^{x^\\prime} = V_{\\mathbf{B}} + (\\hat{\\mathbf{x}}^{x^\\prime} \\log{(\\hat{\\mathbf{x}})})^{x^\\prime}", "derivation": "\\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)} = \\hat{\\mathbf{x}}^{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} \\hat{\\mathbf{x}}^{x^\\prime} and (\\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)})^{x^\\prime} = (\\frac{\\partial}{\\partial x^\\prime} \\hat{\\mathbf{x}}^{x^\\prime})^{x^\\prime} and (\\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)})^{x^\\prime} = (\\hat{\\mathbf{x}}^{x^\\prime} \\log{(\\hat{\\mathbf{x}})})^{x^\\prime} and V_{\\mathbf{B}} + (\\frac{\\partial}{\\partial x^\\prime} \\sigma_{x}{(\\hat{\\mathbf{x}},x^\\prime)})^{x^\\prime} = V_{\\mathbf{B}} + (\\hat{\\mathbf{x}}^{x^\\prime} \\log{(\\hat{\\mathbf{x}})})^{x^\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(F_{c})} = \\cos{(F_{c})}, then derive \\int \\phi_{1}{(F_{c})} dF_{c} = \\rho_f + \\sin{(F_{c})}, then obtain \\int \\cos{(F_{c})} dF_{c} = \\rho_f + \\sin{(F_{c})}", "derivation": "\\phi_{1}{(F_{c})} = \\cos{(F_{c})} and \\int \\phi_{1}{(F_{c})} dF_{c} = \\int \\cos{(F_{c})} dF_{c} and \\int \\phi_{1}{(F_{c})} dF_{c} = \\rho_f + \\sin{(F_{c})} and \\int \\cos{(F_{c})} dF_{c} = \\rho_f + \\sin{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), sin(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), sin(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(u)} = \\cos{(\\sin{(u)})} and \\phi_{2}{(u)} = \\sin{(u)}, then obtain u \\cos{(\\phi_{2}{(u)})} = u \\cos{(\\sin{(u)})}", "derivation": "\\hat{H}_{\\lambda}{(u)} = \\cos{(\\sin{(u)})} and u \\hat{H}_{\\lambda}{(u)} = u \\cos{(\\sin{(u)})} and \\phi_{2}{(u)} = \\sin{(u)} and \\hat{H}_{\\lambda}{(u)} = \\cos{(\\phi_{2}{(u)})} and u \\cos{(\\phi_{2}{(u)})} = u \\cos{(\\sin{(u)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True))))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), cos(sin(Symbol('u', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('u', commutative=True)), cos(Function('\\\\phi_2')(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('u', commutative=True), cos(Function('\\\\phi_2')(Symbol('u', commutative=True)))), Mul(Symbol('u', commutative=True), cos(sin(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = B c_{0} + \\mathbf{J}_f, then obtain B c_{0} + 2 \\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = 3 B c_{0} + 2 \\mathbf{J}_f", "derivation": "\\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = B c_{0} + \\mathbf{J}_f and B c_{0} + \\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = 2 B c_{0} + \\mathbf{J}_f and 2 B c_{0} + \\mathbf{J}_f + \\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = 3 B c_{0} + 2 \\mathbf{J}_f and B c_{0} + 2 \\psi^{*}{(\\mathbf{J}_f,c_{0},B)} = 3 B c_{0} + 2 \\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True), Symbol('B', commutative=True)), Add(Mul(Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 1, "Mul(Symbol('B', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 1, "Add(Mul(Integer(2), Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(3), Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(2), Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(3), Symbol('B', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\Psi,L)} = L - \\Psi, then obtain \\frac{\\partial}{\\partial \\Psi} (- \\frac{\\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)}}{\\Psi})^{L} = \\frac{\\partial}{\\partial \\Psi} (- \\frac{\\frac{\\partial}{\\partial \\Psi} (L - \\Psi)}{\\Psi})^{L}", "derivation": "\\hat{\\mathbf{r}}{(\\Psi,L)} = L - \\Psi and \\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)} = \\frac{\\partial}{\\partial \\Psi} (L - \\Psi) and - \\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)} = - \\frac{\\partial}{\\partial \\Psi} (L - \\Psi) and - \\frac{\\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)}}{\\Psi} = - \\frac{\\frac{\\partial}{\\partial \\Psi} (L - \\Psi)}{\\Psi} and (- \\frac{\\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)}}{\\Psi})^{L} = (- \\frac{\\frac{\\partial}{\\partial \\Psi} (L - \\Psi)}{\\Psi})^{L} and \\frac{\\partial}{\\partial \\Psi} (- \\frac{\\frac{\\partial}{\\partial \\Psi} \\hat{\\mathbf{r}}{(\\Psi,L)}}{\\Psi})^{L} = \\frac{\\partial}{\\partial \\Psi} (- \\frac{\\frac{\\partial}{\\partial \\Psi} (L - \\Psi)}{\\Psi})^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["divide", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('L', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('L', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('L', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('L', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(a^{\\dagger})} = \\sin{(\\log{(a^{\\dagger})})} and Z{(a^{\\dagger})} = \\log{(a^{\\dagger})} + \\sin{(\\log{(a^{\\dagger})})}, then obtain a^{\\dagger} e^{a{(a^{\\dagger})}} = a^{\\dagger} e^{\\sin{(\\log{(a^{\\dagger})})}}", "derivation": "a{(a^{\\dagger})} = \\sin{(\\log{(a^{\\dagger})})} and Z{(a^{\\dagger})} = \\log{(a^{\\dagger})} + \\sin{(\\log{(a^{\\dagger})})} and e^{Z{(a^{\\dagger})}} = a^{\\dagger} e^{\\sin{(\\log{(a^{\\dagger})})}} and Z{(a^{\\dagger})} = a{(a^{\\dagger})} + \\log{(a^{\\dagger})} and a^{\\dagger} e^{a{(a^{\\dagger})}} = a^{\\dagger} e^{\\sin{(\\log{(a^{\\dagger})})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), sin(log(Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('a^{\\\\dagger}', commutative=True)), Add(log(Symbol('a^{\\\\dagger}', commutative=True)), sin(log(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["exp", 2], "Equality(exp(Function('Z')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(sin(log(Symbol('a^{\\\\dagger}', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Z')(Symbol('a^{\\\\dagger}', commutative=True)), Add(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(sin(log(Symbol('a^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(a,A_{y})} = \\frac{\\log{(a)}}{A_{y}} and \\Psi^{\\dagger}{(a,A_{y})} = \\frac{\\log{(a)}}{A_{y}}, then obtain \\frac{\\partial}{\\partial A_{y}} \\Psi^{\\dagger}{(a,A_{y})} = - \\frac{\\log{(a)}}{A_{y}^{2}}", "derivation": "\\operatorname{t_{1}}{(a,A_{y})} = \\frac{\\log{(a)}}{A_{y}} and \\Psi^{\\dagger}{(a,A_{y})} = \\frac{\\log{(a)}}{A_{y}} and \\Psi^{\\dagger}{(a,A_{y})} = \\operatorname{t_{1}}{(a,A_{y})} and \\frac{\\partial}{\\partial A_{y}} \\Psi^{\\dagger}{(a,A_{y})} = \\frac{\\partial}{\\partial A_{y}} \\operatorname{t_{1}}{(a,A_{y})} and \\frac{\\partial}{\\partial A_{y}} \\Psi^{\\dagger}{(a,A_{y})} = \\frac{\\partial}{\\partial A_{y}} \\frac{\\log{(a)}}{A_{y}} and \\frac{\\partial}{\\partial A_{y}} \\Psi^{\\dagger}{(a,A_{y})} = - \\frac{\\log{(a)}}{A_{y}^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), log(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), log(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Function('t_1')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)))"], [["differentiate", 3, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Function('t_1')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), log(Symbol('a', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-2)), log(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\dot{x})} = \\cos{(\\dot{x})} and q{(x^\\prime,\\nabla)} = \\log{(\\nabla - x^\\prime)}, then obtain (- \\operatorname{C_{d}}{(\\dot{x})} + q{(x^\\prime,\\nabla)})^{\\nabla} = (- \\operatorname{C_{d}}{(\\dot{x})} + \\log{(\\nabla - x^\\prime)})^{\\nabla}", "derivation": "\\operatorname{C_{d}}{(\\dot{x})} = \\cos{(\\dot{x})} and q{(x^\\prime,\\nabla)} = \\log{(\\nabla - x^\\prime)} and q{(x^\\prime,\\nabla)} - \\cos{(\\dot{x})} = \\log{(\\nabla - x^\\prime)} - \\cos{(\\dot{x})} and (q{(x^\\prime,\\nabla)} - \\cos{(\\dot{x})})^{\\nabla} = (\\log{(\\nabla - x^\\prime)} - \\cos{(\\dot{x})})^{\\nabla} and (- \\operatorname{C_{d}}{(\\dot{x})} + q{(x^\\prime,\\nabla)})^{\\nabla} = (- \\operatorname{C_{d}}{(\\dot{x})} + \\log{(\\nabla - x^\\prime)})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], ["get_premise", "Equality(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["minus", 2, "cos(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\dot{x}', commutative=True)))), Add(log(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(-1), cos(Symbol('\\\\dot{x}', commutative=True)))))"], [["power", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Add(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\dot{x}', commutative=True)))), Symbol('\\\\nabla', commutative=True)), Pow(Add(log(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(-1), cos(Symbol('\\\\dot{x}', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\dot{x}', commutative=True))), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\dot{x}', commutative=True))), log(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\tilde{g})} = e^{\\tilde{g}}, then obtain (\\tilde{g} + \\varepsilon{(\\tilde{g})}) \\sin{(\\frac{d}{d \\tilde{g}} 1)} = (\\tilde{g} + \\varepsilon{(\\tilde{g})}) \\sin{(\\frac{d}{d \\tilde{g}} \\frac{e^{\\tilde{g}}}{\\varepsilon{(\\tilde{g})}})}", "derivation": "\\varepsilon{(\\tilde{g})} = e^{\\tilde{g}} and 1 = \\frac{e^{\\tilde{g}}}{\\varepsilon{(\\tilde{g})}} and \\frac{d}{d \\tilde{g}} 1 = \\frac{d}{d \\tilde{g}} \\frac{e^{\\tilde{g}}}{\\varepsilon{(\\tilde{g})}} and \\sin{(\\frac{d}{d \\tilde{g}} 1)} = \\sin{(\\frac{d}{d \\tilde{g}} \\frac{e^{\\tilde{g}}}{\\varepsilon{(\\tilde{g})}})} and (\\tilde{g} + \\varepsilon{(\\tilde{g})}) \\sin{(\\frac{d}{d \\tilde{g}} 1)} = (\\tilde{g} + \\varepsilon{(\\tilde{g})}) \\sin{(\\frac{d}{d \\tilde{g}} \\frac{e^{\\tilde{g}}}{\\varepsilon{(\\tilde{g})}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), sin(Derivative(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["times", 4, "Add(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True))), sin(Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))), Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True))), sin(Derivative(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given S{(t_{2},\\hat{p})} = \\log{(- \\hat{p} + t_{2})}, then derive \\int \\frac{S{(t_{2},\\hat{p})}}{\\log{(- \\hat{p} + t_{2})}} dt_{2} = \\mathbb{I} + t_{2}, then derive \\mathbb{I} + t_{2} = k + t_{2}, then obtain \\frac{\\partial}{\\partial t_{2}} (\\mathbb{I} + t_{2} - S{(t_{2},\\hat{p})}) = \\frac{\\partial}{\\partial t_{2}} (k + t_{2} - S{(t_{2},\\hat{p})})", "derivation": "S{(t_{2},\\hat{p})} = \\log{(- \\hat{p} + t_{2})} and \\frac{S{(t_{2},\\hat{p})}}{\\log{(- \\hat{p} + t_{2})}} = 1 and \\int \\frac{S{(t_{2},\\hat{p})}}{\\log{(- \\hat{p} + t_{2})}} dt_{2} = \\int 1 dt_{2} and \\int \\frac{S{(t_{2},\\hat{p})}}{\\log{(- \\hat{p} + t_{2})}} dt_{2} = \\mathbb{I} + t_{2} and \\mathbb{I} + t_{2} = \\int 1 dt_{2} and \\mathbb{I} + t_{2} = k + t_{2} and \\mathbb{I} + t_{2} - S{(t_{2},\\hat{p})} = k + t_{2} - S{(t_{2},\\hat{p})} and \\frac{\\partial}{\\partial t_{2}} (\\mathbb{I} + t_{2} - S{(t_{2},\\hat{p})}) = \\frac{\\partial}{\\partial t_{2}} (k + t_{2} - S{(t_{2},\\hat{p})})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('t_2', commutative=True))))"], [["divide", 1, "log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('t_2', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('t_2', commutative=True))), Integer(-1))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('t_2', commutative=True))), Integer(-1))), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('k', commutative=True), Symbol('t_2', commutative=True)))"], [["minus", 6, "Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Add(Symbol('k', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["differentiate", 7, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('S')(Symbol('t_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(M,f_{E})} = - M + f_{E}, then obtain \\frac{1}{M} = \\frac{(- M + f_{E})^{2}}{M \\rho_{b}^{2}{(M,f_{E})}}", "derivation": "\\rho_{b}{(M,f_{E})} = - M + f_{E} and 1 = \\frac{- M + f_{E}}{\\rho_{b}{(M,f_{E})}} and \\frac{1}{M} = \\frac{- M + f_{E}}{M \\rho_{b}{(M,f_{E})}} and \\frac{- M + f_{E}}{M \\rho_{b}{(M,f_{E})}} = \\frac{(- M + f_{E})^{2}}{M \\rho_{b}^{2}{(M,f_{E})}} and \\frac{1}{M} = \\frac{(- M + f_{E})^{2}}{M \\rho_{b}^{2}{(M,f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)))"], [["divide", 1, "Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["divide", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Symbol('M', commutative=True), Integer(-1)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["times", 2, "Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Integer(2)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('M', commutative=True), Integer(-1)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('f_E', commutative=True)), Integer(2)), Pow(Function('\\\\rho_b')(Symbol('M', commutative=True), Symbol('f_E', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{2})} = e^{e^{A_{2}}}, then obtain A_{2} + \\mathbf{A} = \\int \\frac{e^{e^{A_{2}}}}{\\operatorname{F_{x}}{(A_{2})}} dA_{2}", "derivation": "\\operatorname{F_{x}}{(A_{2})} = e^{e^{A_{2}}} and 1 = \\frac{e^{e^{A_{2}}}}{\\operatorname{F_{x}}{(A_{2})}} and \\int 1 dA_{2} = \\int \\frac{e^{e^{A_{2}}}}{\\operatorname{F_{x}}{(A_{2})}} dA_{2} and A_{2} + \\mathbf{A} = \\int \\frac{e^{e^{A_{2}}}}{\\operatorname{F_{x}}{(A_{2})}} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('A_2', commutative=True)), exp(exp(Symbol('A_2', commutative=True))))"], [["divide", 1, "Function('F_x')(Symbol('A_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('A_2', commutative=True)), Integer(-1)), exp(exp(Symbol('A_2', commutative=True)))))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_2', commutative=True))), Integral(Mul(Pow(Function('F_x')(Symbol('A_2', commutative=True)), Integer(-1)), exp(exp(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Mul(Pow(Function('F_x')(Symbol('A_2', commutative=True)), Integer(-1)), exp(exp(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{p},Z)} = Z + \\hat{p} and n{(\\hat{p},Z)} = \\hat{p} \\mathbb{I}{(\\hat{p},Z)} and \\operatorname{E_{\\lambda}}{(b,\\pi)} = \\pi + b, then obtain \\int \\hat{p} (\\pi + 2 b) \\mathbb{I}{(\\hat{p},Z)} db = \\int \\hat{p} (Z + \\hat{p}) (\\pi + 2 b) db", "derivation": "\\mathbb{I}{(\\hat{p},Z)} = Z + \\hat{p} and n{(\\hat{p},Z)} = \\hat{p} \\mathbb{I}{(\\hat{p},Z)} and \\operatorname{E_{\\lambda}}{(b,\\pi)} = \\pi + b and n{(\\hat{p},Z)} = \\hat{p} (Z + \\hat{p}) and (b + \\operatorname{E_{\\lambda}}{(b,\\pi)}) n{(\\hat{p},Z)} = \\hat{p} (Z + \\hat{p}) (b + \\operatorname{E_{\\lambda}}{(b,\\pi)}) and \\hat{p} (b + \\operatorname{E_{\\lambda}}{(b,\\pi)}) \\mathbb{I}{(\\hat{p},Z)} = \\hat{p} (Z + \\hat{p}) (b + \\operatorname{E_{\\lambda}}{(b,\\pi)}) and \\hat{p} (\\pi + 2 b) \\mathbb{I}{(\\hat{p},Z)} = \\hat{p} (Z + \\hat{p}) (\\pi + 2 b) and \\int \\hat{p} (\\pi + 2 b) \\mathbb{I}{(\\hat{p},Z)} db = \\int \\hat{p} (Z + \\hat{p}) (\\pi + 2 b) db", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))))"], ["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["times", 4, "Add(Symbol('b', commutative=True), Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Symbol('b', commutative=True), Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('b', commutative=True), Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('b', commutative=True), Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('b', commutative=True), Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Symbol('b', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Symbol('b', commutative=True)))))"], [["integrate", 7, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Symbol('b', commutative=True))), Function('\\\\mathbb{I}')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})}, then derive \\frac{d}{d \\tilde{g}} \\operatorname{v_{2}}{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain - \\operatorname{v_{2}}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\operatorname{v_{2}}{(\\tilde{g})} = - \\operatorname{v_{2}}{(\\tilde{g})} + \\cos{(\\tilde{g})}", "derivation": "\\operatorname{v_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\operatorname{v_{2}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\sin{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\operatorname{v_{2}}{(\\tilde{g})} = \\cos{(\\tilde{g})} and - \\operatorname{v_{2}}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\operatorname{v_{2}}{(\\tilde{g})} = - \\operatorname{v_{2}}{(\\tilde{g})} + \\cos{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 3, "Function('v_2')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\tilde{g}', commutative=True))), Derivative(Function('v_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\tilde{g}', commutative=True))), cos(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(c_{0},\\Omega)} = \\sin{(c_{0}^{\\Omega})}, then obtain (\\iint \\operatorname{C_{2}}{(c_{0},\\Omega)} d\\Omega d\\Omega)^{c_{0}} = (\\iint \\sin{(c_{0}^{\\Omega})} d\\Omega d\\Omega)^{c_{0}}", "derivation": "\\operatorname{C_{2}}{(c_{0},\\Omega)} = \\sin{(c_{0}^{\\Omega})} and \\int \\operatorname{C_{2}}{(c_{0},\\Omega)} d\\Omega = \\int \\sin{(c_{0}^{\\Omega})} d\\Omega and \\iint \\operatorname{C_{2}}{(c_{0},\\Omega)} d\\Omega d\\Omega = \\iint \\sin{(c_{0}^{\\Omega})} d\\Omega d\\Omega and (\\iint \\operatorname{C_{2}}{(c_{0},\\Omega)} d\\Omega d\\Omega)^{c_{0}} = (\\iint \\sin{(c_{0}^{\\Omega})} d\\Omega d\\Omega)^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Pow(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Pow(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Pow(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Integral(Function('C_2')(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('c_0', commutative=True)), Pow(Integral(sin(Pow(Symbol('c_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(T,\\mathbf{J}_M)} = T + \\mathbf{J}_M and \\hat{\\mathbf{r}}{(T,\\mathbf{J}_M,v)} = T + \\mathbf{J}_M + v, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_M} (T + \\mathbf{J}_M + v) = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\hat{\\mathbf{r}}{(T,\\mathbf{J}_M,v)}", "derivation": "\\hat{x}_0{(T,\\mathbf{J}_M)} = T + \\mathbf{J}_M and v + \\hat{x}_0{(T,\\mathbf{J}_M)} = T + \\mathbf{J}_M + v and \\frac{\\partial}{\\partial \\mathbf{J}_M} (v + \\hat{x}_0{(T,\\mathbf{J}_M)}) = \\frac{\\partial}{\\partial \\mathbf{J}_M} (T + \\mathbf{J}_M + v) and \\hat{\\mathbf{r}}{(T,\\mathbf{J}_M,v)} = T + \\mathbf{J}_M + v and \\frac{\\partial}{\\partial \\mathbf{J}_M} (v + \\hat{x}_0{(T,\\mathbf{J}_M)}) = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\hat{\\mathbf{r}}{(T,\\mathbf{J}_M,v)} and \\frac{\\partial}{\\partial \\mathbf{J}_M} (T + \\mathbf{J}_M + v) = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\hat{\\mathbf{r}}{(T,\\mathbf{J}_M,v)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Add(Symbol('v', commutative=True), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('v', commutative=True), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(x,F_{H})} = F_{H} + x, then derive \\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})} = 1, then obtain \\frac{\\partial}{\\partial F_{H}} \\frac{\\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})}}{x} = \\frac{d}{d F_{H}} \\frac{1}{x}", "derivation": "\\mathbf{S}{(x,F_{H})} = F_{H} + x and \\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})} = \\frac{\\partial}{\\partial x} (F_{H} + x) and \\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})} = 1 and \\frac{\\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})}}{x} = \\frac{1}{x} and \\frac{\\frac{\\partial}{\\partial x} (F_{H} + x)}{x} = \\frac{1}{x} and \\frac{\\partial}{\\partial F_{H}} \\frac{\\frac{\\partial}{\\partial x} (F_{H} + x)}{x} = \\frac{d}{d F_{H}} \\frac{1}{x} and \\frac{\\partial}{\\partial F_{H}} \\frac{\\frac{\\partial}{\\partial x} \\mathbf{S}{(x,F_{H})}}{x} = \\frac{d}{d F_{H}} \\frac{1}{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('x', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Add(Symbol('F_H', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["differentiate", 5, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Add(Symbol('F_H', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Symbol('x', commutative=True), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{S}')(Symbol('x', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Symbol('x', commutative=True), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(t_{1})} = \\sin{(t_{1})} and \\ddot{x}{(t_{1})} = t_{1}, then obtain - \\sin{(t_{1})} + \\int \\ddot{x}{(t_{1})} d\\ddot{x}{(t_{1})} = - \\sin{(t_{1})} + \\int t_{1} d\\ddot{x}{(t_{1})}", "derivation": "\\operatorname{f^{\\prime}}{(t_{1})} = \\sin{(t_{1})} and \\ddot{x}{(t_{1})} = t_{1} and \\int \\ddot{x}{(t_{1})} dt_{1} = \\int t_{1} dt_{1} and \\int \\ddot{x}{(t_{1})} d\\ddot{x}{(t_{1})} = \\int t_{1} d\\ddot{x}{(t_{1})} and - \\operatorname{f^{\\prime}}{(t_{1})} + \\int \\ddot{x}{(t_{1})} d\\ddot{x}{(t_{1})} = - \\operatorname{f^{\\prime}}{(t_{1})} + \\int t_{1} d\\ddot{x}{(t_{1})} and - \\sin{(t_{1})} + \\int \\ddot{x}{(t_{1})} d\\ddot{x}{(t_{1})} = - \\sin{(t_{1})} + \\int t_{1} d\\ddot{x}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)))), Integral(Symbol('t_1', commutative=True), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)))))"], [["minus", 4, "Function('f^{\\\\prime}')(Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('t_1', commutative=True))), Integral(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True))))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Integral(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True)), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True))))), Add(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Integral(Symbol('t_1', commutative=True), Tuple(Function('\\\\ddot{x}')(Symbol('t_1', commutative=True))))))"]]}, {"prompt": "Given t{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain \\frac{(\\frac{d}{d \\mathbf{s}} t{(\\mathbf{s})})^{\\mathbf{s}}}{- \\ddot{x} + \\log{(\\hbar)}} = \\frac{(\\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}})^{\\mathbf{s}}}{- \\ddot{x} + \\log{(\\hbar)}}", "derivation": "t{(\\mathbf{s})} = e^{\\mathbf{s}} and \\frac{d}{d \\mathbf{s}} t{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} and (\\frac{d}{d \\mathbf{s}} t{(\\mathbf{s})})^{\\mathbf{s}} = (\\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}})^{\\mathbf{s}} and \\frac{(\\frac{d}{d \\mathbf{s}} t{(\\mathbf{s})})^{\\mathbf{s}}}{- \\ddot{x} + \\log{(\\hbar)}} = \\frac{(\\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}})^{\\mathbf{s}}}{- \\ddot{x} + \\log{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Derivative(Function('t')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Integer(-1)), Pow(Derivative(Function('t')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\hbar', commutative=True))), Integer(-1)), Pow(Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(H)} = e^{e^{H}} and \\operatorname{x^{{\\}'}}{(H)} = e^{H}, then derive \\frac{d}{d H} \\operatorname{f_{E}}{(H)} = e^{H} e^{e^{H}}, then obtain \\operatorname{x^{{\\}'}}{(H)} e^{\\operatorname{x^{{\\}'}}{(H)}} = e^{H} e^{e^{H}}", "derivation": "\\operatorname{f_{E}}{(H)} = e^{e^{H}} and \\frac{d}{d H} \\operatorname{f_{E}}{(H)} = \\frac{d}{d H} e^{e^{H}} and \\frac{d}{d H} \\operatorname{f_{E}}{(H)} = e^{H} e^{e^{H}} and \\operatorname{x^{{\\}'}}{(H)} = e^{H} and \\frac{d}{d H} \\operatorname{f_{E}}{(H)} = \\operatorname{x^{{\\}'}}{(H)} e^{\\operatorname{x^{{\\}'}}{(H)}} and \\operatorname{x^{{\\}'}}{(H)} e^{\\operatorname{x^{{\\}'}}{(H)}} = \\frac{d}{d H} e^{e^{H}} and \\operatorname{x^{{\\}'}}{(H)} e^{\\operatorname{x^{{\\}'}}{(H)}} = e^{H} e^{e^{H}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(exp(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('f_E')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Function('x^\\\\prime')(Symbol('H', commutative=True)), exp(Function('x^\\\\prime')(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Function('x^\\\\prime')(Symbol('H', commutative=True)), exp(Function('x^\\\\prime')(Symbol('H', commutative=True)))), Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Function('x^\\\\prime')(Symbol('H', commutative=True)), exp(Function('x^\\\\prime')(Symbol('H', commutative=True)))), Mul(exp(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(J)} = \\log{(\\sin{(J)})}, then obtain (- J - (- J + \\operatorname{n_{1}}{(J)})^{J})^{J} = (- J - (- J + \\log{(\\sin{(J)})})^{J})^{J}", "derivation": "\\operatorname{n_{1}}{(J)} = \\log{(\\sin{(J)})} and - J + \\operatorname{n_{1}}{(J)} = - J + \\log{(\\sin{(J)})} and (- J + \\operatorname{n_{1}}{(J)})^{J} = (- J + \\log{(\\sin{(J)})})^{J} and - (- J + \\operatorname{n_{1}}{(J)})^{J} = - (- J + \\log{(\\sin{(J)})})^{J} and - J - (- J + \\operatorname{n_{1}}{(J)})^{J} = - J - (- J + \\log{(\\sin{(J)})})^{J} and (- J - (- J + \\operatorname{n_{1}}{(J)})^{J})^{J} = (- J - (- J + \\log{(\\sin{(J)})})^{J})^{J}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True))))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('n_1')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('n_1')(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('n_1')(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), Symbol('J', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('n_1')(Symbol('J', commutative=True))), Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('n_1')(Symbol('J', commutative=True))), Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), log(sin(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given y{(n,\\mathbf{A})} = \\mathbf{A} + n, then derive \\int y{(n,\\mathbf{A})} dn = \\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2}, then obtain \\frac{\\cos{((\\int y{(n,\\mathbf{A})} dn)^{4})}}{\\int (\\mathbf{A} + n) dn} = \\frac{\\cos{((\\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2})^{4})}}{\\int (\\mathbf{A} + n) dn}", "derivation": "y{(n,\\mathbf{A})} = \\mathbf{A} + n and \\int y{(n,\\mathbf{A})} dn = \\int (\\mathbf{A} + n) dn and \\int y{(n,\\mathbf{A})} dn = \\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2} and (\\int y{(n,\\mathbf{A})} dn)^{2} = (\\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2})^{2} and (\\int y{(n,\\mathbf{A})} dn)^{4} = (\\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2})^{4} and \\cos{((\\int y{(n,\\mathbf{A})} dn)^{4})} = \\cos{((\\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2})^{4})} and \\frac{\\cos{((\\int y{(n,\\mathbf{A})} dn)^{4})}}{\\int (\\mathbf{A} + n) dn} = \\frac{\\cos{((\\mathbf{A} n + f^{\\prime} + \\frac{n^{2}}{2})^{4})}}{\\int (\\mathbf{A} + n) dn}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))))"], [["power", 3, 2], "Equality(Pow(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(2)), Pow(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(4)), Pow(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Integer(4)))"], [["cos", 5], "Equality(cos(Pow(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(4))), cos(Pow(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Integer(4))))"], [["divide", 6, "Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Mul(cos(Pow(Integral(Function('y')(Symbol('n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(4))), Pow(Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))), Mul(cos(Pow(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Integer(4))), Pow(Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given A{(\\mu,r)} = \\mu^{r} and \\operatorname{g_{\\varepsilon}}{(\\mu,r)} = \\mu^{r}, then obtain \\mu^{- r} \\operatorname{g_{\\varepsilon}}{(\\mu,r)} + \\mu^{- r} = 1 + \\mu^{- r}", "derivation": "A{(\\mu,r)} = \\mu^{r} and \\mu^{- r} A{(\\mu,r)} = 1 and \\operatorname{g_{\\varepsilon}}{(\\mu,r)} = \\mu^{r} and A{(\\mu,r)} = \\operatorname{g_{\\varepsilon}}{(\\mu,r)} and \\mu^{- r} \\operatorname{g_{\\varepsilon}}{(\\mu,r)} = 1 and \\mu^{- r} \\operatorname{g_{\\varepsilon}}{(\\mu,r)} + \\mu^{- r} = 1 + \\mu^{- r}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('A')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True))), Integer(1))"], [["add", 5, "Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True))), Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Add(Integer(1), Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given z{(\\mu_0,J_{\\varepsilon})} = \\frac{\\mu_0}{J_{\\varepsilon}}, then derive \\frac{\\partial}{\\partial \\mu_0} z{(\\mu_0,J_{\\varepsilon})} + \\frac{1}{J_{\\varepsilon}} = \\frac{2}{J_{\\varepsilon}}, then obtain \\int (\\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0}{J_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}}) d\\mu_0 = \\int \\frac{2}{J_{\\varepsilon}} d\\mu_0", "derivation": "z{(\\mu_0,J_{\\varepsilon})} = \\frac{\\mu_0}{J_{\\varepsilon}} and z{(\\mu_0,J_{\\varepsilon})} + \\frac{\\mu_0}{J_{\\varepsilon}} = \\frac{2 \\mu_0}{J_{\\varepsilon}} and \\frac{\\partial}{\\partial \\mu_0} (z{(\\mu_0,J_{\\varepsilon})} + \\frac{\\mu_0}{J_{\\varepsilon}}) = \\frac{\\partial}{\\partial \\mu_0} \\frac{2 \\mu_0}{J_{\\varepsilon}} and \\frac{\\partial}{\\partial \\mu_0} z{(\\mu_0,J_{\\varepsilon})} + \\frac{1}{J_{\\varepsilon}} = \\frac{2}{J_{\\varepsilon}} and \\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0}{J_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}} = \\frac{2}{J_{\\varepsilon}} and \\int (\\frac{\\partial}{\\partial \\mu_0} \\frac{\\mu_0}{J_{\\varepsilon}} + \\frac{1}{J_{\\varepsilon}}) d\\mu_0 = \\int \\frac{2}{J_{\\varepsilon}} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Function('z')(Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('z')(Symbol('\\\\mu_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Derivative(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\phi,L)} = - L + \\phi and U{(\\phi,L)} = - \\frac{\\operatorname{m_{s}}{(\\phi,L)}}{L}, then obtain \\frac{(- \\frac{- L + \\phi}{L})^{- L} (L - \\phi)}{L^{2}} = \\frac{(- \\frac{- L + \\phi}{L})^{- L} U{(\\phi,L)}}{L}", "derivation": "\\operatorname{m_{s}}{(\\phi,L)} = - L + \\phi and - \\frac{\\operatorname{m_{s}}{(\\phi,L)}}{L} = - \\frac{- L + \\phi}{L} and U{(\\phi,L)} = - \\frac{\\operatorname{m_{s}}{(\\phi,L)}}{L} and \\frac{U{(\\phi,L)}}{L} = - \\frac{\\operatorname{m_{s}}{(\\phi,L)}}{L^{2}} and U{(\\phi,L)} = \\frac{L - \\phi}{L} and \\frac{L - \\phi}{L^{2}} = - \\frac{\\operatorname{m_{s}}{(\\phi,L)}}{L^{2}} and \\frac{L - \\phi}{L^{2}} = \\frac{U{(\\phi,L)}}{L} and \\frac{(- \\frac{- L + \\phi}{L})^{- L} (L - \\phi)}{L^{2}} = \\frac{(- \\frac{- L + \\phi}{L})^{- L} U{(\\phi,L)}}{L}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('L', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))))"], [["times", 3, "Pow(Symbol('L', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('U')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-2)), Function('m_s')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('U')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-2)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-2)), Function('m_s')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-2)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('U')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))))"], [["divide", 7, "Pow(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-2)), Pow(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True))), Function('U')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)} = \\log{(G - \\hat{x})}, then obtain (G - \\hat{x}) \\cos^{\\hat{x}}{(\\hat{x} - \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)})} = (G - \\hat{x}) \\cos^{\\hat{x}}{(\\hat{x} - \\log{(G - \\hat{x})})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)} = \\log{(G - \\hat{x})} and - \\hat{x} + \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)} = - \\hat{x} + \\log{(G - \\hat{x})} and \\cos{(\\hat{x} - \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)})} = \\cos{(\\hat{x} - \\log{(G - \\hat{x})})} and \\cos^{\\hat{x}}{(\\hat{x} - \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)})} = \\cos^{\\hat{x}}{(\\hat{x} - \\log{(G - \\hat{x})})} and (G - \\hat{x}) \\cos^{\\hat{x}}{(\\hat{x} - \\operatorname{V_{\\mathbf{E}}}{(\\hat{x},G)})} = (G - \\hat{x}) \\cos^{\\hat{x}}{(\\hat{x} - \\log{(G - \\hat{x})})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('G', commutative=True)), log(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)))))"], [["minus", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), log(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('G', commutative=True))))), cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), log(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))))))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('G', commutative=True))))), Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), log(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))))), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 4, "Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))), Pow(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('G', commutative=True))))), Symbol('\\\\hat{x}', commutative=True))), Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))), Pow(cos(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), log(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))))), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given s{(n_{2})} = e^{n_{2}}, then derive \\int \\frac{s{(n_{2})}}{n_{2}} dn_{2} = \\phi_2 + \\operatorname{Ei}{(n_{2})}, then obtain \\phi_2 + \\operatorname{Ei}{(n_{2})} = M + \\operatorname{Ei}{(n_{2})}", "derivation": "s{(n_{2})} = e^{n_{2}} and \\frac{s{(n_{2})}}{n_{2}} = \\frac{e^{n_{2}}}{n_{2}} and \\int \\frac{s{(n_{2})}}{n_{2}} dn_{2} = \\int \\frac{e^{n_{2}}}{n_{2}} dn_{2} and \\int \\frac{s{(n_{2})}}{n_{2}} dn_{2} = \\phi_2 + \\operatorname{Ei}{(n_{2})} and \\phi_2 + \\operatorname{Ei}{(n_{2})} = \\int \\frac{e^{n_{2}}}{n_{2}} dn_{2} and \\phi_2 + \\operatorname{Ei}{(n_{2})} = M + \\operatorname{Ei}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["divide", 1, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('s')(Symbol('n_2', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), exp(Symbol('n_2', commutative=True))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('s')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('s')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Ei(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Ei(Symbol('n_2', commutative=True))), Integral(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Ei(Symbol('n_2', commutative=True))), Add(Symbol('M', commutative=True), Ei(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(C_{2})} = \\log{(C_{2})}, then obtain 0 = - C_{2} \\frac{d}{d C_{2}} \\operatorname{F_{c}}{(C_{2})} - \\operatorname{F_{c}}{(C_{2})} + \\log{(C_{2})} + 1", "derivation": "\\operatorname{F_{c}}{(C_{2})} = \\log{(C_{2})} and C_{2} \\operatorname{F_{c}}{(C_{2})} = C_{2} \\log{(C_{2})} and \\frac{d}{d C_{2}} C_{2} \\operatorname{F_{c}}{(C_{2})} = \\frac{d}{d C_{2}} C_{2} \\log{(C_{2})} and 0 = - \\frac{d}{d C_{2}} C_{2} \\operatorname{F_{c}}{(C_{2})} + \\frac{d}{d C_{2}} C_{2} \\log{(C_{2})} and 0 = - C_{2} \\frac{d}{d C_{2}} \\operatorname{F_{c}}{(C_{2})} - \\operatorname{F_{c}}{(C_{2})} + \\log{(C_{2})} + 1", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('F_c')(Symbol('C_2', commutative=True))), Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('C_2', commutative=True), Function('F_c')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Mul(Symbol('C_2', commutative=True), Function('F_c')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Mul(Symbol('C_2', commutative=True), Function('F_c')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Derivative(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Derivative(Function('F_c')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Mul(Integer(-1), Function('F_c')(Symbol('C_2', commutative=True))), log(Symbol('C_2', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\frac{\\hat{x}_0}{f^{\\prime}})} and \\operatorname{a^{\\dagger}}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\frac{\\hat{x}_0}{f^{\\prime}})}, then obtain 1 = \\frac{\\operatorname{a^{\\dagger}}{(\\hat{x}_0,f^{\\prime})}}{\\Psi_{nl}{(\\hat{x}_0,f^{\\prime})}}", "derivation": "\\Psi_{nl}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\frac{\\hat{x}_0}{f^{\\prime}})} and 1 = \\frac{\\sin{(\\frac{\\hat{x}_0}{f^{\\prime}})}}{\\Psi_{nl}{(\\hat{x}_0,f^{\\prime})}} and \\operatorname{a^{\\dagger}}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\frac{\\hat{x}_0}{f^{\\prime}})} and 1 = \\frac{\\operatorname{a^{\\dagger}}{(\\hat{x}_0,f^{\\prime})}}{\\Psi_{nl}{(\\hat{x}_0,f^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"], [["divide", 1, "Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), sin(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(c_{0})} = \\cos{(c_{0})} and \\mathbf{J}{(c_{0})} = \\cos^{c_{0}}{(c_{0})}, then obtain \\frac{d^{2}}{d c_{0}^{2}} \\mathbf{J}{(c_{0})} = \\frac{d^{2}}{d c_{0}^{2}} \\nabla^{c_{0}}{(c_{0})}", "derivation": "\\nabla{(c_{0})} = \\cos{(c_{0})} and \\nabla^{c_{0}}{(c_{0})} = \\cos^{c_{0}}{(c_{0})} and \\mathbf{J}{(c_{0})} = \\cos^{c_{0}}{(c_{0})} and \\frac{d}{d c_{0}} \\mathbf{J}{(c_{0})} = \\frac{d}{d c_{0}} \\cos^{c_{0}}{(c_{0})} and \\frac{d}{d c_{0}} \\mathbf{J}{(c_{0})} = \\frac{d}{d c_{0}} \\nabla^{c_{0}}{(c_{0})} and \\frac{d^{2}}{d c_{0}^{2}} \\mathbf{J}{(c_{0})} = \\frac{d^{2}}{d c_{0}^{2}} \\nabla^{c_{0}}{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\nabla')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(2))), Derivative(Pow(Function('\\\\nabla')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(2))))"]]}, {"prompt": "Given B{(g)} = \\sin{(e^{g})}, then derive \\int B{(g)} dg = \\hbar + \\operatorname{Si}{(e^{g})}, then obtain \\iint \\frac{(\\int \\sin{(e^{g})} dg)^{\\hbar}}{g} d\\hbar d\\hbar = \\iint \\frac{(\\hbar + \\operatorname{Si}{(e^{g})})^{\\hbar}}{g} d\\hbar d\\hbar", "derivation": "B{(g)} = \\sin{(e^{g})} and \\int B{(g)} dg = \\int \\sin{(e^{g})} dg and \\int B{(g)} dg = \\hbar + \\operatorname{Si}{(e^{g})} and \\int \\sin{(e^{g})} dg = \\hbar + \\operatorname{Si}{(e^{g})} and (\\int \\sin{(e^{g})} dg)^{\\hbar} = (\\hbar + \\operatorname{Si}{(e^{g})})^{\\hbar} and \\frac{(\\int \\sin{(e^{g})} dg)^{\\hbar}}{g} = \\frac{(\\hbar + \\operatorname{Si}{(e^{g})})^{\\hbar}}{g} and \\int \\frac{(\\int \\sin{(e^{g})} dg)^{\\hbar}}{g} d\\hbar = \\int \\frac{(\\hbar + \\operatorname{Si}{(e^{g})})^{\\hbar}}{g} d\\hbar and \\iint \\frac{(\\int \\sin{(e^{g})} dg)^{\\hbar}}{g} d\\hbar d\\hbar = \\iint \\frac{(\\hbar + \\operatorname{Si}{(e^{g})})^{\\hbar}}{g} d\\hbar d\\hbar", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))), Symbol('\\\\hbar', commutative=True)))"], [["divide", 5, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 6, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["integrate", 7, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Integral(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hbar', commutative=True), Si(exp(Symbol('g', commutative=True)))), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\hat{X},B)} = \\frac{B}{\\hat{X}} and E{(c)} = e^{c} and \\operatorname{A_{z}}{(\\hat{X},B)} = B + \\frac{B}{\\hat{X}}, then obtain \\operatorname{A_{z}}{(\\hat{X},B)} + e^{c} = B + \\frac{B}{\\hat{X}} + e^{c}", "derivation": "\\hat{H}_l{(\\hat{X},B)} = \\frac{B}{\\hat{X}} and E{(c)} = e^{c} and \\operatorname{A_{z}}{(\\hat{X},B)} = B + \\frac{B}{\\hat{X}} and \\operatorname{A_{z}}{(\\hat{X},B)} = B + \\hat{H}_l{(\\hat{X},B)} and \\operatorname{A_{z}}{(\\hat{X},B)} + E{(c)} = B + E{(c)} + \\hat{H}_l{(\\hat{X},B)} and \\operatorname{A_{z}}{(\\hat{X},B)} + e^{c} = B + \\hat{H}_l{(\\hat{X},B)} + e^{c} and \\operatorname{A_{z}}{(\\hat{X},B)} + e^{c} = B + \\frac{B}{\\hat{X}} + e^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('E')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_z')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True))))"], [["add", 4, "Function('E')(Symbol('c', commutative=True))"], "Equality(Add(Function('A_z')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), Function('E')(Symbol('c', commutative=True))), Add(Symbol('B', commutative=True), Function('E')(Symbol('c', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('A_z')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), exp(Symbol('c', commutative=True))), Add(Symbol('B', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), exp(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Function('A_z')(Symbol('\\\\hat{X}', commutative=True), Symbol('B', commutative=True)), exp(Symbol('c', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1))), exp(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\hat{H}{(\\theta_2,s)} = \\int \\theta_2^{s} d\\theta_2, then obtain \\frac{\\int \\hat{H}{(\\theta_2,s)} ds}{\\log{(\\varepsilon)}^{2}} = \\frac{\\iint \\theta_2^{s} d\\theta_2 ds}{\\log{(\\varepsilon)}^{2}}", "derivation": "\\operatorname{P_{e}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\hat{H}{(\\theta_2,s)} = \\int \\theta_2^{s} d\\theta_2 and \\int \\hat{H}{(\\theta_2,s)} ds = \\iint \\theta_2^{s} d\\theta_2 ds and \\frac{\\operatorname{P_{e}}^{4}{(\\varepsilon)} \\int \\hat{H}{(\\theta_2,s)} ds}{\\log{(\\varepsilon)}^{3}} = \\frac{\\operatorname{P_{e}}^{4}{(\\varepsilon)} \\iint \\theta_2^{s} d\\theta_2 ds}{\\log{(\\varepsilon)}^{3}} and \\log{(\\varepsilon)} \\int \\hat{H}{(\\theta_2,s)} ds = \\log{(\\varepsilon)} \\iint \\theta_2^{s} d\\theta_2 ds and \\frac{\\int \\hat{H}{(\\theta_2,s)} ds}{\\log{(\\varepsilon)}^{2}} = \\frac{\\iint \\theta_2^{s} d\\theta_2 ds}{\\log{(\\varepsilon)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Integral(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["divide", 3, "Mul(Pow(Function('P_e')(Symbol('\\\\varepsilon', commutative=True)), Integer(-4)), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(3)))"], "Equality(Mul(Pow(Function('P_e')(Symbol('\\\\varepsilon', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-3)), Integral(Function('\\\\hat{H}')(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Pow(Function('P_e')(Symbol('\\\\varepsilon', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-3)), Integral(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(log(Symbol('\\\\varepsilon', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(log(Symbol('\\\\varepsilon', commutative=True)), Integral(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["divide", 5, "Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(3))"], "Equality(Mul(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Integral(Function('\\\\hat{H}')(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-2)), Integral(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(f,y^{\\prime})} = y^{\\prime} + \\cos{(f)}, then derive \\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})} = - \\sin{(f)}, then obtain - \\frac{\\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})}}{\\sin{(f)}} = 1", "derivation": "\\theta_{1}{(f,y^{\\prime})} = y^{\\prime} + \\cos{(f)} and \\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})} = \\frac{\\partial}{\\partial f} (y^{\\prime} + \\cos{(f)}) and \\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})} = - \\sin{(f)} and (y^{\\prime} + \\cos{(f)}) \\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})} = - (y^{\\prime} + \\cos{(f)}) \\sin{(f)} and - \\frac{\\frac{\\partial}{\\partial f} \\theta_{1}{(f,y^{\\prime})}}{\\sin{(f)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('f', commutative=True))))"], [["times", 3, "Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True)))"], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True))), Derivative(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Add(Symbol('y^{\\\\prime}', commutative=True), cos(Symbol('f', commutative=True))), sin(Symbol('f', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} = \\hat{\\mathbf{x}} + s, then obtain \\frac{d}{d s} - s = \\frac{\\partial}{\\partial s} s (\\hat{\\mathbf{x}} + s - \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} - 1)", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} = \\hat{\\mathbf{x}} + s and 0 = \\hat{\\mathbf{x}} + s - \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} and -1 = \\hat{\\mathbf{x}} + s - \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} - 1 and - s = s (\\hat{\\mathbf{x}} + s - \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} - 1) and \\frac{d}{d s} - s = \\frac{\\partial}{\\partial s} s (\\hat{\\mathbf{x}} + s - \\operatorname{L_{\\varepsilon}}{(\\hat{\\mathbf{x}},s)} - 1)", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True))), Integer(-1)))"], [["times", 3, "Symbol('s', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True))), Integer(-1))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('s', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} = \\log{(\\mathbf{r})}^{\\hat{p}_0}, then obtain - \\hat{p}_0 + 2 \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} - \\log{(\\mathbf{r})}^{\\hat{p}_0} = - \\hat{p}_0 + \\log{(\\mathbf{r})}^{\\hat{p}_0}", "derivation": "\\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} = \\log{(\\mathbf{r})}^{\\hat{p}_0} and - \\hat{p}_0 + \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} = - \\hat{p}_0 + \\log{(\\mathbf{r})}^{\\hat{p}_0} and - \\hat{p}_0 + \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} - \\log{(\\mathbf{r})}^{\\hat{p}_0} = - \\hat{p}_0 and - \\hat{p}_0 + 2 \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} - \\log{(\\mathbf{r})}^{\\hat{p}_0} = - \\hat{p}_0 + \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} and - \\hat{p}_0 + 2 \\operatorname{E_{n}}{(\\hat{p}_0,\\mathbf{r})} - \\log{(\\mathbf{r})}^{\\hat{p}_0} = - \\hat{p}_0 + \\log{(\\mathbf{r})}^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 2, "Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{v},y)} = y \\cos{(\\mathbf{v})} and f{(\\mathbf{v},y)} = y \\cos{(\\mathbf{v})}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} f{(\\mathbf{v},y)} = \\frac{\\partial}{\\partial \\mathbf{v}} y \\cos{(\\mathbf{v})}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{v},y)} = y \\cos{(\\mathbf{v})} and f{(\\mathbf{v},y)} = y \\cos{(\\mathbf{v})} and \\operatorname{C_{2}}{(\\mathbf{v},y)} = f{(\\mathbf{v},y)} and \\frac{\\partial}{\\partial \\mathbf{v}} \\operatorname{C_{2}}{(\\mathbf{v},y)} = \\frac{\\partial}{\\partial \\mathbf{v}} y \\cos{(\\mathbf{v})} and \\frac{\\partial}{\\partial \\mathbf{v}} f{(\\mathbf{v},y)} = \\frac{\\partial}{\\partial \\mathbf{v}} y \\cos{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('y', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('y', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Function('f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Symbol('y', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Function('f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Symbol('y', commutative=True), cos(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(\\psi)} = \\sin{(\\psi)}, then derive \\int G{(\\psi)} d\\psi = L_{\\varepsilon} - \\cos{(\\psi)}, then obtain \\int \\sin{(\\psi)} d\\psi = L_{\\varepsilon} - \\cos{(\\psi)}", "derivation": "G{(\\psi)} = \\sin{(\\psi)} and \\int G{(\\psi)} d\\psi = \\int \\sin{(\\psi)} d\\psi and \\int G{(\\psi)} d\\psi = L_{\\varepsilon} - \\cos{(\\psi)} and \\int \\sin{(\\psi)} d\\psi = L_{\\varepsilon} - \\cos{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given v{(\\mathbf{J}_P)} = \\log{(\\log{(\\mathbf{J}_P)})} and \\operatorname{f_{\\mathbf{v}}}{(s)} = \\log{(s)}, then obtain - \\log{(\\log{(\\mathbf{J}_P)})}^{\\mathbf{J}_P} + \\int \\operatorname{f_{\\mathbf{v}}}{(s)} ds = - \\log{(\\log{(\\mathbf{J}_P)})}^{\\mathbf{J}_P} + \\int \\log{(s)} ds", "derivation": "v{(\\mathbf{J}_P)} = \\log{(\\log{(\\mathbf{J}_P)})} and v^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = \\log{(\\log{(\\mathbf{J}_P)})}^{\\mathbf{J}_P} and \\operatorname{f_{\\mathbf{v}}}{(s)} = \\log{(s)} and \\int \\operatorname{f_{\\mathbf{v}}}{(s)} ds = \\int \\log{(s)} ds and - v^{\\mathbf{J}_P}{(\\mathbf{J}_P)} + \\int \\operatorname{f_{\\mathbf{v}}}{(s)} ds = - v^{\\mathbf{J}_P}{(\\mathbf{J}_P)} + \\int \\log{(s)} ds and - \\log{(\\log{(\\mathbf{J}_P)})}^{\\mathbf{J}_P} + \\int \\operatorname{f_{\\mathbf{v}}}{(s)} ds = - \\log{(\\log{(\\mathbf{J}_P)})}^{\\mathbf{J}_P} + \\int \\log{(s)} ds", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(log(log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["get_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["minus", 4, "Pow(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Pow(log(log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Pow(log(log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(h,A_{2})} = \\cos{(A_{2} + h)}, then derive (\\frac{\\partial}{\\partial A_{2}} \\mathbf{J}_f{(h,A_{2})})^{h} = (- \\sin{(A_{2} + h)})^{h}, then obtain ((\\frac{\\partial}{\\partial A_{2}} \\mathbf{J}_f{(h,A_{2})})^{h})^{h} = ((- \\sin{(A_{2} + h)})^{h})^{h}", "derivation": "\\mathbf{J}_f{(h,A_{2})} = \\cos{(A_{2} + h)} and h + \\mathbf{J}_f{(h,A_{2})} = h + \\cos{(A_{2} + h)} and \\frac{\\partial}{\\partial A_{2}} (h + \\mathbf{J}_f{(h,A_{2})}) = \\frac{\\partial}{\\partial A_{2}} (h + \\cos{(A_{2} + h)}) and (\\frac{\\partial}{\\partial A_{2}} (h + \\mathbf{J}_f{(h,A_{2})}))^{h} = (\\frac{\\partial}{\\partial A_{2}} (h + \\cos{(A_{2} + h)}))^{h} and (\\frac{\\partial}{\\partial A_{2}} \\mathbf{J}_f{(h,A_{2})})^{h} = (- \\sin{(A_{2} + h)})^{h} and ((\\frac{\\partial}{\\partial A_{2}} \\mathbf{J}_f{(h,A_{2})})^{h})^{h} = ((- \\sin{(A_{2} + h)})^{h})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True))), Add(Symbol('h', commutative=True), cos(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Symbol('h', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Symbol('h', commutative=True), cos(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('h', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Derivative(Add(Symbol('h', commutative=True), cos(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["power", 5, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Mul(Integer(-1), sin(Add(Symbol('A_2', commutative=True), Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given l{(I)} = \\sin{(I)}, then obtain 1 = (\\frac{\\sin^{2}{(I)}}{l^{2}{(I)}})^{I}", "derivation": "l{(I)} = \\sin{(I)} and 1 = \\frac{\\sin{(I)}}{l{(I)}} and 1 = (\\frac{\\sin{(I)}}{l{(I)}})^{I} and \\frac{\\sin{(I)}}{l{(I)}} = \\frac{\\sin^{2}{(I)}}{l^{2}{(I)}} and 1 = (\\frac{\\sin^{2}{(I)}}{l^{2}{(I)}})^{I}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["divide", 1, "Function('l')(Symbol('I', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["times", 2, "Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True)))"], "Equality(Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True))), Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-2)), Pow(sin(Symbol('I', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Pow(Mul(Pow(Function('l')(Symbol('I', commutative=True)), Integer(-2)), Pow(sin(Symbol('I', commutative=True)), Integer(2))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + \\mu and \\eta{(\\mu,\\hat{H}_{\\lambda})} = \\iint (\\hat{H}_{\\lambda} - \\mu) d\\mu d\\mu, then obtain \\hat{H}_{\\lambda} + \\eta{(\\mu,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\iint - \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} d\\mu d\\mu", "derivation": "\\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + \\mu and - \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - \\mu and \\int - \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} d\\mu = \\int (\\hat{H}_{\\lambda} - \\mu) d\\mu and \\iint - \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} d\\mu d\\mu = \\iint (\\hat{H}_{\\lambda} - \\mu) d\\mu d\\mu and \\eta{(\\mu,\\hat{H}_{\\lambda})} = \\iint (\\hat{H}_{\\lambda} - \\mu) d\\mu d\\mu and \\hat{H}_{\\lambda} + \\eta{(\\mu,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\iint (\\hat{H}_{\\lambda} - \\mu) d\\mu d\\mu and \\hat{H}_{\\lambda} + \\eta{(\\mu,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\iint - \\phi_{2}{(\\mu,\\hat{H}_{\\lambda})} d\\mu d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["add", 5, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(M,\\varphi,v_{y})} = \\varphi (M + v_{y}), then derive \\frac{\\partial}{\\partial v_{y}} \\mathbf{F}{(M,\\varphi,v_{y})} + 1 = \\varphi + 1, then obtain \\frac{\\partial}{\\partial \\varphi} \\dot{z} (\\frac{\\partial}{\\partial v_{y}} \\varphi (M + v_{y}) + 1) = \\frac{\\partial}{\\partial \\varphi} \\dot{z} (\\varphi + 1)", "derivation": "\\mathbf{F}{(M,\\varphi,v_{y})} = \\varphi (M + v_{y}) and M + v_{y} + \\mathbf{F}{(M,\\varphi,v_{y})} = M + \\varphi (M + v_{y}) + v_{y} and \\frac{\\partial}{\\partial v_{y}} (M + v_{y} + \\mathbf{F}{(M,\\varphi,v_{y})}) = \\frac{\\partial}{\\partial v_{y}} (M + \\varphi (M + v_{y}) + v_{y}) and \\frac{\\partial}{\\partial v_{y}} \\mathbf{F}{(M,\\varphi,v_{y})} + 1 = \\varphi + 1 and \\frac{\\partial}{\\partial v_{y}} \\varphi (M + v_{y}) + 1 = \\varphi + 1 and \\dot{z} (\\frac{\\partial}{\\partial v_{y}} \\varphi (M + v_{y}) + 1) = \\dot{z} (\\varphi + 1) and \\frac{\\partial}{\\partial \\varphi} \\dot{z} (\\frac{\\partial}{\\partial v_{y}} \\varphi (M + v_{y}) + 1) = \\frac{\\partial}{\\partial \\varphi} \\dot{z} (\\varphi + 1)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('M', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))))"], [["add", 1, "Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True), Function('\\\\mathbf{F}')(Symbol('M', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True), Function('\\\\mathbf{F}')(Symbol('M', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('M', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\varphi', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\varphi', commutative=True), Integer(1)))"], [["times", 5, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('M', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\varphi', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})} = F_{N} \\hat{\\mathbf{r}}, then obtain F_{N} + \\frac{\\partial}{\\partial F_{N}} \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})} = F_{N} + \\hat{\\mathbf{r}}", "derivation": "\\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})} = F_{N} \\hat{\\mathbf{r}} and - \\hat{\\mathbf{r}} + \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})} = F_{N} \\hat{\\mathbf{r}} - \\hat{\\mathbf{r}} and \\frac{\\partial}{\\partial F_{N}} (- \\hat{\\mathbf{r}} + \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})}) = \\frac{\\partial}{\\partial F_{N}} (F_{N} \\hat{\\mathbf{r}} - \\hat{\\mathbf{r}}) and F_{N} + \\frac{\\partial}{\\partial F_{N}} (- \\hat{\\mathbf{r}} + \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})}) = F_{N} + \\frac{\\partial}{\\partial F_{N}} (F_{N} \\hat{\\mathbf{r}} - \\hat{\\mathbf{r}}) and F_{N} + \\frac{\\partial}{\\partial F_{N}} \\operatorname{n_{1}}{(\\hat{\\mathbf{r}},F_{N})} = F_{N} + \\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 3, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Symbol('F_N', commutative=True), Derivative(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('F_N', commutative=True), Derivative(Function('n_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Symbol('F_N', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given J{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})}, then derive \\frac{d}{d \\Psi_{\\lambda}} J{(\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda})}, then obtain - J{(\\Psi_{\\lambda})} + \\sin{(\\Psi_{\\lambda})} = - J{(\\Psi_{\\lambda})} - \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}", "derivation": "J{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and \\frac{d}{d \\Psi_{\\lambda}} J{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})} and \\frac{d}{d \\Psi_{\\lambda}} J{(\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda})} and - \\sin{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})} and \\sin{(\\Psi_{\\lambda})} = - \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})} and - J{(\\Psi_{\\lambda})} + \\sin{(\\Psi_{\\lambda})} = - J{(\\Psi_{\\lambda})} - \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 4, "Integer(-1)"], "Equality(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["minus", 5, "Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Function('J')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\theta{(\\mathbf{H},\\varepsilon)} = \\mathbf{H} + \\cos{(\\varepsilon)}, then derive \\int (\\mathbf{H} + \\theta{(\\mathbf{H},\\varepsilon)} + \\cos{(\\varepsilon)}) d\\mathbf{H} = \\hat{x} + \\mathbf{H}^{2} + 2 \\mathbf{H} \\cos{(\\varepsilon)}, then obtain \\int (2 \\mathbf{H} + 2 \\cos{(\\varepsilon)}) d\\mathbf{H} = \\hat{x} + \\mathbf{H}^{2} + 2 \\mathbf{H} \\cos{(\\varepsilon)}", "derivation": "\\theta{(\\mathbf{H},\\varepsilon)} = \\mathbf{H} + \\cos{(\\varepsilon)} and \\mathbf{H} + \\theta{(\\mathbf{H},\\varepsilon)} + \\cos{(\\varepsilon)} = 2 \\mathbf{H} + 2 \\cos{(\\varepsilon)} and \\int (\\mathbf{H} + \\theta{(\\mathbf{H},\\varepsilon)} + \\cos{(\\varepsilon)}) d\\mathbf{H} = \\int (2 \\mathbf{H} + 2 \\cos{(\\varepsilon)}) d\\mathbf{H} and \\int (\\mathbf{H} + \\theta{(\\mathbf{H},\\varepsilon)} + \\cos{(\\varepsilon)}) d\\mathbf{H} = \\hat{x} + \\mathbf{H}^{2} + 2 \\mathbf{H} \\cos{(\\varepsilon)} and \\int (2 \\mathbf{H} + 2 \\cos{(\\varepsilon)}) d\\mathbf{H} = \\hat{x} + \\mathbf{H}^{2} + 2 \\mathbf{H} \\cos{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\theta')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\theta')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\theta')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\nabla,u)} = - u + \\sin{(\\nabla)}, then obtain - \\frac{\\operatorname{n_{1}}^{2}{(\\nabla,u)}}{\\nabla u} = - \\frac{(- u + \\sin{(\\nabla)}) \\operatorname{n_{1}}{(\\nabla,u)}}{\\nabla u}", "derivation": "\\operatorname{n_{1}}{(\\nabla,u)} = - u + \\sin{(\\nabla)} and - \\frac{\\operatorname{n_{1}}{(\\nabla,u)}}{u} = - \\frac{- u + \\sin{(\\nabla)}}{u} and - \\frac{\\operatorname{n_{1}}^{2}{(\\nabla,u)}}{u} = - \\frac{(- u + \\sin{(\\nabla)}) \\operatorname{n_{1}}{(\\nabla,u)}}{u} and - \\frac{\\operatorname{n_{1}}^{2}{(\\nabla,u)}}{\\nabla u} = - \\frac{(- u + \\sin{(\\nabla)}) \\operatorname{n_{1}}{(\\nabla,u)}}{\\nabla u}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))))"], [["times", 2, "Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True))))"], [["divide", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Function('n_1')(Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\phi{(u,F_{H})} = F_{H}^{u}, then obtain \\frac{\\partial^{2}}{\\partial u\\partial F_{H}} \\phi{(u,F_{H})} - 1 = -1 + \\frac{F_{H}^{u} u \\log{(F_{H})}}{F_{H}} + \\frac{F_{H}^{u}}{F_{H}}", "derivation": "\\phi{(u,F_{H})} = F_{H}^{u} and \\frac{\\partial}{\\partial F_{H}} \\phi{(u,F_{H})} = \\frac{\\partial}{\\partial F_{H}} F_{H}^{u} and - u + \\frac{\\partial}{\\partial F_{H}} \\phi{(u,F_{H})} = - u + \\frac{\\partial}{\\partial F_{H}} F_{H}^{u} and \\frac{\\partial}{\\partial u} (- u + \\frac{\\partial}{\\partial F_{H}} \\phi{(u,F_{H})}) = \\frac{\\partial}{\\partial u} (- u + \\frac{\\partial}{\\partial F_{H}} F_{H}^{u}) and \\frac{\\partial^{2}}{\\partial u\\partial F_{H}} \\phi{(u,F_{H})} - 1 = -1 + \\frac{F_{H}^{u} u \\log{(F_{H})}}{F_{H}} + \\frac{F_{H}^{u}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True), log(Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(g)} = \\log{(g)}^{2}, then derive \\frac{d}{d g} \\theta_{1}{(g)} = \\frac{2 \\log{(g)}}{g}, then obtain \\frac{\\log{(g)} + \\int \\frac{d}{d g} \\theta_{1}{(g)} dg}{\\frac{d}{d g} \\dot{x}{(g)} \\log{(g)}} = \\frac{\\log{(g)} + \\int \\frac{2 \\log{(g)}}{g} dg}{\\frac{d}{d g} \\dot{x}{(g)} \\log{(g)}}", "derivation": "\\theta_{1}{(g)} = \\log{(g)}^{2} and \\frac{d}{d g} \\theta_{1}{(g)} = \\frac{d}{d g} \\log{(g)}^{2} and \\frac{d}{d g} \\theta_{1}{(g)} = \\frac{2 \\log{(g)}}{g} and \\int \\frac{d}{d g} \\theta_{1}{(g)} dg = \\int \\frac{2 \\log{(g)}}{g} dg and \\log{(g)} + \\int \\frac{d}{d g} \\theta_{1}{(g)} dg = \\log{(g)} + \\int \\frac{2 \\log{(g)}}{g} dg and \\frac{\\log{(g)} + \\int \\frac{d}{d g} \\theta_{1}{(g)} dg}{\\frac{d}{d g} \\dot{x}{(g)} \\log{(g)}} = \\frac{\\log{(g)} + \\int \\frac{2 \\log{(g)}}{g} dg}{\\frac{d}{d g} \\dot{x}{(g)} \\log{(g)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(2)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(2), Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["add", 4, "log(Symbol('g', commutative=True))"], "Equality(Add(log(Symbol('g', commutative=True)), Integral(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True)))), Add(log(Symbol('g', commutative=True)), Integral(Mul(Integer(2), Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["divide", 5, "Derivative(Mul(Function('\\\\dot{x}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))"], "Equality(Mul(Add(log(Symbol('g', commutative=True)), Integral(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True)))), Pow(Derivative(Mul(Function('\\\\dot{x}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))), Mul(Add(log(Symbol('g', commutative=True)), Integral(Mul(Integer(2), Pow(Symbol('g', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Pow(Derivative(Mul(Function('\\\\dot{x}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{A}{(v_{z})} = \\sin{(\\sin{(v_{z})})}, then obtain \\frac{d}{d v_{z}} 0^{v_{z}} \\tilde{\\infty} = \\frac{d}{d v_{z}} 0^{v_{z}} \\tilde{\\infty} (- \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})})^{v_{z}}", "derivation": "\\mathbf{A}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and 0 = - \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})} and 0^{v_{z}} = (- \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})})^{v_{z}} and 0^{v_{z}} \\tilde{\\infty} = \\tilde{\\infty} (- \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})})^{v_{z}} and 0^{v_{z}} \\tilde{\\infty} = 0^{v_{z}} \\tilde{\\infty} (- \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})})^{v_{z}} and \\frac{d}{d v_{z}} 0^{v_{z}} \\tilde{\\infty} = \\frac{d}{d v_{z}} 0^{v_{z}} \\tilde{\\infty} (- \\mathbf{A}{(v_{z})} + \\sin{(\\sin{(v_{z})})})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True)), sin(sin(Symbol('v_z', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))), sin(sin(Symbol('v_z', commutative=True)))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))), sin(sin(Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True)))"], [["divide", 3, 0], "Equality(Mul(Pow(Integer(0), Symbol('v_z', commutative=True)), zoo), Mul(zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))), sin(sin(Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True))))"], [["times", 4, "Pow(Integer(0), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('v_z', commutative=True)), zoo), Mul(Pow(Integer(0), Symbol('v_z', commutative=True)), zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))), sin(sin(Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True))))"], [["differentiate", 5, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Integer(0), Symbol('v_z', commutative=True)), zoo), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Integer(0), Symbol('v_z', commutative=True)), zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('v_z', commutative=True))), sin(sin(Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(J,r)} = r^{J}, then obtain \\frac{d}{d r} (-1) = \\frac{\\partial}{\\partial r} - \\frac{\\int r^{J} dJ}{\\int \\phi{(J,r)} dJ}", "derivation": "\\phi{(J,r)} = r^{J} and \\int \\phi{(J,r)} dJ = \\int r^{J} dJ and - \\int \\phi{(J,r)} dJ = - \\int r^{J} dJ and -1 = - \\frac{\\int r^{J} dJ}{\\int \\phi{(J,r)} dJ} and \\frac{d}{d r} (-1) = \\frac{\\partial}{\\partial r} - \\frac{\\int r^{J} dJ}{\\int \\phi{(J,r)} dJ}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Symbol('r', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Integer(-1), Integral(Pow(Symbol('r', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Integral(Pow(Symbol('r', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(Pow(Symbol('r', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Function('\\\\phi')(Symbol('J', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})} = \\operatorname{t_{2}}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}}, then obtain \\operatorname{t_{2}}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})}} = 1", "derivation": "\\operatorname{t_{2}}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\operatorname{t_{2}}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} = 1 and \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})} = \\operatorname{t_{2}}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda}} and \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})} = 1 and - \\Psi_{\\lambda} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})} = - \\Psi_{\\lambda} and \\operatorname{t_{2}}{(\\Psi_{\\lambda})} e^{- \\Psi_{\\lambda} \\operatorname{f_{\\mathbf{v}}}{(\\Psi_{\\lambda})}} = 1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('t_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Function('t_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Function('t_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mu_0)} = \\sin{(\\mu_0)} and v{(\\mathbf{J}_f,l)} = \\frac{l}{\\mathbf{J}_f}, then obtain - \\log{(\\operatorname{f^{\\prime}}{(\\mu_0)})} + \\frac{\\partial}{\\partial l} v{(\\mathbf{J}_f,l)} = - \\log{(\\operatorname{f^{\\prime}}{(\\mu_0)})} + \\frac{\\partial}{\\partial l} \\frac{l}{\\mathbf{J}_f}", "derivation": "\\operatorname{f^{\\prime}}{(\\mu_0)} = \\sin{(\\mu_0)} and v{(\\mathbf{J}_f,l)} = \\frac{l}{\\mathbf{J}_f} and \\frac{\\partial}{\\partial l} v{(\\mathbf{J}_f,l)} = \\frac{\\partial}{\\partial l} \\frac{l}{\\mathbf{J}_f} and \\log{(\\operatorname{f^{\\prime}}{(\\mu_0)})} = \\log{(\\sin{(\\mu_0)})} and - \\log{(\\sin{(\\mu_0)})} + \\frac{\\partial}{\\partial l} v{(\\mathbf{J}_f,l)} = - \\log{(\\sin{(\\mu_0)})} + \\frac{\\partial}{\\partial l} \\frac{l}{\\mathbf{J}_f} and - \\log{(\\operatorname{f^{\\prime}}{(\\mu_0)})} + \\frac{\\partial}{\\partial l} v{(\\mathbf{J}_f,l)} = - \\log{(\\operatorname{f^{\\prime}}{(\\mu_0)})} + \\frac{\\partial}{\\partial l} \\frac{l}{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], ["get_premise", "Equality(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True))), log(sin(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 3, "log(sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True)))), Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(sin(Symbol('\\\\mu_0', commutative=True)))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), log(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)))), Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(\\mu)} = e^{\\mu}, then obtain \\frac{d}{d \\mu} (\\int I{(\\mu)} d\\mu + \\int e^{\\mu} d\\mu) = \\frac{d}{d \\mu} 2 \\int e^{\\mu} d\\mu", "derivation": "I{(\\mu)} = e^{\\mu} and \\int I{(\\mu)} d\\mu = \\int e^{\\mu} d\\mu and \\int I{(\\mu)} d\\mu + \\int e^{\\mu} d\\mu = 2 \\int e^{\\mu} d\\mu and \\frac{d}{d \\mu} (\\int I{(\\mu)} d\\mu + \\int e^{\\mu} d\\mu) = \\frac{d}{d \\mu} 2 \\int e^{\\mu} d\\mu", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["add", 2, "Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Integral(Function('I')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Integral(Function('I')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\eta,\\hat{p}_0)} = e^{\\hat{p}_0^{\\eta}}, then obtain - 2 \\eta + \\hat{p}_0 + t{(\\eta,\\hat{p}_0)} = - 2 \\eta + \\hat{p}_0 + e^{\\hat{p}_0^{\\eta}}", "derivation": "t{(\\eta,\\hat{p}_0)} = e^{\\hat{p}_0^{\\eta}} and - \\eta + t{(\\eta,\\hat{p}_0)} = - \\eta + e^{\\hat{p}_0^{\\eta}} and - \\eta + \\hat{p}_0 + t{(\\eta,\\hat{p}_0)} = - \\eta + \\hat{p}_0 + e^{\\hat{p}_0^{\\eta}} and - 2 \\eta + \\hat{p}_0 + t{(\\eta,\\hat{p}_0)} = - 2 \\eta + \\hat{p}_0 + e^{\\hat{p}_0^{\\eta}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('t')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["add", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Function('t')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Function('t')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), exp(Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\mathbf{s}{(\\mathbf{H})} = - \\sin{(\\mathbf{H})}, then obtain \\frac{d}{d \\mathbf{H}} (- \\sin{(\\mathbf{H})})^{\\mathbf{H}} = \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})})^{\\mathbf{H}}", "derivation": "\\mathbf{s}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{s}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})} and (\\frac{d}{d \\mathbf{H}} \\mathbf{s}{(\\mathbf{H})})^{\\mathbf{H}} = (\\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})})^{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\mathbf{s}{(\\mathbf{H})})^{\\mathbf{H}} = \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})})^{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\mathbf{s}{(\\mathbf{H})} = - \\sin{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} (- \\sin{(\\mathbf{H})})^{\\mathbf{H}} = \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(E_{n})} = \\sin{(\\log{(E_{n})})}, then obtain (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) \\frac{d}{d E_{n}} (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) = (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) \\frac{d}{d E_{n}} (- \\log{(E_{n})} + \\sin{(\\log{(E_{n})})})", "derivation": "\\mathbf{M}{(E_{n})} = \\sin{(\\log{(E_{n})})} and \\mathbf{M}{(E_{n})} - \\log{(E_{n})} = - \\log{(E_{n})} + \\sin{(\\log{(E_{n})})} and \\frac{d}{d E_{n}} (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) = \\frac{d}{d E_{n}} (- \\log{(E_{n})} + \\sin{(\\log{(E_{n})})}) and (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) \\frac{d}{d E_{n}} (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) = (\\mathbf{M}{(E_{n})} - \\log{(E_{n})}) \\frac{d}{d E_{n}} (- \\log{(E_{n})} + \\sin{(\\log{(E_{n})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), sin(log(Symbol('E_n', commutative=True))))"], [["minus", 1, "log(Symbol('E_n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('E_n', commutative=True))), sin(log(Symbol('E_n', commutative=True)))))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('E_n', commutative=True))), sin(log(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["times", 3, "Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Derivative(Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Derivative(Add(Mul(Integer(-1), log(Symbol('E_n', commutative=True))), sin(log(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(f^{*},c)} = c - f^{*}, then obtain c + \\frac{\\partial}{\\partial f^{*}} (- c + (\\nabla^{2}{(f^{*},c)})^{c}) = c + \\frac{\\partial}{\\partial f^{*}} (- c + ((c - f^{*}) \\nabla{(f^{*},c)})^{c})", "derivation": "\\nabla{(f^{*},c)} = c - f^{*} and \\nabla^{2}{(f^{*},c)} = (c - f^{*}) \\nabla{(f^{*},c)} and (\\nabla^{2}{(f^{*},c)})^{c} = ((c - f^{*}) \\nabla{(f^{*},c)})^{c} and - c + (\\nabla^{2}{(f^{*},c)})^{c} = - c + ((c - f^{*}) \\nabla{(f^{*},c)})^{c} and \\frac{\\partial}{\\partial f^{*}} (- c + (\\nabla^{2}{(f^{*},c)})^{c}) = \\frac{\\partial}{\\partial f^{*}} (- c + ((c - f^{*}) \\nabla{(f^{*},c)})^{c}) and c + \\frac{\\partial}{\\partial f^{*}} (- c + (\\nabla^{2}{(f^{*},c)})^{c}) = c + \\frac{\\partial}{\\partial f^{*}} (- c + ((c - f^{*}) \\nabla{(f^{*},c)})^{c})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["times", 1, "Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))"], "Equality(Pow(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Integer(2)), Mul(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Integer(2)), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["minus", 3, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Pow(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Integer(2)), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))))"], [["differentiate", 4, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Pow(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Integer(2)), Symbol('c', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Pow(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True)), Integer(2)), Symbol('c', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Symbol('c', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\rho_b,A_{z})} = A_{z} + \\rho_b, then derive \\int \\operatorname{P_{e}}{(\\rho_b,A_{z})} d\\rho_b = A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2}, then obtain 1 = \\frac{A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2}}{\\int \\operatorname{P_{e}}{(\\rho_b,A_{z})} d\\rho_b}", "derivation": "\\operatorname{P_{e}}{(\\rho_b,A_{z})} = A_{z} + \\rho_b and \\int \\operatorname{P_{e}}{(\\rho_b,A_{z})} d\\rho_b = \\int (A_{z} + \\rho_b) d\\rho_b and \\int \\operatorname{P_{e}}{(\\rho_b,A_{z})} d\\rho_b = A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2} and \\int (A_{z} + \\rho_b) d\\rho_b = A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2} and 1 = \\frac{A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2}}{\\int (A_{z} + \\rho_b) d\\rho_b} and 1 = \\frac{A_{z} \\rho_b + \\mathbf{F} + \\frac{\\rho_b^{2}}{2}}{\\int \\operatorname{P_{e}}{(\\rho_b,A_{z})} d\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)))))"], [["divide", 4, "Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Integer(1), Mul(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)))), Pow(Integral(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(1), Mul(Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)))), Pow(Integral(Function('P_e')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\rho{(M,\\theta)} = \\frac{\\partial}{\\partial \\theta} M^{\\theta}, then derive - M^{\\theta} + \\rho{(M,\\theta)} = M^{\\theta} \\log{(M)} - M^{\\theta}, then obtain - M^{\\theta} \\log{(M)} + \\frac{\\partial}{\\partial \\theta} \\rho{(M,\\theta)} = M^{\\theta} \\log{(M)}^{2} - M^{\\theta} \\log{(M)}", "derivation": "\\rho{(M,\\theta)} = \\frac{\\partial}{\\partial \\theta} M^{\\theta} and - M^{\\theta} + \\rho{(M,\\theta)} = - M^{\\theta} + \\frac{\\partial}{\\partial \\theta} M^{\\theta} and - M^{\\theta} + \\rho{(M,\\theta)} = M^{\\theta} \\log{(M)} - M^{\\theta} and \\frac{\\partial}{\\partial \\theta} (- M^{\\theta} + \\rho{(M,\\theta)}) = \\frac{\\partial}{\\partial \\theta} (M^{\\theta} \\log{(M)} - M^{\\theta}) and - M^{\\theta} \\log{(M)} + \\frac{\\partial}{\\partial \\theta} \\rho{(M,\\theta)} = M^{\\theta} \\log{(M)}^{2} - M^{\\theta} \\log{(M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 1, "Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Function('\\\\rho')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Function('\\\\rho')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Function('\\\\rho')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('M', commutative=True))), Derivative(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(log(Symbol('M', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True)), log(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v)} = e^{v}, then derive \\int \\operatorname{M_{E}}{(v)} dv = t_{2} + e^{v}, then obtain \\frac{d}{d t_{2}} \\int e^{v} dv = \\frac{\\partial}{\\partial t_{2}} (t_{2} + e^{v})", "derivation": "\\operatorname{M_{E}}{(v)} = e^{v} and \\int \\operatorname{M_{E}}{(v)} dv = \\int e^{v} dv and \\int \\operatorname{M_{E}}{(v)} dv = t_{2} + e^{v} and \\int \\operatorname{M_{E}}{(v)} dv = t_{2} + \\operatorname{M_{E}}{(v)} and \\int e^{v} dv = t_{2} + e^{v} and \\frac{d}{d t_{2}} \\int e^{v} dv = \\frac{\\partial}{\\partial t_{2}} (t_{2} + e^{v})", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('t_2', commutative=True), exp(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('M_E')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('t_2', commutative=True), Function('M_E')(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('t_2', commutative=True), exp(Symbol('v', commutative=True))))"], [["differentiate", 5, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('t_2', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(Q)} = \\sin{(Q)}, then obtain - \\frac{0^{Q}}{\\mu{(Q)}} - \\mu{(Q)} = - \\mu{(Q)} - \\frac{1}{\\mu{(Q)}}", "derivation": "\\mu{(Q)} = \\sin{(Q)} and 0 = - \\mu{(Q)} + \\sin{(Q)} and 0^{Q} = (- \\mu{(Q)} + \\sin{(Q)})^{Q} and - \\frac{0^{Q}}{\\mu{(Q)}} = - \\frac{(- \\mu{(Q)} + \\sin{(Q)})^{Q}}{\\mu{(Q)}} and - \\frac{0^{Q}}{\\mu{(Q)}} - \\mu{(Q)} = - \\frac{(- \\mu{(Q)} + \\sin{(Q)})^{Q}}{\\mu{(Q)}} - \\mu{(Q)} and - \\frac{(- \\mu{(Q)} + \\sin{(Q)})^{Q}}{\\mu{(Q)}} - \\mu{(Q)} = - \\mu{(Q)} - \\frac{1}{\\mu{(Q)}} and - \\frac{0^{Q}}{\\mu{(Q)}} - \\mu{(Q)} = - \\mu{(Q)} - \\frac{1}{\\mu{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["minus", 1, "Function('\\\\mu')(Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Integer(0), Symbol('Q', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))))"], [["add", 4, "Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{v},s)} = s e^{\\mathbf{v}}, then derive \\frac{\\partial}{\\partial s} \\mathbf{H}{(\\mathbf{v},s)} = e^{\\mathbf{v}}, then obtain \\mathbf{H}{(\\mathbf{v},s)} e^{\\mathbf{v}} = s e^{\\mathbf{v}} \\frac{\\partial}{\\partial s} s e^{\\mathbf{v}}", "derivation": "\\mathbf{H}{(\\mathbf{v},s)} = s e^{\\mathbf{v}} and \\frac{\\partial}{\\partial s} \\mathbf{H}{(\\mathbf{v},s)} = \\frac{\\partial}{\\partial s} s e^{\\mathbf{v}} and \\frac{\\partial}{\\partial s} \\mathbf{H}{(\\mathbf{v},s)} = e^{\\mathbf{v}} and \\frac{\\partial}{\\partial s} s e^{\\mathbf{v}} = e^{\\mathbf{v}} and \\mathbf{H}{(\\mathbf{v},s)} = s \\frac{\\partial}{\\partial s} s e^{\\mathbf{v}} and \\mathbf{H}{(\\mathbf{v},s)} e^{\\mathbf{v}} = s e^{\\mathbf{v}} \\frac{\\partial}{\\partial s} s e^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Derivative(Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["times", 5, "exp(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Mul(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(g,u)} = - g + e^{u}, then obtain u + \\sin{(\\int (- g + \\operatorname{v_{2}}{(g,u)}) du)} = u + \\sin{(\\int (- 2 g + e^{u}) du)}", "derivation": "\\operatorname{v_{2}}{(g,u)} = - g + e^{u} and - g + \\operatorname{v_{2}}{(g,u)} = - 2 g + e^{u} and \\int (- g + \\operatorname{v_{2}}{(g,u)}) du = \\int (- 2 g + e^{u}) du and \\sin{(\\int (- g + \\operatorname{v_{2}}{(g,u)}) du)} = \\sin{(\\int (- 2 g + e^{u}) du)} and u + \\sin{(\\int (- g + \\operatorname{v_{2}}{(g,u)}) du)} = u + \\sin{(\\int (- 2 g + e^{u}) du)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('g', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Symbol('u', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('v_2')(Symbol('g', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), exp(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('v_2')(Symbol('g', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('v_2')(Symbol('g', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), sin(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["add", 4, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), sin(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('v_2')(Symbol('g', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))), Add(Symbol('u', commutative=True), sin(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\lambda)} = \\sin{(\\cos{(\\lambda)})}, then obtain (\\operatorname{A_{2}}{(\\lambda)} + \\sin{(\\cos{(\\lambda)})})^{2} + \\operatorname{A_{2}}{(\\lambda)} = 4 \\operatorname{A_{2}}{(\\lambda)} \\sin{(\\cos{(\\lambda)})} + \\operatorname{A_{2}}{(\\lambda)}", "derivation": "\\operatorname{A_{2}}{(\\lambda)} = \\sin{(\\cos{(\\lambda)})} and \\operatorname{A_{2}}{(\\lambda)} \\sin{(\\cos{(\\lambda)})} = \\sin^{2}{(\\cos{(\\lambda)})} and \\operatorname{A_{2}}{(\\lambda)} + \\sin{(\\cos{(\\lambda)})} = 2 \\sin{(\\cos{(\\lambda)})} and (\\operatorname{A_{2}}{(\\lambda)} + \\sin{(\\cos{(\\lambda)})})^{2} = 4 \\sin^{2}{(\\cos{(\\lambda)})} and (\\operatorname{A_{2}}{(\\lambda)} + \\sin{(\\cos{(\\lambda)})})^{2} = 4 \\operatorname{A_{2}}{(\\lambda)} \\sin{(\\cos{(\\lambda)})} and (\\operatorname{A_{2}}{(\\lambda)} + \\sin{(\\cos{(\\lambda)})})^{2} + \\operatorname{A_{2}}{(\\lambda)} = 4 \\operatorname{A_{2}}{(\\lambda)} \\sin{(\\cos{(\\lambda)})} + \\operatorname{A_{2}}{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "sin(cos(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Pow(sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(2)))"], [["add", 1, "sin(cos(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(2), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Integer(2)), Mul(Integer(4), Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["add", 5, "Function('A_2')(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Pow(Add(Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Integer(2)), Function('A_2')(Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(4), Function('A_2')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True)))), Function('A_2')(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(I)} = \\cos{(I)} and \\phi{(I)} = \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}}, then obtain \\frac{d}{d I} \\phi{(I)} + \\frac{1}{(\\frac{d}{d I} \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}})^{2}} = \\frac{d}{d I} 0 + \\frac{1}{(\\frac{d}{d I} \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}})^{2}}", "derivation": "\\hat{\\mathbf{r}}{(I)} = \\cos{(I)} and \\hat{\\mathbf{r}}{(I)} - \\cos{(I)} = 0 and \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos{(I)}} = 0 and \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}} = 0 and \\phi{(I)} = \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}} and \\frac{d}{d I} \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}} = \\frac{d}{d I} 0 and \\frac{d}{d I} \\phi{(I)} = \\frac{d}{d I} 0 and \\frac{d}{d I} \\phi{(I)} + \\frac{1}{(\\frac{d}{d I} \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}})^{2}} = \\frac{d}{d I} 0 + \\frac{1}{(\\frac{d}{d I} \\frac{\\hat{\\mathbf{r}}{(I)} - \\cos{(I)}}{\\cos^{2}{(I)}})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["minus", 1, "cos(Symbol('I', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Integer(0))"], [["divide", 2, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 3, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('I', commutative=True)), Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))))"], [["differentiate", 4, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('\\\\phi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 7, "Pow(Derivative(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-2))"], "Equality(Add(Derivative(Function('\\\\phi')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Derivative(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-2))), Add(Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Derivative(Mul(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(cos(Symbol('I', commutative=True)), Integer(-2))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\operatorname{A_{1}}{(\\sigma_p)} = \\sin{(\\sigma_p)}, then obtain 2 \\operatorname{A_{1}}{(\\sigma_p)} = \\operatorname{A_{1}}{(\\sigma_p)} + \\operatorname{t_{1}}{(\\sigma_p)}", "derivation": "\\operatorname{t_{1}}{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\operatorname{A_{1}}{(\\sigma_p)} = \\sin{(\\sigma_p)} and 2 \\operatorname{A_{1}}{(\\sigma_p)} = \\operatorname{A_{1}}{(\\sigma_p)} + \\sin{(\\sigma_p)} and 2 \\operatorname{A_{1}}{(\\sigma_p)} = \\operatorname{A_{1}}{(\\sigma_p)} + \\operatorname{t_{1}}{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 2, "Function('A_1')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('A_1')(Symbol('\\\\sigma_p', commutative=True))), Add(Function('A_1')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('A_1')(Symbol('\\\\sigma_p', commutative=True))), Add(Function('A_1')(Symbol('\\\\sigma_p', commutative=True)), Function('t_1')(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{p})} = \\sin{(\\hat{p})}, then obtain \\frac{d}{d \\hat{p}} (\\mu_{0}{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p}) = \\frac{d}{d \\hat{p}} (\\sin{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p})", "derivation": "\\mu_{0}{(\\hat{p})} = \\sin{(\\hat{p})} and \\int \\mu_{0}{(\\hat{p})} d\\hat{p} = \\int \\sin{(\\hat{p})} d\\hat{p} and \\mu_{0}{(\\hat{p})} - \\int \\sin{(\\hat{p})} d\\hat{p} = \\sin{(\\hat{p})} - \\int \\sin{(\\hat{p})} d\\hat{p} and \\mu_{0}{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p} = \\sin{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p} and \\frac{d}{d \\hat{p}} (\\mu_{0}{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p}) = \\frac{d}{d \\hat{p}} (\\sin{(\\hat{p})} - \\int \\mu_{0}{(\\hat{p})} d\\hat{p})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["minus", 1, "Integral(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Add(sin(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Add(sin(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(k,A_{y})} = k^{A_{y}}, then obtain - \\frac{- A_{y} + k^{- A_{y}} c{(k,A_{y})}}{A_{y}} = - \\frac{1 - A_{y}}{A_{y}}", "derivation": "c{(k,A_{y})} = k^{A_{y}} and k^{- A_{y}} c{(k,A_{y})} = 1 and - A_{y} + k^{- A_{y}} c{(k,A_{y})} = 1 - A_{y} and - \\frac{- A_{y} + k^{- A_{y}} c{(k,A_{y})}}{A_{y}} = - \\frac{1 - A_{y}}{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('k', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('A_y', commutative=True)))"], [["divide", 1, "Pow(Symbol('k', commutative=True), Symbol('A_y', commutative=True))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('A_y', commutative=True))), Function('c')(Symbol('k', commutative=True), Symbol('A_y', commutative=True))), Integer(1))"], [["add", 2, "Mul(Integer(-1), Symbol('A_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('A_y', commutative=True))), Function('c')(Symbol('k', commutative=True), Symbol('A_y', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('A_y', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('A_y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('A_y', commutative=True))), Function('c')(Symbol('k', commutative=True), Symbol('A_y', commutative=True))))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-1)), Add(Integer(1), Mul(Integer(-1), Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\theta{(s)} = e^{s}, then derive \\frac{d}{d s} \\theta{(s)} = e^{s}, then obtain \\frac{d}{d s} (\\frac{d}{d s} \\theta{(s)} - 1) \\frac{d}{d s} \\theta{(s)} = \\frac{d}{d s} (e^{s} - 1) \\frac{d}{d s} \\theta{(s)}", "derivation": "\\theta{(s)} = e^{s} and \\theta{(s)} - 1 = e^{s} - 1 and \\frac{d}{d s} (\\theta{(s)} - 1) = \\frac{d}{d s} (e^{s} - 1) and \\theta{(s)} \\frac{d}{d s} (\\theta{(s)} - 1) = \\theta{(s)} \\frac{d}{d s} (e^{s} - 1) and \\frac{d}{d s} \\theta{(s)} = e^{s} and \\theta{(s)} = \\frac{d}{d s} \\theta{(s)} and \\frac{d}{d s} (\\frac{d}{d s} \\theta{(s)} - 1) \\frac{d}{d s} \\theta{(s)} = \\frac{d}{d s} (e^{s} - 1) \\frac{d}{d s} \\theta{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta')(Symbol('s', commutative=True)), Integer(-1)), Add(exp(Symbol('s', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('s', commutative=True)), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('s', commutative=True)), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["divide", 3, "Pow(Function('\\\\theta')(Symbol('s', commutative=True)), Integer(-1))"], "Equality(Mul(Function('\\\\theta')(Symbol('s', commutative=True)), Derivative(Add(Function('\\\\theta')(Symbol('s', commutative=True)), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Function('\\\\theta')(Symbol('s', commutative=True)), Derivative(Add(exp(Symbol('s', commutative=True)), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\theta')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), exp(Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('\\\\theta')(Symbol('s', commutative=True)), Derivative(Function('\\\\theta')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Derivative(Add(Derivative(Function('\\\\theta')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Derivative(Add(exp(Symbol('s', commutative=True)), Integer(-1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(n_{2},k)} = \\frac{\\sin{(n_{2})}}{k}, then derive k (\\frac{\\partial}{\\partial n_{2}} H{(n_{2},k)} + 1) = k (1 + \\frac{\\cos{(n_{2})}}{k}), then obtain k (\\frac{\\partial}{\\partial n_{2}} \\frac{\\sin{(n_{2})}}{k} + 1) = k (1 + \\frac{\\cos{(n_{2})}}{k})", "derivation": "H{(n_{2},k)} = \\frac{\\sin{(n_{2})}}{k} and n_{2} + H{(n_{2},k)} = n_{2} + \\frac{\\sin{(n_{2})}}{k} and k (n_{2} + H{(n_{2},k)}) = k (n_{2} + \\frac{\\sin{(n_{2})}}{k}) and \\frac{\\partial}{\\partial n_{2}} k (n_{2} + H{(n_{2},k)}) = \\frac{\\partial}{\\partial n_{2}} k (n_{2} + \\frac{\\sin{(n_{2})}}{k}) and k (\\frac{\\partial}{\\partial n_{2}} H{(n_{2},k)} + 1) = k (1 + \\frac{\\cos{(n_{2})}}{k}) and k (\\frac{\\partial}{\\partial n_{2}} \\frac{\\sin{(n_{2})}}{k} + 1) = k (1 + \\frac{\\cos{(n_{2})}}{k})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('n_2', commutative=True))))"], [["add", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Symbol('n_2', commutative=True), Function('H')(Symbol('n_2', commutative=True), Symbol('k', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('n_2', commutative=True)))))"], [["divide", 2, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('k', commutative=True), Add(Symbol('n_2', commutative=True), Function('H')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)))), Mul(Symbol('k', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('n_2', commutative=True))))))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('k', commutative=True), Add(Symbol('n_2', commutative=True), Function('H')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('k', commutative=True), Add(Symbol('n_2', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('n_2', commutative=True))))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('k', commutative=True), Add(Derivative(Function('H')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('k', commutative=True), Add(Integer(1), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), cos(Symbol('n_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('k', commutative=True), Add(Derivative(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('k', commutative=True), Add(Integer(1), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), cos(Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(m_{s},Q)} = \\frac{\\partial}{\\partial Q} Q^{m_{s}}, then derive \\operatorname{E_{x}}{(m_{s},Q)} = \\frac{Q^{m_{s}} m_{s}}{Q}, then obtain 1 = \\frac{Q Q^{- m_{s}} \\operatorname{E_{x}}{(m_{s},Q)}}{m_{s}}", "derivation": "\\operatorname{E_{x}}{(m_{s},Q)} = \\frac{\\partial}{\\partial Q} Q^{m_{s}} and 1 = \\frac{\\frac{\\partial}{\\partial Q} Q^{m_{s}}}{\\operatorname{E_{x}}{(m_{s},Q)}} and \\operatorname{E_{x}}{(m_{s},Q)} = \\frac{Q^{m_{s}} m_{s}}{Q} and 1 = \\frac{Q Q^{- m_{s}} \\frac{\\partial}{\\partial Q} Q^{m_{s}}}{m_{s}} and 1 = \\frac{Q Q^{- m_{s}} \\operatorname{E_{x}}{(m_{s},Q)}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('m_s', commutative=True), Symbol('Q', commutative=True)), Derivative(Pow(Symbol('Q', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["divide", 1, "Function('E_x')(Symbol('m_s', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_x')(Symbol('m_s', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('Q', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('E_x')(Symbol('m_s', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Symbol('Q', commutative=True), Pow(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Pow(Symbol('m_s', commutative=True), Integer(-1)), Derivative(Pow(Symbol('Q', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(1), Mul(Symbol('Q', commutative=True), Pow(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('E_x')(Symbol('m_s', commutative=True), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(f^{*})} = e^{f^{*}}, then derive \\int f^{*} \\mathbf{p}{(f^{*})} df^{*} = m + (f^{*} - 1) e^{f^{*}}, then obtain \\int f^{*} \\mathbf{p}{(f^{*})} df^{*} = m + (f^{*} - 1) \\mathbf{p}{(f^{*})}", "derivation": "\\mathbf{p}{(f^{*})} = e^{f^{*}} and f^{*} \\mathbf{p}{(f^{*})} = f^{*} e^{f^{*}} and \\int f^{*} \\mathbf{p}{(f^{*})} df^{*} = \\int f^{*} e^{f^{*}} df^{*} and \\int f^{*} \\mathbf{p}{(f^{*})} df^{*} = m + (f^{*} - 1) e^{f^{*}} and \\int f^{*} \\mathbf{p}{(f^{*})} df^{*} = m + (f^{*} - 1) \\mathbf{p}{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["times", 1, "Symbol('f^*', commutative=True)"], "Equality(Mul(Symbol('f^*', commutative=True), Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True))), Mul(Symbol('f^*', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Symbol('f^*', commutative=True), Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Symbol('f^*', commutative=True), exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('f^*', commutative=True), Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('m', commutative=True), Mul(Add(Symbol('f^*', commutative=True), Integer(-1)), exp(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Symbol('f^*', commutative=True), Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('m', commutative=True), Mul(Add(Symbol('f^*', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given v{(Q)} = \\sin{(Q)}, then obtain \\frac{Q + v{(Q)}}{v{(Q)} \\sin{(Q)}} = \\frac{Q + \\sin{(Q)}}{v{(Q)} \\sin{(Q)}}", "derivation": "v{(Q)} = \\sin{(Q)} and v^{2}{(Q)} = v{(Q)} \\sin{(Q)} and Q + v{(Q)} = Q + \\sin{(Q)} and \\frac{Q + v{(Q)}}{v^{2}{(Q)}} = \\frac{Q + \\sin{(Q)}}{v^{2}{(Q)}} and \\frac{Q + v{(Q)}}{v{(Q)} \\sin{(Q)}} = \\frac{Q + \\sin{(Q)}}{v{(Q)} \\sin{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["times", 1, "Function('v')(Symbol('Q', commutative=True))"], "Equality(Pow(Function('v')(Symbol('Q', commutative=True)), Integer(2)), Mul(Function('v')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["add", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Function('v')(Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))))"], [["divide", 3, "Pow(Function('v')(Symbol('Q', commutative=True)), Integer(2))"], "Equality(Mul(Add(Symbol('Q', commutative=True), Function('v')(Symbol('Q', commutative=True))), Pow(Function('v')(Symbol('Q', commutative=True)), Integer(-2))), Mul(Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('v')(Symbol('Q', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('Q', commutative=True), Function('v')(Symbol('Q', commutative=True))), Pow(Function('v')(Symbol('Q', commutative=True)), Integer(-1)), Pow(sin(Symbol('Q', commutative=True)), Integer(-1))), Mul(Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Pow(Function('v')(Symbol('Q', commutative=True)), Integer(-1)), Pow(sin(Symbol('Q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(s)} = \\cos{(\\cos{(s)})} and \\dot{z}{(s)} = \\frac{\\cos^{s}{(\\cos{(s)})}}{s}, then obtain - \\cos{(\\cos{(s)})} + \\frac{\\operatorname{v_{z}}^{s}{(s)}}{s} = \\dot{z}{(s)} - \\cos{(\\cos{(s)})}", "derivation": "\\operatorname{v_{z}}{(s)} = \\cos{(\\cos{(s)})} and \\operatorname{v_{z}}^{s}{(s)} = \\cos^{s}{(\\cos{(s)})} and \\frac{\\operatorname{v_{z}}^{s}{(s)}}{s} = \\frac{\\cos^{s}{(\\cos{(s)})}}{s} and - \\cos{(\\cos{(s)})} + \\frac{\\operatorname{v_{z}}^{s}{(s)}}{s} = - \\cos{(\\cos{(s)})} + \\frac{\\cos^{s}{(\\cos{(s)})}}{s} and \\dot{z}{(s)} = \\frac{\\cos^{s}{(\\cos{(s)})}}{s} and - \\cos{(\\cos{(s)})} + \\frac{\\operatorname{v_{z}}^{s}{(s)}}{s} = \\dot{z}{(s)} - \\cos{(\\cos{(s)})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('s', commutative=True)), cos(cos(Symbol('s', commutative=True))))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(cos(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["divide", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('s', commutative=True)), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(cos(cos(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["minus", 3, "cos(cos(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(cos(Symbol('s', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))), Add(Mul(Integer(-1), cos(cos(Symbol('s', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(cos(cos(Symbol('s', commutative=True))), Symbol('s', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(cos(cos(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), cos(cos(Symbol('s', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))), Add(Function('\\\\dot{z}')(Symbol('s', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then derive 2 \\frac{d}{d \\mathbf{E}} \\operatorname{y^{\\prime}}{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\operatorname{y^{\\prime}}{(\\mathbf{E})}, then obtain 4 (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2} = (- \\sin{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2}", "derivation": "\\operatorname{y^{\\prime}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and 2 \\operatorname{y^{\\prime}}{(\\mathbf{E})} = \\operatorname{y^{\\prime}}{(\\mathbf{E})} + \\cos{(\\mathbf{E})} and \\frac{d}{d \\mathbf{E}} 2 \\operatorname{y^{\\prime}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} (\\operatorname{y^{\\prime}}{(\\mathbf{E})} + \\cos{(\\mathbf{E})}) and 2 \\frac{d}{d \\mathbf{E}} \\operatorname{y^{\\prime}}{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\operatorname{y^{\\prime}}{(\\mathbf{E})} and 2 \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and 4 (\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2} = (- \\sin{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})})^{2}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["power", 5, 2], "Equality(Mul(Integer(4), Pow(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2))), Pow(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(E)} = \\int e^{E} dE, then derive E + (l + \\operatorname{M_{E}}{(E)} + e^{E}) \\operatorname{M_{E}}{(E)} = E + 2 (l + e^{E}) \\operatorname{M_{E}}{(E)}, then obtain E + (\\mathbb{I} + e^{E}) (\\mathbb{I} + l + 2 e^{E}) = E + 2 (\\mathbb{I} + e^{E}) (l + e^{E})", "derivation": "\\operatorname{M_{E}}{(E)} = \\int e^{E} dE and \\operatorname{M_{E}}{(E)} + \\int e^{E} dE = 2 \\int e^{E} dE and (\\operatorname{M_{E}}{(E)} + \\int e^{E} dE) \\operatorname{M_{E}}{(E)} = 2 \\operatorname{M_{E}}{(E)} \\int e^{E} dE and E + (\\operatorname{M_{E}}{(E)} + \\int e^{E} dE) \\operatorname{M_{E}}{(E)} = E + 2 \\operatorname{M_{E}}{(E)} \\int e^{E} dE and E + (l + \\operatorname{M_{E}}{(E)} + e^{E}) \\operatorname{M_{E}}{(E)} = E + 2 (l + e^{E}) \\operatorname{M_{E}}{(E)} and E + (l + e^{E} + \\int e^{E} dE) \\int e^{E} dE = E + 2 (l + e^{E}) \\int e^{E} dE and E + (\\mathbb{I} + e^{E}) (\\mathbb{I} + l + 2 e^{E}) = E + 2 (\\mathbb{I} + e^{E}) (l + e^{E})", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 1, "Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Add(Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["times", 2, "Function('M_E')(Symbol('E', commutative=True))"], "Equality(Mul(Add(Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Function('M_E')(Symbol('E', commutative=True))), Mul(Integer(2), Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["add", 3, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Mul(Add(Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Function('M_E')(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(2), Function('M_E')(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('E', commutative=True), Mul(Add(Symbol('l', commutative=True), Function('M_E')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Function('M_E')(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(2), Add(Symbol('l', commutative=True), exp(Symbol('E', commutative=True))), Function('M_E')(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('E', commutative=True), Mul(Add(Symbol('l', commutative=True), exp(Symbol('E', commutative=True)), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))), Add(Symbol('E', commutative=True), Mul(Integer(2), Add(Symbol('l', commutative=True), exp(Symbol('E', commutative=True))), Integral(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('E', commutative=True), Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('E', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('l', commutative=True), Mul(Integer(2), exp(Symbol('E', commutative=True)))))), Add(Symbol('E', commutative=True), Mul(Integer(2), Add(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('E', commutative=True))), Add(Symbol('l', commutative=True), exp(Symbol('E', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(v)} = v and \\phi_{1}{(v)} = \\log{(\\operatorname{g_{\\varepsilon}}{(v)})}, then obtain \\frac{(\\phi_{1}{(v)} + \\operatorname{g_{\\varepsilon}}{(v)})^{v}}{v} = \\frac{(\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(v)})^{v}}{v}", "derivation": "\\operatorname{g_{\\varepsilon}}{(v)} = v and \\log{(\\operatorname{g_{\\varepsilon}}{(v)})} = \\log{(v)} and \\operatorname{g_{\\varepsilon}}{(v)} + \\log{(\\operatorname{g_{\\varepsilon}}{(v)})} = \\operatorname{g_{\\varepsilon}}{(v)} + \\log{(v)} and \\phi_{1}{(v)} = \\log{(\\operatorname{g_{\\varepsilon}}{(v)})} and (\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(\\operatorname{g_{\\varepsilon}}{(v)})})^{v} = (\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(v)})^{v} and \\frac{(\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(\\operatorname{g_{\\varepsilon}}{(v)})})^{v}}{v} = \\frac{(\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(v)})^{v}}{v} and \\frac{(\\phi_{1}{(v)} + \\operatorname{g_{\\varepsilon}}{(v)})^{v}}{v} = \\frac{(\\operatorname{g_{\\varepsilon}}{(v)} + \\log{(v)})^{v}}{v}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), Symbol('v', commutative=True))"], [["log", 1], "Equality(log(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True))), log(Symbol('v', commutative=True)))"], [["add", 2, "Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)))), Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('v', commutative=True)), log(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["divide", 5, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)))), Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Function('\\\\phi_1')(Symbol('v', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + \\pi and L{(\\pi,\\hat{\\mathbf{x}})} = - \\frac{\\partial}{\\partial \\pi} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})}, then derive - \\frac{\\partial}{\\partial \\pi} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = -1, then obtain L{(\\pi,\\hat{\\mathbf{x}})} \\frac{\\partial^{- L{(\\pi,\\hat{\\mathbf{x}})}}}{\\partial \\pi^{- L{(\\pi,\\hat{\\mathbf{x}})}}} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = L{(\\pi,\\hat{\\mathbf{x}})}", "derivation": "\\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} + \\pi and - \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} - \\pi and \\frac{\\partial}{\\partial \\pi} - \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial \\pi} (\\hat{\\mathbf{x}} - \\pi) and - \\frac{\\partial}{\\partial \\pi} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = -1 and L{(\\pi,\\hat{\\mathbf{x}})} = - \\frac{\\partial}{\\partial \\pi} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} and L{(\\pi,\\hat{\\mathbf{x}})} = -1 and L{(\\pi,\\hat{\\mathbf{x}})} \\frac{\\partial^{- L{(\\pi,\\hat{\\mathbf{x}})}}}{\\partial \\pi^{- L{(\\pi,\\hat{\\mathbf{x}})}}} \\theta_{1}{(\\pi,\\hat{\\mathbf{x}})} = L{(\\pi,\\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Integer(-1))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('L')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))), Function('L')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\theta)} = \\int e^{\\theta} d\\theta and \\mathbf{P}{(\\theta)} = \\theta + \\hat{\\mathbf{x}}{(\\theta)}, then derive \\mathbf{P}{(\\theta)} = F_{g} + \\theta + e^{\\theta}, then obtain 2 \\mathbf{P}{(\\theta)} = F_{g} + \\theta + \\mathbf{P}{(\\theta)} + e^{\\theta}", "derivation": "\\hat{\\mathbf{x}}{(\\theta)} = \\int e^{\\theta} d\\theta and \\theta + \\hat{\\mathbf{x}}{(\\theta)} = \\theta + \\int e^{\\theta} d\\theta and \\mathbf{P}{(\\theta)} = \\theta + \\hat{\\mathbf{x}}{(\\theta)} and \\mathbf{P}{(\\theta)} = \\theta + \\int e^{\\theta} d\\theta and \\mathbf{P}{(\\theta)} = F_{g} + \\theta + e^{\\theta} and 2 \\mathbf{P}{(\\theta)} = F_{g} + \\theta + \\mathbf{P}{(\\theta)} + e^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True)), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\theta', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True))), Add(Symbol('F_g', commutative=True), Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\psi)} = \\log{(\\psi)}, then obtain \\int \\frac{d}{d \\psi} (2 \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)}) d\\psi - 1 = \\int \\frac{d}{d \\psi} (\\operatorname{F_{x}}{(\\psi)} + 2 \\log{(\\psi)}) d\\psi - 1", "derivation": "\\operatorname{F_{x}}{(\\psi)} = \\log{(\\psi)} and \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)} = 2 \\log{(\\psi)} and 2 \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)} = \\operatorname{F_{x}}{(\\psi)} + 2 \\log{(\\psi)} and \\frac{d}{d \\psi} (2 \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)}) = \\frac{d}{d \\psi} (\\operatorname{F_{x}}{(\\psi)} + 2 \\log{(\\psi)}) and \\int \\frac{d}{d \\psi} (2 \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)}) d\\psi = \\int \\frac{d}{d \\psi} (\\operatorname{F_{x}}{(\\psi)} + 2 \\log{(\\psi)}) d\\psi and \\int \\frac{d}{d \\psi} (2 \\operatorname{F_{x}}{(\\psi)} + \\log{(\\psi)}) d\\psi - 1 = \\int \\frac{d}{d \\psi} (\\operatorname{F_{x}}{(\\psi)} + 2 \\log{(\\psi)}) d\\psi - 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["add", 1, "log(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True))))"], [["add", 2, "Function('F_x')(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Add(Function('F_x')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Function('F_x')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(Add(Function('F_x')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Integral(Derivative(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)), Add(Integral(Derivative(Add(Function('F_x')(Symbol('\\\\psi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} = \\frac{z}{P_{g} \\dot{\\mathbf{r}}} and \\mathbf{J}_f{(P_{g})} = P_{g}, then obtain 2 \\mathbf{J}_f{(P_{g})} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} = P_{g} + \\mathbf{J}_f{(P_{g})} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)}", "derivation": "\\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} = \\frac{z}{P_{g} \\dot{\\mathbf{r}}} and \\mathbf{J}_f{(P_{g})} = P_{g} and \\mathbf{J}_f{(P_{g})} + \\frac{z}{P_{g} \\dot{\\mathbf{r}}} = P_{g} + \\frac{z}{P_{g} \\dot{\\mathbf{r}}} and \\mathbf{J}_f{(P_{g})} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} = P_{g} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} and 2 \\mathbf{J}_f{(P_{g})} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)} = P_{g} + \\mathbf{J}_f{(P_{g})} + \\operatorname{v_{z}}{(P_{g},\\dot{\\mathbf{r}},z)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('P_g', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["add", 2, "Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Add(Symbol('P_g', commutative=True), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True)), Function('v_z')(Symbol('P_g', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True))), Add(Symbol('P_g', commutative=True), Function('v_z')(Symbol('P_g', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True))), Function('v_z')(Symbol('P_g', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True))), Add(Symbol('P_g', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('P_g', commutative=True)), Function('v_z')(Symbol('P_g', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},E_{x})} = \\mathbf{P}^{E_{x}} and \\pi{(\\mathbf{P},E_{x})} = \\mathbf{P}^{E_{x}}, then obtain \\pi{(\\mathbf{P},E_{x})} + \\frac{\\mathbf{P}^{E_{x}}}{E_{x} \\pi{(\\mathbf{P},E_{x})}} = \\pi{(\\mathbf{P},E_{x})} + \\frac{1}{E_{x}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{P},E_{x})} = \\mathbf{P}^{E_{x}} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\mathbf{P},E_{x})}}{E_{x}} = \\frac{\\mathbf{P}^{E_{x}}}{E_{x}} and \\pi{(\\mathbf{P},E_{x})} = \\mathbf{P}^{E_{x}} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\mathbf{P},E_{x})}}{E_{x}} = \\frac{\\pi{(\\mathbf{P},E_{x})}}{E_{x}} and \\frac{\\mathbf{P}^{E_{x}}}{E_{x}} = \\frac{\\pi{(\\mathbf{P},E_{x})}}{E_{x}} and \\frac{\\mathbf{P}^{E_{x}}}{E_{x} \\pi{(\\mathbf{P},E_{x})}} = \\frac{1}{E_{x}} and \\pi{(\\mathbf{P},E_{x})} + \\frac{\\mathbf{P}^{E_{x}}}{E_{x} \\pi{(\\mathbf{P},E_{x})}} = \\pi{(\\mathbf{P},E_{x})} + \\frac{1}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))))"], [["divide", 5, "Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Pow(Symbol('E_x', commutative=True), Integer(-1)))"], [["add", 6, "Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)))), Add(Function('\\\\pi')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(m,\\Psi_{nl})} = \\Psi_{nl} - m, then derive \\int \\dot{y}{(m,\\Psi_{nl})} dm = \\Omega + \\Psi_{nl} m - \\frac{m^{2}}{2}, then obtain \\iint - \\int (\\Psi_{nl} - m) dm dm d\\Psi_{nl} = \\iint - \\int \\dot{y}{(m,\\Psi_{nl})} dm dm d\\Psi_{nl}", "derivation": "\\dot{y}{(m,\\Psi_{nl})} = \\Psi_{nl} - m and \\int \\dot{y}{(m,\\Psi_{nl})} dm = \\int (\\Psi_{nl} - m) dm and \\int \\dot{y}{(m,\\Psi_{nl})} dm = \\Omega + \\Psi_{nl} m - \\frac{m^{2}}{2} and - \\int \\dot{y}{(m,\\Psi_{nl})} dm = - \\Omega - \\Psi_{nl} m + \\frac{m^{2}}{2} and - \\int (\\Psi_{nl} - m) dm = - \\Omega - \\Psi_{nl} m + \\frac{m^{2}}{2} and - \\int (\\Psi_{nl} - m) dm = - \\int \\dot{y}{(m,\\Psi_{nl})} dm and \\int - \\int (\\Psi_{nl} - m) dm dm = \\int - \\int \\dot{y}{(m,\\Psi_{nl})} dm dm and \\iint - \\int (\\Psi_{nl} - m) dm dm d\\Psi_{nl} = \\iint - \\int \\dot{y}{(m,\\Psi_{nl})} dm dm d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('m', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["integrate", 6, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["integrate", 7, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(n_{2},\\mathbf{M})} = \\mathbf{M} + n_{2}, then obtain - n_{2} \\sin{((\\mathbf{M} - \\operatorname{C_{2}}{(n_{2},\\mathbf{M})})^{\\mathbf{M}})} = - n_{2} \\sin{((- n_{2})^{\\mathbf{M}})}", "derivation": "\\operatorname{C_{2}}{(n_{2},\\mathbf{M})} = \\mathbf{M} + n_{2} and - \\mathbf{M} + \\operatorname{C_{2}}{(n_{2},\\mathbf{M})} = n_{2} and \\mathbf{M} - \\operatorname{C_{2}}{(n_{2},\\mathbf{M})} = - n_{2} and (\\mathbf{M} - \\operatorname{C_{2}}{(n_{2},\\mathbf{M})})^{\\mathbf{M}} = (- n_{2})^{\\mathbf{M}} and \\sin{((\\mathbf{M} - \\operatorname{C_{2}}{(n_{2},\\mathbf{M})})^{\\mathbf{M}})} = \\sin{((- n_{2})^{\\mathbf{M}})} and - n_{2} \\sin{((\\mathbf{M} - \\operatorname{C_{2}}{(n_{2},\\mathbf{M})})^{\\mathbf{M}})} = - n_{2} \\sin{((- n_{2})^{\\mathbf{M}})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('n_2', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True))), sin(Pow(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True), sin(Pow(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\psi^*,\\mathbf{S},Q)} = Q - \\mathbf{S} + \\psi^* and \\ddot{x}{(\\psi^*)} = - \\psi^*, then obtain \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S} + \\psi^* + \\ddot{x}{(\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\psi^*,\\mathbf{S},Q)} = Q - \\mathbf{S} + \\psi^* and - \\psi^* + \\operatorname{V_{\\mathbf{B}}}{(\\psi^*,\\mathbf{S},Q)} = Q - \\mathbf{S} and \\frac{\\partial}{\\partial \\psi^*} (- \\psi^* + \\operatorname{V_{\\mathbf{B}}}{(\\psi^*,\\mathbf{S},Q)}) = \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S}) and \\ddot{x}{(\\psi^*)} = - \\psi^* and \\frac{\\partial}{\\partial \\psi^*} (\\operatorname{V_{\\mathbf{B}}}{(\\psi^*,\\mathbf{S},Q)} + \\ddot{x}{(\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S}) and \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S} + \\psi^* + \\ddot{x}{(\\psi^*)}) = \\frac{\\partial}{\\partial \\psi^*} (Q - \\mathbf{S})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Q', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\psi^*', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{B},\\mathbf{g})} = \\mathbf{g}^{\\mathbf{B}} and \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi_{nl}^{\\mathbf{g}}{(\\mathbf{B},\\mathbf{g})}, then obtain \\mathbf{B} + \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\mathbf{B} + \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi_{nl}^{\\mathbf{g}}{(\\mathbf{B},\\mathbf{g})}", "derivation": "\\Psi_{nl}{(\\mathbf{B},\\mathbf{g})} = \\mathbf{g}^{\\mathbf{B}} and \\Psi_{nl}^{\\mathbf{g}}{(\\mathbf{B},\\mathbf{g})} = (\\mathbf{g}^{\\mathbf{B}})^{\\mathbf{g}} and \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi_{nl}^{\\mathbf{g}}{(\\mathbf{B},\\mathbf{g})} and \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{g}^{\\mathbf{B}})^{\\mathbf{g}} and \\mathbf{B} + \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\mathbf{B} + \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{g}^{\\mathbf{B}})^{\\mathbf{g}} and \\mathbf{B} + \\mathbb{I}{(\\mathbf{B},\\mathbf{g})} = \\mathbf{B} + \\frac{\\partial}{\\partial \\mathbf{B}} \\Psi_{nl}^{\\mathbf{g}}{(\\mathbf{B},\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Derivative(Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Derivative(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given U{(\\mathbf{f})} = \\cos{(\\mathbf{f})}, then derive \\frac{d^{2}}{d \\mathbf{f}^{2}} U{(\\mathbf{f})} = - \\cos{(\\mathbf{f})}, then obtain \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\mathbf{f})} = - \\cos{(\\mathbf{f})}", "derivation": "U{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} U{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\cos{(\\mathbf{f})} and \\frac{d^{2}}{d \\mathbf{f}^{2}} U{(\\mathbf{f})} = \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\mathbf{f})} and \\frac{d^{2}}{d \\mathbf{f}^{2}} U{(\\mathbf{f})} = - \\cos{(\\mathbf{f})} and \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\mathbf{f})} = - \\cos{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('U')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(cos(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\rho_f,y)} = - y + \\sin{(\\rho_f)}, then derive \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} = -1, then obtain \\int \\frac{\\partial^{- \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)}}}{\\partial y^{- \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)}}} \\Psi^{\\dagger}{(\\rho_f,y)} d\\rho_f = \\int \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} d\\rho_f", "derivation": "\\Psi^{\\dagger}{(\\rho_f,y)} = - y + \\sin{(\\rho_f)} and \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} = \\frac{\\partial}{\\partial y} (- y + \\sin{(\\rho_f)}) and \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} = -1 and \\int \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} d\\rho_f = \\int (-1) d\\rho_f and \\int \\frac{\\partial^{- \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)}}}{\\partial y^{- \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)}}} \\Psi^{\\dagger}{(\\rho_f,y)} d\\rho_f = \\int \\frac{\\partial}{\\partial y} \\Psi^{\\dagger}{(\\rho_f,y)} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and \\operatorname{E_{n}}{(\\Psi_{nl})} = \\Psi_{nl}, then obtain \\frac{d}{d \\Psi_{nl}} \\operatorname{E_{n}}{(\\Psi_{nl})} \\Psi{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\Psi_{nl} \\Psi{(\\Psi_{nl})}", "derivation": "\\Psi{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and \\operatorname{E_{n}}{(\\Psi_{nl})} = \\Psi_{nl} and \\operatorname{E_{n}}{(\\Psi_{nl})} \\cos{(\\Psi_{nl})} = \\Psi_{nl} \\cos{(\\Psi_{nl})} and \\operatorname{E_{n}}{(\\Psi_{nl})} \\Psi{(\\Psi_{nl})} = \\Psi_{nl} \\Psi{(\\Psi_{nl})} and \\frac{d}{d \\Psi_{nl}} \\operatorname{E_{n}}{(\\Psi_{nl})} \\Psi{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\Psi_{nl} \\Psi{(\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], [["times", 2, "cos(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('E_n')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('E_n')(Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\Psi')(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Mul(Function('E_n')(Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\Psi')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(g)} = \\sin{(\\cos{(g)})}, then obtain \\operatorname{a^{\\dagger}}{(g)} - \\sin{(\\cos{(g)})} - \\cos{(g)} = - \\cos{(g)}", "derivation": "\\operatorname{a^{\\dagger}}{(g)} = \\sin{(\\cos{(g)})} and - g + \\operatorname{a^{\\dagger}}{(g)} = - g + \\sin{(\\cos{(g)})} and \\operatorname{a^{\\dagger}}{(g)} - \\sin{(\\cos{(g)})} = 0 and \\operatorname{a^{\\dagger}}{(g)} - \\sin{(\\cos{(g)})} - \\cos{(g)} = - \\cos{(g)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('a^{\\\\dagger}')(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), sin(cos(Symbol('g', commutative=True))))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('g', commutative=True))))), Integer(0))"], [["minus", 3, "cos(Symbol('g', commutative=True))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('g', commutative=True)))), Mul(Integer(-1), cos(Symbol('g', commutative=True)))), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{p})} = \\log{(\\hat{p})} and r{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain \\hat{p} + \\mu_{0}{(\\hat{p})} + r{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\cos{(\\sigma_p)} = \\hat{p} + \\mu_{0}{(\\hat{p})} + \\cos{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\cos{(\\sigma_p)}", "derivation": "\\mu_{0}{(\\hat{p})} = \\log{(\\hat{p})} and r{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\hat{p} + r{(\\sigma_p)} + \\log{(\\hat{p})} = \\hat{p} + \\log{(\\hat{p})} + \\cos{(\\sigma_p)} and \\hat{p} + \\mu_{0}{(\\hat{p})} + r{(\\sigma_p)} = \\hat{p} + \\mu_{0}{(\\hat{p})} + \\cos{(\\sigma_p)} and \\hat{p} + \\mu_{0}{(\\hat{p})} + r{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\cos{(\\sigma_p)} = \\hat{p} + \\mu_{0}{(\\hat{p})} + \\cos{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\cos{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True)))"], ["get_premise", "Equality(Function('r')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 2, "Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('r')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Function('r')(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 4, "Derivative(cos(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), Function('r')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given y{(z)} = \\cos{(z)}, then obtain \\frac{d}{d z} (\\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} 0) = \\frac{d}{d z} (\\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} (- 2 y{(z)} + 2 \\cos{(z)}))", "derivation": "y{(z)} = \\cos{(z)} and 0 = - y{(z)} + \\cos{(z)} and - y{(z)} = - 2 y{(z)} + \\cos{(z)} and 0 = - 2 y{(z)} + 2 \\cos{(z)} and \\frac{d}{d z} 0 = \\frac{d}{d z} (- 2 y{(z)} + 2 \\cos{(z)}) and \\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} 0 = \\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} (- 2 y{(z)} + 2 \\cos{(z)}) and \\frac{d}{d z} (\\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} 0) = \\frac{d}{d z} (\\sin{(2 y{(z)} - 2 \\cos{(z)})} + \\frac{d}{d z} (- 2 y{(z)} + 2 \\cos{(z)}))", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["minus", 1, "Function('y')(Symbol('z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y')(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('y')(Symbol('z', commutative=True)))"], "Equality(Mul(Integer(-1), Function('y')(Symbol('z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(2), cos(Symbol('z', commutative=True)))))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(2), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["add", 5, "sin(Add(Mul(Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True)))))"], "Equality(Add(sin(Add(Mul(Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True))))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(sin(Add(Mul(Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True))))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(2), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(sin(Add(Mul(Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True))))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(sin(Add(Mul(Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True))))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('z', commutative=True))), Mul(Integer(2), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(I,\\mathbf{v})} = \\log{(I^{\\mathbf{v}})}, then obtain \\log{(I (I \\operatorname{P_{e}}{(I,\\mathbf{v})} - I^{\\mathbf{v}}))}^{\\mathbf{v}} = \\log{(I (I \\log{(I^{\\mathbf{v}})} - I^{\\mathbf{v}}))}^{\\mathbf{v}}", "derivation": "\\operatorname{P_{e}}{(I,\\mathbf{v})} = \\log{(I^{\\mathbf{v}})} and I \\operatorname{P_{e}}{(I,\\mathbf{v})} = I \\log{(I^{\\mathbf{v}})} and I \\operatorname{P_{e}}{(I,\\mathbf{v})} - I^{\\mathbf{v}} = I \\log{(I^{\\mathbf{v}})} - I^{\\mathbf{v}} and I (I \\operatorname{P_{e}}{(I,\\mathbf{v})} - I^{\\mathbf{v}}) = I (I \\log{(I^{\\mathbf{v}})} - I^{\\mathbf{v}}) and \\log{(I (I \\operatorname{P_{e}}{(I,\\mathbf{v})} - I^{\\mathbf{v}}))} = \\log{(I (I \\log{(I^{\\mathbf{v}})} - I^{\\mathbf{v}}))} and \\log{(I (I \\operatorname{P_{e}}{(I,\\mathbf{v})} - I^{\\mathbf{v}}))}^{\\mathbf{v}} = \\log{(I (I \\log{(I^{\\mathbf{v}})} - I^{\\mathbf{v}}))}^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Symbol('I', commutative=True), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))"], [["minus", 2, "Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Mul(Symbol('I', commutative=True), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Symbol('I', commutative=True), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))"], [["times", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))), Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))))"], [["log", 4], "Equality(log(Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))), log(Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))))"], [["power", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(log(Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Mul(Symbol('I', commutative=True), Add(Mul(Symbol('I', commutative=True), log(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\Psi{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain \\Psi{(\\mathbf{A})} + \\operatorname{v_{x}}{(\\mathbf{A})} = 2 \\Psi{(\\mathbf{A})}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\operatorname{v_{x}}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} = 2 \\cos{(\\mathbf{A})} and \\Psi{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\Psi{(\\mathbf{A})} + \\operatorname{v_{x}}{(\\mathbf{A})} = 2 \\Psi{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\Psi')(Symbol('\\\\mathbf{A}', commutative=True)), Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given g{(h)} = e^{h} and \\mathbf{P}{(h)} = \\frac{e^{h}}{g{(h)}}, then obtain (\\mathbf{P}^{2}{(h)})^{h} = 1", "derivation": "g{(h)} = e^{h} and \\mathbf{P}{(h)} = \\frac{e^{h}}{g{(h)}} and \\mathbf{P}{(h)} = 1 and \\mathbf{P}^{2}{(h)} = \\frac{\\mathbf{P}{(h)} e^{h}}{g{(h)}} and \\mathbf{P}^{2}{(h)} = \\mathbf{P}{(h)} and \\mathbf{P}^{2}{(h)} = 1 and (\\mathbf{P}^{2}{(h)})^{h} = 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Mul(Pow(Function('g')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Integer(1))"], [["times", 2, "Function('\\\\mathbf{P}')(Symbol('h', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Pow(Function('g')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Integer(2)), Function('\\\\mathbf{P}')(Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Integer(2)), Integer(1))"], [["power", 6, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{P}')(Symbol('h', commutative=True)), Integer(2)), Symbol('h', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\theta_{1}{(T,\\mu)} = T + \\mu, then obtain ((T + \\mu)^{- T} (- (T + \\mu)^{T} + \\theta_{1}^{T}{(T,\\mu)}))^{T} = 0^{T}", "derivation": "\\theta_{1}{(T,\\mu)} = T + \\mu and \\theta_{1}^{T}{(T,\\mu)} = (T + \\mu)^{T} and - (T + \\mu)^{T} + \\theta_{1}^{T}{(T,\\mu)} = 0 and (T + \\mu)^{- T} (- (T + \\mu)^{T} + \\theta_{1}^{T}{(T,\\mu)}) = 0 and ((T + \\mu)^{- T} (- (T + \\mu)^{T} + \\theta_{1}^{T}{(T,\\mu)}))^{T} = 0^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True)), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True)))"], [["minus", 2, "Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))), Integer(0))"], [["divide", 3, "Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True)))), Integer(0))"], [["power", 4, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('T', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Integer(0), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\delta,\\eta^{\\prime})} = \\sin{(\\delta^{\\eta^{\\prime}})} and \\phi_{1}{(\\delta,\\eta^{\\prime})} = \\int (- \\mu{(\\delta,\\eta^{\\prime})} + \\sin{(\\delta^{\\eta^{\\prime}})}) d\\delta, then obtain \\int 0 d\\delta = \\phi_{1}{(\\delta,\\eta^{\\prime})}", "derivation": "\\mu{(\\delta,\\eta^{\\prime})} = \\sin{(\\delta^{\\eta^{\\prime}})} and 0 = - \\mu{(\\delta,\\eta^{\\prime})} + \\sin{(\\delta^{\\eta^{\\prime}})} and \\int 0 d\\delta = \\int (- \\mu{(\\delta,\\eta^{\\prime})} + \\sin{(\\delta^{\\eta^{\\prime}})}) d\\delta and \\phi_{1}{(\\delta,\\eta^{\\prime})} = \\int (- \\mu{(\\delta,\\eta^{\\prime})} + \\sin{(\\delta^{\\eta^{\\prime}})}) d\\delta and \\int 0 d\\delta = \\phi_{1}{(\\delta,\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["minus", 1, "Function('\\\\mu')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), sin(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), sin(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), sin(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given B{(\\mathbf{J})} = e^{\\mathbf{J}}, then derive \\int B{(\\mathbf{J})} d\\mathbf{J} = \\mathbf{J}_M + e^{\\mathbf{J}}, then obtain \\frac{\\int B{(\\mathbf{J})} d\\mathbf{J} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}}}{\\mathbf{J}} = \\frac{\\mathbf{J}_M + B{(\\mathbf{J})} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}}}{\\mathbf{J}}", "derivation": "B{(\\mathbf{J})} = e^{\\mathbf{J}} and \\int B{(\\mathbf{J})} d\\mathbf{J} = \\int e^{\\mathbf{J}} d\\mathbf{J} and \\int B{(\\mathbf{J})} d\\mathbf{J} = \\mathbf{J}_M + e^{\\mathbf{J}} and \\int B{(\\mathbf{J})} d\\mathbf{J} = \\mathbf{J}_M + B{(\\mathbf{J})} and \\int B{(\\mathbf{J})} d\\mathbf{J} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}} = \\mathbf{J}_M + B{(\\mathbf{J})} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}} and \\frac{\\int B{(\\mathbf{J})} d\\mathbf{J} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}}}{\\mathbf{J}} = \\frac{\\mathbf{J}_M + B{(\\mathbf{J})} - (\\int B{(\\mathbf{J})} d\\mathbf{J})^{\\mathbf{J}}}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('B')(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 4, "Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Add(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(v_{z})} = e^{v_{z}}, then obtain - F_{g} + v_{z} \\int \\operatorname{V_{\\mathbf{B}}}{(v_{z})} dv_{z} - e^{v_{z}} = - F_{g} + v_{z} \\int e^{v_{z}} dv_{z} - e^{v_{z}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(v_{z})} = e^{v_{z}} and \\int \\operatorname{V_{\\mathbf{B}}}{(v_{z})} dv_{z} = \\int e^{v_{z}} dv_{z} and v_{z} \\int \\operatorname{V_{\\mathbf{B}}}{(v_{z})} dv_{z} = v_{z} \\int e^{v_{z}} dv_{z} and - F_{g} + v_{z} \\int \\operatorname{V_{\\mathbf{B}}}{(v_{z})} dv_{z} - e^{v_{z}} = - F_{g} + v_{z} \\int e^{v_{z}} dv_{z} - e^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["times", 2, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Symbol('v_z', commutative=True), Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["minus", 3, "Add(Symbol('F_g', commutative=True), exp(Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Symbol('v_z', commutative=True), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Symbol('v_z', commutative=True), Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(W,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} W \\theta_2 and m{(W,\\theta_2)} = (W + \\frac{\\partial}{\\partial \\theta_2} W \\theta_2) \\mathbf{g}{(W,\\theta_2)}, then derive m{(W,\\theta_2)} = 2 W \\mathbf{g}{(W,\\theta_2)}, then obtain (2 W \\mathbf{g}{(W,\\theta_2)})^{\\theta_2} = ((W + \\mathbf{g}{(W,\\theta_2)}) \\mathbf{g}{(W,\\theta_2)})^{\\theta_2}", "derivation": "\\mathbf{g}{(W,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} W \\theta_2 and W + \\mathbf{g}{(W,\\theta_2)} = W + \\frac{\\partial}{\\partial \\theta_2} W \\theta_2 and m{(W,\\theta_2)} = (W + \\frac{\\partial}{\\partial \\theta_2} W \\theta_2) \\mathbf{g}{(W,\\theta_2)} and m{(W,\\theta_2)} = 2 W \\mathbf{g}{(W,\\theta_2)} and 2 W \\mathbf{g}{(W,\\theta_2)} = (W + \\frac{\\partial}{\\partial \\theta_2} W \\theta_2) \\mathbf{g}{(W,\\theta_2)} and 2 W \\mathbf{g}{(W,\\theta_2)} = (W + \\mathbf{g}{(W,\\theta_2)}) \\mathbf{g}{(W,\\theta_2)} and (2 W \\mathbf{g}{(W,\\theta_2)})^{\\theta_2} = ((W + \\mathbf{g}{(W,\\theta_2)}) \\mathbf{g}{(W,\\theta_2)})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Mul(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["add", 1, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('W', commutative=True), Derivative(Mul(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Add(Symbol('W', commutative=True), Derivative(Mul(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Function('m')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Add(Symbol('W', commutative=True), Derivative(Mul(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(2), Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Add(Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["power", 6, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Integer(2), Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Add(Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Function('\\\\mathbf{g}')(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(M,n_{1},n)} = - M + n^{n_{1}}, then obtain M - n^{n_{1}} + \\dot{x}^{2}{(M,n_{1},n)} = - M \\dot{x}{(M,n_{1},n)} + M - n^{n_{1}} + (M + \\dot{x}{(M,n_{1},n)}) \\dot{x}{(M,n_{1},n)}", "derivation": "\\dot{x}{(M,n_{1},n)} = - M + n^{n_{1}} and \\dot{x}^{2}{(M,n_{1},n)} = (- M + n^{n_{1}}) \\dot{x}{(M,n_{1},n)} and M + \\dot{x}{(M,n_{1},n)} = n^{n_{1}} and \\dot{x}^{2}{(M,n_{1},n)} = - M \\dot{x}{(M,n_{1},n)} + n^{n_{1}} \\dot{x}{(M,n_{1},n)} and \\dot{x}^{2}{(M,n_{1},n)} = - M \\dot{x}{(M,n_{1},n)} + (M + \\dot{x}{(M,n_{1},n)}) \\dot{x}{(M,n_{1},n)} and M - n^{n_{1}} + \\dot{x}^{2}{(M,n_{1},n)} = - M \\dot{x}{(M,n_{1},n)} + M - n^{n_{1}} + (M + \\dot{x}{(M,n_{1},n)}) \\dot{x}{(M,n_{1},n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True))))"], [["times", 1, "Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True))), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('M', commutative=True))"], "Equality(Add(Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True)))"], [["expand", 2], "Equality(Pow(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Mul(Add(Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Symbol('n_1', commutative=True))), Mul(Add(Symbol('M', commutative=True), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True))), Function('\\\\dot{x}')(Symbol('M', commutative=True), Symbol('n_1', commutative=True), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\delta,C)} = C \\delta, then derive (\\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)})^{\\delta} = \\delta^{\\delta}, then obtain C + (\\frac{\\partial}{\\partial C} C \\delta)^{\\delta} - \\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)} = C + \\delta^{\\delta} - \\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)}", "derivation": "\\Psi{(\\delta,C)} = C \\delta and \\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)} = \\frac{\\partial}{\\partial C} C \\delta and (\\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)})^{\\delta} = (\\frac{\\partial}{\\partial C} C \\delta)^{\\delta} and (\\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)})^{\\delta} = \\delta^{\\delta} and (\\frac{\\partial}{\\partial C} C \\delta)^{\\delta} = \\delta^{\\delta} and C + (\\frac{\\partial}{\\partial C} C \\delta)^{\\delta} = C + \\delta^{\\delta} and C + (\\frac{\\partial}{\\partial C} C \\delta)^{\\delta} - \\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)} = C + \\delta^{\\delta} - \\frac{\\partial}{\\partial C} \\Psi{(\\delta,C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["add", 5, "Symbol('C', commutative=True)"], "Equality(Add(Symbol('C', commutative=True), Pow(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True))), Add(Symbol('C', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["minus", 6, "Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Add(Symbol('C', commutative=True), Pow(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))), Add(Symbol('C', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))))"]]}, {"prompt": "Given c{(v_{z})} = \\sin{(v_{z})}, then derive 0 = \\frac{\\cos{(v_{z})}}{c{(v_{z})}} - \\frac{\\sin{(v_{z})} \\frac{d}{d v_{z}} c{(v_{z})}}{c^{2}{(v_{z})}}, then obtain 0 = \\frac{\\cos{(v_{z})}}{c{(v_{z})}} - \\frac{\\frac{d}{d v_{z}} c{(v_{z})}}{c{(v_{z})}}", "derivation": "c{(v_{z})} = \\sin{(v_{z})} and 1 = \\frac{\\sin{(v_{z})}}{c{(v_{z})}} and \\frac{d}{d v_{z}} 1 = \\frac{d}{d v_{z}} \\frac{\\sin{(v_{z})}}{c{(v_{z})}} and 0 = \\frac{\\cos{(v_{z})}}{c{(v_{z})}} - \\frac{\\sin{(v_{z})} \\frac{d}{d v_{z}} c{(v_{z})}}{c^{2}{(v_{z})}} and 0 = \\frac{\\cos{(v_{z})}}{c{(v_{z})}} - \\frac{\\frac{d}{d v_{z}} c{(v_{z})}}{c{(v_{z})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["divide", 1, "Function('c')(Symbol('v_z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-1)), sin(Symbol('v_z', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-1)), sin(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-1)), cos(Symbol('v_z', commutative=True))), Mul(Integer(-1), Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-2)), sin(Symbol('v_z', commutative=True)), Derivative(Function('c')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-1)), cos(Symbol('v_z', commutative=True))), Mul(Integer(-1), Pow(Function('c')(Symbol('v_z', commutative=True)), Integer(-1)), Derivative(Function('c')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{P}{(A_{y})} = e^{A_{y}}, then obtain \\mathbf{P}^{A_{y}}{(A_{y})} - (e^{A_{y}})^{A_{y}} = 0", "derivation": "\\mathbf{P}{(A_{y})} = e^{A_{y}} and \\mathbf{P}^{A_{y}}{(A_{y})} = (e^{A_{y}})^{A_{y}} and 2 \\mathbf{P}^{A_{y}}{(A_{y})} = \\mathbf{P}^{A_{y}}{(A_{y})} + (e^{A_{y}})^{A_{y}} and 2 \\mathbf{P}^{A_{y}}{(A_{y})} - \\frac{1}{\\mathbf{P}^{A_{y}}{(A_{y})} + (e^{A_{y}})^{A_{y}}} = \\mathbf{P}^{A_{y}}{(A_{y})} + (e^{A_{y}})^{A_{y}} - \\frac{1}{\\mathbf{P}^{A_{y}}{(A_{y})} + (e^{A_{y}})^{A_{y}}} and \\mathbf{P}^{A_{y}}{(A_{y})} - (e^{A_{y}})^{A_{y}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["add", 2, "Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))))"], [["minus", 3, "Pow(Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Integer(-1)))), Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Integer(-1)))))"], [["minus", 4, "Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Integer(-1))))"], "Equality(Add(Pow(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))), Integer(0))"]]}, {"prompt": "Given l{(I,A_{x})} = I^{A_{x}} and \\operatorname{J_{\\varepsilon}}{(I,A_{x})} = \\int 2 I^{A_{x}} dA_{x}, then obtain \\frac{\\partial}{\\partial I} \\operatorname{J_{\\varepsilon}}{(I,A_{x})} = \\frac{\\partial}{\\partial I} \\int 2 I^{A_{x}} dA_{x}", "derivation": "l{(I,A_{x})} = I^{A_{x}} and I^{A_{x}} + l{(I,A_{x})} = 2 I^{A_{x}} and \\int (I^{A_{x}} + l{(I,A_{x})}) dA_{x} = \\int 2 I^{A_{x}} dA_{x} and \\operatorname{J_{\\varepsilon}}{(I,A_{x})} = \\int 2 I^{A_{x}} dA_{x} and \\operatorname{J_{\\varepsilon}}{(I,A_{x})} = \\int (I^{A_{x}} + l{(I,A_{x})}) dA_{x} and \\frac{\\partial}{\\partial I} \\int (I^{A_{x}} + l{(I,A_{x})}) dA_{x} = \\frac{\\partial}{\\partial I} \\int 2 I^{A_{x}} dA_{x} and \\frac{\\partial}{\\partial I} \\operatorname{J_{\\varepsilon}}{(I,A_{x})} = \\frac{\\partial}{\\partial I} \\int 2 I^{A_{x}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True)))"], [["add", 1, "Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Add(Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Function('l')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Function('l')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Integral(Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('J_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Integral(Add(Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Function('l')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["differentiate", 3, "Symbol('I', commutative=True)"], "Equality(Derivative(Integral(Add(Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Function('l')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), Pow(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(a)} = \\sin{(a)}, then obtain \\cos{(\\hat{x}_0{(a)} - \\frac{1}{\\sin{(a)}})} = \\cos{(\\sin{(a)} - \\frac{1}{\\sin{(a)}})}", "derivation": "\\hat{x}_0{(a)} = \\sin{(a)} and 1 = \\frac{\\sin{(a)}}{\\hat{x}_0{(a)}} and \\frac{1}{\\sin{(a)}} = \\frac{1}{\\hat{x}_0{(a)}} and \\hat{x}_0{(a)} - \\frac{1}{\\hat{x}_0{(a)}} = \\sin{(a)} - \\frac{1}{\\hat{x}_0{(a)}} and \\hat{x}_0{(a)} - \\frac{1}{\\sin{(a)}} = \\sin{(a)} - \\frac{1}{\\sin{(a)}} and \\cos{(\\hat{x}_0{(a)} - \\frac{1}{\\sin{(a)}})} = \\cos{(\\sin{(a)} - \\frac{1}{\\sin{(a)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["divide", 1, "Function('\\\\hat{x}_0')(Symbol('a', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Integer(-1)), sin(Symbol('a', commutative=True))))"], [["divide", 2, "sin(Symbol('a', commutative=True))"], "Equality(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Integer(-1)))"], [["minus", 1, "Pow(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Integer(-1)))), Add(sin(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('a', commutative=True)), Integer(-1)))), Add(sin(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('a', commutative=True)), Integer(-1)))))"], [["cos", 5], "Equality(cos(Add(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('a', commutative=True)), Integer(-1))))), cos(Add(sin(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('a', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\phi_{2}{(m)} = \\sin{(m)}, then derive \\frac{d}{d m} \\phi_{2}{(m)} = \\cos{(m)}, then obtain \\frac{\\cos{(m)} \\frac{d}{d m} \\phi_{2}{(m)}}{\\phi_{2}^{2}{(m)}} = \\frac{\\cos^{2}{(m)}}{\\phi_{2}^{2}{(m)}}", "derivation": "\\phi_{2}{(m)} = \\sin{(m)} and \\frac{d}{d m} \\phi_{2}{(m)} = \\frac{d}{d m} \\sin{(m)} and \\frac{d}{d m} \\phi_{2}{(m)} = \\cos{(m)} and \\frac{\\frac{d}{d m} \\phi_{2}{(m)}}{\\phi_{2}{(m)}} = \\frac{\\cos{(m)}}{\\phi_{2}{(m)}} and \\frac{\\cos{(m)} \\frac{d}{d m} \\phi_{2}{(m)}}{\\phi_{2}^{2}{(m)}} = \\frac{\\cos^{2}{(m)}}{\\phi_{2}^{2}{(m)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), cos(Symbol('m', commutative=True)))"], [["divide", 3, "Function('\\\\phi_2')(Symbol('m', commutative=True))"], "Equality(Mul(Pow(Function('\\\\phi_2')(Symbol('m', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\phi_2')(Symbol('m', commutative=True)), Integer(-1)), cos(Symbol('m', commutative=True))))"], [["times", 4, "Mul(Pow(Function('\\\\phi_2')(Symbol('m', commutative=True)), Integer(-1)), cos(Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\phi_2')(Symbol('m', commutative=True)), Integer(-2)), cos(Symbol('m', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\phi_2')(Symbol('m', commutative=True)), Integer(-2)), Pow(cos(Symbol('m', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\chi{(\\omega)} = e^{\\cos{(\\omega)}}, then obtain \\chi{(\\omega)} e^{- 2 \\cos{(\\omega)}} = \\frac{1}{\\chi{(\\omega)}}", "derivation": "\\chi{(\\omega)} = e^{\\cos{(\\omega)}} and \\chi{(\\omega)} e^{- \\cos{(\\omega)}} = 1 and \\chi^{2}{(\\omega)} e^{- 2 \\cos{(\\omega)}} = \\chi{(\\omega)} e^{- \\cos{(\\omega)}} and \\chi^{2}{(\\omega)} e^{- 2 \\cos{(\\omega)}} = 1 and \\chi{(\\omega)} e^{- 2 \\cos{(\\omega)}} = \\frac{1}{\\chi{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), exp(cos(Symbol('\\\\omega', commutative=True))))"], [["divide", 1, "exp(cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))))), Integer(1))"], [["times", 2, "Mul(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('\\\\omega', commutative=True))))), Mul(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('\\\\omega', commutative=True))))), Integer(1))"], [["divide", 4, "Function('\\\\chi')(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Integer(2), cos(Symbol('\\\\omega', commutative=True))))), Pow(Function('\\\\chi')(Symbol('\\\\omega', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{A}{(M,\\nabla)} = M \\cos{(\\nabla)}, then obtain (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} + \\int (\\int \\mathbf{A}{(M,\\nabla)} d\\nabla)^{\\nabla} d\\nabla = (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} + \\int (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} d\\nabla", "derivation": "\\mathbf{A}{(M,\\nabla)} = M \\cos{(\\nabla)} and \\int \\mathbf{A}{(M,\\nabla)} d\\nabla = \\int M \\cos{(\\nabla)} d\\nabla and (\\int \\mathbf{A}{(M,\\nabla)} d\\nabla)^{\\nabla} = (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} and \\int (\\int \\mathbf{A}{(M,\\nabla)} d\\nabla)^{\\nabla} d\\nabla = \\int (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} d\\nabla and (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} + \\int (\\int \\mathbf{A}{(M,\\nabla)} d\\nabla)^{\\nabla} d\\nabla = (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} + \\int (\\int M \\cos{(\\nabla)} d\\nabla)^{\\nabla} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 4, "Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Integral(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Integral(Pow(Integral(Mul(Symbol('M', commutative=True), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(b,E_{x})} = E_{x} + b and c{(b,E_{x})} = 2 E_{x} + b, then obtain E_{x} + \\mathbf{E}{(b,E_{x})} - 1 = c{(b,E_{x})} - 1", "derivation": "\\mathbf{E}{(b,E_{x})} = E_{x} + b and E_{x} + \\mathbf{E}{(b,E_{x})} = 2 E_{x} + b and E_{x} + \\mathbf{E}{(b,E_{x})} - 1 = 2 E_{x} + b - 1 and c{(b,E_{x})} = 2 E_{x} + b and E_{x} + \\mathbf{E}{(b,E_{x})} - 1 = c{(b,E_{x})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('b', commutative=True)))"], [["add", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(2), Symbol('E_x', commutative=True)), Symbol('b', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('E_x', commutative=True), Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('E_x', commutative=True)), Symbol('b', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('c')(Symbol('b', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(2), Symbol('E_x', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('E_x', commutative=True), Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Add(Function('c')(Symbol('b', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\rho{(\\hat{p},\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} \\hat{p})}, then obtain \\hat{x} - \\rho{(\\hat{p},\\Psi_{\\lambda})} + \\rho^{\\Psi_{\\lambda}}{(\\hat{p},\\Psi_{\\lambda})} = \\hat{x} - \\rho{(\\hat{p},\\Psi_{\\lambda})} + \\sin^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda} \\hat{p})}", "derivation": "\\rho{(\\hat{p},\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} \\hat{p})} and \\rho^{\\Psi_{\\lambda}}{(\\hat{p},\\Psi_{\\lambda})} = \\sin^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda} \\hat{p})} and \\hat{x} + \\rho^{\\Psi_{\\lambda}}{(\\hat{p},\\Psi_{\\lambda})} = \\hat{x} + \\sin^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda} \\hat{p})} and \\hat{x} - \\rho{(\\hat{p},\\Psi_{\\lambda})} + \\rho^{\\Psi_{\\lambda}}{(\\hat{p},\\Psi_{\\lambda})} = \\hat{x} - \\rho{(\\hat{p},\\Psi_{\\lambda})} + \\sin^{\\Psi_{\\lambda}}{(\\Psi_{\\lambda} \\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Pow(Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Pow(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 3, "Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('\\\\rho')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(sin(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given c{(M)} = \\log{(M)}, then derive \\frac{\\frac{d}{d M} c{(M)}}{M} - \\frac{c{(M)}}{M^{2}} = - \\frac{\\log{(M)}}{M^{2}} + \\frac{1}{M^{2}}, then obtain \\frac{\\frac{d}{d M} \\log{(M)}}{M} - \\frac{\\log{(M)}}{M^{2}} = - \\frac{\\log{(M)}}{M^{2}} + \\frac{1}{M^{2}}", "derivation": "c{(M)} = \\log{(M)} and \\frac{c{(M)}}{M} = \\frac{\\log{(M)}}{M} and \\frac{d}{d M} \\frac{c{(M)}}{M} = \\frac{d}{d M} \\frac{\\log{(M)}}{M} and \\frac{\\frac{d}{d M} c{(M)}}{M} - \\frac{c{(M)}}{M^{2}} = - \\frac{\\log{(M)}}{M^{2}} + \\frac{1}{M^{2}} and \\frac{\\frac{d}{d M} c{(M)}}{M} - \\frac{c{(M)}}{M^{2}} = - \\frac{c{(M)}}{M^{2}} + \\frac{1}{M^{2}} and \\frac{\\frac{d}{d M} \\log{(M)}}{M} - \\frac{\\log{(M)}}{M^{2}} = - \\frac{\\log{(M)}}{M^{2}} + \\frac{1}{M^{2}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["divide", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('c')(Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), log(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('c')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), log(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(Function('c')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), Function('c')(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), log(Symbol('M', commutative=True))), Pow(Symbol('M', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(Function('c')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), Function('c')(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), Function('c')(Symbol('M', commutative=True))), Pow(Symbol('M', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), log(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), log(Symbol('M', commutative=True))), Pow(Symbol('M', commutative=True), Integer(-2))))"]]}, {"prompt": "Given G{(W,v)} = \\log{(\\frac{v}{W})}, then derive \\frac{\\partial}{\\partial W} G{(W,v)} = - \\frac{1}{W}, then obtain - \\frac{\\partial}{\\partial W} G{(W,v)} = \\frac{1}{W}", "derivation": "G{(W,v)} = \\log{(\\frac{v}{W})} and G{(W,v)} - 1 = \\log{(\\frac{v}{W})} - 1 and \\frac{\\partial}{\\partial W} (G{(W,v)} - 1) = \\frac{\\partial}{\\partial W} (\\log{(\\frac{v}{W})} - 1) and \\frac{\\partial}{\\partial W} G{(W,v)} = - \\frac{1}{W} and - \\frac{\\partial}{\\partial W} G{(W,v)} = \\frac{1}{W}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('W', commutative=True), Symbol('v', commutative=True)), log(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('G')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Add(log(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Integer(-1)))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(log(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Integer(-1)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('G')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('G')(Symbol('W', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Pow(Symbol('W', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(x^\\prime)} = \\sin{(x^\\prime)}, then derive \\frac{d}{d x^\\prime} \\Psi^{\\dagger}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain x^\\prime + \\frac{d^{2}}{d (x^\\prime)^{2}} \\Psi^{\\dagger}{(x^\\prime)} = x^\\prime + \\frac{d}{d x^\\prime} \\cos{(x^\\prime)}", "derivation": "\\Psi^{\\dagger}{(x^\\prime)} = \\sin{(x^\\prime)} and \\frac{d}{d x^\\prime} \\Psi^{\\dagger}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(x^\\prime)} and \\frac{d}{d x^\\prime} \\Psi^{\\dagger}{(x^\\prime)} = \\cos{(x^\\prime)} and \\frac{d^{2}}{d (x^\\prime)^{2}} \\Psi^{\\dagger}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\cos{(x^\\prime)} and x^\\prime + \\frac{d^{2}}{d (x^\\prime)^{2}} \\Psi^{\\dagger}{(x^\\prime)} = x^\\prime + \\frac{d}{d x^\\prime} \\cos{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), cos(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["add", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Add(Symbol('x^\\\\prime', commutative=True), Derivative(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\log{(\\mathbb{I})}, then derive \\operatorname{t_{1}}{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}} = 0, then obtain \\frac{d}{d \\mathbb{I}} (\\frac{d}{d \\mathbb{I}} \\log{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}})^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} 0^{\\mathbb{I}}", "derivation": "\\operatorname{t_{1}}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\log{(\\mathbb{I})} and \\operatorname{t_{1}}{(\\mathbb{I})} - \\frac{d}{d \\mathbb{I}} \\log{(\\mathbb{I})} = 0 and \\operatorname{t_{1}}{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}} = 0 and (\\operatorname{t_{1}}{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}})^{\\mathbb{I}} = 0^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} (\\operatorname{t_{1}}{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}})^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} 0^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} (\\frac{d}{d \\mathbb{I}} \\log{(\\mathbb{I})} - \\frac{1}{\\mathbb{I}})^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} 0^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Add(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Integer(0))"], [["power", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Add(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Pow(Add(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(Add(Derivative(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(V,\\psi^*)} = (e^{\\psi^*})^{V} and \\mathbf{J}{(V,\\psi^*)} = \\int \\hat{H}{(V,\\psi^*)} d\\psi^*, then obtain (\\hat{H}{(V,\\psi^*)} + \\mathbf{J}{(V,\\psi^*)})^{\\psi^*} = (\\hat{H}{(V,\\psi^*)} + \\int (e^{\\psi^*})^{V} d\\psi^*)^{\\psi^*}", "derivation": "\\hat{H}{(V,\\psi^*)} = (e^{\\psi^*})^{V} and \\mathbf{J}{(V,\\psi^*)} = \\int \\hat{H}{(V,\\psi^*)} d\\psi^* and \\mathbf{J}{(V,\\psi^*)} + (e^{\\psi^*})^{V} = (e^{\\psi^*})^{V} + \\int \\hat{H}{(V,\\psi^*)} d\\psi^* and \\mathbf{J}{(V,\\psi^*)} + (e^{\\psi^*})^{V} = (e^{\\psi^*})^{V} + \\int (e^{\\psi^*})^{V} d\\psi^* and (\\mathbf{J}{(V,\\psi^*)} + (e^{\\psi^*})^{V})^{\\psi^*} = ((e^{\\psi^*})^{V} + \\int (e^{\\psi^*})^{V} d\\psi^*)^{\\psi^*} and (\\hat{H}{(V,\\psi^*)} + \\mathbf{J}{(V,\\psi^*)})^{\\psi^*} = (\\hat{H}{(V,\\psi^*)} + \\int (e^{\\psi^*})^{V} d\\psi^*)^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True))), Add(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True))), Add(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Integral(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["power", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{J}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Integral(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Function('\\\\hat{H}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Function('\\\\mathbf{J}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Function('\\\\hat{H}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Pow(exp(Symbol('\\\\psi^*', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(S,n_{2})} = \\log{(S + n_{2})}, then obtain 2 S + 3 \\dot{z}{(S,n_{2})} = 2 S + \\dot{z}{(S,n_{2})} + 2 \\log{(S + n_{2})}", "derivation": "\\dot{z}{(S,n_{2})} = \\log{(S + n_{2})} and S + \\dot{z}{(S,n_{2})} = S + \\log{(S + n_{2})} and 2 S + 2 \\dot{z}{(S,n_{2})} = 2 S + \\dot{z}{(S,n_{2})} + \\log{(S + n_{2})} and 2 S + 3 \\dot{z}{(S,n_{2})} = 2 S + 2 \\dot{z}{(S,n_{2})} + \\log{(S + n_{2})} and 2 S + 3 \\dot{z}{(S,n_{2})} = 2 S + \\dot{z}{(S,n_{2})} + 2 \\log{(S + n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)), log(Add(Symbol('S', commutative=True), Symbol('n_2', commutative=True))))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True))), Add(Symbol('S', commutative=True), log(Add(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))))"], [["add", 2, "Add(Symbol('S', commutative=True), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)), log(Add(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))))"], [["add", 3, "Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(3), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True))), log(Add(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('S', commutative=True)), Mul(Integer(3), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Integer(2), Symbol('S', commutative=True)), Function('\\\\dot{z}')(Symbol('S', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(2), log(Add(Symbol('S', commutative=True), Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{X}{(\\phi)} = e^{\\phi}, then derive - e^{\\phi} + \\frac{d}{d \\phi} \\hat{X}{(\\phi)} = 0, then obtain - e^{\\phi} + \\frac{d}{d \\phi} e^{\\phi} = 0", "derivation": "\\hat{X}{(\\phi)} = e^{\\phi} and \\hat{X}{(\\phi)} - e^{\\phi} = 0 and \\frac{d}{d \\phi} (\\hat{X}{(\\phi)} - e^{\\phi}) = \\frac{d}{d \\phi} 0 and - e^{\\phi} + \\frac{d}{d \\phi} \\hat{X}{(\\phi)} = 0 and - e^{\\phi} + \\frac{d}{d \\phi} e^{\\phi} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{X}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True))), Derivative(Function('\\\\hat{X}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True))), Derivative(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(z,G)} = \\frac{G}{z} and \\omega{(z,G)} = \\frac{G}{z} - (\\frac{G}{z})^{G}, then obtain G + \\frac{G}{z} - (\\frac{G}{z})^{G} = G + \\frac{G}{z} - \\phi_{1}^{G}{(z,G)}", "derivation": "\\phi_{1}{(z,G)} = \\frac{G}{z} and \\phi_{1}^{G}{(z,G)} = (\\frac{G}{z})^{G} and - (\\frac{G}{z})^{G} + \\phi_{1}{(z,G)} = \\frac{G}{z} - (\\frac{G}{z})^{G} and G - (\\frac{G}{z})^{G} + \\phi_{1}{(z,G)} = G + \\frac{G}{z} - (\\frac{G}{z})^{G} and \\omega{(z,G)} = \\frac{G}{z} - (\\frac{G}{z})^{G} and G + \\phi_{1}{(z,G)} - \\phi_{1}^{G}{(z,G)} = G + \\frac{G}{z} - \\phi_{1}^{G}{(z,G)} and \\omega{(z,G)} = \\phi_{1}{(z,G)} - \\phi_{1}^{G}{(z,G)} and \\frac{G}{z} - (\\frac{G}{z})^{G} = \\phi_{1}{(z,G)} - \\phi_{1}^{G}{(z,G)} and G + \\frac{G}{z} - (\\frac{G}{z})^{G} = G + \\frac{G}{z} - \\phi_{1}^{G}{(z,G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))"], [["minus", 1, "Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True))), Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))))"], [["add", 3, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True))), Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('G', commutative=True), Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Add(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Add(Symbol('G', commutative=True), Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Symbol('G', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Symbol('G', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\psi,\\omega)} = \\cos{(\\omega^{\\psi})} and \\mathbf{J}{(\\psi,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\operatorname{F_{N}}{(\\psi,\\omega)}, then derive \\frac{\\partial}{\\partial \\omega} \\operatorname{F_{N}}{(\\psi,\\omega)} + 1 = 1 - \\frac{\\omega^{\\psi} \\psi \\sin{(\\omega^{\\psi})}}{\\omega}, then obtain \\mathbf{J}{(\\psi,\\omega)} + 1 = 1 - \\frac{\\omega^{\\psi} \\psi \\sin{(\\omega^{\\psi})}}{\\omega}", "derivation": "\\operatorname{F_{N}}{(\\psi,\\omega)} = \\cos{(\\omega^{\\psi})} and \\omega + \\operatorname{F_{N}}{(\\psi,\\omega)} = \\omega + \\cos{(\\omega^{\\psi})} and \\frac{\\partial}{\\partial \\omega} (\\omega + \\operatorname{F_{N}}{(\\psi,\\omega)}) = \\frac{\\partial}{\\partial \\omega} (\\omega + \\cos{(\\omega^{\\psi})}) and \\frac{\\partial}{\\partial \\omega} \\operatorname{F_{N}}{(\\psi,\\omega)} + 1 = 1 - \\frac{\\omega^{\\psi} \\psi \\sin{(\\omega^{\\psi})}}{\\omega} and \\mathbf{J}{(\\psi,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\operatorname{F_{N}}{(\\psi,\\omega)} and \\mathbf{J}{(\\psi,\\omega)} + 1 = 1 - \\frac{\\omega^{\\psi} \\psi \\sin{(\\omega^{\\psi})}}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('F_N')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\omega', commutative=True), cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\omega', commutative=True), Function('F_N')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\omega', commutative=True), cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('F_N')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True), sin(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Function('F_N')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True), sin(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(l)} = \\cos{(\\sin{(l)})} and V{(l)} = \\sin{(l)}, then obtain \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = \\frac{d}{d l} \\cos{(V{(l)})}", "derivation": "\\operatorname{C_{2}}{(l)} = \\cos{(\\sin{(l)})} and V{(l)} = \\sin{(l)} and \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = \\frac{d}{d l} \\cos{(\\sin{(l)})} and \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = \\frac{d}{d l} \\cos{(V{(l)})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('l', commutative=True)), cos(sin(Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(cos(Function('V')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(c_{0})} = \\int \\log{(c_{0})} dc_{0}, then derive U{(c_{0})} = I + c_{0} \\log{(c_{0})} - c_{0}, then obtain (\\int (I + c_{0} \\log{(c_{0})} - c_{0}) dc_{0})^{c_{0}} = (\\iint \\log{(c_{0})} dc_{0} dc_{0})^{c_{0}}", "derivation": "U{(c_{0})} = \\int \\log{(c_{0})} dc_{0} and \\int U{(c_{0})} dc_{0} = \\iint \\log{(c_{0})} dc_{0} dc_{0} and U{(c_{0})} = I + c_{0} \\log{(c_{0})} - c_{0} and \\int (I + c_{0} \\log{(c_{0})} - c_{0}) dc_{0} = \\iint \\log{(c_{0})} dc_{0} dc_{0} and \\int (I + c_{0} \\log{(c_{0})} - c_{0}) dc_{0} = \\int U{(c_{0})} dc_{0} and (\\int (I + c_{0} \\log{(c_{0})} - c_{0}) dc_{0})^{c_{0}} = (\\int U{(c_{0})} dc_{0})^{c_{0}} and (\\int (I + c_{0} \\log{(c_{0})} - c_{0}) dc_{0})^{c_{0}} = (\\iint \\log{(c_{0})} dc_{0} dc_{0})^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('c_0', commutative=True)), Integral(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('U')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('U')(Symbol('c_0', commutative=True)), Add(Symbol('I', commutative=True), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Add(Symbol('I', commutative=True), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integral(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Add(Symbol('I', commutative=True), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integral(Function('U')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["power", 5, "Symbol('c_0', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('I', commutative=True), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)), Pow(Integral(Function('U')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Integral(Add(Symbol('I', commutative=True), Mul(Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)), Pow(Integral(log(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(f^{*},L)} = (f^{*})^{L}, then obtain (f^{*} + (f^{*})^{L}) (f^{*} + \\hat{p}{(f^{*},L)})^{L} = (f^{*} + (f^{*})^{L}) (f^{*} + (f^{*})^{L})^{L}", "derivation": "\\hat{p}{(f^{*},L)} = (f^{*})^{L} and f^{*} + \\hat{p}{(f^{*},L)} = f^{*} + (f^{*})^{L} and (f^{*} + \\hat{p}{(f^{*},L)})^{L} = (f^{*} + (f^{*})^{L})^{L} and (f^{*} + (f^{*})^{L}) (f^{*} + \\hat{p}{(f^{*},L)})^{L} = (f^{*} + (f^{*})^{L}) (f^{*} + (f^{*})^{L})^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('f^*', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True)))"], [["add", 1, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Function('\\\\hat{p}')(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Symbol('f^*', commutative=True), Function('\\\\hat{p}')(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["times", 3, "Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Pow(Add(Symbol('f^*', commutative=True), Function('\\\\hat{p}')(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Pow(Add(Symbol('f^*', commutative=True), Pow(Symbol('f^*', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\phi{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})}, then obtain \\frac{\\mathbf{H} \\phi^{2}{(A_{2})}}{F_{N} \\log{(A_{2})}} = \\frac{\\mathbf{H} \\phi{(A_{2})} \\frac{d}{d A_{2}} \\log{(A_{2})}}{F_{N} \\log{(A_{2})}}", "derivation": "\\phi{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and \\frac{\\phi{(A_{2})}}{\\log{(A_{2})}} = \\frac{\\frac{d}{d A_{2}} \\log{(A_{2})}}{\\log{(A_{2})}} and \\frac{\\mathbf{H} \\phi{(A_{2})}}{F_{N} \\log{(A_{2})}} = \\frac{\\mathbf{H} \\frac{d}{d A_{2}} \\log{(A_{2})}}{F_{N} \\log{(A_{2})}} and \\frac{\\mathbf{H} \\phi^{2}{(A_{2})}}{F_{N} \\log{(A_{2})}} = \\frac{\\mathbf{H} \\phi{(A_{2})} \\frac{d}{d A_{2}} \\log{(A_{2})}}{F_{N} \\log{(A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('A_2', commutative=True)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["divide", 1, "log(Symbol('A_2', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('A_2', commutative=True)), Pow(log(Symbol('A_2', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["times", 2, "Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\phi')(Symbol('A_2', commutative=True)), Pow(log(Symbol('A_2', commutative=True)), Integer(-1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True), Pow(log(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["times", 3, "Function('\\\\phi')(Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\phi')(Symbol('A_2', commutative=True)), Integer(2)), Pow(log(Symbol('A_2', commutative=True)), Integer(-1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\phi')(Symbol('A_2', commutative=True)), Pow(log(Symbol('A_2', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(\\delta)} = e^{\\delta} and \\operatorname{A_{z}}{(\\delta)} = e^{\\delta}, then obtain ((2 e^{\\delta})^{\\delta})^{\\delta} = ((\\operatorname{A_{z}}{(\\delta)} + e^{\\delta})^{\\delta})^{\\delta}", "derivation": "\\lambda{(\\delta)} = e^{\\delta} and \\operatorname{A_{z}}{(\\delta)} = e^{\\delta} and 2 \\operatorname{A_{z}}{(\\delta)} = \\operatorname{A_{z}}{(\\delta)} + e^{\\delta} and 2 \\operatorname{A_{z}}{(\\delta)} = \\operatorname{A_{z}}{(\\delta)} + \\lambda{(\\delta)} and 2 e^{\\delta} = \\lambda{(\\delta)} + e^{\\delta} and (2 e^{\\delta})^{\\delta} = (\\lambda{(\\delta)} + e^{\\delta})^{\\delta} and \\operatorname{A_{z}}{(\\delta)} = \\lambda{(\\delta)} and (2 e^{\\delta})^{\\delta} = (\\operatorname{A_{z}}{(\\delta)} + e^{\\delta})^{\\delta} and ((2 e^{\\delta})^{\\delta})^{\\delta} = ((\\operatorname{A_{z}}{(\\delta)} + e^{\\delta})^{\\delta})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["add", 2, "Function('A_z')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(2), Function('A_z')(Symbol('\\\\delta', commutative=True))), Add(Function('A_z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('A_z')(Symbol('\\\\delta', commutative=True))), Add(Function('A_z')(Symbol('\\\\delta', commutative=True)), Function('\\\\lambda')(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Add(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["power", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Function('\\\\lambda')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_z')(Symbol('\\\\delta', commutative=True)), Function('\\\\lambda')(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Function('A_z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["power", 8, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(Add(Function('A_z')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(A_{2},B)} = \\log{(- A_{2} + B)} and \\mathbf{J}_P{(A_{2},B)} = - 2 \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)}^{2}, then obtain - (\\operatorname{m_{s}}{(A_{2},B)} + \\log{(- A_{2} + B)}) \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)} = \\mathbf{J}_P{(A_{2},B)}", "derivation": "\\operatorname{m_{s}}{(A_{2},B)} = \\log{(- A_{2} + B)} and \\operatorname{m_{s}}{(A_{2},B)} + \\log{(- A_{2} + B)} = 2 \\log{(- A_{2} + B)} and - (\\operatorname{m_{s}}{(A_{2},B)} + \\log{(- A_{2} + B)}) \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)} = - 2 \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)}^{2} and \\mathbf{J}_P{(A_{2},B)} = - 2 \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)}^{2} and - (\\operatorname{m_{s}}{(A_{2},B)} + \\log{(- A_{2} + B)}) \\operatorname{m_{s}}{(A_{2},B)} \\log{(- A_{2} + B)} = \\mathbf{J}_P{(A_{2},B)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True))))"], "Equality(Add(Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))), Mul(Integer(2), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))), Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))), Mul(Integer(-1), Integer(2), Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Integer(2), Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Add(Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))), Function('m_s')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('B', commutative=True)))), Function('\\\\mathbf{J}_P')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(f)} = \\log{(f)}, then obtain \\frac{\\mathbf{p}{(f)}}{\\mathbf{p}{(f)} + \\log{(f)}} = \\frac{1}{2}", "derivation": "\\mathbf{p}{(f)} = \\log{(f)} and \\mathbf{p}{(f)} + \\log{(f)} = 2 \\log{(f)} and \\frac{\\mathbf{p}{(f)}}{2 \\log{(f)}} = \\frac{1}{2} and \\frac{\\mathbf{p}{(f)}}{\\mathbf{p}{(f)} + \\log{(f)}} = \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["add", 1, "log(Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Mul(Integer(2), log(Symbol('f', commutative=True))))"], [["divide", 1, "Mul(Integer(2), log(Symbol('f', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True))), Rational(1, 2))"]]}, {"prompt": "Given \\dot{y}{(v)} = \\sin{(v)}, then obtain \\frac{\\sin{(2 \\dot{y}{(v)})}}{2 \\dot{y}{(v)}} = \\frac{\\sin{(\\dot{y}{(v)} + \\sin{(v)})}}{2 \\dot{y}{(v)}}", "derivation": "\\dot{y}{(v)} = \\sin{(v)} and 2 \\dot{y}{(v)} = \\dot{y}{(v)} + \\sin{(v)} and \\sin{(2 \\dot{y}{(v)})} = \\sin{(\\dot{y}{(v)} + \\sin{(v)})} and \\frac{\\sin{(2 \\dot{y}{(v)})}}{2 \\dot{y}{(v)}} = \\frac{\\sin{(\\dot{y}{(v)} + \\sin{(v)})}}{2 \\dot{y}{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["add", 1, "Function('\\\\dot{y}')(Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('v', commutative=True))), Add(Function('\\\\dot{y}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('v', commutative=True)))), sin(Add(Function('\\\\dot{y}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), Function('\\\\dot{y}')(Symbol('v', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\dot{y}')(Symbol('v', commutative=True)), Integer(-1)), sin(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('v', commutative=True))))), Mul(Rational(1, 2), Pow(Function('\\\\dot{y}')(Symbol('v', commutative=True)), Integer(-1)), sin(Add(Function('\\\\dot{y}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(p)} = \\sin{(p)}, then obtain - \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 2 = (- 2 \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 1)^{4} - \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 1", "derivation": "\\operatorname{E_{\\lambda}}{(p)} = \\sin{(p)} and 0 = - \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} and 1 = - \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1 and - \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1 = (- \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1)^{2} and 1 = (- \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1)^{2} and \\sin{(p)} + 1 = - \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 1 and \\sin{(p)} + 2 = (- \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1)^{2} + \\sin{(p)} + 1 and \\sin{(p)} + 2 = (- \\operatorname{E_{\\lambda}}{(p)} + \\sin{(p)} + 1)^{4} + \\sin{(p)} + 1 and - \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 2 = (- 2 \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 1)^{4} - \\operatorname{E_{\\lambda}}{(p)} + 2 \\sin{(p)} + 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["minus", 1, "Function('E_{\\\\lambda}')(Symbol('p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)))"], [["times", 3, "Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)), Pow(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Pow(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)), Integer(2)))"], [["add", 3, "sin(Symbol('p', commutative=True))"], "Equality(Add(sin(Symbol('p', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Integer(2), sin(Symbol('p', commutative=True))), Integer(1)))"], [["add", 5, "Add(sin(Symbol('p', commutative=True)), Integer(1))"], "Equality(Add(sin(Symbol('p', commutative=True)), Integer(2)), Add(Pow(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)), Integer(2)), sin(Symbol('p', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(sin(Symbol('p', commutative=True)), Integer(2)), Add(Pow(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True)), Integer(1)), Integer(4)), sin(Symbol('p', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Integer(2), sin(Symbol('p', commutative=True))), Integer(2)), Add(Pow(Add(Mul(Integer(-1), Integer(2), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Integer(2), sin(Symbol('p', commutative=True))), Integer(1)), Integer(4)), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Integer(2), sin(Symbol('p', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})} = \\dot{z} \\mathbf{v}, then obtain (0^{\\dot{z}})^{\\mathbf{v}} = 1", "derivation": "\\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})} = \\dot{z} \\mathbf{v} and 0 = \\dot{z} \\mathbf{v} - \\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})} and 0^{\\dot{z}} = (\\dot{z} \\mathbf{v} - \\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})})^{\\dot{z}} and (0^{\\dot{z}})^{\\mathbf{v}} = ((\\dot{z} \\mathbf{v} - \\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})})^{\\dot{z}})^{\\mathbf{v}} and ((\\dot{z} \\mathbf{v} - \\operatorname{n_{2}}{(\\mathbf{v},\\dot{z})})^{\\dot{z}})^{\\mathbf{v}} = 1 and (0^{\\dot{z}})^{\\mathbf{v}} = 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["minus", 1, "Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('\\\\dot{z}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Pow(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Pow(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Pow(Integer(0), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{A}{(a)} = e^{a}, then obtain \\int \\mathbf{A}{(a)} da - 1 = e^{- \\mathbf{A}{(a)} + e^{a}} \\int e^{a} da - 1", "derivation": "\\mathbf{A}{(a)} = e^{a} and \\int \\mathbf{A}{(a)} da = \\int e^{a} da and 0 = - \\mathbf{A}{(a)} + e^{a} and 1 = e^{- \\mathbf{A}{(a)} + e^{a}} and \\int e^{a} da = e^{- \\mathbf{A}{(a)} + e^{a}} \\int e^{a} da and \\int \\mathbf{A}{(a)} da = e^{- \\mathbf{A}{(a)} + e^{a}} \\int e^{a} da and \\int \\mathbf{A}{(a)} da - 1 = e^{- \\mathbf{A}{(a)} + e^{a}} \\int e^{a} da - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{A}')(Symbol('a', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True))))"], [["exp", 3], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True)))))"], [["times", 4, "Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Mul(exp(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True)))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Mul(exp(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True)))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\mathbf{A}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1)), Add(Mul(exp(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('a', commutative=True))), exp(Symbol('a', commutative=True)))), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{S}{(W)} = \\sin{(W)}, then derive \\int \\mathbf{S}{(W)} dW = v_{1} - \\cos{(W)}, then derive x^\\prime + \\sin^{W}{(W)} - \\cos{(W)} = v_{1} + \\sin^{W}{(W)} - \\cos{(W)}, then obtain \\frac{\\partial}{\\partial W} (x^\\prime + \\sin^{W}{(W)} - \\cos{(W)}) = \\frac{\\partial}{\\partial W} (v_{1} + \\sin^{W}{(W)} - \\cos{(W)})", "derivation": "\\mathbf{S}{(W)} = \\sin{(W)} and \\int \\mathbf{S}{(W)} dW = \\int \\sin{(W)} dW and \\int \\mathbf{S}{(W)} dW = v_{1} - \\cos{(W)} and \\mathbf{S}^{W}{(W)} + \\int \\mathbf{S}{(W)} dW = v_{1} + \\mathbf{S}^{W}{(W)} - \\cos{(W)} and \\sin^{W}{(W)} + \\int \\sin{(W)} dW = v_{1} + \\sin^{W}{(W)} - \\cos{(W)} and x^\\prime + \\sin^{W}{(W)} - \\cos{(W)} = v_{1} + \\sin^{W}{(W)} - \\cos{(W)} and \\frac{\\partial}{\\partial W} (x^\\prime + \\sin^{W}{(W)} - \\cos{(W)}) = \\frac{\\partial}{\\partial W} (v_{1} + \\sin^{W}{(W)} - \\cos{(W)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["add", 3, "Pow(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Symbol('v_1', commutative=True), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Add(Symbol('v_1', commutative=True), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Symbol('x^\\\\prime', commutative=True), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Symbol('v_1', commutative=True), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = P_{e}^{\\varepsilon_0}, then derive \\frac{\\partial}{\\partial P_{e}} \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = \\frac{P_{e}^{\\varepsilon_0} \\varepsilon_0}{P_{e}}, then obtain \\frac{P_{e}^{\\varepsilon_0} \\varepsilon_0}{P_{e}} = \\frac{\\varepsilon_0 \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})}}{P_{e}}", "derivation": "\\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = P_{e}^{\\varepsilon_0} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = \\frac{\\partial}{\\partial P_{e}} P_{e}^{\\varepsilon_0} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = \\frac{P_{e}^{\\varepsilon_0} \\varepsilon_0}{P_{e}} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})} = \\frac{\\varepsilon_0 \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})}}{P_{e}} and \\frac{P_{e}^{\\varepsilon_0} \\varepsilon_0}{P_{e}} = \\frac{\\varepsilon_0 \\operatorname{A_{y}}{(\\varepsilon_0,P_{e})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True), Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True), Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\phi_2,\\mu)} = \\cos{(\\mu + \\phi_2)}, then obtain - \\mu + \\sin{(\\hat{H}_{\\lambda}{(\\phi_2,\\mu)})} - \\sin{(\\cos{(\\mu + \\phi_2)})} = - \\mu", "derivation": "\\hat{H}_{\\lambda}{(\\phi_2,\\mu)} = \\cos{(\\mu + \\phi_2)} and \\sin{(\\hat{H}_{\\lambda}{(\\phi_2,\\mu)})} = \\sin{(\\cos{(\\mu + \\phi_2)})} and - \\mu + \\sin{(\\hat{H}_{\\lambda}{(\\phi_2,\\mu)})} = - \\mu + \\sin{(\\cos{(\\mu + \\phi_2)})} and - \\mu + \\sin{(\\hat{H}_{\\lambda}{(\\phi_2,\\mu)})} - \\sin{(\\cos{(\\mu + \\phi_2)})} = - \\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True))), sin(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), sin(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\phi_2', commutative=True))))))"], [["minus", 3, "sin(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), sin(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\phi_2', commutative=True)))))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(i,\\hat{p}_0)} = \\cos{(\\frac{\\hat{p}_0}{i})} and \\operatorname{f^{*}}{(i,\\hat{p}_0)} = - \\dot{z}{(i,\\hat{p}_0)}, then obtain 0 = \\dot{z}{(i,\\hat{p}_0)} + \\operatorname{f^{*}}{(i,\\hat{p}_0)}", "derivation": "\\dot{z}{(i,\\hat{p}_0)} = \\cos{(\\frac{\\hat{p}_0}{i})} and 0 = - \\dot{z}{(i,\\hat{p}_0)} + \\cos{(\\frac{\\hat{p}_0}{i})} and \\operatorname{f^{*}}{(i,\\hat{p}_0)} = - \\dot{z}{(i,\\hat{p}_0)} and 0 = \\operatorname{f^{*}}{(i,\\hat{p}_0)} + \\cos{(\\frac{\\hat{p}_0}{i})} and 0 = \\dot{z}{(i,\\hat{p}_0)} + \\operatorname{f^{*}}{(i,\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))"], [["minus", 1, "Function('\\\\dot{z}')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), cos(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Function('\\\\dot{z}')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('f^*')(Symbol('i', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(C_{1},r,H)} = C_{1} - H + r and \\operatorname{m_{s}}{(r,C_{1},\\theta_2,H)} = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H)) \\int \\rho_{f}{(C_{1},r,H)} dr, then derive (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H)) \\int \\rho_{f}{(C_{1},r,H)} dr = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H))^{2}, then obtain \\operatorname{m_{s}}{(r,C_{1},\\theta_2,H)} = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H))^{2}", "derivation": "\\rho_{f}{(C_{1},r,H)} = C_{1} - H + r and \\int \\rho_{f}{(C_{1},r,H)} dr = \\int (C_{1} - H + r) dr and (\\int (C_{1} - H + r) dr) \\int \\rho_{f}{(C_{1},r,H)} dr = (\\int (C_{1} - H + r) dr)^{2} and (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H)) \\int \\rho_{f}{(C_{1},r,H)} dr = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H))^{2} and \\operatorname{m_{s}}{(r,C_{1},\\theta_2,H)} = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H)) \\int \\rho_{f}{(C_{1},r,H)} dr and \\operatorname{m_{s}}{(r,C_{1},\\theta_2,H)} = (\\theta_2 + \\frac{r^{2}}{2} + r (C_{1} - H))^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('C_1', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('C_1', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["times", 2, "Integral(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Function('\\\\rho_f')(Symbol('C_1', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('r', commutative=True)))), Pow(Integral(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('r', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))))), Integral(Function('\\\\rho_f')(Symbol('C_1', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('r', commutative=True)))), Pow(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('r', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))))), Integer(2)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('r', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True)), Mul(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('r', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))))), Integral(Function('\\\\rho_f')(Symbol('C_1', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('m_s')(Symbol('r', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('r', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('H', commutative=True))))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)} = A_{2} + \\Omega \\theta_2, then obtain 2 \\pi + 2 y^{\\prime} + \\log{(A_{2} + \\Omega \\theta_2)} + \\log{(\\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)})} = 2 \\pi + 2 y^{\\prime} + 2 \\log{(A_{2} + \\Omega \\theta_2)}", "derivation": "\\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)} = A_{2} + \\Omega \\theta_2 and \\log{(\\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)})} = \\log{(A_{2} + \\Omega \\theta_2)} and \\pi + y^{\\prime} + \\log{(\\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)})} = \\pi + y^{\\prime} + \\log{(A_{2} + \\Omega \\theta_2)} and 2 \\pi + 2 y^{\\prime} + \\log{(A_{2} + \\Omega \\theta_2)} + \\log{(\\operatorname{A_{x}}{(A_{2},\\theta_2,\\Omega)})} = 2 \\pi + 2 y^{\\prime} + 2 \\log{(A_{2} + \\Omega \\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["log", 1], "Equality(log(Function('A_x')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Omega', commutative=True))), log(Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["add", 2, "Add(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True), log(Function('A_x')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True), log(Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True))))))"], [["add", 3, "Add(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True), log(Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)), log(Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)))), log(Function('A_x')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), log(Add(Symbol('A_2', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_2', commutative=True)))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(u)} = \\frac{d}{d u} e^{u}, then derive \\eta^{\\prime}{(u)} e^{u} = e^{2 u}, then obtain \\cos{(\\int e^{u} \\frac{d}{d u} e^{u} du)} = \\cos{(\\int e^{2 u} du)}", "derivation": "\\eta^{\\prime}{(u)} = \\frac{d}{d u} e^{u} and \\eta^{\\prime}{(u)} e^{u} = e^{u} \\frac{d}{d u} e^{u} and \\eta^{\\prime}{(u)} e^{u} = e^{2 u} and \\int \\eta^{\\prime}{(u)} e^{u} du = \\int e^{2 u} du and \\int e^{u} \\frac{d}{d u} e^{u} du = \\int e^{2 u} du and \\cos{(\\int e^{u} \\frac{d}{d u} e^{u} du)} = \\cos{(\\int e^{2 u} du)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 1, "exp(Symbol('u', commutative=True))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))), Mul(exp(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))), exp(Mul(Integer(2), Symbol('u', commutative=True))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(exp(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["cos", 5], "Equality(cos(Integral(Mul(exp(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True)))), cos(Integral(exp(Mul(Integer(2), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given B{(n_{2},f)} = \\log{(f + n_{2})}, then obtain (B^{f}{(n_{2},f)} - \\log{(f + n_{2})}^{f})^{n_{2}} = 0^{n_{2}}", "derivation": "B{(n_{2},f)} = \\log{(f + n_{2})} and B^{f}{(n_{2},f)} = \\log{(f + n_{2})}^{f} and B^{f}{(n_{2},f)} - \\log{(f + n_{2})}^{f} = 0 and (B^{f}{(n_{2},f)} - \\log{(f + n_{2})}^{f})^{n_{2}} = 0^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('n_2', commutative=True), Symbol('f', commutative=True)), log(Add(Symbol('f', commutative=True), Symbol('n_2', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('B')(Symbol('n_2', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Add(Symbol('f', commutative=True), Symbol('n_2', commutative=True))), Symbol('f', commutative=True)))"], [["minus", 2, "Pow(log(Add(Symbol('f', commutative=True), Symbol('n_2', commutative=True))), Symbol('f', commutative=True))"], "Equality(Add(Pow(Function('B')(Symbol('n_2', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Mul(Integer(-1), Pow(log(Add(Symbol('f', commutative=True), Symbol('n_2', commutative=True))), Symbol('f', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Add(Pow(Function('B')(Symbol('n_2', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Mul(Integer(-1), Pow(log(Add(Symbol('f', commutative=True), Symbol('n_2', commutative=True))), Symbol('f', commutative=True)))), Symbol('n_2', commutative=True)), Pow(Integer(0), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(n_{2},l)} = l - n_{2}, then derive \\int \\mathbb{I}{(n_{2},l)} dl = a^{\\dagger} + \\frac{l^{2}}{2} - l n_{2}, then derive \\frac{l^{2}}{2} - l n_{2} + r = a^{\\dagger} + \\frac{l^{2}}{2} - l n_{2}, then obtain \\frac{l^{2}}{2} - l n_{2} + r = \\int (l - n_{2}) dl", "derivation": "\\mathbb{I}{(n_{2},l)} = l - n_{2} and \\int \\mathbb{I}{(n_{2},l)} dl = \\int (l - n_{2}) dl and \\int \\mathbb{I}{(n_{2},l)} dl = a^{\\dagger} + \\frac{l^{2}}{2} - l n_{2} and \\int (l - n_{2}) dl = a^{\\dagger} + \\frac{l^{2}}{2} - l n_{2} and \\frac{l^{2}}{2} - l n_{2} + r = a^{\\dagger} + \\frac{l^{2}}{2} - l n_{2} and \\frac{l^{2}}{2} - l n_{2} + r = \\int (l - n_{2}) dl", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('l', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Symbol('n_2', commutative=True)), Symbol('r', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Symbol('n_2', commutative=True)), Symbol('r', commutative=True)), Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(a,J_{\\varepsilon})} = \\int J_{\\varepsilon} a da, then derive \\int (- J_{\\varepsilon} a + \\rho_{b}{(a,J_{\\varepsilon})}) dJ_{\\varepsilon} = J_{\\varepsilon}^{2} (\\frac{a^{2}}{4} - \\frac{a}{2}) + \\nabla, then obtain \\int (- J_{\\varepsilon} a + \\int J_{\\varepsilon} a da) dJ_{\\varepsilon} = J_{\\varepsilon}^{2} (\\frac{a^{2}}{4} - \\frac{a}{2}) + \\nabla", "derivation": "\\rho_{b}{(a,J_{\\varepsilon})} = \\int J_{\\varepsilon} a da and - J_{\\varepsilon} a + \\rho_{b}{(a,J_{\\varepsilon})} = - J_{\\varepsilon} a + \\int J_{\\varepsilon} a da and \\int (- J_{\\varepsilon} a + \\rho_{b}{(a,J_{\\varepsilon})}) dJ_{\\varepsilon} = \\int (- J_{\\varepsilon} a + \\int J_{\\varepsilon} a da) dJ_{\\varepsilon} and \\int (- J_{\\varepsilon} a + \\rho_{b}{(a,J_{\\varepsilon})}) dJ_{\\varepsilon} = J_{\\varepsilon}^{2} (\\frac{a^{2}}{4} - \\frac{a}{2}) + \\nabla and \\int (- J_{\\varepsilon} a + \\int J_{\\varepsilon} a da) dJ_{\\varepsilon} = J_{\\varepsilon}^{2} (\\frac{a^{2}}{4} - \\frac{a}{2}) + \\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["minus", 1, "Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Function('\\\\rho_b')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Function('\\\\rho_b')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Function('\\\\rho_b')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), Add(Mul(Rational(1, 4), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Symbol('a', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), Add(Mul(Rational(1, 4), Pow(Symbol('a', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Symbol('a', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(v_{y})} = \\sin{(v_{y})}, then obtain \\frac{2 \\operatorname{A_{z}}^{v_{y}}{(v_{y})}}{\\operatorname{A_{z}}^{v_{y}}{(v_{y})} + \\sin^{v_{y}}{(v_{y})}} = 1", "derivation": "\\operatorname{A_{z}}{(v_{y})} = \\sin{(v_{y})} and \\operatorname{A_{z}}^{v_{y}}{(v_{y})} = \\sin^{v_{y}}{(v_{y})} and 2 \\operatorname{A_{z}}^{v_{y}}{(v_{y})} = \\operatorname{A_{z}}^{v_{y}}{(v_{y})} + \\sin^{v_{y}}{(v_{y})} and \\frac{2 \\operatorname{A_{z}}^{v_{y}}{(v_{y})}}{\\operatorname{A_{z}}^{v_{y}}{(v_{y})} + \\sin^{v_{y}}{(v_{y})}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"], [["add", 2, "Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Add(Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))))"], [["divide", 3, "Add(Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Integer(-1)), Pow(Function('A_z')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Integer(1))"]]}, {"prompt": "Given t{(M)} = \\sin{(M)} and \\eta{(\\varepsilon)} = \\log{(\\varepsilon)}, then obtain \\frac{\\eta{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} t{(M)})}} = \\frac{\\log{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} t{(M)})}}", "derivation": "t{(M)} = \\sin{(M)} and \\eta{(\\varepsilon)} = \\log{(\\varepsilon)} and \\eta{(\\varepsilon)} - 1 = \\log{(\\varepsilon)} - 1 and \\frac{\\eta{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} \\sin{(M)})}} = \\frac{\\log{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} \\sin{(M)})}} and \\frac{\\eta{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} t{(M)})}} = \\frac{\\log{(\\varepsilon)} - 1}{\\cos{(\\tilde{\\infty} t{(M)})}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Add(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))"], [["divide", 3, "cos(Mul(zoo, sin(Symbol('M', commutative=True))))"], "Equality(Mul(Add(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Pow(cos(Mul(zoo, sin(Symbol('M', commutative=True)))), Integer(-1))), Mul(Add(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Pow(cos(Mul(zoo, sin(Symbol('M', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Function('\\\\eta')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Pow(cos(Mul(zoo, Function('t')(Symbol('M', commutative=True)))), Integer(-1))), Mul(Add(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Pow(cos(Mul(zoo, Function('t')(Symbol('M', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given v{(a,u)} = a u, then derive \\frac{\\partial}{\\partial a} v{(a,u)} = u, then obtain \\frac{u \\frac{\\partial^{2}}{\\partial a^{2}} a u + 1}{\\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)}} = \\frac{u \\frac{d}{d a} u + 1}{\\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)}}", "derivation": "v{(a,u)} = a u and \\frac{\\partial}{\\partial a} v{(a,u)} = \\frac{\\partial}{\\partial a} a u and \\frac{\\partial}{\\partial a} v{(a,u)} = u and \\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)} = \\frac{d}{d a} u and u \\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)} = u \\frac{d}{d a} u and u \\frac{\\partial^{2}}{\\partial a^{2}} a u = u \\frac{d}{d a} u and u \\frac{\\partial^{2}}{\\partial a^{2}} a u + 1 = u \\frac{d}{d a} u + 1 and \\frac{u \\frac{\\partial^{2}}{\\partial a^{2}} a u + 1}{\\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)}} = \\frac{u \\frac{d}{d a} u + 1}{\\frac{\\partial^{2}}{\\partial a^{2}} v{(a,u)}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('a', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('u', commutative=True))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Symbol('u', commutative=True), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 4, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Mul(Symbol('u', commutative=True), Derivative(Symbol('u', commutative=True), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('u', commutative=True), Derivative(Mul(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Mul(Symbol('u', commutative=True), Derivative(Symbol('u', commutative=True), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["add", 6, 1], "Equality(Add(Mul(Symbol('u', commutative=True), Derivative(Mul(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Integer(1)), Add(Mul(Symbol('u', commutative=True), Derivative(Symbol('u', commutative=True), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(1)))"], [["divide", 7, "Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))"], "Equality(Mul(Add(Mul(Symbol('u', commutative=True), Derivative(Mul(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Integer(1)), Pow(Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Integer(-1))), Mul(Add(Mul(Symbol('u', commutative=True), Derivative(Symbol('u', commutative=True), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(1)), Pow(Derivative(Function('v')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain - \\operatorname{y^{\\prime}}{(\\dot{z})} - \\sin{(\\dot{z})} + \\frac{\\operatorname{y^{\\prime}}{(\\dot{z})} + \\sin{(\\dot{z})}}{\\dot{z}} = - \\operatorname{y^{\\prime}}{(\\dot{z})} - \\sin{(\\dot{z})} + \\frac{2 \\sin{(\\dot{z})}}{\\dot{z}}", "derivation": "\\operatorname{y^{\\prime}}{(\\dot{z})} = \\sin{(\\dot{z})} and \\operatorname{y^{\\prime}}{(\\dot{z})} + \\sin{(\\dot{z})} = 2 \\sin{(\\dot{z})} and \\frac{\\operatorname{y^{\\prime}}{(\\dot{z})} + \\sin{(\\dot{z})}}{\\dot{z}} = \\frac{2 \\sin{(\\dot{z})}}{\\dot{z}} and - \\operatorname{y^{\\prime}}{(\\dot{z})} - \\sin{(\\dot{z})} + \\frac{\\operatorname{y^{\\prime}}{(\\dot{z})} + \\sin{(\\dot{z})}}{\\dot{z}} = - \\operatorname{y^{\\prime}}{(\\dot{z})} - \\sin{(\\dot{z})} + \\frac{2 \\sin{(\\dot{z})}}{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Add(Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 3, "Add(Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Add(Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))))), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), sin(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})}, then derive \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} = (\\frac{1}{\\mathbf{P}})^{\\mathbf{P}}, then obtain (\\frac{1}{\\mathbf{P}})^{\\mathbf{P}} \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} = (\\frac{1}{\\mathbf{P}})^{2 \\mathbf{P}}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})} and \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} = (\\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})})^{\\mathbf{P}} and \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} = (\\frac{1}{\\mathbf{P}})^{\\mathbf{P}} and \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} (\\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})})^{\\mathbf{P}} = (\\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})})^{2 \\mathbf{P}} and (\\frac{d}{d \\mathbf{P}} \\log{(\\mathbf{P})})^{\\mathbf{P}} = (\\frac{1}{\\mathbf{P}})^{\\mathbf{P}} and (\\frac{1}{\\mathbf{P}})^{\\mathbf{P}} \\operatorname{v_{z}}^{\\mathbf{P}}{(\\mathbf{P})} = (\\frac{1}{\\mathbf{P}})^{2 \\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["times", 2, "Pow(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Pow(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Function('v_z')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{t})} = \\log{(\\log{(v_{t})})}, then obtain 1 = \\operatorname{x^{{\\}'}}^{- 2 v_{t}}{(v_{t})} \\log{(\\log{(v_{t})})}^{2 v_{t}}", "derivation": "\\operatorname{x^{{\\}'}}{(v_{t})} = \\log{(\\log{(v_{t})})} and \\operatorname{x^{{\\}'}}^{v_{t}}{(v_{t})} = \\log{(\\log{(v_{t})})}^{v_{t}} and 1 = \\operatorname{x^{{\\}'}}^{- v_{t}}{(v_{t})} \\log{(\\log{(v_{t})})}^{v_{t}} and \\log{(\\log{(v_{t})})}^{v_{t}} = \\operatorname{x^{{\\}'}}^{- v_{t}}{(v_{t})} \\log{(\\log{(v_{t})})}^{2 v_{t}} and 1 = \\operatorname{x^{{\\}'}}^{- 2 v_{t}}{(v_{t})} \\log{(\\log{(v_{t})})}^{2 v_{t}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), log(log(Symbol('v_t', commutative=True))))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(log(log(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["divide", 2, "Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Pow(log(log(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))))"], [["times", 2, "Mul(Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Pow(log(log(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], "Equality(Pow(log(log(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Mul(Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Pow(log(log(Symbol('v_t', commutative=True))), Mul(Integer(2), Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True))), Pow(log(log(Symbol('v_t', commutative=True))), Mul(Integer(2), Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(J)} = e^{J}, then obtain \\int (J + \\hat{x}_0{(J)}) dJ = \\frac{J^{2}}{2} + \\hat{x}_0 + e^{J}", "derivation": "\\hat{x}_0{(J)} = e^{J} and J + \\hat{x}_0{(J)} = J + e^{J} and \\int (J + \\hat{x}_0{(J)}) dJ = \\int (J + e^{J}) dJ and \\int (J + \\hat{x}_0{(J)}) dJ = \\frac{J^{2}}{2} + \\hat{x}_0 + e^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('\\\\hat{x}_0', commutative=True), exp(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(G,p)} = \\sin{(G + p)} and B{(G,p)} = G + p, then obtain (\\frac{\\partial}{\\partial G} (\\operatorname{m_{s}}^{G}{(G,p)})^{G})^{p} = (\\frac{\\partial}{\\partial G} (\\sin^{G}{(B{(G,p)})})^{G})^{p}", "derivation": "\\operatorname{m_{s}}{(G,p)} = \\sin{(G + p)} and B{(G,p)} = G + p and \\operatorname{m_{s}}^{G}{(G,p)} = \\sin^{G}{(G + p)} and (\\operatorname{m_{s}}^{G}{(G,p)})^{G} = (\\sin^{G}{(G + p)})^{G} and \\frac{\\partial}{\\partial G} (\\operatorname{m_{s}}^{G}{(G,p)})^{G} = \\frac{\\partial}{\\partial G} (\\sin^{G}{(G + p)})^{G} and (\\frac{\\partial}{\\partial G} (\\operatorname{m_{s}}^{G}{(G,p)})^{G})^{p} = (\\frac{\\partial}{\\partial G} (\\sin^{G}{(G + p)})^{G})^{p} and (\\frac{\\partial}{\\partial G} (\\operatorname{m_{s}}^{G}{(G,p)})^{G})^{p} = (\\frac{\\partial}{\\partial G} (\\sin^{G}{(B{(G,p)})})^{G})^{p}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), sin(Add(Symbol('G', commutative=True), Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Add(Symbol('G', commutative=True), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Pow(sin(Add(Symbol('G', commutative=True), Symbol('p', commutative=True))), Symbol('G', commutative=True)))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Pow(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(Pow(sin(Add(Symbol('G', commutative=True), Symbol('p', commutative=True))), Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Add(Symbol('G', commutative=True), Symbol('p', commutative=True))), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["power", 5, "Symbol('p', commutative=True)"], "Equality(Pow(Derivative(Pow(Pow(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(Pow(Pow(sin(Add(Symbol('G', commutative=True), Symbol('p', commutative=True))), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Derivative(Pow(Pow(Function('m_s')(Symbol('G', commutative=True), Symbol('p', commutative=True)), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(Pow(Pow(sin(Function('B')(Symbol('G', commutative=True), Symbol('p', commutative=True))), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(t_{1})} = \\cos{(t_{1})}, then derive \\frac{\\int \\operatorname{m_{s}}{(t_{1})} dt_{1}}{g + \\sin{(t_{1})}} = 1, then obtain \\log{(\\frac{b + \\sin{(t_{1})}}{g + \\sin{(t_{1})}})} = 0", "derivation": "\\operatorname{m_{s}}{(t_{1})} = \\cos{(t_{1})} and \\int \\operatorname{m_{s}}{(t_{1})} dt_{1} = \\int \\cos{(t_{1})} dt_{1} and \\frac{\\int \\operatorname{m_{s}}{(t_{1})} dt_{1}}{\\int \\cos{(t_{1})} dt_{1}} = 1 and \\frac{\\int \\operatorname{m_{s}}{(t_{1})} dt_{1}}{g + \\sin{(t_{1})}} = 1 and \\log{(\\frac{\\int \\operatorname{m_{s}}{(t_{1})} dt_{1}}{g + \\sin{(t_{1})}})} = 0 and \\log{(\\frac{\\int \\cos{(t_{1})} dt_{1}}{g + \\sin{(t_{1})}})} = 0 and \\log{(\\frac{b + \\sin{(t_{1})}}{g + \\sin{(t_{1})}})} = 0", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 2, "Integral(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Integral(Function('m_s')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Pow(Integral(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), sin(Symbol('t_1', commutative=True))), Integer(-1)), Integral(Function('m_s')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Integer(1))"], [["log", 4], "Equality(log(Mul(Pow(Add(Symbol('g', commutative=True), sin(Symbol('t_1', commutative=True))), Integer(-1)), Integral(Function('m_s')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Pow(Add(Symbol('g', commutative=True), sin(Symbol('t_1', commutative=True))), Integer(-1)), Integral(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))), Integer(0))"], [["evaluate_integrals", 6], "Equality(log(Mul(Add(Symbol('b', commutative=True), sin(Symbol('t_1', commutative=True))), Pow(Add(Symbol('g', commutative=True), sin(Symbol('t_1', commutative=True))), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)} = \\log{(\\frac{a}{\\phi})}, then derive \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)} = - \\frac{1}{\\phi}, then obtain \\int (- \\frac{1}{\\phi})^{\\phi} d\\phi = \\int (\\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)})^{\\phi} d\\phi", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(a,\\phi)} = \\log{(\\frac{a}{\\phi})} and \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)} = \\frac{\\partial}{\\partial \\phi} \\log{(\\frac{a}{\\phi})} and \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)} = - \\frac{1}{\\phi} and - \\frac{1}{\\phi} = \\frac{\\partial}{\\partial \\phi} \\log{(\\frac{a}{\\phi})} and (- \\frac{1}{\\phi})^{\\phi} = (\\frac{\\partial}{\\partial \\phi} \\log{(\\frac{a}{\\phi})})^{\\phi} and (- \\frac{1}{\\phi})^{\\phi} = (\\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)})^{\\phi} and \\int (- \\frac{1}{\\phi})^{\\phi} d\\phi = \\int (\\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{p}}}{(a,\\phi)})^{\\phi} d\\phi", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Derivative(log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(log(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)))"], [["integrate", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\phi)} = \\log{(\\sin{(\\phi)})}, then obtain (\\frac{d}{d \\phi} \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}})^{2} = \\frac{d}{d \\phi} 1 \\frac{d}{d \\phi} \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}}", "derivation": "\\operatorname{P_{e}}{(\\phi)} = \\log{(\\sin{(\\phi)})} and \\phi + \\operatorname{P_{e}}{(\\phi)} = \\phi + \\log{(\\sin{(\\phi)})} and \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}} = 1 and \\frac{d}{d \\phi} \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}} = \\frac{d}{d \\phi} 1 and (\\frac{d}{d \\phi} \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}})^{2} = \\frac{d}{d \\phi} 1 \\frac{d}{d \\phi} \\frac{\\phi + \\operatorname{P_{e}}{(\\phi)}}{\\phi + \\log{(\\sin{(\\phi)})}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\phi', commutative=True)), log(sin(Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))))"], [["divide", 2, "Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Pow(Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Pow(Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Mul(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Pow(Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Pow(Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Integer(1), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True))), Pow(Add(Symbol('\\\\phi', commutative=True), log(sin(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given E{(\\hat{\\mathbf{r}},E_{\\lambda})} = E_{\\lambda} - \\hat{\\mathbf{r}} and \\mathbb{I}{(\\hat{\\mathbf{r}},E_{\\lambda})} = \\int E^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}},E_{\\lambda})} dE_{\\lambda}, then obtain \\mathbb{I}{(\\hat{\\mathbf{r}},E_{\\lambda})} = \\int (E_{\\lambda} - \\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} dE_{\\lambda}", "derivation": "E{(\\hat{\\mathbf{r}},E_{\\lambda})} = E_{\\lambda} - \\hat{\\mathbf{r}} and E^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}},E_{\\lambda})} = (E_{\\lambda} - \\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} and \\int E^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}},E_{\\lambda})} dE_{\\lambda} = \\int (E_{\\lambda} - \\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} dE_{\\lambda} and \\mathbb{I}{(\\hat{\\mathbf{r}},E_{\\lambda})} = \\int E^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}},E_{\\lambda})} dE_{\\lambda} and \\mathbb{I}{(\\hat{\\mathbf{r}},E_{\\lambda})} = \\int (E_{\\lambda} - \\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Pow(Function('E')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(S,\\mathbf{J})} = S + \\mathbf{J}, then obtain (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) + \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} = (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) + \\frac{\\partial}{\\partial S} (S + \\mathbf{J})", "derivation": "\\phi_{2}{(S,\\mathbf{J})} = S + \\mathbf{J} and \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} = \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) and (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} = (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) and (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} + \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} = (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} + \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) and (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) + \\frac{\\partial}{\\partial S} \\phi_{2}{(S,\\mathbf{J})} = (S + \\mathbf{J}) \\frac{\\partial}{\\partial S} (S + \\mathbf{J}) + \\frac{\\partial}{\\partial S} (S + \\mathbf{J})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 2, "Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["add", 2, "Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], "Equality(Add(Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Mul(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\chi,\\mathbf{J}_P,p)} = - \\mathbf{J}_P + \\frac{p}{\\chi}, then obtain 0^{\\mathbf{J}_P} = (-1 + \\frac{\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)}}{- \\mathbf{J}_P + \\frac{p}{\\chi}})^{\\mathbf{J}_P}", "derivation": "\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)} = - \\mathbf{J}_P + \\frac{p}{\\chi} and \\frac{\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)}}{- \\mathbf{J}_P + \\frac{p}{\\chi}} = 1 and - \\frac{\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)}}{- \\mathbf{J}_P + \\frac{p}{\\chi}} = -1 and 0 = -1 + \\frac{\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)}}{- \\mathbf{J}_P + \\frac{p}{\\chi}} and 0^{\\mathbf{J}_P} = (-1 + \\frac{\\mathbf{D}{(\\chi,\\mathbf{J}_P,p)}}{- \\mathbf{J}_P + \\frac{p}{\\chi}})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True))), Integer(-1))"], [["minus", 3, "Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))"], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))))"], [["power", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)}, then derive \\operatorname{f^{\\prime}}{(\\eta)} = - \\sin{(\\eta)}, then derive \\int \\operatorname{f^{\\prime}}{(\\eta)} d\\eta = a^{\\dagger} + \\cos{(\\eta)}, then derive \\frac{d}{d \\eta} \\int - \\sin{(\\eta)} d\\eta = - \\sin{(\\eta)}, then obtain (\\frac{d}{d \\eta} \\int - \\sin{(\\eta)} d\\eta)^{\\eta} = (- \\sin{(\\eta)})^{\\eta}", "derivation": "\\operatorname{f^{\\prime}}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)} and \\int \\operatorname{f^{\\prime}}{(\\eta)} d\\eta = \\int \\frac{d}{d \\eta} \\cos{(\\eta)} d\\eta and \\operatorname{f^{\\prime}}{(\\eta)} = - \\sin{(\\eta)} and \\int \\operatorname{f^{\\prime}}{(\\eta)} d\\eta = a^{\\dagger} + \\cos{(\\eta)} and \\frac{d}{d \\eta} \\int \\operatorname{f^{\\prime}}{(\\eta)} d\\eta = \\frac{\\partial}{\\partial \\eta} (a^{\\dagger} + \\cos{(\\eta)}) and \\frac{d}{d \\eta} \\int - \\sin{(\\eta)} d\\eta = \\frac{\\partial}{\\partial \\eta} (a^{\\dagger} + \\cos{(\\eta)}) and \\frac{d}{d \\eta} \\int - \\sin{(\\eta)} d\\eta = - \\sin{(\\eta)} and (\\frac{d}{d \\eta} \\int - \\sin{(\\eta)} d\\eta)^{\\eta} = (- \\sin{(\\eta)})^{\\eta}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integral(Function('f^{\\\\prime}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integral(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Integral(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))"], [["power", 7, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Derivative(Integral(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given a{(v_{1})} = e^{v_{1}}, then obtain 1 + \\frac{1}{a{(v_{1})}} = (\\frac{e^{v_{1}}}{a{(v_{1})}})^{v_{1}} (\\frac{e^{v_{1}}}{a{(v_{1})}} + \\frac{1}{a{(v_{1})}})", "derivation": "a{(v_{1})} = e^{v_{1}} and 1 = \\frac{e^{v_{1}}}{a{(v_{1})}} and 1 + \\frac{1}{a{(v_{1})}} = \\frac{e^{v_{1}}}{a{(v_{1})}} + \\frac{1}{a{(v_{1})}} and 1 = (\\frac{e^{v_{1}}}{a{(v_{1})}})^{v_{1}} and \\frac{e^{v_{1}}}{a{(v_{1})}} + \\frac{1}{a{(v_{1})}} = (\\frac{e^{v_{1}}}{a{(v_{1})}})^{v_{1}} (\\frac{e^{v_{1}}}{a{(v_{1})}} + \\frac{1}{a{(v_{1})}}) and 1 + \\frac{1}{a{(v_{1})}} = (\\frac{e^{v_{1}}}{a{(v_{1})}})^{v_{1}} (\\frac{e^{v_{1}}}{a{(v_{1})}} + \\frac{1}{a{(v_{1})}})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["divide", 1, "Function('a')(Symbol('v_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))))"], [["add", 2, "Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], [["times", 4, "Add(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Add(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Integer(1), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1))), Mul(Pow(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Add(Mul(Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)), exp(Symbol('v_1', commutative=True))), Pow(Function('a')(Symbol('v_1', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(b,\\mathbf{H})} = \\cos{(\\mathbf{H}^{b})} and H{(b,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\mathbf{H}^{b})}, then obtain P_{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\operatorname{E_{x}}{(b,\\mathbf{H})} = P_{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\mathbf{H}^{b})}", "derivation": "\\operatorname{E_{x}}{(b,\\mathbf{H})} = \\cos{(\\mathbf{H}^{b})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\operatorname{E_{x}}{(b,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\mathbf{H}^{b})} and H{(b,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\mathbf{H}^{b})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\operatorname{E_{x}}{(b,\\mathbf{H})} = H{(b,\\mathbf{H})} and P_{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\operatorname{E_{x}}{(b,\\mathbf{H})} = P_{g} + H{(b,\\mathbf{H})} and P_{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\operatorname{E_{x}}{(b,\\mathbf{H})} = P_{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\cos{(\\mathbf{H}^{b})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('H')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('E_x')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Function('H')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 4, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Derivative(Function('E_x')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Symbol('P_g', commutative=True), Function('H')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('P_g', commutative=True), Derivative(Function('E_x')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Add(Symbol('P_g', commutative=True), Derivative(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(S,\\omega)} = S^{\\omega} and \\operatorname{t_{2}}{(S,\\omega)} = \\int \\operatorname{F_{c}}{(S,\\omega)} d\\omega, then obtain 0 = - \\operatorname{t_{2}}{(S,\\omega)} + \\int S^{\\omega} d\\omega", "derivation": "\\operatorname{F_{c}}{(S,\\omega)} = S^{\\omega} and \\int \\operatorname{F_{c}}{(S,\\omega)} d\\omega = \\int S^{\\omega} d\\omega and \\operatorname{t_{2}}{(S,\\omega)} = \\int \\operatorname{F_{c}}{(S,\\omega)} d\\omega and \\operatorname{t_{2}}{(S,\\omega)} - \\int S^{\\omega} d\\omega = - \\int S^{\\omega} d\\omega + \\int \\operatorname{F_{c}}{(S,\\omega)} d\\omega and \\operatorname{t_{2}}{(S,\\omega)} = \\int S^{\\omega} d\\omega and 0 = - \\operatorname{t_{2}}{(S,\\omega)} + \\int \\operatorname{F_{c}}{(S,\\omega)} d\\omega and 0 = - \\operatorname{t_{2}}{(S,\\omega)} + \\int S^{\\omega} d\\omega", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('F_c')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Function('t_2')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(Mul(Integer(-1), Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(Function('F_c')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('t_2')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t_2')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True))), Integral(Function('F_c')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t_2')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(n)} = \\sin{(\\cos{(n)})}, then obtain \\int \\frac{d}{d n} \\operatorname{F_{g}}^{n}{(n)} dn = \\int \\frac{d}{d n} \\sin^{n}{(\\cos{(n)})} dn", "derivation": "\\operatorname{F_{g}}{(n)} = \\sin{(\\cos{(n)})} and \\operatorname{F_{g}}^{n}{(n)} = \\sin^{n}{(\\cos{(n)})} and \\frac{d}{d n} \\operatorname{F_{g}}^{n}{(n)} = \\frac{d}{d n} \\sin^{n}{(\\cos{(n)})} and \\int \\frac{d}{d n} \\operatorname{F_{g}}^{n}{(n)} dn = \\int \\frac{d}{d n} \\sin^{n}{(\\cos{(n)})} dn", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('n', commutative=True)), sin(cos(Symbol('n', commutative=True))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(sin(cos(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Function('F_g')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(sin(cos(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('F_g')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(Derivative(Pow(sin(cos(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)} = (- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu}, then obtain \\frac{d^{2}}{d \\mud x^\\prime} 0 = \\frac{\\partial^{2}}{\\partial \\mu\\partial x^\\prime} ((- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu} - Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)})", "derivation": "Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)} = (- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu} and 0 = (- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu} - Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)} and \\frac{d}{d x^\\prime} 0 = \\frac{\\partial}{\\partial x^\\prime} ((- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu} - Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)}) and \\frac{d^{2}}{d \\mud x^\\prime} 0 = \\frac{\\partial^{2}}{\\partial \\mu\\partial x^\\prime} ((- \\hat{\\mathbf{x}} + x^\\prime)^{\\mu} - Z{(x^\\prime,\\hat{\\mathbf{x}},\\mu)})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 1, "Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})} = \\Psi_{nl} + a, then derive (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2}) \\int (\\Psi_{nl} + \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})}) da = (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2})^{2}, then obtain (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2}) \\int (2 \\Psi_{nl} + a) da = (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2})^{2}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})} = \\Psi_{nl} + a and \\Psi_{nl} + \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})} = 2 \\Psi_{nl} + a and \\int (\\Psi_{nl} + \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})}) da = \\int (2 \\Psi_{nl} + a) da and (\\int (\\Psi_{nl} + \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})}) da) \\int (2 \\Psi_{nl} + a) da = (\\int (2 \\Psi_{nl} + a) da)^{2} and (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2}) \\int (\\Psi_{nl} + \\operatorname{V_{\\mathbf{E}}}{(a,\\Psi_{nl})}) da = (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2})^{2} and (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2}) \\int (2 \\Psi_{nl} + a) da = (2 \\Psi_{nl} a + \\pi + \\frac{a^{2}}{2})^{2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)))"], [["add", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["times", 3, "Integral(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(2)))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('a', commutative=True)))), Pow(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)))), Integral(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Pow(Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\pi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a', commutative=True), Integer(2)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{v},m)} = \\mathbf{v} e^{m}, then derive \\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{H}{(\\mathbf{v},m)} = e^{m}, then obtain \\frac{e^{- m} (\\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{v} e^{m})^{m}}{\\mathbf{v}} = \\frac{e^{- m} (e^{m})^{m}}{\\mathbf{v}}", "derivation": "\\mathbf{H}{(\\mathbf{v},m)} = \\mathbf{v} e^{m} and \\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{H}{(\\mathbf{v},m)} = \\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{v} e^{m} and \\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{H}{(\\mathbf{v},m)} = e^{m} and (\\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{H}{(\\mathbf{v},m)})^{m} = (e^{m})^{m} and \\frac{e^{- m} (\\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{H}{(\\mathbf{v},m)})^{m}}{\\mathbf{v}} = \\frac{e^{- m} (e^{m})^{m}}{\\mathbf{v}} and \\frac{e^{- m} (\\frac{\\partial}{\\partial \\mathbf{v}} \\mathbf{v} e^{m})^{m}}{\\mathbf{v}} = \\frac{e^{- m} (e^{m})^{m}}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('m', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('m', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), exp(Symbol('m', commutative=True)))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["divide", 4, "Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('m', commutative=True))), Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('m', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('m', commutative=True))), Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('m', commutative=True))), Pow(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('m', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('m', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('m', commutative=True))), Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = C_{d} a^{\\dagger}, then derive \\frac{\\partial}{\\partial C_{d}} \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = a^{\\dagger}, then obtain - \\frac{\\partial}{\\partial C_{d}} \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = - a^{\\dagger}", "derivation": "\\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = C_{d} a^{\\dagger} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = \\frac{\\partial}{\\partial C_{d}} C_{d} a^{\\dagger} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = a^{\\dagger} and - \\frac{\\partial}{\\partial C_{d}} \\operatorname{A_{2}}{(a^{\\dagger},C_{d})} = - a^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{v},s)} = - s + \\log{(\\mathbf{v})}, then derive (\\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v})^{s} = (\\mathbf{E} + \\mathbf{v} (- s - 1) + \\mathbf{v} \\log{(\\mathbf{v})})^{s}, then obtain \\cos{((\\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v})^{s})} = \\cos{((\\mathbf{E} + \\mathbf{v} (- s - 1) + \\mathbf{v} \\log{(\\mathbf{v})})^{s})}", "derivation": "\\mathbf{g}{(\\mathbf{v},s)} = - s + \\log{(\\mathbf{v})} and \\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v} = \\int (- s + \\log{(\\mathbf{v})}) d\\mathbf{v} and (\\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v})^{s} = (\\int (- s + \\log{(\\mathbf{v})}) d\\mathbf{v})^{s} and (\\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v})^{s} = (\\mathbf{E} + \\mathbf{v} (- s - 1) + \\mathbf{v} \\log{(\\mathbf{v})})^{s} and \\cos{((\\int \\mathbf{g}{(\\mathbf{v},s)} d\\mathbf{v})^{s})} = \\cos{((\\mathbf{E} + \\mathbf{v} (- s - 1) + \\mathbf{v} \\log{(\\mathbf{v})})^{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('s', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('s', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('s', commutative=True)))"], [["cos", 4], "Equality(cos(Pow(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('s', commutative=True))), cos(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(x)} = \\sin{(x)} and \\tilde{g}^*{(x)} = \\mu_{0}{(x)} - \\sin{(x)}, then obtain (-1)^{- x} (\\tilde{g}^*{(x)} - 1) = - (-1)^{- x}", "derivation": "\\mu_{0}{(x)} = \\sin{(x)} and \\mu_{0}{(x)} - \\sin{(x)} = 0 and \\mu_{0}{(x)} - \\sin{(x)} - 1 = -1 and \\tilde{g}^*{(x)} = \\mu_{0}{(x)} - \\sin{(x)} and \\tilde{g}^*{(x)} - 1 = -1 and (\\tilde{g}^*{(x)} - 1) \\sin{(x)} + \\mu_{0}{(x)} + \\tilde{g}^*{(x)} - 1 = \\tilde{g}^*{(x)} - 1 and (\\tilde{g}^*{(x)} - 1)^{- x} ((\\tilde{g}^*{(x)} - 1) \\sin{(x)} + \\mu_{0}{(x)} + \\tilde{g}^*{(x)} - 1) = (\\tilde{g}^*{(x)} - 1) (\\tilde{g}^*{(x)} - 1)^{- x} and (-1)^{- x} (\\mu_{0}{(x)} - \\sin{(x)} - 1) = - (-1)^{- x} and (-1)^{- x} (\\tilde{g}^*{(x)} - 1) = - (-1)^{- x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["minus", 1, "sin(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('x', commutative=True)), Mul(Integer(-1), sin(Symbol('x', commutative=True)))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mu_0')(Symbol('x', commutative=True)), Mul(Integer(-1), sin(Symbol('x', commutative=True))), Integer(-1)), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Add(Function('\\\\mu_0')(Symbol('x', commutative=True)), Mul(Integer(-1), sin(Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), sin(Symbol('x', commutative=True))), Function('\\\\mu_0')(Symbol('x', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)))"], [["divide", 6, "Pow(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Mul(Integer(-1), Symbol('x', commutative=True))), Add(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), sin(Symbol('x', commutative=True))), Function('\\\\mu_0')(Symbol('x', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Pow(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1)), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('x', commutative=True))), Add(Function('\\\\mu_0')(Symbol('x', commutative=True)), Mul(Integer(-1), sin(Symbol('x', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 4], "Equality(Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('x', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(I,B)} = I^{B}, then obtain \\frac{\\partial}{\\partial B} ((I^{B})^{B} + \\mathbf{s}^{B}{(I,B)} - \\frac{1}{\\mathbf{s}{(I,B)}}) = \\frac{\\partial}{\\partial B} (2 (I^{B})^{B} - \\frac{1}{\\mathbf{s}{(I,B)}})", "derivation": "\\mathbf{s}{(I,B)} = I^{B} and \\mathbf{s}^{B}{(I,B)} = (I^{B})^{B} and (I^{B})^{B} + \\mathbf{s}^{B}{(I,B)} = 2 (I^{B})^{B} and (I^{B})^{B} + \\mathbf{s}^{B}{(I,B)} - \\frac{1}{\\mathbf{s}{(I,B)}} = 2 (I^{B})^{B} - \\frac{1}{\\mathbf{s}{(I,B)}} and \\frac{\\partial}{\\partial B} ((I^{B})^{B} + \\mathbf{s}^{B}{(I,B)} - \\frac{1}{\\mathbf{s}{(I,B)}}) = \\frac{\\partial}{\\partial B} (2 (I^{B})^{B} - \\frac{1}{\\mathbf{s}{(I,B)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(-1)))), Add(Mul(Integer(2), Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(-1)))))"], [["differentiate", 4, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(-1)))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Pow(Pow(Symbol('I', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Integer(-1)))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A,\\mathbf{J}_M)} = A - \\mathbf{J}_M and \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = \\int \\Psi_{\\lambda}{(A,\\mathbf{J}_M)} d\\mathbf{J}_M, then obtain (- A + \\mathbf{J}_M + 1) \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = (- A + \\mathbf{J}_M + 1) (A \\mathbf{J}_M + \\mathbf{E} - \\frac{\\mathbf{J}_M^{2}}{2})", "derivation": "\\Psi_{\\lambda}{(A,\\mathbf{J}_M)} = A - \\mathbf{J}_M and \\int \\Psi_{\\lambda}{(A,\\mathbf{J}_M)} d\\mathbf{J}_M = \\int (A - \\mathbf{J}_M) d\\mathbf{J}_M and \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = \\int \\Psi_{\\lambda}{(A,\\mathbf{J}_M)} d\\mathbf{J}_M and \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = \\int (A - \\mathbf{J}_M) d\\mathbf{J}_M and (- A + \\mathbf{J}_M + 1) \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = (- A + \\mathbf{J}_M + 1) \\int (A - \\mathbf{J}_M) d\\mathbf{J}_M and (- A + \\mathbf{J}_M + 1) \\operatorname{A_{1}}{(A,\\mathbf{J}_M)} = (- A + \\mathbf{J}_M + 1) (A \\mathbf{J}_M + \\mathbf{E} - \\frac{\\mathbf{J}_M^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('A_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)), Function('A_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)), Integral(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)), Function('A_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))))))"]]}, {"prompt": "Given x{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain x^{2}{(\\mathbb{I})} - 1 - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}} = x{(\\mathbb{I})} \\cos{(\\mathbb{I})} - 1 - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}}", "derivation": "x{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and x^{2}{(\\mathbb{I})} = x{(\\mathbb{I})} \\cos{(\\mathbb{I})} and x^{2}{(\\mathbb{I})} - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}} = x{(\\mathbb{I})} \\cos{(\\mathbb{I})} - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}} and x^{2}{(\\mathbb{I})} - 1 - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}} = x{(\\mathbb{I})} \\cos{(\\mathbb{I})} - 1 - \\frac{\\cos{(\\mathbb{I})}}{x{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["times", 1, "Function('x')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Mul(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["minus", 3, 1], "Equality(Add(Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Integer(-1), Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1), Mul(Integer(-1), Pow(Function('x')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)} and \\operatorname{v_{z}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)}, then obtain \\frac{d}{d \\mathbf{J}_M} 1 = \\frac{d}{d \\mathbf{J}_M} \\eta^{\\prime}^{- \\mathbf{J}_M}{(\\mathbf{J}_M)} \\operatorname{v_{z}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)}", "derivation": "\\eta^{\\prime}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)} and \\eta^{\\prime}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)}^{\\mathbf{J}_M} and 1 = \\eta^{\\prime}^{- \\mathbf{J}_M}{(\\mathbf{J}_M)} \\log{(\\mathbf{J}_M)}^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} 1 = \\frac{d}{d \\mathbf{J}_M} \\eta^{\\prime}^{- \\mathbf{J}_M}{(\\mathbf{J}_M)} \\log{(\\mathbf{J}_M)}^{\\mathbf{J}_M} and \\operatorname{v_{z}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} 1 = \\frac{d}{d \\mathbf{J}_M} \\eta^{\\prime}^{- \\mathbf{J}_M}{(\\mathbf{J}_M)} \\operatorname{v_{z}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(log(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(log(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(c)} = \\cos{(\\cos{(c)})}, then obtain \\int ((\\operatorname{v_{1}}^{c}{(c)})^{c} + \\operatorname{v_{1}}^{c}{(c)}) dc = \\int ((\\cos^{c}{(\\cos{(c)})})^{c} + \\operatorname{v_{1}}^{c}{(c)}) dc", "derivation": "\\operatorname{v_{1}}{(c)} = \\cos{(\\cos{(c)})} and \\operatorname{v_{1}}^{c}{(c)} = \\cos^{c}{(\\cos{(c)})} and (\\operatorname{v_{1}}^{c}{(c)})^{c} = (\\cos^{c}{(\\cos{(c)})})^{c} and (\\operatorname{v_{1}}^{c}{(c)})^{c} + \\operatorname{v_{1}}^{c}{(c)} = (\\cos^{c}{(\\cos{(c)})})^{c} + \\operatorname{v_{1}}^{c}{(c)} and \\int ((\\operatorname{v_{1}}^{c}{(c)})^{c} + \\operatorname{v_{1}}^{c}{(c)}) dc = \\int ((\\cos^{c}{(\\cos{(c)})})^{c} + \\operatorname{v_{1}}^{c}{(c)}) dc", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('c', commutative=True)), cos(cos(Symbol('c', commutative=True))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(cos(cos(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(cos(cos(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["add", 3, "Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Add(Pow(Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Pow(Pow(cos(cos(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Pow(Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Pow(Pow(cos(cos(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('v_1')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given b{(\\psi,h)} = \\psi h, then obtain (- h + b{(\\psi,h)}) (\\psi h + \\frac{\\psi (- h + b{(\\psi,h)})}{\\psi h - h} - h) = (- h + b{(\\psi,h)}) (\\psi h + \\psi - h)", "derivation": "b{(\\psi,h)} = \\psi h and - h + b{(\\psi,h)} = \\psi h - h and \\frac{- h + b{(\\psi,h)}}{\\psi h - h} = 1 and \\frac{\\psi (- h + b{(\\psi,h)})}{\\psi h - h} = \\psi and \\psi h + \\frac{\\psi (- h + b{(\\psi,h)})}{\\psi h - h} - h = \\psi h + \\psi - h and (- h + b{(\\psi,h)}) (\\psi h + \\frac{\\psi (- h + b{(\\psi,h)})}{\\psi h - h} - h) = (- h + b{(\\psi,h)}) (\\psi h + \\psi - h)", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["divide", 2, "Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Integer(-1))), Integer(1))"], [["times", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Integer(-1))), Symbol('\\\\psi', commutative=True))"], [["add", 4, "Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('b')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given s{(\\varphi^*,v)} = v^{\\varphi^*} and l{(\\delta)} = \\log{(\\delta)}, then obtain (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta})^{2} s^{2}{(\\varphi^*,v)} = (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) (\\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) s^{2}{(\\varphi^*,v)}", "derivation": "s{(\\varphi^*,v)} = v^{\\varphi^*} and l{(\\delta)} = \\log{(\\delta)} and l{(\\delta)} - \\log{(\\delta)}^{2 \\delta} = \\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta} and (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) s{(\\varphi^*,v)} = (\\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) s{(\\varphi^*,v)} and v^{\\varphi^*} (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) = v^{\\varphi^*} (\\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) and v^{\\varphi^*} (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta})^{2} s{(\\varphi^*,v)} = v^{\\varphi^*} (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) (\\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) s{(\\varphi^*,v)} and (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta})^{2} s^{2}{(\\varphi^*,v)} = (l{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) (\\log{(\\delta)} - \\log{(\\delta)}^{2 \\delta}) s^{2}{(\\varphi^*,v)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], ["get_premise", "Equality(Function('l')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["minus", 2, "Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Add(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))))"], [["times", 3, "Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True))), Mul(Add(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))))), Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))))))"], [["times", 5, "Mul(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)))"], "Equality(Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Integer(2)), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Add(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Integer(2)), Pow(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Integer(2))), Mul(Add(Function('l')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Add(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))))), Pow(Function('s')(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\phi{(\\phi_2,y,\\sigma_p)} = \\frac{y^{\\sigma_p}}{\\phi_2}, then obtain \\phi_2 + \\sin^{2}{(2 \\phi{(\\phi_2,y,\\sigma_p)})} = \\phi_2 + \\sin^{2}{(\\phi{(\\phi_2,y,\\sigma_p)} + \\frac{y^{\\sigma_p}}{\\phi_2})}", "derivation": "\\phi{(\\phi_2,y,\\sigma_p)} = \\frac{y^{\\sigma_p}}{\\phi_2} and 2 \\phi{(\\phi_2,y,\\sigma_p)} = \\phi{(\\phi_2,y,\\sigma_p)} + \\frac{y^{\\sigma_p}}{\\phi_2} and \\sin{(2 \\phi{(\\phi_2,y,\\sigma_p)})} = \\sin{(\\phi{(\\phi_2,y,\\sigma_p)} + \\frac{y^{\\sigma_p}}{\\phi_2})} and \\sin^{2}{(2 \\phi{(\\phi_2,y,\\sigma_p)})} = \\sin^{2}{(\\phi{(\\phi_2,y,\\sigma_p)} + \\frac{y^{\\sigma_p}}{\\phi_2})} and \\phi_2 + \\sin^{2}{(2 \\phi{(\\phi_2,y,\\sigma_p)})} = \\phi_2 + \\sin^{2}{(\\phi{(\\phi_2,y,\\sigma_p)} + \\frac{y^{\\sigma_p}}{\\phi_2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 1, "Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), sin(Add(Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))))"], [["power", 3, 2], "Equality(Pow(sin(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Integer(2)), Pow(sin(Add(Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))), Integer(2)))"], [["add", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Pow(sin(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Integer(2))), Add(Symbol('\\\\phi_2', commutative=True), Pow(sin(Add(Function('\\\\phi')(Symbol('\\\\phi_2', commutative=True), Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(S)} = \\log{(S)}, then obtain ((2 \\operatorname{F_{H}}{(S)} - \\log{(S)}) \\int \\log{(S)} dS)^{S} = (\\log{(S)} \\int \\log{(S)} dS)^{S}", "derivation": "\\operatorname{F_{H}}{(S)} = \\log{(S)} and 2 \\operatorname{F_{H}}{(S)} - \\log{(S)} = \\operatorname{F_{H}}{(S)} and 2 \\operatorname{F_{H}}{(S)} - \\log{(S)} = \\log{(S)} and (2 \\operatorname{F_{H}}{(S)} - \\log{(S)}) \\int \\log{(S)} dS = \\log{(S)} \\int \\log{(S)} dS and ((2 \\operatorname{F_{H}}{(S)} - \\log{(S)}) \\int \\log{(S)} dS)^{S} = (\\log{(S)} \\int \\log{(S)} dS)^{S}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Function('F_H')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('S', commutative=True))), Mul(Integer(-1), log(Symbol('S', commutative=True)))), Function('F_H')(Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('S', commutative=True))), Mul(Integer(-1), log(Symbol('S', commutative=True)))), log(Symbol('S', commutative=True)))"], [["times", 3, "Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Function('F_H')(Symbol('S', commutative=True))), Mul(Integer(-1), log(Symbol('S', commutative=True)))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(log(Symbol('S', commutative=True)), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(2), Function('F_H')(Symbol('S', commutative=True))), Mul(Integer(-1), log(Symbol('S', commutative=True)))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)), Pow(Mul(log(Symbol('S', commutative=True)), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\pi{(\\mathbf{J}_P)} = \\sin{(\\cos{(\\mathbf{J}_P)})} and A{(\\mathbf{J}_P)} = \\cos^{\\mathbf{J}_P}{(\\mathbf{J}_P)}, then obtain A{(\\mathbf{J}_P)} = (\\frac{\\sin{(\\cos{(\\mathbf{J}_P)})} \\cos{(\\mathbf{J}_P)}}{\\pi{(\\mathbf{J}_P)}})^{\\mathbf{J}_P}", "derivation": "\\pi{(\\mathbf{J}_P)} = \\sin{(\\cos{(\\mathbf{J}_P)})} and \\pi{(\\mathbf{J}_P)} \\cos{(\\mathbf{J}_P)} = \\sin{(\\cos{(\\mathbf{J}_P)})} \\cos{(\\mathbf{J}_P)} and \\cos{(\\mathbf{J}_P)} = \\frac{\\sin{(\\cos{(\\mathbf{J}_P)})} \\cos{(\\mathbf{J}_P)}}{\\pi{(\\mathbf{J}_P)}} and \\cos^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = (\\frac{\\sin{(\\cos{(\\mathbf{J}_P)})} \\cos{(\\mathbf{J}_P)}}{\\pi{(\\mathbf{J}_P)}})^{\\mathbf{J}_P} and A{(\\mathbf{J}_P)} = \\cos^{\\mathbf{J}_P}{(\\mathbf{J}_P)} and A{(\\mathbf{J}_P)} = (\\frac{\\sin{(\\cos{(\\mathbf{J}_P)})} \\cos{(\\mathbf{J}_P)}}{\\pi{(\\mathbf{J}_P)}})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(sin(cos(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 2, "Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Pow(Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('A')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Pow(Function('\\\\pi')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given h{(t_{1},\\mathbf{D})} = \\cos{(\\mathbf{D} + t_{1})}, then derive \\int h{(t_{1},\\mathbf{D})} d\\mathbf{D} = C_{2} + \\sin{(\\mathbf{D} + t_{1})}, then obtain \\int \\cos{(\\mathbf{D} + t_{1})} d\\mathbf{D} = C_{2} + \\sin{(\\mathbf{D} + t_{1})}", "derivation": "h{(t_{1},\\mathbf{D})} = \\cos{(\\mathbf{D} + t_{1})} and \\int h{(t_{1},\\mathbf{D})} d\\mathbf{D} = \\int \\cos{(\\mathbf{D} + t_{1})} d\\mathbf{D} and \\int h{(t_{1},\\mathbf{D})} d\\mathbf{D} = C_{2} + \\sin{(\\mathbf{D} + t_{1})} and \\int \\cos{(\\mathbf{D} + t_{1})} d\\mathbf{D} = C_{2} + \\sin{(\\mathbf{D} + t_{1})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), cos(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(cos(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('C_2', commutative=True), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('C_2', commutative=True), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(t,\\Psi_{nl})} = \\Psi_{nl} t, then derive \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})} = t, then obtain (t + \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})})^{\\Psi_{nl}} = (2 t)^{\\Psi_{nl}}", "derivation": "\\hat{x}_0{(t,\\Psi_{nl})} = \\Psi_{nl} t and \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} \\Psi_{nl} t and \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})} = t and t + \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})} = 2 t and (t + \\frac{\\partial}{\\partial \\Psi_{nl}} \\hat{x}_0{(t,\\Psi_{nl})})^{\\Psi_{nl}} = (2 t)^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('t', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('t', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('t', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Symbol('t', commutative=True))"], [["minus", 3, "Mul(Integer(-1), Symbol('t', commutative=True))"], "Equality(Add(Symbol('t', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('t', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('t', commutative=True)))"], [["power", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Symbol('t', commutative=True), Derivative(Function('\\\\hat{x}_0')(Symbol('t', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Mul(Integer(2), Symbol('t', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\pi{(\\phi_2,M,r)} = \\frac{M + \\phi_2}{r}, then obtain (M + \\phi_2) (\\pi^{r}{(\\phi_2,M,r)})^{M} = (M + \\phi_2) ((\\frac{M + \\phi_2}{r})^{r})^{M}", "derivation": "\\pi{(\\phi_2,M,r)} = \\frac{M + \\phi_2}{r} and \\pi^{r}{(\\phi_2,M,r)} = (\\frac{M + \\phi_2}{r})^{r} and (\\pi^{r}{(\\phi_2,M,r)})^{M} = ((\\frac{M + \\phi_2}{r})^{r})^{M} and (M + \\phi_2) (\\pi^{r}{(\\phi_2,M,r)})^{M} = (M + \\phi_2) ((\\frac{M + \\phi_2}{r})^{r})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('r', commutative=True)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('r', commutative=True)), Symbol('M', commutative=True)))"], [["times", 3, "Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('M', commutative=True))), Mul(Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('r', commutative=True)), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(k)} = \\sin{(k)}, then derive \\frac{d}{d k} \\operatorname{A_{1}}{(k)} = \\cos{(k)}, then obtain - 2 \\log{(\\cos{(k)} + \\frac{d}{d k} \\operatorname{A_{1}}{(k)})} \\cos{(k)} = - 2 \\log{(\\cos{(k)} + \\frac{d}{d k} \\sin{(k)})} \\cos{(k)}", "derivation": "\\operatorname{A_{1}}{(k)} = \\sin{(k)} and \\frac{d}{d k} \\operatorname{A_{1}}{(k)} = \\frac{d}{d k} \\sin{(k)} and \\frac{d}{d k} \\operatorname{A_{1}}{(k)} = \\cos{(k)} and \\cos{(k)} + \\frac{d}{d k} \\operatorname{A_{1}}{(k)} = 2 \\cos{(k)} and \\log{(\\cos{(k)} + \\frac{d}{d k} \\operatorname{A_{1}}{(k)})} = \\log{(2 \\cos{(k)})} and \\log{(\\cos{(k)} + \\frac{d}{d k} \\sin{(k)})} = \\log{(2 \\cos{(k)})} and \\log{(\\cos{(k)} + \\frac{d}{d k} \\operatorname{A_{1}}{(k)})} = \\log{(\\cos{(k)} + \\frac{d}{d k} \\sin{(k)})} and - 2 \\log{(\\cos{(k)} + \\frac{d}{d k} \\operatorname{A_{1}}{(k)})} \\cos{(k)} = - 2 \\log{(\\cos{(k)} + \\frac{d}{d k} \\sin{(k)})} \\cos{(k)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), cos(Symbol('k', commutative=True)))"], [["add", 3, "cos(Symbol('k', commutative=True))"], "Equality(Add(cos(Symbol('k', commutative=True)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('k', commutative=True))))"], [["log", 4], "Equality(log(Add(cos(Symbol('k', commutative=True)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), log(Mul(Integer(2), cos(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(log(Add(cos(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), log(Mul(Integer(2), cos(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(log(Add(cos(Symbol('k', commutative=True)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), log(Add(cos(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["times", 7, "Mul(Integer(-1), Integer(2), cos(Symbol('k', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), log(Add(cos(Symbol('k', commutative=True)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), cos(Symbol('k', commutative=True))), Mul(Integer(-1), Integer(2), log(Add(cos(Symbol('k', commutative=True)), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), cos(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{P},t)} = \\log{(\\frac{\\mathbf{P}}{t})} and \\lambda{(\\mathbf{P},t)} = \\frac{\\mathbf{P}}{t} + \\mu_{0}{(\\mathbf{P},t)}, then obtain \\lambda{(\\mathbf{P},t)} - \\frac{1}{t} = \\frac{\\mathbf{P}}{t} + \\log{(\\frac{\\mathbf{P}}{t})} - \\frac{1}{t}", "derivation": "\\mu_{0}{(\\mathbf{P},t)} = \\log{(\\frac{\\mathbf{P}}{t})} and \\frac{\\mathbf{P}}{t} + \\mu_{0}{(\\mathbf{P},t)} = \\frac{\\mathbf{P}}{t} + \\log{(\\frac{\\mathbf{P}}{t})} and \\lambda{(\\mathbf{P},t)} = \\frac{\\mathbf{P}}{t} + \\mu_{0}{(\\mathbf{P},t)} and \\frac{\\mathbf{P}}{t} + \\mu_{0}{(\\mathbf{P},t)} - \\frac{1}{t} = \\frac{\\mathbf{P}}{t} + \\log{(\\frac{\\mathbf{P}}{t})} - \\frac{1}{t} and \\lambda{(\\mathbf{P},t)} - \\frac{1}{t} = \\frac{\\mathbf{P}}{t} + \\log{(\\frac{\\mathbf{P}}{t})} - \\frac{1}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["add", 1, "Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True))))"], [["minus", 2, "Pow(Symbol('t', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{X}{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime}, then obtain (\\varphi + 2 \\hat{X}{(f^{\\prime},\\varphi)})^{\\varphi} = (\\varphi f^{\\prime} + \\varphi + \\hat{X}{(f^{\\prime},\\varphi)})^{\\varphi}", "derivation": "\\hat{X}{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime} and \\varphi + \\hat{X}{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime} + \\varphi and \\varphi f^{\\prime} + \\varphi + \\hat{X}{(f^{\\prime},\\varphi)} = 2 \\varphi f^{\\prime} + \\varphi and \\varphi + 2 \\hat{X}{(f^{\\prime},\\varphi)} = 2 \\varphi f^{\\prime} + \\varphi and \\varphi + 2 \\hat{X}{(f^{\\prime},\\varphi)} = \\varphi f^{\\prime} + \\varphi + \\hat{X}{(f^{\\prime},\\varphi)} and (\\varphi + 2 \\hat{X}{(f^{\\prime},\\varphi)})^{\\varphi} = (\\varphi f^{\\prime} + \\varphi + \\hat{X}{(f^{\\prime},\\varphi)})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["power", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\varphi', commutative=True), Function('\\\\hat{X}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\varphi,u)} = \\cos{(\\varphi u)}, then derive \\frac{\\partial}{\\partial u} \\operatorname{r_{0}}{(\\varphi,u)} = - \\varphi \\sin{(\\varphi u)}, then obtain \\varphi \\sin{(\\varphi u)} = - \\frac{\\partial}{\\partial u} \\cos{(\\varphi u)}", "derivation": "\\operatorname{r_{0}}{(\\varphi,u)} = \\cos{(\\varphi u)} and \\frac{\\partial}{\\partial u} \\operatorname{r_{0}}{(\\varphi,u)} = \\frac{\\partial}{\\partial u} \\cos{(\\varphi u)} and \\frac{\\partial}{\\partial u} \\operatorname{r_{0}}{(\\varphi,u)} = - \\varphi \\sin{(\\varphi u)} and - \\frac{\\partial}{\\partial u} \\operatorname{r_{0}}{(\\varphi,u)} = - \\frac{\\partial}{\\partial u} \\cos{(\\varphi u)} and \\varphi \\sin{(\\varphi u)} = - \\frac{\\partial}{\\partial u} \\cos{(\\varphi u)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), sin(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('r_0')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varphi', commutative=True), sin(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Mul(Integer(-1), Derivative(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(a)} = \\log{(a)}, then derive \\int Z{(a)} da = \\hat{H}_{\\lambda} + a \\log{(a)} - a, then obtain Z{(a)} + \\int \\log{(a)} da = \\hat{H}_{\\lambda} + a Z{(a)} - a + Z{(a)}", "derivation": "Z{(a)} = \\log{(a)} and \\int Z{(a)} da = \\int \\log{(a)} da and \\int Z{(a)} da = \\hat{H}_{\\lambda} + a \\log{(a)} - a and \\int \\log{(a)} da = \\hat{H}_{\\lambda} + a \\log{(a)} - a and \\log{(a)} + \\int \\log{(a)} da = \\hat{H}_{\\lambda} + a \\log{(a)} - a + \\log{(a)} and Z{(a)} + \\int \\log{(a)} da = \\hat{H}_{\\lambda} + a Z{(a)} - a + Z{(a)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Z')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["add", 4, "log(Symbol('a', commutative=True))"], "Equality(Add(log(Symbol('a', commutative=True)), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('Z')(Symbol('a', commutative=True)), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Symbol('a', commutative=True), Function('Z')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), Function('Z')(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(n,v_{z})} = n^{v_{z}} and \\rho_{f}{(n_{1})} = \\cos{(n_{1})}, then obtain \\rho_{f}{(n_{1})} + \\iint \\dot{z}{(n,v_{z})} dv_{z} dn = \\cos{(n_{1})} + \\iint \\dot{z}{(n,v_{z})} dv_{z} dn", "derivation": "\\dot{z}{(n,v_{z})} = n^{v_{z}} and \\int \\dot{z}{(n,v_{z})} dv_{z} = \\int n^{v_{z}} dv_{z} and \\iint \\dot{z}{(n,v_{z})} dv_{z} dn = \\iint n^{v_{z}} dv_{z} dn and \\rho_{f}{(n_{1})} = \\cos{(n_{1})} and \\rho_{f}{(n_{1})} + \\iint n^{v_{z}} dv_{z} dn = \\cos{(n_{1})} + \\iint n^{v_{z}} dv_{z} dn and \\rho_{f}{(n_{1})} + \\iint \\dot{z}{(n,v_{z})} dv_{z} dn = \\cos{(n_{1})} + \\iint \\dot{z}{(n,v_{z})} dv_{z} dn", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["add", 4, "Integral(Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), Integral(Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(cos(Symbol('n_1', commutative=True)), Integral(Pow(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(cos(Symbol('n_1', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(b)} = \\cos{(\\log{(b)})}, then obtain - \\frac{\\frac{\\partial}{\\partial b} \\int - \\Omega{(b)} \\log{(\\rho_b)} d\\rho_b}{\\Omega{(b)} \\mathbf{M}{(\\rho_b)}} = - \\frac{\\frac{\\partial}{\\partial b} \\int - \\log{(\\rho_b)} \\cos{(\\log{(b)})} d\\rho_b}{\\Omega{(b)} \\mathbf{M}{(\\rho_b)}}", "derivation": "\\Omega{(b)} = \\cos{(\\log{(b)})} and - \\Omega{(b)} \\log{(\\rho_b)} = - \\log{(\\rho_b)} \\cos{(\\log{(b)})} and \\int - \\Omega{(b)} \\log{(\\rho_b)} d\\rho_b = \\int - \\log{(\\rho_b)} \\cos{(\\log{(b)})} d\\rho_b and \\frac{\\partial}{\\partial b} \\int - \\Omega{(b)} \\log{(\\rho_b)} d\\rho_b = \\frac{\\partial}{\\partial b} \\int - \\log{(\\rho_b)} \\cos{(\\log{(b)})} d\\rho_b and - \\frac{\\frac{\\partial}{\\partial b} \\int - \\Omega{(b)} \\log{(\\rho_b)} d\\rho_b}{\\Omega{(b)} \\mathbf{M}{(\\rho_b)}} = - \\frac{\\frac{\\partial}{\\partial b} \\int - \\log{(\\rho_b)} \\cos{(\\log{(b)})} d\\rho_b}{\\Omega{(b)} \\mathbf{M}{(\\rho_b)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('b', commutative=True)), cos(log(Symbol('b', commutative=True))))"], [["times", 1, "Mul(Integer(-1), log(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\Omega')(Symbol('b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\rho_b', commutative=True)), cos(log(Symbol('b', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\Omega')(Symbol('b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Integer(-1), log(Symbol('\\\\rho_b', commutative=True)), cos(log(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Function('\\\\Omega')(Symbol('b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), log(Symbol('\\\\rho_b', commutative=True)), cos(log(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Integer(-1), Function('\\\\Omega')(Symbol('b', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('b', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Derivative(Integral(Mul(Integer(-1), Function('\\\\Omega')(Symbol('b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('b', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Derivative(Integral(Mul(Integer(-1), log(Symbol('\\\\rho_b', commutative=True)), cos(log(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)} = W + \\mathbf{J} + f_{\\mathbf{v}}, then obtain \\log{(W - f_{\\mathbf{v}})} = \\log{(- \\mathbf{J} - 2 f_{\\mathbf{v}} + \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)})}", "derivation": "\\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)} = W + \\mathbf{J} + f_{\\mathbf{v}} and - W + \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)} = \\mathbf{J} + f_{\\mathbf{v}} and W - \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)} = - \\mathbf{J} - f_{\\mathbf{v}} and W - f_{\\mathbf{v}} = - \\mathbf{J} - 2 f_{\\mathbf{v}} + \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)} and \\log{(W - f_{\\mathbf{v}})} = \\log{(- \\mathbf{J} - 2 f_{\\mathbf{v}} + \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},\\mathbf{J},W)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True))))"], [["log", 4], "Equality(log(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\theta_2)} = e^{\\theta_2}, then obtain \\int \\frac{d}{d \\theta_2} \\int (\\varepsilon_{0}{(\\theta_2)} + e^{\\theta_2}) d\\theta_2 d\\theta_2 = \\int \\frac{d}{d \\theta_2} \\int 2 e^{\\theta_2} d\\theta_2 d\\theta_2", "derivation": "\\varepsilon_{0}{(\\theta_2)} = e^{\\theta_2} and \\varepsilon_{0}{(\\theta_2)} + e^{\\theta_2} = 2 e^{\\theta_2} and \\int (\\varepsilon_{0}{(\\theta_2)} + e^{\\theta_2}) d\\theta_2 = \\int 2 e^{\\theta_2} d\\theta_2 and \\frac{d}{d \\theta_2} \\int (\\varepsilon_{0}{(\\theta_2)} + e^{\\theta_2}) d\\theta_2 = \\frac{d}{d \\theta_2} \\int 2 e^{\\theta_2} d\\theta_2 and \\int \\frac{d}{d \\theta_2} \\int (\\varepsilon_{0}{(\\theta_2)} + e^{\\theta_2}) d\\theta_2 d\\theta_2 = \\int \\frac{d}{d \\theta_2} \\int 2 e^{\\theta_2} d\\theta_2 d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Integral(Mul(Integer(2), exp(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(k)} = \\sin{(k)}, then derive - \\mu_{0}{(k)} + \\int \\mu_{0}{(k)} dk = x^\\prime - \\mu_{0}{(k)} - \\cos{(k)}, then obtain - \\mu_{0}{(k)} + \\int \\sin{(k)} dk = x^\\prime - \\mu_{0}{(k)} - \\cos{(k)}", "derivation": "\\mu_{0}{(k)} = \\sin{(k)} and \\int \\mu_{0}{(k)} dk = \\int \\sin{(k)} dk and - \\mu_{0}{(k)} + \\int \\mu_{0}{(k)} dk = - \\mu_{0}{(k)} + \\int \\sin{(k)} dk and - \\mu_{0}{(k)} + \\int \\mu_{0}{(k)} dk = x^\\prime - \\mu_{0}{(k)} - \\cos{(k)} and - \\mu_{0}{(k)} + \\int \\sin{(k)} dk = x^\\prime - \\mu_{0}{(k)} - \\cos{(k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["minus", 2, "Function('\\\\mu_0')(Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Integral(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Integral(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(x,\\mathbf{v})} = \\frac{x}{\\mathbf{v}}, then derive \\frac{\\mathbf{v} \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{z}{(x,\\mathbf{v})}}{x} + \\frac{\\dot{z}{(x,\\mathbf{v})}}{x} = 0, then obtain \\frac{\\mathbf{v} \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{z}{(x,\\mathbf{v})}}{x} + \\frac{1}{\\mathbf{v}} = 0", "derivation": "\\dot{z}{(x,\\mathbf{v})} = \\frac{x}{\\mathbf{v}} and \\frac{\\mathbf{v} \\dot{z}{(x,\\mathbf{v})}}{x} = 1 and \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v} \\dot{z}{(x,\\mathbf{v})}}{x} = \\frac{d}{d \\mathbf{v}} 1 and \\frac{\\mathbf{v} \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{z}{(x,\\mathbf{v})}}{x} + \\frac{\\dot{z}{(x,\\mathbf{v})}}{x} = 0 and \\frac{\\mathbf{v} \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{x}{\\mathbf{v}}}{x} + \\frac{1}{\\mathbf{v}} = 0 and \\frac{\\mathbf{v} \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{z}{(x,\\mathbf{v})}}{x} + \\frac{1}{\\mathbf{v}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('x', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{z}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Integer(0))"]]}, {"prompt": "Given T{(Q,E_{x})} = Q^{E_{x}}, then obtain Q T{(Q,E_{x})} - \\frac{\\partial}{\\partial Q} (Q^{E_{x}} + T{(Q,E_{x})}) = Q Q^{E_{x}} - \\frac{\\partial}{\\partial Q} (Q^{E_{x}} + T{(Q,E_{x})})", "derivation": "T{(Q,E_{x})} = Q^{E_{x}} and Q^{E_{x}} + T{(Q,E_{x})} = 2 Q^{E_{x}} and Q T{(Q,E_{x})} = Q Q^{E_{x}} and Q T{(Q,E_{x})} - \\frac{\\partial}{\\partial Q} 2 Q^{E_{x}} = Q Q^{E_{x}} - \\frac{\\partial}{\\partial Q} 2 Q^{E_{x}} and Q T{(Q,E_{x})} - \\frac{\\partial}{\\partial Q} (Q^{E_{x}} + T{(Q,E_{x})}) = Q Q^{E_{x}} - \\frac{\\partial}{\\partial Q} (Q^{E_{x}} + T{(Q,E_{x})})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True)))"], [["add", 1, "Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True)), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Symbol('Q', commutative=True), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))))"], [["minus", 3, "Derivative(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Derivative(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))), Add(Mul(Symbol('Q', commutative=True), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Derivative(Mul(Integer(2), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Derivative(Add(Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True)), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))), Add(Mul(Symbol('Q', commutative=True), Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Derivative(Add(Pow(Symbol('Q', commutative=True), Symbol('E_x', commutative=True)), Function('T')(Symbol('Q', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{M}{(F_{N})} = \\sin{(F_{N})}, then derive \\mathbf{M}{(F_{N})} - \\cos{(F_{N})} + \\frac{d}{d F_{N}} \\mathbf{M}{(F_{N})} + 1 = \\mathbf{M}{(F_{N})} + 1, then obtain \\sin{(F_{N})} - \\cos{(F_{N})} + \\frac{d}{d F_{N}} \\sin{(F_{N})} + 1 = \\sin{(F_{N})} + 1", "derivation": "\\mathbf{M}{(F_{N})} = \\sin{(F_{N})} and F_{N} + \\mathbf{M}{(F_{N})} = F_{N} + \\sin{(F_{N})} and F_{N} + \\mathbf{M}{(F_{N})} - \\sin{(F_{N})} = F_{N} and \\frac{d}{d F_{N}} (F_{N} + \\mathbf{M}{(F_{N})} - \\sin{(F_{N})}) = \\frac{d}{d F_{N}} F_{N} and \\mathbf{M}{(F_{N})} + \\frac{d}{d F_{N}} (F_{N} + \\mathbf{M}{(F_{N})} - \\sin{(F_{N})}) = \\mathbf{M}{(F_{N})} + \\frac{d}{d F_{N}} F_{N} and \\mathbf{M}{(F_{N})} - \\cos{(F_{N})} + \\frac{d}{d F_{N}} \\mathbf{M}{(F_{N})} + 1 = \\mathbf{M}{(F_{N})} + 1 and \\sin{(F_{N})} - \\cos{(F_{N})} + \\frac{d}{d F_{N}} \\sin{(F_{N})} + 1 = \\sin{(F_{N})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["minus", 2, "sin(Symbol('F_N', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Symbol('F_N', commutative=True))"], [["differentiate", 3, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 4, "Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Derivative(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)), Add(Function('\\\\mathbf{M}')(Symbol('F_N', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(sin(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True))), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)), Add(sin(Symbol('F_N', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{D}{(n_{2},v)} = \\cos^{v}{(n_{2})} and \\dot{z}{(n_{2},v)} = \\int \\cos^{v}{(n_{2})} dv, then obtain \\cos^{v}{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\int \\cos^{v}{(n_{2})} dv = \\cos^{v}{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\dot{z}{(n_{2},v)}", "derivation": "\\mathbf{D}{(n_{2},v)} = \\cos^{v}{(n_{2})} and \\int \\mathbf{D}{(n_{2},v)} dv = \\int \\cos^{v}{(n_{2})} dv and \\dot{z}{(n_{2},v)} = \\int \\cos^{v}{(n_{2})} dv and \\int \\mathbf{D}{(n_{2},v)} dv = \\dot{z}{(n_{2},v)} and \\frac{\\partial}{\\partial n_{2}} \\int \\mathbf{D}{(n_{2},v)} dv = \\frac{\\partial}{\\partial n_{2}} \\dot{z}{(n_{2},v)} and \\frac{\\partial}{\\partial n_{2}} \\int \\cos^{v}{(n_{2})} dv = \\frac{\\partial}{\\partial n_{2}} \\dot{z}{(n_{2},v)} and \\cos^{v}{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\int \\cos^{v}{(n_{2})} dv = \\cos^{v}{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\dot{z}{(n_{2},v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Integral(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Function('\\\\dot{z}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 4, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Function('\\\\dot{z}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Integral(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Function('\\\\dot{z}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["add", 6, "Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True))"], "Equality(Add(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Derivative(Integral(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('n_2', commutative=True)), Symbol('v', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})} and \\tilde{g}{(\\mathbf{P})} = \\frac{\\log{(\\operatorname{t_{2}}{(\\mathbf{P})})}}{\\log{(\\cos{(\\log{(\\mathbf{P})})})}}, then obtain \\frac{\\frac{d}{d \\mathbf{P}} \\tilde{g}{(\\mathbf{P})}}{u{(\\mathbf{P})}} = \\frac{\\frac{d}{d \\mathbf{P}} 1}{u{(\\mathbf{P})}}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})} and \\tilde{g}{(\\mathbf{P})} = \\frac{\\log{(\\operatorname{t_{2}}{(\\mathbf{P})})}}{\\log{(\\cos{(\\log{(\\mathbf{P})})})}} and \\tilde{g}{(\\mathbf{P})} = 1 and \\frac{d}{d \\mathbf{P}} \\tilde{g}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} 1 and \\frac{\\frac{d}{d \\mathbf{P}} \\tilde{g}{(\\mathbf{P})}}{u{(\\mathbf{P})}} = \\frac{\\frac{d}{d \\mathbf{P}} 1}{u{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)), cos(log(Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(log(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(cos(log(Symbol('\\\\mathbf{P}', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["divide", 4, "Function('u')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Pow(Function('u')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Pow(Function('u')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\Psi,\\hat{X})} = \\Psi + \\log{(\\hat{X})} and \\phi_{1}{(\\Psi,\\hat{X})} = \\frac{\\operatorname{A_{2}}{(\\Psi,\\hat{X})}}{\\Psi}, then derive \\int \\phi_{1}{(\\Psi,\\hat{X})} d\\Psi = \\Psi + f^{*} + \\log{(\\Psi)} \\log{(\\hat{X})}, then obtain \\hat{X} \\int \\phi_{1}{(\\Psi,\\hat{X})} d\\Psi = \\hat{X} (\\Psi + f^{*} + \\log{(\\Psi)} \\log{(\\hat{X})})", "derivation": "\\operatorname{A_{2}}{(\\Psi,\\hat{X})} = \\Psi + \\log{(\\hat{X})} and \\frac{\\operatorname{A_{2}}{(\\Psi,\\hat{X})}}{\\Psi} = \\frac{\\Psi + \\log{(\\hat{X})}}{\\Psi} and \\phi_{1}{(\\Psi,\\hat{X})} = \\frac{\\operatorname{A_{2}}{(\\Psi,\\hat{X})}}{\\Psi} and \\phi_{1}{(\\Psi,\\hat{X})} = \\frac{\\Psi + \\log{(\\hat{X})}}{\\Psi} and \\int \\phi_{1}{(\\Psi,\\hat{X})} d\\Psi = \\int \\frac{\\Psi + \\log{(\\hat{X})}}{\\Psi} d\\Psi and \\int \\phi_{1}{(\\Psi,\\hat{X})} d\\Psi = \\Psi + f^{*} + \\log{(\\Psi)} \\log{(\\hat{X})} and \\hat{X} \\int \\phi_{1}{(\\Psi,\\hat{X})} d\\Psi = \\hat{X} (\\Psi + f^{*} + \\log{(\\Psi)} \\log{(\\hat{X})})", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\hat{X}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\phi_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\hat{X}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True), Mul(log(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))))"], [["times", 6, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Integral(Function('\\\\phi_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\hat{X}', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True), Mul(log(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True))))))"]]}, {"prompt": "Given \\delta{(y^{\\prime},\\theta_2)} = - \\sin{(\\theta_2 - y^{\\prime})} and \\mathbf{r}{(y^{\\prime},\\theta_2)} = \\theta_2 - 2 \\sin{(\\theta_2 - y^{\\prime})}, then obtain (\\theta_2 + \\delta{(y^{\\prime},\\theta_2)} - \\sin{(\\theta_2 - y^{\\prime})})^{y^{\\prime}} = \\mathbf{r}^{y^{\\prime}}{(y^{\\prime},\\theta_2)}", "derivation": "\\delta{(y^{\\prime},\\theta_2)} = - \\sin{(\\theta_2 - y^{\\prime})} and \\theta_2 + \\delta{(y^{\\prime},\\theta_2)} = \\theta_2 - \\sin{(\\theta_2 - y^{\\prime})} and \\theta_2 + \\delta{(y^{\\prime},\\theta_2)} - \\sin{(\\theta_2 - y^{\\prime})} = \\theta_2 - 2 \\sin{(\\theta_2 - y^{\\prime})} and (\\theta_2 + \\delta{(y^{\\prime},\\theta_2)} - \\sin{(\\theta_2 - y^{\\prime})})^{y^{\\prime}} = (\\theta_2 - 2 \\sin{(\\theta_2 - y^{\\prime})})^{y^{\\prime}} and \\mathbf{r}{(y^{\\prime},\\theta_2)} = \\theta_2 - 2 \\sin{(\\theta_2 - y^{\\prime})} and (\\theta_2 + \\delta{(y^{\\prime},\\theta_2)} - \\sin{(\\theta_2 - y^{\\prime})})^{y^{\\prime}} = \\mathbf{r}^{y^{\\prime}}{(y^{\\prime},\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["add", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\delta')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))))"], [["add", 2, "Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\delta')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\delta')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\delta')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given E{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} e^{\\tilde{g}^*}, then derive E{(\\tilde{g}^*)} = e^{\\tilde{g}^*}, then derive \\int E{(\\tilde{g}^*)} d\\tilde{g}^* = G + e^{\\tilde{g}^*}, then obtain \\int E{(\\tilde{g}^*)} d\\tilde{g}^* = G + \\frac{d}{d \\tilde{g}^*} e^{\\tilde{g}^*}", "derivation": "E{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} e^{\\tilde{g}^*} and E{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and e^{\\tilde{g}^*} = \\frac{d}{d \\tilde{g}^*} e^{\\tilde{g}^*} and \\int E{(\\tilde{g}^*)} d\\tilde{g}^* = \\int e^{\\tilde{g}^*} d\\tilde{g}^* and \\int E{(\\tilde{g}^*)} d\\tilde{g}^* = G + e^{\\tilde{g}^*} and \\int E{(\\tilde{g}^*)} d\\tilde{g}^* = G + \\frac{d}{d \\tilde{g}^*} e^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(exp(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(exp(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('E')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('G', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Function('E')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('G', commutative=True), Derivative(exp(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\theta_{2}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain \\int \\frac{\\theta_{2}{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} d\\mathbf{A} = \\int 1 d\\mathbf{A}", "derivation": "\\sigma_{p}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\frac{\\sigma_{p}{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} = 1 and \\int \\frac{\\sigma_{p}{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} d\\mathbf{A} = \\int 1 d\\mathbf{A} and \\theta_{2}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\sigma_{p}{(\\mathbf{A})} = \\theta_{2}{(\\mathbf{A})} and \\int \\frac{\\theta_{2}{(\\mathbf{A})}}{\\sin{(\\mathbf{A})}} d\\mathbf{A} = \\int 1 d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\lambda{(m,\\mathbf{F})} = e^{m^{\\mathbf{F}}} and \\mathbf{J}{(c)} = \\log{(c)}, then obtain - 2 \\mathbf{F} + 2 m \\lambda{(m,\\mathbf{F})} - \\mathbf{J}{(c)} = - 2 \\mathbf{F} + m \\lambda{(m,\\mathbf{F})} + m e^{m^{\\mathbf{F}}} - \\mathbf{J}{(c)}", "derivation": "\\lambda{(m,\\mathbf{F})} = e^{m^{\\mathbf{F}}} and m \\lambda{(m,\\mathbf{F})} = m e^{m^{\\mathbf{F}}} and - \\mathbf{F} + m \\lambda{(m,\\mathbf{F})} = - \\mathbf{F} + m e^{m^{\\mathbf{F}}} and \\mathbf{J}{(c)} = \\log{(c)} and - 2 \\mathbf{F} + 2 m \\lambda{(m,\\mathbf{F})} - \\log{(c)} = - 2 \\mathbf{F} + m \\lambda{(m,\\mathbf{F})} + m e^{m^{\\mathbf{F}}} - \\log{(c)} and - 2 \\mathbf{F} + 2 m \\lambda{(m,\\mathbf{F})} - \\mathbf{J}{(c)} = - 2 \\mathbf{F} + m \\lambda{(m,\\mathbf{F})} + m e^{m^{\\mathbf{F}}} - \\mathbf{J}{(c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('m', commutative=True), exp(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["minus", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('m', commutative=True), exp(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["minus", 3, "Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), log(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), log(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('m', commutative=True), exp(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(-1), log(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('m', commutative=True), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('m', commutative=True), exp(Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\eta{(f_{\\mathbf{p}})} = f_{\\mathbf{p}}, then obtain \\frac{\\partial}{\\partial m} (f_{\\mathbf{p}} + m) ((f_{\\mathbf{p}} + m)^{F_{g}} + \\eta{(f_{\\mathbf{p}})}) = \\frac{\\partial}{\\partial m} (f_{\\mathbf{p}} + m) (f_{\\mathbf{p}} + (f_{\\mathbf{p}} + m)^{F_{g}})", "derivation": "\\eta{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} and (f_{\\mathbf{p}} + m)^{F_{g}} + \\eta{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + (f_{\\mathbf{p}} + m)^{F_{g}} and (f_{\\mathbf{p}} + m) ((f_{\\mathbf{p}} + m)^{F_{g}} + \\eta{(f_{\\mathbf{p}})}) = (f_{\\mathbf{p}} + m) (f_{\\mathbf{p}} + (f_{\\mathbf{p}} + m)^{F_{g}}) and \\frac{\\partial}{\\partial m} (f_{\\mathbf{p}} + m) ((f_{\\mathbf{p}} + m)^{F_{g}} + \\eta{(f_{\\mathbf{p}})}) = \\frac{\\partial}{\\partial m} (f_{\\mathbf{p}} + m) (f_{\\mathbf{p}} + (f_{\\mathbf{p}} + m)^{F_{g}})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\eta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["add", 1, "Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True))"], "Equality(Add(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\eta')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True))))"], [["times", 2, "Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\eta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True)))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\eta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True)), Symbol('F_g', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(v)} = \\frac{d}{d v} e^{v} and \\operatorname{F_{x}}{(v)} = \\frac{d}{d v} e^{v}, then obtain \\frac{d}{d v} \\psi{(v)} = \\frac{d^{2}}{d v^{2}} e^{v}", "derivation": "\\psi{(v)} = \\frac{d}{d v} e^{v} and \\operatorname{F_{x}}{(v)} = \\frac{d}{d v} e^{v} and \\psi{(v)} = \\operatorname{F_{x}}{(v)} and \\frac{d}{d v} \\psi{(v)} = \\frac{d}{d v} \\operatorname{F_{x}}{(v)} and \\frac{d^{2}}{d v^{2}} e^{v} = \\frac{d}{d v} \\operatorname{F_{x}}{(v)} and \\frac{d}{d v} \\psi{(v)} = \\frac{d^{2}}{d v^{2}} e^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('v', commutative=True)), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\psi')(Symbol('v', commutative=True)), Function('F_x')(Symbol('v', commutative=True)))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Function('F_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(2))), Derivative(Function('F_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('\\\\psi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}{(V,\\phi,\\phi_2)} = V \\phi_2 + \\phi and \\dot{x}{(V,\\phi_2)} = V \\phi_2, then derive \\log{(\\frac{\\partial}{\\partial \\phi} \\tilde{g}{(V,\\phi,\\phi_2)})} = 0, then obtain \\frac{\\log{(\\frac{\\partial}{\\partial \\phi} (\\phi + \\dot{x}{(V,\\phi_2)}))}}{V \\phi_2} = 0", "derivation": "\\tilde{g}{(V,\\phi,\\phi_2)} = V \\phi_2 + \\phi and \\dot{x}{(V,\\phi_2)} = V \\phi_2 and \\tilde{g}{(V,\\phi,\\phi_2)} = \\phi + \\dot{x}{(V,\\phi_2)} and \\frac{\\partial}{\\partial \\phi} \\tilde{g}{(V,\\phi,\\phi_2)} = \\frac{\\partial}{\\partial \\phi} (V \\phi_2 + \\phi) and \\log{(\\frac{\\partial}{\\partial \\phi} \\tilde{g}{(V,\\phi,\\phi_2)})} = \\log{(\\frac{\\partial}{\\partial \\phi} (V \\phi_2 + \\phi))} and \\log{(\\frac{\\partial}{\\partial \\phi} \\tilde{g}{(V,\\phi,\\phi_2)})} = 0 and V \\phi_2 + \\phi = \\phi + \\dot{x}{(V,\\phi_2)} and \\log{(\\frac{\\partial}{\\partial \\phi} (V \\phi_2 + \\phi))} = 0 and \\frac{\\log{(\\frac{\\partial}{\\partial \\phi} (V \\phi_2 + \\phi))}}{V \\phi_2} = 0 and \\frac{\\log{(\\frac{\\partial}{\\partial \\phi} (\\phi + \\dot{x}{(V,\\phi_2)}))}}{V \\phi_2} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('V', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\tilde{g}')(Symbol('V', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Function('\\\\dot{x}')(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('V', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["log", 4], "Equality(log(Derivative(Function('\\\\tilde{g}')(Symbol('V', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), log(Derivative(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(log(Derivative(Function('\\\\tilde{g}')(Symbol('V', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Function('\\\\dot{x}')(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(log(Derivative(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Integer(0))"], [["divide", 8, "Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Derivative(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 9, 7], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Derivative(Add(Symbol('\\\\phi', commutative=True), Function('\\\\dot{x}')(Symbol('V', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\Psi{(g)} = \\log{(g)}, then derive - \\log{(g)} + \\frac{d}{d g} \\Psi{(g)} = - \\log{(g)} + \\frac{1}{g}, then obtain - \\Psi{(g)} + \\frac{d}{d g} \\Psi{(g)} = - \\Psi{(g)} + \\frac{1}{g}", "derivation": "\\Psi{(g)} = \\log{(g)} and \\frac{d}{d g} \\Psi{(g)} = \\frac{d}{d g} \\log{(g)} and - \\log{(g)} + \\frac{d}{d g} \\Psi{(g)} = - \\log{(g)} + \\frac{d}{d g} \\log{(g)} and - \\log{(g)} + \\frac{d}{d g} \\Psi{(g)} = - \\log{(g)} + \\frac{1}{g} and - \\log{(g)} + \\frac{d}{d g} \\log{(g)} = - \\log{(g)} + \\frac{1}{g} and - \\Psi{(g)} + \\frac{d}{d g} \\Psi{(g)} = - \\Psi{(g)} + \\frac{1}{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 2, "log(Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('g', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('g', commutative=True))), Pow(Symbol('g', commutative=True), Integer(-1))))"]]}, {"prompt": "Given I{(\\mathbf{H},u)} = \\cos{(\\mathbf{H} u)} and \\operatorname{E_{x}}{(\\mathbf{H},u)} = \\mathbf{H} u, then obtain \\frac{\\partial}{\\partial u} I{(\\mathbf{H},u)} = \\frac{\\partial}{\\partial u} \\cos{(\\operatorname{E_{x}}{(\\mathbf{H},u)})}", "derivation": "I{(\\mathbf{H},u)} = \\cos{(\\mathbf{H} u)} and \\operatorname{E_{x}}{(\\mathbf{H},u)} = \\mathbf{H} u and I{(\\mathbf{H},u)} = \\cos{(\\operatorname{E_{x}}{(\\mathbf{H},u)})} and \\frac{\\partial}{\\partial u} I{(\\mathbf{H},u)} = \\frac{\\partial}{\\partial u} \\cos{(\\operatorname{E_{x}}{(\\mathbf{H},u)})}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True)), cos(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True)), cos(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True))))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Function('E_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\dot{z},\\Omega)} = \\frac{\\Omega}{\\dot{z}}, then obtain \\frac{\\int \\dot{z} (- \\dot{z} + \\varphi{(\\dot{z},\\Omega)}) d\\dot{z}}{\\Omega} = \\frac{\\int \\dot{z} (\\frac{\\Omega}{\\dot{z}} - \\dot{z}) d\\dot{z}}{\\Omega}", "derivation": "\\varphi{(\\dot{z},\\Omega)} = \\frac{\\Omega}{\\dot{z}} and - \\dot{z} + \\varphi{(\\dot{z},\\Omega)} = \\frac{\\Omega}{\\dot{z}} - \\dot{z} and \\dot{z} (- \\dot{z} + \\varphi{(\\dot{z},\\Omega)}) = \\dot{z} (\\frac{\\Omega}{\\dot{z}} - \\dot{z}) and \\int \\dot{z} (- \\dot{z} + \\varphi{(\\dot{z},\\Omega)}) d\\dot{z} = \\int \\dot{z} (\\frac{\\Omega}{\\dot{z}} - \\dot{z}) d\\dot{z} and \\frac{\\int \\dot{z} (- \\dot{z} + \\varphi{(\\dot{z},\\Omega)}) d\\dot{z}}{\\Omega} = \\frac{\\int \\dot{z} (\\frac{\\Omega}{\\dot{z}} - \\dot{z}) d\\dot{z}}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given m{(\\mathbf{D},n)} = \\frac{\\mathbf{D}}{n}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (m{(\\mathbf{D},n)} - \\frac{1}{n}) dn = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (\\frac{\\mathbf{D}}{n} - \\frac{1}{n}) dn", "derivation": "m{(\\mathbf{D},n)} = \\frac{\\mathbf{D}}{n} and m{(\\mathbf{D},n)} - \\frac{1}{n} = \\frac{\\mathbf{D}}{n} - \\frac{1}{n} and \\int (m{(\\mathbf{D},n)} - \\frac{1}{n}) dn = \\int (\\frac{\\mathbf{D}}{n} - \\frac{1}{n}) dn and (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (m{(\\mathbf{D},n)} - \\frac{1}{n}) dn = (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (\\frac{\\mathbf{D}}{n} - \\frac{1}{n}) dn and \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (m{(\\mathbf{D},n)} - \\frac{1}{n}) dn = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + m{(\\mathbf{D},n)} - \\frac{1}{n}) \\int (\\frac{\\mathbf{D}}{n} - \\frac{1}{n}) dn", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True))))"], [["times", 3, "Add(Symbol('\\\\mathbf{D}', commutative=True), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Integral(Add(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Integral(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Integral(Add(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Integral(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(a)} = \\log{(e^{a})} and \\hat{x}{(v_{x},\\mathbf{p})} = \\mathbf{p} + v_{x} and \\mathbf{r}{(a)} = e^{a}, then obtain E{(a)} \\log{(\\frac{\\partial}{\\partial v_{x}} (\\mathbf{p} + v_{x}))}^{- v_{x}} = \\log{(\\mathbf{r}{(a)})} \\log{(\\frac{\\partial}{\\partial v_{x}} (\\mathbf{p} + v_{x}))}^{- v_{x}}", "derivation": "E{(a)} = \\log{(e^{a})} and \\hat{x}{(v_{x},\\mathbf{p})} = \\mathbf{p} + v_{x} and \\mathbf{r}{(a)} = e^{a} and \\frac{\\partial}{\\partial v_{x}} \\hat{x}{(v_{x},\\mathbf{p})} = \\frac{\\partial}{\\partial v_{x}} (\\mathbf{p} + v_{x}) and E{(a)} = \\log{(\\mathbf{r}{(a)})} and E{(a)} \\log{(\\frac{\\partial}{\\partial v_{x}} \\hat{x}{(v_{x},\\mathbf{p})})}^{- v_{x}} = \\log{(\\mathbf{r}{(a)})} \\log{(\\frac{\\partial}{\\partial v_{x}} \\hat{x}{(v_{x},\\mathbf{p})})}^{- v_{x}} and E{(a)} \\log{(\\frac{\\partial}{\\partial v_{x}} (\\mathbf{p} + v_{x}))}^{- v_{x}} = \\log{(\\mathbf{r}{(a)})} \\log{(\\frac{\\partial}{\\partial v_{x}} (\\mathbf{p} + v_{x}))}^{- v_{x}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('a', commutative=True)), log(exp(Symbol('a', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('E')(Symbol('a', commutative=True)), log(Function('\\\\mathbf{r}')(Symbol('a', commutative=True))))"], [["divide", 5, "Pow(log(Derivative(Function('\\\\hat{x}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Symbol('v_x', commutative=True))"], "Equality(Mul(Function('E')(Symbol('a', commutative=True)), Pow(log(Derivative(Function('\\\\hat{x}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Mul(log(Function('\\\\mathbf{r}')(Symbol('a', commutative=True))), Pow(log(Derivative(Function('\\\\hat{x}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('v_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Function('E')(Symbol('a', commutative=True)), Pow(log(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Mul(log(Function('\\\\mathbf{r}')(Symbol('a', commutative=True))), Pow(log(Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(p)} = e^{\\cos{(p)}} and \\operatorname{M_{E}}{(p)} = \\frac{1}{e^{\\cos{(p)}} - \\cos{(p)}}, then obtain \\operatorname{M_{E}}{(p)} - \\Psi_{\\lambda}{(p)} e^{\\cos{(p)}} = - \\Psi_{\\lambda}{(p)} e^{\\cos{(p)}} + \\frac{1}{\\Psi_{\\lambda}{(p)} - \\cos{(p)}}", "derivation": "\\Psi_{\\lambda}{(p)} = e^{\\cos{(p)}} and \\Psi_{\\lambda}{(p)} - \\cos{(p)} = e^{\\cos{(p)}} - \\cos{(p)} and \\Psi_{\\lambda}^{2}{(p)} = \\Psi_{\\lambda}{(p)} e^{\\cos{(p)}} and \\operatorname{M_{E}}{(p)} = \\frac{1}{e^{\\cos{(p)}} - \\cos{(p)}} and \\operatorname{M_{E}}{(p)} = \\frac{1}{\\Psi_{\\lambda}{(p)} - \\cos{(p)}} and \\operatorname{M_{E}}{(p)} - \\Psi_{\\lambda}^{2}{(p)} = - \\Psi_{\\lambda}^{2}{(p)} + \\frac{1}{\\Psi_{\\lambda}{(p)} - \\cos{(p)}} and \\operatorname{M_{E}}{(p)} - \\Psi_{\\lambda}{(p)} e^{\\cos{(p)}} = - \\Psi_{\\lambda}{(p)} e^{\\cos{(p)}} + \\frac{1}{\\Psi_{\\lambda}{(p)} - \\cos{(p)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), exp(cos(Symbol('p', commutative=True))))"], [["minus", 1, "cos(Symbol('p', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Add(exp(cos(Symbol('p', commutative=True))), Mul(Integer(-1), cos(Symbol('p', commutative=True)))))"], [["times", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Integer(2)), Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), exp(cos(Symbol('p', commutative=True)))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('p', commutative=True)), Pow(Add(exp(cos(Symbol('p', commutative=True))), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('M_E')(Symbol('p', commutative=True)), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Integer(-1)))"], [["minus", 5, "Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Integer(2))"], "Equality(Add(Function('M_E')(Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Integer(2))), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Function('M_E')(Symbol('p', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), exp(cos(Symbol('p', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), exp(cos(Symbol('p', commutative=True)))), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(a^{\\dagger})} = \\int \\cos{(a^{\\dagger})} da^{\\dagger}, then derive \\varphi{(a^{\\dagger})} = t + \\sin{(a^{\\dagger})}, then obtain \\cos^{a^{\\dagger}}{(a^{\\dagger})} = (\\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})})^{a^{\\dagger}}", "derivation": "\\varphi{(a^{\\dagger})} = \\int \\cos{(a^{\\dagger})} da^{\\dagger} and - \\omega + \\varphi{(a^{\\dagger})} = - \\omega + \\int \\cos{(a^{\\dagger})} da^{\\dagger} and \\varphi{(a^{\\dagger})} = t + \\sin{(a^{\\dagger})} and - \\omega + t + \\sin{(a^{\\dagger})} = - \\omega + \\int \\cos{(a^{\\dagger})} da^{\\dagger} and - \\omega + t + \\sin{(a^{\\dagger})} = - \\omega + \\varphi{(a^{\\dagger})} and \\frac{\\partial}{\\partial a^{\\dagger}} (- \\omega + t + \\sin{(a^{\\dagger})}) = \\frac{\\partial}{\\partial a^{\\dagger}} (- \\omega + \\varphi{(a^{\\dagger})}) and (\\frac{\\partial}{\\partial a^{\\dagger}} (- \\omega + t + \\sin{(a^{\\dagger})}))^{a^{\\dagger}} = (\\frac{\\partial}{\\partial a^{\\dagger}} (- \\omega + \\varphi{(a^{\\dagger})}))^{a^{\\dagger}} and \\cos^{a^{\\dagger}}{(a^{\\dagger})} = (\\frac{d}{d a^{\\dagger}} \\varphi{(a^{\\dagger})})^{a^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('t', commutative=True), sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('t', commutative=True), sin(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('t', commutative=True), sin(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('t', commutative=True), sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["power", 6, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('t', commutative=True), sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["evaluate_derivatives", 7], "Equality(Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(Function('\\\\varphi')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(Z)} = \\log{(Z)}, then obtain (\\operatorname{t_{2}}{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ)^{Z} = (\\log{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ)^{Z}", "derivation": "\\operatorname{t_{2}}{(Z)} = \\log{(Z)} and \\int \\operatorname{t_{2}}{(Z)} dZ = \\int \\log{(Z)} dZ and \\operatorname{t_{2}}{(Z)} - \\int \\log{(Z)} dZ = \\log{(Z)} - \\int \\log{(Z)} dZ and \\operatorname{t_{2}}{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ = \\log{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ and (\\operatorname{t_{2}}{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ)^{Z} = (\\log{(Z)} - \\int \\operatorname{t_{2}}{(Z)} dZ)^{Z}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["minus", 1, "Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Add(Function('t_2')(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Add(log(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('t_2')(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(Function('t_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Add(log(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(Function('t_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Function('t_2')(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(Function('t_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Symbol('Z', commutative=True)), Pow(Add(log(Symbol('Z', commutative=True)), Mul(Integer(-1), Integral(Function('t_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given s{(\\mathbf{v},\\phi)} = \\mathbf{v} + \\phi, then derive \\int \\frac{s{(\\mathbf{v},\\phi)}}{\\mathbf{v} + \\phi} d\\mathbf{v} = \\mathbf{v} + \\varphi, then obtain \\frac{1}{\\mathbf{v} + \\varphi} = \\frac{\\mathbf{v} + \\phi}{(\\mathbf{v} + \\varphi) s{(\\mathbf{v},\\phi)}}", "derivation": "s{(\\mathbf{v},\\phi)} = \\mathbf{v} + \\phi and \\frac{s{(\\mathbf{v},\\phi)}}{\\mathbf{v} + \\phi} = 1 and 1 = \\frac{\\mathbf{v} + \\phi}{s{(\\mathbf{v},\\phi)}} and \\int \\frac{s{(\\mathbf{v},\\phi)}}{\\mathbf{v} + \\phi} d\\mathbf{v} = \\int 1 d\\mathbf{v} and \\int \\frac{s{(\\mathbf{v},\\phi)}}{\\mathbf{v} + \\phi} d\\mathbf{v} = \\mathbf{v} + \\varphi and \\frac{1}{\\int 1 d\\mathbf{v}} = \\frac{\\mathbf{v} + \\phi}{s{(\\mathbf{v},\\phi)} \\int 1 d\\mathbf{v}} and \\int 1 d\\mathbf{v} = \\mathbf{v} + \\varphi and \\frac{1}{\\mathbf{v} + \\varphi} = \\frac{\\mathbf{v} + \\phi}{(\\mathbf{v} + \\varphi) s{(\\mathbf{v},\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(1))"], [["divide", 1, "Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["divide", 3, "Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1)), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}{(i,r)} = i^{r}, then derive \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)} = \\frac{i^{r} r}{i}, then obtain \\frac{\\partial}{\\partial i} i^{r} + \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)} = 2 \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)}", "derivation": "\\hat{H}{(i,r)} = i^{r} and \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)} = \\frac{\\partial}{\\partial i} i^{r} and \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)} = \\frac{i^{r} r}{i} and \\frac{\\partial}{\\partial i} i^{r} = \\frac{i^{r} r}{i} and \\frac{\\partial}{\\partial i} i^{r} + \\frac{i^{r} r}{i} = \\frac{2 i^{r} r}{i} and \\frac{\\partial}{\\partial i} i^{r} + \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)} = 2 \\frac{\\partial}{\\partial i} \\hat{H}{(i,r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["add", 4, "Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True))"], "Equality(Add(Derivative(Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True))), Mul(Integer(2), Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Pow(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Function('\\\\hat{H}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('\\\\hat{H}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{J},n)} = \\cos{(\\frac{\\mathbf{J}}{n})}, then derive \\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)} = \\frac{\\mathbf{J} \\sin{(\\frac{\\mathbf{J}}{n})}}{n^{2}}, then obtain (\\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)})^{n} = (\\frac{\\mathbf{J} \\sin{(\\frac{\\mathbf{J}}{n})}}{n^{2}})^{n}", "derivation": "\\mathbf{B}{(\\mathbf{J},n)} = \\cos{(\\frac{\\mathbf{J}}{n})} and \\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)} = \\frac{\\partial}{\\partial n} \\cos{(\\frac{\\mathbf{J}}{n})} and (\\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)})^{n} = (\\frac{\\partial}{\\partial n} \\cos{(\\frac{\\mathbf{J}}{n})})^{n} and \\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)} = \\frac{\\mathbf{J} \\sin{(\\frac{\\mathbf{J}}{n})}}{n^{2}} and \\frac{\\partial}{\\partial n} \\cos{(\\frac{\\mathbf{J}}{n})} = \\frac{\\mathbf{J} \\sin{(\\frac{\\mathbf{J}}{n})}}{n^{2}} and (\\frac{\\partial}{\\partial n} \\mathbf{B}{(\\mathbf{J},n)})^{n} = (\\frac{\\mathbf{J} \\sin{(\\frac{\\mathbf{J}}{n})}}{n^{2}})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2)), sin(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2)), sin(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-2)), sin(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then derive \\int \\operatorname{A_{2}}{(\\varepsilon_0)} d\\varepsilon_0 = \\Psi_{\\lambda} - \\cos{(\\varepsilon_0)}, then derive \\frac{d}{d \\varepsilon_0} \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\sin{(\\varepsilon_0)}, then obtain \\frac{d}{d \\varepsilon_0} \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\operatorname{A_{2}}{(\\varepsilon_0)}", "derivation": "\\operatorname{A_{2}}{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\int \\operatorname{A_{2}}{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and \\int \\operatorname{A_{2}}{(\\varepsilon_0)} d\\varepsilon_0 = \\Psi_{\\lambda} - \\cos{(\\varepsilon_0)} and \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\Psi_{\\lambda} - \\cos{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\frac{\\partial}{\\partial \\varepsilon_0} (\\Psi_{\\lambda} - \\cos{(\\varepsilon_0)}) and \\frac{d}{d \\varepsilon_0} \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = \\operatorname{A_{2}}{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(U,Q)} = \\cos{(\\frac{Q}{U})}, then obtain (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\frac{\\partial}{\\partial U} \\tilde{g}^*{(U,Q)} = \\frac{Q (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\sin{(\\frac{Q}{U})}}{U^{2}}", "derivation": "\\tilde{g}^*{(U,Q)} = \\cos{(\\frac{Q}{U})} and \\frac{\\partial}{\\partial U} \\tilde{g}^*{(U,Q)} = \\frac{\\partial}{\\partial U} \\cos{(\\frac{Q}{U})} and (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\frac{\\partial}{\\partial U} \\tilde{g}^*{(U,Q)} = (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\frac{\\partial}{\\partial U} \\cos{(\\frac{Q}{U})} and (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\frac{\\partial}{\\partial U} \\tilde{g}^*{(U,Q)} = \\frac{Q (- \\frac{Q}{U} + \\tilde{g}^*{(U,Q)})^{- U} \\sin{(\\frac{Q}{U})}}{U^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True)), cos(Mul(Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(cos(Mul(Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)))), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Function('\\\\tilde{g}^*')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True))), sin(Mul(Symbol('Q', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and S{(g^{\\prime}_{\\varepsilon})} = \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain 2 \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} S{(g^{\\prime}_{\\varepsilon})} = S^{2}{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(g^{\\prime}_{\\varepsilon})} and 2 \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} = \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} and 2 (\\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})}) \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} = (\\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})})^{2} and S{(g^{\\prime}_{\\varepsilon})} = \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} + \\cos{(g^{\\prime}_{\\varepsilon})} and 2 \\operatorname{P_{e}}{(g^{\\prime}_{\\varepsilon})} S{(g^{\\prime}_{\\varepsilon})} = S^{2}{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Add(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Add(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('S')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Function('P_e')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('S')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Pow(Function('S')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\dot{x},u)} = \\dot{x} + u and \\bar{\\h}{(\\dot{x},u)} = \\dot{x} + u, then derive \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = 1, then obtain - \\operatorname{C_{d}}{(\\dot{x},u)} + \\frac{\\partial^{2}}{\\partial u\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = - \\operatorname{C_{d}}{(\\dot{x},u)} + \\frac{d}{d u} 1", "derivation": "\\operatorname{C_{d}}{(\\dot{x},u)} = \\dot{x} + u and \\bar{\\h}{(\\dot{x},u)} = \\dot{x} + u and \\frac{\\partial}{\\partial \\dot{x}} \\bar{\\h}{(\\dot{x},u)} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} + u) and \\operatorname{C_{d}}{(\\dot{x},u)} = \\bar{\\h}{(\\dot{x},u)} and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} + u) and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = 1 and \\frac{\\partial^{2}}{\\partial u\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = \\frac{d}{d u} 1 and - \\operatorname{C_{d}}{(\\dot{x},u)} + \\frac{\\partial^{2}}{\\partial u\\partial \\dot{x}} \\operatorname{C_{d}}{(\\dot{x},u)} = - \\operatorname{C_{d}}{(\\dot{x},u)} + \\frac{d}{d u} 1", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["minus", 7, "Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True))), Derivative(Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\dot{x}', commutative=True), Symbol('u', commutative=True))), Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given U{(\\psi^*,f^{\\prime},\\mu)} = - f^{\\prime} + \\frac{\\psi^*}{\\mu}, then derive \\frac{\\partial}{\\partial \\mu} U{(\\psi^*,f^{\\prime},\\mu)} + \\frac{\\psi^*}{\\mu} = \\frac{\\psi^*}{\\mu} - \\frac{\\psi^*}{\\mu^{2}}, then obtain \\frac{\\psi^*}{\\mu} - \\frac{\\psi^*}{\\mu^{2}} = \\frac{\\partial}{\\partial \\mu} (- f^{\\prime} + \\frac{\\psi^*}{\\mu}) + \\frac{\\psi^*}{\\mu}", "derivation": "U{(\\psi^*,f^{\\prime},\\mu)} = - f^{\\prime} + \\frac{\\psi^*}{\\mu} and \\frac{\\partial}{\\partial \\mu} U{(\\psi^*,f^{\\prime},\\mu)} = \\frac{\\partial}{\\partial \\mu} (- f^{\\prime} + \\frac{\\psi^*}{\\mu}) and \\frac{\\partial}{\\partial \\mu} U{(\\psi^*,f^{\\prime},\\mu)} + \\frac{\\psi^*}{\\mu} = \\frac{\\partial}{\\partial \\mu} (- f^{\\prime} + \\frac{\\psi^*}{\\mu}) + \\frac{\\psi^*}{\\mu} and \\frac{\\partial}{\\partial \\mu} U{(\\psi^*,f^{\\prime},\\mu)} + \\frac{\\psi^*}{\\mu} = \\frac{\\psi^*}{\\mu} - \\frac{\\psi^*}{\\mu^{2}} and \\frac{\\psi^*}{\\mu} - \\frac{\\psi^*}{\\mu^{2}} = \\frac{\\partial}{\\partial \\mu} (- f^{\\prime} + \\frac{\\psi^*}{\\mu}) + \\frac{\\psi^*}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\psi^*', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\psi^*', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 2, "Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Derivative(Function('U')(Symbol('\\\\psi^*', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Add(Derivative(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('U')(Symbol('\\\\psi^*', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Symbol('\\\\psi^*', commutative=True))), Add(Derivative(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given l{(t)} = \\log{(t)}, then obtain \\frac{d}{d t} (t + l{(t)} - \\log{(t)}) = \\frac{d}{d t} t", "derivation": "l{(t)} = \\log{(t)} and t + l{(t)} = t + \\log{(t)} and t + l{(t)} - \\log{(t)} = t and \\frac{d}{d t} (t + l{(t)} - \\log{(t)}) = \\frac{d}{d t} t", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('l')(Symbol('t', commutative=True))), Add(Symbol('t', commutative=True), log(Symbol('t', commutative=True))))"], [["minus", 2, "log(Symbol('t', commutative=True))"], "Equality(Add(Symbol('t', commutative=True), Function('l')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Symbol('t', commutative=True))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Function('l')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(z^{*})} = \\sin{(\\cos{(z^{*})})}, then obtain \\iint (\\dot{z}^{z^{*}}{(z^{*})})^{z^{*}} dz^{*} dz^{*} = \\iint (\\sin^{z^{*}}{(\\cos{(z^{*})})})^{z^{*}} dz^{*} dz^{*}", "derivation": "\\dot{z}{(z^{*})} = \\sin{(\\cos{(z^{*})})} and \\dot{z}^{z^{*}}{(z^{*})} = \\sin^{z^{*}}{(\\cos{(z^{*})})} and (\\dot{z}^{z^{*}}{(z^{*})})^{z^{*}} = (\\sin^{z^{*}}{(\\cos{(z^{*})})})^{z^{*}} and \\int (\\dot{z}^{z^{*}}{(z^{*})})^{z^{*}} dz^{*} = \\int (\\sin^{z^{*}}{(\\cos{(z^{*})})})^{z^{*}} dz^{*} and \\iint (\\dot{z}^{z^{*}}{(z^{*})})^{z^{*}} dz^{*} dz^{*} = \\iint (\\sin^{z^{*}}{(\\cos{(z^{*})})})^{z^{*}} dz^{*} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('z^*', commutative=True)), sin(cos(Symbol('z^*', commutative=True))))"], [["power", 1, "Symbol('z^*', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(sin(cos(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{z}')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(Pow(sin(cos(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["integrate", 3, "Symbol('z^*', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\dot{z}')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Pow(Pow(sin(cos(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["integrate", 4, "Symbol('z^*', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\dot{z}')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Pow(Pow(sin(cos(Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{v},\\mu)} = \\mathbf{v} + \\mu and \\operatorname{A_{1}}{(\\mathbf{v},\\mu)} = (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})^{2}, then obtain \\frac{\\partial}{\\partial \\mu} \\operatorname{A_{1}}{(\\mathbf{v},\\mu)} = \\frac{\\partial}{\\partial \\mu} \\mu (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})", "derivation": "\\operatorname{C_{d}}{(\\mathbf{v},\\mu)} = \\mathbf{v} + \\mu and - \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)} = \\mu and (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})^{2} = \\mu (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)}) and \\frac{\\partial}{\\partial \\mu} (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})^{2} = \\frac{\\partial}{\\partial \\mu} \\mu (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)}) and \\operatorname{A_{1}}{(\\mathbf{v},\\mu)} = (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})^{2} and \\frac{\\partial}{\\partial \\mu} \\operatorname{A_{1}}{(\\mathbf{v},\\mu)} = \\frac{\\partial}{\\partial \\mu} \\mu (- \\mathbf{v} + \\operatorname{C_{d}}{(\\mathbf{v},\\mu)})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(2)), Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} g_{\\varepsilon}, then derive \\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})} = g_{\\varepsilon}, then obtain g_{\\varepsilon} = \\frac{g_{\\varepsilon}^{2}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} g_{\\varepsilon}}", "derivation": "\\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} g_{\\varepsilon} and \\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})} = g_{\\varepsilon} and g_{\\varepsilon} \\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})} = g_{\\varepsilon}^{2} and g_{\\varepsilon} = \\frac{g_{\\varepsilon}^{2}}{\\operatorname{v_{y}}{(V_{\\mathbf{B}},g_{\\varepsilon})}} and g_{\\varepsilon} = \\frac{g_{\\varepsilon}^{2}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["times", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2)))"], [["divide", 3, "Function('v_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Function('v_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(F_{x})} = \\log{(F_{x})}, then derive F_{x} \\frac{d}{d F_{x}} \\hat{H}_{\\lambda}{(F_{x})} + \\hat{H}_{\\lambda}{(F_{x})} = \\log{(F_{x})} + 1, then obtain \\hat{H}_{\\lambda}{(F_{x})} + 1 = \\log{(F_{x})} + 1", "derivation": "\\hat{H}_{\\lambda}{(F_{x})} = \\log{(F_{x})} and F_{x} \\hat{H}_{\\lambda}{(F_{x})} = F_{x} \\log{(F_{x})} and \\frac{d}{d F_{x}} F_{x} \\hat{H}_{\\lambda}{(F_{x})} = \\frac{d}{d F_{x}} F_{x} \\log{(F_{x})} and F_{x} \\frac{d}{d F_{x}} \\hat{H}_{\\lambda}{(F_{x})} + \\hat{H}_{\\lambda}{(F_{x})} = \\log{(F_{x})} + 1 and F_{x} \\frac{d}{d F_{x}} \\hat{H}_{\\lambda}{(F_{x})} + \\hat{H}_{\\lambda}{(F_{x})} = \\hat{H}_{\\lambda}{(F_{x})} + 1 and \\hat{H}_{\\lambda}{(F_{x})} + 1 = \\log{(F_{x})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], [["times", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_x', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('F_x', commutative=True), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True))), Add(log(Symbol('F_x', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('F_x', commutative=True), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_x', commutative=True)), Integer(1)), Add(log(Symbol('F_x', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\hat{p}{(\\hat{\\mathbf{r}},\\omega)} = \\hat{\\mathbf{r}} e^{\\omega} and \\mathbf{J}_f{(\\hat{\\mathbf{r}},\\omega)} = \\hat{p}{(\\hat{\\mathbf{r}},\\omega)} e^{\\omega}, then obtain \\mathbf{J}_f{(\\hat{\\mathbf{r}},\\omega)} = \\hat{\\mathbf{r}} e^{2 \\omega}", "derivation": "\\hat{p}{(\\hat{\\mathbf{r}},\\omega)} = \\hat{\\mathbf{r}} e^{\\omega} and \\hat{p}{(\\hat{\\mathbf{r}},\\omega)} e^{\\omega} = \\hat{\\mathbf{r}} e^{2 \\omega} and \\mathbf{J}_f{(\\hat{\\mathbf{r}},\\omega)} = \\hat{p}{(\\hat{\\mathbf{r}},\\omega)} e^{\\omega} and \\mathbf{J}_f{(\\hat{\\mathbf{r}},\\omega)} = \\hat{\\mathbf{r}} e^{2 \\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Function('\\\\hat{p}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} = \\int (- C_{2} + \\mathbf{A}) dC_{2}, then obtain \\int (\\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} + 1) dC_{2} = \\frac{C_{2}^{2}}{2} + C_{2} + g^{\\prime}_{\\varepsilon}", "derivation": "\\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} = \\int (- C_{2} + \\mathbf{A}) dC_{2} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\int (- C_{2} + \\mathbf{A}) dC_{2} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} + 1 = \\frac{\\partial}{\\partial \\mathbf{A}} \\int (- C_{2} + \\mathbf{A}) dC_{2} + 1 and \\int (\\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} + 1) dC_{2} = \\int (\\frac{\\partial}{\\partial \\mathbf{A}} \\int (- C_{2} + \\mathbf{A}) dC_{2} + 1) dC_{2} and \\int (\\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{f^{\\prime}}{(C_{2},\\mathbf{A})} + 1) dC_{2} = \\frac{C_{2}^{2}}{2} + C_{2} + g^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Derivative(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Derivative(Function('f^{\\\\prime}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Symbol('C_2', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{s})} = e^{e^{\\mathbf{s}}}, then obtain \\phi_{1}{(\\mathbf{s})} + \\iint \\phi_{1}{(\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\phi_{1}{(\\mathbf{s})} + \\iint e^{e^{\\mathbf{s}}} d\\mathbf{s} d\\mathbf{s}", "derivation": "\\phi_{1}{(\\mathbf{s})} = e^{e^{\\mathbf{s}}} and \\int \\phi_{1}{(\\mathbf{s})} d\\mathbf{s} = \\int e^{e^{\\mathbf{s}}} d\\mathbf{s} and \\iint \\phi_{1}{(\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\iint e^{e^{\\mathbf{s}}} d\\mathbf{s} d\\mathbf{s} and e^{e^{\\mathbf{s}}} + \\iint \\phi_{1}{(\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = e^{e^{\\mathbf{s}}} + \\iint e^{e^{\\mathbf{s}}} d\\mathbf{s} d\\mathbf{s} and \\phi_{1}{(\\mathbf{s})} + \\iint \\phi_{1}{(\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\phi_{1}{(\\mathbf{s})} + \\iint e^{e^{\\mathbf{s}}} d\\mathbf{s} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), exp(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 3, "exp(exp(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{s}', commutative=True)), Integral(exp(exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given k{(u,A,F_{x})} = A^{u} + F_{x}, then obtain (- k{(u,A,F_{x})})^{A} \\frac{\\partial}{\\partial A} k{(u,A,F_{x})} = (- A^{u} - F_{x})^{A} \\frac{\\partial}{\\partial A} k{(u,A,F_{x})}", "derivation": "k{(u,A,F_{x})} = A^{u} + F_{x} and - k{(u,A,F_{x})} = - A^{u} - F_{x} and (- k{(u,A,F_{x})})^{A} = (- A^{u} - F_{x})^{A} and (- k{(u,A,F_{x})})^{A} \\frac{\\partial}{\\partial A} k{(u,A,F_{x})} = (- A^{u} - F_{x})^{A} \\frac{\\partial}{\\partial A} k{(u,A,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Add(Pow(Symbol('A', commutative=True), Symbol('u', commutative=True)), Symbol('F_x', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Symbol('A', commutative=True)))"], [["times", 3, "Derivative(Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Mul(Integer(-1), Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True))), Symbol('A', commutative=True)), Derivative(Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Symbol('A', commutative=True)), Derivative(Function('k')(Symbol('u', commutative=True), Symbol('A', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(z,\\mathbf{S})} = \\mathbf{S}^{z} and u{(\\hat{p})} = \\log{(\\hat{p})}, then obtain \\mathbf{S} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) \\log{(\\hat{p})} + b{(z,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{z} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) \\log{(\\hat{p})}", "derivation": "b{(z,\\mathbf{S})} = \\mathbf{S}^{z} and \\mathbf{S} + b{(z,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{z} and u{(\\hat{p})} = \\log{(\\hat{p})} and \\mathbf{S} - z + b{(z,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{z} - z and \\mathbf{S} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) u{(\\hat{p})} + b{(z,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{z} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) u{(\\hat{p})} and \\mathbf{S} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) \\log{(\\hat{p})} + b{(z,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{z} - z - (\\mathbf{S} + \\mathbf{S}^{z} - z) \\log{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('b')(Symbol('z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True))))"], ["get_premise", "Equality(Function('u')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 2, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Function('b')(Symbol('z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["minus", 4, "Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Function('u')(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Function('u')(Symbol('\\\\hat{p}', commutative=True))), Function('b')(Symbol('z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), Function('u')(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), log(Symbol('\\\\hat{p}', commutative=True))), Function('b')(Symbol('z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), log(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(t_{2})} = \\sin{(t_{2})} and \\operatorname{v_{z}}{(x)} = e^{x}, then obtain \\hat{H}_l{(t_{2})} + e^{x} + \\sin{(t_{2})} = e^{x} + 2 \\sin{(t_{2})}", "derivation": "\\hat{H}_l{(t_{2})} = \\sin{(t_{2})} and \\hat{H}_l{(t_{2})} + \\sin{(t_{2})} = 2 \\sin{(t_{2})} and \\operatorname{v_{z}}{(x)} = e^{x} and \\hat{H}_l{(t_{2})} + \\operatorname{v_{z}}{(x)} + \\sin{(t_{2})} = \\operatorname{v_{z}}{(x)} + 2 \\sin{(t_{2})} and \\hat{H}_l{(t_{2})} + e^{x} + \\sin{(t_{2})} = e^{x} + 2 \\sin{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["add", 1, "sin(Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Mul(Integer(2), sin(Symbol('t_2', commutative=True))))"], ["get_premise", "Equality(Function('v_z')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["add", 2, "Function('v_z')(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('t_2', commutative=True)), Function('v_z')(Symbol('x', commutative=True)), sin(Symbol('t_2', commutative=True))), Add(Function('v_z')(Symbol('x', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('t_2', commutative=True)), exp(Symbol('x', commutative=True)), sin(Symbol('t_2', commutative=True))), Add(exp(Symbol('x', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(p)} = \\log{(p)} and \\hat{x}{(p)} = \\mathbf{B}^{p}{(p)}, then obtain ((\\log{(p)}^{p})^{p})^{p} - \\mathbf{B}^{p}{(p)} + \\int \\mathbf{B}^{p}{(p)} dp = ((\\log{(p)}^{p})^{p})^{p} - \\mathbf{B}^{p}{(p)} + \\int \\log{(p)}^{p} dp", "derivation": "\\mathbf{B}{(p)} = \\log{(p)} and \\mathbf{B}^{p}{(p)} = \\log{(p)}^{p} and \\hat{x}{(p)} = \\mathbf{B}^{p}{(p)} and \\hat{x}{(p)} = \\log{(p)}^{p} and \\int \\hat{x}{(p)} dp = \\int \\log{(p)}^{p} dp and - \\hat{x}{(p)} + \\int \\hat{x}{(p)} dp = - \\hat{x}{(p)} + \\int \\log{(p)}^{p} dp and - \\mathbf{B}^{p}{(p)} + \\int \\mathbf{B}^{p}{(p)} dp = - \\mathbf{B}^{p}{(p)} + \\int \\log{(p)}^{p} dp and ((\\log{(p)}^{p})^{p})^{p} - \\mathbf{B}^{p}{(p)} + \\int \\mathbf{B}^{p}{(p)} dp = ((\\log{(p)}^{p})^{p})^{p} - \\mathbf{B}^{p}{(p)} + \\int \\log{(p)}^{p} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["minus", 5, "Function('\\\\hat{x}')(Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('p', commutative=True))), Integral(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('p', commutative=True))), Integral(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integral(Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integral(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["add", 7, "Pow(Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Pow(Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integral(Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Pow(Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integral(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given l{(t_{2},n_{2})} = \\cos{(n_{2} + t_{2})}, then derive \\frac{\\partial}{\\partial n_{2}} l{(t_{2},n_{2})} = - \\sin{(n_{2} + t_{2})}, then obtain \\frac{\\partial}{\\partial n_{2}} \\cos{(n_{2} + t_{2})} = - \\sin{(n_{2} + t_{2})}", "derivation": "l{(t_{2},n_{2})} = \\cos{(n_{2} + t_{2})} and \\frac{\\partial}{\\partial n_{2}} l{(t_{2},n_{2})} = \\frac{\\partial}{\\partial n_{2}} \\cos{(n_{2} + t_{2})} and \\frac{\\partial}{\\partial n_{2}} l{(t_{2},n_{2})} = - \\sin{(n_{2} + t_{2})} and \\frac{\\partial}{\\partial n_{2}} \\cos{(n_{2} + t_{2})} = - \\sin{(n_{2} + t_{2})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('t_2', commutative=True), Symbol('n_2', commutative=True)), cos(Add(Symbol('n_2', commutative=True), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('t_2', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('n_2', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('l')(Symbol('t_2', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('n_2', commutative=True), Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Add(Symbol('n_2', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('n_2', commutative=True), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given G{(Q)} = \\log{(\\log{(Q)})}, then obtain \\frac{\\partial}{\\partial \\rho_f} \\rho_f \\frac{d}{d Q} \\sin{(G{(Q)})} = \\frac{\\partial}{\\partial \\rho_f} \\rho_f \\frac{d}{d Q} \\sin{(\\log{(\\log{(Q)})})}", "derivation": "G{(Q)} = \\log{(\\log{(Q)})} and \\sin{(G{(Q)})} = \\sin{(\\log{(\\log{(Q)})})} and \\frac{d}{d Q} \\sin{(G{(Q)})} = \\frac{d}{d Q} \\sin{(\\log{(\\log{(Q)})})} and \\rho_f \\frac{d}{d Q} \\sin{(G{(Q)})} = \\rho_f \\frac{d}{d Q} \\sin{(\\log{(\\log{(Q)})})} and \\frac{\\partial}{\\partial \\rho_f} \\rho_f \\frac{d}{d Q} \\sin{(G{(Q)})} = \\frac{\\partial}{\\partial \\rho_f} \\rho_f \\frac{d}{d Q} \\sin{(\\log{(\\log{(Q)})})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))"], [["sin", 1], "Equality(sin(Function('G')(Symbol('Q', commutative=True))), sin(log(log(Symbol('Q', commutative=True)))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(sin(Function('G')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(sin(log(log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Derivative(sin(Function('G')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho_f', commutative=True), Derivative(sin(log(log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\rho_f', commutative=True), Derivative(sin(Function('G')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho_f', commutative=True), Derivative(sin(log(log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(n_{1})} = \\cos{(\\cos{(n_{1})})}, then derive \\frac{d}{d n_{1}} \\operatorname{A_{z}}{(n_{1})} = \\sin{(n_{1})} \\sin{(\\cos{(n_{1})})}, then obtain \\int \\frac{d}{d n_{1}} \\cos{(\\cos{(n_{1})})} dn_{1} = \\int \\sin{(n_{1})} \\sin{(\\cos{(n_{1})})} dn_{1}", "derivation": "\\operatorname{A_{z}}{(n_{1})} = \\cos{(\\cos{(n_{1})})} and \\frac{d}{d n_{1}} \\operatorname{A_{z}}{(n_{1})} = \\frac{d}{d n_{1}} \\cos{(\\cos{(n_{1})})} and \\frac{d}{d n_{1}} \\operatorname{A_{z}}{(n_{1})} = \\sin{(n_{1})} \\sin{(\\cos{(n_{1})})} and \\frac{d}{d n_{1}} \\cos{(\\cos{(n_{1})})} = \\sin{(n_{1})} \\sin{(\\cos{(n_{1})})} and \\int \\frac{d}{d n_{1}} \\cos{(\\cos{(n_{1})})} dn_{1} = \\int \\sin{(n_{1})} \\sin{(\\cos{(n_{1})})} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('n_1', commutative=True)), cos(cos(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(sin(Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(cos(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(sin(Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True)))))"], [["integrate", 4, "Symbol('n_1', commutative=True)"], "Equality(Integral(Derivative(cos(cos(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(sin(Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given L{(\\rho_f,v)} = \\cos{(\\frac{\\rho_f}{v})} and \\operatorname{f^{\\prime}}{(v)} = \\frac{1}{v}, then obtain - v L^{v}{(\\rho_f,v)} + \\operatorname{f^{\\prime}}^{v}{(v)} = - v L^{v}{(\\rho_f,v)} + (\\frac{1}{v})^{v}", "derivation": "L{(\\rho_f,v)} = \\cos{(\\frac{\\rho_f}{v})} and L^{v}{(\\rho_f,v)} = \\cos^{v}{(\\frac{\\rho_f}{v})} and \\operatorname{f^{\\prime}}{(v)} = \\frac{1}{v} and v L^{v}{(\\rho_f,v)} = v \\cos^{v}{(\\frac{\\rho_f}{v})} and \\operatorname{f^{\\prime}}^{v}{(v)} = (\\frac{1}{v})^{v} and - v \\cos^{v}{(\\frac{\\rho_f}{v})} + \\operatorname{f^{\\prime}}^{v}{(v)} = - v \\cos^{v}{(\\frac{\\rho_f}{v})} + (\\frac{1}{v})^{v} and - v L^{v}{(\\rho_f,v)} + \\operatorname{f^{\\prime}}^{v}{(v)} = - v L^{v}{(\\rho_f,v)} + (\\frac{1}{v})^{v}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1)))"], [["times", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Pow(Function('L')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), Pow(cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))), Symbol('v', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True)))"], [["minus", 5, "Mul(Symbol('v', commutative=True), Pow(cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))), Symbol('v', commutative=True))), Pow(Function('f^{\\\\prime}')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(cos(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))), Symbol('v', commutative=True))), Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Function('L')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Pow(Function('f^{\\\\prime}')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Function('L')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f)} = \\sin{(f)}, then obtain \\frac{2 f + \\operatorname{F_{H}}{(f)}}{2 f + \\sin{(f)}} = 1", "derivation": "\\operatorname{F_{H}}{(f)} = \\sin{(f)} and f + \\operatorname{F_{H}}{(f)} = f + \\sin{(f)} and 2 f + \\operatorname{F_{H}}{(f)} = 2 f + \\sin{(f)} and \\frac{2 f + \\operatorname{F_{H}}{(f)}}{2 f + \\sin{(f)}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('F_H')(Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('f', commutative=True))))"], [["add", 2, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('f', commutative=True)), Function('F_H')(Symbol('f', commutative=True))), Add(Mul(Integer(2), Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(2), Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('f', commutative=True)), Function('F_H')(Symbol('f', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given C{(U)} = \\cos{(\\log{(U)})} and \\hat{X}{(U)} = \\cos{(\\log{(U)})}, then obtain \\frac{\\hat{X}{(U)}}{a^{\\dagger} \\sigma_{x}{(a^{\\dagger},\\mathbf{F})} \\cos{(\\log{(U)})}} = \\frac{1}{a^{\\dagger} \\sigma_{x}{(a^{\\dagger},\\mathbf{F})}}", "derivation": "C{(U)} = \\cos{(\\log{(U)})} and \\hat{X}{(U)} = \\cos{(\\log{(U)})} and \\hat{X}{(U)} = C{(U)} and \\frac{\\hat{X}{(U)}}{\\cos{(\\log{(U)})}} = \\frac{C{(U)}}{\\cos{(\\log{(U)})}} and \\frac{\\hat{X}{(U)}}{\\cos{(\\log{(U)})}} = 1 and \\frac{\\hat{X}{(U)}}{a^{\\dagger} \\sigma_{x}{(a^{\\dagger},\\mathbf{F})} \\cos{(\\log{(U)})}} = \\frac{1}{a^{\\dagger} \\sigma_{x}{(a^{\\dagger},\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('U', commutative=True)), cos(log(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True)), cos(log(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Function('C')(Symbol('U', commutative=True)))"], [["divide", 3, "cos(log(Symbol('U', commutative=True)))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Pow(cos(log(Symbol('U', commutative=True))), Integer(-1))), Mul(Function('C')(Symbol('U', commutative=True)), Pow(cos(log(Symbol('U', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Pow(cos(log(Symbol('U', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 5, "Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('U', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Pow(cos(log(Symbol('U', commutative=True))), Integer(-1))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given f{(\\pi)} = \\sin{(e^{\\pi})}, then derive \\frac{d}{d \\pi} f{(\\pi)} = e^{\\pi} \\cos{(e^{\\pi})}, then obtain \\frac{d^{2}}{d \\pi^{2}} \\sin{(e^{\\pi})} - 1 = \\frac{d}{d \\pi} e^{\\pi} \\cos{(e^{\\pi})} - 1", "derivation": "f{(\\pi)} = \\sin{(e^{\\pi})} and \\frac{d}{d \\pi} f{(\\pi)} = \\frac{d}{d \\pi} \\sin{(e^{\\pi})} and \\frac{d}{d \\pi} f{(\\pi)} = e^{\\pi} \\cos{(e^{\\pi})} and \\frac{d^{2}}{d \\pi^{2}} f{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} \\cos{(e^{\\pi})} and \\frac{d^{2}}{d \\pi^{2}} f{(\\pi)} - 1 = \\frac{d}{d \\pi} e^{\\pi} \\cos{(e^{\\pi})} - 1 and \\frac{d^{2}}{d \\pi^{2}} \\sin{(e^{\\pi})} = \\frac{d}{d \\pi} e^{\\pi} \\cos{(e^{\\pi})} and \\frac{d^{2}}{d \\pi^{2}} f{(\\pi)} - 1 = \\frac{d^{2}}{d \\pi^{2}} \\sin{(e^{\\pi})} - 1 and \\frac{d^{2}}{d \\pi^{2}} \\sin{(e^{\\pi})} - 1 = \\frac{d}{d \\pi} e^{\\pi} \\cos{(e^{\\pi})} - 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\pi', commutative=True)), sin(exp(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\pi', commutative=True)), cos(exp(Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(Mul(exp(Symbol('\\\\pi', commutative=True)), cos(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Integer(-1)), Add(Derivative(Mul(exp(Symbol('\\\\pi', commutative=True)), cos(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(Mul(exp(Symbol('\\\\pi', commutative=True)), cos(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Derivative(Function('f')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Integer(-1)), Add(Derivative(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Derivative(sin(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Integer(-1)), Add(Derivative(Mul(exp(Symbol('\\\\pi', commutative=True)), cos(exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given s{(\\hat{p},m_{s})} = \\frac{\\hat{p}}{m_{s}}, then obtain \\frac{d}{d \\hat{p}} \\frac{1}{m_{s}} = \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\hat{p}}{m_{s}^{2} s{(\\hat{p},m_{s})}}", "derivation": "s{(\\hat{p},m_{s})} = \\frac{\\hat{p}}{m_{s}} and m_{s} s{(\\hat{p},m_{s})} = \\hat{p} and 1 = \\frac{\\hat{p}}{m_{s} s{(\\hat{p},m_{s})}} and \\frac{1}{m_{s}} = \\frac{\\hat{p}}{m_{s}^{2} s{(\\hat{p},m_{s})}} and \\frac{d}{d \\hat{p}} \\frac{1}{m_{s}} = \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\hat{p}}{m_{s}^{2} s{(\\hat{p},m_{s})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Symbol('m_s', commutative=True), Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True))), Symbol('\\\\hat{p}', commutative=True))"], [["divide", 2, "Mul(Symbol('m_s', commutative=True), Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Integer(1), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))))"], [["times", 3, "Pow(Symbol('m_s', commutative=True), Integer(-1))"], "Equality(Pow(Symbol('m_s', commutative=True), Integer(-1)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-2)), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Pow(Symbol('m_s', commutative=True), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-2)), Pow(Function('s')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} = \\frac{x^\\prime}{\\hat{H}}, then derive \\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} = \\frac{1}{\\hat{H}}, then obtain \\frac{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)}}{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} + 1} = \\frac{1}{\\hat{H} (\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} + 1)}", "derivation": "\\operatorname{P_{e}}{(\\hat{H},x^\\prime)} = \\frac{x^\\prime}{\\hat{H}} and \\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} \\frac{x^\\prime}{\\hat{H}} and \\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} = \\frac{1}{\\hat{H}} and \\frac{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)}}{\\frac{\\partial}{\\partial x^\\prime} \\frac{x^\\prime}{\\hat{H}} + 1} = \\frac{1}{\\hat{H} (\\frac{\\partial}{\\partial x^\\prime} \\frac{x^\\prime}{\\hat{H}} + 1)} and \\frac{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)}}{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} + 1} = \\frac{1}{\\hat{H} (\\frac{\\partial}{\\partial x^\\prime} \\operatorname{P_{e}}{(\\hat{H},x^\\prime)} + 1)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], [["divide", 3, "Add(Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Pow(Add(Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Add(Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Integer(-1)), Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Add(Derivative(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given a{(\\rho_b,y^{\\prime})} = \\frac{y^{\\prime}}{\\rho_b}, then obtain \\frac{\\rho_b \\cos^{y^{\\prime}}{(a^{\\rho_b}{(\\rho_b,y^{\\prime})})}}{y^{\\prime}} = \\frac{\\rho_b \\cos^{y^{\\prime}}{((\\frac{y^{\\prime}}{\\rho_b})^{\\rho_b})}}{y^{\\prime}}", "derivation": "a{(\\rho_b,y^{\\prime})} = \\frac{y^{\\prime}}{\\rho_b} and a^{\\rho_b}{(\\rho_b,y^{\\prime})} = (\\frac{y^{\\prime}}{\\rho_b})^{\\rho_b} and \\cos{(a^{\\rho_b}{(\\rho_b,y^{\\prime})})} = \\cos{((\\frac{y^{\\prime}}{\\rho_b})^{\\rho_b})} and \\cos^{y^{\\prime}}{(a^{\\rho_b}{(\\rho_b,y^{\\prime})})} = \\cos^{y^{\\prime}}{((\\frac{y^{\\prime}}{\\rho_b})^{\\rho_b})} and \\frac{\\rho_b \\cos^{y^{\\prime}}{(a^{\\rho_b}{(\\rho_b,y^{\\prime})})}}{y^{\\prime}} = \\frac{\\rho_b \\cos^{y^{\\prime}}{((\\frac{y^{\\prime}}{\\rho_b})^{\\rho_b})}}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('a')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), cos(Pow(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(cos(Pow(Function('a')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Pow(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 4, "Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(cos(Pow(Function('a')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(cos(Pow(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\phi_1)} = \\int \\cos{(\\phi_1)} d\\phi_1, then derive \\rho_{f}{(\\phi_1)} = p + \\sin{(\\phi_1)}, then obtain \\int \\rho_{f}^{2}{(\\phi_1)} dp = V_{\\mathbf{E}} + \\frac{p^{2} \\rho_{f}{(\\phi_1)}}{2} + p \\rho_{f}{(\\phi_1)} \\sin{(\\phi_1)}", "derivation": "\\rho_{f}{(\\phi_1)} = \\int \\cos{(\\phi_1)} d\\phi_1 and \\rho_{f}{(\\phi_1)} = p + \\sin{(\\phi_1)} and \\rho_{f}^{2}{(\\phi_1)} = (p + \\sin{(\\phi_1)}) \\rho_{f}{(\\phi_1)} and \\int \\rho_{f}^{2}{(\\phi_1)} dp = \\int (p + \\sin{(\\phi_1)}) \\rho_{f}{(\\phi_1)} dp and \\int \\rho_{f}^{2}{(\\phi_1)} dp = V_{\\mathbf{E}} + \\frac{p^{2} \\rho_{f}{(\\phi_1)}}{2} + p \\rho_{f}{(\\phi_1)} \\sin{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), Integral(cos(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), Add(Symbol('p', commutative=True), sin(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Mul(Add(Symbol('p', commutative=True), sin(Symbol('\\\\phi_1', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Add(Symbol('p', commutative=True), sin(Symbol('\\\\phi_1', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Tuple(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2)), Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('p', commutative=True), Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(A_{1})} = \\sin{(\\log{(A_{1})})}, then obtain \\cos{(\\operatorname{f_{\\mathbf{p}}}{(A_{1})})} = \\cos{(\\frac{\\sin^{2}{(\\log{(A_{1})})}}{\\operatorname{f_{\\mathbf{p}}}{(A_{1})}})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(A_{1})} = \\sin{(\\log{(A_{1})})} and \\operatorname{f_{\\mathbf{p}}}{(A_{1})} \\sin{(\\log{(A_{1})})} = \\sin^{2}{(\\log{(A_{1})})} and \\sin{(\\log{(A_{1})})} = \\frac{\\sin^{2}{(\\log{(A_{1})})}}{\\operatorname{f_{\\mathbf{p}}}{(A_{1})}} and \\operatorname{f_{\\mathbf{p}}}{(A_{1})} = \\frac{\\sin^{2}{(\\log{(A_{1})})}}{\\operatorname{f_{\\mathbf{p}}}{(A_{1})}} and \\cos{(\\operatorname{f_{\\mathbf{p}}}{(A_{1})})} = \\cos{(\\frac{\\sin^{2}{(\\log{(A_{1})})}}{\\operatorname{f_{\\mathbf{p}}}{(A_{1})}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), sin(log(Symbol('A_1', commutative=True))))"], [["times", 1, "sin(log(Symbol('A_1', commutative=True)))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), sin(log(Symbol('A_1', commutative=True)))), Pow(sin(log(Symbol('A_1', commutative=True))), Integer(2)))"], [["divide", 2, "Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True))"], "Equality(sin(log(Symbol('A_1', commutative=True))), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('A_1', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('A_1', commutative=True))), Integer(2))))"], [["cos", 4], "Equality(cos(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True))), cos(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('A_1', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('A_1', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given G{(I,q)} = \\frac{\\sin{(q)}}{I^{2}}, then obtain - \\int 0 dq + \\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq = - \\mathbf{J}_P + \\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq + \\frac{\\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq}{I^{2}}", "derivation": "G{(I,q)} = \\frac{\\sin{(q)}}{I^{2}} and 0 = - G{(I,q)} + \\frac{\\sin{(q)}}{I^{2}} and \\int 0 dq = \\int (- G{(I,q)} + \\frac{\\sin{(q)}}{I^{2}}) dq and - \\int 0 dq = - \\int (- G{(I,q)} + \\frac{\\sin{(q)}}{I^{2}}) dq and - \\int 0 dq + \\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq = \\int I^{2} G{(I,q)} dq - \\int (- G{(I,q)} + \\frac{\\sin{(q)}}{I^{2}}) dq + \\int - \\sin{(q)} dq and - \\int 0 dq + \\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq = - \\mathbf{J}_P + \\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq + \\frac{\\int I^{2} G{(I,q)} dq + \\int - \\sin{(q)} dq}{I^{2}}", "srepr_derivation": [["renaming_premise", "Equality(Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), sin(Symbol('q', commutative=True))))"], [["minus", 1, "Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), sin(Symbol('q', commutative=True)))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), sin(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), sin(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))))"], [["add", 4, "Add(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('q', commutative=True)))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Add(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), sin(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('q', commutative=True)))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), Add(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('G')(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{r}{(f^{\\prime},l)} = \\cos{(l^{f^{\\prime}})}, then derive \\frac{\\partial}{\\partial f^{\\prime}} \\mathbf{r}{(f^{\\prime},l)} = - l^{f^{\\prime}} \\log{(l)} \\sin{(l^{f^{\\prime}})}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(l^{f^{\\prime}})} = - l^{f^{\\prime}} \\log{(l)} \\sin{(l^{f^{\\prime}})}", "derivation": "\\mathbf{r}{(f^{\\prime},l)} = \\cos{(l^{f^{\\prime}})} and l + \\mathbf{r}{(f^{\\prime},l)} = l + \\cos{(l^{f^{\\prime}})} and \\frac{\\partial}{\\partial f^{\\prime}} (l + \\mathbf{r}{(f^{\\prime},l)}) = \\frac{\\partial}{\\partial f^{\\prime}} (l + \\cos{(l^{f^{\\prime}})}) and \\frac{\\partial}{\\partial f^{\\prime}} \\mathbf{r}{(f^{\\prime},l)} = - l^{f^{\\prime}} \\log{(l)} \\sin{(l^{f^{\\prime}})} and \\frac{\\partial}{\\partial f^{\\prime}} \\cos{(l^{f^{\\prime}})} = - l^{f^{\\prime}} \\log{(l)} \\sin{(l^{f^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), cos(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), cos(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('l', commutative=True), Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('l', commutative=True), cos(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('l', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('l', commutative=True)), sin(Pow(Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(V)} = e^{\\sin{(V)}} and \\hat{X}{(V)} = \\sin{(V)}, then obtain \\frac{(\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)})^{2}}{\\operatorname{f^{\\prime}}{(V)}} = \\frac{\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)} \\frac{d}{d V} e^{\\hat{X}{(V)}}}{\\operatorname{f^{\\prime}}{(V)}}", "derivation": "\\operatorname{f^{\\prime}}{(V)} = e^{\\sin{(V)}} and \\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)} = \\frac{d}{d V} e^{\\sin{(V)}} and (\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)})^{2} = \\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)} \\frac{d}{d V} e^{\\sin{(V)}} and \\hat{X}{(V)} = \\sin{(V)} and (\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)})^{2} = \\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)} \\frac{d}{d V} e^{\\hat{X}{(V)}} and \\frac{(\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)})^{2}}{\\operatorname{f^{\\prime}}{(V)}} = \\frac{\\frac{d}{d V} \\operatorname{f^{\\prime}}{(V)} \\frac{d}{d V} e^{\\hat{X}{(V)}}}{\\operatorname{f^{\\prime}}{(V)}}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), exp(sin(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(exp(Function('\\\\hat{X}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["divide", 5, "Function('f^{\\\\prime}')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Integer(-1)), Pow(Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Integer(-1)), Derivative(Function('f^{\\\\prime}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(exp(Function('\\\\hat{X}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)} = \\frac{n_{1}}{A \\mathbf{J}_P}, then obtain e^{q} + \\frac{1}{Z{(q)}} = e^{q} + \\frac{n_{1}}{A \\mathbf{J}_P \\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)} Z{(q)}}", "derivation": "\\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)} = \\frac{n_{1}}{A \\mathbf{J}_P} and \\frac{\\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)}}{A} = \\frac{n_{1}}{A^{2} \\mathbf{J}_P} and \\frac{\\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)}}{A Z{(q)}} = \\frac{n_{1}}{A^{2} \\mathbf{J}_P Z{(q)}} and \\frac{1}{Z{(q)}} = \\frac{n_{1}}{A \\mathbf{J}_P \\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)} Z{(q)}} and e^{q} + \\frac{1}{Z{(q)}} = e^{q} + \\frac{n_{1}}{A \\mathbf{J}_P \\operatorname{F_{g}}{(n_{1},A,\\mathbf{J}_P)} Z{(q)}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["divide", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["divide", 2, "Function('Z')(Symbol('q', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-2)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Pow(Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1))))"], [["add", 4, "exp(Symbol('q', commutative=True))"], "Equality(Add(exp(Symbol('q', commutative=True)), Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1))), Add(exp(Symbol('q', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Pow(Function('F_g')(Symbol('n_1', commutative=True), Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('q', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given J{(v_{y})} = e^{v_{y}}, then obtain \\frac{d}{d v_{y}} (J{(v_{y})} e^{- v_{y}} + 1) = \\frac{d}{d v_{y}} 2", "derivation": "J{(v_{y})} = e^{v_{y}} and J{(v_{y})} e^{- v_{y}} = 1 and J{(v_{y})} e^{- v_{y}} + 1 = 2 and \\frac{d}{d v_{y}} (J{(v_{y})} e^{- v_{y}} + 1) = \\frac{d}{d v_{y}} 2", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["divide", 1, "exp(Symbol('v_y', commutative=True))"], "Equality(Mul(Function('J')(Symbol('v_y', commutative=True)), exp(Mul(Integer(-1), Symbol('v_y', commutative=True)))), Integer(1))"], [["add", 2, 1], "Equality(Add(Mul(Function('J')(Symbol('v_y', commutative=True)), exp(Mul(Integer(-1), Symbol('v_y', commutative=True)))), Integer(1)), Integer(2))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Function('J')(Symbol('v_y', commutative=True)), exp(Mul(Integer(-1), Symbol('v_y', commutative=True)))), Integer(1)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(f)} = \\cos{(f)}, then derive \\int y{(f)} df = l + \\sin{(f)}, then obtain \\iint y{(f)} df dl = \\int (l + \\sin{(f)}) dl", "derivation": "y{(f)} = \\cos{(f)} and \\int y{(f)} df = \\int \\cos{(f)} df and \\int y{(f)} df = l + \\sin{(f)} and \\iint y{(f)} df dl = \\int (l + \\sin{(f)}) dl", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('y')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('l', commutative=True), sin(Symbol('f', commutative=True))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Function('y')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), sin(Symbol('f', commutative=True))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given Q{(\\Psi_{nl})} = \\sin{(e^{\\Psi_{nl}})} and \\hat{H}{(C_{d})} = \\sin{(C_{d})}, then obtain - V_{\\mathbf{B}} \\eta + \\frac{\\hat{H}{(C_{d})}}{Q{(\\Psi_{nl})}} = - V_{\\mathbf{B}} \\eta + \\frac{\\sin{(C_{d})}}{Q{(\\Psi_{nl})}}", "derivation": "Q{(\\Psi_{nl})} = \\sin{(e^{\\Psi_{nl}})} and \\hat{H}{(C_{d})} = \\sin{(C_{d})} and \\frac{\\hat{H}{(C_{d})}}{\\sin{(e^{\\Psi_{nl}})}} = \\frac{\\sin{(C_{d})}}{\\sin{(e^{\\Psi_{nl}})}} and \\frac{\\hat{H}{(C_{d})}}{Q{(\\Psi_{nl})}} = \\frac{\\sin{(C_{d})}}{Q{(\\Psi_{nl})}} and - V_{\\mathbf{B}} \\eta + \\frac{\\hat{H}{(C_{d})}}{Q{(\\Psi_{nl})}} = - V_{\\mathbf{B}} \\eta + \\frac{\\sin{(C_{d})}}{Q{(\\Psi_{nl})}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["divide", 2, "sin(exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('C_d', commutative=True)), Pow(sin(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))), Mul(sin(Symbol('C_d', commutative=True)), Pow(sin(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('Q')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Function('\\\\hat{H}')(Symbol('C_d', commutative=True))), Mul(Pow(Function('Q')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), sin(Symbol('C_d', commutative=True))))"], [["minus", 4, "Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Function('Q')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), Function('\\\\hat{H}')(Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Function('Q')(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), sin(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{s},\\chi)} = \\frac{\\log{(\\chi)}}{\\mathbf{s}}, then derive \\mathbf{s} \\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)} = \\frac{1}{\\chi}, then obtain \\mathbf{s} (\\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)})^{2} = \\frac{\\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)}}{\\chi}", "derivation": "\\Omega{(\\mathbf{s},\\chi)} = \\frac{\\log{(\\chi)}}{\\mathbf{s}} and \\mathbf{s} \\Omega{(\\mathbf{s},\\chi)} = \\log{(\\chi)} and \\frac{\\partial}{\\partial \\chi} \\mathbf{s} \\Omega{(\\mathbf{s},\\chi)} = \\frac{d}{d \\chi} \\log{(\\chi)} and \\mathbf{s} \\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)} = \\frac{1}{\\chi} and \\mathbf{s} (\\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)})^{2} = \\frac{\\frac{\\partial}{\\partial \\chi} \\Omega{(\\mathbf{s},\\chi)}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), log(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))"], [["times", 4, "Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})}, then obtain \\frac{d}{d \\mathbb{I}} \\mathbf{v}^{2}{(\\mathbb{I})} \\log{(e^{\\mathbb{I}})}^{2} = \\frac{d}{d \\mathbb{I}} \\log{(e^{\\mathbb{I}})}^{4}", "derivation": "\\mathbf{v}{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})} and \\mathbf{v}{(\\mathbb{I})} \\log{(e^{\\mathbb{I}})} = \\log{(e^{\\mathbb{I}})}^{2} and \\mathbf{v}^{2}{(\\mathbb{I})} \\log{(e^{\\mathbb{I}})}^{2} = \\log{(e^{\\mathbb{I}})}^{4} and \\frac{d}{d \\mathbb{I}} \\mathbf{v}^{2}{(\\mathbb{I})} \\log{(e^{\\mathbb{I}})}^{2} = \\frac{d}{d \\mathbb{I}} \\log{(e^{\\mathbb{I}})}^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True)), log(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 1, "log(exp(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True)), log(exp(Symbol('\\\\mathbb{I}', commutative=True)))), Pow(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2))), Pow(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(4)))"], [["differentiate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Pow(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(2))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(4)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(\\omega,C_{2})} = C_{2} \\cos{(\\omega)}, then derive v_{y} - \\frac{\\int C_{2} \\cos{(\\omega)} dC_{2} + \\int - \\omega u{(\\omega,C_{2})} dC_{2}}{\\omega} = \\int (C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega}) dC_{2}, then obtain v_{y} - \\frac{\\int C_{2} \\cos{(\\omega)} dC_{2} + \\int - C_{2} \\omega \\cos{(\\omega)} dC_{2}}{\\omega} = \\int (C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega}) dC_{2}", "derivation": "u{(\\omega,C_{2})} = C_{2} \\cos{(\\omega)} and - \\frac{C_{2} \\cos{(\\omega)}}{\\omega} + u{(\\omega,C_{2})} = C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega} and \\int (- \\frac{C_{2} \\cos{(\\omega)}}{\\omega} + u{(\\omega,C_{2})}) dC_{2} = \\int (C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega}) dC_{2} and v_{y} - \\frac{\\int C_{2} \\cos{(\\omega)} dC_{2} + \\int - \\omega u{(\\omega,C_{2})} dC_{2}}{\\omega} = \\int (C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega}) dC_{2} and v_{y} - \\frac{\\int C_{2} \\cos{(\\omega)} dC_{2} + \\int - C_{2} \\omega \\cos{(\\omega)} dC_{2}}{\\omega} = \\int (C_{2} \\cos{(\\omega)} - \\frac{C_{2} \\cos{(\\omega)}}{\\omega}) dC_{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True))), Function('u')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True))), Function('u')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Integral(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Function('u')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))), Integral(Add(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Integral(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))), Integral(Add(Mul(Symbol('C_2', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(J,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}), then derive \\operatorname{v_{2}}^{\\mathbf{f}}{(J,\\mathbf{f})} = 1, then obtain \\mathbf{f} + (\\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}))^{\\mathbf{f}} = \\mathbf{f} + 1", "derivation": "\\operatorname{v_{2}}{(J,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}) and \\operatorname{v_{2}}^{\\mathbf{f}}{(J,\\mathbf{f})} = (\\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}))^{\\mathbf{f}} and \\operatorname{v_{2}}^{\\mathbf{f}}{(J,\\mathbf{f})} = 1 and (\\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}))^{\\mathbf{f}} = 1 and \\mathbf{f} + (\\frac{\\partial}{\\partial \\mathbf{f}} (J + \\mathbf{f}))^{\\mathbf{f}} = \\mathbf{f} + 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\mathbf{f}', commutative=True)), Integer(1))"], [["add", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\mathbf{r}{(\\varphi)} = \\log{(e^{\\varphi})}, then obtain \\frac{\\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)}}{\\mathbf{r}{(\\varphi)} - 1} = 1", "derivation": "\\mathbf{r}{(\\varphi)} = \\log{(e^{\\varphi})} and \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(e^{\\varphi})} and - \\mathbf{r}{(\\varphi)} + \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)} = - \\mathbf{r}{(\\varphi)} + \\frac{d}{d \\varphi} \\log{(e^{\\varphi})} and \\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)} = \\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\log{(e^{\\varphi})} and \\frac{\\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)}}{\\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\log{(e^{\\varphi})}} = 1 and \\frac{\\mathbf{r}{(\\varphi)} - \\frac{d}{d \\varphi} \\mathbf{r}{(\\varphi)}}{\\mathbf{r}{(\\varphi)} - 1} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True))), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True))), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))))"], [["divide", 4, "Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], "Equality(Mul(Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Pow(Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integer(-1)), Add(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))), Integer(1))"]]}, {"prompt": "Given \\lambda{(v_{2})} = e^{v_{2}}, then obtain \\frac{\\lambda^{3}{(v_{2})}}{2 v_{2}} = \\frac{\\lambda{(v_{2})} e^{2 v_{2}}}{2 v_{2}}", "derivation": "\\lambda{(v_{2})} = e^{v_{2}} and \\lambda^{2}{(v_{2})} = \\lambda{(v_{2})} e^{v_{2}} and \\lambda^{3}{(v_{2})} = \\lambda^{2}{(v_{2})} e^{v_{2}} and \\lambda^{3}{(v_{2})} = \\lambda{(v_{2})} e^{2 v_{2}} and \\frac{\\lambda^{3}{(v_{2})}}{2 v_{2}} = \\frac{\\lambda{(v_{2})} e^{2 v_{2}}}{2 v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["times", 1, "Function('\\\\lambda')(Symbol('v_2', commutative=True))"], "Equality(Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(2)), Mul(Function('\\\\lambda')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))))"], [["times", 1, "Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(2)), exp(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(3)), Mul(Function('\\\\lambda')(Symbol('v_2', commutative=True)), exp(Mul(Integer(2), Symbol('v_2', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), Symbol('v_2', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('v_2', commutative=True)), Integer(3))), Mul(Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('v_2', commutative=True)), exp(Mul(Integer(2), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\Omega)} = \\Omega, then derive \\int \\mathbf{g}{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\hat{p}, then derive J_{\\varepsilon} + \\frac{\\Omega^{2}}{2} = \\frac{\\Omega^{2}}{2} + \\hat{p}, then obtain \\int (J_{\\varepsilon} + \\frac{\\Omega^{2}}{2}) d\\hat{p} = \\int (\\frac{\\Omega^{2}}{2} + \\hat{p}) d\\hat{p}", "derivation": "\\mathbf{g}{(\\Omega)} = \\Omega and \\int \\mathbf{g}{(\\Omega)} d\\Omega = \\int \\Omega d\\Omega and \\int \\mathbf{g}{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\hat{p} and \\int \\Omega d\\Omega = \\frac{\\Omega^{2}}{2} + \\hat{p} and J_{\\varepsilon} + \\frac{\\Omega^{2}}{2} = \\frac{\\Omega^{2}}{2} + \\hat{p} and \\int (J_{\\varepsilon} + \\frac{\\Omega^{2}}{2}) d\\hat{p} = \\int (\\frac{\\Omega^{2}}{2} + \\hat{p}) d\\hat{p}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\hat{p}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(F_{c})} = e^{F_{c}}, then derive \\int \\hat{\\mathbf{r}}{(F_{c})} dF_{c} = \\pi + e^{F_{c}}, then obtain \\int e^{F_{c}} dF_{c} = \\pi + e^{F_{c}}", "derivation": "\\hat{\\mathbf{r}}{(F_{c})} = e^{F_{c}} and \\int \\hat{\\mathbf{r}}{(F_{c})} dF_{c} = \\int e^{F_{c}} dF_{c} and \\int \\hat{\\mathbf{r}}{(F_{c})} dF_{c} = \\pi + e^{F_{c}} and \\int e^{F_{c}} dF_{c} = \\pi + e^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\pi', commutative=True), exp(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Symbol('\\\\pi', commutative=True), exp(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(b,G)} = \\frac{\\cos{(G)}}{b} and \\operatorname{F_{g}}{(b,G)} = \\int ((G + \\frac{\\cos{(G)}}{b})^{G} - (G + \\operatorname{E_{n}}{(b,G)})^{G}) db, then obtain \\frac{d}{d b} \\int 0 db = \\frac{\\partial}{\\partial b} \\operatorname{F_{g}}{(b,G)}", "derivation": "\\operatorname{E_{n}}{(b,G)} = \\frac{\\cos{(G)}}{b} and G + \\operatorname{E_{n}}{(b,G)} = G + \\frac{\\cos{(G)}}{b} and (G + \\operatorname{E_{n}}{(b,G)})^{G} = (G + \\frac{\\cos{(G)}}{b})^{G} and 0 = (G + \\frac{\\cos{(G)}}{b})^{G} - (G + \\operatorname{E_{n}}{(b,G)})^{G} and \\int 0 db = \\int ((G + \\frac{\\cos{(G)}}{b})^{G} - (G + \\operatorname{E_{n}}{(b,G)})^{G}) db and \\frac{d}{d b} \\int 0 db = \\frac{\\partial}{\\partial b} \\int ((G + \\frac{\\cos{(G)}}{b})^{G} - (G + \\operatorname{E_{n}}{(b,G)})^{G}) db and \\operatorname{F_{g}}{(b,G)} = \\int ((G + \\frac{\\cos{(G)}}{b})^{G} - (G + \\operatorname{E_{n}}{(b,G)})^{G}) db and \\frac{d}{d b} \\int 0 db = \\frac{\\partial}{\\partial b} \\operatorname{F_{g}}{(b,G)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))), Symbol('G', commutative=True)))"], [["minus", 3, "Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Integral(Add(Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Tuple(Symbol('b', commutative=True))))"], [["differentiate", 5, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integral(Add(Pow(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), cos(Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Tuple(Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Function('F_g')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(v)} = \\sin{(v)}, then derive \\frac{d}{d v} i{(v)} = \\cos{(v)}, then obtain \\cos^{v}{(\\cos{(v)})} = \\cos^{v}{(\\frac{d}{d v} \\sin{(v)})}", "derivation": "i{(v)} = \\sin{(v)} and \\frac{d}{d v} i{(v)} = \\frac{d}{d v} \\sin{(v)} and \\frac{d}{d v} i{(v)} = \\cos{(v)} and \\cos{(\\frac{d}{d v} i{(v)})} = \\cos{(\\frac{d}{d v} \\sin{(v)})} and \\cos^{v}{(\\frac{d}{d v} i{(v)})} = \\cos^{v}{(\\frac{d}{d v} \\sin{(v)})} and \\cos^{v}{(\\cos{(v)})} = \\cos^{v}{(\\frac{d}{d v} \\sin{(v)})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), cos(Symbol('v', commutative=True)))"], [["cos", 2], "Equality(cos(Derivative(Function('i')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), cos(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(cos(Derivative(Function('i')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Symbol('v', commutative=True)), Pow(cos(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(cos(cos(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(cos(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} = - E_{\\lambda} + \\mathbf{H}, then obtain 2 E_{\\lambda} + \\int E_{\\lambda} \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} d\\mathbf{H} = 2 E_{\\lambda} + \\int E_{\\lambda} (- E_{\\lambda} + \\mathbf{H}) d\\mathbf{H}", "derivation": "\\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} = - E_{\\lambda} + \\mathbf{H} and E_{\\lambda} \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} = E_{\\lambda} (- E_{\\lambda} + \\mathbf{H}) and \\int E_{\\lambda} \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} d\\mathbf{H} = \\int E_{\\lambda} (- E_{\\lambda} + \\mathbf{H}) d\\mathbf{H} and E_{\\lambda} + \\int E_{\\lambda} \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} d\\mathbf{H} = E_{\\lambda} + \\int E_{\\lambda} (- E_{\\lambda} + \\mathbf{H}) d\\mathbf{H} and 2 E_{\\lambda} + \\int E_{\\lambda} \\operatorname{r_{0}}{(\\mathbf{H},E_{\\lambda})} d\\mathbf{H} = 2 E_{\\lambda} + \\int E_{\\lambda} (- E_{\\lambda} + \\mathbf{H}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["add", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{A})} = e^{\\mathbf{A}}, then derive \\frac{d}{d \\mathbf{A}} \\hat{H}_{\\lambda}{(\\mathbf{A})} = e^{\\mathbf{A}}, then obtain \\frac{d^{2}}{d \\mathbf{A}^{2}} e^{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\hat{H}_{\\lambda}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\hat{H}_{\\lambda}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} = e^{\\mathbf{A}} and \\frac{d^{2}}{d \\mathbf{A}^{2}} e^{\\mathbf{A}} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(Q)} = \\log{(Q)}, then derive \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ = Q + k, then obtain \\frac{d}{d k} \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ + \\iint 1 dQ dQ = \\iint 1 dQ dQ + 1", "derivation": "\\mathbf{A}{(Q)} = \\log{(Q)} and \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} = 1 and \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ = \\int 1 dQ and \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ = Q + k and \\frac{d}{d k} \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ = \\frac{\\partial}{\\partial k} (Q + k) and \\frac{d}{d k} \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ + \\iint 1 dQ dQ = \\frac{\\partial}{\\partial k} (Q + k) + \\iint 1 dQ dQ and \\frac{d}{d k} \\int \\frac{\\mathbf{A}{(Q)}}{\\log{(Q)}} dQ + \\iint 1 dQ dQ = \\iint 1 dQ dQ + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["divide", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), Symbol('k', commutative=True)))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["add", 5, "Integral(Integer(1), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Derivative(Add(Symbol('Q', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(1)))"]]}, {"prompt": "Given U{(F_{N},y)} = F_{N} - y and \\mathbf{E}{(F_{N},y)} = U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) + \\frac{\\partial}{\\partial y} U{(F_{N},y)} + 1, then derive \\frac{\\partial}{\\partial y} U{(F_{N},y)} = -1, then obtain U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) = \\mathbf{E}{(F_{N},y)}", "derivation": "U{(F_{N},y)} = F_{N} - y and \\frac{\\partial}{\\partial y} U{(F_{N},y)} = \\frac{\\partial}{\\partial y} (F_{N} - y) and \\frac{\\partial}{\\partial y} U{(F_{N},y)} = -1 and -1 = \\frac{\\partial}{\\partial y} (F_{N} - y) and U{(F_{N},y)} - 1 = U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) and U{(F_{N},y)} - 1 = U{(F_{N},y)} + \\frac{\\partial}{\\partial y} U{(F_{N},y)} and U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) = U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) + \\frac{\\partial}{\\partial y} U{(F_{N},y)} + 1 and \\mathbf{E}{(F_{N},y)} = U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) + \\frac{\\partial}{\\partial y} U{(F_{N},y)} + 1 and U{(F_{N},y)} + \\frac{\\partial}{\\partial y} (F_{N} - y) = \\mathbf{E}{(F_{N},y)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["add", 4, "Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Add(Function('U')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), Function('\\\\mathbf{E}')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\lambda,q)} = \\log{(- \\lambda + q)} and M{(\\lambda,q)} = - \\lambda + q, then obtain (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(M{(\\lambda,q)})}^{q} = (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(- \\lambda + q)}^{q}", "derivation": "\\operatorname{y^{\\prime}}{(\\lambda,q)} = \\log{(- \\lambda + q)} and \\operatorname{y^{\\prime}}^{q}{(\\lambda,q)} = \\log{(- \\lambda + q)}^{q} and (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\operatorname{y^{\\prime}}^{q}{(\\lambda,q)} = (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(- \\lambda + q)}^{q} and M{(\\lambda,q)} = - \\lambda + q and (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\operatorname{y^{\\prime}}^{q}{(\\lambda,q)} = (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(M{(\\lambda,q)})}^{q} and (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(M{(\\lambda,q)})}^{q} = (\\lambda + \\operatorname{y^{\\prime}}{(\\lambda,q)}) \\log{(- \\lambda + q)}^{q}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(log(Function('M')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(log(Function('M')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))), Mul(Add(Symbol('\\\\lambda', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('q', commutative=True))), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = \\Omega - \\chi - \\mu_0, then derive \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = -1, then obtain - \\frac{\\Omega + \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)}}{\\mu_0} = - \\frac{\\Omega - 1}{\\mu_0}", "derivation": "\\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = \\Omega - \\chi - \\mu_0 and \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = \\frac{\\partial}{\\partial \\mu_0} (\\Omega - \\chi - \\mu_0) and \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = -1 and \\Omega + \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)} = \\Omega - 1 and - \\frac{\\Omega + \\frac{\\partial}{\\partial \\mu_0} \\operatorname{n_{2}}{(\\mu_0,\\chi,\\Omega)}}{\\mu_0} = - \\frac{\\Omega - 1}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))"], [["add", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Derivative(Function('n_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Derivative(Function('n_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\theta_2,\\hbar)} = \\hbar \\log{(\\theta_2)} and \\theta{(\\theta_2,\\hbar)} = \\hat{\\mathbf{r}}^{\\theta_2}{(\\theta_2,\\hbar)}, then obtain \\theta^{\\theta_2}{(\\theta_2,\\hbar)} = ((\\hbar \\log{(\\theta_2)})^{\\theta_2})^{\\theta_2}", "derivation": "\\hat{\\mathbf{r}}{(\\theta_2,\\hbar)} = \\hbar \\log{(\\theta_2)} and \\hat{\\mathbf{r}}^{\\theta_2}{(\\theta_2,\\hbar)} = (\\hbar \\log{(\\theta_2)})^{\\theta_2} and (\\hat{\\mathbf{r}}^{\\theta_2}{(\\theta_2,\\hbar)})^{\\theta_2} = ((\\hbar \\log{(\\theta_2)})^{\\theta_2})^{\\theta_2} and \\theta{(\\theta_2,\\hbar)} = \\hat{\\mathbf{r}}^{\\theta_2}{(\\theta_2,\\hbar)} and \\theta{(\\theta_2,\\hbar)} = (\\hbar \\log{(\\theta_2)})^{\\theta_2} and (\\hat{\\mathbf{r}}^{\\theta_2}{(\\theta_2,\\hbar)})^{\\theta_2} = \\theta^{\\theta_2}{(\\theta_2,\\hbar)} and \\theta^{\\theta_2}{(\\theta_2,\\hbar)} = ((\\hbar \\log{(\\theta_2)})^{\\theta_2})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\psi,l)} = \\psi - l, then obtain - l + \\frac{\\partial}{\\partial l} \\int \\cos{(\\hat{x}{(\\psi,l)})} dl = - l + \\frac{\\partial}{\\partial l} \\int \\cos{(\\psi - l)} dl", "derivation": "\\hat{x}{(\\psi,l)} = \\psi - l and \\cos{(\\hat{x}{(\\psi,l)})} = \\cos{(\\psi - l)} and \\int \\cos{(\\hat{x}{(\\psi,l)})} dl = \\int \\cos{(\\psi - l)} dl and \\frac{\\partial}{\\partial l} \\int \\cos{(\\hat{x}{(\\psi,l)})} dl = \\frac{\\partial}{\\partial l} \\int \\cos{(\\psi - l)} dl and - l + \\frac{\\partial}{\\partial l} \\int \\cos{(\\hat{x}{(\\psi,l)})} dl = - l + \\frac{\\partial}{\\partial l} \\int \\cos{(\\psi - l)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), cos(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(cos(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(cos(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Integral(cos(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(cos(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 4, "Mul(Integer(-1), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Integral(cos(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Derivative(Integral(cos(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(F_{H})} = \\cos{(F_{H})}, then obtain (\\operatorname{P_{e}}{(F_{H})} + 2 \\cos{(F_{H})})^{3} = 27 \\cos^{3}{(F_{H})}", "derivation": "\\operatorname{P_{e}}{(F_{H})} = \\cos{(F_{H})} and \\operatorname{P_{e}}{(F_{H})} + \\cos{(F_{H})} = 2 \\cos{(F_{H})} and \\operatorname{P_{e}}{(F_{H})} + 2 \\cos{(F_{H})} = 3 \\cos{(F_{H})} and 2 \\operatorname{P_{e}}{(F_{H})} + \\cos{(F_{H})} = 3 \\cos{(F_{H})} and (2 \\operatorname{P_{e}}{(F_{H})} + \\cos{(F_{H})})^{3} = 27 \\cos^{3}{(F_{H})} and \\operatorname{P_{e}}{(F_{H})} + 2 \\cos{(F_{H})} = 2 \\operatorname{P_{e}}{(F_{H})} + \\cos{(F_{H})} and (\\operatorname{P_{e}}{(F_{H})} + 2 \\cos{(F_{H})})^{3} = 27 \\cos^{3}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["add", 1, "cos(Symbol('F_H', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Mul(Integer(2), cos(Symbol('F_H', commutative=True))))"], [["add", 2, "cos(Symbol('F_H', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('F_H', commutative=True)), Mul(Integer(2), cos(Symbol('F_H', commutative=True)))), Mul(Integer(3), cos(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('P_e')(Symbol('F_H', commutative=True))), cos(Symbol('F_H', commutative=True))), Mul(Integer(3), cos(Symbol('F_H', commutative=True))))"], [["power", 4, 3], "Equality(Pow(Add(Mul(Integer(2), Function('P_e')(Symbol('F_H', commutative=True))), cos(Symbol('F_H', commutative=True))), Integer(3)), Mul(Integer(27), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('P_e')(Symbol('F_H', commutative=True)), Mul(Integer(2), cos(Symbol('F_H', commutative=True)))), Add(Mul(Integer(2), Function('P_e')(Symbol('F_H', commutative=True))), cos(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Function('P_e')(Symbol('F_H', commutative=True)), Mul(Integer(2), cos(Symbol('F_H', commutative=True)))), Integer(3)), Mul(Integer(27), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))))"]]}, {"prompt": "Given r{(U)} = \\sin{(U)} and \\operatorname{t_{2}}{(U)} = r{(U)} + \\sin^{2}{(U)}, then obtain \\operatorname{t_{2}}{(U)} = r^{2}{(U)} + r{(U)}", "derivation": "r{(U)} = \\sin{(U)} and 2 r{(U)} - \\sin{(U)} = r{(U)} and 2 r{(U)} - \\sin{(U)} = \\sin{(U)} and (2 r{(U)} - \\sin{(U)})^{2} = r^{2}{(U)} and \\sin^{2}{(U)} = r^{2}{(U)} and r{(U)} + \\sin^{2}{(U)} = r^{2}{(U)} + r{(U)} and \\operatorname{t_{2}}{(U)} = r{(U)} + \\sin^{2}{(U)} and \\operatorname{t_{2}}{(U)} = r^{2}{(U)} + r{(U)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["add", 1, "Add(Function('r')(Symbol('U', commutative=True)), Mul(Integer(-1), sin(Symbol('U', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('r')(Symbol('U', commutative=True))), Mul(Integer(-1), sin(Symbol('U', commutative=True)))), Function('r')(Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Add(Mul(Integer(2), Function('r')(Symbol('U', commutative=True))), Mul(Integer(-1), sin(Symbol('U', commutative=True)))), sin(Symbol('U', commutative=True)))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Integer(2), Function('r')(Symbol('U', commutative=True))), Mul(Integer(-1), sin(Symbol('U', commutative=True)))), Integer(2)), Pow(Function('r')(Symbol('U', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(sin(Symbol('U', commutative=True)), Integer(2)), Pow(Function('r')(Symbol('U', commutative=True)), Integer(2)))"], [["add", 5, "Function('r')(Symbol('U', commutative=True))"], "Equality(Add(Function('r')(Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Integer(2))), Add(Pow(Function('r')(Symbol('U', commutative=True)), Integer(2)), Function('r')(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('U', commutative=True)), Add(Function('r')(Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Function('t_2')(Symbol('U', commutative=True)), Add(Pow(Function('r')(Symbol('U', commutative=True)), Integer(2)), Function('r')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})} = \\frac{I \\mathbf{S}}{\\rho_f} and \\psi{(i,A_{1})} = - A_{1} + e^{i}, then obtain \\frac{1}{- A_{1} + e^{i}} = \\frac{I \\mathbf{S}}{\\rho_f (- A_{1} + e^{i}) \\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})}}", "derivation": "\\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})} = \\frac{I \\mathbf{S}}{\\rho_f} and 1 = \\frac{I \\mathbf{S}}{\\rho_f \\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})}} and \\psi{(i,A_{1})} = - A_{1} + e^{i} and \\frac{1}{\\psi{(i,A_{1})}} = \\frac{I \\mathbf{S}}{\\rho_f \\psi{(i,A_{1})} \\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})}} and \\frac{1}{- A_{1} + e^{i}} = \\frac{I \\mathbf{S}}{\\rho_f (- A_{1} + e^{i}) \\operatorname{v_{y}}{(I,\\rho_f,\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('I', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["divide", 1, "Function('v_y')(Symbol('I', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('v_y')(Symbol('I', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\psi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), exp(Symbol('i', commutative=True))))"], [["divide", 2, "Function('\\\\psi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Pow(Function('\\\\psi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Pow(Function('v_y')(Symbol('I', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), exp(Symbol('i', commutative=True))), Integer(-1)), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), exp(Symbol('i', commutative=True))), Integer(-1)), Pow(Function('v_y')(Symbol('I', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\phi,u)} = \\frac{\\phi}{u}, then obtain J - \\phi (\\int - \\frac{\\phi}{u} du + \\int \\operatorname{J_{\\varepsilon}}{(\\phi,u)} du) - 1 = \\int 0 du - 1", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\phi,u)} = \\frac{\\phi}{u} and 0 = \\frac{\\phi}{u} - \\operatorname{J_{\\varepsilon}}{(\\phi,u)} and 0 = - \\frac{\\phi}{u} + \\operatorname{J_{\\varepsilon}}{(\\phi,u)} and 0 = \\phi (- \\frac{\\phi}{u} + \\operatorname{J_{\\varepsilon}}{(\\phi,u)}) and - \\phi (- \\frac{\\phi}{u} + \\operatorname{J_{\\varepsilon}}{(\\phi,u)}) = 0 and \\int - \\phi (- \\frac{\\phi}{u} + \\operatorname{J_{\\varepsilon}}{(\\phi,u)}) du = \\int 0 du and \\int - \\phi (- \\frac{\\phi}{u} + \\operatorname{J_{\\varepsilon}}{(\\phi,u)}) du - 1 = \\int 0 du - 1 and J - \\phi (\\int - \\frac{\\phi}{u} du + \\int \\operatorname{J_{\\varepsilon}}{(\\phi,u)} du) - 1 = \\int 0 du - 1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["minus", 1, "Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))))"], [["times", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)))), Integer(0))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Integer(0), Tuple(Symbol('u', commutative=True))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Integral(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integer(-1)), Add(Integral(Integer(0), Tuple(Symbol('u', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Add(Integral(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))), Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))), Integer(-1)), Add(Integral(Integer(0), Tuple(Symbol('u', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\lambda{(v_{x})} = e^{v_{x}}, then obtain - \\lambda^{2}{(v_{x})} e^{2 v_{x}} + \\lambda^{2}{(v_{x})} e^{v_{x}} = - \\lambda^{2}{(v_{x})} e^{2 v_{x}} + \\lambda{(v_{x})} e^{2 v_{x}}", "derivation": "\\lambda{(v_{x})} = e^{v_{x}} and \\lambda{(v_{x})} e^{v_{x}} = e^{2 v_{x}} and \\lambda^{2}{(v_{x})} e^{v_{x}} = \\lambda{(v_{x})} e^{2 v_{x}} and \\lambda^{3}{(v_{x})} e^{v_{x}} = \\lambda^{2}{(v_{x})} e^{2 v_{x}} and - \\lambda^{3}{(v_{x})} e^{v_{x}} + \\lambda^{2}{(v_{x})} e^{v_{x}} = - \\lambda^{3}{(v_{x})} e^{v_{x}} + \\lambda{(v_{x})} e^{2 v_{x}} and - \\lambda^{2}{(v_{x})} e^{2 v_{x}} + \\lambda^{2}{(v_{x})} e^{v_{x}} = - \\lambda^{2}{(v_{x})} e^{2 v_{x}} + \\lambda{(v_{x})} e^{2 v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["times", 1, "exp(Symbol('v_x', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), exp(Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["times", 2, "Function('\\\\lambda')(Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Symbol('v_x', commutative=True))), Mul(Function('\\\\lambda')(Symbol('v_x', commutative=True)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))))"], [["times", 2, "Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(3)), exp(Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))))"], [["minus", 3, "Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(3)), exp(Symbol('v_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(3)), exp(Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(3)), exp(Symbol('v_x', commutative=True))), Mul(Function('\\\\lambda')(Symbol('v_x', commutative=True)), exp(Mul(Integer(2), Symbol('v_x', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))), Mul(Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\lambda')(Symbol('v_x', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('v_x', commutative=True)))), Mul(Function('\\\\lambda')(Symbol('v_x', commutative=True)), exp(Mul(Integer(2), Symbol('v_x', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(l)} = \\log{(\\cos{(l)})}, then obtain (\\int (\\frac{\\mathbf{F}{(l)}}{\\cos{(l)}} - \\log{(\\cos{(l)})}) dl)^{l} = (\\int (- \\log{(\\cos{(l)})} + \\frac{\\log{(\\cos{(l)})}}{\\cos{(l)}}) dl)^{l}", "derivation": "\\mathbf{F}{(l)} = \\log{(\\cos{(l)})} and \\frac{\\mathbf{F}{(l)}}{\\cos{(l)}} = \\frac{\\log{(\\cos{(l)})}}{\\cos{(l)}} and \\frac{\\mathbf{F}{(l)}}{\\cos{(l)}} - \\log{(\\cos{(l)})} = - \\log{(\\cos{(l)})} + \\frac{\\log{(\\cos{(l)})}}{\\cos{(l)}} and \\int (\\frac{\\mathbf{F}{(l)}}{\\cos{(l)}} - \\log{(\\cos{(l)})}) dl = \\int (- \\log{(\\cos{(l)})} + \\frac{\\log{(\\cos{(l)})}}{\\cos{(l)}}) dl and (\\int (\\frac{\\mathbf{F}{(l)}}{\\cos{(l)}} - \\log{(\\cos{(l)})}) dl)^{l} = (\\int (- \\log{(\\cos{(l)})} + \\frac{\\log{(\\cos{(l)})}}{\\cos{(l)}}) dl)^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('l', commutative=True)), log(cos(Symbol('l', commutative=True))))"], [["divide", 1, "cos(Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Mul(log(cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1))))"], [["minus", 2, "log(cos(Symbol('l', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{F}')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Mul(Integer(-1), log(cos(Symbol('l', commutative=True))))), Add(Mul(Integer(-1), log(cos(Symbol('l', commutative=True)))), Mul(log(cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1)))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Mul(Function('\\\\mathbf{F}')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Mul(Integer(-1), log(cos(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), log(cos(Symbol('l', commutative=True)))), Mul(log(cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1)))), Tuple(Symbol('l', commutative=True))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Function('\\\\mathbf{F}')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Mul(Integer(-1), log(cos(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), log(cos(Symbol('l', commutative=True)))), Mul(log(cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1)))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\omega)} = e^{\\omega}, then obtain \\frac{d}{d \\omega} e^{- e^{\\omega}} = \\frac{d}{d \\omega} e^{- \\operatorname{V_{\\mathbf{B}}}{(\\omega)}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\omega)} = e^{\\omega} and 0 = - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + e^{\\omega} and - e^{\\omega} = - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} and e^{- e^{\\omega}} = e^{- \\operatorname{V_{\\mathbf{B}}}{(\\omega)}} and \\frac{d}{d \\omega} e^{- e^{\\omega}} = \\frac{d}{d \\omega} e^{- \\operatorname{V_{\\mathbf{B}}}{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))), exp(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), exp(Symbol('\\\\omega', commutative=True)))), exp(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(exp(Mul(Integer(-1), exp(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(k)} = e^{\\sin{(k)}}, then obtain \\varphi^{*}{(k)} e^{- \\sin{(k)}} \\iint \\varphi^{*}{(k)} dk dk = \\iint \\varphi^{*}{(k)} dk dk", "derivation": "\\varphi^{*}{(k)} = e^{\\sin{(k)}} and \\int \\varphi^{*}{(k)} dk = \\int e^{\\sin{(k)}} dk and \\iint \\varphi^{*}{(k)} dk dk = \\iint e^{\\sin{(k)}} dk dk and \\varphi^{*}{(k)} e^{- \\sin{(k)}} = 1 and \\varphi^{*}{(k)} e^{- \\sin{(k)}} \\iint e^{\\sin{(k)}} dk dk = \\iint e^{\\sin{(k)}} dk dk and \\varphi^{*}{(k)} e^{- \\sin{(k)}} \\iint \\varphi^{*}{(k)} dk dk = \\iint \\varphi^{*}{(k)} dk dk", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('k', commutative=True)), exp(sin(Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["divide", 1, "exp(sin(Symbol('k', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('k', commutative=True))))), Integer(1))"], [["times", 4, "Integral(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('k', commutative=True)))), Integral(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integral(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('k', commutative=True)))), Integral(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integral(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given c{(\\theta_1,y)} = \\log{(\\frac{y}{\\theta_1})}, then derive e^{\\int c{(\\theta_1,y)} d\\theta_1} = e^{\\Psi_{nl} + \\theta_1 \\log{(\\frac{y}{\\theta_1})} + \\theta_1}, then obtain (e^{\\int c{(\\theta_1,y)} d\\theta_1})^{\\Psi_{nl}} = (e^{\\Psi_{nl} + \\theta_1 c{(\\theta_1,y)} + \\theta_1})^{\\Psi_{nl}}", "derivation": "c{(\\theta_1,y)} = \\log{(\\frac{y}{\\theta_1})} and \\int c{(\\theta_1,y)} d\\theta_1 = \\int \\log{(\\frac{y}{\\theta_1})} d\\theta_1 and e^{\\int c{(\\theta_1,y)} d\\theta_1} = e^{\\int \\log{(\\frac{y}{\\theta_1})} d\\theta_1} and e^{\\int c{(\\theta_1,y)} d\\theta_1} = e^{\\Psi_{nl} + \\theta_1 \\log{(\\frac{y}{\\theta_1})} + \\theta_1} and (e^{\\int c{(\\theta_1,y)} d\\theta_1})^{\\Psi_{nl}} = (e^{\\Psi_{nl} + \\theta_1 \\log{(\\frac{y}{\\theta_1})} + \\theta_1})^{\\Psi_{nl}} and (e^{\\int c{(\\theta_1,y)} d\\theta_1})^{\\Psi_{nl}} = (e^{\\Psi_{nl} + \\theta_1 c{(\\theta_1,y)} + \\theta_1})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), log(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Integral(log(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), log(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\theta_1', commutative=True))))"], [["power", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(exp(Integral(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), log(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(exp(Integral(Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(exp(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('c')(Symbol('\\\\theta_1', commutative=True), Symbol('y', commutative=True))), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given M{(\\phi_2)} = \\phi_2, then obtain \\cos{(\\int M{(\\phi_2)} dM{(\\phi_2)})} = \\cos{(\\int \\phi_2 dM{(\\phi_2)})}", "derivation": "M{(\\phi_2)} = \\phi_2 and \\int M{(\\phi_2)} d\\phi_2 = \\int \\phi_2 d\\phi_2 and \\int M{(\\phi_2)} dM{(\\phi_2)} = \\int \\phi_2 dM{(\\phi_2)} and \\cos{(\\int M{(\\phi_2)} dM{(\\phi_2)})} = \\cos{(\\int \\phi_2 dM{(\\phi_2)})}", "srepr_derivation": [["renaming_premise", "Equality(Function('M')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('M')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('M')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('M')(Symbol('\\\\phi_2', commutative=True)))), Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Function('M')(Symbol('\\\\phi_2', commutative=True)))))"], [["cos", 3], "Equality(cos(Integral(Function('M')(Symbol('\\\\phi_2', commutative=True)), Tuple(Function('M')(Symbol('\\\\phi_2', commutative=True))))), cos(Integral(Symbol('\\\\phi_2', commutative=True), Tuple(Function('M')(Symbol('\\\\phi_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\pi,C)} = C \\log{(\\pi)}, then derive \\frac{C}{\\pi} + \\operatorname{F_{N}}{(\\pi,C)} = C \\log{(\\pi)} + \\frac{C}{\\pi}, then obtain (\\frac{C}{\\pi} + \\operatorname{F_{N}}{(\\pi,C)}) \\frac{\\partial}{\\partial \\pi} \\operatorname{F_{N}}{(\\pi,C)} = (C \\log{(\\pi)} + \\frac{C}{\\pi}) \\frac{\\partial}{\\partial \\pi} \\operatorname{F_{N}}{(\\pi,C)}", "derivation": "\\operatorname{F_{N}}{(\\pi,C)} = C \\log{(\\pi)} and \\frac{\\partial}{\\partial \\pi} \\operatorname{F_{N}}{(\\pi,C)} = \\frac{\\partial}{\\partial \\pi} C \\log{(\\pi)} and \\operatorname{F_{N}}{(\\pi,C)} + \\frac{\\partial}{\\partial \\pi} C \\log{(\\pi)} = C \\log{(\\pi)} + \\frac{\\partial}{\\partial \\pi} C \\log{(\\pi)} and \\frac{C}{\\pi} + \\operatorname{F_{N}}{(\\pi,C)} = C \\log{(\\pi)} + \\frac{C}{\\pi} and (\\frac{C}{\\pi} + \\operatorname{F_{N}}{(\\pi,C)}) \\frac{\\partial}{\\partial \\pi} C \\log{(\\pi)} = (C \\log{(\\pi)} + \\frac{C}{\\pi}) \\frac{\\partial}{\\partial \\pi} C \\log{(\\pi)} and (\\frac{C}{\\pi} + \\operatorname{F_{N}}{(\\pi,C)}) \\frac{\\partial}{\\partial \\pi} \\operatorname{F_{N}}{(\\pi,C)} = (C \\log{(\\pi)} + \\frac{C}{\\pi}) \\frac{\\partial}{\\partial \\pi} \\operatorname{F_{N}}{(\\pi,C)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Add(Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))"], [["times", 4, "Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True))), Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Derivative(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True))), Derivative(Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('C', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Derivative(Function('F_N')(Symbol('\\\\pi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(\\delta,F_{N})} = \\sin{(F_{N} + \\delta)}, then obtain u{(\\delta,F_{N})} - \\int \\sin{(F_{N} + \\delta)} d\\delta = \\sin{(F_{N} + \\delta)} - \\int \\sin{(F_{N} + \\delta)} d\\delta", "derivation": "u{(\\delta,F_{N})} = \\sin{(F_{N} + \\delta)} and \\int u{(\\delta,F_{N})} d\\delta = \\int \\sin{(F_{N} + \\delta)} d\\delta and u{(\\delta,F_{N})} - \\int u{(\\delta,F_{N})} d\\delta = \\sin{(F_{N} + \\delta)} - \\int u{(\\delta,F_{N})} d\\delta and u{(\\delta,F_{N})} - \\int \\sin{(F_{N} + \\delta)} d\\delta = \\sin{(F_{N} + \\delta)} - \\int \\sin{(F_{N} + \\delta)} d\\delta", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Integral(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Integral(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Add(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('u')(Symbol('\\\\delta', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))), Add(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\psi,i)} = i^{\\psi} and \\operatorname{y^{\\prime}}{(\\psi,i)} = \\frac{i^{\\psi}}{\\operatorname{F_{x}}{(\\psi,i)}}, then obtain 0^{\\psi} = (\\operatorname{y^{\\prime}}{(\\psi,i)} - 1)^{\\psi}", "derivation": "\\operatorname{F_{x}}{(\\psi,i)} = i^{\\psi} and 1 = \\frac{i^{\\psi}}{\\operatorname{F_{x}}{(\\psi,i)}} and 0 = \\frac{i^{\\psi}}{\\operatorname{F_{x}}{(\\psi,i)}} - 1 and 0^{\\psi} = (\\frac{i^{\\psi}}{\\operatorname{F_{x}}{(\\psi,i)}} - 1)^{\\psi} and \\operatorname{y^{\\prime}}{(\\psi,i)} = \\frac{i^{\\psi}}{\\operatorname{F_{x}}{(\\psi,i)}} and 0^{\\psi} = (\\operatorname{y^{\\prime}}{(\\psi,i)} - 1)^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Integer(-1))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Integer(-1)))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Integer(-1)), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('F_x')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Add(Function('y^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('i', commutative=True)), Integer(-1)), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(P_{g})} = \\int \\log{(P_{g})} dP_{g}, then derive \\phi_{2}{(P_{g})} = A_{1} + P_{g} \\log{(P_{g})} - P_{g}, then derive (A_{1} + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}} = (I + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}}, then obtain \\phi_{2}^{A_{1}}{(P_{g})} = (I + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}}", "derivation": "\\phi_{2}{(P_{g})} = \\int \\log{(P_{g})} dP_{g} and \\phi_{2}{(P_{g})} = A_{1} + P_{g} \\log{(P_{g})} - P_{g} and A_{1} + P_{g} \\log{(P_{g})} - P_{g} = \\int \\log{(P_{g})} dP_{g} and (A_{1} + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}} = (\\int \\log{(P_{g})} dP_{g})^{A_{1}} and (A_{1} + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}} = (I + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}} and (\\int \\log{(P_{g})} dP_{g})^{A_{1}} = (I + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}} and \\phi_{2}^{A_{1}}{(P_{g})} = (I + P_{g} \\log{(P_{g})} - P_{g})^{A_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Integral(log(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('A_1', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Integral(log(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Symbol('A_1', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)), Pow(Integral(log(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('A_1', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Symbol('I', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(log(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Symbol('I', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Function('\\\\phi_2')(Symbol('P_g', commutative=True)), Symbol('A_1', commutative=True)), Pow(Add(Symbol('I', commutative=True), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given H{(\\theta)} = \\sin{(\\theta)}, then obtain 0 = - 2 \\sin{(H{(\\theta)} - \\sin{(\\theta)})}", "derivation": "H{(\\theta)} = \\sin{(\\theta)} and \\theta + H{(\\theta)} = \\theta + \\sin{(\\theta)} and 0 = - H{(\\theta)} + \\sin{(\\theta)} and 0 = - \\sin{(H{(\\theta)} - \\sin{(\\theta)})} and - \\sin{(H{(\\theta)} - \\sin{(\\theta)})} = - 2 \\sin{(H{(\\theta)} - \\sin{(\\theta)})} and 0 = - 2 \\sin{(H{(\\theta)} - \\sin{(\\theta)})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('H')(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\theta', commutative=True), Function('H')(Symbol('\\\\theta', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('H')(Symbol('\\\\theta', commutative=True))), sin(Symbol('\\\\theta', commutative=True))))"], [["sin", 3], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('H')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))))))"], [["minus", 4, "sin(Add(Function('H')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))))"], "Equality(Mul(Integer(-1), sin(Add(Function('H')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))))), Mul(Integer(-1), Integer(2), sin(Add(Function('H')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(0), Mul(Integer(-1), Integer(2), sin(Add(Function('H')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))))))"]]}, {"prompt": "Given l{(\\mathbf{J},v_{y})} = \\mathbf{J} \\cos{(v_{y})}, then obtain (\\frac{\\partial}{\\partial v_{y}} \\int l{(\\mathbf{J},v_{y})} dv_{y})^{v_{y}} = (\\frac{\\partial}{\\partial v_{y}} \\int \\mathbf{J} \\cos{(v_{y})} dv_{y})^{v_{y}}", "derivation": "l{(\\mathbf{J},v_{y})} = \\mathbf{J} \\cos{(v_{y})} and \\int l{(\\mathbf{J},v_{y})} dv_{y} = \\int \\mathbf{J} \\cos{(v_{y})} dv_{y} and \\frac{\\partial}{\\partial v_{y}} \\int l{(\\mathbf{J},v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} \\int \\mathbf{J} \\cos{(v_{y})} dv_{y} and (\\frac{\\partial}{\\partial v_{y}} \\int l{(\\mathbf{J},v_{y})} dv_{y})^{v_{y}} = (\\frac{\\partial}{\\partial v_{y}} \\int \\mathbf{J} \\cos{(v_{y})} dv_{y})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integral(Function('l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('l')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} = \\mathbf{f}^{\\mu_0}, then derive \\frac{\\partial}{\\partial \\mu_0} \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} - 1 = \\mathbf{f}^{\\mu_0} \\log{(\\mathbf{f})} - 1, then obtain \\frac{\\partial}{\\partial \\mu_0} \\mathbf{f}^{\\mu_0} - 1 = \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} \\log{(\\mathbf{f})} - 1", "derivation": "\\mathbf{J}_P{(\\mu_0,\\mathbf{f})} = \\mathbf{f}^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mu_0} \\mathbf{f}^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} - 1 = \\frac{\\partial}{\\partial \\mu_0} \\mathbf{f}^{\\mu_0} - 1 and \\frac{\\partial}{\\partial \\mu_0} \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} - 1 = \\mathbf{f}^{\\mu_0} \\log{(\\mathbf{f})} - 1 and \\frac{\\partial}{\\partial \\mu_0} \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} - 1 = \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} \\log{(\\mathbf{f})} - 1 and \\frac{\\partial}{\\partial \\mu_0} \\mathbf{f}^{\\mu_0} - 1 = \\mathbf{J}_P{(\\mu_0,\\mathbf{f})} \\log{(\\mathbf{f})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given r{(F_{c})} = \\sin{(F_{c})}, then obtain \\frac{F_{c} + r{(F_{c})}}{F_{c} + r{(F_{c})} + \\sin{(F_{c})}} = \\frac{F_{c} + \\sin{(F_{c})}}{F_{c} + r{(F_{c})} + \\sin{(F_{c})}}", "derivation": "r{(F_{c})} = \\sin{(F_{c})} and F_{c} + r{(F_{c})} = F_{c} + \\sin{(F_{c})} and F_{c} + r{(F_{c})} + \\sin{(F_{c})} = F_{c} + 2 \\sin{(F_{c})} and \\frac{F_{c} + r{(F_{c})}}{F_{c} + 2 \\sin{(F_{c})}} = \\frac{F_{c} + \\sin{(F_{c})}}{F_{c} + 2 \\sin{(F_{c})}} and \\frac{F_{c} + r{(F_{c})}}{F_{c} + r{(F_{c})} + \\sin{(F_{c})}} = \\frac{F_{c} + \\sin{(F_{c})}}{F_{c} + r{(F_{c})} + \\sin{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["add", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), sin(Symbol('F_c', commutative=True))))"], [["add", 1, "Add(Symbol('F_c', commutative=True), sin(Symbol('F_c', commutative=True)))"], "Equality(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(2), sin(Symbol('F_c', commutative=True)))))"], [["divide", 2, "Add(Symbol('F_c', commutative=True), Mul(Integer(2), sin(Symbol('F_c', commutative=True))))"], "Equality(Mul(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True))), Pow(Add(Symbol('F_c', commutative=True), Mul(Integer(2), sin(Symbol('F_c', commutative=True)))), Integer(-1))), Mul(Add(Symbol('F_c', commutative=True), sin(Symbol('F_c', commutative=True))), Pow(Add(Symbol('F_c', commutative=True), Mul(Integer(2), sin(Symbol('F_c', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True))), Pow(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True))), Integer(-1))), Mul(Add(Symbol('F_c', commutative=True), sin(Symbol('F_c', commutative=True))), Pow(Add(Symbol('F_c', commutative=True), Function('r')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} and \\omega{(M_{E})} = \\sin{(\\cos{(M_{E})})}, then obtain \\frac{g^{\\prime}_{\\varepsilon} \\omega{(M_{E})}}{\\int \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon}} = \\frac{g^{\\prime}_{\\varepsilon} \\sin{(\\cos{(M_{E})})}}{\\int \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(g^{\\prime}_{\\varepsilon})} = \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} and \\omega{(M_{E})} = \\sin{(\\cos{(M_{E})})} and \\frac{g^{\\prime}_{\\varepsilon} \\omega{(M_{E})}}{\\int \\operatorname{f_{\\mathbf{p}}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon}} = \\frac{g^{\\prime}_{\\varepsilon} \\sin{(\\cos{(M_{E})})}}{\\int \\operatorname{f_{\\mathbf{p}}}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon}} and \\frac{g^{\\prime}_{\\varepsilon} \\omega{(M_{E})}}{\\int \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon}} = \\frac{g^{\\prime}_{\\varepsilon} \\sin{(\\cos{(M_{E})})}}{\\int \\cos{(\\cos{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\omega')(Symbol('M_E', commutative=True)), sin(cos(Symbol('M_E', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\omega')(Symbol('M_E', commutative=True)), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(-1))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), sin(cos(Symbol('M_E', commutative=True))), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\omega')(Symbol('M_E', commutative=True)), Pow(Integral(cos(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(-1))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), sin(cos(Symbol('M_E', commutative=True))), Pow(Integral(cos(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given E{(\\phi,\\sigma_p,A_{1})} = A_{1} + \\phi - \\sigma_p, then obtain - A_{1} + \\frac{E^{\\sigma_p}{(\\phi,\\sigma_p,A_{1})}}{\\phi} = - A_{1} + \\frac{(A_{1} + \\phi - \\sigma_p)^{\\sigma_p}}{\\phi}", "derivation": "E{(\\phi,\\sigma_p,A_{1})} = A_{1} + \\phi - \\sigma_p and E^{\\sigma_p}{(\\phi,\\sigma_p,A_{1})} = (A_{1} + \\phi - \\sigma_p)^{\\sigma_p} and \\frac{E^{\\sigma_p}{(\\phi,\\sigma_p,A_{1})}}{\\phi} = \\frac{(A_{1} + \\phi - \\sigma_p)^{\\sigma_p}}{\\phi} and - A_{1} + \\frac{E^{\\sigma_p}{(\\phi,\\sigma_p,A_{1})}}{\\phi} = - A_{1} + \\frac{(A_{1} + \\phi - \\sigma_p)^{\\sigma_p}}{\\phi}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(i,M_{E})} = M_{E} - i, then obtain \\int (3 M_{E} + \\tilde{g}{(i,M_{E})})^{i} di = \\int (4 M_{E} - i)^{i} di", "derivation": "\\tilde{g}{(i,M_{E})} = M_{E} - i and M_{E} + \\tilde{g}{(i,M_{E})} = 2 M_{E} - i and 3 M_{E} + \\tilde{g}{(i,M_{E})} = 4 M_{E} - i and (3 M_{E} + \\tilde{g}{(i,M_{E})})^{i} = (4 M_{E} - i)^{i} and \\int (3 M_{E} + \\tilde{g}{(i,M_{E})})^{i} di = \\int (4 M_{E} - i)^{i} di", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(2), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["add", 2, "Mul(Integer(2), Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Integer(3), Symbol('M_E', commutative=True)), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(4), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Mul(Integer(3), Symbol('M_E', commutative=True)), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(4), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["integrate", 4, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(3), Symbol('M_E', commutative=True)), Function('\\\\tilde{g}')(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Add(Mul(Integer(4), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(z)} = z, then obtain \\int 1 d\\mathbb{I}{(z)} = \\int z^{z} \\mathbb{I}^{- z}{(z)} d\\mathbb{I}{(z)}", "derivation": "\\mathbb{I}{(z)} = z and \\mathbb{I}^{z}{(z)} = z^{z} and 1 = z^{z} \\mathbb{I}^{- z}{(z)} and \\int 1 dz = \\int z^{z} \\mathbb{I}^{- z}{(z)} dz and \\int 1 d\\mathbb{I}{(z)} = \\int z^{z} \\mathbb{I}^{- z}{(z)} d\\mathbb{I}{(z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Symbol('z', commutative=True))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Symbol('z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('z', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)))), Integral(Mul(Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given t{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain e^{- (e^{t{(\\mathbf{E})}} + e^{e^{\\mathbf{E}}})^{2} + 4 e^{2 t{(\\mathbf{E})}}} = 1", "derivation": "t{(\\mathbf{E})} = e^{\\mathbf{E}} and e^{t{(\\mathbf{E})}} = e^{e^{\\mathbf{E}}} and 2 e^{t{(\\mathbf{E})}} = e^{t{(\\mathbf{E})}} + e^{e^{\\mathbf{E}}} and 4 e^{2 t{(\\mathbf{E})}} = (e^{t{(\\mathbf{E})}} + e^{e^{\\mathbf{E}}})^{2} and - (e^{t{(\\mathbf{E})}} + e^{e^{\\mathbf{E}}})^{2} + 4 e^{2 t{(\\mathbf{E})}} = 0 and e^{- (e^{t{(\\mathbf{E})}} + e^{e^{\\mathbf{E}}})^{2} + 4 e^{2 t{(\\mathbf{E})}}} = 1", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 2, "exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Integer(2), exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True)))), Add(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), exp(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))))), Pow(Add(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(2)))"], [["minus", 4, "Pow(Add(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(2))), Mul(Integer(4), exp(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True)))))), Integer(0))"], [["exp", 5], "Equality(exp(Add(Mul(Integer(-1), Pow(Add(exp(Function('t')(Symbol('\\\\mathbf{E}', commutative=True))), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(2))), Mul(Integer(4), exp(Mul(Integer(2), Function('t')(Symbol('\\\\mathbf{E}', commutative=True))))))), Integer(1))"]]}, {"prompt": "Given W{(C,\\mathbf{r})} = - C + \\mathbf{r} and \\operatorname{f_{\\mathbf{v}}}{(C)} = - C, then obtain - \\mathbf{r} = - C - W{(C,\\mathbf{r})}", "derivation": "W{(C,\\mathbf{r})} = - C + \\mathbf{r} and \\operatorname{f_{\\mathbf{v}}}{(C)} = - C and W{(C,\\mathbf{r})} = \\mathbf{r} + \\operatorname{f_{\\mathbf{v}}}{(C)} and - \\mathbf{r} + W{(C,\\mathbf{r})} = \\operatorname{f_{\\mathbf{v}}}{(C)} and - \\mathbf{r} + W{(C,\\mathbf{r})} = - C and - \\mathbf{r} = - C - W{(C,\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True))))"], [["minus", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["minus", 5, "Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('C', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(P_{g},\\mathbf{S},n)} = P_{g} \\mathbf{S} + n and Q{(n,P_{g},\\mathbf{S})} = \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)}, then obtain \\int (P_{g} \\mathbf{S} + n)^{\\mathbf{S}} dn = \\int \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)} dn", "derivation": "\\operatorname{v_{y}}{(P_{g},\\mathbf{S},n)} = P_{g} \\mathbf{S} + n and \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)} = (P_{g} \\mathbf{S} + n)^{\\mathbf{S}} and Q{(n,P_{g},\\mathbf{S})} = \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)} and Q{(n,P_{g},\\mathbf{S})} = (P_{g} \\mathbf{S} + n)^{\\mathbf{S}} and \\int Q{(n,P_{g},\\mathbf{S})} dn = \\int \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)} dn and \\int (P_{g} \\mathbf{S} + n)^{\\mathbf{S}} dn = \\int \\operatorname{v_{y}}^{\\mathbf{S}}{(P_{g},\\mathbf{S},n)} dn", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('n', commutative=True)), Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('n', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('n', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('v_y')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('Q')(Symbol('n', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('n', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Function('v_y')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Function('v_y')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(n_{2})} = \\log{(n_{2})}, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = \\mathbf{f} + n_{2} \\log{(n_{2})} - n_{2}, then obtain - \\mathbf{f} - n_{2} \\log{(n_{2})} + n_{2} + \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = n_{2} \\operatorname{f_{\\mathbf{p}}}{(n_{2})} - n_{2} \\log{(n_{2})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(n_{2})} = \\log{(n_{2})} and \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = \\int \\log{(n_{2})} dn_{2} and \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = \\mathbf{f} + n_{2} \\log{(n_{2})} - n_{2} and \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = \\mathbf{f} + n_{2} \\operatorname{f_{\\mathbf{p}}}{(n_{2})} - n_{2} and - \\mathbf{f} - n_{2} \\log{(n_{2})} + n_{2} + \\int \\operatorname{f_{\\mathbf{p}}}{(n_{2})} dn_{2} = n_{2} \\operatorname{f_{\\mathbf{p}}}{(n_{2})} - n_{2} \\log{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('n_2', commutative=True), log(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('n_2', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["minus", 4, "Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('n_2', commutative=True), log(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True), log(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Symbol('n_2', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True), log(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\chi,\\Psi_{nl})} = \\log{(\\Psi_{nl} + \\chi)}, then obtain - \\Psi_{nl} - \\chi + \\int \\mathbf{v}{(\\chi,\\Psi_{nl})} d\\Psi_{nl} + 1 = \\Psi_{nl} \\log{(\\Psi_{nl} + \\chi)} - 2 \\Psi_{nl} + \\chi \\log{(\\Psi_{nl} + \\chi)} - \\chi + v_{2} + 1", "derivation": "\\mathbf{v}{(\\chi,\\Psi_{nl})} = \\log{(\\Psi_{nl} + \\chi)} and \\int \\mathbf{v}{(\\chi,\\Psi_{nl})} d\\Psi_{nl} = \\int \\log{(\\Psi_{nl} + \\chi)} d\\Psi_{nl} and - \\Psi_{nl} - \\chi + \\int \\mathbf{v}{(\\chi,\\Psi_{nl})} d\\Psi_{nl} = - \\Psi_{nl} - \\chi + \\int \\log{(\\Psi_{nl} + \\chi)} d\\Psi_{nl} and - \\Psi_{nl} - \\chi + \\int \\mathbf{v}{(\\chi,\\Psi_{nl})} d\\Psi_{nl} + 1 = - \\Psi_{nl} - \\chi + \\int \\log{(\\Psi_{nl} + \\chi)} d\\Psi_{nl} + 1 and - \\Psi_{nl} - \\chi + \\int \\mathbf{v}{(\\chi,\\Psi_{nl})} d\\Psi_{nl} + 1 = \\Psi_{nl} \\log{(\\Psi_{nl} + \\chi)} - 2 \\Psi_{nl} + \\chi \\log{(\\Psi_{nl} + \\chi)} - \\chi + v_{2} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('v_2', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\Omega{(V)} = e^{V} and \\dot{y}{(V)} = (e^{V})^{V}, then obtain \\Omega^{V}{(V)} = \\dot{y}{(V)}", "derivation": "\\Omega{(V)} = e^{V} and \\Omega^{V}{(V)} = (e^{V})^{V} and \\dot{y}{(V)} = (e^{V})^{V} and \\Omega^{V}{(V)} = \\dot{y}{(V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('V', commutative=True)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Omega')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Function('\\\\dot{y}')(Symbol('V', commutative=True)))"]]}, {"prompt": "Given a{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and \\mathbf{E}{(v_{x})} = v_{x}^{2} a{(v_{x})} \\int \\cos{(v_{x})} dv_{x}, then obtain \\mathbf{E}^{v_{x}}{(v_{x})} = (v_{x}^{2} a^{2}{(v_{x})})^{v_{x}}", "derivation": "a{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and v_{x} a{(v_{x})} = v_{x} \\int \\cos{(v_{x})} dv_{x} and v_{x}^{2} a{(v_{x})} \\int \\cos{(v_{x})} dv_{x} = v_{x}^{2} (\\int \\cos{(v_{x})} dv_{x})^{2} and \\mathbf{E}{(v_{x})} = v_{x}^{2} a{(v_{x})} \\int \\cos{(v_{x})} dv_{x} and \\mathbf{E}{(v_{x})} = v_{x}^{2} (\\int \\cos{(v_{x})} dv_{x})^{2} and \\mathbf{E}^{v_{x}}{(v_{x})} = (v_{x}^{2} (\\int \\cos{(v_{x})} dv_{x})^{2})^{v_{x}} and \\mathbf{E}^{v_{x}}{(v_{x})} = (v_{x}^{2} a^{2}{(v_{x})})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["times", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Function('a')(Symbol('v_x', commutative=True))), Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["times", 2, "Mul(Symbol('v_x', commutative=True), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Function('a')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Pow(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Function('a')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Pow(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(2))))"], [["power", 5, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Pow(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integer(2))), Symbol('v_x', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Mul(Pow(Symbol('v_x', commutative=True), Integer(2)), Pow(Function('a')(Symbol('v_x', commutative=True)), Integer(2))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(l,\\eta)} = \\eta l, then derive \\frac{\\partial}{\\partial \\eta} \\mathbf{B}{(l,\\eta)} = l, then obtain l^{l} = (\\frac{\\partial}{\\partial \\eta} \\eta l)^{l}", "derivation": "\\mathbf{B}{(l,\\eta)} = \\eta l and \\frac{\\partial}{\\partial \\eta} \\mathbf{B}{(l,\\eta)} = \\frac{\\partial}{\\partial \\eta} \\eta l and \\frac{\\partial}{\\partial \\eta} \\mathbf{B}{(l,\\eta)} = l and l = \\frac{\\partial}{\\partial \\eta} \\eta l and l^{l} = (\\frac{\\partial}{\\partial \\eta} \\eta l)^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('l', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('l', commutative=True), Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Symbol('l', commutative=True), Symbol('l', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{J})} = \\mathbf{J}, then derive \\frac{d}{d \\mathbf{J}} \\operatorname{v_{2}}{(\\mathbf{J})} = 1, then obtain \\frac{d}{d \\mathbf{J}} \\mathbf{J} = 1", "derivation": "\\operatorname{v_{2}}{(\\mathbf{J})} = \\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\operatorname{v_{2}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\operatorname{v_{2}}{(\\mathbf{J})} = 1 and \\frac{d}{d \\mathbf{J}} \\mathbf{J} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{J}', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\mathbf{J}', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(E_{x})} = \\cos{(E_{x})}, then derive A_{z} + E_{x} + \\operatorname{A_{2}}{(E_{x})} = E_{x} + \\Omega + \\cos{(E_{x})}, then obtain \\int (A_{z} + E_{x} + \\cos{(E_{x})}) d\\Omega = \\int (E_{x} + \\Omega + \\cos{(E_{x})}) d\\Omega", "derivation": "\\operatorname{A_{2}}{(E_{x})} = \\cos{(E_{x})} and E_{x} + \\operatorname{A_{2}}{(E_{x})} = E_{x} + \\cos{(E_{x})} and \\frac{d}{d E_{x}} (E_{x} + \\operatorname{A_{2}}{(E_{x})}) = \\frac{d}{d E_{x}} (E_{x} + \\cos{(E_{x})}) and \\int \\frac{d}{d E_{x}} (E_{x} + \\operatorname{A_{2}}{(E_{x})}) dE_{x} = \\int \\frac{d}{d E_{x}} (E_{x} + \\cos{(E_{x})}) dE_{x} and A_{z} + E_{x} + \\operatorname{A_{2}}{(E_{x})} = E_{x} + \\Omega + \\cos{(E_{x})} and A_{z} + E_{x} + \\cos{(E_{x})} = E_{x} + \\Omega + \\cos{(E_{x})} and \\int (A_{z} + E_{x} + \\cos{(E_{x})}) d\\Omega = \\int (E_{x} + \\Omega + \\cos{(E_{x})}) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["add", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Function('A_2')(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), cos(Symbol('E_x', commutative=True))))"], [["differentiate", 2, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Add(Symbol('E_x', commutative=True), Function('A_2')(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Add(Symbol('E_x', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('E_x', commutative=True), Function('A_2')(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(Add(Symbol('E_x', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), Function('A_2')(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('\\\\Omega', commutative=True), cos(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), cos(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('\\\\Omega', commutative=True), cos(Symbol('E_x', commutative=True))))"], [["integrate", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Symbol('\\\\Omega', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(I)} = e^{e^{I}}, then obtain \\sigma_{p}^{3}{(I)} e^{e^{I}} = \\sigma_{p}^{2}{(I)} e^{2 e^{I}}", "derivation": "\\sigma_{p}{(I)} = e^{e^{I}} and \\sigma_{p}{(I)} e^{e^{I}} = e^{2 e^{I}} and \\sigma_{p}^{2}{(I)} e^{2 e^{I}} = e^{4 e^{I}} and \\sigma_{p}^{3}{(I)} e^{e^{I}} = \\sigma_{p}^{2}{(I)} e^{2 e^{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('I', commutative=True)), exp(exp(Symbol('I', commutative=True))))"], [["times", 1, "exp(exp(Symbol('I', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('I', commutative=True)), exp(exp(Symbol('I', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('I', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('I', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('I', commutative=True))))), exp(Mul(Integer(4), exp(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('I', commutative=True)), Integer(3)), exp(exp(Symbol('I', commutative=True)))), Mul(Pow(Function('\\\\sigma_p')(Symbol('I', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given a{(\\mathbf{H},Q,L)} = \\frac{L Q}{\\mathbf{H}}, then obtain (f_{E} - \\int - L d\\mathbf{H} - \\int \\frac{L Q}{\\mathbf{H}} d\\mathbf{H} - \\int - a{(\\mathbf{H},Q,L)} d\\mathbf{H})^{Q} = (\\int L d\\mathbf{H})^{Q}", "derivation": "a{(\\mathbf{H},Q,L)} = \\frac{L Q}{\\mathbf{H}} and L + a{(\\mathbf{H},Q,L)} = \\frac{L Q}{\\mathbf{H}} + L and - \\frac{L Q}{\\mathbf{H}} + L + a{(\\mathbf{H},Q,L)} = L and \\int (- \\frac{L Q}{\\mathbf{H}} + L + a{(\\mathbf{H},Q,L)}) d\\mathbf{H} = \\int L d\\mathbf{H} and (\\int (- \\frac{L Q}{\\mathbf{H}} + L + a{(\\mathbf{H},Q,L)}) d\\mathbf{H})^{Q} = (\\int L d\\mathbf{H})^{Q} and (f_{E} - \\int - L d\\mathbf{H} - \\int \\frac{L Q}{\\mathbf{H}} d\\mathbf{H} - \\int - a{(\\mathbf{H},Q,L)} d\\mathbf{H})^{Q} = (\\int L d\\mathbf{H})^{Q}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('L', commutative=True)))"], [["minus", 2, "Mul(Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('L', commutative=True), Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('L', commutative=True), Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Symbol('L', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Symbol('L', commutative=True), Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('Q', commutative=True)), Pow(Integral(Symbol('L', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('Q', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('L', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('L', commutative=True), Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Function('a')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Q', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Symbol('Q', commutative=True)), Pow(Integral(Symbol('L', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(M,F_{H})} = M^{F_{H}} and W{(M,F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})}, then derive W^{M}{(M,F_{H})} = (M^{F_{H}} \\log{(M)})^{M}, then obtain \\int (\\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})})^{M} dF_{H} = \\int (M^{F_{H}} \\log{(M)})^{M} dF_{H}", "derivation": "\\operatorname{f^{\\prime}}{(M,F_{H})} = M^{F_{H}} and \\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})} = \\frac{\\partial}{\\partial F_{H}} M^{F_{H}} and (\\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})})^{M} = (\\frac{\\partial}{\\partial F_{H}} M^{F_{H}})^{M} and W{(M,F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})} and W^{M}{(M,F_{H})} = (\\frac{\\partial}{\\partial F_{H}} M^{F_{H}})^{M} and W^{M}{(M,F_{H})} = (M^{F_{H}} \\log{(M)})^{M} and \\int W^{M}{(M,F_{H})} dF_{H} = \\int (M^{F_{H}} \\log{(M)})^{M} dF_{H} and \\int (\\frac{\\partial}{\\partial F_{H}} \\operatorname{f^{\\prime}}{(M,F_{H})})^{M} dF_{H} = \\int (M^{F_{H}} \\log{(M)})^{M} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('W')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('W')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Symbol('M', commutative=True)), Pow(Derivative(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Function('W')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), log(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["integrate", 6, "Symbol('F_H', commutative=True)"], "Equality(Integral(Pow(Function('W')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Mul(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), log(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Integral(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('M', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Mul(Pow(Symbol('M', commutative=True), Symbol('F_H', commutative=True)), log(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})} and \\operatorname{F_{x}}{(\\lambda)} = \\cos^{\\lambda}{(\\cos{(\\lambda)})}, then obtain - \\Psi_{nl}{(\\lambda)} + \\Psi_{nl}^{\\lambda}{(\\lambda)} = \\operatorname{F_{x}}{(\\lambda)} - \\Psi_{nl}{(\\lambda)}", "derivation": "\\Psi_{nl}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})} and \\Psi_{nl}^{\\lambda}{(\\lambda)} = \\cos^{\\lambda}{(\\cos{(\\lambda)})} and - \\Psi_{nl}{(\\lambda)} + \\Psi_{nl}^{\\lambda}{(\\lambda)} = - \\Psi_{nl}{(\\lambda)} + \\cos^{\\lambda}{(\\cos{(\\lambda)})} and \\operatorname{F_{x}}{(\\lambda)} = \\cos^{\\lambda}{(\\cos{(\\lambda)})} and - \\Psi_{nl}{(\\lambda)} + \\Psi_{nl}^{\\lambda}{(\\lambda)} = \\operatorname{F_{x}}{(\\lambda)} - \\Psi_{nl}{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["minus", 2, "Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True))), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True))), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\lambda', commutative=True)), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True))), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Add(Function('F_x')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\varepsilon_0,\\Omega)} = \\frac{\\Omega}{\\varepsilon_0} and \\varepsilon{(\\varepsilon_0,\\Omega)} = \\frac{\\Omega}{\\varepsilon_0}, then obtain - \\dot{y}^{2}{(\\varepsilon_0,\\Omega)} + \\varepsilon{(\\varepsilon_0,\\Omega)} = - \\dot{y}^{2}{(\\varepsilon_0,\\Omega)} + \\dot{y}{(\\varepsilon_0,\\Omega)}", "derivation": "\\dot{y}{(\\varepsilon_0,\\Omega)} = \\frac{\\Omega}{\\varepsilon_0} and \\dot{y}^{2}{(\\varepsilon_0,\\Omega)} = \\frac{\\Omega \\dot{y}{(\\varepsilon_0,\\Omega)}}{\\varepsilon_0} and \\varepsilon{(\\varepsilon_0,\\Omega)} = \\frac{\\Omega}{\\varepsilon_0} and \\varepsilon{(\\varepsilon_0,\\Omega)} = \\dot{y}{(\\varepsilon_0,\\Omega)} and - \\frac{\\Omega \\dot{y}{(\\varepsilon_0,\\Omega)}}{\\varepsilon_0} + \\varepsilon{(\\varepsilon_0,\\Omega)} = - \\frac{\\Omega \\dot{y}{(\\varepsilon_0,\\Omega)}}{\\varepsilon_0} + \\dot{y}{(\\varepsilon_0,\\Omega)} and - \\dot{y}^{2}{(\\varepsilon_0,\\Omega)} + \\varepsilon{(\\varepsilon_0,\\Omega)} = - \\dot{y}^{2}{(\\varepsilon_0,\\Omega)} + \\dot{y}{(\\varepsilon_0,\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given W{(r_{0})} = \\cos{(r_{0})}, then derive \\int \\frac{W{(r_{0})}}{r_{0}} dr_{0} = f^{*} - \\log{(r_{0})} + \\frac{\\log{(r_{0}^{2})}}{2} + \\operatorname{Ci}{(r_{0})}, then obtain \\iint \\frac{W{(r_{0})}}{r_{0}} dr_{0} df^{*} = \\int (f^{*} - \\log{(r_{0})} + \\frac{\\log{(r_{0}^{2})}}{2} + \\operatorname{Ci}{(r_{0})}) df^{*}", "derivation": "W{(r_{0})} = \\cos{(r_{0})} and \\frac{W{(r_{0})}}{r_{0}} = \\frac{\\cos{(r_{0})}}{r_{0}} and \\int \\frac{W{(r_{0})}}{r_{0}} dr_{0} = \\int \\frac{\\cos{(r_{0})}}{r_{0}} dr_{0} and \\int \\frac{W{(r_{0})}}{r_{0}} dr_{0} = f^{*} - \\log{(r_{0})} + \\frac{\\log{(r_{0}^{2})}}{2} + \\operatorname{Ci}{(r_{0})} and \\iint \\frac{W{(r_{0})}}{r_{0}} dr_{0} df^{*} = \\int (f^{*} - \\log{(r_{0})} + \\frac{\\log{(r_{0}^{2})}}{2} + \\operatorname{Ci}{(r_{0})}) df^{*}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('W')(Symbol('r_0', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), cos(Symbol('r_0', commutative=True))))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('W')(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), cos(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('W')(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), log(Symbol('r_0', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('r_0', commutative=True), Integer(2)))), Ci(Symbol('r_0', commutative=True))))"], [["integrate", 4, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('W')(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), log(Symbol('r_0', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('r_0', commutative=True), Integer(2)))), Ci(Symbol('r_0', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given L{(\\phi_1,z^{*})} = \\frac{\\sin{(z^{*})}}{\\phi_1}, then derive - \\frac{\\partial}{\\partial \\phi_1} L{(\\phi_1,z^{*})} = \\frac{\\sin{(z^{*})}}{\\phi_1^{2}}, then obtain - \\frac{\\frac{\\partial}{\\partial \\phi_1} \\frac{\\sin{(z^{*})}}{\\phi_1}}{\\phi_1^{2}} = \\frac{\\sin{(z^{*})}}{\\phi_1^{4}}", "derivation": "L{(\\phi_1,z^{*})} = \\frac{\\sin{(z^{*})}}{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} L{(\\phi_1,z^{*})} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\sin{(z^{*})}}{\\phi_1} and - \\frac{\\partial}{\\partial \\phi_1} L{(\\phi_1,z^{*})} = - \\frac{\\partial}{\\partial \\phi_1} \\frac{\\sin{(z^{*})}}{\\phi_1} and - \\frac{\\partial}{\\partial \\phi_1} L{(\\phi_1,z^{*})} = \\frac{\\sin{(z^{*})}}{\\phi_1^{2}} and - \\frac{\\partial}{\\partial \\phi_1} \\frac{\\sin{(z^{*})}}{\\phi_1} = \\frac{\\sin{(z^{*})}}{\\phi_1^{2}} and - \\frac{\\frac{\\partial}{\\partial \\phi_1} \\frac{\\sin{(z^{*})}}{\\phi_1}}{\\phi_1^{2}} = \\frac{\\sin{(z^{*})}}{\\phi_1^{4}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\phi_1', commutative=True), Symbol('z^*', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), sin(Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\phi_1', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\phi_1', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\phi_1', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), sin(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), sin(Symbol('z^*', commutative=True))))"], [["times", 5, "Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-4)), sin(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mathbf{g},\\varepsilon)} = \\mathbf{g} + \\varepsilon, then obtain \\frac{(\\operatorname{g_{\\varepsilon}}^{\\mathbf{g}}{(\\mathbf{g},\\varepsilon)})^{\\mathbf{g}}}{\\varepsilon} = \\frac{((\\mathbf{g} + \\varepsilon)^{\\mathbf{g}})^{\\mathbf{g}}}{\\varepsilon}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mathbf{g},\\varepsilon)} = \\mathbf{g} + \\varepsilon and \\operatorname{g_{\\varepsilon}}^{\\mathbf{g}}{(\\mathbf{g},\\varepsilon)} = (\\mathbf{g} + \\varepsilon)^{\\mathbf{g}} and (\\operatorname{g_{\\varepsilon}}^{\\mathbf{g}}{(\\mathbf{g},\\varepsilon)})^{\\mathbf{g}} = ((\\mathbf{g} + \\varepsilon)^{\\mathbf{g}})^{\\mathbf{g}} and \\frac{(\\operatorname{g_{\\varepsilon}}^{\\mathbf{g}}{(\\mathbf{g},\\varepsilon)})^{\\mathbf{g}}}{\\varepsilon} = \\frac{((\\mathbf{g} + \\varepsilon)^{\\mathbf{g}})^{\\mathbf{g}}}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\theta_1,B)} = \\log{(- B + \\theta_1)}, then obtain \\int \\frac{\\theta{(\\theta_1,B)}}{\\theta_1 \\log{(- B + \\theta_1)}} dB = \\int \\frac{1}{\\theta_1} dB", "derivation": "\\theta{(\\theta_1,B)} = \\log{(- B + \\theta_1)} and \\frac{\\theta{(\\theta_1,B)}}{\\theta_1} = \\frac{\\log{(- B + \\theta_1)}}{\\theta_1} and \\frac{\\theta{(\\theta_1,B)}}{\\theta_1 \\log{(- B + \\theta_1)}} = \\frac{1}{\\theta_1} and \\int \\frac{\\theta{(\\theta_1,B)}}{\\theta_1 \\log{(- B + \\theta_1)}} dB = \\int \\frac{1}{\\theta_1} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), log(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\theta_1', commutative=True)))))"], [["divide", 2, "log(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Tuple(Symbol('B', commutative=True))), Integral(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\pi)} = \\int \\log{(\\pi)} d\\pi and h{(\\pi)} = \\frac{\\mathbf{S}{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} and \\mathbf{g}{(\\pi)} = (\\mathbf{S}{(\\pi)} + h{(\\pi)})^{\\pi}, then obtain \\mathbf{g}{(\\pi)} = (\\mathbf{S}{(\\pi)} + 1)^{\\pi}", "derivation": "\\mathbf{S}{(\\pi)} = \\int \\log{(\\pi)} d\\pi and \\frac{\\mathbf{S}{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} = 1 and h{(\\pi)} = \\frac{\\mathbf{S}{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} and h{(\\pi)} = 1 and \\mathbf{S}{(\\pi)} + h{(\\pi)} = \\mathbf{S}{(\\pi)} + 1 and (\\mathbf{S}{(\\pi)} + h{(\\pi)})^{\\pi} = (\\mathbf{S}{(\\pi)} + 1)^{\\pi} and \\mathbf{g}{(\\pi)} = (\\mathbf{S}{(\\pi)} + h{(\\pi)})^{\\pi} and \\mathbf{g}{(\\pi)} = (\\mathbf{S}{(\\pi)} + 1)^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\pi', commutative=True)), Mul(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('h')(Symbol('\\\\pi', commutative=True)), Integer(1))"], [["add", 4, "Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Function('h')(Symbol('\\\\pi', commutative=True))), Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Integer(1)))"], [["power", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Function('h')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Integer(1)), Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\pi', commutative=True)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Function('h')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\pi', commutative=True)), Pow(Add(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Integer(1)), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(F_{N})} = \\sin{(F_{N})}, then derive \\int (F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + v - \\cos{(F_{N})}, then obtain - \\operatorname{J_{\\varepsilon}}{(F_{N})} + \\int (F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + v - \\operatorname{J_{\\varepsilon}}{(F_{N})} - \\cos{(F_{N})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(F_{N})} = \\sin{(F_{N})} and F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})} = F_{N} + \\sin{(F_{N})} and \\int (F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})}) dF_{N} = \\int (F_{N} + \\sin{(F_{N})}) dF_{N} and \\int (F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + v - \\cos{(F_{N})} and - \\operatorname{J_{\\varepsilon}}{(F_{N})} + \\int (F_{N} + \\operatorname{J_{\\varepsilon}}{(F_{N})}) dF_{N} = \\frac{F_{N}^{2}}{2} + v - \\operatorname{J_{\\varepsilon}}{(F_{N})} - \\cos{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Symbol('F_N', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('F_N', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))))"], [["minus", 4, "Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Symbol('v', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(S,x^\\prime)} = \\frac{x^\\prime}{S}, then obtain \\frac{\\partial}{\\partial x^\\prime} \\frac{S \\mathbf{J}_f{(S,x^\\prime)}}{x^\\prime} = \\frac{d}{d x^\\prime} 1", "derivation": "\\mathbf{J}_f{(S,x^\\prime)} = \\frac{x^\\prime}{S} and x^\\prime \\mathbf{J}_f{(S,x^\\prime)} = \\frac{(x^\\prime)^{2}}{S} and \\frac{S \\mathbf{J}_f{(S,x^\\prime)}}{x^\\prime} = 1 and \\frac{\\partial}{\\partial x^\\prime} \\frac{S \\mathbf{J}_f{(S,x^\\prime)}}{x^\\prime} = \\frac{d}{d x^\\prime} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["divide", 2, "Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))"], "Equality(Mul(Symbol('S', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integer(1))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} = \\frac{\\mathbf{M}}{L_{\\varepsilon}}, then obtain \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\dot{x}{(\\mathbf{M},L_{\\varepsilon})}}{L_{\\varepsilon}} - \\frac{1}{L_{\\varepsilon}^{2}} = \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\mathbf{M}}{L_{\\varepsilon}^{2}} - \\frac{1}{L_{\\varepsilon}^{2}}", "derivation": "\\dot{x}{(\\mathbf{M},L_{\\varepsilon})} = \\frac{\\mathbf{M}}{L_{\\varepsilon}} and \\frac{\\dot{x}{(\\mathbf{M},L_{\\varepsilon})}}{L_{\\varepsilon}} = \\frac{\\mathbf{M}}{L_{\\varepsilon}^{2}} and \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\dot{x}{(\\mathbf{M},L_{\\varepsilon})}}{L_{\\varepsilon}} = \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\mathbf{M}}{L_{\\varepsilon}^{2}} and \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\dot{x}{(\\mathbf{M},L_{\\varepsilon})}}{L_{\\varepsilon}} - \\frac{1}{L_{\\varepsilon}^{2}} = \\dot{x}{(\\mathbf{M},L_{\\varepsilon})} + \\frac{\\mathbf{M}}{L_{\\varepsilon}^{2}} - \\frac{1}{L_{\\varepsilon}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 1, "Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 2, "Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 3, "Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2)))), Add(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given L{(H)} = e^{H} and \\mathbf{P}{(H)} = (e^{H})^{H}, then derive \\frac{d}{d H} \\mathbf{P}{(H)} = (H + \\log{(e^{H})}) (e^{H})^{H}, then obtain H + (H + \\log{(e^{H})}) (e^{H})^{H} + \\log{(e^{H})} = H + (H + \\log{(e^{H})}) \\mathbf{P}{(H)} + \\log{(e^{H})}", "derivation": "L{(H)} = e^{H} and \\mathbf{P}{(H)} = (e^{H})^{H} and \\frac{d}{d H} \\mathbf{P}{(H)} = \\frac{d}{d H} (e^{H})^{H} and \\frac{d}{d H} \\mathbf{P}{(H)} = \\frac{d}{d H} L^{H}{(H)} and \\frac{d}{d H} \\mathbf{P}{(H)} = (H + \\log{(e^{H})}) (e^{H})^{H} and (H + \\log{(e^{H})}) (e^{H})^{H} = \\frac{d}{d H} L^{H}{(H)} and H + (H + \\log{(e^{H})}) (e^{H})^{H} + \\log{(e^{H})} = H + \\log{(e^{H})} + \\frac{d}{d H} L^{H}{(H)} and (H + \\log{(e^{H})}) \\mathbf{P}{(H)} = \\frac{d}{d H} L^{H}{(H)} and H + (H + \\log{(e^{H})}) (e^{H})^{H} + \\log{(e^{H})} = H + (H + \\log{(e^{H})}) \\mathbf{P}{(H)} + \\log{(e^{H})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('H', commutative=True)), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Function('L')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Derivative(Pow(Function('L')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 6, "Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True))))"], "Equality(Add(Symbol('H', commutative=True), Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), log(exp(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True))), Derivative(Pow(Function('L')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Function('\\\\mathbf{P}')(Symbol('H', commutative=True))), Derivative(Pow(Function('L')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Add(Symbol('H', commutative=True), Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Pow(exp(Symbol('H', commutative=True)), Symbol('H', commutative=True))), log(exp(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), Mul(Add(Symbol('H', commutative=True), log(exp(Symbol('H', commutative=True)))), Function('\\\\mathbf{P}')(Symbol('H', commutative=True))), log(exp(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given c{(\\mathbf{B},n)} = \\mathbf{B} n, then derive - \\mathbf{B} + \\frac{\\partial}{\\partial n} c{(\\mathbf{B},n)} = 0, then obtain \\frac{\\partial^{2}}{\\partial n^{2}} (- \\mathbf{B} + \\frac{\\partial}{\\partial n} \\mathbf{B} n) = \\frac{d^{2}}{d n^{2}} 0", "derivation": "c{(\\mathbf{B},n)} = \\mathbf{B} n and \\frac{\\partial}{\\partial n} c{(\\mathbf{B},n)} = \\frac{\\partial}{\\partial n} \\mathbf{B} n and - \\mathbf{B} + \\frac{\\partial}{\\partial n} c{(\\mathbf{B},n)} = - \\mathbf{B} + \\frac{\\partial}{\\partial n} \\mathbf{B} n and - \\mathbf{B} + \\frac{\\partial}{\\partial n} c{(\\mathbf{B},n)} = 0 and - \\mathbf{B} + \\frac{\\partial}{\\partial n} \\mathbf{B} n = 0 and \\frac{\\partial}{\\partial n} (- \\mathbf{B} + \\frac{\\partial}{\\partial n} \\mathbf{B} n) = \\frac{d}{d n} 0 and \\frac{\\partial^{2}}{\\partial n^{2}} (- \\mathbf{B} + \\frac{\\partial}{\\partial n} \\mathbf{B} n) = \\frac{d^{2}}{d n^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('c')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('c')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(2))))"]]}, {"prompt": "Given Z{(n_{1},\\sigma_x)} = \\sigma_x n_{1}, then obtain \\sigma_x^{2} n_{1}^{2} + (- \\sigma_x n_{1} Z{(n_{1},\\sigma_x)})^{\\sigma_x} = \\sigma_x^{2} n_{1}^{2} + (- \\sigma_x^{2} n_{1}^{2})^{\\sigma_x}", "derivation": "Z{(n_{1},\\sigma_x)} = \\sigma_x n_{1} and - \\sigma_x n_{1} Z{(n_{1},\\sigma_x)} = - \\sigma_x^{2} n_{1}^{2} and (- \\sigma_x n_{1} Z{(n_{1},\\sigma_x)})^{\\sigma_x} = (- \\sigma_x^{2} n_{1}^{2})^{\\sigma_x} and \\sigma_x^{2} n_{1}^{2} + (- \\sigma_x n_{1} Z{(n_{1},\\sigma_x)})^{\\sigma_x} = \\sigma_x^{2} n_{1}^{2} + (- \\sigma_x^{2} n_{1}^{2})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('n_1', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2)))"], "Equality(Add(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Pow(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('n_1', commutative=True), Function('Z')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Pow(Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('n_1', commutative=True), Integer(2))), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given k{(n_{1},\\mathbb{I})} = \\log{(\\mathbb{I} + n_{1})} and \\mathbf{J}_M{(n_{1},\\mathbb{I})} = \\log{(\\mathbb{I} + n_{1})}^{n_{1}}, then obtain k^{n_{1}}{(n_{1},\\mathbb{I})} = \\mathbf{J}_M{(n_{1},\\mathbb{I})}", "derivation": "k{(n_{1},\\mathbb{I})} = \\log{(\\mathbb{I} + n_{1})} and k^{n_{1}}{(n_{1},\\mathbb{I})} = \\log{(\\mathbb{I} + n_{1})}^{n_{1}} and \\mathbf{J}_M{(n_{1},\\mathbb{I})} = \\log{(\\mathbb{I} + n_{1})}^{n_{1}} and k^{n_{1}}{(n_{1},\\mathbb{I})} = \\mathbf{J}_M{(n_{1},\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('n_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), log(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n_1', commutative=True))))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('k')(Symbol('n_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('n_1', commutative=True)), Pow(log(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('n_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(log(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('k')(Symbol('n_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('n_1', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('n_1', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(C)} = \\cos{(\\sin{(C)})}, then derive \\frac{d}{d C} \\ddot{x}{(C)} = - \\sin{(\\sin{(C)})} \\cos{(C)}, then obtain - \\sin{(\\sin{(C)})} \\cos{(C)} \\int \\frac{d}{d C} \\cos{(\\sin{(C)})} dC = - \\sin{(\\sin{(C)})} \\cos{(C)} \\int - \\sin{(\\sin{(C)})} \\cos{(C)} dC", "derivation": "\\ddot{x}{(C)} = \\cos{(\\sin{(C)})} and \\frac{d}{d C} \\ddot{x}{(C)} = \\frac{d}{d C} \\cos{(\\sin{(C)})} and \\frac{d}{d C} \\ddot{x}{(C)} = - \\sin{(\\sin{(C)})} \\cos{(C)} and \\frac{d}{d C} \\cos{(\\sin{(C)})} = - \\sin{(\\sin{(C)})} \\cos{(C)} and \\int \\frac{d}{d C} \\cos{(\\sin{(C)})} dC = \\int - \\sin{(\\sin{(C)})} \\cos{(C)} dC and \\frac{d}{d C} \\cos{(\\sin{(C)})} \\int \\frac{d}{d C} \\cos{(\\sin{(C)})} dC = \\frac{d}{d C} \\cos{(\\sin{(C)})} \\int - \\sin{(\\sin{(C)})} \\cos{(C)} dC and - \\sin{(\\sin{(C)})} \\cos{(C)} \\int \\frac{d}{d C} \\cos{(\\sin{(C)})} dC = - \\sin{(\\sin{(C)})} \\cos{(C)} \\int - \\sin{(\\sin{(C)})} \\cos{(C)} dC", "srepr_derivation": [["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["divide", 5, "Pow(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))"], "Equality(Mul(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True)), Integral(Derivative(cos(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True)), Integral(Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(U,\\mu_0)} = - \\sin{(U - \\mu_0)} and f{(\\dot{x},\\hat{H}_{\\lambda})} = \\dot{x} \\cos{(\\hat{H}_{\\lambda})} and \\operatorname{F_{N}}{(U,\\mu_0)} = \\log{(\\mathbf{F}{(U,\\mu_0)})}, then obtain \\dot{x} \\operatorname{F_{N}}{(U,\\mu_0)} \\cos{(\\hat{H}_{\\lambda})} = \\dot{x} \\log{(- \\sin{(U - \\mu_0)})} \\cos{(\\hat{H}_{\\lambda})}", "derivation": "\\mathbf{F}{(U,\\mu_0)} = - \\sin{(U - \\mu_0)} and \\log{(\\mathbf{F}{(U,\\mu_0)})} = \\log{(- \\sin{(U - \\mu_0)})} and f{(\\dot{x},\\hat{H}_{\\lambda})} = \\dot{x} \\cos{(\\hat{H}_{\\lambda})} and \\operatorname{F_{N}}{(U,\\mu_0)} = \\log{(\\mathbf{F}{(U,\\mu_0)})} and \\operatorname{F_{N}}{(U,\\mu_0)} f{(\\dot{x},\\hat{H}_{\\lambda})} = f{(\\dot{x},\\hat{H}_{\\lambda})} \\log{(\\mathbf{F}{(U,\\mu_0)})} and \\operatorname{F_{N}}{(U,\\mu_0)} f{(\\dot{x},\\hat{H}_{\\lambda})} = f{(\\dot{x},\\hat{H}_{\\lambda})} \\log{(- \\sin{(U - \\mu_0)})} and \\dot{x} \\operatorname{F_{N}}{(U,\\mu_0)} \\cos{(\\hat{H}_{\\lambda})} = \\dot{x} \\log{(- \\sin{(U - \\mu_0)})} \\cos{(\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))))"], [["log", 1], "Equality(log(Function('\\\\mathbf{F}')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True))), log(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))))"], ["get_premise", "Equality(Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Function('\\\\mathbf{F}')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["times", 4, "Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Function('\\\\mathbf{F}')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('F_N')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Function('f')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Function('F_N')(Symbol('U', commutative=True), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\dot{x}', commutative=True), log(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\varepsilon)} = \\sin{(\\sin{(\\varepsilon)})}, then obtain \\cos{(\\sin{(2 \\omega{(\\varepsilon)})})} = \\cos{(\\sin{(\\omega{(\\varepsilon)} + \\sin{(\\sin{(\\varepsilon)})})})}", "derivation": "\\omega{(\\varepsilon)} = \\sin{(\\sin{(\\varepsilon)})} and 2 \\omega{(\\varepsilon)} = \\omega{(\\varepsilon)} + \\sin{(\\sin{(\\varepsilon)})} and \\sin{(2 \\omega{(\\varepsilon)})} = \\sin{(\\omega{(\\varepsilon)} + \\sin{(\\sin{(\\varepsilon)})})} and \\cos{(\\sin{(2 \\omega{(\\varepsilon)})})} = \\cos{(\\sin{(\\omega{(\\varepsilon)} + \\sin{(\\sin{(\\varepsilon)})})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True)), sin(sin(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True))), Add(Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True)), sin(sin(Symbol('\\\\varepsilon', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True)))), sin(Add(Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True)), sin(sin(Symbol('\\\\varepsilon', commutative=True))))))"], [["cos", 3], "Equality(cos(sin(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True))))), cos(sin(Add(Function('\\\\omega')(Symbol('\\\\varepsilon', commutative=True)), sin(sin(Symbol('\\\\varepsilon', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(E_{x})} = E_{x}, then obtain \\int - \\sin{(\\operatorname{V_{\\mathbf{E}}}{(E_{x})})} dE_{x} = \\int - \\sin{(E_{x})} dE_{x}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(E_{x})} = E_{x} and \\sin{(\\operatorname{V_{\\mathbf{E}}}{(E_{x})})} = \\sin{(E_{x})} and - \\sin{(\\operatorname{V_{\\mathbf{E}}}{(E_{x})})} = - \\sin{(E_{x})} and \\int - \\sin{(\\operatorname{V_{\\mathbf{E}}}{(E_{x})})} dE_{x} = \\int - \\sin{(E_{x})} dE_{x}", "srepr_derivation": [["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], [["sin", 1], "Equality(sin(Function('V_{\\\\mathbf{E}}')(Symbol('E_x', commutative=True))), sin(Symbol('E_x', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(Function('V_{\\\\mathbf{E}}')(Symbol('E_x', commutative=True)))), Mul(Integer(-1), sin(Symbol('E_x', commutative=True))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Function('V_{\\\\mathbf{E}}')(Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{A})} = \\log{(\\mathbf{A})}, then derive \\frac{d}{d \\mathbf{A}} \\operatorname{f_{E}}{(\\mathbf{A})} = \\frac{1}{\\mathbf{A}}, then obtain \\operatorname{f_{E}}{(\\mathbf{A})} \\frac{d}{d \\mathbf{A}} \\log{(\\mathbf{A})} = \\frac{\\operatorname{f_{E}}{(\\mathbf{A})}}{\\mathbf{A}}", "derivation": "\\operatorname{f_{E}}{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\operatorname{f_{E}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\log{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\operatorname{f_{E}}{(\\mathbf{A})} = \\frac{1}{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\log{(\\mathbf{A})} = \\frac{1}{\\mathbf{A}} and \\operatorname{f_{E}}{(\\mathbf{A})} \\frac{d}{d \\mathbf{A}} \\log{(\\mathbf{A})} = \\frac{\\operatorname{f_{E}}{(\\mathbf{A})}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)))"], [["times", 4, "Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)} = \\Psi \\Psi_{\\lambda}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} = 2 \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)}", "derivation": "\\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)} = \\Psi \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)} = 2 \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} = 2 \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi \\Psi_{\\lambda} - \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{\\prime}}{(\\Psi_{\\lambda},\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["minus", 3, "Derivative(Function('f^{\\\\prime}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Add(Mul(Integer(2), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given S{(A_{2},\\Psi^{\\dagger})} = - A_{2} + \\log{(\\Psi^{\\dagger})}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} S{(A_{2},\\Psi^{\\dagger})} = \\frac{1}{\\Psi^{\\dagger}}, then obtain \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- A_{2} + \\log{(\\Psi^{\\dagger})}) = \\frac{1}{\\Psi^{\\dagger}}", "derivation": "S{(A_{2},\\Psi^{\\dagger})} = - A_{2} + \\log{(\\Psi^{\\dagger})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} S{(A_{2},\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- A_{2} + \\log{(\\Psi^{\\dagger})}) and \\hat{H}_l^{P_{e}} + S{(A_{2},\\Psi^{\\dagger})} = - A_{2} + \\hat{H}_l^{P_{e}} + \\log{(\\Psi^{\\dagger})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\hat{H}_l^{P_{e}} + S{(A_{2},\\Psi^{\\dagger})}) = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- A_{2} + \\hat{H}_l^{P_{e}} + \\log{(\\Psi^{\\dagger})}) and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} S{(A_{2},\\Psi^{\\dagger})} = \\frac{1}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- A_{2} + \\log{(\\Psi^{\\dagger})}) = \\frac{1}{\\Psi^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 1, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('P_e', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('P_e', commutative=True)), Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('P_e', commutative=True)), Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('P_e', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('S')(Symbol('A_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{H})} = \\log{(\\hat{H})}, then derive \\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{1}{\\hat{H}} = 0, then obtain - 0^{\\hat{H}} + \\sin{(\\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{1}{\\hat{H}})} = - 0^{\\hat{H}}", "derivation": "\\operatorname{F_{g}}{(\\hat{H})} = \\log{(\\hat{H})} and \\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\log{(\\hat{H})} and \\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{d}{d \\hat{H}} \\log{(\\hat{H})} = 0 and \\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{1}{\\hat{H}} = 0 and \\sin{(\\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{1}{\\hat{H}})} = 0 and - 0^{\\hat{H}} + \\sin{(\\frac{d}{d \\hat{H}} \\operatorname{F_{g}}{(\\hat{H})} - \\frac{1}{\\hat{H}})} = - 0^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))), Integer(0))"], [["sin", 4], "Equality(sin(Add(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))), Integer(0))"], [["minus", 5, "Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True))), sin(Add(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))))), Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(k,\\mathbf{p})} = (- \\mathbf{p} + k)^{k}, then obtain \\frac{(\\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})})^{2}}{(\\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k})^{2}} = \\frac{\\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})}}{\\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k}}", "derivation": "\\operatorname{F_{c}}{(k,\\mathbf{p})} = (- \\mathbf{p} + k)^{k} and \\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})} = \\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k} and \\frac{\\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})}}{\\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k}} = 1 and \\frac{(\\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})})^{2}}{(\\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k})^{2}} = \\frac{\\frac{\\partial}{\\partial k} \\operatorname{F_{c}}{(k,\\mathbf{p})}}{\\frac{\\partial}{\\partial k} (- \\mathbf{p} + k)^{k}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(1))"], [["times", 3, "Mul(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-2)), Pow(Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('F_c')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(n,F_{x})} = - F_{x} + n, then obtain F_{x} - n + s{(n,F_{x})} \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n) = F_{x} - n + (- F_{x} + n) \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n)", "derivation": "s{(n,F_{x})} = - F_{x} + n and \\frac{\\partial}{\\partial F_{x}} s{(n,F_{x})} = \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n) and s{(n,F_{x})} \\frac{\\partial}{\\partial F_{x}} s{(n,F_{x})} = (- F_{x} + n) \\frac{\\partial}{\\partial F_{x}} s{(n,F_{x})} and s{(n,F_{x})} \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n) = (- F_{x} + n) \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n) and F_{x} - n + s{(n,F_{x})} \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n) = F_{x} - n + (- F_{x} + n) \\frac{\\partial}{\\partial F_{x}} (- F_{x} + n)", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Mul(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Derivative(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Derivative(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True))"], "Equality(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Function('s')(Symbol('n', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given Z{(E_{\\lambda},\\hat{x})} = \\frac{\\log{(E_{\\lambda})}}{\\hat{x}}, then obtain \\int \\frac{\\partial}{\\partial E_{\\lambda}} Z^{E_{\\lambda}}{(E_{\\lambda},\\hat{x})} dE_{\\lambda} = \\int \\frac{\\partial}{\\partial E_{\\lambda}} (\\frac{\\log{(E_{\\lambda})}}{\\hat{x}})^{E_{\\lambda}} dE_{\\lambda}", "derivation": "Z{(E_{\\lambda},\\hat{x})} = \\frac{\\log{(E_{\\lambda})}}{\\hat{x}} and Z^{E_{\\lambda}}{(E_{\\lambda},\\hat{x})} = (\\frac{\\log{(E_{\\lambda})}}{\\hat{x}})^{E_{\\lambda}} and \\frac{\\partial}{\\partial E_{\\lambda}} Z^{E_{\\lambda}}{(E_{\\lambda},\\hat{x})} = \\frac{\\partial}{\\partial E_{\\lambda}} (\\frac{\\log{(E_{\\lambda})}}{\\hat{x}})^{E_{\\lambda}} and \\int \\frac{\\partial}{\\partial E_{\\lambda}} Z^{E_{\\lambda}}{(E_{\\lambda},\\hat{x})} dE_{\\lambda} = \\int \\frac{\\partial}{\\partial E_{\\lambda}} (\\frac{\\log{(E_{\\lambda})}}{\\hat{x}})^{E_{\\lambda}} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Pow(Function('Z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('Z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Derivative(Pow(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(t_{1},i)} = i - t_{1} and \\psi{(t_{1})} = t_{1}, then obtain \\frac{\\partial}{\\partial i} \\frac{\\int (- i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)}) d\\psi{(t_{1})}}{\\int 0 d\\psi{(t_{1})}} = \\frac{d}{d i} 1", "derivation": "\\operatorname{A_{x}}{(t_{1},i)} = i - t_{1} and - i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)} = 0 and \\int (- i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)}) dt_{1} = \\int 0 dt_{1} and \\frac{\\int (- i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)}) dt_{1}}{\\int 0 dt_{1}} = 1 and \\psi{(t_{1})} = t_{1} and \\frac{\\int (- i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)}) d\\psi{(t_{1})}}{\\int 0 d\\psi{(t_{1})}} = 1 and \\frac{\\partial}{\\partial i} \\frac{\\int (- i + t_{1} + \\operatorname{A_{x}}{(t_{1},i)}) d\\psi{(t_{1})}}{\\int 0 d\\psi{(t_{1})}} = \\frac{d}{d i} 1", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["minus", 1, "Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('t_1', commutative=True), Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('t_1', commutative=True), Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 3, "Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Integral(Integer(0), Tuple(Symbol('t_1', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('t_1', commutative=True), Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('t_1', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Integral(Integer(0), Tuple(Function('\\\\psi')(Symbol('t_1', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('t_1', commutative=True), Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True))), Tuple(Function('\\\\psi')(Symbol('t_1', commutative=True))))), Integer(1))"], [["differentiate", 6, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Pow(Integral(Integer(0), Tuple(Function('\\\\psi')(Symbol('t_1', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('t_1', commutative=True), Function('A_x')(Symbol('t_1', commutative=True), Symbol('i', commutative=True))), Tuple(Function('\\\\psi')(Symbol('t_1', commutative=True))))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}, then obtain r + \\frac{d}{d \\mathbb{I}} \\operatorname{E_{x}}{(\\mathbb{I})} = v_{y} + e^{\\mathbb{I}}", "derivation": "\\operatorname{E_{x}}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} \\operatorname{E_{x}}{(\\mathbb{I})} = \\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}} and \\frac{d^{2}}{d \\mathbb{I}^{2}} \\operatorname{E_{x}}{(\\mathbb{I})} = \\frac{d^{3}}{d \\mathbb{I}^{3}} e^{\\mathbb{I}} and \\int \\frac{d^{2}}{d \\mathbb{I}^{2}} \\operatorname{E_{x}}{(\\mathbb{I})} d\\mathbb{I} = \\int \\frac{d^{3}}{d \\mathbb{I}^{3}} e^{\\mathbb{I}} d\\mathbb{I} and r + \\frac{d}{d \\mathbb{I}} \\operatorname{E_{x}}{(\\mathbb{I})} = v_{y} + e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(3))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Derivative(Function('E_x')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(3))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('r', commutative=True), Derivative(Function('E_x')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Add(Symbol('v_y', commutative=True), exp(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{x})} = \\sin{(e^{\\hat{x}})}, then obtain (\\frac{d}{d \\hat{x}} \\int \\operatorname{C_{2}}^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} = (\\frac{d}{d \\hat{x}} \\int \\sin^{\\hat{x}}{(e^{\\hat{x}})} d\\hat{x})^{\\hat{x}}", "derivation": "\\operatorname{C_{2}}{(\\hat{x})} = \\sin{(e^{\\hat{x}})} and \\operatorname{C_{2}}^{\\hat{x}}{(\\hat{x})} = \\sin^{\\hat{x}}{(e^{\\hat{x}})} and \\int \\operatorname{C_{2}}^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\int \\sin^{\\hat{x}}{(e^{\\hat{x}})} d\\hat{x} and \\frac{d}{d \\hat{x}} \\int \\operatorname{C_{2}}^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\frac{d}{d \\hat{x}} \\int \\sin^{\\hat{x}}{(e^{\\hat{x}})} d\\hat{x} and (\\frac{d}{d \\hat{x}} \\int \\operatorname{C_{2}}^{\\hat{x}}{(\\hat{x})} d\\hat{x})^{\\hat{x}} = (\\frac{d}{d \\hat{x}} \\int \\sin^{\\hat{x}}{(e^{\\hat{x}})} d\\hat{x})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{x}', commutative=True)), sin(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(sin(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Pow(Function('C_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(sin(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('C_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Derivative(Integral(Pow(Function('C_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True)), Pow(Derivative(Integral(Pow(sin(exp(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\hat{X})} = \\log{(\\hat{X})} and \\tilde{g}{(\\hat{X})} = \\log{(\\hat{X})} and \\hat{H}_{\\lambda}{(A_{x},F_{N})} = A_{x} F_{N}, then obtain \\frac{\\mathbf{B}{(\\hat{X})}}{\\hat{X} \\hat{H}_{\\lambda}{(A_{x},F_{N})}} = \\frac{\\tilde{g}{(\\hat{X})}}{\\hat{X} \\hat{H}_{\\lambda}{(A_{x},F_{N})}}", "derivation": "\\mathbf{B}{(\\hat{X})} = \\log{(\\hat{X})} and \\frac{\\mathbf{B}{(\\hat{X})}}{\\hat{X}} = \\frac{\\log{(\\hat{X})}}{\\hat{X}} and \\tilde{g}{(\\hat{X})} = \\log{(\\hat{X})} and \\frac{\\mathbf{B}{(\\hat{X})}}{\\hat{X}} = \\frac{\\tilde{g}{(\\hat{X})}}{\\hat{X}} and \\hat{H}_{\\lambda}{(A_{x},F_{N})} = A_{x} F_{N} and \\frac{\\mathbf{B}{(\\hat{X})}}{A_{x} F_{N} \\hat{X}} = \\frac{\\tilde{g}{(\\hat{X})}}{A_{x} F_{N} \\hat{X}} and \\frac{\\mathbf{B}{(\\hat{X})}}{\\hat{X} \\hat{H}_{\\lambda}{(A_{x},F_{N})}} = \\frac{\\tilde{g}{(\\hat{X})}}{\\hat{X} \\hat{H}_{\\lambda}{(A_{x},F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), log(Symbol('\\\\hat{X}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\hat{X}', commutative=True)), log(Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{X}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_x', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('F_N', commutative=True)))"], [["divide", 4, "Mul(Symbol('A_x', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_x', commutative=True), Symbol('F_N', commutative=True)), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_x', commutative=True), Symbol('F_N', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain 1 = (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}}", "derivation": "\\mathbf{M}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})} = 0 and (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = 0^{V_{\\mathbf{B}}} and V_{\\mathbf{B}} (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}) = 0 and (V_{\\mathbf{B}} (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}))^{V_{\\mathbf{B}}} = 0^{V_{\\mathbf{B}}} and 1 = (V_{\\mathbf{B}} (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})}))^{V_{\\mathbf{B}}} and 1 = 0^{V_{\\mathbf{B}}} and 1 = (\\mathbf{M}{(V_{\\mathbf{B}})} - \\sin{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Integer(0), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Integer(0), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(1), Pow(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(1), Pow(Integer(0), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Integer(1), Pow(Add(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given L{(\\eta,v_{t})} = \\frac{\\eta}{v_{t}}, then derive \\frac{\\partial}{\\partial \\eta} L{(\\eta,v_{t})} = \\frac{1}{v_{t}}, then obtain \\cos{(\\frac{\\eta}{v_{t}} + \\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{v_{t}})} = \\cos{(\\frac{\\eta}{v_{t}} + \\frac{1}{v_{t}})}", "derivation": "L{(\\eta,v_{t})} = \\frac{\\eta}{v_{t}} and \\frac{\\partial}{\\partial \\eta} L{(\\eta,v_{t})} = \\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{v_{t}} and \\frac{\\partial}{\\partial \\eta} L{(\\eta,v_{t})} = \\frac{1}{v_{t}} and L{(\\eta,v_{t})} + \\frac{\\partial}{\\partial \\eta} L{(\\eta,v_{t})} = L{(\\eta,v_{t})} + \\frac{1}{v_{t}} and \\frac{\\eta}{v_{t}} + \\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{v_{t}} = \\frac{\\eta}{v_{t}} + \\frac{1}{v_{t}} and \\cos{(\\frac{\\eta}{v_{t}} + \\frac{\\partial}{\\partial \\eta} \\frac{\\eta}{v_{t}})} = \\cos{(\\frac{\\eta}{v_{t}} + \\frac{1}{v_{t}})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Pow(Symbol('v_t', commutative=True), Integer(-1)))"], [["add", 3, "Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Derivative(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Function('L')(Symbol('\\\\eta', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["cos", 5], "Equality(cos(Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))), cos(Add(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Pow(Symbol('v_t', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\rho{(\\lambda)} = \\log{(\\lambda)}, then obtain \\frac{\\rho^{2 \\lambda}{(\\lambda)}}{\\log{(\\lambda)}} = \\frac{\\rho^{\\lambda}{(\\lambda)} \\log{(\\lambda)}^{\\lambda}}{\\log{(\\lambda)}}", "derivation": "\\rho{(\\lambda)} = \\log{(\\lambda)} and \\rho^{\\lambda}{(\\lambda)} = \\log{(\\lambda)}^{\\lambda} and \\frac{\\rho^{\\lambda}{(\\lambda)}}{\\log{(\\lambda)}} = \\frac{\\log{(\\lambda)}^{\\lambda}}{\\log{(\\lambda)}} and \\frac{\\rho^{\\lambda}{(\\lambda)} \\log{(\\lambda)}^{\\lambda}}{\\log{(\\lambda)}} = \\frac{\\log{(\\lambda)}^{2 \\lambda}}{\\log{(\\lambda)}} and \\frac{\\rho^{2 \\lambda}{(\\lambda)}}{\\log{(\\lambda)}} = \\frac{\\rho^{\\lambda}{(\\lambda)} \\log{(\\lambda)}^{\\lambda}}{\\log{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["divide", 2, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["times", 3, "Pow(log(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True))), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\rho')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} = \\int (- \\hat{H}_{\\lambda} + \\varphi^*) d\\hat{H}_{\\lambda}, then obtain \\int - \\hat{H}_{\\lambda} \\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} d\\varphi^* = J + \\frac{\\hat{H}_{\\lambda}^{3} \\varphi^*}{2} - \\frac{\\hat{H}_{\\lambda}^{2} (\\varphi^*)^{2}}{2}", "derivation": "\\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} = \\int (- \\hat{H}_{\\lambda} + \\varphi^*) d\\hat{H}_{\\lambda} and - \\hat{H}_{\\lambda} \\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} \\int (- \\hat{H}_{\\lambda} + \\varphi^*) d\\hat{H}_{\\lambda} and \\int - \\hat{H}_{\\lambda} \\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} d\\varphi^* = \\int - \\hat{H}_{\\lambda} \\int (- \\hat{H}_{\\lambda} + \\varphi^*) d\\hat{H}_{\\lambda} d\\varphi^* and \\int - \\hat{H}_{\\lambda} \\mathbf{D}{(\\varphi^*,\\hat{H}_{\\lambda})} d\\varphi^* = J + \\frac{\\hat{H}_{\\lambda}^{3} \\varphi^*}{2} - \\frac{\\hat{H}_{\\lambda}^{2} (\\varphi^*)^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(3)), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2)), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)))))"]]}, {"prompt": "Given C{(F_{N})} = \\cos{(F_{N})} and \\Psi_{nl}{(F_{N})} = \\frac{1}{\\cos{(F_{N})} - 1}, then obtain (\\Psi_{nl}{(F_{N})} - 1) (\\cos{(F_{N})} - 1) (\\cos{(F_{N})} - 1)^{- F_{N}} = (-1 + \\frac{1}{\\cos{(F_{N})} - 1}) (\\cos{(F_{N})} - 1) (\\cos{(F_{N})} - 1)^{- F_{N}}", "derivation": "C{(F_{N})} = \\cos{(F_{N})} and C{(F_{N})} - 1 = \\cos{(F_{N})} - 1 and \\Psi_{nl}{(F_{N})} = \\frac{1}{\\cos{(F_{N})} - 1} and \\Psi_{nl}{(F_{N})} = \\frac{1}{C{(F_{N})} - 1} and \\Psi_{nl}{(F_{N})} - 1 = -1 + \\frac{1}{C{(F_{N})} - 1} and \\Psi_{nl}{(F_{N})} - 1 = -1 + \\frac{1}{\\cos{(F_{N})} - 1} and (\\Psi_{nl}{(F_{N})} - 1) (\\cos{(F_{N})} - 1) (\\cos{(F_{N})} - 1)^{- F_{N}} = (-1 + \\frac{1}{\\cos{(F_{N})} - 1}) (\\cos{(F_{N})} - 1) (\\cos{(F_{N})} - 1)^{- F_{N}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('C')(Symbol('F_N', commutative=True)), Integer(-1)), Add(cos(Symbol('F_N', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True)), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True)), Pow(Add(Function('C')(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True)), Integer(-1)), Add(Integer(-1), Pow(Add(Function('C')(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True)), Integer(-1)), Add(Integer(-1), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1))))"], [["divide", 6, "Mul(Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1)), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Symbol('F_N', commutative=True)))"], "Equality(Mul(Add(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True)), Integer(-1)), Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Add(Integer(-1), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Integer(-1))), Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Pow(Add(cos(Symbol('F_N', commutative=True)), Integer(-1)), Mul(Integer(-1), Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\psi,h,\\phi)} = \\phi + \\psi h, then obtain - \\sin{(3 \\phi + 3 \\psi h - 4 \\phi{(\\psi,h,\\phi)})} = - \\sin{(\\phi + \\psi h - 2 \\phi{(\\psi,h,\\phi)})}", "derivation": "\\phi{(\\psi,h,\\phi)} = \\phi + \\psi h and - \\phi - \\psi h + \\phi{(\\psi,h,\\phi)} = 0 and - \\phi - \\psi h + 2 \\phi{(\\psi,h,\\phi)} = \\phi{(\\psi,h,\\phi)} and - \\sin{(\\phi + \\psi h - 2 \\phi{(\\psi,h,\\phi)})} = \\sin{(\\phi{(\\psi,h,\\phi)})} and - \\sin{(3 \\phi + 3 \\psi h - 4 \\phi{(\\psi,h,\\phi)})} = - \\sin{(\\phi + \\psi h - 2 \\phi{(\\psi,h,\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(0))"], [["add", 2, "Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)))), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)))))), sin(Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), sin(Add(Mul(Integer(3), Symbol('\\\\phi', commutative=True)), Mul(Integer(3), Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Integer(4), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)))))), Mul(Integer(-1), sin(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\phi')(Symbol('\\\\psi', commutative=True), Symbol('h', commutative=True), Symbol('\\\\phi', commutative=True)))))))"]]}, {"prompt": "Given \\dot{x}{(A_{1})} = \\sin{(A_{1})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\dot{x}{(A_{1})} \\sin{(A_{1})}, then obtain \\frac{d}{d A_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\frac{d}{d A_{1}} \\sin^{2}{(A_{1})}", "derivation": "\\dot{x}{(A_{1})} = \\sin{(A_{1})} and \\dot{x}{(A_{1})} \\sin{(A_{1})} = \\sin^{2}{(A_{1})} and \\dot{x}^{2}{(A_{1})} = \\dot{x}{(A_{1})} \\sin{(A_{1})} and \\dot{x}^{2}{(A_{1})} = \\sin^{2}{(A_{1})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\dot{x}{(A_{1})} \\sin{(A_{1})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\dot{x}^{2}{(A_{1})} and \\frac{d}{d A_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\frac{d}{d A_{1}} \\dot{x}^{2}{(A_{1})} and \\frac{d}{d A_{1}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{1})} = \\frac{d}{d A_{1}} \\sin^{2}{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["times", 1, "sin(Symbol('A_1', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))), Pow(sin(Symbol('A_1', commutative=True)), Integer(2)))"], [["times", 1, "Function('\\\\dot{x}')(Symbol('A_1', commutative=True))"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Integer(2)), Mul(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Integer(2)), Pow(sin(Symbol('A_1', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Integer(2)))"], [["differentiate", 6, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\dot{x}')(Symbol('A_1', commutative=True)), Integer(2)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('A_1', commutative=True)), Integer(2)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(V_{\\mathbf{B}},\\mathbf{g})} = e^{- V_{\\mathbf{B}} + \\mathbf{g}} and \\sigma_{p}{(V_{\\mathbf{B}},\\mathbf{g})} = \\frac{\\int e^{- V_{\\mathbf{B}} + \\mathbf{g}} d\\mathbf{g}}{\\int \\operatorname{v_{z}}{(V_{\\mathbf{B}},\\mathbf{g})} d\\mathbf{g}}, then obtain \\int \\sigma_{p}{(V_{\\mathbf{B}},\\mathbf{g})} d\\mathbf{g} = \\int 1 d\\mathbf{g}", "derivation": "\\operatorname{v_{z}}{(V_{\\mathbf{B}},\\mathbf{g})} = e^{- V_{\\mathbf{B}} + \\mathbf{g}} and \\sigma_{p}{(V_{\\mathbf{B}},\\mathbf{g})} = \\frac{\\int e^{- V_{\\mathbf{B}} + \\mathbf{g}} d\\mathbf{g}}{\\int \\operatorname{v_{z}}{(V_{\\mathbf{B}},\\mathbf{g})} d\\mathbf{g}} and \\sigma_{p}{(V_{\\mathbf{B}},\\mathbf{g})} = 1 and \\int \\sigma_{p}{(V_{\\mathbf{B}},\\mathbf{g})} d\\mathbf{g} = \\int 1 d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Integral(Function('v_z')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)), Integral(exp(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(C_{d},\\mathbf{S})} = - C_{d} + \\mathbf{S}, then obtain - C_{d} \\mathbf{S} \\mathbf{v}^{\\mathbf{S}}{(C_{d},\\mathbf{S})} = - C_{d} \\mathbf{S} (- C_{d} + \\mathbf{S})^{\\mathbf{S}}", "derivation": "\\mathbf{v}{(C_{d},\\mathbf{S})} = - C_{d} + \\mathbf{S} and \\mathbf{v}^{\\mathbf{S}}{(C_{d},\\mathbf{S})} = (- C_{d} + \\mathbf{S})^{\\mathbf{S}} and \\mathbf{S} \\mathbf{v}^{\\mathbf{S}}{(C_{d},\\mathbf{S})} = \\mathbf{S} (- C_{d} + \\mathbf{S})^{\\mathbf{S}} and - C_{d} \\mathbf{S} \\mathbf{v}^{\\mathbf{S}}{(C_{d},\\mathbf{S})} = - C_{d} \\mathbf{S} (- C_{d} + \\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('C_d', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(\\lambda,\\Psi)} = \\cos^{\\lambda}{(\\Psi)}, then obtain \\frac{\\partial}{\\partial \\lambda} \\int (\\dot{y}{(\\lambda,\\Psi)} + \\cos{(\\Psi)}) d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int (\\cos{(\\Psi)} + \\cos^{\\lambda}{(\\Psi)}) d\\lambda", "derivation": "\\dot{y}{(\\lambda,\\Psi)} = \\cos^{\\lambda}{(\\Psi)} and \\dot{y}{(\\lambda,\\Psi)} + \\cos{(\\Psi)} = \\cos{(\\Psi)} + \\cos^{\\lambda}{(\\Psi)} and \\int (\\dot{y}{(\\lambda,\\Psi)} + \\cos{(\\Psi)}) d\\lambda = \\int (\\cos{(\\Psi)} + \\cos^{\\lambda}{(\\Psi)}) d\\lambda and \\frac{\\partial}{\\partial \\lambda} \\int (\\dot{y}{(\\lambda,\\Psi)} + \\cos{(\\Psi)}) d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int (\\cos{(\\Psi)} + \\cos^{\\lambda}{(\\Psi)}) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True))), Add(cos(Symbol('\\\\Psi', commutative=True)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Function('\\\\dot{y}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(cos(Symbol('\\\\Psi', commutative=True)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\dot{y}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Add(cos(Symbol('\\\\Psi', commutative=True)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(Q)} = \\cos{(e^{Q})} and V{(Q)} = \\frac{\\chi{(Q)}}{\\cos{(e^{Q})}}, then obtain \\frac{V{(Q)}}{\\chi{(Q)}} = \\frac{1}{\\chi{(Q)}}", "derivation": "\\chi{(Q)} = \\cos{(e^{Q})} and \\frac{\\chi{(Q)}}{\\cos{(e^{Q})}} = 1 and V{(Q)} = \\frac{\\chi{(Q)}}{\\cos{(e^{Q})}} and V{(Q)} = 1 and \\frac{V{(Q)}}{\\chi{(Q)}} = \\frac{1}{\\chi{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('Q', commutative=True)), cos(exp(Symbol('Q', commutative=True))))"], [["divide", 1, "cos(exp(Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\chi')(Symbol('Q', commutative=True)), Pow(cos(exp(Symbol('Q', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('V')(Symbol('Q', commutative=True)), Mul(Function('\\\\chi')(Symbol('Q', commutative=True)), Pow(cos(exp(Symbol('Q', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('V')(Symbol('Q', commutative=True)), Integer(1))"], [["times", 4, "Pow(Function('\\\\chi')(Symbol('Q', commutative=True)), Integer(-1))"], "Equality(Mul(Function('V')(Symbol('Q', commutative=True)), Pow(Function('\\\\chi')(Symbol('Q', commutative=True)), Integer(-1))), Pow(Function('\\\\chi')(Symbol('Q', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} = C \\hat{p}_0, then obtain (\\int C \\hat{p}_0 d\\hat{p}_0)^{\\hat{p}_0} + (\\int \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} d\\hat{p}_0)^{\\hat{p}_0} = 2 (\\int C \\hat{p}_0 d\\hat{p}_0)^{\\hat{p}_0}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} = C \\hat{p}_0 and \\int \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} d\\hat{p}_0 = \\int C \\hat{p}_0 d\\hat{p}_0 and (\\int \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} d\\hat{p}_0)^{\\hat{p}_0} = (\\int C \\hat{p}_0 d\\hat{p}_0)^{\\hat{p}_0} and (\\int C \\hat{p}_0 d\\hat{p}_0)^{\\hat{p}_0} + (\\int \\operatorname{J_{\\varepsilon}}{(\\hat{p}_0,C)} d\\hat{p}_0)^{\\hat{p}_0} = 2 (\\int C \\hat{p}_0 d\\hat{p}_0)^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["add", 3, "Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{2},\\rho)} = - \\rho + e^{A_{2}} and C{(A_{2},\\rho)} = \\int \\operatorname{P_{g}}{(A_{2},\\rho)} d\\rho, then obtain C^{\\rho}{(A_{2},\\rho)} = (\\int (- \\rho + e^{A_{2}}) d\\rho)^{\\rho}", "derivation": "\\operatorname{P_{g}}{(A_{2},\\rho)} = - \\rho + e^{A_{2}} and \\int \\operatorname{P_{g}}{(A_{2},\\rho)} d\\rho = \\int (- \\rho + e^{A_{2}}) d\\rho and (\\int \\operatorname{P_{g}}{(A_{2},\\rho)} d\\rho)^{\\rho} = (\\int (- \\rho + e^{A_{2}}) d\\rho)^{\\rho} and C{(A_{2},\\rho)} = \\int \\operatorname{P_{g}}{(A_{2},\\rho)} d\\rho and C^{\\rho}{(A_{2},\\rho)} = (\\int (- \\rho + e^{A_{2}}) d\\rho)^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), exp(Symbol('A_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Function('P_g')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('P_g')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('C')(Symbol('A_2', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(I)} = \\sin{(I)}, then derive \\int \\Psi_{\\lambda}{(I)} dI = v_{z} - \\cos{(I)}, then obtain I (v_{z} - \\cos{(I)}) - v_{z} + \\cos{(I)} = I \\int \\Psi_{\\lambda}{(I)} dI - v_{z} + \\cos{(I)}", "derivation": "\\Psi_{\\lambda}{(I)} = \\sin{(I)} and \\int \\Psi_{\\lambda}{(I)} dI = \\int \\sin{(I)} dI and \\int \\Psi_{\\lambda}{(I)} dI = v_{z} - \\cos{(I)} and \\int \\sin{(I)} dI = v_{z} - \\cos{(I)} and I \\int \\sin{(I)} dI = I (v_{z} - \\cos{(I)}) and I \\int \\sin{(I)} dI = I \\int \\Psi_{\\lambda}{(I)} dI and I (v_{z} - \\cos{(I)}) = I \\int \\Psi_{\\lambda}{(I)} dI and I (v_{z} - \\cos{(I)}) - v_{z} + \\cos{(I)} = I \\int \\Psi_{\\lambda}{(I)} dI - v_{z} + \\cos{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True)))))"], [["divide", 4, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('I', commutative=True), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('I', commutative=True), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Symbol('I', commutative=True), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True))))), Mul(Symbol('I', commutative=True), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["minus", 7, "Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True))))"], "Equality(Add(Mul(Symbol('I', commutative=True), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True))))), Mul(Integer(-1), Symbol('v_z', commutative=True)), cos(Symbol('I', commutative=True))), Add(Mul(Symbol('I', commutative=True), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Symbol('v_z', commutative=True)), cos(Symbol('I', commutative=True))))"]]}, {"prompt": "Given i{(\\hat{H}_{\\lambda})} = e^{\\cos{(\\hat{H}_{\\lambda})}}, then obtain \\hat{H}_{\\lambda} \\frac{d}{d \\hat{H}_{\\lambda}} i{(\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} e^{\\cos{(\\hat{H}_{\\lambda})}} \\sin{(\\hat{H}_{\\lambda})}", "derivation": "i{(\\hat{H}_{\\lambda})} = e^{\\cos{(\\hat{H}_{\\lambda})}} and \\frac{d}{d \\hat{H}_{\\lambda}} i{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} e^{\\cos{(\\hat{H}_{\\lambda})}} and \\hat{H}_{\\lambda} \\frac{d}{d \\hat{H}_{\\lambda}} i{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} \\frac{d}{d \\hat{H}_{\\lambda}} e^{\\cos{(\\hat{H}_{\\lambda})}} and \\hat{H}_{\\lambda} \\frac{d}{d \\hat{H}_{\\lambda}} i{(\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} e^{\\cos{(\\hat{H}_{\\lambda})}} \\sin{(\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), exp(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Derivative(Function('i')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Derivative(exp(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Derivative(Function('i')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(C_{1})} = \\sin{(C_{1})}, then derive \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = A_{2} - \\cos{(C_{1})}, then obtain (A_{2} - \\cos{(C_{1})}) \\int \\sin{(C_{1})} dC_{1} = (\\int \\sin{(C_{1})} dC_{1})^{2}", "derivation": "\\operatorname{r_{0}}{(C_{1})} = \\sin{(C_{1})} and \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = \\int \\sin{(C_{1})} dC_{1} and (\\int \\operatorname{r_{0}}{(C_{1})} dC_{1}) \\int \\sin{(C_{1})} dC_{1} = (\\int \\sin{(C_{1})} dC_{1})^{2} and \\int \\operatorname{r_{0}}{(C_{1})} dC_{1} = A_{2} - \\cos{(C_{1})} and (A_{2} - \\cos{(C_{1})}) \\int \\sin{(C_{1})} dC_{1} = (\\int \\sin{(C_{1})} dC_{1})^{2}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Mul(Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Pow(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integer(2)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Pow(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\eta{(\\mathbf{J}_f,c_{0})} = \\mathbf{J}_f - c_{0}, then derive (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\eta{(\\mathbf{J}_f,c_{0})})^{\\mathbf{J}_f} = 1, then obtain (\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - c_{0}))^{\\mathbf{J}_f} = 1", "derivation": "\\eta{(\\mathbf{J}_f,c_{0})} = \\mathbf{J}_f - c_{0} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\eta{(\\mathbf{J}_f,c_{0})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - c_{0}) and (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\eta{(\\mathbf{J}_f,c_{0})})^{\\mathbf{J}_f} = (\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - c_{0}))^{\\mathbf{J}_f} and (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\eta{(\\mathbf{J}_f,c_{0})})^{\\mathbf{J}_f} = 1 and (\\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f - c_{0}))^{\\mathbf{J}_f} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{D}{(n_{2},E_{x})} = \\frac{E_{x}}{n_{2}}, then obtain \\frac{\\partial}{\\partial E_{x}} (n_{2} \\mathbf{D}{(n_{2},E_{x})} - 1) = \\frac{d}{d E_{x}} (E_{x} - 1)", "derivation": "\\mathbf{D}{(n_{2},E_{x})} = \\frac{E_{x}}{n_{2}} and n_{2} \\mathbf{D}{(n_{2},E_{x})} = E_{x} and n_{2} \\mathbf{D}{(n_{2},E_{x})} - 1 = E_{x} - 1 and \\frac{\\partial}{\\partial E_{x}} (n_{2} \\mathbf{D}{(n_{2},E_{x})} - 1) = \\frac{d}{d E_{x}} (E_{x} - 1)", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('n_2', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('n_2', commutative=True), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('n_2', commutative=True), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True))), Integer(-1)), Add(Symbol('E_x', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('n_2', commutative=True), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True))), Integer(-1)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Add(Symbol('E_x', commutative=True), Integer(-1)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(\\omega,c_{0})} = \\omega c_{0} and \\operatorname{A_{2}}{(\\omega,c_{0})} = - \\omega c_{0} + \\mathbf{s}{(\\omega,c_{0})}, then obtain (\\frac{\\operatorname{A_{2}}{(\\omega,c_{0})}}{c_{0}})^{\\omega} = 0^{\\omega}", "derivation": "\\mathbf{s}{(\\omega,c_{0})} = \\omega c_{0} and - \\omega c_{0} + \\mathbf{s}{(\\omega,c_{0})} = 0 and \\frac{- \\omega c_{0} + \\mathbf{s}{(\\omega,c_{0})}}{c_{0}} = 0 and (\\frac{- \\omega c_{0} + \\mathbf{s}{(\\omega,c_{0})}}{c_{0}})^{\\omega} = 0^{\\omega} and \\operatorname{A_{2}}{(\\omega,c_{0})} = - \\omega c_{0} + \\mathbf{s}{(\\omega,c_{0})} and (\\frac{\\operatorname{A_{2}}{(\\omega,c_{0})}}{c_{0}})^{\\omega} = 0^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True))), Integer(0))"], [["divide", 2, "Symbol('c_0', commutative=True)"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\omega', commutative=True), Symbol('c_0', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(y)} = e^{\\cos{(y)}} and \\mathbf{B}{(y)} = e^{\\cos{(y)}} \\cos{(y)}, then derive \\int \\frac{d}{d y} \\sigma_{x}{(y)} \\cos{(y)} dy = S + e^{\\cos{(y)}} \\cos{(y)}, then obtain \\frac{S + e^{\\cos{(y)}} \\cos{(y)}}{V} = \\frac{S + \\mathbf{B}{(y)}}{V}", "derivation": "\\sigma_{x}{(y)} = e^{\\cos{(y)}} and \\sigma_{x}{(y)} \\cos{(y)} = e^{\\cos{(y)}} \\cos{(y)} and \\frac{d}{d y} \\sigma_{x}{(y)} \\cos{(y)} = \\frac{d}{d y} e^{\\cos{(y)}} \\cos{(y)} and \\int \\frac{d}{d y} \\sigma_{x}{(y)} \\cos{(y)} dy = \\int \\frac{d}{d y} e^{\\cos{(y)}} \\cos{(y)} dy and \\mathbf{B}{(y)} = e^{\\cos{(y)}} \\cos{(y)} and \\int \\frac{d}{d y} \\sigma_{x}{(y)} \\cos{(y)} dy = S + e^{\\cos{(y)}} \\cos{(y)} and \\int \\frac{d}{d y} \\sigma_{x}{(y)} \\cos{(y)} dy = S + \\mathbf{B}{(y)} and S + e^{\\cos{(y)}} \\cos{(y)} = S + \\mathbf{B}{(y)} and \\frac{S + e^{\\cos{(y)}} \\cos{(y)}}{V} = \\frac{S + \\mathbf{B}{(y)}}{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('y', commutative=True)), exp(cos(Symbol('y', commutative=True))))"], [["times", 1, "cos(Symbol('y', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('y', commutative=True)), Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Mul(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Add(Symbol('S', commutative=True), Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integral(Derivative(Mul(Function('\\\\sigma_x')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Add(Symbol('S', commutative=True), Function('\\\\mathbf{B}')(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Symbol('S', commutative=True), Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True)))), Add(Symbol('S', commutative=True), Function('\\\\mathbf{B}')(Symbol('y', commutative=True))))"], [["divide", 8, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Mul(exp(cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Function('\\\\mathbf{B}')(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(F_{x})} = \\sin{(\\log{(F_{x})})} and \\mathbf{J}{(F_{x})} = F_{x} + \\operatorname{C_{1}}{(F_{x})}, then obtain \\mathbf{J}^{2}{(F_{x})} = (F_{x} + \\sin{(\\log{(F_{x})})}) \\mathbf{J}{(F_{x})}", "derivation": "\\operatorname{C_{1}}{(F_{x})} = \\sin{(\\log{(F_{x})})} and F_{x} + \\operatorname{C_{1}}{(F_{x})} = F_{x} + \\sin{(\\log{(F_{x})})} and \\mathbf{J}{(F_{x})} = F_{x} + \\operatorname{C_{1}}{(F_{x})} and \\mathbf{J}{(F_{x})} = F_{x} + \\sin{(\\log{(F_{x})})} and \\mathbf{J}^{2}{(F_{x})} = (F_{x} + \\sin{(\\log{(F_{x})})}) \\mathbf{J}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('F_x', commutative=True)), sin(log(Symbol('F_x', commutative=True))))"], [["add", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('C_1')(Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), sin(log(Symbol('F_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Function('C_1')(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}')(Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), sin(log(Symbol('F_x', commutative=True)))))"], [["times", 4, "Function('\\\\mathbf{J}')(Symbol('F_x', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('F_x', commutative=True)), Integer(2)), Mul(Add(Symbol('F_x', commutative=True), sin(log(Symbol('F_x', commutative=True)))), Function('\\\\mathbf{J}')(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} = z^{V_{\\mathbf{B}}}, then obtain \\iint \\operatorname{y^{\\prime}}^{2}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} dV_{\\mathbf{B}} = \\iint z^{V_{\\mathbf{B}}} \\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} dV_{\\mathbf{B}}", "derivation": "\\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} = z^{V_{\\mathbf{B}}} and \\operatorname{y^{\\prime}}^{2}{(V_{\\mathbf{B}},z)} = z^{V_{\\mathbf{B}}} \\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} and \\int \\operatorname{y^{\\prime}}^{2}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} = \\int z^{V_{\\mathbf{B}}} \\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} and \\iint \\operatorname{y^{\\prime}}^{2}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} dV_{\\mathbf{B}} = \\iint z^{V_{\\mathbf{B}}} \\operatorname{y^{\\prime}}{(V_{\\mathbf{B}},z)} dV_{\\mathbf{B}} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Mul(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["integrate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('y^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)} = A_{x} - \\mathbf{J}_M, then obtain \\mathbf{J}_M (\\frac{\\partial}{\\partial A_{x}} \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)} - 1) = 0", "derivation": "\\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)} = A_{x} - \\mathbf{J}_M and - A_{x} + \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)} = - \\mathbf{J}_M and \\frac{\\partial}{\\partial A_{x}} (- A_{x} + \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)}) = \\frac{d}{d A_{x}} - \\mathbf{J}_M and \\mathbf{J}_M \\frac{\\partial}{\\partial A_{x}} (- A_{x} + \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)}) = \\mathbf{J}_M \\frac{d}{d A_{x}} - \\mathbf{J}_M and \\mathbf{J}_M (\\frac{\\partial}{\\partial A_{x}} \\operatorname{c_{0}}{(A_{x},\\mathbf{J}_M)} - 1) = 0", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('c_0')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 2, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('c_0')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('c_0')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Add(Derivative(Function('c_0')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\hat{p}{(S)} = \\sin{(\\log{(S)})}, then obtain \\log{(S)} \\int \\hat{p}{(S)} dS = (\\frac{S \\sin{(\\log{(S)})}}{2} - \\frac{S \\cos{(\\log{(S)})}}{2} + \\mathbf{A}) \\log{(S)}", "derivation": "\\hat{p}{(S)} = \\sin{(\\log{(S)})} and \\int \\hat{p}{(S)} dS = \\int \\sin{(\\log{(S)})} dS and \\log{(S)} \\int \\hat{p}{(S)} dS = \\log{(S)} \\int \\sin{(\\log{(S)})} dS and \\log{(S)} \\int \\hat{p}{(S)} dS = (\\frac{S \\sin{(\\log{(S)})}}{2} - \\frac{S \\cos{(\\log{(S)})}}{2} + \\mathbf{A}) \\log{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('S', commutative=True)), sin(log(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(sin(log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["times", 2, "log(Symbol('S', commutative=True))"], "Equality(Mul(log(Symbol('S', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(log(Symbol('S', commutative=True)), Integral(sin(log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(log(Symbol('S', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Symbol('S', commutative=True), sin(log(Symbol('S', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('S', commutative=True), cos(log(Symbol('S', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})} = - \\mathbf{M} + \\cos{(\\phi_2)}, then obtain \\frac{\\partial}{\\partial \\phi_2} \\frac{\\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})}}{\\phi_2^{2}} = \\frac{\\partial}{\\partial \\phi_2} \\frac{- \\mathbf{M} + \\cos{(\\phi_2)}}{\\phi_2^{2}}", "derivation": "\\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})} = - \\mathbf{M} + \\cos{(\\phi_2)} and \\frac{\\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})}}{\\phi_2} = \\frac{- \\mathbf{M} + \\cos{(\\phi_2)}}{\\phi_2} and \\frac{\\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})}}{\\phi_2^{2}} = \\frac{- \\mathbf{M} + \\cos{(\\phi_2)}}{\\phi_2^{2}} and \\frac{\\partial}{\\partial \\phi_2} \\frac{\\operatorname{n_{1}}{(\\phi_2,\\mathbf{M})}}{\\phi_2^{2}} = \\frac{\\partial}{\\partial \\phi_2} \\frac{- \\mathbf{M} + \\cos{(\\phi_2)}}{\\phi_2^{2}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('n_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))))"], [["times", 2, "Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Function('n_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Function('n_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then derive \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = - \\sin{(\\mathbf{g})}, then obtain - \\mathbf{g} - \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = - \\mathbf{g} + \\sin{(\\mathbf{g})}", "derivation": "v{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = - \\sin{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} = - \\sin{(\\mathbf{g})} and \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})} and \\mathbf{g} + \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})} and - \\mathbf{g} - \\frac{d}{d \\mathbf{g}} v{(\\mathbf{g})} = - \\mathbf{g} + \\sin{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('v')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["times", 6, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Derivative(Function('v')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given h{(\\Omega)} = \\sin{(\\Omega)}, then derive \\int h{(\\Omega)} d\\Omega = \\eta - \\cos{(\\Omega)}, then obtain - \\eta + \\cos{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega = 0", "derivation": "h{(\\Omega)} = \\sin{(\\Omega)} and \\int h{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\int h{(\\Omega)} d\\Omega = \\eta - \\cos{(\\Omega)} and 0 = \\eta - \\cos{(\\Omega)} - \\int h{(\\Omega)} d\\Omega and 0 = \\eta - \\cos{(\\Omega)} - \\int \\sin{(\\Omega)} d\\Omega and - \\eta + \\cos{(\\Omega)} + \\int h{(\\Omega)} d\\Omega = \\int h{(\\Omega)} d\\Omega - \\int \\sin{(\\Omega)} d\\Omega and - \\eta + \\cos{(\\Omega)} + \\int \\sin{(\\Omega)} d\\Omega = 0", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], [["minus", 3, "Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["minus", 5, "Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)), Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Integral(Function('h')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(0))"]]}, {"prompt": "Given W{(\\hat{p}_0)} = \\log{(\\log{(\\hat{p}_0)})}, then derive (\\int W{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\hat{p}_0 \\log{(\\log{(\\hat{p}_0)})} + \\varphi^* - \\operatorname{li}{(\\hat{p}_0)})^{\\hat{p}_0}, then obtain (\\int W{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\hat{p}_0 W{(\\hat{p}_0)} + \\varphi^* - \\operatorname{li}{(\\hat{p}_0)})^{\\hat{p}_0}", "derivation": "W{(\\hat{p}_0)} = \\log{(\\log{(\\hat{p}_0)})} and \\int W{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\log{(\\log{(\\hat{p}_0)})} d\\hat{p}_0 and (\\int W{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\int \\log{(\\log{(\\hat{p}_0)})} d\\hat{p}_0)^{\\hat{p}_0} and (\\int W{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\hat{p}_0 \\log{(\\log{(\\hat{p}_0)})} + \\varphi^* - \\operatorname{li}{(\\hat{p}_0)})^{\\hat{p}_0} and (\\int W{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\hat{p}_0 W{(\\hat{p}_0)} + \\varphi^* - \\operatorname{li}{(\\hat{p}_0)})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), log(log(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(log(log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Integral(log(log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), log(log(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), li(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('W')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), li(Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(c,H)} = H^{c}, then obtain \\sin{(\\frac{\\partial}{\\partial H} \\operatorname{f^{\\prime}}^{H}{(c,H)})} = \\sin{(\\frac{\\partial}{\\partial H} (H^{c})^{H})}", "derivation": "\\operatorname{f^{\\prime}}{(c,H)} = H^{c} and \\operatorname{f^{\\prime}}^{H}{(c,H)} = (H^{c})^{H} and \\frac{\\partial}{\\partial H} \\operatorname{f^{\\prime}}^{H}{(c,H)} = \\frac{\\partial}{\\partial H} (H^{c})^{H} and \\sin{(\\frac{\\partial}{\\partial H} \\operatorname{f^{\\prime}}^{H}{(c,H)})} = \\sin{(\\frac{\\partial}{\\partial H} (H^{c})^{H})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)), Symbol('H', commutative=True)))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('f^{\\\\prime}')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Pow(Function('f^{\\\\prime}')(Symbol('c', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), sin(Derivative(Pow(Pow(Symbol('H', commutative=True), Symbol('c', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(b,W)} = W + b, then derive \\int V{(b,W)} dW = \\frac{W^{2}}{2} + W b + t_{1}, then derive \\frac{W^{2}}{2} + W b + \\mathbf{A} = \\frac{W^{2}}{2} + W b + t_{1}, then obtain W + \\int (W + b) dW = \\frac{W^{2}}{2} + W b + W + \\mathbf{A}", "derivation": "V{(b,W)} = W + b and \\int V{(b,W)} dW = \\int (W + b) dW and \\int V{(b,W)} dW = \\frac{W^{2}}{2} + W b + t_{1} and \\int (W + b) dW = \\frac{W^{2}}{2} + W b + t_{1} and W + \\int (W + b) dW = \\frac{W^{2}}{2} + W b + W + t_{1} and \\frac{W^{2}}{2} + W b + \\mathbf{A} = \\frac{W^{2}}{2} + W b + t_{1} and \\frac{W^{2}}{2} + W b + W + \\mathbf{A} = \\frac{W^{2}}{2} + W b + W + t_{1} and W + \\int (W + b) dW = \\frac{W^{2}}{2} + W b + W + \\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('V')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Add(Symbol('W', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V')(Symbol('b', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('W', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('W', commutative=True))"], "Equality(Add(Symbol('W', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('W', commutative=True), Symbol('t_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 6, "Mul(Integer(-1), Symbol('W', commutative=True))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('W', commutative=True), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Symbol('W', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Mul(Symbol('W', commutative=True), Symbol('b', commutative=True)), Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{J},l)} = e^{\\mathbf{J}^{l}}, then obtain (l - 1) (\\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} - 1) ((- l + \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}})^{\\mathbf{J}})^{- l} = 0", "derivation": "\\operatorname{A_{2}}{(\\mathbf{J},l)} = e^{\\mathbf{J}^{l}} and \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} = 1 and \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} - 1 = 0 and l - 1 = l - \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} and (l - \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}}) (\\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} - 1) = 0 and (l - 1) (\\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} - 1) = 0 and (l - 1) (\\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}} - 1) ((- l + \\operatorname{A_{2}}{(\\mathbf{J},l)} e^{- \\mathbf{J}^{l}})^{\\mathbf{J}})^{- l} = 0", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))"], [["divide", 1, "exp(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)))"], "Equality(Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))), Integer(-1)), Integer(0))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))))"], "Equality(Add(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)))))))"], [["times", 3, "Add(Symbol('l', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))))"], "Equality(Mul(Add(Symbol('l', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)))))), Add(Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))), Integer(-1))), Integer(0))"], [["divide", 6, "Pow(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)))))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('l', commutative=True))"], "Equality(Mul(Add(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))))), Integer(-1)), Pow(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)))))), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{p}_0{(\\lambda)} = e^{\\lambda}, then obtain \\frac{d}{d \\lambda} (\\frac{d}{d \\lambda} \\hat{p}_0{(\\lambda)})^{\\lambda} = \\frac{d}{d \\lambda} (\\frac{d}{d \\lambda} e^{\\lambda})^{\\lambda}", "derivation": "\\hat{p}_0{(\\lambda)} = e^{\\lambda} and \\frac{d}{d \\lambda} \\hat{p}_0{(\\lambda)} = \\frac{d}{d \\lambda} e^{\\lambda} and (\\frac{d}{d \\lambda} \\hat{p}_0{(\\lambda)})^{\\lambda} = (\\frac{d}{d \\lambda} e^{\\lambda})^{\\lambda} and \\frac{d}{d \\lambda} (\\frac{d}{d \\lambda} \\hat{p}_0{(\\lambda)})^{\\lambda} = \\frac{d}{d \\lambda} (\\frac{d}{d \\lambda} e^{\\lambda})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(c_{0},\\dot{y})} = \\cos{(\\dot{y} c_{0})}, then obtain \\sin{((m^{c_{0}}{(c_{0},\\dot{y})})^{\\dot{y}})} = \\sin{((\\cos^{c_{0}}{(\\dot{y} c_{0})})^{\\dot{y}})}", "derivation": "m{(c_{0},\\dot{y})} = \\cos{(\\dot{y} c_{0})} and m^{c_{0}}{(c_{0},\\dot{y})} = \\cos^{c_{0}}{(\\dot{y} c_{0})} and (m^{c_{0}}{(c_{0},\\dot{y})})^{\\dot{y}} = (\\cos^{c_{0}}{(\\dot{y} c_{0})})^{\\dot{y}} and \\sin{((m^{c_{0}}{(c_{0},\\dot{y})})^{\\dot{y}})} = \\sin{((\\cos^{c_{0}}{(\\dot{y} c_{0})})^{\\dot{y}})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('c_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), cos(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('c_0', commutative=True))))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('m')(Symbol('c_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('c_0', commutative=True)), Pow(cos(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Pow(Function('m')(Symbol('c_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(cos(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Pow(Function('m')(Symbol('c_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('c_0', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), sin(Pow(Pow(cos(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('c_0', commutative=True))), Symbol('c_0', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(L)} = \\log{(L)}, then obtain 0 = \\int (\\operatorname{C_{2}}{(L)} + \\log{(L)})^{2} dL - \\int 4 \\operatorname{C_{2}}^{2}{(L)} dL", "derivation": "\\operatorname{C_{2}}{(L)} = \\log{(L)} and 2 \\operatorname{C_{2}}{(L)} = \\operatorname{C_{2}}{(L)} + \\log{(L)} and 4 \\operatorname{C_{2}}^{2}{(L)} = (\\operatorname{C_{2}}{(L)} + \\log{(L)})^{2} and \\int 4 \\operatorname{C_{2}}^{2}{(L)} dL = \\int (\\operatorname{C_{2}}{(L)} + \\log{(L)})^{2} dL and 0 = \\int (\\operatorname{C_{2}}{(L)} + \\log{(L)})^{2} dL - \\int 4 \\operatorname{C_{2}}^{2}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["add", 1, "Function('C_2')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('C_2')(Symbol('L', commutative=True))), Add(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('C_2')(Symbol('L', commutative=True)), Integer(2))), Pow(Add(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Integer(2)))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('C_2')(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True))), Integral(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Integer(2)), Tuple(Symbol('L', commutative=True))))"], [["minus", 4, "Integral(Mul(Integer(4), Pow(Function('C_2')(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True)))"], "Equality(Integer(0), Add(Integral(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Integer(2)), Tuple(Symbol('L', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(4), Pow(Function('C_2')(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{r})} = e^{\\mathbf{r}}, then derive \\frac{d}{d \\mathbf{r}} \\mathbf{J}_f{(\\mathbf{r})} = e^{\\mathbf{r}}, then obtain \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} = e^{\\mathbf{r}}", "derivation": "\\mathbf{J}_f{(\\mathbf{r})} = e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\mathbf{J}_f{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\mathbf{J}_f{(\\mathbf{r})} = e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} = e^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given q{(F_{N})} = \\cos{(e^{F_{N}})}, then obtain 2 \\int q{(F_{N})} dF_{N} = \\theta_2 + \\operatorname{Ci}{(e^{F_{N}})} + \\int q{(F_{N})} dF_{N}", "derivation": "q{(F_{N})} = \\cos{(e^{F_{N}})} and \\int q{(F_{N})} dF_{N} = \\int \\cos{(e^{F_{N}})} dF_{N} and 2 \\int q{(F_{N})} dF_{N} = \\int q{(F_{N})} dF_{N} + \\int \\cos{(e^{F_{N}})} dF_{N} and 2 \\int q{(F_{N})} dF_{N} = \\theta_2 + \\operatorname{Ci}{(e^{F_{N}})} + \\int q{(F_{N})} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('F_N', commutative=True)), cos(exp(Symbol('F_N', commutative=True))))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["add", 2, "Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Symbol('\\\\theta_2', commutative=True), Ci(exp(Symbol('F_N', commutative=True))), Integral(Function('q')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given T{(m_{s},\\ddot{x})} = \\cos{(\\ddot{x} - m_{s})}, then derive \\ddot{x} + \\theta_2 = \\int \\frac{\\cos{(\\ddot{x} - m_{s})}}{T{(m_{s},\\ddot{x})}} d\\ddot{x}, then obtain P_{e} + \\ddot{x} = \\ddot{x} + \\theta_2", "derivation": "T{(m_{s},\\ddot{x})} = \\cos{(\\ddot{x} - m_{s})} and - m_{s} T{(m_{s},\\ddot{x})} = - m_{s} \\cos{(\\ddot{x} - m_{s})} and 1 = \\frac{\\cos{(\\ddot{x} - m_{s})}}{T{(m_{s},\\ddot{x})}} and \\int 1 d\\ddot{x} = \\int \\frac{\\cos{(\\ddot{x} - m_{s})}}{T{(m_{s},\\ddot{x})}} d\\ddot{x} and \\ddot{x} + \\theta_2 = \\int \\frac{\\cos{(\\ddot{x} - m_{s})}}{T{(m_{s},\\ddot{x})}} d\\ddot{x} and \\int 1 d\\ddot{x} = \\ddot{x} + \\theta_2 and P_{e} + \\ddot{x} = \\ddot{x} + \\theta_2", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), cos(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"], [["times", 1, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('m_s', commutative=True), Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True), cos(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"], [["divide", 2, "Mul(Integer(-1), Symbol('m_s', commutative=True), Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Pow(Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Mul(Pow(Function('T')(Symbol('m_s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('P_e', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given C{(c,A_{1})} = A_{1}^{c}, then obtain \\frac{(\\int (c + C{(c,A_{1})}) dc) \\int C{(c,A_{1})} dA_{1}}{A_{1}} = \\frac{(\\int A_{1}^{c} dA_{1}) \\int (c + C{(c,A_{1})}) dc}{A_{1}}", "derivation": "C{(c,A_{1})} = A_{1}^{c} and c + C{(c,A_{1})} = A_{1}^{c} + c and \\int (c + C{(c,A_{1})}) dc = \\int (A_{1}^{c} + c) dc and \\frac{\\int (c + C{(c,A_{1})}) dc}{A_{1}} = \\frac{\\int (A_{1}^{c} + c) dc}{A_{1}} and \\int C{(c,A_{1})} dA_{1} = \\int A_{1}^{c} dA_{1} and \\frac{(\\int (A_{1}^{c} + c) dc) \\int C{(c,A_{1})} dA_{1}}{A_{1}} = \\frac{(\\int A_{1}^{c} dA_{1}) \\int (A_{1}^{c} + c) dc}{A_{1}} and \\frac{(\\int (c + C{(c,A_{1})}) dc) \\int C{(c,A_{1})} dA_{1}}{A_{1}} = \\frac{(\\int A_{1}^{c} dA_{1}) \\int (c + C{(c,A_{1})}) dc}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True))), Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Symbol('c', commutative=True), Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["divide", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Add(Symbol('c', commutative=True), Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('c', commutative=True)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["times", 5, "Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Add(Symbol('c', commutative=True), Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Integral(Pow(Symbol('A_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Symbol('c', commutative=True), Function('C')(Symbol('c', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(f_{\\mathbf{p}},\\psi^*)} = f_{\\mathbf{p}} \\cos{(\\psi^*)}, then obtain 0 = - \\frac{f_{\\mathbf{p}} \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\nabla{(f_{\\mathbf{p}},\\psi^*)}}{\\nabla^{2}{(f_{\\mathbf{p}},\\psi^*)}} + \\frac{1}{\\nabla{(f_{\\mathbf{p}},\\psi^*)}}", "derivation": "\\nabla{(f_{\\mathbf{p}},\\psi^*)} = f_{\\mathbf{p}} \\cos{(\\psi^*)} and \\frac{\\nabla{(f_{\\mathbf{p}},\\psi^*)}}{\\cos{(\\psi^*)}} = f_{\\mathbf{p}} and \\frac{1}{\\cos{(\\psi^*)}} = \\frac{f_{\\mathbf{p}}}{\\nabla{(f_{\\mathbf{p}},\\psi^*)}} and \\frac{d}{d f_{\\mathbf{p}}} \\frac{1}{\\cos{(\\psi^*)}} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{f_{\\mathbf{p}}}{\\nabla{(f_{\\mathbf{p}},\\psi^*)}} and 0 = - \\frac{f_{\\mathbf{p}} \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\nabla{(f_{\\mathbf{p}},\\psi^*)}}{\\nabla^{2}{(f_{\\mathbf{p}},\\psi^*)}} + \\frac{1}{\\nabla{(f_{\\mathbf{p}},\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "cos(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["divide", 2, "Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-2)), Derivative(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Pow(Function('\\\\nabla')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(g)} = \\log{(\\sin{(g)})}, then obtain \\varphi{(g)} = \\varphi{(g)} - \\frac{\\frac{d}{d g} \\int \\varphi{(g)} dg}{\\int \\varphi{(g)} dg} + \\frac{\\frac{d}{d g} \\int \\log{(\\sin{(g)})} dg}{\\int \\varphi{(g)} dg}", "derivation": "\\varphi{(g)} = \\log{(\\sin{(g)})} and \\int \\varphi{(g)} dg = \\int \\log{(\\sin{(g)})} dg and \\frac{d}{d g} \\int \\varphi{(g)} dg = \\frac{d}{d g} \\int \\log{(\\sin{(g)})} dg and \\frac{\\frac{d}{d g} \\int \\varphi{(g)} dg}{\\int \\varphi{(g)} dg} = \\frac{\\frac{d}{d g} \\int \\log{(\\sin{(g)})} dg}{\\int \\varphi{(g)} dg} and 0 = - \\frac{\\frac{d}{d g} \\int \\varphi{(g)} dg}{\\int \\varphi{(g)} dg} + \\frac{\\frac{d}{d g} \\int \\log{(\\sin{(g)})} dg}{\\int \\varphi{(g)} dg} and \\varphi{(g)} = \\varphi{(g)} - \\frac{\\frac{d}{d g} \\int \\varphi{(g)} dg}{\\int \\varphi{(g)} dg} + \\frac{\\frac{d}{d g} \\int \\log{(\\sin{(g)})} dg}{\\int \\varphi{(g)} dg}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('g', commutative=True)), log(sin(Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Derivative(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))))"], [["minus", 4, "Mul(Derivative(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Derivative(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)))))"], [["minus", 5, "Mul(Integer(-1), Function('\\\\varphi')(Symbol('g', commutative=True)))"], "Equality(Function('\\\\varphi')(Symbol('g', commutative=True)), Add(Function('\\\\varphi')(Symbol('g', commutative=True)), Mul(Integer(-1), Derivative(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Derivative(Integral(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('\\\\varphi')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\nabla)} = \\sin{(\\nabla)}, then obtain - \\nabla + \\sin{(\\nabla)} + \\frac{d}{d \\nabla} (- \\nabla + \\operatorname{r_{0}}{(\\nabla)}) = - \\nabla + \\sin{(\\nabla)} + \\frac{d}{d \\nabla} (- \\nabla + \\sin{(\\nabla)})", "derivation": "\\operatorname{r_{0}}{(\\nabla)} = \\sin{(\\nabla)} and - \\nabla + \\operatorname{r_{0}}{(\\nabla)} = - \\nabla + \\sin{(\\nabla)} and \\frac{d}{d \\nabla} (- \\nabla + \\operatorname{r_{0}}{(\\nabla)}) = \\frac{d}{d \\nabla} (- \\nabla + \\sin{(\\nabla)}) and - \\nabla + \\sin{(\\nabla)} + \\frac{d}{d \\nabla} (- \\nabla + \\operatorname{r_{0}}{(\\nabla)}) = - \\nabla + \\sin{(\\nabla)} + \\frac{d}{d \\nabla} (- \\nabla + \\sin{(\\nabla)})", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('r_0')(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('r_0')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('r_0')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\ddot{x}{(i)} = \\sin{(\\log{(i)})} and \\operatorname{v_{1}}{(i)} = \\frac{- \\log{(\\ddot{x}{(i)})} + \\log{(\\sin{(\\log{(i)})})}}{\\sin{(\\log{(i)})}}, then obtain \\frac{d}{d i} (0^{i})^{i} = \\frac{d}{d i} (\\operatorname{v_{1}}^{i}{(i)})^{i}", "derivation": "\\ddot{x}{(i)} = \\sin{(\\log{(i)})} and \\log{(\\ddot{x}{(i)})} = \\log{(\\sin{(\\log{(i)})})} and 0 = - \\log{(\\ddot{x}{(i)})} + \\log{(\\sin{(\\log{(i)})})} and 0 = \\frac{- \\log{(\\ddot{x}{(i)})} + \\log{(\\sin{(\\log{(i)})})}}{\\sin{(\\log{(i)})}} and 0^{i} = (\\frac{- \\log{(\\ddot{x}{(i)})} + \\log{(\\sin{(\\log{(i)})})}}{\\sin{(\\log{(i)})}})^{i} and \\operatorname{v_{1}}{(i)} = \\frac{- \\log{(\\ddot{x}{(i)})} + \\log{(\\sin{(\\log{(i)})})}}{\\sin{(\\log{(i)})}} and 0^{i} = \\operatorname{v_{1}}^{i}{(i)} and (0^{i})^{i} = (\\operatorname{v_{1}}^{i}{(i)})^{i} and \\frac{d}{d i} (0^{i})^{i} = \\frac{d}{d i} (\\operatorname{v_{1}}^{i}{(i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\ddot{x}')(Symbol('i', commutative=True))), log(sin(log(Symbol('i', commutative=True)))))"], [["minus", 2, "log(Function('\\\\ddot{x}')(Symbol('i', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), log(Function('\\\\ddot{x}')(Symbol('i', commutative=True)))), log(sin(log(Symbol('i', commutative=True))))))"], [["divide", 3, "sin(log(Symbol('i', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), log(Function('\\\\ddot{x}')(Symbol('i', commutative=True)))), log(sin(log(Symbol('i', commutative=True))))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Integer(0), Symbol('i', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), log(Function('\\\\ddot{x}')(Symbol('i', commutative=True)))), log(sin(log(Symbol('i', commutative=True))))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('i', commutative=True)), Mul(Add(Mul(Integer(-1), log(Function('\\\\ddot{x}')(Symbol('i', commutative=True)))), log(sin(log(Symbol('i', commutative=True))))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Integer(0), Symbol('i', commutative=True)), Pow(Function('v_1')(Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["power", 7, "Symbol('i', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(Pow(Function('v_1')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 8, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Pow(Integer(0), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('v_1')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(z)} = \\log{(z)} and \\mathbf{D}{(z)} = \\log{(z)}, then obtain L + 2 \\mathbf{D}{(z)} \\int \\eta^{\\prime}{(L)} dL = L + (\\mathbf{D}{(z)} + \\log{(z)}) \\int \\eta^{\\prime}{(L)} dL", "derivation": "\\mathbb{I}{(z)} = \\log{(z)} and \\mathbf{D}{(z)} = \\log{(z)} and 2 \\mathbf{D}{(z)} = \\mathbf{D}{(z)} + \\log{(z)} and 2 \\mathbf{D}{(z)} = \\mathbb{I}{(z)} + \\mathbf{D}{(z)} and \\mathbb{I}{(z)} + \\mathbf{D}{(z)} = \\mathbf{D}{(z)} + \\log{(z)} and (\\mathbb{I}{(z)} + \\mathbf{D}{(z)}) \\int \\eta^{\\prime}{(L)} dL = (\\mathbf{D}{(z)} + \\log{(z)}) \\int \\eta^{\\prime}{(L)} dL and \\mathbf{D}{(z)} = \\mathbb{I}{(z)} and 2 \\mathbf{D}{(z)} \\int \\eta^{\\prime}{(L)} dL = (\\mathbf{D}{(z)} + \\log{(z)}) \\int \\eta^{\\prime}{(L)} dL and L + 2 \\mathbf{D}{(z)} \\int \\eta^{\\prime}{(L)} dL = L + (\\mathbf{D}{(z)} + \\log{(z)}) \\int \\eta^{\\prime}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["add", 2, "Function('\\\\mathbf{D}')(Symbol('z', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('z', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('z', commutative=True))), Add(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Function('\\\\mathbf{D}')(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Function('\\\\mathbf{D}')(Symbol('z', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))))"], [["times", 5, "Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Function('\\\\mathbf{D}')(Symbol('z', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Mul(Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Function('\\\\mathbb{I}')(Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Mul(Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["add", 8, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))), Add(Symbol('L', commutative=True), Mul(Add(Function('\\\\mathbf{D}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\chi)} = e^{\\chi}, then obtain (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} e^{- \\chi} - e^{\\chi}) ((e^{\\chi})^{\\chi})^{- \\chi} = (1 - e^{\\chi}) ((e^{\\chi})^{\\chi})^{- \\chi}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\chi)} = e^{\\chi} and \\operatorname{V_{\\mathbf{B}}}{(\\chi)} e^{- \\chi} = 1 and \\operatorname{V_{\\mathbf{B}}}{(\\chi)} e^{- \\chi} - e^{\\chi} = 1 - e^{\\chi} and \\operatorname{V_{\\mathbf{B}}}^{\\chi}{(\\chi)} = (e^{\\chi})^{\\chi} and (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} e^{- \\chi} - e^{\\chi}) (\\operatorname{V_{\\mathbf{B}}}^{\\chi}{(\\chi)})^{- \\chi} = (1 - e^{\\chi}) (\\operatorname{V_{\\mathbf{B}}}^{\\chi}{(\\chi)})^{- \\chi} and (\\operatorname{V_{\\mathbf{B}}}{(\\chi)} e^{- \\chi} - e^{\\chi}) ((e^{\\chi})^{\\chi})^{- \\chi} = (1 - e^{\\chi}) ((e^{\\chi})^{\\chi})^{- \\chi}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1))"], [["minus", 2, "exp(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 3, "Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Add(Integer(1), Mul(Integer(-1), exp(Symbol('\\\\chi', commutative=True)))), Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(M)} = e^{M}, then obtain (\\hat{\\mathbf{x}}{(M)} + \\int 1 dM) \\iint \\hat{\\mathbf{x}}{(M)} e^{- M} dM dM = (\\hat{\\mathbf{x}}{(M)} + \\int 1 dM) \\iint 1 dM dM", "derivation": "\\hat{\\mathbf{x}}{(M)} = e^{M} and \\hat{\\mathbf{x}}{(M)} e^{- M} = 1 and \\int \\hat{\\mathbf{x}}{(M)} e^{- M} dM = \\int 1 dM and \\iint \\hat{\\mathbf{x}}{(M)} e^{- M} dM dM = \\iint 1 dM dM and (\\hat{\\mathbf{x}}{(M)} + \\int 1 dM) \\iint \\hat{\\mathbf{x}}{(M)} e^{- M} dM dM = (\\hat{\\mathbf{x}}{(M)} + \\int 1 dM) \\iint 1 dM dM", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["divide", 1, "exp(Symbol('M', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integral(Integer(1), Tuple(Symbol('M', commutative=True))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Integer(1), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["times", 4, "Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), Integral(Integer(1), Tuple(Symbol('M', commutative=True))))"], "Equality(Mul(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), Integral(Integer(1), Tuple(Symbol('M', commutative=True)))), Integral(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('M', commutative=True)), Integral(Integer(1), Tuple(Symbol('M', commutative=True)))), Integral(Integer(1), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(x,\\theta_1)} = \\log{(x^{\\theta_1})}, then obtain \\frac{\\partial^{2}}{\\partial \\theta_1^{2}} - \\dot{x}^{2}{(x,\\theta_1)} = \\frac{\\partial^{2}}{\\partial \\theta_1^{2}} - \\dot{x}{(x,\\theta_1)} \\log{(x^{\\theta_1})}", "derivation": "\\dot{x}{(x,\\theta_1)} = \\log{(x^{\\theta_1})} and - \\dot{x}^{2}{(x,\\theta_1)} = - \\dot{x}{(x,\\theta_1)} \\log{(x^{\\theta_1})} and \\frac{\\partial}{\\partial \\theta_1} - \\dot{x}^{2}{(x,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} - \\dot{x}{(x,\\theta_1)} \\log{(x^{\\theta_1})} and \\frac{\\partial^{2}}{\\partial \\theta_1^{2}} - \\dot{x}^{2}{(x,\\theta_1)} = \\frac{\\partial^{2}}{\\partial \\theta_1^{2}} - \\dot{x}{(x,\\theta_1)} \\log{(x^{\\theta_1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Pow(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Pow(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Pow(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Pow(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{r}{(g_{\\varepsilon})} = e^{e^{g_{\\varepsilon}}}, then derive \\frac{d}{d g_{\\varepsilon}} \\mathbf{r}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}}, then obtain \\int \\frac{d}{d g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} dg_{\\varepsilon} = \\int e^{g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} dg_{\\varepsilon}", "derivation": "\\mathbf{r}{(g_{\\varepsilon})} = e^{e^{g_{\\varepsilon}}} and \\frac{d}{d g_{\\varepsilon}} \\mathbf{r}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} and \\frac{d}{d g_{\\varepsilon}} \\mathbf{r}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} and \\frac{d}{d g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} = e^{g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} and \\int \\frac{d}{d g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} dg_{\\varepsilon} = \\int e^{g_{\\varepsilon}} e^{e^{g_{\\varepsilon}}} dg_{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(exp(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given a{(g)} = \\log{(\\cos{(g)})}, then derive 0 = \\frac{d}{d g} \\frac{\\tilde{\\infty} (- a{(g)} + \\log{(\\cos{(g)})})}{a{(g)}}, then obtain 0 = - a{(g)} \\frac{d}{d g} \\frac{\\tilde{\\infty} (- a{(g)} + \\log{(\\cos{(g)})})}{a{(g)}}", "derivation": "a{(g)} = \\log{(\\cos{(g)})} and 0 = - a{(g)} + \\log{(\\cos{(g)})} and 0 = - \\frac{- a{(g)} + \\log{(\\cos{(g)})}}{a{(g)}} and 0 = - \\frac{- a{(g)} + \\log{(\\cos{(g)})}}{a{(g)} \\frac{d}{d g} 0} and \\frac{d}{d g} 0 = \\frac{d}{d g} - \\frac{- a{(g)} + \\log{(\\cos{(g)})}}{a{(g)} \\frac{d}{d g} 0} and 0 = \\frac{d}{d g} \\frac{\\tilde{\\infty} (- a{(g)} + \\log{(\\cos{(g)})})}{a{(g)}} and 0 = - a{(g)} \\frac{d}{d g} \\frac{\\tilde{\\infty} (- a{(g)} + \\log{(\\cos{(g)})})}{a{(g)}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('g', commutative=True)), log(cos(Symbol('g', commutative=True))))"], [["minus", 1, "Function('a')(Symbol('g', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Function('a')(Symbol('g', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))), Pow(Function('a')(Symbol('g', commutative=True)), Integer(-1))))"], [["divide", 3, "Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))), Pow(Function('a')(Symbol('g', commutative=True)), Integer(-1)), Pow(Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))), Pow(Function('a')(Symbol('g', commutative=True)), Integer(-1)), Pow(Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Derivative(Mul(zoo, Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))), Pow(Function('a')(Symbol('g', commutative=True)), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 6, "Mul(Integer(-1), Function('a')(Symbol('g', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Function('a')(Symbol('g', commutative=True)), Derivative(Mul(zoo, Add(Mul(Integer(-1), Function('a')(Symbol('g', commutative=True))), log(cos(Symbol('g', commutative=True)))), Pow(Function('a')(Symbol('g', commutative=True)), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{f},L)} = L - \\mathbf{f}, then obtain L (- \\mathbf{f} + \\operatorname{F_{c}}{(\\mathbf{f},L)}) - L + \\mathbf{f} = L (L - 2 \\mathbf{f}) - L + \\mathbf{f}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{f},L)} = L - \\mathbf{f} and - \\mathbf{f} + \\operatorname{F_{c}}{(\\mathbf{f},L)} = L - 2 \\mathbf{f} and L (- \\mathbf{f} + \\operatorname{F_{c}}{(\\mathbf{f},L)}) = L (L - 2 \\mathbf{f}) and L (- \\mathbf{f} + \\operatorname{F_{c}}{(\\mathbf{f},L)}) - L + \\mathbf{f} = L (L - 2 \\mathbf{f}) - L + \\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)))), Mul(Symbol('L', commutative=True), Add(Symbol('L', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["minus", 3, "Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('L', commutative=True)))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Symbol('L', commutative=True), Add(Symbol('L', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(I,\\tilde{g}^*)} = \\int I \\tilde{g}^* dI, then obtain \\iint \\operatorname{v_{1}}^{I}{(I,\\tilde{g}^*)} dI dI = \\iint (\\int I \\tilde{g}^* dI)^{I} dI dI", "derivation": "\\operatorname{v_{1}}{(I,\\tilde{g}^*)} = \\int I \\tilde{g}^* dI and \\operatorname{v_{1}}^{I}{(I,\\tilde{g}^*)} = (\\int I \\tilde{g}^* dI)^{I} and \\int \\operatorname{v_{1}}^{I}{(I,\\tilde{g}^*)} dI = \\int (\\int I \\tilde{g}^* dI)^{I} dI and \\iint \\operatorname{v_{1}}^{I}{(I,\\tilde{g}^*)} dI dI = \\iint (\\int I \\tilde{g}^* dI)^{I} dI dI", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Mul(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('I', commutative=True)), Pow(Integral(Mul(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Function('v_1')(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Integral(Mul(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Function('v_1')(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Integral(Mul(Symbol('I', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\phi_1,\\Omega)} = \\Omega - \\phi_1, then derive \\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega = \\phi_1 (\\frac{\\Omega^{2}}{2} - \\Omega \\phi_1 + \\mathbf{f}), then obtain - \\frac{v_{x} (\\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega)^{\\phi_1}}{z} = - \\frac{v_{x} (\\phi_1 (\\frac{\\Omega^{2}}{2} - \\Omega \\phi_1 + \\mathbf{f}))^{\\phi_1}}{z}", "derivation": "\\sigma_{x}{(\\phi_1,\\Omega)} = \\Omega - \\phi_1 and \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega = \\int (\\Omega - \\phi_1) d\\Omega and \\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega = \\phi_1 \\int (\\Omega - \\phi_1) d\\Omega and \\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega = \\phi_1 (\\frac{\\Omega^{2}}{2} - \\Omega \\phi_1 + \\mathbf{f}) and (\\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega)^{\\phi_1} = (\\phi_1 (\\frac{\\Omega^{2}}{2} - \\Omega \\phi_1 + \\mathbf{f}))^{\\phi_1} and - \\frac{v_{x} (\\phi_1 \\int \\sigma_{x}{(\\phi_1,\\Omega)} d\\Omega)^{\\phi_1}}{z} = - \\frac{v_{x} (\\phi_1 (\\frac{\\Omega^{2}}{2} - \\Omega \\phi_1 + \\mathbf{f}))^{\\phi_1}}{z}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["times", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\phi_1', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 5, "Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1)), Symbol('z', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_x', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Integral(Function('\\\\sigma_x')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(V)} = \\sin{(V)}, then derive \\frac{d}{d V} \\operatorname{E_{\\lambda}}{(V)} = \\cos{(V)}, then obtain \\frac{d}{d V} \\sin{(V)} = \\cos{(V)}", "derivation": "\\operatorname{E_{\\lambda}}{(V)} = \\sin{(V)} and \\frac{d}{d V} \\operatorname{E_{\\lambda}}{(V)} = \\frac{d}{d V} \\sin{(V)} and \\frac{d}{d V} \\operatorname{E_{\\lambda}}{(V)} = \\cos{(V)} and \\frac{d}{d V} \\sin{(V)} = \\cos{(V)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), cos(Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), cos(Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(t)} = \\log{(t)}, then obtain e^{2} = e^{0^{t} + 1}", "derivation": "\\varepsilon_{0}{(t)} = \\log{(t)} and \\varepsilon_{0}{(t)} - \\log{(t)} = 0 and (\\varepsilon_{0}{(t)} - \\log{(t)})^{t} = 0^{t} and (\\varepsilon_{0}{(t)} - \\log{(t)})^{t} + \\varepsilon_{0}^{t}{(t)} = 0^{t} + \\varepsilon_{0}^{t}{(t)} and 2 = (\\varepsilon_{0}{(t)} - \\log{(t)})^{t} + 1 and 2 = 0^{t} + 1 and e^{2} = e^{0^{t} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["minus", 1, "log(Symbol('t', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Pow(Integer(0), Symbol('t', commutative=True)))"], [["add", 3, "Pow(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Symbol('t', commutative=True))"], "Equality(Add(Pow(Add(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Pow(Integer(0), Symbol('t', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(2), Add(Pow(Add(Function('\\\\varepsilon_0')(Symbol('t', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(2), Add(Pow(Integer(0), Symbol('t', commutative=True)), Integer(1)))"], [["exp", 6], "Equality(exp(Integer(2)), exp(Add(Pow(Integer(0), Symbol('t', commutative=True)), Integer(1))))"]]}, {"prompt": "Given x{(g)} = \\sin{(\\sin{(g)})} and l{(A_{2},z^{*})} = \\cos{(A_{2} z^{*})}, then obtain x{(g)} - \\sin{(g)} + \\sin{(\\sin{(g)})} - \\cos{(A_{2} z^{*})} = - \\sin{(g)} + 2 \\sin{(\\sin{(g)})} - \\cos{(A_{2} z^{*})}", "derivation": "x{(g)} = \\sin{(\\sin{(g)})} and x{(g)} - \\sin{(g)} = - \\sin{(g)} + \\sin{(\\sin{(g)})} and 2 x{(g)} - \\sin{(g)} = x{(g)} - \\sin{(g)} + \\sin{(\\sin{(g)})} and 2 x{(g)} - \\sin{(g)} = - \\sin{(g)} + 2 \\sin{(\\sin{(g)})} and x{(g)} - \\sin{(g)} + \\sin{(\\sin{(g)})} = - \\sin{(g)} + 2 \\sin{(\\sin{(g)})} and l{(A_{2},z^{*})} = \\cos{(A_{2} z^{*})} and - l{(A_{2},z^{*})} + x{(g)} - \\sin{(g)} + \\sin{(\\sin{(g)})} = - l{(A_{2},z^{*})} - \\sin{(g)} + 2 \\sin{(\\sin{(g)})} and x{(g)} - \\sin{(g)} + \\sin{(\\sin{(g)})} - \\cos{(A_{2} z^{*})} = - \\sin{(g)} + 2 \\sin{(\\sin{(g)})} - \\cos{(A_{2} z^{*})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('g', commutative=True)), sin(sin(Symbol('g', commutative=True))))"], [["minus", 1, "sin(Symbol('g', commutative=True))"], "Equality(Add(Function('x')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('g', commutative=True))), sin(sin(Symbol('g', commutative=True)))))"], [["add", 2, "Function('x')(Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('g', commutative=True))), Mul(Integer(-1), sin(Symbol('g', commutative=True)))), Add(Function('x')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True))), sin(sin(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('g', commutative=True))), Mul(Integer(-1), sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('g', commutative=True))), Mul(Integer(2), sin(sin(Symbol('g', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('x')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True))), sin(sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('g', commutative=True))), Mul(Integer(2), sin(sin(Symbol('g', commutative=True))))))"], ["get_premise", "Equality(Function('l')(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))))"], [["minus", 5, "Function('l')(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))), Function('x')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True))), sin(sin(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Function('l')(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), sin(Symbol('g', commutative=True))), Mul(Integer(2), sin(sin(Symbol('g', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Function('x')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Symbol('g', commutative=True))), sin(sin(Symbol('g', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))))), Add(Mul(Integer(-1), sin(Symbol('g', commutative=True))), Mul(Integer(2), sin(sin(Symbol('g', commutative=True)))), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Symbol('z^*', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}_0{(c,\\mathbf{J}_f)} = \\mathbf{J}_f + c and m{(c,\\mathbf{J}_f)} = \\mathbf{J}_f + c, then obtain 1 = e^{- \\mathbf{J}_f - c + m{(c,\\mathbf{J}_f)}}", "derivation": "\\hat{p}_0{(c,\\mathbf{J}_f)} = \\mathbf{J}_f + c and m{(c,\\mathbf{J}_f)} = \\mathbf{J}_f + c and m{(c,\\mathbf{J}_f)} = \\hat{p}_0{(c,\\mathbf{J}_f)} and - \\hat{p}_0{(c,\\mathbf{J}_f)} + m{(c,\\mathbf{J}_f)} = \\mathbf{J}_f + c - \\hat{p}_0{(c,\\mathbf{J}_f)} and \\hat{p}_0{(c,\\mathbf{J}_f)} - m{(c,\\mathbf{J}_f)} = - \\mathbf{J}_f - c + \\hat{p}_0{(c,\\mathbf{J}_f)} and e^{\\hat{p}_0{(c,\\mathbf{J}_f)} - m{(c,\\mathbf{J}_f)}} = e^{- \\mathbf{J}_f - c + \\hat{p}_0{(c,\\mathbf{J}_f)}} and 1 = e^{- \\mathbf{J}_f - c + m{(c,\\mathbf{J}_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 2, "Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["exp", 5], "Equality(exp(Add(Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{p}_0')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('m')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(v_{x})} = \\cos{(v_{x})} and \\operatorname{F_{x}}{(v_{x})} = \\frac{d}{d v_{x}} \\tilde{g}^*{(v_{x})}, then obtain \\int \\operatorname{F_{x}}{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})} dv_{x} = \\int (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} dv_{x}", "derivation": "\\tilde{g}^*{(v_{x})} = \\cos{(v_{x})} and \\frac{d}{d v_{x}} \\tilde{g}^*{(v_{x})} = \\frac{d}{d v_{x}} \\cos{(v_{x})} and \\operatorname{F_{x}}{(v_{x})} = \\frac{d}{d v_{x}} \\tilde{g}^*{(v_{x})} and \\frac{d}{d v_{x}} \\tilde{g}^*{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})} = (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} and \\operatorname{F_{x}}{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})} = (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} and \\int \\operatorname{F_{x}}{(v_{x})} \\frac{d}{d v_{x}} \\cos{(v_{x})} dv_{x} = \\int (\\frac{d}{d v_{x}} \\cos{(v_{x})})^{2} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('v_x', commutative=True)), Derivative(Function('\\\\tilde{g}^*')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["times", 2, "Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\tilde{g}^*')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('F_x')(Symbol('v_x', commutative=True)), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(2)))"], [["integrate", 5, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Function('F_x')(Symbol('v_x', commutative=True)), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Tuple(Symbol('v_x', commutative=True))), Integral(Pow(Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} y, then derive \\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = y, then obtain \\frac{\\partial^{3}}{\\partial y^{2}\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} y = \\frac{d^{2}}{d y^{2}} y", "derivation": "\\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} y and \\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = y and \\frac{\\partial}{\\partial y} \\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = \\frac{d}{d y} y and \\frac{\\partial^{2}}{\\partial y^{2}} \\operatorname{M_{E}}{(y,\\hat{H}_{\\lambda})} = \\frac{d^{2}}{d y^{2}} y and \\frac{\\partial^{3}}{\\partial y^{2}\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} y = \\frac{d^{2}}{d y^{2}} y", "srepr_derivation": [["get_premise", "Equality(Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('y', commutative=True))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Derivative(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(2))), Derivative(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(k)} = \\sin{(k)}, then derive - \\cos{(k)} + 2 \\frac{d}{d k} \\operatorname{v_{t}}{(k)} = \\frac{d}{d k} \\operatorname{v_{t}}{(k)}, then obtain \\int (- 2 \\cos{(k)} + 4 \\frac{d}{d k} \\sin{(k)}) dk = \\int (- \\cos{(k)} + 3 \\frac{d}{d k} \\sin{(k)}) dk", "derivation": "\\operatorname{v_{t}}{(k)} = \\sin{(k)} and 2 \\operatorname{v_{t}}{(k)} = \\operatorname{v_{t}}{(k)} + \\sin{(k)} and 2 \\operatorname{v_{t}}{(k)} - \\sin{(k)} = \\operatorname{v_{t}}{(k)} and \\frac{d}{d k} (2 \\operatorname{v_{t}}{(k)} - \\sin{(k)}) = \\frac{d}{d k} \\operatorname{v_{t}}{(k)} and - \\cos{(k)} + 2 \\frac{d}{d k} \\operatorname{v_{t}}{(k)} = \\frac{d}{d k} \\operatorname{v_{t}}{(k)} and - \\cos{(k)} + 2 \\frac{d}{d k} \\sin{(k)} = \\frac{d}{d k} \\sin{(k)} and - 2 \\cos{(k)} + 4 \\frac{d}{d k} \\sin{(k)} = - \\cos{(k)} + 3 \\frac{d}{d k} \\sin{(k)} and \\int (- 2 \\cos{(k)} + 4 \\frac{d}{d k} \\sin{(k)}) dk = \\int (- \\cos{(k)} + 3 \\frac{d}{d k} \\sin{(k)}) dk", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["add", 1, "Function('v_t')(Symbol('k', commutative=True))"], "Equality(Mul(Integer(2), Function('v_t')(Symbol('k', commutative=True))), Add(Function('v_t')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))))"], [["minus", 2, "sin(Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('v_t')(Symbol('k', commutative=True))), Mul(Integer(-1), sin(Symbol('k', commutative=True)))), Function('v_t')(Symbol('k', commutative=True)))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('v_t')(Symbol('k', commutative=True))), Mul(Integer(-1), sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('v_t')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Mul(Integer(2), Derivative(Function('v_t')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Derivative(Function('v_t')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Mul(Integer(2), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["add", 6, "Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Mul(Integer(2), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), Integer(2), cos(Symbol('k', commutative=True))), Mul(Integer(4), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Mul(Integer(3), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["integrate", 7, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), cos(Symbol('k', commutative=True))), Mul(Integer(4), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Mul(Integer(3), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})}, then derive \\int \\mathbf{f}{(\\mathbf{M})} d\\mathbf{M} = S + \\cos{(\\mathbf{M})}, then obtain \\hat{X} + \\cos{(\\mathbf{M})} = S + \\cos{(\\mathbf{M})}", "derivation": "\\mathbf{f}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} and \\int \\mathbf{f}{(\\mathbf{M})} d\\mathbf{M} = \\int \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} d\\mathbf{M} and \\int \\mathbf{f}{(\\mathbf{M})} d\\mathbf{M} = S + \\cos{(\\mathbf{M})} and \\int \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} d\\mathbf{M} = S + \\cos{(\\mathbf{M})} and \\hat{X} + \\cos{(\\mathbf{M})} = S + \\cos{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given L{(h,\\varepsilon)} = \\varepsilon h, then obtain \\int - \\frac{- h + L{(h,\\varepsilon)}}{h} d\\varepsilon = \\int - \\frac{\\varepsilon h - h}{h} d\\varepsilon", "derivation": "L{(h,\\varepsilon)} = \\varepsilon h and - h + L{(h,\\varepsilon)} = \\varepsilon h - h and - \\frac{- h + L{(h,\\varepsilon)}}{h} = - \\frac{\\varepsilon h - h}{h} and \\int - \\frac{- h + L{(h,\\varepsilon)}}{h} d\\varepsilon = \\int - \\frac{\\varepsilon h - h}{h} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('L')(Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('L')(Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('L')(Symbol('h', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(y^{\\prime})} = \\sin{(y^{\\prime})} and \\mathbf{f}{(\\varphi^*,F_{H})} = \\log{(F_{H} + \\varphi^*)}, then derive \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = E_{x} - \\cos{(y^{\\prime})}, then obtain \\mathbf{f}{(\\varphi^*,F_{H})} + \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = E_{x} + \\mathbf{f}{(\\varphi^*,F_{H})} - \\cos{(y^{\\prime})}", "derivation": "\\operatorname{r_{0}}{(y^{\\prime})} = \\sin{(y^{\\prime})} and \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = \\int \\sin{(y^{\\prime})} dy^{\\prime} and \\mathbf{f}{(\\varphi^*,F_{H})} = \\log{(F_{H} + \\varphi^*)} and \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = E_{x} - \\cos{(y^{\\prime})} and \\log{(F_{H} + \\varphi^*)} + \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = E_{x} + \\log{(F_{H} + \\varphi^*)} - \\cos{(y^{\\prime})} and \\mathbf{f}{(\\varphi^*,F_{H})} + \\int \\operatorname{r_{0}}{(y^{\\prime})} dy^{\\prime} = E_{x} + \\mathbf{f}{(\\varphi^*,F_{H})} - \\cos{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(sin(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), log(Add(Symbol('F_H', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 4, "log(Add(Symbol('F_H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(log(Add(Symbol('F_H', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(Function('r_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('E_x', commutative=True), log(Add(Symbol('F_H', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Integral(Function('r_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('E_x', commutative=True), Function('\\\\mathbf{f}')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Mul(Integer(-1), cos(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{p},x^\\prime)} = \\hat{p} + x^\\prime, then obtain 0^{\\hat{p}} \\hat{p} = \\hat{p} (\\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)})^{\\hat{p}}", "derivation": "\\tilde{g}^*{(\\hat{p},x^\\prime)} = \\hat{p} + x^\\prime and \\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)} = \\tilde{\\infty} (\\hat{p} + x^\\prime) and 0 = \\tilde{\\infty} (\\hat{p} + x^\\prime) + \\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)} and 0 = \\tilde{\\infty} (\\hat{p} + x^\\prime) and 0 = \\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)} and 0^{\\hat{p}} = (\\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)})^{\\hat{p}} and 0^{\\hat{p}} \\hat{p} = \\hat{p} (\\tilde{\\infty} \\tilde{g}^*{(\\hat{p},x^\\prime)})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, 0], "Equality(Mul(zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(zoo, Add(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 2, "Mul(zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Integer(0), Add(Mul(zoo, Add(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integer(0), Mul(Integer(2), zoo, Add(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Mul(zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["power", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["times", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Mul(zoo, Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given q{(l)} = \\cos{(l)}, then obtain (\\int \\frac{q{(l)}}{l} dl)^{l} = (\\int \\frac{\\cos{(l)}}{l} dl)^{l}", "derivation": "q{(l)} = \\cos{(l)} and \\frac{q{(l)}}{l} = \\frac{\\cos{(l)}}{l} and \\int \\frac{q{(l)}}{l} dl = \\int \\frac{\\cos{(l)}}{l} dl and (\\int \\frac{q{(l)}}{l} dl)^{l} = (\\int \\frac{\\cos{(l)}}{l} dl)^{l}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["divide", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('q')(Symbol('l', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), cos(Symbol('l', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('q')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('q')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\hat{\\mathbf{r}},U)} = - U + \\hat{\\mathbf{r}} and V{(\\hat{\\mathbf{r}},U)} = \\iint \\eta{(\\hat{\\mathbf{r}},U)} dU d\\hat{\\mathbf{r}}, then obtain V{(\\hat{\\mathbf{r}},U)} = \\iint (- U + \\hat{\\mathbf{r}}) dU d\\hat{\\mathbf{r}}", "derivation": "\\eta{(\\hat{\\mathbf{r}},U)} = - U + \\hat{\\mathbf{r}} and \\int \\eta{(\\hat{\\mathbf{r}},U)} dU = \\int (- U + \\hat{\\mathbf{r}}) dU and \\iint \\eta{(\\hat{\\mathbf{r}},U)} dU d\\hat{\\mathbf{r}} = \\iint (- U + \\hat{\\mathbf{r}}) dU d\\hat{\\mathbf{r}} and V{(\\hat{\\mathbf{r}},U)} = \\iint \\eta{(\\hat{\\mathbf{r}},U)} dU d\\hat{\\mathbf{r}} and V{(\\hat{\\mathbf{r}},U)} = \\iint (- U + \\hat{\\mathbf{r}}) dU d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('V')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('U', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{B},\\mu)} = \\sin^{\\mathbf{B}}{(\\mu)}, then obtain (\\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)})^{\\mathbf{B}} + \\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)} = ((\\sin^{\\mathbf{B}}{(\\mu)})^{\\mu})^{\\mathbf{B}} + \\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)}", "derivation": "\\phi_{2}{(\\mathbf{B},\\mu)} = \\sin^{\\mathbf{B}}{(\\mu)} and \\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)} = (\\sin^{\\mathbf{B}}{(\\mu)})^{\\mu} and (\\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)})^{\\mathbf{B}} = ((\\sin^{\\mathbf{B}}{(\\mu)})^{\\mu})^{\\mathbf{B}} and (\\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)})^{\\mathbf{B}} + \\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)} = ((\\sin^{\\mathbf{B}}{(\\mu)})^{\\mu})^{\\mathbf{B}} + \\phi_{2}^{\\mu}{(\\mathbf{B},\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Pow(Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 3, "Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Add(Pow(Pow(Pow(sin(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(b,\\eta)} = \\log{(\\eta + b)} and \\mathbf{E}{(v,\\tilde{g})} = \\tilde{g} v and \\mu_{0}{(b,\\eta,v,\\tilde{g})} = \\mathbf{E}{(v,\\tilde{g})} - \\log{(\\eta + b)}, then obtain b \\mu_{0}{(b,\\eta,v,\\tilde{g})} = b (\\tilde{g} v - \\log{(\\eta + b)})", "derivation": "\\tilde{g}{(b,\\eta)} = \\log{(\\eta + b)} and \\mathbf{E}{(v,\\tilde{g})} = \\tilde{g} v and \\mathbf{E}{(v,\\tilde{g})} - \\tilde{g}{(b,\\eta)} = \\tilde{g} v - \\tilde{g}{(b,\\eta)} and \\mathbf{E}{(v,\\tilde{g})} - \\log{(\\eta + b)} = \\tilde{g} v - \\log{(\\eta + b)} and \\mu_{0}{(b,\\eta,v,\\tilde{g})} = \\mathbf{E}{(v,\\tilde{g})} - \\log{(\\eta + b)} and \\mu_{0}{(b,\\eta,v,\\tilde{g})} = \\tilde{g} v - \\log{(\\eta + b)} and b \\mu_{0}{(b,\\eta,v,\\tilde{g})} = b (\\tilde{g} v - \\log{(\\eta + b)})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)))"], [["minus", 2, "Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True))))), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Function('\\\\mathbf{E}')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mu_0')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True))))))"], [["times", 6, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\mu_0')(Symbol('b', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('b', commutative=True), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\eta', commutative=True), Symbol('b', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\eta,A_{z})} = \\cos{(A_{z} - \\eta)}, then derive \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\Omega - \\sin{(A_{z} - \\eta)}, then derive \\frac{\\partial}{\\partial \\eta} \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\cos{(A_{z} - \\eta)}, then obtain \\operatorname{f^{*}}{(\\eta,A_{z})} = \\frac{\\partial}{\\partial \\eta} \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta", "derivation": "\\operatorname{f^{*}}{(\\eta,A_{z})} = \\cos{(A_{z} - \\eta)} and \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\int \\cos{(A_{z} - \\eta)} d\\eta and \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\Omega - \\sin{(A_{z} - \\eta)} and \\frac{\\partial}{\\partial \\eta} \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\frac{\\partial}{\\partial \\eta} (\\Omega - \\sin{(A_{z} - \\eta)}) and \\frac{\\partial}{\\partial \\eta} \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta = \\cos{(A_{z} - \\eta)} and \\operatorname{f^{*}}{(\\eta,A_{z})} = \\frac{\\partial}{\\partial \\eta} \\int \\operatorname{f^{*}}{(\\eta,A_{z})} d\\eta", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), cos(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))))"], [["differentiate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integral(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), cos(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Derivative(Integral(Function('f^*')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(L)} = \\log{(\\log{(L)})} and \\mathbf{g}{(L)} = - L + \\log{(\\log{(L)})}, then obtain \\frac{\\mathbf{g}{(L)}}{\\log{(L)}} + \\log{(\\log{(L)})} = \\frac{- L + \\log{(\\log{(L)})}}{\\log{(L)}} + \\log{(\\log{(L)})}", "derivation": "\\operatorname{x^{{\\}'}}{(L)} = \\log{(\\log{(L)})} and \\mathbf{g}{(L)} = - L + \\log{(\\log{(L)})} and \\frac{\\mathbf{g}{(L)}}{\\log{(L)}} = \\frac{- L + \\log{(\\log{(L)})}}{\\log{(L)}} and \\frac{\\mathbf{g}{(L)}}{\\log{(L)}} + \\operatorname{x^{{\\}'}}{(L)} = \\frac{- L + \\log{(\\log{(L)})}}{\\log{(L)}} + \\operatorname{x^{{\\}'}}{(L)} and \\frac{\\mathbf{g}{(L)}}{\\log{(L)}} + \\log{(\\log{(L)})} = \\frac{- L + \\log{(\\log{(L)})}}{\\log{(L)}} + \\log{(\\log{(L)})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True)))))"], [["divide", 2, "log(Symbol('L', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True)))), Pow(log(Symbol('L', commutative=True)), Integer(-1))))"], [["add", 3, "Function('x^\\\\prime')(Symbol('L', commutative=True))"], "Equality(Add(Mul(Function('\\\\mathbf{g}')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Function('x^\\\\prime')(Symbol('L', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True)))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Function('x^\\\\prime')(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Function('\\\\mathbf{g}')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(-1))), log(log(Symbol('L', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True)))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), log(log(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(A_{x},l)} = - \\sin{(A_{x} - l)} and \\psi^{*}{(A_{x},l)} = - \\sin^{2}{(A_{x} - l)}, then obtain \\dot{y}{(A_{x},l)} \\sin{(A_{x} - l)} + \\psi^{*}{(A_{x},l)} = 2 \\psi^{*}{(A_{x},l)}", "derivation": "\\dot{y}{(A_{x},l)} = - \\sin{(A_{x} - l)} and \\dot{y}{(A_{x},l)} \\sin{(A_{x} - l)} = - \\sin^{2}{(A_{x} - l)} and \\psi^{*}{(A_{x},l)} = - \\sin^{2}{(A_{x} - l)} and \\dot{y}{(A_{x},l)} \\sin{(A_{x} - l)} = \\psi^{*}{(A_{x},l)} and \\dot{y}{(A_{x},l)} \\sin{(A_{x} - l)} + \\psi^{*}{(A_{x},l)} = 2 \\psi^{*}{(A_{x},l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["times", 1, "sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Mul(Integer(-1), Pow(sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)))"], [["add", 4, "Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Mul(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('l', commutative=True))), Mul(Integer(2), Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\hat{H}_l,E_{x})} = E_{x} + e^{\\hat{H}_l}, then obtain \\frac{- \\hat{H}_l + \\phi^{\\hat{H}_l}{(\\hat{H}_l,E_{x})} - 1}{\\hat{H}_l} = \\frac{- \\hat{H}_l + (E_{x} + e^{\\hat{H}_l})^{\\hat{H}_l} - 1}{\\hat{H}_l}", "derivation": "\\phi{(\\hat{H}_l,E_{x})} = E_{x} + e^{\\hat{H}_l} and \\phi^{\\hat{H}_l}{(\\hat{H}_l,E_{x})} = (E_{x} + e^{\\hat{H}_l})^{\\hat{H}_l} and - \\hat{H}_l + \\phi^{\\hat{H}_l}{(\\hat{H}_l,E_{x})} = - \\hat{H}_l + (E_{x} + e^{\\hat{H}_l})^{\\hat{H}_l} and - \\hat{H}_l + \\phi^{\\hat{H}_l}{(\\hat{H}_l,E_{x})} - 1 = - \\hat{H}_l + (E_{x} + e^{\\hat{H}_l})^{\\hat{H}_l} - 1 and \\frac{- \\hat{H}_l + \\phi^{\\hat{H}_l}{(\\hat{H}_l,E_{x})} - 1}{\\hat{H}_l} = \\frac{- \\hat{H}_l + (E_{x} + e^{\\hat{H}_l})^{\\hat{H}_l} - 1}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)))"], [["divide", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and \\dot{x}{(\\mathbf{J}_P)} = \\sin^{2}{(\\mathbf{J}_P)}, then obtain \\operatorname{F_{N}}{(\\mathbf{J}_P)} \\sin{(\\mathbf{J}_P)} + \\operatorname{F_{N}}{(\\mathbf{J}_P)} = \\operatorname{F_{N}}^{2}{(\\mathbf{J}_P)} + \\operatorname{F_{N}}{(\\mathbf{J}_P)}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and \\operatorname{F_{N}}{(\\mathbf{J}_P)} \\sin{(\\mathbf{J}_P)} = \\sin^{2}{(\\mathbf{J}_P)} and \\operatorname{F_{N}}{(\\mathbf{J}_P)} \\sin{(\\mathbf{J}_P)} + \\operatorname{F_{N}}{(\\mathbf{J}_P)} = \\operatorname{F_{N}}{(\\mathbf{J}_P)} + \\sin^{2}{(\\mathbf{J}_P)} and \\dot{x}{(\\mathbf{J}_P)} = \\sin^{2}{(\\mathbf{J}_P)} and \\dot{x}{(\\mathbf{J}_P)} = \\operatorname{F_{N}}^{2}{(\\mathbf{J}_P)} and \\operatorname{F_{N}}^{2}{(\\mathbf{J}_P)} = \\sin^{2}{(\\mathbf{J}_P)} and \\operatorname{F_{N}}{(\\mathbf{J}_P)} \\sin{(\\mathbf{J}_P)} + \\operatorname{F_{N}}{(\\mathbf{J}_P)} = \\operatorname{F_{N}}^{2}{(\\mathbf{J}_P)} + \\operatorname{F_{N}}{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))"], [["add", 2, "Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Mul(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Add(Mul(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Pow(Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2)), Function('F_N')(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given L{(\\hat{x}_0,m)} = e^{\\hat{x}_0 + m} and \\varphi{(\\hat{x}_0,m)} = (e^{\\hat{x}_0 + m})^{m}, then obtain m (L^{m}{(\\hat{x}_0,m)})^{m} = m ((e^{\\hat{x}_0 + m})^{m})^{m}", "derivation": "L{(\\hat{x}_0,m)} = e^{\\hat{x}_0 + m} and L^{m}{(\\hat{x}_0,m)} = (e^{\\hat{x}_0 + m})^{m} and \\varphi{(\\hat{x}_0,m)} = (e^{\\hat{x}_0 + m})^{m} and \\varphi^{m}{(\\hat{x}_0,m)} = ((e^{\\hat{x}_0 + m})^{m})^{m} and \\varphi^{m}{(\\hat{x}_0,m)} = (L^{m}{(\\hat{x}_0,m)})^{m} and (L^{m}{(\\hat{x}_0,m)})^{m} = ((e^{\\hat{x}_0 + m})^{m})^{m} and m (L^{m}{(\\hat{x}_0,m)})^{m} = m ((e^{\\hat{x}_0 + m})^{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Pow(exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["times", 6, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Pow(Pow(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Pow(exp(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(a,u)} = a + u and k{(f_{E})} = f_{E}, then obtain \\frac{k{(f_{E})}}{2 a + 3 u} = \\frac{f_{E}}{2 a + 3 u}", "derivation": "\\operatorname{g_{\\varepsilon}}{(a,u)} = a + u and a + u + \\operatorname{g_{\\varepsilon}}{(a,u)} = 2 a + 2 u and a + 2 u + \\operatorname{g_{\\varepsilon}}{(a,u)} = 2 a + 3 u and k{(f_{E})} = f_{E} and \\frac{k{(f_{E})}}{a + 2 u + \\operatorname{g_{\\varepsilon}}{(a,u)}} = \\frac{f_{E}}{a + 2 u + \\operatorname{g_{\\varepsilon}}{(a,u)}} and \\frac{k{(f_{E})}}{2 a + 3 u} = \\frac{f_{E}}{2 a + 3 u}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True)), Add(Symbol('a', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Add(Symbol('a', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Symbol('a', commutative=True), Symbol('u', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))))"], [["add", 2, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(3), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], [["divide", 4, "Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True))), Integer(-1)), Function('k')(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(Add(Symbol('a', commutative=True), Mul(Integer(2), Symbol('u', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('u', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(3), Symbol('u', commutative=True))), Integer(-1)), Function('k')(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(Add(Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(3), Symbol('u', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{v}{(g_{\\varepsilon})} = g_{\\varepsilon}, then obtain 2 g_{\\varepsilon}^{g_{\\varepsilon}} (g_{\\varepsilon}^{g_{\\varepsilon}} + \\mathbf{v}^{g_{\\varepsilon}}{(g_{\\varepsilon})}) = 4 g_{\\varepsilon}^{2 g_{\\varepsilon}}", "derivation": "\\mathbf{v}{(g_{\\varepsilon})} = g_{\\varepsilon} and \\mathbf{v}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = g_{\\varepsilon}^{g_{\\varepsilon}} and g_{\\varepsilon}^{g_{\\varepsilon}} + \\mathbf{v}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = 2 g_{\\varepsilon}^{g_{\\varepsilon}} and 2 g_{\\varepsilon}^{g_{\\varepsilon}} (g_{\\varepsilon}^{g_{\\varepsilon}} + \\mathbf{v}^{g_{\\varepsilon}}{(g_{\\varepsilon})}) = 4 g_{\\varepsilon}^{2 g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 3, "Mul(Integer(2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(4), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\mathbf{r},v_{t})} = v_{t}^{\\mathbf{r}}, then derive \\frac{\\partial}{\\partial \\mathbf{r}} \\omega{(\\mathbf{r},v_{t})} = v_{t}^{\\mathbf{r}} \\log{(v_{t})}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\omega{(\\mathbf{r},v_{t})} dv_{t} = \\int v_{t}^{\\mathbf{r}} \\log{(v_{t})} dv_{t}", "derivation": "\\omega{(\\mathbf{r},v_{t})} = v_{t}^{\\mathbf{r}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\omega{(\\mathbf{r},v_{t})} = \\frac{\\partial}{\\partial \\mathbf{r}} v_{t}^{\\mathbf{r}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\omega{(\\mathbf{r},v_{t})} = v_{t}^{\\mathbf{r}} \\log{(v_{t})} and \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\omega{(\\mathbf{r},v_{t})} dv_{t} = \\int v_{t}^{\\mathbf{r}} \\log{(v_{t})} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Pow(Symbol('v_t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Mul(Pow(Symbol('v_t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('v_t', commutative=True))))"], [["integrate", 3, "Symbol('v_t', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Pow(Symbol('v_t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(t_{1})} = \\log{(\\cos{(t_{1})})} and n{(t_{1})} = \\frac{d}{d t_{1}} \\log{(\\cos{(t_{1})})}, then obtain n{(t_{1})} \\log{(\\cos{(t_{1})})} = \\log{(\\cos{(t_{1})})} \\frac{d}{d t_{1}} \\log{(\\cos{(t_{1})})}", "derivation": "\\operatorname{v_{z}}{(t_{1})} = \\log{(\\cos{(t_{1})})} and n{(t_{1})} = \\frac{d}{d t_{1}} \\log{(\\cos{(t_{1})})} and n{(t_{1})} \\operatorname{v_{z}}{(t_{1})} = \\operatorname{v_{z}}{(t_{1})} \\frac{d}{d t_{1}} \\log{(\\cos{(t_{1})})} and n{(t_{1})} \\log{(\\cos{(t_{1})})} = \\log{(\\cos{(t_{1})})} \\frac{d}{d t_{1}} \\log{(\\cos{(t_{1})})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('t_1', commutative=True)), log(cos(Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('t_1', commutative=True)), Derivative(log(cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["times", 2, "Function('v_z')(Symbol('t_1', commutative=True))"], "Equality(Mul(Function('n')(Symbol('t_1', commutative=True)), Function('v_z')(Symbol('t_1', commutative=True))), Mul(Function('v_z')(Symbol('t_1', commutative=True)), Derivative(log(cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('n')(Symbol('t_1', commutative=True)), log(cos(Symbol('t_1', commutative=True)))), Mul(log(cos(Symbol('t_1', commutative=True))), Derivative(log(cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(f)} = \\log{(f)}, then obtain \\sin{(\\rho_{b}{(f)} + \\frac{2 \\rho_{b}{(f)}}{\\log{(f)}} - 1)} = \\sin{(\\rho_{b}{(f)} + \\frac{\\rho_{b}{(f)}}{\\log{(f)}})}", "derivation": "\\rho_{b}{(f)} = \\log{(f)} and \\frac{\\rho_{b}{(f)}}{\\log{(f)}} = 1 and \\rho_{b}{(f)} + \\frac{\\rho_{b}{(f)}}{\\log{(f)}} = \\rho_{b}{(f)} + 1 and \\sin{(\\rho_{b}{(f)} + \\frac{\\rho_{b}{(f)}}{\\log{(f)}})} = \\sin{(\\rho_{b}{(f)} + 1)} and \\sin{(\\rho_{b}{(f)} + \\frac{2 \\rho_{b}{(f)}}{\\log{(f)}} - 1)} = \\sin{(\\rho_{b}{(f)} + \\frac{\\rho_{b}{(f)}}{\\log{(f)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["divide", 1, "log(Symbol('f', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Function('\\\\rho_b')(Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Mul(Function('\\\\rho_b')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1)))), Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Integer(1)))"], [["sin", 3], "Equality(sin(Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Mul(Function('\\\\rho_b')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1))))), sin(Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(sin(Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1))), Integer(-1))), sin(Add(Function('\\\\rho_b')(Symbol('f', commutative=True)), Mul(Function('\\\\rho_b')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(P_{e})} = \\cos{(\\sin{(P_{e})})}, then derive \\frac{d}{d P_{e}} \\operatorname{v_{z}}{(P_{e})} = - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})}, then obtain (\\frac{d}{d P_{e}} \\operatorname{v_{z}}{(P_{e})})^{P_{e}} = (- \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})})^{P_{e}}", "derivation": "\\operatorname{v_{z}}{(P_{e})} = \\cos{(\\sin{(P_{e})})} and \\frac{d}{d P_{e}} \\operatorname{v_{z}}{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})} and \\frac{d}{d P_{e}} \\operatorname{v_{z}}{(P_{e})} = - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})} and \\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})} = - \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})} and (\\frac{d}{d P_{e}} \\cos{(\\sin{(P_{e})})})^{P_{e}} = (- \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})})^{P_{e}} and (\\frac{d}{d P_{e}} \\operatorname{v_{z}}{(P_{e})})^{P_{e}} = (- \\sin{(\\sin{(P_{e})})} \\cos{(P_{e})})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('P_e', commutative=True)), cos(sin(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))))"], [["power", 4, "Symbol('P_e', commutative=True)"], "Equality(Pow(Derivative(cos(sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Pow(Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('v_z')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Pow(Mul(Integer(-1), sin(sin(Symbol('P_e', commutative=True))), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\dot{y},p)} = - \\dot{y} + p, then obtain \\frac{\\partial}{\\partial p} (- \\dot{y} + p) + \\int \\frac{\\partial}{\\partial p} \\operatorname{a^{\\dagger}}{(\\dot{y},p)} d\\dot{y} = \\frac{\\partial}{\\partial p} (- \\dot{y} + p) + \\int \\frac{\\partial}{\\partial p} (- \\dot{y} + p) d\\dot{y}", "derivation": "\\operatorname{a^{\\dagger}}{(\\dot{y},p)} = - \\dot{y} + p and \\frac{\\partial}{\\partial p} \\operatorname{a^{\\dagger}}{(\\dot{y},p)} = \\frac{\\partial}{\\partial p} (- \\dot{y} + p) and \\int \\frac{\\partial}{\\partial p} \\operatorname{a^{\\dagger}}{(\\dot{y},p)} d\\dot{y} = \\int \\frac{\\partial}{\\partial p} (- \\dot{y} + p) d\\dot{y} and \\frac{\\partial}{\\partial p} (- \\dot{y} + p) + \\int \\frac{\\partial}{\\partial p} \\operatorname{a^{\\dagger}}{(\\dot{y},p)} d\\dot{y} = \\frac{\\partial}{\\partial p} (- \\dot{y} + p) + \\int \\frac{\\partial}{\\partial p} (- \\dot{y} + p) d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 3, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(F_{g})} = \\cos{(F_{g})}, then obtain (- F_{g} + \\mathbf{P}{(F_{g})}) \\mathbf{P}^{F_{g}}{(F_{g})} = (- F_{g} + \\mathbf{P}{(F_{g})}) \\cos^{F_{g}}{(F_{g})}", "derivation": "\\mathbf{P}{(F_{g})} = \\cos{(F_{g})} and - F_{g} + \\mathbf{P}{(F_{g})} = - F_{g} + \\cos{(F_{g})} and \\mathbf{P}^{F_{g}}{(F_{g})} = \\cos^{F_{g}}{(F_{g})} and (- F_{g} + \\cos{(F_{g})}) \\mathbf{P}^{F_{g}}{(F_{g})} = (- F_{g} + \\cos{(F_{g})}) \\cos^{F_{g}}{(F_{g})} and (- F_{g} + \\mathbf{P}{(F_{g})}) \\mathbf{P}^{F_{g}}{(F_{g})} = (- F_{g} + \\mathbf{P}{(F_{g})}) \\cos^{F_{g}}{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True)))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(cos(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Pow(cos(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('F_g', commutative=True))), Pow(cos(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\mu,\\mathbf{M})} = \\frac{\\mu}{\\mathbf{M}} and \\operatorname{E_{x}}{(\\mu,\\mathbf{M})} = \\frac{\\mu}{\\mathbf{M}} and \\operatorname{F_{x}}{(\\mu,\\mathbf{M})} = \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})}, then obtain \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} + \\rho_{b}^{\\mu}{(\\mu,\\mathbf{M})} = \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} + \\operatorname{F_{x}}{(\\mu,\\mathbf{M})}", "derivation": "\\rho_{b}{(\\mu,\\mathbf{M})} = \\frac{\\mu}{\\mathbf{M}} and \\rho_{b}^{\\mu}{(\\mu,\\mathbf{M})} = (\\frac{\\mu}{\\mathbf{M}})^{\\mu} and \\operatorname{E_{x}}{(\\mu,\\mathbf{M})} = \\frac{\\mu}{\\mathbf{M}} and \\rho_{b}^{\\mu}{(\\mu,\\mathbf{M})} = \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} and \\operatorname{F_{x}}{(\\mu,\\mathbf{M})} = \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} and \\rho_{b}^{\\mu}{(\\mu,\\mathbf{M})} = \\operatorname{F_{x}}{(\\mu,\\mathbf{M})} and \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} + \\rho_{b}^{\\mu}{(\\mu,\\mathbf{M})} = \\operatorname{E_{x}}^{\\mu}{(\\mu,\\mathbf{M})} + \\operatorname{F_{x}}{(\\mu,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)), Function('F_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 6, "Pow(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True))), Add(Pow(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mu', commutative=True)), Function('F_x')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J})} = \\int e^{\\mathbf{J}} d\\mathbf{J}, then obtain \\iint (\\operatorname{F_{H}}{(\\mathbf{J})} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} d\\mathbf{J} = \\iint (\\int e^{\\mathbf{J}} d\\mathbf{J} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} d\\mathbf{J}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J})} = \\int e^{\\mathbf{J}} d\\mathbf{J} and \\operatorname{F_{H}}{(\\mathbf{J})} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}} = \\int e^{\\mathbf{J}} d\\mathbf{J} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}} and \\int (\\operatorname{F_{H}}{(\\mathbf{J})} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} = \\int (\\int e^{\\mathbf{J}} d\\mathbf{J} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} and \\iint (\\operatorname{F_{H}}{(\\mathbf{J})} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} d\\mathbf{J} = \\iint (\\int e^{\\mathbf{J}} d\\mathbf{J} - (\\int e^{\\mathbf{J}} d\\mathbf{J})^{\\mathbf{J}}) d\\mathbf{J} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 1, "Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Add(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Add(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Add(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\mu{(a)} = \\cos{(\\log{(a)})} and \\operatorname{J_{\\varepsilon}}{(a)} = \\cos{(\\log{(a)})}, then obtain \\operatorname{J_{\\varepsilon}}^{6}{(a)} \\mu^{3}{(a)} = \\operatorname{J_{\\varepsilon}}^{9}{(a)}", "derivation": "\\mu{(a)} = \\cos{(\\log{(a)})} and \\mu{(a)} \\cos{(\\log{(a)})} = \\cos^{2}{(\\log{(a)})} and \\operatorname{J_{\\varepsilon}}{(a)} = \\cos{(\\log{(a)})} and \\operatorname{J_{\\varepsilon}}{(a)} \\mu{(a)} \\cos{(\\log{(a)})} = \\operatorname{J_{\\varepsilon}}{(a)} \\cos^{2}{(\\log{(a)})} and \\operatorname{J_{\\varepsilon}}^{2}{(a)} \\mu{(a)} = \\operatorname{J_{\\varepsilon}}^{3}{(a)} and \\operatorname{J_{\\varepsilon}}^{6}{(a)} \\mu^{3}{(a)} = \\operatorname{J_{\\varepsilon}}^{9}{(a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True))))"], [["times", 1, "cos(log(Symbol('a', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True)))), Pow(cos(log(Symbol('a', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True))))"], [["times", 2, "Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Function('\\\\mu')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True)))), Mul(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Integer(2)), Function('\\\\mu')(Symbol('a', commutative=True))), Pow(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Integer(3)))"], [["power", 5, 3], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Integer(6)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(3))), Pow(Function('J_{\\\\varepsilon}')(Symbol('a', commutative=True)), Integer(9)))"]]}, {"prompt": "Given U{(\\hat{\\mathbf{r}})} = e^{e^{\\hat{\\mathbf{r}}}}, then obtain U^{2}{(\\hat{\\mathbf{r}})} e^{3 e^{\\hat{\\mathbf{r}}}} = e^{5 e^{\\hat{\\mathbf{r}}}}", "derivation": "U{(\\hat{\\mathbf{r}})} = e^{e^{\\hat{\\mathbf{r}}}} and U{(\\hat{\\mathbf{r}})} e^{e^{\\hat{\\mathbf{r}}}} = e^{2 e^{\\hat{\\mathbf{r}}}} and U^{2}{(\\hat{\\mathbf{r}})} e^{2 e^{\\hat{\\mathbf{r}}}} = e^{4 e^{\\hat{\\mathbf{r}}}} and U^{2}{(\\hat{\\mathbf{r}})} e^{3 e^{\\hat{\\mathbf{r}}}} = e^{5 e^{\\hat{\\mathbf{r}}}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 1, "exp(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Function('U')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('U')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), exp(Mul(Integer(4), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["times", 3, "exp(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Pow(Function('U')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2)), exp(Mul(Integer(3), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), exp(Mul(Integer(5), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given I{(x^\\prime)} = \\log{(x^\\prime)}, then derive \\frac{d}{d x^\\prime} I{(x^\\prime)} = \\frac{1}{x^\\prime}, then obtain \\frac{1}{x^\\prime} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)}", "derivation": "I{(x^\\prime)} = \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} I{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} I{(x^\\prime)} = \\frac{1}{x^\\prime} and \\frac{1}{x^\\prime} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(z)} = e^{z}, then derive \\int \\operatorname{v_{1}}{(z)} dz = G + e^{z}, then obtain (\\int \\operatorname{v_{1}}{(z)} dz)^{z} = (\\mathbf{J}_M + e^{z})^{z}", "derivation": "\\operatorname{v_{1}}{(z)} = e^{z} and \\int \\operatorname{v_{1}}{(z)} dz = \\int e^{z} dz and \\int \\operatorname{v_{1}}{(z)} dz = G + e^{z} and (\\int \\operatorname{v_{1}}{(z)} dz)^{z} = (G + e^{z})^{z} and (\\int e^{z} dz)^{z} = (G + e^{z})^{z} and (\\int \\operatorname{v_{1}}{(z)} dz)^{z} = (\\int e^{z} dz)^{z} and (\\int \\operatorname{v_{1}}{(z)} dz)^{z} = (\\mathbf{J}_M + e^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('G', commutative=True), exp(Symbol('z', commutative=True))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Integral(Function('v_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('G', commutative=True), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('G', commutative=True), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(Function('v_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Integral(Function('v_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given v{(z,H)} = H - z and k{(z,H)} = - z (H - z), then obtain (\\mathbf{A} + \\frac{\\partial}{\\partial H} k{(z,H)} - \\int z \\frac{\\partial}{\\partial H} v{(z,H)} dz)^{H} = (\\mathbf{A} + \\frac{\\partial}{\\partial H} - z v{(z,H)} - \\int z \\frac{\\partial}{\\partial H} v{(z,H)} dz)^{H}", "derivation": "v{(z,H)} = H - z and k{(z,H)} = - z (H - z) and \\frac{\\partial}{\\partial H} k{(z,H)} = \\frac{\\partial}{\\partial H} - z (H - z) and \\frac{\\partial}{\\partial H} k{(z,H)} = \\frac{\\partial}{\\partial H} - z v{(z,H)} and \\frac{\\partial}{\\partial H} k{(z,H)} + \\int \\frac{\\partial}{\\partial H} - z v{(z,H)} dz = \\frac{\\partial}{\\partial H} - z v{(z,H)} + \\int \\frac{\\partial}{\\partial H} - z v{(z,H)} dz and (\\frac{\\partial}{\\partial H} k{(z,H)} + \\int \\frac{\\partial}{\\partial H} - z v{(z,H)} dz)^{H} = (\\frac{\\partial}{\\partial H} - z v{(z,H)} + \\int \\frac{\\partial}{\\partial H} - z v{(z,H)} dz)^{H} and (\\mathbf{A} + \\frac{\\partial}{\\partial H} k{(z,H)} - \\int z \\frac{\\partial}{\\partial H} v{(z,H)} dz)^{H} = (\\mathbf{A} + \\frac{\\partial}{\\partial H} - z v{(z,H)} - \\int z \\frac{\\partial}{\\partial H} v{(z,H)} dz)^{H}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 4, "Integral(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Derivative(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))), Add(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Derivative(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))), Symbol('H', commutative=True)), Pow(Add(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))), Symbol('H', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Derivative(Function('k')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Symbol('z', commutative=True), Derivative(Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))))), Symbol('H', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Derivative(Mul(Integer(-1), Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Symbol('z', commutative=True), Derivative(Function('v')(Symbol('z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\hat{H},\\theta_2)} = \\hat{H} - \\theta_2, then obtain - \\hat{H} + \\mathbf{H} + \\theta_2 + \\mathbf{B}{(\\hat{H},\\theta_2)} = - \\hat{H} + \\varepsilon_0", "derivation": "\\mathbf{B}{(\\hat{H},\\theta_2)} = \\hat{H} - \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} \\mathbf{B}{(\\hat{H},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\hat{H} - \\theta_2) and \\int \\frac{\\partial}{\\partial \\theta_2} \\mathbf{B}{(\\hat{H},\\theta_2)} d\\theta_2 = \\int \\frac{\\partial}{\\partial \\theta_2} (\\hat{H} - \\theta_2) d\\theta_2 and - \\hat{H} + \\theta_2 + \\int \\frac{\\partial}{\\partial \\theta_2} \\mathbf{B}{(\\hat{H},\\theta_2)} d\\theta_2 = - \\hat{H} + \\theta_2 + \\int \\frac{\\partial}{\\partial \\theta_2} (\\hat{H} - \\theta_2) d\\theta_2 and - \\hat{H} + \\mathbf{H} + \\theta_2 + \\mathbf{B}{(\\hat{H},\\theta_2)} = - \\hat{H} + \\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta_2', commutative=True), Integral(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\theta_2', commutative=True), Integral(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given h{(W)} = \\log{(W)} and \\mathbf{g}{(W)} = \\log{(W)}, then derive \\frac{d}{d W} \\int h{(W)} dW = \\frac{\\partial}{\\partial W} (W \\log{(W)} - W + f_{\\mathbf{p}}), then obtain \\frac{d^{2}}{d W^{2}} \\int h{(W)} dW = W \\frac{d^{2}}{d W^{2}} \\mathbf{g}{(W)} + 2 \\frac{d}{d W} \\mathbf{g}{(W)}", "derivation": "h{(W)} = \\log{(W)} and \\int h{(W)} dW = \\int \\log{(W)} dW and \\mathbf{g}{(W)} = \\log{(W)} and \\frac{d}{d W} \\int h{(W)} dW = \\frac{d}{d W} \\int \\log{(W)} dW and \\frac{d}{d W} \\int h{(W)} dW = \\frac{\\partial}{\\partial W} (W \\log{(W)} - W + f_{\\mathbf{p}}) and \\frac{d}{d W} \\int h{(W)} dW = \\frac{\\partial}{\\partial W} (W \\mathbf{g}{(W)} - W + f_{\\mathbf{p}}) and \\frac{d^{2}}{d W^{2}} \\int h{(W)} dW = \\frac{\\partial^{2}}{\\partial W^{2}} (W \\mathbf{g}{(W)} - W + f_{\\mathbf{p}}) and \\frac{d^{2}}{d W^{2}} \\int h{(W)} dW = W \\frac{d^{2}}{d W^{2}} \\mathbf{g}{(W)} + 2 \\frac{d}{d W} \\mathbf{g}{(W)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('W', commutative=True), Function('\\\\mathbf{g}')(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Integral(Function('h')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(2))), Add(Mul(Symbol('W', commutative=True), Derivative(Function('\\\\mathbf{g}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(Function('\\\\mathbf{g}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))))"]]}, {"prompt": "Given s{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain 3 (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} + (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2} = 2 (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} + 2 (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2}", "derivation": "s{(\\theta_2)} = \\cos{(\\theta_2)} and \\frac{d}{d \\theta_2} s{(\\theta_2)} = \\frac{d}{d \\theta_2} \\cos{(\\theta_2)} and (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} = (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2} and 2 (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} = (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} + (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2} and 3 (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} + (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2} = 2 (\\frac{d}{d \\theta_2} s{(\\theta_2)})^{\\theta_2} + 2 (\\frac{d}{d \\theta_2} \\cos{(\\theta_2)})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], [["add", 3, "Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(2), Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))), Add(Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))))"], [["add", 4, "Add(Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Mul(Integer(3), Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))), Pow(Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(2), Pow(Derivative(Function('s')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(2), Pow(Derivative(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\rho_b,\\hbar)} = \\cos{(\\hbar + \\rho_b)}, then obtain (\\int \\operatorname{n_{2}}{(\\rho_b,\\hbar)} d\\rho_b)^{\\rho_b} = (x^\\prime + \\sin{(\\hbar + \\rho_b)})^{\\rho_b}", "derivation": "\\operatorname{n_{2}}{(\\rho_b,\\hbar)} = \\cos{(\\hbar + \\rho_b)} and \\int \\operatorname{n_{2}}{(\\rho_b,\\hbar)} d\\rho_b = \\int \\cos{(\\hbar + \\rho_b)} d\\rho_b and (\\int \\operatorname{n_{2}}{(\\rho_b,\\hbar)} d\\rho_b)^{\\rho_b} = (\\int \\cos{(\\hbar + \\rho_b)} d\\rho_b)^{\\rho_b} and (\\int \\operatorname{n_{2}}{(\\rho_b,\\hbar)} d\\rho_b)^{\\rho_b} = (x^\\prime + \\sin{(\\hbar + \\rho_b)})^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Integral(Function('n_2')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(cos(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('n_2')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('x^\\\\prime', commutative=True), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\sigma_x,v)} = \\sigma_x v and \\rho_{f}{(\\phi,\\nabla,E_{\\lambda})} = E_{\\lambda}^{\\nabla} \\phi, then obtain \\frac{\\rho_{f}{(\\phi,\\nabla,E_{\\lambda})}}{- v + \\eta^{\\prime}{(\\sigma_x,v)}} = \\frac{E_{\\lambda}^{\\nabla} \\phi}{- v + \\eta^{\\prime}{(\\sigma_x,v)}}", "derivation": "\\eta^{\\prime}{(\\sigma_x,v)} = \\sigma_x v and - v + \\eta^{\\prime}{(\\sigma_x,v)} = \\sigma_x v - v and \\rho_{f}{(\\phi,\\nabla,E_{\\lambda})} = E_{\\lambda}^{\\nabla} \\phi and \\frac{\\rho_{f}{(\\phi,\\nabla,E_{\\lambda})}}{\\sigma_x v - v} = \\frac{E_{\\lambda}^{\\nabla} \\phi}{\\sigma_x v - v} and \\frac{\\rho_{f}{(\\phi,\\nabla,E_{\\lambda})}}{- v + \\eta^{\\prime}{(\\sigma_x,v)}} = \\frac{E_{\\lambda}^{\\nabla} \\phi}{- v + \\eta^{\\prime}{(\\sigma_x,v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\phi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["divide", 3, "Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Integer(-1)), Function('\\\\rho_f')(Symbol('\\\\phi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\phi', commutative=True), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True))), Integer(-1)), Function('\\\\rho_f')(Symbol('\\\\phi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\phi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True), Symbol('v', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given A{(U)} = U, then derive (\\hat{p}_0 + \\frac{A^{2}{(U)}}{2})^{A{(U)}} = (\\int U dA{(U)})^{A{(U)}}, then obtain (\\hat{p}_0 + \\frac{A^{2}{(U)}}{2})^{A{(U)}} = (\\int A{(U)} dA{(U)})^{A{(U)}}", "derivation": "A{(U)} = U and \\int A{(U)} dU = \\int U dU and (\\int A{(U)} dU)^{U} = (\\int U dU)^{U} and \\int A{(U)} dA{(U)} = \\int U dA{(U)} and (\\int A{(U)} dA{(U)})^{A{(U)}} = (\\int U dA{(U)})^{A{(U)}} and (\\hat{p}_0 + \\frac{A^{2}{(U)}}{2})^{A{(U)}} = (\\int U dA{(U)})^{A{(U)}} and (\\hat{p}_0 + \\frac{A^{2}{(U)}}{2})^{A{(U)}} = (\\int A{(U)} dA{(U)})^{A{(U)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('U', commutative=True)), Symbol('U', commutative=True))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('A')(Symbol('U', commutative=True)), Tuple(Function('A')(Symbol('U', commutative=True)))), Integral(Symbol('U', commutative=True), Tuple(Function('A')(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Integral(Function('A')(Symbol('U', commutative=True)), Tuple(Function('A')(Symbol('U', commutative=True)))), Function('A')(Symbol('U', commutative=True))), Pow(Integral(Symbol('U', commutative=True), Tuple(Function('A')(Symbol('U', commutative=True)))), Function('A')(Symbol('U', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Rational(1, 2), Pow(Function('A')(Symbol('U', commutative=True)), Integer(2)))), Function('A')(Symbol('U', commutative=True))), Pow(Integral(Symbol('U', commutative=True), Tuple(Function('A')(Symbol('U', commutative=True)))), Function('A')(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Rational(1, 2), Pow(Function('A')(Symbol('U', commutative=True)), Integer(2)))), Function('A')(Symbol('U', commutative=True))), Pow(Integral(Function('A')(Symbol('U', commutative=True)), Tuple(Function('A')(Symbol('U', commutative=True)))), Function('A')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(Z)} = \\log{(e^{Z})}, then obtain (- A_{2} + Z + \\mathbf{r}{(Z)})^{Z} = (- A_{2} + Z + \\log{(e^{Z})})^{Z}", "derivation": "\\mathbf{r}{(Z)} = \\log{(e^{Z})} and Z + \\mathbf{r}{(Z)} = Z + \\log{(e^{Z})} and - A_{2} + Z + \\mathbf{r}{(Z)} = - A_{2} + Z + \\log{(e^{Z})} and (- A_{2} + Z + \\mathbf{r}{(Z)})^{Z} = (- A_{2} + Z + \\log{(e^{Z})})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('Z', commutative=True)), log(exp(Symbol('Z', commutative=True))))"], [["add", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\mathbf{r}')(Symbol('Z', commutative=True))), Add(Symbol('Z', commutative=True), log(exp(Symbol('Z', commutative=True)))))"], [["minus", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Z', commutative=True), Function('\\\\mathbf{r}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Z', commutative=True), log(exp(Symbol('Z', commutative=True)))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Z', commutative=True), Function('\\\\mathbf{r}')(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Z', commutative=True), log(exp(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and E{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})}, then obtain - f_{\\mathbf{v}} + \\hat{H}_{\\lambda}{(f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + E{(f_{\\mathbf{v}})}", "derivation": "\\hat{H}_{\\lambda}{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and - f_{\\mathbf{v}} + \\hat{H}_{\\lambda}{(f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + \\log{(f_{\\mathbf{v}})} and E{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and - f_{\\mathbf{v}} + \\hat{H}_{\\lambda}{(f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + E{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('E')(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(U)} = \\sin{(U)}, then obtain \\frac{\\frac{d}{d U} 1 - 1}{\\operatorname{C_{d}}{(U)}} = \\frac{\\frac{d}{d U} \\frac{\\sin{(U)}}{\\operatorname{C_{d}}{(U)}} - 1}{\\operatorname{C_{d}}{(U)}}", "derivation": "\\operatorname{C_{d}}{(U)} = \\sin{(U)} and 1 = \\frac{\\sin{(U)}}{\\operatorname{C_{d}}{(U)}} and \\frac{d}{d U} 1 = \\frac{d}{d U} \\frac{\\sin{(U)}}{\\operatorname{C_{d}}{(U)}} and \\frac{d}{d U} 1 - 1 = \\frac{d}{d U} \\frac{\\sin{(U)}}{\\operatorname{C_{d}}{(U)}} - 1 and \\frac{\\frac{d}{d U} 1 - 1}{\\operatorname{C_{d}}{(U)}} = \\frac{\\frac{d}{d U} \\frac{\\sin{(U)}}{\\operatorname{C_{d}}{(U)}} - 1}{\\operatorname{C_{d}}{(U)}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["divide", 1, "Function('C_d')(Symbol('U', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)))"], [["times", 4, "Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1))), Mul(Add(Derivative(Mul(Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Pow(Function('C_d')(Symbol('U', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(F_{c},A_{1})} = F_{c} + \\sin{(A_{1})} and \\psi^{*}{(\\hat{\\mathbf{r}})} = \\log{(e^{\\hat{\\mathbf{r}}})}, then obtain F_{c} + \\psi^{*}{(\\hat{\\mathbf{r}})} + \\sin{(A_{1})} = F_{c} + \\log{(e^{\\hat{\\mathbf{r}}})} + \\sin{(A_{1})}", "derivation": "\\mathbf{r}{(F_{c},A_{1})} = F_{c} + \\sin{(A_{1})} and \\psi^{*}{(\\hat{\\mathbf{r}})} = \\log{(e^{\\hat{\\mathbf{r}}})} and \\mathbf{r}{(F_{c},A_{1})} + \\psi^{*}{(\\hat{\\mathbf{r}})} = \\mathbf{r}{(F_{c},A_{1})} + \\log{(e^{\\hat{\\mathbf{r}}})} and F_{c} + \\psi^{*}{(\\hat{\\mathbf{r}})} + \\sin{(A_{1})} = F_{c} + \\log{(e^{\\hat{\\mathbf{r}}})} + \\sin{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('F_c', commutative=True), sin(Symbol('A_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('A_1', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('A_1', commutative=True)), log(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('F_c', commutative=True), Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('A_1', commutative=True))), Add(Symbol('F_c', commutative=True), log(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), sin(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\dot{z})} = \\log{(\\dot{z})}, then obtain (\\hbar + \\lambda^{\\dot{z}}{(\\dot{z})} + \\frac{1}{k})^{\\hbar} = (\\hbar + \\log{(\\dot{z})}^{\\dot{z}} + \\frac{1}{k})^{\\hbar}", "derivation": "\\lambda{(\\dot{z})} = \\log{(\\dot{z})} and \\lambda^{\\dot{z}}{(\\dot{z})} = \\log{(\\dot{z})}^{\\dot{z}} and \\lambda^{\\dot{z}}{(\\dot{z})} + \\frac{1}{k} = \\log{(\\dot{z})}^{\\dot{z}} + \\frac{1}{k} and \\hbar + \\lambda^{\\dot{z}}{(\\dot{z})} + \\frac{1}{k} = \\hbar + \\log{(\\dot{z})}^{\\dot{z}} + \\frac{1}{k} and (\\hbar + \\lambda^{\\dot{z}}{(\\dot{z})} + \\frac{1}{k})^{\\hbar} = (\\hbar + \\log{(\\dot{z})}^{\\dot{z}} + \\frac{1}{k})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["add", 2, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))), Add(Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["add", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))), Add(Symbol('\\\\hbar', commutative=True), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hbar', commutative=True), Pow(Function('\\\\lambda')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given b{(c,V_{\\mathbf{E}})} = V_{\\mathbf{E}} - c, then obtain V_{\\mathbf{E}} (V_{\\mathbf{E}} - c) + V_{\\mathbf{E}} - 2 c = V_{\\mathbf{E}}^{2} - V_{\\mathbf{E}} c + V_{\\mathbf{E}} - 2 c", "derivation": "b{(c,V_{\\mathbf{E}})} = V_{\\mathbf{E}} - c and V_{\\mathbf{E}} b{(c,V_{\\mathbf{E}})} = V_{\\mathbf{E}} (V_{\\mathbf{E}} - c) and V_{\\mathbf{E}} b{(c,V_{\\mathbf{E}})} - c = V_{\\mathbf{E}} (V_{\\mathbf{E}} - c) - c and V_{\\mathbf{E}} b{(c,V_{\\mathbf{E}})} + V_{\\mathbf{E}} - 2 c = V_{\\mathbf{E}} (V_{\\mathbf{E}} - c) + V_{\\mathbf{E}} - 2 c and V_{\\mathbf{E}} b{(c,V_{\\mathbf{E}})} + V_{\\mathbf{E}} - 2 c = V_{\\mathbf{E}}^{2} - V_{\\mathbf{E}} c + V_{\\mathbf{E}} - 2 c and V_{\\mathbf{E}} (V_{\\mathbf{E}} - c) + V_{\\mathbf{E}} - 2 c = V_{\\mathbf{E}}^{2} - V_{\\mathbf{E}} c + V_{\\mathbf{E}} - 2 c", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["times", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('b')(Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('b')(Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["add", 3, "Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('b')(Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))))"], [["expand", 4], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('b')(Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))), Add(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('c', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))), Add(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('c', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))))"]]}, {"prompt": "Given T{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and \\dot{x}{(\\Omega)} = - \\frac{1}{\\Omega^{2} T{(\\Omega)}}, then derive T{(\\Omega)} = \\frac{1}{\\Omega}, then obtain \\int (\\frac{1}{(\\int - \\frac{1}{\\Omega^{2} T{(\\Omega)}} d\\Omega)^{2}})^{\\Omega} d\\Omega = \\int (\\frac{1}{(\\int - \\frac{1}{\\Omega} d\\Omega)^{2}})^{\\Omega} d\\Omega", "derivation": "T{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and T{(\\Omega)} = \\frac{1}{\\Omega} and \\dot{x}{(\\Omega)} = - \\frac{1}{\\Omega^{2} T{(\\Omega)}} and \\dot{x}{(\\Omega)} = - \\frac{1}{\\Omega} and - \\frac{1}{\\Omega^{2} T{(\\Omega)}} = - \\frac{1}{\\Omega} and \\int - \\frac{1}{\\Omega^{2} T{(\\Omega)}} d\\Omega = \\int - \\frac{1}{\\Omega} d\\Omega and \\frac{1}{(\\int - \\frac{1}{\\Omega^{2} T{(\\Omega)}} d\\Omega)^{2}} = \\frac{1}{(\\int - \\frac{1}{\\Omega} d\\Omega)^{2}} and (\\frac{1}{(\\int - \\frac{1}{\\Omega^{2} T{(\\Omega)}} d\\Omega)^{2}})^{\\Omega} = (\\frac{1}{(\\int - \\frac{1}{\\Omega} d\\Omega)^{2}})^{\\Omega} and \\int (\\frac{1}{(\\int - \\frac{1}{\\Omega^{2} T{(\\Omega)}} d\\Omega)^{2}})^{\\Omega} d\\Omega = \\int (\\frac{1}{(\\int - \\frac{1}{\\Omega} d\\Omega)^{2}})^{\\Omega} d\\Omega", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('\\\\Omega', commutative=True)), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('T')(Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\dot{x}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 6, "Integer(-2)"], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)))"], [["power", 7, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 8, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Pow(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Pow(Pow(Integral(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-2)), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\psi^{*}{(\\hat{\\mathbf{r}})} = \\frac{\\sigma_{p}{(\\hat{\\mathbf{r}})}}{\\cos{(\\hat{\\mathbf{r}})}}, then derive \\frac{d}{d \\hat{\\mathbf{r}}} \\psi^{*}{(\\hat{\\mathbf{r}})} = 0, then obtain \\frac{d}{d \\hat{\\mathbf{r}}} 1 - 1 = -1", "derivation": "\\sigma_{p}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\psi^{*}{(\\hat{\\mathbf{r}})} = \\frac{\\sigma_{p}{(\\hat{\\mathbf{r}})}}{\\cos{(\\hat{\\mathbf{r}})}} and \\psi^{*}{(\\hat{\\mathbf{r}})} = 1 and \\frac{d}{d \\hat{\\mathbf{r}}} \\psi^{*}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} 1 and \\frac{d}{d \\hat{\\mathbf{r}}} \\psi^{*}{(\\hat{\\mathbf{r}})} = 0 and \\frac{d}{d \\hat{\\mathbf{r}}} \\psi^{*}{(\\hat{\\mathbf{r}})} - 1 = -1 and \\frac{d}{d \\hat{\\mathbf{r}}} \\frac{\\sigma_{p}{(\\hat{\\mathbf{r}})}}{\\cos{(\\hat{\\mathbf{r}})}} - 1 = -1 and \\frac{d}{d \\hat{\\mathbf{r}}} 1 - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(0))"], [["minus", 5, 1], "Equality(Add(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Derivative(Mul(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain \\frac{d}{d \\mathbf{J}_P} (\\mathbf{E}{(\\mathbf{J}_P)} + \\cos{(\\mathbf{E}{(\\mathbf{J}_P)})} - \\cos{(e^{\\mathbf{J}_P})}) = \\frac{d}{d \\mathbf{J}_P} \\mathbf{E}{(\\mathbf{J}_P)}", "derivation": "\\mathbf{E}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\cos{(\\mathbf{E}{(\\mathbf{J}_P)})} = \\cos{(e^{\\mathbf{J}_P})} and \\mathbf{E}{(\\mathbf{J}_P)} + \\cos{(\\mathbf{E}{(\\mathbf{J}_P)})} - \\cos{(e^{\\mathbf{J}_P})} = \\mathbf{E}{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} (\\mathbf{E}{(\\mathbf{J}_P)} + \\cos{(\\mathbf{E}{(\\mathbf{J}_P)})} - \\cos{(e^{\\mathbf{J}_P})}) = \\frac{d}{d \\mathbf{J}_P} \\mathbf{E}{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True))), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(C_{1},C_{d})} = C_{1} C_{d}, then derive \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = C_{d}, then obtain - C_{d} + \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = 0", "derivation": "f{(C_{1},C_{d})} = C_{1} C_{d} and \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = \\frac{\\partial}{\\partial C_{1}} C_{1} C_{d} and \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = C_{d} and - \\frac{\\partial}{\\partial C_{1}} C_{1} C_{d} + \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = C_{d} - \\frac{\\partial}{\\partial C_{1}} C_{1} C_{d} and \\frac{\\partial}{\\partial C_{1}} C_{1} C_{d} = C_{d} and - C_{d} + \\frac{\\partial}{\\partial C_{1}} f{(C_{1},C_{d})} = 0", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_d', commutative=True))"], [["minus", 3, "Derivative(Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Derivative(Function('f')(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_d', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Derivative(Function('f')(Symbol('C_1', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given x{(L)} = \\log{(\\sin{(L)})} and \\operatorname{v_{y}}{(L)} = \\log{(\\sin{(L)})}, then obtain (\\operatorname{v_{y}}^{L}{(L)})^{L} = (x^{L}{(L)})^{L}", "derivation": "x{(L)} = \\log{(\\sin{(L)})} and \\operatorname{v_{y}}{(L)} = \\log{(\\sin{(L)})} and \\operatorname{v_{y}}{(L)} = x{(L)} and \\operatorname{v_{y}}^{L}{(L)} = \\log{(\\sin{(L)})}^{L} and x^{L}{(L)} = \\log{(\\sin{(L)})}^{L} and \\operatorname{v_{y}}^{L}{(L)} = x^{L}{(L)} and (\\operatorname{v_{y}}^{L}{(L)})^{L} = (x^{L}{(L)})^{L}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('L', commutative=True)), log(sin(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('L', commutative=True)), log(sin(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_y')(Symbol('L', commutative=True)), Function('x')(Symbol('L', commutative=True)))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(sin(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('x')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(sin(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('v_y')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Function('x')(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["power", 6, "Symbol('L', commutative=True)"], "Equality(Pow(Pow(Function('v_y')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Pow(Function('x')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{v_{t}}{(\\mathbf{P})} = \\frac{\\operatorname{A_{2}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}}, then obtain \\operatorname{A_{2}}{(\\mathbf{P})} + \\operatorname{v_{t}}{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}} = \\operatorname{v_{t}}{(\\mathbf{P})} + \\log{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}}", "derivation": "\\operatorname{A_{2}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{A_{2}}{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}} = \\log{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}} and \\operatorname{A_{2}}{(\\mathbf{P})} + \\frac{\\operatorname{A_{2}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} + \\frac{1}{\\log{(\\mathbf{P})}} = \\frac{\\operatorname{A_{2}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} + \\log{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}} and \\operatorname{v_{t}}{(\\mathbf{P})} = \\frac{\\operatorname{A_{2}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} and \\operatorname{A_{2}}{(\\mathbf{P})} + \\operatorname{v_{t}}{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}} = \\operatorname{v_{t}}{(\\mathbf{P})} + \\log{(\\mathbf{P})} + \\frac{1}{\\log{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))"], "Equality(Add(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Add(log(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"], [["add", 2, "Mul(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)))"], "Equality(Add(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Add(Mul(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), log(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Add(Function('v_t')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(\\Omega)} = \\log{(\\Omega)}, then obtain \\Omega + \\hat{p}{(\\Omega)} + \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega = \\Omega + \\log{(\\Omega)} + \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega", "derivation": "\\hat{p}{(\\Omega)} = \\log{(\\Omega)} and \\Omega + \\hat{p}{(\\Omega)} = \\Omega + \\log{(\\Omega)} and \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega = \\int (\\Omega + \\log{(\\Omega)}) d\\Omega and \\hat{p}{(\\Omega)} - \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega = \\log{(\\Omega)} - \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega and \\hat{p}{(\\Omega)} - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega = \\log{(\\Omega)} - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega and \\Omega + \\hat{p}{(\\Omega)} + \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega = \\Omega + \\log{(\\Omega)} + \\int (\\Omega + \\hat{p}{(\\Omega)}) d\\Omega - \\int (\\Omega + \\log{(\\Omega)}) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["minus", 1, "Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(log(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(log(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given r{(\\pi,F_{x})} = (e^{\\pi})^{F_{x}} and H{(\\pi,F_{x})} = - r{(\\pi,F_{x})}, then obtain H{(\\pi,F_{x})} + r{(\\pi,F_{x})} = 0", "derivation": "r{(\\pi,F_{x})} = (e^{\\pi})^{F_{x}} and H{(\\pi,F_{x})} = - r{(\\pi,F_{x})} and H{(\\pi,F_{x})} = - (e^{\\pi})^{F_{x}} and H{(\\pi,F_{x})} + (e^{\\pi})^{F_{x}} = 0 and H{(\\pi,F_{x})} + r{(\\pi,F_{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('r')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('F_x', commutative=True))))"], [["add", 3, "Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Add(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('F_x', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Function('r')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True))), Integer(0))"]]}, {"prompt": "Given q{(\\sigma_p)} = e^{\\sigma_p}, then obtain - \\frac{(q{(\\sigma_p)} - e^{\\sigma_p}) (e^{\\sigma_p})^{\\sigma_p} \\frac{d}{d \\sigma_p} q^{\\sigma_p}{(\\sigma_p)}}{\\frac{d}{d \\sigma_p} (e^{\\sigma_p})^{\\sigma_p}} = 0", "derivation": "q{(\\sigma_p)} = e^{\\sigma_p} and q^{\\sigma_p}{(\\sigma_p)} = (e^{\\sigma_p})^{\\sigma_p} and q{(\\sigma_p)} - e^{\\sigma_p} = 0 and \\frac{d}{d \\sigma_p} q^{\\sigma_p}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} (e^{\\sigma_p})^{\\sigma_p} and (q{(\\sigma_p)} - e^{\\sigma_p}) \\frac{d}{d \\sigma_p} (e^{\\sigma_p})^{\\sigma_p} = 0 and (q{(\\sigma_p)} - e^{\\sigma_p}) \\frac{d}{d \\sigma_p} q^{\\sigma_p}{(\\sigma_p)} = 0 and - \\frac{(q{(\\sigma_p)} - e^{\\sigma_p}) (e^{\\sigma_p})^{\\sigma_p} \\frac{d}{d \\sigma_p} q^{\\sigma_p}{(\\sigma_p)}}{\\frac{d}{d \\sigma_p} (e^{\\sigma_p})^{\\sigma_p}} = 0", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Pow(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True)))), Derivative(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True)))), Derivative(Pow(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Integer(0))"], [["times", 6, "Mul(Integer(-1), Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Mul(Integer(-1), Add(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_p', commutative=True)))), Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Derivative(Pow(Function('q')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Pow(Derivative(Pow(exp(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\Omega{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain \\frac{d}{d \\dot{z}} \\frac{\\frac{d}{d \\dot{z}} \\Omega{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{d}{d \\dot{z}} \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}}", "derivation": "\\Omega{(\\dot{z})} = \\sin{(\\dot{z})} and \\frac{d}{d \\dot{z}} \\Omega{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\sin{(\\dot{z})} and \\frac{\\frac{d}{d \\dot{z}} \\Omega{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}} and \\frac{d}{d \\dot{z}} \\frac{\\frac{d}{d \\dot{z}} \\Omega{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{d}{d \\dot{z}} \\frac{\\frac{d}{d \\dot{z}} \\sin{(\\dot{z})}}{\\sin{(\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["divide", 2, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(Q)} = \\log{(Q)}, then derive \\int \\operatorname{f_{E}}{(Q)} dQ = Q \\log{(Q)} - Q + \\mathbf{J}_f, then obtain 1 = \\frac{Q \\log{(Q)} + \\mathbf{J}_f}{Q + \\int \\log{(Q)} dQ}", "derivation": "\\operatorname{f_{E}}{(Q)} = \\log{(Q)} and \\int \\operatorname{f_{E}}{(Q)} dQ = \\int \\log{(Q)} dQ and \\int \\operatorname{f_{E}}{(Q)} dQ = Q \\log{(Q)} - Q + \\mathbf{J}_f and Q + \\int \\operatorname{f_{E}}{(Q)} dQ = Q \\log{(Q)} + \\mathbf{J}_f and Q + \\int \\log{(Q)} dQ = Q \\log{(Q)} + \\mathbf{J}_f and \\frac{Q + \\int \\log{(Q)} dQ}{Q + \\int \\operatorname{f_{E}}{(Q)} dQ} = \\frac{Q \\log{(Q)} + \\mathbf{J}_f}{Q + \\int \\operatorname{f_{E}}{(Q)} dQ} and Q + \\int \\log{(Q)} dQ = Q + \\int \\operatorname{f_{E}}{(Q)} dQ and 1 = \\frac{Q \\log{(Q)} + \\mathbf{J}_f}{Q + \\int \\log{(Q)} dQ}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('Q', commutative=True), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 5, "Add(Symbol('Q', commutative=True), Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Add(Symbol('Q', commutative=True), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))), Mul(Pow(Add(Symbol('Q', commutative=True), Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('Q', commutative=True), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('Q', commutative=True), Integral(Function('f_E')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integer(1), Mul(Pow(Add(Symbol('Q', commutative=True), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\mu)} = \\sin{(\\mu)}, then derive \\frac{\\frac{d}{d \\mu} \\theta_{2}{(\\mu)}}{\\mu} - \\frac{\\theta_{2}{(\\mu)}}{\\mu^{2}} = \\frac{\\cos{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}}, then obtain \\frac{\\frac{d}{d \\mu} \\theta_{2}{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}} = \\frac{\\cos{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}}", "derivation": "\\theta_{2}{(\\mu)} = \\sin{(\\mu)} and \\frac{\\theta_{2}{(\\mu)}}{\\mu} = \\frac{\\sin{(\\mu)}}{\\mu} and \\frac{d}{d \\mu} \\frac{\\theta_{2}{(\\mu)}}{\\mu} = \\frac{d}{d \\mu} \\frac{\\sin{(\\mu)}}{\\mu} and \\frac{\\frac{d}{d \\mu} \\theta_{2}{(\\mu)}}{\\mu} - \\frac{\\theta_{2}{(\\mu)}}{\\mu^{2}} = \\frac{\\cos{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}} and \\frac{\\frac{d}{d \\mu} \\theta_{2}{(\\mu)}}{\\mu} - \\frac{\\theta_{2}{(\\mu)}}{\\mu^{2}} = \\frac{\\cos{(\\mu)}}{\\mu} - \\frac{\\theta_{2}{(\\mu)}}{\\mu^{2}} and \\frac{\\frac{d}{d \\mu} \\theta_{2}{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}} = \\frac{\\cos{(\\mu)}}{\\mu} - \\frac{\\sin{(\\mu)}}{\\mu^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), sin(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), sin(Symbol('\\\\mu', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), sin(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given V{(\\hat{\\mathbf{x}},v_{z})} = - v_{z} + \\sin{(\\hat{\\mathbf{x}})}, then obtain (e^{V^{v_{z}}{(\\hat{\\mathbf{x}},v_{z})}})^{\\hat{\\mathbf{x}}} = (e^{(- v_{z} + \\sin{(\\hat{\\mathbf{x}})})^{v_{z}}})^{\\hat{\\mathbf{x}}}", "derivation": "V{(\\hat{\\mathbf{x}},v_{z})} = - v_{z} + \\sin{(\\hat{\\mathbf{x}})} and V^{v_{z}}{(\\hat{\\mathbf{x}},v_{z})} = (- v_{z} + \\sin{(\\hat{\\mathbf{x}})})^{v_{z}} and e^{V^{v_{z}}{(\\hat{\\mathbf{x}},v_{z})}} = e^{(- v_{z} + \\sin{(\\hat{\\mathbf{x}})})^{v_{z}}} and (e^{V^{v_{z}}{(\\hat{\\mathbf{x}},v_{z})}})^{\\hat{\\mathbf{x}}} = (e^{(- v_{z} + \\sin{(\\hat{\\mathbf{x}})})^{v_{z}}})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('v_z', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('V')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('v_z', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(exp(Pow(Function('V')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(exp(Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('v_z', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given c{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain c{(\\rho_f)} \\cos{(\\rho_f)} + \\sin{(\\rho_f)} \\frac{d}{d \\rho_f} c{(\\rho_f)} = 2 \\sin{(\\rho_f)} \\cos{(\\rho_f)}", "derivation": "c{(\\rho_f)} = \\sin{(\\rho_f)} and c{(\\rho_f)} \\sin{(\\rho_f)} = \\sin^{2}{(\\rho_f)} and \\frac{d}{d \\rho_f} c{(\\rho_f)} \\sin{(\\rho_f)} = \\frac{d}{d \\rho_f} \\sin^{2}{(\\rho_f)} and c{(\\rho_f)} \\cos{(\\rho_f)} + \\sin{(\\rho_f)} \\frac{d}{d \\rho_f} c{(\\rho_f)} = 2 \\sin{(\\rho_f)} \\cos{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Function('c')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Mul(Function('c')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('c')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Mul(sin(Symbol('\\\\rho_f', commutative=True)), Derivative(Function('c')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))), Mul(Integer(2), sin(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\psi{(W,E_{\\lambda})} = E_{\\lambda} - W, then obtain (\\int (\\psi{(W,E_{\\lambda})} + 1) dW)^{W} = (\\int (E_{\\lambda} - W + 1) dW)^{W}", "derivation": "\\psi{(W,E_{\\lambda})} = E_{\\lambda} - W and \\psi{(W,E_{\\lambda})} + 1 = E_{\\lambda} - W + 1 and \\int (\\psi{(W,E_{\\lambda})} + 1) dW = \\int (E_{\\lambda} - W + 1) dW and (\\int (\\psi{(W,E_{\\lambda})} + 1) dW)^{W} = (\\int (E_{\\lambda} - W + 1) dW)^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\psi')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('\\\\psi')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\psi')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and \\operatorname{a^{\\dagger}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and k{(\\chi)} = \\operatorname{f_{\\mathbf{p}}}^{\\chi}{(\\chi)}, then obtain \\frac{k{(\\chi)}}{\\chi} = \\frac{\\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)}}{\\chi}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and \\operatorname{f_{\\mathbf{p}}}^{\\chi}{(\\chi)} = \\sin^{\\chi}{(\\sin{(\\chi)})} and \\operatorname{a^{\\dagger}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and \\frac{\\operatorname{f_{\\mathbf{p}}}^{\\chi}{(\\chi)}}{\\chi} = \\frac{\\sin^{\\chi}{(\\sin{(\\chi)})}}{\\chi} and \\frac{\\operatorname{f_{\\mathbf{p}}}^{\\chi}{(\\chi)}}{\\chi} = \\frac{\\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)}}{\\chi} and k{(\\chi)} = \\operatorname{f_{\\mathbf{p}}}^{\\chi}{(\\chi)} and \\frac{k{(\\chi)}}{\\chi} = \\frac{\\operatorname{a^{\\dagger}}^{\\chi}{(\\chi)}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\chi', commutative=True)), sin(sin(Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(sin(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), sin(sin(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(sin(sin(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\chi', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(L)} = \\log{(\\cos{(L)})} and \\operatorname{M_{E}}{(L)} = (- L + \\operatorname{f_{E}}{(L)})^{2}, then obtain (- L + \\log{(\\cos{(L)})})^{2} = (- L + \\operatorname{f_{E}}{(L)}) (- L + \\log{(\\cos{(L)})})", "derivation": "\\operatorname{f_{E}}{(L)} = \\log{(\\cos{(L)})} and - L + \\operatorname{f_{E}}{(L)} = - L + \\log{(\\cos{(L)})} and (- L + \\operatorname{f_{E}}{(L)})^{2} = (- L + \\operatorname{f_{E}}{(L)}) (- L + \\log{(\\cos{(L)})}) and \\operatorname{M_{E}}{(L)} = (- L + \\operatorname{f_{E}}{(L)})^{2} and \\operatorname{M_{E}}{(L)} = (- L + \\operatorname{f_{E}}{(L)}) (- L + \\log{(\\cos{(L)})}) and \\operatorname{M_{E}}{(L)} = (- L + \\log{(\\cos{(L)})})^{2} and (- L + \\log{(\\cos{(L)})})^{2} = (- L + \\operatorname{f_{E}}{(L)}) (- L + \\log{(\\cos{(L)})})", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('L', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('M_E')(Symbol('L', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('M_E')(Symbol('L', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True)))), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('f_E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mu_0,\\mathbf{P})} = \\mathbf{P} + \\mu_0 and \\operatorname{v_{x}}{(\\mu_0,\\mathbf{P})} = (\\mathbf{P} + \\mu_0) \\mathbf{F}{(\\mu_0,\\mathbf{P})}, then obtain \\mathbf{F}^{2}{(\\mu_0,\\mathbf{P})} = \\operatorname{v_{x}}{(\\mu_0,\\mathbf{P})}", "derivation": "\\mathbf{F}{(\\mu_0,\\mathbf{P})} = \\mathbf{P} + \\mu_0 and \\mathbf{F}^{2}{(\\mu_0,\\mathbf{P})} = (\\mathbf{P} + \\mu_0) \\mathbf{F}{(\\mu_0,\\mathbf{P})} and \\operatorname{v_{x}}{(\\mu_0,\\mathbf{P})} = (\\mathbf{P} + \\mu_0) \\mathbf{F}{(\\mu_0,\\mathbf{P})} and \\mathbf{F}^{2}{(\\mu_0,\\mathbf{P})} = \\operatorname{v_{x}}{(\\mu_0,\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Function('v_x')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(\\varepsilon_0,s)} = \\varepsilon_0 + s, then derive \\frac{\\partial}{\\partial s} \\mathbf{S}{(\\varepsilon_0,s)} = 1, then obtain \\frac{\\partial}{\\partial s} (\\varepsilon_0 + s) = 1", "derivation": "\\mathbf{S}{(\\varepsilon_0,s)} = \\varepsilon_0 + s and \\frac{\\partial}{\\partial s} \\mathbf{S}{(\\varepsilon_0,s)} = \\frac{\\partial}{\\partial s} (\\varepsilon_0 + s) and \\frac{\\partial}{\\partial s} \\mathbf{S}{(\\varepsilon_0,s)} = 1 and \\frac{\\partial}{\\partial s} (\\varepsilon_0 + s) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} = \\frac{\\mathbf{f}}{\\hat{H}}, then obtain \\frac{\\partial}{\\partial \\mathbf{f}} \\sin{(2 \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\sin{(\\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} + \\frac{\\mathbf{f}}{\\hat{H}})}", "derivation": "\\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} = \\frac{\\mathbf{f}}{\\hat{H}} and 2 \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} = \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} + \\frac{\\mathbf{f}}{\\hat{H}} and \\sin{(2 \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})})} = \\sin{(\\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} + \\frac{\\mathbf{f}}{\\hat{H}})} and \\frac{\\partial}{\\partial \\mathbf{f}} \\sin{(2 \\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\sin{(\\operatorname{F_{H}}{(\\hat{H},\\mathbf{f})} + \\frac{\\mathbf{f}}{\\hat{H}})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Integer(2), Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Integer(2), Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), sin(Add(Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(sin(Mul(Integer(2), Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(sin(Add(Function('F_H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\mathbf{B},A)} = - \\sin{(A - \\mathbf{B})}, then derive \\frac{\\partial}{\\partial A} h{(\\mathbf{B},A)} = - \\cos{(A - \\mathbf{B})}, then obtain e^{\\frac{\\partial}{\\partial A} - \\sin{(A - \\mathbf{B})}} = e^{- \\cos{(A - \\mathbf{B})}}", "derivation": "h{(\\mathbf{B},A)} = - \\sin{(A - \\mathbf{B})} and \\frac{\\partial}{\\partial A} h{(\\mathbf{B},A)} = \\frac{\\partial}{\\partial A} - \\sin{(A - \\mathbf{B})} and \\frac{\\partial}{\\partial A} h{(\\mathbf{B},A)} = - \\cos{(A - \\mathbf{B})} and e^{\\frac{\\partial}{\\partial A} h{(\\mathbf{B},A)}} = e^{- \\cos{(A - \\mathbf{B})}} and e^{\\frac{\\partial}{\\partial A} - \\sin{(A - \\mathbf{B})}} = e^{- \\cos{(A - \\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))))"], [["exp", 3], "Equality(exp(Derivative(Function('h')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), exp(Mul(Integer(-1), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Derivative(Mul(Integer(-1), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('A', commutative=True), Integer(1)))), exp(Mul(Integer(-1), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))))"]]}, {"prompt": "Given \\sigma_{p}{(x,\\phi)} = \\phi + x, then obtain \\phi x^{2} (- x + \\sigma_{p}{(x,\\phi)}) = \\phi^{2} x^{2}", "derivation": "\\sigma_{p}{(x,\\phi)} = \\phi + x and - x + \\sigma_{p}{(x,\\phi)} = \\phi and - x (- x + \\sigma_{p}{(x,\\phi)}) = - \\phi x and \\phi x^{2} (- x + \\sigma_{p}{(x,\\phi)}) = \\phi^{2} x^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))"], [["times", 2, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('x', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\lambda,\\mathbf{v})} = \\lambda - \\mathbf{v} and \\nabla{(\\lambda,\\mathbf{v})} = \\lambda \\mathbf{v}, then derive \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = \\lambda \\mathbf{v} - \\frac{\\mathbf{v}^{2}}{2} + \\rho, then obtain - \\mathbf{v}^{2} + \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = - \\frac{3 \\mathbf{v}^{2}}{2} + \\rho + \\nabla{(\\lambda,\\mathbf{v})}", "derivation": "\\Psi_{nl}{(\\lambda,\\mathbf{v})} = \\lambda - \\mathbf{v} and \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = \\int (\\lambda - \\mathbf{v}) d\\mathbf{v} and \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = \\lambda \\mathbf{v} - \\frac{\\mathbf{v}^{2}}{2} + \\rho and \\nabla{(\\lambda,\\mathbf{v})} = \\lambda \\mathbf{v} and \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = - \\frac{\\mathbf{v}^{2}}{2} + \\rho + \\nabla{(\\lambda,\\mathbf{v})} and - \\mathbf{v}^{2} + \\int \\Psi_{nl}{(\\lambda,\\mathbf{v})} d\\mathbf{v} = - \\frac{3 \\mathbf{v}^{2}}{2} + \\rho + \\nabla{(\\lambda,\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True), Function('\\\\nabla')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["minus", 5, "Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Integer(-1), Rational(3, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True), Function('\\\\nabla')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\psi{(v)} = e^{v} and A{(v)} = e^{v}, then obtain A{(v)} + \\int \\psi{(v)} dv = A{(v)} + \\int e^{v} dv", "derivation": "\\psi{(v)} = e^{v} and A{(v)} = e^{v} and \\psi{(v)} = A{(v)} and \\frac{\\psi{(v)} e^{v}}{A{(v)}} = e^{v} and \\int \\frac{\\psi{(v)} e^{v}}{A{(v)}} dv = \\int e^{v} dv and \\int \\psi{(v)} dv = \\int e^{v} dv and A{(v)} + \\int \\psi{(v)} dv = A{(v)} + \\int e^{v} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('A')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\psi')(Symbol('v', commutative=True)), Function('A')(Symbol('v', commutative=True)))"], [["divide", 3, "Mul(Function('A')(Symbol('v', commutative=True)), exp(Mul(Integer(-1), Symbol('v', commutative=True))))"], "Equality(Mul(Pow(Function('A')(Symbol('v', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))), exp(Symbol('v', commutative=True)))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Pow(Function('A')(Symbol('v', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Function('\\\\psi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 6, "Function('A')(Symbol('v', commutative=True))"], "Equality(Add(Function('A')(Symbol('v', commutative=True)), Integral(Function('\\\\psi')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Function('A')(Symbol('v', commutative=True)), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\chi,A_{y},v_{z})} = A_{y} + \\chi v_{z}, then obtain - 2 \\chi v_{z} + (\\chi v_{z} + \\operatorname{P_{e}}{(\\chi,A_{y},v_{z})})^{2} = - 2 \\chi v_{z} + (A_{y} + 2 \\chi v_{z})^{2}", "derivation": "\\operatorname{P_{e}}{(\\chi,A_{y},v_{z})} = A_{y} + \\chi v_{z} and \\chi v_{z} + \\operatorname{P_{e}}{(\\chi,A_{y},v_{z})} = A_{y} + 2 \\chi v_{z} and (\\chi v_{z} + \\operatorname{P_{e}}{(\\chi,A_{y},v_{z})})^{2} = (A_{y} + 2 \\chi v_{z})^{2} and - 2 \\chi v_{z} + (\\chi v_{z} + \\operatorname{P_{e}}{(\\chi,A_{y},v_{z})})^{2} = - 2 \\chi v_{z} + (A_{y} + 2 \\chi v_{z})^{2}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('A_y', commutative=True), Mul(Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True)), Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True)), Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True))), Integer(2)), Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))), Integer(2)))"], [["minus", 3, "Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True)), Pow(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True)), Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True))), Integer(2))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True)), Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('v_z', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(n_{2},k)} = k + n_{2} and \\hat{H}{(n_{2},k)} = - n_{2} + \\operatorname{f_{\\mathbf{v}}}{(n_{2},k)}, then obtain \\hat{H}{(n_{2},k)} = k", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(n_{2},k)} = k + n_{2} and - n_{2} + \\operatorname{f_{\\mathbf{v}}}{(n_{2},k)} = k and \\hat{H}{(n_{2},k)} = - n_{2} + \\operatorname{f_{\\mathbf{v}}}{(n_{2},k)} and \\hat{H}{(n_{2},k)} = k", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)), Add(Symbol('k', commutative=True), Symbol('n_2', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{H}')(Symbol('n_2', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))"]]}, {"prompt": "Given t{(i)} = \\frac{d}{d i} \\log{(i)}, then derive t{(i)} = \\frac{1}{i}, then obtain - \\Psi_{nl}{(i)} + \\int t{(i)} di = x - \\Psi_{nl}{(i)} + \\log{(i)}", "derivation": "t{(i)} = \\frac{d}{d i} \\log{(i)} and t{(i)} = \\frac{1}{i} and \\int t{(i)} di = \\int \\frac{1}{i} di and \\int t{(i)} di = \\int \\frac{d}{d i} \\log{(i)} di and - \\Psi_{nl}{(i)} + \\int t{(i)} di = - \\Psi_{nl}{(i)} + \\int \\frac{d}{d i} \\log{(i)} di and - \\Psi_{nl}{(i)} + \\int \\frac{1}{i} di = - \\Psi_{nl}{(i)} + \\int \\frac{d}{d i} \\log{(i)} di and - \\Psi_{nl}{(i)} + \\int t{(i)} di = - \\Psi_{nl}{(i)} + \\int \\frac{1}{i} di and - \\Psi_{nl}{(i)} + \\int t{(i)} di = x - \\Psi_{nl}{(i)} + \\log{(i)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('i', commutative=True)), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t')(Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Function('t')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('t')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))))"], [["minus", 4, "Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Function('t')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Function('t')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Pow(Symbol('i', commutative=True), Integer(-1)), Tuple(Symbol('i', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Function('t')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Symbol('x', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), log(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(l)} = \\cos{(e^{l})}, then obtain ((\\int \\sin{(\\mathbf{r}^{l}{(l)})} dl)^{l})^{l} = ((\\int \\sin{(\\cos^{l}{(e^{l})})} dl)^{l})^{l}", "derivation": "\\mathbf{r}{(l)} = \\cos{(e^{l})} and \\mathbf{r}^{l}{(l)} = \\cos^{l}{(e^{l})} and \\sin{(\\mathbf{r}^{l}{(l)})} = \\sin{(\\cos^{l}{(e^{l})})} and \\int \\sin{(\\mathbf{r}^{l}{(l)})} dl = \\int \\sin{(\\cos^{l}{(e^{l})})} dl and (\\int \\sin{(\\mathbf{r}^{l}{(l)})} dl)^{l} = (\\int \\sin{(\\cos^{l}{(e^{l})})} dl)^{l} and ((\\int \\sin{(\\mathbf{r}^{l}{(l)})} dl)^{l})^{l} = ((\\int \\sin{(\\cos^{l}{(e^{l})})} dl)^{l})^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), cos(exp(Symbol('l', commutative=True))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(cos(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), Symbol('l', commutative=True))), sin(Pow(cos(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True))))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(sin(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(sin(Pow(cos(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Integral(sin(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(sin(Pow(cos(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["power", 5, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Integral(sin(Pow(Function('\\\\mathbf{r}')(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Integral(sin(Pow(cos(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then derive \\int \\phi_{1}{(E_{\\lambda})} dE_{\\lambda} = E_{\\lambda} \\log{(E_{\\lambda})} - E_{\\lambda} + \\mathbf{J}_P, then obtain E_{\\lambda} \\phi_{1}{(E_{\\lambda})} - E_{\\lambda} + \\mathbf{J}_P = \\int \\log{(E_{\\lambda})} dE_{\\lambda}", "derivation": "\\phi_{1}{(E_{\\lambda})} = \\log{(E_{\\lambda})} and \\int \\phi_{1}{(E_{\\lambda})} dE_{\\lambda} = \\int \\log{(E_{\\lambda})} dE_{\\lambda} and \\int \\phi_{1}{(E_{\\lambda})} dE_{\\lambda} = E_{\\lambda} \\log{(E_{\\lambda})} - E_{\\lambda} + \\mathbf{J}_P and \\int \\phi_{1}{(E_{\\lambda})} dE_{\\lambda} = E_{\\lambda} \\phi_{1}{(E_{\\lambda})} - E_{\\lambda} + \\mathbf{J}_P and E_{\\lambda} \\phi_{1}{(E_{\\lambda})} - E_{\\lambda} + \\mathbf{J}_P = \\int \\log{(E_{\\lambda})} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), log(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\phi_1')(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(m,\\mathbf{S})} = \\mathbf{S} + m, then obtain \\int (\\frac{\\partial}{\\partial m} (m + \\phi_{1}{(m,\\mathbf{S})}))^{2} dm = E_{\\lambda} + 4 m", "derivation": "\\phi_{1}{(m,\\mathbf{S})} = \\mathbf{S} + m and m + \\phi_{1}{(m,\\mathbf{S})} = \\mathbf{S} + 2 m and \\frac{\\partial}{\\partial m} (m + \\phi_{1}{(m,\\mathbf{S})}) = \\frac{\\partial}{\\partial m} (\\mathbf{S} + 2 m) and (\\frac{\\partial}{\\partial m} (m + \\phi_{1}{(m,\\mathbf{S})}))^{2} = (\\frac{\\partial}{\\partial m} (\\mathbf{S} + 2 m))^{2} and \\int (\\frac{\\partial}{\\partial m} (m + \\phi_{1}{(m,\\mathbf{S})}))^{2} dm = \\int (\\frac{\\partial}{\\partial m} (\\mathbf{S} + 2 m))^{2} dm and \\int (\\frac{\\partial}{\\partial m} (m + \\phi_{1}{(m,\\mathbf{S})}))^{2} dm = E_{\\lambda} + 4 m", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(2), Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(2), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Add(Symbol('m', commutative=True), Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(2), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Derivative(Add(Symbol('m', commutative=True), Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(2), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Pow(Derivative(Add(Symbol('m', commutative=True), Function('\\\\phi_1')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('m', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(4), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\lambda{(B)} = \\log{(B)}, then obtain B \\lambda^{2}{(B)} - B + \\lambda{(B)} = B \\lambda^{2}{(B)} - B + \\log{(B)}", "derivation": "\\lambda{(B)} = \\log{(B)} and \\lambda^{2}{(B)} = \\lambda{(B)} \\log{(B)} and - B + \\lambda{(B)} = - B + \\log{(B)} and B \\lambda^{2}{(B)} = B \\lambda{(B)} \\log{(B)} and B \\lambda{(B)} \\log{(B)} - B + \\lambda{(B)} = B \\lambda{(B)} \\log{(B)} - B + \\log{(B)} and B \\lambda^{2}{(B)} - B + \\lambda{(B)} = B \\lambda^{2}{(B)} - B + \\log{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], [["times", 1, "Function('\\\\lambda')(Symbol('B', commutative=True))"], "Equality(Pow(Function('\\\\lambda')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\lambda')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"], [["times", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('\\\\lambda')(Symbol('B', commutative=True)), Integer(2))), Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"], [["add", 3, "Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\lambda')(Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Function('\\\\lambda')(Symbol('B', commutative=True)), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('B', commutative=True), Pow(Function('\\\\lambda')(Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\lambda')(Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Pow(Function('\\\\lambda')(Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True)), log(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\nabla{(M)} = e^{\\sin{(M)}}, then obtain - M + \\int 4 dM = - M + \\int (\\nabla{(M)} + e^{\\sin{(M)}})^{\\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}} \\nabla^{- \\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}}{(M)} dM", "derivation": "\\nabla{(M)} = e^{\\sin{(M)}} and 2 \\nabla{(M)} = \\nabla{(M)} + e^{\\sin{(M)}} and 2 = \\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}} and 4 = \\frac{(\\nabla{(M)} + e^{\\sin{(M)}})^{2}}{\\nabla^{2}{(M)}} and \\int 4 dM = \\int \\frac{(\\nabla{(M)} + e^{\\sin{(M)}})^{2}}{\\nabla^{2}{(M)}} dM and \\int 4 dM = \\int (\\nabla{(M)} + e^{\\sin{(M)}})^{\\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}} \\nabla^{- \\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}}{(M)} dM and - M + \\int 4 dM = - M + \\int (\\nabla{(M)} + e^{\\sin{(M)}})^{\\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}} \\nabla^{- \\frac{\\nabla{(M)} + e^{\\sin{(M)}}}{\\nabla{(M)}}}{(M)} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True))))"], [["add", 1, "Function('\\\\nabla')(Symbol('M', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('M', commutative=True))), Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))))"], [["divide", 2, "Function('\\\\nabla')(Symbol('M', commutative=True))"], "Equality(Integer(2), Mul(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-1))))"], [["power", 3, 2], "Equality(Integer(4), Mul(Pow(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Integer(2)), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-2))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Integer(4), Tuple(Symbol('M', commutative=True))), Integral(Mul(Pow(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Integer(2)), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-2))), Tuple(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Integer(4), Tuple(Symbol('M', commutative=True))), Integral(Mul(Pow(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Mul(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-1)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Mul(Integer(-1), Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-1))))), Tuple(Symbol('M', commutative=True))))"], [["minus", 6, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Integral(Integer(4), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Integral(Mul(Pow(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Mul(Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-1)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Mul(Integer(-1), Add(Function('\\\\nabla')(Symbol('M', commutative=True)), exp(sin(Symbol('M', commutative=True)))), Pow(Function('\\\\nabla')(Symbol('M', commutative=True)), Integer(-1))))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and V{(\\mathbf{H})} = \\frac{\\operatorname{A_{2}}{(\\mathbf{H})}}{\\mathbf{H}}, then obtain \\frac{V{(\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\sin{(\\mathbf{H})}}{\\mathbf{H}^{2}}", "derivation": "\\operatorname{A_{2}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\frac{\\operatorname{A_{2}}{(\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\sin{(\\mathbf{H})}}{\\mathbf{H}} and V{(\\mathbf{H})} = \\frac{\\operatorname{A_{2}}{(\\mathbf{H})}}{\\mathbf{H}} and V{(\\mathbf{H})} = \\frac{\\sin{(\\mathbf{H})}}{\\mathbf{H}} and \\frac{V{(\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\sin{(\\mathbf{H})}}{\\mathbf{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('V')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 4, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\chi)} = \\cos{(\\chi)}, then obtain (\\chi + \\operatorname{v_{t}}{(\\chi)} - \\cos{(\\chi)})^{\\chi} = \\chi^{\\chi}", "derivation": "\\operatorname{v_{t}}{(\\chi)} = \\cos{(\\chi)} and \\chi + \\operatorname{v_{t}}{(\\chi)} = \\chi + \\cos{(\\chi)} and \\chi + \\operatorname{v_{t}}{(\\chi)} - \\cos{(\\chi)} = \\chi and (\\chi + \\operatorname{v_{t}}{(\\chi)} - \\cos{(\\chi)})^{\\chi} = \\chi^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('v_t')(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["minus", 2, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('v_t')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\chi', commutative=True), Function('v_t')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\delta{(i)} = \\cos{(e^{i})} and \\theta_{1}{(i)} = \\frac{d}{d i} \\delta{(i)} \\int \\delta{(i)} di, then obtain \\int (\\frac{d}{d i} \\theta_{1}{(i)})^{2} di = \\int (\\frac{d^{2}}{d i^{2}} \\cos{(e^{i})} \\int \\delta{(i)} di)^{2} di", "derivation": "\\delta{(i)} = \\cos{(e^{i})} and \\int \\delta{(i)} di = \\int \\cos{(e^{i})} di and \\theta_{1}{(i)} = \\frac{d}{d i} \\delta{(i)} \\int \\delta{(i)} di and \\theta_{1}{(i)} = \\frac{d}{d i} \\cos{(e^{i})} \\int \\cos{(e^{i})} di and \\theta_{1}{(i)} = \\frac{d}{d i} \\cos{(e^{i})} \\int \\delta{(i)} di and \\frac{d}{d i} \\theta_{1}{(i)} = \\frac{d^{2}}{d i^{2}} \\cos{(e^{i})} \\int \\delta{(i)} di and (\\frac{d}{d i} \\theta_{1}{(i)})^{2} = (\\frac{d^{2}}{d i^{2}} \\cos{(e^{i})} \\int \\delta{(i)} di)^{2} and \\int (\\frac{d}{d i} \\theta_{1}{(i)})^{2} di = \\int (\\frac{d^{2}}{d i^{2}} \\cos{(e^{i})} \\int \\delta{(i)} di)^{2} di", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(cos(exp(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('i', commutative=True)), Derivative(Mul(Function('\\\\delta')(Symbol('i', commutative=True)), Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\theta_1')(Symbol('i', commutative=True)), Derivative(Mul(cos(exp(Symbol('i', commutative=True))), Integral(cos(exp(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\theta_1')(Symbol('i', commutative=True)), Derivative(Mul(cos(exp(Symbol('i', commutative=True))), Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(cos(exp(Symbol('i', commutative=True))), Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(2))))"], [["power", 6, 2], "Equality(Pow(Derivative(Function('\\\\theta_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(cos(exp(Symbol('i', commutative=True))), Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(2))), Integer(2)))"], [["integrate", 7, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\theta_1')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Derivative(Mul(cos(exp(Symbol('i', commutative=True))), Integral(Function('\\\\delta')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(2))), Integer(2)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)}, then derive 0 = - \\mathbf{f}{(\\varepsilon)} + \\frac{1}{\\varepsilon}, then obtain 0 = - \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)} + \\frac{1}{\\varepsilon}", "derivation": "\\mathbf{f}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)} and 0 = - \\mathbf{f}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)} and 0 = - \\mathbf{f}{(\\varepsilon)} + \\frac{1}{\\varepsilon} and 0 = - \\frac{d}{d \\varepsilon} \\log{(\\varepsilon)} + \\frac{1}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\varepsilon', commutative=True)), Derivative(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\mathbf{f}')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\varepsilon', commutative=True))), Derivative(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{J},v_{t})} = \\log{(\\mathbf{J} - v_{t})}, then obtain \\int (\\int \\operatorname{v_{y}}{(\\mathbf{J},v_{t})} dv_{t})^{\\mathbf{J}} d\\mathbf{J} = \\int (\\int \\log{(\\mathbf{J} - v_{t})} dv_{t})^{\\mathbf{J}} d\\mathbf{J}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{J},v_{t})} = \\log{(\\mathbf{J} - v_{t})} and \\int \\operatorname{v_{y}}{(\\mathbf{J},v_{t})} dv_{t} = \\int \\log{(\\mathbf{J} - v_{t})} dv_{t} and (\\int \\operatorname{v_{y}}{(\\mathbf{J},v_{t})} dv_{t})^{\\mathbf{J}} = (\\int \\log{(\\mathbf{J} - v_{t})} dv_{t})^{\\mathbf{J}} and \\int (\\int \\operatorname{v_{y}}{(\\mathbf{J},v_{t})} dv_{t})^{\\mathbf{J}} d\\mathbf{J} = \\int (\\int \\log{(\\mathbf{J} - v_{t})} dv_{t})^{\\mathbf{J}} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Integral(Function('v_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Integral(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Pow(Integral(Function('v_y')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Pow(Integral(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given Q{(T)} = \\sin{(T)}, then obtain \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{Q{(T)}}{T} dT dT)} - \\iint \\frac{\\sin{(T)}}{T} dT dT = \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{\\sin{(T)}}{T} dT dT)} - \\iint \\frac{\\sin{(T)}}{T} dT dT", "derivation": "Q{(T)} = \\sin{(T)} and \\frac{Q{(T)}}{T} = \\frac{\\sin{(T)}}{T} and \\int \\frac{Q{(T)}}{T} dT = \\int \\frac{\\sin{(T)}}{T} dT and \\iint \\frac{Q{(T)}}{T} dT dT = \\iint \\frac{\\sin{(T)}}{T} dT dT and \\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{Q{(T)}}{T} dT dT = \\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{\\sin{(T)}}{T} dT dT and \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{Q{(T)}}{T} dT dT)} = \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{\\sin{(T)}}{T} dT dT)} and \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{Q{(T)}}{T} dT dT)} - \\iint \\frac{\\sin{(T)}}{T} dT dT = \\sin{(\\int \\frac{Q{(T)}}{T} dT + \\iint \\frac{\\sin{(T)}}{T} dT dT)} - \\iint \\frac{\\sin{(T)}}{T} dT dT", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 4, "Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["sin", 5], "Equality(sin(Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), sin(Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))))"], [["minus", 6, "Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(sin(Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(sin(Add(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('Q')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))))"]]}, {"prompt": "Given \\omega{(k,m_{s})} = e^{m_{s}^{k}}, then obtain \\frac{\\partial}{\\partial k} \\omega^{m_{s}}{(k,m_{s})} e^{- m_{s}^{k}} = \\frac{\\partial}{\\partial k} e^{- m_{s}^{k}} (e^{m_{s}^{k}})^{m_{s}}", "derivation": "\\omega{(k,m_{s})} = e^{m_{s}^{k}} and \\omega^{m_{s}}{(k,m_{s})} = (e^{m_{s}^{k}})^{m_{s}} and \\omega^{m_{s}}{(k,m_{s})} e^{- m_{s}^{k}} = e^{- m_{s}^{k}} (e^{m_{s}^{k}})^{m_{s}} and \\frac{\\partial}{\\partial k} \\omega^{m_{s}}{(k,m_{s})} e^{- m_{s}^{k}} = \\frac{\\partial}{\\partial k} e^{- m_{s}^{k}} (e^{m_{s}^{k}})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('k', commutative=True), Symbol('m_s', commutative=True)), exp(Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('k', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(exp(Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))), Symbol('m_s', commutative=True)))"], [["divide", 2, "exp(Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\omega')(Symbol('k', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))))), Mul(exp(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True)))), Pow(exp(Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))), Symbol('m_s', commutative=True))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\omega')(Symbol('k', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True)))), Pow(exp(Pow(Symbol('m_s', commutative=True), Symbol('k', commutative=True))), Symbol('m_s', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\Psi,L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi}, then obtain \\Psi (-1 + \\frac{\\int B{(\\Psi,L_{\\varepsilon})} d\\Psi}{\\int L_{\\varepsilon}^{\\Psi} d\\Psi}) = 0", "derivation": "B{(\\Psi,L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi} and \\int B{(\\Psi,L_{\\varepsilon})} d\\Psi = \\int L_{\\varepsilon}^{\\Psi} d\\Psi and \\frac{\\int B{(\\Psi,L_{\\varepsilon})} d\\Psi}{\\int L_{\\varepsilon}^{\\Psi} d\\Psi} = 1 and -1 + \\frac{\\int B{(\\Psi,L_{\\varepsilon})} d\\Psi}{\\int L_{\\varepsilon}^{\\Psi} d\\Psi} = 0 and \\Psi (-1 + \\frac{\\int B{(\\Psi,L_{\\varepsilon})} d\\Psi}{\\int L_{\\varepsilon}^{\\Psi} d\\Psi}) = 0", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Psi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\Psi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Integral(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Pow(Integral(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\Psi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Integer(1))"], [["minus", 3, 1], "Equality(Add(Integer(-1), Mul(Pow(Integral(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\Psi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))), Integer(0))"], [["times", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Add(Integer(-1), Mul(Pow(Integral(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\Psi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given \\mu_{0}{(v_{2},\\eta)} = e^{- \\eta + v_{2}} and \\operatorname{L_{\\varepsilon}}{(v_{2},\\eta)} = \\mu_{0}^{\\eta}{(v_{2},\\eta)}, then obtain \\operatorname{L_{\\varepsilon}}{(v_{2},\\eta)} + (e^{- \\eta + v_{2}})^{\\eta} = 2 (e^{- \\eta + v_{2}})^{\\eta}", "derivation": "\\mu_{0}{(v_{2},\\eta)} = e^{- \\eta + v_{2}} and \\mu_{0}^{\\eta}{(v_{2},\\eta)} = (e^{- \\eta + v_{2}})^{\\eta} and \\operatorname{L_{\\varepsilon}}{(v_{2},\\eta)} = \\mu_{0}^{\\eta}{(v_{2},\\eta)} and \\operatorname{L_{\\varepsilon}}{(v_{2},\\eta)} + \\mu_{0}^{\\eta}{(v_{2},\\eta)} = 2 \\mu_{0}^{\\eta}{(v_{2},\\eta)} and \\operatorname{L_{\\varepsilon}}{(v_{2},\\eta)} + (e^{- \\eta + v_{2}})^{\\eta} = 2 (e^{- \\eta + v_{2}})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('v_2', commutative=True))))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('v_2', commutative=True))), Symbol('\\\\eta', commutative=True)))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["add", 3, "Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('v_2', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('v_2', commutative=True))), Symbol('\\\\eta', commutative=True))), Mul(Integer(2), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('v_2', commutative=True))), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}} and \\mathbf{v}{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}}, then derive \\dot{\\mathbf{r}}^{F_{x}}{(F_{x})} = (e^{F_{x}})^{F_{x}}, then obtain \\mathbf{v}^{F_{x}}{(F_{x})} = (e^{F_{x}})^{F_{x}}", "derivation": "\\dot{\\mathbf{r}}{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}} and \\mathbf{v}{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}} and \\dot{\\mathbf{r}}^{F_{x}}{(F_{x})} = (\\frac{d}{d F_{x}} e^{F_{x}})^{F_{x}} and \\mathbf{v}{(F_{x})} = \\dot{\\mathbf{r}}{(F_{x})} and \\mathbf{v}^{F_{x}}{(F_{x})} = \\dot{\\mathbf{r}}^{F_{x}}{(F_{x})} and \\dot{\\mathbf{r}}^{F_{x}}{(F_{x})} = (e^{F_{x}})^{F_{x}} and \\mathbf{v}^{F_{x}}{(F_{x})} = (e^{F_{x}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_x', commutative=True)), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{v}')(Symbol('F_x', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_x', commutative=True)))"], [["power", 4, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(n_{1},c)} = c n_{1} and H{(n_{1},c)} = c n_{1}, then obtain \\operatorname{v_{2}}{(n_{1},c)} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)} = c n_{1} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)}", "derivation": "\\operatorname{v_{2}}{(n_{1},c)} = c n_{1} and H{(n_{1},c)} = c n_{1} and H{(n_{1},c)} = \\operatorname{v_{2}}{(n_{1},c)} and H{(n_{1},c)} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)} = c n_{1} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)} and \\operatorname{v_{2}}{(n_{1},c)} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)} = c n_{1} + \\frac{\\partial}{\\partial c} \\operatorname{v_{2}}{(n_{1},c)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('H')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)))"], [["add", 2, "Derivative(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Add(Function('H')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Derivative(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Symbol('c', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Derivative(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Symbol('c', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('v_2')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\ddot{x}{(c_{0})} = \\sin{(c_{0})} and \\hat{\\mathbf{r}}{(c_{0})} = 2 \\sin{(c_{0})}, then obtain \\ddot{x}{(c_{0})} + \\hat{\\mathbf{r}}{(c_{0})} = 2 \\ddot{x}{(c_{0})} + \\sin{(c_{0})}", "derivation": "\\ddot{x}{(c_{0})} = \\sin{(c_{0})} and 2 \\ddot{x}{(c_{0})} = \\ddot{x}{(c_{0})} + \\sin{(c_{0})} and 3 \\ddot{x}{(c_{0})} = 2 \\ddot{x}{(c_{0})} + \\sin{(c_{0})} and 3 \\ddot{x}{(c_{0})} = \\ddot{x}{(c_{0})} + 2 \\sin{(c_{0})} and \\ddot{x}{(c_{0})} + 2 \\sin{(c_{0})} = 2 \\ddot{x}{(c_{0})} + \\sin{(c_{0})} and \\hat{\\mathbf{r}}{(c_{0})} = 2 \\sin{(c_{0})} and \\ddot{x}{(c_{0})} + \\hat{\\mathbf{r}}{(c_{0})} = 2 \\ddot{x}{(c_{0})} + \\sin{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["add", 1, "Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), Add(Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), Add(Mul(Integer(2), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), sin(Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), Add(Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)), Mul(Integer(2), sin(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)), Mul(Integer(2), sin(Symbol('c_0', commutative=True)))), Add(Mul(Integer(2), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), sin(Symbol('c_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c_0', commutative=True)), Mul(Integer(2), sin(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Function('\\\\ddot{x}')(Symbol('c_0', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c_0', commutative=True))), Add(Mul(Integer(2), Function('\\\\ddot{x}')(Symbol('c_0', commutative=True))), sin(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given i{(C,\\hbar)} = \\hbar^{C}, then obtain - \\hbar \\hbar^{C} + (- i{(C,\\hbar)})^{\\hbar} = - \\hbar \\hbar^{C} + (- \\hbar^{C})^{\\hbar}", "derivation": "i{(C,\\hbar)} = \\hbar^{C} and - i{(C,\\hbar)} = - \\hbar^{C} and (- i{(C,\\hbar)})^{\\hbar} = (- \\hbar^{C})^{\\hbar} and \\hbar i{(C,\\hbar)} = \\hbar \\hbar^{C} and - \\hbar i{(C,\\hbar)} = - \\hbar \\hbar^{C} and - \\hbar i{(C,\\hbar)} + (- i{(C,\\hbar)})^{\\hbar} = - \\hbar i{(C,\\hbar)} + (- \\hbar^{C})^{\\hbar} and - \\hbar \\hbar^{C} + (- i{(C,\\hbar)})^{\\hbar} = - \\hbar \\hbar^{C} + (- \\hbar^{C})^{\\hbar}", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Mul(Integer(-1), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))), Pow(Mul(Integer(-1), Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Symbol('C', commutative=True))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given G{(k,\\theta)} = e^{\\frac{k}{\\theta}}, then derive (\\frac{\\partial}{\\partial k} G{(k,\\theta)})^{k} = (\\frac{e^{\\frac{k}{\\theta}}}{\\theta})^{k}, then obtain ((\\frac{\\partial}{\\partial k} G{(k,\\theta)})^{k})^{k} = ((\\frac{e^{\\frac{k}{\\theta}}}{\\theta})^{k})^{k}", "derivation": "G{(k,\\theta)} = e^{\\frac{k}{\\theta}} and G{(k,\\theta)} + \\frac{1}{\\theta} = e^{\\frac{k}{\\theta}} + \\frac{1}{\\theta} and G{(k,\\theta)} + 1 + \\frac{1}{\\theta} = e^{\\frac{k}{\\theta}} + 1 + \\frac{1}{\\theta} and \\frac{\\partial}{\\partial k} (G{(k,\\theta)} + 1 + \\frac{1}{\\theta}) = \\frac{\\partial}{\\partial k} (e^{\\frac{k}{\\theta}} + 1 + \\frac{1}{\\theta}) and (\\frac{\\partial}{\\partial k} (G{(k,\\theta)} + 1 + \\frac{1}{\\theta}))^{k} = (\\frac{\\partial}{\\partial k} (e^{\\frac{k}{\\theta}} + 1 + \\frac{1}{\\theta}))^{k} and (\\frac{\\partial}{\\partial k} G{(k,\\theta)})^{k} = (\\frac{e^{\\frac{k}{\\theta}}}{\\theta})^{k} and ((\\frac{\\partial}{\\partial k} G{(k,\\theta)})^{k})^{k} = ((\\frac{e^{\\frac{k}{\\theta}}}{\\theta})^{k})^{k}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))))"], [["add", 1, "Pow(Symbol('\\\\theta', commutative=True), Integer(-1))"], "Equality(Add(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Add(exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Add(exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Add(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(Add(exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Integer(1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Derivative(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('G')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given Z{(F_{N})} = \\int e^{F_{N}} dF_{N}, then derive Z{(F_{N})} = \\mathbf{S} + e^{F_{N}}, then derive (v_{z} + e^{F_{N}}) e^{- F_{N}} = (\\mathbf{S} + e^{F_{N}}) e^{- F_{N}}, then obtain \\frac{(v_{z} + e^{F_{N}}) e^{- F_{N}}}{\\int \\mathbf{P}{(\\mathbf{f},\\mathbf{g})} d\\mathbf{f}} = \\frac{(\\mathbf{S} + e^{F_{N}}) e^{- F_{N}}}{\\int \\mathbf{P}{(\\mathbf{f},\\mathbf{g})} d\\mathbf{f}}", "derivation": "Z{(F_{N})} = \\int e^{F_{N}} dF_{N} and Z{(F_{N})} = \\mathbf{S} + e^{F_{N}} and Z{(F_{N})} e^{- F_{N}} = (\\mathbf{S} + e^{F_{N}}) e^{- F_{N}} and e^{- F_{N}} \\int e^{F_{N}} dF_{N} = (\\mathbf{S} + e^{F_{N}}) e^{- F_{N}} and (v_{z} + e^{F_{N}}) e^{- F_{N}} = (\\mathbf{S} + e^{F_{N}}) e^{- F_{N}} and \\frac{(v_{z} + e^{F_{N}}) e^{- F_{N}}}{\\int \\mathbf{P}{(\\mathbf{f},\\mathbf{g})} d\\mathbf{f}} = \\frac{(\\mathbf{S} + e^{F_{N}}) e^{- F_{N}}}{\\int \\mathbf{P}{(\\mathbf{f},\\mathbf{g})} d\\mathbf{f}}", "srepr_derivation": [["get_premise", "Equality(Function('Z')(Symbol('F_N', commutative=True)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('Z')(Symbol('F_N', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('F_N', commutative=True))))"], [["divide", 2, "exp(Symbol('F_N', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('F_N', commutative=True)), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(exp(Mul(Integer(-1), Symbol('F_N', commutative=True))), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('v_z', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True)))))"], [["divide", 5, "Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Add(Symbol('v_z', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True))), Pow(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Mul(Integer(-1), Symbol('F_N', commutative=True))), Pow(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(T)} = \\cos{(T)} and \\operatorname{f_{E}}{(T)} = \\cos{(T)}, then obtain \\int 0 dT = \\int (- \\ddot{x}{(T)} + \\cos{(T)}) dT", "derivation": "\\ddot{x}{(T)} = \\cos{(T)} and \\operatorname{f_{E}}{(T)} = \\cos{(T)} and \\operatorname{f_{E}}{(T)} = \\ddot{x}{(T)} and - \\ddot{x}{(T)} + \\operatorname{f_{E}}{(T)} = - \\ddot{x}{(T)} + \\cos{(T)} and \\int (- \\ddot{x}{(T)} + \\operatorname{f_{E}}{(T)}) dT = \\int (- \\ddot{x}{(T)} + \\cos{(T)}) dT and \\int 0 dT = \\int (- \\ddot{x}{(T)} + \\cos{(T)}) dT", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f_E')(Symbol('T', commutative=True)), Function('\\\\ddot{x}')(Symbol('T', commutative=True)))"], [["minus", 2, "Function('\\\\ddot{x}')(Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('T', commutative=True))), Function('f_E')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('T', commutative=True))), Function('f_E')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Integer(0), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('T', commutative=True))), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(U,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{U} and x{(U,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{U}, then obtain \\operatorname{L_{\\varepsilon}}^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} - x{(U,\\mathbf{J}_M)} - x^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} = - x{(U,\\mathbf{J}_M)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(U,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{U} and \\operatorname{L_{\\varepsilon}}^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} = (\\frac{\\mathbf{J}_M}{U})^{\\mathbf{J}_M} and - (\\frac{\\mathbf{J}_M}{U})^{\\mathbf{J}_M} + \\operatorname{L_{\\varepsilon}}^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} = 0 and x{(U,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{U} and - (\\frac{\\mathbf{J}_M}{U})^{\\mathbf{J}_M} + \\operatorname{L_{\\varepsilon}}^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} - \\frac{\\mathbf{J}_M}{U} = - \\frac{\\mathbf{J}_M}{U} and \\operatorname{L_{\\varepsilon}}^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} - x{(U,\\mathbf{J}_M)} - x^{\\mathbf{J}_M}{(U,\\mathbf{J}_M)} = - x{(U,\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 2, "Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Function('L_{\\\\varepsilon}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Function('x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(-1), Pow(Function('x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Integer(-1), Function('x')(Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(k,p)} = \\cos{(k p)}, then obtain \\operatorname{f_{E}}^{k}{(k,p)} \\cos^{p}{(k p)} - \\cos{(k p)} = - \\cos{(k p)} + \\cos^{k}{(k p)} \\cos^{p}{(k p)}", "derivation": "\\operatorname{f_{E}}{(k,p)} = \\cos{(k p)} and \\operatorname{f_{E}}^{p}{(k,p)} = \\cos^{p}{(k p)} and \\operatorname{f_{E}}^{k}{(k,p)} = \\cos^{k}{(k p)} and \\operatorname{f_{E}}^{k}{(k,p)} \\cos^{p}{(k p)} = \\cos^{k}{(k p)} \\cos^{p}{(k p)} and \\operatorname{f_{E}}^{k}{(k,p)} \\operatorname{f_{E}}^{p}{(k,p)} = \\operatorname{f_{E}}^{p}{(k,p)} \\cos^{k}{(k p)} and \\operatorname{f_{E}}^{k}{(k,p)} \\operatorname{f_{E}}^{p}{(k,p)} - \\cos{(k p)} = \\operatorname{f_{E}}^{p}{(k,p)} \\cos^{k}{(k p)} - \\cos{(k p)} and \\operatorname{f_{E}}^{k}{(k,p)} \\cos^{p}{(k p)} - \\cos{(k p)} = - \\cos{(k p)} + \\cos^{k}{(k p)} \\cos^{p}{(k p)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True)))"], [["times", 3, "Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Mul(Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True))))"], [["minus", 5, "cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))))), Add(Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))))), Add(Mul(Integer(-1), cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True)))), Mul(Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True)), Pow(cos(Mul(Symbol('k', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(t)} = e^{t}, then obtain - \\operatorname{v_{2}}{(t)} + \\operatorname{v_{2}}^{t}{(t)} = - \\operatorname{v_{2}}{(t)} - \\operatorname{v_{2}}^{t}{(t)} + 2 (e^{t})^{t}", "derivation": "\\operatorname{v_{2}}{(t)} = e^{t} and \\operatorname{v_{2}}^{t}{(t)} = (e^{t})^{t} and - \\operatorname{v_{2}}{(t)} + \\operatorname{v_{2}}^{t}{(t)} = - \\operatorname{v_{2}}{(t)} + (e^{t})^{t} and \\operatorname{v_{2}}{(t)} + \\operatorname{v_{2}}^{t}{(t)} - (e^{t})^{t} = \\operatorname{v_{2}}{(t)} and (\\operatorname{v_{2}}{(t)} + \\operatorname{v_{2}}^{t}{(t)} - (e^{t})^{t})^{t} - \\operatorname{v_{2}}{(t)} - \\operatorname{v_{2}}^{t}{(t)} + (e^{t})^{t} = - \\operatorname{v_{2}}{(t)} - \\operatorname{v_{2}}^{t}{(t)} + 2 (e^{t})^{t} and - \\operatorname{v_{2}}{(t)} + \\operatorname{v_{2}}^{t}{(t)} = - \\operatorname{v_{2}}{(t)} - \\operatorname{v_{2}}^{t}{(t)} + 2 (e^{t})^{t}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["minus", 2, "Function('v_2')(Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], "Equality(Add(Function('v_2')(Symbol('t', commutative=True)), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))), Function('v_2')(Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Add(Function('v_2')(Symbol('t', commutative=True)), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(2), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('v_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(2), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given S{(z,r,v_{t})} = v_{t}^{r} + z and \\operatorname{v_{y}}{(r,v_{t})} = v_{t}^{r} and q{(v_{t})} = v_{t}, then obtain \\int z S^{z}{(z,r,v_{t})} dq{(v_{t})} = \\int z (z + \\operatorname{v_{y}}{(r,v_{t})})^{z} dq{(v_{t})}", "derivation": "S{(z,r,v_{t})} = v_{t}^{r} + z and S^{z}{(z,r,v_{t})} = (v_{t}^{r} + z)^{z} and \\operatorname{v_{y}}{(r,v_{t})} = v_{t}^{r} and S^{z}{(z,r,v_{t})} = (z + \\operatorname{v_{y}}{(r,v_{t})})^{z} and z S^{z}{(z,r,v_{t})} = z (z + \\operatorname{v_{y}}{(r,v_{t})})^{z} and q{(v_{t})} = v_{t} and \\int z S^{z}{(z,r,v_{t})} dv_{t} = \\int z (z + \\operatorname{v_{y}}{(r,v_{t})})^{z} dv_{t} and \\int z S^{z}{(z,r,v_{t})} dq{(v_{t})} = \\int z (z + \\operatorname{v_{y}}{(r,v_{t})})^{z} dq{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Add(Pow(Symbol('v_t', commutative=True), Symbol('r', commutative=True)), Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Pow(Symbol('v_t', commutative=True), Symbol('r', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Symbol('z', commutative=True), Function('v_y')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Symbol('z', commutative=True)))"], [["times", 4, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Pow(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), Pow(Add(Symbol('z', commutative=True), Function('v_y')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], [["integrate", 5, "Symbol('v_t', commutative=True)"], "Equality(Integral(Mul(Symbol('z', commutative=True), Pow(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Symbol('z', commutative=True), Pow(Add(Symbol('z', commutative=True), Function('v_y')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Symbol('z', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integral(Mul(Symbol('z', commutative=True), Pow(Function('S')(Symbol('z', commutative=True), Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Symbol('z', commutative=True))), Tuple(Function('q')(Symbol('v_t', commutative=True)))), Integral(Mul(Symbol('z', commutative=True), Pow(Add(Symbol('z', commutative=True), Function('v_y')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))), Symbol('z', commutative=True))), Tuple(Function('q')(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\mu_0,\\mathbf{r})} = - \\mathbf{r} + \\mu_0 and \\operatorname{C_{2}}{(\\mu_0,\\mathbf{r})} = - \\varphi{(\\mu_0,\\mathbf{r})} and \\delta{(\\mu_0,\\mathbf{r})} = \\operatorname{C_{2}}^{\\mu_0}{(\\mu_0,\\mathbf{r})}, then obtain (- \\varphi{(\\mu_0,\\mathbf{r})})^{\\mu_0} = \\delta{(\\mu_0,\\mathbf{r})}", "derivation": "\\varphi{(\\mu_0,\\mathbf{r})} = - \\mathbf{r} + \\mu_0 and \\operatorname{C_{2}}{(\\mu_0,\\mathbf{r})} = - \\varphi{(\\mu_0,\\mathbf{r})} and \\operatorname{C_{2}}{(\\mu_0,\\mathbf{r})} = \\mathbf{r} - \\mu_0 and \\operatorname{C_{2}}^{\\mu_0}{(\\mu_0,\\mathbf{r})} = (\\mathbf{r} - \\mu_0)^{\\mu_0} and (- \\varphi{(\\mu_0,\\mathbf{r})})^{\\mu_0} = (\\mathbf{r} - \\mu_0)^{\\mu_0} and \\delta{(\\mu_0,\\mathbf{r})} = \\operatorname{C_{2}}^{\\mu_0}{(\\mu_0,\\mathbf{r})} and \\delta{(\\mu_0,\\mathbf{r})} = (\\mathbf{r} - \\mu_0)^{\\mu_0} and (- \\varphi{(\\mu_0,\\mathbf{r})})^{\\mu_0} = \\delta{(\\mu_0,\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))"], [["power", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('C_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('\\\\delta')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Pow(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Function('\\\\delta')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given M{(q)} = \\frac{d}{d q} \\log{(q)}, then derive M{(q)} = \\frac{1}{q}, then obtain \\frac{d}{d q} (M{(q)} - \\frac{1}{q}) = \\frac{d}{d q} 0", "derivation": "M{(q)} = \\frac{d}{d q} \\log{(q)} and M{(q)} = \\frac{1}{q} and M{(q)} - \\frac{1}{q} = 0 and \\frac{d}{d q} (M{(q)} - \\frac{1}{q}) = \\frac{d}{d q} 0", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('q', commutative=True)), Derivative(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1)))"], [["minus", 2, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Add(Function('M')(Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))), Integer(0))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Function('M')(Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(F_{c},q)} = F_{c} + q, then obtain (- (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q})^{F_{c}} = (- (\\int q dF_{c})^{q})^{F_{c}}", "derivation": "m{(F_{c},q)} = F_{c} + q and - F_{c} + m{(F_{c},q)} = q and \\int (- F_{c} + m{(F_{c},q)}) dF_{c} = \\int q dF_{c} and (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q} = (\\int q dF_{c})^{q} and F_{c} + (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q} = F_{c} + (\\int q dF_{c})^{q} and - F_{c} - (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q} = - F_{c} - (\\int q dF_{c})^{q} and - (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q} = - (\\int q dF_{c})^{q} and (- (\\int (- F_{c} + m{(F_{c},q)}) dF_{c})^{q})^{F_{c}} = (- (\\int q dF_{c})^{q})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True)), Pow(Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('F_c', commutative=True))"], "Equality(Add(Symbol('F_c', commutative=True), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))), Add(Symbol('F_c', commutative=True), Pow(Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True)))))"], [["add", 6, "Symbol('F_c', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))))"], [["power", 7, "Symbol('F_c', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('m')(Symbol('F_c', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))), Symbol('F_c', commutative=True)), Pow(Mul(Integer(-1), Pow(Integral(Symbol('q', commutative=True), Tuple(Symbol('F_c', commutative=True))), Symbol('q', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\phi_2,Q)} = \\frac{\\phi_2}{Q}, then obtain \\varphi{(\\phi_2,Q)} + \\frac{1}{Q} = \\frac{\\phi_2}{Q} + \\frac{1}{Q}", "derivation": "\\varphi{(\\phi_2,Q)} = \\frac{\\phi_2}{Q} and \\frac{\\partial}{\\partial \\phi_2} \\varphi{(\\phi_2,Q)} = \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2}{Q} and \\varphi{(\\phi_2,Q)} + \\frac{\\partial}{\\partial \\phi_2} \\varphi{(\\phi_2,Q)} = \\frac{\\partial}{\\partial \\phi_2} \\varphi{(\\phi_2,Q)} + \\frac{\\phi_2}{Q} and \\varphi{(\\phi_2,Q)} + \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2}{Q} = \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2}{Q} + \\frac{\\phi_2}{Q} and \\varphi{(\\phi_2,Q)} + \\frac{1}{Q} = \\frac{\\phi_2}{Q} + \\frac{1}{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))))"]]}, {"prompt": "Given J{(n_{2})} = \\sin{(n_{2})}, then obtain \\int (\\frac{J{(n_{2})}}{\\sin{(n_{2})}})^{n_{2}} dn_{2} = \\int 1 dn_{2}", "derivation": "J{(n_{2})} = \\sin{(n_{2})} and \\frac{J{(n_{2})}}{\\sin{(n_{2})}} = 1 and (\\frac{J{(n_{2})}}{\\sin{(n_{2})}})^{n_{2}} = 1 and \\int (\\frac{J{(n_{2})}}{\\sin{(n_{2})}})^{n_{2}} dn_{2} = \\int 1 dn_{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["divide", 1, "sin(Symbol('n_2', commutative=True))"], "Equality(Mul(Function('J')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Function('J')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1))), Symbol('n_2', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Pow(Mul(Function('J')(Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given k{(F_{g})} = \\sin{(e^{F_{g}})} and h{(F_{g})} = \\sin{(e^{F_{g}})} and \\mathbf{D}{(F_{g})} = \\frac{d}{d F_{g}} \\sin{(e^{F_{g}})}, then obtain - \\mathbf{D}{(F_{g})} + k^{2}{(F_{g})} = - \\mathbf{D}{(F_{g})} + h{(F_{g})} k{(F_{g})}", "derivation": "k{(F_{g})} = \\sin{(e^{F_{g}})} and k^{2}{(F_{g})} = k{(F_{g})} \\sin{(e^{F_{g}})} and h{(F_{g})} = \\sin{(e^{F_{g}})} and k^{2}{(F_{g})} - \\frac{d}{d F_{g}} \\sin{(e^{F_{g}})} = k{(F_{g})} \\sin{(e^{F_{g}})} - \\frac{d}{d F_{g}} \\sin{(e^{F_{g}})} and \\mathbf{D}{(F_{g})} = \\frac{d}{d F_{g}} \\sin{(e^{F_{g}})} and - \\mathbf{D}{(F_{g})} + k^{2}{(F_{g})} = - \\mathbf{D}{(F_{g})} + k{(F_{g})} \\sin{(e^{F_{g}})} and - \\mathbf{D}{(F_{g})} + k^{2}{(F_{g})} = - \\mathbf{D}{(F_{g})} + h{(F_{g})} k{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('F_g', commutative=True)), sin(exp(Symbol('F_g', commutative=True))))"], [["times", 1, "Function('k')(Symbol('F_g', commutative=True))"], "Equality(Pow(Function('k')(Symbol('F_g', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('F_g', commutative=True)), sin(exp(Symbol('F_g', commutative=True)))))"], ["renaming_premise", "Equality(Function('h')(Symbol('F_g', commutative=True)), sin(exp(Symbol('F_g', commutative=True))))"], [["minus", 2, "Derivative(sin(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('k')(Symbol('F_g', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(sin(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))), Add(Mul(Function('k')(Symbol('F_g', commutative=True)), sin(exp(Symbol('F_g', commutative=True)))), Mul(Integer(-1), Derivative(sin(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('F_g', commutative=True)), Derivative(sin(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('F_g', commutative=True))), Pow(Function('k')(Symbol('F_g', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('F_g', commutative=True))), Mul(Function('k')(Symbol('F_g', commutative=True)), sin(exp(Symbol('F_g', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('F_g', commutative=True))), Pow(Function('k')(Symbol('F_g', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('F_g', commutative=True))), Mul(Function('h')(Symbol('F_g', commutative=True)), Function('k')(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given Q{(U,\\theta)} = U \\theta, then obtain \\log{(- \\frac{- \\theta + Q{(U,\\theta)}}{\\theta})} = \\log{(- \\frac{U \\theta - \\theta}{\\theta})}", "derivation": "Q{(U,\\theta)} = U \\theta and - \\theta + Q{(U,\\theta)} = U \\theta - \\theta and - \\frac{- \\theta + Q{(U,\\theta)}}{\\theta} = - \\frac{U \\theta - \\theta}{\\theta} and \\log{(- \\frac{- \\theta + Q{(U,\\theta)}}{\\theta})} = \\log{(- \\frac{U \\theta - \\theta}{\\theta})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('Q')(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('Q')(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["log", 3], "Equality(log(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('Q')(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True))))), log(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Symbol('U', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = - \\mathbf{J}_f + m^{\\mathbf{E}}, then obtain - m + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = - m - 1", "derivation": "\\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = - \\mathbf{J}_f + m^{\\mathbf{E}} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (- \\mathbf{J}_f + m^{\\mathbf{E}}) and - m + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = - m + \\frac{\\partial}{\\partial \\mathbf{J}_f} (- \\mathbf{J}_f + m^{\\mathbf{E}}) and - m + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\Omega{(\\mathbf{E},m,\\mathbf{J}_f)} = - m - 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["add", 2, "Mul(Integer(-1), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\psi{(\\nabla,Q)} = Q + \\nabla, then derive \\int \\psi{(\\nabla,Q)} dQ = \\frac{Q^{2}}{2} + Q \\nabla + f_{E}, then derive \\cos{(\\frac{\\frac{Q^{2}}{2} + Q \\nabla + t_{2}}{\\frac{Q^{2}}{2} + Q \\nabla + f_{E}})} = \\cos{(1)}, then obtain \\cos{(\\frac{\\frac{Q^{2}}{2} + Q \\nabla + t_{2}}{\\int (Q + \\nabla) dQ})} = \\cos{(1)}", "derivation": "\\psi{(\\nabla,Q)} = Q + \\nabla and \\int \\psi{(\\nabla,Q)} dQ = \\int (Q + \\nabla) dQ and \\int \\psi{(\\nabla,Q)} dQ = \\frac{Q^{2}}{2} + Q \\nabla + f_{E} and \\frac{\\int \\psi{(\\nabla,Q)} dQ}{\\frac{Q^{2}}{2} + Q \\nabla + f_{E}} = 1 and \\frac{\\int (Q + \\nabla) dQ}{\\frac{Q^{2}}{2} + Q \\nabla + f_{E}} = 1 and \\frac{Q^{2}}{2} + Q \\nabla + f_{E} = \\int (Q + \\nabla) dQ and \\cos{(\\frac{\\int (Q + \\nabla) dQ}{\\frac{Q^{2}}{2} + Q \\nabla + f_{E}})} = \\cos{(1)} and \\cos{(\\frac{\\frac{Q^{2}}{2} + Q \\nabla + t_{2}}{\\frac{Q^{2}}{2} + Q \\nabla + f_{E}})} = \\cos{(1)} and \\cos{(\\frac{\\frac{Q^{2}}{2} + Q \\nabla + t_{2}}{\\int (Q + \\nabla) dQ})} = \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\nabla', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\nabla', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\nabla', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)))"], [["divide", 3, "Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)), Integer(-1)), Integral(Function('\\\\psi')(Symbol('\\\\nabla', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)), Integer(-1)), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["cos", 5], "Equality(cos(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)), Integer(-1)), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('Q', commutative=True))))), cos(Integer(1)))"], [["evaluate_integrals", 7], "Equality(cos(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('f_E', commutative=True)), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('t_2', commutative=True)))), cos(Integer(1)))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(cos(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('t_2', commutative=True)), Pow(Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1)))), cos(Integer(1)))"]]}, {"prompt": "Given s{(x)} = \\log{(x)} and \\operatorname{M_{E}}{(x)} = \\log{(x)}, then derive \\int s{(x)} dx = \\mathbb{I} + x \\log{(x)} - x, then obtain \\mathbb{I} + x \\log{(x)} - x = \\int \\log{(x)} dx", "derivation": "s{(x)} = \\log{(x)} and \\operatorname{M_{E}}{(x)} = \\log{(x)} and s{(x)} = \\operatorname{M_{E}}{(x)} and \\int \\operatorname{M_{E}}{(x)} dx = \\int \\log{(x)} dx and \\int s{(x)} dx = \\int \\log{(x)} dx and \\int s{(x)} dx = \\mathbb{I} + x \\log{(x)} - x and \\mathbb{I} + x \\log{(x)} - x = \\int \\log{(x)} dx", "srepr_derivation": [["get_premise", "Equality(Function('s')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('s')(Symbol('x', commutative=True)), Function('M_E')(Symbol('x', commutative=True)))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Function('s')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('s')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(V,v_{x})} = v_{x}^{V}, then obtain \\frac{\\partial}{\\partial V} v_{x}^{V} \\operatorname{f^{\\prime}}^{2}{(V,v_{x})} = \\frac{\\partial}{\\partial V} v_{x}^{2 V} \\operatorname{f^{\\prime}}{(V,v_{x})}", "derivation": "\\operatorname{f^{\\prime}}{(V,v_{x})} = v_{x}^{V} and v_{x}^{V} \\operatorname{f^{\\prime}}{(V,v_{x})} = v_{x}^{2 V} and v_{x}^{V} \\operatorname{f^{\\prime}}^{2}{(V,v_{x})} = v_{x}^{2 V} \\operatorname{f^{\\prime}}{(V,v_{x})} and v_{x}^{2 V} \\operatorname{f^{\\prime}}{(V,v_{x})} = v_{x}^{3 V} and v_{x}^{V} \\operatorname{f^{\\prime}}^{2}{(V,v_{x})} = v_{x}^{3 V} and \\frac{\\partial}{\\partial V} v_{x}^{V} \\operatorname{f^{\\prime}}^{2}{(V,v_{x})} = \\frac{\\partial}{\\partial V} v_{x}^{3 V} and \\frac{\\partial}{\\partial V} v_{x}^{V} \\operatorname{f^{\\prime}}^{2}{(V,v_{x})} = \\frac{\\partial}{\\partial V} v_{x}^{2 V} \\operatorname{f^{\\prime}}{(V,v_{x})}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True))), Pow(Symbol('v_x', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))))"], [["times", 2, "Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Integer(2))), Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True))), Pow(Symbol('v_x', commutative=True), Mul(Integer(3), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Integer(2))), Pow(Symbol('v_x', commutative=True), Mul(Integer(3), Symbol('V', commutative=True))))"], [["differentiate", 5, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Integer(2))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Symbol('v_x', commutative=True), Mul(Integer(3), Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Mul(Pow(Symbol('v_x', commutative=True), Symbol('V', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Integer(2))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(t_{2})} = \\cos{(\\sin{(t_{2})})}, then obtain (\\cos{(\\sin{(t_{2})})} + \\frac{\\cos{(\\sin{(t_{2})})}}{W{(t_{2})}})^{t_{2}} = (\\cos{(\\sin{(t_{2})})} - 1 + \\frac{2 \\cos{(\\sin{(t_{2})})}}{W{(t_{2})}})^{t_{2}}", "derivation": "W{(t_{2})} = \\cos{(\\sin{(t_{2})})} and W^{2}{(t_{2})} = W{(t_{2})} \\cos{(\\sin{(t_{2})})} and 1 = \\frac{\\cos{(\\sin{(t_{2})})}}{W{(t_{2})}} and \\cos{(\\sin{(t_{2})})} + 1 = \\cos{(\\sin{(t_{2})})} + \\frac{\\cos{(\\sin{(t_{2})})}}{W{(t_{2})}} and (\\cos{(\\sin{(t_{2})})} + 1)^{t_{2}} = (\\cos{(\\sin{(t_{2})})} + \\frac{\\cos{(\\sin{(t_{2})})}}{W{(t_{2})}})^{t_{2}} and (\\cos{(\\sin{(t_{2})})} + \\frac{\\cos{(\\sin{(t_{2})})}}{W{(t_{2})}})^{t_{2}} = (\\cos{(\\sin{(t_{2})})} - 1 + \\frac{2 \\cos{(\\sin{(t_{2})})}}{W{(t_{2})}})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('t_2', commutative=True)), cos(sin(Symbol('t_2', commutative=True))))"], [["times", 1, "Function('W')(Symbol('t_2', commutative=True))"], "Equality(Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(2)), Mul(Function('W')(Symbol('t_2', commutative=True)), cos(sin(Symbol('t_2', commutative=True)))))"], [["divide", 2, "Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(-1)), cos(sin(Symbol('t_2', commutative=True)))))"], [["add", 3, "cos(sin(Symbol('t_2', commutative=True)))"], "Equality(Add(cos(sin(Symbol('t_2', commutative=True))), Integer(1)), Add(cos(sin(Symbol('t_2', commutative=True))), Mul(Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(-1)), cos(sin(Symbol('t_2', commutative=True))))))"], [["power", 4, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(cos(sin(Symbol('t_2', commutative=True))), Integer(1)), Symbol('t_2', commutative=True)), Pow(Add(cos(sin(Symbol('t_2', commutative=True))), Mul(Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(-1)), cos(sin(Symbol('t_2', commutative=True))))), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(cos(sin(Symbol('t_2', commutative=True))), Mul(Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(-1)), cos(sin(Symbol('t_2', commutative=True))))), Symbol('t_2', commutative=True)), Pow(Add(cos(sin(Symbol('t_2', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('W')(Symbol('t_2', commutative=True)), Integer(-1)), cos(sin(Symbol('t_2', commutative=True))))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\pi)} = \\cos{(\\pi)}, then derive 0 = - \\sin{(\\pi)} - \\frac{d}{d \\pi} \\operatorname{a^{\\dagger}}{(\\pi)}, then obtain 0 = - 2 \\sin{(\\pi)} - \\frac{d}{d \\pi} \\operatorname{a^{\\dagger}}{(\\pi)} - \\frac{d}{d \\pi} \\cos{(\\pi)}", "derivation": "\\operatorname{a^{\\dagger}}{(\\pi)} = \\cos{(\\pi)} and 0 = - \\operatorname{a^{\\dagger}}{(\\pi)} + \\cos{(\\pi)} and \\frac{d}{d \\pi} 0 = \\frac{d}{d \\pi} (- \\operatorname{a^{\\dagger}}{(\\pi)} + \\cos{(\\pi)}) and 0 = - \\sin{(\\pi)} - \\frac{d}{d \\pi} \\operatorname{a^{\\dagger}}{(\\pi)} and - \\sin{(\\pi)} = - 2 \\sin{(\\pi)} - \\frac{d}{d \\pi} \\operatorname{a^{\\dagger}}{(\\pi)} and 0 = - \\sin{(\\pi)} - \\frac{d}{d \\pi} \\cos{(\\pi)} and 0 = - 2 \\sin{(\\pi)} - \\frac{d}{d \\pi} \\operatorname{a^{\\dagger}}{(\\pi)} - \\frac{d}{d \\pi} \\cos{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["minus", 4, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Integer(2), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(V)} = \\frac{1}{V} and u{(M,s)} = s + e^{M}, then obtain \\frac{\\partial}{\\partial V} \\frac{2 u{(M,s)}}{V} = \\frac{\\partial}{\\partial V} (\\frac{s + e^{M}}{V} + \\frac{u{(M,s)}}{V})", "derivation": "\\operatorname{y^{\\prime}}{(V)} = \\frac{1}{V} and u{(M,s)} = s + e^{M} and u{(M,s)} \\operatorname{y^{\\prime}}{(V)} = (s + e^{M}) \\operatorname{y^{\\prime}}{(V)} and \\frac{u{(M,s)}}{V} = \\frac{s + e^{M}}{V} and \\frac{2 u{(M,s)}}{V} = \\frac{s + e^{M}}{V} + \\frac{u{(M,s)}}{V} and \\frac{\\partial}{\\partial V} \\frac{2 u{(M,s)}}{V} = \\frac{\\partial}{\\partial V} (\\frac{s + e^{M}}{V} + \\frac{u{(M,s)}}{V})", "srepr_derivation": [["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('V', commutative=True)), Pow(Symbol('V', commutative=True), Integer(-1)))"], ["get_premise", "Equality(Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True)), Add(Symbol('s', commutative=True), exp(Symbol('M', commutative=True))))"], [["times", 2, "Function('y^{\\\\prime}')(Symbol('V', commutative=True))"], "Equality(Mul(Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True)), Function('y^{\\\\prime}')(Symbol('V', commutative=True))), Mul(Add(Symbol('s', commutative=True), exp(Symbol('M', commutative=True))), Function('y^{\\\\prime}')(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), exp(Symbol('M', commutative=True)))))"], [["add", 4, "Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Add(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), exp(Symbol('M', commutative=True)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True)))))"], [["differentiate", 5, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), exp(Symbol('M', commutative=True)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('u')(Symbol('M', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(z,\\theta)} = \\frac{\\theta}{z} and \\operatorname{y^{\\prime}}{(z,\\theta)} = \\frac{2 \\theta}{z}, then obtain \\frac{1}{\\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)}} = \\frac{\\theta}{z (\\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)}) \\operatorname{F_{x}}{(z,\\theta)}}", "derivation": "\\operatorname{F_{x}}{(z,\\theta)} = \\frac{\\theta}{z} and \\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)} = \\frac{2 \\theta}{z} and \\operatorname{y^{\\prime}}{(z,\\theta)} = \\frac{2 \\theta}{z} and \\operatorname{y^{\\prime}}{(z,\\theta)} = 2 \\operatorname{F_{x}}{(z,\\theta)} and 2 \\operatorname{F_{x}}{(z,\\theta)} = \\frac{2 \\theta}{z} and \\frac{1}{2 \\operatorname{F_{x}}{(z,\\theta)}} = \\frac{\\theta}{2 z \\operatorname{F_{x}}^{2}{(z,\\theta)}} and \\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)} = 2 \\operatorname{F_{x}}{(z,\\theta)} and \\frac{1}{\\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)}} = \\frac{\\theta}{z (\\frac{\\theta}{z} + \\operatorname{F_{x}}{(z,\\theta)}) \\operatorname{F_{x}}{(z,\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["divide", 5, "Mul(Integer(4), Pow(Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 2), Pow(Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-1)), Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-1)), Pow(Function('F_x')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(J)} = \\frac{d}{d J} \\cos{(J)}, then obtain -1 = - (\\frac{\\frac{d}{d J} \\cos{(J)}}{\\operatorname{g_{\\varepsilon}}{(J)}})^{J}", "derivation": "\\operatorname{g_{\\varepsilon}}{(J)} = \\frac{d}{d J} \\cos{(J)} and 1 = \\frac{\\frac{d}{d J} \\cos{(J)}}{\\operatorname{g_{\\varepsilon}}{(J)}} and 1 = (\\frac{\\frac{d}{d J} \\cos{(J)}}{\\operatorname{g_{\\varepsilon}}{(J)}})^{J} and -1 = - (\\frac{\\frac{d}{d J} \\cos{(J)}}{\\operatorname{g_{\\varepsilon}}{(J)}})^{J}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('J', commutative=True)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["divide", 1, "Function('g_{\\\\varepsilon}')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('J', commutative=True)), Integer(-1)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('J', commutative=True)), Integer(-1)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Symbol('J', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('J', commutative=True)), Integer(-1)), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Symbol('J', commutative=True))))"]]}, {"prompt": "Given s{(S,A_{1})} = A_{1}^{S}, then derive \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} = \\frac{A_{1}^{S} (S \\log{(A_{1})} + 1)}{A_{1}}, then obtain \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} + \\frac{1}{s{(S,A_{1})}} = \\frac{1}{s{(S,A_{1})}} + \\frac{(S \\log{(A_{1})} + 1) s{(S,A_{1})}}{A_{1}}", "derivation": "s{(S,A_{1})} = A_{1}^{S} and \\frac{\\partial}{\\partial A_{1}} s{(S,A_{1})} = \\frac{\\partial}{\\partial A_{1}} A_{1}^{S} and \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} = \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} A_{1}^{S} and \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} = \\frac{A_{1}^{S} (S \\log{(A_{1})} + 1)}{A_{1}} and \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} = \\frac{(S \\log{(A_{1})} + 1) s{(S,A_{1})}}{A_{1}} and \\frac{\\partial^{2}}{\\partial S\\partial A_{1}} s{(S,A_{1})} + \\frac{1}{s{(S,A_{1})}} = \\frac{1}{s{(S,A_{1})}} + \\frac{(S \\log{(A_{1})} + 1) s{(S,A_{1})}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_1', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_1', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('A_1', commutative=True), Symbol('S', commutative=True)), Add(Mul(Symbol('S', commutative=True), log(Symbol('A_1', commutative=True))), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Add(Mul(Symbol('S', commutative=True), log(Symbol('A_1', commutative=True))), Integer(1)), Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True))))"], [["add", 5, "Pow(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))"], "Equality(Add(Derivative(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Add(Pow(Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Add(Mul(Symbol('S', commutative=True), log(Symbol('A_1', commutative=True))), Integer(1)), Function('s')(Symbol('S', commutative=True), Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(r)} = \\int \\cos{(r)} dr and \\mathbf{F}{(r)} = \\cos{(r)}, then obtain \\delta + \\sin{(r)} + \\frac{d}{d r} \\Psi^{r}{(r)} = \\delta + \\sin{(r)} + \\frac{d}{d r} (\\int \\cos{(r)} dr)^{r}", "derivation": "\\Psi{(r)} = \\int \\cos{(r)} dr and \\mathbf{F}{(r)} = \\cos{(r)} and \\int \\mathbf{F}{(r)} dr = \\int \\cos{(r)} dr and \\Psi{(r)} = \\int \\mathbf{F}{(r)} dr and \\Psi^{r}{(r)} = (\\int \\mathbf{F}{(r)} dr)^{r} and \\frac{d}{d r} \\Psi^{r}{(r)} = \\frac{d}{d r} (\\int \\mathbf{F}{(r)} dr)^{r} and \\delta + \\sin{(r)} + \\frac{d}{d r} \\Psi^{r}{(r)} = \\delta + \\sin{(r)} + \\frac{d}{d r} (\\int \\mathbf{F}{(r)} dr)^{r} and \\delta + \\sin{(r)} + \\frac{d}{d r} \\Psi^{r}{(r)} = \\delta + \\sin{(r)} + \\frac{d}{d r} (\\int \\cos{(r)} dr)^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('r', commutative=True)), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\Psi')(Symbol('r', commutative=True)), Integral(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["power", 4, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Integral(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Integral(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["add", 6, "Add(Symbol('\\\\delta', commutative=True), sin(Symbol('r', commutative=True)))"], "Equality(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('r', commutative=True)), Derivative(Pow(Function('\\\\Psi')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), sin(Symbol('r', commutative=True)), Derivative(Pow(Integral(Function('\\\\mathbf{F}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('r', commutative=True)), Derivative(Pow(Function('\\\\Psi')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), sin(Symbol('r', commutative=True)), Derivative(Pow(Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(\\omega)} = \\cos{(\\omega)}, then obtain (\\frac{\\frac{d}{d \\omega} \\theta{(\\omega)}}{\\cos{(\\omega)}})^{\\omega} = (\\frac{\\frac{d}{d \\omega} \\cos{(\\omega)}}{\\cos{(\\omega)}})^{\\omega}", "derivation": "\\theta{(\\omega)} = \\cos{(\\omega)} and \\frac{d}{d \\omega} \\theta{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)} and \\frac{\\frac{d}{d \\omega} \\theta{(\\omega)}}{\\cos{(\\omega)}} = \\frac{\\frac{d}{d \\omega} \\cos{(\\omega)}}{\\cos{(\\omega)}} and (\\frac{\\frac{d}{d \\omega} \\theta{(\\omega)}}{\\cos{(\\omega)}})^{\\omega} = (\\frac{\\frac{d}{d \\omega} \\cos{(\\omega)}}{\\cos{(\\omega)}})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["divide", 2, "cos(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Pow(cos(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})} and v{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})}, then obtain v{(\\sigma_p)} (e^{\\mathbf{S}{(\\sigma_p)}})^{\\sigma_p} = (e^{\\mathbf{S}{(\\sigma_p)}})^{\\sigma_p} \\log{(\\log{(\\sigma_p)})}", "derivation": "\\mathbf{S}{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})} and e^{\\mathbf{S}{(\\sigma_p)}} = \\log{(\\sigma_p)} and v{(\\sigma_p)} = \\log{(\\log{(\\sigma_p)})} and (e^{\\mathbf{S}{(\\sigma_p)}})^{\\sigma_p} = \\log{(\\sigma_p)}^{\\sigma_p} and v{(\\sigma_p)} \\log{(\\sigma_p)}^{\\sigma_p} = \\log{(\\sigma_p)}^{\\sigma_p} \\log{(\\log{(\\sigma_p)})} and v{(\\sigma_p)} (e^{\\mathbf{S}{(\\sigma_p)}})^{\\sigma_p} = (e^{\\mathbf{S}{(\\sigma_p)}})^{\\sigma_p} \\log{(\\log{(\\sigma_p)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True)), log(log(Symbol('\\\\sigma_p', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\sigma_p', commutative=True)), log(log(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(exp(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["times", 3, "Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('v')(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), log(log(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('v')(Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(exp(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), log(log(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(U)} = \\cos{(U)}, then obtain (- \\frac{\\operatorname{f_{\\mathbf{p}}}^{2}{(U)}}{\\cos{(U)}})^{U} = (- \\operatorname{f_{\\mathbf{p}}}{(U)})^{U}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(U)} = \\cos{(U)} and - \\operatorname{f_{\\mathbf{p}}}{(U)} = - \\cos{(U)} and - \\frac{\\operatorname{f_{\\mathbf{p}}}^{2}{(U)}}{\\cos{(U)}} = - \\operatorname{f_{\\mathbf{p}}}{(U)} and (- \\frac{\\operatorname{f_{\\mathbf{p}}}^{2}{(U)}}{\\cos{(U)}})^{U} = (- \\operatorname{f_{\\mathbf{p}}}{(U)})^{U}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True))))"], [["times", 2, "Mul(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Pow(cos(Symbol('U', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(2)), Pow(cos(Symbol('U', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True)), Integer(2)), Pow(cos(Symbol('U', commutative=True)), Integer(-1))), Symbol('U', commutative=True)), Pow(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\rho_f,\\mathbb{I})} = \\frac{\\sin{(\\rho_f)}}{\\mathbb{I}} and \\operatorname{v_{z}}{(\\rho_f,\\mathbb{I})} = \\frac{\\sin{(\\rho_f)}}{\\mathbb{I}}, then obtain \\operatorname{f^{*}}^{\\mathbb{I}}{(\\rho_f,\\mathbb{I})} = (\\frac{\\sin{(\\rho_f)}}{\\mathbb{I}})^{\\mathbb{I}}", "derivation": "\\operatorname{f^{*}}{(\\rho_f,\\mathbb{I})} = \\frac{\\sin{(\\rho_f)}}{\\mathbb{I}} and \\operatorname{v_{z}}{(\\rho_f,\\mathbb{I})} = \\frac{\\sin{(\\rho_f)}}{\\mathbb{I}} and \\operatorname{v_{z}}{(\\rho_f,\\mathbb{I})} = \\operatorname{f^{*}}{(\\rho_f,\\mathbb{I})} and \\operatorname{v_{z}}^{\\mathbb{I}}{(\\rho_f,\\mathbb{I})} = (\\frac{\\sin{(\\rho_f)}}{\\mathbb{I}})^{\\mathbb{I}} and \\operatorname{f^{*}}^{\\mathbb{I}}{(\\rho_f,\\mathbb{I})} = (\\frac{\\sin{(\\rho_f)}}{\\mathbb{I}})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('f^*')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('f^*')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(\\phi,\\mathbf{D})} = \\phi^{\\mathbf{D}} and \\psi^{*}{(\\phi,\\mathbf{D})} = \\phi^{\\mathbf{D}}, then obtain 0 = \\phi^{\\mathbf{D}} - \\psi^{*}{(\\phi,\\mathbf{D})}", "derivation": "\\mathbf{H}{(\\phi,\\mathbf{D})} = \\phi^{\\mathbf{D}} and 0 = \\phi^{\\mathbf{D}} - \\mathbf{H}{(\\phi,\\mathbf{D})} and \\psi^{*}{(\\phi,\\mathbf{D})} = \\phi^{\\mathbf{D}} and \\psi^{*}{(\\phi,\\mathbf{D})} = \\mathbf{H}{(\\phi,\\mathbf{D})} and 0 = \\phi^{\\mathbf{D}} - \\psi^{*}{(\\phi,\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\psi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(0), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\phi,\\mathbf{g})} = - \\mathbf{g} + \\phi and l{(\\phi,\\mathbf{g})} = - \\mathbf{g} + \\phi - \\operatorname{A_{2}}{(\\phi,\\mathbf{g})}, then obtain \\frac{\\frac{d}{d \\phi} 0}{\\phi + y} = \\frac{\\frac{\\partial}{\\partial \\phi} l{(\\phi,\\mathbf{g})}}{\\phi + y}", "derivation": "\\operatorname{A_{2}}{(\\phi,\\mathbf{g})} = - \\mathbf{g} + \\phi and 0 = - \\mathbf{g} + \\phi - \\operatorname{A_{2}}{(\\phi,\\mathbf{g})} and \\frac{d}{d \\phi} 0 = \\frac{\\partial}{\\partial \\phi} (- \\mathbf{g} + \\phi - \\operatorname{A_{2}}{(\\phi,\\mathbf{g})}) and l{(\\phi,\\mathbf{g})} = - \\mathbf{g} + \\phi - \\operatorname{A_{2}}{(\\phi,\\mathbf{g})} and \\frac{\\frac{d}{d \\phi} 0}{\\int 1 d\\phi} = \\frac{\\frac{\\partial}{\\partial \\phi} (- \\mathbf{g} + \\phi - \\operatorname{A_{2}}{(\\phi,\\mathbf{g})})}{\\int 1 d\\phi} and \\frac{\\frac{d}{d \\phi} 0}{\\int 1 d\\phi} = \\frac{\\frac{\\partial}{\\partial \\phi} l{(\\phi,\\mathbf{g})}}{\\int 1 d\\phi} and \\frac{\\frac{d}{d \\phi} 0}{\\phi + y} = \\frac{\\frac{\\partial}{\\partial \\phi} l{(\\phi,\\mathbf{g})}}{\\phi + y}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 3, "Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Pow(Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))), Mul(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Pow(Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Pow(Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))), Mul(Derivative(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Pow(Integral(Integer(1), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Add(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Derivative(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\omega)} = \\log{(\\omega)}, then obtain e^{\\operatorname{A_{x}}{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}} = e^{\\log{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}}", "derivation": "\\operatorname{A_{x}}{(\\omega)} = \\log{(\\omega)} and \\cos{(\\operatorname{A_{x}}{(\\omega)})} = \\cos{(\\log{(\\omega)})} and \\operatorname{A_{x}}{(\\omega)} \\cos{(\\log{(\\omega)})} = \\log{(\\omega)} \\cos{(\\log{(\\omega)})} and \\operatorname{A_{x}}{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})} = \\operatorname{A_{x}}{(\\omega)} \\cos{(\\log{(\\omega)})} and e^{\\operatorname{A_{x}}{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}} = e^{\\operatorname{A_{x}}{(\\omega)} \\cos{(\\log{(\\omega)})}} and e^{\\operatorname{A_{x}}{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}} = e^{\\log{(\\omega)} \\cos{(\\log{(\\omega)})}} and e^{\\operatorname{A_{x}}{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}} = e^{\\log{(\\omega)} \\cos{(\\operatorname{A_{x}}{(\\omega)})}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["cos", 1], "Equality(cos(Function('A_x')(Symbol('\\\\omega', commutative=True))), cos(log(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "cos(log(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True)))), Mul(log(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True)))))"], [["times", 2, "Function('A_x')(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(Function('A_x')(Symbol('\\\\omega', commutative=True)))), Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True)))))"], [["exp", 4], "Equality(exp(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(Function('A_x')(Symbol('\\\\omega', commutative=True))))), exp(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(exp(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(Function('A_x')(Symbol('\\\\omega', commutative=True))))), exp(Mul(log(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(exp(Mul(Function('A_x')(Symbol('\\\\omega', commutative=True)), cos(Function('A_x')(Symbol('\\\\omega', commutative=True))))), exp(Mul(log(Symbol('\\\\omega', commutative=True)), cos(Function('A_x')(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\chi{(l)} = e^{l}, then derive \\frac{d}{d l} \\chi{(l)} = e^{l}, then obtain \\int \\frac{d}{d l} \\chi{(l)} dl = \\int \\frac{d^{2}}{d l^{2}} e^{l} dl", "derivation": "\\chi{(l)} = e^{l} and \\frac{d}{d l} \\chi{(l)} = \\frac{d}{d l} e^{l} and \\frac{d}{d l} \\chi{(l)} = e^{l} and \\frac{d}{d l} e^{l} = e^{l} and \\frac{d}{d l} \\chi{(l)} = \\frac{d^{2}}{d l^{2}} e^{l} and \\int \\frac{d}{d l} \\chi{(l)} dl = \\int \\frac{d^{2}}{d l^{2}} e^{l} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), exp(Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), exp(Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\chi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))))"], [["integrate", 5, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\chi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(2))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\hat{p}_0,\\eta^{\\prime})} = - \\hat{p}_0 + \\log{(\\eta^{\\prime})}, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} \\lambda{(\\hat{p}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\hat{p}_0} (- 2 \\hat{p}_0 - \\lambda{(\\hat{p}_0,\\eta^{\\prime})} + 2 \\log{(\\eta^{\\prime})})", "derivation": "\\lambda{(\\hat{p}_0,\\eta^{\\prime})} = - \\hat{p}_0 + \\log{(\\eta^{\\prime})} and - \\hat{p}_0 + \\lambda{(\\hat{p}_0,\\eta^{\\prime})} + \\log{(\\eta^{\\prime})} = - 2 \\hat{p}_0 + 2 \\log{(\\eta^{\\prime})} and - \\hat{p}_0 + \\log{(\\eta^{\\prime})} = - 2 \\hat{p}_0 - \\lambda{(\\hat{p}_0,\\eta^{\\prime})} + 2 \\log{(\\eta^{\\prime})} and \\lambda{(\\hat{p}_0,\\eta^{\\prime})} = - 2 \\hat{p}_0 - \\lambda{(\\hat{p}_0,\\eta^{\\prime})} + 2 \\log{(\\eta^{\\prime})} and \\frac{\\partial}{\\partial \\hat{p}_0} \\lambda{(\\hat{p}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\hat{p}_0} (- 2 \\hat{p}_0 - \\lambda{(\\hat{p}_0,\\eta^{\\prime})} + 2 \\log{(\\eta^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["minus", 2, "Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{J}_M)} = \\cos{(e^{\\mathbf{J}_M})} and \\operatorname{C_{1}}{(\\mathbf{J}_M)} = \\frac{\\operatorname{E_{n}}{(\\mathbf{J}_M)}}{\\cos{(e^{\\mathbf{J}_M})}}, then obtain (\\frac{d}{d \\mathbf{J}_M} \\int \\operatorname{C_{1}}{(\\mathbf{J}_M)} d\\mathbf{J}_M)^{\\mathbf{J}_M} = (\\frac{d}{d \\mathbf{J}_M} \\int 1 d\\mathbf{J}_M)^{\\mathbf{J}_M}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{J}_M)} = \\cos{(e^{\\mathbf{J}_M})} and \\operatorname{C_{1}}{(\\mathbf{J}_M)} = \\frac{\\operatorname{E_{n}}{(\\mathbf{J}_M)}}{\\cos{(e^{\\mathbf{J}_M})}} and \\operatorname{C_{1}}{(\\mathbf{J}_M)} = 1 and \\int \\operatorname{C_{1}}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\int 1 d\\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\int \\operatorname{C_{1}}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\frac{d}{d \\mathbf{J}_M} \\int 1 d\\mathbf{J}_M and (\\frac{d}{d \\mathbf{J}_M} \\int \\operatorname{C_{1}}{(\\mathbf{J}_M)} d\\mathbf{J}_M)^{\\mathbf{J}_M} = (\\frac{d}{d \\mathbf{J}_M} \\int 1 d\\mathbf{J}_M)^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(exp(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Function('E_n')(Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_1')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Integral(Function('C_1')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('C_1')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given v{(\\phi_1,n_{2})} = \\phi_1 + \\cos{(n_{2})} and \\rho_{b}{(\\phi_1,n_{2})} = v{(\\phi_1,n_{2})} \\cos{(n_{2})}, then obtain \\rho_{b}{(\\phi_1,n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\frac{\\phi_1 + \\cos{(n_{2})}}{\\cos{(n_{2})}} = (\\phi_1 + \\cos{(n_{2})}) \\cos{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\frac{\\phi_1 + \\cos{(n_{2})}}{\\cos{(n_{2})}}", "derivation": "v{(\\phi_1,n_{2})} = \\phi_1 + \\cos{(n_{2})} and v{(\\phi_1,n_{2})} \\cos{(n_{2})} = (\\phi_1 + \\cos{(n_{2})}) \\cos{(n_{2})} and \\rho_{b}{(\\phi_1,n_{2})} = v{(\\phi_1,n_{2})} \\cos{(n_{2})} and \\rho_{b}{(\\phi_1,n_{2})} = (\\phi_1 + \\cos{(n_{2})}) \\cos{(n_{2})} and \\rho_{b}{(\\phi_1,n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\frac{\\phi_1 + \\cos{(n_{2})}}{\\cos{(n_{2})}} = (\\phi_1 + \\cos{(n_{2})}) \\cos{(n_{2})} + \\frac{\\partial}{\\partial n_{2}} \\frac{\\phi_1 + \\cos{(n_{2})}}{\\cos{(n_{2})}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))))"], [["divide", 1, "Pow(cos(Symbol('n_2', commutative=True)), Integer(-1))"], "Equality(Mul(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))), Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), cos(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), cos(Symbol('n_2', commutative=True))))"], [["add", 4, "Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), Pow(cos(Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('n_2', commutative=True)), Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), Pow(cos(Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), cos(Symbol('n_2', commutative=True))), Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), cos(Symbol('n_2', commutative=True))), Pow(cos(Symbol('n_2', commutative=True)), Integer(-1))), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(m)} = e^{m}, then obtain (\\iint (- m + \\eta^{\\prime}{(m)}) dm dm)^{m} = (\\iint (- m + e^{m}) dm dm)^{m}", "derivation": "\\eta^{\\prime}{(m)} = e^{m} and - m + \\eta^{\\prime}{(m)} = - m + e^{m} and \\int (- m + \\eta^{\\prime}{(m)}) dm = \\int (- m + e^{m}) dm and \\iint (- m + \\eta^{\\prime}{(m)}) dm dm = \\iint (- m + e^{m}) dm dm and (\\iint (- m + \\eta^{\\prime}{(m)}) dm dm)^{m} = (\\iint (- m + e^{m}) dm dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\log{(\\hat{H})}, then derive \\mathbf{A}^{2}{(\\hat{H})} = \\frac{\\mathbf{A}{(\\hat{H})}}{\\hat{H}}, then obtain \\int (\\frac{d}{d \\hat{H}} \\log{(\\hat{H})})^{2} d\\hat{H} = \\int \\frac{\\frac{d}{d \\hat{H}} \\log{(\\hat{H})}}{\\hat{H}} d\\hat{H}", "derivation": "\\mathbf{A}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\log{(\\hat{H})} and \\mathbf{A}^{2}{(\\hat{H})} = \\mathbf{A}{(\\hat{H})} \\frac{d}{d \\hat{H}} \\log{(\\hat{H})} and \\mathbf{A}^{2}{(\\hat{H})} = \\frac{\\mathbf{A}{(\\hat{H})}}{\\hat{H}} and (\\frac{d}{d \\hat{H}} \\log{(\\hat{H})})^{2} = \\frac{\\frac{d}{d \\hat{H}} \\log{(\\hat{H})}}{\\hat{H}} and \\int (\\frac{d}{d \\hat{H}} \\log{(\\hat{H})})^{2} d\\hat{H} = \\int \\frac{\\frac{d}{d \\hat{H}} \\log{(\\hat{H})}}{\\hat{H}} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Pow(Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given S{(s,B)} = B^{s} and \\operatorname{n_{1}}{(s,B)} = \\int S{(s,B)} dB, then obtain \\int \\frac{\\partial}{\\partial s} \\int B^{s} dB ds = \\int \\frac{\\partial}{\\partial s} \\operatorname{n_{1}}{(s,B)} ds", "derivation": "S{(s,B)} = B^{s} and \\int S{(s,B)} dB = \\int B^{s} dB and \\operatorname{n_{1}}{(s,B)} = \\int S{(s,B)} dB and \\operatorname{n_{1}}{(s,B)} = \\int B^{s} dB and \\frac{\\partial}{\\partial s} \\int S{(s,B)} dB = \\frac{\\partial}{\\partial s} \\int B^{s} dB and \\frac{\\partial}{\\partial s} \\int S{(s,B)} dB = \\frac{\\partial}{\\partial s} \\operatorname{n_{1}}{(s,B)} and \\frac{\\partial}{\\partial s} \\int B^{s} dB = \\frac{\\partial}{\\partial s} \\operatorname{n_{1}}{(s,B)} and \\int \\frac{\\partial}{\\partial s} \\int B^{s} dB ds = \\int \\frac{\\partial}{\\partial s} \\operatorname{n_{1}}{(s,B)} ds", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('S')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Integral(Function('S')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('n_1')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Integral(Function('S')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('n_1')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('n_1')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["integrate", 7, "Symbol('s', commutative=True)"], "Equality(Integral(Derivative(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))), Integral(Derivative(Function('n_1')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{s},q)} = \\frac{\\cos{(q)}}{\\mathbf{s}}, then obtain - (- \\frac{\\cos^{2}{(q)}}{\\mathbf{s}^{2}})^{- q} \\hat{p}{(\\mathbf{s},q)} \\cos{(q)} = - \\frac{(- \\frac{\\cos^{2}{(q)}}{\\mathbf{s}^{2}})^{- q} \\cos^{2}{(q)}}{\\mathbf{s}}", "derivation": "\\hat{p}{(\\mathbf{s},q)} = \\frac{\\cos{(q)}}{\\mathbf{s}} and - \\hat{p}{(\\mathbf{s},q)} = - \\frac{\\cos{(q)}}{\\mathbf{s}} and - \\frac{\\hat{p}{(\\mathbf{s},q)} \\cos{(q)}}{\\mathbf{s}} = - \\frac{\\cos^{2}{(q)}}{\\mathbf{s}^{2}} and - \\hat{p}{(\\mathbf{s},q)} \\cos{(q)} = - \\frac{\\cos^{2}{(q)}}{\\mathbf{s}} and - (- \\frac{\\cos^{2}{(q)}}{\\mathbf{s}^{2}})^{- q} \\hat{p}{(\\mathbf{s},q)} \\cos{(q)} = - \\frac{(- \\frac{\\cos^{2}{(q)}}{\\mathbf{s}^{2}})^{- q} \\cos^{2}{(q)}}{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-2)), Pow(cos(Symbol('q', commutative=True)), Integer(2))))"], [["times", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(cos(Symbol('q', commutative=True)), Integer(2))))"], [["divide", 4, "Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-2)), Pow(cos(Symbol('q', commutative=True)), Integer(2))), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-2)), Pow(cos(Symbol('q', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('q', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-2)), Pow(cos(Symbol('q', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(cos(Symbol('q', commutative=True)), Integer(2))))"]]}, {"prompt": "Given i{(Z)} = \\log{(Z)}, then derive 2 \\frac{d}{d Z} i{(Z)} = \\frac{d}{d Z} i{(Z)} + \\frac{1}{Z}, then obtain \\frac{2 \\frac{d}{d Z} \\log{(Z)}}{i{(Z)}} = \\frac{\\frac{d}{d Z} \\log{(Z)} + \\frac{1}{Z}}{i{(Z)}}", "derivation": "i{(Z)} = \\log{(Z)} and 2 i{(Z)} = i{(Z)} + \\log{(Z)} and \\frac{d}{d Z} 2 i{(Z)} = \\frac{d}{d Z} (i{(Z)} + \\log{(Z)}) and 2 \\frac{d}{d Z} i{(Z)} = \\frac{d}{d Z} i{(Z)} + \\frac{1}{Z} and 2 \\frac{d}{d Z} \\log{(Z)} = \\frac{d}{d Z} \\log{(Z)} + \\frac{1}{Z} and \\frac{2 \\frac{d}{d Z} \\log{(Z)}}{i{(Z)}} = \\frac{\\frac{d}{d Z} \\log{(Z)} + \\frac{1}{Z}}{i{(Z)}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["add", 1, "Function('i')(Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Function('i')(Symbol('Z', commutative=True))), Add(Function('i')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('i')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Function('i')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('i')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Derivative(Function('i')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["times", 5, "Pow(Function('i')(Symbol('Z', commutative=True)), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Function('i')(Symbol('Z', commutative=True)), Integer(-1)), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Add(Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Symbol('Z', commutative=True), Integer(-1))), Pow(Function('i')(Symbol('Z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given p{(n_{1})} = \\cos{(n_{1})}, then obtain 2 \\cos{(n_{1})} + 1 = p{(n_{1})} + \\cos{(n_{1})} + 1", "derivation": "p{(n_{1})} = \\cos{(n_{1})} and p{(n_{1})} + 1 = \\cos{(n_{1})} + 1 and 2 p{(n_{1})} + 1 = p{(n_{1})} + \\cos{(n_{1})} + 1 and 2 p{(n_{1})} + 1 = 2 \\cos{(n_{1})} + 1 and 2 \\cos{(n_{1})} + 1 = p{(n_{1})} + \\cos{(n_{1})} + 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('p')(Symbol('n_1', commutative=True)), Integer(1)), Add(cos(Symbol('n_1', commutative=True)), Integer(1)))"], [["add", 2, "Function('p')(Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('p')(Symbol('n_1', commutative=True))), Integer(1)), Add(Function('p')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('p')(Symbol('n_1', commutative=True))), Integer(1)), Add(Mul(Integer(2), cos(Symbol('n_1', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(2), cos(Symbol('n_1', commutative=True))), Integer(1)), Add(Function('p')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{r}{(A)} = \\sin{(A)}, then derive A + k = \\int \\frac{\\sin{(A)}}{\\mathbf{r}{(A)}} dA, then obtain A + k = \\int 1 dA", "derivation": "\\mathbf{r}{(A)} = \\sin{(A)} and 1 = \\frac{\\sin{(A)}}{\\mathbf{r}{(A)}} and \\int 1 dA = \\int \\frac{\\sin{(A)}}{\\mathbf{r}{(A)}} dA and A + k = \\int \\frac{\\sin{(A)}}{\\mathbf{r}{(A)}} dA and A + k = \\int 1 dA", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{r}')(Symbol('A', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A', commutative=True)), Integer(-1)), sin(Symbol('A', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A', commutative=True)), Integer(-1)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A', commutative=True), Symbol('k', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('A', commutative=True)), Integer(-1)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('A', commutative=True), Symbol('k', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} e^{\\phi_1}, then obtain (\\phi_1 \\operatorname{c_{0}}{(\\phi_1)})^{\\phi_1} = (\\phi_1 e^{\\phi_1})^{\\phi_1}", "derivation": "\\operatorname{c_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} e^{\\phi_1} and \\phi_1 \\operatorname{c_{0}}{(\\phi_1)} = \\phi_1 \\frac{d}{d \\phi_1} e^{\\phi_1} and (\\phi_1 \\operatorname{c_{0}}{(\\phi_1)})^{\\phi_1} = (\\phi_1 \\frac{d}{d \\phi_1} e^{\\phi_1})^{\\phi_1} and (\\phi_1 \\operatorname{c_{0}}{(\\phi_1)})^{\\phi_1} = (\\phi_1 e^{\\phi_1})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\phi_1', commutative=True)), Derivative(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["times", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('c_0')(Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), Derivative(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\phi_1', commutative=True), Function('c_0')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), Derivative(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Symbol('\\\\phi_1', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Mul(Symbol('\\\\phi_1', commutative=True), Function('c_0')(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\psi,A_{z})} = \\frac{A_{z}}{\\psi} and x{(\\psi,A_{z})} = \\int \\frac{A_{z}}{\\psi^{2}} d\\psi, then obtain x{(\\psi,A_{z})} = \\int \\frac{\\operatorname{F_{H}}{(\\psi,A_{z})}}{\\psi} d\\psi", "derivation": "\\operatorname{F_{H}}{(\\psi,A_{z})} = \\frac{A_{z}}{\\psi} and \\frac{\\operatorname{F_{H}}{(\\psi,A_{z})}}{\\psi} = \\frac{A_{z}}{\\psi^{2}} and \\int \\frac{\\operatorname{F_{H}}{(\\psi,A_{z})}}{\\psi} d\\psi = \\int \\frac{A_{z}}{\\psi^{2}} d\\psi and x{(\\psi,A_{z})} = \\int \\frac{A_{z}}{\\psi^{2}} d\\psi and x{(\\psi,A_{z})} = \\int \\frac{\\operatorname{F_{H}}{(\\psi,A_{z})}}{\\psi} d\\psi", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('\\\\psi', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-2))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-2))), Tuple(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-2))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\sigma_x)} = e^{\\sigma_x}, then obtain (\\hat{\\mathbf{r}}{(\\sigma_x)} e^{\\sigma_x} + e^{\\sigma_x} \\frac{d}{d \\sigma_x} \\hat{\\mathbf{r}}{(\\sigma_x)})^{\\sigma_x} = (2 e^{2 \\sigma_x})^{\\sigma_x}", "derivation": "\\hat{\\mathbf{r}}{(\\sigma_x)} = e^{\\sigma_x} and \\hat{\\mathbf{r}}{(\\sigma_x)} e^{\\sigma_x} = e^{2 \\sigma_x} and \\frac{d}{d \\sigma_x} \\hat{\\mathbf{r}}{(\\sigma_x)} e^{\\sigma_x} = \\frac{d}{d \\sigma_x} e^{2 \\sigma_x} and (\\frac{d}{d \\sigma_x} \\hat{\\mathbf{r}}{(\\sigma_x)} e^{\\sigma_x})^{\\sigma_x} = (\\frac{d}{d \\sigma_x} e^{2 \\sigma_x})^{\\sigma_x} and (\\hat{\\mathbf{r}}{(\\sigma_x)} e^{\\sigma_x} + e^{\\sigma_x} \\frac{d}{d \\sigma_x} \\hat{\\mathbf{r}}{(\\sigma_x)})^{\\sigma_x} = (2 e^{2 \\sigma_x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)), Pow(Derivative(exp(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Mul(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Mul(exp(Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\rho_f,\\mathbf{M})} = \\cos{(\\mathbf{M} - \\rho_f)} and \\operatorname{E_{n}}{(\\rho_f,\\mathbf{M})} = \\mathbf{M} - \\rho_f, then obtain \\frac{\\cos{(\\operatorname{E_{n}}{(\\rho_f,\\mathbf{M})})}}{\\cos{(\\mathbf{M} - \\rho_f)}} = 1", "derivation": "\\mathbf{B}{(\\rho_f,\\mathbf{M})} = \\cos{(\\mathbf{M} - \\rho_f)} and \\frac{\\mathbf{B}{(\\rho_f,\\mathbf{M})}}{\\cos{(\\mathbf{M} - \\rho_f)}} = 1 and \\operatorname{E_{n}}{(\\rho_f,\\mathbf{M})} = \\mathbf{M} - \\rho_f and \\mathbf{B}{(\\rho_f,\\mathbf{M})} = \\cos{(\\operatorname{E_{n}}{(\\rho_f,\\mathbf{M})})} and \\frac{\\cos{(\\operatorname{E_{n}}{(\\rho_f,\\mathbf{M})})}}{\\cos{(\\mathbf{M} - \\rho_f)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))"], [["divide", 1, "cos(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Function('E_n')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(cos(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(-1)), cos(Function('E_n')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(g)} = \\sin{(e^{g})} and \\mathbf{D}{(g)} = \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})}, then obtain \\frac{d^{2}}{d g^{2}} \\operatorname{V_{\\mathbf{E}}}^{2}{(g)} = \\frac{d^{2}}{d g^{2}} \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(g)} = \\sin{(e^{g})} and \\mathbf{D}{(g)} = \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})} and \\mathbf{D}{(g)} = \\operatorname{V_{\\mathbf{E}}}^{2}{(g)} and \\frac{d}{d g} \\mathbf{D}{(g)} = \\frac{d}{d g} \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})} and \\frac{d}{d g} \\operatorname{V_{\\mathbf{E}}}^{2}{(g)} = \\frac{d}{d g} \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})} and \\frac{d^{2}}{d g^{2}} \\operatorname{V_{\\mathbf{E}}}^{2}{(g)} = \\frac{d^{2}}{d g^{2}} \\operatorname{V_{\\mathbf{E}}}{(g)} \\sin{(e^{g})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('g', commutative=True)), Mul(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{D}')(Symbol('g', commutative=True)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('g', commutative=True)"], "Equality(Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(2))), Derivative(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(G)} = \\sin{(\\sin{(G)})} and \\operatorname{f_{\\mathbf{v}}}{(G)} = \\operatorname{f_{E}}^{2}{(G)}, then obtain (\\operatorname{f_{E}}^{2}{(G)} - \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})}) \\operatorname{f_{E}}^{2}{(G)} = 0", "derivation": "\\operatorname{f_{E}}{(G)} = \\sin{(\\sin{(G)})} and \\operatorname{f_{E}}^{2}{(G)} = \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})} and \\operatorname{f_{\\mathbf{v}}}{(G)} = \\operatorname{f_{E}}^{2}{(G)} and \\operatorname{f_{\\mathbf{v}}}{(G)} = \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})} and - \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})} + \\operatorname{f_{\\mathbf{v}}}{(G)} = 0 and (- \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})} + \\operatorname{f_{\\mathbf{v}}}{(G)}) \\operatorname{f_{E}}^{2}{(G)} = 0 and (- \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})} + \\operatorname{f_{\\mathbf{v}}}{(G)}) \\operatorname{f_{\\mathbf{v}}}{(G)} = 0 and (\\operatorname{f_{E}}^{2}{(G)} - \\operatorname{f_{E}}{(G)} \\sin{(\\sin{(G)})}) \\operatorname{f_{E}}^{2}{(G)} = 0", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], [["times", 1, "Function('f_E')(Symbol('G', commutative=True))"], "Equality(Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2)), Mul(Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True)), Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True)), Mul(Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))))"], [["minus", 4, "Mul(Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True))), Integer(0))"], [["times", 5, "Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2))"], "Equality(Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True))), Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('G', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Add(Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2)), Mul(Integer(-1), Function('f_E')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))), Pow(Function('f_E')(Symbol('G', commutative=True)), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(F_{c},t)} = e^{F_{c}^{t}}, then obtain F_{c} (\\frac{\\partial}{\\partial F_{c}} \\operatorname{r_{0}}{(F_{c},t)} - \\frac{\\partial}{\\partial F_{c}} e^{F_{c}^{t}}) = 0", "derivation": "\\operatorname{r_{0}}{(F_{c},t)} = e^{F_{c}^{t}} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{r_{0}}{(F_{c},t)} = \\frac{\\partial}{\\partial F_{c}} e^{F_{c}^{t}} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{r_{0}}{(F_{c},t)} - \\frac{\\partial}{\\partial F_{c}} e^{F_{c}^{t}} = 0 and F_{c} (\\frac{\\partial}{\\partial F_{c}} \\operatorname{r_{0}}{(F_{c},t)} - \\frac{\\partial}{\\partial F_{c}} e^{F_{c}^{t}}) = 0", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), exp(Pow(Symbol('F_c', commutative=True), Symbol('t', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('F_c', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(exp(Pow(Symbol('F_c', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('r_0')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Pow(Symbol('F_c', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))), Integer(0))"], [["times", 3, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Add(Derivative(Function('r_0')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Pow(Symbol('F_c', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given \\phi_{2}{(f^{\\prime},\\chi)} = \\frac{f^{\\prime}}{\\chi}, then obtain \\int (1 + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}}) d\\chi = \\int (((\\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}})^{\\chi})^{\\chi} + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}}) d\\chi", "derivation": "\\phi_{2}{(f^{\\prime},\\chi)} = \\frac{f^{\\prime}}{\\chi} and 1 = \\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}} and 1 = (\\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}})^{\\chi} and 1 = ((\\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}})^{\\chi})^{\\chi} and 1 + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}} = ((\\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}})^{\\chi})^{\\chi} + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}} and \\int (1 + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}}) d\\chi = \\int (((\\frac{f^{\\prime}}{\\chi \\phi_{2}{(f^{\\prime},\\chi)}})^{\\chi})^{\\chi} + \\frac{1}{\\phi_{2}{(f^{\\prime},\\chi)}}) d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True)))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["add", 4, "Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Add(Pow(Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Integer(1), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Pow(Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbf{D},\\Psi_{nl})} = e^{\\Psi_{nl} \\mathbf{D}}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} p{(\\mathbf{D},\\Psi_{nl})} = \\Psi_{nl} e^{\\Psi_{nl} \\mathbf{D}}, then obtain \\Psi_{nl} e^{\\Psi_{nl} \\mathbf{D}} = \\Psi_{nl} p{(\\mathbf{D},\\Psi_{nl})}", "derivation": "p{(\\mathbf{D},\\Psi_{nl})} = e^{\\Psi_{nl} \\mathbf{D}} and \\frac{\\partial}{\\partial \\mathbf{D}} p{(\\mathbf{D},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\mathbf{D}} e^{\\Psi_{nl} \\mathbf{D}} and \\frac{\\partial}{\\partial \\mathbf{D}} p{(\\mathbf{D},\\Psi_{nl})} = \\Psi_{nl} e^{\\Psi_{nl} \\mathbf{D}} and \\frac{\\partial}{\\partial \\mathbf{D}} p{(\\mathbf{D},\\Psi_{nl})} = \\Psi_{nl} p{(\\mathbf{D},\\Psi_{nl})} and \\Psi_{nl} e^{\\Psi_{nl} \\mathbf{D}} = \\Psi_{nl} p{(\\mathbf{D},\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), exp(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\Psi_{nl}', commutative=True), exp(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Function('p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then obtain (- g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})}) (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{2} = (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{3}", "derivation": "\\hat{X}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and - g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})} = - g_{\\varepsilon} + \\log{(g_{\\varepsilon})} and (- g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})}) (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})}) = (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{2} and (- g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})})^{2} (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})}) = (- g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})}) (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{2} and (- g_{\\varepsilon} + \\hat{X}{(g_{\\varepsilon})}) (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{2} = (- g_{\\varepsilon} + \\log{(g_{\\varepsilon})})^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2)), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{X}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(3)))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{p})} = e^{\\hat{p}}, then obtain \\frac{\\varphi^{*}{(\\hat{p})}}{\\int e^{\\hat{p}} d\\hat{p}} = \\frac{e^{\\hat{p}}}{\\int e^{\\hat{p}} d\\hat{p}}", "derivation": "\\varphi^{*}{(\\hat{p})} = e^{\\hat{p}} and \\int \\varphi^{*}{(\\hat{p})} d\\hat{p} = \\int e^{\\hat{p}} d\\hat{p} and \\frac{\\varphi^{*}{(\\hat{p})}}{\\int \\varphi^{*}{(\\hat{p})} d\\hat{p}} = \\frac{e^{\\hat{p}}}{\\int \\varphi^{*}{(\\hat{p})} d\\hat{p}} and \\frac{\\varphi^{*}{(\\hat{p})}}{\\int e^{\\hat{p}} d\\hat{p}} = \\frac{e^{\\hat{p}}}{\\int e^{\\hat{p}} d\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Mul(exp(Symbol('\\\\hat{p}', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\hat{p}', commutative=True)), Pow(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Mul(exp(Symbol('\\\\hat{p}', commutative=True)), Pow(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\sigma_x)} = \\cos{(\\cos{(\\sigma_x)})}, then obtain T = T - \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(- \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(\\sigma_x)} + \\cos{(\\cos{(\\sigma_x)})})}", "derivation": "\\operatorname{f_{E}}{(\\sigma_x)} = \\cos{(\\cos{(\\sigma_x)})} and 0 = - \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(\\cos{(\\sigma_x)})} and \\cos{(\\sigma_x)} = - \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(\\sigma_x)} + \\cos{(\\cos{(\\sigma_x)})} and 0 = - \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(- \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(\\sigma_x)} + \\cos{(\\cos{(\\sigma_x)})})} and T = T - \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(- \\operatorname{f_{E}}{(\\sigma_x)} + \\cos{(\\sigma_x)} + \\cos{(\\cos{(\\sigma_x)})})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 1, "Function('f_E')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["add", 2, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(cos(Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(Add(Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))))"], [["minus", 4, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(Add(Mul(Integer(-1), Function('f_E')(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))))"]]}, {"prompt": "Given \\nabla{(M_{E})} = \\int e^{M_{E}} dM_{E}, then obtain \\frac{M_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}} + \\frac{\\nabla{(M_{E})}}{\\iint e^{M_{E}} dM_{E} dM_{E}} = \\frac{M_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}} + \\frac{\\int e^{M_{E}} dM_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}}", "derivation": "\\nabla{(M_{E})} = \\int e^{M_{E}} dM_{E} and M_{E} + \\nabla{(M_{E})} = M_{E} + \\int e^{M_{E}} dM_{E} and \\frac{M_{E} + \\nabla{(M_{E})}}{\\iint e^{M_{E}} dM_{E} dM_{E}} = \\frac{M_{E} + \\int e^{M_{E}} dM_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}} and \\frac{M_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}} + \\frac{\\nabla{(M_{E})}}{\\iint e^{M_{E}} dM_{E} dM_{E}} = \\frac{M_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}} + \\frac{\\int e^{M_{E}} dM_{E}}{\\iint e^{M_{E}} dM_{E} dM_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('M_E', commutative=True)), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('\\\\nabla')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["divide", 2, "Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Symbol('M_E', commutative=True), Function('\\\\nabla')(Symbol('M_E', commutative=True))), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Add(Symbol('M_E', commutative=True), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))))"], [["expand", 3], "Equality(Add(Mul(Symbol('M_E', commutative=True), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Function('\\\\nabla')(Symbol('M_E', commutative=True)), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1)))), Add(Mul(Symbol('M_E', commutative=True), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\omega{(\\chi,\\dot{y})} = \\log{(\\chi + \\dot{y})}, then obtain \\chi + \\omega{(\\chi,\\dot{y})} + 2 \\log{(\\chi + \\dot{y})} = \\chi + 3 \\log{(\\chi + \\dot{y})}", "derivation": "\\omega{(\\chi,\\dot{y})} = \\log{(\\chi + \\dot{y})} and \\chi + \\omega{(\\chi,\\dot{y})} = \\chi + \\log{(\\chi + \\dot{y})} and \\chi + 2 \\omega{(\\chi,\\dot{y})} + \\log{(\\chi + \\dot{y})} = \\chi + \\omega{(\\chi,\\dot{y})} + 2 \\log{(\\chi + \\dot{y})} and \\chi + 2 \\omega{(\\chi,\\dot{y})} + \\log{(\\chi + \\dot{y})} = \\chi + 3 \\log{(\\chi + \\dot{y})} and \\chi + \\omega{(\\chi,\\dot{y})} + 2 \\log{(\\chi + \\dot{y})} = \\chi + 3 \\log{(\\chi + \\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 2, "Add(Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(3), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\omega')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(3), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{x})} = \\frac{d}{d v_{x}} \\sin{(v_{x})}, then derive \\mathbf{J}_P{(v_{x})} = \\cos{(v_{x})}, then obtain \\cos{(v_{x})} + 1 = \\mathbf{J}_P{(v_{x})} + 1", "derivation": "\\mathbf{J}_P{(v_{x})} = \\frac{d}{d v_{x}} \\sin{(v_{x})} and \\mathbf{J}_P{(v_{x})} + 1 = \\frac{d}{d v_{x}} \\sin{(v_{x})} + 1 and \\mathbf{J}_P{(v_{x})} = \\cos{(v_{x})} and \\cos{(v_{x})} + 1 = \\frac{d}{d v_{x}} \\sin{(v_{x})} + 1 and \\cos{(v_{x})} + 1 = \\mathbf{J}_P{(v_{x})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Derivative(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(cos(Symbol('v_x', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(cos(Symbol('v_x', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\hat{H}_l{(v_{y})} = \\sin{(\\cos{(v_{y})})}, then obtain 2 \\cos{(v_{y})} = (1 + \\frac{\\sin{(\\cos{(v_{y})})}}{\\hat{H}_l{(v_{y})}}) \\cos{(v_{y})}", "derivation": "\\hat{H}_l{(v_{y})} = \\sin{(\\cos{(v_{y})})} and 1 = \\frac{\\sin{(\\cos{(v_{y})})}}{\\hat{H}_l{(v_{y})}} and 2 = 1 + \\frac{\\sin{(\\cos{(v_{y})})}}{\\hat{H}_l{(v_{y})}} and 2 \\cos{(v_{y})} = (1 + \\frac{\\sin{(\\cos{(v_{y})})}}{\\hat{H}_l{(v_{y})}}) \\cos{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('v_y', commutative=True)), sin(cos(Symbol('v_y', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}_l')(Symbol('v_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('v_y', commutative=True)), Integer(-1)), sin(cos(Symbol('v_y', commutative=True)))))"], [["add", 2, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('v_y', commutative=True)), Integer(-1)), sin(cos(Symbol('v_y', commutative=True))))))"], [["times", 3, "cos(Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(2), cos(Symbol('v_y', commutative=True))), Mul(Add(Integer(1), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('v_y', commutative=True)), Integer(-1)), sin(cos(Symbol('v_y', commutative=True))))), cos(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})}, then derive \\mathbf{M}^{2}{(\\mathbf{s})} = \\frac{\\mathbf{M}{(\\mathbf{s})}}{\\mathbf{s}}, then obtain \\frac{\\mathbf{M}{(\\mathbf{s})}}{\\mathbf{s}} = \\mathbf{M}{(\\mathbf{s})} \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})}", "derivation": "\\mathbf{M}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})} and \\mathbf{M}^{2}{(\\mathbf{s})} = \\mathbf{M}{(\\mathbf{s})} \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})} and \\mathbf{M}^{2}{(\\mathbf{s})} = \\frac{\\mathbf{M}{(\\mathbf{s})}}{\\mathbf{s}} and \\frac{\\mathbf{M}{(\\mathbf{s})}}{\\mathbf{s}} = \\mathbf{M}{(\\mathbf{s})} \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{A}{(q)} = \\log{(q)} and \\ddot{x}{(q)} = \\int \\mathbf{A}{(q)} dq, then derive - \\frac{q \\log{(q)} - q + v_{x}}{\\mathbf{A}{(q)}} + \\frac{\\ddot{x}{(q)}}{\\mathbf{A}{(q)}} = 0, then obtain - \\frac{q \\log{(q)} - q + v_{x}}{\\mathbf{A}{(q)}} + \\frac{\\int \\mathbf{A}{(q)} dq}{\\mathbf{A}{(q)}} = 0", "derivation": "\\mathbf{A}{(q)} = \\log{(q)} and \\int \\mathbf{A}{(q)} dq = \\int \\log{(q)} dq and \\frac{\\int \\mathbf{A}{(q)} dq}{\\mathbf{A}{(q)}} = \\frac{\\int \\log{(q)} dq}{\\mathbf{A}{(q)}} and \\ddot{x}{(q)} = \\int \\mathbf{A}{(q)} dq and \\frac{\\int \\mathbf{A}{(q)} dq}{\\mathbf{A}{(q)}} - \\frac{\\int \\log{(q)} dq}{\\mathbf{A}{(q)}} = 0 and \\frac{\\ddot{x}{(q)}}{\\mathbf{A}{(q)}} - \\frac{\\int \\log{(q)} dq}{\\mathbf{A}{(q)}} = 0 and - \\frac{q \\log{(q)} - q + v_{x}}{\\mathbf{A}{(q)}} + \\frac{\\ddot{x}{(q)}}{\\mathbf{A}{(q)}} = 0 and - \\frac{q \\log{(q)} - q + v_{x}}{\\mathbf{A}{(q)}} + \\frac{\\int \\mathbf{A}{(q)} dq}{\\mathbf{A}{(q)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["minus", 3, "Mul(Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], "Equality(Add(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Integer(0))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Add(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1))), Mul(Function('\\\\ddot{x}')(Symbol('q', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Mul(Integer(-1), Add(Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))), Integer(0))"]]}, {"prompt": "Given M{(f)} = \\sin{(f)}, then derive \\int M{(f)} df = a - \\cos{(f)}, then obtain (\\int M{(f)} df)^{f} - \\int \\sin{(f)} df = (a - \\cos{(f)})^{f} - \\int \\sin{(f)} df", "derivation": "M{(f)} = \\sin{(f)} and \\int M{(f)} df = \\int \\sin{(f)} df and \\int M{(f)} df = a - \\cos{(f)} and a - \\cos{(f)} = \\int \\sin{(f)} df and (a - \\cos{(f)})^{f} = (\\int \\sin{(f)} df)^{f} and (a - \\cos{(f)})^{f} - \\int \\sin{(f)} df = - \\int \\sin{(f)} df + (\\int \\sin{(f)} df)^{f} and (\\int M{(f)} df)^{f} - \\int \\sin{(f)} df = - \\int \\sin{(f)} df + (\\int \\sin{(f)} df)^{f} and (\\int M{(f)} df)^{f} - \\int \\sin{(f)} df = (a - \\cos{(f)})^{f} - \\int \\sin{(f)} df", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('M')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('f', commutative=True)))), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Pow(Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["minus", 5, "Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))"], "Equality(Add(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Pow(Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Pow(Integral(Function('M')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Pow(Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Pow(Integral(Function('M')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Add(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{D})} = \\sin{(\\log{(\\mathbf{D})})} and I{(\\mathbf{D})} = \\frac{\\cos{(\\log{(\\mathbf{D})})}}{\\mathbf{D}}, then derive \\frac{d}{d \\mathbf{D}} \\mathbf{s}{(\\mathbf{D})} = \\frac{\\cos{(\\log{(\\mathbf{D})})}}{\\mathbf{D}}, then obtain I{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = - \\cos{(\\log{(\\mathbf{D})})} + \\frac{d}{d \\mathbf{D}} \\mathbf{s}{(\\mathbf{D})}", "derivation": "\\mathbf{s}{(\\mathbf{D})} = \\sin{(\\log{(\\mathbf{D})})} and \\frac{d}{d \\mathbf{D}} \\mathbf{s}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\sin{(\\log{(\\mathbf{D})})} and \\frac{d}{d \\mathbf{D}} \\mathbf{s}{(\\mathbf{D})} = \\frac{\\cos{(\\log{(\\mathbf{D})})}}{\\mathbf{D}} and I{(\\mathbf{D})} = \\frac{\\cos{(\\log{(\\mathbf{D})})}}{\\mathbf{D}} and I{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = - \\cos{(\\log{(\\mathbf{D})})} + \\frac{\\cos{(\\log{(\\mathbf{D})})}}{\\mathbf{D}} and I{(\\mathbf{D})} - \\cos{(\\log{(\\mathbf{D})})} = - \\cos{(\\log{(\\mathbf{D})})} + \\frac{d}{d \\mathbf{D}} \\mathbf{s}{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(log(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(sin(log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["minus", 4, "cos(log(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('I')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('I')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Mul(Integer(-1), cos(log(Symbol('\\\\mathbf{D}', commutative=True)))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(\\phi,\\theta)} = \\phi \\theta and \\operatorname{P_{e}}{(\\phi,\\theta)} = \\phi \\theta, then obtain \\phi + \\theta^{2} + \\omega{(\\phi,\\theta)} = \\phi \\theta + \\phi + \\theta^{2}", "derivation": "\\omega{(\\phi,\\theta)} = \\phi \\theta and \\operatorname{P_{e}}{(\\phi,\\theta)} = \\phi \\theta and \\omega{(\\phi,\\theta)} = \\operatorname{P_{e}}{(\\phi,\\theta)} and \\phi + \\omega{(\\phi,\\theta)} = \\phi + \\operatorname{P_{e}}{(\\phi,\\theta)} and \\phi + \\theta^{2} + \\omega{(\\phi,\\theta)} = \\phi + \\theta^{2} + \\operatorname{P_{e}}{(\\phi,\\theta)} and \\phi + \\theta^{2} + \\omega{(\\phi,\\theta)} = \\phi \\theta + \\phi + \\theta^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\omega')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('\\\\omega')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["add", 4, "Pow(Symbol('\\\\theta', commutative=True), Integer(2))"], "Equality(Add(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('\\\\omega')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('\\\\omega')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})}, then derive \\int \\theta_{1}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + i, then obtain \\int \\theta_{1}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = g^{\\prime}_{\\varepsilon} \\theta_{1}{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + i", "derivation": "\\theta_{1}{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\int \\theta_{1}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\int \\theta_{1}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + i and \\int \\theta_{1}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = g^{\\prime}_{\\varepsilon} \\theta_{1}{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + i", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('i', commutative=True)))"]]}, {"prompt": "Given Z{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then obtain - A_{x} J_{\\varepsilon} + \\int (V_{\\mathbf{E}} + Z{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}} = - A_{x} J_{\\varepsilon} + \\int (V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}}", "derivation": "Z{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and V_{\\mathbf{E}} + Z{(V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})} and \\int (V_{\\mathbf{E}} + Z{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}} = \\int (V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}} and - A_{x} J_{\\varepsilon} + \\int (V_{\\mathbf{E}} + Z{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}} = - A_{x} J_{\\varepsilon} + \\int (V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})}) dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('Z')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('Z')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 3, "Mul(Symbol('A_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('Z')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(y,q)} = y + \\sin{(q)}, then obtain ((y + \\sin{(q)})^{q} + (\\mathbf{H}^{q}{(y,q)})^{y})^{2} = ((y + \\sin{(q)})^{q} + ((y + \\sin{(q)})^{q})^{y}) ((y + \\sin{(q)})^{q} + (\\mathbf{H}^{q}{(y,q)})^{y})", "derivation": "\\mathbf{H}{(y,q)} = y + \\sin{(q)} and \\mathbf{H}^{q}{(y,q)} = (y + \\sin{(q)})^{q} and (\\mathbf{H}^{q}{(y,q)})^{y} = ((y + \\sin{(q)})^{q})^{y} and (y + \\sin{(q)})^{q} + (\\mathbf{H}^{q}{(y,q)})^{y} = (y + \\sin{(q)})^{q} + ((y + \\sin{(q)})^{q})^{y} and ((y + \\sin{(q)})^{q} + (\\mathbf{H}^{q}{(y,q)})^{y})^{2} = ((y + \\sin{(q)})^{q} + ((y + \\sin{(q)})^{q})^{y}) ((y + \\sin{(q)})^{q} + (\\mathbf{H}^{q}{(y,q)})^{y})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('y', commutative=True)))"], [["add", 3, "Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True))"], "Equality(Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('y', commutative=True))), Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('y', commutative=True))))"], [["times", 4, "Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('y', commutative=True)))"], "Equality(Pow(Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('y', commutative=True))), Integer(2)), Mul(Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('y', commutative=True))), Add(Pow(Add(Symbol('y', commutative=True), sin(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Function('\\\\mathbf{H}')(Symbol('y', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(S)} = \\log{(\\sin{(S)})}, then obtain - S + (S + \\log{(\\sin{(S)})}) \\varepsilon_{0}{(S)} - \\log{(\\sin{(S)})} = - S + (S + \\log{(\\sin{(S)})}) \\log{(\\sin{(S)})} - \\log{(\\sin{(S)})}", "derivation": "\\varepsilon_{0}{(S)} = \\log{(\\sin{(S)})} and S + \\varepsilon_{0}{(S)} = S + \\log{(\\sin{(S)})} and (S + \\varepsilon_{0}{(S)}) \\varepsilon_{0}{(S)} = (S + \\varepsilon_{0}{(S)}) \\log{(\\sin{(S)})} and - S + (S + \\varepsilon_{0}{(S)}) \\varepsilon_{0}{(S)} = - S + (S + \\varepsilon_{0}{(S)}) \\log{(\\sin{(S)})} and - S + (S + \\log{(\\sin{(S)})}) \\varepsilon_{0}{(S)} = - S + (S + \\log{(\\sin{(S)})}) \\log{(\\sin{(S)})} and - S + (S + \\log{(\\sin{(S)})}) \\varepsilon_{0}{(S)} - \\log{(\\sin{(S)})} = - S + (S + \\log{(\\sin{(S)})}) \\log{(\\sin{(S)})} - \\log{(\\sin{(S)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('S', commutative=True)), log(sin(Symbol('S', commutative=True))))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), log(sin(Symbol('S', commutative=True)))))"], [["times", 1, "Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), Mul(Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), log(sin(Symbol('S', commutative=True)))))"], [["minus", 3, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), Function('\\\\varepsilon_0')(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), log(sin(Symbol('S', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), log(sin(Symbol('S', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), log(sin(Symbol('S', commutative=True)))), log(sin(Symbol('S', commutative=True))))))"], [["minus", 5, "log(sin(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), log(sin(Symbol('S', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('S', commutative=True))), Mul(Integer(-1), log(sin(Symbol('S', commutative=True))))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Add(Symbol('S', commutative=True), log(sin(Symbol('S', commutative=True)))), log(sin(Symbol('S', commutative=True)))), Mul(Integer(-1), log(sin(Symbol('S', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)}, then derive \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} = - \\sin{(\\hat{p}_0)}, then obtain - \\sin{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)}", "derivation": "\\tilde{g}^*{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} = - \\sin{(\\hat{p}_0)} and - \\sin{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} and - \\sin{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\tilde{g}^*{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True))), Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\hat{p}_0', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(x,A_{y})} = \\frac{A_{y}}{x} and \\dot{\\mathbf{r}}{(x,A_{y})} = - \\frac{A_{y}}{x} + \\frac{\\partial}{\\partial A_{y}} \\frac{A_{y}}{x}, then derive \\frac{\\partial}{\\partial A_{y}} \\hat{\\mathbf{r}}{(x,A_{y})} = \\frac{1}{x}, then obtain \\dot{\\mathbf{r}}{(x,A_{y})} = - \\frac{A_{y}}{x} + \\frac{1}{x}", "derivation": "\\hat{\\mathbf{r}}{(x,A_{y})} = \\frac{A_{y}}{x} and \\frac{\\partial}{\\partial A_{y}} \\hat{\\mathbf{r}}{(x,A_{y})} = \\frac{\\partial}{\\partial A_{y}} \\frac{A_{y}}{x} and \\frac{\\partial}{\\partial A_{y}} \\hat{\\mathbf{r}}{(x,A_{y})} = \\frac{1}{x} and \\frac{\\partial}{\\partial A_{y}} \\frac{A_{y}}{x} = \\frac{1}{x} and \\dot{\\mathbf{r}}{(x,A_{y})} = - \\frac{A_{y}}{x} + \\frac{\\partial}{\\partial A_{y}} \\frac{A_{y}}{x} and \\dot{\\mathbf{r}}{(x,A_{y})} = - \\frac{A_{y}}{x} + \\frac{1}{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('A_y', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Derivative(Mul(Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('A_y', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Pow(Symbol('x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}_0{(J)} = e^{J} and \\operatorname{A_{y}}{(\\ddot{x},l,Z)} = \\frac{Z}{\\ddot{x} l}, then obtain \\frac{\\ddot{x} l \\int (J + \\hat{x}_0{(J)}) dJ}{Z} = \\frac{\\ddot{x} l \\int (J + e^{J}) dJ}{Z}", "derivation": "\\hat{x}_0{(J)} = e^{J} and J + \\hat{x}_0{(J)} = J + e^{J} and \\operatorname{A_{y}}{(\\ddot{x},l,Z)} = \\frac{Z}{\\ddot{x} l} and \\int (J + \\hat{x}_0{(J)}) dJ = \\int (J + e^{J}) dJ and \\frac{\\int (J + \\hat{x}_0{(J)}) dJ}{\\operatorname{A_{y}}{(\\ddot{x},l,Z)}} = \\frac{\\int (J + e^{J}) dJ}{\\operatorname{A_{y}}{(\\ddot{x},l,Z)}} and \\frac{\\ddot{x} l \\int (J + \\hat{x}_0{(J)}) dJ}{Z} = \\frac{\\ddot{x} l \\int (J + e^{J}) dJ}{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))))"], ["get_premise", "Equality(Function('A_y')(Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["divide", 4, "Function('A_y')(Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Function('A_y')(Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Integral(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Function('A_y')(Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Integral(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Integral(Add(Symbol('J', commutative=True), Function('\\\\hat{x}_0')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\ddot{x}', commutative=True), Symbol('l', commutative=True), Integral(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given C{(s)} = \\cos{(e^{s})}, then derive \\frac{\\frac{d}{d s} C{(s)}}{\\cos{(e^{s})}} = - \\frac{e^{s} \\sin{(e^{s})}}{\\cos{(e^{s})}}, then obtain \\cos{(e^{s})} + \\frac{\\frac{d}{d s} \\cos{(e^{s})}}{\\cos{(e^{s})}} = - \\frac{e^{s} \\sin{(e^{s})}}{\\cos{(e^{s})}} + \\cos{(e^{s})}", "derivation": "C{(s)} = \\cos{(e^{s})} and \\frac{d}{d s} C{(s)} = \\frac{d}{d s} \\cos{(e^{s})} and \\frac{\\frac{d}{d s} C{(s)}}{\\cos{(e^{s})}} = \\frac{\\frac{d}{d s} \\cos{(e^{s})}}{\\cos{(e^{s})}} and \\frac{\\frac{d}{d s} C{(s)}}{\\cos{(e^{s})}} = - \\frac{e^{s} \\sin{(e^{s})}}{\\cos{(e^{s})}} and C{(s)} + \\frac{\\frac{d}{d s} C{(s)}}{\\cos{(e^{s})}} = C{(s)} - \\frac{e^{s} \\sin{(e^{s})}}{\\cos{(e^{s})}} and \\cos{(e^{s})} + \\frac{\\frac{d}{d s} \\cos{(e^{s})}}{\\cos{(e^{s})}} = - \\frac{e^{s} \\sin{(e^{s})}}{\\cos{(e^{s})}} + \\cos{(e^{s})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('s', commutative=True)), cos(exp(Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["divide", 2, "cos(exp(Symbol('s', commutative=True)))"], "Equality(Mul(Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)), Derivative(Function('C')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)), Derivative(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)), Derivative(Function('C')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('s', commutative=True)), sin(exp(Symbol('s', commutative=True))), Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1))))"], [["add", 4, "Function('C')(Symbol('s', commutative=True))"], "Equality(Add(Function('C')(Symbol('s', commutative=True)), Mul(Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)), Derivative(Function('C')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Function('C')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)), sin(exp(Symbol('s', commutative=True))), Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(cos(exp(Symbol('s', commutative=True))), Mul(Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1)), Derivative(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True)), sin(exp(Symbol('s', commutative=True))), Pow(cos(exp(Symbol('s', commutative=True))), Integer(-1))), cos(exp(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(Z)} = \\int \\cos{(Z)} dZ and \\Psi^{\\dagger}{(Z)} = \\hat{x}^{Z}{(Z)}, then derive (S + \\sin{(Z)})^{Z} - \\cos{(Z)} = \\Psi^{\\dagger}{(Z)} - \\cos{(Z)}, then obtain \\frac{\\partial}{\\partial S} ((S + \\sin{(Z)})^{Z} - \\cos{(Z)}) = \\frac{d}{d S} (\\Psi^{\\dagger}{(Z)} - \\cos{(Z)})", "derivation": "\\hat{x}{(Z)} = \\int \\cos{(Z)} dZ and \\hat{x}^{Z}{(Z)} = (\\int \\cos{(Z)} dZ)^{Z} and \\Psi^{\\dagger}{(Z)} = \\hat{x}^{Z}{(Z)} and \\Psi^{\\dagger}{(Z)} - \\cos{(Z)} = \\hat{x}^{Z}{(Z)} - \\cos{(Z)} and \\Psi^{\\dagger}{(Z)} = (\\int \\cos{(Z)} dZ)^{Z} and - \\cos{(Z)} + (\\int \\cos{(Z)} dZ)^{Z} = \\hat{x}^{Z}{(Z)} - \\cos{(Z)} and - \\cos{(Z)} + (\\int \\cos{(Z)} dZ)^{Z} = \\Psi^{\\dagger}{(Z)} - \\cos{(Z)} and (S + \\sin{(Z)})^{Z} - \\cos{(Z)} = \\Psi^{\\dagger}{(Z)} - \\cos{(Z)} and \\frac{\\partial}{\\partial S} ((S + \\sin{(Z)})^{Z} - \\cos{(Z)}) = \\frac{d}{d S} (\\Psi^{\\dagger}{(Z)} - \\cos{(Z)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('Z', commutative=True)), Integral(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Integral(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Pow(Function('\\\\hat{x}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["minus", 3, "cos(Symbol('Z', commutative=True))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Add(Pow(Function('\\\\hat{x}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Pow(Integral(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), cos(Symbol('Z', commutative=True))), Pow(Integral(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), Add(Pow(Function('\\\\hat{x}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('Z', commutative=True))), Pow(Integral(cos(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Pow(Add(Symbol('S', commutative=True), sin(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))))"], [["differentiate", 8, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Symbol('S', commutative=True), sin(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\varepsilon)} = \\sin{(\\cos{(\\varepsilon)})}, then obtain ((- \\frac{b{(\\varepsilon)}}{\\tilde{g}})^{\\tilde{g}})^{\\varepsilon} = ((- \\frac{\\sin{(\\cos{(\\varepsilon)})}}{\\tilde{g}})^{\\tilde{g}})^{\\varepsilon}", "derivation": "b{(\\varepsilon)} = \\sin{(\\cos{(\\varepsilon)})} and - \\frac{b{(\\varepsilon)}}{\\tilde{g}} = - \\frac{\\sin{(\\cos{(\\varepsilon)})}}{\\tilde{g}} and (- \\frac{b{(\\varepsilon)}}{\\tilde{g}})^{\\tilde{g}} = (- \\frac{\\sin{(\\cos{(\\varepsilon)})}}{\\tilde{g}})^{\\tilde{g}} and ((- \\frac{b{(\\varepsilon)}}{\\tilde{g}})^{\\tilde{g}})^{\\varepsilon} = ((- \\frac{\\sin{(\\cos{(\\varepsilon)})}}{\\tilde{g}})^{\\tilde{g}})^{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('\\\\varepsilon', commutative=True)), sin(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('b')(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('b')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('b')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given i{(S)} = \\cos{(S)} and \\omega{(S)} = S + \\cos{(S)} and H{(S)} = \\cos{(S)}, then obtain \\frac{d}{d S} \\omega{(S)} = \\frac{d}{d S} (S + \\cos{(S)})", "derivation": "i{(S)} = \\cos{(S)} and S + i{(S)} = S + \\cos{(S)} and \\omega{(S)} = S + \\cos{(S)} and \\omega{(S)} = S + i{(S)} and H{(S)} = \\cos{(S)} and S + i{(S)} = S + H{(S)} and \\frac{d}{d S} (S + i{(S)}) = \\frac{d}{d S} (S + H{(S)}) and \\frac{d}{d S} (S + i{(S)}) = \\frac{d}{d S} (S + \\cos{(S)}) and \\frac{d}{d S} \\omega{(S)} = \\frac{d}{d S} (S + \\cos{(S)})", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('i')(Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('S', commutative=True)), Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\omega')(Symbol('S', commutative=True)), Add(Symbol('S', commutative=True), Function('i')(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Symbol('S', commutative=True), Function('i')(Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), Function('H')(Symbol('S', commutative=True))))"], [["differentiate", 6, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Symbol('S', commutative=True), Function('i')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Function('H')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(Add(Symbol('S', commutative=True), Function('i')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 8, 4], "Equality(Derivative(Function('\\\\omega')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(n,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + n, then obtain \\frac{\\partial}{\\partial n} (V_{\\mathbf{E}} + V{(n,V_{\\mathbf{E}})}) = \\frac{\\partial}{\\partial n} (2 V_{\\mathbf{E}} + n)", "derivation": "V{(n,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + n and 2 V{(n,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + n + V{(n,V_{\\mathbf{E}})} and V_{\\mathbf{E}} + V{(n,V_{\\mathbf{E}})} = 2 V_{\\mathbf{E}} + n and \\frac{\\partial}{\\partial n} (V_{\\mathbf{E}} + V{(n,V_{\\mathbf{E}})}) = \\frac{\\partial}{\\partial n} (2 V_{\\mathbf{E}} + n)", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Integer(2), Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('n', commutative=True), Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('n', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('V')(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\tilde{g}^*)} = \\log{(\\sin{(\\tilde{g}^*)})}, then obtain - \\tilde{g}^* + \\mathbf{g}{(\\tilde{g}^*)} \\log{(\\sin{(\\tilde{g}^*)})}^{3} = - \\tilde{g}^* + \\log{(\\sin{(\\tilde{g}^*)})}^{4}", "derivation": "\\mathbf{g}{(\\tilde{g}^*)} = \\log{(\\sin{(\\tilde{g}^*)})} and \\mathbf{g}{(\\tilde{g}^*)} \\log{(\\sin{(\\tilde{g}^*)})} = \\log{(\\sin{(\\tilde{g}^*)})}^{2} and \\mathbf{g}{(\\tilde{g}^*)} \\log{(\\sin{(\\tilde{g}^*)})}^{3} = \\log{(\\sin{(\\tilde{g}^*)})}^{4} and - \\tilde{g}^* + \\mathbf{g}{(\\tilde{g}^*)} \\log{(\\sin{(\\tilde{g}^*)})}^{3} = - \\tilde{g}^* + \\log{(\\sin{(\\tilde{g}^*)})}^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 1, "log(sin(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), log(sin(Symbol('\\\\tilde{g}^*', commutative=True)))), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2)))"], [["times", 2, "Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(3))), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(4)))"], [["minus", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(3)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(4))))"]]}, {"prompt": "Given p{(A_{x},c_{0},t)} = (A_{x}^{t})^{c_{0}}, then obtain p{(A_{x},c_{0},t)} - \\int (A_{x}^{t})^{c_{0}} dA_{x} + 1 = (A_{x}^{t})^{c_{0}} - \\int (A_{x}^{t})^{c_{0}} dA_{x} + 1", "derivation": "p{(A_{x},c_{0},t)} = (A_{x}^{t})^{c_{0}} and \\int p{(A_{x},c_{0},t)} dA_{x} = \\int (A_{x}^{t})^{c_{0}} dA_{x} and p{(A_{x},c_{0},t)} - \\int p{(A_{x},c_{0},t)} dA_{x} = (A_{x}^{t})^{c_{0}} - \\int p{(A_{x},c_{0},t)} dA_{x} and p{(A_{x},c_{0},t)} - \\int (A_{x}^{t})^{c_{0}} dA_{x} = (A_{x}^{t})^{c_{0}} - \\int (A_{x}^{t})^{c_{0}} dA_{x} and p{(A_{x},c_{0},t)} - \\int (A_{x}^{t})^{c_{0}} dA_{x} + 1 = (A_{x}^{t})^{c_{0}} - \\int (A_{x}^{t})^{c_{0}} dA_{x} + 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["minus", 1, "Integral(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('A_x', commutative=True)))"], "Equality(Add(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Integral(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('A_x', commutative=True))))), Add(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Mul(Integer(-1), Integral(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('A_x', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('A_x', commutative=True))))), Add(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('A_x', commutative=True))))))"], [["add", 4, 1], "Equality(Add(Function('p')(Symbol('A_x', commutative=True), Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Integer(1)), Add(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Mul(Integer(-1), Integral(Pow(Pow(Symbol('A_x', commutative=True), Symbol('t', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\psi{(\\varphi^*,q)} = \\varphi^* + q, then obtain - (\\frac{2 \\psi^{2}{(\\varphi^*,q)}}{(\\varphi^* + q) \\psi{(\\varphi^*,q)} + \\psi^{2}{(\\varphi^*,q)}})^{q} = -1", "derivation": "\\psi{(\\varphi^*,q)} = \\varphi^* + q and \\psi^{2}{(\\varphi^*,q)} = (\\varphi^* + q) \\psi{(\\varphi^*,q)} and 2 \\psi^{2}{(\\varphi^*,q)} = (\\varphi^* + q) \\psi{(\\varphi^*,q)} + \\psi^{2}{(\\varphi^*,q)} and \\frac{2 \\psi^{2}{(\\varphi^*,q)}}{(\\varphi^* + q) \\psi{(\\varphi^*,q)} + \\psi^{2}{(\\varphi^*,q)}} = 1 and (\\frac{2 \\psi^{2}{(\\varphi^*,q)}}{(\\varphi^* + q) \\psi{(\\varphi^*,q)} + \\psi^{2}{(\\varphi^*,q)}})^{q} = 1 and - (\\frac{2 \\psi^{2}{(\\varphi^*,q)}}{(\\varphi^* + q) \\psi{(\\varphi^*,q)} + \\psi^{2}{(\\varphi^*,q)}})^{q} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))"], "Equality(Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))))"], [["add", 2, "Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Add(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))))"], [["divide", 3, "Add(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Integer(-1)), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Integer(1))"], [["power", 4, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Add(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Integer(-1)), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Symbol('q', commutative=True)), Integer(1))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Mul(Integer(2), Pow(Add(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Integer(-1)), Pow(Function('\\\\psi')(Symbol('\\\\varphi^*', commutative=True), Symbol('q', commutative=True)), Integer(2))), Symbol('q', commutative=True))), Integer(-1))"]]}, {"prompt": "Given \\Psi{(F_{x},l,p)} = (\\frac{p}{l})^{F_{x}} and \\dot{z}{(\\theta_2,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2), then obtain \\frac{- \\Psi^{\\dagger} + \\Psi{(F_{x},l,p)}}{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2)} = \\frac{- \\Psi^{\\dagger} + (\\frac{p}{l})^{F_{x}}}{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2)}", "derivation": "\\Psi{(F_{x},l,p)} = (\\frac{p}{l})^{F_{x}} and - \\Psi^{\\dagger} + \\Psi{(F_{x},l,p)} = - \\Psi^{\\dagger} + (\\frac{p}{l})^{F_{x}} and \\dot{z}{(\\theta_2,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2) and \\frac{- \\Psi^{\\dagger} + \\Psi{(F_{x},l,p)}}{\\dot{z}{(\\theta_2,\\Psi^{\\dagger})}} = \\frac{- \\Psi^{\\dagger} + (\\frac{p}{l})^{F_{x}}}{\\dot{z}{(\\theta_2,\\Psi^{\\dagger})}} and \\frac{- \\Psi^{\\dagger} + \\Psi{(F_{x},l,p)}}{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2)} = \\frac{- \\Psi^{\\dagger} + (\\frac{p}{l})^{F_{x}}}{\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} - \\theta_2)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi')(Symbol('F_x', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Symbol('F_x', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\Psi')(Symbol('F_x', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\Psi')(Symbol('F_x', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True))), Pow(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Symbol('F_x', commutative=True))), Pow(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\Psi')(Symbol('F_x', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True))), Pow(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Symbol('F_x', commutative=True))), Pow(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mu{(\\rho_f,v_{t})} = \\rho_f - v_{t}, then derive \\int \\mu{(\\rho_f,v_{t})} dv_{t} = \\rho_f v_{t} - \\frac{v_{t}^{2}}{2} + z^{*}, then obtain (\\int (\\rho_f - v_{t}) dv_{t})^{z^{*}} = (\\rho_f v_{t} - \\frac{v_{t}^{2}}{2} + z^{*})^{z^{*}}", "derivation": "\\mu{(\\rho_f,v_{t})} = \\rho_f - v_{t} and \\int \\mu{(\\rho_f,v_{t})} dv_{t} = \\int (\\rho_f - v_{t}) dv_{t} and \\int \\mu{(\\rho_f,v_{t})} dv_{t} = \\rho_f v_{t} - \\frac{v_{t}^{2}}{2} + z^{*} and (\\int \\mu{(\\rho_f,v_{t})} dv_{t})^{z^{*}} = (\\rho_f v_{t} - \\frac{v_{t}^{2}}{2} + z^{*})^{z^{*}} and (\\int (\\rho_f - v_{t}) dv_{t})^{z^{*}} = (\\rho_f v_{t} - \\frac{v_{t}^{2}}{2} + z^{*})^{z^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_t', commutative=True), Integer(2))), Symbol('z^*', commutative=True)))"], [["power", 3, "Symbol('z^*', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('z^*', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_t', commutative=True), Integer(2))), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Symbol('z^*', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_t', commutative=True), Integer(2))), Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given x{(H)} = \\log{(\\log{(H)})}, then obtain (x{(H)} + 2 \\log{(\\log{(H)})}) \\log{(\\log{(H)})} = 3 \\log{(\\log{(H)})}^{2}", "derivation": "x{(H)} = \\log{(\\log{(H)})} and x{(H)} + \\log{(\\log{(H)})} = 2 \\log{(\\log{(H)})} and 2 x{(H)} + \\log{(\\log{(H)})} = x{(H)} + 2 \\log{(\\log{(H)})} and x{(H)} + 2 \\log{(\\log{(H)})} = 3 \\log{(\\log{(H)})} and 2 x{(H)} + \\log{(\\log{(H)})} = 3 \\log{(\\log{(H)})} and (2 x{(H)} + \\log{(\\log{(H)})}) \\log{(\\log{(H)})} = 3 \\log{(\\log{(H)})}^{2} and (x{(H)} + 2 \\log{(\\log{(H)})}) \\log{(\\log{(H)})} = 3 \\log{(\\log{(H)})}^{2}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], [["add", 1, "log(log(Symbol('H', commutative=True)))"], "Equality(Add(Function('x')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True)))), Mul(Integer(2), log(log(Symbol('H', commutative=True)))))"], [["add", 1, "Add(Function('x')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), log(log(Symbol('H', commutative=True)))), Add(Function('x')(Symbol('H', commutative=True)), Mul(Integer(2), log(log(Symbol('H', commutative=True))))))"], [["add", 1, "Mul(Integer(2), log(log(Symbol('H', commutative=True))))"], "Equality(Add(Function('x')(Symbol('H', commutative=True)), Mul(Integer(2), log(log(Symbol('H', commutative=True))))), Mul(Integer(3), log(log(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), log(log(Symbol('H', commutative=True)))), Mul(Integer(3), log(log(Symbol('H', commutative=True)))))"], [["times", 5, "log(log(Symbol('H', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), log(log(Symbol('H', commutative=True)))), log(log(Symbol('H', commutative=True)))), Mul(Integer(3), Pow(log(log(Symbol('H', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Add(Function('x')(Symbol('H', commutative=True)), Mul(Integer(2), log(log(Symbol('H', commutative=True))))), log(log(Symbol('H', commutative=True)))), Mul(Integer(3), Pow(log(log(Symbol('H', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\rho_{f}{(\\theta,y)} = \\theta + y, then obtain \\frac{- (\\theta + y)^{2} + (\\theta + y) \\rho_{f}{(\\theta,y)}}{(\\theta + y)^{2}} = 0", "derivation": "\\rho_{f}{(\\theta,y)} = \\theta + y and \\rho_{f}^{2}{(\\theta,y)} = (\\theta + y) \\rho_{f}{(\\theta,y)} and (\\theta + y) \\rho_{f}{(\\theta,y)} = (\\theta + y)^{2} and \\rho_{f}^{2}{(\\theta,y)} = (\\theta + y)^{2} and - (\\theta + y)^{2} + \\rho_{f}^{2}{(\\theta,y)} = - (\\theta + y)^{2} + (\\theta + y) \\rho_{f}{(\\theta,y)} and - (\\theta + y)^{2} + \\rho_{f}^{2}{(\\theta,y)} = 0 and - (\\theta + y)^{2} + (\\theta + y) \\rho_{f}{(\\theta,y)} = 0 and \\frac{- (\\theta + y)^{2} + (\\theta + y) \\rho_{f}{(\\theta,y)}}{\\rho_{f}^{2}{(\\theta,y)}} = 0 and \\frac{- (\\theta + y)^{2} + (\\theta + y) \\rho_{f}{(\\theta,y)}}{(\\theta + y)^{2}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2)), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2)))"], [["minus", 2, "Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)))), Integer(0))"], [["divide", 7, "Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)))), Pow(Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-2))), Integer(0))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-2)), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\mu{(H,B)} = \\sin{(\\frac{B}{H})}, then obtain \\mu{(H,B)} + \\theta_{2}{(H,B)} + \\frac{\\partial}{\\partial H} \\mu{(H,B)} = \\theta_{2}{(H,B)} + \\sin{(\\frac{B}{H})} + \\frac{\\partial}{\\partial H} \\mu{(H,B)}", "derivation": "\\mu{(H,B)} = \\sin{(\\frac{B}{H})} and \\frac{\\partial}{\\partial H} \\mu{(H,B)} = \\frac{\\partial}{\\partial H} \\sin{(\\frac{B}{H})} and \\theta_{2}{(H,B)} + \\frac{\\partial}{\\partial H} \\mu{(H,B)} = \\theta_{2}{(H,B)} + \\frac{\\partial}{\\partial H} \\sin{(\\frac{B}{H})} and \\mu{(H,B)} + \\theta_{2}{(H,B)} + \\frac{\\partial}{\\partial H} \\sin{(\\frac{B}{H})} = \\theta_{2}{(H,B)} + \\sin{(\\frac{B}{H})} + \\frac{\\partial}{\\partial H} \\sin{(\\frac{B}{H})} and \\mu{(H,B)} + \\theta_{2}{(H,B)} + \\frac{\\partial}{\\partial H} \\mu{(H,B)} = \\theta_{2}{(H,B)} + \\sin{(\\frac{B}{H})} + \\frac{\\partial}{\\partial H} \\mu{(H,B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["add", 1, "Add(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Derivative(sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Derivative(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Function('\\\\theta_2')(Symbol('H', commutative=True), Symbol('B', commutative=True)), sin(Mul(Symbol('B', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)))), Derivative(Function('\\\\mu')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(E)} = \\sin{(\\sin{(E)})} and \\operatorname{v_{2}}{(\\psi,F_{N},t)} = (\\frac{F_{N}}{\\psi})^{t}, then obtain \\frac{\\partial}{\\partial E} \\operatorname{L_{\\varepsilon}}^{E}{(E)} \\operatorname{v_{2}}{(\\psi,F_{N},t)} = \\frac{\\partial}{\\partial E} \\operatorname{v_{2}}{(\\psi,F_{N},t)} \\sin^{E}{(\\sin{(E)})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(E)} = \\sin{(\\sin{(E)})} and \\operatorname{v_{2}}{(\\psi,F_{N},t)} = (\\frac{F_{N}}{\\psi})^{t} and \\operatorname{L_{\\varepsilon}}^{E}{(E)} = \\sin^{E}{(\\sin{(E)})} and (\\frac{F_{N}}{\\psi})^{t} \\operatorname{L_{\\varepsilon}}^{E}{(E)} = (\\frac{F_{N}}{\\psi})^{t} \\sin^{E}{(\\sin{(E)})} and \\operatorname{L_{\\varepsilon}}^{E}{(E)} \\operatorname{v_{2}}{(\\psi,F_{N},t)} = \\operatorname{v_{2}}{(\\psi,F_{N},t)} \\sin^{E}{(\\sin{(E)})} and \\frac{\\partial}{\\partial E} \\operatorname{L_{\\varepsilon}}^{E}{(E)} \\operatorname{v_{2}}{(\\psi,F_{N},t)} = \\frac{\\partial}{\\partial E} \\operatorname{v_{2}}{(\\psi,F_{N},t)} \\sin^{E}{(\\sin{(E)})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True)), sin(sin(Symbol('E', commutative=True))))"], ["get_premise", "Equality(Function('v_2')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Pow(Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(sin(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["times", 3, "Pow(Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))), Symbol('t', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))), Symbol('t', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Mul(Pow(Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))), Symbol('t', commutative=True)), Pow(sin(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Function('v_2')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('t', commutative=True))), Mul(Function('v_2')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Pow(sin(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["differentiate", 5, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Function('v_2')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Function('v_2')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Pow(sin(sin(Symbol('E', commutative=True))), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(\\sigma_p)} = e^{\\sigma_p}, then obtain 2 \\frac{d}{d \\sigma_p} \\theta^{2}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\theta{(\\sigma_p)} e^{\\sigma_p} + \\frac{d}{d \\sigma_p} \\theta^{2}{(\\sigma_p)}", "derivation": "\\theta{(\\sigma_p)} = e^{\\sigma_p} and \\theta^{2}{(\\sigma_p)} = \\theta{(\\sigma_p)} e^{\\sigma_p} and \\frac{d}{d \\sigma_p} \\theta^{2}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\theta{(\\sigma_p)} e^{\\sigma_p} and 2 \\frac{d}{d \\sigma_p} \\theta^{2}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\theta{(\\sigma_p)} e^{\\sigma_p} + \\frac{d}{d \\sigma_p} \\theta^{2}{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Mul(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Pow(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Pow(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Derivative(Mul(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\theta')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(a)} = \\log{(a)}, then obtain \\int (\\varepsilon_{0}{(a)} - \\log{(a)}) \\varepsilon_{0}{(a)} \\log{(a)} da = \\int 0 da", "derivation": "\\varepsilon_{0}{(a)} = \\log{(a)} and \\varepsilon_{0}{(a)} \\log{(a)} = \\log{(a)}^{2} and \\varepsilon_{0}{(a)} - \\log{(a)} = 0 and (\\varepsilon_{0}{(a)} - \\log{(a)}) \\log{(a)}^{2} = 0 and (\\varepsilon_{0}{(a)} - \\log{(a)}) \\varepsilon_{0}{(a)} \\log{(a)} = 0 and \\int (\\varepsilon_{0}{(a)} - \\log{(a)}) \\varepsilon_{0}{(a)} \\log{(a)} da = \\int 0 da", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["times", 1, "log(Symbol('a', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Pow(log(Symbol('a', commutative=True)), Integer(2)))"], [["minus", 1, "log(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(0))"], [["times", 3, "Pow(log(Symbol('a', commutative=True)), Integer(2))"], "Equality(Mul(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Pow(log(Symbol('a', commutative=True)), Integer(2))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Integer(0))"], [["integrate", 5, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Integer(0), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(z)} = e^{\\sin{(z)}}, then derive e^{\\sin{(z)}} \\cos{(z)} + \\frac{d}{d z} \\mathbf{P}{(z)} = 2 e^{\\sin{(z)}} \\cos{(z)}, then derive \\frac{d}{d z} \\mathbf{P}{(z)} = e^{\\sin{(z)}} \\cos{(z)}, then obtain 2 \\frac{d}{d z} \\mathbf{P}{(z)} = (\\mathbf{P}{(z)} + e^{\\sin{(z)}}) \\cos{(z)}", "derivation": "\\mathbf{P}{(z)} = e^{\\sin{(z)}} and \\mathbf{P}{(z)} + e^{\\sin{(z)}} = 2 e^{\\sin{(z)}} and \\frac{d}{d z} (\\mathbf{P}{(z)} + e^{\\sin{(z)}}) = \\frac{d}{d z} 2 e^{\\sin{(z)}} and e^{\\sin{(z)}} \\cos{(z)} + \\frac{d}{d z} \\mathbf{P}{(z)} = 2 e^{\\sin{(z)}} \\cos{(z)} and \\frac{d}{d z} \\mathbf{P}{(z)} = \\frac{d}{d z} e^{\\sin{(z)}} and e^{\\sin{(z)}} \\cos{(z)} + \\frac{d}{d z} \\mathbf{P}{(z)} = (\\mathbf{P}{(z)} + e^{\\sin{(z)}}) \\cos{(z)} and \\frac{d}{d z} \\mathbf{P}{(z)} = e^{\\sin{(z)}} \\cos{(z)} and 2 \\frac{d}{d z} \\mathbf{P}{(z)} = (\\mathbf{P}{(z)} + e^{\\sin{(z)}}) \\cos{(z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), exp(sin(Symbol('z', commutative=True))))"], [["add", 1, "exp(sin(Symbol('z', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), exp(sin(Symbol('z', commutative=True)))), Mul(Integer(2), exp(sin(Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), exp(sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(exp(sin(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Derivative(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(2), exp(sin(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(exp(sin(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Derivative(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Add(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), exp(sin(Symbol('z', commutative=True)))), cos(Symbol('z', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(exp(sin(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Add(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), exp(sin(Symbol('z', commutative=True)))), cos(Symbol('z', commutative=True))))"]]}, {"prompt": "Given y{(\\hat{x}_0,\\omega)} = \\sin{(\\hat{x}_0 + \\omega)}, then obtain \\int \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{y{(\\hat{x}_0,\\omega)}}{\\omega})} d\\hat{x}_0 = \\int \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{\\sin{(\\hat{x}_0 + \\omega)}}{\\omega})} d\\hat{x}_0", "derivation": "y{(\\hat{x}_0,\\omega)} = \\sin{(\\hat{x}_0 + \\omega)} and \\frac{y{(\\hat{x}_0,\\omega)}}{\\omega} = \\frac{\\sin{(\\hat{x}_0 + \\omega)}}{\\omega} and \\frac{\\partial}{\\partial \\hat{x}_0} \\frac{y{(\\hat{x}_0,\\omega)}}{\\omega} = \\frac{\\partial}{\\partial \\hat{x}_0} \\frac{\\sin{(\\hat{x}_0 + \\omega)}}{\\omega} and \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{y{(\\hat{x}_0,\\omega)}}{\\omega})} = \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{\\sin{(\\hat{x}_0 + \\omega)}}{\\omega})} and \\int \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{y{(\\hat{x}_0,\\omega)}}{\\omega})} d\\hat{x}_0 = \\int \\sin{(\\frac{\\partial}{\\partial \\hat{x}_0} \\frac{\\sin{(\\hat{x}_0 + \\omega)}}{\\omega})} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["divide", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), sin(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(sin(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(sin(Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\chi{(U,\\phi,\\mathbf{E})} = \\frac{\\mathbf{E} \\phi}{U}, then obtain \\chi{(U,\\phi,\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\chi{(U,\\phi,\\mathbf{E})} dU = \\chi{(U,\\phi,\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\frac{\\mathbf{E} \\phi}{U} dU", "derivation": "\\chi{(U,\\phi,\\mathbf{E})} = \\frac{\\mathbf{E} \\phi}{U} and \\int \\chi{(U,\\phi,\\mathbf{E})} dU = \\int \\frac{\\mathbf{E} \\phi}{U} dU and \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\chi{(U,\\phi,\\mathbf{E})} dU = \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\frac{\\mathbf{E} \\phi}{U} dU and \\chi{(U,\\phi,\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\chi{(U,\\phi,\\mathbf{E})} dU = \\chi{(U,\\phi,\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\int \\frac{\\mathbf{E} \\phi}{U} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Integral(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(J,A,r)} = A^{J} + r, then obtain \\frac{2 A^{J} (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})^{2}}{J} = \\frac{2 A^{J} (2 A^{J} + 2 r) (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})}{J}", "derivation": "\\operatorname{c_{0}}{(J,A,r)} = A^{J} + r and A^{J} + r + \\operatorname{c_{0}}{(J,A,r)} = 2 A^{J} + 2 r and \\frac{A^{J} + r + \\operatorname{c_{0}}{(J,A,r)}}{J} = \\frac{2 A^{J} + 2 r}{J} and 2 \\operatorname{c_{0}}{(J,A,r)} = 2 A^{J} + 2 r and \\frac{(- 2 r + 2 \\operatorname{c_{0}}{(J,A,r)}) (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})}{J} = \\frac{(2 A^{J} + 2 r) (- 2 r + 2 \\operatorname{c_{0}}{(J,A,r)})}{J} and \\frac{2 A^{J} (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})}{J} = \\frac{2 A^{J} (2 A^{J} + 2 r)}{J} and \\frac{2 A^{J} (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})^{2}}{J} = \\frac{2 A^{J} (2 A^{J} + 2 r) (A^{J} + r + \\operatorname{c_{0}}{(J,A,r)})}{J}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)), Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True)))"], [["add", 1, "Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True))"], "Equality(Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))))"], [["divide", 2, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True))))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))), Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True)))))"], [["times", 6, "Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True))), Integer(2))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(2), Pow(Symbol('A', commutative=True), Symbol('J', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True))), Add(Pow(Symbol('A', commutative=True), Symbol('J', commutative=True)), Symbol('r', commutative=True), Function('c_0')(Symbol('J', commutative=True), Symbol('A', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given i{(\\omega,k)} = (e^{\\omega})^{k}, then obtain k - i{(\\omega,k)} + \\int i{(\\omega,k)} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega} = k - i{(\\omega,k)} + \\int (e^{\\omega})^{k} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega}", "derivation": "i{(\\omega,k)} = (e^{\\omega})^{k} and \\int i{(\\omega,k)} d\\omega = \\int (e^{\\omega})^{k} d\\omega and k + \\int i{(\\omega,k)} d\\omega = k + \\int (e^{\\omega})^{k} d\\omega and k + \\int i{(\\omega,k)} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega} = k + \\int (e^{\\omega})^{k} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega} and k - i{(\\omega,k)} + \\int i{(\\omega,k)} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega} = k - i{(\\omega,k)} + \\int (e^{\\omega})^{k} d\\omega - \\frac{1}{\\int (e^{\\omega})^{k} d\\omega}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["add", 2, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Integral(Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Symbol('k', commutative=True), Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["minus", 3, "Pow(Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1))"], "Equality(Add(Symbol('k', commutative=True), Integral(Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1)))), Add(Symbol('k', commutative=True), Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1)))))"], [["minus", 4, "Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Mul(Integer(-1), Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))), Integral(Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1)))), Add(Symbol('k', commutative=True), Mul(Integer(-1), Function('i')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))), Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(exp(Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\phi_1)} = \\log{(\\phi_1)}, then derive \\frac{d}{d \\phi_1} \\Psi_{\\lambda}{(\\phi_1)} = \\frac{1}{\\phi_1}, then obtain \\frac{1}{\\phi_1^{2}} = \\frac{\\frac{d}{d \\phi_1} \\log{(\\phi_1)}}{\\phi_1}", "derivation": "\\Psi_{\\lambda}{(\\phi_1)} = \\log{(\\phi_1)} and \\frac{d}{d \\phi_1} \\Psi_{\\lambda}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\log{(\\phi_1)} and \\frac{\\frac{d}{d \\phi_1} \\Psi_{\\lambda}{(\\phi_1)}}{\\phi_1} = \\frac{\\frac{d}{d \\phi_1} \\log{(\\phi_1)}}{\\phi_1} and \\frac{d}{d \\phi_1} \\Psi_{\\lambda}{(\\phi_1)} = \\frac{1}{\\phi_1} and \\frac{1}{\\phi_1^{2}} = \\frac{\\frac{d}{d \\phi_1} \\log{(\\phi_1)}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(r)} = \\sin{(\\cos{(r)})}, then obtain (1 + \\frac{\\sin{(\\cos{(r)})}}{H{(r)}})^{r} = (\\frac{2 \\sin{(\\cos{(r)})}}{H{(r)}})^{r}", "derivation": "H{(r)} = \\sin{(\\cos{(r)})} and 1 = \\frac{\\sin{(\\cos{(r)})}}{H{(r)}} and 1 + \\frac{\\sin{(\\cos{(r)})}}{H{(r)}} = \\frac{2 \\sin{(\\cos{(r)})}}{H{(r)}} and (1 + \\frac{\\sin{(\\cos{(r)})}}{H{(r)}})^{r} = (\\frac{2 \\sin{(\\cos{(r)})}}{H{(r)}})^{r}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('r', commutative=True)), sin(cos(Symbol('r', commutative=True))))"], [["divide", 1, "Function('H')(Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True)))))"], [["add", 2, "Mul(Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True))))"], "Equality(Add(Integer(1), Mul(Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True))))), Mul(Integer(2), Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True)))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True))))), Symbol('r', commutative=True)), Pow(Mul(Integer(2), Pow(Function('H')(Symbol('r', commutative=True)), Integer(-1)), sin(cos(Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\pi{(\\theta)} = \\cos{(\\sin{(\\theta)})} and J{(\\theta)} = \\pi{(\\theta)} + \\cos{(\\sin{(\\theta)})}, then obtain J{(\\theta)} - \\cos{(\\sin{(\\theta)})} = 2 \\pi{(\\theta)} - \\cos{(\\sin{(\\theta)})}", "derivation": "\\pi{(\\theta)} = \\cos{(\\sin{(\\theta)})} and \\pi{(\\theta)} + \\cos{(\\sin{(\\theta)})} = 2 \\cos{(\\sin{(\\theta)})} and J{(\\theta)} = \\pi{(\\theta)} + \\cos{(\\sin{(\\theta)})} and J{(\\theta)} + \\pi{(\\theta)} - 2 \\cos{(\\sin{(\\theta)})} = 2 \\pi{(\\theta)} - \\cos{(\\sin{(\\theta)})} and J{(\\theta)} - \\cos{(\\sin{(\\theta)})} = 2 \\pi{(\\theta)} - \\cos{(\\sin{(\\theta)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), cos(sin(Symbol('\\\\theta', commutative=True))))"], [["add", 1, "cos(sin(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), cos(sin(Symbol('\\\\theta', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('\\\\theta', commutative=True)))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\theta', commutative=True)), Add(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), cos(sin(Symbol('\\\\theta', commutative=True)))))"], [["add", 3, "Add(Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integer(2), cos(sin(Symbol('\\\\theta', commutative=True)))))"], "Equality(Add(Function('J')(Symbol('\\\\theta', commutative=True)), Function('\\\\pi')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integer(2), cos(sin(Symbol('\\\\theta', commutative=True))))), Add(Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), cos(sin(Symbol('\\\\theta', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('J')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('\\\\theta', commutative=True))))), Add(Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), cos(sin(Symbol('\\\\theta', commutative=True))))))"]]}, {"prompt": "Given H{(\\mathbf{s},\\mathbf{S})} = \\mathbf{s} + \\sin{(\\mathbf{S})}, then obtain \\log{(\\int H{(\\mathbf{s},\\mathbf{S})} d\\mathbf{S})}^{\\mathbf{S}} = \\log{(\\int (\\mathbf{s} + \\sin{(\\mathbf{S})}) d\\mathbf{S})}^{\\mathbf{S}}", "derivation": "H{(\\mathbf{s},\\mathbf{S})} = \\mathbf{s} + \\sin{(\\mathbf{S})} and \\int H{(\\mathbf{s},\\mathbf{S})} d\\mathbf{S} = \\int (\\mathbf{s} + \\sin{(\\mathbf{S})}) d\\mathbf{S} and \\log{(\\int H{(\\mathbf{s},\\mathbf{S})} d\\mathbf{S})} = \\log{(\\int (\\mathbf{s} + \\sin{(\\mathbf{S})}) d\\mathbf{S})} and \\log{(\\int H{(\\mathbf{s},\\mathbf{S})} d\\mathbf{S})}^{\\mathbf{S}} = \\log{(\\int (\\mathbf{s} + \\sin{(\\mathbf{S})}) d\\mathbf{S})}^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), log(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(log(Integral(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(log(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(\\varepsilon_0,v_{z})} = \\varepsilon_0 + v_{z}, then obtain \\frac{d}{d v_{z}} 1 = \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 + v_{z})}{\\frac{\\partial}{\\partial \\varepsilon_0} \\mathbf{S}{(\\varepsilon_0,v_{z})}}", "derivation": "\\mathbf{S}{(\\varepsilon_0,v_{z})} = \\varepsilon_0 + v_{z} and \\frac{\\partial}{\\partial \\varepsilon_0} \\mathbf{S}{(\\varepsilon_0,v_{z})} = \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 + v_{z}) and 1 = \\frac{\\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 + v_{z})}{\\frac{\\partial}{\\partial \\varepsilon_0} \\mathbf{S}{(\\varepsilon_0,v_{z})}} and \\frac{d}{d v_{z}} 1 = \\frac{\\partial}{\\partial v_{z}} \\frac{\\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 + v_{z})}{\\frac{\\partial}{\\partial \\varepsilon_0} \\mathbf{S}{(\\varepsilon_0,v_{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(i,\\varphi)} = \\varphi - i, then derive \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} \\operatorname{m_{s}}{(i,\\varphi)}}{c{(\\mathbf{S})}} = - \\frac{\\mathbf{S}}{c{(\\mathbf{S})}}, then obtain - \\frac{\\mathbf{S}}{c{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} (\\varphi - i)}{c{(\\mathbf{S})}}", "derivation": "\\operatorname{m_{s}}{(i,\\varphi)} = \\varphi - i and \\frac{\\partial}{\\partial i} \\operatorname{m_{s}}{(i,\\varphi)} = \\frac{\\partial}{\\partial i} (\\varphi - i) and \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} \\operatorname{m_{s}}{(i,\\varphi)}}{c{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} (\\varphi - i)}{c{(\\mathbf{S})}} and \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} \\operatorname{m_{s}}{(i,\\varphi)}}{c{(\\mathbf{S})}} = - \\frac{\\mathbf{S}}{c{(\\mathbf{S})}} and - \\frac{\\mathbf{S}}{c{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\frac{\\partial}{\\partial i} (\\varphi - i)}{c{(\\mathbf{S})}}", "srepr_derivation": [["get_premise", "Equality(Function('m_s')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Derivative(Function('m_s')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Derivative(Function('m_s')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('c')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given J{(x^\\prime,t)} = t x^\\prime, then obtain \\int (- x^\\prime + (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)}) dx^\\prime = \\int (- x^\\prime + J{(x^\\prime,t)}) dx^\\prime", "derivation": "J{(x^\\prime,t)} = t x^\\prime and - x^\\prime + J{(x^\\prime,t)} = t x^\\prime - x^\\prime and - t x^\\prime + J{(x^\\prime,t)} = 0 and (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} = 0^{x^\\prime} and (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)} = 0^{x^\\prime} J{(x^\\prime,t)} and J{(x^\\prime,t)} = (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)} and - x^\\prime + (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)} = t x^\\prime - x^\\prime and - x^\\prime + (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)} = - x^\\prime + J{(x^\\prime,t)} and \\int (- x^\\prime + (- t x^\\prime + J{(x^\\prime,t)})^{x^\\prime} J{(x^\\prime,t)}) dx^\\prime = \\int (- x^\\prime + J{(x^\\prime,t)}) dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Add(Mul(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 1, "Mul(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Integer(0))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)))"], [["times", 4, "Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Mul(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 8, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(h,A)} = - A + h, then obtain - A + h - (- A + h)^{h} + \\operatorname{f_{\\mathbf{v}}}{(h,A)} - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)} = - 2 A + 2 h - (- A + h)^{h} - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(h,A)} = - A + h and \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)} = (- A + h)^{h} and - (- A + h)^{h} + \\operatorname{f_{\\mathbf{v}}}{(h,A)} = - A + h - (- A + h)^{h} and \\operatorname{f_{\\mathbf{v}}}{(h,A)} - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)} = - A + h - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)} and - A + h - (- A + h)^{h} + \\operatorname{f_{\\mathbf{v}}}{(h,A)} - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)} = - 2 A + 2 h - (- A + h)^{h} - \\operatorname{f_{\\mathbf{v}}}^{h}{(h,A)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 1, "Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Symbol('h', commutative=True)))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('A', commutative=True)), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(M)} = \\cos{(M)}, then obtain 1 - \\operatorname{C_{2}}^{M}{(M)} = 1 - \\cos^{M}{(M)}", "derivation": "\\operatorname{C_{2}}{(M)} = \\cos{(M)} and \\operatorname{C_{2}}^{M}{(M)} = \\cos^{M}{(M)} and \\operatorname{C_{2}}^{M}{(M)} - 1 = \\cos^{M}{(M)} - 1 and 1 - \\operatorname{C_{2}}^{M}{(M)} = 1 - \\cos^{M}{(M)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(-1)), Add(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(-1)))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})} = \\frac{e^{\\mathbf{s}}}{\\dot{z}}, then obtain - \\dot{z}^{2} e^{- \\mathbf{s}} = \\frac{1}{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})}}", "derivation": "\\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})} = \\frac{e^{\\mathbf{s}}}{\\dot{z}} and \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\frac{e^{\\mathbf{s}}}{\\dot{z}} and \\frac{1}{\\frac{\\partial}{\\partial \\dot{z}} \\frac{e^{\\mathbf{s}}}{\\dot{z}}} = \\frac{1}{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})}} and - \\dot{z}^{2} e^{- \\mathbf{s}} = \\frac{1}{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{n_{1}}{(\\mathbf{s},\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Derivative(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Function('n_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Function('n_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)))), Pow(Derivative(Function('n_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given k{(\\theta_2)} = \\log{(\\theta_2)}, then obtain \\frac{\\log{(\\theta_2)}}{k^{2}{(\\theta_2)}} - \\frac{\\log{(\\theta_2)}^{3}}{k^{4}{(\\theta_2)}} = 0", "derivation": "k{(\\theta_2)} = \\log{(\\theta_2)} and 1 = \\frac{\\log{(\\theta_2)}}{k{(\\theta_2)}} and \\frac{1}{k{(\\theta_2)}} = \\frac{\\log{(\\theta_2)}}{k^{2}{(\\theta_2)}} and \\frac{1}{k{(\\theta_2)}} - \\frac{\\log{(\\theta_2)}}{k^{2}{(\\theta_2)}} = 0 and \\frac{\\log{(\\theta_2)}}{k^{2}{(\\theta_2)}} - \\frac{\\log{(\\theta_2)}^{3}}{k^{4}{(\\theta_2)}} = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Function('k')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), log(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 2, "Function('k')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Mul(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-2)), log(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "Mul(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-2)), log(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Mul(Integer(-1), Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-2)), log(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-2)), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Function('k')(Symbol('\\\\theta_2', commutative=True)), Integer(-4)), Pow(log(Symbol('\\\\theta_2', commutative=True)), Integer(3)))), Integer(0))"]]}, {"prompt": "Given Q{(\\hat{p},b,s)} = \\hat{p} (b + s) and \\Psi{(\\hat{p},b,s)} = - \\hat{p} (b + s) + \\hat{p} + Q{(\\hat{p},b,s)}, then obtain (\\Psi{(\\hat{p},b,s)} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db)^{\\hat{p}} = (\\hat{p} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db)^{\\hat{p}}", "derivation": "Q{(\\hat{p},b,s)} = \\hat{p} (b + s) and \\Psi{(\\hat{p},b,s)} = - \\hat{p} (b + s) + \\hat{p} + Q{(\\hat{p},b,s)} and \\Psi{(\\hat{p},b,s)} = \\hat{p} and \\Psi{(\\hat{p},b,s)} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db = \\hat{p} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db and (\\Psi{(\\hat{p},b,s)} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db)^{\\hat{p}} = (\\hat{p} - \\int (\\hat{p} + Q{(\\hat{p},b,s)}) db)^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('b', commutative=True), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('b', commutative=True), Symbol('s', commutative=True))), Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], [["minus", 3, "Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))))))"], [["power", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))))), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Function('Q')(Symbol('\\\\hat{p}', commutative=True), Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))))), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\sin^{v_{z}}{(\\theta_1)}, then obtain \\frac{\\partial^{2}}{\\partial v_{z}\\partial \\theta_1} \\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\frac{(v_{z} \\log{(\\sin{(\\theta_1)})} + 1) \\sin^{v_{z}}{(\\theta_1)} \\cos{(\\theta_1)}}{\\sin{(\\theta_1)}}", "derivation": "\\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\sin^{v_{z}}{(\\theta_1)} and \\frac{\\partial}{\\partial \\theta_1} \\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\frac{\\partial}{\\partial \\theta_1} \\sin^{v_{z}}{(\\theta_1)} and \\frac{\\partial^{2}}{\\partial v_{z}\\partial \\theta_1} \\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\frac{\\partial^{2}}{\\partial v_{z}\\partial \\theta_1} \\sin^{v_{z}}{(\\theta_1)} and \\frac{\\partial^{2}}{\\partial v_{z}\\partial \\theta_1} \\operatorname{E_{n}}{(\\theta_1,v_{z})} = \\frac{(v_{z} \\log{(\\sin{(\\theta_1)})} + 1) \\sin^{v_{z}}{(\\theta_1)} \\cos{(\\theta_1)}}{\\sin{(\\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\theta_1', commutative=True), Symbol('v_z', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\theta_1', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\theta_1', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\theta_1', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\theta_1', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_n')(Symbol('\\\\theta_1', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Mul(Add(Mul(Symbol('v_z', commutative=True), log(sin(Symbol('\\\\theta_1', commutative=True)))), Integer(1)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Symbol('v_z', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(U,L)} = U^{L}, then obtain \\int (L + (\\int L \\dot{\\mathbf{r}}{(U,L)} dU)^{U}) dU = \\int (L + (\\int L U^{L} dU)^{U}) dU", "derivation": "\\dot{\\mathbf{r}}{(U,L)} = U^{L} and L \\dot{\\mathbf{r}}{(U,L)} = L U^{L} and \\int L \\dot{\\mathbf{r}}{(U,L)} dU = \\int L U^{L} dU and (\\int L \\dot{\\mathbf{r}}{(U,L)} dU)^{U} = (\\int L U^{L} dU)^{U} and L + (\\int L \\dot{\\mathbf{r}}{(U,L)} dU)^{U} = L + (\\int L U^{L} dU)^{U} and \\int (L + (\\int L \\dot{\\mathbf{r}}{(U,L)} dU)^{U}) dU = \\int (L + (\\int L U^{L} dU)^{U}) dU", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True)))"], [["times", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Mul(Symbol('L', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["add", 4, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Add(Symbol('L', commutative=True), Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Symbol('L', commutative=True), Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('L', commutative=True), Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('U', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\phi,W,F_{N})} = \\phi (F_{N} - W), then derive \\int \\operatorname{z^{*}}{(\\phi,W,F_{N})} dW = C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2}, then obtain (e^{\\int \\phi (F_{N} - W) dW})^{W} = (e^{C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2}})^{W}", "derivation": "\\operatorname{z^{*}}{(\\phi,W,F_{N})} = \\phi (F_{N} - W) and \\int \\operatorname{z^{*}}{(\\phi,W,F_{N})} dW = \\int \\phi (F_{N} - W) dW and \\int \\operatorname{z^{*}}{(\\phi,W,F_{N})} dW = C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2} and e^{\\int \\operatorname{z^{*}}{(\\phi,W,F_{N})} dW} = e^{C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2}} and (e^{\\int \\operatorname{z^{*}}{(\\phi,W,F_{N})} dW})^{W} = (e^{C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2}})^{W} and (e^{\\int \\phi (F_{N} - W) dW})^{W} = (e^{C + F_{N} W \\phi - \\frac{W^{2} \\phi}{2}})^{W}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('C', commutative=True), Mul(Symbol('F_N', commutative=True), Symbol('W', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Function('z^*')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('W', commutative=True)))), exp(Add(Symbol('C', commutative=True), Mul(Symbol('F_N', commutative=True), Symbol('W', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True)))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(exp(Integral(Function('z^*')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Pow(exp(Add(Symbol('C', commutative=True), Mul(Symbol('F_N', commutative=True), Symbol('W', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True)))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(exp(Integral(Mul(Symbol('\\\\phi', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Pow(exp(Add(Symbol('C', commutative=True), Mul(Symbol('F_N', commutative=True), Symbol('W', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2)), Symbol('\\\\phi', commutative=True)))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(n_{2})} = e^{n_{2}} and \\dot{y}{(n_{2})} = \\mathbf{J}_f{(n_{2})} + e^{n_{2}}, then obtain \\dot{y}{(n_{2})} - 1 = 2 e^{n_{2}} - 1", "derivation": "\\mathbf{J}_f{(n_{2})} = e^{n_{2}} and \\mathbf{J}_f{(n_{2})} + e^{n_{2}} = 2 e^{n_{2}} and \\dot{y}{(n_{2})} = \\mathbf{J}_f{(n_{2})} + e^{n_{2}} and \\dot{y}{(n_{2})} = 2 e^{n_{2}} and \\dot{y}{(n_{2})} - 1 = 2 e^{n_{2}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["add", 1, "exp(Symbol('n_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True))), Mul(Integer(2), exp(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('n_2', commutative=True)), Add(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\dot{y}')(Symbol('n_2', commutative=True)), Mul(Integer(2), exp(Symbol('n_2', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Function('\\\\dot{y}')(Symbol('n_2', commutative=True)), Integer(-1)), Add(Mul(Integer(2), exp(Symbol('n_2', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{D}{(F_{x})} = \\log{(F_{x})}, then obtain \\frac{\\mathbf{D}^{6}{(F_{x})}}{\\log{(F_{x})}^{5}} = \\frac{\\log{(F_{x})}^{3}}{\\mathbf{D}^{2}{(F_{x})}}", "derivation": "\\mathbf{D}{(F_{x})} = \\log{(F_{x})} and \\frac{\\mathbf{D}{(F_{x})}}{\\log{(F_{x})}} = 1 and \\frac{\\mathbf{D}^{2}{(F_{x})}}{\\log{(F_{x})}} = \\mathbf{D}{(F_{x})} and \\frac{\\mathbf{D}^{2}{(F_{x})}}{\\log{(F_{x})}} = \\log{(F_{x})} and \\frac{\\mathbf{D}^{4}{(F_{x})}}{\\log{(F_{x})}^{2}} = \\log{(F_{x})}^{2} and \\frac{\\mathbf{D}^{3}{(F_{x})}}{\\log{(F_{x})}^{2}} = \\frac{\\log{(F_{x})}^{2}}{\\mathbf{D}{(F_{x})}} and \\frac{\\mathbf{D}^{6}{(F_{x})}}{\\log{(F_{x})}^{5}} = \\frac{\\log{(F_{x})}^{3}}{\\mathbf{D}^{2}{(F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], [["divide", 1, "log(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(2)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(2)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), log(Symbol('F_x', commutative=True)))"], [["power", 4, 2], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(4)), Pow(log(Symbol('F_x', commutative=True)), Integer(-2))), Pow(log(Symbol('F_x', commutative=True)), Integer(2)))"], [["times", 5, "Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(3)), Pow(log(Symbol('F_x', commutative=True)), Integer(-2))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(-1)), Pow(log(Symbol('F_x', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(6)), Pow(log(Symbol('F_x', commutative=True)), Integer(-5))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Integer(-2)), Pow(log(Symbol('F_x', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\rho_{f}{(\\hat{x}_0)} = e^{\\hat{x}_0} and \\operatorname{x^{{\\}'}}{(\\hat{x}_0)} = e^{\\hat{x}_0}, then obtain \\frac{\\operatorname{x^{{\\}'}}{(\\hat{x}_0)} + e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}} = \\frac{2 e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}}", "derivation": "\\rho_{f}{(\\hat{x}_0)} = e^{\\hat{x}_0} and \\operatorname{x^{{\\}'}}{(\\hat{x}_0)} = e^{\\hat{x}_0} and \\rho_{f}{(\\hat{x}_0)} = \\operatorname{x^{{\\}'}}{(\\hat{x}_0)} and \\rho_{f}{(\\hat{x}_0)} + e^{\\hat{x}_0} = 2 e^{\\hat{x}_0} and \\frac{\\rho_{f}{(\\hat{x}_0)} + e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}} = \\frac{2 e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}} and \\frac{\\operatorname{x^{{\\}'}}{(\\hat{x}_0)} + e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}} = \\frac{2 e^{\\hat{x}_0}}{\\frac{d}{d B} \\sin{(B)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 4, "Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True))), Pow(Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(2), exp(Symbol('\\\\hat{x}_0', commutative=True)), Pow(Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Function('x^\\\\prime')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True))), Pow(Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(2), exp(Symbol('\\\\hat{x}_0', commutative=True)), Pow(Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given n{(\\omega)} = \\log{(\\omega)}, then obtain \\omega + \\frac{n{(\\omega)}}{n{(\\omega)} - \\log{(\\omega)}^{\\omega}} = \\omega + \\frac{\\log{(\\omega)}}{n{(\\omega)} - \\log{(\\omega)}^{\\omega}}", "derivation": "n{(\\omega)} = \\log{(\\omega)} and n{(\\omega)} - \\log{(\\omega)}^{\\omega} = \\log{(\\omega)} - \\log{(\\omega)}^{\\omega} and \\frac{n{(\\omega)}}{\\log{(\\omega)} - \\log{(\\omega)}^{\\omega}} = \\frac{\\log{(\\omega)}}{\\log{(\\omega)} - \\log{(\\omega)}^{\\omega}} and \\omega + \\frac{n{(\\omega)}}{\\log{(\\omega)} - \\log{(\\omega)}^{\\omega}} = \\omega + \\frac{\\log{(\\omega)}}{\\log{(\\omega)} - \\log{(\\omega)}^{\\omega}} and \\omega + \\frac{n{(\\omega)}}{n{(\\omega)} - \\log{(\\omega)}^{\\omega}} = \\omega + \\frac{\\log{(\\omega)}}{n{(\\omega)} - \\log{(\\omega)}^{\\omega}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["divide", 1, "Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Pow(Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), Function('n')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), log(Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Mul(Pow(Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), Function('n')(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Mul(Pow(Add(log(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), log(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('\\\\omega', commutative=True), Mul(Pow(Add(Function('n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), Function('n')(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\omega', commutative=True), Mul(Pow(Add(Function('n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Integer(-1)), log(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then derive a + \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} = x^\\prime + \\sin{(\\varphi^*)}, then obtain a + \\sin{(\\varphi^*)} = a + \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\frac{d}{d \\varphi^*} \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} and \\int \\frac{d}{d \\varphi^*} \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} d\\varphi^* = \\int \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} d\\varphi^* and a + \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)} = x^\\prime + \\sin{(\\varphi^*)} and a + \\sin{(\\varphi^*)} = x^\\prime + \\sin{(\\varphi^*)} and a + \\sin{(\\varphi^*)} = a + \\operatorname{f_{\\mathbf{v}}}{(\\varphi^*)}", "srepr_derivation": [["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('a', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('a', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('a', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('a', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given s{(g,\\mathbf{D})} = \\frac{\\mathbf{D}}{g}, then derive Q + s{(g,\\mathbf{D})} = \\frac{\\mathbf{D}}{g} + k, then obtain (\\frac{\\mathbf{D}}{g} + k)^{Q} = (Q + \\frac{\\mathbf{D}}{g})^{Q}", "derivation": "s{(g,\\mathbf{D})} = \\frac{\\mathbf{D}}{g} and \\mathbf{D} + s{(g,\\mathbf{D})} = \\mathbf{D} + \\frac{\\mathbf{D}}{g} and \\frac{\\partial}{\\partial g} (\\mathbf{D} + s{(g,\\mathbf{D})}) = \\frac{\\partial}{\\partial g} (\\mathbf{D} + \\frac{\\mathbf{D}}{g}) and \\int \\frac{\\partial}{\\partial g} (\\mathbf{D} + s{(g,\\mathbf{D})}) dg = \\int \\frac{\\partial}{\\partial g} (\\mathbf{D} + \\frac{\\mathbf{D}}{g}) dg and Q + s{(g,\\mathbf{D})} = \\frac{\\mathbf{D}}{g} + k and Q + \\frac{\\mathbf{D}}{g} = \\frac{\\mathbf{D}}{g} + k and (Q + s{(g,\\mathbf{D})})^{Q} = (\\frac{\\mathbf{D}}{g} + k)^{Q} and (Q + s{(g,\\mathbf{D})})^{Q} = (Q + \\frac{\\mathbf{D}}{g})^{Q} and (\\frac{\\mathbf{D}}{g} + k)^{Q} = (Q + \\frac{\\mathbf{D}}{g})^{Q}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('Q', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('k', commutative=True)))"], [["power", 5, "Symbol('Q', commutative=True)"], "Equality(Pow(Add(Symbol('Q', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('Q', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('k', commutative=True)), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Add(Symbol('Q', commutative=True), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('Q', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('k', commutative=True)), Symbol('Q', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\lambda,u)} = \\lambda + u, then derive \\frac{\\partial^{2}}{\\partial u^{2}} \\operatorname{A_{z}}{(\\lambda,u)} = 0, then obtain \\frac{\\partial^{3}}{\\partial u^{3}} \\operatorname{A_{z}}{(\\lambda,u)} = \\frac{d}{d u} 0", "derivation": "\\operatorname{A_{z}}{(\\lambda,u)} = \\lambda + u and \\frac{\\partial}{\\partial u} \\operatorname{A_{z}}{(\\lambda,u)} = \\frac{\\partial}{\\partial u} (\\lambda + u) and \\frac{\\frac{\\partial}{\\partial u} \\operatorname{A_{z}}{(\\lambda,u)}}{\\frac{\\partial}{\\partial u} (\\lambda + u)} = 1 and \\frac{\\partial}{\\partial u} \\frac{\\frac{\\partial}{\\partial u} \\operatorname{A_{z}}{(\\lambda,u)}}{\\frac{\\partial}{\\partial u} (\\lambda + u)} = \\frac{d}{d u} 1 and \\frac{\\partial^{2}}{\\partial u^{2}} \\operatorname{A_{z}}{(\\lambda,u)} = 0 and \\frac{\\partial^{3}}{\\partial u^{3}} \\operatorname{A_{z}}{(\\lambda,u)} = \\frac{d}{d u} 0", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Integer(1))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Pow(Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))), Integer(0))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(3))), Derivative(Integer(0), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(V,f_{\\mathbf{v}})} = e^{\\frac{V}{f_{\\mathbf{v}}}}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\tilde{\\infty}^{V} (V + \\pi{(V,f_{\\mathbf{v}})} - 1) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\tilde{\\infty}^{V} (V + e^{\\frac{V}{f_{\\mathbf{v}}}} - 1)", "derivation": "\\pi{(V,f_{\\mathbf{v}})} = e^{\\frac{V}{f_{\\mathbf{v}}}} and \\pi{(V,f_{\\mathbf{v}})} - 1 = e^{\\frac{V}{f_{\\mathbf{v}}}} - 1 and V + \\pi{(V,f_{\\mathbf{v}})} - 1 = V + e^{\\frac{V}{f_{\\mathbf{v}}}} - 1 and \\tilde{\\infty}^{V} (V + \\pi{(V,f_{\\mathbf{v}})} - 1) = \\tilde{\\infty}^{V} (V + e^{\\frac{V}{f_{\\mathbf{v}}}} - 1) and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\tilde{\\infty}^{V} (V + \\pi{(V,f_{\\mathbf{v}})} - 1) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\tilde{\\infty}^{V} (V + e^{\\frac{V}{f_{\\mathbf{v}}}} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\pi')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), Add(exp(Mul(Symbol('V', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))), Integer(-1)))"], [["add", 2, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('\\\\pi')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), Add(Symbol('V', commutative=True), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))), Integer(-1)))"], [["divide", 3, "Pow(Integer(0), Symbol('V', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Function('\\\\pi')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Mul(Pow(zoo, Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))), Integer(-1))))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Mul(Pow(zoo, Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Function('\\\\pi')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Mul(Pow(zoo, Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), exp(Mul(Symbol('V', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})} = E_{n} + \\rho v_{1}, then obtain E_{n} + n = \\int \\frac{E_{n} + \\rho v_{1}}{\\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})}} dE_{n}", "derivation": "\\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})} = E_{n} + \\rho v_{1} and 1 = \\frac{E_{n} + \\rho v_{1}}{\\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})}} and \\int 1 dE_{n} = \\int \\frac{E_{n} + \\rho v_{1}}{\\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})}} dE_{n} and E_{n} + n = \\int \\frac{E_{n} + \\rho v_{1}}{\\operatorname{a^{\\dagger}}{(E_{n},\\rho,v_{1})}} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('E_n', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 1, "Function('a^{\\\\dagger}')(Symbol('E_n', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('E_n', commutative=True), Mul(Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('E_n', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Add(Symbol('E_n', commutative=True), Mul(Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('E_n', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E_n', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Add(Symbol('E_n', commutative=True), Mul(Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('E_n', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('v_1', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\sigma_x,y)} = \\frac{\\log{(\\sigma_x)}}{y}, then derive \\frac{\\partial}{\\partial y} \\mathbf{F}{(\\sigma_x,y)} = - \\frac{\\log{(\\sigma_x)}}{y^{2}}, then obtain - \\log{(\\sigma_x)} = - y \\mathbf{F}{(\\sigma_x,y)}", "derivation": "\\mathbf{F}{(\\sigma_x,y)} = \\frac{\\log{(\\sigma_x)}}{y} and \\frac{\\partial}{\\partial y} \\mathbf{F}{(\\sigma_x,y)} = \\frac{\\partial}{\\partial y} \\frac{\\log{(\\sigma_x)}}{y} and \\frac{\\partial}{\\partial y} \\mathbf{F}{(\\sigma_x,y)} = - \\frac{\\log{(\\sigma_x)}}{y^{2}} and \\frac{\\partial}{\\partial y} \\mathbf{F}{(\\sigma_x,y)} = - \\frac{\\mathbf{F}{(\\sigma_x,y)}}{y} and - \\frac{\\log{(\\sigma_x)}}{y^{2}} = - \\frac{\\mathbf{F}{(\\sigma_x,y)}}{y} and - \\log{(\\sigma_x)} = - y \\mathbf{F}{(\\sigma_x,y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-2)), log(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-2)), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True))))"], [["divide", 5, "Pow(Symbol('y', commutative=True), Integer(-2))"], "Equality(Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\sigma_x', commutative=True), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(s,v_{t})} = s v_{t}, then obtain (s v_{t} - s) \\iint (- s + \\operatorname{f_{E}}{(s,v_{t})}) ds ds = (s v_{t} - s) \\iint (s v_{t} - s) ds ds", "derivation": "\\operatorname{f_{E}}{(s,v_{t})} = s v_{t} and - s + \\operatorname{f_{E}}{(s,v_{t})} = s v_{t} - s and \\int (- s + \\operatorname{f_{E}}{(s,v_{t})}) ds = \\int (s v_{t} - s) ds and \\iint (- s + \\operatorname{f_{E}}{(s,v_{t})}) ds ds = \\iint (s v_{t} - s) ds ds and \\frac{(s v_{t} - s) \\iint (- s + \\operatorname{f_{E}}{(s,v_{t})}) ds ds}{\\operatorname{f_{E}}{(s,v_{t})}} = \\frac{(s v_{t} - s) \\iint (s v_{t} - s) ds ds}{\\operatorname{f_{E}}{(s,v_{t})}} and (s v_{t} - s) \\iint (- s + \\operatorname{f_{E}}{(s,v_{t})}) ds ds = (s v_{t} - s) \\iint (s v_{t} - s) ds ds", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["times", 4, "Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["divide", 5, "Pow(Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('f_E')(Symbol('s', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Integral(Add(Mul(Symbol('s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(n)} = \\log{(\\log{(n)})}, then obtain \\frac{d}{d n} - (\\mathbf{J}{(n)} + \\log{(n)})^{n} = \\frac{d}{d n} - (\\log{(n)} + \\log{(\\log{(n)})})^{n}", "derivation": "\\mathbf{J}{(n)} = \\log{(\\log{(n)})} and \\mathbf{J}{(n)} + \\log{(n)} = \\log{(n)} + \\log{(\\log{(n)})} and (\\mathbf{J}{(n)} + \\log{(n)})^{n} = (\\log{(n)} + \\log{(\\log{(n)})})^{n} and - (\\mathbf{J}{(n)} + \\log{(n)})^{n} = - (\\log{(n)} + \\log{(\\log{(n)})})^{n} and \\frac{d}{d n} - (\\mathbf{J}{(n)} + \\log{(n)})^{n} = \\frac{d}{d n} - (\\log{(n)} + \\log{(\\log{(n)})})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True))))"], [["add", 1, "log(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Add(log(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True)))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{J}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Add(log(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True)))), Symbol('n', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{J}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Symbol('n', commutative=True))), Mul(Integer(-1), Pow(Add(log(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True)))), Symbol('n', commutative=True))))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{J}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Add(log(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True)))), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})}, then derive e^{\\operatorname{f^{*}}{(L_{\\varepsilon})}} \\frac{d}{d L_{\\varepsilon}} \\operatorname{f^{*}}{(L_{\\varepsilon})} = 1, then obtain - L_{\\varepsilon} \\frac{d}{d L_{\\varepsilon}} \\log{(L_{\\varepsilon})} = -1", "derivation": "\\operatorname{f^{*}}{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})} and e^{\\operatorname{f^{*}}{(L_{\\varepsilon})}} = L_{\\varepsilon} and \\frac{d}{d L_{\\varepsilon}} e^{\\operatorname{f^{*}}{(L_{\\varepsilon})}} = \\frac{d}{d L_{\\varepsilon}} L_{\\varepsilon} and e^{\\operatorname{f^{*}}{(L_{\\varepsilon})}} \\frac{d}{d L_{\\varepsilon}} \\operatorname{f^{*}}{(L_{\\varepsilon})} = 1 and L_{\\varepsilon} \\frac{d}{d L_{\\varepsilon}} \\operatorname{f^{*}}{(L_{\\varepsilon})} = 1 and - L_{\\varepsilon} \\frac{d}{d L_{\\varepsilon}} \\operatorname{f^{*}}{(L_{\\varepsilon})} = -1 and - L_{\\varepsilon} \\frac{d}{d L_{\\varepsilon}} \\log{(L_{\\varepsilon})} = -1", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(exp(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Symbol('L_{\\\\varepsilon}', commutative=True), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True))), Derivative(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(1))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(Function('f^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(-1))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(-1))"]]}, {"prompt": "Given \\rho_{f}{(M,f_{\\mathbf{v}})} = \\int \\frac{M}{f_{\\mathbf{v}}} dM, then obtain 2 \\cdot 0^{M} - \\int \\frac{M}{f_{\\mathbf{v}}} dM = 0^{M} + (- \\rho_{f}{(M,f_{\\mathbf{v}})} + \\int \\frac{M}{f_{\\mathbf{v}}} dM)^{M} - \\int \\frac{M}{f_{\\mathbf{v}}} dM", "derivation": "\\rho_{f}{(M,f_{\\mathbf{v}})} = \\int \\frac{M}{f_{\\mathbf{v}}} dM and 0 = - \\rho_{f}{(M,f_{\\mathbf{v}})} + \\int \\frac{M}{f_{\\mathbf{v}}} dM and 0^{M} = (- \\rho_{f}{(M,f_{\\mathbf{v}})} + \\int \\frac{M}{f_{\\mathbf{v}}} dM)^{M} and 2 \\cdot 0^{M} = 0^{M} + (- \\rho_{f}{(M,f_{\\mathbf{v}})} + \\int \\frac{M}{f_{\\mathbf{v}}} dM)^{M} and 2 \\cdot 0^{M} - \\int \\frac{M}{f_{\\mathbf{v}}} dM = 0^{M} + (- \\rho_{f}{(M,f_{\\mathbf{v}})} + \\int \\frac{M}{f_{\\mathbf{v}}} dM)^{M} - \\int \\frac{M}{f_{\\mathbf{v}}} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True))))"], [["minus", 1, "Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True)))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Integer(0), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True)))), Symbol('M', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Pow(Integer(0), Symbol('M', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Integer(0), Symbol('M', commutative=True))), Add(Pow(Integer(0), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True)))), Symbol('M', commutative=True))))"], [["minus", 4, "Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Integer(0), Symbol('M', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True))))), Add(Pow(Integer(0), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('M', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\mu{(n_{2},\\eta^{\\prime})} = \\eta^{\\prime} + n_{2}, then obtain (\\int 0 d\\eta^{\\prime})^{\\eta^{\\prime}} = (\\int ((\\eta^{\\prime} + n_{2})^{\\eta^{\\prime}} - \\mu^{\\eta^{\\prime}}{(n_{2},\\eta^{\\prime})}) d\\eta^{\\prime})^{\\eta^{\\prime}}", "derivation": "\\mu{(n_{2},\\eta^{\\prime})} = \\eta^{\\prime} + n_{2} and \\mu^{\\eta^{\\prime}}{(n_{2},\\eta^{\\prime})} = (\\eta^{\\prime} + n_{2})^{\\eta^{\\prime}} and 0 = (\\eta^{\\prime} + n_{2})^{\\eta^{\\prime}} - \\mu^{\\eta^{\\prime}}{(n_{2},\\eta^{\\prime})} and \\int 0 d\\eta^{\\prime} = \\int ((\\eta^{\\prime} + n_{2})^{\\eta^{\\prime}} - \\mu^{\\eta^{\\prime}}{(n_{2},\\eta^{\\prime})}) d\\eta^{\\prime} and (\\int 0 d\\eta^{\\prime})^{\\eta^{\\prime}} = (\\int ((\\eta^{\\prime} + n_{2})^{\\eta^{\\prime}} - \\mu^{\\eta^{\\prime}}{(n_{2},\\eta^{\\prime})}) d\\eta^{\\prime})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Integral(Add(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('n_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{S},\\rho)} = \\mathbf{S}^{\\rho}, then obtain \\int e^{\\cos{(\\mathbf{A}{(\\mathbf{S},\\rho)})}} d\\mathbf{S} = \\int e^{\\cos{(\\mathbf{S}^{\\rho})}} d\\mathbf{S}", "derivation": "\\mathbf{A}{(\\mathbf{S},\\rho)} = \\mathbf{S}^{\\rho} and \\cos{(\\mathbf{A}{(\\mathbf{S},\\rho)})} = \\cos{(\\mathbf{S}^{\\rho})} and e^{\\cos{(\\mathbf{A}{(\\mathbf{S},\\rho)})}} = e^{\\cos{(\\mathbf{S}^{\\rho})}} and \\int e^{\\cos{(\\mathbf{A}{(\\mathbf{S},\\rho)})}} d\\mathbf{S} = \\int e^{\\cos{(\\mathbf{S}^{\\rho})}} d\\mathbf{S}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["exp", 2], "Equality(exp(cos(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), exp(cos(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(exp(cos(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(cos(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(A_{1})} = \\log{(A_{1})}, then obtain \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} = \\log{(A_{1})}^{2 A_{1} + \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} - \\log{(A_{1})}^{2 A_{1}}}", "derivation": "\\theta_{1}{(A_{1})} = \\log{(A_{1})} and \\theta_{1}^{A_{1}}{(A_{1})} = \\log{(A_{1})}^{A_{1}} and \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} = \\log{(A_{1})}^{2 A_{1}} and \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} - \\log{(A_{1})}^{2 A_{1}} = 0 and 2 A_{1} + \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} - \\log{(A_{1})}^{2 A_{1}} = 2 A_{1} and \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} = \\log{(A_{1})}^{2 A_{1} + \\theta_{1}^{A_{1}}{(A_{1})} \\log{(A_{1})}^{A_{1}} - \\log{(A_{1})}^{2 A_{1}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["times", 2, "Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Pow(log(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True))))"], [["minus", 3, "Pow(log(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True))))), Integer(0))"], [["add", 4, "Mul(Integer(2), Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True))))), Mul(Integer(2), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Pow(log(Symbol('A_1', commutative=True)), Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Pow(Function('\\\\theta_1')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('A_1', commutative=True)))))))"]]}, {"prompt": "Given f{(\\varepsilon_0,\\mathbf{s})} = \\int (\\mathbf{s} - \\varepsilon_0) d\\varepsilon_0, then obtain \\varepsilon_0^{\\mathbf{s}} = (\\frac{\\varepsilon_0 (\\chi + \\mathbf{s} \\varepsilon_0 - \\frac{\\varepsilon_0^{2}}{2})}{f{(\\varepsilon_0,\\mathbf{s})}})^{\\mathbf{s}}", "derivation": "f{(\\varepsilon_0,\\mathbf{s})} = \\int (\\mathbf{s} - \\varepsilon_0) d\\varepsilon_0 and \\varepsilon_0 f{(\\varepsilon_0,\\mathbf{s})} = \\varepsilon_0 \\int (\\mathbf{s} - \\varepsilon_0) d\\varepsilon_0 and \\varepsilon_0 = \\frac{\\varepsilon_0 \\int (\\mathbf{s} - \\varepsilon_0) d\\varepsilon_0}{f{(\\varepsilon_0,\\mathbf{s})}} and \\varepsilon_0^{\\mathbf{s}} = (\\frac{\\varepsilon_0 \\int (\\mathbf{s} - \\varepsilon_0) d\\varepsilon_0}{f{(\\varepsilon_0,\\mathbf{s})}})^{\\mathbf{s}} and \\varepsilon_0^{\\mathbf{s}} = (\\frac{\\varepsilon_0 (\\chi + \\mathbf{s} \\varepsilon_0 - \\frac{\\varepsilon_0^{2}}{2})}{f{(\\varepsilon_0,\\mathbf{s})}})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["times", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["divide", 2, "Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Symbol('\\\\varepsilon_0', commutative=True), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)))), Pow(Function('f')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(n_{1})} = e^{n_{1}}, then obtain (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) \\frac{d}{d n_{1}} (- n_{1} + e^{n_{1}}) = (- n_{1} + e^{n_{1}}) \\frac{d}{d n_{1}} (- n_{1} + e^{n_{1}})", "derivation": "\\operatorname{F_{H}}{(n_{1})} = e^{n_{1}} and - n_{1} + \\operatorname{F_{H}}{(n_{1})} = - n_{1} + e^{n_{1}} and \\frac{d}{d n_{1}} (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) = \\frac{d}{d n_{1}} (- n_{1} + e^{n_{1}}) and (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) \\frac{d}{d n_{1}} (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) = (- n_{1} + e^{n_{1}}) \\frac{d}{d n_{1}} (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) and (- n_{1} + \\operatorname{F_{H}}{(n_{1})}) \\frac{d}{d n_{1}} (- n_{1} + e^{n_{1}}) = (- n_{1} + e^{n_{1}}) \\frac{d}{d n_{1}} (- n_{1} + e^{n_{1}})", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["minus", 1, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('F_H')(Symbol('n_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(a,C_{d})} = \\frac{\\partial}{\\partial a} a^{C_{d}}, then derive b{(a,C_{d})} = \\frac{C_{d} a^{C_{d}}}{a}, then obtain \\frac{\\int (\\frac{C_{d} a^{C_{d}}}{a} + \\frac{\\partial}{\\partial a} a^{C_{d}}) da}{a} = \\frac{\\int 2 \\frac{\\partial}{\\partial a} a^{C_{d}} da}{a}", "derivation": "b{(a,C_{d})} = \\frac{\\partial}{\\partial a} a^{C_{d}} and b{(a,C_{d})} + \\frac{\\partial}{\\partial a} a^{C_{d}} = 2 \\frac{\\partial}{\\partial a} a^{C_{d}} and b{(a,C_{d})} = \\frac{C_{d} a^{C_{d}}}{a} and \\frac{C_{d} a^{C_{d}}}{a} + \\frac{\\partial}{\\partial a} a^{C_{d}} = 2 \\frac{\\partial}{\\partial a} a^{C_{d}} and \\int (\\frac{C_{d} a^{C_{d}}}{a} + \\frac{\\partial}{\\partial a} a^{C_{d}}) da = \\int 2 \\frac{\\partial}{\\partial a} a^{C_{d}} da and \\frac{\\int (\\frac{C_{d} a^{C_{d}}}{a} + \\frac{\\partial}{\\partial a} a^{C_{d}}) da}{a} = \\frac{\\int 2 \\frac{\\partial}{\\partial a} a^{C_{d}} da}{a}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Add(Function('b')(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('b')(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True))), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True))), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True))), Integral(Mul(Integer(2), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True))))"], [["divide", 5, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True))), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Derivative(Pow(Symbol('a', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(I,\\rho_b)} = I \\rho_b, then derive \\frac{\\partial}{\\partial \\rho_b} \\operatorname{c_{0}}{(I,\\rho_b)} = I, then obtain \\frac{\\partial}{\\partial \\rho_b} I \\rho_b = I", "derivation": "\\operatorname{c_{0}}{(I,\\rho_b)} = I \\rho_b and \\operatorname{c_{0}}{(I,\\rho_b)} + \\frac{1}{\\mathbf{E}} = I \\rho_b + \\frac{1}{\\mathbf{E}} and \\operatorname{c_{0}}{(I,\\rho_b)} - \\log{(A)} + \\frac{1}{\\mathbf{E}} = I \\rho_b - \\log{(A)} + \\frac{1}{\\mathbf{E}} and \\frac{\\partial}{\\partial \\rho_b} (\\operatorname{c_{0}}{(I,\\rho_b)} - \\log{(A)} + \\frac{1}{\\mathbf{E}}) = \\frac{\\partial}{\\partial \\rho_b} (I \\rho_b - \\log{(A)} + \\frac{1}{\\mathbf{E}}) and \\frac{\\partial}{\\partial \\rho_b} \\operatorname{c_{0}}{(I,\\rho_b)} = I and \\frac{\\partial}{\\partial \\rho_b} I \\rho_b = I", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))"], "Equality(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))))"], [["minus", 2, "log(Symbol('A', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True))), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True))), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True))), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), log(Symbol('A', commutative=True))), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('I', commutative=True))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('I', commutative=True))"]]}, {"prompt": "Given \\mathbf{H}{(b,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (\\hbar - b) and v{(t_{2},\\psi^*)} = t_{2}^{\\psi^*}, then obtain (t_{2}^{- \\psi^*} \\frac{\\partial}{\\partial \\hbar} \\mathbf{H}{(b,\\hbar)})^{\\hbar} = (t_{2}^{- \\psi^*} \\frac{\\partial^{2}}{\\partial \\hbar^{2}} (\\hbar - b))^{\\hbar}", "derivation": "\\mathbf{H}{(b,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (\\hbar - b) and \\frac{\\partial}{\\partial \\hbar} \\mathbf{H}{(b,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\hbar^{2}} (\\hbar - b) and v{(t_{2},\\psi^*)} = t_{2}^{\\psi^*} and \\frac{\\frac{\\partial}{\\partial \\hbar} \\mathbf{H}{(b,\\hbar)}}{v{(t_{2},\\psi^*)}} = \\frac{\\frac{\\partial^{2}}{\\partial \\hbar^{2}} (\\hbar - b)}{v{(t_{2},\\psi^*)}} and t_{2}^{- \\psi^*} \\frac{\\partial}{\\partial \\hbar} \\mathbf{H}{(b,\\hbar)} = t_{2}^{- \\psi^*} \\frac{\\partial^{2}}{\\partial \\hbar^{2}} (\\hbar - b) and (t_{2}^{- \\psi^*} \\frac{\\partial}{\\partial \\hbar} \\mathbf{H}{(b,\\hbar)})^{\\hbar} = (t_{2}^{- \\psi^*} \\frac{\\partial^{2}}{\\partial \\hbar^{2}} (\\hbar - b))^{\\hbar}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))"], ["get_premise", "Equality(Function('v')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["divide", 2, "Function('v')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Function('v')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Pow(Function('v')(Symbol('t_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Derivative(Function('\\\\mathbf{H}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Pow(Symbol('t_2', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2)))))"], [["power", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('t_2', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Derivative(Function('\\\\mathbf{H}')(Symbol('b', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Pow(Symbol('t_2', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Derivative(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2)))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f,y^{\\prime})} = (y^{\\prime})^{f} and z{(f,y^{\\prime})} = \\cos{(y^{\\prime} + (y^{\\prime})^{f})}, then obtain \\int \\frac{z{(f,y^{\\prime})}}{y^{\\prime} + (y^{\\prime})^{f}} df = \\int \\frac{\\cos{(y^{\\prime} + (y^{\\prime})^{f})}}{y^{\\prime} + (y^{\\prime})^{f}} df", "derivation": "\\hat{\\mathbf{r}}{(f,y^{\\prime})} = (y^{\\prime})^{f} and y^{\\prime} + \\hat{\\mathbf{r}}{(f,y^{\\prime})} = y^{\\prime} + (y^{\\prime})^{f} and z{(f,y^{\\prime})} = \\cos{(y^{\\prime} + (y^{\\prime})^{f})} and \\frac{z{(f,y^{\\prime})}}{y^{\\prime} + \\hat{\\mathbf{r}}{(f,y^{\\prime})}} = \\frac{\\cos{(y^{\\prime} + (y^{\\prime})^{f})}}{y^{\\prime} + \\hat{\\mathbf{r}}{(f,y^{\\prime})}} and \\frac{z{(f,y^{\\prime})}}{y^{\\prime} + (y^{\\prime})^{f}} = \\frac{\\cos{(y^{\\prime} + (y^{\\prime})^{f})}}{y^{\\prime} + (y^{\\prime})^{f}} and \\int \\frac{z{(f,y^{\\prime})}}{y^{\\prime} + (y^{\\prime})^{f}} df = \\int \\frac{\\cos{(y^{\\prime} + (y^{\\prime})^{f})}}{y^{\\prime} + (y^{\\prime})^{f}} df", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)))"], [["add", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True)))))"], [["divide", 3, "Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Function('z')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Function('z')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))))))"], [["integrate", 5, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), Function('z')(Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), cos(Add(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(P_{g},W,M_{E})} = - M_{E} - P_{g} + W, then obtain \\frac{M_{E} + 2 P_{g} - W + 2 \\operatorname{v_{1}}{(P_{g},W,M_{E})}}{P_{g}} = \\frac{- M_{E} + W}{P_{g}}", "derivation": "\\operatorname{v_{1}}{(P_{g},W,M_{E})} = - M_{E} - P_{g} + W and P_{g} + \\operatorname{v_{1}}{(P_{g},W,M_{E})} = - M_{E} + W and M_{E} + 2 P_{g} - W + 2 \\operatorname{v_{1}}{(P_{g},W,M_{E})} = P_{g} + \\operatorname{v_{1}}{(P_{g},W,M_{E})} and \\frac{P_{g} + \\operatorname{v_{1}}{(P_{g},W,M_{E})}}{P_{g}} = \\frac{- M_{E} + W}{P_{g}} and \\frac{M_{E} + 2 P_{g} - W + 2 \\operatorname{v_{1}}{(P_{g},W,M_{E})}}{P_{g}} = \\frac{- M_{E} + W}{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('W', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('P_g', commutative=True))"], "Equality(Add(Symbol('P_g', commutative=True), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('W', commutative=True)))"], [["add", 2, "Add(Symbol('M_E', commutative=True), Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Add(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True)))), Add(Symbol('P_g', commutative=True), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True))))"], [["divide", 2, "Symbol('P_g', commutative=True)"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Add(Symbol('P_g', commutative=True), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True)))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(2), Function('v_1')(Symbol('P_g', commutative=True), Symbol('W', commutative=True), Symbol('M_E', commutative=True))))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(F_{x})} = \\sin{(\\sin{(F_{x})})}, then obtain \\frac{d}{d F_{x}} (\\frac{d}{d F_{x}} \\mathbf{D}^{F_{x}}{(F_{x})})^{F_{x}} = \\frac{d}{d F_{x}} (\\frac{d}{d F_{x}} \\sin^{F_{x}}{(\\sin{(F_{x})})})^{F_{x}}", "derivation": "\\mathbf{D}{(F_{x})} = \\sin{(\\sin{(F_{x})})} and \\mathbf{D}^{F_{x}}{(F_{x})} = \\sin^{F_{x}}{(\\sin{(F_{x})})} and \\frac{d}{d F_{x}} \\mathbf{D}^{F_{x}}{(F_{x})} = \\frac{d}{d F_{x}} \\sin^{F_{x}}{(\\sin{(F_{x})})} and (\\frac{d}{d F_{x}} \\mathbf{D}^{F_{x}}{(F_{x})})^{F_{x}} = (\\frac{d}{d F_{x}} \\sin^{F_{x}}{(\\sin{(F_{x})})})^{F_{x}} and \\frac{d}{d F_{x}} (\\frac{d}{d F_{x}} \\mathbf{D}^{F_{x}}{(F_{x})})^{F_{x}} = \\frac{d}{d F_{x}} (\\frac{d}{d F_{x}} \\sin^{F_{x}}{(\\sin{(F_{x})})})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), sin(sin(Symbol('F_x', commutative=True))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(sin(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Pow(sin(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Derivative(Pow(sin(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["differentiate", 4, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Pow(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Pow(Derivative(Pow(sin(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then obtain - \\sigma_{x}{(L_{\\varepsilon})} + \\cos{(L_{\\varepsilon})} + \\frac{d}{d L_{\\varepsilon}} (\\sigma_{x}{(L_{\\varepsilon})} - \\cos{(L_{\\varepsilon})}) = - \\sigma_{x}{(L_{\\varepsilon})} + \\cos{(L_{\\varepsilon})} + \\frac{d}{d L_{\\varepsilon}} 0", "derivation": "\\sigma_{x}{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\sigma_{x}{(L_{\\varepsilon})} - \\cos{(L_{\\varepsilon})} = 0 and \\frac{d}{d L_{\\varepsilon}} (\\sigma_{x}{(L_{\\varepsilon})} - \\cos{(L_{\\varepsilon})}) = \\frac{d}{d L_{\\varepsilon}} 0 and - \\sigma_{x}{(L_{\\varepsilon})} + \\cos{(L_{\\varepsilon})} + \\frac{d}{d L_{\\varepsilon}} (\\sigma_{x}{(L_{\\varepsilon})} - \\cos{(L_{\\varepsilon})}) = - \\sigma_{x}{(L_{\\varepsilon})} + \\cos{(L_{\\varepsilon})} + \\frac{d}{d L_{\\varepsilon}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "cos(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 3, "Add(Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True))), cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(Add(Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('L_{\\\\varepsilon}', commutative=True))), cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(Integer(0), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(p)} = \\log{(\\cos{(p)})} and q{(p)} = \\frac{d}{d p} f{(p)}, then obtain e^{- q{(p)}} (\\frac{d}{d p} \\log{(\\cos{(p)})})^{p} = e^{- q{(p)}} (\\frac{d}{d p} f{(p)})^{p}", "derivation": "f{(p)} = \\log{(\\cos{(p)})} and \\frac{d}{d p} f{(p)} = \\frac{d}{d p} \\log{(\\cos{(p)})} and q{(p)} = \\frac{d}{d p} f{(p)} and q^{p}{(p)} = (\\frac{d}{d p} f{(p)})^{p} and q{(p)} = \\frac{d}{d p} \\log{(\\cos{(p)})} and (\\frac{d}{d p} \\log{(\\cos{(p)})})^{p} = (\\frac{d}{d p} f{(p)})^{p} and e^{- q{(p)}} (\\frac{d}{d p} \\log{(\\cos{(p)})})^{p} = e^{- q{(p)}} (\\frac{d}{d p} f{(p)})^{p}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('p', commutative=True)), log(cos(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('p', commutative=True)), Derivative(Function('f')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Function('q')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Derivative(Function('f')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('q')(Symbol('p', commutative=True)), Derivative(log(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(log(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(Function('f')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["divide", 6, "exp(Function('q')(Symbol('p', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Function('q')(Symbol('p', commutative=True)))), Pow(Derivative(log(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True))), Mul(exp(Mul(Integer(-1), Function('q')(Symbol('p', commutative=True)))), Pow(Derivative(Function('f')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\varepsilon)} = \\cos{(\\cos{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\cos{(\\varepsilon)})}, then obtain \\cos^{\\varepsilon}{(\\operatorname{F_{H}}{(\\varepsilon)} + \\cos{(\\varepsilon)} - \\cos{(\\cos{(\\varepsilon)})})} = \\cos^{\\varepsilon}{(\\cos{(\\varepsilon)})}", "derivation": "\\operatorname{F_{H}}{(\\varepsilon)} = \\cos{(\\cos{(\\varepsilon)})} and \\operatorname{F_{H}}{(\\varepsilon)} + \\cos{(\\varepsilon)} = \\cos{(\\varepsilon)} + \\cos{(\\cos{(\\varepsilon)})} and \\operatorname{F_{H}}{(\\varepsilon)} + \\cos{(\\varepsilon)} - \\cos{(\\cos{(\\varepsilon)})} = \\cos{(\\varepsilon)} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\cos{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\operatorname{F_{H}}{(\\varepsilon)} + \\cos{(\\varepsilon)} - \\cos{(\\cos{(\\varepsilon)})})} and \\cos^{\\varepsilon}{(\\operatorname{F_{H}}{(\\varepsilon)} + \\cos{(\\varepsilon)} - \\cos{(\\cos{(\\varepsilon)})})} = \\cos^{\\varepsilon}{(\\cos{(\\varepsilon)})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), cos(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True))), Add(cos(Symbol('\\\\varepsilon', commutative=True)), cos(cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["minus", 2, "cos(cos(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\varepsilon', commutative=True))))), cos(Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(cos(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), Pow(cos(Add(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\varepsilon', commutative=True)))))), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(cos(Add(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\varepsilon', commutative=True)))))), Symbol('\\\\varepsilon', commutative=True)), Pow(cos(cos(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(\\chi)} = \\log{(\\chi)}, then obtain \\int \\frac{\\chi + \\mathbb{I}{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} d\\chi = \\int \\frac{\\chi + \\log{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} d\\chi", "derivation": "\\mathbb{I}{(\\chi)} = \\log{(\\chi)} and \\chi + \\mathbb{I}{(\\chi)} = \\chi + \\log{(\\chi)} and \\frac{\\chi + \\mathbb{I}{(\\chi)}}{\\log{(\\chi)}} = \\frac{\\chi + \\log{(\\chi)}}{\\log{(\\chi)}} and \\frac{\\chi + \\mathbb{I}{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} = \\frac{\\chi + \\log{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} and \\int \\frac{\\chi + \\mathbb{I}{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} d\\chi = \\int \\frac{\\chi + \\log{(\\chi)}}{\\mathbb{I}{(\\chi)} \\log{(\\chi)}} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True))), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["divide", 3, "Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Add(Symbol('\\\\chi', commutative=True), log(Symbol('\\\\chi', commutative=True))), Pow(Function('\\\\mathbb{I}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(S,\\delta,m_{s})} = S + m_{s}^{\\delta}, then derive \\frac{\\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})}}{m_{s}} = \\frac{\\delta m_{s}^{\\delta}}{m_{s}^{2}}, then obtain \\frac{m_{s}^{2}}{(\\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})})^{2}} = \\frac{m_{s}^{4} m_{s}^{- 2 \\delta}}{\\delta^{2}}", "derivation": "\\hat{x}_0{(S,\\delta,m_{s})} = S + m_{s}^{\\delta} and \\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (S + m_{s}^{\\delta}) and \\frac{\\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})}}{m_{s}} = \\frac{\\frac{\\partial}{\\partial m_{s}} (S + m_{s}^{\\delta})}{m_{s}} and \\frac{\\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})}}{m_{s}} = \\frac{\\delta m_{s}^{\\delta}}{m_{s}^{2}} and \\frac{m_{s}^{2}}{(\\frac{\\partial}{\\partial m_{s}} \\hat{x}_0{(S,\\delta,m_{s})})^{2}} = \\frac{m_{s}^{4} m_{s}^{- 2 \\delta}}{\\delta^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('S', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Derivative(Add(Symbol('S', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-2)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["power", 4, "Integer(-2)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(2)), Pow(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-2))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-2)), Pow(Symbol('m_s', commutative=True), Integer(4)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given n{(x)} = \\int \\cos{(x)} dx, then derive n{(x)} = \\varepsilon_0 + \\sin{(x)}, then derive \\theta_2 (n_{2} + \\sin{(x)}) = \\theta_2 (\\varepsilon_0 + \\sin{(x)}), then obtain \\theta_2 (n_{2} + \\sin{(x)}) - n{(x)} = \\theta_2 (\\varepsilon_0 + \\sin{(x)}) - n{(x)}", "derivation": "n{(x)} = \\int \\cos{(x)} dx and n{(x)} = \\varepsilon_0 + \\sin{(x)} and \\int \\cos{(x)} dx = \\varepsilon_0 + \\sin{(x)} and \\theta_2 \\int \\cos{(x)} dx = \\theta_2 (\\varepsilon_0 + \\sin{(x)}) and \\theta_2 (n_{2} + \\sin{(x)}) = \\theta_2 (\\varepsilon_0 + \\sin{(x)}) and \\theta_2 (n_{2} + \\sin{(x)}) - n{(x)} = \\theta_2 (\\varepsilon_0 + \\sin{(x)}) - n{(x)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('x', commutative=True)), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n')(Symbol('x', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('x', commutative=True))))"], [["times", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('x', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('n_2', commutative=True), sin(Symbol('x', commutative=True)))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('x', commutative=True)))))"], [["minus", 5, "Function('n')(Symbol('x', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('n_2', commutative=True), sin(Symbol('x', commutative=True)))), Mul(Integer(-1), Function('n')(Symbol('x', commutative=True)))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('x', commutative=True)))), Mul(Integer(-1), Function('n')(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = \\frac{d}{d \\pi} \\cos{(\\pi)}, then derive \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = - \\sin{(\\pi)}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\pi}{(\\pi)} - \\sin{(\\pi)} = (- \\sin{(\\pi)})^{\\pi} - \\sin{(\\pi)}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = \\frac{d}{d \\pi} \\cos{(\\pi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = - \\sin{(\\pi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\pi}{(\\pi)} = (- \\sin{(\\pi)})^{\\pi} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\pi}{(\\pi)} - \\sin{(\\pi)} = (- \\sin{(\\pi)})^{\\pi} - \\sin{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["add", 3, "Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True)))), Add(Pow(Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(x)} = \\cos{(x)} and \\operatorname{n_{1}}{(x)} = \\int \\mathbf{S}{(x)} dx, then obtain \\int ((\\int \\cos{(x)} dx)^{x} - 2) dx = \\int ((\\int \\mathbf{S}{(x)} dx)^{x} - 2) dx", "derivation": "\\mathbf{S}{(x)} = \\cos{(x)} and \\int \\mathbf{S}{(x)} dx = \\int \\cos{(x)} dx and \\operatorname{n_{1}}{(x)} = \\int \\mathbf{S}{(x)} dx and \\operatorname{n_{1}}^{x}{(x)} = (\\int \\mathbf{S}{(x)} dx)^{x} and \\operatorname{n_{1}}{(x)} = \\int \\cos{(x)} dx and (\\int \\cos{(x)} dx)^{x} = (\\int \\mathbf{S}{(x)} dx)^{x} and (\\int \\cos{(x)} dx)^{x} - 1 = (\\int \\mathbf{S}{(x)} dx)^{x} - 1 and (\\int \\cos{(x)} dx)^{x} - 2 = (\\int \\mathbf{S}{(x)} dx)^{x} - 2 and \\int ((\\int \\cos{(x)} dx)^{x} - 2) dx = \\int ((\\int \\mathbf{S}{(x)} dx)^{x} - 2) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('x', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('n_1')(Symbol('x', commutative=True)), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["minus", 6, 1], "Equality(Add(Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-1)), Add(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-1)))"], [["add", 7, "Integer(-1)"], "Equality(Add(Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-2)), Add(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-2)))"], [["integrate", 8, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-2)), Tuple(Symbol('x', commutative=True))), Integral(Add(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Integer(-2)), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given f{(\\mathbf{r},q)} = \\mathbf{r}^{q}, then derive h + f{(\\mathbf{r},q)} = \\mathbf{r}^{q} + \\mu_0, then obtain (\\mathbf{r}^{q} + h)^{\\mathbf{r}} = (\\mathbf{r}^{q} + \\mu_0)^{\\mathbf{r}}", "derivation": "f{(\\mathbf{r},q)} = \\mathbf{r}^{q} and \\frac{\\partial}{\\partial q} f{(\\mathbf{r},q)} = \\frac{\\partial}{\\partial q} \\mathbf{r}^{q} and \\int \\frac{\\partial}{\\partial q} f{(\\mathbf{r},q)} dq = \\int \\frac{\\partial}{\\partial q} \\mathbf{r}^{q} dq and h + f{(\\mathbf{r},q)} = \\mathbf{r}^{q} + \\mu_0 and \\mathbf{r}^{q} + h = \\mathbf{r}^{q} + \\mu_0 and (\\mathbf{r}^{q} + h)^{\\mathbf{r}} = (\\mathbf{r}^{q} + \\mu_0)^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Derivative(Function('f')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('h', commutative=True), Function('f')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True))), Add(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True)), Add(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["power", 5, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(E_{n},\\hbar)} = \\cos{(\\frac{E_{n}}{\\hbar})} and \\operatorname{P_{g}}{(E_{n},\\hbar)} = 2 \\cos{(\\frac{E_{n}}{\\hbar})}, then obtain E_{n} + 2 \\dot{y}{(E_{n},\\hbar)} = E_{n} + \\operatorname{P_{g}}{(E_{n},\\hbar)}", "derivation": "\\dot{y}{(E_{n},\\hbar)} = \\cos{(\\frac{E_{n}}{\\hbar})} and E_{n} + \\dot{y}{(E_{n},\\hbar)} = E_{n} + \\cos{(\\frac{E_{n}}{\\hbar})} and E_{n} + 2 \\dot{y}{(E_{n},\\hbar)} = E_{n} + \\dot{y}{(E_{n},\\hbar)} + \\cos{(\\frac{E_{n}}{\\hbar})} and E_{n} + 2 \\dot{y}{(E_{n},\\hbar)} = E_{n} + 2 \\cos{(\\frac{E_{n}}{\\hbar})} and \\operatorname{P_{g}}{(E_{n},\\hbar)} = 2 \\cos{(\\frac{E_{n}}{\\hbar})} and E_{n} + 2 \\dot{y}{(E_{n},\\hbar)} = E_{n} + \\operatorname{P_{g}}{(E_{n},\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))"], [["add", 1, "Symbol('E_n', commutative=True)"], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('E_n', commutative=True), cos(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))))"], [["add", 2, "Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Symbol('E_n', commutative=True), Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Symbol('E_n', commutative=True), Mul(Integer(2), cos(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), cos(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Symbol('E_n', commutative=True), Function('P_g')(Symbol('E_n', commutative=True), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(P_{e})} = \\log{(P_{e})}, then obtain (\\frac{\\sigma_{x}^{P_{e}}{(P_{e})}}{P_{e}})^{P_{e}} = (\\frac{\\log{(P_{e})}^{P_{e}}}{P_{e}})^{P_{e}}", "derivation": "\\sigma_{x}{(P_{e})} = \\log{(P_{e})} and \\sigma_{x}^{P_{e}}{(P_{e})} = \\log{(P_{e})}^{P_{e}} and \\frac{\\sigma_{x}^{P_{e}}{(P_{e})}}{P_{e}} = \\frac{\\log{(P_{e})}^{P_{e}}}{P_{e}} and (\\frac{\\sigma_{x}^{P_{e}}{(P_{e})}}{P_{e}})^{P_{e}} = (\\frac{\\log{(P_{e})}^{P_{e}}}{P_{e}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(log(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))"], [["divide", 2, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Function('\\\\sigma_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(log(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))))"], [["power", 3, "Symbol('P_e', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Function('\\\\sigma_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(log(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(F_{c})} = \\cos{(F_{c})} and \\operatorname{E_{n}}{(F_{c})} = \\frac{1}{\\sin{(\\varepsilon{(F_{c})})}}, then obtain \\frac{1}{\\sin{(\\cos{(F_{c})})}} = \\frac{1}{\\sin{(\\varepsilon{(F_{c})})}}", "derivation": "\\varepsilon{(F_{c})} = \\cos{(F_{c})} and \\operatorname{E_{n}}{(F_{c})} = \\frac{1}{\\sin{(\\varepsilon{(F_{c})})}} and \\operatorname{E_{n}}{(F_{c})} = \\frac{1}{\\sin{(\\cos{(F_{c})})}} and \\frac{1}{\\sin{(\\cos{(F_{c})})}} = \\frac{1}{\\sin{(\\varepsilon{(F_{c})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('F_c', commutative=True)), Pow(sin(Function('\\\\varepsilon')(Symbol('F_c', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E_n')(Symbol('F_c', commutative=True)), Pow(sin(cos(Symbol('F_c', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(sin(cos(Symbol('F_c', commutative=True))), Integer(-1)), Pow(sin(Function('\\\\varepsilon')(Symbol('F_c', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given x{(T,\\eta)} = \\sin{(T + \\eta)}, then derive \\frac{\\frac{\\partial}{\\partial T} x{(T,\\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}} = \\frac{\\cos{(T + \\eta)}}{T} - \\frac{\\sin{(T + \\eta)}}{T^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial T} \\sin{(T + \\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}} = \\frac{\\cos{(T + \\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}}", "derivation": "x{(T,\\eta)} = \\sin{(T + \\eta)} and \\frac{x{(T,\\eta)}}{T} = \\frac{\\sin{(T + \\eta)}}{T} and \\frac{\\partial}{\\partial T} \\frac{x{(T,\\eta)}}{T} = \\frac{\\partial}{\\partial T} \\frac{\\sin{(T + \\eta)}}{T} and \\frac{\\frac{\\partial}{\\partial T} x{(T,\\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}} = \\frac{\\cos{(T + \\eta)}}{T} - \\frac{\\sin{(T + \\eta)}}{T^{2}} and \\frac{\\frac{\\partial}{\\partial T} \\sin{(T + \\eta)}}{T} - \\frac{\\sin{(T + \\eta)}}{T^{2}} = \\frac{\\cos{(T + \\eta)}}{T} - \\frac{\\sin{(T + \\eta)}}{T^{2}} and \\frac{\\frac{\\partial}{\\partial T} \\sin{(T + \\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}} = \\frac{\\cos{(T + \\eta)}}{T} - \\frac{x{(T,\\eta)}}{T^{2}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))))), Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Add(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)), Function('x')(Symbol('T', commutative=True), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(C_{d},\\mathbf{A})} = - C_{d} + \\mathbf{A}, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\psi^{*}{(C_{d},\\mathbf{A})} dC_{d} = \\frac{\\partial}{\\partial \\mathbf{A}} (- \\frac{C_{d}^{2}}{2} + C_{d} \\mathbf{A} + L_{\\varepsilon})", "derivation": "\\psi^{*}{(C_{d},\\mathbf{A})} = - C_{d} + \\mathbf{A} and \\int \\psi^{*}{(C_{d},\\mathbf{A})} dC_{d} = \\int (- C_{d} + \\mathbf{A}) dC_{d} and \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\psi^{*}{(C_{d},\\mathbf{A})} dC_{d} = \\frac{\\partial}{\\partial \\mathbf{A}} \\int (- C_{d} + \\mathbf{A}) dC_{d} and \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\psi^{*}{(C_{d},\\mathbf{A})} dC_{d} = \\frac{\\partial}{\\partial \\mathbf{A}} (- \\frac{C_{d}^{2}}{2} + C_{d} \\mathbf{A} + L_{\\varepsilon})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{c},C)} = \\log{(C + F_{c})}, then obtain \\frac{((\\frac{\\partial}{\\partial C} C \\operatorname{y^{\\prime}}{(F_{c},C)})^{C})^{C}}{(C \\log{(C + F_{c})})^{C} + 1} = \\frac{((\\frac{\\partial}{\\partial C} C \\log{(C + F_{c})})^{C})^{C}}{(C \\log{(C + F_{c})})^{C} + 1}", "derivation": "\\operatorname{y^{\\prime}}{(F_{c},C)} = \\log{(C + F_{c})} and C \\operatorname{y^{\\prime}}{(F_{c},C)} = C \\log{(C + F_{c})} and \\frac{\\partial}{\\partial C} C \\operatorname{y^{\\prime}}{(F_{c},C)} = \\frac{\\partial}{\\partial C} C \\log{(C + F_{c})} and (\\frac{\\partial}{\\partial C} C \\operatorname{y^{\\prime}}{(F_{c},C)})^{C} = (\\frac{\\partial}{\\partial C} C \\log{(C + F_{c})})^{C} and ((\\frac{\\partial}{\\partial C} C \\operatorname{y^{\\prime}}{(F_{c},C)})^{C})^{C} = ((\\frac{\\partial}{\\partial C} C \\log{(C + F_{c})})^{C})^{C} and \\frac{((\\frac{\\partial}{\\partial C} C \\operatorname{y^{\\prime}}{(F_{c},C)})^{C})^{C}}{(C \\log{(C + F_{c})})^{C} + 1} = \\frac{((\\frac{\\partial}{\\partial C} C \\log{(C + F_{c})})^{C})^{C}}{(C \\log{(C + F_{c})})^{C} + 1}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True))))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Symbol('C', commutative=True), Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('C', commutative=True), Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Pow(Derivative(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)))"], [["power", 4, "Symbol('C', commutative=True)"], "Equality(Pow(Pow(Derivative(Mul(Symbol('C', commutative=True), Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(Pow(Derivative(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["divide", 5, "Add(Pow(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Symbol('C', commutative=True)), Integer(1))"], "Equality(Mul(Pow(Add(Pow(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Symbol('C', commutative=True)), Integer(1)), Integer(-1)), Pow(Pow(Derivative(Mul(Symbol('C', commutative=True), Function('y^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Pow(Add(Pow(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Symbol('C', commutative=True)), Integer(1)), Integer(-1)), Pow(Pow(Derivative(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then derive \\mathbf{B} \\frac{d}{d \\mathbf{B}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\mathbf{B} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})}, then obtain \\mathbf{B} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} + \\sin{(\\mathbf{B})} = \\mathbf{B} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\mathbf{B} \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\mathbf{B} \\sin{(\\mathbf{B})} and \\mathbf{B} \\frac{d}{d \\mathbf{B}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B})} = \\mathbf{B} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})} and \\mathbf{B} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} + \\sin{(\\mathbf{B})} = \\mathbf{B} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), sin(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(a)} = \\log{(a)}, then derive \\int \\Psi_{nl}{(a)} da = H + a \\log{(a)} - a, then derive - a \\log{(a)} + \\int \\Psi_{nl}{(a)} da = - a + r_{0}, then obtain H - a = - a + r_{0}", "derivation": "\\Psi_{nl}{(a)} = \\log{(a)} and \\int \\Psi_{nl}{(a)} da = \\int \\log{(a)} da and \\int \\Psi_{nl}{(a)} da = H + a \\log{(a)} - a and - a \\log{(a)} + \\int \\Psi_{nl}{(a)} da = - a \\log{(a)} + \\int \\log{(a)} da and - a \\log{(a)} + \\int \\Psi_{nl}{(a)} da = - a + r_{0} and H - a = - a + r_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('H', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["minus", 2, "Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\lambda)} = e^{\\lambda}, then derive \\frac{d}{d \\lambda} \\mathbf{g}{(\\lambda)} = e^{\\lambda}, then obtain \\frac{d}{d \\lambda} e^{\\lambda} = \\frac{d^{2}}{d \\lambda^{2}} e^{\\lambda}", "derivation": "\\mathbf{g}{(\\lambda)} = e^{\\lambda} and \\frac{d}{d \\lambda} \\mathbf{g}{(\\lambda)} = \\frac{d}{d \\lambda} e^{\\lambda} and \\frac{d}{d \\lambda} \\mathbf{g}{(\\lambda)} = e^{\\lambda} and \\frac{d}{d \\lambda} \\mathbf{g}{(\\lambda)} = \\frac{d^{2}}{d \\lambda^{2}} \\mathbf{g}{(\\lambda)} and \\frac{d}{d \\lambda} e^{\\lambda} = \\frac{d^{2}}{d \\lambda^{2}} e^{\\lambda}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), exp(Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(p)} = \\cos{(p)} and \\operatorname{y^{\\prime}}{(p)} = \\cos{(p)}, then obtain \\int 2 \\operatorname{y^{\\prime}}{(p)} dp = \\int (\\operatorname{y^{\\prime}}{(p)} + \\cos{(p)}) dp", "derivation": "\\hat{\\mathbf{x}}{(p)} = \\cos{(p)} and 2 \\hat{\\mathbf{x}}{(p)} = \\hat{\\mathbf{x}}{(p)} + \\cos{(p)} and \\operatorname{y^{\\prime}}{(p)} = \\cos{(p)} and \\int 2 \\hat{\\mathbf{x}}{(p)} dp = \\int (\\hat{\\mathbf{x}}{(p)} + \\cos{(p)}) dp and \\hat{\\mathbf{x}}{(p)} = \\operatorname{y^{\\prime}}{(p)} and \\int 2 \\operatorname{y^{\\prime}}{(p)} dp = \\int (\\operatorname{y^{\\prime}}{(p)} + \\cos{(p)}) dp", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), Function('y^{\\\\prime}')(Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Add(Function('y^{\\\\prime}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\tilde{g})} = \\log{(\\tilde{g})}, then obtain \\mathbf{S}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\mathbf{S}{(\\tilde{g})} = \\log{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\mathbf{S}{(\\tilde{g})}", "derivation": "\\mathbf{S}{(\\tilde{g})} = \\log{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\mathbf{S}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} and \\mathbf{S}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} = \\log{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} and \\mathbf{S}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\mathbf{S}{(\\tilde{g})} = \\log{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\mathbf{S}{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\tilde{g}', commutative=True)), Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(Z,\\tilde{g}^*)} = \\cos{(\\frac{\\tilde{g}^*}{Z})} and \\bar{\\h}{(Z,\\tilde{g}^*)} = G{(Z,\\tilde{g}^*)} - \\frac{1}{Z}, then obtain \\bar{\\h}^{Z}{(Z,\\tilde{g}^*)} = (\\cos{(\\frac{\\tilde{g}^*}{Z})} - \\frac{1}{Z})^{Z}", "derivation": "G{(Z,\\tilde{g}^*)} = \\cos{(\\frac{\\tilde{g}^*}{Z})} and G{(Z,\\tilde{g}^*)} - \\frac{1}{Z} = \\cos{(\\frac{\\tilde{g}^*}{Z})} - \\frac{1}{Z} and \\bar{\\h}{(Z,\\tilde{g}^*)} = G{(Z,\\tilde{g}^*)} - \\frac{1}{Z} and \\bar{\\h}{(Z,\\tilde{g}^*)} = \\cos{(\\frac{\\tilde{g}^*}{Z})} - \\frac{1}{Z} and \\bar{\\h}^{Z}{(Z,\\tilde{g}^*)} = (\\cos{(\\frac{\\tilde{g}^*}{Z})} - \\frac{1}{Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 1, "Pow(Symbol('Z', commutative=True), Integer(-1))"], "Equality(Add(Function('G')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))), Add(cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Function('G')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('Z', commutative=True)), Pow(Add(cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(E_{n},y)} = E_{n} y, then obtain E_{n} y + \\iint 0 dE_{n} dE_{n} = E_{n} y + \\iint (E_{n} y - \\mathbf{r}{(E_{n},y)}) dE_{n} dE_{n}", "derivation": "\\mathbf{r}{(E_{n},y)} = E_{n} y and 0 = E_{n} y - \\mathbf{r}{(E_{n},y)} and \\int 0 dE_{n} = \\int (E_{n} y - \\mathbf{r}{(E_{n},y)}) dE_{n} and \\iint 0 dE_{n} dE_{n} = \\iint (E_{n} y - \\mathbf{r}{(E_{n},y)}) dE_{n} dE_{n} and E_{n} y + \\iint 0 dE_{n} dE_{n} = E_{n} y + \\iint (E_{n} y - \\mathbf{r}{(E_{n},y)}) dE_{n} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True)))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["add", 4, "Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Integral(Integer(0), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Integral(Add(Mul(Symbol('E_n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('E_n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given U{(x,\\varepsilon)} = \\frac{\\varepsilon}{x}, then derive \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon = \\frac{\\partial}{\\partial \\varepsilon} (\\varepsilon + i), then obtain e^{\\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon} = e^{\\frac{\\partial}{\\partial \\varepsilon} (\\varepsilon + i)}", "derivation": "U{(x,\\varepsilon)} = \\frac{\\varepsilon}{x} and \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} = 1 and \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon = \\int 1 d\\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon = \\frac{d}{d \\varepsilon} \\int 1 d\\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon = \\frac{\\partial}{\\partial \\varepsilon} (\\varepsilon + i) and e^{\\frac{\\partial}{\\partial \\varepsilon} \\int \\frac{x U{(x,\\varepsilon)}}{\\varepsilon} d\\varepsilon} = e^{\\frac{\\partial}{\\partial \\varepsilon} (\\varepsilon + i)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["exp", 5], "Equality(exp(Derivative(Integral(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('U')(Symbol('x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), exp(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(\\tilde{g}^*)} = e^{\\tilde{g}^*}, then obtain - \\hat{H}{(\\tilde{g}^*)} + \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* \\hat{H}{(\\tilde{g}^*)} = - \\hat{H}{(\\tilde{g}^*)} + \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* e^{\\tilde{g}^*}", "derivation": "\\hat{H}{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and \\tilde{g}^* \\hat{H}{(\\tilde{g}^*)} = \\tilde{g}^* e^{\\tilde{g}^*} and \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* \\hat{H}{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* e^{\\tilde{g}^*} and - \\hat{H}{(\\tilde{g}^*)} + \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* \\hat{H}{(\\tilde{g}^*)} = - \\hat{H}{(\\tilde{g}^*)} + \\frac{d}{d \\tilde{g}^*} \\tilde{g}^* e^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["minus", 3, "Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))), Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\tilde{g}^*', commutative=True))), Derivative(Mul(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(V,\\delta)} = \\int \\frac{V}{\\delta} d\\delta, then obtain ((\\dot{y}^{V}{(V,\\delta)})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV) \\iint \\frac{V}{\\delta} d\\delta d\\delta = (((\\int \\frac{V}{\\delta} d\\delta)^{V})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV) \\iint \\frac{V}{\\delta} d\\delta d\\delta", "derivation": "\\dot{y}{(V,\\delta)} = \\int \\frac{V}{\\delta} d\\delta and \\dot{y}^{V}{(V,\\delta)} = (\\int \\frac{V}{\\delta} d\\delta)^{V} and (\\dot{y}^{V}{(V,\\delta)})^{V} = ((\\int \\frac{V}{\\delta} d\\delta)^{V})^{V} and (\\dot{y}^{V}{(V,\\delta)})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV = ((\\int \\frac{V}{\\delta} d\\delta)^{V})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV and ((\\dot{y}^{V}{(V,\\delta)})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV) \\iint \\frac{V}{\\delta} d\\delta d\\delta = (((\\int \\frac{V}{\\delta} d\\delta)^{V})^{V} + \\iiint \\frac{V}{\\delta} d\\delta d\\delta dV) \\iint \\frac{V}{\\delta} d\\delta d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('\\\\delta', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('V', commutative=True)), Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('V', commutative=True)))"], [["power", 2, "Symbol('V', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["add", 3, "Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True)))"], "Equality(Add(Pow(Pow(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Pow(Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["times", 4, "Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Add(Pow(Pow(Function('\\\\dot{y}')(Symbol('V', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Pow(Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(x)} = \\sin{(x)}, then obtain \\frac{d}{d x} (\\operatorname{t_{1}}{(x)} + 3 \\sin{(x)}) = \\frac{d}{d x} 4 \\sin{(x)}", "derivation": "\\operatorname{t_{1}}{(x)} = \\sin{(x)} and \\operatorname{t_{1}}{(x)} + \\sin{(x)} = 2 \\sin{(x)} and \\operatorname{t_{1}}{(x)} + 3 \\sin{(x)} = 4 \\sin{(x)} and \\frac{d}{d x} (\\operatorname{t_{1}}{(x)} + 3 \\sin{(x)}) = \\frac{d}{d x} 4 \\sin{(x)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["add", 1, "sin(Symbol('x', commutative=True))"], "Equality(Add(Function('t_1')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(2), sin(Symbol('x', commutative=True))))"], [["add", 2, "Mul(Integer(2), sin(Symbol('x', commutative=True)))"], "Equality(Add(Function('t_1')(Symbol('x', commutative=True)), Mul(Integer(3), sin(Symbol('x', commutative=True)))), Mul(Integer(4), sin(Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Function('t_1')(Symbol('x', commutative=True)), Mul(Integer(3), sin(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Integer(4), sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given p{(\\psi,n_{1})} = \\psi n_{1} and \\operatorname{A_{2}}{(\\psi,n_{1})} = 2 \\psi n_{1}, then obtain (\\psi n_{1} + p{(\\psi,n_{1})})^{\\psi} = (2 p{(\\psi,n_{1})})^{\\psi}", "derivation": "p{(\\psi,n_{1})} = \\psi n_{1} and \\psi n_{1} + p{(\\psi,n_{1})} = 2 \\psi n_{1} and \\operatorname{A_{2}}{(\\psi,n_{1})} = 2 \\psi n_{1} and \\psi n_{1} + p{(\\psi,n_{1})} = \\operatorname{A_{2}}{(\\psi,n_{1})} and \\operatorname{A_{2}}{(\\psi,n_{1})} = 2 p{(\\psi,n_{1})} and \\operatorname{A_{2}}^{\\psi}{(\\psi,n_{1})} = (2 p{(\\psi,n_{1})})^{\\psi} and (\\psi n_{1} + p{(\\psi,n_{1})})^{\\psi} = (2 p{(\\psi,n_{1})})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))), Mul(Integer(2), Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(2), Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))), Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(2), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))))"], [["power", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(2), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(2), Function('p')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(A_{x},W)} = A_{x} W and \\sigma_{p}{(\\omega)} = \\omega, then obtain - A_{x} W + \\sigma_{p}^{\\omega}{(\\omega)} = - A_{x} W + \\omega^{\\omega}", "derivation": "\\mathbf{P}{(A_{x},W)} = A_{x} W and \\sigma_{p}{(\\omega)} = \\omega and \\sigma_{p}^{\\omega}{(\\omega)} = \\omega^{\\omega} and - \\mathbf{P}{(A_{x},W)} + \\sigma_{p}^{\\omega}{(\\omega)} = \\omega^{\\omega} - \\mathbf{P}{(A_{x},W)} and - A_{x} W + \\sigma_{p}^{\\omega}{(\\omega)} = - A_{x} W + \\omega^{\\omega}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["minus", 3, "Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True), Symbol('W', commutative=True))), Pow(Function('\\\\sigma_p')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('A_x', commutative=True), Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given G{(\\phi_2,\\hat{p}_0)} = \\hat{p}_0 + \\phi_2, then derive \\int G{(\\phi_2,\\hat{p}_0)} d\\phi_2 = \\hat{p}_0 \\phi_2 + \\frac{\\phi_2^{2}}{2} + f, then obtain \\int (\\hat{p}_0 + \\phi_2) d\\phi_2 = \\hat{p}_0 \\phi_2 + \\frac{\\phi_2^{2}}{2} + f", "derivation": "G{(\\phi_2,\\hat{p}_0)} = \\hat{p}_0 + \\phi_2 and \\int G{(\\phi_2,\\hat{p}_0)} d\\phi_2 = \\int (\\hat{p}_0 + \\phi_2) d\\phi_2 and \\int G{(\\phi_2,\\hat{p}_0)} d\\phi_2 = \\hat{p}_0 \\phi_2 + \\frac{\\phi_2^{2}}{2} + f and \\int (\\hat{p}_0 + \\phi_2) d\\phi_2 = \\hat{p}_0 \\phi_2 + \\frac{\\phi_2^{2}}{2} + f", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\mu{(T,W)} = \\log{(T)}^{W} and \\varepsilon_{0}{(T,W)} = \\log{(T)}^{W}, then obtain (\\varepsilon_{0}{(T,W)} - \\log{(T)})^{W} = (- \\log{(T)} + \\log{(T)}^{W})^{W}", "derivation": "\\mu{(T,W)} = \\log{(T)}^{W} and \\mu{(T,W)} - \\log{(T)} = - \\log{(T)} + \\log{(T)}^{W} and \\varepsilon_{0}{(T,W)} = \\log{(T)}^{W} and \\mu{(T,W)} - \\log{(T)} = \\varepsilon_{0}{(T,W)} - \\log{(T)} and (\\mu{(T,W)} - \\log{(T)})^{W} = (- \\log{(T)} + \\log{(T)}^{W})^{W} and (\\varepsilon_{0}{(T,W)} - \\log{(T)})^{W} = (- \\log{(T)} + \\log{(T)}^{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('W', commutative=True)))"], [["minus", 1, "log(Symbol('T', commutative=True))"], "Equality(Add(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Add(Function('\\\\varepsilon_0')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} = \\frac{x^\\prime}{\\psi}, then obtain - \\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)}) d\\psi = - \\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\frac{x^\\prime}{\\psi}) d\\psi", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} = \\frac{x^\\prime}{\\psi} and \\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} = \\psi + \\frac{x^\\prime}{\\psi} and \\int (\\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)}) d\\psi = \\int (\\psi + \\frac{x^\\prime}{\\psi}) d\\psi and \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)}) d\\psi = \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\frac{x^\\prime}{\\psi}) d\\psi and - \\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)}) d\\psi = - \\psi + \\operatorname{V_{\\mathbf{B}}}{(\\psi,x^\\prime)} + \\int (\\psi + \\frac{x^\\prime}{\\psi}) d\\psi", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Symbol('\\\\psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Symbol('\\\\psi', commutative=True), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Symbol('\\\\psi', commutative=True), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["add", 3, "Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Add(Symbol('\\\\psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Add(Symbol('\\\\psi', commutative=True), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["minus", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Add(Symbol('\\\\psi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Add(Symbol('\\\\psi', commutative=True), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(J,n)} = J - n, then obtain \\log{(- \\eta + i)} = \\log{(- \\eta + i)} + \\int (J - n) dn - \\int \\tilde{g}{(J,n)} dn", "derivation": "\\tilde{g}{(J,n)} = J - n and \\int \\tilde{g}{(J,n)} dn = \\int (J - n) dn and 0 = \\int (J - n) dn - \\int \\tilde{g}{(J,n)} dn and \\log{(- \\eta + i)} = \\log{(- \\eta + i)} + \\int (J - n) dn - \\int \\tilde{g}{(J,n)} dn", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('n', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))))"], [["add", 3, "log(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('i', commutative=True)))"], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('i', commutative=True))), Add(log(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('i', commutative=True))), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(F_{c})} = \\cos{(\\cos{(F_{c})})}, then obtain \\frac{\\tilde{\\infty} (\\operatorname{F_{N}}{(F_{c})} - \\cos{(\\cos{(F_{c})})})}{\\cos{(F_{c})}} = 0", "derivation": "\\operatorname{F_{N}}{(F_{c})} = \\cos{(\\cos{(F_{c})})} and \\operatorname{F_{N}}{(F_{c})} - \\cos{(\\cos{(F_{c})})} = 0 and \\frac{\\operatorname{F_{N}}{(F_{c})} - \\cos{(\\cos{(F_{c})})}}{\\cos{(F_{c})}} = 0 and \\frac{\\operatorname{F_{N}}{(F_{c})} - \\cos{(\\cos{(F_{c})})}}{\\cos{(F_{c})} \\frac{d}{d F_{c}} 0} = 0 and \\frac{\\tilde{\\infty} (\\operatorname{F_{N}}{(F_{c})} - \\cos{(\\cos{(F_{c})})})}{\\cos{(F_{c})}} = 0", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('F_c', commutative=True)), cos(cos(Symbol('F_c', commutative=True))))"], [["minus", 1, "cos(cos(Symbol('F_c', commutative=True)))"], "Equality(Add(Function('F_N')(Symbol('F_c', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('F_c', commutative=True))))), Integer(0))"], [["divide", 2, "cos(Symbol('F_c', commutative=True))"], "Equality(Mul(Add(Function('F_N')(Symbol('F_c', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('F_c', commutative=True))))), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 3, "Derivative(Integer(0), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('F_N')(Symbol('F_c', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('F_c', commutative=True))))), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1)), Pow(Derivative(Integer(0), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["evaluate_derivatives", 4], "Equality(Mul(zoo, Add(Function('F_N')(Symbol('F_c', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('F_c', commutative=True))))), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\delta{(S)} = e^{S}, then obtain (\\frac{d}{d S} \\int \\delta{(S)} dS)^{S} = (\\frac{\\partial}{\\partial S} (f + e^{S}))^{S}", "derivation": "\\delta{(S)} = e^{S} and \\int \\delta{(S)} dS = \\int e^{S} dS and \\frac{d}{d S} \\int \\delta{(S)} dS = \\frac{d}{d S} \\int e^{S} dS and (\\frac{d}{d S} \\int \\delta{(S)} dS)^{S} = (\\frac{d}{d S} \\int e^{S} dS)^{S} and (\\frac{d}{d S} \\int \\delta{(S)} dS)^{S} = (\\frac{\\partial}{\\partial S} (f + e^{S}))^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\delta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\delta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Integral(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('\\\\delta')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Add(Symbol('f', commutative=True), exp(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\phi,\\mu_0)} = \\mu_0 \\phi, then derive \\frac{\\partial}{\\partial \\mu_0} \\hat{p}_0{(\\phi,\\mu_0)} = \\phi, then obtain - \\phi + \\hat{p}_0{(\\phi,\\mu_0)} = \\mu_0 \\phi - \\phi", "derivation": "\\hat{p}_0{(\\phi,\\mu_0)} = \\mu_0 \\phi and \\frac{\\partial}{\\partial \\mu_0} \\hat{p}_0{(\\phi,\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\phi and \\hat{p}_0{(\\phi,\\mu_0)} - \\frac{\\partial}{\\partial \\mu_0} \\hat{p}_0{(\\phi,\\mu_0)} = \\mu_0 \\phi - \\frac{\\partial}{\\partial \\mu_0} \\hat{p}_0{(\\phi,\\mu_0)} and \\frac{\\partial}{\\partial \\mu_0} \\hat{p}_0{(\\phi,\\mu_0)} = \\phi and - \\phi + \\hat{p}_0{(\\phi,\\mu_0)} = \\mu_0 \\phi - \\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given i{(\\theta_1,\\dot{y})} = \\frac{\\partial}{\\partial \\theta_1} (\\dot{y} - \\theta_1), then derive i^{\\dot{y}}{(\\theta_1,\\dot{y})} = (-1)^{\\dot{y}}, then obtain \\frac{\\partial}{\\partial \\dot{y}} i^{\\dot{y}}{(\\theta_1,\\dot{y})} = \\frac{d}{d \\dot{y}} (-1)^{\\dot{y}}", "derivation": "i{(\\theta_1,\\dot{y})} = \\frac{\\partial}{\\partial \\theta_1} (\\dot{y} - \\theta_1) and i^{\\dot{y}}{(\\theta_1,\\dot{y})} = (\\frac{\\partial}{\\partial \\theta_1} (\\dot{y} - \\theta_1))^{\\dot{y}} and \\frac{\\partial}{\\partial \\dot{y}} i^{\\dot{y}}{(\\theta_1,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} (\\frac{\\partial}{\\partial \\theta_1} (\\dot{y} - \\theta_1))^{\\dot{y}} and i^{\\dot{y}}{(\\theta_1,\\dot{y})} = (-1)^{\\dot{y}} and \\frac{d}{d \\dot{y}} (-1)^{\\dot{y}} = \\frac{\\partial}{\\partial \\dot{y}} (\\frac{\\partial}{\\partial \\theta_1} (\\dot{y} - \\theta_1))^{\\dot{y}} and \\frac{\\partial}{\\partial \\dot{y}} i^{\\dot{y}}{(\\theta_1,\\dot{y})} = \\frac{d}{d \\dot{y}} (-1)^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Pow(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Pow(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Pow(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\ddot{x}{(A_{x},C_{1})} = \\cos{(A_{x}^{C_{1}})} and \\hat{p}_0{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} = - A_{x}^{- C_{1}} \\cos{(A_{x}^{C_{1}})}, then obtain \\hat{p}_0{(\\mathbf{P})} + \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} = \\hat{p}_0{(\\mathbf{P})} - A_{x}^{- C_{1}} \\ddot{x}{(A_{x},C_{1})}", "derivation": "\\ddot{x}{(A_{x},C_{1})} = \\cos{(A_{x}^{C_{1}})} and \\hat{p}_0{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} = - A_{x}^{- C_{1}} \\cos{(A_{x}^{C_{1}})} and \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} = - A_{x}^{- C_{1}} \\ddot{x}{(A_{x},C_{1})} and \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} + \\log{(\\mathbf{P})} = \\log{(\\mathbf{P})} - A_{x}^{- C_{1}} \\ddot{x}{(A_{x},C_{1})} and \\hat{p}_0{(\\mathbf{P})} + \\operatorname{x^{{\\}'}}{(A_{x},C_{1})} = \\hat{p}_0{(\\mathbf{P})} - A_{x}^{- C_{1}} \\ddot{x}{(A_{x},C_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), cos(Pow(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), cos(Pow(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('x^\\\\prime')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('\\\\ddot{x}')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))))"], [["add", 4, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Function('x^\\\\prime')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Add(log(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('\\\\ddot{x}')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), Function('x^\\\\prime')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Function('\\\\ddot{x}')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given Q{(T)} = \\int \\log{(T)} dT, then derive Q{(T)} = T \\log{(T)} - T + y, then obtain b + \\frac{y^{2}}{2} + y (2 T \\log{(T)} - 2 T) = Z + y (2 T \\log{(T)} - 2 T)", "derivation": "Q{(T)} = \\int \\log{(T)} dT and Q{(T)} + \\int \\log{(T)} dT = 2 \\int \\log{(T)} dT and Q{(T)} = T \\log{(T)} - T + y and T \\log{(T)} - T + y + \\int \\log{(T)} dT = 2 \\int \\log{(T)} dT and \\int (T \\log{(T)} - T + y + \\int \\log{(T)} dT) dy = \\int 2 \\int \\log{(T)} dT dy and b + \\frac{y^{2}}{2} + y (2 T \\log{(T)} - 2 T) = Z + y (2 T \\log{(T)} - 2 T)", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('T', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 1, "Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Function('Q')(Symbol('T', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Integer(2), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('Q')(Symbol('T', commutative=True)), Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('y', commutative=True), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Integer(2), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('y', commutative=True), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Integer(2), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('b', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Add(Mul(Integer(2), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('T', commutative=True))))), Add(Symbol('Z', commutative=True), Mul(Symbol('y', commutative=True), Add(Mul(Integer(2), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('T', commutative=True))))))"]]}, {"prompt": "Given M{(P_{e})} = e^{P_{e}}, then derive e^{P_{e}} \\int M{(P_{e})} e^{P_{e}} dP_{e} = (a^{\\dagger} + \\frac{e^{2 P_{e}}}{2}) e^{P_{e}}, then obtain e^{P_{e}} \\int e^{2 P_{e}} dP_{e} = (a^{\\dagger} + \\frac{e^{2 P_{e}}}{2}) e^{P_{e}}", "derivation": "M{(P_{e})} = e^{P_{e}} and M{(P_{e})} e^{P_{e}} = e^{2 P_{e}} and \\int M{(P_{e})} e^{P_{e}} dP_{e} = \\int e^{2 P_{e}} dP_{e} and e^{P_{e}} \\int M{(P_{e})} e^{P_{e}} dP_{e} = e^{P_{e}} \\int e^{2 P_{e}} dP_{e} and e^{P_{e}} \\int M{(P_{e})} e^{P_{e}} dP_{e} = (a^{\\dagger} + \\frac{e^{2 P_{e}}}{2}) e^{P_{e}} and e^{P_{e}} \\int e^{2 P_{e}} dP_{e} = (a^{\\dagger} + \\frac{e^{2 P_{e}}}{2}) e^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["times", 1, "exp(Symbol('P_e', commutative=True))"], "Equality(Mul(Function('M')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), exp(Mul(Integer(2), Symbol('P_e', commutative=True))))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Mul(Function('M')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["times", 3, "exp(Symbol('P_e', commutative=True))"], "Equality(Mul(exp(Symbol('P_e', commutative=True)), Integral(Mul(Function('M')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Mul(exp(Symbol('P_e', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(exp(Symbol('P_e', commutative=True)), Integral(Mul(Function('M')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('P_e', commutative=True))))), exp(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(exp(Symbol('P_e', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('P_e', commutative=True))))), exp(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given i{(\\psi,\\theta_2)} = \\cos{(\\psi \\theta_2)} and \\mu_{0}{(\\psi,\\theta_2)} = \\frac{i{(\\psi,\\theta_2)}}{\\psi \\theta_2}, then obtain - \\mu_{0}^{\\psi}{(\\psi,\\theta_2)} = - (\\frac{\\cos{(\\psi \\theta_2)}}{\\psi \\theta_2})^{\\psi}", "derivation": "i{(\\psi,\\theta_2)} = \\cos{(\\psi \\theta_2)} and \\frac{i{(\\psi,\\theta_2)}}{\\psi \\theta_2} = \\frac{\\cos{(\\psi \\theta_2)}}{\\psi \\theta_2} and \\mu_{0}{(\\psi,\\theta_2)} = \\frac{i{(\\psi,\\theta_2)}}{\\psi \\theta_2} and \\mu_{0}{(\\psi,\\theta_2)} = \\frac{\\cos{(\\psi \\theta_2)}}{\\psi \\theta_2} and \\mu_{0}^{\\psi}{(\\psi,\\theta_2)} = (\\frac{\\cos{(\\psi \\theta_2)}}{\\psi \\theta_2})^{\\psi} and - \\mu_{0}^{\\psi}{(\\psi,\\theta_2)} = - (\\frac{\\cos{(\\psi \\theta_2)}}{\\psi \\theta_2})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\psi', commutative=True)))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given b{(x)} = \\log{(\\sin{(x)})} and p{(x)} = \\frac{d}{d x} b{(x)}, then obtain p{(x)} = \\frac{\\cos{(x)}}{\\sin{(x)}}", "derivation": "b{(x)} = \\log{(\\sin{(x)})} and \\frac{d}{d x} b{(x)} = \\frac{d}{d x} \\log{(\\sin{(x)})} and p{(x)} = \\frac{d}{d x} b{(x)} and p{(x)} = \\frac{d}{d x} \\log{(\\sin{(x)})} and p{(x)} = \\frac{\\cos{(x)}}{\\sin{(x)}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('x', commutative=True)), log(sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(log(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('p')(Symbol('x', commutative=True)), Derivative(Function('b')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('p')(Symbol('x', commutative=True)), Derivative(log(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('p')(Symbol('x', commutative=True)), Mul(Pow(sin(Symbol('x', commutative=True)), Integer(-1)), cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\Omega)} = e^{\\Omega}, then obtain (\\mathbf{F}^{\\Omega}{(\\Omega)})^{- \\Omega} ((\\mathbf{F}^{\\Omega}{(\\Omega)})^{\\Omega})^{\\Omega} = (\\mathbf{F}^{\\Omega}{(\\Omega)})^{- \\Omega} (((e^{\\Omega})^{\\Omega})^{\\Omega})^{\\Omega}", "derivation": "\\mathbf{F}{(\\Omega)} = e^{\\Omega} and \\mathbf{F}^{\\Omega}{(\\Omega)} = (e^{\\Omega})^{\\Omega} and (\\mathbf{F}^{\\Omega}{(\\Omega)})^{\\Omega} = ((e^{\\Omega})^{\\Omega})^{\\Omega} and ((\\mathbf{F}^{\\Omega}{(\\Omega)})^{\\Omega})^{\\Omega} = (((e^{\\Omega})^{\\Omega})^{\\Omega})^{\\Omega} and (\\mathbf{F}^{\\Omega}{(\\Omega)})^{- \\Omega} ((\\mathbf{F}^{\\Omega}{(\\Omega)})^{\\Omega})^{\\Omega} = (\\mathbf{F}^{\\Omega}{(\\Omega)})^{- \\Omega} (((e^{\\Omega})^{\\Omega})^{\\Omega})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["divide", 4, "Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Pow(Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Pow(Pow(Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(i)} = \\sin{(i)}, then obtain \\int \\tilde{g}^*{(i)} di + \\int \\frac{d}{d i} \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} di = \\int \\sin{(i)} di + \\int \\frac{d}{d i} \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} di", "derivation": "\\tilde{g}^*{(i)} = \\sin{(i)} and \\tilde{g}^*^{i}{(i)} = \\sin^{i}{(i)} and \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} = 1 and \\frac{d}{d i} \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} = \\frac{d}{d i} 1 and \\tilde{g}^*{(i)} + \\frac{d}{d i} 1 = \\sin{(i)} + \\frac{d}{d i} 1 and \\int (\\tilde{g}^*{(i)} + \\frac{d}{d i} 1) di = \\int (\\sin{(i)} + \\frac{d}{d i} 1) di and \\int \\tilde{g}^*{(i)} di + \\int \\frac{d}{d i} 1 di = \\int \\sin{(i)} di + \\int \\frac{d}{d i} 1 di and \\int \\tilde{g}^*{(i)} di + \\int \\frac{d}{d i} \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} di = \\int \\sin{(i)} di + \\int \\frac{d}{d i} \\tilde{g}^*^{i}{(i)} \\sin^{- i}{(i)} di", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["divide", 2, "Pow(sin(Symbol('i', commutative=True)), Symbol('i', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))), Integer(1))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(sin(Symbol('i', commutative=True)), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(Add(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True))), Integral(Add(sin(Symbol('i', commutative=True)), Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True))))"], [["expand", 6], "Equality(Add(Integral(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))), Add(Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Integral(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))), Add(Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\nabla)} = \\sin{(\\nabla)}, then derive \\frac{d}{d \\nabla} \\delta{(\\nabla)} = \\cos{(\\nabla)}, then obtain \\frac{d^{2}}{d \\nabla^{2}} \\delta{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)}", "derivation": "\\delta{(\\nabla)} = \\sin{(\\nabla)} and \\frac{d}{d \\nabla} \\delta{(\\nabla)} = \\frac{d}{d \\nabla} \\sin{(\\nabla)} and \\frac{d}{d \\nabla} \\delta{(\\nabla)} = \\cos{(\\nabla)} and \\frac{d^{2}}{d \\nabla^{2}} \\delta{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), cos(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = I^{\\mathbf{P}}, then derive \\frac{\\partial}{\\partial \\mathbf{P}} \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = I^{\\mathbf{P}} \\log{(I)}, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} \\log{(I)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = I^{\\mathbf{P}} and \\frac{\\partial}{\\partial \\mathbf{P}} \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = \\frac{\\partial}{\\partial \\mathbf{P}} I^{\\mathbf{P}} and \\frac{\\partial}{\\partial \\mathbf{P}} \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = I^{\\mathbf{P}} \\log{(I)} and \\frac{\\partial}{\\partial \\mathbf{P}} \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} = \\operatorname{L_{\\varepsilon}}{(I,\\mathbf{P})} \\log{(I)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Mul(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Mul(Function('L_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(c_{0})} = c_{0}, then obtain \\frac{\\partial}{\\partial v_{t}} \\frac{\\bar{\\h}^{c_{0}}{(c_{0})}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}} = \\frac{\\partial}{\\partial v_{t}} \\frac{c_{0}^{c_{0}}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}}", "derivation": "\\bar{\\h}{(c_{0})} = c_{0} and \\bar{\\h}^{c_{0}}{(c_{0})} = c_{0}^{c_{0}} and \\frac{\\bar{\\h}^{c_{0}}{(c_{0})}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}} = \\frac{c_{0}^{c_{0}}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}} and \\frac{\\partial}{\\partial v_{t}} \\frac{\\bar{\\h}^{c_{0}}{(c_{0})}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}} = \\frac{\\partial}{\\partial v_{t}} \\frac{c_{0}^{c_{0}}}{\\frac{\\partial}{\\partial v_{t}} v_{t}^{c_{0}}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)))"], [["divide", 2, "Derivative(Pow(Symbol('v_t', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('\\\\hbar')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Derivative(Pow(Symbol('v_t', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)), Pow(Derivative(Pow(Symbol('v_t', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\hbar')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Derivative(Pow(Symbol('v_t', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)), Pow(Derivative(Pow(Symbol('v_t', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(\\theta_2,A_{z})} = \\frac{\\theta_2}{A_{z}}, then obtain \\frac{\\partial}{\\partial \\theta_2} \\frac{A_{z} \\hat{X}{(\\theta_2,A_{z})} - 1}{A_{z}} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2 - 1}{A_{z}}", "derivation": "\\hat{X}{(\\theta_2,A_{z})} = \\frac{\\theta_2}{A_{z}} and A_{z} \\hat{X}{(\\theta_2,A_{z})} = \\theta_2 and A_{z} \\hat{X}{(\\theta_2,A_{z})} - 1 = \\theta_2 - 1 and \\frac{A_{z} \\hat{X}{(\\theta_2,A_{z})} - 1}{A_{z}} = \\frac{\\theta_2 - 1}{A_{z}} and \\frac{\\partial}{\\partial \\theta_2} \\frac{A_{z} \\hat{X}{(\\theta_2,A_{z})} - 1}{A_{z}} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2 - 1}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Integer(-1)))"], [["times", 3, "Pow(Symbol('A_z', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Integer(-1))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Mul(Symbol('A_z', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\phi_2)} = \\log{(\\phi_2)}, then derive \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 = \\mathbf{P} + \\phi_2, then obtain \\int \\mathbf{P} \\int 1 d\\phi_2 d\\phi_2 = \\int \\mathbf{P} \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 d\\phi_2", "derivation": "\\operatorname{E_{n}}{(\\phi_2)} = \\log{(\\phi_2)} and \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} = 1 and \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 = \\int 1 d\\phi_2 and \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 = \\mathbf{P} + \\phi_2 and \\int 1 d\\phi_2 = \\mathbf{P} + \\phi_2 and \\mathbf{P} \\int 1 d\\phi_2 = \\mathbf{P} (\\mathbf{P} + \\phi_2) and \\mathbf{P} \\int 1 d\\phi_2 = \\mathbf{P} \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 and \\int \\mathbf{P} \\int 1 d\\phi_2 d\\phi_2 = \\int \\mathbf{P} \\int \\frac{\\operatorname{E_{n}}{(\\phi_2)}}{\\log{(\\phi_2)}} d\\phi_2 d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["times", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Integral(Mul(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["integrate", 7, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Integral(Mul(Function('E_n')(Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given n{(J)} = \\cos{(\\cos{(J)})}, then derive \\mathbb{I} + n{(J)} = \\nabla + \\cos{(\\cos{(J)})}, then obtain \\mathbb{I} + n{(J)} = \\nabla + n{(J)}", "derivation": "n{(J)} = \\cos{(\\cos{(J)})} and \\frac{d}{d J} n{(J)} = \\frac{d}{d J} \\cos{(\\cos{(J)})} and \\int \\frac{d}{d J} n{(J)} dJ = \\int \\frac{d}{d J} \\cos{(\\cos{(J)})} dJ and \\mathbb{I} + n{(J)} = \\nabla + \\cos{(\\cos{(J)})} and \\mathbb{I} + n{(J)} = \\nabla + n{(J)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('J', commutative=True)), cos(cos(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Derivative(cos(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('n')(Symbol('J', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), cos(cos(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Function('n')(Symbol('J', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Function('n')(Symbol('J', commutative=True))))"]]}, {"prompt": "Given B{(C_{1})} = e^{C_{1}}, then derive \\int B{(C_{1})} dC_{1} = \\theta_1 + e^{C_{1}}, then obtain \\frac{\\partial}{\\partial C_{1}} (\\theta_1 + e^{C_{1}}) = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1}", "derivation": "B{(C_{1})} = e^{C_{1}} and \\int B{(C_{1})} dC_{1} = \\int e^{C_{1}} dC_{1} and \\frac{d}{d C_{1}} \\int B{(C_{1})} dC_{1} = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1} and \\int B{(C_{1})} dC_{1} = \\theta_1 + e^{C_{1}} and \\frac{\\partial}{\\partial C_{1}} (\\theta_1 + e^{C_{1}}) = \\frac{d}{d C_{1}} \\int e^{C_{1}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('B')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Integral(Function('B')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{E})} = \\mathbf{E}, then obtain \\frac{\\varphi^{\\mathbf{E}} (- \\operatorname{E_{x}}{(\\mathbf{E})})^{\\mathbf{E}}}{\\theta_{2}{(\\mathbf{E},\\varphi)} \\int \\varphi^{\\mathbf{E}} d\\varphi} = \\frac{\\varphi^{\\mathbf{E}} (- \\mathbf{E})^{\\mathbf{E}}}{\\theta_{2}{(\\mathbf{E},\\varphi)} \\int \\varphi^{\\mathbf{E}} d\\varphi}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{E})} = \\mathbf{E} and - \\operatorname{E_{x}}{(\\mathbf{E})} = - \\mathbf{E} and (- \\operatorname{E_{x}}{(\\mathbf{E})})^{\\mathbf{E}} = (- \\mathbf{E})^{\\mathbf{E}} and \\frac{\\varphi^{\\mathbf{E}} (- \\operatorname{E_{x}}{(\\mathbf{E})})^{\\mathbf{E}}}{\\theta_{2}{(\\mathbf{E},\\varphi)} \\int \\varphi^{\\mathbf{E}} d\\varphi} = \\frac{\\varphi^{\\mathbf{E}} (- \\mathbf{E})^{\\mathbf{E}}}{\\theta_{2}{(\\mathbf{E},\\varphi)} \\int \\varphi^{\\mathbf{E}} d\\varphi}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Integral(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Integral(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\hat{p}_0{(\\mathbf{g})} = \\Psi_{nl}^{2}{(\\mathbf{g})} \\cos{(\\mathbf{g})}, then obtain \\hat{p}_0{(\\mathbf{g})} = \\Psi_{nl}{(\\mathbf{g})} \\cos^{2}{(\\mathbf{g})}", "derivation": "\\Psi_{nl}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\Psi_{nl}^{2}{(\\mathbf{g})} = \\Psi_{nl}{(\\mathbf{g})} \\cos{(\\mathbf{g})} and \\Psi_{nl}^{3}{(\\mathbf{g})} = \\Psi_{nl}^{2}{(\\mathbf{g})} \\cos{(\\mathbf{g})} and \\Psi_{nl}^{3}{(\\mathbf{g})} = \\Psi_{nl}{(\\mathbf{g})} \\cos^{2}{(\\mathbf{g})} and \\Psi_{nl}^{2}{(\\mathbf{g})} \\cos{(\\mathbf{g})} = \\Psi_{nl}{(\\mathbf{g})} \\cos^{2}{(\\mathbf{g})} and \\hat{p}_0{(\\mathbf{g})} = \\Psi_{nl}^{2}{(\\mathbf{g})} \\cos{(\\mathbf{g})} and \\hat{p}_0{(\\mathbf{g})} = \\Psi_{nl}{(\\mathbf{g})} \\cos^{2}{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 1, "Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(3)), Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\phi_{2}{(\\mathbf{P})} = - \\mathbf{J}{(\\mathbf{P})}, then obtain - \\phi_{2}{(\\mathbf{P})} - \\cos{(\\mathbf{P})} = 0", "derivation": "\\mathbf{J}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and - \\mathbf{J}{(\\mathbf{P})} = - \\cos{(\\mathbf{P})} and \\phi_{2}{(\\mathbf{P})} = - \\mathbf{J}{(\\mathbf{P})} and \\phi_{2}{(\\mathbf{P})} = - \\cos{(\\mathbf{P})} and - \\phi_{2}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\mathbf{J}{(\\mathbf{P})} - \\phi_{2}{(\\mathbf{P})} - \\cos{(\\mathbf{P})} = \\mathbf{J}{(\\mathbf{P})} and - \\phi_{2}{(\\mathbf{P})} - \\cos{(\\mathbf{P})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True))), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 5, "Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{P}', commutative=True)))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 6, "Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{P}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v)} = e^{v}, then obtain \\int (- v \\operatorname{t_{2}}{(v)} - v e^{v} + (v \\operatorname{t_{2}}{(v)})^{v}) dv = \\int (- v \\operatorname{t_{2}}{(v)} - v e^{v} + (v e^{v})^{v}) dv", "derivation": "\\operatorname{t_{2}}{(v)} = e^{v} and v \\operatorname{t_{2}}{(v)} = v e^{v} and (v \\operatorname{t_{2}}{(v)})^{v} = (v e^{v})^{v} and 2 v \\operatorname{t_{2}}{(v)} = v \\operatorname{t_{2}}{(v)} + v e^{v} and - 2 v \\operatorname{t_{2}}{(v)} + (v \\operatorname{t_{2}}{(v)})^{v} = - 2 v \\operatorname{t_{2}}{(v)} + (v e^{v})^{v} and - v \\operatorname{t_{2}}{(v)} - v e^{v} + (v \\operatorname{t_{2}}{(v)})^{v} = - v \\operatorname{t_{2}}{(v)} - v e^{v} + (v e^{v})^{v} and \\int (- v \\operatorname{t_{2}}{(v)} - v e^{v} + (v \\operatorname{t_{2}}{(v)})^{v}) dv = \\int (- v \\operatorname{t_{2}}{(v)} - v e^{v} + (v e^{v})^{v}) dv", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["times", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["add", 2, "Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Add(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["integrate", 6, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('t_2')(Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Pow(Mul(Symbol('v', commutative=True), exp(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(b,A)} = A b, then obtain \\cos{(A b \\operatorname{c_{0}}{(b,A)})} \\frac{\\partial}{\\partial A} (A^{2} b^{2} + b) = \\cos{(A^{2} b^{2})} \\frac{\\partial}{\\partial A} (A^{2} b^{2} + b)", "derivation": "\\operatorname{c_{0}}{(b,A)} = A b and A b \\operatorname{c_{0}}{(b,A)} = A^{2} b^{2} and A b \\operatorname{c_{0}}{(b,A)} + b = A^{2} b^{2} + b and \\frac{\\partial}{\\partial A} (A b \\operatorname{c_{0}}{(b,A)} + b) = \\frac{\\partial}{\\partial A} (A^{2} b^{2} + b) and \\cos{(A b \\operatorname{c_{0}}{(b,A)})} = \\cos{(A^{2} b^{2})} and \\cos{(A b \\operatorname{c_{0}}{(b,A)})} \\frac{\\partial}{\\partial A} (A b \\operatorname{c_{0}}{(b,A)} + b) = \\cos{(A^{2} b^{2})} \\frac{\\partial}{\\partial A} (A b \\operatorname{c_{0}}{(b,A)} + b) and \\cos{(A b \\operatorname{c_{0}}{(b,A)})} \\frac{\\partial}{\\partial A} (A^{2} b^{2} + b) = \\cos{(A^{2} b^{2})} \\frac{\\partial}{\\partial A} (A^{2} b^{2} + b)", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Mul(Symbol('A', commutative=True), Symbol('b', commutative=True))"], "Equality(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2))))"], [["add", 2, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Symbol('b', commutative=True)), Add(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True)))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True)))), cos(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2)))))"], [["times", 5, "Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Mul(cos(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True)))), Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(cos(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2)))), Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(cos(Mul(Symbol('A', commutative=True), Symbol('b', commutative=True), Function('c_0')(Symbol('b', commutative=True), Symbol('A', commutative=True)))), Derivative(Add(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(cos(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2)))), Derivative(Add(Mul(Pow(Symbol('A', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('b', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(a^{\\dagger},f^{\\prime},\\dot{y})} = f^{\\prime} (\\dot{y} + a^{\\dagger}), then obtain W{(a^{\\dagger},f^{\\prime},\\dot{y})} - \\frac{\\partial}{\\partial \\dot{y}} f^{\\prime} (\\dot{y} + a^{\\dagger}) = f^{\\prime} (\\dot{y} + a^{\\dagger}) - \\frac{\\partial}{\\partial \\dot{y}} f^{\\prime} (\\dot{y} + a^{\\dagger})", "derivation": "W{(a^{\\dagger},f^{\\prime},\\dot{y})} = f^{\\prime} (\\dot{y} + a^{\\dagger}) and \\frac{\\partial}{\\partial \\dot{y}} W{(a^{\\dagger},f^{\\prime},\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} f^{\\prime} (\\dot{y} + a^{\\dagger}) and W{(a^{\\dagger},f^{\\prime},\\dot{y})} - \\frac{\\partial}{\\partial \\dot{y}} W{(a^{\\dagger},f^{\\prime},\\dot{y})} = f^{\\prime} (\\dot{y} + a^{\\dagger}) - \\frac{\\partial}{\\partial \\dot{y}} W{(a^{\\dagger},f^{\\prime},\\dot{y})} and W{(a^{\\dagger},f^{\\prime},\\dot{y})} - \\frac{\\partial}{\\partial \\dot{y}} f^{\\prime} (\\dot{y} + a^{\\dagger}) = f^{\\prime} (\\dot{y} + a^{\\dagger}) - \\frac{\\partial}{\\partial \\dot{y}} f^{\\prime} (\\dot{y} + a^{\\dagger})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))"], "Equality(Add(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\lambda{(x^\\prime,n)} = n x^\\prime and \\dot{x}{(n,\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{n}, then derive \\frac{\\dot{x}{(n,\\varphi^*)}}{\\lambda{(x^\\prime,n)}} = \\frac{(\\varphi^*)^{n} n}{\\varphi^* \\lambda{(x^\\prime,n)}}, then obtain \\frac{\\dot{x}{(n,\\varphi^*)}}{F_{c} \\lambda{(x^\\prime,n)}} = \\frac{(\\varphi^*)^{n} n}{F_{c} \\varphi^* \\lambda{(x^\\prime,n)}}", "derivation": "\\lambda{(x^\\prime,n)} = n x^\\prime and \\dot{x}{(n,\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{n} and \\frac{\\dot{x}{(n,\\varphi^*)}}{n x^\\prime} = \\frac{\\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{n}}{n x^\\prime} and \\frac{\\dot{x}{(n,\\varphi^*)}}{\\lambda{(x^\\prime,n)}} = \\frac{\\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{n}}{\\lambda{(x^\\prime,n)}} and \\frac{\\dot{x}{(n,\\varphi^*)}}{\\lambda{(x^\\prime,n)}} = \\frac{(\\varphi^*)^{n} n}{\\varphi^* \\lambda{(x^\\prime,n)}} and \\frac{\\dot{x}{(n,\\varphi^*)}}{F_{c} \\lambda{(x^\\prime,n)}} = \\frac{(\\varphi^*)^{n} n}{F_{c} \\varphi^* \\lambda{(x^\\prime,n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('n', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('n', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\dot{x}')(Symbol('n', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Function('\\\\dot{x}')(Symbol('n', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True), Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1))))"], [["divide", 5, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('n', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True), Pow(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\varepsilon_0)} = e^{e^{\\varepsilon_0}}, then obtain (- g + 2 \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}} - 1)^{\\varepsilon_0} = (- g + \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}})^{\\varepsilon_0}", "derivation": "\\dot{\\mathbf{r}}{(\\varepsilon_0)} = e^{e^{\\varepsilon_0}} and \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}} = 1 and - g + \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}} = 1 - g and (- g + \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}})^{\\varepsilon_0} = (1 - g)^{\\varepsilon_0} and (- g + 2 \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}} - 1)^{\\varepsilon_0} = (- g + \\dot{\\mathbf{r}}{(\\varepsilon_0)} e^{- e^{\\varepsilon_0}})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))))), Integer(1))"], [["minus", 2, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))), Add(Integer(1), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Symbol('g', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))))), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(u)} = \\log{(u)} and \\mathbf{J}{(u)} = u + \\log{(u)}, then obtain - \\mathbf{M} + \\int (- \\mathbf{M} + \\mathbf{p} + \\mathbf{J}{(u)}) d\\mathbf{M} = - \\mathbf{M} + \\int (- \\mathbf{M} + \\mathbf{p} + u + \\operatorname{f^{*}}{(u)}) d\\mathbf{M}", "derivation": "\\operatorname{f^{*}}{(u)} = \\log{(u)} and u + \\operatorname{f^{*}}{(u)} = u + \\log{(u)} and \\mathbf{J}{(u)} = u + \\log{(u)} and \\mathbf{J}{(u)} = u + \\operatorname{f^{*}}{(u)} and - \\mathbf{M} + \\mathbf{p} + \\mathbf{J}{(u)} = - \\mathbf{M} + \\mathbf{p} + u + \\operatorname{f^{*}}{(u)} and \\int (- \\mathbf{M} + \\mathbf{p} + \\mathbf{J}{(u)}) d\\mathbf{M} = \\int (- \\mathbf{M} + \\mathbf{p} + u + \\operatorname{f^{*}}{(u)}) d\\mathbf{M} and - \\mathbf{M} + \\int (- \\mathbf{M} + \\mathbf{p} + \\mathbf{J}{(u)}) d\\mathbf{M} = - \\mathbf{M} + \\int (- \\mathbf{M} + \\mathbf{p} + u + \\operatorname{f^{*}}{(u)}) d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('f^*')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), log(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), log(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}')(Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Function('f^*')(Symbol('u', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}')(Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True), Function('f^*')(Symbol('u', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}')(Symbol('u', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True), Function('f^*')(Symbol('u', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 6, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}')(Symbol('u', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True), Function('f^*')(Symbol('u', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given z{(C,F_{H})} = \\frac{\\cos{(F_{H})}}{C}, then obtain e^{\\iint (- C + z{(C,F_{H})}) dC dC} = e^{\\iint (- C + \\frac{\\cos{(F_{H})}}{C}) dC dC}", "derivation": "z{(C,F_{H})} = \\frac{\\cos{(F_{H})}}{C} and - C + z{(C,F_{H})} = - C + \\frac{\\cos{(F_{H})}}{C} and \\int (- C + z{(C,F_{H})}) dC = \\int (- C + \\frac{\\cos{(F_{H})}}{C}) dC and \\iint (- C + z{(C,F_{H})}) dC dC = \\iint (- C + \\frac{\\cos{(F_{H})}}{C}) dC dC and e^{\\iint (- C + z{(C,F_{H})}) dC dC} = e^{\\iint (- C + \\frac{\\cos{(F_{H})}}{C}) dC dC}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), cos(Symbol('F_H', commutative=True))))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), cos(Symbol('F_H', commutative=True)))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), cos(Symbol('F_H', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), cos(Symbol('F_H', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), exp(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), cos(Symbol('F_H', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\Omega)} = \\Omega, then derive \\int \\operatorname{A_{1}}{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\mathbf{g}, then obtain \\iint \\Omega d\\Omega d\\mathbf{g} = \\int (\\frac{\\Omega^{2}}{2} + \\mathbf{g}) d\\mathbf{g}", "derivation": "\\operatorname{A_{1}}{(\\Omega)} = \\Omega and \\int \\operatorname{A_{1}}{(\\Omega)} d\\Omega = \\int \\Omega d\\Omega and \\int \\operatorname{A_{1}}{(\\Omega)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\mathbf{g} and \\int \\Omega d\\Omega = \\frac{\\Omega^{2}}{2} + \\mathbf{g} and \\iint \\Omega d\\Omega d\\mathbf{g} = \\int (\\frac{\\Omega^{2}}{2} + \\mathbf{g}) d\\mathbf{g}", "srepr_derivation": [["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\sin{(\\sigma_x)}, then obtain \\int (\\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d}{d \\sigma_x} \\phi{(\\sigma_x)}) d\\mathbf{A} = \\int (\\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d^{2}}{d \\sigma_x^{2}} \\sin{(\\sigma_x)}) d\\mathbf{A}", "derivation": "\\phi{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\sin{(\\sigma_x)} and \\frac{d}{d \\sigma_x} \\phi{(\\sigma_x)} = \\frac{d^{2}}{d \\sigma_x^{2}} \\sin{(\\sigma_x)} and \\sigma_x + \\frac{d}{d \\sigma_x} \\phi{(\\sigma_x)} = \\sigma_x + \\frac{d^{2}}{d \\sigma_x^{2}} \\sin{(\\sigma_x)} and \\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d}{d \\sigma_x} \\phi{(\\sigma_x)} = \\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d^{2}}{d \\sigma_x^{2}} \\sin{(\\sigma_x)} and \\int (\\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d}{d \\sigma_x} \\phi{(\\sigma_x)}) d\\mathbf{A} = \\int (\\sigma_x + \\operatorname{f^{\\prime}}{(V,\\mathbf{A})} + \\frac{d^{2}}{d \\sigma_x^{2}} \\sin{(\\sigma_x)}) d\\mathbf{A}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Derivative(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))))"], [["add", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Derivative(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Add(Symbol('\\\\sigma_x', commutative=True), Derivative(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2)))))"], [["add", 3, "Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Add(Symbol('\\\\sigma_x', commutative=True), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2)))))"], [["integrate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\sigma_x', commutative=True), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Symbol('\\\\sigma_x', commutative=True), Function('f^{\\\\prime}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given b{(c,\\hbar)} = - \\sin{(\\hbar - c)}, then obtain - (\\hbar - c) b^{3}{(c,\\hbar)} \\sin{(\\hbar - c)} + b^{2}{(c,\\hbar)} = (\\hbar - c) b^{4}{(c,\\hbar)} + b^{2}{(c,\\hbar)}", "derivation": "b{(c,\\hbar)} = - \\sin{(\\hbar - c)} and b{(c,\\hbar)} \\sin{(\\hbar - c)} = - \\sin^{2}{(\\hbar - c)} and b{(c,\\hbar)} \\sin{(\\hbar - c)} = - b^{2}{(c,\\hbar)} and - b^{3}{(c,\\hbar)} \\sin{(\\hbar - c)} = b^{4}{(c,\\hbar)} and - (\\hbar - c) b^{3}{(c,\\hbar)} \\sin{(\\hbar - c)} = (\\hbar - c) b^{4}{(c,\\hbar)} and - (\\hbar - c) b^{3}{(c,\\hbar)} \\sin{(\\hbar - c)} + b^{2}{(c,\\hbar)} = (\\hbar - c) b^{4}{(c,\\hbar)} + b^{2}{(c,\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], [["times", 1, "sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], "Equality(Mul(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Mul(Integer(-1), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))))"], [["times", 3, "Mul(Integer(-1), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(-1), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(3)), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(4)))"], [["times", 4, "Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(3)), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Mul(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(4))))"], [["add", 5, "Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(3)), sin(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Add(Mul(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(4))), Pow(Function('b')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))))"]]}, {"prompt": "Given v{(f^{*})} = \\sin{(f^{*})} and \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = v^{f^{*}}{(f^{*})}, then obtain \\frac{d}{d f^{*}} \\sin^{f^{*}}{(f^{*})} = \\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{B}}}{(f^{*})}", "derivation": "v{(f^{*})} = \\sin{(f^{*})} and v^{f^{*}}{(f^{*})} = \\sin^{f^{*}}{(f^{*})} and \\frac{d}{d f^{*}} v^{f^{*}}{(f^{*})} = \\frac{d}{d f^{*}} \\sin^{f^{*}}{(f^{*})} and \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = v^{f^{*}}{(f^{*})} and \\operatorname{V_{\\mathbf{B}}}{(f^{*})} = \\sin^{f^{*}}{(f^{*})} and \\frac{d}{d f^{*}} v^{f^{*}}{(f^{*})} = \\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{B}}}{(f^{*})} and \\frac{d}{d f^{*}} \\sin^{f^{*}}{(f^{*})} = \\frac{d}{d f^{*}} \\operatorname{V_{\\mathbf{B}}}{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('v')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(sin(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["differentiate", 2, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Pow(Function('v')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), Pow(Function('v')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), Pow(sin(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Pow(Function('v')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Pow(sin(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} = \\rho + f_{\\mathbf{v}} and \\dot{\\mathbf{r}}{(f_{\\mathbf{v}},\\rho)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)}, then derive \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} = 1, then obtain \\rho \\dot{\\mathbf{r}}{(f_{\\mathbf{v}},\\rho)} = \\rho", "derivation": "\\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} = \\rho + f_{\\mathbf{v}} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\rho + f_{\\mathbf{v}}) and \\dot{\\mathbf{r}}{(f_{\\mathbf{v}},\\rho)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{F_{g}}{(f_{\\mathbf{v}},\\rho)} = 1 and \\dot{\\mathbf{r}}{(f_{\\mathbf{v}},\\rho)} = 1 and \\rho \\dot{\\mathbf{r}}{(f_{\\mathbf{v}},\\rho)} = \\rho", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Function('F_g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(1))"], [["times", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))"]]}, {"prompt": "Given S{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}}, then obtain \\int \\sin{(S{(a^{\\dagger})} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} da^{\\dagger} = \\int \\sin{(e^{\\sin{(a^{\\dagger})}} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} da^{\\dagger}", "derivation": "S{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}} and S{(a^{\\dagger})} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} = e^{\\sin{(a^{\\dagger})}} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} and S{(a^{\\dagger})} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1 = e^{\\sin{(a^{\\dagger})}} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1 and \\sin{(S{(a^{\\dagger})} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} = \\sin{(e^{\\sin{(a^{\\dagger})}} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} and \\int \\sin{(S{(a^{\\dagger})} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} da^{\\dagger} = \\int \\sin{(e^{\\sin{(a^{\\dagger})}} (e^{\\sin{(a^{\\dagger})}})^{- a^{\\dagger}} + 1)} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('a^{\\\\dagger}', commutative=True)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('S')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Function('S')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1)), Add(Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1)))"], [["sin", 3], "Equality(sin(Add(Mul(Function('S')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1))), sin(Add(Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1))))"], [["integrate", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(sin(Add(Mul(Function('S')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Add(Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Pow(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given c{(W,C)} = e^{\\frac{W}{C}}, then derive \\int c^{C}{(W,C)} dW = \\mathbf{J}_M + (e^{\\frac{W}{C}})^{C}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{C \\int (e^{\\frac{W}{C}})^{C} dW}{W} = \\frac{C}{W}", "derivation": "c{(W,C)} = e^{\\frac{W}{C}} and c^{C}{(W,C)} = (e^{\\frac{W}{C}})^{C} and \\int c^{C}{(W,C)} dW = \\int (e^{\\frac{W}{C}})^{C} dW and \\int c^{C}{(W,C)} dW = \\mathbf{J}_M + (e^{\\frac{W}{C}})^{C} and \\frac{C \\int c^{C}{(W,C)} dW}{W} = \\frac{C (\\mathbf{J}_M + (e^{\\frac{W}{C}})^{C})}{W} and \\frac{C \\int (e^{\\frac{W}{C}})^{C} dW}{W} = \\frac{C (\\mathbf{J}_M + (e^{\\frac{W}{C}})^{C})}{W} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{C \\int (e^{\\frac{W}{C}})^{C} dW}{W} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{C (\\mathbf{J}_M + (e^{\\frac{W}{C}})^{C})}{W} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{C \\int (e^{\\frac{W}{C}})^{C} dW}{W} = \\frac{C}{W}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('W', commutative=True), Symbol('C', commutative=True)), exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('c')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('c')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Pow(Function('c')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))"], "Equality(Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Pow(Function('c')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1)), Integral(Pow(exp(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Symbol('C', commutative=True), Pow(Symbol('W', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{g},H)} = H + \\mathbf{g}, then derive \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{P_{g}}{(\\mathbf{g},H)}}{H + \\mathbf{g}} = \\frac{1}{H + \\mathbf{g}}, then obtain \\frac{1}{H + \\mathbf{g}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} (H + \\mathbf{g})}{H + \\mathbf{g}}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{g},H)} = H + \\mathbf{g} and \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{P_{g}}{(\\mathbf{g},H)} = \\frac{\\partial}{\\partial \\mathbf{g}} (H + \\mathbf{g}) and \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{P_{g}}{(\\mathbf{g},H)}}{H + \\mathbf{g}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} (H + \\mathbf{g})}{H + \\mathbf{g}} and \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{P_{g}}{(\\mathbf{g},H)}}{H + \\mathbf{g}} = \\frac{1}{H + \\mathbf{g}} and \\frac{1}{H + \\mathbf{g}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{g}} (H + \\mathbf{g})}{H + \\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["divide", 2, "Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Mul(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{\\mathbf{r}},v_{x})} = v_{x}^{\\hat{\\mathbf{r}}}, then obtain v_{x}^{- 2 \\hat{\\mathbf{r}}} \\mu_{0}^{2}{(\\hat{\\mathbf{r}},v_{x})} = 1", "derivation": "\\mu_{0}{(\\hat{\\mathbf{r}},v_{x})} = v_{x}^{\\hat{\\mathbf{r}}} and v_{x}^{- \\hat{\\mathbf{r}}} \\mu_{0}{(\\hat{\\mathbf{r}},v_{x})} = 1 and v_{x}^{- 2 \\hat{\\mathbf{r}}} \\mu_{0}{(\\hat{\\mathbf{r}},v_{x})} = v_{x}^{- \\hat{\\mathbf{r}}} and v_{x}^{- 2 \\hat{\\mathbf{r}}} \\mu_{0}^{2}{(\\hat{\\mathbf{r}},v_{x})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "Pow(Symbol('v_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Integer(1))"], [["divide", 2, "Pow(Symbol('v_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True))), Pow(Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\mathbf{H}{(V)} = \\log{(V)}, then derive \\log{(V)} \\frac{d}{d V} \\mathbf{H}{(V)} + \\frac{\\mathbf{H}{(V)}}{V} = \\frac{2 \\log{(V)}}{V}, then obtain \\log{(V)} \\frac{d}{d V} \\log{(V)} + \\frac{\\log{(V)}}{V} = \\log{(V)} \\frac{d}{d V} \\mathbf{H}{(V)} + \\frac{\\mathbf{H}{(V)}}{V}", "derivation": "\\mathbf{H}{(V)} = \\log{(V)} and \\mathbf{H}{(V)} \\log{(V)} = \\log{(V)}^{2} and \\frac{d}{d V} \\mathbf{H}{(V)} \\log{(V)} = \\frac{d}{d V} \\log{(V)}^{2} and \\log{(V)} \\frac{d}{d V} \\mathbf{H}{(V)} + \\frac{\\mathbf{H}{(V)}}{V} = \\frac{2 \\log{(V)}}{V} and \\log{(V)} \\frac{d}{d V} \\log{(V)} + \\frac{\\log{(V)}}{V} = \\frac{2 \\log{(V)}}{V} and \\log{(V)} \\frac{d}{d V} \\log{(V)} + \\frac{\\log{(V)}}{V} = \\log{(V)} \\frac{d}{d V} \\mathbf{H}{(V)} + \\frac{\\mathbf{H}{(V)}}{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["times", 1, "log(Symbol('V', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Pow(log(Symbol('V', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('V', commutative=True)), Integer(2)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(log(Symbol('V', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('V', commutative=True)))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(log(Symbol('V', commutative=True)), Derivative(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(log(Symbol('V', commutative=True)), Derivative(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))), Add(Mul(log(Symbol('V', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = \\frac{P_{e}}{\\mathbf{p}} - \\tilde{g}^*, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = - \\frac{P_{e}}{\\mathbf{p}^{2}}, then obtain - \\tilde{g}^* \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = \\frac{P_{e} \\tilde{g}^*}{\\mathbf{p}^{2}}", "derivation": "\\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = \\frac{P_{e}}{\\mathbf{p}} - \\tilde{g}^* and \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\mathbf{p}} (\\frac{P_{e}}{\\mathbf{p}} - \\tilde{g}^*) and \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = - \\frac{P_{e}}{\\mathbf{p}^{2}} and - \\tilde{g}^* \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{F_{g}}{(P_{e},\\mathbf{p},\\tilde{g}^*)} = \\frac{P_{e} \\tilde{g}^*}{\\mathbf{p}^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('F_g')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-2))))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), Derivative(Function('F_g')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-2)), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\dot{\\mathbf{r}},\\Omega)} = e^{\\Omega + \\dot{\\mathbf{r}}} and \\tilde{g}{(\\dot{\\mathbf{r}},\\Omega)} = \\Omega + \\dot{\\mathbf{r}}, then obtain \\log{(\\lambda{(\\dot{\\mathbf{r}},\\Omega)})} = \\log{(e^{\\tilde{g}{(\\dot{\\mathbf{r}},\\Omega)}})}", "derivation": "\\lambda{(\\dot{\\mathbf{r}},\\Omega)} = e^{\\Omega + \\dot{\\mathbf{r}}} and \\log{(\\lambda{(\\dot{\\mathbf{r}},\\Omega)})} = \\log{(e^{\\Omega + \\dot{\\mathbf{r}}})} and \\tilde{g}{(\\dot{\\mathbf{r}},\\Omega)} = \\Omega + \\dot{\\mathbf{r}} and \\log{(\\lambda{(\\dot{\\mathbf{r}},\\Omega)})} = \\log{(e^{\\tilde{g}{(\\dot{\\mathbf{r}},\\Omega)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Omega', commutative=True)), exp(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\lambda')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Omega', commutative=True))), log(exp(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(log(Function('\\\\lambda')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Omega', commutative=True))), log(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\varphi^*)} = e^{\\cos{(\\varphi^*)}}, then derive \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} = - e^{\\cos{(\\varphi^*)}} \\sin{(\\varphi^*)}, then obtain \\log{(\\int \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} d\\varphi^*)} = \\log{(\\int - \\Omega{(\\varphi^*)} \\sin{(\\varphi^*)} d\\varphi^*)}", "derivation": "\\Omega{(\\varphi^*)} = e^{\\cos{(\\varphi^*)}} and \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\cos{(\\varphi^*)}} and \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} = - e^{\\cos{(\\varphi^*)}} \\sin{(\\varphi^*)} and \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} = - \\Omega{(\\varphi^*)} \\sin{(\\varphi^*)} and \\int \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} d\\varphi^* = \\int - \\Omega{(\\varphi^*)} \\sin{(\\varphi^*)} d\\varphi^* and \\log{(\\int \\frac{d}{d \\varphi^*} \\Omega{(\\varphi^*)} d\\varphi^*)} = \\log{(\\int - \\Omega{(\\varphi^*)} \\sin{(\\varphi^*)} d\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), exp(cos(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["log", 5], "Equality(log(Integral(Derivative(Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), log(Integral(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given U{(C_{1},q)} = \\log{(C_{1} q)} and \\operatorname{P_{g}}{(C_{1},q)} = C_{1} q, then obtain (U{(C_{1},q)} - \\log{(\\operatorname{P_{g}}{(C_{1},q)})})^{2} = 0", "derivation": "U{(C_{1},q)} = \\log{(C_{1} q)} and U{(C_{1},q)} - \\log{(C_{1} q)} = 0 and (U{(C_{1},q)} - \\log{(C_{1} q)})^{2} = 0 and \\operatorname{P_{g}}{(C_{1},q)} = C_{1} q and (U{(C_{1},q)} - \\log{(\\operatorname{P_{g}}{(C_{1},q)})})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))"], [["minus", 1, "log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))"], "Equality(Add(Function('U')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))), Integer(0))"], [["times", 2, "Add(Function('U')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))))"], "Equality(Pow(Add(Function('U')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))), Integer(2)), Integer(0))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Function('U')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), log(Function('P_g')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))), Integer(2)), Integer(0))"]]}, {"prompt": "Given k{(A_{2})} = e^{A_{2}}, then derive \\int k{(A_{2})} dA_{2} = \\ddot{x} + e^{A_{2}}, then obtain \\ddot{x} + e^{A_{2}} = \\int e^{A_{2}} dA_{2}", "derivation": "k{(A_{2})} = e^{A_{2}} and \\int k{(A_{2})} dA_{2} = \\int e^{A_{2}} dA_{2} and \\int k{(A_{2})} dA_{2} = \\ddot{x} + e^{A_{2}} and \\ddot{x} + e^{A_{2}} = \\int e^{A_{2}} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('k')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), exp(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), exp(Symbol('A_2', commutative=True))), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given s{(M_{E})} = \\log{(M_{E})}, then derive 2 s{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} = \\log{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} + \\frac{s{(M_{E})}}{M_{E}}, then obtain 4 s^{2}{(M_{E})} (\\frac{d}{d M_{E}} s{(M_{E})})^{2} = (\\log{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} + \\frac{s{(M_{E})}}{M_{E}})^{2}", "derivation": "s{(M_{E})} = \\log{(M_{E})} and s^{2}{(M_{E})} = s{(M_{E})} \\log{(M_{E})} and \\frac{d}{d M_{E}} s^{2}{(M_{E})} = \\frac{d}{d M_{E}} s{(M_{E})} \\log{(M_{E})} and 2 s{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} = \\log{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} + \\frac{s{(M_{E})}}{M_{E}} and 4 s^{2}{(M_{E})} (\\frac{d}{d M_{E}} s{(M_{E})})^{2} = (\\log{(M_{E})} \\frac{d}{d M_{E}} s{(M_{E})} + \\frac{s{(M_{E})}}{M_{E}})^{2}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["times", 1, "Function('s')(Symbol('M_E', commutative=True))"], "Equality(Pow(Function('s')(Symbol('M_E', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Pow(Function('s')(Symbol('M_E', commutative=True)), Integer(2)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Function('s')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('s')(Symbol('M_E', commutative=True)), Derivative(Function('s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Mul(log(Symbol('M_E', commutative=True)), Derivative(Function('s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('s')(Symbol('M_E', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Function('s')(Symbol('M_E', commutative=True)), Integer(2)), Pow(Derivative(Function('s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2))), Pow(Add(Mul(log(Symbol('M_E', commutative=True)), Derivative(Function('s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('s')(Symbol('M_E', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(A_{x},y^{\\prime})} = \\sin{(A_{x} y^{\\prime})}, then derive \\int \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} dA_{x} = A_{x} + z, then obtain \\frac{\\int 1 dA_{x}}{\\int \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} dA_{x}} = 1", "derivation": "\\operatorname{f^{*}}{(A_{x},y^{\\prime})} = \\sin{(A_{x} y^{\\prime})} and \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} = 1 and \\int \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} dA_{x} = \\int 1 dA_{x} and \\int \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} dA_{x} = A_{x} + z and \\int 1 dA_{x} = A_{x} + z and \\frac{\\int 1 dA_{x}}{A_{x} + z} = 1 and \\frac{\\int 1 dA_{x}}{\\int \\frac{\\operatorname{f^{*}}{(A_{x},y^{\\prime})}}{\\sin{(A_{x} y^{\\prime})}} dA_{x}} = 1", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('f^*')(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Mul(Function('f^*')(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('f^*')(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Symbol('z', commutative=True)))"], [["divide", 5, "Add(Symbol('A_x', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('A_x', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('A_x', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))), Pow(Integral(Mul(Function('f^*')(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Tuple(Symbol('A_x', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\chi)} = \\cos{(\\chi)} and \\mathbf{F}{(\\chi)} = \\frac{\\cos{(\\chi)}}{\\chi}, then obtain (\\frac{\\operatorname{F_{g}}{(\\chi)}}{\\chi})^{\\chi} = \\mathbf{F}^{\\chi}{(\\chi)}", "derivation": "\\operatorname{F_{g}}{(\\chi)} = \\cos{(\\chi)} and \\frac{\\operatorname{F_{g}}{(\\chi)}}{\\chi} = \\frac{\\cos{(\\chi)}}{\\chi} and \\mathbf{F}{(\\chi)} = \\frac{\\cos{(\\chi)}}{\\chi} and (\\frac{\\operatorname{F_{g}}{(\\chi)}}{\\chi})^{\\chi} = (\\frac{\\cos{(\\chi)}}{\\chi})^{\\chi} and (\\frac{\\operatorname{F_{g}}{(\\chi)}}{\\chi})^{\\chi} = \\mathbf{F}^{\\chi}{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), cos(Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), cos(Symbol('\\\\chi', commutative=True))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), cos(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('F_g')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A,B)} = \\frac{B}{A}, then obtain \\frac{\\partial}{\\partial A} ((\\frac{B}{A})^{B} - \\operatorname{A_{x}}^{B}{(A,B)} - \\frac{1}{A}) = \\frac{d}{d A} - \\frac{1}{A}", "derivation": "\\operatorname{A_{x}}{(A,B)} = \\frac{B}{A} and \\operatorname{A_{x}}^{B}{(A,B)} = (\\frac{B}{A})^{B} and - \\operatorname{A_{x}}^{B}{(A,B)} = - (\\frac{B}{A})^{B} and (\\frac{B}{A})^{B} - \\operatorname{A_{x}}^{B}{(A,B)} = 0 and (\\frac{B}{A})^{B} - \\operatorname{A_{x}}^{B}{(A,B)} - \\frac{1}{A} = - \\frac{1}{A} and \\frac{\\partial}{\\partial A} ((\\frac{B}{A})^{B} - \\operatorname{A_{x}}^{B}{(A,B)} - \\frac{1}{A}) = \\frac{d}{d A} - \\frac{1}{A}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["add", 3, "Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)))), Integer(0))"], [["minus", 4, "Pow(Symbol('A', commutative=True), Integer(-1))"], "Equality(Add(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('A_x')(Symbol('A', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(\\pi)} = \\sin{(\\pi)} and C{(\\pi)} = \\sin{(\\pi)}, then obtain C{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)} = \\sin{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)}", "derivation": "Z{(\\pi)} = \\sin{(\\pi)} and C{(\\pi)} = \\sin{(\\pi)} and C{(\\pi)} = Z{(\\pi)} and Z{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)} = \\sin{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)} and C{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)} = \\sin{(\\pi)} + \\frac{d}{d \\pi} C{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('C')(Symbol('\\\\pi', commutative=True)), Function('Z')(Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Derivative(Function('C')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Add(Function('Z')(Symbol('\\\\pi', commutative=True)), Derivative(Function('C')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\pi', commutative=True)), Derivative(Function('C')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('C')(Symbol('\\\\pi', commutative=True)), Derivative(Function('C')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\pi', commutative=True)), Derivative(Function('C')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(c,E_{x})} = E_{x} + c and \\psi{(v_{x})} = \\log{(v_{x})}, then obtain \\int \\frac{v_{x} \\psi{(v_{x})}}{E_{x} + c} dE_{x} = \\int \\frac{v_{x} \\log{(v_{x})}}{E_{x} + c} dE_{x}", "derivation": "r{(c,E_{x})} = E_{x} + c and \\psi{(v_{x})} = \\log{(v_{x})} and \\frac{v_{x} \\psi{(v_{x})}}{r{(c,E_{x})}} = \\frac{v_{x} \\log{(v_{x})}}{r{(c,E_{x})}} and \\int \\frac{v_{x} \\psi{(v_{x})}}{r{(c,E_{x})}} dE_{x} = \\int \\frac{v_{x} \\log{(v_{x})}}{r{(c,E_{x})}} dE_{x} and \\int \\frac{v_{x} \\psi{(v_{x})}}{E_{x} + c} dE_{x} = \\int \\frac{v_{x} \\log{(v_{x})}}{E_{x} + c} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('c', commutative=True)))"], ["get_premise", "Equality(Function('\\\\psi')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["divide", 2, "Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Mul(Symbol('v_x', commutative=True), Function('\\\\psi')(Symbol('v_x', commutative=True)), Pow(Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Mul(Symbol('v_x', commutative=True), Pow(Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Symbol('v_x', commutative=True), Function('\\\\psi')(Symbol('v_x', commutative=True)), Pow(Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Symbol('v_x', commutative=True), Pow(Function('r')(Symbol('c', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Symbol('v_x', commutative=True), Pow(Add(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('v_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Symbol('v_x', commutative=True), Pow(Add(Symbol('E_x', commutative=True), Symbol('c', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then derive \\frac{d}{d \\mathbb{I}} \\rho_{f}{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain e^{\\cos{(\\mathbb{I})}} = e^{\\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})}}", "derivation": "\\rho_{f}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\rho_{f}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\rho_{f}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\cos{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} and e^{\\cos{(\\mathbb{I})}} = e^{\\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(cos(Symbol('\\\\mathbb{I}', commutative=True))), exp(Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}{(\\psi^*,s)} = e^{\\psi^* s}, then obtain (2 \\tilde{g}{(\\psi^*,s)} e^{\\psi^* s})^{\\psi^*} = (2 e^{2 \\psi^* s})^{\\psi^*}", "derivation": "\\tilde{g}{(\\psi^*,s)} = e^{\\psi^* s} and \\tilde{g}{(\\psi^*,s)} + e^{\\psi^* s} = 2 e^{\\psi^* s} and (\\tilde{g}{(\\psi^*,s)} + e^{\\psi^* s}) \\tilde{g}{(\\psi^*,s)} = (\\tilde{g}{(\\psi^*,s)} + e^{\\psi^* s}) e^{\\psi^* s} and ((\\tilde{g}{(\\psi^*,s)} + e^{\\psi^* s}) \\tilde{g}{(\\psi^*,s)})^{\\psi^*} = ((\\tilde{g}{(\\psi^*,s)} + e^{\\psi^* s}) e^{\\psi^* s})^{\\psi^*} and (2 \\tilde{g}{(\\psi^*,s)} e^{\\psi^* s})^{\\psi^*} = (2 e^{2 \\psi^* s})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True))))"], [["add", 1, "exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Mul(Integer(2), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))))"], [["times", 1, "Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True))))"], "Equality(Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True))), Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))))"], [["power", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True), Symbol('s', commutative=True)))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given S{(\\varphi^*)} = \\sin{(\\varphi^*)} and L{(\\varphi^*)} = \\frac{d}{d \\varphi^*} S{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)}, then obtain \\int L{(\\varphi^*)} d\\varphi^* = \\int 2 \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} d\\varphi^*", "derivation": "S{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\frac{d}{d \\varphi^*} S{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} and L{(\\varphi^*)} = \\frac{d}{d \\varphi^*} S{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} and L{(\\varphi^*)} = 2 \\frac{d}{d \\varphi^*} S{(\\varphi^*)} and L{(\\varphi^*)} = 2 \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} and \\int L{(\\varphi^*)} d\\varphi^* = \\int 2 \\frac{d}{d \\varphi^*} \\sin{(\\varphi^*)} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\varphi^*', commutative=True)), Add(Derivative(Function('S')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('L')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Derivative(Function('S')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('L')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(2), Derivative(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\Psi{(v_{t})} = \\log{(v_{t})}, then derive \\int \\Psi{(v_{t})} dv_{t} = v_{t} \\log{(v_{t})} - v_{t} + x^\\prime, then obtain (v_{t} \\Psi{(v_{t})} - v_{t} + x^\\prime)^{v_{t}} = (\\int \\log{(v_{t})} dv_{t})^{v_{t}}", "derivation": "\\Psi{(v_{t})} = \\log{(v_{t})} and \\int \\Psi{(v_{t})} dv_{t} = \\int \\log{(v_{t})} dv_{t} and (\\int \\Psi{(v_{t})} dv_{t})^{v_{t}} = (\\int \\log{(v_{t})} dv_{t})^{v_{t}} and \\int \\Psi{(v_{t})} dv_{t} = v_{t} \\log{(v_{t})} - v_{t} + x^\\prime and \\int \\Psi{(v_{t})} dv_{t} = v_{t} \\Psi{(v_{t})} - v_{t} + x^\\prime and (v_{t} \\Psi{(v_{t})} - v_{t} + x^\\prime)^{v_{t}} = (\\int \\log{(v_{t})} dv_{t})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Mul(Symbol('v_t', commutative=True), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('\\\\Psi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Mul(Symbol('v_t', commutative=True), Function('\\\\Psi')(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Mul(Symbol('v_t', commutative=True), Function('\\\\Psi')(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('v_t', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('v_t', commutative=True)), Pow(Integral(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Q)} = \\sin{(Q)}, then obtain Q \\frac{d}{d Q} \\int \\operatorname{y^{\\prime}}{(Q)} dQ = Q \\frac{\\partial}{\\partial Q} (U - \\cos{(Q)})", "derivation": "\\operatorname{y^{\\prime}}{(Q)} = \\sin{(Q)} and \\int \\operatorname{y^{\\prime}}{(Q)} dQ = \\int \\sin{(Q)} dQ and \\frac{d}{d Q} \\int \\operatorname{y^{\\prime}}{(Q)} dQ = \\frac{d}{d Q} \\int \\sin{(Q)} dQ and Q \\frac{d}{d Q} \\int \\operatorname{y^{\\prime}}{(Q)} dQ = Q \\frac{d}{d Q} \\int \\sin{(Q)} dQ and Q \\frac{d}{d Q} \\int \\operatorname{y^{\\prime}}{(Q)} dQ = Q \\frac{\\partial}{\\partial Q} (U - \\cos{(Q)})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 3, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Integral(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('Q', commutative=True), Derivative(Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Integral(Function('y^{\\\\prime}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('Q', commutative=True), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(m)} = \\int e^{m} dm, then derive \\dot{x}^{m}{(m)} = (v_{x} + e^{m})^{m}, then obtain (v_{x} + e^{m})^{m} = (\\int e^{m} dm)^{m}", "derivation": "\\dot{x}{(m)} = \\int e^{m} dm and \\dot{x}^{m}{(m)} = (\\int e^{m} dm)^{m} and \\dot{x}^{m}{(m)} = (v_{x} + e^{m})^{m} and (v_{x} + e^{m})^{m} = (\\int e^{m} dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\dot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Symbol('v_x', commutative=True), exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('v_x', commutative=True), exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given f{(\\rho_f,x^\\prime)} = (x^\\prime)^{\\rho_f}, then obtain \\frac{\\partial}{\\partial x^\\prime} \\int f{(\\rho_f,x^\\prime)} d\\rho_f - 1 = \\frac{\\partial}{\\partial x^\\prime} \\int (x^\\prime)^{\\rho_f} d\\rho_f - 1", "derivation": "f{(\\rho_f,x^\\prime)} = (x^\\prime)^{\\rho_f} and \\int f{(\\rho_f,x^\\prime)} d\\rho_f = \\int (x^\\prime)^{\\rho_f} d\\rho_f and \\frac{\\partial}{\\partial x^\\prime} \\int f{(\\rho_f,x^\\prime)} d\\rho_f = \\frac{\\partial}{\\partial x^\\prime} \\int (x^\\prime)^{\\rho_f} d\\rho_f and \\frac{\\partial}{\\partial x^\\prime} \\int f{(\\rho_f,x^\\prime)} d\\rho_f - 1 = \\frac{\\partial}{\\partial x^\\prime} \\int (x^\\prime)^{\\rho_f} d\\rho_f - 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\rho_f', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\rho_f', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('f')(Symbol('\\\\rho_f', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Integral(Function('f')(Symbol('\\\\rho_f', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\rho_f)} = \\log{(\\rho_f)}, then obtain \\rho_f (\\frac{\\operatorname{V_{\\mathbf{E}}}{(\\rho_f)}}{\\rho_f} - \\frac{\\log{(\\rho_f)}}{\\rho_f}) = 0", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\rho_f)} = \\log{(\\rho_f)} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\rho_f)}}{\\rho_f} = \\frac{\\log{(\\rho_f)}}{\\rho_f} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\rho_f)}}{\\rho_f} - \\frac{\\log{(\\rho_f)}}{\\rho_f} = 0 and \\rho_f (\\frac{\\operatorname{V_{\\mathbf{E}}}{(\\rho_f)}}{\\rho_f} - \\frac{\\log{(\\rho_f)}}{\\rho_f}) = 0", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), log(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), log(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), log(Symbol('\\\\rho_f', commutative=True)))), Integer(0))"], [["divide", 3, "Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), log(Symbol('\\\\rho_f', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(P_{g})} = e^{P_{g}} and \\theta_{2}{(P_{g})} = e^{P_{g}} + \\frac{d}{d P_{g}} \\Psi^{\\dagger}{(P_{g})}, then derive \\theta_{2}{(P_{g})} = 2 e^{P_{g}}, then obtain 2 e^{P_{g}} = e^{P_{g}} + \\frac{d}{d P_{g}} e^{P_{g}}", "derivation": "\\Psi^{\\dagger}{(P_{g})} = e^{P_{g}} and \\frac{d}{d P_{g}} \\Psi^{\\dagger}{(P_{g})} = \\frac{d}{d P_{g}} e^{P_{g}} and \\theta_{2}{(P_{g})} = e^{P_{g}} + \\frac{d}{d P_{g}} \\Psi^{\\dagger}{(P_{g})} and \\theta_{2}{(P_{g})} = e^{P_{g}} + \\frac{d}{d P_{g}} e^{P_{g}} and \\theta_{2}{(P_{g})} = 2 e^{P_{g}} and 2 e^{P_{g}} = e^{P_{g}} + \\frac{d}{d P_{g}} e^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('P_g', commutative=True)), Add(exp(Symbol('P_g', commutative=True)), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\theta_2')(Symbol('P_g', commutative=True)), Add(exp(Symbol('P_g', commutative=True)), Derivative(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Function('\\\\theta_2')(Symbol('P_g', commutative=True)), Mul(Integer(2), exp(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), exp(Symbol('P_g', commutative=True))), Add(exp(Symbol('P_g', commutative=True)), Derivative(exp(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(A_{x},\\pi)} = A_{x} \\pi, then obtain \\frac{\\partial}{\\partial A_{x}} \\cos{(\\int \\operatorname{v_{1}}{(A_{x},\\pi)} dA_{x})} = \\frac{\\partial}{\\partial A_{x}} \\cos{(\\int A_{x} \\pi dA_{x})}", "derivation": "\\operatorname{v_{1}}{(A_{x},\\pi)} = A_{x} \\pi and \\int \\operatorname{v_{1}}{(A_{x},\\pi)} dA_{x} = \\int A_{x} \\pi dA_{x} and \\cos{(\\int \\operatorname{v_{1}}{(A_{x},\\pi)} dA_{x})} = \\cos{(\\int A_{x} \\pi dA_{x})} and \\frac{\\partial}{\\partial A_{x}} \\cos{(\\int \\operatorname{v_{1}}{(A_{x},\\pi)} dA_{x})} = \\frac{\\partial}{\\partial A_{x}} \\cos{(\\int A_{x} \\pi dA_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Mul(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('v_1')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), cos(Integral(Mul(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(cos(Integral(Function('v_1')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(cos(Integral(Mul(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\hat{p}_0)} = e^{\\hat{p}_0}, then obtain - (\\frac{d}{d \\hat{p}_0} \\lambda^{\\hat{p}_0}{(\\hat{p}_0)})^{\\hat{p}_0} = - (\\frac{d}{d \\hat{p}_0} (e^{\\hat{p}_0})^{\\hat{p}_0})^{\\hat{p}_0}", "derivation": "\\lambda{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\lambda^{\\hat{p}_0}{(\\hat{p}_0)} = (e^{\\hat{p}_0})^{\\hat{p}_0} and \\frac{d}{d \\hat{p}_0} \\lambda^{\\hat{p}_0}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} (e^{\\hat{p}_0})^{\\hat{p}_0} and (\\frac{d}{d \\hat{p}_0} \\lambda^{\\hat{p}_0}{(\\hat{p}_0)})^{\\hat{p}_0} = (\\frac{d}{d \\hat{p}_0} (e^{\\hat{p}_0})^{\\hat{p}_0})^{\\hat{p}_0} and - (\\frac{d}{d \\hat{p}_0} \\lambda^{\\hat{p}_0}{(\\hat{p}_0)})^{\\hat{p}_0} = - (\\frac{d}{d \\hat{p}_0} (e^{\\hat{p}_0})^{\\hat{p}_0})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(exp(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Pow(exp(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(-1), Pow(Derivative(Pow(exp(Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\dot{y})} = \\log{(\\dot{y})}, then obtain \\operatorname{f^{*}}{(\\dot{y})} + \\frac{d}{d \\dot{y}} 1 = \\operatorname{f^{*}}{(\\dot{y})} + \\frac{d}{d \\dot{y}} \\operatorname{f^{*}}^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}}", "derivation": "\\operatorname{f^{*}}{(\\dot{y})} = \\log{(\\dot{y})} and \\operatorname{f^{*}}^{\\dot{y}}{(\\dot{y})} = \\log{(\\dot{y})}^{\\dot{y}} and 1 = \\operatorname{f^{*}}^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}} and \\frac{d}{d \\dot{y}} 1 = \\frac{d}{d \\dot{y}} \\operatorname{f^{*}}^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}} and \\operatorname{f^{*}}{(\\dot{y})} + \\frac{d}{d \\dot{y}} 1 = \\operatorname{f^{*}}{(\\dot{y})} + \\frac{d}{d \\dot{y}} \\operatorname{f^{*}}^{- \\dot{y}}{(\\dot{y})} \\log{(\\dot{y})}^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 2, "Pow(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Integer(-1), Function('f^*')(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Add(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Derivative(Mul(Pow(Function('f^*')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(n)} = \\log{(\\sin{(n)})} and U{(n)} = (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})}, then obtain \\int (- n + \\int \\log{(\\sin{(n)})} dn) \\log{(\\sin{(n)})} dn = \\int (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})} dn", "derivation": "\\operatorname{E_{x}}{(n)} = \\log{(\\sin{(n)})} and \\int \\operatorname{E_{x}}{(n)} dn = \\int \\log{(\\sin{(n)})} dn and - n + \\int \\operatorname{E_{x}}{(n)} dn = - n + \\int \\log{(\\sin{(n)})} dn and (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})} = (- n + \\int \\log{(\\sin{(n)})} dn) \\log{(\\sin{(n)})} and U{(n)} = (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})} and \\int U{(n)} dn = \\int (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})} dn and U{(n)} = (- n + \\int \\log{(\\sin{(n)})} dn) \\log{(\\sin{(n)})} and \\int (- n + \\int \\log{(\\sin{(n)})} dn) \\log{(\\sin{(n)})} dn = \\int (- n + \\int \\operatorname{E_{x}}{(n)} dn) \\log{(\\sin{(n)})} dn", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('n', commutative=True)), log(sin(Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(log(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["minus", 2, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(log(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["times", 3, "log(sin(Symbol('n', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(log(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('U')(Symbol('n', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))))"], [["integrate", 5, "Symbol('n', commutative=True)"], "Equality(Integral(Function('U')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('U')(Symbol('n', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(log(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(log(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Integral(Function('E_x')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), log(sin(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(c,p)} = \\log{(c^{p})} and \\mathbf{f}{(c,p)} = \\frac{1}{\\operatorname{F_{H}}^{p}{(c,p)} + \\log{(c^{p})}}, then obtain \\frac{\\log{(c^{p})}^{- p}}{(\\log{(c^{p})} + \\log{(c^{p})}^{p}) \\operatorname{F_{H}}{(c,p)}} = \\frac{\\log{(c^{p})}^{- p}}{(\\operatorname{F_{H}}^{p}{(c,p)} + \\log{(c^{p})}) \\operatorname{F_{H}}{(c,p)}}", "derivation": "\\operatorname{F_{H}}{(c,p)} = \\log{(c^{p})} and \\mathbf{f}{(c,p)} = \\frac{1}{\\operatorname{F_{H}}^{p}{(c,p)} + \\log{(c^{p})}} and \\mathbf{f}{(c,p)} = \\frac{1}{\\log{(c^{p})} + \\log{(c^{p})}^{p}} and \\frac{1}{\\log{(c^{p})} + \\log{(c^{p})}^{p}} = \\frac{1}{\\operatorname{F_{H}}^{p}{(c,p)} + \\log{(c^{p})}} and \\frac{\\log{(c^{p})}^{- p}}{(\\log{(c^{p})} + \\log{(c^{p})}^{p}) \\operatorname{F_{H}}{(c,p)}} = \\frac{\\log{(c^{p})}^{- p}}{(\\operatorname{F_{H}}^{p}{(c,p)} + \\log{(c^{p})}) \\operatorname{F_{H}}{(c,p)}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Pow(Add(Pow(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Pow(Add(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Integer(-1)), Pow(Add(Pow(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True)))), Integer(-1)))"], [["divide", 4, "Mul(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Add(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Integer(-1)), Pow(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Pow(Add(Pow(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True)))), Integer(-1)), Pow(Function('F_H')(Symbol('c', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(log(Pow(Symbol('c', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(c,v_{z})} = \\int c v_{z} dv_{z}, then derive \\frac{\\partial}{\\partial v_{z}} \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\frac{c^{2} v_{z}}{2}, then obtain v_{z} + \\frac{\\partial}{\\partial v_{z}} \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\frac{c^{2} v_{z}}{2} + v_{z}", "derivation": "\\operatorname{A_{x}}{(c,v_{z})} = \\int c v_{z} dv_{z} and \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\iint c v_{z} dv_{z} dc and \\frac{\\partial}{\\partial v_{z}} \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\frac{\\partial}{\\partial v_{z}} \\iint c v_{z} dv_{z} dc and \\frac{\\partial}{\\partial v_{z}} \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\frac{c^{2} v_{z}}{2} and v_{z} + \\frac{\\partial}{\\partial v_{z}} \\int \\operatorname{A_{x}}{(c,v_{z})} dc = \\frac{c^{2} v_{z}}{2} + v_{z}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Integral(Mul(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Mul(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integral(Function('A_x')(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Integral(Function('A_x')(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_z', commutative=True)))"], [["add", 4, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Derivative(Integral(Function('A_x')(Symbol('c', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(i)} = e^{i} and \\operatorname{x^{{\\}'}}{(i)} = e^{i}, then obtain 0 = \\operatorname{x^{{\\}'}}{(i)} - e^{i}", "derivation": "\\Psi_{nl}{(i)} = e^{i} and 0 = - \\Psi_{nl}{(i)} + e^{i} and \\operatorname{x^{{\\}'}}{(i)} = e^{i} and 0 = - \\Psi_{nl}{(i)} + \\operatorname{x^{{\\}'}}{(i)} and 0 = \\operatorname{x^{{\\}'}}{(i)} - e^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), exp(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Function('x^\\\\prime')(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Function('x^\\\\prime')(Symbol('i', commutative=True)), Mul(Integer(-1), exp(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(x,F_{N})} = F_{N} x, then obtain \\frac{\\partial}{\\partial F_{N}} (\\frac{- \\rho_b + \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)}}{2 \\log{(m)}} + 1) = \\frac{\\partial}{\\partial F_{N}} (\\frac{F_{N} x - \\rho_b + 2 \\log{(m)}}{2 \\log{(m)}} + 1)", "derivation": "\\operatorname{A_{1}}{(x,F_{N})} = F_{N} x and \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)} = F_{N} x + 2 \\log{(m)} and - \\rho_b + \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)} = F_{N} x - \\rho_b + 2 \\log{(m)} and \\frac{- \\rho_b + \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)}}{2 \\log{(m)}} = \\frac{F_{N} x - \\rho_b + 2 \\log{(m)}}{2 \\log{(m)}} and \\frac{- \\rho_b + \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)}}{2 \\log{(m)}} + 1 = \\frac{F_{N} x - \\rho_b + 2 \\log{(m)}}{2 \\log{(m)}} + 1 and \\frac{\\partial}{\\partial F_{N}} (\\frac{- \\rho_b + \\operatorname{A_{1}}{(x,F_{N})} + 2 \\log{(m)}}{2 \\log{(m)}} + 1) = \\frac{\\partial}{\\partial F_{N}} (\\frac{F_{N} x - \\rho_b + 2 \\log{(m)}}{2 \\log{(m)}} + 1)", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Mul(Integer(2), log(Symbol('m', commutative=True)))"], "Equality(Add(Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Add(Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))))"], [["minus", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Add(Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), log(Symbol('m', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Rational(1, 2), Add(Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Integer(1)))"], [["differentiate", 5, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_1')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Add(Mul(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(f^{*})} = \\log{(f^{*})}, then obtain \\phi_{2}{(f^{*})} \\int \\phi_{2}^{f^{*}}{(f^{*})} df^{*} = \\log{(f^{*})} \\int \\phi_{2}^{f^{*}}{(f^{*})} df^{*}", "derivation": "\\phi_{2}{(f^{*})} = \\log{(f^{*})} and \\phi_{2}^{f^{*}}{(f^{*})} = \\log{(f^{*})}^{f^{*}} and \\int \\phi_{2}^{f^{*}}{(f^{*})} df^{*} = \\int \\log{(f^{*})}^{f^{*}} df^{*} and \\phi_{2}{(f^{*})} \\int \\log{(f^{*})}^{f^{*}} df^{*} = \\log{(f^{*})} \\int \\log{(f^{*})}^{f^{*}} df^{*} and \\phi_{2}{(f^{*})} \\int \\phi_{2}^{f^{*}}{(f^{*})} df^{*} = \\log{(f^{*})} \\int \\phi_{2}^{f^{*}}{(f^{*})} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Pow(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["times", 1, "Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(log(Symbol('f^*', commutative=True)), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Integral(Pow(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(log(Symbol('f^*', commutative=True)), Integral(Pow(Function('\\\\phi_2')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})}, then obtain - \\frac{\\Psi_{\\lambda} \\operatorname{v_{y}}{(\\Psi_{\\lambda})}}{\\log{(\\log{(\\Psi_{\\lambda})})}} = - \\Psi_{\\lambda}", "derivation": "\\operatorname{v_{y}}{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})} and \\Psi_{\\lambda} \\operatorname{v_{y}}{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\log{(\\log{(\\Psi_{\\lambda})})} and \\frac{\\Psi_{\\lambda} \\operatorname{v_{y}}{(\\Psi_{\\lambda})}}{\\log{(\\log{(\\Psi_{\\lambda})})}} = \\Psi_{\\lambda} and - \\frac{\\Psi_{\\lambda} \\operatorname{v_{y}}{(\\Psi_{\\lambda})}}{\\log{(\\log{(\\Psi_{\\lambda})})}} = - \\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('v_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["divide", 2, "log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('v_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('v_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given J{(J,f^{\\prime})} = \\frac{\\sin{(f^{\\prime})}}{J}, then obtain (\\frac{\\int (- J{(J,f^{\\prime})} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}})^{J} = (\\frac{\\int (- \\frac{\\sin{(f^{\\prime})}}{J} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}})^{J}", "derivation": "J{(J,f^{\\prime})} = \\frac{\\sin{(f^{\\prime})}}{J} and - J{(J,f^{\\prime})} = - \\frac{\\sin{(f^{\\prime})}}{J} and - J{(J,f^{\\prime})} + \\frac{1}{J} = - \\frac{\\sin{(f^{\\prime})}}{J} + \\frac{1}{J} and \\int (- J{(J,f^{\\prime})} + \\frac{1}{J}) df^{\\prime} = \\int (- \\frac{\\sin{(f^{\\prime})}}{J} + \\frac{1}{J}) df^{\\prime} and \\frac{\\int (- J{(J,f^{\\prime})} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}} = \\frac{\\int (- \\frac{\\sin{(f^{\\prime})}}{J} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}} and (\\frac{\\int (- J{(J,f^{\\prime})} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}})^{J} = (\\frac{\\int (- \\frac{\\sin{(f^{\\prime})}}{J} + \\frac{1}{J}) df^{\\prime}}{\\sin{(f^{\\prime})}})^{J}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "Pow(Symbol('J', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 4, "sin(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('J')(Symbol('J', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('J', commutative=True)), Pow(Mul(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{B},\\theta_1)} = \\frac{e^{\\theta_1}}{\\mathbf{B}} and l{(\\theta_1)} = e^{\\theta_1}, then obtain \\log{(- \\mathbf{B} + \\mathbf{A}{(\\mathbf{B},\\theta_1)})} = \\log{(- \\mathbf{B} + \\frac{l{(\\theta_1)}}{\\mathbf{B}})}", "derivation": "\\mathbf{A}{(\\mathbf{B},\\theta_1)} = \\frac{e^{\\theta_1}}{\\mathbf{B}} and - \\mathbf{B} + \\mathbf{A}{(\\mathbf{B},\\theta_1)} = - \\mathbf{B} + \\frac{e^{\\theta_1}}{\\mathbf{B}} and \\log{(- \\mathbf{B} + \\mathbf{A}{(\\mathbf{B},\\theta_1)})} = \\log{(- \\mathbf{B} + \\frac{e^{\\theta_1}}{\\mathbf{B}})} and l{(\\theta_1)} = e^{\\theta_1} and \\log{(- \\mathbf{B} + \\mathbf{A}{(\\mathbf{B},\\theta_1)})} = \\log{(- \\mathbf{B} + \\frac{l{(\\theta_1)}}{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["log", 2], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Function('l')(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given x{(f)} = \\log{(f)}, then derive \\mathbf{J}_P + x{(f)} + \\log{(f)} = \\mathbf{J} + 2 \\log{(f)}, then obtain \\mathbf{J}_P + 2 \\log{(f)} = \\mathbf{J} + 2 \\log{(f)}", "derivation": "x{(f)} = \\log{(f)} and x{(f)} + \\log{(f)} = 2 \\log{(f)} and \\frac{d}{d f} (x{(f)} + \\log{(f)}) = \\frac{d}{d f} 2 \\log{(f)} and \\int \\frac{d}{d f} (x{(f)} + \\log{(f)}) df = \\int \\frac{d}{d f} 2 \\log{(f)} df and \\mathbf{J}_P + x{(f)} + \\log{(f)} = \\mathbf{J} + 2 \\log{(f)} and \\mathbf{J}_P + 2 \\log{(f)} = \\mathbf{J} + 2 \\log{(f)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["add", 1, "log(Symbol('f', commutative=True))"], "Equality(Add(Function('x')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Mul(Integer(2), log(Symbol('f', commutative=True))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Function('x')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Derivative(Add(Function('x')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integral(Derivative(Mul(Integer(2), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('x')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), log(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(2), log(Symbol('f', commutative=True)))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), log(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(s,v_{1})} = \\frac{s}{v_{1}}, then obtain \\int 0 ds = \\int (- \\frac{s}{v_{1}} + \\operatorname{E_{\\lambda}}{(s,v_{1})}) ds", "derivation": "\\operatorname{E_{\\lambda}}{(s,v_{1})} = \\frac{s}{v_{1}} and - \\operatorname{E_{\\lambda}}{(s,v_{1})} = - \\frac{s}{v_{1}} and 0 = - \\frac{s}{v_{1}} + \\operatorname{E_{\\lambda}}{(s,v_{1})} and \\int 0 ds = \\int (- \\frac{s}{v_{1}} + \\operatorname{E_{\\lambda}}{(s,v_{1})}) ds", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('s', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["minus", 2, "Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('s', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Function('E_{\\\\lambda}')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Function('E_{\\\\lambda}')(Symbol('s', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given I{(M_{E})} = \\sin{(\\cos{(M_{E})})}, then obtain M_{E} I{(M_{E})} - \\int M_{E} \\sin{(\\cos{(M_{E})})} dM_{E} = M_{E} \\sin{(\\cos{(M_{E})})} - \\int M_{E} \\sin{(\\cos{(M_{E})})} dM_{E}", "derivation": "I{(M_{E})} = \\sin{(\\cos{(M_{E})})} and M_{E} I{(M_{E})} = M_{E} \\sin{(\\cos{(M_{E})})} and \\int M_{E} I{(M_{E})} dM_{E} = \\int M_{E} \\sin{(\\cos{(M_{E})})} dM_{E} and M_{E} I{(M_{E})} - \\int M_{E} I{(M_{E})} dM_{E} = M_{E} \\sin{(\\cos{(M_{E})})} - \\int M_{E} I{(M_{E})} dM_{E} and M_{E} I{(M_{E})} - \\int M_{E} \\sin{(\\cos{(M_{E})})} dM_{E} = M_{E} \\sin{(\\cos{(M_{E})})} - \\int M_{E} \\sin{(\\cos{(M_{E})})} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('M_E', commutative=True)), sin(cos(Symbol('M_E', commutative=True))))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Add(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))), Add(Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('M_E', commutative=True), Function('I')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))))), Add(Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('M_E', commutative=True), sin(cos(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(v_{y})} = e^{v_{y}} and A{(v_{y})} = e^{v_{y}}, then obtain \\log{(\\int (A{(v_{y})} + e^{v_{y}})^{2} dv_{y})} = \\log{(\\int 4 e^{2 v_{y}} dv_{y})}", "derivation": "\\hat{x}{(v_{y})} = e^{v_{y}} and \\hat{x}{(v_{y})} + e^{v_{y}} = 2 e^{v_{y}} and (\\hat{x}{(v_{y})} + e^{v_{y}})^{2} = 4 e^{2 v_{y}} and \\int (\\hat{x}{(v_{y})} + e^{v_{y}})^{2} dv_{y} = \\int 4 e^{2 v_{y}} dv_{y} and A{(v_{y})} = e^{v_{y}} and \\log{(\\int (\\hat{x}{(v_{y})} + e^{v_{y}})^{2} dv_{y})} = \\log{(\\int 4 e^{2 v_{y}} dv_{y})} and A{(v_{y})} = \\hat{x}{(v_{y})} and \\log{(\\int (A{(v_{y})} + e^{v_{y}})^{2} dv_{y})} = \\log{(\\int 4 e^{2 v_{y}} dv_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["add", 1, "exp(Symbol('v_y', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Mul(Integer(2), exp(Symbol('v_y', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\hat{x}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('v_y', commutative=True)))))"], [["integrate", 3, "Symbol('v_y', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\hat{x}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Integer(2)), Tuple(Symbol('v_y', commutative=True))), Integral(Mul(Integer(4), exp(Mul(Integer(2), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["log", 4], "Equality(log(Integral(Pow(Add(Function('\\\\hat{x}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Integer(2)), Tuple(Symbol('v_y', commutative=True)))), log(Integral(Mul(Integer(4), exp(Mul(Integer(2), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('A')(Symbol('v_y', commutative=True)), Function('\\\\hat{x}')(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(log(Integral(Pow(Add(Function('A')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True))), Integer(2)), Tuple(Symbol('v_y', commutative=True)))), log(Integral(Mul(Integer(4), exp(Mul(Integer(2), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(h,\\hat{x})} = \\hat{x} h and \\operatorname{E_{\\lambda}}{(h,\\hat{x})} = \\frac{\\partial}{\\partial h} \\mathbf{v}{(h,\\hat{x})}, then obtain \\operatorname{E_{\\lambda}}^{h}{(h,\\hat{x})} = (\\frac{\\partial}{\\partial h} \\hat{x} h)^{h}", "derivation": "\\mathbf{v}{(h,\\hat{x})} = \\hat{x} h and \\frac{\\partial}{\\partial h} \\mathbf{v}{(h,\\hat{x})} = \\frac{\\partial}{\\partial h} \\hat{x} h and \\operatorname{E_{\\lambda}}{(h,\\hat{x})} = \\frac{\\partial}{\\partial h} \\mathbf{v}{(h,\\hat{x})} and \\operatorname{E_{\\lambda}}^{h}{(h,\\hat{x})} = (\\frac{\\partial}{\\partial h} \\mathbf{v}{(h,\\hat{x})})^{h} and \\operatorname{E_{\\lambda}}^{h}{(h,\\hat{x})} = (\\frac{\\partial}{\\partial h} \\hat{x} h)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('h', commutative=True)), Pow(Derivative(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('h', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(A_{x},\\pi)} = - A_{x} + \\cos{(\\pi)}, then obtain \\psi^{*}{(A_{x},\\pi)} + \\frac{\\int \\psi^{*}{(A_{x},\\pi)} d\\pi}{\\cos{(\\pi)}} = - A_{x} + \\cos{(\\pi)} + \\frac{\\int \\psi^{*}{(A_{x},\\pi)} d\\pi}{\\cos{(\\pi)}}", "derivation": "\\psi^{*}{(A_{x},\\pi)} = - A_{x} + \\cos{(\\pi)} and \\int \\psi^{*}{(A_{x},\\pi)} d\\pi = \\int (- A_{x} + \\cos{(\\pi)}) d\\pi and \\frac{\\int \\psi^{*}{(A_{x},\\pi)} d\\pi}{\\cos{(\\pi)}} = \\frac{\\int (- A_{x} + \\cos{(\\pi)}) d\\pi}{\\cos{(\\pi)}} and \\psi^{*}{(A_{x},\\pi)} + \\frac{\\int (- A_{x} + \\cos{(\\pi)}) d\\pi}{\\cos{(\\pi)}} = - A_{x} + \\cos{(\\pi)} + \\frac{\\int (- A_{x} + \\cos{(\\pi)}) d\\pi}{\\cos{(\\pi)}} and \\psi^{*}{(A_{x},\\pi)} + \\frac{\\int \\psi^{*}{(A_{x},\\pi)} d\\pi}{\\cos{(\\pi)}} = - A_{x} + \\cos{(\\pi)} + \\frac{\\int \\psi^{*}{(A_{x},\\pi)} d\\pi}{\\cos{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["add", 1, "Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('\\\\pi', commutative=True)), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('A_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(V,q)} = \\log{(V^{q})}, then obtain - \\sin{(V \\int (- V - V^{q} + \\eta^{\\prime}{(V,q)})^{V} dV)} = - \\sin{(V \\int (- V - V^{q} + \\log{(V^{q})})^{V} dV)}", "derivation": "\\eta^{\\prime}{(V,q)} = \\log{(V^{q})} and - V^{q} + \\eta^{\\prime}{(V,q)} = - V^{q} + \\log{(V^{q})} and - V - V^{q} + \\eta^{\\prime}{(V,q)} = - V - V^{q} + \\log{(V^{q})} and (- V - V^{q} + \\eta^{\\prime}{(V,q)})^{V} = (- V - V^{q} + \\log{(V^{q})})^{V} and \\int (- V - V^{q} + \\eta^{\\prime}{(V,q)})^{V} dV = \\int (- V - V^{q} + \\log{(V^{q})})^{V} dV and - V \\int (- V - V^{q} + \\eta^{\\prime}{(V,q)})^{V} dV = - V \\int (- V - V^{q} + \\log{(V^{q})})^{V} dV and - \\sin{(V \\int (- V - V^{q} + \\eta^{\\prime}{(V,q)})^{V} dV)} = - \\sin{(V \\int (- V - V^{q} + \\log{(V^{q})})^{V} dV)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))))"], [["minus", 1, "Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))))"], [["minus", 2, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Symbol('V', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))), Symbol('V', commutative=True)))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Symbol('V', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('V', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Integer(-1), Symbol('V', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["sin", 6], "Equality(Mul(Integer(-1), sin(Mul(Symbol('V', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V', commutative=True), Symbol('q', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))), Mul(Integer(-1), sin(Mul(Symbol('V', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Symbol('q', commutative=True))), log(Pow(Symbol('V', commutative=True), Symbol('q', commutative=True)))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))))"]]}, {"prompt": "Given \\phi_{2}{(t_{1},A_{1},q)} = \\frac{A_{1}}{t_{1}} - q, then obtain \\int \\frac{\\iint \\phi_{2}{(t_{1},A_{1},q)} dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} dq = \\int \\frac{\\iint (\\frac{A_{1}}{t_{1}} - q) dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} dq", "derivation": "\\phi_{2}{(t_{1},A_{1},q)} = \\frac{A_{1}}{t_{1}} - q and \\int \\phi_{2}{(t_{1},A_{1},q)} dq = \\int (\\frac{A_{1}}{t_{1}} - q) dq and \\iint \\phi_{2}{(t_{1},A_{1},q)} dq dt_{1} = \\iint (\\frac{A_{1}}{t_{1}} - q) dq dt_{1} and \\frac{\\iint \\phi_{2}{(t_{1},A_{1},q)} dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} = \\frac{\\iint (\\frac{A_{1}}{t_{1}} - q) dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} and \\int \\frac{\\iint \\phi_{2}{(t_{1},A_{1},q)} dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} dq = \\int \\frac{\\iint (\\frac{A_{1}}{t_{1}} - q) dq dt_{1}}{\\phi_{2}{(t_{1},A_{1},q)}} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 3, "Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Pow(Function('\\\\phi_2')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(n)} = e^{n}, then derive \\frac{d}{d n} \\operatorname{r_{0}}{(n)} = e^{n}, then obtain \\operatorname{r_{0}}{(n)} = \\frac{d}{d n} \\operatorname{r_{0}}{(n)}", "derivation": "\\operatorname{r_{0}}{(n)} = e^{n} and \\frac{d}{d n} \\operatorname{r_{0}}{(n)} = \\frac{d}{d n} e^{n} and \\frac{d}{d n} \\operatorname{r_{0}}{(n)} = e^{n} and \\frac{d}{d n} e^{n} = e^{n} and \\operatorname{r_{0}}{(n)} = \\frac{d}{d n} e^{n} and \\operatorname{r_{0}}{(n)} = \\frac{d}{d n} \\operatorname{r_{0}}{(n)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), exp(Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), exp(Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('r_0')(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('r_0')(Symbol('n', commutative=True)), Derivative(Function('r_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\phi_1,Z)} = \\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1} and \\operatorname{A_{1}}{(\\phi_1,Z)} = \\int \\frac{\\operatorname{r_{0}}{(\\phi_1,Z)}}{Z} d\\phi_1, then obtain \\frac{\\partial}{\\partial Z} \\operatorname{A_{1}}{(\\phi_1,Z)} = \\frac{\\partial}{\\partial Z} \\int \\frac{\\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1}}{Z} d\\phi_1", "derivation": "\\operatorname{r_{0}}{(\\phi_1,Z)} = \\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1} and \\frac{\\operatorname{r_{0}}{(\\phi_1,Z)}}{Z} = \\frac{\\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1}}{Z} and \\int \\frac{\\operatorname{r_{0}}{(\\phi_1,Z)}}{Z} d\\phi_1 = \\int \\frac{\\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1}}{Z} d\\phi_1 and \\frac{\\partial}{\\partial Z} \\int \\frac{\\operatorname{r_{0}}{(\\phi_1,Z)}}{Z} d\\phi_1 = \\frac{\\partial}{\\partial Z} \\int \\frac{\\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1}}{Z} d\\phi_1 and \\operatorname{A_{1}}{(\\phi_1,Z)} = \\int \\frac{\\operatorname{r_{0}}{(\\phi_1,Z)}}{Z} d\\phi_1 and \\frac{\\partial}{\\partial Z} \\operatorname{A_{1}}{(\\phi_1,Z)} = \\frac{\\partial}{\\partial Z} \\int \\frac{\\frac{\\partial}{\\partial \\phi_1} Z^{\\phi_1}}{Z} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('Z', commutative=True)"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('A_1')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(a)} = e^{a}, then derive \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} = e^{a}, then obtain - a + \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} = - a + e^{a}", "derivation": "\\operatorname{x^{{\\}'}}{(a)} = e^{a} and - a + \\operatorname{x^{{\\}'}}{(a)} = - a + e^{a} and \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} = \\frac{d}{d a} e^{a} and \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} = e^{a} and \\operatorname{x^{{\\}'}}{(a)} = \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} and - a + \\frac{d}{d a} \\operatorname{x^{{\\}'}}{(a)} = - a + e^{a}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('x^\\\\prime')(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), exp(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('x^\\\\prime')(Symbol('a', commutative=True)), Derivative(Function('x^\\\\prime')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Derivative(Function('x^\\\\prime')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), exp(Symbol('a', commutative=True))))"]]}, {"prompt": "Given i{(\\varepsilon_0,F_{N})} = F_{N} + \\cos{(\\varepsilon_0)}, then obtain \\varepsilon_0 (\\frac{- F_{N} - \\cos{(\\varepsilon_0)}}{\\varepsilon_0} + \\frac{i{(\\varepsilon_0,F_{N})}}{\\varepsilon_0}) = 0", "derivation": "i{(\\varepsilon_0,F_{N})} = F_{N} + \\cos{(\\varepsilon_0)} and \\frac{i{(\\varepsilon_0,F_{N})}}{\\varepsilon_0} = \\frac{F_{N} + \\cos{(\\varepsilon_0)}}{\\varepsilon_0} and - \\frac{F_{N} + \\cos{(\\varepsilon_0)}}{\\varepsilon_0} + \\frac{i{(\\varepsilon_0,F_{N})}}{\\varepsilon_0} = 0 and \\varepsilon_0 (- \\frac{F_{N} + \\cos{(\\varepsilon_0)}}{\\varepsilon_0} + \\frac{i{(\\varepsilon_0,F_{N})}}{\\varepsilon_0}) = 0 and \\varepsilon_0 (\\frac{- F_{N} - \\cos{(\\varepsilon_0)}}{\\varepsilon_0} + \\frac{F_{N} + \\cos{(\\varepsilon_0)}}{\\varepsilon_0}) = 0 and \\varepsilon_0 (\\frac{- F_{N} - \\cos{(\\varepsilon_0)}}{\\varepsilon_0} + \\frac{i{(\\varepsilon_0,F_{N})}}{\\varepsilon_0}) = 0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)))), Integer(0))"], [["divide", 3, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(x,U)} = \\frac{x}{U}, then obtain \\frac{\\partial}{\\partial U} (\\operatorname{E_{x}}^{U}{(x,U)} - \\frac{1}{U^{2}}) + 1 = \\frac{\\partial}{\\partial U} ((\\frac{x}{U})^{U} - \\frac{1}{U^{2}}) + 1", "derivation": "\\operatorname{E_{x}}{(x,U)} = \\frac{x}{U} and \\operatorname{E_{x}}^{U}{(x,U)} = (\\frac{x}{U})^{U} and \\operatorname{E_{x}}^{U}{(x,U)} - \\frac{1}{U^{2}} = (\\frac{x}{U})^{U} - \\frac{1}{U^{2}} and \\frac{\\partial}{\\partial U} (\\operatorname{E_{x}}^{U}{(x,U)} - \\frac{1}{U^{2}}) = \\frac{\\partial}{\\partial U} ((\\frac{x}{U})^{U} - \\frac{1}{U^{2}}) and \\frac{\\partial}{\\partial U} (\\operatorname{E_{x}}^{U}{(x,U)} - \\frac{1}{U^{2}}) + 1 = \\frac{\\partial}{\\partial U} ((\\frac{x}{U})^{U} - \\frac{1}{U^{2}}) + 1", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('x', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('x', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Symbol('U', commutative=True)))"], [["minus", 2, "Pow(Symbol('U', commutative=True), Integer(-2))"], "Equality(Add(Pow(Function('E_x')(Symbol('x', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))), Add(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Pow(Function('E_x')(Symbol('x', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(Add(Pow(Function('E_x')(Symbol('x', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(f_{E})} = \\sin{(\\log{(f_{E})})}, then derive \\frac{d}{d f_{E}} \\operatorname{z^{*}}{(f_{E})} = \\frac{\\cos{(\\log{(f_{E})})}}{f_{E}}, then obtain \\frac{d}{d f_{E}} \\sin{(\\log{(f_{E})})} = \\frac{\\cos{(\\log{(f_{E})})}}{f_{E}}", "derivation": "\\operatorname{z^{*}}{(f_{E})} = \\sin{(\\log{(f_{E})})} and \\frac{d}{d f_{E}} \\operatorname{z^{*}}{(f_{E})} = \\frac{d}{d f_{E}} \\sin{(\\log{(f_{E})})} and \\frac{d}{d f_{E}} \\operatorname{z^{*}}{(f_{E})} = \\frac{\\cos{(\\log{(f_{E})})}}{f_{E}} and \\frac{d}{d f_{E}} \\sin{(\\log{(f_{E})})} = \\frac{\\cos{(\\log{(f_{E})})}}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('f_E', commutative=True)), sin(log(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(sin(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), cos(log(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), cos(log(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given i{(y,u)} = u y, then obtain \\int (- u y - i{(y,u)} + (\\int u y du)^{u}) (- u y - i{(y,u)} + (\\int i{(y,u)} du)^{u}) dy = \\int (- u y - i{(y,u)} + (\\int u y du)^{u})^{2} dy", "derivation": "i{(y,u)} = u y and \\int i{(y,u)} du = \\int u y du and (\\int i{(y,u)} du)^{u} = (\\int u y du)^{u} and - u y + (\\int i{(y,u)} du)^{u} = - u y + (\\int u y du)^{u} and - u y - i{(y,u)} + (\\int i{(y,u)} du)^{u} = - u y - i{(y,u)} + (\\int u y du)^{u} and (- u y - i{(y,u)} + (\\int u y du)^{u}) (- u y - i{(y,u)} + (\\int i{(y,u)} du)^{u}) = (- u y - i{(y,u)} + (\\int u y du)^{u})^{2} and \\int (- u y - i{(y,u)} + (\\int u y du)^{u}) (- u y - i{(y,u)} + (\\int i{(y,u)} du)^{u}) dy = \\int (- u y - i{(y,u)} + (\\int u y du)^{u})^{2} dy", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["minus", 3, "Mul(Symbol('u', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["minus", 4, "Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Integer(2)))"], [["integrate", 6, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Integer(2)), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(A,\\nabla)} = A + \\nabla, then derive \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,\\nabla)} = 1, then obtain \\int \\nabla^{A} dA = \\int (\\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + 1)^{A} dA", "derivation": "\\operatorname{v_{t}}{(A,\\nabla)} = A + \\nabla and \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,\\nabla)} = \\frac{\\partial}{\\partial A} (A + \\nabla) and \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,\\nabla)} = 1 and \\nabla + \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,\\nabla)} = \\nabla + 1 and \\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + \\frac{\\partial}{\\partial A} \\operatorname{v_{t}}{(A,\\nabla)} = \\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + 1 and \\nabla = \\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + 1 and \\nabla^{A} = (\\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + 1)^{A} and \\int \\nabla^{A} dA = \\int (\\nabla - \\frac{\\partial}{\\partial A} (A + \\nabla) + 1)^{A} dA", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Derivative(Function('v_t')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Symbol('\\\\nabla', commutative=True), Integer(1)))"], [["minus", 4, "Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Derivative(Function('v_t')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Symbol('\\\\nabla', commutative=True), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(1)))"], [["power", 6, "Symbol('A', commutative=True)"], "Equality(Pow(Symbol('\\\\nabla', commutative=True), Symbol('A', commutative=True)), Pow(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(1)), Symbol('A', commutative=True)))"], [["integrate", 7, "Symbol('A', commutative=True)"], "Equality(Integral(Pow(Symbol('\\\\nabla', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Integer(1)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given q{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then obtain q^{2}{(\\mathbf{E})} \\cos{(\\mathbf{E})} = \\cos^{3}{(\\mathbf{E})}", "derivation": "q{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and q{(\\mathbf{E})} \\cos{(\\mathbf{E})} = \\cos^{2}{(\\mathbf{E})} and q^{2}{(\\mathbf{E})} \\cos{(\\mathbf{E})} = q{(\\mathbf{E})} \\cos^{2}{(\\mathbf{E})} and q{(\\mathbf{E})} \\cos^{2}{(\\mathbf{E})} = \\cos^{3}{(\\mathbf{E})} and q^{2}{(\\mathbf{E})} \\cos{(\\mathbf{E})} = \\cos^{3}{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)))"], [["times", 2, "Function('q')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('q')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{E}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} = \\phi + \\cos{(\\dot{y})} and k{(\\phi,\\dot{y})} = \\phi + \\cos{(\\dot{y})}, then derive \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} = 1, then obtain ((\\phi - \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} + \\cos{(\\dot{y})})^{\\phi})^{\\phi} \\frac{\\partial}{\\partial \\phi} k{(\\phi,\\dot{y})} = ((\\phi - \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} + \\cos{(\\dot{y})})^{\\phi})^{\\phi}", "derivation": "\\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} = \\phi + \\cos{(\\dot{y})} and \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} = \\frac{\\partial}{\\partial \\phi} (\\phi + \\cos{(\\dot{y})}) and \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} = 1 and k{(\\phi,\\dot{y})} = \\phi + \\cos{(\\dot{y})} and k{(\\phi,\\dot{y})} = \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} and \\frac{\\partial}{\\partial \\phi} k{(\\phi,\\dot{y})} = 1 and ((\\phi - \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} + \\cos{(\\dot{y})})^{\\phi})^{\\phi} \\frac{\\partial}{\\partial \\phi} k{(\\phi,\\dot{y})} = ((\\phi - \\operatorname{y^{\\prime}}{(\\phi,\\dot{y})} + \\cos{(\\dot{y})})^{\\phi})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1))"], [["times", 6, "Pow(Pow(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), cos(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Pow(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), cos(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Function('k')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Pow(Pow(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), cos(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\Omega{(v_{x},q)} = e^{- q + v_{x}}, then obtain (- \\frac{\\Omega{(v_{x},q)}}{q})^{v_{x}} - \\mathbf{P}{(\\sigma_x)} = (- \\frac{e^{- q + v_{x}}}{q})^{v_{x}} - \\mathbf{P}{(\\sigma_x)}", "derivation": "\\Omega{(v_{x},q)} = e^{- q + v_{x}} and - \\frac{\\Omega{(v_{x},q)}}{q} = - \\frac{e^{- q + v_{x}}}{q} and (- \\frac{\\Omega{(v_{x},q)}}{q})^{v_{x}} = (- \\frac{e^{- q + v_{x}}}{q})^{v_{x}} and (- \\frac{\\Omega{(v_{x},q)}}{q})^{v_{x}} - \\mathbf{P}{(\\sigma_x)} = (- \\frac{e^{- q + v_{x}}}{q})^{v_{x}} - \\mathbf{P}{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('v_x', commutative=True), Symbol('q', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('v_x', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True)))))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('v_x', commutative=True), Symbol('q', commutative=True))), Symbol('v_x', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)))"], [["minus", 3, "Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Pow(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('v_x', commutative=True), Symbol('q', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)))), Add(Pow(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\pi)} = \\log{(\\cos{(\\pi)})} and \\operatorname{F_{H}}{(\\pi)} = \\log{(\\cos{(\\pi)})}, then obtain \\operatorname{F_{N}}{(\\pi)} - \\cos{(\\pi)} = \\operatorname{F_{H}}{(\\pi)} - \\cos{(\\pi)}", "derivation": "\\operatorname{F_{N}}{(\\pi)} = \\log{(\\cos{(\\pi)})} and \\operatorname{F_{H}}{(\\pi)} = \\log{(\\cos{(\\pi)})} and \\operatorname{F_{N}}{(\\pi)} = \\operatorname{F_{H}}{(\\pi)} and \\operatorname{F_{N}}{(\\pi)} - \\cos{(\\pi)} = \\operatorname{F_{H}}{(\\pi)} - \\cos{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\pi', commutative=True)), log(cos(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\pi', commutative=True)), log(cos(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_N')(Symbol('\\\\pi', commutative=True)), Function('F_H')(Symbol('\\\\pi', commutative=True)))"], [["minus", 3, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('F_N')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Add(Function('F_H')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(s,f_{E})} = (e^{s})^{f_{E}}, then derive \\frac{\\partial}{\\partial s} \\mathbf{B}{(s,f_{E})} = f_{E} (e^{s})^{f_{E}}, then obtain \\frac{\\partial}{\\partial s} \\mathbf{B}{(s,f_{E})} = f_{E} \\mathbf{B}{(s,f_{E})}", "derivation": "\\mathbf{B}{(s,f_{E})} = (e^{s})^{f_{E}} and \\frac{\\partial}{\\partial s} \\mathbf{B}{(s,f_{E})} = \\frac{\\partial}{\\partial s} (e^{s})^{f_{E}} and \\frac{\\partial}{\\partial s} \\mathbf{B}{(s,f_{E})} = f_{E} (e^{s})^{f_{E}} and \\frac{\\partial}{\\partial s} \\mathbf{B}{(s,f_{E})} = f_{E} \\mathbf{B}{(s,f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('s', commutative=True), Symbol('f_E', commutative=True)), Pow(exp(Symbol('s', commutative=True)), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('s', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('s', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('s', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Symbol('f_E', commutative=True), Function('\\\\mathbf{B}')(Symbol('s', commutative=True), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\phi_2)} = e^{\\phi_2}, then obtain 0^{\\phi_2} = (f_{E} + e^{\\phi_2} - \\int \\operatorname{a^{\\dagger}}{(\\phi_2)} d\\phi_2)^{\\phi_2}", "derivation": "\\operatorname{a^{\\dagger}}{(\\phi_2)} = e^{\\phi_2} and \\int \\operatorname{a^{\\dagger}}{(\\phi_2)} d\\phi_2 = \\int e^{\\phi_2} d\\phi_2 and 0 = - \\int \\operatorname{a^{\\dagger}}{(\\phi_2)} d\\phi_2 + \\int e^{\\phi_2} d\\phi_2 and 0^{\\phi_2} = (- \\int \\operatorname{a^{\\dagger}}{(\\phi_2)} d\\phi_2 + \\int e^{\\phi_2} d\\phi_2)^{\\phi_2} and 0^{\\phi_2} = (f_{E} + e^{\\phi_2} - \\int \\operatorname{a^{\\dagger}}{(\\phi_2)} d\\phi_2)^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["minus", 2, "Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["power", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Mul(Integer(-1), Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Symbol('f_E', commutative=True), exp(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})} = \\eta^{\\prime} \\mathbf{E} and \\mathbf{F}{(\\mathbf{E},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbf{E}, then obtain 2 \\eta^{\\prime} \\mathbf{E} (\\eta^{\\prime} \\mathbf{E} + \\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})}) = 4 (\\eta^{\\prime})^{2} \\mathbf{E}^{2}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})} = \\eta^{\\prime} \\mathbf{E} and \\eta^{\\prime} \\mathbf{E} + \\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbf{E} and \\mathbf{F}{(\\mathbf{E},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbf{E} and (\\eta^{\\prime} \\mathbf{E} + \\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})}) \\mathbf{F}{(\\mathbf{E},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbf{E} \\mathbf{F}{(\\mathbf{E},\\eta^{\\prime})} and 2 \\eta^{\\prime} \\mathbf{E} (\\eta^{\\prime} \\mathbf{E} + \\operatorname{A_{1}}{(\\mathbf{E},\\eta^{\\prime})}) = 4 (\\eta^{\\prime})^{2} \\mathbf{E}^{2}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('A_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(4), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(B)} = \\int \\log{(B)} dB, then obtain 0 = 2 (- B + 2 \\int \\log{(B)} dB) \\int \\log{(B)} dB - 2 (- B + \\operatorname{A_{1}}{(B)} + \\int \\log{(B)} dB) \\int \\log{(B)} dB", "derivation": "\\operatorname{A_{1}}{(B)} = \\int \\log{(B)} dB and - B + \\operatorname{A_{1}}{(B)} = - B + \\int \\log{(B)} dB and - B + \\operatorname{A_{1}}{(B)} + \\int \\log{(B)} dB = - B + 2 \\int \\log{(B)} dB and 2 (- B + \\operatorname{A_{1}}{(B)} + \\int \\log{(B)} dB) \\int \\log{(B)} dB = 2 (- B + 2 \\int \\log{(B)} dB) \\int \\log{(B)} dB and 0 = 2 (- B + 2 \\int \\log{(B)} dB) \\int \\log{(B)} dB - 2 (- B + \\operatorname{A_{1}}{(B)} + \\int \\log{(B)} dB) \\int \\log{(B)} dB", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('A_1')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["add", 2, "Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('A_1')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(2), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"], [["times", 3, "Mul(Integer(2), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('A_1')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Integer(2), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(2), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["minus", 4, "Mul(Integer(2), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('A_1')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(2), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('A_1')(Symbol('B', commutative=True)), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integral(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(C)} = e^{C}, then derive (\\int \\operatorname{P_{e}}{(C)} dC)^{C} = (F_{H} + e^{C})^{C}, then obtain (\\int \\operatorname{P_{e}}{(C)} dC)^{C} = (F_{H} + \\operatorname{P_{e}}{(C)})^{C}", "derivation": "\\operatorname{P_{e}}{(C)} = e^{C} and \\int \\operatorname{P_{e}}{(C)} dC = \\int e^{C} dC and (\\int \\operatorname{P_{e}}{(C)} dC)^{C} = (\\int e^{C} dC)^{C} and (\\int \\operatorname{P_{e}}{(C)} dC)^{C} = (F_{H} + e^{C})^{C} and (\\int \\operatorname{P_{e}}{(C)} dC)^{C} = (F_{H} + \\operatorname{P_{e}}{(C)})^{C}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('F_H', commutative=True), exp(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('P_e')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('F_H', commutative=True), Function('P_e')(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(A_{2})} = e^{A_{2}}, then derive \\int \\mathbf{J}{(A_{2})} dA_{2} = A_{x} + e^{A_{2}}, then obtain \\int \\mathbf{J}{(A_{2})} dA_{2} = A_{x} + \\mathbf{J}{(A_{2})}", "derivation": "\\mathbf{J}{(A_{2})} = e^{A_{2}} and \\int \\mathbf{J}{(A_{2})} dA_{2} = \\int e^{A_{2}} dA_{2} and \\int \\mathbf{J}{(A_{2})} dA_{2} = A_{x} + e^{A_{2}} and \\int \\mathbf{J}{(A_{2})} dA_{2} = A_{x} + \\mathbf{J}{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Symbol('A_x', commutative=True), exp(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Symbol('A_x', commutative=True), Function('\\\\mathbf{J}')(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(M,P_{e})} = \\cos{(\\frac{M}{P_{e}})}, then obtain \\frac{P_{e} \\operatorname{r_{0}}^{P_{e}}{(M,P_{e})} \\frac{\\partial}{\\partial M} \\operatorname{r_{0}}{(M,P_{e})}}{\\operatorname{r_{0}}{(M,P_{e})}} = - \\frac{\\sin{(\\frac{M}{P_{e}})} \\cos^{P_{e}}{(\\frac{M}{P_{e}})}}{\\cos{(\\frac{M}{P_{e}})}}", "derivation": "\\operatorname{r_{0}}{(M,P_{e})} = \\cos{(\\frac{M}{P_{e}})} and \\operatorname{r_{0}}^{P_{e}}{(M,P_{e})} = \\cos^{P_{e}}{(\\frac{M}{P_{e}})} and \\frac{\\partial}{\\partial M} \\operatorname{r_{0}}^{P_{e}}{(M,P_{e})} = \\frac{\\partial}{\\partial M} \\cos^{P_{e}}{(\\frac{M}{P_{e}})} and \\frac{P_{e} \\operatorname{r_{0}}^{P_{e}}{(M,P_{e})} \\frac{\\partial}{\\partial M} \\operatorname{r_{0}}{(M,P_{e})}}{\\operatorname{r_{0}}{(M,P_{e})}} = - \\frac{\\sin{(\\frac{M}{P_{e}})} \\cos^{P_{e}}{(\\frac{M}{P_{e}})}}{\\cos{(\\frac{M}{P_{e}})}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), cos(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(cos(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Symbol('P_e', commutative=True)))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(cos(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Symbol('P_e', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('P_e', commutative=True), Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Integer(-1)), Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Derivative(Function('r_0')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Pow(cos(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Integer(-1)), Pow(cos(Mul(Symbol('M', commutative=True), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\omega{(f)} = \\log{(f)}, then obtain 4 \\omega^{f}{(f)} \\log{(f)}^{2 f} = 4 \\log{(f)}^{3 f}", "derivation": "\\omega{(f)} = \\log{(f)} and \\omega^{f}{(f)} = \\log{(f)}^{f} and \\omega^{f}{(f)} + \\log{(f)}^{f} = 2 \\log{(f)}^{f} and 2 (\\omega^{f}{(f)} + \\log{(f)}^{f}) \\omega^{f}{(f)} \\log{(f)}^{f} = 2 (\\omega^{f}{(f)} + \\log{(f)}^{f}) \\log{(f)}^{2 f} and 4 \\omega^{f}{(f)} \\log{(f)}^{2 f} = 4 \\log{(f)}^{3 f}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["add", 2, "Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Mul(Integer(2), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))))"], [["times", 2, "Mul(Integer(2), Add(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], "Equality(Mul(Integer(2), Add(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Mul(Integer(2), Add(Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True))), Pow(log(Symbol('f', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(4), Pow(Function('\\\\omega')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))), Mul(Integer(4), Pow(log(Symbol('f', commutative=True)), Mul(Integer(3), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(A_{y})} = \\sin{(A_{y})} and z{(S,W)} = \\cos{(W^{S})}, then obtain (\\varepsilon{(A_{y})} + z{(S,W)})^{A_{y}} = (z{(S,W)} + \\sin{(A_{y})})^{A_{y}}", "derivation": "\\varepsilon{(A_{y})} = \\sin{(A_{y})} and z{(S,W)} = \\cos{(W^{S})} and \\varepsilon{(A_{y})} + \\cos{(W^{S})} = \\sin{(A_{y})} + \\cos{(W^{S})} and (\\varepsilon{(A_{y})} + \\cos{(W^{S})})^{A_{y}} = (\\sin{(A_{y})} + \\cos{(W^{S})})^{A_{y}} and (\\varepsilon{(A_{y})} + z{(S,W)})^{A_{y}} = (z{(S,W)} + \\sin{(A_{y})})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], ["get_premise", "Equality(Function('z')(Symbol('S', commutative=True), Symbol('W', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True))))"], [["add", 1, "cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True)))), Add(sin(Symbol('A_y', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True)))))"], [["power", 3, "Symbol('A_y', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True)))), Symbol('A_y', commutative=True)), Pow(Add(sin(Symbol('A_y', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('S', commutative=True)))), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), Function('z')(Symbol('S', commutative=True), Symbol('W', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Function('z')(Symbol('S', commutative=True), Symbol('W', commutative=True)), sin(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given q{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and q{(J)} = \\log{(J)} and L{(J,\\mathbf{D})} = (- q{(J)} e^{- \\mathbf{D}})^{J}, then obtain L{(J,\\mathbf{D})} = (- e^{- \\mathbf{D}} \\log{(J)})^{J}", "derivation": "q{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and q{(\\mathbf{D})} e^{- \\mathbf{D}} = e^{- \\mathbf{D}} e^{e^{\\mathbf{D}}} and q{(J)} = \\log{(J)} and - q{(J)} e^{- e^{\\mathbf{D}}} = - e^{- e^{\\mathbf{D}}} \\log{(J)} and - q{(J)} q{(\\mathbf{D})} e^{- \\mathbf{D}} e^{- e^{\\mathbf{D}}} = - q{(\\mathbf{D})} e^{- \\mathbf{D}} e^{- e^{\\mathbf{D}}} \\log{(J)} and - q{(J)} e^{- \\mathbf{D}} = - e^{- \\mathbf{D}} \\log{(J)} and (- q{(J)} e^{- \\mathbf{D}})^{J} = (- e^{- \\mathbf{D}} \\log{(J)})^{J} and L{(J,\\mathbf{D})} = (- q{(J)} e^{- \\mathbf{D}})^{J} and L{(J,\\mathbf{D})} = (- e^{- \\mathbf{D}} \\log{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))))"], ["get_premise", "Equality(Function('q')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], "Equality(Mul(Integer(-1), Function('q')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))))), Mul(Integer(-1), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), log(Symbol('J', commutative=True))))"], [["times", 4, "Mul(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], "Equality(Mul(Integer(-1), Function('q')(Symbol('J', commutative=True)), Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))))), Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), exp(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), log(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Function('q')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), log(Symbol('J', commutative=True))))"], [["power", 6, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('q')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('J', commutative=True)), Pow(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), log(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('J', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Integer(-1), Function('q')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Function('L')(Symbol('J', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), log(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})}, then obtain \\int \\frac{\\int \\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} d\\hat{\\mathbf{r}} = \\int \\frac{\\int \\cos{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} d\\hat{\\mathbf{r}}", "derivation": "\\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\int \\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} = \\int \\cos{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} and \\frac{\\int \\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} = \\frac{\\int \\cos{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} and \\int \\frac{\\int \\operatorname{A_{2}}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} d\\hat{\\mathbf{r}} = \\int \\frac{\\int \\cos{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}}{2 \\cos{(\\hat{\\mathbf{r}})}} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 2, "Mul(Integer(2), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Integral(Function('A_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Rational(1, 2), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Integral(Function('A_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Integral(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given l{(p)} = \\log{(p)} and \\operatorname{P_{g}}{(\\mu)} = e^{\\mu}, then obtain (l^{p}{(p)})^{p} + 2 \\operatorname{P_{g}}{(\\mu)} = (l^{p}{(p)})^{p} + \\operatorname{P_{g}}{(\\mu)} + e^{\\mu}", "derivation": "l{(p)} = \\log{(p)} and l^{p}{(p)} = \\log{(p)}^{p} and (l^{p}{(p)})^{p} = (\\log{(p)}^{p})^{p} and \\operatorname{P_{g}}{(\\mu)} = e^{\\mu} and 2 \\operatorname{P_{g}}{(\\mu)} = \\operatorname{P_{g}}{(\\mu)} + e^{\\mu} and (\\log{(p)}^{p})^{p} + 2 \\operatorname{P_{g}}{(\\mu)} = (\\log{(p)}^{p})^{p} + \\operatorname{P_{g}}{(\\mu)} + e^{\\mu} and (l^{p}{(p)})^{p} + 2 \\operatorname{P_{g}}{(\\mu)} = (l^{p}{(p)})^{p} + \\operatorname{P_{g}}{(\\mu)} + e^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('l')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Pow(Function('l')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["add", 4, "Function('P_g')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(2), Function('P_g')(Symbol('\\\\mu', commutative=True))), Add(Function('P_g')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))))"], [["add", 5, "Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('\\\\mu', commutative=True)))), Add(Pow(Pow(log(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('P_g')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Pow(Pow(Function('l')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('\\\\mu', commutative=True)))), Add(Pow(Pow(Function('l')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('P_g')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(C_{1},\\mathbf{g})} = \\cos{(C_{1} - \\mathbf{g})}, then derive \\int \\psi^{*}{(C_{1},\\mathbf{g})} d\\mathbf{g} = k - \\sin{(C_{1} - \\mathbf{g})}, then obtain - \\sin{(C_{1} - \\mathbf{g})} = - k + \\int \\cos{(C_{1} - \\mathbf{g})} d\\mathbf{g}", "derivation": "\\psi^{*}{(C_{1},\\mathbf{g})} = \\cos{(C_{1} - \\mathbf{g})} and \\int \\psi^{*}{(C_{1},\\mathbf{g})} d\\mathbf{g} = \\int \\cos{(C_{1} - \\mathbf{g})} d\\mathbf{g} and \\int \\psi^{*}{(C_{1},\\mathbf{g})} d\\mathbf{g} = k - \\sin{(C_{1} - \\mathbf{g})} and k - \\sin{(C_{1} - \\mathbf{g})} = \\int \\cos{(C_{1} - \\mathbf{g})} d\\mathbf{g} and - \\sin{(C_{1} - \\mathbf{g})} = - k + \\int \\cos{(C_{1} - \\mathbf{g})} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(cos(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('k', commutative=True), Mul(Integer(-1), sin(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('k', commutative=True), Mul(Integer(-1), sin(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))), Integral(cos(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 4, "Symbol('k', commutative=True)"], "Equality(Mul(Integer(-1), sin(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(cos(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given t{(I,E_{x})} = \\cos{(E_{x} - I)} and \\psi^{*}{(I,E_{x})} = \\frac{t{(I,E_{x})} \\cos{(E_{x} - I)}}{E_{x}}, then obtain 0 = E_{x} (- \\psi^{*}{(I,E_{x})} + \\frac{t{(I,E_{x})} \\cos{(E_{x} - I)}}{E_{x}})", "derivation": "t{(I,E_{x})} = \\cos{(E_{x} - I)} and \\psi^{*}{(I,E_{x})} = \\frac{t{(I,E_{x})} \\cos{(E_{x} - I)}}{E_{x}} and \\psi^{*}{(I,E_{x})} = \\frac{\\cos^{2}{(E_{x} - I)}}{E_{x}} and 0 = - \\psi^{*}{(I,E_{x})} + \\frac{\\cos^{2}{(E_{x} - I)}}{E_{x}} and \\frac{\\cos^{2}{(E_{x} - I)}}{E_{x}} = \\frac{t{(I,E_{x})} \\cos{(E_{x} - I)}}{E_{x}} and 0 = E_{x} (- \\psi^{*}{(I,E_{x})} + \\frac{\\cos^{2}{(E_{x} - I)}}{E_{x}}) and 0 = E_{x} (- \\psi^{*}{(I,E_{x})} + \\frac{t{(I,E_{x})} \\cos{(E_{x} - I)}}{E_{x}})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('t')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Integer(2))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('t')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["divide", 4, "Pow(Symbol('E_x', commutative=True), Integer(-1))"], "Equality(Integer(0), Mul(Symbol('E_x', commutative=True), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Integer(2))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integer(0), Mul(Symbol('E_x', commutative=True), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('I', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('t')(Symbol('I', commutative=True), Symbol('E_x', commutative=True)), cos(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))))"]]}, {"prompt": "Given \\mu{(C_{d},\\Psi_{nl})} = C_{d}^{\\Psi_{nl}} and S{(C_{d})} = C_{d}, then obtain \\int \\mu{(C_{d},\\Psi_{nl})} dS{(C_{d})} = \\int C_{d}^{\\Psi_{nl}} dS{(C_{d})}", "derivation": "\\mu{(C_{d},\\Psi_{nl})} = C_{d}^{\\Psi_{nl}} and \\int \\mu{(C_{d},\\Psi_{nl})} dC_{d} = \\int C_{d}^{\\Psi_{nl}} dC_{d} and S{(C_{d})} = C_{d} and \\int \\mu{(C_{d},\\Psi_{nl})} dS{(C_{d})} = \\int C_{d}^{\\Psi_{nl}} dS{(C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Pow(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\mu')(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Function('S')(Symbol('C_d', commutative=True)))), Integral(Pow(Symbol('C_d', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Function('S')(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain (\\mathbf{M}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}} = (2 \\cos^{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}}", "derivation": "\\mathbf{M}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\mathbf{M}^{\\mathbf{r}}{(\\mathbf{r})} = \\cos^{\\mathbf{r}}{(\\mathbf{r})} and \\mathbf{M}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\mathbf{r})} = 2 \\cos^{\\mathbf{r}}{(\\mathbf{r})} and (\\mathbf{M}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}} = (2 \\cos^{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 2, "Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given p{(F_{N})} = \\sin{(F_{N})} and x{(F_{N})} = F_{N}, then obtain (- F_{N} + p{(F_{N})}) (2 p{(F_{N})} - x{(F_{N})}) = (- F_{N} + \\sin{(F_{N})}) (2 p{(F_{N})} - x{(F_{N})})", "derivation": "p{(F_{N})} = \\sin{(F_{N})} and x{(F_{N})} = F_{N} and p{(F_{N})} - x{(F_{N})} = - x{(F_{N})} + \\sin{(F_{N})} and - F_{N} + p{(F_{N})} = - F_{N} + \\sin{(F_{N})} and (- F_{N} + p{(F_{N})}) (2 p{(F_{N})} - x{(F_{N})}) = (- F_{N} + \\sin{(F_{N})}) (2 p{(F_{N})} - x{(F_{N})})", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["minus", 1, "Function('x')(Symbol('F_N', commutative=True))"], "Equality(Add(Function('p')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Function('x')(Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Function('x')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('p')(Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True))))"], [["times", 4, "Add(Mul(Integer(2), Function('p')(Symbol('F_N', commutative=True))), Mul(Integer(-1), Function('x')(Symbol('F_N', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('p')(Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Function('p')(Symbol('F_N', commutative=True))), Mul(Integer(-1), Function('x')(Symbol('F_N', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Function('p')(Symbol('F_N', commutative=True))), Mul(Integer(-1), Function('x')(Symbol('F_N', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(v_{t})} = \\frac{d}{d v_{t}} \\cos{(v_{t})}, then derive \\int \\operatorname{F_{N}}{(v_{t})} dv_{t} = \\tilde{g}^* + \\cos{(v_{t})}, then obtain (\\int \\frac{d}{d v_{t}} \\cos{(v_{t})} dv_{t})^{v_{t}} = (\\int \\operatorname{F_{N}}{(v_{t})} dv_{t})^{v_{t}}", "derivation": "\\operatorname{F_{N}}{(v_{t})} = \\frac{d}{d v_{t}} \\cos{(v_{t})} and \\int \\operatorname{F_{N}}{(v_{t})} dv_{t} = \\int \\frac{d}{d v_{t}} \\cos{(v_{t})} dv_{t} and \\int \\operatorname{F_{N}}{(v_{t})} dv_{t} = \\tilde{g}^* + \\cos{(v_{t})} and (\\int \\operatorname{F_{N}}{(v_{t})} dv_{t})^{v_{t}} = (\\tilde{g}^* + \\cos{(v_{t})})^{v_{t}} and (\\int \\frac{d}{d v_{t}} \\cos{(v_{t})} dv_{t})^{v_{t}} = (\\tilde{g}^* + \\cos{(v_{t})})^{v_{t}} and (\\int \\frac{d}{d v_{t}} \\cos{(v_{t})} dv_{t})^{v_{t}} = (\\int \\operatorname{F_{N}}{(v_{t})} dv_{t})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('v_t', commutative=True)), Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_N')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), cos(Symbol('v_t', commutative=True))))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Function('F_N')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), cos(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), cos(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integral(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Function('F_N')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\hat{x})} = \\cos{(\\hat{x})} and V{(\\hat{x})} = \\hat{x}, then obtain \\frac{d}{d \\hat{x}} \\frac{V{(\\hat{x})}}{\\cos{(\\hat{x})}} = \\frac{d}{d \\hat{x}} \\frac{\\hat{x}}{\\cos{(\\hat{x})}}", "derivation": "\\phi{(\\hat{x})} = \\cos{(\\hat{x})} and V{(\\hat{x})} = \\hat{x} and \\frac{V{(\\hat{x})}}{\\cos{(\\hat{x})}} = \\frac{\\hat{x}}{\\cos{(\\hat{x})}} and \\frac{V{(\\hat{x})}}{\\phi{(\\hat{x})}} = \\frac{\\hat{x}}{\\phi{(\\hat{x})}} and \\frac{d}{d \\hat{x}} \\frac{V{(\\hat{x})}}{\\phi{(\\hat{x})}} = \\frac{d}{d \\hat{x}} \\frac{\\hat{x}}{\\phi{(\\hat{x})}} and \\frac{d}{d \\hat{x}} \\frac{V{(\\hat{x})}}{\\cos{(\\hat{x})}} = \\frac{d}{d \\hat{x}} \\frac{\\hat{x}}{\\cos{(\\hat{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], [["divide", 2, "cos(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Mul(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Function('V')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(cos(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(z^{*},\\sigma_p)} = (z^{*})^{\\sigma_p}, then obtain \\frac{\\partial}{\\partial \\sigma_p} (z^{*} \\Omega{(z^{*},\\sigma_p)})^{\\sigma_p} = \\frac{\\partial}{\\partial \\sigma_p} (z^{*} (z^{*})^{\\sigma_p})^{\\sigma_p}", "derivation": "\\Omega{(z^{*},\\sigma_p)} = (z^{*})^{\\sigma_p} and z^{*} \\Omega{(z^{*},\\sigma_p)} = z^{*} (z^{*})^{\\sigma_p} and (z^{*} \\Omega{(z^{*},\\sigma_p)})^{\\sigma_p} = (z^{*} (z^{*})^{\\sigma_p})^{\\sigma_p} and \\frac{\\partial}{\\partial \\sigma_p} (z^{*} \\Omega{(z^{*},\\sigma_p)})^{\\sigma_p} = \\frac{\\partial}{\\partial \\sigma_p} (z^{*} (z^{*})^{\\sigma_p})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('z^*', commutative=True), Pow(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Symbol('z^*', commutative=True), Pow(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('z^*', commutative=True), Pow(Symbol('z^*', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(F_{g},\\dot{x})} = \\dot{x} + e^{F_{g}} and \\operatorname{A_{x}}{(F_{g},\\dot{x})} = \\dot{x} + e^{F_{g}}, then obtain \\operatorname{A_{x}}^{2}{(F_{g},\\dot{x})} \\operatorname{v_{y}}^{2}{(F_{g},\\dot{x})} = \\operatorname{A_{x}}^{4}{(F_{g},\\dot{x})}", "derivation": "\\operatorname{v_{y}}{(F_{g},\\dot{x})} = \\dot{x} + e^{F_{g}} and \\operatorname{A_{x}}{(F_{g},\\dot{x})} = \\dot{x} + e^{F_{g}} and (\\dot{x} + e^{F_{g}}) \\operatorname{v_{y}}{(F_{g},\\dot{x})} = (\\dot{x} + e^{F_{g}})^{2} and \\operatorname{A_{x}}{(F_{g},\\dot{x})} \\operatorname{v_{y}}{(F_{g},\\dot{x})} = \\operatorname{A_{x}}^{2}{(F_{g},\\dot{x})} and \\operatorname{A_{x}}^{2}{(F_{g},\\dot{x})} \\operatorname{v_{y}}^{2}{(F_{g},\\dot{x})} = \\operatorname{A_{x}}^{4}{(F_{g},\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('F_g', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('F_g', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('F_g', commutative=True))), Function('v_y')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Pow(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('F_g', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('A_x')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Function('v_y')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Pow(Function('A_x')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)))"], [["power", 4, 2], "Equality(Mul(Pow(Function('A_x')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Pow(Function('v_y')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2))), Pow(Function('A_x')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(4)))"]]}, {"prompt": "Given T{(\\Psi)} = \\cos{(\\Psi)}, then obtain \\frac{- T{(\\Psi)} + T^{\\Psi}{(\\Psi)}}{\\Psi} = \\frac{- T{(\\Psi)} + \\cos^{\\Psi}{(\\Psi)}}{\\Psi}", "derivation": "T{(\\Psi)} = \\cos{(\\Psi)} and T^{\\Psi}{(\\Psi)} = \\cos^{\\Psi}{(\\Psi)} and - T{(\\Psi)} + T^{\\Psi}{(\\Psi)} = - T{(\\Psi)} + \\cos^{\\Psi}{(\\Psi)} and \\frac{- T{(\\Psi)} + T^{\\Psi}{(\\Psi)}}{\\Psi} = \\frac{- T{(\\Psi)} + \\cos^{\\Psi}{(\\Psi)}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('T')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["minus", 2, "Function('T')(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('\\\\Psi', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\Psi', commutative=True))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))))"], [["divide", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\Psi', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\Psi', commutative=True))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(V,A)} = \\frac{\\partial}{\\partial V} A V, then derive 1 = \\frac{A}{\\operatorname{c_{0}}{(V,A)}}, then obtain 1 = \\frac{A}{\\frac{\\partial}{\\partial V} A V}", "derivation": "\\operatorname{c_{0}}{(V,A)} = \\frac{\\partial}{\\partial V} A V and A V \\operatorname{c_{0}}{(V,A)} = A V \\frac{\\partial}{\\partial V} A V and 1 = \\frac{\\frac{\\partial}{\\partial V} A V}{\\operatorname{c_{0}}{(V,A)}} and 1 = \\frac{A}{\\operatorname{c_{0}}{(V,A)}} and 1 = \\frac{A}{\\frac{\\partial}{\\partial V} A V}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Derivative(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["times", 1, "Mul(Symbol('A', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True), Function('c_0')(Symbol('V', commutative=True), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('V', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["divide", 2, "Mul(Symbol('A', commutative=True), Symbol('V', commutative=True), Function('c_0')(Symbol('V', commutative=True), Symbol('A', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('c_0')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(Symbol('A', commutative=True), Pow(Function('c_0')(Symbol('V', commutative=True), Symbol('A', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Symbol('A', commutative=True), Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\operatorname{P_{g}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})}, then obtain \\cos{(\\mathbf{P})} = \\frac{\\operatorname{P_{g}}{(\\mathbf{P})} \\cos{(\\mathbf{P})}}{\\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})}}", "derivation": "\\mathbf{D}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\mathbf{D}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} and \\operatorname{P_{g}}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} and 1 = \\frac{\\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})}}{\\frac{d}{d \\mathbf{P}} \\mathbf{D}{(\\mathbf{P})}} and 1 = \\frac{\\operatorname{P_{g}}{(\\mathbf{P})}}{\\frac{d}{d \\mathbf{P}} \\mathbf{D}{(\\mathbf{P})}} and \\cos{(\\mathbf{P})} = \\frac{\\operatorname{P_{g}}{(\\mathbf{P})} \\cos{(\\mathbf{P})}}{\\frac{d}{d \\mathbf{P}} \\mathbf{D}{(\\mathbf{P})}} and \\cos{(\\mathbf{P})} = \\frac{\\operatorname{P_{g}}{(\\mathbf{P})} \\cos{(\\mathbf{P})}}{\\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1)), Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1))))"], [["times", 5, "cos(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(cos(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(cos(Symbol('\\\\mathbf{P}', commutative=True)), Mul(Function('P_g')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(a,n_{1})} = a - n_{1}, then obtain n_{1} + \\int (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})})^{2} da = n_{1} + \\int (a - 2 n_{1}) (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})}) da", "derivation": "\\operatorname{v_{z}}{(a,n_{1})} = a - n_{1} and - n_{1} + \\operatorname{v_{z}}{(a,n_{1})} = a - 2 n_{1} and (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})})^{2} = (a - 2 n_{1}) (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})}) and \\int (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})})^{2} da = \\int (a - 2 n_{1}) (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})}) da and n_{1} + \\int (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})})^{2} da = n_{1} + \\int (a - 2 n_{1}) (- n_{1} + \\operatorname{v_{z}}{(a,n_{1})}) da", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True))))"], [["divide", 2, "Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True))), Integer(-1))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True))), Integer(2)), Mul(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True)))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True))), Integer(2)), Tuple(Symbol('a', commutative=True))), Integral(Mul(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('a', commutative=True))))"], [["add", 4, "Symbol('n_1', commutative=True)"], "Equality(Add(Symbol('n_1', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True))), Integer(2)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('n_1', commutative=True), Integral(Mul(Add(Symbol('a', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('v_z')(Symbol('a', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\delta{(F_{N})} = \\log{(\\sin{(F_{N})})}, then obtain \\frac{d}{d F_{N}} (\\delta^{2}{(F_{N})} \\log{(\\sin{(F_{N})})} - \\delta^{2}{(F_{N})}) = \\frac{d}{d F_{N}} (- \\delta^{2}{(F_{N})} + \\delta{(F_{N})} \\log{(\\sin{(F_{N})})}^{2})", "derivation": "\\delta{(F_{N})} = \\log{(\\sin{(F_{N})})} and \\delta{(F_{N})} \\log{(\\sin{(F_{N})})} = \\log{(\\sin{(F_{N})})}^{2} and \\delta^{2}{(F_{N})} \\log{(\\sin{(F_{N})})} = \\delta{(F_{N})} \\log{(\\sin{(F_{N})})}^{2} and \\delta^{2}{(F_{N})} \\log{(\\sin{(F_{N})})} - \\delta^{2}{(F_{N})} = - \\delta^{2}{(F_{N})} + \\delta{(F_{N})} \\log{(\\sin{(F_{N})})}^{2} and \\frac{d}{d F_{N}} (\\delta^{2}{(F_{N})} \\log{(\\sin{(F_{N})})} - \\delta^{2}{(F_{N})}) = \\frac{d}{d F_{N}} (- \\delta^{2}{(F_{N})} + \\delta{(F_{N})} \\log{(\\sin{(F_{N})})}^{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_N', commutative=True)), log(sin(Symbol('F_N', commutative=True))))"], [["times", 1, "log(sin(Symbol('F_N', commutative=True)))"], "Equality(Mul(Function('\\\\delta')(Symbol('F_N', commutative=True)), log(sin(Symbol('F_N', commutative=True)))), Pow(log(sin(Symbol('F_N', commutative=True))), Integer(2)))"], [["times", 2, "Function('\\\\delta')(Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), log(sin(Symbol('F_N', commutative=True)))), Mul(Function('\\\\delta')(Symbol('F_N', commutative=True)), Pow(log(sin(Symbol('F_N', commutative=True))), Integer(2))))"], [["minus", 3, "Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2))"], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), log(sin(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2))), Mul(Function('\\\\delta')(Symbol('F_N', commutative=True)), Pow(log(sin(Symbol('F_N', commutative=True))), Integer(2)))))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), log(sin(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2))), Mul(Function('\\\\delta')(Symbol('F_N', commutative=True)), Pow(log(sin(Symbol('F_N', commutative=True))), Integer(2)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(h,y)} = h + y, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(h,y)} dy = \\sigma_p + h y + \\frac{y^{2}}{2}, then obtain \\frac{2 \\int (h + y) dy}{y^{2}} = \\frac{2 (\\sigma_p + h y + \\frac{y^{2}}{2})}{y^{2}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(h,y)} = h + y and \\int \\operatorname{f_{\\mathbf{p}}}{(h,y)} dy = \\int (h + y) dy and \\int \\operatorname{f_{\\mathbf{p}}}{(h,y)} dy = \\sigma_p + h y + \\frac{y^{2}}{2} and \\int (h + y) dy = \\sigma_p + h y + \\frac{y^{2}}{2} and \\frac{2 \\int (h + y) dy}{y^{2}} = \\frac{2 (\\sigma_p + h y + \\frac{y^{2}}{2})}{y^{2}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True), Symbol('y', commutative=True)), Add(Symbol('h', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('h', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('h', commutative=True), Symbol('y', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('h', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('h', commutative=True), Symbol('y', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))))"], [["divide", 4, "Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('y', commutative=True), Integer(-2)), Integral(Add(Symbol('h', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(Integer(2), Pow(Symbol('y', commutative=True), Integer(-2)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('h', commutative=True), Symbol('y', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))))))"]]}, {"prompt": "Given t{(t_{1},\\rho_f)} = \\frac{\\rho_f}{t_{1}}, then obtain \\frac{\\partial}{\\partial t_{1}} t{(t_{1},\\rho_f)} - 1 + \\frac{1}{t_{1}^{2}} = - \\frac{\\rho_f}{t_{1}^{2}} - 1 + \\frac{1}{t_{1}^{2}}", "derivation": "t{(t_{1},\\rho_f)} = \\frac{\\rho_f}{t_{1}} and - t_{1} + t{(t_{1},\\rho_f)} = \\frac{\\rho_f}{t_{1}} - t_{1} and - t_{1} + t{(t_{1},\\rho_f)} - \\frac{1}{t_{1}} = \\frac{\\rho_f}{t_{1}} - t_{1} - \\frac{1}{t_{1}} and \\frac{\\partial}{\\partial t_{1}} (- t_{1} + t{(t_{1},\\rho_f)} - \\frac{1}{t_{1}}) = \\frac{\\partial}{\\partial t_{1}} (\\frac{\\rho_f}{t_{1}} - t_{1} - \\frac{1}{t_{1}}) and \\frac{\\partial}{\\partial t_{1}} t{(t_{1},\\rho_f)} - 1 + \\frac{1}{t_{1}^{2}} = - \\frac{\\rho_f}{t_{1}^{2}} - 1 + \\frac{1}{t_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('t_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('t')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["minus", 2, "Pow(Symbol('t_1', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('t')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)))))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('t')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\rho_f', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('t')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-2))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-2))), Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\hat{x}{(v)} = \\sin{(v)}, then obtain \\frac{v + \\int \\hat{x}{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\sin{(v)} dv dv = \\frac{v + \\int \\sin{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\sin{(v)} dv dv", "derivation": "\\hat{x}{(v)} = \\sin{(v)} and \\int \\hat{x}{(v)} dv = \\int \\sin{(v)} dv and v + \\int \\hat{x}{(v)} dv = v + \\int \\sin{(v)} dv and \\iint \\hat{x}{(v)} dv dv = \\iint \\sin{(v)} dv dv and \\frac{v + \\int \\hat{x}{(v)} dv}{\\sin{(v)}} = \\frac{v + \\int \\sin{(v)} dv}{\\sin{(v)}} and \\frac{v + \\int \\hat{x}{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\hat{x}{(v)} dv dv = \\frac{v + \\int \\sin{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\hat{x}{(v)} dv dv and \\frac{v + \\int \\hat{x}{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\sin{(v)} dv dv = \\frac{v + \\int \\sin{(v)} dv}{\\sin{(v)}} + \\frac{\\partial}{\\partial B} B r - \\iint \\sin{(v)} dv dv", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Symbol('v', commutative=True), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 3, "sin(Symbol('v', commutative=True))"], "Equality(Mul(Add(Symbol('v', commutative=True), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Mul(Add(Symbol('v', commutative=True), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))))"], [["add", 5, "Add(Derivative(Mul(Symbol('B', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], "Equality(Add(Mul(Add(Symbol('v', commutative=True), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))), Add(Mul(Add(Symbol('v', commutative=True), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Add(Symbol('v', commutative=True), Integral(Function('\\\\hat{x}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))), Add(Mul(Add(Symbol('v', commutative=True), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given G{(\\dot{\\mathbf{r}},\\hat{x},A_{1})} = \\dot{\\mathbf{r}}^{A_{1}} + \\hat{x} and \\mathbf{r}{(A_{1},\\dot{\\mathbf{r}},\\hat{x})} = (\\dot{\\mathbf{r}}^{A_{1}} + \\hat{x}) G{(\\dot{\\mathbf{r}},\\hat{x},A_{1})}, then obtain - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + \\mathbf{r}{(A_{1},\\dot{\\mathbf{r}},\\hat{x})} = - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + (\\dot{\\mathbf{r}}^{A_{1}} + \\hat{x})^{2}", "derivation": "G{(\\dot{\\mathbf{r}},\\hat{x},A_{1})} = \\dot{\\mathbf{r}}^{A_{1}} + \\hat{x} and \\mathbf{r}{(A_{1},\\dot{\\mathbf{r}},\\hat{x})} = (\\dot{\\mathbf{r}}^{A_{1}} + \\hat{x}) G{(\\dot{\\mathbf{r}},\\hat{x},A_{1})} and - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + \\mathbf{r}{(A_{1},\\dot{\\mathbf{r}},\\hat{x})} = - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + (\\dot{\\mathbf{r}}^{A_{1}} + \\hat{x}) G{(\\dot{\\mathbf{r}},\\hat{x},A_{1})} and - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + \\mathbf{r}{(A_{1},\\dot{\\mathbf{r}},\\hat{x})} = - \\dot{\\mathbf{r}}^{A_{1}} - \\hat{x} + (\\dot{\\mathbf{r}}^{A_{1}} + \\hat{x})^{2}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))))"], [["minus", 2, "Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{r}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Mul(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{r}')(Symbol('A_1', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and E{(\\mathbf{J})} = \\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})}, then obtain (E^{2}{(\\mathbf{J})})^{2 \\mathbf{J}} = (2 E{(\\mathbf{J})} \\sin{(\\mathbf{J})})^{2 \\mathbf{J}}", "derivation": "\\mathbf{H}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})} = 2 \\sin{(\\mathbf{J})} and (\\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})})^{2} = 2 (\\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})}) \\sin{(\\mathbf{J})} and ((\\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})})^{2})^{\\mathbf{J}} = (2 (\\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})}) \\sin{(\\mathbf{J})})^{\\mathbf{J}} and E{(\\mathbf{J})} = \\mathbf{H}{(\\mathbf{J})} + \\sin{(\\mathbf{J})} and (E^{2}{(\\mathbf{J})})^{\\mathbf{J}} = (2 E{(\\mathbf{J})} \\sin{(\\mathbf{J})})^{\\mathbf{J}} and (E^{2}{(\\mathbf{J})})^{2 \\mathbf{J}} = (2 E{(\\mathbf{J})} \\sin{(\\mathbf{J})})^{2 \\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 2, "Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Pow(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(2)), Mul(Integer(2), Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(2)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Mul(Integer(2), Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), sin(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{J}', commutative=True)), Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Pow(Function('E')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Mul(Integer(2), Function('E')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 6, 2], "Equality(Pow(Pow(Function('E')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Mul(Integer(2), Function('E')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given H{(\\mathbf{r})} = \\sin{(\\sin{(\\mathbf{r})})}, then derive \\int \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} d\\mathbf{r} = P_{e} + \\mathbf{r}, then obtain \\iint \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} d\\mathbf{r} dP_{e} = \\int (P_{e} + \\mathbf{r}) dP_{e}", "derivation": "H{(\\mathbf{r})} = \\sin{(\\sin{(\\mathbf{r})})} and \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} = 1 and \\int \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} d\\mathbf{r} = \\int 1 d\\mathbf{r} and \\int \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} d\\mathbf{r} = P_{e} + \\mathbf{r} and \\iint \\frac{H{(\\mathbf{r})}}{\\sin{(\\sin{(\\mathbf{r})})}} d\\mathbf{r} dP_{e} = \\int (P_{e} + \\mathbf{r}) dP_{e}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{r}', commutative=True)), sin(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Function('H')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Function('H')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('H')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Mul(Function('H')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given Z{(\\phi_2,U,\\dot{z})} = \\frac{U \\dot{z}}{\\phi_2}, then derive - \\frac{\\partial}{\\partial \\dot{z}} Z{(\\phi_2,U,\\dot{z})} = - \\frac{U}{\\phi_2}, then obtain \\frac{\\partial}{\\partial \\phi_2} - \\frac{\\partial}{\\partial \\dot{z}} \\frac{U \\dot{z}}{\\phi_2} = \\frac{\\partial}{\\partial \\phi_2} - \\frac{U}{\\phi_2}", "derivation": "Z{(\\phi_2,U,\\dot{z})} = \\frac{U \\dot{z}}{\\phi_2} and - Z{(\\phi_2,U,\\dot{z})} = - \\frac{U \\dot{z}}{\\phi_2} and \\frac{\\partial}{\\partial \\dot{z}} - Z{(\\phi_2,U,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} - \\frac{U \\dot{z}}{\\phi_2} and - \\frac{\\partial}{\\partial \\dot{z}} Z{(\\phi_2,U,\\dot{z})} = - \\frac{U}{\\phi_2} and - \\frac{\\partial}{\\partial \\dot{z}} \\frac{U \\dot{z}}{\\phi_2} = - \\frac{U}{\\phi_2} and \\frac{\\partial}{\\partial \\phi_2} - \\frac{\\partial}{\\partial \\dot{z}} \\frac{U \\dot{z}}{\\phi_2} = \\frac{\\partial}{\\partial \\phi_2} - \\frac{U}{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('U', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('U', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('U', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{J}_f)} = \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f, then obtain (\\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\dot{x}{(\\mathbf{J}_f)})^{2} = (\\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f)^{2}", "derivation": "\\dot{x}{(\\mathbf{J}_f)} = \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f and \\frac{d}{d \\mathbf{J}_f} \\dot{x}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f and \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\dot{x}{(\\mathbf{J}_f)} = \\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f and (\\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\dot{x}{(\\mathbf{J}_f)})^{2} = (\\frac{d^{2}}{d \\mathbf{J}_f^{2}} \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Derivative(Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{S}{(s)} = e^{s}, then obtain \\int (s \\mathbf{S}{(s)} e^{s} - e^{2 s}) ds = \\delta + \\frac{(2 s - 3) e^{2 s}}{4}", "derivation": "\\mathbf{S}{(s)} = e^{s} and s \\mathbf{S}{(s)} = s e^{s} and s \\mathbf{S}^{2}{(s)} = s \\mathbf{S}{(s)} e^{s} and s \\mathbf{S}{(s)} e^{s} = s e^{2 s} and s \\mathbf{S}{(s)} e^{s} - e^{2 s} = s e^{2 s} - e^{2 s} and \\int (s \\mathbf{S}{(s)} e^{s} - e^{2 s}) ds = \\int (s e^{2 s} - e^{2 s}) ds and \\int (s \\mathbf{S}{(s)} e^{s} - e^{2 s}) ds = \\delta + \\frac{(2 s - 3) e^{2 s}}{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["times", 1, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True))), Mul(Symbol('s', commutative=True), exp(Symbol('s', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{S}')(Symbol('s', commutative=True))"], "Equality(Mul(Symbol('s', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), Integer(2))), Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Symbol('s', commutative=True), exp(Mul(Integer(2), Symbol('s', commutative=True)))))"], [["minus", 4, "exp(Mul(Integer(2), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('s', commutative=True))))), Add(Mul(Symbol('s', commutative=True), exp(Mul(Integer(2), Symbol('s', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('s', commutative=True))))))"], [["integrate", 5, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('s', commutative=True))))), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Symbol('s', commutative=True), exp(Mul(Integer(2), Symbol('s', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('s', commutative=True))))), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Add(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('s', commutative=True))))), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 4), Add(Mul(Integer(2), Symbol('s', commutative=True)), Integer(-3)), exp(Mul(Integer(2), Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbb{I},\\mathbf{p})} = \\frac{e^{\\mathbb{I}}}{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\mathbf{F}{(\\mathbb{I},\\mathbf{p})} = \\frac{e^{\\mathbb{I}}}{\\mathbf{p}}, then obtain 0 = - \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} + \\frac{e^{\\mathbb{I}}}{\\mathbf{p}}", "derivation": "\\mathbf{F}{(\\mathbb{I},\\mathbf{p})} = \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbb{I}} \\mathbf{F}{(\\mathbb{I},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbb{I}} \\mathbf{F}{(\\mathbb{I},\\mathbf{p})} = \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} = \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} and 0 = - \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{e^{\\mathbb{I}}}{\\mathbf{p}} + \\frac{e^{\\mathbb{I}}}{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 4, "Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{B})} = \\cos{(\\mathbf{B})}, then obtain - r - \\sin{(\\mathbf{B})} + \\int \\Psi_{\\lambda}{(\\mathbf{B})} d\\mathbf{B} = 0", "derivation": "\\Psi_{\\lambda}{(\\mathbf{B})} = \\cos{(\\mathbf{B})} and \\int \\Psi_{\\lambda}{(\\mathbf{B})} d\\mathbf{B} = \\int \\cos{(\\mathbf{B})} d\\mathbf{B} and \\int \\Psi_{\\lambda}{(\\mathbf{B})} d\\mathbf{B} - \\int \\cos{(\\mathbf{B})} d\\mathbf{B} = 0 and - r - \\sin{(\\mathbf{B})} + \\int \\Psi_{\\lambda}{(\\mathbf{B})} d\\mathbf{B} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 2, "Integral(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given k{(\\mu,\\mathbf{J}_M)} = \\mathbf{J}_M \\mu, then obtain \\frac{\\partial}{\\partial \\mu} (- (\\mathbf{J}_M \\mu)^{\\mathbf{J}_M} + k^{\\mathbf{J}_M}{(\\mu,\\mathbf{J}_M)}) = \\frac{d}{d \\mu} 0", "derivation": "k{(\\mu,\\mathbf{J}_M)} = \\mathbf{J}_M \\mu and k^{\\mathbf{J}_M}{(\\mu,\\mathbf{J}_M)} = (\\mathbf{J}_M \\mu)^{\\mathbf{J}_M} and - (\\mathbf{J}_M \\mu)^{\\mathbf{J}_M} + k^{\\mathbf{J}_M}{(\\mu,\\mathbf{J}_M)} = 0 and \\frac{\\partial}{\\partial \\mu} (- (\\mathbf{J}_M \\mu)^{\\mathbf{J}_M} + k^{\\mathbf{J}_M}{(\\mu,\\mathbf{J}_M)}) = \\frac{d}{d \\mu} 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 2, "Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('k')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('k')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain - \\mathbf{s} + \\operatorname{g^{\\prime}_{\\varepsilon}}^{16}{(\\mathbf{s})} = - \\mathbf{s} + \\cos^{16}{(\\mathbf{s})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{s})} \\cos{(\\mathbf{s})} = \\cos^{2}{(\\mathbf{s})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(\\mathbf{s})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{s})} \\cos{(\\mathbf{s})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(\\mathbf{s})} = \\cos^{2}{(\\mathbf{s})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{4}{(\\mathbf{s})} = \\cos^{4}{(\\mathbf{s})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{16}{(\\mathbf{s})} = \\cos^{16}{(\\mathbf{s})} and - \\mathbf{s} + \\operatorname{g^{\\prime}_{\\varepsilon}}^{16}{(\\mathbf{s})} = - \\mathbf{s} + \\cos^{16}{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)))"], [["times", 1, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(4)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(4)))"], [["power", 5, 4], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(16)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(16)))"], [["minus", 6, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(16))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(16))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{p}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0, then derive \\tilde{g}^*{(\\hat{p}_0,\\eta^{\\prime})} = \\hat{p}_0, then obtain - \\sin{(\\eta)} + \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0}{\\eta^{\\prime}} = - \\sin{(\\eta)} + \\frac{\\hat{p}_0}{\\eta^{\\prime}}", "derivation": "\\tilde{g}^*{(\\hat{p}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0 and \\tilde{g}^*{(\\hat{p}_0,\\eta^{\\prime})} = \\hat{p}_0 and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0 = \\hat{p}_0 and \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0}{\\eta^{\\prime}} = \\frac{\\hat{p}_0}{\\eta^{\\prime}} and - \\sin{(\\eta)} + \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} \\hat{p}_0}{\\eta^{\\prime}} = - \\sin{(\\eta)} + \\frac{\\hat{p}_0}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))"], [["divide", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 4, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(A_{z})} = e^{A_{z}}, then obtain \\mathbf{J}{(A_{z})} \\iint \\mathbf{J}^{2}{(A_{z})} dA_{z} dA_{z} = \\mathbf{J}{(A_{z})} \\iint \\mathbf{J}{(A_{z})} e^{A_{z}} dA_{z} dA_{z}", "derivation": "\\mathbf{J}{(A_{z})} = e^{A_{z}} and \\mathbf{J}^{2}{(A_{z})} = \\mathbf{J}{(A_{z})} e^{A_{z}} and \\int \\mathbf{J}^{2}{(A_{z})} dA_{z} = \\int \\mathbf{J}{(A_{z})} e^{A_{z}} dA_{z} and \\iint \\mathbf{J}^{2}{(A_{z})} dA_{z} dA_{z} = \\iint \\mathbf{J}{(A_{z})} e^{A_{z}} dA_{z} dA_{z} and e^{A_{z}} \\iint \\mathbf{J}^{2}{(A_{z})} dA_{z} dA_{z} = e^{A_{z}} \\iint \\mathbf{J}{(A_{z})} e^{A_{z}} dA_{z} dA_{z} and \\mathbf{J}{(A_{z})} \\iint \\mathbf{J}^{2}{(A_{z})} dA_{z} dA_{z} = \\mathbf{J}{(A_{z})} \\iint \\mathbf{J}{(A_{z})} e^{A_{z}} dA_{z} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integer(2)), Tuple(Symbol('A_z', commutative=True))), Integral(Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True))))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integer(2)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["times", 4, "exp(Symbol('A_z', commutative=True))"], "Equality(Mul(exp(Symbol('A_z', commutative=True)), Integral(Pow(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integer(2)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(exp(Symbol('A_z', commutative=True)), Integral(Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integral(Pow(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integer(2)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), Integral(Mul(Function('\\\\mathbf{J}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} = - S + \\Psi_{nl} + k, then obtain (- k + \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk = (\\tilde{g}^* + \\frac{k^{2}}{2} + k (- S + \\Psi_{nl}) - k) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk", "derivation": "\\Psi_{\\lambda}{(S,k,\\Psi_{nl})} = - S + \\Psi_{nl} + k and \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk = \\int (- S + \\Psi_{nl} + k) dk and - k + \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk = - k + \\int (- S + \\Psi_{nl} + k) dk and (- k + \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk = (- k + \\int (- S + \\Psi_{nl} + k) dk) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk and (- k + \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk = (\\tilde{g}^* + \\frac{k^{2}}{2} + k (- S + \\Psi_{nl}) - k) \\int \\Psi_{\\lambda}{(S,k,\\Psi_{nl})} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["minus", 2, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["times", 3, "Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2))), Mul(Symbol('k', commutative=True), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('k', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{J}_M)} = \\mathbf{J}_M and m{(f_{\\mathbf{v}},z^{*})} = \\log{(f_{\\mathbf{v}} + z^{*})}, then obtain - \\hat{p}^{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} + m{(f_{\\mathbf{v}},z^{*})} = - \\hat{p}^{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} + \\log{(f_{\\mathbf{v}} + z^{*})}", "derivation": "\\hat{p}{(\\mathbf{J}_M)} = \\mathbf{J}_M and m{(f_{\\mathbf{v}},z^{*})} = \\log{(f_{\\mathbf{v}} + z^{*})} and - \\mathbf{J}_M^{f_{\\mathbf{v}}} + m{(f_{\\mathbf{v}},z^{*})} = - \\mathbf{J}_M^{f_{\\mathbf{v}}} + \\log{(f_{\\mathbf{v}} + z^{*})} and - \\hat{p}^{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} + m{(f_{\\mathbf{v}},z^{*})} = - \\hat{p}^{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} + \\log{(f_{\\mathbf{v}} + z^{*})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], ["get_premise", "Equality(Function('m')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True)), log(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('m')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('m')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Pow(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} = (e^{\\Psi_{\\lambda}})^{\\Omega}, then obtain 1 - \\int (\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}) d\\Psi_{\\lambda} = - \\int (\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}) d\\Psi_{\\lambda} + \\frac{\\Omega + 2 (e^{\\Psi_{\\lambda}})^{\\Omega}}{\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}}", "derivation": "\\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} = (e^{\\Psi_{\\lambda}})^{\\Omega} and \\Omega + \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} = \\Omega + (e^{\\Psi_{\\lambda}})^{\\Omega} and \\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} = \\Omega + \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} + (e^{\\Psi_{\\lambda}})^{\\Omega} and \\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)} = \\Omega + 2 (e^{\\Psi_{\\lambda}})^{\\Omega} and 1 = \\frac{\\Omega + 2 (e^{\\Psi_{\\lambda}})^{\\Omega}}{\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}} and 1 - \\int (\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}) d\\Psi_{\\lambda} = - \\int (\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}) d\\Psi_{\\lambda} + \\frac{\\Omega + 2 (e^{\\Psi_{\\lambda}})^{\\Omega}}{\\Omega + 2 \\mathbf{g}{(\\Psi_{\\lambda},\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True)))))"], [["divide", 4, "Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True))))))"], [["minus", 5, "Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(2), Pow(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Omega', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} = \\theta_1 + c, then obtain (\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} \\log{(\\theta_1 + c)})^{c} = ((\\theta_1 + c) \\log{(\\theta_1 + c)})^{c}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} = \\theta_1 + c and \\log{(\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)})} = \\log{(\\theta_1 + c)} and \\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} \\log{(\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)})} = (\\theta_1 + c) \\log{(\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)})} and (\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} \\log{(\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)})})^{c} = ((\\theta_1 + c) \\log{(\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)})})^{c} and (\\operatorname{L_{\\varepsilon}}{(\\theta_1,c)} \\log{(\\theta_1 + c)})^{c} = ((\\theta_1 + c) \\log{(\\theta_1 + c)})^{c}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))"], [["log", 1], "Equality(log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True))), log(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True))))"], [["times", 1, "log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)), log(Add(Symbol('\\\\theta_1', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given I{(v_{2})} = \\cos{(\\log{(v_{2})})}, then obtain \\frac{\\int (I^{v_{2}}{(v_{2})})^{v_{2}} dv_{2}}{\\sin{(\\dot{z})}} = \\frac{\\int (\\cos^{v_{2}}{(\\log{(v_{2})})})^{v_{2}} dv_{2}}{\\sin{(\\dot{z})}}", "derivation": "I{(v_{2})} = \\cos{(\\log{(v_{2})})} and I^{v_{2}}{(v_{2})} = \\cos^{v_{2}}{(\\log{(v_{2})})} and (I^{v_{2}}{(v_{2})})^{v_{2}} = (\\cos^{v_{2}}{(\\log{(v_{2})})})^{v_{2}} and \\int (I^{v_{2}}{(v_{2})})^{v_{2}} dv_{2} = \\int (\\cos^{v_{2}}{(\\log{(v_{2})})})^{v_{2}} dv_{2} and \\frac{\\int (I^{v_{2}}{(v_{2})})^{v_{2}} dv_{2}}{\\sin{(\\dot{z})}} = \\frac{\\int (\\cos^{v_{2}}{(\\log{(v_{2})})})^{v_{2}} dv_{2}}{\\sin{(\\dot{z})}}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('v_2', commutative=True)), cos(log(Symbol('v_2', commutative=True))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('I')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(cos(log(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"], [["power", 2, "Symbol('v_2', commutative=True)"], "Equality(Pow(Pow(Function('I')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(Pow(cos(log(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(Pow(Pow(Function('I')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Pow(Pow(cos(log(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["divide", 4, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Integral(Pow(Pow(Function('I')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Mul(Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Integral(Pow(Pow(cos(log(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given f{(r,f_{E})} = \\sin{(\\frac{f_{E}}{r})}, then obtain (t + \\sin{(\\frac{f_{E}}{r})} + 1) f^{r}{(r,f_{E})} = (t + \\sin{(\\frac{f_{E}}{r})} + 1) \\sin^{r}{(\\frac{f_{E}}{r})}", "derivation": "f{(r,f_{E})} = \\sin{(\\frac{f_{E}}{r})} and f{(r,f_{E})} + 1 = \\sin{(\\frac{f_{E}}{r})} + 1 and f^{r}{(r,f_{E})} = \\sin^{r}{(\\frac{f_{E}}{r})} and (t + f{(r,f_{E})} + 1) f^{r}{(r,f_{E})} = (t + f{(r,f_{E})} + 1) \\sin^{r}{(\\frac{f_{E}}{r})} and (t + \\sin{(\\frac{f_{E}}{r})} + 1) f^{r}{(r,f_{E})} = (t + \\sin{(\\frac{f_{E}}{r})} + 1) \\sin^{r}{(\\frac{f_{E}}{r})}", "srepr_derivation": [["get_premise", "Equality(Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Add(sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Integer(1)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Pow(sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Symbol('r', commutative=True)))"], [["times", 3, "Add(Symbol('t', commutative=True), Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Integer(1))"], "Equality(Mul(Add(Symbol('t', commutative=True), Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Pow(Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True))), Mul(Add(Symbol('t', commutative=True), Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Pow(sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('t', commutative=True), sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Integer(1)), Pow(Function('f')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True))), Mul(Add(Symbol('t', commutative=True), sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Integer(1)), Pow(sin(Mul(Symbol('f_E', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)))), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(v,m_{s})} = - m_{s} + \\sin{(v)}, then obtain \\int \\frac{\\partial}{\\partial v} \\operatorname{z^{*}}{(v,m_{s})} dm_{s} = \\sigma_p + m_{s} \\cos{(v)}", "derivation": "\\operatorname{z^{*}}{(v,m_{s})} = - m_{s} + \\sin{(v)} and \\frac{\\partial}{\\partial v} \\operatorname{z^{*}}{(v,m_{s})} = \\frac{\\partial}{\\partial v} (- m_{s} + \\sin{(v)}) and \\int \\frac{\\partial}{\\partial v} \\operatorname{z^{*}}{(v,m_{s})} dm_{s} = \\int \\frac{\\partial}{\\partial v} (- m_{s} + \\sin{(v)}) dm_{s} and \\int \\frac{\\partial}{\\partial v} \\operatorname{z^{*}}{(v,m_{s})} dm_{s} = \\sigma_p + m_{s} \\cos{(v)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('v', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('v', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Derivative(Function('z^*')(Symbol('v', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('z^*')(Symbol('v', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('m_s', commutative=True), cos(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})} = \\rho_b + v_{t}, then obtain 2 \\rho_b (\\rho_b + v_{t}) = \\rho_b (2 \\rho_b + 2 v_{t})", "derivation": "\\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})} = \\rho_b + v_{t} and 2 \\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})} = \\rho_b + v_{t} + \\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})} and 2 \\rho_b \\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})} = \\rho_b (\\rho_b + v_{t} + \\operatorname{E_{\\lambda}}{(\\rho_b,v_{t})}) and 2 \\rho_b (\\rho_b + v_{t}) = \\rho_b (2 \\rho_b + 2 v_{t})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)))"], [["add", 1, "Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))))"], [["times", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Add(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Add(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given x{(h)} = \\log{(e^{h})}, then obtain 0 = (- x{(h)} + \\log{(e^{h})}) (x{(h)} - e^{h})", "derivation": "x{(h)} = \\log{(e^{h})} and x{(h)} - e^{h} = - e^{h} + \\log{(e^{h})} and 0 = - x{(h)} + \\log{(e^{h})} and 0 = (- x{(h)} + \\log{(e^{h})}) (- e^{h} + \\log{(e^{h})}) and 0 = (- x{(h)} + \\log{(e^{h})}) (x{(h)} - e^{h})", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('h', commutative=True)), log(exp(Symbol('h', commutative=True))))"], [["minus", 1, "exp(Symbol('h', commutative=True))"], "Equality(Add(Function('x')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True)))))"], [["minus", 2, "Add(Function('x')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x')(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True)))))"], [["times", 3, "Add(Mul(Integer(-1), exp(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True))))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('x')(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('x')(Symbol('h', commutative=True))), log(exp(Symbol('h', commutative=True)))), Add(Function('x')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(t_{1})} = \\log{(t_{1})}, then derive \\frac{d}{d t_{1}} \\mathbf{f}{(t_{1})} = \\frac{1}{t_{1}}, then obtain \\frac{d}{d t_{1}} \\frac{1}{t_{1}} = \\frac{d^{2}}{d t_{1}^{2}} \\log{(t_{1})}", "derivation": "\\mathbf{f}{(t_{1})} = \\log{(t_{1})} and \\frac{d}{d t_{1}} \\mathbf{f}{(t_{1})} = \\frac{d}{d t_{1}} \\log{(t_{1})} and \\frac{d}{d t_{1}} \\mathbf{f}{(t_{1})} = \\frac{1}{t_{1}} and \\frac{1}{t_{1}} = \\frac{d}{d t_{1}} \\log{(t_{1})} and \\frac{d}{d t_{1}} \\frac{1}{t_{1}} = \\frac{d^{2}}{d t_{1}^{2}} \\log{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Symbol('t_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('t_1', commutative=True), Integer(-1)), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Pow(Symbol('t_1', commutative=True), Integer(-1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain - \\mathbf{A} - \\mathbf{J}{(\\mathbf{A})} \\cos{(\\mathbf{A})} = - \\mathbf{A} - \\mathbf{J}{(\\mathbf{A})} \\cos{(\\mathbf{A})} - \\mathbf{J}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}", "derivation": "\\mathbf{J}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and 0 = - \\mathbf{J}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and - \\mathbf{A} = - \\mathbf{A} - \\mathbf{J}{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and - \\mathbf{A} - \\mathbf{J}{(\\mathbf{A})} \\cos{(\\mathbf{A})} = - \\mathbf{A} - \\mathbf{J}{(\\mathbf{A})} \\cos{(\\mathbf{A})} - \\mathbf{J}{(\\mathbf{A})} + \\cos{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{A}', commutative=True))), cos(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given J{(v_{2})} = \\log{(v_{2})}, then derive \\int (J{(v_{2})} + \\log{(v_{2})}) dv_{2} = \\mathbf{J}_f + 2 v_{2} \\log{(v_{2})} - 2 v_{2}, then obtain \\int (\\log{(v_{2})} + \\frac{2 J{(v_{2})} \\log{(v_{2})}}{J{(v_{2})} + \\log{(v_{2})}}) dv_{2} = \\mathbf{J}_f + 2 v_{2} \\log{(v_{2})} - 2 v_{2}", "derivation": "J{(v_{2})} = \\log{(v_{2})} and J{(v_{2})} + \\log{(v_{2})} = 2 \\log{(v_{2})} and \\int (J{(v_{2})} + \\log{(v_{2})}) dv_{2} = \\int 2 \\log{(v_{2})} dv_{2} and (J{(v_{2})} + \\log{(v_{2})}) J{(v_{2})} = 2 J{(v_{2})} \\log{(v_{2})} and \\int (J{(v_{2})} + \\log{(v_{2})}) dv_{2} = \\mathbf{J}_f + 2 v_{2} \\log{(v_{2})} - 2 v_{2} and J{(v_{2})} = \\frac{2 J{(v_{2})} \\log{(v_{2})}}{J{(v_{2})} + \\log{(v_{2})}} and \\int (\\log{(v_{2})} + \\frac{2 J{(v_{2})} \\log{(v_{2})}}{J{(v_{2})} + \\log{(v_{2})}}) dv_{2} = \\mathbf{J}_f + 2 v_{2} \\log{(v_{2})} - 2 v_{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], [["add", 1, "log(Symbol('v_2', commutative=True))"], "Equality(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Mul(Integer(2), log(Symbol('v_2', commutative=True))))"], [["integrate", 2, "Symbol('v_2', commutative=True)"], "Equality(Integral(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Integer(2), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["times", 2, "Function('J')(Symbol('v_2', commutative=True))"], "Equality(Mul(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Function('J')(Symbol('v_2', commutative=True))), Mul(Integer(2), Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(2), Symbol('v_2', commutative=True), log(Symbol('v_2', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))))"], [["divide", 4, "Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], "Equality(Function('J')(Symbol('v_2', commutative=True)), Mul(Integer(2), Pow(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Integer(-1)), Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Integral(Add(log(Symbol('v_2', commutative=True)), Mul(Integer(2), Pow(Add(Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Integer(-1)), Function('J')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(2), Symbol('v_2', commutative=True), log(Symbol('v_2', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given S{(l)} = \\log{(l)} and \\operatorname{v_{2}}{(b,F_{H})} = \\frac{b}{F_{H}}, then obtain - \\frac{F_{H} \\int S{(l)} \\log{(l)} dl}{b} = - \\frac{F_{H} \\int \\log{(l)}^{2} dl}{b}", "derivation": "S{(l)} = \\log{(l)} and S{(l)} \\log{(l)} = \\log{(l)}^{2} and \\operatorname{v_{2}}{(b,F_{H})} = \\frac{b}{F_{H}} and \\int S{(l)} \\log{(l)} dl = \\int \\log{(l)}^{2} dl and - \\frac{\\int S{(l)} \\log{(l)} dl}{\\operatorname{v_{2}}{(b,F_{H})}} = - \\frac{\\int \\log{(l)}^{2} dl}{\\operatorname{v_{2}}{(b,F_{H})}} and - \\frac{F_{H} \\int S{(l)} \\log{(l)} dl}{b} = - \\frac{F_{H} \\int \\log{(l)}^{2} dl}{b}", "srepr_derivation": [["renaming_premise", "Equality(Function('S')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["times", 1, "log(Symbol('l', commutative=True))"], "Equality(Mul(Function('S')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Pow(log(Symbol('l', commutative=True)), Integer(2)))"], ["get_premise", "Equality(Function('v_2')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('b', commutative=True)))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(Function('S')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Function('v_2')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('v_2')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Integral(Mul(Function('S')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(Function('v_2')(Symbol('b', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Mul(Function('S')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\nabla)} = e^{\\nabla}, then derive - \\int \\hat{p}{(\\nabla)} d\\nabla = - \\mathbf{r} - e^{\\nabla}, then obtain - (\\int \\hat{p}{(\\nabla)} d\\nabla)^{2} = (- \\mathbf{r} - e^{\\nabla}) \\int \\hat{p}{(\\nabla)} d\\nabla", "derivation": "\\hat{p}{(\\nabla)} = e^{\\nabla} and \\int \\hat{p}{(\\nabla)} d\\nabla = \\int e^{\\nabla} d\\nabla and - \\int \\hat{p}{(\\nabla)} d\\nabla = - \\int e^{\\nabla} d\\nabla and - \\int \\hat{p}{(\\nabla)} d\\nabla = - \\mathbf{r} - e^{\\nabla} and - \\int \\hat{p}{(\\nabla)} d\\nabla = - \\mathbf{r} - \\hat{p}{(\\nabla)} and - (\\int \\hat{p}{(\\nabla)} d\\nabla)^{2} = (- \\mathbf{r} - \\hat{p}{(\\nabla)}) \\int \\hat{p}{(\\nabla)} d\\nabla and - (\\int e^{\\nabla} d\\nabla)^{2} = (- \\mathbf{r} - e^{\\nabla}) \\int e^{\\nabla} d\\nabla and - (\\int \\hat{p}{(\\nabla)} d\\nabla)^{2} = (- \\mathbf{r} - e^{\\nabla}) \\int \\hat{p}{(\\nabla)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)))))"], [["times", 5, "Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)))), Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\nabla', commutative=True)))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Mul(Integer(-1), Pow(Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\nabla', commutative=True)))), Integral(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\chi,y)} = \\sin{(\\frac{y}{\\chi})}, then obtain \\int (- (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\operatorname{t_{1}}{(\\chi,y)}) d\\chi = \\int (- (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\sin{(\\frac{y}{\\chi})}) d\\chi", "derivation": "\\operatorname{t_{1}}{(\\chi,y)} = \\sin{(\\frac{y}{\\chi})} and - \\operatorname{t_{1}}{(\\chi,y)} = - \\sin{(\\frac{y}{\\chi})} and - (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\operatorname{t_{1}}{(\\chi,y)} = - (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\sin{(\\frac{y}{\\chi})} and \\int (- (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\operatorname{t_{1}}{(\\chi,y)}) d\\chi = \\int (- (\\chi + \\sin{(\\frac{y}{\\chi})})^{\\chi} - \\sin{(\\frac{y}{\\chi})}) d\\chi", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('y', commutative=True)), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))))"], [["minus", 2, "Pow(Add(Symbol('\\\\chi', commutative=True), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\chi', commutative=True), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\chi', commutative=True), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\chi', commutative=True), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\chi', commutative=True), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('y', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\psi)} = \\psi and m{(\\psi)} = \\psi, then obtain \\psi \\varphi^{*}{(\\psi)} - \\psi + \\frac{d}{d \\psi} (- \\varphi^{*}{(\\psi)} + m{(\\psi)}) = \\psi \\varphi^{*}{(\\psi)} - \\psi + \\frac{d}{d \\psi} 0", "derivation": "\\varphi^{*}{(\\psi)} = \\psi and - \\varphi^{*}{(\\psi)} = - \\psi and m{(\\psi)} = \\psi and - \\varphi^{*}{(\\psi)} + m{(\\psi)} = \\psi - \\varphi^{*}{(\\psi)} and - \\psi + m{(\\psi)} = 0 and - \\varphi^{*}{(\\psi)} + m{(\\psi)} = 0 and \\frac{d}{d \\psi} (- \\varphi^{*}{(\\psi)} + m{(\\psi)}) = \\frac{d}{d \\psi} 0 and \\psi \\varphi^{*}{(\\psi)} - \\psi + \\frac{d}{d \\psi} (- \\varphi^{*}{(\\psi)} + m{(\\psi)}) = \\psi \\varphi^{*}{(\\psi)} - \\psi + \\frac{d}{d \\psi} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["minus", 3, "Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Function('m')(Symbol('\\\\psi', commutative=True))), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('m')(Symbol('\\\\psi', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Function('m')(Symbol('\\\\psi', commutative=True))), Integer(0))"], [["differentiate", 6, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Function('m')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["add", 7, "Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Function('m')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(\\hat{p}_0,\\tilde{g}^*)} = \\hat{p}_0 + \\tilde{g}^*, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*) \\int H{(\\hat{p}_0,\\tilde{g}^*)} d\\tilde{g}^* = \\frac{\\partial}{\\partial \\hat{p}_0} (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*)^{2}", "derivation": "H{(\\hat{p}_0,\\tilde{g}^*)} = \\hat{p}_0 + \\tilde{g}^* and \\int H{(\\hat{p}_0,\\tilde{g}^*)} d\\tilde{g}^* = \\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^* and (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*) \\int H{(\\hat{p}_0,\\tilde{g}^*)} d\\tilde{g}^* = (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*)^{2} and \\frac{\\partial}{\\partial \\hat{p}_0} (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*) \\int H{(\\hat{p}_0,\\tilde{g}^*)} d\\tilde{g}^* = \\frac{\\partial}{\\partial \\hat{p}_0} (\\int (\\hat{p}_0 + \\tilde{g}^*) d\\tilde{g}^*)^{2}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 2, "Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Pow(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('H')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(A_{y})} = \\sin{(A_{y})} and \\mu{(A_{y})} = \\sin{(A_{y})}, then obtain M{(A_{y})} + \\mu{(A_{y})} = 2 M{(A_{y})}", "derivation": "M{(A_{y})} = \\sin{(A_{y})} and \\mu{(A_{y})} = \\sin{(A_{y})} and \\mu{(A_{y})} = M{(A_{y})} and M{(A_{y})} + \\mu{(A_{y})} = 2 M{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mu')(Symbol('A_y', commutative=True)), Function('M')(Symbol('A_y', commutative=True)))"], [["add", 3, "Function('M')(Symbol('A_y', commutative=True))"], "Equality(Add(Function('M')(Symbol('A_y', commutative=True)), Function('\\\\mu')(Symbol('A_y', commutative=True))), Mul(Integer(2), Function('M')(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given I{(t_{2},r_{0},F_{H})} = F_{H} + r_{0} + t_{2}, then obtain (- 2 r_{0} + I{(t_{2},r_{0},F_{H})})^{F_{H}} = (F_{H} - r_{0} + t_{2})^{F_{H}}", "derivation": "I{(t_{2},r_{0},F_{H})} = F_{H} + r_{0} + t_{2} and F_{H} + r_{0} + t_{2} + I{(t_{2},r_{0},F_{H})} = 2 F_{H} + 2 r_{0} + 2 t_{2} and F_{H} - r_{0} + t_{2} + I{(t_{2},r_{0},F_{H})} = 2 F_{H} + 2 t_{2} and - 2 r_{0} + I{(t_{2},r_{0},F_{H})} = F_{H} - r_{0} + t_{2} and (- 2 r_{0} + I{(t_{2},r_{0},F_{H})})^{F_{H}} = (F_{H} - r_{0} + t_{2})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 1, "Add(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True), Symbol('t_2', commutative=True), Function('I')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Mul(Integer(2), Symbol('r_0', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Symbol('t_2', commutative=True), Function('I')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))))"], [["minus", 3, "Add(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('r_0', commutative=True)), Function('I')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Symbol('t_2', commutative=True)))"], [["power", 4, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('r_0', commutative=True)), Function('I')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Symbol('t_2', commutative=True)), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\theta{(\\delta)} = \\sin{(\\delta)} and \\operatorname{A_{z}}{(\\delta)} = \\theta{(\\delta)} + \\sin{(\\delta)}, then obtain \\theta{(\\delta)} = - \\theta{(\\delta)} + 2 \\sin{(\\delta)}", "derivation": "\\theta{(\\delta)} = \\sin{(\\delta)} and \\theta{(\\delta)} + \\sin{(\\delta)} = 2 \\sin{(\\delta)} and \\operatorname{A_{z}}{(\\delta)} = \\theta{(\\delta)} + \\sin{(\\delta)} and \\operatorname{A_{z}}{(\\delta)} = 2 \\sin{(\\delta)} and \\operatorname{A_{z}}{(\\delta)} - \\theta{(\\delta)} = - \\theta{(\\delta)} + 2 \\sin{(\\delta)} and \\operatorname{A_{z}}{(\\delta)} = 2 \\theta{(\\delta)} and \\theta{(\\delta)} = - \\theta{(\\delta)} + 2 \\sin{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\delta', commutative=True)), Add(Function('\\\\theta')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('A_z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\delta', commutative=True))))"], [["minus", 4, "Function('\\\\theta')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('A_z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_z')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Function('\\\\theta')(Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given v{(\\mathbf{f})} = \\sin{(\\mathbf{f})}, then obtain \\iint v^{2}{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} = \\iint v{(\\mathbf{f})} \\sin{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}", "derivation": "v{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and v^{2}{(\\mathbf{f})} = v{(\\mathbf{f})} \\sin{(\\mathbf{f})} and \\int v^{2}{(\\mathbf{f})} d\\mathbf{f} = \\int v{(\\mathbf{f})} \\sin{(\\mathbf{f})} d\\mathbf{f} and \\iint v^{2}{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} = \\iint v{(\\mathbf{f})} \\sin{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 1, "Function('v')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Pow(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Mul(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Pow(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Pow(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Function('v')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mu{(b,A_{1})} = \\log{(A_{1})}^{b}, then obtain ((A_{1} \\mu{(b,A_{1})} - b)^{b})^{A_{1}} = ((A_{1} \\log{(A_{1})}^{b} - b)^{b})^{A_{1}}", "derivation": "\\mu{(b,A_{1})} = \\log{(A_{1})}^{b} and A_{1} \\mu{(b,A_{1})} = A_{1} \\log{(A_{1})}^{b} and A_{1} \\mu{(b,A_{1})} - b = A_{1} \\log{(A_{1})}^{b} - b and (A_{1} \\mu{(b,A_{1})} - b)^{b} = (A_{1} \\log{(A_{1})}^{b} - b)^{b} and ((A_{1} \\mu{(b,A_{1})} - b)^{b})^{A_{1}} = ((A_{1} \\log{(A_{1})}^{b} - b)^{b})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('b', commutative=True), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\mu')(Symbol('b', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(log(Symbol('A_1', commutative=True)), Symbol('b', commutative=True))))"], [["minus", 2, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Symbol('A_1', commutative=True), Function('\\\\mu')(Symbol('b', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), Pow(log(Symbol('A_1', commutative=True)), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('A_1', commutative=True), Function('\\\\mu')(Symbol('b', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Mul(Symbol('A_1', commutative=True), Pow(log(Symbol('A_1', commutative=True)), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Symbol('A_1', commutative=True), Function('\\\\mu')(Symbol('b', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Add(Mul(Symbol('A_1', commutative=True), Pow(log(Symbol('A_1', commutative=True)), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given I{(\\mathbf{J},l)} = l \\cos{(\\mathbf{J})}, then obtain - l \\frac{\\partial}{\\partial \\mathbf{J}} I{(\\mathbf{J},l)} = l^{2} \\sin{(\\mathbf{J})}", "derivation": "I{(\\mathbf{J},l)} = l \\cos{(\\mathbf{J})} and \\frac{I{(\\mathbf{J},l)}}{\\cos{(\\mathbf{J})}} = l and - \\frac{I{(\\mathbf{J},l)}}{\\cos{(\\mathbf{J})}} = - l and - l I{(\\mathbf{J},l)} = - l^{2} \\cos{(\\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} - l I{(\\mathbf{J},l)} = \\frac{\\partial}{\\partial \\mathbf{J}} - l^{2} \\cos{(\\mathbf{J})} and - l \\frac{\\partial}{\\partial \\mathbf{J}} I{(\\mathbf{J},l)} = l^{2} \\sin{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 1, "cos(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))), Symbol('l', commutative=True))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('l', commutative=True)))"], [["times", 3, "Mul(Symbol('l', commutative=True), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('l', commutative=True), Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(2)), cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('l', commutative=True), Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(2)), cos(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Symbol('l', commutative=True), Derivative(Function('I')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Pow(Symbol('l', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,v_{2})} = \\psi^* e^{v_{2}} and G{(n_{2})} = \\cos{(\\cos{(n_{2})})}, then obtain \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,v_{2})} + \\cos{(\\cos{(n_{2})})} = \\psi^* e^{v_{2}} + \\cos{(\\cos{(n_{2})})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\psi^*,v_{2})} = \\psi^* e^{v_{2}} and G{(n_{2})} = \\cos{(\\cos{(n_{2})})} and G{(n_{2})} + \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,v_{2})} = \\psi^* e^{v_{2}} + G{(n_{2})} and \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,v_{2})} + \\cos{(\\cos{(n_{2})})} = \\psi^* e^{v_{2}} + \\cos{(\\cos{(n_{2})})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('v_2', commutative=True))))"], ["get_premise", "Equality(Function('G')(Symbol('n_2', commutative=True)), cos(cos(Symbol('n_2', commutative=True))))"], [["add", 1, "Function('G')(Symbol('n_2', commutative=True))"], "Equality(Add(Function('G')(Symbol('n_2', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('v_2', commutative=True))), Function('G')(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)), cos(cos(Symbol('n_2', commutative=True)))), Add(Mul(Symbol('\\\\psi^*', commutative=True), exp(Symbol('v_2', commutative=True))), cos(cos(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(I)} = \\cos{(\\log{(I)})}, then obtain \\frac{(\\frac{\\frac{d}{d I} \\lambda{(I)}}{I \\lambda{(I)}})^{I} \\frac{d}{d I} \\lambda{(I)}}{\\lambda{(I)}} = \\frac{(\\frac{\\frac{d}{d I} \\cos{(\\log{(I)})}}{I \\lambda{(I)}})^{I} \\frac{d}{d I} \\lambda{(I)}}{\\lambda{(I)}}", "derivation": "\\lambda{(I)} = \\cos{(\\log{(I)})} and \\frac{d}{d I} \\lambda{(I)} = \\frac{d}{d I} \\cos{(\\log{(I)})} and \\frac{\\frac{d}{d I} \\lambda{(I)}}{\\lambda{(I)}} = \\frac{\\frac{d}{d I} \\cos{(\\log{(I)})}}{\\lambda{(I)}} and \\frac{\\frac{d}{d I} \\lambda{(I)}}{I \\lambda{(I)}} = \\frac{\\frac{d}{d I} \\cos{(\\log{(I)})}}{I \\lambda{(I)}} and (\\frac{\\frac{d}{d I} \\lambda{(I)}}{I \\lambda{(I)}})^{I} = (\\frac{\\frac{d}{d I} \\cos{(\\log{(I)})}}{I \\lambda{(I)}})^{I} and \\frac{(\\frac{\\frac{d}{d I} \\lambda{(I)}}{I \\lambda{(I)}})^{I} \\frac{d}{d I} \\lambda{(I)}}{\\lambda{(I)}} = \\frac{(\\frac{\\frac{d}{d I} \\cos{(\\log{(I)})}}{I \\lambda{(I)}})^{I} \\frac{d}{d I} \\lambda{(I)}}{\\lambda{(I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('I', commutative=True)), cos(log(Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\lambda')(Symbol('I', commutative=True))"], "Equality(Mul(Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["divide", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Symbol('I', commutative=True)))"], [["times", 5, "Mul(Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Symbol('I', commutative=True)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(cos(log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Symbol('I', commutative=True)), Pow(Function('\\\\lambda')(Symbol('I', commutative=True)), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(f^{\\prime},\\mathbf{f})} = \\sin{(\\mathbf{f} f^{\\prime})} and \\operatorname{g_{\\varepsilon}}{(f^{\\prime},\\mathbf{f})} = \\sin^{f^{\\prime}}{(\\mathbf{f} f^{\\prime})} and \\mathbf{B}{(f^{\\prime},\\mathbf{f})} = a^{f^{\\prime}}{(f^{\\prime},\\mathbf{f})}, then obtain \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},\\mathbf{f})} df^{\\prime} = \\int \\mathbf{B}{(f^{\\prime},\\mathbf{f})} df^{\\prime}", "derivation": "a{(f^{\\prime},\\mathbf{f})} = \\sin{(\\mathbf{f} f^{\\prime})} and a^{f^{\\prime}}{(f^{\\prime},\\mathbf{f})} = \\sin^{f^{\\prime}}{(\\mathbf{f} f^{\\prime})} and \\operatorname{g_{\\varepsilon}}{(f^{\\prime},\\mathbf{f})} = \\sin^{f^{\\prime}}{(\\mathbf{f} f^{\\prime})} and \\mathbf{B}{(f^{\\prime},\\mathbf{f})} = a^{f^{\\prime}}{(f^{\\prime},\\mathbf{f})} and \\mathbf{B}{(f^{\\prime},\\mathbf{f})} = \\sin^{f^{\\prime}}{(\\mathbf{f} f^{\\prime})} and \\operatorname{g_{\\varepsilon}}{(f^{\\prime},\\mathbf{f})} = \\mathbf{B}{(f^{\\prime},\\mathbf{f})} and \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},\\mathbf{f})} df^{\\prime} = \\int \\mathbf{B}{(f^{\\prime},\\mathbf{f})} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('a')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('a')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\mathbf{B}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 6, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given i{(F_{g},y)} = y^{F_{g}}, then derive \\frac{\\partial}{\\partial y} i{(F_{g},y)} = \\frac{F_{g} y^{F_{g}}}{y}, then obtain \\frac{\\partial}{\\partial y} y^{F_{g}} = \\frac{F_{g} y^{F_{g}}}{y}", "derivation": "i{(F_{g},y)} = y^{F_{g}} and \\frac{\\partial}{\\partial y} i{(F_{g},y)} = \\frac{\\partial}{\\partial y} y^{F_{g}} and \\frac{\\partial}{\\partial y} i{(F_{g},y)} = \\frac{F_{g} y^{F_{g}}}{y} and \\frac{\\partial}{\\partial y} i{(F_{g},y)} = \\frac{F_{g} i{(F_{g},y)}}{y} and \\frac{\\partial}{\\partial y} y^{F_{g}} = \\frac{F_{g} y^{F_{g}}}{y}", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('F_g', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('F_g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Symbol('y', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('F_g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('i')(Symbol('F_g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Function('i')(Symbol('F_g', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Symbol('y', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('F_g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\theta_1)} = e^{\\sin{(\\theta_1)}}, then derive \\frac{d}{d \\theta_1} \\mathbf{E}{(\\theta_1)} = e^{\\sin{(\\theta_1)}} \\cos{(\\theta_1)}, then obtain \\int \\frac{d^{2}}{d \\theta_1^{2}} \\mathbf{E}{(\\theta_1)} d\\theta_1 = \\int \\frac{d}{d \\theta_1} e^{\\sin{(\\theta_1)}} \\cos{(\\theta_1)} d\\theta_1", "derivation": "\\mathbf{E}{(\\theta_1)} = e^{\\sin{(\\theta_1)}} and \\frac{d}{d \\theta_1} \\mathbf{E}{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\sin{(\\theta_1)}} and \\frac{d}{d \\theta_1} \\mathbf{E}{(\\theta_1)} = e^{\\sin{(\\theta_1)}} \\cos{(\\theta_1)} and \\frac{d^{2}}{d \\theta_1^{2}} \\mathbf{E}{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\sin{(\\theta_1)}} \\cos{(\\theta_1)} and \\int \\frac{d^{2}}{d \\theta_1^{2}} \\mathbf{E}{(\\theta_1)} d\\theta_1 = \\int \\frac{d}{d \\theta_1} e^{\\sin{(\\theta_1)}} \\cos{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), exp(sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(exp(sin(Symbol('\\\\theta_1', commutative=True))), cos(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))), Derivative(Mul(exp(sin(Symbol('\\\\theta_1', commutative=True))), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(Mul(exp(sin(Symbol('\\\\theta_1', commutative=True))), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\psi^*,v_{2})} = \\psi^* + v_{2}, then obtain \\log{(- 4 \\psi^* - 3 v_{2} + 3 \\Psi{(\\psi^*,v_{2})})} + 1 = \\log{(- \\psi^*)} + 1", "derivation": "\\Psi{(\\psi^*,v_{2})} = \\psi^* + v_{2} and - \\psi^* - v_{2} + \\Psi{(\\psi^*,v_{2})} = 0 and - 2 \\psi^* - v_{2} + \\Psi{(\\psi^*,v_{2})} = - \\psi^* and \\log{(- 2 \\psi^* - v_{2} + \\Psi{(\\psi^*,v_{2})})} = \\log{(- \\psi^*)} and \\log{(- 4 \\psi^* - 3 v_{2} + 3 \\Psi{(\\psi^*,v_{2})})} = \\log{(- 2 \\psi^* - v_{2} + \\Psi{(\\psi^*,v_{2})})} and \\log{(- 4 \\psi^* - 3 v_{2} + 3 \\Psi{(\\psi^*,v_{2})})} = \\log{(- \\psi^*)} and \\log{(- 4 \\psi^* - 3 v_{2} + 3 \\Psi{(\\psi^*,v_{2})})} + 1 = \\log{(- \\psi^*)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))"], [["log", 3], "Equality(log(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)))), log(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(log(Add(Mul(Integer(-1), Integer(4), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('v_2', commutative=True)), Mul(Integer(3), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))))), log(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(log(Add(Mul(Integer(-1), Integer(4), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('v_2', commutative=True)), Mul(Integer(3), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))))), log(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 6, "Integer(-1)"], "Equality(Add(log(Add(Mul(Integer(-1), Integer(4), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('v_2', commutative=True)), Mul(Integer(3), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('v_2', commutative=True))))), Integer(1)), Add(log(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{M}{(v,y)} = \\frac{v}{y} and \\operatorname{P_{e}}{(v,y)} = \\frac{v}{y}, then obtain (\\int (- \\operatorname{P_{e}}{(v,y)} - 1) dy)^{y} = (\\int (- \\frac{v}{y} - 1) dy)^{y}", "derivation": "\\mathbf{M}{(v,y)} = \\frac{v}{y} and - \\mathbf{M}{(v,y)} = - \\frac{v}{y} and - \\mathbf{M}{(v,y)} - 1 = - \\frac{v}{y} - 1 and \\operatorname{P_{e}}{(v,y)} = \\frac{v}{y} and \\operatorname{P_{e}}{(v,y)} = \\mathbf{M}{(v,y)} and - \\operatorname{P_{e}}{(v,y)} - 1 = - \\frac{v}{y} - 1 and \\int (- \\operatorname{P_{e}}{(v,y)} - 1) dy = \\int (- \\frac{v}{y} - 1) dy and (\\int (- \\operatorname{P_{e}}{(v,y)} - 1) dy)^{y} = (\\int (- \\frac{v}{y} - 1) dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('y', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('v', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('P_e')(Symbol('v', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(-1), Function('P_e')(Symbol('v', commutative=True), Symbol('y', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Integer(-1)))"], [["integrate", 6, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('v', commutative=True), Symbol('y', commutative=True))), Integer(-1)), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('y', commutative=True))))"], [["power", 7, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('v', commutative=True), Symbol('y', commutative=True))), Integer(-1)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(Z)} = \\sin{(\\cos{(Z)})}, then obtain \\frac{d}{d Z} \\log{(- \\mathbf{J}_M{(Z)})} = \\frac{d}{d Z} \\log{(- \\sin{(\\cos{(Z)})})}", "derivation": "\\mathbf{J}_M{(Z)} = \\sin{(\\cos{(Z)})} and - \\mathbf{J}_M{(Z)} = - \\sin{(\\cos{(Z)})} and \\log{(- \\mathbf{J}_M{(Z)})} = \\log{(- \\sin{(\\cos{(Z)})})} and \\frac{d}{d Z} \\log{(- \\mathbf{J}_M{(Z)})} = \\frac{d}{d Z} \\log{(- \\sin{(\\cos{(Z)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True)), sin(cos(Symbol('Z', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True))), Mul(Integer(-1), sin(cos(Symbol('Z', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True)))), log(Mul(Integer(-1), sin(cos(Symbol('Z', commutative=True))))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(log(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(log(Mul(Integer(-1), sin(cos(Symbol('Z', commutative=True))))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(\\phi,P_{e})} = P_{e} \\phi and A{(\\phi)} = \\phi, then obtain (\\int \\phi \\phi_{2}{(\\phi,P_{e})} dA{(\\phi)})^{P_{e}} = (\\int P_{e} \\phi^{2} dA{(\\phi)})^{P_{e}}", "derivation": "\\phi_{2}{(\\phi,P_{e})} = P_{e} \\phi and \\phi \\phi_{2}{(\\phi,P_{e})} = P_{e} \\phi^{2} and \\int \\phi \\phi_{2}{(\\phi,P_{e})} d\\phi = \\int P_{e} \\phi^{2} d\\phi and A{(\\phi)} = \\phi and \\int \\phi \\phi_{2}{(\\phi,P_{e})} dA{(\\phi)} = \\int P_{e} \\phi^{2} dA{(\\phi)} and (\\int \\phi \\phi_{2}{(\\phi,P_{e})} dA{(\\phi)})^{P_{e}} = (\\int P_{e} \\phi^{2} dA{(\\phi)})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('P_e', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True))), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(2))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Tuple(Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True))), Tuple(Function('A')(Symbol('\\\\phi', commutative=True)))), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Tuple(Function('A')(Symbol('\\\\phi', commutative=True)))))"], [["power", 5, "Symbol('P_e', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True))), Tuple(Function('A')(Symbol('\\\\phi', commutative=True)))), Symbol('P_e', commutative=True)), Pow(Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Tuple(Function('A')(Symbol('\\\\phi', commutative=True)))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)}, then obtain \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^* = \\frac{\\sin{(\\tilde{g}^*)} \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^*}{\\eta{(\\tilde{g}^*)}}", "derivation": "\\eta{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)} and \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^* = \\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^* and \\eta{(\\tilde{g}^*)} \\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^* = \\sin{(\\tilde{g}^*)} \\int \\sin{(\\tilde{g}^*)} d\\tilde{g}^* and \\eta{(\\tilde{g}^*)} \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^* = \\sin{(\\tilde{g}^*)} \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^* and \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^* = \\frac{\\sin{(\\tilde{g}^*)} \\int \\eta{(\\tilde{g}^*)} d\\tilde{g}^*}{\\eta{(\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 1, "Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["times", 4, "Pow(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Pow(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given f{(E)} = e^{E}, then obtain (e^{- E} e^{f{(E)}})^{E} = (e^{- E} e^{e^{E}})^{E}", "derivation": "f{(E)} = e^{E} and e^{f{(E)}} = e^{e^{E}} and e^{- E} e^{f{(E)}} = e^{- E} e^{e^{E}} and (e^{- E} e^{f{(E)}})^{E} = (e^{- E} e^{e^{E}})^{E}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["exp", 1], "Equality(exp(Function('f')(Symbol('E', commutative=True))), exp(exp(Symbol('E', commutative=True))))"], [["divide", 2, "exp(Symbol('E', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('E', commutative=True))), exp(Function('f')(Symbol('E', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('E', commutative=True))), exp(exp(Symbol('E', commutative=True)))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(exp(Mul(Integer(-1), Symbol('E', commutative=True))), exp(Function('f')(Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Pow(Mul(exp(Mul(Integer(-1), Symbol('E', commutative=True))), exp(exp(Symbol('E', commutative=True)))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given Z{(\\dot{z})} = \\cos{(\\cos{(\\dot{z})})}, then obtain \\int\\limits^{\\dot{z} + Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})}} Z{(\\dot{z})} d\\dot{z} = \\int\\limits^{\\dot{z} + Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})}} \\cos{(\\cos{(\\dot{z})})} d\\dot{z}", "derivation": "Z{(\\dot{z})} = \\cos{(\\cos{(\\dot{z})})} and Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})} = 0 and \\dot{z} + Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})} = \\dot{z} and \\int Z{(\\dot{z})} d\\dot{z} = \\int \\cos{(\\cos{(\\dot{z})})} d\\dot{z} and \\int\\limits^{\\dot{z} + Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})}} Z{(\\dot{z})} d\\dot{z} = \\int\\limits^{\\dot{z} + Z{(\\dot{z})} - \\cos{(\\cos{(\\dot{z})})}} \\cos{(\\cos{(\\dot{z})})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\dot{z}', commutative=True)), cos(cos(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 1, "cos(cos(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\dot{z}', commutative=True))))), Integer(0))"], [["add", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\dot{z}', commutative=True))))), Symbol('\\\\dot{z}', commutative=True))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\dot{z}', commutative=True))))))), Integral(cos(cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Function('Z')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\dot{z}', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\psi^*,g_{\\varepsilon})} = \\psi^* + g_{\\varepsilon} and \\eta^{\\prime}{(\\psi^*,g_{\\varepsilon})} = \\psi^* + g_{\\varepsilon}, then obtain \\eta^{\\prime}^{g_{\\varepsilon}}{(\\psi^*,g_{\\varepsilon})} = (\\psi^* + g_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "\\operatorname{C_{2}}{(\\psi^*,g_{\\varepsilon})} = \\psi^* + g_{\\varepsilon} and \\operatorname{C_{2}}^{g_{\\varepsilon}}{(\\psi^*,g_{\\varepsilon})} = (\\psi^* + g_{\\varepsilon})^{g_{\\varepsilon}} and \\eta^{\\prime}{(\\psi^*,g_{\\varepsilon})} = \\psi^* + g_{\\varepsilon} and \\operatorname{C_{2}}^{g_{\\varepsilon}}{(\\psi^*,g_{\\varepsilon})} = \\eta^{\\prime}^{g_{\\varepsilon}}{(\\psi^*,g_{\\varepsilon})} and \\eta^{\\prime}^{g_{\\varepsilon}}{(\\psi^*,g_{\\varepsilon})} = (\\psi^* + g_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('C_2')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('\\\\psi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\eta{(P_{g})} = e^{P_{g}} and \\operatorname{f_{E}}{(P_{g})} = P_{g} e^{P_{g}}, then obtain 2 \\operatorname{f_{E}}{(P_{g})} \\operatorname{f_{E}}^{P_{g}}{(P_{g})} = 2 (P_{g} e^{P_{g}})^{P_{g}} \\operatorname{f_{E}}{(P_{g})}", "derivation": "\\eta{(P_{g})} = e^{P_{g}} and P_{g} \\eta{(P_{g})} = P_{g} e^{P_{g}} and (P_{g} \\eta{(P_{g})})^{P_{g}} = (P_{g} e^{P_{g}})^{P_{g}} and 2 P_{g} (P_{g} \\eta{(P_{g})})^{P_{g}} \\eta{(P_{g})} = 2 P_{g} (P_{g} e^{P_{g}})^{P_{g}} \\eta{(P_{g})} and \\operatorname{f_{E}}{(P_{g})} = P_{g} e^{P_{g}} and P_{g} \\eta{(P_{g})} = \\operatorname{f_{E}}{(P_{g})} and 2 \\operatorname{f_{E}}{(P_{g})} \\operatorname{f_{E}}^{P_{g}}{(P_{g})} = 2 (P_{g} e^{P_{g}})^{P_{g}} \\operatorname{f_{E}}{(P_{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\eta')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Mul(Symbol('P_g', commutative=True), Function('\\\\eta')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], [["times", 3, "Mul(Integer(2), Symbol('P_g', commutative=True), Function('\\\\eta')(Symbol('P_g', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('P_g', commutative=True), Pow(Mul(Symbol('P_g', commutative=True), Function('\\\\eta')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Function('\\\\eta')(Symbol('P_g', commutative=True))), Mul(Integer(2), Symbol('P_g', commutative=True), Pow(Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Function('\\\\eta')(Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('P_g', commutative=True)), Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\eta')(Symbol('P_g', commutative=True))), Function('f_E')(Symbol('P_g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Integer(2), Function('f_E')(Symbol('P_g', commutative=True)), Pow(Function('f_E')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Function('f_E')(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(i,E_{x})} = \\frac{E_{x}}{i}, then obtain \\frac{E_{x} \\dot{\\mathbf{r}}{(i,E_{x})}}{i} = \\frac{E_{x}^{2}}{i^{2}}", "derivation": "\\dot{\\mathbf{r}}{(i,E_{x})} = \\frac{E_{x}}{i} and \\dot{\\mathbf{r}}^{2}{(i,E_{x})} = \\frac{E_{x} \\dot{\\mathbf{r}}{(i,E_{x})}}{i} and E_{x} \\dot{\\mathbf{r}}{(i,E_{x})} = \\frac{E_{x}^{2}}{i} and \\dot{\\mathbf{r}}^{2}{(i,E_{x})} = \\frac{E_{x}^{2}}{i^{2}} and \\frac{E_{x} \\dot{\\mathbf{r}}{(i,E_{x})}}{i} = \\frac{E_{x}^{2}}{i^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["times", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True))))"], [["times", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('E_x', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('i', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(x,f)} = f \\sin{(x)}, then derive \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)} = f \\cos{(x)}, then obtain 2 \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)} = f \\cos{(x)} + \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)}", "derivation": "\\operatorname{F_{N}}{(x,f)} = f \\sin{(x)} and \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)} = \\frac{\\partial}{\\partial x} f \\sin{(x)} and \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)} = f \\cos{(x)} and 2 \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)} = f \\cos{(x)} + \\frac{\\partial}{\\partial x} \\operatorname{F_{N}}{(x,f)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Symbol('f', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Symbol('f', commutative=True), cos(Symbol('x', commutative=True))))"], [["add", 3, "Derivative(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(Mul(Symbol('f', commutative=True), cos(Symbol('x', commutative=True))), Derivative(Function('F_N')(Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(\\hat{x}_0,\\mathbf{M})} = \\cos^{\\mathbf{M}}{(\\hat{x}_0)} and \\hat{\\mathbf{x}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}, then obtain (\\sin{(\\cos^{\\mathbf{M}}{(\\hat{x}_0)})} + 1) \\hat{\\mathbf{x}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} (\\sin{(\\cos^{\\mathbf{M}}{(\\hat{x}_0)})} + 1)", "derivation": "L{(\\hat{x}_0,\\mathbf{M})} = \\cos^{\\mathbf{M}}{(\\hat{x}_0)} and \\sin{(L{(\\hat{x}_0,\\mathbf{M})})} = \\sin{(\\cos^{\\mathbf{M}}{(\\hat{x}_0)})} and \\hat{\\mathbf{x}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} and (\\sin{(L{(\\hat{x}_0,\\mathbf{M})})} + 1) \\hat{\\mathbf{x}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} (\\sin{(L{(\\hat{x}_0,\\mathbf{M})})} + 1) and (\\sin{(\\cos^{\\mathbf{M}}{(\\hat{x}_0)})} + 1) \\hat{\\mathbf{x}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} (\\sin{(\\cos^{\\mathbf{M}}{(\\hat{x}_0)})} + 1)", "srepr_derivation": [["get_premise", "Equality(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(cos(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), sin(Pow(cos(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], [["times", 3, "Add(sin(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1))"], "Equality(Mul(Add(sin(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(sin(Function('L')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(sin(Pow(cos(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Add(sin(Pow(cos(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},c)} = \\int (- \\hat{\\mathbf{r}} + c) dc and \\tilde{g}^*{(\\hat{\\mathbf{r}},c)} = ((\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{\\hat{\\mathbf{r}}} - \\operatorname{n_{2}}^{c}{(\\hat{\\mathbf{r}},c)}, then obtain \\tilde{g}^*^{c}{(\\hat{\\mathbf{r}},c)} = (((\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{\\hat{\\mathbf{r}}} - (\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{c}", "derivation": "\\operatorname{n_{2}}{(\\hat{\\mathbf{r}},c)} = \\int (- \\hat{\\mathbf{r}} + c) dc and \\operatorname{n_{2}}^{c}{(\\hat{\\mathbf{r}},c)} = (\\int (- \\hat{\\mathbf{r}} + c) dc)^{c} and \\tilde{g}^*{(\\hat{\\mathbf{r}},c)} = ((\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{\\hat{\\mathbf{r}}} - \\operatorname{n_{2}}^{c}{(\\hat{\\mathbf{r}},c)} and \\tilde{g}^*{(\\hat{\\mathbf{r}},c)} = ((\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{\\hat{\\mathbf{r}}} - (\\int (- \\hat{\\mathbf{r}} + c) dc)^{c} and \\tilde{g}^*^{c}{(\\hat{\\mathbf{r}},c)} = (((\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{\\hat{\\mathbf{r}}} - (\\int (- \\hat{\\mathbf{r}} + c) dc)^{c})^{c}", "srepr_derivation": [["get_premise", "Equality(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Add(Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Pow(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Add(Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)))))"], [["power", 4, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given n{(C_{1},\\hbar)} = C_{1} + \\hbar, then obtain \\frac{\\partial^{2}}{\\partial \\hbar\\partial C_{1}} (\\hbar (- \\hbar + n{(C_{1},\\hbar)}))^{\\hbar} = \\frac{\\partial^{2}}{\\partial \\hbar\\partial C_{1}} (C_{1} \\hbar)^{\\hbar}", "derivation": "n{(C_{1},\\hbar)} = C_{1} + \\hbar and - \\hbar + n{(C_{1},\\hbar)} = C_{1} and \\hbar (- \\hbar + n{(C_{1},\\hbar)}) = C_{1} \\hbar and (\\hbar (- \\hbar + n{(C_{1},\\hbar)}))^{\\hbar} = (C_{1} \\hbar)^{\\hbar} and \\frac{\\partial}{\\partial C_{1}} (\\hbar (- \\hbar + n{(C_{1},\\hbar)}))^{\\hbar} = \\frac{\\partial}{\\partial C_{1}} (C_{1} \\hbar)^{\\hbar} and \\frac{\\partial^{2}}{\\partial \\hbar\\partial C_{1}} (\\hbar (- \\hbar + n{(C_{1},\\hbar)}))^{\\hbar} = \\frac{\\partial^{2}}{\\partial \\hbar\\partial C_{1}} (C_{1} \\hbar)^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('C_1', commutative=True))"], [["times", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 4, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('\\\\hbar', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(E_{x})} = e^{\\sin{(E_{x})}}, then obtain 1 = e^{1 - 0^{E_{x}}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(E_{x})} = e^{\\sin{(E_{x})}} and 0 = - \\operatorname{f_{\\mathbf{v}}}{(E_{x})} + e^{\\sin{(E_{x})}} and 0^{E_{x}} = (- \\operatorname{f_{\\mathbf{v}}}{(E_{x})} + e^{\\sin{(E_{x})}})^{E_{x}} and 0 = - 0^{E_{x}} + (- \\operatorname{f_{\\mathbf{v}}}{(E_{x})} + e^{\\sin{(E_{x})}})^{E_{x}} and 0 = 1 - (- \\operatorname{f_{\\mathbf{v}}}{(E_{x})} + e^{\\sin{(E_{x})}})^{E_{x}} and 0 = 1 - 0^{E_{x}} and 1 = e^{1 - 0^{E_{x}}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True)), exp(sin(Symbol('E_x', commutative=True))))"], [["minus", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True))), exp(sin(Symbol('E_x', commutative=True)))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E_x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True))), exp(sin(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)))"], [["minus", 3, "Pow(Integer(0), Symbol('E_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('E_x', commutative=True))), Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True))), exp(sin(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True))), exp(sin(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('E_x', commutative=True)))))"], [["exp", 6], "Equality(Integer(1), exp(Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('E_x', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{x})} = \\sin{(v_{x})}, then derive \\int \\sigma_{p}{(v_{x})} dv_{x} = a^{\\dagger} - \\cos{(v_{x})}, then obtain \\frac{\\int \\sin{(v_{x})} dv_{x}}{a^{\\dagger}} = \\frac{a^{\\dagger} - \\cos{(v_{x})}}{a^{\\dagger}}", "derivation": "\\sigma_{p}{(v_{x})} = \\sin{(v_{x})} and \\int \\sigma_{p}{(v_{x})} dv_{x} = \\int \\sin{(v_{x})} dv_{x} and \\int \\sigma_{p}{(v_{x})} dv_{x} = a^{\\dagger} - \\cos{(v_{x})} and \\frac{\\int \\sigma_{p}{(v_{x})} dv_{x}}{a^{\\dagger}} = \\frac{a^{\\dagger} - \\cos{(v_{x})}}{a^{\\dagger}} and \\frac{\\int \\sin{(v_{x})} dv_{x}}{a^{\\dagger}} = \\frac{a^{\\dagger} - \\cos{(v_{x})}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_p')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('v_x', commutative=True)))))"], [["divide", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Integral(Function('\\\\sigma_p')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('v_x', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('v_x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} and \\operatorname{P_{e}}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}}, then obtain \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{P_{e}}{(\\tilde{g}^*)}}{\\sin{(\\tilde{g}^*)}} = \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{f_{\\mathbf{v}}}{(\\tilde{g}^*)}}{\\sin{(\\tilde{g}^*)}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} and \\operatorname{P_{e}}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} and \\frac{d}{d \\tilde{g}^*} \\operatorname{P_{e}}{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} e^{\\sin{(\\tilde{g}^*)}} and \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{P_{e}}{(\\tilde{g}^*)}}{\\sin{(\\tilde{g}^*)}} = \\frac{\\frac{d}{d \\tilde{g}^*} e^{\\sin{(\\tilde{g}^*)}}}{\\sin{(\\tilde{g}^*)}} and \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{P_{e}}{(\\tilde{g}^*)}}{\\sin{(\\tilde{g}^*)}} = \\frac{\\frac{d}{d \\tilde{g}^*} \\operatorname{f_{\\mathbf{v}}}{(\\tilde{g}^*)}}{\\sin{(\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["divide", 3, "sin(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Derivative(Function('P_e')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Derivative(exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Derivative(Function('P_e')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(h,\\sigma_x)} = \\sigma_x + h, then obtain h + x{(h,\\sigma_x)} - x^{h}{(h,\\sigma_x)} = \\sigma_x + 2 h - x^{h}{(h,\\sigma_x)}", "derivation": "x{(h,\\sigma_x)} = \\sigma_x + h and h + x{(h,\\sigma_x)} = \\sigma_x + 2 h and x^{h}{(h,\\sigma_x)} = (\\sigma_x + h)^{h} and h - (\\sigma_x + h)^{h} + x{(h,\\sigma_x)} = \\sigma_x + 2 h - (\\sigma_x + h)^{h} and h + x{(h,\\sigma_x)} - x^{h}{(h,\\sigma_x)} = \\sigma_x + 2 h - x^{h}{(h,\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 2, "Pow(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('h', commutative=True), Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('h', commutative=True)))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Function('x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\Psi_{nl},u)} = \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}}, then derive \\phi{(\\Psi_{nl},u)} = - \\frac{u}{\\Psi_{nl}^{2}}, then obtain 1 - \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}} = 1 - \\phi{(\\Psi_{nl},u)}", "derivation": "\\phi{(\\Psi_{nl},u)} = \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}} and \\phi{(\\Psi_{nl},u)} = - \\frac{u}{\\Psi_{nl}^{2}} and - \\phi{(\\Psi_{nl},u)} = \\frac{u}{\\Psi_{nl}^{2}} and - \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}} = \\frac{u}{\\Psi_{nl}^{2}} and 1 - \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}} = 1 + \\frac{u}{\\Psi_{nl}^{2}} and 1 - \\phi{(\\Psi_{nl},u)} = 1 + \\frac{u}{\\Psi_{nl}^{2}} and 1 - \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{u}{\\Psi_{nl}} = 1 - \\phi{(\\Psi_{nl},u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('u', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('u', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('u', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('u', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} = \\sin{(\\sin{(\\mathbf{D})})} and \\hat{x}_0{(\\mathbf{D})} = \\iint \\sin{(\\sin{(\\mathbf{D})})} d\\mathbf{D} d\\mathbf{D}, then obtain 1 = \\frac{\\hat{x}_0{(\\mathbf{D})}}{\\iint \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} = \\sin{(\\sin{(\\mathbf{D})})} and \\int \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} d\\mathbf{D} = \\int \\sin{(\\sin{(\\mathbf{D})})} d\\mathbf{D} and \\iint \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D} = \\iint \\sin{(\\sin{(\\mathbf{D})})} d\\mathbf{D} d\\mathbf{D} and 1 = \\frac{\\iint \\sin{(\\sin{(\\mathbf{D})})} d\\mathbf{D} d\\mathbf{D}}{\\iint \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D}} and \\hat{x}_0{(\\mathbf{D})} = \\iint \\sin{(\\sin{(\\mathbf{D})})} d\\mathbf{D} d\\mathbf{D} and 1 = \\frac{\\hat{x}_0{(\\mathbf{D})}}{\\iint \\operatorname{J_{\\varepsilon}}{(\\mathbf{D})} d\\mathbf{D} d\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(sin(sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(sin(sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 3, "Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), Integral(sin(sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{D}', commutative=True)), Integral(sin(sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(1), Mul(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(z)} = \\cos{(z)}, then obtain \\frac{z + k{(z)} - k^{z}{(z)}}{\\int k^{z}{(z)} dz} = \\frac{z - k^{z}{(z)} + \\cos{(z)}}{\\int k^{z}{(z)} dz}", "derivation": "k{(z)} = \\cos{(z)} and k^{z}{(z)} = \\cos^{z}{(z)} and \\int k^{z}{(z)} dz = \\int \\cos^{z}{(z)} dz and z + k{(z)} - \\cos^{z}{(z)} = z + \\cos{(z)} - \\cos^{z}{(z)} and z + k{(z)} - k^{z}{(z)} = z - k^{z}{(z)} + \\cos{(z)} and \\frac{z + k{(z)} - k^{z}{(z)}}{\\int \\cos^{z}{(z)} dz} = \\frac{z - k^{z}{(z)} + \\cos{(z)}}{\\int \\cos^{z}{(z)} dz} and \\frac{z + k{(z)} - k^{z}{(z)}}{\\int k^{z}{(z)} dz} = \\frac{z - k^{z}{(z)} + \\cos{(z)}}{\\int k^{z}{(z)} dz}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], "Equality(Add(Symbol('z', commutative=True), Function('k')(Symbol('z', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))), Add(Symbol('z', commutative=True), cos(Symbol('z', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('z', commutative=True), Function('k')(Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))), Add(Symbol('z', commutative=True), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["divide", 5, "Integral(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Mul(Add(Symbol('z', commutative=True), Function('k')(Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))), Pow(Integral(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))), Mul(Add(Symbol('z', commutative=True), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Pow(Integral(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Add(Symbol('z', commutative=True), Function('k')(Symbol('z', commutative=True)), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))), Pow(Integral(Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))), Mul(Add(Symbol('z', commutative=True), Mul(Integer(-1), Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Pow(Integral(Pow(Function('k')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(\\theta)} = e^{\\theta}, then obtain \\theta (\\mathbb{I}{(\\theta)} - e^{\\theta})^{\\theta} = 0^{\\theta} \\theta", "derivation": "\\mathbb{I}{(\\theta)} = e^{\\theta} and \\mathbb{I}{(\\theta)} - e^{\\theta} = 0 and (\\mathbb{I}{(\\theta)} - e^{\\theta})^{\\theta} = 0^{\\theta} and \\theta (\\mathbb{I}{(\\theta)} - e^{\\theta})^{\\theta} = 0^{\\theta} \\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbb{I}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(Integer(0), Symbol('\\\\theta', commutative=True)))"], [["times", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Add(Function('\\\\mathbb{I}')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = F_{N} m_{s} + i, then obtain 2 \\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = F_{N} m_{s} + i + \\hat{H}_{\\lambda}{(i,m_{s},F_{N})}", "derivation": "\\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = F_{N} m_{s} + i and F_{N} m_{s} + i + \\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = 2 F_{N} m_{s} + 2 i and 2 \\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = 2 F_{N} m_{s} + 2 i and 2 \\hat{H}_{\\lambda}{(i,m_{s},F_{N})} = F_{N} m_{s} + i + \\hat{H}_{\\lambda}{(i,m_{s},F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('m_s', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('i', commutative=True)))"], [["add", 1, "Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('i', commutative=True))"], "Equality(Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('i', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(2), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(2), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('i', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('i', commutative=True), Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given W{(z^{*},G)} = (z^{*})^{G}, then derive \\frac{\\partial}{\\partial z^{*}} W{(z^{*},G)} = \\frac{G (z^{*})^{G}}{z^{*}}, then obtain \\frac{\\partial}{\\partial z^{*}} W{(z^{*},G)} = \\frac{G W{(z^{*},G)}}{z^{*}}", "derivation": "W{(z^{*},G)} = (z^{*})^{G} and \\frac{\\partial}{\\partial z^{*}} W{(z^{*},G)} = \\frac{\\partial}{\\partial z^{*}} (z^{*})^{G} and \\frac{\\partial}{\\partial z^{*}} W{(z^{*},G)} = \\frac{G (z^{*})^{G}}{z^{*}} and \\frac{\\partial}{\\partial z^{*}} W{(z^{*},G)} = \\frac{G W{(z^{*},G)}}{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('z^*', commutative=True), Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Pow(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Symbol('G', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Symbol('G', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('W')(Symbol('z^*', commutative=True), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} = - H + \\frac{\\mathbf{A}}{\\varepsilon_0}, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} - \\frac{\\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} + 1}{H} = \\frac{\\partial}{\\partial \\mathbf{A}} - \\frac{- H + \\frac{\\mathbf{A}}{\\varepsilon_0} + 1}{H}", "derivation": "\\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} = - H + \\frac{\\mathbf{A}}{\\varepsilon_0} and \\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} + 1 = - H + \\frac{\\mathbf{A}}{\\varepsilon_0} + 1 and - \\frac{\\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} + 1}{H} = - \\frac{- H + \\frac{\\mathbf{A}}{\\varepsilon_0} + 1}{H} and \\frac{\\partial}{\\partial \\mathbf{A}} - \\frac{\\varepsilon{(\\varepsilon_0,\\mathbf{A},H)} + 1}{H} = \\frac{\\partial}{\\partial \\mathbf{A}} - \\frac{- H + \\frac{\\mathbf{A}}{\\varepsilon_0} + 1}{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('H', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Integer(1)))"], [["divide", 2, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('H', commutative=True)), Integer(1))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('H', commutative=True)), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(C_{1})} = \\log{(C_{1})}, then derive \\frac{d}{d C_{1}} \\mathbf{B}{(C_{1})} = \\frac{1}{C_{1}}, then obtain \\int \\frac{d}{d C_{1}} \\log{(C_{1})} dC_{1} = \\int \\frac{1}{C_{1}} dC_{1}", "derivation": "\\mathbf{B}{(C_{1})} = \\log{(C_{1})} and \\frac{d}{d C_{1}} \\mathbf{B}{(C_{1})} = \\frac{d}{d C_{1}} \\log{(C_{1})} and \\frac{d}{d C_{1}} \\mathbf{B}{(C_{1})} = \\frac{1}{C_{1}} and \\frac{d}{d C_{1}} \\log{(C_{1})} = \\frac{1}{C_{1}} and \\int \\frac{d}{d C_{1}} \\log{(C_{1})} dC_{1} = \\int \\frac{1}{C_{1}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True)), log(Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(log(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Pow(Symbol('C_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Pow(Symbol('C_1', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(log(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(Symbol('C_1', commutative=True), Integer(-1)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given S{(z,E_{\\lambda})} = z^{E_{\\lambda}} and \\rho{(z,E_{\\lambda})} = \\frac{- E_{\\lambda} + z^{E_{\\lambda}}}{- E_{\\lambda} + S{(z,E_{\\lambda})}}, then obtain (\\frac{\\partial}{\\partial E_{\\lambda}} \\rho{(z,E_{\\lambda})})^{z} = 0^{z}", "derivation": "S{(z,E_{\\lambda})} = z^{E_{\\lambda}} and \\rho{(z,E_{\\lambda})} = \\frac{- E_{\\lambda} + z^{E_{\\lambda}}}{- E_{\\lambda} + S{(z,E_{\\lambda})}} and \\rho{(z,E_{\\lambda})} = 1 and \\frac{\\partial}{\\partial E_{\\lambda}} \\rho{(z,E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} 1 and (\\frac{\\partial}{\\partial E_{\\lambda}} \\rho{(z,E_{\\lambda})})^{z} = (\\frac{d}{d E_{\\lambda}} 1)^{z} and - \\frac{d}{d E_{\\lambda}} 1 + (\\frac{\\partial}{\\partial E_{\\lambda}} \\rho{(z,E_{\\lambda})})^{z} = - \\frac{d}{d E_{\\lambda}} 1 + (\\frac{d}{d E_{\\lambda}} 1)^{z} and (\\frac{\\partial}{\\partial E_{\\lambda}} \\rho{(z,E_{\\lambda})})^{z} = 0^{z}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('S')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["minus", 5, "Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Pow(Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('z', commutative=True))))"], [["evaluate_derivatives", 6], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('z', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Integer(0), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(T)} = \\cos{(T)} and \\operatorname{v_{1}}{(\\mu_0)} = \\log{(e^{\\mu_0})}, then obtain \\frac{\\operatorname{v_{1}}{(\\mu_0)} - e^{\\mu_0}}{\\cos{(T)}} = \\frac{- e^{\\mu_0} + \\log{(e^{\\mu_0})}}{\\cos{(T)}}", "derivation": "\\dot{\\mathbf{r}}{(T)} = \\cos{(T)} and \\operatorname{v_{1}}{(\\mu_0)} = \\log{(e^{\\mu_0})} and \\operatorname{v_{1}}{(\\mu_0)} - e^{\\mu_0} = - e^{\\mu_0} + \\log{(e^{\\mu_0})} and \\frac{\\operatorname{v_{1}}{(\\mu_0)} - e^{\\mu_0}}{\\dot{\\mathbf{r}}{(T)}} = \\frac{- e^{\\mu_0} + \\log{(e^{\\mu_0})}}{\\dot{\\mathbf{r}}{(T)}} and \\frac{\\operatorname{v_{1}}{(\\mu_0)} - e^{\\mu_0}}{\\cos{(T)}} = \\frac{- e^{\\mu_0} + \\log{(e^{\\mu_0})}}{\\cos{(T)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], ["get_premise", "Equality(Function('v_1')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True)))))"], [["divide", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('T', commutative=True))"], "Equality(Mul(Add(Function('v_1')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('T', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('T', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Function('v_1')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(cos(Symbol('T', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), log(exp(Symbol('\\\\mu_0', commutative=True)))), Pow(cos(Symbol('T', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(\\hat{p},p)} = \\frac{\\log{(\\hat{p})}}{p}, then derive \\frac{\\partial}{\\partial \\hat{p}} \\varphi{(\\hat{p},p)} = \\frac{1}{\\hat{p} p}, then obtain \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\log{(\\hat{p})}}{p} + \\frac{\\log{(\\hat{p})}}{p} = \\frac{\\log{(\\hat{p})}}{p} + \\frac{1}{\\hat{p} p}", "derivation": "\\varphi{(\\hat{p},p)} = \\frac{\\log{(\\hat{p})}}{p} and p + \\varphi{(\\hat{p},p)} = p + \\frac{\\log{(\\hat{p})}}{p} and \\frac{\\partial}{\\partial \\hat{p}} (p + \\varphi{(\\hat{p},p)}) = \\frac{\\partial}{\\partial \\hat{p}} (p + \\frac{\\log{(\\hat{p})}}{p}) and \\frac{\\partial}{\\partial \\hat{p}} \\varphi{(\\hat{p},p)} = \\frac{1}{\\hat{p} p} and \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\log{(\\hat{p})}}{p} = \\frac{1}{\\hat{p} p} and \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\log{(\\hat{p})}}{p} + \\frac{\\log{(\\hat{p})}}{p} = \\frac{\\log{(\\hat{p})}}{p} + \\frac{1}{\\hat{p} p}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 1, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Function('\\\\varphi')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True))), Add(Symbol('p', commutative=True), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Symbol('p', commutative=True), Function('\\\\varphi')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('p', commutative=True), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\hat{p}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["add", 5, "Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{B},W)} = \\cos{(W^{\\mathbf{B}})} and E{(\\mathbf{B},W)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)}, then obtain (\\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(W^{\\mathbf{B}})})^{\\mathbf{B}} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)})^{\\mathbf{B}}", "derivation": "\\tilde{g}{(\\mathbf{B},W)} = \\cos{(W^{\\mathbf{B}})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(W^{\\mathbf{B}})} and E{(\\mathbf{B},W)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)} and E^{\\mathbf{B}}{(\\mathbf{B},W)} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)})^{\\mathbf{B}} and E{(\\mathbf{B},W)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(W^{\\mathbf{B}})} and (\\frac{\\partial}{\\partial \\mathbf{B}} \\cos{(W^{\\mathbf{B}})})^{\\mathbf{B}} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\tilde{g}{(\\mathbf{B},W)})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Derivative(cos(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(cos(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(S,F_{x})} = - F_{x} + S and \\delta{(S,F_{x})} = - \\operatorname{P_{e}}{(S,F_{x})}, then obtain - (- F_{x} + S) (F_{x} - S) \\delta{(S,F_{x})} = (F_{x} - S) \\operatorname{P_{e}}^{2}{(S,F_{x})}", "derivation": "\\operatorname{P_{e}}{(S,F_{x})} = - F_{x} + S and - \\operatorname{P_{e}}{(S,F_{x})} = F_{x} - S and - (F_{x} - S) \\operatorname{P_{e}}{(S,F_{x})} = (F_{x} - S)^{2} and - (- F_{x} + S) (F_{x} - S) = (F_{x} - S)^{2} and - (- F_{x} + S) (F_{x} - S) = - (F_{x} - S) \\operatorname{P_{e}}{(S,F_{x})} and \\delta{(S,F_{x})} = - \\operatorname{P_{e}}{(S,F_{x})} and - (F_{x} - S) \\operatorname{P_{e}}{(S,F_{x})} \\delta{(S,F_{x})} = (F_{x} - S) \\operatorname{P_{e}}^{2}{(S,F_{x})} and - (- F_{x} + S) (F_{x} - S) \\delta{(S,F_{x})} = (F_{x} - S) \\operatorname{P_{e}}^{2}{(S,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('S', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))))"], [["times", 2, "Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))), Pow(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('S', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Pow(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('S', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Mul(Integer(-1), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))))"], [["times", 6, "Mul(Integer(-1), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Function('\\\\delta')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))), Mul(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('S', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('\\\\delta')(Symbol('S', commutative=True), Symbol('F_x', commutative=True))), Mul(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Pow(Function('P_e')(Symbol('S', commutative=True), Symbol('F_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})}, then obtain - \\mathbf{J}_P + \\iint \\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} dJ_{\\varepsilon} dJ_{\\varepsilon} = - \\mathbf{J}_P + \\iint \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})} dJ_{\\varepsilon} dJ_{\\varepsilon}", "derivation": "\\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})} and \\int \\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} dJ_{\\varepsilon} = \\int \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})} dJ_{\\varepsilon} and \\iint \\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} dJ_{\\varepsilon} dJ_{\\varepsilon} = \\iint \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})} dJ_{\\varepsilon} dJ_{\\varepsilon} and - \\mathbf{J}_P + \\iint \\mathbf{J}{(J_{\\varepsilon},\\mathbf{J}_P)} dJ_{\\varepsilon} dJ_{\\varepsilon} = - \\mathbf{J}_P + \\iint \\cos{(\\mathbf{J}_P^{J_{\\varepsilon}})} dJ_{\\varepsilon} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(cos(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(cos(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(cos(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mu{(h,\\eta)} = - \\eta + e^{h}, then obtain (\\mu{(h,\\eta)} - 1)^{h} (h + \\mu{(h,\\eta)} - 1) = (- \\eta + e^{h} - 1)^{h} (h + \\mu{(h,\\eta)} - 1)", "derivation": "\\mu{(h,\\eta)} = - \\eta + e^{h} and \\mu{(h,\\eta)} - 1 = - \\eta + e^{h} - 1 and (\\mu{(h,\\eta)} - 1)^{h} = (- \\eta + e^{h} - 1)^{h} and \\frac{(\\mu{(h,\\eta)} - 1)^{h}}{(\\mu{(h,\\eta)} - 1)^{2}} = \\frac{(- \\eta + e^{h} - 1)^{h}}{(\\mu{(h,\\eta)} - 1)^{2}} and \\frac{(\\mu{(h,\\eta)} - 1)^{h} (h + \\mu{(h,\\eta)} - 1)}{(\\mu{(h,\\eta)} - 1)^{2}} = \\frac{(- \\eta + e^{h} - 1)^{h} (h + \\mu{(h,\\eta)} - 1)}{(\\mu{(h,\\eta)} - 1)^{2}} and (\\mu{(h,\\eta)} - 1)^{h} (h + \\mu{(h,\\eta)} - 1) = (- \\eta + e^{h} - 1)^{h} (h + \\mu{(h,\\eta)} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True)), Integer(-1)), Symbol('h', commutative=True)))"], [["divide", 3, "Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(2))"], "Equality(Mul(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(-2)), Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Symbol('h', commutative=True))), Mul(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True)), Integer(-1)), Symbol('h', commutative=True))))"], [["times", 4, "Add(Symbol('h', commutative=True), Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(-2)), Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Mul(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"], [["divide", 5, "Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Integer(-2))"], "Equality(Mul(Pow(Add(Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('h', commutative=True)), Integer(-1)), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Function('\\\\mu')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C,s)} = \\sin{(C - s)}, then derive \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)} = - \\cos{(C - s)}, then obtain \\frac{\\partial}{\\partial s} (- \\operatorname{F_{x}}{(C,s)} - \\cos{(C - s)}) = \\frac{\\partial}{\\partial s} (- \\operatorname{F_{x}}{(C,s)} + \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)})", "derivation": "\\operatorname{F_{x}}{(C,s)} = \\sin{(C - s)} and \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)} = \\frac{\\partial}{\\partial s} \\sin{(C - s)} and \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)} = - \\cos{(C - s)} and - \\sin{(C - s)} + \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)} = - \\sin{(C - s)} + \\frac{\\partial}{\\partial s} \\sin{(C - s)} and - \\sin{(C - s)} - \\cos{(C - s)} = - \\sin{(C - s)} + \\frac{\\partial}{\\partial s} \\sin{(C - s)} and - \\operatorname{F_{x}}{(C,s)} - \\cos{(C - s)} = - \\operatorname{F_{x}}{(C,s)} + \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)} and \\frac{\\partial}{\\partial s} (- \\operatorname{F_{x}}{(C,s)} - \\cos{(C - s)}) = \\frac{\\partial}{\\partial s} (- \\operatorname{F_{x}}{(C,s)} + \\frac{\\partial}{\\partial s} \\operatorname{F_{x}}{(C,s)})", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))))"], [["minus", 2, "sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Derivative(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Derivative(sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Mul(Integer(-1), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))), Add(Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Derivative(sin(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))), Add(Mul(Integer(-1), Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True))), Derivative(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["differentiate", 6, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True))), Derivative(Function('F_x')(Symbol('C', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{S},u)} = (e^{u})^{\\mathbf{S}}, then obtain (\\int (- \\mathbf{S} + \\sin{(\\operatorname{v_{2}}{(\\mathbf{S},u)})}) du)^{u} = (\\int (- \\mathbf{S} + \\sin{((e^{u})^{\\mathbf{S}})}) du)^{u}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{S},u)} = (e^{u})^{\\mathbf{S}} and \\sin{(\\operatorname{v_{2}}{(\\mathbf{S},u)})} = \\sin{((e^{u})^{\\mathbf{S}})} and - \\mathbf{S} + \\sin{(\\operatorname{v_{2}}{(\\mathbf{S},u)})} = - \\mathbf{S} + \\sin{((e^{u})^{\\mathbf{S}})} and \\int (- \\mathbf{S} + \\sin{(\\operatorname{v_{2}}{(\\mathbf{S},u)})}) du = \\int (- \\mathbf{S} + \\sin{((e^{u})^{\\mathbf{S}})}) du and (\\int (- \\mathbf{S} + \\sin{(\\operatorname{v_{2}}{(\\mathbf{S},u)})}) du)^{u} = (\\int (- \\mathbf{S} + \\sin{((e^{u})^{\\mathbf{S}})}) du)^{u}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True))), sin(Pow(exp(Symbol('u', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Pow(exp(Symbol('u', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Pow(exp(Symbol('u', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('u', commutative=True))))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Function('v_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Pow(exp(Symbol('u', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(V,F_{H})} = F_{H} + V, then obtain - F_{H} - V - 2 ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V} + (\\operatorname{E_{\\lambda}}^{2}{(V,F_{H})})^{V} = - F_{H} - V - ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V}", "derivation": "\\operatorname{E_{\\lambda}}{(V,F_{H})} = F_{H} + V and \\operatorname{E_{\\lambda}}^{2}{(V,F_{H})} = (F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})} and (\\operatorname{E_{\\lambda}}^{2}{(V,F_{H})})^{V} = ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V} and - ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V} + (\\operatorname{E_{\\lambda}}^{2}{(V,F_{H})})^{V} = 0 and - F_{H} - V - 2 ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V} + (\\operatorname{E_{\\lambda}}^{2}{(V,F_{H})})^{V} = - F_{H} - V - ((F_{H} + V) \\operatorname{E_{\\lambda}}{(V,F_{H})})^{V}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Integer(2)), Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))))"], [["power", 2, "Symbol('V', commutative=True)"], "Equality(Pow(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Integer(2)), Symbol('V', commutative=True)), Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True)))"], [["minus", 3, "Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True))), Pow(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Integer(2)), Symbol('V', commutative=True))), Integer(0))"], [["minus", 4, "Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True), Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True))), Pow(Pow(Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Integer(2)), Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Mul(Add(Symbol('F_H', commutative=True), Symbol('V', commutative=True)), Function('E_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{\\mathbf{r}},c)} = \\hat{\\mathbf{r}} c, then obtain \\frac{\\partial}{\\partial y} \\frac{y \\mathbb{I}{(\\hat{\\mathbf{r}},c)}}{\\hat{\\mathbf{r}} c \\frac{d}{d y} \\cos{(y)}} = \\frac{d}{d y} \\frac{y}{\\frac{d}{d y} \\cos{(y)}}", "derivation": "\\mathbb{I}{(\\hat{\\mathbf{r}},c)} = \\hat{\\mathbf{r}} c and \\frac{\\mathbb{I}{(\\hat{\\mathbf{r}},c)}}{\\hat{\\mathbf{r}} c} = 1 and \\frac{y \\mathbb{I}{(\\hat{\\mathbf{r}},c)}}{\\hat{\\mathbf{r}} c} = y and \\frac{y \\mathbb{I}{(\\hat{\\mathbf{r}},c)}}{\\hat{\\mathbf{r}} c \\frac{d}{d y} \\cos{(y)}} = \\frac{y}{\\frac{d}{d y} \\cos{(y)}} and \\frac{\\partial}{\\partial y} \\frac{y \\mathbb{I}{(\\hat{\\mathbf{r}},c)}}{\\hat{\\mathbf{r}} c \\frac{d}{d y} \\cos{(y)}} = \\frac{d}{d y} \\frac{y}{\\frac{d}{d y} \\cos{(y)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True))), Integer(1))"], [["times", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True))), Symbol('y', commutative=True))"], [["divide", 3, "Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Pow(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('y', commutative=True), Pow(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('c', commutative=True)), Pow(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Symbol('y', commutative=True), Pow(Derivative(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})} = - \\Psi_{nl} + \\sin{(C_{2})}, then obtain \\cos{(\\frac{1}{(- \\Psi_{nl} + \\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})})^{2}})} = \\cos{(\\frac{1}{(- 2 \\Psi_{nl} + \\sin{(C_{2})})^{2}})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})} = - \\Psi_{nl} + \\sin{(C_{2})} and - \\Psi_{nl} + \\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})} = - 2 \\Psi_{nl} + \\sin{(C_{2})} and \\frac{1}{(- \\Psi_{nl} + \\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})})^{2}} = \\frac{1}{(- 2 \\Psi_{nl} + \\sin{(C_{2})})^{2}} and \\cos{(\\frac{1}{(- \\Psi_{nl} + \\operatorname{f_{\\mathbf{v}}}{(C_{2},\\Psi_{nl})})^{2}})} = \\cos{(\\frac{1}{(- 2 \\Psi_{nl} + \\sin{(C_{2})})^{2}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('C_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('C_2', commutative=True))))"], [["minus", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('C_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('C_2', commutative=True))))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('C_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('C_2', commutative=True))), Integer(-2)))"], [["cos", 3], "Equality(cos(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('C_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-2))), cos(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('C_2', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{D}{(t)} = e^{t}, then derive e^{t} + \\frac{d}{d t} \\mathbf{D}{(t)} = 2 e^{t}, then obtain \\frac{d}{d t} ((e^{t} + \\frac{d}{d t} e^{t})^{2})^{t} = \\frac{d}{d t} (4 e^{2 t})^{t}", "derivation": "\\mathbf{D}{(t)} = e^{t} and \\frac{d}{d t} \\mathbf{D}{(t)} = \\frac{d}{d t} e^{t} and e^{t} + \\frac{d}{d t} \\mathbf{D}{(t)} = e^{t} + \\frac{d}{d t} e^{t} and e^{t} + \\frac{d}{d t} \\mathbf{D}{(t)} = 2 e^{t} and (e^{t} + \\frac{d}{d t} \\mathbf{D}{(t)})^{2} = 4 e^{2 t} and ((e^{t} + \\frac{d}{d t} \\mathbf{D}{(t)})^{2})^{t} = (4 e^{2 t})^{t} and ((e^{t} + \\frac{d}{d t} e^{t})^{2})^{t} = (4 e^{2 t})^{t} and \\frac{d}{d t} ((e^{t} + \\frac{d}{d t} e^{t})^{2})^{t} = \\frac{d}{d t} (4 e^{2 t})^{t}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["add", 2, "exp(Symbol('t', commutative=True))"], "Equality(Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(exp(Symbol('t', commutative=True)), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('t', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('t', commutative=True)))))"], [["power", 5, "Symbol('t', commutative=True)"], "Equality(Pow(Pow(Add(exp(Symbol('t', commutative=True)), Derivative(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(2)), Symbol('t', commutative=True)), Pow(Mul(Integer(4), exp(Mul(Integer(2), Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Pow(Add(exp(Symbol('t', commutative=True)), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(2)), Symbol('t', commutative=True)), Pow(Mul(Integer(4), exp(Mul(Integer(2), Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["differentiate", 7, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(exp(Symbol('t', commutative=True)), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(2)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(4), exp(Mul(Integer(2), Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\rho_b,y^{\\prime})} = \\rho_b y^{\\prime}, then obtain \\frac{\\rho_b + 2 \\psi{(\\rho_b,y^{\\prime})}}{2 \\psi{(\\rho_b,y^{\\prime})}} = \\frac{2 \\rho_b y^{\\prime} + \\rho_b}{2 \\psi{(\\rho_b,y^{\\prime})}}", "derivation": "\\psi{(\\rho_b,y^{\\prime})} = \\rho_b y^{\\prime} and \\rho_b + \\psi{(\\rho_b,y^{\\prime})} = \\rho_b y^{\\prime} + \\rho_b and \\rho_b y^{\\prime} + \\rho_b + \\psi{(\\rho_b,y^{\\prime})} = 2 \\rho_b y^{\\prime} + \\rho_b and \\rho_b + 2 \\psi{(\\rho_b,y^{\\prime})} = 2 \\rho_b y^{\\prime} + \\rho_b and \\frac{\\rho_b + 2 \\psi{(\\rho_b,y^{\\prime})}}{2 \\psi{(\\rho_b,y^{\\prime})}} = \\frac{2 \\rho_b y^{\\prime} + \\rho_b}{2 \\psi{(\\rho_b,y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True), Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 4, "Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Function('\\\\psi')(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(I)} = e^{I}, then obtain \\int \\frac{\\rho_{f}^{2}{(I)} e^{- 2 I}}{2} dI = \\int \\frac{1}{2} dI", "derivation": "\\rho_{f}{(I)} = e^{I} and \\frac{\\rho_{f}{(I)} e^{- I}}{2} = \\frac{1}{2} and \\int \\frac{\\rho_{f}{(I)} e^{- I}}{2} dI = \\int \\frac{1}{2} dI and \\int \\frac{\\rho_{f}^{2}{(I)} e^{- 2 I}}{2} dI = \\int \\frac{\\rho_{f}{(I)} e^{- I}}{2} dI and \\int \\frac{\\rho_{f}^{2}{(I)} e^{- 2 I}}{2} dI = \\int \\frac{1}{2} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["divide", 1, "Mul(Integer(2), exp(Symbol('I', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('\\\\rho_f')(Symbol('I', commutative=True)), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Rational(1, 2))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Function('\\\\rho_f')(Symbol('I', commutative=True)), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Rational(1, 2), Tuple(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Mul(Rational(1, 2), Pow(Function('\\\\rho_f')(Symbol('I', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Rational(1, 2), Function('\\\\rho_f')(Symbol('I', commutative=True)), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Rational(1, 2), Pow(Function('\\\\rho_f')(Symbol('I', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Rational(1, 2), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given C{(p)} = \\cos{(e^{p})}, then derive \\int C{(p)} dp = \\hat{H}_l + \\operatorname{Ci}{(e^{p})}, then derive \\mathbf{B} + \\operatorname{Ci}{(e^{p})} = \\hat{H}_l + \\operatorname{Ci}{(e^{p})}, then obtain \\frac{\\mathbf{B} + \\operatorname{Ci}{(e^{p})}}{\\hat{H}_l + \\operatorname{Ci}{(e^{p})}} = 1", "derivation": "C{(p)} = \\cos{(e^{p})} and \\int C{(p)} dp = \\int \\cos{(e^{p})} dp and \\int C{(p)} dp = \\hat{H}_l + \\operatorname{Ci}{(e^{p})} and \\int \\cos{(e^{p})} dp = \\hat{H}_l + \\operatorname{Ci}{(e^{p})} and \\mathbf{B} + \\operatorname{Ci}{(e^{p})} = \\hat{H}_l + \\operatorname{Ci}{(e^{p})} and \\frac{\\mathbf{B} + \\operatorname{Ci}{(e^{p})}}{\\hat{H}_l + \\operatorname{Ci}{(e^{p})}} = 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('C')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Ci(exp(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Ci(exp(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Ci(exp(Symbol('p', commutative=True)))), Add(Symbol('\\\\hat{H}_l', commutative=True), Ci(exp(Symbol('p', commutative=True)))))"], [["divide", 5, "Add(Symbol('\\\\hat{H}_l', commutative=True), Ci(exp(Symbol('p', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Ci(exp(Symbol('p', commutative=True)))), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Ci(exp(Symbol('p', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\delta{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain \\delta{(\\mathbf{J}_M)} \\mathbf{p}{(\\mathbf{J}_M)} - \\delta{(\\mathbf{J}_M)} = \\delta{(\\mathbf{J}_M)} \\cos{(\\mathbf{J}_M)} - \\delta{(\\mathbf{J}_M)}", "derivation": "\\mathbf{p}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\mathbf{p}^{2}{(\\mathbf{J}_M)} = \\mathbf{p}{(\\mathbf{J}_M)} \\cos{(\\mathbf{J}_M)} and \\delta{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\delta{(\\mathbf{J}_M)} = \\mathbf{p}{(\\mathbf{J}_M)} and \\delta^{2}{(\\mathbf{J}_M)} = \\delta{(\\mathbf{J}_M)} \\cos{(\\mathbf{J}_M)} and \\delta^{2}{(\\mathbf{J}_M)} = \\delta{(\\mathbf{J}_M)} \\mathbf{p}{(\\mathbf{J}_M)} and \\delta{(\\mathbf{J}_M)} \\mathbf{p}{(\\mathbf{J}_M)} = \\delta{(\\mathbf{J}_M)} \\cos{(\\mathbf{J}_M)} and \\delta{(\\mathbf{J}_M)} \\mathbf{p}{(\\mathbf{J}_M)} - \\delta{(\\mathbf{J}_M)} = \\delta{(\\mathbf{J}_M)} \\cos{(\\mathbf{J}_M)} - \\delta{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 4, "Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 7, "Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Mul(Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given b{(\\mathbf{s},\\dot{y})} = \\dot{y} \\mathbf{s}, then obtain 2 \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} + 1", "derivation": "b{(\\mathbf{s},\\dot{y})} = \\dot{y} \\mathbf{s} and \\frac{\\partial}{\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} \\dot{y} \\mathbf{s} and \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} \\dot{y} \\mathbf{s} and 2 \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} \\dot{y} \\mathbf{s} + \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} and 2 \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial \\dot{y}} b{(\\mathbf{s},\\dot{y})} + 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Derivative(Function('b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then obtain 0 = - \\frac{d}{d \\varphi^*} (\\int (\\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)}) d\\varphi^*)^{2} + \\frac{d}{d \\varphi^*} (\\int 2 \\sin{(\\varphi^*)} d\\varphi^*)^{2}", "derivation": "\\operatorname{C_{d}}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)} = 2 \\sin{(\\varphi^*)} and \\int (\\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)}) d\\varphi^* = \\int 2 \\sin{(\\varphi^*)} d\\varphi^* and (\\int (\\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)}) d\\varphi^*)^{2} = (\\int 2 \\sin{(\\varphi^*)} d\\varphi^*)^{2} and \\frac{d}{d \\varphi^*} (\\int (\\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)}) d\\varphi^*)^{2} = \\frac{d}{d \\varphi^*} (\\int 2 \\sin{(\\varphi^*)} d\\varphi^*)^{2} and 0 = - \\frac{d}{d \\varphi^*} (\\int (\\operatorname{C_{d}}{(\\varphi^*)} + \\sin{(\\varphi^*)}) d\\varphi^*)^{2} + \\frac{d}{d \\varphi^*} (\\int 2 \\sin{(\\varphi^*)} d\\varphi^*)^{2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)))"], [["differentiate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Pow(Integral(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Pow(Integral(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Pow(Integral(Add(Function('C_d')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Derivative(Pow(Integral(Mul(Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(r_{0})} = \\log{(r_{0})} and V{(r_{0})} = \\Psi^{\\dagger}{(r_{0})} + \\log{(r_{0})}, then obtain \\frac{d}{d r_{0}} (V{(r_{0})} + 2 \\log{(r_{0})}) = \\frac{d}{d r_{0}} 4 \\log{(r_{0})}", "derivation": "\\Psi^{\\dagger}{(r_{0})} = \\log{(r_{0})} and V{(r_{0})} = \\Psi^{\\dagger}{(r_{0})} + \\log{(r_{0})} and V{(r_{0})} + 2 \\log{(r_{0})} = \\Psi^{\\dagger}{(r_{0})} + 3 \\log{(r_{0})} and \\frac{d}{d r_{0}} (V{(r_{0})} + 2 \\log{(r_{0})}) = \\frac{d}{d r_{0}} (\\Psi^{\\dagger}{(r_{0})} + 3 \\log{(r_{0})}) and \\frac{d}{d r_{0}} (V{(r_{0})} + 2 \\log{(r_{0})}) = \\frac{d}{d r_{0}} 4 \\log{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('r_0', commutative=True)), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True))))"], [["add", 2, "Mul(Integer(2), log(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('V')(Symbol('r_0', commutative=True)), Mul(Integer(2), log(Symbol('r_0', commutative=True)))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('r_0', commutative=True)), Mul(Integer(3), log(Symbol('r_0', commutative=True)))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Function('V')(Symbol('r_0', commutative=True)), Mul(Integer(2), log(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('r_0', commutative=True)), Mul(Integer(3), log(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Function('V')(Symbol('r_0', commutative=True)), Mul(Integer(2), log(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Integer(4), log(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(g)} = e^{g} and \\operatorname{g_{\\varepsilon}}{(g)} = \\delta{(g)} + e^{g}, then obtain (- \\delta{(g)} + \\operatorname{g_{\\varepsilon}}{(g)} - e^{g})^{g} = 0^{g}", "derivation": "\\delta{(g)} = e^{g} and 2 \\delta{(g)} = \\delta{(g)} + e^{g} and \\operatorname{g_{\\varepsilon}}{(g)} = \\delta{(g)} + e^{g} and - \\delta{(g)} + \\operatorname{g_{\\varepsilon}}{(g)} - e^{g} = 0 and - 2 \\delta{(g)} + \\operatorname{g_{\\varepsilon}}{(g)} = 0 and (- 2 \\delta{(g)} + \\operatorname{g_{\\varepsilon}}{(g)})^{g} = 0^{g} and (- \\delta{(g)} + \\operatorname{g_{\\varepsilon}}{(g)} - e^{g})^{g} = 0^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["add", 1, "Function('\\\\delta')(Symbol('g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('g', commutative=True))), Add(Function('\\\\delta')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True)), Add(Function('\\\\delta')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))))"], [["minus", 3, "Add(Function('\\\\delta')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('g', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True)), Mul(Integer(-1), exp(Symbol('g', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('g', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True))), Integer(0))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Function('\\\\delta')(Symbol('g', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Integer(0), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('g', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('g', commutative=True)), Mul(Integer(-1), exp(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Integer(0), Symbol('g', commutative=True)))"]]}, {"prompt": "Given W{(E,\\delta)} = E + \\delta, then obtain - E + W{(E,\\delta)} + \\int W{(E,\\delta)} dE = \\delta + \\int W{(E,\\delta)} dE", "derivation": "W{(E,\\delta)} = E + \\delta and \\int W{(E,\\delta)} dE = \\int (E + \\delta) dE and - E + W{(E,\\delta)} + \\int (E + \\delta) dE = \\delta + \\int (E + \\delta) dE and - E + W{(E,\\delta)} + \\int W{(E,\\delta)} dE = \\delta + \\int W{(E,\\delta)} dE", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Symbol('\\\\delta', commutative=True), Integral(Add(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Integral(Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Symbol('\\\\delta', commutative=True), Integral(Function('W')(Symbol('E', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given E{(s,A_{1})} = \\frac{s}{A_{1}} and \\bar{\\h}{(s,A_{1})} = \\frac{s}{A_{1}}, then obtain \\frac{E{(s,A_{1})} + \\frac{1}{A_{1}}}{s} = \\frac{\\bar{\\h}{(s,A_{1})} + \\frac{1}{A_{1}}}{s}", "derivation": "E{(s,A_{1})} = \\frac{s}{A_{1}} and E{(s,A_{1})} + \\frac{1}{A_{1}} = \\frac{s}{A_{1}} + \\frac{1}{A_{1}} and \\bar{\\h}{(s,A_{1})} = \\frac{s}{A_{1}} and E{(s,A_{1})} + \\frac{1}{A_{1}} = \\bar{\\h}{(s,A_{1})} + \\frac{1}{A_{1}} and \\frac{E{(s,A_{1})} + \\frac{1}{A_{1}}}{s} = \\frac{\\bar{\\h}{(s,A_{1})} + \\frac{1}{A_{1}}}{s}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["add", 1, "Pow(Symbol('A_1', commutative=True), Integer(-1))"], "Equality(Add(Function('E')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('s', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('E')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1))), Add(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1))))"], [["divide", 4, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Function('E')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given z{(\\psi)} = \\sin{(\\psi)}, then obtain \\frac{z{(\\psi)}}{\\int (\\sin{(\\psi)} + 1) d\\psi} = \\frac{\\sin{(\\psi)}}{\\int (\\sin{(\\psi)} + 1) d\\psi}", "derivation": "z{(\\psi)} = \\sin{(\\psi)} and z{(\\psi)} + 1 = \\sin{(\\psi)} + 1 and \\int (z{(\\psi)} + 1) d\\psi = \\int (\\sin{(\\psi)} + 1) d\\psi and \\frac{z{(\\psi)}}{\\int (z{(\\psi)} + 1) d\\psi} = \\frac{\\sin{(\\psi)}}{\\int (z{(\\psi)} + 1) d\\psi} and \\frac{z{(\\psi)}}{\\int (\\sin{(\\psi)} + 1) d\\psi} = \\frac{\\sin{(\\psi)}}{\\int (\\sin{(\\psi)} + 1) d\\psi}", "srepr_derivation": [["get_premise", "Equality(Function('z')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(1)), Add(sin(Symbol('\\\\psi', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(sin(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 1, "Integral(Add(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Function('z')(Symbol('\\\\psi', commutative=True)), Pow(Integral(Add(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))), Mul(sin(Symbol('\\\\psi', commutative=True)), Pow(Integral(Add(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('z')(Symbol('\\\\psi', commutative=True)), Pow(Integral(Add(sin(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))), Mul(sin(Symbol('\\\\psi', commutative=True)), Pow(Integral(Add(sin(Symbol('\\\\psi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(A_{2},c)} = A_{2} + c, then obtain (A_{2} + c)^{c} \\phi_{1}^{2 c}{(A_{2},c)} = (A_{2} + c)^{3 c}", "derivation": "\\phi_{1}{(A_{2},c)} = A_{2} + c and \\phi_{1}^{c}{(A_{2},c)} = (A_{2} + c)^{c} and (A_{2} + c)^{c} \\phi_{1}^{c}{(A_{2},c)} = (A_{2} + c)^{2 c} and (A_{2} + c)^{2 c} \\phi_{1}^{c}{(A_{2},c)} = (A_{2} + c)^{3 c} and (A_{2} + c)^{c} \\phi_{1}^{2 c}{(A_{2},c)} = (A_{2} + c)^{3 c}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["times", 2, "Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))))"], [["times", 3, "Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))), Pow(Function('\\\\phi_1')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(3), Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True)))), Pow(Add(Symbol('A_2', commutative=True), Symbol('c', commutative=True)), Mul(Integer(3), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(f_{\\mathbf{p}})} = e^{\\cos{(f_{\\mathbf{p}})}}, then obtain (\\int \\theta_{2}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}})^{f_{\\mathbf{p}}} = (\\int (e^{\\cos{(f_{\\mathbf{p}})}})^{f_{\\mathbf{p}}} df_{\\mathbf{p}})^{f_{\\mathbf{p}}}", "derivation": "\\theta_{2}{(f_{\\mathbf{p}})} = e^{\\cos{(f_{\\mathbf{p}})}} and \\theta_{2}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} = (e^{\\cos{(f_{\\mathbf{p}})}})^{f_{\\mathbf{p}}} and \\int \\theta_{2}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int (e^{\\cos{(f_{\\mathbf{p}})}})^{f_{\\mathbf{p}}} df_{\\mathbf{p}} and (\\int \\theta_{2}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}})^{f_{\\mathbf{p}}} = (\\int (e^{\\cos{(f_{\\mathbf{p}})}})^{f_{\\mathbf{p}}} df_{\\mathbf{p}})^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(exp(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(exp(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\theta_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Integral(Pow(exp(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\pi{(\\phi_1,J_{\\varepsilon})} = \\frac{J_{\\varepsilon}}{\\phi_1}, then obtain \\int\\limits^{\\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})}} \\pi{(\\phi_1,J_{\\varepsilon})} dJ_{\\varepsilon} - \\frac{1}{\\phi_1} = \\int\\limits^{\\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})}} \\frac{J_{\\varepsilon}}{\\phi_1} dJ_{\\varepsilon} - \\frac{1}{\\phi_1}", "derivation": "\\pi{(\\phi_1,J_{\\varepsilon})} = \\frac{J_{\\varepsilon}}{\\phi_1} and \\int \\pi{(\\phi_1,J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\frac{J_{\\varepsilon}}{\\phi_1} dJ_{\\varepsilon} and \\frac{\\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})}}{J_{\\varepsilon}} = 1 and \\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})} = J_{\\varepsilon} and \\int \\pi{(\\phi_1,J_{\\varepsilon})} dJ_{\\varepsilon} - \\frac{1}{\\phi_1} = \\int \\frac{J_{\\varepsilon}}{\\phi_1} dJ_{\\varepsilon} - \\frac{1}{\\phi_1} and \\int\\limits^{\\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})}} \\pi{(\\phi_1,J_{\\varepsilon})} dJ_{\\varepsilon} - \\frac{1}{\\phi_1} = \\int\\limits^{\\phi_1 \\pi{(\\phi_1,J_{\\varepsilon})}} \\frac{J_{\\varepsilon}}{\\phi_1} dJ_{\\varepsilon} - \\frac{1}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["divide", 1, "Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(1))"], [["divide", 3, "Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["minus", 2, "Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Integral(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\phi,p)} = - \\phi + p, then obtain \\phi \\frac{\\partial}{\\partial p} \\operatorname{F_{H}}{(\\phi,p)} = \\phi", "derivation": "\\operatorname{F_{H}}{(\\phi,p)} = - \\phi + p and \\phi \\operatorname{F_{H}}{(\\phi,p)} = \\phi (- \\phi + p) and \\frac{\\partial}{\\partial p} \\phi \\operatorname{F_{H}}{(\\phi,p)} = \\frac{\\partial}{\\partial p} \\phi (- \\phi + p) and \\phi \\frac{\\partial}{\\partial p} \\operatorname{F_{H}}{(\\phi,p)} = \\phi", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('F_H')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Function('F_H')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\phi', commutative=True), Derivative(Function('F_H')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Symbol('\\\\phi', commutative=True))"]]}, {"prompt": "Given g{(\\theta_2)} = \\cos{(\\cos{(\\theta_2)})} and u{(\\theta_2)} = g{(\\theta_2)} - \\cos{(\\cos{(\\theta_2)})} - 1, then obtain \\int (g{(\\theta_2)} + u{(\\theta_2)} \\cos{(\\cos{(\\theta_2)})} + u{(\\theta_2)}) d\\theta_2 = \\int u{(\\theta_2)} d\\theta_2", "derivation": "g{(\\theta_2)} = \\cos{(\\cos{(\\theta_2)})} and g{(\\theta_2)} - \\cos{(\\cos{(\\theta_2)})} = 0 and g{(\\theta_2)} - \\cos{(\\cos{(\\theta_2)})} - 1 = -1 and \\int (g{(\\theta_2)} - \\cos{(\\cos{(\\theta_2)})} - 1) d\\theta_2 = \\int (-1) d\\theta_2 and u{(\\theta_2)} = g{(\\theta_2)} - \\cos{(\\cos{(\\theta_2)})} - 1 and u{(\\theta_2)} = -1 and \\int (g{(\\theta_2)} + u{(\\theta_2)} \\cos{(\\cos{(\\theta_2)})} + u{(\\theta_2)}) d\\theta_2 = \\int u{(\\theta_2)} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\theta_2', commutative=True)), cos(cos(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "cos(cos(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Function('g')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\theta_2', commutative=True))))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('g')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\theta_2', commutative=True)))), Integer(-1)), Integer(-1))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Function('g')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\theta_2', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\theta_2', commutative=True)), Add(Function('g')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\theta_2', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('u')(Symbol('\\\\theta_2', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integral(Add(Function('g')(Symbol('\\\\theta_2', commutative=True)), Mul(Function('u')(Symbol('\\\\theta_2', commutative=True)), cos(cos(Symbol('\\\\theta_2', commutative=True)))), Function('u')(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Function('u')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\operatorname{t_{2}}{(a)} = e^{a}, then obtain \\operatorname{t_{2}}{(a)} + \\cos^{\\mathbf{S}}{(\\mathbf{S})} = e^{a} + \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "derivation": "\\hat{x}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\hat{x}^{\\mathbf{S}}{(\\mathbf{S})} = \\cos^{\\mathbf{S}}{(\\mathbf{S})} and \\operatorname{t_{2}}{(a)} = e^{a} and \\hat{x}^{\\mathbf{S}}{(\\mathbf{S})} + \\operatorname{t_{2}}{(a)} = \\hat{x}^{\\mathbf{S}}{(\\mathbf{S})} + e^{a} and \\operatorname{t_{2}}{(a)} + \\cos^{\\mathbf{S}}{(\\mathbf{S})} = e^{a} + \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], ["get_premise", "Equality(Function('t_2')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["add", 3, "Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Function('t_2')(Symbol('a', commutative=True))), Add(Pow(Function('\\\\hat{x}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('t_2')(Symbol('a', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Add(exp(Symbol('a', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(i)} = e^{i}, then obtain i (\\varphi^{*}{(i)} e^{- i} - e^{- i}) = i (1 - e^{- i})", "derivation": "\\varphi^{*}{(i)} = e^{i} and \\varphi^{*}{(i)} e^{- i} = 1 and \\varphi^{*}{(i)} e^{- i} - e^{- i} = 1 - e^{- i} and i (\\varphi^{*}{(i)} e^{- i} - e^{- i}) = i (1 - e^{- i})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["divide", 1, "exp(Symbol('i', commutative=True))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('i', commutative=True)), exp(Mul(Integer(-1), Symbol('i', commutative=True)))), Integer(1))"], [["minus", 2, "exp(Mul(Integer(-1), Symbol('i', commutative=True)))"], "Equality(Add(Mul(Function('\\\\varphi^*')(Symbol('i', commutative=True)), exp(Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('i', commutative=True))))), Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('i', commutative=True))))))"], [["times", 3, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Add(Mul(Function('\\\\varphi^*')(Symbol('i', commutative=True)), exp(Mul(Integer(-1), Symbol('i', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('i', commutative=True)))))), Mul(Symbol('i', commutative=True), Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('i', commutative=True)))))))"]]}, {"prompt": "Given \\dot{z}{(A_{y},\\Psi^{\\dagger})} = A_{y} - \\Psi^{\\dagger}, then derive \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} = 1, then derive \\int \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\mathbf{P}, then obtain \\cos{(\\int \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} d\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger} + \\mathbf{P})}", "derivation": "\\dot{z}{(A_{y},\\Psi^{\\dagger})} = A_{y} - \\Psi^{\\dagger} and \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} - \\Psi^{\\dagger}) and \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} = 1 and \\int \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int 1 d\\Psi^{\\dagger} and \\int \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\mathbf{P} and \\cos{(\\int \\frac{\\partial}{\\partial A_{y}} \\dot{z}{(A_{y},\\Psi^{\\dagger})} d\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger} + \\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["cos", 5], "Equality(cos(Integral(Derivative(Function('\\\\dot{z}')(Symbol('A_y', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given x{(\\theta)} = \\log{(\\log{(\\theta)})}, then obtain x^{\\theta}{(\\theta)} \\int 2 x{(\\theta)} d\\theta = \\log{(\\log{(\\theta)})}^{\\theta} \\int 2 x{(\\theta)} d\\theta", "derivation": "x{(\\theta)} = \\log{(\\log{(\\theta)})} and 2 x{(\\theta)} = x{(\\theta)} + \\log{(\\log{(\\theta)})} and x^{\\theta}{(\\theta)} = \\log{(\\log{(\\theta)})}^{\\theta} and \\int 2 x{(\\theta)} d\\theta = \\int (x{(\\theta)} + \\log{(\\log{(\\theta)})}) d\\theta and x^{\\theta}{(\\theta)} \\int (x{(\\theta)} + \\log{(\\log{(\\theta)})}) d\\theta = \\log{(\\log{(\\theta)})}^{\\theta} \\int (x{(\\theta)} + \\log{(\\log{(\\theta)})}) d\\theta and x^{\\theta}{(\\theta)} \\int 2 x{(\\theta)} d\\theta = \\log{(\\log{(\\theta)})}^{\\theta} \\int 2 x{(\\theta)} d\\theta", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Function('x')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True))), Add(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True)))))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(log(log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 3, "Integral(Add(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Function('x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integral(Add(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(log(log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Integral(Add(Function('x')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integral(Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(log(log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Integral(Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(C_{2},f^{\\prime})} = C_{2} + f^{\\prime}, then obtain 2 f^{\\prime} + (C_{2} + f^{\\prime} + 2 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}} = 2 f^{\\prime} + (3 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}}", "derivation": "\\phi_{2}{(C_{2},f^{\\prime})} = C_{2} + f^{\\prime} and C_{2} + f^{\\prime} + 2 \\phi_{2}{(C_{2},f^{\\prime})} = 2 C_{2} + 2 f^{\\prime} + \\phi_{2}{(C_{2},f^{\\prime})} and 3 \\phi_{2}{(C_{2},f^{\\prime})} = 2 C_{2} + 2 f^{\\prime} + \\phi_{2}{(C_{2},f^{\\prime})} and C_{2} + f^{\\prime} + 2 \\phi_{2}{(C_{2},f^{\\prime})} = 3 \\phi_{2}{(C_{2},f^{\\prime})} and (C_{2} + f^{\\prime} + 2 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}} = (3 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}} and 2 f^{\\prime} + (C_{2} + f^{\\prime} + 2 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}} = 2 f^{\\prime} + (3 \\phi_{2}{(C_{2},f^{\\prime})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(3), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(3), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Mul(Integer(3), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 5, "Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True)), Pow(Mul(Integer(3), Function('\\\\phi_2')(Symbol('C_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and W{(\\omega,i)} = e^{- \\omega + i}, then obtain \\frac{\\frac{\\partial}{\\partial \\omega} W{(\\omega,i)}}{\\cos{(\\hat{\\mathbf{r}})}} = \\frac{\\frac{\\partial}{\\partial \\omega} e^{- \\omega + i}}{\\cos{(\\hat{\\mathbf{r}})}}", "derivation": "\\nabla{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and W{(\\omega,i)} = e^{- \\omega + i} and \\frac{\\partial}{\\partial \\omega} W{(\\omega,i)} = \\frac{\\partial}{\\partial \\omega} e^{- \\omega + i} and \\frac{\\frac{\\partial}{\\partial \\omega} W{(\\omega,i)}}{\\nabla{(\\hat{\\mathbf{r}})}} = \\frac{\\frac{\\partial}{\\partial \\omega} e^{- \\omega + i}}{\\nabla{(\\hat{\\mathbf{r}})}} and \\frac{\\frac{\\partial}{\\partial \\omega} W{(\\omega,i)}}{\\cos{(\\hat{\\mathbf{r}})}} = \\frac{\\frac{\\partial}{\\partial \\omega} e^{- \\omega + i}}{\\cos{(\\hat{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], ["get_premise", "Equality(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('i', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["divide", 3, "Function('\\\\nabla')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\nabla')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})} = \\frac{\\hat{H}}{\\mathbf{J}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})} = - \\frac{\\hat{H}}{\\mathbf{J}^{2}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})} = \\frac{\\hat{H}}{\\mathbf{J}} and - \\hat{H} + \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})} = - \\hat{H} + \\frac{\\hat{H}}{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mathbf{J}} (- \\hat{H} + \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})}) = \\frac{\\partial}{\\partial \\mathbf{J}} (- \\hat{H} + \\frac{\\hat{H}}{\\mathbf{J}}) and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},\\mathbf{J})} = - \\frac{\\hat{H}}{\\mathbf{J}^{2}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given i{(g_{\\varepsilon},\\hat{H}_l)} = \\hat{H}_l - g_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial g_{\\varepsilon}} (2 g_{\\varepsilon} + i{(g_{\\varepsilon},\\hat{H}_l)}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\hat{H}_l + g_{\\varepsilon})", "derivation": "i{(g_{\\varepsilon},\\hat{H}_l)} = \\hat{H}_l - g_{\\varepsilon} and g_{\\varepsilon} + i{(g_{\\varepsilon},\\hat{H}_l)} = \\hat{H}_l and 2 g_{\\varepsilon} + i{(g_{\\varepsilon},\\hat{H}_l)} = \\hat{H}_l + g_{\\varepsilon} and \\frac{\\partial}{\\partial g_{\\varepsilon}} (2 g_{\\varepsilon} + i{(g_{\\varepsilon},\\hat{H}_l)}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} (\\hat{H}_l + g_{\\varepsilon})", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('i')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True))"], [["minus", 2, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('i')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('i')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{p})} = \\sin{(\\hat{p})}, then derive \\int \\operatorname{a^{\\dagger}}{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} - \\cos{(\\hat{p})}, then derive \\frac{d}{d \\hat{p}} \\operatorname{a^{\\dagger}}{(\\hat{p})} = \\cos{(\\hat{p})}, then obtain \\int \\operatorname{a^{\\dagger}}{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} - \\frac{d}{d \\hat{p}} \\operatorname{a^{\\dagger}}{(\\hat{p})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{p})} = \\sin{(\\hat{p})} and \\int \\operatorname{a^{\\dagger}}{(\\hat{p})} d\\hat{p} = \\int \\sin{(\\hat{p})} d\\hat{p} and \\int \\operatorname{a^{\\dagger}}{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} - \\cos{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\operatorname{a^{\\dagger}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} and \\frac{d}{d \\hat{p}} \\operatorname{a^{\\dagger}}{(\\hat{p})} = \\cos{(\\hat{p})} and \\int \\operatorname{a^{\\dagger}}{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} - \\frac{d}{d \\hat{p}} \\operatorname{a^{\\dagger}}{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{p}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{X}{(\\hat{X},Q)} = \\cos^{\\hat{X}}{(Q)}, then obtain \\int 2 \\hat{X}{(\\hat{X},Q)} dQ - \\frac{1}{\\int 2 \\hat{X}{(\\hat{X},Q)} dQ} = \\int (\\hat{X}{(\\hat{X},Q)} + \\cos^{\\hat{X}}{(Q)}) dQ - \\frac{1}{\\int 2 \\hat{X}{(\\hat{X},Q)} dQ}", "derivation": "\\hat{X}{(\\hat{X},Q)} = \\cos^{\\hat{X}}{(Q)} and 2 \\hat{X}{(\\hat{X},Q)} = \\hat{X}{(\\hat{X},Q)} + \\cos^{\\hat{X}}{(Q)} and \\int 2 \\hat{X}{(\\hat{X},Q)} dQ = \\int (\\hat{X}{(\\hat{X},Q)} + \\cos^{\\hat{X}}{(Q)}) dQ and \\int 2 \\hat{X}{(\\hat{X},Q)} dQ - \\frac{1}{\\int 2 \\hat{X}{(\\hat{X},Q)} dQ} = \\int (\\hat{X}{(\\hat{X},Q)} + \\cos^{\\hat{X}}{(Q)}) dQ - \\frac{1}{\\int 2 \\hat{X}{(\\hat{X},Q)} dQ}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 1, "Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Add(Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["minus", 3, "Pow(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integer(-1)))), Add(Integral(Add(Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\hat{X}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{x}_0,M_{E})} = \\frac{\\partial}{\\partial \\hat{x}_0} M_{E} \\hat{x}_0, then derive \\operatorname{z^{*}}{(\\hat{x}_0,M_{E})} = M_{E}, then obtain - M_{E} + \\frac{\\partial}{\\partial \\hat{x}_0} M_{E} \\hat{x}_0 = 0", "derivation": "\\operatorname{z^{*}}{(\\hat{x}_0,M_{E})} = \\frac{\\partial}{\\partial \\hat{x}_0} M_{E} \\hat{x}_0 and \\operatorname{z^{*}}{(\\hat{x}_0,M_{E})} = M_{E} and \\frac{\\partial}{\\partial \\hat{x}_0} M_{E} \\hat{x}_0 = M_{E} and - M_{E} + \\frac{\\partial}{\\partial \\hat{x}_0} M_{E} \\hat{x}_0 = 0", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True)), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Symbol('M_E', commutative=True))"], [["minus", 3, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\phi{(A_{2},C)} = \\int (- A_{2} + C) dA_{2}, then derive \\frac{\\phi{(A_{2},C)}}{- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda}} = 1, then obtain \\frac{(- \\frac{A_{2}^{2}}{2} + A_{2} C + r)^{2}}{(- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda})^{2}} = 1", "derivation": "\\phi{(A_{2},C)} = \\int (- A_{2} + C) dA_{2} and \\frac{\\phi{(A_{2},C)}}{\\int (- A_{2} + C) dA_{2}} = 1 and \\frac{\\phi{(A_{2},C)}}{- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda}} = 1 and \\frac{\\int (- A_{2} + C) dA_{2}}{- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda}} = 1 and \\frac{(\\int (- A_{2} + C) dA_{2})^{2}}{(- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda})^{2}} = 1 and \\frac{(- \\frac{A_{2}^{2}}{2} + A_{2} C + r)^{2}}{(- \\frac{A_{2}^{2}}{2} + A_{2} C + \\hat{H}_{\\lambda})^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["divide", 1, "Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('A_2', commutative=True)))"], "Equality(Mul(Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Integer(1))"], [["power", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-2)), Pow(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integer(2))), Integer(1))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-2)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Symbol('r', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\mathbf{A}{(\\varphi)} = \\sin{(\\varphi)}, then derive \\frac{d}{d \\varphi} \\mathbf{A}{(\\varphi)} = \\cos{(\\varphi)}, then obtain \\frac{d}{d \\varphi} \\sin{(\\varphi)} = \\cos{(\\varphi)}", "derivation": "\\mathbf{A}{(\\varphi)} = \\sin{(\\varphi)} and \\frac{d}{d \\varphi} \\mathbf{A}{(\\varphi)} = \\frac{d}{d \\varphi} \\sin{(\\varphi)} and \\frac{d}{d \\varphi} \\mathbf{A}{(\\varphi)} = \\cos{(\\varphi)} and \\frac{d}{d \\varphi} \\sin{(\\varphi)} = \\cos{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), cos(Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), cos(Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(F_{N},m_{s})} = F_{N} m_{s} and \\operatorname{E_{\\lambda}}{(F_{N},m_{s})} = F_{N} m_{s}, then derive \\frac{\\partial}{\\partial F_{N}} \\operatorname{t_{1}}{(F_{N},m_{s})} + 1 = m_{s} + 1, then obtain \\frac{\\partial}{\\partial F_{N}} \\operatorname{E_{\\lambda}}{(F_{N},m_{s})} + 1 = m_{s} + 1", "derivation": "\\operatorname{t_{1}}{(F_{N},m_{s})} = F_{N} m_{s} and F_{N} + \\operatorname{t_{1}}{(F_{N},m_{s})} = F_{N} m_{s} + F_{N} and \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\operatorname{t_{1}}{(F_{N},m_{s})}) = \\frac{\\partial}{\\partial F_{N}} (F_{N} m_{s} + F_{N}) and \\frac{\\partial}{\\partial F_{N}} \\operatorname{t_{1}}{(F_{N},m_{s})} + 1 = m_{s} + 1 and \\operatorname{E_{\\lambda}}{(F_{N},m_{s})} = F_{N} m_{s} and \\frac{\\partial}{\\partial F_{N}} F_{N} m_{s} + 1 = m_{s} + 1 and \\frac{\\partial}{\\partial F_{N}} \\operatorname{E_{\\lambda}}{(F_{N},m_{s})} = \\frac{\\partial}{\\partial F_{N}} F_{N} m_{s} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{E_{\\lambda}}{(F_{N},m_{s})} + 1 = m_{s} + 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('t_1')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('F_N', commutative=True)))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Function('t_1')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t_1')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)), Add(Symbol('m_s', commutative=True), Integer(1)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)), Add(Symbol('m_s', commutative=True), Integer(1)))"], [["differentiate", 5, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Derivative(Function('E_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)), Add(Symbol('m_s', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\hat{x}_0{(L,A_{1})} = A_{1} + \\sin{(L)}, then obtain \\frac{\\partial}{\\partial A_{1}} 2 \\hat{x}_0^{L}{(L,A_{1})} = \\frac{\\partial}{\\partial A_{1}} ((A_{1} + \\sin{(L)})^{L} + \\hat{x}_0^{L}{(L,A_{1})})", "derivation": "\\hat{x}_0{(L,A_{1})} = A_{1} + \\sin{(L)} and \\hat{x}_0^{L}{(L,A_{1})} = (A_{1} + \\sin{(L)})^{L} and 2 \\hat{x}_0^{L}{(L,A_{1})} = (A_{1} + \\sin{(L)})^{L} + \\hat{x}_0^{L}{(L,A_{1})} and \\frac{\\partial}{\\partial A_{1}} 2 \\hat{x}_0^{L}{(L,A_{1})} = \\frac{\\partial}{\\partial A_{1}} ((A_{1} + \\sin{(L)})^{L} + \\hat{x}_0^{L}{(L,A_{1})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), sin(Symbol('L', commutative=True))))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), sin(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["add", 2, "Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True))), Add(Pow(Add(Symbol('A_1', commutative=True), sin(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True))))"], [["differentiate", 3, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Symbol('A_1', commutative=True), sin(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('L', commutative=True), Symbol('A_1', commutative=True)), Symbol('L', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(A)} = \\int \\log{(A)} dA, then obtain \\int \\cos{(\\int \\frac{\\int \\operatorname{C_{d}}{(A)} dA}{A \\operatorname{C_{d}}{(A)}} dA)} dA = \\int \\cos{(\\int \\frac{\\iint \\log{(A)} dA dA}{A \\operatorname{C_{d}}{(A)}} dA)} dA", "derivation": "\\operatorname{C_{d}}{(A)} = \\int \\log{(A)} dA and \\int \\operatorname{C_{d}}{(A)} dA = \\iint \\log{(A)} dA dA and \\frac{\\int \\operatorname{C_{d}}{(A)} dA}{A \\operatorname{C_{d}}{(A)}} = \\frac{\\iint \\log{(A)} dA dA}{A \\operatorname{C_{d}}{(A)}} and \\int \\frac{\\int \\operatorname{C_{d}}{(A)} dA}{A \\operatorname{C_{d}}{(A)}} dA = \\int \\frac{\\iint \\log{(A)} dA dA}{A \\operatorname{C_{d}}{(A)}} dA and \\cos{(\\int \\frac{\\int \\operatorname{C_{d}}{(A)} dA}{A \\operatorname{C_{d}}{(A)}} dA)} = \\cos{(\\int \\frac{\\iint \\log{(A)} dA dA}{A \\operatorname{C_{d}}{(A)}} dA)} and \\int \\cos{(\\int \\frac{\\int \\operatorname{C_{d}}{(A)} dA}{A \\operatorname{C_{d}}{(A)}} dA)} dA = \\int \\cos{(\\int \\frac{\\iint \\log{(A)} dA dA}{A \\operatorname{C_{d}}{(A)}} dA)} dA", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('A', commutative=True)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["divide", 2, "Mul(Symbol('A', commutative=True), Function('C_d')(Symbol('A', commutative=True)))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(Function('C_d')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["integrate", 3, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(Function('C_d')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True))), Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True))))"], [["cos", 4], "Equality(cos(Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(Function('C_d')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True)))), cos(Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True)))))"], [["integrate", 5, "Symbol('A', commutative=True)"], "Equality(Integral(cos(Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(Function('C_d')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True))), Integral(cos(Integral(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('C_d')(Symbol('A', commutative=True)), Integer(-1)), Integral(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\varepsilon_0)} = e^{\\sin{(\\varepsilon_0)}} and \\operatorname{E_{x}}{(\\varepsilon_0)} = 2 e^{\\sin{(\\varepsilon_0)}}, then obtain - \\varepsilon_0 + \\operatorname{E_{x}}{(\\varepsilon_0)} - \\frac{d}{d \\varepsilon_0} \\omega{(\\varepsilon_0)} = - \\varepsilon_0 + \\omega{(\\varepsilon_0)} + e^{\\sin{(\\varepsilon_0)}} - \\frac{d}{d \\varepsilon_0} \\omega{(\\varepsilon_0)}", "derivation": "\\omega{(\\varepsilon_0)} = e^{\\sin{(\\varepsilon_0)}} and \\omega{(\\varepsilon_0)} + e^{\\sin{(\\varepsilon_0)}} = 2 e^{\\sin{(\\varepsilon_0)}} and \\operatorname{E_{x}}{(\\varepsilon_0)} = 2 e^{\\sin{(\\varepsilon_0)}} and - \\varepsilon_0 + \\operatorname{E_{x}}{(\\varepsilon_0)} = - \\varepsilon_0 + 2 e^{\\sin{(\\varepsilon_0)}} and - \\varepsilon_0 + \\operatorname{E_{x}}{(\\varepsilon_0)} = - \\varepsilon_0 + \\omega{(\\varepsilon_0)} + e^{\\sin{(\\varepsilon_0)}} and - \\varepsilon_0 + \\operatorname{E_{x}}{(\\varepsilon_0)} - \\frac{d}{d \\varepsilon_0} \\omega{(\\varepsilon_0)} = - \\varepsilon_0 + \\omega{(\\varepsilon_0)} + e^{\\sin{(\\varepsilon_0)}} - \\frac{d}{d \\varepsilon_0} \\omega{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), exp(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "exp(sin(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), exp(sin(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Integer(2), exp(sin(Symbol('\\\\varepsilon_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), exp(sin(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), exp(sin(Symbol('\\\\varepsilon_0', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), exp(sin(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 5, "Derivative(Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), exp(sin(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))))"]]}, {"prompt": "Given g{(\\nabla,\\tilde{g})} = \\tilde{g}^{\\nabla}, then obtain \\int \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + g{(\\nabla,\\tilde{g})}))} d\\tilde{g} = \\int \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + \\tilde{g}^{\\nabla}))} d\\tilde{g}", "derivation": "g{(\\nabla,\\tilde{g})} = \\tilde{g}^{\\nabla} and - \\tilde{g} + g{(\\nabla,\\tilde{g})} = - \\tilde{g} + \\tilde{g}^{\\nabla} and \\tilde{g}^{- \\nabla} (- \\tilde{g} + g{(\\nabla,\\tilde{g})}) = \\tilde{g}^{- \\nabla} (- \\tilde{g} + \\tilde{g}^{\\nabla}) and \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + g{(\\nabla,\\tilde{g})}))} = \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + \\tilde{g}^{\\nabla}))} and \\int \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + g{(\\nabla,\\tilde{g})}))} d\\tilde{g} = \\int \\log{(\\tilde{g}^{- \\nabla} (- \\tilde{g} + \\tilde{g}^{\\nabla}))} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('g')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('g')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["log", 3], "Equality(log(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('g')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))), log(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(log(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('g')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\nabla', commutative=True))))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then derive \\int \\operatorname{C_{2}}{(\\mathbf{D})} d\\mathbf{D} = E_{n} - \\cos{(\\mathbf{D})}, then obtain E_{n} + \\operatorname{C_{2}}{(\\mathbf{D})} - \\cos{(\\mathbf{D})} = E_{n} + \\sin{(\\mathbf{D})} - \\cos{(\\mathbf{D})}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\int \\operatorname{C_{2}}{(\\mathbf{D})} d\\mathbf{D} = \\int \\sin{(\\mathbf{D})} d\\mathbf{D} and \\operatorname{C_{2}}{(\\mathbf{D})} + \\int \\sin{(\\mathbf{D})} d\\mathbf{D} = \\sin{(\\mathbf{D})} + \\int \\sin{(\\mathbf{D})} d\\mathbf{D} and \\int \\operatorname{C_{2}}{(\\mathbf{D})} d\\mathbf{D} = E_{n} - \\cos{(\\mathbf{D})} and \\int \\sin{(\\mathbf{D})} d\\mathbf{D} = E_{n} - \\cos{(\\mathbf{D})} and E_{n} + \\operatorname{C_{2}}{(\\mathbf{D})} - \\cos{(\\mathbf{D})} = E_{n} + \\sin{(\\mathbf{D})} - \\cos{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 1, "Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Function('C_2')(Symbol('\\\\mathbf{D}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Add(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('E_n', commutative=True), Function('C_2')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Symbol('E_n', commutative=True), sin(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\varphi^*,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\frac{\\varphi^*}{A_{1}}, then derive \\mathbf{p}{(\\varphi^*,A_{1})} = - \\frac{\\varphi^*}{A_{1}^{2}}, then obtain - \\frac{\\varphi^* (\\frac{\\mathbf{p}{(\\varphi^*,A_{1})}}{\\varphi^*} - \\frac{\\varphi^*}{A_{1}^{2}})}{A_{1}^{2}} = - \\frac{\\varphi^* (- \\frac{\\varphi^*}{A_{1}^{2}} - \\frac{1}{A_{1}^{2}})}{A_{1}^{2}}", "derivation": "\\mathbf{p}{(\\varphi^*,A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\frac{\\varphi^*}{A_{1}} and \\mathbf{p}{(\\varphi^*,A_{1})} = - \\frac{\\varphi^*}{A_{1}^{2}} and \\frac{\\mathbf{p}{(\\varphi^*,A_{1})}}{\\varphi^*} = - \\frac{1}{A_{1}^{2}} and \\frac{\\mathbf{p}{(\\varphi^*,A_{1})}}{\\varphi^*} - \\frac{\\varphi^*}{A_{1}^{2}} = - \\frac{\\varphi^*}{A_{1}^{2}} - \\frac{1}{A_{1}^{2}} and - \\frac{\\varphi^* (\\frac{\\mathbf{p}{(\\varphi^*,A_{1})}}{\\varphi^*} - \\frac{\\varphi^*}{A_{1}^{2}})}{A_{1}^{2}} = - \\frac{\\varphi^* (- \\frac{\\varphi^*}{A_{1}^{2}} - \\frac{1}{A_{1}^{2}})}{A_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Derivative(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)))))"], [["times", 4, "Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True), Add(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2)), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-2))))))"]]}, {"prompt": "Given \\mu{(\\mu_0,G)} = \\frac{\\mu_0}{G}, then obtain \\frac{\\partial^{2}}{\\partial G\\partial \\mu_0} (- \\mu_0 + \\mu{(\\mu_0,G)}) = \\frac{\\partial^{2}}{\\partial G\\partial \\mu_0} (- \\mu_0 + \\frac{\\mu_0}{G})", "derivation": "\\mu{(\\mu_0,G)} = \\frac{\\mu_0}{G} and - \\mu_0 + \\mu{(\\mu_0,G)} = - \\mu_0 + \\frac{\\mu_0}{G} and \\frac{\\partial}{\\partial \\mu_0} (- \\mu_0 + \\mu{(\\mu_0,G)}) = \\frac{\\partial}{\\partial \\mu_0} (- \\mu_0 + \\frac{\\mu_0}{G}) and \\frac{\\partial^{2}}{\\partial G\\partial \\mu_0} (- \\mu_0 + \\mu{(\\mu_0,G)}) = \\frac{\\partial^{2}}{\\partial G\\partial \\mu_0} (- \\mu_0 + \\frac{\\mu_0}{G})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\mu')(Symbol('\\\\mu_0', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)} = - \\hat{\\mathbf{r}} + \\varepsilon + r_{0}, then obtain \\frac{\\partial}{\\partial \\varepsilon} (\\hat{p}_0^{\\hat{\\mathbf{r}}}{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)})^{r_{0}} = \\frac{\\partial}{\\partial \\varepsilon} ((- \\hat{\\mathbf{r}} + \\varepsilon + r_{0})^{\\hat{\\mathbf{r}}})^{r_{0}}", "derivation": "\\hat{p}_0{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)} = - \\hat{\\mathbf{r}} + \\varepsilon + r_{0} and \\hat{p}_0^{\\hat{\\mathbf{r}}}{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)} = (- \\hat{\\mathbf{r}} + \\varepsilon + r_{0})^{\\hat{\\mathbf{r}}} and (\\hat{p}_0^{\\hat{\\mathbf{r}}}{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)})^{r_{0}} = ((- \\hat{\\mathbf{r}} + \\varepsilon + r_{0})^{\\hat{\\mathbf{r}}})^{r_{0}} and \\frac{\\partial}{\\partial \\varepsilon} (\\hat{p}_0^{\\hat{\\mathbf{r}}}{(r_{0},\\hat{\\mathbf{r}},\\varepsilon)})^{r_{0}} = \\frac{\\partial}{\\partial \\varepsilon} ((- \\hat{\\mathbf{r}} + \\varepsilon + r_{0})^{\\hat{\\mathbf{r}}})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('r_0', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\hat{p}_0')(Symbol('r_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(t_{1})} = e^{t_{1}}, then derive \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})} = e^{t_{1}}, then obtain \\frac{d^{2}}{d t_{1}^{2}} \\operatorname{F_{c}}{(t_{1})} = \\operatorname{F_{c}}{(t_{1})}", "derivation": "\\operatorname{F_{c}}{(t_{1})} = e^{t_{1}} and \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})} = \\frac{d}{d t_{1}} e^{t_{1}} and \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})} = e^{t_{1}} and \\operatorname{F_{c}}{(t_{1})} = \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})} and \\operatorname{F_{c}}{(t_{1})} = \\frac{d}{d t_{1}} e^{t_{1}} and \\frac{d^{2}}{d t_{1}^{2}} \\operatorname{F_{c}}{(t_{1})} = \\frac{d}{d t_{1}} e^{t_{1}} and \\frac{d^{2}}{d t_{1}^{2}} \\operatorname{F_{c}}{(t_{1})} = \\operatorname{F_{c}}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(exp(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), exp(Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('F_c')(Symbol('t_1', commutative=True)), Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('F_c')(Symbol('t_1', commutative=True)), Derivative(exp(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(2))), Derivative(exp(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(2))), Function('F_c')(Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = \\mathbf{P}^{r} and \\hat{H}_l{(\\mathbf{P},r)} = \\mathbf{P}^{r} + 1, then obtain \\hat{H}_l{(\\mathbf{P},r)} - 1 + \\mathbf{P}^{- r} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = \\hat{H}_l{(\\mathbf{P},r)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = \\mathbf{P}^{r} and \\mathbf{P}^{- r} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = 1 and \\mathbf{P}^{r} + \\mathbf{P}^{- r} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = \\mathbf{P}^{r} + 1 and \\hat{H}_l{(\\mathbf{P},r)} = \\mathbf{P}^{r} + 1 and \\hat{H}_l{(\\mathbf{P},r)} - 1 + \\mathbf{P}^{- r} \\operatorname{J_{\\varepsilon}}{(\\mathbf{P},r)} = \\hat{H}_l{(\\mathbf{P},r)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))), Integer(1))"], [["add", 2, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))), Add(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Add(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(u)} = \\sin{(e^{u})}, then obtain \\frac{d}{d u} (u + \\operatorname{t_{2}}{(u)})^{3} (u + \\sin{(e^{u})}) = \\frac{d}{d u} (u + \\sin{(e^{u})})^{4}", "derivation": "\\operatorname{t_{2}}{(u)} = \\sin{(e^{u})} and u + \\operatorname{t_{2}}{(u)} = u + \\sin{(e^{u})} and (u + \\operatorname{t_{2}}{(u)}) (u + \\sin{(e^{u})}) = (u + \\sin{(e^{u})})^{2} and (u + \\operatorname{t_{2}}{(u)})^{2} (u + \\sin{(e^{u})})^{2} = (u + \\sin{(e^{u})})^{4} and (u + \\operatorname{t_{2}}{(u)})^{3} (u + \\sin{(e^{u})}) = (u + \\operatorname{t_{2}}{(u)})^{2} (u + \\sin{(e^{u})})^{2} and (u + \\operatorname{t_{2}}{(u)})^{3} (u + \\sin{(e^{u})}) = (u + \\sin{(e^{u})})^{4} and \\frac{d}{d u} (u + \\operatorname{t_{2}}{(u)})^{3} (u + \\sin{(e^{u})}) = \\frac{d}{d u} (u + \\sin{(e^{u})})^{4}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('u', commutative=True)), sin(exp(Symbol('u', commutative=True))))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))))"], [["times", 2, "Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True))))"], "Equality(Mul(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True))))), Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(2)))"], [["power", 3, 2], "Equality(Mul(Pow(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Integer(2)), Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(2))), Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(4)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Integer(3)), Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True))))), Mul(Pow(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Integer(2)), Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Integer(3)), Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True))))), Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(4)))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Integer(3)), Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True))))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('u', commutative=True), sin(exp(Symbol('u', commutative=True)))), Integer(4)), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{p},Z)} = e^{\\hat{p}^{Z}}, then obtain - \\hat{p}^{- Z} (- \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} \\operatorname{F_{N}}{(\\hat{p},Z)}) = - \\hat{p}^{- Z} (- \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} e^{\\hat{p}^{Z}})", "derivation": "\\operatorname{F_{N}}{(\\hat{p},Z)} = e^{\\hat{p}^{Z}} and \\frac{\\partial}{\\partial Z} \\operatorname{F_{N}}{(\\hat{p},Z)} = \\frac{\\partial}{\\partial Z} e^{\\hat{p}^{Z}} and - \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} \\operatorname{F_{N}}{(\\hat{p},Z)} = - \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} e^{\\hat{p}^{Z}} and - \\hat{p}^{- Z} (- \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} \\operatorname{F_{N}}{(\\hat{p},Z)}) = - \\hat{p}^{- Z} (- \\operatorname{F_{N}}{(\\hat{p},Z)} + \\frac{\\partial}{\\partial Z} e^{\\hat{p}^{Z}})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["minus", 2, "Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Derivative(exp(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Derivative(exp(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(B)} = \\frac{d}{d B} e^{B} and E{(B)} = \\frac{d}{d B} e^{B}, then derive E{(B)} = e^{B}, then obtain \\frac{\\operatorname{v_{x}}{(B)}}{B} = \\frac{\\frac{d}{d B} E{(B)}}{B}", "derivation": "\\operatorname{v_{x}}{(B)} = \\frac{d}{d B} e^{B} and \\frac{\\operatorname{v_{x}}{(B)}}{B} = \\frac{\\frac{d}{d B} e^{B}}{B} and E{(B)} = \\frac{d}{d B} e^{B} and E{(B)} = e^{B} and \\frac{\\operatorname{v_{x}}{(B)}}{B} = \\frac{\\frac{d}{d B} E{(B)}}{B}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('B', commutative=True)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('v_x')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('E')(Symbol('B', commutative=True)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('E')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('v_x')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(\\lambda)} = \\sin{(\\sin{(\\lambda)})}, then obtain -1 = - \\frac{\\sin^{2}{(\\sin{(\\lambda)})}}{C^{2}{(\\lambda)}}", "derivation": "C{(\\lambda)} = \\sin{(\\sin{(\\lambda)})} and \\frac{C{(\\lambda)}}{\\sin{(\\sin{(\\lambda)})}} = 1 and -1 = - \\frac{\\sin{(\\sin{(\\lambda)})}}{C{(\\lambda)}} and \\frac{\\sin{(\\sin{(\\lambda)})}}{C{(\\lambda)}} = \\frac{\\sin^{2}{(\\sin{(\\lambda)})}}{C^{2}{(\\lambda)}} and -1 = - \\frac{\\sin^{2}{(\\sin{(\\lambda)})}}{C^{2}{(\\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\lambda', commutative=True)), sin(sin(Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('C')(Symbol('\\\\lambda', commutative=True)), Pow(sin(sin(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 2, "Mul(Integer(-1), Function('C')(Symbol('\\\\lambda', commutative=True)), Pow(sin(sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('C')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\lambda', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Function('C')(Symbol('\\\\lambda', commutative=True)), Pow(sin(sin(Symbol('\\\\lambda', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Function('C')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Function('C')(Symbol('\\\\lambda', commutative=True)), Integer(-2)), Pow(sin(sin(Symbol('\\\\lambda', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('C')(Symbol('\\\\lambda', commutative=True)), Integer(-2)), Pow(sin(sin(Symbol('\\\\lambda', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\nabla)} = \\cos{(\\nabla)} and \\hat{H}_l{(\\nabla)} = \\frac{d}{d \\nabla} \\operatorname{M_{E}}{(\\nabla)}, then obtain \\hat{H}_l{(\\nabla)} + \\hat{H}_l^{\\nabla}{(\\nabla)} = \\hat{H}_l{(\\nabla)} + (\\frac{d}{d \\nabla} \\cos{(\\nabla)})^{\\nabla}", "derivation": "\\operatorname{M_{E}}{(\\nabla)} = \\cos{(\\nabla)} and \\frac{d}{d \\nabla} \\operatorname{M_{E}}{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)} and \\hat{H}_l{(\\nabla)} = \\frac{d}{d \\nabla} \\operatorname{M_{E}}{(\\nabla)} and \\hat{H}_l{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)} and \\hat{H}_l^{\\nabla}{(\\nabla)} = (\\frac{d}{d \\nabla} \\cos{(\\nabla)})^{\\nabla} and \\hat{H}_l{(\\nabla)} + \\hat{H}_l^{\\nabla}{(\\nabla)} = \\hat{H}_l{(\\nabla)} + (\\frac{d}{d \\nabla} \\cos{(\\nabla)})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Derivative(Function('M_E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True)))"], [["add", 5, "Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))), Add(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Pow(Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} = t_{1}, then obtain m t_{1}^{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} = m t_{1}^{3}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} = t_{1} and m t_{1} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} = m t_{1}^{2} and m t_{1} \\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(t_{1})} = m t_{1}^{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} and m t_{1}^{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(t_{1})} = m t_{1}^{3}", "srepr_derivation": [["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["times", 1, "Mul(Symbol('m', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Mul(Symbol('m', commutative=True), Symbol('t_1', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(2))))"], [["times", 1, "Mul(Symbol('m', commutative=True), Symbol('t_1', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True)))"], "Equality(Mul(Symbol('m', commutative=True), Symbol('t_1', commutative=True), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True)), Integer(2))), Mul(Symbol('m', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(2)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('m', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(2)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('t_1', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(h)} = \\log{(h)}, then obtain \\frac{\\partial}{\\partial \\hat{p}} (\\eta \\operatorname{E_{\\lambda}}{(h)} - \\frac{1}{\\hat{p}})^{h} = \\frac{\\partial}{\\partial \\hat{p}} (\\eta \\log{(h)} - \\frac{1}{\\hat{p}})^{h}", "derivation": "\\operatorname{E_{\\lambda}}{(h)} = \\log{(h)} and \\eta \\operatorname{E_{\\lambda}}{(h)} = \\eta \\log{(h)} and \\eta \\operatorname{E_{\\lambda}}{(h)} - \\frac{1}{\\hat{p}} = \\eta \\log{(h)} - \\frac{1}{\\hat{p}} and (\\eta \\operatorname{E_{\\lambda}}{(h)} - \\frac{1}{\\hat{p}})^{h} = (\\eta \\log{(h)} - \\frac{1}{\\hat{p}})^{h} and \\frac{\\partial}{\\partial \\hat{p}} (\\eta \\operatorname{E_{\\lambda}}{(h)} - \\frac{1}{\\hat{p}})^{h} = \\frac{\\partial}{\\partial \\hat{p}} (\\eta \\log{(h)} - \\frac{1}{\\hat{p}})^{h}", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["times", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Function('E_{\\\\lambda}')(Symbol('h', commutative=True))), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('h', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\eta', commutative=True), Function('E_{\\\\lambda}')(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\eta', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\eta', commutative=True), Function('E_{\\\\lambda}')(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))), Symbol('h', commutative=True)), Pow(Add(Mul(Symbol('\\\\eta', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))), Symbol('h', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Symbol('\\\\eta', commutative=True), Function('E_{\\\\lambda}')(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('\\\\eta', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\Psi)} = \\log{(\\Psi)}, then obtain \\tilde{\\infty} \\int B{(\\Psi)} d\\Psi = \\tilde{\\infty} \\int \\log{(\\Psi)} d\\Psi", "derivation": "B{(\\Psi)} = \\log{(\\Psi)} and \\int B{(\\Psi)} d\\Psi = \\int \\log{(\\Psi)} d\\Psi and \\frac{\\int B{(\\Psi)} d\\Psi}{B{(\\Psi)} - \\log{(\\Psi)}} = \\frac{\\int \\log{(\\Psi)} d\\Psi}{B{(\\Psi)} - \\log{(\\Psi)}} and \\tilde{\\infty} \\int B{(\\Psi)} d\\Psi = \\tilde{\\infty} \\int \\log{(\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Add(Function('B')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))))"], "Equality(Mul(Pow(Add(Function('B')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), Integer(-1)), Integral(Function('B')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Add(Function('B')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), Integer(-1)), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(zoo, Integral(Function('B')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(zoo, Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{x})} = \\cos{(\\sin{(v_{x})})} and \\mathbf{v}{(v_{x})} = \\cos{(\\sin{(v_{x})})}, then obtain - v_{x} \\mathbf{v}{(v_{x})} + \\mathbf{v}{(v_{x})} = - v_{x} \\mathbf{v}{(v_{x})} + \\operatorname{F_{H}}{(v_{x})}", "derivation": "\\operatorname{F_{H}}{(v_{x})} = \\cos{(\\sin{(v_{x})})} and \\mathbf{v}{(v_{x})} = \\cos{(\\sin{(v_{x})})} and \\mathbf{v}{(v_{x})} = \\operatorname{F_{H}}{(v_{x})} and - v_{x} \\cos{(\\sin{(v_{x})})} + \\mathbf{v}{(v_{x})} = - v_{x} \\cos{(\\sin{(v_{x})})} + \\operatorname{F_{H}}{(v_{x})} and - v_{x} \\mathbf{v}{(v_{x})} + \\mathbf{v}{(v_{x})} = - v_{x} \\mathbf{v}{(v_{x})} + \\operatorname{F_{H}}{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_x', commutative=True)), cos(sin(Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True)), cos(sin(Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True)), Function('F_H')(Symbol('v_x', commutative=True)))"], [["minus", 3, "Mul(Symbol('v_x', commutative=True), cos(sin(Symbol('v_x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), cos(sin(Symbol('v_x', commutative=True)))), Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True), cos(sin(Symbol('v_x', commutative=True)))), Function('F_H')(Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True))), Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\mathbf{v}')(Symbol('v_x', commutative=True))), Function('F_H')(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} = (e^{\\dot{z}})^{T}, then obtain \\int \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT + \\iint (e^{\\dot{z}})^{T} dT d\\dot{z} = \\int (e^{\\dot{z}})^{T} dT + \\iint (e^{\\dot{z}})^{T} dT d\\dot{z}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} = (e^{\\dot{z}})^{T} and \\int \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT = \\int (e^{\\dot{z}})^{T} dT and \\iint \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT d\\dot{z} = \\iint (e^{\\dot{z}})^{T} dT d\\dot{z} and \\int \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT + \\iint \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT d\\dot{z} = \\int (e^{\\dot{z}})^{T} dT + \\iint \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT d\\dot{z} and \\int \\operatorname{f_{\\mathbf{p}}}{(T,\\dot{z})} dT + \\iint (e^{\\dot{z}})^{T} dT d\\dot{z} = \\int (e^{\\dot{z}})^{T} dT + \\iint (e^{\\dot{z}})^{T} dT d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 2, "Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Add(Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Add(Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Pow(exp(Symbol('\\\\dot{z}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{1},L_{\\varepsilon})} = e^{\\frac{L_{\\varepsilon}}{v_{1}}}, then derive \\int \\mathbf{f}{(v_{1},L_{\\varepsilon})} dL_{\\varepsilon} = E + v_{1} e^{\\frac{L_{\\varepsilon}}{v_{1}}}, then obtain E + v_{1} \\mathbf{f}{(v_{1},L_{\\varepsilon})} = E + v_{1} e^{\\frac{L_{\\varepsilon}}{v_{1}}}", "derivation": "\\mathbf{f}{(v_{1},L_{\\varepsilon})} = e^{\\frac{L_{\\varepsilon}}{v_{1}}} and \\int \\mathbf{f}{(v_{1},L_{\\varepsilon})} dL_{\\varepsilon} = \\int e^{\\frac{L_{\\varepsilon}}{v_{1}}} dL_{\\varepsilon} and \\int \\mathbf{f}{(v_{1},L_{\\varepsilon})} dL_{\\varepsilon} = E + v_{1} e^{\\frac{L_{\\varepsilon}}{v_{1}}} and \\int \\mathbf{f}{(v_{1},L_{\\varepsilon})} dL_{\\varepsilon} = E + v_{1} \\mathbf{f}{(v_{1},L_{\\varepsilon})} and E + v_{1} \\mathbf{f}{(v_{1},L_{\\varepsilon})} = E + v_{1} e^{\\frac{L_{\\varepsilon}}{v_{1}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('v_1', commutative=True), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('E', commutative=True), Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Symbol('v_1', commutative=True), exp(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},\\Omega)} = \\Omega \\hat{\\mathbf{x}}, then obtain \\frac{\\operatorname{F_{g}}^{2}{(\\hat{\\mathbf{x}},\\Omega)}}{\\Omega} = \\Omega \\hat{\\mathbf{x}}^{2}", "derivation": "\\operatorname{F_{g}}{(\\hat{\\mathbf{x}},\\Omega)} = \\Omega \\hat{\\mathbf{x}} and \\hat{\\mathbf{x}} \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},\\Omega)} = \\Omega \\hat{\\mathbf{x}}^{2} and \\frac{\\operatorname{F_{g}}^{2}{(\\hat{\\mathbf{x}},\\Omega)}}{\\Omega} = \\hat{\\mathbf{x}} \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},\\Omega)} and \\frac{\\operatorname{F_{g}}^{2}{(\\hat{\\mathbf{x}},\\Omega)}}{\\Omega} = \\Omega \\hat{\\mathbf{x}}^{2}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2))))"], [["divide", 2, "Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given L{(f_{\\mathbf{p}},m,\\dot{z})} = - f_{\\mathbf{p}} + m^{\\dot{z}}, then obtain f_{\\mathbf{p}} - m^{\\dot{z}} + \\int (- f_{\\mathbf{p}} + m^{\\dot{z}}) L{(f_{\\mathbf{p}},m,\\dot{z})} dm = f_{\\mathbf{p}} - m^{\\dot{z}} + \\int (- f_{\\mathbf{p}} + m^{\\dot{z}})^{2} dm", "derivation": "L{(f_{\\mathbf{p}},m,\\dot{z})} = - f_{\\mathbf{p}} + m^{\\dot{z}} and (- f_{\\mathbf{p}} + m^{\\dot{z}}) L{(f_{\\mathbf{p}},m,\\dot{z})} = (- f_{\\mathbf{p}} + m^{\\dot{z}})^{2} and \\int (- f_{\\mathbf{p}} + m^{\\dot{z}}) L{(f_{\\mathbf{p}},m,\\dot{z})} dm = \\int (- f_{\\mathbf{p}} + m^{\\dot{z}})^{2} dm and f_{\\mathbf{p}} - m^{\\dot{z}} + \\int (- f_{\\mathbf{p}} + m^{\\dot{z}}) L{(f_{\\mathbf{p}},m,\\dot{z})} dm = f_{\\mathbf{p}} - m^{\\dot{z}} + \\int (- f_{\\mathbf{p}} + m^{\\dot{z}})^{2} dm", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integer(2)), Tuple(Symbol('m', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Function('L')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('m', commutative=True)))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integer(2)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{x}_0,\\varepsilon_0)} = e^{\\hat{x}_0 \\varepsilon_0}, then obtain (\\operatorname{y^{\\prime}}^{\\varepsilon_0}{(\\hat{x}_0,\\varepsilon_0)})^{\\hat{x}_0} (e^{\\hat{x}_0 \\varepsilon_0})^{- \\varepsilon_0} = ((e^{\\hat{x}_0 \\varepsilon_0})^{\\varepsilon_0})^{\\hat{x}_0} (e^{\\hat{x}_0 \\varepsilon_0})^{- \\varepsilon_0}", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{x}_0,\\varepsilon_0)} = e^{\\hat{x}_0 \\varepsilon_0} and \\operatorname{y^{\\prime}}^{\\varepsilon_0}{(\\hat{x}_0,\\varepsilon_0)} = (e^{\\hat{x}_0 \\varepsilon_0})^{\\varepsilon_0} and (\\operatorname{y^{\\prime}}^{\\varepsilon_0}{(\\hat{x}_0,\\varepsilon_0)})^{\\hat{x}_0} = ((e^{\\hat{x}_0 \\varepsilon_0})^{\\varepsilon_0})^{\\hat{x}_0} and (\\operatorname{y^{\\prime}}^{\\varepsilon_0}{(\\hat{x}_0,\\varepsilon_0)})^{\\hat{x}_0} (e^{\\hat{x}_0 \\varepsilon_0})^{- \\varepsilon_0} = ((e^{\\hat{x}_0 \\varepsilon_0})^{\\varepsilon_0})^{\\hat{x}_0} (e^{\\hat{x}_0 \\varepsilon_0})^{- \\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 3, "Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Pow(Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Pow(Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(exp(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given v{(\\Omega)} = \\cos{(\\Omega)}, then derive \\int v{(\\Omega)} d\\Omega = \\mathbf{v} + \\sin{(\\Omega)}, then obtain \\frac{1}{v{(\\Omega)}} = \\frac{(\\frac{\\mathbf{v} + \\sin{(\\Omega)}}{\\int \\cos{(\\Omega)} d\\Omega})^{\\mathbf{v}}}{v{(\\Omega)}}", "derivation": "v{(\\Omega)} = \\cos{(\\Omega)} and \\int v{(\\Omega)} d\\Omega = \\int \\cos{(\\Omega)} d\\Omega and \\int v{(\\Omega)} d\\Omega = \\mathbf{v} + \\sin{(\\Omega)} and \\int \\cos{(\\Omega)} d\\Omega = \\mathbf{v} + \\sin{(\\Omega)} and 1 = \\frac{\\mathbf{v} + \\sin{(\\Omega)}}{\\int \\cos{(\\Omega)} d\\Omega} and 1 = (\\frac{\\mathbf{v} + \\sin{(\\Omega)}}{\\int \\cos{(\\Omega)} d\\Omega})^{\\mathbf{v}} and \\frac{1}{v{(\\Omega)}} = \\frac{(\\frac{\\mathbf{v} + \\sin{(\\Omega)}}{\\int \\cos{(\\Omega)} d\\Omega})^{\\mathbf{v}}}{v{(\\Omega)}}", "srepr_derivation": [["get_premise", "Equality(Function('v')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))))"], [["power", 5, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 6, "Function('v')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('v')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Mul(Pow(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('v')(Symbol('\\\\Omega', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given W{(\\mathbf{J}_M,H)} = H \\mathbf{J}_M and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_M,H)} = W{(\\mathbf{J}_M,H)} + \\frac{W{(\\mathbf{J}_M,H)}}{\\mathbf{J}_M}, then obtain \\frac{\\partial}{\\partial H} \\frac{W{(\\mathbf{J}_M,H)} + \\frac{W{(\\mathbf{J}_M,H)}}{\\mathbf{J}_M}}{H \\mathbf{J}_M + H} = \\frac{d}{d H} 1", "derivation": "W{(\\mathbf{J}_M,H)} = H \\mathbf{J}_M and \\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_M,H)} = W{(\\mathbf{J}_M,H)} + \\frac{W{(\\mathbf{J}_M,H)}}{\\mathbf{J}_M} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_M,H)}}{H + W{(\\mathbf{J}_M,H)}} = \\frac{W{(\\mathbf{J}_M,H)} + \\frac{W{(\\mathbf{J}_M,H)}}{\\mathbf{J}_M}}{H + W{(\\mathbf{J}_M,H)}} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_M,H)}}{H \\mathbf{J}_M + H} = 1 and \\frac{\\partial}{\\partial H} \\frac{\\operatorname{J_{\\varepsilon}}{(\\mathbf{J}_M,H)}}{H \\mathbf{J}_M + H} = \\frac{d}{d H} 1 and \\frac{\\partial}{\\partial H} \\frac{W{(\\mathbf{J}_M,H)} + \\frac{W{(\\mathbf{J}_M,H)}}{\\mathbf{J}_M}}{H \\mathbf{J}_M + H} = \\frac{d}{d H} 1", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Add(Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)))))"], [["divide", 2, "Add(Symbol('H', commutative=True), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Add(Symbol('H', commutative=True), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Add(Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))), Integer(1))"], [["differentiate", 4, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Mul(Pow(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Add(Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('W')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(h,\\mathbf{S})} = e^{\\mathbf{S} - h}, then derive - e^{\\mathbf{S} - h} e^{e^{\\mathbf{S} - h}} + e^{\\operatorname{f^{*}}{(h,\\mathbf{S})}} \\frac{\\partial}{\\partial \\mathbf{S}} \\operatorname{f^{*}}{(h,\\mathbf{S})} = 0, then obtain - e^{\\mathbf{S} - h} e^{e^{\\mathbf{S} - h}} + e^{e^{\\mathbf{S} - h}} \\frac{\\partial}{\\partial \\mathbf{S}} e^{\\mathbf{S} - h} = 0", "derivation": "\\operatorname{f^{*}}{(h,\\mathbf{S})} = e^{\\mathbf{S} - h} and e^{\\operatorname{f^{*}}{(h,\\mathbf{S})}} = e^{e^{\\mathbf{S} - h}} and \\frac{\\partial}{\\partial \\mathbf{S}} e^{\\operatorname{f^{*}}{(h,\\mathbf{S})}} = \\frac{\\partial}{\\partial \\mathbf{S}} e^{e^{\\mathbf{S} - h}} and \\frac{\\partial}{\\partial \\mathbf{S}} e^{\\operatorname{f^{*}}{(h,\\mathbf{S})}} - \\frac{\\partial}{\\partial \\mathbf{S}} e^{e^{\\mathbf{S} - h}} = 0 and - e^{\\mathbf{S} - h} e^{e^{\\mathbf{S} - h}} + e^{\\operatorname{f^{*}}{(h,\\mathbf{S})}} \\frac{\\partial}{\\partial \\mathbf{S}} \\operatorname{f^{*}}{(h,\\mathbf{S})} = 0 and - e^{\\mathbf{S} - h} e^{e^{\\mathbf{S} - h}} + e^{e^{\\mathbf{S} - h}} \\frac{\\partial}{\\partial \\mathbf{S}} e^{\\mathbf{S} - h} = 0", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["exp", 1], "Equality(exp(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(exp(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(exp(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))), Mul(exp(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Derivative(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))))), Mul(exp(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True))))), Derivative(exp(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given b{(S)} = \\sin{(\\cos{(S)})}, then obtain \\frac{b{(S)}}{\\int \\sin{(\\cos{(S)})} dS} = \\frac{\\sin{(\\cos{(S)})}}{\\int \\sin{(\\cos{(S)})} dS}", "derivation": "b{(S)} = \\sin{(\\cos{(S)})} and \\int b{(S)} dS = \\int \\sin{(\\cos{(S)})} dS and \\frac{b{(S)}}{\\int b{(S)} dS} = \\frac{\\sin{(\\cos{(S)})}}{\\int b{(S)} dS} and \\frac{b{(S)}}{\\int \\sin{(\\cos{(S)})} dS} = \\frac{\\sin{(\\cos{(S)})}}{\\int \\sin{(\\cos{(S)})} dS}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('S', commutative=True)), sin(cos(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('b')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(sin(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["divide", 1, "Integral(Function('b')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Function('b')(Symbol('S', commutative=True)), Pow(Integral(Function('b')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))), Mul(sin(cos(Symbol('S', commutative=True))), Pow(Integral(Function('b')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('b')(Symbol('S', commutative=True)), Pow(Integral(sin(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integer(-1))), Mul(sin(cos(Symbol('S', commutative=True))), Pow(Integral(sin(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} = g^{\\prime}_{\\varepsilon} t_{1}, then obtain t_{1} + \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} + \\log{(\\hat{H}_l)}^{\\hat{H}_l} = g^{\\prime}_{\\varepsilon} t_{1} + t_{1} + \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "derivation": "\\operatorname{C_{1}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} = g^{\\prime}_{\\varepsilon} t_{1} and \\operatorname{C_{1}}^{\\hat{H}_l}{(\\hat{H}_l)} + \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} = g^{\\prime}_{\\varepsilon} t_{1} + \\operatorname{C_{1}}^{\\hat{H}_l}{(\\hat{H}_l)} and \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} + \\log{(\\hat{H}_l)}^{\\hat{H}_l} = g^{\\prime}_{\\varepsilon} t_{1} + \\log{(\\hat{H}_l)}^{\\hat{H}_l} and t_{1} + \\dot{z}{(g^{\\prime}_{\\varepsilon},t_{1})} + \\log{(\\hat{H}_l)}^{\\hat{H}_l} = g^{\\prime}_{\\varepsilon} t_{1} + t_{1} + \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)))"], [["add", 2, "Pow(Function('C_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Pow(Function('C_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True))), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('C_1')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 4, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given Q{(n_{1},\\theta)} = \\theta^{n_{1}} and \\operatorname{V_{\\mathbf{E}}}{(\\theta)} = \\theta, then obtain \\frac{\\iint Q{(n_{1},\\theta)} dn_{1} dn_{1}}{\\theta \\iint \\theta^{n_{1}} dn_{1} dn_{1}} = \\frac{1}{\\theta}", "derivation": "Q{(n_{1},\\theta)} = \\theta^{n_{1}} and \\int Q{(n_{1},\\theta)} dn_{1} = \\int \\theta^{n_{1}} dn_{1} and \\operatorname{V_{\\mathbf{E}}}{(\\theta)} = \\theta and \\iint Q{(n_{1},\\theta)} dn_{1} dn_{1} = \\iint \\theta^{n_{1}} dn_{1} dn_{1} and \\frac{\\iint Q{(n_{1},\\theta)} dn_{1} dn_{1}}{\\operatorname{V_{\\mathbf{E}}}{(\\theta)}} = \\frac{\\iint \\theta^{n_{1}} dn_{1} dn_{1}}{\\operatorname{V_{\\mathbf{E}}}{(\\theta)}} and \\frac{\\iint Q{(n_{1},\\theta)} dn_{1} dn_{1}}{\\theta} = \\frac{\\iint \\theta^{n_{1}} dn_{1} dn_{1}}{\\theta} and \\frac{\\iint Q{(n_{1},\\theta)} dn_{1} dn_{1}}{\\theta \\iint \\theta^{n_{1}} dn_{1} dn_{1}} = \\frac{1}{\\theta}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))"], [["integrate", 2, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["divide", 4, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True)), Integer(-1)), Integral(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["divide", 6, "Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Integral(Pow(Symbol('\\\\theta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integer(-1)), Integral(Function('Q')(Symbol('n_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\hat{p}_0{(\\nabla,P_{e},q)} = P_{e} + \\nabla + q, then derive \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,P_{e},q)} = 1, then obtain \\frac{\\partial}{\\partial \\nabla} (P_{e} + \\nabla + q) = 1", "derivation": "\\hat{p}_0{(\\nabla,P_{e},q)} = P_{e} + \\nabla + q and \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,P_{e},q)} = \\frac{\\partial}{\\partial \\nabla} (P_{e} + \\nabla + q) and \\frac{\\partial}{\\partial \\nabla} \\hat{p}_0{(\\nabla,P_{e},q)} = 1 and \\frac{\\partial}{\\partial \\nabla} (P_{e} + \\nabla + q) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('P_e', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('P_e', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\theta_1}, then derive \\mathbf{J}_P{(\\theta_1)} = e^{\\theta_1}, then obtain (e^{\\theta_1})^{\\theta_1} = (\\frac{d}{d \\theta_1} e^{\\theta_1})^{\\theta_1}", "derivation": "\\mathbf{J}_P{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\theta_1} and \\mathbf{J}_P^{\\theta_1}{(\\theta_1)} = (\\frac{d}{d \\theta_1} e^{\\theta_1})^{\\theta_1} and \\mathbf{J}_P{(\\theta_1)} = e^{\\theta_1} and \\mathbf{J}_P^{\\theta_1}{(\\theta_1)} = (\\frac{d}{d \\theta_1} \\mathbf{J}_P{(\\theta_1)})^{\\theta_1} and (e^{\\theta_1})^{\\theta_1} = (\\frac{d}{d \\theta_1} e^{\\theta_1})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta_1', commutative=True)), Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})}, then derive \\mathbf{E} + \\eta^{\\prime}{(x^\\prime)} = P_{e} + \\log{(\\sin{(x^\\prime)})}, then obtain \\mathbf{E} + \\eta^{\\prime}{(x^\\prime)} = P_{e} + \\eta^{\\prime}{(x^\\prime)}", "derivation": "\\eta^{\\prime}{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\eta^{\\prime}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} and \\int \\frac{d}{d x^\\prime} \\eta^{\\prime}{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} dx^\\prime and \\mathbf{E} + \\eta^{\\prime}{(x^\\prime)} = P_{e} + \\log{(\\sin{(x^\\prime)})} and \\mathbf{E} + \\eta^{\\prime}{(x^\\prime)} = P_{e} + \\eta^{\\prime}{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), log(sin(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(log(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('P_e', commutative=True), log(sin(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('P_e', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(v_{x})} = v_{x}, then obtain \\cos{(v_{x} + \\operatorname{E_{n}}^{2}{(v_{x})})} = \\cos{(v_{x} \\operatorname{E_{n}}{(v_{x})} + v_{x})}", "derivation": "\\operatorname{E_{n}}{(v_{x})} = v_{x} and \\operatorname{E_{n}}^{2}{(v_{x})} = v_{x} \\operatorname{E_{n}}{(v_{x})} and v_{x} + \\operatorname{E_{n}}^{2}{(v_{x})} = v_{x} \\operatorname{E_{n}}{(v_{x})} + v_{x} and \\cos{(v_{x} + \\operatorname{E_{n}}^{2}{(v_{x})})} = \\cos{(v_{x} \\operatorname{E_{n}}{(v_{x})} + v_{x})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], [["times", 1, "Function('E_n')(Symbol('v_x', commutative=True))"], "Equality(Pow(Function('E_n')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Symbol('v_x', commutative=True), Function('E_n')(Symbol('v_x', commutative=True))))"], [["add", 2, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Pow(Function('E_n')(Symbol('v_x', commutative=True)), Integer(2))), Add(Mul(Symbol('v_x', commutative=True), Function('E_n')(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["cos", 3], "Equality(cos(Add(Symbol('v_x', commutative=True), Pow(Function('E_n')(Symbol('v_x', commutative=True)), Integer(2)))), cos(Add(Mul(Symbol('v_x', commutative=True), Function('E_n')(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given V{(f^{\\prime},E_{x})} = \\frac{E_{x}}{f^{\\prime}}, then derive - \\frac{\\partial}{\\partial E_{x}} V{(f^{\\prime},E_{x})} = - \\frac{1}{f^{\\prime}}, then obtain V{(f^{\\prime},E_{x})} - \\frac{\\partial}{\\partial E_{x}} V{(f^{\\prime},E_{x})} + 1 = V{(f^{\\prime},E_{x})} + 1 - \\frac{1}{f^{\\prime}}", "derivation": "V{(f^{\\prime},E_{x})} = \\frac{E_{x}}{f^{\\prime}} and - V{(f^{\\prime},E_{x})} = - \\frac{E_{x}}{f^{\\prime}} and \\frac{\\partial}{\\partial E_{x}} - V{(f^{\\prime},E_{x})} = \\frac{\\partial}{\\partial E_{x}} - \\frac{E_{x}}{f^{\\prime}} and - \\frac{\\partial}{\\partial E_{x}} V{(f^{\\prime},E_{x})} = - \\frac{1}{f^{\\prime}} and V{(f^{\\prime},E_{x})} - \\frac{\\partial}{\\partial E_{x}} V{(f^{\\prime},E_{x})} + 1 = V{(f^{\\prime},E_{x})} + 1 - \\frac{1}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["add", 4, "Add(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Integer(1))"], "Equality(Add(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Derivative(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1)), Add(Function('V')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_x', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\delta{(\\psi,\\varphi)} = \\frac{\\log{(\\psi)}}{\\varphi}, then obtain \\frac{1}{\\varphi^{2}} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\delta{(\\psi,\\varphi)}}{\\psi} = \\frac{1}{\\varphi^{2}} - \\frac{\\log{(\\psi)}}{\\psi \\varphi^{2}}", "derivation": "\\delta{(\\psi,\\varphi)} = \\frac{\\log{(\\psi)}}{\\varphi} and \\frac{\\delta{(\\psi,\\varphi)}}{\\psi} = \\frac{\\log{(\\psi)}}{\\psi \\varphi} and - \\frac{1}{\\varphi} + \\frac{\\delta{(\\psi,\\varphi)}}{\\psi} = - \\frac{1}{\\varphi} + \\frac{\\log{(\\psi)}}{\\psi \\varphi} and \\frac{\\partial}{\\partial \\varphi} (- \\frac{1}{\\varphi} + \\frac{\\delta{(\\psi,\\varphi)}}{\\psi}) = \\frac{\\partial}{\\partial \\varphi} (- \\frac{1}{\\varphi} + \\frac{\\log{(\\psi)}}{\\psi \\varphi}) and \\frac{1}{\\varphi^{2}} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\delta{(\\psi,\\varphi)}}{\\psi} = \\frac{1}{\\varphi^{2}} - \\frac{\\log{(\\psi)}}{\\psi \\varphi^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))))"], [["divide", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Pow(Symbol('\\\\varphi', commutative=True), Integer(-2)), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Add(Pow(Symbol('\\\\varphi', commutative=True), Integer(-2)), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(-2)), log(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(P_{e},t)} = P_{e} t and \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} = P_{e} t and \\operatorname{F_{N}}{(P_{e},t)} = \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} - \\frac{1}{P_{e}}, then obtain \\operatorname{F_{N}}{(P_{e},t)} = P_{e} t - \\frac{1}{P_{e}}", "derivation": "\\operatorname{x^{{\\}'}}{(P_{e},t)} = P_{e} t and \\operatorname{x^{{\\}'}}{(P_{e},t)} - \\frac{1}{P_{e}} = P_{e} t - \\frac{1}{P_{e}} and \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} = P_{e} t and \\operatorname{x^{{\\}'}}{(P_{e},t)} - \\frac{1}{P_{e}} = \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} - \\frac{1}{P_{e}} and \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} - \\frac{1}{P_{e}} = P_{e} t - \\frac{1}{P_{e}} and \\operatorname{F_{N}}{(P_{e},t)} = \\operatorname{f_{\\mathbf{v}}}{(P_{e},t)} - \\frac{1}{P_{e}} and \\operatorname{F_{N}}{(P_{e},t)} = P_{e} t - \\frac{1}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('P_e', commutative=True), Symbol('t', commutative=True)))"], [["minus", 1, "Pow(Symbol('P_e', commutative=True), Integer(-1))"], "Equality(Add(Function('x^\\\\prime')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(Mul(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('P_e', commutative=True), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('x^\\\\prime')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))), Add(Mul(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('F_N')(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Add(Mul(Symbol('P_e', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('P_e', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\Psi{(T,p)} = p \\cos{(T)} and \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} = \\delta + \\mathbf{J}_M, then obtain \\delta + \\mathbf{J}_M + \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} - \\int p \\cos{(T)} dp = 2 \\delta + 2 \\mathbf{J}_M - \\int p \\cos{(T)} dp", "derivation": "\\Psi{(T,p)} = p \\cos{(T)} and \\int \\Psi{(T,p)} dp = \\int p \\cos{(T)} dp and \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} = \\delta + \\mathbf{J}_M and \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} - \\int \\Psi{(T,p)} dp = \\delta + \\mathbf{J}_M - \\int \\Psi{(T,p)} dp and \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} - \\int p \\cos{(T)} dp = \\delta + \\mathbf{J}_M - \\int p \\cos{(T)} dp and \\delta + \\mathbf{J}_M + \\operatorname{y^{\\prime}}{(\\delta,\\mathbf{J}_M)} - \\int p \\cos{(T)} dp = 2 \\delta + 2 \\mathbf{J}_M - \\int p \\cos{(T)} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('p', commutative=True))))"], ["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 3, "Integral(Function('\\\\Psi')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\Psi')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Integral(Function('\\\\Psi')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('p', commutative=True))))), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('p', commutative=True))))))"], [["add", 5, "Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('p', commutative=True))))), Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('p', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\varphi^{*}{(\\omega)} = \\cos{(\\log{(\\omega)})}, then derive \\frac{d}{d \\omega} \\varphi^{*}{(\\omega)} + \\frac{1}{\\omega} = - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} + \\frac{1}{\\omega}, then obtain \\int (\\frac{d}{d \\omega} \\varphi^{*}{(\\omega)} + \\frac{1}{\\omega}) d\\omega = \\int (- \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} + \\frac{1}{\\omega}) d\\omega", "derivation": "\\varphi^{*}{(\\omega)} = \\cos{(\\log{(\\omega)})} and \\varphi^{*}{(\\omega)} + \\log{(\\omega)} = \\log{(\\omega)} + \\cos{(\\log{(\\omega)})} and \\frac{d}{d \\omega} (\\varphi^{*}{(\\omega)} + \\log{(\\omega)}) = \\frac{d}{d \\omega} (\\log{(\\omega)} + \\cos{(\\log{(\\omega)})}) and \\frac{d}{d \\omega} \\varphi^{*}{(\\omega)} + \\frac{1}{\\omega} = - \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} + \\frac{1}{\\omega} and \\int (\\frac{d}{d \\omega} \\varphi^{*}{(\\omega)} + \\frac{1}{\\omega}) d\\omega = \\int (- \\frac{\\sin{(\\log{(\\omega)})}}{\\omega} + \\frac{1}{\\omega}) d\\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True))))"], [["add", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True))), Add(log(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(log(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(log(Symbol('\\\\omega', commutative=True)))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(log(Symbol('\\\\omega', commutative=True)))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(M)} = \\log{(M)}, then obtain \\frac{d^{2}}{d M^{2}} \\int 2 dM = \\frac{d^{2}}{d M^{2}} \\int (1 + \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}}) dM", "derivation": "\\operatorname{v_{z}}{(M)} = \\log{(M)} and 1 = \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}} and 2 = 1 + \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}} and \\int 2 dM = \\int (1 + \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}}) dM and \\frac{d}{d M} \\int 2 dM = \\frac{d}{d M} \\int (1 + \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}}) dM and \\frac{d^{2}}{d M^{2}} \\int 2 dM = \\frac{d^{2}}{d M^{2}} \\int (1 + \\frac{\\log{(M)}}{\\operatorname{v_{z}}{(M)}}) dM", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["divide", 1, "Function('v_z')(Symbol('M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_z')(Symbol('M', commutative=True)), Integer(-1)), log(Symbol('M', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('v_z')(Symbol('M', commutative=True)), Integer(-1)), log(Symbol('M', commutative=True)))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Integer(2), Tuple(Symbol('M', commutative=True))), Integral(Add(Integer(1), Mul(Pow(Function('v_z')(Symbol('M', commutative=True)), Integer(-1)), log(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Integer(2), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(Add(Integer(1), Mul(Pow(Function('v_z')(Symbol('M', commutative=True)), Integer(-1)), log(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Integer(2), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(2))), Derivative(Integral(Add(Integer(1), Mul(Pow(Function('v_z')(Symbol('M', commutative=True)), Integer(-1)), log(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{r}{(\\eta,F_{H})} = e^{F_{H}^{\\eta}}, then obtain \\frac{\\mathbf{r}^{2}{(\\eta,F_{H})}}{\\eta} = \\frac{\\mathbf{r}{(\\eta,F_{H})} e^{F_{H}^{\\eta}}}{\\eta}", "derivation": "\\mathbf{r}{(\\eta,F_{H})} = e^{F_{H}^{\\eta}} and \\frac{\\mathbf{r}{(\\eta,F_{H})}}{\\eta} = \\frac{e^{F_{H}^{\\eta}}}{\\eta} and \\frac{\\mathbf{r}{(\\eta,F_{H})} e^{F_{H}^{\\eta}}}{\\eta} = \\frac{e^{2 F_{H}^{\\eta}}}{\\eta} and \\frac{\\mathbf{r}^{2}{(\\eta,F_{H})}}{\\eta} = \\frac{\\mathbf{r}{(\\eta,F_{H})} e^{F_{H}^{\\eta}}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('F_H', commutative=True)), exp(Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["times", 1, "Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('F_H', commutative=True)), exp(Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('F_H', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('F_H', commutative=True)), exp(Pow(Symbol('F_H', commutative=True), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(m_{s})} = \\cos{(\\log{(m_{s})})}, then obtain \\int (- m_{s} + \\operatorname{f_{\\mathbf{p}}}^{m_{s}}{(m_{s})}) dm_{s} = \\int (- m_{s} + \\cos^{m_{s}}{(\\log{(m_{s})})}) dm_{s}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(m_{s})} = \\cos{(\\log{(m_{s})})} and \\operatorname{f_{\\mathbf{p}}}^{m_{s}}{(m_{s})} = \\cos^{m_{s}}{(\\log{(m_{s})})} and - m_{s} + \\operatorname{f_{\\mathbf{p}}}^{m_{s}}{(m_{s})} = - m_{s} + \\cos^{m_{s}}{(\\log{(m_{s})})} and \\int (- m_{s} + \\operatorname{f_{\\mathbf{p}}}^{m_{s}}{(m_{s})}) dm_{s} = \\int (- m_{s} + \\cos^{m_{s}}{(\\log{(m_{s})})}) dm_{s}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('m_s', commutative=True)), cos(log(Symbol('m_s', commutative=True))))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(cos(log(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(cos(log(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))))"], [["integrate", 3, "Symbol('m_s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(cos(log(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given Q{(M_{E},\\varepsilon)} = \\varepsilon^{M_{E}} and c{(M_{E},\\varepsilon)} = \\varepsilon^{M_{E}}, then obtain \\Psi{(q)} c{(M_{E},\\varepsilon)} = Q{(M_{E},\\varepsilon)} \\Psi{(q)}", "derivation": "Q{(M_{E},\\varepsilon)} = \\varepsilon^{M_{E}} and c{(M_{E},\\varepsilon)} = \\varepsilon^{M_{E}} and \\Psi{(q)} c{(M_{E},\\varepsilon)} = \\varepsilon^{M_{E}} \\Psi{(q)} and \\Psi{(q)} c{(M_{E},\\varepsilon)} = Q{(M_{E},\\varepsilon)} \\Psi{(q)}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('M_E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('M_E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('M_E', commutative=True)))"], [["times", 2, "Function('\\\\Psi')(Symbol('q', commutative=True))"], "Equality(Mul(Function('\\\\Psi')(Symbol('q', commutative=True)), Function('c')(Symbol('M_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('M_E', commutative=True)), Function('\\\\Psi')(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\Psi')(Symbol('q', commutative=True)), Function('c')(Symbol('M_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Function('Q')(Symbol('M_E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\Psi')(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\eta^{\\prime})} = \\cos{(\\log{(\\eta^{\\prime})})}, then obtain \\frac{\\Psi{(\\eta^{\\prime})} \\Psi^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\Psi{(\\eta^{\\prime})} \\cos^{\\eta^{\\prime}}{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}}", "derivation": "\\Psi{(\\eta^{\\prime})} = \\cos{(\\log{(\\eta^{\\prime})})} and \\frac{\\Psi{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\cos{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} and \\Psi^{\\eta^{\\prime}}{(\\eta^{\\prime})} = \\cos^{\\eta^{\\prime}}{(\\log{(\\eta^{\\prime})})} and \\frac{\\Psi^{\\eta^{\\prime}}{(\\eta^{\\prime})} \\cos{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} = \\frac{\\cos{(\\log{(\\eta^{\\prime})})} \\cos^{\\eta^{\\prime}}{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}} and \\frac{\\Psi{(\\eta^{\\prime})} \\Psi^{\\eta^{\\prime}}{(\\eta^{\\prime})}}{\\eta^{\\prime}} = \\frac{\\Psi{(\\eta^{\\prime})} \\cos^{\\eta^{\\prime}}{(\\log{(\\eta^{\\prime})})}}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Pow(cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(cos(log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\varepsilon_0,\\omega)} = - \\varepsilon_0 + e^{\\omega}, then obtain ((- \\varepsilon_0 + e^{\\omega})^{\\omega} + \\mathbf{p}^{\\omega}{(\\varepsilon_0,\\omega)})^{\\varepsilon_0} = (2 (- \\varepsilon_0 + e^{\\omega})^{\\omega})^{\\varepsilon_0}", "derivation": "\\mathbf{p}{(\\varepsilon_0,\\omega)} = - \\varepsilon_0 + e^{\\omega} and \\mathbf{p}^{\\omega}{(\\varepsilon_0,\\omega)} = (- \\varepsilon_0 + e^{\\omega})^{\\omega} and (- \\varepsilon_0 + e^{\\omega})^{\\omega} + \\mathbf{p}^{\\omega}{(\\varepsilon_0,\\omega)} = 2 (- \\varepsilon_0 + e^{\\omega})^{\\omega} and ((- \\varepsilon_0 + e^{\\omega})^{\\omega} + \\mathbf{p}^{\\omega}{(\\varepsilon_0,\\omega)})^{\\varepsilon_0} = (2 (- \\varepsilon_0 + e^{\\omega})^{\\omega})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["add", 2, "Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(A_{x})} = \\log{(A_{x})}, then obtain (\\int \\theta_{2}^{A_{x}}{(A_{x})} dA_{x})^{A_{x}} = (\\int \\log{(A_{x})}^{A_{x}} dA_{x})^{A_{x}}", "derivation": "\\theta_{2}{(A_{x})} = \\log{(A_{x})} and \\theta_{2}^{A_{x}}{(A_{x})} = \\log{(A_{x})}^{A_{x}} and \\int \\theta_{2}^{A_{x}}{(A_{x})} dA_{x} = \\int \\log{(A_{x})}^{A_{x}} dA_{x} and (\\int \\theta_{2}^{A_{x}}{(A_{x})} dA_{x})^{A_{x}} = (\\int \\log{(A_{x})}^{A_{x}} dA_{x})^{A_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_2')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_2')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\theta_2')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integral(Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(k,y^{\\prime})} = \\frac{k}{y^{\\prime}} and n{(k,y^{\\prime})} = \\int (\\operatorname{r_{0}}{(k,y^{\\prime})} + 1)^{k} dk, then obtain \\frac{k}{y^{\\prime}} + n{(k,y^{\\prime})} = \\frac{k}{y^{\\prime}} + \\int (\\frac{k}{y^{\\prime}} + 1)^{k} dk", "derivation": "\\operatorname{r_{0}}{(k,y^{\\prime})} = \\frac{k}{y^{\\prime}} and \\operatorname{r_{0}}{(k,y^{\\prime})} + 1 = \\frac{k}{y^{\\prime}} + 1 and (\\operatorname{r_{0}}{(k,y^{\\prime})} + 1)^{k} = (\\frac{k}{y^{\\prime}} + 1)^{k} and \\int (\\operatorname{r_{0}}{(k,y^{\\prime})} + 1)^{k} dk = \\int (\\frac{k}{y^{\\prime}} + 1)^{k} dk and n{(k,y^{\\prime})} = \\int (\\operatorname{r_{0}}{(k,y^{\\prime})} + 1)^{k} dk and n{(k,y^{\\prime})} = \\int (\\frac{k}{y^{\\prime}} + 1)^{k} dk and \\frac{k}{y^{\\prime}} + n{(k,y^{\\prime})} = \\frac{k}{y^{\\prime}} + \\int (\\frac{k}{y^{\\prime}} + 1)^{k} dk", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('r_0')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integer(1)))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Function('r_0')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Pow(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integer(1)), Symbol('k', commutative=True)))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Add(Function('r_0')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Pow(Add(Function('r_0')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('n')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Pow(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["add", 6, "Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Function('n')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integral(Pow(Add(Mul(Symbol('k', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\theta,Q)} = \\theta + e^{Q}, then obtain \\theta + (2 \\mathbf{B}{(\\theta,Q)})^{Q} - \\mathbf{B}{(\\theta,Q)} + e^{Q} = \\theta + (\\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q})^{Q} - \\mathbf{B}{(\\theta,Q)} + e^{Q}", "derivation": "\\mathbf{B}{(\\theta,Q)} = \\theta + e^{Q} and 2 \\mathbf{B}{(\\theta,Q)} = \\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q} and (2 \\mathbf{B}{(\\theta,Q)})^{Q} = (\\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q})^{Q} and (2 \\mathbf{B}{(\\theta,Q)})^{Q} - \\mathbf{B}{(\\theta,Q)} = (\\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q})^{Q} - \\mathbf{B}{(\\theta,Q)} and \\theta + (2 \\mathbf{B}{(\\theta,Q)})^{Q} + e^{Q} = \\theta + (\\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q})^{Q} + e^{Q} and \\theta + (2 \\mathbf{B}{(\\theta,Q)})^{Q} - \\mathbf{B}{(\\theta,Q)} + e^{Q} = \\theta + (\\theta + \\mathbf{B}{(\\theta,Q)} + e^{Q})^{Q} - \\mathbf{B}{(\\theta,Q)} + e^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('\\\\theta', commutative=True), exp(Symbol('Q', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["minus", 3, "Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Pow(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)))), Add(Pow(Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)))))"], [["add", 4, "Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Pow(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Pow(Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["minus", 5, "Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Pow(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), exp(Symbol('Q', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Pow(Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True), Symbol('Q', commutative=True))), exp(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x}{\\phi_1} and \\mathbf{J}_P{(v_{1})} = q^{v_{1}}{(v_{1})}, then obtain (\\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)})^{\\sigma_x} + \\mathbf{J}_P{(v_{1})} = (\\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)})^{\\sigma_x} + q^{v_{1}}{(v_{1})}", "derivation": "\\operatorname{n_{1}}{(\\phi_1,\\sigma_x)} = \\frac{\\sigma_x}{\\phi_1} and \\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)} = \\sigma_x + \\frac{\\sigma_x}{\\phi_1} and (\\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)})^{\\sigma_x} = (\\sigma_x + \\frac{\\sigma_x}{\\phi_1})^{\\sigma_x} and \\mathbf{J}_P{(v_{1})} = q^{v_{1}}{(v_{1})} and (\\sigma_x + \\frac{\\sigma_x}{\\phi_1})^{\\sigma_x} + \\mathbf{J}_P{(v_{1})} = (\\sigma_x + \\frac{\\sigma_x}{\\phi_1})^{\\sigma_x} + q^{v_{1}}{(v_{1})} and (\\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)})^{\\sigma_x} + \\mathbf{J}_P{(v_{1})} = (\\sigma_x + \\operatorname{n_{1}}{(\\phi_1,\\sigma_x)})^{\\sigma_x} + q^{v_{1}}{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('n_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Function('n_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True)), Pow(Function('q')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["add", 4, "Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True))), Add(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Function('q')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Function('n_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_1', commutative=True))), Add(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Function('n_1')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Function('q')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})}, then obtain \\cos{(\\hat{\\mathbf{r}})} + \\frac{d}{d \\hat{\\mathbf{r}}} \\mathbf{J}_M{(\\hat{\\mathbf{r}})} = 2 \\cos{(\\hat{\\mathbf{r}})}", "derivation": "\\mathbf{J}_M{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and \\mathbf{J}_M{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})} = 2 \\sin{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} (\\mathbf{J}_M{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})}) = \\frac{d}{d \\hat{\\mathbf{r}}} 2 \\sin{(\\hat{\\mathbf{r}})} and \\cos{(\\hat{\\mathbf{r}})} + \\frac{d}{d \\hat{\\mathbf{r}}} \\mathbf{J}_M{(\\hat{\\mathbf{r}})} = 2 \\cos{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(n,\\Omega)} = \\Omega + n, then obtain (\\frac{1}{\\Omega + n})^{n} = (\\frac{(\\Omega + n)^{2 \\Omega} \\hat{p}^{- 2 \\Omega}{(n,\\Omega)}}{\\Omega + n})^{n}", "derivation": "\\hat{p}{(n,\\Omega)} = \\Omega + n and \\hat{p}^{\\Omega}{(n,\\Omega)} = (\\Omega + n)^{\\Omega} and \\frac{1}{\\Omega + n} = \\frac{(\\Omega + n)^{\\Omega} \\hat{p}^{- \\Omega}{(n,\\Omega)}}{\\Omega + n} and (\\frac{1}{\\Omega + n})^{n} = (\\frac{(\\Omega + n)^{\\Omega} \\hat{p}^{- \\Omega}{(n,\\Omega)}}{\\Omega + n})^{n} and (\\frac{(\\Omega + n)^{\\Omega} \\hat{p}^{- \\Omega}{(n,\\Omega)}}{\\Omega + n})^{n} = (\\frac{(\\Omega + n)^{2 \\Omega} \\hat{p}^{- 2 \\Omega}{(n,\\Omega)}}{\\Omega + n})^{n} and (\\frac{1}{\\Omega + n})^{n} = (\\frac{(\\Omega + n)^{2 \\Omega} \\hat{p}^{- 2 \\Omega}{(n,\\Omega)}}{\\Omega + n})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["divide", 2, "Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Symbol('n', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)))), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} = \\frac{\\log{(f_{\\mathbf{v}})}}{s}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} ds + \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds + \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} = \\frac{\\log{(f_{\\mathbf{v}})}}{s} and \\int \\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} ds = \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} ds = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\operatorname{V_{\\mathbf{B}}}{(f_{\\mathbf{v}},s)} ds + \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds + \\int \\frac{\\log{(f_{\\mathbf{v}})}}{s} ds", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["add", 3, "Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True)))), Add(Derivative(Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integral(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\rho,H,M_{E})} = \\frac{H M_{E}}{\\rho}, then obtain \\frac{H M_{E} \\varepsilon^{H}{(\\rho,H,M_{E})}}{\\rho^{2}} = \\frac{H M_{E} (\\frac{H M_{E}}{\\rho})^{H}}{\\rho^{2}}", "derivation": "\\varepsilon{(\\rho,H,M_{E})} = \\frac{H M_{E}}{\\rho} and \\frac{\\varepsilon{(\\rho,H,M_{E})}}{\\rho} = \\frac{H M_{E}}{\\rho^{2}} and \\varepsilon^{H}{(\\rho,H,M_{E})} = (\\frac{H M_{E}}{\\rho})^{H} and \\frac{\\varepsilon{(\\rho,H,M_{E})} \\varepsilon^{H}{(\\rho,H,M_{E})}}{\\rho} = \\frac{(\\frac{H M_{E}}{\\rho})^{H} \\varepsilon{(\\rho,H,M_{E})}}{\\rho} and \\frac{H M_{E} \\varepsilon^{H}{(\\rho,H,M_{E})}}{\\rho^{2}} = \\frac{H M_{E} (\\frac{H M_{E}}{\\rho})^{H}}{\\rho^{2}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('\\\\rho', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Symbol('H', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Symbol('H', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Symbol('H', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('\\\\varepsilon')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Symbol('H', commutative=True))), Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Mul(Symbol('H', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\lambda{(V,b)} = e^{V + b}, then derive \\frac{\\frac{\\partial}{\\partial V} \\lambda{(V,b)}}{b} = \\frac{e^{V + b}}{b}, then obtain V - \\lambda{(V,b)} + \\frac{\\frac{\\partial}{\\partial V} e^{V + b}}{b} = V - \\lambda{(V,b)} + \\frac{e^{V + b}}{b}", "derivation": "\\lambda{(V,b)} = e^{V + b} and \\frac{\\partial}{\\partial V} \\lambda{(V,b)} = \\frac{\\partial}{\\partial V} e^{V + b} and \\frac{\\frac{\\partial}{\\partial V} \\lambda{(V,b)}}{b} = \\frac{\\frac{\\partial}{\\partial V} e^{V + b}}{b} and \\frac{\\frac{\\partial}{\\partial V} \\lambda{(V,b)}}{b} = \\frac{e^{V + b}}{b} and \\frac{\\frac{\\partial}{\\partial V} \\lambda{(V,b)}}{b} = \\frac{\\lambda{(V,b)}}{b} and \\frac{\\frac{\\partial}{\\partial V} e^{V + b}}{b} = \\frac{e^{V + b}}{b} and V + \\frac{\\frac{\\partial}{\\partial V} e^{V + b}}{b} = V + \\frac{e^{V + b}}{b} and V - \\lambda{(V,b)} + \\frac{\\frac{\\partial}{\\partial V} e^{V + b}}{b} = V - \\lambda{(V,b)} + \\frac{e^{V + b}}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True)), exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True)))))"], [["add", 6, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))), Add(Symbol('V', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))))))"], [["minus", 7, "Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Symbol('V', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))), Add(Symbol('V', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('V', commutative=True), Symbol('b', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), exp(Add(Symbol('V', commutative=True), Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(f^{\\prime},\\rho)} = - \\rho + \\log{(f^{\\prime})}, then obtain f^{\\prime} - \\frac{(- \\rho + \\log{(f^{\\prime})}) \\rho_{f}{(f^{\\prime},\\rho)}}{\\rho} = f^{\\prime} - \\frac{(- \\rho + \\log{(f^{\\prime})})^{2}}{\\rho}", "derivation": "\\rho_{f}{(f^{\\prime},\\rho)} = - \\rho + \\log{(f^{\\prime})} and (- \\rho + \\log{(f^{\\prime})}) \\rho_{f}{(f^{\\prime},\\rho)} = (- \\rho + \\log{(f^{\\prime})})^{2} and - \\frac{(- \\rho + \\log{(f^{\\prime})}) \\rho_{f}{(f^{\\prime},\\rho)}}{\\rho} = - \\frac{(- \\rho + \\log{(f^{\\prime})})^{2}}{\\rho} and f^{\\prime} - \\frac{(- \\rho + \\log{(f^{\\prime})}) \\rho_{f}{(f^{\\prime},\\rho)}}{\\rho} = f^{\\prime} - \\frac{(- \\rho + \\log{(f^{\\prime})})^{2}}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Integer(2))))"], [["add", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Function('\\\\rho_f')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\theta{(C_{2},\\Psi,C)} = C + C_{2} + \\Psi, then derive \\int \\theta{(C_{2},\\Psi,C)} dC_{2} = \\frac{C_{2}^{2}}{2} + C_{2} (C + \\Psi) + \\mu_0, then obtain C + \\int (C + C_{2} + \\Psi) dC_{2} = C + \\frac{C_{2}^{2}}{2} + C_{2} (C + \\Psi) + \\mu_0", "derivation": "\\theta{(C_{2},\\Psi,C)} = C + C_{2} + \\Psi and \\int \\theta{(C_{2},\\Psi,C)} dC_{2} = \\int (C + C_{2} + \\Psi) dC_{2} and \\int \\theta{(C_{2},\\Psi,C)} dC_{2} = \\frac{C_{2}^{2}}{2} + C_{2} (C + \\Psi) + \\mu_0 and \\int (C + C_{2} + \\Psi) dC_{2} = \\frac{C_{2}^{2}}{2} + C_{2} (C + \\Psi) + \\mu_0 and C + \\int (C + C_{2} + \\Psi) dC_{2} = C + \\frac{C_{2}^{2}}{2} + C_{2} (C + \\Psi) + \\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Symbol('C', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta')(Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Mul(Symbol('C_2', commutative=True), Add(Symbol('C', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('C', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Mul(Symbol('C_2', commutative=True), Add(Symbol('C', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["add", 4, "Symbol('C', commutative=True)"], "Equality(Add(Symbol('C', commutative=True), Integral(Add(Symbol('C', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2))), Mul(Symbol('C_2', commutative=True), Add(Symbol('C', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(J)} = \\cos{(J)}, then derive \\int \\mu_{0}{(J)} dJ = v_{y} + \\sin{(J)}, then obtain 1 = \\frac{v_{y} + \\sin{(J)}}{\\int \\cos{(J)} dJ}", "derivation": "\\mu_{0}{(J)} = \\cos{(J)} and \\int \\mu_{0}{(J)} dJ = \\int \\cos{(J)} dJ and \\int \\mu_{0}{(J)} dJ = v_{y} + \\sin{(J)} and \\frac{\\int \\mu_{0}{(J)} dJ}{\\int \\cos{(J)} dJ} = \\frac{v_{y} + \\sin{(J)}}{\\int \\cos{(J)} dJ} and 1 = \\frac{v_{y} + \\sin{(J)}}{\\int \\cos{(J)} dJ}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu_0')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Symbol('v_y', commutative=True), sin(Symbol('J', commutative=True))))"], [["divide", 3, "Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mu_0')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Mul(Add(Symbol('v_y', commutative=True), sin(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Add(Symbol('v_y', commutative=True), sin(Symbol('J', commutative=True))), Pow(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(z^{*})} = \\cos{(\\sin{(z^{*})})}, then obtain (\\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})})^{z^{*}} + \\mathbf{p}{(z^{*})} = (\\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})})^{z^{*}} + \\cos{(\\sin{(z^{*})})}", "derivation": "\\mathbf{p}{(z^{*})} = \\cos{(\\sin{(z^{*})})} and \\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})} = 0 and (\\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})})^{z^{*}} = 0^{z^{*}} and 0^{z^{*}} + \\mathbf{p}{(z^{*})} = 0^{z^{*}} + \\cos{(\\sin{(z^{*})})} and (\\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})})^{z^{*}} + \\mathbf{p}{(z^{*})} = (\\mathbf{p}{(z^{*})} - \\cos{(\\sin{(z^{*})})})^{z^{*}} + \\cos{(\\sin{(z^{*})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True))))"], [["minus", 1, "cos(sin(Symbol('z^*', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('z^*', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('z^*', commutative=True))))), Symbol('z^*', commutative=True)), Pow(Integer(0), Symbol('z^*', commutative=True)))"], [["add", 1, "Pow(Integer(0), Symbol('z^*', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('z^*', commutative=True)), Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True))), Add(Pow(Integer(0), Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('z^*', commutative=True))))), Symbol('z^*', commutative=True)), Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True))), Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('z^*', commutative=True))))), Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given G{(b)} = e^{\\cos{(b)}} and \\mathbf{S}{(b)} = e^{- \\cos{(b)}}, then obtain \\frac{d}{d b} \\cos{(G{(b)} \\mathbf{S}{(b)})} = \\frac{d}{d b} \\cos{(1)}", "derivation": "G{(b)} = e^{\\cos{(b)}} and G{(b)} e^{- \\cos{(b)}} = 1 and \\cos{(G{(b)} e^{- \\cos{(b)}})} = \\cos{(1)} and \\frac{d}{d b} \\cos{(G{(b)} e^{- \\cos{(b)}})} = \\frac{d}{d b} \\cos{(1)} and \\mathbf{S}{(b)} = e^{- \\cos{(b)}} and \\frac{d}{d b} \\cos{(G{(b)} \\mathbf{S}{(b)})} = \\frac{d}{d b} \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True))))"], [["divide", 1, "exp(cos(Symbol('b', commutative=True)))"], "Equality(Mul(Function('G')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True))))), Integer(1))"], [["cos", 2], "Equality(cos(Mul(Function('G')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True)))))), cos(Integer(1)))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(cos(Mul(Function('G')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True)))))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(cos(Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(cos(Mul(Function('G')(Symbol('b', commutative=True)), Function('\\\\mathbf{S}')(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(cos(Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(B)} = \\frac{d}{d B} e^{B}, then obtain \\iint ((\\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}})^{B} + I{(B)}) dB dB = \\iint (I{(B)} + (e^{- B})^{B}) dB dB", "derivation": "I{(B)} = \\frac{d}{d B} e^{B} and \\frac{I{(B)}}{\\frac{d}{d B} e^{B}} = 1 and \\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}} = e^{- B} and (\\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}})^{B} = (e^{- B})^{B} and (\\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}})^{B} + I{(B)} = I{(B)} + (e^{- B})^{B} and \\int ((\\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}})^{B} + I{(B)}) dB = \\int (I{(B)} + (e^{- B})^{B}) dB and \\iint ((\\frac{I{(B)} e^{- B}}{\\frac{d}{d B} e^{B}})^{B} + I{(B)}) dB dB = \\iint (I{(B)} + (e^{- B})^{B}) dB dB", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('B', commutative=True)), Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Mul(Function('I')(Symbol('B', commutative=True)), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["divide", 2, "exp(Symbol('B', commutative=True))"], "Equality(Mul(Function('I')(Symbol('B', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), exp(Mul(Integer(-1), Symbol('B', commutative=True))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Mul(Function('I')(Symbol('B', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Symbol('B', commutative=True)), Pow(exp(Mul(Integer(-1), Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["add", 4, "Function('I')(Symbol('B', commutative=True))"], "Equality(Add(Pow(Mul(Function('I')(Symbol('B', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Symbol('B', commutative=True)), Function('I')(Symbol('B', commutative=True))), Add(Function('I')(Symbol('B', commutative=True)), Pow(exp(Mul(Integer(-1), Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["integrate", 5, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Pow(Mul(Function('I')(Symbol('B', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Symbol('B', commutative=True)), Function('I')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Add(Function('I')(Symbol('B', commutative=True)), Pow(exp(Mul(Integer(-1), Symbol('B', commutative=True))), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["integrate", 6, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Pow(Mul(Function('I')(Symbol('B', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))), Pow(Derivative(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Symbol('B', commutative=True)), Function('I')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Function('I')(Symbol('B', commutative=True)), Pow(exp(Mul(Integer(-1), Symbol('B', commutative=True))), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given n{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} and a{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})}, then obtain (\\frac{1}{f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} = (\\frac{d}{d f_{\\mathbf{p}}} a{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}}", "derivation": "n{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} and a{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and n{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} a{(f_{\\mathbf{p}})} and n^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} = (\\frac{d}{d f_{\\mathbf{p}}} a{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} and (\\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} = (\\frac{d}{d f_{\\mathbf{p}}} a{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}} and (\\frac{1}{f_{\\mathbf{p}}})^{f_{\\mathbf{p}}} = (\\frac{d}{d f_{\\mathbf{p}}} a{(f_{\\mathbf{p}})})^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('n')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Function('n')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Derivative(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Derivative(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Derivative(Function('a')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given C{(p)} = \\cos{(\\cos{(p)})}, then obtain \\sin{(p)} \\sin{(\\cos{(p)})} + 3 \\frac{d}{d p} C{(p)} = 2 \\sin{(p)} \\sin{(\\cos{(p)})} + 2 \\frac{d}{d p} C{(p)}", "derivation": "C{(p)} = \\cos{(\\cos{(p)})} and 2 C{(p)} = C{(p)} + \\cos{(\\cos{(p)})} and 3 C{(p)} + \\cos{(\\cos{(p)})} = 2 C{(p)} + 2 \\cos{(\\cos{(p)})} and \\frac{d}{d p} (3 C{(p)} + \\cos{(\\cos{(p)})}) = \\frac{d}{d p} (2 C{(p)} + 2 \\cos{(\\cos{(p)})}) and \\sin{(p)} \\sin{(\\cos{(p)})} + 3 \\frac{d}{d p} C{(p)} = 2 \\sin{(p)} \\sin{(\\cos{(p)})} + 2 \\frac{d}{d p} C{(p)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('p', commutative=True)), cos(cos(Symbol('p', commutative=True))))"], [["add", 1, "Function('C')(Symbol('p', commutative=True))"], "Equality(Mul(Integer(2), Function('C')(Symbol('p', commutative=True))), Add(Function('C')(Symbol('p', commutative=True)), cos(cos(Symbol('p', commutative=True)))))"], [["add", 2, "Add(Function('C')(Symbol('p', commutative=True)), cos(cos(Symbol('p', commutative=True))))"], "Equality(Add(Mul(Integer(3), Function('C')(Symbol('p', commutative=True))), cos(cos(Symbol('p', commutative=True)))), Add(Mul(Integer(2), Function('C')(Symbol('p', commutative=True))), Mul(Integer(2), cos(cos(Symbol('p', commutative=True))))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(3), Function('C')(Symbol('p', commutative=True))), cos(cos(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Function('C')(Symbol('p', commutative=True))), Mul(Integer(2), cos(cos(Symbol('p', commutative=True))))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(sin(Symbol('p', commutative=True)), sin(cos(Symbol('p', commutative=True)))), Mul(Integer(3), Derivative(Function('C')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Add(Mul(Integer(2), sin(Symbol('p', commutative=True)), sin(cos(Symbol('p', commutative=True)))), Mul(Integer(2), Derivative(Function('C')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\bar{\\h}{(H,\\tilde{g}^*)} = \\int (H + \\tilde{g}^*) dH and \\operatorname{A_{z}}{(H,\\tilde{g}^*)} = H + \\tilde{g}^*, then obtain \\frac{\\operatorname{A_{z}}{(H,\\tilde{g}^*)}}{\\int (H + \\tilde{g}^*) dH} - 1 = \\frac{H + \\tilde{g}^*}{\\int (H + \\tilde{g}^*) dH} - 1", "derivation": "\\bar{\\h}{(H,\\tilde{g}^*)} = \\int (H + \\tilde{g}^*) dH and \\operatorname{A_{z}}{(H,\\tilde{g}^*)} = H + \\tilde{g}^* and \\frac{\\operatorname{A_{z}}{(H,\\tilde{g}^*)}}{\\bar{\\h}{(H,\\tilde{g}^*)}} = \\frac{H + \\tilde{g}^*}{\\bar{\\h}{(H,\\tilde{g}^*)}} and \\frac{\\operatorname{A_{z}}{(H,\\tilde{g}^*)}}{\\bar{\\h}{(H,\\tilde{g}^*)}} - 1 = \\frac{H + \\tilde{g}^*}{\\bar{\\h}{(H,\\tilde{g}^*)}} - 1 and \\frac{\\operatorname{A_{z}}{(H,\\tilde{g}^*)}}{\\int (H + \\tilde{g}^*) dH} - 1 = \\frac{H + \\tilde{g}^*}{\\int (H + \\tilde{g}^*) dH} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["divide", 2, "Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Function('A_z')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Mul(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Function('A_z')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Integer(-1)), Add(Mul(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Function('A_z')(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Integer(-1)), Add(Mul(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Integer(-1)))"]]}, {"prompt": "Given \\mu{(G,Z)} = - G + Z, then obtain (\\int\\limits^{G + \\mu{(G,Z)}} \\mu{(G,Z)} dZ)^{G} = (\\int\\limits^{G + \\mu{(G,Z)}} (- G + Z) dZ)^{G}", "derivation": "\\mu{(G,Z)} = - G + Z and G + \\mu{(G,Z)} = Z and \\int \\mu{(G,Z)} dZ = \\int (- G + Z) dZ and (\\int \\mu{(G,Z)} dZ)^{G} = (\\int (- G + Z) dZ)^{G} and (\\int\\limits^{G + \\mu{(G,Z)}} \\mu{(G,Z)} dZ)^{G} = (\\int\\limits^{G + \\mu{(G,Z)}} (- G + Z) dZ)^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('Z', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('G', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Add(Symbol('G', commutative=True), Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True))))), Symbol('G', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Add(Symbol('G', commutative=True), Function('\\\\mu')(Symbol('G', commutative=True), Symbol('Z', commutative=True))))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given Z{(\\phi,E_{x})} = \\phi + \\log{(E_{x})}, then obtain \\frac{\\partial}{\\partial \\phi} \\frac{\\int Z{(\\phi,E_{x})} d\\phi}{\\phi} - \\int (\\phi + \\log{(E_{x})}) d\\phi = \\frac{\\partial}{\\partial \\phi} \\frac{\\int (\\phi + \\log{(E_{x})}) d\\phi}{\\phi} - \\int (\\phi + \\log{(E_{x})}) d\\phi", "derivation": "Z{(\\phi,E_{x})} = \\phi + \\log{(E_{x})} and \\int Z{(\\phi,E_{x})} d\\phi = \\int (\\phi + \\log{(E_{x})}) d\\phi and \\frac{\\int Z{(\\phi,E_{x})} d\\phi}{\\phi} = \\frac{\\int (\\phi + \\log{(E_{x})}) d\\phi}{\\phi} and \\frac{\\partial}{\\partial \\phi} \\frac{\\int Z{(\\phi,E_{x})} d\\phi}{\\phi} = \\frac{\\partial}{\\partial \\phi} \\frac{\\int (\\phi + \\log{(E_{x})}) d\\phi}{\\phi} and \\frac{\\partial}{\\partial \\phi} \\frac{\\int Z{(\\phi,E_{x})} d\\phi}{\\phi} - \\int (\\phi + \\log{(E_{x})}) d\\phi = \\frac{\\partial}{\\partial \\phi} \\frac{\\int (\\phi + \\log{(E_{x})}) d\\phi}{\\phi} - \\int (\\phi + \\log{(E_{x})}) d\\phi", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\phi', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["divide", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('\\\\phi', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('\\\\phi', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 4, "Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('\\\\phi', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))), Add(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given r{(\\psi)} = \\psi, then derive \\frac{d}{d \\psi} r{(\\psi)} = 1, then obtain \\frac{d}{d \\psi} (\\frac{d}{d \\psi} r{(\\psi)} + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi}) = \\frac{d}{d \\psi} (1 + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi})", "derivation": "r{(\\psi)} = \\psi and \\frac{d}{d \\psi} r{(\\psi)} = \\frac{d}{d \\psi} \\psi and \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi} = 1 and \\frac{d}{d \\psi} r{(\\psi)} = 1 and \\frac{d}{d \\psi} r{(\\psi)} + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi} = 1 + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi} and \\frac{d}{d \\psi} (\\frac{d}{d \\psi} r{(\\psi)} + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi}) = \\frac{d}{d \\psi} (1 + \\frac{\\frac{d}{d \\psi} r{(\\psi)}}{\\frac{d}{d \\psi} \\psi})", "srepr_derivation": [["renaming_premise", "Equality(Function('r')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], "Equality(Add(Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Add(Integer(1), Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))))"], [["differentiate", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(m_{s})} = e^{m_{s}}, then derive \\int \\lambda{(m_{s})} dm_{s} = z^{*} + e^{m_{s}}, then obtain \\frac{z^{*} + \\lambda{(m_{s})}}{m_{s}} = \\frac{z^{*} + e^{m_{s}}}{m_{s}}", "derivation": "\\lambda{(m_{s})} = e^{m_{s}} and \\int \\lambda{(m_{s})} dm_{s} = \\int e^{m_{s}} dm_{s} and \\int \\lambda{(m_{s})} dm_{s} = z^{*} + e^{m_{s}} and \\int \\lambda{(m_{s})} dm_{s} = z^{*} + \\lambda{(m_{s})} and \\frac{\\int \\lambda{(m_{s})} dm_{s}}{m_{s}} = \\frac{z^{*} + e^{m_{s}}}{m_{s}} and \\frac{z^{*} + \\lambda{(m_{s})}}{m_{s}} = \\frac{z^{*} + e^{m_{s}}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('z^*', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\lambda')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('z^*', commutative=True), Function('\\\\lambda')(Symbol('m_s', commutative=True))))"], [["divide", 3, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), exp(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), Function('\\\\lambda')(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), exp(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(l)} = \\sin{(l)}, then obtain 1 = \\frac{(\\int \\operatorname{E_{x}}{(l)} \\sin{(l)} dl)^{2}}{(\\int \\operatorname{E_{x}}^{2}{(l)} dl)^{2}}", "derivation": "\\operatorname{E_{x}}{(l)} = \\sin{(l)} and \\operatorname{E_{x}}^{2}{(l)} = \\operatorname{E_{x}}{(l)} \\sin{(l)} and \\int \\operatorname{E_{x}}^{2}{(l)} dl = \\int \\operatorname{E_{x}}{(l)} \\sin{(l)} dl and (\\int \\operatorname{E_{x}}^{2}{(l)} dl)^{2} = (\\int \\operatorname{E_{x}}{(l)} \\sin{(l)} dl)^{2} and 1 = \\frac{(\\int \\operatorname{E_{x}}{(l)} \\sin{(l)} dl)^{2}}{(\\int \\operatorname{E_{x}}^{2}{(l)} dl)^{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('l', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('l', commutative=True)), Integer(2)), Mul(Function('E_x')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Function('E_x')(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Function('E_x')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Pow(Function('E_x')(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integer(2)), Pow(Integral(Mul(Function('E_x')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(2)))"], [["divide", 4, "Pow(Integral(Pow(Function('E_x')(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integer(2))"], "Equality(Integer(1), Mul(Pow(Integral(Mul(Function('E_x')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(2)), Pow(Integral(Pow(Function('E_x')(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} = \\frac{M_{E} - g}{v_{z}}, then obtain - g = - g (0^{g})^{g}", "derivation": "\\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} = \\frac{M_{E} - g}{v_{z}} and \\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} - \\frac{M_{E} - g}{v_{z}} = 0 and (\\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} - \\frac{M_{E} - g}{v_{z}})^{g} = 0^{g} and (\\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} + \\frac{- M_{E} + g}{v_{z}})^{g} = 0^{g} and 1 = (\\operatorname{f^{\\prime}}{(g,M_{E},v_{z})} - \\frac{M_{E} - g}{v_{z}})^{g} and 1 = 0^{g} and 1 = (0^{g})^{g} and - g = - g (0^{g})^{g}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('M_E', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))"], [["minus", 1, "Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('M_E', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('M_E', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))), Symbol('g', commutative=True)), Pow(Integer(0), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Add(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('M_E', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Integer(0), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Add(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('M_E', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(1), Pow(Integer(0), Symbol('g', commutative=True)))"], [["power", 6, "Symbol('g', commutative=True)"], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["times", 7, "Mul(Integer(-1), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True), Pow(Pow(Integer(0), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then obtain 0 = - \\int \\frac{d}{d \\mathbb{I}} \\operatorname{v_{t}}{(\\mathbb{I})} d\\mathbb{I} + \\int \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} d\\mathbb{I}", "derivation": "\\operatorname{v_{t}}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} \\operatorname{v_{t}}{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} and \\int \\frac{d}{d \\mathbb{I}} \\operatorname{v_{t}}{(\\mathbb{I})} d\\mathbb{I} = \\int \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} d\\mathbb{I} and 0 = - \\int \\frac{d}{d \\mathbb{I}} \\operatorname{v_{t}}{(\\mathbb{I})} d\\mathbb{I} + \\int \\frac{d}{d \\mathbb{I}} \\sin{(\\mathbb{I})} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Derivative(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 3, "Integral(Derivative(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Derivative(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Integral(Derivative(sin(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\hat{p}_0,A_{z})} = - A_{z} + \\hat{p}_0, then derive \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{g}{(\\hat{p}_0,A_{z})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\hat{p}_0\\partial A_{z}} \\mathbf{g}{(\\hat{p}_0,A_{z})} = 0", "derivation": "\\mathbf{g}{(\\hat{p}_0,A_{z})} = - A_{z} + \\hat{p}_0 and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{g}{(\\hat{p}_0,A_{z})} = \\frac{\\partial}{\\partial \\hat{p}_0} (- A_{z} + \\hat{p}_0) and \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{g}{(\\hat{p}_0,A_{z})} = 1 and \\frac{\\partial^{2}}{\\partial A_{z}\\partial \\hat{p}_0} \\mathbf{g}{(\\hat{p}_0,A_{z})} = \\frac{d}{d A_{z}} 1 and \\frac{\\partial^{2}}{\\partial \\hat{p}_0\\partial A_{z}} \\mathbf{g}{(\\hat{p}_0,A_{z})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given U{(v)} = \\sin{(\\sin{(v)})} and \\operatorname{v_{t}}{(v)} = \\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})}, then obtain \\operatorname{v_{t}}{(v)} \\frac{d}{d v} \\sin{(\\sin{(v)})} = (\\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})}) \\frac{d}{d v} \\sin{(\\sin{(v)})}", "derivation": "U{(v)} = \\sin{(\\sin{(v)})} and \\frac{d}{d v} U{(v)} = \\frac{d}{d v} \\sin{(\\sin{(v)})} and 2 \\frac{d}{d v} U{(v)} = \\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})} and 2 \\frac{d}{d v} U{(v)} \\frac{d}{d v} \\sin{(\\sin{(v)})} = (\\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})}) \\frac{d}{d v} \\sin{(\\sin{(v)})} and \\operatorname{v_{t}}{(v)} = \\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})} and \\operatorname{v_{t}}{(v)} = 2 \\frac{d}{d v} U{(v)} and \\operatorname{v_{t}}{(v)} \\frac{d}{d v} \\sin{(\\sin{(v)})} = (\\frac{d}{d v} U{(v)} + \\frac{d}{d v} \\sin{(\\sin{(v)})}) \\frac{d}{d v} \\sin{(\\sin{(v)})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('v', commutative=True)), sin(sin(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Add(Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('v', commutative=True)), Add(Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('v_t')(Symbol('v', commutative=True)), Mul(Integer(2), Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Function('v_t')(Symbol('v', commutative=True)), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Add(Derivative(Function('U')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(\\varphi^*,F_{H})} = \\frac{F_{H}}{\\varphi^*}, then obtain \\nabla{(\\varphi^*,F_{H})} - \\sin{(\\sin{(\\frac{F_{H}}{\\varphi^*})})} - 1 = \\frac{F_{H}}{\\varphi^*} - \\sin{(\\sin{(\\frac{F_{H}}{\\varphi^*})})} - 1", "derivation": "\\nabla{(\\varphi^*,F_{H})} = \\frac{F_{H}}{\\varphi^*} and \\sin{(\\nabla{(\\varphi^*,F_{H})})} = \\sin{(\\frac{F_{H}}{\\varphi^*})} and \\nabla{(\\varphi^*,F_{H})} - \\sin{(\\sin{(\\nabla{(\\varphi^*,F_{H})})})} = \\frac{F_{H}}{\\varphi^*} - \\sin{(\\sin{(\\nabla{(\\varphi^*,F_{H})})})} and \\nabla{(\\varphi^*,F_{H})} - \\sin{(\\sin{(\\nabla{(\\varphi^*,F_{H})})})} - 1 = \\frac{F_{H}}{\\varphi^*} - \\sin{(\\sin{(\\nabla{(\\varphi^*,F_{H})})})} - 1 and \\nabla{(\\varphi^*,F_{H})} - \\sin{(\\sin{(\\frac{F_{H}}{\\varphi^*})})} - 1 = \\frac{F_{H}}{\\varphi^*} - \\sin{(\\sin{(\\frac{F_{H}}{\\varphi^*})})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))"], [["sin", 1], "Equality(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True))), sin(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))"], [["minus", 1, "sin(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True))))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)))))), Add(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), sin(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)))))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True))))), Integer(-1)), Add(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), sin(sin(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True))))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(sin(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))), Integer(-1)), Add(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Mul(Integer(-1), sin(sin(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)))))), Integer(-1)))"]]}, {"prompt": "Given y{(S,L_{\\varepsilon})} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} and x{(S,L_{\\varepsilon})} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon}, then obtain x{(S,L_{\\varepsilon})} - (\\int S^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} - (\\int S^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}}", "derivation": "y{(S,L_{\\varepsilon})} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} and y^{L_{\\varepsilon}}{(S,L_{\\varepsilon})} = (\\int S^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} and x{(S,L_{\\varepsilon})} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} and x{(S,L_{\\varepsilon})} - y^{L_{\\varepsilon}}{(S,L_{\\varepsilon})} = - y^{L_{\\varepsilon}}{(S,L_{\\varepsilon})} + \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} and x{(S,L_{\\varepsilon})} - (\\int S^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} = \\int S^{L_{\\varepsilon}} dL_{\\varepsilon} - (\\int S^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('y')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Pow(Function('y')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('x')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('y')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('x')(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(Symbol('S', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given m{(J)} = \\int \\sin{(J)} dJ, then derive \\frac{d}{d J} - m{(J)} = \\frac{\\partial}{\\partial J} (- \\mathbf{F} + \\cos{(J)}), then obtain (- \\frac{\\frac{d}{d J} m{(J)}}{\\cos{(J)}})^{\\mathbf{F}} = (- \\frac{\\sin{(J)}}{\\cos{(J)}})^{\\mathbf{F}}", "derivation": "m{(J)} = \\int \\sin{(J)} dJ and - m{(J)} = - \\int \\sin{(J)} dJ and \\frac{d}{d J} - m{(J)} = \\frac{d}{d J} - \\int \\sin{(J)} dJ and \\frac{d}{d J} - m{(J)} = \\frac{\\partial}{\\partial J} (- \\mathbf{F} + \\cos{(J)}) and \\frac{\\frac{d}{d J} - m{(J)}}{\\cos{(J)}} = \\frac{\\frac{\\partial}{\\partial J} (- \\mathbf{F} + \\cos{(J)})}{\\cos{(J)}} and (\\frac{\\frac{d}{d J} - m{(J)}}{\\cos{(J)}})^{\\mathbf{F}} = (\\frac{\\frac{\\partial}{\\partial J} (- \\mathbf{F} + \\cos{(J)})}{\\cos{(J)}})^{\\mathbf{F}} and (- \\frac{\\frac{d}{d J} m{(J)}}{\\cos{(J)}})^{\\mathbf{F}} = (- \\frac{\\sin{(J)}}{\\cos{(J)}})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('J', commutative=True)), Integral(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('m')(Symbol('J', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('m')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Mul(Integer(-1), Function('m')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["divide", 4, "cos(Symbol('J', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('J', commutative=True)), Integer(-1)), Derivative(Mul(Integer(-1), Function('m')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('J', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('J', commutative=True)), Integer(-1)), Derivative(Mul(Integer(-1), Function('m')(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(cos(Symbol('J', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Mul(Integer(-1), Pow(cos(Symbol('J', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True)), Pow(cos(Symbol('J', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given r{(\\hat{H}_l,l)} = - l + \\cos{(\\hat{H}_l)}, then derive \\frac{\\partial}{\\partial \\hat{H}_l} r{(\\hat{H}_l,l)} = - \\sin{(\\hat{H}_l)}, then obtain \\frac{\\partial}{\\partial \\hat{H}_l} (- l + \\cos{(\\hat{H}_l)}) = - \\sin{(\\hat{H}_l)}", "derivation": "r{(\\hat{H}_l,l)} = - l + \\cos{(\\hat{H}_l)} and \\frac{\\partial}{\\partial \\hat{H}_l} r{(\\hat{H}_l,l)} = \\frac{\\partial}{\\partial \\hat{H}_l} (- l + \\cos{(\\hat{H}_l)}) and \\frac{\\partial}{\\partial \\hat{H}_l} r{(\\hat{H}_l,l)} = - \\sin{(\\hat{H}_l)} and \\frac{\\partial}{\\partial \\hat{H}_l} (- l + \\cos{(\\hat{H}_l)}) = - \\sin{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(f,T)} = T + \\sin{(f)} and p{(f,T)} = \\log{(\\frac{\\operatorname{F_{g}}{(f,T)}}{T + \\sin{(f)}})}, then derive \\frac{\\partial}{\\partial T} p{(f,T)} = 0, then obtain \\log{(\\tilde{g}^{r})} + \\frac{\\partial}{\\partial T} p{(f,T)} + \\frac{1}{T + \\sin{(f)}} = \\log{(\\tilde{g}^{r})} + \\frac{1}{T + \\sin{(f)}}", "derivation": "\\operatorname{F_{g}}{(f,T)} = T + \\sin{(f)} and p{(f,T)} = \\log{(\\frac{\\operatorname{F_{g}}{(f,T)}}{T + \\sin{(f)}})} and p{(f,T)} = 0 and \\frac{\\partial}{\\partial T} p{(f,T)} = \\frac{d}{d T} 0 and \\frac{\\partial}{\\partial T} p{(f,T)} = 0 and \\log{(\\tilde{g}^{r})} + \\frac{\\partial}{\\partial T} p{(f,T)} + \\frac{1}{\\operatorname{F_{g}}{(f,T)}} = \\log{(\\tilde{g}^{r})} + \\frac{1}{\\operatorname{F_{g}}{(f,T)}} and \\log{(\\tilde{g}^{r})} + \\frac{\\partial}{\\partial T} p{(f,T)} + \\frac{1}{T + \\sin{(f)}} = \\log{(\\tilde{g}^{r})} + \\frac{1}{T + \\sin{(f)}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), sin(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), log(Mul(Pow(Add(Symbol('T', commutative=True), sin(Symbol('f', commutative=True))), Integer(-1)), Function('F_g')(Symbol('f', commutative=True), Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(0))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(0))"], [["add", 5, "Add(log(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('r', commutative=True))), Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(-1)))"], "Equality(Add(log(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('r', commutative=True))), Derivative(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Add(log(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('r', commutative=True))), Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(log(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('r', commutative=True))), Derivative(Function('p')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Add(Symbol('T', commutative=True), sin(Symbol('f', commutative=True))), Integer(-1))), Add(log(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('r', commutative=True))), Pow(Add(Symbol('T', commutative=True), sin(Symbol('f', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\Psi_{nl})} = \\log{(e^{\\Psi_{nl}})} and n{(\\Psi_{nl})} = \\frac{\\eta^{\\prime}{(\\Psi_{nl})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}}, then obtain n{(\\Psi_{nl})} = \\frac{\\log{(e^{\\Psi_{nl}})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}}", "derivation": "\\eta^{\\prime}{(\\Psi_{nl})} = \\log{(e^{\\Psi_{nl}})} and \\frac{\\eta^{\\prime}{(\\Psi_{nl})}}{\\Psi_{nl}} = \\frac{\\log{(e^{\\Psi_{nl}})}}{\\Psi_{nl}} and \\frac{\\eta^{\\prime}{(\\Psi_{nl})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}} = \\frac{\\log{(e^{\\Psi_{nl}})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}} and n{(\\Psi_{nl})} = \\frac{\\eta^{\\prime}{(\\Psi_{nl})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}} and n{(\\Psi_{nl})} = \\frac{\\log{(e^{\\Psi_{nl}})}}{\\Psi_{nl}} + \\frac{1}{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True)), log(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["add", 2, "Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('n')(Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), log(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C_{1})} = \\sin{(C_{1})}, then derive \\int \\operatorname{F_{x}}{(C_{1})} dC_{1} = \\theta - \\cos{(C_{1})}, then derive \\mathbf{J}_M - \\cos{(C_{1})} = \\theta - \\cos{(C_{1})}, then obtain (\\int \\operatorname{F_{x}}{(C_{1})} dC_{1})^{C_{1}} = (\\mathbf{J}_M - \\cos{(C_{1})})^{C_{1}}", "derivation": "\\operatorname{F_{x}}{(C_{1})} = \\sin{(C_{1})} and \\int \\operatorname{F_{x}}{(C_{1})} dC_{1} = \\int \\sin{(C_{1})} dC_{1} and \\int \\operatorname{F_{x}}{(C_{1})} dC_{1} = \\theta - \\cos{(C_{1})} and \\int \\sin{(C_{1})} dC_{1} = \\theta - \\cos{(C_{1})} and (\\int \\operatorname{F_{x}}{(C_{1})} dC_{1})^{C_{1}} = (\\theta - \\cos{(C_{1})})^{C_{1}} and \\mathbf{J}_M - \\cos{(C_{1})} = \\theta - \\cos{(C_{1})} and (\\int \\operatorname{F_{x}}{(C_{1})} dC_{1})^{C_{1}} = (\\mathbf{J}_M - \\cos{(C_{1})})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Integral(Function('F_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Integral(Function('F_x')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(n_{2})} = \\log{(n_{2})}, then obtain 0 = n_{2} \\log{(n_{2})} - n_{2} + t_{2} - \\int \\mathbf{H}{(n_{2})} dn_{2}", "derivation": "\\mathbf{H}{(n_{2})} = \\log{(n_{2})} and \\int \\mathbf{H}{(n_{2})} dn_{2} = \\int \\log{(n_{2})} dn_{2} and 0 = - \\int \\mathbf{H}{(n_{2})} dn_{2} + \\int \\log{(n_{2})} dn_{2} and 0 = n_{2} \\log{(n_{2})} - n_{2} + t_{2} - \\int \\mathbf{H}{(n_{2})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Integral(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Mul(Symbol('n_2', commutative=True), log(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('t_2', commutative=True), Mul(Integer(-1), Integral(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(\\psi^*,\\mu)} = - \\mu + \\psi^*, then obtain - \\mu + (- \\mu + \\psi^*)^{2} \\mu_{0}^{2}{(\\psi^*,\\mu)} = - \\mu + (- \\mu + \\psi^*)^{3} \\mu_{0}{(\\psi^*,\\mu)}", "derivation": "\\mu_{0}{(\\psi^*,\\mu)} = - \\mu + \\psi^* and \\mu_{0}^{2}{(\\psi^*,\\mu)} = (- \\mu + \\psi^*) \\mu_{0}{(\\psi^*,\\mu)} and \\mu_{0}^{4}{(\\psi^*,\\mu)} = (- \\mu + \\psi^*)^{2} \\mu_{0}^{2}{(\\psi^*,\\mu)} and (- \\mu + \\psi^*)^{2} \\mu_{0}^{2}{(\\psi^*,\\mu)} = (- \\mu + \\psi^*)^{3} \\mu_{0}{(\\psi^*,\\mu)} and - \\mu + (- \\mu + \\psi^*)^{2} \\mu_{0}^{2}{(\\psi^*,\\mu)} = - \\mu + (- \\mu + \\psi^*)^{3} \\mu_{0}{(\\psi^*,\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(4)), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Pow(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Pow(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integer(3)), Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["minus", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Pow(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Integer(3)), Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\Psi)} = \\cos{(\\Psi)} and z{(C_{2})} = e^{C_{2}}, then obtain \\frac{z{(C_{2})} - \\cos{(\\Psi)}}{\\cos{(\\Psi)}} = \\frac{e^{C_{2}} - \\cos{(\\Psi)}}{\\cos{(\\Psi)}}", "derivation": "\\operatorname{A_{1}}{(\\Psi)} = \\cos{(\\Psi)} and z{(C_{2})} = e^{C_{2}} and - \\operatorname{A_{1}}{(\\Psi)} + z{(C_{2})} = - \\operatorname{A_{1}}{(\\Psi)} + e^{C_{2}} and - \\frac{- \\operatorname{A_{1}}{(\\Psi)} + z{(C_{2})}}{\\operatorname{A_{1}}{(\\Psi)}} = - \\frac{- \\operatorname{A_{1}}{(\\Psi)} + e^{C_{2}}}{\\operatorname{A_{1}}{(\\Psi)}} and \\frac{- \\operatorname{A_{1}}{(\\Psi)} + z{(C_{2})}}{\\operatorname{A_{1}}{(\\Psi)}} = \\frac{- \\operatorname{A_{1}}{(\\Psi)} + e^{C_{2}}}{\\operatorname{A_{1}}{(\\Psi)}} and \\frac{z{(C_{2})} - \\cos{(\\Psi)}}{\\cos{(\\Psi)}} = \\frac{e^{C_{2}} - \\cos{(\\Psi)}}{\\cos{(\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], ["get_premise", "Equality(Function('z')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["minus", 2, "Function('A_1')(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), Function('z')(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('C_2', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), Function('z')(Symbol('C_2', commutative=True))), Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('C_2', commutative=True))), Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), Function('z')(Symbol('C_2', commutative=True))), Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('C_2', commutative=True))), Pow(Function('A_1')(Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Function('z')(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(Add(exp(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Pow(cos(Symbol('\\\\Psi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\bar{\\h}{(i)} = e^{i}, then obtain \\cos^{i}{(2 e^{i} + \\log{(\\bar{\\h}{(i)})})} = \\cos^{i}{(2 e^{i} + \\log{(e^{i})})}", "derivation": "\\bar{\\h}{(i)} = e^{i} and \\bar{\\h}{(i)} + e^{i} = 2 e^{i} and \\log{(\\bar{\\h}{(i)})} = \\log{(e^{i})} and \\bar{\\h}{(i)} + e^{i} + \\log{(\\bar{\\h}{(i)})} = \\bar{\\h}{(i)} + e^{i} + \\log{(e^{i})} and \\cos{(\\bar{\\h}{(i)} + e^{i} + \\log{(\\bar{\\h}{(i)})})} = \\cos{(\\bar{\\h}{(i)} + e^{i} + \\log{(e^{i})})} and \\cos{(2 e^{i} + \\log{(\\bar{\\h}{(i)})})} = \\cos{(2 e^{i} + \\log{(e^{i})})} and \\cos^{i}{(2 e^{i} + \\log{(\\bar{\\h}{(i)})})} = \\cos^{i}{(2 e^{i} + \\log{(e^{i})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["add", 1, "exp(Symbol('i', commutative=True))"], "Equality(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True))), Mul(Integer(2), exp(Symbol('i', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\hbar')(Symbol('i', commutative=True))), log(exp(Symbol('i', commutative=True))))"], [["add", 3, "Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)), log(Function('\\\\hbar')(Symbol('i', commutative=True)))), Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)), log(exp(Symbol('i', commutative=True)))))"], [["cos", 4], "Equality(cos(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)), log(Function('\\\\hbar')(Symbol('i', commutative=True))))), cos(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)), log(exp(Symbol('i', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(cos(Add(Mul(Integer(2), exp(Symbol('i', commutative=True))), log(Function('\\\\hbar')(Symbol('i', commutative=True))))), cos(Add(Mul(Integer(2), exp(Symbol('i', commutative=True))), log(exp(Symbol('i', commutative=True))))))"], [["power", 6, "Symbol('i', commutative=True)"], "Equality(Pow(cos(Add(Mul(Integer(2), exp(Symbol('i', commutative=True))), log(Function('\\\\hbar')(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), Pow(cos(Add(Mul(Integer(2), exp(Symbol('i', commutative=True))), log(exp(Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\hbar)} = e^{\\hbar}, then obtain \\hat{x}{(\\hbar)} + \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} d\\hbar = \\hat{x}{(\\hbar)} + \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar", "derivation": "\\hat{x}{(\\hbar)} = e^{\\hbar} and \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} = \\frac{d}{d \\hbar} e^{\\hbar} and \\int \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} d\\hbar = \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar and \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} d\\hbar = \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar and \\hat{x}{(\\hbar)} + \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} d\\hbar = \\hat{x}{(\\hbar)} + \\frac{d}{d \\hbar} \\hat{x}{(\\hbar)} + \\int \\frac{d}{d \\hbar} e^{\\hbar} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 3, "Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["add", 4, "Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integral(Derivative(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\dot{\\mathbf{r}},\\mathbf{J}_M)} = - \\dot{\\mathbf{r}} + \\mathbf{J}_M, then derive \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\theta_{2}{(\\dot{\\mathbf{r}},\\mathbf{J}_M)} = -1, then obtain \\frac{1}{\\mathbf{J}_M} = - \\frac{\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\mathbf{J}_M)}{\\mathbf{J}_M}", "derivation": "\\theta_{2}{(\\dot{\\mathbf{r}},\\mathbf{J}_M)} = - \\dot{\\mathbf{r}} + \\mathbf{J}_M and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\theta_{2}{(\\dot{\\mathbf{r}},\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\mathbf{J}_M) and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\theta_{2}{(\\dot{\\mathbf{r}},\\mathbf{J}_M)} = -1 and -1 = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\mathbf{J}_M) and \\frac{1}{\\mathbf{J}_M} = - \\frac{\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (- \\dot{\\mathbf{r}} + \\mathbf{J}_M)}{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(C_{d},\\theta_2)} = e^{C_{d} + \\theta_2} and \\mathbf{B}{(C_{d},\\theta_2)} = e^{C_{d} + \\theta_2}, then obtain \\frac{\\int \\operatorname{f_{E}}{(C_{d},\\theta_2)} dC_{d}}{C_{d}} = \\frac{\\int \\mathbf{B}{(C_{d},\\theta_2)} dC_{d}}{C_{d}}", "derivation": "\\operatorname{f_{E}}{(C_{d},\\theta_2)} = e^{C_{d} + \\theta_2} and \\mathbf{B}{(C_{d},\\theta_2)} = e^{C_{d} + \\theta_2} and \\int \\mathbf{B}{(C_{d},\\theta_2)} dC_{d} = \\int e^{C_{d} + \\theta_2} dC_{d} and \\operatorname{f_{E}}{(C_{d},\\theta_2)} = \\mathbf{B}{(C_{d},\\theta_2)} and \\int \\operatorname{f_{E}}{(C_{d},\\theta_2)} dC_{d} = \\int e^{C_{d} + \\theta_2} dC_{d} and \\int \\operatorname{f_{E}}{(C_{d},\\theta_2)} dC_{d} = \\int \\mathbf{B}{(C_{d},\\theta_2)} dC_{d} and \\frac{\\int \\operatorname{f_{E}}{(C_{d},\\theta_2)} dC_{d}}{C_{d}} = \\frac{\\int \\mathbf{B}{(C_{d},\\theta_2)} dC_{d}}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(exp(Add(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_E')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('f_E')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(exp(Add(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Function('f_E')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 6, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Function('f_E')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{B}')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and b{(\\mathbf{D})} = \\frac{1}{\\sin{(\\mathbf{D})}}, then obtain \\frac{- \\mathbf{D} + \\frac{\\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}}}{\\sin{(\\mathbf{D})}} = \\frac{2 - \\mathbf{D}}{\\sin{(\\mathbf{D})}}", "derivation": "\\hat{H}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})} = 2 \\sin{(\\mathbf{D})} and \\frac{\\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}} = 2 and - \\mathbf{D} + \\frac{\\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}} = 2 - \\mathbf{D} and b{(\\mathbf{D})} = \\frac{1}{\\sin{(\\mathbf{D})}} and (- \\mathbf{D} + \\frac{\\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}}) b{(\\mathbf{D})} = (2 - \\mathbf{D}) b{(\\mathbf{D})} and \\frac{- \\mathbf{D} + \\frac{\\hat{H}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}}}{\\sin{(\\mathbf{D})}} = \\frac{2 - \\mathbf{D}}{\\sin{(\\mathbf{D})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 2, "sin(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Integer(2))"], [["minus", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))), Add(Integer(2), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))"], [["times", 4, "Function('b')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))), Function('b')(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Add(Integer(2), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('b')(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Mul(Add(Integer(2), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given B{(\\mu_0,\\chi)} = \\chi^{\\mu_0}, then obtain \\frac{\\int B{(\\mu_0,\\chi)} d\\mu_0}{\\chi^{\\mu_0} + \\mu_0} = \\frac{\\int \\chi^{\\mu_0} d\\mu_0}{\\chi^{\\mu_0} + \\mu_0}", "derivation": "B{(\\mu_0,\\chi)} = \\chi^{\\mu_0} and \\int B{(\\mu_0,\\chi)} d\\mu_0 = \\int \\chi^{\\mu_0} d\\mu_0 and \\mu_0 + B{(\\mu_0,\\chi)} = \\chi^{\\mu_0} + \\mu_0 and \\frac{\\int B{(\\mu_0,\\chi)} d\\mu_0}{\\mu_0 + B{(\\mu_0,\\chi)}} = \\frac{\\int \\chi^{\\mu_0} d\\mu_0}{\\mu_0 + B{(\\mu_0,\\chi)}} and \\frac{\\int B{(\\mu_0,\\chi)} d\\mu_0}{\\chi^{\\mu_0} + \\mu_0} = \\frac{\\int \\chi^{\\mu_0} d\\mu_0}{\\chi^{\\mu_0} + \\mu_0}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 2, "Add(Symbol('\\\\mu_0', commutative=True), Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Function('B')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given k{(\\mu_0,\\chi)} = \\chi + e^{\\mu_0}, then obtain \\sin{(k^{\\chi}{(\\mu_0,\\chi)} e^{- \\mu_0} - 1)} = \\sin{((\\chi + e^{\\mu_0})^{\\chi} e^{- \\mu_0} - 1)}", "derivation": "k{(\\mu_0,\\chi)} = \\chi + e^{\\mu_0} and k^{\\chi}{(\\mu_0,\\chi)} = (\\chi + e^{\\mu_0})^{\\chi} and k^{\\chi}{(\\mu_0,\\chi)} e^{- \\mu_0} = (\\chi + e^{\\mu_0})^{\\chi} e^{- \\mu_0} and k^{\\chi}{(\\mu_0,\\chi)} e^{- \\mu_0} - 1 = (\\chi + e^{\\mu_0})^{\\chi} e^{- \\mu_0} - 1 and \\sin{(k^{\\chi}{(\\mu_0,\\chi)} e^{- \\mu_0} - 1)} = \\sin{((\\chi + e^{\\mu_0})^{\\chi} e^{- \\mu_0} - 1)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["divide", 2, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Function('k')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 3, 1], "Equality(Add(Mul(Pow(Function('k')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(-1)), Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(-1)))"], [["sin", 4], "Equality(sin(Add(Mul(Pow(Function('k')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(-1))), sin(Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\chi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\Psi_{\\lambda},F_{N})} = F_{N} + \\cos{(\\Psi_{\\lambda})} and \\mathbf{g}{(z,A_{y},p)} = z (- A_{y} + p), then obtain \\frac{\\mathbf{g}{(z,A_{y},p)}}{F_{N} + \\cos{(\\Psi_{\\lambda})}} = \\frac{z (- A_{y} + p)}{F_{N} + \\cos{(\\Psi_{\\lambda})}}", "derivation": "\\operatorname{C_{d}}{(\\Psi_{\\lambda},F_{N})} = F_{N} + \\cos{(\\Psi_{\\lambda})} and \\mathbf{g}{(z,A_{y},p)} = z (- A_{y} + p) and \\frac{\\mathbf{g}{(z,A_{y},p)}}{\\operatorname{C_{d}}{(\\Psi_{\\lambda},F_{N})}} = \\frac{z (- A_{y} + p)}{\\operatorname{C_{d}}{(\\Psi_{\\lambda},F_{N})}} and \\frac{\\mathbf{g}{(z,A_{y},p)}}{F_{N} + \\cos{(\\Psi_{\\lambda})}} = \\frac{z (- A_{y} + p)}{F_{N} + \\cos{(\\Psi_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A_y', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('p', commutative=True))))"], [["divide", 2, "Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A_y', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('p', commutative=True)), Pow(Function('C_d')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A_y', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Symbol('F_N', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given n{(\\mu_0)} = \\cos{(e^{\\mu_0})}, then obtain n{(\\mu_0)} + e^{\\mu_0} \\sin{(e^{\\mu_0})} - e^{\\mu_0} = e^{\\mu_0} \\sin{(e^{\\mu_0})} - e^{\\mu_0} + \\cos{(e^{\\mu_0})}", "derivation": "n{(\\mu_0)} = \\cos{(e^{\\mu_0})} and n{(\\mu_0)} - e^{\\mu_0} = - e^{\\mu_0} + \\cos{(e^{\\mu_0})} and \\frac{d}{d \\mu_0} n{(\\mu_0)} = \\frac{d}{d \\mu_0} \\cos{(e^{\\mu_0})} and n{(\\mu_0)} - e^{\\mu_0} - \\frac{d}{d \\mu_0} n{(\\mu_0)} = - e^{\\mu_0} + \\cos{(e^{\\mu_0})} - \\frac{d}{d \\mu_0} n{(\\mu_0)} and n{(\\mu_0)} - e^{\\mu_0} - \\frac{d}{d \\mu_0} \\cos{(e^{\\mu_0})} = - e^{\\mu_0} + \\cos{(e^{\\mu_0})} - \\frac{d}{d \\mu_0} \\cos{(e^{\\mu_0})} and n{(\\mu_0)} + e^{\\mu_0} \\sin{(e^{\\mu_0})} - e^{\\mu_0} = e^{\\mu_0} \\sin{(e^{\\mu_0})} - e^{\\mu_0} + \\cos{(e^{\\mu_0})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mu_0', commutative=True)), cos(exp(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), cos(exp(Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('n')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Derivative(Function('n')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), cos(exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Derivative(Function('n')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Derivative(cos(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), cos(exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Derivative(cos(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(exp(Symbol('\\\\mu_0', commutative=True)), sin(exp(Symbol('\\\\mu_0', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(exp(Symbol('\\\\mu_0', commutative=True)), sin(exp(Symbol('\\\\mu_0', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), cos(exp(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given s{(\\mathbf{J})} = \\cos{(\\cos{(\\mathbf{J})})}, then obtain \\frac{(\\mathbf{J} + s{(\\mathbf{J})})^{2 \\mathbf{J}}}{s^{2}{(\\mathbf{J})}} = \\frac{(\\mathbf{J} + s{(\\mathbf{J})})^{\\mathbf{J}} (\\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})})^{\\mathbf{J}}}{s^{2}{(\\mathbf{J})}}", "derivation": "s{(\\mathbf{J})} = \\cos{(\\cos{(\\mathbf{J})})} and \\mathbf{J} + s{(\\mathbf{J})} = \\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})} and (\\mathbf{J} + s{(\\mathbf{J})})^{\\mathbf{J}} = (\\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})})^{\\mathbf{J}} and \\frac{(\\mathbf{J} + s{(\\mathbf{J})})^{\\mathbf{J}}}{s{(\\mathbf{J})}} = \\frac{(\\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})})^{\\mathbf{J}}}{s{(\\mathbf{J})}} and \\frac{(\\mathbf{J} + s{(\\mathbf{J})})^{2 \\mathbf{J}}}{s^{2}{(\\mathbf{J})}} = \\frac{(\\mathbf{J} + s{(\\mathbf{J})})^{\\mathbf{J}} (\\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})})^{\\mathbf{J}}}{s^{2}{(\\mathbf{J})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), cos(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 3, "Function('s')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1))))"], [["times", 4, "Mul(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-2))), Mul(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('s')(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(W)} = \\cos{(W)}, then obtain (\\int \\operatorname{v_{z}}{(W)} dW)^{2} = (\\rho + \\sin{(W)}) \\int \\operatorname{v_{z}}{(W)} dW", "derivation": "\\operatorname{v_{z}}{(W)} = \\cos{(W)} and \\int \\operatorname{v_{z}}{(W)} dW = \\int \\cos{(W)} dW and (\\int \\operatorname{v_{z}}{(W)} dW)^{2} = (\\int \\operatorname{v_{z}}{(W)} dW) \\int \\cos{(W)} dW and (\\int \\operatorname{v_{z}}{(W)} dW)^{2} = (\\rho + \\sin{(W)}) \\int \\operatorname{v_{z}}{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["times", 2, "Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Pow(Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)), Mul(Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\rho', commutative=True), sin(Symbol('W', commutative=True))), Integral(Function('v_z')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(x^\\prime,\\theta_2)} = \\theta_2 x^\\prime, then obtain (\\operatorname{F_{x}}{(x^\\prime,\\theta_2)} + \\frac{\\operatorname{F_{x}}{(x^\\prime,\\theta_2)}}{x^\\prime}) \\int \\theta_2 dx^\\prime = (\\theta_2 + \\operatorname{F_{x}}{(x^\\prime,\\theta_2)}) \\int \\theta_2 dx^\\prime", "derivation": "\\operatorname{F_{x}}{(x^\\prime,\\theta_2)} = \\theta_2 x^\\prime and \\frac{\\operatorname{F_{x}}{(x^\\prime,\\theta_2)}}{x^\\prime} = \\theta_2 and \\operatorname{F_{x}}{(x^\\prime,\\theta_2)} + \\frac{\\operatorname{F_{x}}{(x^\\prime,\\theta_2)}}{x^\\prime} = \\theta_2 + \\operatorname{F_{x}}{(x^\\prime,\\theta_2)} and (\\operatorname{F_{x}}{(x^\\prime,\\theta_2)} + \\frac{\\operatorname{F_{x}}{(x^\\prime,\\theta_2)}}{x^\\prime}) \\int \\theta_2 dx^\\prime = (\\theta_2 + \\operatorname{F_{x}}{(x^\\prime,\\theta_2)}) \\int \\theta_2 dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], [["add", 2, "Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('\\\\theta_2', commutative=True), Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["times", 3, "Integral(Symbol('\\\\theta_2', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Add(Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Integral(Symbol('\\\\theta_2', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('F_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_2', commutative=True))), Integral(Symbol('\\\\theta_2', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given t{(f^{*})} = \\log{(\\sin{(f^{*})})}, then obtain t{(f^{*})} + 2 \\log{(\\sin{(f^{*})})} = 2 t{(f^{*})} + \\log{(\\sin{(f^{*})})}", "derivation": "t{(f^{*})} = \\log{(\\sin{(f^{*})})} and t{(f^{*})} + \\log{(\\sin{(f^{*})})} = 2 \\log{(\\sin{(f^{*})})} and t{(f^{*})} + 2 \\log{(\\sin{(f^{*})})} = 3 \\log{(\\sin{(f^{*})})} and 2 t{(f^{*})} + \\log{(\\sin{(f^{*})})} = 3 \\log{(\\sin{(f^{*})})} and t{(f^{*})} + 2 \\log{(\\sin{(f^{*})})} = 2 t{(f^{*})} + \\log{(\\sin{(f^{*})})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('f^*', commutative=True)), log(sin(Symbol('f^*', commutative=True))))"], [["add", 1, "log(sin(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('t')(Symbol('f^*', commutative=True)), log(sin(Symbol('f^*', commutative=True)))), Mul(Integer(2), log(sin(Symbol('f^*', commutative=True)))))"], [["add", 1, "Mul(Integer(2), log(sin(Symbol('f^*', commutative=True))))"], "Equality(Add(Function('t')(Symbol('f^*', commutative=True)), Mul(Integer(2), log(sin(Symbol('f^*', commutative=True))))), Mul(Integer(3), log(sin(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('t')(Symbol('f^*', commutative=True))), log(sin(Symbol('f^*', commutative=True)))), Mul(Integer(3), log(sin(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('t')(Symbol('f^*', commutative=True)), Mul(Integer(2), log(sin(Symbol('f^*', commutative=True))))), Add(Mul(Integer(2), Function('t')(Symbol('f^*', commutative=True))), log(sin(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(g)} = \\cos{(e^{g})}, then derive \\frac{d}{d g} \\operatorname{F_{c}}{(g)} = - e^{g} \\sin{(e^{g})}, then obtain \\int \\frac{d}{d g} \\cos{(e^{g})} dg = \\int \\frac{d}{d g} \\operatorname{F_{c}}{(g)} dg", "derivation": "\\operatorname{F_{c}}{(g)} = \\cos{(e^{g})} and \\frac{d}{d g} \\operatorname{F_{c}}{(g)} = \\frac{d}{d g} \\cos{(e^{g})} and \\frac{d}{d g} \\operatorname{F_{c}}{(g)} = - e^{g} \\sin{(e^{g})} and \\int \\frac{d}{d g} \\operatorname{F_{c}}{(g)} dg = \\int - e^{g} \\sin{(e^{g})} dg and \\int \\frac{d}{d g} \\cos{(e^{g})} dg = \\int - e^{g} \\sin{(e^{g})} dg and \\int \\frac{d}{d g} \\cos{(e^{g})} dg = \\int \\frac{d}{d g} \\operatorname{F_{c}}{(g)} dg", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('g', commutative=True)), cos(exp(Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Derivative(Function('F_c')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Derivative(cos(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))), Integral(Derivative(Function('F_c')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(y^{\\prime})} = e^{\\sin{(y^{\\prime})}} and c{(y^{\\prime})} = \\frac{1}{\\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime}}, then obtain \\frac{d}{d y^{\\prime}} \\frac{1}{\\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime}} = \\frac{d}{d y^{\\prime}} \\frac{1}{\\int e^{\\sin{(y^{\\prime})}} dy^{\\prime}}", "derivation": "\\mathbf{f}{(y^{\\prime})} = e^{\\sin{(y^{\\prime})}} and \\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime} = \\int e^{\\sin{(y^{\\prime})}} dy^{\\prime} and c{(y^{\\prime})} = \\frac{1}{\\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime}} and c{(y^{\\prime})} = \\frac{1}{\\int e^{\\sin{(y^{\\prime})}} dy^{\\prime}} and \\frac{1}{\\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime}} = \\frac{1}{\\int e^{\\sin{(y^{\\prime})}} dy^{\\prime}} and \\frac{d}{d y^{\\prime}} \\frac{1}{\\int \\mathbf{f}{(y^{\\prime})} dy^{\\prime}} = \\frac{d}{d y^{\\prime}} \\frac{1}{\\int e^{\\sin{(y^{\\prime})}} dy^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), exp(sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(exp(sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Integral(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('c')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Integral(exp(sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Pow(Integral(exp(sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"], [["differentiate", 5, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{f}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Integral(exp(sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(U)} = \\sin{(U)}, then derive \\int \\pi{(U)} dU = \\pi - \\cos{(U)}, then obtain \\pi - \\cos{(U)} + \\int \\pi{(U)} dU - 1 = 2 \\int \\pi{(U)} dU - 1", "derivation": "\\pi{(U)} = \\sin{(U)} and \\int \\pi{(U)} dU = \\int \\sin{(U)} dU and \\int \\pi{(U)} dU + \\int \\sin{(U)} dU = 2 \\int \\sin{(U)} dU and \\int \\pi{(U)} dU = \\pi - \\cos{(U)} and \\pi - \\cos{(U)} + \\int \\sin{(U)} dU = 2 \\int \\sin{(U)} dU and \\pi - \\cos{(U)} + \\int \\pi{(U)} dU = 2 \\int \\pi{(U)} dU and \\pi - \\cos{(U)} + \\int \\pi{(U)} dU - 1 = 2 \\int \\pi{(U)} dU - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["add", 2, "Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True))), Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True))), Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Integral(Function('\\\\pi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given y{(\\Psi^{\\dagger})} = \\sin{(\\sin{(\\Psi^{\\dagger})})} and \\mathbf{g}{(\\Psi^{\\dagger})} = 2 y{(\\Psi^{\\dagger})}, then obtain \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{g}{(\\Psi^{\\dagger})} = 2 \\frac{d}{d \\Psi^{\\dagger}} y{(\\Psi^{\\dagger})}", "derivation": "y{(\\Psi^{\\dagger})} = \\sin{(\\sin{(\\Psi^{\\dagger})})} and 2 y{(\\Psi^{\\dagger})} = y{(\\Psi^{\\dagger})} + \\sin{(\\sin{(\\Psi^{\\dagger})})} and \\mathbf{g}{(\\Psi^{\\dagger})} = 2 y{(\\Psi^{\\dagger})} and \\mathbf{g}{(\\Psi^{\\dagger})} = y{(\\Psi^{\\dagger})} + \\sin{(\\sin{(\\Psi^{\\dagger})})} and \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{g}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} (y{(\\Psi^{\\dagger})} + \\sin{(\\sin{(\\Psi^{\\dagger})})}) and \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{g}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} 2 y{(\\Psi^{\\dagger})} and \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{g}{(\\Psi^{\\dagger})} = 2 \\frac{d}{d \\Psi^{\\dagger}} y{(\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 1, "Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(2), Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(2), Derivative(Function('y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(h)} = \\cos{(\\cos{(h)})} and \\varphi{(h)} = 0^{h}, then obtain 0^{h} (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{- h} + 1 = 2 \\cdot 0^{h} (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{- h}", "derivation": "\\operatorname{f_{E}}{(h)} = \\cos{(\\cos{(h)})} and 0 = - \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})} and 0^{h} = (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{h} and \\varphi{(h)} = 0^{h} and \\varphi{(h)} = (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{h} and 1 = \\frac{0^{h}}{\\varphi{(h)}} and 1 = 0^{h} (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{- h} and 0^{h} (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{- h} + 1 = 2 \\cdot 0^{h} (- \\operatorname{f_{E}}{(h)} + \\cos{(\\cos{(h)})})^{- h}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('h', commutative=True)), cos(cos(Symbol('h', commutative=True))))"], [["minus", 1, "Function('f_E')(Symbol('h', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('h', commutative=True)), Pow(Integer(0), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\varphi')(Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["divide", 4, "Function('\\\\varphi')(Symbol('h', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Function('\\\\varphi')(Symbol('h', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integer(1), Mul(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["add", 7, "Mul(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Mul(Integer(-1), Symbol('h', commutative=True))))"], "Equality(Add(Mul(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Mul(Integer(-1), Symbol('h', commutative=True)))), Integer(1)), Mul(Integer(2), Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_E')(Symbol('h', commutative=True))), cos(cos(Symbol('h', commutative=True)))), Mul(Integer(-1), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(t)} = \\cos{(t)}, then obtain \\frac{t + \\operatorname{f_{\\mathbf{v}}}{(t)}}{\\int (t + \\cos{(t)}) dt} = \\frac{t + \\cos{(t)}}{\\int (t + \\cos{(t)}) dt}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(t)} = \\cos{(t)} and t + \\operatorname{f_{\\mathbf{v}}}{(t)} = t + \\cos{(t)} and \\int (t + \\operatorname{f_{\\mathbf{v}}}{(t)}) dt = \\int (t + \\cos{(t)}) dt and \\frac{t + \\operatorname{f_{\\mathbf{v}}}{(t)}}{\\int (t + \\operatorname{f_{\\mathbf{v}}}{(t)}) dt} = \\frac{t + \\cos{(t)}}{\\int (t + \\operatorname{f_{\\mathbf{v}}}{(t)}) dt} and \\frac{t + \\operatorname{f_{\\mathbf{v}}}{(t)}}{\\int (t + \\cos{(t)}) dt} = \\frac{t + \\cos{(t)}}{\\int (t + \\cos{(t)}) dt}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["divide", 2, "Integral(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Pow(Integral(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integer(-1))), Mul(Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))), Pow(Integral(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('t', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('t', commutative=True))), Pow(Integral(Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integer(-1))), Mul(Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))), Pow(Integral(Add(Symbol('t', commutative=True), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{f}{(A_{1})} = \\cos{(A_{1})}, then derive \\int \\mathbf{f}{(A_{1})} dA_{1} = m + \\sin{(A_{1})}, then obtain - A_{1} + m + \\sin{(A_{1})} = - A_{1} + \\int \\cos{(A_{1})} dA_{1}", "derivation": "\\mathbf{f}{(A_{1})} = \\cos{(A_{1})} and \\int \\mathbf{f}{(A_{1})} dA_{1} = \\int \\cos{(A_{1})} dA_{1} and \\int \\mathbf{f}{(A_{1})} dA_{1} = m + \\sin{(A_{1})} and m + \\sin{(A_{1})} = \\int \\cos{(A_{1})} dA_{1} and - A_{1} + m + \\sin{(A_{1})} = - A_{1} + \\int \\cos{(A_{1})} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('m', commutative=True), sin(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('m', commutative=True), sin(Symbol('A_1', commutative=True))), Integral(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["minus", 4, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('m', commutative=True), sin(Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Integral(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given V{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})}, then obtain V{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{2} = \\log{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{2}", "derivation": "V{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})} and \\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\log{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and V{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}}) \\int \\log{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\log{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}}) \\int \\log{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and V{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{2} = \\log{(V_{\\mathbf{E}})} + (\\int V{(V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{2}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["add", 1, "Mul(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], "Equality(Add(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))), Add(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(2))), Add(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Integral(Function('V')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given i{(C_{d},\\rho)} = C_{d} + \\rho, then obtain m (C_{d} + \\rho) + \\int m i{(C_{d},\\rho)} dm = m (C_{d} + \\rho) + \\int m (C_{d} + \\rho) dm", "derivation": "i{(C_{d},\\rho)} = C_{d} + \\rho and m i{(C_{d},\\rho)} = m (C_{d} + \\rho) and \\int m i{(C_{d},\\rho)} dm = \\int m (C_{d} + \\rho) dm and m i{(C_{d},\\rho)} + \\int m i{(C_{d},\\rho)} dm = m i{(C_{d},\\rho)} + \\int m (C_{d} + \\rho) dm and m (C_{d} + \\rho) + \\int m i{(C_{d},\\rho)} dm = m (C_{d} + \\rho) + \\int m (C_{d} + \\rho) dm", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["times", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["add", 3, "Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True)))), Add(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Function('i')(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True)))), Add(Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('m', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(l)} = \\cos{(l)}, then obtain - l - \\cos{(l)} \\cos^{l}{(l)} + \\int \\Psi{(l)} dl = - l - \\cos{(l)} \\cos^{l}{(l)} + \\int \\cos{(l)} dl", "derivation": "\\Psi{(l)} = \\cos{(l)} and \\int \\Psi{(l)} dl = \\int \\cos{(l)} dl and - l + \\int \\Psi{(l)} dl = - l + \\int \\cos{(l)} dl and \\Psi{(l)} \\cos^{l}{(l)} = \\cos{(l)} \\cos^{l}{(l)} and - l - \\Psi{(l)} \\cos^{l}{(l)} + \\int \\Psi{(l)} dl = - l - \\Psi{(l)} \\cos^{l}{(l)} + \\int \\cos{(l)} dl and - l - \\cos{(l)} \\cos^{l}{(l)} + \\int \\Psi{(l)} dl = - l - \\cos{(l)} \\cos^{l}{(l)} + \\int \\cos{(l)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["minus", 2, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('\\\\Psi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["times", 1, "Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\Psi')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Mul(cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\Psi')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(Function('\\\\Psi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(Function('\\\\Psi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(-1), cos(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given J{(V)} = \\frac{1}{V} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,y)} = \\frac{J{(V)} - \\frac{1}{V}}{y}, then obtain \\cos{(\\frac{1}{y})} = \\cos{(\\frac{J{(V)} - \\frac{1}{V}}{y} + \\frac{1}{y})}", "derivation": "J{(V)} = \\frac{1}{V} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,y)} = \\frac{J{(V)} - \\frac{1}{V}}{y} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,y)} + \\frac{1}{y} = \\frac{J{(V)} - \\frac{1}{V}}{y} + \\frac{1}{y} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(V,y)} + \\frac{1}{y} = \\frac{1}{y} and \\frac{1}{y} = \\frac{J{(V)} - \\frac{1}{V}}{y} + \\frac{1}{y} and \\cos{(\\frac{1}{y})} = \\cos{(\\frac{J{(V)} - \\frac{1}{V}}{y} + \\frac{1}{y})}", "srepr_derivation": [["renaming_premise", "Equality(Function('J')(Symbol('V', commutative=True)), Pow(Symbol('V', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('J')(Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))))))"], [["add", 2, "Pow(Symbol('y', commutative=True), Integer(-1))"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('J')(Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))))), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))), Pow(Symbol('y', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('J')(Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))))), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["cos", 5], "Equality(cos(Pow(Symbol('y', commutative=True), Integer(-1))), cos(Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('J')(Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))))), Pow(Symbol('y', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\varepsilon_0,V)} = \\frac{\\varepsilon_0}{V}, then obtain (\\frac{\\log{(\\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0})}}{V})^{V} + \\frac{1}{\\varepsilon_0} = 0^{V} + \\frac{1}{\\varepsilon_0}", "derivation": "\\operatorname{v_{x}}{(\\varepsilon_0,V)} = \\frac{\\varepsilon_0}{V} and \\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0} = 1 and \\log{(\\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0})} = 0 and \\frac{\\log{(\\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0})}}{V} = 0 and (\\frac{\\log{(\\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0})}}{V})^{V} = 0^{V} and (\\frac{\\log{(\\frac{V \\operatorname{v_{x}}{(\\varepsilon_0,V)}}{\\varepsilon_0})}}{V})^{V} + \\frac{1}{\\varepsilon_0} = 0^{V} + \\frac{1}{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True))), Integer(1))"], [["log", 2], "Equality(log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)))), Integer(0))"], [["divide", 3, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True))))), Symbol('V', commutative=True)), Pow(Integer(0), Symbol('V', commutative=True)))"], [["add", 5, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Add(Pow(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('v_x')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True))))), Symbol('V', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Add(Pow(Integer(0), Symbol('V', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\eta{(C_{2})} = e^{C_{2}}, then obtain \\int (\\frac{d}{d C_{2}} \\eta{(C_{2})} e^{- C_{2}})^{C_{2}} dC_{2} = \\int (\\frac{d}{d C_{2}} 1)^{C_{2}} dC_{2}", "derivation": "\\eta{(C_{2})} = e^{C_{2}} and \\eta{(C_{2})} e^{- C_{2}} = 1 and \\frac{d}{d C_{2}} \\eta{(C_{2})} e^{- C_{2}} = \\frac{d}{d C_{2}} 1 and (\\frac{d}{d C_{2}} \\eta{(C_{2})} e^{- C_{2}})^{C_{2}} = (\\frac{d}{d C_{2}} 1)^{C_{2}} and \\int (\\frac{d}{d C_{2}} \\eta{(C_{2})} e^{- C_{2}})^{C_{2}} dC_{2} = \\int (\\frac{d}{d C_{2}} 1)^{C_{2}} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["divide", 1, "exp(Symbol('C_2', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('C_2', commutative=True)), exp(Mul(Integer(-1), Symbol('C_2', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\eta')(Symbol('C_2', commutative=True)), exp(Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('C_2', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('\\\\eta')(Symbol('C_2', commutative=True)), exp(Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)))"], [["integrate", 4, "Symbol('C_2', commutative=True)"], "Equality(Integral(Pow(Derivative(Mul(Function('\\\\eta')(Symbol('C_2', commutative=True)), exp(Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Pow(Derivative(Integer(1), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(t_{2})} = \\cos{(t_{2})}, then obtain (\\hat{\\mathbf{x}}{(t_{2})} \\cos{(t_{2})} - \\cos{(t_{2})})^{t_{2}} = (\\cos^{2}{(t_{2})} - \\cos{(t_{2})})^{t_{2}}", "derivation": "\\hat{\\mathbf{x}}{(t_{2})} = \\cos{(t_{2})} and \\hat{\\mathbf{x}}{(t_{2})} \\cos{(t_{2})} = \\cos^{2}{(t_{2})} and \\hat{\\mathbf{x}}{(t_{2})} \\cos{(t_{2})} - \\cos{(t_{2})} = \\cos^{2}{(t_{2})} - \\cos{(t_{2})} and (\\hat{\\mathbf{x}}{(t_{2})} \\cos{(t_{2})} - \\cos{(t_{2})})^{t_{2}} = (\\cos^{2}{(t_{2})} - \\cos{(t_{2})})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True)))"], [["times", 1, "cos(Symbol('t_2', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True))), Pow(cos(Symbol('t_2', commutative=True)), Integer(2)))"], [["minus", 2, "cos(Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True))), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Add(Pow(cos(Symbol('t_2', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))))"], [["power", 3, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_2', commutative=True)), cos(Symbol('t_2', commutative=True))), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Symbol('t_2', commutative=True)), Pow(Add(Pow(cos(Symbol('t_2', commutative=True)), Integer(2)), Mul(Integer(-1), cos(Symbol('t_2', commutative=True)))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\Psi)} = \\log{(\\log{(\\Psi)})}, then obtain - \\mu{(\\Psi)} + \\log{(\\log{(\\Psi)})} = 0", "derivation": "\\mu{(\\Psi)} = \\log{(\\log{(\\Psi)})} and \\mu{(\\Psi)} - \\log{(\\Psi)} = - \\log{(\\Psi)} + \\log{(\\log{(\\Psi)})} and - \\mu{(\\Psi)} + \\log{(\\Psi)} = \\log{(\\Psi)} - \\log{(\\log{(\\Psi)})} and - \\mu{(\\Psi)} + \\log{(\\log{(\\Psi)})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\Psi', commutative=True)), log(log(Symbol('\\\\Psi', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('\\\\mu')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))), log(log(Symbol('\\\\Psi', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))), Add(log(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\Psi', commutative=True))))))"], [["minus", 3, "Add(log(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\Psi', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True))), log(log(Symbol('\\\\Psi', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(c)} = \\sin{(\\sin{(c)})}, then obtain I (\\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\operatorname{J_{\\varepsilon}}{(c)}) = I (\\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\sin{(\\sin{(c)})})", "derivation": "\\operatorname{J_{\\varepsilon}}{(c)} = \\sin{(\\sin{(c)})} and \\operatorname{J_{\\varepsilon}}{(c)} \\sin{(\\sin{(c)})} = \\sin^{2}{(\\sin{(c)})} and \\frac{d}{d c} \\operatorname{J_{\\varepsilon}}{(c)} = \\frac{d}{d c} \\sin{(\\sin{(c)})} and \\operatorname{J_{\\varepsilon}}{(c)} \\sin{(\\sin{(c)})} + \\frac{d}{d c} \\operatorname{J_{\\varepsilon}}{(c)} = \\operatorname{J_{\\varepsilon}}{(c)} \\sin{(\\sin{(c)})} + \\frac{d}{d c} \\sin{(\\sin{(c)})} and \\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\operatorname{J_{\\varepsilon}}{(c)} = \\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\sin{(\\sin{(c)})} and I (\\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\operatorname{J_{\\varepsilon}}{(c)}) = I (\\sin^{2}{(\\sin{(c)})} + \\frac{d}{d c} \\sin{(\\sin{(c)})})", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))))"], [["times", 1, "sin(sin(Symbol('c', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True)))), Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["add", 3, "Mul(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True))))"], "Equality(Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True)))), Derivative(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), sin(sin(Symbol('c', commutative=True)))), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["times", 5, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Add(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))), Mul(Symbol('I', commutative=True), Add(Pow(sin(sin(Symbol('c', commutative=True))), Integer(2)), Derivative(sin(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(n_{2},c,E)} = \\frac{E}{c} + n_{2}, then derive \\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)} = 1, then obtain (\\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)})^{n_{2}} = 1", "derivation": "\\operatorname{f^{*}}{(n_{2},c,E)} = \\frac{E}{c} + n_{2} and \\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)} = \\frac{\\partial}{\\partial n_{2}} (\\frac{E}{c} + n_{2}) and \\frac{\\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)}}{\\frac{\\partial}{\\partial n_{2}} (\\frac{E}{c} + n_{2})} = 1 and \\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)} = 1 and (\\frac{\\partial}{\\partial n_{2}} \\operatorname{f^{*}}{(n_{2},c,E)})^{n_{2}} = 1", "srepr_derivation": [["get_premise", "Equality(Function('f^*')(Symbol('n_2', commutative=True), Symbol('c', commutative=True), Symbol('E', commutative=True)), Add(Mul(Symbol('E', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('n_2', commutative=True), Symbol('c', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('f^*')(Symbol('n_2', commutative=True), Symbol('c', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f^*')(Symbol('n_2', commutative=True), Symbol('c', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Function('f^*')(Symbol('n_2', commutative=True), Symbol('c', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},C)} = A_{2} + C, then obtain - A_{2} - C + 4 \\mathbf{S}^{2}{(A_{2},C)} = - A_{2} - C + (A_{2} + C + \\mathbf{S}{(A_{2},C)})^{2}", "derivation": "\\mathbf{S}{(A_{2},C)} = A_{2} + C and 2 \\mathbf{S}{(A_{2},C)} = A_{2} + C + \\mathbf{S}{(A_{2},C)} and 4 \\mathbf{S}^{2}{(A_{2},C)} = (A_{2} + C + \\mathbf{S}{(A_{2},C)})^{2} and - A_{2} - C + 4 \\mathbf{S}^{2}{(A_{2},C)} = - A_{2} - C + (A_{2} + C + \\mathbf{S}{(A_{2},C)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('C', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))), Add(Symbol('A_2', commutative=True), Symbol('C', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integer(2))), Pow(Add(Symbol('A_2', commutative=True), Symbol('C', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))), Integer(2)))"], [["minus", 3, "Add(Symbol('A_2', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(4), Pow(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Add(Symbol('A_2', commutative=True), Symbol('C', commutative=True), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\phi_1,\\dot{x})} = \\dot{x}^{\\phi_1}, then obtain \\tilde{g}^*^{- \\phi_1}{(\\phi_1,\\dot{x})} \\log{(\\tilde{g}^*^{2}{(\\phi_1,\\dot{x})})} = \\tilde{g}^*^{- \\phi_1}{(\\phi_1,\\dot{x})} \\log{(\\dot{x}^{\\phi_1} \\tilde{g}^*{(\\phi_1,\\dot{x})})}", "derivation": "\\tilde{g}^*{(\\phi_1,\\dot{x})} = \\dot{x}^{\\phi_1} and \\tilde{g}^*^{2}{(\\phi_1,\\dot{x})} = \\dot{x}^{\\phi_1} \\tilde{g}^*{(\\phi_1,\\dot{x})} and \\log{(\\tilde{g}^*^{2}{(\\phi_1,\\dot{x})})} = \\log{(\\dot{x}^{\\phi_1} \\tilde{g}^*{(\\phi_1,\\dot{x})})} and \\tilde{g}^*^{- \\phi_1}{(\\phi_1,\\dot{x})} \\log{(\\tilde{g}^*^{2}{(\\phi_1,\\dot{x})})} = \\tilde{g}^*^{- \\phi_1}{(\\phi_1,\\dot{x})} \\log{(\\dot{x}^{\\phi_1} \\tilde{g}^*{(\\phi_1,\\dot{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["log", 2], "Equality(log(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2))), log(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["divide", 3, "Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), log(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), log(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\dot{x}', commutative=True))))))"]]}, {"prompt": "Given x{(A,\\hat{H}_l)} = A \\hat{H}_l, then obtain \\frac{\\partial^{2}}{\\partial A\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} x{(A,\\hat{H}_l)})^{\\hat{H}_l} = \\frac{\\partial^{2}}{\\partial A\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} A \\hat{H}_l)^{\\hat{H}_l}", "derivation": "x{(A,\\hat{H}_l)} = A \\hat{H}_l and \\frac{\\partial}{\\partial \\hat{H}_l} x{(A,\\hat{H}_l)} = \\frac{\\partial}{\\partial \\hat{H}_l} A \\hat{H}_l and (\\frac{\\partial}{\\partial \\hat{H}_l} x{(A,\\hat{H}_l)})^{\\hat{H}_l} = (\\frac{\\partial}{\\partial \\hat{H}_l} A \\hat{H}_l)^{\\hat{H}_l} and \\frac{\\partial}{\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} x{(A,\\hat{H}_l)})^{\\hat{H}_l} = \\frac{\\partial}{\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} A \\hat{H}_l)^{\\hat{H}_l} and \\frac{\\partial^{2}}{\\partial A\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} x{(A,\\hat{H}_l)})^{\\hat{H}_l} = \\frac{\\partial^{2}}{\\partial A\\partial \\hat{H}_l} (\\frac{\\partial}{\\partial \\hat{H}_l} A \\hat{H}_l)^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Derivative(Function('x')(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('x')(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('x')(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(b)} = \\cos{(b)}, then obtain \\frac{\\frac{d}{d b} (- 2 \\mathbf{p}{(b)} + \\cos{(b)} - 1)}{- 2 \\mathbf{p}{(b)} + 2 \\cos{(b)}} = \\frac{\\frac{d}{d b} (- \\cos{(b)} - 1)}{- 2 \\mathbf{p}{(b)} + 2 \\cos{(b)}}", "derivation": "\\mathbf{p}{(b)} = \\cos{(b)} and - \\mathbf{p}{(b)} = - \\cos{(b)} and - \\mathbf{p}{(b)} - 1 = - \\cos{(b)} - 1 and \\frac{d}{d b} (- \\mathbf{p}{(b)} - 1) = \\frac{d}{d b} (- \\cos{(b)} - 1) and 2 \\mathbf{p}{(b)} - \\cos{(b)} = \\mathbf{p}{(b)} and \\frac{d}{d b} (- 2 \\mathbf{p}{(b)} + \\cos{(b)} - 1) = \\frac{d}{d b} (- \\cos{(b)} - 1) and \\frac{\\frac{d}{d b} (- 2 \\mathbf{p}{(b)} + \\cos{(b)} - 1)}{- 2 \\mathbf{p}{(b)} + 2 \\cos{(b)}} = \\frac{\\frac{d}{d b} (- \\cos{(b)} - 1)}{- 2 \\mathbf{p}{(b)} + 2 \\cos{(b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Mul(Integer(-1), cos(Symbol('b', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["minus", 1, "Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Function('\\\\mathbf{p}')(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True)), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 6, "Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Mul(Integer(2), cos(Symbol('b', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Mul(Integer(2), cos(Symbol('b', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True)), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Mul(Integer(2), cos(Symbol('b', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(y^{\\prime},v_{1})} = \\sin{((y^{\\prime})^{v_{1}})}, then obtain \\int y^{\\prime} dy^{\\prime} = \\int (y^{\\prime} - u{(y^{\\prime},v_{1})} + \\sin{((y^{\\prime})^{v_{1}})}) dy^{\\prime}", "derivation": "u{(y^{\\prime},v_{1})} = \\sin{((y^{\\prime})^{v_{1}})} and 0 = - u{(y^{\\prime},v_{1})} + \\sin{((y^{\\prime})^{v_{1}})} and y^{\\prime} = y^{\\prime} - u{(y^{\\prime},v_{1})} + \\sin{((y^{\\prime})^{v_{1}})} and \\int y^{\\prime} dy^{\\prime} = \\int (y^{\\prime} - u{(y^{\\prime},v_{1})} + \\sin{((y^{\\prime})^{v_{1}})}) dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)), sin(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True))))"], [["minus", 1, "Function('u')(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('u')(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True))), sin(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)))))"], [["add", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Symbol('y^{\\\\prime}', commutative=True), Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('u')(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True))), sin(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)))))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Symbol('y^{\\\\prime}', commutative=True), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('u')(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True))), sin(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{S},H)} = \\mathbf{S}^{H}, then obtain (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)}) (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)})^{H} = (H + (\\mathbf{S}^{H})^{\\mathbf{S}})^{H} (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)})", "derivation": "\\phi_{1}{(\\mathbf{S},H)} = \\mathbf{S}^{H} and \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)} = (\\mathbf{S}^{H})^{\\mathbf{S}} and H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)} = H + (\\mathbf{S}^{H})^{\\mathbf{S}} and (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)})^{H} = (H + (\\mathbf{S}^{H})^{\\mathbf{S}})^{H} and (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)}) (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)})^{H} = (H + (\\mathbf{S}^{H})^{\\mathbf{S}})^{H} (H + \\phi_{1}^{\\mathbf{S}}{(\\mathbf{S},H)})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 2, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('H', commutative=True), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('H', commutative=True)))"], [["times", 4, "Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('H', commutative=True))), Mul(Pow(Add(Symbol('H', commutative=True), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(A_{x},v_{2})} = \\frac{v_{2}}{A_{x}}, then obtain (- \\frac{2 A_{x} \\mathbf{B}{(A_{x},v_{2})}}{v_{2}})^{A_{x}} = (-2)^{A_{x}}", "derivation": "\\mathbf{B}{(A_{x},v_{2})} = \\frac{v_{2}}{A_{x}} and - \\mathbf{B}{(A_{x},v_{2})} = - \\frac{v_{2}}{A_{x}} and - \\mathbf{B}{(A_{x},v_{2})} - \\frac{v_{2}}{A_{x}} = - \\frac{2 v_{2}}{A_{x}} and \\frac{A_{x} (- \\mathbf{B}{(A_{x},v_{2})} - \\frac{v_{2}}{A_{x}})}{v_{2}} = -2 and - \\frac{2 A_{x} \\mathbf{B}{(A_{x},v_{2})}}{v_{2}} = -2 and (- \\frac{2 A_{x} \\mathbf{B}{(A_{x},v_{2})}}{v_{2}})^{A_{x}} = (-2)^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))"], "Equality(Mul(Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))), Integer(-2))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Integer(2), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True))), Integer(-2))"], [["power", 5, "Symbol('A_x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integer(2), Symbol('A_x', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('A_x', commutative=True), Symbol('v_2', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integer(-2), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\rho,A_{1})} = \\rho \\cos{(A_{1})}, then derive \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = u + \\sin{(A_{1})}, then obtain \\frac{\\partial}{\\partial A_{1}} \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = \\cos{(A_{1})}", "derivation": "\\hat{x}{(\\rho,A_{1})} = \\rho \\cos{(A_{1})} and \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} = \\cos{(A_{1})} and \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = \\int \\cos{(A_{1})} dA_{1} and \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = u + \\sin{(A_{1})} and \\frac{\\partial}{\\partial A_{1}} \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = \\frac{\\partial}{\\partial A_{1}} (u + \\sin{(A_{1})}) and \\frac{\\partial}{\\partial A_{1}} \\int \\frac{\\hat{x}{(\\rho,A_{1})}}{\\rho} dA_{1} = \\cos{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('A_1', commutative=True))))"], [["divide", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True))), cos(Symbol('A_1', commutative=True)))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('u', commutative=True), sin(Symbol('A_1', commutative=True))))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('u', commutative=True), sin(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), cos(Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(I,y^{\\prime})} = I y^{\\prime}, then obtain \\cos{(1 - \\frac{\\mathbf{A}{(I,y^{\\prime})}}{y^{\\prime}})} = \\cos{(I - 1)}", "derivation": "\\mathbf{A}{(I,y^{\\prime})} = I y^{\\prime} and \\frac{\\mathbf{A}{(I,y^{\\prime})}}{y^{\\prime}} = I and -1 + \\frac{\\mathbf{A}{(I,y^{\\prime})}}{y^{\\prime}} = I - 1 and \\cos{(1 - \\frac{\\mathbf{A}{(I,y^{\\prime})}}{y^{\\prime}})} = \\cos{(I - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('I', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('I', commutative=True), Integer(-1)))"], [["cos", 3], "Equality(cos(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))), cos(Add(Symbol('I', commutative=True), Integer(-1))))"]]}, {"prompt": "Given E{(s,\\ddot{x})} = \\ddot{x} - s, then derive (\\int E{(s,\\ddot{x})} d\\ddot{x} + 1)^{\\ddot{x}} = (\\frac{\\ddot{x}^{2}}{2} - \\ddot{x} s + i + 1)^{\\ddot{x}}, then obtain (\\frac{\\ddot{x}^{2}}{2} - \\ddot{x} s + i + 1)^{\\ddot{x}} = (\\int (\\ddot{x} - s) d\\ddot{x} + 1)^{\\ddot{x}}", "derivation": "E{(s,\\ddot{x})} = \\ddot{x} - s and \\int E{(s,\\ddot{x})} d\\ddot{x} = \\int (\\ddot{x} - s) d\\ddot{x} and \\int E{(s,\\ddot{x})} d\\ddot{x} + 1 = \\int (\\ddot{x} - s) d\\ddot{x} + 1 and (\\int E{(s,\\ddot{x})} d\\ddot{x} + 1)^{\\ddot{x}} = (\\int (\\ddot{x} - s) d\\ddot{x} + 1)^{\\ddot{x}} and (\\int E{(s,\\ddot{x})} d\\ddot{x} + 1)^{\\ddot{x}} = (\\frac{\\ddot{x}^{2}}{2} - \\ddot{x} s + i + 1)^{\\ddot{x}} and (\\frac{\\ddot{x}^{2}}{2} - \\ddot{x} s + i + 1)^{\\ddot{x}} = (\\int (\\ddot{x} - s) d\\ddot{x} + 1)^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('E')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Add(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)))"], [["power", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Add(Integral(Function('E')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Integral(Function('E')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)), Symbol('i', commutative=True), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)), Symbol('i', commutative=True), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Integral(Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})}, then derive - \\mathbf{A} + \\psi^{*}{(\\mathbf{A})} = - \\mathbf{A} + \\cos{(\\mathbf{A})}, then obtain - \\mathbf{A} + \\cos{(\\mathbf{A})} - 1 = - \\mathbf{A} + \\psi^{*}{(\\mathbf{A})} - 1", "derivation": "\\psi^{*}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and - \\mathbf{A} + \\psi^{*}{(\\mathbf{A})} = - \\mathbf{A} + \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and - \\mathbf{A} + \\psi^{*}{(\\mathbf{A})} = - \\mathbf{A} + \\cos{(\\mathbf{A})} and - \\mathbf{A} + \\cos{(\\mathbf{A})} = - \\mathbf{A} + \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and - \\mathbf{A} + \\cos{(\\mathbf{A})} - 1 = - \\mathbf{A} + \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} - 1 and - \\mathbf{A} + \\cos{(\\mathbf{A})} - 1 = - \\mathbf{A} + \\psi^{*}{(\\mathbf{A})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["minus", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(c,l)} = l + \\sin{(c)}, then derive c \\frac{\\partial}{\\partial l} \\operatorname{E_{x}}{(c,l)} + \\int (v_{t} v_{x})^{t} dt = c + \\int (v_{t} v_{x})^{t} dt, then obtain c \\frac{\\partial}{\\partial l} \\operatorname{E_{x}}{(c,l)} + \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) + \\int (v_{t} v_{x})^{t} dt = c + \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) + \\int (v_{t} v_{x})^{t} dt", "derivation": "\\operatorname{E_{x}}{(c,l)} = l + \\sin{(c)} and c \\operatorname{E_{x}}{(c,l)} = c (l + \\sin{(c)}) and \\frac{\\partial}{\\partial l} c \\operatorname{E_{x}}{(c,l)} = \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) and \\frac{\\partial}{\\partial l} c \\operatorname{E_{x}}{(c,l)} + \\int (v_{t} v_{x})^{t} dt = \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) + \\int (v_{t} v_{x})^{t} dt and c \\frac{\\partial}{\\partial l} \\operatorname{E_{x}}{(c,l)} + \\int (v_{t} v_{x})^{t} dt = c + \\int (v_{t} v_{x})^{t} dt and c \\frac{\\partial}{\\partial l} \\operatorname{E_{x}}{(c,l)} + \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) + \\int (v_{t} v_{x})^{t} dt = c + \\frac{\\partial}{\\partial l} c (l + \\sin{(c)}) + \\int (v_{t} v_{x})^{t} dt", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True))))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Symbol('c', commutative=True), Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Integer(-1), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], "Equality(Add(Derivative(Mul(Symbol('c', commutative=True), Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('c', commutative=True), Derivative(Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('c', commutative=True), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["add", 5, "Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('c', commutative=True), Derivative(Function('E_x')(Symbol('c', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('c', commutative=True), Derivative(Mul(Symbol('c', commutative=True), Add(Symbol('l', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Pow(Mul(Symbol('v_t', commutative=True), Symbol('v_x', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})} = \\mathbf{H} \\sigma_x, then obtain \\int - \\sin{(1 - \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\sigma_x})} d\\sigma_x = \\int \\sin{(\\mathbf{H} - 1)} d\\sigma_x", "derivation": "\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})} = \\mathbf{H} \\sigma_x and \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\mathbf{H} \\sigma_x} = 1 and \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\sigma_x} = \\mathbf{H} and -1 + \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\sigma_x} = \\mathbf{H} - 1 and - \\sin{(1 - \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\sigma_x})} = \\sin{(\\mathbf{H} - 1)} and \\int - \\sin{(1 - \\frac{\\operatorname{r_{0}}{(\\sigma_x,\\mathbf{H})}}{\\sigma_x})} d\\sigma_x = \\int \\sin{(\\mathbf{H} - 1)} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(1))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))"], [["minus", 3, 1], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))), sin(Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(V)} = e^{V}, then obtain \\rho_{b}{(V)} + \\int \\rho_{b}{(V)} dV = \\Psi_{\\lambda} + \\rho_{b}{(V)} + e^{V}", "derivation": "\\rho_{b}{(V)} = e^{V} and \\int \\rho_{b}{(V)} dV = \\int e^{V} dV and \\rho_{b}{(V)} + \\int \\rho_{b}{(V)} dV = \\rho_{b}{(V)} + \\int e^{V} dV and \\rho_{b}{(V)} + \\int \\rho_{b}{(V)} dV = \\Psi_{\\lambda} + \\rho_{b}{(V)} + e^{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 2, "Function('\\\\rho_b')(Symbol('V', commutative=True))"], "Equality(Add(Function('\\\\rho_b')(Symbol('V', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Function('\\\\rho_b')(Symbol('V', commutative=True)), Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('\\\\rho_b')(Symbol('V', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\rho_b')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\phi,\\mathbf{A},p)} = \\phi^{\\mathbf{A}} p, then obtain \\dot{x}{(\\phi,\\mathbf{A},p)} + \\frac{\\partial}{\\partial \\phi} \\phi^{\\mathbf{A}} p = \\phi^{\\mathbf{A}} p + \\frac{\\partial}{\\partial \\phi} \\phi^{\\mathbf{A}} p", "derivation": "\\dot{x}{(\\phi,\\mathbf{A},p)} = \\phi^{\\mathbf{A}} p and \\frac{\\partial}{\\partial \\phi} \\dot{x}{(\\phi,\\mathbf{A},p)} = \\frac{\\partial}{\\partial \\phi} \\phi^{\\mathbf{A}} p and \\dot{x}{(\\phi,\\mathbf{A},p)} + \\frac{\\partial}{\\partial \\phi} \\dot{x}{(\\phi,\\mathbf{A},p)} = \\phi^{\\mathbf{A}} p + \\frac{\\partial}{\\partial \\phi} \\dot{x}{(\\phi,\\mathbf{A},p)} and \\dot{x}{(\\phi,\\mathbf{A},p)} + \\frac{\\partial}{\\partial \\phi} \\phi^{\\mathbf{A}} p = \\phi^{\\mathbf{A}} p + \\frac{\\partial}{\\partial \\phi} \\phi^{\\mathbf{A}} p", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(E,I)} = E I and \\operatorname{v_{1}}{(I)} = I^{2}, then obtain E^{2} I^{2} = E^{2} \\operatorname{v_{1}}{(I)}", "derivation": "\\hat{H}_{\\lambda}{(E,I)} = E I and E I \\hat{H}_{\\lambda}{(E,I)} = E^{2} I^{2} and \\operatorname{v_{1}}{(I)} = I^{2} and E I \\hat{H}_{\\lambda}{(E,I)} = E^{2} \\operatorname{v_{1}}{(I)} and E^{2} I^{2} = E^{2} \\operatorname{v_{1}}{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('I', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('I', commutative=True)))"], [["times", 1, "Mul(Symbol('E', commutative=True), Symbol('I', commutative=True))"], "Equality(Mul(Symbol('E', commutative=True), Symbol('I', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('I', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('E', commutative=True), Symbol('I', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Function('v_1')(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Pow(Symbol('E', commutative=True), Integer(2)), Function('v_1')(Symbol('I', commutative=True))))"]]}, {"prompt": "Given m{(M,\\hat{H})} = M + \\hat{H}, then derive \\frac{\\partial}{\\partial \\hat{H}} m{(M,\\hat{H})} = 1, then obtain \\int \\frac{\\frac{\\partial}{\\partial \\hat{H}} (M + \\hat{H})}{\\hat{H}} dM = \\int \\frac{1}{\\hat{H}} dM", "derivation": "m{(M,\\hat{H})} = M + \\hat{H} and \\frac{\\partial}{\\partial \\hat{H}} m{(M,\\hat{H})} = \\frac{\\partial}{\\partial \\hat{H}} (M + \\hat{H}) and \\frac{\\partial}{\\partial \\hat{H}} m{(M,\\hat{H})} = 1 and \\frac{\\frac{\\partial}{\\partial \\hat{H}} m{(M,\\hat{H})}}{\\hat{H}} = \\frac{1}{\\hat{H}} and \\frac{\\frac{\\partial}{\\partial \\hat{H}} (M + \\hat{H})}{\\hat{H}} = \\frac{1}{\\hat{H}} and \\int \\frac{\\frac{\\partial}{\\partial \\hat{H}} (M + \\hat{H})}{\\hat{H}} dM = \\int \\frac{1}{\\hat{H}} dM", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m')(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Derivative(Function('m')(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], [["integrate", 5, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True))), Integral(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(b,v)} = - b + v, then derive \\frac{\\partial}{\\partial v} \\operatorname{F_{x}}{(b,v)} - 1 = 0, then obtain \\frac{\\partial}{\\partial v} (- b + v) - 1 = 0", "derivation": "\\operatorname{F_{x}}{(b,v)} = - b + v and - v + \\operatorname{F_{x}}{(b,v)} = - b and \\frac{\\partial}{\\partial v} (- v + \\operatorname{F_{x}}{(b,v)}) = \\frac{d}{d v} - b and \\frac{\\partial}{\\partial v} \\operatorname{F_{x}}{(b,v)} - 1 = 0 and \\frac{\\partial}{\\partial v} (- b + v) - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('b', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('F_x')(Symbol('b', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('F_x')(Symbol('b', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('b', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('F_x')(Symbol('b', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(S,C_{1})} = - C_{1} + S, then derive \\frac{- S + \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} - 1}{S} = -1, then obtain - \\rho_b + \\frac{- S + \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1}{S} = - \\rho_b - 1", "derivation": "\\operatorname{v_{1}}{(S,C_{1})} = - C_{1} + S and \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} = \\frac{\\partial}{\\partial S} (- C_{1} + S) and \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} - 1 = \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1 and - S + \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} - 1 = - S + \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1 and \\frac{- S + \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} - 1}{S} = \\frac{- S + \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1}{S} and \\frac{- S + \\frac{\\partial}{\\partial S} \\operatorname{v_{1}}{(S,C_{1})} - 1}{S} = -1 and \\frac{- S + \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1}{S} = -1 and - \\rho_b + \\frac{- S + \\frac{\\partial}{\\partial S} (- C_{1} + S) - 1}{S} = - \\rho_b - 1", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"], [["add", 3, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 4, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('v_1')(Symbol('S', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Integer(-1))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Integer(-1))"], [["minus", 7, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\delta,\\nabla,u)} = \\frac{\\delta \\nabla}{u}, then obtain - \\frac{\\int \\operatorname{t_{2}}{(\\delta,\\nabla,u)} du}{\\int \\frac{\\delta \\nabla}{u} du} = -1", "derivation": "\\operatorname{t_{2}}{(\\delta,\\nabla,u)} = \\frac{\\delta \\nabla}{u} and \\int \\operatorname{t_{2}}{(\\delta,\\nabla,u)} du = \\int \\frac{\\delta \\nabla}{u} du and \\frac{\\int \\operatorname{t_{2}}{(\\delta,\\nabla,u)} du}{\\int \\frac{\\delta \\nabla}{u} du} = 1 and - \\frac{\\int \\operatorname{t_{2}}{(\\delta,\\nabla,u)} du}{\\int \\frac{\\delta \\nabla}{u} du} = -1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))))"], [["divide", 2, "Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Integer(1))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Function('t_2')(Symbol('\\\\delta', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Integer(-1))"]]}, {"prompt": "Given \\hat{H}{(t_{1})} = \\sin{(t_{1})}, then obtain t_{1} + \\hat{H}{(t_{1})} + \\sin{(t_{1})} = t_{1} + 2 \\sin{(t_{1})}", "derivation": "\\hat{H}{(t_{1})} = \\sin{(t_{1})} and t_{1} + \\hat{H}{(t_{1})} = t_{1} + \\sin{(t_{1})} and t_{1} + 2 \\hat{H}{(t_{1})} = t_{1} + \\hat{H}{(t_{1})} + \\sin{(t_{1})} and t_{1} + 2 \\hat{H}{(t_{1})} = t_{1} + 2 \\sin{(t_{1})} and t_{1} + \\hat{H}{(t_{1})} + \\sin{(t_{1})} = t_{1} + 2 \\sin{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], [["add", 1, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True))), Add(Symbol('t_1', commutative=True), sin(Symbol('t_1', commutative=True))))"], [["add", 1, "Add(Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True)))"], "Equality(Add(Symbol('t_1', commutative=True), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('t_1', commutative=True)))), Add(Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('t_1', commutative=True), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('t_1', commutative=True)))), Add(Symbol('t_1', commutative=True), Mul(Integer(2), sin(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True))), Add(Symbol('t_1', commutative=True), Mul(Integer(2), sin(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given m{(\\varepsilon_0,p)} = \\sin{(\\varepsilon_0 p)}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} (- m{(\\varepsilon_0,p)} + \\int m{(\\varepsilon_0,p)} d\\varepsilon_0) = \\frac{\\partial}{\\partial \\varepsilon_0} (- m{(\\varepsilon_0,p)} + \\int \\sin{(\\varepsilon_0 p)} d\\varepsilon_0)", "derivation": "m{(\\varepsilon_0,p)} = \\sin{(\\varepsilon_0 p)} and \\int m{(\\varepsilon_0,p)} d\\varepsilon_0 = \\int \\sin{(\\varepsilon_0 p)} d\\varepsilon_0 and - m{(\\varepsilon_0,p)} + \\int m{(\\varepsilon_0,p)} d\\varepsilon_0 = - m{(\\varepsilon_0,p)} + \\int \\sin{(\\varepsilon_0 p)} d\\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} (- m{(\\varepsilon_0,p)} + \\int m{(\\varepsilon_0,p)} d\\varepsilon_0) = \\frac{\\partial}{\\partial \\varepsilon_0} (- m{(\\varepsilon_0,p)} + \\int \\sin{(\\varepsilon_0 p)} d\\varepsilon_0)", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True)), sin(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 2, "Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Integral(Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Add(Mul(Integer(-1), Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Integral(sin(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Integral(Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('m')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Integral(sin(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{f},\\varphi^*)} = \\cos^{\\mathbf{f}}{(\\varphi^*)}, then obtain (\\frac{\\partial}{\\partial \\varphi^*} \\int \\mu_{0}{(\\mathbf{f},\\varphi^*)} d\\mathbf{f})^{\\varphi^*} = (\\frac{\\partial}{\\partial \\varphi^*} \\int \\cos^{\\mathbf{f}}{(\\varphi^*)} d\\mathbf{f})^{\\varphi^*}", "derivation": "\\mu_{0}{(\\mathbf{f},\\varphi^*)} = \\cos^{\\mathbf{f}}{(\\varphi^*)} and \\int \\mu_{0}{(\\mathbf{f},\\varphi^*)} d\\mathbf{f} = \\int \\cos^{\\mathbf{f}}{(\\varphi^*)} d\\mathbf{f} and \\frac{\\partial}{\\partial \\varphi^*} \\int \\mu_{0}{(\\mathbf{f},\\varphi^*)} d\\mathbf{f} = \\frac{\\partial}{\\partial \\varphi^*} \\int \\cos^{\\mathbf{f}}{(\\varphi^*)} d\\mathbf{f} and (\\frac{\\partial}{\\partial \\varphi^*} \\int \\mu_{0}{(\\mathbf{f},\\varphi^*)} d\\mathbf{f})^{\\varphi^*} = (\\frac{\\partial}{\\partial \\varphi^*} \\int \\cos^{\\mathbf{f}}{(\\varphi^*)} d\\mathbf{f})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\mu_0')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)), Pow(Derivative(Integral(Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(S,\\ddot{x})} = S \\ddot{x}, then derive \\frac{\\partial}{\\partial \\ddot{x}} \\operatorname{P_{e}}{(S,\\ddot{x})} + 1 = S + 1, then obtain \\frac{\\partial}{\\partial \\ddot{x}} S \\ddot{x} + 1 = S + 1", "derivation": "\\operatorname{P_{e}}{(S,\\ddot{x})} = S \\ddot{x} and \\ddot{x} + \\operatorname{P_{e}}{(S,\\ddot{x})} = S \\ddot{x} + \\ddot{x} and \\frac{\\partial}{\\partial \\ddot{x}} (\\ddot{x} + \\operatorname{P_{e}}{(S,\\ddot{x})}) = \\frac{\\partial}{\\partial \\ddot{x}} (S \\ddot{x} + \\ddot{x}) and \\frac{\\partial}{\\partial \\ddot{x}} \\operatorname{P_{e}}{(S,\\ddot{x})} + 1 = S + 1 and \\frac{\\partial}{\\partial \\ddot{x}} S \\ddot{x} + 1 = S + 1", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["add", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\ddot{x}', commutative=True), Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Integer(1)), Add(Symbol('S', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('S', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Integer(1)), Add(Symbol('S', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(v_{x})} = \\log{(v_{x})}, then derive a + v_{x} - 1 = \\int \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} dv_{x} - 1, then obtain a + v_{x} - 1 = \\int 1 dv_{x} - 1", "derivation": "\\operatorname{C_{2}}{(v_{x})} = \\log{(v_{x})} and 1 = \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} and \\int 1 dv_{x} = \\int \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} dv_{x} and \\int 1 dv_{x} - \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} = \\int \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} dv_{x} - \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} and \\int 1 dv_{x} - 1 = \\int \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} dv_{x} - 1 and a + v_{x} - 1 = \\int \\frac{\\log{(v_{x})}}{\\operatorname{C_{2}}{(v_{x})}} dv_{x} - 1 and a + v_{x} - 1 = \\int 1 dv_{x} - 1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["divide", 1, "Function('C_2')(Symbol('v_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"], [["minus", 3, "Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True)))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('v_x', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True)))), Add(Integral(Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Integer(1), Tuple(Symbol('v_x', commutative=True))), Integer(-1)), Add(Integral(Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('a', commutative=True), Symbol('v_x', commutative=True), Integer(-1)), Add(Integral(Mul(Pow(Function('C_2')(Symbol('v_x', commutative=True)), Integer(-1)), log(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('a', commutative=True), Symbol('v_x', commutative=True), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('v_x', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given I{(\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}})}, then derive \\int I{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\mu + \\sin{(\\dot{\\mathbf{r}})}, then obtain \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (A_{x} + \\sin{(\\dot{\\mathbf{r}})}) = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\mu + \\sin{(\\dot{\\mathbf{r}})})", "derivation": "I{(\\dot{\\mathbf{r}})} = \\cos{(\\dot{\\mathbf{r}})} and \\int I{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\cos{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\int I{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\mu + \\sin{(\\dot{\\mathbf{r}})} and \\int \\cos{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\mu + \\sin{(\\dot{\\mathbf{r}})} and \\frac{d}{d \\dot{\\mathbf{r}}} \\int \\cos{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\mu + \\sin{(\\dot{\\mathbf{r}})}) and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (A_{x} + \\sin{(\\dot{\\mathbf{r}})}) = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\mu + \\sin{(\\dot{\\mathbf{r}})})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integral(cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = \\phi_2 + e^{\\dot{y}}, then derive \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = e^{\\dot{y}}, then obtain 0 = \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + e^{\\dot{y}})) - \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})}", "derivation": "\\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = \\phi_2 + e^{\\dot{y}} and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + e^{\\dot{y}}) and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = e^{\\dot{y}} and e^{\\dot{y}} = \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + e^{\\dot{y}}) and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + e^{\\dot{y}})) and 0 = \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + \\frac{\\partial}{\\partial \\dot{y}} (\\phi_2 + e^{\\dot{y}})) - \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{1}}{(\\phi_2,\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\dot{y}', commutative=True)), Derivative(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_2', commutative=True), Derivative(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Symbol('\\\\phi_2', commutative=True), Derivative(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('A_1')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(g)} = e^{g} + \\int e^{g} dg, then obtain - \\int \\mathbf{J}_M{(g)} dg - \\frac{\\dot{y} + 2 e^{g}}{\\dot{y} + e^{g}} = - L - 2 e^{g} - \\frac{\\dot{y} + 2 e^{g}}{\\dot{y} + e^{g}}", "derivation": "\\mathbf{J}_M{(g)} = e^{g} + \\int e^{g} dg and \\int \\mathbf{J}_M{(g)} dg = \\int (e^{g} + \\int e^{g} dg) dg and - \\int \\mathbf{J}_M{(g)} dg = - \\int (e^{g} + \\int e^{g} dg) dg and - \\frac{e^{g} + \\int e^{g} dg}{\\int e^{g} dg} - \\int \\mathbf{J}_M{(g)} dg = - \\frac{e^{g} + \\int e^{g} dg}{\\int e^{g} dg} - \\int (e^{g} + \\int e^{g} dg) dg and - \\int \\mathbf{J}_M{(g)} dg - \\frac{\\dot{y} + 2 e^{g}}{\\dot{y} + e^{g}} = - L - 2 e^{g} - \\frac{\\dot{y} + 2 e^{g}}{\\dot{y} + e^{g}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('g', commutative=True)), Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\mathbf{J}_M')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))))"], [["minus", 3, "Mul(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Function('\\\\mathbf{J}_M')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))), Add(Mul(Integer(-1), Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Pow(Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Add(exp(Symbol('g', commutative=True)), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{J}_M')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{y}', commutative=True), exp(Symbol('g', commutative=True))), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(2), exp(Symbol('g', commutative=True)))))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{y}', commutative=True), exp(Symbol('g', commutative=True))), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(2), exp(Symbol('g', commutative=True)))))))"]]}, {"prompt": "Given J{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain \\frac{d^{2}}{d \\mathbb{I}^{2}} J{(\\mathbb{I})} = e^{\\mathbb{I}}", "derivation": "J{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} J{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d^{2}}{d \\mathbb{I}^{2}} J{(\\mathbb{I})} = \\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}} and \\frac{d^{2}}{d \\mathbb{I}^{2}} J{(\\mathbb{I})} = e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('J')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), exp(Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(F_{H})} = \\sin{(F_{H})}, then derive 2 \\dot{z}{(F_{H})} \\frac{d}{d F_{H}} \\dot{z}{(F_{H})} = \\dot{z}{(F_{H})} \\cos{(F_{H})} + \\sin{(F_{H})} \\frac{d}{d F_{H}} \\dot{z}{(F_{H})}, then obtain 2 \\sin{(F_{H})} \\frac{d}{d F_{H}} \\sin{(F_{H})} = \\sin{(F_{H})} \\cos{(F_{H})} + \\sin{(F_{H})} \\frac{d}{d F_{H}} \\sin{(F_{H})}", "derivation": "\\dot{z}{(F_{H})} = \\sin{(F_{H})} and \\dot{z}^{2}{(F_{H})} = \\dot{z}{(F_{H})} \\sin{(F_{H})} and \\frac{d}{d F_{H}} \\dot{z}^{2}{(F_{H})} = \\frac{d}{d F_{H}} \\dot{z}{(F_{H})} \\sin{(F_{H})} and 2 \\dot{z}{(F_{H})} \\frac{d}{d F_{H}} \\dot{z}{(F_{H})} = \\dot{z}{(F_{H})} \\cos{(F_{H})} + \\sin{(F_{H})} \\frac{d}{d F_{H}} \\dot{z}{(F_{H})} and 2 \\sin{(F_{H})} \\frac{d}{d F_{H}} \\sin{(F_{H})} = \\sin{(F_{H})} \\cos{(F_{H})} + \\sin{(F_{H})} \\frac{d}{d F_{H}} \\sin{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["times", 1, "Function('\\\\dot{z}')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), Integer(2)), Mul(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), Integer(2)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Mul(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Mul(sin(Symbol('F_H', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), sin(Symbol('F_H', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Mul(sin(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))), Mul(sin(Symbol('F_H', commutative=True)), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then obtain \\frac{\\cos{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}^{2}{(V_{\\mathbf{E}})}} = \\frac{\\cos^{2}{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}^{3}{(V_{\\mathbf{E}})}}", "derivation": "\\operatorname{F_{H}}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and 1 = \\frac{\\cos{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}{(V_{\\mathbf{E}})}} and \\frac{1}{\\operatorname{F_{H}}{(V_{\\mathbf{E}})}} = \\frac{\\cos{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}^{2}{(V_{\\mathbf{E}})}} and \\frac{\\cos{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}^{2}{(V_{\\mathbf{E}})}} = \\frac{\\cos^{2}{(V_{\\mathbf{E}})}}{\\operatorname{F_{H}}^{3}{(V_{\\mathbf{E}})}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["divide", 1, "Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 2, "Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))"], "Equality(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Mul(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 3, "Mul(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Mul(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Pow(Function('F_H')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-3)), Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(n_{2},A_{2})} = A_{2} - n_{2} and S{(n_{2},A_{2})} = \\mathbf{J}{(n_{2},A_{2})} - 1, then obtain v_{z} (- n_{2} + S{(n_{2},A_{2})}) = v_{z} (A_{2} - 2 n_{2} - 1)", "derivation": "\\mathbf{J}{(n_{2},A_{2})} = A_{2} - n_{2} and S{(n_{2},A_{2})} = \\mathbf{J}{(n_{2},A_{2})} - 1 and - n_{2} + S{(n_{2},A_{2})} = - n_{2} + \\mathbf{J}{(n_{2},A_{2})} - 1 and - n_{2} + S{(n_{2},A_{2})} = A_{2} - 2 n_{2} - 1 and v_{z} (- n_{2} + S{(n_{2},A_{2})}) = v_{z} (A_{2} - 2 n_{2} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)), Add(Function('\\\\mathbf{J}')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)), Integer(-1)))"], [["minus", 2, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('S')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\mathbf{J}')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('S')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_2', commutative=True)), Integer(-1)))"], [["times", 4, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('S')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)))), Mul(Symbol('v_z', commutative=True), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('n_2', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})} = f_{\\mathbf{p}} + y^{\\prime}, then obtain (e^{f_{\\mathbf{p}} \\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})}})^{y^{\\prime}} = (e^{f_{\\mathbf{p}} (f_{\\mathbf{p}} + y^{\\prime})})^{y^{\\prime}}", "derivation": "\\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})} = f_{\\mathbf{p}} + y^{\\prime} and f_{\\mathbf{p}} \\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})} = f_{\\mathbf{p}} (f_{\\mathbf{p}} + y^{\\prime}) and e^{f_{\\mathbf{p}} \\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})}} = e^{f_{\\mathbf{p}} (f_{\\mathbf{p}} + y^{\\prime})} and (e^{f_{\\mathbf{p}} \\operatorname{n_{1}}{(f_{\\mathbf{p}},y^{\\prime})}})^{y^{\\prime}} = (e^{f_{\\mathbf{p}} (f_{\\mathbf{p}} + y^{\\prime})})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('n_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["exp", 2], "Equality(exp(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('n_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), exp(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(exp(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('n_1')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Pow(exp(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A,\\varphi^*)} = e^{A \\varphi^*}, then obtain (\\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int \\operatorname{E_{n}}{(A,\\varphi^*)} d\\varphi^*)}) (\\int e^{A \\varphi^*} d\\varphi^*)^{- A} = (\\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int e^{A \\varphi^*} d\\varphi^*)}) (\\int e^{A \\varphi^*} d\\varphi^*)^{- A}", "derivation": "\\operatorname{E_{n}}{(A,\\varphi^*)} = e^{A \\varphi^*} and \\int \\operatorname{E_{n}}{(A,\\varphi^*)} d\\varphi^* = \\int e^{A \\varphi^*} d\\varphi^* and \\log{(\\int \\operatorname{E_{n}}{(A,\\varphi^*)} d\\varphi^*)} = \\log{(\\int e^{A \\varphi^*} d\\varphi^*)} and \\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int \\operatorname{E_{n}}{(A,\\varphi^*)} d\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int e^{A \\varphi^*} d\\varphi^*)} and (\\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int \\operatorname{E_{n}}{(A,\\varphi^*)} d\\varphi^*)}) (\\int e^{A \\varphi^*} d\\varphi^*)^{- A} = (\\frac{\\partial}{\\partial \\varphi^*} \\log{(\\int e^{A \\varphi^*} d\\varphi^*)}) (\\int e^{A \\varphi^*} d\\varphi^*)^{- A}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('E_n')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), log(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(log(Integral(Function('E_n')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(log(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["divide", 4, "Pow(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('A', commutative=True))"], "Equality(Mul(Derivative(log(Integral(Function('E_n')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Pow(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)))), Mul(Derivative(log(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Pow(Integral(exp(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given A{(\\theta_2,L)} = L + \\log{(\\theta_2)} and \\Psi_{nl}{(\\theta_2,L)} = L + \\log{(\\theta_2)} - \\int A{(\\theta_2,L)} dL, then obtain \\frac{\\partial}{\\partial \\theta_2} \\Psi_{nl}{(\\theta_2,L)} = \\frac{\\partial}{\\partial \\theta_2} (- \\frac{L^{2}}{2} - L \\log{(\\theta_2)} + L - f_{\\mathbf{v}} + \\log{(\\theta_2)})", "derivation": "A{(\\theta_2,L)} = L + \\log{(\\theta_2)} and \\int A{(\\theta_2,L)} dL = \\int (L + \\log{(\\theta_2)}) dL and \\Psi_{nl}{(\\theta_2,L)} = L + \\log{(\\theta_2)} - \\int A{(\\theta_2,L)} dL and \\Psi_{nl}{(\\theta_2,L)} = L + \\log{(\\theta_2)} - \\int (L + \\log{(\\theta_2)}) dL and \\frac{\\partial}{\\partial \\theta_2} \\Psi_{nl}{(\\theta_2,L)} = \\frac{\\partial}{\\partial \\theta_2} (L + \\log{(\\theta_2)} - \\int (L + \\log{(\\theta_2)}) dL) and \\frac{\\partial}{\\partial \\theta_2} \\Psi_{nl}{(\\theta_2,L)} = \\frac{\\partial}{\\partial \\theta_2} (- \\frac{L^{2}}{2} - L \\log{(\\theta_2)} + L - f_{\\mathbf{v}} + \\log{(\\theta_2)})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Integral(Function('A')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('L', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('L', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\theta_2', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('L', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('L', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(x)} = \\sin{(e^{x})}, then obtain \\frac{d}{d x} 1 = \\frac{d}{d x} (\\operatorname{f_{E}}^{x}{(x)} \\sin{(e^{x})})^{- x} (\\sin{(e^{x})} \\sin^{x}{(e^{x})})^{x}", "derivation": "\\operatorname{f_{E}}{(x)} = \\sin{(e^{x})} and \\operatorname{f_{E}}^{x}{(x)} = \\sin^{x}{(e^{x})} and \\operatorname{f_{E}}^{x}{(x)} \\sin{(e^{x})} = \\sin{(e^{x})} \\sin^{x}{(e^{x})} and (\\operatorname{f_{E}}^{x}{(x)} \\sin{(e^{x})})^{x} = (\\sin{(e^{x})} \\sin^{x}{(e^{x})})^{x} and 1 = (\\operatorname{f_{E}}^{x}{(x)} \\sin{(e^{x})})^{- x} (\\sin{(e^{x})} \\sin^{x}{(e^{x})})^{x} and \\frac{d}{d x} 1 = \\frac{d}{d x} (\\operatorname{f_{E}}^{x}{(x)} \\sin{(e^{x})})^{- x} (\\sin{(e^{x})} \\sin^{x}{(e^{x})})^{x}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(exp(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["times", 2, "sin(exp(Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True)))), Mul(sin(exp(Symbol('x', commutative=True))), Pow(sin(exp(Symbol('x', commutative=True))), Symbol('x', commutative=True))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Mul(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), Pow(Mul(sin(exp(Symbol('x', commutative=True))), Pow(sin(exp(Symbol('x', commutative=True))), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["divide", 4, "Pow(Mul(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True)))), Symbol('x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True)))), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Mul(sin(exp(Symbol('x', commutative=True))), Pow(sin(exp(Symbol('x', commutative=True))), Symbol('x', commutative=True))), Symbol('x', commutative=True))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Pow(Function('f_E')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), sin(exp(Symbol('x', commutative=True)))), Mul(Integer(-1), Symbol('x', commutative=True))), Pow(Mul(sin(exp(Symbol('x', commutative=True))), Pow(sin(exp(Symbol('x', commutative=True))), Symbol('x', commutative=True))), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\theta,A_{1})} = A_{1} - \\theta and y{(\\theta,A_{1})} = A_{1} - \\theta, then obtain \\mathbf{J}_P{(\\theta)} \\varepsilon{(\\theta,A_{1})} = \\frac{\\mathbf{J}_P{(\\theta)} \\varepsilon{(\\theta,A_{1})} y{(\\theta,A_{1})}}{A_{1} - \\theta}", "derivation": "\\varepsilon{(\\theta,A_{1})} = A_{1} - \\theta and \\varepsilon^{2}{(\\theta,A_{1})} = (A_{1} - \\theta) \\varepsilon{(\\theta,A_{1})} and y{(\\theta,A_{1})} = A_{1} - \\theta and \\varepsilon^{2}{(\\theta,A_{1})} = \\varepsilon{(\\theta,A_{1})} y{(\\theta,A_{1})} and \\frac{\\mathbf{J}_P{(\\theta)} \\varepsilon^{2}{(\\theta,A_{1})}}{A_{1} - \\theta} = \\frac{\\mathbf{J}_P{(\\theta)} \\varepsilon{(\\theta,A_{1})} y{(\\theta,A_{1})}}{A_{1} - \\theta} and \\mathbf{J}_P{(\\theta)} \\varepsilon{(\\theta,A_{1})} = \\frac{\\mathbf{J}_P{(\\theta)} \\varepsilon{(\\theta,A_{1})} y{(\\theta,A_{1})}}{A_{1} - \\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Integer(2)), Mul(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Integer(2)), Mul(Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Function('y')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))))"], [["times", 4, "Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\theta', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Function('y')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\theta', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\theta', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True)), Function('y')(Symbol('\\\\theta', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(r)} = \\cos{(r)}, then obtain 1 = (\\frac{\\operatorname{v_{z}}{(r)} + \\cos{(r)}}{2 \\operatorname{v_{z}}{(r)}})^{r}", "derivation": "\\operatorname{v_{z}}{(r)} = \\cos{(r)} and 2 \\operatorname{v_{z}}{(r)} = \\operatorname{v_{z}}{(r)} + \\cos{(r)} and 1 = \\frac{\\operatorname{v_{z}}{(r)} + \\cos{(r)}}{2 \\operatorname{v_{z}}{(r)}} and 1 = (\\frac{\\operatorname{v_{z}}{(r)} + \\cos{(r)}}{2 \\operatorname{v_{z}}{(r)}})^{r}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["add", 1, "Function('v_z')(Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Function('v_z')(Symbol('r', commutative=True))), Add(Function('v_z')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('v_z')(Symbol('r', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('v_z')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Pow(Function('v_z')(Symbol('r', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Integer(1), Pow(Mul(Rational(1, 2), Add(Function('v_z')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Pow(Function('v_z')(Symbol('r', commutative=True)), Integer(-1))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(n)} = \\cos{(n)}, then derive \\int \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} dn = f_{\\mathbf{p}} + n, then obtain \\frac{d}{d f_{\\mathbf{p}}} \\int \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} dn = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (f_{\\mathbf{p}} + n)", "derivation": "\\operatorname{E_{x}}{(n)} = \\cos{(n)} and \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} = 1 and \\int \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} dn = \\int 1 dn and \\int \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} dn = f_{\\mathbf{p}} + n and \\frac{d}{d f_{\\mathbf{p}}} \\int \\frac{\\operatorname{E_{x}}{(n)}}{\\cos{(n)}} dn = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (f_{\\mathbf{p}} + n)", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["divide", 1, "cos(Symbol('n', commutative=True))"], "Equality(Mul(Function('E_x')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Function('E_x')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Tuple(Symbol('n', commutative=True))), Integral(Integer(1), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('E_x')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Tuple(Symbol('n', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('E_x')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(v_{2})} = \\cos{(v_{2})}, then obtain 3 \\omega{(v_{2})} + \\cos{(v_{2})} = 4 \\omega{(v_{2})}", "derivation": "\\omega{(v_{2})} = \\cos{(v_{2})} and \\omega{(v_{2})} + \\cos{(v_{2})} = 2 \\cos{(v_{2})} and 2 \\omega{(v_{2})} + 2 \\cos{(v_{2})} = \\omega{(v_{2})} + 3 \\cos{(v_{2})} and 3 \\omega{(v_{2})} + \\cos{(v_{2})} = \\omega{(v_{2})} + 3 \\cos{(v_{2})} and 3 \\omega{(v_{2})} + \\cos{(v_{2})} = 2 \\omega{(v_{2})} + 2 \\cos{(v_{2})} and 2 \\omega{(v_{2})} = 2 \\cos{(v_{2})} and 3 \\omega{(v_{2})} + \\cos{(v_{2})} = 4 \\omega{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["add", 1, "cos(Symbol('v_2', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Mul(Integer(2), cos(Symbol('v_2', commutative=True))))"], [["add", 2, "Add(Function('\\\\omega')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\omega')(Symbol('v_2', commutative=True))), Mul(Integer(2), cos(Symbol('v_2', commutative=True)))), Add(Function('\\\\omega')(Symbol('v_2', commutative=True)), Mul(Integer(3), cos(Symbol('v_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('\\\\omega')(Symbol('v_2', commutative=True))), cos(Symbol('v_2', commutative=True))), Add(Function('\\\\omega')(Symbol('v_2', commutative=True)), Mul(Integer(3), cos(Symbol('v_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(3), Function('\\\\omega')(Symbol('v_2', commutative=True))), cos(Symbol('v_2', commutative=True))), Add(Mul(Integer(2), Function('\\\\omega')(Symbol('v_2', commutative=True))), Mul(Integer(2), cos(Symbol('v_2', commutative=True)))))"], [["minus", 4, "Add(Function('\\\\omega')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('v_2', commutative=True))), Mul(Integer(2), cos(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(3), Function('\\\\omega')(Symbol('v_2', commutative=True))), cos(Symbol('v_2', commutative=True))), Mul(Integer(4), Function('\\\\omega')(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given A{(y)} = \\cos{(y)} and \\mathbf{p}{(y)} = \\cos{(y)}, then obtain 0 = - \\mathbf{p}{(y)} + \\cos{(y)}", "derivation": "A{(y)} = \\cos{(y)} and \\mathbf{p}{(y)} = \\cos{(y)} and \\mathbf{p}{(y)} = A{(y)} and A{(y)} + \\mathbf{p}{(y)} = 2 A{(y)} and 0 = A{(y)} - \\mathbf{p}{(y)} and 0 = - \\mathbf{p}{(y)} + \\cos{(y)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Function('A')(Symbol('y', commutative=True)))"], [["add", 3, "Function('A')(Symbol('y', commutative=True))"], "Equality(Add(Function('A')(Symbol('y', commutative=True)), Function('\\\\mathbf{p}')(Symbol('y', commutative=True))), Mul(Integer(2), Function('A')(Symbol('y', commutative=True))))"], [["minus", 4, "Add(Function('A')(Symbol('y', commutative=True)), Function('\\\\mathbf{p}')(Symbol('y', commutative=True)))"], "Equality(Integer(0), Add(Function('A')(Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))"]]}, {"prompt": "Given Z{(m_{s})} = \\log{(m_{s})} and p{(m_{s})} = \\log{(m_{s})}, then obtain \\frac{d}{d m_{s}} \\log{(m_{s})}^{m_{s}} = \\frac{d}{d m_{s}} Z^{m_{s}}{(m_{s})}", "derivation": "Z{(m_{s})} = \\log{(m_{s})} and p{(m_{s})} = \\log{(m_{s})} and p^{m_{s}}{(m_{s})} = \\log{(m_{s})}^{m_{s}} and p^{m_{s}}{(m_{s})} = Z^{m_{s}}{(m_{s})} and \\frac{d}{d m_{s}} p^{m_{s}}{(m_{s})} = \\frac{d}{d m_{s}} Z^{m_{s}}{(m_{s})} and \\frac{d}{d m_{s}} \\log{(m_{s})}^{m_{s}} = \\frac{d}{d m_{s}} Z^{m_{s}}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], ["renaming_premise", "Equality(Function('p')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(log(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Function('Z')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["differentiate", 4, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Pow(Function('p')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Pow(Function('Z')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Pow(log(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Pow(Function('Z')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(M)} = \\sin{(M)}, then derive \\frac{d}{d M} \\operatorname{A_{z}}{(M)} = \\cos{(M)}, then obtain \\sin{(M)} + \\cos{(M)} = \\sin{(M)} + \\frac{d}{d M} \\operatorname{A_{z}}{(M)}", "derivation": "\\operatorname{A_{z}}{(M)} = \\sin{(M)} and \\frac{d}{d M} \\operatorname{A_{z}}{(M)} = \\frac{d}{d M} \\sin{(M)} and \\frac{d}{d M} \\operatorname{A_{z}}{(M)} = \\cos{(M)} and \\frac{d}{d M} \\sin{(M)} = \\cos{(M)} and \\sin{(M)} + \\frac{d}{d M} \\sin{(M)} = \\sin{(M)} + \\cos{(M)} and \\sin{(M)} + \\frac{d}{d M} \\sin{(M)} = \\sin{(M)} + \\frac{d}{d M} \\operatorname{A_{z}}{(M)} and \\sin{(M)} + \\cos{(M)} = \\sin{(M)} + \\frac{d}{d M} \\operatorname{A_{z}}{(M)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), cos(Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), cos(Symbol('M', commutative=True)))"], [["add", 4, "sin(Symbol('M', commutative=True))"], "Equality(Add(sin(Symbol('M', commutative=True)), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(sin(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(sin(Symbol('M', commutative=True)), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(sin(Symbol('M', commutative=True)), Derivative(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(sin(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Add(sin(Symbol('M', commutative=True)), Derivative(Function('A_z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho{(\\eta,v_{y})} = \\frac{\\sin{(v_{y})}}{\\eta}, then obtain \\frac{\\partial}{\\partial P_{g}} (- P_{g} - \\mathbf{s} + \\rho{(\\eta,v_{y})}) = \\frac{\\partial}{\\partial P_{g}} (- P_{g} - \\mathbf{s} + \\frac{\\sin{(v_{y})}}{\\eta})", "derivation": "\\rho{(\\eta,v_{y})} = \\frac{\\sin{(v_{y})}}{\\eta} and - \\mathbf{s} + \\rho{(\\eta,v_{y})} = - \\mathbf{s} + \\frac{\\sin{(v_{y})}}{\\eta} and - P_{g} - \\mathbf{s} + \\rho{(\\eta,v_{y})} = - P_{g} - \\mathbf{s} + \\frac{\\sin{(v_{y})}}{\\eta} and \\frac{\\partial}{\\partial P_{g}} (- P_{g} - \\mathbf{s} + \\rho{(\\eta,v_{y})}) = \\frac{\\partial}{\\partial P_{g}} (- P_{g} - \\mathbf{s} + \\frac{\\sin{(v_{y})}}{\\eta})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True)))))"], [["minus", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True)))))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(U,v_{2})} = U^{v_{2}} and \\operatorname{f^{\\prime}}{(U,v_{2})} = v_{2} V{(U,v_{2})}, then obtain \\operatorname{f^{\\prime}}^{v_{2}}{(U,v_{2})} = (U^{v_{2}} v_{2})^{v_{2}}", "derivation": "V{(U,v_{2})} = U^{v_{2}} and v_{2} V{(U,v_{2})} = U^{v_{2}} v_{2} and \\operatorname{f^{\\prime}}{(U,v_{2})} = v_{2} V{(U,v_{2})} and \\operatorname{f^{\\prime}}{(U,v_{2})} = U^{v_{2}} v_{2} and \\operatorname{f^{\\prime}}^{v_{2}}{(U,v_{2})} = (U^{v_{2}} v_{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('v_2', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('v_2', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('U', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\psi,\\hat{X})} = \\frac{\\psi}{\\hat{X}}, then obtain - \\hat{X} = - \\frac{\\hat{X} (\\eta{(\\psi,\\hat{X})} + \\frac{\\psi}{\\hat{X}})}{2 \\eta{(\\psi,\\hat{X})}}", "derivation": "\\eta{(\\psi,\\hat{X})} = \\frac{\\psi}{\\hat{X}} and 2 \\eta{(\\psi,\\hat{X})} = \\eta{(\\psi,\\hat{X})} + \\frac{\\psi}{\\hat{X}} and 2 \\hat{X} \\eta{(\\psi,\\hat{X})} = \\hat{X} (\\eta{(\\psi,\\hat{X})} + \\frac{\\psi}{\\hat{X}}) and \\hat{X} = \\frac{\\hat{X} (\\eta{(\\psi,\\hat{X})} + \\frac{\\psi}{\\hat{X}})}{2 \\eta{(\\psi,\\hat{X})}} and - \\hat{X} = - \\frac{\\hat{X} (\\eta{(\\psi,\\hat{X})} + \\frac{\\psi}{\\hat{X}})}{2 \\eta{(\\psi,\\hat{X})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1))"], "Equality(Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True), Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Symbol('\\\\hat{X}', commutative=True), Add(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\hat{X}', commutative=True), Add(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Pow(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\hat{X}', commutative=True), Add(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Pow(Function('\\\\eta')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given v{(k)} = \\log{(k)}, then obtain \\iint \\frac{k v{(k)} - v{(k)}}{k \\log{(k)} - k} dk dk = \\iint \\frac{k \\log{(k)} - v{(k)}}{k \\log{(k)} - k} dk dk", "derivation": "v{(k)} = \\log{(k)} and k v{(k)} = k \\log{(k)} and k v{(k)} - v{(k)} = k \\log{(k)} - v{(k)} and \\frac{k v{(k)} - v{(k)}}{k \\log{(k)} - k} = \\frac{k \\log{(k)} - v{(k)}}{k \\log{(k)} - k} and \\int \\frac{k v{(k)} - v{(k)}}{k \\log{(k)} - k} dk = \\int \\frac{k \\log{(k)} - v{(k)}}{k \\log{(k)} - k} dk and \\iint \\frac{k v{(k)} - v{(k)}}{k \\log{(k)} - k} dk dk = \\iint \\frac{k \\log{(k)} - v{(k)}}{k \\log{(k)} - k} dk dk", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('v')(Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))))"], [["minus", 2, "Function('v')(Symbol('k', commutative=True))"], "Equality(Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True)))), Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True)))))"], [["divide", 3, "Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True)))), Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1)), Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True))))))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True)))), Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1))), Tuple(Symbol('k', commutative=True))), Integral(Mul(Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1)), Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True)))), Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Pow(Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integer(-1)), Add(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Function('v')(Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given f{(a)} = \\int \\cos{(a)} da, then derive \\frac{f{(a)} + \\cos{(a)}}{i + \\sin{(a)}} = \\frac{i + \\sin{(a)} + \\cos{(a)}}{i + \\sin{(a)}}, then obtain \\frac{\\cos{(a)} + \\int \\cos{(a)} da}{i + \\sin{(a)}} = \\frac{i + \\sin{(a)} + \\cos{(a)}}{i + \\sin{(a)}}", "derivation": "f{(a)} = \\int \\cos{(a)} da and f{(a)} + \\cos{(a)} = \\cos{(a)} + \\int \\cos{(a)} da and \\frac{f{(a)} + \\cos{(a)}}{\\int \\cos{(a)} da} = \\frac{\\cos{(a)} + \\int \\cos{(a)} da}{\\int \\cos{(a)} da} and \\frac{f{(a)} + \\cos{(a)}}{i + \\sin{(a)}} = \\frac{i + \\sin{(a)} + \\cos{(a)}}{i + \\sin{(a)}} and \\frac{\\cos{(a)} + \\int \\cos{(a)} da}{i + \\sin{(a)}} = \\frac{i + \\sin{(a)} + \\cos{(a)}}{i + \\sin{(a)}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('a', commutative=True)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["add", 1, "cos(Symbol('a', commutative=True))"], "Equality(Add(Function('f')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Add(cos(Symbol('a', commutative=True)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["divide", 2, "Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Mul(Add(Function('f')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Pow(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))), Mul(Add(cos(Symbol('a', commutative=True)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Pow(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Add(Function('f')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))), Mul(Pow(Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Add(cos(Symbol('a', commutative=True)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))), Mul(Pow(Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Add(Symbol('i', commutative=True), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given c{(E,B)} = \\frac{E}{B}, then obtain \\frac{\\partial}{\\partial B} \\frac{2 \\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} = \\frac{\\partial}{\\partial B} (\\frac{\\int \\frac{E}{B} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} + \\frac{\\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB})", "derivation": "c{(E,B)} = \\frac{E}{B} and \\int c{(E,B)} dB = \\int \\frac{E}{B} dB and \\frac{\\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} = \\frac{\\int \\frac{E}{B} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} and \\frac{2 \\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} = \\frac{\\int \\frac{E}{B} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} + \\frac{\\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} and \\frac{\\partial}{\\partial B} \\frac{2 \\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} = \\frac{\\partial}{\\partial B} (\\frac{\\int \\frac{E}{B} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB} + \\frac{\\int c{(E,B)} dB}{- c{(E,B)} + \\int \\frac{E}{B} dB})", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["add", 3, "Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"], [["differentiate", 4, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E', commutative=True)), Tuple(Symbol('B', commutative=True)))), Integer(-1)), Integral(Function('c')(Symbol('E', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})}, then obtain \\cos{(g_{\\varepsilon})} \\frac{d}{d g_{\\varepsilon}} 1 = \\cos{(g_{\\varepsilon})} \\frac{d}{d g_{\\varepsilon}} (\\frac{\\cos{(g_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(g_{\\varepsilon})}})^{g_{\\varepsilon}}", "derivation": "\\hat{\\mathbf{x}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and 1 = \\frac{\\cos{(g_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(g_{\\varepsilon})}} and 1 = (\\frac{\\cos{(g_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(g_{\\varepsilon})}})^{g_{\\varepsilon}} and \\frac{d}{d g_{\\varepsilon}} 1 = \\frac{d}{d g_{\\varepsilon}} (\\frac{\\cos{(g_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(g_{\\varepsilon})}})^{g_{\\varepsilon}} and \\cos{(g_{\\varepsilon})} \\frac{d}{d g_{\\varepsilon}} 1 = \\cos{(g_{\\varepsilon})} \\frac{d}{d g_{\\varepsilon}} (\\frac{\\cos{(g_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(g_{\\varepsilon})}})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["times", 4, "cos(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Pow(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(n_{2},\\varphi)} = \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}), then derive \\operatorname{f_{\\mathbf{v}}}{(n_{2},\\varphi)} = 1, then obtain \\frac{(\\varphi \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}))^{n_{2}}}{\\sin{(\\omega)} - 1} = \\frac{\\varphi^{n_{2}}}{\\sin{(\\omega)} - 1}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(n_{2},\\varphi)} = \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}) and \\operatorname{f_{\\mathbf{v}}}{(n_{2},\\varphi)} = 1 and \\varphi \\operatorname{f_{\\mathbf{v}}}{(n_{2},\\varphi)} = \\varphi and \\varphi \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}) = \\varphi and (\\varphi \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}))^{n_{2}} = \\varphi^{n_{2}} and \\frac{(\\varphi \\frac{\\partial}{\\partial n_{2}} (\\varphi + n_{2}))^{n_{2}}}{\\sin{(\\omega)} - 1} = \\frac{\\varphi^{n_{2}}}{\\sin{(\\omega)} - 1}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1))"], [["times", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('\\\\varphi', commutative=True))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 5, "Add(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Mul(Symbol('\\\\varphi', commutative=True), Derivative(Add(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True)), Pow(Add(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)), Integer(-1))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Pow(Add(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(F_{g},\\mu,y^{\\prime})} = \\frac{F_{g} + y^{\\prime}}{\\mu}, then obtain 1 - \\frac{F_{g} + y^{\\prime}}{\\mu} = 1 + \\frac{- F_{g} - y^{\\prime}}{\\mu}", "derivation": "\\mathbf{B}{(F_{g},\\mu,y^{\\prime})} = \\frac{F_{g} + y^{\\prime}}{\\mu} and \\mathbf{B}{(F_{g},\\mu,y^{\\prime})} - 1 = -1 + \\frac{F_{g} + y^{\\prime}}{\\mu} and 1 - \\mathbf{B}{(F_{g},\\mu,y^{\\prime})} = 1 - \\frac{F_{g} + y^{\\prime}}{\\mu} and 1 - \\mathbf{B}{(F_{g},\\mu,y^{\\prime})} = 1 + \\frac{- F_{g} - y^{\\prime}}{\\mu} and 1 - \\frac{F_{g} + y^{\\prime}}{\\mu} = 1 + \\frac{- F_{g} - y^{\\prime}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(m_{s})} = \\log{(m_{s})}, then derive \\int \\theta_{1}{(m_{s})} dm_{s} = \\omega + m_{s} \\log{(m_{s})} - m_{s}, then obtain \\int \\log{(m_{s})} dm_{s} = \\omega + m_{s} \\theta_{1}{(m_{s})} - m_{s}", "derivation": "\\theta_{1}{(m_{s})} = \\log{(m_{s})} and \\int \\theta_{1}{(m_{s})} dm_{s} = \\int \\log{(m_{s})} dm_{s} and \\int \\theta_{1}{(m_{s})} dm_{s} = \\omega + m_{s} \\log{(m_{s})} - m_{s} and \\int \\theta_{1}{(m_{s})} dm_{s} = \\omega + m_{s} \\theta_{1}{(m_{s})} - m_{s} and \\int \\log{(m_{s})} dm_{s} = \\omega + m_{s} \\theta_{1}{(m_{s})} - m_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(log(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('m_s', commutative=True), log(Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\theta_1')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('m_s', commutative=True), Function('\\\\theta_1')(Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Mul(Symbol('m_s', commutative=True), Function('\\\\theta_1')(Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\psi^*)} = \\sin{(\\psi^*)} and \\theta{(\\pi)} = \\sin{(\\pi)}, then obtain 2 \\sin{(\\psi^*)} + \\int 2 \\theta{(\\pi)} \\sin{(\\psi^*)} d\\psi^* = 2 \\sin{(\\psi^*)} + \\int 2 \\sin{(\\pi)} \\sin{(\\psi^*)} d\\psi^*", "derivation": "\\mathbf{P}{(\\psi^*)} = \\sin{(\\psi^*)} and \\mathbf{P}{(\\psi^*)} + \\sin{(\\psi^*)} = 2 \\sin{(\\psi^*)} and \\theta{(\\pi)} = \\sin{(\\pi)} and (\\mathbf{P}{(\\psi^*)} + \\sin{(\\psi^*)}) \\theta{(\\pi)} = (\\mathbf{P}{(\\psi^*)} + \\sin{(\\psi^*)}) \\sin{(\\pi)} and 2 \\theta{(\\pi)} \\sin{(\\psi^*)} = 2 \\sin{(\\pi)} \\sin{(\\psi^*)} and \\int 2 \\theta{(\\pi)} \\sin{(\\psi^*)} d\\psi^* = \\int 2 \\sin{(\\pi)} \\sin{(\\psi^*)} d\\psi^* and 2 \\sin{(\\psi^*)} + \\int 2 \\theta{(\\pi)} \\sin{(\\psi^*)} d\\psi^* = 2 \\sin{(\\psi^*)} + \\int 2 \\sin{(\\pi)} \\sin{(\\psi^*)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["divide", 3, "Pow(Add(Function('\\\\mathbf{P}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Integer(-1))"], "Equality(Mul(Add(Function('\\\\mathbf{P}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Function('\\\\theta')(Symbol('\\\\pi', commutative=True))), Mul(Add(Function('\\\\mathbf{P}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), sin(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 6, "Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(C_{1},\\mathbf{M})} = \\frac{\\mathbf{M}}{C_{1}}, then obtain (\\int (- \\mathbf{M} - \\tilde{g}{(C_{1},\\mathbf{M})}) dC_{1})^{C_{1}} = (\\int (- \\mathbf{M} - \\frac{\\mathbf{M}}{C_{1}}) dC_{1})^{C_{1}}", "derivation": "\\tilde{g}{(C_{1},\\mathbf{M})} = \\frac{\\mathbf{M}}{C_{1}} and - \\tilde{g}{(C_{1},\\mathbf{M})} = - \\frac{\\mathbf{M}}{C_{1}} and - \\mathbf{M} - \\tilde{g}{(C_{1},\\mathbf{M})} = - \\mathbf{M} - \\frac{\\mathbf{M}}{C_{1}} and \\int (- \\mathbf{M} - \\tilde{g}{(C_{1},\\mathbf{M})}) dC_{1} = \\int (- \\mathbf{M} - \\frac{\\mathbf{M}}{C_{1}}) dC_{1} and (\\int (- \\mathbf{M} - \\tilde{g}{(C_{1},\\mathbf{M})}) dC_{1})^{C_{1}} = (\\int (- \\mathbf{M} - \\frac{\\mathbf{M}}{C_{1}}) dC_{1})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["power", 4, "Symbol('C_1', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{v})} = e^{\\mathbf{v}}, then derive \\frac{d}{d \\mathbf{v}} \\mathbf{J}_f{(\\mathbf{v})} + 1 = e^{\\mathbf{v}} + 1, then obtain \\mathbf{J}_f{(\\mathbf{v})} + 1 = e^{\\mathbf{v}} + 1", "derivation": "\\mathbf{J}_f{(\\mathbf{v})} = e^{\\mathbf{v}} and \\mathbf{v} + \\mathbf{J}_f{(\\mathbf{v})} = \\mathbf{v} + e^{\\mathbf{v}} and \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + \\mathbf{J}_f{(\\mathbf{v})}) = \\frac{d}{d \\mathbf{v}} (\\mathbf{v} + e^{\\mathbf{v}}) and \\frac{d}{d \\mathbf{v}} \\mathbf{J}_f{(\\mathbf{v})} + 1 = e^{\\mathbf{v}} + 1 and \\frac{d}{d \\mathbf{v}} \\mathbf{J}_f{(\\mathbf{v})} + 1 = \\mathbf{J}_f{(\\mathbf{v})} + 1 and \\mathbf{J}_f{(\\mathbf{v})} + 1 = e^{\\mathbf{v}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Integer(1)), Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)), Add(exp(Symbol('\\\\mathbf{v}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given Z{(\\phi_2)} = \\log{(\\phi_2)}, then derive (\\frac{d}{d \\phi_2} Z{(\\phi_2)})^{2} = \\frac{\\frac{d}{d \\phi_2} Z{(\\phi_2)}}{\\phi_2}, then obtain e^{(\\frac{d}{d \\phi_2} \\log{(\\phi_2)})^{2}} = e^{\\frac{\\frac{d}{d \\phi_2} \\log{(\\phi_2)}}{\\phi_2}}", "derivation": "Z{(\\phi_2)} = \\log{(\\phi_2)} and \\frac{d}{d \\phi_2} Z{(\\phi_2)} = \\frac{d}{d \\phi_2} \\log{(\\phi_2)} and (\\frac{d}{d \\phi_2} Z{(\\phi_2)})^{2} = \\frac{d}{d \\phi_2} Z{(\\phi_2)} \\frac{d}{d \\phi_2} \\log{(\\phi_2)} and (\\frac{d}{d \\phi_2} Z{(\\phi_2)})^{2} = \\frac{\\frac{d}{d \\phi_2} Z{(\\phi_2)}}{\\phi_2} and (\\frac{d}{d \\phi_2} \\log{(\\phi_2)})^{2} = \\frac{\\frac{d}{d \\phi_2} \\log{(\\phi_2)}}{\\phi_2} and e^{(\\frac{d}{d \\phi_2} \\log{(\\phi_2)})^{2}} = e^{\\frac{\\frac{d}{d \\phi_2} \\log{(\\phi_2)}}{\\phi_2}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(Function('Z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["exp", 5], "Equality(exp(Pow(Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2))), exp(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{B})} = e^{\\mathbf{B}}, then derive \\frac{\\int \\mathbf{F}{(\\mathbf{B})} d\\mathbf{B}}{x^\\prime} = \\frac{\\theta_1 + e^{\\mathbf{B}}}{x^\\prime}, then obtain \\frac{\\int e^{\\mathbf{B}} d\\mathbf{B}}{x^\\prime} = \\frac{\\theta_1 + e^{\\mathbf{B}}}{x^\\prime}", "derivation": "\\mathbf{F}{(\\mathbf{B})} = e^{\\mathbf{B}} and \\int \\mathbf{F}{(\\mathbf{B})} d\\mathbf{B} = \\int e^{\\mathbf{B}} d\\mathbf{B} and \\frac{\\int \\mathbf{F}{(\\mathbf{B})} d\\mathbf{B}}{x^\\prime} = \\frac{\\int e^{\\mathbf{B}} d\\mathbf{B}}{x^\\prime} and \\frac{\\int \\mathbf{F}{(\\mathbf{B})} d\\mathbf{B}}{x^\\prime} = \\frac{\\theta_1 + e^{\\mathbf{B}}}{x^\\prime} and \\frac{\\int e^{\\mathbf{B}} d\\mathbf{B}}{x^\\prime} = \\frac{\\theta_1 + e^{\\mathbf{B}}}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(exp(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given H{(v_{2})} = \\cos{(e^{v_{2}})}, then obtain H{(v_{2})} + e^{- v_{2}} \\cos{(e^{v_{2}})} + e^{- v_{2}} = \\cos{(e^{v_{2}})} + e^{- v_{2}} \\cos{(e^{v_{2}})} + e^{- v_{2}}", "derivation": "H{(v_{2})} = \\cos{(e^{v_{2}})} and H{(v_{2})} e^{- v_{2}} = e^{- v_{2}} \\cos{(e^{v_{2}})} and H{(v_{2})} + H{(v_{2})} e^{- v_{2}} = H{(v_{2})} e^{- v_{2}} + \\cos{(e^{v_{2}})} and H{(v_{2})} + e^{- v_{2}} \\cos{(e^{v_{2}})} = \\cos{(e^{v_{2}})} + e^{- v_{2}} \\cos{(e^{v_{2}})} and H{(v_{2})} + e^{- v_{2}} \\cos{(e^{v_{2}})} + e^{- v_{2}} = \\cos{(e^{v_{2}})} + e^{- v_{2}} \\cos{(e^{v_{2}})} + e^{- v_{2}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_2', commutative=True)), cos(exp(Symbol('v_2', commutative=True))))"], [["divide", 1, "exp(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('H')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), cos(exp(Symbol('v_2', commutative=True)))))"], [["add", 1, "Mul(Function('H')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))))"], "Equality(Add(Function('H')(Symbol('v_2', commutative=True)), Mul(Function('H')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True))))), Add(Mul(Function('H')(Symbol('v_2', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))), cos(exp(Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('H')(Symbol('v_2', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), cos(exp(Symbol('v_2', commutative=True))))), Add(cos(exp(Symbol('v_2', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), cos(exp(Symbol('v_2', commutative=True))))))"], [["add", 4, "exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))"], "Equality(Add(Function('H')(Symbol('v_2', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), cos(exp(Symbol('v_2', commutative=True)))), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))), Add(cos(exp(Symbol('v_2', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('v_2', commutative=True))), cos(exp(Symbol('v_2', commutative=True)))), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{J}_M)} = \\sin{(\\sin{(\\mathbf{J}_M)})} and i{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then derive \\frac{d}{d \\mathbf{J}_M} \\mathbf{s}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} \\cos{(\\sin{(\\mathbf{J}_M)})}, then obtain \\frac{d}{d \\mathbf{J}_M} \\sin{(\\sin{(\\mathbf{J}_M)})} = i{(\\mathbf{J}_M)} \\cos{(\\sin{(\\mathbf{J}_M)})}", "derivation": "\\mathbf{s}{(\\mathbf{J}_M)} = \\sin{(\\sin{(\\mathbf{J}_M)})} and \\frac{d}{d \\mathbf{J}_M} \\mathbf{s}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\sin{(\\mathbf{J}_M)})} and \\frac{d}{d \\mathbf{J}_M} \\mathbf{s}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} \\cos{(\\sin{(\\mathbf{J}_M)})} and i{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\sin{(\\sin{(\\mathbf{J}_M)})} = \\cos{(\\mathbf{J}_M)} \\cos{(\\sin{(\\mathbf{J}_M)})} and \\frac{d}{d \\mathbf{J}_M} \\sin{(\\sin{(\\mathbf{J}_M)})} = i{(\\mathbf{J}_M)} \\cos{(\\sin{(\\mathbf{J}_M)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(sin(sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Mul(Function('i')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given l{(n_{1},v_{y})} = v_{y}^{n_{1}} and \\operatorname{v_{y}}{(l,\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P^{l})}, then obtain (v_{y}^{n_{1}} \\operatorname{v_{y}}{(l,\\mathbf{J}_P)})^{l} = (v_{y}^{n_{1}} \\log{(\\mathbf{J}_P^{l})})^{l}", "derivation": "l{(n_{1},v_{y})} = v_{y}^{n_{1}} and \\operatorname{v_{y}}{(l,\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P^{l})} and l{(n_{1},v_{y})} \\operatorname{v_{y}}{(l,\\mathbf{J}_P)} = l{(n_{1},v_{y})} \\log{(\\mathbf{J}_P^{l})} and v_{y}^{n_{1}} \\operatorname{v_{y}}{(l,\\mathbf{J}_P)} = v_{y}^{n_{1}} \\log{(\\mathbf{J}_P^{l})} and (v_{y}^{n_{1}} \\operatorname{v_{y}}{(l,\\mathbf{J}_P)})^{l} = (v_{y}^{n_{1}} \\log{(\\mathbf{J}_P^{l})})^{l}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('n_1', commutative=True), Symbol('v_y', commutative=True)), Pow(Symbol('v_y', commutative=True), Symbol('n_1', commutative=True)))"], ["get_premise", "Equality(Function('v_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('l', commutative=True))))"], [["times", 2, "Function('l')(Symbol('n_1', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Function('l')(Symbol('n_1', commutative=True), Symbol('v_y', commutative=True)), Function('v_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Function('l')(Symbol('n_1', commutative=True), Symbol('v_y', commutative=True)), log(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Symbol('n_1', commutative=True)), Function('v_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('v_y', commutative=True), Symbol('n_1', commutative=True)), log(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('l', commutative=True)))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v_y', commutative=True), Symbol('n_1', commutative=True)), Function('v_y')(Symbol('l', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('v_y', commutative=True), Symbol('n_1', commutative=True)), log(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('l', commutative=True)))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\Omega)} = \\log{(\\Omega)}, then derive \\frac{d}{d \\Omega} \\operatorname{f_{E}}{(\\Omega)} = \\frac{1}{\\Omega}, then obtain (\\frac{1}{\\Omega})^{\\Omega} = (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega}", "derivation": "\\operatorname{f_{E}}{(\\Omega)} = \\log{(\\Omega)} and \\frac{d}{d \\Omega} \\operatorname{f_{E}}{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and \\frac{d}{d \\Omega} \\operatorname{f_{E}}{(\\Omega)} = \\frac{1}{\\Omega} and \\frac{1}{\\Omega} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and (\\frac{1}{\\Omega})^{\\Omega} = (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\chi{(\\omega,\\mathbf{J}_P)} = \\sin{(\\frac{\\omega}{\\mathbf{J}_P})}, then derive \\int \\chi{(\\omega,\\mathbf{J}_P)} d\\mathbf{J}_P = \\lambda + \\mathbf{J}_P \\sin{(\\frac{\\omega}{\\mathbf{J}_P})} - \\frac{\\omega \\log{(\\frac{\\omega^{2}}{\\mathbf{J}_P^{2}})}}{2} + \\omega \\log{(\\frac{\\omega}{\\mathbf{J}_P})} - \\omega \\operatorname{Ci}{(\\frac{\\omega}{\\mathbf{J}_P})}, then obtain \\int \\chi{(\\omega,\\mathbf{J}_P)} d\\mathbf{J}_P = \\lambda + \\mathbf{J}_P \\chi{(\\omega,\\mathbf{J}_P)} - \\frac{\\omega \\log{(\\frac{\\omega^{2}}{\\mathbf{J}_P^{2}})}}{2} + \\omega \\log{(\\frac{\\omega}{\\mathbf{J}_P})} - \\omega \\operatorname{Ci}{(\\frac{\\omega}{\\mathbf{J}_P})}", "derivation": "\\chi{(\\omega,\\mathbf{J}_P)} = \\sin{(\\frac{\\omega}{\\mathbf{J}_P})} and \\int \\chi{(\\omega,\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\sin{(\\frac{\\omega}{\\mathbf{J}_P})} d\\mathbf{J}_P and \\int \\chi{(\\omega,\\mathbf{J}_P)} d\\mathbf{J}_P = \\lambda + \\mathbf{J}_P \\sin{(\\frac{\\omega}{\\mathbf{J}_P})} - \\frac{\\omega \\log{(\\frac{\\omega^{2}}{\\mathbf{J}_P^{2}})}}{2} + \\omega \\log{(\\frac{\\omega}{\\mathbf{J}_P})} - \\omega \\operatorname{Ci}{(\\frac{\\omega}{\\mathbf{J}_P})} and \\int \\chi{(\\omega,\\mathbf{J}_P)} d\\mathbf{J}_P = \\lambda + \\mathbf{J}_P \\chi{(\\omega,\\mathbf{J}_P)} - \\frac{\\omega \\log{(\\frac{\\omega^{2}}{\\mathbf{J}_P^{2}})}}{2} + \\omega \\log{(\\frac{\\omega}{\\mathbf{J}_P})} - \\omega \\operatorname{Ci}{(\\frac{\\omega}{\\mathbf{J}_P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\omega', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))))), Mul(Symbol('\\\\omega', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Ci(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\chi')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\omega', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))))), Mul(Symbol('\\\\omega', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Ci(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given H{(\\lambda,n)} = \\lambda \\cos{(n)}, then derive \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - \\lambda \\sin{(n)}, then obtain - \\lambda \\sin{(n)} + 2 \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - 2 \\lambda \\sin{(n)} + \\frac{\\partial}{\\partial n} H{(\\lambda,n)}", "derivation": "H{(\\lambda,n)} = \\lambda \\cos{(n)} and \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = \\frac{\\partial}{\\partial n} \\lambda \\cos{(n)} and \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - \\lambda \\sin{(n)} and - \\lambda \\sin{(n)} = \\frac{\\partial}{\\partial n} \\lambda \\cos{(n)} and \\frac{\\partial}{\\partial n} \\lambda \\cos{(n)} + \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - \\lambda \\sin{(n)} + \\frac{\\partial}{\\partial n} \\lambda \\cos{(n)} and - \\lambda \\sin{(n)} + \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - 2 \\lambda \\sin{(n)} and - \\lambda \\sin{(n)} + 2 \\frac{\\partial}{\\partial n} H{(\\lambda,n)} = - 2 \\lambda \\sin{(n)} + \\frac{\\partial}{\\partial n} H{(\\lambda,n)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))), Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))))"], [["add", 6, "Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))), Mul(Integer(2), Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True), sin(Symbol('n', commutative=True))), Derivative(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(n)} = \\cos{(n)}, then derive (\\frac{d}{d n} \\operatorname{c_{0}}{(n)})^{n} = (- \\sin{(n)})^{n}, then obtain \\frac{d}{d n} (- \\sin{(n)})^{n} = \\frac{d}{d n} (\\frac{d}{d n} \\cos{(n)})^{n}", "derivation": "\\operatorname{c_{0}}{(n)} = \\cos{(n)} and \\frac{d}{d n} \\operatorname{c_{0}}{(n)} = \\frac{d}{d n} \\cos{(n)} and (\\frac{d}{d n} \\operatorname{c_{0}}{(n)})^{n} = (\\frac{d}{d n} \\cos{(n)})^{n} and (\\frac{d}{d n} \\operatorname{c_{0}}{(n)})^{n} = (- \\sin{(n)})^{n} and (- \\sin{(n)})^{n} = (\\frac{d}{d n} \\cos{(n)})^{n} and \\frac{d}{d n} (- \\sin{(n)})^{n} = \\frac{d}{d n} (\\frac{d}{d n} \\cos{(n)})^{n}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('c_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('c_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Mul(Integer(-1), sin(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), sin(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(E,v)} = E + v and \\hat{\\mathbf{r}}{(E,v)} = - v + \\operatorname{f^{\\prime}}^{v}{(E,v)}, then obtain \\int (- v + \\operatorname{f^{\\prime}}^{v}{(E,v)})^{v} dv = \\int (- v + (E + v)^{v})^{v} dv", "derivation": "\\operatorname{f^{\\prime}}{(E,v)} = E + v and \\operatorname{f^{\\prime}}^{v}{(E,v)} = (E + v)^{v} and \\hat{\\mathbf{r}}{(E,v)} = - v + \\operatorname{f^{\\prime}}^{v}{(E,v)} and \\hat{\\mathbf{r}}{(E,v)} = - v + (E + v)^{v} and \\hat{\\mathbf{r}}^{v}{(E,v)} = (- v + (E + v)^{v})^{v} and (- v + \\operatorname{f^{\\prime}}^{v}{(E,v)})^{v} = (- v + (E + v)^{v})^{v} and \\int (- v + \\operatorname{f^{\\prime}}^{v}{(E,v)})^{v} dv = \\int (- v + (E + v)^{v})^{v} dv", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Add(Symbol('E', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Add(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Add(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Add(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["integrate", 6, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Pow(Add(Symbol('E', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(y,H)} = H^{y} and \\mathbf{E}{(E_{n},c,\\varphi^*)} = (\\frac{c}{\\varphi^*})^{E_{n}}, then obtain (- H + \\operatorname{C_{2}}{(y,H)})^{y} - \\mathbf{E}{(E_{n},c,\\varphi^*)} = (- H + H^{y})^{y} - \\mathbf{E}{(E_{n},c,\\varphi^*)}", "derivation": "\\operatorname{C_{2}}{(y,H)} = H^{y} and - H + \\operatorname{C_{2}}{(y,H)} = - H + H^{y} and \\mathbf{E}{(E_{n},c,\\varphi^*)} = (\\frac{c}{\\varphi^*})^{E_{n}} and (- H + \\operatorname{C_{2}}{(y,H)})^{y} = (- H + H^{y})^{y} and - (\\frac{c}{\\varphi^*})^{E_{n}} + (- H + \\operatorname{C_{2}}{(y,H)})^{y} = - (\\frac{c}{\\varphi^*})^{E_{n}} + (- H + H^{y})^{y} and (- H + \\operatorname{C_{2}}{(y,H)})^{y} - \\mathbf{E}{(E_{n},c,\\varphi^*)} = (- H + H^{y})^{y} - \\mathbf{E}{(E_{n},c,\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('y', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('y', commutative=True)))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('C_2')(Symbol('y', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('y', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('E_n', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Symbol('E_n', commutative=True)))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('C_2')(Symbol('y', commutative=True), Symbol('H', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["minus", 4, "Pow(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Symbol('E_n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Symbol('E_n', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('C_2')(Symbol('y', commutative=True), Symbol('H', commutative=True))), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Symbol('E_n', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('C_2')(Symbol('y', commutative=True), Symbol('H', commutative=True))), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('E_n', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('E_n', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given I{(z^{*})} = \\sin{(\\sin{(z^{*})})}, then obtain \\int (I{(z^{*})} + \\sin{(z^{*})}) dz^{*} + \\int e^{I{(z^{*})} + \\sin{(z^{*})}} dz^{*} = \\int (I{(z^{*})} + \\sin{(z^{*})}) dz^{*} + \\int e^{\\sin{(z^{*})} + \\sin{(\\sin{(z^{*})})}} dz^{*}", "derivation": "I{(z^{*})} = \\sin{(\\sin{(z^{*})})} and I{(z^{*})} + \\sin{(z^{*})} = \\sin{(z^{*})} + \\sin{(\\sin{(z^{*})})} and e^{I{(z^{*})} + \\sin{(z^{*})}} = e^{\\sin{(z^{*})} + \\sin{(\\sin{(z^{*})})}} and \\int e^{I{(z^{*})} + \\sin{(z^{*})}} dz^{*} = \\int e^{\\sin{(z^{*})} + \\sin{(\\sin{(z^{*})})}} dz^{*} and \\int (I{(z^{*})} + \\sin{(z^{*})}) dz^{*} + \\int e^{I{(z^{*})} + \\sin{(z^{*})}} dz^{*} = \\int (I{(z^{*})} + \\sin{(z^{*})}) dz^{*} + \\int e^{\\sin{(z^{*})} + \\sin{(\\sin{(z^{*})})}} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('z^*', commutative=True)), sin(sin(Symbol('z^*', commutative=True))))"], [["add", 1, "sin(Symbol('z^*', commutative=True))"], "Equality(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Add(sin(Symbol('z^*', commutative=True)), sin(sin(Symbol('z^*', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))), exp(Add(sin(Symbol('z^*', commutative=True)), sin(sin(Symbol('z^*', commutative=True))))))"], [["integrate", 3, "Symbol('z^*', commutative=True)"], "Equality(Integral(exp(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True))), Integral(exp(Add(sin(Symbol('z^*', commutative=True)), sin(sin(Symbol('z^*', commutative=True))))), Tuple(Symbol('z^*', commutative=True))))"], [["add", 4, "Integral(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))"], "Equality(Add(Integral(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(exp(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True)))), Add(Integral(Add(Function('I')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(exp(Add(sin(Symbol('z^*', commutative=True)), sin(sin(Symbol('z^*', commutative=True))))), Tuple(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given y{(\\hat{x}_0,h)} = \\hat{x}_0 + h and \\hat{\\mathbf{r}}{(\\hat{x}_0,h)} = \\hat{x}_0 + 2 h and \\sigma_{x}{(h,\\hat{x}_0)} = h + y{(\\hat{x}_0,h)}, then obtain \\sigma_{x}{(h,\\hat{x}_0)} = \\hat{\\mathbf{r}}{(\\hat{x}_0,h)}", "derivation": "y{(\\hat{x}_0,h)} = \\hat{x}_0 + h and h + y{(\\hat{x}_0,h)} = \\hat{x}_0 + 2 h and \\hat{\\mathbf{r}}{(\\hat{x}_0,h)} = \\hat{x}_0 + 2 h and \\hat{\\mathbf{r}}{(\\hat{x}_0,h)} = h + y{(\\hat{x}_0,h)} and \\sigma_{x}{(h,\\hat{x}_0)} = h + y{(\\hat{x}_0,h)} and \\sigma_{x}{(h,\\hat{x}_0)} = \\hat{\\mathbf{r}}{(\\hat{x}_0,h)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Add(Symbol('h', commutative=True), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(a,\\eta^{\\prime})} = \\sin^{\\eta^{\\prime}}{(a)}, then obtain 1 = \\frac{\\frac{\\eta^{\\prime} \\sin^{\\eta^{\\prime}}{(a)} \\cos{(a)}}{\\sin{(a)}} - \\sin{(a)}}{- \\sin{(a)} + \\frac{\\partial}{\\partial a} \\operatorname{C_{d}}{(a,\\eta^{\\prime})}}", "derivation": "\\operatorname{C_{d}}{(a,\\eta^{\\prime})} = \\sin^{\\eta^{\\prime}}{(a)} and \\frac{\\partial}{\\partial a} \\operatorname{C_{d}}{(a,\\eta^{\\prime})} = \\frac{\\partial}{\\partial a} \\sin^{\\eta^{\\prime}}{(a)} and - \\sin{(a)} + \\frac{\\partial}{\\partial a} \\operatorname{C_{d}}{(a,\\eta^{\\prime})} = - \\sin{(a)} + \\frac{\\partial}{\\partial a} \\sin^{\\eta^{\\prime}}{(a)} and 1 = \\frac{- \\sin{(a)} + \\frac{\\partial}{\\partial a} \\sin^{\\eta^{\\prime}}{(a)}}{- \\sin{(a)} + \\frac{\\partial}{\\partial a} \\operatorname{C_{d}}{(a,\\eta^{\\prime})}} and 1 = \\frac{\\frac{\\eta^{\\prime} \\sin^{\\eta^{\\prime}}{(a)} \\cos{(a)}}{\\sin{(a)}} - \\sin{(a)}}{- \\sin{(a)} + \\frac{\\partial}{\\partial a} \\operatorname{C_{d}}{(a,\\eta^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('a', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 2, "sin(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Pow(sin(Symbol('a', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["divide", 3, "Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Pow(sin(Symbol('a', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Mul(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Pow(sin(Symbol('a', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('a', commutative=True))), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Pow(Add(Mul(Integer(-1), sin(Symbol('a', commutative=True))), Derivative(Function('C_d')(Symbol('a', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(\\psi,\\rho_b,y^{\\prime})} = \\psi \\rho_b y^{\\prime}, then derive \\frac{\\partial}{\\partial \\psi} \\theta_{2}{(\\psi,\\rho_b,y^{\\prime})} - 1 = \\rho_b y^{\\prime} - 1, then obtain \\frac{\\partial}{\\partial \\psi} \\psi \\rho_b y^{\\prime} - 1 = \\rho_b y^{\\prime} - 1", "derivation": "\\theta_{2}{(\\psi,\\rho_b,y^{\\prime})} = \\psi \\rho_b y^{\\prime} and - \\psi + \\theta_{2}{(\\psi,\\rho_b,y^{\\prime})} = \\psi \\rho_b y^{\\prime} - \\psi and \\frac{\\partial}{\\partial \\psi} (- \\psi + \\theta_{2}{(\\psi,\\rho_b,y^{\\prime})}) = \\frac{\\partial}{\\partial \\psi} (\\psi \\rho_b y^{\\prime} - \\psi) and \\frac{\\partial}{\\partial \\psi} \\theta_{2}{(\\psi,\\rho_b,y^{\\prime})} - 1 = \\rho_b y^{\\prime} - 1 and \\frac{\\partial}{\\partial \\psi} \\psi \\rho_b y^{\\prime} - 1 = \\rho_b y^{\\prime} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\rho_{f}{(V,x^\\prime)} = V x^\\prime, then obtain \\frac{\\partial}{\\partial x^\\prime} (e^{\\rho_{f}^{V}{(V,x^\\prime)}})^{x^\\prime} = \\frac{\\partial}{\\partial x^\\prime} (e^{(V x^\\prime)^{V}})^{x^\\prime}", "derivation": "\\rho_{f}{(V,x^\\prime)} = V x^\\prime and \\rho_{f}^{V}{(V,x^\\prime)} = (V x^\\prime)^{V} and e^{\\rho_{f}^{V}{(V,x^\\prime)}} = e^{(V x^\\prime)^{V}} and (e^{\\rho_{f}^{V}{(V,x^\\prime)}})^{x^\\prime} = (e^{(V x^\\prime)^{V}})^{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} (e^{\\rho_{f}^{V}{(V,x^\\prime)}})^{x^\\prime} = \\frac{\\partial}{\\partial x^\\prime} (e^{(V x^\\prime)^{V}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True)), Pow(Mul(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))), exp(Pow(Mul(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(exp(Pow(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(exp(Pow(Mul(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(exp(Pow(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(exp(Pow(Mul(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('V', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(n,\\sigma_p)} = \\sigma_p - n, then obtain (\\int (\\varphi^{*}^{2}{(n,\\sigma_p)} - 1) dn)^{2} = (\\int ((\\sigma_p - n) \\varphi^{*}{(n,\\sigma_p)} - 1) dn)^{2}", "derivation": "\\varphi^{*}{(n,\\sigma_p)} = \\sigma_p - n and \\varphi^{*}^{2}{(n,\\sigma_p)} = (\\sigma_p - n) \\varphi^{*}{(n,\\sigma_p)} and \\varphi^{*}^{2}{(n,\\sigma_p)} - 1 = (\\sigma_p - n) \\varphi^{*}{(n,\\sigma_p)} - 1 and \\int (\\varphi^{*}^{2}{(n,\\sigma_p)} - 1) dn = \\int ((\\sigma_p - n) \\varphi^{*}{(n,\\sigma_p)} - 1) dn and (\\int (\\varphi^{*}^{2}{(n,\\sigma_p)} - 1) dn)^{2} = (\\int ((\\sigma_p - n) \\varphi^{*}{(n,\\sigma_p)} - 1) dn)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["times", 1, "Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Integer(-1)), Add(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Integer(-1)), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Tuple(Symbol('n', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Pow(Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Integer(-1)), Tuple(Symbol('n', commutative=True))), Integer(2)), Pow(Integral(Add(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Function('\\\\varphi^*')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Tuple(Symbol('n', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\varphi)} = \\log{(\\sin{(\\varphi)})}, then obtain \\frac{d}{d \\varphi} 0^{\\varphi} = \\frac{d}{d \\varphi} 1", "derivation": "\\mathbf{J}_f{(\\varphi)} = \\log{(\\sin{(\\varphi)})} and 0 = - \\mathbf{J}_f{(\\varphi)} + \\log{(\\sin{(\\varphi)})} and 0^{\\varphi} = (- \\mathbf{J}_f{(\\varphi)} + \\log{(\\sin{(\\varphi)})})^{\\varphi} and \\frac{d}{d \\varphi} 0^{\\varphi} = \\frac{d}{d \\varphi} (- \\mathbf{J}_f{(\\varphi)} + \\log{(\\sin{(\\varphi)})})^{\\varphi} and \\frac{d}{d \\varphi} (- \\mathbf{J}_f{(\\varphi)} + \\log{(\\sin{(\\varphi)})})^{\\varphi} = \\frac{d}{d \\varphi} 1 and \\frac{d}{d \\varphi} 0^{\\varphi} = \\frac{d}{d \\varphi} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True)), log(sin(Symbol('\\\\varphi', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True))), log(sin(Symbol('\\\\varphi', commutative=True)))))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True))), log(sin(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True))), log(sin(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True))), log(sin(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\lambda,F_{x})} = F_{x} \\lambda, then obtain - F_{x} \\lambda + \\frac{\\partial}{\\partial F_{x}} F_{x} \\lambda B{(\\lambda,F_{x})} = - F_{x} \\lambda + \\frac{\\partial}{\\partial F_{x}} F_{x}^{2} \\lambda^{2}", "derivation": "B{(\\lambda,F_{x})} = F_{x} \\lambda and F_{x} \\lambda B{(\\lambda,F_{x})} = F_{x}^{2} \\lambda^{2} and \\frac{\\partial}{\\partial F_{x}} F_{x} \\lambda B{(\\lambda,F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x}^{2} \\lambda^{2} and - F_{x} \\lambda + \\frac{\\partial}{\\partial F_{x}} F_{x} \\lambda B{(\\lambda,F_{x})} = - F_{x} \\lambda + \\frac{\\partial}{\\partial F_{x}} F_{x}^{2} \\lambda^{2}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Function('B')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Function('B')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Function('B')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(c,\\hat{X})} = e^{\\hat{X} - c} and \\chi{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then obtain \\chi{(a^{\\dagger})} e^{\\hat{X} - c} \\frac{\\partial}{\\partial c} e^{\\hat{X} - c} = e^{\\hat{X} - c} \\cos{(a^{\\dagger})} \\frac{\\partial}{\\partial c} e^{\\hat{X} - c}", "derivation": "\\hat{H}_{\\lambda}{(c,\\hat{X})} = e^{\\hat{X} - c} and \\frac{\\partial}{\\partial c} \\hat{H}_{\\lambda}{(c,\\hat{X})} = \\frac{\\partial}{\\partial c} e^{\\hat{X} - c} and \\chi{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\chi{(a^{\\dagger})} e^{\\hat{X} - c} \\frac{\\partial}{\\partial c} \\hat{H}_{\\lambda}{(c,\\hat{X})} = e^{\\hat{X} - c} \\cos{(a^{\\dagger})} \\frac{\\partial}{\\partial c} \\hat{H}_{\\lambda}{(c,\\hat{X})} and \\chi{(a^{\\dagger})} e^{\\hat{X} - c} \\frac{\\partial}{\\partial c} e^{\\hat{X} - c} = e^{\\hat{X} - c} \\cos{(a^{\\dagger})} \\frac{\\partial}{\\partial c} e^{\\hat{X} - c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\chi')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 3, "Mul(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], "Equality(Mul(Function('\\\\chi')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), cos(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\chi')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Derivative(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), cos(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(exp(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\hat{H}_l,C)} = C \\hat{H}_l, then obtain ((C \\hat{H}_l (- \\hat{H}_l + \\operatorname{c_{0}}{(\\hat{H}_l,C)}))^{\\hat{H}_l})^{\\hat{H}_l} = ((C \\hat{H}_l (C \\hat{H}_l - \\hat{H}_l))^{\\hat{H}_l})^{\\hat{H}_l}", "derivation": "\\operatorname{c_{0}}{(\\hat{H}_l,C)} = C \\hat{H}_l and - \\hat{H}_l + \\operatorname{c_{0}}{(\\hat{H}_l,C)} = C \\hat{H}_l - \\hat{H}_l and C \\hat{H}_l (- \\hat{H}_l + \\operatorname{c_{0}}{(\\hat{H}_l,C)}) = C \\hat{H}_l (C \\hat{H}_l - \\hat{H}_l) and (C \\hat{H}_l (- \\hat{H}_l + \\operatorname{c_{0}}{(\\hat{H}_l,C)}))^{\\hat{H}_l} = (C \\hat{H}_l (C \\hat{H}_l - \\hat{H}_l))^{\\hat{H}_l} and ((C \\hat{H}_l (- \\hat{H}_l + \\operatorname{c_{0}}{(\\hat{H}_l,C)}))^{\\hat{H}_l})^{\\hat{H}_l} = ((C \\hat{H}_l (C \\hat{H}_l - \\hat{H}_l))^{\\hat{H}_l})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('c_0')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 2, "Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('c_0')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('c_0')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('C', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('c_0')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('C', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\lambda{(\\mathbf{g})} = (\\operatorname{f^{\\prime}}^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}}, then obtain \\lambda^{\\mathbf{g}}{(\\mathbf{g})} = ((\\sin^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}})^{\\mathbf{g}}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\operatorname{f^{\\prime}}^{\\mathbf{g}}{(\\mathbf{g})} = \\sin^{\\mathbf{g}}{(\\mathbf{g})} and (\\operatorname{f^{\\prime}}^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} = (\\sin^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} and \\lambda{(\\mathbf{g})} = (\\operatorname{f^{\\prime}}^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} and \\lambda{(\\mathbf{g})} = (\\sin^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}} and \\lambda^{\\mathbf{g}}{(\\mathbf{g})} = ((\\sin^{\\mathbf{g}}{(\\mathbf{g})})^{\\mathbf{g}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Pow(Pow(sin(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\theta_2,m_{s})} = e^{\\theta_2 - m_{s}}, then obtain (- \\theta_2 + e^{\\theta_2 - m_{s}}) \\operatorname{t_{2}}{(\\theta_2,m_{s})} = (- \\theta_2 + e^{\\theta_2 - m_{s}}) e^{\\theta_2 - m_{s}}", "derivation": "\\operatorname{t_{2}}{(\\theta_2,m_{s})} = e^{\\theta_2 - m_{s}} and - \\theta_2 + \\operatorname{t_{2}}{(\\theta_2,m_{s})} = - \\theta_2 + e^{\\theta_2 - m_{s}} and (- \\theta_2 + \\operatorname{t_{2}}{(\\theta_2,m_{s})}) \\operatorname{t_{2}}{(\\theta_2,m_{s})} = (- \\theta_2 + \\operatorname{t_{2}}{(\\theta_2,m_{s})}) e^{\\theta_2 - m_{s}} and (- \\theta_2 + e^{\\theta_2 - m_{s}}) \\operatorname{t_{2}}{(\\theta_2,m_{s})} = (- \\theta_2 + e^{\\theta_2 - m_{s}}) e^{\\theta_2 - m_{s}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True)), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"], [["minus", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True))), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True))), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))), Function('t_2')(Symbol('\\\\theta_2', commutative=True), Symbol('m_s', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))), exp(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given \\varphi^{*}{(W,i)} = - W + i and \\dot{x}{(W,i)} = \\int (- W + i + 1) dW, then obtain \\frac{\\frac{\\partial}{\\partial W} \\dot{x}{(W,i)}}{q} = \\frac{\\frac{\\partial}{\\partial W} \\int (- W + i + 1) dW}{q}", "derivation": "\\varphi^{*}{(W,i)} = - W + i and \\varphi^{*}{(W,i)} + 1 = - W + i + 1 and \\int (\\varphi^{*}{(W,i)} + 1) dW = \\int (- W + i + 1) dW and \\dot{x}{(W,i)} = \\int (- W + i + 1) dW and \\int (\\varphi^{*}{(W,i)} + 1) dW = \\dot{x}{(W,i)} and \\frac{\\partial}{\\partial W} \\int (\\varphi^{*}{(W,i)} + 1) dW = \\frac{\\partial}{\\partial W} \\int (- W + i + 1) dW and \\frac{\\frac{\\partial}{\\partial W} \\int (\\varphi^{*}{(W,i)} + 1) dW}{q} = \\frac{\\frac{\\partial}{\\partial W} \\int (- W + i + 1) dW}{q} and \\frac{\\frac{\\partial}{\\partial W} \\dot{x}{(W,i)}}{q} = \\frac{\\frac{\\partial}{\\partial W} \\int (- W + i + 1) dW}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["divide", 6, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Integral(Add(Function('\\\\varphi^*')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('i', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(f_{\\mathbf{v}},\\delta)} = - \\delta + f_{\\mathbf{v}}, then obtain \\delta + 1 = \\delta + (\\frac{- \\delta + f_{\\mathbf{v}}}{r{(f_{\\mathbf{v}},\\delta)}})^{\\delta}", "derivation": "r{(f_{\\mathbf{v}},\\delta)} = - \\delta + f_{\\mathbf{v}} and 1 = \\frac{- \\delta + f_{\\mathbf{v}}}{r{(f_{\\mathbf{v}},\\delta)}} and 1 = (\\frac{- \\delta + f_{\\mathbf{v}}}{r{(f_{\\mathbf{v}},\\delta)}})^{\\delta} and \\delta + 1 = \\delta + (\\frac{- \\delta + f_{\\mathbf{v}}}{r{(f_{\\mathbf{v}},\\delta)}})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 1, "Function('r')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('r')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integer(1), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('r')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))), Symbol('\\\\delta', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Integer(1)), Add(Symbol('\\\\delta', commutative=True), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('r')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(f^{*},\\varphi)} = \\frac{\\varphi}{f^{*}}, then obtain \\int (\\operatorname{A_{2}}{(f^{*},\\varphi)} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1) df^{*} = \\int (\\frac{\\varphi}{f^{*}} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1) df^{*}", "derivation": "\\operatorname{A_{2}}{(f^{*},\\varphi)} = \\frac{\\varphi}{f^{*}} and \\operatorname{A_{2}}{(f^{*},\\varphi)} - 1 = \\frac{\\varphi}{f^{*}} - 1 and \\operatorname{A_{2}}{(f^{*},\\varphi)} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1 = \\frac{\\varphi}{f^{*}} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1 and \\int (\\operatorname{A_{2}}{(f^{*},\\varphi)} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1) df^{*} = \\int (\\frac{\\varphi}{f^{*}} - \\frac{\\partial}{\\partial f^{*}} (\\frac{\\varphi}{f^{*}} - 1) - 1) df^{*}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)))"], [["minus", 2, "Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('f^*', commutative=True), Integer(1)))"], "Equality(Add(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(-1)))"], [["integrate", 3, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)} = (G \\hat{H})^{\\varepsilon}, then obtain (G (G \\hat{H})^{\\varepsilon})^{\\hat{H}} + \\frac{(G \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)})^{\\hat{H}}}{G} = (G (G \\hat{H})^{\\varepsilon})^{\\hat{H}} + \\frac{(G (G \\hat{H})^{\\varepsilon})^{\\hat{H}}}{G}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)} = (G \\hat{H})^{\\varepsilon} and G \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)} = G (G \\hat{H})^{\\varepsilon} and (G \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)})^{\\hat{H}} = (G (G \\hat{H})^{\\varepsilon})^{\\hat{H}} and \\frac{(G \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)})^{\\hat{H}}}{G} = \\frac{(G (G \\hat{H})^{\\varepsilon})^{\\hat{H}}}{G} and (G (G \\hat{H})^{\\varepsilon})^{\\hat{H}} + \\frac{(G \\operatorname{f_{\\mathbf{v}}}{(\\hat{H},G,\\varepsilon)})^{\\hat{H}}}{G} = (G (G \\hat{H})^{\\varepsilon})^{\\hat{H}} + \\frac{(G (G \\hat{H})^{\\varepsilon})^{\\hat{H}}}{G}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Symbol('G', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 3, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Mul(Symbol('G', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True))))"], [["add", 4, "Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Mul(Symbol('G', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))), Add(Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Mul(Symbol('G', commutative=True), Pow(Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(h,E_{x})} = E_{x}^{h}, then obtain h = h (E_{x}^{h})^{E_{x}} \\mathbf{F}^{- E_{x}}{(h,E_{x})}", "derivation": "\\mathbf{F}{(h,E_{x})} = E_{x}^{h} and \\mathbf{F}^{E_{x}}{(h,E_{x})} = (E_{x}^{h})^{E_{x}} and h \\mathbf{F}^{E_{x}}{(h,E_{x})} = h (E_{x}^{h})^{E_{x}} and h = h (E_{x}^{h})^{E_{x}} \\mathbf{F}^{- E_{x}}{(h,E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)), Symbol('E_x', commutative=True)))"], [["times", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Mul(Symbol('h', commutative=True), Pow(Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)), Symbol('E_x', commutative=True))))"], [["divide", 3, "Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], "Equality(Symbol('h', commutative=True), Mul(Symbol('h', commutative=True), Pow(Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)), Symbol('E_x', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C_{1},u)} = C_{1} + u, then derive \\frac{\\partial}{\\partial u} \\operatorname{F_{x}}{(C_{1},u)} = 1, then obtain (\\frac{\\partial}{\\partial u} (C_{1} + u))^{C_{1}} = 1", "derivation": "\\operatorname{F_{x}}{(C_{1},u)} = C_{1} + u and \\frac{\\partial}{\\partial u} \\operatorname{F_{x}}{(C_{1},u)} = \\frac{\\partial}{\\partial u} (C_{1} + u) and \\frac{\\frac{\\partial}{\\partial u} \\operatorname{F_{x}}{(C_{1},u)}}{\\frac{\\partial}{\\partial u} (C_{1} + u)} = 1 and \\frac{\\partial}{\\partial u} \\operatorname{F_{x}}{(C_{1},u)} = 1 and \\frac{\\partial}{\\partial u} (C_{1} + u) = 1 and (\\frac{\\partial}{\\partial u} (C_{1} + u))^{C_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('F_x')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('F_x')(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))"], [["power", 5, "Symbol('C_1', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(s)} = \\log{(s)}, then obtain - s + \\operatorname{x^{{\\}'}}{(s)} \\log{(s)} + \\operatorname{x^{{\\}'}}{(s)} = - s + \\operatorname{x^{{\\}'}}{(s)} + \\log{(s)}^{2}", "derivation": "\\operatorname{x^{{\\}'}}{(s)} = \\log{(s)} and \\operatorname{x^{{\\}'}}{(s)} \\log{(s)} = \\log{(s)}^{2} and \\operatorname{x^{{\\}'}}{(s)} \\log{(s)} + \\log{(s)} = \\log{(s)}^{2} + \\log{(s)} and - s + \\operatorname{x^{{\\}'}}{(s)} \\log{(s)} + \\log{(s)} = - s + \\log{(s)}^{2} + \\log{(s)} and - s + \\operatorname{x^{{\\}'}}{(s)} = - s + \\log{(s)} and - s + \\operatorname{x^{{\\}'}}{(s)} \\log{(s)} + \\operatorname{x^{{\\}'}}{(s)} = - s + \\operatorname{x^{{\\}'}}{(s)} + \\log{(s)}^{2}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["times", 1, "log(Symbol('s', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Pow(log(Symbol('s', commutative=True)), Integer(2)))"], [["add", 2, "log(Symbol('s', commutative=True))"], "Equality(Add(Mul(Function('x^\\\\prime')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Add(Pow(log(Symbol('s', commutative=True)), Integer(2)), log(Symbol('s', commutative=True))))"], [["minus", 3, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Function('x^\\\\prime')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Pow(log(Symbol('s', commutative=True)), Integer(2)), log(Symbol('s', commutative=True))))"], [["minus", 1, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('x^\\\\prime')(Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Function('x^\\\\prime')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Function('x^\\\\prime')(Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('x^\\\\prime')(Symbol('s', commutative=True)), Pow(log(Symbol('s', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain \\hat{\\mathbf{x}}{(\\mathbf{H})} - e^{\\mathbf{H}} = 0", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{H})} = e^{\\mathbf{H}} and \\hat{\\mathbf{x}}^{2}{(\\mathbf{H})} = \\hat{\\mathbf{x}}{(\\mathbf{H})} e^{\\mathbf{H}} and - \\hat{\\mathbf{x}}{(\\mathbf{H})} e^{\\mathbf{H}} + \\hat{\\mathbf{x}}{(\\mathbf{H})} = - \\hat{\\mathbf{x}}{(\\mathbf{H})} e^{\\mathbf{H}} + e^{\\mathbf{H}} and \\hat{\\mathbf{x}}^{2}{(\\mathbf{H})} - \\hat{\\mathbf{x}}{(\\mathbf{H})} e^{\\mathbf{H}} + \\hat{\\mathbf{x}}{(\\mathbf{H})} - e^{\\mathbf{H}} = \\hat{\\mathbf{x}}^{2}{(\\mathbf{H})} - \\hat{\\mathbf{x}}{(\\mathbf{H})} e^{\\mathbf{H}} and \\hat{\\mathbf{x}}{(\\mathbf{H})} - e^{\\mathbf{H}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 1, "Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = \\log{(V_{\\mathbf{E}} \\dot{y})}, then obtain - V_{\\mathbf{E}} \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} + \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = - V_{\\mathbf{E}} \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} + \\log{(V_{\\mathbf{E}} \\dot{y})}", "derivation": "\\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = \\log{(V_{\\mathbf{E}} \\dot{y})} and V_{\\mathbf{E}} \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = V_{\\mathbf{E}} \\log{(V_{\\mathbf{E}} \\dot{y})} and - V_{\\mathbf{E}} \\log{(V_{\\mathbf{E}} \\dot{y})} + \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = - V_{\\mathbf{E}} \\log{(V_{\\mathbf{E}} \\dot{y})} + \\log{(V_{\\mathbf{E}} \\dot{y})} and - V_{\\mathbf{E}} \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} + \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} = - V_{\\mathbf{E}} \\dot{z}{(V_{\\mathbf{E}},\\dot{y})} + \\log{(V_{\\mathbf{E}} \\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["times", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["minus", 1, "Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\dot{z}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(n_{1})} = \\sin{(\\cos{(n_{1})})}, then obtain \\frac{d}{d n_{1}} \\operatorname{C_{d}}{(n_{1})} - 1 = - \\sin{(n_{1})} \\cos{(\\cos{(n_{1})})} - 1", "derivation": "\\operatorname{C_{d}}{(n_{1})} = \\sin{(\\cos{(n_{1})})} and - n_{1} + \\operatorname{C_{d}}{(n_{1})} = - n_{1} + \\sin{(\\cos{(n_{1})})} and \\frac{d}{d n_{1}} (- n_{1} + \\operatorname{C_{d}}{(n_{1})}) = \\frac{d}{d n_{1}} (- n_{1} + \\sin{(\\cos{(n_{1})})}) and \\frac{d}{d n_{1}} \\operatorname{C_{d}}{(n_{1})} - 1 = - \\sin{(n_{1})} \\cos{(\\cos{(n_{1})})} - 1", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True))))"], [["minus", 1, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('C_d')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True)))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('C_d')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), sin(cos(Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('C_d')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('n_1', commutative=True)), cos(cos(Symbol('n_1', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\mu_{0}{(P_{e})} = e^{P_{e}}, then obtain - e^{P_{e}} + e^{\\frac{d}{d P_{e}} \\int \\mu_{0}{(P_{e})} dP_{e}} = - e^{P_{e}} + e^{\\frac{d}{d P_{e}} \\int e^{P_{e}} dP_{e}}", "derivation": "\\mu_{0}{(P_{e})} = e^{P_{e}} and \\int \\mu_{0}{(P_{e})} dP_{e} = \\int e^{P_{e}} dP_{e} and \\frac{d}{d P_{e}} \\int \\mu_{0}{(P_{e})} dP_{e} = \\frac{d}{d P_{e}} \\int e^{P_{e}} dP_{e} and e^{\\frac{d}{d P_{e}} \\int \\mu_{0}{(P_{e})} dP_{e}} = e^{\\frac{d}{d P_{e}} \\int e^{P_{e}} dP_{e}} and - e^{P_{e}} + e^{\\frac{d}{d P_{e}} \\int \\mu_{0}{(P_{e})} dP_{e}} = - e^{P_{e}} + e^{\\frac{d}{d P_{e}} \\int e^{P_{e}} dP_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mu_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Integral(Function('\\\\mu_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1)))), exp(Derivative(Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["minus", 4, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('P_e', commutative=True))), exp(Derivative(Integral(Function('\\\\mu_0')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('P_e', commutative=True))), exp(Derivative(Integral(exp(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))))"]]}, {"prompt": "Given b{(I)} = e^{I}, then obtain \\log{(- 2 b{(I)} + 2 \\int 0 dI)} = \\log{(- 2 b{(I)} + \\int 0 dI + \\int (- b{(I)} + e^{I}) dI)}", "derivation": "b{(I)} = e^{I} and 0 = - b{(I)} + e^{I} and \\int 0 dI = \\int (- b{(I)} + e^{I}) dI and - e^{I} + \\int 0 dI = - e^{I} + \\int (- b{(I)} + e^{I}) dI and - 2 e^{I} + 2 \\int 0 dI = - 2 e^{I} + \\int 0 dI + \\int (- b{(I)} + e^{I}) dI and \\log{(- 2 e^{I} + 2 \\int 0 dI)} = \\log{(- 2 e^{I} + \\int 0 dI + \\int (- b{(I)} + e^{I}) dI)} and \\log{(- 2 b{(I)} + 2 \\int 0 dI)} = \\log{(- 2 b{(I)} + \\int 0 dI + \\int (- b{(I)} + e^{I}) dI)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["minus", 1, "Function('b')(Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["minus", 3, "exp(Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["add", 4, "Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), exp(Symbol('I', commutative=True))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))), Add(Mul(Integer(-1), Integer(2), exp(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Integer(2), exp(Symbol('I', commutative=True))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('I', commutative=True)))))), log(Add(Mul(Integer(-1), Integer(2), exp(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(log(Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('I', commutative=True))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('I', commutative=True)))))), log(Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('b')(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{r}{(\\dot{y})} = e^{\\dot{y}}, then derive \\int \\mathbf{r}{(\\dot{y})} d\\dot{y} = x + e^{\\dot{y}}, then obtain x + e^{\\dot{y}} = \\int e^{\\dot{y}} d\\dot{y}", "derivation": "\\mathbf{r}{(\\dot{y})} = e^{\\dot{y}} and \\int \\mathbf{r}{(\\dot{y})} d\\dot{y} = \\int e^{\\dot{y}} d\\dot{y} and \\int \\mathbf{r}{(\\dot{y})} d\\dot{y} = x + e^{\\dot{y}} and x + e^{\\dot{y}} = \\int e^{\\dot{y}} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('x', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('x', commutative=True), exp(Symbol('\\\\dot{y}', commutative=True))), Integral(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(S)} = \\cos{(\\cos{(S)})}, then obtain \\frac{d}{d S} \\frac{\\theta_{2}{(S)} - \\cos{(\\cos{(S)})}}{\\cos{(S)}} = \\frac{d}{d S} 0", "derivation": "\\theta_{2}{(S)} = \\cos{(\\cos{(S)})} and \\theta_{2}{(S)} - \\cos{(\\cos{(S)})} = 0 and \\frac{\\theta_{2}{(S)} - \\cos{(\\cos{(S)})}}{\\cos{(S)}} = 0 and \\frac{d}{d S} \\frac{\\theta_{2}{(S)} - \\cos{(\\cos{(S)})}}{\\cos{(S)}} = \\frac{d}{d S} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('S', commutative=True)), cos(cos(Symbol('S', commutative=True))))"], [["minus", 1, "cos(cos(Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\theta_2')(Symbol('S', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('S', commutative=True))))), Integer(0))"], [["divide", 2, "cos(Symbol('S', commutative=True))"], "Equality(Mul(Add(Function('\\\\theta_2')(Symbol('S', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('S', commutative=True))))), Pow(cos(Symbol('S', commutative=True)), Integer(-1))), Integer(0))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\theta_2')(Symbol('S', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('S', commutative=True))))), Pow(cos(Symbol('S', commutative=True)), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(n_{2},A_{x},\\nabla)} = \\frac{- A_{x} + \\nabla}{n_{2}} and \\operatorname{F_{c}}{(\\nabla)} = \\nabla, then obtain A_{x} \\nabla (\\frac{- A_{x} + \\nabla}{n_{2}} + \\frac{A_{x} - \\nabla}{n_{2}}) = 0", "derivation": "\\operatorname{C_{2}}{(n_{2},A_{x},\\nabla)} = \\frac{- A_{x} + \\nabla}{n_{2}} and \\operatorname{C_{2}}{(n_{2},A_{x},\\nabla)} - \\frac{- A_{x} + \\nabla}{n_{2}} = 0 and \\operatorname{C_{2}}{(n_{2},A_{x},\\nabla)} + \\frac{A_{x} - \\nabla}{n_{2}} = 0 and \\operatorname{F_{c}}{(\\nabla)} = \\nabla and \\frac{- A_{x} + \\nabla}{n_{2}} + \\frac{A_{x} - \\nabla}{n_{2}} = 0 and A_{x} (\\frac{- A_{x} + \\nabla}{n_{2}} + \\frac{A_{x} - \\nabla}{n_{2}}) \\operatorname{F_{c}}{(\\nabla)} = 0 and A_{x} \\nabla (\\frac{- A_{x} + \\nabla}{n_{2}} + \\frac{A_{x} - \\nabla}{n_{2}}) = 0", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('n_2', commutative=True), Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('C_2')(Symbol('n_2', commutative=True), Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Add(Function('C_2')(Symbol('n_2', commutative=True), Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Integer(0))"], [["times", 5, "Mul(Symbol('A_x', commutative=True), Function('F_c')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Symbol('A_x', commutative=True), Add(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Function('F_c')(Symbol('\\\\nabla', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Add(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given y{(i)} = \\cos{(i)} and c{(i)} = \\cos{(i)}, then obtain \\frac{(\\frac{1}{y{(i)}})^{i} c{(i)} y{(i)}}{\\cos{(i)}} = \\frac{(\\frac{\\cos{(i)}}{y^{2}{(i)}})^{i} c{(i)} y{(i)}}{\\cos{(i)}}", "derivation": "y{(i)} = \\cos{(i)} and c{(i)} = \\cos{(i)} and c{(i)} = y{(i)} and \\frac{1}{y{(i)}} = \\frac{\\cos{(i)}}{c{(i)} y{(i)}} and \\frac{1}{y{(i)}} = \\frac{\\cos{(i)}}{y^{2}{(i)}} and (\\frac{1}{y{(i)}})^{i} = (\\frac{\\cos{(i)}}{y^{2}{(i)}})^{i} and \\frac{(\\frac{1}{y{(i)}})^{i} c{(i)} y{(i)}}{\\cos{(i)}} = \\frac{(\\frac{\\cos{(i)}}{y^{2}{(i)}})^{i} c{(i)} y{(i)}}{\\cos{(i)}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('c')(Symbol('i', commutative=True)), Function('y')(Symbol('i', commutative=True)))"], [["divide", 2, "Mul(Function('c')(Symbol('i', commutative=True)), Function('y')(Symbol('i', commutative=True)))"], "Equality(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), Mul(Pow(Function('c')(Symbol('i', commutative=True)), Integer(-1)), Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), cos(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), Mul(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-2)), cos(Symbol('i', commutative=True))))"], [["power", 5, "Symbol('i', commutative=True)"], "Equality(Pow(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), Symbol('i', commutative=True)), Pow(Mul(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-2)), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["divide", 6, "Mul(Pow(Function('c')(Symbol('i', commutative=True)), Integer(-1)), Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), cos(Symbol('i', commutative=True)))"], "Equality(Mul(Pow(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-1)), Symbol('i', commutative=True)), Function('c')(Symbol('i', commutative=True)), Function('y')(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Mul(Pow(Mul(Pow(Function('y')(Symbol('i', commutative=True)), Integer(-2)), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Function('c')(Symbol('i', commutative=True)), Function('y')(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{r})} = \\cos{(\\sin{(\\mathbf{r})})}, then obtain \\int (\\frac{d}{d \\mathbf{r}} \\mathbf{S}{(\\mathbf{r})} + 1) d\\mathbf{r} = \\int (\\frac{d}{d \\mathbf{r}} \\cos{(\\sin{(\\mathbf{r})})} + 1) d\\mathbf{r}", "derivation": "\\mathbf{S}{(\\mathbf{r})} = \\cos{(\\sin{(\\mathbf{r})})} and \\frac{d}{d \\mathbf{r}} \\mathbf{S}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\cos{(\\sin{(\\mathbf{r})})} and \\frac{d}{d \\mathbf{r}} \\mathbf{S}{(\\mathbf{r})} + 1 = \\frac{d}{d \\mathbf{r}} \\cos{(\\sin{(\\mathbf{r})})} + 1 and \\int (\\frac{d}{d \\mathbf{r}} \\mathbf{S}{(\\mathbf{r})} + 1) d\\mathbf{r} = \\int (\\frac{d}{d \\mathbf{r}} \\cos{(\\sin{(\\mathbf{r})})} + 1) d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Add(Derivative(cos(sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given n{(f^{*})} = \\frac{d}{d f^{*}} \\log{(f^{*})}, then derive n{(f^{*})} = \\frac{1}{f^{*}}, then obtain (1 + \\frac{1}{f^{*}}) (\\log{(f^{*})} + \\frac{d}{d f^{*}} \\log{(f^{*})} + 1) = (\\frac{d}{d f^{*}} \\log{(f^{*})} + 1) (\\log{(f^{*})} + \\frac{d}{d f^{*}} \\log{(f^{*})} + 1)", "derivation": "n{(f^{*})} = \\frac{d}{d f^{*}} \\log{(f^{*})} and n{(f^{*})} + 1 = \\frac{d}{d f^{*}} \\log{(f^{*})} + 1 and n{(f^{*})} = \\frac{1}{f^{*}} and 1 + \\frac{1}{f^{*}} = \\frac{d}{d f^{*}} \\log{(f^{*})} + 1 and (1 + \\frac{1}{f^{*}}) (n{(f^{*})} + \\log{(f^{*})} + 1) = (\\frac{d}{d f^{*}} \\log{(f^{*})} + 1) (n{(f^{*})} + \\log{(f^{*})} + 1) and (1 + \\frac{1}{f^{*}}) (n{(f^{*})} + \\log{(f^{*})} + 1) = (n{(f^{*})} + 1) (n{(f^{*})} + \\log{(f^{*})} + 1) and (1 + \\frac{1}{f^{*}}) (\\log{(f^{*})} + \\frac{d}{d f^{*}} \\log{(f^{*})} + 1) = (\\frac{d}{d f^{*}} \\log{(f^{*})} + 1) (\\log{(f^{*})} + \\frac{d}{d f^{*}} \\log{(f^{*})} + 1)", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('f^*', commutative=True)), Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('n')(Symbol('f^*', commutative=True)), Integer(1)), Add(Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('n')(Symbol('f^*', commutative=True)), Pow(Symbol('f^*', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Integer(1), Pow(Symbol('f^*', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)))"], [["times", 4, "Add(Function('n')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)), Integer(1))"], "Equality(Mul(Add(Integer(1), Pow(Symbol('f^*', commutative=True), Integer(-1))), Add(Function('n')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)), Integer(1))), Mul(Add(Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)), Add(Function('n')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Integer(1), Pow(Symbol('f^*', commutative=True), Integer(-1))), Add(Function('n')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)), Integer(1))), Mul(Add(Function('n')(Symbol('f^*', commutative=True)), Integer(1)), Add(Function('n')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Integer(1), Pow(Symbol('f^*', commutative=True), Integer(-1))), Add(log(Symbol('f^*', commutative=True)), Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1))), Mul(Add(Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1)), Add(log(Symbol('f^*', commutative=True)), Derivative(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1))))"]]}, {"prompt": "Given c{(h)} = e^{h}, then obtain 1 = \\frac{\\frac{d}{d h} (e^{h})^{h}}{\\frac{d}{d h} c^{h}{(h)}}", "derivation": "c{(h)} = e^{h} and c^{h}{(h)} = (e^{h})^{h} and \\frac{d}{d h} c^{h}{(h)} = \\frac{d}{d h} (e^{h})^{h} and 1 = \\frac{\\frac{d}{d h} (e^{h})^{h}}{\\frac{d}{d h} c^{h}{(h)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('c')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Function('c')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Pow(Function('c')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Pow(Function('c')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)), Derivative(Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(f)} = \\log{(f)} and \\operatorname{E_{n}}{(f)} = (\\operatorname{n_{2}}{(f)} + \\log{(f)})^{2}, then obtain \\log{(\\operatorname{E_{n}}{(f)})} = \\log{(4 \\operatorname{n_{2}}^{2}{(f)})}", "derivation": "\\operatorname{n_{2}}{(f)} = \\log{(f)} and 2 \\operatorname{n_{2}}{(f)} = \\operatorname{n_{2}}{(f)} + \\log{(f)} and 2 (\\operatorname{n_{2}}{(f)} + \\log{(f)}) \\operatorname{n_{2}}{(f)} = (\\operatorname{n_{2}}{(f)} + \\log{(f)})^{2} and \\operatorname{E_{n}}{(f)} = (\\operatorname{n_{2}}{(f)} + \\log{(f)})^{2} and \\operatorname{E_{n}}{(f)} = 2 (\\operatorname{n_{2}}{(f)} + \\log{(f)}) \\operatorname{n_{2}}{(f)} and \\operatorname{E_{n}}{(f)} = 4 \\operatorname{n_{2}}^{2}{(f)} and \\log{(\\operatorname{E_{n}}{(f)})} = \\log{(4 \\operatorname{n_{2}}^{2}{(f)})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["add", 1, "Function('n_2')(Symbol('f', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('f', commutative=True))), Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))))"], [["times", 2, "Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Function('n_2')(Symbol('f', commutative=True))), Pow(Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('f', commutative=True)), Pow(Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('E_n')(Symbol('f', commutative=True)), Mul(Integer(2), Add(Function('n_2')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Function('n_2')(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('E_n')(Symbol('f', commutative=True)), Mul(Integer(4), Pow(Function('n_2')(Symbol('f', commutative=True)), Integer(2))))"], [["log", 6], "Equality(log(Function('E_n')(Symbol('f', commutative=True))), log(Mul(Integer(4), Pow(Function('n_2')(Symbol('f', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given p{(Z,\\varepsilon)} = Z + \\varepsilon and \\sigma_{x}{(f^{\\prime},J_{\\varepsilon})} = J_{\\varepsilon} f^{\\prime}, then obtain - J_{\\varepsilon} f^{\\prime} + p{(Z,\\varepsilon)} = - J_{\\varepsilon} f^{\\prime} + 2 Z + 2 \\varepsilon - p{(Z,\\varepsilon)}", "derivation": "p{(Z,\\varepsilon)} = Z + \\varepsilon and \\sigma_{x}{(f^{\\prime},J_{\\varepsilon})} = J_{\\varepsilon} f^{\\prime} and - \\sigma_{x}{(f^{\\prime},J_{\\varepsilon})} + p{(Z,\\varepsilon)} = Z + \\varepsilon - \\sigma_{x}{(f^{\\prime},J_{\\varepsilon})} and - J_{\\varepsilon} f^{\\prime} + p{(Z,\\varepsilon)} = - J_{\\varepsilon} f^{\\prime} + Z + \\varepsilon and - J_{\\varepsilon} f^{\\prime} + Z = - J_{\\varepsilon} f^{\\prime} + 2 Z + \\varepsilon - p{(Z,\\varepsilon)} and - J_{\\varepsilon} f^{\\prime} + p{(Z,\\varepsilon)} = - J_{\\varepsilon} f^{\\prime} + 2 Z + 2 \\varepsilon - p{(Z,\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], ["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('f^{\\\\prime}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Function('p')(Symbol('Z', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})} = \\sin{(\\dot{x} f^{\\prime})} and B{(f^{\\prime},\\dot{x})} = \\dot{x} f^{\\prime}, then obtain (- \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})})^{f^{\\prime}} = (- \\sin{(\\dot{x} f^{\\prime})})^{f^{\\prime}}", "derivation": "\\operatorname{M_{E}}{(f^{\\prime},\\dot{x})} = \\sin{(\\dot{x} f^{\\prime})} and B{(f^{\\prime},\\dot{x})} = \\dot{x} f^{\\prime} and \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})} = \\sin{(B{(f^{\\prime},\\dot{x})})} and - \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})} = - \\sin{(\\dot{x} f^{\\prime})} and - \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})} = - \\sin{(B{(f^{\\prime},\\dot{x})})} and - \\sin{(B{(f^{\\prime},\\dot{x})})} = - \\sin{(\\dot{x} f^{\\prime})} and (- \\sin{(B{(f^{\\prime},\\dot{x})})})^{f^{\\prime}} = (- \\sin{(\\dot{x} f^{\\prime})})^{f^{\\prime}} and (- \\operatorname{M_{E}}{(f^{\\prime},\\dot{x})})^{f^{\\prime}} = (- \\sin{(\\dot{x} f^{\\prime})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('M_E')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), sin(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('M_E')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Function('M_E')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), sin(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), sin(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["power", 6, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), sin(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Mul(Integer(-1), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Pow(Mul(Integer(-1), Function('M_E')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Mul(Integer(-1), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(b,Z)} = Z^{b} and \\mathbf{J}_f{(b,Z)} = Z^{b}, then obtain \\operatorname{A_{2}}^{Z}{(b,Z)} \\mathbf{J}_f^{Z}{(b,Z)} + \\cos{((Z^{b})^{2 Z})} = \\mathbf{J}_f^{2 Z}{(b,Z)} + \\cos{((Z^{b})^{2 Z})}", "derivation": "\\operatorname{A_{2}}{(b,Z)} = Z^{b} and \\operatorname{A_{2}}^{Z}{(b,Z)} = (Z^{b})^{Z} and (Z^{b})^{Z} \\operatorname{A_{2}}^{Z}{(b,Z)} = (Z^{b})^{2 Z} and \\mathbf{J}_f{(b,Z)} = Z^{b} and \\operatorname{A_{2}}^{Z}{(b,Z)} \\mathbf{J}_f^{Z}{(b,Z)} = \\mathbf{J}_f^{2 Z}{(b,Z)} and \\operatorname{A_{2}}^{Z}{(b,Z)} \\mathbf{J}_f^{Z}{(b,Z)} + \\cos{((Z^{b})^{2 Z})} = \\mathbf{J}_f^{2 Z}{(b,Z)} + \\cos{((Z^{b})^{2 Z})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 2, "Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('A_2')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('A_2')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["add", 5, "cos(Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], "Equality(Add(Mul(Pow(Function('A_2')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), cos(Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))), Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), cos(Pow(Pow(Symbol('Z', commutative=True), Symbol('b', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))))"]]}, {"prompt": "Given \\rho_{b}{(\\lambda,\\delta)} = \\lambda \\cos{(\\delta)}, then obtain (\\frac{d}{d \\lambda} \\int 0 d\\delta)^{\\lambda} = (\\frac{\\partial}{\\partial \\lambda} \\int (\\lambda \\cos{(\\delta)} - \\rho_{b}{(\\lambda,\\delta)}) d\\delta)^{\\lambda}", "derivation": "\\rho_{b}{(\\lambda,\\delta)} = \\lambda \\cos{(\\delta)} and 0 = \\lambda \\cos{(\\delta)} - \\rho_{b}{(\\lambda,\\delta)} and \\int 0 d\\delta = \\int (\\lambda \\cos{(\\delta)} - \\rho_{b}{(\\lambda,\\delta)}) d\\delta and \\frac{d}{d \\lambda} \\int 0 d\\delta = \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda \\cos{(\\delta)} - \\rho_{b}{(\\lambda,\\delta)}) d\\delta and (\\frac{d}{d \\lambda} \\int 0 d\\delta)^{\\lambda} = (\\frac{\\partial}{\\partial \\lambda} \\int (\\lambda \\cos{(\\delta)} - \\rho_{b}{(\\lambda,\\delta)}) d\\delta)^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(Integral(Add(Mul(Symbol('\\\\lambda', commutative=True), cos(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(r_{0},\\eta)} = \\cos{(r_{0}^{\\eta})} and h{(r_{0},\\eta)} = \\cos{(r_{0}^{\\eta})}, then obtain \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\eta)} = \\frac{\\partial}{\\partial r_{0}} h{(r_{0},\\eta)}", "derivation": "\\hat{x}_0{(r_{0},\\eta)} = \\cos{(r_{0}^{\\eta})} and h{(r_{0},\\eta)} = \\cos{(r_{0}^{\\eta})} and \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\eta)} = \\frac{\\partial}{\\partial r_{0}} \\cos{(r_{0}^{\\eta})} and \\frac{\\partial}{\\partial r_{0}} \\hat{x}_0{(r_{0},\\eta)} = \\frac{\\partial}{\\partial r_{0}} h{(r_{0},\\eta)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Pow(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Pow(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Function('h')(Symbol('r_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(\\mathbf{J}_f,z^{*})} = \\mathbf{J}_f + z^{*}, then obtain \\frac{\\partial}{\\partial z^{*}} 4 (\\mathbf{J}_f + z^{*})^{2} = \\frac{\\partial}{\\partial z^{*}} 4 L^{2}{(\\mathbf{J}_f,z^{*})}", "derivation": "L{(\\mathbf{J}_f,z^{*})} = \\mathbf{J}_f + z^{*} and 2 L{(\\mathbf{J}_f,z^{*})} = \\mathbf{J}_f + z^{*} + L{(\\mathbf{J}_f,z^{*})} and 4 L^{2}{(\\mathbf{J}_f,z^{*})} = (\\mathbf{J}_f + z^{*} + L{(\\mathbf{J}_f,z^{*})})^{2} and 4 (\\mathbf{J}_f + z^{*})^{2} = (2 \\mathbf{J}_f + 2 z^{*})^{2} and \\frac{\\partial}{\\partial z^{*}} 4 (\\mathbf{J}_f + z^{*})^{2} = \\frac{\\partial}{\\partial z^{*}} (2 \\mathbf{J}_f + 2 z^{*})^{2} and 4 L^{2}{(\\mathbf{J}_f,z^{*})} = (2 \\mathbf{J}_f + 2 z^{*})^{2} and \\frac{\\partial}{\\partial z^{*}} 4 (\\mathbf{J}_f + z^{*})^{2} = \\frac{\\partial}{\\partial z^{*}} 4 L^{2}{(\\mathbf{J}_f,z^{*})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)))"], [["add", 1, "Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(2), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True), Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Integer(2)))"], [["differentiate", 4, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Integer(2)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Integer(4), Pow(Function('L')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(L)} = \\log{(L)}, then obtain \\hat{p}_0^{4}{(L)} \\log{(L)} = \\hat{p}_0^{3}{(L)} \\log{(L)}^{2}", "derivation": "\\hat{p}_0{(L)} = \\log{(L)} and \\hat{p}_0{(L)} \\log{(L)} = \\log{(L)}^{2} and \\hat{p}_0^{2}{(L)} \\log{(L)} = \\hat{p}_0{(L)} \\log{(L)}^{2} and \\hat{p}_0{(L)} \\log{(L)}^{2} = \\log{(L)}^{3} and \\hat{p}_0^{2}{(L)} \\log{(L)} = \\log{(L)}^{3} and \\hat{p}_0^{2}{(L)} \\log{(L)}^{3} = \\hat{p}_0{(L)} \\log{(L)}^{4} and \\hat{p}_0^{3}{(L)} \\log{(L)}^{2} = \\hat{p}_0{(L)} \\log{(L)}^{4} and \\hat{p}_0^{4}{(L)} \\log{(L)} = \\hat{p}_0{(L)} \\log{(L)}^{4} and \\hat{p}_0^{4}{(L)} \\log{(L)} = \\hat{p}_0^{3}{(L)} \\log{(L)}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["times", 1, "log(Symbol('L', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(2)))"], [["times", 2, "Function('\\\\hat{p}_0')(Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(2)), log(Symbol('L', commutative=True))), Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(2))), Pow(log(Symbol('L', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(2)), log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(3)))"], [["times", 4, "Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(2)), Pow(log(Symbol('L', commutative=True)), Integer(3))), Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(4))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(3)), Pow(log(Symbol('L', commutative=True)), Integer(2))), Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(4))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(4)), log(Symbol('L', commutative=True))), Mul(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(4))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(4)), log(Symbol('L', commutative=True))), Mul(Pow(Function('\\\\hat{p}_0')(Symbol('L', commutative=True)), Integer(3)), Pow(log(Symbol('L', commutative=True)), Integer(2))))"]]}, {"prompt": "Given x{(v_{1})} = e^{v_{1}}, then derive \\frac{d}{d v_{1}} x{(v_{1})} + 1 = e^{v_{1}} + 1, then obtain (\\frac{d}{d v_{1}} e^{v_{1}} + 1) (v_{1} + \\frac{d}{d v_{1}} e^{v_{1}} + 1) = (\\frac{d}{d v_{1}} e^{v_{1}} + 1) (v_{1} + e^{v_{1}} + 1)", "derivation": "x{(v_{1})} = e^{v_{1}} and v_{1} + x{(v_{1})} = v_{1} + e^{v_{1}} and \\frac{d}{d v_{1}} (v_{1} + x{(v_{1})}) = \\frac{d}{d v_{1}} (v_{1} + e^{v_{1}}) and \\frac{d}{d v_{1}} x{(v_{1})} + 1 = e^{v_{1}} + 1 and v_{1} + \\frac{d}{d v_{1}} x{(v_{1})} + 1 = v_{1} + e^{v_{1}} + 1 and (\\frac{d}{d v_{1}} x{(v_{1})} + 1) (v_{1} + \\frac{d}{d v_{1}} x{(v_{1})} + 1) = (\\frac{d}{d v_{1}} x{(v_{1})} + 1) (v_{1} + e^{v_{1}} + 1) and (\\frac{d}{d v_{1}} e^{v_{1}} + 1) (v_{1} + \\frac{d}{d v_{1}} e^{v_{1}} + 1) = (\\frac{d}{d v_{1}} e^{v_{1}} + 1) (v_{1} + e^{v_{1}} + 1)", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('x')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Symbol('v_1', commutative=True), Function('x')(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('v_1', commutative=True)), Integer(1)))"], [["add", 4, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True)), Integer(1)))"], [["times", 5, "Add(Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('v_1', commutative=True), Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Mul(Add(Derivative(Function('x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True)), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('v_1', commutative=True), Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1))), Mul(Add(Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Symbol('v_1', commutative=True), exp(Symbol('v_1', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} = \\frac{\\psi^*}{J_{\\varepsilon}}, then obtain \\int (2 \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 2) d\\psi^* = \\int (2 + \\frac{2 \\psi^*}{J_{\\varepsilon}}) d\\psi^*", "derivation": "\\hat{p}_0{(\\psi^*,J_{\\varepsilon})} = \\frac{\\psi^*}{J_{\\varepsilon}} and \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 1 = 1 + \\frac{\\psi^*}{J_{\\varepsilon}} and \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 1 + \\frac{\\psi^*}{J_{\\varepsilon}} = 1 + \\frac{2 \\psi^*}{J_{\\varepsilon}} and 2 \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 1 = 1 + \\frac{2 \\psi^*}{J_{\\varepsilon}} and 2 \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 2 = 2 + \\frac{2 \\psi^*}{J_{\\varepsilon}} and \\int (2 \\hat{p}_0{(\\psi^*,J_{\\varepsilon})} + 2) d\\psi^* = \\int (2 + \\frac{2 \\psi^*}{J_{\\varepsilon}}) d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(1), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Add(Integer(1), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(2)), Add(Integer(2), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\psi^*', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Integer(2), Mul(Integer(2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = \\mathbf{B} (- \\phi + a^{\\dagger}), then derive \\frac{\\partial}{\\partial \\phi} \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = - \\mathbf{B}, then obtain - \\phi + \\frac{\\partial}{\\partial \\phi} \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = - \\mathbf{B} - \\phi", "derivation": "\\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = \\mathbf{B} (- \\phi + a^{\\dagger}) and \\frac{\\partial}{\\partial \\phi} \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = \\frac{\\partial}{\\partial \\phi} \\mathbf{B} (- \\phi + a^{\\dagger}) and \\frac{\\partial}{\\partial \\phi} \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = - \\mathbf{B} and - \\phi + \\frac{\\partial}{\\partial \\phi} \\mathbf{J}_P{(\\mathbf{B},\\phi,a^{\\dagger})} = - \\mathbf{B} - \\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(U)} = e^{U}, then derive \\frac{d}{d U} \\operatorname{y^{\\prime}}{(U)} = e^{U}, then obtain \\frac{d}{d U} \\operatorname{y^{\\prime}}{(U)} = \\operatorname{y^{\\prime}}{(U)}", "derivation": "\\operatorname{y^{\\prime}}{(U)} = e^{U} and \\frac{d}{d U} \\operatorname{y^{\\prime}}{(U)} = \\frac{d}{d U} e^{U} and \\frac{d}{d U} \\operatorname{y^{\\prime}}{(U)} = e^{U} and \\frac{d}{d U} e^{U} = e^{U} and \\frac{d}{d U} \\operatorname{y^{\\prime}}{(U)} = \\operatorname{y^{\\prime}}{(U)}", "srepr_derivation": [["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), exp(Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), exp(Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Function('y^{\\\\prime}')(Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(g_{\\varepsilon})} = e^{\\cos{(g_{\\varepsilon})}}, then obtain \\frac{d}{d g_{\\varepsilon}} (\\int 0 dg_{\\varepsilon})^{g_{\\varepsilon}} = \\frac{d}{d g_{\\varepsilon}} (\\int (- \\mathbf{B}{(g_{\\varepsilon})} + e^{\\cos{(g_{\\varepsilon})}}) dg_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "\\mathbf{B}{(g_{\\varepsilon})} = e^{\\cos{(g_{\\varepsilon})}} and 0 = - \\mathbf{B}{(g_{\\varepsilon})} + e^{\\cos{(g_{\\varepsilon})}} and \\int 0 dg_{\\varepsilon} = \\int (- \\mathbf{B}{(g_{\\varepsilon})} + e^{\\cos{(g_{\\varepsilon})}}) dg_{\\varepsilon} and (\\int 0 dg_{\\varepsilon})^{g_{\\varepsilon}} = (\\int (- \\mathbf{B}{(g_{\\varepsilon})} + e^{\\cos{(g_{\\varepsilon})}}) dg_{\\varepsilon})^{g_{\\varepsilon}} and \\frac{d}{d g_{\\varepsilon}} (\\int 0 dg_{\\varepsilon})^{g_{\\varepsilon}} = \\frac{d}{d g_{\\varepsilon}} (\\int (- \\mathbf{B}{(g_{\\varepsilon})} + e^{\\cos{(g_{\\varepsilon})}}) dg_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True))), exp(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True))), exp(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True))), exp(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Pow(Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('g_{\\\\varepsilon}', commutative=True))), exp(cos(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(v_{x},\\mathbf{S},M_{E})} = (M_{E} \\mathbf{S})^{v_{x}}, then obtain (\\int \\frac{\\partial}{\\partial v_{x}} Z{(v_{x},\\mathbf{S},M_{E})} d\\mathbf{S})^{\\mathbf{S}} = (\\int \\frac{\\partial}{\\partial v_{x}} (M_{E} \\mathbf{S})^{v_{x}} d\\mathbf{S})^{\\mathbf{S}}", "derivation": "Z{(v_{x},\\mathbf{S},M_{E})} = (M_{E} \\mathbf{S})^{v_{x}} and \\frac{\\partial}{\\partial v_{x}} Z{(v_{x},\\mathbf{S},M_{E})} = \\frac{\\partial}{\\partial v_{x}} (M_{E} \\mathbf{S})^{v_{x}} and \\int \\frac{\\partial}{\\partial v_{x}} Z{(v_{x},\\mathbf{S},M_{E})} d\\mathbf{S} = \\int \\frac{\\partial}{\\partial v_{x}} (M_{E} \\mathbf{S})^{v_{x}} d\\mathbf{S} and (\\int \\frac{\\partial}{\\partial v_{x}} Z{(v_{x},\\mathbf{S},M_{E})} d\\mathbf{S})^{\\mathbf{S}} = (\\int \\frac{\\partial}{\\partial v_{x}} (M_{E} \\mathbf{S})^{v_{x}} d\\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Pow(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Derivative(Function('Z')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Derivative(Pow(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('Z')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(Derivative(Pow(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)}, then obtain - \\cos{(\\hat{p}_0)} = - \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)}}{\\cos{(\\hat{p}_0)}} - \\cos{(\\hat{p}_0)} + 1", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)} = \\cos{(\\hat{p}_0)} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)}}{\\cos{(\\hat{p}_0)}} = 1 and 0 = - \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)}}{\\cos{(\\hat{p}_0)}} + 1 and - \\cos{(\\hat{p}_0)} = - \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\hat{p}_0)}}{\\cos{(\\hat{p}_0)}} - \\cos{(\\hat{p}_0)} + 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{p}_0', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{p}_0', commutative=True)), Pow(cos(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{p}_0', commutative=True)), Pow(cos(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{p}_0', commutative=True)), Pow(cos(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Integer(1)))"], [["minus", 3, "cos(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{p}_0', commutative=True)), Pow(cos(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\Psi_{nl}{(x^\\prime)} = e^{x^\\prime}, then obtain (\\frac{1}{x^\\prime})^{x^\\prime} = (\\frac{e^{x^\\prime}}{x^\\prime \\Psi_{nl}{(x^\\prime)}})^{x^\\prime}", "derivation": "\\Psi_{nl}{(x^\\prime)} = e^{x^\\prime} and \\frac{\\Psi_{nl}{(x^\\prime)}}{x^\\prime} = \\frac{e^{x^\\prime}}{x^\\prime} and \\frac{1}{x^\\prime} = \\frac{e^{x^\\prime}}{x^\\prime \\Psi_{nl}{(x^\\prime)}} and (\\frac{1}{x^\\prime})^{x^\\prime} = (\\frac{e^{x^\\prime}}{x^\\prime \\Psi_{nl}{(x^\\prime)}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 2, "Function('\\\\Psi_{nl}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\Psi_{nl}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), exp(Symbol('x^\\\\prime', commutative=True))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('\\\\Psi_{nl}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), exp(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\psi{(s,\\eta^{\\prime})} = s^{\\eta^{\\prime}}, then obtain \\frac{(\\eta^{\\prime} \\psi{(s,\\eta^{\\prime})})^{\\eta^{\\prime}}}{s} = \\frac{(\\eta^{\\prime} s^{\\eta^{\\prime}})^{\\eta^{\\prime}}}{s}", "derivation": "\\psi{(s,\\eta^{\\prime})} = s^{\\eta^{\\prime}} and \\eta^{\\prime} \\psi{(s,\\eta^{\\prime})} = \\eta^{\\prime} s^{\\eta^{\\prime}} and (\\eta^{\\prime} \\psi{(s,\\eta^{\\prime})})^{\\eta^{\\prime}} = (\\eta^{\\prime} s^{\\eta^{\\prime}})^{\\eta^{\\prime}} and \\frac{(\\eta^{\\prime} \\psi{(s,\\eta^{\\prime})})^{\\eta^{\\prime}}}{s} = \\frac{(\\eta^{\\prime} s^{\\eta^{\\prime}})^{\\eta^{\\prime}}}{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["times", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 3, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\psi')(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(C_{1})} = \\cos{(\\sin{(C_{1})})}, then obtain \\frac{d}{d C_{1}} (\\frac{\\operatorname{A_{z}}{(C_{1})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}} = \\frac{d}{d C_{1}} (\\frac{\\cos{(\\sin{(C_{1})})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}}", "derivation": "\\operatorname{A_{z}}{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\frac{\\operatorname{A_{z}}{(C_{1})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}} = \\frac{\\cos{(\\sin{(C_{1})})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}} and (\\frac{\\operatorname{A_{z}}{(C_{1})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}} = (\\frac{\\cos{(\\sin{(C_{1})})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}} and \\frac{d}{d C_{1}} (\\frac{\\operatorname{A_{z}}{(C_{1})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}} = \\frac{d}{d C_{1}} (\\frac{\\cos{(\\sin{(C_{1})})}}{\\frac{d}{d C_{1}} \\operatorname{A_{z}}{(C_{1})}})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('C_1', commutative=True)), cos(sin(Symbol('C_1', commutative=True))))"], [["divide", 1, "Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Mul(Function('A_z')(Symbol('C_1', commutative=True)), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Mul(cos(sin(Symbol('C_1', commutative=True))), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Mul(Function('A_z')(Symbol('C_1', commutative=True)), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Symbol('C_1', commutative=True)), Pow(Mul(cos(sin(Symbol('C_1', commutative=True))), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Symbol('C_1', commutative=True)))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('A_z')(Symbol('C_1', commutative=True)), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Mul(cos(sin(Symbol('C_1', commutative=True))), Pow(Derivative(Function('A_z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(S,\\mathbf{D})} = \\sin{(S + \\mathbf{D})}, then obtain \\frac{\\mathbf{D} \\hat{p}_0^{\\mathbf{D}}{(S,\\mathbf{D})} \\frac{\\partial}{\\partial S} \\hat{p}_0{(S,\\mathbf{D})}}{\\hat{p}_0{(S,\\mathbf{D})}} = \\frac{\\mathbf{D} \\sin^{\\mathbf{D}}{(S + \\mathbf{D})} \\cos{(S + \\mathbf{D})}}{\\sin{(S + \\mathbf{D})}}", "derivation": "\\hat{p}_0{(S,\\mathbf{D})} = \\sin{(S + \\mathbf{D})} and \\hat{p}_0^{\\mathbf{D}}{(S,\\mathbf{D})} = \\sin^{\\mathbf{D}}{(S + \\mathbf{D})} and \\frac{\\partial}{\\partial S} \\hat{p}_0^{\\mathbf{D}}{(S,\\mathbf{D})} = \\frac{\\partial}{\\partial S} \\sin^{\\mathbf{D}}{(S + \\mathbf{D})} and \\frac{\\mathbf{D} \\hat{p}_0^{\\mathbf{D}}{(S,\\mathbf{D})} \\frac{\\partial}{\\partial S} \\hat{p}_0{(S,\\mathbf{D})}}{\\hat{p}_0{(S,\\mathbf{D})}} = \\frac{\\mathbf{D} \\sin^{\\mathbf{D}}{(S + \\mathbf{D})} \\cos{(S + \\mathbf{D})}}{\\sin{(S + \\mathbf{D})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), sin(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(sin(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Pow(sin(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), Pow(sin(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(z)} = \\cos{(z)}, then derive \\int \\operatorname{E_{x}}{(z)} dz = \\mathbf{J}_P + \\sin{(z)}, then obtain \\frac{\\int \\cos{(z)} dz}{\\sin{(z)}} = \\frac{\\mathbf{J}_P + \\sin{(z)}}{\\sin{(z)}}", "derivation": "\\operatorname{E_{x}}{(z)} = \\cos{(z)} and \\int \\operatorname{E_{x}}{(z)} dz = \\int \\cos{(z)} dz and \\int \\operatorname{E_{x}}{(z)} dz = \\mathbf{J}_P + \\sin{(z)} and \\frac{\\int \\operatorname{E_{x}}{(z)} dz}{\\sin{(z)}} = \\frac{\\mathbf{J}_P + \\sin{(z)}}{\\sin{(z)}} and \\frac{\\int \\cos{(z)} dz}{\\sin{(z)}} = \\frac{\\mathbf{J}_P + \\sin{(z)}}{\\sin{(z)}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Symbol('z', commutative=True))))"], [["divide", 3, "sin(Symbol('z', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('z', commutative=True)), Integer(-1)), Integral(Function('E_x')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Symbol('z', commutative=True))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(sin(Symbol('z', commutative=True)), Integer(-1)), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Symbol('z', commutative=True))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given a{(\\mathbf{D})} = \\cos{(\\mathbf{D})}, then obtain \\frac{2 a^{\\mathbf{D}}{(\\mathbf{D})}}{\\mathbf{D}} = \\frac{a^{\\mathbf{D}}{(\\mathbf{D})} + \\cos^{\\mathbf{D}}{(\\mathbf{D})}}{\\mathbf{D}}", "derivation": "a{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and a^{\\mathbf{D}}{(\\mathbf{D})} = \\cos^{\\mathbf{D}}{(\\mathbf{D})} and 2 a^{\\mathbf{D}}{(\\mathbf{D})} = a^{\\mathbf{D}}{(\\mathbf{D})} + \\cos^{\\mathbf{D}}{(\\mathbf{D})} and \\frac{2 a^{\\mathbf{D}}{(\\mathbf{D})}}{\\mathbf{D}} = \\frac{a^{\\mathbf{D}}{(\\mathbf{D})} + \\cos^{\\mathbf{D}}{(\\mathbf{D})}}{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 2, "Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Add(Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Pow(Function('a')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(A,\\nabla)} = \\frac{\\nabla}{A}, then obtain (\\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A})^{2} = \\frac{4 \\nabla^{2}}{A^{2}}", "derivation": "\\operatorname{t_{1}}{(A,\\nabla)} = \\frac{\\nabla}{A} and \\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A} = \\frac{2 \\nabla}{A} and (\\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A})^{2} = \\frac{2 \\nabla (\\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A})}{A} and \\frac{\\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A}}{A} = \\frac{2 \\nabla}{A^{2}} and (\\operatorname{t_{1}}{(A,\\nabla)} + \\frac{\\nabla}{A})^{2} = \\frac{4 \\nabla^{2}}{A^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))"], [["times", 2, "Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))"], "Equality(Pow(Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True))), Integer(2)), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True), Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))))"], [["divide", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-2)), Symbol('\\\\nabla', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Function('t_1')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Symbol('A', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} = \\varepsilon_0 - n_{1}, then obtain \\int \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} d\\varepsilon_0 = \\frac{\\varepsilon_0^{2}}{2} - \\varepsilon_0 n_{1}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} = \\varepsilon_0 - n_{1} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} d\\varepsilon_0 = \\int (\\varepsilon_0 - n_{1}) d\\varepsilon_0 and \\iint \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} d\\varepsilon_0 dn_{1} = \\iint (\\varepsilon_0 - n_{1}) d\\varepsilon_0 dn_{1} and \\frac{\\partial}{\\partial n_{1}} \\iint \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} d\\varepsilon_0 dn_{1} = \\frac{\\partial}{\\partial n_{1}} \\iint (\\varepsilon_0 - n_{1}) d\\varepsilon_0 dn_{1} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\varepsilon_0,n_{1})} d\\varepsilon_0 = \\frac{\\varepsilon_0^{2}}{2} - \\varepsilon_0 n_{1}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 2, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given f{(t_{1},\\theta_2,v_{2})} = (\\frac{t_{1}}{\\theta_2})^{v_{2}}, then obtain - \\frac{\\partial^{2}}{\\partial v_{2}^{2}} f{(t_{1},\\theta_2,v_{2})} = - (\\frac{t_{1}}{\\theta_2})^{v_{2}} \\log{(\\frac{t_{1}}{\\theta_2})}^{2}", "derivation": "f{(t_{1},\\theta_2,v_{2})} = (\\frac{t_{1}}{\\theta_2})^{v_{2}} and - f{(t_{1},\\theta_2,v_{2})} = - (\\frac{t_{1}}{\\theta_2})^{v_{2}} and \\frac{\\partial}{\\partial v_{2}} - f{(t_{1},\\theta_2,v_{2})} = \\frac{\\partial}{\\partial v_{2}} - (\\frac{t_{1}}{\\theta_2})^{v_{2}} and \\frac{\\partial^{2}}{\\partial v_{2}^{2}} - f{(t_{1},\\theta_2,v_{2})} = \\frac{\\partial^{2}}{\\partial v_{2}^{2}} - (\\frac{t_{1}}{\\theta_2})^{v_{2}} and - \\frac{\\partial^{2}}{\\partial v_{2}^{2}} f{(t_{1},\\theta_2,v_{2})} = - (\\frac{t_{1}}{\\theta_2})^{v_{2}} \\log{(\\frac{t_{1}}{\\theta_2})}^{2}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Symbol('v_2', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Derivative(Function('f')(Symbol('t_1', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)), Symbol('v_2', commutative=True)), Pow(log(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mu_{0}{(C_{1},B)} = \\log{(C_{1}^{B})} and \\operatorname{v_{y}}{(f_{\\mathbf{v}})} = \\cos{(\\cos{(f_{\\mathbf{v}})})}, then obtain \\frac{\\sin{(\\operatorname{v_{y}}{(f_{\\mathbf{v}})})}}{C_{1} + \\log{(C_{1}^{B})}} = \\frac{\\sin{(\\cos{(\\cos{(f_{\\mathbf{v}})})})}}{C_{1} + \\log{(C_{1}^{B})}}", "derivation": "\\mu_{0}{(C_{1},B)} = \\log{(C_{1}^{B})} and C_{1} + \\mu_{0}{(C_{1},B)} = C_{1} + \\log{(C_{1}^{B})} and \\operatorname{v_{y}}{(f_{\\mathbf{v}})} = \\cos{(\\cos{(f_{\\mathbf{v}})})} and \\sin{(\\operatorname{v_{y}}{(f_{\\mathbf{v}})})} = \\sin{(\\cos{(\\cos{(f_{\\mathbf{v}})})})} and \\frac{\\sin{(\\operatorname{v_{y}}{(f_{\\mathbf{v}})})}}{C_{1} + \\mu_{0}{(C_{1},B)}} = \\frac{\\sin{(\\cos{(\\cos{(f_{\\mathbf{v}})})})}}{C_{1} + \\mu_{0}{(C_{1},B)}} and \\frac{\\sin{(\\operatorname{v_{y}}{(f_{\\mathbf{v}})})}}{C_{1} + \\log{(C_{1}^{B})}} = \\frac{\\sin{(\\cos{(\\cos{(f_{\\mathbf{v}})})})}}{C_{1} + \\log{(C_{1}^{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('C_1', commutative=True), Symbol('B', commutative=True))))"], [["add", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\mu_0')(Symbol('C_1', commutative=True), Symbol('B', commutative=True))), Add(Symbol('C_1', commutative=True), log(Pow(Symbol('C_1', commutative=True), Symbol('B', commutative=True)))))"], ["get_premise", "Equality(Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["sin", 3], "Equality(sin(Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), sin(cos(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["divide", 4, "Add(Symbol('C_1', commutative=True), Function('\\\\mu_0')(Symbol('C_1', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Function('\\\\mu_0')(Symbol('C_1', commutative=True), Symbol('B', commutative=True))), Integer(-1)), sin(Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(Pow(Add(Symbol('C_1', commutative=True), Function('\\\\mu_0')(Symbol('C_1', commutative=True), Symbol('B', commutative=True))), Integer(-1)), sin(cos(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), log(Pow(Symbol('C_1', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), sin(Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(Pow(Add(Symbol('C_1', commutative=True), log(Pow(Symbol('C_1', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), sin(cos(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(n_{2})} = \\cos{(n_{2})}, then obtain - J_{\\varepsilon} + \\operatorname{g_{\\varepsilon}}{(n_{2})} - \\sin{(n_{2})} + \\frac{d}{d n_{2}} n_{2} \\int \\rho_{f}{(n_{2})} dn_{2} = - J_{\\varepsilon} + \\operatorname{g_{\\varepsilon}}{(n_{2})} - \\sin{(n_{2})} + \\frac{d}{d n_{2}} n_{2} \\int \\cos{(n_{2})} dn_{2}", "derivation": "\\rho_{f}{(n_{2})} = \\cos{(n_{2})} and \\int \\rho_{f}{(n_{2})} dn_{2} = \\int \\cos{(n_{2})} dn_{2} and n_{2} \\int \\rho_{f}{(n_{2})} dn_{2} = n_{2} \\int \\cos{(n_{2})} dn_{2} and \\frac{d}{d n_{2}} n_{2} \\int \\rho_{f}{(n_{2})} dn_{2} = \\frac{d}{d n_{2}} n_{2} \\int \\cos{(n_{2})} dn_{2} and - J_{\\varepsilon} + \\operatorname{g_{\\varepsilon}}{(n_{2})} - \\sin{(n_{2})} + \\frac{d}{d n_{2}} n_{2} \\int \\rho_{f}{(n_{2})} dn_{2} = - J_{\\varepsilon} + \\operatorname{g_{\\varepsilon}}{(n_{2})} - \\sin{(n_{2})} + \\frac{d}{d n_{2}} n_{2} \\int \\cos{(n_{2})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["times", 2, "Symbol('n_2', commutative=True)"], "Equality(Mul(Symbol('n_2', commutative=True), Integral(Function('\\\\rho_f')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Mul(Symbol('n_2', commutative=True), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('n_2', commutative=True), Integral(Function('\\\\rho_f')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('n_2', commutative=True), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["minus", 4, "Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('n_2', commutative=True))), sin(Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Derivative(Mul(Symbol('n_2', commutative=True), Integral(Function('\\\\rho_f')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Derivative(Mul(Symbol('n_2', commutative=True), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{g})} = \\mathbf{g}, then derive \\frac{d}{d \\mathbf{g}} \\operatorname{v_{1}}{(\\mathbf{g})} = 1, then obtain \\frac{d}{d \\mathbf{g}} \\mathbf{g} = 1", "derivation": "\\operatorname{v_{1}}{(\\mathbf{g})} = \\mathbf{g} and \\frac{d}{d \\mathbf{g}} \\operatorname{v_{1}}{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\mathbf{g} and \\frac{d}{d \\mathbf{g}} \\operatorname{v_{1}}{(\\mathbf{g})} = 1 and \\frac{d}{d \\mathbf{g}} \\mathbf{g} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\mu_{0}{(\\phi_1,l)} = \\phi_1 + l, then obtain \\frac{\\mu_{0}{(\\phi_1,l)}}{(\\phi_1 + l) \\mu_{0}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}} = \\frac{\\phi_1 + l}{(\\phi_1 + l) \\mu_{0}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}}", "derivation": "\\mu_{0}{(\\phi_1,l)} = \\phi_1 + l and \\mu_{0}^{2}{(\\phi_1,l)} = (\\phi_1 + l) \\mu_{0}{(\\phi_1,l)} and \\mu_{0}^{2}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)} = \\phi_1 + l + \\mu_{0}^{2}{(\\phi_1,l)} and \\frac{\\mu_{0}{(\\phi_1,l)}}{\\phi_1 + l + \\mu_{0}^{2}{(\\phi_1,l)}} = \\frac{\\phi_1 + l}{\\phi_1 + l + \\mu_{0}^{2}{(\\phi_1,l)}} and \\frac{\\mu_{0}{(\\phi_1,l)}}{\\mu_{0}^{2}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}} = \\frac{\\phi_1 + l}{\\mu_{0}^{2}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}} and \\frac{\\mu_{0}{(\\phi_1,l)}}{(\\phi_1 + l) \\mu_{0}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}} = \\frac{\\phi_1 + l}{(\\phi_1 + l) \\mu_{0}{(\\phi_1,l)} + \\mu_{0}{(\\phi_1,l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))))"], [["add", 1, "Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2))))"], [["divide", 1, "Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2))), Integer(-1)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Pow(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Integer(-1)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Pow(Add(Mul(Add(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Function('\\\\mu_0')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given i{(\\omega)} = \\omega, then obtain 2 (\\mathbf{r} \\omega + i{(\\omega)})^{2} = (\\mathbf{r} \\omega + \\omega) (\\mathbf{r} \\omega + i{(\\omega)}) + (\\mathbf{r} \\omega + i{(\\omega)})^{2}", "derivation": "i{(\\omega)} = \\omega and \\mathbf{r} \\omega + i{(\\omega)} = \\mathbf{r} \\omega + \\omega and (\\mathbf{r} \\omega + i{(\\omega)})^{2} = (\\mathbf{r} \\omega + \\omega) (\\mathbf{r} \\omega + i{(\\omega)}) and 2 (\\mathbf{r} \\omega + i{(\\omega)})^{2} = (\\mathbf{r} \\omega + \\omega) (\\mathbf{r} \\omega + i{(\\omega)}) + (\\mathbf{r} \\omega + i{(\\omega)})^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('i')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["add", 1, "Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["times", 2, "Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True)))"], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True))), Integer(2)), Mul(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True)))))"], [["add", 3, "Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True))), Integer(2))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True))), Integer(2))), Add(Mul(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True)))), Pow(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('i')(Symbol('\\\\omega', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{H},\\theta_1)} = \\hat{H} + \\theta_1, then derive \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)} = 1, then obtain \\log{(\\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)})}^{\\hat{H}} = 0^{\\hat{H}}", "derivation": "\\operatorname{F_{g}}{(\\hat{H},\\theta_1)} = \\hat{H} + \\theta_1 and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} + \\theta_1) and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)} = 1 and \\log{(\\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)})} = 0 and \\log{(\\frac{\\partial}{\\partial \\hat{H}} \\operatorname{F_{g}}{(\\hat{H},\\theta_1)})}^{\\hat{H}} = 0^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"], [["log", 3], "Equality(log(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(0))"], [["power", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(log(Derivative(Function('F_g')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\eta{(u)} = \\sin{(u)}, then obtain - \\eta{(u)} + \\sin{(u)} = 0", "derivation": "\\eta{(u)} = \\sin{(u)} and - u + \\eta{(u)} = - u + \\sin{(u)} and \\eta{(u)} - \\sin{(u)} = 0 and - \\eta{(u)} + \\sin{(u)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\eta')(Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], "Equality(Add(Function('\\\\eta')(Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Integer(0))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{p}{(z^{*},\\mathbf{F})} = \\log{(\\mathbf{F} z^{*})}, then obtain 0 = - \\frac{\\mathbf{p}{(z^{*},\\mathbf{F})}}{\\log{(\\mathbf{F} z^{*})}} + 1", "derivation": "\\mathbf{p}{(z^{*},\\mathbf{F})} = \\log{(\\mathbf{F} z^{*})} and \\frac{\\mathbf{p}{(z^{*},\\mathbf{F})}}{\\log{(\\mathbf{F} z^{*})}} = 1 and \\frac{\\mathbf{p}{(z^{*},\\mathbf{F})}}{\\log{(\\mathbf{F} z^{*})}} - 1 = 0 and 0 = - \\frac{\\mathbf{p}{(z^{*},\\mathbf{F})}}{\\log{(\\mathbf{F} z^{*})}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True))))"], [["divide", 1, "log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Mul(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True))), Integer(-1))), Integer(-1)), Integer(0))"], [["minus", 3, "Add(Mul(Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True))), Integer(-1))), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('z^*', commutative=True))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\theta{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and \\operatorname{v_{t}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})}, then obtain (\\theta^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}} = (\\operatorname{v_{t}}^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}}", "derivation": "\\theta{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and \\theta^{E_{\\lambda}}{(E_{\\lambda})} = \\sin^{E_{\\lambda}}{(E_{\\lambda})} and \\operatorname{v_{t}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and \\theta^{E_{\\lambda}}{(E_{\\lambda})} = \\operatorname{v_{t}}^{E_{\\lambda}}{(E_{\\lambda})} and (\\theta^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}} = (\\operatorname{v_{t}}^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(sin(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Pow(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given b{(\\rho_b,v_{t})} = \\log{(v_{t})}^{\\rho_b}, then derive e^{\\int b{(\\rho_b,v_{t})} dv_{t}} = e^{x + (- \\log{(v_{t})})^{- \\rho_b} \\log{(v_{t})}^{\\rho_b} \\Gamma(\\rho_b + 1, - \\log{(v_{t})})}, then obtain e^{\\int b{(\\rho_b,v_{t})} dv_{t}} = e^{x + (- \\log{(v_{t})})^{- \\rho_b} b{(\\rho_b,v_{t})} \\Gamma(\\rho_b + 1, - \\log{(v_{t})})}", "derivation": "b{(\\rho_b,v_{t})} = \\log{(v_{t})}^{\\rho_b} and \\int b{(\\rho_b,v_{t})} dv_{t} = \\int \\log{(v_{t})}^{\\rho_b} dv_{t} and e^{\\int b{(\\rho_b,v_{t})} dv_{t}} = e^{\\int \\log{(v_{t})}^{\\rho_b} dv_{t}} and e^{\\int b{(\\rho_b,v_{t})} dv_{t}} = e^{x + (- \\log{(v_{t})})^{- \\rho_b} \\log{(v_{t})}^{\\rho_b} \\Gamma(\\rho_b + 1, - \\log{(v_{t})})} and e^{\\int b{(\\rho_b,v_{t})} dv_{t}} = e^{x + (- \\log{(v_{t})})^{- \\rho_b} b{(\\rho_b,v_{t})} \\Gamma(\\rho_b + 1, - \\log{(v_{t})})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Pow(log(Symbol('v_t', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Pow(log(Symbol('v_t', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), exp(Integral(Pow(log(Symbol('v_t', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), exp(Add(Symbol('x', commutative=True), Mul(Pow(Mul(Integer(-1), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Pow(log(Symbol('v_t', commutative=True)), Symbol('\\\\rho_b', commutative=True)), uppergamma(Add(Symbol('\\\\rho_b', commutative=True), Integer(1)), Mul(Integer(-1), log(Symbol('v_t', commutative=True))))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(exp(Integral(Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), exp(Add(Symbol('x', commutative=True), Mul(Pow(Mul(Integer(-1), log(Symbol('v_t', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Function('b')(Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), uppergamma(Add(Symbol('\\\\rho_b', commutative=True), Integer(1)), Mul(Integer(-1), log(Symbol('v_t', commutative=True))))))))"]]}, {"prompt": "Given \\hat{H}_l{(C)} = \\log{(e^{C})}, then obtain \\log{(e^{- \\hat{H}_l{(C)}})} = \\log{(e^{C} e^{2 C - 4 \\hat{H}_l{(C)}})}", "derivation": "\\hat{H}_l{(C)} = \\log{(e^{C})} and - \\hat{H}_l{(C)} = - 2 \\hat{H}_l{(C)} + \\log{(e^{C})} and e^{- \\hat{H}_l{(C)}} = e^{C - 2 \\hat{H}_l{(C)}} and \\log{(e^{- \\hat{H}_l{(C)}})} = \\log{(e^{C - 2 \\hat{H}_l{(C)}})} and \\log{(e^{C - 2 \\hat{H}_l{(C)}})} = \\log{(e^{C} e^{2 C - 4 \\hat{H}_l{(C)}})} and \\log{(e^{- \\hat{H}_l{(C)}})} = \\log{(e^{C} e^{2 C - 4 \\hat{H}_l{(C)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('C', commutative=True)), log(exp(Symbol('C', commutative=True))))"], [["minus", 1, "Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('C', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))), log(exp(Symbol('C', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('C', commutative=True)))), exp(Add(Symbol('C', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))))))"], [["log", 3], "Equality(log(exp(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))))), log(exp(Add(Symbol('C', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('C', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(exp(Add(Symbol('C', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('C', commutative=True)))))), log(Mul(exp(Symbol('C', commutative=True)), exp(Add(Mul(Integer(2), Symbol('C', commutative=True)), Mul(Integer(-1), Integer(4), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(log(exp(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))))), log(Mul(exp(Symbol('C', commutative=True)), exp(Add(Mul(Integer(2), Symbol('C', commutative=True)), Mul(Integer(-1), Integer(4), Function('\\\\hat{H}_l')(Symbol('C', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = \\mathbf{J}_M + \\mu_0, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = 1, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + \\mu_0) + \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + \\mu_0) + 1", "derivation": "\\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = \\mathbf{J}_M + \\mu_0 and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + \\mu_0) and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = 1 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + \\mu_0) + \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mu_0,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + \\mu_0) + 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Function('v_2')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given J{(\\phi,n)} = \\phi + n and y{(U,\\mu)} = \\frac{\\sin{(\\mu)}}{U}, then obtain - (\\phi + J{(\\phi,n)}) y{(U,\\mu)} = - \\frac{(\\phi + J{(\\phi,n)}) \\sin{(\\mu)}}{U}", "derivation": "J{(\\phi,n)} = \\phi + n and \\phi + J{(\\phi,n)} = 2 \\phi + n and y{(U,\\mu)} = \\frac{\\sin{(\\mu)}}{U} and - y{(U,\\mu)} = - \\frac{\\sin{(\\mu)}}{U} and - (2 \\phi + n) y{(U,\\mu)} = - \\frac{(2 \\phi + n) \\sin{(\\mu)}}{U} and - (\\phi + J{(\\phi,n)}) y{(U,\\mu)} = - \\frac{(\\phi + J{(\\phi,n)}) \\sin{(\\mu)}}{U}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\phi', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('J')(Symbol('\\\\phi', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True)), Symbol('n', commutative=True)))"], ["get_premise", "Equality(Function('y')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('y')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), sin(Symbol('\\\\mu', commutative=True))))"], [["times", 4, "Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True)), Symbol('n', commutative=True))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True)), Symbol('n', commutative=True)), Function('y')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\phi', commutative=True)), Symbol('n', commutative=True)), sin(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Add(Symbol('\\\\phi', commutative=True), Function('J')(Symbol('\\\\phi', commutative=True), Symbol('n', commutative=True))), Function('y')(Symbol('U', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Function('J')(Symbol('\\\\phi', commutative=True), Symbol('n', commutative=True))), sin(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v_{x},A_{y})} = A_{y} + v_{x}, then obtain (3 \\operatorname{M_{E}}{(v_{x},A_{y})})^{A_{y}} = (3 A_{y} + 3 v_{x})^{A_{y}}", "derivation": "\\operatorname{M_{E}}{(v_{x},A_{y})} = A_{y} + v_{x} and A_{y} + v_{x} + \\operatorname{M_{E}}{(v_{x},A_{y})} = 2 A_{y} + 2 v_{x} and 2 A_{y} + 2 v_{x} + \\operatorname{M_{E}}{(v_{x},A_{y})} = 3 A_{y} + 3 v_{x} and 2 \\operatorname{M_{E}}{(v_{x},A_{y})} = 2 A_{y} + 2 v_{x} and 3 \\operatorname{M_{E}}{(v_{x},A_{y})} = 3 A_{y} + 3 v_{x} and (3 \\operatorname{M_{E}}{(v_{x},A_{y})})^{A_{y}} = (3 A_{y} + 3 v_{x})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 1, "Add(Symbol('A_y', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Symbol('A_y', commutative=True), Symbol('v_x', commutative=True), Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["add", 2, "Add(Symbol('A_y', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True)), Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Integer(3), Symbol('A_y', commutative=True)), Mul(Integer(3), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(3), Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Integer(3), Symbol('A_y', commutative=True)), Mul(Integer(3), Symbol('v_x', commutative=True))))"], [["power", 5, "Symbol('A_y', commutative=True)"], "Equality(Pow(Mul(Integer(3), Function('M_E')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(3), Symbol('A_y', commutative=True)), Mul(Integer(3), Symbol('v_x', commutative=True))), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given f{(\\dot{z},\\mathbf{M})} = \\frac{\\cos{(\\dot{z})}}{\\mathbf{M}}, then obtain \\frac{f^{2}{(\\dot{z},\\mathbf{M})}}{\\mathbf{M}} = \\frac{\\cos^{2}{(\\dot{z})}}{\\mathbf{M}^{3}}", "derivation": "f{(\\dot{z},\\mathbf{M})} = \\frac{\\cos{(\\dot{z})}}{\\mathbf{M}} and \\frac{f{(\\dot{z},\\mathbf{M})}}{\\mathbf{M}} = \\frac{\\cos{(\\dot{z})}}{\\mathbf{M}^{2}} and \\frac{f{(\\dot{z},\\mathbf{M})} \\cos{(\\dot{z})}}{\\mathbf{M}^{2}} = \\frac{\\cos^{2}{(\\dot{z})}}{\\mathbf{M}^{3}} and \\frac{f^{2}{(\\dot{z},\\mathbf{M})}}{\\mathbf{M}} = \\frac{f{(\\dot{z},\\mathbf{M})} \\cos{(\\dot{z})}}{\\mathbf{M}^{2}} and \\frac{f^{2}{(\\dot{z},\\mathbf{M})}}{\\mathbf{M}} = \\frac{\\cos^{2}{(\\dot{z})}}{\\mathbf{M}^{3}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-2)), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), cos(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-2)), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-3)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-2)), Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-3)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(x)} = \\log{(\\cos{(x)})}, then obtain 2 x \\operatorname{n_{1}}{(x)} \\log{(\\cos{(x)})} = x (\\operatorname{n_{1}}{(x)} + \\log{(\\cos{(x)})}) \\log{(\\cos{(x)})}", "derivation": "\\operatorname{n_{1}}{(x)} = \\log{(\\cos{(x)})} and x \\operatorname{n_{1}}{(x)} = x \\log{(\\cos{(x)})} and 2 \\operatorname{n_{1}}{(x)} = \\operatorname{n_{1}}{(x)} + \\log{(\\cos{(x)})} and 2 x \\operatorname{n_{1}}^{2}{(x)} = x (\\operatorname{n_{1}}{(x)} + \\log{(\\cos{(x)})}) \\operatorname{n_{1}}{(x)} and 2 x \\operatorname{n_{1}}{(x)} \\log{(\\cos{(x)})} = x (\\operatorname{n_{1}}{(x)} + \\log{(\\cos{(x)})}) \\log{(\\cos{(x)})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('n_1')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), log(cos(Symbol('x', commutative=True)))))"], [["add", 1, "Function('n_1')(Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('x', commutative=True))), Add(Function('n_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True)))))"], [["times", 3, "Mul(Symbol('x', commutative=True), Function('n_1')(Symbol('x', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('x', commutative=True), Pow(Function('n_1')(Symbol('x', commutative=True)), Integer(2))), Mul(Symbol('x', commutative=True), Add(Function('n_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True)))), Function('n_1')(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Symbol('x', commutative=True), Function('n_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True)))), Mul(Symbol('x', commutative=True), Add(Function('n_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True)))), log(cos(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(n_{2})} = \\log{(e^{n_{2}})} and \\mathbf{g}{(n_{2})} = \\frac{(- n_{2} + \\log{(e^{n_{2}})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}}, then obtain \\int \\mathbf{g}^{n_{2}}{(n_{2})} dn_{2} = \\int (\\frac{(- n_{2} + \\mathbf{D}{(n_{2})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}})^{n_{2}} dn_{2}", "derivation": "\\mathbf{D}{(n_{2})} = \\log{(e^{n_{2}})} and - n_{2} + \\mathbf{D}{(n_{2})} = - n_{2} + \\log{(e^{n_{2}})} and \\mathbf{g}{(n_{2})} = \\frac{(- n_{2} + \\log{(e^{n_{2}})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}} and \\mathbf{g}{(n_{2})} = \\frac{(- n_{2} + \\mathbf{D}{(n_{2})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}} and \\mathbf{g}^{n_{2}}{(n_{2})} = (\\frac{(- n_{2} + \\mathbf{D}{(n_{2})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}})^{n_{2}} and \\int \\mathbf{g}^{n_{2}}{(n_{2})} dn_{2} = \\int (\\frac{(- n_{2} + \\mathbf{D}{(n_{2})})^{n_{2}} - \\mathbf{D}{(n_{2})}}{n_{2}})^{n_{2}} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), log(exp(Symbol('n_2', commutative=True))))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), log(exp(Symbol('n_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('n_2', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), log(exp(Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{g}')(Symbol('n_2', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))))))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))))), Symbol('n_2', commutative=True)))"], [["integrate", 5, "Symbol('n_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{g}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Pow(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True))))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(S,h)} = S h, then obtain \\cos{(\\hat{H}_{\\lambda}{(S,h)})} \\frac{\\partial}{\\partial S} \\hat{H}_{\\lambda}{(S,h)} = \\cos{(\\hat{H}_{\\lambda}{(S,h)})} \\frac{\\partial}{\\partial S} S h", "derivation": "\\hat{H}_{\\lambda}{(S,h)} = S h and \\frac{\\partial}{\\partial S} \\hat{H}_{\\lambda}{(S,h)} = \\frac{\\partial}{\\partial S} S h and \\cos{(\\hat{H}_{\\lambda}{(S,h)})} = \\cos{(S h)} and \\cos{(S h)} \\frac{\\partial}{\\partial S} \\hat{H}_{\\lambda}{(S,h)} = \\cos{(S h)} \\frac{\\partial}{\\partial S} S h and \\cos{(\\hat{H}_{\\lambda}{(S,h)})} \\frac{\\partial}{\\partial S} \\hat{H}_{\\lambda}{(S,h)} = \\cos{(\\hat{H}_{\\lambda}{(S,h)})} \\frac{\\partial}{\\partial S} S h", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["cos", 1], "Equality(cos(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True))), cos(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True))))"], [["times", 2, "cos(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(cos(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(cos(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(cos(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(cos(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('S', commutative=True), Symbol('h', commutative=True))), Derivative(Mul(Symbol('S', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(v,v_{z})} = \\frac{v_{z}}{v}, then obtain \\mathbf{p}{(v,v_{z})} - \\int v_{z} \\mathbf{p}{(v,v_{z})} dv_{z} = - \\int v_{z} \\mathbf{p}{(v,v_{z})} dv_{z} + \\frac{v_{z}}{v}", "derivation": "\\mathbf{p}{(v,v_{z})} = \\frac{v_{z}}{v} and v_{z} \\mathbf{p}{(v,v_{z})} = \\frac{v_{z}^{2}}{v} and \\int v_{z} \\mathbf{p}{(v,v_{z})} dv_{z} = \\int \\frac{v_{z}^{2}}{v} dv_{z} and \\mathbf{p}{(v,v_{z})} - \\int \\frac{v_{z}^{2}}{v} dv_{z} = - \\int \\frac{v_{z}^{2}}{v} dv_{z} + \\frac{v_{z}}{v} and \\mathbf{p}{(v,v_{z})} - \\int v_{z} \\mathbf{p}{(v,v_{z})} dv_{z} = - \\int v_{z} \\mathbf{p}{(v,v_{z})} dv_{z} + \\frac{v_{z}}{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["times", 1, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Symbol('v_z', commutative=True), Integer(2))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Mul(Symbol('v_z', commutative=True), Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Symbol('v_z', commutative=True), Integer(2))), Tuple(Symbol('v_z', commutative=True))))"], [["minus", 1, "Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Symbol('v_z', commutative=True), Integer(2))), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Symbol('v_z', commutative=True), Integer(2))), Tuple(Symbol('v_z', commutative=True))))), Add(Mul(Integer(-1), Integral(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Symbol('v_z', commutative=True), Integer(2))), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('v_z', commutative=True), Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))), Add(Mul(Integer(-1), Integral(Mul(Symbol('v_z', commutative=True), Function('\\\\mathbf{p}')(Symbol('v', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(M)} = M, then obtain (\\frac{1}{M} + \\frac{\\theta_{1}^{4}{(M)}}{M^{4}})^{4} = (1 + \\frac{1}{M})^{4}", "derivation": "\\theta_{1}{(M)} = M and \\frac{\\theta_{1}{(M)}}{M} = 1 and \\frac{\\theta_{1}^{2}{(M)}}{M^{2}} = \\frac{\\theta_{1}{(M)}}{M} and \\frac{\\theta_{1}^{2}{(M)}}{M^{2}} = 1 and \\frac{\\theta_{1}^{4}{(M)}}{M^{4}} = 1 and \\frac{1}{M} + \\frac{\\theta_{1}^{4}{(M)}}{M^{4}} = 1 + \\frac{1}{M} and (\\frac{1}{M} + \\frac{\\theta_{1}^{4}{(M)}}{M^{4}})^{4} = (1 + \\frac{1}{M})^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('M', commutative=True)), Symbol('M', commutative=True))"], [["divide", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('M', commutative=True))), Integer(1))"], [["times", 2, "Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('M', commutative=True)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-2)), Pow(Function('\\\\theta_1')(Symbol('M', commutative=True)), Integer(2))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-2)), Pow(Function('\\\\theta_1')(Symbol('M', commutative=True)), Integer(2))), Integer(1))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-4)), Pow(Function('\\\\theta_1')(Symbol('M', commutative=True)), Integer(4))), Integer(1))"], [["add", 5, "Pow(Symbol('M', commutative=True), Integer(-1))"], "Equality(Add(Pow(Symbol('M', commutative=True), Integer(-1)), Mul(Pow(Symbol('M', commutative=True), Integer(-4)), Pow(Function('\\\\theta_1')(Symbol('M', commutative=True)), Integer(4)))), Add(Integer(1), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["power", 6, 4], "Equality(Pow(Add(Pow(Symbol('M', commutative=True), Integer(-1)), Mul(Pow(Symbol('M', commutative=True), Integer(-4)), Pow(Function('\\\\theta_1')(Symbol('M', commutative=True)), Integer(4)))), Integer(4)), Pow(Add(Integer(1), Pow(Symbol('M', commutative=True), Integer(-1))), Integer(4)))"]]}, {"prompt": "Given \\Psi_{nl}{(Z)} = \\log{(Z)} and z{(Z)} = \\Psi_{nl}{(Z)} - \\log{(Z)}, then obtain \\frac{d}{d Z} z^{Z}{(Z)} = \\frac{d}{d Z} (\\Psi_{nl}{(Z)} - \\log{(Z)})^{Z}", "derivation": "\\Psi_{nl}{(Z)} = \\log{(Z)} and z{(Z)} = \\Psi_{nl}{(Z)} - \\log{(Z)} and z{(Z)} = 0 and z^{Z}{(Z)} = 0^{Z} and \\frac{d}{d Z} z^{Z}{(Z)} = \\frac{d}{d Z} 0^{Z} and \\frac{d}{d Z} (\\Psi_{nl}{(Z)} - \\log{(Z)})^{Z} = \\frac{d}{d Z} 0^{Z} and \\frac{d}{d Z} z^{Z}{(Z)} = \\frac{d}{d Z} (\\Psi_{nl}{(Z)} - \\log{(Z)})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('Z', commutative=True)), Add(Function('\\\\Psi_{nl}')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('z')(Symbol('Z', commutative=True)), Integer(0))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('z')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Integer(0), Symbol('Z', commutative=True)))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Function('z')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Pow(Function('z')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(v_{1})} = \\cos{(v_{1})}, then obtain \\int \\frac{v_{1} + \\bar{\\h}{(v_{1})}}{v_{1}} dv_{1} = \\int \\frac{v_{1} + \\cos{(v_{1})}}{v_{1}} dv_{1}", "derivation": "\\bar{\\h}{(v_{1})} = \\cos{(v_{1})} and v_{1} + \\bar{\\h}{(v_{1})} = v_{1} + \\cos{(v_{1})} and \\frac{v_{1} + \\bar{\\h}{(v_{1})}}{v_{1}} = \\frac{v_{1} + \\cos{(v_{1})}}{v_{1}} and \\int \\frac{v_{1} + \\bar{\\h}{(v_{1})}}{v_{1}} dv_{1} = \\int \\frac{v_{1} + \\cos{(v_{1})}}{v_{1}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\hbar')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))))"], [["divide", 2, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), Function('\\\\hbar')(Symbol('v_1', commutative=True)))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True)))))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), Function('\\\\hbar')(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given L{(\\lambda)} = e^{\\lambda} and m{(\\lambda)} = \\int L^{\\lambda}{(\\lambda)} d\\lambda, then obtain \\frac{\\frac{d}{d \\lambda} m{(\\lambda)}}{\\frac{d}{d \\lambda} \\int (e^{\\lambda})^{\\lambda} d\\lambda} = 1", "derivation": "L{(\\lambda)} = e^{\\lambda} and L^{\\lambda}{(\\lambda)} = (e^{\\lambda})^{\\lambda} and \\int L^{\\lambda}{(\\lambda)} d\\lambda = \\int (e^{\\lambda})^{\\lambda} d\\lambda and m{(\\lambda)} = \\int L^{\\lambda}{(\\lambda)} d\\lambda and m{(\\lambda)} = \\int (e^{\\lambda})^{\\lambda} d\\lambda and \\frac{d}{d \\lambda} m{(\\lambda)} = \\frac{d}{d \\lambda} \\int (e^{\\lambda})^{\\lambda} d\\lambda and \\frac{\\frac{d}{d \\lambda} m{(\\lambda)}}{\\frac{d}{d \\lambda} \\int (e^{\\lambda})^{\\lambda} d\\lambda} = 1", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Pow(Function('L')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\lambda', commutative=True)), Integral(Pow(Function('L')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('m')(Symbol('\\\\lambda', commutative=True)), Integral(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["divide", 6, "Derivative(Integral(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('m')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Pow(Derivative(Integral(Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\tilde{g}{(f^{*},i)} = - f^{*} + i, then obtain \\tilde{g}{(f^{*},i)} \\int \\tilde{g}{(f^{*},i)} df^{*} = (- f^{*} + i) \\int \\tilde{g}{(f^{*},i)} df^{*}", "derivation": "\\tilde{g}{(f^{*},i)} = - f^{*} + i and \\int \\tilde{g}{(f^{*},i)} df^{*} = \\int (- f^{*} + i) df^{*} and \\tilde{g}{(f^{*},i)} \\int (- f^{*} + i) df^{*} = (- f^{*} + i) \\int (- f^{*} + i) df^{*} and \\tilde{g}{(f^{*},i)} \\int \\tilde{g}{(f^{*},i)} df^{*} = (- f^{*} + i) \\int \\tilde{g}{(f^{*},i)} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["times", 1, "Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True)))"], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Integral(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('i', commutative=True)), Integral(Function('\\\\tilde{g}')(Symbol('f^*', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\theta,\\mathbf{p})} = \\mathbf{p} \\theta, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{C_{d}}{(\\theta,\\mathbf{p})} = \\theta, then obtain \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} \\theta = \\theta", "derivation": "\\operatorname{C_{d}}{(\\theta,\\mathbf{p})} = \\mathbf{p} \\theta and \\theta + \\operatorname{C_{d}}{(\\theta,\\mathbf{p})} = \\mathbf{p} \\theta + \\theta and \\frac{\\partial}{\\partial \\mathbf{p}} (\\theta + \\operatorname{C_{d}}{(\\theta,\\mathbf{p})}) = \\frac{\\partial}{\\partial \\mathbf{p}} (\\mathbf{p} \\theta + \\theta) and \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{C_{d}}{(\\theta,\\mathbf{p})} = \\theta and \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} \\theta = \\theta", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('C_d')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\theta', commutative=True), Function('C_d')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('C_d')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True))"]]}, {"prompt": "Given g{(f_{E})} = \\int \\sin{(f_{E})} df_{E} and \\ddot{x}{(S,f_{E})} = (S - \\cos{(f_{E})})^{2}, then derive g{(f_{E})} = S - \\cos{(f_{E})}, then obtain \\frac{S + \\ddot{x}{(S,f_{E})}}{\\cos{(f_{E})} + (\\int \\sin{(f_{E})} df_{E})^{2}} = \\frac{S + (\\int \\sin{(f_{E})} df_{E})^{2}}{\\cos{(f_{E})} + (\\int \\sin{(f_{E})} df_{E})^{2}}", "derivation": "g{(f_{E})} = \\int \\sin{(f_{E})} df_{E} and g{(f_{E})} = S - \\cos{(f_{E})} and \\ddot{x}{(S,f_{E})} = (S - \\cos{(f_{E})})^{2} and \\ddot{x}{(S,f_{E})} = g^{2}{(f_{E})} and S + \\ddot{x}{(S,f_{E})} = S + g^{2}{(f_{E})} and \\frac{S + \\ddot{x}{(S,f_{E})}}{g^{2}{(f_{E})} + \\cos{(f_{E})}} = \\frac{S + g^{2}{(f_{E})}}{g^{2}{(f_{E})} + \\cos{(f_{E})}} and \\frac{S + \\ddot{x}{(S,f_{E})}}{\\cos{(f_{E})} + (\\int \\sin{(f_{E})} df_{E})^{2}} = \\frac{S + (\\int \\sin{(f_{E})} df_{E})^{2}}{\\cos{(f_{E})} + (\\int \\sin{(f_{E})} df_{E})^{2}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('f_E', commutative=True)), Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('g')(Symbol('f_E', commutative=True)), Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\ddot{x}')(Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2)))"], [["add", 4, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('\\\\ddot{x}')(Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Add(Symbol('S', commutative=True), Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2))))"], [["divide", 5, "Add(Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2)), cos(Symbol('f_E', commutative=True)))"], "Equality(Mul(Add(Symbol('S', commutative=True), Function('\\\\ddot{x}')(Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Pow(Add(Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2)), cos(Symbol('f_E', commutative=True))), Integer(-1))), Mul(Add(Symbol('S', commutative=True), Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2))), Pow(Add(Pow(Function('g')(Symbol('f_E', commutative=True)), Integer(2)), cos(Symbol('f_E', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Symbol('S', commutative=True), Function('\\\\ddot{x}')(Symbol('S', commutative=True), Symbol('f_E', commutative=True))), Pow(Add(cos(Symbol('f_E', commutative=True)), Pow(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integer(2))), Integer(-1))), Mul(Add(Symbol('S', commutative=True), Pow(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integer(2))), Pow(Add(cos(Symbol('f_E', commutative=True)), Pow(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given \\rho{(u,A_{1})} = A_{1} + u and \\operatorname{M_{E}}{(A_{1},u)} = \\frac{\\partial}{\\partial u} (u + \\rho{(u,A_{1})}), then derive \\operatorname{M_{E}}{(A_{1},u)} = 2, then obtain \\frac{\\operatorname{M_{E}}{(A_{1},u)} \\rho{(u,A_{1})}}{A_{1} \\operatorname{M_{E}}{(A_{1},u)} + u \\operatorname{M_{E}}{(A_{1},u)}} = 1", "derivation": "\\rho{(u,A_{1})} = A_{1} + u and u + \\rho{(u,A_{1})} = A_{1} + 2 u and 2 \\rho{(u,A_{1})} = A_{1} + u + \\rho{(u,A_{1})} and \\frac{2 \\rho{(u,A_{1})}}{A_{1} + u + \\rho{(u,A_{1})}} = 1 and \\frac{\\partial}{\\partial u} (u + \\rho{(u,A_{1})}) = \\frac{\\partial}{\\partial u} (A_{1} + 2 u) and \\operatorname{M_{E}}{(A_{1},u)} = \\frac{\\partial}{\\partial u} (u + \\rho{(u,A_{1})}) and \\operatorname{M_{E}}{(A_{1},u)} = \\frac{\\partial}{\\partial u} (A_{1} + 2 u) and \\operatorname{M_{E}}{(A_{1},u)} = 2 and \\frac{2 \\rho{(u,A_{1})}}{2 A_{1} + 2 u} = 1 and \\frac{\\operatorname{M_{E}}{(A_{1},u)} \\rho{(u,A_{1})}}{A_{1} \\operatorname{M_{E}}{(A_{1},u)} + u \\operatorname{M_{E}}{(A_{1},u)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))))"], [["add", 1, "Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))))"], [["divide", 3, "Add(Symbol('A_1', commutative=True), Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Symbol('A_1', commutative=True), Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Integer(-1)), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True)), Derivative(Add(Symbol('u', commutative=True), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))), Integer(-1)), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Integer(1))"], [["substitute_RHS_for_LHS", 9, 8], "Equality(Mul(Pow(Add(Mul(Symbol('A_1', commutative=True), Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True)))), Integer(-1)), Function('M_E')(Symbol('A_1', commutative=True), Symbol('u', commutative=True)), Function('\\\\rho')(Symbol('u', commutative=True), Symbol('A_1', commutative=True))), Integer(1))"]]}, {"prompt": "Given Z{(x,\\tilde{g}^*)} = - \\tilde{g}^* + \\cos{(x)}, then obtain \\tilde{g}^* + 2 Z{(x,\\tilde{g}^*)} = Z{(x,\\tilde{g}^*)} + \\cos{(x)}", "derivation": "Z{(x,\\tilde{g}^*)} = - \\tilde{g}^* + \\cos{(x)} and - x + 2 Z{(x,\\tilde{g}^*)} = - \\tilde{g}^* - x + Z{(x,\\tilde{g}^*)} + \\cos{(x)} and 2 Z{(x,\\tilde{g}^*)} = - \\tilde{g}^* + Z{(x,\\tilde{g}^*)} + \\cos{(x)} and \\tilde{g}^* + 2 Z{(x,\\tilde{g}^*)} = Z{(x,\\tilde{g}^*)} + \\cos{(x)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('x', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(2), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('x', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('x', commutative=True))))"], [["add", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(2), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Function('Z')(Symbol('x', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\varphi{(c)} = e^{c}, then obtain \\frac{d}{d c} (\\varphi{(c)} + \\int e^{c} dc - 1) e^{- c} = \\frac{d}{d c} (e^{c} + \\int e^{c} dc - 1) e^{- c}", "derivation": "\\varphi{(c)} = e^{c} and \\int \\varphi{(c)} dc = \\int e^{c} dc and \\varphi{(c)} + \\int e^{c} dc = e^{c} + \\int e^{c} dc and \\varphi{(c)} + \\int e^{c} dc - 1 = e^{c} + \\int e^{c} dc - 1 and (\\varphi{(c)} + \\int e^{c} dc - 1) e^{- c} = (e^{c} + \\int e^{c} dc - 1) e^{- c} and (\\varphi{(c)} + \\int \\varphi{(c)} dc - 1) e^{- c} = (e^{c} + \\int \\varphi{(c)} dc - 1) e^{- c} and \\frac{d}{d c} (\\varphi{(c)} + \\int \\varphi{(c)} dc - 1) e^{- c} = \\frac{d}{d c} (e^{c} + \\int \\varphi{(c)} dc - 1) e^{- c} and \\frac{d}{d c} (\\varphi{(c)} + \\int e^{c} dc - 1) e^{- c} = \\frac{d}{d c} (e^{c} + \\int e^{c} dc - 1) e^{- c}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 1, "Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(exp(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), Add(exp(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)))"], [["divide", 4, "exp(Symbol('c', commutative=True))"], "Equality(Mul(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Add(exp(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(Function('\\\\varphi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Add(exp(Symbol('c', commutative=True)), Integral(Function('\\\\varphi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["differentiate", 6, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(Function('\\\\varphi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Add(exp(Symbol('c', commutative=True)), Integral(Function('\\\\varphi')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Derivative(Mul(Add(Function('\\\\varphi')(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Add(exp(Symbol('c', commutative=True)), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(-1)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(x^\\prime,E_{x})} = \\frac{\\partial}{\\partial E_{x}} (E_{x} - x^\\prime), then derive e^{x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})}} = e^{x^\\prime}, then obtain \\int e^{x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})}} dx^\\prime = \\int e^{x^\\prime} dx^\\prime", "derivation": "\\operatorname{C_{2}}{(x^\\prime,E_{x})} = \\frac{\\partial}{\\partial E_{x}} (E_{x} - x^\\prime) and \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})} = (\\frac{\\partial}{\\partial E_{x}} (E_{x} - x^\\prime))^{E_{x}} and x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})} = x^\\prime (\\frac{\\partial}{\\partial E_{x}} (E_{x} - x^\\prime))^{E_{x}} and e^{x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})}} = e^{x^\\prime (\\frac{\\partial}{\\partial E_{x}} (E_{x} - x^\\prime))^{E_{x}}} and e^{x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})}} = e^{x^\\prime} and \\int e^{x^\\prime \\operatorname{C_{2}}^{E_{x}}{(x^\\prime,E_{x})}} dx^\\prime = \\int e^{x^\\prime} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)))"], [["times", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), Pow(Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))), exp(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(exp(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))), exp(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(exp(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Function('C_2')(Symbol('x^\\\\prime', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\eta,s)} = \\frac{e^{s}}{\\eta}, then derive \\frac{\\partial}{\\partial \\eta} \\operatorname{t_{1}}{(\\eta,s)} - \\frac{e^{s}}{\\eta^{2}} = - \\frac{2 e^{s}}{\\eta^{2}}, then obtain \\frac{\\partial}{\\partial \\eta} \\operatorname{t_{1}}{(\\eta,s)} - \\frac{\\operatorname{t_{1}}{(\\eta,s)}}{\\eta} = - \\frac{2 \\operatorname{t_{1}}{(\\eta,s)}}{\\eta}", "derivation": "\\operatorname{t_{1}}{(\\eta,s)} = \\frac{e^{s}}{\\eta} and \\operatorname{t_{1}}{(\\eta,s)} + \\frac{e^{s}}{\\eta} = \\frac{2 e^{s}}{\\eta} and \\frac{\\partial}{\\partial \\eta} (\\operatorname{t_{1}}{(\\eta,s)} + \\frac{e^{s}}{\\eta}) = \\frac{\\partial}{\\partial \\eta} \\frac{2 e^{s}}{\\eta} and \\frac{\\partial}{\\partial \\eta} \\operatorname{t_{1}}{(\\eta,s)} - \\frac{e^{s}}{\\eta^{2}} = - \\frac{2 e^{s}}{\\eta^{2}} and \\frac{\\partial}{\\partial \\eta} \\operatorname{t_{1}}{(\\eta,s)} - \\frac{\\operatorname{t_{1}}{(\\eta,s)}}{\\eta} = - \\frac{2 \\operatorname{t_{1}}{(\\eta,s)}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))"], "Equality(Add(Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-2)), exp(Symbol('s', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-2)), exp(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\eta', commutative=True), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\eta{(n)} = \\sin{(n)}, then derive \\frac{d}{d n} \\eta{(n)} = \\cos{(n)}, then obtain \\sin{(n)} + \\frac{d}{d n} \\eta{(n)} = \\sin{(n)} + \\frac{d}{d n} \\sin{(n)}", "derivation": "\\eta{(n)} = \\sin{(n)} and \\frac{d}{d n} \\eta{(n)} = \\frac{d}{d n} \\sin{(n)} and \\frac{d}{d n} \\eta{(n)} = \\cos{(n)} and \\frac{d}{d n} \\sin{(n)} = \\cos{(n)} and \\sin{(n)} + \\frac{d}{d n} \\eta{(n)} = \\sin{(n)} + \\cos{(n)} and \\sin{(n)} + \\frac{d}{d n} \\eta{(n)} = \\sin{(n)} + \\frac{d}{d n} \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), cos(Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), cos(Symbol('n', commutative=True)))"], [["add", 3, "sin(Symbol('n', commutative=True))"], "Equality(Add(sin(Symbol('n', commutative=True)), Derivative(Function('\\\\eta')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(sin(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(sin(Symbol('n', commutative=True)), Derivative(Function('\\\\eta')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(sin(Symbol('n', commutative=True)), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu_{0}{(x^\\prime,\\mathbf{S})} = - \\mathbf{S} + \\sin{(x^\\prime)} and c{(x^\\prime)} = 3 \\sin{(x^\\prime)}, then obtain x^\\prime (- 2 \\mathbf{S} - x^\\prime + 3 \\sin{(x^\\prime)}) = x^\\prime (- 2 \\mathbf{S} - x^\\prime + c{(x^\\prime)})", "derivation": "\\mu_{0}{(x^\\prime,\\mathbf{S})} = - \\mathbf{S} + \\sin{(x^\\prime)} and - \\mathbf{S} + \\mu_{0}{(x^\\prime,\\mathbf{S})} + \\sin{(x^\\prime)} = - 2 \\mathbf{S} + 2 \\sin{(x^\\prime)} and - \\mathbf{S} + \\mu_{0}{(x^\\prime,\\mathbf{S})} + 2 \\sin{(x^\\prime)} = - 2 \\mathbf{S} + 3 \\sin{(x^\\prime)} and c{(x^\\prime)} = 3 \\sin{(x^\\prime)} and - \\mathbf{S} + \\mu_{0}{(x^\\prime,\\mathbf{S})} + 2 \\sin{(x^\\prime)} = - 2 \\mathbf{S} + c{(x^\\prime)} and - \\mathbf{S} - x^\\prime + \\mu_{0}{(x^\\prime,\\mathbf{S})} + 2 \\sin{(x^\\prime)} = - 2 \\mathbf{S} - x^\\prime + c{(x^\\prime)} and - 2 \\mathbf{S} - x^\\prime + 3 \\sin{(x^\\prime)} = - 2 \\mathbf{S} - x^\\prime + c{(x^\\prime)} and x^\\prime (- 2 \\mathbf{S} - x^\\prime + 3 \\sin{(x^\\prime)}) = x^\\prime (- 2 \\mathbf{S} - x^\\prime + c{(x^\\prime)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), sin(Symbol('x^\\\\prime', commutative=True)))))"], [["add", 2, "sin(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), sin(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(3), sin(Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(3), sin(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), sin(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Function('c')(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), sin(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('c')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(3), sin(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('c')(Symbol('x^\\\\prime', commutative=True))))"], [["times", 7, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(3), sin(Symbol('x^\\\\prime', commutative=True))))), Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('c')(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given S{(c)} = e^{c}, then derive \\int S{(c)} dc = M + e^{c}, then obtain \\eta^{\\prime} + e^{c} = M + S{(c)}", "derivation": "S{(c)} = e^{c} and \\int S{(c)} dc = \\int e^{c} dc and \\int S{(c)} dc = M + e^{c} and \\int S{(c)} dc = M + S{(c)} and \\int e^{c} dc = M + S{(c)} and \\eta^{\\prime} + e^{c} = M + S{(c)}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('M', commutative=True), exp(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('M', commutative=True), Function('S')(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('M', commutative=True), Function('S')(Symbol('c', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('c', commutative=True))), Add(Symbol('M', commutative=True), Function('S')(Symbol('c', commutative=True))))"]]}, {"prompt": "Given U{(F_{H})} = \\cos{(e^{F_{H}})}, then derive \\int U{(F_{H})} dF_{H} = F_{N} + \\operatorname{Ci}{(e^{F_{H}})}, then obtain \\frac{\\partial^{2}}{\\partial F_{H}^{2}} (F_{N} + \\operatorname{Ci}{(e^{F_{H}})}) = \\frac{d^{2}}{d F_{H}^{2}} \\int \\cos{(e^{F_{H}})} dF_{H}", "derivation": "U{(F_{H})} = \\cos{(e^{F_{H}})} and \\int U{(F_{H})} dF_{H} = \\int \\cos{(e^{F_{H}})} dF_{H} and \\frac{d}{d F_{H}} \\int U{(F_{H})} dF_{H} = \\frac{d}{d F_{H}} \\int \\cos{(e^{F_{H}})} dF_{H} and \\int U{(F_{H})} dF_{H} = F_{N} + \\operatorname{Ci}{(e^{F_{H}})} and \\frac{d^{2}}{d F_{H}^{2}} \\int U{(F_{H})} dF_{H} = \\frac{d^{2}}{d F_{H}^{2}} \\int \\cos{(e^{F_{H}})} dF_{H} and \\frac{\\partial^{2}}{\\partial F_{H}^{2}} (F_{N} + \\operatorname{Ci}{(e^{F_{H}})}) = \\frac{d^{2}}{d F_{H}^{2}} \\int \\cos{(e^{F_{H}})} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('F_H', commutative=True)), cos(exp(Symbol('F_H', commutative=True))))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('U')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(cos(exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integral(Function('U')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integral(cos(exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('U')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Add(Symbol('F_N', commutative=True), Ci(exp(Symbol('F_H', commutative=True)))))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integral(Function('U')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(2))), Derivative(Integral(cos(exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Ci(exp(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(2))), Derivative(Integral(cos(exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\eta^{\\prime}{(I)} = \\log{(I)}, then obtain \\frac{d}{d I} \\eta^{\\prime}{(I)} \\log{(I)} + \\int \\log{(I)} dI = \\frac{d}{d I} \\log{(I)}^{2} + \\int \\log{(I)} dI", "derivation": "\\eta^{\\prime}{(I)} = \\log{(I)} and \\eta^{\\prime}{(I)} \\log{(I)} = \\log{(I)}^{2} and \\frac{d}{d I} \\eta^{\\prime}{(I)} \\log{(I)} = \\frac{d}{d I} \\log{(I)}^{2} and \\int \\eta^{\\prime}{(I)} dI = \\int \\log{(I)} dI and \\frac{d}{d I} \\eta^{\\prime}{(I)} \\log{(I)} + \\int \\eta^{\\prime}{(I)} dI = \\frac{d}{d I} \\log{(I)}^{2} + \\int \\eta^{\\prime}{(I)} dI and \\frac{d}{d I} \\eta^{\\prime}{(I)} \\log{(I)} + \\int \\log{(I)} dI = \\frac{d}{d I} \\log{(I)}^{2} + \\int \\log{(I)} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["times", 1, "log(Symbol('I', commutative=True))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), Pow(log(Symbol('I', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('I', commutative=True)), Integer(2)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 3, "Integral(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(Derivative(Pow(log(Symbol('I', commutative=True)), Integer(2)), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(Derivative(Pow(log(Symbol('I', commutative=True)), Integer(2)), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given b{(Z)} = \\cos{(\\log{(Z)})} and \\mathbf{r}{(W,\\mathbf{F})} = \\frac{e^{\\mathbf{F}}}{W}, then obtain \\mathbf{r}{(W,\\mathbf{F})} - b^{Z}{(Z)} \\log{(Z)} \\cos^{Z}{(\\log{(Z)})} = - b^{Z}{(Z)} \\log{(Z)} \\cos^{Z}{(\\log{(Z)})} + \\frac{e^{\\mathbf{F}}}{W}", "derivation": "b{(Z)} = \\cos{(\\log{(Z)})} and b^{Z}{(Z)} = \\cos^{Z}{(\\log{(Z)})} and b^{2 Z}{(Z)} = b^{Z}{(Z)} \\cos^{Z}{(\\log{(Z)})} and \\mathbf{r}{(W,\\mathbf{F})} = \\frac{e^{\\mathbf{F}}}{W} and \\mathbf{r}{(W,\\mathbf{F})} - b^{2 Z}{(Z)} \\log{(Z)} = - b^{2 Z}{(Z)} \\log{(Z)} + \\frac{e^{\\mathbf{F}}}{W} and \\mathbf{r}{(W,\\mathbf{F})} - b^{Z}{(Z)} \\log{(Z)} \\cos^{Z}{(\\log{(Z)})} = - b^{Z}{(Z)} \\log{(Z)} \\cos^{Z}{(\\log{(Z)})} + \\frac{e^{\\mathbf{F}}}{W}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(cos(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["times", 2, "Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Pow(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Mul(Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(cos(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Pow(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)), Pow(cos(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)), Pow(cos(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(H)} = e^{H} and \\operatorname{M_{E}}{(H)} = \\int \\operatorname{A_{z}}{(H)} dH, then derive \\frac{d}{d H} \\operatorname{A_{z}}{(H)} = e^{H}, then obtain \\frac{d}{d H} \\operatorname{A_{z}}{(H)} - \\int \\operatorname{A_{z}}{(H)} dH = e^{H} - \\int \\operatorname{A_{z}}{(H)} dH", "derivation": "\\operatorname{A_{z}}{(H)} = e^{H} and \\operatorname{M_{E}}{(H)} = \\int \\operatorname{A_{z}}{(H)} dH and \\frac{d}{d H} \\operatorname{A_{z}}{(H)} = \\frac{d}{d H} e^{H} and \\frac{d}{d H} \\operatorname{A_{z}}{(H)} = e^{H} and - \\operatorname{M_{E}}{(H)} + \\frac{d}{d H} \\operatorname{A_{z}}{(H)} = - \\operatorname{M_{E}}{(H)} + e^{H} and \\frac{d}{d H} \\operatorname{A_{z}}{(H)} - \\int \\operatorname{A_{z}}{(H)} dH = e^{H} - \\int \\operatorname{A_{z}}{(H)} dH", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('H', commutative=True)), Integral(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), exp(Symbol('H', commutative=True)))"], [["minus", 4, "Function('M_E')(Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('H', commutative=True))), Derivative(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('M_E')(Symbol('H', commutative=True))), exp(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Derivative(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(exp(Symbol('H', commutative=True)), Mul(Integer(-1), Integral(Function('A_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\psi,\\psi^*)} = \\frac{\\psi}{\\psi^*} and \\mu{(\\psi^*)} = \\psi^*, then obtain - \\frac{\\psi}{\\psi^*} + \\int \\Psi_{nl}{(\\psi,\\psi^*)} d\\mu{(\\psi^*)} = - \\frac{\\psi}{\\psi^*} + \\int \\frac{\\psi}{\\psi^*} d\\mu{(\\psi^*)}", "derivation": "\\Psi_{nl}{(\\psi,\\psi^*)} = \\frac{\\psi}{\\psi^*} and \\int \\Psi_{nl}{(\\psi,\\psi^*)} d\\psi^* = \\int \\frac{\\psi}{\\psi^*} d\\psi^* and \\mu{(\\psi^*)} = \\psi^* and \\int \\Psi_{nl}{(\\psi,\\psi^*)} d\\mu{(\\psi^*)} = \\int \\frac{\\psi}{\\psi^*} d\\mu{(\\psi^*)} and - \\frac{\\psi}{\\psi^*} + \\int \\Psi_{nl}{(\\psi,\\psi^*)} d\\mu{(\\psi^*)} = - \\frac{\\psi}{\\psi^*} + \\int \\frac{\\psi}{\\psi^*} d\\mu{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Integral(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(\\hat{H}_l)} = \\cos{(\\log{(\\hat{H}_l)})}, then derive \\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l = C + \\frac{\\hat{H}_l \\sin{(\\log{(\\hat{H}_l)})}}{2} + \\frac{\\hat{H}_l \\cos{(\\log{(\\hat{H}_l)})}}{2}, then obtain \\log{(\\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l)} = \\log{(C + \\frac{\\hat{H}_l \\hat{x}{(\\hat{H}_l)}}{2} + \\frac{\\hat{H}_l \\sin{(\\log{(\\hat{H}_l)})}}{2})}", "derivation": "\\hat{x}{(\\hat{H}_l)} = \\cos{(\\log{(\\hat{H}_l)})} and \\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l = \\int \\cos{(\\log{(\\hat{H}_l)})} d\\hat{H}_l and \\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l = C + \\frac{\\hat{H}_l \\sin{(\\log{(\\hat{H}_l)})}}{2} + \\frac{\\hat{H}_l \\cos{(\\log{(\\hat{H}_l)})}}{2} and \\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l = C + \\frac{\\hat{H}_l \\hat{x}{(\\hat{H}_l)}}{2} + \\frac{\\hat{H}_l \\sin{(\\log{(\\hat{H}_l)})}}{2} and \\log{(\\int \\hat{x}{(\\hat{H}_l)} d\\hat{H}_l)} = \\log{(C + \\frac{\\hat{H}_l \\hat{x}{(\\hat{H}_l)}}{2} + \\frac{\\hat{H}_l \\sin{(\\log{(\\hat{H}_l)})}}{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(log(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(cos(log(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), sin(log(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), cos(log(Symbol('\\\\hat{H}_l', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), sin(log(Symbol('\\\\hat{H}_l', commutative=True))))))"], [["log", 4], "Equality(log(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), log(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Rational(1, 2), Symbol('\\\\hat{H}_l', commutative=True), sin(log(Symbol('\\\\hat{H}_l', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{E}{(M,\\phi)} = \\sin{(\\frac{\\phi}{M})}, then obtain - \\phi + \\mathbf{E}{(M,\\phi)} + \\mathbf{E}^{\\phi}{(M,\\phi)} = - \\phi + \\mathbf{E}{(M,\\phi)} + \\sin^{\\phi}{(\\frac{\\phi}{M})}", "derivation": "\\mathbf{E}{(M,\\phi)} = \\sin{(\\frac{\\phi}{M})} and \\mathbf{E}^{\\phi}{(M,\\phi)} = \\sin^{\\phi}{(\\frac{\\phi}{M})} and - \\phi + \\mathbf{E}^{\\phi}{(M,\\phi)} = - \\phi + \\sin^{\\phi}{(\\frac{\\phi}{M})} and - \\phi + \\mathbf{E}{(M,\\phi)} + \\mathbf{E}^{\\phi}{(M,\\phi)} = - \\phi + \\mathbf{E}{(M,\\phi)} + \\sin^{\\phi}{(\\frac{\\phi}{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["minus", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))))"], [["add", 3, "Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('M', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\mu_0,\\dot{z},\\mathbf{f})} = (\\dot{z} + \\mu_0)^{\\mathbf{f}}, then obtain - \\dot{z} (\\dot{z} - (\\dot{z} + \\mu_0)^{\\mathbf{f}} + \\phi{(\\mu_0,\\dot{z},\\mathbf{f})}) = - \\dot{z}^{2}", "derivation": "\\phi{(\\mu_0,\\dot{z},\\mathbf{f})} = (\\dot{z} + \\mu_0)^{\\mathbf{f}} and \\dot{z} + \\phi{(\\mu_0,\\dot{z},\\mathbf{f})} = \\dot{z} + (\\dot{z} + \\mu_0)^{\\mathbf{f}} and \\dot{z} - (\\dot{z} + \\mu_0)^{\\mathbf{f}} + \\phi{(\\mu_0,\\dot{z},\\mathbf{f})} = \\dot{z} and - \\dot{z} (\\dot{z} - (\\dot{z} + \\mu_0)^{\\mathbf{f}} + \\phi{(\\mu_0,\\dot{z},\\mathbf{f})}) = - \\dot{z}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 2, "Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\dot{z}', commutative=True))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{A})} = e^{\\mathbf{A}}, then derive \\frac{d^{2}}{d \\mathbf{A}^{2}} \\phi_{1}{(\\mathbf{A})} = e^{\\mathbf{A}}, then obtain \\frac{d^{2}}{d \\mathbf{A}^{2}} e^{\\mathbf{A}} = e^{\\mathbf{A}}", "derivation": "\\phi_{1}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\phi_{1}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and \\frac{d^{2}}{d \\mathbf{A}^{2}} \\phi_{1}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} e^{\\mathbf{A}} and \\frac{d^{2}}{d \\mathbf{A}^{2}} \\phi_{1}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d^{2}}{d \\mathbf{A}^{2}} e^{\\mathbf{A}} = e^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), exp(Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given U{(L)} = \\sin{(L)} and \\operatorname{P_{e}}{(L)} = - L + U{(L)}, then obtain \\int \\frac{- L + U{(L)}}{- L + \\sin{(L)}} dL = \\int 1 dL", "derivation": "U{(L)} = \\sin{(L)} and - L + U{(L)} = - L + \\sin{(L)} and \\frac{- L + U{(L)}}{- L + \\sin{(L)}} = 1 and \\operatorname{P_{e}}{(L)} = - L + U{(L)} and \\frac{\\operatorname{P_{e}}{(L)}}{- L + \\sin{(L)}} = 1 and \\int \\frac{\\operatorname{P_{e}}{(L)}}{- L + \\sin{(L)}} dL = \\int 1 dL and \\int \\frac{- L + U{(L)}}{- L + \\sin{(L)}} dL = \\int 1 dL", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('U')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('U')(Symbol('L', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('U')(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Integer(-1)), Function('P_e')(Symbol('L', commutative=True))), Integer(1))"], [["integrate", 5, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Integer(-1)), Function('P_e')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('U')(Symbol('L', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Integer(-1))), Tuple(Symbol('L', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(C_{1})} = \\cos{(C_{1})} and L{(C_{1})} = - \\cos{(C_{1})}, then obtain (- L{(C_{1})})^{C_{1}} = \\Psi_{\\lambda}^{C_{1}}{(C_{1})}", "derivation": "\\Psi_{\\lambda}{(C_{1})} = \\cos{(C_{1})} and L{(C_{1})} = - \\cos{(C_{1})} and - L{(C_{1})} = \\cos{(C_{1})} and - L{(C_{1})} = \\Psi_{\\lambda}{(C_{1})} and (- L{(C_{1})})^{C_{1}} = \\Psi_{\\lambda}^{C_{1}}{(C_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('C_1', commutative=True)), Mul(Integer(-1), cos(Symbol('C_1', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('L')(Symbol('C_1', commutative=True))), cos(Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(-1), Function('L')(Symbol('C_1', commutative=True))), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)))"], [["power", 4, "Symbol('C_1', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('L')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given t{(\\varepsilon_0,M)} = \\varepsilon_0 \\sin{(M)}, then obtain \\int\\limits^{\\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}} \\frac{M + \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}}{M} d\\varepsilon_0 = \\int\\limits^{\\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}} \\frac{M + \\varepsilon_0}{M} d\\varepsilon_0", "derivation": "t{(\\varepsilon_0,M)} = \\varepsilon_0 \\sin{(M)} and \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}} = \\varepsilon_0 and M + \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}} = M + \\varepsilon_0 and \\frac{M + \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}}{M} = \\frac{M + \\varepsilon_0}{M} and \\int \\frac{M + \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}}{M} d\\varepsilon_0 = \\int \\frac{M + \\varepsilon_0}{M} d\\varepsilon_0 and \\int\\limits^{\\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}} \\frac{M + \\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}}{M} d\\varepsilon_0 = \\int\\limits^{\\frac{t{(\\varepsilon_0,M)}}{\\sin{(M)}}} \\frac{M + \\varepsilon_0}{M} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), sin(Symbol('M', commutative=True))))"], [["divide", 1, "sin(Symbol('M', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))), Symbol('\\\\varepsilon_0', commutative=True))"], [["add", 2, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1)))), Add(Symbol('M', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 3, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Mul(Function('t')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\dot{y}{(t_{2},z,\\tilde{g}^*)} = t_{2} (\\tilde{g}^* - z), then obtain 0 = t_{2} + \\frac{\\partial}{\\partial z} \\dot{y}{(t_{2},z,\\tilde{g}^*)}", "derivation": "\\dot{y}{(t_{2},z,\\tilde{g}^*)} = t_{2} (\\tilde{g}^* - z) and 0 = t_{2} (\\tilde{g}^* - z) - \\dot{y}{(t_{2},z,\\tilde{g}^*)} and 0 = - t_{2} (\\tilde{g}^* - z) + \\dot{y}{(t_{2},z,\\tilde{g}^*)} and \\frac{d}{d z} 0 = \\frac{\\partial}{\\partial z} (- t_{2} (\\tilde{g}^* - z) + \\dot{y}{(t_{2},z,\\tilde{g}^*)}) and 0 = t_{2} + \\frac{\\partial}{\\partial z} \\dot{y}{(t_{2},z,\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('t_2', commutative=True), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["minus", 1, "Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('t_2', commutative=True), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('t_2', commutative=True), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True), Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Symbol('t_2', commutative=True), Derivative(Function('\\\\dot{y}')(Symbol('t_2', commutative=True), Symbol('z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(\\mathbf{M})} = e^{e^{\\mathbf{M}}} and \\rho_{b}{(\\mathbf{M})} = e^{x{(\\mathbf{M})}}, then obtain \\frac{d}{d \\mathbf{M}} e^{e^{e^{\\mathbf{M}}}} = \\frac{d}{d \\mathbf{M}} e^{x{(\\mathbf{M})}}", "derivation": "x{(\\mathbf{M})} = e^{e^{\\mathbf{M}}} and e^{x{(\\mathbf{M})}} = e^{e^{e^{\\mathbf{M}}}} and \\rho_{b}{(\\mathbf{M})} = e^{x{(\\mathbf{M})}} and \\rho_{b}{(\\mathbf{M})} = e^{e^{e^{\\mathbf{M}}}} and \\frac{d}{d \\mathbf{M}} \\rho_{b}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} e^{x{(\\mathbf{M})}} and \\frac{d}{d \\mathbf{M}} e^{e^{e^{\\mathbf{M}}}} = \\frac{d}{d \\mathbf{M}} e^{x{(\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{M}', commutative=True)), exp(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('x')(Symbol('\\\\mathbf{M}', commutative=True))), exp(exp(exp(Symbol('\\\\mathbf{M}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Function('x')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True)), exp(exp(exp(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Function('x')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(exp(exp(exp(Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Function('x')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\phi_2,\\tilde{g}^*)} = \\cos{(\\phi_2 + \\tilde{g}^*)}, then obtain 2 = \\frac{\\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)} + \\mathbf{H}{(\\phi_2,\\tilde{g}^*)} \\cos{(\\phi_2 + \\tilde{g}^*)}}{\\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)}}", "derivation": "\\mathbf{H}{(\\phi_2,\\tilde{g}^*)} = \\cos{(\\phi_2 + \\tilde{g}^*)} and \\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)} = \\mathbf{H}{(\\phi_2,\\tilde{g}^*)} \\cos{(\\phi_2 + \\tilde{g}^*)} and 2 \\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)} = \\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)} + \\mathbf{H}{(\\phi_2,\\tilde{g}^*)} \\cos{(\\phi_2 + \\tilde{g}^*)} and 2 = \\frac{\\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)} + \\mathbf{H}{(\\phi_2,\\tilde{g}^*)} \\cos{(\\phi_2 + \\tilde{g}^*)}}{\\mathbf{H}^{2}{(\\phi_2,\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["add", 2, "Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))), Add(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))))"], [["divide", 3, "Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))"], "Equality(Integer(2), Mul(Add(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{D}{(\\phi,\\Psi)} = \\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi), then derive \\mathbf{D}{(\\phi,\\Psi)} = -1, then obtain \\frac{\\cos{(\\frac{\\partial^{2}}{\\partial \\Psi\\partial \\phi} (\\Psi - \\phi))}}{\\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi)} = \\frac{\\cos{(\\frac{d}{d \\Psi} (-1))}}{\\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi)}", "derivation": "\\mathbf{D}{(\\phi,\\Psi)} = \\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi) and \\mathbf{D}{(\\phi,\\Psi)} = -1 and \\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi) = -1 and \\frac{\\partial^{2}}{\\partial \\Psi\\partial \\phi} (\\Psi - \\phi) = \\frac{d}{d \\Psi} (-1) and \\cos{(\\frac{\\partial^{2}}{\\partial \\Psi\\partial \\phi} (\\Psi - \\phi))} = \\cos{(\\frac{d}{d \\Psi} (-1))} and \\frac{\\cos{(\\frac{\\partial^{2}}{\\partial \\Psi\\partial \\phi} (\\Psi - \\phi))}}{\\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi)} = \\frac{\\cos{(\\frac{d}{d \\Psi} (-1))}}{\\frac{\\partial}{\\partial \\phi} (\\Psi - \\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi', commutative=True)), Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), cos(Derivative(Integer(-1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["divide", 5, "Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Mul(cos(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))), Mul(cos(Derivative(Integer(-1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given V{(M,A_{z})} = \\frac{\\partial}{\\partial M} (A_{z} - M), then derive \\frac{V{(M,A_{z})}}{A_{z} - M} = - \\frac{1}{A_{z} - M}, then obtain \\frac{\\partial}{\\partial A_{z}} (M + \\frac{V{(M,A_{z})}}{A_{z} - M}) = \\frac{\\partial}{\\partial A_{z}} (M - \\frac{1}{A_{z} - M})", "derivation": "V{(M,A_{z})} = \\frac{\\partial}{\\partial M} (A_{z} - M) and \\frac{V{(M,A_{z})}}{A_{z} - M} = \\frac{\\frac{\\partial}{\\partial M} (A_{z} - M)}{A_{z} - M} and \\frac{V{(M,A_{z})}}{A_{z} - M} = - \\frac{1}{A_{z} - M} and \\frac{\\frac{\\partial}{\\partial M} (A_{z} - M)}{A_{z} - M} = - \\frac{1}{A_{z} - M} and M + \\frac{\\frac{\\partial}{\\partial M} (A_{z} - M)}{A_{z} - M} = M - \\frac{1}{A_{z} - M} and M + \\frac{V{(M,A_{z})}}{A_{z} - M} = M - \\frac{1}{A_{z} - M} and \\frac{\\partial}{\\partial A_{z}} (M + \\frac{V{(M,A_{z})}}{A_{z} - M}) = \\frac{\\partial}{\\partial A_{z}} (M - \\frac{1}{A_{z} - M})", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)), Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 1, "Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Function('V')(Symbol('M', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Function('V')(Symbol('M', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1))))"], [["add", 4, "Symbol('M', commutative=True)"], "Equality(Add(Symbol('M', commutative=True), Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('M', commutative=True), Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Function('V')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)))))"], [["differentiate", 6, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('M', commutative=True), Mul(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)), Function('V')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))), Integer(-1)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(t_{2},x)} = t_{2}^{x}, then obtain (t_{2}^{x})^{x} + 2 \\mathbf{D}{(t_{2},x)} = t_{2}^{x} + (t_{2}^{x})^{x} + \\mathbf{D}{(t_{2},x)}", "derivation": "\\mathbf{D}{(t_{2},x)} = t_{2}^{x} and \\mathbf{D}^{x}{(t_{2},x)} = (t_{2}^{x})^{x} and 2 \\mathbf{D}{(t_{2},x)} = t_{2}^{x} + \\mathbf{D}{(t_{2},x)} and 2 \\mathbf{D}{(t_{2},x)} + \\mathbf{D}^{x}{(t_{2},x)} = t_{2}^{x} + \\mathbf{D}{(t_{2},x)} + \\mathbf{D}^{x}{(t_{2},x)} and (t_{2}^{x})^{x} + 2 \\mathbf{D}{(t_{2},x)} = t_{2}^{x} + (t_{2}^{x})^{x} + \\mathbf{D}{(t_{2},x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True))), Add(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True))))"], [["add", 3, "Pow(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Add(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True)))), Add(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Pow(Pow(Symbol('t_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Function('\\\\mathbf{D}')(Symbol('t_2', commutative=True), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\Psi)} = \\log{(\\Psi)}, then obtain 0 = - \\frac{\\Psi + \\operatorname{g_{\\varepsilon}}{(\\Psi)}}{\\Psi + \\log{(\\Psi)}} + 1", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\Psi)} = \\log{(\\Psi)} and \\Psi + \\operatorname{g_{\\varepsilon}}{(\\Psi)} = \\Psi + \\log{(\\Psi)} and \\frac{\\Psi + \\operatorname{g_{\\varepsilon}}{(\\Psi)}}{\\Psi + \\log{(\\Psi)}} = 1 and 0 = - \\frac{\\Psi + \\operatorname{g_{\\varepsilon}}{(\\Psi)}}{\\Psi + \\log{(\\Psi)}} + 1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["add", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True))), Pow(Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Mul(Add(Symbol('\\\\Psi', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True))), Pow(Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Symbol('\\\\Psi', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\Psi', commutative=True))), Pow(Add(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{J}_M{(k,v_{z})} = k v_{z}, then derive 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} = v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})}, then obtain 0 = \\int (v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})}) dv_{z} - \\int 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} dv_{z}", "derivation": "\\mathbf{J}_M{(k,v_{z})} = k v_{z} and \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} = \\frac{\\partial}{\\partial k} k v_{z} and 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} = \\frac{\\partial}{\\partial k} k v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} and 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} = v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} and \\int 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} dv_{z} = \\int (v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})}) dv_{z} and 0 = \\int (v_{z} + \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})}) dv_{z} - \\int 2 \\frac{\\partial}{\\partial k} \\mathbf{J}_M{(k,v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('k', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Symbol('v_z', commutative=True), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('v_z', commutative=True)"], "Equality(Integral(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('v_z', commutative=True), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))))"], [["minus", 5, "Integral(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Symbol('v_z', commutative=True), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(f)} = \\cos{(\\sin{(f)})}, then derive f \\frac{d}{d f} \\varepsilon{(f)} + \\varepsilon{(f)} = - f \\sin{(\\sin{(f)})} \\cos{(f)} + \\cos{(\\sin{(f)})}, then obtain f \\frac{d}{d f} \\cos{(\\sin{(f)})} + \\cos{(\\sin{(f)})} = - f \\sin{(\\sin{(f)})} \\cos{(f)} + \\cos{(\\sin{(f)})}", "derivation": "\\varepsilon{(f)} = \\cos{(\\sin{(f)})} and f \\varepsilon{(f)} = f \\cos{(\\sin{(f)})} and \\frac{d}{d f} f \\varepsilon{(f)} = \\frac{d}{d f} f \\cos{(\\sin{(f)})} and f \\frac{d}{d f} \\varepsilon{(f)} + \\varepsilon{(f)} = - f \\sin{(\\sin{(f)})} \\cos{(f)} + \\cos{(\\sin{(f)})} and f \\frac{d}{d f} \\cos{(\\sin{(f)})} + \\cos{(\\sin{(f)})} = - f \\sin{(\\sin{(f)})} \\cos{(f)} + \\cos{(\\sin{(f)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('f', commutative=True)), cos(sin(Symbol('f', commutative=True))))"], [["times", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Function('\\\\varepsilon')(Symbol('f', commutative=True))), Mul(Symbol('f', commutative=True), cos(sin(Symbol('f', commutative=True)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Symbol('f', commutative=True), Function('\\\\varepsilon')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Symbol('f', commutative=True), cos(sin(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('f', commutative=True), Derivative(Function('\\\\varepsilon')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Function('\\\\varepsilon')(Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True), sin(sin(Symbol('f', commutative=True))), cos(Symbol('f', commutative=True))), cos(sin(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('f', commutative=True), Derivative(cos(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), cos(sin(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('f', commutative=True), sin(sin(Symbol('f', commutative=True))), cos(Symbol('f', commutative=True))), cos(sin(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\eta)} = e^{e^{\\eta}}, then obtain (\\frac{e^{e^{\\eta}}}{\\eta})^{\\eta} + \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta} = (\\frac{e^{e^{\\eta}}}{\\eta})^{\\eta} + \\frac{e^{e^{\\eta}}}{\\eta}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\eta)} = e^{e^{\\eta}} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta} = \\frac{e^{e^{\\eta}}}{\\eta} and (\\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta})^{\\eta} = (\\frac{e^{e^{\\eta}}}{\\eta})^{\\eta} and (\\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta})^{\\eta} + \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta} = (\\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta})^{\\eta} + \\frac{e^{e^{\\eta}}}{\\eta} and (\\frac{e^{e^{\\eta}}}{\\eta})^{\\eta} + \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\eta)}}{\\eta} = (\\frac{e^{e^{\\eta}}}{\\eta})^{\\eta} + \\frac{e^{e^{\\eta}}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), exp(exp(Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True)))))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)))"], [["add", 2, "Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)))), Add(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)))), Add(Pow(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(exp(Symbol('\\\\eta', commutative=True))))))"]]}, {"prompt": "Given T{(\\Psi_{\\lambda},u)} = \\Psi_{\\lambda} - u, then obtain \\int \\frac{\\partial}{\\partial u} T^{3}{(\\Psi_{\\lambda},u)} du = \\int \\frac{\\partial}{\\partial u} (\\Psi_{\\lambda} - u) T^{2}{(\\Psi_{\\lambda},u)} du", "derivation": "T{(\\Psi_{\\lambda},u)} = \\Psi_{\\lambda} - u and T^{2}{(\\Psi_{\\lambda},u)} = (\\Psi_{\\lambda} - u) T{(\\Psi_{\\lambda},u)} and (\\Psi_{\\lambda} - u) T^{2}{(\\Psi_{\\lambda},u)} = (\\Psi_{\\lambda} - u)^{2} T{(\\Psi_{\\lambda},u)} and T^{3}{(\\Psi_{\\lambda},u)} = (\\Psi_{\\lambda} - u) T^{2}{(\\Psi_{\\lambda},u)} and \\frac{\\partial}{\\partial u} T^{3}{(\\Psi_{\\lambda},u)} = \\frac{\\partial}{\\partial u} (\\Psi_{\\lambda} - u) T^{2}{(\\Psi_{\\lambda},u)} and \\int \\frac{\\partial}{\\partial u} T^{3}{(\\Psi_{\\lambda},u)} du = \\int \\frac{\\partial}{\\partial u} (\\Psi_{\\lambda} - u) T^{2}{(\\Psi_{\\lambda},u)} du", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["times", 1, "Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True))"], "Equality(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Integer(2)), Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(3)), Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(2))))"], [["differentiate", 4, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(3)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(3)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\lambda,I)} = \\frac{I}{\\lambda} + \\lambda, then obtain \\int (- \\frac{I}{\\lambda} - \\lambda) dI + \\int - \\operatorname{F_{g}}{(\\lambda,I)} dI = 2 \\int (- \\frac{I}{\\lambda} - \\lambda) dI", "derivation": "\\operatorname{F_{g}}{(\\lambda,I)} = \\frac{I}{\\lambda} + \\lambda and - \\operatorname{F_{g}}{(\\lambda,I)} = - \\frac{I}{\\lambda} - \\lambda and \\int - \\operatorname{F_{g}}{(\\lambda,I)} dI = \\int (- \\frac{I}{\\lambda} - \\lambda) dI and \\int (- \\frac{I}{\\lambda} - \\lambda) dI + \\int - \\operatorname{F_{g}}{(\\lambda,I)} dI = 2 \\int (- \\frac{I}{\\lambda} - \\lambda) dI", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\lambda', commutative=True), Symbol('I', commutative=True)), Add(Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_g')(Symbol('\\\\lambda', commutative=True), Symbol('I', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('F_g')(Symbol('\\\\lambda', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), Function('F_g')(Symbol('\\\\lambda', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Mul(Integer(2), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\eta{(y,m_{s})} = m_{s} + y, then obtain \\eta{(y,m_{s})} \\int (\\eta{(y,m_{s})} + 1) dm_{s} = (\\frac{m_{s}^{2}}{2} + m_{s} (y + 1) + t_{1}) \\eta{(y,m_{s})}", "derivation": "\\eta{(y,m_{s})} = m_{s} + y and \\eta{(y,m_{s})} + 1 = m_{s} + y + 1 and \\int (\\eta{(y,m_{s})} + 1) dm_{s} = \\int (m_{s} + y + 1) dm_{s} and (m_{s} + y) \\int (\\eta{(y,m_{s})} + 1) dm_{s} = (m_{s} + y) \\int (m_{s} + y + 1) dm_{s} and \\eta{(y,m_{s})} \\int (\\eta{(y,m_{s})} + 1) dm_{s} = \\eta{(y,m_{s})} \\int (m_{s} + y + 1) dm_{s} and \\eta{(y,m_{s})} \\int (\\eta{(y,m_{s})} + 1) dm_{s} = (\\frac{m_{s}^{2}}{2} + m_{s} (y + 1) + t_{1}) \\eta{(y,m_{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True))))"], [["divide", 3, "Pow(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True)), Integral(Add(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Tuple(Symbol('m_s', commutative=True)))), Mul(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True)), Integral(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integral(Add(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Tuple(Symbol('m_s', commutative=True)))), Mul(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integral(Add(Symbol('m_s', commutative=True), Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integral(Add(Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True)), Integer(1)), Tuple(Symbol('m_s', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('m_s', commutative=True), Integer(2))), Mul(Symbol('m_s', commutative=True), Add(Symbol('y', commutative=True), Integer(1))), Symbol('t_1', commutative=True)), Function('\\\\eta')(Symbol('y', commutative=True), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\hat{x},A_{1})} = A_{1}^{\\hat{x}}, then obtain - A_{1}^{\\hat{x}} \\chi{(\\hat{x},A_{1})} + \\chi{(\\hat{x},A_{1})} = - A_{1}^{\\hat{x}} \\chi{(\\hat{x},A_{1})} + A_{1}^{\\hat{x}}", "derivation": "\\chi{(\\hat{x},A_{1})} = A_{1}^{\\hat{x}} and \\chi^{2}{(\\hat{x},A_{1})} = A_{1}^{\\hat{x}} \\chi{(\\hat{x},A_{1})} and - \\chi^{2}{(\\hat{x},A_{1})} + \\chi{(\\hat{x},A_{1})} = A_{1}^{\\hat{x}} - \\chi^{2}{(\\hat{x},A_{1})} and - A_{1}^{\\hat{x}} \\chi{(\\hat{x},A_{1})} + \\chi{(\\hat{x},A_{1})} = - A_{1}^{\\hat{x}} \\chi{(\\hat{x},A_{1})} + A_{1}^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Integer(2)), Mul(Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))))"], [["minus", 1, "Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))), Add(Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))), Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_1', commutative=True))), Pow(Symbol('A_1', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\eta)} = \\cos{(\\eta)}, then obtain \\int \\mathbb{I}{(\\eta)} d\\eta + \\iint \\mathbb{I}{(\\eta)} d\\eta d\\eta = \\int \\cos{(\\eta)} d\\eta + \\iint \\mathbb{I}{(\\eta)} d\\eta d\\eta", "derivation": "\\mathbb{I}{(\\eta)} = \\cos{(\\eta)} and \\int \\mathbb{I}{(\\eta)} d\\eta = \\int \\cos{(\\eta)} d\\eta and \\iint \\mathbb{I}{(\\eta)} d\\eta d\\eta = \\iint \\cos{(\\eta)} d\\eta d\\eta and \\int \\mathbb{I}{(\\eta)} d\\eta + \\iint \\cos{(\\eta)} d\\eta d\\eta = \\int \\cos{(\\eta)} d\\eta + \\iint \\cos{(\\eta)} d\\eta d\\eta and \\int \\mathbb{I}{(\\eta)} d\\eta + \\iint \\mathbb{I}{(\\eta)} d\\eta d\\eta = \\int \\cos{(\\eta)} d\\eta + \\iint \\mathbb{I}{(\\eta)} d\\eta d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(A_{2},\\phi)} = e^{A_{2} - \\phi}, then derive \\frac{\\partial}{\\partial A_{2}} \\mathbf{M}{(A_{2},\\phi)} = e^{A_{2} - \\phi}, then obtain \\frac{\\partial^{2}}{\\partial A_{2}^{2}} \\mathbf{M}{(A_{2},\\phi)} = \\frac{\\partial}{\\partial A_{2}} e^{A_{2} - \\phi}", "derivation": "\\mathbf{M}{(A_{2},\\phi)} = e^{A_{2} - \\phi} and \\frac{\\partial}{\\partial A_{2}} \\mathbf{M}{(A_{2},\\phi)} = \\frac{\\partial}{\\partial A_{2}} e^{A_{2} - \\phi} and \\frac{\\partial}{\\partial A_{2}} \\mathbf{M}{(A_{2},\\phi)} = e^{A_{2} - \\phi} and \\frac{\\partial^{2}}{\\partial A_{2}^{2}} \\mathbf{M}{(A_{2},\\phi)} = \\frac{\\partial}{\\partial A_{2}} e^{A_{2} - \\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('A_2', commutative=True), Symbol('\\\\phi', commutative=True)), exp(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('A_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('A_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), exp(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('A_2', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(2))), Derivative(exp(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(n_{1},\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + n_{1} and \\operatorname{y^{\\prime}}{(n_{1})} = n_{1}, then derive \\frac{\\partial}{\\partial n_{1}} G{(n_{1},\\hat{\\mathbf{x}})} = 1, then obtain \\frac{\\partial}{\\partial \\operatorname{y^{\\prime}}{(n_{1})}} (\\hat{\\mathbf{x}} + \\operatorname{y^{\\prime}}{(n_{1})}) = 1", "derivation": "G{(n_{1},\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + n_{1} and \\frac{\\partial}{\\partial n_{1}} G{(n_{1},\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial n_{1}} (\\hat{\\mathbf{x}} + n_{1}) and \\frac{\\partial}{\\partial n_{1}} G{(n_{1},\\hat{\\mathbf{x}})} = 1 and \\operatorname{y^{\\prime}}{(n_{1})} = n_{1} and \\frac{\\partial}{\\partial n_{1}} (\\hat{\\mathbf{x}} + n_{1}) = 1 and \\frac{\\partial}{\\partial \\operatorname{y^{\\prime}}{(n_{1})}} (\\hat{\\mathbf{x}} + \\operatorname{y^{\\prime}}{(n_{1})}) = 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('n_1', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(1))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('y^{\\\\prime}')(Symbol('n_1', commutative=True))), Tuple(Function('y^{\\\\prime}')(Symbol('n_1', commutative=True)), Integer(1))), Integer(1))"]]}, {"prompt": "Given H{(a^{\\dagger},\\mathbf{D})} = \\mathbf{D} a^{\\dagger}, then obtain - 2 \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} - 1 = - 4 \\mathbf{D} a^{\\dagger} + 3 H{(a^{\\dagger},\\mathbf{D})} - 1", "derivation": "H{(a^{\\dagger},\\mathbf{D})} = \\mathbf{D} a^{\\dagger} and - \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} = 0 and - 2 \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} = - \\mathbf{D} a^{\\dagger} and - 2 \\mathbf{D} a^{\\dagger} + 2 H{(a^{\\dagger},\\mathbf{D})} = 0 and - 2 \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} - 1 = - H{(a^{\\dagger},\\mathbf{D})} - 1 and - \\mathbf{D} a^{\\dagger} - 1 = - H{(a^{\\dagger},\\mathbf{D})} - 1 and - \\mathbf{D} a^{\\dagger} - 1 = - 2 \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} - 1 and - 2 \\mathbf{D} a^{\\dagger} + H{(a^{\\dagger},\\mathbf{D})} - 1 = - 4 \\mathbf{D} a^{\\dagger} + 3 H{(a^{\\dagger},\\mathbf{D})} - 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(0))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Integer(0))"], [["minus", 4, "Add(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(4), Symbol('\\\\mathbf{D}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(3), Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given a{(l)} = \\log{(\\sin{(l)})}, then obtain \\frac{\\sin{(l)} \\frac{d}{d l} a{(l)}}{\\cos{(l)}} = 1", "derivation": "a{(l)} = \\log{(\\sin{(l)})} and \\frac{d}{d l} a{(l)} = \\frac{d}{d l} \\log{(\\sin{(l)})} and \\frac{\\frac{d}{d l} a{(l)}}{\\frac{d}{d l} \\log{(\\sin{(l)})}} = 1 and \\frac{\\sin{(l)} \\frac{d}{d l} a{(l)}}{\\cos{(l)}} = 1", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('l', commutative=True)), log(sin(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(log(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Pow(Derivative(log(sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Mul(sin(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(-1)), Derivative(Function('a')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given k{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\psi^{*}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then obtain \\frac{- V_{\\mathbf{B}} + k{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{- V_{\\mathbf{B}} + \\psi^{*}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}}", "derivation": "k{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and - V_{\\mathbf{B}} + k{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} + \\log{(V_{\\mathbf{B}})} and \\frac{- V_{\\mathbf{B}} + k{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{- V_{\\mathbf{B}} + \\log{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} and \\psi^{*}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and - V_{\\mathbf{B}} + k{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} + \\psi^{*}{(V_{\\mathbf{B}})} and - V_{\\mathbf{B}} + \\log{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} + \\psi^{*}{(V_{\\mathbf{B}})} and \\frac{- V_{\\mathbf{B}} + k{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{- V_{\\mathbf{B}} + \\psi^{*}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["divide", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\psi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\psi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\psi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(a)} = e^{a} and \\mathbf{g}{(a)} = (\\frac{d}{d a} e^{a})^{2}, then obtain (\\frac{d}{d a} \\mathbf{g}{(a)})^{a} = (\\frac{d}{d a} \\frac{d}{d a} \\operatorname{v_{t}}{(a)} \\frac{d}{d a} e^{a})^{a}", "derivation": "\\operatorname{v_{t}}{(a)} = e^{a} and \\frac{d}{d a} \\operatorname{v_{t}}{(a)} = \\frac{d}{d a} e^{a} and \\frac{d}{d a} \\operatorname{v_{t}}{(a)} \\frac{d}{d a} e^{a} = (\\frac{d}{d a} e^{a})^{2} and \\mathbf{g}{(a)} = (\\frac{d}{d a} e^{a})^{2} and \\frac{d}{d a} \\mathbf{g}{(a)} = \\frac{d}{d a} (\\frac{d}{d a} e^{a})^{2} and \\frac{d}{d a} \\mathbf{g}{(a)} = \\frac{d}{d a} \\frac{d}{d a} \\operatorname{v_{t}}{(a)} \\frac{d}{d a} e^{a} and (\\frac{d}{d a} \\mathbf{g}{(a)})^{a} = (\\frac{d}{d a} \\frac{d}{d a} \\operatorname{v_{t}}{(a)} \\frac{d}{d a} e^{a})^{a}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 2, "Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('v_t')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('a', commutative=True)), Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Derivative(Function('v_t')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)), Pow(Derivative(Mul(Derivative(Function('v_t')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(x)} = e^{e^{x}}, then obtain (- \\operatorname{F_{c}}{(x)} e^{- x} + e^{- x} e^{e^{x}}) \\iint \\operatorname{F_{c}}{(x)} dx dx = (- \\operatorname{F_{c}}{(x)} e^{- x} + e^{- x} e^{e^{x}}) \\iint e^{e^{x}} dx dx", "derivation": "\\operatorname{F_{c}}{(x)} = e^{e^{x}} and \\int \\operatorname{F_{c}}{(x)} dx = \\int e^{e^{x}} dx and \\iint \\operatorname{F_{c}}{(x)} dx dx = \\iint e^{e^{x}} dx dx and (- \\operatorname{F_{c}}{(x)} e^{- x} + e^{- x} e^{e^{x}}) \\iint \\operatorname{F_{c}}{(x)} dx dx = (- \\operatorname{F_{c}}{(x)} e^{- x} + e^{- x} e^{e^{x}}) \\iint e^{e^{x}} dx dx", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('x', commutative=True)), exp(exp(Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Function('F_c')(Symbol('x', commutative=True)), exp(Mul(Integer(-1), Symbol('x', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('x', commutative=True))), exp(exp(Symbol('x', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_c')(Symbol('x', commutative=True)), exp(Mul(Integer(-1), Symbol('x', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('x', commutative=True))), exp(exp(Symbol('x', commutative=True))))), Integral(Function('F_c')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('F_c')(Symbol('x', commutative=True)), exp(Mul(Integer(-1), Symbol('x', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('x', commutative=True))), exp(exp(Symbol('x', commutative=True))))), Integral(exp(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(A_{y})} = A_{y}, then obtain \\frac{d}{d A_{y}} 0 = \\frac{d}{d A_{y}} - \\frac{A_{y}^{2} \\mathbf{J}_M{(A_{y})} - \\mathbf{J}_M^{3}{(A_{y})}}{\\mathbf{J}_M^{3}{(A_{y})}}", "derivation": "\\mathbf{J}_M{(A_{y})} = A_{y} and \\mathbf{J}_M^{2}{(A_{y})} = A_{y} \\mathbf{J}_M{(A_{y})} and \\mathbf{J}_M^{3}{(A_{y})} = A_{y} \\mathbf{J}_M^{2}{(A_{y})} and \\mathbf{J}_M^{3}{(A_{y})} = A_{y}^{2} \\mathbf{J}_M{(A_{y})} and 0 = A_{y}^{2} \\mathbf{J}_M{(A_{y})} - \\mathbf{J}_M^{3}{(A_{y})} and 0 = - \\frac{A_{y}^{2} \\mathbf{J}_M{(A_{y})} - \\mathbf{J}_M^{3}{(A_{y})}}{\\mathbf{J}_M^{3}{(A_{y})}} and \\frac{d}{d A_{y}} 0 = \\frac{d}{d A_{y}} - \\frac{A_{y}^{2} \\mathbf{J}_M{(A_{y})} - \\mathbf{J}_M^{3}{(A_{y})}}{\\mathbf{J}_M^{3}{(A_{y})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["times", 1, "Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(2)), Mul(Symbol('A_y', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)), Mul(Symbol('A_y', commutative=True), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)), Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))))"], [["minus", 4, "Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)))))"], [["divide", 5, "Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)))), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(-3))))"], [["differentiate", 6, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(3)))), Pow(Function('\\\\mathbf{J}_M')(Symbol('A_y', commutative=True)), Integer(-3))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(E,\\dot{z})} = \\frac{E}{\\dot{z}}, then obtain (e^{\\int \\hat{p}_0{(E,\\dot{z})} d\\dot{z}})^{E} = (e^{\\int \\frac{E}{\\dot{z}} d\\dot{z}})^{E}", "derivation": "\\hat{p}_0{(E,\\dot{z})} = \\frac{E}{\\dot{z}} and \\int \\hat{p}_0{(E,\\dot{z})} d\\dot{z} = \\int \\frac{E}{\\dot{z}} d\\dot{z} and e^{\\int \\hat{p}_0{(E,\\dot{z})} d\\dot{z}} = e^{\\int \\frac{E}{\\dot{z}} d\\dot{z}} and (e^{\\int \\hat{p}_0{(E,\\dot{z})} d\\dot{z}})^{E} = (e^{\\int \\frac{E}{\\dot{z}} d\\dot{z}})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), exp(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(exp(Integral(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Symbol('E', commutative=True)), Pow(exp(Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\mathbf{J}_M{(J)} = \\log{(J)}, then obtain \\frac{\\mathbf{J}_M{(J)} - \\log{(\\hat{H}_l)}}{\\log{(J)} - \\log{(\\hat{H}_l)}} + \\log{(J)} = \\log{(J)} + 1", "derivation": "\\mathbf{J}_P{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\mathbf{J}_M{(J)} = \\log{(J)} and \\mathbf{J}_M{(J)} - \\log{(\\hat{H}_l)} = \\log{(J)} - \\log{(\\hat{H}_l)} and \\frac{\\mathbf{J}_M{(J)} - \\log{(\\hat{H}_l)}}{- \\mathbf{J}_P{(\\hat{H}_l)} + \\log{(J)}} = \\frac{\\log{(J)} - \\log{(\\hat{H}_l)}}{- \\mathbf{J}_P{(\\hat{H}_l)} + \\log{(J)}} and \\frac{\\mathbf{J}_M{(J)} - \\mathbf{J}_P{(\\hat{H}_l)}}{- \\mathbf{J}_P{(\\hat{H}_l)} + \\log{(J)}} = 1 and \\frac{\\mathbf{J}_M{(J)} - \\log{(\\hat{H}_l)}}{\\log{(J)} - \\log{(\\hat{H}_l)}} = 1 and \\frac{\\mathbf{J}_M{(J)} - \\log{(\\hat{H}_l)}}{\\log{(J)} - \\log{(\\hat{H}_l)}} + \\log{(J)} = \\log{(J)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["minus", 2, "log(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Add(log(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('J', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('J', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('J', commutative=True))), Integer(-1)), Add(log(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('J', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Add(log(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(-1))), Integer(1))"], [["add", 6, "log(Symbol('J', commutative=True))"], "Equality(Add(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Pow(Add(log(Symbol('J', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(-1))), log(Symbol('J', commutative=True))), Add(log(Symbol('J', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} = \\chi + \\mathbf{M}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\chi} \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} = 0", "derivation": "\\operatorname{v_{y}}{(\\chi,\\mathbf{M})} = \\chi + \\mathbf{M} and \\frac{\\partial}{\\partial \\chi} \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} = \\frac{\\partial}{\\partial \\chi} (\\chi + \\mathbf{M}) and \\frac{\\partial}{\\partial \\chi} \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} - \\frac{\\log{(v)}^{v}}{v} = \\frac{\\partial}{\\partial \\chi} (\\chi + \\mathbf{M}) - \\frac{\\log{(v)}^{v}}{v} and \\frac{\\partial}{\\partial \\mathbf{M}} (\\frac{\\partial}{\\partial \\chi} \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} - \\frac{\\log{(v)}^{v}}{v}) = \\frac{\\partial}{\\partial \\mathbf{M}} (\\frac{\\partial}{\\partial \\chi} (\\chi + \\mathbf{M}) - \\frac{\\log{(v)}^{v}}{v}) and \\frac{\\partial^{2}}{\\partial \\mathbf{M}\\partial \\chi} \\operatorname{v_{y}}{(\\chi,\\mathbf{M})} = 0", "srepr_derivation": [["get_premise", "Equality(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], "Equality(Add(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))), Add(Derivative(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\pi{(E_{n},r)} = r \\cos{(E_{n})}, then obtain (\\frac{\\partial}{\\partial r} \\pi^{r}{(E_{n},r)})^{E_{n}} = (\\frac{\\partial}{\\partial r} (r \\cos{(E_{n})})^{r})^{E_{n}}", "derivation": "\\pi{(E_{n},r)} = r \\cos{(E_{n})} and \\pi^{r}{(E_{n},r)} = (r \\cos{(E_{n})})^{r} and \\frac{\\partial}{\\partial r} \\pi^{r}{(E_{n},r)} = \\frac{\\partial}{\\partial r} (r \\cos{(E_{n})})^{r} and (\\frac{\\partial}{\\partial r} \\pi^{r}{(E_{n},r)})^{E_{n}} = (\\frac{\\partial}{\\partial r} (r \\cos{(E_{n})})^{r})^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('E_n', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('r', commutative=True), cos(Symbol('E_n', commutative=True))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('E_n', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Mul(Symbol('r', commutative=True), cos(Symbol('E_n', commutative=True))), Symbol('r', commutative=True)))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\pi')(Symbol('E_n', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('r', commutative=True), cos(Symbol('E_n', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\pi')(Symbol('E_n', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('E_n', commutative=True)), Pow(Derivative(Pow(Mul(Symbol('r', commutative=True), cos(Symbol('E_n', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(F_{H})} = \\frac{d}{d F_{H}} \\log{(F_{H})}, then derive \\operatorname{v_{1}}{(F_{H})} = \\frac{1}{F_{H}}, then obtain F_{H} + \\frac{d}{d F_{H}} (F_{H} + \\frac{1}{F_{H}}) = F_{H} + \\frac{d}{d F_{H}} (F_{H} + \\frac{d}{d F_{H}} \\log{(F_{H})})", "derivation": "\\operatorname{v_{1}}{(F_{H})} = \\frac{d}{d F_{H}} \\log{(F_{H})} and F_{H} + \\operatorname{v_{1}}{(F_{H})} = F_{H} + \\frac{d}{d F_{H}} \\log{(F_{H})} and \\frac{d}{d F_{H}} (F_{H} + \\operatorname{v_{1}}{(F_{H})}) = \\frac{d}{d F_{H}} (F_{H} + \\frac{d}{d F_{H}} \\log{(F_{H})}) and \\operatorname{v_{1}}{(F_{H})} = \\frac{1}{F_{H}} and \\frac{d}{d F_{H}} (F_{H} + \\frac{1}{F_{H}}) = \\frac{d}{d F_{H}} (F_{H} + \\frac{d}{d F_{H}} \\log{(F_{H})}) and F_{H} + \\frac{d}{d F_{H}} (F_{H} + \\frac{1}{F_{H}}) = F_{H} + \\frac{d}{d F_{H}} (F_{H} + \\frac{d}{d F_{H}} \\log{(F_{H})})", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('F_H', commutative=True)), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["add", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Function('v_1')(Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Add(Symbol('F_H', commutative=True), Function('v_1')(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_1')(Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('F_H', commutative=True), Pow(Symbol('F_H', commutative=True), Integer(-1))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["add", 5, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Derivative(Add(Symbol('F_H', commutative=True), Pow(Symbol('F_H', commutative=True), Integer(-1))), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Symbol('F_H', commutative=True), Derivative(Add(Symbol('F_H', commutative=True), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(a)} = \\log{(a)}, then obtain \\Psi_{nl}{(a)} + \\frac{d}{d a} (\\Psi_{nl}{(a)} + \\log{(a)}) = \\log{(a)} + \\frac{d}{d a} (\\Psi_{nl}{(a)} + \\log{(a)})", "derivation": "\\Psi_{nl}{(a)} = \\log{(a)} and \\Psi_{nl}{(a)} + \\log{(a)} = 2 \\log{(a)} and \\frac{d}{d a} (\\Psi_{nl}{(a)} + \\log{(a)}) = \\frac{d}{d a} 2 \\log{(a)} and \\Psi_{nl}{(a)} + \\frac{d}{d a} 2 \\log{(a)} = \\log{(a)} + \\frac{d}{d a} 2 \\log{(a)} and \\Psi_{nl}{(a)} + \\frac{d}{d a} (\\Psi_{nl}{(a)} + \\log{(a)}) = \\log{(a)} + \\frac{d}{d a} (\\Psi_{nl}{(a)} + \\log{(a)})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["add", 1, "log(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Mul(Integer(2), log(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Mul(Integer(2), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Derivative(Mul(Integer(2), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(log(Symbol('a', commutative=True)), Derivative(Mul(Integer(2), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(log(Symbol('a', commutative=True)), Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(v_{t})} = v_{t}, then obtain (v_{t} + x{(v_{t})}) (M{(v_{t})} + e^{v_{t}}) = (v_{t} + x{(v_{t})}) (v_{t} + e^{v_{t}})", "derivation": "M{(v_{t})} = v_{t} and e^{M{(v_{t})}} = e^{v_{t}} and M{(v_{t})} + e^{v_{t}} = v_{t} + e^{v_{t}} and M{(v_{t})} + e^{M{(v_{t})}} = v_{t} + e^{M{(v_{t})}} and (v_{t} + x{(v_{t})}) (M{(v_{t})} + e^{M{(v_{t})}}) = (v_{t} + x{(v_{t})}) (v_{t} + e^{M{(v_{t})}}) and (v_{t} + x{(v_{t})}) (M{(v_{t})} + e^{v_{t}}) = (v_{t} + x{(v_{t})}) (v_{t} + e^{v_{t}})", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], [["exp", 1], "Equality(exp(Function('M')(Symbol('v_t', commutative=True))), exp(Symbol('v_t', commutative=True)))"], [["add", 1, "exp(Symbol('v_t', commutative=True))"], "Equality(Add(Function('M')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('M')(Symbol('v_t', commutative=True)), exp(Function('M')(Symbol('v_t', commutative=True)))), Add(Symbol('v_t', commutative=True), exp(Function('M')(Symbol('v_t', commutative=True)))))"], [["times", 4, "Add(Symbol('v_t', commutative=True), Function('x')(Symbol('v_t', commutative=True)))"], "Equality(Mul(Add(Symbol('v_t', commutative=True), Function('x')(Symbol('v_t', commutative=True))), Add(Function('M')(Symbol('v_t', commutative=True)), exp(Function('M')(Symbol('v_t', commutative=True))))), Mul(Add(Symbol('v_t', commutative=True), Function('x')(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Function('M')(Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Symbol('v_t', commutative=True), Function('x')(Symbol('v_t', commutative=True))), Add(Function('M')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))), Mul(Add(Symbol('v_t', commutative=True), Function('x')(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(c_{0})} = \\sin{(c_{0})}, then obtain (- \\cos{(c_{0})} + \\frac{d}{d c_{0}} \\varepsilon_{0}{(c_{0})})^{c_{0}} = 0^{c_{0}}", "derivation": "\\varepsilon_{0}{(c_{0})} = \\sin{(c_{0})} and \\varepsilon_{0}{(c_{0})} - \\sin{(c_{0})} = 0 and \\frac{d}{d c_{0}} (\\varepsilon_{0}{(c_{0})} - \\sin{(c_{0})}) = \\frac{d}{d c_{0}} 0 and (\\frac{d}{d c_{0}} (\\varepsilon_{0}{(c_{0})} - \\sin{(c_{0})}))^{c_{0}} = (\\frac{d}{d c_{0}} 0)^{c_{0}} and (- \\cos{(c_{0})} + \\frac{d}{d c_{0}} \\varepsilon_{0}{(c_{0})})^{c_{0}} = 0^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["minus", 1, "sin(Symbol('c_0', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('c_0', commutative=True)), Mul(Integer(-1), sin(Symbol('c_0', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varepsilon_0')(Symbol('c_0', commutative=True)), Mul(Integer(-1), sin(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\varepsilon_0')(Symbol('c_0', commutative=True)), Mul(Integer(-1), sin(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Symbol('c_0', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('c_0', commutative=True), Integer(1))), Symbol('c_0', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Derivative(Function('\\\\varepsilon_0')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Symbol('c_0', commutative=True)), Pow(Integer(0), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\delta,\\sigma_p)} = \\sigma_p^{\\delta}, then obtain (\\frac{\\sigma_p^{\\delta} \\operatorname{P_{g}}{(\\delta,\\sigma_p)}}{\\sigma_p})^{2 \\sigma_p} = (\\frac{\\sigma_p^{2 \\delta}}{\\sigma_p})^{2 \\sigma_p}", "derivation": "\\operatorname{P_{g}}{(\\delta,\\sigma_p)} = \\sigma_p^{\\delta} and \\frac{\\sigma_p^{\\delta} \\operatorname{P_{g}}{(\\delta,\\sigma_p)}}{\\sigma_p} = \\frac{\\sigma_p^{2 \\delta}}{\\sigma_p} and (\\frac{\\sigma_p^{\\delta} \\operatorname{P_{g}}{(\\delta,\\sigma_p)}}{\\sigma_p})^{\\sigma_p} = (\\frac{\\sigma_p^{2 \\delta}}{\\sigma_p})^{\\sigma_p} and (\\frac{\\sigma_p^{\\delta} \\operatorname{P_{g}}{(\\delta,\\sigma_p)}}{\\sigma_p})^{2 \\sigma_p} = (\\frac{\\sigma_p^{2 \\delta}}{\\sigma_p})^{2 \\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\delta', commutative=True)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\delta', commutative=True)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\delta', commutative=True)), Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True))), Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))), Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given I{(m_{s})} = \\log{(m_{s})}, then derive \\frac{d}{d m_{s}} I{(m_{s})} - 1 = -1 + \\frac{1}{m_{s}}, then obtain - \\frac{\\log{(\\frac{d}{d m_{s}} I{(m_{s})} - 1)}}{m_{s}} = - \\frac{\\log{(-1 + \\frac{1}{m_{s}})}}{m_{s}}", "derivation": "I{(m_{s})} = \\log{(m_{s})} and - m_{s} + I{(m_{s})} = - m_{s} + \\log{(m_{s})} and \\frac{d}{d m_{s}} (- m_{s} + I{(m_{s})}) = \\frac{d}{d m_{s}} (- m_{s} + \\log{(m_{s})}) and \\frac{d}{d m_{s}} I{(m_{s})} - 1 = -1 + \\frac{1}{m_{s}} and \\frac{d}{d m_{s}} \\log{(m_{s})} - 1 = -1 + \\frac{1}{m_{s}} and \\log{(\\frac{d}{d m_{s}} \\log{(m_{s})} - 1)} = \\log{(-1 + \\frac{1}{m_{s}})} and \\log{(\\frac{d}{d m_{s}} I{(m_{s})} - 1)} = \\log{(-1 + \\frac{1}{m_{s}})} and - \\frac{\\log{(\\frac{d}{d m_{s}} I{(m_{s})} - 1)}}{m_{s}} = - \\frac{\\log{(-1 + \\frac{1}{m_{s}})}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], [["minus", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('I')(Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('I')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('I')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["log", 5], "Equality(log(Add(Derivative(log(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), log(Add(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(log(Add(Derivative(Function('I')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), log(Add(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)))))"], [["divide", 7, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), log(Add(Derivative(Function('I')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), log(Add(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\eta{(t,E_{\\lambda})} = \\cos^{t}{(E_{\\lambda})} and \\mathbf{J}_f{(\\mathbf{D},\\hbar)} = \\int \\hbar \\mathbf{D} d\\mathbf{D}, then obtain E_{\\lambda} \\eta{(t,E_{\\lambda})} - \\int \\hbar \\mathbf{D} d\\mathbf{D} = E_{\\lambda} \\cos^{t}{(E_{\\lambda})} - \\int \\hbar \\mathbf{D} d\\mathbf{D}", "derivation": "\\eta{(t,E_{\\lambda})} = \\cos^{t}{(E_{\\lambda})} and E_{\\lambda} \\eta{(t,E_{\\lambda})} = E_{\\lambda} \\cos^{t}{(E_{\\lambda})} and \\mathbf{J}_f{(\\mathbf{D},\\hbar)} = \\int \\hbar \\mathbf{D} d\\mathbf{D} and E_{\\lambda} \\eta{(t,E_{\\lambda})} - \\mathbf{J}_f{(\\mathbf{D},\\hbar)} = E_{\\lambda} \\cos^{t}{(E_{\\lambda})} - \\mathbf{J}_f{(\\mathbf{D},\\hbar)} and E_{\\lambda} \\eta{(t,E_{\\lambda})} - \\int \\hbar \\mathbf{D} d\\mathbf{D} = E_{\\lambda} \\cos^{t}{(E_{\\lambda})} - \\int \\hbar \\mathbf{D} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t', commutative=True)))"], [["times", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(cos(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(cos(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('t', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(cos(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(F_{N})} = \\int \\cos{(F_{N})} dF_{N}, then derive \\operatorname{F_{H}}{(F_{N})} = G + \\sin{(F_{N})}, then obtain \\frac{d}{d F_{N}} \\operatorname{F_{H}}{(F_{N})} = \\frac{d}{d F_{N}} \\int \\cos{(F_{N})} dF_{N}", "derivation": "\\operatorname{F_{H}}{(F_{N})} = \\int \\cos{(F_{N})} dF_{N} and \\operatorname{F_{H}}{(F_{N})} = G + \\sin{(F_{N})} and G + \\sin{(F_{N})} = \\int \\cos{(F_{N})} dF_{N} and \\frac{d}{d F_{N}} \\operatorname{F_{H}}{(F_{N})} = \\frac{\\partial}{\\partial F_{N}} (G + \\sin{(F_{N})}) and \\frac{d}{d F_{N}} \\operatorname{F_{H}}{(F_{N})} = \\frac{d}{d F_{N}} \\int \\cos{(F_{N})} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('F_N', commutative=True)), Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('F_H')(Symbol('F_N', commutative=True)), Add(Symbol('G', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('G', commutative=True), sin(Symbol('F_N', commutative=True))), Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Symbol('G', commutative=True), sin(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Function('F_H')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(f^{*})} = e^{f^{*}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f^{*},f_{E})} = f_{E} + H{(f^{*})}, then derive f_{E} + H{(f^{*})} = A_{2} + e^{f^{*}}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(f^{*},f_{E})} = f_{E} + e^{f^{*}}", "derivation": "H{(f^{*})} = e^{f^{*}} and \\frac{d}{d f^{*}} H{(f^{*})} = \\frac{d}{d f^{*}} e^{f^{*}} and \\int \\frac{d}{d f^{*}} H{(f^{*})} df^{*} = \\int \\frac{d}{d f^{*}} e^{f^{*}} df^{*} and f_{E} + H{(f^{*})} = A_{2} + e^{f^{*}} and f_{E} + e^{f^{*}} = A_{2} + e^{f^{*}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f^{*},f_{E})} = f_{E} + H{(f^{*})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f^{*},f_{E})} = A_{2} + e^{f^{*}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f^{*},f_{E})} = f_{E} + e^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Derivative(Function('H')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))), Integral(Derivative(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('f_E', commutative=True), Function('H')(Symbol('f^*', commutative=True))), Add(Symbol('A_2', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('f_E', commutative=True), exp(Symbol('f^*', commutative=True))), Add(Symbol('A_2', commutative=True), exp(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f_E', commutative=True), Function('H')(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('A_2', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f_E', commutative=True), exp(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(W)} = \\frac{d}{d W} \\log{(W)}, then derive (\\frac{W \\frac{d}{d W} \\operatorname{C_{d}}{(W)}}{\\operatorname{C_{d}}{(W)}} + \\log{(\\operatorname{C_{d}}{(W)})}) \\operatorname{C_{d}}^{W}{(W)} = (\\log{(\\frac{1}{W})} - 1) (\\frac{1}{W})^{W}, then obtain (\\frac{W \\frac{d^{2}}{d W^{2}} \\log{(W)}}{\\operatorname{C_{d}}{(W)}} + \\log{(\\operatorname{C_{d}}{(W)})}) \\operatorname{C_{d}}^{W}{(W)} = (\\log{(\\frac{1}{W})} - 1) (\\frac{1}{W})^{W}", "derivation": "\\operatorname{C_{d}}{(W)} = \\frac{d}{d W} \\log{(W)} and \\operatorname{C_{d}}^{W}{(W)} = (\\frac{d}{d W} \\log{(W)})^{W} and \\frac{d}{d W} \\operatorname{C_{d}}^{W}{(W)} = \\frac{d}{d W} (\\frac{d}{d W} \\log{(W)})^{W} and \\frac{d}{d W} \\operatorname{C_{d}}{(W)} = \\frac{d^{2}}{d W^{2}} \\log{(W)} and (\\frac{W \\frac{d}{d W} \\operatorname{C_{d}}{(W)}}{\\operatorname{C_{d}}{(W)}} + \\log{(\\operatorname{C_{d}}{(W)})}) \\operatorname{C_{d}}^{W}{(W)} = (\\log{(\\frac{1}{W})} - 1) (\\frac{1}{W})^{W} and (\\frac{W \\frac{d^{2}}{d W^{2}} \\log{(W)}}{\\operatorname{C_{d}}{(W)}} + \\log{(\\operatorname{C_{d}}{(W)})}) \\operatorname{C_{d}}^{W}{(W)} = (\\log{(\\frac{1}{W})} - 1) (\\frac{1}{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('W', commutative=True)), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Function('C_d')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('W', commutative=True), Pow(Function('C_d')(Symbol('W', commutative=True)), Integer(-1)), Derivative(Function('C_d')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), log(Function('C_d')(Symbol('W', commutative=True)))), Pow(Function('C_d')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Add(log(Pow(Symbol('W', commutative=True), Integer(-1))), Integer(-1)), Pow(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Mul(Symbol('W', commutative=True), Pow(Function('C_d')(Symbol('W', commutative=True)), Integer(-1)), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))), log(Function('C_d')(Symbol('W', commutative=True)))), Pow(Function('C_d')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Add(log(Pow(Symbol('W', commutative=True), Integer(-1))), Integer(-1)), Pow(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('W', commutative=True))))"]]}, {"prompt": "Given u{(n)} = e^{e^{n}}, then derive - e^{n} e^{e^{n}} + \\frac{d}{d n} u{(n)} = 0, then derive - \\theta - u{(n)} + e^{e^{n}} = - \\int 0 dn, then obtain - \\theta = - \\int 0 dn", "derivation": "u{(n)} = e^{e^{n}} and n + u{(n)} = n + e^{e^{n}} and u{(n)} - e^{e^{n}} = 0 and \\frac{d}{d n} (u{(n)} - e^{e^{n}}) = \\frac{d}{d n} 0 and - e^{n} e^{e^{n}} + \\frac{d}{d n} u{(n)} = 0 and \\int (- e^{n} e^{e^{n}} + \\frac{d}{d n} u{(n)}) dn = \\int 0 dn and - \\int (- e^{n} e^{e^{n}} + \\frac{d}{d n} u{(n)}) dn = - \\int 0 dn and - \\theta - u{(n)} + e^{e^{n}} = - \\int 0 dn and - \\theta = - \\int 0 dn", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('n', commutative=True)), exp(exp(Symbol('n', commutative=True))))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('u')(Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), exp(exp(Symbol('n', commutative=True)))))"], [["minus", 2, "Add(Symbol('n', commutative=True), exp(exp(Symbol('n', commutative=True))))"], "Equality(Add(Function('u')(Symbol('n', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('n', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('u')(Symbol('n', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True)), exp(exp(Symbol('n', commutative=True)))), Derivative(Function('u')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 5, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True)), exp(exp(Symbol('n', commutative=True)))), Derivative(Function('u')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))), Integral(Integer(0), Tuple(Symbol('n', commutative=True))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True)), exp(exp(Symbol('n', commutative=True)))), Derivative(Function('u')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True)))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('n', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('n', commutative=True))), exp(exp(Symbol('n', commutative=True)))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 1], "Equality(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(E)} = \\log{(E)}, then derive \\frac{d^{2}}{d E^{2}} \\tilde{g}{(E)} = - \\frac{1}{E^{2}}, then obtain \\frac{\\frac{d^{2}}{d E^{2}} \\log{(E)}}{E} + \\frac{1}{E^{3}} = 0", "derivation": "\\tilde{g}{(E)} = \\log{(E)} and \\frac{d}{d E} \\tilde{g}{(E)} = \\frac{d}{d E} \\log{(E)} and \\frac{d^{2}}{d E^{2}} \\tilde{g}{(E)} = \\frac{d^{2}}{d E^{2}} \\log{(E)} and \\frac{d^{2}}{d E^{2}} \\tilde{g}{(E)} = - \\frac{1}{E^{2}} and \\frac{d^{2}}{d E^{2}} \\log{(E)} = - \\frac{1}{E^{2}} and \\frac{\\frac{d^{2}}{d E^{2}} \\log{(E)}}{E} = - \\frac{1}{E^{3}} and \\frac{\\frac{d^{2}}{d E^{2}} \\log{(E)}}{E} + \\frac{1}{E^{3}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-2))))"], [["divide", 5, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-3))))"], [["minus", 6, "Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-3)))"], "Equality(Add(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2)))), Pow(Symbol('E', commutative=True), Integer(-3))), Integer(0))"]]}, {"prompt": "Given \\mathbf{S}{(\\psi^*,\\chi,t)} = \\frac{- \\psi^* + t}{\\chi} and \\mathbf{A}{(\\psi^*)} = - \\psi^*, then obtain e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} \\int e^{\\frac{- \\psi^* + t}{\\chi}} dt = e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} \\int e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} dt", "derivation": "\\mathbf{S}{(\\psi^*,\\chi,t)} = \\frac{- \\psi^* + t}{\\chi} and \\mathbf{A}{(\\psi^*)} = - \\psi^* and \\mathbf{S}{(\\psi^*,\\chi,t)} = \\frac{t + \\mathbf{A}{(\\psi^*)}}{\\chi} and \\frac{- \\psi^* + t}{\\chi} = \\frac{t + \\mathbf{A}{(\\psi^*)}}{\\chi} and e^{\\frac{- \\psi^* + t}{\\chi}} = e^{\\frac{t + \\mathbf{A}{(\\psi^*)}}{\\chi}} and e^{\\frac{- \\psi^* + t}{\\chi}} = e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} and \\int e^{\\frac{- \\psi^* + t}{\\chi}} dt = \\int e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} dt and e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} \\int e^{\\frac{- \\psi^* + t}{\\chi}} dt = e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} \\int e^{\\mathbf{S}{(\\psi^*,\\chi,t)}} dt", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True)))))"], [["exp", 4], "Equality(exp(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True)))), exp(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\psi^*', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(exp(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True)))), exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 6, "Symbol('t', commutative=True)"], "Equality(Integral(exp(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["times", 7, "exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True)))"], "Equality(Mul(exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True))), Integral(exp(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))), Mul(exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True))), Integral(exp(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(L)} = \\sin{(L)}, then obtain \\mathbf{J}^{L}{(L)} \\sin^{- L}{(L)} \\int \\mathbf{J}^{L}{(L)} dL = \\mathbf{J}^{L}{(L)} \\sin^{- L}{(L)} \\int \\sin^{L}{(L)} dL", "derivation": "\\mathbf{J}{(L)} = \\sin{(L)} and \\mathbf{J}^{L}{(L)} = \\sin^{L}{(L)} and \\int \\mathbf{J}^{L}{(L)} dL = \\int \\sin^{L}{(L)} dL and \\sin^{- L}{(L)} \\int \\mathbf{J}^{L}{(L)} dL = \\sin^{- L}{(L)} \\int \\sin^{L}{(L)} dL and \\mathbf{J}^{L}{(L)} \\sin^{- L}{(L)} \\int \\mathbf{J}^{L}{(L)} dL = \\mathbf{J}^{L}{(L)} \\sin^{- L}{(L)} \\int \\sin^{L}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(sin(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["divide", 3, "Pow(sin(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))), Integral(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Mul(Pow(sin(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))), Integral(Pow(sin(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["times", 4, "Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))), Integral(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(sin(Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True))), Integral(Pow(sin(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\phi,M_{E})} = M_{E} \\phi and \\delta{(\\phi)} = \\phi, then obtain \\delta^{\\phi}{(\\phi)} + \\frac{\\partial}{\\partial \\phi} \\operatorname{v_{y}}{(\\phi,M_{E})} = \\phi^{\\phi} + \\frac{\\partial}{\\partial \\phi} \\operatorname{v_{y}}{(\\phi,M_{E})}", "derivation": "\\operatorname{v_{y}}{(\\phi,M_{E})} = M_{E} \\phi and \\frac{\\partial}{\\partial \\phi} \\operatorname{v_{y}}{(\\phi,M_{E})} = \\frac{\\partial}{\\partial \\phi} M_{E} \\phi and \\delta{(\\phi)} = \\phi and \\delta^{\\phi}{(\\phi)} = \\phi^{\\phi} and \\delta^{\\phi}{(\\phi)} + \\frac{\\partial}{\\partial \\phi} M_{E} \\phi = \\phi^{\\phi} + \\frac{\\partial}{\\partial \\phi} M_{E} \\phi and \\delta^{\\phi}{(\\phi)} + \\frac{\\partial}{\\partial \\phi} \\operatorname{v_{y}}{(\\phi,M_{E})} = \\phi^{\\phi} + \\frac{\\partial}{\\partial \\phi} \\operatorname{v_{y}}{(\\phi,M_{E})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], [["power", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 4, "Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('\\\\delta')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Derivative(Function('v_y')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi', commutative=True)), Derivative(Function('v_y')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given q{(L_{\\varepsilon},B)} = \\frac{B}{L_{\\varepsilon}} and m{(g,t_{1})} = \\frac{g}{t_{1}}, then obtain \\frac{g}{t_{1}} + m{(g,t_{1})} + \\int q{(L_{\\varepsilon},B)} dB = \\frac{2 g}{t_{1}} + \\int q{(L_{\\varepsilon},B)} dB", "derivation": "q{(L_{\\varepsilon},B)} = \\frac{B}{L_{\\varepsilon}} and \\int q{(L_{\\varepsilon},B)} dB = \\int \\frac{B}{L_{\\varepsilon}} dB and m{(g,t_{1})} = \\frac{g}{t_{1}} and \\frac{g}{t_{1}} + m{(g,t_{1})} = \\frac{2 g}{t_{1}} and \\frac{g}{t_{1}} + m{(g,t_{1})} + \\int \\frac{B}{L_{\\varepsilon}} dB = \\frac{2 g}{t_{1}} + \\int \\frac{B}{L_{\\varepsilon}} dB and \\frac{g}{t_{1}} + m{(g,t_{1})} + \\int q{(L_{\\varepsilon},B)} dB = \\frac{2 g}{t_{1}} + \\int q{(L_{\\varepsilon},B)} dB", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('q')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Mul(Symbol('B', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))))"], ["get_premise", "Equality(Function('m')(Symbol('g', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["add", 3, "Mul(Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Function('m')(Symbol('g', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(2), Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["add", 4, "Integral(Mul(Symbol('B', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Function('m')(Symbol('g', commutative=True), Symbol('t_1', commutative=True)), Integral(Mul(Symbol('B', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(2), Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integral(Mul(Symbol('B', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Function('m')(Symbol('g', commutative=True), Symbol('t_1', commutative=True)), Integral(Function('q')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(2), Symbol('g', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Integral(Function('q')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\pi,\\chi)} = \\chi + \\pi, then obtain 2 \\pi \\mathbf{s}{(\\pi,\\chi)} = \\pi (\\chi + \\pi + \\mathbf{s}{(\\pi,\\chi)})", "derivation": "\\mathbf{s}{(\\pi,\\chi)} = \\chi + \\pi and \\chi + \\pi + \\mathbf{s}{(\\pi,\\chi)} = 2 \\chi + 2 \\pi and 2 \\mathbf{s}{(\\pi,\\chi)} = 2 \\chi + 2 \\pi and 2 \\pi \\mathbf{s}{(\\pi,\\chi)} = \\pi (2 \\chi + 2 \\pi) and 2 \\pi \\mathbf{s}{(\\pi,\\chi)} = \\pi (\\chi + \\pi + \\mathbf{s}{(\\pi,\\chi)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True))))"], [["times", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given y{(W)} = e^{W}, then derive \\frac{d}{d W} y{(W)} = e^{W}, then obtain \\frac{d^{2}}{d W^{2}} e^{W} = - e^{W} + \\frac{d}{d W} e^{W} + \\frac{d^{2}}{d W^{2}} e^{W}", "derivation": "y{(W)} = e^{W} and \\frac{d}{d W} y{(W)} = \\frac{d}{d W} e^{W} and 0 = - y{(W)} + e^{W} and \\frac{d}{d W} y{(W)} = e^{W} and \\frac{d}{d W} e^{W} = - y{(W)} + e^{W} + \\frac{d}{d W} e^{W} and \\frac{d^{2}}{d W^{2}} y{(W)} = - y{(W)} + \\frac{d}{d W} y{(W)} + \\frac{d^{2}}{d W^{2}} y{(W)} and \\frac{d^{2}}{d W^{2}} e^{W} = - e^{W} + \\frac{d}{d W} e^{W} + \\frac{d^{2}}{d W^{2}} e^{W}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["minus", 1, "Function('y')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y')(Symbol('W', commutative=True))), exp(Symbol('W', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), exp(Symbol('W', commutative=True)))"], [["add", 3, "Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Add(Mul(Integer(-1), Function('y')(Symbol('W', commutative=True))), exp(Symbol('W', commutative=True)), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('y')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Add(Mul(Integer(-1), Function('y')(Symbol('W', commutative=True))), Derivative(Function('y')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Function('y')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Add(Mul(Integer(-1), exp(Symbol('W', commutative=True))), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(L)} = \\log{(L)} and \\tilde{g}{(L)} = - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\operatorname{C_{2}}^{L}{(L)}, then obtain - L - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\operatorname{C_{2}}^{L}{(L)} = - L - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\log{(L)}^{L}", "derivation": "\\operatorname{C_{2}}{(L)} = \\log{(L)} and \\operatorname{C_{2}}^{L}{(L)} = \\log{(L)}^{L} and \\operatorname{C_{2}}{(L)} \\log{(L)} + \\operatorname{C_{2}}^{L}{(L)} = \\operatorname{C_{2}}{(L)} \\log{(L)} + \\log{(L)}^{L} and - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\operatorname{C_{2}}^{L}{(L)} = - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\log{(L)}^{L} and \\tilde{g}{(L)} = - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\operatorname{C_{2}}^{L}{(L)} and \\tilde{g}{(L)} = - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\log{(L)}^{L} and - L + \\tilde{g}{(L)} = - L - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\log{(L)}^{L} and - L - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\operatorname{C_{2}}^{L}{(L)} = - L - \\operatorname{C_{2}}{(L)} \\log{(L)} - \\log{(L)}^{L}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["add", 2, "Mul(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Pow(Function('C_2')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Add(Mul(Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Add(Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Add(Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], [["minus", 6, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\tilde{g}')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(J)} = e^{J} and \\varepsilon{(J)} = (e^{J})^{J}, then obtain \\varepsilon{(J)} = \\psi^{*}^{J}{(J)}", "derivation": "\\psi^{*}{(J)} = e^{J} and \\psi^{*}^{J}{(J)} = (e^{J})^{J} and \\varepsilon{(J)} = (e^{J})^{J} and \\varepsilon{(J)} = \\psi^{*}^{J}{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varepsilon')(Symbol('J', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given A{(W)} = e^{e^{W}}, then obtain (\\frac{0^{W} A^{W}{(W)}}{W})^{W} = (\\frac{0^{W} (e^{e^{W}})^{W}}{W})^{W}", "derivation": "A{(W)} = e^{e^{W}} and A^{W}{(W)} = (e^{e^{W}})^{W} and 0 = - A^{W}{(W)} + (e^{e^{W}})^{W} and 0^{W} = (- A^{W}{(W)} + (e^{e^{W}})^{W})^{W} and \\frac{(- A^{W}{(W)} + (e^{e^{W}})^{W})^{W} A^{W}{(W)}}{W} = \\frac{(- A^{W}{(W)} + (e^{e^{W}})^{W})^{W} (e^{e^{W}})^{W}}{W} and \\frac{0^{W} A^{W}{(W)}}{W} = \\frac{0^{W} (e^{e^{W}})^{W}}{W} and (\\frac{0^{W} A^{W}{(W)}}{W})^{W} = (\\frac{0^{W} (e^{e^{W}})^{W}}{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('W', commutative=True)), exp(exp(Symbol('W', commutative=True))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["minus", 2, "Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Symbol('W', commutative=True), Integer(-1)), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"], [["power", 6, "Symbol('W', commutative=True)"], "Equality(Pow(Mul(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Function('A')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Mul(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Symbol('W', commutative=True), Integer(-1)), Pow(exp(exp(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{s},\\eta)} = \\cos{(\\eta^{\\mathbf{s}})} and E{(\\mathbf{s},\\eta)} = - \\operatorname{F_{g}}{(\\mathbf{s},\\eta)}, then obtain \\iint \\frac{E{(\\mathbf{s},\\eta)}}{\\operatorname{F_{g}}{(\\mathbf{s},\\eta)}} d\\mathbf{s} d\\eta = \\iint (-1) d\\mathbf{s} d\\eta", "derivation": "\\operatorname{F_{g}}{(\\mathbf{s},\\eta)} = \\cos{(\\eta^{\\mathbf{s}})} and E{(\\mathbf{s},\\eta)} = - \\operatorname{F_{g}}{(\\mathbf{s},\\eta)} and E{(\\mathbf{s},\\eta)} = - \\cos{(\\eta^{\\mathbf{s}})} and \\frac{E{(\\mathbf{s},\\eta)}}{\\cos{(\\eta^{\\mathbf{s}})}} = -1 and \\frac{E{(\\mathbf{s},\\eta)}}{\\operatorname{F_{g}}{(\\mathbf{s},\\eta)}} = -1 and \\int \\frac{E{(\\mathbf{s},\\eta)}}{\\operatorname{F_{g}}{(\\mathbf{s},\\eta)}} d\\mathbf{s} = \\int (-1) d\\mathbf{s} and \\iint \\frac{E{(\\mathbf{s},\\eta)}}{\\operatorname{F_{g}}{(\\mathbf{s},\\eta)}} d\\mathbf{s} d\\eta = \\iint (-1) d\\mathbf{s} d\\eta", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["divide", 3, "cos(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(cos(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Integer(-1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('F_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(-1))"], [["integrate", 5, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Mul(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('F_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 6, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Function('E')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('F_g')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mu{(v_{z})} = \\cos{(e^{v_{z}})}, then obtain - \\mu{(v_{z})} e^{- v_{z}} \\cos^{v_{z}}{(e^{v_{z}})} + \\mu^{v_{z}}{(v_{z})} = - \\mu{(v_{z})} e^{- v_{z}} \\cos^{v_{z}}{(e^{v_{z}})} + \\cos^{v_{z}}{(e^{v_{z}})}", "derivation": "\\mu{(v_{z})} = \\cos{(e^{v_{z}})} and \\mu{(v_{z})} e^{- v_{z}} = e^{- v_{z}} \\cos{(e^{v_{z}})} and \\mu^{v_{z}}{(v_{z})} = \\cos^{v_{z}}{(e^{v_{z}})} and \\mu^{v_{z}}{(v_{z})} - e^{- v_{z}} \\cos{(e^{v_{z}})} \\cos^{v_{z}}{(e^{v_{z}})} = \\cos^{v_{z}}{(e^{v_{z}})} - e^{- v_{z}} \\cos{(e^{v_{z}})} \\cos^{v_{z}}{(e^{v_{z}})} and - \\mu{(v_{z})} e^{- v_{z}} \\cos^{v_{z}}{(e^{v_{z}})} + \\mu^{v_{z}}{(v_{z})} = - \\mu{(v_{z})} e^{- v_{z}} \\cos^{v_{z}}{(e^{v_{z}})} + \\cos^{v_{z}}{(e^{v_{z}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('v_z', commutative=True)), cos(exp(Symbol('v_z', commutative=True))))"], [["divide", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('v_z', commutative=True)), exp(Mul(Integer(-1), Symbol('v_z', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), cos(exp(Symbol('v_z', commutative=True)))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["minus", 3, "Mul(exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), cos(exp(Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mu')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), cos(exp(Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))), Add(Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), cos(exp(Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('v_z', commutative=True)), exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Pow(Function('\\\\mu')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('v_z', commutative=True)), exp(Mul(Integer(-1), Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Pow(cos(exp(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given u{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}, then obtain \\frac{d}{d \\eta^{\\prime}} (3 u{(\\eta^{\\prime})} - 4 \\cos{(\\eta^{\\prime})}) = \\frac{d}{d \\eta^{\\prime}} (u{(\\eta^{\\prime})} - 2 \\cos{(\\eta^{\\prime})})", "derivation": "u{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and u{(\\eta^{\\prime})} - \\cos{(\\eta^{\\prime})} = 0 and u{(\\eta^{\\prime})} - 2 \\cos{(\\eta^{\\prime})} = - \\cos{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} (u{(\\eta^{\\prime})} - 2 \\cos{(\\eta^{\\prime})}) = \\frac{d}{d \\eta^{\\prime}} - \\cos{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} (3 u{(\\eta^{\\prime})} - 4 \\cos{(\\eta^{\\prime})}) = \\frac{d}{d \\eta^{\\prime}} (u{(\\eta^{\\prime})} - 2 \\cos{(\\eta^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integer(0))"], [["minus", 2, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(3), Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Function('u')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(v)} = \\cos{(v)}, then obtain \\frac{d}{d v} ((\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\theta{(v)} + 3)) = \\frac{d}{d v} ((\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\cos{(v)} + 3))", "derivation": "\\theta{(v)} = \\cos{(v)} and \\theta{(v)} + 1 = \\cos{(v)} + 1 and (\\theta{(v)} + 1)^{v} = (\\cos{(v)} + 1)^{v} and \\theta{(v)} + 2 = \\cos{(v)} + 2 and \\theta{(v)} + 3 = \\cos{(v)} + 3 and \\frac{d}{d v} (\\theta{(v)} + 3) = \\frac{d}{d v} (\\cos{(v)} + 3) and (\\theta{(v)} + 1)^{v} + \\frac{d}{d v} (\\theta{(v)} + 3) = (\\theta{(v)} + 1)^{v} + \\frac{d}{d v} (\\cos{(v)} + 3) and (\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\theta{(v)} + 3) = (\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\cos{(v)} + 3) and \\frac{d}{d v} ((\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\theta{(v)} + 3)) = \\frac{d}{d v} ((\\cos{(v)} + 1)^{v} + \\frac{d}{d v} (\\cos{(v)} + 3))", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(1)), Add(cos(Symbol('v', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Pow(Add(cos(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(2)), Add(cos(Symbol('v', commutative=True)), Integer(2)))"], [["add", 4, 1], "Equality(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(3)), Add(cos(Symbol('v', commutative=True)), Integer(3)))"], [["differentiate", 5, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 6, "Pow(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True))"], "Equality(Add(Pow(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Pow(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(cos(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Pow(Add(cos(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Pow(Add(cos(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(cos(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["differentiate", 8, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Pow(Add(cos(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(Function('\\\\theta')(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Pow(Add(cos(Symbol('v', commutative=True)), Integer(1)), Symbol('v', commutative=True)), Derivative(Add(cos(Symbol('v', commutative=True)), Integer(3)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{B},\\theta)} = \\sin{(\\mathbf{B} \\theta)} and \\hat{H}_l{(\\mathbf{B},\\theta)} = \\mathbf{B} \\theta, then obtain \\sin{(\\hat{H}_l{(\\mathbf{B},\\theta)})} = \\sin{(\\mathbf{B} \\theta)}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{B},\\theta)} = \\sin{(\\mathbf{B} \\theta)} and \\hat{H}_l{(\\mathbf{B},\\theta)} = \\mathbf{B} \\theta and \\operatorname{F_{x}}{(\\mathbf{B},\\theta)} = \\sin{(\\hat{H}_l{(\\mathbf{B},\\theta)})} and \\sin{(\\hat{H}_l{(\\mathbf{B},\\theta)})} = \\sin{(\\mathbf{B} \\theta)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), sin(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(sin(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))), sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\psi,I)} = \\log{(I + \\psi)} and k{(\\psi,I)} = \\int \\theta{(\\psi,I)} d\\psi, then obtain \\log{(k{(\\psi,I)})} + 1 = \\log{(\\int \\theta{(\\psi,I)} d\\psi)} + 1", "derivation": "\\theta{(\\psi,I)} = \\log{(I + \\psi)} and k{(\\psi,I)} = \\int \\theta{(\\psi,I)} d\\psi and k{(\\psi,I)} = \\int \\log{(I + \\psi)} d\\psi and \\log{(k{(\\psi,I)})} = \\log{(\\int \\log{(I + \\psi)} d\\psi)} and \\log{(\\int \\theta{(\\psi,I)} d\\psi)} = \\log{(\\int \\log{(I + \\psi)} d\\psi)} and \\log{(k{(\\psi,I)})} + 1 = \\log{(\\int \\log{(I + \\psi)} d\\psi)} + 1 and \\log{(k{(\\psi,I)})} + 1 = \\log{(\\int \\theta{(\\psi,I)} d\\psi)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), log(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Integral(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('k')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Integral(log(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["log", 3], "Equality(log(Function('k')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True))), log(Integral(log(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(log(Integral(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), log(Integral(log(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(log(Function('k')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True))), Integer(1)), Add(log(Integral(log(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(log(Function('k')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True))), Integer(1)), Add(log(Integral(Function('\\\\theta')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\Psi)} = \\sin{(\\Psi)}, then derive \\frac{d}{d \\Psi} \\mathbf{J}_P{(\\Psi)} = \\cos{(\\Psi)}, then obtain \\cos{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)}", "derivation": "\\mathbf{J}_P{(\\Psi)} = \\sin{(\\Psi)} and \\frac{d}{d \\Psi} \\mathbf{J}_P{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)} and \\frac{d}{d \\Psi} \\mathbf{J}_P{(\\Psi)} = \\cos{(\\Psi)} and \\cos{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), cos(Symbol('\\\\Psi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\Psi', commutative=True)), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(\\sigma_p)} = \\sigma_p, then derive E_{n} + \\frac{L^{2}{(\\sigma_p)}}{2} = \\int \\sigma_p dL{(\\sigma_p)}, then obtain (E_{n} + \\frac{\\sigma_p^{2}}{2})^{2} = (\\int \\sigma_p d\\sigma_p)^{2}", "derivation": "L{(\\sigma_p)} = \\sigma_p and \\int L{(\\sigma_p)} d\\sigma_p = \\int \\sigma_p d\\sigma_p and \\int L{(\\sigma_p)} dL{(\\sigma_p)} = \\int \\sigma_p dL{(\\sigma_p)} and E_{n} + \\frac{L^{2}{(\\sigma_p)}}{2} = \\int \\sigma_p dL{(\\sigma_p)} and E_{n} + \\frac{\\sigma_p^{2}}{2} = \\int \\sigma_p d\\sigma_p and (E_{n} + \\frac{\\sigma_p^{2}}{2})^{2} = (\\int \\sigma_p d\\sigma_p)^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Symbol('\\\\sigma_p', commutative=True), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Function('L')(Symbol('\\\\sigma_p', commutative=True)))), Integral(Symbol('\\\\sigma_p', commutative=True), Tuple(Function('L')(Symbol('\\\\sigma_p', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Function('L')(Symbol('\\\\sigma_p', commutative=True)), Integer(2)))), Integral(Symbol('\\\\sigma_p', commutative=True), Tuple(Function('L')(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)))), Integral(Symbol('\\\\sigma_p', commutative=True), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 5, 2], "Equality(Pow(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)))), Integer(2)), Pow(Integral(Symbol('\\\\sigma_p', commutative=True), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\hat{x}_0{(y^{\\prime})} = e^{y^{\\prime}} and \\rho_{f}{(g,y^{\\prime})} = g + e^{y^{\\prime}}, then derive \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = g + e^{y^{\\prime}}, then obtain \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = \\rho_{f}{(g,y^{\\prime})}", "derivation": "\\hat{x}_0{(y^{\\prime})} = e^{y^{\\prime}} and \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = \\int e^{y^{\\prime}} dy^{\\prime} and \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = g + e^{y^{\\prime}} and \\rho_{f}{(g,y^{\\prime})} = g + e^{y^{\\prime}} and \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = g + \\hat{x}_0{(y^{\\prime})} and \\rho_{f}{(g,y^{\\prime})} = g + \\hat{x}_0{(y^{\\prime})} and \\int \\hat{x}_0{(y^{\\prime})} dy^{\\prime} = \\rho_{f}{(g,y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('g', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('g', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('g', commutative=True), Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('g', commutative=True), Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given l{(z,t)} = \\cos{(t^{z})}, then derive \\frac{\\partial}{\\partial z} l{(z,t)} = - t^{z} \\log{(t)} \\sin{(t^{z})}, then obtain \\frac{- t^{z} \\log{(t)} \\sin{(t^{z})} - \\frac{\\partial}{\\partial z} \\cos{(t^{z})}}{\\operatorname{v_{2}}{(E,\\mathbf{D})}} = 0", "derivation": "l{(z,t)} = \\cos{(t^{z})} and \\frac{\\partial}{\\partial z} l{(z,t)} = \\frac{\\partial}{\\partial z} \\cos{(t^{z})} and \\frac{\\partial}{\\partial z} l{(z,t)} - 1 = \\frac{\\partial}{\\partial z} \\cos{(t^{z})} - 1 and \\frac{\\partial}{\\partial z} l{(z,t)} - \\frac{\\partial}{\\partial z} \\cos{(t^{z})} = 0 and \\frac{\\partial}{\\partial z} l{(z,t)} = - t^{z} \\log{(t)} \\sin{(t^{z})} and - t^{z} \\log{(t)} \\sin{(t^{z})} - \\frac{\\partial}{\\partial z} \\cos{(t^{z})} = 0 and \\frac{- t^{z} \\log{(t)} \\sin{(t^{z})} - \\frac{\\partial}{\\partial z} \\cos{(t^{z})}}{\\operatorname{v_{2}}{(E,\\mathbf{D})}} = 0", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('z', commutative=True), Symbol('t', commutative=True)), cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('l')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 3, "Add(Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Function('l')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('l')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)), log(Symbol('t', commutative=True)), sin(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)), log(Symbol('t', commutative=True)), sin(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)))), Mul(Integer(-1), Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))), Integer(0))"], [["divide", 6, "Function('v_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)), log(Symbol('t', commutative=True)), sin(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True)))), Mul(Integer(-1), Derivative(cos(Pow(Symbol('t', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))), Pow(Function('v_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\phi_2,M)} = M \\phi_2, then derive \\frac{\\phi_2 \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} \\frac{\\partial}{\\partial M} \\operatorname{f_{E}}{(\\phi_2,M)}}{\\operatorname{f_{E}}{(\\phi_2,M)}} = \\frac{\\phi_2 (M \\phi_2)^{\\phi_2}}{M}, then obtain \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2 \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} \\frac{\\partial}{\\partial M} \\operatorname{f_{E}}{(\\phi_2,M)}}{\\operatorname{f_{E}}{(\\phi_2,M)}} = \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2 (M \\phi_2)^{\\phi_2}}{M}", "derivation": "\\operatorname{f_{E}}{(\\phi_2,M)} = M \\phi_2 and \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} = (M \\phi_2)^{\\phi_2} and \\frac{\\partial}{\\partial M} \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} = \\frac{\\partial}{\\partial M} (M \\phi_2)^{\\phi_2} and \\frac{\\phi_2 \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} \\frac{\\partial}{\\partial M} \\operatorname{f_{E}}{(\\phi_2,M)}}{\\operatorname{f_{E}}{(\\phi_2,M)}} = \\frac{\\phi_2 (M \\phi_2)^{\\phi_2}}{M} and \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2 \\operatorname{f_{E}}^{\\phi_2}{(\\phi_2,M)} \\frac{\\partial}{\\partial M} \\operatorname{f_{E}}{(\\phi_2,M)}}{\\operatorname{f_{E}}{(\\phi_2,M)}} = \\frac{\\partial}{\\partial \\phi_2} \\frac{\\phi_2 (M \\phi_2)^{\\phi_2}}{M}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Derivative(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Derivative(Function('f_E')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(\\dot{\\mathbf{r}},f^{\\prime})} = \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}} and \\hat{H}_l{(f^{\\prime})} = f^{\\prime}, then obtain \\hat{H}_l{(f^{\\prime})} - \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}} = f^{\\prime} - \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}}", "derivation": "\\phi_{2}{(\\dot{\\mathbf{r}},f^{\\prime})} = \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}} and \\hat{H}_l{(f^{\\prime})} = f^{\\prime} and \\hat{H}_l{(f^{\\prime})} - \\phi_{2}{(\\dot{\\mathbf{r}},f^{\\prime})} = f^{\\prime} - \\phi_{2}{(\\dot{\\mathbf{r}},f^{\\prime})} and \\hat{H}_l{(f^{\\prime})} - \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}} = f^{\\prime} - \\frac{f^{\\prime}}{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["minus", 2, "Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\hat{x}_0,\\theta_1)} = \\log{(\\hat{x}_0 + \\theta_1)}, then obtain \\frac{\\hat{x}_0 (\\theta_{2}{(\\hat{x}_0,\\theta_1)} - \\log{(\\hat{x}_0 + \\theta_1)})}{4 \\theta_{2}^{2}{(\\hat{x}_0,\\theta_1)}} = 0", "derivation": "\\theta_{2}{(\\hat{x}_0,\\theta_1)} = \\log{(\\hat{x}_0 + \\theta_1)} and 2 \\theta_{2}{(\\hat{x}_0,\\theta_1)} = \\theta_{2}{(\\hat{x}_0,\\theta_1)} + \\log{(\\hat{x}_0 + \\theta_1)} and \\theta_{2}{(\\hat{x}_0,\\theta_1)} - \\log{(\\hat{x}_0 + \\theta_1)} = 0 and \\frac{\\theta_{2}{(\\hat{x}_0,\\theta_1)} - \\log{(\\hat{x}_0 + \\theta_1)}}{2 \\theta_{2}{(\\hat{x}_0,\\theta_1)}} = 0 and \\frac{\\hat{x}_0 (\\theta_{2}{(\\hat{x}_0,\\theta_1)} - \\log{(\\hat{x}_0 + \\theta_1)})}{2 \\theta_{2}{(\\hat{x}_0,\\theta_1)}} = 0 and \\frac{\\hat{x}_0 (\\theta_{2}{(\\hat{x}_0,\\theta_1)} - \\log{(\\hat{x}_0 + \\theta_1)})}{4 \\theta_{2}^{2}{(\\hat{x}_0,\\theta_1)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["add", 1, "Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 2, "Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Integer(0))"], [["divide", 3, "Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Pow(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Integer(0))"], [["times", 4, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Rational(1, 2), Symbol('\\\\hat{x}_0', commutative=True), Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Pow(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 5, "Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Rational(1, 4), Symbol('\\\\hat{x}_0', commutative=True), Add(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Pow(Function('\\\\theta_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-2))), Integer(0))"]]}, {"prompt": "Given \\varepsilon{(s,\\Psi_{nl})} = \\Psi_{nl} s, then obtain \\Psi_{nl} s^{2} - \\Psi_{nl} s + s \\varepsilon{(s,\\Psi_{nl})} = 2 \\Psi_{nl} s^{2} - \\Psi_{nl} s", "derivation": "\\varepsilon{(s,\\Psi_{nl})} = \\Psi_{nl} s and s \\varepsilon{(s,\\Psi_{nl})} = \\Psi_{nl} s^{2} and \\Psi_{nl} s^{2} + s \\varepsilon{(s,\\Psi_{nl})} = 2 \\Psi_{nl} s^{2} and \\Psi_{nl} s^{2} - \\Psi_{nl} s + s \\varepsilon{(s,\\Psi_{nl})} = 2 \\Psi_{nl} s^{2} - \\Psi_{nl} s", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('s', commutative=True)))"], [["times", 1, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Function('\\\\varepsilon')(Symbol('s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2))))"], [["add", 2, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2)))"], "Equality(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Symbol('s', commutative=True), Function('\\\\varepsilon')(Symbol('s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2))))"], [["minus", 3, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Function('\\\\varepsilon')(Symbol('s', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(x)} = \\log{(\\sin{(x)})}, then obtain (-1)^{x} = (- \\frac{\\log{(\\sin{(x)})}}{\\operatorname{M_{E}}{(x)}})^{x}", "derivation": "\\operatorname{M_{E}}{(x)} = \\log{(\\sin{(x)})} and 1 = \\frac{\\log{(\\sin{(x)})}}{\\operatorname{M_{E}}{(x)}} and -1 = - \\frac{\\log{(\\sin{(x)})}}{\\operatorname{M_{E}}{(x)}} and (-1)^{x} = (- \\frac{\\log{(\\sin{(x)})}}{\\operatorname{M_{E}}{(x)}})^{x}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('x', commutative=True)), log(sin(Symbol('x', commutative=True))))"], [["divide", 1, "Function('M_E')(Symbol('x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('M_E')(Symbol('x', commutative=True)), Integer(-1)), log(sin(Symbol('x', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True)), Integer(-1)), log(sin(Symbol('x', commutative=True)))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('x', commutative=True)), Pow(Mul(Integer(-1), Pow(Function('M_E')(Symbol('x', commutative=True)), Integer(-1)), log(sin(Symbol('x', commutative=True)))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given z{(\\hat{x},\\hbar)} = \\hat{x} + \\hbar, then derive \\frac{\\partial}{\\partial \\hbar} \\int z{(\\hat{x},\\hbar)} d\\hat{x} = \\frac{\\partial}{\\partial \\hbar} (\\frac{\\hat{x}^{2}}{2} + \\hat{x} \\hbar + c), then obtain \\frac{\\partial}{\\partial \\hbar} \\int (\\hat{x} + \\hbar) d\\hat{x} = \\frac{\\partial}{\\partial \\hbar} (\\frac{\\hat{x}^{2}}{2} + \\hat{x} \\hbar + c)", "derivation": "z{(\\hat{x},\\hbar)} = \\hat{x} + \\hbar and \\int z{(\\hat{x},\\hbar)} d\\hat{x} = \\int (\\hat{x} + \\hbar) d\\hat{x} and \\frac{\\partial}{\\partial \\hbar} \\int z{(\\hat{x},\\hbar)} d\\hat{x} = \\frac{\\partial}{\\partial \\hbar} \\int (\\hat{x} + \\hbar) d\\hat{x} and \\frac{\\partial}{\\partial \\hbar} \\int z{(\\hat{x},\\hbar)} d\\hat{x} = \\frac{\\partial}{\\partial \\hbar} (\\frac{\\hat{x}^{2}}{2} + \\hat{x} \\hbar + c) and \\frac{\\partial}{\\partial \\hbar} \\int (\\hat{x} + \\hbar) d\\hat{x} = \\frac{\\partial}{\\partial \\hbar} (\\frac{\\hat{x}^{2}}{2} + \\hat{x} \\hbar + c)", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Integral(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(i)} = i, then obtain 2 \\mathbf{f}{(i)} \\frac{d}{d i} \\mathbf{f}{(i)} = i \\frac{d}{d i} \\mathbf{f}{(i)} + \\mathbf{f}{(i)}", "derivation": "\\mathbf{f}{(i)} = i and \\mathbf{f}^{2}{(i)} = i \\mathbf{f}{(i)} and \\frac{d}{d i} \\mathbf{f}^{2}{(i)} = \\frac{d}{d i} i \\mathbf{f}{(i)} and 2 \\mathbf{f}{(i)} \\frac{d}{d i} \\mathbf{f}{(i)} = i \\frac{d}{d i} \\mathbf{f}{(i)} + \\mathbf{f}{(i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Symbol('i', commutative=True))"], [["times", 1, "Function('\\\\mathbf{f}')(Symbol('i', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Integer(2)), Mul(Symbol('i', commutative=True), Function('\\\\mathbf{f}')(Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Integer(2)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Symbol('i', commutative=True), Function('\\\\mathbf{f}')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Symbol('i', commutative=True), Derivative(Function('\\\\mathbf{f}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Function('\\\\mathbf{f}')(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\omega{(y,\\dot{y})} = \\dot{y} y, then derive \\frac{\\partial}{\\partial y} \\omega{(y,\\dot{y})} = \\dot{y}, then obtain \\iint (\\dot{y} y - 2 \\dot{y} + \\omega{(y,\\dot{y})}) d\\dot{y} dy = \\iint (2 \\dot{y} y - 2 \\dot{y}) d\\dot{y} dy", "derivation": "\\omega{(y,\\dot{y})} = \\dot{y} y and \\frac{\\partial}{\\partial y} \\omega{(y,\\dot{y})} = \\frac{\\partial}{\\partial y} \\dot{y} y and \\frac{\\partial}{\\partial y} \\omega{(y,\\dot{y})} = \\dot{y} and \\omega{(y,\\dot{y})} - \\frac{\\partial}{\\partial y} \\omega{(y,\\dot{y})} = \\dot{y} y - \\frac{\\partial}{\\partial y} \\omega{(y,\\dot{y})} and - \\dot{y} + \\omega{(y,\\dot{y})} = \\dot{y} y - \\dot{y} and \\dot{y} y - 2 \\dot{y} + \\omega{(y,\\dot{y})} = 2 \\dot{y} y - 2 \\dot{y} and \\int (\\dot{y} y - 2 \\dot{y} + \\omega{(y,\\dot{y})}) d\\dot{y} = \\int (2 \\dot{y} y - 2 \\dot{y}) d\\dot{y} and \\iint (\\dot{y} y - 2 \\dot{y} + \\omega{(y,\\dot{y})}) d\\dot{y} dy = \\iint (2 \\dot{y} y - 2 \\dot{y}) d\\dot{y} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True))"], [["minus", 1, "Derivative(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], [["add", 5, "Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 6, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 7, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\omega')(Symbol('y', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} = V_{\\mathbf{B}} - W and \\operatorname{t_{1}}{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}}, then obtain W + 2 \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} + \\operatorname{t_{1}}{(V_{\\mathbf{B}})} - 1 = V_{\\mathbf{B}} - W - 1", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} = V_{\\mathbf{B}} - W and - V_{\\mathbf{B}} + W + \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} = 0 and \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} - 1 = V_{\\mathbf{B}} - W - 1 and - V_{\\mathbf{B}} + W + 2 \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} - 1 = \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} - 1 and - V_{\\mathbf{B}} + W + 2 \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} - 1 = V_{\\mathbf{B}} - W - 1 and \\operatorname{t_{1}}{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} and W + 2 \\operatorname{V_{\\mathbf{E}}}{(V_{\\mathbf{B}},W)} + \\operatorname{t_{1}}{(V_{\\mathbf{B}})} - 1 = V_{\\mathbf{B}} - W - 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["minus", 1, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('W', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True))), Integer(0))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(-1)))"], [["add", 2, "Add(Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('W', commutative=True), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True))), Integer(-1)), Add(Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('W', commutative=True), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True))), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('W', commutative=True))), Function('t_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(f^{\\prime},m)} = f^{\\prime} m and \\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})} = f^{\\prime}, then obtain \\frac{(f^{\\prime} m - m) \\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})}}{\\Psi^{\\dagger}} = \\frac{f^{\\prime} (f^{\\prime} m - m)}{\\Psi^{\\dagger}}", "derivation": "\\operatorname{v_{t}}{(f^{\\prime},m)} = f^{\\prime} m and - m + \\operatorname{v_{t}}{(f^{\\prime},m)} = f^{\\prime} m - m and \\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})} = f^{\\prime} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})}}{\\Psi^{\\dagger}} = \\frac{f^{\\prime}}{\\Psi^{\\dagger}} and \\frac{(- m + \\operatorname{v_{t}}{(f^{\\prime},m)}) \\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})}}{\\Psi^{\\dagger}} = \\frac{f^{\\prime} (- m + \\operatorname{v_{t}}{(f^{\\prime},m)})}{\\Psi^{\\dagger}} and \\frac{(f^{\\prime} m - m) \\operatorname{V_{\\mathbf{E}}}{(f^{\\prime})}}{\\Psi^{\\dagger}} = \\frac{f^{\\prime} (f^{\\prime} m - m)}{\\Psi^{\\dagger}}", "srepr_derivation": [["get_premise", "Equality(Function('v_t')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_t')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["divide", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_t')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_t')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_t')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} = \\hat{x}_0^{\\varepsilon_0}, then obtain (\\frac{\\partial}{\\partial \\hat{x}_0} \\int \\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} d\\varepsilon_0) \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0 = (\\frac{\\partial}{\\partial \\hat{x}_0} \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0) \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0", "derivation": "\\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} = \\hat{x}_0^{\\varepsilon_0} and \\int \\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} d\\varepsilon_0 = \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0 and \\frac{\\partial}{\\partial \\hat{x}_0} \\int \\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} d\\varepsilon_0 = \\frac{\\partial}{\\partial \\hat{x}_0} \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0 and (\\frac{\\partial}{\\partial \\hat{x}_0} \\int \\operatorname{A_{1}}{(\\hat{x}_0,\\varepsilon_0)} d\\varepsilon_0) \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0 = (\\frac{\\partial}{\\partial \\hat{x}_0} \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0) \\int \\hat{x}_0^{\\varepsilon_0} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Integral(Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["times", 3, "Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Derivative(Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\sigma_x,v_{y})} = - \\sigma_x + v_{y}, then obtain \\sigma_x = \\frac{\\sigma_x \\int (- \\sigma_x + 2 v_{y} + 1) d\\sigma_x}{\\int (v_{y} + \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1) d\\sigma_x}", "derivation": "\\operatorname{F_{H}}{(\\sigma_x,v_{y})} = - \\sigma_x + v_{y} and \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1 = - \\sigma_x + v_{y} + 1 and v_{y} + \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1 = - \\sigma_x + 2 v_{y} + 1 and \\int (v_{y} + \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1) d\\sigma_x = \\int (- \\sigma_x + 2 v_{y} + 1) d\\sigma_x and \\sigma_x \\int (v_{y} + \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1) d\\sigma_x = \\sigma_x \\int (- \\sigma_x + 2 v_{y} + 1) d\\sigma_x and \\sigma_x = \\frac{\\sigma_x \\int (- \\sigma_x + 2 v_{y} + 1) d\\sigma_x}{\\int (v_{y} + \\operatorname{F_{H}}{(\\sigma_x,v_{y})} + 1) d\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_y', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_y', commutative=True), Integer(1)))"], [["add", 2, "Symbol('v_y', commutative=True)"], "Equality(Add(Symbol('v_y', commutative=True), Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)), Integer(1)))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Symbol('v_y', commutative=True), Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Add(Symbol('v_y', commutative=True), Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["divide", 5, "Integral(Add(Symbol('v_y', commutative=True), Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Symbol('\\\\sigma_x', commutative=True), Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Pow(Integral(Add(Symbol('v_y', commutative=True), Function('F_H')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_y', commutative=True)), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given M{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain \\frac{M^{3}{(E_{\\lambda})}}{\\log{(E_{\\lambda})}^{3}} = \\frac{M{(E_{\\lambda})}}{\\log{(E_{\\lambda})}}", "derivation": "M{(E_{\\lambda})} = \\log{(E_{\\lambda})} and 1 = \\frac{\\log{(E_{\\lambda})}}{M{(E_{\\lambda})}} and \\frac{M{(E_{\\lambda})}}{\\log{(E_{\\lambda})}} = 1 and \\log{(E_{\\lambda})} = \\frac{\\log{(E_{\\lambda})}^{2}}{M{(E_{\\lambda})}} and \\frac{M^{2}{(E_{\\lambda})}}{\\log{(E_{\\lambda})}^{2}} = 1 and \\frac{M^{3}{(E_{\\lambda})}}{\\log{(E_{\\lambda})}^{3}} = \\frac{M{(E_{\\lambda})}}{\\log{(E_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Function('M')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 1, "log(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))"], "Equality(log(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-2))), Integer(1))"], [["times", 5, "Mul(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Integer(3)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-3))), Mul(Function('M')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(log(Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})} = \\dot{z} \\theta_2 l, then obtain \\frac{(\\dot{z} + \\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})}) \\cos{(\\dot{z} \\theta_2 l + \\dot{z})}}{\\dot{z} \\theta_2 l} = \\frac{(\\dot{z} \\theta_2 l + \\dot{z}) \\cos{(\\dot{z} \\theta_2 l + \\dot{z})}}{\\dot{z} \\theta_2 l}", "derivation": "\\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})} = \\dot{z} \\theta_2 l and \\dot{z} + \\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})} = \\dot{z} \\theta_2 l + \\dot{z} and \\frac{\\dot{z} + \\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})}}{\\dot{z} \\theta_2 l} = \\frac{\\dot{z} \\theta_2 l + \\dot{z}}{\\dot{z} \\theta_2 l} and \\frac{(\\dot{z} + \\operatorname{A_{x}}{(\\theta_2,l,\\dot{z})}) \\cos{(\\dot{z} \\theta_2 l + \\dot{z})}}{\\dot{z} \\theta_2 l} = \\frac{(\\dot{z} \\theta_2 l + \\dot{z}) \\cos{(\\dot{z} \\theta_2 l + \\dot{z})}}{\\dot{z} \\theta_2 l}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('A_x')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Function('A_x')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["times", 3, "cos(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Function('A_x')(Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), cos(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), cos(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('l', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given c{(s)} = \\cos{(e^{s})} and \\hat{x}_0{(s)} = \\iint c{(s)} ds ds, then obtain \\frac{d}{d s} \\hat{x}_0{(s)} = \\frac{d}{d s} \\iint \\cos{(e^{s})} ds ds", "derivation": "c{(s)} = \\cos{(e^{s})} and \\int c{(s)} ds = \\int \\cos{(e^{s})} ds and \\iint c{(s)} ds ds = \\iint \\cos{(e^{s})} ds ds and \\hat{x}_0{(s)} = \\iint c{(s)} ds ds and \\hat{x}_0{(s)} = \\iint \\cos{(e^{s})} ds ds and \\frac{d}{d s} \\hat{x}_0{(s)} = \\frac{d}{d s} \\iint \\cos{(e^{s})} ds ds", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('s', commutative=True)), cos(exp(Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('c')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Function('c')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('s', commutative=True)), Integral(Function('c')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\hat{x}_0')(Symbol('s', commutative=True)), Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 5, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(\\phi_2)} = \\sin{(\\phi_2)}, then obtain \\cos{(0^{\\phi_2} \\phi_2)} = \\cos{(\\phi_2)}", "derivation": "\\omega{(\\phi_2)} = \\sin{(\\phi_2)} and 0 = - \\omega{(\\phi_2)} + \\sin{(\\phi_2)} and 0^{\\phi_2} = (- \\omega{(\\phi_2)} + \\sin{(\\phi_2)})^{\\phi_2} and 0^{\\phi_2} \\phi_2 = \\phi_2 (- \\omega{(\\phi_2)} + \\sin{(\\phi_2)})^{\\phi_2} and \\phi_2 (- \\omega{(\\phi_2)} + \\sin{(\\phi_2)})^{\\phi_2} = \\phi_2 and \\cos{(0^{\\phi_2} \\phi_2)} = \\cos{(\\phi_2 (- \\omega{(\\phi_2)} + \\sin{(\\phi_2)})^{\\phi_2})} and \\cos{(0^{\\phi_2} \\phi_2)} = \\cos{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["times", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], [["cos", 4], "Equality(cos(Mul(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), cos(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), sin(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(cos(Mul(Pow(Integer(0), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(C_{d})} = \\cos{(C_{d})}, then obtain (\\frac{d}{d C_{d}} \\operatorname{P_{e}}{(C_{d})})^{C_{d}} = (- \\sin{(C_{d})})^{C_{d}}", "derivation": "\\operatorname{P_{e}}{(C_{d})} = \\cos{(C_{d})} and \\frac{d}{d C_{d}} \\operatorname{P_{e}}{(C_{d})} = \\frac{d}{d C_{d}} \\cos{(C_{d})} and (\\frac{d}{d C_{d}} \\operatorname{P_{e}}{(C_{d})})^{C_{d}} = (\\frac{d}{d C_{d}} \\cos{(C_{d})})^{C_{d}} and (\\frac{d}{d C_{d}} \\operatorname{P_{e}}{(C_{d})})^{C_{d}} = (- \\sin{(C_{d})})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Derivative(Function('P_e')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Pow(Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('P_e')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{J}_P,\\sigma_p)} = \\sigma_p + \\sin{(\\mathbf{J}_P)} and \\operatorname{y^{\\prime}}{(\\mathbf{J}_P,\\sigma_p)} = \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{y^{\\prime}}{(\\mathbf{J}_P,\\sigma_p)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{J}_P,\\sigma_p)} = \\sigma_p + \\sin{(\\mathbf{J}_P)} and \\cos{(\\operatorname{A_{y}}{(\\mathbf{J}_P,\\sigma_p)})} = \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\cos{(\\operatorname{A_{y}}{(\\mathbf{J}_P,\\sigma_p)})} = \\frac{\\partial}{\\partial \\mathbf{J}_P} \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})} and \\operatorname{y^{\\prime}}{(\\mathbf{J}_P,\\sigma_p)} = \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})} and \\operatorname{y^{\\prime}}{(\\mathbf{J}_P,\\sigma_p)} = \\cos{(\\operatorname{A_{y}}{(\\mathbf{J}_P,\\sigma_p)})} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{y^{\\prime}}{(\\mathbf{J}_P,\\sigma_p)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} \\cos{(\\sigma_p + \\sin{(\\mathbf{J}_P)})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["cos", 1], "Equality(cos(Function('A_y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True))), cos(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(cos(Function('A_y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True)), cos(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True)), cos(Function('A_y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{A},\\Omega)} = \\Omega \\mathbf{A}, then obtain \\mathbf{A} (\\Omega \\mathbf{A} + \\hat{x}_0{(\\mathbf{A},\\Omega)}) \\hat{x}_0{(\\mathbf{A},\\Omega)} = 2 \\Omega \\mathbf{A}^{2} \\hat{x}_0{(\\mathbf{A},\\Omega)}", "derivation": "\\hat{x}_0{(\\mathbf{A},\\Omega)} = \\Omega \\mathbf{A} and \\Omega \\mathbf{A} + \\hat{x}_0{(\\mathbf{A},\\Omega)} = 2 \\Omega \\mathbf{A} and (\\Omega \\mathbf{A} + \\hat{x}_0{(\\mathbf{A},\\Omega)}) \\hat{x}_0{(\\mathbf{A},\\Omega)} = 2 \\Omega \\mathbf{A} \\hat{x}_0{(\\mathbf{A},\\Omega)} and \\mathbf{A} (\\Omega \\mathbf{A} + \\hat{x}_0{(\\mathbf{A},\\Omega)}) \\hat{x}_0{(\\mathbf{A},\\Omega)} = 2 \\Omega \\mathbf{A}^{2} \\hat{x}_0{(\\mathbf{A},\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 2, "Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["divide", 3, "Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(A_{x})} = \\cos{(A_{x})}, then obtain (A_{x} \\cos^{A_{x}}{(\\hat{x}_0{(A_{x})})})^{A_{x}} = (A_{x} \\cos^{A_{x}}{(\\cos{(A_{x})})})^{A_{x}}", "derivation": "\\hat{x}_0{(A_{x})} = \\cos{(A_{x})} and \\cos{(\\hat{x}_0{(A_{x})})} = \\cos{(\\cos{(A_{x})})} and \\cos^{A_{x}}{(\\hat{x}_0{(A_{x})})} = \\cos^{A_{x}}{(\\cos{(A_{x})})} and A_{x} \\cos^{A_{x}}{(\\hat{x}_0{(A_{x})})} = A_{x} \\cos^{A_{x}}{(\\cos{(A_{x})})} and (A_{x} \\cos^{A_{x}}{(\\hat{x}_0{(A_{x})})})^{A_{x}} = (A_{x} \\cos^{A_{x}}{(\\cos{(A_{x})})})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True))), cos(cos(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(cos(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(cos(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["times", 3, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Pow(cos(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Mul(Symbol('A_x', commutative=True), Pow(cos(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], [["power", 4, "Symbol('A_x', commutative=True)"], "Equality(Pow(Mul(Symbol('A_x', commutative=True), Pow(cos(Function('\\\\hat{x}_0')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Mul(Symbol('A_x', commutative=True), Pow(cos(cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given U{(\\chi)} = \\log{(\\chi)} and \\theta_{1}{(E_{\\lambda},\\theta)} = \\frac{\\theta}{E_{\\lambda}}, then obtain - U{(\\chi)} + \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} \\log{(\\chi)} + \\log{(\\chi)} = (\\frac{\\theta}{E_{\\lambda}})^{\\theta} \\log{(\\chi)}", "derivation": "U{(\\chi)} = \\log{(\\chi)} and 0 = - U{(\\chi)} + \\log{(\\chi)} and \\theta_{1}{(E_{\\lambda},\\theta)} = \\frac{\\theta}{E_{\\lambda}} and \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} = (\\frac{\\theta}{E_{\\lambda}})^{\\theta} and \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} \\log{(\\chi)} = (\\frac{\\theta}{E_{\\lambda}})^{\\theta} \\log{(\\chi)} and \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} \\log{(\\chi)} = - U{(\\chi)} + \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} \\log{(\\chi)} + \\log{(\\chi)} and - U{(\\chi)} + \\theta_{1}^{\\theta}{(E_{\\lambda},\\theta)} \\log{(\\chi)} + \\log{(\\chi)} = (\\frac{\\theta}{E_{\\lambda}})^{\\theta} \\log{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Function('U')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('U')(Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["times", 4, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Mul(Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Mul(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Function('U')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Function('U')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Function('\\\\theta_1')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True))), Mul(Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(q)} = \\sin{(q)}, then derive 1 = \\frac{F_{c} - \\cos{(q)}}{\\int \\operatorname{L_{\\varepsilon}}{(q)} dq}, then derive 0 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\int \\sin{(q)} dq}, then obtain 0 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\dot{z} - \\cos{(q)}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(q)} = \\sin{(q)} and \\int \\operatorname{L_{\\varepsilon}}{(q)} dq = \\int \\sin{(q)} dq and 1 = \\frac{\\int \\sin{(q)} dq}{\\int \\operatorname{L_{\\varepsilon}}{(q)} dq} and 1 = \\frac{F_{c} - \\cos{(q)}}{\\int \\operatorname{L_{\\varepsilon}}{(q)} dq} and \\frac{d}{d F_{c}} 1 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\int \\operatorname{L_{\\varepsilon}}{(q)} dq} and \\frac{d}{d F_{c}} 1 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\int \\sin{(q)} dq} and 0 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\int \\sin{(q)} dq} and 0 = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c} - \\cos{(q)}}{\\dot{z} - \\cos{(q)}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["divide", 2, "Integral(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1)), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))))"], [["differentiate", 4, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Pow(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Derivative(Mul(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Pow(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_integrals", 7], "Equality(Integer(0), Derivative(Mul(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('q', commutative=True)))), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(C_{2})} = \\log{(C_{2})}, then obtain M{(C_{2})} + M^{C_{2}}{(C_{2})} + \\log{(C_{2})} = M{(C_{2})} + \\log{(C_{2})} + \\log{(C_{2})}^{C_{2}}", "derivation": "M{(C_{2})} = \\log{(C_{2})} and M^{C_{2}}{(C_{2})} = \\log{(C_{2})}^{C_{2}} and 2 M{(C_{2})} = M{(C_{2})} + \\log{(C_{2})} and 2 M{(C_{2})} + M^{C_{2}}{(C_{2})} = 2 M{(C_{2})} + \\log{(C_{2})}^{C_{2}} and M{(C_{2})} + M^{C_{2}}{(C_{2})} + \\log{(C_{2})} = M{(C_{2})} + \\log{(C_{2})} + \\log{(C_{2})}^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('M')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["add", 1, "Function('M')(Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(2), Function('M')(Symbol('C_2', commutative=True))), Add(Function('M')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('M')(Symbol('C_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('M')(Symbol('C_2', commutative=True))), Pow(Function('M')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Mul(Integer(2), Function('M')(Symbol('C_2', commutative=True))), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('M')(Symbol('C_2', commutative=True)), Pow(Function('M')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True))), Add(Function('M')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given B{(\\psi^*)} = \\psi^*, then obtain \\frac{2 (B{(\\psi^*)} - \\int \\psi^* d\\psi^*)}{(\\psi^*)^{2}} = \\frac{2 (\\psi^* - \\int \\psi^* d\\psi^*)}{(\\psi^*)^{2}}", "derivation": "B{(\\psi^*)} = \\psi^* and \\int B{(\\psi^*)} d\\psi^* = \\int \\psi^* d\\psi^* and B{(\\psi^*)} - \\int B{(\\psi^*)} d\\psi^* = \\psi^* - \\int B{(\\psi^*)} d\\psi^* and B{(\\psi^*)} - \\int \\psi^* d\\psi^* = \\psi^* - \\int \\psi^* d\\psi^* and \\frac{2 (B{(\\psi^*)} - \\int \\psi^* d\\psi^*)}{(\\psi^*)^{2}} = \\frac{2 (\\psi^* - \\int \\psi^* d\\psi^*)}{(\\psi^*)^{2}}", "srepr_derivation": [["renaming_premise", "Equality(Function('B')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 1, "Integral(Function('B')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Function('B')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integral(Function('B')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Integral(Function('B')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('B')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))))))"], [["divide", 4, "Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Add(Function('B')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True)))))), Mul(Integer(2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True)))))))"]]}, {"prompt": "Given \\rho{(T,\\mathbf{f})} = \\int (- T + \\mathbf{f}) d\\mathbf{f} and \\rho_{b}{(\\sigma_x,v_{t})} = \\cos{(\\sigma_x + v_{t})}, then derive - \\rho{(T,\\mathbf{f})} = T \\mathbf{f} - \\hat{p} - \\frac{\\mathbf{f}^{2}}{2}, then obtain - \\rho{(T,\\mathbf{f})} \\rho_{b}{(\\sigma_x,v_{t})} = (T \\mathbf{f} - \\hat{p} - \\frac{\\mathbf{f}^{2}}{2}) \\rho_{b}{(\\sigma_x,v_{t})}", "derivation": "\\rho{(T,\\mathbf{f})} = \\int (- T + \\mathbf{f}) d\\mathbf{f} and - \\rho{(T,\\mathbf{f})} = - \\int (- T + \\mathbf{f}) d\\mathbf{f} and - \\rho{(T,\\mathbf{f})} = T \\mathbf{f} - \\hat{p} - \\frac{\\mathbf{f}^{2}}{2} and \\rho_{b}{(\\sigma_x,v_{t})} = \\cos{(\\sigma_x + v_{t})} and - \\rho{(T,\\mathbf{f})} \\cos{(\\sigma_x + v_{t})} = (T \\mathbf{f} - \\hat{p} - \\frac{\\mathbf{f}^{2}}{2}) \\cos{(\\sigma_x + v_{t})} and - \\rho{(T,\\mathbf{f})} \\rho_{b}{(\\sigma_x,v_{t})} = (T \\mathbf{f} - \\hat{p} - \\frac{\\mathbf{f}^{2}}{2}) \\rho_{b}{(\\sigma_x,v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho')(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Integer(-1), Function('\\\\rho')(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)))))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True)), cos(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True))))"], [["times", 3, "cos(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\rho')(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True)))), Mul(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)))), cos(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Function('\\\\rho')(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True))), Mul(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)))), Function('\\\\rho_b')(Symbol('\\\\sigma_x', commutative=True), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\mathbf{M},H)} = \\log{(H - \\mathbf{M})}, then derive \\int \\phi{(\\mathbf{M},H)} d\\mathbf{M} = - H \\log{(- H + \\mathbf{M})} + \\Psi_{nl} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M}, then derive - H \\log{(- H + \\mathbf{M})} + \\Psi_{nl} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} = - H \\log{(- H + \\mathbf{M})} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} + \\phi_1, then obtain - H \\log{(- H + \\mathbf{M})} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} + \\phi_1 = \\int \\log{(H - \\mathbf{M})} d\\mathbf{M}", "derivation": "\\phi{(\\mathbf{M},H)} = \\log{(H - \\mathbf{M})} and \\int \\phi{(\\mathbf{M},H)} d\\mathbf{M} = \\int \\log{(H - \\mathbf{M})} d\\mathbf{M} and \\int \\phi{(\\mathbf{M},H)} d\\mathbf{M} = - H \\log{(- H + \\mathbf{M})} + \\Psi_{nl} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} and - H \\log{(- H + \\mathbf{M})} + \\Psi_{nl} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} = \\int \\log{(H - \\mathbf{M})} d\\mathbf{M} and - H \\log{(- H + \\mathbf{M})} + \\Psi_{nl} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} = - H \\log{(- H + \\mathbf{M})} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} + \\phi_1 and - H \\log{(- H + \\mathbf{M})} + \\mathbf{M} \\log{(H - \\mathbf{M})} - \\mathbf{M} + \\phi_1 = \\int \\log{(H - \\mathbf{M})} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('H', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Integral(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Integral(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given A{(t)} = \\cos{(t)}, then derive (\\frac{d}{d t} A{(t)} - 1)^{t} = (- \\sin{(t)} - 1)^{t}, then obtain \\int ((\\frac{d}{d t} \\cos{(t)} - 1)^{t} - \\sin{(t)}) dt = \\int ((- \\sin{(t)} - 1)^{t} - \\sin{(t)}) dt", "derivation": "A{(t)} = \\cos{(t)} and - t + A{(t)} = - t + \\cos{(t)} and \\frac{d}{d t} (- t + A{(t)}) = \\frac{d}{d t} (- t + \\cos{(t)}) and (\\frac{d}{d t} (- t + A{(t)}))^{t} = (\\frac{d}{d t} (- t + \\cos{(t)}))^{t} and (\\frac{d}{d t} A{(t)} - 1)^{t} = (- \\sin{(t)} - 1)^{t} and (\\frac{d}{d t} \\cos{(t)} - 1)^{t} = (- \\sin{(t)} - 1)^{t} and (\\frac{d}{d t} \\cos{(t)} - 1)^{t} - \\sin{(t)} = (- \\sin{(t)} - 1)^{t} - \\sin{(t)} and \\int ((\\frac{d}{d t} \\cos{(t)} - 1)^{t} - \\sin{(t)}) dt = \\int ((- \\sin{(t)} - 1)^{t} - \\sin{(t)}) dt", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["minus", 1, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('A')(Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('A')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('A')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('A')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Integer(-1)), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Integer(-1)), Symbol('t', commutative=True)))"], [["add", 6, "Mul(Integer(-1), sin(Symbol('t', commutative=True)))"], "Equality(Add(Pow(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Integer(-1)), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))))"], [["integrate", 7, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Pow(Add(Derivative(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Add(Pow(Add(Mul(Integer(-1), sin(Symbol('t', commutative=True))), Integer(-1)), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\varphi{(Q)} = e^{Q}, then obtain \\log{(\\hat{H}_l \\varphi{(Q)} + \\varphi{(Q)})} = \\log{(\\hat{H}_l e^{Q} + \\varphi{(Q)})}", "derivation": "\\varphi{(Q)} = e^{Q} and \\hat{H}_l \\varphi{(Q)} = \\hat{H}_l e^{Q} and \\hat{H}_l \\varphi{(Q)} + \\varphi{(Q)} = \\hat{H}_l e^{Q} + \\varphi{(Q)} and \\log{(\\hat{H}_l \\varphi{(Q)} + \\varphi{(Q)})} = \\log{(\\hat{H}_l e^{Q} + \\varphi{(Q)})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\varphi')(Symbol('Q', commutative=True))), Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(Symbol('Q', commutative=True))))"], [["add", 2, "Function('\\\\varphi')(Symbol('Q', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\varphi')(Symbol('Q', commutative=True))), Function('\\\\varphi')(Symbol('Q', commutative=True))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(Symbol('Q', commutative=True))), Function('\\\\varphi')(Symbol('Q', commutative=True))))"], [["log", 3], "Equality(log(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\varphi')(Symbol('Q', commutative=True))), Function('\\\\varphi')(Symbol('Q', commutative=True)))), log(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(Symbol('Q', commutative=True))), Function('\\\\varphi')(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)} = \\hat{p}_0 \\mathbf{p}, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)} = \\hat{p}_0, then obtain (\\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)})^{\\hat{p}_0} = \\hat{p}_0^{\\hat{p}_0}", "derivation": "\\mathbf{r}{(\\mathbf{p},\\hat{p}_0)} = \\hat{p}_0 \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)} = \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{p}_0 \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)} = \\hat{p}_0 and (\\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)})^{\\hat{p}_0} = (\\frac{\\partial}{\\partial \\mathbf{p}} \\hat{p}_0 \\mathbf{p})^{\\hat{p}_0} and \\frac{\\partial}{\\partial \\mathbf{p}} \\hat{p}_0 \\mathbf{p} = \\hat{p}_0 and (\\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{r}{(\\mathbf{p},\\hat{p}_0)})^{\\hat{p}_0} = \\hat{p}_0^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\rho{(g,G)} = \\sin{(G^{g})}, then obtain \\tilde{\\infty} \\rho^{G}{(g,G)} = \\tilde{\\infty} \\sin^{G}{(G^{g})}", "derivation": "\\rho{(g,G)} = \\sin{(G^{g})} and \\rho^{G}{(g,G)} = \\sin^{G}{(G^{g})} and \\frac{\\rho{(g,G)}}{C_{d}} = \\frac{\\sin{(G^{g})}}{C_{d}} and \\frac{\\rho^{G}{(g,G)}}{- \\frac{\\rho{(g,G)}}{C_{d}} + \\frac{\\sin{(G^{g})}}{C_{d}}} = \\frac{\\sin^{G}{(G^{g})}}{- \\frac{\\rho{(g,G)}}{C_{d}} + \\frac{\\sin{(G^{g})}}{C_{d}}} and \\tilde{\\infty} \\rho^{G}{(g,G)} = \\tilde{\\infty} \\sin^{G}{(G^{g})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True)), sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))), Symbol('G', commutative=True)))"], [["divide", 1, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True)))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))))), Integer(-1)), Pow(Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))))), Integer(-1)), Pow(sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(zoo, Pow(Function('\\\\rho')(Symbol('g', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(zoo, Pow(sin(Pow(Symbol('G', commutative=True), Symbol('g', commutative=True))), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(T,\\theta)} = e^{\\theta^{T}}, then derive \\frac{\\partial}{\\partial \\theta} \\dot{z}{(T,\\theta)} = \\frac{T \\theta^{T} e^{\\theta^{T}}}{\\theta}, then obtain \\frac{\\partial}{\\partial \\theta} e^{\\theta^{T}} = \\frac{T \\theta^{T} e^{\\theta^{T}}}{\\theta}", "derivation": "\\dot{z}{(T,\\theta)} = e^{\\theta^{T}} and \\frac{\\partial}{\\partial \\theta} \\dot{z}{(T,\\theta)} = \\frac{\\partial}{\\partial \\theta} e^{\\theta^{T}} and \\frac{\\partial}{\\partial \\theta} \\dot{z}{(T,\\theta)} = \\frac{T \\theta^{T} e^{\\theta^{T}}}{\\theta} and \\frac{\\partial}{\\partial \\theta} e^{\\theta^{T}} = \\frac{T \\theta^{T} e^{\\theta^{T}}}{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), exp(Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('T', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True)), exp(Pow(Symbol('\\\\theta', commutative=True), Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(c,\\theta_2)} = \\cos{(\\frac{c}{\\theta_2})}, then derive \\frac{\\partial}{\\partial c} \\mathbf{A}{(c,\\theta_2)} = - \\frac{\\sin{(\\frac{c}{\\theta_2})}}{\\theta_2}, then obtain - \\frac{\\partial}{\\partial c} \\cos{(\\frac{c}{\\theta_2})} = \\frac{\\sin{(\\frac{c}{\\theta_2})}}{\\theta_2}", "derivation": "\\mathbf{A}{(c,\\theta_2)} = \\cos{(\\frac{c}{\\theta_2})} and \\frac{\\partial}{\\partial c} \\mathbf{A}{(c,\\theta_2)} = \\frac{\\partial}{\\partial c} \\cos{(\\frac{c}{\\theta_2})} and \\frac{\\partial}{\\partial c} \\mathbf{A}{(c,\\theta_2)} = - \\frac{\\sin{(\\frac{c}{\\theta_2})}}{\\theta_2} and \\frac{\\partial}{\\partial c} \\cos{(\\frac{c}{\\theta_2})} = - \\frac{\\sin{(\\frac{c}{\\theta_2})}}{\\theta_2} and - \\frac{\\partial}{\\partial c} \\cos{(\\frac{c}{\\theta_2})} = \\frac{\\sin{(\\frac{c}{\\theta_2})}}{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True)))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(cos(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given I{(C_{1},\\dot{y})} = \\dot{y}^{C_{1}}, then obtain \\dot{y}^{C_{1}} + \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + I{(C_{1},\\dot{y})}) = \\dot{y}^{C_{1}} + \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + \\dot{y}^{C_{1}})", "derivation": "I{(C_{1},\\dot{y})} = \\dot{y}^{C_{1}} and - C_{1} + I{(C_{1},\\dot{y})} = - C_{1} + \\dot{y}^{C_{1}} and \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + I{(C_{1},\\dot{y})}) = \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + \\dot{y}^{C_{1}}) and \\dot{y}^{C_{1}} + \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + I{(C_{1},\\dot{y})}) = \\dot{y}^{C_{1}} + \\frac{\\partial}{\\partial \\dot{y}} (- C_{1} + \\dot{y}^{C_{1}})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True)))"], [["minus", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["add", 3, "Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Add(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(i,z)} = e^{i + z}, then obtain (2 i + 2 z)^{2} \\hat{p}_0^{4}{(i,z)} = (2 i + 2 z)^{2} e^{4 i + 4 z}", "derivation": "\\hat{p}_0{(i,z)} = e^{i + z} and \\hat{p}_0{(i,z)} e^{i + z} = e^{2 i + 2 z} and \\hat{p}_0^{2}{(i,z)} = e^{2 i + 2 z} and \\hat{p}_0^{2}{(i,z)} = \\hat{p}_0{(i,z)} e^{i + z} and (2 i + 2 z) \\hat{p}_0{(i,z)} e^{i + z} = (2 i + 2 z) e^{2 i + 2 z} and (2 i + 2 z) \\hat{p}_0^{2}{(i,z)} = (2 i + 2 z) e^{2 i + 2 z} and (2 i + 2 z)^{2} \\hat{p}_0^{4}{(i,z)} = (2 i + 2 z)^{2} e^{4 i + 4 z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))"], [["times", 1, "exp(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))), exp(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(2)), exp(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(2)), Mul(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Mul(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), exp(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), Pow(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(2))), Mul(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), exp(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))))))"], [["power", 6, 2], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), Integer(2)), Pow(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(4))), Mul(Pow(Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True))), Integer(2)), exp(Add(Mul(Integer(4), Symbol('i', commutative=True)), Mul(Integer(4), Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\lambda,B)} = B \\lambda, then derive B \\lambda + \\frac{\\frac{\\partial}{\\partial B} \\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} - \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B^{2} \\lambda} = B \\lambda, then obtain \\frac{1}{(B \\lambda + \\frac{\\frac{\\partial}{\\partial B} \\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} - \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B^{2} \\lambda})^{2}} = \\frac{1}{B^{2} \\lambda^{2}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\lambda,B)} = B \\lambda and \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} = 1 and \\frac{\\partial}{\\partial B} \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} = \\frac{d}{d B} 1 and B \\lambda + \\frac{\\partial}{\\partial B} \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} = B \\lambda + \\frac{d}{d B} 1 and B \\lambda + \\frac{\\frac{\\partial}{\\partial B} \\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} - \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B^{2} \\lambda} = B \\lambda and \\frac{1}{(B \\lambda + \\frac{\\frac{\\partial}{\\partial B} \\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B \\lambda} - \\frac{\\operatorname{a^{\\dagger}}{(\\lambda,B)}}{B^{2} \\lambda})^{2}} = \\frac{1}{B^{2} \\lambda^{2}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Integer(1), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True)))), Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["power", 5, "Integer(-2)"], "Equality(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\lambda', commutative=True), Symbol('B', commutative=True)))), Integer(-2)), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{B},\\mu_0)} = \\mathbf{B}^{\\mu_0}, then obtain \\frac{\\partial}{\\partial \\mu_0} (- \\int \\mathbf{B}^{\\mu_0} d\\mu_0 + \\int \\hat{x}{(\\mathbf{B},\\mu_0)} d\\mu_0) = \\frac{d}{d \\mu_0} 0", "derivation": "\\hat{x}{(\\mathbf{B},\\mu_0)} = \\mathbf{B}^{\\mu_0} and \\int \\hat{x}{(\\mathbf{B},\\mu_0)} d\\mu_0 = \\int \\mathbf{B}^{\\mu_0} d\\mu_0 and \\hat{x}{(\\mathbf{B},\\mu_0)} + \\int \\hat{x}{(\\mathbf{B},\\mu_0)} d\\mu_0 = \\hat{x}{(\\mathbf{B},\\mu_0)} + \\int \\mathbf{B}^{\\mu_0} d\\mu_0 and - \\int \\mathbf{B}^{\\mu_0} d\\mu_0 + \\int \\hat{x}{(\\mathbf{B},\\mu_0)} d\\mu_0 = 0 and \\frac{\\partial}{\\partial \\mu_0} (- \\int \\mathbf{B}^{\\mu_0} d\\mu_0 + \\int \\hat{x}{(\\mathbf{B},\\mu_0)} d\\mu_0) = \\frac{d}{d \\mu_0} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 3, "Add(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integral(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(n_{1})} = \\log{(n_{1})}, then obtain \\int (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) (\\mathbf{H}{(n_{1})} + \\log{(n_{1})}) dn_{1} = \\int 2 (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) \\log{(n_{1})} dn_{1}", "derivation": "\\mathbf{H}{(n_{1})} = \\log{(n_{1})} and \\mathbf{H}{(n_{1})} + \\log{(n_{1})} = 2 \\log{(n_{1})} and (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) (\\mathbf{H}{(n_{1})} + \\log{(n_{1})}) = 2 (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) \\log{(n_{1})} and \\int (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) (\\mathbf{H}{(n_{1})} + \\log{(n_{1})}) dn_{1} = \\int 2 (- \\mathbf{H}{(n_{1})} + \\log{(n_{1})}) \\log{(n_{1})} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))"], [["add", 1, "log(Symbol('n_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True))), Mul(Integer(2), log(Symbol('n_1', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))), Mul(Integer(2), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))))"], [["integrate", 3, "Symbol('n_1', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Integer(2), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))), log(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given v{(s)} = e^{e^{s}}, then obtain (v{(s)} + e^{e^{s}}) e^{s} - 3 v{(s)} + e^{e^{s}} = (v{(s)} + e^{e^{s}}) e^{s} - 5 v{(s)} + 3 e^{e^{s}}", "derivation": "v{(s)} = e^{e^{s}} and 2 v{(s)} = v{(s)} + e^{e^{s}} and 2 v{(s)} - e^{e^{s}} = v{(s)} and 4 v{(s)} - 2 e^{e^{s}} = 2 v{(s)} and - 2 v{(s)} + 2 e^{e^{s}} = - 3 v{(s)} + 3 e^{e^{s}} and - 4 v{(s)} + 4 e^{e^{s}} = - 3 v{(s)} + 3 e^{e^{s}} and 2 v{(s)} e^{s} - 4 v{(s)} + 4 e^{e^{s}} = 2 v{(s)} e^{s} - 3 v{(s)} + 3 e^{e^{s}} and 2 v{(s)} e^{s} - 6 v{(s)} + 4 e^{e^{s}} = 2 v{(s)} e^{s} - 5 v{(s)} + 3 e^{e^{s}} and (v{(s)} + e^{e^{s}}) e^{s} - 3 v{(s)} + e^{e^{s}} = (v{(s)} + e^{e^{s}}) e^{s} - 5 v{(s)} + 3 e^{e^{s}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True))))"], [["add", 1, "Function('v')(Symbol('s', commutative=True))"], "Equality(Mul(Integer(2), Function('v')(Symbol('s', commutative=True))), Add(Function('v')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True)))))"], [["minus", 2, "exp(exp(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('s', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('s', commutative=True))))), Function('v')(Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(4), Function('v')(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), exp(exp(Symbol('s', commutative=True))))), Mul(Integer(2), Function('v')(Symbol('s', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(4), Function('v')(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), exp(exp(Symbol('s', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('v')(Symbol('s', commutative=True))), Mul(Integer(2), exp(exp(Symbol('s', commutative=True))))), Add(Mul(Integer(-1), Integer(3), Function('v')(Symbol('s', commutative=True))), Mul(Integer(3), exp(exp(Symbol('s', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(4), Function('v')(Symbol('s', commutative=True))), Mul(Integer(4), exp(exp(Symbol('s', commutative=True))))), Add(Mul(Integer(-1), Integer(3), Function('v')(Symbol('s', commutative=True))), Mul(Integer(3), exp(exp(Symbol('s', commutative=True))))))"], [["add", 6, "Mul(Integer(2), Function('v')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(4), Function('v')(Symbol('s', commutative=True))), Mul(Integer(4), exp(exp(Symbol('s', commutative=True))))), Add(Mul(Integer(2), Function('v')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(3), Function('v')(Symbol('s', commutative=True))), Mul(Integer(3), exp(exp(Symbol('s', commutative=True))))))"], [["minus", 7, "Mul(Integer(2), Function('v')(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(6), Function('v')(Symbol('s', commutative=True))), Mul(Integer(4), exp(exp(Symbol('s', commutative=True))))), Add(Mul(Integer(2), Function('v')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(5), Function('v')(Symbol('s', commutative=True))), Mul(Integer(3), exp(exp(Symbol('s', commutative=True))))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Add(Mul(Add(Function('v')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True)))), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(3), Function('v')(Symbol('s', commutative=True))), exp(exp(Symbol('s', commutative=True)))), Add(Mul(Add(Function('v')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True)))), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Integer(5), Function('v')(Symbol('s', commutative=True))), Mul(Integer(3), exp(exp(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\theta)} = \\cos{(\\theta)} and \\mathbf{B}{(\\theta)} = \\cos{(\\theta)}, then derive \\int \\dot{\\mathbf{r}}{(\\theta)} d\\theta = \\mathbf{r} + \\sin{(\\theta)}, then obtain \\int \\mathbf{B}{(\\theta)} d\\theta = \\mathbf{r} + \\sin{(\\theta)}", "derivation": "\\dot{\\mathbf{r}}{(\\theta)} = \\cos{(\\theta)} and \\int \\dot{\\mathbf{r}}{(\\theta)} d\\theta = \\int \\cos{(\\theta)} d\\theta and \\mathbf{B}{(\\theta)} = \\cos{(\\theta)} and \\int \\dot{\\mathbf{r}}{(\\theta)} d\\theta = \\mathbf{r} + \\sin{(\\theta)} and \\mathbf{B}{(\\theta)} = \\dot{\\mathbf{r}}{(\\theta)} and \\int \\mathbf{B}{(\\theta)} d\\theta = \\mathbf{r} + \\sin{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given A{(E_{n},\\hat{H})} = \\cos{(E_{n} - \\hat{H})} and \\hat{H}_l{(E_{n},\\hat{H})} = \\cos{(E_{n} - \\hat{H})}, then obtain \\hat{H} + \\frac{A{(E_{n},\\hat{H})}}{\\hat{H}_l{(E_{n},\\hat{H})}} = \\hat{H} + 1", "derivation": "A{(E_{n},\\hat{H})} = \\cos{(E_{n} - \\hat{H})} and \\frac{A{(E_{n},\\hat{H})}}{\\cos{(E_{n} - \\hat{H})}} = 1 and \\hat{H} + \\frac{A{(E_{n},\\hat{H})}}{\\cos{(E_{n} - \\hat{H})}} = \\hat{H} + 1 and \\hat{H}_l{(E_{n},\\hat{H})} = \\cos{(E_{n} - \\hat{H})} and \\hat{H} + \\frac{A{(E_{n},\\hat{H})}}{\\hat{H}_l{(E_{n},\\hat{H})}} = \\hat{H} + 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))))"], [["divide", 1, "cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))))"], "Equality(Mul(Function('A')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Function('A')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)))), Add(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Function('A')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))), Add(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(S,r)} = \\cos{(S + r)}, then obtain (\\operatorname{v_{2}}{(S,r)} + \\int \\operatorname{v_{2}}^{S}{(S,r)} dS)^{S} = (\\operatorname{v_{2}}{(S,r)} + \\int \\cos^{S}{(S + r)} dS)^{S}", "derivation": "\\operatorname{v_{2}}{(S,r)} = \\cos{(S + r)} and \\operatorname{v_{2}}^{S}{(S,r)} = \\cos^{S}{(S + r)} and \\int \\operatorname{v_{2}}^{S}{(S,r)} dS = \\int \\cos^{S}{(S + r)} dS and \\cos{(S + r)} + \\int \\operatorname{v_{2}}^{S}{(S,r)} dS = \\cos{(S + r)} + \\int \\cos^{S}{(S + r)} dS and \\operatorname{v_{2}}{(S,r)} + \\int \\operatorname{v_{2}}^{S}{(S,r)} dS = \\operatorname{v_{2}}{(S,r)} + \\int \\cos^{S}{(S + r)} dS and (\\operatorname{v_{2}}{(S,r)} + \\int \\operatorname{v_{2}}^{S}{(S,r)} dS)^{S} = (\\operatorname{v_{2}}{(S,r)} + \\int \\cos^{S}{(S + r)} dS)^{S}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('S', commutative=True)), Pow(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Symbol('S', commutative=True)))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["add", 3, "cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True)))"], "Equality(Add(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Integral(Pow(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Integral(Pow(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Integral(Pow(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Integral(Pow(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["power", 5, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Integral(Pow(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)), Pow(Add(Function('v_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Integral(Pow(cos(Add(Symbol('S', commutative=True), Symbol('r', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive \\frac{d}{d \\mathbf{p}} \\phi_{1}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain e^{\\mathbf{p}} + \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} = 2 \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}}", "derivation": "\\phi_{1}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\phi_{1}{(\\mathbf{p})} + e^{\\mathbf{p}} = 2 e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} \\phi_{1}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} \\phi_{1}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\phi_{1}{(\\mathbf{p})} + \\frac{d}{d \\mathbf{p}} \\phi_{1}{(\\mathbf{p})} = 2 \\frac{d}{d \\mathbf{p}} \\phi_{1}{(\\mathbf{p})} and e^{\\mathbf{p}} + \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}} = 2 \\frac{d}{d \\mathbf{p}} e^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(exp(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(v)} = \\sin{(v)}, then obtain \\int ((\\mathbf{p}{(v)} - \\sin{(v)})^{2} + \\mathbf{p}{(v)} - \\sin{(v)}) dv = \\int (\\mathbf{p}{(v)} - \\sin{(v)})^{2} dv", "derivation": "\\mathbf{p}{(v)} = \\sin{(v)} and \\mathbf{p}{(v)} - \\sin{(v)} = 0 and (\\mathbf{p}{(v)} - \\sin{(v)})^{2} + \\mathbf{p}{(v)} - \\sin{(v)} = (\\mathbf{p}{(v)} - \\sin{(v)})^{2} and \\int ((\\mathbf{p}{(v)} - \\sin{(v)})^{2} + \\mathbf{p}{(v)} - \\sin{(v)}) dv = \\int (\\mathbf{p}{(v)} - \\sin{(v)})^{2} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["minus", 1, "sin(Symbol('v', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(0))"], [["add", 2, "Pow(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(2))"], "Equality(Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(2)), Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Pow(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(2)))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Pow(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(2)), Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Function('\\\\mathbf{p}')(Symbol('v', commutative=True)), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), Integer(2)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{f},B)} = - \\mathbf{f} + \\sin{(B)}, then obtain \\frac{\\frac{\\partial}{\\partial B} \\Omega{(\\mathbf{f},B)}}{- \\mathbf{f} + \\sin{(B)}} = \\frac{\\cos{(B)}}{- \\mathbf{f} + \\sin{(B)}}", "derivation": "\\Omega{(\\mathbf{f},B)} = - \\mathbf{f} + \\sin{(B)} and \\mathbf{f} + \\Omega{(\\mathbf{f},B)} = \\sin{(B)} and \\frac{\\partial}{\\partial B} (\\mathbf{f} + \\Omega{(\\mathbf{f},B)}) = \\frac{d}{d B} \\sin{(B)} and \\frac{\\frac{\\partial}{\\partial B} (\\mathbf{f} + \\Omega{(\\mathbf{f},B)})}{- \\mathbf{f} + \\sin{(B)}} = \\frac{\\frac{d}{d B} \\sin{(B)}}{- \\mathbf{f} + \\sin{(B)}} and \\frac{\\frac{\\partial}{\\partial B} \\Omega{(\\mathbf{f},B)}}{- \\mathbf{f} + \\sin{(B)}} = \\frac{\\cos{(B)}}{- \\mathbf{f} + \\sin{(B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('B', commutative=True))), sin(Symbol('B', commutative=True)))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True))), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True))), Integer(-1)), Derivative(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True))), Integer(-1)), Derivative(Function('\\\\Omega')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('B', commutative=True))), Integer(-1)), cos(Symbol('B', commutative=True))))"]]}, {"prompt": "Given A{(\\pi)} = \\cos{(\\pi)}, then obtain \\int \\frac{A^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi + \\int \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi = 2 \\int \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi", "derivation": "A{(\\pi)} = \\cos{(\\pi)} and A^{\\pi}{(\\pi)} = \\cos^{\\pi}{(\\pi)} and \\frac{A^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} = \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} and \\int \\frac{A^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi = \\int \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi and \\int \\frac{A^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi + \\int \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi = 2 \\int \\frac{\\cos^{\\pi}{(\\pi)}}{\\frac{d}{d \\pi} A{(\\pi)}} d\\pi", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["divide", 2, "Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('A')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Pow(Function('A')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 4, "Integral(Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Function('A')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Integer(2), Integral(Mul(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Function('A')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} = - \\ddot{x} + \\eta + l, then derive - l + \\int \\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} d\\ddot{x} = - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\eta + l) + \\pi - l, then obtain - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\eta + l) + \\pi - l = - l + \\int (- \\ddot{x} + \\eta + l) d\\ddot{x}", "derivation": "\\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} = - \\ddot{x} + \\eta + l and \\int \\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} d\\ddot{x} = \\int (- \\ddot{x} + \\eta + l) d\\ddot{x} and - l + \\int \\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} d\\ddot{x} = - l + \\int (- \\ddot{x} + \\eta + l) d\\ddot{x} and - l + \\int \\operatorname{A_{x}}{(l,\\eta,\\ddot{x})} d\\ddot{x} = - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\eta + l) + \\pi - l and - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\eta + l) + \\pi - l = - l + \\int (- \\ddot{x} + \\eta + l) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["minus", 2, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('A_x')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('A_x')(Symbol('l', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True))), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\eta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given B{(v_{1},A_{z})} = A_{z} + v_{1}, then obtain (\\frac{B{(v_{1},A_{z})}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} + \\int \\frac{A_{z} + v_{1}}{v_{1}} dA_{z} = (\\frac{A_{z} + v_{1}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} + \\int \\frac{A_{z} + v_{1}}{v_{1}} dA_{z}", "derivation": "B{(v_{1},A_{z})} = A_{z} + v_{1} and \\frac{B{(v_{1},A_{z})}}{v_{1}} = \\frac{A_{z} + v_{1}}{v_{1}} and \\frac{B{(v_{1},A_{z})}}{v_{1}} - \\frac{1}{v_{1}} = \\frac{A_{z} + v_{1}}{v_{1}} - \\frac{1}{v_{1}} and (\\frac{B{(v_{1},A_{z})}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} = (\\frac{A_{z} + v_{1}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} and (\\frac{B{(v_{1},A_{z})}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} + \\int \\frac{A_{z} + v_{1}}{v_{1}} dA_{z} = (\\frac{A_{z} + v_{1}}{v_{1}} - \\frac{1}{v_{1}})^{v_{1}} + \\int \\frac{A_{z} + v_{1}}{v_{1}} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)))"], [["divide", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('B')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))))"], [["minus", 2, "Pow(Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('B')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('B')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('v_1', commutative=True)), Pow(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('v_1', commutative=True)))"], [["add", 4, "Integral(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('A_z', commutative=True)))"], "Equality(Add(Pow(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('B')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('v_1', commutative=True)), Integral(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('A_z', commutative=True)))), Add(Pow(Add(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('v_1', commutative=True)), Integral(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C_{d},L)} = \\int \\frac{L}{C_{d}} dC_{d}, then obtain (2 - \\operatorname{F_{x}}{(C_{d},L)})^{C_{d}} = (- \\operatorname{F_{x}}{(C_{d},L)} + 1 + \\frac{\\int \\frac{L}{C_{d}} dC_{d}}{\\operatorname{F_{x}}{(C_{d},L)}})^{C_{d}}", "derivation": "\\operatorname{F_{x}}{(C_{d},L)} = \\int \\frac{L}{C_{d}} dC_{d} and 1 = \\frac{\\int \\frac{L}{C_{d}} dC_{d}}{\\operatorname{F_{x}}{(C_{d},L)}} and 1 - \\operatorname{F_{x}}{(C_{d},L)} = - \\operatorname{F_{x}}{(C_{d},L)} + \\frac{\\int \\frac{L}{C_{d}} dC_{d}}{\\operatorname{F_{x}}{(C_{d},L)}} and 2 - \\operatorname{F_{x}}{(C_{d},L)} = - \\operatorname{F_{x}}{(C_{d},L)} + 1 + \\frac{\\int \\frac{L}{C_{d}} dC_{d}}{\\operatorname{F_{x}}{(C_{d},L)}} and (2 - \\operatorname{F_{x}}{(C_{d},L)})^{C_{d}} = (- \\operatorname{F_{x}}{(C_{d},L)} + 1 + \\frac{\\int \\frac{L}{C_{d}} dC_{d}}{\\operatorname{F_{x}}{(C_{d},L)}})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 1, "Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["minus", 2, "Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('C_d', commutative=True))))))"], [["add", 3, 1], "Equality(Add(Integer(2), Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))), Integer(1), Mul(Pow(Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('C_d', commutative=True))))))"], [["power", 4, "Symbol('C_d', commutative=True)"], "Equality(Pow(Add(Integer(2), Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))), Symbol('C_d', commutative=True)), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))), Integer(1), Mul(Pow(Function('F_x')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('C_d', commutative=True))))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^*, then derive \\operatorname{y^{\\prime}}{(\\varphi^*)} = \\eta^{\\prime} - \\cos{(\\varphi^*)}, then derive (q - \\cos{(\\varphi^*)})^{\\eta^{\\prime}} = (\\eta^{\\prime} - \\cos{(\\varphi^*)})^{\\eta^{\\prime}}, then obtain ((q - \\cos{(\\varphi^*)})^{\\eta^{\\prime}})^{\\eta^{\\prime}} = ((\\eta^{\\prime} - \\cos{(\\varphi^*)})^{\\eta^{\\prime}})^{\\eta^{\\prime}}", "derivation": "\\operatorname{y^{\\prime}}{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^* and \\operatorname{y^{\\prime}}{(\\varphi^*)} = \\eta^{\\prime} - \\cos{(\\varphi^*)} and \\int \\sin{(\\varphi^*)} d\\varphi^* = \\eta^{\\prime} - \\cos{(\\varphi^*)} and (\\int \\sin{(\\varphi^*)} d\\varphi^*)^{\\eta^{\\prime}} = (\\eta^{\\prime} - \\cos{(\\varphi^*)})^{\\eta^{\\prime}} and (q - \\cos{(\\varphi^*)})^{\\eta^{\\prime}} = (\\eta^{\\prime} - \\cos{(\\varphi^*)})^{\\eta^{\\prime}} and ((q - \\cos{(\\varphi^*)})^{\\eta^{\\prime}})^{\\eta^{\\prime}} = ((\\eta^{\\prime} - \\cos{(\\varphi^*)})^{\\eta^{\\prime}})^{\\eta^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\varphi^*', commutative=True)), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 5, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})} = - F_{c} + f^{*}, then obtain (\\int \\log{(- \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})})} dF_{c})^{F_{c}} = (\\int \\log{(F_{c} - f^{*})} dF_{c})^{F_{c}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})} = - F_{c} + f^{*} and - \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})} = F_{c} - f^{*} and \\log{(- \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})})} = \\log{(F_{c} - f^{*})} and \\int \\log{(- \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})})} dF_{c} = \\int \\log{(F_{c} - f^{*})} dF_{c} and (\\int \\log{(- \\operatorname{g_{\\varepsilon}}{(f^{*},F_{c})})} dF_{c})^{F_{c}} = (\\int \\log{(F_{c} - f^{*})} dF_{c})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('F_c', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('f^*', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["log", 2], "Equality(log(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('F_c', commutative=True)))), log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(log(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Integral(log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Tuple(Symbol('F_c', commutative=True))))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integral(log(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('f^*', commutative=True), Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\omega{(\\rho_f,\\mu_0,\\eta)} = \\eta + \\mu_0 + \\rho_f, then derive \\frac{\\partial}{\\partial \\mu_0} \\omega{(\\rho_f,\\mu_0,\\eta)} = 1, then obtain \\mu_0 \\int (\\eta + \\mu_0 + \\rho_f) d\\rho_f + \\frac{\\partial}{\\partial \\mu_0} \\omega{(\\rho_f,\\mu_0,\\eta)} = \\mu_0 \\int (\\eta + \\mu_0 + \\rho_f) d\\rho_f + 1", "derivation": "\\omega{(\\rho_f,\\mu_0,\\eta)} = \\eta + \\mu_0 + \\rho_f and \\frac{\\partial}{\\partial \\mu_0} \\omega{(\\rho_f,\\mu_0,\\eta)} = \\frac{\\partial}{\\partial \\mu_0} (\\eta + \\mu_0 + \\rho_f) and \\frac{\\partial}{\\partial \\mu_0} \\omega{(\\rho_f,\\mu_0,\\eta)} = 1 and \\mu_0 \\int (\\eta + \\mu_0 + \\rho_f) d\\rho_f + \\frac{\\partial}{\\partial \\mu_0} \\omega{(\\rho_f,\\mu_0,\\eta)} = \\mu_0 \\int (\\eta + \\mu_0 + \\rho_f) d\\rho_f + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Mul(Symbol('\\\\mu_0', commutative=True), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Derivative(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given U{(g,Z)} = Z g, then obtain \\frac{\\int U{(g,Z)} dg}{Z} = \\frac{\\int Z g dg}{Z}", "derivation": "U{(g,Z)} = Z g and \\frac{\\partial}{\\partial g} U{(g,Z)} = \\frac{\\partial}{\\partial g} Z g and \\int U{(g,Z)} dg = \\int Z g dg and \\frac{\\int U{(g,Z)} dg}{\\frac{\\partial}{\\partial g} U{(g,Z)}} = \\frac{\\int Z g dg}{\\frac{\\partial}{\\partial g} U{(g,Z)}} and \\frac{\\int U{(g,Z)} dg}{\\frac{\\partial}{\\partial g} Z g} = \\frac{\\int Z g dg}{\\frac{\\partial}{\\partial g} Z g} and \\frac{\\int U{(g,Z)} dg}{Z} = \\frac{\\int Z g dg}{Z}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["divide", 3, "Derivative(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Integral(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Derivative(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Derivative(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Integral(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Derivative(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Integral(Function('U')(Symbol('g', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Integral(Mul(Symbol('Z', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\pi)} = \\cos{(\\pi)} and \\operatorname{z^{*}}{(\\pi)} = - \\Omega{(\\pi)}, then obtain - \\Omega{(\\pi)} - 2 \\cos{(\\pi)} = - 2 \\Omega{(\\pi)} - \\cos{(\\pi)}", "derivation": "\\Omega{(\\pi)} = \\cos{(\\pi)} and - \\cos{(\\pi)} = - \\Omega{(\\pi)} and \\operatorname{z^{*}}{(\\pi)} = - \\Omega{(\\pi)} and \\operatorname{z^{*}}{(\\pi)} = - \\cos{(\\pi)} and - \\Omega{(\\pi)} - \\cos{(\\pi)} = - 2 \\Omega{(\\pi)} and - \\Omega{(\\pi)} + \\operatorname{z^{*}}{(\\pi)} - \\cos{(\\pi)} = - 2 \\Omega{(\\pi)} + \\operatorname{z^{*}}{(\\pi)} and - \\Omega{(\\pi)} - 2 \\cos{(\\pi)} = - 2 \\Omega{(\\pi)} - \\cos{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('z^*')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integer(2), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))))"], [["add", 5, "Function('z^*')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))), Function('z^*')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))), Function('z^*')(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Omega')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given t{(g,n_{1})} = n_{1} \\log{(g)} and \\operatorname{P_{e}}{(g)} = \\frac{d}{d n_{1}} g, then obtain \\operatorname{P_{e}}{(g)} = \\frac{\\partial}{\\partial n_{1}} e^{\\frac{t{(g,n_{1})}}{n_{1}}}", "derivation": "t{(g,n_{1})} = n_{1} \\log{(g)} and \\frac{t{(g,n_{1})}}{n_{1}} = \\log{(g)} and e^{\\frac{t{(g,n_{1})}}{n_{1}}} = g and \\frac{\\partial}{\\partial n_{1}} e^{\\frac{t{(g,n_{1})}}{n_{1}}} = \\frac{d}{d n_{1}} g and \\operatorname{P_{e}}{(g)} = \\frac{d}{d n_{1}} g and \\operatorname{P_{e}}{(g)} = \\frac{\\partial}{\\partial n_{1}} e^{\\frac{t{(g,n_{1})}}{n_{1}}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)), Mul(Symbol('n_1', commutative=True), log(Symbol('g', commutative=True))))"], [["divide", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('t')(Symbol('g', commutative=True), Symbol('n_1', commutative=True))), log(Symbol('g', commutative=True)))"], [["exp", 2], "Equality(exp(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('t')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)))), Symbol('g', commutative=True))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(exp(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('t')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Symbol('g', commutative=True), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('g', commutative=True)), Derivative(Symbol('g', commutative=True), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('P_e')(Symbol('g', commutative=True)), Derivative(exp(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('t')(Symbol('g', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(\\hat{x}_0)} = e^{\\hat{x}_0}, then derive 2 \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)} = e^{\\hat{x}_0} + \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)}, then obtain 2 \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)} = \\hat{x}{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)}", "derivation": "\\hat{x}{(\\hat{x}_0)} = e^{\\hat{x}_0} and 2 \\hat{x}{(\\hat{x}_0)} = \\hat{x}{(\\hat{x}_0)} + e^{\\hat{x}_0} and \\frac{d}{d \\hat{x}_0} 2 \\hat{x}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} (\\hat{x}{(\\hat{x}_0)} + e^{\\hat{x}_0}) and 2 \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)} = e^{\\hat{x}_0} + \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)} and 2 \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)} = \\hat{x}{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\hat{x}{(\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True))), Add(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(T,\\mathbf{A})} = T + \\mathbf{A}, then obtain (\\int (T + \\mathbf{A}) \\operatorname{A_{x}}^{3}{(T,\\mathbf{A})} dT)^{T} = (\\int (T + \\mathbf{A})^{2} \\operatorname{A_{x}}^{2}{(T,\\mathbf{A})} dT)^{T}", "derivation": "\\operatorname{A_{x}}{(T,\\mathbf{A})} = T + \\mathbf{A} and (T + \\mathbf{A}) \\operatorname{A_{x}}{(T,\\mathbf{A})} = (T + \\mathbf{A})^{2} and (T + \\mathbf{A})^{2} \\operatorname{A_{x}}^{2}{(T,\\mathbf{A})} = (T + \\mathbf{A})^{4} and (T + \\mathbf{A}) \\operatorname{A_{x}}^{3}{(T,\\mathbf{A})} = (T + \\mathbf{A})^{2} \\operatorname{A_{x}}^{2}{(T,\\mathbf{A})} and \\int (T + \\mathbf{A}) \\operatorname{A_{x}}^{3}{(T,\\mathbf{A})} dT = \\int (T + \\mathbf{A})^{2} \\operatorname{A_{x}}^{2}{(T,\\mathbf{A})} dT and (\\int (T + \\mathbf{A}) \\operatorname{A_{x}}^{3}{(T,\\mathbf{A})} dT)^{T} = (\\int (T + \\mathbf{A})^{2} \\operatorname{A_{x}}^{2}{(T,\\mathbf{A})} dT)^{T}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(3))), Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(3))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Tuple(Symbol('T', commutative=True))))"], [["power", 5, "Symbol('T', commutative=True)"], "Equality(Pow(Integral(Mul(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(3))), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True)), Pow(Integral(Mul(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Pow(Function('A_x')(Symbol('T', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(r)} = \\cos{(r)}, then obtain \\operatorname{E_{x}}{(r)} \\int \\operatorname{E_{x}}{(r)} dr = \\operatorname{E_{x}}{(r)} \\int \\cos{(r)} dr", "derivation": "\\operatorname{E_{x}}{(r)} = \\cos{(r)} and \\int \\operatorname{E_{x}}{(r)} dr = \\int \\cos{(r)} dr and \\cos{(r)} \\int \\operatorname{E_{x}}{(r)} dr = \\cos{(r)} \\int \\cos{(r)} dr and \\operatorname{E_{x}}{(r)} \\int \\operatorname{E_{x}}{(r)} dr = \\operatorname{E_{x}}{(r)} \\int \\cos{(r)} dr", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["times", 2, "cos(Symbol('r', commutative=True))"], "Equality(Mul(cos(Symbol('r', commutative=True)), Integral(Function('E_x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(cos(Symbol('r', commutative=True)), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('E_x')(Symbol('r', commutative=True)), Integral(Function('E_x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Function('E_x')(Symbol('r', commutative=True)), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given t{(E,F_{H})} = - E + F_{H}, then derive \\int t{(E,F_{H})} dF_{H} = - E F_{H} + \\frac{F_{H}^{2}}{2} + t_{2}, then derive - E + F_{H} = \\frac{\\partial}{\\partial F_{H}} \\int (- E + F_{H}) dF_{H}, then obtain t{(E,F_{H})} = \\frac{\\partial}{\\partial F_{H}} (- E F_{H} + \\frac{F_{H}^{2}}{2} + \\mathbf{H})", "derivation": "t{(E,F_{H})} = - E + F_{H} and \\int t{(E,F_{H})} dF_{H} = \\int (- E + F_{H}) dF_{H} and \\int t{(E,F_{H})} dF_{H} = - E F_{H} + \\frac{F_{H}^{2}}{2} + t_{2} and - E F_{H} + \\frac{F_{H}^{2}}{2} + t_{2} = \\int (- E + F_{H}) dF_{H} and \\frac{\\partial}{\\partial F_{H}} (- E F_{H} + \\frac{F_{H}^{2}}{2} + t_{2}) = \\frac{\\partial}{\\partial F_{H}} \\int (- E + F_{H}) dF_{H} and - E + F_{H} = \\frac{\\partial}{\\partial F_{H}} \\int (- E + F_{H}) dF_{H} and t{(E,F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\int (- E + F_{H}) dF_{H} and t{(E,F_{H})} = \\frac{\\partial}{\\partial F_{H}} (- E F_{H} + \\frac{F_{H}^{2}}{2} + \\mathbf{H})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('t')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('t_2', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('t_2', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Derivative(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 6], "Equality(Function('t')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Derivative(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_integrals", 7], "Equality(Function('t')(Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('F_H', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{p},\\omega)} = \\frac{\\omega}{\\hat{p}}, then derive \\omega \\frac{\\partial}{\\partial \\hat{p}} \\tilde{g}^*{(\\hat{p},\\omega)} = - \\frac{\\omega^{2}}{\\hat{p}^{2}}, then obtain \\int \\omega \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\omega}{\\hat{p}} d\\hat{p} = \\int - \\frac{\\omega^{2}}{\\hat{p}^{2}} d\\hat{p}", "derivation": "\\tilde{g}^*{(\\hat{p},\\omega)} = \\frac{\\omega}{\\hat{p}} and \\omega \\tilde{g}^*{(\\hat{p},\\omega)} = \\frac{\\omega^{2}}{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} \\omega \\tilde{g}^*{(\\hat{p},\\omega)} = \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\omega^{2}}{\\hat{p}} and \\omega \\frac{\\partial}{\\partial \\hat{p}} \\tilde{g}^*{(\\hat{p},\\omega)} = - \\frac{\\omega^{2}}{\\hat{p}^{2}} and \\int \\omega \\frac{\\partial}{\\partial \\hat{p}} \\tilde{g}^*{(\\hat{p},\\omega)} d\\hat{p} = \\int - \\frac{\\omega^{2}}{\\hat{p}^{2}} d\\hat{p} and \\int \\omega \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\omega}{\\hat{p}} d\\hat{p} = \\int - \\frac{\\omega^{2}}{\\hat{p}^{2}} d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\omega', commutative=True), Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\omega', commutative=True), Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Symbol('\\\\omega', commutative=True), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(C_{1},\\mu)} = C_{1} - \\mu and \\operatorname{t_{1}}{(C_{1},\\mu)} = C_{1} - \\mu, then obtain - 2 C_{1} + (C_{1} - \\mu)^{C_{1}} = - 2 C_{1} + \\operatorname{t_{1}}^{C_{1}}{(C_{1},\\mu)}", "derivation": "\\varphi^{*}{(C_{1},\\mu)} = C_{1} - \\mu and \\varphi^{*}^{C_{1}}{(C_{1},\\mu)} = (C_{1} - \\mu)^{C_{1}} and \\operatorname{t_{1}}{(C_{1},\\mu)} = C_{1} - \\mu and \\varphi^{*}^{C_{1}}{(C_{1},\\mu)} = \\operatorname{t_{1}}^{C_{1}}{(C_{1},\\mu)} and (C_{1} - \\mu)^{C_{1}} = \\operatorname{t_{1}}^{C_{1}}{(C_{1},\\mu)} and - 2 C_{1} + (C_{1} - \\mu)^{C_{1}} = - 2 C_{1} + \\operatorname{t_{1}}^{C_{1}}{(C_{1},\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\varphi^*')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)), Pow(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Symbol('C_1', commutative=True)), Pow(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))"], [["minus", 5, "Mul(Integer(2), Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Pow(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(M_{E})} = M_{E}, then obtain \\int (- m + \\frac{d}{d M_{E}} \\int \\hat{X}{(M_{E})} dM_{E}) dm = \\int (- m + \\frac{d}{d M_{E}} \\int M_{E} dM_{E}) dm", "derivation": "\\hat{X}{(M_{E})} = M_{E} and \\int \\hat{X}{(M_{E})} dM_{E} = \\int M_{E} dM_{E} and \\frac{d}{d M_{E}} \\int \\hat{X}{(M_{E})} dM_{E} = \\frac{d}{d M_{E}} \\int M_{E} dM_{E} and - m + \\frac{d}{d M_{E}} \\int \\hat{X}{(M_{E})} dM_{E} = - m + \\frac{d}{d M_{E}} \\int M_{E} dM_{E} and \\int (- m + \\frac{d}{d M_{E}} \\int \\hat{X}{(M_{E})} dM_{E}) dm = \\int (- m + \\frac{d}{d M_{E}} \\int M_{E} dM_{E}) dm", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{X}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Integral(Function('\\\\hat{X}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Integral(Function('\\\\hat{X}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Derivative(Integral(Symbol('M_E', commutative=True), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given c{(B)} = e^{B}, then obtain \\frac{B c^{B}{(B)} + c^{B}{(B)}}{c{(B)}} = \\frac{B c^{B}{(B)} + (e^{B})^{B}}{c{(B)}}", "derivation": "c{(B)} = e^{B} and c^{B}{(B)} = (e^{B})^{B} and B c^{B}{(B)} = B (e^{B})^{B} and B (e^{B})^{B} + c^{B}{(B)} = B (e^{B})^{B} + (e^{B})^{B} and \\frac{B (e^{B})^{B} + c^{B}{(B)}}{c{(B)}} = \\frac{B (e^{B})^{B} + (e^{B})^{B}}{c{(B)}} and \\frac{B c^{B}{(B)} + c^{B}{(B)}}{c{(B)}} = \\frac{B c^{B}{(B)} + (e^{B})^{B}}{c{(B)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["times", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["add", 2, "Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], "Equality(Add(Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["divide", 4, "Function('c')(Symbol('B', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('B', commutative=True), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Mul(Symbol('B', commutative=True), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('B', commutative=True), Pow(Function('c')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(exp(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Pow(Function('c')(Symbol('B', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given k{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain \\frac{d}{d \\mathbb{I}} 2 k{(\\mathbb{I})} e^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} 2 e^{2 \\mathbb{I}}", "derivation": "k{(\\mathbb{I})} = e^{\\mathbb{I}} and k{(\\mathbb{I})} + e^{\\mathbb{I}} = 2 e^{\\mathbb{I}} and (k{(\\mathbb{I})} + e^{\\mathbb{I}}) k{(\\mathbb{I})} = (k{(\\mathbb{I})} + e^{\\mathbb{I}}) e^{\\mathbb{I}} and 2 k{(\\mathbb{I})} e^{\\mathbb{I}} = 2 e^{2 \\mathbb{I}} and \\frac{d}{d \\mathbb{I}} 2 k{(\\mathbb{I})} e^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} 2 e^{2 \\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 1, "Add(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Add(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Function('k')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Add(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\nabla)} = e^{\\nabla}, then obtain \\sin{(\\dot{x}{(\\nabla)} + \\dot{x}{(\\nabla)} e^{- \\nabla} + e^{- \\nabla})} = \\sin{(e^{\\nabla} + 1 + e^{- \\nabla})}", "derivation": "\\dot{x}{(\\nabla)} = e^{\\nabla} and \\dot{x}{(\\nabla)} e^{- \\nabla} = 1 and \\dot{x}{(\\nabla)} + e^{- \\nabla} = e^{\\nabla} + e^{- \\nabla} and \\dot{x}{(\\nabla)} + 1 + e^{- \\nabla} = e^{\\nabla} + 1 + e^{- \\nabla} and \\dot{x}{(\\nabla)} e^{- \\nabla} + e^{\\nabla} + e^{- \\nabla} = e^{\\nabla} + 1 + e^{- \\nabla} and \\dot{x}{(\\nabla)} + \\dot{x}{(\\nabla)} e^{- \\nabla} + e^{- \\nabla} = \\dot{x}{(\\nabla)} + 1 + e^{- \\nabla} and \\sin{(\\dot{x}{(\\nabla)} + 1 + e^{- \\nabla})} = \\sin{(e^{\\nabla} + 1 + e^{- \\nabla})} and \\sin{(\\dot{x}{(\\nabla)} + \\dot{x}{(\\nabla)} e^{- \\nabla} + e^{- \\nabla})} = \\sin{(e^{\\nabla} + 1 + e^{- \\nabla})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Integer(1))"], [["add", 1, "exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Add(exp(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["add", 3, 1], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Add(exp(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["add", 2, "Add(exp(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], "Equality(Add(Mul(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), exp(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Add(exp(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), sin(Add(exp(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(sin(Add(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), Mul(Function('\\\\dot{x}')(Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), sin(Add(exp(Symbol('\\\\nabla', commutative=True)), Integer(1), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(c_{0},s)} = \\log{(\\frac{s}{c_{0}})}, then obtain - 2 c_{0} + \\frac{\\mu_{0}{(c_{0},s)}}{c_{0}} - \\frac{\\log{(\\frac{s}{c_{0}})}}{c_{0}} = - 2 c_{0}", "derivation": "\\mu_{0}{(c_{0},s)} = \\log{(\\frac{s}{c_{0}})} and \\frac{\\mu_{0}{(c_{0},s)}}{c_{0}} = \\frac{\\log{(\\frac{s}{c_{0}})}}{c_{0}} and \\frac{\\mu_{0}{(c_{0},s)}}{c_{0}} - \\frac{\\log{(\\frac{s}{c_{0}})}}{c_{0}} = 0 and - c_{0} + \\frac{\\mu_{0}{(c_{0},s)}}{c_{0}} - \\frac{\\log{(\\frac{s}{c_{0}})}}{c_{0}} = - c_{0} and - 2 c_{0} + \\frac{\\mu_{0}{(c_{0},s)}}{c_{0}} - \\frac{\\log{(\\frac{s}{c_{0}})}}{c_{0}} = - 2 c_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('c_0', commutative=True), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["divide", 1, "Symbol('c_0', commutative=True)"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True)))))"], [["minus", 2, "Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True))))), Integer(0))"], [["minus", 3, "Symbol('c_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True))))), Mul(Integer(-1), Symbol('c_0', commutative=True)))"], [["minus", 4, "Symbol('c_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('c_0', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Symbol('s', commutative=True))))), Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(m_{s})} = \\int e^{m_{s}} dm_{s}, then derive \\mathbf{F}{(m_{s})} = \\rho_b + e^{m_{s}}, then obtain 2 \\mathbf{F}{(m_{s})} + 1 = 2 \\rho_b + 2 e^{m_{s}} + 1", "derivation": "\\mathbf{F}{(m_{s})} = \\int e^{m_{s}} dm_{s} and \\mathbf{F}{(m_{s})} = \\rho_b + e^{m_{s}} and \\rho_b + e^{m_{s}} = \\int e^{m_{s}} dm_{s} and \\mathbf{F}{(m_{s})} + 1 = \\int e^{m_{s}} dm_{s} + 1 and \\rho_b + \\mathbf{F}{(m_{s})} + e^{m_{s}} + 1 = \\rho_b + e^{m_{s}} + \\int e^{m_{s}} dm_{s} + 1 and \\mathbf{F}{(m_{s})} + \\int e^{m_{s}} dm_{s} + 1 = 2 \\int e^{m_{s}} dm_{s} + 1 and 2 \\mathbf{F}{(m_{s})} + 1 = 2 \\int e^{m_{s}} dm_{s} + 1 and 2 \\mathbf{F}{(m_{s})} + 1 = 2 \\rho_b + 2 e^{m_{s}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True)), Integer(1)), Add(Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integer(1)))"], [["add", 4, "Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('m_s', commutative=True)))"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)), Integer(1)), Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('m_s', commutative=True)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integer(1)), Add(Mul(Integer(2), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True))), Integer(1)), Add(Mul(Integer(2), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('m_s', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), exp(Symbol('m_s', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\psi{(\\mathbf{A},A_{1})} = \\frac{\\cos{(A_{1})}}{\\mathbf{A}}, then obtain (\\int \\psi{(\\mathbf{A},A_{1})} dA_{1})^{2} = (\\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1})^{2}", "derivation": "\\psi{(\\mathbf{A},A_{1})} = \\frac{\\cos{(A_{1})}}{\\mathbf{A}} and \\int \\psi{(\\mathbf{A},A_{1})} dA_{1} = \\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1} and \\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1} = \\frac{(\\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1})^{2}}{\\int \\psi{(\\mathbf{A},A_{1})} dA_{1}} and \\int \\psi{(\\mathbf{A},A_{1})} dA_{1} = \\frac{(\\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1})^{2}}{\\int \\psi{(\\mathbf{A},A_{1})} dA_{1}} and (\\int \\psi{(\\mathbf{A},A_{1})} dA_{1})^{2} = (\\int \\frac{\\cos{(A_{1})}}{\\mathbf{A}} dA_{1})^{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"], [["divide", 2, "Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integer(-1)), Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integer(2)), Pow(Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integer(2)), Pow(Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integer(-1))))"], [["times", 4, "Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integer(2)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\chi{(r_{0},\\mathbf{J}_f)} = \\mathbf{J}_f r_{0}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} \\chi{(r_{0},\\mathbf{J}_f)} = r_{0}, then obtain \\dot{x} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f r_{0} = \\dot{x} r_{0}", "derivation": "\\chi{(r_{0},\\mathbf{J}_f)} = \\mathbf{J}_f r_{0} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\chi{(r_{0},\\mathbf{J}_f)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f r_{0} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\chi{(r_{0},\\mathbf{J}_f)} = r_{0} and \\dot{x} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\chi{(r_{0},\\mathbf{J}_f)} = \\dot{x} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f r_{0} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f r_{0} = r_{0} and \\dot{x} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\chi{(r_{0},\\mathbf{J}_f)} = \\dot{x} r_{0} and \\dot{x} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f r_{0} = \\dot{x} r_{0}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\chi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["times", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Function('\\\\chi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Function('\\\\chi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given Z{(z,\\Omega)} = \\frac{\\Omega}{z}, then obtain - \\log{(\\pi + Z{(z,\\Omega)})} + \\frac{Z{(z,\\Omega)}}{z} = \\frac{\\Omega}{z^{2}} - \\log{(\\pi + Z{(z,\\Omega)})}", "derivation": "Z{(z,\\Omega)} = \\frac{\\Omega}{z} and \\frac{\\partial}{\\partial z} Z{(z,\\Omega)} = \\frac{\\partial}{\\partial z} \\frac{\\Omega}{z} and \\int \\frac{\\partial}{\\partial z} Z{(z,\\Omega)} dz = \\int \\frac{\\partial}{\\partial z} \\frac{\\Omega}{z} dz and \\frac{Z{(z,\\Omega)}}{z} = \\frac{\\Omega}{z^{2}} and - \\log{(\\int \\frac{\\partial}{\\partial z} \\frac{\\Omega}{z} dz)} + \\frac{Z{(z,\\Omega)}}{z} = \\frac{\\Omega}{z^{2}} - \\log{(\\int \\frac{\\partial}{\\partial z} \\frac{\\Omega}{z} dz)} and - \\log{(\\int \\frac{\\partial}{\\partial z} Z{(z,\\Omega)} dz)} + \\frac{Z{(z,\\Omega)}}{z} = \\frac{\\Omega}{z^{2}} - \\log{(\\int \\frac{\\partial}{\\partial z} Z{(z,\\Omega)} dz)} and - \\log{(\\pi + Z{(z,\\Omega)})} + \\frac{Z{(z,\\Omega)}}{z} = \\frac{\\Omega}{z^{2}} - \\log{(\\pi + Z{(z,\\Omega)})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Derivative(Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))))"], [["divide", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-2))))"], [["minus", 4, "log(Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))))"], "Equality(Add(Mul(Integer(-1), log(Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-2))), Mul(Integer(-1), log(Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), log(Integral(Derivative(Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-2))), Mul(Integer(-1), log(Integral(Derivative(Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('\\\\pi', commutative=True), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True))))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('z', commutative=True), Integer(-2))), Mul(Integer(-1), log(Add(Symbol('\\\\pi', commutative=True), Function('Z')(Symbol('z', commutative=True), Symbol('\\\\Omega', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}}, then derive \\operatorname{v_{2}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\operatorname{v_{2}}{(J_{\\varepsilon})} (e^{J_{\\varepsilon}})^{J_{\\varepsilon}} = e^{J_{\\varepsilon}} (e^{J_{\\varepsilon}})^{J_{\\varepsilon}}", "derivation": "\\operatorname{v_{2}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and \\operatorname{v_{2}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and e^{J_{\\varepsilon}} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and \\operatorname{v_{2}}{(J_{\\varepsilon})} (\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})^{J_{\\varepsilon}} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} (\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})^{J_{\\varepsilon}} and \\operatorname{v_{2}}{(J_{\\varepsilon})} (e^{J_{\\varepsilon}})^{J_{\\varepsilon}} = e^{J_{\\varepsilon}} (e^{J_{\\varepsilon}})^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["times", 1, "Pow(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('v_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('v_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(H)} = \\log{(H)}, then obtain (\\frac{d}{d H} \\operatorname{C_{2}}{(H)})^{H} = (\\frac{1}{H})^{H}", "derivation": "\\operatorname{C_{2}}{(H)} = \\log{(H)} and \\frac{d}{d H} \\operatorname{C_{2}}{(H)} = \\frac{d}{d H} \\log{(H)} and (\\frac{d}{d H} \\operatorname{C_{2}}{(H)})^{H} = (\\frac{d}{d H} \\log{(H)})^{H} and (\\frac{d}{d H} \\operatorname{C_{2}}{(H)})^{H} = (\\frac{1}{H})^{H}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Function('C_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('C_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True)))"]]}, {"prompt": "Given H{(a)} = \\sin{(a)} and \\mathbf{M}{(\\dot{x},\\varphi)} = - \\sin{(\\dot{x} - \\varphi)}, then obtain - a + \\mathbf{M}{(\\dot{x},\\varphi)} + \\int H{(a)} da = - a - \\sin{(\\dot{x} - \\varphi)} + \\int H{(a)} da", "derivation": "H{(a)} = \\sin{(a)} and \\int H{(a)} da = \\int \\sin{(a)} da and \\mathbf{M}{(\\dot{x},\\varphi)} = - \\sin{(\\dot{x} - \\varphi)} and \\mathbf{M}{(\\dot{x},\\varphi)} + \\int \\sin{(a)} da = - \\sin{(\\dot{x} - \\varphi)} + \\int \\sin{(a)} da and \\mathbf{M}{(\\dot{x},\\varphi)} + \\int H{(a)} da = - \\sin{(\\dot{x} - \\varphi)} + \\int H{(a)} da and - a + \\mathbf{M}{(\\dot{x},\\varphi)} + \\int H{(a)} da = - a - \\sin{(\\dot{x} - \\varphi)} + \\int H{(a)} da", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))))"], [["add", 3, "Integral(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))), Integral(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Function('H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))), Integral(Function('H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["minus", 5, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Function('H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))), Integral(Function('H')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(t)} = e^{t}, then obtain - \\operatorname{t_{2}}{(t)} - \\operatorname{t_{2}}^{t}{(t)} + \\iint \\operatorname{t_{2}}{(t)} dt dt = - \\operatorname{t_{2}}{(t)} - \\operatorname{t_{2}}^{t}{(t)} + \\iint e^{t} dt dt", "derivation": "\\operatorname{t_{2}}{(t)} = e^{t} and \\int \\operatorname{t_{2}}{(t)} dt = \\int e^{t} dt and \\iint \\operatorname{t_{2}}{(t)} dt dt = \\iint e^{t} dt dt and - \\operatorname{t_{2}}{(t)} - \\operatorname{t_{2}}^{t}{(t)} + \\iint \\operatorname{t_{2}}{(t)} dt dt = - \\operatorname{t_{2}}{(t)} - \\operatorname{t_{2}}^{t}{(t)} + \\iint e^{t} dt dt", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["minus", 3, "Add(Function('t_2')(Symbol('t', commutative=True)), Pow(Function('t_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('t_2')(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('t_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Integral(Function('t_2')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('t_2')(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Function('t_2')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(a,x^\\prime)} = \\log{(a^{x^\\prime})} and \\pi{(a,x^\\prime)} = \\log{(a^{x^\\prime})}^{a}, then obtain \\pi{(a,x^\\prime)} = \\operatorname{f_{\\mathbf{v}}}^{a}{(a,x^\\prime)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(a,x^\\prime)} = \\log{(a^{x^\\prime})} and \\operatorname{f_{\\mathbf{v}}}^{a}{(a,x^\\prime)} = \\log{(a^{x^\\prime})}^{a} and \\pi{(a,x^\\prime)} = \\log{(a^{x^\\prime})}^{a} and \\pi{(a,x^\\prime)} = \\operatorname{f_{\\mathbf{v}}}^{a}{(a,x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Pow(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('a', commutative=True)), Pow(log(Pow(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(log(Pow(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\pi')(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('f_{\\\\mathbf{v}}')(Symbol('a', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(F_{c},v_{y})} = F_{c} - v_{y}, then obtain (v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})}) (F_{c} - v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})}) = F_{c} (F_{c} - v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})})", "derivation": "\\operatorname{A_{x}}{(F_{c},v_{y})} = F_{c} - v_{y} and v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})} = F_{c} and F_{c} - v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})} = 2 F_{c} - 2 v_{y} and (2 F_{c} - 2 v_{y}) (v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})}) = F_{c} (2 F_{c} - 2 v_{y}) and (v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})}) (F_{c} - v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})}) = F_{c} (F_{c} - v_{y} + \\operatorname{A_{x}}{(F_{c},v_{y})})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('v_y', commutative=True))"], "Equality(Add(Symbol('v_y', commutative=True), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Symbol('F_c', commutative=True))"], [["add", 1, "Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)))"], "Equality(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True))))"], [["times", 2, "Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True))), Add(Symbol('v_y', commutative=True), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('F_c', commutative=True), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('v_y', commutative=True), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('F_c', commutative=True), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('A_x')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(F_{c},\\dot{y})} = F_{c} \\dot{y}, then obtain - 2 \\int F_{c} \\dot{y} d\\dot{y} + 2 \\int \\varepsilon{(F_{c},\\dot{y})} d\\dot{y} = 0", "derivation": "\\varepsilon{(F_{c},\\dot{y})} = F_{c} \\dot{y} and \\int \\varepsilon{(F_{c},\\dot{y})} d\\dot{y} = \\int F_{c} \\dot{y} d\\dot{y} and - \\int F_{c} \\dot{y} d\\dot{y} + \\int \\varepsilon{(F_{c},\\dot{y})} d\\dot{y} = 0 and - 2 \\int F_{c} \\dot{y} d\\dot{y} + \\int \\varepsilon{(F_{c},\\dot{y})} d\\dot{y} = - \\int F_{c} \\dot{y} d\\dot{y} and - 2 \\int F_{c} \\dot{y} d\\dot{y} + 2 \\int \\varepsilon{(F_{c},\\dot{y})} d\\dot{y} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Integral(Function('\\\\varepsilon')(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Integer(0))"], [["minus", 3, "Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Integral(Function('\\\\varepsilon')(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(2), Integral(Function('\\\\varepsilon')(Symbol('F_c', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{D}{(t,C)} = \\frac{\\partial}{\\partial C} (C - t) and \\mathbf{p}{(t,C)} = \\frac{\\partial}{\\partial C} (C - t), then obtain -1 = \\mathbf{D}{(t,C)} - \\mathbf{p}{(t,C)} - 1", "derivation": "\\mathbf{D}{(t,C)} = \\frac{\\partial}{\\partial C} (C - t) and \\mathbf{D}{(t,C)} + 1 = \\frac{\\partial}{\\partial C} (C - t) + 1 and - \\mathbf{D}{(t,C)} - 1 = - \\frac{\\partial}{\\partial C} (C - t) - 1 and \\mathbf{p}{(t,C)} = \\frac{\\partial}{\\partial C} (C - t) and - \\mathbf{D}{(t,C)} - 1 = - \\mathbf{p}{(t,C)} - 1 and -1 = \\mathbf{D}{(t,C)} - \\mathbf{p}{(t,C)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True)), Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True)), Integer(1)), Add(Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(1)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('t', commutative=True), Symbol('C', commutative=True)), Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('t', commutative=True), Symbol('C', commutative=True))), Integer(-1)))"], [["add", 5, "Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True))"], "Equality(Integer(-1), Add(Function('\\\\mathbf{D}')(Symbol('t', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('t', commutative=True), Symbol('C', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{X},v_{1})} = \\frac{\\partial}{\\partial \\hat{X}} v_{1}^{\\hat{X}}, then derive v_{1} \\mathbf{J}_M{(\\hat{X},v_{1})} = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})}, then obtain v_{1} \\mathbf{J}_M{(\\hat{X},v_{1})} - 1 = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})} - 1", "derivation": "\\mathbf{J}_M{(\\hat{X},v_{1})} = \\frac{\\partial}{\\partial \\hat{X}} v_{1}^{\\hat{X}} and v_{1} \\mathbf{J}_M{(\\hat{X},v_{1})} = v_{1} \\frac{\\partial}{\\partial \\hat{X}} v_{1}^{\\hat{X}} and v_{1} \\mathbf{J}_M{(\\hat{X},v_{1})} = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})} and v_{1} \\frac{\\partial}{\\partial \\hat{X}} v_{1}^{\\hat{X}} = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})} and v_{1} \\frac{\\partial}{\\partial \\hat{X}} v_{1}^{\\hat{X}} - 1 = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})} - 1 and v_{1} \\mathbf{J}_M{(\\hat{X},v_{1})} - 1 = v_{1} v_{1}^{\\hat{X}} \\log{(v_{1})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_1', commutative=True)), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["times", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('v_1', commutative=True), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('v_1', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Mul(Symbol('v_1', commutative=True), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('v_1', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_1', commutative=True))), Integer(-1)), Add(Mul(Symbol('v_1', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), log(Symbol('v_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(M,l)} = M + l, then obtain M l^{2} \\operatorname{F_{N}}^{2}{(M,l)} = M l^{2} (M + l)^{2}", "derivation": "\\operatorname{F_{N}}{(M,l)} = M + l and M \\operatorname{F_{N}}{(M,l)} = M (M + l) and M (M + l) \\operatorname{F_{N}}{(M,l)} = M (M + l)^{2} and M \\operatorname{F_{N}}^{2}{(M,l)} = M (M + l) \\operatorname{F_{N}}{(M,l)} and M \\operatorname{F_{N}}^{2}{(M,l)} = M (M + l)^{2} and M l^{2} \\operatorname{F_{N}}^{2}{(M,l)} = M l^{2} (M + l)^{2}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True)), Add(Symbol('M', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('M', commutative=True), Add(Symbol('M', commutative=True), Symbol('l', commutative=True))))"], [["times", 2, "Add(Symbol('M', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Symbol('M', commutative=True), Add(Symbol('M', commutative=True), Symbol('l', commutative=True)), Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('M', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('M', commutative=True), Pow(Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))), Mul(Symbol('M', commutative=True), Add(Symbol('M', commutative=True), Symbol('l', commutative=True)), Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('M', commutative=True), Pow(Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))), Mul(Symbol('M', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))))"], [["times", 5, "Pow(Symbol('l', commutative=True), Integer(2))"], "Equality(Mul(Symbol('M', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('F_N')(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))), Mul(Symbol('M', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)), Pow(Add(Symbol('M', commutative=True), Symbol('l', commutative=True)), Integer(2))))"]]}, {"prompt": "Given r{(y)} = \\log{(y)} and x{(y)} = \\iint \\log{(y)} dy dy, then obtain \\int \\frac{d}{d y} x{(y)} dy = \\int \\frac{d}{d y} \\iint \\log{(y)} dy dy dy", "derivation": "r{(y)} = \\log{(y)} and \\int r{(y)} dy = \\int \\log{(y)} dy and \\iint r{(y)} dy dy = \\iint \\log{(y)} dy dy and x{(y)} = \\iint \\log{(y)} dy dy and x{(y)} = \\iint r{(y)} dy dy and \\frac{d}{d y} x{(y)} = \\frac{d}{d y} \\iint r{(y)} dy dy and \\int \\frac{d}{d y} x{(y)} dy = \\int \\frac{d}{d y} \\iint r{(y)} dy dy dy and \\int \\frac{d}{d y} x{(y)} dy = \\int \\frac{d}{d y} \\iint \\log{(y)} dy dy dy", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('r')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Function('r')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('y', commutative=True)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('x')(Symbol('y', commutative=True)), Integral(Function('r')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["differentiate", 5, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integral(Function('r')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Integral(Function('r')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integral(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given q{(M,v_{x},n)} = (n^{v_{x}})^{M}, then obtain \\frac{\\partial}{\\partial v_{x}} ((\\frac{\\partial}{\\partial M} \\int q{(M,v_{x},n)} dM)^{n} + 1) = \\frac{\\partial}{\\partial v_{x}} ((\\frac{\\partial}{\\partial M} \\int (n^{v_{x}})^{M} dM)^{n} + 1)", "derivation": "q{(M,v_{x},n)} = (n^{v_{x}})^{M} and \\int q{(M,v_{x},n)} dM = \\int (n^{v_{x}})^{M} dM and \\frac{\\partial}{\\partial M} \\int q{(M,v_{x},n)} dM = \\frac{\\partial}{\\partial M} \\int (n^{v_{x}})^{M} dM and (\\frac{\\partial}{\\partial M} \\int q{(M,v_{x},n)} dM)^{n} = (\\frac{\\partial}{\\partial M} \\int (n^{v_{x}})^{M} dM)^{n} and (\\frac{\\partial}{\\partial M} \\int q{(M,v_{x},n)} dM)^{n} + 1 = (\\frac{\\partial}{\\partial M} \\int (n^{v_{x}})^{M} dM)^{n} + 1 and \\frac{\\partial}{\\partial v_{x}} ((\\frac{\\partial}{\\partial M} \\int q{(M,v_{x},n)} dM)^{n} + 1) = \\frac{\\partial}{\\partial v_{x}} ((\\frac{\\partial}{\\partial M} \\int (n^{v_{x}})^{M} dM)^{n} + 1)", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(Integral(Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["add", 4, 1], "Equality(Add(Pow(Derivative(Integral(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1)), Add(Pow(Derivative(Integral(Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1)))"], [["differentiate", 5, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Pow(Derivative(Integral(Function('q')(Symbol('M', commutative=True), Symbol('v_x', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Pow(Derivative(Integral(Pow(Pow(Symbol('n', commutative=True), Symbol('v_x', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(f_{\\mathbf{v}})} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}, then derive \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{D}{(f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\psi^* + e^{f_{\\mathbf{v}}}), then obtain \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{D}{(f_{\\mathbf{v}})} - \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\psi^* + e^{f_{\\mathbf{v}}}) - \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}", "derivation": "\\mathbf{D}{(f_{\\mathbf{v}})} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{D}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{D}{(f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\psi^* + e^{f_{\\mathbf{v}}}) and \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{D}{(f_{\\mathbf{v}})} - \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\psi^* + e^{f_{\\mathbf{v}}}) - \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["minus", 3, "Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\mathbf{D}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Mul(Integer(-1), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))), Add(Derivative(Add(Symbol('\\\\psi^*', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Mul(Integer(-1), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(k)} = e^{\\sin{(k)}} and \\mathbf{D}{(k)} = \\cos{(e^{\\sin{(k)}})} \\frac{d}{d k} e^{\\sin{(k)}}, then derive \\cos{(\\Omega{(k)})} \\frac{d}{d k} \\Omega{(k)} = e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})}, then obtain \\frac{d}{d k} \\mathbf{D}{(k)} = \\frac{d}{d k} e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})}", "derivation": "\\Omega{(k)} = e^{\\sin{(k)}} and \\sin{(\\Omega{(k)})} = \\sin{(e^{\\sin{(k)}})} and \\frac{d}{d k} \\sin{(\\Omega{(k)})} = \\frac{d}{d k} \\sin{(e^{\\sin{(k)}})} and \\cos{(\\Omega{(k)})} \\frac{d}{d k} \\Omega{(k)} = e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})} and \\cos{(e^{\\sin{(k)}})} \\frac{d}{d k} e^{\\sin{(k)}} = e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})} and \\mathbf{D}{(k)} = \\cos{(e^{\\sin{(k)}})} \\frac{d}{d k} e^{\\sin{(k)}} and \\mathbf{D}{(k)} = e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})} and \\frac{d}{d k} \\mathbf{D}{(k)} = \\frac{d}{d k} e^{\\sin{(k)}} \\cos{(k)} \\cos{(e^{\\sin{(k)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('k', commutative=True)), exp(sin(Symbol('k', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\Omega')(Symbol('k', commutative=True))), sin(exp(sin(Symbol('k', commutative=True)))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(sin(Function('\\\\Omega')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(exp(sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Function('\\\\Omega')(Symbol('k', commutative=True))), Derivative(Function('\\\\Omega')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(exp(sin(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True)), cos(exp(sin(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(cos(exp(sin(Symbol('k', commutative=True)))), Derivative(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(exp(sin(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True)), cos(exp(sin(Symbol('k', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('k', commutative=True)), Mul(cos(exp(sin(Symbol('k', commutative=True)))), Derivative(exp(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\mathbf{D}')(Symbol('k', commutative=True)), Mul(exp(sin(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True)), cos(exp(sin(Symbol('k', commutative=True))))))"], [["differentiate", 7, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(exp(sin(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True)), cos(exp(sin(Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(F_{N})} = e^{F_{N}}, then obtain \\mathbf{J}_f{(F_{N})} + 2 e^{F_{N}} = 2 \\mathbf{J}_f{(F_{N})} + e^{F_{N}}", "derivation": "\\mathbf{J}_f{(F_{N})} = e^{F_{N}} and 2 \\mathbf{J}_f{(F_{N})} = \\mathbf{J}_f{(F_{N})} + e^{F_{N}} and 3 \\mathbf{J}_f{(F_{N})} = 2 \\mathbf{J}_f{(F_{N})} + e^{F_{N}} and 3 \\mathbf{J}_f{(F_{N})} = \\mathbf{J}_f{(F_{N})} + 2 e^{F_{N}} and \\mathbf{J}_f{(F_{N})} + 2 e^{F_{N}} = 2 \\mathbf{J}_f{(F_{N})} + e^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), Add(Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), Add(Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)), Mul(Integer(2), exp(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True)), Mul(Integer(2), exp(Symbol('F_N', commutative=True)))), Add(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\Psi,\\eta^{\\prime})} = \\Psi \\eta^{\\prime} and \\pi{(\\phi_1,E)} = E \\phi_1, then obtain (\\frac{\\partial}{\\partial E} \\pi{(\\phi_1,E)})^{\\phi_1} - 1 = (\\frac{\\partial}{\\partial E} E \\phi_1)^{\\phi_1} - 1", "derivation": "\\operatorname{t_{1}}{(\\Psi,\\eta^{\\prime})} = \\Psi \\eta^{\\prime} and \\pi{(\\phi_1,E)} = E \\phi_1 and \\frac{\\partial}{\\partial E} \\pi{(\\phi_1,E)} = \\frac{\\partial}{\\partial E} E \\phi_1 and (\\frac{\\partial}{\\partial E} \\pi{(\\phi_1,E)})^{\\phi_1} = (\\frac{\\partial}{\\partial E} E \\phi_1)^{\\phi_1} and (\\frac{\\partial}{\\partial E} \\pi{(\\phi_1,E)})^{\\phi_1} - \\frac{\\operatorname{t_{1}}{(\\Psi,\\eta^{\\prime})}}{\\Psi \\eta^{\\prime}} = (\\frac{\\partial}{\\partial E} E \\phi_1)^{\\phi_1} - \\frac{\\operatorname{t_{1}}{(\\Psi,\\eta^{\\prime})}}{\\Psi \\eta^{\\prime}} and (\\frac{\\partial}{\\partial E} \\pi{(\\phi_1,E)})^{\\phi_1} - 1 = (\\frac{\\partial}{\\partial E} E \\phi_1)^{\\phi_1} - 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Symbol('E', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Mul(Symbol('E', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], [["minus", 4, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Pow(Derivative(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Pow(Derivative(Mul(Symbol('E', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Pow(Derivative(Function('\\\\pi')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Integer(-1)), Add(Pow(Derivative(Mul(Symbol('E', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\rho_b,t,y)} = - t + \\frac{y}{\\rho_b}, then derive \\int 2 \\mathbf{J}_M{(\\rho_b,t,y)} dt = q + \\frac{\\int y dt + \\int - \\rho_b t dt + \\int \\rho_b \\mathbf{J}_M{(\\rho_b,t,y)} dt}{\\rho_b}, then obtain 1 = \\frac{q + \\frac{\\int y dt + \\int - \\rho_b t dt + \\int \\rho_b \\mathbf{J}_M{(\\rho_b,t,y)} dt}{\\rho_b}}{\\int 2 \\mathbf{J}_M{(\\rho_b,t,y)} dt}", "derivation": "\\mathbf{J}_M{(\\rho_b,t,y)} = - t + \\frac{y}{\\rho_b} and 2 \\mathbf{J}_M{(\\rho_b,t,y)} = - t + \\mathbf{J}_M{(\\rho_b,t,y)} + \\frac{y}{\\rho_b} and \\int 2 \\mathbf{J}_M{(\\rho_b,t,y)} dt = \\int (- t + \\mathbf{J}_M{(\\rho_b,t,y)} + \\frac{y}{\\rho_b}) dt and \\int 2 \\mathbf{J}_M{(\\rho_b,t,y)} dt = q + \\frac{\\int y dt + \\int - \\rho_b t dt + \\int \\rho_b \\mathbf{J}_M{(\\rho_b,t,y)} dt}{\\rho_b} and 1 = \\frac{q + \\frac{\\int y dt + \\int - \\rho_b t dt + \\int \\rho_b \\mathbf{J}_M{(\\rho_b,t,y)} dt}{\\rho_b}}{\\int 2 \\mathbf{J}_M{(\\rho_b,t,y)} dt}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True))), Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Integral(Symbol('y', commutative=True), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True)))))))"], [["divide", 4, "Integral(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Integral(Symbol('y', commutative=True), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True)))))), Pow(Integral(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(A_{1})} = \\sin{(A_{1})}, then obtain (\\frac{d}{d A_{1}} \\varepsilon{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}})^{A_{1}} = (\\frac{d}{d A_{1}} \\sin{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}})^{A_{1}}", "derivation": "\\varepsilon{(A_{1})} = \\sin{(A_{1})} and \\frac{d}{d A_{1}} \\varepsilon{(A_{1})} = \\frac{d}{d A_{1}} \\sin{(A_{1})} and \\frac{d}{d A_{1}} \\varepsilon{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}} = \\frac{d}{d A_{1}} \\sin{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}} and (\\frac{d}{d A_{1}} \\varepsilon{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}})^{A_{1}} = (\\frac{d}{d A_{1}} \\sin{(A_{1})} - \\frac{1}{\\varepsilon{(A_{1})}})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(sin(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Integer(-1)))), Add(Derivative(sin(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Integer(-1)))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Integer(-1)))), Symbol('A_1', commutative=True)), Pow(Add(Derivative(sin(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Function('\\\\varepsilon')(Symbol('A_1', commutative=True)), Integer(-1)))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given s{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain s{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} = e^{\\mathbf{E}} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}}", "derivation": "s{(\\mathbf{E})} = e^{\\mathbf{E}} and \\frac{d}{d \\mathbf{E}} s{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} and s{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} s{(\\mathbf{E})} = e^{\\mathbf{E}} \\frac{d}{d \\mathbf{E}} s{(\\mathbf{E})} and s{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} = e^{\\mathbf{E}} \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('s')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(z,r)} = \\cos^{r}{(z)} and \\operatorname{F_{H}}{(z,r)} = (\\operatorname{A_{1}}^{r}{(z,r)})^{r}, then obtain \\operatorname{A_{1}}^{r}{(z,r)} - \\operatorname{F_{H}}{(z,r)} = (\\cos^{r}{(z)})^{r} - \\operatorname{F_{H}}{(z,r)}", "derivation": "\\operatorname{A_{1}}{(z,r)} = \\cos^{r}{(z)} and \\operatorname{A_{1}}^{r}{(z,r)} = (\\cos^{r}{(z)})^{r} and (\\operatorname{A_{1}}^{r}{(z,r)})^{r} = ((\\cos^{r}{(z)})^{r})^{r} and - ((\\cos^{r}{(z)})^{r})^{r} + \\operatorname{A_{1}}^{r}{(z,r)} = (\\cos^{r}{(z)})^{r} - ((\\cos^{r}{(z)})^{r})^{r} and \\operatorname{F_{H}}{(z,r)} = (\\operatorname{A_{1}}^{r}{(z,r)})^{r} and \\operatorname{F_{H}}{(z,r)} = ((\\cos^{r}{(z)})^{r})^{r} and \\operatorname{A_{1}}^{r}{(z,r)} - \\operatorname{F_{H}}{(z,r)} = (\\cos^{r}{(z)})^{r} - \\operatorname{F_{H}}{(z,r)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["minus", 2, "Pow(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True))), Pow(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True))), Add(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Pow(Pow(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('F_H')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Pow(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Pow(Function('A_1')(Symbol('z', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), Function('F_H')(Symbol('z', commutative=True), Symbol('r', commutative=True)))), Add(Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(-1), Function('F_H')(Symbol('z', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given Q{(P_{e})} = \\sin{(P_{e})} and \\operatorname{F_{g}}{(P_{e})} = - Q{(P_{e})} \\sin{(P_{e})} + Q{(P_{e})}, then obtain \\frac{\\operatorname{F_{g}}{(P_{e})}}{\\sin^{2}{(P_{e})}} = \\frac{- \\sin^{2}{(P_{e})} + \\sin{(P_{e})}}{\\sin^{2}{(P_{e})}}", "derivation": "Q{(P_{e})} = \\sin{(P_{e})} and \\operatorname{F_{g}}{(P_{e})} = - Q{(P_{e})} \\sin{(P_{e})} + Q{(P_{e})} and \\frac{\\operatorname{F_{g}}{(P_{e})}}{Q{(P_{e})} \\sin{(P_{e})}} = \\frac{- Q{(P_{e})} \\sin{(P_{e})} + Q{(P_{e})}}{Q{(P_{e})} \\sin{(P_{e})}} and \\frac{\\operatorname{F_{g}}{(P_{e})}}{Q^{2}{(P_{e})}} = \\frac{- Q^{2}{(P_{e})} + Q{(P_{e})}}{Q^{2}{(P_{e})}} and \\frac{\\operatorname{F_{g}}{(P_{e})}}{\\sin^{2}{(P_{e})}} = \\frac{- \\sin^{2}{(P_{e})} + \\sin{(P_{e})}}{\\sin^{2}{(P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('P_e', commutative=True)), Add(Mul(Integer(-1), Function('Q')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))), Function('Q')(Symbol('P_e', commutative=True))))"], [["divide", 2, "Mul(Function('Q')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], "Equality(Mul(Function('F_g')(Symbol('P_e', commutative=True)), Pow(Function('Q')(Symbol('P_e', commutative=True)), Integer(-1)), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Function('Q')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))), Function('Q')(Symbol('P_e', commutative=True))), Pow(Function('Q')(Symbol('P_e', commutative=True)), Integer(-1)), Pow(sin(Symbol('P_e', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('F_g')(Symbol('P_e', commutative=True)), Pow(Function('Q')(Symbol('P_e', commutative=True)), Integer(-2))), Mul(Add(Mul(Integer(-1), Pow(Function('Q')(Symbol('P_e', commutative=True)), Integer(2))), Function('Q')(Symbol('P_e', commutative=True))), Pow(Function('Q')(Symbol('P_e', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('F_g')(Symbol('P_e', commutative=True)), Pow(sin(Symbol('P_e', commutative=True)), Integer(-2))), Mul(Add(Mul(Integer(-1), Pow(sin(Symbol('P_e', commutative=True)), Integer(2))), sin(Symbol('P_e', commutative=True))), Pow(sin(Symbol('P_e', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given Z{(S,\\mathbf{J}_P)} = \\cos{(S \\mathbf{J}_P)}, then derive \\frac{\\partial}{\\partial S} Z{(S,\\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(S \\mathbf{J}_P)}, then obtain \\frac{\\partial}{\\partial S} Z{(S,\\mathbf{J}_P)} \\frac{\\partial}{\\partial S} \\cos{(S \\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(S \\mathbf{J}_P)} \\frac{\\partial}{\\partial S} \\cos{(S \\mathbf{J}_P)}", "derivation": "Z{(S,\\mathbf{J}_P)} = \\cos{(S \\mathbf{J}_P)} and \\frac{\\partial}{\\partial S} Z{(S,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial S} \\cos{(S \\mathbf{J}_P)} and \\frac{\\partial}{\\partial S} Z{(S,\\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(S \\mathbf{J}_P)} and \\frac{\\partial}{\\partial S} Z{(S,\\mathbf{J}_P)} \\frac{\\partial}{\\partial S} \\cos{(S \\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(S \\mathbf{J}_P)} \\frac{\\partial}{\\partial S} \\cos{(S \\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["times", 3, "Derivative(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Derivative(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mathbb{I},p)} = - \\mathbb{I} + p and M{(B,M_{E})} = M_{E}^{B}, then obtain - \\operatorname{C_{1}}^{2}{(\\mathbb{I},p)} + M{(B,M_{E})} = M_{E}^{B} - \\operatorname{C_{1}}^{2}{(\\mathbb{I},p)}", "derivation": "\\operatorname{C_{1}}{(\\mathbb{I},p)} = - \\mathbb{I} + p and M{(B,M_{E})} = M_{E}^{B} and - (- \\mathbb{I} + p)^{2} + M{(B,M_{E})} = M_{E}^{B} - (- \\mathbb{I} + p)^{2} and - \\operatorname{C_{1}}^{2}{(\\mathbb{I},p)} + M{(B,M_{E})} = M_{E}^{B} - \\operatorname{C_{1}}^{2}{(\\mathbb{I},p)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('M')(Symbol('B', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('B', commutative=True)))"], [["minus", 2, "Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('p', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('p', commutative=True)), Integer(2))), Function('M')(Symbol('B', commutative=True), Symbol('M_E', commutative=True))), Add(Pow(Symbol('M_E', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('p', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)), Integer(2))), Function('M')(Symbol('B', commutative=True), Symbol('M_E', commutative=True))), Add(Pow(Symbol('M_E', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('C_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('p', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(U)} = e^{U} and a{(U)} = - ((e^{U})^{U})^{U}, then obtain (- ((e^{U})^{U})^{U} + (e^{U})^{U}) (\\hat{\\mathbf{r}}^{U}{(U)} + a{(U)}) = (- ((e^{U})^{U})^{U} + (e^{U})^{U}) (a{(U)} + (e^{U})^{U})", "derivation": "\\hat{\\mathbf{r}}{(U)} = e^{U} and \\hat{\\mathbf{r}}^{U}{(U)} = (e^{U})^{U} and - ((e^{U})^{U})^{U} + \\hat{\\mathbf{r}}^{U}{(U)} = - ((e^{U})^{U})^{U} + (e^{U})^{U} and a{(U)} = - ((e^{U})^{U})^{U} and \\hat{\\mathbf{r}}^{U}{(U)} + a{(U)} = a{(U)} + (e^{U})^{U} and (- ((e^{U})^{U})^{U} + (e^{U})^{U}) (\\hat{\\mathbf{r}}^{U}{(U)} + a{(U)}) = (- ((e^{U})^{U})^{U} + (e^{U})^{U}) (a{(U)} + (e^{U})^{U})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["minus", 2, "Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('a')(Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Function('a')(Symbol('U', commutative=True))), Add(Function('a')(Symbol('U', commutative=True)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Add(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Function('a')(Symbol('U', commutative=True)))), Mul(Add(Mul(Integer(-1), Pow(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Add(Function('a')(Symbol('U', commutative=True)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given n{(S,M)} = M S, then obtain n^{3}{(S,M)} = M S n^{2}{(S,M)}", "derivation": "n{(S,M)} = M S and n^{2}{(S,M)} = M S n{(S,M)} and M S n^{2}{(S,M)} = M^{2} S^{2} n{(S,M)} and n^{3}{(S,M)} = M S n^{2}{(S,M)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('S', commutative=True)))"], [["times", 1, "Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True))"], "Equality(Pow(Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True)), Integer(2)), Mul(Symbol('M', commutative=True), Symbol('S', commutative=True), Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True))))"], [["times", 2, "Mul(Symbol('M', commutative=True), Symbol('S', commutative=True))"], "Equality(Mul(Symbol('M', commutative=True), Symbol('S', commutative=True), Pow(Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True)), Integer(2))), Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('S', commutative=True), Integer(2)), Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True)), Integer(3)), Mul(Symbol('M', commutative=True), Symbol('S', commutative=True), Pow(Function('n')(Symbol('S', commutative=True), Symbol('M', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{D}{(\\psi^*)} = \\cos{(\\psi^*)}, then obtain (\\mathbf{D}^{2}{(\\psi^*)} - \\mathbf{D}{(\\psi^*)} \\cos{(\\psi^*)} + \\cos{(\\psi^*)})^{2} = \\cos^{2}{(\\psi^*)}", "derivation": "\\mathbf{D}{(\\psi^*)} = \\cos{(\\psi^*)} and \\mathbf{D}^{2}{(\\psi^*)} = \\mathbf{D}{(\\psi^*)} \\cos{(\\psi^*)} and \\mathbf{D}^{2}{(\\psi^*)} - \\mathbf{D}{(\\psi^*)} \\cos{(\\psi^*)} = 0 and \\mathbf{D}^{2}{(\\psi^*)} - \\mathbf{D}{(\\psi^*)} \\cos{(\\psi^*)} + \\cos{(\\psi^*)} = \\cos{(\\psi^*)} and (\\mathbf{D}^{2}{(\\psi^*)} - \\mathbf{D}{(\\psi^*)} \\cos{(\\psi^*)} + \\cos{(\\psi^*)})^{2} = \\cos^{2}{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 2, "Mul(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))), Integer(0))"], [["add", 3, "cos(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))), cos(Symbol('\\\\psi^*', commutative=True))), cos(Symbol('\\\\psi^*', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))), cos(Symbol('\\\\psi^*', commutative=True))), Integer(2)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\psi^{*}{(T,t)} = e^{\\frac{t}{T}}, then obtain - t + \\psi^{*}{(T,t)} + \\frac{- \\psi^{*}{(T,t)} + \\int \\psi^{*}{(T,t)} dT}{T} = - t + \\psi^{*}{(T,t)} + \\frac{- \\psi^{*}{(T,t)} + \\int e^{\\frac{t}{T}} dT}{T}", "derivation": "\\psi^{*}{(T,t)} = e^{\\frac{t}{T}} and \\int \\psi^{*}{(T,t)} dT = \\int e^{\\frac{t}{T}} dT and - \\psi^{*}{(T,t)} + \\int \\psi^{*}{(T,t)} dT = - \\psi^{*}{(T,t)} + \\int e^{\\frac{t}{T}} dT and \\frac{- \\psi^{*}{(T,t)} + \\int \\psi^{*}{(T,t)} dT}{T} = \\frac{- \\psi^{*}{(T,t)} + \\int e^{\\frac{t}{T}} dT}{T} and - t + \\frac{- \\psi^{*}{(T,t)} + \\int \\psi^{*}{(T,t)} dT}{T} = - t + \\frac{- \\psi^{*}{(T,t)} + \\int e^{\\frac{t}{T}} dT}{T} and - t + \\psi^{*}{(T,t)} + \\frac{- \\psi^{*}{(T,t)} + \\int \\psi^{*}{(T,t)} dT}{T} = - t + \\psi^{*}{(T,t)} + \\frac{- \\psi^{*}{(T,t)} + \\int e^{\\frac{t}{T}} dT}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["minus", 2, "Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["divide", 3, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True))))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('T', commutative=True))))))"], [["minus", 4, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True)))))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('T', commutative=True)))))))"], [["add", 5, "Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('T', commutative=True)))))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('T', commutative=True), Symbol('t', commutative=True))), Integral(exp(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('T', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{1})} = \\log{(v_{1})} and f{(v_{1})} = - \\log{(v_{1})}, then obtain - f{(v_{1})} - \\log{(v_{1})} = 0", "derivation": "\\mathbf{f}{(v_{1})} = \\log{(v_{1})} and \\mathbf{f}{(v_{1})} - \\log{(v_{1})} = 0 and f{(v_{1})} = - \\log{(v_{1})} and f{(v_{1})} = - \\mathbf{f}{(v_{1})} and - f{(v_{1})} = \\mathbf{f}{(v_{1})} and - f{(v_{1})} - \\log{(v_{1})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["minus", 1, "log(Symbol('v_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('f')(Symbol('v_1', commutative=True)), Mul(Integer(-1), log(Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('f')(Symbol('v_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f')(Symbol('v_1', commutative=True))), Function('\\\\mathbf{f}')(Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Mul(Integer(-1), Function('f')(Symbol('v_1', commutative=True))), Mul(Integer(-1), log(Symbol('v_1', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)} = \\hbar + f_{\\mathbf{v}} and \\mathbf{H}{(\\hbar)} = 2 \\hbar, then obtain \\frac{\\int \\mathbf{H}{(\\hbar)} d\\hbar}{2 \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}} = \\frac{\\int 2 \\hbar d\\hbar}{2 \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}}", "derivation": "\\mathbf{J}{(f_{\\mathbf{v}},\\hbar)} = \\hbar + f_{\\mathbf{v}} and \\mathbf{H}{(\\hbar)} = 2 \\hbar and \\int \\mathbf{H}{(\\hbar)} d\\hbar = \\int 2 \\hbar d\\hbar and \\frac{\\int \\mathbf{H}{(\\hbar)} d\\hbar}{\\hbar + f_{\\mathbf{v}} + \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}} = \\frac{\\int 2 \\hbar d\\hbar}{\\hbar + f_{\\mathbf{v}} + \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}} and \\frac{\\int \\mathbf{H}{(\\hbar)} d\\hbar}{2 \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}} = \\frac{\\int 2 \\hbar d\\hbar}{2 \\mathbf{J}{(f_{\\mathbf{v}},\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\hbar', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Integral(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{J}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(L)} = \\log{(L)}, then derive (\\frac{L \\frac{d}{d L} \\operatorname{F_{H}}{(L)}}{\\operatorname{F_{H}}{(L)}} + \\log{(\\operatorname{F_{H}}{(L)})}) \\operatorname{F_{H}}^{L}{(L)} = (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L}, then obtain (\\frac{L \\frac{d}{d L} \\operatorname{F_{H}}{(L)}}{\\operatorname{F_{H}}{(L)}} + \\log{(\\operatorname{F_{H}}{(L)})}) \\operatorname{F_{H}}^{L}{(L)} = (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\operatorname{F_{H}}^{L}{(L)}", "derivation": "\\operatorname{F_{H}}{(L)} = \\log{(L)} and \\operatorname{F_{H}}^{L}{(L)} = \\log{(L)}^{L} and \\frac{d}{d L} \\operatorname{F_{H}}^{L}{(L)} = \\frac{d}{d L} \\log{(L)}^{L} and (\\frac{L \\frac{d}{d L} \\operatorname{F_{H}}{(L)}}{\\operatorname{F_{H}}{(L)}} + \\log{(\\operatorname{F_{H}}{(L)})}) \\operatorname{F_{H}}^{L}{(L)} = (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L} and (\\frac{L \\frac{d}{d L} \\operatorname{F_{H}}{(L)}}{\\operatorname{F_{H}}{(L)}} + \\log{(\\operatorname{F_{H}}{(L)})}) \\operatorname{F_{H}}^{L}{(L)} = (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\operatorname{F_{H}}^{L}{(L)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Function('F_H')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('L', commutative=True), Pow(Function('F_H')(Symbol('L', commutative=True)), Integer(-1)), Derivative(Function('F_H')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(Function('F_H')(Symbol('L', commutative=True)))), Pow(Function('F_H')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Add(log(log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('L', commutative=True), Pow(Function('F_H')(Symbol('L', commutative=True)), Integer(-1)), Derivative(Function('F_H')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(Function('F_H')(Symbol('L', commutative=True)))), Pow(Function('F_H')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Add(log(log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Pow(Function('F_H')(Symbol('L', commutative=True)), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\varphi{(v_{2},\\varphi)} = \\varphi + \\sin{(v_{2})}, then obtain 0 = - \\frac{\\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})}}{\\varphi}", "derivation": "\\varphi{(v_{2},\\varphi)} = \\varphi + \\sin{(v_{2})} and 0 = \\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})} and 0 = \\frac{\\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})}}{\\varphi} and \\sin{(v_{2})} = \\sin{(v_{2})} + \\frac{\\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})}}{\\varphi} and \\varphi{(v_{2},\\varphi)} = \\varphi + \\sin{(v_{2})} + \\frac{\\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})}}{\\varphi} and 0 = - \\frac{\\varphi - \\varphi{(v_{2},\\varphi)} + \\sin{(v_{2})}}{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('v_2', commutative=True))))"], [["minus", 1, "Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), sin(Symbol('v_2', commutative=True))))"], [["divide", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), sin(Symbol('v_2', commutative=True)))))"], [["add", 3, "sin(Symbol('v_2', commutative=True))"], "Equality(sin(Symbol('v_2', commutative=True)), Add(sin(Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), sin(Symbol('v_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), sin(Symbol('v_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integer(0), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), sin(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\Psi)} = \\sin{(\\log{(\\Psi)})}, then obtain ((\\frac{d}{d \\Psi} \\operatorname{v_{x}}^{\\Psi}{(\\Psi)})^{\\Psi})^{\\Psi} = ((\\frac{d}{d \\Psi} \\sin^{\\Psi}{(\\log{(\\Psi)})})^{\\Psi})^{\\Psi}", "derivation": "\\operatorname{v_{x}}{(\\Psi)} = \\sin{(\\log{(\\Psi)})} and \\operatorname{v_{x}}^{\\Psi}{(\\Psi)} = \\sin^{\\Psi}{(\\log{(\\Psi)})} and \\frac{d}{d \\Psi} \\operatorname{v_{x}}^{\\Psi}{(\\Psi)} = \\frac{d}{d \\Psi} \\sin^{\\Psi}{(\\log{(\\Psi)})} and (\\frac{d}{d \\Psi} \\operatorname{v_{x}}^{\\Psi}{(\\Psi)})^{\\Psi} = (\\frac{d}{d \\Psi} \\sin^{\\Psi}{(\\log{(\\Psi)})})^{\\Psi} and ((\\frac{d}{d \\Psi} \\operatorname{v_{x}}^{\\Psi}{(\\Psi)})^{\\Psi})^{\\Psi} = ((\\frac{d}{d \\Psi} \\sin^{\\Psi}{(\\log{(\\Psi)})})^{\\Psi})^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\Psi', commutative=True)), sin(log(Symbol('\\\\Psi', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(sin(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Pow(Function('v_x')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Pow(sin(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('v_x')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Pow(sin(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)))"], [["power", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Pow(Derivative(Pow(Function('v_x')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Derivative(Pow(sin(log(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(G)} = \\cos{(\\sin{(G)})} and \\mathbb{I}{(G)} = \\frac{\\operatorname{J_{\\varepsilon}}^{3}{(G)}}{\\cos^{4}{(\\sin{(G)})}} + \\sin{(G)}, then obtain \\mathbb{I}{(G)} = \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos^{2}{(\\sin{(G)})}} + \\sin{(G)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(G)} = \\cos{(\\sin{(G)})} and \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos{(\\sin{(G)})}} = 1 and \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos^{2}{(\\sin{(G)})}} = \\frac{1}{\\cos{(\\sin{(G)})}} and \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos^{2}{(\\sin{(G)})}} + \\sin{(G)} = \\sin{(G)} + \\frac{1}{\\cos{(\\sin{(G)})}} and \\frac{\\operatorname{J_{\\varepsilon}}^{3}{(G)}}{\\cos^{4}{(\\sin{(G)})}} + \\sin{(G)} = \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos^{2}{(\\sin{(G)})}} + \\sin{(G)} and \\mathbb{I}{(G)} = \\frac{\\operatorname{J_{\\varepsilon}}^{3}{(G)}}{\\cos^{4}{(\\sin{(G)})}} + \\sin{(G)} and \\mathbb{I}{(G)} = \\frac{\\operatorname{J_{\\varepsilon}}{(G)}}{\\cos^{2}{(\\sin{(G)})}} + \\sin{(G)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), cos(sin(Symbol('G', commutative=True))))"], [["divide", 1, "cos(sin(Symbol('G', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Pow(cos(sin(Symbol('G', commutative=True))), Integer(-1))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-2))), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-1)))"], [["add", 3, "sin(Symbol('G', commutative=True))"], "Equality(Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-2))), sin(Symbol('G', commutative=True))), Add(sin(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Integer(3)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-4))), sin(Symbol('G', commutative=True))), Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-2))), sin(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Add(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Integer(3)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-4))), sin(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\mathbb{I}')(Symbol('G', commutative=True)), Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('G', commutative=True)), Pow(cos(sin(Symbol('G', commutative=True))), Integer(-2))), sin(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\ddot{x})} = \\log{(e^{\\ddot{x}})}, then obtain \\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})} + \\int (\\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})}) d\\ddot{x} = \\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})} + \\int 2 \\log{(e^{\\ddot{x}})} d\\ddot{x}", "derivation": "\\theta{(\\ddot{x})} = \\log{(e^{\\ddot{x}})} and \\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})} = 2 \\log{(e^{\\ddot{x}})} and \\int (\\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})}) d\\ddot{x} = \\int 2 \\log{(e^{\\ddot{x}})} d\\ddot{x} and \\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})} + \\int (\\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})}) d\\ddot{x} = \\theta{(\\ddot{x})} + \\log{(e^{\\ddot{x}})} + \\int 2 \\log{(e^{\\ddot{x}})} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 1, "log(exp(Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True)))), Mul(Integer(2), log(exp(Symbol('\\\\ddot{x}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Integer(2), log(exp(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 3, "Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True))))"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Add(Function('\\\\theta')(Symbol('\\\\ddot{x}', commutative=True)), log(exp(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Integer(2), log(exp(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given l{(F_{c},\\hat{X})} = \\frac{\\cos{(F_{c})}}{\\hat{X}} and \\operatorname{A_{1}}{(F_{c})} = F_{c}, then obtain F_{c} + 1 = F_{c} + (\\frac{\\cos{(F_{c})}}{\\hat{X} l{(F_{c},\\hat{X})}})^{\\hat{X}}", "derivation": "l{(F_{c},\\hat{X})} = \\frac{\\cos{(F_{c})}}{\\hat{X}} and 1 = \\frac{\\cos{(F_{c})}}{\\hat{X} l{(F_{c},\\hat{X})}} and 1 = (\\frac{\\cos{(F_{c})}}{\\hat{X} l{(F_{c},\\hat{X})}})^{\\hat{X}} and \\operatorname{A_{1}}{(F_{c})} = F_{c} and \\operatorname{A_{1}}{(F_{c})} + 1 = (\\frac{\\cos{(F_{c})}}{\\hat{X} l{(F_{c},\\hat{X})}})^{\\hat{X}} + \\operatorname{A_{1}}{(F_{c})} and F_{c} + 1 = F_{c} + (\\frac{\\cos{(F_{c})}}{\\hat{X} l{(F_{c},\\hat{X})}})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), cos(Symbol('F_c', commutative=True))))"], [["divide", 1, "Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), cos(Symbol('F_c', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), cos(Symbol('F_c', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["add", 3, "Function('A_1')(Symbol('F_c', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('F_c', commutative=True)), Integer(1)), Add(Pow(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), cos(Symbol('F_c', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Function('A_1')(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('F_c', commutative=True), Integer(1)), Add(Symbol('F_c', commutative=True), Pow(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('F_c', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), cos(Symbol('F_c', commutative=True))), Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\sigma_x,u)} = \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u, then obtain \\int (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\hat{x}{(\\sigma_x,u)} d\\sigma_x = \\int (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u d\\sigma_x", "derivation": "\\hat{x}{(\\sigma_x,u)} = \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u and - u + \\hat{x}{(\\sigma_x,u)} = - u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u and (- u + \\hat{x}{(\\sigma_x,u)}) \\hat{x}{(\\sigma_x,u)} = (- u + \\hat{x}{(\\sigma_x,u)}) \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u and (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\hat{x}{(\\sigma_x,u)} = (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u and \\int (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\hat{x}{(\\sigma_x,u)} d\\sigma_x = \\int (- u + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u) \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x u d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Function('\\\\hat{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(i,\\mathbf{p})} = \\mathbf{p} i and C{(n_{1},v_{x})} = n_{1} v_{x}, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{E_{n}}{(i,\\mathbf{p})} = i, then obtain n_{1} v_{x} + \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} i = i + n_{1} v_{x}", "derivation": "\\operatorname{E_{n}}{(i,\\mathbf{p})} = \\mathbf{p} i and \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{E_{n}}{(i,\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} i and \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{E_{n}}{(i,\\mathbf{p})} = i and \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} i = i and C{(n_{1},v_{x})} = n_{1} v_{x} and C{(n_{1},v_{x})} + \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} i = i + C{(n_{1},v_{x})} and n_{1} v_{x} + \\frac{\\partial}{\\partial \\mathbf{p}} \\mathbf{p} i = i + n_{1} v_{x}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('i', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('i', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('i', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('i', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Symbol('i', commutative=True))"], ["get_premise", "Equality(Function('C')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 4, "Function('C')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Function('C')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Function('C')(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Mul(Symbol('n_1', commutative=True), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given W{(\\rho)} = \\log{(\\log{(\\rho)})}, then derive \\frac{d}{d \\rho} W{(\\rho)} = \\frac{1}{\\rho \\log{(\\rho)}}, then obtain \\frac{\\frac{d}{d \\rho} \\log{(\\log{(\\rho)})}}{\\rho} = \\frac{1}{\\rho^{2} \\log{(\\rho)}}", "derivation": "W{(\\rho)} = \\log{(\\log{(\\rho)})} and \\frac{d}{d \\rho} W{(\\rho)} = \\frac{d}{d \\rho} \\log{(\\log{(\\rho)})} and \\frac{d}{d \\rho} W{(\\rho)} = \\frac{1}{\\rho \\log{(\\rho)}} and \\frac{\\frac{d}{d \\rho} W{(\\rho)}}{\\rho} = \\frac{1}{\\rho^{2} \\log{(\\rho)}} and \\frac{\\frac{d}{d \\rho} \\log{(\\log{(\\rho)})}}{\\rho} = \\frac{1}{\\rho^{2} \\log{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\rho', commutative=True)), log(log(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["times", 3, "Pow(Symbol('\\\\rho', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Derivative(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(log(Symbol('\\\\rho', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{x}{(V)} = \\sin{(\\log{(V)})}, then obtain (V + \\sigma_{x}{(V)} - \\log{(V)}) \\sin{(\\log{(V)})} = (V - \\log{(V)} + \\sin{(\\log{(V)})}) \\sin{(\\log{(V)})}", "derivation": "\\sigma_{x}{(V)} = \\sin{(\\log{(V)})} and V + \\sigma_{x}{(V)} = V + \\sin{(\\log{(V)})} and V + \\sigma_{x}{(V)} - \\log{(V)} = V - \\log{(V)} + \\sin{(\\log{(V)})} and (V + \\sigma_{x}{(V)} - \\log{(V)}) \\sin{(\\log{(V)})} = (V - \\log{(V)} + \\sin{(\\log{(V)})}) \\sin{(\\log{(V)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True))))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('\\\\sigma_x')(Symbol('V', commutative=True))), Add(Symbol('V', commutative=True), sin(log(Symbol('V', commutative=True)))))"], [["minus", 2, "log(Symbol('V', commutative=True))"], "Equality(Add(Symbol('V', commutative=True), Function('\\\\sigma_x')(Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))), Add(Symbol('V', commutative=True), Mul(Integer(-1), log(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))))"], [["times", 3, "sin(log(Symbol('V', commutative=True)))"], "Equality(Mul(Add(Symbol('V', commutative=True), Function('\\\\sigma_x')(Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))), sin(log(Symbol('V', commutative=True)))), Mul(Add(Symbol('V', commutative=True), Mul(Integer(-1), log(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))), sin(log(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(\\mathbf{D})} = \\mathbf{D}, then obtain \\log{(v_{1} (\\mathbf{D} - v_{1} + \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})}))}^{v_{1}} = \\log{(v_{1} (\\mathbf{D} + \\mathbf{D}^{\\mathbf{D}} - v_{1}))}^{v_{1}}", "derivation": "\\hat{X}{(\\mathbf{D})} = \\mathbf{D} and \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})} = \\mathbf{D}^{\\mathbf{D}} and \\mathbf{D} - v_{1} + \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})} = \\mathbf{D} + \\mathbf{D}^{\\mathbf{D}} - v_{1} and v_{1} (\\mathbf{D} - v_{1} + \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})}) = v_{1} (\\mathbf{D} + \\mathbf{D}^{\\mathbf{D}} - v_{1}) and \\log{(v_{1} (\\mathbf{D} - v_{1} + \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})}))} = \\log{(v_{1} (\\mathbf{D} + \\mathbf{D}^{\\mathbf{D}} - v_{1}))} and \\log{(v_{1} (\\mathbf{D} - v_{1} + \\hat{X}^{\\mathbf{D}}{(\\mathbf{D})}))}^{v_{1}} = \\log{(v_{1} (\\mathbf{D} + \\mathbf{D}^{\\mathbf{D}} - v_{1}))}^{v_{1}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('v_1', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["times", 3, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"], [["log", 4], "Equality(log(Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))), log(Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))))"], [["power", 5, "Symbol('v_1', commutative=True)"], "Equality(Pow(log(Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('v_1', commutative=True)), Pow(log(Mul(Symbol('v_1', commutative=True), Add(Symbol('\\\\mathbf{D}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varphi^*,u)} = \\log{(\\frac{u}{\\varphi^*})}, then obtain \\int \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\operatorname{A_{x}}{(\\varphi^*,u)}}{u} du = \\int \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\log{(\\frac{u}{\\varphi^*})}}{u} du", "derivation": "\\operatorname{A_{x}}{(\\varphi^*,u)} = \\log{(\\frac{u}{\\varphi^*})} and \\frac{\\partial}{\\partial u} \\operatorname{A_{x}}{(\\varphi^*,u)} = \\frac{\\partial}{\\partial u} \\log{(\\frac{u}{\\varphi^*})} and \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\operatorname{A_{x}}{(\\varphi^*,u)}}{u} = \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\log{(\\frac{u}{\\varphi^*})}}{u} and \\int \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\operatorname{A_{x}}{(\\varphi^*,u)}}{u} du = \\int \\frac{\\varphi^* \\frac{\\partial}{\\partial u} \\log{(\\frac{u}{\\varphi^*})}}{u} du", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('u', commutative=True))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Function('A_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(log(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Function('A_x')(Symbol('\\\\varphi^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(log(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given V{(g)} = \\sin{(e^{g})}, then obtain - e^{g} + \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V^{2}{(g)} = - e^{g} + \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V{(g)} \\sin{(e^{g})}", "derivation": "V{(g)} = \\sin{(e^{g})} and (V{(g)} - \\sin{(e^{g})}) V{(g)} = (V{(g)} - \\sin{(e^{g})}) \\sin{(e^{g})} and (V{(g)} - \\sin{(e^{g})})^{2} V^{2}{(g)} = (V{(g)} - \\sin{(e^{g})})^{2} V{(g)} \\sin{(e^{g})} and \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V^{2}{(g)} = \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V{(g)} \\sin{(e^{g})} and - e^{g} + \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V^{2}{(g)} = - e^{g} + \\frac{d}{d g} (V{(g)} - \\sin{(e^{g})})^{2} V{(g)} \\sin{(e^{g})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True))))"], [["times", 1, "Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True)))))"], "Equality(Mul(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Function('V')(Symbol('g', commutative=True))), Mul(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), sin(exp(Symbol('g', commutative=True)))))"], [["times", 2, "Mul(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Function('V')(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Pow(Function('V')(Symbol('g', commutative=True)), Integer(2))), Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Function('V')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Pow(Function('V')(Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Function('V')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 4, "exp(Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), Derivative(Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Pow(Function('V')(Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('g', commutative=True))), Derivative(Mul(Pow(Add(Function('V')(Symbol('g', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('g', commutative=True))))), Integer(2)), Function('V')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\delta,\\mathbf{f})} = \\delta + \\mathbf{f}, then obtain \\frac{(\\delta + \\mathbf{f}) \\dot{x}{(\\delta,\\mathbf{f})}}{\\delta} = \\frac{(\\delta + \\mathbf{f})^{2}}{\\delta}", "derivation": "\\dot{x}{(\\delta,\\mathbf{f})} = \\delta + \\mathbf{f} and \\frac{\\dot{x}{(\\delta,\\mathbf{f})}}{\\delta} = \\frac{\\delta + \\mathbf{f}}{\\delta} and \\frac{\\dot{x}^{2}{(\\delta,\\mathbf{f})}}{\\delta} = \\frac{(\\delta + \\mathbf{f}) \\dot{x}{(\\delta,\\mathbf{f})}}{\\delta} and \\frac{(\\delta + \\mathbf{f}) \\dot{x}{(\\delta,\\mathbf{f})}}{\\delta} = \\frac{(\\delta + \\mathbf{f})^{2}}{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 2, "Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\phi_1)} = \\log{(\\log{(\\phi_1)})}, then obtain 3 \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} + 2 \\frac{d}{d \\phi_1} \\log{(\\log{(\\phi_1)})}", "derivation": "\\operatorname{r_{0}}{(\\phi_1)} = \\log{(\\log{(\\phi_1)})} and \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\log{(\\log{(\\phi_1)})} and 2 \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} + \\frac{d}{d \\phi_1} \\log{(\\log{(\\phi_1)})} and 3 \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} = 2 \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} + \\frac{d}{d \\phi_1} \\log{(\\log{(\\phi_1)})} and 3 \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\operatorname{r_{0}}{(\\phi_1)} + 2 \\frac{d}{d \\phi_1} \\log{(\\log{(\\phi_1)})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), log(log(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Add(Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))"], "Equality(Mul(Integer(3), Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Add(Mul(Integer(2), Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Derivative(log(log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(3), Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Add(Derivative(Function('r_0')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Integer(2), Derivative(log(log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\tilde{g}{(n_{2})} = \\cos{(n_{2})}, then derive \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} = - \\sin{(n_{2})}, then obtain e^{0^{n_{2}}} = e^{(\\sin{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})})^{n_{2}}}", "derivation": "\\tilde{g}{(n_{2})} = \\cos{(n_{2})} and \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} = \\frac{d}{d n_{2}} \\cos{(n_{2})} and \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} = - \\sin{(n_{2})} and 0 = - \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})} and 0^{n_{2}} = (- \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})})^{n_{2}} and e^{0^{n_{2}}} = e^{(- \\frac{d}{d n_{2}} \\tilde{g}{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})})^{n_{2}}} and e^{0^{n_{2}}} = e^{(\\sin{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})})^{n_{2}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))))"], [["minus", 2, "Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Integer(0), Symbol('n_2', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Integer(0), Symbol('n_2', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\tilde{g}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(exp(Pow(Integer(0), Symbol('n_2', commutative=True))), exp(Pow(Add(sin(Symbol('n_2', commutative=True)), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given n{(P_{e})} = \\sin{(P_{e})} and \\operatorname{E_{n}}{(P_{e})} = n^{2}{(P_{e})}, then obtain \\int n^{2}{(P_{e})} dP_{e} = \\int n{(P_{e})} \\sin{(P_{e})} dP_{e}", "derivation": "n{(P_{e})} = \\sin{(P_{e})} and n^{2}{(P_{e})} = n{(P_{e})} \\sin{(P_{e})} and \\operatorname{E_{n}}{(P_{e})} = n^{2}{(P_{e})} and \\operatorname{E_{n}}{(P_{e})} = n{(P_{e})} \\sin{(P_{e})} and \\int \\operatorname{E_{n}}{(P_{e})} dP_{e} = \\int n{(P_{e})} \\sin{(P_{e})} dP_{e} and \\int n^{2}{(P_{e})} dP_{e} = \\int n{(P_{e})} \\sin{(P_{e})} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["times", 1, "Function('n')(Symbol('P_e', commutative=True))"], "Equality(Pow(Function('n')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('n')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('P_e', commutative=True)), Pow(Function('n')(Symbol('P_e', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('E_n')(Symbol('P_e', commutative=True)), Mul(Function('n')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Mul(Function('n')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Pow(Function('n')(Symbol('P_e', commutative=True)), Integer(2)), Tuple(Symbol('P_e', commutative=True))), Integral(Mul(Function('n')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}, then obtain (\\frac{\\sigma_{p}{(\\eta^{\\prime})} - \\cos{(\\eta^{\\prime})}}{\\int \\sigma_{p}{(\\eta^{\\prime})} d\\eta^{\\prime}})^{\\eta^{\\prime}} = 0^{\\eta^{\\prime}}", "derivation": "\\sigma_{p}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and \\sigma_{p}{(\\eta^{\\prime})} - \\cos{(\\eta^{\\prime})} = 0 and \\frac{\\sigma_{p}{(\\eta^{\\prime})} - \\cos{(\\eta^{\\prime})}}{\\int \\sigma_{p}{(\\eta^{\\prime})} d\\eta^{\\prime}} = 0 and (\\frac{\\sigma_{p}{(\\eta^{\\prime})} - \\cos{(\\eta^{\\prime})}}{\\int \\sigma_{p}{(\\eta^{\\prime})} d\\eta^{\\prime}})^{\\eta^{\\prime}} = 0^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integer(0))"], [["divide", 2, "Integral(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))), Integer(0))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Pow(Integral(Function('\\\\sigma_p')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(-1))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Integer(0), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(S,\\sigma_p)} = \\cos{(S + \\sigma_p)}, then obtain (\\frac{\\partial}{\\partial S} (- S + \\operatorname{C_{d}}{(S,\\sigma_p)})^{S})^{S} = (\\frac{\\partial}{\\partial S} (- S + \\cos{(S + \\sigma_p)})^{S})^{S}", "derivation": "\\operatorname{C_{d}}{(S,\\sigma_p)} = \\cos{(S + \\sigma_p)} and - S + \\operatorname{C_{d}}{(S,\\sigma_p)} = - S + \\cos{(S + \\sigma_p)} and (- S + \\operatorname{C_{d}}{(S,\\sigma_p)})^{S} = (- S + \\cos{(S + \\sigma_p)})^{S} and \\frac{\\partial}{\\partial S} (- S + \\operatorname{C_{d}}{(S,\\sigma_p)})^{S} = \\frac{\\partial}{\\partial S} (- S + \\cos{(S + \\sigma_p)})^{S} and (\\frac{\\partial}{\\partial S} (- S + \\operatorname{C_{d}}{(S,\\sigma_p)})^{S})^{S} = (\\frac{\\partial}{\\partial S} (- S + \\cos{(S + \\sigma_p)})^{S})^{S}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 1, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('C_d')(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('C_d')(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('S', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Symbol('S', commutative=True)))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('C_d')(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('C_d')(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\varphi^*,M_{E})} = (\\varphi^*)^{M_{E}}, then obtain \\sin^{\\varphi^*}{(\\log{(\\mathbf{A}{(\\varphi^*,M_{E})})})} = \\sin^{\\varphi^*}{(\\log{((\\varphi^*)^{M_{E}})})}", "derivation": "\\mathbf{A}{(\\varphi^*,M_{E})} = (\\varphi^*)^{M_{E}} and \\log{(\\mathbf{A}{(\\varphi^*,M_{E})})} = \\log{((\\varphi^*)^{M_{E}})} and \\sin{(\\log{(\\mathbf{A}{(\\varphi^*,M_{E})})})} = \\sin{(\\log{((\\varphi^*)^{M_{E}})})} and \\sin^{\\varphi^*}{(\\log{(\\mathbf{A}{(\\varphi^*,M_{E})})})} = \\sin^{\\varphi^*}{(\\log{((\\varphi^*)^{M_{E}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{A}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True))), log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True))))"], [["sin", 2], "Equality(sin(log(Function('\\\\mathbf{A}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))), sin(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))))"], [["power", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(sin(log(Function('\\\\mathbf{A}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)), Pow(sin(log(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given L{(U)} = \\log{(U)}, then obtain \\frac{d}{d U} L{(U)} \\iint L{(U)} dU dU = \\frac{d}{d U} \\log{(U)} \\iint L{(U)} dU dU", "derivation": "L{(U)} = \\log{(U)} and \\int L{(U)} dU = \\int \\log{(U)} dU and \\frac{d}{d U} L{(U)} = \\frac{d}{d U} \\log{(U)} and \\iint L{(U)} dU dU = \\iint \\log{(U)} dU dU and \\frac{d}{d U} L{(U)} \\iint \\log{(U)} dU dU = \\frac{d}{d U} \\log{(U)} \\iint \\log{(U)} dU dU and \\frac{d}{d U} L{(U)} \\iint L{(U)} dU dU = \\frac{d}{d U} \\log{(U)} \\iint L{(U)} dU dU", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["times", 3, "Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Derivative(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Derivative(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Function('L')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given r{(t_{1})} = e^{e^{t_{1}}}, then obtain \\int \\frac{d}{d t_{1}} (r{(t_{1})} + r{(t_{1})} e^{- e^{t_{1}}}) dt_{1} = \\int \\frac{d}{d t_{1}} (r{(t_{1})} + 1) dt_{1}", "derivation": "r{(t_{1})} = e^{e^{t_{1}}} and r{(t_{1})} e^{- e^{t_{1}}} = 1 and r{(t_{1})} + r{(t_{1})} e^{- e^{t_{1}}} = r{(t_{1})} + 1 and \\frac{d}{d t_{1}} (r{(t_{1})} + r{(t_{1})} e^{- e^{t_{1}}}) = \\frac{d}{d t_{1}} (r{(t_{1})} + 1) and \\int \\frac{d}{d t_{1}} (r{(t_{1})} + r{(t_{1})} e^{- e^{t_{1}}}) dt_{1} = \\int \\frac{d}{d t_{1}} (r{(t_{1})} + 1) dt_{1}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('t_1', commutative=True)), exp(exp(Symbol('t_1', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('t_1', commutative=True)))"], "Equality(Mul(Function('r')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('t_1', commutative=True))))), Integer(1))"], [["add", 2, "Function('r')(Symbol('t_1', commutative=True))"], "Equality(Add(Function('r')(Symbol('t_1', commutative=True)), Mul(Function('r')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('t_1', commutative=True)))))), Add(Function('r')(Symbol('t_1', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('t_1', commutative=True)), Mul(Function('r')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('t_1', commutative=True)))))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Function('r')(Symbol('t_1', commutative=True)), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('t_1', commutative=True)"], "Equality(Integral(Derivative(Add(Function('r')(Symbol('t_1', commutative=True)), Mul(Function('r')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('t_1', commutative=True)))))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))), Integral(Derivative(Add(Function('r')(Symbol('t_1', commutative=True)), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given B{(U)} = \\log{(U)}, then derive \\mathbf{P} + B{(U)} = a^{\\dagger} + \\log{(U)}, then obtain \\mathbf{P} + \\log{(U)} = \\mathbf{P} + B{(U)}", "derivation": "B{(U)} = \\log{(U)} and \\frac{d}{d U} B{(U)} = \\frac{d}{d U} \\log{(U)} and \\int \\frac{d}{d U} B{(U)} dU = \\int \\frac{d}{d U} \\log{(U)} dU and \\mathbf{P} + B{(U)} = a^{\\dagger} + \\log{(U)} and \\mathbf{P} + \\log{(U)} = a^{\\dagger} + \\log{(U)} and \\mathbf{P} + \\log{(U)} = \\mathbf{P} + B{(U)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('B')(Symbol('U', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('U', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Function('B')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\theta,C)} = \\log{(C + \\theta)}, then derive \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)} = \\frac{1}{C + \\theta}, then obtain (\\frac{\\log{(C + \\theta)}}{C + \\theta})^{\\theta} = (\\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)})^{\\theta}", "derivation": "\\sigma_{x}{(\\theta,C)} = \\log{(C + \\theta)} and \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)} = \\frac{\\partial}{\\partial C} \\log{(C + \\theta)} and \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)} = \\frac{1}{C + \\theta} and \\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)} = \\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\log{(C + \\theta)} and \\frac{\\log{(C + \\theta)}}{C + \\theta} = \\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\log{(C + \\theta)} and \\frac{\\log{(C + \\theta)}}{C + \\theta} = \\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)} and (\\frac{\\log{(C + \\theta)}}{C + \\theta})^{\\theta} = (\\log{(C + \\theta)} \\frac{\\partial}{\\partial C} \\sigma_{x}{(\\theta,C)})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)))"], [["times", 2, "log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)), log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))), Mul(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)), log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))), Mul(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)), log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(Mul(log(Add(Symbol('C', commutative=True), Symbol('\\\\theta', commutative=True))), Derivative(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\mu_0,y^{\\prime})} = \\log{(\\mu_0 + y^{\\prime})}, then obtain - \\log{(\\mu_0 + y^{\\prime})} + \\frac{\\log{(\\mu_0 + y^{\\prime})}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}{(\\mu_0,y^{\\prime})}} = - \\log{(\\mu_0 + y^{\\prime})} + \\frac{\\log{(\\mu_0 + y^{\\prime})}^{2}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}^{2}{(\\mu_0,y^{\\prime})}}", "derivation": "\\mathbf{A}{(\\mu_0,y^{\\prime})} = \\log{(\\mu_0 + y^{\\prime})} and 1 = \\frac{\\log{(\\mu_0 + y^{\\prime})}}{\\mathbf{A}{(\\mu_0,y^{\\prime})}} and \\frac{1}{\\mu_0 + y^{\\prime}} = \\frac{\\log{(\\mu_0 + y^{\\prime})}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}{(\\mu_0,y^{\\prime})}} and - \\log{(\\mu_0 + y^{\\prime})} + \\frac{1}{\\mu_0 + y^{\\prime}} = - \\log{(\\mu_0 + y^{\\prime})} + \\frac{\\log{(\\mu_0 + y^{\\prime})}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}{(\\mu_0,y^{\\prime})}} and - \\log{(\\mu_0 + y^{\\prime})} + \\frac{\\log{(\\mu_0 + y^{\\prime})}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}{(\\mu_0,y^{\\prime})}} = - \\log{(\\mu_0 + y^{\\prime})} + \\frac{\\log{(\\mu_0 + y^{\\prime})}^{2}}{(\\mu_0 + y^{\\prime}) \\mathbf{A}^{2}{(\\mu_0,y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["divide", 2, "Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["minus", 3, "log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-2)), Pow(log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})}, then obtain \\lambda \\hat{\\mathbf{x}}^{2}{(\\lambda)} = \\lambda \\cos^{2}{(\\cos{(\\lambda)})}", "derivation": "\\hat{\\mathbf{x}}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})} and \\lambda \\hat{\\mathbf{x}}{(\\lambda)} = \\lambda \\cos{(\\cos{(\\lambda)})} and \\lambda \\hat{\\mathbf{x}}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} = \\lambda \\cos^{2}{(\\cos{(\\lambda)})} and \\lambda \\hat{\\mathbf{x}}^{2}{(\\lambda)} = \\lambda \\hat{\\mathbf{x}}{(\\lambda)} \\cos{(\\cos{(\\lambda)})} and \\lambda \\hat{\\mathbf{x}}^{2}{(\\lambda)} = \\lambda \\cos^{2}{(\\cos{(\\lambda)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), cos(cos(Symbol('\\\\lambda', commutative=True)))))"], [["times", 1, "Mul(Symbol('\\\\lambda', commutative=True), cos(cos(Symbol('\\\\lambda', commutative=True))))"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))), Mul(Symbol('\\\\lambda', commutative=True), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Pow(cos(cos(Symbol('\\\\lambda', commutative=True))), Integer(2))))"]]}, {"prompt": "Given g{(\\phi_1,\\omega,\\mathbf{P})} = \\frac{\\omega \\phi_1}{\\mathbf{P}} and \\operatorname{A_{x}}{(\\omega)} = \\omega^{2} and k{(\\omega)} = \\omega^{2}, then obtain \\int \\frac{\\phi_1 k{(\\omega)}}{\\mathbf{P}} d\\phi_1 = \\int \\frac{\\phi_1 \\operatorname{A_{x}}{(\\omega)}}{\\mathbf{P}} d\\phi_1", "derivation": "g{(\\phi_1,\\omega,\\mathbf{P})} = \\frac{\\omega \\phi_1}{\\mathbf{P}} and \\omega g{(\\phi_1,\\omega,\\mathbf{P})} = \\frac{\\omega^{2} \\phi_1}{\\mathbf{P}} and \\operatorname{A_{x}}{(\\omega)} = \\omega^{2} and \\omega g{(\\phi_1,\\omega,\\mathbf{P})} = \\frac{\\phi_1 \\operatorname{A_{x}}{(\\omega)}}{\\mathbf{P}} and \\frac{\\omega^{2} \\phi_1}{\\mathbf{P}} = \\frac{\\phi_1 \\operatorname{A_{x}}{(\\omega)}}{\\mathbf{P}} and \\int \\frac{\\omega^{2} \\phi_1}{\\mathbf{P}} d\\phi_1 = \\int \\frac{\\phi_1 \\operatorname{A_{x}}{(\\omega)}}{\\mathbf{P}} d\\phi_1 and k{(\\omega)} = \\omega^{2} and \\int \\frac{\\phi_1 k{(\\omega)}}{\\mathbf{P}} d\\phi_1 = \\int \\frac{\\phi_1 \\operatorname{A_{x}}{(\\omega)}}{\\mathbf{P}} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('g')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('g')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('A_x')(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Symbol('\\\\phi_1', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('A_x')(Symbol('\\\\omega', commutative=True))))"], [["integrate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(2)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('A_x')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('k')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('A_x')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\Psi{(m,\\hat{x})} = \\frac{m}{\\hat{x}} and Q{(m,\\hat{x})} = \\frac{m}{\\hat{x}}, then obtain \\int \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} \\Psi{(m,\\hat{x})}}{\\hat{x}} dm = \\int \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} Q{(m,\\hat{x})}}{\\hat{x}} dm", "derivation": "\\Psi{(m,\\hat{x})} = \\frac{m}{\\hat{x}} and Q{(m,\\hat{x})} = \\frac{m}{\\hat{x}} and \\Psi{(m,\\hat{x})} = Q{(m,\\hat{x})} and \\frac{\\partial}{\\partial \\hat{x}} \\Psi{(m,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} Q{(m,\\hat{x})} and \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} \\Psi{(m,\\hat{x})}}{\\hat{x}} = \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} Q{(m,\\hat{x})}}{\\hat{x}} and \\int \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} \\Psi{(m,\\hat{x})}}{\\hat{x}} dm = \\int \\frac{m Q{(m,\\hat{x})} \\frac{\\partial}{\\partial \\hat{x}} Q{(m,\\hat{x})}}{\\hat{x}} dm", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["times", 4, "Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('m', commutative=True), Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('Q')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given z{(E,y)} = \\frac{\\sin{(y)}}{E} and \\theta_{1}{(y)} = \\cos{(y)}, then derive \\frac{\\partial}{\\partial y} z{(E,y)} = \\frac{\\cos{(y)}}{E}, then obtain (\\frac{\\theta_{1}{(y)}}{E})^{E} = (\\frac{\\partial}{\\partial y} \\frac{\\sin{(y)}}{E})^{E}", "derivation": "z{(E,y)} = \\frac{\\sin{(y)}}{E} and \\frac{\\partial}{\\partial y} z{(E,y)} = \\frac{\\partial}{\\partial y} \\frac{\\sin{(y)}}{E} and \\frac{\\partial}{\\partial y} z{(E,y)} = \\frac{\\cos{(y)}}{E} and (\\frac{\\partial}{\\partial y} z{(E,y)})^{E} = (\\frac{\\partial}{\\partial y} \\frac{\\sin{(y)}}{E})^{E} and (\\frac{\\cos{(y)}}{E})^{E} = (\\frac{\\partial}{\\partial y} \\frac{\\sin{(y)}}{E})^{E} and \\theta_{1}{(y)} = \\cos{(y)} and (\\frac{\\theta_{1}{(y)}}{E})^{E} = (\\frac{\\partial}{\\partial y} \\frac{\\sin{(y)}}{E})^{E}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('y', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('z')(Symbol('E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('y', commutative=True))), Symbol('E', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('y', commutative=True))), Symbol('E', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(\\sigma_p)} = \\log{(\\sigma_p)}, then derive \\frac{d}{d \\sigma_p} \\rho_{b}{(\\sigma_p)} = \\frac{1}{\\sigma_p}, then obtain - (\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)})^{\\sigma_p} = - (\\frac{1}{\\sigma_p})^{\\sigma_p}", "derivation": "\\rho_{b}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} \\rho_{b}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} \\rho_{b}{(\\sigma_p)} = \\frac{1}{\\sigma_p} and (\\frac{d}{d \\sigma_p} \\rho_{b}{(\\sigma_p)})^{\\sigma_p} = (\\frac{1}{\\sigma_p})^{\\sigma_p} and (\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)})^{\\sigma_p} = (\\frac{1}{\\sigma_p})^{\\sigma_p} and - (\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)})^{\\sigma_p} = - (\\frac{1}{\\sigma_p})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho_b')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Pow(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mu{(F_{N})} = - F_{N}, then obtain \\frac{d}{d F_{N}} 1 = \\frac{d}{d F_{N}} \\frac{16 \\mu^{4}{(F_{N})}}{(- F_{N} + \\mu{(F_{N})})^{4}}", "derivation": "\\mu{(F_{N})} = - F_{N} and - F_{N} + \\mu{(F_{N})} = - 2 F_{N} and \\frac{1}{(- F_{N} + \\mu{(F_{N})})^{2}} = \\frac{1}{4 F_{N}^{2}} and \\frac{1}{4 \\mu^{2}{(F_{N})}} = \\frac{1}{4 F_{N}^{2}} and \\frac{1}{(- F_{N} + \\mu{(F_{N})})^{2}} = \\frac{1}{4 \\mu^{2}{(F_{N})}} and 1 = \\frac{(- F_{N} + \\mu{(F_{N})})^{2}}{4 \\mu^{2}{(F_{N})}} and 1 = \\frac{16 \\mu^{4}{(F_{N})}}{(- F_{N} + \\mu{(F_{N})})^{4}} and \\frac{d}{d F_{N}} 1 = \\frac{d}{d F_{N}} \\frac{16 \\mu^{4}{(F_{N})}}{(- F_{N} + \\mu{(F_{N})})^{4}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('F_N', commutative=True)))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(-2)), Mul(Rational(1, 4), Pow(Symbol('F_N', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Rational(1, 4), Pow(Function('\\\\mu')(Symbol('F_N', commutative=True)), Integer(-2))), Mul(Rational(1, 4), Pow(Symbol('F_N', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(-2)), Mul(Rational(1, 4), Pow(Function('\\\\mu')(Symbol('F_N', commutative=True)), Integer(-2))))"], [["divide", 5, "Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(-2))"], "Equality(Integer(1), Mul(Rational(1, 4), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(2)), Pow(Function('\\\\mu')(Symbol('F_N', commutative=True)), Integer(-2))))"], [["power", 6, "Integer(-2)"], "Equality(Integer(1), Mul(Integer(16), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(-4)), Pow(Function('\\\\mu')(Symbol('F_N', commutative=True)), Integer(4))))"], [["differentiate", 7, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Integer(16), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\mu')(Symbol('F_N', commutative=True))), Integer(-4)), Pow(Function('\\\\mu')(Symbol('F_N', commutative=True)), Integer(4))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(z)} = e^{z}, then obtain 2 \\operatorname{f_{E}}{(z)} + e^{z} = \\operatorname{f_{E}}{(z)} + 2 e^{z}", "derivation": "\\operatorname{f_{E}}{(z)} = e^{z} and \\operatorname{f_{E}}{(z)} + e^{z} = 2 e^{z} and \\operatorname{f_{E}}{(z)} + 2 e^{z} = 3 e^{z} and 2 \\operatorname{f_{E}}{(z)} + e^{z} = 3 e^{z} and 2 \\operatorname{f_{E}}{(z)} + e^{z} = \\operatorname{f_{E}}{(z)} + 2 e^{z}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["add", 1, "exp(Symbol('z', commutative=True))"], "Equality(Add(Function('f_E')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True))), Mul(Integer(2), exp(Symbol('z', commutative=True))))"], [["add", 1, "Mul(Integer(2), exp(Symbol('z', commutative=True)))"], "Equality(Add(Function('f_E')(Symbol('z', commutative=True)), Mul(Integer(2), exp(Symbol('z', commutative=True)))), Mul(Integer(3), exp(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('f_E')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Mul(Integer(3), exp(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('f_E')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Add(Function('f_E')(Symbol('z', commutative=True)), Mul(Integer(2), exp(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f_{\\mathbf{p}},\\Psi)} = \\frac{\\partial}{\\partial \\Psi} \\frac{\\Psi}{f_{\\mathbf{p}}}, then derive \\int \\operatorname{E_{n}}{(f_{\\mathbf{p}},\\Psi)} df_{\\mathbf{p}} = T + \\log{(f_{\\mathbf{p}})}, then obtain T + \\log{(f_{\\mathbf{p}})} = \\int \\frac{\\partial}{\\partial \\Psi} \\frac{\\Psi}{f_{\\mathbf{p}}} df_{\\mathbf{p}}", "derivation": "\\operatorname{E_{n}}{(f_{\\mathbf{p}},\\Psi)} = \\frac{\\partial}{\\partial \\Psi} \\frac{\\Psi}{f_{\\mathbf{p}}} and \\int \\operatorname{E_{n}}{(f_{\\mathbf{p}},\\Psi)} df_{\\mathbf{p}} = \\int \\frac{\\partial}{\\partial \\Psi} \\frac{\\Psi}{f_{\\mathbf{p}}} df_{\\mathbf{p}} and \\int \\operatorname{E_{n}}{(f_{\\mathbf{p}},\\Psi)} df_{\\mathbf{p}} = T + \\log{(f_{\\mathbf{p}})} and T + \\log{(f_{\\mathbf{p}})} = \\int \\frac{\\partial}{\\partial \\Psi} \\frac{\\Psi}{f_{\\mathbf{p}}} df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('T', commutative=True), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('T', commutative=True), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(n_{2},E_{\\lambda},J_{\\varepsilon})} = (- E_{\\lambda} + J_{\\varepsilon})^{n_{2}} and \\mu{(E_{\\lambda})} = - E_{\\lambda}, then obtain - \\operatorname{v_{y}}{(A_{x})} + e^{(J_{\\varepsilon} + \\mu{(E_{\\lambda})})^{n_{2}}} = - \\operatorname{v_{y}}{(A_{x})} + e^{(- E_{\\lambda} + J_{\\varepsilon})^{n_{2}}}", "derivation": "\\operatorname{v_{t}}{(n_{2},E_{\\lambda},J_{\\varepsilon})} = (- E_{\\lambda} + J_{\\varepsilon})^{n_{2}} and e^{\\operatorname{v_{t}}{(n_{2},E_{\\lambda},J_{\\varepsilon})}} = e^{(- E_{\\lambda} + J_{\\varepsilon})^{n_{2}}} and \\mu{(E_{\\lambda})} = - E_{\\lambda} and \\operatorname{v_{t}}{(n_{2},E_{\\lambda},J_{\\varepsilon})} = (J_{\\varepsilon} + \\mu{(E_{\\lambda})})^{n_{2}} and e^{(J_{\\varepsilon} + \\mu{(E_{\\lambda})})^{n_{2}}} = e^{(- E_{\\lambda} + J_{\\varepsilon})^{n_{2}}} and - \\operatorname{v_{y}}{(A_{x})} + e^{(J_{\\varepsilon} + \\mu{(E_{\\lambda})})^{n_{2}}} = - \\operatorname{v_{y}}{(A_{x})} + e^{(- E_{\\lambda} + J_{\\varepsilon})^{n_{2}}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('n_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_2', commutative=True)))"], [["exp", 1], "Equality(exp(Function('v_t')(Symbol('n_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_t')(Symbol('n_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\mu')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(exp(Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\mu')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n_2', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_2', commutative=True))))"], [["minus", 5, "Function('v_y')(Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True))), exp(Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\mu')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(E,\\mathbf{F})} = \\sin{(\\frac{\\mathbf{F}}{E})} and \\phi{(E,\\mathbf{F})} = \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E}, then obtain - \\mathbf{F} + \\frac{\\operatorname{A_{2}}{(E,\\mathbf{F})}}{E} = - \\mathbf{F} + \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E}", "derivation": "\\operatorname{A_{2}}{(E,\\mathbf{F})} = \\sin{(\\frac{\\mathbf{F}}{E})} and \\frac{\\operatorname{A_{2}}{(E,\\mathbf{F})}}{E} = \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E} and \\phi{(E,\\mathbf{F})} = \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E} and \\frac{\\operatorname{A_{2}}{(E,\\mathbf{F})}}{E} = \\phi{(E,\\mathbf{F})} and - \\mathbf{F} + \\phi{(E,\\mathbf{F})} = - \\mathbf{F} + \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E} and - \\mathbf{F} + \\frac{\\operatorname{A_{2}}{(E,\\mathbf{F})}}{E} = - \\mathbf{F} + \\frac{\\sin{(\\frac{\\mathbf{F}}{E})}}{E}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 1, "Pow(Symbol('E', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\phi')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))))"]]}, {"prompt": "Given B{(\\dot{y},M_{E})} = M_{E} \\dot{y}, then derive \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})} = \\dot{y}, then obtain \\frac{\\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y} + \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})}}{\\dot{y}} = \\frac{\\dot{y} + \\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y}}{\\dot{y}}", "derivation": "B{(\\dot{y},M_{E})} = M_{E} \\dot{y} and \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})} = \\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y} and \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})} = \\dot{y} and \\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y} + \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})} = \\dot{y} + \\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y} and \\frac{\\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y} + \\frac{\\partial}{\\partial M_{E}} B{(\\dot{y},M_{E})}}{\\dot{y}} = \\frac{\\dot{y} + \\frac{\\partial}{\\partial M_{E}} M_{E} \\dot{y}}{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True))"], [["add", 3, "Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Function('B')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{y}', commutative=True), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["divide", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Function('B')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\omega)} = \\sin{(\\omega)}, then derive \\int \\operatorname{v_{t}}{(\\omega)} d\\omega = b - \\cos{(\\omega)}, then obtain \\frac{d}{d \\omega} \\int (\\int \\sin{(\\omega)} d\\omega)^{\\omega} d\\omega = \\frac{d}{d \\omega} \\int (\\int \\operatorname{v_{t}}{(\\omega)} d\\omega)^{\\omega} d\\omega", "derivation": "\\operatorname{v_{t}}{(\\omega)} = \\sin{(\\omega)} and \\int \\operatorname{v_{t}}{(\\omega)} d\\omega = \\int \\sin{(\\omega)} d\\omega and \\int \\operatorname{v_{t}}{(\\omega)} d\\omega = b - \\cos{(\\omega)} and (\\int \\operatorname{v_{t}}{(\\omega)} d\\omega)^{\\omega} = (b - \\cos{(\\omega)})^{\\omega} and \\int (\\int \\operatorname{v_{t}}{(\\omega)} d\\omega)^{\\omega} d\\omega = \\int (b - \\cos{(\\omega)})^{\\omega} d\\omega and \\frac{d}{d \\omega} \\int (\\int \\operatorname{v_{t}}{(\\omega)} d\\omega)^{\\omega} d\\omega = \\frac{\\partial}{\\partial \\omega} \\int (b - \\cos{(\\omega)})^{\\omega} d\\omega and \\frac{d}{d \\omega} \\int (\\int \\sin{(\\omega)} d\\omega)^{\\omega} d\\omega = \\frac{\\partial}{\\partial \\omega} \\int (b - \\cos{(\\omega)})^{\\omega} d\\omega and \\frac{d}{d \\omega} \\int (\\int \\sin{(\\omega)} d\\omega)^{\\omega} d\\omega = \\frac{d}{d \\omega} \\int (\\int \\operatorname{v_{t}}{(\\omega)} d\\omega)^{\\omega} d\\omega", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)))"], [["integrate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integral(Pow(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Integral(Pow(Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(Integral(Pow(Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Pow(Integral(Function('v_t')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(v_{1})} = \\sin{(v_{1})}, then obtain 2 A{(v_{1})} + \\sin{(v_{1})} + \\frac{d^{2}}{d v_{1}^{2}} (2 A{(v_{1})} + \\sin{(v_{1})}) = 2 A{(v_{1})} + \\sin{(v_{1})} + \\frac{d^{2}}{d v_{1}^{2}} 3 \\sin{(v_{1})}", "derivation": "A{(v_{1})} = \\sin{(v_{1})} and A{(v_{1})} + \\sin{(v_{1})} = 2 \\sin{(v_{1})} and A{(v_{1})} + 2 \\sin{(v_{1})} = 3 \\sin{(v_{1})} and 2 A{(v_{1})} + \\sin{(v_{1})} = 3 \\sin{(v_{1})} and \\frac{d}{d v_{1}} (2 A{(v_{1})} + \\sin{(v_{1})}) = \\frac{d}{d v_{1}} 3 \\sin{(v_{1})} and \\frac{d^{2}}{d v_{1}^{2}} (2 A{(v_{1})} + \\sin{(v_{1})}) = \\frac{d^{2}}{d v_{1}^{2}} 3 \\sin{(v_{1})} and 2 A{(v_{1})} + \\sin{(v_{1})} + \\frac{d^{2}}{d v_{1}^{2}} (2 A{(v_{1})} + \\sin{(v_{1})}) = 2 A{(v_{1})} + \\sin{(v_{1})} + \\frac{d^{2}}{d v_{1}^{2}} 3 \\sin{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["add", 1, "sin(Symbol('v_1', commutative=True))"], "Equality(Add(Function('A')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True))), Mul(Integer(2), sin(Symbol('v_1', commutative=True))))"], [["add", 1, "Mul(Integer(2), sin(Symbol('v_1', commutative=True)))"], "Equality(Add(Function('A')(Symbol('v_1', commutative=True)), Mul(Integer(2), sin(Symbol('v_1', commutative=True)))), Mul(Integer(3), sin(Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), Mul(Integer(3), sin(Symbol('v_1', commutative=True))))"], [["differentiate", 4, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Integer(3), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(2))), Derivative(Mul(Integer(3), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(2))))"], [["add", 6, "Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True)), Derivative(Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(2)))), Add(Mul(Integer(2), Function('A')(Symbol('v_1', commutative=True))), sin(Symbol('v_1', commutative=True)), Derivative(Mul(Integer(3), sin(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\ddot{x}{(g,\\eta)} = \\eta - g, then derive \\frac{\\partial}{\\partial g} \\ddot{x}{(g,\\eta)} = -1, then obtain - g + \\int \\eta dg + \\int \\frac{\\partial}{\\partial g} (\\eta - g) dg = - g + \\int (-1) dg + \\int \\eta dg", "derivation": "\\ddot{x}{(g,\\eta)} = \\eta - g and \\frac{\\partial}{\\partial g} \\ddot{x}{(g,\\eta)} = \\frac{\\partial}{\\partial g} (\\eta - g) and \\frac{\\partial}{\\partial g} \\ddot{x}{(g,\\eta)} = -1 and \\eta + \\frac{\\partial}{\\partial g} \\ddot{x}{(g,\\eta)} = \\eta - 1 and \\eta + \\frac{\\partial}{\\partial g} (\\eta - g) = \\eta - 1 and \\int (\\eta + \\frac{\\partial}{\\partial g} (\\eta - g)) dg = \\int (\\eta - 1) dg and \\int \\eta dg + \\int \\frac{\\partial}{\\partial g} (\\eta - g) dg = \\int (-1) dg + \\int \\eta dg and - g + \\int \\eta dg + \\int \\frac{\\partial}{\\partial g} (\\eta - g) dg = - g + \\int (-1) dg + \\int \\eta dg", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))"], [["add", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Symbol('\\\\eta', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\eta', commutative=True), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Symbol('\\\\eta', commutative=True), Integer(-1)))"], [["integrate", 5, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\eta', commutative=True), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('\\\\eta', commutative=True), Integer(-1)), Tuple(Symbol('g', commutative=True))))"], [["expand", 6], "Equality(Add(Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('g', commutative=True))), Integral(Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True)))), Add(Integral(Integer(-1), Tuple(Symbol('g', commutative=True))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('g', commutative=True)))))"], [["minus", 7, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('g', commutative=True))), Integral(Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Integer(-1), Tuple(Symbol('g', commutative=True))), Integral(Symbol('\\\\eta', commutative=True), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\chi)} = \\cos{(\\chi)}, then obtain - \\sin{(\\chi)} + \\frac{d}{d \\chi} \\theta_{1}{(\\chi)} + 1 = 1 - 2 \\sin{(\\chi)}", "derivation": "\\theta_{1}{(\\chi)} = \\cos{(\\chi)} and \\theta_{1}{(\\chi)} + \\cos{(\\chi)} = 2 \\cos{(\\chi)} and \\chi + \\theta_{1}{(\\chi)} + \\cos{(\\chi)} = \\chi + 2 \\cos{(\\chi)} and \\frac{d}{d \\chi} (\\chi + \\theta_{1}{(\\chi)} + \\cos{(\\chi)}) = \\frac{d}{d \\chi} (\\chi + 2 \\cos{(\\chi)}) and - \\sin{(\\chi)} + \\frac{d}{d \\chi} \\theta_{1}{(\\chi)} + 1 = 1 - 2 \\sin{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\theta_1')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), cos(Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Function('\\\\theta_1')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))), Derivative(Function('\\\\theta_1')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given H{(t_{2},M)} = t_{2}^{M}, then obtain - M + H{(t_{2},M)} + \\int \\frac{\\partial}{\\partial t_{2}} H{(t_{2},M)} dM = - M + H{(t_{2},M)} + \\int \\frac{\\partial}{\\partial t_{2}} t_{2}^{M} dM", "derivation": "H{(t_{2},M)} = t_{2}^{M} and \\frac{\\partial}{\\partial t_{2}} H{(t_{2},M)} = \\frac{\\partial}{\\partial t_{2}} t_{2}^{M} and \\int \\frac{\\partial}{\\partial t_{2}} H{(t_{2},M)} dM = \\int \\frac{\\partial}{\\partial t_{2}} t_{2}^{M} dM and - M + t_{2}^{M} + \\int \\frac{\\partial}{\\partial t_{2}} H{(t_{2},M)} dM = - M + t_{2}^{M} + \\int \\frac{\\partial}{\\partial t_{2}} t_{2}^{M} dM and - M + H{(t_{2},M)} + \\int \\frac{\\partial}{\\partial t_{2}} H{(t_{2},M)} dM = - M + H{(t_{2},M)} + \\int \\frac{\\partial}{\\partial t_{2}} t_{2}^{M} dM", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Integral(Derivative(Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Integral(Derivative(Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Integral(Derivative(Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('H')(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Integral(Derivative(Pow(Symbol('t_2', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(k,C_{d})} = \\frac{C_{d}}{k} and y{(k,C_{d})} = \\frac{C_{d}}{k}, then obtain 0 = - \\frac{C_{d}}{k} + y{(k,C_{d})}", "derivation": "\\mathbf{J}_M{(k,C_{d})} = \\frac{C_{d}}{k} and \\mathbf{J}_M{(k,C_{d})} + \\frac{1}{k} = \\frac{C_{d}}{k} + \\frac{1}{k} and y{(k,C_{d})} = \\frac{C_{d}}{k} and 0 = \\frac{C_{d}}{k} - \\mathbf{J}_M{(k,C_{d})} and 0 = - \\mathbf{J}_M{(k,C_{d})} + y{(k,C_{d})} and 0 = - \\frac{C_{d}}{k} + y{(k,C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1))), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Pow(Symbol('k', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('y')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["minus", 2, "Add(Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)), Pow(Symbol('k', commutative=True), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('C_d', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))), Function('y')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('y')(Symbol('k', commutative=True), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given y{(\\tilde{g})} = \\cos{(\\tilde{g})} and S{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain \\frac{d}{d \\tilde{g}} (S{(\\tilde{g})} + \\cos{(\\tilde{g})}) = \\frac{d}{d \\tilde{g}} 2 \\cos{(\\tilde{g})}", "derivation": "y{(\\tilde{g})} = \\cos{(\\tilde{g})} and y{(\\tilde{g})} + \\cos{(\\tilde{g})} = 2 \\cos{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} (y{(\\tilde{g})} + \\cos{(\\tilde{g})}) = \\frac{d}{d \\tilde{g}} 2 \\cos{(\\tilde{g})} and S{(\\tilde{g})} = \\cos{(\\tilde{g})} and S{(\\tilde{g})} = y{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} (S{(\\tilde{g})} + \\cos{(\\tilde{g})}) = \\frac{d}{d \\tilde{g}} 2 \\cos{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('y')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Add(Function('y')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('S')(Symbol('\\\\tilde{g}', commutative=True)), Function('y')(Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Add(Function('S')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mu,\\chi)} = \\chi + 2 \\mu, then derive \\frac{\\frac{\\partial}{\\partial \\chi} \\mathbf{p}{(\\mu,\\chi)}}{\\mu} = \\frac{1}{\\mu}, then obtain \\frac{\\frac{\\partial}{\\partial \\chi} (\\chi + 2 \\mu)}{\\mu} = \\frac{1}{\\mu}", "derivation": "\\mathbf{p}{(\\mu,\\chi)} = \\chi + 2 \\mu and \\frac{\\mathbf{p}{(\\mu,\\chi)}}{\\mu} = \\frac{\\chi + 2 \\mu}{\\mu} and \\frac{\\partial}{\\partial \\chi} \\frac{\\mathbf{p}{(\\mu,\\chi)}}{\\mu} = \\frac{\\partial}{\\partial \\chi} \\frac{\\chi + 2 \\mu}{\\mu} and \\frac{\\frac{\\partial}{\\partial \\chi} \\mathbf{p}{(\\mu,\\chi)}}{\\mu} = \\frac{1}{\\mu} and \\frac{\\frac{\\partial}{\\partial \\chi} (\\chi + 2 \\mu)}{\\mu} = \\frac{1}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["divide", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(2), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\varphi^{*}{(y^{\\prime})} = \\sin{(y^{\\prime})}, then obtain \\int (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\varphi^{*}{(y^{\\prime})} dy^{\\prime} = \\int (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\sin{(y^{\\prime})} dy^{\\prime}", "derivation": "\\varphi^{*}{(y^{\\prime})} = \\sin{(y^{\\prime})} and y^{\\prime} + \\varphi^{*}{(y^{\\prime})} = y^{\\prime} + \\sin{(y^{\\prime})} and (y^{\\prime} + \\sin{(y^{\\prime})}) \\varphi^{*}{(y^{\\prime})} = (y^{\\prime} + \\sin{(y^{\\prime})}) \\sin{(y^{\\prime})} and (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\varphi^{*}{(y^{\\prime})} = (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\sin{(y^{\\prime})} and \\int (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\varphi^{*}{(y^{\\prime})} dy^{\\prime} = \\int (y^{\\prime} + \\varphi^{*}{(y^{\\prime})}) \\sin{(y^{\\prime})} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 1, "Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\varphi^*')(Symbol('y^{\\\\prime}', commutative=True))), sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\delta)} = \\sin{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} = \\sin{(\\delta)}, then obtain - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{z^{*}}{(\\delta)} + \\operatorname{z^{*}}^{\\delta}{(\\delta)} = - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{z^{*}}{(\\delta)} + \\sin^{\\delta}{(\\delta)}", "derivation": "\\operatorname{v_{1}}{(\\delta)} = \\sin{(\\delta)} and \\operatorname{v_{1}}^{\\delta}{(\\delta)} = \\sin^{\\delta}{(\\delta)} and - (\\sin^{\\delta}{(\\delta)})^{\\delta} + \\operatorname{v_{1}}^{\\delta}{(\\delta)} = - (\\sin^{\\delta}{(\\delta)})^{\\delta} + \\sin^{\\delta}{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} = \\sin{(\\delta)} and - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{v_{1}}{(\\delta)} + \\operatorname{v_{1}}^{\\delta}{(\\delta)} = - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{v_{1}}{(\\delta)} + \\sin^{\\delta}{(\\delta)} and \\operatorname{z^{*}}{(\\delta)} = \\operatorname{v_{1}}{(\\delta)} and - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{z^{*}}{(\\delta)} + \\operatorname{z^{*}}^{\\delta}{(\\delta)} = - (\\sin^{\\delta}{(\\delta)})^{\\delta} - \\operatorname{z^{*}}{(\\delta)} + \\sin^{\\delta}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["minus", 2, "Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Pow(Function('v_1')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["minus", 3, "Function('v_1')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('v_1')(Symbol('\\\\delta', commutative=True))), Pow(Function('v_1')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('v_1')(Symbol('\\\\delta', commutative=True))), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True)), Function('v_1')(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('z^*')(Symbol('\\\\delta', commutative=True))), Pow(Function('z^*')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('z^*')(Symbol('\\\\delta', commutative=True))), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(f_{E},p)} = \\frac{\\cos{(f_{E})}}{p}, then obtain p \\Psi^{\\dagger}^{3}{(f_{E},p)} = \\frac{\\Psi^{\\dagger}{(f_{E},p)} \\cos^{2}{(f_{E})}}{p}", "derivation": "\\Psi^{\\dagger}{(f_{E},p)} = \\frac{\\cos{(f_{E})}}{p} and \\Psi^{\\dagger}^{2}{(f_{E},p)} = \\frac{\\Psi^{\\dagger}{(f_{E},p)} \\cos{(f_{E})}}{p} and \\Psi^{\\dagger}^{3}{(f_{E},p)} = \\frac{\\Psi^{\\dagger}^{2}{(f_{E},p)} \\cos{(f_{E})}}{p} and \\Psi^{\\dagger}^{3}{(f_{E},p)} = \\frac{\\Psi^{\\dagger}{(f_{E},p)} \\cos^{2}{(f_{E})}}{p^{2}} and p \\Psi^{\\dagger}^{3}{(f_{E},p)} = \\frac{\\Psi^{\\dagger}{(f_{E},p)} \\cos^{2}{(f_{E})}}{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), cos(Symbol('f_E', commutative=True))))"], [["times", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(2)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), cos(Symbol('f_E', commutative=True))))"], [["times", 1, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(3)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(2)), cos(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(3)), Mul(Pow(Symbol('p', commutative=True), Integer(-2)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Integer(2))))"], [["divide", 4, "Pow(Symbol('p', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('p', commutative=True), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Integer(3))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('p', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Integer(2))))"]]}, {"prompt": "Given s{(\\dot{\\mathbf{r}},p)} = \\dot{\\mathbf{r}}^{p}, then derive k + s{(\\dot{\\mathbf{r}},p)} = \\dot{\\mathbf{r}}^{p} + u, then obtain (\\dot{\\mathbf{r}}^{p} + k)^{k} = (k + s{(\\dot{\\mathbf{r}},p)})^{k}", "derivation": "s{(\\dot{\\mathbf{r}},p)} = \\dot{\\mathbf{r}}^{p} and \\frac{\\partial}{\\partial p} s{(\\dot{\\mathbf{r}},p)} = \\frac{\\partial}{\\partial p} \\dot{\\mathbf{r}}^{p} and \\int \\frac{\\partial}{\\partial p} s{(\\dot{\\mathbf{r}},p)} dp = \\int \\frac{\\partial}{\\partial p} \\dot{\\mathbf{r}}^{p} dp and k + s{(\\dot{\\mathbf{r}},p)} = \\dot{\\mathbf{r}}^{p} + u and \\dot{\\mathbf{r}}^{p} + k = \\dot{\\mathbf{r}}^{p} + u and \\dot{\\mathbf{r}}^{p} + k = k + s{(\\dot{\\mathbf{r}},p)} and (\\dot{\\mathbf{r}}^{p} + k)^{k} = (k + s{(\\dot{\\mathbf{r}},p)})^{k}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('k', commutative=True), Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Add(Symbol('k', commutative=True), Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Symbol('k', commutative=True), Function('s')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\delta{(U,\\hat{H})} = \\cos^{\\hat{H}}{(U)}, then obtain (\\cos{(U)} \\iint \\delta{(U,\\hat{H})} dU d\\hat{H})^{U} = (\\cos{(U)} \\iint \\cos^{\\hat{H}}{(U)} dU d\\hat{H})^{U}", "derivation": "\\delta{(U,\\hat{H})} = \\cos^{\\hat{H}}{(U)} and \\int \\delta{(U,\\hat{H})} dU = \\int \\cos^{\\hat{H}}{(U)} dU and \\iint \\delta{(U,\\hat{H})} dU d\\hat{H} = \\iint \\cos^{\\hat{H}}{(U)} dU d\\hat{H} and \\cos{(U)} \\iint \\delta{(U,\\hat{H})} dU d\\hat{H} = \\cos{(U)} \\iint \\cos^{\\hat{H}}{(U)} dU d\\hat{H} and (\\cos{(U)} \\iint \\delta{(U,\\hat{H})} dU d\\hat{H})^{U} = (\\cos{(U)} \\iint \\cos^{\\hat{H}}{(U)} dU d\\hat{H})^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(cos(Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Pow(cos(Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 3, "cos(Symbol('U', commutative=True))"], "Equality(Mul(cos(Symbol('U', commutative=True)), Integral(Function('\\\\delta')(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(cos(Symbol('U', commutative=True)), Integral(Pow(cos(Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(cos(Symbol('U', commutative=True)), Integral(Function('\\\\delta')(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Symbol('U', commutative=True)), Pow(Mul(cos(Symbol('U', commutative=True)), Integral(Pow(cos(Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given A{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\log{(E_{\\lambda})} and \\operatorname{v_{2}}{(E_{\\lambda})} = \\log{(E_{\\lambda})}, then obtain (\\frac{d}{d E_{\\lambda}} \\log{(E_{\\lambda})})^{E_{\\lambda}} = A^{E_{\\lambda}}{(E_{\\lambda})}", "derivation": "A{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\log{(E_{\\lambda})} and \\operatorname{v_{2}}{(E_{\\lambda})} = \\log{(E_{\\lambda})} and \\frac{d}{d E_{\\lambda}} \\operatorname{v_{2}}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\log{(E_{\\lambda})} and \\frac{d}{d E_{\\lambda}} \\operatorname{v_{2}}{(E_{\\lambda})} = A{(E_{\\lambda})} and (\\frac{d}{d E_{\\lambda}} \\operatorname{v_{2}}{(E_{\\lambda})})^{E_{\\lambda}} = A^{E_{\\lambda}}{(E_{\\lambda})} and (\\frac{d}{d E_{\\lambda}} \\log{(E_{\\lambda})})^{E_{\\lambda}} = A^{E_{\\lambda}}{(E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('E_{\\\\lambda}', commutative=True)), Derivative(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('E_{\\\\lambda}', commutative=True)), log(Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('v_2')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Function('A')(Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Function('v_2')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('A')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Derivative(log(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('A')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given A{(\\chi)} = \\sin{(\\cos{(\\chi)})}, then obtain \\int \\frac{d}{d \\chi} (\\frac{d}{d \\chi} A{(\\chi)})^{\\chi} d\\chi = \\int \\frac{d}{d \\chi} (\\frac{d}{d \\chi} \\sin{(\\cos{(\\chi)})})^{\\chi} d\\chi", "derivation": "A{(\\chi)} = \\sin{(\\cos{(\\chi)})} and \\frac{d}{d \\chi} A{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\cos{(\\chi)})} and (\\frac{d}{d \\chi} A{(\\chi)})^{\\chi} = (\\frac{d}{d \\chi} \\sin{(\\cos{(\\chi)})})^{\\chi} and \\frac{d}{d \\chi} (\\frac{d}{d \\chi} A{(\\chi)})^{\\chi} = \\frac{d}{d \\chi} (\\frac{d}{d \\chi} \\sin{(\\cos{(\\chi)})})^{\\chi} and \\int \\frac{d}{d \\chi} (\\frac{d}{d \\chi} A{(\\chi)})^{\\chi} d\\chi = \\int \\frac{d}{d \\chi} (\\frac{d}{d \\chi} \\sin{(\\cos{(\\chi)})})^{\\chi} d\\chi", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\chi', commutative=True)), sin(cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Derivative(Function('A')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Pow(Derivative(sin(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('A')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(Derivative(sin(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Pow(Derivative(Function('A')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(Pow(Derivative(sin(cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(v_{x})} = e^{v_{x}}, then derive 0 = e^{v_{x}} - \\frac{d}{d v_{x}} \\hat{x}_0{(v_{x})}, then obtain 0 = e^{v_{x}} - \\frac{d}{d v_{x}} e^{v_{x}}", "derivation": "\\hat{x}_0{(v_{x})} = e^{v_{x}} and 0 = - \\hat{x}_0{(v_{x})} + e^{v_{x}} and \\frac{d}{d v_{x}} 0 = \\frac{d}{d v_{x}} (- \\hat{x}_0{(v_{x})} + e^{v_{x}}) and 0 = e^{v_{x}} - \\frac{d}{d v_{x}} \\hat{x}_0{(v_{x})} and 0 = e^{v_{x}} - \\frac{d}{d v_{x}} e^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["minus", 1, "Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(exp(Symbol('v_x', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(exp(Symbol('v_x', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\Psi)} = \\sin{(\\Psi)}, then obtain \\operatorname{M_{E}}{(\\Psi)} - \\int \\sin{(\\Psi)} d\\Psi = \\sin{(\\Psi)} - \\int \\sin{(\\Psi)} d\\Psi", "derivation": "\\operatorname{M_{E}}{(\\Psi)} = \\sin{(\\Psi)} and \\int \\operatorname{M_{E}}{(\\Psi)} d\\Psi = \\int \\sin{(\\Psi)} d\\Psi and \\operatorname{M_{E}}{(\\Psi)} - \\int \\operatorname{M_{E}}{(\\Psi)} d\\Psi = \\sin{(\\Psi)} - \\int \\operatorname{M_{E}}{(\\Psi)} d\\Psi and \\operatorname{M_{E}}{(\\Psi)} - \\int \\sin{(\\Psi)} d\\Psi = \\sin{(\\Psi)} - \\int \\sin{(\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["minus", 1, "Integral(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integral(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))), Add(sin(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integral(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('M_E')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))), Add(sin(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and i{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}}, then obtain \\mathbf{J}_P{(f_{\\mathbf{p}})} - e^{f_{\\mathbf{p}}} = 0", "derivation": "\\mathbf{J}_P{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and i{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and i{(f_{\\mathbf{p}})} = \\mathbf{J}_P{(f_{\\mathbf{p}})} and i{(f_{\\mathbf{p}})} - e^{f_{\\mathbf{p}}} = 0 and \\mathbf{J}_P{(f_{\\mathbf{p}})} - e^{f_{\\mathbf{p}}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('i')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["minus", 2, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Function('i')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given u{(p)} = \\sin{(\\cos{(p)})}, then obtain u{(p)} - \\int u{(p)} dp = \\sin{(\\cos{(p)})} - \\int u{(p)} dp", "derivation": "u{(p)} = \\sin{(\\cos{(p)})} and \\int u{(p)} dp = \\int \\sin{(\\cos{(p)})} dp and u{(p)} - \\int \\sin{(\\cos{(p)})} dp = \\sin{(\\cos{(p)})} - \\int \\sin{(\\cos{(p)})} dp and u{(p)} - \\int u{(p)} dp = \\sin{(\\cos{(p)})} - \\int u{(p)} dp", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('p', commutative=True)), sin(cos(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('u')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(sin(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["minus", 1, "Integral(sin(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Function('u')(Symbol('p', commutative=True)), Mul(Integer(-1), Integral(sin(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))), Add(sin(cos(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(sin(cos(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('u')(Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Function('u')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Add(sin(cos(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('u')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(A,E_{\\lambda})} = E_{\\lambda} + \\sin{(A)} and \\sigma_{p}{(A,E_{\\lambda})} = - \\operatorname{L_{\\varepsilon}}{(A,E_{\\lambda})}, then obtain \\sigma_{p}^{A}{(A,E_{\\lambda})} = (- E_{\\lambda} - \\sin{(A)})^{A}", "derivation": "\\operatorname{L_{\\varepsilon}}{(A,E_{\\lambda})} = E_{\\lambda} + \\sin{(A)} and \\sigma_{p}{(A,E_{\\lambda})} = - \\operatorname{L_{\\varepsilon}}{(A,E_{\\lambda})} and \\sigma_{p}{(A,E_{\\lambda})} = - E_{\\lambda} - \\sin{(A)} and \\sigma_{p}^{A}{(A,E_{\\lambda})} = (- E_{\\lambda} - \\sin{(A)})^{A}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('A', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('A', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\sigma_p')(Symbol('A', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('A', commutative=True)))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('A', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\psi{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then derive \\int \\psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\varphi^* - \\cos{(\\hat{\\mathbf{x}})}, then obtain \\frac{d}{d \\varphi^*} \\int \\psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^* - \\cos{(\\hat{\\mathbf{x}})})", "derivation": "\\psi{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\int \\psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} and \\int \\psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\varphi^* - \\cos{(\\hat{\\mathbf{x}})} and \\frac{d}{d \\varphi^*} \\int \\psi{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^* - \\cos{(\\hat{\\mathbf{x}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{p},\\hbar)} = \\mathbf{p} + \\sin{(\\hbar)} and \\mathbf{J}_M{(f)} = \\cos{(f)}, then obtain \\mathbf{p} + \\mathbf{J}_M{(f)} + \\sin{(\\hbar)} = 2 \\mathbf{p} + \\mathbf{J}_M{(f)} - \\mathbf{J}_f{(\\mathbf{p},\\hbar)} + 2 \\sin{(\\hbar)}", "derivation": "\\mathbf{J}_f{(\\mathbf{p},\\hbar)} = \\mathbf{p} + \\sin{(\\hbar)} and 0 = \\mathbf{p} - \\mathbf{J}_f{(\\mathbf{p},\\hbar)} + \\sin{(\\hbar)} and \\mathbf{J}_M{(f)} = \\cos{(f)} and \\mathbf{p} + \\mathbf{J}_M{(f)} + \\sin{(\\hbar)} = \\mathbf{p} + \\sin{(\\hbar)} + \\cos{(f)} and \\cos{(f)} = \\mathbf{p} - \\mathbf{J}_f{(\\mathbf{p},\\hbar)} + \\sin{(\\hbar)} + \\cos{(f)} and \\mathbf{p} + \\mathbf{J}_M{(f)} + \\sin{(\\hbar)} = 2 \\mathbf{p} - \\mathbf{J}_f{(\\mathbf{p},\\hbar)} + 2 \\sin{(\\hbar)} + \\cos{(f)} and \\mathbf{p} + \\mathbf{J}_M{(f)} + \\sin{(\\hbar)} = 2 \\mathbf{p} + \\mathbf{J}_M{(f)} - \\mathbf{J}_f{(\\mathbf{p},\\hbar)} + 2 \\sin{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["add", 3, "Add(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\hbar', commutative=True)), cos(Symbol('f', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), cos(Symbol('f', commutative=True)))"], "Equality(cos(Symbol('f', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)), cos(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hbar', commutative=True))), cos(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), sin(Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(l,r)} = \\frac{r}{l}, then obtain \\frac{\\partial}{\\partial l} \\frac{(\\mathbf{D}^{2}{(l,r)})^{r}}{l} = \\frac{\\partial}{\\partial l} \\frac{(\\frac{r \\mathbf{D}{(l,r)}}{l})^{r}}{l}", "derivation": "\\mathbf{D}{(l,r)} = \\frac{r}{l} and \\mathbf{D}^{2}{(l,r)} = \\frac{r \\mathbf{D}{(l,r)}}{l} and (\\mathbf{D}^{2}{(l,r)})^{r} = (\\frac{r \\mathbf{D}{(l,r)}}{l})^{r} and \\frac{(\\mathbf{D}^{2}{(l,r)})^{r}}{l} = \\frac{(\\frac{r \\mathbf{D}{(l,r)}}{l})^{r}}{l} and \\frac{\\partial}{\\partial l} \\frac{(\\mathbf{D}^{2}{(l,r)})^{r}}{l} = \\frac{\\partial}{\\partial l} \\frac{(\\frac{r \\mathbf{D}{(l,r)}}{l})^{r}}{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Integer(2)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('r', commutative=True), Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('r', commutative=True), Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["divide", 3, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('r', commutative=True), Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('r', commutative=True), Function('\\\\mathbf{D}')(Symbol('l', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})} = L_{\\varepsilon} + y^{\\prime}, then derive \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})} = 1, then obtain e^{\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + y^{\\prime})} = e", "derivation": "\\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})} = L_{\\varepsilon} + y^{\\prime} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + y^{\\prime}) and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})} = 1 and e^{\\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{F_{N}}{(y^{\\prime},L_{\\varepsilon})}} = e and e^{\\frac{\\partial}{\\partial L_{\\varepsilon}} (L_{\\varepsilon} + y^{\\prime})} = e", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["exp", 3], "Equality(exp(Derivative(Function('F_N')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), E)"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Derivative(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), E)"]]}, {"prompt": "Given \\mathbf{s}{(c_{0})} = \\cos{(c_{0})}, then obtain 0^{c_{0}} + \\mathbf{s}{(c_{0})} - \\cos{(c_{0})} = 0^{c_{0}}", "derivation": "\\mathbf{s}{(c_{0})} = \\cos{(c_{0})} and \\mathbf{s}{(c_{0})} - \\cos{(c_{0})} = 0 and (\\mathbf{s}{(c_{0})} - \\cos{(c_{0})})^{c_{0}} = 0^{c_{0}} and (\\mathbf{s}{(c_{0})} - \\cos{(c_{0})})^{c_{0}} + \\mathbf{s}{(c_{0})} - \\cos{(c_{0})} = (\\mathbf{s}{(c_{0})} - \\cos{(c_{0})})^{c_{0}} and 0^{c_{0}} + \\mathbf{s}{(c_{0})} - \\cos{(c_{0})} = 0^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["minus", 1, "cos(Symbol('c_0', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('c_0', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)), Pow(Integer(0), Symbol('c_0', commutative=True)))"], [["add", 2, "Pow(Add(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True))"], "Equality(Add(Pow(Add(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Pow(Add(Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Integer(0), Symbol('c_0', commutative=True)), Function('\\\\mathbf{s}')(Symbol('c_0', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True)))), Pow(Integer(0), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given W{(s,G)} = \\frac{e^{s}}{G}, then obtain - \\int (\\frac{e^{s}}{G})^{G} dG + \\frac{W^{G}{(s,G)} - e^{s} - \\frac{e^{s}}{G}}{G} = - \\int (\\frac{e^{s}}{G})^{G} dG + \\frac{(\\frac{e^{s}}{G})^{G} - e^{s} - \\frac{e^{s}}{G}}{G}", "derivation": "W{(s,G)} = \\frac{e^{s}}{G} and W^{G}{(s,G)} = (\\frac{e^{s}}{G})^{G} and W^{G}{(s,G)} - e^{s} - \\frac{e^{s}}{G} = (\\frac{e^{s}}{G})^{G} - e^{s} - \\frac{e^{s}}{G} and \\frac{W^{G}{(s,G)} - e^{s} - \\frac{e^{s}}{G}}{G} = \\frac{(\\frac{e^{s}}{G})^{G} - e^{s} - \\frac{e^{s}}{G}}{G} and - \\int (\\frac{e^{s}}{G})^{G} dG + \\frac{W^{G}{(s,G)} - e^{s} - \\frac{e^{s}}{G}}{G} = - \\int (\\frac{e^{s}}{G})^{G} dG + \\frac{(\\frac{e^{s}}{G})^{G} - e^{s} - \\frac{e^{s}}{G}}{G}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('s', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('W')(Symbol('s', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)))"], [["minus", 2, "Add(exp(Symbol('s', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))"], "Equality(Add(Pow(Function('W')(Symbol('s', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))), Add(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))))"], [["times", 3, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Pow(Function('W')(Symbol('s', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))))))"], [["minus", 4, "Integral(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Pow(Function('W')(Symbol('s', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))))), Add(Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True))), Symbol('G', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), exp(Symbol('s', commutative=True)))))))"]]}, {"prompt": "Given S{(J)} = e^{J}, then derive \\int S{(J)} dJ = B + e^{J}, then obtain B = - e^{J} + \\int e^{J} dJ", "derivation": "S{(J)} = e^{J} and \\int S{(J)} dJ = \\int e^{J} dJ and \\int S{(J)} dJ = B + e^{J} and B + e^{J} = \\int e^{J} dJ and B - S{(J)} + e^{J} = - S{(J)} + \\int e^{J} dJ and B = - e^{J} + \\int e^{J} dJ", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('S')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('S')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Symbol('B', commutative=True), exp(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('B', commutative=True), exp(Symbol('J', commutative=True))), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["minus", 4, "Function('S')(Symbol('J', commutative=True))"], "Equality(Add(Symbol('B', commutative=True), Mul(Integer(-1), Function('S')(Symbol('J', commutative=True))), exp(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Function('S')(Symbol('J', commutative=True))), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Symbol('B', commutative=True), Add(Mul(Integer(-1), exp(Symbol('J', commutative=True))), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(V)} = \\log{(V)}, then obtain \\iint \\frac{\\operatorname{a^{\\dagger}}{(V)}}{V} dV dV = \\iint \\frac{\\log{(V)}}{V} dV dV", "derivation": "\\operatorname{a^{\\dagger}}{(V)} = \\log{(V)} and \\frac{\\operatorname{a^{\\dagger}}{(V)}}{V} = \\frac{\\log{(V)}}{V} and \\int \\frac{\\operatorname{a^{\\dagger}}{(V)}}{V} dV = \\int \\frac{\\log{(V)}}{V} dV and \\iint \\frac{\\operatorname{a^{\\dagger}}{(V)}}{V} dV dV = \\iint \\frac{\\log{(V)}}{V} dV dV", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} = \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}}, then obtain \\frac{\\int (\\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} - \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}}) d\\mathbf{v}}{g^{\\prime}_{\\varepsilon}} = \\frac{\\int 0 d\\mathbf{v}}{g^{\\prime}_{\\varepsilon}}", "derivation": "\\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} = \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}} and \\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} - \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}} = 0 and \\int (\\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} - \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}}) d\\mathbf{v} = \\int 0 d\\mathbf{v} and \\frac{\\int (\\operatorname{v_{x}}{(g^{\\prime}_{\\varepsilon},\\mathbf{v})} - \\frac{g^{\\prime}_{\\varepsilon}}{\\mathbf{v}}) d\\mathbf{v}}{g^{\\prime}_{\\varepsilon}} = \\frac{\\int 0 d\\mathbf{v}}{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Add(Function('v_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(Add(Function('v_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(t_{1},J)} = \\frac{t_{1}}{J}, then obtain (- \\dot{z}{(t_{1},J)} - 1)^{t_{1}} = (-1 - \\frac{t_{1}}{J})^{t_{1}}", "derivation": "\\dot{z}{(t_{1},J)} = \\frac{t_{1}}{J} and \\dot{z}{(t_{1},J)} + 1 = 1 + \\frac{t_{1}}{J} and - \\dot{z}{(t_{1},J)} - 1 = -1 - \\frac{t_{1}}{J} and (- \\dot{z}{(t_{1},J)} - 1)^{t_{1}} = (-1 - \\frac{t_{1}}{J})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('t_1', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\dot{z}')(Symbol('t_1', commutative=True), Symbol('J', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('t_1', commutative=True), Symbol('J', commutative=True))), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('t_1', commutative=True), Symbol('J', commutative=True))), Integer(-1)), Symbol('t_1', commutative=True)), Pow(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given y{(\\psi,\\mathbf{E},u)} = \\mathbf{E} \\psi u, then obtain \\mathbf{E}^{2} u^{2} \\int (u + y{(\\psi,\\mathbf{E},u)}) du = \\mathbf{E}^{2} u^{2} \\int (\\mathbf{E} \\psi u + u) du", "derivation": "y{(\\psi,\\mathbf{E},u)} = \\mathbf{E} \\psi u and u + y{(\\psi,\\mathbf{E},u)} = \\mathbf{E} \\psi u + u and \\int (u + y{(\\psi,\\mathbf{E},u)}) du = \\int (\\mathbf{E} \\psi u + u) du and \\mathbf{E} u \\int (u + y{(\\psi,\\mathbf{E},u)}) du = \\mathbf{E} u \\int (\\mathbf{E} \\psi u + u) du and \\mathbf{E}^{2} u^{2} \\int (u + y{(\\psi,\\mathbf{E},u)}) du = \\mathbf{E}^{2} u^{2} \\int (\\mathbf{E} \\psi u + u) du", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Symbol('u', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True), Integral(Add(Symbol('u', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True), Integral(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["times", 4, "Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Pow(Symbol('u', commutative=True), Integer(2)), Integral(Add(Symbol('u', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Pow(Symbol('u', commutative=True), Integer(2)), Integral(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(P_{g})} = \\log{(P_{g})} and y{(E_{n})} = \\cos{(E_{n})}, then obtain (P_{g} \\mathbf{f}{(P_{g})})^{P_{g}} y^{E_{n}}{(E_{n})} = (P_{g} \\mathbf{f}{(P_{g})})^{P_{g}} \\cos^{E_{n}}{(E_{n})}", "derivation": "\\mathbf{f}{(P_{g})} = \\log{(P_{g})} and P_{g} \\mathbf{f}{(P_{g})} = P_{g} \\log{(P_{g})} and (P_{g} \\mathbf{f}{(P_{g})})^{P_{g}} = (P_{g} \\log{(P_{g})})^{P_{g}} and y{(E_{n})} = \\cos{(E_{n})} and y^{E_{n}}{(E_{n})} = \\cos^{E_{n}}{(E_{n})} and (P_{g} \\log{(P_{g})})^{P_{g}} y^{E_{n}}{(E_{n})} = (P_{g} \\log{(P_{g})})^{P_{g}} \\cos^{E_{n}}{(E_{n})} and (P_{g} \\mathbf{f}{(P_{g})})^{P_{g}} y^{E_{n}}{(E_{n})} = (P_{g} \\mathbf{f}{(P_{g})})^{P_{g}} \\cos^{E_{n}}{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True)))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\mathbf{f}')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Mul(Symbol('P_g', commutative=True), Function('\\\\mathbf{f}')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], ["get_premise", "Equality(Function('y')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["power", 4, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('y')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)))"], [["times", 5, "Pow(Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Function('y')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))), Mul(Pow(Mul(Symbol('P_g', commutative=True), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Mul(Symbol('P_g', commutative=True), Function('\\\\mathbf{f}')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Function('y')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))), Mul(Pow(Mul(Symbol('P_g', commutative=True), Function('\\\\mathbf{f}')(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(n_{2})} = \\sin{(\\cos{(n_{2})})} and \\operatorname{E_{n}}{(n_{2})} = \\cos{(n_{2})}, then obtain \\int \\sin{(\\operatorname{E_{n}}{(n_{2})})} dn_{2} = \\int \\sin{(\\cos{(n_{2})})} dn_{2}", "derivation": "\\dot{\\mathbf{r}}{(n_{2})} = \\sin{(\\cos{(n_{2})})} and \\int \\dot{\\mathbf{r}}{(n_{2})} dn_{2} = \\int \\sin{(\\cos{(n_{2})})} dn_{2} and \\operatorname{E_{n}}{(n_{2})} = \\cos{(n_{2})} and \\dot{\\mathbf{r}}{(n_{2})} = \\sin{(\\operatorname{E_{n}}{(n_{2})})} and \\int \\sin{(\\operatorname{E_{n}}{(n_{2})})} dn_{2} = \\int \\sin{(\\cos{(n_{2})})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True)), sin(cos(Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(sin(cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True)), sin(Function('E_n')(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(sin(Function('E_n')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(sin(cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(F_{g},t_{1})} = \\frac{t_{1}}{F_{g}}, then derive \\int \\frac{\\partial}{\\partial t_{1}} F_{g} \\varphi^{*}{(F_{g},t_{1})} dF_{g} = \\int 1 dF_{g}, then obtain \\int \\frac{d}{d t_{1}} t_{1} dF_{g} = \\int 1 dF_{g}", "derivation": "\\varphi^{*}{(F_{g},t_{1})} = \\frac{t_{1}}{F_{g}} and F_{g} \\varphi^{*}{(F_{g},t_{1})} = t_{1} and \\frac{\\partial}{\\partial t_{1}} F_{g} \\varphi^{*}{(F_{g},t_{1})} = \\frac{d}{d t_{1}} t_{1} and \\int \\frac{\\partial}{\\partial t_{1}} F_{g} \\varphi^{*}{(F_{g},t_{1})} dF_{g} = \\int \\frac{d}{d t_{1}} t_{1} dF_{g} and \\int \\frac{\\partial}{\\partial t_{1}} F_{g} \\varphi^{*}{(F_{g},t_{1})} dF_{g} = \\int 1 dF_{g} and \\int \\frac{d}{d t_{1}} t_{1} dF_{g} = \\int 1 dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["times", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_g', commutative=True), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('F_g', commutative=True), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))), Integral(Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integral(Derivative(Mul(Symbol('F_g', commutative=True), Function('\\\\varphi^*')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(r_{0},F_{H},\\Omega)} = \\frac{F_{H} - r_{0}}{\\Omega}, then obtain ((\\frac{\\Omega \\hat{H}{(r_{0},F_{H},\\Omega)}}{F_{H} - r_{0}})^{r_{0}})^{r_{0}} + 1 = 2", "derivation": "\\hat{H}{(r_{0},F_{H},\\Omega)} = \\frac{F_{H} - r_{0}}{\\Omega} and \\frac{\\Omega \\hat{H}{(r_{0},F_{H},\\Omega)}}{F_{H} - r_{0}} = 1 and (\\frac{\\Omega \\hat{H}{(r_{0},F_{H},\\Omega)}}{F_{H} - r_{0}})^{r_{0}} = 1 and ((\\frac{\\Omega \\hat{H}{(r_{0},F_{H},\\Omega)}}{F_{H} - r_{0}})^{r_{0}})^{r_{0}} = 1 and ((\\frac{\\Omega \\hat{H}{(r_{0},F_{H},\\Omega)}}{F_{H} - r_{0}})^{r_{0}})^{r_{0}} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["divide", 1, "Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(1))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('r_0', commutative=True)), Integer(1))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Integer(1))"], [["add", 4, 1], "Equality(Add(Pow(Pow(Mul(Symbol('\\\\Omega', commutative=True), Pow(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('r_0', commutative=True), Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Integer(1)), Integer(2))"]]}, {"prompt": "Given \\mathbf{P}{(z,v_{y})} = \\cos{(v_{y} - z)} and \\hat{H}{(\\hat{x}_0,\\theta_2)} = \\hat{x}_0 \\theta_2, then obtain z + \\hat{H}{(\\hat{x}_0,\\theta_2)} + \\frac{\\cos{(v_{y} - z)}}{v_{y} - z} = \\hat{x}_0 \\theta_2 + z + \\frac{\\cos{(v_{y} - z)}}{v_{y} - z}", "derivation": "\\mathbf{P}{(z,v_{y})} = \\cos{(v_{y} - z)} and \\frac{\\mathbf{P}{(z,v_{y})}}{v_{y} - z} = \\frac{\\cos{(v_{y} - z)}}{v_{y} - z} and \\hat{H}{(\\hat{x}_0,\\theta_2)} = \\hat{x}_0 \\theta_2 and z + \\hat{H}{(\\hat{x}_0,\\theta_2)} = \\hat{x}_0 \\theta_2 + z and z + \\hat{H}{(\\hat{x}_0,\\theta_2)} + \\frac{\\mathbf{P}{(z,v_{y})}}{v_{y} - z} = \\hat{x}_0 \\theta_2 + z + \\frac{\\mathbf{P}{(z,v_{y})}}{v_{y} - z} and z + \\hat{H}{(\\hat{x}_0,\\theta_2)} + \\frac{\\cos{(v_{y} - z)}}{v_{y} - z} = \\hat{x}_0 \\theta_2 + z + \\frac{\\cos{(v_{y} - z)}}{v_{y} - z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('v_y', commutative=True)), cos(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["divide", 1, "Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), cos(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('z', commutative=True))"], "Equality(Add(Symbol('z', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('z', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('v_y', commutative=True)))"], "Equality(Add(Symbol('z', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('v_y', commutative=True)))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('z', commutative=True), Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('z', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), cos(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('z', commutative=True), Mul(Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), cos(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\dot{z})} = \\log{(\\dot{z})}, then obtain 4 \\dot{z} \\dot{\\mathbf{r}}{(\\dot{z})} - 4 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{z} (3 \\dot{\\mathbf{r}}{(\\dot{z})} + \\log{(\\dot{z})}) - 4 \\dot{\\mathbf{r}}{(\\dot{z})}", "derivation": "\\dot{\\mathbf{r}}{(\\dot{z})} = \\log{(\\dot{z})} and 2 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{\\mathbf{r}}{(\\dot{z})} + \\log{(\\dot{z})} and 4 \\dot{\\mathbf{r}}{(\\dot{z})} = 3 \\dot{\\mathbf{r}}{(\\dot{z})} + \\log{(\\dot{z})} and 4 \\dot{z} \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{z} (3 \\dot{\\mathbf{r}}{(\\dot{z})} + \\log{(\\dot{z})}) and 4 \\dot{z} \\dot{\\mathbf{r}}{(\\dot{z})} - 4 \\dot{\\mathbf{r}}{(\\dot{z})} = \\dot{z} (3 \\dot{\\mathbf{r}}{(\\dot{z})} + \\log{(\\dot{z})}) - 4 \\dot{\\mathbf{r}}{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), log(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Integer(4), Symbol('\\\\dot{z}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), log(Symbol('\\\\dot{z}', commutative=True)))))"], [["minus", 4, "Mul(Integer(4), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Mul(Integer(4), Symbol('\\\\dot{z}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Integer(4), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Mul(Integer(3), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True))), log(Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(-1), Integer(4), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})}, then derive \\psi^* + \\omega{(\\sigma_x)} = h + \\sin{(\\cos{(\\sigma_x)})}, then obtain h + \\sin{(\\cos{(\\sigma_x)})} = h + \\omega{(\\sigma_x)}", "derivation": "\\omega{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})} and \\frac{d}{d \\sigma_x} \\omega{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\sin{(\\cos{(\\sigma_x)})} and \\int \\frac{d}{d \\sigma_x} \\omega{(\\sigma_x)} d\\sigma_x = \\int \\frac{d}{d \\sigma_x} \\sin{(\\cos{(\\sigma_x)})} d\\sigma_x and \\psi^* + \\omega{(\\sigma_x)} = h + \\sin{(\\cos{(\\sigma_x)})} and \\psi^* + \\sin{(\\cos{(\\sigma_x)})} = h + \\sin{(\\cos{(\\sigma_x)})} and \\psi^* + \\omega{(\\sigma_x)} = h + \\omega{(\\sigma_x)} and h + \\sin{(\\cos{(\\sigma_x)})} = h + \\omega{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True)), sin(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(sin(cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('h', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\psi^*', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))), Add(Symbol('h', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('h', commutative=True), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Symbol('h', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))), Add(Symbol('h', commutative=True), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(f_{E})} = \\cos{(f_{E})}, then derive - \\sin{(f_{E})} + \\frac{d}{d f_{E}} \\operatorname{A_{1}}{(f_{E})} = - 2 \\sin{(f_{E})}, then obtain \\frac{d}{d f_{E}} (- \\sin{(f_{E})} + \\frac{d}{d f_{E}} \\operatorname{A_{1}}{(f_{E})}) = \\frac{d}{d f_{E}} - 2 \\sin{(f_{E})}", "derivation": "\\operatorname{A_{1}}{(f_{E})} = \\cos{(f_{E})} and \\operatorname{A_{1}}{(f_{E})} + \\cos{(f_{E})} = 2 \\cos{(f_{E})} and \\frac{d}{d f_{E}} (\\operatorname{A_{1}}{(f_{E})} + \\cos{(f_{E})}) = \\frac{d}{d f_{E}} 2 \\cos{(f_{E})} and - \\sin{(f_{E})} + \\frac{d}{d f_{E}} \\operatorname{A_{1}}{(f_{E})} = - 2 \\sin{(f_{E})} and \\frac{d}{d f_{E}} (- \\sin{(f_{E})} + \\frac{d}{d f_{E}} \\operatorname{A_{1}}{(f_{E})}) = \\frac{d}{d f_{E}} - 2 \\sin{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["add", 1, "cos(Symbol('f_E', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Mul(Integer(2), cos(Symbol('f_E', commutative=True))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Function('A_1')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('f_E', commutative=True))), Derivative(Function('A_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('f_E', commutative=True))))"], [["differentiate", 4, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), sin(Symbol('f_E', commutative=True))), Derivative(Function('A_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), sin(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(F_{N})} = \\sin{(F_{N})}, then obtain - \\frac{\\frac{\\delta^{2}{(F_{N})}}{\\sin{(F_{N})}} - 2 \\sin{(F_{N})}}{2 \\sin{(F_{N})}} = \\frac{1}{2}", "derivation": "\\delta{(F_{N})} = \\sin{(F_{N})} and \\delta{(F_{N})} - \\sin{(F_{N})} = 0 and \\frac{\\delta{(F_{N})}}{\\sin{(F_{N})}} = 1 and \\frac{\\delta^{2}{(F_{N})}}{\\sin{(F_{N})}} = \\delta{(F_{N})} and \\frac{\\delta^{2}{(F_{N})}}{\\sin{(F_{N})}} - \\sin{(F_{N})} = 0 and \\frac{\\delta^{2}{(F_{N})}}{\\sin{(F_{N})}} - 2 \\sin{(F_{N})} = - \\sin{(F_{N})} and - \\frac{\\frac{\\delta^{2}{(F_{N})}}{\\sin{(F_{N})}} - 2 \\sin{(F_{N})}}{2 \\sin{(F_{N})}} = \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["minus", 1, "sin(Symbol('F_N', commutative=True))"], "Equality(Add(Function('\\\\delta')(Symbol('F_N', commutative=True)), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(0))"], [["divide", 1, "sin(Symbol('F_N', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('F_N', commutative=True)), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Integer(1))"], [["times", 3, "Function('\\\\delta')(Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Function('\\\\delta')(Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('F_N', commutative=True)))), Integer(0))"], [["minus", 5, "sin(Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), sin(Symbol('F_N', commutative=True)))), Mul(Integer(-1), sin(Symbol('F_N', commutative=True))))"], [["divide", 6, "Mul(Integer(-1), Integer(2), sin(Symbol('F_N', commutative=True)))"], "Equality(Mul(Integer(-1), Rational(1, 2), Add(Mul(Pow(Function('\\\\delta')(Symbol('F_N', commutative=True)), Integer(2)), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), sin(Symbol('F_N', commutative=True)))), Pow(sin(Symbol('F_N', commutative=True)), Integer(-1))), Rational(1, 2))"]]}, {"prompt": "Given V{(k)} = \\cos{(k)}, then derive (\\frac{d^{2}}{d k^{2}} V{(k)})^{k} = (- \\cos{(k)})^{k}, then obtain (\\frac{d^{2}}{d k^{2}} \\cos{(k)})^{k} = (- \\cos{(k)})^{k}", "derivation": "V{(k)} = \\cos{(k)} and k + V{(k)} = k + \\cos{(k)} and \\frac{d}{d k} (k + V{(k)}) = \\frac{d}{d k} (k + \\cos{(k)}) and \\frac{d^{2}}{d k^{2}} (k + V{(k)}) = \\frac{d^{2}}{d k^{2}} (k + \\cos{(k)}) and (\\frac{d^{2}}{d k^{2}} (k + V{(k)}))^{k} = (\\frac{d^{2}}{d k^{2}} (k + \\cos{(k)}))^{k} and (\\frac{d^{2}}{d k^{2}} V{(k)})^{k} = (- \\cos{(k)})^{k} and (\\frac{d^{2}}{d k^{2}} \\cos{(k)})^{k} = (- \\cos{(k)})^{k}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('V')(Symbol('k', commutative=True))), Add(Symbol('k', commutative=True), cos(Symbol('k', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Symbol('k', commutative=True), Function('V')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Symbol('k', commutative=True), Function('V')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))), Derivative(Add(Symbol('k', commutative=True), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('k', commutative=True), Function('V')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))), Symbol('k', commutative=True)), Pow(Derivative(Add(Symbol('k', commutative=True), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))), Symbol('k', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Derivative(Function('V')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))), Symbol('k', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))), Symbol('k', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given z{(x,\\rho)} = - \\rho + x, then obtain \\int (- \\rho + x - z{(x,\\rho)})^{\\rho} dx = \\int 1 dx", "derivation": "z{(x,\\rho)} = - \\rho + x and 0 = - \\rho + x - z{(x,\\rho)} and 0^{\\rho} = (- \\rho + x - z{(x,\\rho)})^{\\rho} and \\int 0^{\\rho} dx = \\int (- \\rho + x - z{(x,\\rho)})^{\\rho} dx and \\int (- \\rho + x - z{(x,\\rho)})^{\\rho} dx = \\int 1 dx", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('x', commutative=True)))"], [["minus", 1, "Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('x', commutative=True), Mul(Integer(-1), Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('x', commutative=True), Mul(Integer(-1), Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('x', commutative=True), Mul(Integer(-1), Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('x', commutative=True), Mul(Integer(-1), Function('z')(Symbol('x', commutative=True), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Integer(1), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\psi{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\operatorname{E_{x}}{(\\varepsilon_0)} = - \\sin{(\\varepsilon_0)}, then derive \\frac{d}{d \\varepsilon_0} \\psi{(\\varepsilon_0)} = - \\sin{(\\varepsilon_0)}, then obtain \\frac{\\operatorname{E_{x}}{(\\varepsilon_0)}}{\\varepsilon_0} = - \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0}", "derivation": "\\psi{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\psi{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} \\cos{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\psi{(\\varepsilon_0)} = - \\sin{(\\varepsilon_0)} and \\frac{\\frac{d}{d \\varepsilon_0} \\psi{(\\varepsilon_0)}}{\\varepsilon_0} = - \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0} and \\operatorname{E_{x}}{(\\varepsilon_0)} = - \\sin{(\\varepsilon_0)} and \\frac{d}{d \\varepsilon_0} \\psi{(\\varepsilon_0)} = \\operatorname{E_{x}}{(\\varepsilon_0)} and \\frac{\\operatorname{E_{x}}{(\\varepsilon_0)}}{\\varepsilon_0} = - \\frac{\\sin{(\\varepsilon_0)}}{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Function('\\\\psi')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(i)} = e^{i}, then obtain m_{s} + \\frac{d}{d i} \\int \\cos{(\\operatorname{E_{n}}{(i)})} di = m_{s} + \\frac{d}{d i} \\int \\cos{(e^{i})} di", "derivation": "\\operatorname{E_{n}}{(i)} = e^{i} and \\cos{(\\operatorname{E_{n}}{(i)})} = \\cos{(e^{i})} and \\int \\cos{(\\operatorname{E_{n}}{(i)})} di = \\int \\cos{(e^{i})} di and \\frac{d}{d i} \\int \\cos{(\\operatorname{E_{n}}{(i)})} di = \\frac{d}{d i} \\int \\cos{(e^{i})} di and m_{s} + \\frac{d}{d i} \\int \\cos{(\\operatorname{E_{n}}{(i)})} di = m_{s} + \\frac{d}{d i} \\int \\cos{(e^{i})} di", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["cos", 1], "Equality(cos(Function('E_n')(Symbol('i', commutative=True))), cos(exp(Symbol('i', commutative=True))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(cos(Function('E_n')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(cos(exp(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(cos(Function('E_n')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(cos(exp(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["add", 4, "Symbol('m_s', commutative=True)"], "Equality(Add(Symbol('m_s', commutative=True), Derivative(Integral(cos(Function('E_n')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Symbol('m_s', commutative=True), Derivative(Integral(cos(exp(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(\\hat{x})} = \\sin{(\\sin{(\\hat{x})})}, then obtain \\iint \\chi^{\\hat{x}}{(\\hat{x})} d\\hat{x} d\\hat{x} = \\iint \\sin^{\\hat{x}}{(\\sin{(\\hat{x})})} d\\hat{x} d\\hat{x}", "derivation": "\\chi{(\\hat{x})} = \\sin{(\\sin{(\\hat{x})})} and \\chi^{\\hat{x}}{(\\hat{x})} = \\sin^{\\hat{x}}{(\\sin{(\\hat{x})})} and \\int \\chi^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\int \\sin^{\\hat{x}}{(\\sin{(\\hat{x})})} d\\hat{x} and \\iint \\chi^{\\hat{x}}{(\\hat{x})} d\\hat{x} d\\hat{x} = \\iint \\sin^{\\hat{x}}{(\\sin{(\\hat{x})})} d\\hat{x} d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True)), sin(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(sin(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(sin(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\chi')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(sin(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(t_{2})} = \\sin{(t_{2})}, then obtain \\frac{\\frac{d}{d t_{2}} (t_{2} + 2 \\operatorname{C_{d}}{(t_{2})})}{\\frac{d}{d t_{2}} (t_{2} + 2 \\sin{(t_{2})})} = 1", "derivation": "\\operatorname{C_{d}}{(t_{2})} = \\sin{(t_{2})} and t_{2} + \\operatorname{C_{d}}{(t_{2})} = t_{2} + \\sin{(t_{2})} and t_{2} + 2 \\operatorname{C_{d}}{(t_{2})} = t_{2} + \\operatorname{C_{d}}{(t_{2})} + \\sin{(t_{2})} and t_{2} + 2 \\operatorname{C_{d}}{(t_{2})} = t_{2} + 2 \\sin{(t_{2})} and \\frac{d}{d t_{2}} (t_{2} + 2 \\operatorname{C_{d}}{(t_{2})}) = \\frac{d}{d t_{2}} (t_{2} + 2 \\sin{(t_{2})}) and \\frac{\\frac{d}{d t_{2}} (t_{2} + 2 \\operatorname{C_{d}}{(t_{2})})}{\\frac{d}{d t_{2}} (t_{2} + 2 \\sin{(t_{2})})} = 1", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Function('C_d')(Symbol('t_2', commutative=True))), Add(Symbol('t_2', commutative=True), sin(Symbol('t_2', commutative=True))))"], [["add", 2, "Function('C_d')(Symbol('t_2', commutative=True))"], "Equality(Add(Symbol('t_2', commutative=True), Mul(Integer(2), Function('C_d')(Symbol('t_2', commutative=True)))), Add(Symbol('t_2', commutative=True), Function('C_d')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('t_2', commutative=True), Mul(Integer(2), Function('C_d')(Symbol('t_2', commutative=True)))), Add(Symbol('t_2', commutative=True), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Symbol('t_2', commutative=True), Mul(Integer(2), Function('C_d')(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('t_2', commutative=True), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Add(Symbol('t_2', commutative=True), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('t_2', commutative=True), Mul(Integer(2), Function('C_d')(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('t_2', commutative=True), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\rho_{f}{(v_{y},B)} = B^{v_{y}}, then obtain - v_{y} + \\frac{\\partial}{\\partial v_{y}} \\rho_{f}{(v_{y},B)} = B^{v_{y}} \\log{(B)} - v_{y}", "derivation": "\\rho_{f}{(v_{y},B)} = B^{v_{y}} and \\frac{\\partial}{\\partial v_{y}} \\rho_{f}{(v_{y},B)} = \\frac{\\partial}{\\partial v_{y}} B^{v_{y}} and - v_{y} + \\frac{\\partial}{\\partial v_{y}} \\rho_{f}{(v_{y},B)} = - v_{y} + \\frac{\\partial}{\\partial v_{y}} B^{v_{y}} and - v_{y} + \\frac{\\partial}{\\partial v_{y}} \\rho_{f}{(v_{y},B)} = B^{v_{y}} \\log{(B)} - v_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('v_y', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('v_y', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Pow(Symbol('B', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('v_y', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Derivative(Pow(Symbol('B', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('v_y', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('B', commutative=True), Symbol('v_y', commutative=True)), log(Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})} = \\sigma_p y^{\\prime} - f_{E}, then obtain - \\frac{\\frac{\\partial}{\\partial \\sigma_p} (- f_{E} + \\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})})}{f_{E}} = - \\frac{\\frac{\\partial}{\\partial \\sigma_p} (\\sigma_p y^{\\prime} - 2 f_{E})}{f_{E}}", "derivation": "\\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})} = \\sigma_p y^{\\prime} - f_{E} and - f_{E} + \\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})} = \\sigma_p y^{\\prime} - 2 f_{E} and \\frac{\\partial}{\\partial \\sigma_p} (- f_{E} + \\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})}) = \\frac{\\partial}{\\partial \\sigma_p} (\\sigma_p y^{\\prime} - 2 f_{E}) and - \\frac{\\frac{\\partial}{\\partial \\sigma_p} (- f_{E} + \\ddot{x}{(y^{\\prime},\\sigma_p,f_{E})})}{f_{E}} = - \\frac{\\frac{\\partial}{\\partial \\sigma_p} (\\sigma_p y^{\\prime} - 2 f_{E})}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True))), Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), Symbol('f_E', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('f_E', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\ddot{x}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f_E', commutative=True), Integer(-1)), Derivative(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{M},Q)} = \\int (Q + \\mathbf{M}) dQ, then derive \\sigma_{p}{(\\mathbf{M},Q)} = \\frac{Q^{2}}{2} + Q \\mathbf{M} + r, then derive \\sigma_{p}{(\\mathbf{M},Q)} = \\frac{Q^{2}}{2} + \\phi_2 + \\int \\mathbf{M} dQ, then derive \\frac{Q^{2}}{2} + Q \\mathbf{M} + z = \\frac{Q^{2}}{2} + Q \\mathbf{M} + r, then obtain \\frac{Q^{2}}{2} + Q \\mathbf{M} + z = \\frac{Q^{2}}{2} + \\phi_2 + \\int \\mathbf{M} dQ", "derivation": "\\sigma_{p}{(\\mathbf{M},Q)} = \\int (Q + \\mathbf{M}) dQ and \\sigma_{p}{(\\mathbf{M},Q)} = \\frac{Q^{2}}{2} + Q \\mathbf{M} + r and \\sigma_{p}{(\\mathbf{M},Q)} = \\int Q dQ + \\int \\mathbf{M} dQ and \\int (Q + \\mathbf{M}) dQ = \\frac{Q^{2}}{2} + Q \\mathbf{M} + r and \\sigma_{p}{(\\mathbf{M},Q)} = \\frac{Q^{2}}{2} + \\phi_2 + \\int \\mathbf{M} dQ and \\frac{Q^{2}}{2} + Q \\mathbf{M} + z = \\frac{Q^{2}}{2} + Q \\mathbf{M} + r and \\frac{Q^{2}}{2} + Q \\mathbf{M} + r = \\frac{Q^{2}}{2} + \\phi_2 + \\int \\mathbf{M} dQ and \\frac{Q^{2}}{2} + Q \\mathbf{M} + z = \\frac{Q^{2}}{2} + \\phi_2 + \\int \\mathbf{M} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('r', commutative=True)))"], [["expand", 1], "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('Q', commutative=True)), Add(Integral(Symbol('Q', commutative=True), Tuple(Symbol('Q', commutative=True))), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('r', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('\\\\phi_2', commutative=True), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('z', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('r', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('\\\\phi_2', commutative=True), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('z', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('\\\\phi_2', commutative=True), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\mu,\\mathbf{H})} = \\mathbf{H} + \\mu, then obtain \\frac{\\int 4 (\\mathbf{H} + \\mu)^{2} d\\mu}{4 \\dot{z}^{2}{(\\mu,\\mathbf{H})}} = \\frac{\\int 2 (\\mathbf{H} + \\mu) (2 \\mathbf{H} + 2 \\mu) d\\mu}{4 \\dot{z}^{2}{(\\mu,\\mathbf{H})}}", "derivation": "\\dot{z}{(\\mu,\\mathbf{H})} = \\mathbf{H} + \\mu and \\mathbf{H} + \\mu + \\dot{z}{(\\mu,\\mathbf{H})} = 2 \\mathbf{H} + 2 \\mu and 2 \\dot{z}{(\\mu,\\mathbf{H})} = 2 \\mathbf{H} + 2 \\mu and 2 \\dot{z}{(\\mu,\\mathbf{H})} = \\mathbf{H} + \\mu + \\dot{z}{(\\mu,\\mathbf{H})} and 4 \\dot{z}^{2}{(\\mu,\\mathbf{H})} = 2 (\\mathbf{H} + \\mu + \\dot{z}{(\\mu,\\mathbf{H})}) \\dot{z}{(\\mu,\\mathbf{H})} and 4 (\\mathbf{H} + \\mu)^{2} = 2 (\\mathbf{H} + \\mu) (2 \\mathbf{H} + 2 \\mu) and \\int 4 (\\mathbf{H} + \\mu)^{2} d\\mu = \\int 2 (\\mathbf{H} + \\mu) (2 \\mathbf{H} + 2 \\mu) d\\mu and \\frac{\\int 4 (\\mathbf{H} + \\mu)^{2} d\\mu}{4 \\dot{z}^{2}{(\\mu,\\mathbf{H})}} = \\frac{\\int 2 (\\mathbf{H} + \\mu) (2 \\mathbf{H} + 2 \\mu) d\\mu}{4 \\dot{z}^{2}{(\\mu,\\mathbf{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 4, "Mul(Integer(2), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Integer(2), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Integer(2), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Integer(2), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["divide", 7, "Mul(Integer(4), Pow(Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 4), Pow(Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-2)), Integral(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Rational(1, 4), Pow(Function('\\\\dot{z}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-2)), Integral(Mul(Integer(2), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(S)} = \\cos{(S)}, then derive 1 + \\frac{\\frac{d}{d S} \\theta_{1}{(S)}}{S} - \\frac{\\theta_{1}{(S)}}{S^{2}} = 1 - \\frac{\\sin{(S)}}{S} - \\frac{\\cos{(S)}}{S^{2}}, then obtain (1 + \\frac{\\frac{d}{d S} \\theta_{1}{(S)}}{S} - \\frac{\\theta_{1}{(S)}}{S^{2}}) \\theta_{1}^{S}{(S)} = (1 - \\frac{\\sin{(S)}}{S} - \\frac{\\cos{(S)}}{S^{2}}) \\theta_{1}^{S}{(S)}", "derivation": "\\theta_{1}{(S)} = \\cos{(S)} and \\frac{\\theta_{1}{(S)}}{S} = \\frac{\\cos{(S)}}{S} and S + \\frac{\\theta_{1}{(S)}}{S} = S + \\frac{\\cos{(S)}}{S} and \\frac{d}{d S} (S + \\frac{\\theta_{1}{(S)}}{S}) = \\frac{d}{d S} (S + \\frac{\\cos{(S)}}{S}) and 1 + \\frac{\\frac{d}{d S} \\theta_{1}{(S)}}{S} - \\frac{\\theta_{1}{(S)}}{S^{2}} = 1 - \\frac{\\sin{(S)}}{S} - \\frac{\\cos{(S)}}{S^{2}} and (1 + \\frac{\\frac{d}{d S} \\theta_{1}{(S)}}{S} - \\frac{\\theta_{1}{(S)}}{S^{2}}) \\theta_{1}^{S}{(S)} = (1 - \\frac{\\sin{(S)}}{S} - \\frac{\\cos{(S)}}{S^{2}}) \\theta_{1}^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True))))"], [["add", 2, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('S', commutative=True)))), Add(Symbol('S', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True)))))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(1), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), Function('\\\\theta_1')(Symbol('S', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), cos(Symbol('S', commutative=True)))))"], [["times", 5, "Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))"], "Equality(Mul(Add(Integer(1), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), Function('\\\\theta_1')(Symbol('S', commutative=True)))), Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Mul(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)), cos(Symbol('S', commutative=True)))), Pow(Function('\\\\theta_1')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given S{(\\phi_2,A_{z})} = A_{z} \\phi_2, then derive (C_{2} + \\log{(A_{z})})^{\\phi_2} = (\\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z})^{\\phi_2}, then obtain (C_{2} + \\log{(A_{z})})^{\\phi_2} - S{(\\phi_2,A_{z})} = - S{(\\phi_2,A_{z})} + (\\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z})^{\\phi_2}", "derivation": "S{(\\phi_2,A_{z})} = A_{z} \\phi_2 and \\frac{S{(\\phi_2,A_{z})}}{A_{z}} = \\phi_2 and \\frac{1}{A_{z}} = \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} and \\int \\frac{1}{A_{z}} dA_{z} = \\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z} and (\\int \\frac{1}{A_{z}} dA_{z})^{\\phi_2} = (\\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z})^{\\phi_2} and (C_{2} + \\log{(A_{z})})^{\\phi_2} = (\\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z})^{\\phi_2} and (C_{2} + \\log{(A_{z})})^{\\phi_2} - S{(\\phi_2,A_{z})} = - S{(\\phi_2,A_{z})} + (\\int \\frac{\\phi_2}{S{(\\phi_2,A_{z})}} dA_{z})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True))"], [["divide", 2, "Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Pow(Symbol('A_z', commutative=True), Integer(-1)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Symbol('A_z', commutative=True), Integer(-1)), Tuple(Symbol('A_z', commutative=True))), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))), Tuple(Symbol('A_z', commutative=True))))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Integral(Pow(Symbol('A_z', commutative=True), Integer(-1)), Tuple(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))), Tuple(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('C_2', commutative=True), log(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))), Tuple(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 6, "Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Add(Pow(Add(Symbol('C_2', commutative=True), log(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True))), Pow(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('S')(Symbol('\\\\phi_2', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))), Tuple(Symbol('A_z', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given a{(r_{0},\\pi,L)} = - L - \\pi + r_{0}, then derive \\frac{\\partial}{\\partial L} a{(r_{0},\\pi,L)} = -1, then obtain \\frac{\\frac{\\partial}{\\partial L} (- L - \\pi + r_{0})}{- L - \\pi + r_{0}} = - \\frac{1}{- L - \\pi + r_{0}}", "derivation": "a{(r_{0},\\pi,L)} = - L - \\pi + r_{0} and \\frac{\\partial}{\\partial L} a{(r_{0},\\pi,L)} = \\frac{\\partial}{\\partial L} (- L - \\pi + r_{0}) and \\frac{\\partial}{\\partial L} a{(r_{0},\\pi,L)} = -1 and \\frac{\\frac{\\partial}{\\partial L} a{(r_{0},\\pi,L)}}{- L - \\pi + r_{0}} = - \\frac{1}{- L - \\pi + r_{0}} and \\frac{\\frac{\\partial}{\\partial L} (- L - \\pi + r_{0})}{- L - \\pi + r_{0}} = - \\frac{1}{- L - \\pi + r_{0}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Integer(-1)), Derivative(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('r_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then derive \\int \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} d\\mathbf{J}_P = \\Omega + \\operatorname{Si}{(2 e^{\\mathbf{J}_P})}, then obtain \\int \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} d\\mathbf{J}_P = \\Omega + \\operatorname{Si}{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})}", "derivation": "\\operatorname{f_{E}}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} = 2 e^{\\mathbf{J}_P} and \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} = \\sin{(2 e^{\\mathbf{J}_P})} and \\int \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} d\\mathbf{J}_P = \\int \\sin{(2 e^{\\mathbf{J}_P})} d\\mathbf{J}_P and \\int \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} d\\mathbf{J}_P = \\Omega + \\operatorname{Si}{(2 e^{\\mathbf{J}_P})} and \\int \\sin{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})} d\\mathbf{J}_P = \\Omega + \\operatorname{Si}{(\\operatorname{f_{E}}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["sin", 2], "Equality(sin(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), sin(Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(sin(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(sin(Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(sin(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Si(Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(sin(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Si(Add(Function('f_E')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(J,i)} = J + i, then obtain -1 + \\sin{(1)} = \\sin{((\\frac{- J - i + \\operatorname{v_{2}}{(J,i)}}{\\operatorname{v_{2}}{(J,i)}})^{J})} - 1", "derivation": "\\operatorname{v_{2}}{(J,i)} = J + i and - J - i + \\operatorname{v_{2}}{(J,i)} = 0 and \\frac{- J - i + \\operatorname{v_{2}}{(J,i)}}{\\operatorname{v_{2}}{(J,i)}} = 0 and (\\frac{- J - i + \\operatorname{v_{2}}{(J,i)}}{\\operatorname{v_{2}}{(J,i)}})^{J} = 0^{J} and (- \\frac{J}{\\operatorname{v_{2}}{(J,i)}} - \\frac{i}{\\operatorname{v_{2}}{(J,i)}} + 1)^{J} = 0^{J} and \\sin{((- \\frac{J}{\\operatorname{v_{2}}{(J,i)}} - \\frac{i}{\\operatorname{v_{2}}{(J,i)}} + 1)^{J})} = \\sin{(0^{J})} and \\sin{(1)} = \\sin{((\\frac{- J - i + \\operatorname{v_{2}}{(J,i)}}{\\operatorname{v_{2}}{(J,i)}})^{J})} and -1 + \\sin{(1)} = \\sin{((\\frac{- J - i + \\operatorname{v_{2}}{(J,i)}}{\\operatorname{v_{2}}{(J,i)}})^{J})} - 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Add(Symbol('J', commutative=True), Symbol('i', commutative=True)))"], [["minus", 1, "Add(Symbol('J', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))), Integer(0))"], [["divide", 2, "Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Integer(0))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Symbol('J', commutative=True)), Pow(Integer(0), Symbol('J', commutative=True)))"], [["expand", 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('i', commutative=True), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Integer(1)), Symbol('J', commutative=True)), Pow(Integer(0), Symbol('J', commutative=True)))"], [["sin", 5], "Equality(sin(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('i', commutative=True), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Integer(1)), Symbol('J', commutative=True))), sin(Pow(Integer(0), Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(sin(Integer(1)), sin(Pow(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Symbol('J', commutative=True))))"], [["minus", 7, 1], "Equality(Add(Integer(-1), sin(Integer(1))), Add(sin(Pow(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True))), Pow(Function('v_2')(Symbol('J', commutative=True), Symbol('i', commutative=True)), Integer(-1))), Symbol('J', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given S{(\\Omega)} = e^{\\Omega}, then obtain \\frac{Z S{(\\Omega)} e^{- \\Omega} \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} + (e^{\\Omega} \\int S{(\\Omega)} d\\Omega)^{\\Omega} = \\frac{Z \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} + (e^{\\Omega} \\int S{(\\Omega)} d\\Omega)^{\\Omega}", "derivation": "S{(\\Omega)} = e^{\\Omega} and \\int S{(\\Omega)} d\\Omega = \\int e^{\\Omega} d\\Omega and S{(\\Omega)} \\int e^{\\Omega} d\\Omega = e^{\\Omega} \\int e^{\\Omega} d\\Omega and S{(\\Omega)} \\int S{(\\Omega)} d\\Omega = e^{\\Omega} \\int S{(\\Omega)} d\\Omega and Z S{(\\Omega)} \\int S{(\\Omega)} d\\Omega = Z e^{\\Omega} \\int S{(\\Omega)} d\\Omega and Z S{(\\Omega)} \\int e^{\\Omega} d\\Omega = Z e^{\\Omega} \\int e^{\\Omega} d\\Omega and \\frac{Z S{(\\Omega)} e^{- \\Omega} \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} = \\frac{Z \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} and \\frac{Z S{(\\Omega)} e^{- \\Omega} \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} + (e^{\\Omega} \\int S{(\\Omega)} d\\Omega)^{\\Omega} = \\frac{Z \\int e^{\\Omega} d\\Omega}{\\int S{(\\Omega)} d\\Omega} + (e^{\\Omega} \\int S{(\\Omega)} d\\Omega)^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Function('S')(Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('S')(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["times", 4, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Function('S')(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('Z', commutative=True), exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('Z', commutative=True), Function('S')(Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('Z', commutative=True), exp(Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["divide", 6, "Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(Symbol('Z', commutative=True), Function('S')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Pow(Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('Z', commutative=True), Pow(Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["add", 7, "Pow(Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Symbol('Z', commutative=True), Function('S')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Pow(Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Pow(Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Pow(Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Pow(Mul(exp(Symbol('\\\\Omega', commutative=True)), Integral(Function('S')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})}, then obtain \\Psi^{\\dagger} \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} + 1) = \\Psi^{\\dagger} \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})} + 1)", "derivation": "\\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})} and \\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} + 1 = \\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})} + 1 and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} + 1) = \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})} + 1) and \\Psi^{\\dagger} \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\dot{z}{(\\hat{\\mathbf{x}},\\Psi^{\\dagger})} + 1) = \\Psi^{\\dagger} \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\cos{(\\Psi^{\\dagger} - \\hat{\\mathbf{x}})} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(1)), Add(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Add(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Derivative(Add(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Derivative(Add(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(F_{g})} = F_{g} and \\omega{(\\sigma_x)} = \\sigma_x, then derive - \\psi^* - \\omega{(\\sigma_x)} + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\omega{(\\sigma_x)} + 1, then obtain - \\psi^* - \\sigma_x + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\sigma_x + 1", "derivation": "\\delta{(F_{g})} = F_{g} and \\omega{(\\sigma_x)} = \\sigma_x and \\frac{d}{d F_{g}} \\delta{(F_{g})} = \\frac{d}{d F_{g}} F_{g} and - \\psi^* - \\sigma_x + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\sigma_x + \\frac{d}{d F_{g}} F_{g} and - \\psi^* - \\omega{(\\sigma_x)} + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\omega{(\\sigma_x)} + \\frac{d}{d F_{g}} F_{g} and - \\psi^* - \\omega{(\\sigma_x)} + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\omega{(\\sigma_x)} + 1 and - \\psi^* - \\sigma_x + \\frac{d}{d F_{g}} \\delta{(F_{g})} = - \\psi^* - \\sigma_x + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\delta')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Integer(1)))"]]}, {"prompt": "Given u{(\\psi,A_{1})} = A_{1} \\log{(\\psi)} and \\tilde{g}{(\\mathbf{B},W,F_{H})} = \\mathbf{B} (F_{H} + W), then obtain \\tilde{g}{(\\mathbf{B},W,F_{H})} \\int (u{(\\psi,A_{1})} - \\log{(\\psi)}) d\\psi = \\tilde{g}{(\\mathbf{B},W,F_{H})} \\int (A_{1} \\log{(\\psi)} - \\log{(\\psi)}) d\\psi", "derivation": "u{(\\psi,A_{1})} = A_{1} \\log{(\\psi)} and u{(\\psi,A_{1})} - \\log{(\\psi)} = A_{1} \\log{(\\psi)} - \\log{(\\psi)} and \\tilde{g}{(\\mathbf{B},W,F_{H})} = \\mathbf{B} (F_{H} + W) and \\int (u{(\\psi,A_{1})} - \\log{(\\psi)}) d\\psi = \\int (A_{1} \\log{(\\psi)} - \\log{(\\psi)}) d\\psi and \\mathbf{B} (F_{H} + W) \\int (u{(\\psi,A_{1})} - \\log{(\\psi)}) d\\psi = \\mathbf{B} (F_{H} + W) \\int (A_{1} \\log{(\\psi)} - \\log{(\\psi)}) d\\psi and \\tilde{g}{(\\mathbf{B},W,F_{H})} \\int (u{(\\psi,A_{1})} - \\log{(\\psi)}) d\\psi = \\tilde{g}{(\\mathbf{B},W,F_{H})} \\int (A_{1} \\log{(\\psi)} - \\log{(\\psi)}) d\\psi", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('W', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["times", 4, "Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('W', commutative=True)), Integral(Add(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Add(Symbol('F_H', commutative=True), Symbol('W', commutative=True)), Integral(Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('F_H', commutative=True)), Integral(Add(Function('u')(Symbol('\\\\psi', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('F_H', commutative=True)), Integral(Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given J{(y^{\\prime},\\mathbf{D})} = \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{D} + y^{\\prime}), then derive J{(y^{\\prime},\\mathbf{D})} = 1, then derive J^{2}{(y^{\\prime},\\mathbf{D})} = J{(y^{\\prime},\\mathbf{D})}, then obtain J^{2}{(y^{\\prime},\\mathbf{D})} = 1", "derivation": "J{(y^{\\prime},\\mathbf{D})} = \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{D} + y^{\\prime}) and J{(y^{\\prime},\\mathbf{D})} = 1 and J^{2}{(y^{\\prime},\\mathbf{D})} = J{(y^{\\prime},\\mathbf{D})} \\frac{\\partial}{\\partial y^{\\prime}} (\\mathbf{D} + y^{\\prime}) and J^{2}{(y^{\\prime},\\mathbf{D})} = J{(y^{\\prime},\\mathbf{D})} and J^{2}{(y^{\\prime},\\mathbf{D})} = 1", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1))"], [["times", 1, "Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Pow(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2)), Mul(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2)), Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('J')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2)), Integer(1))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\operatorname{y^{\\prime}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain - \\operatorname{y^{\\prime}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and - \\operatorname{P_{g}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} and \\operatorname{y^{\\prime}}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and - \\operatorname{P_{g}}{(\\mathbf{A})} = - \\operatorname{y^{\\prime}}{(\\mathbf{A})} and - \\operatorname{y^{\\prime}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given I{(\\mathbf{g})} = \\cos{(\\log{(\\mathbf{g})})}, then obtain \\frac{d}{d \\mathbf{g}} 1 = \\frac{d}{d \\mathbf{g}} (\\frac{\\cos{(\\log{(\\mathbf{g})})}}{I{(\\mathbf{g})}})^{\\mathbf{g}}", "derivation": "I{(\\mathbf{g})} = \\cos{(\\log{(\\mathbf{g})})} and 1 = \\frac{\\cos{(\\log{(\\mathbf{g})})}}{I{(\\mathbf{g})}} and 1 = (\\frac{\\cos{(\\log{(\\mathbf{g})})}}{I{(\\mathbf{g})}})^{\\mathbf{g}} and \\frac{d}{d \\mathbf{g}} 1 = \\frac{d}{d \\mathbf{g}} (\\frac{\\cos{(\\log{(\\mathbf{g})})}}{I{(\\mathbf{g})}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{g}', commutative=True)), cos(log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["divide", 1, "Function('I')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), cos(log(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('I')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), cos(log(Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('I')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), cos(log(Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(f^{*},x^\\prime)} = f^{*} x^\\prime, then obtain 2 \\frac{\\partial}{\\partial x^\\prime} M{(f^{*},x^\\prime)} = f^{*} + \\frac{\\partial}{\\partial x^\\prime} M{(f^{*},x^\\prime)}", "derivation": "M{(f^{*},x^\\prime)} = f^{*} x^\\prime and 2 M{(f^{*},x^\\prime)} = f^{*} x^\\prime + M{(f^{*},x^\\prime)} and \\frac{\\partial}{\\partial x^\\prime} 2 M{(f^{*},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} (f^{*} x^\\prime + M{(f^{*},x^\\prime)}) and 2 \\frac{\\partial}{\\partial x^\\prime} M{(f^{*},x^\\prime)} = f^{*} + \\frac{\\partial}{\\partial x^\\prime} M{(f^{*},x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Integer(2), Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Symbol('f^*', commutative=True), Derivative(Function('M')(Symbol('f^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{H}{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})} and \\hat{x}{(V_{\\mathbf{E}})} = \\mathbf{H}{(V_{\\mathbf{E}})} + \\log{(V_{\\mathbf{E}})}, then obtain \\frac{\\hat{x}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}} = 2", "derivation": "\\mathbf{H}{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})} and \\hat{x}{(V_{\\mathbf{E}})} = \\mathbf{H}{(V_{\\mathbf{E}})} + \\log{(V_{\\mathbf{E}})} and \\frac{\\hat{x}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}} = \\frac{\\mathbf{H}{(V_{\\mathbf{E}})} + \\log{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}} and \\frac{\\hat{x}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}} = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Function('\\\\mathbf{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["divide", 2, "log(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\mathbf{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Integer(2))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(q,B)} = B - q, then derive (\\int \\operatorname{v_{y}}{(q,B)} dq)^{B} = (B q - \\frac{q^{2}}{2} + s)^{B}, then obtain \\frac{\\partial}{\\partial B} (\\int \\operatorname{v_{y}}{(q,B)} dq)^{2 B} = \\frac{\\partial}{\\partial B} (B q - \\frac{q^{2}}{2} + s)^{2 B}", "derivation": "\\operatorname{v_{y}}{(q,B)} = B - q and \\int \\operatorname{v_{y}}{(q,B)} dq = \\int (B - q) dq and (\\int \\operatorname{v_{y}}{(q,B)} dq)^{B} = (\\int (B - q) dq)^{B} and (\\int \\operatorname{v_{y}}{(q,B)} dq)^{B} = (B q - \\frac{q^{2}}{2} + s)^{B} and (\\int \\operatorname{v_{y}}{(q,B)} dq)^{2 B} = (B q - \\frac{q^{2}}{2} + s)^{2 B} and \\frac{\\partial}{\\partial B} (\\int \\operatorname{v_{y}}{(q,B)} dq)^{2 B} = \\frac{\\partial}{\\partial B} (B q - \\frac{q^{2}}{2} + s)^{2 B}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('B', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Mul(Symbol('B', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('s', commutative=True)), Symbol('B', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Integral(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('q', commutative=True))), Mul(Integer(2), Symbol('B', commutative=True))), Pow(Add(Mul(Symbol('B', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('s', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))))"], [["differentiate", 5, "Symbol('B', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('v_y')(Symbol('q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('q', commutative=True))), Mul(Integer(2), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('s', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(c,\\hat{H}_l)} = \\frac{\\hat{H}_l}{c}, then derive \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)} = - \\frac{\\hat{H}_l}{c^{2}}, then obtain - \\frac{\\hat{H}_l}{c^{2}} - c = - c + \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)}", "derivation": "\\mathbf{M}{(c,\\hat{H}_l)} = \\frac{\\hat{H}_l}{c} and \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)} = \\frac{\\partial}{\\partial c} \\frac{\\hat{H}_l}{c} and - c + \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)} = - c + \\frac{\\partial}{\\partial c} \\frac{\\hat{H}_l}{c} and \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)} = - \\frac{\\hat{H}_l}{c^{2}} and - \\frac{\\hat{H}_l}{c^{2}} - c = - c + \\frac{\\partial}{\\partial c} \\frac{\\hat{H}_l}{c} and - \\frac{\\hat{H}_l}{c^{2}} - c = - c + \\frac{\\partial}{\\partial c} \\mathbf{M}{(c,\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{v},F_{c},\\sigma_x)} = F_{c} \\mathbf{v} + \\sigma_x, then derive \\frac{\\partial}{\\partial F_{c}} \\varphi{(\\mathbf{v},F_{c},\\sigma_x)} = \\mathbf{v}, then obtain \\varphi{(\\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x),F_{c},\\sigma_x)} = F_{c} \\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x) + \\sigma_x", "derivation": "\\varphi{(\\mathbf{v},F_{c},\\sigma_x)} = F_{c} \\mathbf{v} + \\sigma_x and \\frac{\\partial}{\\partial F_{c}} \\varphi{(\\mathbf{v},F_{c},\\sigma_x)} = \\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x) and \\frac{\\partial}{\\partial F_{c}} \\varphi{(\\mathbf{v},F_{c},\\sigma_x)} = \\mathbf{v} and \\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x) = \\mathbf{v} and \\varphi{(\\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x),F_{c},\\sigma_x)} = F_{c} \\frac{\\partial}{\\partial F_{c}} (F_{c} \\mathbf{v} + \\sigma_x) + \\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\varphi')(Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Symbol('F_c', commutative=True), Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given s{(C,\\mathbf{M})} = \\mathbf{M}^{C} and \\Psi_{nl}{(C)} = - C, then obtain \\mathbf{M}^{- C} \\Psi_{nl}{(C)} s{(C,\\mathbf{M})} = - C", "derivation": "s{(C,\\mathbf{M})} = \\mathbf{M}^{C} and \\mathbf{M}^{- C} s{(C,\\mathbf{M})} = 1 and \\Psi_{nl}{(C)} = - C and \\mathbf{M}^{- C} \\Psi_{nl}{(C)} s{(C,\\mathbf{M})} = \\Psi_{nl}{(C)} and \\mathbf{M}^{- C} \\Psi_{nl}{(C)} s{(C,\\mathbf{M})} = - C", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('C', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('C', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('C', commutative=True))), Function('s')(Symbol('C', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["times", 2, "Function('\\\\Psi_{nl}')(Symbol('C', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('C', commutative=True))), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True)), Function('s')(Symbol('C', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('C', commutative=True))), Function('\\\\Psi_{nl}')(Symbol('C', commutative=True)), Function('s')(Symbol('C', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(z^{*},f,t)} = t (f - z^{*}) and z{(f,z^{*})} = f - z^{*}, then obtain - t (f - z^{*}) + \\cos{(\\operatorname{f_{E}}{(z^{*},f,t)})} = - t (f - z^{*}) + \\cos{(t z{(f,z^{*})})}", "derivation": "\\operatorname{f_{E}}{(z^{*},f,t)} = t (f - z^{*}) and z{(f,z^{*})} = f - z^{*} and \\operatorname{f_{E}}{(z^{*},f,t)} = t z{(f,z^{*})} and t z{(f,z^{*})} = t (f - z^{*}) and \\cos{(t z{(f,z^{*})})} = \\cos{(t (f - z^{*}))} and \\cos{(\\operatorname{f_{E}}{(z^{*},f,t)})} = \\cos{(t (f - z^{*}))} and - t (f - z^{*}) + \\cos{(\\operatorname{f_{E}}{(z^{*},f,t)})} = - t (f - z^{*}) + \\cos{(t (f - z^{*}))} and - t (f - z^{*}) + \\cos{(\\operatorname{f_{E}}{(z^{*},f,t)})} = - t (f - z^{*}) + \\cos{(t z{(f,z^{*})})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('z^*', commutative=True), Symbol('f', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('f', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_E')(Symbol('z^*', commutative=True), Symbol('f', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), Function('z')(Symbol('f', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Mul(Symbol('t', commutative=True), Function('z')(Symbol('f', commutative=True), Symbol('z^*', commutative=True))), Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["cos", 4], "Equality(cos(Mul(Symbol('t', commutative=True), Function('z')(Symbol('f', commutative=True), Symbol('z^*', commutative=True)))), cos(Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(cos(Function('f_E')(Symbol('z^*', commutative=True), Symbol('f', commutative=True), Symbol('t', commutative=True))), cos(Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["minus", 6, "Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), cos(Function('f_E')(Symbol('z^*', commutative=True), Symbol('f', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), cos(Mul(Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), cos(Function('f_E')(Symbol('z^*', commutative=True), Symbol('f', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('t', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), cos(Mul(Symbol('t', commutative=True), Function('z')(Symbol('f', commutative=True), Symbol('z^*', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(t)} = \\cos{(t)} and A{(t)} = - \\hat{H}_{\\lambda}{(t)} + \\cos{(t)} + 1, then obtain A{(t)} - 1 + \\frac{\\frac{d}{d t} (1 - A{(t)})}{\\frac{d}{d t} 0} = A{(t)}", "derivation": "\\hat{H}_{\\lambda}{(t)} = \\cos{(t)} and 0 = - \\hat{H}_{\\lambda}{(t)} + \\cos{(t)} and \\hat{H}_{\\lambda}{(t)} - \\cos{(t)} = 0 and \\frac{d}{d t} (\\hat{H}_{\\lambda}{(t)} - \\cos{(t)}) = \\frac{d}{d t} 0 and \\frac{\\frac{d}{d t} (\\hat{H}_{\\lambda}{(t)} - \\cos{(t)})}{\\frac{d}{d t} 0} = 1 and - \\hat{H}_{\\lambda}{(t)} + \\cos{(t)} + \\frac{\\frac{d}{d t} (\\hat{H}_{\\lambda}{(t)} - \\cos{(t)})}{\\frac{d}{d t} 0} = - \\hat{H}_{\\lambda}{(t)} + \\cos{(t)} + 1 and A{(t)} = - \\hat{H}_{\\lambda}{(t)} + \\cos{(t)} + 1 and A{(t)} - 1 + \\frac{\\frac{d}{d t} (1 - A{(t)})}{\\frac{d}{d t} 0} = A{(t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["minus", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(1))"], [["add", 5, "Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True)), Mul(Pow(Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('A')(Symbol('t', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('t', commutative=True))), cos(Symbol('t', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Function('A')(Symbol('t', commutative=True)), Integer(-1), Mul(Pow(Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Integer(1), Mul(Integer(-1), Function('A')(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))), Function('A')(Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\phi_1,C_{d})} = C_{d} \\phi_1, then derive \\int \\frac{\\operatorname{A_{z}}{(\\phi_1,C_{d})}}{\\phi_1} dC_{d} = \\frac{C_{d}^{2}}{2} + x, then obtain (\\int C_{d} dC_{d})^{2} = (\\frac{C_{d}^{2}}{2} + x)^{2}", "derivation": "\\operatorname{A_{z}}{(\\phi_1,C_{d})} = C_{d} \\phi_1 and \\frac{\\operatorname{A_{z}}{(\\phi_1,C_{d})}}{\\phi_1} = C_{d} and \\int \\frac{\\operatorname{A_{z}}{(\\phi_1,C_{d})}}{\\phi_1} dC_{d} = \\int C_{d} dC_{d} and \\int \\frac{\\operatorname{A_{z}}{(\\phi_1,C_{d})}}{\\phi_1} dC_{d} = \\frac{C_{d}^{2}}{2} + x and (\\int \\frac{\\operatorname{A_{z}}{(\\phi_1,C_{d})}}{\\phi_1} dC_{d})^{2} = (\\frac{C_{d}^{2}}{2} + x)^{2} and (\\int C_{d} dC_{d})^{2} = (\\frac{C_{d}^{2}}{2} + x)^{2}", "srepr_derivation": [["get_premise", "Equality(Function('A_z')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('x', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('A_z')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integer(2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('x', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))), Integer(2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('x', commutative=True)), Integer(2)))"]]}, {"prompt": "Given x{(\\hat{p})} = \\cos{(\\hat{p})} and \\Omega{(\\delta)} = - \\delta, then obtain \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + x^{\\hat{p}}{(\\hat{p})} - 1) - 1 = \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + \\cos^{\\hat{p}}{(\\hat{p})} - 1) - 1", "derivation": "x{(\\hat{p})} = \\cos{(\\hat{p})} and x^{\\hat{p}}{(\\hat{p})} = \\cos^{\\hat{p}}{(\\hat{p})} and - \\delta + x^{\\hat{p}}{(\\hat{p})} = - \\delta + \\cos^{\\hat{p}}{(\\hat{p})} and - \\delta + x^{\\hat{p}}{(\\hat{p})} - 1 = - \\delta + \\cos^{\\hat{p}}{(\\hat{p})} - 1 and \\Omega{(\\delta)} = - \\delta and \\Omega{(\\delta)} + x^{\\hat{p}}{(\\hat{p})} - 1 = \\Omega{(\\delta)} + \\cos^{\\hat{p}}{(\\hat{p})} - 1 and \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + x^{\\hat{p}}{(\\hat{p})} - 1) = \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + \\cos^{\\hat{p}}{(\\hat{p})} - 1) and \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + x^{\\hat{p}}{(\\hat{p})} - 1) - 1 = \\frac{\\partial}{\\partial \\delta} (\\Omega{(\\delta)} + \\cos^{\\hat{p}}{(\\hat{p})} - 1) - 1", "srepr_derivation": [["get_premise", "Equality(Function('x')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)))"], [["differentiate", 6, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["add", 7, "Integer(-1)"], "Equality(Add(Derivative(Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(Function('x')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\varphi^{*}{(h,\\delta,E_{x})} = E_{x} - \\delta + h, then derive \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x} = - E_{x} + \\hbar, then derive f = \\hbar, then obtain f = E_{x} + \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x}", "derivation": "\\varphi^{*}{(h,\\delta,E_{x})} = E_{x} - \\delta + h and \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} = \\frac{\\partial}{\\partial \\delta} (E_{x} - \\delta + h) and \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x} = \\int \\frac{\\partial}{\\partial \\delta} (E_{x} - \\delta + h) dE_{x} and \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x} = - E_{x} + \\hbar and E_{x} + \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x} = \\hbar and E_{x} + \\int \\frac{\\partial}{\\partial \\delta} (E_{x} - \\delta + h) dE_{x} = \\hbar and f = \\hbar and f = E_{x} + \\int \\frac{\\partial}{\\partial \\delta} \\varphi^{*}{(h,\\delta,E_{x})} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('E_x', commutative=True))"], "Equality(Add(Symbol('E_x', commutative=True), Integral(Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True)))), Symbol('\\\\hbar', commutative=True))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('E_x', commutative=True), Integral(Derivative(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True)))), Symbol('\\\\hbar', commutative=True))"], [["evaluate_integrals", 6], "Equality(Symbol('f', commutative=True), Symbol('\\\\hbar', commutative=True))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Symbol('f', commutative=True), Add(Symbol('E_x', commutative=True), Integral(Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(F_{N},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\frac{F_{N}}{F_{H}}, then derive \\tilde{g}^*{(F_{N},F_{H})} = - \\frac{F_{N}}{F_{H}^{2}}, then obtain F_{H} + \\frac{\\partial}{\\partial F_{N}} \\tilde{g}^*{(F_{N},F_{H})} = F_{H} + \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{F_{H}^{2}}", "derivation": "\\tilde{g}^*{(F_{N},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\frac{F_{N}}{F_{H}} and \\tilde{g}^*{(F_{N},F_{H})} = - \\frac{F_{N}}{F_{H}^{2}} and \\frac{\\partial}{\\partial F_{N}} \\tilde{g}^*{(F_{N},F_{H})} = \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{F_{H}^{2}} and F_{H} + \\frac{\\partial}{\\partial F_{N}} \\tilde{g}^*{(F_{N},F_{H})} = F_{H} + \\frac{\\partial}{\\partial F_{N}} - \\frac{F_{N}}{F_{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True), Symbol('F_H', commutative=True)), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True), Symbol('F_H', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-2)), Symbol('F_N', commutative=True)))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-2)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 3, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Derivative(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Symbol('F_H', commutative=True), Derivative(Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-2)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(m,\\Psi_{nl})} = \\Psi_{nl} m and \\operatorname{v_{x}}{(m,\\Psi_{nl})} = \\Psi_{nl} m, then obtain \\frac{\\partial}{\\partial \\Psi_{nl}} (- m + \\operatorname{v_{x}}{(m,\\Psi_{nl})}) = \\frac{\\partial}{\\partial \\Psi_{nl}} (- m + \\varphi{(m,\\Psi_{nl})})", "derivation": "\\varphi{(m,\\Psi_{nl})} = \\Psi_{nl} m and \\operatorname{v_{x}}{(m,\\Psi_{nl})} = \\Psi_{nl} m and - m + \\operatorname{v_{x}}{(m,\\Psi_{nl})} = \\Psi_{nl} m - m and - m + \\operatorname{v_{x}}{(m,\\Psi_{nl})} = - m + \\varphi{(m,\\Psi_{nl})} and \\Psi_{nl} m - m = - m + \\varphi{(m,\\Psi_{nl})} and \\frac{\\partial}{\\partial \\Psi_{nl}} (- m + \\operatorname{v_{x}}{(m,\\Psi_{nl})}) = \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi_{nl} m - m) and \\frac{\\partial}{\\partial \\Psi_{nl}} (- m + \\operatorname{v_{x}}{(m,\\Psi_{nl})}) = \\frac{\\partial}{\\partial \\Psi_{nl}} (- m + \\varphi{(m,\\Psi_{nl})})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)))"], [["minus", 2, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_x')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_x')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_x')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('v_x')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('m', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(\\lambda,\\varepsilon)} = \\frac{\\lambda}{\\varepsilon}, then obtain 0 = \\frac{\\lambda}{\\varepsilon^{2}} + \\frac{\\partial}{\\partial \\varepsilon} q{(\\lambda,\\varepsilon)}", "derivation": "q{(\\lambda,\\varepsilon)} = \\frac{\\lambda}{\\varepsilon} and 0 = \\frac{\\lambda}{\\varepsilon} - q{(\\lambda,\\varepsilon)} and 0 = - \\frac{\\lambda}{\\varepsilon} + q{(\\lambda,\\varepsilon)} and \\frac{d}{d \\varepsilon} 0 = \\frac{\\partial}{\\partial \\varepsilon} (- \\frac{\\lambda}{\\varepsilon} + q{(\\lambda,\\varepsilon)}) and 0 = \\frac{\\lambda}{\\varepsilon^{2}} + \\frac{\\partial}{\\partial \\varepsilon} q{(\\lambda,\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["minus", 1, "Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Mul(Integer(-1), Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-2))), Derivative(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(\\sigma_p)} = \\sin{(\\sigma_p)}, then derive \\int \\sigma_p I{(\\sigma_p)} d\\sigma_p = \\mathbf{J}_M - \\sigma_p \\cos{(\\sigma_p)} + \\sin{(\\sigma_p)}, then obtain \\int \\sigma_p \\sin{(\\sigma_p)} d\\sigma_p = \\mathbf{J}_M - \\sigma_p \\cos{(\\sigma_p)} + \\sin{(\\sigma_p)}", "derivation": "I{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\sigma_p I{(\\sigma_p)} = \\sigma_p \\sin{(\\sigma_p)} and \\int \\sigma_p I{(\\sigma_p)} d\\sigma_p = \\int \\sigma_p \\sin{(\\sigma_p)} d\\sigma_p and \\int \\sigma_p I{(\\sigma_p)} d\\sigma_p = \\mathbf{J}_M - \\sigma_p \\cos{(\\sigma_p)} + \\sin{(\\sigma_p)} and \\int \\sigma_p \\sin{(\\sigma_p)} d\\sigma_p = \\mathbf{J}_M - \\sigma_p \\cos{(\\sigma_p)} + \\sin{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('I')(Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('I')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('I')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{P},\\mathbf{g})} = \\mathbf{P} \\mathbf{g}, then obtain \\int \\mathbf{g}^{2} \\operatorname{F_{H}}^{2}{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g} + 1 = \\int \\mathbf{P}^{2} \\mathbf{g}^{4} d\\mathbf{g} + 1", "derivation": "\\operatorname{F_{H}}{(\\mathbf{P},\\mathbf{g})} = \\mathbf{P} \\mathbf{g} and \\mathbf{g} \\operatorname{F_{H}}{(\\mathbf{P},\\mathbf{g})} = \\mathbf{P} \\mathbf{g}^{2} and \\mathbf{g}^{2} \\operatorname{F_{H}}^{2}{(\\mathbf{P},\\mathbf{g})} = \\mathbf{P}^{2} \\mathbf{g}^{4} and \\int \\mathbf{g}^{2} \\operatorname{F_{H}}^{2}{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g} = \\int \\mathbf{P}^{2} \\mathbf{g}^{4} d\\mathbf{g} and \\int \\mathbf{g}^{2} \\operatorname{F_{H}}^{2}{(\\mathbf{P},\\mathbf{g})} d\\mathbf{g} + 1 = \\int \\mathbf{P}^{2} \\mathbf{g}^{4} d\\mathbf{g} + 1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('F_H')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)), Pow(Function('F_H')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(4))))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)), Pow(Function('F_H')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(4))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 4, 1], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)), Pow(Function('F_H')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(4))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{D},E_{x})} = E_{x} + \\mathbf{D} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{D},E_{x})} = E_{x} + \\mathbf{D}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} - \\mathbf{D} + \\operatorname{J_{\\varepsilon}}{(\\mathbf{D},E_{x})}) = \\frac{d}{d \\mathbf{D}} 0", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{D},E_{x})} = E_{x} + \\mathbf{D} and - E_{x} - \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{D},E_{x})} = 0 and \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} - \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{D},E_{x})}) = \\frac{d}{d \\mathbf{D}} 0 and \\operatorname{J_{\\varepsilon}}{(\\mathbf{D},E_{x})} = E_{x} + \\mathbf{D} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{D},E_{x})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{D},E_{x})} and \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} - \\mathbf{D} + \\operatorname{J_{\\varepsilon}}{(\\mathbf{D},E_{x})}) = \\frac{d}{d \\mathbf{D}} 0", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(f,\\nabla)} = \\nabla f, then derive - \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\hat{H}{(f,\\nabla)} = \\nabla - \\hat{H}{(f,\\nabla)}, then obtain \\frac{- \\nabla f + \\frac{\\partial}{\\partial f} \\nabla f}{\\nabla} = \\frac{- \\nabla f + \\nabla}{\\nabla}", "derivation": "\\hat{H}{(f,\\nabla)} = \\nabla f and \\frac{\\partial}{\\partial f} \\hat{H}{(f,\\nabla)} = \\frac{\\partial}{\\partial f} \\nabla f and - \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\hat{H}{(f,\\nabla)} = - \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\nabla f and - \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\hat{H}{(f,\\nabla)} = \\nabla - \\hat{H}{(f,\\nabla)} and - \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\nabla f = \\nabla - \\hat{H}{(f,\\nabla)} and \\frac{- \\hat{H}{(f,\\nabla)} + \\frac{\\partial}{\\partial f} \\nabla f}{\\nabla} = \\frac{\\nabla - \\hat{H}{(f,\\nabla)}}{\\nabla} and \\frac{- \\nabla f + \\frac{\\partial}{\\partial f} \\nabla f}{\\nabla} = \\frac{- \\nabla f + \\nabla}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))), Derivative(Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))), Derivative(Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["divide", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('f', commutative=True), Symbol('\\\\nabla', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Derivative(Mul(Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\rho_f,v)} = \\sin^{v}{(\\rho_f)}, then obtain \\rho_f \\int 0 dv = \\rho_f \\int (- \\mathbb{I}{(\\rho_f,v)} + \\sin^{v}{(\\rho_f)}) dv", "derivation": "\\mathbb{I}{(\\rho_f,v)} = \\sin^{v}{(\\rho_f)} and 0 = - \\mathbb{I}{(\\rho_f,v)} + \\sin^{v}{(\\rho_f)} and \\int 0 dv = \\int (- \\mathbb{I}{(\\rho_f,v)} + \\sin^{v}{(\\rho_f)}) dv and \\rho_f \\int 0 dv = \\rho_f \\int (- \\mathbb{I}{(\\rho_f,v)} + \\sin^{v}{(\\rho_f)}) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('v', commutative=True)))"], [["minus", 1, "Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('v', commutative=True))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["times", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Integral(Integer(0), Tuple(Symbol('v', commutative=True)))), Mul(Symbol('\\\\rho_f', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('v', commutative=True))), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l}, then derive \\phi_{2}{(\\hat{H}_l)} = e^{\\hat{H}_l}, then obtain \\phi_{2}^{\\hat{H}_l}{(\\hat{H}_l)} + \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} = \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} + (\\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l})^{\\hat{H}_l}", "derivation": "\\phi_{2}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and \\phi_{2}{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\phi_{2}^{\\hat{H}_l}{(\\hat{H}_l)} = (\\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l})^{\\hat{H}_l} and \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} = e^{\\hat{H}_l} and \\phi_{2}^{\\hat{H}_l}{(\\hat{H}_l)} = (\\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l})^{\\hat{H}_l} and \\phi_{2}^{\\hat{H}_l}{(\\hat{H}_l)} + \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} = \\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l} + (\\frac{d^{2}}{d \\hat{H}_l^{2}} e^{\\hat{H}_l})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 5, "Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)))"], "Equality(Add(Pow(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)))), Add(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Pow(Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\hat{H})} = \\sin{(\\hat{H})} and \\phi_{1}{(C_{1},v_{y},u)} = \\frac{C_{1}}{u v_{y}}, then obtain \\phi{(\\hat{H})} + \\sin{(\\hat{H})} + \\int \\phi_{1}{(C_{1},v_{y},u)} dC_{1} = \\phi{(\\hat{H})} + \\sin{(\\hat{H})} + \\int \\frac{C_{1}}{u v_{y}} dC_{1}", "derivation": "\\phi{(\\hat{H})} = \\sin{(\\hat{H})} and 2 \\phi{(\\hat{H})} = \\phi{(\\hat{H})} + \\sin{(\\hat{H})} and \\phi_{1}{(C_{1},v_{y},u)} = \\frac{C_{1}}{u v_{y}} and \\int \\phi_{1}{(C_{1},v_{y},u)} dC_{1} = \\int \\frac{C_{1}}{u v_{y}} dC_{1} and 2 \\phi{(\\hat{H})} + \\int \\phi_{1}{(C_{1},v_{y},u)} dC_{1} = 2 \\phi{(\\hat{H})} + \\int \\frac{C_{1}}{u v_{y}} dC_{1} and \\phi{(\\hat{H})} + \\sin{(\\hat{H})} + \\int \\phi_{1}{(C_{1},v_{y},u)} dC_{1} = \\phi{(\\hat{H})} + \\sin{(\\hat{H})} + \\int \\frac{C_{1}}{u v_{y}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True))), Add(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\phi_1')(Symbol('C_1', commutative=True), Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('C_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Symbol('v_y', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('C_1', commutative=True), Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Symbol('C_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Symbol('v_y', commutative=True), Integer(-1))), Tuple(Symbol('C_1', commutative=True))))"], [["add", 4, "Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('C_1', commutative=True), Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Symbol('C_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Symbol('v_y', commutative=True), Integer(-1))), Tuple(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('C_1', commutative=True), Symbol('v_y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)), Integral(Mul(Symbol('C_1', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Symbol('v_y', commutative=True), Integer(-1))), Tuple(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\dot{z},m_{s})} = \\frac{m_{s}}{\\dot{z}}, then obtain (\\int (\\delta{(\\dot{z},m_{s})} - \\frac{m_{s}}{\\dot{z}}) d\\dot{z})^{\\dot{z}} = (\\int 0 d\\dot{z})^{\\dot{z}}", "derivation": "\\delta{(\\dot{z},m_{s})} = \\frac{m_{s}}{\\dot{z}} and \\delta{(\\dot{z},m_{s})} - \\frac{m_{s}}{\\dot{z}} = 0 and \\int (\\delta{(\\dot{z},m_{s})} - \\frac{m_{s}}{\\dot{z}}) d\\dot{z} = \\int 0 d\\dot{z} and (\\int (\\delta{(\\dot{z},m_{s})} - \\frac{m_{s}}{\\dot{z}}) d\\dot{z})^{\\dot{z}} = (\\int 0 d\\dot{z})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\dot{z}', commutative=True), Symbol('m_s', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('m_s', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('m_s', commutative=True))"], "Equality(Add(Function('\\\\delta')(Symbol('\\\\dot{z}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('m_s', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Function('\\\\delta')(Symbol('\\\\dot{z}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\delta')(Symbol('\\\\dot{z}', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given z{(S)} = \\cos{(S)}, then obtain \\cos^{S}{(S)} = (- S + z{(S)}) (- z{(S)} + \\cos{(S)}) \\cos^{- S}{(S)} + \\cos^{S}{(S)}", "derivation": "z{(S)} = \\cos{(S)} and 0 = - z{(S)} + \\cos{(S)} and z^{S}{(S)} = \\cos^{S}{(S)} and 0 = (- S + z{(S)}) (- z{(S)} + \\cos{(S)}) z^{- S}{(S)} and \\cos^{S}{(S)} = (- S + z{(S)}) (- z{(S)} + \\cos{(S)}) z^{- S}{(S)} + \\cos^{S}{(S)} and z^{S}{(S)} = (- S + z{(S)}) (- z{(S)} + \\cos{(S)}) z^{- S}{(S)} + z^{S}{(S)} and \\cos^{S}{(S)} = (- S + z{(S)}) (- z{(S)} + \\cos{(S)}) \\cos^{- S}{(S)} + \\cos^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["minus", 1, "Function('z')(Symbol('S', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('z')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["divide", 2, "Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('z')(Symbol('S', commutative=True))), Integer(-1)), Pow(Function('z')(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('z')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Function('z')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Pow(Function('z')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)))))"], [["add", 4, "Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True))"], "Equality(Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('z')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Function('z')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Pow(Function('z')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)))), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('z')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('z')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Function('z')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Pow(Function('z')(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)))), Pow(Function('z')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Add(Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('z')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Function('z')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Pow(cos(Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)))), Pow(cos(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given T{(r_{0},Z)} = Z r_{0}, then obtain \\frac{\\partial}{\\partial r_{0}} (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} T^{Z}{(r_{0},Z)} = \\frac{\\partial}{\\partial r_{0}} (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} (Z r_{0})^{Z}", "derivation": "T{(r_{0},Z)} = Z r_{0} and T^{Z}{(r_{0},Z)} = (Z r_{0})^{Z} and \\frac{\\partial}{\\partial r_{0}} T^{Z}{(r_{0},Z)} = \\frac{\\partial}{\\partial r_{0}} (Z r_{0})^{Z} and \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} T^{Z}{(r_{0},Z)} = \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} (Z r_{0})^{Z} and (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} T^{Z}{(r_{0},Z)} = (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} (Z r_{0})^{Z} and \\frac{\\partial}{\\partial r_{0}} (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} T^{Z}{(r_{0},Z)} = \\frac{\\partial}{\\partial r_{0}} (Z r_{0})^{Z} \\frac{\\partial^{2}}{\\partial Z\\partial r_{0}} (Z r_{0})^{Z}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Pow(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["times", 4, "Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Derivative(Pow(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Derivative(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Derivative(Pow(Function('T')(Symbol('r_0', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Derivative(Pow(Mul(Symbol('Z', commutative=True), Symbol('r_0', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then derive \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime}, then obtain - \\log{(\\mathbf{P})} + \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime} - \\log{(\\mathbf{P})}", "derivation": "\\mathbf{J}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\int \\log{(\\mathbf{P})} d\\mathbf{P} and \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime} and - \\mathbf{J}{(\\mathbf{P})} + \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime} - \\mathbf{J}{(\\mathbf{P})} and - \\log{(\\mathbf{P})} + \\int \\log{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime} - \\log{(\\mathbf{P})} and - \\log{(\\mathbf{P})} + \\int \\mathbf{J}{(\\mathbf{P})} d\\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} + f^{\\prime} - \\log{(\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 3, "Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True))), Integral(log(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\rho_b,\\theta,q)} = \\frac{\\rho_b^{q}}{\\theta} and U{(\\rho_b,\\theta,q)} = \\iint \\frac{\\rho_b^{q}}{\\theta} d\\theta d\\theta, then obtain \\frac{\\iint \\mathbf{H}{(\\rho_b,\\theta,q)} d\\theta d\\theta}{\\frac{\\rho_b^{q}}{\\theta} - 1} = \\frac{U{(\\rho_b,\\theta,q)}}{\\frac{\\rho_b^{q}}{\\theta} - 1}", "derivation": "\\mathbf{H}{(\\rho_b,\\theta,q)} = \\frac{\\rho_b^{q}}{\\theta} and \\int \\mathbf{H}{(\\rho_b,\\theta,q)} d\\theta = \\int \\frac{\\rho_b^{q}}{\\theta} d\\theta and \\iint \\mathbf{H}{(\\rho_b,\\theta,q)} d\\theta d\\theta = \\iint \\frac{\\rho_b^{q}}{\\theta} d\\theta d\\theta and U{(\\rho_b,\\theta,q)} = \\iint \\frac{\\rho_b^{q}}{\\theta} d\\theta d\\theta and \\iint \\mathbf{H}{(\\rho_b,\\theta,q)} d\\theta d\\theta = U{(\\rho_b,\\theta,q)} and \\frac{\\iint \\mathbf{H}{(\\rho_b,\\theta,q)} d\\theta d\\theta}{\\frac{\\rho_b^{q}}{\\theta} - 1} = \\frac{U{(\\rho_b,\\theta,q)}}{\\frac{\\rho_b^{q}}{\\theta} - 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Integral(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Function('U')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)))"], [["divide", 5, "Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Integer(-1)), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Integer(-1)), Integer(-1)), Function('U')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\lambda,E_{\\lambda})} = E_{\\lambda} + \\lambda, then derive \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(\\lambda,E_{\\lambda})} = 1, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\lambda) = 1", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\lambda,E_{\\lambda})} = E_{\\lambda} + \\lambda and \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(\\lambda,E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\lambda) and \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(\\lambda,E_{\\lambda})} = 1 and \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\lambda) = 1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\omega)} = e^{\\sin{(\\omega)}}, then obtain \\frac{\\log{(\\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)})} + 1}{\\sin{(\\omega)}} = \\frac{\\log{(e^{\\sin{(\\omega)}} - \\sin{(\\omega)})} + 1}{\\sin{(\\omega)}}", "derivation": "\\operatorname{E_{n}}{(\\omega)} = e^{\\sin{(\\omega)}} and \\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)} = e^{\\sin{(\\omega)}} - \\sin{(\\omega)} and \\log{(\\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)})} = \\log{(e^{\\sin{(\\omega)}} - \\sin{(\\omega)})} and \\log{(\\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)})} + 1 = \\log{(e^{\\sin{(\\omega)}} - \\sin{(\\omega)})} + 1 and \\frac{\\log{(\\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)})} + 1}{\\sin{(\\omega)}} = \\frac{\\log{(e^{\\sin{(\\omega)}} - \\sin{(\\omega)})} + 1}{\\sin{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('E_n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True)))), Add(exp(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True)))))"], [["log", 2], "Equality(log(Add(Function('E_n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))), log(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(log(Add(Function('E_n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))), Integer(1)), Add(log(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))), Integer(1)))"], [["divide", 4, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Add(log(Add(Function('E_n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))), Integer(1)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Add(log(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))), Integer(1)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(Q)} = e^{Q}, then derive \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} = e^{Q}, then obtain e^{\\int \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} dQ} = e^{\\int \\frac{d^{2}}{d Q^{2}} e^{Q} dQ}", "derivation": "\\operatorname{v_{y}}{(Q)} = e^{Q} and \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} = \\frac{d}{d Q} e^{Q} and \\int \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} dQ = \\int \\frac{d}{d Q} e^{Q} dQ and e^{\\int \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} dQ} = e^{\\int \\frac{d}{d Q} e^{Q} dQ} and \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} = e^{Q} and \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} = \\frac{d^{2}}{d Q^{2}} \\operatorname{v_{y}}{(Q)} and \\frac{d}{d Q} e^{Q} = \\frac{d^{2}}{d Q^{2}} e^{Q} and \\int \\frac{d}{d Q} e^{Q} dQ = \\int \\frac{d^{2}}{d Q^{2}} e^{Q} dQ and e^{\\int \\frac{d}{d Q} \\operatorname{v_{y}}{(Q)} dQ} = e^{\\int \\frac{d^{2}}{d Q^{2}} e^{Q} dQ}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), exp(Integral(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), exp(Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))))"], [["integrate", 7, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Tuple(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 8], "Equality(exp(Integral(Derivative(Function('v_y')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True)))), exp(Integral(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} = v_{1} + \\frac{f_{\\mathbf{v}}}{E}, then obtain \\log{(\\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} + 1)}^{f_{\\mathbf{v}}} = \\log{(v_{1} + 1 + \\frac{f_{\\mathbf{v}}}{E})}^{f_{\\mathbf{v}}}", "derivation": "\\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} = v_{1} + \\frac{f_{\\mathbf{v}}}{E} and \\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} + 1 = v_{1} + 1 + \\frac{f_{\\mathbf{v}}}{E} and \\log{(\\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} + 1)} = \\log{(v_{1} + 1 + \\frac{f_{\\mathbf{v}}}{E})} and \\log{(\\ddot{x}{(f_{\\mathbf{v}},E,v_{1})} + 1)}^{f_{\\mathbf{v}}} = \\log{(v_{1} + 1 + \\frac{f_{\\mathbf{v}}}{E})}^{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('v_1', commutative=True), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Integer(1)), Add(Symbol('v_1', commutative=True), Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["log", 2], "Equality(log(Add(Function('\\\\ddot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Integer(1))), log(Add(Symbol('v_1', commutative=True), Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["power", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(log(Add(Function('\\\\ddot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E', commutative=True), Symbol('v_1', commutative=True)), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(log(Add(Symbol('v_1', commutative=True), Integer(1), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"]]}, {"prompt": "Given \\eta{(B)} = e^{B}, then derive \\int \\frac{\\eta{(B)}}{B} dB = T + \\operatorname{Ei}{(B)}, then derive (\\hat{x} + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} = (T + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)}, then obtain (\\hat{x} + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} = \\sigma_{p}{(B)} \\int \\frac{e^{B}}{B} dB", "derivation": "\\eta{(B)} = e^{B} and \\frac{\\eta{(B)}}{B} = \\frac{e^{B}}{B} and \\int \\frac{\\eta{(B)}}{B} dB = \\int \\frac{e^{B}}{B} dB and \\int \\frac{\\eta{(B)}}{B} dB = T + \\operatorname{Ei}{(B)} and \\sigma_{p}{(B)} \\int \\frac{\\eta{(B)}}{B} dB = (T + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} and \\sigma_{p}{(B)} \\int \\frac{e^{B}}{B} dB = (T + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} and (\\hat{x} + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} = (T + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} and (\\hat{x} + \\operatorname{Ei}{(B)}) \\sigma_{p}{(B)} = \\sigma_{p}{(B)} \\int \\frac{e^{B}}{B} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), exp(Symbol('B', commutative=True))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Add(Symbol('T', commutative=True), Ei(Symbol('B', commutative=True))))"], [["times", 4, "Function('\\\\sigma_p')(Symbol('B', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('B', commutative=True)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Mul(Add(Symbol('T', commutative=True), Ei(Symbol('B', commutative=True))), Function('\\\\sigma_p')(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('\\\\sigma_p')(Symbol('B', commutative=True)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Mul(Add(Symbol('T', commutative=True), Ei(Symbol('B', commutative=True))), Function('\\\\sigma_p')(Symbol('B', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Ei(Symbol('B', commutative=True))), Function('\\\\sigma_p')(Symbol('B', commutative=True))), Mul(Add(Symbol('T', commutative=True), Ei(Symbol('B', commutative=True))), Function('\\\\sigma_p')(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Ei(Symbol('B', commutative=True))), Function('\\\\sigma_p')(Symbol('B', commutative=True))), Mul(Function('\\\\sigma_p')(Symbol('B', commutative=True)), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given x{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})}, then derive x{(\\eta^{\\prime})} - 1 = \\cos{(\\eta^{\\prime})} - 1, then obtain \\int \\frac{d}{d \\eta^{\\prime}} (\\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} - 1) d\\eta^{\\prime} = \\int \\frac{d}{d \\eta^{\\prime}} (\\cos{(\\eta^{\\prime})} - 1) d\\eta^{\\prime}", "derivation": "x{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} and x{(\\eta^{\\prime})} - 1 = \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} - 1 and x{(\\eta^{\\prime})} - 1 = \\cos{(\\eta^{\\prime})} - 1 and \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} - 1 = \\cos{(\\eta^{\\prime})} - 1 and \\frac{d}{d \\eta^{\\prime}} (\\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} - 1) = \\frac{d}{d \\eta^{\\prime}} (\\cos{(\\eta^{\\prime})} - 1) and \\int \\frac{d}{d \\eta^{\\prime}} (\\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} - 1) d\\eta^{\\prime} = \\int \\frac{d}{d \\eta^{\\prime}} (\\cos{(\\eta^{\\prime})} - 1) d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Add(Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(Add(Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Derivative(Add(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(b,l)} = \\frac{\\partial}{\\partial l} b l, then derive \\operatorname{g^{\\prime}_{\\varepsilon}}{(b,l)} = b, then obtain \\int b db = \\int \\frac{\\partial}{\\partial l} b l db", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(b,l)} = \\frac{\\partial}{\\partial l} b l and \\operatorname{g^{\\prime}_{\\varepsilon}}{(b,l)} = b and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(b,l)} db = \\int \\frac{\\partial}{\\partial l} b l db and \\int b db = \\int \\frac{\\partial}{\\partial l} b l db", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Derivative(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Mul(Symbol('b', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})} = e^{F_{x} + \\hat{\\mathbf{x}}}, then obtain \\int e^{- F_{x} - \\hat{\\mathbf{x}}} e^{F_{x} + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} = \\int \\frac{e^{F_{x} + \\hat{\\mathbf{x}}}}{\\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})}} d\\hat{\\mathbf{x}}", "derivation": "\\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})} = e^{F_{x} + \\hat{\\mathbf{x}}} and 1 = \\frac{e^{F_{x} + \\hat{\\mathbf{x}}}}{\\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})}} and \\int 1 d\\hat{\\mathbf{x}} = \\int \\frac{e^{F_{x} + \\hat{\\mathbf{x}}}}{\\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})}} d\\hat{\\mathbf{x}} and \\int 1 d\\hat{\\mathbf{x}} = \\int e^{- F_{x} - \\hat{\\mathbf{x}}} e^{F_{x} + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} and \\int e^{- F_{x} - \\hat{\\mathbf{x}}} e^{F_{x} + \\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} = \\int \\frac{e^{F_{x} + \\hat{\\mathbf{x}}}}{\\operatorname{v_{t}}{(\\hat{\\mathbf{x}},F_{x})}} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('F_x', commutative=True)), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["divide", 1, "Function('v_t')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_t')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('F_x', commutative=True)), Integer(-1)), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(Pow(Function('v_t')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('F_x', commutative=True)), Integer(-1)), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(Pow(Function('v_t')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('F_x', commutative=True)), Integer(-1)), exp(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given U{(Z)} = Z and \\operatorname{M_{E}}{(Z)} = Z U{(Z)}, then obtain Z \\operatorname{M_{E}}{(Z)} = \\operatorname{M_{E}}{(Z)} U{(Z)}", "derivation": "U{(Z)} = Z and U^{2}{(Z)} = Z U{(Z)} and U^{3}{(Z)} = Z U^{2}{(Z)} and U^{3}{(Z)} = Z^{2} U{(Z)} and Z^{2} U{(Z)} = Z U^{2}{(Z)} and \\operatorname{M_{E}}{(Z)} = Z U{(Z)} and Z \\operatorname{M_{E}}{(Z)} = \\operatorname{M_{E}}{(Z)} U{(Z)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], [["times", 1, "Function('U')(Symbol('Z', commutative=True))"], "Equality(Pow(Function('U')(Symbol('Z', commutative=True)), Integer(2)), Mul(Symbol('Z', commutative=True), Function('U')(Symbol('Z', commutative=True))))"], [["times", 1, "Pow(Function('U')(Symbol('Z', commutative=True)), Integer(2))"], "Equality(Pow(Function('U')(Symbol('Z', commutative=True)), Integer(3)), Mul(Symbol('Z', commutative=True), Pow(Function('U')(Symbol('Z', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('U')(Symbol('Z', commutative=True)), Integer(3)), Mul(Pow(Symbol('Z', commutative=True), Integer(2)), Function('U')(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(2)), Function('U')(Symbol('Z', commutative=True))), Mul(Symbol('Z', commutative=True), Pow(Function('U')(Symbol('Z', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Function('U')(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Symbol('Z', commutative=True), Function('M_E')(Symbol('Z', commutative=True))), Mul(Function('M_E')(Symbol('Z', commutative=True)), Function('U')(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{y})} = \\sin{(A_{y})}, then derive \\frac{d}{d A_{y}} \\operatorname{A_{x}}{(A_{y})} = \\cos{(A_{y})}, then obtain \\frac{\\operatorname{A_{x}}{(A_{y})}}{\\cos^{2}{(A_{y})}} = \\frac{\\sin{(A_{y})}}{\\cos^{2}{(A_{y})}}", "derivation": "\\operatorname{A_{x}}{(A_{y})} = \\sin{(A_{y})} and \\frac{d}{d A_{y}} \\operatorname{A_{x}}{(A_{y})} = \\frac{d}{d A_{y}} \\sin{(A_{y})} and \\frac{d}{d A_{y}} \\operatorname{A_{x}}{(A_{y})} = \\cos{(A_{y})} and \\frac{d}{d A_{y}} \\sin{(A_{y})} = \\cos{(A_{y})} and \\frac{\\operatorname{A_{x}}{(A_{y})}}{\\frac{d}{d A_{y}} \\sin{(A_{y})}} = \\frac{\\sin{(A_{y})}}{\\frac{d}{d A_{y}} \\sin{(A_{y})}} and \\frac{\\operatorname{A_{x}}{(A_{y})}}{\\cos{(A_{y})}} = \\frac{\\sin{(A_{y})}}{\\cos{(A_{y})}} and \\frac{\\operatorname{A_{x}}{(A_{y})}}{\\cos^{2}{(A_{y})}} = \\frac{\\sin{(A_{y})}}{\\cos^{2}{(A_{y})}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(sin(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), cos(Symbol('A_y', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), cos(Symbol('A_y', commutative=True)))"], [["divide", 1, "Derivative(sin(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Mul(Function('A_x')(Symbol('A_y', commutative=True)), Pow(Derivative(sin(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Symbol('A_y', commutative=True)), Pow(Derivative(sin(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('A_x')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Mul(sin(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))))"], [["divide", 6, "cos(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('A_x')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-2))), Mul(sin(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(v_{2})} = \\log{(v_{2})}, then derive \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{1}{v_{2}}, then obtain \\operatorname{t_{1}}{(v_{2})} \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{\\operatorname{t_{1}}{(v_{2})}}{v_{2}}", "derivation": "\\operatorname{A_{x}}{(v_{2})} = \\log{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{1}{v_{2}} and \\operatorname{t_{1}}{(v_{2})} \\frac{d}{d v_{2}} \\operatorname{A_{x}}{(v_{2})} = \\frac{\\operatorname{t_{1}}{(v_{2})}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Pow(Symbol('v_2', commutative=True), Integer(-1)))"], [["times", 3, "Function('t_1')(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('v_2', commutative=True)), Derivative(Function('A_x')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\delta{(f^{*})} = e^{f^{*}}, then derive \\int \\delta{(f^{*})} df^{*} = T + e^{f^{*}}, then obtain - \\delta{(f^{*})} + \\int \\delta{(f^{*})} df^{*} = T", "derivation": "\\delta{(f^{*})} = e^{f^{*}} and \\int \\delta{(f^{*})} df^{*} = \\int e^{f^{*}} df^{*} and \\int \\delta{(f^{*})} df^{*} = T + e^{f^{*}} and - \\delta{(f^{*})} + \\int \\delta{(f^{*})} df^{*} = T - \\delta{(f^{*})} + e^{f^{*}} and - \\delta{(f^{*})} + \\int \\delta{(f^{*})} df^{*} = T", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('T', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["minus", 3, "Function('\\\\delta')(Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('f^*', commutative=True))), Integral(Function('\\\\delta')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('f^*', commutative=True))), exp(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('f^*', commutative=True))), Integral(Function('\\\\delta')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Symbol('T', commutative=True))"]]}, {"prompt": "Given f{(J)} = \\sin{(J)}, then obtain - J - f{(J)} + \\frac{d^{2}}{d J^{2}} f{(J)} = - J - f{(J)} + \\frac{d^{2}}{d J^{2}} \\sin{(J)}", "derivation": "f{(J)} = \\sin{(J)} and \\frac{d}{d J} f{(J)} = \\frac{d}{d J} \\sin{(J)} and \\frac{d^{2}}{d J^{2}} f{(J)} = \\frac{d^{2}}{d J^{2}} \\sin{(J)} and - J - f{(J)} + \\frac{d^{2}}{d J^{2}} f{(J)} = - J - f{(J)} + \\frac{d^{2}}{d J^{2}} \\sin{(J)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))))"], [["minus", 3, "Add(Symbol('J', commutative=True), Function('f')(Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('J', commutative=True))), Derivative(Function('f')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('J', commutative=True))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)} = B^{\\mathbf{J}}, then obtain (B + B^{\\mathbf{J}})^{B} (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)}) = (B + B^{\\mathbf{J}}) (B + B^{\\mathbf{J}})^{B}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)} = B^{\\mathbf{J}} and B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)} = B + B^{\\mathbf{J}} and (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)})^{B} = (B + B^{\\mathbf{J}})^{B} and (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)}) (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)})^{B} = (B + B^{\\mathbf{J}}) (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)})^{B} and (B + B^{\\mathbf{J}})^{B} (B + \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J},B)}) = (B + B^{\\mathbf{J}}) (B + B^{\\mathbf{J}})^{B}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 1, "Symbol('B', commutative=True)"], "Equality(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('B', commutative=True)))"], [["times", 2, "Pow(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Symbol('B', commutative=True))"], "Equality(Mul(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Pow(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)))), Mul(Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Add(Symbol('B', commutative=True), Pow(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\delta)} = e^{\\delta}, then derive \\frac{d}{d \\delta} \\psi^{*}{(\\delta)} = e^{\\delta}, then obtain (\\frac{d}{d \\delta} \\psi^{*}{(\\delta)})^{\\delta} = (\\frac{d^{2}}{d \\delta^{2}} \\psi^{*}{(\\delta)})^{\\delta}", "derivation": "\\psi^{*}{(\\delta)} = e^{\\delta} and \\frac{d}{d \\delta} \\psi^{*}{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta} and \\frac{d}{d \\delta} \\psi^{*}{(\\delta)} = e^{\\delta} and (\\frac{d}{d \\delta} \\psi^{*}{(\\delta)})^{\\delta} = (\\frac{d}{d \\delta} e^{\\delta})^{\\delta} and (\\frac{d}{d \\delta} \\psi^{*}{(\\delta)})^{\\delta} = (\\frac{d^{2}}{d \\delta^{2}} \\psi^{*}{(\\delta)})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), exp(Symbol('\\\\delta', commutative=True)))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\chi{(\\mu_0)} = e^{\\mu_0}, then obtain \\log{(\\frac{d}{d \\mu_0} \\frac{\\chi{(\\mu_0)}}{\\mu_0})} = \\log{(\\frac{d}{d \\mu_0} \\frac{e^{\\mu_0}}{\\mu_0})}", "derivation": "\\chi{(\\mu_0)} = e^{\\mu_0} and \\frac{\\chi{(\\mu_0)}}{\\mu_0} = \\frac{e^{\\mu_0}}{\\mu_0} and \\frac{d}{d \\mu_0} \\frac{\\chi{(\\mu_0)}}{\\mu_0} = \\frac{d}{d \\mu_0} \\frac{e^{\\mu_0}}{\\mu_0} and \\log{(\\frac{d}{d \\mu_0} \\frac{\\chi{(\\mu_0)}}{\\mu_0})} = \\log{(\\frac{d}{d \\mu_0} \\frac{e^{\\mu_0}}{\\mu_0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), log(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(E,\\sigma_p)} = E + \\sigma_p and \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} = J_{\\varepsilon} \\mathbf{J}_M, then obtain \\sigma_p + \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} - \\frac{1}{E} = J_{\\varepsilon} \\mathbf{J}_M + \\sigma_p - \\frac{1}{E}", "derivation": "\\mathbf{F}{(E,\\sigma_p)} = E + \\sigma_p and \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} = J_{\\varepsilon} \\mathbf{J}_M and \\sigma_p + \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} = J_{\\varepsilon} \\mathbf{J}_M + \\sigma_p and \\sigma_p + \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} - \\frac{E + \\sigma_p}{E \\mathbf{F}{(E,\\sigma_p)}} = J_{\\varepsilon} \\mathbf{J}_M + \\sigma_p - \\frac{E + \\sigma_p}{E \\mathbf{F}{(E,\\sigma_p)}} and \\sigma_p + \\eta^{\\prime}{(\\mathbf{J}_M,J_{\\varepsilon})} - \\frac{1}{E} = J_{\\varepsilon} \\mathbf{J}_M + \\sigma_p - \\frac{1}{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)))"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('\\\\mathbf{F}')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\mu,f^{\\prime})} = \\mu + \\log{(f^{\\prime})}, then obtain \\frac{\\partial}{\\partial \\mu} (\\mu \\sin{(\\operatorname{A_{x}}{(\\mu,f^{\\prime})})} + \\mu + \\log{(f^{\\prime})}) = \\frac{\\partial}{\\partial \\mu} (\\mu \\sin{(\\mu + \\log{(f^{\\prime})})} + \\mu + \\log{(f^{\\prime})})", "derivation": "\\operatorname{A_{x}}{(\\mu,f^{\\prime})} = \\mu + \\log{(f^{\\prime})} and \\sin{(\\operatorname{A_{x}}{(\\mu,f^{\\prime})})} = \\sin{(\\mu + \\log{(f^{\\prime})})} and \\mu \\sin{(\\operatorname{A_{x}}{(\\mu,f^{\\prime})})} = \\mu \\sin{(\\mu + \\log{(f^{\\prime})})} and \\mu \\sin{(\\operatorname{A_{x}}{(\\mu,f^{\\prime})})} + \\mu + \\log{(f^{\\prime})} = \\mu \\sin{(\\mu + \\log{(f^{\\prime})})} + \\mu + \\log{(f^{\\prime})} and \\frac{\\partial}{\\partial \\mu} (\\mu \\sin{(\\operatorname{A_{x}}{(\\mu,f^{\\prime})})} + \\mu + \\log{(f^{\\prime})}) = \\frac{\\partial}{\\partial \\mu} (\\mu \\sin{(\\mu + \\log{(f^{\\prime})})} + \\mu + \\log{(f^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["sin", 1], "Equality(sin(Function('A_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), sin(Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True)))))"], [["times", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), sin(Function('A_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))))"], [["add", 3, "Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), sin(Function('A_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\mu', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))), Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\mu', commutative=True), sin(Function('A_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mu', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))), Symbol('\\\\mu', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(W)} = \\log{(W)}, then obtain \\frac{d}{d W} \\int W B{(W)} dW = \\frac{d}{d W} \\int W \\log{(W)} dW", "derivation": "B{(W)} = \\log{(W)} and W B{(W)} = W \\log{(W)} and \\int W B{(W)} dW = \\int W \\log{(W)} dW and \\frac{d}{d W} \\int W B{(W)} dW = \\frac{d}{d W} \\int W \\log{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('B')(Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Symbol('W', commutative=True), Function('B')(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('W', commutative=True), Function('B')(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(b,G)} = (e^{G})^{b}, then obtain G (1 + \\frac{(e^{G})^{b}}{U{(b,G)}}) + U{(b,G)} + \\frac{(e^{G})^{b}}{U{(b,G)}} = \\frac{2 G (e^{G})^{b}}{U{(b,G)}} + U{(b,G)} + \\frac{(e^{G})^{b}}{U{(b,G)}}", "derivation": "U{(b,G)} = (e^{G})^{b} and 1 = \\frac{(e^{G})^{b}}{U{(b,G)}} and U{(b,G)} + 1 = U{(b,G)} + \\frac{(e^{G})^{b}}{U{(b,G)}} and 1 + \\frac{(e^{G})^{b}}{U{(b,G)}} = \\frac{2 (e^{G})^{b}}{U{(b,G)}} and G (1 + \\frac{(e^{G})^{b}}{U{(b,G)}}) = \\frac{2 G (e^{G})^{b}}{U{(b,G)}} and G (1 + \\frac{(e^{G})^{b}}{U{(b,G)}}) + U{(b,G)} + 1 = \\frac{2 G (e^{G})^{b}}{U{(b,G)}} + U{(b,G)} + 1 and G (1 + \\frac{(e^{G})^{b}}{U{(b,G)}}) + U{(b,G)} + \\frac{(e^{G})^{b}}{U{(b,G)}} = \\frac{2 G (e^{G})^{b}}{U{(b,G)}} + U{(b,G)} + \\frac{(e^{G})^{b}}{U{(b,G)}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))"], [["divide", 1, "Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))"], [["add", 2, "Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(1)), Add(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))))"], [["add", 2, "Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))"], "Equality(Add(Integer(1), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))), Mul(Integer(2), Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))"], [["times", 4, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Add(Integer(1), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))), Mul(Integer(2), Symbol('G', commutative=True), Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))"], [["add", 5, "Add(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(1))"], "Equality(Add(Mul(Symbol('G', commutative=True), Add(Integer(1), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))), Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(1)), Add(Mul(Integer(2), Symbol('G', commutative=True), Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))), Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Symbol('G', commutative=True), Add(Integer(1), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))))), Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))), Add(Mul(Integer(2), Symbol('G', commutative=True), Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True))), Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Function('U')(Symbol('b', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(exp(Symbol('G', commutative=True)), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(F_{x})} = \\cos{(F_{x})}, then obtain \\cos{(F_{x})} + \\frac{d}{d F_{x}} \\sigma_{x}{(F_{x})} - 1 = - \\sin{(F_{x})} + \\cos{(F_{x})} - 1", "derivation": "\\sigma_{x}{(F_{x})} = \\cos{(F_{x})} and \\frac{d}{d F_{x}} \\sigma_{x}{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})} and \\cos{(F_{x})} + \\frac{d}{d F_{x}} \\sigma_{x}{(F_{x})} - 1 = \\cos{(F_{x})} + \\frac{d}{d F_{x}} \\cos{(F_{x})} - 1 and \\cos{(F_{x})} + \\frac{d}{d F_{x}} \\sigma_{x}{(F_{x})} - 1 = - \\sin{(F_{x})} + \\cos{(F_{x})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["add", 2, "Add(cos(Symbol('F_x', commutative=True)), Integer(-1))"], "Equality(Add(cos(Symbol('F_x', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('F_x', commutative=True)), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('F_x', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\phi_{2}{(\\varphi)} = \\sin{(\\varphi)} and \\phi{(\\varphi)} = \\varphi, then obtain - \\varphi + (\\phi{(\\varphi)} + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} - \\sin{(\\varphi)} = - \\varphi + (\\varphi + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} - \\sin{(\\varphi)}", "derivation": "\\phi_{2}{(\\varphi)} = \\sin{(\\varphi)} and e^{\\phi_{2}{(\\varphi)}} = e^{\\sin{(\\varphi)}} and \\phi{(\\varphi)} = \\varphi and \\phi{(\\varphi)} + \\sin{(\\varphi)} = \\varphi + \\sin{(\\varphi)} and \\phi{(\\varphi)} + \\phi_{2}{(\\varphi)} = \\varphi + \\phi_{2}{(\\varphi)} and (\\phi{(\\varphi)} + \\phi_{2}{(\\varphi)}) e^{\\phi_{2}{(\\varphi)}} = (\\varphi + \\phi_{2}{(\\varphi)}) e^{\\phi_{2}{(\\varphi)}} and (\\phi{(\\varphi)} + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} = (\\varphi + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} and - \\varphi + (\\phi{(\\varphi)} + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} - \\sin{(\\varphi)} = - \\varphi + (\\varphi + \\phi_{2}{(\\varphi)}) e^{\\sin{(\\varphi)}} - \\sin{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(sin(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], [["add", 3, "sin(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))))"], [["times", 5, "exp(Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Add(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True)))), Mul(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Add(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(sin(Symbol('\\\\varphi', commutative=True)))), Mul(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(sin(Symbol('\\\\varphi', commutative=True)))))"], [["minus", 7, "Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Add(Function('\\\\phi')(Symbol('\\\\varphi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(sin(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\varphi', commutative=True))), exp(sin(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\varphi)} = \\cos{(\\log{(\\varphi)})}, then derive 2 \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}{(\\varphi)} = \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}{(\\varphi)} - \\frac{\\sin{(\\log{(\\varphi)})}}{\\varphi}, then obtain 2 \\frac{d}{d \\varphi} \\cos{(\\log{(\\varphi)})} = \\frac{d}{d \\varphi} \\cos{(\\log{(\\varphi)})} - \\frac{\\sin{(\\log{(\\varphi)})}}{\\varphi}", "derivation": "\\operatorname{y^{\\prime}}{(\\varphi)} = \\cos{(\\log{(\\varphi)})} and 2 \\operatorname{y^{\\prime}}{(\\varphi)} = \\operatorname{y^{\\prime}}{(\\varphi)} + \\cos{(\\log{(\\varphi)})} and \\frac{d}{d \\varphi} 2 \\operatorname{y^{\\prime}}{(\\varphi)} = \\frac{d}{d \\varphi} (\\operatorname{y^{\\prime}}{(\\varphi)} + \\cos{(\\log{(\\varphi)})}) and 2 \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}{(\\varphi)} = \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}{(\\varphi)} - \\frac{\\sin{(\\log{(\\varphi)})}}{\\varphi} and 2 \\frac{d}{d \\varphi} \\cos{(\\log{(\\varphi)})} = \\frac{d}{d \\varphi} \\cos{(\\log{(\\varphi)})} - \\frac{\\sin{(\\log{(\\varphi)})}}{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), cos(log(Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))), Add(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), cos(log(Symbol('\\\\varphi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), cos(log(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(log(Symbol('\\\\varphi', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(cos(log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Derivative(cos(log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(log(Symbol('\\\\varphi', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\delta)} = \\log{(\\delta)} and \\tilde{g}^*{(\\Omega)} = \\log{(\\Omega)}, then derive A_{y} + (\\log{(\\delta)} + 1) \\int \\tilde{g}^*{(\\Omega)} d\\Omega = \\Omega (- \\log{(\\delta)} - 1) + x^\\prime + (\\Omega \\log{(\\delta)} + \\Omega) \\log{(\\Omega)}, then obtain A_{y} + (\\dot{\\mathbf{r}}{(\\delta)} + 1) \\int \\tilde{g}^*{(\\Omega)} d\\Omega = \\Omega (- \\dot{\\mathbf{r}}{(\\delta)} - 1) + x^\\prime + (\\Omega \\dot{\\mathbf{r}}{(\\delta)} + \\Omega) \\log{(\\Omega)}", "derivation": "\\dot{\\mathbf{r}}{(\\delta)} = \\log{(\\delta)} and \\tilde{g}^*{(\\Omega)} = \\log{(\\Omega)} and \\delta \\tilde{g}^*{(\\Omega)} \\log{(\\delta)} = \\delta \\log{(\\Omega)} \\log{(\\delta)} and \\frac{\\partial}{\\partial \\delta} \\delta \\tilde{g}^*{(\\Omega)} \\log{(\\delta)} = \\frac{\\partial}{\\partial \\delta} \\delta \\log{(\\Omega)} \\log{(\\delta)} and \\int \\frac{\\partial}{\\partial \\delta} \\delta \\tilde{g}^*{(\\Omega)} \\log{(\\delta)} d\\Omega = \\int \\frac{\\partial}{\\partial \\delta} \\delta \\log{(\\Omega)} \\log{(\\delta)} d\\Omega and A_{y} + (\\log{(\\delta)} + 1) \\int \\tilde{g}^*{(\\Omega)} d\\Omega = \\Omega (- \\log{(\\delta)} - 1) + x^\\prime + (\\Omega \\log{(\\delta)} + \\Omega) \\log{(\\Omega)} and A_{y} + (\\dot{\\mathbf{r}}{(\\delta)} + 1) \\int \\tilde{g}^*{(\\Omega)} d\\Omega = \\Omega (- \\dot{\\mathbf{r}}{(\\delta)} - 1) + x^\\prime + (\\Omega \\dot{\\mathbf{r}}{(\\delta)} + \\Omega) \\log{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('A_y', commutative=True), Mul(Add(log(Symbol('\\\\delta', commutative=True)), Integer(1)), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Mul(Symbol('\\\\Omega', commutative=True), Add(Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True))), Integer(-1))), Symbol('x^\\\\prime', commutative=True), Mul(Add(Mul(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('A_y', commutative=True), Mul(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True)), Integer(1)), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Mul(Symbol('\\\\Omega', commutative=True), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True))), Integer(-1))), Symbol('x^\\\\prime', commutative=True), Mul(Add(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\delta', commutative=True))), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\tilde{g}^*)} = \\log{(e^{\\tilde{g}^*})}, then derive \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} = 1, then obtain \\frac{d}{d \\tilde{g}^*} \\log{(e^{\\tilde{g}^*})} = 1", "derivation": "\\varphi{(\\tilde{g}^*)} = \\log{(e^{\\tilde{g}^*})} and \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\log{(e^{\\tilde{g}^*})} and \\frac{d}{d \\tilde{g}^*} \\varphi{(\\tilde{g}^*)} = 1 and \\frac{d}{d \\tilde{g}^*} \\log{(e^{\\tilde{g}^*})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), log(exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given V{(s)} = e^{e^{s}}, then obtain 1 + \\frac{V{(s)}}{s + e^{e^{s}}} = 1 + \\frac{e^{e^{s}}}{s + e^{e^{s}}}", "derivation": "V{(s)} = e^{e^{s}} and s + V{(s)} = s + e^{e^{s}} and \\frac{V{(s)}}{s + V{(s)}} = \\frac{e^{e^{s}}}{s + V{(s)}} and 1 + \\frac{V{(s)}}{s + V{(s)}} = 1 + \\frac{e^{e^{s}}}{s + V{(s)}} and 1 + \\frac{V{(s)}}{s + e^{e^{s}}} = 1 + \\frac{e^{e^{s}}}{s + e^{e^{s}}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('s', commutative=True)), exp(exp(Symbol('s', commutative=True))))"], [["add", 1, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), exp(exp(Symbol('s', commutative=True)))))"], [["divide", 1, "Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True))), Integer(-1)), Function('V')(Symbol('s', commutative=True))), Mul(Pow(Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True))), Integer(-1)), exp(exp(Symbol('s', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True))), Integer(-1)), Function('V')(Symbol('s', commutative=True)))), Add(Integer(1), Mul(Pow(Add(Symbol('s', commutative=True), Function('V')(Symbol('s', commutative=True))), Integer(-1)), exp(exp(Symbol('s', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integer(1), Mul(Pow(Add(Symbol('s', commutative=True), exp(exp(Symbol('s', commutative=True)))), Integer(-1)), Function('V')(Symbol('s', commutative=True)))), Add(Integer(1), Mul(Pow(Add(Symbol('s', commutative=True), exp(exp(Symbol('s', commutative=True)))), Integer(-1)), exp(exp(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given l{(\\mathbf{P})} = \\sin{(\\cos{(\\mathbf{P})})}, then obtain \\int l{(\\mathbf{P})} d\\mathbf{P} - \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P} l{(\\mathbf{P})}} = \\int \\sin{(\\cos{(\\mathbf{P})})} d\\mathbf{P} - \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P} l{(\\mathbf{P})}}", "derivation": "l{(\\mathbf{P})} = \\sin{(\\cos{(\\mathbf{P})})} and \\frac{l{(\\mathbf{P})}}{\\mathbf{P}} = \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P}} and \\frac{1}{\\mathbf{P}} = \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P} l{(\\mathbf{P})}} and \\int l{(\\mathbf{P})} d\\mathbf{P} = \\int \\sin{(\\cos{(\\mathbf{P})})} d\\mathbf{P} and \\int l{(\\mathbf{P})} d\\mathbf{P} - \\frac{1}{\\mathbf{P}} = \\int \\sin{(\\cos{(\\mathbf{P})})} d\\mathbf{P} - \\frac{1}{\\mathbf{P}} and \\int l{(\\mathbf{P})} d\\mathbf{P} - \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P} l{(\\mathbf{P})}} = \\int \\sin{(\\cos{(\\mathbf{P})})} d\\mathbf{P} - \\frac{\\sin{(\\cos{(\\mathbf{P})})}}{\\mathbf{P} l{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('l')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 2, "Function('l')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(sin(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 4, "Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))), Add(Integral(sin(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Integral(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True))))), Add(Integral(sin(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given Q{(m)} = e^{m}, then obtain \\frac{d}{d m} 8 Q^{3}{(m)} = \\frac{d}{d m} \\frac{16 Q^{4}{(m)}}{Q{(m)} + e^{m}}", "derivation": "Q{(m)} = e^{m} and 2 Q{(m)} = Q{(m)} + e^{m} and 4 Q^{2}{(m)} = (Q{(m)} + e^{m})^{2} and \\frac{4 Q^{2}{(m)}}{Q{(m)} + e^{m}} = Q{(m)} + e^{m} and 2 Q{(m)} = \\frac{4 Q^{2}{(m)}}{Q{(m)} + e^{m}} and 8 Q^{3}{(m)} = \\frac{16 Q^{4}{(m)}}{Q{(m)} + e^{m}} and \\frac{d}{d m} 8 Q^{3}{(m)} = \\frac{d}{d m} \\frac{16 Q^{4}{(m)}}{Q{(m)} + e^{m}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["add", 1, "Function('Q')(Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Function('Q')(Symbol('m', commutative=True))), Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(2))), Pow(Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Integer(2)))"], [["divide", 3, "Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(2))), Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Integer(2), Function('Q')(Symbol('m', commutative=True))), Mul(Integer(4), Pow(Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(2))))"], [["times", 5, "Mul(Integer(4), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(8), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(3))), Mul(Integer(16), Pow(Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(4))))"], [["differentiate", 6, "Symbol('m', commutative=True)"], "Equality(Derivative(Mul(Integer(8), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(3))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Integer(16), Pow(Add(Function('Q')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True)), Integer(4))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(\\mu,\\rho)} = \\frac{\\rho}{\\mu}, then obtain ((\\rho_{f}^{\\mu}{(\\mu,\\rho)})^{\\rho})^{\\mu} - \\rho_{f}^{\\mu}{(\\mu,\\rho)} = (((\\frac{\\rho}{\\mu})^{\\mu})^{\\rho})^{\\mu} - \\rho_{f}^{\\mu}{(\\mu,\\rho)}", "derivation": "\\rho_{f}{(\\mu,\\rho)} = \\frac{\\rho}{\\mu} and \\rho_{f}^{\\mu}{(\\mu,\\rho)} = (\\frac{\\rho}{\\mu})^{\\mu} and (\\rho_{f}^{\\mu}{(\\mu,\\rho)})^{\\rho} = ((\\frac{\\rho}{\\mu})^{\\mu})^{\\rho} and ((\\rho_{f}^{\\mu}{(\\mu,\\rho)})^{\\rho})^{\\mu} = (((\\frac{\\rho}{\\mu})^{\\mu})^{\\rho})^{\\mu} and ((\\rho_{f}^{\\mu}{(\\mu,\\rho)})^{\\rho})^{\\mu} - \\rho_{f}^{\\mu}{(\\mu,\\rho)} = (((\\frac{\\rho}{\\mu})^{\\mu})^{\\rho})^{\\mu} - \\rho_{f}^{\\mu}{(\\mu,\\rho)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 4, "Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)))), Add(Pow(Pow(Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\delta)} = \\log{(\\log{(\\delta)})}, then obtain - \\frac{\\rho_{f}{(\\delta)} - \\log{(\\delta)}}{\\log{(\\delta)} \\frac{d}{d \\delta} \\rho_{f}{(\\delta)}} = - \\frac{- \\log{(\\delta)} + \\log{(\\log{(\\delta)})}}{\\log{(\\delta)} \\frac{d}{d \\delta} \\rho_{f}{(\\delta)}}", "derivation": "\\rho_{f}{(\\delta)} = \\log{(\\log{(\\delta)})} and \\rho_{f}{(\\delta)} - \\log{(\\delta)} = - \\log{(\\delta)} + \\log{(\\log{(\\delta)})} and - \\frac{\\rho_{f}{(\\delta)} - \\log{(\\delta)}}{\\log{(\\delta)}} = - \\frac{- \\log{(\\delta)} + \\log{(\\log{(\\delta)})}}{\\log{(\\delta)}} and - \\frac{\\rho_{f}{(\\delta)} - \\log{(\\delta)}}{\\log{(\\delta)} \\frac{d}{d \\delta} \\rho_{f}{(\\delta)}} = - \\frac{- \\log{(\\delta)} + \\log{(\\log{(\\delta)})}}{\\log{(\\delta)} \\frac{d}{d \\delta} \\rho_{f}{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), log(log(Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True))), log(log(Symbol('\\\\delta', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True))), log(log(Symbol('\\\\delta', commutative=True)))), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["divide", 3, "Derivative(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Add(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True))), log(log(Symbol('\\\\delta', commutative=True)))), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\rho_f')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{x})} = e^{\\hat{x}} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{x})} = \\int \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} d\\hat{x}, then derive \\int \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} d\\hat{x} = L_{\\varepsilon} + \\operatorname{Ei}{(\\hat{x})}, then obtain \\log{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{x})})} = \\log{(L_{\\varepsilon} + \\operatorname{Ei}{(\\hat{x})})}", "derivation": "\\operatorname{P_{g}}{(\\hat{x})} = e^{\\hat{x}} and \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} = \\frac{e^{\\hat{x}}}{\\hat{x}} and \\int \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} d\\hat{x} = \\int \\frac{e^{\\hat{x}}}{\\hat{x}} d\\hat{x} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{x})} = \\int \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} d\\hat{x} and \\int \\frac{\\operatorname{P_{g}}{(\\hat{x})}}{\\hat{x}} d\\hat{x} = L_{\\varepsilon} + \\operatorname{Ei}{(\\hat{x})} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{x})} = L_{\\varepsilon} + \\operatorname{Ei}{(\\hat{x})} and \\log{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{x})})} = \\log{(L_{\\varepsilon} + \\operatorname{Ei}{(\\hat{x})})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Ei(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Ei(Symbol('\\\\hat{x}', commutative=True))))"], [["log", 6], "Equality(log(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True))), log(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Ei(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given r{(\\rho_b,Q)} = Q - \\rho_b, then obtain - r{(\\rho_b,Q)} + \\int \\sin{(r{(\\rho_b,Q)})} d\\rho_b = - r{(\\rho_b,Q)} + \\int \\sin{(Q - \\rho_b)} d\\rho_b", "derivation": "r{(\\rho_b,Q)} = Q - \\rho_b and \\sin{(r{(\\rho_b,Q)})} = \\sin{(Q - \\rho_b)} and \\int \\sin{(r{(\\rho_b,Q)})} d\\rho_b = \\int \\sin{(Q - \\rho_b)} d\\rho_b and - r{(\\rho_b,Q)} + \\int \\sin{(r{(\\rho_b,Q)})} d\\rho_b = - r{(\\rho_b,Q)} + \\int \\sin{(Q - \\rho_b)} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["sin", 1], "Equality(sin(Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(sin(Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))), Integral(sin(Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))), Add(Mul(Integer(-1), Function('r')(Symbol('\\\\rho_b', commutative=True), Symbol('Q', commutative=True))), Integral(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} = - F_{g} + e^{\\theta_1}, then derive \\theta_1 + i = \\int - \\frac{2 F_{g} - 2 e^{\\theta_1}}{2 (- F_{g} + e^{\\theta_1})} d\\theta_1, then obtain \\int 1 d\\theta_1 = \\theta_1 + i", "derivation": "\\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} = - F_{g} + e^{\\theta_1} and - \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} = F_{g} - e^{\\theta_1} and - 2 \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} = F_{g} - \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} - e^{\\theta_1} and 1 = - \\frac{F_{g} - \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)} - e^{\\theta_1}}{2 \\operatorname{y^{\\prime}}{(F_{g},\\theta_1)}} and 1 = - \\frac{2 F_{g} - 2 e^{\\theta_1}}{2 (- F_{g} + e^{\\theta_1})} and \\int 1 d\\theta_1 = \\int - \\frac{2 F_{g} - 2 e^{\\theta_1}}{2 (- F_{g} + e^{\\theta_1})} d\\theta_1 and \\theta_1 + i = \\int - \\frac{2 F_{g} - 2 e^{\\theta_1}}{2 (- F_{g} + e^{\\theta_1})} d\\theta_1 and \\int 1 d\\theta_1 = \\theta_1 + i", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 2, "Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Integer(2), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Integer(1), Mul(Integer(-1), Rational(1, 2), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))), Pow(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Integer(-1), Rational(1, 2), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True))))))"], [["integrate", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Integer(-1), Rational(1, 2), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True))))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True)), Integral(Mul(Integer(-1), Rational(1, 2), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\theta_1', commutative=True))))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(\\pi,q)} = \\pi + q, then derive \\int \\dot{x}{(\\pi,q)} dq = \\pi q + \\frac{q^{2}}{2} + z, then derive 2 \\int \\dot{x}{(\\pi,q)} dq = \\pi q + \\rho_b + \\frac{q^{2}}{2} + \\int \\dot{x}{(\\pi,q)} dq, then obtain 2 \\pi q + q^{2} + 2 z = 2 \\pi q + \\rho_b + q^{2} + z", "derivation": "\\dot{x}{(\\pi,q)} = \\pi + q and \\int \\dot{x}{(\\pi,q)} dq = \\int (\\pi + q) dq and \\int \\dot{x}{(\\pi,q)} dq = \\pi q + \\frac{q^{2}}{2} + z and 2 \\int \\dot{x}{(\\pi,q)} dq = \\int (\\pi + q) dq + \\int \\dot{x}{(\\pi,q)} dq and 2 \\int \\dot{x}{(\\pi,q)} dq = \\pi q + \\rho_b + \\frac{q^{2}}{2} + \\int \\dot{x}{(\\pi,q)} dq and 2 \\pi q + q^{2} + 2 z = 2 \\pi q + \\rho_b + q^{2} + z", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["add", 2, "Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Integral(Add(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\rho_b', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), Integral(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(2)), Mul(Integer(2), Symbol('z', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('q', commutative=True), Integer(2)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(h,r)} = \\log{(h + r)} and \\operatorname{F_{x}}{(h,r)} = \\log{(h + r)}^{h} and \\tilde{g}{(h,r)} = \\log{(h + r)}, then obtain \\log{(h + r)}^{h} - \\log{(h + r)}^{r} = \\tilde{g}^{h}{(h,r)} - \\log{(h + r)}^{r}", "derivation": "\\operatorname{F_{g}}{(h,r)} = \\log{(h + r)} and \\operatorname{F_{g}}^{h}{(h,r)} = \\log{(h + r)}^{h} and \\operatorname{F_{x}}{(h,r)} = \\log{(h + r)}^{h} and \\operatorname{F_{x}}{(h,r)} = \\operatorname{F_{g}}^{h}{(h,r)} and \\tilde{g}{(h,r)} = \\log{(h + r)} and \\operatorname{F_{g}}^{h}{(h,r)} = \\tilde{g}^{h}{(h,r)} and \\operatorname{F_{x}}{(h,r)} = \\tilde{g}^{h}{(h,r)} and \\log{(h + r)}^{h} = \\tilde{g}^{h}{(h,r)} and \\log{(h + r)}^{h} - \\log{(h + r)}^{r} = \\tilde{g}^{h}{(h,r)} - \\log{(h + r)}^{r}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('h', commutative=True), Symbol('r', commutative=True)), log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)), Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('F_x')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(Function('F_g')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('r', commutative=True)), log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('F_g')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Function('F_x')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('h', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 8, "Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True))"], "Equality(Add(Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))), Add(Pow(Function('\\\\tilde{g}')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(log(Add(Symbol('h', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given z{(E_{\\lambda},\\sigma_p)} = \\sigma_p^{E_{\\lambda}}, then obtain 4 \\sigma_p^{2 E_{\\lambda}} = 2 \\sigma_p^{E_{\\lambda}} (\\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)})", "derivation": "z{(E_{\\lambda},\\sigma_p)} = \\sigma_p^{E_{\\lambda}} and \\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)} = 2 \\sigma_p^{E_{\\lambda}} and (\\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)})^{2} = 2 \\sigma_p^{E_{\\lambda}} (\\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)}) and (\\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)})^{2} = 4 \\sigma_p^{2 E_{\\lambda}} and 4 \\sigma_p^{2 E_{\\lambda}} = 2 \\sigma_p^{E_{\\lambda}} (\\sigma_p^{E_{\\lambda}} + z{(E_{\\lambda},\\sigma_p)})", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 2, "Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Pow(Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(2)), Mul(Integer(2), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(4), Pow(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\chi{(k,\\pi)} = \\pi^{k}, then derive \\frac{\\partial}{\\partial \\pi} \\chi{(k,\\pi)} = \\frac{\\pi^{k} k}{\\pi}, then obtain (\\frac{\\partial}{\\partial k} \\frac{\\pi^{k} k}{\\pi})^{k} = (\\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\chi{(k,\\pi)})^{k}", "derivation": "\\chi{(k,\\pi)} = \\pi^{k} and \\frac{\\partial}{\\partial \\pi} \\chi{(k,\\pi)} = \\frac{\\partial}{\\partial \\pi} \\pi^{k} and \\frac{\\partial}{\\partial \\pi} \\chi{(k,\\pi)} = \\frac{\\pi^{k} k}{\\pi} and \\frac{\\pi^{k} k}{\\pi} = \\frac{\\partial}{\\partial \\pi} \\pi^{k} and \\frac{\\partial}{\\partial k} \\frac{\\pi^{k} k}{\\pi} = \\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\pi^{k} and \\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\chi{(k,\\pi)} = \\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\pi^{k} and \\frac{\\partial}{\\partial k} \\frac{\\pi^{k} k}{\\pi} = \\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\chi{(k,\\pi)} and (\\frac{\\partial}{\\partial k} \\frac{\\pi^{k} k}{\\pi})^{k} = (\\frac{\\partial^{2}}{\\partial k\\partial \\pi} \\chi{(k,\\pi)})^{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 7, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(Function('\\\\chi')(Symbol('k', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given b{(m_{s})} = \\frac{d}{d m_{s}} \\cos{(m_{s})}, then obtain \\cos{(m_{s})} + \\frac{d}{d m_{s}} \\frac{b{(m_{s})}}{\\frac{d}{d m_{s}} \\cos{(m_{s})}} = \\cos{(m_{s})} + \\frac{d}{d m_{s}} 1", "derivation": "b{(m_{s})} = \\frac{d}{d m_{s}} \\cos{(m_{s})} and \\frac{b{(m_{s})}}{\\frac{d}{d m_{s}} \\cos{(m_{s})}} = 1 and \\frac{d}{d m_{s}} \\frac{b{(m_{s})}}{\\frac{d}{d m_{s}} \\cos{(m_{s})}} = \\frac{d}{d m_{s}} 1 and \\cos{(m_{s})} + \\frac{d}{d m_{s}} \\frac{b{(m_{s})}}{\\frac{d}{d m_{s}} \\cos{(m_{s})}} = \\cos{(m_{s})} + \\frac{d}{d m_{s}} 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('m_s', commutative=True)), Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))"], "Equality(Mul(Function('b')(Symbol('m_s', commutative=True)), Pow(Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Mul(Function('b')(Symbol('m_s', commutative=True)), Pow(Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["add", 3, "cos(Symbol('m_s', commutative=True))"], "Equality(Add(cos(Symbol('m_s', commutative=True)), Derivative(Mul(Function('b')(Symbol('m_s', commutative=True)), Pow(Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(cos(Symbol('m_s', commutative=True)), Derivative(Integer(1), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(\\theta,J_{\\varepsilon})} = J_{\\varepsilon} \\theta, then obtain \\log{(\\frac{\\theta V^{\\theta}{(\\theta,J_{\\varepsilon})} \\frac{\\partial}{\\partial J_{\\varepsilon}} V{(\\theta,J_{\\varepsilon})}}{V{(\\theta,J_{\\varepsilon})}})} = \\log{(\\frac{\\theta (J_{\\varepsilon} \\theta)^{\\theta}}{J_{\\varepsilon}})}", "derivation": "V{(\\theta,J_{\\varepsilon})} = J_{\\varepsilon} \\theta and V^{\\theta}{(\\theta,J_{\\varepsilon})} = (J_{\\varepsilon} \\theta)^{\\theta} and \\frac{\\partial}{\\partial J_{\\varepsilon}} V^{\\theta}{(\\theta,J_{\\varepsilon})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} \\theta)^{\\theta} and \\log{(\\frac{\\partial}{\\partial J_{\\varepsilon}} V^{\\theta}{(\\theta,J_{\\varepsilon})})} = \\log{(\\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} \\theta)^{\\theta})} and \\log{(\\frac{\\theta V^{\\theta}{(\\theta,J_{\\varepsilon})} \\frac{\\partial}{\\partial J_{\\varepsilon}} V{(\\theta,J_{\\varepsilon})}}{V{(\\theta,J_{\\varepsilon})}})} = \\log{(\\frac{\\theta (J_{\\varepsilon} \\theta)^{\\theta}}{J_{\\varepsilon}})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Pow(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Pow(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), log(Derivative(Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(log(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Pow(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta', commutative=True)), Derivative(Function('V')(Symbol('\\\\theta', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True), Pow(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then derive \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} + 1 = \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} + 1", "derivation": "\\operatorname{M_{E}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} + 1 = \\sin{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)} + 1 and \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} + 1 = \\sin{(\\mathbf{J}_M)} + \\cos{(\\mathbf{J}_M)} + 1 and \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} + 1 = \\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{M_{E}}{(\\mathbf{J}_M)} + 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M_E')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 3, "Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], "Equality(Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('M_E')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1)), Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1)), Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1)), Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('M_E')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given G{(\\mathbf{E},\\tilde{g})} = \\log{(\\mathbf{E} - \\tilde{g})}, then obtain \\int (2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int G{(\\mathbf{E},\\tilde{g})} d\\mathbf{E}) d\\tilde{g} = \\int (2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int \\log{(\\mathbf{E} - \\tilde{g})} d\\mathbf{E}) d\\tilde{g}", "derivation": "G{(\\mathbf{E},\\tilde{g})} = \\log{(\\mathbf{E} - \\tilde{g})} and \\int G{(\\mathbf{E},\\tilde{g})} d\\mathbf{E} = \\int \\log{(\\mathbf{E} - \\tilde{g})} d\\mathbf{E} and 2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int G{(\\mathbf{E},\\tilde{g})} d\\mathbf{E} = 2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int \\log{(\\mathbf{E} - \\tilde{g})} d\\mathbf{E} and \\int (2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int G{(\\mathbf{E},\\tilde{g})} d\\mathbf{E}) d\\tilde{g} = \\int (2 \\log{(\\mathbf{E} - \\tilde{g})} + \\int \\log{(\\mathbf{E} - \\tilde{g})} d\\mathbf{E}) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 2, "Mul(Integer(2), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))))"], "Equality(Add(Mul(Integer(2), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Integral(Function('G')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(2), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Integral(Function('G')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Mul(Integer(2), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(t,\\mathbf{S})} = \\mathbf{S} + t, then obtain (- (\\mathbf{S} + t)^{t} + \\operatorname{F_{c}}{(t,\\mathbf{S})})^{t} = (\\mathbf{S} + t - (\\mathbf{S} + t)^{t})^{t}", "derivation": "\\operatorname{F_{c}}{(t,\\mathbf{S})} = \\mathbf{S} + t and \\operatorname{F_{c}}^{t}{(t,\\mathbf{S})} = (\\mathbf{S} + t)^{t} and \\operatorname{F_{c}}{(t,\\mathbf{S})} - \\operatorname{F_{c}}^{t}{(t,\\mathbf{S})} = \\mathbf{S} + t - \\operatorname{F_{c}}^{t}{(t,\\mathbf{S})} and - (\\mathbf{S} + t)^{t} + \\operatorname{F_{c}}{(t,\\mathbf{S})} = \\mathbf{S} + t - (\\mathbf{S} + t)^{t} and (- (\\mathbf{S} + t)^{t} + \\operatorname{F_{c}}{(t,\\mathbf{S})})^{t} = (\\mathbf{S} + t - (\\mathbf{S} + t)^{t})^{t}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('t', commutative=True)), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["minus", 1, "Pow(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('t', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('t', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))))"], [["power", 4, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('F_c')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\dot{\\mathbf{r}},s)} = - \\sin{(\\dot{\\mathbf{r}} - s)}, then obtain - \\frac{- \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)} + \\mathbf{P}^{s}{(\\dot{\\mathbf{r}},s)}}{s} = - \\frac{(- \\sin{(\\dot{\\mathbf{r}} - s)})^{s} - \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)}}{s}", "derivation": "\\mathbf{P}{(\\dot{\\mathbf{r}},s)} = - \\sin{(\\dot{\\mathbf{r}} - s)} and \\mathbf{P}^{s}{(\\dot{\\mathbf{r}},s)} = (- \\sin{(\\dot{\\mathbf{r}} - s)})^{s} and - \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)} + \\mathbf{P}^{s}{(\\dot{\\mathbf{r}},s)} = (- \\sin{(\\dot{\\mathbf{r}} - s)})^{s} - \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)} and - \\frac{- \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)} + \\mathbf{P}^{s}{(\\dot{\\mathbf{r}},s)}}{s} = - \\frac{(- \\sin{(\\dot{\\mathbf{r}} - s)})^{s} - \\mathbf{P}^{2}{(\\dot{\\mathbf{r}},s)}}{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Symbol('s', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Add(Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Integer(2)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('s', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)))), Mul(Integer(-1), Pow(Symbol('s', commutative=True), Integer(-1)), Add(Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True)), Integer(2))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})} = C_{d} \\mathbf{F} - \\mathbf{S}, then derive \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})} = C_{d}, then obtain C_{d}^{\\mathbf{F}} = (\\frac{\\partial}{\\partial \\mathbf{F}} (C_{d} \\mathbf{F} - \\mathbf{S}))^{\\mathbf{F}}", "derivation": "\\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})} = C_{d} \\mathbf{F} - \\mathbf{S} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} (C_{d} \\mathbf{F} - \\mathbf{S}) and (\\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})})^{\\mathbf{F}} = (\\frac{\\partial}{\\partial \\mathbf{F}} (C_{d} \\mathbf{F} - \\mathbf{S}))^{\\mathbf{F}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{z^{*}}{(C_{d},\\mathbf{S},\\mathbf{F})} = C_{d} and C_{d}^{\\mathbf{F}} = (\\frac{\\partial}{\\partial \\mathbf{F}} (C_{d} \\mathbf{F} - \\mathbf{S}))^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Derivative(Function('z^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('C_d', commutative=True))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(Add(Mul(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\theta_1,z^{*})} = \\theta_1 + z^{*} and \\phi_{2}{(\\theta_1,z^{*})} = 2 \\varepsilon{(\\theta_1,z^{*})}, then obtain 2 (\\theta_1 + z^{*}) (3 \\theta_1 + z^{*}) - 2 \\varepsilon{(\\theta_1,z^{*})} = (2 \\theta_1 + 2 z^{*}) (3 \\theta_1 + z^{*}) - 2 \\varepsilon{(\\theta_1,z^{*})}", "derivation": "\\varepsilon{(\\theta_1,z^{*})} = \\theta_1 + z^{*} and \\phi_{2}{(\\theta_1,z^{*})} = 2 \\varepsilon{(\\theta_1,z^{*})} and \\phi_{2}{(\\theta_1,z^{*})} = 2 \\theta_1 + 2 z^{*} and 2 \\varepsilon{(\\theta_1,z^{*})} = 2 \\theta_1 + 2 z^{*} and 2 (\\theta_1 - z^{*} + 2 \\varepsilon{(\\theta_1,z^{*})}) \\varepsilon{(\\theta_1,z^{*})} = (2 \\theta_1 + 2 z^{*}) (\\theta_1 - z^{*} + 2 \\varepsilon{(\\theta_1,z^{*})}) and 2 (\\theta_1 + z^{*}) (3 \\theta_1 + z^{*}) = (2 \\theta_1 + 2 z^{*}) (3 \\theta_1 + z^{*}) and 2 (\\theta_1 + z^{*}) (3 \\theta_1 + z^{*}) - 2 \\varepsilon{(\\theta_1,z^{*})} = (2 \\theta_1 + 2 z^{*}) (3 \\theta_1 + z^{*}) - 2 \\varepsilon{(\\theta_1,z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\phi_2')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["times", 4, "Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True))))"], "Equality(Mul(Integer(2), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)))), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Add(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Symbol('z^*', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Symbol('z^*', commutative=True))))"], [["add", 6, "Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)))"], "Equality(Add(Mul(Integer(2), Add(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Symbol('z^*', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)))), Add(Mul(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('z^*', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\theta_1', commutative=True)), Symbol('z^*', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('\\\\theta_1', commutative=True), Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(P_{g})} = \\sin{(P_{g})}, then obtain ((\\operatorname{A_{y}}{(P_{g})} - \\sin{(P_{g})})^{P_{g}})^{P_{g}} = (0^{P_{g}})^{P_{g}}", "derivation": "\\operatorname{A_{y}}{(P_{g})} = \\sin{(P_{g})} and \\operatorname{A_{y}}{(P_{g})} - \\sin{(P_{g})} = 0 and (\\operatorname{A_{y}}{(P_{g})} - \\sin{(P_{g})})^{P_{g}} = 0^{P_{g}} and ((\\operatorname{A_{y}}{(P_{g})} - \\sin{(P_{g})})^{P_{g}})^{P_{g}} = (0^{P_{g}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["minus", 1, "sin(Symbol('P_g', commutative=True))"], "Equality(Add(Function('A_y')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Add(Function('A_y')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)), Pow(Integer(0), Symbol('P_g', commutative=True)))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Pow(Add(Function('A_y')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(Pow(Integer(0), Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(b,m)} = b m, then obtain (\\frac{3}{2 b m})^{b} = (\\frac{3}{2 \\operatorname{y^{\\prime}}{(b,m)}})^{b}", "derivation": "\\operatorname{y^{\\prime}}{(b,m)} = b m and 2 \\operatorname{y^{\\prime}}^{2}{(b,m)} = 2 b m \\operatorname{y^{\\prime}}{(b,m)} and 3 \\operatorname{y^{\\prime}}{(b,m)} = b m + 2 \\operatorname{y^{\\prime}}{(b,m)} and \\frac{3}{2 b m} = \\frac{b m + 2 \\operatorname{y^{\\prime}}{(b,m)}}{2 b m \\operatorname{y^{\\prime}}{(b,m)}} and \\frac{3}{2 b m} = \\frac{b m + 2 \\operatorname{y^{\\prime}}{(b,m)}}{2 \\operatorname{y^{\\prime}}^{2}{(b,m)}} and (\\frac{3}{2 b m})^{b} = (\\frac{b m + 2 \\operatorname{y^{\\prime}}{(b,m)}}{2 \\operatorname{y^{\\prime}}^{2}{(b,m)}})^{b} and (\\frac{3}{2 b m})^{b} = (\\frac{3}{2 \\operatorname{y^{\\prime}}{(b,m)}})^{b}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], [["times", 1, "Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('b', commutative=True), Symbol('m', commutative=True), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], "Equality(Mul(Integer(3), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True))), Add(Mul(Symbol('b', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), Symbol('b', commutative=True), Symbol('m', commutative=True), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], "Equality(Mul(Rational(3, 2), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Symbol('b', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))), Pow(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Rational(3, 2), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Symbol('b', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))), Pow(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Integer(-2))))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Mul(Rational(3, 2), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('b', commutative=True)), Pow(Mul(Rational(1, 2), Add(Mul(Symbol('b', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)))), Pow(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Integer(-2))), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Mul(Rational(3, 2), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('b', commutative=True)), Pow(Mul(Rational(3, 2), Pow(Function('y^{\\\\prime}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Integer(-1))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(m_{s},\\Omega)} = \\Omega - m_{s}, then obtain T (- T + \\hat{\\mathbf{x}}{(m_{s},\\Omega)})^{\\Omega} = T (- T + \\Omega - m_{s})^{\\Omega}", "derivation": "\\hat{\\mathbf{x}}{(m_{s},\\Omega)} = \\Omega - m_{s} and - T + \\hat{\\mathbf{x}}{(m_{s},\\Omega)} = - T + \\Omega - m_{s} and (- T + \\hat{\\mathbf{x}}{(m_{s},\\Omega)})^{\\Omega} = (- T + \\Omega - m_{s})^{\\Omega} and T (- T + \\hat{\\mathbf{x}}{(m_{s},\\Omega)})^{\\Omega} = T (- T + \\Omega - m_{s})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["times", 3, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\Omega)} = \\cos{(e^{\\Omega})} and z{(\\Omega)} = \\cos{(e^{\\Omega})}, then obtain \\cos{((- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)})^{\\Omega})} = \\cos{((- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} z{(\\Omega)})^{\\Omega})}", "derivation": "\\hat{\\mathbf{r}}{(\\Omega)} = \\cos{(e^{\\Omega})} and \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)} = \\frac{d}{d \\Omega} \\cos{(e^{\\Omega})} and z{(\\Omega)} = \\cos{(e^{\\Omega})} and \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)} = \\frac{d}{d \\Omega} z{(\\Omega)} and - \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)} = - \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} z{(\\Omega)} and (- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)})^{\\Omega} = (- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} z{(\\Omega)})^{\\Omega} and \\cos{((- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} \\hat{\\mathbf{r}}{(\\Omega)})^{\\Omega})} = \\cos{((- \\sin{(e^{\\Omega})} + \\frac{d}{d \\Omega} z{(\\Omega)})^{\\Omega})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\Omega', commutative=True)), cos(exp(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 4, "sin(exp(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)))"], [["cos", 6], "Equality(cos(Pow(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True))), cos(Pow(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\Omega', commutative=True)))), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given c{(V,f^{\\prime})} = e^{- V + f^{\\prime}} and J{(I)} = e^{\\sin{(I)}}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} J{(I)} \\int 0 dV = \\frac{\\partial}{\\partial f^{\\prime}} J{(I)} \\int (- c{(V,f^{\\prime})} + e^{- V + f^{\\prime}}) dV", "derivation": "c{(V,f^{\\prime})} = e^{- V + f^{\\prime}} and 0 = - c{(V,f^{\\prime})} + e^{- V + f^{\\prime}} and \\int 0 dV = \\int (- c{(V,f^{\\prime})} + e^{- V + f^{\\prime}}) dV and J{(I)} = e^{\\sin{(I)}} and e^{\\sin{(I)}} \\int 0 dV = e^{\\sin{(I)}} \\int (- c{(V,f^{\\prime})} + e^{- V + f^{\\prime}}) dV and J{(I)} \\int 0 dV = J{(I)} \\int (- c{(V,f^{\\prime})} + e^{- V + f^{\\prime}}) dV and \\frac{\\partial}{\\partial f^{\\prime}} J{(I)} \\int 0 dV = \\frac{\\partial}{\\partial f^{\\prime}} J{(I)} \\int (- c{(V,f^{\\prime})} + e^{- V + f^{\\prime}}) dV", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('V', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], ["get_premise", "Equality(Function('J')(Symbol('I', commutative=True)), exp(sin(Symbol('I', commutative=True))))"], [["times", 3, "exp(sin(Symbol('I', commutative=True)))"], "Equality(Mul(exp(sin(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))), Mul(exp(sin(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('J')(Symbol('I', commutative=True)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))), Mul(Function('J')(Symbol('I', commutative=True)), Integral(Add(Mul(Integer(-1), Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('V', commutative=True)))))"], [["differentiate", 6, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Function('J')(Symbol('I', commutative=True)), Integral(Integer(0), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Function('J')(Symbol('I', commutative=True)), Integral(Add(Mul(Integer(-1), Function('c')(Symbol('V', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('V', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(F_{H})} = \\sin{(F_{H})}, then obtain ((\\frac{d}{d F_{H}} \\int \\operatorname{F_{N}}{(F_{H})} dF_{H})^{F_{H}})^{F_{H}} = ((\\frac{d}{d F_{H}} \\int \\sin{(F_{H})} dF_{H})^{F_{H}})^{F_{H}}", "derivation": "\\operatorname{F_{N}}{(F_{H})} = \\sin{(F_{H})} and \\int \\operatorname{F_{N}}{(F_{H})} dF_{H} = \\int \\sin{(F_{H})} dF_{H} and \\frac{d}{d F_{H}} \\int \\operatorname{F_{N}}{(F_{H})} dF_{H} = \\frac{d}{d F_{H}} \\int \\sin{(F_{H})} dF_{H} and (\\frac{d}{d F_{H}} \\int \\operatorname{F_{N}}{(F_{H})} dF_{H})^{F_{H}} = (\\frac{d}{d F_{H}} \\int \\sin{(F_{H})} dF_{H})^{F_{H}} and ((\\frac{d}{d F_{H}} \\int \\operatorname{F_{N}}{(F_{H})} dF_{H})^{F_{H}})^{F_{H}} = ((\\frac{d}{d F_{H}} \\int \\sin{(F_{H})} dF_{H})^{F_{H}})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integral(Function('F_N')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('F_N')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Pow(Derivative(Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)))"], [["power", 4, "Symbol('F_H', commutative=True)"], "Equality(Pow(Pow(Derivative(Integral(Function('F_N')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(Pow(Derivative(Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given g{(V_{\\mathbf{B}},Q)} = Q V_{\\mathbf{B}}, then obtain V_{\\mathbf{B}} + 2 = \\frac{Q V_{\\mathbf{B}}}{g{(V_{\\mathbf{B}},Q)}} + V_{\\mathbf{B}} + 1", "derivation": "g{(V_{\\mathbf{B}},Q)} = Q V_{\\mathbf{B}} and 1 = \\frac{Q V_{\\mathbf{B}}}{g{(V_{\\mathbf{B}},Q)}} and 2 = \\frac{Q V_{\\mathbf{B}}}{g{(V_{\\mathbf{B}},Q)}} + 1 and V_{\\mathbf{B}} + 2 = \\frac{Q V_{\\mathbf{B}}}{g{(V_{\\mathbf{B}},Q)}} + V_{\\mathbf{B}} + 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 1, "Function('g')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Symbol('Q', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Function('g')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('Q', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Function('g')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Integer(1)))"], [["add", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(2)), Add(Mul(Symbol('Q', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Function('g')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\mathbf{P}{(x^\\prime)} = e^{x^\\prime}, then derive \\frac{d}{d x^\\prime} \\mathbf{P}{(x^\\prime)} = e^{x^\\prime}, then obtain \\frac{d}{d x^\\prime} e^{x^\\prime} = e^{x^\\prime}", "derivation": "\\mathbf{P}{(x^\\prime)} = e^{x^\\prime} and \\frac{d}{d x^\\prime} \\mathbf{P}{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} and \\frac{d}{d x^\\prime} \\mathbf{P}{(x^\\prime)} = e^{x^\\prime} and \\frac{d}{d x^\\prime} e^{x^\\prime} = e^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), exp(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), exp(Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(t_{1})} = \\sin{(\\log{(t_{1})})} and \\phi_{2}{(t_{1})} = \\log{(t_{1})}, then obtain \\int (- \\sigma_{x}{(t_{1})} + \\sin{(\\phi_{2}{(t_{1})})})^{t_{1}} dt_{1} = \\int 0^{t_{1}} dt_{1}", "derivation": "\\sigma_{x}{(t_{1})} = \\sin{(\\log{(t_{1})})} and \\phi_{2}{(t_{1})} = \\log{(t_{1})} and \\sigma_{x}{(t_{1})} = \\sin{(\\phi_{2}{(t_{1})})} and \\sin{(\\log{(t_{1})})} = \\sin{(\\phi_{2}{(t_{1})})} and - \\sin{(\\phi_{2}{(t_{1})})} + \\sin{(\\log{(t_{1})})} = 0 and - \\sigma_{x}{(t_{1})} + \\sin{(\\log{(t_{1})})} = 0 and (- \\sigma_{x}{(t_{1})} + \\sin{(\\log{(t_{1})})})^{t_{1}} = 0^{t_{1}} and (- \\sigma_{x}{(t_{1})} + \\sin{(\\phi_{2}{(t_{1})})})^{t_{1}} = 0^{t_{1}} and \\int (- \\sigma_{x}{(t_{1})} + \\sin{(\\phi_{2}{(t_{1})})})^{t_{1}} dt_{1} = \\int 0^{t_{1}} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('t_1', commutative=True)), sin(log(Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\sigma_x')(Symbol('t_1', commutative=True)), sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(sin(log(Symbol('t_1', commutative=True))), sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True))))"], [["minus", 4, "sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True)))), sin(log(Symbol('t_1', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_1', commutative=True))), sin(log(Symbol('t_1', commutative=True)))), Integer(0))"], [["power", 6, "Symbol('t_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_1', commutative=True))), sin(log(Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Pow(Integer(0), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_1', commutative=True))), sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Pow(Integer(0), Symbol('t_1', commutative=True)))"], [["integrate", 8, "Symbol('t_1', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('t_1', commutative=True))), sin(Function('\\\\phi_2')(Symbol('t_1', commutative=True)))), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Pow(Integer(0), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(a,A_{1})} = \\sin{(A_{1} a)}, then obtain \\frac{\\int \\ddot{x}{(a,A_{1})} dA_{1}}{A_{1} a} = \\frac{\\int \\sin{(A_{1} a)} dA_{1}}{A_{1} a}", "derivation": "\\ddot{x}{(a,A_{1})} = \\sin{(A_{1} a)} and \\int \\ddot{x}{(a,A_{1})} dA_{1} = \\int \\sin{(A_{1} a)} dA_{1} and \\frac{\\int \\ddot{x}{(a,A_{1})} dA_{1}}{a} = \\frac{\\int \\sin{(A_{1} a)} dA_{1}}{a} and \\frac{\\int \\ddot{x}{(a,A_{1})} dA_{1}}{A_{1} a} = \\frac{\\int \\sin{(A_{1} a)} dA_{1}}{A_{1} a}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('A_1', commutative=True)), sin(Mul(Symbol('A_1', commutative=True), Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(sin(Mul(Symbol('A_1', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"], [["divide", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(sin(Mul(Symbol('A_1', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('A_1', commutative=True)))))"], [["divide", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('a', commutative=True), Integer(-1)), Integral(sin(Mul(Symbol('A_1', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given g{(\\mathbb{I},\\eta)} = \\mathbb{I}^{\\eta} and \\psi^{*}{(\\mathbb{I},\\eta)} = \\mathbb{I}^{\\eta}, then obtain 2 g{(\\mathbb{I},\\eta)} = \\psi^{*}{(\\mathbb{I},\\eta)} + g{(\\mathbb{I},\\eta)}", "derivation": "g{(\\mathbb{I},\\eta)} = \\mathbb{I}^{\\eta} and 2 g{(\\mathbb{I},\\eta)} = \\mathbb{I}^{\\eta} + g{(\\mathbb{I},\\eta)} and \\psi^{*}{(\\mathbb{I},\\eta)} = \\mathbb{I}^{\\eta} and 2 g{(\\mathbb{I},\\eta)} = \\psi^{*}{(\\mathbb{I},\\eta)} + g{(\\mathbb{I},\\eta)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["add", 1, "Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Function('\\\\psi^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True)), Function('g')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\pi)} = e^{\\pi} and \\mathbf{S}{(\\pi)} = - \\operatorname{V_{\\mathbf{E}}}{(\\pi)} + e^{\\pi} and L{(\\pi)} = e^{\\pi}, then obtain \\mathbf{S}{(\\pi)} = - 2 L{(\\pi)} + 2 e^{\\pi}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\pi)} = e^{\\pi} and 0 = - \\operatorname{V_{\\mathbf{E}}}{(\\pi)} + e^{\\pi} and \\mathbf{S}{(\\pi)} = - \\operatorname{V_{\\mathbf{E}}}{(\\pi)} + e^{\\pi} and - \\operatorname{V_{\\mathbf{E}}}{(\\pi)} = - 2 \\operatorname{V_{\\mathbf{E}}}{(\\pi)} + e^{\\pi} and \\mathbf{S}{(\\pi)} = - 2 \\operatorname{V_{\\mathbf{E}}}{(\\pi)} + 2 e^{\\pi} and L{(\\pi)} = e^{\\pi} and L{(\\pi)} = \\operatorname{V_{\\mathbf{E}}}{(\\pi)} and \\mathbf{S}{(\\pi)} = - 2 L{(\\pi)} + 2 e^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))), exp(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))), exp(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))), exp(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True)))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Function('L')(Symbol('\\\\pi', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\varphi^*,F_{x})} = F_{x} \\varphi^*, then derive \\frac{\\partial}{\\partial F_{x}} \\operatorname{f_{E}}{(\\varphi^*,F_{x})} = \\varphi^*, then obtain \\frac{\\partial}{\\partial \\varphi^*} (\\frac{\\partial}{\\partial F_{x}} F_{x} \\varphi^*)^{F_{x}} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{F_{x}}", "derivation": "\\operatorname{f_{E}}{(\\varphi^*,F_{x})} = F_{x} \\varphi^* and \\frac{\\partial}{\\partial F_{x}} \\operatorname{f_{E}}{(\\varphi^*,F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x} \\varphi^* and \\frac{\\partial}{\\partial F_{x}} \\operatorname{f_{E}}{(\\varphi^*,F_{x})} = \\varphi^* and \\frac{\\partial}{\\partial F_{x}} F_{x} \\varphi^* = \\varphi^* and (\\frac{\\partial}{\\partial F_{x}} F_{x} \\varphi^*)^{F_{x}} = (\\varphi^*)^{F_{x}} and \\frac{\\partial}{\\partial \\varphi^*} (\\frac{\\partial}{\\partial F_{x}} F_{x} \\varphi^*)^{F_{x}} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\varphi^*', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True))"], [["power", 4, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('F_x', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} = t^{f_{\\mathbf{v}}}, then obtain \\cos{(\\frac{\\partial}{\\partial t} \\int \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} df_{\\mathbf{v}})} = \\cos{(\\frac{\\partial}{\\partial t} \\int t^{f_{\\mathbf{v}}} df_{\\mathbf{v}})}", "derivation": "\\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} = t^{f_{\\mathbf{v}}} and \\int \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} df_{\\mathbf{v}} = \\int t^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\frac{\\partial}{\\partial t} \\int \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} df_{\\mathbf{v}} = \\frac{\\partial}{\\partial t} \\int t^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\cos{(\\frac{\\partial}{\\partial t} \\int \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},t)} df_{\\mathbf{v}})} = \\cos{(\\frac{\\partial}{\\partial t} \\int t^{f_{\\mathbf{v}}} df_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Pow(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), cos(Derivative(Integral(Pow(Symbol('t', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{B},A_{1})} = \\frac{\\mathbf{B}}{A_{1}}, then obtain (\\operatorname{M_{E}}{(\\mathbf{B},A_{1})} + \\frac{\\mathbf{B}}{A_{1}})^{A_{1}} \\operatorname{M_{E}}{(\\mathbf{B},A_{1})} = (\\frac{2 \\mathbf{B}}{A_{1}})^{A_{1}} \\operatorname{M_{E}}{(\\mathbf{B},A_{1})}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{B},A_{1})} = \\frac{\\mathbf{B}}{A_{1}} and \\operatorname{M_{E}}{(\\mathbf{B},A_{1})} + \\frac{\\mathbf{B}}{A_{1}} = \\frac{2 \\mathbf{B}}{A_{1}} and (\\operatorname{M_{E}}{(\\mathbf{B},A_{1})} + \\frac{\\mathbf{B}}{A_{1}})^{A_{1}} = (\\frac{2 \\mathbf{B}}{A_{1}})^{A_{1}} and (\\operatorname{M_{E}}{(\\mathbf{B},A_{1})} + \\frac{\\mathbf{B}}{A_{1}})^{A_{1}} \\operatorname{M_{E}}{(\\mathbf{B},A_{1})} = (\\frac{2 \\mathbf{B}}{A_{1}})^{A_{1}} \\operatorname{M_{E}}{(\\mathbf{B},A_{1})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('A_1', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('A_1', commutative=True)))"], [["times", 3, "Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Add(Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('A_1', commutative=True)), Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Mul(Integer(2), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('A_1', commutative=True)), Function('M_E')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given r{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain \\frac{\\frac{d}{d \\mathbf{B}} (r{(\\mathbf{B})} + e^{\\mathbf{B}})}{\\mathbf{B}^{2}} = \\frac{\\frac{d}{d \\mathbf{B}} 2 e^{\\mathbf{B}}}{\\mathbf{B}^{2}}", "derivation": "r{(\\mathbf{B})} = e^{\\mathbf{B}} and r{(\\mathbf{B})} + e^{\\mathbf{B}} = 2 e^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} (r{(\\mathbf{B})} + e^{\\mathbf{B}}) = \\frac{d}{d \\mathbf{B}} 2 e^{\\mathbf{B}} and \\frac{\\frac{d}{d \\mathbf{B}} (r{(\\mathbf{B})} + e^{\\mathbf{B}})}{\\mathbf{B}} = \\frac{\\frac{d}{d \\mathbf{B}} 2 e^{\\mathbf{B}}}{\\mathbf{B}} and \\frac{\\frac{d}{d \\mathbf{B}} (r{(\\mathbf{B})} + e^{\\mathbf{B}})}{\\mathbf{B}^{2}} = \\frac{\\frac{d}{d \\mathbf{B}} 2 e^{\\mathbf{B}}}{\\mathbf{B}^{2}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Function('r')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Derivative(Add(Function('r')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Derivative(Mul(Integer(2), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["times", 4, "Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Derivative(Add(Function('r')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Derivative(Mul(Integer(2), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(f^{\\prime})} = \\int \\log{(f^{\\prime})} df^{\\prime}, then derive \\operatorname{A_{2}}{(f^{\\prime})} = f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} + f_{E}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} + f_{E} - \\log{(f^{\\prime})}) = \\frac{d}{d f^{\\prime}} (- \\log{(f^{\\prime})} + \\int \\log{(f^{\\prime})} df^{\\prime})", "derivation": "\\operatorname{A_{2}}{(f^{\\prime})} = \\int \\log{(f^{\\prime})} df^{\\prime} and \\operatorname{A_{2}}{(f^{\\prime})} - \\log{(f^{\\prime})} = - \\log{(f^{\\prime})} + \\int \\log{(f^{\\prime})} df^{\\prime} and \\operatorname{A_{2}}{(f^{\\prime})} = f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} + f_{E} and \\frac{d}{d f^{\\prime}} (\\operatorname{A_{2}}{(f^{\\prime})} - \\log{(f^{\\prime})}) = \\frac{d}{d f^{\\prime}} (- \\log{(f^{\\prime})} + \\int \\log{(f^{\\prime})} df^{\\prime}) and \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} + f_{E} - \\log{(f^{\\prime})}) = \\frac{d}{d f^{\\prime}} (- \\log{(f^{\\prime})} + \\int \\log{(f^{\\prime})} df^{\\prime})", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "log(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True)))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\rho)} = \\rho, then derive (P_{e} + \\frac{\\hat{p}_0^{2}{(\\rho)}}{2})^{\\hat{p}_0{(\\rho)}} = (\\int \\rho d\\hat{p}_0{(\\rho)})^{\\hat{p}_0{(\\rho)}}, then obtain (P_{e} + \\frac{\\rho^{2}}{2})^{\\rho} = (W + \\frac{\\rho^{2}}{2})^{\\rho}", "derivation": "\\hat{p}_0{(\\rho)} = \\rho and \\int \\hat{p}_0{(\\rho)} d\\rho = \\int \\rho d\\rho and (\\int \\hat{p}_0{(\\rho)} d\\rho)^{\\rho} = (\\int \\rho d\\rho)^{\\rho} and (\\int \\hat{p}_0{(\\rho)} d\\hat{p}_0{(\\rho)})^{\\hat{p}_0{(\\rho)}} = (\\int \\rho d\\hat{p}_0{(\\rho)})^{\\hat{p}_0{(\\rho)}} and (P_{e} + \\frac{\\hat{p}_0^{2}{(\\rho)}}{2})^{\\hat{p}_0{(\\rho)}} = (\\int \\rho d\\hat{p}_0{(\\rho)})^{\\hat{p}_0{(\\rho)}} and (P_{e} + \\frac{\\rho^{2}}{2})^{\\rho} = (\\int \\rho d\\rho)^{\\rho} and (P_{e} + \\frac{\\rho^{2}}{2})^{\\rho} = (W + \\frac{\\rho^{2}}{2})^{\\rho}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Tuple(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))), Pow(Integral(Symbol('\\\\rho', commutative=True), Tuple(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('P_e', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(2)))), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))), Pow(Integral(Symbol('\\\\rho', commutative=True), Tuple(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Symbol('P_e', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(2)))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('P_e', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(2)))), Symbol('\\\\rho', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(2)))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(A,\\eta,A_{1})} = - \\eta + \\frac{A_{1}}{A} and J{(\\eta)} = - \\eta, then derive \\frac{\\partial}{\\partial A} \\mathbf{F}{(A,\\eta,A_{1})} + \\frac{A_{1}}{A^{2}} = 0, then obtain \\frac{\\partial}{\\partial A} (J{(\\eta)} + \\frac{A_{1}}{A}) + \\frac{A_{1}}{A^{2}} = 0", "derivation": "\\mathbf{F}{(A,\\eta,A_{1})} = - \\eta + \\frac{A_{1}}{A} and \\eta + \\mathbf{F}{(A,\\eta,A_{1})} - \\frac{A_{1}}{A} = 0 and \\frac{\\partial}{\\partial A} (\\eta + \\mathbf{F}{(A,\\eta,A_{1})} - \\frac{A_{1}}{A}) = \\frac{d}{d A} 0 and \\frac{\\partial}{\\partial A} \\mathbf{F}{(A,\\eta,A_{1})} + \\frac{A_{1}}{A^{2}} = 0 and \\frac{\\partial}{\\partial A} (- \\eta + \\frac{A_{1}}{A}) + \\frac{A_{1}}{A^{2}} = 0 and J{(\\eta)} = - \\eta and \\frac{\\partial}{\\partial A} (J{(\\eta)} + \\frac{A_{1}}{A}) + \\frac{A_{1}}{A^{2}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True)))"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{F}')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{F}')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Pow(Symbol('A', commutative=True), Integer(-2)), Symbol('A_1', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Pow(Symbol('A', commutative=True), Integer(-2)), Symbol('A_1', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Derivative(Add(Function('J')(Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('A_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Pow(Symbol('A', commutative=True), Integer(-2)), Symbol('A_1', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{D}{(\\phi_1)} = \\sin{(\\phi_1)} and \\mathbf{S}{(\\phi_1)} = \\phi_1, then obtain \\frac{\\mathbf{D}{(\\phi_1)} \\sin{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\phi_1}{F_{c}} = \\frac{\\sin^{2}{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\phi_1}{F_{c}}", "derivation": "\\mathbf{D}{(\\phi_1)} = \\sin{(\\phi_1)} and \\mathbf{D}{(\\phi_1)} \\sin{(\\phi_1)} = \\sin^{2}{(\\phi_1)} and \\frac{\\mathbf{D}{(\\phi_1)} \\sin{(\\phi_1)}}{\\pi{(F_{c},H)}} = \\frac{\\sin^{2}{(\\phi_1)}}{\\pi{(F_{c},H)}} and \\mathbf{S}{(\\phi_1)} = \\phi_1 and \\frac{\\mathbf{D}{(\\phi_1)} \\sin{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\mathbf{S}{(\\phi_1)}}{F_{c}} = \\frac{\\sin^{2}{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\mathbf{S}{(\\phi_1)}}{F_{c}} and \\frac{\\mathbf{D}{(\\phi_1)} \\sin{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\phi_1}{F_{c}} = \\frac{\\sin^{2}{(\\phi_1)}}{\\pi{(F_{c},H)}} - \\frac{\\phi_1}{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(2)))"], [["divide", 2, "Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\phi_1', commutative=True)), Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\phi_1', commutative=True)), Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True)))), Add(Mul(Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\phi_1', commutative=True)), Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Pow(Function('\\\\pi')(Symbol('F_c', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(b)} = \\sin{(b)}, then obtain \\cos{(b)} \\frac{d}{d b} \\mathbf{p}{(b)} = \\cos^{2}{(b)}", "derivation": "\\mathbf{p}{(b)} = \\sin{(b)} and \\frac{d}{d b} \\mathbf{p}{(b)} = \\frac{d}{d b} \\sin{(b)} and \\frac{d}{d b} \\mathbf{p}{(b)} \\frac{d}{d b} \\sin{(b)} = (\\frac{d}{d b} \\sin{(b)})^{2} and \\cos{(b)} \\frac{d}{d b} \\mathbf{p}{(b)} = \\cos^{2}{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 2, "Derivative(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Symbol('b', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Pow(cos(Symbol('b', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{v}{(v,t)} = t v and s{(t,v)} = (- t v + (t v)^{v}) \\mathbf{v}^{- v}{(v,t)}, then obtain s{(t,v)} = (- t v + \\mathbf{v}^{v}{(v,t)}) \\mathbf{v}^{- v}{(v,t)}", "derivation": "\\mathbf{v}{(v,t)} = t v and \\mathbf{v}^{v}{(v,t)} = (t v)^{v} and - t v + \\mathbf{v}^{v}{(v,t)} = - t v + (t v)^{v} and (- t v + \\mathbf{v}^{v}{(v,t)}) \\mathbf{v}^{- v}{(v,t)} = (- t v + (t v)^{v}) \\mathbf{v}^{- v}{(v,t)} and s{(t,v)} = (- t v + (t v)^{v}) \\mathbf{v}^{- v}{(v,t)} and s{(t,v)} = (- t v + \\mathbf{v}^{v}{(v,t)}) \\mathbf{v}^{- v}{(v,t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('t', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Symbol('v', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["minus", 2, "Mul(Symbol('t', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], [["divide", 3, "Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Symbol('v', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Symbol('v', commutative=True))), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))))"], ["renaming_premise", "Equality(Function('s')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('s')(Symbol('t', commutative=True), Symbol('v', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Symbol('v', commutative=True))), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given G{(t_{2},E)} = E + t_{2} and \\varphi{(t_{2})} = t_{2}, then derive \\sin{(G{(t_{2},E)} + \\int \\varphi{(t_{2})} dt_{2})} = \\sin{(g_{\\varepsilon} + \\frac{t_{2}^{2}}{2} + G{(t_{2},E)})}, then obtain \\sin{(E + t_{2} + \\int t_{2} dt_{2})} = \\sin{(E + g_{\\varepsilon} + \\frac{t_{2}^{2}}{2} + t_{2})}", "derivation": "G{(t_{2},E)} = E + t_{2} and \\varphi{(t_{2})} = t_{2} and \\int \\varphi{(t_{2})} dt_{2} = \\int t_{2} dt_{2} and G{(t_{2},E)} + \\int \\varphi{(t_{2})} dt_{2} = G{(t_{2},E)} + \\int t_{2} dt_{2} and \\sin{(G{(t_{2},E)} + \\int \\varphi{(t_{2})} dt_{2})} = \\sin{(G{(t_{2},E)} + \\int t_{2} dt_{2})} and \\sin{(G{(t_{2},E)} + \\int \\varphi{(t_{2})} dt_{2})} = \\sin{(g_{\\varepsilon} + \\frac{t_{2}^{2}}{2} + G{(t_{2},E)})} and \\sin{(E + t_{2} + \\int \\varphi{(t_{2})} dt_{2})} = \\sin{(E + g_{\\varepsilon} + \\frac{t_{2}^{2}}{2} + t_{2})} and \\sin{(E + t_{2} + \\int t_{2} dt_{2})} = \\sin{(E + g_{\\varepsilon} + \\frac{t_{2}^{2}}{2} + t_{2})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Symbol('t_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True))))"], [["add", 3, "Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True))"], "Equality(Add(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Integral(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Integral(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), sin(Add(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(sin(Add(Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)), Integral(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), sin(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Function('G')(Symbol('t_2', commutative=True), Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(sin(Add(Symbol('E', commutative=True), Symbol('t_2', commutative=True), Integral(Function('\\\\varphi')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), sin(Add(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(sin(Add(Symbol('E', commutative=True), Symbol('t_2', commutative=True), Integral(Symbol('t_2', commutative=True), Tuple(Symbol('t_2', commutative=True))))), sin(Add(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_2', commutative=True), Integer(2))), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given C{(Q,Z)} = - Q + Z and \\mathbf{E}{(h,\\theta)} = \\frac{h}{\\theta}, then obtain Q + C{(Q,Z)} - \\mathbf{E}{(h,\\theta)} = Z - \\mathbf{E}{(h,\\theta)}", "derivation": "C{(Q,Z)} = - Q + Z and Q + C{(Q,Z)} = Z and \\mathbf{E}{(h,\\theta)} = \\frac{h}{\\theta} and Q + C{(Q,Z)} - \\frac{h}{\\theta} = Z - \\frac{h}{\\theta} and Q + C{(Q,Z)} - \\mathbf{E}{(h,\\theta)} = Z - \\mathbf{E}{(h,\\theta)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Function('C')(Symbol('Q', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], [["minus", 2, "Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('h', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Function('C')(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('h', commutative=True))), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('Q', commutative=True), Function('C')(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(f_{E})} = e^{f_{E}}, then obtain - \\operatorname{E_{x}}^{2}{(f_{E})} e^{2 f_{E}} + \\operatorname{E_{x}}^{f_{E}}{(f_{E})} = - \\operatorname{E_{x}}^{2}{(f_{E})} e^{2 f_{E}} + (e^{f_{E}})^{f_{E}}", "derivation": "\\operatorname{E_{x}}{(f_{E})} = e^{f_{E}} and \\operatorname{E_{x}}^{f_{E}}{(f_{E})} = (e^{f_{E}})^{f_{E}} and \\operatorname{E_{x}}{(f_{E})} e^{f_{E}} = e^{2 f_{E}} and \\operatorname{E_{x}}^{2}{(f_{E})} e^{2 f_{E}} = e^{4 f_{E}} and \\operatorname{E_{x}}^{f_{E}}{(f_{E})} - e^{4 f_{E}} = - e^{4 f_{E}} + (e^{f_{E}})^{f_{E}} and - \\operatorname{E_{x}}^{2}{(f_{E})} e^{2 f_{E}} + \\operatorname{E_{x}}^{f_{E}}{(f_{E})} = - \\operatorname{E_{x}}^{2}{(f_{E})} e^{2 f_{E}} + (e^{f_{E}})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["times", 1, "exp(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('E_x')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True))), exp(Mul(Integer(2), Symbol('f_E', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('E_x')(Symbol('f_E', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('f_E', commutative=True)))), exp(Mul(Integer(4), Symbol('f_E', commutative=True))))"], [["minus", 2, "exp(Mul(Integer(4), Symbol('f_E', commutative=True)))"], "Equality(Add(Pow(Function('E_x')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(4), Symbol('f_E', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(4), Symbol('f_E', commutative=True)))), Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('f_E', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('f_E', commutative=True)))), Pow(Function('E_x')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('f_E', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('f_E', commutative=True)))), Pow(exp(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given v{(V)} = e^{V}, then obtain r + e^{V} + \\int v{(V)} dV = 2 r + 2 e^{V}", "derivation": "v{(V)} = e^{V} and \\int v{(V)} dV = \\int e^{V} dV and \\int v{(V)} dV + \\int e^{V} dV = 2 \\int e^{V} dV and r + e^{V} + \\int v{(V)} dV = 2 r + 2 e^{V}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('v')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 2, "Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))"], "Equality(Add(Integral(Function('v')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('r', commutative=True), exp(Symbol('V', commutative=True)), Integral(Function('v')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), exp(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(n_{2},\\mathbf{D})} = n_{2} + \\cos{(\\mathbf{D})}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\rho_{b}{(n_{2},\\mathbf{D})} dn_{2} d\\mathbf{D} = \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\int (n_{2} + \\cos{(\\mathbf{D})}) dn_{2} d\\mathbf{D}", "derivation": "\\rho_{b}{(n_{2},\\mathbf{D})} = n_{2} + \\cos{(\\mathbf{D})} and \\int \\rho_{b}{(n_{2},\\mathbf{D})} dn_{2} = \\int (n_{2} + \\cos{(\\mathbf{D})}) dn_{2} and \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\rho_{b}{(n_{2},\\mathbf{D})} dn_{2} = \\frac{\\partial}{\\partial \\mathbf{D}} \\int (n_{2} + \\cos{(\\mathbf{D})}) dn_{2} and \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\rho_{b}{(n_{2},\\mathbf{D})} dn_{2} d\\mathbf{D} = \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\int (n_{2} + \\cos{(\\mathbf{D})}) dn_{2} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('n_2', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Symbol('n_2', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\rho_b')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('n_2', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\rho_b')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Derivative(Integral(Add(Symbol('n_2', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(U)} = \\sin{(U)}, then obtain \\cos{(\\log{(\\sin{(U)})} - \\frac{d}{d U} \\operatorname{F_{x}}{(U)})} = \\cos{(\\log{(\\sin{(U)})} - \\frac{d}{d U} \\sin{(U)})}", "derivation": "\\operatorname{F_{x}}{(U)} = \\sin{(U)} and \\log{(\\operatorname{F_{x}}{(U)})} = \\log{(\\sin{(U)})} and \\frac{d}{d U} \\operatorname{F_{x}}{(U)} = \\frac{d}{d U} \\sin{(U)} and - \\log{(\\operatorname{F_{x}}{(U)})} + \\frac{d}{d U} \\operatorname{F_{x}}{(U)} = - \\log{(\\operatorname{F_{x}}{(U)})} + \\frac{d}{d U} \\sin{(U)} and - \\log{(\\sin{(U)})} + \\frac{d}{d U} \\operatorname{F_{x}}{(U)} = - \\log{(\\sin{(U)})} + \\frac{d}{d U} \\sin{(U)} and \\cos{(\\log{(\\sin{(U)})} - \\frac{d}{d U} \\operatorname{F_{x}}{(U)})} = \\cos{(\\log{(\\sin{(U)})} - \\frac{d}{d U} \\sin{(U)})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["log", 1], "Equality(log(Function('F_x')(Symbol('U', commutative=True))), log(sin(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 3, "log(Function('F_x')(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Function('F_x')(Symbol('U', commutative=True)))), Derivative(Function('F_x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Function('F_x')(Symbol('U', commutative=True)))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), log(sin(Symbol('U', commutative=True)))), Derivative(Function('F_x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(sin(Symbol('U', commutative=True)))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["cos", 5], "Equality(cos(Add(log(sin(Symbol('U', commutative=True))), Mul(Integer(-1), Derivative(Function('F_x')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))), cos(Add(log(sin(Symbol('U', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given i{(\\theta)} = \\log{(\\theta)} and \\mathbf{f}{(\\theta)} = - \\log{(\\theta)} + \\log{(\\theta)}^{\\theta}, then obtain \\mathbf{f}{(\\theta)} + \\log{(\\theta)}^{\\theta} = - \\log{(\\theta)} + 2 \\log{(\\theta)}^{\\theta}", "derivation": "i{(\\theta)} = \\log{(\\theta)} and i^{\\theta}{(\\theta)} = \\log{(\\theta)}^{\\theta} and \\mathbf{f}{(\\theta)} = - \\log{(\\theta)} + \\log{(\\theta)}^{\\theta} and \\mathbf{f}{(\\theta)} + i^{\\theta}{(\\theta)} = i^{\\theta}{(\\theta)} - \\log{(\\theta)} + \\log{(\\theta)}^{\\theta} and \\mathbf{f}{(\\theta)} + i^{\\theta}{(\\theta)} = 2 i^{\\theta}{(\\theta)} - \\log{(\\theta)} and \\mathbf{f}{(\\theta)} + \\log{(\\theta)}^{\\theta} = - \\log{(\\theta)} + 2 \\log{(\\theta)}^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["add", 3, "Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Add(Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(2), Pow(Function('i')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\theta', commutative=True)), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Pow(log(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(n)} = e^{n}, then obtain \\operatorname{F_{x}}{(n)} \\frac{d}{d n} \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}} = e^{n} \\frac{d}{d n} \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}}", "derivation": "\\operatorname{F_{x}}{(n)} = e^{n} and 1 = \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}} and \\frac{d}{d n} 1 = \\frac{d}{d n} \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}} and \\operatorname{F_{x}}{(n)} \\frac{d}{d n} 1 = e^{n} \\frac{d}{d n} 1 and \\operatorname{F_{x}}{(n)} \\frac{d}{d n} \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}} = e^{n} \\frac{d}{d n} \\frac{e^{n}}{\\operatorname{F_{x}}{(n)}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["divide", 1, "Function('F_x')(Symbol('n', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_x')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('F_x')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Function('F_x')(Symbol('n', commutative=True)), Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(exp(Symbol('n', commutative=True)), Derivative(Integer(1), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('F_x')(Symbol('n', commutative=True)), Derivative(Mul(Pow(Function('F_x')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(exp(Symbol('n', commutative=True)), Derivative(Mul(Pow(Function('F_x')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(\\hat{H},\\hat{x})} = \\hat{H} + \\hat{x} and \\mathbb{I}{(\\hat{H},\\hat{x})} = 2 \\varphi{(\\hat{H},\\hat{x})}, then obtain 2 \\hat{H} + 2 \\hat{x} = 2 \\varphi{(\\hat{H},\\hat{x})}", "derivation": "\\varphi{(\\hat{H},\\hat{x})} = \\hat{H} + \\hat{x} and \\mathbb{I}{(\\hat{H},\\hat{x})} = 2 \\varphi{(\\hat{H},\\hat{x})} and \\mathbb{I}{(\\hat{H},\\hat{x})} = 2 \\hat{H} + 2 \\hat{x} and 2 \\hat{H} + 2 \\hat{x} = 2 \\varphi{(\\hat{H},\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(2), Function('\\\\varphi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(x,\\mathbf{f})} = \\log{(x)}^{\\mathbf{f}} and J{(x,\\mathbf{f})} = \\log{(x)}^{\\mathbf{f}}, then obtain \\log{(x)}^{\\mathbf{f}} = \\frac{\\log{(x)}^{2 \\mathbf{f}}}{J{(x,\\mathbf{f})}}", "derivation": "\\phi_{1}{(x,\\mathbf{f})} = \\log{(x)}^{\\mathbf{f}} and J{(x,\\mathbf{f})} = \\log{(x)}^{\\mathbf{f}} and \\frac{J{(x,\\mathbf{f})} \\log{(x)}^{\\mathbf{f}}}{x} = \\frac{\\log{(x)}^{2 \\mathbf{f}}}{x} and \\frac{J{(x,\\mathbf{f})} \\log{(x)}^{\\mathbf{f}}}{\\phi_{1}{(x,\\mathbf{f})}} = \\frac{\\log{(x)}^{2 \\mathbf{f}}}{\\phi_{1}{(x,\\mathbf{f})}} and J{(x,\\mathbf{f})} = \\phi_{1}{(x,\\mathbf{f})} and \\log{(x)}^{\\mathbf{f}} = \\frac{\\log{(x)}^{2 \\mathbf{f}}}{J{(x,\\mathbf{f})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('J')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(log(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Function('J')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Function('\\\\phi_1')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Pow(log(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('J')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\phi_1')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(log(Symbol('x', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Function('J')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Pow(log(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\varphi)} = \\cos{(\\varphi)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\varphi)} = \\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}}, then obtain (\\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}})^{\\varphi} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\varphi}{(\\varphi)} = 1", "derivation": "\\operatorname{f^{*}}{(\\varphi)} = \\cos{(\\varphi)} and \\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}} = 1 and (\\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}})^{\\varphi} = 1 and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\varphi)} = \\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\varphi)} = 1 and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\varphi}{(\\varphi)} = 1 and (\\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}})^{\\varphi} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\varphi}{(\\varphi)} = (\\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}})^{\\varphi} and (\\frac{\\operatorname{f^{*}}{(\\varphi)}}{\\cos{(\\varphi)}})^{\\varphi} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\varphi}{(\\varphi)} = 1", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True)), Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True)), Integer(1))"], [["power", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Integer(1))"], [["times", 6, "Pow(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Pow(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Mul(Pow(Mul(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{x},\\ddot{x})} = \\ddot{x} + \\hat{x}, then obtain \\frac{\\partial}{\\partial \\hat{x}} (2 \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}} = \\frac{\\partial}{\\partial \\hat{x}} (\\ddot{x} + \\hat{x} + \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}}", "derivation": "\\mathbf{p}{(\\hat{x},\\ddot{x})} = \\ddot{x} + \\hat{x} and 2 \\mathbf{p}{(\\hat{x},\\ddot{x})} = \\ddot{x} + \\hat{x} + \\mathbf{p}{(\\hat{x},\\ddot{x})} and (2 \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}} = (\\ddot{x} + \\hat{x} + \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}} and \\frac{\\partial}{\\partial \\hat{x}} (2 \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}} = \\frac{\\partial}{\\partial \\hat{x}} (\\ddot{x} + \\hat{x} + \\mathbf{p}{(\\hat{x},\\ddot{x})})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(f_{E})} = \\log{(f_{E})}, then obtain (0^{f_{E}})^{f_{E}} - 2 \\log{(f_{E})} = 1 - 2 \\log{(f_{E})}", "derivation": "U{(f_{E})} = \\log{(f_{E})} and 0 = - U{(f_{E})} + \\log{(f_{E})} and 0 = U{(f_{E})} - \\log{(f_{E})} and 0^{f_{E}} = (U{(f_{E})} - \\log{(f_{E})})^{f_{E}} and (0^{f_{E}})^{f_{E}} = ((U{(f_{E})} - \\log{(f_{E})})^{f_{E}})^{f_{E}} and ((U{(f_{E})} - \\log{(f_{E})})^{f_{E}})^{f_{E}} = 1 and (0^{f_{E}})^{f_{E}} = 1 and (0^{f_{E}})^{f_{E}} - 2 U{(f_{E})} = 1 - 2 U{(f_{E})} and (0^{f_{E}})^{f_{E}} - 2 \\log{(f_{E})} = 1 - 2 \\log{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["minus", 1, "Function('U')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('U')(Symbol('f_E', commutative=True))), log(Symbol('f_E', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('U')(Symbol('f_E', commutative=True)), Mul(Integer(-1), log(Symbol('f_E', commutative=True)))))"], [["power", 3, "Symbol('f_E', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f_E', commutative=True)), Pow(Add(Function('U')(Symbol('f_E', commutative=True)), Mul(Integer(-1), log(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(Pow(Add(Function('U')(Symbol('f_E', commutative=True)), Mul(Integer(-1), log(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Add(Function('U')(Symbol('f_E', commutative=True)), Mul(Integer(-1), log(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Pow(Integer(0), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Integer(1))"], [["add", 7, "Mul(Integer(-1), Integer(2), Function('U')(Symbol('f_E', commutative=True)))"], "Equality(Add(Pow(Pow(Integer(0), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Integer(2), Function('U')(Symbol('f_E', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Integer(2), Function('U')(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Add(Pow(Pow(Integer(0), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('f_E', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Integer(2), log(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given v{(\\rho)} = e^{\\rho}, then obtain \\mathbf{E} + \\rho = \\int (v{(\\rho)} e^{- \\rho})^{\\rho} d\\rho", "derivation": "v{(\\rho)} = e^{\\rho} and 1 = \\frac{e^{\\rho}}{v{(\\rho)}} and e^{\\rho} = \\frac{e^{2 \\rho}}{v{(\\rho)}} and v{(\\rho)} = \\frac{e^{2 \\rho}}{v{(\\rho)}} and 1 = (\\frac{e^{\\rho}}{v{(\\rho)}})^{\\rho} and \\int 1 d\\rho = \\int (\\frac{e^{\\rho}}{v{(\\rho)}})^{\\rho} d\\rho and \\int 1 d\\rho = \\int (v{(\\rho)} e^{- \\rho})^{\\rho} d\\rho and \\mathbf{E} + \\rho = \\int (v{(\\rho)} e^{- \\rho})^{\\rho} d\\rho", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Function('v')(Symbol('\\\\rho', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\rho', commutative=True))"], "Equality(exp(Symbol('\\\\rho', commutative=True)), Mul(Pow(Function('v')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('v')(Symbol('\\\\rho', commutative=True)), Mul(Pow(Function('v')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\rho', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('v')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["integrate", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Pow(Mul(Pow(Function('v')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Pow(Mul(Function('v')(Symbol('\\\\rho', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Pow(Mul(Function('v')(Symbol('\\\\rho', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\theta)} = \\sin{(\\theta)}, then obtain \\cos{(\\int \\frac{(e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}})^{\\theta}}{\\theta} d\\theta)} = \\cos{(\\int \\frac{1}{\\theta} d\\theta)}", "derivation": "\\operatorname{t_{1}}{(\\theta)} = \\sin{(\\theta)} and \\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)} = 0 and e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}} = 1 and (e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}})^{\\theta} = 1 and \\frac{(e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}})^{\\theta}}{\\theta} = \\frac{1}{\\theta} and \\int \\frac{(e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}})^{\\theta}}{\\theta} d\\theta = \\int \\frac{1}{\\theta} d\\theta and \\cos{(\\int \\frac{(e^{\\operatorname{t_{1}}{(\\theta)} - \\sin{(\\theta)}})^{\\theta}}{\\theta} d\\theta)} = \\cos{(\\int \\frac{1}{\\theta} d\\theta)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True))))), Integer(1))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(exp(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True)), Integer(1))"], [["divide", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(exp(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True))), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(exp(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["cos", 6], "Equality(cos(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(exp(Add(Function('t_1')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), cos(Integral(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given k{(S)} = \\log{(S)}, then derive \\frac{d^{2}}{d S^{2}} k{(S)} = - \\frac{1}{S^{2}}, then obtain \\frac{d}{d S} (- S + \\frac{d^{2}}{d S^{2}} k{(S)}) = \\frac{d}{d S} (- S - \\frac{1}{S^{2}})", "derivation": "k{(S)} = \\log{(S)} and \\frac{d}{d S} k{(S)} = \\frac{d}{d S} \\log{(S)} and \\frac{d^{2}}{d S^{2}} k{(S)} = \\frac{d^{2}}{d S^{2}} \\log{(S)} and - S + \\frac{d^{2}}{d S^{2}} k{(S)} = - S + \\frac{d^{2}}{d S^{2}} \\log{(S)} and \\frac{d^{2}}{d S^{2}} k{(S)} = - \\frac{1}{S^{2}} and \\frac{d^{2}}{d S^{2}} \\log{(S)} = - \\frac{1}{S^{2}} and - S + \\frac{d^{2}}{d S^{2}} k{(S)} = - S - \\frac{1}{S^{2}} and \\frac{d}{d S} (- S + \\frac{d^{2}}{d S^{2}} k{(S)}) = \\frac{d}{d S} (- S - \\frac{1}{S^{2}})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))))"], [["minus", 3, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)))))"], [["differentiate", 7, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Derivative(Function('k')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(2)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-2)))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\varepsilon)} = \\log{(\\varepsilon)}, then derive \\int C{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + c, then obtain \\int \\log{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + c", "derivation": "C{(\\varepsilon)} = \\log{(\\varepsilon)} and \\int C{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\varepsilon)} d\\varepsilon and \\int C{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + c and \\int \\log{(\\varepsilon)} d\\varepsilon = \\varepsilon \\log{(\\varepsilon)} - \\varepsilon + c", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})} = \\mathbf{D} + v_{x}, then obtain \\int \\frac{\\partial}{\\partial v_{x}} \\frac{- \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})}}{\\mathbf{D}} dv_{x} = \\int \\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{\\mathbf{D}} dv_{x}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})} = \\mathbf{D} + v_{x} and - \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})} = v_{x} and \\frac{- \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})}}{\\mathbf{D}} = \\frac{v_{x}}{\\mathbf{D}} and \\frac{\\partial}{\\partial v_{x}} \\frac{- \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})}}{\\mathbf{D}} = \\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{\\mathbf{D}} and \\int \\frac{\\partial}{\\partial v_{x}} \\frac{- \\mathbf{D} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{x},\\mathbf{D})}}{\\mathbf{D}} dv_{x} = \\int \\frac{\\partial}{\\partial v_{x}} \\frac{v_{x}}{\\mathbf{D}} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_x', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('v_x', commutative=True))"], [["divide", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\chi)} = \\log{(\\chi)}, then obtain (\\hat{p}^{\\chi}{(\\chi)})^{- \\chi} \\hat{p}{(\\chi)} = (\\hat{p}^{\\chi}{(\\chi)})^{- \\chi} \\log{(\\chi)}", "derivation": "\\hat{p}{(\\chi)} = \\log{(\\chi)} and \\hat{p}^{\\chi}{(\\chi)} = \\log{(\\chi)}^{\\chi} and (\\log{(\\chi)}^{\\chi})^{- \\chi} \\hat{p}{(\\chi)} = (\\log{(\\chi)}^{\\chi})^{- \\chi} \\log{(\\chi)} and (\\hat{p}^{\\chi}{(\\chi)})^{- \\chi} \\hat{p}{(\\chi)} = (\\hat{p}^{\\chi}{(\\chi)})^{- \\chi} \\log{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Pow(Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Pow(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Pow(Function('\\\\hat{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), log(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\theta_1,\\Omega)} = \\sin{(\\Omega^{\\theta_1})} and \\Psi^{\\dagger}{(\\theta_1,\\Omega)} = \\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})}, then obtain (\\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})})^{\\Omega} = \\Omega^{\\Omega}", "derivation": "\\rho_{f}{(\\theta_1,\\Omega)} = \\sin{(\\Omega^{\\theta_1})} and \\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})} = 0 and \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})} = 1 and \\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})} = \\Omega and \\Psi^{\\dagger}{(\\theta_1,\\Omega)} = \\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})} and \\Psi^{\\dagger}^{\\Omega}{(\\theta_1,\\Omega)} = (\\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})})^{\\Omega} and \\Psi^{\\dagger}^{\\Omega}{(\\theta_1,\\Omega)} = \\Omega^{\\Omega} and (\\Omega \\cos{(\\rho_{f}{(\\theta_1,\\Omega)} - \\sin{(\\Omega^{\\theta_1})})})^{\\Omega} = \\Omega^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))), Integer(0))"], [["cos", 2], "Equality(cos(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))))), Integer(1))"], [["times", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), cos(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))))), Symbol('\\\\Omega', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), cos(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), cos(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Mul(Symbol('\\\\Omega', commutative=True), cos(Add(Function('\\\\rho_f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta_1', commutative=True))))))), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given f{(\\chi,v_{1})} = \\int \\chi^{v_{1}} d\\chi, then obtain (\\chi f{(\\chi,v_{1})} \\int \\chi^{v_{1}} d\\chi)^{\\chi} = (\\chi (\\int \\chi^{v_{1}} d\\chi)^{2})^{\\chi}", "derivation": "f{(\\chi,v_{1})} = \\int \\chi^{v_{1}} d\\chi and \\chi f{(\\chi,v_{1})} = \\chi \\int \\chi^{v_{1}} d\\chi and \\chi f{(\\chi,v_{1})} \\int \\chi^{v_{1}} d\\chi = \\chi (\\int \\chi^{v_{1}} d\\chi)^{2} and (\\chi f{(\\chi,v_{1})} \\int \\chi^{v_{1}} d\\chi)^{\\chi} = (\\chi (\\int \\chi^{v_{1}} d\\chi)^{2})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('f')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["times", 2, "Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('f')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), Pow(Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\chi', commutative=True), Function('f')(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Pow(Integral(Pow(Symbol('\\\\chi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{J}_P)} = \\cos{(\\sin{(\\mathbf{J}_P)})} and \\mathbf{p}{(\\mathbf{J}_P)} = \\cos{(\\sin{(\\mathbf{J}_P)})}, then obtain \\frac{\\int \\frac{\\mathbf{p}{(\\mathbf{J}_P)}}{\\cos{(\\sin{(\\mathbf{J}_P)})}} d\\mathbf{J}_P}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}} = \\frac{\\int 1 d\\mathbf{J}_P}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{J}_P)} = \\cos{(\\sin{(\\mathbf{J}_P)})} and \\mathbf{p}{(\\mathbf{J}_P)} = \\cos{(\\sin{(\\mathbf{J}_P)})} and \\frac{\\mathbf{p}{(\\mathbf{J}_P)}}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}} = \\frac{\\cos{(\\sin{(\\mathbf{J}_P)})}}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}} and \\frac{\\mathbf{p}{(\\mathbf{J}_P)}}{\\cos{(\\sin{(\\mathbf{J}_P)})}} = 1 and \\int \\frac{\\mathbf{p}{(\\mathbf{J}_P)}}{\\cos{(\\sin{(\\mathbf{J}_P)})}} d\\mathbf{J}_P = \\int 1 d\\mathbf{J}_P and \\frac{\\int \\frac{\\mathbf{p}{(\\mathbf{J}_P)}}{\\cos{(\\sin{(\\mathbf{J}_P)})}} d\\mathbf{J}_P}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}} = \\frac{\\int 1 d\\mathbf{J}_P}{\\operatorname{v_{1}}{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 2, "Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))), Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 5, "Pow(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(cos(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Pow(Function('v_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} = \\frac{\\hat{X}}{\\hat{\\mathbf{x}}} and b{(\\psi,x^\\prime)} = \\psi x^\\prime, then obtain b{(\\psi,x^\\prime)} + \\int \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} d\\hat{X} = \\psi x^\\prime + \\int \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} d\\hat{X}", "derivation": "\\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} = \\frac{\\hat{X}}{\\hat{\\mathbf{x}}} and b{(\\psi,x^\\prime)} = \\psi x^\\prime and \\int \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} d\\hat{X} = \\int \\frac{\\hat{X}}{\\hat{\\mathbf{x}}} d\\hat{X} and b{(\\psi,x^\\prime)} + \\int \\frac{\\hat{X}}{\\hat{\\mathbf{x}}} d\\hat{X} = \\psi x^\\prime + \\int \\frac{\\hat{X}}{\\hat{\\mathbf{x}}} d\\hat{X} and b{(\\psi,x^\\prime)} + \\int \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} d\\hat{X} = \\psi x^\\prime + \\int \\operatorname{P_{g}}{(\\hat{\\mathbf{x}},\\hat{X})} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('b')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["add", 2, "Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Function('b')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('b')(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Function('P_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Function('P_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\mu_0)} = e^{e^{\\mu_0}}, then obtain \\psi^{\\mu_0}{(\\mu_0)} e^{- e^{\\mu_0}} - \\int \\psi{(\\mu_0)} d\\mu_0 = - \\int \\psi{(\\mu_0)} d\\mu_0 + e^{- e^{\\mu_0}} (e^{e^{\\mu_0}})^{\\mu_0}", "derivation": "\\psi{(\\mu_0)} = e^{e^{\\mu_0}} and \\int \\psi{(\\mu_0)} d\\mu_0 = \\int e^{e^{\\mu_0}} d\\mu_0 and \\psi^{\\mu_0}{(\\mu_0)} = (e^{e^{\\mu_0}})^{\\mu_0} and \\psi^{\\mu_0}{(\\mu_0)} e^{- e^{\\mu_0}} = e^{- e^{\\mu_0}} (e^{e^{\\mu_0}})^{\\mu_0} and \\psi^{\\mu_0}{(\\mu_0)} e^{- e^{\\mu_0}} - \\int e^{e^{\\mu_0}} d\\mu_0 = - \\int e^{e^{\\mu_0}} d\\mu_0 + e^{- e^{\\mu_0}} (e^{e^{\\mu_0}})^{\\mu_0} and \\psi^{\\mu_0}{(\\mu_0)} e^{- e^{\\mu_0}} - \\int \\psi{(\\mu_0)} d\\mu_0 = - \\int \\psi{(\\mu_0)} d\\mu_0 + e^{- e^{\\mu_0}} (e^{e^{\\mu_0}})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), exp(exp(Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(exp(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(exp(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 3, "exp(exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))))), Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(exp(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"], [["minus", 4, "Integral(exp(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))))), Mul(Integer(-1), Integral(exp(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))), Add(Mul(Integer(-1), Integral(exp(exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(exp(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Pow(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))))), Mul(Integer(-1), Integral(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Add(Mul(Integer(-1), Integral(Function('\\\\psi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(exp(exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)} = x^\\prime \\cos{(\\mathbf{E})}, then obtain \\int \\frac{\\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)}}{x^\\prime \\cos{(\\mathbf{E})}} d\\mathbf{E} = \\mathbf{E} + r", "derivation": "\\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)} = x^\\prime \\cos{(\\mathbf{E})} and \\frac{\\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)}}{x^\\prime \\cos{(\\mathbf{E})}} = 1 and \\int \\frac{\\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)}}{x^\\prime \\cos{(\\mathbf{E})}} d\\mathbf{E} = \\int 1 d\\mathbf{E} and \\int \\frac{\\operatorname{A_{x}}{(\\mathbf{E},x^\\prime)}}{x^\\prime \\cos{(\\mathbf{E})}} d\\mathbf{E} = \\mathbf{E} + r", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["divide", 1, "Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{f})} = \\mathbf{f}, then obtain (((\\mathbf{v}^{\\mathbf{f}}{(\\mathbf{f})})^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}} = (((\\mathbf{f}^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}}", "derivation": "\\mathbf{v}{(\\mathbf{f})} = \\mathbf{f} and \\mathbf{v}^{\\mathbf{f}}{(\\mathbf{f})} = \\mathbf{f}^{\\mathbf{f}} and (\\mathbf{v}^{\\mathbf{f}}{(\\mathbf{f})})^{\\mathbf{f}} = (\\mathbf{f}^{\\mathbf{f}})^{\\mathbf{f}} and ((\\mathbf{v}^{\\mathbf{f}}{(\\mathbf{f})})^{\\mathbf{f}})^{\\mathbf{f}} = ((\\mathbf{f}^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}} and (((\\mathbf{v}^{\\mathbf{f}}{(\\mathbf{f})})^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}} = (((\\mathbf{f}^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}})^{\\mathbf{f}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Pow(Pow(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Pow(Pow(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(m)} = \\log{(m)}, then obtain \\int \\mathbf{J}_M{(m)} dm - \\int \\mathbf{J}_M^{m}{(m)} dm = - \\int \\mathbf{J}_M^{m}{(m)} dm + \\int \\log{(m)} dm", "derivation": "\\mathbf{J}_M{(m)} = \\log{(m)} and \\mathbf{J}_M^{m}{(m)} = \\log{(m)}^{m} and \\int \\mathbf{J}_M^{m}{(m)} dm = \\int \\log{(m)}^{m} dm and \\int \\mathbf{J}_M{(m)} dm = \\int \\log{(m)} dm and \\int \\mathbf{J}_M{(m)} dm - \\int \\log{(m)}^{m} dm = \\int \\log{(m)} dm - \\int \\log{(m)}^{m} dm and \\int \\mathbf{J}_M{(m)} dm - \\int \\mathbf{J}_M^{m}{(m)} dm = - \\int \\mathbf{J}_M^{m}{(m)} dm + \\int \\log{(m)} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["minus", 4, "Integral(Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given M{(\\psi^*,G)} = G - \\psi^*, then obtain \\frac{- G + \\psi^* - M{(\\psi^*,G)}}{G} = \\frac{- 2 G + 2 \\psi^*}{G}", "derivation": "M{(\\psi^*,G)} = G - \\psi^* and - M{(\\psi^*,G)} = - G + \\psi^* and - G + \\psi^* - M{(\\psi^*,G)} = - 2 G + 2 \\psi^* and \\frac{- G + \\psi^* - M{(\\psi^*,G)}}{G} = \\frac{- 2 G + 2 \\psi^*}{G}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\psi^*', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('M')(Symbol('\\\\psi^*', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\psi^*', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["divide", 3, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\psi^*', commutative=True), Symbol('G', commutative=True))))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(H)} = \\sin{(\\sin{(H)})}, then obtain 0^{H} - \\int \\cos{(\\sin{(\\sin{(H)})})} dH = ((- \\operatorname{L_{\\varepsilon}}{(H)} + \\sin{(\\sin{(H)})}) \\operatorname{L_{\\varepsilon}}{(H)})^{H} - \\int \\cos{(\\sin{(\\sin{(H)})})} dH", "derivation": "\\operatorname{L_{\\varepsilon}}{(H)} = \\sin{(\\sin{(H)})} and 0 = - \\operatorname{L_{\\varepsilon}}{(H)} + \\sin{(\\sin{(H)})} and 0 = (- \\operatorname{L_{\\varepsilon}}{(H)} + \\sin{(\\sin{(H)})}) \\operatorname{L_{\\varepsilon}}{(H)} and 0^{H} = ((- \\operatorname{L_{\\varepsilon}}{(H)} + \\sin{(\\sin{(H)})}) \\operatorname{L_{\\varepsilon}}{(H)})^{H} and 0^{H} - \\int \\cos{(\\sin{(\\sin{(H)})})} dH = ((- \\operatorname{L_{\\varepsilon}}{(H)} + \\sin{(\\sin{(H)})}) \\operatorname{L_{\\varepsilon}}{(H)})^{H} - \\int \\cos{(\\sin{(\\sin{(H)})})} dH", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), sin(sin(Symbol('H', commutative=True)))))"], [["times", 2, "Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), sin(sin(Symbol('H', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Integer(0), Symbol('H', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), sin(sin(Symbol('H', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["minus", 4, "Integral(cos(sin(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Pow(Integer(0), Symbol('H', commutative=True)), Mul(Integer(-1), Integral(cos(sin(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))), Add(Pow(Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), sin(sin(Symbol('H', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Mul(Integer(-1), Integral(cos(sin(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given a{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\varphi^{*}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})}, then obtain \\varphi^{*}{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})} = (\\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})})^{2}", "derivation": "a{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\varphi^{*}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})} and \\varphi^{*}{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})} and \\varphi^{*}{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})} = (\\frac{d}{d \\mathbf{S}} a{(\\mathbf{S})})^{2}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Derivative(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Pow(Derivative(Function('a')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given r{(F_{c},t)} = \\int \\frac{F_{c}}{t} dt, then obtain - \\frac{F_{c}}{t} + r{(F_{c},t)} + \\int \\frac{F_{c}}{t} dt = - \\frac{F_{c}}{t} + 2 \\int \\frac{F_{c}}{t} dt", "derivation": "r{(F_{c},t)} = \\int \\frac{F_{c}}{t} dt and - \\frac{F_{c}}{t} + r{(F_{c},t)} = - \\frac{F_{c}}{t} + \\int \\frac{F_{c}}{t} dt and - \\frac{F_{c}}{t} + 2 r{(F_{c},t)} = - \\frac{F_{c}}{t} + r{(F_{c},t)} + \\int \\frac{F_{c}}{t} dt and - \\frac{F_{c}}{t} + 2 r{(F_{c},t)} = - \\frac{F_{c}}{t} + 2 \\int \\frac{F_{c}}{t} dt and - \\frac{F_{c}}{t} + r{(F_{c},t)} + \\int \\frac{F_{c}}{t} dt = - \\frac{F_{c}}{t} + 2 \\int \\frac{F_{c}}{t} dt", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True))))"], [["minus", 1, "Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True)))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(2), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(2), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(2), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('r')(Symbol('F_c', commutative=True), Symbol('t', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Mul(Integer(2), Integral(Mul(Symbol('F_c', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Tuple(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\operatorname{V_{\\mathbf{E}}}{(a^{\\dagger})} = \\int 1 da^{\\dagger}, then obtain \\int \\operatorname{V_{\\mathbf{E}}}{(a^{\\dagger})} da^{\\dagger} = \\iint \\operatorname{P_{g}}^{- a^{\\dagger}}{(a^{\\dagger})} \\sin^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} da^{\\dagger}", "derivation": "\\operatorname{P_{g}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\operatorname{P_{g}}^{a^{\\dagger}}{(a^{\\dagger})} = \\sin^{a^{\\dagger}}{(a^{\\dagger})} and 1 = \\operatorname{P_{g}}^{- a^{\\dagger}}{(a^{\\dagger})} \\sin^{a^{\\dagger}}{(a^{\\dagger})} and \\int 1 da^{\\dagger} = \\int \\operatorname{P_{g}}^{- a^{\\dagger}}{(a^{\\dagger})} \\sin^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} and \\operatorname{V_{\\mathbf{E}}}{(a^{\\dagger})} = \\int 1 da^{\\dagger} and \\operatorname{V_{\\mathbf{E}}}{(a^{\\dagger})} = \\int \\operatorname{P_{g}}^{- a^{\\dagger}}{(a^{\\dagger})} \\sin^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} and \\int \\operatorname{V_{\\mathbf{E}}}{(a^{\\dagger})} da^{\\dagger} = \\iint \\operatorname{P_{g}}^{- a^{\\dagger}}{(a^{\\dagger})} \\sin^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 2, "Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Mul(Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 6, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(x^\\prime,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime), then obtain x^\\prime + (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\operatorname{A_{x}}{(x^\\prime,\\theta)}}{\\theta})^{x^\\prime} = x^\\prime + (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime)}{\\theta})^{x^\\prime}", "derivation": "\\operatorname{A_{x}}{(x^\\prime,\\theta)} = \\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime) and \\frac{\\operatorname{A_{x}}{(x^\\prime,\\theta)}}{\\theta} = \\frac{\\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime)}{\\theta} and \\frac{\\partial}{\\partial x^\\prime} \\frac{\\operatorname{A_{x}}{(x^\\prime,\\theta)}}{\\theta} = \\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime)}{\\theta} and (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\operatorname{A_{x}}{(x^\\prime,\\theta)}}{\\theta})^{x^\\prime} = (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime)}{\\theta})^{x^\\prime} and x^\\prime + (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\operatorname{A_{x}}{(x^\\prime,\\theta)}}{\\theta})^{x^\\prime} = x^\\prime + (\\frac{\\partial}{\\partial x^\\prime} \\frac{\\frac{\\partial}{\\partial \\theta} (\\theta - x^\\prime)}{\\theta})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Pow(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Function('A_x')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Pow(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given H{(y^{\\prime},u)} = \\cos{(\\frac{u}{y^{\\prime}})} and p{(y^{\\prime},u)} = - \\cos{(\\frac{u}{y^{\\prime}})}, then obtain \\frac{p{(y^{\\prime},u)}}{u} = - \\frac{H{(y^{\\prime},u)}}{u}", "derivation": "H{(y^{\\prime},u)} = \\cos{(\\frac{u}{y^{\\prime}})} and p{(y^{\\prime},u)} = - \\cos{(\\frac{u}{y^{\\prime}})} and p{(y^{\\prime},u)} = - H{(y^{\\prime},u)} and \\frac{p{(y^{\\prime},u)}}{u} = - \\frac{H{(y^{\\prime},u)}}{u}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('u', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True))))"], [["divide", 3, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Function('H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{p},E_{x})} = E_{x} \\hat{p}, then obtain \\frac{d}{d E_{x}} 0 = \\frac{\\partial}{\\partial E_{x}} - \\frac{\\hat{p} (E_{x} - \\frac{\\tilde{g}^*{(\\hat{p},E_{x})}}{\\hat{p}})}{\\tilde{g}^*{(\\hat{p},E_{x})}}", "derivation": "\\tilde{g}^*{(\\hat{p},E_{x})} = E_{x} \\hat{p} and \\frac{\\tilde{g}^*{(\\hat{p},E_{x})}}{\\hat{p}} = E_{x} and 0 = E_{x} - \\frac{\\tilde{g}^*{(\\hat{p},E_{x})}}{\\hat{p}} and 0 = - \\frac{\\hat{p} (E_{x} - \\frac{\\tilde{g}^*{(\\hat{p},E_{x})}}{\\hat{p}})}{\\tilde{g}^*{(\\hat{p},E_{x})}} and \\frac{d}{d E_{x}} 0 = \\frac{\\partial}{\\partial E_{x}} - \\frac{\\hat{p} (E_{x} - \\frac{\\tilde{g}^*{(\\hat{p},E_{x})}}{\\hat{p}})}{\\tilde{g}^*{(\\hat{p},E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))"], [["minus", 2, "Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\chi{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then derive \\eta^{\\prime} + n = \\int \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} d\\eta^{\\prime}, then obtain \\int (\\eta^{\\prime} + n) dn = \\iint \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} d\\eta^{\\prime} dn", "derivation": "\\chi{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and 1 = \\frac{\\sin{(\\eta^{\\prime})}}{\\chi{(\\eta^{\\prime})}} and \\frac{\\chi^{2}{(\\eta^{\\prime})}}{\\sin{(\\eta^{\\prime})}} = \\chi{(\\eta^{\\prime})} and 1 = \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} and \\int 1 d\\eta^{\\prime} = \\int \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} d\\eta^{\\prime} and \\eta^{\\prime} + n = \\int \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} d\\eta^{\\prime} and \\int (\\eta^{\\prime} + n) dn = \\iint \\frac{\\sin^{2}{(\\eta^{\\prime})}}{\\chi^{2}{(\\eta^{\\prime})}} d\\eta^{\\prime} dn", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 1, "Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))), Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["integrate", 6, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Pow(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(v_{2})} = \\sin{(v_{2})}, then obtain v_{1} + \\operatorname{n_{2}}{(v_{2})} = l + \\sin{(v_{2})}", "derivation": "\\operatorname{n_{2}}{(v_{2})} = \\sin{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{n_{2}}{(v_{2})} = \\frac{d}{d v_{2}} \\sin{(v_{2})} and \\int \\frac{d}{d v_{2}} \\operatorname{n_{2}}{(v_{2})} dv_{2} = \\int \\frac{d}{d v_{2}} \\sin{(v_{2})} dv_{2} and \\frac{d}{d v_{2}} \\int \\frac{d}{d v_{2}} \\operatorname{n_{2}}{(v_{2})} dv_{2} = \\frac{d}{d v_{2}} \\int \\frac{d}{d v_{2}} \\sin{(v_{2})} dv_{2} and \\int \\frac{d}{d v_{2}} \\int \\frac{d}{d v_{2}} \\operatorname{n_{2}}{(v_{2})} dv_{2} dv_{2} = \\int \\frac{d}{d v_{2}} \\int \\frac{d}{d v_{2}} \\sin{(v_{2})} dv_{2} dv_{2} and v_{1} + \\operatorname{n_{2}}{(v_{2})} = l + \\sin{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('v_2', commutative=True)"], "Equality(Integral(Derivative(Function('n_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Integral(Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('n_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Derivative(Integral(Derivative(Function('n_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Integral(Derivative(Integral(Derivative(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('v_1', commutative=True), Function('n_2')(Symbol('v_2', commutative=True))), Add(Symbol('l', commutative=True), sin(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given x{(\\psi,\\lambda,m)} = - \\lambda - \\psi + m and E{(\\psi)} = - \\psi, then obtain E^{\\psi}{(\\psi)} \\int \\lambda d\\lambda = (- \\psi)^{\\psi} \\int \\lambda d\\lambda", "derivation": "x{(\\psi,\\lambda,m)} = - \\lambda - \\psi + m and 1 = \\frac{- \\lambda - \\psi + m}{x{(\\psi,\\lambda,m)}} and E{(\\psi)} = - \\psi and \\lambda = \\frac{\\lambda (- \\lambda - \\psi + m)}{x{(\\psi,\\lambda,m)}} and \\int \\lambda d\\lambda = \\int \\frac{\\lambda (- \\lambda - \\psi + m)}{x{(\\psi,\\lambda,m)}} d\\lambda and E^{\\psi}{(\\psi)} = (- \\psi)^{\\psi} and E^{\\psi}{(\\psi)} \\int \\frac{\\lambda (- \\lambda - \\psi + m)}{x{(\\psi,\\lambda,m)}} d\\lambda = (- \\psi)^{\\psi} \\int \\frac{\\lambda (- \\lambda - \\psi + m)}{x{(\\psi,\\lambda,m)}} d\\lambda and E^{\\psi}{(\\psi)} \\int \\lambda d\\lambda = (- \\psi)^{\\psi} \\int \\lambda d\\lambda", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)))"], [["divide", 1, "Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))"], [["times", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["times", 6, "Integral(Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Function('E')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True)), Pow(Function('x')(Symbol('\\\\psi', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Pow(Function('E')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(v)} = \\log{(v)}, then obtain 3 \\frac{d}{d v} \\operatorname{P_{e}}{(v)} = \\frac{d}{d v} \\operatorname{P_{e}}{(v)} + 2 \\frac{d}{d v} \\log{(v)}", "derivation": "\\operatorname{P_{e}}{(v)} = \\log{(v)} and \\frac{d}{d v} \\operatorname{P_{e}}{(v)} = \\frac{d}{d v} \\log{(v)} and 2 \\frac{d}{d v} \\operatorname{P_{e}}{(v)} = \\frac{d}{d v} \\operatorname{P_{e}}{(v)} + \\frac{d}{d v} \\log{(v)} and 3 \\frac{d}{d v} \\operatorname{P_{e}}{(v)} = 2 \\frac{d}{d v} \\operatorname{P_{e}}{(v)} + \\frac{d}{d v} \\log{(v)} and 3 \\frac{d}{d v} \\operatorname{P_{e}}{(v)} = \\frac{d}{d v} \\operatorname{P_{e}}{(v)} + 2 \\frac{d}{d v} \\log{(v)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["add", 2, "Mul(Integer(2), Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], "Equality(Mul(Integer(3), Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Mul(Integer(2), Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(3), Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Derivative(Function('P_e')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(2), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))))"]]}, {"prompt": "Given l{(v_{t})} = e^{v_{t}}, then obtain \\int 0 dv_{t} + \\int (l{(v_{t})} e^{- v_{t}} - 1) dv_{t} = 2 \\int 0 dv_{t}", "derivation": "l{(v_{t})} = e^{v_{t}} and l{(v_{t})} e^{- v_{t}} = 1 and l{(v_{t})} e^{- v_{t}} - 1 = 0 and \\int (l{(v_{t})} e^{- v_{t}} - 1) dv_{t} = \\int 0 dv_{t} and \\int 0 dv_{t} + \\int (l{(v_{t})} e^{- v_{t}} - 1) dv_{t} = 2 \\int 0 dv_{t}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["divide", 1, "exp(Symbol('v_t', commutative=True))"], "Equality(Mul(Function('l')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Mul(Function('l')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(-1)), Integer(0))"], [["integrate", 3, "Symbol('v_t', commutative=True)"], "Equality(Integral(Add(Mul(Function('l')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(-1)), Tuple(Symbol('v_t', commutative=True))), Integral(Integer(0), Tuple(Symbol('v_t', commutative=True))))"], [["add", 4, "Integral(Integer(0), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('v_t', commutative=True))), Integral(Add(Mul(Function('l')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Integer(-1)), Tuple(Symbol('v_t', commutative=True)))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\pi,t)} = e^{\\frac{\\pi}{t}}, then derive \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = \\frac{e^{\\frac{\\pi}{t}}}{t}, then obtain - \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = - \\frac{e^{\\frac{\\pi}{t}}}{t}", "derivation": "\\operatorname{A_{z}}{(\\pi,t)} = e^{\\frac{\\pi}{t}} and \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = \\frac{\\partial}{\\partial \\pi} e^{\\frac{\\pi}{t}} and \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = \\frac{e^{\\frac{\\pi}{t}}}{t} and \\frac{\\partial}{\\partial \\pi} e^{\\frac{\\pi}{t}} = \\frac{e^{\\frac{\\pi}{t}}}{t} and - \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = - \\frac{\\partial}{\\partial \\pi} e^{\\frac{\\pi}{t}} and - \\frac{\\partial}{\\partial \\pi} \\operatorname{A_{z}}{(\\pi,t)} = - \\frac{e^{\\frac{\\pi}{t}}}{t}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\theta{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\cos{(\\Psi_{nl})}, then derive \\frac{d}{d \\Psi_{nl}} \\theta{(\\Psi_{nl})} = - \\cos{(\\Psi_{nl})}, then obtain - \\cos{(\\Psi_{nl})} = \\frac{d^{2}}{d \\Psi_{nl}^{2}} \\cos{(\\Psi_{nl})}", "derivation": "\\theta{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\cos{(\\Psi_{nl})} and \\frac{d}{d \\Psi_{nl}} \\theta{(\\Psi_{nl})} = \\frac{d^{2}}{d \\Psi_{nl}^{2}} \\cos{(\\Psi_{nl})} and \\frac{d}{d \\Psi_{nl}} \\theta{(\\Psi_{nl})} = - \\cos{(\\Psi_{nl})} and - \\cos{(\\Psi_{nl})} = \\frac{d^{2}}{d \\Psi_{nl}^{2}} \\cos{(\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Derivative(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2))))"]]}, {"prompt": "Given V{(\\mathbb{I},F_{N})} = F_{N} + \\mathbb{I}, then obtain \\frac{(\\lambda + 2 V{(\\mathbb{I},F_{N})}) (F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})})}{2 V{(\\mathbb{I},F_{N})}} = \\frac{(F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})})^{2}}{2 V{(\\mathbb{I},F_{N})}}", "derivation": "V{(\\mathbb{I},F_{N})} = F_{N} + \\mathbb{I} and \\lambda + V{(\\mathbb{I},F_{N})} = F_{N} + \\lambda + \\mathbb{I} and \\lambda + 2 V{(\\mathbb{I},F_{N})} = F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})} and \\frac{\\lambda + 2 V{(\\mathbb{I},F_{N})}}{2 V{(\\mathbb{I},F_{N})}} = \\frac{F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})}}{2 V{(\\mathbb{I},F_{N})}} and \\frac{(\\lambda + 2 V{(\\mathbb{I},F_{N})}) (F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})})}{2 V{(\\mathbb{I},F_{N})}} = \\frac{(F_{N} + \\lambda + \\mathbb{I} + V{(\\mathbb{I},F_{N})})^{2}}{2 V{(\\mathbb{I},F_{N})}}", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 2, "Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)))), Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)))), Pow(Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))), Pow(Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"], [["times", 4, "Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)))), Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))), Pow(Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True))), Integer(2)), Pow(Function('V')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given g{(W,i,\\hat{H}_{\\lambda})} = - W + \\frac{\\hat{H}_{\\lambda}}{i}, then obtain \\log{((- \\frac{\\hat{H}_{\\lambda}}{i} + g{(W,i,\\hat{H}_{\\lambda})})^{W})} = \\log{((- W)^{W})}", "derivation": "g{(W,i,\\hat{H}_{\\lambda})} = - W + \\frac{\\hat{H}_{\\lambda}}{i} and - \\frac{\\hat{H}_{\\lambda}}{i} + g{(W,i,\\hat{H}_{\\lambda})} = - W and (- \\frac{\\hat{H}_{\\lambda}}{i} + g{(W,i,\\hat{H}_{\\lambda})})^{W} = (- W)^{W} and \\log{((- \\frac{\\hat{H}_{\\lambda}}{i} + g{(W,i,\\hat{H}_{\\lambda})})^{W})} = \\log{((- W)^{W})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('W', commutative=True), Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))"], [["minus", 1, "Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('g')(Symbol('W', commutative=True), Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('g')(Symbol('W', commutative=True), Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('W', commutative=True)), Pow(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["log", 3], "Equality(log(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('g')(Symbol('W', commutative=True), Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('W', commutative=True))), log(Pow(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('W', commutative=True))))"]]}, {"prompt": "Given v{(q)} = e^{q}, then obtain (\\frac{0^{q}}{q})^{q} = (\\frac{1}{q})^{q}", "derivation": "v{(q)} = e^{q} and 0 = - v{(q)} + e^{q} and 0^{q} = (- v{(q)} + e^{q})^{q} and \\frac{0^{q}}{q} = \\frac{(- v{(q)} + e^{q})^{q}}{q} and \\frac{(- v{(q)} + e^{q})^{q}}{q} = \\frac{1}{q} and (\\frac{(- v{(q)} + e^{q})^{q}}{q})^{q} = (\\frac{1}{q})^{q} and (\\frac{0^{q}}{q})^{q} = (\\frac{1}{q})^{q}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["minus", 1, "Function('v')(Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Integer(0), Symbol('q', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["divide", 3, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Pow(Symbol('q', commutative=True), Integer(-1)))"], [["power", 5, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Mul(Pow(Integer(0), Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Symbol('q', commutative=True)), Pow(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(\\tilde{g}^*)} = e^{\\tilde{g}^*}, then obtain \\frac{e^{\\tilde{g}^*} \\int e^{- \\tilde{g}^*} d\\tilde{g}^*}{\\mathbf{g}{(\\tilde{g}^*)}} = \\int \\frac{1}{\\mathbf{g}{(\\tilde{g}^*)}} d\\tilde{g}^*", "derivation": "\\mathbf{g}{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and e^{- \\tilde{g}^*} = \\frac{1}{\\mathbf{g}{(\\tilde{g}^*)}} and \\int e^{- \\tilde{g}^*} d\\tilde{g}^* = \\int \\frac{1}{\\mathbf{g}{(\\tilde{g}^*)}} d\\tilde{g}^* and 1 = \\frac{e^{\\tilde{g}^*}}{\\mathbf{g}{(\\tilde{g}^*)}} and \\int e^{- \\tilde{g}^*} d\\tilde{g}^* = \\frac{e^{\\tilde{g}^*} \\int e^{- \\tilde{g}^*} d\\tilde{g}^*}{\\mathbf{g}{(\\tilde{g}^*)}} and \\frac{e^{\\tilde{g}^*} \\int e^{- \\tilde{g}^*} d\\tilde{g}^*}{\\mathbf{g}{(\\tilde{g}^*)}} = \\int \\frac{1}{\\mathbf{g}{(\\tilde{g}^*)}} d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["divide", 1, "Mul(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["times", 4, "Integral(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Integral(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(exp(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Integral(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(E_{n})} = \\log{(E_{n})}, then obtain 0 = - 4 \\mathbf{D}{(E_{n})} + 4 \\log{(E_{n})}", "derivation": "\\mathbf{D}{(E_{n})} = \\log{(E_{n})} and 0 = - \\mathbf{D}{(E_{n})} + \\log{(E_{n})} and 2 \\mathbf{D}{(E_{n})} - \\log{(E_{n})} = \\mathbf{D}{(E_{n})} and 0 = - 2 \\mathbf{D}{(E_{n})} + 2 \\log{(E_{n})} and 0 = - 4 \\mathbf{D}{(E_{n})} + 4 \\log{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))), log(Symbol('E_n', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))), log(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))), Mul(Integer(-1), log(Symbol('E_n', commutative=True)))), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(4), Function('\\\\mathbf{D}')(Symbol('E_n', commutative=True))), Mul(Integer(4), log(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} = - \\mathbf{A} + \\frac{E_{n}}{B}, then obtain - \\int (- \\mathbf{A} + \\frac{E_{n}}{B}) d\\mathbf{A} + \\int \\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} d\\mathbf{A} = 0", "derivation": "\\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} = - \\mathbf{A} + \\frac{E_{n}}{B} and \\int \\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} d\\mathbf{A} = \\int (- \\mathbf{A} + \\frac{E_{n}}{B}) d\\mathbf{A} and \\int \\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} d\\mathbf{A} + \\frac{\\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)}}{- \\mathbf{A} + \\frac{E_{n}}{B}} = \\int (- \\mathbf{A} + \\frac{E_{n}}{B}) d\\mathbf{A} + \\frac{\\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)}}{- \\mathbf{A} + \\frac{E_{n}}{B}} and - \\int (- \\mathbf{A} + \\frac{E_{n}}{B}) d\\mathbf{A} + \\int \\operatorname{C_{1}}{(E_{n},\\mathbf{A},B)} d\\mathbf{A} = 0", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 2, "Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Integer(-1)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)))"], "Equality(Add(Integral(Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Integer(-1)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Integer(-1)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)))))"], [["minus", 3, "Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Integer(-1)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Integral(Function('C_1')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{H}{(A_{x})} = \\cos{(A_{x})}, then obtain - A_{x} \\cos{(A_{x})} + (- A_{x} + \\hat{H}{(A_{x})})^{A_{x}} = - A_{x} \\cos{(A_{x})} + (- A_{x} + \\cos{(A_{x})})^{A_{x}}", "derivation": "\\hat{H}{(A_{x})} = \\cos{(A_{x})} and - A_{x} + \\hat{H}{(A_{x})} = - A_{x} + \\cos{(A_{x})} and (- A_{x} + \\hat{H}{(A_{x})})^{A_{x}} = (- A_{x} + \\cos{(A_{x})})^{A_{x}} and A_{x} \\hat{H}{(A_{x})} = A_{x} \\cos{(A_{x})} and - A_{x} \\hat{H}{(A_{x})} + (- A_{x} + \\hat{H}{(A_{x})})^{A_{x}} = - A_{x} \\hat{H}{(A_{x})} + (- A_{x} + \\cos{(A_{x})})^{A_{x}} and - A_{x} \\cos{(A_{x})} + (- A_{x} + \\hat{H}{(A_{x})})^{A_{x}} = - A_{x} \\cos{(A_{x})} + (- A_{x} + \\cos{(A_{x})})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["minus", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["times", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Mul(Symbol('A_x', commutative=True), cos(Symbol('A_x', commutative=True))))"], [["minus", 3, "Mul(Symbol('A_x', commutative=True), Function('\\\\hat{H}')(Symbol('A_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), cos(Symbol('A_x', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('\\\\hat{H}')(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), cos(Symbol('A_x', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then derive \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = x - \\cos{(a^{\\dagger})}, then obtain \\frac{\\sin{(a^{\\dagger})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger}}{\\operatorname{v_{x}}{(a^{\\dagger})}} = \\frac{(x - \\cos{(a^{\\dagger})}) \\sin{(a^{\\dagger})}}{\\operatorname{v_{x}}{(a^{\\dagger})}}", "derivation": "\\operatorname{v_{x}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = \\int \\sin{(a^{\\dagger})} da^{\\dagger} and \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = x - \\cos{(a^{\\dagger})} and \\sin{(a^{\\dagger})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = (x - \\cos{(a^{\\dagger})}) \\sin{(a^{\\dagger})} and \\frac{\\sin{(a^{\\dagger})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger}}{\\operatorname{v_{x}}{(a^{\\dagger})}} = \\frac{(x - \\cos{(a^{\\dagger})}) \\sin{(a^{\\dagger})}}{\\operatorname{v_{x}}{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('x', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["times", 3, "sin(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(sin(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Add(Symbol('x', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))), sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 4, "Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), sin(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Add(Symbol('x', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Pow(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), sin(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(v_{y})} = v_{y}, then obtain \\sin{(\\operatorname{n_{2}}{(v_{y})} \\sin{(\\operatorname{n_{2}}{(v_{y})})})} = \\sin{(\\operatorname{n_{2}}{(v_{y})} \\sin{(v_{y})})}", "derivation": "\\operatorname{n_{2}}{(v_{y})} = v_{y} and \\sin{(\\operatorname{n_{2}}{(v_{y})})} = \\sin{(v_{y})} and \\operatorname{n_{2}}{(v_{y})} \\sin{(\\operatorname{n_{2}}{(v_{y})})} = \\operatorname{n_{2}}{(v_{y})} \\sin{(v_{y})} and \\sin{(\\operatorname{n_{2}}{(v_{y})} \\sin{(\\operatorname{n_{2}}{(v_{y})})})} = \\sin{(\\operatorname{n_{2}}{(v_{y})} \\sin{(v_{y})})}", "srepr_derivation": [["renaming_premise", "Equality(Function('n_2')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], [["sin", 1], "Equality(sin(Function('n_2')(Symbol('v_y', commutative=True))), sin(Symbol('v_y', commutative=True)))"], [["times", 2, "Function('n_2')(Symbol('v_y', commutative=True))"], "Equality(Mul(Function('n_2')(Symbol('v_y', commutative=True)), sin(Function('n_2')(Symbol('v_y', commutative=True)))), Mul(Function('n_2')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["sin", 3], "Equality(sin(Mul(Function('n_2')(Symbol('v_y', commutative=True)), sin(Function('n_2')(Symbol('v_y', commutative=True))))), sin(Mul(Function('n_2')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given u{(E)} = \\cos{(\\log{(E)})}, then derive \\frac{d}{d E} u{(E)} = - \\frac{\\sin{(\\log{(E)})}}{E}, then derive C_{d} + \\cos{(\\log{(E)})} = \\int - \\frac{\\sin{(\\log{(E)})}}{E} dE, then obtain \\frac{\\partial}{\\partial E} (C_{d} + \\cos{(\\log{(E)})}) = \\frac{d}{d E} \\int - \\frac{\\sin{(\\log{(E)})}}{E} dE", "derivation": "u{(E)} = \\cos{(\\log{(E)})} and \\frac{d}{d E} u{(E)} = \\frac{d}{d E} \\cos{(\\log{(E)})} and \\frac{d}{d E} u{(E)} = - \\frac{\\sin{(\\log{(E)})}}{E} and \\frac{d}{d E} \\cos{(\\log{(E)})} = - \\frac{\\sin{(\\log{(E)})}}{E} and \\int \\frac{d}{d E} \\cos{(\\log{(E)})} dE = \\int - \\frac{\\sin{(\\log{(E)})}}{E} dE and C_{d} + \\cos{(\\log{(E)})} = \\int - \\frac{\\sin{(\\log{(E)})}}{E} dE and \\frac{\\partial}{\\partial E} (C_{d} + \\cos{(\\log{(E)})}) = \\frac{d}{d E} \\int - \\frac{\\sin{(\\log{(E)})}}{E} dE", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('E', commutative=True)), cos(log(Symbol('E', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Derivative(cos(log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('C_d', commutative=True), cos(log(Symbol('E', commutative=True)))), Integral(Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 6, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Symbol('C_d', commutative=True), cos(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), sin(log(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\sin{(g^{\\prime}_{\\varepsilon})}, then derive \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\mathbf{p}{(g^{\\prime}_{\\varepsilon})} = - \\sin{(g^{\\prime}_{\\varepsilon})}, then obtain \\log{(\\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\sin{(g^{\\prime}_{\\varepsilon})})} = \\log{(- \\sin{(g^{\\prime}_{\\varepsilon})})}", "derivation": "\\mathbf{p}{(g^{\\prime}_{\\varepsilon})} = \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\sin{(g^{\\prime}_{\\varepsilon})} and \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\mathbf{p}{(g^{\\prime}_{\\varepsilon})} = \\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\sin{(g^{\\prime}_{\\varepsilon})} and \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\mathbf{p}{(g^{\\prime}_{\\varepsilon})} = - \\sin{(g^{\\prime}_{\\varepsilon})} and \\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\sin{(g^{\\prime}_{\\varepsilon})} = - \\sin{(g^{\\prime}_{\\varepsilon})} and \\log{(\\frac{d^{2}}{d (g^{\\prime}_{\\varepsilon})^{2}} \\sin{(g^{\\prime}_{\\varepsilon})})} = \\log{(- \\sin{(g^{\\prime}_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["log", 4], "Equality(log(Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2)))), log(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given c{(F_{c},f_{E})} = F_{c} - f_{E}, then derive \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c} = F_{c}^{2} - 2 F_{c} f_{E} + G, then obtain (\\int (2 F_{c} - 2 f_{E}) dF_{c}) \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c} = (F_{c}^{2} - 2 F_{c} f_{E} + G) \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c}", "derivation": "c{(F_{c},f_{E})} = F_{c} - f_{E} and F_{c} - f_{E} + c{(F_{c},f_{E})} = 2 F_{c} - 2 f_{E} and \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c} = \\int (2 F_{c} - 2 f_{E}) dF_{c} and \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c} = F_{c}^{2} - 2 F_{c} f_{E} + G and \\int (2 F_{c} - 2 f_{E}) dF_{c} = F_{c}^{2} - 2 F_{c} f_{E} + G and (\\int (2 F_{c} - 2 f_{E}) dF_{c}) \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c} = (F_{c}^{2} - 2 F_{c} f_{E} + G) \\int (F_{c} - f_{E} + c{(F_{c},f_{E})}) dF_{c}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["add", 1, "Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)))"], "Equality(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True), Symbol('f_E', commutative=True)), Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True), Symbol('f_E', commutative=True)), Symbol('G', commutative=True)))"], [["times", 5, "Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True)))"], "Equality(Mul(Integral(Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True)))), Mul(Add(Pow(Symbol('F_c', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('F_c', commutative=True), Symbol('f_E', commutative=True)), Symbol('G', commutative=True)), Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('F_c', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(U)} = e^{U} and t{(U)} = - \\mathbf{J}_P{(U)}, then obtain t{(U)} = - e^{U}", "derivation": "\\mathbf{J}_P{(U)} = e^{U} and \\mathbf{J}_P{(U)} - e^{U} = 0 and t{(U)} = - \\mathbf{J}_P{(U)} and - e^{U} = - \\mathbf{J}_P{(U)} and t{(U)} = - e^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["minus", 1, "exp(Symbol('U', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(Symbol('U', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('t')(Symbol('U', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Symbol('U', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('t')(Symbol('U', commutative=True)), Mul(Integer(-1), exp(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(\\phi)} = \\phi, then derive \\mu_{0}{(\\phi)} \\int \\mu_{0}{(\\phi)} d\\phi = (\\mathbf{F} + \\frac{\\phi^{2}}{2}) \\mu_{0}{(\\phi)}, then obtain \\phi \\int \\phi d\\phi = \\phi (\\mathbf{F} + \\frac{\\phi^{2}}{2})", "derivation": "\\mu_{0}{(\\phi)} = \\phi and \\int \\mu_{0}{(\\phi)} d\\phi = \\int \\phi d\\phi and \\mu_{0}{(\\phi)} \\int \\mu_{0}{(\\phi)} d\\phi = \\mu_{0}{(\\phi)} \\int \\phi d\\phi and \\mu_{0}{(\\phi)} \\int \\mu_{0}{(\\phi)} d\\phi = (\\mathbf{F} + \\frac{\\phi^{2}}{2}) \\mu_{0}{(\\phi)} and \\phi \\int \\phi d\\phi = \\phi (\\mathbf{F} + \\frac{\\phi^{2}}{2})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))))"], [["times", 2, "Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))), Function('\\\\mu_0')(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\phi', commutative=True), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Symbol('\\\\phi', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2))))))"]]}, {"prompt": "Given s{(g,y)} = \\sin{(g y)}, then derive g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)} = g y + g \\cos{(g y)} + s{(g,y)}, then obtain ((g y + g \\cos{(g y)} + s{(g,y)}) (g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)}))^{y} = ((g y + g \\cos{(g y)} + s{(g,y)})^{2})^{y}", "derivation": "s{(g,y)} = \\sin{(g y)} and \\frac{\\partial}{\\partial y} s{(g,y)} = \\frac{\\partial}{\\partial y} \\sin{(g y)} and g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)} = g y + s{(g,y)} + \\frac{\\partial}{\\partial y} \\sin{(g y)} and g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)} = g y + g \\cos{(g y)} + s{(g,y)} and (g y + g \\cos{(g y)} + s{(g,y)}) (g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)}) = (g y + g \\cos{(g y)} + s{(g,y)})^{2} and ((g y + g \\cos{(g y)} + s{(g,y)}) (g y + s{(g,y)} + \\frac{\\partial}{\\partial y} s{(g,y)}))^{y} = ((g y + g \\cos{(g y)} + s{(g,y)})^{2})^{y}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), sin(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["add", 2, "Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Derivative(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Derivative(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True))))"], [["times", 4, "Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Derivative(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Pow(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True))), Integer(2)))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Derivative(Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Symbol('y', commutative=True)), Pow(Pow(Add(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g', commutative=True), cos(Mul(Symbol('g', commutative=True), Symbol('y', commutative=True)))), Function('s')(Symbol('g', commutative=True), Symbol('y', commutative=True))), Integer(2)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\mu,\\tilde{g}^*)} = \\mu^{\\tilde{g}^*}, then obtain 1 = \\frac{\\mu^{2 \\tilde{g}^*}}{\\varphi^{*}^{2}{(\\mu,\\tilde{g}^*)}}", "derivation": "\\varphi^{*}{(\\mu,\\tilde{g}^*)} = \\mu^{\\tilde{g}^*} and \\varphi^{*}^{2}{(\\mu,\\tilde{g}^*)} = \\mu^{\\tilde{g}^*} \\varphi^{*}{(\\mu,\\tilde{g}^*)} and \\varphi^{*}^{4}{(\\mu,\\tilde{g}^*)} = \\mu^{2 \\tilde{g}^*} \\varphi^{*}^{2}{(\\mu,\\tilde{g}^*)} and 1 = \\frac{\\mu^{2 \\tilde{g}^*}}{\\varphi^{*}^{2}{(\\mu,\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(4)), Mul(Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2))))"], [["divide", 3, "Pow(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(4))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mu', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\phi_{2}{(\\hat{x})} = \\log{(e^{\\hat{x}})}, then derive 0 = 1 - \\frac{d}{d \\hat{x}} \\phi_{2}{(\\hat{x})}, then obtain 0 = - \\frac{d^{2}}{d \\hat{x}^{2}} \\phi_{2}{(\\hat{x})}", "derivation": "\\phi_{2}{(\\hat{x})} = \\log{(e^{\\hat{x}})} and 0 = - \\phi_{2}{(\\hat{x})} + \\log{(e^{\\hat{x}})} and \\frac{d}{d \\hat{x}} 0 = \\frac{d}{d \\hat{x}} (- \\phi_{2}{(\\hat{x})} + \\log{(e^{\\hat{x}})}) and 0 = 1 - \\frac{d}{d \\hat{x}} \\phi_{2}{(\\hat{x})} and 0 = 1 - \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})} and \\frac{d}{d \\hat{x}} 0 = \\frac{d}{d \\hat{x}} (1 - \\frac{d}{d \\hat{x}} \\log{(e^{\\hat{x}})}) and \\frac{d}{d \\hat{x}} 0 = \\frac{d}{d \\hat{x}} (1 - \\frac{d}{d \\hat{x}} \\phi_{2}{(\\hat{x})}) and 0 = - \\frac{d^{2}}{d \\hat{x}^{2}} \\phi_{2}{(\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True)), log(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True))), log(exp(Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True))), log(exp(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))))"], [["differentiate", 5, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Derivative(log(exp(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\dot{y}{(\\pi)} = \\sin{(\\log{(\\pi)})}, then derive \\int \\dot{y}{(\\pi)} d\\pi = \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2} + \\sigma_p, then obtain \\int \\sin{(\\log{(\\pi)})} d\\pi = \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2} + \\sigma_p", "derivation": "\\dot{y}{(\\pi)} = \\sin{(\\log{(\\pi)})} and \\int \\dot{y}{(\\pi)} d\\pi = \\int \\sin{(\\log{(\\pi)})} d\\pi and \\int \\dot{y}{(\\pi)} d\\pi = \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2} + \\sigma_p and \\int \\sin{(\\log{(\\pi)})} d\\pi = \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2} + \\sigma_p", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\pi', commutative=True)), sin(log(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(sin(log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\pi', commutative=True), sin(log(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\pi', commutative=True), cos(log(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\pi', commutative=True), sin(log(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\pi', commutative=True), cos(log(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(x^\\prime)} = \\cos{(x^\\prime)}, then derive \\int \\operatorname{y^{\\prime}}{(x^\\prime)} dx^\\prime = F_{N} + \\sin{(x^\\prime)}, then obtain F_{N} + \\sin{(x^\\prime)} = M_{E} + \\sin{(x^\\prime)}", "derivation": "\\operatorname{y^{\\prime}}{(x^\\prime)} = \\cos{(x^\\prime)} and \\int \\operatorname{y^{\\prime}}{(x^\\prime)} dx^\\prime = \\int \\cos{(x^\\prime)} dx^\\prime and \\int \\operatorname{y^{\\prime}}{(x^\\prime)} dx^\\prime = F_{N} + \\sin{(x^\\prime)} and F_{N} + \\sin{(x^\\prime)} = \\int \\cos{(x^\\prime)} dx^\\prime and F_{N} + \\sin{(x^\\prime)} = M_{E} + \\sin{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('F_N', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_N', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_N', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('M_E', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\hat{x})} = e^{\\hat{x}}, then obtain \\frac{1}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}} = \\operatorname{g_{\\varepsilon}}^{-1 - \\frac{e^{\\hat{x}}}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}}}{(\\hat{x})} e^{\\hat{x}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\hat{x})} = e^{\\hat{x}} and \\hat{x} \\operatorname{g_{\\varepsilon}}{(\\hat{x})} = \\hat{x} e^{\\hat{x}} and 1 = \\frac{e^{\\hat{x}}}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}} and \\frac{1}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}} = \\frac{e^{\\hat{x}}}{\\operatorname{g_{\\varepsilon}}^{2}{(\\hat{x})}} and 2 = 1 + \\frac{e^{\\hat{x}}}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}} and \\frac{1}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}} = \\operatorname{g_{\\varepsilon}}^{-1 - \\frac{e^{\\hat{x}}}{\\operatorname{g_{\\varepsilon}}{(\\hat{x})}}}{(\\hat{x})} e^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\hat{x}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 3, "Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-2)), exp(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Add(Integer(-1), Mul(Integer(-1), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), exp(Symbol('\\\\hat{x}', commutative=True))))), exp(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given S{(\\Psi)} = \\sin{(\\Psi)}, then obtain 0 = \\frac{S{(\\Psi)} - \\sin{(\\Psi)}}{\\int - \\sin{(\\Psi)} d\\Psi}", "derivation": "S{(\\Psi)} = \\sin{(\\Psi)} and 0 = - S{(\\Psi)} + \\sin{(\\Psi)} and - S{(\\Psi)} = - \\sin{(\\Psi)} and - \\sin{(\\Psi)} = S{(\\Psi)} - 2 \\sin{(\\Psi)} and - S{(\\Psi)} = S{(\\Psi)} - 2 \\sin{(\\Psi)} and S{(\\Psi)} = - S{(\\Psi)} + 2 \\sin{(\\Psi)} and 0 = S{(\\Psi)} - \\sin{(\\Psi)} and 0 = \\frac{S{(\\Psi)} - \\sin{(\\Psi)}}{\\int - \\sin{(\\Psi)} d\\Psi}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('S')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\Psi', commutative=True))), sin(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('S')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Function('S')(Symbol('\\\\Psi', commutative=True))), sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Add(Function('S')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Function('S')(Symbol('\\\\Psi', commutative=True))), Add(Function('S')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\Psi', commutative=True)))))"], [["times", 5, "Integer(-1)"], "Equality(Function('S')(Symbol('\\\\Psi', commutative=True)), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Integer(0), Add(Function('S')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))))"], [["divide", 7, "Integral(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Integer(0), Mul(Add(Function('S')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True)))), Pow(Integral(Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(f_{\\mathbf{v}},n)} = f_{\\mathbf{v}} + n, then derive \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{r_{0}}{(f_{\\mathbf{v}},n)} = 1, then obtain (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + n))^{n} = 1", "derivation": "\\operatorname{r_{0}}{(f_{\\mathbf{v}},n)} = f_{\\mathbf{v}} + n and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{r_{0}}{(f_{\\mathbf{v}},n)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + n) and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{r_{0}}{(f_{\\mathbf{v}},n)} = 1 and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + n) = 1 and (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + n))^{n} = 1", "srepr_derivation": [["get_premise", "Equality(Function('r_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1))"]]}, {"prompt": "Given y{(t_{1})} = \\int \\log{(t_{1})} dt_{1}, then derive \\frac{d}{d t_{1}} y{(t_{1})} = \\frac{\\partial}{\\partial t_{1}} (\\rho_f + t_{1} \\log{(t_{1})} - t_{1}), then obtain \\frac{d}{d t_{1}} \\int \\log{(t_{1})} dt_{1} = \\frac{\\partial}{\\partial t_{1}} (\\rho_f + t_{1} \\log{(t_{1})} - t_{1})", "derivation": "y{(t_{1})} = \\int \\log{(t_{1})} dt_{1} and \\frac{d}{d t_{1}} y{(t_{1})} = \\frac{d}{d t_{1}} \\int \\log{(t_{1})} dt_{1} and \\frac{d}{d t_{1}} y{(t_{1})} = \\frac{\\partial}{\\partial t_{1}} (\\rho_f + t_{1} \\log{(t_{1})} - t_{1}) and \\frac{d}{d t_{1}} \\int \\log{(t_{1})} dt_{1} = \\frac{\\partial}{\\partial t_{1}} (\\rho_f + t_{1} \\log{(t_{1})} - t_{1})", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t_1', commutative=True)), Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{D},n)} = \\mathbf{D} + n, then derive \\frac{\\mathbf{D}^{2}}{2} + a = \\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D}, then obtain - \\frac{\\frac{\\mathbf{D} (- n + \\varphi{(\\mathbf{D},n)})}{2} + a}{n} = - \\frac{\\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D}}{n}", "derivation": "\\varphi{(\\mathbf{D},n)} = \\mathbf{D} + n and - n + \\varphi{(\\mathbf{D},n)} = \\mathbf{D} and \\mathbf{D} = 2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)} and \\mathbf{D} (- n + \\varphi{(\\mathbf{D},n)}) = \\mathbf{D}^{2} and \\int \\mathbf{D} d\\mathbf{D} = \\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D} and \\frac{\\mathbf{D}^{2}}{2} + a = \\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D} and \\frac{\\mathbf{D} (- n + \\varphi{(\\mathbf{D},n)})}{2} + a = \\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D} and - \\frac{\\frac{\\mathbf{D} (- n + \\varphi{(\\mathbf{D},n)})}{2} + a}{n} = - \\frac{\\int (2 \\mathbf{D} + n - \\varphi{(\\mathbf{D},n)}) d\\mathbf{D}}{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))"], [["add", 2, "Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True))))"], "Equality(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))))"], [["times", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)))"], [["integrate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Symbol('\\\\mathbf{D}', commutative=True), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Symbol('a', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Symbol('a', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 7, "Mul(Integer(-1), Symbol('n', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{D}', commutative=True), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Symbol('a', commutative=True))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(M_{E})} = \\sin{(M_{E})}, then derive \\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1 = \\cos{(M_{E})} - 1, then obtain \\frac{d}{d M_{E}} (\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1) = \\frac{d}{d M_{E}} (\\cos{(M_{E})} - 1)", "derivation": "\\operatorname{E_{x}}{(M_{E})} = \\sin{(M_{E})} and \\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(M_{E})} and \\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1 = \\frac{d}{d M_{E}} \\sin{(M_{E})} - 1 and \\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1 = \\cos{(M_{E})} - 1 and \\frac{d}{d M_{E}} \\sin{(M_{E})} - 1 = \\cos{(M_{E})} - 1 and \\frac{d}{d M_{E}} (\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1) = \\frac{d}{d M_{E}} (\\frac{d}{d M_{E}} \\sin{(M_{E})} - 1) and \\frac{d}{d M_{E}} (\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} - 1) = \\frac{d}{d M_{E}} (\\cos{(M_{E})} - 1)", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('M_E', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('M_E', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Add(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('M_E', commutative=True)), Integer(-1)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(\\theta,\\ddot{x})} = \\ddot{x} + \\theta, then obtain e^{2} = e^{\\frac{2 \\ddot{x} + 2 \\theta}{J{(\\theta,\\ddot{x})}}}", "derivation": "J{(\\theta,\\ddot{x})} = \\ddot{x} + \\theta and \\ddot{x} + \\theta + J{(\\theta,\\ddot{x})} = 2 \\ddot{x} + 2 \\theta and \\frac{\\ddot{x} + \\theta + J{(\\theta,\\ddot{x})}}{J{(\\theta,\\ddot{x})}} = \\frac{2 \\ddot{x} + 2 \\theta}{J{(\\theta,\\ddot{x})}} and e^{\\frac{\\ddot{x} + \\theta + J{(\\theta,\\ddot{x})}}{J{(\\theta,\\ddot{x})}}} = e^{\\frac{2 \\ddot{x} + 2 \\theta}{J{(\\theta,\\ddot{x})}}} and e^{2} = e^{\\frac{2 \\ddot{x} + 2 \\theta}{J{(\\theta,\\ddot{x})}}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta', commutative=True), Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))))"], [["divide", 2, "Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta', commutative=True), Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Pow(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))))"], [["exp", 3], "Equality(exp(Mul(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta', commutative=True), Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)))), exp(Mul(Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Pow(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(exp(Integer(2)), exp(Mul(Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Pow(Function('J')(Symbol('\\\\theta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given n{(A)} = \\sin{(A)}, then obtain \\frac{d}{d A} \\frac{n{(A)} + \\sin{(A)}}{2 \\sin{(A)}} = \\frac{d}{d A} 1", "derivation": "n{(A)} = \\sin{(A)} and n{(A)} + \\sin{(A)} = 2 \\sin{(A)} and \\frac{n{(A)} + \\sin{(A)}}{2 \\sin{(A)}} = 1 and \\frac{d}{d A} \\frac{n{(A)} + \\sin{(A)}}{2 \\sin{(A)}} = \\frac{d}{d A} 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["add", 1, "sin(Symbol('A', commutative=True))"], "Equality(Add(Function('n')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Mul(Integer(2), sin(Symbol('A', commutative=True))))"], [["divide", 2, "Mul(Integer(2), sin(Symbol('A', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('n')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Add(Function('n')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(-1))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(A_{x},\\mu,M_{E})} = A_{x} + M_{E} + \\mu, then obtain \\int (M_{E} \\mathbf{v}{(A_{x},\\mu,M_{E})} - M_{E}) d\\mu = \\int (M_{E} (A_{x} + M_{E} + \\mu) - M_{E}) d\\mu", "derivation": "\\mathbf{v}{(A_{x},\\mu,M_{E})} = A_{x} + M_{E} + \\mu and M_{E} \\mathbf{v}{(A_{x},\\mu,M_{E})} = M_{E} (A_{x} + M_{E} + \\mu) and M_{E} \\mathbf{v}{(A_{x},\\mu,M_{E})} - M_{E} = M_{E} (A_{x} + M_{E} + \\mu) - M_{E} and \\int (M_{E} \\mathbf{v}{(A_{x},\\mu,M_{E})} - M_{E}) d\\mu = \\int (M_{E} (A_{x} + M_{E} + \\mu) - M_{E}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('M_E', commutative=True), Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('M_E', commutative=True), Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Mul(Symbol('M_E', commutative=True), Add(Symbol('A_x', commutative=True), Symbol('M_E', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given x{(\\theta)} = \\cos{(\\theta)}, then obtain \\frac{d}{d \\theta} \\frac{- \\theta + 2 x{(\\theta)}}{- \\theta + \\cos{(\\theta)}} = \\frac{d}{d \\theta} \\frac{- \\theta + 2 \\cos{(\\theta)}}{- \\theta + \\cos{(\\theta)}}", "derivation": "x{(\\theta)} = \\cos{(\\theta)} and - \\theta + x{(\\theta)} = - \\theta + \\cos{(\\theta)} and - \\theta + x{(\\theta)} + \\cos{(\\theta)} = - \\theta + 2 \\cos{(\\theta)} and \\frac{- \\theta + x{(\\theta)} + \\cos{(\\theta)}}{- \\theta + \\cos{(\\theta)}} = \\frac{- \\theta + 2 \\cos{(\\theta)}}{- \\theta + \\cos{(\\theta)}} and \\frac{- \\theta + 2 x{(\\theta)}}{- \\theta + x{(\\theta)}} = \\frac{- \\theta + 2 \\cos{(\\theta)}}{- \\theta + x{(\\theta)}} and \\frac{- \\theta + 2 x{(\\theta)}}{- \\theta + \\cos{(\\theta)}} = \\frac{- \\theta + 2 \\cos{(\\theta)}}{- \\theta + \\cos{(\\theta)}} and \\frac{d}{d \\theta} \\frac{- \\theta + 2 x{(\\theta)}}{- \\theta + \\cos{(\\theta)}} = \\frac{d}{d \\theta} \\frac{- \\theta + 2 \\cos{(\\theta)}}{- \\theta + \\cos{(\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('x')(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('x')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('x')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('x')(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('x')(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))))"], [["differentiate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), Function('x')(Symbol('\\\\theta', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\mu)} = \\int \\cos{(\\mu)} d\\mu, then derive e^{\\varphi{(\\mu)}} = e^{\\mathbf{f} + \\sin{(\\mu)}}, then obtain \\varphi{(\\mu)} e^{\\int \\cos{(\\mu)} d\\mu} = \\varphi{(\\mu)} e^{\\mathbf{f} + \\sin{(\\mu)}}", "derivation": "\\varphi{(\\mu)} = \\int \\cos{(\\mu)} d\\mu and e^{\\varphi{(\\mu)}} = e^{\\int \\cos{(\\mu)} d\\mu} and e^{\\varphi{(\\mu)}} = e^{\\mathbf{f} + \\sin{(\\mu)}} and e^{\\int \\cos{(\\mu)} d\\mu} = e^{\\mathbf{f} + \\sin{(\\mu)}} and \\varphi{(\\mu)} e^{\\int \\cos{(\\mu)} d\\mu} = \\varphi{(\\mu)} e^{\\mathbf{f} + \\sin{(\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mu', commutative=True)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\varphi')(Symbol('\\\\mu', commutative=True))), exp(Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(exp(Function('\\\\varphi')(Symbol('\\\\mu', commutative=True))), exp(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), exp(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('\\\\mu', commutative=True)))))"], [["times", 4, "Function('\\\\varphi')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\mu', commutative=True)), exp(Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), Mul(Function('\\\\varphi')(Symbol('\\\\mu', commutative=True)), exp(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\rho_{f}{(V,\\mathbf{H})} = V + \\mathbf{H}, then obtain (\\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})}) (2 \\frac{\\partial}{\\partial \\mathbf{H}} \\rho_{f}{(V,\\mathbf{H})} + 2) - 2 = 4 V + 8 \\mathbf{H} - 2", "derivation": "\\rho_{f}{(V,\\mathbf{H})} = V + \\mathbf{H} and \\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})} = V + 2 \\mathbf{H} and (\\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})})^{2} = (V + 2 \\mathbf{H})^{2} and - 2 \\mathbf{H} + (\\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})})^{2} = - 2 \\mathbf{H} + (V + 2 \\mathbf{H})^{2} and \\frac{\\partial}{\\partial \\mathbf{H}} (- 2 \\mathbf{H} + (\\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})})^{2}) = \\frac{\\partial}{\\partial \\mathbf{H}} (- 2 \\mathbf{H} + (V + 2 \\mathbf{H})^{2}) and (\\mathbf{H} + \\rho_{f}{(V,\\mathbf{H})}) (2 \\frac{\\partial}{\\partial \\mathbf{H}} \\rho_{f}{(V,\\mathbf{H})} + 2) - 2 = 4 V + 8 \\mathbf{H} - 2", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('V', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)), Pow(Add(Symbol('V', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)))"], [["minus", 3, "Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('V', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('V', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(2), Derivative(Function('\\\\rho_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Integer(2))), Integer(-2)), Add(Mul(Integer(4), Symbol('V', commutative=True)), Mul(Integer(8), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\dot{z},\\theta)} = \\sin{(\\frac{\\theta}{\\dot{z}})}, then obtain (\\int \\operatorname{n_{2}}{(\\dot{z},\\theta)} d\\theta + \\frac{\\theta}{\\dot{z}})^{\\dot{z}} = (\\int \\sin{(\\frac{\\theta}{\\dot{z}})} d\\theta + \\frac{\\theta}{\\dot{z}})^{\\dot{z}}", "derivation": "\\operatorname{n_{2}}{(\\dot{z},\\theta)} = \\sin{(\\frac{\\theta}{\\dot{z}})} and \\int \\operatorname{n_{2}}{(\\dot{z},\\theta)} d\\theta = \\int \\sin{(\\frac{\\theta}{\\dot{z}})} d\\theta and \\int \\operatorname{n_{2}}{(\\dot{z},\\theta)} d\\theta + \\frac{\\theta}{\\dot{z}} = \\int \\sin{(\\frac{\\theta}{\\dot{z}})} d\\theta + \\frac{\\theta}{\\dot{z}} and (\\int \\operatorname{n_{2}}{(\\dot{z},\\theta)} d\\theta + \\frac{\\theta}{\\dot{z}})^{\\dot{z}} = (\\int \\sin{(\\frac{\\theta}{\\dot{z}})} d\\theta + \\frac{\\theta}{\\dot{z}})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta', commutative=True)), sin(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(sin(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Integral(Function('n_2')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Add(Integral(sin(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Add(Integral(Function('n_2')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Integral(sin(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(t_{1})} = e^{t_{1}} and \\mathbf{H}{(t_{1},\\hat{H},Q)} = Q \\hat{H} \\operatorname{y^{\\prime}}{(t_{1})} e^{t_{1}}, then obtain t_{1} + \\mathbf{H}{(t_{1},\\hat{H},Q)} + \\int \\log{(Q \\hat{H})} dQ = Q \\hat{H} \\operatorname{y^{\\prime}}^{2}{(t_{1})} + t_{1} + \\int \\log{(Q \\hat{H})} dQ", "derivation": "\\operatorname{y^{\\prime}}{(t_{1})} = e^{t_{1}} and \\mathbf{H}{(t_{1},\\hat{H},Q)} = Q \\hat{H} \\operatorname{y^{\\prime}}{(t_{1})} e^{t_{1}} and t_{1} + \\mathbf{H}{(t_{1},\\hat{H},Q)} = Q \\hat{H} \\operatorname{y^{\\prime}}{(t_{1})} e^{t_{1}} + t_{1} and t_{1} + \\mathbf{H}{(t_{1},\\hat{H},Q)} = Q \\hat{H} \\operatorname{y^{\\prime}}^{2}{(t_{1})} + t_{1} and t_{1} + \\mathbf{H}{(t_{1},\\hat{H},Q)} + \\int \\log{(Q \\hat{H})} dQ = Q \\hat{H} \\operatorname{y^{\\prime}}^{2}{(t_{1})} + t_{1} + \\int \\log{(Q \\hat{H})} dQ", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True), Function('y^{\\\\prime}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True))))"], [["add", 2, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True), Function('y^{\\\\prime}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True)), Integer(2))), Symbol('t_1', commutative=True)))"], [["add", 4, "Integral(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('Q', commutative=True)), Integral(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True)), Integer(2))), Symbol('t_1', commutative=True), Integral(log(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{M})} = \\sin{(\\mathbf{M})}, then derive \\frac{d}{d \\mathbf{M}} \\Psi^{\\dagger}{(\\mathbf{M})} = \\cos{(\\mathbf{M})}, then obtain \\mathbf{p} + \\cos{(\\mathbf{M})} = v_{x} + \\cos{(\\mathbf{M})}", "derivation": "\\Psi^{\\dagger}{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} \\Psi^{\\dagger}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} \\Psi^{\\dagger}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\cos{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} = \\frac{d^{2}}{d \\mathbf{M}^{2}} \\sin{(\\mathbf{M})} and \\int \\frac{d}{d \\mathbf{M}} \\cos{(\\mathbf{M})} d\\mathbf{M} = \\int \\frac{d^{2}}{d \\mathbf{M}^{2}} \\sin{(\\mathbf{M})} d\\mathbf{M} and \\mathbf{p} + \\cos{(\\mathbf{M})} = v_{x} + \\cos{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))))"], [["integrate", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Derivative(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('v_x', commutative=True), cos(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})}, then derive \\frac{d}{d f_{\\mathbf{p}}} \\mathbf{F}{(f_{\\mathbf{p}})} = \\frac{1}{f_{\\mathbf{p}}}, then obtain \\iint \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint \\frac{1}{f_{\\mathbf{p}}} df_{\\mathbf{p}} df_{\\mathbf{p}}", "derivation": "\\mathbf{F}{(f_{\\mathbf{p}})} = \\log{(f_{\\mathbf{p}})} and \\frac{d}{d f_{\\mathbf{p}}} \\mathbf{F}{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} and \\frac{d}{d f_{\\mathbf{p}}} \\mathbf{F}{(f_{\\mathbf{p}})} = \\frac{1}{f_{\\mathbf{p}}} and \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} = \\frac{1}{f_{\\mathbf{p}}} and \\int \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int \\frac{1}{f_{\\mathbf{p}}} df_{\\mathbf{p}} and \\iint \\frac{d}{d f_{\\mathbf{p}}} \\log{(f_{\\mathbf{p}})} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint \\frac{1}{f_{\\mathbf{p}}} df_{\\mathbf{p}} df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 5, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Derivative(log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\hat{x},p)} = - \\hat{x} + p, then obtain \\frac{\\partial}{\\partial \\hat{x}} \\rho_{f}{(\\hat{x},p)} - 1 = -2", "derivation": "\\rho_{f}{(\\hat{x},p)} = - \\hat{x} + p and - \\hat{x} + p + \\rho_{f}{(\\hat{x},p)} = - 2 \\hat{x} + 2 p and 2 \\rho_{f}{(\\hat{x},p)} = - 2 \\hat{x} + 2 p and \\frac{\\partial}{\\partial \\hat{x}} 2 \\rho_{f}{(\\hat{x},p)} = \\frac{\\partial}{\\partial \\hat{x}} (- 2 \\hat{x} + 2 p) and - \\hat{x} + p + \\rho_{f}{(\\hat{x},p)} = 2 \\rho_{f}{(\\hat{x},p)} and \\frac{\\partial}{\\partial \\hat{x}} (- \\hat{x} + p + \\rho_{f}{(\\hat{x},p)}) = \\frac{\\partial}{\\partial \\hat{x}} (- 2 \\hat{x} + 2 p) and \\frac{\\partial}{\\partial \\hat{x}} \\rho_{f}{(\\hat{x},p)} - 1 = -2", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('p', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('p', commutative=True), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('p', commutative=True), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('p', commutative=True), Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Function('\\\\rho_f')(Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)}, then obtain - \\int - \\mathbf{J}_M d\\mathbf{J}_M = - \\int - \\mathbf{J}_M d\\mathbf{J}_M - 1 + \\frac{\\log{(\\mathbf{J}_M)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)} = \\log{(\\mathbf{J}_M)} and 1 = \\frac{\\log{(\\mathbf{J}_M)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)}} and - \\mathbf{J}_M = - \\mathbf{J}_M - 1 + \\frac{\\log{(\\mathbf{J}_M)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)}} and 0 = -1 + \\frac{\\log{(\\mathbf{J}_M)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)}} and - \\int - \\mathbf{J}_M d\\mathbf{J}_M = - \\int - \\mathbf{J}_M d\\mathbf{J}_M - 1 + \\frac{\\log{(\\mathbf{J}_M)}}{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{J}_M)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["divide", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["minus", 4, "Integral(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(-1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given l{(F_{x})} = \\log{(F_{x})} and \\mathbf{F}{(F_{x})} = \\frac{\\log{(F_{x})}}{l{(F_{x})}}, then obtain \\frac{(\\frac{\\mathbf{F}{(F_{x})}}{\\log{(F_{x})}} + 1) \\log{(F_{x})}}{\\mathbf{F}{(F_{x})}} = \\frac{(1 + \\frac{1}{\\log{(F_{x})}}) \\log{(F_{x})}}{\\mathbf{F}{(F_{x})}}", "derivation": "l{(F_{x})} = \\log{(F_{x})} and \\mathbf{F}{(F_{x})} = \\frac{\\log{(F_{x})}}{l{(F_{x})}} and \\mathbf{F}{(F_{x})} = 1 and \\frac{\\mathbf{F}{(F_{x})}}{\\log{(F_{x})}} = \\frac{1}{\\log{(F_{x})}} and \\frac{\\mathbf{F}{(F_{x})}}{\\log{(F_{x})}} + 1 = 1 + \\frac{1}{\\log{(F_{x})}} and \\frac{(\\frac{\\mathbf{F}{(F_{x})}}{\\log{(F_{x})}} + 1) \\log{(F_{x})}}{\\mathbf{F}{(F_{x})}} = \\frac{(1 + \\frac{1}{\\log{(F_{x})}}) \\log{(F_{x})}}{\\mathbf{F}{(F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Mul(Pow(Function('l')(Symbol('F_x', commutative=True)), Integer(-1)), log(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Integer(1))"], [["divide", 3, "log(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Pow(log(Symbol('F_x', commutative=True)), Integer(-1)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Integer(1)), Add(Integer(1), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))))"], [["divide", 5, "Mul(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(Mul(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Integer(1)), Pow(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Integer(-1)), log(Symbol('F_x', commutative=True))), Mul(Add(Integer(1), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))), Pow(Function('\\\\mathbf{F}')(Symbol('F_x', commutative=True)), Integer(-1)), log(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given k{(\\Psi,\\sigma_p)} = \\frac{\\partial}{\\partial \\Psi} \\Psi \\sigma_p, then derive k{(\\Psi,\\sigma_p)} = \\sigma_p, then derive \\frac{\\partial}{\\partial \\sigma_p} k{(\\Psi,\\sigma_p)} = 1, then obtain (\\frac{\\partial}{\\partial \\sigma_p} k{(\\Psi,\\sigma_p)})^{\\sigma_p} = 1", "derivation": "k{(\\Psi,\\sigma_p)} = \\frac{\\partial}{\\partial \\Psi} \\Psi \\sigma_p and k{(\\Psi,\\sigma_p)} = \\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} k{(\\Psi,\\sigma_p)} = \\frac{d}{d \\sigma_p} \\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} k{(\\Psi,\\sigma_p)} = 1 and (\\frac{\\partial}{\\partial \\sigma_p} k{(\\Psi,\\sigma_p)})^{\\sigma_p} = 1", "srepr_derivation": [["get_premise", "Equality(Function('k')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('k')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Symbol('\\\\sigma_p', commutative=True), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('k')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Function('k')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\hat{x}_0{(S,F_{N})} = F_{N} - S and \\mathbf{A}{(S,F_{N})} = \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,F_{N})}, then derive \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,F_{N})} = -1, then obtain \\frac{\\partial^{- \\mathbf{A}{(S,F_{N})}}}{\\partial S^{- \\mathbf{A}{(S,F_{N})}}} \\hat{x}_0{(S,F_{N})} = \\mathbf{A}{(S,F_{N})}", "derivation": "\\hat{x}_0{(S,F_{N})} = F_{N} - S and \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,F_{N})} = \\frac{\\partial}{\\partial S} (F_{N} - S) and \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,F_{N})} = -1 and \\mathbf{A}{(S,F_{N})} = \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,F_{N})} and \\mathbf{A}{(S,F_{N})} = -1 and \\frac{\\partial^{- \\mathbf{A}{(S,F_{N})}}}{\\partial S^{- \\mathbf{A}{(S,F_{N})}}} \\hat{x}_0{(S,F_{N})} = \\mathbf{A}{(S,F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{A}')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('S', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('S', commutative=True), Symbol('F_N', commutative=True))))), Function('\\\\mathbf{A}')(Symbol('S', commutative=True), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(t_{1},\\mathbf{A})} = \\mathbf{A} t_{1}, then derive \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} = \\mathbf{A}, then obtain - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (C_{d} + f_{\\mathbf{v}}) + \\int \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} dt_{1} = - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (C_{d} + f_{\\mathbf{v}}) + \\int \\mathbf{A} dt_{1}", "derivation": "\\mathbf{S}{(t_{1},\\mathbf{A})} = \\mathbf{A} t_{1} and \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} = \\frac{\\partial}{\\partial t_{1}} \\mathbf{A} t_{1} and \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} = \\mathbf{A} and \\int \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} dt_{1} = \\int \\mathbf{A} dt_{1} and - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (C_{d} + f_{\\mathbf{v}}) + \\int \\frac{\\partial}{\\partial t_{1}} \\mathbf{S}{(t_{1},\\mathbf{A})} dt_{1} = - \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (C_{d} + f_{\\mathbf{v}}) + \\int \\mathbf{A} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Symbol('\\\\mathbf{A}', commutative=True))"], [["integrate", 3, "Symbol('t_1', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{S}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))), Integral(Symbol('\\\\mathbf{A}', commutative=True), Tuple(Symbol('t_1', commutative=True))))"], [["minus", 4, "Derivative(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), Integral(Derivative(Function('\\\\mathbf{S}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Derivative(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), Integral(Symbol('\\\\mathbf{A}', commutative=True), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\delta{(F_{N})} = \\sin{(F_{N})}, then obtain \\frac{d}{d F_{N}} (- F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\delta{(F_{N})})}) = \\frac{d}{d F_{N}} (- F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\sin{(F_{N})})})", "derivation": "\\delta{(F_{N})} = \\sin{(F_{N})} and F_{N} \\delta{(F_{N})} = F_{N} \\sin{(F_{N})} and \\log{(F_{N} \\delta{(F_{N})})} = \\log{(F_{N} \\sin{(F_{N})})} and - F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\delta{(F_{N})})} = - F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\sin{(F_{N})})} and \\frac{d}{d F_{N}} (- F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\delta{(F_{N})})}) = \\frac{d}{d F_{N}} (- F_{N} \\sin{(F_{N})} + \\log{(F_{N} \\sin{(F_{N})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["times", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Function('\\\\delta')(Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["log", 2], "Equality(log(Mul(Symbol('F_N', commutative=True), Function('\\\\delta')(Symbol('F_N', commutative=True)))), log(Mul(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Mul(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))), log(Mul(Symbol('F_N', commutative=True), Function('\\\\delta')(Symbol('F_N', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))), log(Mul(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))))))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))), log(Mul(Symbol('F_N', commutative=True), Function('\\\\delta')(Symbol('F_N', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))), log(Mul(Symbol('F_N', commutative=True), sin(Symbol('F_N', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(A_{2},\\eta^{\\prime})} = A_{2} \\eta^{\\prime}, then derive - A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{P}{(A_{2},\\eta^{\\prime})} = 0, then obtain A_{2} \\eta^{\\prime} (- A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} A_{2} \\eta^{\\prime}) = 0", "derivation": "\\mathbf{P}{(A_{2},\\eta^{\\prime})} = A_{2} \\eta^{\\prime} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{P}{(A_{2},\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} A_{2} \\eta^{\\prime} and - A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{P}{(A_{2},\\eta^{\\prime})} = - A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} A_{2} \\eta^{\\prime} and - A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{P}{(A_{2},\\eta^{\\prime})} = 0 and A_{2} \\eta^{\\prime} (- A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{P}{(A_{2},\\eta^{\\prime})}) = 0 and A_{2} \\eta^{\\prime} (- A_{2} + \\frac{\\partial}{\\partial \\eta^{\\prime}} A_{2} \\eta^{\\prime}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Integer(0))"], [["times", 4, "Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\dot{x}{(A_{z})} = \\cos{(A_{z})}, then derive \\frac{d}{d A_{z}} \\dot{x}{(A_{z})} = - \\sin{(A_{z})}, then obtain - \\sin{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})}", "derivation": "\\dot{x}{(A_{z})} = \\cos{(A_{z})} and \\frac{d}{d A_{z}} \\dot{x}{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})} and \\frac{d}{d A_{z}} \\dot{x}{(A_{z})} = - \\sin{(A_{z})} and - \\sin{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('A_z', commutative=True))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(M,\\Omega)} = \\int (M - \\Omega) dM, then derive B + \\frac{M^{2}}{2} - M \\Omega + \\operatorname{f^{\\prime}}{(M,\\Omega)} = 2 B + M^{2} - 2 M \\Omega, then derive \\frac{\\partial}{\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\int (M - \\Omega) dM) = - 2 M, then obtain \\frac{\\partial^{2}}{\\partial B\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\int (M - \\Omega) dM) = \\frac{d}{d B} - 2 M", "derivation": "\\operatorname{f^{\\prime}}{(M,\\Omega)} = \\int (M - \\Omega) dM and \\operatorname{f^{\\prime}}{(M,\\Omega)} + \\int (M - \\Omega) dM = 2 \\int (M - \\Omega) dM and B + \\frac{M^{2}}{2} - M \\Omega + \\operatorname{f^{\\prime}}{(M,\\Omega)} = 2 B + M^{2} - 2 M \\Omega and \\frac{\\partial}{\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\operatorname{f^{\\prime}}{(M,\\Omega)}) = \\frac{\\partial}{\\partial \\Omega} (2 B + M^{2} - 2 M \\Omega) and \\frac{\\partial}{\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\int (M - \\Omega) dM) = \\frac{\\partial}{\\partial \\Omega} (2 B + M^{2} - 2 M \\Omega) and \\frac{\\partial}{\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\int (M - \\Omega) dM) = - 2 M and \\frac{\\partial^{2}}{\\partial B\\partial \\Omega} (B + \\frac{M^{2}}{2} - M \\Omega + \\int (M - \\Omega) dM) = \\frac{d}{d B} - 2 M", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["add", 1, "Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(2), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Pow(Symbol('M', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('f^{\\\\prime}')(Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Pow(Symbol('M', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Pow(Symbol('M', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)))"], [["differentiate", 6, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(i,C)} = C + i, then obtain \\int (- C - i - \\cos{((C + i)^{i})} + \\cos{(\\sigma_{p}^{i}{(i,C)})}) dC = \\int (- C - i) dC", "derivation": "\\sigma_{p}{(i,C)} = C + i and \\sigma_{p}^{i}{(i,C)} = (C + i)^{i} and \\cos{(\\sigma_{p}^{i}{(i,C)})} = \\cos{((C + i)^{i})} and - \\cos{((C + i)^{i})} + \\cos{(\\sigma_{p}^{i}{(i,C)})} = 0 and - C - i - \\cos{((C + i)^{i})} + \\cos{(\\sigma_{p}^{i}{(i,C)})} = - C - i and \\int (- C - i - \\cos{((C + i)^{i})} + \\cos{(\\sigma_{p}^{i}{(i,C)})}) dC = \\int (- C - i) dC", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('i', commutative=True)))"], [["power", 1, "Symbol('i', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Symbol('i', commutative=True))), cos(Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True))))"], [["minus", 3, "cos(Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))), cos(Pow(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Symbol('i', commutative=True)))), Integer(0))"], [["minus", 4, "Add(Symbol('C', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), cos(Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))), cos(Pow(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["integrate", 5, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), cos(Pow(Add(Symbol('C', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))), cos(Pow(Function('\\\\sigma_p')(Symbol('i', commutative=True), Symbol('C', commutative=True)), Symbol('i', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{d},\\rho_b)} = \\rho_b^{C_{d}}, then obtain \\rho_b \\Psi^{\\dagger}{(C_{d},\\rho_b)} - \\rho_b^{C_{d}} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}} = \\rho_b \\rho_b^{C_{d}} - \\rho_b^{C_{d}} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}}", "derivation": "\\Psi^{\\dagger}{(C_{d},\\rho_b)} = \\rho_b^{C_{d}} and \\rho_b \\Psi^{\\dagger}{(C_{d},\\rho_b)} = \\rho_b \\rho_b^{C_{d}} and \\rho_b \\Psi^{\\dagger}{(C_{d},\\rho_b)} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}} = \\rho_b \\rho_b^{C_{d}} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}} and \\rho_b \\Psi^{\\dagger}{(C_{d},\\rho_b)} - \\rho_b^{C_{d}} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}} = \\rho_b \\rho_b^{C_{d}} - \\rho_b^{C_{d}} + \\frac{1}{\\Psi^{\\dagger}{(C_{d},\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True))))"], [["add", 2, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('C_d', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v,n)} = n - v and a{(V,A_{2})} = \\cos{(\\frac{A_{2}}{V})}, then obtain 0 = \\frac{\\cos{(a{(V,A_{2})} - \\cos{(\\frac{A_{2}}{V})})} - 1}{\\operatorname{F_{c}}{(v,n)}}", "derivation": "\\operatorname{F_{c}}{(v,n)} = n - v and a{(V,A_{2})} = \\cos{(\\frac{A_{2}}{V})} and 0 = - a{(V,A_{2})} + \\cos{(\\frac{A_{2}}{V})} and 1 = \\cos{(a{(V,A_{2})} - \\cos{(\\frac{A_{2}}{V})})} and 0 = \\cos{(a{(V,A_{2})} - \\cos{(\\frac{A_{2}}{V})})} - 1 and 0 = \\frac{\\cos{(a{(V,A_{2})} - \\cos{(\\frac{A_{2}}{V})})} - 1}{n - v} and 0 = \\frac{\\cos{(a{(V,A_{2})} - \\cos{(\\frac{A_{2}}{V})})} - 1}{\\operatorname{F_{c}}{(v,n)}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], ["get_premise", "Equality(Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1)))))"], [["minus", 2, "Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True))), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))"], [["cos", 3], "Equality(Integer(1), cos(Add(Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))))"], [["minus", 4, 1], "Equality(Integer(0), Add(cos(Add(Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))), Integer(-1)))"], [["divide", 5, "Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Integer(-1)), Add(cos(Add(Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Mul(Add(cos(Add(Function('a')(Symbol('V', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('A_2', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))))), Integer(-1)), Pow(Function('F_c')(Symbol('v', commutative=True), Symbol('n', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given B{(x)} = \\frac{d}{d x} e^{x}, then derive B{(x)} = e^{x}, then obtain \\frac{d}{d x} B{(x)} = \\frac{d^{2}}{d x^{2}} e^{x}", "derivation": "B{(x)} = \\frac{d}{d x} e^{x} and B{(x)} = e^{x} and \\frac{d}{d x} e^{x} = e^{x} and B{(x)} = \\frac{d}{d x} B{(x)} and \\frac{d}{d x} B{(x)} = \\frac{d}{d x} e^{x} and \\frac{d}{d x} B{(x)} = \\frac{d^{2}}{d x^{2}} e^{x}", "srepr_derivation": [["get_premise", "Equality(Function('B')(Symbol('x', commutative=True)), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('B')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), exp(Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('B')(Symbol('x', commutative=True)), Derivative(Function('B')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Derivative(Function('B')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('B')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and \\operatorname{f_{E}}{(\\lambda)} = \\frac{\\log{(\\sin{(\\lambda)})}}{\\operatorname{m_{s}}{(\\lambda)}}, then obtain \\frac{d}{d \\lambda} 0 = \\frac{d}{d \\lambda} \\log{(\\operatorname{f_{E}}{(\\lambda)})}", "derivation": "\\operatorname{m_{s}}{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and \\lambda \\operatorname{m_{s}}{(\\lambda)} = \\lambda \\log{(\\sin{(\\lambda)})} and 1 = \\frac{\\log{(\\sin{(\\lambda)})}}{\\operatorname{m_{s}}{(\\lambda)}} and 0 = \\log{(\\frac{\\log{(\\sin{(\\lambda)})}}{\\operatorname{m_{s}}{(\\lambda)}})} and \\frac{d}{d \\lambda} 0 = \\frac{d}{d \\lambda} \\log{(\\frac{\\log{(\\sin{(\\lambda)})}}{\\operatorname{m_{s}}{(\\lambda)}})} and \\operatorname{f_{E}}{(\\lambda)} = \\frac{\\log{(\\sin{(\\lambda)})}}{\\operatorname{m_{s}}{(\\lambda)}} and \\frac{d}{d \\lambda} 0 = \\frac{d}{d \\lambda} \\log{(\\operatorname{f_{E}}{(\\lambda)})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\lambda', commutative=True)), log(sin(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('m_s')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), log(sin(Symbol('\\\\lambda', commutative=True)))))"], [["divide", 2, "Mul(Symbol('\\\\lambda', commutative=True), Function('m_s')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('m_s')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\lambda', commutative=True)))))"], [["log", 3], "Equality(Integer(0), log(Mul(Pow(Function('m_s')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\lambda', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Function('m_s')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\lambda', commutative=True))))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Function('m_s')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(log(Function('f_E')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\mathbf{A},u)} = \\mathbf{A}^{u} and \\operatorname{v_{x}}{(\\mathbf{A},u)} = \\mathbf{A}^{u}, then obtain \\operatorname{v_{x}}^{u}{(\\mathbf{A},u)} - \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{u} = (\\mathbf{A}^{u})^{u} - \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{u}", "derivation": "V{(\\mathbf{A},u)} = \\mathbf{A}^{u} and \\frac{\\partial}{\\partial \\mathbf{A}} V{(\\mathbf{A},u)} = \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{u} and \\operatorname{v_{x}}{(\\mathbf{A},u)} = \\mathbf{A}^{u} and \\operatorname{v_{x}}^{u}{(\\mathbf{A},u)} = (\\mathbf{A}^{u})^{u} and \\operatorname{v_{x}}^{u}{(\\mathbf{A},u)} - \\frac{\\partial}{\\partial \\mathbf{A}} V{(\\mathbf{A},u)} = (\\mathbf{A}^{u})^{u} - \\frac{\\partial}{\\partial \\mathbf{A}} V{(\\mathbf{A},u)} and \\operatorname{v_{x}}^{u}{(\\mathbf{A},u)} - \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{u} = (\\mathbf{A}^{u})^{u} - \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{u}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["minus", 4, "Derivative(Function('V')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Derivative(Function('V')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))), Add(Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Derivative(Function('V')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Function('v_x')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))), Add(Pow(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\lambda{(f_{E})} = \\sin{(f_{E})} and y{(f_{E})} = \\sin^{2}{(f_{E})} + \\sin^{f_{E}}{(f_{E})}, then obtain \\lambda^{f_{E}}{(f_{E})} + \\sin^{2}{(f_{E})} = \\lambda^{2}{(f_{E})} + \\lambda^{f_{E}}{(f_{E})}", "derivation": "\\lambda{(f_{E})} = \\sin{(f_{E})} and \\lambda^{f_{E}}{(f_{E})} = \\sin^{f_{E}}{(f_{E})} and \\lambda^{f_{E}}{(f_{E})} + \\sin^{2}{(f_{E})} = \\sin^{2}{(f_{E})} + \\sin^{f_{E}}{(f_{E})} and y{(f_{E})} = \\sin^{2}{(f_{E})} + \\sin^{f_{E}}{(f_{E})} and y{(f_{E})} = \\lambda^{f_{E}}{(f_{E})} + \\sin^{2}{(f_{E})} and y{(f_{E})} = \\lambda^{2}{(f_{E})} + \\lambda^{f_{E}}{(f_{E})} and \\lambda^{f_{E}}{(f_{E})} + \\sin^{2}{(f_{E})} = \\lambda^{2}{(f_{E})} + \\lambda^{f_{E}}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["add", 2, "Pow(sin(Symbol('f_E', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(2))), Add(Pow(sin(Symbol('f_E', commutative=True)), Integer(2)), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('f_E', commutative=True)), Add(Pow(sin(Symbol('f_E', commutative=True)), Integer(2)), Pow(sin(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('y')(Symbol('f_E', commutative=True)), Add(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('y')(Symbol('f_E', commutative=True)), Add(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Integer(2)), Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(2))), Add(Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Integer(2)), Pow(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(t_{1})} = \\frac{d}{d t_{1}} \\log{(t_{1})}, then derive \\operatorname{z^{*}}{(t_{1})} = \\frac{1}{t_{1}}, then obtain \\operatorname{z^{*}}{(t_{1})} + \\frac{d}{d t_{1}} \\log{(t_{1})} + \\frac{1}{t_{1}} = \\frac{d}{d t_{1}} \\log{(t_{1})} + \\frac{2}{t_{1}}", "derivation": "\\operatorname{z^{*}}{(t_{1})} = \\frac{d}{d t_{1}} \\log{(t_{1})} and \\operatorname{z^{*}}{(t_{1})} = \\frac{1}{t_{1}} and \\operatorname{z^{*}}{(t_{1})} + \\frac{d}{d t_{1}} \\log{(t_{1})} = \\frac{d}{d t_{1}} \\log{(t_{1})} + \\frac{1}{t_{1}} and \\frac{1}{t_{1}} = \\frac{d}{d t_{1}} \\log{(t_{1})} and \\operatorname{z^{*}}{(t_{1})} + \\frac{1}{t_{1}} = \\frac{2}{t_{1}} and \\operatorname{z^{*}}{(t_{1})} + \\frac{d}{d t_{1}} \\log{(t_{1})} + \\frac{1}{t_{1}} = \\frac{d}{d t_{1}} \\log{(t_{1})} + \\frac{2}{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('t_1', commutative=True)), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('z^*')(Symbol('t_1', commutative=True)), Pow(Symbol('t_1', commutative=True), Integer(-1)))"], [["add", 2, "Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Add(Function('z^*')(Symbol('t_1', commutative=True)), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('t_1', commutative=True), Integer(-1)), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('z^*')(Symbol('t_1', commutative=True)), Pow(Symbol('t_1', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["add", 5, "Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Add(Function('z^*')(Symbol('t_1', commutative=True)), Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Symbol('t_1', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Integer(2), Pow(Symbol('t_1', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(F_{c})} = \\cos{(\\log{(F_{c})})}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(F_{c})} dF_{c} = \\frac{F_{c} \\sin{(\\log{(F_{c})})}}{2} + \\frac{F_{c} \\cos{(\\log{(F_{c})})}}{2} + t, then obtain \\int \\cos{(\\log{(F_{c})})} dF_{c} = \\frac{F_{c} \\sin{(\\log{(F_{c})})}}{2} + \\frac{F_{c} \\cos{(\\log{(F_{c})})}}{2} + t", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(F_{c})} = \\cos{(\\log{(F_{c})})} and \\int \\operatorname{f_{\\mathbf{v}}}{(F_{c})} dF_{c} = \\int \\cos{(\\log{(F_{c})})} dF_{c} and \\int \\operatorname{f_{\\mathbf{v}}}{(F_{c})} dF_{c} = \\frac{F_{c} \\sin{(\\log{(F_{c})})}}{2} + \\frac{F_{c} \\cos{(\\log{(F_{c})})}}{2} + t and \\int \\cos{(\\log{(F_{c})})} dF_{c} = \\frac{F_{c} \\sin{(\\log{(F_{c})})}}{2} + \\frac{F_{c} \\cos{(\\log{(F_{c})})}}{2} + t", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('F_c', commutative=True)), cos(log(Symbol('F_c', commutative=True))))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(cos(log(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Symbol('F_c', commutative=True), sin(log(Symbol('F_c', commutative=True)))), Mul(Rational(1, 2), Symbol('F_c', commutative=True), cos(log(Symbol('F_c', commutative=True)))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(log(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Symbol('F_c', commutative=True), sin(log(Symbol('F_c', commutative=True)))), Mul(Rational(1, 2), Symbol('F_c', commutative=True), cos(log(Symbol('F_c', commutative=True)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(v_{2})} = \\log{(v_{2})}, then obtain (\\log{(v_{2})}^{2 v_{2}})^{v_{2}} = ((- \\operatorname{E_{x}}^{v_{2}}{(v_{2})} + 2 \\log{(v_{2})}^{v_{2}})^{2})^{v_{2}}", "derivation": "\\operatorname{E_{x}}{(v_{2})} = \\log{(v_{2})} and \\operatorname{E_{x}}^{v_{2}}{(v_{2})} = \\log{(v_{2})}^{v_{2}} and 0 = - \\operatorname{E_{x}}^{v_{2}}{(v_{2})} + \\log{(v_{2})}^{v_{2}} and \\log{(v_{2})}^{v_{2}} = - \\operatorname{E_{x}}^{v_{2}}{(v_{2})} + 2 \\log{(v_{2})}^{v_{2}} and \\log{(v_{2})}^{2 v_{2}} = (- \\operatorname{E_{x}}^{v_{2}}{(v_{2})} + 2 \\log{(v_{2})}^{v_{2}})^{2} and (\\log{(v_{2})}^{2 v_{2}})^{v_{2}} = ((- \\operatorname{E_{x}}^{v_{2}}{(v_{2})} + 2 \\log{(v_{2})}^{v_{2}})^{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], [["minus", 2, "Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))))"], [["add", 3, "Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(2), Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(log(Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(2), Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))), Integer(2)))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(Pow(log(Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(2), Pow(log(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))), Integer(2)), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)}, then derive \\frac{d}{d \\hat{x}_0} \\dot{x}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)}, then obtain \\cos{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\sin{(\\hat{x}_0)}", "derivation": "\\dot{x}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\dot{x}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\sin{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\dot{x}{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)} and \\cos{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\sin{(\\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), cos(Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and J{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\mathbf{H}{(\\mathbf{P})}, then obtain \\frac{J{(\\mathbf{P})}}{\\mathbf{H}{(\\mathbf{P})}} = \\frac{\\frac{d}{d \\mathbf{P}} \\mathbf{H}{(\\mathbf{P})}}{\\mathbf{H}{(\\mathbf{P})}}", "derivation": "\\mathbf{H}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and J{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\mathbf{H}{(\\mathbf{P})} and \\frac{J{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} = \\frac{\\frac{d}{d \\mathbf{P}} \\mathbf{H}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} and \\frac{J{(\\mathbf{P})}}{\\mathbf{H}{(\\mathbf{P})}} = \\frac{\\frac{d}{d \\mathbf{P}} \\mathbf{H}{(\\mathbf{P})}}{\\mathbf{H}{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Function('J')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('J')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(n_{1},g_{\\varepsilon})} = g_{\\varepsilon} - n_{1}, then obtain \\int - g_{\\varepsilon} g{(n_{1},g_{\\varepsilon})} dg_{\\varepsilon} = \\int - g_{\\varepsilon} (g_{\\varepsilon} - n_{1}) dg_{\\varepsilon}", "derivation": "g{(n_{1},g_{\\varepsilon})} = g_{\\varepsilon} - n_{1} and - g{(n_{1},g_{\\varepsilon})} = - g_{\\varepsilon} + n_{1} and - g_{\\varepsilon} g{(n_{1},g_{\\varepsilon})} = g_{\\varepsilon} (- g_{\\varepsilon} + n_{1}) and \\int - g_{\\varepsilon} g{(n_{1},g_{\\varepsilon})} dg_{\\varepsilon} = \\int g_{\\varepsilon} (- g_{\\varepsilon} + n_{1}) dg_{\\varepsilon} and \\int - g_{\\varepsilon} (g_{\\varepsilon} - n_{1}) dg_{\\varepsilon} = \\int g_{\\varepsilon} (- g_{\\varepsilon} + n_{1}) dg_{\\varepsilon} and \\int - g_{\\varepsilon} g{(n_{1},g_{\\varepsilon})} dg_{\\varepsilon} = \\int - g_{\\varepsilon} (g_{\\varepsilon} - n_{1}) dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('n_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('g')(Symbol('n_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True)))"], [["times", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('g')(Symbol('n_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('g')(Symbol('n_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Function('g')(Symbol('n_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(f_{\\mathbf{v}},v_{y})} = \\log{(f_{\\mathbf{v}} v_{y})}, then obtain (\\int f_{\\mathbf{v}} v_{y} \\dot{x}{(f_{\\mathbf{v}},v_{y})} dv_{y})^{v_{y}} = (\\int f_{\\mathbf{v}} v_{y} \\log{(f_{\\mathbf{v}} v_{y})} dv_{y})^{v_{y}}", "derivation": "\\dot{x}{(f_{\\mathbf{v}},v_{y})} = \\log{(f_{\\mathbf{v}} v_{y})} and f_{\\mathbf{v}} v_{y} \\dot{x}{(f_{\\mathbf{v}},v_{y})} = f_{\\mathbf{v}} v_{y} \\log{(f_{\\mathbf{v}} v_{y})} and \\int f_{\\mathbf{v}} v_{y} \\dot{x}{(f_{\\mathbf{v}},v_{y})} dv_{y} = \\int f_{\\mathbf{v}} v_{y} \\log{(f_{\\mathbf{v}} v_{y})} dv_{y} and (\\int f_{\\mathbf{v}} v_{y} \\dot{x}{(f_{\\mathbf{v}},v_{y})} dv_{y})^{v_{y}} = (\\int f_{\\mathbf{v}} v_{y} \\log{(f_{\\mathbf{v}} v_{y})} dv_{y})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True))))"], [["times", 1, "Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), Function('\\\\dot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True)))))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), Function('\\\\dot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), Function('\\\\dot{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True), log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(G,\\mathbf{F})} = G \\mathbf{F}, then derive G \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{c_{0}}{(G,\\mathbf{F})} + G \\operatorname{c_{0}}{(G,\\mathbf{F})} = 2 G^{2} \\mathbf{F}, then obtain G \\operatorname{c_{0}}{(G,\\mathbf{F})} + \\operatorname{c_{0}}{(G,\\mathbf{F})} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{c_{0}}{(G,\\mathbf{F})} = 2 G \\operatorname{c_{0}}{(G,\\mathbf{F})}", "derivation": "\\operatorname{c_{0}}{(G,\\mathbf{F})} = G \\mathbf{F} and \\mathbf{F} \\operatorname{c_{0}}{(G,\\mathbf{F})} = G \\mathbf{F}^{2} and G \\mathbf{F} \\operatorname{c_{0}}{(G,\\mathbf{F})} = G^{2} \\mathbf{F}^{2} and \\frac{\\partial}{\\partial \\mathbf{F}} G \\mathbf{F} \\operatorname{c_{0}}{(G,\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} G^{2} \\mathbf{F}^{2} and G \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{c_{0}}{(G,\\mathbf{F})} + G \\operatorname{c_{0}}{(G,\\mathbf{F})} = 2 G^{2} \\mathbf{F} and G^{2} \\mathbf{F} + G \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} G \\mathbf{F} = 2 G^{2} \\mathbf{F} and G \\operatorname{c_{0}}{(G,\\mathbf{F})} + \\operatorname{c_{0}}{(G,\\mathbf{F})} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{c_{0}}{(G,\\mathbf{F})} = 2 G \\operatorname{c_{0}}{(G,\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"], [["times", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Derivative(Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Symbol('G', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(2)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Pow(Symbol('G', commutative=True), Integer(2)), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))), Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(2)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Symbol('G', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))), Mul(Integer(2), Symbol('G', commutative=True), Function('c_0')(Symbol('G', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\hat{X},f_{E})} = \\hat{X} - f_{E}, then derive \\int \\omega{(\\hat{X},f_{E})} d\\hat{X} = \\Psi + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E}, then derive \\Psi^{\\dagger} + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} = \\Psi + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E}, then obtain \\Psi^{\\dagger} + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} = \\int (\\hat{X} - f_{E}) d\\hat{X}", "derivation": "\\omega{(\\hat{X},f_{E})} = \\hat{X} - f_{E} and \\int \\omega{(\\hat{X},f_{E})} d\\hat{X} = \\int (\\hat{X} - f_{E}) d\\hat{X} and \\int \\omega{(\\hat{X},f_{E})} d\\hat{X} = \\Psi + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} and \\int (\\hat{X} - f_{E}) d\\hat{X} = \\Psi + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} and \\Psi^{\\dagger} + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} = \\Psi + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} and \\Psi^{\\dagger} + \\frac{\\hat{X}^{2}}{2} - \\hat{X} f_{E} = \\int (\\hat{X} - f_{E}) d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f_E', commutative=True))), Integral(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = \\sin{(A_{2})}, then obtain - A_{2} \\sin{(A_{2})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = - A_{2} \\sin{(A_{2})} + \\sin{(A_{2})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = \\sin{(A_{2})} and A_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = A_{2} \\sin{(A_{2})} and - A_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = - A_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} + \\sin{(A_{2})} and - A_{2} \\sin{(A_{2})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{2})} = - A_{2} \\sin{(A_{2})} + \\sin{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["times", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), sin(Symbol('A_2', commutative=True))))"], [["minus", 1, "Mul(Symbol('A_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True))), sin(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True), sin(Symbol('A_2', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True), sin(Symbol('A_2', commutative=True))), sin(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given G{(\\mathbf{F})} = \\int e^{\\mathbf{F}} d\\mathbf{F}, then derive \\cos{(G{(\\mathbf{F})})} = \\cos{(C + e^{\\mathbf{F}})}, then obtain - \\cos{(G{(\\mathbf{F})})} + \\cos{(\\int e^{\\mathbf{F}} d\\mathbf{F})} = \\cos{(C + e^{\\mathbf{F}})} - \\cos{(G{(\\mathbf{F})})}", "derivation": "G{(\\mathbf{F})} = \\int e^{\\mathbf{F}} d\\mathbf{F} and \\cos{(G{(\\mathbf{F})})} = \\cos{(\\int e^{\\mathbf{F}} d\\mathbf{F})} and \\cos{(G{(\\mathbf{F})})} = \\cos{(C + e^{\\mathbf{F}})} and \\cos{(\\int e^{\\mathbf{F}} d\\mathbf{F})} = \\cos{(C + e^{\\mathbf{F}})} and - \\cos{(G{(\\mathbf{F})})} + \\cos{(\\int e^{\\mathbf{F}} d\\mathbf{F})} = \\cos{(C + e^{\\mathbf{F}})} - \\cos{(G{(\\mathbf{F})})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{F}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["cos", 1], "Equality(cos(Function('G')(Symbol('\\\\mathbf{F}', commutative=True))), cos(Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(cos(Function('G')(Symbol('\\\\mathbf{F}', commutative=True))), cos(Add(Symbol('C', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(cos(Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), cos(Add(Symbol('C', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["minus", 4, "cos(Function('G')(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Function('G')(Symbol('\\\\mathbf{F}', commutative=True)))), cos(Integral(exp(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))), Add(cos(Add(Symbol('C', commutative=True), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(-1), cos(Function('G')(Symbol('\\\\mathbf{F}', commutative=True))))))"]]}, {"prompt": "Given G{(t_{2},v_{t})} = - t_{2} + e^{v_{t}}, then derive \\frac{\\partial}{\\partial v_{t}} G{(t_{2},v_{t})} = e^{v_{t}}, then derive V + e^{v_{t}} = n_{2} + e^{v_{t}}, then obtain V + \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}}) = n_{2} + \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}})", "derivation": "G{(t_{2},v_{t})} = - t_{2} + e^{v_{t}} and \\frac{\\partial}{\\partial v_{t}} G{(t_{2},v_{t})} = \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}}) and \\frac{\\partial}{\\partial v_{t}} G{(t_{2},v_{t})} = e^{v_{t}} and \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}}) = e^{v_{t}} and \\int \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}}) dv_{t} = \\int e^{v_{t}} dv_{t} and V + e^{v_{t}} = n_{2} + e^{v_{t}} and V + \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}}) = n_{2} + \\frac{\\partial}{\\partial v_{t}} (- t_{2} + e^{v_{t}})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), exp(Symbol('v_t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), exp(Symbol('v_t', commutative=True)))"], [["integrate", 4, "Symbol('v_t', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('V', commutative=True), exp(Symbol('v_t', commutative=True))), Add(Symbol('n_2', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('V', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Add(Symbol('n_2', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{H}_l,v_{t})} = v_{t} \\sin{(\\hat{H}_l)} and \\hat{\\mathbf{x}}{(A_{y})} = e^{A_{y}}, then obtain \\hat{\\mathbf{x}}{(A_{y})} + \\int v_{t} \\sin{(\\hat{H}_l)} dv_{t} = e^{A_{y}} + \\int v_{t} \\sin{(\\hat{H}_l)} dv_{t}", "derivation": "\\mathbb{I}{(\\hat{H}_l,v_{t})} = v_{t} \\sin{(\\hat{H}_l)} and \\int \\mathbb{I}{(\\hat{H}_l,v_{t})} dv_{t} = \\int v_{t} \\sin{(\\hat{H}_l)} dv_{t} and \\hat{\\mathbf{x}}{(A_{y})} = e^{A_{y}} and \\hat{\\mathbf{x}}{(A_{y})} + \\int \\mathbb{I}{(\\hat{H}_l,v_{t})} dv_{t} = e^{A_{y}} + \\int \\mathbb{I}{(\\hat{H}_l,v_{t})} dv_{t} and \\hat{\\mathbf{x}}{(A_{y})} + \\int v_{t} \\sin{(\\hat{H}_l)} dv_{t} = e^{A_{y}} + \\int v_{t} \\sin{(\\hat{H}_l)} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('v_t', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Symbol('v_t', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["add", 3, "Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_y', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(exp(Symbol('A_y', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A_y', commutative=True)), Integral(Mul(Symbol('v_t', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('v_t', commutative=True)))), Add(exp(Symbol('A_y', commutative=True)), Integral(Mul(Symbol('v_t', commutative=True), sin(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\hat{p})} = \\cos{(\\hat{p})}, then derive I + \\frac{\\hat{p}^{2}}{2} = \\int (\\hat{p} - \\eta^{\\prime}{(\\hat{p})} + \\cos{(\\hat{p})}) d\\hat{p}, then obtain \\hat{p}^{2} + \\int (I + \\frac{\\hat{p}^{2}}{2}) dI = \\hat{p}^{2} + \\iint \\hat{p} d\\hat{p} dI", "derivation": "\\eta^{\\prime}{(\\hat{p})} = \\cos{(\\hat{p})} and \\hat{p} = \\hat{p} - \\eta^{\\prime}{(\\hat{p})} + \\cos{(\\hat{p})} and \\int \\hat{p} d\\hat{p} = \\int (\\hat{p} - \\eta^{\\prime}{(\\hat{p})} + \\cos{(\\hat{p})}) d\\hat{p} and I + \\frac{\\hat{p}^{2}}{2} = \\int (\\hat{p} - \\eta^{\\prime}{(\\hat{p})} + \\cos{(\\hat{p})}) d\\hat{p} and I + \\frac{\\hat{p}^{2}}{2} = \\int \\hat{p} d\\hat{p} and \\int (I + \\frac{\\hat{p}^{2}}{2}) dI = \\iint \\hat{p} d\\hat{p} dI and \\hat{p}^{2} + \\int (I + \\frac{\\hat{p}^{2}}{2}) dI = \\hat{p}^{2} + \\iint \\hat{p} d\\hat{p} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Symbol('\\\\hat{p}', commutative=True), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Symbol('\\\\hat{p}', commutative=True), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True))), cos(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Integral(Symbol('\\\\hat{p}', commutative=True), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 5, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Tuple(Symbol('I', commutative=True))), Integral(Symbol('\\\\hat{p}', commutative=True), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 6, "Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)), Integral(Add(Symbol('I', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Tuple(Symbol('I', commutative=True)))), Add(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2)), Integral(Symbol('\\\\hat{p}', commutative=True), Tuple(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(m_{s})} = \\log{(m_{s})}, then derive \\int \\Omega{(m_{s})} dm_{s} = \\mathbf{M} + m_{s} \\log{(m_{s})} - m_{s}, then obtain (\\int \\Omega{(m_{s})} dm_{s})^{\\mathbf{M}} = (\\mathbf{M} + m_{s} \\log{(m_{s})} - m_{s})^{\\mathbf{M}}", "derivation": "\\Omega{(m_{s})} = \\log{(m_{s})} and \\int \\Omega{(m_{s})} dm_{s} = \\int \\log{(m_{s})} dm_{s} and \\int \\Omega{(m_{s})} dm_{s} = \\mathbf{M} + m_{s} \\log{(m_{s})} - m_{s} and (\\int \\Omega{(m_{s})} dm_{s})^{\\mathbf{M}} = (\\mathbf{M} + m_{s} \\log{(m_{s})} - m_{s})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('m_s', commutative=True)), log(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(log(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Symbol('m_s', commutative=True), log(Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Omega')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Symbol('m_s', commutative=True), log(Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\Psi_{nl})} = \\sin{(\\Psi_{nl})}, then derive \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})}, then obtain \\int 0 d\\Psi_{nl} = \\int (\\cos{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})}) d\\Psi_{nl}", "derivation": "\\operatorname{v_{1}}{(\\Psi_{nl})} = \\sin{(\\Psi_{nl})} and \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\sin{(\\Psi_{nl})} and \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and 0 = \\cos{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})} and \\int 0 d\\Psi_{nl} = \\int (\\cos{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\operatorname{v_{1}}{(\\Psi_{nl})}) d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), sin(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 3, "Derivative(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\psi{(g,n)} = g n, then obtain (\\frac{\\partial}{\\partial g} \\psi{(g,n)})^{n} = n^{n}", "derivation": "\\psi{(g,n)} = g n and \\frac{\\partial}{\\partial g} \\psi{(g,n)} = \\frac{\\partial}{\\partial g} g n and (\\frac{\\partial}{\\partial g} \\psi{(g,n)})^{n} = (\\frac{\\partial}{\\partial g} g n)^{n} and (\\frac{\\partial}{\\partial g} \\psi{(g,n)})^{n} = n^{n}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('g', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('g', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\psi')(Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(Mul(Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\psi')(Symbol('g', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(m_{s})} = \\cos{(m_{s})}, then obtain \\frac{(\\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1) \\cos{(m_{s})}}{m_{s}} = \\frac{\\cos{(m_{s})}}{m_{s}}", "derivation": "\\operatorname{P_{e}}{(m_{s})} = \\cos{(m_{s})} and \\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} = 0 and \\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1 = 1 and \\frac{\\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1}{m_{s}} = \\frac{1}{m_{s}} and - \\frac{\\cos{(m_{s})}}{m_{s}} = - \\frac{\\cos{(m_{s})}}{m_{s} (\\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1)} and - \\frac{(\\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1) \\cos{(m_{s})}}{m_{s}} = - \\frac{\\cos{(m_{s})}}{m_{s}} and \\frac{(\\operatorname{P_{e}}{(m_{s})} - \\cos{(m_{s})} + 1) \\cos{(m_{s})}}{m_{s}} = \\frac{\\cos{(m_{s})}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('m_s', commutative=True)), cos(Symbol('m_s', commutative=True)))"], [["minus", 1, "cos(Symbol('m_s', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1)), Integer(1))"], [["divide", 3, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1))), Pow(Symbol('m_s', commutative=True), Integer(-1)))"], [["divide", 4, "Mul(Integer(-1), Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1)), Pow(cos(Symbol('m_s', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), cos(Symbol('m_s', commutative=True))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1)), Integer(-1)), cos(Symbol('m_s', commutative=True))))"], [["times", 5, "Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1))"], "Equality(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1)), cos(Symbol('m_s', commutative=True))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), cos(Symbol('m_s', commutative=True))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Function('P_e')(Symbol('m_s', commutative=True)), Mul(Integer(-1), cos(Symbol('m_s', commutative=True))), Integer(1)), cos(Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), cos(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mu,F_{H})} = \\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu), then derive \\frac{\\partial}{\\partial F_{H}} \\Psi^{\\dagger}{(\\mu,F_{H})} = 0, then obtain \\frac{\\partial}{\\partial F_{H}} (\\Psi^{\\dagger}{(\\mu,F_{H})} - 1) = \\frac{\\partial}{\\partial F_{H}} (\\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) + \\frac{\\partial}{\\partial F_{H}} \\Psi^{\\dagger}{(\\mu,F_{H})} - 1)", "derivation": "\\Psi^{\\dagger}{(\\mu,F_{H})} = \\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) and \\Psi^{\\dagger}{(\\mu,F_{H})} - 1 = \\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) - 1 and \\frac{\\partial}{\\partial F_{H}} (\\Psi^{\\dagger}{(\\mu,F_{H})} - 1) = \\frac{\\partial}{\\partial F_{H}} (\\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) - 1) and \\frac{\\partial}{\\partial F_{H}} \\Psi^{\\dagger}{(\\mu,F_{H})} = 0 and \\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) + \\frac{\\partial}{\\partial F_{H}} \\Psi^{\\dagger}{(\\mu,F_{H})} = \\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) and \\frac{\\partial}{\\partial F_{H}} (\\Psi^{\\dagger}{(\\mu,F_{H})} - 1) = \\frac{\\partial}{\\partial F_{H}} (\\frac{\\partial}{\\partial \\mu} (F_{H} - \\mu) + \\frac{\\partial}{\\partial F_{H}} \\Psi^{\\dagger}{(\\mu,F_{H})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Add(Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(0))"], [["add", 4, "Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(C_{d})} = C_{d}, then derive \\int s{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + \\eta, then obtain \\int C_{d}^{- C_{d}} s{(C_{d})} s^{C_{d}}{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + \\eta", "derivation": "s{(C_{d})} = C_{d} and s^{C_{d}}{(C_{d})} = C_{d}^{C_{d}} and \\int s{(C_{d})} dC_{d} = \\int C_{d} dC_{d} and C_{d}^{- C_{d}} s{(C_{d})} s^{C_{d}}{(C_{d})} = s{(C_{d})} and \\int s{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + \\eta and \\int C_{d}^{- C_{d}} s{(C_{d})} s^{C_{d}}{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + \\eta", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('s')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('s')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)), Pow(Function('s')(Symbol('C_d', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('s')(Symbol('C_d', commutative=True)), Pow(Function('s')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Function('s')(Symbol('C_d', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integral(Function('s')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Mul(Pow(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))), Function('s')(Symbol('C_d', commutative=True)), Pow(Function('s')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(v_{2})} = \\cos{(v_{2})}, then obtain - \\operatorname{F_{N}}{(v_{2})} - \\cos{(v_{2})} + \\frac{d}{d v_{2}} 2 \\operatorname{F_{N}}{(v_{2})} = - \\operatorname{F_{N}}{(v_{2})} - \\cos{(v_{2})} + \\frac{d}{d v_{2}} (\\operatorname{F_{N}}{(v_{2})} + \\cos{(v_{2})})", "derivation": "\\operatorname{F_{N}}{(v_{2})} = \\cos{(v_{2})} and 2 \\operatorname{F_{N}}{(v_{2})} = \\operatorname{F_{N}}{(v_{2})} + \\cos{(v_{2})} and \\frac{d}{d v_{2}} 2 \\operatorname{F_{N}}{(v_{2})} = \\frac{d}{d v_{2}} (\\operatorname{F_{N}}{(v_{2})} + \\cos{(v_{2})}) and - \\operatorname{F_{N}}{(v_{2})} - \\cos{(v_{2})} + \\frac{d}{d v_{2}} 2 \\operatorname{F_{N}}{(v_{2})} = - \\operatorname{F_{N}}{(v_{2})} - \\cos{(v_{2})} + \\frac{d}{d v_{2}} (\\operatorname{F_{N}}{(v_{2})} + \\cos{(v_{2})})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["add", 1, "Function('F_N')(Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(2), Function('F_N')(Symbol('v_2', commutative=True))), Add(Function('F_N')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('F_N')(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Function('F_N')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["minus", 3, "Add(Function('F_N')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_N')(Symbol('v_2', commutative=True))), Mul(Integer(-1), cos(Symbol('v_2', commutative=True))), Derivative(Mul(Integer(2), Function('F_N')(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('F_N')(Symbol('v_2', commutative=True))), Mul(Integer(-1), cos(Symbol('v_2', commutative=True))), Derivative(Add(Function('F_N')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(\\phi_2)} = \\phi_2, then derive \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} + 1 = 2, then obtain \\frac{d}{d \\phi_2} \\delta^{\\frac{d}{d \\phi_2} \\phi_2 + 1}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 \\delta{(\\phi_2)}", "derivation": "\\delta{(\\phi_2)} = \\phi_2 and \\delta^{2}{(\\phi_2)} = \\phi_2 \\delta{(\\phi_2)} and \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 and \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} + 1 = \\frac{d}{d \\phi_2} \\phi_2 + 1 and \\frac{d}{d \\phi_2} \\delta^{2}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 \\delta{(\\phi_2)} and \\frac{d}{d \\phi_2} \\delta{(\\phi_2)} + 1 = 2 and \\delta^{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)} + 1}{(\\phi_2)} = \\phi_2 \\delta{(\\phi_2)} and \\delta^{2}{(\\phi_2)} = \\delta^{\\frac{d}{d \\phi_2} \\delta{(\\phi_2)} + 1}{(\\phi_2)} and \\delta^{2}{(\\phi_2)} = \\delta^{\\frac{d}{d \\phi_2} \\phi_2 + 1}{(\\phi_2)} and \\frac{d}{d \\phi_2} \\delta^{\\frac{d}{d \\phi_2} \\phi_2 + 1}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\phi_2 \\delta{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["times", 1, "Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Mul(Symbol('\\\\phi_2', commutative=True), Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Add(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('\\\\phi_2', commutative=True), Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 7], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Add(Derivative(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Add(Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 9], "Equality(Derivative(Pow(Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True)), Add(Derivative(Symbol('\\\\phi_2', commutative=True), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Function('\\\\delta')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(k)} = \\sin{(\\log{(k)})}, then derive (\\frac{d}{d k} \\operatorname{P_{e}}{(k)})^{k} = (\\frac{\\cos{(\\log{(k)})}}{k})^{k}, then obtain \\int (\\frac{\\cos{(\\log{(k)})}}{k})^{k} dk = \\int (\\frac{d}{d k} \\sin{(\\log{(k)})})^{k} dk", "derivation": "\\operatorname{P_{e}}{(k)} = \\sin{(\\log{(k)})} and \\frac{d}{d k} \\operatorname{P_{e}}{(k)} = \\frac{d}{d k} \\sin{(\\log{(k)})} and (\\frac{d}{d k} \\operatorname{P_{e}}{(k)})^{k} = (\\frac{d}{d k} \\sin{(\\log{(k)})})^{k} and (\\frac{d}{d k} \\operatorname{P_{e}}{(k)})^{k} = (\\frac{\\cos{(\\log{(k)})}}{k})^{k} and (\\frac{\\cos{(\\log{(k)})}}{k})^{k} = (\\frac{d}{d k} \\sin{(\\log{(k)})})^{k} and \\int (\\frac{\\cos{(\\log{(k)})}}{k})^{k} dk = \\int (\\frac{d}{d k} \\sin{(\\log{(k)})})^{k} dk", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('k', commutative=True)), sin(log(Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Function('P_e')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(sin(log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('P_e')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), cos(log(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), cos(log(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Pow(Derivative(sin(log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), cos(log(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Derivative(sin(log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = \\hat{H} - \\hat{p} + \\phi_2, then derive \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = \\frac{d}{d \\hat{H}} 1", "derivation": "\\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = \\hat{H} - \\hat{p} + \\phi_2 and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - \\hat{p} + \\phi_2) and \\frac{\\partial}{\\partial \\hat{H}} \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = 1 and \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} \\operatorname{n_{2}}{(\\phi_2,\\hat{p},\\hat{H})} = \\frac{d}{d \\hat{H}} 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(V,B)} = B - V and \\varphi^{*}{(E_{\\lambda})} = e^{\\sin{(E_{\\lambda})}}, then obtain \\int (\\varphi^{*}{(E_{\\lambda})} + \\frac{1}{\\mu{(V,B)}}) dE_{\\lambda} = \\int (e^{\\sin{(E_{\\lambda})}} + \\frac{1}{\\mu{(V,B)}}) dE_{\\lambda}", "derivation": "\\mu{(V,B)} = B - V and \\varphi^{*}{(E_{\\lambda})} = e^{\\sin{(E_{\\lambda})}} and \\varphi^{*}{(E_{\\lambda})} + \\frac{1}{B - V} = e^{\\sin{(E_{\\lambda})}} + \\frac{1}{B - V} and \\varphi^{*}{(E_{\\lambda})} + \\frac{1}{\\mu{(V,B)}} = e^{\\sin{(E_{\\lambda})}} + \\frac{1}{\\mu{(V,B)}} and \\int (\\varphi^{*}{(E_{\\lambda})} + \\frac{1}{\\mu{(V,B)}}) dE_{\\lambda} = \\int (e^{\\sin{(E_{\\lambda})}} + \\frac{1}{\\mu{(V,B)}}) dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('E_{\\\\lambda}', commutative=True)), exp(sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 2, "Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Integer(-1))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Integer(-1))), Add(exp(sin(Symbol('E_{\\\\lambda}', commutative=True))), Pow(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\varphi^*')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('\\\\mu')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Integer(-1))), Add(exp(sin(Symbol('E_{\\\\lambda}', commutative=True))), Pow(Function('\\\\mu')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi^*')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('\\\\mu')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(exp(sin(Symbol('E_{\\\\lambda}', commutative=True))), Pow(Function('\\\\mu')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given a{(g_{\\varepsilon})} = \\sin{(g_{\\varepsilon})} and \\mathbf{r}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} 2 \\sin{(g_{\\varepsilon})}, then obtain - r_{0} + \\mathbf{r}{(g_{\\varepsilon})} = - r_{0} + \\frac{d}{d g_{\\varepsilon}} (a{(g_{\\varepsilon})} + \\sin{(g_{\\varepsilon})})", "derivation": "a{(g_{\\varepsilon})} = \\sin{(g_{\\varepsilon})} and a{(g_{\\varepsilon})} + \\sin{(g_{\\varepsilon})} = 2 \\sin{(g_{\\varepsilon})} and \\frac{d}{d g_{\\varepsilon}} (a{(g_{\\varepsilon})} + \\sin{(g_{\\varepsilon})}) = \\frac{d}{d g_{\\varepsilon}} 2 \\sin{(g_{\\varepsilon})} and \\mathbf{r}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} 2 \\sin{(g_{\\varepsilon})} and \\mathbf{r}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} (a{(g_{\\varepsilon})} + \\sin{(g_{\\varepsilon})}) and - r_{0} + \\mathbf{r}{(g_{\\varepsilon})} = - r_{0} + \\frac{d}{d g_{\\varepsilon}} (a{(g_{\\varepsilon})} + \\sin{(g_{\\varepsilon})})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "sin(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('a')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Function('a')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Mul(Integer(2), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Add(Function('a')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 5, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\mathbf{r}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Function('a')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{f}{(c,\\mathbf{g})} = e^{\\mathbf{g} + c}, then derive \\frac{\\partial}{\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = e^{\\mathbf{g} + c}, then obtain \\frac{\\partial^{3}}{\\partial \\mathbf{g}^{2}\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} e^{\\mathbf{g} + c}", "derivation": "\\mathbf{f}{(c,\\mathbf{g})} = e^{\\mathbf{g} + c} and \\frac{\\partial}{\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = \\frac{\\partial}{\\partial c} e^{\\mathbf{g} + c} and \\frac{\\partial}{\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = e^{\\mathbf{g} + c} and \\frac{\\partial^{2}}{\\partial \\mathbf{g}\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} e^{\\mathbf{g} + c} and \\frac{\\partial^{3}}{\\partial \\mathbf{g}^{2}\\partial c} \\mathbf{f}{(c,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} e^{\\mathbf{g} + c}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Derivative(exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\omega{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain - \\cos{(\\hat{H}_l)} + \\frac{d}{d \\hat{H}_l} \\omega{(\\hat{H}_l)} = 0", "derivation": "\\omega{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\omega{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} = 0 and \\omega{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} - 1 = -1 and \\frac{d}{d \\hat{H}_l} (\\omega{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} - 1) = \\frac{d}{d \\hat{H}_l} (-1) and - \\cos{(\\hat{H}_l)} + \\frac{d}{d \\hat{H}_l} \\omega{(\\hat{H}_l)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)), Integer(-1))"], [["differentiate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hat{H}_l', commutative=True))), Derivative(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{s}{(\\theta_1)} = \\log{(\\sin{(\\theta_1)})}, then obtain \\cos{(\\frac{e^{- \\mathbf{s}{(\\theta_1)}}}{\\mathbb{I}})} = \\cos{(\\frac{1}{\\mathbb{I} \\sin{(\\theta_1)}})}", "derivation": "\\mathbf{s}{(\\theta_1)} = \\log{(\\sin{(\\theta_1)})} and - \\mathbf{s}{(\\theta_1)} = - \\log{(\\sin{(\\theta_1)})} and e^{- \\mathbf{s}{(\\theta_1)}} = \\frac{1}{\\sin{(\\theta_1)}} and \\frac{e^{- \\mathbf{s}{(\\theta_1)}}}{\\mathbb{I}} = \\frac{1}{\\mathbb{I} \\sin{(\\theta_1)}} and \\cos{(\\frac{e^{- \\mathbf{s}{(\\theta_1)}}}{\\mathbb{I}})} = \\cos{(\\frac{1}{\\mathbb{I} \\sin{(\\theta_1)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), log(sin(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), log(sin(Symbol('\\\\theta_1', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)))"], [["divide", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True))))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"], [["cos", 4], "Equality(cos(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)))))), cos(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given Z{(r,\\Psi)} = \\sin{(\\Psi + r)}, then derive \\frac{\\frac{\\partial}{\\partial r} Z{(r,\\Psi)}}{\\Psi} = \\frac{\\cos{(\\Psi + r)}}{\\Psi}, then obtain \\frac{\\frac{\\partial}{\\partial r} \\sin{(\\Psi + r)}}{\\Psi} = \\frac{\\cos{(\\Psi + r)}}{\\Psi}", "derivation": "Z{(r,\\Psi)} = \\sin{(\\Psi + r)} and \\frac{\\partial}{\\partial r} Z{(r,\\Psi)} = \\frac{\\partial}{\\partial r} \\sin{(\\Psi + r)} and \\frac{\\frac{\\partial}{\\partial r} Z{(r,\\Psi)}}{\\Psi} = \\frac{\\frac{\\partial}{\\partial r} \\sin{(\\Psi + r)}}{\\Psi} and \\frac{\\frac{\\partial}{\\partial r} Z{(r,\\Psi)}}{\\Psi} = \\frac{\\cos{(\\Psi + r)}}{\\Psi} and \\frac{\\frac{\\partial}{\\partial r} \\sin{(\\Psi + r)}}{\\Psi} = \\frac{\\cos{(\\Psi + r)}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('r', commutative=True), Symbol('\\\\Psi', commutative=True)), sin(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('r', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Function('Z')(Symbol('r', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(Function('Z')(Symbol('r', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Derivative(sin(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\theta,\\varepsilon)} = \\varepsilon^{\\theta}, then obtain \\iint \\mathbf{J}^{2}{(\\theta,\\varepsilon)} d\\theta d\\theta = \\iint \\varepsilon^{2 \\theta} d\\theta d\\theta", "derivation": "\\mathbf{J}{(\\theta,\\varepsilon)} = \\varepsilon^{\\theta} and \\varepsilon \\mathbf{J}{(\\theta,\\varepsilon)} = \\varepsilon \\varepsilon^{\\theta} and \\varepsilon \\varepsilon^{\\theta} \\mathbf{J}{(\\theta,\\varepsilon)} = \\varepsilon \\varepsilon^{2 \\theta} and \\varepsilon \\mathbf{J}^{2}{(\\theta,\\varepsilon)} = \\varepsilon \\varepsilon^{\\theta} \\mathbf{J}{(\\theta,\\varepsilon)} and \\varepsilon \\mathbf{J}^{2}{(\\theta,\\varepsilon)} = \\varepsilon \\varepsilon^{2 \\theta} and \\mathbf{J}^{2}{(\\theta,\\varepsilon)} = \\varepsilon^{2 \\theta} and \\int \\mathbf{J}^{2}{(\\theta,\\varepsilon)} d\\theta = \\int \\varepsilon^{2 \\theta} d\\theta and \\iint \\mathbf{J}^{2}{(\\theta,\\varepsilon)} d\\theta d\\theta = \\iint \\varepsilon^{2 \\theta} d\\theta d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('\\\\theta', commutative=True)))))"], [["divide", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('\\\\theta', commutative=True))))"], [["integrate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 7, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(2), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v_{2},C)} = C + v_{2} and \\operatorname{F_{g}}{(v_{2})} = v_{2} and \\operatorname{n_{1}}{(v_{2},C)} = \\frac{v_{2}}{C + v_{2}}, then obtain (\\frac{\\operatorname{F_{g}}{(v_{2})}}{C + v_{2}})^{C} = \\operatorname{n_{1}}^{C}{(v_{2},C)}", "derivation": "\\operatorname{y^{\\prime}}{(v_{2},C)} = C + v_{2} and \\operatorname{F_{g}}{(v_{2})} = v_{2} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{\\operatorname{y^{\\prime}}{(v_{2},C)}} = \\frac{v_{2}}{\\operatorname{y^{\\prime}}{(v_{2},C)}} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{C + v_{2}} = \\frac{v_{2}}{C + v_{2}} and \\operatorname{n_{1}}{(v_{2},C)} = \\frac{v_{2}}{C + v_{2}} and \\frac{\\operatorname{F_{g}}{(v_{2})}}{C + v_{2}} = \\operatorname{n_{1}}{(v_{2},C)} and (\\frac{\\operatorname{F_{g}}{(v_{2})}}{C + v_{2}})^{C} = \\operatorname{n_{1}}^{C}{(v_{2},C)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], [["divide", 2, "Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('C', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('v_2', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1))), Mul(Symbol('v_2', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Function('F_g')(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Pow(Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('v_2', commutative=True), Pow(Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Function('F_g')(Symbol('v_2', commutative=True))), Function('n_1')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)))"], [["power", 6, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Function('F_g')(Symbol('v_2', commutative=True))), Symbol('C', commutative=True)), Pow(Function('n_1')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(z,v_{1})} = \\frac{\\cos{(v_{1})}}{z}, then obtain \\psi^{*}{(z,v_{1})} + 2 \\int \\psi^{*}{(z,v_{1})} dv_{1} = \\psi^{*}{(z,v_{1})} + 2 \\int \\frac{\\cos{(v_{1})}}{z} dv_{1}", "derivation": "\\psi^{*}{(z,v_{1})} = \\frac{\\cos{(v_{1})}}{z} and \\int \\psi^{*}{(z,v_{1})} dv_{1} = \\int \\frac{\\cos{(v_{1})}}{z} dv_{1} and \\psi^{*}{(z,v_{1})} + \\int \\psi^{*}{(z,v_{1})} dv_{1} = \\psi^{*}{(z,v_{1})} + \\int \\frac{\\cos{(v_{1})}}{z} dv_{1} and \\psi^{*}{(z,v_{1})} + \\int \\frac{\\cos{(v_{1})}}{z} dv_{1} + \\int \\psi^{*}{(z,v_{1})} dv_{1} = \\psi^{*}{(z,v_{1})} + 2 \\int \\frac{\\cos{(v_{1})}}{z} dv_{1} and \\psi^{*}{(z,v_{1})} + 2 \\int \\psi^{*}{(z,v_{1})} dv_{1} = \\psi^{*}{(z,v_{1})} + 2 \\int \\frac{\\cos{(v_{1})}}{z} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["add", 2, "Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Integral(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))))"], [["add", 3, "Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Add(Function('\\\\psi^*')(Symbol('z', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), cos(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{H},\\mathbf{M})} = \\frac{\\mathbf{M}}{\\hat{H}}, then derive \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{A}{(\\hat{H},\\mathbf{M})} d\\hat{H} = P_{g} + \\log{(\\hat{H})}, then obtain Q + \\log{(\\hat{H})} = P_{g} + \\log{(\\hat{H})}", "derivation": "\\mathbf{A}{(\\hat{H},\\mathbf{M})} = \\frac{\\mathbf{M}}{\\hat{H}} and \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{A}{(\\hat{H},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\hat{H}} and \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{A}{(\\hat{H},\\mathbf{M})} d\\hat{H} = \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\hat{H}} d\\hat{H} and \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{A}{(\\hat{H},\\mathbf{M})} d\\hat{H} = P_{g} + \\log{(\\hat{H})} and \\int \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M}}{\\hat{H}} d\\hat{H} = P_{g} + \\log{(\\hat{H})} and Q + \\log{(\\hat{H})} = P_{g} + \\log{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('P_g', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('P_g', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('Q', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('P_g', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\theta_1,\\dot{z},r)} = \\theta_1 (\\dot{z} + r), then obtain \\iint \\frac{\\theta_{2}{(\\theta_1,\\dot{z},r)}}{r} d\\dot{z} dr - \\frac{1}{r} = \\iint \\frac{\\theta_1 (\\dot{z} + r)}{r} d\\dot{z} dr - \\frac{1}{r}", "derivation": "\\theta_{2}{(\\theta_1,\\dot{z},r)} = \\theta_1 (\\dot{z} + r) and \\frac{\\theta_{2}{(\\theta_1,\\dot{z},r)}}{r} = \\frac{\\theta_1 (\\dot{z} + r)}{r} and \\int \\frac{\\theta_{2}{(\\theta_1,\\dot{z},r)}}{r} d\\dot{z} = \\int \\frac{\\theta_1 (\\dot{z} + r)}{r} d\\dot{z} and \\iint \\frac{\\theta_{2}{(\\theta_1,\\dot{z},r)}}{r} d\\dot{z} dr = \\iint \\frac{\\theta_1 (\\dot{z} + r)}{r} d\\dot{z} dr and \\iint \\frac{\\theta_{2}{(\\theta_1,\\dot{z},r)}}{r} d\\dot{z} dr - \\frac{1}{r} = \\iint \\frac{\\theta_1 (\\dot{z} + r)}{r} d\\dot{z} dr - \\frac{1}{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))))"], [["divide", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["minus", 4, "Pow(Symbol('r', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{p},P_{g})} = - \\hat{p} + \\log{(P_{g})} and \\operatorname{n_{1}}{(\\mathbf{D},z^{*})} = \\sin{(\\mathbf{D} + z^{*})}, then obtain \\hat{p} + \\operatorname{n_{1}}{(\\mathbf{D},z^{*})} - \\frac{\\partial}{\\partial P_{g}} (- \\hat{p} + \\log{(P_{g})}) = \\hat{p} + \\sin{(\\mathbf{D} + z^{*})} - \\frac{\\partial}{\\partial P_{g}} (- \\hat{p} + \\log{(P_{g})})", "derivation": "\\mu_{0}{(\\hat{p},P_{g})} = - \\hat{p} + \\log{(P_{g})} and \\frac{\\partial}{\\partial P_{g}} \\mu_{0}{(\\hat{p},P_{g})} = \\frac{\\partial}{\\partial P_{g}} (- \\hat{p} + \\log{(P_{g})}) and \\operatorname{n_{1}}{(\\mathbf{D},z^{*})} = \\sin{(\\mathbf{D} + z^{*})} and \\hat{p} + \\operatorname{n_{1}}{(\\mathbf{D},z^{*})} - \\frac{\\partial}{\\partial P_{g}} \\mu_{0}{(\\hat{p},P_{g})} = \\hat{p} + \\sin{(\\mathbf{D} + z^{*})} - \\frac{\\partial}{\\partial P_{g}} \\mu_{0}{(\\hat{p},P_{g})} and \\hat{p} + \\operatorname{n_{1}}{(\\mathbf{D},z^{*})} - \\frac{\\partial}{\\partial P_{g}} (- \\hat{p} + \\log{(P_{g})}) = \\hat{p} + \\sin{(\\mathbf{D} + z^{*})} - \\frac{\\partial}{\\partial P_{g}} (- \\hat{p} + \\log{(P_{g})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('P_g', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('P_g', commutative=True))))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True)), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('\\\\hat{p}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), log(Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(M,\\mathbf{B})} = - M + e^{\\mathbf{B}}, then obtain \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- M + \\operatorname{A_{x}}{(M,\\mathbf{B})}) d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- 2 M + e^{\\mathbf{B}}) d\\mathbf{B}", "derivation": "\\operatorname{A_{x}}{(M,\\mathbf{B})} = - M + e^{\\mathbf{B}} and - M + \\operatorname{A_{x}}{(M,\\mathbf{B})} = - 2 M + e^{\\mathbf{B}} and \\int (- M + \\operatorname{A_{x}}{(M,\\mathbf{B})}) d\\mathbf{B} = \\int (- 2 M + e^{\\mathbf{B}}) d\\mathbf{B} and \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- M + \\operatorname{A_{x}}{(M,\\mathbf{B})}) d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- 2 M + e^{\\mathbf{B}}) d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 1, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('A_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('A_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('A_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} = \\mathbf{p} \\cos{(\\eta)}, then derive \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 1 = - \\mathbf{p} \\sin{(\\eta)} - 1, then obtain 2 \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 2 = - \\mathbf{p} \\sin{(\\eta)} + \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 2", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} = \\mathbf{p} \\cos{(\\eta)} and - \\eta + \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} = - \\eta + \\mathbf{p} \\cos{(\\eta)} and \\frac{\\partial}{\\partial \\eta} (- \\eta + \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})}) = \\frac{\\partial}{\\partial \\eta} (- \\eta + \\mathbf{p} \\cos{(\\eta)}) and \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 1 = - \\mathbf{p} \\sin{(\\eta)} - 1 and 2 \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 2 = - \\mathbf{p} \\sin{(\\eta)} + \\frac{\\partial}{\\partial \\eta} \\operatorname{V_{\\mathbf{E}}}{(\\eta,\\mathbf{p})} - 2", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), cos(Symbol('\\\\eta', commutative=True))))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), cos(Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), cos(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Integer(-1)))"], [["add", 4, "Add(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Integer(2), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-2)))"]]}, {"prompt": "Given \\varphi{(x,\\pi)} = \\frac{\\partial}{\\partial \\pi} (\\pi + x), then derive \\varphi{(x,\\pi)} = 1, then derive - x + \\varphi{(x,\\pi)} = 1 - x, then obtain (- x + \\varphi{(x,\\pi)} + \\frac{\\partial}{\\partial \\pi} (\\pi + x) - 1)^{x} = (1 - x)^{x}", "derivation": "\\varphi{(x,\\pi)} = \\frac{\\partial}{\\partial \\pi} (\\pi + x) and \\varphi{(x,\\pi)} = 1 and - x + \\varphi{(x,\\pi)} = - x + \\frac{\\partial}{\\partial \\pi} (\\pi + x) and \\frac{\\partial}{\\partial \\pi} (\\pi + x) = 1 and \\varphi{(x,\\pi)} + \\frac{\\partial}{\\partial \\pi} (\\pi + x) = \\varphi{(x,\\pi)} + 1 and - x + \\varphi{(x,\\pi)} = 1 - x and \\varphi{(x,\\pi)} + \\frac{\\partial}{\\partial \\pi} (\\pi + x) - 1 = \\varphi{(x,\\pi)} and (- x + \\varphi{(x,\\pi)})^{x} = (1 - x)^{x} and (- x + \\varphi{(x,\\pi)} + \\frac{\\partial}{\\partial \\pi} (\\pi + x) - 1)^{x} = (1 - x)^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["minus", 5, 1], "Equality(Add(Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)), Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["power", 6, "Symbol('x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('x', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\varphi')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Add(Symbol('\\\\pi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)), Symbol('x', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given r{(g,\\mathbf{s})} = \\mathbf{s} + g, then obtain \\frac{(- g + \\frac{\\partial}{\\partial \\mathbf{s}} r{(g,\\mathbf{s})} - 1) r{(g,\\mathbf{s})}}{g} = - r{(g,\\mathbf{s})}", "derivation": "r{(g,\\mathbf{s})} = \\mathbf{s} + g and - \\mathbf{s} - g + r{(g,\\mathbf{s})} = 0 and \\frac{\\partial}{\\partial \\mathbf{s}} (- \\mathbf{s} - g + r{(g,\\mathbf{s})}) = \\frac{d}{d \\mathbf{s}} 0 and - g + \\frac{\\partial}{\\partial \\mathbf{s}} (- \\mathbf{s} - g + r{(g,\\mathbf{s})}) = - g + \\frac{d}{d \\mathbf{s}} 0 and \\frac{(- g + \\frac{\\partial}{\\partial \\mathbf{s}} (- \\mathbf{s} - g + r{(g,\\mathbf{s})})) r{(g,\\mathbf{s})}}{g} = \\frac{(- g + \\frac{d}{d \\mathbf{s}} 0) r{(g,\\mathbf{s})}}{g} and \\frac{(- g + \\frac{\\partial}{\\partial \\mathbf{s}} r{(g,\\mathbf{s})} - 1) r{(g,\\mathbf{s})}}{g} = - r{(g,\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["times", 4, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Integer(-1)), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Function('r')(Symbol('g', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given s{(p,\\hat{H}_l,A_{1})} = - A_{1} + \\frac{\\hat{H}_l}{p}, then obtain p \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} + \\log{(s{(p,\\hat{H}_l,A_{1})})} = p \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} + \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})}", "derivation": "s{(p,\\hat{H}_l,A_{1})} = - A_{1} + \\frac{\\hat{H}_l}{p} and \\log{(s{(p,\\hat{H}_l,A_{1})})} = \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} and p \\log{(s{(p,\\hat{H}_l,A_{1})})} = p \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} and p \\log{(s{(p,\\hat{H}_l,A_{1})})} + \\log{(s{(p,\\hat{H}_l,A_{1})})} = p \\log{(s{(p,\\hat{H}_l,A_{1})})} + \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} and p \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} + \\log{(s{(p,\\hat{H}_l,A_{1})})} = p \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})} + \\log{(- A_{1} + \\frac{\\hat{H}_l}{p})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))"], [["log", 1], "Equality(log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True))), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))))"], [["divide", 2, "Pow(Symbol('p', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('p', commutative=True), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)))), Mul(Symbol('p', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))))"], [["add", 2, "Mul(Symbol('p', commutative=True), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True))))"], "Equality(Add(Mul(Symbol('p', commutative=True), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)))), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Symbol('p', commutative=True), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('p', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))), log(Function('s')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Symbol('p', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given E{(\\pi)} = \\sin{(\\log{(\\pi)})}, then obtain \\pi + 2 E{(\\pi)} + 2 \\sin{(\\log{(\\pi)})} = \\pi + 4 \\sin{(\\log{(\\pi)})}", "derivation": "E{(\\pi)} = \\sin{(\\log{(\\pi)})} and \\pi + E{(\\pi)} = \\pi + \\sin{(\\log{(\\pi)})} and \\pi + 2 E{(\\pi)} = \\pi + E{(\\pi)} + \\sin{(\\log{(\\pi)})} and \\pi + 2 E{(\\pi)} = \\pi + 2 \\sin{(\\log{(\\pi)})} and \\pi + 2 E{(\\pi)} + 2 \\sin{(\\log{(\\pi)})} = \\pi + 4 \\sin{(\\log{(\\pi)})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\pi', commutative=True)), sin(log(Symbol('\\\\pi', commutative=True))))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('E')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(log(Symbol('\\\\pi', commutative=True)))))"], [["add", 2, "Function('E')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Function('E')(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Function('E')(Symbol('\\\\pi', commutative=True)), sin(log(Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Function('E')(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), sin(log(Symbol('\\\\pi', commutative=True))))))"], [["add", 4, "Mul(Integer(2), sin(log(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Function('E')(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), sin(log(Symbol('\\\\pi', commutative=True))))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(4), sin(log(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(U,H)} = H e^{U} and \\operatorname{A_{y}}{(U)} = e^{U}, then obtain - \\frac{e^{- U} \\frac{\\partial}{\\partial H} H e^{U}}{H} = - \\frac{e^{- U} \\frac{\\partial}{\\partial H} H \\operatorname{A_{y}}{(U)}}{H}", "derivation": "\\operatorname{M_{E}}{(U,H)} = H e^{U} and \\frac{\\partial}{\\partial H} \\operatorname{M_{E}}{(U,H)} = \\frac{\\partial}{\\partial H} H e^{U} and - \\frac{\\partial}{\\partial H} \\operatorname{M_{E}}{(U,H)} = - \\frac{\\partial}{\\partial H} H e^{U} and \\operatorname{A_{y}}{(U)} = e^{U} and - \\frac{\\partial}{\\partial H} \\operatorname{M_{E}}{(U,H)} = - \\frac{\\partial}{\\partial H} H \\operatorname{A_{y}}{(U)} and - \\frac{\\partial}{\\partial H} H e^{U} = - \\frac{\\partial}{\\partial H} H \\operatorname{A_{y}}{(U)} and - \\frac{e^{- U} \\frac{\\partial}{\\partial H} H e^{U}}{H} = - \\frac{e^{- U} \\frac{\\partial}{\\partial H} H \\operatorname{A_{y}}{(U)}}{H}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('U', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('U', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('M_E')(Symbol('U', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Derivative(Function('M_E')(Symbol('U', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Symbol('H', commutative=True), Function('A_y')(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Derivative(Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Symbol('H', commutative=True), Function('A_y')(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["divide", 6, "Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Mul(Symbol('H', commutative=True), exp(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Mul(Symbol('H', commutative=True), Function('A_y')(Symbol('U', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(c)} = \\sin{(c)}, then obtain \\frac{\\hat{H}_{\\lambda}^{c}{(c)}}{c + \\hat{H}_{\\lambda}{(c)}} = \\frac{\\sin^{c}{(c)}}{c + \\hat{H}_{\\lambda}{(c)}}", "derivation": "\\hat{H}_{\\lambda}{(c)} = \\sin{(c)} and \\hat{H}_{\\lambda}^{c}{(c)} = \\sin^{c}{(c)} and c + \\hat{H}_{\\lambda}{(c)} = c + \\sin{(c)} and \\frac{\\hat{H}_{\\lambda}^{c}{(c)}}{c + \\sin{(c)}} = \\frac{\\sin^{c}{(c)}}{c + \\sin{(c)}} and \\frac{\\hat{H}_{\\lambda}^{c}{(c)}}{c + \\hat{H}_{\\lambda}{(c)}} = \\frac{\\sin^{c}{(c)}}{c + \\hat{H}_{\\lambda}{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True))), Add(Symbol('c', commutative=True), sin(Symbol('c', commutative=True))))"], [["divide", 2, "Add(Symbol('c', commutative=True), sin(Symbol('c', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('c', commutative=True), sin(Symbol('c', commutative=True))), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Pow(Add(Symbol('c', commutative=True), sin(Symbol('c', commutative=True))), Integer(-1)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True))), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Pow(Add(Symbol('c', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('c', commutative=True))), Integer(-1)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True))))"]]}, {"prompt": "Given c{(\\mathbf{P},r)} = \\log{(\\mathbf{P} r)} and \\sigma_{x}{(\\mathbf{P},r)} = \\log{(\\mathbf{P} r)}, then obtain \\frac{c^{2}{(\\mathbf{P},r)}}{\\mathbf{P} r} = \\frac{\\sigma_{x}{(\\mathbf{P},r)} c{(\\mathbf{P},r)}}{\\mathbf{P} r}", "derivation": "c{(\\mathbf{P},r)} = \\log{(\\mathbf{P} r)} and \\frac{c{(\\mathbf{P},r)}}{\\mathbf{P} r} = \\frac{\\log{(\\mathbf{P} r)}}{\\mathbf{P} r} and \\frac{c{(\\mathbf{P},r)} \\log{(\\mathbf{P} r)}}{\\mathbf{P} r} = \\frac{\\log{(\\mathbf{P} r)}^{2}}{\\mathbf{P} r} and \\frac{c^{2}{(\\mathbf{P},r)}}{\\mathbf{P} r} = \\frac{c{(\\mathbf{P},r)} \\log{(\\mathbf{P} r)}}{\\mathbf{P} r} and \\sigma_{x}{(\\mathbf{P},r)} = \\log{(\\mathbf{P} r)} and \\frac{c^{2}{(\\mathbf{P},r)}}{\\mathbf{P} r} = \\frac{\\sigma_{x}{(\\mathbf{P},r)} c{(\\mathbf{P},r)}}{\\mathbf{P} r}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))))"], [["times", 2, "log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Pow(log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True)), Function('c')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(I,\\hat{H})} = \\cos{(\\hat{H}^{I})} and \\operatorname{t_{2}}{(I,\\hat{H})} = \\hat{H}^{I}, then obtain 0^{\\hat{H}} = ((-1 + \\frac{\\cos{(\\operatorname{t_{2}}{(I,\\hat{H})})}}{\\mathbf{J}_P{(I,\\hat{H})}}) \\mathbf{J}_P{(I,\\hat{H})})^{\\hat{H}}", "derivation": "\\mathbf{J}_P{(I,\\hat{H})} = \\cos{(\\hat{H}^{I})} and 1 = \\frac{\\cos{(\\hat{H}^{I})}}{\\mathbf{J}_P{(I,\\hat{H})}} and 0 = -1 + \\frac{\\cos{(\\hat{H}^{I})}}{\\mathbf{J}_P{(I,\\hat{H})}} and 0 = (-1 + \\frac{\\cos{(\\hat{H}^{I})}}{\\mathbf{J}_P{(I,\\hat{H})}}) \\mathbf{J}_P{(I,\\hat{H})} and 0^{\\hat{H}} = ((-1 + \\frac{\\cos{(\\hat{H}^{I})}}{\\mathbf{J}_P{(I,\\hat{H})}}) \\mathbf{J}_P{(I,\\hat{H})})^{\\hat{H}} and \\operatorname{t_{2}}{(I,\\hat{H})} = \\hat{H}^{I} and 0^{\\hat{H}} = ((-1 + \\frac{\\cos{(\\operatorname{t_{2}}{(I,\\hat{H})})}}{\\mathbf{J}_P{(I,\\hat{H})}}) \\mathbf{J}_P{(I,\\hat{H})})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), cos(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True)))))"], [["minus", 2, 1], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True))))))"], [["times", 3, "Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(0), Mul(Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True))))), Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True))))), Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Add(Integer(-1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), cos(Function('t_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))))), Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\delta{(M,\\Psi)} = - M + \\Psi, then obtain - \\int \\Psi (- M + \\Psi) d\\Psi + \\int \\delta{(M,\\Psi)} d\\Psi = - \\int \\Psi (- M + \\Psi) d\\Psi + \\int (- M + \\Psi) d\\Psi", "derivation": "\\delta{(M,\\Psi)} = - M + \\Psi and \\int \\delta{(M,\\Psi)} d\\Psi = \\int (- M + \\Psi) d\\Psi and \\Psi \\delta{(M,\\Psi)} = \\Psi (- M + \\Psi) and \\int \\Psi \\delta{(M,\\Psi)} d\\Psi = \\int \\Psi (- M + \\Psi) d\\Psi and - \\int \\Psi \\delta{(M,\\Psi)} d\\Psi + \\int \\delta{(M,\\Psi)} d\\Psi = - \\int \\Psi \\delta{(M,\\Psi)} d\\Psi + \\int (- M + \\Psi) d\\Psi and - \\int \\Psi (- M + \\Psi) d\\Psi + \\int \\delta{(M,\\Psi)} d\\Psi = - \\int \\Psi (- M + \\Psi) d\\Psi + \\int (- M + \\Psi) d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Integral(Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Integral(Function('\\\\delta')(Symbol('M', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given k{(x^\\prime)} = \\log{(x^\\prime)}, then obtain e^{2 \\int 0 dx^\\prime} = 1", "derivation": "k{(x^\\prime)} = \\log{(x^\\prime)} and k{(x^\\prime)} - \\log{(x^\\prime)} = 0 and (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} = 0 and \\int (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} dx^\\prime = \\int 0 dx^\\prime and 2 \\int (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} dx^\\prime = \\int 0 dx^\\prime + \\int (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} dx^\\prime and e^{2 \\int (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} dx^\\prime} = e^{\\int (k{(x^\\prime)} - \\log{(x^\\prime)}) k{(x^\\prime)} dx^\\prime} and e^{2 \\int 0 dx^\\prime} = 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Integer(0))"], [["times", 2, "Function('k')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["add", 4, "Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["exp", 5], "Equality(exp(Mul(Integer(2), Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))), exp(Integral(Mul(Add(Function('k')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Function('k')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(exp(Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\theta_{2}{(\\sigma_p,L_{\\varepsilon})} = \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon}, then obtain \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\theta_{2}{(\\sigma_p,L_{\\varepsilon})} - \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon} = 0", "derivation": "\\theta_{2}{(\\sigma_p,L_{\\varepsilon})} = \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon} and \\frac{\\partial}{\\partial \\sigma_p} \\theta_{2}{(\\sigma_p,L_{\\varepsilon})} = \\frac{\\partial}{\\partial \\sigma_p} \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon} and \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\theta_{2}{(\\sigma_p,L_{\\varepsilon})} = \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon} and \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\theta_{2}{(\\sigma_p,L_{\\varepsilon})} - \\frac{\\partial^{2}}{\\partial L_{\\varepsilon}\\partial \\sigma_p} \\int L_{\\varepsilon} \\sigma_p dL_{\\varepsilon} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\theta_2')(Symbol('\\\\sigma_p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\mu{(v)} = \\log{(v)} and t{(v_{x},G)} = G^{v_{x}}, then obtain \\frac{(t{(v_{x},G)} + \\frac{\\log{(v)}}{v})^{v_{x}}}{\\mu{(v)}} = \\frac{(G^{v_{x}} + \\frac{\\log{(v)}}{v})^{v_{x}}}{\\mu{(v)}}", "derivation": "\\mu{(v)} = \\log{(v)} and \\frac{\\mu{(v)}}{v} = \\frac{\\log{(v)}}{v} and t{(v_{x},G)} = G^{v_{x}} and t{(v_{x},G)} + \\frac{\\mu{(v)}}{v} = G^{v_{x}} + \\frac{\\mu{(v)}}{v} and (t{(v_{x},G)} + \\frac{\\mu{(v)}}{v})^{v_{x}} = (G^{v_{x}} + \\frac{\\mu{(v)}}{v})^{v_{x}} and \\frac{(t{(v_{x},G)} + \\frac{\\mu{(v)}}{v})^{v_{x}}}{\\mu{(v)}} = \\frac{(G^{v_{x}} + \\frac{\\mu{(v)}}{v})^{v_{x}}}{\\mu{(v)}} and \\frac{(t{(v_{x},G)} + \\frac{\\log{(v)}}{v})^{v_{x}}}{\\mu{(v)}} = \\frac{(G^{v_{x}} + \\frac{\\log{(v)}}{v})^{v_{x}}}{\\mu{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["divide", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('v', commutative=True))))"], ["get_premise", "Equality(Function('t')(Symbol('v_x', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 3, "Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))"], "Equality(Add(Function('t')(Symbol('v_x', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))), Add(Pow(Symbol('G', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))))"], [["power", 4, "Symbol('v_x', commutative=True)"], "Equality(Pow(Add(Function('t')(Symbol('v_x', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Add(Pow(Symbol('G', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)))"], [["divide", 5, "Function('\\\\mu')(Symbol('v', commutative=True))"], "Equality(Mul(Pow(Add(Function('t')(Symbol('v_x', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Function('\\\\mu')(Symbol('v', commutative=True)), Integer(-1))), Mul(Pow(Add(Pow(Symbol('G', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Function('\\\\mu')(Symbol('v', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Add(Function('t')(Symbol('v_x', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Function('\\\\mu')(Symbol('v', commutative=True)), Integer(-1))), Mul(Pow(Add(Pow(Symbol('G', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), log(Symbol('v', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Function('\\\\mu')(Symbol('v', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given V{(\\mathbf{M})} = \\cos{(e^{\\mathbf{M}})}, then obtain - V{(\\mathbf{M})} e^{- \\mathbf{M}} + e^{- \\mathbf{M}} \\frac{d}{d \\mathbf{M}} V{(\\mathbf{M})} = - \\sin{(e^{\\mathbf{M}})} - e^{- \\mathbf{M}} \\cos{(e^{\\mathbf{M}})}", "derivation": "V{(\\mathbf{M})} = \\cos{(e^{\\mathbf{M}})} and V{(\\mathbf{M})} e^{- \\mathbf{M}} = e^{- \\mathbf{M}} \\cos{(e^{\\mathbf{M}})} and \\frac{d}{d \\mathbf{M}} V{(\\mathbf{M})} e^{- \\mathbf{M}} = \\frac{d}{d \\mathbf{M}} e^{- \\mathbf{M}} \\cos{(e^{\\mathbf{M}})} and - V{(\\mathbf{M})} e^{- \\mathbf{M}} + e^{- \\mathbf{M}} \\frac{d}{d \\mathbf{M}} V{(\\mathbf{M})} = - \\sin{(e^{\\mathbf{M}})} - e^{- \\mathbf{M}} \\cos{(e^{\\mathbf{M}})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{M}', commutative=True)), cos(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('V')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), cos(exp(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Mul(Function('V')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), cos(exp(Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('V')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), cos(exp(Symbol('\\\\mathbf{M}', commutative=True))))))"]]}, {"prompt": "Given E{(p,F_{x},\\varepsilon)} = - F_{x} + \\varepsilon + p, then obtain \\sin{(\\frac{\\partial}{\\partial p} (F_{x} - \\varepsilon - p + E{(p,F_{x},\\varepsilon)}))} = \\sin{(\\frac{d}{d p} 0)}", "derivation": "E{(p,F_{x},\\varepsilon)} = - F_{x} + \\varepsilon + p and F_{x} - \\varepsilon - p + E{(p,F_{x},\\varepsilon)} = 0 and \\frac{\\partial}{\\partial p} (F_{x} - \\varepsilon - p + E{(p,F_{x},\\varepsilon)}) = \\frac{d}{d p} 0 and \\sin{(\\frac{\\partial}{\\partial p} (F_{x} - \\varepsilon - p + E{(p,F_{x},\\varepsilon)}))} = \\sin{(\\frac{d}{d p} 0)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('p', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)), Function('E')(Symbol('p', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)), Function('E')(Symbol('p', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)), Function('E')(Symbol('p', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), sin(Derivative(Integer(0), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} = f_{E} (\\hbar + \\mathbf{J}_P), then derive - V_{\\mathbf{E}} - \\frac{\\hbar^{2} f_{E}}{2} - \\hbar \\mathbf{J}_P f_{E} + \\int \\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} d\\hbar = 0, then obtain - V_{\\mathbf{E}} - \\frac{\\hbar^{2} f_{E}}{2} - \\hbar \\mathbf{J}_P f_{E} + \\int f_{E} (\\hbar + \\mathbf{J}_P) d\\hbar = 0", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} = f_{E} (\\hbar + \\mathbf{J}_P) and \\int \\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} d\\hbar = \\int f_{E} (\\hbar + \\mathbf{J}_P) d\\hbar and - \\int f_{E} (\\hbar + \\mathbf{J}_P) d\\hbar + \\int \\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} d\\hbar = 0 and - V_{\\mathbf{E}} - \\frac{\\hbar^{2} f_{E}}{2} - \\hbar \\mathbf{J}_P f_{E} + \\int \\operatorname{F_{H}}{(\\mathbf{J}_P,\\hbar,f_{E})} d\\hbar = 0 and - V_{\\mathbf{E}} - \\frac{\\hbar^{2} f_{E}}{2} - \\hbar \\mathbf{J}_P f_{E} + \\int f_{E} (\\hbar + \\mathbf{J}_P) d\\hbar = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))), Integral(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Integral(Function('F_H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('f_E', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(x^\\prime)} = \\int \\sin{(x^\\prime)} dx^\\prime, then derive \\frac{d}{d x^\\prime} \\operatorname{v_{t}}{(x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} (\\mathbf{s} - \\cos{(x^\\prime)}), then obtain \\frac{\\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime}{\\sin{(x^\\prime)}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{s} - \\cos{(x^\\prime)})}{\\sin{(x^\\prime)}}", "derivation": "\\operatorname{v_{t}}{(x^\\prime)} = \\int \\sin{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} \\operatorname{v_{t}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} \\operatorname{v_{t}}{(x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} (\\mathbf{s} - \\cos{(x^\\prime)}) and \\frac{\\frac{d}{d x^\\prime} \\operatorname{v_{t}}{(x^\\prime)}}{\\sin{(x^\\prime)}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{s} - \\cos{(x^\\prime)})}{\\sin{(x^\\prime)}} and \\frac{\\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime}{\\sin{(x^\\prime)}} = \\frac{\\frac{\\partial}{\\partial x^\\prime} (\\mathbf{s} - \\cos{(x^\\prime)})}{\\sin{(x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('x^\\\\prime', commutative=True)), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('v_t')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 3, "sin(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Function('v_t')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{E}{(P_{g})} = \\log{(\\sin{(P_{g})})} and \\operatorname{M_{E}}{(P_{g})} = \\frac{d}{d P_{g}} \\mathbf{E}^{2}{(P_{g})}, then obtain \\operatorname{M_{E}}{(P_{g})} - \\frac{d}{d P_{g}} \\log{(\\sin{(P_{g})})}^{2} = 0", "derivation": "\\mathbf{E}{(P_{g})} = \\log{(\\sin{(P_{g})})} and \\mathbf{E}^{2}{(P_{g})} = \\mathbf{E}{(P_{g})} \\log{(\\sin{(P_{g})})} and \\frac{d}{d P_{g}} \\mathbf{E}^{2}{(P_{g})} = \\frac{d}{d P_{g}} \\mathbf{E}{(P_{g})} \\log{(\\sin{(P_{g})})} and \\operatorname{M_{E}}{(P_{g})} = \\frac{d}{d P_{g}} \\mathbf{E}^{2}{(P_{g})} and \\operatorname{M_{E}}{(P_{g})} = \\frac{d}{d P_{g}} \\mathbf{E}{(P_{g})} \\log{(\\sin{(P_{g})})} and \\operatorname{M_{E}}{(P_{g})} - \\frac{d}{d P_{g}} \\mathbf{E}{(P_{g})} \\log{(\\sin{(P_{g})})} = 0 and \\operatorname{M_{E}}{(P_{g})} - \\frac{d}{d P_{g}} \\log{(\\sin{(P_{g})})}^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True)))))"], [["differentiate", 2, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), Integer(2)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('P_g', commutative=True)), Derivative(Pow(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), Integer(2)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('M_E')(Symbol('P_g', commutative=True)), Derivative(Mul(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Mul(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1)))"], "Equality(Add(Function('M_E')(Symbol('P_g', commutative=True)), Mul(Integer(-1), Derivative(Mul(Function('\\\\mathbf{E}')(Symbol('P_g', commutative=True)), log(sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Function('M_E')(Symbol('P_g', commutative=True)), Mul(Integer(-1), Derivative(Pow(log(sin(Symbol('P_g', commutative=True))), Integer(2)), Tuple(Symbol('P_g', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\rho{(f,x^\\prime)} = \\frac{\\log{(x^\\prime)}}{f}, then obtain (x^\\prime \\rho{(f,x^\\prime)})^{f} (\\frac{x^\\prime \\log{(x^\\prime)}}{f})^{- f} = 1", "derivation": "\\rho{(f,x^\\prime)} = \\frac{\\log{(x^\\prime)}}{f} and x^\\prime \\rho{(f,x^\\prime)} = \\frac{x^\\prime \\log{(x^\\prime)}}{f} and (x^\\prime \\rho{(f,x^\\prime)})^{f} = (\\frac{x^\\prime \\log{(x^\\prime)}}{f})^{f} and \\frac{(x^\\prime \\rho{(f,x^\\prime)})^{f} \\log{(x^\\prime)}}{f} = \\frac{(\\frac{x^\\prime \\log{(x^\\prime)}}{f})^{f} \\log{(x^\\prime)}}{f} and - \\frac{(x^\\prime \\rho{(f,x^\\prime)})^{f} (- \\hat{X} + \\frac{\\log{(x^\\prime)}}{f}) \\log{(x^\\prime)}}{\\hat{X} f} = - \\frac{(\\frac{x^\\prime \\log{(x^\\prime)}}{f})^{f} (- \\hat{X} + \\frac{\\log{(x^\\prime)}}{f}) \\log{(x^\\prime)}}{\\hat{X} f} and (x^\\prime \\rho{(f,x^\\prime)})^{f} (\\frac{x^\\prime \\log{(x^\\prime)}}{f})^{- f} = 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True))))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True)))))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True)))), log(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True)))), log(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), log(Symbol('x^\\\\prime', commutative=True)))), log(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\rho')(Symbol('f', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('f', commutative=True)), Pow(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Symbol('f', commutative=True)))), Integer(1))"]]}, {"prompt": "Given Z{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})}, then derive \\int Z{(L_{\\varepsilon})} dL_{\\varepsilon} = g - \\cos{(L_{\\varepsilon})}, then obtain \\int \\sin{(L_{\\varepsilon})} dL_{\\varepsilon} = g - \\cos{(L_{\\varepsilon})}", "derivation": "Z{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\int Z{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\sin{(L_{\\varepsilon})} dL_{\\varepsilon} and \\int Z{(L_{\\varepsilon})} dL_{\\varepsilon} = g - \\cos{(L_{\\varepsilon})} and \\int \\sin{(L_{\\varepsilon})} dL_{\\varepsilon} = g - \\cos{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('g', commutative=True), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('g', commutative=True), Mul(Integer(-1), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given T{(x,\\hat{H}_l)} = - \\hat{H}_l + x, then obtain \\sin{(x + \\frac{T{(x,\\hat{H}_l)}}{- \\hat{H}_l + x})} = \\sin{(x + 1)}", "derivation": "T{(x,\\hat{H}_l)} = - \\hat{H}_l + x and \\frac{T{(x,\\hat{H}_l)}}{- \\hat{H}_l + x} = 1 and x + \\frac{T{(x,\\hat{H}_l)}}{- \\hat{H}_l + x} = x + 1 and \\sin{(x + \\frac{T{(x,\\hat{H}_l)}}{- \\hat{H}_l + x})} = \\sin{(x + 1)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('x', commutative=True)), Integer(-1)), Function('T')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(1))"], [["add", 2, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('x', commutative=True)), Integer(-1)), Function('T')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Add(Symbol('x', commutative=True), Integer(1)))"], [["sin", 3], "Equality(sin(Add(Symbol('x', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('x', commutative=True)), Integer(-1)), Function('T')(Symbol('x', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))), sin(Add(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\operatorname{z^{*}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain 0 = \\dot{\\mathbf{r}}{(\\mathbf{p})} - \\cos{(\\mathbf{p})}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\operatorname{z^{*}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\operatorname{z^{*}}{(\\mathbf{p})} = \\dot{\\mathbf{r}}{(\\mathbf{p})} and \\operatorname{z^{*}}{(\\mathbf{p})} - \\cos{(\\mathbf{p})} = \\dot{\\mathbf{r}}{(\\mathbf{p})} - \\cos{(\\mathbf{p})} and 0 = \\dot{\\mathbf{r}}{(\\mathbf{p})} - \\cos{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('z^*')(Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 3, "cos(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('z^*')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given G{(\\mathbf{v},p)} = \\mathbf{v} - p, then obtain - \\int G{(\\mathbf{v},p)} d\\mathbf{v} = - M - \\frac{\\mathbf{v}^{2}}{2} + \\mathbf{v} p", "derivation": "G{(\\mathbf{v},p)} = \\mathbf{v} - p and \\int G{(\\mathbf{v},p)} d\\mathbf{v} = \\int (\\mathbf{v} - p) d\\mathbf{v} and - \\int G{(\\mathbf{v},p)} d\\mathbf{v} = - \\int (\\mathbf{v} - p) d\\mathbf{v} and - \\int G{(\\mathbf{v},p)} d\\mathbf{v} = - M - \\frac{\\mathbf{v}^{2}}{2} + \\mathbf{v} p", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('G')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(t)} = \\frac{d}{d t} \\sin{(t)}, then derive \\mathbf{J}_f{(t)} = \\cos{(t)}, then obtain \\frac{\\sin{(t)} + \\cos{(t)} \\frac{d}{d t} \\sin{(t)}}{\\frac{d}{d t} \\sin{(t)}} = \\frac{\\sin{(t)} + (\\frac{d}{d t} \\sin{(t)})^{2}}{\\frac{d}{d t} \\sin{(t)}}", "derivation": "\\mathbf{J}_f{(t)} = \\frac{d}{d t} \\sin{(t)} and \\mathbf{J}_f{(t)} \\frac{d}{d t} \\sin{(t)} = (\\frac{d}{d t} \\sin{(t)})^{2} and \\mathbf{J}_f{(t)} \\frac{d}{d t} \\sin{(t)} + \\sin{(t)} = \\sin{(t)} + (\\frac{d}{d t} \\sin{(t)})^{2} and \\frac{\\mathbf{J}_f{(t)} \\frac{d}{d t} \\sin{(t)} + \\sin{(t)}}{\\frac{d}{d t} \\sin{(t)}} = \\frac{\\sin{(t)} + (\\frac{d}{d t} \\sin{(t)})^{2}}{\\frac{d}{d t} \\sin{(t)}} and \\mathbf{J}_f{(t)} = \\cos{(t)} and \\frac{\\sin{(t)} + \\cos{(t)} \\frac{d}{d t} \\sin{(t)}}{\\frac{d}{d t} \\sin{(t)}} = \\frac{\\sin{(t)} + (\\frac{d}{d t} \\sin{(t)})^{2}}{\\frac{d}{d t} \\sin{(t)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True)), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["times", 1, "Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True)), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(2)))"], [["add", 2, "sin(Symbol('t', commutative=True))"], "Equality(Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True)), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), sin(Symbol('t', commutative=True))), Add(sin(Symbol('t', commutative=True)), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(2))))"], [["divide", 3, "Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True)), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), sin(Symbol('t', commutative=True))), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))), Mul(Add(sin(Symbol('t', commutative=True)), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Add(sin(Symbol('t', commutative=True)), Mul(cos(Symbol('t', commutative=True)), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))), Mul(Add(sin(Symbol('t', commutative=True)), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given l{(C_{d})} = C_{d}, then obtain - 3 C_{d}^{C_{d}} + l{(C_{d})} = C_{d} - 3 C_{d}^{C_{d}}", "derivation": "l{(C_{d})} = C_{d} and l^{C_{d}}{(C_{d})} = C_{d}^{C_{d}} and - C_{d}^{C_{d}} + l{(C_{d})} = C_{d} - C_{d}^{C_{d}} and - C_{d}^{C_{d}} + l{(C_{d})} - l^{C_{d}}{(C_{d})} = C_{d} - C_{d}^{C_{d}} - l^{C_{d}}{(C_{d})} and - 2 C_{d}^{C_{d}} + l{(C_{d})} = C_{d} - 2 C_{d}^{C_{d}} and - 3 C_{d}^{C_{d}} + l{(C_{d})} = C_{d} - 3 C_{d}^{C_{d}}", "srepr_derivation": [["renaming_premise", "Equality(Function('l')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('l')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))"], [["minus", 1, "Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Function('l')(Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))))"], [["minus", 3, "Pow(Function('l')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Function('l')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Pow(Function('l')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Mul(Integer(-1), Pow(Function('l')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Function('l')(Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Integer(2), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))))"], [["minus", 5, "Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(3), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Function('l')(Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Integer(3), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} = \\frac{\\mathbf{F} + n}{\\mathbf{s}} and k{(v_{z})} = v_{z}, then derive \\int \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} d\\mathbf{F} = \\frac{\\mathbf{F}^{2}}{2 \\mathbf{s}} + \\frac{\\mathbf{F} n}{\\mathbf{s}} + v_{z}, then obtain \\int \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} d\\mathbf{F} = \\frac{\\mathbf{F}^{2}}{2 \\mathbf{s}} + \\frac{\\mathbf{F} n}{\\mathbf{s}} + k{(v_{z})}", "derivation": "\\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} = \\frac{\\mathbf{F} + n}{\\mathbf{s}} and \\int \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} d\\mathbf{F} = \\int \\frac{\\mathbf{F} + n}{\\mathbf{s}} d\\mathbf{F} and \\int \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} d\\mathbf{F} = \\frac{\\mathbf{F}^{2}}{2 \\mathbf{s}} + \\frac{\\mathbf{F} n}{\\mathbf{s}} + v_{z} and k{(v_{z})} = v_{z} and \\int \\mathbf{S}{(\\mathbf{s},n,\\mathbf{F})} d\\mathbf{F} = \\frac{\\mathbf{F}^{2}}{2 \\mathbf{s}} + \\frac{\\mathbf{F} n}{\\mathbf{s}} + k{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Function('k')(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given U{(\\psi,\\phi_2)} = e^{- \\phi_2 + \\psi}, then derive F_{N} + \\frac{\\partial}{\\partial \\psi} U{(\\psi,\\phi_2)} = \\mathbf{p} + e^{- \\phi_2 + \\psi}, then obtain F_{N} + \\phi_2 + \\frac{\\partial}{\\partial \\psi} U{(\\psi,\\phi_2)} = \\mathbf{p} + \\phi_2 + e^{- \\phi_2 + \\psi}", "derivation": "U{(\\psi,\\phi_2)} = e^{- \\phi_2 + \\psi} and \\frac{\\partial}{\\partial \\phi_2} U{(\\psi,\\phi_2)} = \\frac{\\partial}{\\partial \\phi_2} e^{- \\phi_2 + \\psi} and \\frac{\\partial^{2}}{\\partial \\psi\\partial \\phi_2} U{(\\psi,\\phi_2)} = \\frac{\\partial^{2}}{\\partial \\psi\\partial \\phi_2} e^{- \\phi_2 + \\psi} and \\int \\frac{\\partial^{2}}{\\partial \\psi\\partial \\phi_2} U{(\\psi,\\phi_2)} d\\phi_2 = \\int \\frac{\\partial^{2}}{\\partial \\psi\\partial \\phi_2} e^{- \\phi_2 + \\psi} d\\phi_2 and F_{N} + \\frac{\\partial}{\\partial \\psi} U{(\\psi,\\phi_2)} = \\mathbf{p} + e^{- \\phi_2 + \\psi} and F_{N} + \\phi_2 + \\frac{\\partial}{\\partial \\psi} U{(\\psi,\\phi_2)} = \\mathbf{p} + \\phi_2 + e^{- \\phi_2 + \\psi}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_N', commutative=True), Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{p}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Symbol('\\\\phi_2', commutative=True), Derivative(Function('U')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\phi_2', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\dot{z})} = \\cos{(\\dot{z})}, then obtain \\int \\frac{d}{d \\dot{z}} (\\operatorname{v_{y}}{(\\dot{z})} + \\cos{(\\dot{z})}) d\\dot{z} = \\int \\frac{d}{d \\dot{z}} 2 \\cos{(\\dot{z})} d\\dot{z}", "derivation": "\\operatorname{v_{y}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\operatorname{v_{y}}{(\\dot{z})} + \\cos{(\\dot{z})} = 2 \\cos{(\\dot{z})} and \\frac{d}{d \\dot{z}} (\\operatorname{v_{y}}{(\\dot{z})} + \\cos{(\\dot{z})}) = \\frac{d}{d \\dot{z}} 2 \\cos{(\\dot{z})} and \\int \\frac{d}{d \\dot{z}} (\\operatorname{v_{y}}{(\\dot{z})} + \\cos{(\\dot{z})}) d\\dot{z} = \\int \\frac{d}{d \\dot{z}} 2 \\cos{(\\dot{z})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Derivative(Add(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Derivative(Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\dot{z})} = e^{\\dot{z}}, then obtain \\mathbf{J}_P^{2}{(\\dot{z})} e^{- 2 \\dot{z}} = 1", "derivation": "\\mathbf{J}_P{(\\dot{z})} = e^{\\dot{z}} and \\mathbf{J}_P{(\\dot{z})} e^{- \\dot{z}} = 1 and \\mathbf{J}_P^{2}{(\\dot{z})} e^{- \\dot{z}} = \\mathbf{J}_P{(\\dot{z})} and \\mathbf{J}_P^{2}{(\\dot{z})} e^{- 2 \\dot{z}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))), Integer(1))"], [["times", 1, "Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))), Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given l{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\operatorname{M_{E}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then obtain \\operatorname{M_{E}}^{3}{(\\mathbf{v})} l{(\\mathbf{v})} = \\operatorname{M_{E}}^{4}{(\\mathbf{v})}", "derivation": "l{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and l{(\\mathbf{v})} \\sin{(\\mathbf{v})} = \\sin^{2}{(\\mathbf{v})} and l{(\\mathbf{v})} \\sin^{3}{(\\mathbf{v})} = \\sin^{4}{(\\mathbf{v})} and \\operatorname{M_{E}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\operatorname{M_{E}}^{3}{(\\mathbf{v})} l{(\\mathbf{v})} = \\operatorname{M_{E}}^{4}{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Function('l')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["times", 2, "Pow(sin(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))"], "Equality(Mul(Function('l')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{v}', commutative=True)), Integer(3))), Pow(sin(Symbol('\\\\mathbf{v}', commutative=True)), Integer(4)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(3)), Function('l')(Symbol('\\\\mathbf{v}', commutative=True))), Pow(Function('M_E')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(4)))"]]}, {"prompt": "Given \\hat{x}_0{(\\rho_b)} = e^{\\rho_b}, then obtain \\hat{x}_0^{4}{(\\rho_b)} e^{- 4 \\rho_b} = \\hat{x}_0^{2}{(\\rho_b)} e^{- 2 \\rho_b}", "derivation": "\\hat{x}_0{(\\rho_b)} = e^{\\rho_b} and \\hat{x}_0{(\\rho_b)} e^{- 2 \\rho_b} = e^{- \\rho_b} and \\hat{x}_0^{2}{(\\rho_b)} e^{- 2 \\rho_b} = \\hat{x}_0{(\\rho_b)} e^{- \\rho_b} and \\hat{x}_0^{4}{(\\rho_b)} e^{- 4 \\rho_b} = \\hat{x}_0^{2}{(\\rho_b)} e^{- 2 \\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["divide", 1, "exp(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["times", 2, "Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True)))), Mul(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), Integer(4)), exp(Mul(Integer(-1), Integer(4), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\nabla)} = e^{\\nabla}, then obtain e^{\\nabla} \\sin{(3 e^{\\nabla})} = e^{\\nabla} \\sin{(2 \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla})}", "derivation": "\\operatorname{v_{2}}{(\\nabla)} = e^{\\nabla} and \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla} = 2 e^{\\nabla} and \\operatorname{v_{2}}{(\\nabla)} + 2 e^{\\nabla} = 3 e^{\\nabla} and \\sin{(\\operatorname{v_{2}}{(\\nabla)} + 2 e^{\\nabla})} = \\sin{(3 e^{\\nabla})} and 2 \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla} = 3 e^{\\nabla} and \\sin{(\\operatorname{v_{2}}{(\\nabla)} + 2 e^{\\nabla})} = \\sin{(2 \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla})} and \\sin{(3 e^{\\nabla})} = \\sin{(2 \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla})} and e^{\\nabla} \\sin{(3 e^{\\nabla})} = e^{\\nabla} \\sin{(2 \\operatorname{v_{2}}{(\\nabla)} + e^{\\nabla})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('v_2')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('v_2')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(3), exp(Symbol('\\\\nabla', commutative=True))))"], [["sin", 3], "Equality(sin(Add(Function('v_2')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))), sin(Mul(Integer(3), exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('v_2')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))), Mul(Integer(3), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(sin(Add(Function('v_2')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))), sin(Add(Mul(Integer(2), Function('v_2')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(sin(Mul(Integer(3), exp(Symbol('\\\\nabla', commutative=True)))), sin(Add(Mul(Integer(2), Function('v_2')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True)))))"], [["times", 7, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\nabla', commutative=True)), sin(Mul(Integer(3), exp(Symbol('\\\\nabla', commutative=True))))), Mul(exp(Symbol('\\\\nabla', commutative=True)), sin(Add(Mul(Integer(2), Function('v_2')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(t)} = \\log{(t)} and \\operatorname{t_{2}}{(t)} = \\int \\operatorname{L_{\\varepsilon}}{(t)} dt, then derive \\operatorname{t_{2}}{(t)} = \\chi + t \\log{(t)} - t, then obtain \\frac{\\partial}{\\partial t} \\sin^{t}{(\\chi + t \\log{(t)} - t)} = \\frac{d}{d t} \\sin^{t}{(\\int \\operatorname{L_{\\varepsilon}}{(t)} dt)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(t)} = \\log{(t)} and \\operatorname{t_{2}}{(t)} = \\int \\operatorname{L_{\\varepsilon}}{(t)} dt and \\sin{(\\operatorname{t_{2}}{(t)})} = \\sin{(\\int \\operatorname{L_{\\varepsilon}}{(t)} dt)} and \\operatorname{t_{2}}{(t)} = \\int \\log{(t)} dt and \\sin^{t}{(\\operatorname{t_{2}}{(t)})} = \\sin^{t}{(\\int \\operatorname{L_{\\varepsilon}}{(t)} dt)} and \\operatorname{t_{2}}{(t)} = \\chi + t \\log{(t)} - t and \\frac{d}{d t} \\sin^{t}{(\\operatorname{t_{2}}{(t)})} = \\frac{d}{d t} \\sin^{t}{(\\int \\operatorname{L_{\\varepsilon}}{(t)} dt)} and \\frac{\\partial}{\\partial t} \\sin^{t}{(\\chi + t \\log{(t)} - t)} = \\frac{d}{d t} \\sin^{t}{(\\int \\operatorname{L_{\\varepsilon}}{(t)} dt)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('t', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["sin", 2], "Equality(sin(Function('t_2')(Symbol('t', commutative=True))), sin(Integral(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('t_2')(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(sin(Function('t_2')(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(sin(Integral(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Function('t_2')(Symbol('t', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["differentiate", 5, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(sin(Function('t_2')(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(sin(Integral(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Derivative(Pow(sin(Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(sin(Integral(Function('L_{\\\\varepsilon}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(f_{\\mathbf{p}},p)} = \\cos^{f_{\\mathbf{p}}}{(p)}, then obtain 2 \\cos{(p)} + 2 \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp = 2 \\cos{(p)} + \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp + \\int \\cos^{f_{\\mathbf{p}}}{(p)} dp", "derivation": "\\hat{H}{(f_{\\mathbf{p}},p)} = \\cos^{f_{\\mathbf{p}}}{(p)} and \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp = \\int \\cos^{f_{\\mathbf{p}}}{(p)} dp and \\cos{(p)} + \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp = \\cos{(p)} + \\int \\cos^{f_{\\mathbf{p}}}{(p)} dp and 2 \\cos{(p)} + 2 \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp = 2 \\cos{(p)} + \\int \\hat{H}{(f_{\\mathbf{p}},p)} dp + \\int \\cos^{f_{\\mathbf{p}}}{(p)} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(cos(Symbol('p', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "cos(Symbol('p', commutative=True))"], "Equality(Add(cos(Symbol('p', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(cos(Symbol('p', commutative=True)), Integral(Pow(cos(Symbol('p', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["add", 3, "Add(cos(Symbol('p', commutative=True)), Integral(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Add(Mul(Integer(2), cos(Symbol('p', commutative=True))), Mul(Integer(2), Integral(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Add(Mul(Integer(2), cos(Symbol('p', commutative=True))), Integral(Function('\\\\hat{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(cos(Symbol('p', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given U{(I,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + I), then obtain \\int - \\frac{U{(I,C_{2})} + 1}{C_{2}} dC_{2} = \\int - \\frac{\\frac{\\partial}{\\partial C_{2}} (- C_{2} + I) + 1}{C_{2}} dC_{2}", "derivation": "U{(I,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + I) and U{(I,C_{2})} + 1 = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + I) + 1 and - \\frac{U{(I,C_{2})} + 1}{C_{2}} = - \\frac{\\frac{\\partial}{\\partial C_{2}} (- C_{2} + I) + 1}{C_{2}} and \\int - \\frac{U{(I,C_{2})} + 1}{C_{2}} dC_{2} = \\int - \\frac{\\frac{\\partial}{\\partial C_{2}} (- C_{2} + I) + 1}{C_{2}} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('U')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Integer(1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1)))"], [["divide", 2, "Mul(Integer(-1), Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Function('U')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Integer(1))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1))))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Function('U')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Integer(1))), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(z^{*})} = \\log{(\\sin{(z^{*})})}, then derive \\frac{d}{d z^{*}} \\operatorname{c_{0}}{(z^{*})} = \\frac{\\cos{(z^{*})}}{\\sin{(z^{*})}}, then obtain \\frac{\\cos{(z^{*})} \\frac{d}{d z^{*}} \\operatorname{c_{0}}{(z^{*})}}{\\sin{(z^{*})}} = \\frac{\\cos^{2}{(z^{*})}}{\\sin^{2}{(z^{*})}}", "derivation": "\\operatorname{c_{0}}{(z^{*})} = \\log{(\\sin{(z^{*})})} and \\frac{d}{d z^{*}} \\operatorname{c_{0}}{(z^{*})} = \\frac{d}{d z^{*}} \\log{(\\sin{(z^{*})})} and \\frac{d}{d z^{*}} \\operatorname{c_{0}}{(z^{*})} = \\frac{\\cos{(z^{*})}}{\\sin{(z^{*})}} and \\frac{\\cos{(z^{*})} \\frac{d}{d z^{*}} \\operatorname{c_{0}}{(z^{*})}}{\\sin{(z^{*})}} = \\frac{\\cos^{2}{(z^{*})}}{\\sin^{2}{(z^{*})}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('z^*', commutative=True)), log(sin(Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(sin(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c_0')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True))))"], [["times", 3, "Mul(Pow(sin(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True)))"], "Equality(Mul(Pow(sin(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True)), Derivative(Function('c_0')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('z^*', commutative=True)), Integer(-2)), Pow(cos(Symbol('z^*', commutative=True)), Integer(2))))"]]}, {"prompt": "Given z{(\\omega,\\mathbf{F},p)} = p (\\mathbf{F} + \\omega), then obtain p (\\mathbf{F} + \\omega) z{(\\omega,\\mathbf{F},p)} + p (\\mathbf{F} + \\omega) + z{(\\omega,\\mathbf{F},p)} = 2 p (\\mathbf{F} + \\omega) + z^{2}{(\\omega,\\mathbf{F},p)}", "derivation": "z{(\\omega,\\mathbf{F},p)} = p (\\mathbf{F} + \\omega) and z^{2}{(\\omega,\\mathbf{F},p)} = p (\\mathbf{F} + \\omega) z{(\\omega,\\mathbf{F},p)} and p (\\mathbf{F} + \\omega) + z^{2}{(\\omega,\\mathbf{F},p)} = p (\\mathbf{F} + \\omega) z{(\\omega,\\mathbf{F},p)} + p (\\mathbf{F} + \\omega) and p (\\mathbf{F} + \\omega) + z^{2}{(\\omega,\\mathbf{F},p)} + z{(\\omega,\\mathbf{F},p)} = 2 p (\\mathbf{F} + \\omega) + z^{2}{(\\omega,\\mathbf{F},p)} and p (\\mathbf{F} + \\omega) z{(\\omega,\\mathbf{F},p)} + p (\\mathbf{F} + \\omega) + z{(\\omega,\\mathbf{F},p)} = 2 p (\\mathbf{F} + \\omega) + z^{2}{(\\omega,\\mathbf{F},p)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))"], "Equality(Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2)), Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))))"], [["add", 2, "Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2))), Add(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["add", 1, "Add(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2)), Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(2), Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)), Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(2), Symbol('p', commutative=True), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Function('z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Symbol('p', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)} = G g^{\\prime}_{\\varepsilon}, then derive \\frac{\\frac{\\partial}{\\partial G} \\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)}}{g^{\\prime}_{\\varepsilon}} = 1, then obtain \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} G g^{\\prime}_{\\varepsilon}}{g^{\\prime}_{\\varepsilon}} = \\frac{d}{d G} 1", "derivation": "\\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)} = G g^{\\prime}_{\\varepsilon} and \\frac{\\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)}}{g^{\\prime}_{\\varepsilon}} = G and \\frac{\\partial}{\\partial G} \\frac{\\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)}}{g^{\\prime}_{\\varepsilon}} = \\frac{d}{d G} G and \\frac{\\frac{\\partial}{\\partial G} \\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)}}{g^{\\prime}_{\\varepsilon}} = 1 and \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} \\mathbf{v}{(g^{\\prime}_{\\varepsilon},G)}}{g^{\\prime}_{\\varepsilon}} = \\frac{d}{d G} 1 and \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial G} G g^{\\prime}_{\\varepsilon}}{g^{\\prime}_{\\varepsilon}} = \\frac{d}{d G} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Integer(1))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{v}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('G', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(v_{z},\\tilde{g}^*)} = e^{\\frac{\\tilde{g}^*}{v_{z}}} and \\hat{x}_0{(v_{z},\\tilde{g}^*)} = \\frac{\\tilde{g}^*}{v_{z}}, then obtain (e^{\\frac{\\tilde{g}^*}{v_{z}}})^{v_{z}} = (e^{\\hat{x}_0{(v_{z},\\tilde{g}^*)}})^{v_{z}}", "derivation": "\\dot{\\mathbf{r}}{(v_{z},\\tilde{g}^*)} = e^{\\frac{\\tilde{g}^*}{v_{z}}} and \\dot{\\mathbf{r}}^{v_{z}}{(v_{z},\\tilde{g}^*)} = (e^{\\frac{\\tilde{g}^*}{v_{z}}})^{v_{z}} and \\hat{x}_0{(v_{z},\\tilde{g}^*)} = \\frac{\\tilde{g}^*}{v_{z}} and e^{\\hat{x}_0{(v_{z},\\tilde{g}^*)}} = e^{\\frac{\\tilde{g}^*}{v_{z}}} and \\dot{\\mathbf{r}}^{v_{z}}{(v_{z},\\tilde{g}^*)} = (e^{\\hat{x}_0{(v_{z},\\tilde{g}^*)}})^{v_{z}} and (e^{\\frac{\\tilde{g}^*}{v_{z}}})^{v_{z}} = (e^{\\hat{x}_0{(v_{z},\\tilde{g}^*)}})^{v_{z}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), exp(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('v_z', commutative=True)), Pow(exp(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)))), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["exp", 3], "Equality(exp(Function('\\\\hat{x}_0')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), exp(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('v_z', commutative=True)), Pow(exp(Function('\\\\hat{x}_0')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(exp(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)))), Symbol('v_z', commutative=True)), Pow(exp(Function('\\\\hat{x}_0')(Symbol('v_z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} = \\log{(\\mathbf{E} + \\pi)} and \\operatorname{A_{z}}{(\\pi)} = \\pi, then obtain - \\pi + \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\pi = \\mathbf{E} \\log{(\\mathbf{E} + \\pi)} + \\pi \\log{(\\mathbf{E} + \\pi)} - 2 \\pi + v_{y}", "derivation": "\\operatorname{f^{*}}{(\\pi,\\mathbf{E})} = \\log{(\\mathbf{E} + \\pi)} and \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\pi = \\int \\log{(\\mathbf{E} + \\pi)} d\\pi and \\operatorname{A_{z}}{(\\pi)} = \\pi and \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\operatorname{A_{z}}{(\\pi)} = \\int \\log{(\\mathbf{E} + \\pi)} d\\operatorname{A_{z}}{(\\pi)} and - \\operatorname{A_{z}}{(\\pi)} + \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\operatorname{A_{z}}{(\\pi)} = - \\operatorname{A_{z}}{(\\pi)} + \\int \\log{(\\mathbf{E} + \\pi)} d\\operatorname{A_{z}}{(\\pi)} and - \\pi + \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\pi = - \\pi + \\int \\log{(\\mathbf{E} + \\pi)} d\\pi and - \\pi + \\int \\operatorname{f^{*}}{(\\pi,\\mathbf{E})} d\\pi = \\mathbf{E} \\log{(\\mathbf{E} + \\pi)} + \\pi \\log{(\\mathbf{E} + \\pi)} - 2 \\pi + v_{y}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Function('A_z')(Symbol('\\\\pi', commutative=True)))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Function('A_z')(Symbol('\\\\pi', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Function('A_z')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\pi', commutative=True))), Integral(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Function('A_z')(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\pi', commutative=True))), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Function('A_z')(Symbol('\\\\pi', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(Function('f^*')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), log(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True)), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(t)} = \\sin{(\\cos{(t)})} and \\mathbf{D}{(t)} = \\sin{(\\cos{(t)})}, then obtain (\\frac{\\hat{X}{(t)}}{\\mathbf{D}{(t)}})^{t} + \\mathbf{D}{(t)} = \\mathbf{D}{(t)} + 1", "derivation": "\\hat{X}{(t)} = \\sin{(\\cos{(t)})} and \\frac{\\hat{X}{(t)}}{\\sin{(\\cos{(t)})}} = 1 and (\\frac{\\hat{X}{(t)}}{\\sin{(\\cos{(t)})}})^{t} = 1 and (\\frac{\\hat{X}{(t)}}{\\sin{(\\cos{(t)})}})^{t} + \\sin{(\\cos{(t)})} = \\sin{(\\cos{(t)})} + 1 and \\mathbf{D}{(t)} = \\sin{(\\cos{(t)})} and (\\frac{\\hat{X}{(t)}}{\\mathbf{D}{(t)}})^{t} + \\mathbf{D}{(t)} = \\mathbf{D}{(t)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('t', commutative=True)), sin(cos(Symbol('t', commutative=True))))"], [["divide", 1, "sin(cos(Symbol('t', commutative=True)))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{X}')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Symbol('t', commutative=True)), Integer(1))"], [["add", 3, "sin(cos(Symbol('t', commutative=True)))"], "Equality(Add(Pow(Mul(Function('\\\\hat{X}')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Symbol('t', commutative=True)), sin(cos(Symbol('t', commutative=True)))), Add(sin(cos(Symbol('t', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), sin(cos(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Mul(Function('\\\\hat{X}')(Symbol('t', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Integer(-1))), Symbol('t', commutative=True)), Function('\\\\mathbf{D}')(Symbol('t', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('t', commutative=True)), Integer(1)))"]]}, {"prompt": "Given a{(r_{0},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0}), then derive a{(r_{0},\\hat{H}_{\\lambda})} = 1, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0}) = 1", "derivation": "a{(r_{0},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0}) and a^{2}{(r_{0},\\hat{H}_{\\lambda})} = a{(r_{0},\\hat{H}_{\\lambda})} \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0}) and \\frac{a{(r_{0},\\hat{H}_{\\lambda})}}{\\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0})} = 1 and a{(r_{0},\\hat{H}_{\\lambda})} = 1 and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} + r_{0}) = 1", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 1, "Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"], [["divide", 2, "Mul(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], "Equality(Mul(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Function('a')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\varepsilon{(t_{1})} = \\log{(t_{1})} and x{(J)} = \\log{(e^{J})}, then obtain \\frac{x{(J)} - \\log{(e^{J})}}{(\\int \\varepsilon{(t_{1})} dt_{1}) \\iint \\log{(t_{1})} dt_{1} dt_{1}} = 0", "derivation": "\\varepsilon{(t_{1})} = \\log{(t_{1})} and x{(J)} = \\log{(e^{J})} and \\varepsilon{(t_{1})} + x{(J)} = \\varepsilon{(t_{1})} + \\log{(e^{J})} and x{(J)} + \\log{(t_{1})} = \\log{(t_{1})} + \\log{(e^{J})} and x{(J)} - \\log{(e^{J})} = 0 and \\frac{x{(J)} - \\log{(e^{J})}}{\\iint \\log{(t_{1})} dt_{1} dt_{1}} = 0 and \\frac{x{(J)} - \\log{(e^{J})}}{(\\int \\varepsilon{(t_{1})} dt_{1}) \\iint \\log{(t_{1})} dt_{1} dt_{1}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], ["get_premise", "Equality(Function('x')(Symbol('J', commutative=True)), log(exp(Symbol('J', commutative=True))))"], [["add", 2, "Function('\\\\varepsilon')(Symbol('t_1', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), Function('x')(Symbol('J', commutative=True))), Add(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), log(exp(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('x')(Symbol('J', commutative=True)), log(Symbol('t_1', commutative=True))), Add(log(Symbol('t_1', commutative=True)), log(exp(Symbol('J', commutative=True)))))"], [["minus", 4, "Add(log(Symbol('t_1', commutative=True)), log(exp(Symbol('J', commutative=True))))"], "Equality(Add(Function('x')(Symbol('J', commutative=True)), Mul(Integer(-1), log(exp(Symbol('J', commutative=True))))), Integer(0))"], [["divide", 5, "Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Add(Function('x')(Symbol('J', commutative=True)), Mul(Integer(-1), log(exp(Symbol('J', commutative=True))))), Pow(Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Integer(0))"], [["divide", 6, "Integral(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Add(Function('x')(Symbol('J', commutative=True)), Mul(Integer(-1), log(exp(Symbol('J', commutative=True))))), Pow(Integral(Function('\\\\varepsilon')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1)), Pow(Integral(log(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given A{(x,m_{s})} = m_{s} + x, then derive \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} = 1, then obtain m_{s} + (m_{s} + 1) \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} + 1 = 2 m_{s} + 2", "derivation": "A{(x,m_{s})} = m_{s} + x and \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (m_{s} + x) and \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} = 1 and (m_{s} + 1) \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} = m_{s} + 1 and m_{s} + (m_{s} + 1) \\frac{\\partial}{\\partial m_{s}} A{(x,m_{s})} + 1 = 2 m_{s} + 2", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Symbol('m_s', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Add(Symbol('m_s', commutative=True), Integer(1))"], "Equality(Mul(Add(Symbol('m_s', commutative=True), Integer(1)), Derivative(Function('A')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Symbol('m_s', commutative=True), Integer(1)))"], [["add", 4, "Add(Symbol('m_s', commutative=True), Integer(1))"], "Equality(Add(Symbol('m_s', commutative=True), Mul(Add(Symbol('m_s', commutative=True), Integer(1)), Derivative(Function('A')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Integer(2), Symbol('m_s', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{H})} = \\cos{(F_{H})}, then obtain \\operatorname{y^{\\prime}}^{F_{H}}{(F_{H})} + 2 \\cos^{F_{H}}{(F_{H})} = 3 \\cos^{F_{H}}{(F_{H})}", "derivation": "\\operatorname{y^{\\prime}}{(F_{H})} = \\cos{(F_{H})} and \\operatorname{y^{\\prime}}^{F_{H}}{(F_{H})} = \\cos^{F_{H}}{(F_{H})} and \\operatorname{y^{\\prime}}^{F_{H}}{(F_{H})} + \\cos^{F_{H}}{(F_{H})} = 2 \\cos^{F_{H}}{(F_{H})} and \\operatorname{y^{\\prime}}^{F_{H}}{(F_{H})} + 2 \\cos^{F_{H}}{(F_{H})} = 3 \\cos^{F_{H}}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["add", 2, "Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], "Equality(Add(Pow(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Integer(2), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["add", 3, "Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], "Equality(Add(Pow(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))), Mul(Integer(3), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given W{(U,\\Psi)} = \\frac{\\partial}{\\partial U} \\frac{\\Psi}{U} and \\operatorname{t_{2}}{(U,\\Psi)} = W^{\\Psi}{(U,\\Psi)}, then derive W^{\\Psi}{(U,\\Psi)} = (- \\frac{\\Psi}{U^{2}})^{\\Psi}, then obtain \\operatorname{t_{2}}{(U,\\Psi)} - 1 = (- \\frac{\\Psi}{U^{2}})^{\\Psi} - 1", "derivation": "W{(U,\\Psi)} = \\frac{\\partial}{\\partial U} \\frac{\\Psi}{U} and W^{\\Psi}{(U,\\Psi)} = (\\frac{\\partial}{\\partial U} \\frac{\\Psi}{U})^{\\Psi} and \\operatorname{t_{2}}{(U,\\Psi)} = W^{\\Psi}{(U,\\Psi)} and W^{\\Psi}{(U,\\Psi)} = (- \\frac{\\Psi}{U^{2}})^{\\Psi} and \\operatorname{t_{2}}{(U,\\Psi)} = (\\frac{\\partial}{\\partial U} \\frac{\\Psi}{U})^{\\Psi} and \\operatorname{t_{2}}{(U,\\Psi)} - 1 = (\\frac{\\partial}{\\partial U} \\frac{\\Psi}{U})^{\\Psi} - 1 and \\operatorname{t_{2}}{(U,\\Psi)} - 1 = W^{\\Psi}{(U,\\Psi)} - 1 and \\operatorname{t_{2}}{(U,\\Psi)} - 1 = (- \\frac{\\Psi}{U^{2}})^{\\Psi} - 1", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('t_2')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)))"], [["add", 5, "Integer(-1)"], "Equality(Add(Function('t_2')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Add(Pow(Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Function('t_2')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Add(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Function('t_2')(Symbol('U', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Add(Pow(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(S,\\mathbf{H})} = \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}}, then derive \\operatorname{f^{\\prime}}{(S,\\mathbf{H})} = \\frac{1}{\\mathbf{H}}, then obtain \\operatorname{f^{\\prime}}{(S,\\frac{1}{\\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}}})} - \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}} = 0", "derivation": "\\operatorname{f^{\\prime}}{(S,\\mathbf{H})} = \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}} and \\operatorname{f^{\\prime}}{(S,\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{1}{\\mathbf{H}} = \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}} and \\operatorname{f^{\\prime}}{(S,\\frac{1}{\\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}}})} = \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}} and \\operatorname{f^{\\prime}}{(S,\\frac{1}{\\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}}})} - \\frac{\\partial}{\\partial S} \\frac{S}{\\mathbf{H}} = 0", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Pow(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('S', commutative=True), Pow(Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\theta_{2}{(v_{1})} = e^{e^{v_{1}}} and \\pi{(v_{1})} = (\\frac{d}{d v_{1}} \\theta_{2}{(v_{1})})^{v_{1}}, then derive \\frac{d}{d v_{1}} \\theta_{2}{(v_{1})} = e^{v_{1}} e^{e^{v_{1}}}, then obtain \\pi{(v_{1})} = (e^{v_{1}} e^{e^{v_{1}}})^{v_{1}}", "derivation": "\\theta_{2}{(v_{1})} = e^{e^{v_{1}}} and \\frac{d}{d v_{1}} \\theta_{2}{(v_{1})} = \\frac{d}{d v_{1}} e^{e^{v_{1}}} and \\frac{d}{d v_{1}} \\theta_{2}{(v_{1})} = e^{v_{1}} e^{e^{v_{1}}} and (\\frac{d}{d v_{1}} \\theta_{2}{(v_{1})})^{v_{1}} = (e^{v_{1}} e^{e^{v_{1}}})^{v_{1}} and \\pi{(v_{1})} = (\\frac{d}{d v_{1}} \\theta_{2}{(v_{1})})^{v_{1}} and \\pi{(v_{1})} = (e^{v_{1}} e^{e^{v_{1}}})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(exp(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\theta_2')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Mul(exp(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('v_1', commutative=True)), Pow(Derivative(Function('\\\\theta_2')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\pi')(Symbol('v_1', commutative=True)), Pow(Mul(exp(Symbol('v_1', commutative=True)), exp(exp(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given M{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})}, then derive M{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})}, then obtain - \\cos{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})}", "derivation": "M{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} and M{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})} = - \\cos{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} and M{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})} and - \\cos{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon})} - \\cos{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 1, "cos(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\dot{\\mathbf{r}},z)} = \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z, then derive \\mathbf{H}{(\\dot{\\mathbf{r}},z)} = \\dot{\\mathbf{r}}, then obtain \\varphi + \\frac{\\mathbf{H}^{2}{(\\dot{\\mathbf{r}},z)}}{2} = \\int \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z d\\mathbf{H}{(\\dot{\\mathbf{r}},z)}", "derivation": "\\mathbf{H}{(\\dot{\\mathbf{r}},z)} = \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z and \\int \\mathbf{H}{(\\dot{\\mathbf{r}},z)} d\\dot{\\mathbf{r}} = \\int \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z d\\dot{\\mathbf{r}} and \\mathbf{H}{(\\dot{\\mathbf{r}},z)} = \\dot{\\mathbf{r}} and \\int \\mathbf{H}{(\\dot{\\mathbf{r}},z)} d\\mathbf{H}{(\\dot{\\mathbf{r}},z)} = \\int \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z d\\mathbf{H}{(\\dot{\\mathbf{r}},z)} and \\varphi + \\frac{\\mathbf{H}^{2}{(\\dot{\\mathbf{r}},z)}}{2} = \\int \\frac{\\partial}{\\partial z} \\dot{\\mathbf{r}} z d\\mathbf{H}{(\\dot{\\mathbf{r}},z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)))), Integral(Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Integer(2)))), Integral(Derivative(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Function('\\\\mathbf{H}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\phi_2)} = e^{\\phi_2} and \\omega{(\\mathbf{J},y^{\\prime})} = \\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})}, then obtain \\frac{\\sin{(\\phi_2 + \\hat{\\mathbf{r}}{(\\phi_2)})}}{\\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})}} = \\frac{\\sin{(\\phi_2 + e^{\\phi_2})}}{\\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})}}", "derivation": "\\hat{\\mathbf{r}}{(\\phi_2)} = e^{\\phi_2} and \\phi_2 + \\hat{\\mathbf{r}}{(\\phi_2)} = \\phi_2 + e^{\\phi_2} and \\omega{(\\mathbf{J},y^{\\prime})} = \\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})} and \\sin{(\\phi_2 + \\hat{\\mathbf{r}}{(\\phi_2)})} = \\sin{(\\phi_2 + e^{\\phi_2})} and \\frac{\\sin{(\\phi_2 + \\hat{\\mathbf{r}}{(\\phi_2)})}}{\\omega{(\\mathbf{J},y^{\\prime})}} = \\frac{\\sin{(\\phi_2 + e^{\\phi_2})}}{\\omega{(\\mathbf{J},y^{\\prime})}} and \\frac{\\sin{(\\phi_2 + \\hat{\\mathbf{r}}{(\\phi_2)})}}{\\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})}} = \\frac{\\sin{(\\phi_2 + e^{\\phi_2})}}{\\cos{(\\frac{y^{\\prime}}{\\mathbf{J}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))"], ["get_premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))))"], [["sin", 2], "Equality(sin(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)))), sin(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True)))))"], [["divide", 4, "Function('\\\\omega')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True))))), Mul(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(sin(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)))), Pow(cos(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Mul(sin(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True)))), Pow(cos(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\phi_1)} = \\cos{(\\cos{(\\phi_1)})}, then derive \\frac{d}{d \\phi_1} \\operatorname{C_{1}}{(\\phi_1)} = \\sin{(\\phi_1)} \\sin{(\\cos{(\\phi_1)})}, then obtain \\frac{d}{d \\phi_1} \\cos{(\\cos{(\\phi_1)})} = \\sin{(\\phi_1)} \\sin{(\\cos{(\\phi_1)})}", "derivation": "\\operatorname{C_{1}}{(\\phi_1)} = \\cos{(\\cos{(\\phi_1)})} and \\frac{d}{d \\phi_1} \\operatorname{C_{1}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\cos{(\\cos{(\\phi_1)})} and \\frac{d}{d \\phi_1} \\operatorname{C_{1}}{(\\phi_1)} = \\sin{(\\phi_1)} \\sin{(\\cos{(\\phi_1)})} and \\frac{d}{d \\phi_1} \\cos{(\\cos{(\\phi_1)})} = \\sin{(\\phi_1)} \\sin{(\\cos{(\\phi_1)})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\phi_1', commutative=True)), cos(cos(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(sin(Symbol('\\\\phi_1', commutative=True)), sin(cos(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(sin(Symbol('\\\\phi_1', commutative=True)), sin(cos(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given q{(\\varphi^*)} = e^{- \\varphi^*}, then derive \\frac{d}{d \\varphi^*} q{(\\varphi^*)} = - e^{- \\varphi^*}, then obtain e^{\\frac{d}{d \\varphi^*} e^{- \\varphi^*}} = e^{- q{(\\varphi^*)}}", "derivation": "q{(\\varphi^*)} = e^{- \\varphi^*} and \\frac{d}{d \\varphi^*} q{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{- \\varphi^*} and \\frac{d}{d \\varphi^*} q{(\\varphi^*)} = - e^{- \\varphi^*} and \\frac{d}{d \\varphi^*} q{(\\varphi^*)} = - q{(\\varphi^*)} and - e^{- \\varphi^*} = - q{(\\varphi^*)} and \\frac{d}{d \\varphi^*} e^{- \\varphi^*} = - e^{- \\varphi^*} and e^{\\frac{d}{d \\varphi^*} e^{- \\varphi^*}} = e^{- e^{- \\varphi^*}} and e^{\\frac{d}{d \\varphi^*} e^{- \\varphi^*}} = e^{- q{(\\varphi^*)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('q')(Symbol('\\\\varphi^*', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('q')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Integer(-1), Function('q')(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Function('q')(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"], [["exp", 6], "Equality(exp(Derivative(exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), exp(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(exp(Derivative(exp(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Function('q')(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(c,v_{t})} = - c + v_{t} and v{(A_{2},m_{s})} = A_{2}^{m_{s}}, then obtain \\frac{- c - \\mathbf{A}{(c,v_{t})} + v{(A_{2},m_{s})}}{A_{2}^{m_{s}} - v_{t}} = \\frac{A_{2}^{m_{s}} - c - \\mathbf{A}{(c,v_{t})}}{A_{2}^{m_{s}} - v_{t}}", "derivation": "\\mathbf{A}{(c,v_{t})} = - c + v_{t} and c + \\mathbf{A}{(c,v_{t})} = v_{t} and v{(A_{2},m_{s})} = A_{2}^{m_{s}} and - v_{t} + v{(A_{2},m_{s})} = A_{2}^{m_{s}} - v_{t} and - c - \\mathbf{A}{(c,v_{t})} + v{(A_{2},m_{s})} = A_{2}^{m_{s}} - c - \\mathbf{A}{(c,v_{t})} and \\frac{- c - \\mathbf{A}{(c,v_{t})} + v{(A_{2},m_{s})}}{A_{2}^{m_{s}} - v_{t}} = \\frac{A_{2}^{m_{s}} - c - \\mathbf{A}{(c,v_{t})}}{A_{2}^{m_{s}} - v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('v_t', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))"], ["get_premise", "Equality(Function('v')(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)))"], [["minus", 3, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('v')(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True))), Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True))), Function('v')(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True))), Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True)))))"], [["divide", 5, "Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)))"], "Equality(Mul(Pow(Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True))), Function('v')(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)))), Mul(Pow(Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Integer(-1)), Add(Pow(Symbol('A_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given y{(f^{*},\\mu_0)} = \\mu_0 f^{*}, then obtain - \\sin{((\\mu_0 f^{*})^{\\mu_0})} + \\frac{\\partial}{\\partial \\mu_0} \\sin{(y^{\\mu_0}{(f^{*},\\mu_0)})} = - \\sin{((\\mu_0 f^{*})^{\\mu_0})} + \\frac{\\partial}{\\partial \\mu_0} \\sin{((\\mu_0 f^{*})^{\\mu_0})}", "derivation": "y{(f^{*},\\mu_0)} = \\mu_0 f^{*} and y^{\\mu_0}{(f^{*},\\mu_0)} = (\\mu_0 f^{*})^{\\mu_0} and \\sin{(y^{\\mu_0}{(f^{*},\\mu_0)})} = \\sin{((\\mu_0 f^{*})^{\\mu_0})} and \\frac{\\partial}{\\partial \\mu_0} \\sin{(y^{\\mu_0}{(f^{*},\\mu_0)})} = \\frac{\\partial}{\\partial \\mu_0} \\sin{((\\mu_0 f^{*})^{\\mu_0})} and - \\sin{((\\mu_0 f^{*})^{\\mu_0})} + \\frac{\\partial}{\\partial \\mu_0} \\sin{(y^{\\mu_0}{(f^{*},\\mu_0)})} = - \\sin{((\\mu_0 f^{*})^{\\mu_0})} + \\frac{\\partial}{\\partial \\mu_0} \\sin{((\\mu_0 f^{*})^{\\mu_0})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('y')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('y')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(sin(Pow(Function('y')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 4, "sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Derivative(sin(Pow(Function('y')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Derivative(sin(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})}, then obtain \\frac{d}{d f_{\\mathbf{v}}} 0 = \\frac{d}{d f_{\\mathbf{v}}} (- 2 \\hat{H}_l{(f_{\\mathbf{v}})} + 2 \\cos{(f_{\\mathbf{v}})})", "derivation": "\\hat{H}_l{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})} and 0 = - \\hat{H}_l{(f_{\\mathbf{v}})} + \\cos{(f_{\\mathbf{v}})} and - \\hat{H}_l{(f_{\\mathbf{v}})} = - 2 \\hat{H}_l{(f_{\\mathbf{v}})} + \\cos{(f_{\\mathbf{v}})} and 0 = - 2 \\hat{H}_l{(f_{\\mathbf{v}})} + 2 \\cos{(f_{\\mathbf{v}})} and \\frac{d}{d f_{\\mathbf{v}}} 0 = \\frac{d}{d f_{\\mathbf{v}}} (- 2 \\hat{H}_l{(f_{\\mathbf{v}})} + 2 \\cos{(f_{\\mathbf{v}})})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["minus", 2, "Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(2), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}_l')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(2), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(\\mu,\\varphi)} = \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu}, then derive 2 \\mu - \\varphi^{\\mu} \\log{(\\varphi)} - \\nabla{(\\mu,\\varphi)} = 2 \\mu - 2 \\varphi^{\\mu} \\log{(\\varphi)}, then obtain 2 \\mu - \\varphi^{\\mu} \\log{(\\varphi)} - \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} = 2 \\mu - 2 \\varphi^{\\mu} \\log{(\\varphi)}", "derivation": "\\nabla{(\\mu,\\varphi)} = \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} and - \\mu + \\nabla{(\\mu,\\varphi)} = - \\mu + \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} and \\mu - \\nabla{(\\mu,\\varphi)} = \\mu - \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} and 2 \\mu - \\nabla{(\\mu,\\varphi)} - \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} = 2 \\mu - 2 \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} and 2 \\mu - \\varphi^{\\mu} \\log{(\\varphi)} - \\nabla{(\\mu,\\varphi)} = 2 \\mu - 2 \\varphi^{\\mu} \\log{(\\varphi)} and 2 \\mu - \\varphi^{\\mu} \\log{(\\varphi)} - \\frac{\\partial}{\\partial \\mu} \\varphi^{\\mu} = 2 \\mu - 2 \\varphi^{\\mu} \\log{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Function('\\\\nabla')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\theta_1,C)} = C - \\theta_1, then obtain \\frac{(C \\hat{H}_l^{\\theta_1}{(\\theta_1,C)})^{C}}{C - \\theta_1} - 1 = \\frac{(C (C - \\theta_1)^{\\theta_1})^{C}}{C - \\theta_1} - 1", "derivation": "\\hat{H}_l{(\\theta_1,C)} = C - \\theta_1 and \\hat{H}_l^{\\theta_1}{(\\theta_1,C)} = (C - \\theta_1)^{\\theta_1} and C \\hat{H}_l^{\\theta_1}{(\\theta_1,C)} = C (C - \\theta_1)^{\\theta_1} and (C \\hat{H}_l^{\\theta_1}{(\\theta_1,C)})^{C} = (C (C - \\theta_1)^{\\theta_1})^{C} and \\frac{(C \\hat{H}_l^{\\theta_1}{(\\theta_1,C)})^{C}}{C - \\theta_1} = \\frac{(C (C - \\theta_1)^{\\theta_1})^{C}}{C - \\theta_1} and \\frac{(C \\hat{H}_l^{\\theta_1}{(\\theta_1,C)})^{C}}{C - \\theta_1} - 1 = \\frac{(C (C - \\theta_1)^{\\theta_1})^{C}}{C - \\theta_1} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"], [["times", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('C', commutative=True), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)))"], [["divide", 4, "Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Mul(Pow(Mul(Symbol('C', commutative=True), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(-1))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Mul(Pow(Mul(Symbol('C', commutative=True), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_1', commutative=True), Symbol('C', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Integer(-1)), Add(Mul(Pow(Mul(Symbol('C', commutative=True), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(-1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(p)} = \\cos{(e^{p})}, then obtain \\iint (\\int \\operatorname{x^{{\\}'}}{(p)} dp + \\frac{1}{p}) dp dp = \\iint (\\int \\cos{(e^{p})} dp + \\frac{1}{p}) dp dp", "derivation": "\\operatorname{x^{{\\}'}}{(p)} = \\cos{(e^{p})} and \\int \\operatorname{x^{{\\}'}}{(p)} dp = \\int \\cos{(e^{p})} dp and \\int \\operatorname{x^{{\\}'}}{(p)} dp + \\frac{1}{p} = \\int \\cos{(e^{p})} dp + \\frac{1}{p} and \\int (\\int \\operatorname{x^{{\\}'}}{(p)} dp + \\frac{1}{p}) dp = \\int (\\int \\cos{(e^{p})} dp + \\frac{1}{p}) dp and \\iint (\\int \\operatorname{x^{{\\}'}}{(p)} dp + \\frac{1}{p}) dp dp = \\iint (\\int \\cos{(e^{p})} dp + \\frac{1}{p}) dp dp", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Pow(Symbol('p', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('x^\\\\prime')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))), Add(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Integral(Function('x^\\\\prime')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('p', commutative=True))), Integral(Add(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Integral(Function('x^\\\\prime')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Add(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Pow(Symbol('p', commutative=True), Integer(-1))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given z{(A_{y},\\mathbf{S})} = A_{y} + \\cos{(\\mathbf{S})}, then obtain - A_{y} + E + z{(A_{y},\\mathbf{S})} = \\int 0 dA_{y}", "derivation": "z{(A_{y},\\mathbf{S})} = A_{y} + \\cos{(\\mathbf{S})} and \\frac{\\partial}{\\partial A_{y}} z{(A_{y},\\mathbf{S})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} + \\cos{(\\mathbf{S})}) and - \\frac{\\partial}{\\partial A_{y}} (A_{y} + \\cos{(\\mathbf{S})}) + \\frac{\\partial}{\\partial A_{y}} z{(A_{y},\\mathbf{S})} = 0 and \\int (- \\frac{\\partial}{\\partial A_{y}} (A_{y} + \\cos{(\\mathbf{S})}) + \\frac{\\partial}{\\partial A_{y}} z{(A_{y},\\mathbf{S})}) dA_{y} = \\int 0 dA_{y} and - A_{y} + E + z{(A_{y},\\mathbf{S})} = \\int 0 dA_{y}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('A_y', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('A_y', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Derivative(Function('z')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 3, "Symbol('A_y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Derivative(Function('z')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Tuple(Symbol('A_y', commutative=True))), Integral(Integer(0), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('E', commutative=True), Function('z')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integral(Integer(0), Tuple(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given r{(u)} = \\log{(u)}, then obtain \\int r{(u)} \\log{(u)} \\log{(u)}^{u} du = \\int \\log{(u)}^{2} \\log{(u)}^{u} du", "derivation": "r{(u)} = \\log{(u)} and r^{u}{(u)} = \\log{(u)}^{u} and r{(u)} \\log{(u)} = \\log{(u)}^{2} and r{(u)} r^{u}{(u)} \\log{(u)} = r^{u}{(u)} \\log{(u)}^{2} and r{(u)} \\log{(u)} \\log{(u)}^{u} = \\log{(u)}^{2} \\log{(u)}^{u} and \\int r{(u)} \\log{(u)} \\log{(u)}^{u} du = \\int \\log{(u)}^{2} \\log{(u)}^{u} du", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('r')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(log(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["times", 1, "log(Symbol('u', commutative=True))"], "Equality(Mul(Function('r')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(log(Symbol('u', commutative=True)), Integer(2)))"], [["times", 3, "Pow(Function('r')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Function('r')(Symbol('u', commutative=True)), Pow(Function('r')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Mul(Pow(Function('r')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(log(Symbol('u', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('r')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)), Pow(log(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Pow(log(Symbol('u', commutative=True)), Integer(2)), Pow(log(Symbol('u', commutative=True)), Symbol('u', commutative=True))))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Function('r')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)), Pow(log(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Pow(log(Symbol('u', commutative=True)), Integer(2)), Pow(log(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(F_{g})} = \\int \\sin{(F_{g})} dF_{g}, then obtain \\frac{\\frac{d}{d F_{g}} \\operatorname{A_{z}}{(F_{g})}}{\\frac{\\partial}{\\partial F_{g}} (\\Omega - \\cos{(F_{g})})} = 1", "derivation": "\\operatorname{A_{z}}{(F_{g})} = \\int \\sin{(F_{g})} dF_{g} and \\frac{d}{d F_{g}} \\operatorname{A_{z}}{(F_{g})} = \\frac{d}{d F_{g}} \\int \\sin{(F_{g})} dF_{g} and \\frac{\\frac{d}{d F_{g}} \\operatorname{A_{z}}{(F_{g})}}{\\frac{d}{d F_{g}} \\int \\sin{(F_{g})} dF_{g}} = 1 and \\frac{\\frac{d}{d F_{g}} \\operatorname{A_{z}}{(F_{g})}}{\\frac{\\partial}{\\partial F_{g}} (\\Omega - \\cos{(F_{g})})} = 1", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('A_z')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Pow(Derivative(Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Derivative(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('A_z')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{D}{(E_{\\lambda},\\mathbf{g})} = E_{\\lambda} + \\mathbf{g}, then obtain (\\iint (- E_{\\lambda} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})}) d\\mathbf{g} d\\mathbf{g})^{E_{\\lambda}} = (\\iint \\mathbf{g} d\\mathbf{g} d\\mathbf{g})^{E_{\\lambda}}", "derivation": "\\mathbf{D}{(E_{\\lambda},\\mathbf{g})} = E_{\\lambda} + \\mathbf{g} and - E_{\\lambda} - \\mathbf{g} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})} = 0 and - E_{\\lambda} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})} = \\mathbf{g} and \\int (- E_{\\lambda} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})}) d\\mathbf{g} = \\int \\mathbf{g} d\\mathbf{g} and \\iint (- E_{\\lambda} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})}) d\\mathbf{g} d\\mathbf{g} = \\iint \\mathbf{g} d\\mathbf{g} d\\mathbf{g} and (\\iint (- E_{\\lambda} + \\mathbf{D}{(E_{\\lambda},\\mathbf{g})}) d\\mathbf{g} d\\mathbf{g})^{E_{\\lambda}} = (\\iint \\mathbf{g} d\\mathbf{g} d\\mathbf{g})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Integer(0))"], [["add", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))"], [["integrate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given U{(c,\\mathbf{P})} = \\cos{(\\mathbf{P} - c)}, then obtain - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})} + U{(c,\\mathbf{P})} - \\cos{(\\mathbf{P} - c)} = - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})}", "derivation": "U{(c,\\mathbf{P})} = \\cos{(\\mathbf{P} - c)} and (\\mathbf{P} - c) U{(c,\\mathbf{P})} = (\\mathbf{P} - c) \\cos{(\\mathbf{P} - c)} and - (\\mathbf{P} - c) U{(c,\\mathbf{P})} - (\\mathbf{P} - c) \\cos{(\\mathbf{P} - c)} + U{(c,\\mathbf{P})} = - (\\mathbf{P} - c) U{(c,\\mathbf{P})} - (\\mathbf{P} - c) \\cos{(\\mathbf{P} - c)} + \\cos{(\\mathbf{P} - c)} and - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})} + U{(c,\\mathbf{P})} = - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})} + \\cos{(\\mathbf{P} - c)} and - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})} + U{(c,\\mathbf{P})} - \\cos{(\\mathbf{P} - c)} = - 2 (\\mathbf{P} - c) U{(c,\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["times", 1, "Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], [["minus", 1, "Add(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], [["minus", 4, "cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))), Mul(Integer(-1), Integer(2), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Function('U')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given c{(\\mathbb{I},\\dot{z},q)} = \\frac{\\dot{z}^{\\mathbb{I}}}{q}, then obtain q (2 q + c{(\\mathbb{I},\\dot{z},q)}) = q (\\frac{\\dot{z}^{\\mathbb{I}}}{q} + 2 q)", "derivation": "c{(\\mathbb{I},\\dot{z},q)} = \\frac{\\dot{z}^{\\mathbb{I}}}{q} and q + c{(\\mathbb{I},\\dot{z},q)} = \\frac{\\dot{z}^{\\mathbb{I}}}{q} + q and 2 q + c{(\\mathbb{I},\\dot{z},q)} = \\frac{\\dot{z}^{\\mathbb{I}}}{q} + 2 q and q (2 q + c{(\\mathbb{I},\\dot{z},q)}) = q (\\frac{\\dot{z}^{\\mathbb{I}}}{q} + 2 q)", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True))), Add(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Symbol('q', commutative=True)))"], [["add", 2, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('q', commutative=True)), Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True))), Add(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('q', commutative=True))))"], [["divide", 3, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('q', commutative=True), Add(Mul(Integer(2), Symbol('q', commutative=True)), Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)))), Mul(Symbol('q', commutative=True), Add(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given A{(\\dot{x},M_{E})} = \\int \\dot{x}^{M_{E}} dM_{E} and \\ddot{x}{(M_{E})} = M_{E}, then obtain - M_{E} + A{(\\dot{x},M_{E})} = - M_{E} + \\int \\dot{x}^{M_{E}} dM_{E}", "derivation": "A{(\\dot{x},M_{E})} = \\int \\dot{x}^{M_{E}} dM_{E} and \\ddot{x}{(M_{E})} = M_{E} and A{(\\dot{x},M_{E})} - \\ddot{x}{(M_{E})} = - \\ddot{x}{(M_{E})} + \\int \\dot{x}^{M_{E}} dM_{E} and - M_{E} + A{(\\dot{x},M_{E})} = - M_{E} + \\int \\dot{x}^{M_{E}} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True)), Integral(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["minus", 1, "Function('\\\\ddot{x}')(Symbol('M_E', commutative=True))"], "Equality(Add(Function('A')(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('M_E', commutative=True))), Integral(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('A')(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Integral(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(y,p)} = p y, then derive - p y + p \\frac{\\partial}{\\partial p} \\ddot{x}{(y,p)} = 0, then obtain - p y + p \\frac{\\partial}{\\partial p} p y = 0", "derivation": "\\ddot{x}{(y,p)} = p y and \\frac{\\partial}{\\partial p} \\ddot{x}{(y,p)} = \\frac{\\partial}{\\partial p} p y and p \\frac{\\partial}{\\partial p} \\ddot{x}{(y,p)} = p \\frac{\\partial}{\\partial p} p y and - p y + p \\frac{\\partial}{\\partial p} \\ddot{x}{(y,p)} = - p y + p \\frac{\\partial}{\\partial p} p y and - p y + p \\frac{\\partial}{\\partial p} \\ddot{x}{(y,p)} = 0 and - p y + p \\frac{\\partial}{\\partial p} p y = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Symbol('p', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["times", 2, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Symbol('p', commutative=True), Derivative(Mul(Symbol('p', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["minus", 3, "Mul(Symbol('p', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('p', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('p', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('p', commutative=True), Derivative(Mul(Symbol('p', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('p', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('p', commutative=True), Derivative(Mul(Symbol('p', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{S})} = \\mathbf{S}, then derive 2 \\operatorname{F_{c}}{(\\mathbf{S})} \\int \\operatorname{F_{c}}{(\\mathbf{S})} d\\mathbf{S} = 2 (\\Psi_{nl} + \\frac{\\mathbf{S}^{2}}{2}) \\operatorname{F_{c}}{(\\mathbf{S})}, then obtain \\frac{\\mathbf{S}^{2}}{2} + 2 \\mathbf{S} \\int \\mathbf{S} d\\mathbf{S} = \\frac{\\mathbf{S}^{2}}{2} + 2 \\mathbf{S} (\\Psi_{nl} + \\frac{\\mathbf{S}^{2}}{2})", "derivation": "\\operatorname{F_{c}}{(\\mathbf{S})} = \\mathbf{S} and \\int \\operatorname{F_{c}}{(\\mathbf{S})} d\\mathbf{S} = \\int \\mathbf{S} d\\mathbf{S} and 2 \\operatorname{F_{c}}{(\\mathbf{S})} \\int \\operatorname{F_{c}}{(\\mathbf{S})} d\\mathbf{S} = 2 \\operatorname{F_{c}}{(\\mathbf{S})} \\int \\mathbf{S} d\\mathbf{S} and 2 \\operatorname{F_{c}}{(\\mathbf{S})} \\int \\operatorname{F_{c}}{(\\mathbf{S})} d\\mathbf{S} = 2 (\\Psi_{nl} + \\frac{\\mathbf{S}^{2}}{2}) \\operatorname{F_{c}}{(\\mathbf{S})} and 2 \\mathbf{S} \\int \\mathbf{S} d\\mathbf{S} = 2 \\mathbf{S} (\\Psi_{nl} + \\frac{\\mathbf{S}^{2}}{2}) and \\frac{\\mathbf{S}^{2}}{2} + 2 \\mathbf{S} \\int \\mathbf{S} d\\mathbf{S} = \\frac{\\mathbf{S}^{2}}{2} + 2 \\mathbf{S} (\\Psi_{nl} + \\frac{\\mathbf{S}^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Symbol('\\\\mathbf{S}', commutative=True), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 2, "Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(Symbol('\\\\mathbf{S}', commutative=True), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)))), Function('F_c')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Integral(Symbol('\\\\mathbf{S}', commutative=True), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))))))"], [["add", 5, "Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Integral(Symbol('\\\\mathbf{S}', commutative=True), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)))))))"]]}, {"prompt": "Given \\mathbf{E}{(J)} = \\log{(J)} and g{(\\rho_b)} = e^{\\rho_b}, then obtain \\int g{(\\rho_b)} \\log{(J)} d\\rho_b = \\int e^{\\rho_b} \\log{(J)} d\\rho_b", "derivation": "\\mathbf{E}{(J)} = \\log{(J)} and g{(\\rho_b)} = e^{\\rho_b} and \\mathbf{E}{(J)} g{(\\rho_b)} = \\mathbf{E}{(J)} e^{\\rho_b} and \\int \\mathbf{E}{(J)} g{(\\rho_b)} d\\rho_b = \\int \\mathbf{E}{(J)} e^{\\rho_b} d\\rho_b and \\int g{(\\rho_b)} \\log{(J)} d\\rho_b = \\int e^{\\rho_b} \\log{(J)} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], ["get_premise", "Equality(Function('g')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{E}')(Symbol('J', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('J', commutative=True)), Function('g')(Symbol('\\\\rho_b', commutative=True))), Mul(Function('\\\\mathbf{E}')(Symbol('J', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{E}')(Symbol('J', commutative=True)), Function('g')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Function('\\\\mathbf{E}')(Symbol('J', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Function('g')(Symbol('\\\\rho_b', commutative=True)), log(Symbol('J', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(exp(Symbol('\\\\rho_b', commutative=True)), log(Symbol('J', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{t})} = \\cos{(v_{t})}, then derive \\int \\mathbf{E}{(v_{t})} dv_{t} = A + \\sin{(v_{t})}, then obtain 2 \\int \\mathbf{E}{(v_{t})} dv_{t} = A + \\sin{(v_{t})} + \\int \\mathbf{E}{(v_{t})} dv_{t}", "derivation": "\\mathbf{E}{(v_{t})} = \\cos{(v_{t})} and \\int \\mathbf{E}{(v_{t})} dv_{t} = \\int \\cos{(v_{t})} dv_{t} and \\int \\mathbf{E}{(v_{t})} dv_{t} = A + \\sin{(v_{t})} and A + \\sin{(v_{t})} + \\int \\mathbf{E}{(v_{t})} dv_{t} = A + \\sin{(v_{t})} + \\int \\cos{(v_{t})} dv_{t} and 2 \\int \\mathbf{E}{(v_{t})} dv_{t} = \\int \\mathbf{E}{(v_{t})} dv_{t} + \\int \\cos{(v_{t})} dv_{t} and A + \\sin{(v_{t})} = \\int \\cos{(v_{t})} dv_{t} and 2 \\int \\mathbf{E}{(v_{t})} dv_{t} = A + \\sin{(v_{t})} + \\int \\mathbf{E}{(v_{t})} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True))))"], [["add", 2, "Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True)))"], "Equality(Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True)), Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True)), Integral(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True))), Integral(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(Symbol('A', commutative=True), sin(Symbol('v_t', commutative=True)), Integral(Function('\\\\mathbf{E}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(u)} = \\log{(u)}, then obtain e^{\\frac{d}{d u} 2 \\operatorname{L_{\\varepsilon}}^{2}{(u)}} = e^{\\frac{d}{d u} (\\operatorname{L_{\\varepsilon}}^{2}{(u)} + \\operatorname{L_{\\varepsilon}}{(u)} \\log{(u)})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(u)} = \\log{(u)} and \\operatorname{L_{\\varepsilon}}^{2}{(u)} = \\operatorname{L_{\\varepsilon}}{(u)} \\log{(u)} and 2 \\operatorname{L_{\\varepsilon}}^{2}{(u)} = \\operatorname{L_{\\varepsilon}}^{2}{(u)} + \\operatorname{L_{\\varepsilon}}{(u)} \\log{(u)} and \\frac{d}{d u} 2 \\operatorname{L_{\\varepsilon}}^{2}{(u)} = \\frac{d}{d u} (\\operatorname{L_{\\varepsilon}}^{2}{(u)} + \\operatorname{L_{\\varepsilon}}{(u)} \\log{(u)}) and e^{\\frac{d}{d u} 2 \\operatorname{L_{\\varepsilon}}^{2}{(u)}} = e^{\\frac{d}{d u} (\\operatorname{L_{\\varepsilon}}^{2}{(u)} + \\operatorname{L_{\\varepsilon}}{(u)} \\log{(u)})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["times", 1, "Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True))"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))))"], [["add", 2, "Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2))), Add(Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(Derivative(Mul(Integer(2), Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1)))), exp(Derivative(Add(Pow(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('L_{\\\\varepsilon}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(k)} = \\sin{(k)}, then obtain (\\sin{(k)} + \\sin^{k}{(\\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)})}) \\sin{(k)} = (0^{k} + \\sin{(k)}) \\sin{(k)}", "derivation": "\\varphi^{*}{(k)} = \\sin{(k)} and \\varphi^{*}^{k}{(k)} = \\sin^{k}{(k)} and \\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)} = 0 and \\sin{(\\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)})} = 0 and \\sin^{k}{(\\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)})} = 0^{k} and \\sin{(k)} + \\sin^{k}{(\\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)})} = 0^{k} + \\sin{(k)} and (\\sin{(k)} + \\sin^{k}{(\\varphi^{*}^{k}{(k)} - \\sin^{k}{(k)})}) \\sin{(k)} = (0^{k} + \\sin{(k)}) \\sin{(k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 2, "Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Add(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Integer(0))"], [["sin", 3], "Equality(sin(Add(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(sin(Add(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True))))), Symbol('k', commutative=True)), Pow(Integer(0), Symbol('k', commutative=True)))"], [["minus", 5, "Mul(Integer(-1), sin(Symbol('k', commutative=True)))"], "Equality(Add(sin(Symbol('k', commutative=True)), Pow(sin(Add(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True))))), Symbol('k', commutative=True))), Add(Pow(Integer(0), Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))))"], [["times", 6, "sin(Symbol('k', commutative=True))"], "Equality(Mul(Add(sin(Symbol('k', commutative=True)), Pow(sin(Add(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('k', commutative=True)), Symbol('k', commutative=True))))), Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Mul(Add(Pow(Integer(0), Symbol('k', commutative=True)), sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\dot{z})} = \\log{(e^{\\dot{z}})}, then obtain (\\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\mathbf{M}{(\\dot{z})})^{\\dot{z}} = (\\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\log{(e^{\\dot{z}})})^{\\dot{z}}", "derivation": "\\mathbf{M}{(\\dot{z})} = \\log{(e^{\\dot{z}})} and \\frac{d}{d \\dot{z}} \\mathbf{M}{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\log{(e^{\\dot{z}})} and \\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\mathbf{M}{(\\dot{z})} = \\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\log{(e^{\\dot{z}})} and (\\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\mathbf{M}{(\\dot{z})})^{\\dot{z}} = (\\log{(e^{\\dot{z}})} \\frac{d}{d \\dot{z}} \\log{(e^{\\dot{z}})})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True)), log(exp(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["times", 2, "log(exp(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(log(exp(Symbol('\\\\dot{z}', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(log(exp(Symbol('\\\\dot{z}', commutative=True))), Derivative(log(exp(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Mul(log(exp(Symbol('\\\\dot{z}', commutative=True))), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Symbol('\\\\dot{z}', commutative=True)), Pow(Mul(log(exp(Symbol('\\\\dot{z}', commutative=True))), Derivative(log(exp(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(A_{y},\\hat{p}_0)} = A_{y} \\sin{(\\hat{p}_0)}, then obtain 1 + \\frac{\\int \\mathbf{r}{(A_{y},\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}} = 1 + \\frac{\\int A_{y} \\sin{(\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}}", "derivation": "\\mathbf{r}{(A_{y},\\hat{p}_0)} = A_{y} \\sin{(\\hat{p}_0)} and \\int \\mathbf{r}{(A_{y},\\hat{p}_0)} dA_{y} = \\int A_{y} \\sin{(\\hat{p}_0)} dA_{y} and \\frac{\\int \\mathbf{r}{(A_{y},\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}} = \\frac{\\int A_{y} \\sin{(\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}} and 1 + \\frac{\\int \\mathbf{r}{(A_{y},\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}} = 1 + \\frac{\\int A_{y} \\sin{(\\hat{p}_0)} dA_{y}}{A_{y} \\sin{(\\hat{p}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(Mul(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], [["divide", 2, "Mul(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Integral(Mul(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('A_y', commutative=True)))))"], [["add", 3, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('A_y', commutative=True))))), Add(Integer(1), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Integral(Mul(Symbol('A_y', commutative=True), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('A_y', commutative=True))))))"]]}, {"prompt": "Given T{(G)} = \\sin{(\\log{(G)})}, then obtain \\frac{d}{d G} \\int (T{(G)} - \\sin{(\\log{(G)})}) dG = \\frac{d}{d G} \\int 0 dG", "derivation": "T{(G)} = \\sin{(\\log{(G)})} and T{(G)} - \\sin{(\\log{(G)})} = 0 and \\int (T{(G)} - \\sin{(\\log{(G)})}) dG = \\int 0 dG and T{(G)} - \\sin{(\\log{(G)})} + \\int (T{(G)} - \\sin{(\\log{(G)})}) dG = T{(G)} - \\sin{(\\log{(G)})} + \\int 0 dG and \\frac{d}{d G} (T{(G)} - \\sin{(\\log{(G)})} + \\int (T{(G)} - \\sin{(\\log{(G)})}) dG) = \\frac{d}{d G} (T{(G)} - \\sin{(\\log{(G)})} + \\int 0 dG) and \\frac{d}{d G} \\int (T{(G)} - \\sin{(\\log{(G)})}) dG = \\frac{d}{d G} \\int 0 dG", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('G', commutative=True)), sin(log(Symbol('G', commutative=True))))"], [["minus", 1, "sin(log(Symbol('G', commutative=True)))"], "Equality(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True))), Integral(Integer(0), Tuple(Symbol('G', commutative=True))))"], [["add", 3, "Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True)))))"], "Equality(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True)))), Integral(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True)))), Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True)))), Integral(Integer(0), Tuple(Symbol('G', commutative=True)))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True)))), Integral(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True)))), Integral(Integer(0), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Integral(Add(Function('T')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(log(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}, then obtain \\log{(\\sin{(\\hat{x})})} \\frac{d}{d \\hat{x}} \\operatorname{n_{2}}{(\\hat{x})} = \\frac{\\log{(\\sin{(\\hat{x})})} \\cos{(\\hat{x})}}{\\sin{(\\hat{x})}}", "derivation": "\\operatorname{n_{2}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})} and \\frac{d}{d \\hat{x}} \\operatorname{n_{2}}{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(\\sin{(\\hat{x})})} and \\log{(\\sin{(\\hat{x})})} \\frac{d}{d \\hat{x}} \\operatorname{n_{2}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})} \\frac{d}{d \\hat{x}} \\log{(\\sin{(\\hat{x})})} and \\log{(\\sin{(\\hat{x})})} \\frac{d}{d \\hat{x}} \\operatorname{n_{2}}{(\\hat{x})} = \\frac{\\log{(\\sin{(\\hat{x})})} \\cos{(\\hat{x})}}{\\sin{(\\hat{x})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["times", 2, "log(sin(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Derivative(Function('n_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Derivative(log(sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Derivative(Function('n_2')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(m)} = e^{\\cos{(m)}}, then obtain \\int \\log{(\\frac{\\operatorname{f^{*}}{(m)} e^{\\cos{(m)}}}{2 \\cos{(m)}})} dm = \\int \\log{(\\frac{e^{2 \\cos{(m)}}}{2 \\cos{(m)}})} dm", "derivation": "\\operatorname{f^{*}}{(m)} = e^{\\cos{(m)}} and \\operatorname{f^{*}}{(m)} e^{\\cos{(m)}} = e^{2 \\cos{(m)}} and \\frac{\\operatorname{f^{*}}{(m)} e^{\\cos{(m)}}}{2 \\cos{(m)}} = \\frac{e^{2 \\cos{(m)}}}{2 \\cos{(m)}} and \\log{(\\frac{\\operatorname{f^{*}}{(m)} e^{\\cos{(m)}}}{2 \\cos{(m)}})} = \\log{(\\frac{e^{2 \\cos{(m)}}}{2 \\cos{(m)}})} and \\int \\log{(\\frac{\\operatorname{f^{*}}{(m)} e^{\\cos{(m)}}}{2 \\cos{(m)}})} dm = \\int \\log{(\\frac{e^{2 \\cos{(m)}}}{2 \\cos{(m)}})} dm", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('m', commutative=True)), exp(cos(Symbol('m', commutative=True))))"], [["times", 1, "exp(cos(Symbol('m', commutative=True)))"], "Equality(Mul(Function('f^*')(Symbol('m', commutative=True)), exp(cos(Symbol('m', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('m', commutative=True)))))"], [["divide", 2, "Mul(Integer(2), cos(Symbol('m', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('f^*')(Symbol('m', commutative=True)), exp(cos(Symbol('m', commutative=True))), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Mul(Rational(1, 2), exp(Mul(Integer(2), cos(Symbol('m', commutative=True)))), Pow(cos(Symbol('m', commutative=True)), Integer(-1))))"], [["log", 3], "Equality(log(Mul(Rational(1, 2), Function('f^*')(Symbol('m', commutative=True)), exp(cos(Symbol('m', commutative=True))), Pow(cos(Symbol('m', commutative=True)), Integer(-1)))), log(Mul(Rational(1, 2), exp(Mul(Integer(2), cos(Symbol('m', commutative=True)))), Pow(cos(Symbol('m', commutative=True)), Integer(-1)))))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(log(Mul(Rational(1, 2), Function('f^*')(Symbol('m', commutative=True)), exp(cos(Symbol('m', commutative=True))), Pow(cos(Symbol('m', commutative=True)), Integer(-1)))), Tuple(Symbol('m', commutative=True))), Integral(log(Mul(Rational(1, 2), exp(Mul(Integer(2), cos(Symbol('m', commutative=True)))), Pow(cos(Symbol('m', commutative=True)), Integer(-1)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(h,\\ddot{x})} = e^{\\ddot{x} + h}, then derive - \\mathbf{p} - e^{\\ddot{x} + h} + \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh = 0, then obtain - \\mathbf{p} - \\Psi^{\\dagger}{(h,\\ddot{x})} + \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh = 0", "derivation": "\\Psi^{\\dagger}{(h,\\ddot{x})} = e^{\\ddot{x} + h} and \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh = \\int e^{\\ddot{x} + h} dh and \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh - \\int e^{\\ddot{x} + h} dh = 0 and - \\mathbf{p} - e^{\\ddot{x} + h} + \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh = 0 and - \\mathbf{p} - \\Psi^{\\dagger}{(h,\\ddot{x})} + \\int \\Psi^{\\dagger}{(h,\\ddot{x})} dh = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), exp(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(exp(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["minus", 2, "Integral(exp(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('h', commutative=True))), Mul(Integer(-1), Integral(exp(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('h', commutative=True)))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('h', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{A}{(V,\\mu_0)} = \\int V \\mu_0 dV, then derive \\mathbf{f} + \\frac{\\int - \\frac{\\mu_0}{V} dV + \\int \\frac{\\mathbf{A}{(V,\\mu_0)}}{V} dV}{\\mu_0} = \\frac{V^{2}}{4} + m - \\log{(V)}, then obtain \\frac{\\partial}{\\partial \\mu_0} (\\mathbf{f} + \\frac{\\int - \\frac{\\mu_0}{V} dV + \\int \\frac{\\mathbf{A}{(V,\\mu_0)}}{V} dV}{\\mu_0}) = \\frac{\\partial}{\\partial \\mu_0} (\\frac{V^{2}}{4} + m - \\log{(V)})", "derivation": "\\mathbf{A}{(V,\\mu_0)} = \\int V \\mu_0 dV and \\frac{\\mathbf{A}{(V,\\mu_0)}}{V \\mu_0} = \\frac{\\int V \\mu_0 dV}{V \\mu_0} and - \\frac{1}{V} + \\frac{\\mathbf{A}{(V,\\mu_0)}}{V \\mu_0} = - \\frac{1}{V} + \\frac{\\int V \\mu_0 dV}{V \\mu_0} and \\int (- \\frac{1}{V} + \\frac{\\mathbf{A}{(V,\\mu_0)}}{V \\mu_0}) dV = \\int (- \\frac{1}{V} + \\frac{\\int V \\mu_0 dV}{V \\mu_0}) dV and \\mathbf{f} + \\frac{\\int - \\frac{\\mu_0}{V} dV + \\int \\frac{\\mathbf{A}{(V,\\mu_0)}}{V} dV}{\\mu_0} = \\frac{V^{2}}{4} + m - \\log{(V)} and \\frac{\\partial}{\\partial \\mu_0} (\\mathbf{f} + \\frac{\\int - \\frac{\\mu_0}{V} dV + \\int \\frac{\\mathbf{A}{(V,\\mu_0)}}{V} dV}{\\mu_0}) = \\frac{\\partial}{\\partial \\mu_0} (\\frac{V^{2}}{4} + m - \\log{(V)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integral(Mul(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["divide", 1, "Mul(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["minus", 2, "Pow(Symbol('V', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True))))))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True))))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('V', commutative=True)))))), Add(Mul(Rational(1, 4), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('m', commutative=True), Mul(Integer(-1), log(Symbol('V', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('V', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('V', commutative=True)))))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 4), Pow(Symbol('V', commutative=True), Integer(2))), Symbol('m', commutative=True), Mul(Integer(-1), log(Symbol('V', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(T,\\varphi)} = T \\varphi and \\operatorname{c_{0}}{(B,\\tilde{g}^*)} = - B + \\tilde{g}^*, then obtain T \\varphi \\operatorname{c_{0}}{(B,\\tilde{g}^*)} + \\int \\hat{p}{(T,\\varphi)} dT = T \\varphi (- B + \\tilde{g}^*) + \\int \\hat{p}{(T,\\varphi)} dT", "derivation": "\\hat{p}{(T,\\varphi)} = T \\varphi and \\int \\hat{p}{(T,\\varphi)} dT = \\int T \\varphi dT and \\operatorname{c_{0}}{(B,\\tilde{g}^*)} = - B + \\tilde{g}^* and \\hat{p}{(T,\\varphi)} \\operatorname{c_{0}}{(B,\\tilde{g}^*)} = (- B + \\tilde{g}^*) \\hat{p}{(T,\\varphi)} and T \\varphi \\operatorname{c_{0}}{(B,\\tilde{g}^*)} = T \\varphi (- B + \\tilde{g}^*) and T \\varphi \\operatorname{c_{0}}{(B,\\tilde{g}^*)} + \\int T \\varphi dT = T \\varphi (- B + \\tilde{g}^*) + \\int T \\varphi dT and T \\varphi \\operatorname{c_{0}}{(B,\\tilde{g}^*)} + \\int \\hat{p}{(T,\\varphi)} dT = T \\varphi (- B + \\tilde{g}^*) + \\int \\hat{p}{(T,\\varphi)} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True))))"], ["get_premise", "Equality(Function('c_0')(Symbol('B', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 3, "Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('c_0')(Symbol('B', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Function('c_0')(Symbol('B', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 5, "Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Function('c_0')(Symbol('B', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Function('c_0')(Symbol('B', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(U)} = \\cos{(e^{U})}, then derive \\int \\hat{X}{(U)} dU = A + \\operatorname{Ci}{(e^{U})}, then obtain \\int (\\int \\cos{(e^{U})} dU)^{U} dU = \\int (A + \\operatorname{Ci}{(e^{U})})^{U} dU", "derivation": "\\hat{X}{(U)} = \\cos{(e^{U})} and \\int \\hat{X}{(U)} dU = \\int \\cos{(e^{U})} dU and \\int \\hat{X}{(U)} dU = A + \\operatorname{Ci}{(e^{U})} and (\\int \\hat{X}{(U)} dU)^{U} = (A + \\operatorname{Ci}{(e^{U})})^{U} and \\int (\\int \\hat{X}{(U)} dU)^{U} dU = \\int (A + \\operatorname{Ci}{(e^{U})})^{U} dU and \\int (\\int \\cos{(e^{U})} dU)^{U} dU = \\int (A + \\operatorname{Ci}{(e^{U})})^{U} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True)), cos(exp(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('A', commutative=True), Ci(exp(Symbol('U', commutative=True)))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Add(Symbol('A', commutative=True), Ci(exp(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\hat{X}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Symbol('A', commutative=True), Ci(exp(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Pow(Integral(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Symbol('A', commutative=True), Ci(exp(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)} = \\frac{\\varepsilon}{\\Psi \\dot{z}}, then obtain - \\frac{\\dot{z}^{2} \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)}}{\\varepsilon} = - \\frac{\\dot{z}}{\\Psi}", "derivation": "\\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)} = \\frac{\\varepsilon}{\\Psi \\dot{z}} and \\Psi \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)} = \\frac{\\varepsilon}{\\dot{z}} and \\Psi \\dot{z} \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)} = \\varepsilon and \\frac{\\dot{z}^{2} \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)}}{\\varepsilon} = \\frac{\\dot{z}}{\\Psi} and - \\frac{\\dot{z}^{2} \\mathbf{D}{(\\dot{z},\\Psi,\\varepsilon)}}{\\varepsilon} = - \\frac{\\dot{z}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 2, "Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\dot{z}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))"], [["divide", 3, "Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True)))"]]}, {"prompt": "Given S{(E)} = 2 E and \\operatorname{E_{\\lambda}}{(E,c_{0})} = 2 E - c_{0}, then obtain E - c_{0} - (2 E - c_{0})^{2} - \\frac{S{(E)}}{c_{0}} = E - \\frac{2 E}{c_{0}} - c_{0} - (2 E - c_{0})^{2}", "derivation": "S{(E)} = 2 E and \\operatorname{E_{\\lambda}}{(E,c_{0})} = 2 E - c_{0} and - \\frac{S{(E)}}{c_{0}} = - \\frac{2 E}{c_{0}} and - (2 E - c_{0}) \\operatorname{E_{\\lambda}}{(E,c_{0})} - \\frac{S{(E)}}{c_{0}} = - \\frac{2 E}{c_{0}} - (2 E - c_{0}) \\operatorname{E_{\\lambda}}{(E,c_{0})} and E - c_{0} - (2 E - c_{0}) \\operatorname{E_{\\lambda}}{(E,c_{0})} - \\frac{S{(E)}}{c_{0}} = E - \\frac{2 E}{c_{0}} - c_{0} - (2 E - c_{0}) \\operatorname{E_{\\lambda}}{(E,c_{0})} and E - c_{0} - (2 E - c_{0})^{2} - \\frac{S{(E)}}{c_{0}} = E - \\frac{2 E}{c_{0}} - c_{0} - (2 E - c_{0})^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('S')(Symbol('E', commutative=True)), Mul(Integer(2), Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)))))"], [["add", 4, "Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('E_{\\\\lambda}')(Symbol('E', commutative=True), Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Integer(-1)), Function('S')(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Integer(2), Symbol('E', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\rho_{f}{(n_{2},H)} = \\int (H + n_{2}) dH, then derive \\rho_{f}{(n_{2},H)} = \\frac{H^{2}}{2} + H n_{2} + P_{e}, then obtain \\frac{\\rho_{f}^{2}{(n_{2},H)}}{P_{e}^{2}} = \\frac{(\\frac{H^{2}}{2} + H n_{2} + P_{e})^{2}}{P_{e}^{2}}", "derivation": "\\rho_{f}{(n_{2},H)} = \\int (H + n_{2}) dH and \\rho_{f}{(n_{2},H)} = \\frac{H^{2}}{2} + H n_{2} + P_{e} and \\int (H + n_{2}) dH = \\frac{H^{2}}{2} + H n_{2} + P_{e} and \\frac{\\int (H + n_{2}) dH}{P_{e}} = \\frac{\\frac{H^{2}}{2} + H n_{2} + P_{e}}{P_{e}} and \\frac{(\\int (H + n_{2}) dH)^{2}}{P_{e}^{2}} = \\frac{(\\frac{H^{2}}{2} + H n_{2} + P_{e})^{2}}{P_{e}^{2}} and \\frac{\\rho_{f}^{2}{(n_{2},H)}}{P_{e}^{2}} = \\frac{(\\frac{H^{2}}{2} + H n_{2} + P_{e})^{2}}{P_{e}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('n_2', commutative=True), Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\rho_f')(Symbol('n_2', commutative=True), Symbol('H', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Symbol('P_e', commutative=True)))"], [["divide", 3, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Integral(Add(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Symbol('P_e', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-2)), Pow(Integral(Add(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(2))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Symbol('P_e', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-2)), Pow(Function('\\\\rho_f')(Symbol('n_2', commutative=True), Symbol('H', commutative=True)), Integer(2))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('n_2', commutative=True)), Symbol('P_e', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)} = \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}}, then obtain \\int (\\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)}) d\\mathbf{E} = \\int (\\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}}) d\\mathbf{E}", "derivation": "\\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)} = \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}} and \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}} and \\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)} = \\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}} and \\int (\\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\tilde{g}{(v_{x},\\mathbf{E},\\hat{p}_0)}) d\\mathbf{E} = \\int (\\hat{p}_0 + \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{\\hat{p}_0 v_{x}}) d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(M,r_{0},y)} = \\frac{r_{0} + y}{M}, then obtain \\frac{d^{2}}{d y^{2}} 0^{y} = \\frac{\\partial^{2}}{\\partial y^{2}} (- \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} + \\frac{r_{0} + y}{M r_{0}})^{y}", "derivation": "\\operatorname{A_{1}}{(M,r_{0},y)} = \\frac{r_{0} + y}{M} and \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} = \\frac{r_{0} + y}{M r_{0}} and 0 = - \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} + \\frac{r_{0} + y}{M r_{0}} and 0^{y} = (- \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} + \\frac{r_{0} + y}{M r_{0}})^{y} and \\frac{d}{d y} 0^{y} = \\frac{\\partial}{\\partial y} (- \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} + \\frac{r_{0} + y}{M r_{0}})^{y} and \\frac{d^{2}}{d y^{2}} 0^{y} = \\frac{\\partial^{2}}{\\partial y^{2}} (- \\frac{\\operatorname{A_{1}}{(M,r_{0},y)}}{r_{0}} + \\frac{r_{0} + y}{M r_{0}})^{y}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True))))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True)))))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Integer(0), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Derivative(Pow(Add(Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('A_1')(Symbol('M', commutative=True), Symbol('r_0', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\theta_2,\\dot{z})} = \\log{(\\dot{z})}^{\\theta_2} and \\varphi{(\\theta_2,\\dot{z})} = \\operatorname{C_{1}}{(\\theta_2,\\dot{z})} \\log{(\\dot{z})}^{\\theta_2}, then obtain \\int (\\varphi{(\\theta_2,\\dot{z})} + \\log{(\\dot{z})}^{- 2 \\theta_2}) d\\dot{z} = \\int (\\log{(\\dot{z})}^{2 \\theta_2} + \\log{(\\dot{z})}^{- 2 \\theta_2}) d\\dot{z}", "derivation": "\\operatorname{C_{1}}{(\\theta_2,\\dot{z})} = \\log{(\\dot{z})}^{\\theta_2} and \\operatorname{C_{1}}{(\\theta_2,\\dot{z})} \\log{(\\dot{z})}^{\\theta_2} = \\log{(\\dot{z})}^{2 \\theta_2} and \\varphi{(\\theta_2,\\dot{z})} = \\operatorname{C_{1}}{(\\theta_2,\\dot{z})} \\log{(\\dot{z})}^{\\theta_2} and \\varphi{(\\theta_2,\\dot{z})} = \\log{(\\dot{z})}^{2 \\theta_2} and \\varphi{(\\theta_2,\\dot{z})} + \\log{(\\dot{z})}^{- 2 \\theta_2} = \\log{(\\dot{z})}^{2 \\theta_2} + \\log{(\\dot{z})}^{- 2 \\theta_2} and \\int (\\varphi{(\\theta_2,\\dot{z})} + \\log{(\\dot{z})}^{- 2 \\theta_2}) d\\dot{z} = \\int (\\log{(\\dot{z})}^{2 \\theta_2} + \\log{(\\dot{z})}^{- 2 \\theta_2}) d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Function('C_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Function('C_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\varphi')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], [["add", 4, "Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)))), Add(Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)}, then derive \\mathbf{g}{(\\nabla)} = - \\sin{(\\nabla)}, then obtain \\mathbf{g}{(\\nabla)} + 1 = \\mathbf{g}{(\\nabla)} - \\frac{\\sin{(\\nabla)}}{\\mathbf{g}{(\\nabla)}}", "derivation": "\\mathbf{g}{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)} and 1 = \\frac{\\frac{d}{d \\nabla} \\cos{(\\nabla)}}{\\mathbf{g}{(\\nabla)}} and \\mathbf{g}{(\\nabla)} = - \\sin{(\\nabla)} and \\mathbf{g}{(\\nabla)} + 1 = \\mathbf{g}{(\\nabla)} + \\frac{\\frac{d}{d \\nabla} \\cos{(\\nabla)}}{\\mathbf{g}{(\\nabla)}} and - \\sin{(\\nabla)} = \\frac{d}{d \\nabla} \\cos{(\\nabla)} and \\mathbf{g}{(\\nabla)} + 1 = \\mathbf{g}{(\\nabla)} - \\frac{\\sin{(\\nabla)}}{\\mathbf{g}{(\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["divide", 1, "Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))), Derivative(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\tilde{g})} = \\tilde{g}, then derive \\hat{H}_l + \\nabla{(\\tilde{g})} = \\tilde{g} + i, then obtain \\hat{H}_l + \\tilde{g} = \\tilde{g} + i", "derivation": "\\nabla{(\\tilde{g})} = \\tilde{g} and \\frac{d}{d \\tilde{g}} \\nabla{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\tilde{g} and \\int \\frac{d}{d \\tilde{g}} \\nabla{(\\tilde{g})} d\\tilde{g} = \\int \\frac{d}{d \\tilde{g}} \\tilde{g} d\\tilde{g} and \\hat{H}_l + \\nabla{(\\tilde{g})} = \\tilde{g} + i and \\hat{H}_l + \\tilde{g} = \\tilde{g} + i", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Symbol('\\\\tilde{g}', commutative=True), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Derivative(Symbol('\\\\tilde{g}', commutative=True), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\nabla')(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\delta{(n_{2})} = \\int \\cos{(n_{2})} dn_{2}, then derive 0 = q - \\delta{(n_{2})} + \\sin{(n_{2})}, then obtain \\int 0 dn_{2} = \\int (q + \\sin{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) dn_{2}", "derivation": "\\delta{(n_{2})} = \\int \\cos{(n_{2})} dn_{2} and 0 = - \\delta{(n_{2})} + \\int \\cos{(n_{2})} dn_{2} and 0 = q - \\delta{(n_{2})} + \\sin{(n_{2})} and \\int 0 dn_{2} = \\int (q - \\delta{(n_{2})} + \\sin{(n_{2})}) dn_{2} and \\int 0 dn_{2} = \\int (q + \\sin{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["minus", 1, "Function('\\\\delta')(Symbol('n_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('n_2', commutative=True))), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(0), Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('n_2', commutative=True))), sin(Symbol('n_2', commutative=True))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('n_2', commutative=True))), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(0), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Symbol('q', commutative=True), sin(Symbol('n_2', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{v},m)} = \\mathbf{v} - m and \\omega{(\\mathbf{v},m)} = \\mathbf{v} - m, then obtain - \\mathbf{v} + 2 m + \\omega{(\\mathbf{v},m)} = m", "derivation": "\\operatorname{C_{2}}{(\\mathbf{v},m)} = \\mathbf{v} - m and - \\mathbf{v} + m + \\operatorname{C_{2}}{(\\mathbf{v},m)} = 0 and \\omega{(\\mathbf{v},m)} = \\mathbf{v} - m and \\omega{(\\mathbf{v},m)} = \\operatorname{C_{2}}{(\\mathbf{v},m)} and - \\mathbf{v} + m + \\omega{(\\mathbf{v},m)} = 0 and - \\mathbf{v} + 2 m + \\omega{(\\mathbf{v},m)} = m", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('m', commutative=True), Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('m', commutative=True), Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True))), Integer(0))"], [["minus", 5, "Mul(Integer(-1), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(2), Symbol('m', commutative=True)), Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{1},B)} = B^{A_{1}}, then obtain - 2 B^{A_{1}} + 1 - B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} = - 2 B^{A_{1}} + 3 - 3 B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)}", "derivation": "\\operatorname{P_{g}}{(A_{1},B)} = B^{A_{1}} and B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} = 1 and B^{A_{1}} + B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} = B^{A_{1}} + 1 and - B^{A_{1}} = - B^{A_{1}} + 1 - B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} and - B^{A_{1}} = - B^{A_{1}} + 2 - 2 B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} and - 2 B^{A_{1}} + 1 - B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)} = - 2 B^{A_{1}} + 3 - 3 B^{- A_{1}} \\operatorname{P_{g}}{(A_{1},B)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True)))"], [["divide", 1, "Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["add", 2, "Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Add(Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))), Add(Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True)), Integer(1)))"], [["minus", 2, "Add(Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Integer(2), Mul(Integer(-1), Integer(2), Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))))"], [["add", 5, "Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Integer(3), Mul(Integer(-1), Integer(3), Pow(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('A_1', commutative=True))), Function('P_g')(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\theta_1,G)} = - G + \\theta_1, then obtain - (- G + \\theta_1)^{G} - \\frac{- G + \\theta_1}{G} = - (- G + \\theta_1)^{G} + \\frac{G - \\theta_1}{G}", "derivation": "\\Psi^{\\dagger}{(\\theta_1,G)} = - G + \\theta_1 and - \\frac{\\Psi^{\\dagger}{(\\theta_1,G)}}{G} = - \\frac{- G + \\theta_1}{G} and \\frac{\\Psi^{\\dagger}{(\\theta_1,G)}}{G} = \\frac{- G + \\theta_1}{G} and - \\frac{- G + \\theta_1}{G} = \\frac{G - \\theta_1}{G} and - \\Psi^{\\dagger}^{G}{(\\theta_1,G)} - \\frac{- G + \\theta_1}{G} = - \\Psi^{\\dagger}^{G}{(\\theta_1,G)} + \\frac{G - \\theta_1}{G} and - (- G + \\theta_1)^{G} - \\frac{- G + \\theta_1}{G} = - (- G + \\theta_1)^{G} + \\frac{G - \\theta_1}{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 4, "Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta_1', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(\\theta_2)} = \\int \\sin{(\\theta_2)} d\\theta_2, then derive \\frac{d}{d \\theta_2} \\hat{p}{(\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\tilde{g}^* - \\cos{(\\theta_2)}), then obtain \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 = \\sin{(\\theta_2)}", "derivation": "\\hat{p}{(\\theta_2)} = \\int \\sin{(\\theta_2)} d\\theta_2 and \\frac{d}{d \\theta_2} \\hat{p}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 and \\frac{d}{d \\theta_2} \\hat{p}{(\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\tilde{g}^* - \\cos{(\\theta_2)}) and \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 = \\frac{\\partial}{\\partial \\theta_2} (\\tilde{g}^* - \\cos{(\\theta_2)}) and \\frac{d}{d \\theta_2} \\int \\sin{(\\theta_2)} d\\theta_2 = \\sin{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), sin(Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} = \\frac{\\mathbf{M}}{x^\\prime}, then obtain - \\sin{(\\sin{(r + \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime)})} = - \\sin{(\\sin{(r + \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime)})}", "derivation": "\\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} = \\frac{\\mathbf{M}}{x^\\prime} and \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime = \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime and r + \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime = r + \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime and \\sin{(r + \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime)} = \\sin{(r + \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime)} and \\sin{(\\sin{(r + \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime)})} = \\sin{(\\sin{(r + \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime)})} and - \\sin{(\\sin{(r + \\int \\operatorname{A_{y}}{(x^\\prime,\\mathbf{M})} dx^\\prime)})} = - \\sin{(\\sin{(r + \\int \\frac{\\mathbf{M}}{x^\\prime} dx^\\prime)})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('r', commutative=True))"], "Equality(Add(Symbol('r', commutative=True), Integral(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Symbol('r', commutative=True), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["sin", 3], "Equality(sin(Add(Symbol('r', commutative=True), Integral(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), sin(Add(Symbol('r', commutative=True), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))))))"], [["sin", 4], "Equality(sin(sin(Add(Symbol('r', commutative=True), Integral(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))), sin(sin(Add(Symbol('r', commutative=True), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True)))))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(sin(Add(Symbol('r', commutative=True), Integral(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))), Mul(Integer(-1), sin(sin(Add(Symbol('r', commutative=True), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))))))))"]]}, {"prompt": "Given n{(\\mathbf{B},\\varphi)} = \\mathbf{B} + \\cos{(\\varphi)} and \\operatorname{v_{1}}{(T)} = \\sin{(T)}, then obtain \\int n{(\\mathbf{B},\\varphi)} d\\varphi + \\frac{\\operatorname{v_{1}}{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}} = \\int n{(\\mathbf{B},\\varphi)} d\\varphi + \\frac{\\sin{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}}", "derivation": "n{(\\mathbf{B},\\varphi)} = \\mathbf{B} + \\cos{(\\varphi)} and \\operatorname{v_{1}}{(T)} = \\sin{(T)} and \\frac{\\operatorname{v_{1}}{(T)}}{n{(\\mathbf{B},\\varphi)}} = \\frac{\\sin{(T)}}{n{(\\mathbf{B},\\varphi)}} and \\frac{\\operatorname{v_{1}}{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}} = \\frac{\\sin{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}} and \\int n{(\\mathbf{B},\\varphi)} d\\varphi + \\frac{\\operatorname{v_{1}}{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}} = \\int n{(\\mathbf{B},\\varphi)} d\\varphi + \\frac{\\sin{(T)}}{\\mathbf{B} + \\cos{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))))"], ["get_premise", "Equality(Function('v_1')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["divide", 2, "Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('v_1')(Symbol('T', commutative=True))), Mul(Pow(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Function('v_1')(Symbol('T', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["add", 4, "Integral(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Integral(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Function('v_1')(Symbol('T', commutative=True)))), Add(Integral(Function('n')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), cos(Symbol('\\\\varphi', commutative=True))), Integer(-1)), sin(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = \\dot{x} \\cos{(A_{x})}, then obtain A_{x} \\frac{\\partial}{\\partial A_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = - A_{x} \\dot{x} \\sin{(A_{x})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = \\dot{x} \\cos{(A_{x})} and \\frac{\\partial}{\\partial A_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = \\frac{\\partial}{\\partial A_{x}} \\dot{x} \\cos{(A_{x})} and A_{x} \\frac{\\partial}{\\partial A_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = A_{x} \\frac{\\partial}{\\partial A_{x}} \\dot{x} \\cos{(A_{x})} and A_{x} \\frac{\\partial}{\\partial A_{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{x},\\dot{x})} = - A_{x} \\dot{x} \\sin{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('A_x', commutative=True))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["times", 2, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), cos(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('A_x', commutative=True), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('\\\\dot{x}', commutative=True), sin(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x} \\mathbf{g}}{h}, then derive \\frac{\\partial}{\\partial \\mathbf{g}} r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x}}{h}, then obtain - \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{F_{x} \\mathbf{g}}{h} + \\frac{\\partial}{\\partial \\mathbf{g}} r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x}}{h} - \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{F_{x} \\mathbf{g}}{h}", "derivation": "r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x} \\mathbf{g}}{h} and \\frac{\\partial}{\\partial \\mathbf{g}} r{(h,\\mathbf{g},F_{x})} = \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{F_{x} \\mathbf{g}}{h} and \\frac{\\partial}{\\partial \\mathbf{g}} r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x}}{h} and - \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{F_{x} \\mathbf{g}}{h} + \\frac{\\partial}{\\partial \\mathbf{g}} r{(h,\\mathbf{g},F_{x})} = \\frac{F_{x}}{h} - \\frac{\\partial}{\\partial \\mathbf{g}} \\frac{F_{x} \\mathbf{g}}{h}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('h', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('h', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('h', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Mul(Symbol('F_x', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["minus", 3, "Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Derivative(Function('r')(Symbol('h', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\theta{(c)} = e^{c}, then obtain (\\frac{d}{d c} (\\theta{(c)} - \\theta^{c}{(c)}))^{c} = (\\frac{d}{d c} (- \\theta^{c}{(c)} + e^{c}))^{c}", "derivation": "\\theta{(c)} = e^{c} and \\theta^{c}{(c)} = (e^{c})^{c} and \\theta{(c)} - \\theta^{c}{(c)} = - \\theta^{c}{(c)} + e^{c} and \\theta{(c)} - (e^{c})^{c} = e^{c} - (e^{c})^{c} and \\frac{d}{d c} (\\theta{(c)} - (e^{c})^{c}) = \\frac{d}{d c} (e^{c} - (e^{c})^{c}) and \\frac{d}{d c} (\\theta{(c)} - \\theta^{c}{(c)}) = \\frac{d}{d c} (- \\theta^{c}{(c)} + e^{c}) and (\\frac{d}{d c} (\\theta{(c)} - \\theta^{c}{(c)}))^{c} = (\\frac{d}{d c} (- \\theta^{c}{(c)} + e^{c}))^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), exp(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\theta')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Add(exp(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)))))"], [["differentiate", 4, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 6, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\theta')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('c', commutative=True)), Symbol('c', commutative=True))), exp(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(\\pi,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\pi}, then obtain (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi}) (- \\mathbf{J}_P + \\pi + \\bar{\\h}{(\\pi,\\mathbf{J}_P)}) = (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi}) (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi} + \\pi)", "derivation": "\\bar{\\h}{(\\pi,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{\\pi} and - \\mathbf{J}_P + \\bar{\\h}{(\\pi,\\mathbf{J}_P)} = - \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi} and - \\mathbf{J}_P + \\pi + \\bar{\\h}{(\\pi,\\mathbf{J}_P)} = - \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi} + \\pi and (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi}) (- \\mathbf{J}_P + \\pi + \\bar{\\h}{(\\pi,\\mathbf{J}_P)}) = (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi}) (- \\mathbf{J}_P + \\frac{\\mathbf{J}_P}{\\pi} + \\pi)", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))"], [["add", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\pi', commutative=True), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Symbol('\\\\pi', commutative=True)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\pi', commutative=True), Function('\\\\hbar')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\varepsilon_0)} = e^{\\varepsilon_0}, then obtain (\\frac{\\operatorname{A_{y}}{(\\varepsilon_0)}}{2 \\varepsilon_0})^{\\varepsilon_0} (\\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0})^{\\varepsilon_0} = (\\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0})^{2 \\varepsilon_0}", "derivation": "\\operatorname{A_{y}}{(\\varepsilon_0)} = e^{\\varepsilon_0} and \\frac{\\operatorname{A_{y}}{(\\varepsilon_0)}}{2 \\varepsilon_0} = \\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0} and (\\frac{\\operatorname{A_{y}}{(\\varepsilon_0)}}{2 \\varepsilon_0})^{\\varepsilon_0} = (\\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0})^{\\varepsilon_0} and (\\frac{\\operatorname{A_{y}}{(\\varepsilon_0)}}{2 \\varepsilon_0})^{\\varepsilon_0} (\\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0})^{\\varepsilon_0} = (\\frac{e^{\\varepsilon_0}}{2 \\varepsilon_0})^{2 \\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 3, "Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))), Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given q{(G)} = \\sin{(G)}, then obtain q{(G)} \\sin{(\\sin{(G)})} = \\sin{(G)} \\sin{(\\sin{(G)})}", "derivation": "q{(G)} = \\sin{(G)} and \\sin{(q{(G)})} = \\sin{(\\sin{(G)})} and q{(G)} \\sin{(q{(G)})} = \\sin{(G)} \\sin{(q{(G)})} and q{(G)} \\sin{(\\sin{(G)})} = \\sin{(G)} \\sin{(\\sin{(G)})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["sin", 1], "Equality(sin(Function('q')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True))))"], [["times", 1, "sin(Function('q')(Symbol('G', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('G', commutative=True)), sin(Function('q')(Symbol('G', commutative=True)))), Mul(sin(Symbol('G', commutative=True)), sin(Function('q')(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('q')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Mul(sin(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\eta)} = \\sin{(e^{\\eta})}, then obtain \\iint (A + \\operatorname{f_{\\mathbf{v}}}{(\\eta)}) e^{- \\eta} d\\eta d\\eta = \\iint (A + \\sin{(e^{\\eta})}) e^{- \\eta} d\\eta d\\eta", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\eta)} = \\sin{(e^{\\eta})} and A + \\operatorname{f_{\\mathbf{v}}}{(\\eta)} = A + \\sin{(e^{\\eta})} and (A + \\operatorname{f_{\\mathbf{v}}}{(\\eta)}) e^{- \\eta} = (A + \\sin{(e^{\\eta})}) e^{- \\eta} and \\int (A + \\operatorname{f_{\\mathbf{v}}}{(\\eta)}) e^{- \\eta} d\\eta = \\int (A + \\sin{(e^{\\eta})}) e^{- \\eta} d\\eta and \\iint (A + \\operatorname{f_{\\mathbf{v}}}{(\\eta)}) e^{- \\eta} d\\eta d\\eta = \\iint (A + \\sin{(e^{\\eta})}) e^{- \\eta} d\\eta d\\eta", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True)), sin(exp(Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True))), Add(Symbol('A', commutative=True), sin(exp(Symbol('\\\\eta', commutative=True)))))"], [["divide", 2, "exp(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Add(Symbol('A', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Mul(Add(Symbol('A', commutative=True), sin(exp(Symbol('\\\\eta', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('A', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Mul(Add(Symbol('A', commutative=True), sin(exp(Symbol('\\\\eta', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('A', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Mul(Add(Symbol('A', commutative=True), sin(exp(Symbol('\\\\eta', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(l)} = \\cos{(l)}, then obtain - \\operatorname{V_{\\mathbf{E}}}^{l}{(l)} \\int \\operatorname{V_{\\mathbf{E}}}{(l)} dl = - \\cos^{l}{(l)} \\int \\operatorname{V_{\\mathbf{E}}}{(l)} dl", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(l)} = \\cos{(l)} and \\int \\operatorname{V_{\\mathbf{E}}}{(l)} dl = \\int \\cos{(l)} dl and \\operatorname{V_{\\mathbf{E}}}^{l}{(l)} = \\cos^{l}{(l)} and - \\operatorname{V_{\\mathbf{E}}}^{l}{(l)} = - \\cos^{l}{(l)} and - \\operatorname{V_{\\mathbf{E}}}^{l}{(l)} \\int \\cos{(l)} dl = - \\cos^{l}{(l)} \\int \\cos{(l)} dl and - \\operatorname{V_{\\mathbf{E}}}^{l}{(l)} \\int \\operatorname{V_{\\mathbf{E}}}{(l)} dl = - \\cos^{l}{(l)} \\int \\operatorname{V_{\\mathbf{E}}}{(l)} dl", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["times", 4, "Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})} = - F_{H} + \\mathbb{I} \\phi_2, then obtain \\frac{\\phi_2 \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2,\\mathbb{I},F_{H})} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})}}{\\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})}} = \\frac{\\phi_2^{2} (- F_{H} + \\mathbb{I} \\phi_2)^{\\phi_2}}{- F_{H} + \\mathbb{I} \\phi_2}", "derivation": "\\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})} = - F_{H} + \\mathbb{I} \\phi_2 and \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2,\\mathbb{I},F_{H})} = (- F_{H} + \\mathbb{I} \\phi_2)^{\\phi_2} and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2,\\mathbb{I},F_{H})} = \\frac{\\partial}{\\partial \\mathbb{I}} (- F_{H} + \\mathbb{I} \\phi_2)^{\\phi_2} and \\frac{\\phi_2 \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2,\\mathbb{I},F_{H})} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})}}{\\operatorname{t_{2}}{(\\phi_2,\\mathbb{I},F_{H})}} = \\frac{\\phi_2^{2} (- F_{H} + \\mathbb{I} \\phi_2)^{\\phi_2}}{- F_{H} + \\mathbb{I} \\phi_2}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Derivative(Function('t_2')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(A)} = \\log{(A)}, then derive \\frac{d}{d A} \\dot{y}{(A)} = \\frac{1}{A}, then obtain \\log{(A)} \\frac{d}{d A} \\dot{y}{(A)} = \\frac{\\log{(A)}}{A}", "derivation": "\\dot{y}{(A)} = \\log{(A)} and \\frac{d}{d A} \\dot{y}{(A)} = \\frac{d}{d A} \\log{(A)} and \\frac{d}{d A} \\dot{y}{(A)} = \\frac{1}{A} and \\dot{y}{(A)} \\frac{d}{d A} \\dot{y}{(A)} = \\frac{\\dot{y}{(A)}}{A} and \\log{(A)} \\frac{d}{d A} \\log{(A)} = \\frac{\\log{(A)}}{A} and \\log{(A)} \\frac{d}{d A} \\dot{y}{(A)} = \\frac{\\log{(A)}}{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Pow(Symbol('A', commutative=True), Integer(-1)))"], [["times", 3, "Function('\\\\dot{y}')(Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(log(Symbol('A', commutative=True)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), log(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(log(Symbol('A', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), log(Symbol('A', commutative=True))))"]]}, {"prompt": "Given I{(Q)} = \\cos{(Q)} and i{(l,Q)} = l + \\cos{(Q)}, then obtain - l + \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint i{(l,Q)} dl dl) dl = - l + \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint (l + I{(Q)}) dl dl) dl", "derivation": "I{(Q)} = \\cos{(Q)} and i{(l,Q)} = l + \\cos{(Q)} and \\int i{(l,Q)} dl = \\int (l + \\cos{(Q)}) dl and \\int i{(l,Q)} dl = \\int (l + I{(Q)}) dl and \\iint i{(l,Q)} dl dl = \\iint (l + I{(Q)}) dl dl and (\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint i{(l,Q)} dl dl = (\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint (l + I{(Q)}) dl dl and \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint i{(l,Q)} dl dl) dl = \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint (l + I{(Q)}) dl dl) dl and - l + \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint i{(l,Q)} dl dl) dl = - l + \\int ((\\int m{(l,\\varphi)} dl)^{\\varphi} + \\iint (l + I{(Q)}) dl dl) dl", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('l', commutative=True), cos(Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), cos(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Function('I')(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Function('I')(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["add", 5, "Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Add(Symbol('l', commutative=True), Function('I')(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["integrate", 6, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Integral(Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Add(Symbol('l', commutative=True), Function('I')(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))))"], [["minus", 7, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Function('i')(Symbol('l', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Add(Pow(Integral(Function('m')(Symbol('l', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\varphi', commutative=True)), Integral(Add(Symbol('l', commutative=True), Function('I')(Symbol('Q', commutative=True))), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(V,u)} = u^{V} and \\hat{p}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}, then obtain \\frac{\\hat{H}_{\\lambda}}{\\hat{p}{(\\hat{H}_{\\lambda})}} = \\frac{\\hat{H}_{\\lambda} u^{V}}{\\operatorname{C_{1}}{(V,u)} \\hat{p}{(\\hat{H}_{\\lambda})}}", "derivation": "\\operatorname{C_{1}}{(V,u)} = u^{V} and \\frac{\\operatorname{C_{1}}{(V,u)}}{\\hat{H}_{\\lambda}} = \\frac{u^{V}}{\\hat{H}_{\\lambda}} and \\hat{p}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and \\frac{\\operatorname{C_{1}}{(V,u)}}{\\hat{p}{(\\hat{H}_{\\lambda})}} = \\frac{u^{V}}{\\hat{p}{(\\hat{H}_{\\lambda})}} and \\frac{\\hat{H}_{\\lambda}}{\\hat{p}{(\\hat{H}_{\\lambda})}} = \\frac{\\hat{H}_{\\lambda} u^{V}}{\\operatorname{C_{1}}{(V,u)} \\hat{p}{(\\hat{H}_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('V', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('V', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Function('C_1')(Symbol('V', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('C_1')(Symbol('V', commutative=True), Symbol('u', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('u', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["divide", 4, "Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Function('C_1')(Symbol('V', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('u', commutative=True), Symbol('V', commutative=True)), Pow(Function('C_1')(Symbol('V', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(C_{1})} = e^{\\cos{(C_{1})}}, then obtain \\frac{\\frac{d^{2}}{d C_{1}^{2}} \\rho_{f}{(C_{1})}}{\\cos{(C_{1})}} = \\frac{\\frac{d^{2}}{d C_{1}^{2}} e^{\\cos{(C_{1})}}}{\\cos{(C_{1})}}", "derivation": "\\rho_{f}{(C_{1})} = e^{\\cos{(C_{1})}} and \\frac{d}{d C_{1}} \\rho_{f}{(C_{1})} = \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}} and \\frac{d^{2}}{d C_{1}^{2}} \\rho_{f}{(C_{1})} = \\frac{d^{2}}{d C_{1}^{2}} e^{\\cos{(C_{1})}} and \\frac{\\frac{d^{2}}{d C_{1}^{2}} \\rho_{f}{(C_{1})}}{\\cos{(C_{1})}} = \\frac{\\frac{d^{2}}{d C_{1}^{2}} e^{\\cos{(C_{1})}}}{\\cos{(C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), exp(cos(Symbol('C_1', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2))), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(2))))"], [["divide", 3, "cos(Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('C_1', commutative=True)), Integer(-1)), Derivative(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2)))), Mul(Pow(cos(Symbol('C_1', commutative=True)), Integer(-1)), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given G{(\\lambda)} = \\log{(\\lambda)}, then derive \\int G{(\\lambda)} d\\lambda = \\chi + \\lambda \\log{(\\lambda)} - \\lambda, then obtain (G{(\\lambda)} + \\int \\log{(\\lambda)} d\\lambda) (\\chi + \\lambda \\log{(\\lambda)} - \\lambda) = (G{(\\lambda)} + \\int \\log{(\\lambda)} d\\lambda) \\int \\log{(\\lambda)} d\\lambda", "derivation": "G{(\\lambda)} = \\log{(\\lambda)} and \\int G{(\\lambda)} d\\lambda = \\int \\log{(\\lambda)} d\\lambda and \\int G{(\\lambda)} d\\lambda = \\chi + \\lambda \\log{(\\lambda)} - \\lambda and \\chi + \\lambda \\log{(\\lambda)} - \\lambda = \\int \\log{(\\lambda)} d\\lambda and (G{(\\lambda)} + \\int \\log{(\\lambda)} d\\lambda) (\\chi + \\lambda \\log{(\\lambda)} - \\lambda) = (G{(\\lambda)} + \\int \\log{(\\lambda)} d\\lambda) \\int \\log{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 4, "Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], "Equality(Mul(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Mul(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Integral(log(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(n)} = e^{n} and z{(n)} = n + \\frac{\\rho_{f}^{n}{(n)}}{n}, then obtain \\frac{d}{d n} \\int z{(n)} dn = \\frac{d}{d n} \\int (n + \\frac{(e^{n})^{n}}{n}) dn", "derivation": "\\rho_{f}{(n)} = e^{n} and \\rho_{f}^{n}{(n)} = (e^{n})^{n} and \\frac{\\rho_{f}^{n}{(n)}}{n} = \\frac{(e^{n})^{n}}{n} and n + \\frac{\\rho_{f}^{n}{(n)}}{n} = n + \\frac{(e^{n})^{n}}{n} and z{(n)} = n + \\frac{\\rho_{f}^{n}{(n)}}{n} and z{(n)} = n + \\frac{(e^{n})^{n}}{n} and \\int z{(n)} dn = \\int (n + \\frac{(e^{n})^{n}}{n}) dn and \\frac{d}{d n} \\int z{(n)} dn = \\frac{d}{d n} \\int (n + \\frac{(e^{n})^{n}}{n}) dn", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["divide", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('\\\\rho_f')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True))))"], [["add", 3, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('\\\\rho_f')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Function('\\\\rho_f')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('z')(Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True)))))"], [["integrate", 6, "Symbol('n', commutative=True)"], "Equality(Integral(Function('z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["differentiate", 7, "Symbol('n', commutative=True)"], "Equality(Derivative(Integral(Function('z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('n', commutative=True), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Pow(exp(Symbol('n', commutative=True)), Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(t,\\Omega)} = e^{\\Omega - t}, then obtain (\\frac{\\partial}{\\partial t} (2 \\rho_{f}{(t,\\Omega)} - 1))^{\\Omega} = (\\frac{\\partial}{\\partial t} (2 e^{\\Omega - t} - 1))^{\\Omega}", "derivation": "\\rho_{f}{(t,\\Omega)} = e^{\\Omega - t} and \\rho_{f}{(t,\\Omega)} - 1 = e^{\\Omega - t} - 1 and 2 \\rho_{f}{(t,\\Omega)} - 1 = \\rho_{f}{(t,\\Omega)} + e^{\\Omega - t} - 1 and 2 \\rho_{f}{(t,\\Omega)} - 1 = 2 e^{\\Omega - t} - 1 and \\frac{\\partial}{\\partial t} (2 \\rho_{f}{(t,\\Omega)} - 1) = \\frac{\\partial}{\\partial t} (2 e^{\\Omega - t} - 1) and (\\frac{\\partial}{\\partial t} (2 \\rho_{f}{(t,\\Omega)} - 1))^{\\Omega} = (\\frac{\\partial}{\\partial t} (2 e^{\\Omega - t} - 1))^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Integer(-1)))"], [["add", 2, "Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1)), Add(Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1)), Add(Mul(Integer(2), exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Integer(-1)))"], [["differentiate", 4, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Integer(-1)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Add(Mul(Integer(2), exp(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))), Integer(-1)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given k{(l,B)} = \\log{(l)}^{B}, then obtain - B + k{(l,B)} + \\frac{\\partial}{\\partial l} l \\log{(l)}^{B} = - B + \\log{(l)}^{B} + \\frac{\\partial}{\\partial l} l \\log{(l)}^{B}", "derivation": "k{(l,B)} = \\log{(l)}^{B} and - B + k{(l,B)} = - B + \\log{(l)}^{B} and l k{(l,B)} = l \\log{(l)}^{B} and - B + k{(l,B)} + \\frac{\\partial}{\\partial l} l k{(l,B)} = - B + \\log{(l)}^{B} + \\frac{\\partial}{\\partial l} l k{(l,B)} and - B + k{(l,B)} + \\frac{\\partial}{\\partial l} l \\log{(l)}^{B} = - B + \\log{(l)}^{B} + \\frac{\\partial}{\\partial l} l \\log{(l)}^{B}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True)))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('l', commutative=True), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True))))"], [["add", 2, "Derivative(Mul(Symbol('l', commutative=True), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('k')(Symbol('l', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Pow(log(Symbol('l', commutative=True)), Symbol('B', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given B{(V,v_{x})} = \\int \\frac{v_{x}}{V} dv_{x}, then derive \\int \\frac{\\partial}{\\partial v_{x}} B{(V,v_{x})} dV = \\mathbf{P} + v_{x} \\log{(V)}, then obtain \\int \\frac{\\partial}{\\partial v_{x}} \\int \\frac{v_{x}}{V} dv_{x} dV = \\mathbf{P} + v_{x} \\log{(V)}", "derivation": "B{(V,v_{x})} = \\int \\frac{v_{x}}{V} dv_{x} and \\frac{\\partial}{\\partial v_{x}} B{(V,v_{x})} = \\frac{\\partial}{\\partial v_{x}} \\int \\frac{v_{x}}{V} dv_{x} and \\int \\frac{\\partial}{\\partial v_{x}} B{(V,v_{x})} dV = \\int \\frac{\\partial}{\\partial v_{x}} \\int \\frac{v_{x}}{V} dv_{x} dV and \\int \\frac{\\partial}{\\partial v_{x}} B{(V,v_{x})} dV = \\mathbf{P} + v_{x} \\log{(V)} and \\int \\frac{\\partial}{\\partial v_{x}} \\int \\frac{v_{x}}{V} dv_{x} dV = \\mathbf{P} + v_{x} \\log{(V)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('B')(Symbol('V', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Derivative(Integral(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given v{(v_{y})} = e^{v_{y}}, then obtain 0 = v{(v_{y})} - e^{v_{y}}", "derivation": "v{(v_{y})} = e^{v_{y}} and 0 = - v{(v_{y})} + e^{v_{y}} and e^{v_{y}} = - v{(v_{y})} + 2 e^{v_{y}} and v{(v_{y})} = - v{(v_{y})} + 2 e^{v_{y}} and 0 = v{(v_{y})} - e^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["minus", 1, "Function('v')(Symbol('v_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v')(Symbol('v_y', commutative=True))), exp(Symbol('v_y', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Function('v')(Symbol('v_y', commutative=True))), exp(Symbol('v_y', commutative=True)))"], "Equality(exp(Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Function('v')(Symbol('v_y', commutative=True))), Mul(Integer(2), exp(Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('v')(Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Function('v')(Symbol('v_y', commutative=True))), Mul(Integer(2), exp(Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Add(Function('v')(Symbol('v_y', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)} = Z \\hat{\\mathbf{x}}, then obtain \\frac{(e^{\\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)}})^{\\hat{\\mathbf{x}}}}{\\hat{\\mathbf{x}}} = \\frac{(e^{Z \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}}}{\\hat{\\mathbf{x}}}", "derivation": "\\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)} = Z \\hat{\\mathbf{x}} and e^{\\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)}} = e^{Z \\hat{\\mathbf{x}}} and (e^{\\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)}})^{\\hat{\\mathbf{x}}} = (e^{Z \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}} and \\frac{(e^{\\operatorname{F_{N}}{(\\hat{\\mathbf{x}},Z)}})^{\\hat{\\mathbf{x}}}}{\\hat{\\mathbf{x}}} = \\frac{(e^{Z \\hat{\\mathbf{x}}})^{\\hat{\\mathbf{x}}}}{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('F_N')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True))), exp(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(exp(Function('F_N')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(exp(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["divide", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Pow(exp(Function('F_N')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Pow(exp(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given Q{(V_{\\mathbf{B}},m)} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m, then derive Q{(V_{\\mathbf{B}},m)} = m, then obtain \\frac{m^{m} - 2 (m^{m})^{m} + Q^{m}{(V_{\\mathbf{B}},m)}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m} = \\frac{2 m^{m} - 2 (m^{m})^{m}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m}", "derivation": "Q{(V_{\\mathbf{B}},m)} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m and Q{(V_{\\mathbf{B}},m)} = m and Q^{m}{(V_{\\mathbf{B}},m)} = m^{m} and - 2 (m^{m})^{m} + Q^{m}{(V_{\\mathbf{B}},m)} = m^{m} - 2 (m^{m})^{m} and m^{m} - 2 (m^{m})^{m} + Q^{m}{(V_{\\mathbf{B}},m)} = 2 m^{m} - 2 (m^{m})^{m} and \\frac{m^{m} - 2 (m^{m})^{m} + Q^{m}{(V_{\\mathbf{B}},m)}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m} = \\frac{2 m^{m} - 2 (m^{m})^{m}}{\\frac{\\partial}{\\partial V_{\\mathbf{B}}} V_{\\mathbf{B}} m}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Add(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["add", 4, "Pow(Symbol('m', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["divide", 5, "Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Pow(Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('m', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Pow(Derivative(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(p)} = e^{p} and \\sigma_{p}{(p)} = (p + \\Psi_{nl}{(p)}) (\\Psi_{nl}{(p)} + e^{p}), then obtain \\sigma_{p}{(p)} = 2 (p + e^{p}) e^{p}", "derivation": "\\Psi_{nl}{(p)} = e^{p} and p + \\Psi_{nl}{(p)} = p + e^{p} and \\Psi_{nl}{(p)} + e^{p} = 2 e^{p} and (p + \\Psi_{nl}{(p)}) (\\Psi_{nl}{(p)} + e^{p}) = 2 (p + \\Psi_{nl}{(p)}) e^{p} and (p + e^{p}) (\\Psi_{nl}{(p)} + e^{p}) = 2 (p + e^{p}) e^{p} and \\sigma_{p}{(p)} = (p + \\Psi_{nl}{(p)}) (\\Psi_{nl}{(p)} + e^{p}) and \\sigma_{p}{(p)} = (p + e^{p}) (\\Psi_{nl}{(p)} + e^{p}) and \\sigma_{p}{(p)} = 2 (p + e^{p}) e^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["add", 1, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Add(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))))"], [["add", 1, "exp(Symbol('p', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True))), Mul(Integer(2), exp(Symbol('p', commutative=True))))"], [["times", 3, "Add(Symbol('p', commutative=True), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)))"], "Equality(Mul(Add(Symbol('p', commutative=True), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))), Mul(Integer(2), Add(Symbol('p', commutative=True), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), exp(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))), Mul(Integer(2), Add(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), exp(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('p', commutative=True)), Mul(Add(Symbol('p', commutative=True), Function('\\\\Psi_{nl}')(Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Function('\\\\sigma_p')(Symbol('p', commutative=True)), Mul(Add(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Function('\\\\sigma_p')(Symbol('p', commutative=True)), Mul(Integer(2), Add(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), exp(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(y^{\\prime})} = \\sin{(\\sin{(y^{\\prime})})}, then obtain - \\sin{(y^{\\prime})} + 2 \\sin{(\\sin{(y^{\\prime})})} = \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} + \\sin{(\\sin{(y^{\\prime})})}", "derivation": "\\Psi_{\\lambda}{(y^{\\prime})} = \\sin{(\\sin{(y^{\\prime})})} and \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} = - \\sin{(y^{\\prime})} + \\sin{(\\sin{(y^{\\prime})})} and 2 \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} = \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} + \\sin{(\\sin{(y^{\\prime})})} and 2 \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} = - \\sin{(y^{\\prime})} + 2 \\sin{(\\sin{(y^{\\prime})})} and - \\sin{(y^{\\prime})} + 2 \\sin{(\\sin{(y^{\\prime})})} = \\Psi_{\\lambda}{(y^{\\prime})} - \\sin{(y^{\\prime})} + \\sin{(\\sin{(y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), sin(sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "sin(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), sin(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 1, "Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), sin(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(2), sin(sin(Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(2), sin(sin(Symbol('y^{\\\\prime}', commutative=True))))), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), sin(sin(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given M{(B)} = e^{B}, then obtain \\frac{\\int M{(B)} dB}{B} + \\frac{\\int e^{B} dB}{B} = \\frac{2 \\int e^{B} dB}{B}", "derivation": "M{(B)} = e^{B} and \\int M{(B)} dB = \\int e^{B} dB and \\frac{\\int M{(B)} dB}{B} = \\frac{\\int e^{B} dB}{B} and \\frac{\\int M{(B)} dB}{B} + \\frac{\\int e^{B} dB}{B} = \\frac{2 \\int e^{B} dB}{B}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["divide", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Mul(Integer(2), Pow(Symbol('B', commutative=True), Integer(-1)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(y^{\\prime},L)} = L y^{\\prime}, then obtain y^{\\prime} \\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)} - ((L y^{\\prime})^{L})^{L} = y^{\\prime} (L y^{\\prime})^{L} - ((L y^{\\prime})^{L})^{L}", "derivation": "\\operatorname{g_{\\varepsilon}}{(y^{\\prime},L)} = L y^{\\prime} and \\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)} = (L y^{\\prime})^{L} and (\\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)})^{L} = ((L y^{\\prime})^{L})^{L} and y^{\\prime} \\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)} = y^{\\prime} (L y^{\\prime})^{L} and y^{\\prime} \\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)} - (\\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)})^{L} = y^{\\prime} (L y^{\\prime})^{L} - (\\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)})^{L} and y^{\\prime} \\operatorname{g_{\\varepsilon}}^{L}{(y^{\\prime},L)} - ((L y^{\\prime})^{L})^{L} = y^{\\prime} (L y^{\\prime})^{L} - ((L y^{\\prime})^{L})^{L}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True)))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["times", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True))))"], [["minus", 4, "Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Add(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Add(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Add(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True))), Mul(Integer(-1), Pow(Pow(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)} = \\pi e^{f_{\\mathbf{p}}}, then obtain \\frac{\\partial^{2}}{\\partial f_{\\mathbf{p}}\\partial \\pi} (\\pi e^{f_{\\mathbf{p}}} + \\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)}) = \\frac{\\partial^{2}}{\\partial f_{\\mathbf{p}}\\partial \\pi} 2 \\pi e^{f_{\\mathbf{p}}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)} = \\pi e^{f_{\\mathbf{p}}} and \\pi e^{f_{\\mathbf{p}}} + \\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)} = 2 \\pi e^{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial \\pi} (\\pi e^{f_{\\mathbf{p}}} + \\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)}) = \\frac{\\partial}{\\partial \\pi} 2 \\pi e^{f_{\\mathbf{p}}} and \\frac{\\partial^{2}}{\\partial f_{\\mathbf{p}}\\partial \\pi} (\\pi e^{f_{\\mathbf{p}}} + \\operatorname{V_{\\mathbf{E}}}{(f_{\\mathbf{p}},\\pi)}) = \\frac{\\partial^{2}}{\\partial f_{\\mathbf{p}}\\partial \\pi} 2 \\pi e^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\pi', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(\\mu_0)} = \\mu_0, then obtain \\frac{(\\mu_0 + J{(\\mu_0)})^{\\mu_0}}{2 J{(\\mu_0)} \\int \\mu_0 dJ{(\\mu_0)}} = \\frac{(2 \\mu_0)^{\\mu_0}}{2 J{(\\mu_0)} \\int \\mu_0 dJ{(\\mu_0)}}", "derivation": "J{(\\mu_0)} = \\mu_0 and \\mu_0 + J{(\\mu_0)} = 2 \\mu_0 and \\int J{(\\mu_0)} d\\mu_0 = \\int \\mu_0 d\\mu_0 and 2 \\mu_0 \\int J{(\\mu_0)} d\\mu_0 = 2 \\mu_0 \\int \\mu_0 d\\mu_0 and 2 J{(\\mu_0)} \\int J{(\\mu_0)} dJ{(\\mu_0)} = 2 J{(\\mu_0)} \\int \\mu_0 dJ{(\\mu_0)} and (\\mu_0 + J{(\\mu_0)})^{\\mu_0} = (2 \\mu_0)^{\\mu_0} and \\frac{(\\mu_0 + J{(\\mu_0)})^{\\mu_0}}{2 J{(\\mu_0)} \\int J{(\\mu_0)} dJ{(\\mu_0)}} = \\frac{(2 \\mu_0)^{\\mu_0}}{2 J{(\\mu_0)} \\int J{(\\mu_0)} dJ{(\\mu_0)}} and \\frac{(\\mu_0 + J{(\\mu_0)})^{\\mu_0}}{2 J{(\\mu_0)} \\int \\mu_0 dJ{(\\mu_0)}} = \\frac{(2 \\mu_0)^{\\mu_0}}{2 J{(\\mu_0)} \\int \\mu_0 dJ{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('J')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Symbol('\\\\mu_0', commutative=True), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["times", 3, "Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Integral(Symbol('\\\\mu_0', commutative=True), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Function('J')(Symbol('\\\\mu_0', commutative=True)), Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True))))), Mul(Integer(2), Function('J')(Symbol('\\\\mu_0', commutative=True)), Integral(Symbol('\\\\mu_0', commutative=True), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True))))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('J')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 6, "Mul(Integer(2), Function('J')(Symbol('\\\\mu_0', commutative=True)), Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True)))))"], "Equality(Mul(Rational(1, 2), Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('J')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Function('J')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True)))), Integer(-1))), Mul(Rational(1, 2), Pow(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Function('J')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Integral(Function('J')(Symbol('\\\\mu_0', commutative=True)), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Rational(1, 2), Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('J')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Function('J')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Integral(Symbol('\\\\mu_0', commutative=True), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True)))), Integer(-1))), Mul(Rational(1, 2), Pow(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Function('J')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Integral(Symbol('\\\\mu_0', commutative=True), Tuple(Function('J')(Symbol('\\\\mu_0', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(V_{\\mathbf{B}})} = \\sin{(e^{V_{\\mathbf{B}}})} and I{(V_{\\mathbf{B}})} = e^{V_{\\mathbf{B}}}, then obtain \\frac{\\operatorname{f_{\\mathbf{p}}}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{\\sin{(I{(V_{\\mathbf{B}})})}}{V_{\\mathbf{B}}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(V_{\\mathbf{B}})} = \\sin{(e^{V_{\\mathbf{B}}})} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{\\sin{(e^{V_{\\mathbf{B}}})}}{V_{\\mathbf{B}}} and I{(V_{\\mathbf{B}})} = e^{V_{\\mathbf{B}}} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{\\sin{(I{(V_{\\mathbf{B}})})}}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["divide", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), sin(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), sin(Function('I')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(q)} = e^{q}, then obtain 2 \\operatorname{C_{1}}{(q)} + 2 e^{q} - \\sin{(\\mathbf{M})} = \\operatorname{C_{1}}{(q)} + 3 e^{q} - \\sin{(\\mathbf{M})}", "derivation": "\\operatorname{C_{1}}{(q)} = e^{q} and \\operatorname{C_{1}}{(q)} + e^{q} = 2 e^{q} and 2 \\operatorname{C_{1}}{(q)} + 2 e^{q} = \\operatorname{C_{1}}{(q)} + 3 e^{q} and 3 \\operatorname{C_{1}}{(q)} + e^{q} = \\operatorname{C_{1}}{(q)} + 3 e^{q} and 2 \\operatorname{C_{1}}{(q)} + 2 e^{q} = 3 \\operatorname{C_{1}}{(q)} + e^{q} and 3 \\operatorname{C_{1}}{(q)} + e^{q} - \\sin{(\\mathbf{M})} = \\operatorname{C_{1}}{(q)} + 3 e^{q} - \\sin{(\\mathbf{M})} and 2 \\operatorname{C_{1}}{(q)} + 2 e^{q} - \\sin{(\\mathbf{M})} = \\operatorname{C_{1}}{(q)} + 3 e^{q} - \\sin{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["add", 1, "exp(Symbol('q', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(Integer(2), exp(Symbol('q', commutative=True))))"], [["add", 2, "Add(Function('C_1')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('C_1')(Symbol('q', commutative=True))), Mul(Integer(2), exp(Symbol('q', commutative=True)))), Add(Function('C_1')(Symbol('q', commutative=True)), Mul(Integer(3), exp(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('C_1')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))), Add(Function('C_1')(Symbol('q', commutative=True)), Mul(Integer(3), exp(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(2), Function('C_1')(Symbol('q', commutative=True))), Mul(Integer(2), exp(Symbol('q', commutative=True)))), Add(Mul(Integer(3), Function('C_1')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True))))"], [["minus", 4, "sin(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(3), Function('C_1')(Symbol('q', commutative=True))), exp(Symbol('q', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Function('C_1')(Symbol('q', commutative=True)), Mul(Integer(3), exp(Symbol('q', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(2), Function('C_1')(Symbol('q', commutative=True))), Mul(Integer(2), exp(Symbol('q', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Function('C_1')(Symbol('q', commutative=True)), Mul(Integer(3), exp(Symbol('q', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(E,s)} = \\sin{(\\frac{s}{E})}, then obtain 3 \\operatorname{v_{z}}{(E,s)} + \\sin{(\\frac{s}{E})} - \\frac{4 s}{E} = 2 \\operatorname{v_{z}}{(E,s)} + 2 \\sin{(\\frac{s}{E})} - \\frac{4 s}{E}", "derivation": "\\operatorname{v_{z}}{(E,s)} = \\sin{(\\frac{s}{E})} and \\operatorname{v_{z}}{(E,s)} - \\frac{s}{E} = \\sin{(\\frac{s}{E})} - \\frac{s}{E} and 2 \\operatorname{v_{z}}{(E,s)} - \\frac{2 s}{E} = \\operatorname{v_{z}}{(E,s)} + \\sin{(\\frac{s}{E})} - \\frac{2 s}{E} and 3 \\operatorname{v_{z}}{(E,s)} + \\sin{(\\frac{s}{E})} - \\frac{4 s}{E} = 2 \\operatorname{v_{z}}{(E,s)} + 2 \\sin{(\\frac{s}{E})} - \\frac{4 s}{E}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Add(sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["add", 2, "Add(Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Add(Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["add", 3, "Add(Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True))), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(4), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Add(Mul(Integer(2), Function('v_z')(Symbol('E', commutative=True), Symbol('s', commutative=True))), Mul(Integer(2), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Mul(Integer(-1), Integer(4), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain (- \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}})^{\\mathbf{E}} e^{- \\mathbf{E}} = e^{- \\mathbf{E}}", "derivation": "\\mathbf{p}{(\\mathbf{E})} = e^{\\mathbf{E}} and 0 = - \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}} and 0^{\\mathbf{E}} = (- \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}})^{\\mathbf{E}} and 0^{\\mathbf{E}} e^{- \\mathbf{E}} = (- \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}})^{\\mathbf{E}} e^{- \\mathbf{E}} and (- \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}})^{\\mathbf{E}} = 1 and (- \\mathbf{p}{(\\mathbf{E})} + e^{\\mathbf{E}})^{\\mathbf{E}} e^{- \\mathbf{E}} = e^{- \\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["divide", 3, "exp(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Integer(1))"], [["times", 5, "exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(x,W)} = e^{W + x} and \\tilde{g}^*{(x,W)} = x e^{W + x}, then obtain - \\psi^* i + \\log{(\\tilde{g}^*{(x,W)})} + 1 = - \\psi^* i + \\log{(x \\mu_{0}{(x,W)})} + 1", "derivation": "\\mu_{0}{(x,W)} = e^{W + x} and x \\mu_{0}{(x,W)} = x e^{W + x} and \\log{(x \\mu_{0}{(x,W)})} = \\log{(x e^{W + x})} and \\tilde{g}^*{(x,W)} = x e^{W + x} and \\log{(x \\mu_{0}{(x,W)})} = \\log{(\\tilde{g}^*{(x,W)})} and - \\psi^* i + \\log{(x \\mu_{0}{(x,W)})} = - \\psi^* i + \\log{(x e^{W + x})} and - \\psi^* i + \\log{(\\tilde{g}^*{(x,W)})} = - \\psi^* i + \\log{(x e^{W + x})} and - \\psi^* i + \\log{(\\tilde{g}^*{(x,W)})} = - \\psi^* i + \\log{(x \\mu_{0}{(x,W)})} and - \\psi^* i + \\log{(\\tilde{g}^*{(x,W)})} + 1 = - \\psi^* i + \\log{(x \\mu_{0}{(x,W)})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True)), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('x', commutative=True), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True)))), log(Mul(Symbol('x', commutative=True), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('x', commutative=True), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True)))), log(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('W', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('x', commutative=True), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('x', commutative=True), exp(Add(Symbol('W', commutative=True), Symbol('x', commutative=True)))))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True))))))"], [["minus", 8, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('W', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('x', commutative=True), Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('W', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given S{(f_{\\mathbf{v}},A)} = f_{\\mathbf{v}}^{A}, then derive (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} S{(f_{\\mathbf{v}},A)})^{f_{\\mathbf{v}}} = (\\frac{A f_{\\mathbf{v}}^{A}}{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}}, then obtain (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} f_{\\mathbf{v}}^{A})^{f_{\\mathbf{v}}} = (\\frac{A f_{\\mathbf{v}}^{A}}{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}}", "derivation": "S{(f_{\\mathbf{v}},A)} = f_{\\mathbf{v}}^{A} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} S{(f_{\\mathbf{v}},A)} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} f_{\\mathbf{v}}^{A} and (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} S{(f_{\\mathbf{v}},A)})^{f_{\\mathbf{v}}} = (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} f_{\\mathbf{v}}^{A})^{f_{\\mathbf{v}}} and (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} S{(f_{\\mathbf{v}},A)})^{f_{\\mathbf{v}}} = (\\frac{A f_{\\mathbf{v}}^{A}}{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} and (\\frac{\\partial}{\\partial f_{\\mathbf{v}}} f_{\\mathbf{v}}^{A})^{f_{\\mathbf{v}}} = (\\frac{A f_{\\mathbf{v}}^{A}}{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Derivative(Function('S')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Derivative(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('S')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('A', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\theta_2)} = \\sin{(\\theta_2)} and \\sigma_{x}{(\\theta_2)} = \\sin^{\\theta_2}{(\\theta_2)}, then obtain \\sigma_{x}^{\\theta_2}{(\\theta_2)} = (\\sin^{\\theta_2}{(\\theta_2)})^{\\theta_2}", "derivation": "\\operatorname{m_{s}}{(\\theta_2)} = \\sin{(\\theta_2)} and \\operatorname{m_{s}}^{\\theta_2}{(\\theta_2)} = \\sin^{\\theta_2}{(\\theta_2)} and \\sigma_{x}{(\\theta_2)} = \\sin^{\\theta_2}{(\\theta_2)} and \\sigma_{x}{(\\theta_2)} = \\operatorname{m_{s}}^{\\theta_2}{(\\theta_2)} and (\\operatorname{m_{s}}^{\\theta_2}{(\\theta_2)})^{\\theta_2} = (\\sin^{\\theta_2}{(\\theta_2)})^{\\theta_2} and \\sigma_{x}^{\\theta_2}{(\\theta_2)} = (\\sin^{\\theta_2}{(\\theta_2)})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta_2', commutative=True)), Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta_2', commutative=True)), Pow(Function('m_s')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Pow(Function('m_s')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\psi^*,E)} = \\cos{((\\psi^*)^{E})}, then obtain \\frac{\\partial}{\\partial E} \\int (\\psi^*)^{- E} d\\psi^* = \\frac{\\partial}{\\partial E} \\int \\frac{(\\psi^*)^{- E} \\cos{((\\psi^*)^{E})}}{\\mathbf{J}_f{(\\psi^*,E)}} d\\psi^*", "derivation": "\\mathbf{J}_f{(\\psi^*,E)} = \\cos{((\\psi^*)^{E})} and 1 = \\frac{\\cos{((\\psi^*)^{E})}}{\\mathbf{J}_f{(\\psi^*,E)}} and (\\psi^*)^{- E} = \\frac{(\\psi^*)^{- E} \\cos{((\\psi^*)^{E})}}{\\mathbf{J}_f{(\\psi^*,E)}} and \\int (\\psi^*)^{- E} d\\psi^* = \\int \\frac{(\\psi^*)^{- E} \\cos{((\\psi^*)^{E})}}{\\mathbf{J}_f{(\\psi^*,E)}} d\\psi^* and \\frac{\\partial}{\\partial E} \\int (\\psi^*)^{- E} d\\psi^* = \\frac{\\partial}{\\partial E} \\int \\frac{(\\psi^*)^{- E} \\cos{((\\psi^*)^{E})}}{\\mathbf{J}_f{(\\psi^*,E)}} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)), cos(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)))))"], [["divide", 2, "Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True))"], "Equality(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('E', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(b)} = \\cos{(b)}, then obtain - b + \\int 2 W{(b)} \\cos{(b)} db + \\int W{(b)} db = - b + \\int W{(b)} db + \\int 2 \\cos^{2}{(b)} db", "derivation": "W{(b)} = \\cos{(b)} and W{(b)} + \\cos{(b)} = 2 \\cos{(b)} and (W{(b)} + \\cos{(b)}) W{(b)} = (W{(b)} + \\cos{(b)}) \\cos{(b)} and (W{(b)} + \\cos{(b)}) W{(b)} + W{(b)} = (W{(b)} + \\cos{(b)}) \\cos{(b)} + W{(b)} and 2 W{(b)} \\cos{(b)} + W{(b)} = W{(b)} + 2 \\cos^{2}{(b)} and \\int (2 W{(b)} \\cos{(b)} + W{(b)}) db = \\int (W{(b)} + 2 \\cos^{2}{(b)}) db and \\int 2 W{(b)} \\cos{(b)} db + \\int W{(b)} db = \\int W{(b)} db + \\int 2 \\cos^{2}{(b)} db and - b + \\int 2 W{(b)} \\cos{(b)} db + \\int W{(b)} db = - b + \\int W{(b)} db + \\int 2 \\cos^{2}{(b)} db", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["add", 1, "cos(Symbol('b', commutative=True))"], "Equality(Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Mul(Integer(2), cos(Symbol('b', commutative=True))))"], [["times", 1, "Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], "Equality(Mul(Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))), Mul(Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True))))"], [["add", 3, "Function('W')(Symbol('b', commutative=True))"], "Equality(Add(Mul(Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))), Add(Mul(Add(Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))), Add(Function('W')(Symbol('b', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('b', commutative=True)), Integer(2)))))"], [["integrate", 5, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Function('W')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Add(Function('W')(Symbol('b', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('b', commutative=True)), Integer(2)))), Tuple(Symbol('b', commutative=True))))"], [["expand", 6], "Equality(Add(Integral(Mul(Integer(2), Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Function('W')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Integral(Function('W')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Mul(Integer(2), Pow(cos(Symbol('b', commutative=True)), Integer(2))), Tuple(Symbol('b', commutative=True)))))"], [["minus", 7, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Integral(Mul(Integer(2), Function('W')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Function('W')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Integral(Function('W')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Mul(Integer(2), Pow(cos(Symbol('b', commutative=True)), Integer(2))), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(a^{\\dagger},\\Psi_{\\lambda})} = \\Psi_{\\lambda} + a^{\\dagger}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{v_{x}}{(a^{\\dagger},\\Psi_{\\lambda})} = 1, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} (\\Psi_{\\lambda} + a^{\\dagger}) = 1", "derivation": "\\operatorname{v_{x}}{(a^{\\dagger},\\Psi_{\\lambda})} = \\Psi_{\\lambda} + a^{\\dagger} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{v_{x}}{(a^{\\dagger},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial a^{\\dagger}} (\\Psi_{\\lambda} + a^{\\dagger}) and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{v_{x}}{(a^{\\dagger},\\Psi_{\\lambda})} = 1 and \\frac{\\partial}{\\partial a^{\\dagger}} (\\Psi_{\\lambda} + a^{\\dagger}) = 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(F_{N},q)} = - F_{N} + q, then obtain 2 \\int 2 \\operatorname{C_{d}}{(F_{N},q)} dF_{N} = \\int (- F_{N} + q + \\operatorname{C_{d}}{(F_{N},q)}) dF_{N} + \\int 2 \\operatorname{C_{d}}{(F_{N},q)} dF_{N}", "derivation": "\\operatorname{C_{d}}{(F_{N},q)} = - F_{N} + q and 2 \\operatorname{C_{d}}{(F_{N},q)} = - F_{N} + q + \\operatorname{C_{d}}{(F_{N},q)} and \\int 2 \\operatorname{C_{d}}{(F_{N},q)} dF_{N} = \\int (- F_{N} + q + \\operatorname{C_{d}}{(F_{N},q)}) dF_{N} and 2 \\int 2 \\operatorname{C_{d}}{(F_{N},q)} dF_{N} = \\int (- F_{N} + q + \\operatorname{C_{d}}{(F_{N},q)}) dF_{N} + \\int 2 \\operatorname{C_{d}}{(F_{N},q)} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('q', commutative=True)))"], [["add", 1, "Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('q', commutative=True), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('q', commutative=True), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["add", 3, "Integral(Mul(Integer(2), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Mul(Integer(2), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('q', commutative=True), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Integer(2), Function('C_d')(Symbol('F_N', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mu,p)} = \\log{(\\frac{\\mu}{p})} and \\Psi_{nl}{(\\mu,p)} = \\frac{\\mu}{p}, then obtain \\mathbf{J}_f^{\\mu}{(\\mu,p)} \\log{(\\frac{\\mu}{p})}^{\\mu} = \\mathbf{J}_f^{\\mu}{(\\mu,p)} \\log{(\\Psi_{nl}{(\\mu,p)})}^{\\mu}", "derivation": "\\mathbf{J}_f{(\\mu,p)} = \\log{(\\frac{\\mu}{p})} and \\Psi_{nl}{(\\mu,p)} = \\frac{\\mu}{p} and \\mathbf{J}_f^{\\mu}{(\\mu,p)} = \\log{(\\frac{\\mu}{p})}^{\\mu} and \\mathbf{J}_f^{\\mu}{(\\mu,p)} = \\log{(\\Psi_{nl}{(\\mu,p)})}^{\\mu} and \\log{(\\frac{\\mu}{p})}^{\\mu} = \\log{(\\Psi_{nl}{(\\mu,p)})}^{\\mu} and \\frac{p \\log{(\\frac{\\mu}{p})}^{\\mu}}{\\mu} = \\frac{p \\log{(\\Psi_{nl}{(\\mu,p)})}^{\\mu}}{\\mu} and \\mathbf{J}_f^{\\mu}{(\\mu,p)} \\log{(\\frac{\\mu}{p})}^{\\mu} = \\mathbf{J}_f^{\\mu}{(\\mu,p)} \\log{(\\Psi_{nl}{(\\mu,p)})}^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), log(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(log(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('\\\\mu', commutative=True)), Pow(log(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["divide", 5, "Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True), Pow(log(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True), Pow(log(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["times", 6, "Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given k{(\\hat{H}_l,\\eta^{\\prime})} = \\sin{(\\eta^{\\prime} + \\hat{H}_l)}, then obtain \\int \\frac{\\partial}{\\partial \\hat{H}_l} k^{2}{(\\hat{H}_l,\\eta^{\\prime})} d\\hat{H}_l = \\int \\frac{\\partial}{\\partial \\hat{H}_l} k{(\\hat{H}_l,\\eta^{\\prime})} \\sin{(\\eta^{\\prime} + \\hat{H}_l)} d\\hat{H}_l", "derivation": "k{(\\hat{H}_l,\\eta^{\\prime})} = \\sin{(\\eta^{\\prime} + \\hat{H}_l)} and k^{2}{(\\hat{H}_l,\\eta^{\\prime})} = k{(\\hat{H}_l,\\eta^{\\prime})} \\sin{(\\eta^{\\prime} + \\hat{H}_l)} and \\frac{\\partial}{\\partial \\hat{H}_l} k^{2}{(\\hat{H}_l,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\hat{H}_l} k{(\\hat{H}_l,\\eta^{\\prime})} \\sin{(\\eta^{\\prime} + \\hat{H}_l)} and \\int \\frac{\\partial}{\\partial \\hat{H}_l} k^{2}{(\\hat{H}_l,\\eta^{\\prime})} d\\hat{H}_l = \\int \\frac{\\partial}{\\partial \\hat{H}_l} k{(\\hat{H}_l,\\eta^{\\prime})} \\sin{(\\eta^{\\prime} + \\hat{H}_l)} d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 1, "Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Pow(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Derivative(Mul(Function('k')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given G{(y^{\\prime})} = \\log{(y^{\\prime})}, then derive \\frac{d}{d y^{\\prime}} G{(y^{\\prime})} = \\frac{1}{y^{\\prime}}, then obtain \\frac{1}{y^{\\prime}} = \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})}", "derivation": "G{(y^{\\prime})} = \\log{(y^{\\prime})} and \\frac{d}{d y^{\\prime}} G{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})} and \\frac{d}{d y^{\\prime}} G{(y^{\\prime})} = \\frac{1}{y^{\\prime}} and \\frac{1}{y^{\\prime}} = \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Derivative(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(n)} = \\sin{(n)}, then derive \\frac{d^{2}}{d n^{2}} \\phi{(n)} = - \\sin{(n)}, then obtain 0 = \\phi{(n)} - \\sin{(n)}", "derivation": "\\phi{(n)} = \\sin{(n)} and \\frac{d}{d n} \\phi{(n)} = \\frac{d}{d n} \\sin{(n)} and \\frac{d^{2}}{d n^{2}} \\phi{(n)} = \\frac{d^{2}}{d n^{2}} \\sin{(n)} and \\frac{d^{2}}{d n^{2}} \\phi{(n)} = - \\sin{(n)} and \\frac{d^{2}}{d n^{2}} \\phi{(n)} = - \\phi{(n)} and - \\phi{(n)} = - \\sin{(n)} and 0 = \\phi{(n)} - \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\phi')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))), Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True))))"], [["minus", 6, "Mul(Integer(-1), Function('\\\\phi')(Symbol('n', commutative=True)))"], "Equality(Integer(0), Add(Function('\\\\phi')(Symbol('n', commutative=True)), Mul(Integer(-1), sin(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given k{(\\mathbf{F})} = \\sin{(e^{\\mathbf{F}})}, then obtain 3 k^{2}{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})} = (k{(\\mathbf{F})} + \\sin{(e^{\\mathbf{F}})}) k{(\\mathbf{F})} + k^{2}{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})}", "derivation": "k{(\\mathbf{F})} = \\sin{(e^{\\mathbf{F}})} and 2 k{(\\mathbf{F})} = k{(\\mathbf{F})} + \\sin{(e^{\\mathbf{F}})} and 2 k^{2}{(\\mathbf{F})} = (k{(\\mathbf{F})} + \\sin{(e^{\\mathbf{F}})}) k{(\\mathbf{F})} and 2 k^{2}{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})} = (k{(\\mathbf{F})} + \\sin{(e^{\\mathbf{F}})}) k{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})} and 3 k^{2}{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})} = (k{(\\mathbf{F})} + \\sin{(e^{\\mathbf{F}})}) k{(\\mathbf{F})} + k^{2}{(\\mathbf{F})} - k{(\\mathbf{F})} - \\sin{(e^{\\mathbf{F}})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "Function('k')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["times", 2, "Function('k')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 3, "Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Pow(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))), Add(Mul(Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))))"], [["add", 4, "Pow(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(3), Pow(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))), Add(Mul(Add(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), sin(exp(Symbol('\\\\mathbf{F}', commutative=True)))), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('k')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), Mul(Integer(-1), Function('k')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{F}', commutative=True))))))"]]}, {"prompt": "Given s{(M_{E})} = \\sin{(e^{M_{E}})} and \\operatorname{A_{2}}{(M_{E})} = - M_{E}, then obtain (s{(M_{E})} - \\sin{(e^{- \\operatorname{A_{2}}{(M_{E})}})})^{\\operatorname{A_{2}}{(M_{E})}} = 1", "derivation": "s{(M_{E})} = \\sin{(e^{M_{E}})} and s{(M_{E})} - \\sin{(e^{M_{E}})} = 0 and (s{(M_{E})} - \\sin{(e^{M_{E}})})^{M_{E}} = 0^{M_{E}} and (s{(M_{E})} - \\sin{(e^{M_{E}})})^{M_{E}} e^{- M_{E}} = 0^{M_{E}} e^{- M_{E}} and 1 = (s{(M_{E})} - \\sin{(e^{M_{E}})})^{M_{E}} and \\operatorname{A_{2}}{(M_{E})} = - M_{E} and (s{(M_{E})} - \\sin{(e^{M_{E}})})^{- M_{E}} = 1 and (s{(M_{E})} - \\sin{(e^{- \\operatorname{A_{2}}{(M_{E})}})})^{\\operatorname{A_{2}}{(M_{E})}} = 1", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('M_E', commutative=True)), sin(exp(Symbol('M_E', commutative=True))))"], [["minus", 1, "sin(exp(Symbol('M_E', commutative=True)))"], "Equality(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Symbol('M_E', commutative=True)), Pow(Integer(0), Symbol('M_E', commutative=True)))"], [["divide", 3, "exp(Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Mul(Pow(Integer(0), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)))"], [["divide", 5, "Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Symbol('M_E', commutative=True))"], "Equality(Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('M_E', commutative=True))))), Mul(Integer(-1), Symbol('M_E', commutative=True))), Integer(1))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Add(Function('s')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(exp(Mul(Integer(-1), Function('A_2')(Symbol('M_E', commutative=True))))))), Function('A_2')(Symbol('M_E', commutative=True))), Integer(1))"]]}, {"prompt": "Given r{(\\hat{x}_0,\\mathbf{s})} = \\hat{x}_0 \\mathbf{s} - \\mathbf{s}^{2}, then obtain \\int (\\hat{x}_0 \\mathbf{s} - \\mathbf{s}^{2}) d\\hat{x}_0 = \\int - \\mathbf{s}^{2} d\\hat{x}_0 + \\int \\hat{x}_0 \\mathbf{s} d\\hat{x}_0", "derivation": "r{(\\hat{x}_0,\\mathbf{s})} = \\hat{x}_0 \\mathbf{s} - \\mathbf{s}^{2} and \\int r{(\\hat{x}_0,\\mathbf{s})} d\\hat{x}_0 = \\int (\\hat{x}_0 \\mathbf{s} - \\mathbf{s}^{2}) d\\hat{x}_0 and \\int r{(\\hat{x}_0,\\mathbf{s})} d\\hat{x}_0 = \\int - \\mathbf{s}^{2} d\\hat{x}_0 + \\int \\hat{x}_0 \\mathbf{s} d\\hat{x}_0 and \\int (\\hat{x}_0 \\mathbf{s} - \\mathbf{s}^{2}) d\\hat{x}_0 = \\int - \\mathbf{s}^{2} d\\hat{x}_0 + \\int \\hat{x}_0 \\mathbf{s} d\\hat{x}_0", "srepr_derivation": [["renaming_premise", "Equality(Function('r')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)))))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('r')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('r')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Integral(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Integral(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{g},z,H)} = H + \\mathbf{g} - z, then obtain 27 (H + \\mathbf{g} - z)^{3} = (3 H + 3 \\mathbf{g} - 3 z)^{3}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{g},z,H)} = H + \\mathbf{g} - z and 2 \\operatorname{F_{c}}{(\\mathbf{g},z,H)} = H + \\mathbf{g} - z + \\operatorname{F_{c}}{(\\mathbf{g},z,H)} and 3 \\operatorname{F_{c}}{(\\mathbf{g},z,H)} = H + \\mathbf{g} - z + 2 \\operatorname{F_{c}}{(\\mathbf{g},z,H)} and 3 \\operatorname{F_{c}}{(\\mathbf{g},z,H)} = 2 H + 2 \\mathbf{g} - 2 z + \\operatorname{F_{c}}{(\\mathbf{g},z,H)} and 27 \\operatorname{F_{c}}^{3}{(\\mathbf{g},z,H)} = (2 H + 2 \\mathbf{g} - 2 z + \\operatorname{F_{c}}{(\\mathbf{g},z,H)})^{3} and 27 (H + \\mathbf{g} - z)^{3} = (3 H + 3 \\mathbf{g} - 3 z)^{3}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["add", 1, "Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))))"], [["add", 2, "Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Integer(3), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(2), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))))"], [["power", 4, 3], "Equality(Mul(Integer(27), Pow(Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True)), Integer(3))), Pow(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)), Function('F_c')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True), Symbol('H', commutative=True))), Integer(3)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(27), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(3))), Pow(Add(Mul(Integer(3), Symbol('H', commutative=True)), Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('z', commutative=True))), Integer(3)))"]]}, {"prompt": "Given I{(\\ddot{x})} = \\cos{(\\ddot{x})} and \\mathbf{g}{(x,A)} = \\frac{x}{A}, then obtain (I{(\\ddot{x})} - \\cos{(\\ddot{x})} + 1) \\mathbf{g}^{- A}{(x,A)} = \\mathbf{g}^{- A}{(x,A)}", "derivation": "I{(\\ddot{x})} = \\cos{(\\ddot{x})} and I{(\\ddot{x})} - \\cos{(\\ddot{x})} = 0 and \\mathbf{g}{(x,A)} = \\frac{x}{A} and I{(\\ddot{x})} - \\cos{(\\ddot{x})} + 1 = 1 and (\\frac{x}{A})^{- A} (I{(\\ddot{x})} - \\cos{(\\ddot{x})} + 1) = (\\frac{x}{A})^{- A} and (I{(\\ddot{x})} - \\cos{(\\ddot{x})} + 1) \\mathbf{g}^{- A}{(x,A)} = \\mathbf{g}^{- A}{(x,A)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('I')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('I')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Integer(1))"], [["divide", 4, "Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Symbol('A', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))), Add(Function('I')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True))), Integer(1))), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Function('I')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True))), Integer(1)), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)))), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(h)} = e^{h}, then obtain \\frac{e^{h}}{\\mathbf{E}{(h)}} + \\frac{1}{\\mathbf{E}^{2}{(h)}} = \\frac{1}{\\mathbf{E}^{2}{(h)}} + \\frac{e^{4 h}}{\\mathbf{E}^{4}{(h)}}", "derivation": "\\mathbf{E}{(h)} = e^{h} and 1 = \\frac{e^{h}}{\\mathbf{E}{(h)}} and \\frac{e^{h}}{\\mathbf{E}{(h)}} = \\frac{e^{2 h}}{\\mathbf{E}^{2}{(h)}} and \\frac{e^{h}}{\\mathbf{E}{(h)}} + \\frac{1}{\\mathbf{E}^{2}{(h)}} = \\frac{e^{2 h}}{\\mathbf{E}^{2}{(h)}} + \\frac{1}{\\mathbf{E}^{2}{(h)}} and \\frac{e^{2 h}}{\\mathbf{E}^{2}{(h)}} + \\frac{1}{\\mathbf{E}^{2}{(h)}} = \\frac{1}{\\mathbf{E}^{2}{(h)}} + \\frac{e^{4 h}}{\\mathbf{E}^{4}{(h)}} and \\frac{e^{h}}{\\mathbf{E}{(h)}} + \\frac{1}{\\mathbf{E}^{2}{(h)}} = \\frac{1}{\\mathbf{E}^{2}{(h)}} + \\frac{e^{4 h}}{\\mathbf{E}^{4}{(h)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{E}')(Symbol('h', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))))"], [["times", 2, "Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('h', commutative=True)))))"], [["add", 3, "Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2))"], "Equality(Add(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2))), Add(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2))), Add(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2)), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-4)), exp(Mul(Integer(4), Symbol('h', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-1)), exp(Symbol('h', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2))), Add(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-2)), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True)), Integer(-4)), exp(Mul(Integer(4), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(n)} = e^{\\cos{(n)}}, then derive \\frac{d}{d n} \\operatorname{v_{z}}{(n)} = - e^{\\cos{(n)}} \\sin{(n)}, then obtain - \\operatorname{v_{z}}{(n)} \\sin{(n)} - \\sin{(n)} = - e^{\\cos{(n)}} \\sin{(n)} - \\sin{(n)}", "derivation": "\\operatorname{v_{z}}{(n)} = e^{\\cos{(n)}} and \\frac{d}{d n} \\operatorname{v_{z}}{(n)} = \\frac{d}{d n} e^{\\cos{(n)}} and \\frac{d}{d n} \\operatorname{v_{z}}{(n)} = - e^{\\cos{(n)}} \\sin{(n)} and \\frac{d}{d n} \\operatorname{v_{z}}{(n)} = - \\operatorname{v_{z}}{(n)} \\sin{(n)} and - \\operatorname{v_{z}}{(n)} \\sin{(n)} = \\frac{d}{d n} e^{\\cos{(n)}} and - \\operatorname{v_{z}}{(n)} \\sin{(n)} - \\sin{(n)} = - \\sin{(n)} + \\frac{d}{d n} e^{\\cos{(n)}} and - e^{\\cos{(n)}} \\sin{(n)} = \\frac{d}{d n} e^{\\cos{(n)}} and - \\operatorname{v_{z}}{(n)} \\sin{(n)} - \\sin{(n)} = - e^{\\cos{(n)}} \\sin{(n)} - \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('n', commutative=True)), exp(cos(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('v_z')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Function('v_z')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Function('v_z')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Derivative(exp(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 5, "sin(Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_z')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('n', commutative=True))), Derivative(exp(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Derivative(exp(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Mul(Integer(-1), Function('v_z')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), exp(cos(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(n,\\hbar)} = \\hbar + n and C{(\\hbar)} = \\hbar, then obtain \\int \\hat{\\mathbf{x}}^{\\hbar}{(n,\\hbar)} dC{(\\hbar)} = \\int (\\hbar + n)^{\\hbar} dC{(\\hbar)}", "derivation": "\\hat{\\mathbf{x}}{(n,\\hbar)} = \\hbar + n and \\hat{\\mathbf{x}}^{\\hbar}{(n,\\hbar)} = (\\hbar + n)^{\\hbar} and C{(\\hbar)} = \\hbar and \\int \\hat{\\mathbf{x}}^{\\hbar}{(n,\\hbar)} d\\hbar = \\int (\\hbar + n)^{\\hbar} d\\hbar and \\int \\hat{\\mathbf{x}}^{\\hbar}{(n,\\hbar)} dC{(\\hbar)} = \\int (\\hbar + n)^{\\hbar} dC{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Function('C')(Symbol('\\\\hbar', commutative=True)))), Integral(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Function('C')(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(F_{c},\\Psi_{nl})} = \\cos{(F_{c} + \\Psi_{nl})}, then obtain \\int 2 \\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} d\\Psi_{nl} = \\int (\\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} + \\cos^{F_{c}}{(F_{c} + \\Psi_{nl})}) d\\Psi_{nl}", "derivation": "\\hat{p}_0{(F_{c},\\Psi_{nl})} = \\cos{(F_{c} + \\Psi_{nl})} and \\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} = \\cos^{F_{c}}{(F_{c} + \\Psi_{nl})} and 2 \\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} = \\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} + \\cos^{F_{c}}{(F_{c} + \\Psi_{nl})} and \\int 2 \\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} d\\Psi_{nl} = \\int (\\hat{p}_0^{F_{c}}{(F_{c},\\Psi_{nl})} + \\cos^{F_{c}}{(F_{c} + \\Psi_{nl})}) d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True)), Pow(cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('F_c', commutative=True)))"], [["add", 2, "Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True))), Add(Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True)), Pow(cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('F_c', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(Pow(Function('\\\\hat{p}_0')(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('F_c', commutative=True)), Pow(cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\dot{\\mathbf{r}},A_{y})} = A_{y} + \\log{(\\dot{\\mathbf{r}})}, then derive \\int (- A_{y} + \\rho{(\\dot{\\mathbf{r}},A_{y})}) d\\dot{\\mathbf{r}} = \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})} - \\dot{\\mathbf{r}} + \\hat{X}, then obtain \\int \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})} - \\dot{\\mathbf{r}} + \\hat{X}", "derivation": "\\rho{(\\dot{\\mathbf{r}},A_{y})} = A_{y} + \\log{(\\dot{\\mathbf{r}})} and - A_{y} + \\rho{(\\dot{\\mathbf{r}},A_{y})} = \\log{(\\dot{\\mathbf{r}})} and \\int (- A_{y} + \\rho{(\\dot{\\mathbf{r}},A_{y})}) d\\dot{\\mathbf{r}} = \\int \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\int (- A_{y} + \\rho{(\\dot{\\mathbf{r}},A_{y})}) d\\dot{\\mathbf{r}} = \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})} - \\dot{\\mathbf{r}} + \\hat{X} and \\int \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\dot{\\mathbf{r}} \\log{(\\dot{\\mathbf{r}})} - \\dot{\\mathbf{r}} + \\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["minus", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_y', commutative=True))), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\rho')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given Z{(\\mu_0,v_{t})} = \\sin{(\\mu_0 v_{t})}, then derive \\frac{\\partial}{\\partial \\mu_0} Z{(\\mu_0,v_{t})} = v_{t} \\cos{(\\mu_0 v_{t})}, then obtain - v_{t} \\cos{(\\mu_0 v_{t})} + Z{(\\mu_0,v_{t})} = - v_{t} \\cos{(\\mu_0 v_{t})} + \\sin{(\\mu_0 v_{t})}", "derivation": "Z{(\\mu_0,v_{t})} = \\sin{(\\mu_0 v_{t})} and \\frac{\\partial}{\\partial \\mu_0} Z{(\\mu_0,v_{t})} = \\frac{\\partial}{\\partial \\mu_0} \\sin{(\\mu_0 v_{t})} and Z{(\\mu_0,v_{t})} - \\frac{\\partial}{\\partial \\mu_0} \\sin{(\\mu_0 v_{t})} = \\sin{(\\mu_0 v_{t})} - \\frac{\\partial}{\\partial \\mu_0} \\sin{(\\mu_0 v_{t})} and \\frac{\\partial}{\\partial \\mu_0} Z{(\\mu_0,v_{t})} = v_{t} \\cos{(\\mu_0 v_{t})} and Z{(\\mu_0,v_{t})} - \\frac{\\partial}{\\partial \\mu_0} Z{(\\mu_0,v_{t})} = \\sin{(\\mu_0 v_{t})} - \\frac{\\partial}{\\partial \\mu_0} Z{(\\mu_0,v_{t})} and - v_{t} \\cos{(\\mu_0 v_{t})} + Z{(\\mu_0,v_{t})} = - v_{t} \\cos{(\\mu_0 v_{t})} + \\sin{(\\mu_0 v_{t})}", "srepr_derivation": [["get_premise", "Equality(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Add(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Add(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), Derivative(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(Symbol('v_t', commutative=True), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Derivative(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))), Add(sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), Derivative(Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)))), Function('Z')(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)))), sin(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given z{(V_{\\mathbf{B}})} = \\int \\cos{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}, then derive z{(V_{\\mathbf{B}})} = \\sigma_p + \\sin{(V_{\\mathbf{B}})}, then derive \\sigma_p + \\sin{(V_{\\mathbf{B}})} = \\dot{x} + \\sin{(V_{\\mathbf{B}})}, then obtain \\frac{\\partial}{\\partial \\dot{x}} (\\sigma_p + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})}", "derivation": "z{(V_{\\mathbf{B}})} = \\int \\cos{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and z{(V_{\\mathbf{B}})} = \\sigma_p + \\sin{(V_{\\mathbf{B}})} and \\sigma_p + \\sin{(V_{\\mathbf{B}})} = \\int \\cos{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and \\sigma_p + \\sin{(V_{\\mathbf{B}})} = \\dot{x} + \\sin{(V_{\\mathbf{B}})} and (\\sigma_p + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})} = (\\dot{x} + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})} and \\frac{\\partial}{\\partial \\dot{x}} (\\sigma_p + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} + \\sin{(V_{\\mathbf{B}})}) z^{2}{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 4, "Pow(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(Function('z')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(t_{1},\\delta)} = \\int \\delta^{t_{1}} d\\delta, then obtain (\\int \\frac{\\operatorname{F_{H}}{(t_{1},\\delta)}}{\\delta} d\\delta) \\int \\operatorname{F_{H}}{(t_{1},\\delta)} d\\delta = (\\int \\frac{\\int \\delta^{t_{1}} d\\delta}{\\delta} d\\delta) \\int \\operatorname{F_{H}}{(t_{1},\\delta)} d\\delta", "derivation": "\\operatorname{F_{H}}{(t_{1},\\delta)} = \\int \\delta^{t_{1}} d\\delta and \\frac{\\operatorname{F_{H}}{(t_{1},\\delta)}}{\\delta} = \\frac{\\int \\delta^{t_{1}} d\\delta}{\\delta} and \\int \\operatorname{F_{H}}{(t_{1},\\delta)} d\\delta = \\iint \\delta^{t_{1}} d\\delta d\\delta and \\int \\frac{\\operatorname{F_{H}}{(t_{1},\\delta)}}{\\delta} d\\delta = \\int \\frac{\\int \\delta^{t_{1}} d\\delta}{\\delta} d\\delta and (\\int \\frac{\\operatorname{F_{H}}{(t_{1},\\delta)}}{\\delta} d\\delta) \\iint \\delta^{t_{1}} d\\delta d\\delta = (\\int \\frac{\\int \\delta^{t_{1}} d\\delta}{\\delta} d\\delta) \\iint \\delta^{t_{1}} d\\delta d\\delta and (\\int \\frac{\\operatorname{F_{H}}{(t_{1},\\delta)}}{\\delta} d\\delta) \\int \\operatorname{F_{H}}{(t_{1},\\delta)} d\\delta = (\\int \\frac{\\int \\delta^{t_{1}} d\\delta}{\\delta} d\\delta) \\int \\operatorname{F_{H}}{(t_{1},\\delta)} d\\delta", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True)), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 4, "Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Integral(Pow(Symbol('\\\\delta', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('F_H')(Symbol('t_1', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given m{(b)} = \\cos{(\\sin{(b)})}, then obtain - m^{4}{(b)} = - m{(b)} \\cos^{3}{(\\sin{(b)})}", "derivation": "m{(b)} = \\cos{(\\sin{(b)})} and m^{2}{(b)} = m{(b)} \\cos{(\\sin{(b)})} and - m^{3}{(b)} \\cos{(\\sin{(b)})} = - m^{2}{(b)} \\cos^{2}{(\\sin{(b)})} and - m^{3}{(b)} \\cos{(\\sin{(b)})} = - m{(b)} \\cos^{3}{(\\sin{(b)})} and - m^{4}{(b)} = - m^{2}{(b)} \\cos^{2}{(\\sin{(b)})} and - m^{2}{(b)} \\cos^{2}{(\\sin{(b)})} = - m{(b)} \\cos^{3}{(\\sin{(b)})} and - m^{4}{(b)} = - m{(b)} \\cos^{3}{(\\sin{(b)})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True))))"], [["times", 1, "Function('m')(Symbol('b', commutative=True))"], "Equality(Pow(Function('m')(Symbol('b', commutative=True)), Integer(2)), Mul(Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('m')(Symbol('b', commutative=True)), cos(sin(Symbol('b', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(3)), cos(sin(Symbol('b', commutative=True)))), Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(3)), cos(sin(Symbol('b', commutative=True)))), Mul(Integer(-1), Function('m')(Symbol('b', commutative=True)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(4))), Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(2))), Mul(Integer(-1), Function('m')(Symbol('b', commutative=True)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(-1), Pow(Function('m')(Symbol('b', commutative=True)), Integer(4))), Mul(Integer(-1), Function('m')(Symbol('b', commutative=True)), Pow(cos(sin(Symbol('b', commutative=True))), Integer(3))))"]]}, {"prompt": "Given \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})}, then obtain 1 = \\frac{\\log{(g^{\\prime}_{\\varepsilon})}}{2 \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} - \\log{(g^{\\prime}_{\\varepsilon})}}", "derivation": "\\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} - \\log{(g^{\\prime}_{\\varepsilon})} = 0 and 2 \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} - \\log{(g^{\\prime}_{\\varepsilon})} = \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} and 1 = \\frac{\\log{(g^{\\prime}_{\\varepsilon})}}{\\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})}} and 1 = \\frac{\\log{(g^{\\prime}_{\\varepsilon})}}{2 \\mathbf{J}_P{(g^{\\prime}_{\\varepsilon})} - \\log{(g^{\\prime}_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["add", 2, "Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(2), Function('\\\\mathbf{J}_P')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Integer(-1)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(n_{1},\\sigma_p)} = \\log{(n_{1})}^{\\sigma_p}, then obtain \\operatorname{A_{2}}{(n_{1},\\sigma_p)} \\int \\operatorname{A_{2}}{(n_{1},\\sigma_p)} d\\sigma_p = \\log{(n_{1})}^{\\sigma_p} \\int \\operatorname{A_{2}}{(n_{1},\\sigma_p)} d\\sigma_p", "derivation": "\\operatorname{A_{2}}{(n_{1},\\sigma_p)} = \\log{(n_{1})}^{\\sigma_p} and \\int \\operatorname{A_{2}}{(n_{1},\\sigma_p)} d\\sigma_p = \\int \\log{(n_{1})}^{\\sigma_p} d\\sigma_p and \\operatorname{A_{2}}{(n_{1},\\sigma_p)} \\int \\log{(n_{1})}^{\\sigma_p} d\\sigma_p = \\log{(n_{1})}^{\\sigma_p} \\int \\log{(n_{1})}^{\\sigma_p} d\\sigma_p and \\operatorname{A_{2}}{(n_{1},\\sigma_p)} \\int \\operatorname{A_{2}}{(n_{1},\\sigma_p)} d\\sigma_p = \\log{(n_{1})}^{\\sigma_p} \\int \\operatorname{A_{2}}{(n_{1},\\sigma_p)} d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 1, "Integral(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integral(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(log(Symbol('n_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integral(Function('A_2')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(v_{1})} = \\cos{(v_{1})}, then obtain \\frac{(\\frac{d}{d v_{1}} \\Psi^{\\dagger}{(v_{1})})^{v_{1}}}{\\Psi^{\\dagger}{(v_{1})}} = \\frac{(\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}}}{\\Psi^{\\dagger}{(v_{1})}}", "derivation": "\\Psi^{\\dagger}{(v_{1})} = \\cos{(v_{1})} and \\frac{d}{d v_{1}} \\Psi^{\\dagger}{(v_{1})} = \\frac{d}{d v_{1}} \\cos{(v_{1})} and (\\frac{d}{d v_{1}} \\Psi^{\\dagger}{(v_{1})})^{v_{1}} = (\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}} and \\frac{(\\frac{d}{d v_{1}} \\Psi^{\\dagger}{(v_{1})})^{v_{1}}}{\\Psi^{\\dagger}{(v_{1})}} = \\frac{(\\frac{d}{d v_{1}} \\cos{(v_{1})})^{v_{1}}}{\\Psi^{\\dagger}{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"], [["divide", 3, "Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(Derivative(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{P},v_{t})} = \\mathbf{P} + v_{t}, then derive \\frac{\\partial}{\\partial \\mathbf{P}} \\sigma_{p}{(\\mathbf{P},v_{t})} = 1, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} + v_{t}) - 1 = 0", "derivation": "\\sigma_{p}{(\\mathbf{P},v_{t})} = \\mathbf{P} + v_{t} and \\frac{\\partial}{\\partial \\mathbf{P}} \\sigma_{p}{(\\mathbf{P},v_{t})} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} + v_{t}) and \\frac{\\partial}{\\partial \\mathbf{P}} \\sigma_{p}{(\\mathbf{P},v_{t})} = 1 and \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} + v_{t}) = 1 and - \\mathbf{P} - v_{t} + \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} + v_{t}) = - \\mathbf{P} - v_{t} + 1 and \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} + v_{t}) - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Integer(1)))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Integer(1))"], "Equality(Add(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} = \\frac{P_{e}}{P_{g}}, then obtain - \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\int (\\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\frac{1}{P_{g}}) dP_{g} = - \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\int (\\frac{P_{e}}{P_{g}} + \\frac{1}{P_{g}}) dP_{g}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} = \\frac{P_{e}}{P_{g}} and \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\frac{1}{P_{g}} = \\frac{P_{e}}{P_{g}} + \\frac{1}{P_{g}} and \\int (\\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\frac{1}{P_{g}}) dP_{g} = \\int (\\frac{P_{e}}{P_{g}} + \\frac{1}{P_{g}}) dP_{g} and - \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\int (\\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\frac{1}{P_{g}}) dP_{g} = - \\operatorname{V_{\\mathbf{B}}}{(P_{g},P_{e})} + \\int (\\frac{P_{e}}{P_{g}} + \\frac{1}{P_{g}}) dP_{g}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('P_g', commutative=True), Integer(-1))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_g', commutative=True), Integer(-1))), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Pow(Symbol('P_g', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_g', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Pow(Symbol('P_g', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 3, "Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True))), Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_g', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('P_g', commutative=True), Symbol('P_e', commutative=True))), Integral(Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Pow(Symbol('P_g', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(l)} = \\log{(l)}, then obtain \\frac{d}{d l} \\operatorname{F_{H}}{(l)} - \\frac{1}{l} = 0", "derivation": "\\operatorname{F_{H}}{(l)} = \\log{(l)} and \\frac{d}{d l} \\operatorname{F_{H}}{(l)} = \\frac{d}{d l} \\log{(l)} and \\frac{d}{d l} \\operatorname{F_{H}}{(l)} - \\frac{d}{d l} \\log{(l)} = 0 and \\frac{d}{d l} \\operatorname{F_{H}}{(l)} - \\frac{1}{l} = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('F_H')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('F_H')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\pi{(\\phi_2,\\hbar)} = \\phi_2 e^{\\hbar} and \\varphi{(\\phi_2)} = \\phi_2, then derive \\frac{\\partial}{\\partial \\hbar} \\pi{(\\phi_2,\\hbar)} = \\phi_2 e^{\\hbar}, then derive \\phi_2 e^{\\hbar} = \\frac{\\partial}{\\partial \\hbar} \\phi_2 e^{\\hbar}, then obtain \\varphi{(\\phi_2)} e^{\\hbar} = \\frac{\\partial}{\\partial \\hbar} \\varphi{(\\phi_2)} e^{\\hbar}", "derivation": "\\pi{(\\phi_2,\\hbar)} = \\phi_2 e^{\\hbar} and \\varphi{(\\phi_2)} = \\phi_2 and \\frac{\\partial}{\\partial \\hbar} \\pi{(\\phi_2,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\phi_2 e^{\\hbar} and \\frac{\\partial}{\\partial \\hbar} \\pi{(\\phi_2,\\hbar)} = \\phi_2 e^{\\hbar} and \\frac{\\partial}{\\partial \\hbar} \\pi{(\\phi_2,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\hbar^{2}} \\pi{(\\phi_2,\\hbar)} and \\frac{\\partial}{\\partial \\hbar} \\phi_2 e^{\\hbar} = \\frac{\\partial^{2}}{\\partial \\hbar^{2}} \\phi_2 e^{\\hbar} and \\phi_2 e^{\\hbar} = \\frac{\\partial}{\\partial \\hbar} \\phi_2 e^{\\hbar} and \\varphi{(\\phi_2)} e^{\\hbar} = \\frac{\\partial}{\\partial \\hbar} \\varphi{(\\phi_2)} e^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Function('\\\\pi')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(2))))"], [["evaluate_derivatives", 6], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Derivative(Mul(Function('\\\\varphi')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(a)} = \\log{(a)}, then obtain (\\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} 0)^{a} = (\\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} (- \\mu{(a)} + \\log{(a)}))^{a}", "derivation": "\\mu{(a)} = \\log{(a)} and 0 = - \\mu{(a)} + \\log{(a)} and \\frac{d}{d a} 0 = \\frac{d}{d a} (- \\mu{(a)} + \\log{(a)}) and \\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} 0 = \\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} (- \\mu{(a)} + \\log{(a)}) and (\\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} 0)^{a} = (\\sin{(\\mu{(a)} - \\log{(a)})} + \\frac{d}{d a} (- \\mu{(a)} + \\log{(a)}))^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["minus", 1, "Function('\\\\mu')(Symbol('a', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Integer(-1), sin(Add(Function('\\\\mu')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))))"], "Equality(Add(sin(Add(Function('\\\\mu')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(sin(Add(Function('\\\\mu')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Add(sin(Add(Function('\\\\mu')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))), Derivative(Integer(0), Tuple(Symbol('a', commutative=True), Integer(1)))), Symbol('a', commutative=True)), Pow(Add(sin(Add(Function('\\\\mu')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('a', commutative=True))), log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given g{(\\pi)} = e^{\\pi} and \\lambda{(\\pi)} = \\pi g{(\\pi)}, then obtain \\frac{d}{d \\pi} (\\pi g{(\\pi)} - \\pi e^{\\pi}) = \\frac{d}{d \\pi} 0", "derivation": "g{(\\pi)} = e^{\\pi} and \\pi g{(\\pi)} = \\pi e^{\\pi} and \\lambda{(\\pi)} = \\pi g{(\\pi)} and \\lambda{(\\pi)} = \\pi e^{\\pi} and \\pi g{(\\pi)} - \\lambda{(\\pi)} = \\pi e^{\\pi} - \\lambda{(\\pi)} and \\frac{d}{d \\pi} (\\pi g{(\\pi)} - \\lambda{(\\pi)}) = \\frac{d}{d \\pi} (\\pi e^{\\pi} - \\lambda{(\\pi)}) and \\frac{d}{d \\pi} (\\pi g{(\\pi)} - \\pi e^{\\pi}) = \\frac{d}{d \\pi} 0", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('g')(Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Function('g')(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Function('\\\\lambda')(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('g')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), Function('g')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Add(Mul(Symbol('\\\\pi', commutative=True), Function('g')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(\\Psi_{\\lambda},\\mathbf{f})} = \\mathbf{f}^{\\Psi_{\\lambda}}, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},\\mathbf{f})} = \\mathbf{f}^{\\Psi_{\\lambda}} \\log{(\\mathbf{f})}, then obtain \\Psi_{\\lambda} \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{f}^{\\Psi_{\\lambda}} = \\Psi_{\\lambda} \\mathbf{f}^{\\Psi_{\\lambda}} \\log{(\\mathbf{f})}", "derivation": "f{(\\Psi_{\\lambda},\\mathbf{f})} = \\mathbf{f}^{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},\\mathbf{f})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{f}^{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},\\mathbf{f})} = \\mathbf{f}^{\\Psi_{\\lambda}} \\log{(\\mathbf{f})} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{f}^{\\Psi_{\\lambda}} = \\mathbf{f}^{\\Psi_{\\lambda}} \\log{(\\mathbf{f})} and \\Psi_{\\lambda} \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{f}^{\\Psi_{\\lambda}} = \\Psi_{\\lambda} \\mathbf{f}^{\\Psi_{\\lambda}} \\log{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(F_{g},\\phi_2)} = F_{g} \\phi_2, then obtain - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\theta_{2}{(F_{g},\\phi_2)} = - F_{g} + \\phi_2", "derivation": "\\theta_{2}{(F_{g},\\phi_2)} = F_{g} \\phi_2 and \\frac{\\partial}{\\partial F_{g}} \\theta_{2}{(F_{g},\\phi_2)} = \\frac{\\partial}{\\partial F_{g}} F_{g} \\phi_2 and - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\theta_{2}{(F_{g},\\phi_2)} = - F_{g} + \\frac{\\partial}{\\partial F_{g}} F_{g} \\phi_2 and - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\theta_{2}{(F_{g},\\phi_2)} = - F_{g} + \\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('F_g', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\hat{x})} = e^{\\hat{x}}, then derive \\sin{(1)} = \\sin{(\\frac{e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}})}, then obtain \\frac{d}{d \\hat{x}} \\sin{(1)} = \\frac{d}{d \\hat{x}} \\sin{(\\frac{e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}})}", "derivation": "\\psi^{*}{(\\hat{x})} = e^{\\hat{x}} and \\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})} = \\frac{d}{d \\hat{x}} e^{\\hat{x}} and 1 = \\frac{\\frac{d}{d \\hat{x}} e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}} and \\sin{(1)} = \\sin{(\\frac{\\frac{d}{d \\hat{x}} e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}})} and \\sin{(1)} = \\sin{(\\frac{e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}})} and \\frac{d}{d \\hat{x}} \\sin{(1)} = \\frac{d}{d \\hat{x}} \\sin{(\\frac{e^{\\hat{x}}}{\\frac{d}{d \\hat{x}} \\psi^{*}{(\\hat{x})}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(sin(Integer(1)), sin(Mul(Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(sin(Integer(1)), sin(Mul(exp(Symbol('\\\\hat{x}', commutative=True)), Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1)))))"], [["differentiate", 5, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(sin(Integer(1)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(sin(Mul(exp(Symbol('\\\\hat{x}', commutative=True)), Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(L)} = \\log{(L)} and \\operatorname{v_{2}}{(n_{1})} = n_{1}, then derive \\int c{(L)} dL = L \\log{(L)} - L + n_{1}, then obtain (\\int \\log{(L)} dL)^{\\operatorname{v_{2}}{(n_{1})}} = (L c{(L)} - L + \\operatorname{v_{2}}{(n_{1})})^{\\operatorname{v_{2}}{(n_{1})}}", "derivation": "c{(L)} = \\log{(L)} and \\int c{(L)} dL = \\int \\log{(L)} dL and \\int c{(L)} dL = L \\log{(L)} - L + n_{1} and \\int \\log{(L)} dL = L \\log{(L)} - L + n_{1} and (\\int \\log{(L)} dL)^{n_{1}} = (L \\log{(L)} - L + n_{1})^{n_{1}} and \\operatorname{v_{2}}{(n_{1})} = n_{1} and (\\int \\log{(L)} dL)^{n_{1}} = (L c{(L)} - L + n_{1})^{n_{1}} and (\\int \\log{(L)} dL)^{\\operatorname{v_{2}}{(n_{1})}} = (L c{(L)} - L + \\operatorname{v_{2}}{(n_{1})})^{\\operatorname{v_{2}}{(n_{1})}}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('c')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('n_1', commutative=True)))"], [["power", 4, "Symbol('n_1', commutative=True)"], "Equality(Pow(Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Symbol('L', commutative=True), Function('c')(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Function('v_2')(Symbol('n_1', commutative=True))), Pow(Add(Mul(Symbol('L', commutative=True), Function('c')(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Function('v_2')(Symbol('n_1', commutative=True))), Function('v_2')(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given z{(E,\\dot{x})} = E \\dot{x}, then obtain - 2 E^{2} \\dot{x} + \\dot{x} + \\int \\dot{x} \\int z{(E,\\dot{x})} d\\dot{x} dE = - 2 E^{2} \\dot{x} + \\dot{x} + \\int \\dot{x} \\int E \\dot{x} d\\dot{x} dE", "derivation": "z{(E,\\dot{x})} = E \\dot{x} and \\int z{(E,\\dot{x})} d\\dot{x} = \\int E \\dot{x} d\\dot{x} and \\dot{x} \\int z{(E,\\dot{x})} d\\dot{x} = \\dot{x} \\int E \\dot{x} d\\dot{x} and \\int \\dot{x} \\int z{(E,\\dot{x})} d\\dot{x} dE = \\int \\dot{x} \\int E \\dot{x} d\\dot{x} dE and \\dot{x} + \\int \\dot{x} \\int z{(E,\\dot{x})} d\\dot{x} dE = \\dot{x} + \\int \\dot{x} \\int E \\dot{x} d\\dot{x} dE and - 2 E^{2} \\dot{x} + \\dot{x} + \\int \\dot{x} \\int z{(E,\\dot{x})} d\\dot{x} dE = - 2 E^{2} \\dot{x} + \\dot{x} + \\int \\dot{x} \\int E \\dot{x} d\\dot{x} dE", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["add", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True)))))"], [["minus", 5, "Mul(Integer(2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Function('z')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(2)), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\hat{x},v_{z})} = \\frac{\\partial}{\\partial v_{z}} (- \\hat{x} + v_{z}) and \\varepsilon{(\\hat{x},v_{z})} = - \\hat{x} + v_{z}, then derive 1 = \\frac{\\partial}{\\partial v_{z}} \\varepsilon{(\\hat{x},v_{z})}, then obtain 1 = \\operatorname{v_{1}}{(\\hat{x},v_{z})}", "derivation": "\\operatorname{v_{1}}{(\\hat{x},v_{z})} = \\frac{\\partial}{\\partial v_{z}} (- \\hat{x} + v_{z}) and \\varepsilon{(\\hat{x},v_{z})} = - \\hat{x} + v_{z} and \\operatorname{v_{1}}{(\\hat{x},v_{z})} = \\frac{\\partial}{\\partial v_{z}} \\varepsilon{(\\hat{x},v_{z})} and \\frac{\\partial}{\\partial v_{z}} (- \\hat{x} + v_{z}) = \\frac{\\partial}{\\partial v_{z}} \\varepsilon{(\\hat{x},v_{z})} and 1 = \\frac{\\partial}{\\partial v_{z}} \\varepsilon{(\\hat{x},v_{z})} and 1 = \\operatorname{v_{1}}{(\\hat{x},v_{z})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Derivative(Function('\\\\varepsilon')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Function('v_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mu)} = \\sin{(\\mu)}, then obtain 0 = - \\int \\operatorname{F_{g}}{(\\mu)} d\\mu + \\int \\sin{(\\mu)} d\\mu", "derivation": "\\operatorname{F_{g}}{(\\mu)} = \\sin{(\\mu)} and \\int \\operatorname{F_{g}}{(\\mu)} d\\mu = \\int \\sin{(\\mu)} d\\mu and \\cos{(\\mu)} + \\int \\operatorname{F_{g}}{(\\mu)} d\\mu = \\cos{(\\mu)} + \\int \\sin{(\\mu)} d\\mu and 0 = - \\int \\operatorname{F_{g}}{(\\mu)} d\\mu + \\int \\sin{(\\mu)} d\\mu", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(cos(Symbol('\\\\mu', commutative=True)), Integral(Function('F_g')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(cos(Symbol('\\\\mu', commutative=True)), Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["minus", 3, "Add(cos(Symbol('\\\\mu', commutative=True)), Integral(Function('F_g')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('F_g')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\delta,L)} = e^{L - \\delta}, then derive \\frac{\\partial}{\\partial \\delta} \\tilde{g}{(\\delta,L)} = - e^{L - \\delta}, then obtain - e^{L - \\delta} = - \\tilde{g}{(\\delta,L)}", "derivation": "\\tilde{g}{(\\delta,L)} = e^{L - \\delta} and \\frac{\\partial}{\\partial \\delta} \\tilde{g}{(\\delta,L)} = \\frac{\\partial}{\\partial \\delta} e^{L - \\delta} and \\frac{\\partial}{\\partial \\delta} \\tilde{g}{(\\delta,L)} = - e^{L - \\delta} and \\frac{\\partial}{\\partial \\delta} \\tilde{g}{(\\delta,L)} = - \\tilde{g}{(\\delta,L)} and \\frac{\\partial}{\\partial \\delta} e^{L - \\delta} = - \\tilde{g}{(\\delta,L)} and - e^{L - \\delta} = - \\tilde{g}{(\\delta,L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True)), exp(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), exp(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\delta', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given V{(v_{x})} = \\int \\cos{(v_{x})} dv_{x}, then derive V{(v_{x})} = W + \\sin{(v_{x})}, then derive \\sigma_x + \\sin{(v_{x})} = W + \\sin{(v_{x})}, then obtain \\frac{\\partial}{\\partial \\sigma_x} (W + \\sin{(v_{x})}) = \\frac{d}{d \\sigma_x} \\int \\cos{(v_{x})} dv_{x}", "derivation": "V{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and V{(v_{x})} = W + \\sin{(v_{x})} and \\int \\cos{(v_{x})} dv_{x} = W + \\sin{(v_{x})} and \\sigma_x + \\sin{(v_{x})} = W + \\sin{(v_{x})} and \\sigma_x + \\sin{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + \\sin{(v_{x})}) = \\frac{d}{d \\sigma_x} \\int \\cos{(v_{x})} dv_{x} and \\frac{\\partial}{\\partial \\sigma_x} (W + \\sin{(v_{x})}) = \\frac{d}{d \\sigma_x} \\int \\cos{(v_{x})} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('V')(Symbol('v_x', commutative=True)), Add(Symbol('W', commutative=True), sin(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('W', commutative=True), sin(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('v_x', commutative=True))), Add(Symbol('W', commutative=True), sin(Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('v_x', commutative=True))), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Add(Symbol('W', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(V,v_{2})} = V e^{v_{2}}, then obtain (\\frac{(V + \\mathbf{S}{(V,v_{2})}) e^{- v_{2}}}{V v_{2}})^{v_{2}} = (\\frac{(V e^{v_{2}} + V) e^{- v_{2}}}{V v_{2}})^{v_{2}}", "derivation": "\\mathbf{S}{(V,v_{2})} = V e^{v_{2}} and V + \\mathbf{S}{(V,v_{2})} = V e^{v_{2}} + V and v_{2} \\mathbf{S}{(V,v_{2})} = V v_{2} e^{v_{2}} and \\frac{V + \\mathbf{S}{(V,v_{2})}}{v_{2} \\mathbf{S}{(V,v_{2})}} = \\frac{V e^{v_{2}} + V}{v_{2} \\mathbf{S}{(V,v_{2})}} and (\\frac{V + \\mathbf{S}{(V,v_{2})}}{v_{2} \\mathbf{S}{(V,v_{2})}})^{v_{2}} = (\\frac{V e^{v_{2}} + V}{v_{2} \\mathbf{S}{(V,v_{2})}})^{v_{2}} and (\\frac{(V + \\mathbf{S}{(V,v_{2})}) e^{- v_{2}}}{V v_{2}})^{v_{2}} = (\\frac{(V e^{v_{2}} + V) e^{- v_{2}}}{V v_{2}})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('V', commutative=True), exp(Symbol('v_2', commutative=True))))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Symbol('V', commutative=True), exp(Symbol('v_2', commutative=True))), Symbol('V', commutative=True)))"], [["times", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True))), Mul(Symbol('V', commutative=True), Symbol('v_2', commutative=True), exp(Symbol('v_2', commutative=True))))"], [["divide", 2, "Mul(Symbol('v_2', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True))), Pow(Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Symbol('V', commutative=True), exp(Symbol('v_2', commutative=True))), Symbol('V', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True))), Pow(Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Symbol('V', commutative=True), exp(Symbol('v_2', commutative=True))), Symbol('V', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Function('\\\\mathbf{S}')(Symbol('V', commutative=True), Symbol('v_2', commutative=True))), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Symbol('V', commutative=True), exp(Symbol('v_2', commutative=True))), Symbol('V', commutative=True)), exp(Mul(Integer(-1), Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})} = M_{E} \\hat{\\mathbf{r}} \\hat{x}_0, then obtain \\iint (- \\hat{x}_0 + \\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})}) d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}} = \\iint (M_{E} \\hat{\\mathbf{r}} \\hat{x}_0 - \\hat{x}_0) d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}}", "derivation": "\\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})} = M_{E} \\hat{\\mathbf{r}} \\hat{x}_0 and - \\hat{x}_0 + \\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})} = M_{E} \\hat{\\mathbf{r}} \\hat{x}_0 - \\hat{x}_0 and \\int (- \\hat{x}_0 + \\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})}) d\\hat{\\mathbf{r}} = \\int (M_{E} \\hat{\\mathbf{r}} \\hat{x}_0 - \\hat{x}_0) d\\hat{\\mathbf{r}} and \\iint (- \\hat{x}_0 + \\Omega{(\\hat{x}_0,\\hat{\\mathbf{r}},M_{E})}) d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}} = \\iint (M_{E} \\hat{\\mathbf{r}} \\hat{x}_0 - \\hat{x}_0) d\\hat{\\mathbf{r}} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given L{(\\theta_1)} = \\sin{(\\sin{(\\theta_1)})} and s{(\\theta_1)} = \\int \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} d\\theta_1, then obtain \\int \\frac{d}{d \\theta_1} L{(\\theta_1)} d\\theta_1 = s{(\\theta_1)}", "derivation": "L{(\\theta_1)} = \\sin{(\\sin{(\\theta_1)})} and \\frac{d}{d \\theta_1} L{(\\theta_1)} = \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} and \\int \\frac{d}{d \\theta_1} L{(\\theta_1)} d\\theta_1 = \\int \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} d\\theta_1 and s{(\\theta_1)} = \\int \\frac{d}{d \\theta_1} \\sin{(\\sin{(\\theta_1)})} d\\theta_1 and \\int \\frac{d}{d \\theta_1} L{(\\theta_1)} d\\theta_1 = s{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta_1', commutative=True)), sin(sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Function('L')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(sin(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\theta_1', commutative=True)), Integral(Derivative(sin(sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Derivative(Function('L')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Function('s')(Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)} = \\sin{(\\hat{H}_l - \\mu_0)} and \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} = \\frac{1}{- \\hat{H}_l + \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)}}, then obtain 0 = \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} - \\frac{1}{- \\hat{H}_l + \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)} = \\sin{(\\hat{H}_l - \\mu_0)} and \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} = \\frac{1}{- \\hat{H}_l + \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)}} and \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} = \\frac{1}{- \\hat{H}_l + \\sin{(\\hat{H}_l - \\mu_0)}} and 0 = - \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} + \\frac{1}{- \\hat{H}_l + \\sin{(\\hat{H}_l - \\mu_0)}} and 0 = - \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} + \\frac{1}{- \\hat{H}_l + \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)}} and 0 = \\operatorname{a^{\\dagger}}{(\\mu_0,\\hat{H}_l)} - \\frac{1}{- \\hat{H}_l + \\operatorname{J_{\\varepsilon}}{(\\mu_0,\\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))), Integer(-1)))"], [["minus", 3, "Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1))))"], [["times", 5, "Integer(-1)"], "Equality(Integer(0), Add(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\lambda{(T)} = \\sin{(T)} and Z{(P_{g})} = P_{g}, then obtain - \\frac{P_{g} \\lambda{(T)}}{Z{(P_{g})}} - \\sin{(T)} = - 2 \\sin{(T)}", "derivation": "\\lambda{(T)} = \\sin{(T)} and \\lambda{(T)} + \\sin{(T)} = 2 \\sin{(T)} and - \\lambda{(T)} - \\sin{(T)} = - 2 \\sin{(T)} and Z{(P_{g})} = P_{g} and - Z{(P_{g})} \\sin{(T)} = - P_{g} \\sin{(T)} and - \\sin{(T)} = - \\frac{P_{g} \\sin{(T)}}{Z{(P_{g})}} and - \\lambda{(T)} = - \\frac{P_{g} \\lambda{(T)}}{Z{(P_{g})}} and - \\frac{P_{g} \\lambda{(T)}}{Z{(P_{g})}} - \\sin{(T)} = - 2 \\sin{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["add", 1, "sin(Symbol('T', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True))), Mul(Integer(2), sin(Symbol('T', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('T', commutative=True))), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Mul(Integer(-1), Integer(2), sin(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["times", 4, "Mul(Integer(-1), sin(Symbol('T', commutative=True)))"], "Equality(Mul(Integer(-1), Function('Z')(Symbol('P_g', commutative=True)), sin(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True), sin(Symbol('T', commutative=True))))"], [["divide", 5, "Function('Z')(Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True), Pow(Function('Z')(Symbol('P_g', commutative=True)), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(-1), Function('\\\\lambda')(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('P_g', commutative=True), Pow(Function('Z')(Symbol('P_g', commutative=True)), Integer(-1)), Function('\\\\lambda')(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 7], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True), Pow(Function('Z')(Symbol('P_g', commutative=True)), Integer(-1)), Function('\\\\lambda')(Symbol('T', commutative=True))), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Mul(Integer(-1), Integer(2), sin(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{v})} = \\log{(\\mathbf{v})}, then derive \\mathbf{v} \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} (\\mathbf{v} \\log{(\\mathbf{v})} - \\mathbf{v} + q), then obtain \\mathbf{v} \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} (\\mathbf{v} \\mathbf{A}{(\\mathbf{v})} - \\mathbf{v} + q)", "derivation": "\\mathbf{A}{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\int \\log{(\\mathbf{v})} d\\mathbf{v} and \\mathbf{v} \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} \\int \\log{(\\mathbf{v})} d\\mathbf{v} and \\mathbf{v} \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} (\\mathbf{v} \\log{(\\mathbf{v})} - \\mathbf{v} + q) and \\mathbf{v} \\int \\mathbf{A}{(\\mathbf{v})} d\\mathbf{v} = \\mathbf{v} (\\mathbf{v} \\mathbf{A}{(\\mathbf{v})} - \\mathbf{v} + q)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Integral(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{E})} = \\cos{(e^{\\mathbf{E}})} and \\theta_{2}{(\\mathbf{E})} = \\cos{(e^{\\mathbf{E}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} - \\frac{A_{z} \\theta_{1}{(\\mathbf{E})}}{\\mathbf{v}} = \\frac{\\partial}{\\partial \\mathbf{E}} - \\frac{A_{z} \\theta_{2}{(\\mathbf{E})}}{\\mathbf{v}}", "derivation": "\\theta_{1}{(\\mathbf{E})} = \\cos{(e^{\\mathbf{E}})} and - \\frac{\\theta_{1}{(\\mathbf{E})}}{\\mathbf{v}} = - \\frac{\\cos{(e^{\\mathbf{E}})}}{\\mathbf{v}} and \\theta_{2}{(\\mathbf{E})} = \\cos{(e^{\\mathbf{E}})} and - \\frac{\\theta_{1}{(\\mathbf{E})}}{\\mathbf{v}} = - \\frac{\\theta_{2}{(\\mathbf{E})}}{\\mathbf{v}} and - \\frac{A_{z} \\theta_{1}{(\\mathbf{E})}}{\\mathbf{v}} = - \\frac{A_{z} \\theta_{2}{(\\mathbf{E})}}{\\mathbf{v}} and \\frac{\\partial}{\\partial \\mathbf{E}} - \\frac{A_{z} \\theta_{1}{(\\mathbf{E})}}{\\mathbf{v}} = \\frac{\\partial}{\\partial \\mathbf{E}} - \\frac{A_{z} \\theta_{2}{(\\mathbf{E})}}{\\mathbf{v}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{E}', commutative=True)), cos(exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mathbf{E}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True)), cos(exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True))))"], [["times", 4, "Symbol('A_z', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('A_z', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(\\mathbf{E},\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\hat{H}_{\\lambda} \\delta{(\\mathbf{E},\\hat{H}_{\\lambda})}}{\\mathbf{E}} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\hat{H}_{\\lambda} \\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})}}{\\mathbf{E}}", "derivation": "\\delta{(\\mathbf{E},\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})} and \\frac{\\delta{(\\mathbf{E},\\hat{H}_{\\lambda})}}{\\mathbf{E}} = \\frac{\\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})}}{\\mathbf{E}} and \\frac{\\hat{H}_{\\lambda} \\delta{(\\mathbf{E},\\hat{H}_{\\lambda})}}{\\mathbf{E}} = \\frac{\\hat{H}_{\\lambda} \\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})}}{\\mathbf{E}} and \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\hat{H}_{\\lambda} \\delta{(\\mathbf{E},\\hat{H}_{\\lambda})}}{\\mathbf{E}} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\hat{H}_{\\lambda} \\sin{(\\hat{H}_{\\lambda}^{\\mathbf{E}})}}{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))))"], [["times", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(T,p)} = (p^{T})^{T}, then obtain \\int \\frac{\\partial^{2}}{\\partial T^{2}} \\operatorname{C_{d}}{(T,p)} dp = \\int \\frac{\\partial^{2}}{\\partial T^{2}} (p^{T})^{T} dp", "derivation": "\\operatorname{C_{d}}{(T,p)} = (p^{T})^{T} and \\frac{\\partial}{\\partial T} \\operatorname{C_{d}}{(T,p)} = \\frac{\\partial}{\\partial T} (p^{T})^{T} and \\frac{\\partial^{2}}{\\partial T^{2}} \\operatorname{C_{d}}{(T,p)} = \\frac{\\partial^{2}}{\\partial T^{2}} (p^{T})^{T} and \\int \\frac{\\partial^{2}}{\\partial T^{2}} \\operatorname{C_{d}}{(T,p)} dp = \\int \\frac{\\partial^{2}}{\\partial T^{2}} (p^{T})^{T} dp", "srepr_derivation": [["renaming_premise", "Equality(Function('C_d')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Pow(Pow(Symbol('p', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('p', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Derivative(Pow(Pow(Symbol('p', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Tuple(Symbol('p', commutative=True))), Integral(Derivative(Pow(Pow(Symbol('p', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(J,\\phi,v)} = (v^{J})^{\\phi} and \\operatorname{v_{y}}{(J,\\phi,v)} = - \\phi + \\rho_{f}{(J,\\phi,v)}, then obtain (- \\phi + (v^{J})^{\\phi})^{v} = (- \\phi + \\rho_{f}{(J,\\phi,v)})^{v}", "derivation": "\\rho_{f}{(J,\\phi,v)} = (v^{J})^{\\phi} and - \\phi + \\rho_{f}{(J,\\phi,v)} = - \\phi + (v^{J})^{\\phi} and \\operatorname{v_{y}}{(J,\\phi,v)} = - \\phi + \\rho_{f}{(J,\\phi,v)} and \\operatorname{v_{y}}^{v}{(J,\\phi,v)} = (- \\phi + \\rho_{f}{(J,\\phi,v)})^{v} and \\operatorname{v_{y}}{(J,\\phi,v)} = - \\phi + (v^{J})^{\\phi} and (- \\phi + (v^{J})^{\\phi})^{v} = (- \\phi + \\rho_{f}{(J,\\phi,v)})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Symbol('J', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Symbol('J', commutative=True)), Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_y')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Symbol('J', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Symbol('J', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(m_{s},\\mathbf{A})} = \\mathbf{A} + m_{s}, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\mathbf{f}{(m_{s},\\mathbf{A})}}{\\mathbf{A} + m_{s}} \\int \\mathbf{f}{(m_{s},\\mathbf{A})} dm_{s} = \\frac{d}{d \\mathbf{A}} 1 \\int \\mathbf{f}{(m_{s},\\mathbf{A})} dm_{s}", "derivation": "\\mathbf{f}{(m_{s},\\mathbf{A})} = \\mathbf{A} + m_{s} and \\frac{\\mathbf{f}{(m_{s},\\mathbf{A})}}{\\mathbf{A} + m_{s}} = 1 and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\mathbf{f}{(m_{s},\\mathbf{A})}}{\\mathbf{A} + m_{s}} = \\frac{d}{d \\mathbf{A}} 1 and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\mathbf{f}{(m_{s},\\mathbf{A})}}{\\mathbf{A} + m_{s}} \\int \\mathbf{f}{(m_{s},\\mathbf{A})} dm_{s} = \\frac{d}{d \\mathbf{A}} 1 \\int \\mathbf{f}{(m_{s},\\mathbf{A})} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["times", 3, "Integral(Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m_s', commutative=True)))"], "Equality(Mul(Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{f}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given p{(\\mathbf{J}_f,\\hat{X})} = e^{\\frac{\\mathbf{J}_f}{\\hat{X}}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_f} \\log{(\\sin{(p{(\\mathbf{J}_f,\\hat{X})})})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\log{(\\sin{(e^{\\frac{\\mathbf{J}_f}{\\hat{X}}})})}", "derivation": "p{(\\mathbf{J}_f,\\hat{X})} = e^{\\frac{\\mathbf{J}_f}{\\hat{X}}} and \\sin{(p{(\\mathbf{J}_f,\\hat{X})})} = \\sin{(e^{\\frac{\\mathbf{J}_f}{\\hat{X}}})} and \\log{(\\sin{(p{(\\mathbf{J}_f,\\hat{X})})})} = \\log{(\\sin{(e^{\\frac{\\mathbf{J}_f}{\\hat{X}}})})} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\log{(\\sin{(p{(\\mathbf{J}_f,\\hat{X})})})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\log{(\\sin{(e^{\\frac{\\mathbf{J}_f}{\\hat{X}}})})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), exp(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["sin", 1], "Equality(sin(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), sin(exp(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["log", 2], "Equality(log(sin(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), log(sin(exp(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(log(sin(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(log(sin(exp(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(v,F_{x})} = - v + \\sin{(F_{x})}, then obtain (- v + \\sin{(F_{x})})^{2} \\hat{x}^{4}{(v,F_{x})} = (- v + \\sin{(F_{x})})^{5} \\hat{x}{(v,F_{x})}", "derivation": "\\hat{x}{(v,F_{x})} = - v + \\sin{(F_{x})} and \\hat{x}^{2}{(v,F_{x})} = (- v + \\sin{(F_{x})}) \\hat{x}{(v,F_{x})} and \\hat{x}^{4}{(v,F_{x})} = (- v + \\sin{(F_{x})})^{2} \\hat{x}^{2}{(v,F_{x})} and (- v + \\sin{(F_{x})})^{2} \\hat{x}^{2}{(v,F_{x})} = (- v + \\sin{(F_{x})})^{3} \\hat{x}{(v,F_{x})} and \\hat{x}^{4}{(v,F_{x})} = (- v + \\sin{(F_{x})})^{3} \\hat{x}{(v,F_{x})} and (- v + \\sin{(F_{x})})^{2} \\hat{x}^{4}{(v,F_{x})} = (- v + \\sin{(F_{x})})^{5} \\hat{x}{(v,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))))"], [["times", 1, "Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(4)), Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(2)), Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(2)), Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(3)), Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(4)), Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(3)), Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True))))"], [["times", 5, "Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(2))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(2)), Pow(Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True)), Integer(4))), Mul(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Symbol('F_x', commutative=True))), Integer(5)), Function('\\\\hat{x}')(Symbol('v', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(a)} = \\sin{(a)} and H{(a)} = \\sin{(a)}, then obtain \\int a ((\\frac{\\mathbf{E}{(a)}}{H{(a)}})^{a})^{a} da = \\int a da", "derivation": "\\mathbf{E}{(a)} = \\sin{(a)} and \\frac{\\mathbf{E}{(a)}}{\\sin{(a)}} = 1 and (\\frac{\\mathbf{E}{(a)}}{\\sin{(a)}})^{a} = 1 and H{(a)} = \\sin{(a)} and (\\frac{\\mathbf{E}{(a)}}{H{(a)}})^{a} = 1 and ((\\frac{\\mathbf{E}{(a)}}{H{(a)}})^{a})^{a} = 1 and a ((\\frac{\\mathbf{E}{(a)}}{H{(a)}})^{a})^{a} = a and \\int a ((\\frac{\\mathbf{E}{(a)}}{H{(a)}})^{a})^{a} da = \\int a da", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["divide", 1, "sin(Symbol('a', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(-1))), Symbol('a', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('H')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Pow(Function('H')(Symbol('a', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Integer(1))"], [["power", 5, "Symbol('a', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Function('H')(Symbol('a', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Integer(1))"], [["times", 6, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Pow(Pow(Mul(Pow(Function('H')(Symbol('a', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Symbol('a', commutative=True))"], [["integrate", 7, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Symbol('a', commutative=True), Pow(Pow(Mul(Pow(Function('H')(Symbol('a', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Symbol('a', commutative=True), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain \\int (\\cos{(\\mathbf{s})} + \\frac{\\hat{p}{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} d\\mathbf{s} = \\int (\\cos{(\\mathbf{s})} + \\frac{\\cos{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} d\\mathbf{s}", "derivation": "\\hat{p}{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and \\frac{\\hat{p}{(\\mathbf{s})}}{\\mathbf{s}} = \\frac{\\cos{(\\mathbf{s})}}{\\mathbf{s}} and \\cos{(\\mathbf{s})} + \\frac{\\hat{p}{(\\mathbf{s})}}{\\mathbf{s}} = \\cos{(\\mathbf{s})} + \\frac{\\cos{(\\mathbf{s})}}{\\mathbf{s}} and (\\cos{(\\mathbf{s})} + \\frac{\\hat{p}{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} = (\\cos{(\\mathbf{s})} + \\frac{\\cos{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} and \\int (\\cos{(\\mathbf{s})} + \\frac{\\hat{p}{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} d\\mathbf{s} = \\int (\\cos{(\\mathbf{s})} + \\frac{\\cos{(\\mathbf{s})}}{\\mathbf{s}})^{\\mathbf{s}} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True)))), Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Pow(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Pow(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(F_{H})} = \\cos{(F_{H})}, then obtain \\frac{d}{d F_{H}} - F_{H} \\operatorname{F_{N}}^{3}{(F_{H})} = \\frac{d}{d F_{H}} - F_{H} \\cos^{3}{(F_{H})}", "derivation": "\\operatorname{F_{N}}{(F_{H})} = \\cos{(F_{H})} and \\operatorname{F_{N}}^{2}{(F_{H})} = \\operatorname{F_{N}}{(F_{H})} \\cos{(F_{H})} and \\operatorname{F_{N}}^{4}{(F_{H})} = \\operatorname{F_{N}}^{2}{(F_{H})} \\cos^{2}{(F_{H})} and \\operatorname{F_{N}}^{2}{(F_{H})} \\cos^{2}{(F_{H})} = \\operatorname{F_{N}}{(F_{H})} \\cos^{3}{(F_{H})} and \\operatorname{F_{N}}^{4}{(F_{H})} = \\operatorname{F_{N}}{(F_{H})} \\cos^{3}{(F_{H})} and \\operatorname{F_{N}}^{3}{(F_{H})} = \\cos^{3}{(F_{H})} and - F_{H} \\operatorname{F_{N}}^{3}{(F_{H})} = - F_{H} \\cos^{3}{(F_{H})} and \\frac{d}{d F_{H}} - F_{H} \\operatorname{F_{N}}^{3}{(F_{H})} = \\frac{d}{d F_{H}} - F_{H} \\cos^{3}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["times", 1, "Function('F_N')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(2)), Mul(Function('F_N')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(4)), Mul(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(2)), Pow(cos(Symbol('F_H', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(2)), Pow(cos(Symbol('F_H', commutative=True)), Integer(2))), Mul(Function('F_N')(Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(4)), Mul(Function('F_N')(Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))))"], [["divide", 5, "Function('F_N')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(3)), Pow(cos(Symbol('F_H', commutative=True)), Integer(3)))"], [["times", 6, "Mul(Integer(-1), Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(3))), Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))))"], [["differentiate", 7, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Function('F_N')(Symbol('F_H', commutative=True)), Integer(3))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(cos(Symbol('F_H', commutative=True)), Integer(3))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})} = \\mathbf{A} + \\mathbf{P}, then obtain (\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})})^{2} = 1", "derivation": "\\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})} = \\mathbf{A} + \\mathbf{P} and \\mathbf{P} + \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})} = \\mathbf{A} + 2 \\mathbf{P} and \\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{P} + \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})}) = \\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{A} + 2 \\mathbf{P}) and (\\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{P} + \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})}))^{2} = (\\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{A} + 2 \\mathbf{P}))^{2} and (\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{J}_f{(\\mathbf{P},\\mathbf{A})})^{2} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(2)), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)} = \\frac{n}{\\hat{x}}, then obtain \\int (\\frac{\\hat{x} \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)}}{n} + \\frac{1}{n}) d\\hat{x} = Q + \\hat{x} (1 + \\frac{1}{n})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)} = \\frac{n}{\\hat{x}} and \\frac{\\hat{x} \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)}}{n} = 1 and \\frac{\\hat{x} \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)}}{n} + \\frac{1}{n} = 1 + \\frac{1}{n} and \\int (\\frac{\\hat{x} \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)}}{n} + \\frac{1}{n}) d\\hat{x} = \\int (1 + \\frac{1}{n}) d\\hat{x} and \\int (\\frac{\\hat{x} \\operatorname{f_{\\mathbf{p}}}{(\\hat{x},n)}}{n} + \\frac{1}{n}) d\\hat{x} = Q + \\hat{x} (1 + \\frac{1}{n})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('n', commutative=True))"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('n', commutative=True))), Integer(1))"], [["add", 2, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Integer(1), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Integer(1), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\hat{x}', commutative=True), Add(Integer(1), Pow(Symbol('n', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\lambda{(v_{y})} = \\cos{(v_{y})}, then obtain \\frac{2 \\lambda{(v_{y})}}{\\cos{(v_{y})}} - 1 + \\frac{1}{\\cos{(v_{y})}} = 1 + \\frac{1}{\\cos{(v_{y})}}", "derivation": "\\lambda{(v_{y})} = \\cos{(v_{y})} and \\lambda{(v_{y})} \\cos{(v_{y})} = \\cos^{2}{(v_{y})} and \\frac{\\lambda{(v_{y})}}{\\cos{(v_{y})}} = 1 and \\frac{2 \\lambda{(v_{y})}}{\\cos{(v_{y})}} = \\frac{\\lambda{(v_{y})}}{\\cos{(v_{y})}} + 1 and \\frac{\\lambda{(v_{y})}}{\\cos{(v_{y})}} + \\frac{1}{\\cos{(v_{y})}} = 1 + \\frac{1}{\\cos{(v_{y})}} and \\frac{2 \\lambda{(v_{y})}}{\\cos{(v_{y})}} - 1 + \\frac{1}{\\cos{(v_{y})}} = 1 + \\frac{1}{\\cos{(v_{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["times", 1, "cos(Symbol('v_y', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Pow(cos(Symbol('v_y', commutative=True)), Integer(2)))"], [["divide", 2, "Pow(cos(Symbol('v_y', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Integer(1))"], [["add", 3, "Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Add(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Integer(1)))"], [["add", 3, "Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Add(Integer(1), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(2), Function('\\\\lambda')(Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Integer(-1), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))), Add(Integer(1), Pow(cos(Symbol('v_y', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(Q,L_{\\varepsilon})} = Q^{L_{\\varepsilon}}, then derive \\frac{\\partial}{\\partial Q} \\operatorname{n_{1}}{(Q,L_{\\varepsilon})} = \\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q}, then obtain \\int (\\frac{\\partial}{\\partial Q} Q^{L_{\\varepsilon}})^{L_{\\varepsilon}} dQ = \\int (\\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q})^{L_{\\varepsilon}} dQ", "derivation": "\\operatorname{n_{1}}{(Q,L_{\\varepsilon})} = Q^{L_{\\varepsilon}} and \\frac{\\partial}{\\partial Q} \\operatorname{n_{1}}{(Q,L_{\\varepsilon})} = \\frac{\\partial}{\\partial Q} Q^{L_{\\varepsilon}} and \\frac{\\partial}{\\partial Q} \\operatorname{n_{1}}{(Q,L_{\\varepsilon})} = \\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q} and \\frac{\\partial}{\\partial Q} Q^{L_{\\varepsilon}} = \\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q} and (\\frac{\\partial}{\\partial Q} Q^{L_{\\varepsilon}})^{L_{\\varepsilon}} = (\\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q})^{L_{\\varepsilon}} and \\int (\\frac{\\partial}{\\partial Q} Q^{L_{\\varepsilon}})^{L_{\\varepsilon}} dQ = \\int (\\frac{L_{\\varepsilon} Q^{L_{\\varepsilon}}}{Q})^{L_{\\varepsilon}} dQ", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 4, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Derivative(Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Derivative(Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(n_{1},p)} = (e^{n_{1}})^{p}, then derive \\frac{\\partial^{2}}{\\partial n_{1}^{2}} \\operatorname{z^{*}}{(n_{1},p)} = p^{2} (e^{n_{1}})^{p}, then obtain p^{2} (e^{n_{1}})^{p} = \\frac{\\partial^{2}}{\\partial n_{1}^{2}} (e^{n_{1}})^{p}", "derivation": "\\operatorname{z^{*}}{(n_{1},p)} = (e^{n_{1}})^{p} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{z^{*}}{(n_{1},p)} = \\frac{\\partial}{\\partial n_{1}} (e^{n_{1}})^{p} and \\frac{\\partial^{2}}{\\partial n_{1}^{2}} \\operatorname{z^{*}}{(n_{1},p)} = \\frac{\\partial^{2}}{\\partial n_{1}^{2}} (e^{n_{1}})^{p} and \\frac{\\partial^{2}}{\\partial n_{1}^{2}} \\operatorname{z^{*}}{(n_{1},p)} = p^{2} (e^{n_{1}})^{p} and p^{2} (e^{n_{1}})^{p} = \\frac{\\partial^{2}}{\\partial n_{1}^{2}} (e^{n_{1}})^{p}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))), Derivative(Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('z^*')(Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))), Mul(Pow(Symbol('p', commutative=True), Integer(2)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(2)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True))), Derivative(Pow(exp(Symbol('n_1', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given E{(J,M_{E})} = (e^{J})^{M_{E}}, then obtain ((- E{(J,M_{E})} + (e^{J})^{M_{E}})^{J} + 1) E{(J,M_{E})} = 2 E{(J,M_{E})}", "derivation": "E{(J,M_{E})} = (e^{J})^{M_{E}} and 0 = - E{(J,M_{E})} + (e^{J})^{M_{E}} and 0^{J} = (- E{(J,M_{E})} + (e^{J})^{M_{E}})^{J} and 0^{J} + 1 = (- E{(J,M_{E})} + (e^{J})^{M_{E}})^{J} + 1 and (0^{J} + 1) E{(J,M_{E})} = ((- E{(J,M_{E})} + (e^{J})^{M_{E}})^{J} + 1) E{(J,M_{E})} and ((- E{(J,M_{E})} + (e^{J})^{M_{E}})^{J} + 1) E{(J,M_{E})} = 2 E{(J,M_{E})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True)))"], [["minus", 1, "Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Integer(0), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True))), Symbol('J', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Pow(Integer(0), Symbol('J', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True))), Symbol('J', commutative=True)), Integer(1)))"], [["times", 4, "Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Add(Pow(Integer(0), Symbol('J', commutative=True)), Integer(1)), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Mul(Add(Pow(Add(Mul(Integer(-1), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True))), Symbol('J', commutative=True)), Integer(1)), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Pow(Add(Mul(Integer(-1), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Symbol('M_E', commutative=True))), Symbol('J', commutative=True)), Integer(1)), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(2), Function('E')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(t_{2},\\Omega)} = \\cos{(\\Omega + t_{2})}, then derive \\frac{\\partial}{\\partial t_{2}} \\operatorname{V_{\\mathbf{B}}}{(t_{2},\\Omega)} = - \\sin{(\\Omega + t_{2})}, then derive \\int - \\sin{(\\Omega + t_{2})} d\\Omega = \\Psi^{\\dagger} + \\cos{(\\Omega + t_{2})}, then obtain \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\int - \\sin{(\\Omega + t_{2})} d\\Omega = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\cos{(\\Omega + t_{2})})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(t_{2},\\Omega)} = \\cos{(\\Omega + t_{2})} and \\frac{\\partial}{\\partial t_{2}} \\operatorname{V_{\\mathbf{B}}}{(t_{2},\\Omega)} = \\frac{\\partial}{\\partial t_{2}} \\cos{(\\Omega + t_{2})} and \\frac{\\partial}{\\partial t_{2}} \\operatorname{V_{\\mathbf{B}}}{(t_{2},\\Omega)} = - \\sin{(\\Omega + t_{2})} and - \\sin{(\\Omega + t_{2})} = \\frac{\\partial}{\\partial t_{2}} \\cos{(\\Omega + t_{2})} and \\int - \\sin{(\\Omega + t_{2})} d\\Omega = \\int \\frac{\\partial}{\\partial t_{2}} \\cos{(\\Omega + t_{2})} d\\Omega and \\int - \\sin{(\\Omega + t_{2})} d\\Omega = \\Psi^{\\dagger} + \\cos{(\\Omega + t_{2})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\int - \\sin{(\\Omega + t_{2})} d\\Omega = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\cos{(\\Omega + t_{2})})", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))), Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(J,A_{x})} = J^{A_{x}}, then obtain \\frac{\\partial^{2}}{\\partial J\\partial A_{x}} H^{A_{x}}{(J,A_{x})} = \\frac{\\partial^{2}}{\\partial J\\partial A_{x}} (J^{A_{x}})^{A_{x}}", "derivation": "H{(J,A_{x})} = J^{A_{x}} and H^{A_{x}}{(J,A_{x})} = (J^{A_{x}})^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} H^{A_{x}}{(J,A_{x})} = \\frac{\\partial}{\\partial A_{x}} (J^{A_{x}})^{A_{x}} and \\frac{\\partial^{2}}{\\partial J\\partial A_{x}} H^{A_{x}}{(J,A_{x})} = \\frac{\\partial^{2}}{\\partial J\\partial A_{x}} (J^{A_{x}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('H')(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["differentiate", 2, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Pow(Function('H')(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Function('H')(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('J', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(M,y,u)} = (M y)^{u}, then obtain u + (M y + \\mathbf{S}{(M,y,u)})^{2} + \\mathbf{S}{(M,y,u)} = u + (M y)^{u} + (M y + \\mathbf{S}{(M,y,u)})^{2}", "derivation": "\\mathbf{S}{(M,y,u)} = (M y)^{u} and M y + \\mathbf{S}{(M,y,u)} = M y + (M y)^{u} and (M y + (M y)^{u}) (M y + \\mathbf{S}{(M,y,u)}) + \\mathbf{S}{(M,y,u)} = (M y)^{u} + (M y + (M y)^{u}) (M y + \\mathbf{S}{(M,y,u)}) and u + (M y + (M y)^{u}) (M y + \\mathbf{S}{(M,y,u)}) + \\mathbf{S}{(M,y,u)} = u + (M y)^{u} + (M y + (M y)^{u}) (M y + \\mathbf{S}{(M,y,u)}) and u + (M y + \\mathbf{S}{(M,y,u)})^{2} + \\mathbf{S}{(M,y,u)} = u + (M y)^{u} + (M y + \\mathbf{S}{(M,y,u)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Symbol('M', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))))"], [["add", 1, "Mul(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))))"], "Equality(Add(Mul(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Add(Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Mul(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))))))"], [["add", 3, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Mul(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Mul(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True))), Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('u', commutative=True), Pow(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Integer(2)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), Pow(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Symbol('u', commutative=True)), Pow(Add(Mul(Symbol('M', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{S}')(Symbol('M', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True))), Integer(2))))"]]}, {"prompt": "Given J{(t_{1})} = \\log{(t_{1})} and \\mathbf{J}_M{(t_{1})} = \\log{(t_{1})}, then obtain (\\mathbf{J}_M^{t_{1}}{(t_{1})})^{t_{1}} - J^{t_{1}}{(t_{1})} = (J^{t_{1}}{(t_{1})})^{t_{1}} - J^{t_{1}}{(t_{1})}", "derivation": "J{(t_{1})} = \\log{(t_{1})} and \\mathbf{J}_M{(t_{1})} = \\log{(t_{1})} and \\mathbf{J}_M^{t_{1}}{(t_{1})} = \\log{(t_{1})}^{t_{1}} and \\mathbf{J}_M^{t_{1}}{(t_{1})} = J^{t_{1}}{(t_{1})} and (\\mathbf{J}_M^{t_{1}}{(t_{1})})^{t_{1}} = (J^{t_{1}}{(t_{1})})^{t_{1}} and (\\mathbf{J}_M^{t_{1}}{(t_{1})})^{t_{1}} - J^{t_{1}}{(t_{1})} = (J^{t_{1}}{(t_{1})})^{t_{1}} - J^{t_{1}}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["power", 2, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(log(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["power", 4, "Symbol('t_1', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 5, "Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))), Add(Pow(Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(t_{2})} = e^{t_{2}}, then obtain (- t_{2} + \\operatorname{m_{s}}^{t_{2}}{(t_{2})}) e^{t_{2}} = (- t_{2} + (e^{t_{2}})^{t_{2}}) e^{t_{2}}", "derivation": "\\operatorname{m_{s}}{(t_{2})} = e^{t_{2}} and \\operatorname{m_{s}}^{t_{2}}{(t_{2})} = (e^{t_{2}})^{t_{2}} and - t_{2} + \\operatorname{m_{s}}^{t_{2}}{(t_{2})} = - t_{2} + (e^{t_{2}})^{t_{2}} and (- t_{2} + \\operatorname{m_{s}}^{t_{2}}{(t_{2})}) e^{t_{2}} = (- t_{2} + (e^{t_{2}})^{t_{2}}) e^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 2, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Pow(Function('m_s')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["times", 3, "exp(Symbol('t_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Pow(Function('m_s')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given G{(a)} = \\cos{(a)}, then obtain \\frac{d}{d a} (G^{a}{(a)} - \\cos^{a}{(a)} + 1) = \\frac{d}{d a} 1", "derivation": "G{(a)} = \\cos{(a)} and G^{a}{(a)} = \\cos^{a}{(a)} and G^{a}{(a)} + \\cos{(a)} = \\cos{(a)} + \\cos^{a}{(a)} and G^{a}{(a)} - \\cos^{a}{(a)} = 0 and G^{a}{(a)} - \\cos^{a}{(a)} + 1 = 1 and \\frac{d}{d a} (G^{a}{(a)} - \\cos^{a}{(a)} + 1) = \\frac{d}{d a} 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('G')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], [["add", 2, "cos(Symbol('a', commutative=True))"], "Equality(Add(Pow(Function('G')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Add(cos(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], [["minus", 3, "Add(cos(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], "Equality(Add(Pow(Function('G')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Integer(0))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Pow(Function('G')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Integer(1)), Integer(1))"], [["differentiate", 5, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Pow(Function('G')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,x)} = E^{x} and \\eta^{\\prime}{(E,x)} = E^{x} and \\mathbf{v}{(E,x)} = E^{x}, then obtain \\frac{\\partial}{\\partial x} (- x + \\eta^{\\prime}{(E,x)}) = \\frac{\\partial}{\\partial x} (- x + \\mathbf{v}{(E,x)})", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(E,x)} = E^{x} and - x + \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,x)} = E^{x} - x and \\eta^{\\prime}{(E,x)} = E^{x} and \\eta^{\\prime}{(E,x)} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,x)} and - x + \\eta^{\\prime}{(E,x)} = E^{x} - x and \\frac{\\partial}{\\partial x} (- x + \\eta^{\\prime}{(E,x)}) = \\frac{\\partial}{\\partial x} (E^{x} - x) and \\mathbf{v}{(E,x)} = E^{x} and \\frac{\\partial}{\\partial x} (- x + \\eta^{\\prime}{(E,x)}) = \\frac{\\partial}{\\partial x} (- x + \\mathbf{v}{(E,x)})", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('x', commutative=True))), Add(Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('x', commutative=True))), Add(Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Pow(Symbol('E', commutative=True), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('\\\\mathbf{v}')(Symbol('E', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(\\hat{x}_0)} = \\hat{x}_0, then obtain \\iint 0 d\\hat{x}_0 d\\hat{x}_0 = \\iint (\\hat{x}_0 - \\theta{(\\hat{x}_0)}) d\\hat{x}_0 d\\hat{x}_0", "derivation": "\\theta{(\\hat{x}_0)} = \\hat{x}_0 and 0 = \\hat{x}_0 - \\theta{(\\hat{x}_0)} and \\int 0 d\\hat{x}_0 = \\int (\\hat{x}_0 - \\theta{(\\hat{x}_0)}) d\\hat{x}_0 and \\iint 0 d\\hat{x}_0 d\\hat{x}_0 = \\iint (\\hat{x}_0 - \\theta{(\\hat{x}_0)}) d\\hat{x}_0 d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], [["minus", 1, "Function('\\\\theta')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\hat{x}_0', commutative=True)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\hat{x}_0', commutative=True)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(T,C)} = C + \\cos{(T)}, then obtain \\mathbf{s}{(T,C)} + \\int (- C + T + \\mathbf{s}{(T,C)}) dT = \\frac{T^{2}}{2} + \\mathbb{I} + \\mathbf{s}{(T,C)} + \\sin{(T)}", "derivation": "\\mathbf{s}{(T,C)} = C + \\cos{(T)} and T + \\mathbf{s}{(T,C)} = C + T + \\cos{(T)} and - C + T + \\mathbf{s}{(T,C)} = T + \\cos{(T)} and \\int (- C + T + \\mathbf{s}{(T,C)}) dT = \\int (T + \\cos{(T)}) dT and \\mathbf{s}{(T,C)} + \\int (- C + T + \\mathbf{s}{(T,C)}) dT = \\mathbf{s}{(T,C)} + \\int (T + \\cos{(T)}) dT and \\mathbf{s}{(T,C)} + \\int (- C + T + \\mathbf{s}{(T,C)}) dT = \\frac{T^{2}}{2} + \\mathbb{I} + \\mathbf{s}{(T,C)} + \\sin{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), cos(Symbol('T', commutative=True))))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))), Add(Symbol('C', commutative=True), Symbol('T', commutative=True), cos(Symbol('T', commutative=True))))"], [["minus", 2, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('T', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))), Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('T', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('T', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True)), Integral(Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('T', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Symbol('\\\\mathbb{I}', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('C', commutative=True)), sin(Symbol('T', commutative=True))))"]]}, {"prompt": "Given s{(r)} = \\sin{(r)}, then obtain \\frac{\\frac{d^{2}}{d r^{2}} \\int s^{r}{(r)} dr}{s^{r}{(r)} + \\int \\sin^{r}{(r)} dr} = \\frac{\\frac{d^{2}}{d r^{2}} \\int \\sin^{r}{(r)} dr}{s^{r}{(r)} + \\int \\sin^{r}{(r)} dr}", "derivation": "s{(r)} = \\sin{(r)} and s^{r}{(r)} = \\sin^{r}{(r)} and \\int s^{r}{(r)} dr = \\int \\sin^{r}{(r)} dr and \\frac{d}{d r} \\int s^{r}{(r)} dr = \\frac{d}{d r} \\int \\sin^{r}{(r)} dr and \\frac{d^{2}}{d r^{2}} \\int s^{r}{(r)} dr = \\frac{d^{2}}{d r^{2}} \\int \\sin^{r}{(r)} dr and \\frac{\\frac{d^{2}}{d r^{2}} \\int s^{r}{(r)} dr}{s^{r}{(r)} + \\int \\sin^{r}{(r)} dr} = \\frac{\\frac{d^{2}}{d r^{2}} \\int \\sin^{r}{(r)} dr}{s^{r}{(r)} + \\int \\sin^{r}{(r)} dr}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["divide", 5, "Add(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], "Equality(Mul(Pow(Add(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Integer(-1)), Derivative(Integral(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(2)))), Mul(Pow(Add(Pow(Function('s')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Integer(-1)), Derivative(Integral(Pow(sin(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then derive \\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} + \\cos{(\\mathbf{B})}, then obtain (\\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})})^{\\mathbf{B}} = (\\sin{(\\mathbf{B})} + \\cos{(\\mathbf{B})})^{\\mathbf{B}}", "derivation": "\\mathbf{s}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} and \\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} and \\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} + \\cos{(\\mathbf{B})} and (\\sin{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\mathbf{s}{(\\mathbf{B})})^{\\mathbf{B}} = (\\sin{(\\mathbf{B})} + \\cos{(\\mathbf{B})})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["add", 2, "sin(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(sin(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(A_{2})} = \\cos{(A_{2})} and \\mathbf{D}{(A_{2})} = \\cos{(A_{2})}, then derive \\frac{d}{d A_{2}} \\mathbf{D}{(A_{2})} = - \\sin{(A_{2})}, then obtain 0 = - \\sin{(A_{2})} - \\frac{d}{d A_{2}} \\operatorname{E_{\\lambda}}{(A_{2})}", "derivation": "\\operatorname{E_{\\lambda}}{(A_{2})} = \\cos{(A_{2})} and \\mathbf{D}{(A_{2})} = \\cos{(A_{2})} and \\operatorname{E_{\\lambda}}{(A_{2})} = \\mathbf{D}{(A_{2})} and \\frac{d}{d A_{2}} \\mathbf{D}{(A_{2})} = \\frac{d}{d A_{2}} \\cos{(A_{2})} and \\frac{d}{d A_{2}} \\mathbf{D}{(A_{2})} = - \\sin{(A_{2})} and \\frac{d}{d A_{2}} \\mathbf{D}{(A_{2})} - \\frac{d}{d A_{2}} \\cos{(A_{2})} = - \\sin{(A_{2})} - \\frac{d}{d A_{2}} \\cos{(A_{2})} and 0 = - \\sin{(A_{2})} - \\frac{d}{d A_{2}} \\mathbf{D}{(A_{2})} and 0 = - \\sin{(A_{2})} - \\frac{d}{d A_{2}} \\operatorname{E_{\\lambda}}{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_2', commutative=True))))"], [["minus", 5, "Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{D}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Mul(Integer(-1), Derivative(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\psi,l)} = \\log{(\\psi + l)}, then obtain (\\psi \\eta^{\\prime}{(\\psi,l)} + l \\log{(\\psi + l)})^{\\psi} = (\\psi \\log{(\\psi + l)} + l \\log{(\\psi + l)})^{\\psi}", "derivation": "\\eta^{\\prime}{(\\psi,l)} = \\log{(\\psi + l)} and l \\eta^{\\prime}{(\\psi,l)} = l \\log{(\\psi + l)} and \\psi \\eta^{\\prime}{(\\psi,l)} = \\psi \\log{(\\psi + l)} and \\psi \\eta^{\\prime}{(\\psi,l)} + l \\eta^{\\prime}{(\\psi,l)} = \\psi \\log{(\\psi + l)} + l \\eta^{\\prime}{(\\psi,l)} and \\psi \\eta^{\\prime}{(\\psi,l)} + l \\log{(\\psi + l)} = \\psi \\log{(\\psi + l)} + l \\log{(\\psi + l)} and (\\psi \\eta^{\\prime}{(\\psi,l)} + l \\log{(\\psi + l)})^{\\psi} = (\\psi \\log{(\\psi + l)} + l \\log{(\\psi + l)})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))))"], [["times", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))))"], [["add", 3, "Mul(Symbol('l', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))), Add(Mul(Symbol('\\\\psi', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))))), Add(Mul(Symbol('\\\\psi', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))))))"], [["power", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Symbol('\\\\psi', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi', commutative=True), Symbol('l', commutative=True))))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given A{(f^{*},\\phi)} = \\phi f^{*} and t{(f^{*},\\phi)} = \\phi f^{*}, then obtain (\\int \\phi f^{*} d\\phi)^{f^{*}} + \\int t{(f^{*},\\phi)} d\\phi = \\int \\phi f^{*} d\\phi + (\\int \\phi f^{*} d\\phi)^{f^{*}}", "derivation": "A{(f^{*},\\phi)} = \\phi f^{*} and t{(f^{*},\\phi)} = \\phi f^{*} and A{(f^{*},\\phi)} = t{(f^{*},\\phi)} and \\int A{(f^{*},\\phi)} d\\phi = \\int \\phi f^{*} d\\phi and \\int t{(f^{*},\\phi)} d\\phi = \\int \\phi f^{*} d\\phi and (\\int \\phi f^{*} d\\phi)^{f^{*}} + \\int t{(f^{*},\\phi)} d\\phi = \\int \\phi f^{*} d\\phi + (\\int \\phi f^{*} d\\phi)^{f^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Function('t')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('t')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 5, "Pow(Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('f^*', commutative=True))"], "Equality(Add(Pow(Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('f^*', commutative=True)), Integral(Function('t')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbb{I})} = e^{e^{\\mathbb{I}}}, then derive \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} + e^{\\mathbb{I}} \\frac{d}{d \\mathbb{I}} \\sigma_{p}{(\\mathbb{I})} = e^{2 \\mathbb{I}} e^{e^{\\mathbb{I}}} + e^{\\mathbb{I}} e^{e^{\\mathbb{I}}}, then obtain \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} + e^{\\mathbb{I}} \\frac{d}{d \\mathbb{I}} \\sigma_{p}{(\\mathbb{I})} = \\sigma_{p}{(\\mathbb{I})} e^{2 \\mathbb{I}} + \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}}", "derivation": "\\sigma_{p}{(\\mathbb{I})} = e^{e^{\\mathbb{I}}} and \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} = e^{\\mathbb{I}} e^{e^{\\mathbb{I}}} and \\frac{d}{d \\mathbb{I}} \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} e^{e^{\\mathbb{I}}} and \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} + e^{\\mathbb{I}} \\frac{d}{d \\mathbb{I}} \\sigma_{p}{(\\mathbb{I})} = e^{2 \\mathbb{I}} e^{e^{\\mathbb{I}}} + e^{\\mathbb{I}} e^{e^{\\mathbb{I}}} and \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}} + e^{\\mathbb{I}} \\frac{d}{d \\mathbb{I}} \\sigma_{p}{(\\mathbb{I})} = \\sigma_{p}{(\\mathbb{I})} e^{2 \\mathbb{I}} + \\sigma_{p}{(\\mathbb{I})} e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(exp(Symbol('\\\\mathbb{I}', commutative=True)), exp(exp(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('\\\\mathbb{I}', commutative=True)), exp(exp(Symbol('\\\\mathbb{I}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))), Add(Mul(exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True))), exp(exp(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(exp(Symbol('\\\\mathbb{I}', commutative=True)), exp(exp(Symbol('\\\\mathbb{I}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True))), Mul(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))), Add(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then derive 0 = - \\sin{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\operatorname{m_{s}}{(a^{\\dagger})}, then obtain 0 = - \\sin{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})}", "derivation": "\\operatorname{m_{s}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\operatorname{m_{s}}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})} and 0 = - \\frac{d}{d a^{\\dagger}} \\operatorname{m_{s}}{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})} and 0 = - \\sin{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\operatorname{m_{s}}{(a^{\\dagger})} and 0 = - \\sin{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('m_s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('m_s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(Function('m_s')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{p}{(\\nabla,B)} = \\cos{(B - \\nabla)}, then obtain \\frac{\\int - \\hat{p}{(\\nabla,B)} dB}{\\cos{(B - \\nabla)}} = \\frac{\\int - \\cos{(B - \\nabla)} dB}{\\cos{(B - \\nabla)}}", "derivation": "\\hat{p}{(\\nabla,B)} = \\cos{(B - \\nabla)} and - \\hat{p}{(\\nabla,B)} = - \\cos{(B - \\nabla)} and \\int - \\hat{p}{(\\nabla,B)} dB = \\int - \\cos{(B - \\nabla)} dB and \\frac{\\int - \\hat{p}{(\\nabla,B)} dB}{\\cos{(B - \\nabla)}} = \\frac{\\int - \\cos{(B - \\nabla)} dB}{\\cos{(B - \\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True)), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Tuple(Symbol('B', commutative=True))))"], [["divide", 3, "cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Pow(cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Integral(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\nabla', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Mul(Pow(cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Integral(Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given b{(g)} = \\cos{(g)} and \\dot{x}{(g)} = g \\cos{(g)}, then obtain g \\cos{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1} = g b{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1}", "derivation": "b{(g)} = \\cos{(g)} and g b{(g)} = g \\cos{(g)} and \\dot{x}{(g)} = g \\cos{(g)} and \\dot{x}{(g)} = g b{(g)} and \\dot{x}{(g)} - 1 = g b{(g)} - 1 and g \\cos{(g)} - 1 = g b{(g)} - 1 and g \\cos{(g)} - 1 = \\dot{x}{(g)} - 1 and g \\cos{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1} = \\dot{x}{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1} and g \\cos{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1} = g b{(g)} - 1 + \\frac{\\frac{g \\cos{(g)}}{y} - \\frac{1}{b{(g)}}}{g \\cos{(g)} - 1}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('b')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\dot{x}')(Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Function('b')(Symbol('g', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Function('\\\\dot{x}')(Symbol('g', commutative=True)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Function('b')(Symbol('g', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Function('b')(Symbol('g', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Add(Function('\\\\dot{x}')(Symbol('g', commutative=True)), Integer(-1)))"], [["add", 7, "Mul(Pow(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('b')(Symbol('g', commutative=True)), Integer(-1)))))"], "Equality(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1), Mul(Pow(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('b')(Symbol('g', commutative=True)), Integer(-1)))))), Add(Function('\\\\dot{x}')(Symbol('g', commutative=True)), Integer(-1), Mul(Pow(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('b')(Symbol('g', commutative=True)), Integer(-1)))))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1), Mul(Pow(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('b')(Symbol('g', commutative=True)), Integer(-1)))))), Add(Mul(Symbol('g', commutative=True), Function('b')(Symbol('g', commutative=True))), Integer(-1), Mul(Pow(Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Symbol('g', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('b')(Symbol('g', commutative=True)), Integer(-1)))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(F_{g})} = \\sin{(\\log{(F_{g})})}, then obtain ((\\mathbf{J}_M^{F_{g}}{(F_{g})})^{F_{g}} \\mathbf{J}_M{(F_{g})})^{F_{g}} = ((\\sin^{F_{g}}{(\\log{(F_{g})})})^{F_{g}} \\mathbf{J}_M{(F_{g})})^{F_{g}}", "derivation": "\\mathbf{J}_M{(F_{g})} = \\sin{(\\log{(F_{g})})} and \\mathbf{J}_M^{F_{g}}{(F_{g})} = \\sin^{F_{g}}{(\\log{(F_{g})})} and (\\mathbf{J}_M^{F_{g}}{(F_{g})})^{F_{g}} = (\\sin^{F_{g}}{(\\log{(F_{g})})})^{F_{g}} and (\\mathbf{J}_M^{F_{g}}{(F_{g})})^{F_{g}} \\mathbf{J}_M{(F_{g})} = (\\sin^{F_{g}}{(\\log{(F_{g})})})^{F_{g}} \\mathbf{J}_M{(F_{g})} and ((\\mathbf{J}_M^{F_{g}}{(F_{g})})^{F_{g}} \\mathbf{J}_M{(F_{g})})^{F_{g}} = ((\\sin^{F_{g}}{(\\log{(F_{g})})})^{F_{g}} \\mathbf{J}_M{(F_{g})})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), sin(log(Symbol('F_g', commutative=True))))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(sin(log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Pow(Pow(sin(log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)))"], [["times", 3, "Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True))), Mul(Pow(Pow(sin(log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True))))"], [["power", 4, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Pow(Mul(Pow(Pow(sin(log(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{J}_f,f_{E})} = \\mathbf{J}_f - f_{E}, then derive \\int (f_{E} + \\mathbf{M}{(\\mathbf{J}_f,f_{E})}) d\\mathbf{J}_f = \\hat{p}_0 + \\frac{\\mathbf{J}_f^{2}}{2}, then obtain \\int \\mathbf{J}_f d\\mathbf{J}_f = \\hat{p}_0 + \\frac{\\mathbf{J}_f^{2}}{2}", "derivation": "\\mathbf{M}{(\\mathbf{J}_f,f_{E})} = \\mathbf{J}_f - f_{E} and f_{E} + \\mathbf{M}{(\\mathbf{J}_f,f_{E})} = \\mathbf{J}_f and \\int (f_{E} + \\mathbf{M}{(\\mathbf{J}_f,f_{E})}) d\\mathbf{J}_f = \\int \\mathbf{J}_f d\\mathbf{J}_f and \\int (f_{E} + \\mathbf{M}{(\\mathbf{J}_f,f_{E})}) d\\mathbf{J}_f = \\hat{p}_0 + \\frac{\\mathbf{J}_f^{2}}{2} and \\int \\mathbf{J}_f d\\mathbf{J}_f = \\hat{p}_0 + \\frac{\\mathbf{J}_f^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('f_E', commutative=True))"], "Equality(Add(Symbol('f_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f_E', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Symbol('f_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Symbol('\\\\mathbf{J}_f', commutative=True), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('f_E', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Symbol('\\\\mathbf{J}_f', commutative=True), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\nabla{(k,\\rho)} = e^{k^{\\rho}}, then obtain \\frac{\\partial}{\\partial k} \\int \\nabla{(k,\\rho)} dk = \\frac{\\partial}{\\partial k} (F_{N} + \\frac{e^{- \\frac{i \\pi}{\\rho}} \\Gamma(\\frac{1}{\\rho}) \\gamma(\\frac{1}{\\rho}, k^{\\rho} e^{i \\pi})}{\\rho^{2} \\Gamma(1 + \\frac{1}{\\rho})})", "derivation": "\\nabla{(k,\\rho)} = e^{k^{\\rho}} and \\int \\nabla{(k,\\rho)} dk = \\int e^{k^{\\rho}} dk and \\frac{\\partial}{\\partial k} \\int \\nabla{(k,\\rho)} dk = \\frac{\\partial}{\\partial k} \\int e^{k^{\\rho}} dk and \\frac{\\partial}{\\partial k} \\int \\nabla{(k,\\rho)} dk = \\frac{\\partial}{\\partial k} (F_{N} + \\frac{e^{- \\frac{i \\pi}{\\rho}} \\Gamma(\\frac{1}{\\rho}) \\gamma(\\frac{1}{\\rho}, k^{\\rho} e^{i \\pi})}{\\rho^{2} \\Gamma(1 + \\frac{1}{\\rho})})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), exp(Pow(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Integral(exp(Pow(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('F_N', commutative=True), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), exp(Mul(Integer(-1), I, pi, Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), gamma(Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Pow(gamma(Add(Integer(1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Integer(-1)), lowergamma(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Mul(Pow(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), exp_polar(Mul(I, pi)))))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(\\varepsilon_0,g_{\\varepsilon})} = \\varepsilon_0 + g_{\\varepsilon}, then obtain \\frac{\\frac{\\partial}{\\partial \\varepsilon_0} \\hat{H}{(\\varepsilon_0,g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{1}{g_{\\varepsilon}}", "derivation": "\\hat{H}{(\\varepsilon_0,g_{\\varepsilon})} = \\varepsilon_0 + g_{\\varepsilon} and \\frac{\\hat{H}{(\\varepsilon_0,g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{\\varepsilon_0 + g_{\\varepsilon}}{g_{\\varepsilon}} and \\frac{\\partial}{\\partial \\varepsilon_0} \\frac{\\hat{H}{(\\varepsilon_0,g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{\\partial}{\\partial \\varepsilon_0} \\frac{\\varepsilon_0 + g_{\\varepsilon}}{g_{\\varepsilon}} and \\frac{\\frac{\\partial}{\\partial \\varepsilon_0} \\hat{H}{(\\varepsilon_0,g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{1}{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given x{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain \\frac{d}{d x^\\prime} \\int \\sin{(x{(x^\\prime)})} dx^\\prime = \\frac{d}{d x^\\prime} \\int \\sin{(\\cos{(x^\\prime)})} dx^\\prime", "derivation": "x{(x^\\prime)} = \\cos{(x^\\prime)} and \\sin{(x{(x^\\prime)})} = \\sin{(\\cos{(x^\\prime)})} and \\int \\sin{(x{(x^\\prime)})} dx^\\prime = \\int \\sin{(\\cos{(x^\\prime)})} dx^\\prime and \\frac{d}{d x^\\prime} \\int \\sin{(x{(x^\\prime)})} dx^\\prime = \\frac{d}{d x^\\prime} \\int \\sin{(\\cos{(x^\\prime)})} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["sin", 1], "Equality(sin(Function('x')(Symbol('x^\\\\prime', commutative=True))), sin(cos(Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(sin(Function('x')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(sin(Function('x')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(sin(cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\hat{H},\\mathbf{f})} = \\hat{H} \\mathbf{f}, then obtain (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} - H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} = (\\hat{H} \\mathbf{f})^{\\mathbf{f}} (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) - H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})}", "derivation": "H{(\\hat{H},\\mathbf{f})} = \\hat{H} \\mathbf{f} and H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} = (\\hat{H} \\mathbf{f})^{\\mathbf{f}} and (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} = (\\hat{H} \\mathbf{f})^{\\mathbf{f}} (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) and (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} - H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})} = (\\hat{H} \\mathbf{f})^{\\mathbf{f}} (- \\dot{z} - \\mathbf{f} + h{(\\hat{H}_l,\\dot{z})}) - H^{\\mathbf{f}}{(\\hat{H},\\mathbf{f})}", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True)))))"], [["minus", 3, "Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(-1), Pow(Function('H')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(a^{\\dagger})} = \\sin{(\\sin{(a^{\\dagger})})}, then obtain (\\hat{x}_0{(a^{\\dagger})} - \\sin{(\\sin{(a^{\\dagger})})})^{a^{\\dagger}} + \\sin{(\\sin{(a^{\\dagger})})} = 0^{a^{\\dagger}} + \\sin{(\\sin{(a^{\\dagger})})}", "derivation": "\\hat{x}_0{(a^{\\dagger})} = \\sin{(\\sin{(a^{\\dagger})})} and \\hat{x}_0{(a^{\\dagger})} - \\sin{(\\sin{(a^{\\dagger})})} = 0 and (\\hat{x}_0{(a^{\\dagger})} - \\sin{(\\sin{(a^{\\dagger})})})^{a^{\\dagger}} = 0^{a^{\\dagger}} and (\\hat{x}_0{(a^{\\dagger})} - \\sin{(\\sin{(a^{\\dagger})})})^{a^{\\dagger}} + \\sin{(\\sin{(a^{\\dagger})})} = 0^{a^{\\dagger}} + \\sin{(\\sin{(a^{\\dagger})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('a^{\\\\dagger}', commutative=True)), sin(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 1, "sin(sin(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('a^{\\\\dagger}', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{x}_0')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('a^{\\\\dagger}', commutative=True))))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 3, "sin(sin(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Pow(Add(Function('\\\\hat{x}_0')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('a^{\\\\dagger}', commutative=True))))), Symbol('a^{\\\\dagger}', commutative=True)), sin(sin(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Pow(Integer(0), Symbol('a^{\\\\dagger}', commutative=True)), sin(sin(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(P_{g})} = \\cos{(P_{g})}, then obtain - P_{g} + \\iint (P_{g} + \\mu_{0}^{P_{g}}{(P_{g})}) dP_{g} dP_{g} = - P_{g} + \\iint (P_{g} + \\cos^{P_{g}}{(P_{g})}) dP_{g} dP_{g}", "derivation": "\\mu_{0}{(P_{g})} = \\cos{(P_{g})} and \\mu_{0}^{P_{g}}{(P_{g})} = \\cos^{P_{g}}{(P_{g})} and P_{g} + \\mu_{0}^{P_{g}}{(P_{g})} = P_{g} + \\cos^{P_{g}}{(P_{g})} and \\int (P_{g} + \\mu_{0}^{P_{g}}{(P_{g})}) dP_{g} = \\int (P_{g} + \\cos^{P_{g}}{(P_{g})}) dP_{g} and \\iint (P_{g} + \\mu_{0}^{P_{g}}{(P_{g})}) dP_{g} dP_{g} = \\iint (P_{g} + \\cos^{P_{g}}{(P_{g})}) dP_{g} dP_{g} and - P_{g} + \\iint (P_{g} + \\mu_{0}^{P_{g}}{(P_{g})}) dP_{g} dP_{g} = - P_{g} + \\iint (P_{g} + \\cos^{P_{g}}{(P_{g})}) dP_{g} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True)))"], [["power", 1, "Symbol('P_g', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)))"], [["add", 2, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Pow(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Add(Symbol('P_g', commutative=True), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))))"], [["integrate", 3, "Symbol('P_g', commutative=True)"], "Equality(Integral(Add(Symbol('P_g', commutative=True), Pow(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["integrate", 4, "Symbol('P_g', commutative=True)"], "Equality(Integral(Add(Symbol('P_g', commutative=True), Pow(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["minus", 5, "Symbol('P_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(Add(Symbol('P_g', commutative=True), Pow(Function('\\\\mu_0')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Integral(Add(Symbol('P_g', commutative=True), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})} = F_{g} + \\hat{H}_{\\lambda} and \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})} = 2 \\hat{H}_{\\lambda}, then obtain e^{F_{g} + \\hat{H}_{\\lambda} + \\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})}} = e^{2 F_{g} + \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}", "derivation": "\\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})} = F_{g} + \\hat{H}_{\\lambda} and F_{g} + \\hat{H}_{\\lambda} + \\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})} = 2 F_{g} + 2 \\hat{H}_{\\lambda} and \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})} = 2 \\hat{H}_{\\lambda} and F_{g} + \\hat{H}_{\\lambda} + \\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})} = 2 F_{g} + \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})} and e^{F_{g} + \\hat{H}_{\\lambda} + \\hat{\\mathbf{x}}{(F_{g},\\hat{H}_{\\lambda})}} = e^{2 F_{g} + \\operatorname{m_{s}}{(\\hat{H}_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["add", 1, "Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["exp", 4], "Equality(exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), exp(Add(Mul(Integer(2), Symbol('F_g', commutative=True)), Function('m_s')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given z{(\\pi,\\Omega)} = \\int \\pi^{\\Omega} d\\Omega and \\operatorname{F_{g}}{(\\pi)} = \\pi, then obtain \\pi - \\pi^{\\Omega} + z{(\\pi,\\Omega)} - \\int \\pi^{\\Omega} d\\Omega = \\pi - \\pi^{\\Omega}", "derivation": "z{(\\pi,\\Omega)} = \\int \\pi^{\\Omega} d\\Omega and \\operatorname{F_{g}}{(\\pi)} = \\pi and - \\pi^{\\Omega} + \\operatorname{F_{g}}{(\\pi)} = \\pi - \\pi^{\\Omega} and z{(\\pi,\\Omega)} - \\int \\pi^{\\Omega} d\\Omega = 0 and - \\pi^{\\Omega} + \\operatorname{F_{g}}{(\\pi)} + z{(\\pi,\\Omega)} - \\int \\pi^{\\Omega} d\\Omega = - \\pi^{\\Omega} + \\operatorname{F_{g}}{(\\pi)} and \\pi - \\pi^{\\Omega} + z{(\\pi,\\Omega)} - \\int \\pi^{\\Omega} d\\Omega = \\pi - \\pi^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["add", 2, "Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('F_g')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["minus", 1, "Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Integer(0))"], [["add", 4, "Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('F_g')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('F_g')(Symbol('\\\\pi', commutative=True)), Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('F_g')(Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\hat{x}_0,\\ddot{x})} = \\log{(\\frac{\\hat{x}_0}{\\ddot{x}})} and \\operatorname{C_{1}}{(\\hat{x}_0,\\ddot{x})} = \\int \\frac{\\varphi{(\\hat{x}_0,\\ddot{x})}}{\\log{(\\frac{\\hat{x}_0}{\\ddot{x}})}} d\\hat{x}_0, then derive \\operatorname{C_{1}}{(\\hat{x}_0,\\ddot{x})} = \\hat{x}_0 + \\tilde{g}^*, then obtain \\int \\frac{\\varphi{(\\hat{x}_0,\\ddot{x})}}{\\log{(\\frac{\\hat{x}_0}{\\ddot{x}})}} d\\hat{x}_0 = \\hat{x}_0 + \\tilde{g}^*", "derivation": "\\varphi{(\\hat{x}_0,\\ddot{x})} = \\log{(\\frac{\\hat{x}_0}{\\ddot{x}})} and \\operatorname{C_{1}}{(\\hat{x}_0,\\ddot{x})} = \\int \\frac{\\varphi{(\\hat{x}_0,\\ddot{x})}}{\\log{(\\frac{\\hat{x}_0}{\\ddot{x}})}} d\\hat{x}_0 and \\operatorname{C_{1}}{(\\hat{x}_0,\\ddot{x})} = \\int 1 d\\hat{x}_0 and \\operatorname{C_{1}}{(\\hat{x}_0,\\ddot{x})} = \\hat{x}_0 + \\tilde{g}^* and \\int \\frac{\\varphi{(\\hat{x}_0,\\ddot{x})}}{\\log{(\\frac{\\hat{x}_0}{\\ddot{x}})}} d\\hat{x}_0 = \\hat{x}_0 + \\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Mul(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Function('C_1')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(m,G)} = - m + \\sin{(G)}, then obtain \\frac{\\int (- m + \\operatorname{f_{\\mathbf{p}}}{(m,G)}) dm}{- m + \\sin{(G)}} = \\frac{\\int (- 2 m + \\sin{(G)}) dm}{- m + \\sin{(G)}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(m,G)} = - m + \\sin{(G)} and - m + \\operatorname{f_{\\mathbf{p}}}{(m,G)} = - 2 m + \\sin{(G)} and \\int (- m + \\operatorname{f_{\\mathbf{p}}}{(m,G)}) dm = \\int (- 2 m + \\sin{(G)}) dm and \\frac{\\int (- m + \\operatorname{f_{\\mathbf{p}}}{(m,G)}) dm}{- m + \\sin{(G)}} = \\frac{\\int (- 2 m + \\sin{(G)}) dm}{- m + \\sin{(G)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('m', commutative=True)), sin(Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(t)} = \\sin{(t)} and u{(t)} = \\sin{(t)}, then derive \\frac{d}{d t} \\varphi^{*}{(t)} = \\cos{(t)}, then obtain (2 \\cos{(t)})^{t} = (\\cos{(t)} + \\frac{d}{d t} u{(t)})^{t}", "derivation": "\\varphi^{*}{(t)} = \\sin{(t)} and \\frac{d}{d t} \\varphi^{*}{(t)} = \\frac{d}{d t} \\sin{(t)} and \\frac{d}{d t} \\varphi^{*}{(t)} = \\cos{(t)} and u{(t)} = \\sin{(t)} and \\varphi^{*}{(t)} = u{(t)} and \\frac{d}{d t} \\varphi^{*}{(t)} = \\frac{d}{d t} u{(t)} and \\cos{(t)} + \\frac{d}{d t} \\varphi^{*}{(t)} = \\cos{(t)} + \\frac{d}{d t} u{(t)} and (\\cos{(t)} + \\frac{d}{d t} \\varphi^{*}{(t)})^{t} = (\\cos{(t)} + \\frac{d}{d t} u{(t)})^{t} and (2 \\cos{(t)})^{t} = (\\cos{(t)} + \\frac{d}{d t} u{(t)})^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), cos(Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Function('u')(Symbol('t', commutative=True)))"], [["differentiate", 5, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Function('u')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["add", 6, "cos(Symbol('t', commutative=True))"], "Equality(Add(cos(Symbol('t', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Add(cos(Symbol('t', commutative=True)), Derivative(Function('u')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["power", 7, "Symbol('t', commutative=True)"], "Equality(Pow(Add(cos(Symbol('t', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Symbol('t', commutative=True)), Pow(Add(cos(Symbol('t', commutative=True)), Derivative(Function('u')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 3], "Equality(Pow(Mul(Integer(2), cos(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Add(cos(Symbol('t', commutative=True)), Derivative(Function('u')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})} = A_{z}^{\\mathbf{J}_f}, then obtain \\int (- A_{z} + A_{z}^{\\mathbf{J}_f}) d\\mathbf{J}_f + \\int (- A_{z} + \\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})}) d\\mathbf{J}_f = 2 \\int (- A_{z} + A_{z}^{\\mathbf{J}_f}) d\\mathbf{J}_f", "derivation": "\\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})} = A_{z}^{\\mathbf{J}_f} and - A_{z} + \\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})} = - A_{z} + A_{z}^{\\mathbf{J}_f} and \\int (- A_{z} + \\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})}) d\\mathbf{J}_f = \\int (- A_{z} + A_{z}^{\\mathbf{J}_f}) d\\mathbf{J}_f and \\int (- A_{z} + A_{z}^{\\mathbf{J}_f}) d\\mathbf{J}_f + \\int (- A_{z} + \\operatorname{v_{x}}{(\\mathbf{J}_f,A_{z})}) d\\mathbf{J}_f = 2 \\int (- A_{z} + A_{z}^{\\mathbf{J}_f}) d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 1, "Symbol('A_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(F_{g})} = e^{F_{g}}, then obtain e^{\\mathbf{p}{(F_{g})} + \\int 1 dF_{g}} = e^{e^{F_{g}} + \\int 1 dF_{g}}", "derivation": "\\mathbf{p}{(F_{g})} = e^{F_{g}} and 1 = \\frac{e^{F_{g}}}{\\mathbf{p}{(F_{g})}} and \\int 1 dF_{g} = \\int \\frac{e^{F_{g}}}{\\mathbf{p}{(F_{g})}} dF_{g} and \\mathbf{p}{(F_{g})} + \\int \\frac{e^{F_{g}}}{\\mathbf{p}{(F_{g})}} dF_{g} = e^{F_{g}} + \\int \\frac{e^{F_{g}}}{\\mathbf{p}{(F_{g})}} dF_{g} and \\mathbf{p}{(F_{g})} + \\int 1 dF_{g} = e^{F_{g}} + \\int 1 dF_{g} and e^{\\mathbf{p}{(F_{g})} + \\int 1 dF_{g}} = e^{e^{F_{g}} + \\int 1 dF_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integer(-1)), exp(Symbol('F_g', commutative=True))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integer(-1)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["add", 1, "Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integer(-1)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integer(-1)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))), Add(exp(Symbol('F_g', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integer(-1)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True)))), Add(exp(Symbol('F_g', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True)))))"], [["exp", 5], "Equality(exp(Add(Function('\\\\mathbf{p}')(Symbol('F_g', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))), exp(Add(exp(Symbol('F_g', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))))"]]}, {"prompt": "Given b{(\\theta_1,\\mathbf{J})} = - \\mathbf{J} + \\theta_1 and u{(\\theta_1,\\mathbf{J})} = \\theta_1 + b{(\\theta_1,\\mathbf{J})}, then obtain (- \\mathbf{J} + 2 \\theta_1) b{(\\theta_1,\\mathbf{J})} = (\\theta_1 + b{(\\theta_1,\\mathbf{J})}) b{(\\theta_1,\\mathbf{J})}", "derivation": "b{(\\theta_1,\\mathbf{J})} = - \\mathbf{J} + \\theta_1 and \\theta_1 + b{(\\theta_1,\\mathbf{J})} = - \\mathbf{J} + 2 \\theta_1 and u{(\\theta_1,\\mathbf{J})} = \\theta_1 + b{(\\theta_1,\\mathbf{J})} and u{(\\theta_1,\\mathbf{J})} = - \\mathbf{J} + 2 \\theta_1 and (- \\mathbf{J} + \\theta_1) u{(\\theta_1,\\mathbf{J})} = (- \\mathbf{J} + \\theta_1) (\\theta_1 + b{(\\theta_1,\\mathbf{J})}) and b{(\\theta_1,\\mathbf{J})} u{(\\theta_1,\\mathbf{J})} = (\\theta_1 + b{(\\theta_1,\\mathbf{J})}) b{(\\theta_1,\\mathbf{J})} and (- \\mathbf{J} + 2 \\theta_1) b{(\\theta_1,\\mathbf{J})} = (\\theta_1 + b{(\\theta_1,\\mathbf{J})}) b{(\\theta_1,\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('u')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Function('u')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Function('u')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True))), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Function('b')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(t)} = \\cos{(t)} and \\operatorname{m_{s}}{(\\tilde{g})} = e^{\\tilde{g}}, then obtain \\operatorname{m_{s}}{(\\tilde{g})} - \\frac{\\operatorname{m_{s}}{(\\tilde{g})}}{\\cos^{2}{(t)}} = \\operatorname{m_{s}}{(\\tilde{g})} - \\frac{e^{\\tilde{g}}}{\\cos^{2}{(t)}}", "derivation": "\\mathbf{E}{(t)} = \\cos{(t)} and \\operatorname{m_{s}}{(\\tilde{g})} = e^{\\tilde{g}} and - \\frac{\\operatorname{m_{s}}{(\\tilde{g})}}{\\mathbf{E}{(t)} \\cos{(t)}} = - \\frac{e^{\\tilde{g}}}{\\mathbf{E}{(t)} \\cos{(t)}} and - \\frac{\\operatorname{m_{s}}{(\\tilde{g})}}{\\mathbf{E}^{2}{(t)}} = - \\frac{e^{\\tilde{g}}}{\\mathbf{E}^{2}{(t)}} and - \\frac{\\operatorname{m_{s}}{(\\tilde{g})}}{\\cos^{2}{(t)}} = - \\frac{e^{\\tilde{g}}}{\\cos^{2}{(t)}} and \\operatorname{m_{s}}{(\\tilde{g})} - \\frac{\\operatorname{m_{s}}{(\\tilde{g})}}{\\cos^{2}{(t)}} = \\operatorname{m_{s}}{(\\tilde{g})} - \\frac{e^{\\tilde{g}}}{\\cos^{2}{(t)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], ["get_premise", "Equality(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), Integer(-1)), Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), Integer(-2)), Function('m_s')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{E}')(Symbol('t', commutative=True)), Integer(-2)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-2))), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-2))))"], [["add", 5, "Function('m_s')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-2)))), Add(Function('m_s')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given u{(\\psi^*)} = \\log{(\\psi^*)} and \\operatorname{m_{s}}{(\\psi^*)} = u{(\\psi^*)} \\log{(\\psi^*)}, then obtain - \\operatorname{m_{s}}{(\\psi^*)} + \\int \\operatorname{m_{s}}{(\\psi^*)} d\\psi^* = - \\operatorname{m_{s}}{(\\psi^*)} + \\int \\log{(\\psi^*)}^{2} d\\psi^*", "derivation": "u{(\\psi^*)} = \\log{(\\psi^*)} and u{(\\psi^*)} \\log{(\\psi^*)} = \\log{(\\psi^*)}^{2} and \\operatorname{m_{s}}{(\\psi^*)} = u{(\\psi^*)} \\log{(\\psi^*)} and \\operatorname{m_{s}}{(\\psi^*)} = \\log{(\\psi^*)}^{2} and \\int \\operatorname{m_{s}}{(\\psi^*)} d\\psi^* = \\int \\log{(\\psi^*)}^{2} d\\psi^* and - \\operatorname{m_{s}}{(\\psi^*)} + \\int \\operatorname{m_{s}}{(\\psi^*)} d\\psi^* = - \\operatorname{m_{s}}{(\\psi^*)} + \\int \\log{(\\psi^*)}^{2} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "log(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('u')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Mul(Function('u')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"], [["integrate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 5, "Function('m_s')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\psi^*', commutative=True))), Integral(Function('m_s')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(M)} = \\log{(M)}, then derive \\frac{d}{d M} \\theta_{2}{(M)} = \\frac{1}{M}, then obtain \\log{(M)} \\frac{d}{d M} \\log{(M)} = \\log{(M)} \\frac{d}{d M} \\theta_{2}{(M)}", "derivation": "\\theta_{2}{(M)} = \\log{(M)} and \\frac{d}{d M} \\theta_{2}{(M)} = \\frac{d}{d M} \\log{(M)} and \\frac{d}{d M} \\theta_{2}{(M)} = \\frac{1}{M} and \\frac{1}{M} = \\frac{d}{d M} \\log{(M)} and \\frac{\\log{(M)}}{M} = \\log{(M)} \\frac{d}{d M} \\log{(M)} and \\frac{\\log{(M)}}{M} = \\log{(M)} \\frac{d}{d M} \\theta_{2}{(M)} and \\log{(M)} \\frac{d}{d M} \\log{(M)} = \\log{(M)} \\frac{d}{d M} \\theta_{2}{(M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 4, "log(Symbol('M', commutative=True))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), log(Symbol('M', commutative=True))), Mul(log(Symbol('M', commutative=True)), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), log(Symbol('M', commutative=True))), Mul(log(Symbol('M', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(log(Symbol('M', commutative=True)), Derivative(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(log(Symbol('M', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(t)} = \\log{(t)}, then obtain \\pi^{2}{(t)} + \\int 1 dt = \\log{(t)}^{2} + \\int 1 dt", "derivation": "\\pi{(t)} = \\log{(t)} and \\pi^{2}{(t)} = \\pi{(t)} \\log{(t)} and 1 = \\frac{\\log{(t)}}{\\pi{(t)}} and \\pi{(t)} \\log{(t)} = \\log{(t)}^{2} and \\pi^{2}{(t)} = \\log{(t)}^{2} and \\pi^{2}{(t)} + \\int 1 dt = \\log{(t)}^{2} + \\int 1 dt", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["times", 1, "Function('\\\\pi')(Symbol('t', commutative=True))"], "Equality(Pow(Function('\\\\pi')(Symbol('t', commutative=True)), Integer(2)), Mul(Function('\\\\pi')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\pi')(Symbol('t', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('\\\\pi')(Symbol('t', commutative=True)), Integer(-1)), log(Symbol('t', commutative=True))))"], [["times", 3, "Mul(Function('\\\\pi')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('\\\\pi')(Symbol('t', commutative=True)), Integer(2)), Pow(log(Symbol('t', commutative=True)), Integer(2)))"], [["add", 5, "Integral(Integer(1), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Pow(Function('\\\\pi')(Symbol('t', commutative=True)), Integer(2)), Integral(Integer(1), Tuple(Symbol('t', commutative=True)))), Add(Pow(log(Symbol('t', commutative=True)), Integer(2)), Integral(Integer(1), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\nabla,C)} = \\cos^{C}{(\\nabla)} and m{(C_{1})} = \\sin{(C_{1})}, then obtain (- \\varepsilon_{0}{(\\nabla,C)} + m{(C_{1})})^{C} = (- \\varepsilon_{0}{(\\nabla,C)} + \\sin{(C_{1})})^{C}", "derivation": "\\varepsilon_{0}{(\\nabla,C)} = \\cos^{C}{(\\nabla)} and m{(C_{1})} = \\sin{(C_{1})} and - \\varepsilon_{0}{(\\nabla,C)} + m{(C_{1})} = - \\varepsilon_{0}{(\\nabla,C)} + \\sin{(C_{1})} and m{(C_{1})} - \\cos^{C}{(\\nabla)} = \\sin{(C_{1})} - \\cos^{C}{(\\nabla)} and (m{(C_{1})} - \\cos^{C}{(\\nabla)})^{C} = (\\sin{(C_{1})} - \\cos^{C}{(\\nabla)})^{C} and (- \\varepsilon_{0}{(\\nabla,C)} + m{(C_{1})})^{C} = (- \\varepsilon_{0}{(\\nabla,C)} + \\sin{(C_{1})})^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True)), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('C', commutative=True)))"], ["get_premise", "Equality(Function('m')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["minus", 2, "Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), Function('m')(Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), sin(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('m')(Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('C', commutative=True)))), Add(sin(Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('C', commutative=True)))))"], [["power", 4, "Symbol('C', commutative=True)"], "Equality(Pow(Add(Function('m')(Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('C', commutative=True)))), Symbol('C', commutative=True)), Pow(Add(sin(Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\nabla', commutative=True)), Symbol('C', commutative=True)))), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), Function('m')(Symbol('C_1', commutative=True))), Symbol('C', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\nabla', commutative=True), Symbol('C', commutative=True))), sin(Symbol('C_1', commutative=True))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given G{(x,\\mu)} = \\cos{(\\mu - x)}, then derive \\int G{(x,\\mu)} d\\mu = f_{\\mathbf{p}} + \\sin{(\\mu - x)}, then derive \\mathbf{p} + \\sin{(\\mu - x)} = f_{\\mathbf{p}} + \\sin{(\\mu - x)}, then obtain \\mathbf{p} + \\sin{(\\mu - x)} = \\int \\cos{(\\mu - x)} d\\mu", "derivation": "G{(x,\\mu)} = \\cos{(\\mu - x)} and \\int G{(x,\\mu)} d\\mu = \\int \\cos{(\\mu - x)} d\\mu and \\int G{(x,\\mu)} d\\mu = f_{\\mathbf{p}} + \\sin{(\\mu - x)} and \\int \\cos{(\\mu - x)} d\\mu = f_{\\mathbf{p}} + \\sin{(\\mu - x)} and \\mathbf{p} + \\sin{(\\mu - x)} = f_{\\mathbf{p}} + \\sin{(\\mu - x)} and \\mathbf{p} + \\sin{(\\mu - x)} = \\int \\cos{(\\mu - x)} d\\mu", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('G')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(cos(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('x', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))), Integral(cos(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(v_{t})} = \\frac{d}{d v_{t}} e^{v_{t}}, then derive \\dot{y}{(v_{t})} = e^{v_{t}}, then obtain (\\frac{d^{3}}{d v_{t}^{3}} e^{v_{t}})^{v_{t}} = (\\frac{d}{d v_{t}} e^{v_{t}})^{v_{t}}", "derivation": "\\dot{y}{(v_{t})} = \\frac{d}{d v_{t}} e^{v_{t}} and \\dot{y}{(v_{t})} = e^{v_{t}} and \\frac{d}{d v_{t}} \\dot{y}{(v_{t})} = \\frac{d}{d v_{t}} e^{v_{t}} and (\\frac{d}{d v_{t}} \\dot{y}{(v_{t})})^{v_{t}} = (\\frac{d}{d v_{t}} e^{v_{t}})^{v_{t}} and e^{v_{t}} = \\frac{d}{d v_{t}} e^{v_{t}} and \\dot{y}{(v_{t})} = \\frac{d^{2}}{d v_{t}^{2}} e^{v_{t}} and (\\frac{d^{3}}{d v_{t}^{3}} e^{v_{t}})^{v_{t}} = (\\frac{d}{d v_{t}} e^{v_{t}})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)), Pow(Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('v_t', commutative=True)), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(3))), Symbol('v_t', commutative=True)), Pow(Derivative(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given a{(q)} = \\cos{(\\sin{(q)})}, then obtain \\frac{- q - a{(q)}}{\\sin{(q)}} = \\frac{- q - \\cos{(\\sin{(q)})}}{\\sin{(q)}}", "derivation": "a{(q)} = \\cos{(\\sin{(q)})} and q + a{(q)} = q + \\cos{(\\sin{(q)})} and \\frac{q + a{(q)}}{\\sin{(q)}} = \\frac{q + \\cos{(\\sin{(q)})}}{\\sin{(q)}} and - \\frac{q + a{(q)}}{\\sin{(q)}} = - \\frac{q + \\cos{(\\sin{(q)})}}{\\sin{(q)}} and \\frac{- q - a{(q)}}{\\sin{(q)}} = \\frac{- q - \\cos{(\\sin{(q)})}}{\\sin{(q)}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('q', commutative=True)), cos(sin(Symbol('q', commutative=True))))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('a')(Symbol('q', commutative=True))), Add(Symbol('q', commutative=True), cos(sin(Symbol('q', commutative=True)))))"], [["divide", 2, "sin(Symbol('q', commutative=True))"], "Equality(Mul(Add(Symbol('q', commutative=True), Function('a')(Symbol('q', commutative=True))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Mul(Add(Symbol('q', commutative=True), cos(sin(Symbol('q', commutative=True)))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Symbol('q', commutative=True), Function('a')(Symbol('q', commutative=True))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('q', commutative=True), cos(sin(Symbol('q', commutative=True)))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Function('a')(Symbol('q', commutative=True)))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('q', commutative=True))))), Pow(sin(Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{F}{(n)} = \\log{(n)}, then derive \\frac{d}{d n} \\mathbf{F}{(n)} + \\frac{1}{n} = \\frac{2}{n}, then obtain 2 \\log{(n)} + \\frac{d}{d n} \\log{(n)} + \\frac{1}{n} = 2 \\log{(n)} + \\frac{2}{n}", "derivation": "\\mathbf{F}{(n)} = \\log{(n)} and \\mathbf{F}{(n)} + \\log{(n)} = 2 \\log{(n)} and \\frac{d}{d n} (\\mathbf{F}{(n)} + \\log{(n)}) = \\frac{d}{d n} 2 \\log{(n)} and \\frac{d}{d n} \\mathbf{F}{(n)} + \\frac{1}{n} = \\frac{2}{n} and \\frac{d}{d n} \\log{(n)} + \\frac{1}{n} = \\frac{2}{n} and 2 \\log{(n)} + \\frac{d}{d n} \\log{(n)} + \\frac{1}{n} = 2 \\log{(n)} + \\frac{2}{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["add", 1, "log(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Mul(Integer(2), log(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{F}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["add", 5, "Mul(Integer(2), log(Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(2), log(Symbol('n', commutative=True))), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Mul(Integer(2), log(Symbol('n', commutative=True))), Mul(Integer(2), Pow(Symbol('n', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given v{(I,T)} = (e^{I})^{T}, then obtain v{(I,T)} + v^{T}{(I,T)} e^{I} = ((e^{I})^{T})^{T} e^{I} + v{(I,T)}", "derivation": "v{(I,T)} = (e^{I})^{T} and v^{T}{(I,T)} = ((e^{I})^{T})^{T} and v^{T}{(I,T)} e^{I} = ((e^{I})^{T})^{T} e^{I} and v{(I,T)} + v^{T}{(I,T)} e^{I} = ((e^{I})^{T})^{T} e^{I} + v{(I,T)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True)), Pow(exp(Symbol('I', commutative=True)), Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(exp(Symbol('I', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["times", 2, "exp(Symbol('I', commutative=True))"], "Equality(Mul(Pow(Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Symbol('I', commutative=True))), Mul(Pow(Pow(exp(Symbol('I', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Symbol('I', commutative=True))))"], [["add", 3, "Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Symbol('I', commutative=True)))), Add(Mul(Pow(Pow(exp(Symbol('I', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)), exp(Symbol('I', commutative=True))), Function('v')(Symbol('I', commutative=True), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\lambda,\\dot{z})} = - \\dot{z} + \\lambda, then obtain \\lambda (\\frac{4 \\dot{z} - 4 \\lambda + 4 \\varepsilon{(\\lambda,\\dot{z})}}{\\lambda})^{\\dot{z}} = 0^{\\dot{z}} \\lambda", "derivation": "\\varepsilon{(\\lambda,\\dot{z})} = - \\dot{z} + \\lambda and \\dot{z} - \\lambda + \\varepsilon{(\\lambda,\\dot{z})} = 0 and \\dot{z} - \\lambda + 2 \\varepsilon{(\\lambda,\\dot{z})} = \\varepsilon{(\\lambda,\\dot{z})} and 2 \\dot{z} - 2 \\lambda + 2 \\varepsilon{(\\lambda,\\dot{z})} = 0 and \\frac{2 \\dot{z} - 2 \\lambda + 2 \\varepsilon{(\\lambda,\\dot{z})}}{\\lambda} = 0 and \\frac{4 \\dot{z} - 4 \\lambda + 4 \\varepsilon{(\\lambda,\\dot{z})}}{\\lambda} = 0 and (\\frac{4 \\dot{z} - 4 \\lambda + 4 \\varepsilon{(\\lambda,\\dot{z})}}{\\lambda})^{\\dot{z}} = 0^{\\dot{z}} and \\lambda (\\frac{4 \\dot{z} - 4 \\lambda + 4 \\varepsilon{(\\lambda,\\dot{z})}}{\\lambda})^{\\dot{z}} = 0^{\\dot{z}} \\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Integer(0))"], [["add", 1, "Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Integer(0))"], [["divide", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(4), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('\\\\lambda', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True))))), Integer(0))"], [["power", 6, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(4), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('\\\\lambda', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True))))), Symbol('\\\\dot{z}', commutative=True)), Pow(Integer(0), Symbol('\\\\dot{z}', commutative=True)))"], [["times", 7, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Mul(Integer(4), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Integer(4), Symbol('\\\\lambda', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{z}', commutative=True))))), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(b,\\dot{z})} = \\dot{z} + b, then obtain - 2 \\dot{z} \\operatorname{r_{0}}{(b,\\dot{z})} + ((\\dot{z} + b + \\mathbf{F}{(b,\\dot{z})})^{b})^{b} = - 2 \\dot{z} \\operatorname{r_{0}}{(b,\\dot{z})} + ((2 \\mathbf{F}{(b,\\dot{z})})^{b})^{b}", "derivation": "\\mathbf{F}{(b,\\dot{z})} = \\dot{z} + b and \\dot{z} + b + \\mathbf{F}{(b,\\dot{z})} = 2 \\dot{z} + 2 b and 2 \\mathbf{F}{(b,\\dot{z})} = 2 \\dot{z} + 2 b and \\dot{z} + b + \\mathbf{F}{(b,\\dot{z})} = 2 \\mathbf{F}{(b,\\dot{z})} and (\\dot{z} + b + \\mathbf{F}{(b,\\dot{z})})^{b} = (2 \\mathbf{F}{(b,\\dot{z})})^{b} and ((\\dot{z} + b + \\mathbf{F}{(b,\\dot{z})})^{b})^{b} = ((2 \\mathbf{F}{(b,\\dot{z})})^{b})^{b} and - 2 \\dot{z} \\operatorname{r_{0}}{(b,\\dot{z})} + ((\\dot{z} + b + \\mathbf{F}{(b,\\dot{z})})^{b})^{b} = - 2 \\dot{z} \\operatorname{r_{0}}{(b,\\dot{z})} + ((2 \\mathbf{F}{(b,\\dot{z})})^{b})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Pow(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["minus", 6, "Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True), Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True), Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Pow(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('b', commutative=True), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{z}', commutative=True), Function('r_0')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Pow(Pow(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True))))"]]}, {"prompt": "Given m{(\\dot{y})} = \\sin{(\\cos{(\\dot{y})})}, then derive - (\\sin{(\\dot{y})} \\cos{(\\cos{(\\dot{y})})} + \\frac{d}{d \\dot{y}} m{(\\dot{y})}) \\sin{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = 0, then obtain \\frac{d}{d \\dot{y}} - (\\sin{(\\dot{y})} \\cos{(\\cos{(\\dot{y})})} + \\frac{d}{d \\dot{y}} m{(\\dot{y})}) \\sin{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = \\frac{d}{d \\dot{y}} 0", "derivation": "m{(\\dot{y})} = \\sin{(\\cos{(\\dot{y})})} and m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})} = 0 and \\cos{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = 1 and \\frac{d}{d \\dot{y}} \\cos{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = \\frac{d}{d \\dot{y}} 1 and - (\\sin{(\\dot{y})} \\cos{(\\cos{(\\dot{y})})} + \\frac{d}{d \\dot{y}} m{(\\dot{y})}) \\sin{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = 0 and \\frac{d}{d \\dot{y}} - (\\sin{(\\dot{y})} \\cos{(\\cos{(\\dot{y})})} + \\frac{d}{d \\dot{y}} m{(\\dot{y})}) \\sin{(m{(\\dot{y})} - \\sin{(\\cos{(\\dot{y})})})} = \\frac{d}{d \\dot{y}} 0", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\dot{y}', commutative=True)), sin(cos(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 1, "sin(cos(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\dot{y}', commutative=True))))), Integer(0))"], [["cos", 2], "Equality(cos(Add(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\dot{y}', commutative=True)))))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(cos(Add(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\dot{y}', commutative=True)))))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Add(Mul(sin(Symbol('\\\\dot{y}', commutative=True)), cos(cos(Symbol('\\\\dot{y}', commutative=True)))), Derivative(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), sin(Add(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\dot{y}', commutative=True))))))), Integer(0))"], [["differentiate", 5, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Mul(sin(Symbol('\\\\dot{y}', commutative=True)), cos(cos(Symbol('\\\\dot{y}', commutative=True)))), Derivative(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), sin(Add(Function('m')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\dot{y}', commutative=True))))))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)} = e^{H + \\mathbb{I}}, then obtain (\\frac{H \\operatorname{V_{\\mathbf{E}}}^{H}{(\\mathbb{I},H)} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)}}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)}})^{H} = (H e^{- H - \\mathbb{I}} e^{H + \\mathbb{I}} (e^{H + \\mathbb{I}})^{H})^{H}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)} = e^{H + \\mathbb{I}} and \\operatorname{V_{\\mathbf{E}}}^{H}{(\\mathbb{I},H)} = (e^{H + \\mathbb{I}})^{H} and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{V_{\\mathbf{E}}}^{H}{(\\mathbb{I},H)} = \\frac{\\partial}{\\partial \\mathbb{I}} (e^{H + \\mathbb{I}})^{H} and (\\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{V_{\\mathbf{E}}}^{H}{(\\mathbb{I},H)})^{H} = (\\frac{\\partial}{\\partial \\mathbb{I}} (e^{H + \\mathbb{I}})^{H})^{H} and (\\frac{H \\operatorname{V_{\\mathbf{E}}}^{H}{(\\mathbb{I},H)} \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)}}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbb{I},H)}})^{H} = (H e^{- H - \\mathbb{I}} e^{H + \\mathbb{I}} (e^{H + \\mathbb{I}})^{H})^{H}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('H', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Derivative(Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Symbol('H', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Mul(Symbol('H', commutative=True), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), exp(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))), exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('H', commutative=True))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\delta{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\delta{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then derive G + \\sin{(\\mathbf{H})} = \\chi + \\sin{(\\mathbf{H})}, then obtain G + \\delta{(\\mathbf{H})} - \\int \\cos{(\\mathbf{H})} d\\mathbf{H} = \\chi + \\delta{(\\mathbf{H})} - \\int \\cos{(\\mathbf{H})} d\\mathbf{H}", "derivation": "\\delta{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\delta{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\delta{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\int \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} d\\mathbf{H} = \\int \\cos{(\\mathbf{H})} d\\mathbf{H} and G + \\sin{(\\mathbf{H})} = \\chi + \\sin{(\\mathbf{H})} and G + \\delta{(\\mathbf{H})} = \\chi + \\delta{(\\mathbf{H})} and G + \\delta{(\\mathbf{H})} - \\int \\cos{(\\mathbf{H})} d\\mathbf{H} = \\chi + \\delta{(\\mathbf{H})} - \\int \\cos{(\\mathbf{H})} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('G', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('G', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 7, "Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Symbol('\\\\chi', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\mathbf{J}_f,z)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f^{z}, then derive \\operatorname{y^{\\prime}}{(\\mathbf{J}_f,z)} = \\frac{\\mathbf{J}_f^{z} z}{\\mathbf{J}_f}, then obtain \\operatorname{y^{\\prime}}^{z}{(\\mathbf{J}_f,z)} = (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f^{z})^{z}", "derivation": "\\operatorname{y^{\\prime}}{(\\mathbf{J}_f,z)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f^{z} and \\operatorname{y^{\\prime}}{(\\mathbf{J}_f,z)} = \\frac{\\mathbf{J}_f^{z} z}{\\mathbf{J}_f} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f^{z} = \\frac{\\mathbf{J}_f^{z} z}{\\mathbf{J}_f} and \\operatorname{y^{\\prime}}^{z}{(\\mathbf{J}_f,z)} = (\\frac{\\mathbf{J}_f^{z} z}{\\mathbf{J}_f})^{z} and \\operatorname{y^{\\prime}}^{z}{(\\mathbf{J}_f,z)} = (\\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Derivative(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given z{(\\Omega)} = \\sin{(\\Omega)}, then derive \\frac{d}{d \\Omega} z{(\\Omega)} = \\cos{(\\Omega)}, then obtain \\Omega + z{(\\Omega)} \\frac{d}{d \\Omega} z{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} = \\Omega + z{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{2}", "derivation": "z{(\\Omega)} = \\sin{(\\Omega)} and \\frac{d}{d \\Omega} z{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} z{(\\Omega)} = \\cos{(\\Omega)} and \\frac{d}{d \\Omega} z{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} = \\cos{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} and (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{2} = \\cos{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} z{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} = (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{2} and z{(\\Omega)} \\frac{d}{d \\Omega} z{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} = z{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{2} and \\Omega + z{(\\Omega)} \\frac{d}{d \\Omega} z{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} = \\Omega + z{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{2}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), cos(Symbol('\\\\Omega', commutative=True)))"], [["times", 3, "Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(2)), Mul(cos(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(2)))"], [["times", 6, "Function('z')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('z')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Function('z')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(2))))"], [["add", 7, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Function('z')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('z')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Function('z')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(2)))))"]]}, {"prompt": "Given t{(t_{2},\\eta^{\\prime})} = \\eta^{\\prime} \\sin{(t_{2})}, then obtain - \\eta^{\\prime} \\sin{(t_{2})} + t^{\\eta^{\\prime}}{(t_{2},\\eta^{\\prime})} + 1 = - \\eta^{\\prime} \\sin{(t_{2})} + (\\eta^{\\prime} \\sin{(t_{2})})^{\\eta^{\\prime}} + 1", "derivation": "t{(t_{2},\\eta^{\\prime})} = \\eta^{\\prime} \\sin{(t_{2})} and t^{\\eta^{\\prime}}{(t_{2},\\eta^{\\prime})} = (\\eta^{\\prime} \\sin{(t_{2})})^{\\eta^{\\prime}} and t^{\\eta^{\\prime}}{(t_{2},\\eta^{\\prime})} + 1 = (\\eta^{\\prime} \\sin{(t_{2})})^{\\eta^{\\prime}} + 1 and - \\eta^{\\prime} \\sin{(t_{2})} + t^{\\eta^{\\prime}}{(t_{2},\\eta^{\\prime})} + 1 = - \\eta^{\\prime} \\sin{(t_{2})} + (\\eta^{\\prime} \\sin{(t_{2})})^{\\eta^{\\prime}} + 1", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('t_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('t')(Symbol('t_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 2, 1], "Equality(Add(Pow(Function('t')(Symbol('t_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)), Add(Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))), Pow(Function('t')(Symbol('t_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))), Pow(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Symbol('t_2', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})}, then derive 2 \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}, then obtain 2 \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} and 2 \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} and 2 \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\operatorname{g_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}} and 2 \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(2), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\chi{(t,\\dot{y})} = \\frac{\\dot{y}}{t}, then obtain \\int (1 - \\dot{y}) dt = \\int (- \\dot{y} + 2 (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t} - 1) dt", "derivation": "\\chi{(t,\\dot{y})} = \\frac{\\dot{y}}{t} and 1 = \\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}} and 1 = (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t} and 1 - \\dot{y} = - \\dot{y} + (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t} and \\int (1 - \\dot{y}) dt = \\int (- \\dot{y} + (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t}) dt and \\int (- \\dot{y} + (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t}) dt = \\int (- \\dot{y} + 2 (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t} - 1) dt and \\int (1 - \\dot{y}) dt = \\int (- \\dot{y} + 2 (\\frac{\\dot{y}}{t \\chi{(t,\\dot{y})}})^{t} - 1) dt", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["divide", 1, "Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True)))"], [["minus", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True))))"], [["integrate", 4, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True))), Integer(-1)), Tuple(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('t', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('t', commutative=True))), Integer(-1)), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\dot{x})} = \\cos{(\\dot{x})}, then derive \\int \\frac{\\operatorname{F_{N}}{(\\dot{x})}}{\\cos{(\\dot{x})}} d\\dot{x} = \\dot{x} + \\hat{p}, then obtain \\dot{x} + t = \\dot{x} + \\hat{p}", "derivation": "\\operatorname{F_{N}}{(\\dot{x})} = \\cos{(\\dot{x})} and \\frac{\\operatorname{F_{N}}{(\\dot{x})}}{\\cos{(\\dot{x})}} = 1 and \\int \\frac{\\operatorname{F_{N}}{(\\dot{x})}}{\\cos{(\\dot{x})}} d\\dot{x} = \\int 1 d\\dot{x} and \\int \\frac{\\operatorname{F_{N}}{(\\dot{x})}}{\\cos{(\\dot{x})}} d\\dot{x} = \\dot{x} + \\hat{p} and \\int 1 d\\dot{x} = \\dot{x} + \\hat{p} and \\dot{x} + t = \\dot{x} + \\hat{p}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('\\\\dot{x}', commutative=True)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Mul(Function('F_N')(Symbol('\\\\dot{x}', commutative=True)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('F_N')(Symbol('\\\\dot{x}', commutative=True)), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\psi,m)} = \\psi - m, then derive \\mathbf{H} + \\psi - \\psi^{\\psi} - m + \\dot{y}{(\\psi,m)} = \\mathbf{g} + 2 \\psi - \\psi^{\\psi} - m, then obtain \\mathbf{H} - \\psi^{\\psi} + 2 \\dot{y}{(\\psi,m)} = \\mathbf{g} + 2 \\psi - \\psi^{\\psi} - m", "derivation": "\\dot{y}{(\\psi,m)} = \\psi - m and m + \\dot{y}{(\\psi,m)} = \\psi and \\frac{\\partial}{\\partial \\psi} (m + \\dot{y}{(\\psi,m)}) = \\frac{d}{d \\psi} \\psi and \\int \\frac{\\partial}{\\partial \\psi} (m + \\dot{y}{(\\psi,m)}) d\\psi = \\int \\frac{d}{d \\psi} \\psi d\\psi and \\psi - \\psi^{\\psi} - m + \\int \\frac{\\partial}{\\partial \\psi} (m + \\dot{y}{(\\psi,m)}) d\\psi = \\psi - \\psi^{\\psi} - m + \\int \\frac{d}{d \\psi} \\psi d\\psi and \\mathbf{H} + \\psi - \\psi^{\\psi} - m + \\dot{y}{(\\psi,m)} = \\mathbf{g} + 2 \\psi - \\psi^{\\psi} - m and \\mathbf{H} - \\psi^{\\psi} + 2 \\dot{y}{(\\psi,m)} = \\mathbf{g} + 2 \\psi - \\psi^{\\psi} - m", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))), Symbol('\\\\psi', commutative=True))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('m', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('m', commutative=True))"], "Equality(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)), Integral(Derivative(Add(Symbol('m', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)), Integral(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('\\\\psi', commutative=True), Symbol('m', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(A)} = \\log{(A)} and \\operatorname{A_{2}}{(A)} = \\log{(A)}, then obtain (- \\dot{y}{(A)} + \\log{(A)}) \\dot{y}^{A}{(A)} = (- \\dot{y}{(A)} + \\log{(A)}) \\log{(A)}^{A}", "derivation": "\\dot{y}{(A)} = \\log{(A)} and \\dot{y}^{A}{(A)} = \\log{(A)}^{A} and \\operatorname{A_{2}}{(A)} = \\log{(A)} and \\dot{y}^{A}{(A)} = \\operatorname{A_{2}}^{A}{(A)} and (- \\dot{y}{(A)} + \\log{(A)}) \\dot{y}^{A}{(A)} = (- \\dot{y}{(A)} + \\log{(A)}) \\operatorname{A_{2}}^{A}{(A)} and (- \\dot{y}{(A)} + \\log{(A)}) \\dot{y}^{A}{(A)} = (- \\dot{y}{(A)} + \\log{(A)}) \\log{(A)}^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Function('A_2')(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["times", 4, "Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('A', commutative=True))), log(Symbol('A', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('A', commutative=True))), log(Symbol('A', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('A', commutative=True))), log(Symbol('A', commutative=True))), Pow(Function('A_2')(Symbol('A', commutative=True)), Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('A', commutative=True))), log(Symbol('A', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('A', commutative=True))), log(Symbol('A', commutative=True))), Pow(log(Symbol('A', commutative=True)), Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\rho)} = \\int e^{\\rho} d\\rho, then derive \\pi{(\\rho)} = E_{\\lambda} + e^{\\rho}, then derive \\tilde{g} + e^{\\rho} = E_{\\lambda} + e^{\\rho}, then obtain (E_{\\lambda} + e^{\\rho}) (\\tilde{g} + e^{\\rho}) = (\\tilde{g} + e^{\\rho})^{2}", "derivation": "\\pi{(\\rho)} = \\int e^{\\rho} d\\rho and \\pi{(\\rho)} = E_{\\lambda} + e^{\\rho} and \\int e^{\\rho} d\\rho = E_{\\lambda} + e^{\\rho} and \\pi{(\\rho)} \\int e^{\\rho} d\\rho = (\\int e^{\\rho} d\\rho)^{2} and \\tilde{g} + e^{\\rho} = E_{\\lambda} + e^{\\rho} and \\int e^{\\rho} d\\rho = \\tilde{g} + e^{\\rho} and (\\tilde{g} + e^{\\rho}) \\pi{(\\rho)} = (\\tilde{g} + e^{\\rho})^{2} and (E_{\\lambda} + e^{\\rho}) (\\tilde{g} + e^{\\rho}) = (\\tilde{g} + e^{\\rho})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\pi')(Symbol('\\\\rho', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('\\\\rho', commutative=True)), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Pow(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Function('\\\\pi')(Symbol('\\\\rho', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True)))), Pow(Add(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\rho', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(A_{2},B)} = A_{2} B and U{(A_{2},B)} = A_{2} B, then obtain A_{2} + \\frac{U{(A_{2},B)}}{B^{2}} = A_{2} + \\frac{A_{2}}{B}", "derivation": "\\operatorname{E_{\\lambda}}{(A_{2},B)} = A_{2} B and \\frac{\\operatorname{E_{\\lambda}}{(A_{2},B)}}{B} = A_{2} and - \\frac{\\operatorname{E_{\\lambda}}{(A_{2},B)}}{B} = - A_{2} and \\frac{\\operatorname{E_{\\lambda}}{(A_{2},B)}}{B^{2}} = \\frac{A_{2}}{B} and U{(A_{2},B)} = A_{2} B and A_{2} + \\frac{\\operatorname{E_{\\lambda}}{(A_{2},B)}}{B^{2}} = A_{2} + \\frac{A_{2}}{B} and \\operatorname{E_{\\lambda}}{(A_{2},B)} = U{(A_{2},B)} and A_{2} + \\frac{U{(A_{2},B)}}{B^{2}} = A_{2} + \\frac{A_{2}}{B}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True))), Symbol('A_2', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('U')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))"], [["add", 4, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))), Add(Symbol('A_2', commutative=True), Mul(Symbol('A_2', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)), Function('U')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Symbol('A_2', commutative=True), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Function('U')(Symbol('A_2', commutative=True), Symbol('B', commutative=True)))), Add(Symbol('A_2', commutative=True), Mul(Symbol('A_2', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})} = \\sin{(\\ddot{x} \\mathbf{B})}, then obtain (\\ddot{x} \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})})^{\\mathbf{B}} - \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})} = (\\ddot{x} \\sin{(\\ddot{x} \\mathbf{B})})^{\\mathbf{B}} - \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})}", "derivation": "\\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})} = \\sin{(\\ddot{x} \\mathbf{B})} and \\ddot{x} \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})} = \\ddot{x} \\sin{(\\ddot{x} \\mathbf{B})} and (\\ddot{x} \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})})^{\\mathbf{B}} = (\\ddot{x} \\sin{(\\ddot{x} \\mathbf{B})})^{\\mathbf{B}} and (\\ddot{x} \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})})^{\\mathbf{B}} - \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})} = (\\ddot{x} \\sin{(\\ddot{x} \\mathbf{B})})^{\\mathbf{B}} - \\operatorname{A_{x}}{(\\mathbf{B},\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('\\\\ddot{x}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 3, "Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Add(Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('A_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given C{(V,\\mathbf{J}_P)} = \\mathbf{J}_P + e^{V}, then derive \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = T + V, then obtain T \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = T \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{V}) dV", "derivation": "C{(V,\\mathbf{J}_P)} = \\mathbf{J}_P + e^{V} and \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{V}) and \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{V}) dV and \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = T + V and \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{V}) dV = T + V and T \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = T (T + V) and T \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} C{(V,\\mathbf{J}_P)} dV = T \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + e^{V}) dV", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Add(Symbol('T', commutative=True), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Add(Symbol('T', commutative=True), Symbol('V', commutative=True)))"], [["times", 4, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Integral(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True)))), Mul(Symbol('T', commutative=True), Add(Symbol('T', commutative=True), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Symbol('T', commutative=True), Integral(Derivative(Function('C')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True)))), Mul(Symbol('T', commutative=True), Integral(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(c_{0})} = e^{c_{0}} and y{(\\pi)} = \\sin{(\\cos{(\\pi)})}, then obtain (- \\operatorname{F_{x}}^{2}{(c_{0})} + y{(\\pi)}) \\operatorname{F_{x}}^{3}{(c_{0})} = (- \\operatorname{F_{x}}^{2}{(c_{0})} + \\sin{(\\cos{(\\pi)})}) \\operatorname{F_{x}}^{3}{(c_{0})}", "derivation": "\\operatorname{F_{x}}{(c_{0})} = e^{c_{0}} and y{(\\pi)} = \\sin{(\\cos{(\\pi)})} and - \\operatorname{F_{x}}{(c_{0})} e^{c_{0}} + y{(\\pi)} = - \\operatorname{F_{x}}{(c_{0})} e^{c_{0}} + \\sin{(\\cos{(\\pi)})} and (- \\operatorname{F_{x}}{(c_{0})} e^{c_{0}} + y{(\\pi)}) \\operatorname{F_{x}}^{2}{(c_{0})} e^{c_{0}} = (- \\operatorname{F_{x}}{(c_{0})} e^{c_{0}} + \\sin{(\\cos{(\\pi)})}) \\operatorname{F_{x}}^{2}{(c_{0})} e^{c_{0}} and (- \\operatorname{F_{x}}^{2}{(c_{0})} + y{(\\pi)}) \\operatorname{F_{x}}^{3}{(c_{0})} = (- \\operatorname{F_{x}}^{2}{(c_{0})} + \\sin{(\\cos{(\\pi)})}) \\operatorname{F_{x}}^{3}{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], ["get_premise", "Equality(Function('y')(Symbol('\\\\pi', commutative=True)), sin(cos(Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Mul(Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True))), Function('y')(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True))), sin(cos(Symbol('\\\\pi', commutative=True)))))"], [["times", 3, "Mul(Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(2)), exp(Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True))), Function('y')(Symbol('\\\\pi', commutative=True))), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(2)), exp(Symbol('c_0', commutative=True))), Mul(Add(Mul(Integer(-1), Function('F_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True))), sin(cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(2)), exp(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Integer(-1), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(2))), Function('y')(Symbol('\\\\pi', commutative=True))), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(3))), Mul(Add(Mul(Integer(-1), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(2))), sin(cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('F_x')(Symbol('c_0', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(Q)} = e^{Q}, then obtain (\\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} + 2 \\frac{d^{2}}{d Q^{2}} \\hat{\\mathbf{r}}{(Q)} + \\frac{d^{3}}{d Q^{3}} \\hat{\\mathbf{r}}{(Q)}) e^{Q} = 4 e^{2 Q}", "derivation": "\\hat{\\mathbf{r}}{(Q)} = e^{Q} and \\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} = \\frac{d}{d Q} e^{Q} and \\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} \\frac{d}{d Q} e^{Q} = (\\frac{d}{d Q} e^{Q})^{2} and \\frac{d}{d Q} \\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} \\frac{d}{d Q} e^{Q} = \\frac{d}{d Q} (\\frac{d}{d Q} e^{Q})^{2} and \\frac{d^{2}}{d Q^{2}} \\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} \\frac{d}{d Q} e^{Q} = \\frac{d^{2}}{d Q^{2}} (\\frac{d}{d Q} e^{Q})^{2} and (\\frac{d}{d Q} \\hat{\\mathbf{r}}{(Q)} + 2 \\frac{d^{2}}{d Q^{2}} \\hat{\\mathbf{r}}{(Q)} + \\frac{d^{3}}{d Q^{3}} \\hat{\\mathbf{r}}{(Q)}) e^{Q} = 4 e^{2 Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 2, "Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Tuple(Symbol('Q', commutative=True), Integer(2))), Derivative(Pow(Derivative(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('Q', commutative=True), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(2), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2)))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(3)))), exp(Symbol('Q', commutative=True))), Mul(Integer(4), exp(Mul(Integer(2), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\eta)} = \\cos{(\\eta)} and \\operatorname{P_{e}}{(\\eta)} = \\mathbf{E}{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)}, then obtain \\operatorname{P_{e}}{(\\eta)} = \\cos{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)}", "derivation": "\\mathbf{E}{(\\eta)} = \\cos{(\\eta)} and \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)} = \\frac{d}{d \\eta} \\cos{(\\eta)} and \\mathbf{E}{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} = \\cos{(\\eta)} \\frac{d}{d \\eta} \\cos{(\\eta)} and \\mathbf{E}{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)} = \\cos{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)} and \\operatorname{P_{e}}{(\\eta)} = \\mathbf{E}{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)} and \\operatorname{P_{e}}{(\\eta)} = \\cos{(\\eta)} \\frac{d}{d \\eta} \\mathbf{E}{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["times", 1, "Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\eta', commutative=True)), Derivative(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\eta', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\eta', commutative=True)), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('P_e')(Symbol('\\\\eta', commutative=True)), Mul(cos(Symbol('\\\\eta', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})} = L_{\\varepsilon} \\hat{H}_l \\mathbf{F}, then obtain \\frac{4 T^{2}{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l^{2}} = (L_{\\varepsilon} \\mathbf{F} + \\frac{T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l})^{2}", "derivation": "T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})} = L_{\\varepsilon} \\hat{H}_l \\mathbf{F} and \\frac{T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l} = L_{\\varepsilon} \\mathbf{F} and \\frac{2 T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l} = L_{\\varepsilon} \\mathbf{F} + \\frac{T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l} and \\frac{4 T^{2}{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l^{2}} = (L_{\\varepsilon} \\mathbf{F} + \\frac{T{(\\mathbf{F},\\hat{H}_l,L_{\\varepsilon})}}{\\hat{H}_l})^{2}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-2)), Pow(Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2))), Pow(Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given E{(I,\\hat{x})} = \\cos{(I + \\hat{x})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} = - I + E{(I,\\hat{x})}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} = - E{(I,\\hat{x})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} + \\cos{(I + \\hat{x})}", "derivation": "E{(I,\\hat{x})} = \\cos{(I + \\hat{x})} and - I + E{(I,\\hat{x})} = - I + \\cos{(I + \\hat{x})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} = - I + E{(I,\\hat{x})} and - E{(I,\\hat{x})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} = - I and \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} = - E{(I,\\hat{x})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(I,\\hat{x})} + \\cos{(I + \\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)), cos(Add(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), cos(Add(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 3, "Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Function('E')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)), cos(Add(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(x,\\hat{X})} = \\hat{X} x, then derive \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})} = x, then obtain (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\hat{X} x) (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})}) = (\\hat{X} + x) (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})})", "derivation": "\\operatorname{P_{e}}{(x,\\hat{X})} = \\hat{X} x and \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})} = \\frac{\\partial}{\\partial \\hat{X}} \\hat{X} x and \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})} = x and \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})} = \\hat{X} + x and \\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\hat{X} x = \\hat{X} + x and (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\hat{X} x) (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})}) = (\\hat{X} + x) (\\hat{X} + \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{P_{e}}{(x,\\hat{X})})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Symbol('x', commutative=True))"], [["add", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)))"], [["times", 5, "Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('x', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Derivative(Function('P_e')(Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})} = \\frac{\\mathbf{J}_f z^{*}}{g^{\\prime}_{\\varepsilon}}, then obtain \\log{(g^{\\prime}_{\\varepsilon} \\frac{\\partial}{\\partial z^{*}} l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})})} = \\log{(\\mathbf{J}_f)}", "derivation": "l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})} = \\frac{\\mathbf{J}_f z^{*}}{g^{\\prime}_{\\varepsilon}} and g^{\\prime}_{\\varepsilon} l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})} = \\mathbf{J}_f z^{*} and \\frac{\\partial}{\\partial z^{*}} g^{\\prime}_{\\varepsilon} l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})} = \\frac{\\partial}{\\partial z^{*}} \\mathbf{J}_f z^{*} and \\log{(\\frac{\\partial}{\\partial z^{*}} g^{\\prime}_{\\varepsilon} l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})})} = \\log{(\\frac{\\partial}{\\partial z^{*}} \\mathbf{J}_f z^{*})} and \\log{(g^{\\prime}_{\\varepsilon} \\frac{\\partial}{\\partial z^{*}} l{(g^{\\prime}_{\\varepsilon},\\mathbf{J}_f,z^{*})})} = \\log{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)))"], [["divide", 1, "Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))), log(Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(log(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Derivative(Function('l')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(z,\\theta)} = \\theta + z, then obtain \\int (- \\theta + \\mathbf{P}{(z,\\theta)}) dz = \\int z dz", "derivation": "\\mathbf{P}{(z,\\theta)} = \\theta + z and \\theta + \\mathbf{P}{(z,\\theta)} = 2 \\theta + z and - \\theta + \\mathbf{P}{(z,\\theta)} = z and \\int (- \\theta + \\mathbf{P}{(z,\\theta)}) dz = \\int z dz", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Symbol('z', commutative=True)))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), Symbol('z', commutative=True)))"], [["minus", 2, "Mul(Integer(2), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Symbol('z', commutative=True))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{P}')(Symbol('z', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Symbol('z', commutative=True), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain \\rho_{f}{(x^\\prime)} \\frac{d}{d x^\\prime} (2 \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)}) = \\rho_{f}{(x^\\prime)} \\frac{d}{d x^\\prime} (\\rho_{f}{(x^\\prime)} + 2 \\cos{(x^\\prime)})", "derivation": "\\rho_{f}{(x^\\prime)} = \\cos{(x^\\prime)} and \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)} = 2 \\cos{(x^\\prime)} and \\rho_{f}{(x^\\prime)} + 2 \\cos{(x^\\prime)} = 3 \\cos{(x^\\prime)} and 2 \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)} = 3 \\cos{(x^\\prime)} and 2 \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)} = \\rho_{f}{(x^\\prime)} + 2 \\cos{(x^\\prime)} and \\frac{d}{d x^\\prime} (2 \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)}) = \\frac{d}{d x^\\prime} (\\rho_{f}{(x^\\prime)} + 2 \\cos{(x^\\prime)}) and \\rho_{f}{(x^\\prime)} \\frac{d}{d x^\\prime} (2 \\rho_{f}{(x^\\prime)} + \\cos{(x^\\prime)}) = \\rho_{f}{(x^\\prime)} \\frac{d}{d x^\\prime} (\\rho_{f}{(x^\\prime)} + 2 \\cos{(x^\\prime)})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), cos(Symbol('x^\\\\prime', commutative=True))))"], [["add", 2, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), cos(Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(3), cos(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(3), cos(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Add(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), cos(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["times", 6, "Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Function('\\\\rho_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\omega)} = \\int \\cos{(\\omega)} d\\omega, then derive \\operatorname{f_{E}}{(\\omega)} = \\mathbf{J} + \\sin{(\\omega)}, then obtain \\int \\frac{(\\mathbf{J} + \\sin{(\\omega)}) \\int \\cos{(\\omega)} d\\omega}{\\omega^{2}} d\\omega = \\int \\frac{(\\int \\cos{(\\omega)} d\\omega)^{2}}{\\omega^{2}} d\\omega", "derivation": "\\operatorname{f_{E}}{(\\omega)} = \\int \\cos{(\\omega)} d\\omega and \\frac{\\operatorname{f_{E}}{(\\omega)}}{\\omega} = \\frac{\\int \\cos{(\\omega)} d\\omega}{\\omega} and \\frac{\\operatorname{f_{E}}{(\\omega)} \\int \\cos{(\\omega)} d\\omega}{\\omega^{2}} = \\frac{(\\int \\cos{(\\omega)} d\\omega)^{2}}{\\omega^{2}} and \\int \\frac{\\operatorname{f_{E}}{(\\omega)} \\int \\cos{(\\omega)} d\\omega}{\\omega^{2}} d\\omega = \\int \\frac{(\\int \\cos{(\\omega)} d\\omega)^{2}}{\\omega^{2}} d\\omega and \\operatorname{f_{E}}{(\\omega)} = \\mathbf{J} + \\sin{(\\omega)} and \\int \\frac{(\\mathbf{J} + \\sin{(\\omega)}) \\int \\cos{(\\omega)} d\\omega}{\\omega^{2}} d\\omega = \\int \\frac{(\\int \\cos{(\\omega)} d\\omega)^{2}}{\\omega^{2}} d\\omega", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\omega', commutative=True)), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["times", 2, "Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('f_E')(Symbol('\\\\omega', commutative=True)), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Pow(Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('f_E')(Symbol('\\\\omega', commutative=True)), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Pow(Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('f_E')(Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Pow(Integral(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\chi)} = \\int \\cos{(\\chi)} d\\chi, then derive \\mathbf{p}{(\\chi)} = \\rho_b + \\sin{(\\chi)}, then derive e^{\\rho_b + \\sin{(\\chi)}} = e^{n + \\sin{(\\chi)}}, then obtain \\int \\frac{\\frac{\\partial}{\\partial \\chi} e^{\\rho_b + \\sin{(\\chi)}}}{E} d\\chi = \\int \\frac{\\frac{\\partial}{\\partial \\chi} e^{n + \\sin{(\\chi)}}}{E} d\\chi", "derivation": "\\mathbf{p}{(\\chi)} = \\int \\cos{(\\chi)} d\\chi and \\mathbf{p}{(\\chi)} = \\rho_b + \\sin{(\\chi)} and \\rho_b + \\sin{(\\chi)} = \\int \\cos{(\\chi)} d\\chi and e^{\\rho_b + \\sin{(\\chi)}} = e^{\\int \\cos{(\\chi)} d\\chi} and e^{\\rho_b + \\sin{(\\chi)}} = e^{n + \\sin{(\\chi)}} and \\frac{\\partial}{\\partial \\chi} e^{\\rho_b + \\sin{(\\chi)}} = \\frac{\\partial}{\\partial \\chi} e^{n + \\sin{(\\chi)}} and \\frac{\\frac{\\partial}{\\partial \\chi} e^{\\rho_b + \\sin{(\\chi)}}}{E} = \\frac{\\frac{\\partial}{\\partial \\chi} e^{n + \\sin{(\\chi)}}}{E} and \\int \\frac{\\frac{\\partial}{\\partial \\chi} e^{\\rho_b + \\sin{(\\chi)}}}{E} d\\chi = \\int \\frac{\\frac{\\partial}{\\partial \\chi} e^{n + \\sin{(\\chi)}}}{E} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True))), Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), exp(Integral(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(exp(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), exp(Add(Symbol('n', commutative=True), sin(Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(exp(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('n', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["divide", 6, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('n', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["integrate", 7, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('\\\\rho_b', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(exp(Add(Symbol('n', commutative=True), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given m{(V,C_{1})} = C_{1} + V and \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain C_{1} + \\frac{\\operatorname{v_{t}}{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}} = C_{1} + \\frac{\\log{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}}", "derivation": "m{(V,C_{1})} = C_{1} + V and \\operatorname{v_{t}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\frac{\\operatorname{v_{t}}{(\\hat{H}_{\\lambda})}}{C_{1} + V} = \\frac{\\log{(\\hat{H}_{\\lambda})}}{C_{1} + V} and \\frac{\\operatorname{v_{t}}{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}} = \\frac{\\log{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}} and C_{1} + \\frac{\\operatorname{v_{t}}{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}} = C_{1} + \\frac{\\log{(\\hat{H}_{\\lambda})}}{m{(V,C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('V', commutative=True), Symbol('C_1', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('V', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["times", 2, "Pow(Add(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Integer(-1)), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('V', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('m')(Symbol('V', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Pow(Function('m')(Symbol('V', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), Symbol('C_1', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Mul(Pow(Function('m')(Symbol('V', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), Function('v_t')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Pow(Function('m')(Symbol('V', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\tilde{g},y)} = y^{\\tilde{g}}, then obtain \\log{(\\frac{\\operatorname{E_{n}}^{2}{(\\tilde{g},y)}}{y})} = \\log{(\\frac{y^{\\tilde{g}} \\operatorname{E_{n}}{(\\tilde{g},y)}}{y})}", "derivation": "\\operatorname{E_{n}}{(\\tilde{g},y)} = y^{\\tilde{g}} and \\operatorname{E_{n}}^{2}{(\\tilde{g},y)} = y^{\\tilde{g}} \\operatorname{E_{n}}{(\\tilde{g},y)} and \\frac{\\operatorname{E_{n}}^{2}{(\\tilde{g},y)}}{y} = \\frac{y^{\\tilde{g}} \\operatorname{E_{n}}{(\\tilde{g},y)}}{y} and \\log{(\\frac{\\operatorname{E_{n}}^{2}{(\\tilde{g},y)}}{y})} = \\log{(\\frac{y^{\\tilde{g}} \\operatorname{E_{n}}{(\\tilde{g},y)}}{y})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Pow(Symbol('y', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True))))"], [["divide", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True))))"], [["log", 3], "Equality(log(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True)), Integer(2)))), log(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('E_n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(A_{x})} = \\cos{(A_{x})}, then obtain e^{\\frac{\\cos{(A_{x})}}{A_{x}}} = e^{\\frac{\\cos^{2}{(A_{x})}}{A_{x} \\mu_{0}{(A_{x})}}}", "derivation": "\\mu_{0}{(A_{x})} = \\cos{(A_{x})} and \\frac{1}{A_{x}} = \\frac{\\cos{(A_{x})}}{A_{x} \\mu_{0}{(A_{x})}} and \\frac{\\cos{(A_{x})}}{A_{x}} = \\frac{\\cos^{2}{(A_{x})}}{A_{x} \\mu_{0}{(A_{x})}} and e^{\\frac{\\cos{(A_{x})}}{A_{x}}} = e^{\\frac{\\cos^{2}{(A_{x})}}{A_{x} \\mu_{0}{(A_{x})}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["divide", 1, "Mul(Symbol('A_x', commutative=True), Function('\\\\mu_0')(Symbol('A_x', commutative=True)))"], "Equality(Pow(Symbol('A_x', commutative=True), Integer(-1)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Function('\\\\mu_0')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True))))"], [["times", 2, "cos(Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), cos(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Function('\\\\mu_0')(Symbol('A_x', commutative=True)), Integer(-1)), Pow(cos(Symbol('A_x', commutative=True)), Integer(2))))"], [["exp", 3], "Equality(exp(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), cos(Symbol('A_x', commutative=True)))), exp(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Function('\\\\mu_0')(Symbol('A_x', commutative=True)), Integer(-1)), Pow(cos(Symbol('A_x', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\Psi{(\\Omega)} = e^{\\Omega}, then obtain - e^{\\Omega} + (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int \\Psi{(\\Omega)} d\\Omega = - e^{\\Omega} + (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int e^{\\Omega} d\\Omega", "derivation": "\\Psi{(\\Omega)} = e^{\\Omega} and \\int \\Psi{(\\Omega)} d\\Omega = \\int e^{\\Omega} d\\Omega and (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int \\Psi{(\\Omega)} d\\Omega = (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int e^{\\Omega} d\\Omega and - e^{\\Omega} + (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int \\Psi{(\\Omega)} d\\Omega = - e^{\\Omega} + (\\int \\Psi{(\\Omega)} d\\Omega)^{2} + \\int e^{\\Omega} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Pow(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2))"], "Equality(Add(Pow(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)), Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Pow(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["minus", 3, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Pow(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)), Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Pow(Integral(Function('\\\\Psi')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(2)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)}, then derive \\Omega + \\phi^{\\Omega}{(\\Omega)} = \\Omega + (\\frac{1}{\\Omega})^{\\Omega}, then obtain (\\Omega + \\phi^{\\Omega}{(\\Omega)})^{\\Omega} - (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{- \\Omega} = (\\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega})^{\\Omega} - (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{- \\Omega}", "derivation": "\\phi{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and \\phi^{\\Omega}{(\\Omega)} = (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega} and \\Omega + \\phi^{\\Omega}{(\\Omega)} = \\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega} and \\Omega + \\phi^{\\Omega}{(\\Omega)} = \\Omega + (\\frac{1}{\\Omega})^{\\Omega} and \\Omega + (\\frac{1}{\\Omega})^{\\Omega} = \\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega} and (\\Omega + (\\frac{1}{\\Omega})^{\\Omega})^{\\Omega} = (\\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega})^{\\Omega} and (\\Omega + \\phi^{\\Omega}{(\\Omega)})^{\\Omega} = (\\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega})^{\\Omega} and (\\Omega + \\phi^{\\Omega}{(\\Omega)})^{\\Omega} - (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{- \\Omega} = (\\Omega + (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{\\Omega})^{\\Omega} - (\\frac{d}{d \\Omega} \\log{(\\Omega)})^{- \\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["minus", 7, "Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Add(Pow(Add(Symbol('\\\\Omega', commutative=True), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\pi{(S,E_{x})} = E_{x} \\cos{(S)} and \\mathbf{f}{(S,E_{x})} = \\pi{(S,E_{x})} + \\int \\pi{(S,E_{x})} dS, then obtain \\pi{(S,E_{x})} + \\int E_{x} \\cos{(S)} dS = E_{x} \\cos{(S)} + \\int E_{x} \\cos{(S)} dS", "derivation": "\\pi{(S,E_{x})} = E_{x} \\cos{(S)} and \\int \\pi{(S,E_{x})} dS = \\int E_{x} \\cos{(S)} dS and \\pi{(S,E_{x})} + \\int \\pi{(S,E_{x})} dS = E_{x} \\cos{(S)} + \\int \\pi{(S,E_{x})} dS and \\mathbf{f}{(S,E_{x})} = \\pi{(S,E_{x})} + \\int \\pi{(S,E_{x})} dS and \\mathbf{f}{(S,E_{x})} = \\pi{(S,E_{x})} + \\int E_{x} \\cos{(S)} dS and \\mathbf{f}{(S,E_{x})} = E_{x} \\cos{(S)} + \\int \\pi{(S,E_{x})} dS and \\pi{(S,E_{x})} + \\int E_{x} \\cos{(S)} dS = E_{x} \\cos{(S)} + \\int \\pi{(S,E_{x})} dS and \\pi{(S,E_{x})} + \\int E_{x} \\cos{(S)} dS = E_{x} \\cos{(S)} + \\int E_{x} \\cos{(S)} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["add", 1, "Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Add(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Add(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Integral(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{f}')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Integral(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Integral(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Add(Function('\\\\pi')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Integral(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Integral(Mul(Symbol('E_x', commutative=True), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(G)} = \\sin{(\\sin{(G)})}, then obtain \\frac{d}{d G} (- \\varepsilon_{0}{(G)} + \\cos{(\\int \\varepsilon_{0}{(G)} dG)}) = \\frac{d}{d G} (- \\varepsilon_{0}{(G)} + \\cos{(\\int \\sin{(\\sin{(G)})} dG)})", "derivation": "\\varepsilon_{0}{(G)} = \\sin{(\\sin{(G)})} and \\int \\varepsilon_{0}{(G)} dG = \\int \\sin{(\\sin{(G)})} dG and \\cos{(\\int \\varepsilon_{0}{(G)} dG)} = \\cos{(\\int \\sin{(\\sin{(G)})} dG)} and - \\varepsilon_{0}{(G)} + \\cos{(\\int \\varepsilon_{0}{(G)} dG)} = - \\varepsilon_{0}{(G)} + \\cos{(\\int \\sin{(\\sin{(G)})} dG)} and \\frac{d}{d G} (- \\varepsilon_{0}{(G)} + \\cos{(\\int \\varepsilon_{0}{(G)} dG)}) = \\frac{d}{d G} (- \\varepsilon_{0}{(G)} + \\cos{(\\int \\sin{(\\sin{(G)})} dG)})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\varepsilon_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), cos(Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)))))"], [["minus", 3, "Function('\\\\varepsilon_0')(Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('G', commutative=True))), cos(Integral(Function('\\\\varepsilon_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))), Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('G', commutative=True))), cos(Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('G', commutative=True))), cos(Integral(Function('\\\\varepsilon_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('G', commutative=True))), cos(Integral(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(\\theta,\\mathbf{M})} = e^{\\mathbf{M} - \\theta}, then obtain \\frac{\\partial}{\\partial \\mathbf{M}} (- a^{\\dagger} + \\int g{(\\theta,\\mathbf{M})} d\\theta) = \\frac{\\partial}{\\partial \\mathbf{M}} (- a^{\\dagger} + \\int e^{\\mathbf{M} - \\theta} d\\theta)", "derivation": "g{(\\theta,\\mathbf{M})} = e^{\\mathbf{M} - \\theta} and \\int g{(\\theta,\\mathbf{M})} d\\theta = \\int e^{\\mathbf{M} - \\theta} d\\theta and - a^{\\dagger} + \\int g{(\\theta,\\mathbf{M})} d\\theta = - a^{\\dagger} + \\int e^{\\mathbf{M} - \\theta} d\\theta and \\frac{\\partial}{\\partial \\mathbf{M}} (- a^{\\dagger} + \\int g{(\\theta,\\mathbf{M})} d\\theta) = \\frac{\\partial}{\\partial \\mathbf{M}} (- a^{\\dagger} + \\int e^{\\mathbf{M} - \\theta} d\\theta)", "srepr_derivation": [["get_premise", "Equality(Function('g')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('g')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(exp(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["minus", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('g')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(exp(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('g')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(exp(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\pi,\\eta,A_{1})} = \\pi (A_{1} - \\eta), then obtain A_{1} + \\pi (A_{1} - \\eta) + b{(\\pi,\\eta,A_{1})} - 1 = A_{1} + 2 \\pi (A_{1} - \\eta) - 1", "derivation": "b{(\\pi,\\eta,A_{1})} = \\pi (A_{1} - \\eta) and b{(\\pi,\\eta,A_{1})} - 1 = \\pi (A_{1} - \\eta) - 1 and A_{1} + b{(\\pi,\\eta,A_{1})} - 1 = A_{1} + \\pi (A_{1} - \\eta) - 1 and A_{1} + \\pi (A_{1} - \\eta) + b{(\\pi,\\eta,A_{1})} - 1 = A_{1} + 2 \\pi (A_{1} - \\eta) - 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Integer(-1)))"], [["add", 2, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Function('b')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Add(Symbol('A_1', commutative=True), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Integer(-1)))"], [["add", 3, "Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Function('b')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), Add(Symbol('A_1', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given y{(C_{2})} = \\log{(C_{2})}, then obtain ((y{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})})^{C_{2}} = ((\\log{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})})^{C_{2}}", "derivation": "y{(C_{2})} = \\log{(C_{2})} and y^{C_{2}}{(C_{2})} = \\log{(C_{2})}^{C_{2}} and y{(C_{2})} - y^{C_{2}}{(C_{2})} = - y^{C_{2}}{(C_{2})} + \\log{(C_{2})} and y{(C_{2})} - \\log{(C_{2})}^{C_{2}} = \\log{(C_{2})} - \\log{(C_{2})}^{C_{2}} and (y{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})} = (\\log{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})} and ((y{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})})^{C_{2}} = ((\\log{(C_{2})} - \\log{(C_{2})}^{C_{2}}) y^{C_{2}}{(C_{2})})^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["minus", 1, "Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Add(Function('y')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), log(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('y')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Add(log(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))"], [["times", 4, "Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Mul(Add(Function('y')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Mul(Add(log(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["power", 5, "Symbol('C_2', commutative=True)"], "Equality(Pow(Mul(Add(Function('y')(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Pow(Mul(Add(log(Symbol('C_2', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Pow(Function('y')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(P_{g},f)} = f \\log{(P_{g})} and k{(P_{g},f)} = - f \\log{(P_{g})} + \\varepsilon{(P_{g},f)}, then obtain - f \\log{(P_{g})} + (\\varepsilon{(P_{g},f)} + \\log{(P_{g})}) k{(P_{g},f)} = - f \\log{(P_{g})}", "derivation": "\\varepsilon{(P_{g},f)} = f \\log{(P_{g})} and k{(P_{g},f)} = - f \\log{(P_{g})} + \\varepsilon{(P_{g},f)} and (\\varepsilon{(P_{g},f)} + \\log{(P_{g})}) k{(P_{g},f)} = (- f \\log{(P_{g})} + \\varepsilon{(P_{g},f)}) (\\varepsilon{(P_{g},f)} + \\log{(P_{g})}) and (\\varepsilon{(P_{g},f)} + \\log{(P_{g})}) k{(P_{g},f)} = 0 and - f \\log{(P_{g})} + (\\varepsilon{(P_{g},f)} + \\log{(P_{g})}) k{(P_{g},f)} = - f \\log{(P_{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), log(Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('P_g', commutative=True))), Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True))))"], [["times", 2, "Add(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), log(Symbol('P_g', commutative=True)))"], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), log(Symbol('P_g', commutative=True))), Function('k')(Symbol('P_g', commutative=True), Symbol('f', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('P_g', commutative=True))), Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True))), Add(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), log(Symbol('P_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), log(Symbol('P_g', commutative=True))), Function('k')(Symbol('P_g', commutative=True), Symbol('f', commutative=True))), Integer(0))"], [["add", 4, "Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('P_g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('P_g', commutative=True))), Mul(Add(Function('\\\\varepsilon')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), log(Symbol('P_g', commutative=True))), Function('k')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)))), Mul(Integer(-1), Symbol('f', commutative=True), log(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\mu)} = \\mu, then obtain ((- \\frac{\\int (\\mu + \\hat{x}{(\\mu)}) d\\hat{x}{(\\mu)}}{\\mu})^{\\mu})^{\\mu} = ((- \\frac{\\int 2 \\mu d\\hat{x}{(\\mu)}}{\\mu})^{\\mu})^{\\mu}", "derivation": "\\hat{x}{(\\mu)} = \\mu and \\mu + \\hat{x}{(\\mu)} = 2 \\mu and \\int (\\mu + \\hat{x}{(\\mu)}) d\\mu = \\int 2 \\mu d\\mu and \\int (\\mu + \\hat{x}{(\\mu)}) d\\hat{x}{(\\mu)} = \\int 2 \\mu d\\hat{x}{(\\mu)} and - \\frac{\\int (\\mu + \\hat{x}{(\\mu)}) d\\hat{x}{(\\mu)}}{\\mu} = - \\frac{\\int 2 \\mu d\\hat{x}{(\\mu)}}{\\mu} and (- \\frac{\\int (\\mu + \\hat{x}{(\\mu)}) d\\hat{x}{(\\mu)}}{\\mu})^{\\mu} = (- \\frac{\\int 2 \\mu d\\hat{x}{(\\mu)}}{\\mu})^{\\mu} and ((- \\frac{\\int (\\mu + \\hat{x}{(\\mu)}) d\\hat{x}{(\\mu)}}{\\mu})^{\\mu})^{\\mu} = ((- \\frac{\\int 2 \\mu d\\hat{x}{(\\mu)}}{\\mu})^{\\mu})^{\\mu}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Symbol('\\\\mu', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True)))), Integral(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True)))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))))"], [["power", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)))"], [["power", 6, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(C_{d})} = e^{C_{d}}, then obtain \\frac{d}{d C_{d}} (- C_{d} + e^{\\operatorname{t_{1}}{(C_{d})}}) = \\frac{d}{d C_{d}} (- C_{d} + e^{e^{C_{d}}})", "derivation": "\\operatorname{t_{1}}{(C_{d})} = e^{C_{d}} and e^{\\operatorname{t_{1}}{(C_{d})}} = e^{e^{C_{d}}} and - C_{d} + e^{\\operatorname{t_{1}}{(C_{d})}} = - C_{d} + e^{e^{C_{d}}} and \\frac{d}{d C_{d}} (- C_{d} + e^{\\operatorname{t_{1}}{(C_{d})}}) = \\frac{d}{d C_{d}} (- C_{d} + e^{e^{C_{d}}})", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["exp", 1], "Equality(exp(Function('t_1')(Symbol('C_d', commutative=True))), exp(exp(Symbol('C_d', commutative=True))))"], [["minus", 2, "Symbol('C_d', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), exp(Function('t_1')(Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True)))))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), exp(Function('t_1')(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(f_{\\mathbf{p}},a^{\\dagger})} = a^{\\dagger} + f_{\\mathbf{p}}, then obtain \\iint (a^{\\dagger} + f_{\\mathbf{p}}) G{(f_{\\mathbf{p}},a^{\\dagger})} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint (a^{\\dagger} + f_{\\mathbf{p}})^{2} df_{\\mathbf{p}} df_{\\mathbf{p}}", "derivation": "G{(f_{\\mathbf{p}},a^{\\dagger})} = a^{\\dagger} + f_{\\mathbf{p}} and (a^{\\dagger} + f_{\\mathbf{p}}) G{(f_{\\mathbf{p}},a^{\\dagger})} = (a^{\\dagger} + f_{\\mathbf{p}})^{2} and \\int (a^{\\dagger} + f_{\\mathbf{p}}) G{(f_{\\mathbf{p}},a^{\\dagger})} df_{\\mathbf{p}} = \\int (a^{\\dagger} + f_{\\mathbf{p}})^{2} df_{\\mathbf{p}} and \\iint (a^{\\dagger} + f_{\\mathbf{p}}) G{(f_{\\mathbf{p}},a^{\\dagger})} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint (a^{\\dagger} + f_{\\mathbf{p}})^{2} df_{\\mathbf{p}} df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\phi_1)} = \\log{(\\cos{(\\phi_1)})}, then obtain 0 = \\frac{2 (- 4 \\operatorname{M_{E}}{(\\phi_1)} + 4 \\log{(\\cos{(\\phi_1)})}) (- \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})})}{\\operatorname{M_{E}}{(\\phi_1)}}", "derivation": "\\operatorname{M_{E}}{(\\phi_1)} = \\log{(\\cos{(\\phi_1)})} and 0 = - \\operatorname{M_{E}}{(\\phi_1)} + \\log{(\\cos{(\\phi_1)})} and \\log{(\\cos{(\\phi_1)})} = - \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})} and 0 = - 2 \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})} and 0 = 2 (- 2 \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})}) \\log{(\\cos{(\\phi_1)})} and 0 = 2 (- 4 \\operatorname{M_{E}}{(\\phi_1)} + 4 \\log{(\\cos{(\\phi_1)})}) (- \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})}) and 0 = \\frac{2 (- 4 \\operatorname{M_{E}}{(\\phi_1)} + 4 \\log{(\\cos{(\\phi_1)})}) (- \\operatorname{M_{E}}{(\\phi_1)} + 2 \\log{(\\cos{(\\phi_1)})})}{\\operatorname{M_{E}}{(\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\phi_1', commutative=True)), log(cos(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Function('M_E')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), log(cos(Symbol('\\\\phi_1', commutative=True)))))"], [["add", 2, "log(cos(Symbol('\\\\phi_1', commutative=True)))"], "Equality(log(cos(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True))))))"], [["times", 4, "Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True))))"], "Equality(Integer(0), Mul(Integer(2), Add(Mul(Integer(-1), Integer(2), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True))))), log(cos(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Mul(Integer(2), Add(Mul(Integer(-1), Integer(4), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(4), log(cos(Symbol('\\\\phi_1', commutative=True))))), Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True)))))))"], [["divide", 6, "Function('M_E')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(0), Mul(Integer(2), Add(Mul(Integer(-1), Integer(4), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(4), log(cos(Symbol('\\\\phi_1', commutative=True))))), Add(Mul(Integer(-1), Function('M_E')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\phi_1', commutative=True))))), Pow(Function('M_E')(Symbol('\\\\phi_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)} = \\sin{(\\log{(z)})}, then derive 0 = \\frac{\\sin{(\\log{(z)})} \\frac{d}{d z} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}{\\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(z)}} - \\frac{\\cos{(\\log{(z)})}}{z \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}, then obtain 0 = \\frac{\\frac{d}{d z} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}} - \\frac{\\cos{(\\log{(z)})}}{z \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(z)} = \\sin{(\\log{(z)})} and -1 = - \\frac{\\sin{(\\log{(z)})}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}} and \\frac{d}{d z} (-1) = \\frac{d}{d z} - \\frac{\\sin{(\\log{(z)})}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}} and 0 = \\frac{\\sin{(\\log{(z)})} \\frac{d}{d z} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}{\\operatorname{g^{\\prime}_{\\varepsilon}}^{2}{(z)}} - \\frac{\\cos{(\\log{(z)})}}{z \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}} and 0 = \\frac{\\frac{d}{d z} \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}} - \\frac{\\cos{(\\log{(z)})}}{z \\operatorname{g^{\\prime}_{\\varepsilon}}{(z)}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), sin(log(Symbol('z', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-1)), sin(log(Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-1)), sin(log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-2)), sin(log(Symbol('z', commutative=True))), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-1)), cos(log(Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('z', commutative=True)), Integer(-1)), cos(log(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given S{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain - \\frac{(\\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}})^{\\mathbf{s}}}{\\cos{(\\mathbf{s})}} - \\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}} = - \\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}} - \\frac{1}{\\cos{(\\mathbf{s})}}", "derivation": "S{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and \\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}} = 1 and (\\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}})^{\\mathbf{s}} = 1 and - \\frac{(\\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}})^{\\mathbf{s}}}{\\cos{(\\mathbf{s})}} = - \\frac{1}{\\cos{(\\mathbf{s})}} and - \\frac{(\\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}})^{\\mathbf{s}}}{\\cos{(\\mathbf{s})}} - \\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}} = - \\frac{S{(\\mathbf{s})}}{\\cos{(\\mathbf{s})}} - \\frac{1}{\\cos{(\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"], [["divide", 3, "Mul(Integer(-1), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Mul(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], [["minus", 4, "Mul(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given b{(J,\\hat{x})} = J + \\hat{x} and \\operatorname{t_{2}}{(\\hat{x})} = e^{\\hat{x}}, then obtain (b^{\\hat{x}}{(J,\\hat{x})})^{J} (e^{\\hat{x}})^{\\hat{x}} = ((J + \\hat{x})^{\\hat{x}})^{J} (e^{\\hat{x}})^{\\hat{x}}", "derivation": "b{(J,\\hat{x})} = J + \\hat{x} and b^{\\hat{x}}{(J,\\hat{x})} = (J + \\hat{x})^{\\hat{x}} and \\operatorname{t_{2}}{(\\hat{x})} = e^{\\hat{x}} and (b^{\\hat{x}}{(J,\\hat{x})})^{J} = ((J + \\hat{x})^{\\hat{x}})^{J} and \\operatorname{t_{2}}^{\\hat{x}}{(\\hat{x})} = (e^{\\hat{x}})^{\\hat{x}} and (b^{\\hat{x}}{(J,\\hat{x})})^{J} \\operatorname{t_{2}}^{\\hat{x}}{(\\hat{x})} = ((J + \\hat{x})^{\\hat{x}})^{J} \\operatorname{t_{2}}^{\\hat{x}}{(\\hat{x})} and (b^{\\hat{x}}{(J,\\hat{x})})^{J} (e^{\\hat{x}})^{\\hat{x}} = ((J + \\hat{x})^{\\hat{x}})^{J} (e^{\\hat{x}})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], ["get_premise", "Equality(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Add(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(exp(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 4, "Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)), Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Pow(Add(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)), Pow(Function('t_2')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Pow(Function('b')(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Pow(Add(Symbol('J', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = A_{1} + F_{H} - \\phi, then derive - A_{1} + \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = - A_{1} - 1, then obtain - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi) + \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} + 1 = - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi)", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = A_{1} + F_{H} - \\phi and \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi) and - A_{1} + \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi) and - A_{1} + \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} = - A_{1} - 1 and - A_{1} - 1 = - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi) and - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi) + \\frac{\\partial}{\\partial \\phi} \\operatorname{f_{\\mathbf{v}}}{(\\phi,F_{H},A_{1})} + 1 = - A_{1} + \\frac{\\partial}{\\partial \\phi} (A_{1} + F_{H} - \\phi)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(\\theta_1,\\mathbf{D},A_{1})} = A_{1} \\mathbf{D} \\theta_1, then obtain (e^{A_{1} \\mathbf{D} \\theta_1 \\lambda{(\\theta_1,\\mathbf{D},A_{1})}})^{\\mathbf{D}} = (e^{A_{1}^{2} \\mathbf{D}^{2} \\theta_1^{2}})^{\\mathbf{D}}", "derivation": "\\lambda{(\\theta_1,\\mathbf{D},A_{1})} = A_{1} \\mathbf{D} \\theta_1 and A_{1} \\mathbf{D} \\theta_1 \\lambda{(\\theta_1,\\mathbf{D},A_{1})} = A_{1}^{2} \\mathbf{D}^{2} \\theta_1^{2} and e^{A_{1} \\mathbf{D} \\theta_1 \\lambda{(\\theta_1,\\mathbf{D},A_{1})}} = e^{A_{1}^{2} \\mathbf{D}^{2} \\theta_1^{2}} and (e^{A_{1} \\mathbf{D} \\theta_1 \\lambda{(\\theta_1,\\mathbf{D},A_{1})}})^{\\mathbf{D}} = (e^{A_{1}^{2} \\mathbf{D}^{2} \\theta_1^{2}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True), Function('\\\\lambda')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2))))"], [["exp", 2], "Equality(exp(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True), Function('\\\\lambda')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_1', commutative=True)))), exp(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(exp(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\theta_1', commutative=True), Function('\\\\lambda')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_1', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(exp(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(I,A_{2})} = \\log{(A_{2} - I)}, then obtain \\log{(((A_{2} - I) \\operatorname{z^{*}}{(I,A_{2})})^{I} (A_{2} - I))} = \\log{(((A_{2} - I) \\log{(A_{2} - I)})^{I} (A_{2} - I))}", "derivation": "\\operatorname{z^{*}}{(I,A_{2})} = \\log{(A_{2} - I)} and (A_{2} - I) \\operatorname{z^{*}}{(I,A_{2})} = (A_{2} - I) \\log{(A_{2} - I)} and ((A_{2} - I) \\operatorname{z^{*}}{(I,A_{2})})^{I} = ((A_{2} - I) \\log{(A_{2} - I)})^{I} and ((A_{2} - I) \\operatorname{z^{*}}{(I,A_{2})})^{I} (A_{2} - I) = ((A_{2} - I) \\log{(A_{2} - I)})^{I} (A_{2} - I) and \\log{(((A_{2} - I) \\operatorname{z^{*}}{(I,A_{2})})^{I} (A_{2} - I))} = \\log{(((A_{2} - I) \\log{(A_{2} - I)})^{I} (A_{2} - I))}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('I', commutative=True), Symbol('A_2', commutative=True)), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["times", 1, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))"], "Equality(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Function('z^*')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Function('z^*')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Symbol('I', commutative=True)), Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Symbol('I', commutative=True)))"], [["times", 3, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))"], "Equality(Mul(Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Function('z^*')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Symbol('I', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Symbol('I', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["log", 4], "Equality(log(Mul(Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Function('z^*')(Symbol('I', commutative=True), Symbol('A_2', commutative=True))), Symbol('I', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), log(Mul(Pow(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), log(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Symbol('I', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbb{I},v_{y})} = \\mathbb{I} + v_{y}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} \\log{(2 v_{y} (\\mathbb{I} + v_{y}))} = \\frac{\\partial}{\\partial \\mathbb{I}} \\log{(v_{y} (2 \\mathbb{I} + 2 v_{y}))}", "derivation": "\\mathbf{D}{(\\mathbb{I},v_{y})} = \\mathbb{I} + v_{y} and \\mathbb{I} + v_{y} + \\mathbf{D}{(\\mathbb{I},v_{y})} = 2 \\mathbb{I} + 2 v_{y} and v_{y} (\\mathbb{I} + v_{y} + \\mathbf{D}{(\\mathbb{I},v_{y})}) = v_{y} (2 \\mathbb{I} + 2 v_{y}) and 2 v_{y} \\mathbf{D}{(\\mathbb{I},v_{y})} = v_{y} (2 \\mathbb{I} + 2 v_{y}) and 2 v_{y} (\\mathbb{I} + v_{y}) = v_{y} (2 \\mathbb{I} + 2 v_{y}) and \\log{(2 v_{y} (\\mathbb{I} + v_{y}))} = \\log{(v_{y} (2 \\mathbb{I} + 2 v_{y}))} and \\frac{\\partial}{\\partial \\mathbb{I}} \\log{(2 v_{y} (\\mathbb{I} + v_{y}))} = \\frac{\\partial}{\\partial \\mathbb{I}} \\log{(v_{y} (2 \\mathbb{I} + 2 v_{y}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))))"], [["times", 2, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('v_y', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Symbol('v_y', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Symbol('v_y', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)))))"], [["log", 5], "Equality(log(Mul(Integer(2), Symbol('v_y', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True)))), log(Mul(Symbol('v_y', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))))))"], [["differentiate", 6, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(log(Mul(Integer(2), Symbol('v_y', commutative=True), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('v_y', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(v_{t})} = \\cos{(e^{v_{t}})}, then obtain \\sin{(\\frac{n{(v_{t})}}{\\cos{(e^{v_{t}})}} + 1 - \\frac{2}{\\cos{(e^{v_{t}})}})} = \\sin{(2 - \\frac{2}{\\cos{(e^{v_{t}})}})}", "derivation": "n{(v_{t})} = \\cos{(e^{v_{t}})} and \\frac{n{(v_{t})}}{\\cos{(e^{v_{t}})}} = 1 and \\frac{n{(v_{t})}}{\\cos{(e^{v_{t}})}} - \\frac{1}{\\cos{(e^{v_{t}})}} = 1 - \\frac{1}{\\cos{(e^{v_{t}})}} and \\frac{n{(v_{t})}}{\\cos{(e^{v_{t}})}} + 1 - \\frac{2}{\\cos{(e^{v_{t}})}} = 2 - \\frac{2}{\\cos{(e^{v_{t}})}} and \\sin{(\\frac{n{(v_{t})}}{\\cos{(e^{v_{t}})}} + 1 - \\frac{2}{\\cos{(e^{v_{t}})}})} = \\sin{(2 - \\frac{2}{\\cos{(e^{v_{t}})}})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('v_t', commutative=True)), cos(exp(Symbol('v_t', commutative=True))))"], [["divide", 1, "cos(exp(Symbol('v_t', commutative=True)))"], "Equality(Mul(Function('n')(Symbol('v_t', commutative=True)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Function('n')(Symbol('v_t', commutative=True)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1)))))"], [["add", 3, "Add(Integer(1), Mul(Integer(-1), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))))"], "Equality(Add(Mul(Function('n')(Symbol('v_t', commutative=True)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))), Integer(1), Mul(Integer(-1), Integer(2), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1)))), Add(Integer(2), Mul(Integer(-1), Integer(2), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1)))))"], [["sin", 4], "Equality(sin(Add(Mul(Function('n')(Symbol('v_t', commutative=True)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))), Integer(1), Mul(Integer(-1), Integer(2), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))))), sin(Add(Integer(2), Mul(Integer(-1), Integer(2), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(z^{*},q)} = \\frac{\\cos{(q)}}{z^{*}}, then derive \\frac{\\partial}{\\partial z^{*}} \\operatorname{n_{1}}{(z^{*},q)} = - \\frac{\\cos{(q)}}{(z^{*})^{2}}, then obtain - \\frac{\\operatorname{n_{1}}{(z^{*},q)}}{z^{*}} = - \\frac{\\cos{(q)}}{(z^{*})^{2}}", "derivation": "\\operatorname{n_{1}}{(z^{*},q)} = \\frac{\\cos{(q)}}{z^{*}} and \\frac{\\partial}{\\partial z^{*}} \\operatorname{n_{1}}{(z^{*},q)} = \\frac{\\partial}{\\partial z^{*}} \\frac{\\cos{(q)}}{z^{*}} and \\frac{\\partial}{\\partial z^{*}} \\operatorname{n_{1}}{(z^{*},q)} = - \\frac{\\cos{(q)}}{(z^{*})^{2}} and - \\frac{\\cos{(q)}}{(z^{*})^{2}} = \\frac{\\partial}{\\partial z^{*}} \\frac{\\cos{(q)}}{z^{*}} and - \\frac{\\operatorname{n_{1}}{(z^{*},q)}}{z^{*}} = \\frac{\\partial}{\\partial z^{*}} \\operatorname{n_{1}}{(z^{*},q)} and - \\frac{\\operatorname{n_{1}}{(z^{*},q)}}{z^{*}} = - \\frac{\\cos{(q)}}{(z^{*})^{2}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-2)), cos(Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-2)), cos(Symbol('q', commutative=True))), Derivative(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), cos(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True))), Derivative(Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('n_1')(Symbol('z^*', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-2)), cos(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\dot{z})} = \\dot{z}, then derive \\mathbf{S} + \\operatorname{v_{y}}{(\\dot{z})} = \\dot{z} + v, then obtain \\operatorname{v_{y}}{(\\dot{z} + \\mathbf{S} - v)} = \\dot{z} + \\mathbf{S} - v", "derivation": "\\operatorname{v_{y}}{(\\dot{z})} = \\dot{z} and \\frac{d}{d \\dot{z}} \\operatorname{v_{y}}{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\dot{z} and \\int \\frac{d}{d \\dot{z}} \\operatorname{v_{y}}{(\\dot{z})} d\\dot{z} = \\int \\frac{d}{d \\dot{z}} \\dot{z} d\\dot{z} and \\mathbf{S} + \\operatorname{v_{y}}{(\\dot{z})} = \\dot{z} + v and \\dot{z} + \\mathbf{S} = \\dot{z} + v and \\dot{z} + \\mathbf{S} - v = \\dot{z} and \\operatorname{v_{y}}{(\\dot{z} + \\mathbf{S} - v)} = \\dot{z} + \\mathbf{S} - v", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Derivative(Function('v_y')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('v_y')(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('v', commutative=True)))"], [["minus", 5, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Symbol('\\\\dot{z}', commutative=True))"], [["substitute_RHS_for_LHS", 1, 6], "Equality(Function('v_y')(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"]]}, {"prompt": "Given J{(\\sigma_x,\\mathbf{M})} = \\mathbf{M} - \\sigma_x, then obtain \\frac{\\partial}{\\partial \\mathbf{M}} (J{(\\sigma_x,\\mathbf{M})} + \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + J{(\\sigma_x,\\mathbf{M})})) = \\frac{\\partial}{\\partial \\mathbf{M}} (J{(\\sigma_x,\\mathbf{M})} + \\frac{d}{d \\sigma_x} \\mathbf{M})", "derivation": "J{(\\sigma_x,\\mathbf{M})} = \\mathbf{M} - \\sigma_x and \\sigma_x + J{(\\sigma_x,\\mathbf{M})} = \\mathbf{M} and \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + J{(\\sigma_x,\\mathbf{M})}) = \\frac{d}{d \\sigma_x} \\mathbf{M} and J{(\\sigma_x,\\mathbf{M})} + \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + J{(\\sigma_x,\\mathbf{M})}) = J{(\\sigma_x,\\mathbf{M})} + \\frac{d}{d \\sigma_x} \\mathbf{M} and \\frac{\\partial}{\\partial \\mathbf{M}} (J{(\\sigma_x,\\mathbf{M})} + \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + J{(\\sigma_x,\\mathbf{M})})) = \\frac{\\partial}{\\partial \\mathbf{M}} (J{(\\sigma_x,\\mathbf{M})} + \\frac{d}{d \\sigma_x} \\mathbf{M})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["add", 3, "Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Add(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(U,G)} = \\frac{\\partial}{\\partial U} (- G + U) and \\operatorname{f^{*}}{(U,G)} = \\frac{\\partial}{\\partial G} y{(U,G)}, then derive \\operatorname{f^{*}}{(U,G)} = 0, then obtain \\frac{\\operatorname{f^{*}}^{U}{(U,G)}}{\\frac{\\partial}{\\partial U} (- G + U)} = \\frac{0^{U}}{\\frac{\\partial}{\\partial U} (- G + U)}", "derivation": "y{(U,G)} = \\frac{\\partial}{\\partial U} (- G + U) and \\frac{\\partial}{\\partial G} y{(U,G)} = \\frac{\\partial^{2}}{\\partial G\\partial U} (- G + U) and \\operatorname{f^{*}}{(U,G)} = \\frac{\\partial}{\\partial G} y{(U,G)} and \\operatorname{f^{*}}{(U,G)} = \\frac{\\partial^{2}}{\\partial G\\partial U} (- G + U) and \\operatorname{f^{*}}{(U,G)} = 0 and \\operatorname{f^{*}}^{U}{(U,G)} = 0^{U} and \\frac{\\operatorname{f^{*}}^{U}{(U,G)}}{y{(U,G)}} = \\frac{0^{U}}{y{(U,G)}} and \\frac{\\operatorname{f^{*}}^{U}{(U,G)}}{\\frac{\\partial}{\\partial U} (- G + U)} = \\frac{0^{U}}{\\frac{\\partial}{\\partial U} (- G + U)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Derivative(Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Integer(0))"], [["power", 5, "Symbol('U', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Pow(Integer(0), Symbol('U', commutative=True)))"], [["divide", 6, "Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Pow(Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Integer(-1))), Mul(Pow(Integer(0), Symbol('U', commutative=True)), Pow(Function('y')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Pow(Function('f^*')(Symbol('U', commutative=True), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Integer(0), Symbol('U', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(M_{E})} = \\log{(\\log{(M_{E})})}, then obtain \\frac{(\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})})^{M_{E}}}{\\operatorname{E_{x}}{(M_{E})}} = \\frac{(\\frac{d}{d M_{E}} \\log{(\\log{(M_{E})})})^{M_{E}}}{\\operatorname{E_{x}}{(M_{E})}}", "derivation": "\\operatorname{E_{x}}{(M_{E})} = \\log{(\\log{(M_{E})})} and \\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})} = \\frac{d}{d M_{E}} \\log{(\\log{(M_{E})})} and (\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})})^{M_{E}} = (\\frac{d}{d M_{E}} \\log{(\\log{(M_{E})})})^{M_{E}} and \\frac{(\\frac{d}{d M_{E}} \\operatorname{E_{x}}{(M_{E})})^{M_{E}}}{\\operatorname{E_{x}}{(M_{E})}} = \\frac{(\\frac{d}{d M_{E}} \\log{(\\log{(M_{E})})})^{M_{E}}}{\\operatorname{E_{x}}{(M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('M_E', commutative=True)), log(log(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(log(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True)), Pow(Derivative(log(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True)))"], [["divide", 3, "Function('E_x')(Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Function('E_x')(Symbol('M_E', commutative=True)), Integer(-1)), Pow(Derivative(Function('E_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True))), Mul(Pow(Function('E_x')(Symbol('M_E', commutative=True)), Integer(-1)), Pow(Derivative(log(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(F_{N})} = \\log{(F_{N})} and f{(F_{N})} = (\\operatorname{v_{t}}{(F_{N})} + \\log{(F_{N})})^{2} \\operatorname{v_{t}}{(F_{N})}, then obtain \\int (f^{F_{N}}{(F_{N})} - \\operatorname{v_{t}}^{4}{(F_{N})}) dF_{N} = \\int ((4 \\operatorname{v_{t}}^{3}{(F_{N})})^{F_{N}} - \\operatorname{v_{t}}^{4}{(F_{N})}) dF_{N}", "derivation": "\\operatorname{v_{t}}{(F_{N})} = \\log{(F_{N})} and f{(F_{N})} = (\\operatorname{v_{t}}{(F_{N})} + \\log{(F_{N})})^{2} \\operatorname{v_{t}}{(F_{N})} and f{(F_{N})} = 4 \\operatorname{v_{t}}^{3}{(F_{N})} and f^{F_{N}}{(F_{N})} = (4 \\operatorname{v_{t}}^{3}{(F_{N})})^{F_{N}} and f^{F_{N}}{(F_{N})} - \\operatorname{v_{t}}^{4}{(F_{N})} = (4 \\operatorname{v_{t}}^{3}{(F_{N})})^{F_{N}} - \\operatorname{v_{t}}^{4}{(F_{N})} and \\int (f^{F_{N}}{(F_{N})} - \\operatorname{v_{t}}^{4}{(F_{N})}) dF_{N} = \\int ((4 \\operatorname{v_{t}}^{3}{(F_{N})})^{F_{N}} - \\operatorname{v_{t}}^{4}{(F_{N})}) dF_{N}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('F_N', commutative=True)), Mul(Pow(Add(Function('v_t')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Integer(2)), Function('v_t')(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f')(Symbol('F_N', commutative=True)), Mul(Integer(4), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(3))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('f')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(Mul(Integer(4), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(3))), Symbol('F_N', commutative=True)))"], [["minus", 4, "Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(4))"], "Equality(Add(Pow(Function('f')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(4)))), Add(Pow(Mul(Integer(4), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(3))), Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(4)))))"], [["integrate", 5, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Pow(Function('f')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(4)))), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Pow(Mul(Integer(4), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(3))), Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Function('v_t')(Symbol('F_N', commutative=True)), Integer(4)))), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\mu{(E)} = \\log{(e^{E})}, then derive \\int \\mu{(E)} dE = \\frac{E^{2}}{2} + f_{\\mathbf{p}}, then derive 0 = \\frac{E^{2}}{2} + \\mathbf{v} - \\int \\mu{(E)} dE, then obtain (0^{\\mathbf{v}})^{\\mathbf{v}} = ((\\mathbf{v} - f_{\\mathbf{p}})^{\\mathbf{v}})^{\\mathbf{v}}", "derivation": "\\mu{(E)} = \\log{(e^{E})} and \\int \\mu{(E)} dE = \\int \\log{(e^{E})} dE and 0 = - \\int \\mu{(E)} dE + \\int \\log{(e^{E})} dE and \\int \\mu{(E)} dE = \\frac{E^{2}}{2} + f_{\\mathbf{p}} and 0 = \\frac{E^{2}}{2} + \\mathbf{v} - \\int \\mu{(E)} dE and 0 = \\mathbf{v} - f_{\\mathbf{p}} and 0^{\\mathbf{v}} = (\\mathbf{v} - f_{\\mathbf{p}})^{\\mathbf{v}} and (0^{\\mathbf{v}})^{\\mathbf{v}} = ((\\mathbf{v} - f_{\\mathbf{p}})^{\\mathbf{v}})^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True)), log(exp(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mu')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\mu')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Integral(log(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2))), Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Integral(Function('\\\\mu')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 6, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["power", 7, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given M{(\\mathbf{v})} = \\sin{(e^{\\mathbf{v}})} and \\operatorname{v_{y}}{(\\mathbf{v})} = e^{\\mathbf{v}}, then obtain \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{M^{2}{(\\mathbf{v})}} = \\frac{e^{\\mathbf{v}}}{M^{2}{(\\mathbf{v})}}", "derivation": "M{(\\mathbf{v})} = \\sin{(e^{\\mathbf{v}})} and \\operatorname{v_{y}}{(\\mathbf{v})} = e^{\\mathbf{v}} and \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\sin{(e^{\\mathbf{v}})}} = \\frac{e^{\\mathbf{v}}}{\\sin{(e^{\\mathbf{v}})}} and \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\sin^{2}{(e^{\\mathbf{v}})}} = \\frac{e^{\\mathbf{v}}}{\\sin^{2}{(e^{\\mathbf{v}})}} and \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{M^{2}{(\\mathbf{v})}} = \\frac{e^{\\mathbf{v}}}{M^{2}{(\\mathbf{v})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{v}', commutative=True)), sin(exp(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 2, "sin(exp(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Mul(exp(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))))"], [["times", 3, "Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))"], "Equality(Mul(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-2))), Mul(exp(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('M')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-2)), Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Function('M')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-2)), exp(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given b{(y)} = \\cos{(y)}, then derive \\int b{(y)} dy = \\hat{x} + \\sin{(y)}, then obtain (2 \\hat{x} + \\sin{(y)})^{y} = (\\hat{x} + \\int \\cos{(y)} dy)^{y}", "derivation": "b{(y)} = \\cos{(y)} and \\int b{(y)} dy = \\int \\cos{(y)} dy and \\int b{(y)} dy = \\hat{x} + \\sin{(y)} and \\hat{x} + \\int b{(y)} dy = \\hat{x} + \\int \\cos{(y)} dy and 2 \\hat{x} + \\sin{(y)} = \\hat{x} + \\int \\cos{(y)} dy and 2 \\hat{x} + \\sin{(y)} = \\hat{x} + \\int b{(y)} dy and (2 \\hat{x} + \\sin{(y)})^{y} = (\\hat{x} + \\int b{(y)} dy)^{y} and (2 \\hat{x} + \\sin{(y)})^{y} = (\\hat{x} + \\int \\cos{(y)} dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('b')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('y', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Integral(Function('b')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('\\\\hat{x}', commutative=True), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('y', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('y', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Integral(Function('b')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["power", 6, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Integral(Function('b')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(P_{g},L)} = L P_{g}, then derive \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL = \\frac{\\partial}{\\partial L} (F_{c} + \\frac{L^{2}}{2}), then obtain \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL - 1 = \\frac{\\partial}{\\partial L} (F_{c} + \\frac{L^{2}}{2}) - 1", "derivation": "\\hat{p}_0{(P_{g},L)} = L P_{g} and \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} = \\frac{\\partial}{\\partial P_{g}} L P_{g} and \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL = \\int \\frac{\\partial}{\\partial P_{g}} L P_{g} dL and \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL = \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} L P_{g} dL and \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL = \\frac{\\partial}{\\partial L} (F_{c} + \\frac{L^{2}}{2}) and \\frac{\\partial}{\\partial L} \\int \\frac{\\partial}{\\partial P_{g}} \\hat{p}_0{(P_{g},L)} dL - 1 = \\frac{\\partial}{\\partial L} (F_{c} + \\frac{L^{2}}{2}) - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('P_g', commutative=True)))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Mul(Symbol('L', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Symbol('L', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Derivative(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 5, 1], "Equality(Add(Derivative(Integral(Derivative(Function('\\\\hat{p}_0')(Symbol('P_g', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2)))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\varepsilon_0,F_{g})} = e^{- F_{g} + \\varepsilon_0}, then obtain (F_{g} - \\varepsilon_0 - \\operatorname{t_{2}}{(\\varepsilon_0,F_{g})})^{\\varepsilon_0} = (F_{g} - \\varepsilon_0 - e^{- F_{g} + \\varepsilon_0})^{\\varepsilon_0}", "derivation": "\\operatorname{t_{2}}{(\\varepsilon_0,F_{g})} = e^{- F_{g} + \\varepsilon_0} and - F_{g} + \\varepsilon_0 + \\operatorname{t_{2}}{(\\varepsilon_0,F_{g})} = - F_{g} + \\varepsilon_0 + e^{- F_{g} + \\varepsilon_0} and F_{g} - \\varepsilon_0 - \\operatorname{t_{2}}{(\\varepsilon_0,F_{g})} = F_{g} - \\varepsilon_0 - e^{- F_{g} + \\varepsilon_0} and (F_{g} - \\varepsilon_0 - \\operatorname{t_{2}}{(\\varepsilon_0,F_{g})})^{\\varepsilon_0} = (F_{g} - \\varepsilon_0 - e^{- F_{g} + \\varepsilon_0})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_g', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_g', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_g', commutative=True)))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_g', commutative=True)))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given Q{(\\hat{\\mathbf{r}},\\eta)} = - \\eta + \\hat{\\mathbf{r}}, then obtain (\\hat{\\mathbf{r}} + \\cos{(\\eta + Q{(\\hat{\\mathbf{r}},\\eta)})})^{\\eta} = (\\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}})})^{\\eta}", "derivation": "Q{(\\hat{\\mathbf{r}},\\eta)} = - \\eta + \\hat{\\mathbf{r}} and \\eta + Q{(\\hat{\\mathbf{r}},\\eta)} = \\hat{\\mathbf{r}} and \\cos{(\\eta + Q{(\\hat{\\mathbf{r}},\\eta)})} = \\cos{(\\hat{\\mathbf{r}})} and \\hat{\\mathbf{r}} + \\cos{(\\eta + Q{(\\hat{\\mathbf{r}},\\eta)})} = \\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}})} and (\\hat{\\mathbf{r}} + \\cos{(\\eta + Q{(\\hat{\\mathbf{r}},\\eta)})})^{\\eta} = (\\hat{\\mathbf{r}} + \\cos{(\\hat{\\mathbf{r}})})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('Q')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\eta', commutative=True), Function('Q')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta', commutative=True)))), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["add", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Add(Symbol('\\\\eta', commutative=True), Function('Q')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta', commutative=True))))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["power", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Add(Symbol('\\\\eta', commutative=True), Function('Q')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\eta', commutative=True))))), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(q)} = \\cos{(q)}, then obtain \\operatorname{t_{1}}{(q)} \\int \\operatorname{t_{1}}{(q)} dq + \\int \\cos{(q)} dq = \\operatorname{t_{1}}{(q)} \\int \\cos{(q)} dq + \\int \\cos{(q)} dq", "derivation": "\\operatorname{t_{1}}{(q)} = \\cos{(q)} and \\int \\operatorname{t_{1}}{(q)} dq = \\int \\cos{(q)} dq and \\operatorname{t_{1}}{(q)} \\int \\operatorname{t_{1}}{(q)} dq = \\operatorname{t_{1}}{(q)} \\int \\cos{(q)} dq and \\operatorname{t_{1}}{(q)} \\int \\operatorname{t_{1}}{(q)} dq + \\int \\cos{(q)} dq = \\operatorname{t_{1}}{(q)} \\int \\cos{(q)} dq + \\int \\cos{(q)} dq", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["times", 2, "Function('t_1')(Symbol('q', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('q', commutative=True)), Integral(Function('t_1')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Function('t_1')(Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["add", 3, "Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Function('t_1')(Symbol('q', commutative=True)), Integral(Function('t_1')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Mul(Function('t_1')(Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given E{(\\hat{p},m_{s})} = m_{s}^{\\hat{p}}, then obtain m_{s}^{\\hat{p}} + \\sin{(\\frac{E^{m_{s}}{(\\hat{p},m_{s})}}{m_{s}})} = m_{s}^{\\hat{p}} + \\sin{(\\frac{(m_{s}^{\\hat{p}})^{m_{s}}}{m_{s}})}", "derivation": "E{(\\hat{p},m_{s})} = m_{s}^{\\hat{p}} and E^{m_{s}}{(\\hat{p},m_{s})} = (m_{s}^{\\hat{p}})^{m_{s}} and \\frac{E^{m_{s}}{(\\hat{p},m_{s})}}{m_{s}} = \\frac{(m_{s}^{\\hat{p}})^{m_{s}}}{m_{s}} and \\sin{(\\frac{E^{m_{s}}{(\\hat{p},m_{s})}}{m_{s}})} = \\sin{(\\frac{(m_{s}^{\\hat{p}})^{m_{s}}}{m_{s}})} and m_{s}^{\\hat{p}} + \\sin{(\\frac{E^{m_{s}}{(\\hat{p},m_{s})}}{m_{s}})} = m_{s}^{\\hat{p}} + \\sin{(\\frac{(m_{s}^{\\hat{p}})^{m_{s}}}{m_{s}})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('m_s', commutative=True)))"], [["divide", 2, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('m_s', commutative=True))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))), sin(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('m_s', commutative=True)))))"], [["add", 4, "Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), sin(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))))), Add(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), sin(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given q{(T)} = \\cos{(T)}, then obtain e^{\\cos{(\\frac{d}{d T} (T + q{(T)} - \\cos{(T)}))}} = e^{\\cos{(\\frac{d}{d T} T)}}", "derivation": "q{(T)} = \\cos{(T)} and q{(T)} - \\cos{(T)} = 0 and T + q{(T)} - \\cos{(T)} = T and \\frac{d}{d T} (T + q{(T)} - \\cos{(T)}) = \\frac{d}{d T} T and \\cos{(\\frac{d}{d T} (T + q{(T)} - \\cos{(T)}))} = \\cos{(\\frac{d}{d T} T)} and e^{\\cos{(\\frac{d}{d T} (T + q{(T)} - \\cos{(T)}))}} = e^{\\cos{(\\frac{d}{d T} T)}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["minus", 1, "cos(Symbol('T', commutative=True))"], "Equality(Add(Function('q')(Symbol('T', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Integer(0))"], [["add", 2, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('q')(Symbol('T', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Symbol('T', commutative=True))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Symbol('T', commutative=True), Function('q')(Symbol('T', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Symbol('T', commutative=True), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Add(Symbol('T', commutative=True), Function('q')(Symbol('T', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))), cos(Derivative(Symbol('T', commutative=True), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["exp", 5], "Equality(exp(cos(Derivative(Add(Symbol('T', commutative=True), Function('q')(Symbol('T', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))), exp(cos(Derivative(Symbol('T', commutative=True), Tuple(Symbol('T', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\ddot{x}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain \\frac{L_{\\varepsilon} e^{\\ddot{x}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}}}}{\\operatorname{x^{{\\}'}}{(\\rho)}} = \\frac{L_{\\varepsilon}}{\\operatorname{x^{{\\}'}}{(\\rho)}}", "derivation": "\\ddot{x}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\ddot{x}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}} = 0 and e^{\\ddot{x}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}}} = 1 and \\frac{L_{\\varepsilon} e^{\\ddot{x}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}}}}{\\operatorname{x^{{\\}'}}{(\\rho)}} = \\frac{L_{\\varepsilon}}{\\operatorname{x^{{\\}'}}{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "exp(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Function('\\\\ddot{x}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))), Integer(1))"], [["divide", 3, "Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Function('x^\\\\prime')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Add(Function('\\\\ddot{x}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Function('x^\\\\prime')(Symbol('\\\\rho', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(S,\\pi)} = \\sin{(\\pi^{S})}, then derive \\pi^{S} \\log{(\\pi)} + 2 \\frac{\\partial}{\\partial S} \\operatorname{r_{0}}{(S,\\pi)} = \\pi^{S} \\log{(\\pi)} \\cos{(\\pi^{S})} + \\pi^{S} \\log{(\\pi)} + \\frac{\\partial}{\\partial S} \\operatorname{r_{0}}{(S,\\pi)}, then obtain \\pi^{S} \\log{(\\pi)} + 2 \\frac{\\partial}{\\partial S} \\sin{(\\pi^{S})} = \\pi^{S} \\log{(\\pi)} \\cos{(\\pi^{S})} + \\pi^{S} \\log{(\\pi)} + \\frac{\\partial}{\\partial S} \\sin{(\\pi^{S})}", "derivation": "\\operatorname{r_{0}}{(S,\\pi)} = \\sin{(\\pi^{S})} and 2 \\operatorname{r_{0}}{(S,\\pi)} = \\operatorname{r_{0}}{(S,\\pi)} + \\sin{(\\pi^{S})} and \\pi^{S} + 2 \\operatorname{r_{0}}{(S,\\pi)} = \\pi^{S} + \\operatorname{r_{0}}{(S,\\pi)} + \\sin{(\\pi^{S})} and \\frac{\\partial}{\\partial S} (\\pi^{S} + 2 \\operatorname{r_{0}}{(S,\\pi)}) = \\frac{\\partial}{\\partial S} (\\pi^{S} + \\operatorname{r_{0}}{(S,\\pi)} + \\sin{(\\pi^{S})}) and \\pi^{S} \\log{(\\pi)} + 2 \\frac{\\partial}{\\partial S} \\operatorname{r_{0}}{(S,\\pi)} = \\pi^{S} \\log{(\\pi)} \\cos{(\\pi^{S})} + \\pi^{S} \\log{(\\pi)} + \\frac{\\partial}{\\partial S} \\operatorname{r_{0}}{(S,\\pi)} and \\pi^{S} \\log{(\\pi)} + 2 \\frac{\\partial}{\\partial S} \\sin{(\\pi^{S})} = \\pi^{S} \\log{(\\pi)} \\cos{(\\pi^{S})} + \\pi^{S} \\log{(\\pi)} + \\frac{\\partial}{\\partial S} \\sin{(\\pi^{S})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True))))"], [["add", 1, "Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(2), Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)))))"], [["add", 2, "Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), Mul(Integer(2), Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)))))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), Mul(Integer(2), Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Derivative(Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Derivative(Function('r_0')(Symbol('S', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Derivative(sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Derivative(sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(t_{2},k)} = \\cos{(k^{t_{2}})}, then obtain \\hat{x}_0{(t_{2},k)} \\int \\hat{x}_0{(t_{2},k)} dt_{2} = \\hat{x}_0{(t_{2},k)} \\int \\cos{(k^{t_{2}})} dt_{2}", "derivation": "\\hat{x}_0{(t_{2},k)} = \\cos{(k^{t_{2}})} and \\int \\hat{x}_0{(t_{2},k)} dt_{2} = \\int \\cos{(k^{t_{2}})} dt_{2} and \\cos{(k^{t_{2}})} \\int \\hat{x}_0{(t_{2},k)} dt_{2} = \\cos{(k^{t_{2}})} \\int \\cos{(k^{t_{2}})} dt_{2} and \\hat{x}_0{(t_{2},k)} \\int \\hat{x}_0{(t_{2},k)} dt_{2} = \\hat{x}_0{(t_{2},k)} \\int \\cos{(k^{t_{2}})} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["times", 2, "cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Mul(cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))), Integral(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))), Integral(cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Integral(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Function('\\\\hat{x}_0')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Integral(cos(Pow(Symbol('k', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\mu{(T,\\theta_2)} = - T + \\theta_2, then derive - \\frac{\\partial}{\\partial \\theta_2} \\mu{(T,\\theta_2)} = -1, then obtain - \\mu{(T,\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\mu{(T,\\theta_2)} = - \\mu{(T,\\theta_2)}", "derivation": "\\mu{(T,\\theta_2)} = - T + \\theta_2 and - \\mu{(T,\\theta_2)} = T - \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} - \\mu{(T,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (T - \\theta_2) and - \\frac{\\partial}{\\partial \\theta_2} \\mu{(T,\\theta_2)} = -1 and - \\mu{(T,\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\mu{(T,\\theta_2)} = - \\mu{(T,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Integer(-1))"], [["times", 4, "Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given f{(\\varphi)} = \\sin{(\\varphi)} and q{(\\varphi)} = \\frac{\\sin{(\\varphi)}}{\\varphi}, then obtain ((\\frac{f{(\\varphi)}}{\\varphi})^{\\varphi})^{\\varphi} = ((\\frac{\\sin{(\\varphi)}}{\\varphi})^{\\varphi})^{\\varphi}", "derivation": "f{(\\varphi)} = \\sin{(\\varphi)} and \\frac{f{(\\varphi)}}{\\varphi} = \\frac{\\sin{(\\varphi)}}{\\varphi} and q{(\\varphi)} = \\frac{\\sin{(\\varphi)}}{\\varphi} and q{(\\varphi)} = \\frac{f{(\\varphi)}}{\\varphi} and q^{\\varphi}{(\\varphi)} = (\\frac{\\sin{(\\varphi)}}{\\varphi})^{\\varphi} and (q^{\\varphi}{(\\varphi)})^{\\varphi} = ((\\frac{\\sin{(\\varphi)}}{\\varphi})^{\\varphi})^{\\varphi} and ((\\frac{f{(\\varphi)}}{\\varphi})^{\\varphi})^{\\varphi} = ((\\frac{\\sin{(\\varphi)}}{\\varphi})^{\\varphi})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('q')(Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\varphi', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["power", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Function('q')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), sin(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(f)} = \\frac{d}{d f} \\cos{(f)} and \\operatorname{t_{2}}{(f)} = - \\operatorname{m_{s}}{(f)}, then obtain \\frac{d^{2}}{d f^{2}} 0 = \\frac{d^{2}}{d f^{2}} (\\operatorname{m_{s}}{(f)} + \\operatorname{t_{2}}{(f)})", "derivation": "\\operatorname{m_{s}}{(f)} = \\frac{d}{d f} \\cos{(f)} and 0 = - \\operatorname{m_{s}}{(f)} + \\frac{d}{d f} \\cos{(f)} and \\frac{d}{d f} 0 = \\frac{d}{d f} (- \\operatorname{m_{s}}{(f)} + \\frac{d}{d f} \\cos{(f)}) and \\frac{d^{2}}{d f^{2}} 0 = \\frac{d^{2}}{d f^{2}} (- \\operatorname{m_{s}}{(f)} + \\frac{d}{d f} \\cos{(f)}) and \\operatorname{t_{2}}{(f)} = - \\operatorname{m_{s}}{(f)} and \\frac{d^{2}}{d f^{2}} 0 = \\frac{d^{2}}{d f^{2}} (\\operatorname{t_{2}}{(f)} + \\frac{d}{d f} \\cos{(f)}) and \\frac{d^{2}}{d f^{2}} 0 = \\frac{d^{2}}{d f^{2}} (\\operatorname{m_{s}}{(f)} + \\operatorname{t_{2}}{(f)})", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('f', commutative=True)), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["minus", 1, "Function('m_s')(Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m_s')(Symbol('f', commutative=True))), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('f', commutative=True))), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('f', commutative=True))), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Tuple(Symbol('f', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('f', commutative=True)), Mul(Integer(-1), Function('m_s')(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Add(Function('t_2')(Symbol('f', commutative=True)), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Tuple(Symbol('f', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Add(Function('m_s')(Symbol('f', commutative=True)), Function('t_2')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\dot{z}{(\\phi_2,f)} = \\phi_2 - f, then obtain -1 = \\frac{- \\phi_2 + f}{\\phi_2 - f}", "derivation": "\\dot{z}{(\\phi_2,f)} = \\phi_2 - f and - \\dot{z}{(\\phi_2,f)} = - \\phi_2 + f and - \\frac{\\dot{z}{(\\phi_2,f)}}{\\phi_2 - f} = \\frac{- \\phi_2 + f}{\\phi_2 - f} and -1 = \\frac{- \\phi_2 + f}{\\phi_2 - f}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)))"], [["divide", 2, "Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Integer(-1)), Function('\\\\dot{z}')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(-1), Mul(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Pow(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(A_{x},C_{d})} = e^{A_{x} - C_{d}}, then obtain \\int \\frac{e^{A_{x} - C_{d}}}{\\operatorname{F_{H}}{(A_{x},C_{d})}} dC_{d} = \\int e^{- A_{x} + C_{d}} e^{A_{x} - C_{d}} dC_{d}", "derivation": "\\operatorname{F_{H}}{(A_{x},C_{d})} = e^{A_{x} - C_{d}} and 1 = \\frac{e^{A_{x} - C_{d}}}{\\operatorname{F_{H}}{(A_{x},C_{d})}} and \\int 1 dC_{d} = \\int \\frac{e^{A_{x} - C_{d}}}{\\operatorname{F_{H}}{(A_{x},C_{d})}} dC_{d} and \\int 1 dC_{d} = \\int e^{- A_{x} + C_{d}} e^{A_{x} - C_{d}} dC_{d} and \\int \\frac{e^{A_{x} - C_{d}}}{\\operatorname{F_{H}}{(A_{x},C_{d})}} dC_{d} = \\int e^{- A_{x} + C_{d}} e^{A_{x} - C_{d}} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('A_x', commutative=True), Symbol('C_d', commutative=True)), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True)))))"], [["divide", 1, "Function('F_H')(Symbol('A_x', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_H')(Symbol('A_x', commutative=True), Symbol('C_d', commutative=True)), Integer(-1)), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))))))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Pow(Function('F_H')(Symbol('A_x', commutative=True), Symbol('C_d', commutative=True)), Integer(-1)), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Integer(1), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('C_d', commutative=True))), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Pow(Function('F_H')(Symbol('A_x', commutative=True), Symbol('C_d', commutative=True)), Integer(-1)), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('C_d', commutative=True))), exp(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given B{(\\hat{H})} = e^{\\hat{H}}, then obtain \\int \\sin{(\\int \\frac{d}{d \\hat{H}} B{(\\hat{H})} d\\hat{H})} d\\hat{H} = \\int \\sin{(\\int \\frac{d}{d \\hat{H}} e^{\\hat{H}} d\\hat{H})} d\\hat{H}", "derivation": "B{(\\hat{H})} = e^{\\hat{H}} and \\frac{d}{d \\hat{H}} B{(\\hat{H})} = \\frac{d}{d \\hat{H}} e^{\\hat{H}} and \\int \\frac{d}{d \\hat{H}} B{(\\hat{H})} d\\hat{H} = \\int \\frac{d}{d \\hat{H}} e^{\\hat{H}} d\\hat{H} and \\sin{(\\int \\frac{d}{d \\hat{H}} B{(\\hat{H})} d\\hat{H})} = \\sin{(\\int \\frac{d}{d \\hat{H}} e^{\\hat{H}} d\\hat{H})} and \\int \\sin{(\\int \\frac{d}{d \\hat{H}} B{(\\hat{H})} d\\hat{H})} d\\hat{H} = \\int \\sin{(\\int \\frac{d}{d \\hat{H}} e^{\\hat{H}} d\\hat{H})} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Derivative(Function('B')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), sin(Integral(Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(sin(Integral(Derivative(Function('B')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(sin(Integral(Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hbar)} = \\cos{(\\hbar)} and n{(\\hbar)} = \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar}, then obtain \\frac{d}{d \\hbar} \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar} = \\frac{d}{d \\hbar} n{(\\hbar)}", "derivation": "\\operatorname{t_{2}}{(\\hbar)} = \\cos{(\\hbar)} and \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar} = \\frac{\\cos{(\\hbar)}}{\\hbar} and \\frac{d}{d \\hbar} \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar} = \\frac{d}{d \\hbar} \\frac{\\cos{(\\hbar)}}{\\hbar} and n{(\\hbar)} = \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar} and n{(\\hbar)} = \\frac{\\cos{(\\hbar)}}{\\hbar} and \\frac{d}{d \\hbar} \\frac{\\operatorname{t_{2}}{(\\hbar)}}{\\hbar} = \\frac{d}{d \\hbar} n{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('n')(Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Function('n')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(I,s)} = \\frac{\\partial}{\\partial s} (I - s), then derive \\log{(\\int \\operatorname{F_{c}}{(I,s)} dI)} = \\log{(- I + \\dot{y})}, then obtain I \\log{(\\int \\operatorname{F_{c}}{(I,s)} dI)} = I \\log{(- I + \\dot{y})}", "derivation": "\\operatorname{F_{c}}{(I,s)} = \\frac{\\partial}{\\partial s} (I - s) and \\int \\operatorname{F_{c}}{(I,s)} dI = \\int \\frac{\\partial}{\\partial s} (I - s) dI and \\log{(\\int \\operatorname{F_{c}}{(I,s)} dI)} = \\log{(\\int \\frac{\\partial}{\\partial s} (I - s) dI)} and \\log{(\\int \\operatorname{F_{c}}{(I,s)} dI)} = \\log{(- I + \\dot{y})} and I \\log{(\\int \\operatorname{F_{c}}{(I,s)} dI)} = I \\log{(- I + \\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('F_c')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True)))), log(Integral(Derivative(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('F_c')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["times", 4, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), log(Integral(Function('F_c')(Symbol('I', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('I', commutative=True))))), Mul(Symbol('I', commutative=True), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\hbar,\\mathbf{F})} = \\frac{\\mathbf{F}}{\\hbar}, then obtain \\frac{- (\\frac{\\mathbf{F}}{\\hbar})^{\\mathbf{F}} + \\theta^{\\mathbf{F}}{(\\hbar,\\mathbf{F})}}{\\frac{\\partial}{\\partial \\mathbf{F}} (- \\mathbf{F} + \\frac{\\mathbf{F}}{\\hbar})} = 0", "derivation": "\\theta{(\\hbar,\\mathbf{F})} = \\frac{\\mathbf{F}}{\\hbar} and \\theta^{\\mathbf{F}}{(\\hbar,\\mathbf{F})} = (\\frac{\\mathbf{F}}{\\hbar})^{\\mathbf{F}} and - (\\frac{\\mathbf{F}}{\\hbar})^{\\mathbf{F}} + \\theta^{\\mathbf{F}}{(\\hbar,\\mathbf{F})} = 0 and \\frac{- (\\frac{\\mathbf{F}}{\\hbar})^{\\mathbf{F}} + \\theta^{\\mathbf{F}}{(\\hbar,\\mathbf{F})}}{\\frac{\\partial}{\\partial \\mathbf{F}} (- \\mathbf{F} + \\frac{\\mathbf{F}}{\\hbar})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 2, "Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Integer(0))"], [["divide", 3, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('\\\\theta')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\dot{z},F_{H})} = F_{H} - \\dot{z}, then derive \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{m_{s}}{(\\dot{z},F_{H})} = -1, then obtain - \\dot{z} = \\dot{z} \\frac{\\partial}{\\partial \\dot{z}} (F_{H} - \\dot{z})", "derivation": "\\operatorname{m_{s}}{(\\dot{z},F_{H})} = F_{H} - \\dot{z} and \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{m_{s}}{(\\dot{z},F_{H})} = \\frac{\\partial}{\\partial \\dot{z}} (F_{H} - \\dot{z}) and \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{m_{s}}{(\\dot{z},F_{H})} = -1 and -1 = \\frac{\\partial}{\\partial \\dot{z}} (F_{H} - \\dot{z}) and - \\dot{z} = \\dot{z} \\frac{\\partial}{\\partial \\dot{z}} (F_{H} - \\dot{z})", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\dot{z}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\dot{z}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('\\\\dot{z}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["times", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Derivative(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B}^{\\phi_2})} and \\operatorname{M_{E}}{(\\phi_2,\\mathbf{B})} = \\frac{\\partial}{\\partial \\phi_2} \\log{(\\mathbf{B}^{\\phi_2})}, then derive \\frac{\\partial}{\\partial \\phi_2} s{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B})}, then obtain \\operatorname{M_{E}}^{\\mathbf{B}}{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B})}^{\\mathbf{B}}", "derivation": "s{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B}^{\\phi_2})} and \\frac{\\partial}{\\partial \\phi_2} s{(\\phi_2,\\mathbf{B})} = \\frac{\\partial}{\\partial \\phi_2} \\log{(\\mathbf{B}^{\\phi_2})} and \\frac{\\partial}{\\partial \\phi_2} s{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B})} and \\frac{\\partial}{\\partial \\phi_2} \\log{(\\mathbf{B}^{\\phi_2})} = \\log{(\\mathbf{B})} and \\operatorname{M_{E}}{(\\phi_2,\\mathbf{B})} = \\frac{\\partial}{\\partial \\phi_2} \\log{(\\mathbf{B}^{\\phi_2})} and \\operatorname{M_{E}}^{\\mathbf{B}}{(\\phi_2,\\mathbf{B})} = (\\frac{\\partial}{\\partial \\phi_2} \\log{(\\mathbf{B}^{\\phi_2})})^{\\mathbf{B}} and \\operatorname{M_{E}}^{\\mathbf{B}}{(\\phi_2,\\mathbf{B})} = \\log{(\\mathbf{B})}^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), log(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), log(Symbol('\\\\mathbf{B}', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Derivative(log(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(log(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Function('M_E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(log(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\varphi{(A_{2})} = \\log{(A_{2})} and E{(A_{2})} = - 2 \\varphi{(A_{2})}, then obtain \\frac{d}{d A_{2}} 2 \\log{(A_{2})} = \\frac{d}{d A_{2}} (E{(A_{2})} + 4 \\log{(A_{2})})", "derivation": "\\varphi{(A_{2})} = \\log{(A_{2})} and \\varphi{(A_{2})} + \\log{(A_{2})} = 2 \\log{(A_{2})} and 2 \\varphi{(A_{2})} + \\log{(A_{2})} = \\varphi{(A_{2})} + 2 \\log{(A_{2})} and \\log{(A_{2})} = - \\varphi{(A_{2})} + 2 \\log{(A_{2})} and 2 \\log{(A_{2})} = - 2 \\varphi{(A_{2})} + 4 \\log{(A_{2})} and E{(A_{2})} = - 2 \\varphi{(A_{2})} and 2 \\log{(A_{2})} = E{(A_{2})} + 4 \\log{(A_{2})} and \\frac{d}{d A_{2}} 2 \\log{(A_{2})} = \\frac{d}{d A_{2}} (E{(A_{2})} + 4 \\log{(A_{2})})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["add", 1, "log(Symbol('A_2', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))), Mul(Integer(2), log(Symbol('A_2', commutative=True))))"], [["add", 1, "Add(Function('\\\\varphi')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\varphi')(Symbol('A_2', commutative=True))), log(Symbol('A_2', commutative=True))), Add(Function('\\\\varphi')(Symbol('A_2', commutative=True)), Mul(Integer(2), log(Symbol('A_2', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), Function('\\\\varphi')(Symbol('A_2', commutative=True)))"], "Equality(log(Symbol('A_2', commutative=True)), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('A_2', commutative=True))), Mul(Integer(2), log(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(2), log(Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\varphi')(Symbol('A_2', commutative=True))), Mul(Integer(4), log(Symbol('A_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('E')(Symbol('A_2', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varphi')(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Integer(2), log(Symbol('A_2', commutative=True))), Add(Function('E')(Symbol('A_2', commutative=True)), Mul(Integer(4), log(Symbol('A_2', commutative=True)))))"], [["differentiate", 7, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Integer(2), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Function('E')(Symbol('A_2', commutative=True)), Mul(Integer(4), log(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(b)} = e^{b} and \\hat{\\mathbf{x}}{(b)} = e^{b}, then obtain \\int \\operatorname{r_{0}}^{b}{(b)} (e^{b})^{b} db = \\int \\operatorname{r_{0}}^{2 b}{(b)} db", "derivation": "\\operatorname{r_{0}}{(b)} = e^{b} and \\hat{\\mathbf{x}}{(b)} = e^{b} and \\hat{\\mathbf{x}}^{b}{(b)} = (e^{b})^{b} and \\hat{\\mathbf{x}}^{b}{(b)} (e^{b})^{b} = (e^{b})^{2 b} and \\hat{\\mathbf{x}}^{b}{(b)} \\operatorname{r_{0}}^{b}{(b)} = \\operatorname{r_{0}}^{2 b}{(b)} and \\operatorname{r_{0}}^{b}{(b)} (e^{b})^{b} = \\operatorname{r_{0}}^{2 b}{(b)} and \\int \\operatorname{r_{0}}^{b}{(b)} (e^{b})^{b} db = \\int \\operatorname{r_{0}}^{2 b}{(b)} db", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["times", 3, "Pow(exp(Symbol('b', commutative=True)), Symbol('b', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(exp(Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Function('r_0')(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Function('r_0')(Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Function('r_0')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Pow(Function('r_0')(Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))))"], [["integrate", 6, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Pow(Function('r_0')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(exp(Symbol('b', commutative=True)), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Pow(Function('r_0')(Symbol('b', commutative=True)), Mul(Integer(2), Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\theta_1 \\varepsilon_0, then derive 0 = \\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)}, then obtain \\int 0 d\\theta_1 = \\int \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)}) d\\theta_1", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\theta_1 \\varepsilon_0 and 0 = - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)} + \\frac{\\partial}{\\partial \\theta_1} \\theta_1 \\varepsilon_0 and 0 = \\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)} and \\frac{d}{d \\varepsilon_0} 0 = \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)}) and \\int \\frac{d}{d \\varepsilon_0} 0 d\\theta_1 = \\int \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)}) d\\theta_1 and \\int 0 d\\theta_1 = \\int \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 - \\operatorname{g_{\\varepsilon}}{(\\varepsilon_0,\\theta_1)}) d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["minus", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True))), Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(t_{2},\\mathbf{J}_M)} = t_{2}^{\\mathbf{J}_M}, then obtain \\frac{\\frac{\\partial}{\\partial t_{2}} \\mathbf{M}{(t_{2},\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{t_{2}^{\\mathbf{J}_M}}{t_{2}}", "derivation": "\\mathbf{M}{(t_{2},\\mathbf{J}_M)} = t_{2}^{\\mathbf{J}_M} and \\frac{\\mathbf{M}{(t_{2},\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{t_{2}^{\\mathbf{J}_M}}{\\mathbf{J}_M} and \\frac{\\partial}{\\partial t_{2}} \\frac{\\mathbf{M}{(t_{2},\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{\\partial}{\\partial t_{2}} \\frac{t_{2}^{\\mathbf{J}_M}}{\\mathbf{J}_M} and \\frac{\\frac{\\partial}{\\partial t_{2}} \\mathbf{M}{(t_{2},\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{t_{2}^{\\mathbf{J}_M}}{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{M}')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given Z{(\\mathbf{p})} = \\log{(\\mathbf{p})}, then obtain \\int Z^{\\mathbf{p}}{(\\mathbf{p})} \\log{(\\mathbf{p})}^{- \\mathbf{p}} d\\mathbf{p} = \\int 1 d\\mathbf{p}", "derivation": "Z{(\\mathbf{p})} = \\log{(\\mathbf{p})} and Z^{\\mathbf{p}}{(\\mathbf{p})} = \\log{(\\mathbf{p})}^{\\mathbf{p}} and Z^{\\mathbf{p}}{(\\mathbf{p})} \\log{(\\mathbf{p})}^{- \\mathbf{p}} = 1 and \\int Z^{\\mathbf{p}}{(\\mathbf{p})} \\log{(\\mathbf{p})}^{- \\mathbf{p}} d\\mathbf{p} = \\int 1 d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["divide", 2, "Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(log(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(n_{2},E_{x})} = E_{x} + n_{2} and \\operatorname{f^{\\prime}}{(n_{2},E_{x})} = \\int \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + n_{2}) dE_{x} dn_{2}, then obtain \\int \\frac{\\partial}{\\partial E_{x}} \\int \\phi_{2}{(n_{2},E_{x})} dE_{x} dn_{2} = \\operatorname{f^{\\prime}}{(n_{2},E_{x})}", "derivation": "\\phi_{2}{(n_{2},E_{x})} = E_{x} + n_{2} and \\int \\phi_{2}{(n_{2},E_{x})} dE_{x} = \\int (E_{x} + n_{2}) dE_{x} and \\frac{\\partial}{\\partial E_{x}} \\int \\phi_{2}{(n_{2},E_{x})} dE_{x} = \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + n_{2}) dE_{x} and \\int \\frac{\\partial}{\\partial E_{x}} \\int \\phi_{2}{(n_{2},E_{x})} dE_{x} dn_{2} = \\int \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + n_{2}) dE_{x} dn_{2} and \\operatorname{f^{\\prime}}{(n_{2},E_{x})} = \\int \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + n_{2}) dE_{x} dn_{2} and \\int \\frac{\\partial}{\\partial E_{x}} \\int \\phi_{2}{(n_{2},E_{x})} dE_{x} dn_{2} = \\operatorname{f^{\\prime}}{(n_{2},E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["differentiate", 2, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\phi_2')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))), Integral(Derivative(Integral(Add(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Integral(Derivative(Integral(Add(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Derivative(Integral(Function('\\\\phi_2')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))), Function('f^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given \\phi{(S,I)} = e^{I S}, then obtain \\frac{\\partial}{\\partial S} \\int (\\int (\\phi{(S,I)} - e^{I S}) dI)^{I} dI = \\frac{d}{d S} \\int (\\int 0 dI)^{I} dI", "derivation": "\\phi{(S,I)} = e^{I S} and \\phi{(S,I)} - e^{I S} = 0 and \\int (\\phi{(S,I)} - e^{I S}) dI = \\int 0 dI and (\\int (\\phi{(S,I)} - e^{I S}) dI)^{I} = (\\int 0 dI)^{I} and \\int (\\int (\\phi{(S,I)} - e^{I S}) dI)^{I} dI = \\int (\\int 0 dI)^{I} dI and \\frac{\\partial}{\\partial S} \\int (\\int (\\phi{(S,I)} - e^{I S}) dI)^{I} dI = \\frac{d}{d S} \\int (\\int 0 dI)^{I} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))"], [["minus", 1, "exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))), Tuple(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["integrate", 4, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Integral(Add(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["differentiate", 5, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Pow(Integral(Add(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('I', commutative=True), Symbol('S', commutative=True))))), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(Pow(Integral(Integer(0), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(J,\\theta_2)} = \\theta_2^{J}, then derive \\frac{\\partial}{\\partial \\theta_2} \\rho_{f}{(J,\\theta_2)} = \\frac{J \\theta_2^{J}}{\\theta_2}, then obtain (\\frac{J \\theta_2^{J}}{\\theta_2})^{\\theta_2} = (\\frac{\\partial}{\\partial \\theta_2} \\theta_2^{J})^{\\theta_2}", "derivation": "\\rho_{f}{(J,\\theta_2)} = \\theta_2^{J} and \\frac{\\partial}{\\partial \\theta_2} \\rho_{f}{(J,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\theta_2^{J} and \\frac{\\partial}{\\partial \\theta_2} \\rho_{f}{(J,\\theta_2)} = \\frac{J \\theta_2^{J}}{\\theta_2} and (\\frac{\\partial}{\\partial \\theta_2} \\rho_{f}{(J,\\theta_2)})^{\\theta_2} = (\\frac{\\partial}{\\partial \\theta_2} \\theta_2^{J})^{\\theta_2} and (\\frac{J \\theta_2^{J}}{\\theta_2})^{\\theta_2} = (\\frac{\\partial}{\\partial \\theta_2} \\theta_2^{J})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho_f')(Symbol('J', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(P_{e})} = \\cos{(\\sin{(P_{e})})} and \\hat{H}_{\\lambda}{(P_{e})} = \\frac{\\tilde{g}^*{(P_{e})}}{\\cos{(\\sin{(P_{e})})}}, then obtain \\hat{H}_{\\lambda}^{P_{e}}{(P_{e})} + \\frac{1}{\\cos{(\\sin{(P_{e})})}} = 1 + \\frac{1}{\\cos{(\\sin{(P_{e})})}}", "derivation": "\\tilde{g}^*{(P_{e})} = \\cos{(\\sin{(P_{e})})} and \\hat{H}_{\\lambda}{(P_{e})} = \\frac{\\tilde{g}^*{(P_{e})}}{\\cos{(\\sin{(P_{e})})}} and \\hat{H}_{\\lambda}^{P_{e}}{(P_{e})} = (\\frac{\\tilde{g}^*{(P_{e})}}{\\cos{(\\sin{(P_{e})})}})^{P_{e}} and \\hat{H}_{\\lambda}^{P_{e}}{(P_{e})} = 1 and \\hat{H}_{\\lambda}^{P_{e}}{(P_{e})} + \\frac{1}{\\cos{(\\sin{(P_{e})})}} = 1 + \\frac{1}{\\cos{(\\sin{(P_{e})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('P_e', commutative=True)), cos(sin(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('P_e', commutative=True)), Mul(Function('\\\\tilde{g}^*')(Symbol('P_e', commutative=True)), Pow(cos(sin(Symbol('P_e', commutative=True))), Integer(-1))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(Mul(Function('\\\\tilde{g}^*')(Symbol('P_e', commutative=True)), Pow(cos(sin(Symbol('P_e', commutative=True))), Integer(-1))), Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Integer(1))"], [["add", 4, "Pow(cos(sin(Symbol('P_e', commutative=True))), Integer(-1))"], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(cos(sin(Symbol('P_e', commutative=True))), Integer(-1))), Add(Integer(1), Pow(cos(sin(Symbol('P_e', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\chi{(\\mu)} = \\cos{(e^{\\mu})}, then obtain \\frac{d}{d \\mu} (\\chi^{\\mu}{(\\mu)})^{\\mu} = \\frac{d}{d \\mu} (\\cos^{\\mu}{(e^{\\mu})})^{\\mu}", "derivation": "\\chi{(\\mu)} = \\cos{(e^{\\mu})} and \\chi^{\\mu}{(\\mu)} = \\cos^{\\mu}{(e^{\\mu})} and (\\chi^{\\mu}{(\\mu)})^{\\mu} = (\\cos^{\\mu}{(e^{\\mu})})^{\\mu} and \\frac{d}{d \\mu} (\\chi^{\\mu}{(\\mu)})^{\\mu} = \\frac{d}{d \\mu} (\\cos^{\\mu}{(e^{\\mu})})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mu', commutative=True)), cos(exp(Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(cos(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Function('\\\\chi')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(cos(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\chi')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Pow(cos(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\mu)} = \\cos{(e^{\\mu})}, then obtain (- C{(\\mu)} + \\cos{(e^{\\mu})})^{\\mu} = 1", "derivation": "C{(\\mu)} = \\cos{(e^{\\mu})} and 0 = - C{(\\mu)} + \\cos{(e^{\\mu})} and - C{(\\mu)} = - 2 C{(\\mu)} + \\cos{(e^{\\mu})} and 0^{\\mu} = (- C{(\\mu)} + \\cos{(e^{\\mu})})^{\\mu} and 0^{\\mu} = (- 2 C{(\\mu)} + 2 \\cos{(e^{\\mu})})^{\\mu} and (- C{(\\mu)} + \\cos{(e^{\\mu})})^{\\mu} = 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mu', commutative=True)), cos(exp(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Function('C')(Symbol('\\\\mu', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C')(Symbol('\\\\mu', commutative=True))), cos(exp(Symbol('\\\\mu', commutative=True)))))"], [["minus", 2, "Function('C')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Function('C')(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('C')(Symbol('\\\\mu', commutative=True))), cos(exp(Symbol('\\\\mu', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Integer(-1), Function('C')(Symbol('\\\\mu', commutative=True))), cos(exp(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integer(0), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Function('C')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), cos(exp(Symbol('\\\\mu', commutative=True))))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Mul(Integer(-1), Function('C')(Symbol('\\\\mu', commutative=True))), cos(exp(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Integer(1))"]]}, {"prompt": "Given W{(\\psi,\\delta)} = \\delta + \\psi, then derive 2 (\\int W{(\\psi,\\delta)} d\\delta)^{2} = 2 (Q + \\frac{\\delta^{2}}{2} + \\delta \\psi) \\int W{(\\psi,\\delta)} d\\delta, then obtain 2 (\\int (\\delta + \\psi) d\\delta) \\int W{(\\psi,\\delta)} d\\delta = 2 (Q + \\frac{\\delta^{2}}{2} + \\delta \\psi) \\int W{(\\psi,\\delta)} d\\delta", "derivation": "W{(\\psi,\\delta)} = \\delta + \\psi and \\int W{(\\psi,\\delta)} d\\delta = \\int (\\delta + \\psi) d\\delta and (\\int W{(\\psi,\\delta)} d\\delta)^{2} = (\\int (\\delta + \\psi) d\\delta) \\int W{(\\psi,\\delta)} d\\delta and 2 (\\int W{(\\psi,\\delta)} d\\delta)^{2} = 2 (\\int (\\delta + \\psi) d\\delta) \\int W{(\\psi,\\delta)} d\\delta and 2 (\\int W{(\\psi,\\delta)} d\\delta)^{2} = 2 (Q + \\frac{\\delta^{2}}{2} + \\delta \\psi) \\int W{(\\psi,\\delta)} d\\delta and 2 (\\int (\\delta + \\psi) d\\delta) \\int W{(\\psi,\\delta)} d\\delta = 2 (Q + \\frac{\\delta^{2}}{2} + \\delta \\psi) \\int W{(\\psi,\\delta)} d\\delta", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 2, "Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Pow(Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(2)), Mul(Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["divide", 3, "Rational(1, 2)"], "Equality(Mul(Integer(2), Pow(Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(2))), Mul(Integer(2), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), Pow(Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(2))), Mul(Integer(2), Add(Symbol('Q', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True))), Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Integral(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integer(2), Add(Symbol('Q', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2))), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\psi', commutative=True))), Integral(Function('W')(Symbol('\\\\psi', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(F_{g},C,\\mathbf{F})} = F_{g} + \\mathbf{F}^{C}, then obtain \\frac{\\partial}{\\partial F_{g}} (2 \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}} = \\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{F}^{C} + \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}}", "derivation": "\\hat{H}{(F_{g},C,\\mathbf{F})} = F_{g} + \\mathbf{F}^{C} and 2 \\hat{H}{(F_{g},C,\\mathbf{F})} = F_{g} + \\mathbf{F}^{C} + \\hat{H}{(F_{g},C,\\mathbf{F})} and (2 \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}} = (F_{g} + \\mathbf{F}^{C} + \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}} and \\frac{\\partial}{\\partial F_{g}} (2 \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}} = \\frac{\\partial}{\\partial F_{g}} (F_{g} + \\mathbf{F}^{C} + \\hat{H}{(F_{g},C,\\mathbf{F})})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C', commutative=True))))"], [["add", 1, "Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C', commutative=True)), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 2, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('F_g', commutative=True)), Pow(Add(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C', commutative=True)), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('F_g', commutative=True)))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('C', commutative=True)), Function('\\\\hat{H}')(Symbol('F_g', commutative=True), Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(\\rho_f)} = \\cos{(\\sin{(\\rho_f)})}, then obtain (\\frac{J^{\\rho_f}{(\\rho_f)}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}})^{\\rho_f} = (\\frac{\\cos^{\\rho_f}{(\\sin{(\\rho_f)})}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}})^{\\rho_f}", "derivation": "J{(\\rho_f)} = \\cos{(\\sin{(\\rho_f)})} and J^{\\rho_f}{(\\rho_f)} = \\cos^{\\rho_f}{(\\sin{(\\rho_f)})} and \\frac{J^{\\rho_f}{(\\rho_f)}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}} = \\frac{\\cos^{\\rho_f}{(\\sin{(\\rho_f)})}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}} and (\\frac{J^{\\rho_f}{(\\rho_f)}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}})^{\\rho_f} = (\\frac{\\cos^{\\rho_f}{(\\sin{(\\rho_f)})}}{- J{(\\rho_f)} + \\cos{(\\sin{(\\rho_f)})}})^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\rho_f', commutative=True)), cos(sin(Symbol('\\\\rho_f', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(cos(sin(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(-1), Function('J')(Symbol('\\\\rho_f', commutative=True))), cos(sin(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('\\\\rho_f', commutative=True))), cos(sin(Symbol('\\\\rho_f', commutative=True)))), Integer(-1)), Pow(Function('J')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('\\\\rho_f', commutative=True))), cos(sin(Symbol('\\\\rho_f', commutative=True)))), Integer(-1)), Pow(cos(sin(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))))"], [["power", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('\\\\rho_f', commutative=True))), cos(sin(Symbol('\\\\rho_f', commutative=True)))), Integer(-1)), Pow(Function('J')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('\\\\rho_f', commutative=True))), cos(sin(Symbol('\\\\rho_f', commutative=True)))), Integer(-1)), Pow(cos(sin(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given z{(\\hat{x},\\varepsilon)} = - \\hat{x} + \\varepsilon, then obtain \\frac{\\varepsilon - z{(\\hat{x},\\varepsilon)}}{F_{c} z{(\\hat{x},\\varepsilon)}} = \\frac{\\hat{x}}{F_{c} z{(\\hat{x},\\varepsilon)}}", "derivation": "z{(\\hat{x},\\varepsilon)} = - \\hat{x} + \\varepsilon and - \\varepsilon + z{(\\hat{x},\\varepsilon)} = - \\hat{x} and \\frac{- \\varepsilon + z{(\\hat{x},\\varepsilon)}}{F_{c}} = - \\frac{\\hat{x}}{F_{c}} and \\frac{- \\varepsilon + z{(\\hat{x},\\varepsilon)}}{F_{c}} = - \\frac{\\varepsilon - z{(\\hat{x},\\varepsilon)}}{F_{c}} and - \\frac{\\varepsilon - z{(\\hat{x},\\varepsilon)}}{F_{c}} = - \\frac{\\hat{x}}{F_{c}} and \\frac{\\varepsilon - z{(\\hat{x},\\varepsilon)}}{F_{c} z{(\\hat{x},\\varepsilon)}} = \\frac{\\hat{x}}{F_{c} z{(\\hat{x},\\varepsilon)}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 2, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True))))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 5, "Mul(Integer(-1), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Pow(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True), Pow(Function('z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given k{(g,\\phi)} = \\phi + g, then obtain (- g + k{(g,\\phi)})^{g} - k{(g,\\phi)} = \\phi^{g} - k{(g,\\phi)}", "derivation": "k{(g,\\phi)} = \\phi + g and - g + k{(g,\\phi)} = \\phi and (- g + k{(g,\\phi)})^{g} = \\phi^{g} and (- g + k{(g,\\phi)})^{g} - k{(g,\\phi)} = \\phi^{g} - k{(g,\\phi)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('g', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)))"], [["minus", 3, "Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('g', commutative=True)), Mul(Integer(-1), Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)))), Add(Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Function('k')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}, then derive \\sigma_p \\ddot{x}{(\\sigma_p)} = 1, then obtain - \\ddot{x}{(\\sigma_p)} + e^{\\sigma_p \\ddot{x}{(\\sigma_p)}} = e - \\ddot{x}{(\\sigma_p)}", "derivation": "\\ddot{x}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\sigma_p \\ddot{x}{(\\sigma_p)} = \\sigma_p \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\sigma_p \\ddot{x}{(\\sigma_p)} = 1 and e^{\\sigma_p \\ddot{x}{(\\sigma_p)}} = e and - \\ddot{x}{(\\sigma_p)} + e^{\\sigma_p \\ddot{x}{(\\sigma_p)}} = e - \\ddot{x}{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))), Integer(1))"], [["exp", 3], "Equality(exp(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True)))), E)"], [["minus", 4, "Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))), exp(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True))))), Add(E, Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given W{(\\omega,L)} = \\frac{L}{\\omega}, then derive \\frac{\\partial}{\\partial L} W{(\\omega,L)} = \\frac{1}{\\omega}, then obtain \\frac{\\partial^{2}}{\\partial L^{2}} W{(\\omega,L)} = \\frac{d}{d L} \\frac{1}{\\omega}", "derivation": "W{(\\omega,L)} = \\frac{L}{\\omega} and \\frac{\\partial}{\\partial L} W{(\\omega,L)} = \\frac{\\partial}{\\partial L} \\frac{L}{\\omega} and \\frac{\\partial}{\\partial L} W{(\\omega,L)} = \\frac{1}{\\omega} and \\frac{\\partial^{2}}{\\partial L^{2}} W{(\\omega,L)} = \\frac{d}{d L} \\frac{1}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(W)} = \\cos{(\\sin{(W)})}, then obtain - \\frac{\\varphi^{*}{(W)}}{\\int \\cos{(\\sin{(W)})} dW} = - \\frac{\\cos{(\\sin{(W)})}}{\\int \\cos{(\\sin{(W)})} dW}", "derivation": "\\varphi^{*}{(W)} = \\cos{(\\sin{(W)})} and \\int \\varphi^{*}{(W)} dW = \\int \\cos{(\\sin{(W)})} dW and \\frac{\\varphi^{*}{(W)}}{\\int \\varphi^{*}{(W)} dW} = \\frac{\\cos{(\\sin{(W)})}}{\\int \\varphi^{*}{(W)} dW} and - \\frac{\\varphi^{*}{(W)}}{\\int \\varphi^{*}{(W)} dW} = - \\frac{\\cos{(\\sin{(W)})}}{\\int \\varphi^{*}{(W)} dW} and - \\frac{\\varphi^{*}{(W)}}{\\int \\cos{(\\sin{(W)})} dW} = - \\frac{\\cos{(\\sin{(W)})}}{\\int \\cos{(\\sin{(W)})} dW}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('W', commutative=True)), cos(sin(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(cos(sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1))), Mul(cos(sin(Symbol('W', commutative=True))), Pow(Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('W', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(sin(Symbol('W', commutative=True))), Pow(Integral(Function('\\\\varphi^*')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('W', commutative=True)), Pow(Integral(cos(sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(sin(Symbol('W', commutative=True))), Pow(Integral(cos(sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\rho_b,\\mathbf{H})} = e^{\\mathbf{H} \\rho_b} and \\hat{x}{(\\rho_b,\\mathbf{H})} = - \\mathbf{H} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\mathbf{H})}, then obtain (- \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b}) \\hat{x}{(\\rho_b,\\mathbf{H})} = (- \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b})^{2}", "derivation": "\\dot{\\mathbf{r}}{(\\rho_b,\\mathbf{H})} = e^{\\mathbf{H} \\rho_b} and - \\mathbf{H} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\mathbf{H})} = - \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b} and \\hat{x}{(\\rho_b,\\mathbf{H})} = - \\mathbf{H} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\mathbf{H})} and \\hat{x}{(\\rho_b,\\mathbf{H})} = - \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b} and (- \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b}) \\hat{x}{(\\rho_b,\\mathbf{H})} = (- \\mathbf{H} \\rho_b + e^{\\mathbf{H} \\rho_b})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{x}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Function('\\\\hat{x}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\psi,E_{n})} = E_{n} + \\psi, then obtain \\frac{\\operatorname{F_{H}}{(\\psi,E_{n})}}{\\frac{\\partial}{\\partial \\psi} 2 \\operatorname{F_{H}}{(\\psi,E_{n})}} = \\frac{E_{n} + \\psi}{\\frac{\\partial}{\\partial \\psi} 2 \\operatorname{F_{H}}{(\\psi,E_{n})}}", "derivation": "\\operatorname{F_{H}}{(\\psi,E_{n})} = E_{n} + \\psi and 2 \\operatorname{F_{H}}{(\\psi,E_{n})} = E_{n} + \\psi + \\operatorname{F_{H}}{(\\psi,E_{n})} and \\frac{\\operatorname{F_{H}}{(\\psi,E_{n})}}{\\frac{\\partial}{\\partial \\psi} (E_{n} + \\psi + \\operatorname{F_{H}}{(\\psi,E_{n})})} = \\frac{E_{n} + \\psi}{\\frac{\\partial}{\\partial \\psi} (E_{n} + \\psi + \\operatorname{F_{H}}{(\\psi,E_{n})})} and \\frac{\\operatorname{F_{H}}{(\\psi,E_{n})}}{\\frac{\\partial}{\\partial \\psi} 2 \\operatorname{F_{H}}{(\\psi,E_{n})}} = \\frac{E_{n} + \\psi}{\\frac{\\partial}{\\partial \\psi} 2 \\operatorname{F_{H}}{(\\psi,E_{n})}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))"], "Equality(Mul(Integer(2), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))))"], [["divide", 1, "Derivative(Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))"], "Equality(Mul(Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True)), Pow(Derivative(Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Derivative(Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('E_n', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('F_H')(Symbol('\\\\psi', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(r_{0})} = \\cos{(\\cos{(r_{0})})}, then obtain r_{0} + \\frac{d}{d r_{0}} \\Omega{(r_{0})} = r_{0} + \\sin{(r_{0})} \\sin{(\\cos{(r_{0})})}", "derivation": "\\Omega{(r_{0})} = \\cos{(\\cos{(r_{0})})} and \\frac{d}{d r_{0}} \\Omega{(r_{0})} = \\frac{d}{d r_{0}} \\cos{(\\cos{(r_{0})})} and r_{0} + \\frac{d}{d r_{0}} \\Omega{(r_{0})} = r_{0} + \\frac{d}{d r_{0}} \\cos{(\\cos{(r_{0})})} and r_{0} + \\frac{d}{d r_{0}} \\Omega{(r_{0})} = r_{0} + \\sin{(r_{0})} \\sin{(\\cos{(r_{0})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('r_0', commutative=True)), cos(cos(Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["add", 2, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Derivative(Function('\\\\Omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Symbol('r_0', commutative=True), Derivative(cos(cos(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('r_0', commutative=True), Derivative(Function('\\\\Omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Symbol('r_0', commutative=True), Mul(sin(Symbol('r_0', commutative=True)), sin(cos(Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\ddot{x}{(g,c_{0})} = c_{0}^{g}, then obtain \\frac{\\partial^{2}}{\\partial g\\partial c_{0}} \\int \\ddot{x}{(g,c_{0})} dc_{0} = \\frac{\\partial^{2}}{\\partial g\\partial c_{0}} \\int c_{0}^{g} dc_{0}", "derivation": "\\ddot{x}{(g,c_{0})} = c_{0}^{g} and \\int \\ddot{x}{(g,c_{0})} dc_{0} = \\int c_{0}^{g} dc_{0} and \\frac{\\partial}{\\partial c_{0}} \\int \\ddot{x}{(g,c_{0})} dc_{0} = \\frac{\\partial}{\\partial c_{0}} \\int c_{0}^{g} dc_{0} and \\frac{\\partial^{2}}{\\partial g\\partial c_{0}} \\int \\ddot{x}{(g,c_{0})} dc_{0} = \\frac{\\partial^{2}}{\\partial g\\partial c_{0}} \\int c_{0}^{g} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\ddot{x}')(Symbol('g', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('c_0', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\psi^*,t)} = (\\psi^*)^{t}, then obtain \\frac{\\int \\varphi{(\\psi^*,t)} dt}{\\iint \\varphi{(\\psi^*,t)} dt d\\psi^*} = \\frac{\\int (\\psi^*)^{t} dt}{\\iint \\varphi{(\\psi^*,t)} dt d\\psi^*}", "derivation": "\\varphi{(\\psi^*,t)} = (\\psi^*)^{t} and \\int \\varphi{(\\psi^*,t)} dt = \\int (\\psi^*)^{t} dt and \\iint \\varphi{(\\psi^*,t)} dt d\\psi^* = \\iint (\\psi^*)^{t} dt d\\psi^* and \\frac{\\int \\varphi{(\\psi^*,t)} dt}{\\iint (\\psi^*)^{t} dt d\\psi^*} = \\frac{\\int (\\psi^*)^{t} dt}{\\iint (\\psi^*)^{t} dt d\\psi^*} and \\frac{\\int \\varphi{(\\psi^*,t)} dt}{\\iint \\varphi{(\\psi^*,t)} dt d\\psi^*} = \\frac{\\int (\\psi^*)^{t} dt}{\\iint \\varphi{(\\psi^*,t)} dt d\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 2, "Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(-1))), Mul(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(-1))), Mul(Integral(Pow(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Pow(Integral(Function('\\\\varphi')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\psi,I)} = I + \\psi, then derive \\int \\mathbf{A}{(\\psi,I)} dI = \\frac{I^{2}}{2} + I \\psi + \\dot{\\mathbf{r}}, then obtain \\frac{\\partial}{\\partial I} (\\frac{I^{2}}{2} + I \\psi + \\dot{\\mathbf{r}}) = \\frac{\\partial}{\\partial I} \\int (I + \\psi) dI", "derivation": "\\mathbf{A}{(\\psi,I)} = I + \\psi and \\int \\mathbf{A}{(\\psi,I)} dI = \\int (I + \\psi) dI and \\int \\mathbf{A}{(\\psi,I)} dI = \\frac{I^{2}}{2} + I \\psi + \\dot{\\mathbf{r}} and \\frac{I^{2}}{2} + I \\psi + \\dot{\\mathbf{r}} = \\int (I + \\psi) dI and \\frac{\\partial}{\\partial I} (\\frac{I^{2}}{2} + I \\psi + \\dot{\\mathbf{r}}) = \\frac{\\partial}{\\partial I} \\int (I + \\psi) dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\psi', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integral(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["differentiate", 4, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(v)} = \\frac{d}{d v} \\sin{(v)}, then derive \\operatorname{f^{*}}^{2}{(v)} = \\operatorname{f^{*}}{(v)} \\cos{(v)}, then derive v \\operatorname{f^{*}}^{2}{(v)} = v \\operatorname{f^{*}}{(v)} \\cos{(v)}, then obtain v \\operatorname{f^{*}}{(v)} (\\frac{d}{d v} \\sin{(v)})^{3} = v \\operatorname{f^{*}}^{2}{(v)} (\\frac{d}{d v} \\sin{(v)})^{2}", "derivation": "\\operatorname{f^{*}}{(v)} = \\frac{d}{d v} \\sin{(v)} and \\operatorname{f^{*}}^{2}{(v)} = \\operatorname{f^{*}}{(v)} \\frac{d}{d v} \\sin{(v)} and \\operatorname{f^{*}}^{2}{(v)} = \\operatorname{f^{*}}{(v)} \\cos{(v)} and v \\operatorname{f^{*}}^{2}{(v)} = v \\operatorname{f^{*}}{(v)} \\frac{d}{d v} \\sin{(v)} and v \\operatorname{f^{*}}^{2}{(v)} = v \\operatorname{f^{*}}{(v)} \\cos{(v)} and (\\frac{d}{d v} \\sin{(v)})^{2} = \\cos{(v)} \\frac{d}{d v} \\sin{(v)} and v \\operatorname{f^{*}}{(v)} (\\frac{d}{d v} \\sin{(v)})^{3} = v \\operatorname{f^{*}}{(v)} \\cos{(v)} (\\frac{d}{d v} \\sin{(v)})^{2} and v \\operatorname{f^{*}}{(v)} (\\frac{d}{d v} \\sin{(v)})^{3} = v \\operatorname{f^{*}}^{2}{(v)} (\\frac{d}{d v} \\sin{(v)})^{2}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 1, "Function('f^*')(Symbol('v', commutative=True))"], "Equality(Pow(Function('f^*')(Symbol('v', commutative=True)), Integer(2)), Mul(Function('f^*')(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('f^*')(Symbol('v', commutative=True)), Integer(2)), Mul(Function('f^*')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))))"], [["times", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Pow(Function('f^*')(Symbol('v', commutative=True)), Integer(2))), Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('v', commutative=True), Pow(Function('f^*')(Symbol('v', commutative=True)), Integer(2))), Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2)), Mul(cos(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["times", 6, "Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(3))), Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)), Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Symbol('v', commutative=True), Function('f^*')(Symbol('v', commutative=True)), Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(3))), Mul(Symbol('v', commutative=True), Pow(Function('f^*')(Symbol('v', commutative=True)), Integer(2)), Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(v_{z})} = e^{v_{z}}, then obtain \\frac{d}{d v_{z}} (- v_{z} e^{v_{z}} + \\Psi^{\\dagger}{(v_{z})}) = \\frac{d}{d v_{z}} (- v_{z} e^{v_{z}} + e^{v_{z}})", "derivation": "\\Psi^{\\dagger}{(v_{z})} = e^{v_{z}} and v_{z} \\Psi^{\\dagger}{(v_{z})} = v_{z} e^{v_{z}} and - v_{z} \\Psi^{\\dagger}{(v_{z})} + \\Psi^{\\dagger}{(v_{z})} = - v_{z} \\Psi^{\\dagger}{(v_{z})} + e^{v_{z}} and - v_{z} e^{v_{z}} + \\Psi^{\\dagger}{(v_{z})} = - v_{z} e^{v_{z}} + e^{v_{z}} and \\frac{d}{d v_{z}} (- v_{z} e^{v_{z}} + \\Psi^{\\dagger}{(v_{z})}) = \\frac{d}{d v_{z}} (- v_{z} e^{v_{z}} + e^{v_{z}})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["times", 1, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), Mul(Symbol('v_z', commutative=True), exp(Symbol('v_z', commutative=True))))"], [["minus", 1, "Mul(Symbol('v_z', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), exp(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True), exp(Symbol('v_z', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True), exp(Symbol('v_z', commutative=True))), exp(Symbol('v_z', commutative=True))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True), exp(Symbol('v_z', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True), exp(Symbol('v_z', commutative=True))), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(t_{2})} = \\int e^{t_{2}} dt_{2}, then derive \\mathbf{H}{(t_{2})} + e^{t_{2}} = l + 2 e^{t_{2}}, then obtain e^{t_{2}} + \\int e^{t_{2}} dt_{2} = l + 2 e^{t_{2}}", "derivation": "\\mathbf{H}{(t_{2})} = \\int e^{t_{2}} dt_{2} and \\mathbf{H}{(t_{2})} + e^{t_{2}} = e^{t_{2}} + \\int e^{t_{2}} dt_{2} and \\mathbf{H}{(t_{2})} + e^{t_{2}} = l + 2 e^{t_{2}} and e^{t_{2}} + \\int e^{t_{2}} dt_{2} = l + 2 e^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["add", 1, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Add(exp(Symbol('t_2', commutative=True)), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(2), exp(Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(exp(Symbol('t_2', commutative=True)), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Integer(2), exp(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(z,\\varphi)} = \\varphi^{z}, then obtain 1 = \\frac{\\varphi^{z}}{\\sigma_{p}{(z,\\varphi)}}", "derivation": "\\sigma_{p}{(z,\\varphi)} = \\varphi^{z} and - \\varphi + \\sigma_{p}{(z,\\varphi)} = - \\varphi + \\varphi^{z} and (- \\varphi + \\sigma_{p}{(z,\\varphi)})^{z} = (- \\varphi + \\varphi^{z})^{z} and (- \\varphi + \\varphi^{z})^{- z} \\sigma_{p}{(z,\\varphi)} = \\varphi^{z} (- \\varphi + \\varphi^{z})^{- z} and (- \\varphi + \\sigma_{p}{(z,\\varphi)})^{- z} \\sigma_{p}{(z,\\varphi)} = \\varphi^{z} (- \\varphi + \\sigma_{p}{(z,\\varphi)})^{- z} and (- \\varphi + \\varphi^{z})^{z} (- \\varphi + \\sigma_{p}{(z,\\varphi)})^{- z} = \\frac{\\varphi^{z} (- \\varphi + \\varphi^{z})^{z} (- \\varphi + \\sigma_{p}{(z,\\varphi)})^{- z}}{\\sigma_{p}{(z,\\varphi)}} and 1 = \\frac{\\varphi^{z}}{\\sigma_{p}{(z,\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True)))"], [["minus", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["divide", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["divide", 5, "Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Pow(Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('z', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given v{(b,\\rho_f)} = \\rho_f + b, then obtain - \\frac{(- b + v{(b,\\rho_f)})^{\\rho_f}}{b} = - \\frac{\\rho_f^{\\rho_f}}{b}", "derivation": "v{(b,\\rho_f)} = \\rho_f + b and - b + v{(b,\\rho_f)} = \\rho_f and (- b + v{(b,\\rho_f)})^{\\rho_f} = \\rho_f^{\\rho_f} and - \\frac{(- b + v{(b,\\rho_f)})^{\\rho_f}}{b} = - \\frac{\\rho_f^{\\rho_f}}{b}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('b', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('b', commutative=True)))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('v')(Symbol('b', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))"], [["power", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('v')(Symbol('b', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('v')(Symbol('b', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('b', commutative=True), Integer(-1))))"]]}, {"prompt": "Given u{(m,F_{x})} = - F_{x} + m and \\operatorname{x^{{\\}'}}{(m,F_{x})} = - 2 F_{x} + m, then obtain \\int 2 m \\operatorname{x^{{\\}'}}{(m,F_{x})} dF_{x} = \\int 2 m (- 2 F_{x} + m) dF_{x}", "derivation": "u{(m,F_{x})} = - F_{x} + m and - F_{x} + u{(m,F_{x})} = - 2 F_{x} + m and \\operatorname{x^{{\\}'}}{(m,F_{x})} = - 2 F_{x} + m and \\operatorname{x^{{\\}'}}{(m,F_{x})} = - F_{x} + u{(m,F_{x})} and 2 m \\operatorname{x^{{\\}'}}{(m,F_{x})} = 2 m (- F_{x} + u{(m,F_{x})}) and 2 m \\operatorname{x^{{\\}'}}{(m,F_{x})} = 2 m (- 2 F_{x} + m) and \\int 2 m \\operatorname{x^{{\\}'}}{(m,F_{x})} dF_{x} = \\int 2 m (- 2 F_{x} + m) dF_{x}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('m', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('m', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('u')(Symbol('m', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('m', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('x^\\\\prime')(Symbol('m', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('u')(Symbol('m', commutative=True), Symbol('F_x', commutative=True))))"], [["times", 4, "Mul(Integer(2), Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Symbol('m', commutative=True), Function('x^\\\\prime')(Symbol('m', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('u')(Symbol('m', commutative=True), Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Symbol('m', commutative=True), Function('x^\\\\prime')(Symbol('m', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(2), Symbol('m', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('m', commutative=True))))"], [["integrate", 6, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Integer(2), Symbol('m', commutative=True), Function('x^\\\\prime')(Symbol('m', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Integer(2), Symbol('m', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then derive \\int \\mathbf{M}{(\\varepsilon)} d\\varepsilon = t + \\sin{(\\varepsilon)}, then derive (t + \\sin{(\\varepsilon)}) (z + \\sin{(\\varepsilon)}) = (z + \\sin{(\\varepsilon)})^{2}, then obtain (z + \\sin{(\\varepsilon)}) \\int \\cos{(\\varepsilon)} d\\varepsilon = (z + \\sin{(\\varepsilon)})^{2}", "derivation": "\\mathbf{M}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\int \\mathbf{M}{(\\varepsilon)} d\\varepsilon = \\int \\cos{(\\varepsilon)} d\\varepsilon and \\int \\mathbf{M}{(\\varepsilon)} d\\varepsilon = t + \\sin{(\\varepsilon)} and (\\int \\mathbf{M}{(\\varepsilon)} d\\varepsilon) \\int \\cos{(\\varepsilon)} d\\varepsilon = (\\int \\cos{(\\varepsilon)} d\\varepsilon)^{2} and t + \\sin{(\\varepsilon)} = \\int \\cos{(\\varepsilon)} d\\varepsilon and (t + \\sin{(\\varepsilon)}) \\int \\cos{(\\varepsilon)} d\\varepsilon = (\\int \\cos{(\\varepsilon)} d\\varepsilon)^{2} and (t + \\sin{(\\varepsilon)}) (z + \\sin{(\\varepsilon)}) = (z + \\sin{(\\varepsilon)})^{2} and (z + \\sin{(\\varepsilon)}) \\int \\cos{(\\varepsilon)} d\\varepsilon = (z + \\sin{(\\varepsilon)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('t', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 2, "Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Pow(Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('t', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('t', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Pow(Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integer(2)))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Symbol('t', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('z', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True)))), Pow(Add(Symbol('z', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Add(Symbol('z', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Integral(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Pow(Add(Symbol('z', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2)))"]]}, {"prompt": "Given i{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\mathbf{J}_M{(\\Psi^{\\dagger},Q)} = \\sin{(Q + \\Psi^{\\dagger})}, then obtain \\hat{p}_0 + \\mathbf{J}_M{(\\Psi^{\\dagger},Q)} - i{(\\hat{p}_0)} = \\hat{p}_0 - i{(\\hat{p}_0)} + \\sin{(Q + \\Psi^{\\dagger})}", "derivation": "i{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\mathbf{J}_M{(\\Psi^{\\dagger},Q)} = \\sin{(Q + \\Psi^{\\dagger})} and \\hat{p}_0 + \\mathbf{J}_M{(\\Psi^{\\dagger},Q)} - e^{\\hat{p}_0} = \\hat{p}_0 - e^{\\hat{p}_0} + \\sin{(Q + \\Psi^{\\dagger})} and \\hat{p}_0 + \\mathbf{J}_M{(\\Psi^{\\dagger},Q)} - i{(\\hat{p}_0)} = \\hat{p}_0 - i{(\\hat{p}_0)} + \\sin{(Q + \\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), sin(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), sin(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Function('i')(Symbol('\\\\hat{p}_0', commutative=True))), sin(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given b{(n_{2},G,\\varepsilon)} = - \\varepsilon + n_{2}^{G}, then obtain \\cos{(b{(n_{2},G,\\varepsilon)})} + \\cos^{n_{2}}{(b{(n_{2},G,\\varepsilon)})} + \\frac{1}{- \\varepsilon + n_{2}^{G}} = \\cos^{n_{2}}{(\\varepsilon - n_{2}^{G})} + \\cos{(b{(n_{2},G,\\varepsilon)})} + \\frac{1}{- \\varepsilon + n_{2}^{G}}", "derivation": "b{(n_{2},G,\\varepsilon)} = - \\varepsilon + n_{2}^{G} and \\cos{(b{(n_{2},G,\\varepsilon)})} = \\cos{(\\varepsilon - n_{2}^{G})} and \\cos^{n_{2}}{(b{(n_{2},G,\\varepsilon)})} = \\cos^{n_{2}}{(\\varepsilon - n_{2}^{G})} and \\cos{(b{(n_{2},G,\\varepsilon)})} + \\cos^{n_{2}}{(b{(n_{2},G,\\varepsilon)})} = \\cos^{n_{2}}{(\\varepsilon - n_{2}^{G})} + \\cos{(b{(n_{2},G,\\varepsilon)})} and \\cos{(b{(n_{2},G,\\varepsilon)})} + \\cos^{n_{2}}{(b{(n_{2},G,\\varepsilon)})} + \\frac{1}{- \\varepsilon + n_{2}^{G}} = \\cos^{n_{2}}{(\\varepsilon - n_{2}^{G})} + \\cos{(b{(n_{2},G,\\varepsilon)})} + \\frac{1}{- \\varepsilon + n_{2}^{G}}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))))"], [["cos", 1], "Equality(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), cos(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))))))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('n_2', commutative=True)), Pow(cos(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))))), Symbol('n_2', commutative=True)))"], [["add", 3, "cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('n_2', commutative=True))), Add(Pow(cos(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))))), Symbol('n_2', commutative=True)), cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 4, "Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))), Integer(-1))"], "Equality(Add(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('n_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Add(Pow(cos(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))))), Symbol('n_2', commutative=True)), cos(Function('b')(Symbol('n_2', commutative=True), Symbol('G', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('G', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(x,l,v_{y})} = (l x)^{v_{y}}, then obtain 1 = \\frac{x + \\frac{(l x)^{v_{y}}}{\\mathbf{D}{(x,l,v_{y})}}}{x + 1}", "derivation": "\\mathbf{D}{(x,l,v_{y})} = (l x)^{v_{y}} and 1 = \\frac{(l x)^{v_{y}}}{\\mathbf{D}{(x,l,v_{y})}} and x + 1 = x + \\frac{(l x)^{v_{y}}}{\\mathbf{D}{(x,l,v_{y})}} and 1 = \\frac{x + \\frac{(l x)^{v_{y}}}{\\mathbf{D}{(x,l,v_{y})}}}{x + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('x', commutative=True), Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Pow(Mul(Symbol('l', commutative=True), Symbol('x', commutative=True)), Symbol('v_y', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{D}')(Symbol('x', commutative=True), Symbol('l', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Symbol('l', commutative=True), Symbol('x', commutative=True)), Symbol('v_y', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('x', commutative=True), Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Integer(1)), Add(Symbol('x', commutative=True), Mul(Pow(Mul(Symbol('l', commutative=True), Symbol('x', commutative=True)), Symbol('v_y', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('x', commutative=True), Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)))))"], [["divide", 3, "Add(Symbol('x', commutative=True), Integer(1))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('x', commutative=True), Integer(1)), Integer(-1)), Add(Symbol('x', commutative=True), Mul(Pow(Mul(Symbol('l', commutative=True), Symbol('x', commutative=True)), Symbol('v_y', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('x', commutative=True), Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\sigma_{x}{(\\nabla,v)} = \\sin{(\\nabla + v)}, then obtain \\frac{\\partial}{\\partial \\nabla} \\log{(v (\\sigma_{x}{(\\nabla,v)} + \\sin{(\\nabla + v)}))} = \\frac{\\partial}{\\partial \\nabla} \\log{(2 v \\sin{(\\nabla + v)})}", "derivation": "\\sigma_{x}{(\\nabla,v)} = \\sin{(\\nabla + v)} and \\sigma_{x}{(\\nabla,v)} + \\sin{(\\nabla + v)} = 2 \\sin{(\\nabla + v)} and v (\\sigma_{x}{(\\nabla,v)} + \\sin{(\\nabla + v)}) = 2 v \\sin{(\\nabla + v)} and \\log{(v (\\sigma_{x}{(\\nabla,v)} + \\sin{(\\nabla + v)}))} = \\log{(2 v \\sin{(\\nabla + v)})} and \\frac{\\partial}{\\partial \\nabla} \\log{(v (\\sigma_{x}{(\\nabla,v)} + \\sin{(\\nabla + v)}))} = \\frac{\\partial}{\\partial \\nabla} \\log{(2 v \\sin{(\\nabla + v)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))))"], [["add", 1, "sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))), Mul(Integer(2), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))))"], [["times", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Add(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))))), Mul(Integer(2), Symbol('v', commutative=True), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))))"], [["log", 3], "Equality(log(Mul(Symbol('v', commutative=True), Add(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))))), log(Mul(Integer(2), Symbol('v', commutative=True), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(log(Mul(Symbol('v', commutative=True), Add(Function('\\\\sigma_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(log(Mul(Integer(2), Symbol('v', commutative=True), sin(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True))))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(F_{x})} = \\sin{(F_{x})} and \\chi{(F_{x})} = (\\frac{d}{d F_{x}} \\mathbf{J}_P{(F_{x})})^{F_{x}}, then obtain \\chi{(F_{x})} = \\cos^{F_{x}}{(F_{x})}", "derivation": "\\mathbf{J}_P{(F_{x})} = \\sin{(F_{x})} and \\frac{d}{d F_{x}} \\mathbf{J}_P{(F_{x})} = \\frac{d}{d F_{x}} \\sin{(F_{x})} and (\\frac{d}{d F_{x}} \\mathbf{J}_P{(F_{x})})^{F_{x}} = (\\frac{d}{d F_{x}} \\sin{(F_{x})})^{F_{x}} and \\chi{(F_{x})} = (\\frac{d}{d F_{x}} \\mathbf{J}_P{(F_{x})})^{F_{x}} and \\chi{(F_{x})} = (\\frac{d}{d F_{x}} \\sin{(F_{x})})^{F_{x}} and \\chi{(F_{x})} = \\cos^{F_{x}}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('F_x', commutative=True)), Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\chi')(Symbol('F_x', commutative=True)), Pow(Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\chi')(Symbol('F_x', commutative=True)), Pow(cos(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and T{(F_{x})} = \\sin{(F_{x})}, then obtain 0^{\\mathbf{r}} + T{(F_{x})} = 0^{\\mathbf{r}} + T{(F_{x})} - \\rho_{f}{(\\mathbf{r})} + \\cos{(\\mathbf{r})}", "derivation": "\\rho_{f}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and 0 = - \\rho_{f}{(\\mathbf{r})} + \\cos{(\\mathbf{r})} and T{(F_{x})} = \\sin{(F_{x})} and \\sin{(F_{x})} = - \\rho_{f}{(\\mathbf{r})} + \\sin{(F_{x})} + \\cos{(\\mathbf{r})} and T{(F_{x})} = T{(F_{x})} - \\rho_{f}{(\\mathbf{r})} + \\cos{(\\mathbf{r})} and 0^{\\mathbf{r}} + T{(F_{x})} = 0^{\\mathbf{r}} + T{(F_{x})} - \\rho_{f}{(\\mathbf{r})} + \\cos{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 1, "Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True))), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], ["get_premise", "Equality(Function('T')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["add", 2, "sin(Symbol('F_x', commutative=True))"], "Equality(sin(Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True))), sin(Symbol('F_x', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('T')(Symbol('F_x', commutative=True)), Add(Function('T')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True))), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 5, "Pow(Integer(0), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('\\\\mathbf{r}', commutative=True)), Function('T')(Symbol('F_x', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\mathbf{r}', commutative=True)), Function('T')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\mathbf{r}', commutative=True))), cos(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = \\cos{(f^{\\prime})}, then derive \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = - \\sin{(f^{\\prime})}, then obtain \\cos{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} = \\cos{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = - \\sin{(f^{\\prime})} and \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} = \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})} - \\sin{(f^{\\prime})} and \\cos{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} = - \\sin{(f^{\\prime})} + \\cos{(f^{\\prime})} and \\cos{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} = \\cos{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\operatorname{f_{\\mathbf{v}}}{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 3, "Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('f^{\\\\prime}', commutative=True)), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(cos(Symbol('f^{\\\\prime}', commutative=True)), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(cos(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(a)} = \\cos{(a)}, then derive \\int \\varphi{(a)} da = \\mathbf{v} + \\sin{(a)}, then obtain \\frac{d}{d a} \\int \\varphi{(a)} da = \\cos{(a)}", "derivation": "\\varphi{(a)} = \\cos{(a)} and \\int \\varphi{(a)} da = \\int \\cos{(a)} da and \\int \\varphi{(a)} da = \\mathbf{v} + \\sin{(a)} and \\frac{d}{d a} \\int \\varphi{(a)} da = \\frac{\\partial}{\\partial a} (\\mathbf{v} + \\sin{(a)}) and \\frac{d}{d a} \\int \\varphi{(a)} da = \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('a', commutative=True))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varphi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\varphi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), cos(Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(t)} = e^{t} + (e^{t})^{t} and H{(t)} = e^{t} + (e^{t})^{t}, then obtain - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + 2 H{(t)} + e^{t} = - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + H{(t)} + 2 e^{t} + (e^{t})^{t}", "derivation": "\\mathbf{J}{(t)} = e^{t} + (e^{t})^{t} and H{(t)} = e^{t} + (e^{t})^{t} and 2 \\mathbf{J}{(t)} = \\mathbf{J}{(t)} + e^{t} + (e^{t})^{t} and \\mathbf{J}{(t)} = H{(t)} and 2 H{(t)} = H{(t)} + e^{t} + (e^{t})^{t} and - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + 2 H{(t)} = - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + H{(t)} + e^{t} + (e^{t})^{t} and - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + 2 H{(t)} + e^{t} = - (\\mathbf{J}{(t)} + e^{t} + (e^{t})^{t}) (e^{t})^{t} + H{(t)} + 2 e^{t} + (e^{t})^{t}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), Add(exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('t', commutative=True)), Add(exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('t', commutative=True))), Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), Function('H')(Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Function('H')(Symbol('t', commutative=True))), Add(Function('H')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["minus", 5, "Mul(Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(2), Function('H')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('H')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"], [["add", 6, "exp(Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Integer(2), Function('H')(Symbol('t', commutative=True))), exp(Symbol('t', commutative=True))), Add(Mul(Integer(-1), Add(Function('\\\\mathbf{J}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Function('H')(Symbol('t', commutative=True)), Mul(Integer(2), exp(Symbol('t', commutative=True))), Pow(exp(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} = f^{\\prime} (G + t_{1}), then obtain \\int (G - f^{\\prime} + 2 \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)}) df^{\\prime} = \\int (G + f^{\\prime} (G + t_{1}) - f^{\\prime} + \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)}) df^{\\prime}", "derivation": "\\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} = f^{\\prime} (G + t_{1}) and - f^{\\prime} + \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} = f^{\\prime} (G + t_{1}) - f^{\\prime} and - f^{\\prime} + 2 \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} = f^{\\prime} (G + t_{1}) - f^{\\prime} + \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} and G - f^{\\prime} + 2 \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} = G + f^{\\prime} (G + t_{1}) - f^{\\prime} + \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)} and \\int (G - f^{\\prime} + 2 \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)}) df^{\\prime} = \\int (G + f^{\\prime} (G + t_{1}) - f^{\\prime} + \\operatorname{f^{*}}{(t_{1},f^{\\prime},G)}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('G', commutative=True), Symbol('t_1', commutative=True))))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('G', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('G', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('G', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True))))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('G', commutative=True), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('f^*')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(t_{2},p)} = - t_{2} + \\log{(p)} and \\omega{(t_{2},p)} = \\frac{- t_{2} + \\log{(p)}}{p}, then obtain 1 + \\frac{2 \\operatorname{f^{\\prime}}{(t_{2},p)}}{p} = \\omega{(t_{2},p)} + 1 + \\frac{\\operatorname{f^{\\prime}}{(t_{2},p)}}{p}", "derivation": "\\operatorname{f^{\\prime}}{(t_{2},p)} = - t_{2} + \\log{(p)} and \\frac{\\operatorname{f^{\\prime}}{(t_{2},p)}}{p} = \\frac{- t_{2} + \\log{(p)}}{p} and 1 + \\frac{\\operatorname{f^{\\prime}}{(t_{2},p)}}{p} = 1 + \\frac{- t_{2} + \\log{(p)}}{p} and 1 + \\frac{2 \\operatorname{f^{\\prime}}{(t_{2},p)}}{p} = 1 + \\frac{- t_{2} + \\log{(p)}}{p} + \\frac{\\operatorname{f^{\\prime}}{(t_{2},p)}}{p} and \\omega{(t_{2},p)} = \\frac{- t_{2} + \\log{(p)}}{p} and 1 + \\frac{2 \\operatorname{f^{\\prime}}{(t_{2},p)}}{p} = \\omega{(t_{2},p)} + 1 + \\frac{\\operatorname{f^{\\prime}}{(t_{2},p)}}{p}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('p', commutative=True))))"], [["divide", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('p', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('p', commutative=True))))))"], [["add", 3, "Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(2), Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('p', commutative=True)))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Integer(1), Mul(Integer(2), Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))), Add(Function('\\\\omega')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)), Integer(1), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('t_2', commutative=True), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(v_{t})} = e^{v_{t}}, then obtain - \\frac{d}{d v_{t}} \\mathbf{D}{(v_{t})} + \\int (v_{t} + \\mathbf{D}{(v_{t})}) dv_{t} - 1 = - \\frac{d}{d v_{t}} \\mathbf{D}{(v_{t})} + \\int (v_{t} + e^{v_{t}}) dv_{t} - 1", "derivation": "\\mathbf{D}{(v_{t})} = e^{v_{t}} and v_{t} + \\mathbf{D}{(v_{t})} = v_{t} + e^{v_{t}} and \\int (v_{t} + \\mathbf{D}{(v_{t})}) dv_{t} = \\int (v_{t} + e^{v_{t}}) dv_{t} and - \\frac{d}{d v_{t}} (v_{t} + \\mathbf{D}{(v_{t})}) + \\int (v_{t} + \\mathbf{D}{(v_{t})}) dv_{t} = - \\frac{d}{d v_{t}} (v_{t} + \\mathbf{D}{(v_{t})}) + \\int (v_{t} + e^{v_{t}}) dv_{t} and - \\frac{d}{d v_{t}} \\mathbf{D}{(v_{t})} + \\int (v_{t} + \\mathbf{D}{(v_{t})}) dv_{t} - 1 = - \\frac{d}{d v_{t}} \\mathbf{D}{(v_{t})} + \\int (v_{t} + e^{v_{t}}) dv_{t} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["add", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integral(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["minus", 3, "Derivative(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integral(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), Derivative(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integral(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integral(Add(Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integral(Add(Symbol('v_t', commutative=True), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{M})} = e^{\\mathbf{M}}, then derive \\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})} = \\sin{(e^{\\mathbf{M}})}, then obtain 1 = \\frac{\\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})}}{\\sin{(\\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}})}}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} and \\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})} = \\sin{(\\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}})} and \\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})} = \\sin{(e^{\\mathbf{M}})} and 1 = \\frac{\\sin{(e^{\\mathbf{M}})}}{\\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})}} and 1 = \\frac{\\sin{(e^{\\mathbf{M}})}}{\\sin{(\\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}})}} and 1 = \\frac{\\sin{(\\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})})}}{\\sin{(\\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}})}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), sin(Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), sin(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 4, "Pow(sin(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Integer(-1))"], "Equality(Integer(1), Mul(sin(exp(Symbol('\\\\mathbf{M}', commutative=True))), Pow(sin(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(1), Mul(sin(exp(Symbol('\\\\mathbf{M}', commutative=True))), Pow(sin(Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integer(1), Mul(sin(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Pow(sin(Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Integer(-1))))"]]}, {"prompt": "Given \\dot{x}{(a^{\\dagger})} = \\log{(e^{a^{\\dagger}})}, then derive \\frac{d}{d a^{\\dagger}} \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = \\frac{\\partial}{\\partial a^{\\dagger}} (E + \\frac{(a^{\\dagger})^{2}}{2}), then derive \\frac{d}{d a^{\\dagger}} \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = a^{\\dagger}, then obtain a^{\\dagger} = \\frac{d}{d a^{\\dagger}} \\int \\log{(e^{a^{\\dagger}})} da^{\\dagger}", "derivation": "\\dot{x}{(a^{\\dagger})} = \\log{(e^{a^{\\dagger}})} and \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = \\int \\log{(e^{a^{\\dagger}})} da^{\\dagger} and \\frac{d}{d a^{\\dagger}} \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = \\frac{d}{d a^{\\dagger}} \\int \\log{(e^{a^{\\dagger}})} da^{\\dagger} and \\frac{d}{d a^{\\dagger}} \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = \\frac{\\partial}{\\partial a^{\\dagger}} (E + \\frac{(a^{\\dagger})^{2}}{2}) and \\frac{d}{d a^{\\dagger}} \\int \\dot{x}{(a^{\\dagger})} da^{\\dagger} = a^{\\dagger} and a^{\\dagger} = \\frac{d}{d a^{\\dagger}} \\int \\log{(e^{a^{\\dagger}})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), log(exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(log(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\dot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integral(log(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\dot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\dot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Symbol('a^{\\\\dagger}', commutative=True), Derivative(Integral(log(exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(S,k)} = \\log{(S k)}, then derive - \\int \\hat{p}_0{(S,k)} dS = - S \\log{(S k)} + S - \\phi, then obtain - \\int \\log{(S k)} dS = - S \\hat{p}_0{(S,k)} + S - \\phi", "derivation": "\\hat{p}_0{(S,k)} = \\log{(S k)} and \\int \\hat{p}_0{(S,k)} dS = \\int \\log{(S k)} dS and - \\int \\hat{p}_0{(S,k)} dS = - \\int \\log{(S k)} dS and - \\int \\hat{p}_0{(S,k)} dS = - S \\log{(S k)} + S - \\phi and - \\int \\log{(S k)} dS = - S \\log{(S k)} + S - \\phi and - \\int \\log{(S k)} dS = - S \\hat{p}_0{(S,k)} + S - \\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('k', commutative=True)), log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Integral(log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True)))), Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Integral(log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True), log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True)))), Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Integral(log(Mul(Symbol('S', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Function('\\\\hat{p}_0')(Symbol('S', commutative=True), Symbol('k', commutative=True))), Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given y{(r,F_{x})} = \\frac{\\partial}{\\partial r} (F_{x} + r), then derive y^{F_{x}}{(r,F_{x})} = 1, then obtain 2 = y^{F_{x}}{(r,F_{x})} + 1", "derivation": "y{(r,F_{x})} = \\frac{\\partial}{\\partial r} (F_{x} + r) and y^{F_{x}}{(r,F_{x})} = (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} and y^{F_{x}}{(r,F_{x})} = 1 and y^{F_{x}}{(r,F_{x})} + (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} = (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} + 1 and 2 y^{F_{x}}{(r,F_{x})} = y^{F_{x}}{(r,F_{x})} + 1 and 2 (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} = (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} + 1 and 2 (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} = y^{F_{x}}{(r,F_{x})} + (\\frac{\\partial}{\\partial r} (F_{x} + r))^{F_{x}} and 2 = y^{F_{x}}{(r,F_{x})} + 1", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Integer(1))"], [["add", 3, "Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True))"], "Equality(Add(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True))), Add(Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Add(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True))), Add(Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Integer(2), Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True))), Add(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Derivative(Add(Symbol('F_x', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('F_x', commutative=True))))"], [["evaluate_derivatives", 7], "Equality(Integer(2), Add(Pow(Function('y')(Symbol('r', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(A_{z})} = \\sin{(\\sin{(A_{z})})}, then obtain \\Psi^{\\dagger}^{7}{(A_{z})} \\sin^{15}{(\\sin{(A_{z})})} = \\Psi^{\\dagger}^{5}{(A_{z})} \\sin^{17}{(\\sin{(A_{z})})}", "derivation": "\\Psi^{\\dagger}{(A_{z})} = \\sin{(\\sin{(A_{z})})} and \\Psi^{\\dagger}{(A_{z})} \\sin{(\\sin{(A_{z})})} = \\sin^{2}{(\\sin{(A_{z})})} and \\Psi^{\\dagger}{(A_{z})} \\sin^{3}{(\\sin{(A_{z})})} = \\sin^{4}{(\\sin{(A_{z})})} and \\Psi^{\\dagger}^{2}{(A_{z})} \\sin^{4}{(\\sin{(A_{z})})} = \\Psi^{\\dagger}{(A_{z})} \\sin^{5}{(\\sin{(A_{z})})} and \\Psi^{\\dagger}^{3}{(A_{z})} \\sin^{3}{(\\sin{(A_{z})})} = \\Psi^{\\dagger}{(A_{z})} \\sin^{5}{(\\sin{(A_{z})})} and \\Psi^{\\dagger}^{7}{(A_{z})} \\sin^{15}{(\\sin{(A_{z})})} = \\Psi^{\\dagger}^{5}{(A_{z})} \\sin^{17}{(\\sin{(A_{z})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), sin(sin(Symbol('A_z', commutative=True))))"], [["times", 1, "sin(sin(Symbol('A_z', commutative=True)))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), sin(sin(Symbol('A_z', commutative=True)))), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(2)))"], [["times", 2, "Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(3))), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(4)))"], [["times", 2, "Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(3)))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Integer(2)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(4))), Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(5))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Integer(3)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(3))), Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(5))))"], [["times", 5, "Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Integer(4)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(12)))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Integer(7)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(15))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_z', commutative=True)), Integer(5)), Pow(sin(sin(Symbol('A_z', commutative=True))), Integer(17))))"]]}, {"prompt": "Given \\phi_{1}{(\\chi,\\mathbf{A})} = - \\chi + \\mathbf{A}, then obtain - \\mathbf{A} + \\int \\phi_{1}{(\\chi,\\mathbf{A})} d\\mathbf{A} = - \\mathbf{A} + \\int (- \\chi + \\mathbf{A}) d\\mathbf{A}", "derivation": "\\phi_{1}{(\\chi,\\mathbf{A})} = - \\chi + \\mathbf{A} and \\int \\phi_{1}{(\\chi,\\mathbf{A})} d\\mathbf{A} = \\int (- \\chi + \\mathbf{A}) d\\mathbf{A} and \\chi - \\mathbf{A} + \\int \\phi_{1}{(\\chi,\\mathbf{A})} d\\mathbf{A} = \\chi - \\mathbf{A} + \\int (- \\chi + \\mathbf{A}) d\\mathbf{A} and - \\mathbf{A} + \\int \\phi_{1}{(\\chi,\\mathbf{A})} d\\mathbf{A} = - \\mathbf{A} + \\int (- \\chi + \\mathbf{A}) d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["minus", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(u)} = \\log{(u)}, then obtain 2 \\varphi^{*}{(u)} \\frac{d}{d u} \\varphi^{*}{(u)} = \\log{(u)} \\frac{d}{d u} \\varphi^{*}{(u)} + \\frac{\\varphi^{*}{(u)}}{u}", "derivation": "\\varphi^{*}{(u)} = \\log{(u)} and \\varphi^{*}^{2}{(u)} = \\varphi^{*}{(u)} \\log{(u)} and \\frac{d}{d u} \\varphi^{*}^{2}{(u)} = \\frac{d}{d u} \\varphi^{*}{(u)} \\log{(u)} and 2 \\varphi^{*}{(u)} \\frac{d}{d u} \\varphi^{*}{(u)} = \\log{(u)} \\frac{d}{d u} \\varphi^{*}{(u)} + \\frac{\\varphi^{*}{(u)}}{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["times", 1, "Function('\\\\varphi^*')(Symbol('u', commutative=True))"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\varphi^*')(Symbol('u', commutative=True)), Integer(2)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\varphi^*')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('u', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Mul(log(Symbol('u', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(x)} = \\frac{d}{d x} \\sin{(x)}, then obtain \\frac{d}{d x} (- 2 x \\operatorname{E_{n}}{(x)} + 2 x \\frac{d}{d x} \\sin{(x)})^{x} = \\frac{d}{d x} 1", "derivation": "\\operatorname{E_{n}}{(x)} = \\frac{d}{d x} \\sin{(x)} and x \\operatorname{E_{n}}{(x)} = x \\frac{d}{d x} \\sin{(x)} and 0 = - x \\operatorname{E_{n}}{(x)} + x \\frac{d}{d x} \\sin{(x)} and 0^{x} = (- x \\operatorname{E_{n}}{(x)} + x \\frac{d}{d x} \\sin{(x)})^{x} and - x \\operatorname{E_{n}}{(x)} = - 2 x \\operatorname{E_{n}}{(x)} + x \\frac{d}{d x} \\sin{(x)} and 0^{x} = (- 2 x \\operatorname{E_{n}}{(x)} + 2 x \\frac{d}{d x} \\sin{(x)})^{x} and (- 2 x \\operatorname{E_{n}}{(x)} + 2 x \\frac{d}{d x} \\sin{(x)})^{x} = 1 and \\frac{d}{d x} (- 2 x \\operatorname{E_{n}}{(x)} + 2 x \\frac{d}{d x} \\sin{(x)})^{x} = \\frac{d}{d x} 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('x', commutative=True)), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["minus", 2, "Mul(Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Symbol('x', commutative=True)))"], [["minus", 3, "Mul(Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Integer(0), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Symbol('x', commutative=True)), Integer(1))"], [["differentiate", 7, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True), Function('E_n')(Symbol('x', commutative=True))), Mul(Integer(2), Symbol('x', commutative=True), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\mathbf{g})} = e^{\\mathbf{g}}, then obtain \\sin^{\\mathbf{g}}{(V{(\\mathbf{g})} - e^{\\mathbf{g}})} = 1", "derivation": "V{(\\mathbf{g})} = e^{\\mathbf{g}} and 0 = - V{(\\mathbf{g})} + e^{\\mathbf{g}} and 0 = - \\sin{(V{(\\mathbf{g})} - e^{\\mathbf{g}})} and 0^{\\mathbf{g}} = (- \\sin{(V{(\\mathbf{g})} - e^{\\mathbf{g}})})^{\\mathbf{g}} and \\sin{(V{(\\mathbf{g})} - e^{\\mathbf{g}})} = 0 and \\sin^{\\mathbf{g}}{(V{(\\mathbf{g})} - e^{\\mathbf{g}})} = 0^{\\mathbf{g}} and \\sin^{\\mathbf{g}}{(V{(\\mathbf{g})} - e^{\\mathbf{g}})} = 1", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\mathbf{g}', commutative=True)))"], [["minus", 1, "Function('V')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{g}', commutative=True))), exp(Symbol('\\\\mathbf{g}', commutative=True))))"], [["sin", 2], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True)))))))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True)))))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["add", 3, "sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True)))))"], "Equality(sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))))), Integer(0))"], [["power", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Pow(sin(Add(Function('V')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{g}', commutative=True))))), Symbol('\\\\mathbf{g}', commutative=True)), Integer(1))"]]}, {"prompt": "Given u{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})}, then obtain \\frac{u{(\\Psi_{nl})} \\cos{(\\Psi_{nl})}}{\\int 1 d\\Psi_{nl}} = \\frac{\\cos^{2}{(\\Psi_{nl})}}{\\int 1 d\\Psi_{nl}}", "derivation": "u{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and u{(\\Psi_{nl})} \\cos{(\\Psi_{nl})} = \\cos^{2}{(\\Psi_{nl})} and \\frac{u{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} = 1 and \\int \\frac{u{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} d\\Psi_{nl} = \\int 1 d\\Psi_{nl} and \\frac{u{(\\Psi_{nl})} \\cos{(\\Psi_{nl})}}{\\int \\frac{u{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} d\\Psi_{nl}} = \\frac{\\cos^{2}{(\\Psi_{nl})}}{\\int \\frac{u{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} d\\Psi_{nl}} and \\frac{u{(\\Psi_{nl})} \\cos{(\\Psi_{nl})}}{\\int 1 d\\Psi_{nl}} = \\frac{\\cos^{2}{(\\Psi_{nl})}}{\\int 1 d\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)))"], [["divide", 1, "cos(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 2, "Integral(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Integral(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))), Mul(Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Pow(Integral(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('u')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))), Mul(Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(x^\\prime)} = e^{x^\\prime}, then obtain 1 = \\frac{\\sin{(\\int \\frac{d}{d x^\\prime} e^{x^\\prime} dx^\\prime)}}{\\sin{(\\int \\frac{d}{d x^\\prime} \\varepsilon{(x^\\prime)} dx^\\prime)}}", "derivation": "\\varepsilon{(x^\\prime)} = e^{x^\\prime} and \\frac{d}{d x^\\prime} \\varepsilon{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} and \\int \\frac{d}{d x^\\prime} \\varepsilon{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} e^{x^\\prime} dx^\\prime and \\sin{(\\int \\frac{d}{d x^\\prime} \\varepsilon{(x^\\prime)} dx^\\prime)} = \\sin{(\\int \\frac{d}{d x^\\prime} e^{x^\\prime} dx^\\prime)} and 1 = \\frac{\\sin{(\\int \\frac{d}{d x^\\prime} e^{x^\\prime} dx^\\prime)}}{\\sin{(\\int \\frac{d}{d x^\\prime} \\varepsilon{(x^\\prime)} dx^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Derivative(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True)))), sin(Integral(Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["divide", 4, "sin(Integral(Derivative(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], "Equality(Integer(1), Mul(Pow(sin(Integral(Derivative(Function('\\\\varepsilon')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Integer(-1)), sin(Integral(Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))))"]]}, {"prompt": "Given m{(\\mathbf{D},H)} = H - \\mathbf{D}, then obtain H \\mathbf{D} + \\frac{- H + m{(\\mathbf{D},H)}}{\\int m{(\\mathbf{D},H)} dH} = H \\mathbf{D} - \\frac{\\mathbf{D}}{\\int m{(\\mathbf{D},H)} dH}", "derivation": "m{(\\mathbf{D},H)} = H - \\mathbf{D} and \\int m{(\\mathbf{D},H)} dH = \\int (H - \\mathbf{D}) dH and - H + m{(\\mathbf{D},H)} = - \\mathbf{D} and \\frac{- H + m{(\\mathbf{D},H)}}{\\int (H - \\mathbf{D}) dH} = - \\frac{\\mathbf{D}}{\\int (H - \\mathbf{D}) dH} and \\frac{- H + m{(\\mathbf{D},H)}}{\\int m{(\\mathbf{D},H)} dH} = - \\frac{\\mathbf{D}}{\\int m{(\\mathbf{D},H)} dH} and H \\mathbf{D} + \\frac{- H + m{(\\mathbf{D},H)}}{\\int m{(\\mathbf{D},H)} dH} = H \\mathbf{D} - \\frac{\\mathbf{D}}{\\int m{(\\mathbf{D},H)} dH}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))"], [["divide", 3, "Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True))), Pow(Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True))), Pow(Integral(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Integral(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["minus", 5, "Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True))), Pow(Integral(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1)))), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Integral(Function('m')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given q{(E,g_{\\varepsilon},v_{2})} = - E + g_{\\varepsilon} + v_{2} and n{(E,g_{\\varepsilon},v_{2})} = - E + g_{\\varepsilon} + v_{2}, then obtain n{(E,g_{\\varepsilon},v_{2})} - q{(E,g_{\\varepsilon},v_{2})} = \\frac{(- E + g_{\\varepsilon} + v_{2})^{2}}{q{(E,g_{\\varepsilon},v_{2})}} - q{(E,g_{\\varepsilon},v_{2})}", "derivation": "q{(E,g_{\\varepsilon},v_{2})} = - E + g_{\\varepsilon} + v_{2} and - E + g_{\\varepsilon} + v_{2} = \\frac{(- E + g_{\\varepsilon} + v_{2})^{2}}{q{(E,g_{\\varepsilon},v_{2})}} and n{(E,g_{\\varepsilon},v_{2})} = - E + g_{\\varepsilon} + v_{2} and n{(E,g_{\\varepsilon},v_{2})} = \\frac{(- E + g_{\\varepsilon} + v_{2})^{2}}{q{(E,g_{\\varepsilon},v_{2})}} and n{(E,g_{\\varepsilon},v_{2})} - q{(E,g_{\\varepsilon},v_{2})} = \\frac{(- E + g_{\\varepsilon} + v_{2})^{2}}{q{(E,g_{\\varepsilon},v_{2})}} - q{(E,g_{\\varepsilon},v_{2})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Mul(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Pow(Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(2)), Pow(Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('n')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(2)), Pow(Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["minus", 4, "Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Function('n')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(2)), Pow(Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('q')(Symbol('E', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(l)} = \\log{(l)}, then obtain (\\bar{\\h}{(l)} + \\log{(l)})^{2} + 5 \\bar{\\h}^{2}{(l)} = 2 (\\bar{\\h}{(l)} + \\log{(l)})^{2} + \\bar{\\h}^{2}{(l)}", "derivation": "\\bar{\\h}{(l)} = \\log{(l)} and 2 \\bar{\\h}{(l)} = \\bar{\\h}{(l)} + \\log{(l)} and 4 \\bar{\\h}^{2}{(l)} = (\\bar{\\h}{(l)} + \\log{(l)})^{2} and 5 \\bar{\\h}^{2}{(l)} = (\\bar{\\h}{(l)} + \\log{(l)})^{2} + \\bar{\\h}^{2}{(l)} and 9 \\bar{\\h}^{2}{(l)} = (\\bar{\\h}{(l)} + \\log{(l)})^{2} + 5 \\bar{\\h}^{2}{(l)} and 9 \\bar{\\h}^{2}{(l)} = 2 (\\bar{\\h}{(l)} + \\log{(l)})^{2} + \\bar{\\h}^{2}{(l)} and (\\bar{\\h}{(l)} + \\log{(l)})^{2} + 5 \\bar{\\h}^{2}{(l)} = 2 (\\bar{\\h}{(l)} + \\log{(l)})^{2} + \\bar{\\h}^{2}{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["add", 1, "Function('\\\\hbar')(Symbol('l', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hbar')(Symbol('l', commutative=True))), Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))), Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2)))"], [["add", 3, "Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))"], "Equality(Mul(Integer(5), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))), Add(Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2)), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Integer(4), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(9), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))), Add(Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2)), Mul(Integer(5), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(9), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))), Add(Mul(Integer(2), Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2))), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2)), Mul(Integer(5), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2)))), Add(Mul(Integer(2), Pow(Add(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Integer(2))), Pow(Function('\\\\hbar')(Symbol('l', commutative=True)), Integer(2))))"]]}, {"prompt": "Given T{(f_{\\mathbf{p}},C_{1},\\Omega)} = \\frac{f_{\\mathbf{p}}^{C_{1}}}{\\Omega} and \\mathbb{I}{(f_{\\mathbf{p}},C_{1},\\Omega)} = T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)}, then obtain (T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)})^{C_{1}} = \\mathbb{I}^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)}", "derivation": "T{(f_{\\mathbf{p}},C_{1},\\Omega)} = \\frac{f_{\\mathbf{p}}^{C_{1}}}{\\Omega} and T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)} = (\\frac{f_{\\mathbf{p}}^{C_{1}}}{\\Omega})^{C_{1}} and \\mathbb{I}{(f_{\\mathbf{p}},C_{1},\\Omega)} = T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)} and (T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)})^{C_{1}} = ((\\frac{f_{\\mathbf{p}}^{C_{1}}}{\\Omega})^{C_{1}})^{C_{1}} and \\mathbb{I}{(f_{\\mathbf{p}},C_{1},\\Omega)} = (\\frac{f_{\\mathbf{p}}^{C_{1}}}{\\Omega})^{C_{1}} and (T^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)})^{C_{1}} = \\mathbb{I}^{C_{1}}{(f_{\\mathbf{p}},C_{1},\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('T')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('C_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Function('T')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('C_1', commutative=True)))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Pow(Function('T')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Pow(Function('T')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given p{(F_{g})} = \\sin{(F_{g})}, then obtain \\int p{(F_{g})} \\int \\sin{(F_{g})} dF_{g} dF_{g} = \\int \\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g} dF_{g}", "derivation": "p{(F_{g})} = \\sin{(F_{g})} and \\int p{(F_{g})} dF_{g} = \\int \\sin{(F_{g})} dF_{g} and p{(F_{g})} \\int p{(F_{g})} dF_{g} = \\sin{(F_{g})} \\int p{(F_{g})} dF_{g} and p{(F_{g})} \\int \\sin{(F_{g})} dF_{g} = \\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g} and \\int p{(F_{g})} \\int \\sin{(F_{g})} dF_{g} dF_{g} = \\int \\sin{(F_{g})} \\int \\sin{(F_{g})} dF_{g} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('p')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["times", 1, "Integral(Function('p')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Mul(Function('p')(Symbol('F_g', commutative=True)), Integral(Function('p')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(sin(Symbol('F_g', commutative=True)), Integral(Function('p')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('p')(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(sin(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Function('p')(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(sin(Symbol('F_g', commutative=True)), Integral(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given x{(\\varepsilon_0)} = \\sin{(\\sin{(\\varepsilon_0)})} and \\rho_{b}{(\\varepsilon_0)} = \\varepsilon_0, then obtain \\int 4 x^{2}{(\\varepsilon_0)} d\\rho_{b}{(\\varepsilon_0)} = \\int (x{(\\varepsilon_0)} + \\sin{(\\sin{(\\varepsilon_0)})})^{2} d\\rho_{b}{(\\varepsilon_0)}", "derivation": "x{(\\varepsilon_0)} = \\sin{(\\sin{(\\varepsilon_0)})} and 2 x{(\\varepsilon_0)} = x{(\\varepsilon_0)} + \\sin{(\\sin{(\\varepsilon_0)})} and \\rho_{b}{(\\varepsilon_0)} = \\varepsilon_0 and 4 x^{2}{(\\varepsilon_0)} = (x{(\\varepsilon_0)} + \\sin{(\\sin{(\\varepsilon_0)})})^{2} and \\int 4 x^{2}{(\\varepsilon_0)} d\\varepsilon_0 = \\int (x{(\\varepsilon_0)} + \\sin{(\\sin{(\\varepsilon_0)})})^{2} d\\varepsilon_0 and \\int 4 x^{2}{(\\varepsilon_0)} d\\rho_{b}{(\\varepsilon_0)} = \\int (x{(\\varepsilon_0)} + \\sin{(\\sin{(\\varepsilon_0)})})^{2} d\\rho_{b}{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Function('x')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(sin(Symbol('\\\\varepsilon_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2))), Pow(Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(sin(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(2)))"], [["integrate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Pow(Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(sin(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(2)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Mul(Integer(4), Pow(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2))), Tuple(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True)))), Integral(Pow(Add(Function('x')(Symbol('\\\\varepsilon_0', commutative=True)), sin(sin(Symbol('\\\\varepsilon_0', commutative=True)))), Integer(2)), Tuple(Function('\\\\rho_b')(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(U)} = \\sin{(U)}, then derive \\int \\Psi_{\\lambda}{(U)} dU = \\phi - \\cos{(U)}, then obtain 0 = - \\phi + \\cos{(U)} + \\int \\Psi_{\\lambda}{(U)} dU", "derivation": "\\Psi_{\\lambda}{(U)} = \\sin{(U)} and \\int \\Psi_{\\lambda}{(U)} dU = \\int \\sin{(U)} dU and 0 = - \\int \\Psi_{\\lambda}{(U)} dU + \\int \\sin{(U)} dU and \\int \\Psi_{\\lambda}{(U)} dU = \\phi - \\cos{(U)} and 0 = - \\phi + \\cos{(U)} + \\int \\sin{(U)} dU and 0 = - \\phi + \\cos{(U)} + \\int \\Psi_{\\lambda}{(U)} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('U', commutative=True)), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\theta_2,f_{E})} = - f_{E} + \\sin{(\\theta_2)}, then obtain - \\sin{(e^{\\mathbf{E}})} + \\cos{(h{(\\mathbf{E})})} = - f_{E} - \\varepsilon_{0}{(\\theta_2,f_{E})} + \\sin{(\\theta_2)} - \\sin{(e^{\\mathbf{E}})} + \\cos{(h{(\\mathbf{E})})}", "derivation": "\\varepsilon_{0}{(\\theta_2,f_{E})} = - f_{E} + \\sin{(\\theta_2)} and 0 = - f_{E} - \\varepsilon_{0}{(\\theta_2,f_{E})} + \\sin{(\\theta_2)} and - \\sin{(e^{\\mathbf{E}})} = - f_{E} - \\varepsilon_{0}{(\\theta_2,f_{E})} + \\sin{(\\theta_2)} - \\sin{(e^{\\mathbf{E}})} and - \\sin{(e^{\\mathbf{E}})} + \\cos{(h{(\\mathbf{E})})} = - f_{E} - \\varepsilon_{0}{(\\theta_2,f_{E})} + \\sin{(\\theta_2)} - \\sin{(e^{\\mathbf{E}})} + \\cos{(h{(\\mathbf{E})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True), Symbol('f_E', commutative=True))), sin(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 2, "sin(exp(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True), Symbol('f_E', commutative=True))), sin(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["add", 3, "cos(Function('h')(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{E}', commutative=True)))), cos(Function('h')(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True), Symbol('f_E', commutative=True))), sin(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('\\\\mathbf{E}', commutative=True)))), cos(Function('h')(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\varepsilon_0,\\varphi^*,E_{x})} = E_{x} \\varepsilon_0 + \\varphi^* and \\operatorname{v_{y}}{(\\varepsilon_0,\\varphi^*,E_{x})} = E_{x} \\varepsilon_0 + \\varphi^*, then obtain \\frac{\\partial}{\\partial \\varphi^*} (\\lambda{(\\varepsilon_0,\\varphi^*,E_{x})} + \\operatorname{v_{y}}{(\\varepsilon_0,\\varphi^*,E_{x})}) = \\frac{\\partial}{\\partial \\varphi^*} (2 E_{x} \\varepsilon_0 + 2 \\varphi^*)", "derivation": "\\lambda{(\\varepsilon_0,\\varphi^*,E_{x})} = E_{x} \\varepsilon_0 + \\varphi^* and E_{x} \\varepsilon_0 + \\varphi^* + \\lambda{(\\varepsilon_0,\\varphi^*,E_{x})} = 2 E_{x} \\varepsilon_0 + 2 \\varphi^* and \\frac{\\partial}{\\partial \\varphi^*} (E_{x} \\varepsilon_0 + \\varphi^* + \\lambda{(\\varepsilon_0,\\varphi^*,E_{x})}) = \\frac{\\partial}{\\partial \\varphi^*} (2 E_{x} \\varepsilon_0 + 2 \\varphi^*) and \\operatorname{v_{y}}{(\\varepsilon_0,\\varphi^*,E_{x})} = E_{x} \\varepsilon_0 + \\varphi^* and \\frac{\\partial}{\\partial \\varphi^*} (\\lambda{(\\varepsilon_0,\\varphi^*,E_{x})} + \\operatorname{v_{y}}{(\\varepsilon_0,\\varphi^*,E_{x})}) = \\frac{\\partial}{\\partial \\varphi^*} (2 E_{x} \\varepsilon_0 + 2 \\varphi^*)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True)), Function('v_y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('E_x', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(C,F_{H})} = \\cos{(C - F_{H})} and y{(b,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(b)}, then obtain \\sin{(\\int \\frac{\\eta{(C,F_{H})}}{y{(b,\\Psi_{\\lambda})}} dC)} = \\sin{(\\int \\frac{\\cos{(C - F_{H})}}{y{(b,\\Psi_{\\lambda})}} dC)}", "derivation": "\\eta{(C,F_{H})} = \\cos{(C - F_{H})} and \\frac{\\eta{(C,F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} = \\frac{\\cos{(C - F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} and \\int \\frac{\\eta{(C,F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} dC = \\int \\frac{\\cos{(C - F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} dC and y{(b,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(b)} and \\sin{(\\int \\frac{\\eta{(C,F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} dC)} = \\sin{(\\int \\frac{\\cos{(C - F_{H})}}{\\Psi_{\\lambda} + \\cos{(b)}} dC)} and \\sin{(\\int \\frac{\\eta{(C,F_{H})}}{y{(b,\\Psi_{\\lambda})}} dC)} = \\sin{(\\int \\frac{\\cos{(C - F_{H})}}{y{(b,\\Psi_{\\lambda})}} dC)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))))"], [["divide", 1, "Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Tuple(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('b', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), Function('\\\\eta')(Symbol('C', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('C', commutative=True)))), sin(Integral(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('b', commutative=True))), Integer(-1)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(sin(Integral(Mul(Function('\\\\eta')(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Pow(Function('y')(Symbol('b', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True)))), sin(Integral(Mul(Pow(Function('y')(Symbol('b', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), cos(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varepsilon_0,\\mathbf{B})} = \\log{(\\mathbf{B} + \\varepsilon_0)} and k{(\\varepsilon_0,\\mathbf{B})} = \\mathbf{B} + \\varepsilon_0, then obtain (\\operatorname{A_{2}}^{\\mathbf{B}}{(\\varepsilon_0,\\mathbf{B})})^{\\mathbf{B}} = (\\log{(\\mathbf{B} + \\varepsilon_0)}^{\\mathbf{B}})^{\\mathbf{B}}", "derivation": "\\operatorname{A_{2}}{(\\varepsilon_0,\\mathbf{B})} = \\log{(\\mathbf{B} + \\varepsilon_0)} and k{(\\varepsilon_0,\\mathbf{B})} = \\mathbf{B} + \\varepsilon_0 and \\operatorname{A_{2}}{(\\varepsilon_0,\\mathbf{B})} = \\log{(k{(\\varepsilon_0,\\mathbf{B})})} and \\operatorname{A_{2}}^{\\mathbf{B}}{(\\varepsilon_0,\\mathbf{B})} = \\log{(k{(\\varepsilon_0,\\mathbf{B})})}^{\\mathbf{B}} and \\operatorname{A_{2}}^{\\mathbf{B}}{(\\varepsilon_0,\\mathbf{B})} = \\log{(\\mathbf{B} + \\varepsilon_0)}^{\\mathbf{B}} and (\\operatorname{A_{2}}^{\\mathbf{B}}{(\\varepsilon_0,\\mathbf{B})})^{\\mathbf{B}} = (\\log{(\\mathbf{B} + \\varepsilon_0)}^{\\mathbf{B}})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), log(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), log(Function('k')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(log(Function('k')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(log(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 5, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Pow(log(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(n_{1})} = e^{n_{1}}, then derive \\int \\mathbf{s}{(n_{1})} dn_{1} = i + e^{n_{1}}, then obtain - g^{\\prime}_{\\varepsilon} + i + \\mathbf{s}{(n_{1})} = - g^{\\prime}_{\\varepsilon} + i + e^{n_{1}}", "derivation": "\\mathbf{s}{(n_{1})} = e^{n_{1}} and \\int \\mathbf{s}{(n_{1})} dn_{1} = \\int e^{n_{1}} dn_{1} and \\int \\mathbf{s}{(n_{1})} dn_{1} = i + e^{n_{1}} and i + e^{n_{1}} = \\int e^{n_{1}} dn_{1} and i + \\mathbf{s}{(n_{1})} = \\int e^{n_{1}} dn_{1} and - g^{\\prime}_{\\varepsilon} + i + \\mathbf{s}{(n_{1})} = - g^{\\prime}_{\\varepsilon} + \\int e^{n_{1}} dn_{1} and - g^{\\prime}_{\\varepsilon} + i + \\mathbf{s}{(n_{1})} = - g^{\\prime}_{\\varepsilon} + i + e^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Add(Symbol('i', commutative=True), exp(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('i', commutative=True), exp(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('i', commutative=True), Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True))), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["minus", 5, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('i', commutative=True), Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('i', commutative=True), Function('\\\\mathbf{s}')(Symbol('n_1', commutative=True))), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('i', commutative=True), exp(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\phi_1,\\hat{H}_l)} = \\hat{H}_l + \\phi_1 and V{(\\phi_1,\\hat{H}_l)} = (\\hat{H}_l + \\phi_1) \\operatorname{A_{2}}{(\\phi_1,\\hat{H}_l)}, then obtain V^{\\hat{H}_l}{(\\phi_1,\\hat{H}_l)} = ((\\hat{H}_l + \\phi_1)^{2})^{\\hat{H}_l}", "derivation": "\\operatorname{A_{2}}{(\\phi_1,\\hat{H}_l)} = \\hat{H}_l + \\phi_1 and V{(\\phi_1,\\hat{H}_l)} = (\\hat{H}_l + \\phi_1) \\operatorname{A_{2}}{(\\phi_1,\\hat{H}_l)} and V{(\\phi_1,\\hat{H}_l)} = (\\hat{H}_l + \\phi_1)^{2} and V^{\\hat{H}_l}{(\\phi_1,\\hat{H}_l)} = ((\\hat{H}_l + \\phi_1)^{2})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('A_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('V')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(2)))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\hat{p}{(\\mathbf{D})} = \\operatorname{C_{1}}{(\\mathbf{D})} \\sin{(\\mathbf{D})}, then obtain \\frac{d^{2}}{d \\mathbf{D}^{2}} \\sin{(\\mathbf{D})} = \\frac{d^{2}}{d \\mathbf{D}^{2}} \\operatorname{C_{1}}{(\\mathbf{D})}", "derivation": "\\operatorname{C_{1}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\hat{p}{(\\mathbf{D})} = \\operatorname{C_{1}}{(\\mathbf{D})} \\sin{(\\mathbf{D})} and \\frac{\\hat{p}{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}} = \\operatorname{C_{1}}{(\\mathbf{D})} and \\frac{\\hat{p}{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}} = \\sin{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} \\frac{\\hat{p}{(\\mathbf{D})}}{\\sin{(\\mathbf{D})}} = \\frac{d}{d \\mathbf{D}} \\operatorname{C_{1}}{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} \\sin{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\operatorname{C_{1}}{(\\mathbf{D})} and \\frac{d^{2}}{d \\mathbf{D}^{2}} \\sin{(\\mathbf{D})} = \\frac{d^{2}}{d \\mathbf{D}^{2}} \\operatorname{C_{1}}{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{p}')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))), Derivative(Function('C_1')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} = V_{\\mathbf{E}} (B + \\mathbf{r}) and \\operatorname{A_{z}}{(\\mathbf{r},V_{\\mathbf{E}},B)} = \\int (B V_{\\mathbf{E}} + V_{\\mathbf{E}} \\mathbf{r}) d\\mathbf{r}, then obtain \\int \\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} d\\mathbf{r} = \\operatorname{A_{z}}{(\\mathbf{r},V_{\\mathbf{E}},B)}", "derivation": "\\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} = V_{\\mathbf{E}} (B + \\mathbf{r}) and \\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} = B V_{\\mathbf{E}} + V_{\\mathbf{E}} \\mathbf{r} and \\int \\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} d\\mathbf{r} = \\int (B V_{\\mathbf{E}} + V_{\\mathbf{E}} \\mathbf{r}) d\\mathbf{r} and \\operatorname{A_{z}}{(\\mathbf{r},V_{\\mathbf{E}},B)} = \\int (B V_{\\mathbf{E}} + V_{\\mathbf{E}} \\mathbf{r}) d\\mathbf{r} and \\int \\phi{(\\mathbf{r},V_{\\mathbf{E}},B)} d\\mathbf{r} = \\operatorname{A_{z}}{(\\mathbf{r},V_{\\mathbf{E}},B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["expand", 1], "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), Add(Mul(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Add(Mul(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), Integral(Add(Mul(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Function('A_z')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} = \\sin{(\\hat{H}_l - \\pi)} and y{(\\pi,\\hat{H}_l)} = \\hat{H}_l - \\pi, then obtain e^{\\pi \\sin{(\\hat{H}_l - \\pi)} + \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}} = e^{2 \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}}", "derivation": "\\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} = \\sin{(\\hat{H}_l - \\pi)} and \\pi \\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} = \\pi \\sin{(\\hat{H}_l - \\pi)} and \\pi \\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} + \\pi \\sin{(\\hat{H}_l - \\pi)} = 2 \\pi \\sin{(\\hat{H}_l - \\pi)} and e^{\\pi \\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} + \\pi \\sin{(\\hat{H}_l - \\pi)}} = e^{2 \\pi \\sin{(\\hat{H}_l - \\pi)}} and y{(\\pi,\\hat{H}_l)} = \\hat{H}_l - \\pi and e^{\\pi \\dot{\\mathbf{r}}{(\\pi,\\hat{H}_l)} + \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}} = e^{2 \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}} and e^{\\pi \\sin{(\\hat{H}_l - \\pi)} + \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}} = e^{2 \\pi \\sin{(y{(\\pi,\\hat{H}_l)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))))"], [["add", 2, "Mul(Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))), Mul(Integer(2), Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))))"], [["exp", 3], "Equality(exp(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))))), exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(exp(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), sin(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))), exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True), sin(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(exp(Add(Mul(Symbol('\\\\pi', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))), Mul(Symbol('\\\\pi', commutative=True), sin(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))), exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True), sin(Function('y')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(\\hat{x},z^{*})} = \\cos^{\\hat{x}}{(z^{*})}, then obtain - \\cos{(z^{*})} + (\\frac{\\partial}{\\partial \\hat{x}} \\hat{p}{(\\hat{x},z^{*})})^{\\hat{x}} = - \\cos{(z^{*})} + (\\frac{\\partial}{\\partial \\hat{x}} \\cos^{\\hat{x}}{(z^{*})})^{\\hat{x}}", "derivation": "\\hat{p}{(\\hat{x},z^{*})} = \\cos^{\\hat{x}}{(z^{*})} and \\frac{\\partial}{\\partial \\hat{x}} \\hat{p}{(\\hat{x},z^{*})} = \\frac{\\partial}{\\partial \\hat{x}} \\cos^{\\hat{x}}{(z^{*})} and (\\frac{\\partial}{\\partial \\hat{x}} \\hat{p}{(\\hat{x},z^{*})})^{\\hat{x}} = (\\frac{\\partial}{\\partial \\hat{x}} \\cos^{\\hat{x}}{(z^{*})})^{\\hat{x}} and - \\cos{(z^{*})} + (\\frac{\\partial}{\\partial \\hat{x}} \\hat{p}{(\\hat{x},z^{*})})^{\\hat{x}} = - \\cos{(z^{*})} + (\\frac{\\partial}{\\partial \\hat{x}} \\cos^{\\hat{x}}{(z^{*})})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('z^*', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True)), Pow(Derivative(Pow(cos(Symbol('z^*', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 3, "cos(Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('z^*', commutative=True))), Pow(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), cos(Symbol('z^*', commutative=True))), Pow(Derivative(Pow(cos(Symbol('z^*', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given T{(G)} = \\cos{(G)} and \\operatorname{v_{z}}{(G)} = 0^{G}, then obtain - \\frac{- T{(G)} + \\operatorname{v_{z}}{(G)} + \\cos{(G)}}{G} = - \\frac{- T{(G)} + \\cos{(G)} + 1}{G}", "derivation": "T{(G)} = \\cos{(G)} and 0 = - T{(G)} + \\cos{(G)} and 0^{G} = (- T{(G)} + \\cos{(G)})^{G} and \\operatorname{v_{z}}{(G)} = 0^{G} and \\operatorname{v_{z}}{(G)} = 1 and - T{(G)} + \\operatorname{v_{z}}{(G)} + \\cos{(G)} = - T{(G)} + \\cos{(G)} + 1 and - \\frac{- T{(G)} + \\operatorname{v_{z}}{(G)} + \\cos{(G)}}{G} = - \\frac{- T{(G)} + \\cos{(G)} + 1}{G}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["minus", 1, "Function('T')(Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Integer(0), Symbol('G', commutative=True)), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('G', commutative=True)), Pow(Integer(0), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('v_z')(Symbol('G', commutative=True)), Integer(1))"], [["add", 5, "Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), Function('v_z')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True)), Integer(1)))"], [["divide", 6, "Mul(Integer(-1), Symbol('G', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), Function('v_z')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('T')(Symbol('G', commutative=True))), cos(Symbol('G', commutative=True)), Integer(1))))"]]}, {"prompt": "Given E{(\\mathbf{f})} = \\log{(\\sin{(\\mathbf{f})})}, then obtain \\int (E^{\\mathbf{f}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} d\\mathbf{f} = \\int (2 \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} d\\mathbf{f}", "derivation": "E{(\\mathbf{f})} = \\log{(\\sin{(\\mathbf{f})})} and E^{\\mathbf{f}}{(\\mathbf{f})} = \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}} and E^{\\mathbf{f}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}} = 2 \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}} and (E^{\\mathbf{f}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} = (2 \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} and \\int (E^{\\mathbf{f}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} d\\mathbf{f} = \\int (2 \\log{(\\sin{(\\mathbf{f})})}^{\\mathbf{f}})^{\\mathbf{f}} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{f}', commutative=True)), log(sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 2, "Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Add(Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Pow(Add(Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Pow(Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\mathbf{F})} = e^{\\mathbf{F}} and H{(n)} = \\cos{(n)}, then obtain -1 + \\frac{H{(n)} e^{\\mathbf{F}}}{\\mathbf{F}} = -1 + \\frac{e^{\\mathbf{F}} \\cos{(n)}}{\\mathbf{F}}", "derivation": "\\mu{(\\mathbf{F})} = e^{\\mathbf{F}} and H{(n)} = \\cos{(n)} and \\frac{H{(n)} \\mu{(\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\mu{(\\mathbf{F})} \\cos{(n)}}{\\mathbf{F}} and \\frac{H{(n)} e^{\\mathbf{F}}}{\\mathbf{F}} = \\frac{e^{\\mathbf{F}} \\cos{(n)}}{\\mathbf{F}} and -1 + \\frac{H{(n)} e^{\\mathbf{F}}}{\\mathbf{F}} = -1 + \\frac{e^{\\mathbf{F}} \\cos{(n)}}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], ["get_premise", "Equality(Function('H')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('H')(Symbol('n', commutative=True)), Function('\\\\mu')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('H')(Symbol('n', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('n', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('H')(Symbol('n', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given m{(\\phi_1)} = \\phi_1, then derive \\frac{d}{d \\phi_1} m{(\\phi_1)} = 1, then obtain \\frac{d}{d \\phi_1} \\phi_1 = 1", "derivation": "m{(\\phi_1)} = \\phi_1 and \\frac{m{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\phi_1} = \\frac{\\phi_1}{\\frac{d}{d \\phi_1} \\phi_1} and \\frac{m{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\phi_1} - \\frac{1}{\\frac{d}{d \\phi_1} \\phi_1} = \\frac{\\phi_1}{\\frac{d}{d \\phi_1} \\phi_1} - \\frac{1}{\\frac{d}{d \\phi_1} \\phi_1} and \\frac{d}{d \\phi_1} (\\frac{m{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\phi_1} - \\frac{1}{\\frac{d}{d \\phi_1} \\phi_1}) = \\frac{d}{d \\phi_1} (\\frac{\\phi_1}{\\frac{d}{d \\phi_1} \\phi_1} - \\frac{1}{\\frac{d}{d \\phi_1} \\phi_1}) and \\frac{d}{d \\phi_1} m{(\\phi_1)} = 1 and \\frac{d}{d \\phi_1} \\phi_1 = 1", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], [["divide", 1, "Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))"], "Equality(Mul(Function('m')(Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 2, "Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Function('m')(Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))))"], [["differentiate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Add(Mul(Function('m')(Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('m')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given T{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})}, then derive 0 = - \\sin{(T{(\\mathbf{g})} + \\sin{(\\mathbf{g})})}, then obtain 0 = - \\frac{\\sin{(T{(\\mathbf{g})} + \\sin{(\\mathbf{g})})}}{(\\mathbf{g} - \\sin{(\\mathbf{g})})^{2}}", "derivation": "T{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and \\mathbf{g} + T{(\\mathbf{g})} = \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and 0 = - T{(\\mathbf{g})} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and 0 = - \\sin{(T{(\\mathbf{g})} - \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})})} and 0 = - \\sin{(T{(\\mathbf{g})} + \\sin{(\\mathbf{g})})} and 0 = - \\frac{\\sin{(T{(\\mathbf{g})} + \\sin{(\\mathbf{g})})}}{(\\mathbf{g} - 2 \\sin{(\\mathbf{g})} - \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})})^{2}} and 0 = - \\frac{\\sin{(T{(\\mathbf{g})} + \\sin{(\\mathbf{g})})}}{(\\mathbf{g} - \\sin{(\\mathbf{g})})^{2}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{g}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Function('T')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["minus", 2, "Add(Symbol('\\\\mathbf{g}', commutative=True), Function('T')(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\mathbf{g}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('T')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('T')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True))))))"], [["times", 5, "Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))), Integer(-2))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))), Integer(-2)), sin(Add(Function('T')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True))))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))), Integer(-2)), sin(Add(Function('T')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(S)} = \\frac{d}{d S} \\cos{(S)}, then derive S \\operatorname{E_{n}}{(S)} = - S \\sin{(S)}, then obtain (- S \\sin{(S)} + \\operatorname{E_{n}}{(S)})^{S} = (- S \\sin{(S)} + \\frac{d}{d S} \\cos{(S)})^{S}", "derivation": "\\operatorname{E_{n}}{(S)} = \\frac{d}{d S} \\cos{(S)} and S \\operatorname{E_{n}}{(S)} = S \\frac{d}{d S} \\cos{(S)} and S \\operatorname{E_{n}}{(S)} = - S \\sin{(S)} and - S \\sin{(S)} = S \\frac{d}{d S} \\cos{(S)} and S \\operatorname{E_{n}}{(S)} + \\operatorname{E_{n}}{(S)} = S \\operatorname{E_{n}}{(S)} + \\frac{d}{d S} \\cos{(S)} and S \\frac{d}{d S} \\cos{(S)} + \\operatorname{E_{n}}{(S)} = S \\frac{d}{d S} \\cos{(S)} + \\frac{d}{d S} \\cos{(S)} and (S \\frac{d}{d S} \\cos{(S)} + \\operatorname{E_{n}}{(S)})^{S} = (S \\frac{d}{d S} \\cos{(S)} + \\frac{d}{d S} \\cos{(S)})^{S} and (- S \\sin{(S)} + \\operatorname{E_{n}}{(S)})^{S} = (- S \\sin{(S)} + \\frac{d}{d S} \\cos{(S)})^{S}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('S', commutative=True)), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('E_n')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('S', commutative=True), Function('E_n')(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["add", 1, "Mul(Symbol('S', commutative=True), Function('E_n')(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Symbol('S', commutative=True), Function('E_n')(Symbol('S', commutative=True))), Function('E_n')(Symbol('S', commutative=True))), Add(Mul(Symbol('S', commutative=True), Function('E_n')(Symbol('S', commutative=True))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Function('E_n')(Symbol('S', commutative=True))), Add(Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Function('E_n')(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Add(Mul(Symbol('S', commutative=True), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Function('E_n')(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True), sin(Symbol('S', commutative=True))), Derivative(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\rho_f,\\mathbf{S})} = \\mathbf{S} \\rho_f and \\hat{X}{(\\rho_f,\\mathbf{S})} = \\mathbf{S} \\rho_f - \\rho_f, then obtain (\\mathbf{S} \\rho_f - \\rho_f)^{\\mathbf{S}} = \\hat{X}^{\\mathbf{S}}{(\\rho_f,\\mathbf{S})}", "derivation": "\\operatorname{F_{g}}{(\\rho_f,\\mathbf{S})} = \\mathbf{S} \\rho_f and - \\rho_f + \\operatorname{F_{g}}{(\\rho_f,\\mathbf{S})} = \\mathbf{S} \\rho_f - \\rho_f and (- \\rho_f + \\operatorname{F_{g}}{(\\rho_f,\\mathbf{S})})^{\\mathbf{S}} = (\\mathbf{S} \\rho_f - \\rho_f)^{\\mathbf{S}} and \\hat{X}{(\\rho_f,\\mathbf{S})} = \\mathbf{S} \\rho_f - \\rho_f and (- \\rho_f + \\operatorname{F_{g}}{(\\rho_f,\\mathbf{S})})^{\\mathbf{S}} = \\hat{X}^{\\mathbf{S}}{(\\rho_f,\\mathbf{S})} and (\\mathbf{S} \\rho_f - \\rho_f)^{\\mathbf{S}} = \\hat{X}^{\\mathbf{S}}{(\\rho_f,\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('F_g')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('F_g')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('F_g')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(y)} = \\log{(y)}, then derive \\varepsilon - y \\log{(y)} + y = \\int - \\hat{H}_l{(y)} dy, then obtain \\varepsilon - y \\log{(y)} + y = v_{2} - y \\log{(y)} + y", "derivation": "\\hat{H}_l{(y)} = \\log{(y)} and - \\log{(y)} = - \\hat{H}_l{(y)} and \\int - \\log{(y)} dy = \\int - \\hat{H}_l{(y)} dy and \\varepsilon - y \\log{(y)} + y = \\int - \\hat{H}_l{(y)} dy and \\varepsilon - y \\log{(y)} + y = \\int - \\log{(y)} dy and \\varepsilon - y \\log{(y)} + y = v_{2} - y \\log{(y)} + y", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["minus", 1, "Add(Function('\\\\hat{H}_l')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), log(Symbol('y', commutative=True))), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Integer(-1), log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integral(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integral(Mul(Integer(-1), log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{H})} = \\mathbf{H}, then obtain (\\frac{d}{d \\mathbf{H}} (\\int \\mathbf{H} d\\mathbf{H} + \\int \\operatorname{t_{1}}{(\\mathbf{H})} d\\mathbf{H}))^{\\mathbf{H}} = (\\frac{d}{d \\mathbf{H}} 2 \\int \\mathbf{H} d\\mathbf{H})^{\\mathbf{H}}", "derivation": "\\operatorname{t_{1}}{(\\mathbf{H})} = \\mathbf{H} and \\int \\operatorname{t_{1}}{(\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H} d\\mathbf{H} and \\int \\mathbf{H} d\\mathbf{H} + \\int \\operatorname{t_{1}}{(\\mathbf{H})} d\\mathbf{H} = 2 \\int \\mathbf{H} d\\mathbf{H} and \\frac{d}{d \\mathbf{H}} (\\int \\mathbf{H} d\\mathbf{H} + \\int \\operatorname{t_{1}}{(\\mathbf{H})} d\\mathbf{H}) = \\frac{d}{d \\mathbf{H}} 2 \\int \\mathbf{H} d\\mathbf{H} and (\\frac{d}{d \\mathbf{H}} (\\int \\mathbf{H} d\\mathbf{H} + \\int \\operatorname{t_{1}}{(\\mathbf{H})} d\\mathbf{H}))^{\\mathbf{H}} = (\\frac{d}{d \\mathbf{H}} 2 \\int \\mathbf{H} d\\mathbf{H})^{\\mathbf{H}}", "srepr_derivation": [["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 2, "Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Function('t_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Integer(2), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Function('t_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Derivative(Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Function('t_1')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(Mul(Integer(2), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\pi)} = \\cos{(\\pi)} and \\mu{(\\pi)} = \\frac{\\phi{(\\pi)}}{\\pi}, then derive \\int \\mu{(\\pi)} d\\pi = z^{*} - \\log{(\\pi)} + \\frac{\\log{(\\pi^{2})}}{2} + \\operatorname{Ci}{(\\pi)}, then obtain \\frac{\\pi \\int \\mu{(\\pi)} d\\pi}{\\phi{(\\pi)}} = \\frac{\\pi (z^{*} - \\log{(\\pi)} + \\frac{\\log{(\\pi^{2})}}{2} + \\operatorname{Ci}{(\\pi)})}{\\phi{(\\pi)}}", "derivation": "\\phi{(\\pi)} = \\cos{(\\pi)} and \\frac{\\phi{(\\pi)}}{\\pi} = \\frac{\\cos{(\\pi)}}{\\pi} and \\mu{(\\pi)} = \\frac{\\phi{(\\pi)}}{\\pi} and \\mu{(\\pi)} = \\frac{\\cos{(\\pi)}}{\\pi} and \\int \\mu{(\\pi)} d\\pi = \\int \\frac{\\cos{(\\pi)}}{\\pi} d\\pi and \\int \\mu{(\\pi)} d\\pi = z^{*} - \\log{(\\pi)} + \\frac{\\log{(\\pi^{2})}}{2} + \\operatorname{Ci}{(\\pi)} and \\frac{\\pi \\int \\mu{(\\pi)} d\\pi}{\\phi{(\\pi)}} = \\frac{\\pi (z^{*} - \\log{(\\pi)} + \\frac{\\log{(\\pi^{2})}}{2} + \\operatorname{Ci}{(\\pi)})}{\\phi{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mu')(Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), log(Symbol('\\\\pi', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('\\\\pi', commutative=True), Integer(2)))), Ci(Symbol('\\\\pi', commutative=True))))"], [["divide", 6, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\pi', commutative=True)), Integer(-1)), Integral(Function('\\\\mu')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), log(Symbol('\\\\pi', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('\\\\pi', commutative=True), Integer(2)))), Ci(Symbol('\\\\pi', commutative=True))), Pow(Function('\\\\phi')(Symbol('\\\\pi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given s{(Z)} = \\sin{(Z)} and V{(Z)} = \\sin{(Z)}, then obtain \\int \\frac{d}{d Z} \\sin{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} dZ = \\int \\frac{d}{d Z} V{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} dZ", "derivation": "s{(Z)} = \\sin{(Z)} and V{(Z)} = \\sin{(Z)} and s{(Z)} = V{(Z)} and \\frac{d}{d Z} s{(Z)} = \\frac{d}{d Z} V{(Z)} and \\frac{d}{d Z} \\sin{(Z)} = \\frac{d}{d Z} V{(Z)} and \\frac{d}{d Z} \\sin{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} = \\frac{d}{d Z} V{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} and \\int \\frac{d}{d Z} \\sin{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} dZ = \\int \\frac{d}{d Z} V{(Z)} \\frac{d}{d Z} \\sin^{Z}{(Z)} dZ", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('s')(Symbol('Z', commutative=True)), Function('V')(Symbol('Z', commutative=True)))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["times", 5, "Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Derivative(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["integrate", 6, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Derivative(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Derivative(Function('V')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(u)} = \\cos{(u)} and \\operatorname{v_{t}}{(u)} = \\frac{1}{\\cos{(u)}}, then obtain \\frac{u}{\\mathbf{f}{(u)}} - \\frac{\\mathbf{f}{(u)}}{\\cos{(u)}} = \\frac{u}{\\cos{(u)}} - \\frac{\\mathbf{f}{(u)}}{\\cos{(u)}}", "derivation": "\\mathbf{f}{(u)} = \\cos{(u)} and \\operatorname{v_{t}}{(u)} = \\frac{1}{\\cos{(u)}} and u \\operatorname{v_{t}}{(u)} = \\frac{u}{\\cos{(u)}} and \\operatorname{v_{t}}{(u)} = \\frac{1}{\\mathbf{f}{(u)}} and \\frac{u}{\\mathbf{f}{(u)}} = \\frac{u}{\\cos{(u)}} and \\frac{u}{\\mathbf{f}{(u)}} - \\frac{\\mathbf{f}{(u)}}{\\cos{(u)}} = \\frac{u}{\\cos{(u)}} - \\frac{\\mathbf{f}{(u)}}{\\cos{(u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))"], [["times", 2, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('v_t')(Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Pow(cos(Symbol('u', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_t')(Symbol('u', commutative=True)), Pow(Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Integer(-1))), Mul(Symbol('u', commutative=True), Pow(cos(Symbol('u', commutative=True)), Integer(-1))))"], [["minus", 5, "Mul(Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Symbol('u', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))), Add(Mul(Symbol('u', commutative=True), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta}, then derive \\operatorname{F_{N}}{(\\delta)} = e^{\\delta}, then derive \\operatorname{F_{N}}^{\\delta}{(\\delta)} = (e^{\\delta})^{\\delta}, then obtain (\\frac{d}{d \\delta} \\operatorname{F_{N}}{(\\delta)})^{\\delta} = (e^{\\delta})^{\\delta}", "derivation": "\\operatorname{F_{N}}{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta} and \\operatorname{F_{N}}{(\\delta)} = e^{\\delta} and \\operatorname{F_{N}}{(\\delta)} = \\frac{d}{d \\delta} \\operatorname{F_{N}}{(\\delta)} and \\operatorname{F_{N}}^{\\delta}{(\\delta)} = (\\frac{d}{d \\delta} e^{\\delta})^{\\delta} and \\frac{d}{d \\delta} e^{\\delta} = \\frac{d^{2}}{d \\delta^{2}} e^{\\delta} and \\operatorname{F_{N}}^{\\delta}{(\\delta)} = (\\frac{d^{2}}{d \\delta^{2}} e^{\\delta})^{\\delta} and \\operatorname{F_{N}}^{\\delta}{(\\delta)} = (e^{\\delta})^{\\delta} and (\\frac{d}{d \\delta} \\operatorname{F_{N}}{(\\delta)})^{\\delta} = (e^{\\delta})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_N')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_N')(Symbol('\\\\delta', commutative=True)), Derivative(Function('F_N')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('F_N')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))), Symbol('\\\\delta', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Function('F_N')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(exp(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Pow(Derivative(Function('F_N')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(exp(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} = \\frac{e^{\\tilde{g}}}{\\phi_2}, then obtain \\iint \\phi_2 \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} d\\phi_2 d\\phi_2 = \\frac{\\phi_2^{2} e^{\\tilde{g}}}{2}", "derivation": "\\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} = \\frac{e^{\\tilde{g}}}{\\phi_2} and \\phi_2 \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} = e^{\\tilde{g}} and \\int \\phi_2 \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} d\\phi_2 = \\int e^{\\tilde{g}} d\\phi_2 and \\iint \\phi_2 \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} d\\phi_2 d\\phi_2 = \\iint e^{\\tilde{g}} d\\phi_2 d\\phi_2 and \\frac{\\partial}{\\partial \\tilde{g}} \\iint \\phi_2 \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} d\\phi_2 d\\phi_2 = \\frac{\\partial}{\\partial \\tilde{g}} \\iint e^{\\tilde{g}} d\\phi_2 d\\phi_2 and \\iint \\phi_2 \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{m_{s}}{(\\phi_2,\\tilde{g})} d\\phi_2 d\\phi_2 = \\frac{\\phi_2^{2} e^{\\tilde{g}}}{2}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Derivative(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2)), exp(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given k{(\\mathbf{D},\\dot{x})} = \\dot{x} - \\mathbf{D} and J{(x^\\prime,v_{y})} = \\frac{x^\\prime}{v_{y}}, then obtain \\frac{v_{y} k^{\\mathbf{D}}{(\\mathbf{D},\\dot{x})}}{x^\\prime} = \\frac{v_{y} (\\dot{x} - \\mathbf{D})^{\\mathbf{D}}}{x^\\prime}", "derivation": "k{(\\mathbf{D},\\dot{x})} = \\dot{x} - \\mathbf{D} and k^{\\mathbf{D}}{(\\mathbf{D},\\dot{x})} = (\\dot{x} - \\mathbf{D})^{\\mathbf{D}} and J{(x^\\prime,v_{y})} = \\frac{x^\\prime}{v_{y}} and \\frac{k^{\\mathbf{D}}{(\\mathbf{D},\\dot{x})}}{J{(x^\\prime,v_{y})}} = \\frac{(\\dot{x} - \\mathbf{D})^{\\mathbf{D}}}{J{(x^\\prime,v_{y})}} and \\frac{v_{y} k^{\\mathbf{D}}{(\\mathbf{D},\\dot{x})}}{x^\\prime} = \\frac{v_{y} (\\dot{x} - \\mathbf{D})^{\\mathbf{D}}}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], ["get_premise", "Equality(Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 2, "Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Pow(Function('k')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('J')(Symbol('x^\\\\prime', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('v_y', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('k')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('v_y', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\nabla)} = e^{\\nabla}, then derive e^{\\nabla} + \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} = 2 e^{\\nabla}, then obtain \\hat{H}_l{(\\nabla)} + \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} = 2 \\hat{H}_l{(\\nabla)}", "derivation": "\\hat{H}_l{(\\nabla)} = e^{\\nabla} and \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} = \\frac{d}{d \\nabla} e^{\\nabla} and \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} + \\frac{d}{d \\nabla} e^{\\nabla} = 2 \\frac{d}{d \\nabla} e^{\\nabla} and e^{\\nabla} + \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} = 2 e^{\\nabla} and \\hat{H}_l{(\\nabla)} + \\frac{d}{d \\nabla} \\hat{H}_l{(\\nabla)} = 2 \\hat{H}_l{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["add", 2, "Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('\\\\nabla', commutative=True)), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\eta{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\phi_{1}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain e^{- \\mathbf{J}_P} e^{e^{\\mathbf{J}_P}} = e^{- \\mathbf{J}_P} e^{\\phi_{1}{(\\mathbf{J}_P)}}", "derivation": "\\eta{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\phi_{1}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\eta{(\\mathbf{J}_P)} = \\phi_{1}{(\\mathbf{J}_P)} and e^{\\eta{(\\mathbf{J}_P)}} = e^{\\phi_{1}{(\\mathbf{J}_P)}} and e^{- \\mathbf{J}_P} e^{\\eta{(\\mathbf{J}_P)}} = e^{- \\mathbf{J}_P} e^{\\phi_{1}{(\\mathbf{J}_P)}} and e^{- \\mathbf{J}_P} e^{e^{\\mathbf{J}_P}} = e^{- \\mathbf{J}_P} e^{\\phi_{1}{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["exp", 3], "Equality(exp(Function('\\\\eta')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 4, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Function('\\\\eta')(Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), exp(exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Function('\\\\phi_1')(Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(u)} = \\sin{(\\cos{(u)})}, then derive \\sigma_p + \\mathbf{A}{(u)} = f^{\\prime} + \\sin{(\\cos{(u)})}, then obtain \\sigma_p + \\sin{(\\cos{(u)})} = f^{\\prime} + \\sin{(\\cos{(u)})}", "derivation": "\\mathbf{A}{(u)} = \\sin{(\\cos{(u)})} and \\frac{d}{d u} \\mathbf{A}{(u)} = \\frac{d}{d u} \\sin{(\\cos{(u)})} and \\int \\frac{d}{d u} \\mathbf{A}{(u)} du = \\int \\frac{d}{d u} \\sin{(\\cos{(u)})} du and \\sigma_p + \\mathbf{A}{(u)} = f^{\\prime} + \\sin{(\\cos{(u)})} and \\sigma_p + \\sin{(\\cos{(u)})} = f^{\\prime} + \\sin{(\\cos{(u)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), sin(cos(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(sin(cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('u', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), sin(cos(Symbol('u', commutative=True)))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\psi{(E_{n})} = e^{E_{n}} and \\operatorname{A_{z}}{(f_{\\mathbf{p}},z^{*})} = z^{*} + \\log{(f_{\\mathbf{p}})}, then obtain z^{*} + \\psi{(E_{n})} + \\log{(f_{\\mathbf{p}})} = z^{*} + e^{E_{n}} + \\log{(f_{\\mathbf{p}})}", "derivation": "\\psi{(E_{n})} = e^{E_{n}} and \\operatorname{A_{z}}{(f_{\\mathbf{p}},z^{*})} = z^{*} + \\log{(f_{\\mathbf{p}})} and \\operatorname{A_{z}}{(f_{\\mathbf{p}},z^{*})} + \\psi{(E_{n})} = \\operatorname{A_{z}}{(f_{\\mathbf{p}},z^{*})} + e^{E_{n}} and z^{*} + \\psi{(E_{n})} + \\log{(f_{\\mathbf{p}})} = z^{*} + e^{E_{n}} + \\log{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], ["get_premise", "Equality(Function('A_z')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('z^*', commutative=True), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Function('A_z')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Add(Function('A_z')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\psi')(Symbol('E_n', commutative=True))), Add(Function('A_z')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('z^*', commutative=True)), exp(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('z^*', commutative=True), Function('\\\\psi')(Symbol('E_n', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('z^*', commutative=True), exp(Symbol('E_n', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(k,\\eta^{\\prime})} = \\frac{k}{\\eta^{\\prime}}, then obtain \\int \\frac{\\partial}{\\partial k} \\operatorname{v_{t}}{(k,\\eta^{\\prime})} d\\eta^{\\prime} = f^{\\prime} + \\log{(\\eta^{\\prime})}", "derivation": "\\operatorname{v_{t}}{(k,\\eta^{\\prime})} = \\frac{k}{\\eta^{\\prime}} and \\frac{\\partial}{\\partial k} \\operatorname{v_{t}}{(k,\\eta^{\\prime})} = \\frac{\\partial}{\\partial k} \\frac{k}{\\eta^{\\prime}} and \\int \\frac{\\partial}{\\partial k} \\operatorname{v_{t}}{(k,\\eta^{\\prime})} d\\eta^{\\prime} = \\int \\frac{\\partial}{\\partial k} \\frac{k}{\\eta^{\\prime}} d\\eta^{\\prime} and \\int \\frac{\\partial}{\\partial k} \\operatorname{v_{t}}{(k,\\eta^{\\prime})} d\\eta^{\\prime} = f^{\\prime} + \\log{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('k', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('k', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(Function('v_t')(Symbol('k', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('v_t')(Symbol('k', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given U{(\\mathbf{P},E_{\\lambda})} = \\mathbf{P} + \\sin{(E_{\\lambda})}, then derive \\int U{(\\mathbf{P},E_{\\lambda})} d\\mathbf{P} = \\hbar + \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})}, then obtain \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})} + z = \\hbar + \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})}", "derivation": "U{(\\mathbf{P},E_{\\lambda})} = \\mathbf{P} + \\sin{(E_{\\lambda})} and \\int U{(\\mathbf{P},E_{\\lambda})} d\\mathbf{P} = \\int (\\mathbf{P} + \\sin{(E_{\\lambda})}) d\\mathbf{P} and \\int U{(\\mathbf{P},E_{\\lambda})} d\\mathbf{P} = \\hbar + \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})} and \\int (\\mathbf{P} + \\sin{(E_{\\lambda})}) d\\mathbf{P} = \\hbar + \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})} and \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})} + z = \\hbar + \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} \\sin{(E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('z', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(E,v_{t})} = v_{t}^{E}, then obtain (E + \\Omega{(E,v_{t})})^{E} + \\int (E + \\Omega{(E,v_{t})}) dE = (E + \\Omega{(E,v_{t})})^{E} + \\int (E + v_{t}^{E}) dE", "derivation": "\\Omega{(E,v_{t})} = v_{t}^{E} and E + \\Omega{(E,v_{t})} = E + v_{t}^{E} and (E + \\Omega{(E,v_{t})})^{E} = (E + v_{t}^{E})^{E} and \\int (E + \\Omega{(E,v_{t})}) dE = \\int (E + v_{t}^{E}) dE and (E + v_{t}^{E})^{E} + \\int (E + \\Omega{(E,v_{t})}) dE = (E + v_{t}^{E})^{E} + \\int (E + v_{t}^{E}) dE and (E + \\Omega{(E,v_{t})})^{E} + \\int (E + \\Omega{(E,v_{t})}) dE = (E + \\Omega{(E,v_{t})})^{E} + \\int (E + v_{t}^{E}) dE", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True)))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Symbol('E', commutative=True)), Pow(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["add", 4, "Pow(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))"], "Equality(Add(Pow(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Integral(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Pow(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Integral(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Symbol('E', commutative=True)), Integral(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Pow(Add(Symbol('E', commutative=True), Function('\\\\Omega')(Symbol('E', commutative=True), Symbol('v_t', commutative=True))), Symbol('E', commutative=True)), Integral(Add(Symbol('E', commutative=True), Pow(Symbol('v_t', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\psi^*)} = \\sin{(\\psi^*)}, then derive 1 = \\frac{\\cos{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}}, then derive \\omega + \\psi^* = \\int \\frac{\\cos{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}} d\\psi^*, then obtain \\int 1 d\\psi^* = \\omega + \\psi^*", "derivation": "\\operatorname{F_{x}}{(\\psi^*)} = \\sin{(\\psi^*)} and \\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\sin{(\\psi^*)} and 1 = \\frac{\\frac{d}{d \\psi^*} \\sin{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}} and 1 = \\frac{\\cos{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}} and \\int 1 d\\psi^* = \\int \\frac{\\cos{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}} d\\psi^* and \\omega + \\psi^* = \\int \\frac{\\cos{(\\psi^*)}}{\\frac{d}{d \\psi^*} \\operatorname{F_{x}}{(\\psi^*)}} d\\psi^* and \\int 1 d\\psi^* = \\omega + \\psi^*", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(cos(Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(cos(Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integral(Mul(cos(Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Function('F_x')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given U{(t_{1})} = t_{1}, then obtain \\frac{\\int \\frac{U^{3}{(t_{1})}}{t_{1}} dt_{1}}{t_{1} U^{3}{(t_{1})}} = \\frac{\\int U^{2}{(t_{1})} dt_{1}}{t_{1} U^{3}{(t_{1})}}", "derivation": "U{(t_{1})} = t_{1} and U^{2}{(t_{1})} = t_{1} U{(t_{1})} and U^{4}{(t_{1})} = t_{1} U^{3}{(t_{1})} and \\frac{U^{3}{(t_{1})}}{t_{1}} = U^{2}{(t_{1})} and \\int \\frac{U^{3}{(t_{1})}}{t_{1}} dt_{1} = \\int U^{2}{(t_{1})} dt_{1} and \\frac{\\int \\frac{U^{3}{(t_{1})}}{t_{1}} dt_{1}}{t_{1} U^{3}{(t_{1})}} = \\frac{\\int U^{2}{(t_{1})} dt_{1}}{t_{1} U^{3}{(t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["times", 1, "Function('U')(Symbol('t_1', commutative=True))"], "Equality(Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(2)), Mul(Symbol('t_1', commutative=True), Function('U')(Symbol('t_1', commutative=True))))"], [["times", 2, "Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(2))"], "Equality(Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(4)), Mul(Symbol('t_1', commutative=True), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(3))))"], [["divide", 3, "Mul(Symbol('t_1', commutative=True), Function('U')(Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(3))), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(2)))"], [["integrate", 4, "Symbol('t_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(3))), Tuple(Symbol('t_1', commutative=True))), Integral(Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(2)), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 5, "Mul(Symbol('t_1', commutative=True), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(3)))"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(-3)), Integral(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(3))), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(-3)), Integral(Pow(Function('U')(Symbol('t_1', commutative=True)), Integer(2)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(g)} = \\int \\cos{(g)} dg, then derive \\dot{z}{(g)} = f + \\sin{(g)}, then derive \\dot{z}{(g)} + \\cos{(g)} = \\mu + \\sin{(g)} + \\cos{(g)}, then obtain e^{f + \\sin{(g)} + \\cos{(g)}} = e^{\\mu + \\sin{(g)} + \\cos{(g)}}", "derivation": "\\dot{z}{(g)} = \\int \\cos{(g)} dg and \\dot{z}{(g)} + \\cos{(g)} = \\cos{(g)} + \\int \\cos{(g)} dg and \\dot{z}{(g)} = f + \\sin{(g)} and \\int \\cos{(g)} dg = f + \\sin{(g)} and \\dot{z}{(g)} + \\cos{(g)} = \\mu + \\sin{(g)} + \\cos{(g)} and \\cos{(g)} + \\int \\cos{(g)} dg = \\mu + \\sin{(g)} + \\cos{(g)} and \\dot{z}{(g)} + \\cos{(g)} = f + \\sin{(g)} + \\cos{(g)} and e^{\\cos{(g)} + \\int \\cos{(g)} dg} = e^{\\mu + \\sin{(g)} + \\cos{(g)}} and f + \\sin{(g)} + \\cos{(g)} = \\cos{(g)} + \\int \\cos{(g)} dg and e^{f + \\sin{(g)} + \\cos{(g)}} = e^{\\mu + \\sin{(g)} + \\cos{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["add", 1, "cos(Symbol('g', commutative=True))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(cos(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True)), Add(Symbol('f', commutative=True), sin(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\dot{z}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(Symbol('\\\\mu', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(cos(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('\\\\dot{z}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))))"], [["exp", 6], "Equality(exp(Add(cos(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))), exp(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 7], "Equality(Add(Symbol('f', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(cos(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 9], "Equality(exp(Add(Symbol('f', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))), exp(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} = - F_{g} + f^{\\prime} + t_{1}, then derive \\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1} = f + \\frac{t_{1}^{2}}{2} + t_{1} (- F_{g} + f^{\\prime}), then obtain (\\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1})^{2} = (\\int (- F_{g} + f^{\\prime} + t_{1}) dt_{1})^{2}", "derivation": "\\mathbf{S}{(F_{g},t_{1},f^{\\prime})} = - F_{g} + f^{\\prime} + t_{1} and \\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1} = \\int (- F_{g} + f^{\\prime} + t_{1}) dt_{1} and \\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1} = f + \\frac{t_{1}^{2}}{2} + t_{1} (- F_{g} + f^{\\prime}) and \\int (- F_{g} + f^{\\prime} + t_{1}) dt_{1} = f + \\frac{t_{1}^{2}}{2} + t_{1} (- F_{g} + f^{\\prime}) and (\\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1})^{2} = (f + \\frac{t_{1}^{2}}{2} + t_{1} (- F_{g} + f^{\\prime}))^{2} and (\\int \\mathbf{S}{(F_{g},t_{1},f^{\\prime})} dt_{1})^{2} = (\\int (- F_{g} + f^{\\prime} + t_{1}) dt_{1})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Mul(Symbol('t_1', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Mul(Symbol('t_1', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2)), Pow(Add(Symbol('f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Mul(Symbol('t_1', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(Function('\\\\mathbf{S}')(Symbol('F_g', commutative=True), Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2)), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('f^{\\\\prime}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(B,A_{1})} = A_{1} + B, then obtain B (A_{1} + B) \\operatorname{f^{\\prime}}{(B,A_{1})} = B (A_{1} + B)^{2}", "derivation": "\\operatorname{f^{\\prime}}{(B,A_{1})} = A_{1} + B and B \\operatorname{f^{\\prime}}{(B,A_{1})} = B (A_{1} + B) and B \\operatorname{f^{\\prime}}^{2}{(B,A_{1})} = B (A_{1} + B) \\operatorname{f^{\\prime}}{(B,A_{1})} and B (A_{1} + B) \\operatorname{f^{\\prime}}{(B,A_{1})} = B (A_{1} + B)^{2}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('B', commutative=True)))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('B', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('B', commutative=True))))"], [["times", 2, "Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Symbol('B', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('B', commutative=True)), Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('B', commutative=True), Add(Symbol('A_1', commutative=True), Symbol('B', commutative=True)), Function('f^{\\\\prime}')(Symbol('B', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('B', commutative=True), Pow(Add(Symbol('A_1', commutative=True), Symbol('B', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(T)} = \\cos{(T)}, then derive \\dot{z} + \\operatorname{v_{t}}{(T)} = \\mathbf{F} + \\cos{(T)}, then obtain (\\dot{z} + \\operatorname{v_{t}}{(T)})^{\\mathbf{F}} = (\\mathbf{F} + \\cos{(T)})^{\\mathbf{F}}", "derivation": "\\operatorname{v_{t}}{(T)} = \\cos{(T)} and \\frac{d}{d T} \\operatorname{v_{t}}{(T)} = \\frac{d}{d T} \\cos{(T)} and \\int \\frac{d}{d T} \\operatorname{v_{t}}{(T)} dT = \\int \\frac{d}{d T} \\cos{(T)} dT and \\dot{z} + \\operatorname{v_{t}}{(T)} = \\mathbf{F} + \\cos{(T)} and (\\dot{z} + \\operatorname{v_{t}}{(T)})^{\\mathbf{F}} = (\\mathbf{F} + \\cos{(T)})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Function('v_t')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Function('v_t')(Symbol('T', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), cos(Symbol('T', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Function('v_t')(Symbol('T', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), cos(Symbol('T', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(C_{d})} = \\sin{(\\sin{(C_{d})})}, then obtain \\cos{(\\int - \\sin{(\\sin{(C_{d})})} dC_{d} + \\int \\sin{(\\sin{(C_{d})})} dC_{d})} = 1", "derivation": "\\operatorname{m_{s}}{(C_{d})} = \\sin{(\\sin{(C_{d})})} and \\operatorname{m_{s}}{(C_{d})} - \\sin{(\\sin{(C_{d})})} = 0 and \\int (\\operatorname{m_{s}}{(C_{d})} - \\sin{(\\sin{(C_{d})})}) dC_{d} = \\int 0 dC_{d} and \\cos{(\\int (\\operatorname{m_{s}}{(C_{d})} - \\sin{(\\sin{(C_{d})})}) dC_{d})} = 1 and \\cos{(\\int \\operatorname{m_{s}}{(C_{d})} dC_{d} + \\int - \\sin{(\\sin{(C_{d})})} dC_{d})} = 1 and \\cos{(\\int - \\sin{(\\sin{(C_{d})})} dC_{d} + \\int \\sin{(\\sin{(C_{d})})} dC_{d})} = 1", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('C_d', commutative=True)), sin(sin(Symbol('C_d', commutative=True))))"], [["minus", 1, "sin(sin(Symbol('C_d', commutative=True)))"], "Equality(Add(Function('m_s')(Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('C_d', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Function('m_s')(Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True))), Integral(Integer(0), Tuple(Symbol('C_d', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Add(Function('m_s')(Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Integer(1))"], [["expand", 4], "Equality(cos(Add(Integral(Function('m_s')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Integer(-1), sin(sin(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(cos(Add(Integral(Mul(Integer(-1), sin(sin(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))), Integral(sin(sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(P_{g})} = e^{\\sin{(P_{g})}} and \\varphi{(P_{g})} = \\sin{(P_{g})}, then obtain 0 = - (e^{\\varphi{(P_{g})}} - e^{\\sin{(P_{g})}}) e^{\\sin{(P_{g})}}", "derivation": "\\operatorname{A_{1}}{(P_{g})} = e^{\\sin{(P_{g})}} and \\operatorname{A_{1}}{(P_{g})} - \\sin{(P_{g})} = e^{\\sin{(P_{g})}} - \\sin{(P_{g})} and - \\operatorname{A_{1}}{(P_{g})} = - e^{\\sin{(P_{g})}} and 0 = - \\operatorname{A_{1}}{(P_{g})} + e^{\\sin{(P_{g})}} and 0 = - (- \\operatorname{A_{1}}{(P_{g})} + e^{\\sin{(P_{g})}}) \\operatorname{A_{1}}{(P_{g})} and \\varphi{(P_{g})} = \\sin{(P_{g})} and 0 = - (- \\operatorname{A_{1}}{(P_{g})} + e^{\\varphi{(P_{g})}}) \\operatorname{A_{1}}{(P_{g})} and 0 = - (e^{\\varphi{(P_{g})}} - e^{\\sin{(P_{g})}}) e^{\\sin{(P_{g})}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('P_g', commutative=True)), exp(sin(Symbol('P_g', commutative=True))))"], [["minus", 1, "sin(Symbol('P_g', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Add(exp(sin(Symbol('P_g', commutative=True))), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('A_1')(Symbol('P_g', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('P_g', commutative=True)))))"], [["minus", 2, "Add(Function('A_1')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_1')(Symbol('P_g', commutative=True))), exp(sin(Symbol('P_g', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Function('A_1')(Symbol('P_g', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('A_1')(Symbol('P_g', commutative=True))), exp(sin(Symbol('P_g', commutative=True)))), Function('A_1')(Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('A_1')(Symbol('P_g', commutative=True))), exp(Function('\\\\varphi')(Symbol('P_g', commutative=True)))), Function('A_1')(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Integer(0), Mul(Integer(-1), Add(exp(Function('\\\\varphi')(Symbol('P_g', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('P_g', commutative=True))))), exp(sin(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\phi_1,\\mathbf{B})} = \\mathbf{B} \\phi_1 and \\rho_{f}{(\\phi_1,\\mathbf{B})} = (\\mathbf{B} \\phi_1)^{\\mathbf{B}} \\operatorname{C_{d}}^{\\mathbf{B}}{(\\phi_1,\\mathbf{B})} and \\pi{(\\mathbf{B})} = 2 \\mathbf{B}, then obtain (\\mathbf{B} \\phi_1)^{\\pi{(\\mathbf{B})}} = (\\mathbf{B} \\phi_1)^{\\mathbf{B}} \\operatorname{C_{d}}^{\\mathbf{B}}{(\\phi_1,\\mathbf{B})}", "derivation": "\\operatorname{C_{d}}{(\\phi_1,\\mathbf{B})} = \\mathbf{B} \\phi_1 and \\operatorname{C_{d}}^{\\mathbf{B}}{(\\phi_1,\\mathbf{B})} = (\\mathbf{B} \\phi_1)^{\\mathbf{B}} and \\rho_{f}{(\\phi_1,\\mathbf{B})} = (\\mathbf{B} \\phi_1)^{\\mathbf{B}} \\operatorname{C_{d}}^{\\mathbf{B}}{(\\phi_1,\\mathbf{B})} and \\rho_{f}{(\\phi_1,\\mathbf{B})} = (\\mathbf{B} \\phi_1)^{2 \\mathbf{B}} and \\pi{(\\mathbf{B})} = 2 \\mathbf{B} and \\rho_{f}{(\\phi_1,\\mathbf{B})} = (\\mathbf{B} \\phi_1)^{\\pi{(\\mathbf{B})}} and (\\mathbf{B} \\phi_1)^{\\pi{(\\mathbf{B})}} = (\\mathbf{B} \\phi_1)^{\\mathbf{B}} \\operatorname{C_{d}}^{\\mathbf{B}}{(\\phi_1,\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mu)} = \\sin{(\\mu)}, then obtain \\int (- \\sin{(\\mu)} + 2 \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)}) d\\mu = \\int (- \\sin{(\\mu)} + \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} + \\frac{d}{d \\mu} \\sin{(\\mu)}) d\\mu", "derivation": "\\operatorname{C_{2}}{(\\mu)} = \\sin{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} = \\frac{d}{d \\mu} \\sin{(\\mu)} and 2 \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} = \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} + \\frac{d}{d \\mu} \\sin{(\\mu)} and - \\sin{(\\mu)} + 2 \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} = - \\sin{(\\mu)} + \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} + \\frac{d}{d \\mu} \\sin{(\\mu)} and \\int (- \\sin{(\\mu)} + 2 \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)}) d\\mu = \\int (- \\sin{(\\mu)} + \\frac{d}{d \\mu} \\operatorname{C_{2}}{(\\mu)} + \\frac{d}{d \\mu} \\sin{(\\mu)}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["minus", 3, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))), Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\mu', commutative=True))), Derivative(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given r{(r_{0})} = e^{\\sin{(r_{0})}}, then obtain (e^{\\int r{(r_{0})} e^{- \\sin{(r_{0})}} dr_{0}})^{r_{0}} = (e^{\\int 1 dr_{0}})^{r_{0}}", "derivation": "r{(r_{0})} = e^{\\sin{(r_{0})}} and r{(r_{0})} e^{- \\sin{(r_{0})}} = 1 and \\int r{(r_{0})} e^{- \\sin{(r_{0})}} dr_{0} = \\int 1 dr_{0} and e^{\\int r{(r_{0})} e^{- \\sin{(r_{0})}} dr_{0}} = e^{\\int 1 dr_{0}} and (e^{\\int r{(r_{0})} e^{- \\sin{(r_{0})}} dr_{0}})^{r_{0}} = (e^{\\int 1 dr_{0}})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('r_0', commutative=True)), exp(sin(Symbol('r_0', commutative=True))))"], [["divide", 1, "exp(sin(Symbol('r_0', commutative=True)))"], "Equality(Mul(Function('r')(Symbol('r_0', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('r_0', commutative=True))))), Integer(1))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Mul(Function('r')(Symbol('r_0', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True))), Integral(Integer(1), Tuple(Symbol('r_0', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Mul(Function('r')(Symbol('r_0', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True)))), exp(Integral(Integer(1), Tuple(Symbol('r_0', commutative=True)))))"], [["power", 4, "Symbol('r_0', commutative=True)"], "Equality(Pow(exp(Integral(Mul(Function('r')(Symbol('r_0', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)), Pow(exp(Integral(Integer(1), Tuple(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(f^{\\prime},r)} = - f^{\\prime} + r and t{(f^{\\prime})} = - f^{\\prime} and \\dot{y}{(f^{\\prime},r)} = (r + t{(f^{\\prime})})^{f^{\\prime}}, then obtain \\dot{z}^{f^{\\prime}}{(f^{\\prime},r)} = \\dot{y}{(f^{\\prime},r)}", "derivation": "\\dot{z}{(f^{\\prime},r)} = - f^{\\prime} + r and \\dot{z}^{f^{\\prime}}{(f^{\\prime},r)} = (- f^{\\prime} + r)^{f^{\\prime}} and t{(f^{\\prime})} = - f^{\\prime} and \\dot{z}^{f^{\\prime}}{(f^{\\prime},r)} = (r + t{(f^{\\prime})})^{f^{\\prime}} and \\dot{y}{(f^{\\prime},r)} = (r + t{(f^{\\prime})})^{f^{\\prime}} and \\dot{z}^{f^{\\prime}}{(f^{\\prime},r)} = \\dot{y}{(f^{\\prime},r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('r', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\dot{z}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('r', commutative=True), Function('t')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)), Pow(Add(Symbol('r', commutative=True), Function('t')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('\\\\dot{z}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\dot{y}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(S)} = e^{S}, then obtain \\operatorname{A_{2}}{(S)} e^{- S} = 1", "derivation": "\\operatorname{A_{2}}{(S)} = e^{S} and \\operatorname{A_{2}}{(S)} e^{S} = e^{2 S} and S \\operatorname{A_{2}}{(S)} e^{S} = S e^{2 S} and \\operatorname{A_{2}}{(S)} e^{- S} = 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["times", 1, "exp(Symbol('S', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), exp(Mul(Integer(2), Symbol('S', commutative=True))))"], [["times", 2, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('A_2')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), exp(Mul(Integer(2), Symbol('S', commutative=True)))))"], [["divide", 3, "Mul(Symbol('S', commutative=True), exp(Mul(Integer(2), Symbol('S', commutative=True))))"], "Equality(Mul(Function('A_2')(Symbol('S', commutative=True)), exp(Mul(Integer(-1), Symbol('S', commutative=True)))), Integer(1))"]]}, {"prompt": "Given E{(\\eta,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} - \\eta)}, then obtain L_{\\varepsilon} + \\int (E{(\\eta,L_{\\varepsilon})} + \\sin{(L_{\\varepsilon} - \\eta)}) dL_{\\varepsilon} = L_{\\varepsilon} + \\int 2 \\sin{(L_{\\varepsilon} - \\eta)} dL_{\\varepsilon}", "derivation": "E{(\\eta,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} - \\eta)} and E{(\\eta,L_{\\varepsilon})} + \\sin{(L_{\\varepsilon} - \\eta)} = 2 \\sin{(L_{\\varepsilon} - \\eta)} and \\int (E{(\\eta,L_{\\varepsilon})} + \\sin{(L_{\\varepsilon} - \\eta)}) dL_{\\varepsilon} = \\int 2 \\sin{(L_{\\varepsilon} - \\eta)} dL_{\\varepsilon} and L_{\\varepsilon} + \\int (E{(\\eta,L_{\\varepsilon})} + \\sin{(L_{\\varepsilon} - \\eta)}) dL_{\\varepsilon} = L_{\\varepsilon} + \\int 2 \\sin{(L_{\\varepsilon} - \\eta)} dL_{\\varepsilon}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["add", 1, "sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))"], "Equality(Add(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Mul(Integer(2), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(2), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["add", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Add(Function('E')(Symbol('\\\\eta', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Mul(Integer(2), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\varphi^*,y)} = y \\log{(\\varphi^*)} and \\operatorname{J_{\\varepsilon}}{(\\varphi^*,y)} = 2 y \\log{(\\varphi^*)}, then obtain 2 y \\log{(\\varphi^*)} + \\operatorname{J_{\\varepsilon}}{(\\varphi^*,y)} = 3 y \\log{(\\varphi^*)} + \\theta{(\\varphi^*,y)}", "derivation": "\\theta{(\\varphi^*,y)} = y \\log{(\\varphi^*)} and y \\log{(\\varphi^*)} + \\theta{(\\varphi^*,y)} = 2 y \\log{(\\varphi^*)} and \\operatorname{J_{\\varepsilon}}{(\\varphi^*,y)} = 2 y \\log{(\\varphi^*)} and y \\log{(\\varphi^*)} + \\operatorname{J_{\\varepsilon}}{(\\varphi^*,y)} + \\theta{(\\varphi^*,y)} = 3 y \\log{(\\varphi^*)} + \\theta{(\\varphi^*,y)} and 2 y \\log{(\\varphi^*)} + \\operatorname{J_{\\varepsilon}}{(\\varphi^*,y)} = 3 y \\log{(\\varphi^*)} + \\theta{(\\varphi^*,y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "Mul(Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Mul(Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True))), Mul(Integer(2), Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 3, "Add(Mul(Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True)), Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(3), Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(2), Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(3), Symbol('y', commutative=True), log(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\ddot{x})} = \\log{(\\ddot{x})}, then derive \\int \\omega{(\\ddot{x})} d\\ddot{x} = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\theta, then obtain (\\int \\omega{(\\ddot{x})} d\\ddot{x})^{\\theta} = (\\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\theta)^{\\theta}", "derivation": "\\omega{(\\ddot{x})} = \\log{(\\ddot{x})} and \\int \\omega{(\\ddot{x})} d\\ddot{x} = \\int \\log{(\\ddot{x})} d\\ddot{x} and \\int \\omega{(\\ddot{x})} d\\ddot{x} = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\theta and (\\int \\omega{(\\ddot{x})} d\\ddot{x})^{\\theta} = (\\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\theta)^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\omega')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given Q{(g^{\\prime}_{\\varepsilon},\\phi_2)} = \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2}, then derive \\int (- \\phi_2 + Q{(g^{\\prime}_{\\varepsilon},\\phi_2)}) d\\phi_2 = \\mathbf{F} - \\frac{\\phi_2^{2}}{2} + g^{\\prime}_{\\varepsilon} \\log{(\\phi_2)}, then obtain \\mathbf{F} - \\frac{\\phi_2^{2}}{2} + g^{\\prime}_{\\varepsilon} \\log{(\\phi_2)} = \\int (- \\phi_2 + \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2}) d\\phi_2", "derivation": "Q{(g^{\\prime}_{\\varepsilon},\\phi_2)} = \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2} and - \\phi_2 + Q{(g^{\\prime}_{\\varepsilon},\\phi_2)} = - \\phi_2 + \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2} and \\int (- \\phi_2 + Q{(g^{\\prime}_{\\varepsilon},\\phi_2)}) d\\phi_2 = \\int (- \\phi_2 + \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2}) d\\phi_2 and \\int (- \\phi_2 + Q{(g^{\\prime}_{\\varepsilon},\\phi_2)}) d\\phi_2 = \\mathbf{F} - \\frac{\\phi_2^{2}}{2} + g^{\\prime}_{\\varepsilon} \\log{(\\phi_2)} and \\mathbf{F} - \\frac{\\phi_2^{2}}{2} + g^{\\prime}_{\\varepsilon} \\log{(\\phi_2)} = \\int (- \\phi_2 + \\frac{g^{\\prime}_{\\varepsilon}}{\\phi_2}) d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('Q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('Q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('Q')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('\\\\phi_2', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}}, then derive \\operatorname{A_{x}}{(n_{2})} = e^{n_{2}}, then obtain (\\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}})^{4} = (\\frac{d^{4}}{d n_{2}^{4}} e^{n_{2}})^{4}", "derivation": "\\operatorname{A_{x}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\operatorname{A_{x}}{(n_{2})} = e^{n_{2}} and e^{n_{2}} = \\frac{d}{d n_{2}} e^{n_{2}} and \\operatorname{A_{x}}{(n_{2})} = \\frac{d}{d n_{2}} \\operatorname{A_{x}}{(n_{2})} and \\frac{d}{d n_{2}} e^{n_{2}} = \\frac{d^{2}}{d n_{2}^{2}} e^{n_{2}} and \\frac{d^{2}}{d n_{2}^{2}} e^{n_{2}} = \\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}} and \\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}} = \\frac{d^{4}}{d n_{2}^{4}} e^{n_{2}} and (\\frac{d^{3}}{d n_{2}^{3}} e^{n_{2}})^{4} = (\\frac{d^{4}}{d n_{2}^{4}} e^{n_{2}})^{4}", "srepr_derivation": [["get_premise", "Equality(Function('A_x')(Symbol('n_2', commutative=True)), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_x')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(exp(Symbol('n_2', commutative=True)), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_x')(Symbol('n_2', commutative=True)), Derivative(Function('A_x')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(2))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(3))))"], [["differentiate", 6, "Symbol('n_2', commutative=True)"], "Equality(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(3))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(4))))"], [["power", 7, 4], "Equality(Pow(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(3))), Integer(4)), Pow(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(4))), Integer(4)))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{S},y)} = \\sin{(\\mathbf{S} - y)}, then derive (\\mathbf{S} - y) \\frac{\\partial}{\\partial y} \\phi_{2}{(\\mathbf{S},y)} - \\phi_{2}{(\\mathbf{S},y)} = - (\\mathbf{S} - y) \\cos{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)}, then obtain (\\mathbf{S} - y) \\frac{\\partial}{\\partial y} \\sin{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)} = - (\\mathbf{S} - y) \\cos{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)}", "derivation": "\\phi_{2}{(\\mathbf{S},y)} = \\sin{(\\mathbf{S} - y)} and (\\mathbf{S} - y) \\phi_{2}{(\\mathbf{S},y)} = (\\mathbf{S} - y) \\sin{(\\mathbf{S} - y)} and \\frac{\\partial}{\\partial y} (\\mathbf{S} - y) \\phi_{2}{(\\mathbf{S},y)} = \\frac{\\partial}{\\partial y} (\\mathbf{S} - y) \\sin{(\\mathbf{S} - y)} and (\\mathbf{S} - y) \\frac{\\partial}{\\partial y} \\phi_{2}{(\\mathbf{S},y)} - \\phi_{2}{(\\mathbf{S},y)} = - (\\mathbf{S} - y) \\cos{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)} and (\\mathbf{S} - y) \\frac{\\partial}{\\partial y} \\sin{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)} = - (\\mathbf{S} - y) \\cos{(\\mathbf{S} - y)} - \\sin{(\\mathbf{S} - y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["times", 1, "Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Function('\\\\phi_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))), Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Function('\\\\phi_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Derivative(sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))), Add(Mul(Integer(-1), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{p})} = \\cos{(e^{\\mathbf{p}})}, then derive \\int \\operatorname{f^{\\prime}}{(\\mathbf{p})} d\\mathbf{p} = \\chi + \\operatorname{Ci}{(e^{\\mathbf{p}})}, then obtain \\int \\cos{(e^{\\mathbf{p}})} d\\mathbf{p} = \\chi + \\operatorname{Ci}{(e^{\\mathbf{p}})}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{p})} = \\cos{(e^{\\mathbf{p}})} and \\int \\operatorname{f^{\\prime}}{(\\mathbf{p})} d\\mathbf{p} = \\int \\cos{(e^{\\mathbf{p}})} d\\mathbf{p} and \\int \\operatorname{f^{\\prime}}{(\\mathbf{p})} d\\mathbf{p} = \\chi + \\operatorname{Ci}{(e^{\\mathbf{p}})} and \\int \\cos{(e^{\\mathbf{p}})} d\\mathbf{p} = \\chi + \\operatorname{Ci}{(e^{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Ci(exp(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Ci(exp(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\psi,p)} = \\frac{\\log{(\\psi)}}{p}, then obtain \\frac{\\Psi^{\\psi}{(\\psi,p)}}{- \\psi + \\frac{\\log{(\\psi)}}{p}} = \\frac{(\\frac{\\log{(\\psi)}}{p})^{\\psi}}{- \\psi + \\frac{\\log{(\\psi)}}{p}}", "derivation": "\\Psi{(\\psi,p)} = \\frac{\\log{(\\psi)}}{p} and - \\psi + \\Psi{(\\psi,p)} = - \\psi + \\frac{\\log{(\\psi)}}{p} and \\Psi^{\\psi}{(\\psi,p)} = (\\frac{\\log{(\\psi)}}{p})^{\\psi} and \\frac{\\Psi^{\\psi}{(\\psi,p)}}{- \\psi + \\Psi{(\\psi,p)}} = \\frac{(\\frac{\\log{(\\psi)}}{p})^{\\psi}}{- \\psi + \\Psi{(\\psi,p)}} and \\frac{\\Psi^{\\psi}{(\\psi,p)}}{- \\psi + \\frac{\\log{(\\psi)}}{p}} = \\frac{(\\frac{\\log{(\\psi)}}{p})^{\\psi}}{- \\psi + \\frac{\\log{(\\psi)}}{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True)))))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True))), Integer(-1)), Pow(Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Pow(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True)))), Integer(-1)), Pow(Function('\\\\Psi')(Symbol('\\\\psi', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Pow(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), log(Symbol('\\\\psi', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\eta{(\\mathbf{S},p)} = \\log{(p)}^{\\mathbf{S}}, then obtain e^{(- p + \\eta{(\\mathbf{S},p)})^{\\mathbf{S}}} = e^{(- p + \\log{(p)}^{\\mathbf{S}})^{\\mathbf{S}}}", "derivation": "\\eta{(\\mathbf{S},p)} = \\log{(p)}^{\\mathbf{S}} and - p + \\eta{(\\mathbf{S},p)} = - p + \\log{(p)}^{\\mathbf{S}} and (- p + \\eta{(\\mathbf{S},p)})^{\\mathbf{S}} = (- p + \\log{(p)}^{\\mathbf{S}})^{\\mathbf{S}} and e^{(- p + \\eta{(\\mathbf{S},p)})^{\\mathbf{S}}} = e^{(- p + \\log{(p)}^{\\mathbf{S}})^{\\mathbf{S}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 1, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(log(Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\int - \\frac{- \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\cos{(\\varepsilon)}}{\\cos{(\\varepsilon)}} d\\varepsilon = \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\int 0 d\\varepsilon", "derivation": "\\operatorname{f^{\\prime}}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\operatorname{f^{\\prime}}{(\\varepsilon)} - \\cos{(\\varepsilon)} = 0 and - \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\cos{(\\varepsilon)} = 0 and - \\frac{- \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\cos{(\\varepsilon)}}{\\cos{(\\varepsilon)}} = 0 and \\int - \\frac{- \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\cos{(\\varepsilon)}}{\\cos{(\\varepsilon)}} d\\varepsilon = \\int 0 d\\varepsilon and \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\int - \\frac{- \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\cos{(\\varepsilon)}}{\\cos{(\\varepsilon)}} d\\varepsilon = \\operatorname{f^{\\prime}}{(\\varepsilon)} + \\int 0 d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))), Integer(0))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), cos(Symbol('\\\\varepsilon', commutative=True))), Integer(0))"], [["divide", 3, "Mul(Integer(-1), cos(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), cos(Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Integer(0))"], [["integrate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), cos(Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 5, "Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integral(Mul(Integer(-1), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), cos(Symbol('\\\\varepsilon', commutative=True))), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Function('f^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(E_{n})} = \\cos{(\\sin{(E_{n})})}, then obtain - \\mathbf{B}{(E_{n})} \\cos{(\\sin{(E_{n})})} + \\cos^{2}{(\\sin{(E_{n})})} = 0", "derivation": "\\mathbf{B}{(E_{n})} = \\cos{(\\sin{(E_{n})})} and \\mathbf{B}^{2}{(E_{n})} = \\mathbf{B}{(E_{n})} \\cos{(\\sin{(E_{n})})} and \\mathbf{B}{(E_{n})} \\cos{(\\sin{(E_{n})})} = \\cos^{2}{(\\sin{(E_{n})})} and \\mathbf{B}^{2}{(E_{n})} = \\cos^{2}{(\\sin{(E_{n})})} and \\mathbf{B}^{2}{(E_{n})} - \\cos^{2}{(\\sin{(E_{n})})} = 0 and \\mathbf{B}^{2}{(E_{n})} - \\mathbf{B}{(E_{n})} \\cos{(\\sin{(E_{n})})} = 0 and - \\mathbf{B}{(E_{n})} \\cos{(\\sin{(E_{n})})} + \\cos^{2}{(\\sin{(E_{n})})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))))"], [["times", 1, "cos(sin(Symbol('E_n', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))), Pow(cos(sin(Symbol('E_n', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('E_n', commutative=True))), Integer(2)))"], [["minus", 4, "Pow(cos(sin(Symbol('E_n', commutative=True))), Integer(2))"], "Equality(Add(Pow(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(cos(sin(Symbol('E_n', commutative=True))), Integer(2)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))), Pow(cos(sin(Symbol('E_n', commutative=True))), Integer(2))), Integer(0))"]]}, {"prompt": "Given p{(\\theta_1)} = \\log{(\\theta_1)}, then obtain (\\frac{d}{d \\theta_1} p{(\\theta_1)})^{\\theta_1} (\\frac{d}{d \\theta_1} \\log{(\\theta_1)})^{- \\theta_1} = 1", "derivation": "p{(\\theta_1)} = \\log{(\\theta_1)} and \\frac{d}{d \\theta_1} p{(\\theta_1)} = \\frac{d}{d \\theta_1} \\log{(\\theta_1)} and (\\frac{d}{d \\theta_1} p{(\\theta_1)})^{\\theta_1} = (\\frac{d}{d \\theta_1} \\log{(\\theta_1)})^{\\theta_1} and (\\frac{d}{d \\theta_1} p{(\\theta_1)})^{\\theta_1} (\\frac{d}{d \\theta_1} \\log{(\\theta_1)})^{- \\theta_1} = 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\theta_1', commutative=True)), log(Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Derivative(Function('p')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 3, "Pow(Derivative(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Pow(Derivative(Function('p')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(b)} = \\sin{(\\log{(b)})}, then obtain \\int (\\int \\frac{d}{d b} \\operatorname{F_{c}}{(b)} db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db) db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db = \\int 0 db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db", "derivation": "\\operatorname{F_{c}}{(b)} = \\sin{(\\log{(b)})} and \\frac{d}{d b} \\operatorname{F_{c}}{(b)} = \\frac{d}{d b} \\sin{(\\log{(b)})} and \\int \\frac{d}{d b} \\operatorname{F_{c}}{(b)} db = \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db and \\int \\frac{d}{d b} \\operatorname{F_{c}}{(b)} db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db = 0 and \\int (\\int \\frac{d}{d b} \\operatorname{F_{c}}{(b)} db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db) db = \\int 0 db and \\int (\\int \\frac{d}{d b} \\operatorname{F_{c}}{(b)} db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db) db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db = \\int 0 db - \\int \\frac{d}{d b} \\sin{(\\log{(b)})} db", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('b', commutative=True)), sin(log(Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Derivative(Function('F_c')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], [["minus", 3, "Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Integral(Derivative(Function('F_c')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Integral(Derivative(Function('F_c')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))), Tuple(Symbol('b', commutative=True))), Integral(Integer(0), Tuple(Symbol('b', commutative=True))))"], [["add", 5, "Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], "Equality(Add(Integral(Add(Integral(Derivative(Function('F_c')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))), Add(Integral(Integer(0), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(Derivative(sin(log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\hat{H})} = \\hat{H} and \\operatorname{v_{t}}{(\\hat{H})} = \\sin{(e^{\\hat{H}})}, then obtain \\frac{d}{d \\hat{H}} \\operatorname{v_{t}}{(\\hat{H})} = e^{\\operatorname{n_{1}}{(\\hat{H})}} \\cos{(e^{\\operatorname{n_{1}}{(\\hat{H})}})} \\frac{d}{d \\hat{H}} \\operatorname{n_{1}}{(\\hat{H})}", "derivation": "\\operatorname{n_{1}}{(\\hat{H})} = \\hat{H} and e^{\\operatorname{n_{1}}{(\\hat{H})}} = e^{\\hat{H}} and \\operatorname{v_{t}}{(\\hat{H})} = \\sin{(e^{\\hat{H}})} and \\operatorname{v_{t}}{(\\hat{H})} = \\sin{(e^{\\operatorname{n_{1}}{(\\hat{H})}})} and \\frac{d}{d \\hat{H}} \\operatorname{v_{t}}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\sin{(e^{\\operatorname{n_{1}}{(\\hat{H})}})} and \\frac{d}{d \\hat{H}} \\operatorname{v_{t}}{(\\hat{H})} = e^{\\operatorname{n_{1}}{(\\hat{H})}} \\cos{(e^{\\operatorname{n_{1}}{(\\hat{H})}})} \\frac{d}{d \\hat{H}} \\operatorname{n_{1}}{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], [["exp", 1], "Equality(exp(Function('n_1')(Symbol('\\\\hat{H}', commutative=True))), exp(Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\hat{H}', commutative=True)), sin(exp(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('v_t')(Symbol('\\\\hat{H}', commutative=True)), sin(exp(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(sin(exp(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('v_t')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(exp(Function('n_1')(Symbol('\\\\hat{H}', commutative=True))), cos(exp(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)))), Derivative(Function('n_1')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(F_{c})} = F_{c}, then derive \\int M{(F_{c})} dF_{c} = \\frac{F_{c}^{2}}{2} + \\mathbf{f}, then derive \\frac{F_{c}^{2}}{2} + \\mathbf{f} = \\frac{F_{c}^{2}}{2} + \\dot{\\mathbf{r}}, then obtain - M^{2}{(F_{c})} + \\int M{(F_{c})} dM{(F_{c})} = \\dot{\\mathbf{r}} - \\frac{M^{2}{(F_{c})}}{2}", "derivation": "M{(F_{c})} = F_{c} and \\int M{(F_{c})} dF_{c} = \\int F_{c} dF_{c} and \\int M{(F_{c})} dF_{c} = \\frac{F_{c}^{2}}{2} + \\mathbf{f} and \\frac{F_{c}^{2}}{2} + \\mathbf{f} = \\int F_{c} dF_{c} and \\frac{F_{c}^{2}}{2} + \\mathbf{f} = \\frac{F_{c}^{2}}{2} + \\dot{\\mathbf{r}} and \\int M{(F_{c})} dF_{c} = \\frac{F_{c}^{2}}{2} + \\dot{\\mathbf{r}} and - F_{c}^{2} + \\int M{(F_{c})} dF_{c} = - \\frac{F_{c}^{2}}{2} + \\dot{\\mathbf{r}} and - M^{2}{(F_{c})} + \\int M{(F_{c})} dM{(F_{c})} = \\dot{\\mathbf{r}} - \\frac{M^{2}{(F_{c})}}{2}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('M')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Function('M')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["minus", 6, "Pow(Symbol('F_c', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(2))), Integral(Function('M')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('M')(Symbol('F_c', commutative=True)), Integer(2))), Integral(Function('M')(Symbol('F_c', commutative=True)), Tuple(Function('M')(Symbol('F_c', commutative=True))))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Function('M')(Symbol('F_c', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\nabla)} = \\sin{(\\cos{(\\nabla)})}, then obtain (\\operatorname{m_{s}}^{2}{(\\nabla)} \\sin{(\\cos{(\\nabla)})})^{\\nabla} = (\\operatorname{m_{s}}{(\\nabla)} \\sin^{2}{(\\cos{(\\nabla)})})^{\\nabla}", "derivation": "\\operatorname{m_{s}}{(\\nabla)} = \\sin{(\\cos{(\\nabla)})} and \\operatorname{m_{s}}{(\\nabla)} \\sin{(\\cos{(\\nabla)})} = \\sin^{2}{(\\cos{(\\nabla)})} and \\operatorname{m_{s}}{(\\nabla)} \\sin^{2}{(\\cos{(\\nabla)})} = \\sin^{3}{(\\cos{(\\nabla)})} and \\operatorname{m_{s}}^{2}{(\\nabla)} \\sin{(\\cos{(\\nabla)})} = \\sin^{3}{(\\cos{(\\nabla)})} and \\operatorname{m_{s}}^{2}{(\\nabla)} \\sin{(\\cos{(\\nabla)})} = \\operatorname{m_{s}}{(\\nabla)} \\sin^{2}{(\\cos{(\\nabla)})} and (\\operatorname{m_{s}}^{2}{(\\nabla)} \\sin{(\\cos{(\\nabla)})})^{\\nabla} = (\\operatorname{m_{s}}{(\\nabla)} \\sin^{2}{(\\cos{(\\nabla)})})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\nabla', commutative=True)), sin(cos(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "sin(cos(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), sin(cos(Symbol('\\\\nabla', commutative=True)))), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(2)))"], [["times", 2, "sin(cos(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(2))), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), sin(cos(Symbol('\\\\nabla', commutative=True)))), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(3)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), sin(cos(Symbol('\\\\nabla', commutative=True)))), Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(2))))"], [["power", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Mul(Pow(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Integer(2)), sin(cos(Symbol('\\\\nabla', commutative=True)))), Symbol('\\\\nabla', commutative=True)), Pow(Mul(Function('m_s')(Symbol('\\\\nabla', commutative=True)), Pow(sin(cos(Symbol('\\\\nabla', commutative=True))), Integer(2))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(m)} = \\sin{(m)} and g{(m)} = (-1 + \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m})^{m}, then obtain g{(m)} = (-1 + \\frac{\\sin^{3}{(m)}}{m})^{m}", "derivation": "\\eta^{\\prime}{(m)} = \\sin{(m)} and \\frac{\\eta^{\\prime}{(m)} \\sin{(m)}}{m} = \\frac{\\sin^{2}{(m)}}{m} and \\frac{\\eta^{\\prime}{(m)} \\sin^{2}{(m)}}{m} = \\frac{\\sin^{3}{(m)}}{m} and \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m} = \\frac{\\eta^{\\prime}{(m)} \\sin^{2}{(m)}}{m} and \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m} = \\frac{\\sin^{3}{(m)}}{m} and -1 + \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m} = -1 + \\frac{\\sin^{3}{(m)}}{m} and (-1 + \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m})^{m} = (-1 + \\frac{\\sin^{3}{(m)}}{m})^{m} and g{(m)} = (-1 + \\frac{\\eta^{\\prime}^{2}{(m)} \\sin{(m)}}{m})^{m} and g{(m)} = (-1 + \\frac{\\sin^{3}{(m)}}{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('m', commutative=True), Integer(-1)), sin(Symbol('m', commutative=True)))"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(2))))"], [["times", 2, "sin(Symbol('m', commutative=True))"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(2))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), sin(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), sin(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(3))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), sin(Symbol('m', commutative=True)))), Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(3)))))"], [["power", 6, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(3)))), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('m', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Function('g')(Symbol('m', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Pow(sin(Symbol('m', commutative=True)), Integer(3)))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given E{(\\psi^*,E_{\\lambda})} = E_{\\lambda} - \\psi^*, then obtain e^{(E_{\\lambda} - \\psi^* E{(\\psi^*,E_{\\lambda})})^{E_{\\lambda}}} = e^{(E_{\\lambda} - \\psi^* (E_{\\lambda} - \\psi^*))^{E_{\\lambda}}}", "derivation": "E{(\\psi^*,E_{\\lambda})} = E_{\\lambda} - \\psi^* and - \\psi^* E{(\\psi^*,E_{\\lambda})} = - \\psi^* (E_{\\lambda} - \\psi^*) and E_{\\lambda} - \\psi^* E{(\\psi^*,E_{\\lambda})} = E_{\\lambda} - \\psi^* (E_{\\lambda} - \\psi^*) and (E_{\\lambda} - \\psi^* E{(\\psi^*,E_{\\lambda})})^{E_{\\lambda}} = (E_{\\lambda} - \\psi^* (E_{\\lambda} - \\psi^*))^{E_{\\lambda}} and e^{(E_{\\lambda} - \\psi^* E{(\\psi^*,E_{\\lambda})})^{E_{\\lambda}}} = e^{(E_{\\lambda} - \\psi^* (E_{\\lambda} - \\psi^*))^{E_{\\lambda}}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\psi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Function('E')(Symbol('\\\\psi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))))"], [["add", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Function('E')(Symbol('\\\\psi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))))"], [["power", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Function('E')(Symbol('\\\\psi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Function('E')(Symbol('\\\\psi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('E_{\\\\lambda}', commutative=True))), exp(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))), Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(F_{N},v_{t})} = e^{\\frac{F_{N}}{v_{t}}}, then derive - \\frac{\\partial}{\\partial v_{t}} \\hat{H}_{\\lambda}{(F_{N},v_{t})} = \\frac{F_{N} e^{\\frac{F_{N}}{v_{t}}}}{v_{t}^{2}}, then obtain - \\frac{\\partial}{\\partial v_{t}} \\hat{H}_{\\lambda}{(F_{N},v_{t})} = \\frac{F_{N} \\hat{H}_{\\lambda}{(F_{N},v_{t})}}{v_{t}^{2}}", "derivation": "\\hat{H}_{\\lambda}{(F_{N},v_{t})} = e^{\\frac{F_{N}}{v_{t}}} and - \\hat{H}_{\\lambda}{(F_{N},v_{t})} = - e^{\\frac{F_{N}}{v_{t}}} and \\frac{\\partial}{\\partial v_{t}} - \\hat{H}_{\\lambda}{(F_{N},v_{t})} = \\frac{\\partial}{\\partial v_{t}} - e^{\\frac{F_{N}}{v_{t}}} and - \\frac{\\partial}{\\partial v_{t}} \\hat{H}_{\\lambda}{(F_{N},v_{t})} = \\frac{F_{N} e^{\\frac{F_{N}}{v_{t}}}}{v_{t}^{2}} and - \\frac{\\partial}{\\partial v_{t}} \\hat{H}_{\\lambda}{(F_{N},v_{t})} = \\frac{F_{N} \\hat{H}_{\\lambda}{(F_{N},v_{t})}}{v_{t}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True)), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-2)), exp(Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Mul(Symbol('F_N', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-2)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('F_N', commutative=True), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(b,\\mathbf{J})} = \\mathbf{J} - b, then obtain - b + \\int (\\operatorname{V_{\\mathbf{B}}}^{b}{(b,\\mathbf{J})})^{\\mathbf{J}} db = - b + \\int ((\\mathbf{J} - b)^{b})^{\\mathbf{J}} db", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(b,\\mathbf{J})} = \\mathbf{J} - b and \\operatorname{V_{\\mathbf{B}}}^{b}{(b,\\mathbf{J})} = (\\mathbf{J} - b)^{b} and (\\operatorname{V_{\\mathbf{B}}}^{b}{(b,\\mathbf{J})})^{\\mathbf{J}} = ((\\mathbf{J} - b)^{b})^{\\mathbf{J}} and \\int (\\operatorname{V_{\\mathbf{B}}}^{b}{(b,\\mathbf{J})})^{\\mathbf{J}} db = \\int ((\\mathbf{J} - b)^{b})^{\\mathbf{J}} db and - b + \\int (\\operatorname{V_{\\mathbf{B}}}^{b}{(b,\\mathbf{J})})^{\\mathbf{J}} db = - b + \\int ((\\mathbf{J} - b)^{b})^{\\mathbf{J}} db", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('b', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Pow(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Integral(Pow(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Integral(Pow(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(C,\\hat{x})} = - C + \\hat{x}, then obtain ((\\hat{x} \\varepsilon_{0}{(C,\\hat{x})})^{C})^{\\hat{x}} = ((\\hat{x} (- C + \\hat{x}))^{C})^{\\hat{x}}", "derivation": "\\varepsilon_{0}{(C,\\hat{x})} = - C + \\hat{x} and \\hat{x} \\varepsilon_{0}{(C,\\hat{x})} = \\hat{x} (- C + \\hat{x}) and (\\hat{x} \\varepsilon_{0}{(C,\\hat{x})})^{C} = (\\hat{x} (- C + \\hat{x}))^{C} and ((\\hat{x} \\varepsilon_{0}{(C,\\hat{x})})^{C})^{\\hat{x}} = ((\\hat{x} (- C + \\hat{x}))^{C})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Symbol('C', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Function('\\\\varepsilon_0')(Symbol('C', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True))), Symbol('C', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given U{(f_{\\mathbf{p}},S)} = \\frac{f_{\\mathbf{p}}}{S}, then obtain (\\frac{\\partial}{\\partial f_{\\mathbf{p}}} S U{(f_{\\mathbf{p}},S)})^{S} = (\\frac{d}{d f_{\\mathbf{p}}} f_{\\mathbf{p}})^{S}", "derivation": "U{(f_{\\mathbf{p}},S)} = \\frac{f_{\\mathbf{p}}}{S} and S U{(f_{\\mathbf{p}},S)} = f_{\\mathbf{p}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} S U{(f_{\\mathbf{p}},S)} = \\frac{d}{d f_{\\mathbf{p}}} f_{\\mathbf{p}} and (\\frac{\\partial}{\\partial f_{\\mathbf{p}}} S U{(f_{\\mathbf{p}},S)})^{S} = (\\frac{d}{d f_{\\mathbf{p}}} f_{\\mathbf{p}})^{S}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('S', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('U')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('S', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Mul(Symbol('S', commutative=True), Function('U')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Symbol('f_{\\\\mathbf{p}}', commutative=True), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('S', commutative=True), Function('U')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Symbol('f_{\\\\mathbf{p}}', commutative=True), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\rho{(E_{x})} = \\cos{(E_{x})}, then obtain 2 (\\rho{(E_{x})} + \\cos{(E_{x})}) \\cos{(E_{x})} - 4 \\cos^{2}{(E_{x})} = 0", "derivation": "\\rho{(E_{x})} = \\cos{(E_{x})} and \\rho{(E_{x})} + \\cos{(E_{x})} = 2 \\cos{(E_{x})} and 2 (\\rho{(E_{x})} + \\cos{(E_{x})}) \\cos{(E_{x})} = 4 \\cos^{2}{(E_{x})} and 2 (\\rho{(E_{x})} + \\cos{(E_{x})}) \\cos{(E_{x})} - 4 \\cos^{2}{(E_{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["add", 1, "cos(Symbol('E_x', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), Mul(Integer(2), cos(Symbol('E_x', commutative=True))))"], [["times", 2, "Mul(Integer(2), cos(Symbol('E_x', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('\\\\rho')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), cos(Symbol('E_x', commutative=True))), Mul(Integer(4), Pow(cos(Symbol('E_x', commutative=True)), Integer(2))))"], [["minus", 3, "Mul(Integer(4), Pow(cos(Symbol('E_x', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Integer(2), Add(Function('\\\\rho')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), cos(Symbol('E_x', commutative=True))), Mul(Integer(-1), Integer(4), Pow(cos(Symbol('E_x', commutative=True)), Integer(2)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{g}{(v_{2})} = \\log{(v_{2})} and \\rho_{b}{(v_{2})} = \\mathbf{g}{(v_{2})} \\log{(v_{2})}, then obtain \\log{(v_{2})}^{2} = \\mathbf{g}^{2}{(v_{2})}", "derivation": "\\mathbf{g}{(v_{2})} = \\log{(v_{2})} and \\rho_{b}{(v_{2})} = \\mathbf{g}{(v_{2})} \\log{(v_{2})} and \\rho_{b}{(v_{2})} = \\mathbf{g}^{2}{(v_{2})} and \\rho_{b}{(v_{2})} = \\log{(v_{2})}^{2} and \\log{(v_{2})}^{2} = \\mathbf{g}^{2}{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('v_2', commutative=True)), Mul(Function('\\\\mathbf{g}')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\rho_b')(Symbol('v_2', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('v_2', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\rho_b')(Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(log(Symbol('v_2', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{g}')(Symbol('v_2', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})} = \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})}, then obtain \\log{(\\mathbf{r})} = \\log{(\\frac{\\mathbf{r} \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})}}{\\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})}})}", "derivation": "\\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})} = \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})} and \\mathbf{r} \\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})} = \\mathbf{r} \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})} and \\mathbf{r} = \\frac{\\mathbf{r} \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})}}{\\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})}} and \\log{(\\mathbf{r})} = \\log{(\\frac{\\mathbf{r} \\cos{(\\hat{\\mathbf{r}}^{\\mathbf{r}})}}{\\mathbf{D}{(\\hat{\\mathbf{r}},\\mathbf{r})}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), cos(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), cos(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["divide", 2, "Function('\\\\mathbf{D}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Symbol('\\\\mathbf{r}', commutative=True), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["log", 3], "Equality(log(Symbol('\\\\mathbf{r}', commutative=True)), log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))))"]]}, {"prompt": "Given U{(E,r)} = \\log{(r^{E})}, then derive \\frac{\\partial}{\\partial E} U{(E,r)} = \\log{(r)}, then obtain \\int (U{(E,r)} + \\frac{\\partial}{\\partial E} U{(E,r)}) dr = \\int (U{(E,r)} + \\log{(r)}) dr", "derivation": "U{(E,r)} = \\log{(r^{E})} and \\frac{\\partial}{\\partial E} U{(E,r)} = \\frac{\\partial}{\\partial E} \\log{(r^{E})} and \\frac{\\partial}{\\partial E} U{(E,r)} = \\log{(r)} and U{(E,r)} + \\frac{\\partial}{\\partial E} U{(E,r)} = U{(E,r)} + \\log{(r)} and \\int (U{(E,r)} + \\frac{\\partial}{\\partial E} U{(E,r)}) dr = \\int (U{(E,r)} + \\log{(r)}) dr", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), log(Pow(Symbol('r', commutative=True), Symbol('E', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('r', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), log(Symbol('r', commutative=True)))"], [["add", 3, "Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Derivative(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Derivative(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Tuple(Symbol('r', commutative=True))), Integral(Add(Function('U')(Symbol('E', commutative=True), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given t{(\\dot{\\mathbf{r}},v)} = v^{\\dot{\\mathbf{r}}} and S{(m,\\sigma_p)} = \\sigma_p + m, then obtain \\int - v^{- \\dot{\\mathbf{r}}} S{(m,\\sigma_p)} dv - \\frac{1}{t{(\\dot{\\mathbf{r}},v)}} = \\int - v^{- \\dot{\\mathbf{r}}} (\\sigma_p + m) dv - \\frac{1}{t{(\\dot{\\mathbf{r}},v)}}", "derivation": "t{(\\dot{\\mathbf{r}},v)} = v^{\\dot{\\mathbf{r}}} and S{(m,\\sigma_p)} = \\sigma_p + m and - \\frac{S{(m,\\sigma_p)}}{t{(\\dot{\\mathbf{r}},v)}} = - \\frac{\\sigma_p + m}{t{(\\dot{\\mathbf{r}},v)}} and - v^{- \\dot{\\mathbf{r}}} S{(m,\\sigma_p)} = - v^{- \\dot{\\mathbf{r}}} (\\sigma_p + m) and \\int - v^{- \\dot{\\mathbf{r}}} S{(m,\\sigma_p)} dv = \\int - v^{- \\dot{\\mathbf{r}}} (\\sigma_p + m) dv and \\int - v^{- \\dot{\\mathbf{r}}} S{(m,\\sigma_p)} dv - \\frac{1}{t{(\\dot{\\mathbf{r}},v)}} = \\int - v^{- \\dot{\\mathbf{r}}} (\\sigma_p + m) dv - \\frac{1}{t{(\\dot{\\mathbf{r}},v)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["get_premise", "Equality(Function('S')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)))"], "Equality(Mul(Integer(-1), Function('S')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True)), Pow(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('S')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('S')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["minus", 5, "Pow(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Integer(-1))"], "Equality(Add(Integral(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('S')(Symbol('m', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Integer(-1)))), Add(Integral(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Function('t')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\chi,\\mathbf{J}_M,M_{E})} = \\frac{\\chi^{\\mathbf{J}_M}}{M_{E}}, then obtain ((\\operatorname{C_{2}}^{\\mathbf{J}_M}{(\\chi,\\mathbf{J}_M,M_{E})})^{M_{E}})^{\\chi} = (((\\frac{\\chi^{\\mathbf{J}_M}}{M_{E}})^{\\mathbf{J}_M})^{M_{E}})^{\\chi}", "derivation": "\\operatorname{C_{2}}{(\\chi,\\mathbf{J}_M,M_{E})} = \\frac{\\chi^{\\mathbf{J}_M}}{M_{E}} and \\operatorname{C_{2}}^{\\mathbf{J}_M}{(\\chi,\\mathbf{J}_M,M_{E})} = (\\frac{\\chi^{\\mathbf{J}_M}}{M_{E}})^{\\mathbf{J}_M} and (\\operatorname{C_{2}}^{\\mathbf{J}_M}{(\\chi,\\mathbf{J}_M,M_{E})})^{M_{E}} = ((\\frac{\\chi^{\\mathbf{J}_M}}{M_{E}})^{\\mathbf{J}_M})^{M_{E}} and ((\\operatorname{C_{2}}^{\\mathbf{J}_M}{(\\chi,\\mathbf{J}_M,M_{E})})^{M_{E}})^{\\chi} = (((\\frac{\\chi^{\\mathbf{J}_M}}{M_{E}})^{\\mathbf{J}_M})^{M_{E}})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('M_E', commutative=True)))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Pow(Pow(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given a{(\\hat{p})} = e^{\\hat{p}} and \\mathbf{E}{(\\hat{p})} = \\int 0 d\\hat{p}, then obtain \\mathbf{E}{(\\hat{p})} = \\int \\hat{p} (a{(\\hat{p})} - e^{\\hat{p}}) d\\hat{p}", "derivation": "a{(\\hat{p})} = e^{\\hat{p}} and a{(\\hat{p})} - e^{\\hat{p}} = 0 and \\hat{p} (a{(\\hat{p})} - e^{\\hat{p}}) = 0 and \\int \\hat{p} (a{(\\hat{p})} - e^{\\hat{p}}) d\\hat{p} = \\int 0 d\\hat{p} and \\mathbf{E}{(\\hat{p})} = \\int 0 d\\hat{p} and \\mathbf{E}{(\\hat{p})} = \\int \\hat{p} (a{(\\hat{p})} - e^{\\hat{p}}) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Function('a')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True)))), Integer(0))"], [["times", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Function('a')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))))), Integer(0))"], [["integrate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Function('a')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}', commutative=True)), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Add(Function('a')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(L)} = e^{L} and \\operatorname{E_{\\lambda}}{(L)} = \\frac{d}{d L} \\operatorname{P_{e}}{(L)}, then obtain \\operatorname{E_{\\lambda}}{(L)} - 1 = \\frac{d}{d L} e^{L} - 1", "derivation": "\\operatorname{P_{e}}{(L)} = e^{L} and \\operatorname{E_{\\lambda}}{(L)} = \\frac{d}{d L} \\operatorname{P_{e}}{(L)} and \\operatorname{E_{\\lambda}}{(L)} = \\frac{d}{d L} e^{L} and \\operatorname{E_{\\lambda}}{(L)} - 1 = \\frac{d}{d L} e^{L} - 1", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('L', commutative=True)), Derivative(Function('P_e')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E_{\\\\lambda}')(Symbol('L', commutative=True)), Derivative(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('L', commutative=True)), Integer(-1)), Add(Derivative(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given V{(a,f_{\\mathbf{v}})} = a + f_{\\mathbf{v}}, then obtain \\int (f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}}) da = \\int (f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} + \\frac{a + f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} - 1}{V{(a,f_{\\mathbf{v}})}} - 1) da", "derivation": "V{(a,f_{\\mathbf{v}})} = a + f_{\\mathbf{v}} and 1 = \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} and f_{\\mathbf{v}} + 1 = f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} and \\int (f_{\\mathbf{v}} + 1) da = \\int (f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}}) da and \\int (f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}}) da = \\int (f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} + \\frac{a + f_{\\mathbf{v}} + \\frac{a + f_{\\mathbf{v}}}{V{(a,f_{\\mathbf{v}})}} - 1}{V{(a,f_{\\mathbf{v}})}} - 1) da", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 1, "Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))))"], [["add", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)))), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Add(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Integer(-1)), Pow(Function('V')(Symbol('a', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given t{(g)} = \\log{(\\sin{(g)})}, then derive 0 = \\frac{\\cos{(g)}}{t{(g)} \\sin{(g)}} - \\frac{\\log{(\\sin{(g)})} \\frac{d}{d g} t{(g)}}{t^{2}{(g)}}, then obtain \\frac{d}{d g} 0 = \\frac{d}{d g} (- \\frac{\\frac{d}{d g} \\log{(\\sin{(g)})}}{\\log{(\\sin{(g)})}} + \\frac{\\cos{(g)}}{\\log{(\\sin{(g)})} \\sin{(g)}})", "derivation": "t{(g)} = \\log{(\\sin{(g)})} and 1 = \\frac{\\log{(\\sin{(g)})}}{t{(g)}} and \\frac{d}{d g} 1 = \\frac{d}{d g} \\frac{\\log{(\\sin{(g)})}}{t{(g)}} and 0 = \\frac{\\cos{(g)}}{t{(g)} \\sin{(g)}} - \\frac{\\log{(\\sin{(g)})} \\frac{d}{d g} t{(g)}}{t^{2}{(g)}} and 0 = - \\frac{\\frac{d}{d g} \\log{(\\sin{(g)})}}{\\log{(\\sin{(g)})}} + \\frac{\\cos{(g)}}{\\log{(\\sin{(g)})} \\sin{(g)}} and \\frac{d}{d g} 0 = \\frac{d}{d g} (- \\frac{\\frac{d}{d g} \\log{(\\sin{(g)})}}{\\log{(\\sin{(g)})}} + \\frac{\\cos{(g)}}{\\log{(\\sin{(g)})} \\sin{(g)}})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('g', commutative=True)), log(sin(Symbol('g', commutative=True))))"], [["divide", 1, "Function('t')(Symbol('g', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t')(Symbol('g', commutative=True)), Integer(-1)), log(sin(Symbol('g', commutative=True)))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('t')(Symbol('g', commutative=True)), Integer(-1)), log(sin(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('t')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Function('t')(Symbol('g', commutative=True)), Integer(-2)), log(sin(Symbol('g', commutative=True))), Derivative(Function('t')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(log(sin(Symbol('g', commutative=True))), Integer(-1)), Derivative(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(log(sin(Symbol('g', commutative=True))), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True)))))"], [["differentiate", 5, "Symbol('g', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(log(sin(Symbol('g', commutative=True))), Integer(-1)), Derivative(log(sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(log(sin(Symbol('g', commutative=True))), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(E,I)} = \\log{(E)}^{I}, then obtain (- b{(E,I)} + \\log{(E)}^{I})^{E} + 1 = 2", "derivation": "b{(E,I)} = \\log{(E)}^{I} and 0 = - b{(E,I)} + \\log{(E)}^{I} and 0^{E} = (- b{(E,I)} + \\log{(E)}^{I})^{E} and 0^{E} + (- b{(E,I)} + \\log{(E)}^{I})^{E} = 2 (- b{(E,I)} + \\log{(E)}^{I})^{E} and (- b{(E,I)} + \\log{(E)}^{I})^{E} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True)), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True)))"], [["minus", 1, "Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))), Symbol('E', commutative=True)))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))), Symbol('E', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))), Symbol('E', commutative=True))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))), Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('b')(Symbol('E', commutative=True), Symbol('I', commutative=True))), Pow(log(Symbol('E', commutative=True)), Symbol('I', commutative=True))), Symbol('E', commutative=True)), Integer(1)), Integer(2))"]]}, {"prompt": "Given \\rho_{f}{(A_{2},\\chi)} = - A_{2} + \\chi, then obtain \\frac{\\rho_{f}^{2}{(A_{2},\\chi)}}{A_{2}^{2} \\chi} = \\frac{(- A_{2} + \\chi) \\rho_{f}{(A_{2},\\chi)}}{A_{2}^{2} \\chi}", "derivation": "\\rho_{f}{(A_{2},\\chi)} = - A_{2} + \\chi and \\frac{\\rho_{f}{(A_{2},\\chi)}}{A_{2}} = \\frac{- A_{2} + \\chi}{A_{2}} and \\frac{\\rho_{f}^{2}{(A_{2},\\chi)}}{A_{2}^{2}} = \\frac{(- A_{2} + \\chi) \\rho_{f}{(A_{2},\\chi)}}{A_{2}^{2}} and \\frac{\\rho_{f}^{2}{(A_{2},\\chi)}}{A_{2}^{2} \\chi} = \\frac{(- A_{2} + \\chi) \\rho_{f}{(A_{2},\\chi)}}{A_{2}^{2} \\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)), Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["divide", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)), Function('\\\\rho_f')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(M,A_{x})} = \\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}}, then derive \\hat{x}^{A_{x}}{(M,A_{x})} = (- \\frac{M}{A_{x}^{2}})^{A_{x}}, then obtain 2 \\hat{x}^{A_{x}}{(M,A_{x})} = \\hat{x}^{A_{x}}{(M,A_{x})} + (\\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}})^{A_{x}}", "derivation": "\\hat{x}{(M,A_{x})} = \\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}} and \\hat{x}^{A_{x}}{(M,A_{x})} = (\\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}})^{A_{x}} and \\hat{x}^{A_{x}}{(M,A_{x})} = (- \\frac{M}{A_{x}^{2}})^{A_{x}} and (\\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}})^{A_{x}} = (- \\frac{M}{A_{x}^{2}})^{A_{x}} and 2 \\hat{x}^{A_{x}}{(M,A_{x})} = (- \\frac{M}{A_{x}^{2}})^{A_{x}} + \\hat{x}^{A_{x}}{(M,A_{x})} and 2 \\hat{x}^{A_{x}}{(M,A_{x})} = \\hat{x}^{A_{x}}{(M,A_{x})} + (\\frac{\\partial}{\\partial A_{x}} \\frac{M}{A_{x}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-2)), Symbol('M', commutative=True)), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-2)), Symbol('M', commutative=True)), Symbol('A_x', commutative=True)))"], [["add", 3, "Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Add(Pow(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-2)), Symbol('M', commutative=True)), Symbol('A_x', commutative=True)), Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Add(Pow(Function('\\\\hat{x}')(Symbol('M', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('M', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(v)} = \\sin{(v)}, then derive \\int \\frac{\\cos{(\\varepsilon{(v)})}}{\\cos{(\\sin{(v)})}} dv = \\mathbf{P} + v, then obtain \\frac{\\int 1 dv}{\\mathbf{P}} = \\frac{\\mathbf{P} + v}{\\mathbf{P}}", "derivation": "\\varepsilon{(v)} = \\sin{(v)} and \\cos{(\\varepsilon{(v)})} = \\cos{(\\sin{(v)})} and \\frac{\\cos{(\\varepsilon{(v)})}}{\\cos{(\\sin{(v)})}} = 1 and \\int \\frac{\\cos{(\\varepsilon{(v)})}}{\\cos{(\\sin{(v)})}} dv = \\int 1 dv and \\int \\frac{\\cos{(\\varepsilon{(v)})}}{\\cos{(\\sin{(v)})}} dv = \\mathbf{P} + v and \\frac{\\int \\frac{\\cos{(\\varepsilon{(v)})}}{\\cos{(\\sin{(v)})}} dv}{\\mathbf{P}} = \\frac{\\mathbf{P} + v}{\\mathbf{P}} and \\frac{\\int 1 dv}{\\mathbf{P}} = \\frac{\\mathbf{P} + v}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\varepsilon')(Symbol('v', commutative=True))), cos(sin(Symbol('v', commutative=True))))"], [["divide", 2, "cos(sin(Symbol('v', commutative=True)))"], "Equality(Mul(cos(Function('\\\\varepsilon')(Symbol('v', commutative=True))), Pow(cos(sin(Symbol('v', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(cos(Function('\\\\varepsilon')(Symbol('v', commutative=True))), Pow(cos(sin(Symbol('v', commutative=True))), Integer(-1))), Tuple(Symbol('v', commutative=True))), Integral(Integer(1), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(cos(Function('\\\\varepsilon')(Symbol('v', commutative=True))), Pow(cos(sin(Symbol('v', commutative=True))), Integer(-1))), Tuple(Symbol('v', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v', commutative=True)))"], [["divide", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Integral(Mul(cos(Function('\\\\varepsilon')(Symbol('v', commutative=True))), Pow(cos(sin(Symbol('v', commutative=True))), Integer(-1))), Tuple(Symbol('v', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\rho)} = \\log{(\\log{(\\rho)})}, then obtain \\int \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\operatorname{a^{\\dagger}}{(\\rho)}) dA_{z} = \\int \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\log{(\\log{(\\rho)})}) dA_{z}", "derivation": "\\operatorname{a^{\\dagger}}{(\\rho)} = \\log{(\\log{(\\rho)})} and - A_{z} + \\operatorname{a^{\\dagger}}{(\\rho)} = - A_{z} + \\log{(\\log{(\\rho)})} and - 2 A_{z} + \\operatorname{a^{\\dagger}}{(\\rho)} = - 2 A_{z} + \\log{(\\log{(\\rho)})} and \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\operatorname{a^{\\dagger}}{(\\rho)}) = \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\log{(\\log{(\\rho)})}) and \\int \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\operatorname{a^{\\dagger}}{(\\rho)}) dA_{z} = \\int \\frac{\\partial}{\\partial A_{z}} (- 2 A_{z} + \\log{(\\log{(\\rho)})}) dA_{z}", "srepr_derivation": [["get_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\rho', commutative=True)), log(log(Symbol('\\\\rho', commutative=True))))"], [["minus", 1, "Symbol('A_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), log(log(Symbol('\\\\rho', commutative=True)))))"], [["minus", 2, "Symbol('A_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), log(log(Symbol('\\\\rho', commutative=True)))))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), log(log(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('A_z', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), log(log(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(n,E_{x})} = E_{x} - n, then obtain \\int (n + \\phi_{1}{(n,E_{x})})^{2} dn = E_{x} (\\int n dn + \\int \\phi_{1}{(n,E_{x})} dn) + m_{s}", "derivation": "\\phi_{1}{(n,E_{x})} = E_{x} - n and n + \\phi_{1}{(n,E_{x})} = E_{x} and (n + \\phi_{1}{(n,E_{x})})^{2} = E_{x} (n + \\phi_{1}{(n,E_{x})}) and \\int (n + \\phi_{1}{(n,E_{x})})^{2} dn = \\int E_{x} (n + \\phi_{1}{(n,E_{x})}) dn and \\int (n + \\phi_{1}{(n,E_{x})})^{2} dn = E_{x} (\\int n dn + \\int \\phi_{1}{(n,E_{x})} dn) + m_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))"], [["times", 2, "Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Pow(Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(2)), Mul(Symbol('E_x', commutative=True), Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(2)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Symbol('E_x', commutative=True), Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Add(Symbol('n', commutative=True), Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(2)), Tuple(Symbol('n', commutative=True))), Add(Mul(Symbol('E_x', commutative=True), Add(Integral(Symbol('n', commutative=True), Tuple(Symbol('n', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('n', commutative=True))))), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given \\psi{(a)} = \\log{(a)}, then derive \\int \\psi{(a)} da = a \\log{(a)} - a + n, then derive J_{\\varepsilon} + a \\log{(a)} - a = a \\log{(a)} - a + n, then obtain J_{\\varepsilon} + a \\psi{(a)} - a = a \\psi{(a)} - a + n", "derivation": "\\psi{(a)} = \\log{(a)} and \\int \\psi{(a)} da = \\int \\log{(a)} da and \\int \\psi{(a)} da = a \\log{(a)} - a + n and \\int \\log{(a)} da = a \\log{(a)} - a + n and J_{\\varepsilon} + a \\log{(a)} - a = a \\log{(a)} - a + n and J_{\\varepsilon} + a \\psi{(a)} - a = a \\psi{(a)} - a + n", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('n', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Add(Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('a', commutative=True), Function('\\\\psi')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Add(Mul(Symbol('a', commutative=True), Function('\\\\psi')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given C{(v_{2})} = \\sin{(v_{2})}, then derive \\mathbf{A} + v_{2} = \\int \\frac{\\sin{(v_{2})}}{C{(v_{2})}} dv_{2}, then derive \\frac{d}{d v_{2}} \\int 1 dv_{2} = 1, then derive \\frac{\\partial}{\\partial v_{2}} (\\eta + v_{2}) = 1, then obtain - (\\frac{1}{v_{2}})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (\\eta + v_{2}) = 1 - (\\frac{1}{v_{2}})^{v_{2}}", "derivation": "C{(v_{2})} = \\sin{(v_{2})} and 1 = \\frac{\\sin{(v_{2})}}{C{(v_{2})}} and \\int 1 dv_{2} = \\int \\frac{\\sin{(v_{2})}}{C{(v_{2})}} dv_{2} and \\frac{d}{d v_{2}} \\int 1 dv_{2} = \\frac{d}{d v_{2}} \\int \\frac{\\sin{(v_{2})}}{C{(v_{2})}} dv_{2} and \\mathbf{A} + v_{2} = \\int \\frac{\\sin{(v_{2})}}{C{(v_{2})}} dv_{2} and \\frac{d}{d v_{2}} \\int 1 dv_{2} = \\frac{\\partial}{\\partial v_{2}} (\\mathbf{A} + v_{2}) and \\frac{d}{d v_{2}} \\int 1 dv_{2} = 1 and \\frac{\\partial}{\\partial v_{2}} (\\eta + v_{2}) = 1 and - (\\frac{1}{v_{2}})^{v_{2}} + \\frac{\\partial}{\\partial v_{2}} (\\eta + v_{2}) = 1 - (\\frac{1}{v_{2}})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["divide", 1, "Function('C')(Symbol('v_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True))))"], [["integrate", 2, "Symbol('v_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Pow(Function('C')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Function('C')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_2', commutative=True)), Integral(Mul(Pow(Function('C')(Symbol('v_2', commutative=True)), Integer(-1)), sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1))"], [["evaluate_integrals", 7], "Equality(Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1))"], [["minus", 8, "Pow(Pow(Symbol('v_2', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('v_2', commutative=True), Integer(-1)), Symbol('v_2', commutative=True))), Derivative(Add(Symbol('\\\\eta', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Pow(Pow(Symbol('v_2', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(W)} = \\cos{(W)}, then obtain \\frac{d}{d W} \\operatorname{t_{1}}{(W)} \\cos^{W}{(W)} \\frac{d}{d W} \\cos{(W)} \\cos^{W}{(W)} = (\\frac{d}{d W} \\cos{(W)} \\cos^{W}{(W)})^{2}", "derivation": "\\operatorname{t_{1}}{(W)} = \\cos{(W)} and \\operatorname{t_{1}}^{W}{(W)} = \\cos^{W}{(W)} and \\operatorname{t_{1}}{(W)} \\operatorname{t_{1}}^{W}{(W)} = \\operatorname{t_{1}}^{W}{(W)} \\cos{(W)} and \\frac{d}{d W} \\operatorname{t_{1}}{(W)} \\operatorname{t_{1}}^{W}{(W)} = \\frac{d}{d W} \\operatorname{t_{1}}^{W}{(W)} \\cos{(W)} and \\frac{d}{d W} \\operatorname{t_{1}}{(W)} \\operatorname{t_{1}}^{W}{(W)} \\frac{d}{d W} \\operatorname{t_{1}}^{W}{(W)} \\cos{(W)} = (\\frac{d}{d W} \\operatorname{t_{1}}^{W}{(W)} \\cos{(W)})^{2} and \\frac{d}{d W} \\operatorname{t_{1}}{(W)} \\cos^{W}{(W)} \\frac{d}{d W} \\cos{(W)} \\cos^{W}{(W)} = (\\frac{d}{d W} \\cos{(W)} \\cos^{W}{(W)})^{2}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["times", 1, "Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('W', commutative=True)), Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Function('t_1')(Symbol('W', commutative=True)), Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Mul(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Function('t_1')(Symbol('W', commutative=True)), Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Pow(Derivative(Mul(Pow(Function('t_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Derivative(Mul(Function('t_1')(Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Pow(Derivative(Mul(cos(Symbol('W', commutative=True)), Pow(cos(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then derive \\int \\operatorname{E_{x}}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\mathbf{J}_P \\log{(\\mathbf{J}_P)} - \\mathbf{J}_P + v, then obtain - \\mathbf{J}_P \\log{(\\mathbf{J}_P)} + \\mathbf{J}_P - v + \\operatorname{E_{x}}{(\\mathbf{J}_P)} = - \\mathbf{J}_P \\log{(\\mathbf{J}_P)} + \\mathbf{J}_P - v + \\log{(\\mathbf{J}_P)}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\int \\operatorname{E_{x}}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\log{(\\mathbf{J}_P)} d\\mathbf{J}_P and \\operatorname{E_{x}}{(\\mathbf{J}_P)} - \\int \\operatorname{E_{x}}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\log{(\\mathbf{J}_P)} - \\int \\operatorname{E_{x}}{(\\mathbf{J}_P)} d\\mathbf{J}_P and \\int \\operatorname{E_{x}}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\mathbf{J}_P \\log{(\\mathbf{J}_P)} - \\mathbf{J}_P + v and - \\mathbf{J}_P \\log{(\\mathbf{J}_P)} + \\mathbf{J}_P - v + \\operatorname{E_{x}}{(\\mathbf{J}_P)} = - \\mathbf{J}_P \\log{(\\mathbf{J}_P)} + \\mathbf{J}_P - v + \\log{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["minus", 1, "Integral(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integral(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))), Add(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integral(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Function('E_x')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(F_{x})} = \\sin{(F_{x})}, then derive \\int \\operatorname{A_{z}}{(F_{x})} dF_{x} = \\hat{p}_0 - \\cos{(F_{x})}, then derive t_{2} - \\cos{(F_{x})} = \\hat{p}_0 - \\cos{(F_{x})}, then obtain (t_{2} - \\cos{(F_{x})})^{t_{2}} = (\\int \\operatorname{A_{z}}{(F_{x})} dF_{x})^{t_{2}}", "derivation": "\\operatorname{A_{z}}{(F_{x})} = \\sin{(F_{x})} and \\int \\operatorname{A_{z}}{(F_{x})} dF_{x} = \\int \\sin{(F_{x})} dF_{x} and \\int \\operatorname{A_{z}}{(F_{x})} dF_{x} = \\hat{p}_0 - \\cos{(F_{x})} and \\cos{(F_{x})} + \\int \\operatorname{A_{z}}{(F_{x})} dF_{x} = \\hat{p}_0 and \\int \\sin{(F_{x})} dF_{x} = \\hat{p}_0 - \\cos{(F_{x})} and t_{2} - \\cos{(F_{x})} = \\hat{p}_0 - \\cos{(F_{x})} and (t_{2} - \\cos{(F_{x})})^{t_{2}} = (\\hat{p}_0 - \\cos{(F_{x})})^{t_{2}} and (t_{2} - \\cos{(F_{x})})^{t_{2}} = (\\int \\operatorname{A_{z}}{(F_{x})} dF_{x})^{t_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('A_z')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_z')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))))"], [["add", 3, "cos(Symbol('F_x', commutative=True))"], "Equality(Add(cos(Symbol('F_x', commutative=True)), Integral(Function('A_z')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('t_2', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))))"], [["power", 6, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Symbol('t_2', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Symbol('t_2', commutative=True)), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Add(Symbol('t_2', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Symbol('t_2', commutative=True)), Pow(Integral(Function('A_z')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given B{(H)} = e^{\\cos{(H)}} and \\operatorname{m_{s}}{(H)} = H + B{(H)}, then obtain \\frac{d}{d H} (\\operatorname{m_{s}}{(H)} + \\cos{(H)}) = \\frac{d}{d H} (H + B{(H)} + \\cos{(H)})", "derivation": "B{(H)} = e^{\\cos{(H)}} and H + B{(H)} = H + e^{\\cos{(H)}} and H + B{(H)} + \\cos{(H)} = H + e^{\\cos{(H)}} + \\cos{(H)} and \\operatorname{m_{s}}{(H)} = H + B{(H)} and \\operatorname{m_{s}}{(H)} + \\cos{(H)} = H + e^{\\cos{(H)}} + \\cos{(H)} and \\frac{d}{d H} (\\operatorname{m_{s}}{(H)} + \\cos{(H)}) = \\frac{d}{d H} (H + e^{\\cos{(H)}} + \\cos{(H)}) and \\frac{d}{d H} (\\operatorname{m_{s}}{(H)} + \\cos{(H)}) = \\frac{d}{d H} (H + B{(H)} + \\cos{(H)})", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('H', commutative=True)), exp(cos(Symbol('H', commutative=True))))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('B')(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), exp(cos(Symbol('H', commutative=True)))))"], [["add", 2, "cos(Symbol('H', commutative=True))"], "Equality(Add(Symbol('H', commutative=True), Function('B')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), exp(cos(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Function('B')(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('m_s')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), exp(cos(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Function('m_s')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), exp(cos(Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Add(Function('m_s')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Function('B')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(\\phi_1,F_{N})} = \\sin^{\\phi_1}{(F_{N})} and z{(\\phi_1,F_{N})} = \\int \\sin^{\\phi_1}{(F_{N})} d\\phi_1, then obtain z{(\\phi_1,F_{N})} - \\log{(c)} = - \\log{(c)} + \\int \\sin^{\\phi_1}{(F_{N})} d\\phi_1", "derivation": "\\hat{p}{(\\phi_1,F_{N})} = \\sin^{\\phi_1}{(F_{N})} and \\int \\hat{p}{(\\phi_1,F_{N})} d\\phi_1 = \\int \\sin^{\\phi_1}{(F_{N})} d\\phi_1 and z{(\\phi_1,F_{N})} = \\int \\sin^{\\phi_1}{(F_{N})} d\\phi_1 and z{(\\phi_1,F_{N})} = \\int \\hat{p}{(\\phi_1,F_{N})} d\\phi_1 and z{(\\phi_1,F_{N})} - \\log{(c)} = - \\log{(c)} + \\int \\hat{p}{(\\phi_1,F_{N})} d\\phi_1 and z{(\\phi_1,F_{N})} - \\log{(c)} = - \\log{(c)} + \\int \\sin^{\\phi_1}{(F_{N})} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Pow(sin(Symbol('F_N', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(sin(Symbol('F_N', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Integral(Pow(sin(Symbol('F_N', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 4, "log(Symbol('c', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), log(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('c', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), log(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('c', commutative=True))), Integral(Pow(sin(Symbol('F_N', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given z{(b)} = \\cos{(b)} and t{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}, then derive t{(\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda})}, then obtain - \\frac{\\sin{(\\Psi_{\\lambda})}}{z^{2}{(b)}} = \\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{z^{2}{(b)}}", "derivation": "z{(b)} = \\cos{(b)} and z^{2}{(b)} = z{(b)} \\cos{(b)} and t{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})} and t{(\\Psi_{\\lambda})} = - \\sin{(\\Psi_{\\lambda})} and \\frac{t{(\\Psi_{\\lambda})}}{z{(b)} \\cos{(b)}} = \\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{z{(b)} \\cos{(b)}} and - \\frac{\\sin{(\\Psi_{\\lambda})}}{z{(b)} \\cos{(b)}} = \\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{z{(b)} \\cos{(b)}} and - \\frac{\\sin{(\\Psi_{\\lambda})}}{z^{2}{(b)}} = \\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{z^{2}{(b)}}", "srepr_derivation": [["get_premise", "Equality(Function('z')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["times", 1, "Function('z')(Symbol('b', commutative=True))"], "Equality(Pow(Function('z')(Symbol('b', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True))))"], ["get_premise", "Equality(Function('t')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('t')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Mul(Function('z')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], "Equality(Mul(Function('t')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Function('z')(Symbol('b', commutative=True)), Integer(-1)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Mul(Pow(Function('z')(Symbol('b', commutative=True)), Integer(-1)), Pow(cos(Symbol('b', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Function('z')(Symbol('b', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Mul(Pow(Function('z')(Symbol('b', commutative=True)), Integer(-1)), Pow(cos(Symbol('b', commutative=True)), Integer(-1)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Integer(-1), Pow(Function('z')(Symbol('b', commutative=True)), Integer(-2)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Function('z')(Symbol('b', commutative=True)), Integer(-2)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\bar{\\h}{(f)} = \\sin{(f)}, then obtain (\\frac{1}{\\sin{(f)}})^{f} e^{\\bar{\\h}{(f)}} = (\\frac{1}{\\sin{(f)}})^{f} e^{\\sin{(f)}}", "derivation": "\\bar{\\h}{(f)} = \\sin{(f)} and \\frac{\\bar{\\h}{(f)}}{\\sin{(f)}} = 1 and \\frac{1}{\\sin{(f)}} = \\frac{1}{\\bar{\\h}{(f)}} and e^{\\bar{\\h}{(f)}} = e^{\\sin{(f)}} and (\\frac{1}{\\bar{\\h}{(f)}})^{f} e^{\\bar{\\h}{(f)}} = (\\frac{1}{\\bar{\\h}{(f)}})^{f} e^{\\sin{(f)}} and (\\frac{1}{\\sin{(f)}})^{f} e^{\\bar{\\h}{(f)}} = (\\frac{1}{\\sin{(f)}})^{f} e^{\\sin{(f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["divide", 1, "sin(Symbol('f', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Function('\\\\hbar')(Symbol('f', commutative=True))"], "Equality(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('f', commutative=True)), Integer(-1)))"], [["exp", 1], "Equality(exp(Function('\\\\hbar')(Symbol('f', commutative=True))), exp(sin(Symbol('f', commutative=True))))"], [["times", 4, "Pow(Pow(Function('\\\\hbar')(Symbol('f', commutative=True)), Integer(-1)), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\hbar')(Symbol('f', commutative=True)), Integer(-1)), Symbol('f', commutative=True)), exp(Function('\\\\hbar')(Symbol('f', commutative=True)))), Mul(Pow(Pow(Function('\\\\hbar')(Symbol('f', commutative=True)), Integer(-1)), Symbol('f', commutative=True)), exp(sin(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Symbol('f', commutative=True)), exp(Function('\\\\hbar')(Symbol('f', commutative=True)))), Mul(Pow(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Symbol('f', commutative=True)), exp(sin(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(E_{\\lambda})} = \\sin{(E_{\\lambda})}, then derive \\frac{d}{d E_{\\lambda}} \\mathbf{p}{(E_{\\lambda})} = \\cos{(E_{\\lambda})}, then obtain e^{\\cos{(E_{\\lambda})}} = e^{\\frac{d}{d E_{\\lambda}} \\sin{(E_{\\lambda})}}", "derivation": "\\mathbf{p}{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and \\frac{d}{d E_{\\lambda}} \\mathbf{p}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\sin{(E_{\\lambda})} and \\frac{d}{d E_{\\lambda}} \\mathbf{p}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\cos{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\sin{(E_{\\lambda})} and e^{\\cos{(E_{\\lambda})}} = e^{\\frac{d}{d E_{\\lambda}} \\sin{(E_{\\lambda})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('E_{\\\\lambda}', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(sin(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('E_{\\\\lambda}', commutative=True)), Derivative(sin(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(cos(Symbol('E_{\\\\lambda}', commutative=True))), exp(Derivative(sin(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(n)} = \\sin{(n)} and J{(n)} = \\sin{(n)}, then obtain ((2 \\hat{\\mathbf{r}}{(n)} - \\sin{(n)})^{n})^{n} - J{(n)} \\sin{(n)} = (J^{n}{(n)})^{n} - J{(n)} \\sin{(n)}", "derivation": "\\hat{\\mathbf{r}}{(n)} = \\sin{(n)} and \\hat{\\mathbf{r}}^{n}{(n)} = \\sin^{n}{(n)} and (\\hat{\\mathbf{r}}^{n}{(n)})^{n} = (\\sin^{n}{(n)})^{n} and 2 \\hat{\\mathbf{r}}{(n)} - \\sin{(n)} = \\hat{\\mathbf{r}}{(n)} and J{(n)} = \\sin{(n)} and (\\hat{\\mathbf{r}}^{n}{(n)})^{n} = (J^{n}{(n)})^{n} and (\\hat{\\mathbf{r}}^{n}{(n)})^{n} - J{(n)} \\sin{(n)} = (J^{n}{(n)})^{n} - J{(n)} \\sin{(n)} and ((2 \\hat{\\mathbf{r}}{(n)} - \\sin{(n)})^{n})^{n} - J{(n)} \\sin{(n)} = (J^{n}{(n)})^{n} - J{(n)} \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), sin(Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True)))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Pow(Function('J')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["add", 6, "Mul(Integer(-1), Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], "Equality(Add(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))), Add(Pow(Pow(Function('J')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Pow(Pow(Add(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('n', commutative=True))), Mul(Integer(-1), sin(Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))), Add(Pow(Pow(Function('J')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(C_{1},v)} = v^{C_{1}}, then obtain - \\frac{\\int (v + \\operatorname{v_{x}}{(C_{1},v)}) dv - \\frac{1}{v}}{v} + \\frac{1}{v} = - \\frac{\\int (v + v^{C_{1}}) dv - \\frac{1}{v}}{v} + \\frac{1}{v}", "derivation": "\\operatorname{v_{x}}{(C_{1},v)} = v^{C_{1}} and v + \\operatorname{v_{x}}{(C_{1},v)} = v + v^{C_{1}} and \\int (v + \\operatorname{v_{x}}{(C_{1},v)}) dv = \\int (v + v^{C_{1}}) dv and \\int (v + \\operatorname{v_{x}}{(C_{1},v)}) dv - \\frac{1}{v} = \\int (v + v^{C_{1}}) dv - \\frac{1}{v} and - \\frac{\\int (v + \\operatorname{v_{x}}{(C_{1},v)}) dv - \\frac{1}{v}}{v} = - \\frac{\\int (v + v^{C_{1}}) dv - \\frac{1}{v}}{v} and - \\frac{\\int (v + \\operatorname{v_{x}}{(C_{1},v)}) dv - \\frac{1}{v}}{v} + \\frac{1}{v} = - \\frac{\\int (v + v^{C_{1}}) dv - \\frac{1}{v}}{v} + \\frac{1}{v}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True)))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Symbol('v', commutative=True), Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["minus", 3, "Pow(Symbol('v', commutative=True), Integer(-1))"], "Equality(Add(Integral(Add(Symbol('v', commutative=True), Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Add(Integral(Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))))"], [["times", 4, "Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('v', commutative=True), Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1))))))"], [["minus", 5, "Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('v', commutative=True), Function('v_x')(Symbol('C_1', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1))))), Pow(Symbol('v', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1))))), Pow(Symbol('v', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain \\cos{(x^\\prime)} + 1 = \\cos{(x^\\prime)} + \\frac{\\int x^\\prime \\cos{(x^\\prime)} dx^\\prime}{\\int x^\\prime \\operatorname{c_{0}}{(x^\\prime)} dx^\\prime}", "derivation": "\\operatorname{c_{0}}{(x^\\prime)} = \\cos{(x^\\prime)} and x^\\prime \\operatorname{c_{0}}{(x^\\prime)} = x^\\prime \\cos{(x^\\prime)} and \\int x^\\prime \\operatorname{c_{0}}{(x^\\prime)} dx^\\prime = \\int x^\\prime \\cos{(x^\\prime)} dx^\\prime and 1 = \\frac{\\int x^\\prime \\cos{(x^\\prime)} dx^\\prime}{\\int x^\\prime \\operatorname{c_{0}}{(x^\\prime)} dx^\\prime} and \\cos{(x^\\prime)} + 1 = \\cos{(x^\\prime)} + \\frac{\\int x^\\prime \\cos{(x^\\prime)} dx^\\prime}{\\int x^\\prime \\operatorname{c_{0}}{(x^\\prime)} dx^\\prime}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('c_0')(Symbol('x^\\\\prime', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('c_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 3, "Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('c_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('c_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["add", 4, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(cos(Symbol('x^\\\\prime', commutative=True)), Integer(1)), Add(cos(Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Function('c_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{d})} = \\sin{(C_{d})}, then obtain C_{d} + \\frac{d}{d C_{d}} \\Psi^{\\dagger}{(C_{d})} \\sin{(C_{d})} - \\frac{d}{d C_{d}} \\sin^{2}{(C_{d})} = C_{d}", "derivation": "\\Psi^{\\dagger}{(C_{d})} = \\sin{(C_{d})} and \\Psi^{\\dagger}{(C_{d})} \\sin{(C_{d})} = \\sin^{2}{(C_{d})} and \\frac{d}{d C_{d}} \\Psi^{\\dagger}{(C_{d})} \\sin{(C_{d})} = \\frac{d}{d C_{d}} \\sin^{2}{(C_{d})} and \\frac{d}{d C_{d}} \\Psi^{\\dagger}{(C_{d})} \\sin{(C_{d})} - \\frac{d}{d C_{d}} \\sin^{2}{(C_{d})} = 0 and C_{d} + \\frac{d}{d C_{d}} \\Psi^{\\dagger}{(C_{d})} \\sin{(C_{d})} - \\frac{d}{d C_{d}} \\sin^{2}{(C_{d})} = C_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["times", 1, "sin(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Pow(sin(Symbol('C_d', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('C_d', commutative=True)), Integer(2)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Pow(sin(Symbol('C_d', commutative=True)), Integer(2)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(sin(Symbol('C_d', commutative=True)), Integer(2)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Integer(0))"], [["add", 4, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Derivative(Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(sin(Symbol('C_d', commutative=True)), Integer(2)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Symbol('C_d', commutative=True))"]]}, {"prompt": "Given H{(C_{1},\\mu)} = C_{1} \\mu, then obtain - (C_{1} \\mu)^{C_{1}} + H{(C_{1},\\mu)} = C_{1} \\mu - (C_{1} \\mu)^{C_{1}}", "derivation": "H{(C_{1},\\mu)} = C_{1} \\mu and H^{C_{1}}{(C_{1},\\mu)} = (C_{1} \\mu)^{C_{1}} and H{(C_{1},\\mu)} - H^{C_{1}}{(C_{1},\\mu)} = C_{1} \\mu - H^{C_{1}}{(C_{1},\\mu)} and - (C_{1} \\mu)^{C_{1}} + H{(C_{1},\\mu)} = C_{1} \\mu - (C_{1} \\mu)^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)), Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))"], [["minus", 1, "Pow(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True))"], "Equality(Add(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))), Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True))), Function('H')(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('C_1', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given y{(\\rho_b,z)} = \\rho_b z, then obtain \\rho_b + y^{\\rho_b}{(\\rho_b,z)} + 1 = \\rho_b + (\\rho_b z)^{\\rho_b} + 1", "derivation": "y{(\\rho_b,z)} = \\rho_b z and y^{\\rho_b}{(\\rho_b,z)} = (\\rho_b z)^{\\rho_b} and \\rho_b + y^{\\rho_b}{(\\rho_b,z)} = \\rho_b + (\\rho_b z)^{\\rho_b} and \\rho_b + y^{\\rho_b}{(\\rho_b,z)} + 1 = \\rho_b + (\\rho_b z)^{\\rho_b} + 1", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Pow(Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Pow(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Pow(Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Integer(1)), Add(Symbol('\\\\rho_b', commutative=True), Pow(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(r_{0})} = \\cos{(r_{0})} and \\nabla{(y,\\hat{X})} = - \\sin{(\\hat{X} - y)}, then obtain \\frac{r_{0} + \\nabla{(y,\\hat{X})} + \\operatorname{f_{\\mathbf{v}}}{(r_{0})}}{y} = \\frac{r_{0} + \\operatorname{f_{\\mathbf{v}}}{(r_{0})} - \\sin{(\\hat{X} - y)}}{y}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(r_{0})} = \\cos{(r_{0})} and \\nabla{(y,\\hat{X})} = - \\sin{(\\hat{X} - y)} and r_{0} + \\nabla{(y,\\hat{X})} + \\cos{(r_{0})} = r_{0} - \\sin{(\\hat{X} - y)} + \\cos{(r_{0})} and \\frac{r_{0} + \\nabla{(y,\\hat{X})} + \\cos{(r_{0})}}{y} = \\frac{r_{0} - \\sin{(\\hat{X} - y)} + \\cos{(r_{0})}}{y} and \\frac{r_{0} + \\nabla{(y,\\hat{X})} + \\operatorname{f_{\\mathbf{v}}}{(r_{0})}}{y} = \\frac{r_{0} + \\operatorname{f_{\\mathbf{v}}}{(r_{0})} - \\sin{(\\hat{X} - y)}}{y}", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], ["get_premise", "Equality(Function('\\\\nabla')(Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["add", 2, "Add(Symbol('r_0', commutative=True), cos(Symbol('r_0', commutative=True)))"], "Equality(Add(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), cos(Symbol('r_0', commutative=True))))"], [["divide", 3, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), cos(Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Function('\\\\nabla')(Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('r_0', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('r_0', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(Z,\\hbar)} = \\sin{(Z \\hbar)}, then obtain \\operatorname{A_{z}}^{3}{(Z,\\hbar)} \\sin{(Z \\hbar)} = \\operatorname{A_{z}}^{2}{(Z,\\hbar)} \\sin^{2}{(Z \\hbar)}", "derivation": "\\operatorname{A_{z}}{(Z,\\hbar)} = \\sin{(Z \\hbar)} and \\operatorname{A_{z}}{(Z,\\hbar)} \\sin{(Z \\hbar)} = \\sin^{2}{(Z \\hbar)} and \\operatorname{A_{z}}^{2}{(Z,\\hbar)} \\sin^{2}{(Z \\hbar)} = \\operatorname{A_{z}}{(Z,\\hbar)} \\sin^{3}{(Z \\hbar)} and \\operatorname{A_{z}}^{3}{(Z,\\hbar)} \\sin{(Z \\hbar)} = \\operatorname{A_{z}}{(Z,\\hbar)} \\sin^{3}{(Z \\hbar)} and \\operatorname{A_{z}}^{3}{(Z,\\hbar)} \\sin{(Z \\hbar)} = \\operatorname{A_{z}}^{2}{(Z,\\hbar)} \\sin^{2}{(Z \\hbar)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["times", 1, "sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)))), Pow(sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))))"], "Equality(Mul(Pow(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(2))), Mul(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(3)), sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(3)), sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Function('A_z')(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('Z', commutative=True), Symbol('\\\\hbar', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{D})} = e^{e^{\\mathbf{D}}}, then obtain \\sin{((\\hat{H}{(\\mathbf{D})} - e^{\\mathbf{D}})^{\\mathbf{D}})} = \\sin{((- e^{\\mathbf{D}} + e^{e^{\\mathbf{D}}})^{\\mathbf{D}})}", "derivation": "\\hat{H}{(\\mathbf{D})} = e^{e^{\\mathbf{D}}} and \\hat{H}{(\\mathbf{D})} - e^{\\mathbf{D}} = - e^{\\mathbf{D}} + e^{e^{\\mathbf{D}}} and (\\hat{H}{(\\mathbf{D})} - e^{\\mathbf{D}})^{\\mathbf{D}} = (- e^{\\mathbf{D}} + e^{e^{\\mathbf{D}}})^{\\mathbf{D}} and \\sin{((\\hat{H}{(\\mathbf{D})} - e^{\\mathbf{D}})^{\\mathbf{D}})} = \\sin{((- e^{\\mathbf{D}} + e^{e^{\\mathbf{D}}})^{\\mathbf{D}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), exp(exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))), exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))), exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Add(Function('\\\\hat{H}')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True))), sin(Pow(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))), exp(exp(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(I,\\theta)} = e^{I + \\theta}, then derive y + \\operatorname{y^{\\prime}}{(I,\\theta)} = \\delta + e^{I + \\theta}, then obtain y + \\operatorname{y^{\\prime}}{(I,\\theta)} - e^{I + \\theta} = y", "derivation": "\\operatorname{y^{\\prime}}{(I,\\theta)} = e^{I + \\theta} and \\frac{\\partial}{\\partial \\theta} \\operatorname{y^{\\prime}}{(I,\\theta)} = \\frac{\\partial}{\\partial \\theta} e^{I + \\theta} and \\int \\frac{\\partial}{\\partial \\theta} \\operatorname{y^{\\prime}}{(I,\\theta)} d\\theta = \\int \\frac{\\partial}{\\partial \\theta} e^{I + \\theta} d\\theta and y + \\operatorname{y^{\\prime}}{(I,\\theta)} = \\delta + e^{I + \\theta} and y + e^{I + \\theta} = \\delta + e^{I + \\theta} and y + \\operatorname{y^{\\prime}}{(I,\\theta)} = y + e^{I + \\theta} and y + \\operatorname{y^{\\prime}}{(I,\\theta)} - \\frac{\\partial}{\\partial \\theta} e^{I + \\theta} = y + e^{I + \\theta} - \\frac{\\partial}{\\partial \\theta} e^{I + \\theta} and y + \\operatorname{y^{\\prime}}{(I,\\theta)} - e^{I + \\theta} = y", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('y', commutative=True), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('y', commutative=True), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)))), Add(Symbol('\\\\delta', commutative=True), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('y', commutative=True), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('y', commutative=True), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["minus", 6, "Derivative(exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Add(Symbol('y', commutative=True), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Derivative(exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Add(Symbol('y', commutative=True), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Derivative(exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 7], "Equality(Add(Symbol('y', commutative=True), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('I', commutative=True), Symbol('\\\\theta', commutative=True))))), Symbol('y', commutative=True))"]]}, {"prompt": "Given \\mathbf{B}{(W)} = \\cos{(W)}, then derive A_{2}^{W} = (\\int (- \\mathbf{B}{(W)} + \\cos{(W)})^{W} dW)^{W}, then derive A_{2}^{W} = (g^{\\prime}_{\\varepsilon})^{W}, then obtain \\frac{\\partial}{\\partial A_{2}} A_{2}^{W} = \\frac{\\partial}{\\partial A_{2}} (g^{\\prime}_{\\varepsilon})^{W}", "derivation": "\\mathbf{B}{(W)} = \\cos{(W)} and 0 = - \\mathbf{B}{(W)} + \\cos{(W)} and 0^{W} = (- \\mathbf{B}{(W)} + \\cos{(W)})^{W} and \\int 0^{W} dW = \\int (- \\mathbf{B}{(W)} + \\cos{(W)})^{W} dW and (\\int 0^{W} dW)^{W} = (\\int (- \\mathbf{B}{(W)} + \\cos{(W)})^{W} dW)^{W} and A_{2}^{W} = (\\int (- \\mathbf{B}{(W)} + \\cos{(W)})^{W} dW)^{W} and A_{2}^{W} = (\\int 0^{W} dW)^{W} and A_{2}^{W} = (g^{\\prime}_{\\varepsilon})^{W} and \\frac{\\partial}{\\partial A_{2}} A_{2}^{W} = \\frac{\\partial}{\\partial A_{2}} (g^{\\prime}_{\\varepsilon})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Pow(Integer(0), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Symbol('A_2', commutative=True), Symbol('W', commutative=True)), Pow(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Symbol('A_2', commutative=True), Symbol('W', commutative=True)), Pow(Integral(Pow(Integer(0), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Pow(Symbol('A_2', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 8, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Pow(Symbol('A_2', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then derive \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = t + e^{L_{\\varepsilon}}, then obtain t + z{(L_{\\varepsilon})} + \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = 2 t + 2 z{(L_{\\varepsilon})}", "derivation": "z{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int e^{L_{\\varepsilon}} dL_{\\varepsilon} and \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = t + e^{L_{\\varepsilon}} and \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = t + z{(L_{\\varepsilon})} and t + \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = 2 t + z{(L_{\\varepsilon})} and t + z{(L_{\\varepsilon})} + \\int z{(L_{\\varepsilon})} dL_{\\varepsilon} = 2 t + 2 z{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('t', commutative=True), Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["add", 4, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Integral(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["add", 5, "Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('t', commutative=True), Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(2), Function('z')(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then derive \\log{(\\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda} \\log{(\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + y)}, then obtain \\log{(\\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda} \\eta{(\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + y)}", "derivation": "\\eta{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = \\int \\log{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and \\log{(\\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} = \\log{(\\int \\log{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} and \\log{(\\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda} \\log{(\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + y)} and \\log{(\\int \\eta{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda} \\eta{(\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), log(Integral(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), log(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(log(Integral(Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), log(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\eta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(A_{2},\\eta^{\\prime})} = A_{2} - \\eta^{\\prime}, then obtain - \\frac{\\mathbf{E}{(A_{2},\\eta^{\\prime})}}{- A_{2} + \\eta^{\\prime}} = 1", "derivation": "\\mathbf{E}{(A_{2},\\eta^{\\prime})} = A_{2} - \\eta^{\\prime} and - \\mathbf{E}{(A_{2},\\eta^{\\prime})} = - A_{2} + \\eta^{\\prime} and -1 = \\frac{- A_{2} + \\eta^{\\prime}}{\\mathbf{E}{(A_{2},\\eta^{\\prime})}} and - \\frac{\\mathbf{E}{(A_{2},\\eta^{\\prime})}}{- A_{2} + \\eta^{\\prime}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["divide", 2, "Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(-1), Mul(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\rho_{b}{(E_{x})} = \\frac{d}{d E_{x}} e^{E_{x}}, then derive \\int \\rho_{b}{(E_{x})} dE_{x} = \\theta_2 + e^{E_{x}}, then obtain e^{E_{x}} \\int \\rho_{b}{(E_{x})} dE_{x} = (\\theta_2 + e^{E_{x}}) e^{E_{x}}", "derivation": "\\rho_{b}{(E_{x})} = \\frac{d}{d E_{x}} e^{E_{x}} and \\int \\rho_{b}{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} e^{E_{x}} dE_{x} and \\int \\rho_{b}{(E_{x})} dE_{x} = \\theta_2 + e^{E_{x}} and e^{E_{x}} \\int \\rho_{b}{(E_{x})} dE_{x} = e^{E_{x}} \\int \\frac{d}{d E_{x}} e^{E_{x}} dE_{x} and \\theta_2 + e^{E_{x}} = \\int \\frac{d}{d E_{x}} e^{E_{x}} dE_{x} and e^{E_{x}} \\int \\rho_{b}{(E_{x})} dE_{x} = (\\theta_2 + e^{E_{x}}) e^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('E_x', commutative=True)), Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho_b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), exp(Symbol('E_x', commutative=True))))"], [["times", 2, "exp(Symbol('E_x', commutative=True))"], "Equality(Mul(exp(Symbol('E_x', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Mul(exp(Symbol('E_x', commutative=True)), Integral(Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\theta_2', commutative=True), exp(Symbol('E_x', commutative=True))), Integral(Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(exp(Symbol('E_x', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Mul(Add(Symbol('\\\\theta_2', commutative=True), exp(Symbol('E_x', commutative=True))), exp(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given I{(G)} = e^{\\sin{(G)}} and \\operatorname{n_{1}}{(A,\\dot{y})} = A + \\dot{y}, then obtain - 2 I{(G)} + \\operatorname{n_{1}}{(A,\\dot{y})} = A + \\dot{y} - 2 I{(G)}", "derivation": "I{(G)} = e^{\\sin{(G)}} and \\operatorname{n_{1}}{(A,\\dot{y})} = A + \\dot{y} and \\operatorname{n_{1}}{(A,\\dot{y})} - e^{\\sin{(G)}} = A + \\dot{y} - e^{\\sin{(G)}} and \\operatorname{n_{1}}{(A,\\dot{y})} - 2 e^{\\sin{(G)}} = A + \\dot{y} - 2 e^{\\sin{(G)}} and - 2 I{(G)} + \\operatorname{n_{1}}{(A,\\dot{y})} = A + \\dot{y} - 2 I{(G)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('G', commutative=True)), exp(sin(Symbol('G', commutative=True))))"], ["get_premise", "Equality(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 2, "Mul(Integer(-1), exp(sin(Symbol('G', commutative=True))))"], "Equality(Add(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('G', commutative=True))))), Add(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), exp(sin(Symbol('G', commutative=True))))))"], [["add", 3, "Mul(Integer(-1), exp(sin(Symbol('G', commutative=True))))"], "Equality(Add(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Integer(2), exp(sin(Symbol('G', commutative=True))))), Add(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Integer(2), exp(sin(Symbol('G', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Function('I')(Symbol('G', commutative=True))), Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('A', commutative=True), Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Integer(2), Function('I')(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\psi^*)} = \\frac{d}{d \\psi^*} \\sin{(\\psi^*)}, then derive \\mathbf{J}_P{(\\psi^*)} = \\cos{(\\psi^*)}, then obtain \\frac{d}{d \\psi^*} 1 = \\frac{d}{d \\psi^*} (\\frac{\\cos{(\\psi^*)}}{\\mathbf{J}_P{(\\psi^*)}})^{\\psi^*}", "derivation": "\\mathbf{J}_P{(\\psi^*)} = \\frac{d}{d \\psi^*} \\sin{(\\psi^*)} and 1 = \\frac{\\frac{d}{d \\psi^*} \\sin{(\\psi^*)}}{\\mathbf{J}_P{(\\psi^*)}} and \\mathbf{J}_P{(\\psi^*)} = \\cos{(\\psi^*)} and \\frac{d}{d \\psi^*} \\sin{(\\psi^*)} = \\cos{(\\psi^*)} and 1 = \\frac{\\cos{(\\psi^*)}}{\\mathbf{J}_P{(\\psi^*)}} and 1 = (\\frac{\\cos{(\\psi^*)}}{\\mathbf{J}_P{(\\psi^*)}})^{\\psi^*} and \\frac{d}{d \\psi^*} 1 = \\frac{d}{d \\psi^*} (\\frac{\\cos{(\\psi^*)}}{\\mathbf{J}_P{(\\psi^*)}})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), Derivative(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["divide", 1, "Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), cos(Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi^*', commutative=True))))"], [["power", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 6, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(l)} = \\log{(l)}, then obtain \\int (C{(l)} + \\log{(l)} + 1) dl = \\int (2 \\log{(l)} + 1) dl", "derivation": "C{(l)} = \\log{(l)} and C{(l)} + 1 = \\log{(l)} + 1 and 2 C{(l)} + 1 = C{(l)} + \\log{(l)} + 1 and 2 C{(l)} + 1 = 2 \\log{(l)} + 1 and C{(l)} + \\log{(l)} + 1 = 2 \\log{(l)} + 1 and \\int (C{(l)} + \\log{(l)} + 1) dl = \\int (2 \\log{(l)} + 1) dl", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('C')(Symbol('l', commutative=True)), Integer(1)), Add(log(Symbol('l', commutative=True)), Integer(1)))"], [["add", 1, "Add(Function('C')(Symbol('l', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(2), Function('C')(Symbol('l', commutative=True))), Integer(1)), Add(Function('C')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('C')(Symbol('l', commutative=True))), Integer(1)), Add(Mul(Integer(2), log(Symbol('l', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('C')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)), Integer(1)), Add(Mul(Integer(2), log(Symbol('l', commutative=True))), Integer(1)))"], [["integrate", 5, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Function('C')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)), Integer(1)), Tuple(Symbol('l', commutative=True))), Integral(Add(Mul(Integer(2), log(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(M,u)} = M u and \\operatorname{V_{\\mathbf{B}}}{(M,u)} = M u, then obtain 1 = \\frac{\\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(M,u)}}{M u} du}{\\int \\frac{\\mathbf{J}_M{(M,u)}}{M u} du}", "derivation": "\\mathbf{J}_M{(M,u)} = M u and \\operatorname{V_{\\mathbf{B}}}{(M,u)} = M u and \\mathbf{J}_M{(M,u)} = \\operatorname{V_{\\mathbf{B}}}{(M,u)} and \\frac{\\mathbf{J}_M{(M,u)}}{M u} = \\frac{\\operatorname{V_{\\mathbf{B}}}{(M,u)}}{M u} and \\int \\frac{\\mathbf{J}_M{(M,u)}}{M u} du = \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(M,u)}}{M u} du and \\int \\frac{\\mathbf{J}_M{(M,u)}}{M u} du = \\int 1 du and \\frac{\\int \\frac{\\mathbf{J}_M{(M,u)}}{M u} du}{\\int 1 du} = \\frac{\\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(M,u)}}{M u} du}{\\int 1 du} and 1 = \\frac{\\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(M,u)}}{M u} du}{\\int \\frac{\\mathbf{J}_M{(M,u)}}{M u} du}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True)))"], [["divide", 3, "Mul(Symbol('M', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"], [["divide", 5, "Integral(Integer(1), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Integral(Integer(1), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Integral(Integer(1), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integer(1), Mul(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Pow(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{v}{(a^{\\dagger},\\delta)} = \\sin{(\\delta + a^{\\dagger})} and \\pi{(a^{\\dagger})} = a^{\\dagger}, then obtain \\cos{(\\int \\mathbf{v}{(a^{\\dagger},\\delta)} d\\pi{(a^{\\dagger})})} = \\cos{(\\int \\sin{(\\delta + a^{\\dagger})} d\\pi{(a^{\\dagger})})}", "derivation": "\\mathbf{v}{(a^{\\dagger},\\delta)} = \\sin{(\\delta + a^{\\dagger})} and \\int \\mathbf{v}{(a^{\\dagger},\\delta)} da^{\\dagger} = \\int \\sin{(\\delta + a^{\\dagger})} da^{\\dagger} and \\pi{(a^{\\dagger})} = a^{\\dagger} and \\cos{(\\int \\mathbf{v}{(a^{\\dagger},\\delta)} da^{\\dagger})} = \\cos{(\\int \\sin{(\\delta + a^{\\dagger})} da^{\\dagger})} and \\cos{(\\int \\mathbf{v}{(a^{\\dagger},\\delta)} d\\pi{(a^{\\dagger})})} = \\cos{(\\int \\sin{(\\delta + a^{\\dagger})} d\\pi{(a^{\\dagger})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), sin(Add(Symbol('\\\\delta', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Add(Symbol('\\\\delta', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), cos(Integral(sin(Add(Symbol('\\\\delta', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(cos(Integral(Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Function('\\\\pi')(Symbol('a^{\\\\dagger}', commutative=True))))), cos(Integral(sin(Add(Symbol('\\\\delta', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Function('\\\\pi')(Symbol('a^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\varphi (\\hat{\\mathbf{r}} + \\mathbf{A}), then derive \\frac{\\partial}{\\partial \\varphi} z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\hat{\\mathbf{r}} + \\mathbf{A}, then obtain \\mathbf{A} + z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\mathbf{A} + \\varphi \\frac{\\partial}{\\partial \\varphi} z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})}", "derivation": "z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\varphi (\\hat{\\mathbf{r}} + \\mathbf{A}) and \\mathbf{A} + z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\mathbf{A} + \\varphi (\\hat{\\mathbf{r}} + \\mathbf{A}) and \\frac{\\partial}{\\partial \\varphi} z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\frac{\\partial}{\\partial \\varphi} \\varphi (\\hat{\\mathbf{r}} + \\mathbf{A}) and \\frac{\\partial}{\\partial \\varphi} z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\hat{\\mathbf{r}} + \\mathbf{A} and \\mathbf{A} + z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})} = \\mathbf{A} + \\varphi \\frac{\\partial}{\\partial \\varphi} z{(\\hat{\\mathbf{r}},\\varphi,\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Derivative(Function('z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(v_{2})} = \\cos{(v_{2})}, then obtain 0 = \\frac{(- \\frac{\\mathbf{J}_M{(v_{2})}}{v_{2}} + \\frac{\\cos{(v_{2})}}{v_{2}}) \\cos{(v_{2})}}{v_{2}}", "derivation": "\\mathbf{J}_M{(v_{2})} = \\cos{(v_{2})} and \\frac{\\mathbf{J}_M{(v_{2})}}{v_{2}} = \\frac{\\cos{(v_{2})}}{v_{2}} and 0 = - \\frac{\\mathbf{J}_M{(v_{2})}}{v_{2}} + \\frac{\\cos{(v_{2})}}{v_{2}} and 0 = \\frac{(- \\frac{\\mathbf{J}_M{(v_{2})}}{v_{2}} + \\frac{\\cos{(v_{2})}}{v_{2}}) \\cos{(v_{2})}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["divide", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('v_2', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('v_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('v_2', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('v_2', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Symbol('v_2', commutative=True)))), cos(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mu,\\chi)} = \\chi + \\cos{(\\mu)}, then derive (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)}) \\int \\mu \\Psi^{\\dagger}{(\\mu,\\chi)} d\\chi = (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)})^{2}, then obtain (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)}) \\int \\mu (\\chi + \\cos{(\\mu)}) d\\chi = (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)})^{2}", "derivation": "\\Psi^{\\dagger}{(\\mu,\\chi)} = \\chi + \\cos{(\\mu)} and \\mu \\Psi^{\\dagger}{(\\mu,\\chi)} = \\mu (\\chi + \\cos{(\\mu)}) and \\int \\mu \\Psi^{\\dagger}{(\\mu,\\chi)} d\\chi = \\int \\mu (\\chi + \\cos{(\\mu)}) d\\chi and (\\int \\mu (\\chi + \\cos{(\\mu)}) d\\chi) \\int \\mu \\Psi^{\\dagger}{(\\mu,\\chi)} d\\chi = (\\int \\mu (\\chi + \\cos{(\\mu)}) d\\chi)^{2} and (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)}) \\int \\mu \\Psi^{\\dagger}{(\\mu,\\chi)} d\\chi = (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)})^{2} and (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)}) \\int \\mu (\\chi + \\cos{(\\mu)}) d\\chi = (F_{g} + \\frac{\\chi^{2} \\mu}{2} + \\chi \\mu \\cos{(\\mu)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 3, "Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Integral(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Integral(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\Psi_{nl}{(M,E)} = M^{E} and \\hat{\\mathbf{r}}{(M,E)} = M^{E} and \\theta_{2}{(E,M)} = (E + \\Psi_{nl}{(M,E)})^{E}, then obtain \\hat{\\mathbf{r}}{(M,E)} + 2 \\theta_{2}{(E,M)} = M^{E} + 2 \\theta_{2}{(E,M)}", "derivation": "\\Psi_{nl}{(M,E)} = M^{E} and \\hat{\\mathbf{r}}{(M,E)} = M^{E} and \\hat{\\mathbf{r}}{(M,E)} = \\Psi_{nl}{(M,E)} and E + \\hat{\\mathbf{r}}{(M,E)} = E + \\Psi_{nl}{(M,E)} and (E + \\hat{\\mathbf{r}}{(M,E)})^{E} + \\hat{\\mathbf{r}}{(M,E)} = M^{E} + (E + \\hat{\\mathbf{r}}{(M,E)})^{E} and \\theta_{2}{(E,M)} = (E + \\Psi_{nl}{(M,E)})^{E} and (E + \\Psi_{nl}{(M,E)})^{E} + \\hat{\\mathbf{r}}{(M,E)} = M^{E} + (E + \\Psi_{nl}{(M,E)})^{E} and 2 (E + \\Psi_{nl}{(M,E)})^{E} + \\hat{\\mathbf{r}}{(M,E)} = M^{E} + 2 (E + \\Psi_{nl}{(M,E)})^{E} and \\hat{\\mathbf{r}}{(M,E)} + 2 \\theta_{2}{(E,M)} = M^{E} + 2 \\theta_{2}{(E,M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True)))"], [["add", 3, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))))"], [["add", 2, "Pow(Add(Symbol('E', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))"], "Equality(Add(Pow(Add(Symbol('E', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)), Pow(Add(Symbol('E', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('M', commutative=True)), Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)), Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["add", 7, "Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('E', commutative=True), Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('E', commutative=True)), Mul(Integer(2), Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('M', commutative=True)))), Add(Pow(Symbol('M', commutative=True), Symbol('E', commutative=True)), Mul(Integer(2), Function('\\\\theta_2')(Symbol('E', commutative=True), Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(m)} = \\cos{(m)}, then obtain \\frac{\\int \\operatorname{n_{2}}{(m)} dm}{\\operatorname{n_{2}}{(m)}} = \\frac{\\int \\cos{(m)} dm}{\\operatorname{n_{2}}{(m)}}", "derivation": "\\operatorname{n_{2}}{(m)} = \\cos{(m)} and \\int \\operatorname{n_{2}}{(m)} dm = \\int \\cos{(m)} dm and \\frac{\\int \\operatorname{n_{2}}{(m)} dm}{\\cos{(m)}} = \\frac{\\int \\cos{(m)} dm}{\\cos{(m)}} and \\frac{\\int \\operatorname{n_{2}}{(m)} dm}{\\operatorname{n_{2}}{(m)}} = \\frac{\\int \\cos{(m)} dm}{\\operatorname{n_{2}}{(m)}}", "srepr_derivation": [["get_premise", "Equality(Function('n_2')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["divide", 2, "cos(Symbol('m', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('m', commutative=True)), Integer(-1)), Integral(Function('n_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(cos(Symbol('m', commutative=True)), Integer(-1)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('n_2')(Symbol('m', commutative=True)), Integer(-1)), Integral(Function('n_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Function('n_2')(Symbol('m', commutative=True)), Integer(-1)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(m,v_{t})} = \\cos^{v_{t}}{(m)} and E{(m,v_{t})} = (\\cos^{v_{t}}{(m)})^{m}, then obtain (\\int \\mathbf{H}^{m}{(m,v_{t})} dv_{t})^{v_{t}} = (\\int E{(m,v_{t})} dv_{t})^{v_{t}}", "derivation": "\\mathbf{H}{(m,v_{t})} = \\cos^{v_{t}}{(m)} and \\mathbf{H}^{m}{(m,v_{t})} = (\\cos^{v_{t}}{(m)})^{m} and E{(m,v_{t})} = (\\cos^{v_{t}}{(m)})^{m} and \\mathbf{H}^{m}{(m,v_{t})} = E{(m,v_{t})} and \\int \\mathbf{H}^{m}{(m,v_{t})} dv_{t} = \\int E{(m,v_{t})} dv_{t} and (\\int \\mathbf{H}^{m}{(m,v_{t})} dv_{t})^{v_{t}} = (\\int E{(m,v_{t})} dv_{t})^{v_{t}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Pow(Pow(cos(Symbol('m', commutative=True)), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)), Function('E')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 4, "Symbol('v_t', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Function('E')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["power", 5, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integral(Function('E')(Symbol('m', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given S{(A_{1})} = e^{A_{1}}, then obtain - A_{1} + S^{2}{(A_{1})} e^{2 A_{1}} = - A_{1} + S{(A_{1})} e^{3 A_{1}}", "derivation": "S{(A_{1})} = e^{A_{1}} and S^{2}{(A_{1})} = S{(A_{1})} e^{A_{1}} and S^{4}{(A_{1})} = S^{2}{(A_{1})} e^{2 A_{1}} and - A_{1} + S^{4}{(A_{1})} = - A_{1} + S^{2}{(A_{1})} e^{2 A_{1}} and - A_{1} + S^{2}{(A_{1})} e^{2 A_{1}} = - A_{1} + S{(A_{1})} e^{3 A_{1}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True)))"], [["times", 1, "Function('S')(Symbol('A_1', commutative=True))"], "Equality(Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(2)), Mul(Function('S')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(4)), Mul(Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('A_1', commutative=True)))))"], [["minus", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(4))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('A_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Pow(Function('S')(Symbol('A_1', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('A_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Function('S')(Symbol('A_1', commutative=True)), exp(Mul(Integer(3), Symbol('A_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{J})} = \\sin{(\\mathbf{J})}, then obtain \\frac{\\partial}{\\partial \\varepsilon} (\\int \\mathbf{H}^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}}) = \\frac{\\partial}{\\partial \\varepsilon} (\\int \\sin^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}})", "derivation": "\\mathbf{H}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\mathbf{H}^{\\mathbf{J}}{(\\mathbf{J})} = \\sin^{\\mathbf{J}}{(\\mathbf{J})} and \\int \\mathbf{H}^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} = \\int \\sin^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} and \\int \\mathbf{H}^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}} = \\int \\sin^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}} and \\frac{\\partial}{\\partial \\varepsilon} (\\int \\mathbf{H}^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}}) = \\frac{\\partial}{\\partial \\varepsilon} (\\int \\sin^{\\mathbf{J}}{(\\mathbf{J})} d\\mathbf{J} - \\frac{1}{\\log{(\\varepsilon)}})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 3, "Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))"], "Equality(Add(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))), Add(Integral(Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))))"], [["differentiate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Integral(Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)} = \\int \\lambda^{\\mathbf{J}_P} d\\lambda, then obtain \\int (\\frac{\\partial}{\\partial \\lambda} \\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)})^{\\lambda} d\\lambda = \\int (\\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\mathbf{J}_P} d\\lambda)^{\\lambda} d\\lambda", "derivation": "\\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)} = \\int \\lambda^{\\mathbf{J}_P} d\\lambda and \\frac{\\partial}{\\partial \\lambda} \\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\mathbf{J}_P} d\\lambda and (\\frac{\\partial}{\\partial \\lambda} \\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)})^{\\lambda} = (\\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\mathbf{J}_P} d\\lambda)^{\\lambda} and \\int (\\frac{\\partial}{\\partial \\lambda} \\operatorname{c_{0}}{(\\lambda,\\mathbf{J}_P)})^{\\lambda} d\\lambda = \\int (\\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\mathbf{J}_P} d\\lambda)^{\\lambda} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Derivative(Function('c_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('c_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Pow(Derivative(Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(W,Z)} = \\sin{(W + Z)}, then obtain (\\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1)^{Z})^{W} = (\\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + 2)^{Z})^{W}", "derivation": "\\operatorname{A_{x}}{(W,Z)} = \\sin{(W + Z)} and \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} = 1 and - W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1 = - W - Z + 2 and (- W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1)^{Z} = (- W - Z + 2)^{Z} and \\frac{\\partial}{\\partial Z} (- W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1)^{Z} = \\frac{\\partial}{\\partial Z} (- W - Z + 2)^{Z} and \\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1)^{Z} = \\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + 2)^{Z} and (\\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + \\frac{\\operatorname{A_{x}}{(W,Z)}}{\\sin{(W + Z)}} + 1)^{Z})^{W} = (\\frac{\\partial^{2}}{\\partial W\\partial Z} (- W - Z + 2)^{Z})^{W}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))))"], [["divide", 1, "sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(2)))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1)), Symbol('Z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(2)), Symbol('Z', commutative=True)))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(2)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(2)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 6, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Function('A_x')(Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Add(Symbol('W', commutative=True), Symbol('Z', commutative=True))), Integer(-1))), Integer(1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(2)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(S,g)} = S + g, then derive \\sin{(\\frac{\\frac{\\partial}{\\partial g} \\operatorname{y^{\\prime}}{(S,g)}}{g})} = \\sin{(\\frac{1}{g})}, then obtain \\frac{\\partial}{\\partial S} \\sin{(\\frac{\\frac{\\partial}{\\partial g} (S + g)}{g})} = \\frac{d}{d S} \\sin{(\\frac{1}{g})}", "derivation": "\\operatorname{y^{\\prime}}{(S,g)} = S + g and \\frac{\\partial}{\\partial g} \\operatorname{y^{\\prime}}{(S,g)} = \\frac{\\partial}{\\partial g} (S + g) and \\frac{\\frac{\\partial}{\\partial g} \\operatorname{y^{\\prime}}{(S,g)}}{g} = \\frac{\\frac{\\partial}{\\partial g} (S + g)}{g} and \\sin{(\\frac{\\frac{\\partial}{\\partial g} \\operatorname{y^{\\prime}}{(S,g)}}{g})} = \\sin{(\\frac{\\frac{\\partial}{\\partial g} (S + g)}{g})} and \\sin{(\\frac{\\frac{\\partial}{\\partial g} \\operatorname{y^{\\prime}}{(S,g)}}{g})} = \\sin{(\\frac{1}{g})} and \\sin{(\\frac{\\frac{\\partial}{\\partial g} (S + g)}{g})} = \\sin{(\\frac{1}{g})} and \\frac{\\partial}{\\partial S} \\sin{(\\frac{\\frac{\\partial}{\\partial g} (S + g)}{g})} = \\frac{d}{d S} \\sin{(\\frac{1}{g})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Add(Symbol('S', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Add(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), sin(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Add(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(sin(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), sin(Pow(Symbol('g', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(sin(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Add(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), sin(Pow(Symbol('g', commutative=True), Integer(-1))))"], [["differentiate", 6, "Symbol('S', commutative=True)"], "Equality(Derivative(sin(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Derivative(Add(Symbol('S', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{s},W,\\dot{y})} = W \\dot{y}^{\\mathbf{s}}, then obtain \\dot{y}^{- \\mathbf{s}} (- W \\dot{y}^{\\mathbf{s}} + W) = \\dot{y}^{- \\mathbf{s}} (W - \\mu_{0}{(\\mathbf{s},W,\\dot{y})})", "derivation": "\\mu_{0}{(\\mathbf{s},W,\\dot{y})} = W \\dot{y}^{\\mathbf{s}} and 0 = W \\dot{y}^{\\mathbf{s}} - \\mu_{0}{(\\mathbf{s},W,\\dot{y})} and W = W \\dot{y}^{\\mathbf{s}} + W - \\mu_{0}{(\\mathbf{s},W,\\dot{y})} and - W \\dot{y}^{\\mathbf{s}} + W = W - \\mu_{0}{(\\mathbf{s},W,\\dot{y})} and \\dot{y}^{- \\mathbf{s}} (- W \\dot{y}^{\\mathbf{s}} + W) = \\dot{y}^{- \\mathbf{s}} (W - \\mu_{0}{(\\mathbf{s},W,\\dot{y})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 2, "Symbol('W', commutative=True)"], "Equality(Symbol('W', commutative=True), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["divide", 4, "Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('W', commutative=True))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given u{(S)} = e^{S}, then obtain (\\int u^{S}{(S)} dS) \\int (e^{S})^{S} dS = (\\int (e^{S})^{S} dS)^{2}", "derivation": "u{(S)} = e^{S} and u^{S}{(S)} = (e^{S})^{S} and \\int u^{S}{(S)} dS = \\int (e^{S})^{S} dS and (\\int u^{S}{(S)} dS) \\int (e^{S})^{S} dS = (\\int (e^{S})^{S} dS)^{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('u')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(exp(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Function('u')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(exp(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["times", 3, "Integral(Pow(exp(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Integral(Pow(Function('u')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(exp(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Pow(Integral(Pow(exp(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{H}{(s)} = \\int \\log{(s)} ds, then derive \\frac{d}{d s} \\mathbf{H}{(s)} = \\frac{\\partial}{\\partial s} (\\phi + s \\log{(s)} - s), then obtain - s + \\frac{d}{d s} \\mathbf{H}{(s)} = - s + \\frac{\\partial}{\\partial s} (\\phi + s \\log{(s)} - s)", "derivation": "\\mathbf{H}{(s)} = \\int \\log{(s)} ds and \\frac{d}{d s} \\mathbf{H}{(s)} = \\frac{d}{d s} \\int \\log{(s)} ds and \\frac{d}{d s} \\mathbf{H}{(s)} = \\frac{\\partial}{\\partial s} (\\phi + s \\log{(s)} - s) and - s + \\frac{d}{d s} \\mathbf{H}{(s)} = - s + \\frac{\\partial}{\\partial s} (\\phi + s \\log{(s)} - s)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('s', commutative=True)), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Derivative(Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('s', commutative=True), log(Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(I)} = \\cos{(I)} and \\chi{(I)} = \\hat{H}_l{(I)} + \\cos{(I)}, then obtain \\hat{H}_l{(I)} - \\sin{(I)} + \\cos{(I)} = \\chi{(I)} - \\sin{(I)}", "derivation": "\\hat{H}_l{(I)} = \\cos{(I)} and \\chi{(I)} = \\hat{H}_l{(I)} + \\cos{(I)} and \\chi{(I)} = 2 \\cos{(I)} and \\chi{(I)} + \\frac{d}{d I} \\cos{(I)} = 2 \\cos{(I)} + \\frac{d}{d I} \\cos{(I)} and \\hat{H}_l{(I)} + \\cos{(I)} + \\frac{d}{d I} \\cos{(I)} = 2 \\cos{(I)} + \\frac{d}{d I} \\cos{(I)} and \\hat{H}_l{(I)} + \\cos{(I)} + \\frac{d}{d I} \\cos{(I)} = \\chi{(I)} + \\frac{d}{d I} \\cos{(I)} and \\hat{H}_l{(I)} - \\sin{(I)} + \\cos{(I)} = \\chi{(I)} - \\sin{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('I', commutative=True)), Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('I', commutative=True)), Mul(Integer(2), cos(Symbol('I', commutative=True))))"], [["add", 3, "Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\chi')(Symbol('I', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(2), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(2), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Function('\\\\chi')(Symbol('I', commutative=True)), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True))), cos(Symbol('I', commutative=True))), Add(Function('\\\\chi')(Symbol('I', commutative=True)), Mul(Integer(-1), sin(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(A_{x},\\rho_b)} = \\rho_b^{A_{x}}, then obtain - ((\\rho_b^{A_{x}})^{A_{x}})^{2 \\rho_b} + ((\\rho_b^{A_{x}})^{A_{x}})^{\\rho_b} (\\dot{y}^{A_{x}}{(A_{x},\\rho_b)})^{\\rho_b} = 0", "derivation": "\\dot{y}{(A_{x},\\rho_b)} = \\rho_b^{A_{x}} and \\dot{y}^{A_{x}}{(A_{x},\\rho_b)} = (\\rho_b^{A_{x}})^{A_{x}} and (\\dot{y}^{A_{x}}{(A_{x},\\rho_b)})^{\\rho_b} = ((\\rho_b^{A_{x}})^{A_{x}})^{\\rho_b} and ((\\rho_b^{A_{x}})^{A_{x}})^{\\rho_b} (\\dot{y}^{A_{x}}{(A_{x},\\rho_b)})^{\\rho_b} = ((\\rho_b^{A_{x}})^{A_{x}})^{2 \\rho_b} and - ((\\rho_b^{A_{x}})^{A_{x}})^{2 \\rho_b} + ((\\rho_b^{A_{x}})^{A_{x}})^{\\rho_b} (\\dot{y}^{A_{x}}{(A_{x},\\rho_b)})^{\\rho_b} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Pow(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["times", 3, "Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 4, "Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Pow(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Function('\\\\dot{y}')(Symbol('A_x', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('A_x', commutative=True)), Symbol('\\\\rho_b', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\operatorname{E_{x}}{(\\mathbf{F})} = \\mathbf{F} + \\phi_{2}{(\\mathbf{F})}, then obtain \\mathbf{F} - 2 \\phi_{2}{(\\mathbf{F})} + \\sin{(\\mathbf{F})} = \\mathbf{F} - \\phi_{2}{(\\mathbf{F})}", "derivation": "\\phi_{2}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\mathbf{F} + \\phi_{2}{(\\mathbf{F})} = \\mathbf{F} + \\sin{(\\mathbf{F})} and \\operatorname{E_{x}}{(\\mathbf{F})} = \\mathbf{F} + \\phi_{2}{(\\mathbf{F})} and \\operatorname{E_{x}}{(\\mathbf{F})} = \\mathbf{F} + \\sin{(\\mathbf{F})} and \\operatorname{E_{x}}{(\\mathbf{F})} - 2 \\sin{(\\mathbf{F})} = \\mathbf{F} + \\phi_{2}{(\\mathbf{F})} - 2 \\sin{(\\mathbf{F})} and \\operatorname{E_{x}}{(\\mathbf{F})} - 2 \\phi_{2}{(\\mathbf{F})} = \\mathbf{F} - \\phi_{2}{(\\mathbf{F})} and \\mathbf{F} - 2 \\phi_{2}{(\\mathbf{F})} + \\sin{(\\mathbf{F})} = \\mathbf{F} - \\phi_{2}{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_x')(Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 3, "Mul(Integer(2), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Function('E_x')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('E_x')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True))), sin(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given b{(B)} = e^{B}, then derive \\int b{(B)} dB = v_{x} + e^{B}, then obtain \\cos^{B}{(\\int e^{B} dB)} = \\cos^{B}{(v_{x} + e^{B})}", "derivation": "b{(B)} = e^{B} and \\int b{(B)} dB = \\int e^{B} dB and \\int b{(B)} dB = v_{x} + e^{B} and \\cos{(\\int b{(B)} dB)} = \\cos{(v_{x} + e^{B})} and \\cos{(\\int e^{B} dB)} = \\cos{(v_{x} + e^{B})} and \\cos^{B}{(\\int e^{B} dB)} = \\cos^{B}{(v_{x} + e^{B})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('b')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('v_x', commutative=True), exp(Symbol('B', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Function('b')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), cos(Add(Symbol('v_x', commutative=True), exp(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), cos(Add(Symbol('v_x', commutative=True), exp(Symbol('B', commutative=True)))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(cos(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Pow(cos(Add(Symbol('v_x', commutative=True), exp(Symbol('B', commutative=True)))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} = \\Psi_{\\lambda} - \\hat{H}_l, then obtain \\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} = -1", "derivation": "\\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} = \\Psi_{\\lambda} - \\hat{H}_l and \\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} + 1 = \\Psi_{\\lambda} - \\hat{H}_l + 1 and \\frac{\\partial}{\\partial \\hat{H}_l} (\\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} + 1) = \\frac{\\partial}{\\partial \\hat{H}_l} (\\Psi_{\\lambda} - \\hat{H}_l + 1) and \\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{A_{y}}{(\\Psi_{\\lambda},\\hat{H}_l)} = -1", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_y')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(P_{g},n)} = \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g}}{n}, then obtain (2 \\operatorname{F_{N}}{(P_{g},n)})^{P_{g}} = (\\operatorname{F_{N}}{(P_{g},n)} + \\frac{1}{n})^{P_{g}}", "derivation": "\\operatorname{F_{N}}{(P_{g},n)} = \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g}}{n} and 2 \\operatorname{F_{N}}{(P_{g},n)} = \\operatorname{F_{N}}{(P_{g},n)} + \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g}}{n} and (2 \\operatorname{F_{N}}{(P_{g},n)})^{P_{g}} = (\\operatorname{F_{N}}{(P_{g},n)} + \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g}}{n})^{P_{g}} and (2 \\operatorname{F_{N}}{(P_{g},n)})^{P_{g}} = (\\operatorname{F_{N}}{(P_{g},n)} + \\frac{1}{n})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["add", 1, "Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Add(Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Symbol('P_g', commutative=True)), Pow(Add(Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Symbol('P_g', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Mul(Integer(2), Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Symbol('P_g', commutative=True)), Pow(Add(Function('F_N')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(r,\\mathbf{g})} = \\frac{\\mathbf{g}}{r}, then derive - \\frac{\\partial}{\\partial r} \\dot{y}{(r,\\mathbf{g})} = \\frac{\\mathbf{g}}{r^{2}}, then obtain \\frac{\\dot{y}{(r,\\mathbf{g})}}{r} = - \\frac{\\partial}{\\partial r} \\frac{\\mathbf{g}}{r}", "derivation": "\\dot{y}{(r,\\mathbf{g})} = \\frac{\\mathbf{g}}{r} and - \\dot{y}{(r,\\mathbf{g})} = - \\frac{\\mathbf{g}}{r} and \\frac{\\partial}{\\partial r} - \\dot{y}{(r,\\mathbf{g})} = \\frac{\\partial}{\\partial r} - \\frac{\\mathbf{g}}{r} and - \\frac{\\partial}{\\partial r} \\dot{y}{(r,\\mathbf{g})} = \\frac{\\mathbf{g}}{r^{2}} and - \\frac{\\partial}{\\partial r} \\frac{\\mathbf{g}}{r} = \\frac{\\mathbf{g}}{r^{2}} and - \\frac{\\partial}{\\partial r} \\dot{y}{(r,\\mathbf{g})} = \\frac{\\dot{y}{(r,\\mathbf{g})}}{r} and \\frac{\\dot{y}{(r,\\mathbf{g})}}{r} = \\frac{\\mathbf{g}}{r^{2}} and \\frac{\\dot{y}{(r,\\mathbf{g})}}{r} = - \\frac{\\partial}{\\partial r} \\frac{\\mathbf{g}}{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(E,\\mathbf{v},W)} = \\frac{W}{E \\mathbf{v}}, then derive \\int (\\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}}) dW = \\frac{W}{\\mathbf{v}} + m + \\frac{W^{2}}{2 E \\mathbf{v}}, then obtain \\cos{(\\int (\\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}}) dW)} = \\cos{(\\frac{W}{\\mathbf{v}} + m + \\frac{W^{2}}{2 E \\mathbf{v}})}", "derivation": "\\operatorname{A_{2}}{(E,\\mathbf{v},W)} = \\frac{W}{E \\mathbf{v}} and \\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}} = \\frac{1}{\\mathbf{v}} + \\frac{W}{E \\mathbf{v}} and \\int (\\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}}) dW = \\int (\\frac{1}{\\mathbf{v}} + \\frac{W}{E \\mathbf{v}}) dW and \\int (\\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}}) dW = \\frac{W}{\\mathbf{v}} + m + \\frac{W^{2}}{2 E \\mathbf{v}} and \\cos{(\\int (\\operatorname{A_{2}}{(E,\\mathbf{v},W)} + \\frac{1}{\\mathbf{v}}) dW)} = \\cos{(\\frac{W}{\\mathbf{v}} + m + \\frac{W^{2}}{2 E \\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))"], "Equality(Add(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Add(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('W', commutative=True))), Integral(Add(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], [["cos", 4], "Equality(cos(Integral(Add(Function('A_2')(Symbol('E', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Tuple(Symbol('W', commutative=True)))), cos(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given t{(\\mu_0,T)} = e^{T - \\mu_0}, then derive \\frac{\\partial}{\\partial T} t{(\\mu_0,T)} = e^{T - \\mu_0}, then obtain - \\frac{\\frac{\\partial}{\\partial T} t{(\\mu_0,T)} + 1}{\\mu_0} = - \\frac{e^{T - \\mu_0} + 1}{\\mu_0}", "derivation": "t{(\\mu_0,T)} = e^{T - \\mu_0} and \\frac{\\partial}{\\partial T} t{(\\mu_0,T)} = \\frac{\\partial}{\\partial T} e^{T - \\mu_0} and \\frac{\\partial}{\\partial T} t{(\\mu_0,T)} = e^{T - \\mu_0} and \\frac{\\partial}{\\partial T} t{(\\mu_0,T)} + 1 = e^{T - \\mu_0} + 1 and - \\frac{\\frac{\\partial}{\\partial T} t{(\\mu_0,T)} + 1}{\\mu_0} = - \\frac{e^{T - \\mu_0} + 1}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('t')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Add(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(1)))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Derivative(Function('t')(Symbol('\\\\mu_0', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Integer(1))))"]]}, {"prompt": "Given c{(t_{2},A_{z})} = \\frac{t_{2}}{A_{z}}, then obtain - (\\frac{t_{2}}{A_{z}})^{t_{2}} + c{(t_{2},A_{z})} - \\frac{t_{2}}{A_{z}} = - (\\frac{t_{2}}{A_{z}})^{t_{2}}", "derivation": "c{(t_{2},A_{z})} = \\frac{t_{2}}{A_{z}} and c^{t_{2}}{(t_{2},A_{z})} = (\\frac{t_{2}}{A_{z}})^{t_{2}} and c{(t_{2},A_{z})} - c^{t_{2}}{(t_{2},A_{z})} = - c^{t_{2}}{(t_{2},A_{z})} + \\frac{t_{2}}{A_{z}} and - (\\frac{t_{2}}{A_{z}})^{t_{2}} + c{(t_{2},A_{z})} = - (\\frac{t_{2}}{A_{z}})^{t_{2}} + \\frac{t_{2}}{A_{z}} and - (\\frac{t_{2}}{A_{z}})^{t_{2}} + c{(t_{2},A_{z})} - \\frac{t_{2}}{A_{z}} = - (\\frac{t_{2}}{A_{z}})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Symbol('t_2', commutative=True)), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 1, "Pow(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Symbol('t_2', commutative=True))"], "Equality(Add(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), Pow(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Symbol('t_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["minus", 4, "Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Function('c')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(n_{2},\\mathbf{g})} = \\sin{(\\mathbf{g} + n_{2})}, then obtain \\int (\\int \\operatorname{F_{N}}{(n_{2},\\mathbf{g})} d\\mathbf{g})^{n_{2}} dn_{2} = \\int (\\int \\sin{(\\mathbf{g} + n_{2})} d\\mathbf{g})^{n_{2}} dn_{2}", "derivation": "\\operatorname{F_{N}}{(n_{2},\\mathbf{g})} = \\sin{(\\mathbf{g} + n_{2})} and \\int \\operatorname{F_{N}}{(n_{2},\\mathbf{g})} d\\mathbf{g} = \\int \\sin{(\\mathbf{g} + n_{2})} d\\mathbf{g} and (\\int \\operatorname{F_{N}}{(n_{2},\\mathbf{g})} d\\mathbf{g})^{n_{2}} = (\\int \\sin{(\\mathbf{g} + n_{2})} d\\mathbf{g})^{n_{2}} and \\int (\\int \\operatorname{F_{N}}{(n_{2},\\mathbf{g})} d\\mathbf{g})^{n_{2}} dn_{2} = \\int (\\int \\sin{(\\mathbf{g} + n_{2})} d\\mathbf{g})^{n_{2}} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), sin(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Integral(Function('F_N')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('n_2', commutative=True)), Pow(Integral(sin(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('n_2', commutative=True)))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Pow(Integral(Function('F_N')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Pow(Integral(sin(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain - e^{L_{\\varepsilon}} + \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = 0", "derivation": "\\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}} and \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} - \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}} = 0 and - e^{L_{\\varepsilon}} + \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{a^{\\dagger}}{(L_{\\varepsilon})} = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["minus", 3, "Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Function('a^{\\\\dagger}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Integer(-1), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))), Integer(0))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Derivative(Function('a^{\\\\dagger}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)))), Integer(0))"]]}, {"prompt": "Given \\pi{(n_{2},x)} = n_{2} x and \\operatorname{P_{g}}{(n_{2},x)} = n_{2} x, then obtain x + \\operatorname{P_{g}}{(n_{2},x)} + \\pi{(n_{2},x)} = x + 2 \\operatorname{P_{g}}{(n_{2},x)}", "derivation": "\\pi{(n_{2},x)} = n_{2} x and x + \\pi{(n_{2},x)} = n_{2} x + x and \\operatorname{P_{g}}{(n_{2},x)} = n_{2} x and x + \\pi{(n_{2},x)} = x + \\operatorname{P_{g}}{(n_{2},x)} and x + \\operatorname{P_{g}}{(n_{2},x)} + \\pi{(n_{2},x)} = x + 2 \\operatorname{P_{g}}{(n_{2},x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('n_2', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('n_2', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('\\\\pi')(Symbol('n_2', commutative=True), Symbol('x', commutative=True))), Add(Mul(Symbol('n_2', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('n_2', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('n_2', commutative=True), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('x', commutative=True), Function('\\\\pi')(Symbol('n_2', commutative=True), Symbol('x', commutative=True))), Add(Symbol('x', commutative=True), Function('P_g')(Symbol('n_2', commutative=True), Symbol('x', commutative=True))))"], [["add", 4, "Function('P_g')(Symbol('n_2', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Symbol('x', commutative=True), Function('P_g')(Symbol('n_2', commutative=True), Symbol('x', commutative=True)), Function('\\\\pi')(Symbol('n_2', commutative=True), Symbol('x', commutative=True))), Add(Symbol('x', commutative=True), Mul(Integer(2), Function('P_g')(Symbol('n_2', commutative=True), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\delta)} = \\delta, then derive \\frac{d}{d \\delta} \\nabla{(\\delta)} = 1, then obtain \\log{(\\hat{\\mathbf{x}} + f_{E})} \\frac{d}{d \\delta} \\delta = \\log{(\\hat{\\mathbf{x}} + f_{E})}", "derivation": "\\nabla{(\\delta)} = \\delta and \\frac{d}{d \\delta} \\nabla{(\\delta)} = \\frac{d}{d \\delta} \\delta and \\frac{d}{d \\delta} \\nabla{(\\delta)} = 1 and \\log{(\\hat{\\mathbf{x}} + f_{E})} \\frac{d}{d \\delta} \\nabla{(\\delta)} = \\log{(\\hat{\\mathbf{x}} + f_{E})} and \\log{(\\hat{\\mathbf{x}} + f_{E})} \\frac{d}{d \\delta} \\delta = \\log{(\\hat{\\mathbf{x}} + f_{E})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "log(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(log(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f_E', commutative=True))), Derivative(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), log(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(log(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f_E', commutative=True))), Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), log(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(x^\\prime)} = \\sin{(\\sin{(x^\\prime)})}, then obtain \\frac{\\frac{d}{d x^\\prime} 0}{\\mathbf{s}{(x^\\prime)} \\frac{d}{d x^\\prime} (- \\mathbf{s}{(x^\\prime)} + \\sin{(\\sin{(x^\\prime)})})} = \\frac{1}{\\mathbf{s}{(x^\\prime)}}", "derivation": "\\mathbf{s}{(x^\\prime)} = \\sin{(\\sin{(x^\\prime)})} and 0 = - \\mathbf{s}{(x^\\prime)} + \\sin{(\\sin{(x^\\prime)})} and \\frac{d}{d x^\\prime} 0 = \\frac{d}{d x^\\prime} (- \\mathbf{s}{(x^\\prime)} + \\sin{(\\sin{(x^\\prime)})}) and \\frac{\\frac{d}{d x^\\prime} 0}{\\mathbf{s}{(x^\\prime)}} = \\frac{\\frac{d}{d x^\\prime} (- \\mathbf{s}{(x^\\prime)} + \\sin{(\\sin{(x^\\prime)})})}{\\mathbf{s}{(x^\\prime)}} and \\frac{\\frac{d}{d x^\\prime} 0}{\\mathbf{s}{(x^\\prime)} \\frac{d}{d x^\\prime} (- \\mathbf{s}{(x^\\prime)} + \\sin{(\\sin{(x^\\prime)})})} = \\frac{1}{\\mathbf{s}{(x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), sin(sin(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), sin(sin(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), sin(sin(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 3, "Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), sin(sin(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["divide", 4, "Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), sin(sin(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True))), sin(sin(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Pow(Function('\\\\mathbf{s}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\sin{(\\sigma_x)}, then obtain 0 = \\cos{(\\sigma_x)} - \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})}", "derivation": "\\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\sin{(\\sigma_x)} and 0 = \\Psi^{\\dagger} - \\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})} + \\sin{(\\sigma_x)} and \\frac{d}{d \\sigma_x} 0 = \\frac{\\partial}{\\partial \\sigma_x} (\\Psi^{\\dagger} - \\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})} + \\sin{(\\sigma_x)}) and 0 = \\cos{(\\sigma_x)} - \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{M_{E}}{(\\sigma_x,\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 1, "Function('M_E')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('M_E')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('M_E')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), sin(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(cos(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Derivative(Function('M_E')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\delta)} = \\cos{(\\delta)}, then obtain \\operatorname{v_{x}}^{2}{(\\delta)} = \\operatorname{v_{x}}{(\\delta)} \\frac{d}{d \\delta} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\delta)}", "derivation": "\\operatorname{v_{x}}{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\delta)} = \\cos{(\\delta)} and \\operatorname{v_{x}}{(\\delta)} = \\frac{d}{d \\delta} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\delta)} and \\operatorname{v_{x}}^{2}{(\\delta)} = \\operatorname{v_{x}}{(\\delta)} \\frac{d}{d \\delta} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\delta', commutative=True)), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_x')(Symbol('\\\\delta', commutative=True)), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["times", 3, "Function('v_x')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('v_x')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Function('v_x')(Symbol('\\\\delta', commutative=True)), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(\\hat{\\mathbf{x}},\\varphi)} = \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}, then derive \\frac{\\partial}{\\partial \\varphi} \\pi{(\\hat{\\mathbf{x}},\\varphi)} = - \\frac{\\hat{\\mathbf{x}} \\sin{(\\varphi)} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}}{\\cos{(\\varphi)}}, then obtain - \\frac{\\hat{\\mathbf{x}} \\sin{(\\varphi)} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}}{\\cos{(\\varphi)}} = \\frac{\\partial}{\\partial \\varphi} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}", "derivation": "\\pi{(\\hat{\\mathbf{x}},\\varphi)} = \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)} and \\frac{\\partial}{\\partial \\varphi} \\pi{(\\hat{\\mathbf{x}},\\varphi)} = \\frac{\\partial}{\\partial \\varphi} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)} and \\frac{\\partial}{\\partial \\varphi} \\pi{(\\hat{\\mathbf{x}},\\varphi)} = - \\frac{\\hat{\\mathbf{x}} \\sin{(\\varphi)} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}}{\\cos{(\\varphi)}} and - \\frac{\\hat{\\mathbf{x}} \\sin{(\\varphi)} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}}{\\cos{(\\varphi)}} = \\frac{\\partial}{\\partial \\varphi} \\cos^{\\hat{\\mathbf{x}}}{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\varphi)} = \\log{(e^{\\varphi})}, then obtain \\int (C{(\\varphi)} - e^{\\varphi} - \\log{(e^{\\varphi})}) d\\varphi = \\int - e^{\\varphi} d\\varphi", "derivation": "C{(\\varphi)} = \\log{(e^{\\varphi})} and C{(\\varphi)} - e^{\\varphi} = - e^{\\varphi} + \\log{(e^{\\varphi})} and C{(\\varphi)} - e^{\\varphi} - \\log{(e^{\\varphi})} = - e^{\\varphi} and \\int (C{(\\varphi)} - e^{\\varphi} - \\log{(e^{\\varphi})}) d\\varphi = \\int - e^{\\varphi} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\varphi', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('C')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True))), log(exp(Symbol('\\\\varphi', commutative=True)))))"], [["minus", 2, "log(exp(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Function('C')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), log(exp(Symbol('\\\\varphi', commutative=True))))), Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Function('C')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), log(exp(Symbol('\\\\varphi', commutative=True))))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given I{(z^{*},v_{z})} = - v_{z} + z^{*}, then derive \\int I{(z^{*},v_{z})} dz^{*} = \\mathbf{s} - v_{z} z^{*} + \\frac{(z^{*})^{2}}{2}, then obtain \\mathbf{s} + \\int I{(z^{*},v_{z})} dz^{*} = 2 \\mathbf{s} - v_{z} z^{*} + \\frac{(z^{*})^{2}}{2}", "derivation": "I{(z^{*},v_{z})} = - v_{z} + z^{*} and \\int I{(z^{*},v_{z})} dz^{*} = \\int (- v_{z} + z^{*}) dz^{*} and \\int I{(z^{*},v_{z})} dz^{*} = \\mathbf{s} - v_{z} z^{*} + \\frac{(z^{*})^{2}}{2} and \\mathbf{s} + \\int I{(z^{*},v_{z})} dz^{*} = 2 \\mathbf{s} - v_{z} z^{*} + \\frac{(z^{*})^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('z^*', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('I')(Symbol('z^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('I')(Symbol('z^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True), Symbol('z^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"], [["add", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Integral(Function('I')(Symbol('z^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True), Symbol('z^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('z^*', commutative=True), Integer(2)))))"]]}, {"prompt": "Given h{(\\rho_f,Q)} = \\frac{e^{Q}}{\\rho_f} and \\mathbf{E}{(\\rho_f,Q)} = \\rho_f h{(\\rho_f,Q)} e^{- Q}, then obtain \\int \\mathbf{E}^{Q}{(\\rho_f,Q)} dQ = \\int 1 dQ", "derivation": "h{(\\rho_f,Q)} = \\frac{e^{Q}}{\\rho_f} and \\mathbf{E}{(\\rho_f,Q)} = \\rho_f h{(\\rho_f,Q)} e^{- Q} and \\mathbf{E}^{Q}{(\\rho_f,Q)} = (\\rho_f h{(\\rho_f,Q)} e^{- Q})^{Q} and \\mathbf{E}^{Q}{(\\rho_f,Q)} = 1 and \\int \\mathbf{E}^{Q}{(\\rho_f,Q)} dQ = \\int 1 dQ", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), exp(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Function('h')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Mul(Symbol('\\\\rho_f', commutative=True), Function('h')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\rho_f', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Integer(1), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then obtain \\frac{\\int \\operatorname{F_{H}}{(\\mathbf{J}_M)} d\\mathbf{J}_M}{A_{x} - \\cos{(\\mathbf{J}_M)}} = 1", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and \\int \\operatorname{F_{H}}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\frac{\\int \\operatorname{F_{H}}{(\\mathbf{J}_M)} d\\mathbf{J}_M}{\\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M} = 1 and \\frac{\\int \\operatorname{F_{H}}{(\\mathbf{J}_M)} d\\mathbf{J}_M}{A_{x} - \\cos{(\\mathbf{J}_M)}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Integral(Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(-1)), Integral(Function('F_H')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi_2)} = \\cos{(\\cos{(\\phi_2)})}, then obtain - \\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)} + \\frac{1}{\\cos{(\\phi_2)}} = - \\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)} + \\frac{\\cos{(\\cos{(\\phi_2)})}}{\\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)}}", "derivation": "\\dot{\\mathbf{r}}{(\\phi_2)} = \\cos{(\\cos{(\\phi_2)})} and \\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)} = \\cos{(\\phi_2)} \\cos{(\\cos{(\\phi_2)})} and \\frac{1}{\\cos{(\\phi_2)}} = \\frac{\\cos{(\\cos{(\\phi_2)})}}{\\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)}} and - \\cos{(\\phi_2)} \\cos{(\\cos{(\\phi_2)})} + \\frac{1}{\\cos{(\\phi_2)}} = - \\cos{(\\phi_2)} \\cos{(\\cos{(\\phi_2)})} + \\frac{\\cos{(\\cos{(\\phi_2)})}}{\\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)}} and - \\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)} + \\frac{1}{\\cos{(\\phi_2)}} = - \\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)} + \\frac{\\cos{(\\cos{(\\phi_2)})}}{\\dot{\\mathbf{r}}{(\\phi_2)} \\cos{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(cos(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True)))))"], [["divide", 1, "Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(cos(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 3, "Mul(cos(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True)))), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True)), cos(cos(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(cos(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(cos(Symbol('\\\\phi_2', commutative=True))))))"]]}, {"prompt": "Given h{(J)} = J, then obtain (\\frac{d}{d J} h{(J)})^{J} - 1 = (\\frac{d}{d J} J)^{J} - 1", "derivation": "h{(J)} = J and \\frac{d}{d J} h{(J)} = \\frac{d}{d J} J and (\\frac{d}{d J} h{(J)})^{J} = (\\frac{d}{d J} J)^{J} and (\\frac{d}{d J} h{(J)})^{J} - 1 = (\\frac{d}{d J} J)^{J} - 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Function('h')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["minus", 3, 1], "Equality(Add(Pow(Derivative(Function('h')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Integer(-1)), Add(Pow(Derivative(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\dot{y})} = \\sin{(\\dot{y})}, then obtain \\frac{d}{d \\dot{y}} \\operatorname{f_{\\mathbf{v}}}{(\\dot{y})} + 1 = \\cos{(\\dot{y})} + 1", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\dot{y})} = \\sin{(\\dot{y})} and \\dot{y} + \\operatorname{f_{\\mathbf{v}}}{(\\dot{y})} = \\dot{y} + \\sin{(\\dot{y})} and \\frac{d}{d \\dot{y}} (\\dot{y} + \\operatorname{f_{\\mathbf{v}}}{(\\dot{y})}) = \\frac{d}{d \\dot{y}} (\\dot{y} + \\sin{(\\dot{y})}) and \\frac{d}{d \\dot{y}} \\operatorname{f_{\\mathbf{v}}}{(\\dot{y})} + 1 = \\cos{(\\dot{y})} + 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\psi^{*}{(g,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + \\log{(g)} and \\operatorname{J_{\\varepsilon}}{(g,\\hat{\\mathbf{r}})} = \\psi^{*}^{g}{(g,\\hat{\\mathbf{r}})}, then obtain \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{J_{\\varepsilon}}{(g,\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(g)})^{g}", "derivation": "\\psi^{*}{(g,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + \\log{(g)} and \\psi^{*}^{g}{(g,\\hat{\\mathbf{r}})} = (\\hat{\\mathbf{r}} + \\log{(g)})^{g} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\psi^{*}^{g}{(g,\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(g)})^{g} and \\operatorname{J_{\\varepsilon}}{(g,\\hat{\\mathbf{r}})} = \\psi^{*}^{g}{(g,\\hat{\\mathbf{r}})} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{J_{\\varepsilon}}{(g,\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} + \\log{(g)})^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(\\phi,Z)} = Z \\phi and \\mathbf{A}{(\\phi,Z)} = \\frac{\\partial}{\\partial \\phi} (\\phi + \\hat{H}{(\\phi,Z)}), then obtain \\frac{\\frac{\\partial}{\\partial \\phi} (Z \\phi + \\phi)}{Z + 1} = \\frac{\\frac{\\partial}{\\partial \\phi} (\\phi + \\hat{H}{(\\phi,Z)})}{Z + 1}", "derivation": "\\hat{H}{(\\phi,Z)} = Z \\phi and \\mathbf{A}{(\\phi,Z)} = \\frac{\\partial}{\\partial \\phi} (\\phi + \\hat{H}{(\\phi,Z)}) and \\mathbf{A}{(\\phi,Z)} = \\frac{\\partial}{\\partial \\phi} (Z \\phi + \\phi) and \\frac{\\mathbf{A}{(\\phi,Z)}}{Z + 1} = \\frac{\\frac{\\partial}{\\partial \\phi} (\\phi + \\hat{H}{(\\phi,Z)})}{Z + 1} and \\frac{\\frac{\\partial}{\\partial \\phi} (Z \\phi + \\phi)}{Z + 1} = \\frac{\\frac{\\partial}{\\partial \\phi} (\\phi + \\hat{H}{(\\phi,Z)})}{Z + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('\\\\phi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["divide", 2, "Add(Symbol('Z', commutative=True), Integer(1))"], "Equality(Mul(Pow(Add(Symbol('Z', commutative=True), Integer(1)), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Add(Symbol('Z', commutative=True), Integer(1)), Integer(-1)), Derivative(Add(Symbol('\\\\phi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('Z', commutative=True), Integer(1)), Integer(-1)), Derivative(Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('Z', commutative=True), Integer(1)), Integer(-1)), Derivative(Add(Symbol('\\\\phi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(P_{g},z^{*})} = P_{g} + z^{*}, then obtain \\frac{\\partial}{\\partial z^{*}} \\iint \\operatorname{E_{n}}{(P_{g},z^{*})} dP_{g} dz^{*} + 1 = \\frac{\\partial}{\\partial z^{*}} \\iint (P_{g} + z^{*}) dP_{g} dz^{*} + 1", "derivation": "\\operatorname{E_{n}}{(P_{g},z^{*})} = P_{g} + z^{*} and \\int \\operatorname{E_{n}}{(P_{g},z^{*})} dP_{g} = \\int (P_{g} + z^{*}) dP_{g} and \\iint \\operatorname{E_{n}}{(P_{g},z^{*})} dP_{g} dz^{*} = \\iint (P_{g} + z^{*}) dP_{g} dz^{*} and \\frac{\\partial}{\\partial z^{*}} \\iint \\operatorname{E_{n}}{(P_{g},z^{*})} dP_{g} dz^{*} = \\frac{\\partial}{\\partial z^{*}} \\iint (P_{g} + z^{*}) dP_{g} dz^{*} and \\frac{\\partial}{\\partial z^{*}} \\iint \\operatorname{E_{n}}{(P_{g},z^{*})} dP_{g} dz^{*} + 1 = \\frac{\\partial}{\\partial z^{*}} \\iint (P_{g} + z^{*}) dP_{g} dz^{*} + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["differentiate", 3, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Integral(Function('E_n')(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Integral(Function('E_n')(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Add(Symbol('P_g', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('P_g', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\sigma_{p}{(F_{N},y)} = \\cos{(F_{N}^{y})}, then obtain - F_{N} + 2 \\sigma_{p}{(F_{N},y)} = - F_{N} + 2 \\cos{(F_{N}^{y})}", "derivation": "\\sigma_{p}{(F_{N},y)} = \\cos{(F_{N}^{y})} and - F_{N} + \\sigma_{p}{(F_{N},y)} = - F_{N} + \\cos{(F_{N}^{y})} and - F_{N} + 2 \\sigma_{p}{(F_{N},y)} = - F_{N} + \\sigma_{p}{(F_{N},y)} + \\cos{(F_{N}^{y})} and - F_{N} + 2 \\sigma_{p}{(F_{N},y)} = - F_{N} + 2 \\cos{(F_{N}^{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), cos(Pow(Symbol('F_N', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), cos(Pow(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)), cos(Pow(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('F_N', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(2), cos(Pow(Symbol('F_N', commutative=True), Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\cos{(y^{\\prime})}, then derive \\int \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = \\dot{z} + \\cos{(y^{\\prime})}, then obtain \\log{(\\int \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime})} = \\log{(\\dot{z} + \\cos{(y^{\\prime})})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\cos{(y^{\\prime})} and \\int \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = \\int \\frac{d}{d y^{\\prime}} \\cos{(y^{\\prime})} dy^{\\prime} and \\int \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime} = \\dot{z} + \\cos{(y^{\\prime})} and \\log{(\\int \\operatorname{g_{\\varepsilon}}{(y^{\\prime})} dy^{\\prime})} = \\log{(\\dot{z} + \\cos{(y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(cos(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["log", 3], "Equality(log(Integral(Function('g_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), log(Add(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(A_{z})} = \\cos{(A_{z})}, then derive \\int \\mu_{0}{(A_{z})} dA_{z} = \\mathbf{B} + \\sin{(A_{z})}, then obtain F_{x} + (\\int \\cos{(A_{z})} dA_{z})^{A_{z}} = F_{x} + (\\mathbf{B} + \\sin{(A_{z})})^{A_{z}}", "derivation": "\\mu_{0}{(A_{z})} = \\cos{(A_{z})} and \\int \\mu_{0}{(A_{z})} dA_{z} = \\int \\cos{(A_{z})} dA_{z} and \\int \\mu_{0}{(A_{z})} dA_{z} = \\mathbf{B} + \\sin{(A_{z})} and \\int \\cos{(A_{z})} dA_{z} = \\mathbf{B} + \\sin{(A_{z})} and (\\int \\cos{(A_{z})} dA_{z})^{A_{z}} = (\\mathbf{B} + \\sin{(A_{z})})^{A_{z}} and F_{x} + (\\int \\cos{(A_{z})} dA_{z})^{A_{z}} = F_{x} + (\\mathbf{B} + \\sin{(A_{z})})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu_0')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["power", 4, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integral(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["add", 5, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Pow(Integral(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))), Add(Symbol('F_x', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\psi,r_{0})} = r_{0}^{\\psi} and \\mathbf{E}{(h,H,B)} = B^{h} - H, then obtain \\frac{\\frac{\\partial}{\\partial r_{0}} (\\mathbf{D}{(\\psi,r_{0})} - 1)}{B^{h} - H} = \\frac{\\frac{\\partial}{\\partial r_{0}} (r_{0}^{\\psi} - 1)}{B^{h} - H}", "derivation": "\\mathbf{D}{(\\psi,r_{0})} = r_{0}^{\\psi} and \\mathbf{E}{(h,H,B)} = B^{h} - H and \\mathbf{D}{(\\psi,r_{0})} - 1 = r_{0}^{\\psi} - 1 and \\frac{\\partial}{\\partial r_{0}} (\\mathbf{D}{(\\psi,r_{0})} - 1) = \\frac{\\partial}{\\partial r_{0}} (r_{0}^{\\psi} - 1) and \\frac{\\frac{\\partial}{\\partial r_{0}} (\\mathbf{D}{(\\psi,r_{0})} - 1)}{\\mathbf{E}{(h,H,B)}} = \\frac{\\frac{\\partial}{\\partial r_{0}} (r_{0}^{\\psi} - 1)}{\\mathbf{E}{(h,H,B)}} and \\frac{\\frac{\\partial}{\\partial r_{0}} (\\mathbf{D}{(\\psi,r_{0})} - 1)}{B^{h} - H} = \\frac{\\frac{\\partial}{\\partial r_{0}} (r_{0}^{\\psi} - 1)}{B^{h} - H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('\\\\psi', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('H', commutative=True), Symbol('B', commutative=True)), Add(Pow(Symbol('B', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["divide", 4, "Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('H', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('H', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('H', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Derivative(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Pow(Symbol('B', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Integer(-1)), Derivative(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Pow(Add(Pow(Symbol('B', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Integer(-1)), Derivative(Add(Pow(Symbol('r_0', commutative=True), Symbol('\\\\psi', commutative=True)), Integer(-1)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(C)} = \\cos{(C)} and \\rho_{f}{(C)} = \\cos{(C)}, then derive C \\frac{d}{d C} \\operatorname{V_{\\mathbf{B}}}{(C)} + \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\frac{d}{d C} \\rho_{f}{(C)} + \\rho_{f}{(C)}, then obtain C \\frac{d}{d C} \\operatorname{V_{\\mathbf{B}}}{(C)} + \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\frac{d}{d C} \\cos{(C)} + \\cos{(C)}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(C)} = \\cos{(C)} and C \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\cos{(C)} and \\rho_{f}{(C)} = \\cos{(C)} and \\operatorname{V_{\\mathbf{B}}}{(C)} = \\rho_{f}{(C)} and C \\rho_{f}{(C)} = C \\cos{(C)} and C \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\rho_{f}{(C)} and \\frac{d}{d C} C \\operatorname{V_{\\mathbf{B}}}{(C)} = \\frac{d}{d C} C \\rho_{f}{(C)} and C \\frac{d}{d C} \\operatorname{V_{\\mathbf{B}}}{(C)} + \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\frac{d}{d C} \\rho_{f}{(C)} + \\rho_{f}{(C)} and C \\frac{d}{d C} \\operatorname{V_{\\mathbf{B}}}{(C)} + \\operatorname{V_{\\mathbf{B}}}{(C)} = C \\frac{d}{d C} \\cos{(C)} + \\cos{(C)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), cos(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), Function('\\\\rho_f')(Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\rho_f')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), cos(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Symbol('C', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), Function('\\\\rho_f')(Symbol('C', commutative=True))))"], [["differentiate", 6, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Symbol('C', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Function('\\\\rho_f')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Add(Mul(Symbol('C', commutative=True), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), Derivative(Function('\\\\rho_f')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Function('\\\\rho_f')(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 3], "Equality(Add(Mul(Symbol('C', commutative=True), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), Derivative(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), cos(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(p)} = \\log{(p)}, then obtain \\frac{2 \\rho_{f}{(p)} + 2 \\log{(p)}}{3 \\rho_{f}{(p)} + \\log{(p)}} = 1", "derivation": "\\rho_{f}{(p)} = \\log{(p)} and 2 \\rho_{f}{(p)} = \\rho_{f}{(p)} + \\log{(p)} and 3 \\rho_{f}{(p)} + \\log{(p)} = 2 \\rho_{f}{(p)} + 2 \\log{(p)} and \\frac{3 \\rho_{f}{(p)} + \\log{(p)}}{2 \\rho_{f}{(p)} + 2 \\log{(p)}} = 1 and 3 \\rho_{f}{(p)} + \\log{(p)} = \\rho_{f}{(p)} + 3 \\log{(p)} and \\rho_{f}{(p)} + 3 \\log{(p)} = 2 \\rho_{f}{(p)} + 2 \\log{(p)} and \\frac{\\rho_{f}{(p)} + 3 \\log{(p)}}{2 \\rho_{f}{(p)} + 2 \\log{(p)}} = 1 and \\frac{\\rho_{f}{(p)} + 3 \\log{(p)}}{3 \\rho_{f}{(p)} + \\log{(p)}} = 1 and \\frac{2 \\rho_{f}{(p)} + 2 \\log{(p)}}{3 \\rho_{f}{(p)} + \\log{(p)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["add", 1, "Function('\\\\rho_f')(Symbol('p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True))))"], [["add", 2, "Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('p', commutative=True))), log(Symbol('p', commutative=True))), Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True)))), Integer(-1)), Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('p', commutative=True))), log(Symbol('p', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('p', commutative=True))), log(Symbol('p', commutative=True))), Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), Mul(Integer(3), log(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), Mul(Integer(3), log(Symbol('p', commutative=True)))), Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), Mul(Integer(3), log(Symbol('p', commutative=True)))), Pow(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True)))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Add(Function('\\\\rho_f')(Symbol('p', commutative=True)), Mul(Integer(3), log(Symbol('p', commutative=True)))), Pow(Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('p', commutative=True))), log(Symbol('p', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('p', commutative=True))), Mul(Integer(2), log(Symbol('p', commutative=True)))), Pow(Add(Mul(Integer(3), Function('\\\\rho_f')(Symbol('p', commutative=True))), log(Symbol('p', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{B}{(k,C_{d},P_{e})} = \\frac{C_{d} P_{e}}{k}, then obtain C_{d} P_{e} (2 k + \\mathbf{B}{(k,C_{d},P_{e})}) = C_{d} P_{e} (\\frac{C_{d} P_{e}}{k} + 2 k)", "derivation": "\\mathbf{B}{(k,C_{d},P_{e})} = \\frac{C_{d} P_{e}}{k} and k + \\mathbf{B}{(k,C_{d},P_{e})} = \\frac{C_{d} P_{e}}{k} + k and 2 k + \\mathbf{B}{(k,C_{d},P_{e})} = \\frac{C_{d} P_{e}}{k} + 2 k and \\frac{C_{d} P_{e} (2 k + \\mathbf{B}{(k,C_{d},P_{e})})}{k} = \\frac{C_{d} P_{e} (\\frac{C_{d} P_{e}}{k} + 2 k)}{k} and C_{d} P_{e} (2 k + \\mathbf{B}{(k,C_{d},P_{e})}) = C_{d} P_{e} (\\frac{C_{d} P_{e}}{k} + 2 k)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('C_d', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('C_d', commutative=True), Symbol('P_e', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('k', commutative=True)))"], [["add", 2, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('C_d', commutative=True), Symbol('P_e', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('k', commutative=True))))"], [["times", 3, "Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('C_d', commutative=True), Symbol('P_e', commutative=True)))), Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('k', commutative=True)))))"], [["divide", 4, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Add(Mul(Integer(2), Symbol('k', commutative=True)), Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('C_d', commutative=True), Symbol('P_e', commutative=True)))), Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Add(Mul(Symbol('C_d', commutative=True), Symbol('P_e', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Mul(Integer(2), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(a,v_{y})} = \\log{(\\frac{a}{v_{y}})}, then obtain (\\frac{\\partial}{\\partial a} (\\mathbf{v}^{a}{(a,v_{y})} + \\log{(\\frac{a}{v_{y}})}))^{a} = (\\frac{\\partial}{\\partial a} (\\log{(\\frac{a}{v_{y}})} + \\log{(\\frac{a}{v_{y}})}^{a}))^{a}", "derivation": "\\mathbf{v}{(a,v_{y})} = \\log{(\\frac{a}{v_{y}})} and \\mathbf{v}^{a}{(a,v_{y})} = \\log{(\\frac{a}{v_{y}})}^{a} and \\mathbf{v}^{a}{(a,v_{y})} + \\log{(\\frac{a}{v_{y}})} = \\log{(\\frac{a}{v_{y}})} + \\log{(\\frac{a}{v_{y}})}^{a} and \\frac{\\partial}{\\partial a} (\\mathbf{v}^{a}{(a,v_{y})} + \\log{(\\frac{a}{v_{y}})}) = \\frac{\\partial}{\\partial a} (\\log{(\\frac{a}{v_{y}})} + \\log{(\\frac{a}{v_{y}})}^{a}) and (\\frac{\\partial}{\\partial a} (\\mathbf{v}^{a}{(a,v_{y})} + \\log{(\\frac{a}{v_{y}})}))^{a} = (\\frac{\\partial}{\\partial a} (\\log{(\\frac{a}{v_{y}})} + \\log{(\\frac{a}{v_{y}})}^{a}))^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('v_y', commutative=True)), log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True)), Pow(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Symbol('a', commutative=True)))"], [["add", 2, "log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))))"], "Equality(Add(Pow(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True)), log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))))), Add(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Pow(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Symbol('a', commutative=True))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Pow(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True)), log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Pow(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Derivative(Add(Pow(Function('\\\\mathbf{v}')(Symbol('a', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True)), log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1))))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)), Pow(Derivative(Add(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Pow(log(Mul(Symbol('a', commutative=True), Pow(Symbol('v_y', commutative=True), Integer(-1)))), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\log{(C_{2} \\mathbf{s})}, then derive \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\frac{1}{C_{2}}, then obtain 2 \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} + \\frac{1}{C_{2}}", "derivation": "\\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\log{(C_{2} \\mathbf{s})} and \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\frac{\\partial}{\\partial C_{2}} \\log{(C_{2} \\mathbf{s})} and \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\frac{1}{C_{2}} and \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} + \\frac{\\partial}{\\partial C_{2}} \\log{(C_{2} \\mathbf{s})} = \\frac{\\partial}{\\partial C_{2}} \\log{(C_{2} \\mathbf{s})} + \\frac{1}{C_{2}} and 2 \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} = \\frac{\\partial}{\\partial C_{2}} \\dot{\\mathbf{r}}{(C_{2},\\mathbf{s})} + \\frac{1}{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('C_2', commutative=True), Integer(-1)))"], [["add", 3, "Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('C_2', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('C_2', commutative=True), Integer(-1))))"]]}, {"prompt": "Given x{(\\mathbf{P},z^{*})} = \\mathbf{P} z^{*}, then derive \\mathbf{P} z^{*} \\frac{\\partial}{\\partial z^{*}} x{(\\mathbf{P},z^{*})} + \\mathbf{P} x{(\\mathbf{P},z^{*})} = 2 \\mathbf{P}^{2} z^{*}, then obtain \\mathbf{P} x{(\\mathbf{P},z^{*})} + x{(\\mathbf{P},z^{*})} \\frac{\\partial}{\\partial z^{*}} x{(\\mathbf{P},z^{*})} = 2 \\mathbf{P} x{(\\mathbf{P},z^{*})}", "derivation": "x{(\\mathbf{P},z^{*})} = \\mathbf{P} z^{*} and \\mathbf{P} z^{*} x{(\\mathbf{P},z^{*})} = \\mathbf{P}^{2} (z^{*})^{2} and \\frac{\\partial}{\\partial z^{*}} \\mathbf{P} z^{*} x{(\\mathbf{P},z^{*})} = \\frac{\\partial}{\\partial z^{*}} \\mathbf{P}^{2} (z^{*})^{2} and \\mathbf{P} z^{*} \\frac{\\partial}{\\partial z^{*}} x{(\\mathbf{P},z^{*})} + \\mathbf{P} x{(\\mathbf{P},z^{*})} = 2 \\mathbf{P}^{2} z^{*} and \\mathbf{P} x{(\\mathbf{P},z^{*})} + x{(\\mathbf{P},z^{*})} \\frac{\\partial}{\\partial z^{*}} x{(\\mathbf{P},z^{*})} = 2 \\mathbf{P} x{(\\mathbf{P},z^{*})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True), Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True), Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Pow(Symbol('z^*', commutative=True), Integer(2))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True), Derivative(Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2)), Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))), Mul(Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Derivative(Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True), Function('x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given n{(\\Psi,B)} = B - \\Psi, then obtain \\frac{\\partial}{\\partial \\Psi} n{(\\Psi,B)} \\frac{\\partial}{\\partial B} n{(\\Psi,B)} = \\frac{\\partial}{\\partial \\Psi} n{(\\Psi,B)} \\frac{\\partial}{\\partial B} (B - \\Psi)", "derivation": "n{(\\Psi,B)} = B - \\Psi and \\frac{\\partial}{\\partial B} n{(\\Psi,B)} = \\frac{\\partial}{\\partial B} (B - \\Psi) and n{(\\Psi,B)} \\frac{\\partial}{\\partial B} n{(\\Psi,B)} = n{(\\Psi,B)} \\frac{\\partial}{\\partial B} (B - \\Psi) and \\frac{\\partial}{\\partial \\Psi} n{(\\Psi,B)} \\frac{\\partial}{\\partial B} n{(\\Psi,B)} = \\frac{\\partial}{\\partial \\Psi} n{(\\Psi,B)} \\frac{\\partial}{\\partial B} (B - \\Psi)", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["times", 2, "Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Derivative(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Derivative(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Function('n')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\lambda,\\mathbb{I})} = e^{- \\lambda + \\mathbb{I}} and \\operatorname{n_{1}}{(\\lambda,\\mathbb{I})} = e^{- \\lambda + \\mathbb{I}}, then obtain \\operatorname{J_{\\varepsilon}}^{\\mathbb{I}}{(\\lambda,\\mathbb{I})} \\operatorname{n_{1}}^{- \\mathbb{I}}{(\\lambda,\\mathbb{I})} = 1", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\lambda,\\mathbb{I})} = e^{- \\lambda + \\mathbb{I}} and \\operatorname{n_{1}}{(\\lambda,\\mathbb{I})} = e^{- \\lambda + \\mathbb{I}} and \\operatorname{J_{\\varepsilon}}^{\\mathbb{I}}{(\\lambda,\\mathbb{I})} = (e^{- \\lambda + \\mathbb{I}})^{\\mathbb{I}} and \\operatorname{J_{\\varepsilon}}^{\\mathbb{I}}{(\\lambda,\\mathbb{I})} = \\operatorname{n_{1}}^{\\mathbb{I}}{(\\lambda,\\mathbb{I})} and \\operatorname{J_{\\varepsilon}}^{\\mathbb{I}}{(\\lambda,\\mathbb{I})} \\operatorname{n_{1}}^{- \\mathbb{I}}{(\\lambda,\\mathbb{I})} = 1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('n_1')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 4, "Pow(Function('n_1')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('n_1')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given f{(t_{2},\\theta_1)} = \\log{(\\theta_1 t_{2})}, then obtain ((- t_{2} + f{(t_{2},\\theta_1)})^{\\theta_1} (- t_{2} + \\log{(\\theta_1 t_{2})})^{- \\theta_1})^{t_{2}} = 1", "derivation": "f{(t_{2},\\theta_1)} = \\log{(\\theta_1 t_{2})} and - t_{2} + f{(t_{2},\\theta_1)} = - t_{2} + \\log{(\\theta_1 t_{2})} and (- t_{2} + f{(t_{2},\\theta_1)})^{\\theta_1} = (- t_{2} + \\log{(\\theta_1 t_{2})})^{\\theta_1} and (- t_{2} + f{(t_{2},\\theta_1)})^{\\theta_1} (- t_{2} + \\log{(\\theta_1 t_{2})})^{- \\theta_1} = 1 and ((- t_{2} + f{(t_{2},\\theta_1)})^{\\theta_1} (- t_{2} + \\log{(\\theta_1 t_{2})})^{- \\theta_1})^{t_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True))))"], [["minus", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 3, "Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))), Integer(1))"], [["power", 4, "Symbol('t_2', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))), Symbol('t_2', commutative=True)), Integer(1))"]]}, {"prompt": "Given s{(n_{1},\\mathbf{v})} = \\mathbf{v} + e^{n_{1}}, then derive \\frac{\\partial}{\\partial n_{1}} s{(n_{1},\\mathbf{v})} = e^{n_{1}}, then obtain 0 = \\mathbf{v} - s{(n_{1},\\mathbf{v})} + \\frac{\\partial}{\\partial n_{1}} s{(n_{1},\\mathbf{v})}", "derivation": "s{(n_{1},\\mathbf{v})} = \\mathbf{v} + e^{n_{1}} and 0 = \\mathbf{v} - s{(n_{1},\\mathbf{v})} + e^{n_{1}} and \\frac{\\partial}{\\partial n_{1}} s{(n_{1},\\mathbf{v})} = \\frac{\\partial}{\\partial n_{1}} (\\mathbf{v} + e^{n_{1}}) and \\frac{\\partial}{\\partial n_{1}} s{(n_{1},\\mathbf{v})} = e^{n_{1}} and 0 = \\mathbf{v} - s{(n_{1},\\mathbf{v})} + \\frac{\\partial}{\\partial n_{1}} s{(n_{1},\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('n_1', commutative=True))))"], [["minus", 1, "Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), exp(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), exp(Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(0), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Derivative(Function('s')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(f,L)} = - f + e^{L}, then obtain (\\dot{x}{(f,L)} - 1 + e)^{L} = (\\dot{x}{(f,L)} + e^{(- f - \\dot{x}{(f,L)} + e^{L} + 1)^{L}} - 1)^{L}", "derivation": "\\dot{x}{(f,L)} = - f + e^{L} and 1 = - f - \\dot{x}{(f,L)} + e^{L} + 1 and 1 = (- f - \\dot{x}{(f,L)} + e^{L} + 1)^{L} and e = e^{(- f - \\dot{x}{(f,L)} + e^{L} + 1)^{L}} and \\dot{x}{(f,L)} - 1 + e = \\dot{x}{(f,L)} + e^{(- f - \\dot{x}{(f,L)} + e^{L} + 1)^{L}} - 1 and (\\dot{x}{(f,L)} - 1 + e)^{L} = (\\dot{x}{(f,L)} + e^{(- f - \\dot{x}{(f,L)} + e^{L} + 1)^{L}} - 1)^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), exp(Symbol('L', commutative=True))))"], [["minus", 1, "Add(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(-1))"], "Equality(Integer(1), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))), exp(Symbol('L', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Integer(1), Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))), exp(Symbol('L', commutative=True)), Integer(1)), Symbol('L', commutative=True)))"], [["exp", 3], "Equality(E, exp(Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))), exp(Symbol('L', commutative=True)), Integer(1)), Symbol('L', commutative=True))))"], [["minus", 4, "Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(-1), E), Add(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), exp(Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))), exp(Symbol('L', commutative=True)), Integer(1)), Symbol('L', commutative=True))), Integer(-1)))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), Integer(-1), E), Symbol('L', commutative=True)), Pow(Add(Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True)), exp(Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('f', commutative=True), Symbol('L', commutative=True))), exp(Symbol('L', commutative=True)), Integer(1)), Symbol('L', commutative=True))), Integer(-1)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given r{(S)} = \\sin{(e^{S})} and v{(S)} = r{(S)} - e^{S}, then obtain \\frac{(r{(S)} - e^{S})^{S}}{r{(S)} - 1} = \\frac{v^{S}{(S)}}{r{(S)} - 1}", "derivation": "r{(S)} = \\sin{(e^{S})} and r{(S)} - e^{S} = - e^{S} + \\sin{(e^{S})} and (r{(S)} - e^{S})^{S} = (- e^{S} + \\sin{(e^{S})})^{S} and \\frac{(r{(S)} - e^{S})^{S}}{r{(S)} - 1} = \\frac{(- e^{S} + \\sin{(e^{S})})^{S}}{r{(S)} - 1} and v{(S)} = r{(S)} - e^{S} and v{(S)} = - e^{S} + \\sin{(e^{S})} and \\frac{(r{(S)} - e^{S})^{S}}{r{(S)} - 1} = \\frac{v^{S}{(S)}}{r{(S)} - 1}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('S', commutative=True)), sin(exp(Symbol('S', commutative=True))))"], [["minus", 1, "exp(Symbol('S', commutative=True))"], "Equality(Add(Function('r')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), sin(exp(Symbol('S', commutative=True)))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Function('r')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True)), Pow(Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), sin(exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"], [["divide", 3, "Add(Function('r')(Symbol('S', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Function('r')(Symbol('S', commutative=True)), Integer(-1)), Integer(-1)), Pow(Add(Function('r')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True))), Mul(Pow(Add(Function('r')(Symbol('S', commutative=True)), Integer(-1)), Integer(-1)), Pow(Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), sin(exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('S', commutative=True)), Add(Function('r')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('v')(Symbol('S', commutative=True)), Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), sin(exp(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Add(Function('r')(Symbol('S', commutative=True)), Integer(-1)), Integer(-1)), Pow(Add(Function('r')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True))), Mul(Pow(Add(Function('r')(Symbol('S', commutative=True)), Integer(-1)), Integer(-1)), Pow(Function('v')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(x^\\prime,s)} = \\log{(x^\\prime)}^{s}, then obtain \\int (\\frac{\\int \\operatorname{c_{0}}{(x^\\prime,s)} dx^\\prime}{\\int \\log{(x^\\prime)}^{s} dx^\\prime} + 1) dx^\\prime = \\int 2 dx^\\prime", "derivation": "\\operatorname{c_{0}}{(x^\\prime,s)} = \\log{(x^\\prime)}^{s} and \\int \\operatorname{c_{0}}{(x^\\prime,s)} dx^\\prime = \\int \\log{(x^\\prime)}^{s} dx^\\prime and \\frac{\\int \\operatorname{c_{0}}{(x^\\prime,s)} dx^\\prime}{\\int \\log{(x^\\prime)}^{s} dx^\\prime} = 1 and \\frac{\\int \\operatorname{c_{0}}{(x^\\prime,s)} dx^\\prime}{\\int \\log{(x^\\prime)}^{s} dx^\\prime} + 1 = 2 and \\int (\\frac{\\int \\operatorname{c_{0}}{(x^\\prime,s)} dx^\\prime}{\\int \\log{(x^\\prime)}^{s} dx^\\prime} + 1) dx^\\prime = \\int 2 dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('x^\\\\prime', commutative=True), Symbol('s', commutative=True)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('x^\\\\prime', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 2, "Integral(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Integral(Function('c_0')(Symbol('x^\\\\prime', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integral(Function('c_0')(Symbol('x^\\\\prime', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1)), Integer(2))"], [["integrate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Add(Mul(Integral(Function('c_0')(Symbol('x^\\\\prime', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Pow(Integral(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Integer(2), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given Z{(\\chi,\\tilde{g})} = \\chi - \\tilde{g} and \\psi{(\\chi,\\tilde{g})} = \\int (\\chi - \\tilde{g})^{\\chi} d\\chi, then obtain \\frac{\\psi{(\\chi,\\tilde{g})}}{Z{(\\chi,\\tilde{g})}} = \\frac{\\int Z^{\\chi}{(\\chi,\\tilde{g})} d\\chi}{Z{(\\chi,\\tilde{g})}}", "derivation": "Z{(\\chi,\\tilde{g})} = \\chi - \\tilde{g} and Z^{\\chi}{(\\chi,\\tilde{g})} = (\\chi - \\tilde{g})^{\\chi} and \\int Z^{\\chi}{(\\chi,\\tilde{g})} d\\chi = \\int (\\chi - \\tilde{g})^{\\chi} d\\chi and \\psi{(\\chi,\\tilde{g})} = \\int (\\chi - \\tilde{g})^{\\chi} d\\chi and \\psi{(\\chi,\\tilde{g})} = \\int Z^{\\chi}{(\\chi,\\tilde{g})} d\\chi and \\frac{\\psi{(\\chi,\\tilde{g})}}{Z{(\\chi,\\tilde{g})}} = \\frac{\\int Z^{\\chi}{(\\chi,\\tilde{g})} d\\chi}{Z{(\\chi,\\tilde{g})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 5, "Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Integral(Pow(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\pi{(J,\\phi,E_{\\lambda})} = (- E_{\\lambda} + \\phi)^{J}, then obtain - \\frac{J (- E_{\\lambda} + \\phi)^{J}}{- E_{\\lambda} + \\phi} + \\frac{\\partial}{\\partial E_{\\lambda}} \\pi{(J,\\phi,E_{\\lambda})} + 2 = - \\frac{2 J (- E_{\\lambda} + \\phi)^{J}}{- E_{\\lambda} + \\phi} + 2", "derivation": "\\pi{(J,\\phi,E_{\\lambda})} = (- E_{\\lambda} + \\phi)^{J} and E_{\\lambda} + \\pi{(J,\\phi,E_{\\lambda})} = E_{\\lambda} + (- E_{\\lambda} + \\phi)^{J} and \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\pi{(J,\\phi,E_{\\lambda})}) = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + (- E_{\\lambda} + \\phi)^{J}) and \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + (- E_{\\lambda} + \\phi)^{J}) + \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + \\pi{(J,\\phi,E_{\\lambda})}) = 2 \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} + (- E_{\\lambda} + \\phi)^{J}) and - \\frac{J (- E_{\\lambda} + \\phi)^{J}}{- E_{\\lambda} + \\phi} + \\frac{\\partial}{\\partial E_{\\lambda}} \\pi{(J,\\phi,E_{\\lambda})} + 2 = - \\frac{2 J (- E_{\\lambda} + \\phi)^{J}}{- E_{\\lambda} + \\phi} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Derivative(Function('\\\\pi')(Symbol('J', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(2)), Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\phi', commutative=True)), Symbol('J', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\eta^{\\prime}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and \\operatorname{t_{2}}{(v_{z})} = \\sin{(v_{z})}, then obtain 1 + \\frac{\\sin{(\\sin{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}} = 1 + \\frac{\\sin{(\\operatorname{t_{2}}{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}}", "derivation": "\\eta^{\\prime}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and \\frac{\\eta^{\\prime}{(v_{z})}}{\\sin{(v_{z})}} = \\frac{\\sin{(\\sin{(v_{z})})}}{\\sin{(v_{z})}} and \\operatorname{t_{2}}{(v_{z})} = \\sin{(v_{z})} and \\frac{\\eta^{\\prime}{(v_{z})}}{\\operatorname{t_{2}}{(v_{z})}} = \\frac{\\sin{(\\operatorname{t_{2}}{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}} and \\frac{\\sin{(\\sin{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}} = \\frac{\\sin{(\\operatorname{t_{2}}{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}} and 1 + \\frac{\\sin{(\\sin{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}} = 1 + \\frac{\\sin{(\\operatorname{t_{2}}{(v_{z})})}}{\\operatorname{t_{2}}{(v_{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('v_z', commutative=True)), sin(sin(Symbol('v_z', commutative=True))))"], [["divide", 1, "sin(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('v_z', commutative=True)), Pow(sin(Symbol('v_z', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('v_z', commutative=True)), Integer(-1)), sin(sin(Symbol('v_z', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('v_z', commutative=True)), Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1))), Mul(Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1)), sin(Function('t_2')(Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1)), sin(sin(Symbol('v_z', commutative=True)))), Mul(Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1)), sin(Function('t_2')(Symbol('v_z', commutative=True)))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1)), sin(sin(Symbol('v_z', commutative=True))))), Add(Integer(1), Mul(Pow(Function('t_2')(Symbol('v_z', commutative=True)), Integer(-1)), sin(Function('t_2')(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{H}{(g,\\pi,A)} = \\frac{\\pi + g}{A}, then obtain ((\\pi + g) \\frac{\\partial^{2}}{\\partial g\\partial A} \\mathbf{H}{(g,\\pi,A)} + \\frac{\\partial}{\\partial A} \\mathbf{H}{(g,\\pi,A)})^{\\pi} = (- \\frac{2 (\\pi + g)}{A^{2}})^{\\pi}", "derivation": "\\mathbf{H}{(g,\\pi,A)} = \\frac{\\pi + g}{A} and (\\pi + g) \\mathbf{H}{(g,\\pi,A)} = \\frac{(\\pi + g)^{2}}{A} and \\frac{\\partial}{\\partial g} (\\pi + g) \\mathbf{H}{(g,\\pi,A)} = \\frac{\\partial}{\\partial g} \\frac{(\\pi + g)^{2}}{A} and \\frac{\\partial^{2}}{\\partial A\\partial g} (\\pi + g) \\mathbf{H}{(g,\\pi,A)} = \\frac{\\partial^{2}}{\\partial A\\partial g} \\frac{(\\pi + g)^{2}}{A} and (\\frac{\\partial^{2}}{\\partial A\\partial g} (\\pi + g) \\mathbf{H}{(g,\\pi,A)})^{\\pi} = (\\frac{\\partial^{2}}{\\partial A\\partial g} \\frac{(\\pi + g)^{2}}{A})^{\\pi} and ((\\pi + g) \\frac{\\partial^{2}}{\\partial g\\partial A} \\mathbf{H}{(g,\\pi,A)} + \\frac{\\partial}{\\partial A} \\mathbf{H}{(g,\\pi,A)})^{\\pi} = (- \\frac{2 (\\pi + g)}{A^{2}})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Integer(2))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Symbol('\\\\pi', commutative=True)), Pow(Mul(Integer(-1), Integer(2), Pow(Symbol('A', commutative=True), Integer(-2)), Add(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True))), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\psi{(I,a^{\\dagger})} = I a^{\\dagger} and L{(a^{\\dagger},I)} = - a^{\\dagger} + \\frac{\\psi{(I,a^{\\dagger})}}{I a^{\\dagger}}, then obtain L{(a^{\\dagger},I)} = 1 - a^{\\dagger}", "derivation": "\\psi{(I,a^{\\dagger})} = I a^{\\dagger} and \\frac{\\psi{(I,a^{\\dagger})}}{I a^{\\dagger}} = 1 and - a^{\\dagger} + \\frac{\\psi{(I,a^{\\dagger})}}{I a^{\\dagger}} = 1 - a^{\\dagger} and L{(a^{\\dagger},I)} = - a^{\\dagger} + \\frac{\\psi{(I,a^{\\dagger})}}{I a^{\\dagger}} and L{(a^{\\dagger},I)} = 1 - a^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Mul(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integer(1))"], [["minus", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('I', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('L')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('I', commutative=True)), Add(Integer(1), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given m{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then obtain 2 = e^{m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}} + 1", "derivation": "m{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and 0 = - m{(\\mathbf{E})} + \\cos{(\\mathbf{E})} and 0 = m{(\\mathbf{E})} - \\cos{(\\mathbf{E})} and \\mathbf{E} = \\mathbf{E} + m{(\\mathbf{E})} - \\cos{(\\mathbf{E})} and 1 = e^{m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}} and 1 + \\frac{\\mathbf{E} + m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}}{\\mathbf{E}} = e^{m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}} + \\frac{\\mathbf{E} + m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}}{\\mathbf{E}} and 2 = e^{m{(\\mathbf{E})} - \\cos{(\\mathbf{E})}} + 1", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Function('m')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m')(Symbol('\\\\mathbf{E}', commutative=True))), cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Symbol('\\\\mathbf{E}', commutative=True), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["exp", 3], "Equality(Integer(1), exp(Add(Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["add", 5, "Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))), Add(exp(Add(Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integer(2), Add(exp(Add(Function('m')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True))))), Integer(1)))"]]}, {"prompt": "Given \\lambda{(M_{E})} = \\int \\log{(M_{E})} dM_{E}, then derive - M_{E} \\log{(M_{E})} + M_{E} - \\mathbf{H} + \\lambda{(M_{E})} = 0, then obtain \\lambda{(M_{E})} + \\int (- M_{E} \\log{(M_{E})} + M_{E} - \\mathbf{H} + \\lambda{(M_{E})}) dM_{E} - \\int \\log{(M_{E})} dM_{E} = \\lambda{(M_{E})} + \\int 0 dM_{E} - \\int \\log{(M_{E})} dM_{E}", "derivation": "\\lambda{(M_{E})} = \\int \\log{(M_{E})} dM_{E} and \\lambda{(M_{E})} - \\int \\log{(M_{E})} dM_{E} = 0 and - M_{E} \\log{(M_{E})} + M_{E} - \\mathbf{H} + \\lambda{(M_{E})} = 0 and \\int (- M_{E} \\log{(M_{E})} + M_{E} - \\mathbf{H} + \\lambda{(M_{E})}) dM_{E} = \\int 0 dM_{E} and \\lambda{(M_{E})} + \\int (- M_{E} \\log{(M_{E})} + M_{E} - \\mathbf{H} + \\lambda{(M_{E})}) dM_{E} - \\int \\log{(M_{E})} dM_{E} = \\lambda{(M_{E})} + \\int 0 dM_{E} - \\int \\log{(M_{E})} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('M_E', commutative=True)), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["minus", 1, "Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\lambda')(Symbol('M_E', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\lambda')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True))))"], [["add", 4, "Add(Function('\\\\lambda')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], "Equality(Add(Function('\\\\lambda')(Symbol('M_E', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\lambda')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), Add(Function('\\\\lambda')(Symbol('M_E', commutative=True)), Integral(Integer(0), Tuple(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))))"]]}, {"prompt": "Given \\lambda{(\\theta)} = \\sin{(\\cos{(\\theta)})} and \\Psi^{\\dagger}{(\\theta)} = \\lambda^{\\theta}{(\\theta)} and \\operatorname{E_{n}}{(\\theta)} = \\sin^{\\theta}{(\\cos{(\\theta)})}, then obtain 0 = - (- \\operatorname{E_{n}}{(\\theta)} + \\lambda^{\\theta}{(\\theta)}) \\operatorname{E_{n}}{(\\theta)}", "derivation": "\\lambda{(\\theta)} = \\sin{(\\cos{(\\theta)})} and \\Psi^{\\dagger}{(\\theta)} = \\lambda^{\\theta}{(\\theta)} and \\Psi^{\\dagger}{(\\theta)} = \\sin^{\\theta}{(\\cos{(\\theta)})} and 0 = - \\Psi^{\\dagger}{(\\theta)} + \\sin^{\\theta}{(\\cos{(\\theta)})} and 0 = - \\Psi^{\\dagger}{(\\theta)} + \\lambda^{\\theta}{(\\theta)} and 0 = - (- \\Psi^{\\dagger}{(\\theta)} + \\lambda^{\\theta}{(\\theta)}) \\Psi^{\\dagger}{(\\theta)} and \\operatorname{E_{n}}{(\\theta)} = \\sin^{\\theta}{(\\cos{(\\theta)})} and \\operatorname{E_{n}}{(\\theta)} = \\Psi^{\\dagger}{(\\theta)} and 0 = - (- \\operatorname{E_{n}}{(\\theta)} + \\lambda^{\\theta}{(\\theta)}) \\operatorname{E_{n}}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), sin(cos(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)), Pow(sin(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["minus", 3, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True))), Pow(sin(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\theta', commutative=True)), Pow(sin(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Function('E_n')(Symbol('\\\\theta', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('E_n')(Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\lambda')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Function('E_n')(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\varphi{(C_{d})} = \\log{(C_{d})} and f{(\\hat{H},\\Psi)} = \\Psi \\hat{H}, then obtain \\frac{f{(\\hat{H},\\Psi)}}{\\varphi{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}} = \\frac{\\Psi \\hat{H}}{\\varphi{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}}", "derivation": "\\varphi{(C_{d})} = \\log{(C_{d})} and f{(\\hat{H},\\Psi)} = \\Psi \\hat{H} and \\frac{f{(\\hat{H},\\Psi)}}{\\log{(C_{d})}} = \\frac{\\Psi \\hat{H}}{\\log{(C_{d})}} and \\frac{f{(\\hat{H},\\Psi)}}{\\log{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}} = \\frac{\\Psi \\hat{H}}{\\log{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}} and \\frac{f{(\\hat{H},\\Psi)}}{\\log{(C_{d})}} - \\frac{1}{\\log{(C_{d})}} = \\frac{\\Psi \\hat{H}}{\\log{(C_{d})}} - \\frac{1}{\\log{(C_{d})}} and \\frac{f{(\\hat{H},\\Psi)}}{\\varphi{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}} = \\frac{\\Psi \\hat{H}}{\\varphi{(C_{d})}} - \\frac{1}{\\varphi{(C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], ["get_premise", "Equality(Function('f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 2, "log(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))))"], [["minus", 3, "Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Function('f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('C_d', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(log(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(log(Symbol('C_d', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1)), Function('f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\psi)} = e^{\\psi} and \\operatorname{c_{0}}{(\\psi)} = e^{\\psi}, then obtain \\cos{((\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\mathbf{s}^{\\psi}{(\\psi)})} = \\cos{((\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\operatorname{c_{0}}^{\\psi}{(\\psi)})}", "derivation": "\\mathbf{s}{(\\psi)} = e^{\\psi} and \\mathbf{s}^{\\psi}{(\\psi)} = (e^{\\psi})^{\\psi} and \\operatorname{c_{0}}{(\\psi)} = e^{\\psi} and \\mathbf{s}^{\\psi}{(\\psi)} = \\operatorname{c_{0}}^{\\psi}{(\\psi)} and (\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\mathbf{s}^{\\psi}{(\\psi)} = (\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\operatorname{c_{0}}^{\\psi}{(\\psi)} and \\cos{((\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\mathbf{s}^{\\psi}{(\\psi)})} = \\cos{((\\mathbf{s}^{\\psi}{(\\psi)})^{- \\psi} \\operatorname{c_{0}}^{\\psi}{(\\psi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(exp(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Function('c_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["divide", 4, "Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))), Mul(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Pow(Function('c_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["cos", 5], "Equality(cos(Mul(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))), cos(Mul(Pow(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Pow(Function('c_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain \\frac{d}{d \\hat{H}_l} (\\phi_{2}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} - 1) = \\frac{d}{d \\hat{H}_l} (-1)", "derivation": "\\phi_{2}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\phi_{2}{(\\hat{H}_l)} + 1 = \\sin{(\\hat{H}_l)} + 1 and \\phi_{2}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} = 0 and \\phi_{2}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} - 1 = -1 and \\frac{d}{d \\hat{H}_l} (\\phi_{2}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} - 1) = \\frac{d}{d \\hat{H}_l} (-1)", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)), Add(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)))"], [["minus", 2, "Add(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(1))"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(0))"], [["minus", 3, 1], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)), Integer(-1))"], [["differentiate", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Function('\\\\phi_2')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(\\mu,\\varphi^*)} = - \\mu + \\varphi^*, then obtain \\int \\frac{-1 + \\frac{\\eta{(\\mu,\\varphi^*)}}{- \\mu + \\varphi^*}}{- \\mu + \\varphi^*} d\\varphi^* = \\int 0 d\\varphi^*", "derivation": "\\eta{(\\mu,\\varphi^*)} = - \\mu + \\varphi^* and \\frac{\\eta{(\\mu,\\varphi^*)}}{- \\mu + \\varphi^*} = 1 and -1 + \\frac{\\eta{(\\mu,\\varphi^*)}}{- \\mu + \\varphi^*} = 0 and \\frac{-1 + \\frac{\\eta{(\\mu,\\varphi^*)}}{- \\mu + \\varphi^*}}{- \\mu + \\varphi^*} = 0 and \\int \\frac{-1 + \\frac{\\eta{(\\mu,\\varphi^*)}}{- \\mu + \\varphi^*}}{- \\mu + \\varphi^*} d\\varphi^* = \\int 0 d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Integer(0))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Integer(0))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\mu{(E)} = \\log{(E)}, then obtain \\log{(E \\mu{(E)} + E \\log{(E)})}^{2} = \\log{(2 E \\log{(E)})}^{2}", "derivation": "\\mu{(E)} = \\log{(E)} and E \\mu{(E)} = E \\log{(E)} and E \\mu{(E)} + E \\log{(E)} = 2 E \\log{(E)} and \\log{(E \\mu{(E)} + E \\log{(E)})} = \\log{(2 E \\log{(E)})} and \\log{(E \\mu{(E)} + E \\log{(E)})}^{2} = \\log{(2 E \\log{(E)})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["times", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('\\\\mu')(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))))"], [["add", 2, "Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Symbol('E', commutative=True), Function('\\\\mu')(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True)))), Mul(Integer(2), Symbol('E', commutative=True), log(Symbol('E', commutative=True))))"], [["log", 3], "Equality(log(Add(Mul(Symbol('E', commutative=True), Function('\\\\mu')(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))))), log(Mul(Integer(2), Symbol('E', commutative=True), log(Symbol('E', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(log(Add(Mul(Symbol('E', commutative=True), Function('\\\\mu')(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))))), Integer(2)), Pow(log(Mul(Integer(2), Symbol('E', commutative=True), log(Symbol('E', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{s}{(c,f^{\\prime})} = - c + f^{\\prime}, then derive \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = \\frac{\\partial}{\\partial c} (\\nabla - c f^{\\prime} + \\frac{(f^{\\prime})^{2}}{2}), then derive \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = - f^{\\prime}, then obtain \\mathbf{s}{(c,f^{\\prime})} + \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = - f^{\\prime} + \\mathbf{s}{(c,f^{\\prime})}", "derivation": "\\mathbf{s}{(c,f^{\\prime})} = - c + f^{\\prime} and \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = \\int (- c + f^{\\prime}) df^{\\prime} and \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = \\frac{\\partial}{\\partial c} \\int (- c + f^{\\prime}) df^{\\prime} and \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = \\frac{\\partial}{\\partial c} (\\nabla - c f^{\\prime} + \\frac{(f^{\\prime})^{2}}{2}) and \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = - f^{\\prime} and \\mathbf{s}{(c,f^{\\prime})} + \\frac{\\partial}{\\partial c} \\int \\mathbf{s}{(c,f^{\\prime})} df^{\\prime} = - f^{\\prime} + \\mathbf{s}{(c,f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 5, "Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Integral(Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{s}')(Symbol('c', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given H{(a^{\\dagger},t)} = - a^{\\dagger} + t, then obtain e^{\\frac{\\partial^{2}}{\\partial t^{2}} H{(a^{\\dagger},t)}} = e^{\\frac{\\partial^{2}}{\\partial t^{2}} (- a^{\\dagger} + t)}", "derivation": "H{(a^{\\dagger},t)} = - a^{\\dagger} + t and \\frac{\\partial}{\\partial t} H{(a^{\\dagger},t)} = \\frac{\\partial}{\\partial t} (- a^{\\dagger} + t) and \\frac{\\partial^{2}}{\\partial t^{2}} H{(a^{\\dagger},t)} = \\frac{\\partial^{2}}{\\partial t^{2}} (- a^{\\dagger} + t) and e^{\\frac{\\partial^{2}}{\\partial t^{2}} H{(a^{\\dagger},t)}} = e^{\\frac{\\partial^{2}}{\\partial t^{2}} (- a^{\\dagger} + t)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2))))"], [["exp", 3], "Equality(exp(Derivative(Function('H')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2)))), exp(Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\mu_{0}{(\\nabla,\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P - \\nabla)}, then obtain \\nabla + \\eta^{\\prime}^{2}{(\\sigma_x)} - \\eta^{\\prime}{(\\sigma_x)} \\cos{(\\sigma_x)} - \\sin{(\\mathbf{J}_P - \\nabla)} = \\nabla - \\sin{(\\mathbf{J}_P - \\nabla)}", "derivation": "\\eta^{\\prime}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\eta^{\\prime}^{2}{(\\sigma_x)} = \\eta^{\\prime}{(\\sigma_x)} \\cos{(\\sigma_x)} and \\eta^{\\prime}^{2}{(\\sigma_x)} - \\eta^{\\prime}{(\\sigma_x)} \\cos{(\\sigma_x)} = 0 and \\mu_{0}{(\\nabla,\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P - \\nabla)} and \\nabla + \\eta^{\\prime}^{2}{(\\sigma_x)} - \\eta^{\\prime}{(\\sigma_x)} \\cos{(\\sigma_x)} - \\mu_{0}{(\\nabla,\\mathbf{J}_P)} = \\nabla - \\mu_{0}{(\\nabla,\\mathbf{J}_P)} and \\nabla + \\eta^{\\prime}^{2}{(\\sigma_x)} - \\eta^{\\prime}{(\\sigma_x)} \\cos{(\\sigma_x)} - \\sin{(\\mathbf{J}_P - \\nabla)} = \\nabla - \\sin{(\\mathbf{J}_P - \\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('\\\\nabla', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{r}{(v_{1},q)} = \\sin{(\\frac{q}{v_{1}})}, then obtain - v_{1} + (\\mathbf{r}{(v_{1},q)} - \\frac{1}{v_{1}})^{q} = - v_{1} + (\\sin{(\\frac{q}{v_{1}})} - \\frac{1}{v_{1}})^{q}", "derivation": "\\mathbf{r}{(v_{1},q)} = \\sin{(\\frac{q}{v_{1}})} and \\mathbf{r}{(v_{1},q)} - \\frac{1}{v_{1}} = \\sin{(\\frac{q}{v_{1}})} - \\frac{1}{v_{1}} and (\\mathbf{r}{(v_{1},q)} - \\frac{1}{v_{1}})^{q} = (\\sin{(\\frac{q}{v_{1}})} - \\frac{1}{v_{1}})^{q} and - v_{1} + (\\mathbf{r}{(v_{1},q)} - \\frac{1}{v_{1}})^{q} = - v_{1} + (\\sin{(\\frac{q}{v_{1}})} - \\frac{1}{v_{1}})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), sin(Mul(Symbol('q', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))))"], [["minus", 1, "Pow(Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Add(sin(Mul(Symbol('q', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('q', commutative=True)), Pow(Add(sin(Mul(Symbol('q', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('q', commutative=True)))"], [["minus", 3, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Add(Function('\\\\mathbf{r}')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Pow(Add(sin(Mul(Symbol('q', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\rho{(y,u)} = u y, then obtain \\int (u y + y + 2 \\rho{(y,u)}) du = \\int (2 u y + y + \\rho{(y,u)}) du", "derivation": "\\rho{(y,u)} = u y and 2 \\rho{(y,u)} = u y + \\rho{(y,u)} and y + 2 \\rho{(y,u)} = u y + y + \\rho{(y,u)} and y + 3 \\rho{(y,u)} = u y + y + 2 \\rho{(y,u)} and y + 3 \\rho{(y,u)} = 2 u y + y + \\rho{(y,u)} and u y + y + 2 \\rho{(y,u)} = 2 u y + y + \\rho{(y,u)} and \\int (u y + y + 2 \\rho{(y,u)}) du = \\int (2 u y + y + \\rho{(y,u)}) du", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))))"], [["add", 2, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))))"], [["add", 3, "Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(3), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(3), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 6, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Mul(Integer(2), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('u', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho')(Symbol('y', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\delta,T,\\mathbf{M})} = T - \\delta + \\mathbf{M}, then obtain (\\delta (T - \\delta + \\mathbf{M}) + \\delta \\phi{(\\delta,T,\\mathbf{M})})^{\\delta} + \\phi^{\\delta}{(\\delta,T,\\mathbf{M})} = (2 \\delta (T - \\delta + \\mathbf{M}))^{\\delta} + \\phi^{\\delta}{(\\delta,T,\\mathbf{M})}", "derivation": "\\phi{(\\delta,T,\\mathbf{M})} = T - \\delta + \\mathbf{M} and \\delta \\phi{(\\delta,T,\\mathbf{M})} = \\delta (T - \\delta + \\mathbf{M}) and \\phi^{\\delta}{(\\delta,T,\\mathbf{M})} = (T - \\delta + \\mathbf{M})^{\\delta} and \\delta (T - \\delta + \\mathbf{M}) + \\delta \\phi{(\\delta,T,\\mathbf{M})} = 2 \\delta (T - \\delta + \\mathbf{M}) and (\\delta (T - \\delta + \\mathbf{M}) + \\delta \\phi{(\\delta,T,\\mathbf{M})})^{\\delta} = (2 \\delta (T - \\delta + \\mathbf{M}))^{\\delta} and (\\delta (T - \\delta + \\mathbf{M}) + \\delta \\phi{(\\delta,T,\\mathbf{M})})^{\\delta} + (T - \\delta + \\mathbf{M})^{\\delta} = (2 \\delta (T - \\delta + \\mathbf{M}))^{\\delta} + (T - \\delta + \\mathbf{M})^{\\delta} and (\\delta (T - \\delta + \\mathbf{M}) + \\delta \\phi{(\\delta,T,\\mathbf{M})})^{\\delta} + \\phi^{\\delta}{(\\delta,T,\\mathbf{M})} = (2 \\delta (T - \\delta + \\mathbf{M}))^{\\delta} + \\phi^{\\delta}{(\\delta,T,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(2), Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["add", 5, "Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Pow(Mul(Integer(2), Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Pow(Mul(Integer(2), Symbol('\\\\delta', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given G{(\\varphi^*,Q)} = \\frac{\\partial}{\\partial \\varphi^*} (Q + \\varphi^*), then derive Q G{(\\varphi^*,Q)} + 1 = Q + 1, then obtain \\int (Q G{(\\varphi^*,Q)} + 1) dQ = \\frac{Q^{2}}{2} + Q + \\mathbf{M}", "derivation": "G{(\\varphi^*,Q)} = \\frac{\\partial}{\\partial \\varphi^*} (Q + \\varphi^*) and Q G{(\\varphi^*,Q)} = Q \\frac{\\partial}{\\partial \\varphi^*} (Q + \\varphi^*) and Q G{(\\varphi^*,Q)} + 1 = Q \\frac{\\partial}{\\partial \\varphi^*} (Q + \\varphi^*) + 1 and Q G{(\\varphi^*,Q)} + 1 = Q + 1 and \\int (Q G{(\\varphi^*,Q)} + 1) dQ = \\int (Q + 1) dQ and \\int (Q G{(\\varphi^*,Q)} + 1) dQ = \\frac{Q^{2}}{2} + Q + \\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Derivative(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), Derivative(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Integer(1)), Add(Mul(Symbol('Q', commutative=True), Derivative(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Integer(1)), Add(Symbol('Q', commutative=True), Integer(1)))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('Q', commutative=True), Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Integer(1)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Mul(Symbol('Q', commutative=True), Function('G')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True))), Integer(1)), Tuple(Symbol('Q', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(E_{x})} = \\log{(E_{x})}, then obtain - (\\log{(E_{x})}^{E_{x}})^{E_{x}} + \\mathbf{J}_f^{E_{x}}{(E_{x})} = - (\\log{(E_{x})}^{E_{x}})^{E_{x}} + \\log{(E_{x})}^{E_{x}}", "derivation": "\\mathbf{J}_f{(E_{x})} = \\log{(E_{x})} and \\mathbf{J}_f^{E_{x}}{(E_{x})} = \\log{(E_{x})}^{E_{x}} and (\\mathbf{J}_f^{E_{x}}{(E_{x})})^{E_{x}} = (\\log{(E_{x})}^{E_{x}})^{E_{x}} and - (\\mathbf{J}_f^{E_{x}}{(E_{x})})^{E_{x}} + \\mathbf{J}_f^{E_{x}}{(E_{x})} = - (\\mathbf{J}_f^{E_{x}}{(E_{x})})^{E_{x}} + \\log{(E_{x})}^{E_{x}} and - (\\log{(E_{x})}^{E_{x}})^{E_{x}} + \\mathbf{J}_f^{E_{x}}{(E_{x})} = - (\\log{(E_{x})}^{E_{x}})^{E_{x}} + \\log{(E_{x})}^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["minus", 2, "Pow(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Pow(log(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given M{(h,\\sigma_x)} = h^{\\sigma_x} and \\mathbf{J}_P{(h,\\sigma_x)} = 2 M{(h,\\sigma_x)}, then obtain - h^{\\sigma_x} + \\mathbf{J}_P{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x = - h^{\\sigma_x} + 2 M{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x", "derivation": "M{(h,\\sigma_x)} = h^{\\sigma_x} and \\mathbf{J}_P{(h,\\sigma_x)} = 2 M{(h,\\sigma_x)} and \\mathbf{J}_P{(h,\\sigma_x)} = 2 h^{\\sigma_x} and 2 M{(h,\\sigma_x)} = 2 h^{\\sigma_x} and \\mathbf{J}_P{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x = 2 h^{\\sigma_x} \\int M{(h,\\sigma_x)} d\\sigma_x and \\mathbf{J}_P{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x = 2 M{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x and - h^{\\sigma_x} + \\mathbf{J}_P{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x = - h^{\\sigma_x} + 2 M{(h,\\sigma_x)} \\int M{(h,\\sigma_x)} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 3, "Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(2), Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["minus", 6, "Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Function('\\\\mathbf{J}_P')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(2), Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integral(Function('M')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(r_{0},E_{x})} = \\log{(E_{x}^{r_{0}})}, then derive \\frac{\\partial}{\\partial r_{0}} \\operatorname{A_{z}}{(r_{0},E_{x})} = \\log{(E_{x})}, then obtain \\frac{\\partial}{\\partial r_{0}} \\log{(E_{x}^{r_{0}})} = \\log{(E_{x})}", "derivation": "\\operatorname{A_{z}}{(r_{0},E_{x})} = \\log{(E_{x}^{r_{0}})} and - E_{x} + \\operatorname{A_{z}}{(r_{0},E_{x})} = - E_{x} + \\log{(E_{x}^{r_{0}})} and \\frac{\\partial}{\\partial r_{0}} (- E_{x} + \\operatorname{A_{z}}{(r_{0},E_{x})}) = \\frac{\\partial}{\\partial r_{0}} (- E_{x} + \\log{(E_{x}^{r_{0}})}) and \\frac{\\partial}{\\partial r_{0}} \\operatorname{A_{z}}{(r_{0},E_{x})} = \\log{(E_{x})} and \\frac{\\partial}{\\partial r_{0}} \\log{(E_{x}^{r_{0}})} = \\log{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('r_0', commutative=True), Symbol('E_x', commutative=True)), log(Pow(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True))))"], [["minus", 1, "Symbol('E_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('A_z')(Symbol('r_0', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), log(Pow(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True)))))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('A_z')(Symbol('r_0', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), log(Pow(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_z')(Symbol('r_0', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), log(Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(Pow(Symbol('E_x', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), log(Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{p})} = \\cos{(\\hat{p})}, then obtain (\\hat{p} + \\mathbf{r}{(\\hat{p})}) \\mathbf{r}{(\\hat{p})} \\cos{(\\hat{p})} = (\\hat{p} + \\mathbf{r}{(\\hat{p})}) \\cos^{2}{(\\hat{p})}", "derivation": "\\mathbf{r}{(\\hat{p})} = \\cos{(\\hat{p})} and \\mathbf{r}{(\\hat{p})} \\cos{(\\hat{p})} = \\cos^{2}{(\\hat{p})} and \\hat{p} + \\mathbf{r}{(\\hat{p})} = \\hat{p} + \\cos{(\\hat{p})} and (\\hat{p} + \\cos{(\\hat{p})}) \\mathbf{r}{(\\hat{p})} \\cos{(\\hat{p})} = (\\hat{p} + \\cos{(\\hat{p})}) \\cos^{2}{(\\hat{p})} and (\\hat{p} + \\mathbf{r}{(\\hat{p})}) \\mathbf{r}{(\\hat{p})} \\cos{(\\hat{p})} = (\\hat{p} + \\mathbf{r}{(\\hat{p})}) \\cos^{2}{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True))), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(2)))"], [["add", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\hat{p}', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{p}', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Add(Symbol('\\\\hat{p}', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True)), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}', commutative=True))), Pow(cos(Symbol('\\\\hat{p}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given n{(\\lambda,C)} = e^{C \\lambda} and \\operatorname{A_{z}}{(\\lambda,C)} = \\frac{\\partial}{\\partial C} e^{C \\lambda}, then obtain C \\operatorname{A_{z}}{(\\lambda,C)} n{(\\lambda,C)} = C \\operatorname{A_{z}}{(\\lambda,C)} e^{C \\lambda}", "derivation": "n{(\\lambda,C)} = e^{C \\lambda} and \\frac{\\partial}{\\partial C} n{(\\lambda,C)} = \\frac{\\partial}{\\partial C} e^{C \\lambda} and n{(\\lambda,C)} \\frac{\\partial}{\\partial C} n{(\\lambda,C)} = e^{C \\lambda} \\frac{\\partial}{\\partial C} n{(\\lambda,C)} and C n{(\\lambda,C)} \\frac{\\partial}{\\partial C} n{(\\lambda,C)} = C e^{C \\lambda} \\frac{\\partial}{\\partial C} n{(\\lambda,C)} and \\operatorname{A_{z}}{(\\lambda,C)} = \\frac{\\partial}{\\partial C} e^{C \\lambda} and \\frac{\\partial}{\\partial C} n{(\\lambda,C)} = \\operatorname{A_{z}}{(\\lambda,C)} and C \\operatorname{A_{z}}{(\\lambda,C)} n{(\\lambda,C)} = C \\operatorname{A_{z}}{(\\lambda,C)} e^{C \\lambda}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Mul(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))), Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('C', commutative=True), exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))), Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Derivative(exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Symbol('C', commutative=True), Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Function('n')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), Function('A_z')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), exp(Mul(Symbol('C', commutative=True), Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)}, then derive \\frac{\\phi{(\\mu_0)}}{\\mu_0} = \\frac{1}{\\mu_0^{2}}, then obtain (\\frac{\\frac{d}{d \\mu_0} \\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} = (\\frac{1}{\\mu_0^{2}})^{\\mu_0}", "derivation": "\\phi{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and \\phi{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)} = (\\frac{d}{d \\mu_0} \\log{(\\mu_0)})^{2} and \\frac{\\phi{(\\mu_0)}}{\\mu_0} = \\frac{1}{\\mu_0^{2}} and (\\frac{\\phi{(\\mu_0)}}{\\mu_0})^{\\mu_0} = (\\frac{1}{\\mu_0^{2}})^{\\mu_0} and \\frac{\\frac{d}{d \\mu_0} \\log{(\\mu_0)}}{\\mu_0} = \\frac{1}{\\mu_0^{2}} and \\frac{\\phi{(\\mu_0)}}{\\mu_0} = \\frac{\\frac{d}{d \\mu_0} \\log{(\\mu_0)}}{\\mu_0} and (\\frac{\\frac{d}{d \\mu_0} \\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} = (\\frac{1}{\\mu_0^{2}})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["times", 1, "Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Pow(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True))), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)))"], [["power", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given a{(z)} = \\cos{(z)}, then derive a{(z)} \\cos{(z)} + \\int a{(z)} dz = b + a{(z)} \\cos{(z)} + \\sin{(z)}, then derive b + a^{2}{(z)} + \\sin{(z)} = \\Omega + a^{2}{(z)} + \\sin{(z)}, then obtain \\Omega + a^{2}{(z)} + \\sin{(z)} = a^{2}{(z)} + \\int \\cos{(z)} dz", "derivation": "a{(z)} = \\cos{(z)} and \\int a{(z)} dz = \\int \\cos{(z)} dz and a{(z)} \\cos{(z)} + \\int a{(z)} dz = a{(z)} \\cos{(z)} + \\int \\cos{(z)} dz and a{(z)} \\cos{(z)} + \\int a{(z)} dz = b + a{(z)} \\cos{(z)} + \\sin{(z)} and b + a{(z)} \\cos{(z)} + \\sin{(z)} = a{(z)} \\cos{(z)} + \\int \\cos{(z)} dz and b + \\sin{(z)} + \\cos^{2}{(z)} = \\cos^{2}{(z)} + \\int \\cos{(z)} dz and b + a^{2}{(z)} + \\sin{(z)} = a^{2}{(z)} + \\int \\cos{(z)} dz and b + a^{2}{(z)} + \\sin{(z)} = \\Omega + a^{2}{(z)} + \\sin{(z)} and \\Omega + a^{2}{(z)} + \\sin{(z)} = a^{2}{(z)} + \\int \\cos{(z)} dz", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('a')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["add", 2, "Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Integral(Function('a')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Integral(Function('a')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Symbol('b', commutative=True), Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('b', commutative=True), Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))), Add(Mul(Function('a')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('b', commutative=True), sin(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(2))), Add(Pow(cos(Symbol('z', commutative=True)), Integer(2)), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('b', commutative=True), Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), sin(Symbol('z', commutative=True))), Add(Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('b', commutative=True), Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), sin(Symbol('z', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), sin(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Add(Symbol('\\\\Omega', commutative=True), Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), sin(Symbol('z', commutative=True))), Add(Pow(Function('a')(Symbol('z', commutative=True)), Integer(2)), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f^{*},M_{E})} = M_{E} + f^{*}, then derive \\int \\operatorname{E_{n}}{(f^{*},M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} f^{*} + v_{z}, then obtain \\log{(\\int (M_{E} + f^{*}) dM_{E})} = \\log{(\\frac{M_{E}^{2}}{2} + M_{E} f^{*} + v_{z})}", "derivation": "\\operatorname{E_{n}}{(f^{*},M_{E})} = M_{E} + f^{*} and \\int \\operatorname{E_{n}}{(f^{*},M_{E})} dM_{E} = \\int (M_{E} + f^{*}) dM_{E} and \\int \\operatorname{E_{n}}{(f^{*},M_{E})} dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} f^{*} + v_{z} and \\int (M_{E} + f^{*}) dM_{E} = \\frac{M_{E}^{2}}{2} + M_{E} f^{*} + v_{z} and \\log{(\\int (M_{E} + f^{*}) dM_{E})} = \\log{(\\frac{M_{E}^{2}}{2} + M_{E} f^{*} + v_{z})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f^*', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('f^*', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('f^*', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Symbol('v_z', commutative=True)))"], [["log", 4], "Equality(log(Integral(Add(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), log(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Symbol('M_E', commutative=True), Symbol('f^*', commutative=True)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given c{(\\rho)} = \\cos{(\\rho)}, then derive \\frac{d}{d \\rho} c{(\\rho)} = - \\sin{(\\rho)}, then obtain - \\frac{c{(\\rho)}}{\\sin{(\\rho)}} = - \\frac{\\cos{(\\rho)}}{\\sin{(\\rho)}}", "derivation": "c{(\\rho)} = \\cos{(\\rho)} and \\frac{d}{d \\rho} c{(\\rho)} = \\frac{d}{d \\rho} \\cos{(\\rho)} and \\frac{d}{d \\rho} c{(\\rho)} = - \\sin{(\\rho)} and \\frac{c{(\\rho)}}{\\frac{d}{d \\rho} \\cos{(\\rho)}} = \\frac{\\cos{(\\rho)}}{\\frac{d}{d \\rho} \\cos{(\\rho)}} and \\frac{d}{d \\rho} \\cos{(\\rho)} = - \\sin{(\\rho)} and - \\frac{c{(\\rho)}}{\\sin{(\\rho)}} = - \\frac{\\cos{(\\rho)}}{\\sin{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))))"], [["divide", 1, "Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))"], "Equality(Mul(Function('c')(Symbol('\\\\rho', commutative=True)), Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1))), Mul(cos(Symbol('\\\\rho', commutative=True)), Pow(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), Function('c')(Symbol('\\\\rho', commutative=True)), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('\\\\rho', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given E{(B)} = \\int e^{B} dB, then derive E^{B}{(B)} = (v_{z} + e^{B})^{B}, then derive - B + E^{B}{(B)} = - B + (\\omega + e^{B})^{B}, then obtain (- B + (\\omega + e^{B})^{B})^{\\omega} = (- B + (v_{z} + e^{B})^{B})^{\\omega}", "derivation": "E{(B)} = \\int e^{B} dB and E^{B}{(B)} = (\\int e^{B} dB)^{B} and E^{B}{(B)} = (v_{z} + e^{B})^{B} and - B + E^{B}{(B)} = - B + (\\int e^{B} dB)^{B} and - B + E^{B}{(B)} = - B + (\\omega + e^{B})^{B} and (v_{z} + e^{B})^{B} = (\\int e^{B} dB)^{B} and - B + E^{B}{(B)} = - B + (v_{z} + e^{B})^{B} and - B + (\\omega + e^{B})^{B} = - B + (v_{z} + e^{B})^{B} and (- B + (\\omega + e^{B})^{B})^{\\omega} = (- B + (v_{z} + e^{B})^{B})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('B', commutative=True)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('E')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('E')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["minus", 2, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Function('E')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Function('E')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('\\\\omega', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Function('E')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('\\\\omega', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["power", 8, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('\\\\omega', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} = \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}} and \\operatorname{P_{g}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} + \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}}, then obtain \\frac{\\operatorname{P_{g}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})}}{- \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} + \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}}} = 0", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} = \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}} and \\operatorname{P_{g}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} + \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}} and \\operatorname{P_{g}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} = 0 and \\frac{\\operatorname{P_{g}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})}}{- \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\dot{\\mathbf{r}},J_{\\varepsilon})} + \\frac{\\cos{(J_{\\varepsilon})}}{\\dot{\\mathbf{r}}}} = 0", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_g')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(0))"], [["divide", 3, "Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(-1)), Function('P_g')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Integer(0))"]]}, {"prompt": "Given W{(A)} = \\cos{(A)}, then derive \\int W{(A)} dA = \\hat{p} + \\sin{(A)}, then obtain \\int \\cos{(A)} dA = \\hat{p} + \\sin{(A)}", "derivation": "W{(A)} = \\cos{(A)} and \\int W{(A)} dA = \\int \\cos{(A)} dA and \\int W{(A)} dA = \\hat{p} + \\sin{(A)} and \\int \\cos{(A)} dA = \\hat{p} + \\sin{(A)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('W')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{f},F_{H})} = \\mathbf{f}^{F_{H}}, then obtain v{(\\sigma_p)} + (\\int \\operatorname{m_{s}}{(\\mathbf{f},F_{H})} dF_{H})^{\\mathbf{f}} = v{(\\sigma_p)} + (\\int \\mathbf{f}^{F_{H}} dF_{H})^{\\mathbf{f}}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{f},F_{H})} = \\mathbf{f}^{F_{H}} and \\int \\operatorname{m_{s}}{(\\mathbf{f},F_{H})} dF_{H} = \\int \\mathbf{f}^{F_{H}} dF_{H} and (\\int \\operatorname{m_{s}}{(\\mathbf{f},F_{H})} dF_{H})^{\\mathbf{f}} = (\\int \\mathbf{f}^{F_{H}} dF_{H})^{\\mathbf{f}} and v{(\\sigma_p)} + (\\int \\operatorname{m_{s}}{(\\mathbf{f},F_{H})} dF_{H})^{\\mathbf{f}} = v{(\\sigma_p)} + (\\int \\mathbf{f}^{F_{H}} dF_{H})^{\\mathbf{f}}", "srepr_derivation": [["get_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Integral(Function('m_s')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integral(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 3, "Function('v')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('v')(Symbol('\\\\sigma_p', commutative=True)), Pow(Integral(Function('m_s')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))), Add(Function('v')(Symbol('\\\\sigma_p', commutative=True)), Pow(Integral(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(T,\\Omega)} = T - \\Omega, then obtain \\int T d\\Omega + \\int - \\Omega d\\Omega + \\int 2 \\operatorname{A_{z}}{(T,\\Omega)} d\\Omega = \\int 2 T d\\Omega + \\int - 2 \\Omega d\\Omega + \\int \\operatorname{A_{z}}{(T,\\Omega)} d\\Omega", "derivation": "\\operatorname{A_{z}}{(T,\\Omega)} = T - \\Omega and T - \\Omega + 2 \\operatorname{A_{z}}{(T,\\Omega)} = 2 T - 2 \\Omega + \\operatorname{A_{z}}{(T,\\Omega)} and \\int (T - \\Omega + 2 \\operatorname{A_{z}}{(T,\\Omega)}) d\\Omega = \\int (2 T - 2 \\Omega + \\operatorname{A_{z}}{(T,\\Omega)}) d\\Omega and \\int T d\\Omega + \\int - \\Omega d\\Omega + \\int 2 \\operatorname{A_{z}}{(T,\\Omega)} d\\Omega = \\int 2 T d\\Omega + \\int - 2 \\Omega d\\Omega + \\int \\operatorname{A_{z}}{(T,\\Omega)} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["expand", 3], "Equality(Add(Integral(Symbol('T', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(2), Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Integral(Mul(Integer(2), Symbol('T', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Function('A_z')(Symbol('T', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(l)} = \\log{(e^{l})}, then obtain \\frac{d}{d l} \\frac{\\int \\operatorname{t_{1}}{(l)} dl}{\\int \\log{(e^{l})} dl} = \\frac{d}{d l} 1", "derivation": "\\operatorname{t_{1}}{(l)} = \\log{(e^{l})} and \\int \\operatorname{t_{1}}{(l)} dl = \\int \\log{(e^{l})} dl and \\frac{\\int \\operatorname{t_{1}}{(l)} dl}{\\int \\log{(e^{l})} dl} = 1 and \\frac{d}{d l} \\frac{\\int \\operatorname{t_{1}}{(l)} dl}{\\int \\log{(e^{l})} dl} = \\frac{d}{d l} 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('l', commutative=True)), log(exp(Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(log(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["divide", 2, "Integral(log(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Integral(Function('t_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Pow(Integral(log(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('t_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Pow(Integral(log(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(-1))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(B,\\mathbf{M},\\dot{y})} = \\dot{y} + \\mathbf{M}^{B} and \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\mathbf{g}), then derive \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} = 1, then obtain \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} \\delta^{\\mathbf{M}}{(B,\\mathbf{M},\\dot{y})} = \\delta^{\\mathbf{M}}{(B,\\mathbf{M},\\dot{y})}", "derivation": "\\delta{(B,\\mathbf{M},\\dot{y})} = \\dot{y} + \\mathbf{M}^{B} and \\delta^{\\mathbf{M}}{(B,\\mathbf{M},\\dot{y})} = (\\dot{y} + \\mathbf{M}^{B})^{\\mathbf{M}} and \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\mathbf{g}) and \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} = 1 and (\\dot{y} + \\mathbf{M}^{B})^{\\mathbf{M}} \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} = (\\dot{y} + \\mathbf{M}^{B})^{\\mathbf{M}} and \\operatorname{A_{z}}{(\\dot{z},\\mathbf{g})} \\delta^{\\mathbf{M}}{(B,\\mathbf{M},\\dot{y})} = \\delta^{\\mathbf{M}}{(B,\\mathbf{M},\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('B', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('B', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('B', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], ["get_premise", "Equality(Function('A_z')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('A_z')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(1))"], [["times", 4, "Pow(Add(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Function('A_z')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('A_z')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('\\\\delta')(Symbol('B', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('\\\\delta')(Symbol('B', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(g,V_{\\mathbf{E}})} = \\log{(- V_{\\mathbf{E}} + g)}, then obtain (- \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}})^{2 g} = (- \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}})^{g} (- \\frac{\\log{(- V_{\\mathbf{E}} + g)}}{V_{\\mathbf{E}}})^{g}", "derivation": "\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})} = \\log{(- V_{\\mathbf{E}} + g)} and - \\operatorname{F_{x}}{(g,V_{\\mathbf{E}})} = - \\log{(- V_{\\mathbf{E}} + g)} and - \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}} = - \\frac{\\log{(- V_{\\mathbf{E}} + g)}}{V_{\\mathbf{E}}} and (- \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}})^{g} = (- \\frac{\\log{(- V_{\\mathbf{E}} + g)}}{V_{\\mathbf{E}}})^{g} and (- \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}})^{2 g} = (- \\frac{\\operatorname{F_{x}}{(g,V_{\\mathbf{E}})}}{V_{\\mathbf{E}}})^{g} (- \\frac{\\log{(- V_{\\mathbf{E}} + g)}}{V_{\\mathbf{E}}})^{g}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('g', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('g', commutative=True)))))"], [["divide", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('g', commutative=True)))))"], [["power", 3, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["times", 4, "Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('g', commutative=True))"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(2), Symbol('g', commutative=True))), Mul(Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('F_x')(Symbol('g', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('g', commutative=True)))), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{s},c_{0})} = \\frac{c_{0}}{\\mathbf{s}}, then obtain (- F_{g} + \\theta_{1}{(\\mathbf{s},c_{0})}) \\int \\frac{c_{0}}{\\mathbf{s}} dc_{0} = (- F_{g} + \\frac{c_{0}}{\\mathbf{s}}) \\int \\frac{c_{0}}{\\mathbf{s}} dc_{0}", "derivation": "\\theta_{1}{(\\mathbf{s},c_{0})} = \\frac{c_{0}}{\\mathbf{s}} and \\int \\theta_{1}{(\\mathbf{s},c_{0})} dc_{0} = \\int \\frac{c_{0}}{\\mathbf{s}} dc_{0} and - F_{g} + \\theta_{1}{(\\mathbf{s},c_{0})} = - F_{g} + \\frac{c_{0}}{\\mathbf{s}} and (- F_{g} + \\theta_{1}{(\\mathbf{s},c_{0})}) \\int \\theta_{1}{(\\mathbf{s},c_{0})} dc_{0} = (- F_{g} + \\frac{c_{0}}{\\mathbf{s}}) \\int \\theta_{1}{(\\mathbf{s},c_{0})} dc_{0} and (- F_{g} + \\theta_{1}{(\\mathbf{s},c_{0})}) \\int \\frac{c_{0}}{\\mathbf{s}} dc_{0} = (- F_{g} + \\frac{c_{0}}{\\mathbf{s}}) \\int \\frac{c_{0}}{\\mathbf{s}} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))))"], [["times", 3, "Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True))), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c_0', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given T{(y^{\\prime},\\Omega)} = \\Omega + y^{\\prime} and \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} = \\Omega + y^{\\prime}, then obtain \\frac{\\partial}{\\partial \\Omega} (- \\Omega - y^{\\prime} - \\int (\\Omega + y^{\\prime}) dy^{\\prime}) = \\frac{\\partial}{\\partial \\Omega} (- \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime})", "derivation": "T{(y^{\\prime},\\Omega)} = \\Omega + y^{\\prime} and - T{(y^{\\prime},\\Omega)} = - \\Omega - y^{\\prime} and - T{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} = - \\Omega - y^{\\prime} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} and \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} = \\Omega + y^{\\prime} and - T{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} = - \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} and - \\Omega - y^{\\prime} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} = - \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime} and \\frac{\\partial}{\\partial \\Omega} (- \\Omega - y^{\\prime} - \\int (\\Omega + y^{\\prime}) dy^{\\prime}) = \\frac{\\partial}{\\partial \\Omega} (- \\operatorname{F_{H}}{(y^{\\prime},\\Omega)} - \\int (\\Omega + y^{\\prime}) dy^{\\prime})", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), Function('F_H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), Function('F_H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))))"], [["differentiate", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v_{2},C)} = C - v_{2}, then obtain \\frac{\\partial}{\\partial v_{2}} (- (- C)^{C} + (- v_{2} - \\operatorname{f_{E}}{(v_{2},C)})^{C}) = \\frac{d}{d v_{2}} 0", "derivation": "\\operatorname{f_{E}}{(v_{2},C)} = C - v_{2} and - \\operatorname{f_{E}}{(v_{2},C)} = - C + v_{2} and - v_{2} - \\operatorname{f_{E}}{(v_{2},C)} = - C and (- v_{2} - \\operatorname{f_{E}}{(v_{2},C)})^{C} = (- C)^{C} and - (- C)^{C} + (- v_{2} - \\operatorname{f_{E}}{(v_{2},C)})^{C} = 0 and \\frac{\\partial}{\\partial v_{2}} (- (- C)^{C} + (- v_{2} - \\operatorname{f_{E}}{(v_{2},C)})^{C}) = \\frac{d}{d v_{2}} 0", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v_2', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)))), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)))), Symbol('C', commutative=True)), Pow(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["minus", 4, "Pow(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('C', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)))), Symbol('C', commutative=True))), Integer(0))"], [["differentiate", 5, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('C', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)))), Symbol('C', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(v,L)} = L v and \\mu{(v_{t})} = \\sin{(v_{t})}, then obtain \\frac{\\partial}{\\partial v} (- \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + \\int L v dL + \\int \\operatorname{A_{z}}{(v,L)} dL) = \\frac{\\partial}{\\partial v} (- \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + 2 \\int L v dL)", "derivation": "\\operatorname{A_{z}}{(v,L)} = L v and \\mu{(v_{t})} = \\sin{(v_{t})} and \\int \\operatorname{A_{z}}{(v,L)} dL = \\int L v dL and \\sin{(v_{t})} + \\int L v dL + \\int \\operatorname{A_{z}}{(v,L)} dL = \\sin{(v_{t})} + 2 \\int L v dL and \\mu{(v_{t})} + \\int L v dL + \\int \\operatorname{A_{z}}{(v,L)} dL = \\mu{(v_{t})} + 2 \\int L v dL and - \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + \\int L v dL + \\int \\operatorname{A_{z}}{(v,L)} dL = - \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + 2 \\int L v dL and \\frac{\\partial}{\\partial v} (- \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + \\int L v dL + \\int \\operatorname{A_{z}}{(v,L)} dL) = \\frac{\\partial}{\\partial v} (- \\operatorname{A_{z}}{(v,L)} + \\mu{(v_{t})} + 2 \\int L v dL)", "srepr_derivation": [["get_premise", "Equality(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["add", 3, "Add(sin(Symbol('v_t', commutative=True)), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))"], "Equality(Add(sin(Symbol('v_t', commutative=True)), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(sin(Symbol('v_t', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\mu')(Symbol('v_t', commutative=True)), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Function('\\\\mu')(Symbol('v_t', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))))"], [["minus", 5, "Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True))), Function('\\\\mu')(Symbol('v_t', commutative=True)), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True))), Function('\\\\mu')(Symbol('v_t', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))))"], [["differentiate", 6, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True))), Function('\\\\mu')(Symbol('v_t', commutative=True)), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('A_z')(Symbol('v', commutative=True), Symbol('L', commutative=True))), Function('\\\\mu')(Symbol('v_t', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('L', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('L', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then derive \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}, then obtain \\frac{d}{d \\mathbf{A}} \\int \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} d\\mathbf{A} = \\frac{d}{d \\mathbf{A}} \\int - \\sin{(\\mathbf{A})} d\\mathbf{A}", "derivation": "\\phi{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} and \\int \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} d\\mathbf{A} = \\int - \\sin{(\\mathbf{A})} d\\mathbf{A} and \\frac{d}{d \\mathbf{A}} \\int \\frac{d}{d \\mathbf{A}} \\phi{(\\mathbf{A})} d\\mathbf{A} = \\frac{d}{d \\mathbf{A}} \\int - \\sin{(\\mathbf{A})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('\\\\phi')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(y,p)} = y \\log{(p)} and \\varphi^{*}{(y,p)} = \\frac{\\partial}{\\partial y} \\int y \\log{(p)} dp and n{(z,\\hbar)} = \\hbar + z, then obtain \\frac{n{(z,\\hbar)}}{\\varphi^{*}{(y,p)}} = \\frac{\\hbar + z}{\\varphi^{*}{(y,p)}}", "derivation": "V{(y,p)} = y \\log{(p)} and \\int V{(y,p)} dp = \\int y \\log{(p)} dp and \\frac{\\partial}{\\partial y} \\int V{(y,p)} dp = \\frac{\\partial}{\\partial y} \\int y \\log{(p)} dp and \\varphi^{*}{(y,p)} = \\frac{\\partial}{\\partial y} \\int y \\log{(p)} dp and \\frac{\\partial}{\\partial y} \\int V{(y,p)} dp = \\varphi^{*}{(y,p)} and n{(z,\\hbar)} = \\hbar + z and \\frac{n{(z,\\hbar)}}{\\frac{\\partial}{\\partial y} \\int V{(y,p)} dp} = \\frac{\\hbar + z}{\\frac{\\partial}{\\partial y} \\int V{(y,p)} dp} and \\frac{n{(z,\\hbar)}}{\\varphi^{*}{(y,p)}} = \\frac{\\hbar + z}{\\varphi^{*}{(y,p)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('y', commutative=True), log(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Symbol('y', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('y', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Derivative(Integral(Mul(Symbol('y', commutative=True), log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Function('\\\\varphi^*')(Symbol('y', commutative=True), Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('n')(Symbol('z', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('z', commutative=True)))"], [["times", 6, "Pow(Derivative(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))"], "Equality(Mul(Function('n')(Symbol('z', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('\\\\hbar', commutative=True), Symbol('z', commutative=True)), Pow(Derivative(Integral(Function('V')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Function('\\\\varphi^*')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Function('n')(Symbol('z', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Add(Symbol('\\\\hbar', commutative=True), Symbol('z', commutative=True)), Pow(Function('\\\\varphi^*')(Symbol('y', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\psi,f_{\\mathbf{p}})} = \\sin{(\\psi^{f_{\\mathbf{p}}})} and \\operatorname{E_{x}}{(\\psi,f_{\\mathbf{p}})} = \\psi \\mathbf{H}{(\\psi,f_{\\mathbf{p}})} \\sin{(\\psi^{f_{\\mathbf{p}}})}, then obtain e^{\\psi \\mathbf{H}{(\\psi,f_{\\mathbf{p}})} \\sin{(\\psi^{f_{\\mathbf{p}}})}} = e^{\\psi \\sin^{2}{(\\psi^{f_{\\mathbf{p}}})}}", "derivation": "\\mathbf{H}{(\\psi,f_{\\mathbf{p}})} = \\sin{(\\psi^{f_{\\mathbf{p}}})} and \\operatorname{E_{x}}{(\\psi,f_{\\mathbf{p}})} = \\psi \\mathbf{H}{(\\psi,f_{\\mathbf{p}})} \\sin{(\\psi^{f_{\\mathbf{p}}})} and \\operatorname{E_{x}}{(\\psi,f_{\\mathbf{p}})} = \\psi \\sin^{2}{(\\psi^{f_{\\mathbf{p}}})} and e^{\\operatorname{E_{x}}{(\\psi,f_{\\mathbf{p}})}} = e^{\\psi \\sin^{2}{(\\psi^{f_{\\mathbf{p}}})}} and e^{\\psi \\mathbf{H}{(\\psi,f_{\\mathbf{p}})} \\sin{(\\psi^{f_{\\mathbf{p}}})}} = e^{\\psi \\sin^{2}{(\\psi^{f_{\\mathbf{p}}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2))))"], [["exp", 3], "Equality(exp(Function('E_x')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), exp(Mul(Symbol('\\\\psi', commutative=True), Pow(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))), exp(Mul(Symbol('\\\\psi', commutative=True), Pow(sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\mu_{0}{(i)} = \\cos{(i)}, then obtain \\int (2 \\mu_{0}{(i)})^{i} di = \\int (\\mu_{0}{(i)} + \\cos{(i)})^{i} di", "derivation": "\\mu_{0}{(i)} = \\cos{(i)} and 2 \\mu_{0}{(i)} = \\mu_{0}{(i)} + \\cos{(i)} and (2 \\mu_{0}{(i)})^{i} = (\\mu_{0}{(i)} + \\cos{(i)})^{i} and \\int (2 \\mu_{0}{(i)})^{i} di = \\int (\\mu_{0}{(i)} + \\cos{(i)})^{i} di", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["add", 1, "Function('\\\\mu_0')(Symbol('i', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mu_0')(Symbol('i', commutative=True))), Add(Function('\\\\mu_0')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mu_0')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Function('\\\\mu_0')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(2), Function('\\\\mu_0')(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Add(Function('\\\\mu_0')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(I,\\mathbf{F})} = \\cos{(I - \\mathbf{F})}, then obtain k{(Q)} \\frac{\\partial^{2}}{\\partial I^{2}} \\mathbf{P}{(I,\\mathbf{F})} = k{(Q)} \\frac{\\partial^{2}}{\\partial I^{2}} \\cos{(I - \\mathbf{F})}", "derivation": "\\mathbf{P}{(I,\\mathbf{F})} = \\cos{(I - \\mathbf{F})} and \\frac{\\partial}{\\partial I} \\mathbf{P}{(I,\\mathbf{F})} = \\frac{\\partial}{\\partial I} \\cos{(I - \\mathbf{F})} and \\frac{\\partial^{2}}{\\partial I^{2}} \\mathbf{P}{(I,\\mathbf{F})} = \\frac{\\partial^{2}}{\\partial I^{2}} \\cos{(I - \\mathbf{F})} and k{(Q)} \\frac{\\partial^{2}}{\\partial I^{2}} \\mathbf{P}{(I,\\mathbf{F})} = k{(Q)} \\frac{\\partial^{2}}{\\partial I^{2}} \\cos{(I - \\mathbf{F})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Derivative(cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(2))))"], [["times", 3, "Function('k')(Symbol('Q', commutative=True))"], "Equality(Mul(Function('k')(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2)))), Mul(Function('k')(Symbol('Q', commutative=True)), Derivative(cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{t})} = - v_{t}, then obtain \\frac{- \\cos{(v_{t})} + \\cos{(\\mathbf{f}{(v_{t})})}}{\\cos{(\\mathbf{f}{(v_{t})})}} = 0", "derivation": "\\mathbf{f}{(v_{t})} = - v_{t} and \\cos{(\\mathbf{f}{(v_{t})})} = \\cos{(v_{t})} and - \\cos{(v_{t})} + \\cos{(\\mathbf{f}{(v_{t})})} = 0 and \\frac{- \\cos{(v_{t})} + \\cos{(\\mathbf{f}{(v_{t})})}}{\\cos{(\\mathbf{f}{(v_{t})})}} = 0", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True))), cos(Symbol('v_t', commutative=True)))"], [["minus", 2, "cos(Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), cos(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True)))), Integer(0))"], [["divide", 3, "cos(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), cos(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True)))), Pow(cos(Function('\\\\mathbf{f}')(Symbol('v_t', commutative=True))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\phi_{2}{(f^{\\prime})} = \\sin{(\\cos{(f^{\\prime})})} and \\pi{(f^{\\prime})} = f^{\\prime} \\sin{(\\cos{(f^{\\prime})})}, then obtain f^{\\prime} \\phi_{2}{(f^{\\prime})} - f^{\\prime} + \\operatorname{f_{E}}{(\\Omega)} + \\sin{(\\Omega)} = - f^{\\prime} + \\pi{(f^{\\prime})} + \\operatorname{f_{E}}{(\\Omega)} + \\sin{(\\Omega)}", "derivation": "\\phi_{2}{(f^{\\prime})} = \\sin{(\\cos{(f^{\\prime})})} and f^{\\prime} \\phi_{2}{(f^{\\prime})} = f^{\\prime} \\sin{(\\cos{(f^{\\prime})})} and \\pi{(f^{\\prime})} = f^{\\prime} \\sin{(\\cos{(f^{\\prime})})} and f^{\\prime} \\phi_{2}{(f^{\\prime})} - f^{\\prime} = f^{\\prime} \\sin{(\\cos{(f^{\\prime})})} - f^{\\prime} and f^{\\prime} \\phi_{2}{(f^{\\prime})} - f^{\\prime} + \\sin{(\\Omega)} = f^{\\prime} \\sin{(\\cos{(f^{\\prime})})} - f^{\\prime} + \\sin{(\\Omega)} and f^{\\prime} \\phi_{2}{(f^{\\prime})} - f^{\\prime} + \\sin{(\\Omega)} = - f^{\\prime} + \\pi{(f^{\\prime})} + \\sin{(\\Omega)} and f^{\\prime} \\phi_{2}{(f^{\\prime})} - f^{\\prime} + \\operatorname{f_{E}}{(\\Omega)} + \\sin{(\\Omega)} = - f^{\\prime} + \\pi{(f^{\\prime})} + \\operatorname{f_{E}}{(\\Omega)} + \\sin{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True)), sin(cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('f^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 4, "sin(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('f^{\\\\prime}', commutative=True), sin(cos(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\pi')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))))"], [["add", 6, "Function('f_E')(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Mul(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\phi_2')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('f_E')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\pi')(Symbol('f^{\\\\prime}', commutative=True)), Function('f_E')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given g{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} = - g_{\\varepsilon}, then obtain - g_{\\varepsilon} + g{(g_{\\varepsilon})} + 1 = - g_{\\varepsilon} + e^{g_{\\varepsilon}} + 1", "derivation": "g{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and - g_{\\varepsilon} + g{(g_{\\varepsilon})} = - g_{\\varepsilon} + e^{g_{\\varepsilon}} and \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} = - g_{\\varepsilon} and \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} + g{(g_{\\varepsilon})} = \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} and \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} + g{(g_{\\varepsilon})} + 1 = \\operatorname{f_{\\mathbf{v}}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} + 1 and - g_{\\varepsilon} + g{(g_{\\varepsilon})} + 1 = - g_{\\varepsilon} + e^{g_{\\varepsilon}} + 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Function('f_{\\\\mathbf{v}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('g')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given Q{(G)} = \\cos{(e^{G})} and \\dot{y}{(G)} = e^{G}, then obtain \\frac{d}{d G} Q{(G)} = \\frac{d}{d G} \\cos{(\\dot{y}{(G)})}", "derivation": "Q{(G)} = \\cos{(e^{G})} and \\dot{y}{(G)} = e^{G} and \\frac{d}{d G} Q{(G)} = \\frac{d}{d G} \\cos{(e^{G})} and \\frac{d}{d G} Q{(G)} = \\frac{d}{d G} \\cos{(\\dot{y}{(G)})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('Q')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Function('\\\\dot{y}')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(v_{x},F_{c})} = \\frac{e^{F_{c}}}{v_{x}} and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} = e^{F_{c}}, then obtain \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(F_{c})}}{v_{x}} dF_{c} = \\int \\frac{e^{F_{c}}}{v_{x}} dF_{c}", "derivation": "\\operatorname{J_{\\varepsilon}}{(v_{x},F_{c})} = \\frac{e^{F_{c}}}{v_{x}} and \\int \\operatorname{J_{\\varepsilon}}{(v_{x},F_{c})} dF_{c} = \\int \\frac{e^{F_{c}}}{v_{x}} dF_{c} and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} = e^{F_{c}} and \\operatorname{J_{\\varepsilon}}{(v_{x},F_{c})} = \\frac{\\operatorname{V_{\\mathbf{B}}}{(F_{c})}}{v_{x}} and \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(F_{c})}}{v_{x}} dF_{c} = \\int \\frac{e^{F_{c}}}{v_{x}} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('J_{\\\\varepsilon}')(Symbol('v_x', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given r{(F_{g},\\phi_1)} = F_{g} \\phi_1, then derive \\frac{\\partial}{\\partial \\phi_1} r{(F_{g},\\phi_1)} = F_{g}, then derive \\frac{\\partial^{2}}{\\partial \\phi_1^{2}} r{(F_{g},\\phi_1)} = 0, then obtain \\frac{\\frac{\\partial^{3}}{\\partial F_{g}\\partial \\phi_1^{2}} F_{g} \\phi_1}{\\hat{x}} = \\frac{\\frac{d}{d F_{g}} 0}{\\hat{x}}", "derivation": "r{(F_{g},\\phi_1)} = F_{g} \\phi_1 and \\frac{\\partial}{\\partial \\phi_1} r{(F_{g},\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} F_{g} \\phi_1 and \\frac{\\partial}{\\partial \\phi_1} r{(F_{g},\\phi_1)} = F_{g} and \\frac{\\partial^{2}}{\\partial \\phi_1^{2}} r{(F_{g},\\phi_1)} = \\frac{d}{d \\phi_1} F_{g} and \\frac{\\partial^{2}}{\\partial \\phi_1^{2}} r{(F_{g},\\phi_1)} = 0 and \\frac{\\partial^{2}}{\\partial \\phi_1^{2}} F_{g} \\phi_1 = 0 and \\frac{\\partial^{3}}{\\partial F_{g}\\partial \\phi_1^{2}} F_{g} \\phi_1 = \\frac{d}{d F_{g}} 0 and \\frac{\\frac{\\partial^{3}}{\\partial F_{g}\\partial \\phi_1^{2}} F_{g} \\phi_1}{\\hat{x}} = \\frac{\\frac{d}{d F_{g}} 0}{\\hat{x}}", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('F_g', commutative=True))"], [["differentiate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2))), Derivative(Symbol('F_g', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('r')(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2))), Integer(0))"], [["differentiate", 6, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["divide", 7, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(2)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{1}{(m)} = \\log{(m)}, then obtain \\frac{d}{d m} (m + \\theta_{1}{(m)}) = \\frac{d}{d m} (m + \\log{(m)}) + \\frac{- \\theta_{1}{(m)} + \\log{(m)}}{m}", "derivation": "\\theta_{1}{(m)} = \\log{(m)} and m + \\theta_{1}{(m)} = m + \\log{(m)} and 0 = - \\theta_{1}{(m)} + \\log{(m)} and \\frac{d}{d m} (m + \\theta_{1}{(m)}) = \\frac{d}{d m} (m + \\log{(m)}) and 0 = \\frac{- \\theta_{1}{(m)} + \\log{(m)}}{m} and \\frac{d}{d m} (m + \\log{(m)}) = \\frac{d}{d m} (m + \\log{(m)}) + \\frac{- \\theta_{1}{(m)} + \\log{(m)}}{m} and \\frac{d}{d m} (m + \\theta_{1}{(m)}) = \\frac{d}{d m} (m + \\log{(m)}) + \\frac{- \\theta_{1}{(m)} + \\log{(m)}}{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\theta_1')(Symbol('m', commutative=True))), Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))))"], [["minus", 1, "Function('\\\\theta_1')(Symbol('m', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('m', commutative=True))), log(Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\theta_1')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('m', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('m', commutative=True))), log(Symbol('m', commutative=True)))))"], [["add", 5, "Derivative(Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Derivative(Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Add(Derivative(Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('m', commutative=True))), log(Symbol('m', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\theta_1')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Add(Derivative(Add(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('m', commutative=True))), log(Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(S,\\mathbf{A})} = S^{\\mathbf{A}}, then obtain 0 = S^{\\mathbf{A}} (S^{2 \\mathbf{A}} - S^{\\mathbf{A}} \\hat{H}{(S,\\mathbf{A})}) \\hat{H}{(S,\\mathbf{A})}", "derivation": "\\hat{H}{(S,\\mathbf{A})} = S^{\\mathbf{A}} and S^{\\mathbf{A}} \\hat{H}{(S,\\mathbf{A})} = S^{2 \\mathbf{A}} and 0 = S^{2 \\mathbf{A}} - S^{\\mathbf{A}} \\hat{H}{(S,\\mathbf{A})} and 0 = S^{\\mathbf{A}} (S^{2 \\mathbf{A}} - S^{\\mathbf{A}} \\hat{H}{(S,\\mathbf{A})}) \\hat{H}{(S,\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 1, "Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(0), Add(Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(t_{1},v_{t})} = t_{1} + v_{t}, then obtain - 0^{v_{t}} + v_{t} + 1 = v_{t}", "derivation": "\\bar{\\h}{(t_{1},v_{t})} = t_{1} + v_{t} and - t_{1} - v_{t} + \\bar{\\h}{(t_{1},v_{t})} = 0 and (- t_{1} - v_{t} + \\bar{\\h}{(t_{1},v_{t})})^{v_{t}} = 0^{v_{t}} and - 0^{v_{t}} + (- t_{1} - v_{t} + \\bar{\\h}{(t_{1},v_{t})})^{v_{t}} = 0 and 1 - (- t_{1} - v_{t} + \\bar{\\h}{(t_{1},v_{t})})^{v_{t}} = 0 and 1 - 0^{v_{t}} = 0 and - 0^{v_{t}} + v_{t} + 1 = v_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Add(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\hbar')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True))), Integer(0))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\hbar')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Integer(0), Symbol('v_t', commutative=True)))"], [["minus", 3, "Pow(Integer(0), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('v_t', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\hbar')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\hbar')(Symbol('t_1', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('v_t', commutative=True)))), Integer(0))"], [["add", 6, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True), Integer(1)), Symbol('v_t', commutative=True))"]]}, {"prompt": "Given \\tilde{g}{(\\hat{H}_l,\\chi)} = \\cos{(\\chi^{\\hat{H}_l})}, then obtain (\\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\tilde{g}{(\\hat{H}_l,\\chi)} d\\chi)^{\\hat{H}_l} = (\\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\cos{(\\chi^{\\hat{H}_l})} d\\chi)^{\\hat{H}_l}", "derivation": "\\tilde{g}{(\\hat{H}_l,\\chi)} = \\cos{(\\chi^{\\hat{H}_l})} and \\int \\tilde{g}{(\\hat{H}_l,\\chi)} d\\chi = \\int \\cos{(\\chi^{\\hat{H}_l})} d\\chi and \\chi + \\int \\tilde{g}{(\\hat{H}_l,\\chi)} d\\chi = \\chi + \\int \\cos{(\\chi^{\\hat{H}_l})} d\\chi and \\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\tilde{g}{(\\hat{H}_l,\\chi)} d\\chi = \\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\cos{(\\chi^{\\hat{H}_l})} d\\chi and (\\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\tilde{g}{(\\hat{H}_l,\\chi)} d\\chi)^{\\hat{H}_l} = (\\chi + \\tilde{g}{(\\hat{H}_l,\\chi)} + \\int \\cos{(\\chi^{\\hat{H}_l})} d\\chi)^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), cos(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(cos(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Integral(cos(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["add", 3, "Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(cos(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["power", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\chi', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(cos(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(I,Q)} = I^{Q}, then obtain - \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int \\operatorname{n_{1}}{(I,Q)} dI = - \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int I^{Q} dI", "derivation": "\\operatorname{n_{1}}{(I,Q)} = I^{Q} and \\int \\operatorname{n_{1}}{(I,Q)} dI = \\int I^{Q} dI and \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int \\operatorname{n_{1}}{(I,Q)} dI = \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int I^{Q} dI and - \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int \\operatorname{n_{1}}{(I,Q)} dI = - \\operatorname{n_{1}}^{- Q}{(I,Q)} \\int I^{Q} dI", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["divide", 2, "Pow(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))), Integral(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))), Integral(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Function('n_1')(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))), Integral(Pow(Symbol('I', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(E)} = \\sin{(E)}, then obtain - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{1}{\\operatorname{r_{0}}{(E)}} = - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{\\sin{(E)}}{\\operatorname{r_{0}}^{2}{(E)}}", "derivation": "\\operatorname{r_{0}}{(E)} = \\sin{(E)} and \\frac{1}{\\operatorname{r_{0}}{(E)}} = \\frac{\\sin{(E)}}{\\operatorname{r_{0}}^{2}{(E)}} and - \\operatorname{r_{0}}^{2}{(E)} + \\frac{1}{\\operatorname{r_{0}}{(E)}} = - \\operatorname{r_{0}}^{2}{(E)} + \\frac{\\sin{(E)}}{\\operatorname{r_{0}}^{2}{(E)}} and - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{\\sin{(E)}}{\\operatorname{r_{0}}^{2}{(E)}} = - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{\\sin^{3}{(E)}}{\\operatorname{r_{0}}^{4}{(E)}} and - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{1}{\\operatorname{r_{0}}{(E)}} = - \\frac{\\operatorname{r_{0}}^{4}{(E)}}{\\sin^{2}{(E)}} + \\frac{\\sin{(E)}}{\\operatorname{r_{0}}^{2}{(E)}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["divide", 1, "Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(2))"], "Equality(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-1)), Mul(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-2)), sin(Symbol('E', commutative=True))))"], [["minus", 2, "Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(2))), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(2))), Mul(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-2)), sin(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(4)), Pow(sin(Symbol('E', commutative=True)), Integer(-2))), Mul(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-2)), sin(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(4)), Pow(sin(Symbol('E', commutative=True)), Integer(-2))), Mul(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-4)), Pow(sin(Symbol('E', commutative=True)), Integer(3)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(4)), Pow(sin(Symbol('E', commutative=True)), Integer(-2))), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(4)), Pow(sin(Symbol('E', commutative=True)), Integer(-2))), Mul(Pow(Function('r_0')(Symbol('E', commutative=True)), Integer(-2)), sin(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)}, then derive \\operatorname{F_{c}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)}, then obtain \\int \\cos{(\\hat{H}_l)} d\\hat{H}_l = \\int \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} d\\hat{H}_l", "derivation": "\\operatorname{F_{c}}{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} and \\operatorname{F_{c}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\cos{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} and \\int \\cos{(\\hat{H}_l)} d\\hat{H}_l = \\int \\frac{d}{d \\hat{H}_l} \\sin{(\\hat{H}_l)} d\\hat{H}_l", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(sin(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(sin(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(cos(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Derivative(sin(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given r{(r_{0},v)} = r_{0} v, then derive \\frac{\\partial}{\\partial v} r{(r_{0},v)} = r_{0}, then obtain r{(\\frac{\\partial}{\\partial v} r_{0} v,v)} - \\frac{\\partial}{\\partial v} r_{0} v + 1 = v \\frac{\\partial}{\\partial v} r_{0} v - \\frac{\\partial}{\\partial v} r_{0} v + 1", "derivation": "r{(r_{0},v)} = r_{0} v and \\frac{\\partial}{\\partial v} r{(r_{0},v)} = \\frac{\\partial}{\\partial v} r_{0} v and \\frac{\\partial}{\\partial v} r{(r_{0},v)} = r_{0} and r_{0} = \\frac{\\partial}{\\partial v} r_{0} v and r{(r_{0},v)} + 1 = r_{0} v + 1 and - r_{0} + r{(r_{0},v)} + 1 = r_{0} v - r_{0} + 1 and r{(\\frac{\\partial}{\\partial v} r_{0} v,v)} - \\frac{\\partial}{\\partial v} r_{0} v + 1 = v \\frac{\\partial}{\\partial v} r_{0} v - \\frac{\\partial}{\\partial v} r_{0} v + 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('r')(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Integer(1)), Add(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Integer(1)))"], [["minus", 5, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('r')(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Integer(1)), Add(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('r')(Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('v', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Symbol('v', commutative=True), Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Symbol('r_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given B{(\\mu)} = \\sin{(\\mu)} and a{(\\mu)} = \\frac{1}{\\sin{(\\mu)}}, then obtain B{(\\mu)} a{(\\mu)} + a{(\\mu)} \\sin{(\\mu)} - a{(\\mu)} - 1 = a{(\\mu)} \\sin{(\\mu)} - a{(\\mu)}", "derivation": "B{(\\mu)} = \\sin{(\\mu)} and \\frac{B{(\\mu)}}{\\sin{(\\mu)}} = 1 and \\frac{B{(\\mu)}}{\\sin{(\\mu)}} - \\frac{1}{\\sin{(\\mu)}} = 1 - \\frac{1}{\\sin{(\\mu)}} and a{(\\mu)} = \\frac{1}{\\sin{(\\mu)}} and B{(\\mu)} a{(\\mu)} - a{(\\mu)} = 1 - a{(\\mu)} and a{(\\mu)} \\sin{(\\mu)} - a{(\\mu)} = 1 - a{(\\mu)} and B{(\\mu)} a{(\\mu)} + a{(\\mu)} \\sin{(\\mu)} - a{(\\mu)} - 1 = a{(\\mu)} \\sin{(\\mu)} - a{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('B')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('B')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1)))))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('\\\\mu', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Function('B')(Symbol('\\\\mu', commutative=True)), Function('a')(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Function('a')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Function('B')(Symbol('\\\\mu', commutative=True)), Function('a')(Symbol('\\\\mu', commutative=True))), Mul(Function('a')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True))), Integer(-1)), Add(Mul(Function('a')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(k)} = \\log{(\\log{(k)})} and \\operatorname{x^{{\\}'}}{(k)} = \\log{(\\log{(k)})}^{2}, then obtain \\frac{d}{d k} \\varphi^{*}^{2}{(k)} = \\frac{d}{d k} \\varphi^{*}{(k)} \\log{(\\log{(k)})}", "derivation": "\\varphi^{*}{(k)} = \\log{(\\log{(k)})} and \\varphi^{*}{(k)} \\log{(\\log{(k)})} = \\log{(\\log{(k)})}^{2} and \\operatorname{x^{{\\}'}}{(k)} = \\log{(\\log{(k)})}^{2} and \\operatorname{x^{{\\}'}}{(k)} = \\varphi^{*}^{2}{(k)} and \\operatorname{x^{{\\}'}}{(k)} = \\varphi^{*}{(k)} \\log{(\\log{(k)})} and \\frac{d}{d k} \\operatorname{x^{{\\}'}}{(k)} = \\frac{d}{d k} \\varphi^{*}{(k)} \\log{(\\log{(k)})} and \\frac{d}{d k} \\varphi^{*}^{2}{(k)} = \\frac{d}{d k} \\varphi^{*}{(k)} \\log{(\\log{(k)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('k', commutative=True)), log(log(Symbol('k', commutative=True))))"], [["times", 1, "log(log(Symbol('k', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), log(log(Symbol('k', commutative=True)))), Pow(log(log(Symbol('k', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('k', commutative=True)), Pow(log(log(Symbol('k', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('x^\\\\prime')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('x^\\\\prime')(Symbol('k', commutative=True)), Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), log(log(Symbol('k', commutative=True)))))"], [["differentiate", 5, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), log(log(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Pow(Function('\\\\varphi^*')(Symbol('k', commutative=True)), Integer(2)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\varphi^*')(Symbol('k', commutative=True)), log(log(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(f^{*},G)} = G f^{*}, then obtain - \\frac{\\eta^{\\prime}{(f^{*},G)}}{G f^{*} (f^{*} + \\frac{\\eta^{\\prime}{(f^{*},G)}}{G})} = - \\frac{1}{f^{*} + \\frac{\\eta^{\\prime}{(f^{*},G)}}{G}}", "derivation": "\\eta^{\\prime}{(f^{*},G)} = G f^{*} and \\frac{\\eta^{\\prime}{(f^{*},G)}}{G} = f^{*} and f^{*} + \\frac{\\eta^{\\prime}{(f^{*},G)}}{G} = 2 f^{*} and \\frac{\\eta^{\\prime}{(f^{*},G)}}{G f^{*}} = 1 and - \\frac{\\eta^{\\prime}{(f^{*},G)}}{G f^{*}} = -1 and - \\frac{\\eta^{\\prime}{(f^{*},G)}}{2 G (f^{*})^{2}} = - \\frac{1}{2 f^{*}} and - \\frac{\\eta^{\\prime}{(f^{*},G)}}{G f^{*} (f^{*} + \\frac{\\eta^{\\prime}{(f^{*},G)}}{G})} = - \\frac{1}{f^{*} + \\frac{\\eta^{\\prime}{(f^{*},G)}}{G}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('f^*', commutative=True)))"], [["divide", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True))), Symbol('f^*', commutative=True))"], [["add", 2, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True)))), Mul(Integer(2), Symbol('f^*', commutative=True)))"], [["divide", 2, "Symbol('f^*', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True))), Integer(1))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True))), Integer(-1))"], [["divide", 5, "Mul(Integer(2), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-2)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True)))), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('G', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\pi{(x^\\prime)} = \\cos{(\\sin{(x^\\prime)})}, then derive (\\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1)^{x^\\prime} = (c_{0} + x^\\prime - 1)^{x^\\prime}, then obtain \\int (\\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1)^{x^\\prime} dx^\\prime = \\int (c_{0} + x^\\prime - 1)^{x^\\prime} dx^\\prime", "derivation": "\\pi{(x^\\prime)} = \\cos{(\\sin{(x^\\prime)})} and \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} = 1 and \\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime = \\int 1 dx^\\prime and \\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1 = \\int 1 dx^\\prime - 1 and (\\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1)^{x^\\prime} = (\\int 1 dx^\\prime - 1)^{x^\\prime} and (\\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1)^{x^\\prime} = (c_{0} + x^\\prime - 1)^{x^\\prime} and \\int (\\int \\frac{\\pi{(x^\\prime)}}{\\cos{(\\sin{(x^\\prime)})}} dx^\\prime - 1)^{x^\\prime} dx^\\prime = \\int (c_{0} + x^\\prime - 1)^{x^\\prime} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), cos(sin(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 1, "cos(sin(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Integral(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Integral(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Integral(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Integral(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Symbol('c_0', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Pow(Add(Integral(Mul(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(sin(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(Add(Symbol('c_0', commutative=True), Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(g,\\Psi_{\\lambda})} = \\Psi_{\\lambda} g and \\hat{\\mathbf{r}}{(c,A_{y})} = A_{y} e^{c}, then obtain \\operatorname{V_{\\mathbf{B}}}^{g}{(g,\\Psi_{\\lambda})} + \\hat{\\mathbf{r}}{(c,A_{y})} = A_{y} e^{c} + \\operatorname{V_{\\mathbf{B}}}^{g}{(g,\\Psi_{\\lambda})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(g,\\Psi_{\\lambda})} = \\Psi_{\\lambda} g and \\operatorname{V_{\\mathbf{B}}}^{g}{(g,\\Psi_{\\lambda})} = (\\Psi_{\\lambda} g)^{g} and \\hat{\\mathbf{r}}{(c,A_{y})} = A_{y} e^{c} and (\\Psi_{\\lambda} g)^{g} + \\hat{\\mathbf{r}}{(c,A_{y})} = A_{y} e^{c} + (\\Psi_{\\lambda} g)^{g} and \\operatorname{V_{\\mathbf{B}}}^{g}{(g,\\Psi_{\\lambda})} + \\hat{\\mathbf{r}}{(c,A_{y})} = A_{y} e^{c} + \\operatorname{V_{\\mathbf{B}}}^{g}{(g,\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('g', commutative=True)))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), exp(Symbol('c', commutative=True))))"], [["add", 3, "Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), exp(Symbol('c', commutative=True))), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('g', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('A_y', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), exp(Symbol('c', commutative=True))), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\psi{(I)} = \\log{(I)}, then obtain - \\frac{I + \\psi^{2}{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi^{2}{(I)}}{I} = - \\frac{I + \\psi^{2}{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi{(I)} \\log{(I)}}{I}", "derivation": "\\psi{(I)} = \\log{(I)} and \\psi^{2}{(I)} = \\psi{(I)} \\log{(I)} and I + \\psi^{2}{(I)} = I + \\psi{(I)} \\log{(I)} and \\frac{\\psi^{2}{(I)}}{I} = \\frac{\\psi{(I)} \\log{(I)}}{I} and - \\frac{I + \\psi{(I)} \\log{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi^{2}{(I)}}{I} = - \\frac{I + \\psi{(I)} \\log{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi{(I)} \\log{(I)}}{I} and - \\frac{I + \\psi^{2}{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi^{2}{(I)}}{I} = - \\frac{I + \\psi^{2}{(I)}}{\\psi{(I)} \\log{(I)}} + \\frac{\\psi{(I)} \\log{(I)}}{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["times", 1, "Function('\\\\psi')(Symbol('I', commutative=True))"], "Equality(Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2)), Mul(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))))"], [["add", 2, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2))), Add(Symbol('I', commutative=True), Mul(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))))"], [["divide", 2, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))))"], [["minus", 4, "Mul(Add(Symbol('I', commutative=True), Mul(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('I', commutative=True), Mul(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Add(Symbol('I', commutative=True), Mul(Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Add(Symbol('I', commutative=True), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2))), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Add(Symbol('I', commutative=True), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(2))), Pow(Function('\\\\psi')(Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Integer(-1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given i{(\\psi^*,\\dot{z})} = \\dot{z} - \\psi^*, then obtain ((\\dot{z} - \\psi^*)^{\\psi^*} + i^{\\psi^*}{(\\psi^*,\\dot{z})}) \\phi_{1}{(\\theta)} = 2 (\\dot{z} - \\psi^*)^{\\psi^*} \\phi_{1}{(\\theta)}", "derivation": "i{(\\psi^*,\\dot{z})} = \\dot{z} - \\psi^* and i^{\\psi^*}{(\\psi^*,\\dot{z})} = (\\dot{z} - \\psi^*)^{\\psi^*} and (\\dot{z} - \\psi^*)^{\\psi^*} + i^{\\psi^*}{(\\psi^*,\\dot{z})} = 2 (\\dot{z} - \\psi^*)^{\\psi^*} and ((\\dot{z} - \\psi^*)^{\\psi^*} + i^{\\psi^*}{(\\psi^*,\\dot{z})}) \\phi_{1}{(\\theta)} = 2 (\\dot{z} - \\psi^*)^{\\psi^*} \\phi_{1}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["add", 2, "Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Function('i')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))))"], [["times", 3, "Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Function('i')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{E},V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\mathbf{E}, then derive \\int \\Omega{(\\mathbf{E},V_{\\mathbf{E}})} d\\mathbf{E} = V_{\\mathbf{E}} \\mathbf{E} + \\chi + \\frac{\\mathbf{E}^{2}}{2}, then obtain \\int (V_{\\mathbf{E}} + \\mathbf{E}) d\\mathbf{E} = V_{\\mathbf{E}} \\mathbf{E} + \\chi + \\frac{\\mathbf{E}^{2}}{2}", "derivation": "\\Omega{(\\mathbf{E},V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\mathbf{E} and \\int \\Omega{(\\mathbf{E},V_{\\mathbf{E}})} d\\mathbf{E} = \\int (V_{\\mathbf{E}} + \\mathbf{E}) d\\mathbf{E} and \\int \\Omega{(\\mathbf{E},V_{\\mathbf{E}})} d\\mathbf{E} = V_{\\mathbf{E}} \\mathbf{E} + \\chi + \\frac{\\mathbf{E}^{2}}{2} and \\int (V_{\\mathbf{E}} + \\mathbf{E}) d\\mathbf{E} = V_{\\mathbf{E}} \\mathbf{E} + \\chi + \\frac{\\mathbf{E}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\chi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\chi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given x{(\\mathbf{S})} = \\sin{(\\log{(\\mathbf{S})})}, then obtain \\int \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} x{(\\mathbf{S})}) d\\mathbf{S} = \\int \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} \\sin{(\\log{(\\mathbf{S})})}) d\\mathbf{S}", "derivation": "x{(\\mathbf{S})} = \\sin{(\\log{(\\mathbf{S})})} and \\frac{d}{d \\mathbf{S}} x{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\sin{(\\log{(\\mathbf{S})})} and - \\sigma_x + \\frac{d}{d \\mathbf{S}} x{(\\mathbf{S})} = - \\sigma_x + \\frac{d}{d \\mathbf{S}} \\sin{(\\log{(\\mathbf{S})})} and \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} x{(\\mathbf{S})}) = \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} \\sin{(\\log{(\\mathbf{S})})}) and \\int \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} x{(\\mathbf{S})}) d\\mathbf{S} = \\int \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + \\frac{d}{d \\mathbf{S}} \\sin{(\\log{(\\mathbf{S})})}) d\\mathbf{S}", "srepr_derivation": [["get_premise", "Equality(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), sin(log(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(sin(log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(sin(log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(sin(log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('x')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(sin(log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given a{(B)} = \\sin{(B)}, then derive \\int a{(B)} dB = \\hat{p} - \\cos{(B)}, then obtain \\frac{\\hat{p} - \\cos{(B)}}{\\sin{(B)}} = \\frac{\\int \\sin{(B)} dB}{\\sin{(B)}}", "derivation": "a{(B)} = \\sin{(B)} and \\int a{(B)} dB = \\int \\sin{(B)} dB and \\frac{\\int a{(B)} dB}{\\sin{(B)}} = \\frac{\\int \\sin{(B)} dB}{\\sin{(B)}} and \\int a{(B)} dB = \\hat{p} - \\cos{(B)} and \\frac{\\hat{p} - \\cos{(B)}}{\\sin{(B)}} = \\frac{\\int \\sin{(B)} dB}{\\sin{(B)}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('a')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["divide", 2, "sin(Symbol('B', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('B', commutative=True)), Integer(-1)), Integral(Function('a')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(Pow(sin(Symbol('B', commutative=True)), Integer(-1)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Pow(sin(Symbol('B', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('B', commutative=True)), Integer(-1)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(v_{t})} = \\log{(v_{t})}, then derive \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} = \\frac{1}{v_{t}}, then obtain \\iint \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} dv_{t} dv_{t} = \\iint \\frac{1}{v_{t}} dv_{t} dv_{t}", "derivation": "\\operatorname{n_{2}}{(v_{t})} = \\log{(v_{t})} and \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} = \\frac{d}{d v_{t}} \\log{(v_{t})} and \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} = \\frac{1}{v_{t}} and \\int \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} dv_{t} = \\int \\frac{1}{v_{t}} dv_{t} and \\iint \\frac{d}{d v_{t}} \\operatorname{n_{2}}{(v_{t})} dv_{t} dv_{t} = \\iint \\frac{1}{v_{t}} dv_{t} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('v_t', commutative=True)), log(Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(log(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Pow(Symbol('v_t', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('v_t', commutative=True)"], "Equality(Integral(Derivative(Function('n_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True))), Integral(Pow(Symbol('v_t', commutative=True), Integer(-1)), Tuple(Symbol('v_t', commutative=True))))"], [["integrate", 4, "Symbol('v_t', commutative=True)"], "Equality(Integral(Derivative(Function('n_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Pow(Symbol('v_t', commutative=True), Integer(-1)), Tuple(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given Q{(E_{n})} = \\sin{(E_{n})}, then obtain \\log{(- \\eta^{\\prime} + 2 Q{(E_{n})} + \\sin{(E_{n})})} = \\log{(- \\eta^{\\prime} + 3 \\sin{(E_{n})})}", "derivation": "Q{(E_{n})} = \\sin{(E_{n})} and Q{(E_{n})} + \\sin{(E_{n})} = 2 \\sin{(E_{n})} and - \\eta^{\\prime} + Q{(E_{n})} = - \\eta^{\\prime} + \\sin{(E_{n})} and - \\eta^{\\prime} + 2 Q{(E_{n})} + \\sin{(E_{n})} = - \\eta^{\\prime} + Q{(E_{n})} + 2 \\sin{(E_{n})} and - \\eta^{\\prime} + 2 Q{(E_{n})} + \\sin{(E_{n})} = - \\eta^{\\prime} + 3 \\sin{(E_{n})} and \\log{(- \\eta^{\\prime} + 2 Q{(E_{n})} + \\sin{(E_{n})})} = \\log{(- \\eta^{\\prime} + 3 \\sin{(E_{n})})}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["add", 1, "sin(Symbol('E_n', commutative=True))"], "Equality(Add(Function('Q')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Mul(Integer(2), sin(Symbol('E_n', commutative=True))))"], [["minus", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('Q')(Symbol('E_n', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('E_n', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('Q')(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('Q')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('Q')(Symbol('E_n', commutative=True)), Mul(Integer(2), sin(Symbol('E_n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('Q')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(3), sin(Symbol('E_n', commutative=True)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('Q')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(3), sin(Symbol('E_n', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(I,\\mu)} = \\frac{\\partial}{\\partial \\mu} (- I + \\mu) and \\mathbf{E}{(I,\\mu)} = - I + \\mu, then obtain \\int \\operatorname{P_{e}}{(I,\\mu)} d\\mu = \\rho_f + \\mathbf{E}{(I,\\mu)}", "derivation": "\\operatorname{P_{e}}{(I,\\mu)} = \\frac{\\partial}{\\partial \\mu} (- I + \\mu) and \\mathbf{E}{(I,\\mu)} = - I + \\mu and \\operatorname{P_{e}}{(I,\\mu)} = \\frac{\\partial}{\\partial \\mu} \\mathbf{E}{(I,\\mu)} and \\int \\operatorname{P_{e}}{(I,\\mu)} d\\mu = \\int \\frac{\\partial}{\\partial \\mu} \\mathbf{E}{(I,\\mu)} d\\mu and \\int \\operatorname{P_{e}}{(I,\\mu)} d\\mu = \\rho_f + \\mathbf{E}{(I,\\mu)}", "srepr_derivation": [["get_premise", "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Derivative(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('P_e')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\pi)} = \\int \\log{(\\pi)} d\\pi, then derive C_{2} + \\pi \\log{(\\pi)} + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} = 2 C_{2} + 2 \\pi \\log{(\\pi)} - \\pi + 2 \\log{(\\pi)}, then obtain C_{2} + \\pi \\log{(\\pi)} + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} = C_{2} + Q + 2 \\pi \\log{(\\pi)} - \\pi + 2 \\log{(\\pi)}", "derivation": "\\mathbf{v}{(\\pi)} = \\int \\log{(\\pi)} d\\pi and \\pi + \\mathbf{v}{(\\pi)} = \\pi + \\int \\log{(\\pi)} d\\pi and \\pi + \\mathbf{v}{(\\pi)} + \\log{(\\pi)} = \\pi + \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi and \\pi + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi = \\pi + 2 \\log{(\\pi)} + 2 \\int \\log{(\\pi)} d\\pi and C_{2} + \\pi \\log{(\\pi)} + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} = 2 C_{2} + 2 \\pi \\log{(\\pi)} - \\pi + 2 \\log{(\\pi)} and C_{2} + \\pi \\log{(\\pi)} + 2 \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi = 2 C_{2} + 2 \\pi \\log{(\\pi)} - \\pi + 2 \\log{(\\pi)} and C_{2} + \\pi \\log{(\\pi)} + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} = C_{2} + \\pi \\log{(\\pi)} + 2 \\log{(\\pi)} + \\int \\log{(\\pi)} d\\pi and C_{2} + \\pi \\log{(\\pi)} + \\mathbf{v}{(\\pi)} + 2 \\log{(\\pi)} = C_{2} + Q + 2 \\pi \\log{(\\pi)} - \\pi + 2 \\log{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["add", 2, "log(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["add", 3, "Add(log(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))), Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True))), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Function('\\\\mathbf{v}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))), Add(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given z{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})}, then obtain \\frac{d}{d \\dot{\\mathbf{r}}} 1 = \\frac{d}{d \\dot{\\mathbf{r}}} \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} 1}{\\frac{d}{d \\dot{\\mathbf{r}}} \\frac{z{(\\dot{\\mathbf{r}})}}{\\log{(\\dot{\\mathbf{r}})}}}", "derivation": "z{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})} and \\frac{z{(\\dot{\\mathbf{r}})}}{\\log{(\\dot{\\mathbf{r}})}} = 1 and \\frac{d}{d \\dot{\\mathbf{r}}} \\frac{z{(\\dot{\\mathbf{r}})}}{\\log{(\\dot{\\mathbf{r}})}} = \\frac{d}{d \\dot{\\mathbf{r}}} 1 and 1 = \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} 1}{\\frac{d}{d \\dot{\\mathbf{r}}} \\frac{z{(\\dot{\\mathbf{r}})}}{\\log{(\\dot{\\mathbf{r}})}}} and \\frac{d}{d \\dot{\\mathbf{r}}} 1 = \\frac{d}{d \\dot{\\mathbf{r}}} \\frac{\\frac{d}{d \\dot{\\mathbf{r}}} 1}{\\frac{d}{d \\dot{\\mathbf{r}}} \\frac{z{(\\dot{\\mathbf{r}})}}{\\log{(\\dot{\\mathbf{r}})}}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Mul(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Mul(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Derivative(Mul(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Derivative(Integer(1), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Pow(Derivative(Mul(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(t)} = \\sin{(e^{t})} and \\operatorname{a^{\\dagger}}{(t)} = e^{t}, then obtain \\Omega{(t)} - \\operatorname{a^{\\dagger}}{(t)} = - \\operatorname{a^{\\dagger}}{(t)} + \\sin{(\\operatorname{a^{\\dagger}}{(t)})}", "derivation": "\\Omega{(t)} = \\sin{(e^{t})} and \\Omega{(t)} - e^{t} = - e^{t} + \\sin{(e^{t})} and \\operatorname{a^{\\dagger}}{(t)} = e^{t} and \\Omega{(t)} - \\operatorname{a^{\\dagger}}{(t)} = - \\operatorname{a^{\\dagger}}{(t)} + \\sin{(\\operatorname{a^{\\dagger}}{(t)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('t', commutative=True)), sin(exp(Symbol('t', commutative=True))))"], [["minus", 1, "exp(Symbol('t', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('t', commutative=True)), Mul(Integer(-1), exp(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('t', commutative=True))), sin(exp(Symbol('t', commutative=True)))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\Omega')(Symbol('t', commutative=True)), Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('t', commutative=True))), sin(Function('a^{\\\\dagger}')(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given z{(\\sigma_p)} = \\sin{(\\sigma_p)}, then obtain (z{(\\sigma_p)} \\sin{(\\sigma_p)} + \\sin{(\\sigma_p)})^{2} = (\\sin^{2}{(\\sigma_p)} + \\sin{(\\sigma_p)})^{2}", "derivation": "z{(\\sigma_p)} = \\sin{(\\sigma_p)} and z{(\\sigma_p)} \\sin{(\\sigma_p)} = \\sin^{2}{(\\sigma_p)} and z{(\\sigma_p)} \\sin{(\\sigma_p)} + \\sin{(\\sigma_p)} = \\sin^{2}{(\\sigma_p)} + \\sin{(\\sigma_p)} and (z{(\\sigma_p)} \\sin{(\\sigma_p)} + \\sin{(\\sigma_p)})^{2} = (\\sin^{2}{(\\sigma_p)} + \\sin{(\\sigma_p)})^{2}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(2)))"], [["add", 2, "sin(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))), Add(Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))), Integer(2)), Pow(Add(Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(2)), sin(Symbol('\\\\sigma_p', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\omega,\\delta)} = \\delta \\omega, then obtain - 2 \\omega + 2 (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega} = - 2 \\omega + (\\int \\delta \\omega d\\omega)^{\\omega} + (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega}", "derivation": "\\hat{\\mathbf{x}}{(\\omega,\\delta)} = \\delta \\omega and \\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega = \\int \\delta \\omega d\\omega and (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega} = (\\int \\delta \\omega d\\omega)^{\\omega} and - \\omega + (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega} = - \\omega + (\\int \\delta \\omega d\\omega)^{\\omega} and - 2 \\omega + 2 (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega} = - 2 \\omega + (\\int \\delta \\omega d\\omega)^{\\omega} + (\\int \\hat{\\mathbf{x}}{(\\omega,\\delta)} d\\omega)^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["minus", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\omega', commutative=True)), Pow(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(A_{2})} = e^{A_{2}}, then derive - \\frac{d}{d A_{2}} \\operatorname{z^{*}}{(A_{2})} = - e^{A_{2}}, then obtain - \\operatorname{z^{*}}{(A_{2})} = - \\frac{d}{d A_{2}} e^{A_{2}}", "derivation": "\\operatorname{z^{*}}{(A_{2})} = e^{A_{2}} and \\frac{d}{d A_{2}} \\operatorname{z^{*}}{(A_{2})} = \\frac{d}{d A_{2}} e^{A_{2}} and - \\frac{d}{d A_{2}} \\operatorname{z^{*}}{(A_{2})} = - \\frac{d}{d A_{2}} e^{A_{2}} and - \\frac{d}{d A_{2}} \\operatorname{z^{*}}{(A_{2})} = - e^{A_{2}} and - \\frac{d}{d A_{2}} \\operatorname{z^{*}}{(A_{2})} = - \\operatorname{z^{*}}{(A_{2})} and - \\operatorname{z^{*}}{(A_{2})} = - \\frac{d}{d A_{2}} e^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('z^*')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('z^*')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Function('z^*')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Integer(-1), Function('z^*')(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Function('z^*')(Symbol('A_2', commutative=True))), Mul(Integer(-1), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(r_{0})} = \\cos{(r_{0})}, then obtain \\frac{d}{d r_{0}} (- \\operatorname{f^{*}}{(r_{0})} + \\frac{\\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}}) = \\frac{d}{d r_{0}} (- \\operatorname{f^{*}}{(r_{0})} - 1 + \\frac{2 \\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}})", "derivation": "\\operatorname{f^{*}}{(r_{0})} = \\cos{(r_{0})} and 1 = \\frac{\\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}} and 1 - \\operatorname{f^{*}}{(r_{0})} = - \\operatorname{f^{*}}{(r_{0})} + \\frac{\\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}} and \\frac{d}{d r_{0}} (1 - \\operatorname{f^{*}}{(r_{0})}) = \\frac{d}{d r_{0}} (- \\operatorname{f^{*}}{(r_{0})} + \\frac{\\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}}) and \\frac{d}{d r_{0}} (- \\operatorname{f^{*}}{(r_{0})} + \\frac{\\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}}) = \\frac{d}{d r_{0}} (- \\operatorname{f^{*}}{(r_{0})} - 1 + \\frac{2 \\cos{(r_{0})}}{\\operatorname{f^{*}}{(r_{0})}})", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["divide", 1, "Function('f^*')(Symbol('r_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f^*')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True))))"], [["minus", 2, "Function('f^*')(Symbol('r_0', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True))), Mul(Pow(Function('f^*')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True)))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True))), Mul(Pow(Function('f^*')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True))), Mul(Pow(Function('f^*')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f^*')(Symbol('r_0', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('f^*')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(I)} = \\int \\sin{(I)} dI and \\mathbf{J}{(I)} = \\phi^{I}{(I)}, then derive \\phi^{I}{(I)} = (g_{\\varepsilon} - \\cos{(I)})^{I}, then obtain \\log{(\\mathbf{J}{(I)})} = \\log{((g_{\\varepsilon} - \\cos{(I)})^{I})}", "derivation": "\\phi{(I)} = \\int \\sin{(I)} dI and \\phi^{I}{(I)} = (\\int \\sin{(I)} dI)^{I} and \\mathbf{J}{(I)} = \\phi^{I}{(I)} and \\phi^{I}{(I)} = (g_{\\varepsilon} - \\cos{(I)})^{I} and \\mathbf{J}{(I)} = (\\int \\sin{(I)} dI)^{I} and \\log{(\\mathbf{J}{(I)})} = \\log{((\\int \\sin{(I)} dI)^{I})} and (\\int \\sin{(I)} dI)^{I} = (g_{\\varepsilon} - \\cos{(I)})^{I} and \\log{(\\mathbf{J}{(I)})} = \\log{((g_{\\varepsilon} - \\cos{(I)})^{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('I', commutative=True)), Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('I', commutative=True)), Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}')(Symbol('I', commutative=True)), Pow(Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["log", 5], "Equality(log(Function('\\\\mathbf{J}')(Symbol('I', commutative=True))), log(Pow(Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(log(Function('\\\\mathbf{J}')(Symbol('I', commutative=True))), log(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\hat{\\mathbf{r}},p)} = \\cos{(\\hat{\\mathbf{r}} - p)}, then derive A_{z} + \\theta{(\\hat{\\mathbf{r}},p)} = h + \\cos{(\\hat{\\mathbf{r}} - p)}, then obtain h + \\cos{(\\hat{\\mathbf{r}} - p)} = h + \\theta{(\\hat{\\mathbf{r}},p)}", "derivation": "\\theta{(\\hat{\\mathbf{r}},p)} = \\cos{(\\hat{\\mathbf{r}} - p)} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\theta{(\\hat{\\mathbf{r}},p)} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}} - p)} and \\int \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\theta{(\\hat{\\mathbf{r}},p)} d\\hat{\\mathbf{r}} = \\int \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}} - p)} d\\hat{\\mathbf{r}} and A_{z} + \\theta{(\\hat{\\mathbf{r}},p)} = h + \\cos{(\\hat{\\mathbf{r}} - p)} and A_{z} + \\theta{(\\hat{\\mathbf{r}},p)} = h + \\theta{(\\hat{\\mathbf{r}},p)} and h + \\cos{(\\hat{\\mathbf{r}} - p)} = h + \\theta{(\\hat{\\mathbf{r}},p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_z', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))), Add(Symbol('h', commutative=True), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('A_z', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))), Add(Symbol('h', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('h', commutative=True), cos(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))), Add(Symbol('h', commutative=True), Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(C)} = e^{C}, then obtain \\frac{d}{d C} \\int (\\operatorname{F_{c}}{(C)} + \\int e^{C} dC) dC = \\frac{d}{d C} \\int (e^{C} + \\int e^{C} dC) dC", "derivation": "\\operatorname{F_{c}}{(C)} = e^{C} and \\int \\operatorname{F_{c}}{(C)} dC = \\int e^{C} dC and \\operatorname{F_{c}}{(C)} + \\int \\operatorname{F_{c}}{(C)} dC = e^{C} + \\int \\operatorname{F_{c}}{(C)} dC and \\operatorname{F_{c}}{(C)} + \\int e^{C} dC = e^{C} + \\int e^{C} dC and \\int (\\operatorname{F_{c}}{(C)} + \\int e^{C} dC) dC = \\int (e^{C} + \\int e^{C} dC) dC and \\frac{d}{d C} \\int (\\operatorname{F_{c}}{(C)} + \\int e^{C} dC) dC = \\frac{d}{d C} \\int (e^{C} + \\int e^{C} dC) dC", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["add", 1, "Integral(Function('F_c')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))"], "Equality(Add(Function('F_c')(Symbol('C', commutative=True)), Integral(Function('F_c')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(exp(Symbol('C', commutative=True)), Integral(Function('F_c')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('F_c')(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(exp(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('F_c')(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Add(exp(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 5, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Add(Function('F_c')(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Add(exp(Symbol('C', commutative=True)), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(C_{d},\\theta_2)} = \\cos{(C_{d} - \\theta_2)}, then obtain \\tilde{g}^*{(C_{d},\\theta_2)} + \\int 0 d\\theta_2 + 1 = \\tilde{g}^*{(C_{d},\\theta_2)} + \\int (- \\tilde{g}^*{(C_{d},\\theta_2)} + \\cos{(C_{d} - \\theta_2)}) d\\theta_2 + 1", "derivation": "\\tilde{g}^*{(C_{d},\\theta_2)} = \\cos{(C_{d} - \\theta_2)} and 0 = - \\tilde{g}^*{(C_{d},\\theta_2)} + \\cos{(C_{d} - \\theta_2)} and \\int 0 d\\theta_2 = \\int (- \\tilde{g}^*{(C_{d},\\theta_2)} + \\cos{(C_{d} - \\theta_2)}) d\\theta_2 and \\tilde{g}^*{(C_{d},\\theta_2)} + \\int 0 d\\theta_2 = \\tilde{g}^*{(C_{d},\\theta_2)} + \\int (- \\tilde{g}^*{(C_{d},\\theta_2)} + \\cos{(C_{d} - \\theta_2)}) d\\theta_2 and \\tilde{g}^*{(C_{d},\\theta_2)} + \\int 0 d\\theta_2 + 1 = \\tilde{g}^*{(C_{d},\\theta_2)} + \\int (- \\tilde{g}^*{(C_{d},\\theta_2)} + \\cos{(C_{d} - \\theta_2)}) d\\theta_2 + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 1, "Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)), Add(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\theta_2', commutative=True))), cos(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\bar{\\h}{(\\nabla,p)} = \\nabla p, then obtain (\\nabla p)^{\\nabla} \\bar{\\h}{(\\nabla,p)} = \\nabla p (\\nabla p)^{\\nabla}", "derivation": "\\bar{\\h}{(\\nabla,p)} = \\nabla p and \\bar{\\h}^{\\nabla}{(\\nabla,p)} = (\\nabla p)^{\\nabla} and \\bar{\\h}{(\\nabla,p)} \\bar{\\h}^{\\nabla}{(\\nabla,p)} = \\nabla p \\bar{\\h}^{\\nabla}{(\\nabla,p)} and (\\nabla p)^{\\nabla} \\bar{\\h}{(\\nabla,p)} = \\nabla p (\\nabla p)^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True), Pow(Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True)), Function('\\\\hbar')(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True), Pow(Mul(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\ddot{x},E_{x})} = E_{x} + \\ddot{x}, then obtain 0 = (E_{x} + \\ddot{x}) \\int (E_{x} + \\ddot{x})^{\\ddot{x}} dE_{x} - (E_{x} + \\ddot{x}) \\int \\mathbf{F}^{\\ddot{x}}{(\\ddot{x},E_{x})} dE_{x}", "derivation": "\\mathbf{F}{(\\ddot{x},E_{x})} = E_{x} + \\ddot{x} and \\mathbf{F}^{\\ddot{x}}{(\\ddot{x},E_{x})} = (E_{x} + \\ddot{x})^{\\ddot{x}} and \\int \\mathbf{F}^{\\ddot{x}}{(\\ddot{x},E_{x})} dE_{x} = \\int (E_{x} + \\ddot{x})^{\\ddot{x}} dE_{x} and (E_{x} + \\ddot{x}) \\int \\mathbf{F}^{\\ddot{x}}{(\\ddot{x},E_{x})} dE_{x} = (E_{x} + \\ddot{x}) \\int (E_{x} + \\ddot{x})^{\\ddot{x}} dE_{x} and 0 = (E_{x} + \\ddot{x}) \\int (E_{x} + \\ddot{x})^{\\ddot{x}} dE_{x} - (E_{x} + \\ddot{x}) \\int \\mathbf{F}^{\\ddot{x}}{(\\ddot{x},E_{x})} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["power", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["times", 3, "Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["minus", 4, "Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], "Equality(Integer(0), Add(Mul(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Mul(Integer(-1), Add(Symbol('E_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('E_x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A)} = \\sin{(\\log{(A)})}, then obtain \\frac{\\operatorname{P_{e}}{(A)} \\operatorname{P_{e}}^{A}{(A)}}{\\log{(A)}} = \\frac{\\operatorname{P_{e}}{(A)} \\sin^{A}{(\\log{(A)})}}{\\log{(A)}}", "derivation": "\\operatorname{P_{e}}{(A)} = \\sin{(\\log{(A)})} and \\operatorname{P_{e}}^{A}{(A)} = \\sin^{A}{(\\log{(A)})} and \\operatorname{P_{e}}{(A)} \\operatorname{P_{e}}^{A}{(A)} = \\operatorname{P_{e}}{(A)} \\sin^{A}{(\\log{(A)})} and \\frac{\\operatorname{P_{e}}{(A)} \\operatorname{P_{e}}^{A}{(A)}}{\\log{(A)}} = \\frac{\\operatorname{P_{e}}{(A)} \\sin^{A}{(\\log{(A)})}}{\\log{(A)}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A', commutative=True)), sin(log(Symbol('A', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(sin(log(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["times", 2, "Function('P_e')(Symbol('A', commutative=True))"], "Equality(Mul(Function('P_e')(Symbol('A', commutative=True)), Pow(Function('P_e')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Function('P_e')(Symbol('A', commutative=True)), Pow(sin(log(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"], [["divide", 3, "log(Symbol('A', commutative=True))"], "Equality(Mul(Function('P_e')(Symbol('A', commutative=True)), Pow(Function('P_e')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(log(Symbol('A', commutative=True)), Integer(-1))), Mul(Function('P_e')(Symbol('A', commutative=True)), Pow(log(Symbol('A', commutative=True)), Integer(-1)), Pow(sin(log(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{p})} = \\frac{1}{\\hat{p}}, then obtain \\operatorname{E_{n}}{(\\hat{p})} \\int (- C + \\operatorname{E_{n}}^{\\hat{p}}{(\\hat{p})}) d\\hat{p} = \\operatorname{E_{n}}{(\\hat{p})} \\int (- C + (\\frac{1}{\\hat{p}})^{\\hat{p}}) d\\hat{p}", "derivation": "\\operatorname{E_{n}}{(\\hat{p})} = \\frac{1}{\\hat{p}} and \\operatorname{E_{n}}^{\\hat{p}}{(\\hat{p})} = (\\frac{1}{\\hat{p}})^{\\hat{p}} and - C + \\operatorname{E_{n}}^{\\hat{p}}{(\\hat{p})} = - C + (\\frac{1}{\\hat{p}})^{\\hat{p}} and \\int (- C + \\operatorname{E_{n}}^{\\hat{p}}{(\\hat{p})}) d\\hat{p} = \\int (- C + (\\frac{1}{\\hat{p}})^{\\hat{p}}) d\\hat{p} and \\operatorname{E_{n}}{(\\hat{p})} \\int (- C + \\operatorname{E_{n}}^{\\hat{p}}{(\\hat{p})}) d\\hat{p} = \\operatorname{E_{n}}{(\\hat{p})} \\int (- C + (\\frac{1}{\\hat{p}})^{\\hat{p}}) d\\hat{p}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 2, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 4, "Pow(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Integer(-1))"], "Equality(Mul(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Function('E_n')(Symbol('\\\\hat{p}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\varphi,\\sigma_p)} = \\frac{\\varphi}{\\sigma_p} and \\mathbf{p}{(\\varphi,\\sigma_p)} = \\frac{\\varphi}{\\sigma_p}, then obtain \\mathbf{J}_f{(\\varphi,\\sigma_p)} \\mathbf{p}{(\\varphi,\\sigma_p)} = \\frac{\\varphi^{2}}{\\sigma_p^{2}}", "derivation": "\\mathbf{J}_f{(\\varphi,\\sigma_p)} = \\frac{\\varphi}{\\sigma_p} and \\mathbf{p}{(\\varphi,\\sigma_p)} = \\frac{\\varphi}{\\sigma_p} and \\mathbf{J}_f{(\\varphi,\\sigma_p)} \\mathbf{p}{(\\varphi,\\sigma_p)} = \\frac{\\varphi \\mathbf{p}{(\\varphi,\\sigma_p)}}{\\sigma_p} and \\mathbf{J}_f{(\\varphi,\\sigma_p)} = \\mathbf{p}{(\\varphi,\\sigma_p)} and \\frac{\\varphi \\mathbf{J}_f{(\\varphi,\\sigma_p)}}{\\sigma_p} = \\frac{\\varphi^{2}}{\\sigma_p^{2}} and \\frac{\\varphi \\mathbf{p}{(\\varphi,\\sigma_p)}}{\\sigma_p} = \\frac{\\varphi^{2}}{\\sigma_p^{2}} and \\mathbf{J}_f{(\\varphi,\\sigma_p)} \\mathbf{p}{(\\varphi,\\sigma_p)} = \\frac{\\varphi^{2}}{\\sigma_p^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-2)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-2)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-2)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\rho{(E_{n},C_{d},B)} = (E_{n}^{B})^{C_{d}} and \\operatorname{M_{E}}{(F_{N})} = \\cos{(\\log{(F_{N})})}, then obtain ((E_{n}^{B})^{C_{d}} + \\operatorname{M_{E}}{(F_{N})}) (E_{n}^{B})^{C_{d}} = ((E_{n}^{B})^{C_{d}} + \\cos{(\\log{(F_{N})})}) (E_{n}^{B})^{C_{d}}", "derivation": "\\rho{(E_{n},C_{d},B)} = (E_{n}^{B})^{C_{d}} and \\operatorname{M_{E}}{(F_{N})} = \\cos{(\\log{(F_{N})})} and (E_{n}^{B})^{C_{d}} + \\operatorname{M_{E}}{(F_{N})} = (E_{n}^{B})^{C_{d}} + \\cos{(\\log{(F_{N})})} and ((E_{n}^{B})^{C_{d}} + \\operatorname{M_{E}}{(F_{N})}) \\rho{(E_{n},C_{d},B)} = ((E_{n}^{B})^{C_{d}} + \\cos{(\\log{(F_{N})})}) \\rho{(E_{n},C_{d},B)} and ((E_{n}^{B})^{C_{d}} + \\operatorname{M_{E}}{(F_{N})}) (E_{n}^{B})^{C_{d}} = ((E_{n}^{B})^{C_{d}} + \\cos{(\\log{(F_{N})})}) (E_{n}^{B})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('E_n', commutative=True), Symbol('C_d', commutative=True), Symbol('B', commutative=True)), Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)))"], ["get_premise", "Equality(Function('M_E')(Symbol('F_N', commutative=True)), cos(log(Symbol('F_N', commutative=True))))"], [["add", 2, "Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), Function('M_E')(Symbol('F_N', commutative=True))), Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), cos(log(Symbol('F_N', commutative=True)))))"], [["times", 3, "Function('\\\\rho')(Symbol('E_n', commutative=True), Symbol('C_d', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), Function('M_E')(Symbol('F_N', commutative=True))), Function('\\\\rho')(Symbol('E_n', commutative=True), Symbol('C_d', commutative=True), Symbol('B', commutative=True))), Mul(Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), cos(log(Symbol('F_N', commutative=True)))), Function('\\\\rho')(Symbol('E_n', commutative=True), Symbol('C_d', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), Function('M_E')(Symbol('F_N', commutative=True))), Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True))), Mul(Add(Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True)), cos(log(Symbol('F_N', commutative=True)))), Pow(Pow(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given W{(U,\\varepsilon)} = \\cos{(U^{\\varepsilon})}, then obtain (\\cos^{U}{(U^{\\varepsilon})})^{- U} \\int \\frac{\\partial}{\\partial U} W^{U}{(U,\\varepsilon)} d\\varepsilon = (\\cos^{U}{(U^{\\varepsilon})})^{- U} \\int \\frac{\\partial}{\\partial U} \\cos^{U}{(U^{\\varepsilon})} d\\varepsilon", "derivation": "W{(U,\\varepsilon)} = \\cos{(U^{\\varepsilon})} and W^{U}{(U,\\varepsilon)} = \\cos^{U}{(U^{\\varepsilon})} and \\frac{\\partial}{\\partial U} W^{U}{(U,\\varepsilon)} = \\frac{\\partial}{\\partial U} \\cos^{U}{(U^{\\varepsilon})} and \\int \\frac{\\partial}{\\partial U} W^{U}{(U,\\varepsilon)} d\\varepsilon = \\int \\frac{\\partial}{\\partial U} \\cos^{U}{(U^{\\varepsilon})} d\\varepsilon and (\\cos^{U}{(U^{\\varepsilon})})^{- U} \\int \\frac{\\partial}{\\partial U} W^{U}{(U,\\varepsilon)} d\\varepsilon = (\\cos^{U}{(U^{\\varepsilon})})^{- U} \\int \\frac{\\partial}{\\partial U} \\cos^{U}{(U^{\\varepsilon})} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('U', commutative=True)), Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 4, "Pow(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Derivative(Pow(Function('W')(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Pow(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Derivative(Pow(cos(Pow(Symbol('U', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(F_{N})} = \\int \\cos{(F_{N})} dF_{N}, then derive \\dot{y}{(F_{N})} = E_{\\lambda} + \\sin{(F_{N})}, then obtain 2 (u + \\sin{(F_{N})})^{F_{N}} = (E_{\\lambda} + \\sin{(F_{N})})^{F_{N}} + (u + \\sin{(F_{N})})^{F_{N}}", "derivation": "\\dot{y}{(F_{N})} = \\int \\cos{(F_{N})} dF_{N} and \\dot{y}{(F_{N})} = E_{\\lambda} + \\sin{(F_{N})} and \\int \\cos{(F_{N})} dF_{N} = E_{\\lambda} + \\sin{(F_{N})} and (\\int \\cos{(F_{N})} dF_{N})^{F_{N}} = (E_{\\lambda} + \\sin{(F_{N})})^{F_{N}} and 2 (\\int \\cos{(F_{N})} dF_{N})^{F_{N}} = (E_{\\lambda} + \\sin{(F_{N})})^{F_{N}} + (\\int \\cos{(F_{N})} dF_{N})^{F_{N}} and 2 (u + \\sin{(F_{N})})^{F_{N}} = (E_{\\lambda} + \\sin{(F_{N})})^{F_{N}} + (u + \\sin{(F_{N})})^{F_{N}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('F_N', commutative=True)), Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{y}')(Symbol('F_N', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"], [["add", 4, "Pow(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Pow(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))), Add(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Mul(Integer(2), Pow(Add(Symbol('u', commutative=True), sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))), Add(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Add(Symbol('u', commutative=True), sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(F_{x})} = \\cos{(e^{F_{x}})}, then obtain (\\int (\\mathbf{D}{(F_{x})} + \\cos{(e^{F_{x}})}) dF_{x})^{F_{x}} = (\\int 2 \\cos{(e^{F_{x}})} dF_{x})^{F_{x}}", "derivation": "\\mathbf{D}{(F_{x})} = \\cos{(e^{F_{x}})} and \\mathbf{D}{(F_{x})} + \\cos{(e^{F_{x}})} = 2 \\cos{(e^{F_{x}})} and \\int (\\mathbf{D}{(F_{x})} + \\cos{(e^{F_{x}})}) dF_{x} = \\int 2 \\cos{(e^{F_{x}})} dF_{x} and (\\int (\\mathbf{D}{(F_{x})} + \\cos{(e^{F_{x}})}) dF_{x})^{F_{x}} = (\\int 2 \\cos{(e^{F_{x}})} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), cos(exp(Symbol('F_x', commutative=True))))"], [["add", 1, "cos(exp(Symbol('F_x', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), cos(exp(Symbol('F_x', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('F_x', commutative=True)))))"], [["integrate", 2, "Symbol('F_x', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), cos(exp(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Integer(2), cos(exp(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\mathbf{D}')(Symbol('F_x', commutative=True)), cos(exp(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Integral(Mul(Integer(2), cos(exp(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(E,W,\\hat{H}_l)} = (\\frac{\\hat{H}_l}{W})^{E}, then obtain \\frac{\\partial}{\\partial W} \\int \\hat{H}_l^{2} (\\mathbf{r}{(E,W,\\hat{H}_l)} - 1) dW = \\frac{\\partial}{\\partial W} \\int \\hat{H}_l^{2} ((\\frac{\\hat{H}_l}{W})^{E} - 1) dW", "derivation": "\\mathbf{r}{(E,W,\\hat{H}_l)} = (\\frac{\\hat{H}_l}{W})^{E} and \\mathbf{r}{(E,W,\\hat{H}_l)} - 1 = (\\frac{\\hat{H}_l}{W})^{E} - 1 and \\hat{H}_l (\\mathbf{r}{(E,W,\\hat{H}_l)} - 1) = \\hat{H}_l ((\\frac{\\hat{H}_l}{W})^{E} - 1) and \\hat{H}_l^{2} (\\mathbf{r}{(E,W,\\hat{H}_l)} - 1) = \\hat{H}_l^{2} ((\\frac{\\hat{H}_l}{W})^{E} - 1) and \\int \\hat{H}_l^{2} (\\mathbf{r}{(E,W,\\hat{H}_l)} - 1) dW = \\int \\hat{H}_l^{2} ((\\frac{\\hat{H}_l}{W})^{E} - 1) dW and \\frac{\\partial}{\\partial W} \\int \\hat{H}_l^{2} (\\mathbf{r}{(E,W,\\hat{H}_l)} - 1) dW = \\frac{\\partial}{\\partial W} \\int \\hat{H}_l^{2} ((\\frac{\\hat{H}_l}{W})^{E} - 1) dW", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Add(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)), Integer(-1)))"], [["times", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Add(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)), Integer(-1))))"], [["times", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Tuple(Symbol('W', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)), Integer(-1))), Tuple(Symbol('W', commutative=True))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True), Symbol('W', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1))), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)), Add(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('E', commutative=True)), Integer(-1))), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(a,M_{E})} = M_{E} a, then obtain \\frac{\\mathbf{s}^{2}{(a,M_{E})}}{M_{E}^{2} a^{2}} = 1", "derivation": "\\mathbf{s}{(a,M_{E})} = M_{E} a and \\mathbf{s}^{2}{(a,M_{E})} = M_{E} a \\mathbf{s}{(a,M_{E})} and \\mathbf{s}^{4}{(a,M_{E})} = M_{E}^{2} a^{2} \\mathbf{s}^{2}{(a,M_{E})} and \\frac{\\mathbf{s}^{2}{(a,M_{E})}}{M_{E}^{2} a^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('a', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Integer(2)), Mul(Symbol('M_E', commutative=True), Symbol('a', commutative=True), Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Integer(4)), Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Pow(Symbol('a', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Integer(2))))"], [["divide", 3, "Mul(Pow(Symbol('M_E', commutative=True), Integer(2)), Pow(Symbol('a', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Symbol('a', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{s}')(Symbol('a', commutative=True), Symbol('M_E', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(c)} = e^{c}, then obtain \\frac{- \\operatorname{P_{e}}{(c)} - e^{c}}{\\operatorname{P_{e}}{(c)}} = - \\frac{\\operatorname{P_{e}}{(c)} + e^{c}}{\\operatorname{P_{e}}{(c)}}", "derivation": "\\operatorname{P_{e}}{(c)} = e^{c} and \\operatorname{P_{e}}{(c)} + e^{c} = 2 e^{c} and \\frac{\\operatorname{P_{e}}{(c)} + e^{c}}{\\operatorname{P_{e}}{(c)}} = \\frac{2 e^{c}}{\\operatorname{P_{e}}{(c)}} and - \\frac{\\operatorname{P_{e}}{(c)} + e^{c}}{\\operatorname{P_{e}}{(c)}} = - \\frac{2 e^{c}}{\\operatorname{P_{e}}{(c)}} and \\frac{- \\operatorname{P_{e}}{(c)} - e^{c}}{\\operatorname{P_{e}}{(c)}} = - \\frac{2 e^{c}}{\\operatorname{P_{e}}{(c)}} and \\frac{- \\operatorname{P_{e}}{(c)} - e^{c}}{\\operatorname{P_{e}}{(c)}} = - \\frac{\\operatorname{P_{e}}{(c)} + e^{c}}{\\operatorname{P_{e}}{(c)}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["add", 1, "exp(Symbol('c', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Mul(Integer(2), exp(Symbol('c', commutative=True))))"], [["divide", 2, "Function('P_e')(Symbol('c', commutative=True))"], "Equality(Mul(Add(Function('P_e')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1)), exp(Symbol('c', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Function('P_e')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1)), exp(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Function('P_e')(Symbol('c', commutative=True))), Mul(Integer(-1), exp(Symbol('c', commutative=True)))), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1)), exp(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Function('P_e')(Symbol('c', commutative=True))), Mul(Integer(-1), exp(Symbol('c', commutative=True)))), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Function('P_e')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True))), Pow(Function('P_e')(Symbol('c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\dot{z})} = \\sin{(\\dot{z})} and q{(\\dot{z})} = \\dot{z}, then obtain \\frac{d}{d \\dot{z}} \\frac{q{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{d}{d \\dot{z}} \\frac{\\dot{z}}{\\sin{(\\dot{z})}}", "derivation": "\\mathbf{J}_f{(\\dot{z})} = \\sin{(\\dot{z})} and q{(\\dot{z})} = \\dot{z} and \\frac{q{(\\dot{z})}}{\\mathbf{J}_f{(\\dot{z})}} = \\frac{\\dot{z}}{\\mathbf{J}_f{(\\dot{z})}} and \\frac{q{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{\\dot{z}}{\\sin{(\\dot{z})}} and \\frac{d}{d \\dot{z}} \\frac{q{(\\dot{z})}}{\\sin{(\\dot{z})}} = \\frac{d}{d \\dot{z}} \\frac{\\dot{z}}{\\sin{(\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))"], [["divide", 2, "Function('\\\\mathbf{J}_f')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Function('q')(Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('q')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Function('q')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} = L_{\\varepsilon} \\sin{(Q)}, then obtain ((\\frac{\\partial}{\\partial Q} \\int \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} dL_{\\varepsilon})^{Q})^{L_{\\varepsilon}} = ((\\frac{\\partial}{\\partial Q} \\int L_{\\varepsilon} \\sin{(Q)} dL_{\\varepsilon})^{Q})^{L_{\\varepsilon}}", "derivation": "\\operatorname{f_{E}}{(Q,L_{\\varepsilon})} = L_{\\varepsilon} \\sin{(Q)} and \\int \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} dL_{\\varepsilon} = \\int L_{\\varepsilon} \\sin{(Q)} dL_{\\varepsilon} and \\frac{\\partial}{\\partial Q} \\int \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} dL_{\\varepsilon} = \\frac{\\partial}{\\partial Q} \\int L_{\\varepsilon} \\sin{(Q)} dL_{\\varepsilon} and (\\frac{\\partial}{\\partial Q} \\int \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} dL_{\\varepsilon})^{Q} = (\\frac{\\partial}{\\partial Q} \\int L_{\\varepsilon} \\sin{(Q)} dL_{\\varepsilon})^{Q} and ((\\frac{\\partial}{\\partial Q} \\int \\operatorname{f_{E}}{(Q,L_{\\varepsilon})} dL_{\\varepsilon})^{Q})^{L_{\\varepsilon}} = ((\\frac{\\partial}{\\partial Q} \\int L_{\\varepsilon} \\sin{(Q)} dL_{\\varepsilon})^{Q})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), sin(Symbol('Q', commutative=True))))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Function('f_E')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('f_E')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["power", 4, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Pow(Derivative(Integral(Function('f_E')(Symbol('Q', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Pow(Derivative(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given G{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then obtain (\\frac{d}{d \\mathbf{D}} G{(\\mathbf{D})} - 1)^{\\mathbf{D}} = (\\cos{(\\mathbf{D})} - 1)^{\\mathbf{D}}", "derivation": "G{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and - \\mathbf{D} + G{(\\mathbf{D})} = - \\mathbf{D} + \\sin{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} (- \\mathbf{D} + G{(\\mathbf{D})}) = \\frac{d}{d \\mathbf{D}} (- \\mathbf{D} + \\sin{(\\mathbf{D})}) and (\\frac{d}{d \\mathbf{D}} (- \\mathbf{D} + G{(\\mathbf{D})}))^{\\mathbf{D}} = (\\frac{d}{d \\mathbf{D}} (- \\mathbf{D} + \\sin{(\\mathbf{D})}))^{\\mathbf{D}} and (\\frac{d}{d \\mathbf{D}} G{(\\mathbf{D})} - 1)^{\\mathbf{D}} = (\\cos{(\\mathbf{D})} - 1)^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('G')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('G')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('G')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('G')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(cos(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given W{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\operatorname{A_{2}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} W{(f^{\\prime})}, then obtain \\operatorname{A_{2}}{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})} = 2 \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})}", "derivation": "W{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} W{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})} and \\operatorname{A_{2}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} W{(f^{\\prime})} and \\operatorname{A_{2}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})} and \\operatorname{A_{2}}{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})} = 2 \\frac{d}{d f^{\\prime}} \\sin{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('W')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 4, "Derivative(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Add(Function('A_2')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(t_{2})} = \\int \\cos{(t_{2})} dt_{2}, then derive 2 I{(t_{2})} = 2 \\hat{H} + 2 \\sin{(t_{2})}, then obtain (\\hat{H} + 2 \\int \\cos{(t_{2})} dt_{2})^{3} = (3 \\hat{H} + 2 \\sin{(t_{2})})^{3}", "derivation": "I{(t_{2})} = \\int \\cos{(t_{2})} dt_{2} and 2 I{(t_{2})} = I{(t_{2})} + \\int \\cos{(t_{2})} dt_{2} and I{(t_{2})} + \\int \\cos{(t_{2})} dt_{2} = 2 \\int \\cos{(t_{2})} dt_{2} and 2 I{(t_{2})} = 2 \\int \\cos{(t_{2})} dt_{2} and 2 I{(t_{2})} = 2 \\hat{H} + 2 \\sin{(t_{2})} and \\hat{H} + 2 I{(t_{2})} = 3 \\hat{H} + 2 \\sin{(t_{2})} and (\\hat{H} + 2 I{(t_{2})})^{3} = (3 \\hat{H} + 2 \\sin{(t_{2})})^{3} and (\\hat{H} + 2 \\int \\cos{(t_{2})} dt_{2})^{3} = (3 \\hat{H} + 2 \\sin{(t_{2})})^{3}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["add", 1, "Function('I')(Symbol('t_2', commutative=True))"], "Equality(Mul(Integer(2), Function('I')(Symbol('t_2', commutative=True))), Add(Function('I')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["add", 1, "Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Add(Function('I')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('I')(Symbol('t_2', commutative=True))), Mul(Integer(2), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), Function('I')(Symbol('t_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))))"], [["add", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(2), Function('I')(Symbol('t_2', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))))"], [["power", 6, 3], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(2), Function('I')(Symbol('t_2', commutative=True)))), Integer(3)), Pow(Add(Mul(Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))), Integer(3)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(2), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))), Integer(3)), Pow(Add(Mul(Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), sin(Symbol('t_2', commutative=True)))), Integer(3)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\hat{p},q)} = \\int (\\hat{p} + q) d\\hat{p}, then derive \\mathbf{J}_P{(\\hat{p},q)} = \\frac{\\hat{p}^{2}}{2} + \\hat{p} q + \\nabla, then obtain \\frac{\\mathbf{J}_P{(\\hat{p},q)}}{\\hat{p} + q} = \\frac{\\frac{\\hat{p}^{2}}{2} + \\hat{p} q + \\nabla}{\\hat{p} + q}", "derivation": "\\mathbf{J}_P{(\\hat{p},q)} = \\int (\\hat{p} + q) d\\hat{p} and \\frac{\\mathbf{J}_P{(\\hat{p},q)}}{\\hat{p} + q} = \\frac{\\int (\\hat{p} + q) d\\hat{p}}{\\hat{p} + q} and \\mathbf{J}_P{(\\hat{p},q)} = \\frac{\\hat{p}^{2}}{2} + \\hat{p} q + \\nabla and \\int (\\hat{p} + q) d\\hat{p} = \\frac{\\hat{p}^{2}}{2} + \\hat{p} q + \\nabla and \\frac{\\mathbf{J}_P{(\\hat{p},q)}}{\\hat{p} + q} = \\frac{\\frac{\\hat{p}^{2}}{2} + \\hat{p} q + \\nabla}{\\hat{p} + q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(G)} = \\sin{(G)}, then obtain \\frac{(2 \\operatorname{v_{1}}{(G)} + 1)^{G}}{\\sin{(G)}} = \\frac{(2 \\sin{(G)} + 1)^{G}}{\\sin{(G)}}", "derivation": "\\operatorname{v_{1}}{(G)} = \\sin{(G)} and \\operatorname{v_{1}}{(G)} + 1 = \\sin{(G)} + 1 and 2 \\operatorname{v_{1}}{(G)} + 1 = \\operatorname{v_{1}}{(G)} + \\sin{(G)} + 1 and (2 \\operatorname{v_{1}}{(G)} + 1)^{G} = (\\operatorname{v_{1}}{(G)} + \\sin{(G)} + 1)^{G} and \\frac{(2 \\operatorname{v_{1}}{(G)} + 1)^{G}}{\\sin{(G)}} = \\frac{(\\operatorname{v_{1}}{(G)} + \\sin{(G)} + 1)^{G}}{\\sin{(G)}} and \\frac{(2 \\operatorname{v_{1}}{(G)} + 1)^{G}}{\\sin{(G)}} = \\frac{(2 \\sin{(G)} + 1)^{G}}{\\sin{(G)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_1')(Symbol('G', commutative=True)), Integer(1)), Add(sin(Symbol('G', commutative=True)), Integer(1)))"], [["add", 2, "Function('v_1')(Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('v_1')(Symbol('G', commutative=True))), Integer(1)), Add(Function('v_1')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)), Integer(1)))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('v_1')(Symbol('G', commutative=True))), Integer(1)), Symbol('G', commutative=True)), Pow(Add(Function('v_1')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)), Integer(1)), Symbol('G', commutative=True)))"], [["divide", 4, "sin(Symbol('G', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Function('v_1')(Symbol('G', commutative=True))), Integer(1)), Symbol('G', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Integer(-1))), Mul(Pow(Add(Function('v_1')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)), Integer(1)), Symbol('G', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Integer(2), Function('v_1')(Symbol('G', commutative=True))), Integer(1)), Symbol('G', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(2), sin(Symbol('G', commutative=True))), Integer(1)), Symbol('G', commutative=True)), Pow(sin(Symbol('G', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given s{(\\mathbf{p},z,C)} = (C z)^{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial C} s{(\\mathbf{p},z,C)} - 1 = -1 + \\frac{\\mathbf{p} (C z)^{\\mathbf{p}}}{C}, then obtain \\frac{\\partial}{\\partial C} s{(\\mathbf{p},z,C)} - 1 = -1 + \\frac{\\mathbf{p} s{(\\mathbf{p},z,C)}}{C}", "derivation": "s{(\\mathbf{p},z,C)} = (C z)^{\\mathbf{p}} and - C + s{(\\mathbf{p},z,C)} = - C + (C z)^{\\mathbf{p}} and - C + s{(\\mathbf{p},z,C)} + 1 = - C + (C z)^{\\mathbf{p}} + 1 and \\frac{\\partial}{\\partial C} (- C + s{(\\mathbf{p},z,C)} + 1) = \\frac{\\partial}{\\partial C} (- C + (C z)^{\\mathbf{p}} + 1) and \\frac{\\partial}{\\partial C} s{(\\mathbf{p},z,C)} - 1 = -1 + \\frac{\\mathbf{p} (C z)^{\\mathbf{p}}}{C} and \\frac{\\partial}{\\partial C} s{(\\mathbf{p},z,C)} - 1 = -1 + \\frac{\\mathbf{p} s{(\\mathbf{p},z,C)}}{C}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)), Integer(1)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Integer(1)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Mul(Symbol('C', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Function('s')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z', commutative=True), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)} = U - Z - \\dot{\\mathbf{r}}, then obtain -1 = - (\\frac{U - Z - \\dot{\\mathbf{r}}}{\\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)}})^{Z}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)} = U - Z - \\dot{\\mathbf{r}} and 1 = \\frac{U - Z - \\dot{\\mathbf{r}}}{\\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)}} and 1 = (\\frac{U - Z - \\dot{\\mathbf{r}}}{\\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)}})^{Z} and -1 = - (\\frac{U - Z - \\dot{\\mathbf{r}}}{\\operatorname{V_{\\mathbf{B}}}{(U,\\dot{\\mathbf{r}},Z)}})^{Z}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["divide", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Integer(1), Pow(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Mul(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(P_{e})} = \\log{(P_{e})} and \\Psi{(P_{e})} = - \\mathbf{M}{(P_{e})}, then obtain 0 = - \\frac{d}{d P_{e}} 0 + \\frac{d}{d P_{e}} (\\Psi{(P_{e})} + \\log{(P_{e})})", "derivation": "\\mathbf{M}{(P_{e})} = \\log{(P_{e})} and 0 = - \\mathbf{M}{(P_{e})} + \\log{(P_{e})} and \\frac{d}{d P_{e}} 0 = \\frac{d}{d P_{e}} (- \\mathbf{M}{(P_{e})} + \\log{(P_{e})}) and 0 = - \\frac{d}{d P_{e}} 0 + \\frac{d}{d P_{e}} (- \\mathbf{M}{(P_{e})} + \\log{(P_{e})}) and \\Psi{(P_{e})} = - \\mathbf{M}{(P_{e})} and 0 = - \\frac{d}{d P_{e}} 0 + \\frac{d}{d P_{e}} (\\Psi{(P_{e})} + \\log{(P_{e})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True))), log(Symbol('P_e', commutative=True))))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True))), log(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Integer(0), Tuple(Symbol('P_e', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True))), log(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('P_e', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Integer(0), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Derivative(Add(Function('\\\\Psi')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(c_{0})} = \\cos{(c_{0})}, then derive - \\frac{\\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})}}{\\sin{(c_{0})}} = 1, then obtain - \\frac{\\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})}}{\\sin^{2}{(c_{0})}} = \\frac{1}{\\sin{(c_{0})}}", "derivation": "\\operatorname{C_{d}}{(c_{0})} = \\cos{(c_{0})} and \\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})} = \\frac{d}{d c_{0}} \\cos{(c_{0})} and \\frac{\\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})}}{\\frac{d}{d c_{0}} \\cos{(c_{0})}} = 1 and - \\frac{\\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})}}{\\sin{(c_{0})}} = 1 and - \\frac{\\frac{d}{d c_{0}} \\operatorname{C_{d}}{(c_{0})}}{\\sin^{2}{(c_{0})}} = \\frac{1}{\\sin{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('C_d')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(sin(Symbol('c_0', commutative=True)), Integer(-1)), Derivative(Function('C_d')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(1))"], [["times", 4, "Pow(sin(Symbol('c_0', commutative=True)), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(sin(Symbol('c_0', commutative=True)), Integer(-2)), Derivative(Function('C_d')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Pow(sin(Symbol('c_0', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda} - \\mathbf{M})}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})} = - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{M})}", "derivation": "\\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda} - \\mathbf{M})} and \\mathbf{M} + \\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})} = \\mathbf{M} + \\cos{(\\hat{H}_{\\lambda} - \\mathbf{M})} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\mathbf{M} + \\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})}) = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\mathbf{M} + \\cos{(\\hat{H}_{\\lambda} - \\mathbf{M})}) and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\mathbf{P}{(\\mathbf{M},\\hat{H}_{\\lambda})} = - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))))"]]}, {"prompt": "Given \\bar{\\h}{(c)} = \\sin{(c)}, then obtain (- \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}})^{c} = (- \\frac{\\bar{\\h}^{- \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}}}{(c)} \\sin^{2}{(c)}}{\\bar{\\h}{(c)}})^{c}", "derivation": "\\bar{\\h}{(c)} = \\sin{(c)} and 1 = \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}} and -1 = - \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}} and (-1)^{c} = (- \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}})^{c} and (- \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}})^{c} = (- \\frac{\\bar{\\h}^{- \\frac{\\sin{(c)}}{\\bar{\\h}{(c)}}}{(c)} \\sin^{2}{(c)}}{\\bar{\\h}{(c)}})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["divide", 1, "Function('\\\\hbar')(Symbol('c', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('c', commutative=True)), Integer(-1)), sin(Symbol('c', commutative=True)))), Pow(sin(Symbol('c', commutative=True)), Integer(2))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{J})} = e^{\\mathbf{J}}, then derive \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = c_{0} + e^{\\mathbf{J}}, then obtain - c_{0} + \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = \\mathbf{s}{(\\mathbf{J})}", "derivation": "\\mathbf{s}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = \\int e^{\\mathbf{J}} d\\mathbf{J} and \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = c_{0} + e^{\\mathbf{J}} and \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = c_{0} + \\mathbf{s}{(\\mathbf{J})} and - c_{0} + \\int \\mathbf{s}{(\\mathbf{J})} d\\mathbf{J} = \\mathbf{s}{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 4, "Symbol('c_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mu_0,y^{\\prime})} = \\mu_0 \\sin{(y^{\\prime})}, then derive \\frac{\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{F_{N}}{(\\mu_0,y^{\\prime})}}{\\mu_0} = \\cos{(y^{\\prime})}, then obtain (\\frac{\\frac{\\partial}{\\partial y^{\\prime}} \\mu_0 \\sin{(y^{\\prime})}}{\\mu_0})^{\\mu_0} = \\cos^{\\mu_0}{(y^{\\prime})}", "derivation": "\\operatorname{F_{N}}{(\\mu_0,y^{\\prime})} = \\mu_0 \\sin{(y^{\\prime})} and \\frac{\\operatorname{F_{N}}{(\\mu_0,y^{\\prime})}}{\\mu_0} = \\sin{(y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\frac{\\operatorname{F_{N}}{(\\mu_0,y^{\\prime})}}{\\mu_0} = \\frac{d}{d y^{\\prime}} \\sin{(y^{\\prime})} and \\frac{\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{F_{N}}{(\\mu_0,y^{\\prime})}}{\\mu_0} = \\cos{(y^{\\prime})} and (\\frac{\\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{F_{N}}{(\\mu_0,y^{\\prime})}}{\\mu_0})^{\\mu_0} = \\cos^{\\mu_0}{(y^{\\prime})} and (\\frac{\\frac{\\partial}{\\partial y^{\\prime}} \\mu_0 \\sin{(y^{\\prime})}}{\\mu_0})^{\\mu_0} = \\cos^{\\mu_0}{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('F_N')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('\\\\mu_0', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(T)} = e^{T} and L{(\\varphi^*,\\rho)} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{\\rho}, then obtain - L{(\\varphi^*,\\rho)} + \\hat{\\mathbf{x}}^{T}{(T)} - \\frac{d}{d T} (e^{T})^{T} = - L{(\\varphi^*,\\rho)} + (e^{T})^{T} - \\frac{d}{d T} (e^{T})^{T}", "derivation": "\\hat{\\mathbf{x}}{(T)} = e^{T} and L{(\\varphi^*,\\rho)} = \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{\\rho} and \\hat{\\mathbf{x}}^{T}{(T)} = (e^{T})^{T} and \\hat{\\mathbf{x}}^{T}{(T)} - \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{\\rho} = (e^{T})^{T} - \\frac{\\partial}{\\partial \\varphi^*} (\\varphi^*)^{\\rho} and - L{(\\varphi^*,\\rho)} + \\hat{\\mathbf{x}}^{T}{(T)} = - L{(\\varphi^*,\\rho)} + (e^{T})^{T} and - L{(\\varphi^*,\\rho)} + \\hat{\\mathbf{x}}^{T}{(T)} - \\frac{d}{d T} (e^{T})^{T} = - L{(\\varphi^*,\\rho)} + (e^{T})^{T} - \\frac{d}{d T} (e^{T})^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["minus", 3, "Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Add(Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["minus", 5, "Derivative(Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(exp(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\nabla{(E)} = \\sin{(E)} and \\mu_{0}{(a,U,A_{1})} = \\frac{a}{A_{1} U}, then obtain (\\mu_{0}{(a,U,A_{1})} + \\frac{1}{E \\sin{(E)}})^{a} = (\\frac{1}{E \\sin{(E)}} + \\frac{a}{A_{1} U})^{a}", "derivation": "\\nabla{(E)} = \\sin{(E)} and \\frac{\\nabla{(E)}}{\\sin{(E)}} = 1 and \\frac{\\nabla{(E)}}{E \\sin{(E)}} = \\frac{1}{E} and \\frac{1}{E \\sin{(E)}} = \\frac{1}{E \\nabla{(E)}} and \\mu_{0}{(a,U,A_{1})} = \\frac{a}{A_{1} U} and \\mu_{0}{(a,U,A_{1})} + \\frac{1}{E \\nabla{(E)}} = \\frac{1}{E \\nabla{(E)}} + \\frac{a}{A_{1} U} and (\\mu_{0}{(a,U,A_{1})} + \\frac{1}{E \\nabla{(E)}})^{a} = (\\frac{1}{E \\nabla{(E)}} + \\frac{a}{A_{1} U})^{a} and (\\mu_{0}{(a,U,A_{1})} + \\frac{1}{E \\sin{(E)}})^{a} = (\\frac{1}{E \\sin{(E)}} + \\frac{a}{A_{1} U})^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["divide", 1, "sin(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('E', commutative=True)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('E', commutative=True)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Pow(Symbol('E', commutative=True), Integer(-1)))"], [["divide", 3, "Function('\\\\nabla')(Symbol('E', commutative=True))"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('U', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a', commutative=True)))"], [["add", 5, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('U', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1)))), Add(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('U', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1)))), Symbol('a', commutative=True)), Pow(Add(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Add(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('U', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(sin(Symbol('E', commutative=True)), Integer(-1)))), Symbol('a', commutative=True)), Pow(Add(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(g,f)} = \\frac{\\partial}{\\partial f} (f + g), then derive \\mathbf{s}{(g,f)} = 1, then derive \\frac{\\partial}{\\partial f} \\mathbf{s}{(g,f)} = 0, then obtain - \\mathbf{A} + \\frac{d}{d f} 1 = - \\mathbf{A}", "derivation": "\\mathbf{s}{(g,f)} = \\frac{\\partial}{\\partial f} (f + g) and \\mathbf{s}{(g,f)} = 1 and \\frac{\\partial}{\\partial f} \\mathbf{s}{(g,f)} = \\frac{d}{d f} 1 and \\frac{\\partial}{\\partial f} \\mathbf{s}{(g,f)} = 0 and \\frac{d}{d f} 1 = 0 and - \\mathbf{A} + \\frac{d}{d f} 1 = - \\mathbf{A}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('g', commutative=True), Symbol('f', commutative=True)), Derivative(Add(Symbol('f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('g', commutative=True), Symbol('f', commutative=True)), Integer(1))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('g', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('g', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(0))"], [["add", 5, "Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(Z,G)} = G + Z and \\hat{H}_{\\lambda}{(G,Z)} = \\theta_{1}^{Z}{(Z,G)}, then obtain \\int \\frac{\\partial}{\\partial G} \\hat{H}_{\\lambda}{(G,Z)} dZ = \\int \\frac{\\partial}{\\partial G} (G + Z)^{Z} dZ", "derivation": "\\theta_{1}{(Z,G)} = G + Z and \\theta_{1}^{Z}{(Z,G)} = (G + Z)^{Z} and \\hat{H}_{\\lambda}{(G,Z)} = \\theta_{1}^{Z}{(Z,G)} and \\hat{H}_{\\lambda}{(G,Z)} = (G + Z)^{Z} and \\frac{\\partial}{\\partial G} \\hat{H}_{\\lambda}{(G,Z)} = \\frac{\\partial}{\\partial G} (G + Z)^{Z} and \\frac{\\partial}{\\partial G} \\hat{H}_{\\lambda}{(G,Z)} = \\frac{\\partial}{\\partial G} \\theta_{1}^{Z}{(Z,G)} and \\int \\frac{\\partial}{\\partial G} \\hat{H}_{\\lambda}{(G,Z)} dZ = \\int \\frac{\\partial}{\\partial G} \\theta_{1}^{Z}{(Z,G)} dZ and \\int \\frac{\\partial}{\\partial G} \\hat{H}_{\\lambda}{(G,Z)} dZ = \\int \\frac{\\partial}{\\partial G} (G + Z)^{Z} dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('Z', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Pow(Add(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True), Symbol('G', commutative=True)), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Pow(Add(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True), Symbol('G', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integral(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Pow(Add(Symbol('G', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(i)} = \\log{(\\sin{(i)})}, then obtain \\bar{\\h}^{2}{(i)} \\log{(\\sin{(i)})}^{2} = \\bar{\\h}{(i)} \\log{(\\sin{(i)})}^{3}", "derivation": "\\bar{\\h}{(i)} = \\log{(\\sin{(i)})} and \\bar{\\h}^{2}{(i)} = \\bar{\\h}{(i)} \\log{(\\sin{(i)})} and \\bar{\\h}^{4}{(i)} = \\bar{\\h}^{2}{(i)} \\log{(\\sin{(i)})}^{2} and \\bar{\\h}^{2}{(i)} \\log{(\\sin{(i)})}^{2} = \\bar{\\h}{(i)} \\log{(\\sin{(i)})}^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True))))"], [["times", 1, "Function('\\\\hbar')(Symbol('i', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('i', commutative=True)), Integer(2)), Mul(Function('\\\\hbar')(Symbol('i', commutative=True)), log(sin(Symbol('i', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\hbar')(Symbol('i', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\hbar')(Symbol('i', commutative=True)), Integer(2)), Pow(log(sin(Symbol('i', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('\\\\hbar')(Symbol('i', commutative=True)), Integer(2)), Pow(log(sin(Symbol('i', commutative=True))), Integer(2))), Mul(Function('\\\\hbar')(Symbol('i', commutative=True)), Pow(log(sin(Symbol('i', commutative=True))), Integer(3))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi_2)} = e^{\\phi_2}, then obtain \\frac{\\frac{d}{d \\phi_2} \\int \\operatorname{t_{2}}{(\\phi_2)} d\\phi_2}{\\operatorname{t_{2}}{(\\phi_2)}} = \\frac{\\frac{\\partial}{\\partial \\phi_2} (t + e^{\\phi_2})}{\\operatorname{t_{2}}{(\\phi_2)}}", "derivation": "\\operatorname{t_{2}}{(\\phi_2)} = e^{\\phi_2} and \\int \\operatorname{t_{2}}{(\\phi_2)} d\\phi_2 = \\int e^{\\phi_2} d\\phi_2 and \\frac{d}{d \\phi_2} \\int \\operatorname{t_{2}}{(\\phi_2)} d\\phi_2 = \\frac{d}{d \\phi_2} \\int e^{\\phi_2} d\\phi_2 and \\frac{\\frac{d}{d \\phi_2} \\int \\operatorname{t_{2}}{(\\phi_2)} d\\phi_2}{\\operatorname{t_{2}}{(\\phi_2)}} = \\frac{\\frac{d}{d \\phi_2} \\int e^{\\phi_2} d\\phi_2}{\\operatorname{t_{2}}{(\\phi_2)}} and \\frac{\\frac{d}{d \\phi_2} \\int \\operatorname{t_{2}}{(\\phi_2)} d\\phi_2}{\\operatorname{t_{2}}{(\\phi_2)}} = \\frac{\\frac{\\partial}{\\partial \\phi_2} (t + e^{\\phi_2})}{\\operatorname{t_{2}}{(\\phi_2)}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Integral(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["divide", 3, "Function('t_2')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Derivative(Integral(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Derivative(Integral(exp(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Derivative(Integral(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Derivative(Add(Symbol('t', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(\\phi_1,f)} = \\frac{\\phi_1}{f}, then obtain (\\log{(\\int (\\frac{\\phi_1}{f} + 1) d\\phi_1)}) \\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1 = (\\log{(\\int (\\frac{\\phi_1}{f} + 1) d\\phi_1)}) \\int (\\frac{\\phi_1}{f} + 1) d\\phi_1", "derivation": "\\nabla{(\\phi_1,f)} = \\frac{\\phi_1}{f} and \\nabla{(\\phi_1,f)} + 1 = \\frac{\\phi_1}{f} + 1 and \\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1 = \\int (\\frac{\\phi_1}{f} + 1) d\\phi_1 and \\log{(\\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1)} = \\log{(\\int (\\frac{\\phi_1}{f} + 1) d\\phi_1)} and (\\log{(\\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1)}) \\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1 = (\\log{(\\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1)}) \\int (\\frac{\\phi_1}{f} + 1) d\\phi_1 and (\\log{(\\int (\\frac{\\phi_1}{f} + 1) d\\phi_1)}) \\int (\\nabla{(\\phi_1,f)} + 1) d\\phi_1 = (\\log{(\\int (\\frac{\\phi_1}{f} + 1) d\\phi_1)}) \\int (\\frac{\\phi_1}{f} + 1) d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["log", 3], "Equality(log(Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), log(Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["times", 3, "log(Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], "Equality(Mul(log(Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(log(Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(log(Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integral(Add(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True), Symbol('f', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(log(Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integral(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\dot{z},\\mathbf{M})} = \\int \\dot{z} \\mathbf{M} d\\dot{z}, then obtain (\\frac{\\varphi^{\\dot{z}}{(\\dot{z},\\mathbf{M})}}{\\varphi{(\\dot{z},\\mathbf{M})}})^{\\mathbf{M}} = (\\frac{(\\int \\dot{z} \\mathbf{M} d\\dot{z})^{\\dot{z}}}{\\varphi{(\\dot{z},\\mathbf{M})}})^{\\mathbf{M}}", "derivation": "\\varphi{(\\dot{z},\\mathbf{M})} = \\int \\dot{z} \\mathbf{M} d\\dot{z} and \\varphi^{\\dot{z}}{(\\dot{z},\\mathbf{M})} = (\\int \\dot{z} \\mathbf{M} d\\dot{z})^{\\dot{z}} and \\frac{\\varphi^{\\dot{z}}{(\\dot{z},\\mathbf{M})}}{\\varphi{(\\dot{z},\\mathbf{M})}} = \\frac{(\\int \\dot{z} \\mathbf{M} d\\dot{z})^{\\dot{z}}}{\\varphi{(\\dot{z},\\mathbf{M})}} and (\\frac{\\varphi^{\\dot{z}}{(\\dot{z},\\mathbf{M})}}{\\varphi{(\\dot{z},\\mathbf{M})}})^{\\mathbf{M}} = (\\frac{(\\int \\dot{z} \\mathbf{M} d\\dot{z})^{\\dot{z}}}{\\varphi{(\\dot{z},\\mathbf{M})}})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Pow(Function('\\\\varphi')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\psi,\\mathbf{r})} = \\sin{(\\frac{\\psi}{\\mathbf{r}})}, then obtain 0 = \\tilde{g}^*{(\\psi,\\mathbf{r})} - \\sin{(\\frac{\\psi}{\\mathbf{r}})}", "derivation": "\\tilde{g}^*{(\\psi,\\mathbf{r})} = \\sin{(\\frac{\\psi}{\\mathbf{r}})} and \\tilde{g}^*{(\\psi,\\mathbf{r})} + \\sin{(\\frac{\\psi}{\\mathbf{r}})} = 2 \\sin{(\\frac{\\psi}{\\mathbf{r}})} and - \\tilde{g}^*{(\\psi,\\mathbf{r})} - \\sin{(\\frac{\\psi}{\\mathbf{r}})} = - 2 \\sin{(\\frac{\\psi}{\\mathbf{r}})} and 0 = \\tilde{g}^*{(\\psi,\\mathbf{r})} - \\sin{(\\frac{\\psi}{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))"], [["add", 1, "sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))), Mul(Integer(2), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))), Mul(Integer(-1), Integer(2), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)))))"], "Equality(Integer(0), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(E_{\\lambda},\\tilde{g},B)} = (\\frac{\\tilde{g}}{B})^{E_{\\lambda}} and L{(B,E_{\\lambda},\\tilde{g})} = (\\frac{\\tilde{g}}{B})^{E_{\\lambda}} \\operatorname{f^{*}}{(E_{\\lambda},\\tilde{g},B)}, then obtain \\int L{(B,E_{\\lambda},\\tilde{g})} dB = \\int \\operatorname{f^{*}}^{2}{(E_{\\lambda},\\tilde{g},B)} dB", "derivation": "\\operatorname{f^{*}}{(E_{\\lambda},\\tilde{g},B)} = (\\frac{\\tilde{g}}{B})^{E_{\\lambda}} and L{(B,E_{\\lambda},\\tilde{g})} = (\\frac{\\tilde{g}}{B})^{E_{\\lambda}} \\operatorname{f^{*}}{(E_{\\lambda},\\tilde{g},B)} and L{(B,E_{\\lambda},\\tilde{g})} = \\operatorname{f^{*}}^{2}{(E_{\\lambda},\\tilde{g},B)} and \\int L{(B,E_{\\lambda},\\tilde{g})} dB = \\int \\operatorname{f^{*}}^{2}{(E_{\\lambda},\\tilde{g},B)} dB", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Function('f^*')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('L')(Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Function('f^*')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Function('L')(Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Function('f^*')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('B', commutative=True)), Integer(2)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(v,p)} = p e^{v} and \\mathbf{M}{(v,p)} = v (p e^{v})^{p}, then obtain \\frac{\\mathbf{M}{(v,p)}}{\\mathbf{v}{(v,p)}} = \\frac{v (p e^{v})^{p}}{\\mathbf{v}{(v,p)}}", "derivation": "\\mathbf{v}{(v,p)} = p e^{v} and \\mathbf{v}^{p}{(v,p)} = (p e^{v})^{p} and v \\mathbf{v}^{p}{(v,p)} = v (p e^{v})^{p} and \\mathbf{M}{(v,p)} = v (p e^{v})^{p} and \\mathbf{M}{(v,p)} = v \\mathbf{v}^{p}{(v,p)} and \\frac{\\mathbf{M}{(v,p)}}{\\mathbf{v}{(v,p)}} = \\frac{v \\mathbf{v}^{p}{(v,p)}}{\\mathbf{v}{(v,p)}} and \\frac{\\mathbf{M}{(v,p)}}{\\mathbf{v}{(v,p)}} = \\frac{v (p e^{v})^{p}}{\\mathbf{v}{(v,p)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), exp(Symbol('v', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Mul(Symbol('p', commutative=True), exp(Symbol('v', commutative=True))), Symbol('p', commutative=True)))"], [["times", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Mul(Symbol('v', commutative=True), Pow(Mul(Symbol('p', commutative=True), exp(Symbol('v', commutative=True))), Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Mul(Symbol('p', commutative=True), exp(Symbol('v', commutative=True))), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["divide", 5, "Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Mul(Symbol('v', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Mul(Symbol('v', commutative=True), Pow(Mul(Symbol('p', commutative=True), exp(Symbol('v', commutative=True))), Symbol('p', commutative=True)), Pow(Function('\\\\mathbf{v}')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(g,z)} = g^{z}, then obtain 4 \\Psi_{\\lambda}{(g,z)} (\\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} = (\\frac{\\partial}{\\partial z} g^{z} + \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} \\Psi_{\\lambda}{(g,z)}", "derivation": "\\Psi_{\\lambda}{(g,z)} = g^{z} and \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)} = \\frac{\\partial}{\\partial z} g^{z} and 2 \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)} = \\frac{\\partial}{\\partial z} g^{z} + \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)} and 4 (\\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} = (\\frac{\\partial}{\\partial z} g^{z} + \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} and 4 \\Psi_{\\lambda}{(g,z)} (\\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} = (\\frac{\\partial}{\\partial z} g^{z} + \\frac{\\partial}{\\partial z} \\Psi_{\\lambda}{(g,z)})^{2} \\Psi_{\\lambda}{(g,z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Derivative(Pow(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(2))), Pow(Add(Derivative(Pow(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(2)))"], [["times", 4, "Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Integer(4), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Add(Derivative(Pow(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(2)), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\omega)} = \\sin{(\\omega)} and c{(\\omega)} = \\frac{d}{d \\omega} \\sin{(\\omega)}, then derive c{(\\omega)} = \\cos{(\\omega)}, then obtain \\omega \\operatorname{E_{x}}{(\\omega)} - \\operatorname{E_{x}}{(\\omega)} - \\cos{(\\omega)} = \\omega \\sin{(\\omega)} - \\operatorname{E_{x}}{(\\omega)} - \\cos{(\\omega)}", "derivation": "\\operatorname{E_{x}}{(\\omega)} = \\sin{(\\omega)} and \\omega \\operatorname{E_{x}}{(\\omega)} = \\omega \\sin{(\\omega)} and \\omega \\operatorname{E_{x}}{(\\omega)} - \\frac{d}{d \\omega} \\sin{(\\omega)} = \\omega \\sin{(\\omega)} - \\frac{d}{d \\omega} \\sin{(\\omega)} and c{(\\omega)} = \\frac{d}{d \\omega} \\sin{(\\omega)} and c{(\\omega)} = \\cos{(\\omega)} and \\omega \\operatorname{E_{x}}{(\\omega)} - c{(\\omega)} = \\omega \\sin{(\\omega)} - c{(\\omega)} and \\omega \\operatorname{E_{x}}{(\\omega)} - \\cos{(\\omega)} = \\omega \\sin{(\\omega)} - \\cos{(\\omega)} and \\omega \\operatorname{E_{x}}{(\\omega)} - \\operatorname{E_{x}}{(\\omega)} - \\cos{(\\omega)} = \\omega \\sin{(\\omega)} - \\operatorname{E_{x}}{(\\omega)} - \\cos{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\omega', commutative=True)), Derivative(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Function('c')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('c')(Symbol('\\\\omega', commutative=True)))), Add(Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('c')(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Add(Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"], [["minus", 7, "Function('E_x')(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Add(Mul(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('E_x')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain \\mathbf{r}{(\\dot{x})} \\int \\mathbf{r}{(\\dot{x})} d\\dot{x} = \\mathbf{r}{(\\dot{x})} \\int \\sin{(\\dot{x})} d\\dot{x}", "derivation": "\\mathbf{r}{(\\dot{x})} = \\sin{(\\dot{x})} and \\int \\mathbf{r}{(\\dot{x})} d\\dot{x} = \\int \\sin{(\\dot{x})} d\\dot{x} and \\sin{(\\dot{x})} \\int \\mathbf{r}{(\\dot{x})} d\\dot{x} = \\sin{(\\dot{x})} \\int \\sin{(\\dot{x})} d\\dot{x} and \\mathbf{r}{(\\dot{x})} \\int \\mathbf{r}{(\\dot{x})} d\\dot{x} = \\mathbf{r}{(\\dot{x})} \\int \\sin{(\\dot{x})} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "sin(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\dot{x}', commutative=True)), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Function('\\\\mathbf{r}')(Symbol('\\\\dot{x}', commutative=True)), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(\\nabla,\\chi,p)} = \\frac{\\nabla + p}{\\chi} and \\mathbf{M}{(\\chi,\\nabla,p)} = \\int \\mathbf{A}{(\\nabla,\\chi,p)} dp, then derive \\int \\mathbf{A}{(\\nabla,\\chi,p)} dp = u + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi}, then obtain \\mathbf{M}{(\\chi,\\nabla,p)} = \\psi^* + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi}", "derivation": "\\mathbf{A}{(\\nabla,\\chi,p)} = \\frac{\\nabla + p}{\\chi} and \\int \\mathbf{A}{(\\nabla,\\chi,p)} dp = \\int \\frac{\\nabla + p}{\\chi} dp and \\mathbf{M}{(\\chi,\\nabla,p)} = \\int \\mathbf{A}{(\\nabla,\\chi,p)} dp and \\int \\mathbf{A}{(\\nabla,\\chi,p)} dp = u + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi} and \\int \\frac{\\nabla + p}{\\chi} dp = u + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi} and \\mathbf{M}{(\\chi,\\nabla,p)} = u + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi} and \\mathbf{M}{(\\chi,\\nabla,p)} = \\int \\frac{\\nabla + p}{\\chi} dp and \\mathbf{M}{(\\chi,\\nabla,p)} = \\psi^* + \\frac{\\nabla p}{\\chi} + \\frac{p^{2}}{2 \\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Add(Symbol('u', commutative=True), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True), Symbol('p', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\Omega{(A_{1})} = \\int e^{A_{1}} dA_{1}, then derive \\Omega{(A_{1})} - e^{A_{1}} = \\mathbf{E}, then obtain \\mathbf{E} \\Omega{(A_{1})} e^{- A_{1}} = \\mathbf{E} e^{- A_{1}} \\int e^{A_{1}} dA_{1}", "derivation": "\\Omega{(A_{1})} = \\int e^{A_{1}} dA_{1} and \\Omega{(A_{1})} - e^{A_{1}} = - e^{A_{1}} + \\int e^{A_{1}} dA_{1} and \\Omega{(A_{1})} - e^{A_{1}} = \\mathbf{E} and (\\Omega{(A_{1})} - e^{A_{1}}) \\Omega{(A_{1})} = (\\Omega{(A_{1})} - e^{A_{1}}) \\int e^{A_{1}} dA_{1} and (\\Omega{(A_{1})} - e^{A_{1}}) \\Omega{(A_{1})} e^{- A_{1}} = (\\Omega{(A_{1})} - e^{A_{1}}) e^{- A_{1}} \\int e^{A_{1}} dA_{1} and \\mathbf{E} \\Omega{(A_{1})} e^{- A_{1}} = \\mathbf{E} e^{- A_{1}} \\int e^{A_{1}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["minus", 1, "exp(Symbol('A_1', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Symbol('\\\\mathbf{E}', commutative=True))"], [["times", 1, "Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True))))"], "Equality(Mul(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Function('\\\\Omega')(Symbol('A_1', commutative=True))), Mul(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["divide", 4, "exp(Symbol('A_1', commutative=True))"], "Equality(Mul(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Function('\\\\Omega')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), Symbol('A_1', commutative=True)))), Mul(Add(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), exp(Mul(Integer(-1), Symbol('A_1', commutative=True))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\Omega')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), Symbol('A_1', commutative=True)))), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Mul(Integer(-1), Symbol('A_1', commutative=True))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(T,W)} = \\int (T - W) dT, then derive - \\operatorname{F_{g}}{(T,W)} = - \\frac{T^{2}}{2} + T W - \\mathbf{p}, then obtain \\frac{\\partial}{\\partial T} - \\int (T - W) dT = \\frac{\\partial}{\\partial T} (- \\frac{T^{2}}{2} + T W - \\mathbf{p})", "derivation": "\\operatorname{F_{g}}{(T,W)} = \\int (T - W) dT and - \\operatorname{F_{g}}{(T,W)} = - \\int (T - W) dT and - \\operatorname{F_{g}}{(T,W)} = - \\frac{T^{2}}{2} + T W - \\mathbf{p} and \\frac{\\partial}{\\partial T} - \\operatorname{F_{g}}{(T,W)} = \\frac{\\partial}{\\partial T} (- \\frac{T^{2}}{2} + T W - \\mathbf{p}) and \\frac{\\partial}{\\partial T} - \\int (T - W) dT = \\frac{\\partial}{\\partial T} (- \\frac{T^{2}}{2} + T W - \\mathbf{p})", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('T', commutative=True), Symbol('W', commutative=True)), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_g')(Symbol('T', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Integer(-1), Function('F_g')(Symbol('T', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('F_g')(Symbol('T', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Integer(-1), Integral(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(f^{*},C_{2})} = C_{2} + f^{*}, then obtain \\frac{\\partial}{\\partial C_{2}} \\log{(2 u^{C_{2}}{(f^{*},C_{2})})} = \\frac{\\partial}{\\partial C_{2}} \\log{((C_{2} + f^{*})^{C_{2}} + u^{C_{2}}{(f^{*},C_{2})})}", "derivation": "u{(f^{*},C_{2})} = C_{2} + f^{*} and u^{C_{2}}{(f^{*},C_{2})} = (C_{2} + f^{*})^{C_{2}} and 2 u^{C_{2}}{(f^{*},C_{2})} = (C_{2} + f^{*})^{C_{2}} + u^{C_{2}}{(f^{*},C_{2})} and \\log{(2 u^{C_{2}}{(f^{*},C_{2})})} = \\log{((C_{2} + f^{*})^{C_{2}} + u^{C_{2}}{(f^{*},C_{2})})} and \\frac{\\partial}{\\partial C_{2}} \\log{(2 u^{C_{2}}{(f^{*},C_{2})})} = \\frac{\\partial}{\\partial C_{2}} \\log{((C_{2} + f^{*})^{C_{2}} + u^{C_{2}}{(f^{*},C_{2})})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Symbol('f^*', commutative=True)), Symbol('C_2', commutative=True)))"], [["add", 2, "Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Add(Pow(Add(Symbol('C_2', commutative=True), Symbol('f^*', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["log", 3], "Equality(log(Mul(Integer(2), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), log(Add(Pow(Add(Symbol('C_2', commutative=True), Symbol('f^*', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))))"], [["differentiate", 4, "Symbol('C_2', commutative=True)"], "Equality(Derivative(log(Mul(Integer(2), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(log(Add(Pow(Add(Symbol('C_2', commutative=True), Symbol('f^*', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('u')(Symbol('f^*', commutative=True), Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(P_{e},\\Psi^{\\dagger})} = \\frac{\\Psi^{\\dagger}}{P_{e}}, then obtain (\\frac{\\Psi^{\\dagger}}{P_{e}})^{\\Psi^{\\dagger}} + \\operatorname{M_{E}}{(P_{e},\\Psi^{\\dagger})} = (\\frac{\\Psi^{\\dagger}}{P_{e}})^{\\Psi^{\\dagger}} + \\frac{\\Psi^{\\dagger}}{P_{e}}", "derivation": "\\operatorname{M_{E}}{(P_{e},\\Psi^{\\dagger})} = \\frac{\\Psi^{\\dagger}}{P_{e}} and \\operatorname{M_{E}}^{\\Psi^{\\dagger}}{(P_{e},\\Psi^{\\dagger})} = (\\frac{\\Psi^{\\dagger}}{P_{e}})^{\\Psi^{\\dagger}} and \\operatorname{M_{E}}{(P_{e},\\Psi^{\\dagger})} + \\operatorname{M_{E}}^{\\Psi^{\\dagger}}{(P_{e},\\Psi^{\\dagger})} = \\operatorname{M_{E}}^{\\Psi^{\\dagger}}{(P_{e},\\Psi^{\\dagger})} + \\frac{\\Psi^{\\dagger}}{P_{e}} and (\\frac{\\Psi^{\\dagger}}{P_{e}})^{\\Psi^{\\dagger}} + \\operatorname{M_{E}}{(P_{e},\\Psi^{\\dagger})} = (\\frac{\\Psi^{\\dagger}}{P_{e}})^{\\Psi^{\\dagger}} + \\frac{\\Psi^{\\dagger}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["add", 1, "Pow(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Pow(Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('M_E')(Symbol('P_e', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given k{(T,\\mathbf{S},p)} = T \\mathbf{S} p, then obtain T \\mathbf{S} p (- p + (\\mathbf{S} + k{(T,\\mathbf{S},p)})^{p} (T \\mathbf{S} p + \\mathbf{S})^{- p}) = T \\mathbf{S} p (1 - p)", "derivation": "k{(T,\\mathbf{S},p)} = T \\mathbf{S} p and \\mathbf{S} + k{(T,\\mathbf{S},p)} = T \\mathbf{S} p + \\mathbf{S} and (\\mathbf{S} + k{(T,\\mathbf{S},p)})^{p} = (T \\mathbf{S} p + \\mathbf{S})^{p} and (\\mathbf{S} + k{(T,\\mathbf{S},p)})^{p} (T \\mathbf{S} p + \\mathbf{S})^{- p} = 1 and - p + (\\mathbf{S} + k{(T,\\mathbf{S},p)})^{p} (T \\mathbf{S} p + \\mathbf{S})^{- p} = 1 - p and T \\mathbf{S} p (- p + (\\mathbf{S} + k{(T,\\mathbf{S},p)})^{p} (T \\mathbf{S} p + \\mathbf{S})^{- p}) = T \\mathbf{S} p (1 - p)", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('p', commutative=True)))"], [["divide", 3, "Pow(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)))), Integer(1))"], [["minus", 4, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["times", 5, "Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('k')(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True)))))), Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('p', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(B)} = \\cos{(B)}, then obtain \\frac{d}{d B} (- 3 \\mathbf{B}{(B)} + 3 \\cos{(B)}) = \\frac{d}{d B} \\frac{- \\mathbf{B}{(B)} + \\cos{(B)}}{\\cos{(B)}}", "derivation": "\\mathbf{B}{(B)} = \\cos{(B)} and 0 = - \\mathbf{B}{(B)} + \\cos{(B)} and - \\mathbf{B}{(B)} + \\cos{(B)} = - 2 \\mathbf{B}{(B)} + 2 \\cos{(B)} and 0 = - 2 \\mathbf{B}{(B)} + 2 \\cos{(B)} and - \\mathbf{B}{(B)} = - 3 \\mathbf{B}{(B)} + 2 \\cos{(B)} and 0 = \\frac{- 2 \\mathbf{B}{(B)} + 2 \\cos{(B)}}{\\cos{(B)}} and 0 = \\frac{- \\mathbf{B}{(B)} + \\cos{(B)}}{\\cos{(B)}} and 0 = - 3 \\mathbf{B}{(B)} + 3 \\cos{(B)} and \\frac{d}{d B} 0 = \\frac{d}{d B} \\frac{- \\mathbf{B}{(B)} + \\cos{(B)}}{\\cos{(B)}} and \\frac{d}{d B} 0 = \\frac{d}{d B} (- 3 \\mathbf{B}{(B)} + 3 \\cos{(B)}) and \\frac{d}{d B} (- 3 \\mathbf{B}{(B)} + 3 \\cos{(B)}) = \\frac{d}{d B} \\frac{- \\mathbf{B}{(B)} + \\cos{(B)}}{\\cos{(B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(2), cos(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(2), cos(Symbol('B', commutative=True)))))"], [["minus", 4, "Function('\\\\mathbf{B}')(Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(2), cos(Symbol('B', commutative=True)))))"], [["divide", 4, "cos(Symbol('B', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(2), cos(Symbol('B', commutative=True)))), Pow(cos(Symbol('B', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(3), cos(Symbol('B', commutative=True)))))"], [["differentiate", 7, "Symbol('B', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(-1))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["differentiate", 8, "Symbol('B', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(3), cos(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 9, "10"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(3), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), Mul(Integer(3), cos(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('B', commutative=True))), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(-1))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\mathbf{J}_P)} = \\log{(e^{\\mathbf{J}_P})}, then obtain \\frac{d}{d \\mathbf{J}_P} (b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} b{(\\mathbf{J}_P)}) = \\frac{d}{d \\mathbf{J}_P} (b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} \\log{(e^{\\mathbf{J}_P})})", "derivation": "b{(\\mathbf{J}_P)} = \\log{(e^{\\mathbf{J}_P})} and \\frac{d}{d \\mathbf{J}_P} b{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\log{(e^{\\mathbf{J}_P})} and b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} b{(\\mathbf{J}_P)} = b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} \\log{(e^{\\mathbf{J}_P})} and \\frac{d}{d \\mathbf{J}_P} (b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} b{(\\mathbf{J}_P)}) = \\frac{d}{d \\mathbf{J}_P} (b{(\\mathbf{J}_P)} + \\frac{d}{d \\mathbf{J}_P} \\log{(e^{\\mathbf{J}_P})})", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["add", 2, "Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Add(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(log(exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Function('b')(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(log(exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{g},u)} = \\frac{u}{\\mathbf{g}}, then obtain \\frac{2 \\hat{\\mathbf{r}}{(\\mathbf{g},u)}}{u} = \\frac{2}{\\mathbf{g}}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{g},u)} = \\frac{u}{\\mathbf{g}} and \\hat{\\mathbf{r}}{(\\mathbf{g},u)} + \\frac{u}{\\mathbf{g}} = \\frac{2 u}{\\mathbf{g}} and \\frac{\\hat{\\mathbf{r}}{(\\mathbf{g},u)} + \\frac{u}{\\mathbf{g}}}{u} = \\frac{2}{\\mathbf{g}} and \\frac{2 \\hat{\\mathbf{r}}{(\\mathbf{g},u)}}{u} = \\frac{2}{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('u', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["divide", 2, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Symbol('u', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{E}{(T,\\Psi_{nl})} = \\log{(T + \\Psi_{nl})} and \\tilde{g}^*{(\\phi,\\chi)} = \\chi \\phi, then derive \\frac{\\partial}{\\partial T} \\mathbf{E}{(T,\\Psi_{nl})} = \\frac{1}{T + \\Psi_{nl}}, then obtain \\frac{\\partial}{\\partial T} \\log{(T + \\Psi_{nl})} = \\frac{1}{T + \\Psi_{nl}}", "derivation": "\\mathbf{E}{(T,\\Psi_{nl})} = \\log{(T + \\Psi_{nl})} and \\tilde{g}^*{(\\phi,\\chi)} = \\chi \\phi and \\mathbf{E}{(T,\\Psi_{nl})} + \\tilde{g}^*{(\\phi,\\chi)} = \\tilde{g}^*{(\\phi,\\chi)} + \\log{(T + \\Psi_{nl})} and \\chi \\phi + \\mathbf{E}{(T,\\Psi_{nl})} = \\chi \\phi + \\log{(T + \\Psi_{nl})} and \\frac{\\partial}{\\partial T} (\\chi \\phi + \\mathbf{E}{(T,\\Psi_{nl})}) = \\frac{\\partial}{\\partial T} (\\chi \\phi + \\log{(T + \\Psi_{nl})}) and \\frac{\\partial}{\\partial T} \\mathbf{E}{(T,\\Psi_{nl})} = \\frac{1}{T + \\Psi_{nl}} and \\frac{\\partial}{\\partial T} \\log{(T + \\Psi_{nl})} = \\frac{1}{T + \\Psi_{nl}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\chi', commutative=True)), log(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), log(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), log(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(log(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given p{(b,f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}^{b}}, then obtain b (- b e^{f_{\\mathbf{v}}^{b}} + p{(b,f_{\\mathbf{v}})}) e^{f_{\\mathbf{v}}^{b}} = b (- b e^{f_{\\mathbf{v}}^{b}} + e^{f_{\\mathbf{v}}^{b}}) e^{f_{\\mathbf{v}}^{b}}", "derivation": "p{(b,f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}^{b}} and b p{(b,f_{\\mathbf{v}})} = b e^{f_{\\mathbf{v}}^{b}} and - b p{(b,f_{\\mathbf{v}})} + p{(b,f_{\\mathbf{v}})} = - b p{(b,f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}^{b}} and b (- b p{(b,f_{\\mathbf{v}})} + p{(b,f_{\\mathbf{v}})}) p{(b,f_{\\mathbf{v}})} = b (- b p{(b,f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}^{b}}) p{(b,f_{\\mathbf{v}})} and b (- b e^{f_{\\mathbf{v}}^{b}} + p{(b,f_{\\mathbf{v}})}) e^{f_{\\mathbf{v}}^{b}} = b (- b e^{f_{\\mathbf{v}}^{b}} + e^{f_{\\mathbf{v}}^{b}}) e^{f_{\\mathbf{v}}^{b}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True))))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Symbol('b', commutative=True), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))))"], [["minus", 1, "Mul(Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))))"], [["times", 3, "Mul(Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))), Function('p')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))), Mul(Symbol('b', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))), exp(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(M)} = \\cos{(M)}, then obtain \\operatorname{V_{\\mathbf{E}}}^{4}{(M)} = \\cos^{4}{(M)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(M)} = \\cos{(M)} and \\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} = \\cos^{2}{(M)} and \\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} + \\operatorname{V_{\\mathbf{E}}}{(M)} - \\cos^{2}{(M)} = \\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} - \\cos^{2}{(M)} + \\cos{(M)} and (\\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} + \\operatorname{V_{\\mathbf{E}}}{(M)} - \\cos^{2}{(M)})^{2} = (\\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} - \\cos^{2}{(M)} + \\cos{(M)})^{2} and (\\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} + \\operatorname{V_{\\mathbf{E}}}{(M)} - \\cos^{2}{(M)})^{4} = (\\operatorname{V_{\\mathbf{E}}}{(M)} \\cos{(M)} - \\cos^{2}{(M)} + \\cos{(M)})^{4} and \\operatorname{V_{\\mathbf{E}}}^{4}{(M)} = \\cos^{4}{(M)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["times", 1, "cos(Symbol('M', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Pow(cos(Symbol('M', commutative=True)), Integer(2)))"], [["add", 1, "Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2))))"], "Equality(Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2)))), Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2))), cos(Symbol('M', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2)))), Integer(2)), Pow(Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2))), cos(Symbol('M', commutative=True))), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2)))), Integer(4)), Pow(Add(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('M', commutative=True)), Integer(2))), cos(Symbol('M', commutative=True))), Integer(4)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True)), Integer(4)), Pow(cos(Symbol('M', commutative=True)), Integer(4)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(F_{c})} = \\cos{(e^{F_{c}})}, then obtain - e^{F_{c}} + \\frac{d^{2}}{d F_{c}^{2}} \\operatorname{v_{y}}{(F_{c})} = - (e^{F_{c}} \\cos{(e^{F_{c}})} + \\sin{(e^{F_{c}})} + 1) e^{F_{c}}", "derivation": "\\operatorname{v_{y}}{(F_{c})} = \\cos{(e^{F_{c}})} and \\operatorname{v_{y}}{(F_{c})} - e^{F_{c}} = - e^{F_{c}} + \\cos{(e^{F_{c}})} and \\frac{d}{d F_{c}} (\\operatorname{v_{y}}{(F_{c})} - e^{F_{c}}) = \\frac{d}{d F_{c}} (- e^{F_{c}} + \\cos{(e^{F_{c}})}) and \\frac{d^{2}}{d F_{c}^{2}} (\\operatorname{v_{y}}{(F_{c})} - e^{F_{c}}) = \\frac{d^{2}}{d F_{c}^{2}} (- e^{F_{c}} + \\cos{(e^{F_{c}})}) and - e^{F_{c}} + \\frac{d^{2}}{d F_{c}^{2}} \\operatorname{v_{y}}{(F_{c})} = - (e^{F_{c}} \\cos{(e^{F_{c}})} + \\sin{(e^{F_{c}})} + 1) e^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('F_c', commutative=True)), cos(exp(Symbol('F_c', commutative=True))))"], [["minus", 1, "exp(Symbol('F_c', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('F_c', commutative=True)), Mul(Integer(-1), exp(Symbol('F_c', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('F_c', commutative=True))), cos(exp(Symbol('F_c', commutative=True)))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('F_c', commutative=True)), Mul(Integer(-1), exp(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('F_c', commutative=True))), cos(exp(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('F_c', commutative=True)), Mul(Integer(-1), exp(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), exp(Symbol('F_c', commutative=True))), cos(exp(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('F_c', commutative=True))), Derivative(Function('v_y')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(2)))), Mul(Integer(-1), Add(Mul(exp(Symbol('F_c', commutative=True)), cos(exp(Symbol('F_c', commutative=True)))), sin(exp(Symbol('F_c', commutative=True))), Integer(1)), exp(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(E,\\mathbf{F})} = \\frac{\\mathbf{F}}{E}, then obtain \\frac{\\frac{\\partial}{\\partial E} \\rho_{f}{(E,\\mathbf{F})} - \\frac{\\mathbf{F}}{E}}{\\mathbf{F}} = \\frac{\\frac{\\partial}{\\partial E} \\frac{\\mathbf{F}}{E} - \\frac{\\mathbf{F}}{E}}{\\mathbf{F}}", "derivation": "\\rho_{f}{(E,\\mathbf{F})} = \\frac{\\mathbf{F}}{E} and \\frac{\\partial}{\\partial E} \\rho_{f}{(E,\\mathbf{F})} = \\frac{\\partial}{\\partial E} \\frac{\\mathbf{F}}{E} and \\frac{\\partial}{\\partial E} \\rho_{f}{(E,\\mathbf{F})} - \\frac{\\mathbf{F}}{E} = \\frac{\\partial}{\\partial E} \\frac{\\mathbf{F}}{E} - \\frac{\\mathbf{F}}{E} and \\frac{\\frac{\\partial}{\\partial E} \\rho_{f}{(E,\\mathbf{F})} - \\frac{\\mathbf{F}}{E}}{\\mathbf{F}} = \\frac{\\frac{\\partial}{\\partial E} \\frac{\\mathbf{F}}{E} - \\frac{\\mathbf{F}}{E}}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Derivative(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))), Add(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Derivative(Function('\\\\rho_f')(Symbol('E', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\rho)} = e^{\\rho}, then obtain \\frac{\\rho^{2} e^{- \\rho} \\int \\frac{\\hat{p}_0^{2}{(\\rho)}}{\\rho^{2}} d\\rho}{\\hat{p}_0{(\\rho)}} = \\frac{\\rho^{2} e^{- \\rho} \\int \\frac{\\hat{p}_0{(\\rho)} e^{\\rho}}{\\rho^{2}} d\\rho}{\\hat{p}_0{(\\rho)}}", "derivation": "\\hat{p}_0{(\\rho)} = e^{\\rho} and \\frac{\\hat{p}_0{(\\rho)}}{\\rho} = \\frac{e^{\\rho}}{\\rho} and \\frac{\\hat{p}_0^{2}{(\\rho)}}{\\rho^{2}} = \\frac{\\hat{p}_0{(\\rho)} e^{\\rho}}{\\rho^{2}} and \\int \\frac{\\hat{p}_0^{2}{(\\rho)}}{\\rho^{2}} d\\rho = \\int \\frac{\\hat{p}_0{(\\rho)} e^{\\rho}}{\\rho^{2}} d\\rho and \\frac{\\rho^{2} e^{- \\rho} \\int \\frac{\\hat{p}_0^{2}{(\\rho)}}{\\rho^{2}} d\\rho}{\\hat{p}_0{(\\rho)}} = \\frac{\\rho^{2} e^{- \\rho} \\int \\frac{\\hat{p}_0{(\\rho)} e^{\\rho}}{\\rho^{2}} d\\rho}{\\hat{p}_0{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('\\\\rho', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(2))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(2)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(2))), Tuple(Symbol('\\\\rho', commutative=True)))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(2)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})}, then derive \\hat{p}{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0, then obtain 2 \\hat{p}{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0", "derivation": "\\hat{p}{(\\Psi_{nl})} = \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} and \\hat{p}{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} = 0 and 2 \\hat{p}{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} = \\hat{p}{(\\Psi_{nl})} and \\hat{p}{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0 and 2 \\hat{p}{(\\Psi_{nl})} - \\frac{d}{d \\Psi_{nl}} \\log{(\\Psi_{nl})} - \\frac{1}{\\Psi_{nl}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True)), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Integer(0))"], [["add", 1, "Add(Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))), Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), Derivative(log(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(V,T)} = - V + \\cos{(T)} and \\operatorname{v_{x}}{(V,T)} = \\frac{\\partial}{\\partial V} (- V + \\cos{(T)}), then derive \\operatorname{v_{x}}{(V,T)} = -1, then obtain \\frac{\\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)}}{\\int \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} dT} = - \\frac{1}{\\int \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} dT}", "derivation": "\\operatorname{F_{H}}{(V,T)} = - V + \\cos{(T)} and \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} = \\frac{\\partial}{\\partial V} (- V + \\cos{(T)}) and \\operatorname{v_{x}}{(V,T)} = \\frac{\\partial}{\\partial V} (- V + \\cos{(T)}) and \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} = \\operatorname{v_{x}}{(V,T)} and \\operatorname{v_{x}}{(V,T)} = -1 and \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} = -1 and \\frac{\\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)}}{\\int \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} dT} = - \\frac{1}{\\int \\frac{\\partial}{\\partial V} \\operatorname{F_{H}}{(V,T)} dT}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('T', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('T', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Integer(-1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1))"], [["divide", 6, "Integral(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Pow(Integral(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Derivative(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(T)} = \\frac{d}{d T} e^{T}, then derive \\mathbf{p}{(T)} = e^{T}, then obtain (- \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int \\frac{d}{d T} e^{T} dT)^{T} = (- \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int e^{T} dT)^{T}", "derivation": "\\mathbf{p}{(T)} = \\frac{d}{d T} e^{T} and \\mathbf{p}{(T)} = e^{T} and \\int \\mathbf{p}{(T)} dT = \\int e^{T} dT and \\frac{d}{d T} \\int \\mathbf{p}{(T)} dT = \\frac{d}{d T} \\int e^{T} dT and \\frac{d}{d T} \\int \\frac{d}{d T} e^{T} dT = \\frac{d}{d T} \\int e^{T} dT and - \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int \\frac{d}{d T} e^{T} dT = - \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int e^{T} dT and (- \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int \\frac{d}{d T} e^{T} dT)^{T} = (- \\frac{d^{2}}{d T^{2}} e^{T} + \\frac{d}{d T} \\int e^{T} dT)^{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))), Derivative(Integral(Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))), Derivative(Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))), Derivative(Integral(Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))), Derivative(Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given q{(E,E_{x})} = E E_{x}, then obtain - q{(E,E_{x})} \\int \\frac{q{(E,E_{x})}}{E_{x}} dE_{x} = - E E_{x} \\int \\frac{q{(E,E_{x})}}{E_{x}} dE_{x}", "derivation": "q{(E,E_{x})} = E E_{x} and \\frac{q{(E,E_{x})}}{E_{x}} = E and - q{(E,E_{x})} = - E E_{x} and \\int \\frac{q{(E,E_{x})}}{E_{x}} dE_{x} = \\int E dE_{x} and - q{(E,E_{x})} \\int E dE_{x} = - E E_{x} \\int E dE_{x} and - q{(E,E_{x})} \\int \\frac{q{(E,E_{x})}}{E_{x}} dE_{x} = - E E_{x} \\int \\frac{q{(E,E_{x})}}{E_{x}} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('E_x', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True))), Symbol('E', commutative=True))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True), Symbol('E_x', commutative=True)))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integral(Symbol('E', commutative=True), Tuple(Symbol('E_x', commutative=True))))"], [["times", 3, "Integral(Symbol('E', commutative=True), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Mul(Integer(-1), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True)), Integral(Symbol('E', commutative=True), Tuple(Symbol('E_x', commutative=True)))), Mul(Integer(-1), Symbol('E', commutative=True), Symbol('E_x', commutative=True), Integral(Symbol('E', commutative=True), Tuple(Symbol('E_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True)), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))), Mul(Integer(-1), Symbol('E', commutative=True), Symbol('E_x', commutative=True), Integral(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('q')(Symbol('E', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = \\frac{g_{\\varepsilon}}{y^{\\prime}}, then derive \\frac{\\partial}{\\partial g_{\\varepsilon}} \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = \\frac{1}{y^{\\prime}}, then obtain y^{\\prime} \\frac{\\partial}{\\partial g_{\\varepsilon}} \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = 1", "derivation": "\\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = \\frac{g_{\\varepsilon}}{y^{\\prime}} and \\frac{\\partial}{\\partial g_{\\varepsilon}} \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = \\frac{\\partial}{\\partial g_{\\varepsilon}} \\frac{g_{\\varepsilon}}{y^{\\prime}} and \\frac{\\partial}{\\partial g_{\\varepsilon}} \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = \\frac{1}{y^{\\prime}} and y^{\\prime} \\frac{\\partial}{\\partial g_{\\varepsilon}} \\tilde{g}{(g_{\\varepsilon},y^{\\prime})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))"], [["divide", 3, "Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given E{(F_{g},T)} = e^{F_{g} - T}, then derive - e^{F_{g} - T} + \\frac{\\partial}{\\partial T} E{(F_{g},T)} = - 2 e^{F_{g} - T}, then obtain - 2 e^{F_{g} - T} + \\frac{\\partial}{\\partial T} E{(F_{g},T)} + \\frac{\\partial}{\\partial T} e^{F_{g} - T} = - 3 e^{F_{g} - T} + \\frac{\\partial}{\\partial T} e^{F_{g} - T}", "derivation": "E{(F_{g},T)} = e^{F_{g} - T} and \\frac{\\partial}{\\partial T} E{(F_{g},T)} = \\frac{\\partial}{\\partial T} e^{F_{g} - T} and - e^{F_{g} - T} + \\frac{\\partial}{\\partial T} E{(F_{g},T)} = - e^{F_{g} - T} + \\frac{\\partial}{\\partial T} e^{F_{g} - T} and - e^{F_{g} - T} + \\frac{\\partial}{\\partial T} E{(F_{g},T)} = - 2 e^{F_{g} - T} and - 2 e^{F_{g} - T} + \\frac{\\partial}{\\partial T} E{(F_{g},T)} + \\frac{\\partial}{\\partial T} e^{F_{g} - T} = - 3 e^{F_{g} - T} + \\frac{\\partial}{\\partial T} e^{F_{g} - T}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('F_g', commutative=True), Symbol('T', commutative=True)), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('F_g', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))"], "Equality(Add(Mul(Integer(-1), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(Function('E')(Symbol('F_g', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(Function('E')(Symbol('F_g', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))))"], [["add", 4, "Add(Mul(Integer(-1), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Integer(2), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(Function('E')(Symbol('F_g', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(3), exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Derivative(exp(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(\\dot{y})} = \\cos{(e^{\\dot{y}})}, then derive \\int u{(\\dot{y})} d\\dot{y} = m + \\operatorname{Ci}{(e^{\\dot{y}})}, then obtain \\operatorname{Ci}{(e^{\\dot{y}})} \\iint u{(\\dot{y})} d\\dot{y} dm - \\int u{(\\dot{y})} d\\dot{y} = \\operatorname{Ci}{(e^{\\dot{y}})} \\int (m + \\operatorname{Ci}{(e^{\\dot{y}})}) dm - \\int u{(\\dot{y})} d\\dot{y}", "derivation": "u{(\\dot{y})} = \\cos{(e^{\\dot{y}})} and \\int u{(\\dot{y})} d\\dot{y} = \\int \\cos{(e^{\\dot{y}})} d\\dot{y} and \\int u{(\\dot{y})} d\\dot{y} = m + \\operatorname{Ci}{(e^{\\dot{y}})} and \\iint u{(\\dot{y})} d\\dot{y} dm = \\int (m + \\operatorname{Ci}{(e^{\\dot{y}})}) dm and \\operatorname{Ci}{(e^{\\dot{y}})} \\iint u{(\\dot{y})} d\\dot{y} dm = \\operatorname{Ci}{(e^{\\dot{y}})} \\int (m + \\operatorname{Ci}{(e^{\\dot{y}})}) dm and \\operatorname{Ci}{(e^{\\dot{y}})} \\iint u{(\\dot{y})} d\\dot{y} dm - \\int u{(\\dot{y})} d\\dot{y} = \\operatorname{Ci}{(e^{\\dot{y}})} \\int (m + \\operatorname{Ci}{(e^{\\dot{y}})}) dm - \\int u{(\\dot{y})} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\dot{y}', commutative=True)), cos(exp(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('m', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Symbol('m', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["times", 4, "Ci(exp(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Ci(exp(Symbol('\\\\dot{y}', commutative=True))), Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Ci(exp(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Symbol('m', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('m', commutative=True)))))"], [["minus", 5, "Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Ci(exp(Symbol('\\\\dot{y}', commutative=True))), Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Integer(-1), Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))), Add(Mul(Ci(exp(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Symbol('m', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('m', commutative=True)))), Mul(Integer(-1), Integral(Function('u')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{s},v)} = \\mathbf{s} v, then obtain \\int (- F_{c} + (v \\eta^{\\prime}{(\\mathbf{s},v)})^{\\mathbf{s}})^{F_{c}} dF_{c} = \\int (- F_{c} + (\\mathbf{s} v^{2})^{\\mathbf{s}})^{F_{c}} dF_{c}", "derivation": "\\eta^{\\prime}{(\\mathbf{s},v)} = \\mathbf{s} v and v \\eta^{\\prime}{(\\mathbf{s},v)} = \\mathbf{s} v^{2} and (v \\eta^{\\prime}{(\\mathbf{s},v)})^{\\mathbf{s}} = (\\mathbf{s} v^{2})^{\\mathbf{s}} and - F_{c} + (v \\eta^{\\prime}{(\\mathbf{s},v)})^{\\mathbf{s}} = - F_{c} + (\\mathbf{s} v^{2})^{\\mathbf{s}} and (- F_{c} + (v \\eta^{\\prime}{(\\mathbf{s},v)})^{\\mathbf{s}})^{F_{c}} = (- F_{c} + (\\mathbf{s} v^{2})^{\\mathbf{s}})^{F_{c}} and \\int (- F_{c} + (v \\eta^{\\prime}{(\\mathbf{s},v)})^{\\mathbf{s}})^{F_{c}} dF_{c} = \\int (- F_{c} + (\\mathbf{s} v^{2})^{\\mathbf{s}})^{F_{c}} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True)))"], [["times", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Symbol('v', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 3, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('v', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('v', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('F_c', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('F_c', commutative=True)))"], [["integrate", 5, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('v', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given t{(g_{\\varepsilon},\\mathbf{r},L)} = \\frac{- L + g_{\\varepsilon}}{\\mathbf{r}}, then obtain 1 = \\frac{(- L + g_{\\varepsilon})^{2}}{\\mathbf{r}^{2} t^{2}{(g_{\\varepsilon},\\mathbf{r},L)}}", "derivation": "t{(g_{\\varepsilon},\\mathbf{r},L)} = \\frac{- L + g_{\\varepsilon}}{\\mathbf{r}} and 1 = \\frac{- L + g_{\\varepsilon}}{\\mathbf{r} t{(g_{\\varepsilon},\\mathbf{r},L)}} and \\frac{- L + g_{\\varepsilon}}{\\mathbf{r} t{(g_{\\varepsilon},\\mathbf{r},L)}} = \\frac{(- L + g_{\\varepsilon})^{2}}{\\mathbf{r}^{2} t^{2}{(g_{\\varepsilon},\\mathbf{r},L)}} and 1 = \\frac{(- L + g_{\\varepsilon})^{2}}{\\mathbf{r}^{2} t^{2}{(g_{\\varepsilon},\\mathbf{r},L)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 1, "Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(Function('t')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('L', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given r{(A_{x},C_{1})} = A_{x} + C_{1}, then obtain (- 2 C_{1} + r{(A_{x},C_{1})})^{4} r^{2}{(A_{x},C_{1})} = (A_{x} - C_{1})^{4} r^{2}{(A_{x},C_{1})}", "derivation": "r{(A_{x},C_{1})} = A_{x} + C_{1} and - C_{1} + r{(A_{x},C_{1})} = A_{x} and - A_{x} C_{1} - 2 C_{1} + r{(A_{x},C_{1})} = - A_{x} C_{1} + A_{x} - C_{1} and - 2 C_{1} + r{(A_{x},C_{1})} = A_{x} - C_{1} and \\frac{1}{(- 2 C_{1} + r{(A_{x},C_{1})})^{2}} = \\frac{1}{(A_{x} - C_{1})^{2}} and \\frac{1}{(- 2 C_{1} + r{(A_{x},C_{1})})^{2} r{(A_{x},C_{1})}} = \\frac{1}{(A_{x} - C_{1})^{2} r{(A_{x},C_{1})}} and (- 2 C_{1} + r{(A_{x},C_{1})})^{4} r^{2}{(A_{x},C_{1})} = (A_{x} - C_{1})^{4} r^{2}{(A_{x},C_{1})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)))"], [["minus", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Symbol('A_x', commutative=True))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))))"], [["power", 4, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Integer(-2)), Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Integer(-2)))"], [["divide", 5, "Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Integer(-2)), Pow(Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Integer(-2)), Pow(Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Integer(-1))))"], [["power", 6, "Integer(-2)"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True))), Integer(4)), Pow(Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True))), Integer(4)), Pow(Function('r')(Symbol('A_x', commutative=True), Symbol('C_1', commutative=True)), Integer(2))))"]]}, {"prompt": "Given s{(i,v)} = \\int v^{i} di and \\operatorname{z^{*}}{(i,v)} = v^{i}, then obtain s{(i,v)} + \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} \\operatorname{z^{*}}{(i,v)} = \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} \\operatorname{z^{*}}{(i,v)} + \\int v^{i} di", "derivation": "s{(i,v)} = \\int v^{i} di and \\operatorname{z^{*}}{(i,v)} = v^{i} and v^{i} + s{(i,v)} = v^{i} + \\int v^{i} di and s{(i,v)} + \\operatorname{z^{*}}{(i,v)} = \\operatorname{z^{*}}{(i,v)} + \\int v^{i} di and s{(i,v)} + \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} v^{i} = \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} v^{i} + \\int v^{i} di and v^{i} + s{(i,v)} + \\frac{\\partial}{\\partial i} v^{i} = v^{i} + \\frac{\\partial}{\\partial i} v^{i} + \\int v^{i} di and s{(i,v)} + \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} \\operatorname{z^{*}}{(i,v)} = \\operatorname{z^{*}}{(i,v)} + \\frac{\\partial}{\\partial i} \\operatorname{z^{*}}{(i,v)} + \\int v^{i} di", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)))"], [["add", 1, "Pow(Symbol('v', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True))), Add(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True))), Add(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["add", 4, "Derivative(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Add(Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Derivative(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Derivative(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Derivative(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Derivative(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Function('s')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Derivative(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Derivative(Function('z^*')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Pow(Symbol('v', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(M_{E})} = e^{M_{E}}, then derive \\int \\mathbf{F}{(M_{E})} dM_{E} = F_{N} + e^{M_{E}}, then obtain F_{N} \\int e^{M_{E}} dM_{E} = F_{N} (F_{N} + \\mathbf{F}{(M_{E})})", "derivation": "\\mathbf{F}{(M_{E})} = e^{M_{E}} and \\int \\mathbf{F}{(M_{E})} dM_{E} = \\int e^{M_{E}} dM_{E} and \\int \\mathbf{F}{(M_{E})} dM_{E} = F_{N} + e^{M_{E}} and F_{N} \\int \\mathbf{F}{(M_{E})} dM_{E} = F_{N} (F_{N} + e^{M_{E}}) and F_{N} \\int e^{M_{E}} dM_{E} = F_{N} (F_{N} + e^{M_{E}}) and F_{N} \\int \\mathbf{F}{(M_{E})} dM_{E} = F_{N} (F_{N} + \\mathbf{F}{(M_{E})}) and F_{N} (F_{N} + \\mathbf{F}{(M_{E})}) = F_{N} (F_{N} + e^{M_{E}}) and F_{N} \\int e^{M_{E}} dM_{E} = F_{N} (F_{N} + \\mathbf{F}{(M_{E})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('M_E', commutative=True))))"], [["times", 3, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Integral(Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), exp(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('F_N', commutative=True), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), exp(Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('F_N', commutative=True), Integral(Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), exp(Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Mul(Symbol('F_N', commutative=True), Integral(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(M)} = \\sin{(M)}, then derive \\int M \\mathbb{I}{(M)} dM = J_{\\varepsilon} - M \\cos{(M)} + \\sin{(M)}, then obtain (\\int M \\mathbb{I}{(M)} dM) \\int M \\sin{(M)} dM = (J_{\\varepsilon} - M \\cos{(M)} + \\mathbb{I}{(M)}) \\int M \\sin{(M)} dM", "derivation": "\\mathbb{I}{(M)} = \\sin{(M)} and M \\mathbb{I}{(M)} = M \\sin{(M)} and \\int M \\mathbb{I}{(M)} dM = \\int M \\sin{(M)} dM and \\int M \\mathbb{I}{(M)} dM = J_{\\varepsilon} - M \\cos{(M)} + \\sin{(M)} and \\int M \\mathbb{I}{(M)} dM = J_{\\varepsilon} - M \\cos{(M)} + \\mathbb{I}{(M)} and (\\int M \\mathbb{I}{(M)} dM) \\int M \\sin{(M)} dM = (J_{\\varepsilon} - M \\cos{(M)} + \\mathbb{I}{(M)}) \\int M \\sin{(M)} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Symbol('M', commutative=True), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('M', commutative=True), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), cos(Symbol('M', commutative=True))), sin(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Symbol('M', commutative=True), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), cos(Symbol('M', commutative=True))), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))))"], [["times", 5, "Integral(Mul(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Integral(Mul(Symbol('M', commutative=True), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True), cos(Symbol('M', commutative=True))), Function('\\\\mathbb{I}')(Symbol('M', commutative=True))), Integral(Mul(Symbol('M', commutative=True), sin(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{1})} = \\int \\sin{(A_{1})} dA_{1}, then derive \\operatorname{A_{x}}{(A_{1})} = m_{s} - \\cos{(A_{1})}, then obtain \\operatorname{A_{x}}^{A_{1}}{(A_{1})} + \\cos{(A_{1})} = (m_{s} - \\cos{(A_{1})})^{A_{1}} + \\cos{(A_{1})}", "derivation": "\\operatorname{A_{x}}{(A_{1})} = \\int \\sin{(A_{1})} dA_{1} and \\operatorname{A_{x}}{(A_{1})} = m_{s} - \\cos{(A_{1})} and \\operatorname{A_{x}}^{A_{1}}{(A_{1})} = (m_{s} - \\cos{(A_{1})})^{A_{1}} and \\operatorname{A_{x}}^{A_{1}}{(A_{1})} + \\cos{(A_{1})} = (m_{s} - \\cos{(A_{1})})^{A_{1}} + \\cos{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_1', commutative=True)), Integral(sin(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('A_x')(Symbol('A_1', commutative=True)), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"], [["add", 3, "cos(Symbol('A_1', commutative=True))"], "Equality(Add(Pow(Function('A_x')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True))), Add(Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given n{(\\eta)} = \\cos{(\\eta)}, then obtain \\eta + \\frac{d}{d \\eta} (\\eta n{(\\eta)} + \\eta) = \\eta + \\frac{d}{d \\eta} (\\eta \\cos{(\\eta)} + \\eta)", "derivation": "n{(\\eta)} = \\cos{(\\eta)} and \\eta n{(\\eta)} = \\eta \\cos{(\\eta)} and \\eta n{(\\eta)} + \\eta = \\eta \\cos{(\\eta)} + \\eta and \\frac{d}{d \\eta} (\\eta n{(\\eta)} + \\eta) = \\frac{d}{d \\eta} (\\eta \\cos{(\\eta)} + \\eta) and \\eta + \\frac{d}{d \\eta} (\\eta n{(\\eta)} + \\eta) = \\eta + \\frac{d}{d \\eta} (\\eta \\cos{(\\eta)} + \\eta)", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["times", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Function('n')(Symbol('\\\\eta', commutative=True))), Mul(Symbol('\\\\eta', commutative=True), cos(Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\eta', commutative=True), Function('n')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Add(Mul(Symbol('\\\\eta', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\eta', commutative=True), Function('n')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\eta', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Symbol('\\\\eta', commutative=True), Derivative(Add(Mul(Symbol('\\\\eta', commutative=True), Function('n')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Symbol('\\\\eta', commutative=True), Derivative(Add(Mul(Symbol('\\\\eta', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho{(A_{1})} = \\sin{(A_{1})} and \\nabla{(A_{1})} = \\sin{(A_{1})}, then obtain \\frac{A_{1} + \\int 0 dA_{1}}{\\sin{(A_{1})}} = \\frac{A_{1} + \\int (- \\nabla{(A_{1})} + \\sin{(A_{1})}) dA_{1}}{\\sin{(A_{1})}}", "derivation": "\\rho{(A_{1})} = \\sin{(A_{1})} and 0 = - \\rho{(A_{1})} + \\sin{(A_{1})} and \\int 0 dA_{1} = \\int (- \\rho{(A_{1})} + \\sin{(A_{1})}) dA_{1} and \\nabla{(A_{1})} = \\sin{(A_{1})} and \\rho{(A_{1})} = \\nabla{(A_{1})} and A_{1} + \\int 0 dA_{1} = A_{1} + \\int (- \\rho{(A_{1})} + \\sin{(A_{1})}) dA_{1} and \\frac{A_{1} + \\int 0 dA_{1}}{\\sin{(A_{1})}} = \\frac{A_{1} + \\int (- \\rho{(A_{1})} + \\sin{(A_{1})}) dA_{1}}{\\sin{(A_{1})}} and \\frac{A_{1} + \\int 0 dA_{1}}{\\sin{(A_{1})}} = \\frac{A_{1} + \\int (- \\nabla{(A_{1})} + \\sin{(A_{1})}) dA_{1}}{\\sin{(A_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["minus", 1, "Function('\\\\rho')(Symbol('A_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\rho')(Symbol('A_1', commutative=True)), Function('\\\\nabla')(Symbol('A_1', commutative=True)))"], [["add", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Integral(Integer(0), Tuple(Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))))"], [["divide", 6, "sin(Symbol('A_1', commutative=True))"], "Equality(Mul(Add(Symbol('A_1', commutative=True), Integral(Integer(0), Tuple(Symbol('A_1', commutative=True)))), Pow(sin(Symbol('A_1', commutative=True)), Integer(-1))), Mul(Add(Symbol('A_1', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), Pow(sin(Symbol('A_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Add(Symbol('A_1', commutative=True), Integral(Integer(0), Tuple(Symbol('A_1', commutative=True)))), Pow(sin(Symbol('A_1', commutative=True)), Integer(-1))), Mul(Add(Symbol('A_1', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), Pow(sin(Symbol('A_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given q{(P_{g},n)} = \\sin{(\\frac{P_{g}}{n})}, then obtain 1 = (\\frac{\\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n}}{q{(P_{g},n)} + \\frac{1}{n}})^{P_{g}}", "derivation": "q{(P_{g},n)} = \\sin{(\\frac{P_{g}}{n})} and q{(P_{g},n)} + \\frac{1}{n} = \\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n} and (q{(P_{g},n)} + \\frac{1}{n}) (\\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n}) = (\\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n})^{2} and 1 = \\frac{\\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n}}{q{(P_{g},n)} + \\frac{1}{n}} and 1 = (\\frac{\\sin{(\\frac{P_{g}}{n})} + \\frac{1}{n}}{q{(P_{g},n)} + \\frac{1}{n}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["add", 1, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["times", 2, "Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1)))"], "Equality(Mul(Add(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(2)))"], [["divide", 3, "Mul(Add(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1))))"], "Equality(Integer(1), Mul(Pow(Add(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(-1)), Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('P_g', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('q')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(-1)), Add(sin(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{S},M_{E})} = - M_{E} + \\mathbf{S}, then derive \\frac{\\partial}{\\partial M_{E}} \\varepsilon{(\\mathbf{S},M_{E})} = -1, then obtain \\int \\frac{\\partial}{\\partial M_{E}} (- M_{E} + \\mathbf{S}) dM_{E} = \\int (-1) dM_{E}", "derivation": "\\varepsilon{(\\mathbf{S},M_{E})} = - M_{E} + \\mathbf{S} and \\frac{\\partial}{\\partial M_{E}} \\varepsilon{(\\mathbf{S},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (- M_{E} + \\mathbf{S}) and \\frac{\\partial}{\\partial M_{E}} \\varepsilon{(\\mathbf{S},M_{E})} = -1 and \\frac{\\partial}{\\partial M_{E}} (- M_{E} + \\mathbf{S}) = -1 and \\int \\frac{\\partial}{\\partial M_{E}} (- M_{E} + \\mathbf{S}) dM_{E} = \\int (-1) dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(-1))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(-1), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given i{(n_{2},\\dot{z})} = - \\dot{z} + \\cos{(n_{2})} and \\Psi_{nl}{(n_{2})} = - \\sin{(n_{2})}, then derive \\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})} = - \\sin{(n_{2})}, then obtain (\\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})})^{n_{2}} = \\Psi_{nl}^{n_{2}}{(n_{2})}", "derivation": "i{(n_{2},\\dot{z})} = - \\dot{z} + \\cos{(n_{2})} and \\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})} = \\frac{\\partial}{\\partial n_{2}} (- \\dot{z} + \\cos{(n_{2})}) and \\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})} = - \\sin{(n_{2})} and \\Psi_{nl}{(n_{2})} = - \\sin{(n_{2})} and \\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})} = \\Psi_{nl}{(n_{2})} and (\\frac{\\partial}{\\partial n_{2}} i{(n_{2},\\dot{z})})^{n_{2}} = \\Psi_{nl}^{n_{2}}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('i')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Function('\\\\Psi_{nl}')(Symbol('n_2', commutative=True)))"], [["power", 5, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Function('i')(Symbol('n_2', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(U,F_{c})} = \\sin{(F_{c} U)}, then obtain U (- U \\operatorname{t_{1}}{(U,F_{c})} + \\operatorname{t_{1}}{(U,F_{c})}) \\sin{(F_{c} U)} = U (- U \\operatorname{t_{1}}{(U,F_{c})} + \\sin{(F_{c} U)}) \\sin{(F_{c} U)}", "derivation": "\\operatorname{t_{1}}{(U,F_{c})} = \\sin{(F_{c} U)} and U \\operatorname{t_{1}}{(U,F_{c})} = U \\sin{(F_{c} U)} and - U \\sin{(F_{c} U)} + \\operatorname{t_{1}}{(U,F_{c})} = - U \\sin{(F_{c} U)} + \\sin{(F_{c} U)} and - U \\operatorname{t_{1}}{(U,F_{c})} + \\operatorname{t_{1}}{(U,F_{c})} = - U \\operatorname{t_{1}}{(U,F_{c})} + \\sin{(F_{c} U)} and U (- U \\operatorname{t_{1}}{(U,F_{c})} + \\operatorname{t_{1}}{(U,F_{c})}) \\sin{(F_{c} U)} = U (- U \\operatorname{t_{1}}{(U,F_{c})} + \\sin{(F_{c} U)}) \\sin{(F_{c} U)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True)), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True))))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), Mul(Symbol('U', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))))"], [["minus", 1, "Mul(Symbol('U', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))))"], [["times", 4, "Mul(Symbol('U', commutative=True), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True))))"], "Equality(Mul(Symbol('U', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))), Mul(Symbol('U', commutative=True), Add(Mul(Integer(-1), Symbol('U', commutative=True), Function('t_1')(Symbol('U', commutative=True), Symbol('F_c', commutative=True))), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))), sin(Mul(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given Z{(J,M_{E})} = J - M_{E}, then obtain (\\int U Z^{J}{(J,M_{E})} dM_{E})^{U} = (\\int U (J - M_{E})^{J} dM_{E})^{U}", "derivation": "Z{(J,M_{E})} = J - M_{E} and Z^{J}{(J,M_{E})} = (J - M_{E})^{J} and U Z^{J}{(J,M_{E})} = U (J - M_{E})^{J} and \\int U Z^{J}{(J,M_{E})} dM_{E} = \\int U (J - M_{E})^{J} dM_{E} and (\\int U Z^{J}{(J,M_{E})} dM_{E})^{U} = (\\int U (J - M_{E})^{J} dM_{E})^{U}", "srepr_derivation": [["get_premise", "Equality(Function('Z')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Symbol('J', commutative=True)))"], [["divide", 2, "Pow(Symbol('U', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('U', commutative=True), Pow(Function('Z')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Symbol('J', commutative=True))), Mul(Symbol('U', commutative=True), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Symbol('U', commutative=True), Pow(Function('Z')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('U', commutative=True), Pow(Function('Z')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Mul(Symbol('U', commutative=True), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(\\rho_f,z^{*},\\mathbf{J})} = \\frac{\\rho_f}{\\mathbf{J} z^{*}}, then derive \\frac{\\partial}{\\partial \\rho_f} \\mathbb{I}{(\\rho_f,z^{*},\\mathbf{J})} = \\frac{1}{\\mathbf{J} z^{*}}, then obtain \\log{(\\frac{\\partial}{\\partial \\rho_f} \\frac{\\rho_f}{\\mathbf{J} z^{*}})} = \\log{(\\frac{1}{\\mathbf{J} z^{*}})}", "derivation": "\\mathbb{I}{(\\rho_f,z^{*},\\mathbf{J})} = \\frac{\\rho_f}{\\mathbf{J} z^{*}} and \\frac{\\partial}{\\partial \\rho_f} \\mathbb{I}{(\\rho_f,z^{*},\\mathbf{J})} = \\frac{\\partial}{\\partial \\rho_f} \\frac{\\rho_f}{\\mathbf{J} z^{*}} and \\frac{\\partial}{\\partial \\rho_f} \\mathbb{I}{(\\rho_f,z^{*},\\mathbf{J})} = \\frac{1}{\\mathbf{J} z^{*}} and \\frac{\\partial}{\\partial \\rho_f} \\frac{\\rho_f}{\\mathbf{J} z^{*}} = \\frac{1}{\\mathbf{J} z^{*}} and \\log{(\\frac{\\partial}{\\partial \\rho_f} \\frac{\\rho_f}{\\mathbf{J} z^{*}})} = \\log{(\\frac{1}{\\mathbf{J} z^{*}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\rho_f', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["log", 4], "Equality(log(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), log(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(I)} = I and c{(\\pi,M_{E})} = \\sin^{M_{E}}{(\\pi)}, then obtain - 2 I - (\\int (\\sin^{M_{E}}{(\\pi)})^{\\pi} d\\pi)^{\\pi} + (\\int c^{\\pi}{(\\pi,M_{E})} d\\pi)^{\\pi} = - 2 I", "derivation": "\\operatorname{C_{1}}{(I)} = I and c{(\\pi,M_{E})} = \\sin^{M_{E}}{(\\pi)} and c^{\\pi}{(\\pi,M_{E})} = (\\sin^{M_{E}}{(\\pi)})^{\\pi} and \\int c^{\\pi}{(\\pi,M_{E})} d\\pi = \\int (\\sin^{M_{E}}{(\\pi)})^{\\pi} d\\pi and (\\int c^{\\pi}{(\\pi,M_{E})} d\\pi)^{\\pi} = (\\int (\\sin^{M_{E}}{(\\pi)})^{\\pi} d\\pi)^{\\pi} and - I - \\operatorname{C_{1}}{(I)} - (\\int (\\sin^{M_{E}}{(\\pi)})^{\\pi} d\\pi)^{\\pi} + (\\int c^{\\pi}{(\\pi,M_{E})} d\\pi)^{\\pi} = - I - \\operatorname{C_{1}}{(I)} and - 2 I - (\\int (\\sin^{M_{E}}{(\\pi)})^{\\pi} d\\pi)^{\\pi} + (\\int c^{\\pi}{(\\pi,M_{E})} d\\pi)^{\\pi} = - 2 I", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], ["get_premise", "Equality(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Pow(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Pow(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["minus", 5, "Add(Symbol('I', commutative=True), Function('C_1')(Symbol('I', commutative=True)), Pow(Integral(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('I', commutative=True))), Mul(Integer(-1), Pow(Integral(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Pow(Integral(Pow(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Pow(Integral(Pow(Function('c')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(r_{0},\\rho)} = \\cos{(\\frac{\\rho}{r_{0}})}, then obtain (- \\rho - r_{0} + \\operatorname{E_{n}}{(r_{0},\\rho)}) (- \\rho - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})}) = (- \\rho - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})})^{2}", "derivation": "\\operatorname{E_{n}}{(r_{0},\\rho)} = \\cos{(\\frac{\\rho}{r_{0}})} and - r_{0} + \\operatorname{E_{n}}{(r_{0},\\rho)} = - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})} and - \\rho - r_{0} + \\operatorname{E_{n}}{(r_{0},\\rho)} = - \\rho - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})} and (- \\rho - r_{0} + \\operatorname{E_{n}}{(r_{0},\\rho)}) (- \\rho - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})}) = (- \\rho - r_{0} + \\cos{(\\frac{\\rho}{r_{0}})})^{2}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('E_n')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))))"], [["minus", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('E_n')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('E_n')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1)))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))), Integer(2)))"]]}, {"prompt": "Given \\eta{(a^{\\dagger},\\mathbf{A})} = \\mathbf{A} + a^{\\dagger}, then obtain (\\int \\frac{d}{d \\mathbf{A}} 2 d\\mathbf{A})^{\\mathbf{A}} = (\\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} + 1) d\\mathbf{A})^{\\mathbf{A}}", "derivation": "\\eta{(a^{\\dagger},\\mathbf{A})} = \\mathbf{A} + a^{\\dagger} and 1 = \\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} and 2 = \\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} + 1 and \\frac{d}{d \\mathbf{A}} 2 = \\frac{\\partial}{\\partial \\mathbf{A}} (\\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} + 1) and \\int \\frac{d}{d \\mathbf{A}} 2 d\\mathbf{A} = \\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} + 1) d\\mathbf{A} and (\\int \\frac{d}{d \\mathbf{A}} 2 d\\mathbf{A})^{\\mathbf{A}} = (\\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\frac{\\mathbf{A} + a^{\\dagger}}{\\eta{(a^{\\dagger},\\mathbf{A})}} + 1) d\\mathbf{A})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Integer(1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integer(2), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Integer(2), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Derivative(Add(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Integral(Derivative(Integer(2), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(Derivative(Add(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\eta')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\omega,t)} = - t + \\cos{(\\omega)}, then derive (\\frac{\\partial}{\\partial t} \\operatorname{t_{2}}{(\\omega,t)})^{t} = (-1)^{t}, then obtain t + (\\frac{\\partial}{\\partial t} (- t + \\cos{(\\omega)}))^{t} = (-1)^{t} + t", "derivation": "\\operatorname{t_{2}}{(\\omega,t)} = - t + \\cos{(\\omega)} and \\frac{\\partial}{\\partial t} \\operatorname{t_{2}}{(\\omega,t)} = \\frac{\\partial}{\\partial t} (- t + \\cos{(\\omega)}) and (\\frac{\\partial}{\\partial t} \\operatorname{t_{2}}{(\\omega,t)})^{t} = (\\frac{\\partial}{\\partial t} (- t + \\cos{(\\omega)}))^{t} and (\\frac{\\partial}{\\partial t} \\operatorname{t_{2}}{(\\omega,t)})^{t} = (-1)^{t} and (\\frac{\\partial}{\\partial t} (- t + \\cos{(\\omega)}))^{t} = (-1)^{t} and t + (\\frac{\\partial}{\\partial t} (- t + \\cos{(\\omega)}))^{t} = (-1)^{t} + t", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Function('t_2')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('t_2')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Integer(-1), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Integer(-1), Symbol('t', commutative=True)))"], [["minus", 5, "Mul(Integer(-1), Symbol('t', commutative=True))"], "Equality(Add(Symbol('t', commutative=True), Pow(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True))), Add(Pow(Integer(-1), Symbol('t', commutative=True)), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\omega{(C_{1},\\mathbf{E})} = \\frac{\\mathbf{E}}{C_{1}}, then obtain \\omega{(C_{1},\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{C_{1}} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{C_{1}} + \\frac{\\mathbf{E}}{C_{1}}", "derivation": "\\omega{(C_{1},\\mathbf{E})} = \\frac{\\mathbf{E}}{C_{1}} and \\frac{\\partial}{\\partial \\mathbf{E}} \\omega{(C_{1},\\mathbf{E})} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{C_{1}} and \\omega{(C_{1},\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\omega{(C_{1},\\mathbf{E})} = \\frac{\\partial}{\\partial \\mathbf{E}} \\omega{(C_{1},\\mathbf{E})} + \\frac{\\mathbf{E}}{C_{1}} and \\omega{(C_{1},\\mathbf{E})} + \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{C_{1}} = \\frac{\\partial}{\\partial \\mathbf{E}} \\frac{\\mathbf{E}}{C_{1}} + \\frac{\\mathbf{E}}{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\omega')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\hat{p},G)} = G + \\hat{p} and a{(t_{1},g_{\\varepsilon})} = g_{\\varepsilon} + t_{1}, then derive \\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1 = 0, then obtain a{(t_{1},g_{\\varepsilon})} \\int 0 d\\hat{p} = (g_{\\varepsilon} + t_{1}) \\int 0 d\\hat{p}", "derivation": "\\chi{(\\hat{p},G)} = G + \\hat{p} and \\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} = \\frac{\\partial}{\\partial G} (G + \\hat{p}) and \\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1 = \\frac{\\partial}{\\partial G} (G + \\hat{p}) - 1 and \\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1 = 0 and a{(t_{1},g_{\\varepsilon})} = g_{\\varepsilon} + t_{1} and \\int (\\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1) d\\hat{p} = \\int 0 d\\hat{p} and a{(t_{1},g_{\\varepsilon})} \\int (\\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1) d\\hat{p} = (g_{\\varepsilon} + t_{1}) \\int (\\frac{\\partial}{\\partial G} \\chi{(\\hat{p},G)} - 1) d\\hat{p} and a{(t_{1},g_{\\varepsilon})} \\int 0 d\\hat{p} = (g_{\\varepsilon} + t_{1}) \\int 0 d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('G', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], ["get_premise", "Equality(Function('a')(Symbol('t_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["times", 5, "Integral(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Function('a')(Symbol('t_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Integral(Add(Derivative(Function('\\\\chi')(Symbol('\\\\hat{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Function('a')(Symbol('t_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(W)} = \\log{(\\log{(W)})} and \\operatorname{v_{x}}{(W)} = \\log{(W)}, then obtain (- W + \\mathbf{p}{(W)})^{W} \\operatorname{v_{x}}{(W)} = (- W + \\mathbf{p}{(W)})^{W} \\log{(W)}", "derivation": "\\mathbf{p}{(W)} = \\log{(\\log{(W)})} and - W + \\mathbf{p}{(W)} = - W + \\log{(\\log{(W)})} and (- W + \\mathbf{p}{(W)})^{W} = (- W + \\log{(\\log{(W)})})^{W} and \\operatorname{v_{x}}{(W)} = \\log{(W)} and (- W + \\log{(\\log{(W)})})^{W} \\operatorname{v_{x}}{(W)} = (- W + \\log{(\\log{(W)})})^{W} \\log{(W)} and (- W + \\mathbf{p}{(W)})^{W} \\operatorname{v_{x}}{(W)} = (- W + \\mathbf{p}{(W)})^{W} \\log{(W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True))))"], [["minus", 1, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["times", 4, "Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Function('v_x')(Symbol('W', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), log(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Function('v_x')(Symbol('W', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True))), Symbol('W', commutative=True)), log(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(E,q)} = e^{E q} and \\mathbf{v}{(E,q)} = - 2 q + \\operatorname{c_{0}}{(E,q)} + e^{E q}, then obtain \\mathbf{v}{(E,q)} = - 2 q + 2 e^{E q}", "derivation": "\\operatorname{c_{0}}{(E,q)} = e^{E q} and - q + \\operatorname{c_{0}}{(E,q)} = - q + e^{E q} and - 2 q + \\operatorname{c_{0}}{(E,q)} + e^{E q} = - 2 q + 2 e^{E q} and \\mathbf{v}{(E,q)} = - 2 q + \\operatorname{c_{0}}{(E,q)} + e^{E q} and \\mathbf{v}{(E,q)} = - 2 q + 2 e^{E q}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('E', commutative=True), Symbol('q', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True))))"], [["minus", 1, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('c_0')(Symbol('E', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Function('c_0')(Symbol('E', commutative=True), Symbol('q', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Function('c_0')(Symbol('E', commutative=True), Symbol('q', commutative=True)), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{v}')(Symbol('E', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('q', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('E', commutative=True), Symbol('q', commutative=True))))))"]]}, {"prompt": "Given c{(t_{2},\\mathbf{p})} = \\frac{t_{2}}{\\mathbf{p}} and \\mathbf{D}{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain t_{2} + \\frac{t_{2} \\mathbf{D}{(\\sigma_p)}}{\\mathbf{p}} = t_{2} + \\frac{t_{2} \\log{(\\sigma_p)}}{\\mathbf{p}}", "derivation": "c{(t_{2},\\mathbf{p})} = \\frac{t_{2}}{\\mathbf{p}} and \\mathbf{D}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\mathbf{D}{(\\sigma_p)} c{(t_{2},\\mathbf{p})} = c{(t_{2},\\mathbf{p})} \\log{(\\sigma_p)} and \\frac{t_{2} \\mathbf{D}{(\\sigma_p)}}{\\mathbf{p}} = \\frac{t_{2} \\log{(\\sigma_p)}}{\\mathbf{p}} and t_{2} + \\frac{t_{2} \\mathbf{D}{(\\sigma_p)}}{\\mathbf{p}} = t_{2} + \\frac{t_{2} \\log{(\\sigma_p)}}{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["times", 2, "Function('c')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\sigma_p', commutative=True)), Function('c')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Function('c')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), log(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 4, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\sigma_p', commutative=True)))), Add(Symbol('t_2', commutative=True), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True), log(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given J{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then obtain 0 = (- J^{2}{(\\mathbf{J}_M)} + J{(\\mathbf{J}_M)} \\sin{(\\mathbf{J}_M)})^{2}", "derivation": "J{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and J^{2}{(\\mathbf{J}_M)} = J{(\\mathbf{J}_M)} \\sin{(\\mathbf{J}_M)} and 0 = - J^{2}{(\\mathbf{J}_M)} + J{(\\mathbf{J}_M)} \\sin{(\\mathbf{J}_M)} and 0 = (- J^{2}{(\\mathbf{J}_M)} + J{(\\mathbf{J}_M)} \\sin{(\\mathbf{J}_M)})^{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 1, "Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Pow(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)), Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 2, "Pow(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["power", 3, 2], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Function('J')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\theta_{1}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain \\cos^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)}", "derivation": "\\theta_{1}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\theta_{1}^{\\varepsilon}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and \\theta_{1}^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = \\theta_{1}^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and \\cos^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = \\cos^{\\varepsilon}{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["add", 3, "Pow(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Pow(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Pow(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(Q,\\Psi^{\\dagger})} = (\\Psi^{\\dagger})^{Q}, then obtain \\frac{\\partial}{\\partial Q} (\\Psi^{\\dagger})^{Q} \\mathbf{g}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} = \\frac{\\partial}{\\partial Q} (\\Psi^{\\dagger})^{2 Q} e^{- \\cos{(C_{d})}}", "derivation": "\\mathbf{g}{(Q,\\Psi^{\\dagger})} = (\\Psi^{\\dagger})^{Q} and \\mathbf{g}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} = (\\Psi^{\\dagger})^{Q} e^{- \\cos{(C_{d})}} and \\mathbf{g}^{2}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} = (\\Psi^{\\dagger})^{Q} \\mathbf{g}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} and (\\Psi^{\\dagger})^{Q} \\mathbf{g}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} = (\\Psi^{\\dagger})^{2 Q} e^{- \\cos{(C_{d})}} and \\frac{\\partial}{\\partial Q} (\\Psi^{\\dagger})^{Q} \\mathbf{g}{(Q,\\Psi^{\\dagger})} e^{- \\cos{(C_{d})}} = \\frac{\\partial}{\\partial Q} (\\Psi^{\\dagger})^{2 Q} e^{- \\cos{(C_{d})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)))"], [["divide", 1, "exp(cos(Symbol('C_d', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))))"], [["times", 2, "Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(2), Symbol('Q', commutative=True))), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('Q', commutative=True)), Function('\\\\mathbf{g}')(Symbol('Q', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(2), Symbol('Q', commutative=True))), exp(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(a,l)} = e^{\\frac{l}{a}} and \\mathbf{M}{(a,l)} = \\frac{\\partial}{\\partial a} \\hat{H}_{\\lambda}^{a}{(a,l)}, then obtain \\frac{\\partial}{\\partial l} \\mathbf{M}{(a,l)} = \\frac{\\partial^{2}}{\\partial l\\partial a} (e^{\\frac{l}{a}})^{a}", "derivation": "\\hat{H}_{\\lambda}{(a,l)} = e^{\\frac{l}{a}} and \\mathbf{M}{(a,l)} = \\frac{\\partial}{\\partial a} \\hat{H}_{\\lambda}^{a}{(a,l)} and \\frac{\\partial}{\\partial l} \\mathbf{M}{(a,l)} = \\frac{\\partial^{2}}{\\partial l\\partial a} \\hat{H}_{\\lambda}^{a}{(a,l)} and \\frac{\\partial}{\\partial l} \\mathbf{M}{(a,l)} = \\frac{\\partial^{2}}{\\partial l\\partial a} (e^{\\frac{l}{a}})^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), exp(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Derivative(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('a', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(exp(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(n)} = \\cos{(n)}, then derive \\frac{\\frac{Q^{2}}{2} + \\frac{Q \\hat{p}{(n)}}{\\cos{(n)}} + \\mathbf{S}}{\\cos{(n)}} = \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{F}}{\\cos{(n)}}, then obtain \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{S}}{\\cos{(n)}} = \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{F}}{\\cos{(n)}}", "derivation": "\\hat{p}{(n)} = \\cos{(n)} and \\frac{\\hat{p}{(n)}}{\\cos{(n)}} = 1 and Q + \\frac{\\hat{p}{(n)}}{\\cos{(n)}} = Q + 1 and \\int (Q + \\frac{\\hat{p}{(n)}}{\\cos{(n)}}) dQ = \\int (Q + 1) dQ and \\frac{\\int (Q + \\frac{\\hat{p}{(n)}}{\\cos{(n)}}) dQ}{\\cos{(n)}} = \\frac{\\int (Q + 1) dQ}{\\cos{(n)}} and \\frac{\\frac{Q^{2}}{2} + \\frac{Q \\hat{p}{(n)}}{\\cos{(n)}} + \\mathbf{S}}{\\cos{(n)}} = \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{F}}{\\cos{(n)}} and \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{S}}{\\cos{(n)}} = \\frac{\\frac{Q^{2}}{2} + Q + \\mathbf{F}}{\\cos{(n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["divide", 1, "cos(Symbol('n', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1)))), Add(Symbol('Q', commutative=True), Integer(1)))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1)))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True))))"], [["divide", 4, "cos(Symbol('n', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('n', commutative=True)), Integer(-1)), Integral(Add(Symbol('Q', commutative=True), Mul(Function('\\\\hat{p}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1)))), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(cos(Symbol('n', commutative=True)), Integer(-1)), Integral(Add(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Symbol('Q', commutative=True), Function('\\\\hat{p}')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given H{(\\mathbf{J})} = \\log{(\\sin{(\\mathbf{J})})}, then obtain H^{4}{(\\mathbf{J})} + \\frac{H^{4}{(\\mathbf{J})}}{\\log{(\\sin{(\\mathbf{J})})}^{3}} = H^{4}{(\\mathbf{J})} + H{(\\mathbf{J})}", "derivation": "H{(\\mathbf{J})} = \\log{(\\sin{(\\mathbf{J})})} and H^{2}{(\\mathbf{J})} = H{(\\mathbf{J})} \\log{(\\sin{(\\mathbf{J})})} and \\frac{H^{2}{(\\mathbf{J})}}{\\log{(\\sin{(\\mathbf{J})})}} = H{(\\mathbf{J})} and \\frac{H^{4}{(\\mathbf{J})}}{\\log{(\\sin{(\\mathbf{J})})}^{2}} = H^{2}{(\\mathbf{J})} and \\frac{H^{4}{(\\mathbf{J})}}{\\log{(\\sin{(\\mathbf{J})})}^{3}} = H{(\\mathbf{J})} and H^{4}{(\\mathbf{J})} + \\frac{H^{4}{(\\mathbf{J})}}{\\log{(\\sin{(\\mathbf{J})})}^{3}} = H^{4}{(\\mathbf{J})} + H{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), log(sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 1, "Function('H')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), log(sin(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 2, "log(sin(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Pow(log(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Function('H')(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4)), Pow(log(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-2))), Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4)), Pow(log(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-3))), Function('H')(Symbol('\\\\mathbf{J}', commutative=True)))"], [["add", 5, "Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4))"], "Equality(Add(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4)), Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4)), Pow(log(sin(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-3)))), Add(Pow(Function('H')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(4)), Function('H')(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\phi_2,\\varphi)} = - \\phi_2 + \\varphi, then derive \\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1 = \\phi_2 - \\varphi - 2, then obtain \\int (\\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1) d\\phi_2 = \\int (\\phi_2 - \\varphi - 2) d\\phi_2", "derivation": "\\mathbf{B}{(\\phi_2,\\varphi)} = - \\phi_2 + \\varphi and \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} = \\frac{\\partial}{\\partial \\phi_2} (- \\phi_2 + \\varphi) and \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1 = \\frac{\\partial}{\\partial \\phi_2} (- \\phi_2 + \\varphi) - 1 and \\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1 = \\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} (- \\phi_2 + \\varphi) - 1 and \\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1 = \\phi_2 - \\varphi - 2 and \\int (\\phi_2 - \\varphi + \\frac{\\partial}{\\partial \\phi_2} \\mathbf{B}{(\\phi_2,\\varphi)} - 1) d\\phi_2 = \\int (\\phi_2 - \\varphi - 2) d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integer(-2)))"], [["integrate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Integer(-2)), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given u{(t_{2},g_{\\varepsilon})} = t_{2} + e^{g_{\\varepsilon}}, then obtain 4 u^{2}{(t_{2},g_{\\varepsilon})} = 4 (t_{2} + e^{g_{\\varepsilon}})^{2}", "derivation": "u{(t_{2},g_{\\varepsilon})} = t_{2} + e^{g_{\\varepsilon}} and 2 u{(t_{2},g_{\\varepsilon})} = t_{2} + u{(t_{2},g_{\\varepsilon})} + e^{g_{\\varepsilon}} and 4 u^{2}{(t_{2},g_{\\varepsilon})} = (t_{2} + u{(t_{2},g_{\\varepsilon})} + e^{g_{\\varepsilon}})^{2} and 4 (t_{2} + e^{g_{\\varepsilon}})^{2} = (2 t_{2} + 2 e^{g_{\\varepsilon}})^{2} and 4 u^{2}{(t_{2},g_{\\varepsilon})} = (2 t_{2} + 2 e^{g_{\\varepsilon}})^{2} and 4 u^{2}{(t_{2},g_{\\varepsilon})} = 4 (t_{2} + e^{g_{\\varepsilon}})^{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('t_2', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('t_2', commutative=True), Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2))), Pow(Add(Symbol('t_2', commutative=True), Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('t_2', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('t_2', commutative=True)), Mul(Integer(2), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('t_2', commutative=True)), Mul(Integer(2), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(4), Pow(Function('u')(Symbol('t_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Integer(4), Pow(Add(Symbol('t_2', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\dot{x}{(C_{d},\\mathbf{F})} = - \\mathbf{F} + \\log{(C_{d})}, then obtain \\mathbf{F} + ((- \\dot{x}{(C_{d},\\mathbf{F})})^{C_{d}})^{\\mathbf{F}} = \\mathbf{F} + ((\\mathbf{F} - \\log{(C_{d})})^{C_{d}})^{\\mathbf{F}}", "derivation": "\\dot{x}{(C_{d},\\mathbf{F})} = - \\mathbf{F} + \\log{(C_{d})} and - \\dot{x}{(C_{d},\\mathbf{F})} = \\mathbf{F} - \\log{(C_{d})} and (- \\dot{x}{(C_{d},\\mathbf{F})})^{C_{d}} = (\\mathbf{F} - \\log{(C_{d})})^{C_{d}} and ((- \\dot{x}{(C_{d},\\mathbf{F})})^{C_{d}})^{\\mathbf{F}} = ((\\mathbf{F} - \\log{(C_{d})})^{C_{d}})^{\\mathbf{F}} and \\mathbf{F} + ((- \\dot{x}{(C_{d},\\mathbf{F})})^{C_{d}})^{\\mathbf{F}} = \\mathbf{F} + ((\\mathbf{F} - \\log{(C_{d})})^{C_{d}})^{\\mathbf{F}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('C_d', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), log(Symbol('C_d', commutative=True)))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('C_d', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), log(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), log(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Pow(Pow(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Pow(Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), log(Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given B{(\\hat{X},q)} = \\hat{X} q, then derive \\frac{\\partial}{\\partial \\hat{X}} B{(\\hat{X},q)} - 1 = q - 1, then obtain \\frac{\\frac{\\partial}{\\partial \\hat{X}} B{(\\hat{X},q)} - 1}{B{(\\hat{X},q)}} = \\frac{q - 1}{B{(\\hat{X},q)}}", "derivation": "B{(\\hat{X},q)} = \\hat{X} q and - \\hat{X} + B{(\\hat{X},q)} = \\hat{X} q - \\hat{X} and \\frac{\\partial}{\\partial \\hat{X}} (- \\hat{X} + B{(\\hat{X},q)}) = \\frac{\\partial}{\\partial \\hat{X}} (\\hat{X} q - \\hat{X}) and \\frac{\\partial}{\\partial \\hat{X}} B{(\\hat{X},q)} - 1 = q - 1 and \\frac{\\frac{\\partial}{\\partial \\hat{X}} B{(\\hat{X},q)} - 1}{B{(\\hat{X},q)}} = \\frac{q - 1}{B{(\\hat{X},q)}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True))), Add(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('q', commutative=True), Integer(-1)))"], [["divide", 4, "Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Add(Derivative(Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(-1)), Pow(Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Mul(Add(Symbol('q', commutative=True), Integer(-1)), Pow(Function('B')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given H{(U,y)} = \\cos{(\\frac{y}{U})} and \\hat{X}{(U,y)} = \\iint H{(U,y)} dy dy, then obtain (\\iint \\cos{(\\frac{y}{U})} dy dy)^{y} = (\\iint H{(U,y)} dy dy)^{y}", "derivation": "H{(U,y)} = \\cos{(\\frac{y}{U})} and \\int H{(U,y)} dy = \\int \\cos{(\\frac{y}{U})} dy and \\iint H{(U,y)} dy dy = \\iint \\cos{(\\frac{y}{U})} dy dy and \\hat{X}{(U,y)} = \\iint H{(U,y)} dy dy and \\hat{X}{(U,y)} = \\iint \\cos{(\\frac{y}{U})} dy dy and \\hat{X}^{y}{(U,y)} = (\\iint H{(U,y)} dy dy)^{y} and (\\iint \\cos{(\\frac{y}{U})} dy dy)^{y} = (\\iint H{(U,y)} dy dy)^{y}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Integral(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Integral(cos(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["power", 4, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Integral(cos(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Function('H')(Symbol('U', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\dot{z},p)} = p^{\\dot{z}}, then obtain \\frac{\\partial}{\\partial p} (- p + \\frac{\\partial}{\\partial p} \\varphi^{*}{(\\dot{z},p)}) = \\frac{\\partial}{\\partial p} (- p + \\frac{\\partial}{\\partial p} p^{\\dot{z}})", "derivation": "\\varphi^{*}{(\\dot{z},p)} = p^{\\dot{z}} and \\frac{\\partial}{\\partial p} \\varphi^{*}{(\\dot{z},p)} = \\frac{\\partial}{\\partial p} p^{\\dot{z}} and - p + \\frac{\\partial}{\\partial p} \\varphi^{*}{(\\dot{z},p)} = - p + \\frac{\\partial}{\\partial p} p^{\\dot{z}} and \\frac{\\partial}{\\partial p} (- p + \\frac{\\partial}{\\partial p} \\varphi^{*}{(\\dot{z},p)}) = \\frac{\\partial}{\\partial p} (- p + \\frac{\\partial}{\\partial p} p^{\\dot{z}})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Symbol('p', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Pow(Symbol('p', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Pow(Symbol('p', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(C_{1})} = \\sin{(\\cos{(C_{1})})}, then obtain \\frac{\\partial}{\\partial C_{1}} (E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(r{(C_{1})})}) = \\frac{\\partial}{\\partial C_{1}} (E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(\\sin{(\\cos{(C_{1})})})})", "derivation": "r{(C_{1})} = \\sin{(\\cos{(C_{1})})} and \\sin{(r{(C_{1})})} = \\sin{(\\sin{(\\cos{(C_{1})})})} and \\sin^{C_{1}}{(r{(C_{1})})} = \\sin^{C_{1}}{(\\sin{(\\cos{(C_{1})})})} and E + \\sin^{C_{1}}{(r{(C_{1})})} = E + \\sin^{C_{1}}{(\\sin{(\\cos{(C_{1})})})} and E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(r{(C_{1})})} = E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(\\sin{(\\cos{(C_{1})})})} and \\frac{\\partial}{\\partial C_{1}} (E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(r{(C_{1})})}) = \\frac{\\partial}{\\partial C_{1}} (E + e^{E + \\sin{(E)}} + \\sin^{C_{1}}{(\\sin{(\\cos{(C_{1})})})})", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('C_1', commutative=True)), sin(cos(Symbol('C_1', commutative=True))))"], [["sin", 1], "Equality(sin(Function('r')(Symbol('C_1', commutative=True))), sin(sin(cos(Symbol('C_1', commutative=True)))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(sin(Function('r')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(sin(sin(cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))"], [["add", 3, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Pow(sin(Function('r')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True))), Add(Symbol('E', commutative=True), Pow(sin(sin(cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))))"], [["add", 4, "exp(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))))"], "Equality(Add(Symbol('E', commutative=True), exp(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Pow(sin(Function('r')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True))), Add(Symbol('E', commutative=True), exp(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Pow(sin(sin(cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))))"], [["differentiate", 5, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Symbol('E', commutative=True), exp(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Pow(sin(Function('r')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), exp(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True)))), Pow(sin(sin(cos(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\mathbf{J}_M,\\chi)} = \\chi \\mathbf{J}_M, then derive \\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} b{(\\mathbf{J}_M,\\chi)} = \\mathbf{J}_M^{2}, then obtain \\sin{(\\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} \\chi \\mathbf{J}_M - e^{f_{\\mathbf{p}}})} = \\sin{(\\mathbf{J}_M^{2} - e^{f_{\\mathbf{p}}})}", "derivation": "b{(\\mathbf{J}_M,\\chi)} = \\chi \\mathbf{J}_M and \\mathbf{J}_M b{(\\mathbf{J}_M,\\chi)} = \\chi \\mathbf{J}_M^{2} and \\frac{\\partial}{\\partial \\chi} \\mathbf{J}_M b{(\\mathbf{J}_M,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\chi \\mathbf{J}_M^{2} and \\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} b{(\\mathbf{J}_M,\\chi)} = \\mathbf{J}_M^{2} and \\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} \\chi \\mathbf{J}_M = \\mathbf{J}_M^{2} and \\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} \\chi \\mathbf{J}_M - e^{f_{\\mathbf{p}}} = \\mathbf{J}_M^{2} - e^{f_{\\mathbf{p}}} and \\sin{(\\mathbf{J}_M \\frac{\\partial}{\\partial \\chi} \\chi \\mathbf{J}_M - e^{f_{\\mathbf{p}}})} = \\sin{(\\mathbf{J}_M^{2} - e^{f_{\\mathbf{p}}})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('b')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('b')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Function('b')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)))"], [["minus", 5, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["sin", 6], "Equality(sin(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))), sin(Add(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2)), Mul(Integer(-1), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(v_{2})} = e^{v_{2}}, then derive \\int \\operatorname{F_{g}}{(v_{2})} dv_{2} = y + e^{v_{2}}, then obtain \\int \\operatorname{F_{g}}{(v_{2})} dv_{2} = y + \\operatorname{F_{g}}{(v_{2})}", "derivation": "\\operatorname{F_{g}}{(v_{2})} = e^{v_{2}} and \\int \\operatorname{F_{g}}{(v_{2})} dv_{2} = \\int e^{v_{2}} dv_{2} and \\int \\operatorname{F_{g}}{(v_{2})} dv_{2} = y + e^{v_{2}} and \\int \\operatorname{F_{g}}{(v_{2})} dv_{2} = y + \\operatorname{F_{g}}{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('y', commutative=True), exp(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('F_g')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('y', commutative=True), Function('F_g')(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given u{(x^\\prime,\\dot{z})} = \\dot{z} x^\\prime, then obtain \\frac{\\partial}{\\partial \\dot{z}} u^{x^\\prime}{(x^\\prime,\\dot{z})} \\cos{(u{(x^\\prime,\\dot{z})})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} x^\\prime)^{x^\\prime} \\cos{(u{(x^\\prime,\\dot{z})})}", "derivation": "u{(x^\\prime,\\dot{z})} = \\dot{z} x^\\prime and \\cos{(u{(x^\\prime,\\dot{z})})} = \\cos{(\\dot{z} x^\\prime)} and u^{x^\\prime}{(x^\\prime,\\dot{z})} = (\\dot{z} x^\\prime)^{x^\\prime} and u^{x^\\prime}{(x^\\prime,\\dot{z})} \\cos{(\\dot{z} x^\\prime)} = (\\dot{z} x^\\prime)^{x^\\prime} \\cos{(\\dot{z} x^\\prime)} and \\frac{\\partial}{\\partial \\dot{z}} u^{x^\\prime}{(x^\\prime,\\dot{z})} \\cos{(\\dot{z} x^\\prime)} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} x^\\prime)^{x^\\prime} \\cos{(\\dot{z} x^\\prime)} and \\frac{\\partial}{\\partial \\dot{z}} u^{x^\\prime}{(x^\\prime,\\dot{z})} \\cos{(u{(x^\\prime,\\dot{z})})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} x^\\prime)^{x^\\prime} \\cos{(u{(x^\\prime,\\dot{z})})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["cos", 1], "Equality(cos(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True))), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["times", 3, "cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Mul(Pow(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Function('u')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\psi)} = e^{\\psi} and h{(\\psi)} = \\int e^{2 \\psi} d\\psi, then obtain \\sin{(\\int h{(\\psi)} d\\psi)} = \\sin{(\\iint \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} d\\psi d\\psi)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\psi)} = e^{\\psi} and \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} = \\operatorname{g_{\\varepsilon}}{(\\psi)} e^{\\psi} and \\operatorname{g_{\\varepsilon}}{(\\psi)} e^{\\psi} = e^{2 \\psi} and \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} = e^{2 \\psi} and \\int \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} d\\psi = \\int e^{2 \\psi} d\\psi and h{(\\psi)} = \\int e^{2 \\psi} d\\psi and h{(\\psi)} = \\int \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} d\\psi and \\int h{(\\psi)} d\\psi = \\iint \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} d\\psi d\\psi and \\sin{(\\int h{(\\psi)} d\\psi)} = \\sin{(\\iint \\operatorname{g_{\\varepsilon}}^{2}{(\\psi)} d\\psi d\\psi)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["times", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True))"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), Mul(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\psi', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Function('h')(Symbol('\\\\psi', commutative=True)), Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["integrate", 7, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["sin", 8], "Equality(sin(Integral(Function('h')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), sin(Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given W{(r)} = \\log{(r)} and \\theta{(r)} = r W{(r)}, then derive \\int W{(r)} dr = \\hat{H} + r \\log{(r)} - r, then obtain (\\hat{H} - r + \\theta{(r)}) (\\hat{H} + r W{(r)} - r) = (\\hat{H} + r W{(r)} - r) \\int W{(r)} dr", "derivation": "W{(r)} = \\log{(r)} and \\int W{(r)} dr = \\int \\log{(r)} dr and \\int W{(r)} dr = \\hat{H} + r \\log{(r)} - r and \\int W{(r)} dr = \\hat{H} + r W{(r)} - r and \\theta{(r)} = r W{(r)} and \\int W{(r)} dr = \\hat{H} - r + \\theta{(r)} and \\hat{H} - r + \\theta{(r)} = \\int \\log{(r)} dr and (\\hat{H} - r + \\theta{(r)}) (\\hat{H} + r W{(r)} - r) = (\\hat{H} + r W{(r)} - r) \\int \\log{(r)} dr and (\\hat{H} - r + \\theta{(r)}) (\\hat{H} + r W{(r)} - r) = (\\hat{H} + r W{(r)} - r) \\int W{(r)} dr", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('W')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('W')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('r', commutative=True)), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('W')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('r', commutative=True))), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["times", 7, "Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('r', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)))), Mul(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 2], "Equality(Mul(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('r', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)))), Mul(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Symbol('r', commutative=True), Function('W')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))), Integral(Function('W')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(v_{2},A_{y})} = e^{v_{2}^{A_{y}}} and x{(v_{2},A_{y})} = v_{2}^{A_{y}}, then obtain - \\operatorname{A_{1}}{(v_{2},A_{y})} + x{(v_{2},A_{y})} = v_{2}^{A_{y}} - \\operatorname{A_{1}}{(v_{2},A_{y})}", "derivation": "\\operatorname{A_{1}}{(v_{2},A_{y})} = e^{v_{2}^{A_{y}}} and x{(v_{2},A_{y})} = v_{2}^{A_{y}} and \\operatorname{A_{1}}{(v_{2},A_{y})} = e^{x{(v_{2},A_{y})}} and x{(v_{2},A_{y})} - e^{x{(v_{2},A_{y})}} = v_{2}^{A_{y}} - e^{x{(v_{2},A_{y})}} and - \\operatorname{A_{1}}{(v_{2},A_{y})} + x{(v_{2},A_{y})} = v_{2}^{A_{y}} - \\operatorname{A_{1}}{(v_{2},A_{y})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_1')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), exp(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))))"], [["minus", 2, "exp(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)))"], "Equality(Add(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), exp(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))))), Add(Pow(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), exp(Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('A_1')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))), Function('x')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))), Add(Pow(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(E_{n},\\eta^{\\prime})} = (\\eta^{\\prime})^{E_{n}}, then obtain \\int (E_{n} + 2 \\mathbf{g}{(E_{n},\\eta^{\\prime})}) dE_{n} = \\int (E_{n} + (\\eta^{\\prime})^{E_{n}} + \\mathbf{g}{(E_{n},\\eta^{\\prime})}) dE_{n}", "derivation": "\\mathbf{g}{(E_{n},\\eta^{\\prime})} = (\\eta^{\\prime})^{E_{n}} and E_{n} + \\mathbf{g}{(E_{n},\\eta^{\\prime})} = E_{n} + (\\eta^{\\prime})^{E_{n}} and E_{n} + 2 \\mathbf{g}{(E_{n},\\eta^{\\prime})} = E_{n} + (\\eta^{\\prime})^{E_{n}} + \\mathbf{g}{(E_{n},\\eta^{\\prime})} and \\int (E_{n} + 2 \\mathbf{g}{(E_{n},\\eta^{\\prime})}) dE_{n} = \\int (E_{n} + (\\eta^{\\prime})^{E_{n}} + \\mathbf{g}{(E_{n},\\eta^{\\prime})}) dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)))"], [["add", 1, "Symbol('E_n', commutative=True)"], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Symbol('E_n', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Function('\\\\mathbf{g}')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given L{(\\sigma_x,u)} = - \\sigma_x + u, then obtain \\frac{\\int \\frac{(- \\sigma_x + u)^{2}}{L^{2}{(\\sigma_x,u)}} d\\sigma_x}{- \\sigma_x + u} = \\frac{\\int 1 d\\sigma_x}{- \\sigma_x + u}", "derivation": "L{(\\sigma_x,u)} = - \\sigma_x + u and L^{2}{(\\sigma_x,u)} = (- \\sigma_x + u) L{(\\sigma_x,u)} and \\frac{L^{2}{(\\sigma_x,u)}}{- \\sigma_x + u} = L{(\\sigma_x,u)} and \\frac{L{(\\sigma_x,u)}}{- \\sigma_x + u} = 1 and \\frac{L^{2}{(\\sigma_x,u)}}{(- \\sigma_x + u)^{2}} = 1 and \\frac{L^{2}{(\\sigma_x,u)}}{- \\sigma_x + u} = - \\sigma_x + u and \\frac{(- \\sigma_x + u)^{2}}{L^{2}{(\\sigma_x,u)}} = 1 and \\int \\frac{(- \\sigma_x + u)^{2}}{L^{2}{(\\sigma_x,u)}} d\\sigma_x = \\int 1 d\\sigma_x and \\frac{\\int \\frac{(- \\sigma_x + u)^{2}}{L^{2}{(\\sigma_x,u)}} d\\sigma_x}{- \\sigma_x + u} = \\frac{\\int 1 d\\sigma_x}{- \\sigma_x + u}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)))"], [["times", 1, "Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))"], "Equality(Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(2))), Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)))"], [["times", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-2)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(2))), Integer(1))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(2)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(-2))), Integer(1))"], [["integrate", 7, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(2)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 8, "Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(2)), Pow(Function('L')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('u', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(T)} = \\sin{(\\cos{(T)})} and \\ddot{x}{(T)} = \\sin^{T}{(\\cos{(T)})}, then obtain T + \\int \\operatorname{g_{\\varepsilon}}^{T}{(T)} dT = T + \\int \\ddot{x}{(T)} dT", "derivation": "\\operatorname{g_{\\varepsilon}}{(T)} = \\sin{(\\cos{(T)})} and \\operatorname{g_{\\varepsilon}}^{T}{(T)} = \\sin^{T}{(\\cos{(T)})} and \\ddot{x}{(T)} = \\sin^{T}{(\\cos{(T)})} and \\operatorname{g_{\\varepsilon}}^{T}{(T)} = \\ddot{x}{(T)} and \\int \\operatorname{g_{\\varepsilon}}^{T}{(T)} dT = \\int \\ddot{x}{(T)} dT and T + \\int \\operatorname{g_{\\varepsilon}}^{T}{(T)} dT = T + \\int \\ddot{x}{(T)} dT", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), sin(cos(Symbol('T', commutative=True))))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(sin(cos(Symbol('T', commutative=True))), Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Pow(sin(cos(Symbol('T', commutative=True))), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Function('\\\\ddot{x}')(Symbol('T', commutative=True)))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 5, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Symbol('T', commutative=True), Integral(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\omega{(Z)} = \\log{(\\log{(Z)})}, then obtain \\frac{d}{d Z} 0 + \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(\\log{(Z)})}) = 2 \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(\\log{(Z)})})", "derivation": "\\omega{(Z)} = \\log{(\\log{(Z)})} and \\omega{(Z)} - \\log{(Z)} = - \\log{(Z)} + \\log{(\\log{(Z)})} and 0 = - \\omega{(Z)} + \\log{(\\log{(Z)})} and \\log{(Z)} = - \\omega{(Z)} + \\log{(Z)} + \\log{(\\log{(Z)})} and 0 = - \\omega{(Z)} + \\log{(- \\omega{(Z)} + \\log{(Z)} + \\log{(\\log{(Z)})})} and \\frac{d}{d Z} 0 = \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(- \\omega{(Z)} + \\log{(Z)} + \\log{(\\log{(Z)})})}) and \\frac{d}{d Z} 0 = \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(\\log{(Z)})}) and \\frac{d}{d Z} 0 + \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(\\log{(Z)})}) = 2 \\frac{d}{d Z} (- \\omega{(Z)} + \\log{(\\log{(Z)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True))))"], [["minus", 1, "log(Symbol('Z', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))))"], [["minus", 2, "Add(Function('\\\\omega')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))))"], [["minus", 1, "Add(Function('\\\\omega')(Symbol('Z', commutative=True)), Mul(Integer(-1), log(Symbol('Z', commutative=True))))"], "Equality(log(Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True)))))))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True)))))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Integer(0), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 7, "Derivative(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('Z', commutative=True))), log(log(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{x}{(k)} = \\log{(k)}, then obtain \\frac{d}{d k} (- \\frac{d}{d k} k \\sigma_{x}{(k)} + \\frac{d}{d k} k \\log{(k)}) = \\frac{d}{d k} 0", "derivation": "\\sigma_{x}{(k)} = \\log{(k)} and k \\sigma_{x}{(k)} = k \\log{(k)} and \\frac{d}{d k} k \\sigma_{x}{(k)} = \\frac{d}{d k} k \\log{(k)} and \\frac{d}{d k} k \\sigma_{x}{(k)} - \\frac{d}{d k} k \\log{(k)} = 0 and - \\frac{d}{d k} k \\sigma_{x}{(k)} + \\frac{d}{d k} k \\log{(k)} = 0 and \\frac{d}{d k} (- \\frac{d}{d k} k \\sigma_{x}{(k)} + \\frac{d}{d k} k \\log{(k)}) = \\frac{d}{d k} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('\\\\sigma_x')(Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Symbol('k', commutative=True), Function('\\\\sigma_x')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('k', commutative=True), Function('\\\\sigma_x')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))), Integer(0))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('k', commutative=True), Function('\\\\sigma_x')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 5, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Derivative(Mul(Symbol('k', commutative=True), Function('\\\\sigma_x')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain L_{\\varepsilon}^{2} \\varphi^{*}{(L_{\\varepsilon})} e^{L_{\\varepsilon}} = L_{\\varepsilon}^{2} e^{2 L_{\\varepsilon}}", "derivation": "\\varphi^{*}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and 1 = \\frac{e^{L_{\\varepsilon}}}{\\varphi^{*}{(L_{\\varepsilon})}} and L_{\\varepsilon} = \\frac{L_{\\varepsilon} e^{L_{\\varepsilon}}}{\\varphi^{*}{(L_{\\varepsilon})}} and L_{\\varepsilon} \\varphi^{*}{(L_{\\varepsilon})} = L_{\\varepsilon} e^{L_{\\varepsilon}} and L_{\\varepsilon}^{2} \\varphi^{*}{(L_{\\varepsilon})} e^{L_{\\varepsilon}} = L_{\\varepsilon}^{2} e^{2 L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["divide", 3, "Pow(Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))"], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 4, "Mul(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Function('\\\\varphi^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(f^{\\prime},M_{E})} = f^{\\prime} + \\cos{(M_{E})}, then obtain (\\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})}) \\mathbf{S}{(f^{\\prime},M_{E})})^{2} = (\\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})})^{2})^{2}", "derivation": "\\mathbf{S}{(f^{\\prime},M_{E})} = f^{\\prime} + \\cos{(M_{E})} and (f^{\\prime} + \\cos{(M_{E})}) \\mathbf{S}{(f^{\\prime},M_{E})} = (f^{\\prime} + \\cos{(M_{E})})^{2} and \\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})}) \\mathbf{S}{(f^{\\prime},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})})^{2} and (\\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})}) \\mathbf{S}{(f^{\\prime},M_{E})})^{2} = (\\frac{\\partial}{\\partial M_{E}} (f^{\\prime} + \\cos{(M_{E})})^{2})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))))"], [["times", 1, "Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Function('\\\\mathbf{S}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True))), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Function('\\\\mathbf{S}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Integer(2)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Mul(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Function('\\\\mathbf{S}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('M_E', commutative=True))), Integer(2)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\psi)} = \\cos{(\\psi)}, then derive \\int \\operatorname{C_{d}}{(\\psi)} d\\psi = g_{\\varepsilon} + \\sin{(\\psi)}, then derive \\pi + \\sin{(\\psi)} = g_{\\varepsilon} + \\sin{(\\psi)}, then obtain \\psi \\sin{(\\psi)} (\\int \\operatorname{C_{d}}{(\\psi)} d\\psi)^{2} = \\psi (\\pi + \\sin{(\\psi)}) \\sin{(\\psi)} \\int \\operatorname{C_{d}}{(\\psi)} d\\psi", "derivation": "\\operatorname{C_{d}}{(\\psi)} = \\cos{(\\psi)} and \\int \\operatorname{C_{d}}{(\\psi)} d\\psi = \\int \\cos{(\\psi)} d\\psi and \\int \\operatorname{C_{d}}{(\\psi)} d\\psi = g_{\\varepsilon} + \\sin{(\\psi)} and \\int \\cos{(\\psi)} d\\psi = g_{\\varepsilon} + \\sin{(\\psi)} and \\pi + \\sin{(\\psi)} = g_{\\varepsilon} + \\sin{(\\psi)} and \\int \\operatorname{C_{d}}{(\\psi)} d\\psi = \\pi + \\sin{(\\psi)} and \\sin{(\\psi)} (\\int \\operatorname{C_{d}}{(\\psi)} d\\psi)^{2} = (\\pi + \\sin{(\\psi)}) \\sin{(\\psi)} \\int \\operatorname{C_{d}}{(\\psi)} d\\psi and \\psi \\sin{(\\psi)} (\\int \\operatorname{C_{d}}{(\\psi)} d\\psi)^{2} = \\psi (\\pi + \\sin{(\\psi)}) \\sin{(\\psi)} \\int \\operatorname{C_{d}}{(\\psi)} d\\psi", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi', commutative=True))))"], [["times", 6, "Mul(sin(Symbol('\\\\psi', commutative=True)), Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], "Equality(Mul(sin(Symbol('\\\\psi', commutative=True)), Pow(Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2))), Mul(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi', commutative=True))), sin(Symbol('\\\\psi', commutative=True)), Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["times", 7, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), sin(Symbol('\\\\psi', commutative=True)), Pow(Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2))), Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\pi', commutative=True), sin(Symbol('\\\\psi', commutative=True))), sin(Symbol('\\\\psi', commutative=True)), Integral(Function('C_d')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(y)} = \\int e^{y} dy, then derive \\lambda{(y)} = F_{x} + e^{y}, then derive \\log{(F_{x} + e^{y})} = \\log{(L_{\\varepsilon} + e^{y})}, then obtain \\frac{\\partial}{\\partial L_{\\varepsilon}} \\log{(F_{x} + e^{y})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\log{(L_{\\varepsilon} + e^{y})}", "derivation": "\\lambda{(y)} = \\int e^{y} dy and \\lambda{(y)} = F_{x} + e^{y} and F_{x} + e^{y} = \\int e^{y} dy and \\log{(F_{x} + e^{y})} = \\log{(\\int e^{y} dy)} and \\log{(F_{x} + e^{y})} = \\log{(L_{\\varepsilon} + e^{y})} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\log{(F_{x} + e^{y})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\log{(L_{\\varepsilon} + e^{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('y', commutative=True)), Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\lambda')(Symbol('y', commutative=True)), Add(Symbol('F_x', commutative=True), exp(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('F_x', commutative=True), exp(Symbol('y', commutative=True))), Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["log", 3], "Equality(log(Add(Symbol('F_x', commutative=True), exp(Symbol('y', commutative=True)))), log(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(log(Add(Symbol('F_x', commutative=True), exp(Symbol('y', commutative=True)))), log(Add(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('y', commutative=True)))))"], [["differentiate", 5, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(log(Add(Symbol('F_x', commutative=True), exp(Symbol('y', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(log(Add(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('y', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(J,\\ddot{x})} = \\ddot{x} + e^{J}, then obtain \\int (3 \\ddot{x} + \\operatorname{v_{t}}{(J,\\ddot{x})}) d\\ddot{x} = \\int (4 \\ddot{x} + e^{J}) d\\ddot{x}", "derivation": "\\operatorname{v_{t}}{(J,\\ddot{x})} = \\ddot{x} + e^{J} and \\ddot{x} + \\operatorname{v_{t}}{(J,\\ddot{x})} = 2 \\ddot{x} + e^{J} and 3 \\ddot{x} + \\operatorname{v_{t}}{(J,\\ddot{x})} = 4 \\ddot{x} + e^{J} and \\int (3 \\ddot{x} + \\operatorname{v_{t}}{(J,\\ddot{x})}) d\\ddot{x} = \\int (4 \\ddot{x} + e^{J}) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), exp(Symbol('J', commutative=True))))"], [["add", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('J', commutative=True))))"], [["add", 2, "Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\ddot{x}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(4), Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(3), Symbol('\\\\ddot{x}', commutative=True)), Function('v_t')(Symbol('J', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Mul(Integer(4), Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('J', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(k,m)} = m^{k} and \\varphi{(\\mathbf{J},\\Omega)} = \\Omega \\mathbf{J}, then obtain \\int (\\mathbf{v}^{m}{(k,m)} \\varphi{(\\mathbf{J},\\Omega)})^{m} d\\mathbf{J} = \\int (\\Omega \\mathbf{J} \\mathbf{v}^{m}{(k,m)})^{m} d\\mathbf{J}", "derivation": "\\mathbf{v}{(k,m)} = m^{k} and \\mathbf{v}^{m}{(k,m)} = (m^{k})^{m} and \\varphi{(\\mathbf{J},\\Omega)} = \\Omega \\mathbf{J} and (m^{k})^{m} \\varphi{(\\mathbf{J},\\Omega)} = \\Omega \\mathbf{J} (m^{k})^{m} and \\mathbf{v}^{m}{(k,m)} \\varphi{(\\mathbf{J},\\Omega)} = \\Omega \\mathbf{J} \\mathbf{v}^{m}{(k,m)} and (\\mathbf{v}^{m}{(k,m)} \\varphi{(\\mathbf{J},\\Omega)})^{m} = (\\Omega \\mathbf{J} \\mathbf{v}^{m}{(k,m)})^{m} and \\int (\\mathbf{v}^{m}{(k,m)} \\varphi{(\\mathbf{J},\\Omega)})^{m} d\\mathbf{J} = \\int (\\Omega \\mathbf{J} \\mathbf{v}^{m}{(k,m)})^{m} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Symbol('m', commutative=True), Symbol('k', commutative=True)), Symbol('m', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 3, "Pow(Pow(Symbol('m', commutative=True), Symbol('k', commutative=True)), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Pow(Symbol('m', commutative=True), Symbol('k', commutative=True)), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Pow(Symbol('m', commutative=True), Symbol('k', commutative=True)), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))))"], [["power", 5, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["integrate", 6, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Pow(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given h{(\\mu_0,M_{E})} = M_{E} \\mu_0, then obtain \\frac{\\partial}{\\partial M_{E}} (M_{E} \\mu_0 + (\\frac{h{(\\mu_0,M_{E})}}{M_{E} \\mu_0})^{\\mu_0}) = \\frac{\\partial}{\\partial M_{E}} (M_{E} \\mu_0 + 1)", "derivation": "h{(\\mu_0,M_{E})} = M_{E} \\mu_0 and \\frac{h{(\\mu_0,M_{E})}}{M_{E} \\mu_0} = 1 and (\\frac{h{(\\mu_0,M_{E})}}{M_{E} \\mu_0})^{\\mu_0} = 1 and M_{E} \\mu_0 + (\\frac{h{(\\mu_0,M_{E})}}{M_{E} \\mu_0})^{\\mu_0} = M_{E} \\mu_0 + 1 and \\frac{\\partial}{\\partial M_{E}} (M_{E} \\mu_0 + (\\frac{h{(\\mu_0,M_{E})}}{M_{E} \\mu_0})^{\\mu_0}) = \\frac{\\partial}{\\partial M_{E}} (M_{E} \\mu_0 + 1)", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Integer(1))"], [["add", 3, "Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(1)))"], [["differentiate", 4, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = (J_{\\varepsilon} + y)^{G}, then derive \\frac{\\partial}{\\partial G} \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = (J_{\\varepsilon} + y)^{G} \\log{(J_{\\varepsilon} + y)}, then obtain \\frac{\\partial}{\\partial G} \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} \\log{(J_{\\varepsilon} + y)}", "derivation": "\\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = (J_{\\varepsilon} + y)^{G} and \\frac{\\partial}{\\partial G} \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = \\frac{\\partial}{\\partial G} (J_{\\varepsilon} + y)^{G} and \\frac{\\partial}{\\partial G} \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = (J_{\\varepsilon} + y)^{G} \\log{(J_{\\varepsilon} + y)} and \\frac{\\partial}{\\partial G} \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} = \\operatorname{E_{n}}{(G,J_{\\varepsilon},y)} \\log{(J_{\\varepsilon} + y)}", "srepr_derivation": [["get_premise", "Equality(Function('E_n')(Symbol('G', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('G', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('G', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Pow(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Symbol('G', commutative=True)), log(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('E_n')(Symbol('G', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Function('E_n')(Symbol('G', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), log(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(E_{n})} = E_{n}, then derive \\mathbf{f} + \\frac{\\dot{z}^{2}{(E_{n})}}{2} = \\int E_{n} d\\dot{z}{(E_{n})}, then derive \\frac{E_{n}^{2}}{2} + \\mathbf{f} = \\frac{E_{n}^{2}}{2} + p, then obtain \\int (\\frac{E_{n}^{2}}{2} + \\mathbf{f}) dE_{n} = \\int (\\frac{E_{n}^{2}}{2} + p) dE_{n}", "derivation": "\\dot{z}{(E_{n})} = E_{n} and \\int \\dot{z}{(E_{n})} dE_{n} = \\int E_{n} dE_{n} and \\int \\dot{z}{(E_{n})} d\\dot{z}{(E_{n})} = \\int E_{n} d\\dot{z}{(E_{n})} and \\mathbf{f} + \\frac{\\dot{z}^{2}{(E_{n})}}{2} = \\int E_{n} d\\dot{z}{(E_{n})} and \\frac{E_{n}^{2}}{2} + \\mathbf{f} = \\int E_{n} dE_{n} and \\frac{E_{n}^{2}}{2} + \\mathbf{f} = \\frac{E_{n}^{2}}{2} + p and \\int (\\frac{E_{n}^{2}}{2} + \\mathbf{f}) dE_{n} = \\int (\\frac{E_{n}^{2}}{2} + p) dE_{n}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Symbol('E_n', commutative=True), Tuple(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)), Tuple(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)))), Integral(Symbol('E_n', commutative=True), Tuple(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)), Integer(2)))), Integral(Symbol('E_n', commutative=True), Tuple(Function('\\\\dot{z}')(Symbol('E_n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Symbol('E_n', commutative=True), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('p', commutative=True)))"], [["integrate", 6, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('p', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given h{(\\mathbf{A},v_{y})} = \\mathbf{A} + v_{y} and \\operatorname{E_{\\lambda}}{(\\pi)} = \\int \\sin{(\\pi)} d\\pi, then obtain \\frac{\\operatorname{E_{\\lambda}}{(\\pi)}}{h{(\\mathbf{A},v_{y})}} = \\frac{f_{E} - \\cos{(\\pi)}}{h{(\\mathbf{A},v_{y})}}", "derivation": "h{(\\mathbf{A},v_{y})} = \\mathbf{A} + v_{y} and \\operatorname{E_{\\lambda}}{(\\pi)} = \\int \\sin{(\\pi)} d\\pi and \\frac{\\operatorname{E_{\\lambda}}{(\\pi)}}{\\mathbf{A} + v_{y}} = \\frac{\\int \\sin{(\\pi)} d\\pi}{\\mathbf{A} + v_{y}} and \\frac{\\operatorname{E_{\\lambda}}{(\\pi)}}{h{(\\mathbf{A},v_{y})}} = \\frac{\\int \\sin{(\\pi)} d\\pi}{h{(\\mathbf{A},v_{y})}} and \\frac{\\operatorname{E_{\\lambda}}{(\\pi)}}{h{(\\mathbf{A},v_{y})}} = \\frac{f_{E} - \\cos{(\\pi)}}{h{(\\mathbf{A},v_{y})}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)))"], ["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True)), Pow(Function('h')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))), Mul(Pow(Function('h')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True)), Pow(Function('h')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))), Mul(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('h')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given Q{(a,v_{1})} = \\cos^{a}{(v_{1})}, then obtain - (Q{(a,v_{1})} - \\cos{(v_{1})}) Q{(a,v_{1})} - \\cos^{a}{(v_{1})} = - (- \\cos{(v_{1})} + \\cos^{a}{(v_{1})}) Q{(a,v_{1})} - \\cos^{a}{(v_{1})}", "derivation": "Q{(a,v_{1})} = \\cos^{a}{(v_{1})} and Q{(a,v_{1})} - \\cos{(v_{1})} = - \\cos{(v_{1})} + \\cos^{a}{(v_{1})} and (Q{(a,v_{1})} - \\cos{(v_{1})}) Q{(a,v_{1})} = (- \\cos{(v_{1})} + \\cos^{a}{(v_{1})}) Q{(a,v_{1})} and - (Q{(a,v_{1})} - \\cos{(v_{1})}) Q{(a,v_{1})} = - (- \\cos{(v_{1})} + \\cos^{a}{(v_{1})}) Q{(a,v_{1})} and - (Q{(a,v_{1})} - \\cos{(v_{1})}) Q{(a,v_{1})} - \\cos^{a}{(v_{1})} = - (- \\cos{(v_{1})} + \\cos^{a}{(v_{1})}) Q{(a,v_{1})} - \\cos^{a}{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True)))"], [["minus", 1, "cos(Symbol('v_1', commutative=True))"], "Equality(Add(Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True))))"], [["times", 2, "Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Add(Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Mul(Add(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))))"], [["minus", 4, "Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Add(Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Add(Mul(Integer(-1), cos(Symbol('v_1', commutative=True))), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True))), Function('Q')(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('v_1', commutative=True)), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(E_{n})} = \\frac{d}{d E_{n}} \\sin{(E_{n})}, then derive \\operatorname{v_{1}}{(E_{n})} = \\cos{(E_{n})}, then obtain \\operatorname{v_{1}}^{E_{n}}{(E_{n})} + \\frac{\\cos^{E_{n}}{(E_{n})}}{E_{n}} = \\operatorname{v_{1}}^{E_{n}}{(E_{n})} + \\frac{(\\frac{d}{d E_{n}} \\sin{(E_{n})})^{E_{n}}}{E_{n}}", "derivation": "\\operatorname{v_{1}}{(E_{n})} = \\frac{d}{d E_{n}} \\sin{(E_{n})} and \\operatorname{v_{1}}^{E_{n}}{(E_{n})} = (\\frac{d}{d E_{n}} \\sin{(E_{n})})^{E_{n}} and \\operatorname{v_{1}}{(E_{n})} = \\cos{(E_{n})} and \\cos^{E_{n}}{(E_{n})} = (\\frac{d}{d E_{n}} \\sin{(E_{n})})^{E_{n}} and \\frac{\\cos^{E_{n}}{(E_{n})}}{E_{n}} = \\frac{(\\frac{d}{d E_{n}} \\sin{(E_{n})})^{E_{n}}}{E_{n}} and \\operatorname{v_{1}}^{E_{n}}{(E_{n})} + \\frac{\\cos^{E_{n}}{(E_{n})}}{E_{n}} = \\operatorname{v_{1}}^{E_{n}}{(E_{n})} + \\frac{(\\frac{d}{d E_{n}} \\sin{(E_{n})})^{E_{n}}}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('E_n', commutative=True)), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('v_1')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))"], [["divide", 4, "Symbol('E_n', commutative=True)"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True))))"], [["add", 5, "Pow(Function('v_1')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], "Equality(Add(Pow(Function('v_1')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(cos(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)))), Add(Pow(Function('v_1')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given s{(r_{0})} = \\cos{(\\log{(r_{0})})}, then obtain 2 r_{0} s^{2}{(r_{0})} = 2 r_{0} s{(r_{0})} \\cos{(\\log{(r_{0})})}", "derivation": "s{(r_{0})} = \\cos{(\\log{(r_{0})})} and s{(r_{0})} + \\cos{(\\log{(r_{0})})} = 2 \\cos{(\\log{(r_{0})})} and r_{0} s{(r_{0})} = r_{0} \\cos{(\\log{(r_{0})})} and r_{0} (s{(r_{0})} + \\cos{(\\log{(r_{0})})}) s{(r_{0})} = r_{0} (s{(r_{0})} + \\cos{(\\log{(r_{0})})}) \\cos{(\\log{(r_{0})})} and 2 r_{0} s{(r_{0})} \\cos{(\\log{(r_{0})})} = 2 r_{0} \\cos^{2}{(\\log{(r_{0})})} and 2 r_{0} s^{2}{(r_{0})} = 2 r_{0} s{(r_{0})} \\cos{(\\log{(r_{0})})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True))))"], [["add", 1, "cos(log(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True)))), Mul(Integer(2), cos(log(Symbol('r_0', commutative=True)))))"], [["times", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Function('s')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), cos(log(Symbol('r_0', commutative=True)))))"], [["times", 3, "Add(Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True))))"], "Equality(Mul(Symbol('r_0', commutative=True), Add(Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True)))), Function('s')(Symbol('r_0', commutative=True))), Mul(Symbol('r_0', commutative=True), Add(Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True)))), cos(log(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Symbol('r_0', commutative=True), Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True)))), Mul(Integer(2), Symbol('r_0', commutative=True), Pow(cos(log(Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Symbol('r_0', commutative=True), Pow(Function('s')(Symbol('r_0', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('r_0', commutative=True), Function('s')(Symbol('r_0', commutative=True)), cos(log(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(r_{0})} = e^{e^{r_{0}}}, then obtain \\int \\frac{d}{d r_{0}} (\\operatorname{F_{N}}{(r_{0})} + e^{e^{r_{0}}})^{2} dr_{0} = \\int \\frac{d}{d r_{0}} 4 e^{2 e^{r_{0}}} dr_{0}", "derivation": "\\operatorname{F_{N}}{(r_{0})} = e^{e^{r_{0}}} and \\operatorname{F_{N}}{(r_{0})} + e^{e^{r_{0}}} = 2 e^{e^{r_{0}}} and (\\operatorname{F_{N}}{(r_{0})} + e^{e^{r_{0}}})^{2} = 4 e^{2 e^{r_{0}}} and \\frac{d}{d r_{0}} (\\operatorname{F_{N}}{(r_{0})} + e^{e^{r_{0}}})^{2} = \\frac{d}{d r_{0}} 4 e^{2 e^{r_{0}}} and \\int \\frac{d}{d r_{0}} (\\operatorname{F_{N}}{(r_{0})} + e^{e^{r_{0}}})^{2} dr_{0} = \\int \\frac{d}{d r_{0}} 4 e^{2 e^{r_{0}}} dr_{0}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('r_0', commutative=True)), exp(exp(Symbol('r_0', commutative=True))))"], [["add", 1, "exp(exp(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('F_N')(Symbol('r_0', commutative=True)), exp(exp(Symbol('r_0', commutative=True)))), Mul(Integer(2), exp(exp(Symbol('r_0', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Add(Function('F_N')(Symbol('r_0', commutative=True)), exp(exp(Symbol('r_0', commutative=True)))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), exp(Symbol('r_0', commutative=True))))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Pow(Add(Function('F_N')(Symbol('r_0', commutative=True)), exp(exp(Symbol('r_0', commutative=True)))), Integer(2)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Integer(4), exp(Mul(Integer(2), exp(Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('r_0', commutative=True)"], "Equality(Integral(Derivative(Pow(Add(Function('F_N')(Symbol('r_0', commutative=True)), exp(exp(Symbol('r_0', commutative=True)))), Integer(2)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))), Integral(Derivative(Mul(Integer(4), exp(Mul(Integer(2), exp(Symbol('r_0', commutative=True))))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given y{(\\mathbf{s})} = \\log{(\\mathbf{s})} and \\bar{\\h}{(\\dot{\\mathbf{r}},G)} = \\frac{\\dot{\\mathbf{r}}}{G}, then obtain - \\frac{\\bar{\\h}^{- G}{(\\dot{\\mathbf{r}},G)}}{G} = (- y{(\\mathbf{s})} + \\log{(\\mathbf{s})} - \\frac{1}{G}) \\bar{\\h}^{- G}{(\\dot{\\mathbf{r}},G)}", "derivation": "y{(\\mathbf{s})} = \\log{(\\mathbf{s})} and 0 = - y{(\\mathbf{s})} + \\log{(\\mathbf{s})} and \\bar{\\h}{(\\dot{\\mathbf{r}},G)} = \\frac{\\dot{\\mathbf{r}}}{G} and - \\frac{1}{G} = - y{(\\mathbf{s})} + \\log{(\\mathbf{s})} - \\frac{1}{G} and - \\frac{(\\frac{\\dot{\\mathbf{r}}}{G})^{- G}}{G} = (\\frac{\\dot{\\mathbf{r}}}{G})^{- G} (- y{(\\mathbf{s})} + \\log{(\\mathbf{s})} - \\frac{1}{G}) and - \\frac{\\bar{\\h}^{- G}{(\\dot{\\mathbf{r}},G)}}{G} = (- y{(\\mathbf{s})} + \\log{(\\mathbf{s})} - \\frac{1}{G}) \\bar{\\h}^{- G}{(\\dot{\\mathbf{r}},G)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), log(Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Function('y')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y')(Symbol('\\\\mathbf{s}', commutative=True))), log(Symbol('\\\\mathbf{s}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["minus", 2, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Function('y')(Symbol('\\\\mathbf{s}', commutative=True))), log(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))))"], [["divide", 4, "Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('G', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Function('y')(Symbol('\\\\mathbf{s}', commutative=True))), log(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('y')(Symbol('\\\\mathbf{s}', commutative=True))), log(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))), Pow(Function('\\\\hbar')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(I,H)} = \\log{(H - I)}, then obtain - I + \\frac{(- I + \\dot{x}{(I,H)}) \\dot{x}{(I,H)}}{- I + \\log{(H - I)}} = - I + \\log{(H - I)}", "derivation": "\\dot{x}{(I,H)} = \\log{(H - I)} and - I + \\dot{x}{(I,H)} = - I + \\log{(H - I)} and \\frac{- I + \\dot{x}{(I,H)}}{- I + \\log{(H - I)}} = 1 and \\frac{(- I + \\dot{x}{(I,H)}) \\dot{x}{(I,H)}}{- I + \\log{(H - I)}} = \\dot{x}{(I,H)} and - I + \\frac{(- I + \\dot{x}{(I,H)}) \\dot{x}{(I,H)}}{- I + \\log{(H - I)}} = - I + \\log{(H - I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Integer(-1))), Integer(1))"], [["times", 3, "Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Integer(-1)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Integer(-1)), Function('\\\\dot{x}')(Symbol('I', commutative=True), Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\psi,n_{1})} = \\psi n_{1}, then obtain \\frac{\\frac{d}{d n_{1}} 1}{\\int \\operatorname{m_{s}}{(\\psi,n_{1})} d\\psi} = \\frac{\\frac{\\partial}{\\partial n_{1}} \\frac{\\psi n_{1}}{\\operatorname{m_{s}}{(\\psi,n_{1})}}}{\\int \\operatorname{m_{s}}{(\\psi,n_{1})} d\\psi}", "derivation": "\\operatorname{m_{s}}{(\\psi,n_{1})} = \\psi n_{1} and 1 = \\frac{\\psi n_{1}}{\\operatorname{m_{s}}{(\\psi,n_{1})}} and \\frac{d}{d n_{1}} 1 = \\frac{\\partial}{\\partial n_{1}} \\frac{\\psi n_{1}}{\\operatorname{m_{s}}{(\\psi,n_{1})}} and \\frac{\\frac{d}{d n_{1}} 1}{\\int \\operatorname{m_{s}}{(\\psi,n_{1})} d\\psi} = \\frac{\\frac{\\partial}{\\partial n_{1}} \\frac{\\psi n_{1}}{\\operatorname{m_{s}}{(\\psi,n_{1})}}}{\\int \\operatorname{m_{s}}{(\\psi,n_{1})} d\\psi}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)))"], [["divide", 1, "Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True), Pow(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True), Pow(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Integral(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))), Mul(Derivative(Mul(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True), Pow(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Integral(Function('m_s')(Symbol('\\\\psi', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given Q{(H,B)} = B + H, then obtain \\int (B + H) \\frac{\\partial}{\\partial H} 2 Q{(H,B)} dH = \\int (B + H) \\frac{\\partial}{\\partial H} (B + H + Q{(H,B)}) dH", "derivation": "Q{(H,B)} = B + H and B + H + Q{(H,B)} = 2 B + 2 H and 2 Q{(H,B)} = 2 B + 2 H and B + H + Q{(H,B)} = 2 Q{(H,B)} and \\frac{\\partial}{\\partial H} (B + H + Q{(H,B)}) = \\frac{\\partial}{\\partial H} (2 B + 2 H) and \\frac{\\partial}{\\partial H} 2 Q{(H,B)} = \\frac{\\partial}{\\partial H} (2 B + 2 H) and \\frac{\\partial}{\\partial H} 2 Q{(H,B)} = \\frac{\\partial}{\\partial H} (B + H + Q{(H,B)}) and (B + H) \\frac{\\partial}{\\partial H} 2 Q{(H,B)} = (B + H) \\frac{\\partial}{\\partial H} (B + H + Q{(H,B)}) and \\int (B + H) \\frac{\\partial}{\\partial H} 2 Q{(H,B)} dH = \\int (B + H) \\frac{\\partial}{\\partial H} (B + H + Q{(H,B)}) dH", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('H', commutative=True)))"], [["add", 1, "Add(Symbol('B', commutative=True), Symbol('H', commutative=True))"], "Equality(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["times", 7, "Add(Symbol('B', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Add(Symbol('B', commutative=True), Symbol('H', commutative=True)), Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Symbol('B', commutative=True), Symbol('H', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["integrate", 8, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('B', commutative=True), Symbol('H', commutative=True)), Derivative(Mul(Integer(2), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Add(Symbol('B', commutative=True), Symbol('H', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Symbol('H', commutative=True), Function('Q')(Symbol('H', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\varphi{(h,r)} = \\sin^{r}{(h)}, then obtain \\frac{(h + 2 \\varphi{(h,r)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}} = \\frac{(h + \\varphi{(h,r)} + \\sin^{r}{(h)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}}", "derivation": "\\varphi{(h,r)} = \\sin^{r}{(h)} and h + \\varphi{(h,r)} = h + \\sin^{r}{(h)} and h + 2 \\varphi{(h,r)} = h + \\varphi{(h,r)} + \\sin^{r}{(h)} and h + 2 \\varphi{(h,r)} = h + 2 \\sin^{r}{(h)} and \\frac{(h + 2 \\varphi{(h,r)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}} = \\frac{(h + 2 \\sin^{r}{(h)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}} and h + \\varphi{(h,r)} + \\sin^{r}{(h)} = h + 2 \\sin^{r}{(h)} and \\frac{(h + 2 \\varphi{(h,r)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}} = \\frac{(h + \\varphi{(h,r)} + \\sin^{r}{(h)}) \\sin^{r}{(h)}}{\\varphi{(h,r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True))), Add(Symbol('h', commutative=True), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))))"], [["add", 2, "Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Add(Symbol('h', commutative=True), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Add(Symbol('h', commutative=True), Mul(Integer(2), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True)))))"], [["times", 4, "Mul(Pow(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True)))"], "Equality(Mul(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Pow(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))), Mul(Add(Symbol('h', commutative=True), Mul(Integer(2), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True)))), Pow(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('h', commutative=True), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))), Add(Symbol('h', commutative=True), Mul(Integer(2), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Add(Symbol('h', commutative=True), Mul(Integer(2), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)))), Pow(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))), Mul(Add(Symbol('h', commutative=True), Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))), Pow(Function('\\\\varphi')(Symbol('h', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{y})} = \\cos{(v_{y})}, then obtain - 2 v_{y} + \\operatorname{f^{\\prime}}{(v_{y})} - \\cos{(v_{y})} = - 2 v_{y}", "derivation": "\\operatorname{f^{\\prime}}{(v_{y})} = \\cos{(v_{y})} and - v_{y} + \\operatorname{f^{\\prime}}{(v_{y})} = - v_{y} + \\cos{(v_{y})} and - 2 v_{y} + \\operatorname{f^{\\prime}}{(v_{y})} + \\cos{(v_{y})} = - 2 v_{y} + 2 \\cos{(v_{y})} and - 2 v_{y} + \\operatorname{f^{\\prime}}{(v_{y})} - \\cos{(v_{y})} = - 2 v_{y}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('f^{\\\\prime}')(Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)), Function('f^{\\\\prime}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)), Mul(Integer(2), cos(Symbol('v_y', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), cos(Symbol('v_y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)), Function('f^{\\\\prime}')(Symbol('v_y', commutative=True)), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given u{(c_{0})} = \\sin{(e^{c_{0}})}, then obtain - \\mathbf{p} + \\frac{d}{d c_{0}} (u{(c_{0})} - e^{c_{0}}) = - \\mathbf{p} + \\frac{d}{d c_{0}} (- e^{c_{0}} + \\sin{(e^{c_{0}})})", "derivation": "u{(c_{0})} = \\sin{(e^{c_{0}})} and u{(c_{0})} - e^{c_{0}} = - e^{c_{0}} + \\sin{(e^{c_{0}})} and \\frac{d}{d c_{0}} (u{(c_{0})} - e^{c_{0}}) = \\frac{d}{d c_{0}} (- e^{c_{0}} + \\sin{(e^{c_{0}})}) and - \\mathbf{p} + \\frac{d}{d c_{0}} (u{(c_{0})} - e^{c_{0}}) = - \\mathbf{p} + \\frac{d}{d c_{0}} (- e^{c_{0}} + \\sin{(e^{c_{0}})})", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('c_0', commutative=True)), sin(exp(Symbol('c_0', commutative=True))))"], [["minus", 1, "exp(Symbol('c_0', commutative=True))"], "Equality(Add(Function('u')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), sin(exp(Symbol('c_0', commutative=True)))))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Function('u')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), sin(exp(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Add(Function('u')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Add(Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), sin(exp(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{p}{(P_{g},A)} = \\cos{(A - P_{g})}, then derive \\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g} = \\hat{p} - \\cos{(A - P_{g})}, then obtain e^{\\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g}} = e^{\\hat{p} - \\sigma_{p}{(P_{g},A)}}", "derivation": "\\sigma_{p}{(P_{g},A)} = \\cos{(A - P_{g})} and \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} = \\frac{\\partial}{\\partial A} \\cos{(A - P_{g})} and \\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g} = \\int \\frac{\\partial}{\\partial A} \\cos{(A - P_{g})} dP_{g} and \\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g} = \\hat{p} - \\cos{(A - P_{g})} and \\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g} = \\hat{p} - \\sigma_{p}{(P_{g},A)} and e^{\\int \\frac{\\partial}{\\partial A} \\sigma_{p}{(P_{g},A)} dP_{g}} = e^{\\hat{p} - \\sigma_{p}{(P_{g},A)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Integral(Derivative(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('P_g', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Derivative(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)))))"], [["exp", 5], "Equality(exp(Integral(Derivative(Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True)))), exp(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_g', commutative=True), Symbol('A', commutative=True))))))"]]}, {"prompt": "Given C{(t_{1},C_{1},\\pi)} = t_{1} (C_{1} - \\pi), then obtain 0 = \\frac{\\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi) - \\frac{\\partial}{\\partial C_{1}} C{(t_{1},C_{1},\\pi)}}{\\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi)}", "derivation": "C{(t_{1},C_{1},\\pi)} = t_{1} (C_{1} - \\pi) and \\frac{\\partial}{\\partial C_{1}} C{(t_{1},C_{1},\\pi)} = \\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi) and 0 = \\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi) - \\frac{\\partial}{\\partial C_{1}} C{(t_{1},C_{1},\\pi)} and 0 = \\frac{\\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi) - \\frac{\\partial}{\\partial C_{1}} C{(t_{1},C_{1},\\pi)}}{\\frac{\\partial}{\\partial C_{1}} t_{1} (C_{1} - \\pi)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('t_1', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('t_1', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('C')(Symbol('t_1', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('C')(Symbol('t_1', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))))"], [["divide", 3, "Derivative(Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Add(Derivative(Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('C')(Symbol('t_1', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))), Pow(Derivative(Mul(Symbol('t_1', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\pi)} = \\sin{(\\pi)}, then obtain \\frac{d}{d \\pi} \\mathbf{H}^{\\pi}{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\sin^{\\pi}{(\\pi)} = \\frac{d}{d \\pi} \\sin^{\\pi}{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\sin^{\\pi}{(\\pi)}", "derivation": "\\mathbf{H}{(\\pi)} = \\sin{(\\pi)} and \\mathbf{H}^{\\pi}{(\\pi)} = \\sin^{\\pi}{(\\pi)} and \\frac{d}{d \\pi} \\mathbf{H}^{\\pi}{(\\pi)} = \\frac{d}{d \\pi} \\sin^{\\pi}{(\\pi)} and \\frac{d^{2}}{d \\pi^{2}} \\mathbf{H}^{\\pi}{(\\pi)} = \\frac{d^{2}}{d \\pi^{2}} \\sin^{\\pi}{(\\pi)} and \\frac{d}{d \\pi} \\mathbf{H}^{\\pi}{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\mathbf{H}^{\\pi}{(\\pi)} = \\frac{d^{2}}{d \\pi^{2}} \\mathbf{H}^{\\pi}{(\\pi)} + \\frac{d}{d \\pi} \\sin^{\\pi}{(\\pi)} and \\frac{d}{d \\pi} \\mathbf{H}^{\\pi}{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\sin^{\\pi}{(\\pi)} = \\frac{d}{d \\pi} \\sin^{\\pi}{(\\pi)} + \\frac{d^{2}}{d \\pi^{2}} \\sin^{\\pi}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))))"], [["add", 3, "Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))), Add(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))), Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))), Add(Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\rho_{f}{(b,a^{\\dagger})} = \\frac{\\log{(b)}}{a^{\\dagger}}, then obtain (2 \\rho_{f}{(b,a^{\\dagger})})^{2 b} + 2 \\rho_{f}{(b,a^{\\dagger})} = (\\rho_{f}{(b,a^{\\dagger})} + \\frac{\\log{(b)}}{a^{\\dagger}})^{2 b} + 2 \\rho_{f}{(b,a^{\\dagger})}", "derivation": "\\rho_{f}{(b,a^{\\dagger})} = \\frac{\\log{(b)}}{a^{\\dagger}} and 2 \\rho_{f}{(b,a^{\\dagger})} = \\rho_{f}{(b,a^{\\dagger})} + \\frac{\\log{(b)}}{a^{\\dagger}} and (2 \\rho_{f}{(b,a^{\\dagger})})^{b} = (\\rho_{f}{(b,a^{\\dagger})} + \\frac{\\log{(b)}}{a^{\\dagger}})^{b} and (2 \\rho_{f}{(b,a^{\\dagger})})^{2 b} = (\\rho_{f}{(b,a^{\\dagger})} + \\frac{\\log{(b)}}{a^{\\dagger}})^{2 b} and (2 \\rho_{f}{(b,a^{\\dagger})})^{2 b} + 2 \\rho_{f}{(b,a^{\\dagger})} = (\\rho_{f}{(b,a^{\\dagger})} + \\frac{\\log{(b)}}{a^{\\dagger}})^{2 b} + 2 \\rho_{f}{(b,a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), log(Symbol('b', commutative=True))))"], [["add", 1, "Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), log(Symbol('b', commutative=True)))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), log(Symbol('b', commutative=True)))), Symbol('b', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), Symbol('b', commutative=True))), Pow(Add(Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), log(Symbol('b', commutative=True)))), Mul(Integer(2), Symbol('b', commutative=True))))"], [["add", 4, "Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), Symbol('b', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Add(Pow(Add(Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), log(Symbol('b', commutative=True)))), Mul(Integer(2), Symbol('b', commutative=True))), Mul(Integer(2), Function('\\\\rho_f')(Symbol('b', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given Q{(r_{0},c)} = c + r_{0}, then derive \\frac{\\partial}{\\partial r_{0}} Q{(r_{0},c)} = 1, then obtain - c - r_{0} + \\frac{\\partial}{\\partial r_{0}} (c + r_{0}) = - c - r_{0} + 1", "derivation": "Q{(r_{0},c)} = c + r_{0} and \\frac{\\partial}{\\partial r_{0}} Q{(r_{0},c)} = \\frac{\\partial}{\\partial r_{0}} (c + r_{0}) and \\frac{\\partial}{\\partial r_{0}} Q{(r_{0},c)} = 1 and \\frac{\\partial}{\\partial r_{0}} (c + r_{0}) = 1 and - Q{(r_{0},c)} + \\frac{\\partial}{\\partial r_{0}} (c + r_{0}) = 1 - Q{(r_{0},c)} and - c - r_{0} + \\frac{\\partial}{\\partial r_{0}} (c + r_{0}) = - c - r_{0} + 1", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Symbol('c', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('c', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True))), Derivative(Add(Symbol('c', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('Q')(Symbol('r_0', commutative=True), Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('c', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\nabla{(H)} = \\sin{(H)}, then obtain \\int (\\nabla{(H)} - \\sin{(H)}) dH + (\\int (\\nabla{(H)} - \\sin{(H)}) dH)^{H} = (\\int 0 dH)^{H} + \\int (\\nabla{(H)} - \\sin{(H)}) dH", "derivation": "\\nabla{(H)} = \\sin{(H)} and \\nabla{(H)} - \\sin{(H)} = 0 and \\int (\\nabla{(H)} - \\sin{(H)}) dH = \\int 0 dH and (\\int (\\nabla{(H)} - \\sin{(H)}) dH)^{H} = (\\int 0 dH)^{H} and \\int (\\nabla{(H)} - \\sin{(H)}) dH + (\\int (\\nabla{(H)} - \\sin{(H)}) dH)^{H} = (\\int 0 dH)^{H} + \\int (\\nabla{(H)} - \\sin{(H)}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["minus", 1, "sin(Symbol('H', commutative=True))"], "Equality(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integral(Integer(0), Tuple(Symbol('H', commutative=True))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["add", 4, "Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Pow(Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))), Add(Pow(Integral(Integer(0), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integral(Add(Function('\\\\nabla')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(k,y)} = y^{k} and W{(I)} = e^{I}, then obtain \\frac{(k + \\operatorname{P_{g}}{(k,y)}) W{(I)}}{k + y^{k}} = W{(I)}", "derivation": "\\operatorname{P_{g}}{(k,y)} = y^{k} and k + \\operatorname{P_{g}}{(k,y)} = k + y^{k} and \\frac{k + \\operatorname{P_{g}}{(k,y)}}{k + y^{k}} = 1 and W{(I)} = e^{I} and \\frac{(k + \\operatorname{P_{g}}{(k,y)}) e^{I}}{k + y^{k}} = e^{I} and \\frac{(k + \\operatorname{P_{g}}{(k,y)}) W{(I)}}{k + y^{k}} = W{(I)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('k', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True)))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('P_g')(Symbol('k', commutative=True), Symbol('y', commutative=True))), Add(Symbol('k', commutative=True), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True))))"], [["divide", 2, "Add(Symbol('k', commutative=True), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('k', commutative=True), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Add(Symbol('k', commutative=True), Function('P_g')(Symbol('k', commutative=True), Symbol('y', commutative=True)))), Integer(1))"], ["get_premise", "Equality(Function('W')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["times", 3, "exp(Symbol('I', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('k', commutative=True), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Add(Symbol('k', commutative=True), Function('P_g')(Symbol('k', commutative=True), Symbol('y', commutative=True))), exp(Symbol('I', commutative=True))), exp(Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('k', commutative=True), Pow(Symbol('y', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Add(Symbol('k', commutative=True), Function('P_g')(Symbol('k', commutative=True), Symbol('y', commutative=True))), Function('W')(Symbol('I', commutative=True))), Function('W')(Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain \\frac{\\operatorname{F_{N}}{(\\mathbf{P})} e^{\\mathbf{P}}}{\\mathbf{H}^{2}{(t_{1},y^{\\prime})}} = \\frac{e^{2 \\mathbf{P}}}{\\mathbf{H}^{2}{(t_{1},y^{\\prime})}}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\operatorname{F_{N}}{(\\mathbf{P})} e^{\\mathbf{P}} = e^{2 \\mathbf{P}} and \\frac{\\operatorname{F_{N}}{(\\mathbf{P})} e^{\\mathbf{P}}}{\\mathbf{H}{(t_{1},y^{\\prime})}} = \\frac{e^{2 \\mathbf{P}}}{\\mathbf{H}{(t_{1},y^{\\prime})}} and \\frac{\\operatorname{F_{N}}{(\\mathbf{P})} e^{\\mathbf{P}}}{\\mathbf{H}^{2}{(t_{1},y^{\\prime})}} = \\frac{e^{2 \\mathbf{P}}}{\\mathbf{H}^{2}{(t_{1},y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["times", 3, "Pow(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))"], "Equality(Mul(Function('F_N')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-2)), exp(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('t_1', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)} = - \\mathbf{B} + \\varphi, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)} = -1, then obtain (\\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)})^{\\mathbf{B}} = (-1)^{\\mathbf{B}}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{B},\\varphi)} = - \\mathbf{B} + \\varphi and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)} = \\frac{\\partial}{\\partial \\mathbf{B}} (- \\mathbf{B} + \\varphi) and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)} = -1 and \\frac{\\partial}{\\partial \\mathbf{B}} (- \\mathbf{B} + \\varphi) = -1 and (\\frac{\\partial}{\\partial \\mathbf{B}} (- \\mathbf{B} + \\varphi))^{\\mathbf{B}} = (-1)^{\\mathbf{B}} and (\\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{F_{x}}{(\\mathbf{B},\\varphi)})^{\\mathbf{B}} = (-1)^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))"], [["power", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(V,\\mathbf{f})} = \\frac{\\partial}{\\partial V} (V + \\mathbf{f}), then derive \\mathbf{J}_f{(V,\\mathbf{f})} + 1 = 2, then obtain \\frac{\\partial}{\\partial V} (\\frac{\\partial}{\\partial V} (V + \\mathbf{f}) + 1) = \\frac{d}{d V} 2", "derivation": "\\mathbf{J}_f{(V,\\mathbf{f})} = \\frac{\\partial}{\\partial V} (V + \\mathbf{f}) and \\mathbf{J}_f{(V,\\mathbf{f})} + \\frac{\\partial}{\\partial V} (V + \\mathbf{f}) = 2 \\frac{\\partial}{\\partial V} (V + \\mathbf{f}) and \\mathbf{J}_f{(V,\\mathbf{f})} + 1 = 2 and \\frac{\\partial}{\\partial V} (V + \\mathbf{f}) + 1 = 2 and \\frac{\\partial}{\\partial V} (\\frac{\\partial}{\\partial V} (V + \\mathbf{f}) + 1) = \\frac{d}{d V} 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(\\mathbf{J}_P)} = \\mathbf{J}_P, then derive \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = 1, then obtain \\operatorname{c_{0}}{(\\mathbf{J}_P,\\hat{x})} \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = \\operatorname{c_{0}}{(\\mathbf{J}_P,\\hat{x})}", "derivation": "\\mu{(\\mathbf{J}_P)} = \\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = 1 and \\operatorname{c_{0}}{(\\mathbf{J}_P,\\hat{x})} \\frac{d}{d \\mathbf{J}_P} \\mu{(\\mathbf{J}_P)} = \\operatorname{c_{0}}{(\\mathbf{J}_P,\\hat{x})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Function('c_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('c_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Function('c_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(A_{2},\\mathbf{P})} = \\mathbf{P}^{A_{2}} and \\hat{H}_l{(A_{2},\\mathbf{P})} = \\frac{\\int \\mathbf{P}^{A_{2}} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}}, then obtain \\frac{\\int \\hat{x}_0{(A_{2},\\mathbf{P})} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}} = \\hat{H}_l{(A_{2},\\mathbf{P})}", "derivation": "\\hat{x}_0{(A_{2},\\mathbf{P})} = \\mathbf{P}^{A_{2}} and \\int \\hat{x}_0{(A_{2},\\mathbf{P})} d\\mathbf{P} = \\int \\mathbf{P}^{A_{2}} d\\mathbf{P} and \\frac{\\int \\hat{x}_0{(A_{2},\\mathbf{P})} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}} = \\frac{\\int \\mathbf{P}^{A_{2}} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}} and \\hat{H}_l{(A_{2},\\mathbf{P})} = \\frac{\\int \\mathbf{P}^{A_{2}} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}} and \\frac{\\int \\hat{x}_0{(A_{2},\\mathbf{P})} d\\mathbf{P}}{\\hat{x}_0{(A_{2},\\mathbf{P})}} = \\hat{H}_l{(A_{2},\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 2, "Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Integral(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given C{(v)} = \\log{(v)}, then obtain \\frac{d}{d v} \\log{(v)} = \\frac{d}{d v} C{(v)}", "derivation": "C{(v)} = \\log{(v)} and v C{(v)} = v \\log{(v)} and v C{(v)} + C{(v)} = v C{(v)} + \\log{(v)} and 2 v C{(v)} + \\log{(v)} = v C{(v)} + v \\log{(v)} + \\log{(v)} and \\log{(v)} = - v C{(v)} + v \\log{(v)} + \\log{(v)} and \\frac{d}{d v} \\log{(v)} = \\frac{d}{d v} (- v C{(v)} + v \\log{(v)} + \\log{(v)}) and v C{(v)} + C{(v)} = v \\log{(v)} + \\log{(v)} and \\frac{d}{d v} \\log{(v)} = \\frac{d}{d v} C{(v)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["times", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), log(Symbol('v', commutative=True))))"], [["add", 1, "Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True)))"], "Equality(Add(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Function('C')(Symbol('v', commutative=True))), Add(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))))"], [["add", 2, "Add(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), log(Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))), Add(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), log(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True)))"], "Equality(log(Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), log(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))))"], [["differentiate", 5, "Symbol('v', commutative=True)"], "Equality(Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), log(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Symbol('v', commutative=True), Function('C')(Symbol('v', commutative=True))), Function('C')(Symbol('v', commutative=True))), Add(Mul(Symbol('v', commutative=True), log(Symbol('v', commutative=True))), log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Function('C')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(E_{n},\\rho_f,\\mathbf{J}_f)} = - \\mathbf{J}_f + \\frac{\\rho_f}{E_{n}} and \\mathbf{S}{(\\hat{x})} = \\log{(\\hat{x})}, then obtain - 2 \\mathbf{J}_f + \\mathbf{S}{(\\hat{x})} + \\frac{2 \\rho_f}{E_{n}} = - 2 \\mathbf{J}_f + \\log{(\\hat{x})} + \\frac{2 \\rho_f}{E_{n}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(E_{n},\\rho_f,\\mathbf{J}_f)} = - \\mathbf{J}_f + \\frac{\\rho_f}{E_{n}} and \\mathbf{S}{(\\hat{x})} = \\log{(\\hat{x})} and - \\mathbf{J}_f + \\mathbf{S}{(\\hat{x})} + \\operatorname{f_{\\mathbf{p}}}{(E_{n},\\rho_f,\\mathbf{J}_f)} + \\frac{\\rho_f}{E_{n}} = - \\mathbf{J}_f + \\operatorname{f_{\\mathbf{p}}}{(E_{n},\\rho_f,\\mathbf{J}_f)} + \\log{(\\hat{x})} + \\frac{\\rho_f}{E_{n}} and - 2 \\mathbf{J}_f + \\mathbf{S}{(\\hat{x})} + \\frac{2 \\rho_f}{E_{n}} = - 2 \\mathbf{J}_f + \\log{(\\hat{x})} + \\frac{2 \\rho_f}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('E_n', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_n', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_n', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('E_n', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(2), Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\eta)} = \\sin{(\\eta)} and \\mathbf{H}{(\\eta)} = \\sin{(\\eta)} \\int \\sin{(\\eta)} d\\eta, then obtain \\cos{(\\mathbf{H}{(\\eta)})} = \\cos{(\\sin{(\\eta)} \\int \\mathbb{I}{(\\eta)} d\\eta)}", "derivation": "\\mathbb{I}{(\\eta)} = \\sin{(\\eta)} and \\int \\mathbb{I}{(\\eta)} d\\eta = \\int \\sin{(\\eta)} d\\eta and \\mathbf{H}{(\\eta)} = \\sin{(\\eta)} \\int \\sin{(\\eta)} d\\eta and \\mathbf{H}{(\\eta)} = \\sin{(\\eta)} \\int \\mathbb{I}{(\\eta)} d\\eta and \\cos{(\\mathbf{H}{(\\eta)})} = \\cos{(\\sin{(\\eta)} \\int \\mathbb{I}{(\\eta)} d\\eta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(sin(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True)), Mul(sin(Symbol('\\\\eta', commutative=True)), Integral(sin(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True)), Mul(sin(Symbol('\\\\eta', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["cos", 4], "Equality(cos(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True))), cos(Mul(sin(Symbol('\\\\eta', commutative=True)), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))))"]]}, {"prompt": "Given \\hat{X}{(W,G)} = G^{W}, then obtain - \\frac{\\cos{(- \\hat{X}{(W,G)} + G^{- W} \\hat{X}{(W,G)} + G^{- W})}}{\\hat{X}{(W,G)}} = - \\frac{\\cos{(- \\hat{X}{(W,G)} + 1 + G^{- W})}}{\\hat{X}{(W,G)}}", "derivation": "\\hat{X}{(W,G)} = G^{W} and G^{- W} \\hat{X}{(W,G)} = 1 and - \\hat{X}{(W,G)} + G^{- W} \\hat{X}{(W,G)} + G^{- W} = - \\hat{X}{(W,G)} + 1 + G^{- W} and \\cos{(- \\hat{X}{(W,G)} + G^{- W} \\hat{X}{(W,G)} + G^{- W})} = \\cos{(- \\hat{X}{(W,G)} + 1 + G^{- W})} and - \\frac{\\cos{(- \\hat{X}{(W,G)} + G^{- W} \\hat{X}{(W,G)} + G^{- W})}}{\\hat{X}{(W,G)}} = - \\frac{\\cos{(- \\hat{X}{(W,G)} + 1 + G^{- W})}}{\\hat{X}{(W,G)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('W', commutative=True)))"], [["divide", 1, "Pow(Symbol('G', commutative=True), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Integer(1))"], [["minus", 2, "Add(Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Integer(1), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), cos(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Integer(1), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))))"], [["divide", 4, "Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Integer(-1)), cos(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))), Mul(Integer(-1), Pow(Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Integer(-1)), cos(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('W', commutative=True), Symbol('G', commutative=True))), Integer(1), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(y,\\mathbf{A})} = \\mathbf{A} - y and \\mathbf{B}{(\\mathbf{A},y)} = - y (\\mathbf{A} - y) + \\operatorname{E_{x}}{(y,\\mathbf{A})}, then obtain - y (\\mathbf{A} - y) (- y (\\mathbf{A} - y) + \\operatorname{E_{x}}{(y,\\mathbf{A})} + \\mathbf{B}{(\\mathbf{A},y)}) = - y (\\mathbf{A} - y) (\\mathbf{A} - 2 y (\\mathbf{A} - y) - y + \\operatorname{E_{x}}{(y,\\mathbf{A})})", "derivation": "\\operatorname{E_{x}}{(y,\\mathbf{A})} = \\mathbf{A} - y and \\mathbf{B}{(\\mathbf{A},y)} = - y (\\mathbf{A} - y) + \\operatorname{E_{x}}{(y,\\mathbf{A})} and \\mathbf{B}{(\\mathbf{A},y)} = \\mathbf{A} - y (\\mathbf{A} - y) - y and - y (\\mathbf{A} - y) + \\operatorname{E_{x}}{(y,\\mathbf{A})} + \\mathbf{B}{(\\mathbf{A},y)} = \\mathbf{A} - 2 y (\\mathbf{A} - y) - y + \\operatorname{E_{x}}{(y,\\mathbf{A})} and - y (\\mathbf{A} - y) (- y (\\mathbf{A} - y) + \\operatorname{E_{x}}{(y,\\mathbf{A})} + \\mathbf{B}{(\\mathbf{A},y)}) = - y (\\mathbf{A} - y) (\\mathbf{A} - 2 y (\\mathbf{A} - y) - y + \\operatorname{E_{x}}{(y,\\mathbf{A})})", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('y', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Mul(Integer(-1), Symbol('y', commutative=True)), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(-1), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('y', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Mul(Integer(-1), Symbol('y', commutative=True)), Function('E_x')(Symbol('y', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(l,\\hat{X})} = - \\hat{X} + l, then obtain \\frac{\\partial}{\\partial \\hat{X}} \\tilde{g}^*{(l,\\hat{X})} - 1 = 2 \\frac{\\partial}{\\partial \\hat{X}} \\tilde{g}^*{(l,\\hat{X})}", "derivation": "\\tilde{g}^*{(l,\\hat{X})} = - \\hat{X} + l and - \\hat{X} + l + \\tilde{g}^*{(l,\\hat{X})} = - 2 \\hat{X} + 2 l and 2 \\tilde{g}^*{(l,\\hat{X})} = - 2 \\hat{X} + 2 l and - \\hat{X} + l + \\tilde{g}^*{(l,\\hat{X})} = 2 \\tilde{g}^*{(l,\\hat{X})} and - \\hat{X} + l + \\tilde{g}^*{(l,\\hat{X})} + 1 = 2 \\tilde{g}^*{(l,\\hat{X})} + 1 and \\frac{\\partial}{\\partial \\hat{X}} (- \\hat{X} + l + \\tilde{g}^*{(l,\\hat{X})} + 1) = \\frac{\\partial}{\\partial \\hat{X}} (2 \\tilde{g}^*{(l,\\hat{X})} + 1) and \\frac{\\partial}{\\partial \\hat{X}} \\tilde{g}^*{(l,\\hat{X})} - 1 = 2 \\frac{\\partial}{\\partial \\hat{X}} \\tilde{g}^*{(l,\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(2), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(1)), Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(1)))"], [["differentiate", 5, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('l', commutative=True), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(-1)), Mul(Integer(2), Derivative(Function('\\\\tilde{g}^*')(Symbol('l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{f}{(M,y)} = \\log{(M - y)}, then obtain (\\frac{\\mathbf{f}{(M,y)}}{y})^{M} - (\\frac{\\mathbf{f}{(M,y)}}{y})^{y} = - (\\frac{\\mathbf{f}{(M,y)}}{y})^{y} + (\\frac{\\log{(M - y)}}{y})^{M}", "derivation": "\\mathbf{f}{(M,y)} = \\log{(M - y)} and \\frac{\\mathbf{f}{(M,y)}}{y} = \\frac{\\log{(M - y)}}{y} and (\\frac{\\mathbf{f}{(M,y)}}{y})^{y} = (\\frac{\\log{(M - y)}}{y})^{y} and (\\frac{\\mathbf{f}{(M,y)}}{y})^{M} = (\\frac{\\log{(M - y)}}{y})^{M} and (\\frac{\\mathbf{f}{(M,y)}}{y})^{M} - (\\frac{\\log{(M - y)}}{y})^{y} = (\\frac{\\log{(M - y)}}{y})^{M} - (\\frac{\\log{(M - y)}}{y})^{y} and (\\frac{\\mathbf{f}{(M,y)}}{y})^{M} - (\\frac{\\mathbf{f}{(M,y)}}{y})^{y} = - (\\frac{\\mathbf{f}{(M,y)}}{y})^{y} + (\\frac{\\log{(M - y)}}{y})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('y', commutative=True)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('M', commutative=True)))"], [["minus", 4, "Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('y', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('y', commutative=True)))), Add(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('M', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))), Pow(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Add(Symbol('M', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\nabla{(t_{2},I,m_{s})} = \\frac{I t_{2}}{m_{s}}, then derive \\frac{\\partial}{\\partial I} \\nabla{(t_{2},I,m_{s})} = \\frac{t_{2}}{m_{s}}, then obtain t_{2} + \\frac{\\partial^{3}}{\\partial t_{2}^{2}\\partial I} \\nabla{(t_{2},I,m_{s})} = t_{2} + \\frac{\\partial^{2}}{\\partial t_{2}^{2}} \\frac{t_{2}}{m_{s}}", "derivation": "\\nabla{(t_{2},I,m_{s})} = \\frac{I t_{2}}{m_{s}} and \\frac{\\partial}{\\partial I} \\nabla{(t_{2},I,m_{s})} = \\frac{\\partial}{\\partial I} \\frac{I t_{2}}{m_{s}} and \\frac{\\partial}{\\partial I} \\nabla{(t_{2},I,m_{s})} = \\frac{t_{2}}{m_{s}} and \\frac{\\partial^{2}}{\\partial t_{2}\\partial I} \\nabla{(t_{2},I,m_{s})} = \\frac{\\partial}{\\partial t_{2}} \\frac{t_{2}}{m_{s}} and \\frac{\\partial^{3}}{\\partial t_{2}^{2}\\partial I} \\nabla{(t_{2},I,m_{s})} = \\frac{\\partial^{2}}{\\partial t_{2}^{2}} \\frac{t_{2}}{m_{s}} and t_{2} + \\frac{\\partial^{3}}{\\partial t_{2}^{2}\\partial I} \\nabla{(t_{2},I,m_{s})} = t_{2} + \\frac{\\partial^{2}}{\\partial t_{2}^{2}} \\frac{t_{2}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)), Tuple(Symbol('t_2', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(2))))"], [["add", 5, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Derivative(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)), Tuple(Symbol('t_2', commutative=True), Integer(2)))), Add(Symbol('t_2', commutative=True), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given m{(t)} = e^{t}, then derive \\int m{(t)} dt = I + e^{t}, then derive - \\mathbf{H} + t - e^{t} + \\int m{(t)} dt = I - \\mathbf{H} + t, then obtain - \\mathbf{H} + t - 2 e^{t} + \\int e^{t} dt = I - \\mathbf{H} + t - e^{t}", "derivation": "m{(t)} = e^{t} and \\int m{(t)} dt = \\int e^{t} dt and \\int m{(t)} dt = I + e^{t} and t + \\int m{(t)} dt - \\int e^{t} dt = I + t + e^{t} - \\int e^{t} dt and - \\mathbf{H} + t - e^{t} + \\int m{(t)} dt = I - \\mathbf{H} + t and - \\mathbf{H} + t - 2 e^{t} + \\int m{(t)} dt = I - \\mathbf{H} + t - e^{t} and - \\mathbf{H} + t - 2 e^{t} + \\int e^{t} dt = I - \\mathbf{H} + t - e^{t}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('m')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('I', commutative=True), exp(Symbol('t', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('t', commutative=True)), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], "Equality(Add(Symbol('t', commutative=True), Integral(Function('m')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Symbol('I', commutative=True), Symbol('t', commutative=True), exp(Symbol('t', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), exp(Symbol('t', commutative=True))), Integral(Function('m')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True)))"], [["add", 5, "Mul(Integer(-1), exp(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Integer(2), exp(Symbol('t', commutative=True))), Integral(Function('m')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), exp(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), Integer(2), exp(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('t', commutative=True), Mul(Integer(-1), exp(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\rho{(F_{x},p)} = - F_{x} + p, then obtain - \\rho{(F_{x},p)} - 2 \\int (- F_{x} + p)^{p} dp + 2 \\int \\rho^{p}{(F_{x},p)} dp = - \\rho{(F_{x},p)}", "derivation": "\\rho{(F_{x},p)} = - F_{x} + p and \\rho^{p}{(F_{x},p)} = (- F_{x} + p)^{p} and \\int \\rho^{p}{(F_{x},p)} dp = \\int (- F_{x} + p)^{p} dp and - \\int (- F_{x} + p)^{p} dp + \\int \\rho^{p}{(F_{x},p)} dp = 0 and - 2 \\int (- F_{x} + p)^{p} dp + \\int \\rho^{p}{(F_{x},p)} dp = - \\int (- F_{x} + p)^{p} dp and - 2 \\int (- F_{x} + p)^{p} dp + 2 \\int \\rho^{p}{(F_{x},p)} dp = 0 and F_{x} - p - 2 \\int (- F_{x} + p)^{p} dp + 2 \\int \\rho^{p}{(F_{x},p)} dp = F_{x} - p and - \\rho{(F_{x},p)} - 2 \\int (- F_{x} + p)^{p} dp + 2 \\int \\rho^{p}{(F_{x},p)} dp = - \\rho{(F_{x},p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["minus", 3, "Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Integer(0))"], [["minus", 4, "Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(-1), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Integer(2), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Integer(0))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Integer(2), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Integer(2), Integral(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(Pow(Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Integer(-1), Function('\\\\rho')(Symbol('F_x', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given S{(z)} = e^{z}, then obtain (S{(z)} \\int e^{z} dz + 2 S{(z)} - e^{z})^{z} = (S{(z)} \\int e^{z} dz + e^{z})^{z}", "derivation": "S{(z)} = e^{z} and \\int S{(z)} dz = \\int e^{z} dz and S{(z)} \\int S{(z)} dz = S{(z)} \\int e^{z} dz and S{(z)} \\int S{(z)} dz + S{(z)} = S{(z)} \\int S{(z)} dz + e^{z} and S{(z)} \\int e^{z} dz + S{(z)} = S{(z)} \\int e^{z} dz + e^{z} and S{(z)} \\int e^{z} dz + S{(z)} - e^{z} = S{(z)} \\int e^{z} dz and S{(z)} \\int e^{z} dz + 2 S{(z)} - e^{z} = S{(z)} \\int e^{z} dz + S{(z)} and (S{(z)} \\int e^{z} dz + 2 S{(z)} - e^{z})^{z} = (S{(z)} \\int e^{z} dz + S{(z)})^{z} and (S{(z)} \\int e^{z} dz + 2 S{(z)} - e^{z})^{z} = (S{(z)} \\int e^{z} dz + e^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('S')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 2, "Function('S')(Symbol('z', commutative=True))"], "Equality(Mul(Function('S')(Symbol('z', commutative=True)), Integral(Function('S')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["add", 1, "Mul(Function('S')(Symbol('z', commutative=True)), Integral(Function('S')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], "Equality(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(Function('S')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Function('S')(Symbol('z', commutative=True))), Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(Function('S')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), exp(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Function('S')(Symbol('z', commutative=True))), Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), exp(Symbol('z', commutative=True))))"], [["minus", 5, "exp(Symbol('z', commutative=True))"], "Equality(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Function('S')(Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Integer(2), Function('S')(Symbol('z', commutative=True))), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Function('S')(Symbol('z', commutative=True))))"], [["power", 7, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Integer(2), Function('S')(Symbol('z', commutative=True))), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Function('S')(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Pow(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Integer(2), Function('S')(Symbol('z', commutative=True))), Mul(Integer(-1), exp(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Add(Mul(Function('S')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(f,\\dot{y})} = - \\dot{y} + f, then obtain \\frac{(- f + \\mathbf{r}{(f,\\dot{y})} + 1) \\mathbf{r}{(f,\\dot{y})}}{- \\dot{y} + f} = \\frac{(- f + \\frac{- \\dot{y} + f}{\\mathbf{r}{(f,\\dot{y})}} + \\mathbf{r}{(f,\\dot{y})}) \\mathbf{r}{(f,\\dot{y})}}{- \\dot{y} + f}", "derivation": "\\mathbf{r}{(f,\\dot{y})} = - \\dot{y} + f and 1 = \\frac{- \\dot{y} + f}{\\mathbf{r}{(f,\\dot{y})}} and \\mathbf{r}{(f,\\dot{y})} + 1 = \\frac{- \\dot{y} + f}{\\mathbf{r}{(f,\\dot{y})}} + \\mathbf{r}{(f,\\dot{y})} and - f + \\mathbf{r}{(f,\\dot{y})} + 1 = - f + \\frac{- \\dot{y} + f}{\\mathbf{r}{(f,\\dot{y})}} + \\mathbf{r}{(f,\\dot{y})} and \\frac{(- f + \\mathbf{r}{(f,\\dot{y})} + 1) \\mathbf{r}{(f,\\dot{y})}}{- \\dot{y} + f} = \\frac{(- f + \\frac{- \\dot{y} + f}{\\mathbf{r}{(f,\\dot{y})}} + \\mathbf{r}{(f,\\dot{y})}) \\mathbf{r}{(f,\\dot{y})}}{- \\dot{y} + f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["add", 2, "Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 3, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["divide", 4, "Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('f', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('\\\\mathbf{r}')(Symbol('f', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})} = e^{- P_{g} + g_{\\varepsilon}}, then obtain (\\int (- g_{\\varepsilon} + \\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})}) dP_{g})^{P_{g}} = (\\int (- g_{\\varepsilon} + e^{- P_{g} + g_{\\varepsilon}}) dP_{g})^{P_{g}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})} = e^{- P_{g} + g_{\\varepsilon}} and - g_{\\varepsilon} + \\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})} = - g_{\\varepsilon} + e^{- P_{g} + g_{\\varepsilon}} and \\int (- g_{\\varepsilon} + \\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})}) dP_{g} = \\int (- g_{\\varepsilon} + e^{- P_{g} + g_{\\varepsilon}}) dP_{g} and (\\int (- g_{\\varepsilon} + \\operatorname{f_{\\mathbf{p}}}{(P_{g},g_{\\varepsilon})}) dP_{g})^{P_{g}} = (\\int (- g_{\\varepsilon} + e^{- P_{g} + g_{\\varepsilon}}) dP_{g})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('P_g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('P_g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('P_g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('P_g', commutative=True))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('P_g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(A_{2})} = \\sin{(A_{2})}, then obtain \\frac{\\int \\mathbf{p}{(A_{2})} \\sin{(A_{2})} dA_{2}}{\\mathbf{p}^{2}{(A_{2})}} = \\frac{\\int \\sin^{2}{(A_{2})} dA_{2}}{\\mathbf{p}^{2}{(A_{2})}}", "derivation": "\\mathbf{p}{(A_{2})} = \\sin{(A_{2})} and \\mathbf{p}{(A_{2})} \\sin{(A_{2})} = \\sin^{2}{(A_{2})} and \\int \\mathbf{p}{(A_{2})} \\sin{(A_{2})} dA_{2} = \\int \\sin^{2}{(A_{2})} dA_{2} and \\frac{\\int \\mathbf{p}{(A_{2})} \\sin{(A_{2})} dA_{2}}{\\mathbf{p}{(A_{2})} \\sin{(A_{2})}} = \\frac{\\int \\sin^{2}{(A_{2})} dA_{2}}{\\mathbf{p}{(A_{2})} \\sin{(A_{2})}} and \\frac{\\int \\mathbf{p}{(A_{2})} \\sin{(A_{2})} dA_{2}}{\\mathbf{p}^{2}{(A_{2})}} = \\frac{\\int \\sin^{2}{(A_{2})} dA_{2}}{\\mathbf{p}^{2}{(A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["times", 1, "sin(Symbol('A_2', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Pow(sin(Symbol('A_2', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Integral(Pow(sin(Symbol('A_2', commutative=True)), Integer(2)), Tuple(Symbol('A_2', commutative=True))))"], [["divide", 3, "Mul(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('A_2', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True)))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('A_2', commutative=True)), Integer(-1)), Integral(Pow(sin(Symbol('A_2', commutative=True)), Integer(2)), Tuple(Symbol('A_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), Integer(-2)), Integral(Mul(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True)))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('A_2', commutative=True)), Integer(-2)), Integral(Pow(sin(Symbol('A_2', commutative=True)), Integer(2)), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given A{(M_{E})} = \\log{(M_{E})}, then obtain \\log{(M_{E})}^{M_{E}} + \\int A{(M_{E})} dM_{E} = \\log{(M_{E})}^{M_{E}} + \\int \\log{(M_{E})} dM_{E}", "derivation": "A{(M_{E})} = \\log{(M_{E})} and A^{M_{E}}{(M_{E})} = \\log{(M_{E})}^{M_{E}} and \\int A{(M_{E})} dM_{E} = \\int \\log{(M_{E})} dM_{E} and A^{M_{E}}{(M_{E})} + \\int A{(M_{E})} dM_{E} = A^{M_{E}}{(M_{E})} + \\int \\log{(M_{E})} dM_{E} and \\log{(M_{E})}^{M_{E}} + \\int A{(M_{E})} dM_{E} = \\log{(M_{E})}^{M_{E}} + \\int \\log{(M_{E})} dM_{E}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["power", 1, "Symbol('M_E', commutative=True)"], "Equality(Pow(Function('A')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(log(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('A')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["add", 3, "Pow(Function('A')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], "Equality(Add(Pow(Function('A')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Integral(Function('A')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Pow(Function('A')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(log(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Integral(Function('A')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Pow(log(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given m{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then obtain (m{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}} = (\\cos{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}}", "derivation": "m{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and m^{\\mathbf{s}}{(\\mathbf{s})} = \\cos^{\\mathbf{s}}{(\\mathbf{s})} and m{(\\mathbf{s})} - m^{\\mathbf{s}}{(\\mathbf{s})} = - m^{\\mathbf{s}}{(\\mathbf{s})} + \\cos{(\\mathbf{s})} and m{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})} = \\cos{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})} and (m{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}} = (\\cos{(\\mathbf{s})} - \\cos^{\\mathbf{s}}{(\\mathbf{s})})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Pow(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["power", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Function('m')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(cos(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(k)} = \\cos{(k)}, then obtain \\int (\\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} + 1) \\cos{(k)} dk = \\int 2 \\cos{(k)} dk", "derivation": "\\hat{H}_{\\lambda}{(k)} = \\cos{(k)} and \\hat{H}_{\\lambda}{(k)} + \\cos{(k)} = 2 \\cos{(k)} and \\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} = 1 and \\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} + 1 = 2 and \\hat{H}_{\\lambda}{(k)} + \\cos{(k)} = (\\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} + 1) \\cos{(k)} and (\\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} + 1) \\cos{(k)} = 2 \\cos{(k)} and \\int (\\frac{\\hat{H}_{\\lambda}{(k)}}{\\cos{(k)}} + 1) \\cos{(k)} dk = \\int 2 \\cos{(k)} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["add", 1, "cos(Symbol('k', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Mul(Integer(2), cos(Symbol('k', commutative=True))))"], [["divide", 1, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Mul(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1)), cos(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1)), cos(Symbol('k', commutative=True))), Mul(Integer(2), cos(Symbol('k', commutative=True))))"], [["integrate", 6, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1)), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\Psi,\\Omega)} = \\frac{\\log{(\\Psi)}}{\\Omega}, then derive \\frac{\\int \\lambda{(\\Psi,\\Omega)} d\\Psi}{\\Omega} = \\frac{W + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega}, then obtain \\frac{\\hat{H}_{\\lambda} + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega} = \\frac{W + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega}", "derivation": "\\lambda{(\\Psi,\\Omega)} = \\frac{\\log{(\\Psi)}}{\\Omega} and \\int \\lambda{(\\Psi,\\Omega)} d\\Psi = \\int \\frac{\\log{(\\Psi)}}{\\Omega} d\\Psi and \\frac{\\int \\lambda{(\\Psi,\\Omega)} d\\Psi}{\\Omega} = \\frac{\\int \\frac{\\log{(\\Psi)}}{\\Omega} d\\Psi}{\\Omega} and \\frac{\\int \\lambda{(\\Psi,\\Omega)} d\\Psi}{\\Omega} = \\frac{W + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega} and \\frac{\\int \\frac{\\log{(\\Psi)}}{\\Omega} d\\Psi}{\\Omega} = \\frac{W + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega} and \\frac{\\hat{H}_{\\lambda} + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega} = \\frac{W + \\frac{\\Psi \\log{(\\Psi)}}{\\Omega} - \\frac{\\Psi}{\\Omega}}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given c{(\\hat{H},A_{2})} = A_{2} \\hat{H}, then obtain \\hat{H} + \\frac{\\partial}{\\partial A_{2}} c{(\\hat{H},A_{2})} - 1 = 2 \\hat{H} - 1", "derivation": "c{(\\hat{H},A_{2})} = A_{2} \\hat{H} and - A_{2} + c{(\\hat{H},A_{2})} = A_{2} \\hat{H} - A_{2} and A_{2} \\hat{H} - A_{2} + c{(\\hat{H},A_{2})} = 2 A_{2} \\hat{H} - A_{2} and \\frac{\\partial}{\\partial A_{2}} (A_{2} \\hat{H} - A_{2} + c{(\\hat{H},A_{2})}) = \\frac{\\partial}{\\partial A_{2}} (2 A_{2} \\hat{H} - A_{2}) and \\hat{H} + \\frac{\\partial}{\\partial A_{2}} c{(\\hat{H},A_{2})} - 1 = 2 \\hat{H} - 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))))"], [["add", 2, "Mul(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(2), Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(A)} = e^{A}, then derive \\chi \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA = \\chi (\\hat{H} + e^{A}), then obtain \\cos{(\\chi \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA)} = \\cos{(\\chi (\\hat{H} + e^{A}))}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(A)} = e^{A} and \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA = \\int e^{A} dA and \\chi \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA = \\chi \\int e^{A} dA and \\chi \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA = \\chi (\\hat{H} + e^{A}) and \\cos{(\\chi \\int \\operatorname{V_{\\mathbf{E}}}{(A)} dA)} = \\cos{(\\chi (\\hat{H} + e^{A}))}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('A', commutative=True)))))"], [["cos", 4], "Equality(cos(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('V_{\\\\mathbf{E}}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))), cos(Mul(Symbol('\\\\chi', commutative=True), Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('A', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(Z,p)} = (e^{p})^{Z} and \\hat{p}_0{(p)} = e^{p}, then obtain \\hat{p}_0^{Z}{(p)} = (e^{p})^{Z}", "derivation": "\\tilde{g}^*{(Z,p)} = (e^{p})^{Z} and \\hat{p}_0{(p)} = e^{p} and \\tilde{g}^*{(Z,p)} = \\hat{p}_0^{Z}{(p)} and \\hat{p}_0^{Z}{(p)} = (e^{p})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('Z', commutative=True), Symbol('p', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('p', commutative=True)), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('p', commutative=True)), Symbol('Z', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(S,T)} = S + T, then derive \\frac{\\partial}{\\partial T} \\phi_{2}{(S,T)} + 1 = 2, then obtain S + T (\\frac{\\partial}{\\partial T} \\phi_{2}{(S,T)} + 1) = S + 2 T", "derivation": "\\phi_{2}{(S,T)} = S + T and T + \\phi_{2}{(S,T)} = S + 2 T and \\frac{\\partial}{\\partial T} (T + \\phi_{2}{(S,T)}) = \\frac{\\partial}{\\partial T} (S + 2 T) and \\frac{\\partial}{\\partial T} \\phi_{2}{(S,T)} + 1 = 2 and T + \\phi_{2}{(S,T)} = S + T (\\frac{\\partial}{\\partial T} \\phi_{2}{(S,T)} + 1) and S + T (\\frac{\\partial}{\\partial T} \\phi_{2}{(S,T)} + 1) = S + 2 T", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True)), Add(Symbol('S', commutative=True), Symbol('T', commutative=True)))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True))), Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('T', commutative=True))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Symbol('T', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Symbol('T', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('T', commutative=True), Add(Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Symbol('S', commutative=True), Mul(Symbol('T', commutative=True), Add(Derivative(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1)))), Add(Symbol('S', commutative=True), Mul(Integer(2), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(M,v,t)} = - M + \\frac{v}{t} and \\hat{x}_0{(M)} = - M, then obtain (\\frac{\\partial}{\\partial M} \\frac{\\sigma_{p}{(M,v,t)}}{t})^{M} = (\\frac{\\partial}{\\partial M} \\frac{\\hat{x}_0{(M)} + \\frac{v}{t}}{t})^{M}", "derivation": "\\sigma_{p}{(M,v,t)} = - M + \\frac{v}{t} and \\frac{\\sigma_{p}{(M,v,t)}}{t} = \\frac{- M + \\frac{v}{t}}{t} and \\frac{\\partial}{\\partial M} \\frac{\\sigma_{p}{(M,v,t)}}{t} = \\frac{\\partial}{\\partial M} \\frac{- M + \\frac{v}{t}}{t} and (\\frac{\\partial}{\\partial M} \\frac{\\sigma_{p}{(M,v,t)}}{t})^{M} = (\\frac{\\partial}{\\partial M} \\frac{- M + \\frac{v}{t}}{t})^{M} and \\hat{x}_0{(M)} = - M and (\\frac{\\partial}{\\partial M} \\frac{\\sigma_{p}{(M,v,t)}}{t})^{M} = (\\frac{\\partial}{\\partial M} \\frac{\\hat{x}_0{(M)} + \\frac{v}{t}}{t})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('M', commutative=True), Symbol('v', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"], [["divide", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('M', commutative=True), Symbol('v', commutative=True), Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('M', commutative=True), Symbol('v', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('M', commutative=True), Symbol('v', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('M', commutative=True), Symbol('v', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Function('\\\\hat{x}_0')(Symbol('M', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('v', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(n,u)} = e^{- n + u}, then obtain - \\frac{(\\frac{\\int \\cos{(\\phi_{2}{(n,u)})} du}{2 u})^{n}}{2 n} = - \\frac{(\\frac{r + \\operatorname{Ci}{(e^{- n} e^{u})}}{2 u})^{n}}{2 n}", "derivation": "\\phi_{2}{(n,u)} = e^{- n + u} and \\cos{(\\phi_{2}{(n,u)})} = \\cos{(e^{- n + u})} and \\int \\cos{(\\phi_{2}{(n,u)})} du = \\int \\cos{(e^{- n + u})} du and \\frac{\\int \\cos{(\\phi_{2}{(n,u)})} du}{2 u} = \\frac{\\int \\cos{(e^{- n + u})} du}{2 u} and (\\frac{\\int \\cos{(\\phi_{2}{(n,u)})} du}{2 u})^{n} = (\\frac{\\int \\cos{(e^{- n + u})} du}{2 u})^{n} and - \\frac{(\\frac{\\int \\cos{(\\phi_{2}{(n,u)})} du}{2 u})^{n}}{2 n} = - \\frac{(\\frac{\\int \\cos{(e^{- n + u})} du}{2 u})^{n}}{2 n} and - \\frac{(\\frac{\\int \\cos{(\\phi_{2}{(n,u)})} du}{2 u})^{n}}{2 n} = - \\frac{(\\frac{r + \\operatorname{Ci}{(e^{- n} e^{u})}}{2 u})^{n}}{2 n}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), cos(exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(cos(exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Symbol('u', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))))"], [["power", 4, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Symbol('n', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), Symbol('n', commutative=True)))"], [["divide", 5, "Mul(Integer(-1), Integer(2), Symbol('n', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Symbol('n', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(exp(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True)))), Symbol('n', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(cos(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Symbol('n', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(-1)), Pow(Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-1)), Add(Symbol('r', commutative=True), Ci(Mul(exp(Mul(Integer(-1), Symbol('n', commutative=True))), exp(Symbol('u', commutative=True)))))), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\delta{(\\hat{H}_l)} = \\hat{H}_l, then obtain \\hat{H}_l^{\\hat{H}_l} + (\\delta^{\\hat{H}_l}{(\\hat{H}_l)})^{\\hat{H}_l} = \\hat{H}_l^{\\hat{H}_l} + (\\hat{H}_l^{\\hat{H}_l})^{\\hat{H}_l}", "derivation": "\\delta{(\\hat{H}_l)} = \\hat{H}_l and \\delta^{\\hat{H}_l}{(\\hat{H}_l)} = \\hat{H}_l^{\\hat{H}_l} and (\\delta^{\\hat{H}_l}{(\\hat{H}_l)})^{\\hat{H}_l} = (\\hat{H}_l^{\\hat{H}_l})^{\\hat{H}_l} and \\hat{H}_l^{\\hat{H}_l} + (\\delta^{\\hat{H}_l}{(\\hat{H}_l)})^{\\hat{H}_l} = \\hat{H}_l^{\\hat{H}_l} + (\\hat{H}_l^{\\hat{H}_l})^{\\hat{H}_l}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Pow(Function('\\\\delta')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 3, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Function('\\\\delta')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Add(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(L)} = \\log{(\\log{(L)})}, then obtain \\mathbf{p}{(L)} \\log{(L)} \\frac{d}{d L} \\mathbf{p}{(L)} = \\log{(L)} \\log{(\\log{(L)})} \\frac{d}{d L} \\mathbf{p}{(L)}", "derivation": "\\mathbf{p}{(L)} = \\log{(\\log{(L)})} and \\frac{d}{d L} \\mathbf{p}{(L)} = \\frac{d}{d L} \\log{(\\log{(L)})} and \\mathbf{p}{(L)} \\log{(L)} = \\log{(L)} \\log{(\\log{(L)})} and \\mathbf{p}{(L)} \\log{(L)} \\frac{d}{d L} \\log{(\\log{(L)})} = \\log{(L)} \\log{(\\log{(L)})} \\frac{d}{d L} \\log{(\\log{(L)})} and \\mathbf{p}{(L)} \\log{(L)} \\frac{d}{d L} \\mathbf{p}{(L)} = \\log{(L)} \\log{(\\log{(L)})} \\frac{d}{d L} \\mathbf{p}{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["times", 1, "log(Symbol('L', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(log(Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True)))))"], [["times", 3, "Derivative(log(log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Derivative(log(log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(log(Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True))), Derivative(log(log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(log(Symbol('L', commutative=True)), log(log(Symbol('L', commutative=True))), Derivative(Function('\\\\mathbf{p}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(\\sigma_x,\\mathbf{S},\\Omega)} = \\frac{\\Omega - \\sigma_x}{\\mathbf{S}}, then obtain \\mathbf{S} (\\Omega a{(\\sigma_x,\\mathbf{S},\\Omega)} + \\sigma_x) = \\mathbf{S} (\\frac{\\Omega (\\Omega - \\sigma_x)}{\\mathbf{S}} + \\sigma_x)", "derivation": "a{(\\sigma_x,\\mathbf{S},\\Omega)} = \\frac{\\Omega - \\sigma_x}{\\mathbf{S}} and \\Omega a{(\\sigma_x,\\mathbf{S},\\Omega)} = \\frac{\\Omega (\\Omega - \\sigma_x)}{\\mathbf{S}} and \\Omega a{(\\sigma_x,\\mathbf{S},\\Omega)} + \\sigma_x = \\frac{\\Omega (\\Omega - \\sigma_x)}{\\mathbf{S}} + \\sigma_x and \\mathbf{S} (\\Omega a{(\\sigma_x,\\mathbf{S},\\Omega)} + \\sigma_x) = \\mathbf{S} (\\frac{\\Omega (\\Omega - \\sigma_x)}{\\mathbf{S}} + \\sigma_x)", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('a')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Function('a')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Function('a')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\theta{(f,\\mathbf{H})} = \\mathbf{H} + f and \\psi{(f,\\mathbf{H})} = (\\mathbf{H} + f)^{2}, then obtain \\frac{\\theta^{2}{(f,\\mathbf{H})}}{f} = \\frac{\\psi{(f,\\mathbf{H})}}{f}", "derivation": "\\theta{(f,\\mathbf{H})} = \\mathbf{H} + f and \\frac{\\theta{(f,\\mathbf{H})}}{f} = \\frac{\\mathbf{H} + f}{f} and \\frac{\\theta^{2}{(f,\\mathbf{H})}}{f} = \\frac{(\\mathbf{H} + f) \\theta{(f,\\mathbf{H})}}{f} and \\frac{(\\mathbf{H} + f) \\theta{(f,\\mathbf{H})}}{f} = \\frac{(\\mathbf{H} + f)^{2}}{f} and \\psi{(f,\\mathbf{H})} = (\\mathbf{H} + f)^{2} and \\frac{(\\mathbf{H} + f) \\theta{(f,\\mathbf{H})}}{f} = \\frac{\\psi{(f,\\mathbf{H})}}{f} and \\frac{\\theta^{2}{(f,\\mathbf{H})}}{f} = \\frac{\\psi{(f,\\mathbf{H})}}{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True))))"], [["times", 2, "Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)), Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)), Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f', commutative=True)), Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('\\\\theta')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('f', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})} = (e^{P_{g}})^{\\hat{H}}, then derive (\\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}} = (\\hat{H} (e^{P_{g}})^{\\hat{H}})^{\\hat{H}}, then obtain (\\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}} = (\\hat{H} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})} = (e^{P_{g}})^{\\hat{H}} and \\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})} = \\frac{\\partial}{\\partial P_{g}} (e^{P_{g}})^{\\hat{H}} and (\\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}} = (\\frac{\\partial}{\\partial P_{g}} (e^{P_{g}})^{\\hat{H}})^{\\hat{H}} and (\\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}} = (\\hat{H} (e^{P_{g}})^{\\hat{H}})^{\\hat{H}} and (\\frac{\\partial}{\\partial P_{g}} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}} = (\\hat{H} \\operatorname{a^{\\dagger}}{(\\hat{H},P_{g})})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('P_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)), Pow(Derivative(Pow(exp(Symbol('P_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Pow(exp(Symbol('P_g', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\sigma_x)} = e^{e^{\\sigma_x}}, then derive \\frac{d}{d \\sigma_x} \\operatorname{L_{\\varepsilon}}{(\\sigma_x)} = e^{\\sigma_x} e^{e^{\\sigma_x}}, then obtain \\frac{d}{d \\sigma_x} \\cos{(\\frac{d}{d \\sigma_x} e^{e^{\\sigma_x}})} = \\frac{d}{d \\sigma_x} \\cos{(e^{\\sigma_x} e^{e^{\\sigma_x}})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\sigma_x)} = e^{e^{\\sigma_x}} and \\frac{d}{d \\sigma_x} \\operatorname{L_{\\varepsilon}}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} e^{e^{\\sigma_x}} and \\frac{d}{d \\sigma_x} \\operatorname{L_{\\varepsilon}}{(\\sigma_x)} = e^{\\sigma_x} e^{e^{\\sigma_x}} and \\frac{d}{d \\sigma_x} e^{e^{\\sigma_x}} = e^{\\sigma_x} e^{e^{\\sigma_x}} and \\cos{(\\frac{d}{d \\sigma_x} e^{e^{\\sigma_x}})} = \\cos{(e^{\\sigma_x} e^{e^{\\sigma_x}})} and \\frac{d}{d \\sigma_x} \\cos{(\\frac{d}{d \\sigma_x} e^{e^{\\sigma_x}})} = \\frac{d}{d \\sigma_x} \\cos{(e^{\\sigma_x} e^{e^{\\sigma_x}})}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\sigma_x', commutative=True)), exp(exp(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\sigma_x', commutative=True)), exp(exp(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\sigma_x', commutative=True)), exp(exp(Symbol('\\\\sigma_x', commutative=True)))))"], [["cos", 4], "Equality(cos(Derivative(exp(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), cos(Mul(exp(Symbol('\\\\sigma_x', commutative=True)), exp(exp(Symbol('\\\\sigma_x', commutative=True))))))"], [["differentiate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(cos(Derivative(exp(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(cos(Mul(exp(Symbol('\\\\sigma_x', commutative=True)), exp(exp(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\rho_f)} = \\sin{(\\rho_f)} and \\operatorname{C_{2}}{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain \\rho_f + 1 = \\rho_f + \\frac{\\operatorname{C_{2}}{(\\rho_f)}}{\\sin{(\\rho_f)}}", "derivation": "\\rho_{b}{(\\rho_f)} = \\sin{(\\rho_f)} and \\operatorname{C_{2}}{(\\rho_f)} = \\sin{(\\rho_f)} and \\rho_f + \\operatorname{C_{2}}{(\\rho_f)} = \\rho_f + \\sin{(\\rho_f)} and 1 = \\frac{\\sin{(\\rho_f)}}{\\rho_{b}{(\\rho_f)}} and 1 = \\frac{\\operatorname{C_{2}}{(\\rho_f)}}{\\rho_{b}{(\\rho_f)}} and 1 = \\frac{\\operatorname{C_{2}}{(\\rho_f)}}{\\sin{(\\rho_f)}} and \\rho_f + \\operatorname{C_{2}}{(\\rho_f)} - \\sin{(\\rho_f)} + 1 = \\rho_f + \\operatorname{C_{2}}{(\\rho_f)} + \\frac{\\operatorname{C_{2}}{(\\rho_f)}}{\\sin{(\\rho_f)}} - \\sin{(\\rho_f)} and \\rho_f + 1 = \\rho_f + \\frac{\\operatorname{C_{2}}{(\\rho_f)}}{\\sin{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["add", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Function('C_2')(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 1, "Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Mul(Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Mul(Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"], [["minus", 6, "Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('\\\\rho_f', commutative=True))), sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\rho_f', commutative=True))), Integer(1)), Add(Symbol('\\\\rho_f', commutative=True), Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Mul(Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Integer(1)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Function('C_2')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\dot{y}{(\\hat{X},\\Psi_{nl})} = - \\Psi_{nl} + \\hat{X}, then obtain (\\frac{\\dot{y}{(\\hat{X},\\Psi_{nl})}}{- \\Psi_{nl} + \\hat{X}})^{\\hat{X}} = 1", "derivation": "\\dot{y}{(\\hat{X},\\Psi_{nl})} = - \\Psi_{nl} + \\hat{X} and \\frac{\\dot{y}{(\\hat{X},\\Psi_{nl})}}{\\Psi_{nl}} = \\frac{- \\Psi_{nl} + \\hat{X}}{\\Psi_{nl}} and \\frac{\\dot{y}{(\\hat{X},\\Psi_{nl})}}{- \\Psi_{nl} + \\hat{X}} = 1 and (\\frac{\\dot{y}{(\\hat{X},\\Psi_{nl})}}{- \\Psi_{nl} + \\hat{X}})^{\\hat{X}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\phi{(\\sigma_x)} = \\log{(\\sigma_x)}, then derive - B - \\sigma_x + \\phi{(\\sigma_x)} = - B - \\sigma_x + \\log{(\\sigma_x)}, then obtain B + \\sigma_x - \\phi{(\\sigma_x)} = B + \\sigma_x - \\log{(\\sigma_x)}", "derivation": "\\phi{(\\sigma_x)} = \\log{(\\sigma_x)} and \\phi{(\\sigma_x)} - \\int 1 d\\sigma_x = \\log{(\\sigma_x)} - \\int 1 d\\sigma_x and - B - \\sigma_x + \\phi{(\\sigma_x)} = - B - \\sigma_x + \\log{(\\sigma_x)} and B + \\sigma_x - \\phi{(\\sigma_x)} = B + \\sigma_x - \\log{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True))))), Add(log(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\sigma_x', commutative=True))))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)))), Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given I{(C,L_{\\varepsilon})} = C L_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial C} (- C L_{\\varepsilon} + C I{(C,L_{\\varepsilon})} + C + I{(C,L_{\\varepsilon})}) = \\frac{\\partial}{\\partial C} (C^{2} L_{\\varepsilon} + C)", "derivation": "I{(C,L_{\\varepsilon})} = C L_{\\varepsilon} and C I{(C,L_{\\varepsilon})} = C^{2} L_{\\varepsilon} and C I{(C,L_{\\varepsilon})} + C = C^{2} L_{\\varepsilon} + C and - C L_{\\varepsilon} + I{(C,L_{\\varepsilon})} = 0 and \\frac{\\partial}{\\partial C} (C I{(C,L_{\\varepsilon})} + C) = \\frac{\\partial}{\\partial C} (C^{2} L_{\\varepsilon} + C) and - C L_{\\varepsilon} + C I{(C,L_{\\varepsilon})} + I{(C,L_{\\varepsilon})} = C I{(C,L_{\\varepsilon})} and \\frac{\\partial}{\\partial C} (- C L_{\\varepsilon} + C I{(C,L_{\\varepsilon})} + C + I{(C,L_{\\varepsilon})}) = \\frac{\\partial}{\\partial C} (C^{2} L_{\\varepsilon} + C)", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('C', commutative=True)), Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('C', commutative=True)))"], [["minus", 1, "Mul(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 4, "Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('C', commutative=True), Function('I')(Symbol('C', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\psi^*,f)} = \\cos{(\\psi^* f)}, then obtain \\tilde{g}^*^{5}{(\\psi^*,f)} \\cos{(\\psi^* f)} = \\tilde{g}^*^{4}{(\\psi^*,f)} \\cos^{2}{(\\psi^* f)}", "derivation": "\\tilde{g}^*{(\\psi^*,f)} = \\cos{(\\psi^* f)} and \\tilde{g}^*^{2}{(\\psi^*,f)} = \\tilde{g}^*{(\\psi^*,f)} \\cos{(\\psi^* f)} and \\tilde{g}^*^{3}{(\\psi^*,f)} = \\tilde{g}^*^{2}{(\\psi^*,f)} \\cos{(\\psi^* f)} and \\tilde{g}^*^{3}{(\\psi^*,f)} = \\tilde{g}^*{(\\psi^*,f)} \\cos^{2}{(\\psi^* f)} and \\tilde{g}^*^{2}{(\\psi^*,f)} \\cos{(\\psi^* f)} = \\tilde{g}^*{(\\psi^*,f)} \\cos^{2}{(\\psi^* f)} and \\tilde{g}^*^{5}{(\\psi^*,f)} \\cos{(\\psi^* f)} = \\tilde{g}^*^{4}{(\\psi^*,f)} \\cos^{2}{(\\psi^* f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))))"], [["times", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)))))"], [["times", 2, "Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(2)), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(3)), Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Pow(cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(2)), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)))), Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Pow(cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))), Integer(2))))"], [["times", 5, "Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(3))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(5)), cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integer(4)), Pow(cos(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(f^{\\prime})} = \\cos{(\\sin{(f^{\\prime})})}, then obtain - \\operatorname{C_{1}}{(f^{\\prime})} - \\sin{(f^{\\prime})} = - \\operatorname{C_{1}}{(f^{\\prime})} - 2 \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} + 2 \\cos{(\\sin{(f^{\\prime})})}", "derivation": "\\operatorname{v_{y}}{(f^{\\prime})} = \\cos{(\\sin{(f^{\\prime})})} and 0 = - \\operatorname{v_{y}}{(f^{\\prime})} + \\cos{(\\sin{(f^{\\prime})})} and - \\sin{(f^{\\prime})} = - \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} + \\cos{(\\sin{(f^{\\prime})})} and - \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} = - 2 \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} + \\cos{(\\sin{(f^{\\prime})})} and - \\sin{(f^{\\prime})} = - 2 \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} + 2 \\cos{(\\sin{(f^{\\prime})})} and - \\operatorname{C_{1}}{(f^{\\prime})} - \\sin{(f^{\\prime})} = - \\operatorname{C_{1}}{(f^{\\prime})} - 2 \\operatorname{v_{y}}{(f^{\\prime})} - \\sin{(f^{\\prime})} + 2 \\cos{(\\sin{(f^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), cos(sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), cos(sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 2, "sin(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), cos(sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 3, "Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), cos(sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), cos(sin(Symbol('f^{\\\\prime}', commutative=True))))))"], [["minus", 5, "Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Integer(2), Function('v_y')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(2), cos(sin(Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(z^{*},q)} = z^{*} \\log{(q)}, then derive \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)} = \\log{(q)}, then obtain \\tilde{g}^* (- \\log{(q)} + \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)}) = 0", "derivation": "\\dot{y}{(z^{*},q)} = z^{*} \\log{(q)} and \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)} = \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)} and \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)} = \\log{(q)} and \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)} = \\log{(q)} and - \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)} + \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)} = \\log{(q)} - \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)} and \\tilde{g}^* (- \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)} + \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)}) = \\tilde{g}^* (\\log{(q)} - \\frac{\\partial}{\\partial z^{*}} z^{*} \\log{(q)}) and \\tilde{g}^* (- \\log{(q)} + \\frac{\\partial}{\\partial z^{*}} \\dot{y}{(z^{*},q)}) = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), log(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), log(Symbol('q', commutative=True)))"], [["minus", 3, "Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Derivative(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(log(Symbol('q', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))))"], [["times", 5, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Derivative(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Add(log(Symbol('q', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('z^*', commutative=True), log(Symbol('q', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Derivative(Function('\\\\dot{y}')(Symbol('z^*', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\omega{(m,\\mathbf{f})} = - m + e^{\\mathbf{f}} and \\operatorname{J_{\\varepsilon}}{(m)} = - m, then obtain \\int (\\omega{(m,\\mathbf{f})} - 1) d\\mathbf{f} = \\int (\\operatorname{J_{\\varepsilon}}{(m)} + e^{\\mathbf{f}} - 1) d\\mathbf{f}", "derivation": "\\omega{(m,\\mathbf{f})} = - m + e^{\\mathbf{f}} and \\operatorname{J_{\\varepsilon}}{(m)} = - m and \\omega{(m,\\mathbf{f})} = \\operatorname{J_{\\varepsilon}}{(m)} + e^{\\mathbf{f}} and \\omega{(m,\\mathbf{f})} - \\frac{\\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)}}{\\frac{d}{d m} - m} = \\operatorname{J_{\\varepsilon}}{(m)} + e^{\\mathbf{f}} - \\frac{\\frac{d}{d m} \\operatorname{J_{\\varepsilon}}{(m)}}{\\frac{d}{d m} - m} and \\omega{(m,\\mathbf{f})} - 1 = - m + e^{\\mathbf{f}} - 1 and \\int (\\omega{(m,\\mathbf{f})} - 1) d\\mathbf{f} = \\int (- m + e^{\\mathbf{f}} - 1) d\\mathbf{f} and \\int (\\omega{(m,\\mathbf{f})} - 1) d\\mathbf{f} = \\int (\\operatorname{J_{\\varepsilon}}{(m)} + e^{\\mathbf{f}} - 1) d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 3, "Mul(Pow(Derivative(Mul(Integer(-1), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Pow(Derivative(Mul(Integer(-1), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))), Add(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Pow(Derivative(Mul(Integer(-1), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integral(Add(Function('\\\\omega')(Symbol('m', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Add(Function('J_{\\\\varepsilon}')(Symbol('m', commutative=True)), exp(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given g{(J)} = \\sin{(\\cos{(J)})}, then obtain \\frac{d}{d J} g{(J)} g^{- J}{(J)} \\sin^{J}{(\\cos{(J)})} = \\frac{d}{d J} g^{- J}{(J)} \\sin{(\\cos{(J)})} \\sin^{J}{(\\cos{(J)})}", "derivation": "g{(J)} = \\sin{(\\cos{(J)})} and g^{J}{(J)} = \\sin^{J}{(\\cos{(J)})} and g{(J)} g^{J}{(J)} = g^{J}{(J)} \\sin{(\\cos{(J)})} and g{(J)} \\sin^{J}{(\\cos{(J)})} = \\sin{(\\cos{(J)})} \\sin^{J}{(\\cos{(J)})} and g{(J)} g^{- J}{(J)} \\sin^{J}{(\\cos{(J)})} = g^{- J}{(J)} \\sin{(\\cos{(J)})} \\sin^{J}{(\\cos{(J)})} and \\frac{d}{d J} g{(J)} g^{- J}{(J)} \\sin^{J}{(\\cos{(J)})} = \\frac{d}{d J} g^{- J}{(J)} \\sin{(\\cos{(J)})} \\sin^{J}{(\\cos{(J)})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('J', commutative=True)), sin(cos(Symbol('J', commutative=True))))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('g')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["times", 1, "Pow(Function('g')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Function('g')(Symbol('J', commutative=True)), Pow(Function('g')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Pow(Function('g')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), sin(cos(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('g')(Symbol('J', commutative=True)), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(sin(cos(Symbol('J', commutative=True))), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["divide", 4, "Pow(Function('g')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Function('g')(Symbol('J', commutative=True)), Pow(Function('g')(Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Mul(Pow(Function('g')(Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), sin(cos(Symbol('J', commutative=True))), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Function('g')(Symbol('J', commutative=True)), Pow(Function('g')(Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('g')(Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True))), sin(cos(Symbol('J', commutative=True))), Pow(sin(cos(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(c,\\mathbf{J}_f)} = c^{\\mathbf{J}_f}, then obtain (1 - s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f} = (0^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f}", "derivation": "s{(c,\\mathbf{J}_f)} = c^{\\mathbf{J}_f} and - c^{\\mathbf{J}_f} + s{(c,\\mathbf{J}_f)} = 0 and (- c^{\\mathbf{J}_f} + s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f} = 0^{\\mathbf{J}_f} and (- c^{\\mathbf{J}_f} + s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)} = 0^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)} and 1 - s{(c,\\mathbf{J}_f)} = (- c^{\\mathbf{J}_f} + s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)} and 1 - s{(c,\\mathbf{J}_f)} = 0^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)} and (1 - s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f} = (0^{\\mathbf{J}_f} - s{(c,\\mathbf{J}_f)})^{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 1, "Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 3, "Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["power", 6, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Add(Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('c', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\rho_b)} = \\log{(\\rho_b)}, then derive (\\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b)^{\\rho_b} = (\\mathbf{D} + \\rho_b \\log{(\\rho_b)} - \\rho_b)^{\\rho_b}, then obtain (\\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b)^{\\rho_b} = (\\mathbf{D} + \\rho_b \\operatorname{a^{\\dagger}}{(\\rho_b)} - \\rho_b)^{\\rho_b}", "derivation": "\\operatorname{a^{\\dagger}}{(\\rho_b)} = \\log{(\\rho_b)} and \\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b = \\int \\log{(\\rho_b)} d\\rho_b and (\\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b)^{\\rho_b} = (\\int \\log{(\\rho_b)} d\\rho_b)^{\\rho_b} and (\\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b)^{\\rho_b} = (\\mathbf{D} + \\rho_b \\log{(\\rho_b)} - \\rho_b)^{\\rho_b} and (\\int \\operatorname{a^{\\dagger}}{(\\rho_b)} d\\rho_b)^{\\rho_b} = (\\mathbf{D} + \\rho_b \\operatorname{a^{\\dagger}}{(\\rho_b)} - \\rho_b)^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(log(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(log(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('\\\\rho_b', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(h,E_{x})} = E_{x}^{h}, then obtain \\operatorname{C_{d}}^{4}{(h,E_{x})} = E_{x}^{3 h} \\operatorname{C_{d}}{(h,E_{x})}", "derivation": "\\operatorname{C_{d}}{(h,E_{x})} = E_{x}^{h} and \\operatorname{C_{d}}^{2}{(h,E_{x})} = E_{x}^{h} \\operatorname{C_{d}}{(h,E_{x})} and \\operatorname{C_{d}}^{4}{(h,E_{x})} = E_{x}^{2 h} \\operatorname{C_{d}}^{2}{(h,E_{x})} and E_{x}^{2 h} \\operatorname{C_{d}}^{2}{(h,E_{x})} = E_{x}^{3 h} \\operatorname{C_{d}}{(h,E_{x})} and \\operatorname{C_{d}}^{4}{(h,E_{x})} = E_{x}^{3 h} \\operatorname{C_{d}}{(h,E_{x})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)))"], [["times", 1, "Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Pow(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Symbol('h', commutative=True)), Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Integer(4)), Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))), Pow(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(2), Symbol('h', commutative=True))), Pow(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(3), Symbol('h', commutative=True))), Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True)), Integer(4)), Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(3), Symbol('h', commutative=True))), Function('C_d')(Symbol('h', commutative=True), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given Z{(E_{n},L)} = \\log{(E_{n} + L)}, then obtain - 2 E_{n} - 2 L + 2 (E_{n} + L) Z{(E_{n},L)} - \\log{(E_{n} + L)} = - 2 E_{n} - 2 L + (E_{n} + L) (Z{(E_{n},L)} + \\log{(E_{n} + L)}) - \\log{(E_{n} + L)}", "derivation": "Z{(E_{n},L)} = \\log{(E_{n} + L)} and 2 Z{(E_{n},L)} = Z{(E_{n},L)} + \\log{(E_{n} + L)} and 2 (E_{n} + L) Z{(E_{n},L)} = (E_{n} + L) (Z{(E_{n},L)} + \\log{(E_{n} + L)}) and - E_{n} - L + 2 (E_{n} + L) Z{(E_{n},L)} - \\log{(E_{n} + L)} = - E_{n} - L + (E_{n} + L) (Z{(E_{n},L)} + \\log{(E_{n} + L)}) - \\log{(E_{n} + L)} and - 2 E_{n} - 2 L + 2 (E_{n} + L) Z{(E_{n},L)} - \\log{(E_{n} + L)} = - 2 E_{n} - 2 L + (E_{n} + L) (Z{(E_{n},L)} + \\log{(E_{n} + L)}) - \\log{(E_{n} + L)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))"], [["add", 1, "Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Add(Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)))))"], [["times", 2, "Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Mul(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Add(Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))))"], [["minus", 3, "Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Add(Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))), Mul(Integer(-1), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))))"], [["minus", 4, "Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('L', commutative=True)), Mul(Integer(2), Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('L', commutative=True)), Mul(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Add(Function('Z')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))), Mul(Integer(-1), log(Add(Symbol('E_n', commutative=True), Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}}, then derive 0 = e^{\\sin{(a^{\\dagger})}} \\cos{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})}, then obtain \\int \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} da^{\\dagger} = \\int e^{\\sin{(a^{\\dagger})}} \\cos{(a^{\\dagger})} da^{\\dagger}", "derivation": "\\phi_{2}{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}} and \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{\\sin{(a^{\\dagger})}} and 0 = - \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} e^{\\sin{(a^{\\dagger})}} and 0 = e^{\\sin{(a^{\\dagger})}} \\cos{(a^{\\dagger})} - \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} = e^{\\sin{(a^{\\dagger})}} \\cos{(a^{\\dagger})} and \\int \\frac{d}{d a^{\\dagger}} \\phi_{2}{(a^{\\dagger})} da^{\\dagger} = \\int e^{\\sin{(a^{\\dagger})}} \\cos{(a^{\\dagger})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), exp(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Derivative(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), cos(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))))"], [["minus", 4, "Mul(Integer(-1), Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(exp(sin(Symbol('a^{\\\\dagger}', commutative=True))), cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(W)} = e^{e^{W}} and \\operatorname{f_{E}}{(\\hat{X},v_{y})} = \\log{(\\hat{X}^{v_{y}})}, then obtain \\int \\frac{\\varepsilon_{0}{(W)} \\log{(\\hat{X}^{v_{y}})}}{\\operatorname{f_{E}}{(\\hat{X},v_{y})}} dW = \\int e^{e^{W}} dW", "derivation": "\\varepsilon_{0}{(W)} = e^{e^{W}} and \\operatorname{f_{E}}{(\\hat{X},v_{y})} = \\log{(\\hat{X}^{v_{y}})} and \\varepsilon_{0}{(W)} \\operatorname{f_{E}}{(\\hat{X},v_{y})} = \\varepsilon_{0}{(W)} \\log{(\\hat{X}^{v_{y}})} and \\varepsilon_{0}{(W)} = \\frac{\\varepsilon_{0}{(W)} \\log{(\\hat{X}^{v_{y}})}}{\\operatorname{f_{E}}{(\\hat{X},v_{y})}} and \\frac{\\varepsilon_{0}{(W)} \\log{(\\hat{X}^{v_{y}})}}{\\operatorname{f_{E}}{(\\hat{X},v_{y})}} = e^{e^{W}} and \\int \\frac{\\varepsilon_{0}{(W)} \\log{(\\hat{X}^{v_{y}})}}{\\operatorname{f_{E}}{(\\hat{X},v_{y})}} dW = \\int e^{e^{W}} dW", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), exp(exp(Symbol('W', commutative=True))))"], ["get_premise", "Equality(Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), log(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True))))"], [["times", 2, "Function('\\\\varepsilon_0')(Symbol('W', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True))), Mul(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), log(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)))))"], [["divide", 3, "Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Mul(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Pow(Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Pow(Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)))), exp(exp(Symbol('W', commutative=True))))"], [["integrate", 5, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varepsilon_0')(Symbol('W', commutative=True)), Pow(Function('f_E')(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(exp(exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\psi^*)} = \\cos{(\\psi^*)}, then obtain - \\int (- \\psi^* + \\operatorname{C_{d}}{(\\psi^*)}) d\\psi^* = - \\int (- \\psi^* + \\cos{(\\psi^*)}) d\\psi^*", "derivation": "\\operatorname{C_{d}}{(\\psi^*)} = \\cos{(\\psi^*)} and - \\psi^* + \\operatorname{C_{d}}{(\\psi^*)} = - \\psi^* + \\cos{(\\psi^*)} and \\int (- \\psi^* + \\operatorname{C_{d}}{(\\psi^*)}) d\\psi^* = \\int (- \\psi^* + \\cos{(\\psi^*)}) d\\psi^* and - \\int (- \\psi^* + \\operatorname{C_{d}}{(\\psi^*)}) d\\psi^* = - \\int (- \\psi^* + \\cos{(\\psi^*)}) d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('C_d')(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('C_d')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Function('C_d')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{s},z)} = \\frac{\\cos{(z)}}{\\mathbf{s}} and \\Psi{(\\mathbf{s},z)} = \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz, then obtain \\mathbf{s}^{2} \\Psi^{2}{(\\mathbf{s},z)} = \\mathbf{s}^{2} \\Psi{(\\mathbf{s},z)} \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz", "derivation": "\\mathbf{p}{(\\mathbf{s},z)} = \\frac{\\cos{(z)}}{\\mathbf{s}} and \\int \\mathbf{p}{(\\mathbf{s},z)} dz = \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz and \\Psi{(\\mathbf{s},z)} = \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz and \\Psi{(\\mathbf{s},z)} = \\int \\mathbf{p}{(\\mathbf{s},z)} dz and \\mathbf{s} \\Psi{(\\mathbf{s},z)} = \\mathbf{s} \\int \\mathbf{p}{(\\mathbf{s},z)} dz and \\mathbf{s} \\Psi{(\\mathbf{s},z)} = \\mathbf{s} \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz and \\mathbf{s}^{2} \\Psi^{2}{(\\mathbf{s},z)} = \\mathbf{s}^{2} \\Psi{(\\mathbf{s},z)} \\int \\frac{\\cos{(z)}}{\\mathbf{s}} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Integral(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Integral(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))))"], [["times", 6, "Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)), Pow(Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)), Function('\\\\Psi')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(y,\\hbar)} = e^{\\hbar + y}, then derive \\frac{\\partial}{\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = e^{\\hbar + y}, then obtain \\frac{\\partial^{2}}{\\partial \\hbar\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = \\frac{\\partial^{3}}{\\partial \\hbar\\partial y^{2}} \\operatorname{F_{N}}{(y,\\hbar)}", "derivation": "\\operatorname{F_{N}}{(y,\\hbar)} = e^{\\hbar + y} and \\frac{\\partial}{\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = \\frac{\\partial}{\\partial y} e^{\\hbar + y} and \\frac{\\partial}{\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = e^{\\hbar + y} and \\frac{\\partial^{2}}{\\partial \\hbar\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\hbar\\partial y} e^{\\hbar + y} and \\frac{\\partial^{2}}{\\partial \\hbar\\partial y} \\operatorname{F_{N}}{(y,\\hbar)} = \\frac{\\partial^{3}}{\\partial \\hbar\\partial y^{2}} \\operatorname{F_{N}}{(y,\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), exp(Add(Symbol('\\\\hbar', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\hbar', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), exp(Add(Symbol('\\\\hbar', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\hbar', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Function('F_N')(Symbol('y', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{D},\\chi)} = \\chi^{\\mathbf{D}}, then obtain \\frac{\\partial}{\\partial \\chi} \\chi^{- \\mathbf{D}} \\rho_{b}{(\\mathbf{D},\\chi)} = \\frac{d}{d \\chi} 1", "derivation": "\\rho_{b}{(\\mathbf{D},\\chi)} = \\chi^{\\mathbf{D}} and \\rho_{b}^{\\chi}{(\\mathbf{D},\\chi)} = (\\chi^{\\mathbf{D}})^{\\chi} and \\rho_{b}{(\\mathbf{D},\\chi)} \\rho_{b}^{\\chi}{(\\mathbf{D},\\chi)} = \\chi^{\\mathbf{D}} \\rho_{b}^{\\chi}{(\\mathbf{D},\\chi)} and (\\chi^{\\mathbf{D}})^{\\chi} \\rho_{b}{(\\mathbf{D},\\chi)} = \\chi^{\\mathbf{D}} (\\chi^{\\mathbf{D}})^{\\chi} and \\chi^{- \\mathbf{D}} \\rho_{b}{(\\mathbf{D},\\chi)} = 1 and \\frac{\\partial}{\\partial \\chi} \\chi^{- \\mathbf{D}} \\rho_{b}{(\\mathbf{D},\\chi)} = \\frac{d}{d \\chi} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\chi', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(1))"], [["differentiate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Function('\\\\rho_b')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mu_0)} = \\log{(e^{\\mu_0})}, then obtain \\frac{d}{d \\mu_0} (\\operatorname{f^{\\prime}}{(\\mu_0)} \\log{(e^{\\mu_0})})^{\\mu_0} = \\frac{d}{d \\mu_0} (\\log{(e^{\\mu_0})}^{2})^{\\mu_0}", "derivation": "\\operatorname{f^{\\prime}}{(\\mu_0)} = \\log{(e^{\\mu_0})} and \\operatorname{f^{\\prime}}{(\\mu_0)} \\log{(e^{\\mu_0})} = \\log{(e^{\\mu_0})}^{2} and (\\operatorname{f^{\\prime}}{(\\mu_0)} \\log{(e^{\\mu_0})})^{\\mu_0} = (\\log{(e^{\\mu_0})}^{2})^{\\mu_0} and \\frac{d}{d \\mu_0} (\\operatorname{f^{\\prime}}{(\\mu_0)} \\log{(e^{\\mu_0})})^{\\mu_0} = \\frac{d}{d \\mu_0} (\\log{(e^{\\mu_0})}^{2})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "log(exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))), Pow(log(exp(Symbol('\\\\mu_0', commutative=True))), Integer(2)))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(log(exp(Symbol('\\\\mu_0', commutative=True))), Integer(2)), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), log(exp(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(Pow(log(exp(Symbol('\\\\mu_0', commutative=True))), Integer(2)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f^{\\prime},A_{z})} = f^{\\prime} e^{A_{z}} and \\operatorname{m_{s}}{(\\hat{p},F_{N})} = \\sin{(F_{N} + \\hat{p})}, then obtain \\frac{\\operatorname{m_{s}}{(\\hat{p},F_{N})} e^{- A_{z}}}{f^{\\prime}} = \\frac{e^{- A_{z}} \\sin{(F_{N} + \\hat{p})}}{f^{\\prime}}", "derivation": "\\operatorname{E_{n}}{(f^{\\prime},A_{z})} = f^{\\prime} e^{A_{z}} and \\operatorname{m_{s}}{(\\hat{p},F_{N})} = \\sin{(F_{N} + \\hat{p})} and \\frac{\\operatorname{m_{s}}{(\\hat{p},F_{N})}}{\\operatorname{E_{n}}{(f^{\\prime},A_{z})}} = \\frac{\\sin{(F_{N} + \\hat{p})}}{\\operatorname{E_{n}}{(f^{\\prime},A_{z})}} and \\frac{\\operatorname{m_{s}}{(\\hat{p},F_{N})} e^{- A_{z}}}{f^{\\prime}} = \\frac{e^{- A_{z}} \\sin{(F_{N} + \\hat{p})}}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('A_z', commutative=True))))"], ["get_premise", "Equality(Function('m_s')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_N', commutative=True)), sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 2, "Function('E_n')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(Function('E_n')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_z', commutative=True)), Integer(-1)), Function('m_s')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Function('E_n')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_z', commutative=True)), Integer(-1)), sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_N', commutative=True)), exp(Mul(Integer(-1), Symbol('A_z', commutative=True)))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('A_z', commutative=True))), sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given T{(\\theta_1,\\varphi^*)} = \\cos{(\\theta_1 + \\varphi^*)}, then obtain (- 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)})^{2} = 2 (- 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)})^{2}", "derivation": "T{(\\theta_1,\\varphi^*)} = \\cos{(\\theta_1 + \\varphi^*)} and 0 = - T{(\\theta_1,\\varphi^*)} + \\cos{(\\theta_1 + \\varphi^*)} and - T{(\\theta_1,\\varphi^*)} + \\cos{(\\theta_1 + \\varphi^*)} = - 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)} and 0 = - 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)} and 0 = (- 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)})^{2} and (- 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)})^{2} = 2 (- 2 T{(\\theta_1,\\varphi^*)} + 2 \\cos{(\\theta_1 + \\varphi^*)})^{2}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 1, "Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))))"], [["power", 4, 2], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))), Integer(2)))"], [["add", 5, "Pow(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))), Integer(2))"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))), Integer(2)), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Add(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))))), Integer(2))))"]]}, {"prompt": "Given M{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain \\frac{d}{d \\rho_f} 1 = \\frac{d}{d \\rho_f} \\frac{\\sin{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}}{M{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}}", "derivation": "M{(\\rho_f)} = \\sin{(\\rho_f)} and M{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)} = \\sin{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)} and 1 = \\frac{\\sin{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}}{M{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}} and \\frac{d}{d \\rho_f} 1 = \\frac{d}{d \\rho_f} \\frac{\\sin{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}}{M{(\\rho_f)} + \\sin^{\\rho_f}{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Function('M')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Add(sin(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))))"], [["divide", 2, "Add(Function('M')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Function('M')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Integer(-1)), Add(sin(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Function('M')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Integer(-1)), Add(sin(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\phi_2,\\dot{y})} = \\log{(\\dot{y} + \\phi_2)}, then obtain \\frac{\\dot{y} \\frac{\\partial}{\\partial \\phi_2} \\frac{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}{\\dot{y}}}{\\mathbf{J}_f{(\\phi_2,\\dot{y})}} = \\frac{\\dot{y} \\frac{\\partial}{\\partial \\phi_2} \\frac{\\log{(\\dot{y} + \\phi_2)}}{\\dot{y}}}{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}", "derivation": "\\mathbf{J}_f{(\\phi_2,\\dot{y})} = \\log{(\\dot{y} + \\phi_2)} and \\frac{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}{\\dot{y}} = \\frac{\\log{(\\dot{y} + \\phi_2)}}{\\dot{y}} and \\frac{\\partial}{\\partial \\phi_2} \\frac{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}{\\dot{y}} = \\frac{\\partial}{\\partial \\phi_2} \\frac{\\log{(\\dot{y} + \\phi_2)}}{\\dot{y}} and \\frac{\\dot{y} \\frac{\\partial}{\\partial \\phi_2} \\frac{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}{\\dot{y}}}{\\mathbf{J}_f{(\\phi_2,\\dot{y})}} = \\frac{\\dot{y} \\frac{\\partial}{\\partial \\phi_2} \\frac{\\log{(\\dot{y} + \\phi_2)}}{\\dot{y}}}{\\mathbf{J}_f{(\\phi_2,\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(v_{t})} = \\cos{(v_{t})}, then derive \\frac{d}{d v_{t}} \\varphi{(v_{t})} - 1 = - \\sin{(v_{t})} - 1, then obtain \\frac{d}{d v_{t}} \\cos{(v_{t})} - 1 = - \\sin{(v_{t})} - 1", "derivation": "\\varphi{(v_{t})} = \\cos{(v_{t})} and \\frac{d}{d v_{t}} \\varphi{(v_{t})} = \\frac{d}{d v_{t}} \\cos{(v_{t})} and \\frac{d}{d v_{t}} \\varphi{(v_{t})} - 1 = \\frac{d}{d v_{t}} \\cos{(v_{t})} - 1 and \\frac{d}{d v_{t}} \\varphi{(v_{t})} - 1 = - \\sin{(v_{t})} - 1 and \\frac{d}{d v_{t}} \\cos{(v_{t})} - 1 = - \\sin{(v_{t})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\varphi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\varphi')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(cos(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('v_t', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given r{(r_{0},G)} = \\log{(G r_{0})}, then obtain 0 = - \\frac{\\partial}{\\partial G} r{(r_{0},G)} + \\frac{1}{G}", "derivation": "r{(r_{0},G)} = \\log{(G r_{0})} and r_{0} + r{(r_{0},G)} = r_{0} + \\log{(G r_{0})} and r_{0} = r_{0} - r{(r_{0},G)} + \\log{(G r_{0})} and \\frac{d}{d G} r_{0} = \\frac{\\partial}{\\partial G} (r_{0} - r{(r_{0},G)} + \\log{(G r_{0})}) and 0 = - \\frac{\\partial}{\\partial G} r{(r_{0},G)} + \\frac{1}{G}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True)), log(Mul(Symbol('G', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True))), Add(Symbol('r_0', commutative=True), log(Mul(Symbol('G', commutative=True), Symbol('r_0', commutative=True)))))"], [["minus", 2, "Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True))"], "Equality(Symbol('r_0', commutative=True), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True))), log(Mul(Symbol('G', commutative=True), Symbol('r_0', commutative=True)))))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(Symbol('r_0', commutative=True), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True))), log(Mul(Symbol('G', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('r')(Symbol('r_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Pow(Symbol('G', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(I)} = \\cos{(\\cos{(I)})}, then obtain 0 = - (I + \\operatorname{F_{c}}{(I)}) \\cos{(\\cos{(I)})} + (I + \\cos{(\\cos{(I)})}) \\cos{(\\cos{(I)})}", "derivation": "\\operatorname{F_{c}}{(I)} = \\cos{(\\cos{(I)})} and I + \\operatorname{F_{c}}{(I)} = I + \\cos{(\\cos{(I)})} and (I + \\operatorname{F_{c}}{(I)}) \\cos{(\\cos{(I)})} = (I + \\cos{(\\cos{(I)})}) \\cos{(\\cos{(I)})} and 0 = - (I + \\operatorname{F_{c}}{(I)}) \\cos{(\\cos{(I)})} + (I + \\cos{(\\cos{(I)})}) \\cos{(\\cos{(I)})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('I', commutative=True)), cos(cos(Symbol('I', commutative=True))))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('F_c')(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), cos(cos(Symbol('I', commutative=True)))))"], [["times", 2, "cos(cos(Symbol('I', commutative=True)))"], "Equality(Mul(Add(Symbol('I', commutative=True), Function('F_c')(Symbol('I', commutative=True))), cos(cos(Symbol('I', commutative=True)))), Mul(Add(Symbol('I', commutative=True), cos(cos(Symbol('I', commutative=True)))), cos(cos(Symbol('I', commutative=True)))))"], [["minus", 3, "Mul(Add(Symbol('I', commutative=True), Function('F_c')(Symbol('I', commutative=True))), cos(cos(Symbol('I', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Symbol('I', commutative=True), Function('F_c')(Symbol('I', commutative=True))), cos(cos(Symbol('I', commutative=True)))), Mul(Add(Symbol('I', commutative=True), cos(cos(Symbol('I', commutative=True)))), cos(cos(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})} = F_{H} + V_{\\mathbf{B}}, then obtain (- F_{H} + \\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})})^{F_{H}} \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H} + V_{\\mathbf{B}}}{F_{H}} = V_{\\mathbf{B}}^{F_{H}} \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H} + V_{\\mathbf{B}}}{F_{H}}", "derivation": "\\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})} = F_{H} + V_{\\mathbf{B}} and - F_{H} + \\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})} = V_{\\mathbf{B}} and (- F_{H} + \\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})})^{F_{H}} = V_{\\mathbf{B}}^{F_{H}} and (- F_{H} + \\operatorname{A_{y}}{(V_{\\mathbf{B}},F_{H})})^{F_{H}} \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H} + V_{\\mathbf{B}}}{F_{H}} = V_{\\mathbf{B}}^{F_{H}} \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H} + V_{\\mathbf{B}}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('A_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('A_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True)))"], [["times", 3, "Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('A_y')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('F_H', commutative=True)), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} = \\dot{\\mathbf{r}}^{i} + u, then obtain (\\dot{\\mathbf{r}}^{i} + u + \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)})^{\\dot{\\mathbf{r}}} = (2 \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)})^{\\dot{\\mathbf{r}}}", "derivation": "\\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} = \\dot{\\mathbf{r}}^{i} + u and \\dot{\\mathbf{r}}^{i} + u + \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} = 2 \\dot{\\mathbf{r}}^{i} + 2 u and 2 \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} = 2 \\dot{\\mathbf{r}}^{i} + 2 u and \\dot{\\mathbf{r}}^{i} + u + \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} = 2 \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)} and (\\dot{\\mathbf{r}}^{i} + u + \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)})^{\\dot{\\mathbf{r}}} = (2 \\varepsilon_{0}{(\\dot{\\mathbf{r}},i,u)})^{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True)), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True)), Symbol('u', commutative=True)))"], [["add", 1, "Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True)), Symbol('u', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))))"], [["power", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True)), Symbol('u', commutative=True), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = E + \\mathbb{I}, then derive \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = 0, then obtain \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (E + \\mathbb{I}) = 0", "derivation": "\\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = E + \\mathbb{I} and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = \\frac{\\partial}{\\partial \\mathbb{I}} (E + \\mathbb{I}) and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (E + \\mathbb{I}) and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\operatorname{a^{\\dagger}}{(E,\\mathbb{I})} = 0 and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (E + \\mathbb{I}) = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(Add(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Symbol('E', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given t{(v_{x},\\mathbf{S})} = \\frac{\\mathbf{S}}{v_{x}} and \\operatorname{n_{2}}{(v_{x},\\mathbf{S})} = - \\frac{\\mathbf{S}}{v_{x}^{2}}, then derive \\frac{\\partial}{\\partial v_{x}} t{(v_{x},\\mathbf{S})} = - \\frac{\\mathbf{S}}{v_{x}^{2}}, then obtain \\operatorname{n_{2}}{(v_{x},\\mathbf{S})} = \\frac{\\partial}{\\partial v_{x}} \\frac{\\mathbf{S}}{v_{x}}", "derivation": "t{(v_{x},\\mathbf{S})} = \\frac{\\mathbf{S}}{v_{x}} and \\frac{\\partial}{\\partial v_{x}} t{(v_{x},\\mathbf{S})} = \\frac{\\partial}{\\partial v_{x}} \\frac{\\mathbf{S}}{v_{x}} and \\frac{\\partial}{\\partial v_{x}} t{(v_{x},\\mathbf{S})} = - \\frac{\\mathbf{S}}{v_{x}^{2}} and \\operatorname{n_{2}}{(v_{x},\\mathbf{S})} = - \\frac{\\mathbf{S}}{v_{x}^{2}} and \\frac{\\partial}{\\partial v_{x}} t{(v_{x},\\mathbf{S})} = \\operatorname{n_{2}}{(v_{x},\\mathbf{S})} and \\operatorname{n_{2}}{(v_{x},\\mathbf{S})} = \\frac{\\partial}{\\partial v_{x}} \\frac{\\mathbf{S}}{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-2))))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('t')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(p)} = \\cos{(p)} and \\operatorname{E_{x}}{(p)} = \\cos{(p)}, then obtain 0 = - \\sin{(\\operatorname{E_{x}}{(p)} - \\cos{(p)})}", "derivation": "G{(p)} = \\cos{(p)} and 0 = - G{(p)} + \\cos{(p)} and \\operatorname{E_{x}}{(p)} = \\cos{(p)} and G{(p)} = \\operatorname{E_{x}}{(p)} and 0 = - \\operatorname{E_{x}}{(p)} + \\cos{(p)} and 0 = - \\sin{(\\operatorname{E_{x}}{(p)} - \\cos{(p)})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["minus", 1, "Function('G')(Symbol('p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('G')(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('G')(Symbol('p', commutative=True)), Function('E_x')(Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E_x')(Symbol('p', commutative=True))), cos(Symbol('p', commutative=True))))"], [["sin", 5], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('E_x')(Symbol('p', commutative=True)), Mul(Integer(-1), cos(Symbol('p', commutative=True)))))))"]]}, {"prompt": "Given v{(I)} = \\cos{(I)}, then derive \\frac{d}{d I} v{(I)} = - \\sin{(I)}, then obtain (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I} = (- \\sin{(I)} - \\cos{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I}", "derivation": "v{(I)} = \\cos{(I)} and \\frac{d}{d I} v{(I)} = \\frac{d}{d I} \\cos{(I)} and \\frac{d}{d I} v{(I)} = - \\sin{(I)} and - v{(I)} + \\frac{d}{d I} v{(I)} = - v{(I)} - \\sin{(I)} and - \\cos{(I)} + \\frac{d}{d I} \\cos{(I)} = - \\sin{(I)} - \\cos{(I)} and - \\cos{(I)} + \\frac{d}{d I} v{(I)} = - \\sin{(I)} - \\cos{(I)} and (- \\cos{(I)} + \\frac{d}{d I} v{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I} = (- \\sin{(I)} - \\cos{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I} and (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I} = (- \\sin{(I)} - \\cos{(I)}) (- \\cos{(I)} + \\frac{d}{d I} \\cos{(I)})^{- I}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('I', commutative=True))))"], [["minus", 3, "Function('v')(Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v')(Symbol('I', commutative=True))), Derivative(Function('v')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('v')(Symbol('I', commutative=True))), Mul(Integer(-1), sin(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('I', commutative=True))), Mul(Integer(-1), cos(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(Function('v')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('I', commutative=True))), Mul(Integer(-1), cos(Symbol('I', commutative=True)))))"], [["divide", 6, "Pow(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Symbol('I', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(Function('v')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Pow(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Add(Mul(Integer(-1), sin(Symbol('I', commutative=True))), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Pow(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Add(Mul(Integer(-1), sin(Symbol('I', commutative=True))), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Pow(Add(Mul(Integer(-1), cos(Symbol('I', commutative=True))), Derivative(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})} = \\frac{\\mathbf{B}}{f} - \\mathbf{f} and L{(\\omega,t_{1})} = \\cos{(\\omega^{t_{1}})}, then obtain \\frac{f \\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})}}{L{(\\omega,t_{1})}} = \\frac{f (\\frac{\\mathbf{B}}{f} - \\mathbf{f})}{L{(\\omega,t_{1})}}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})} = \\frac{\\mathbf{B}}{f} - \\mathbf{f} and f \\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})} = f (\\frac{\\mathbf{B}}{f} - \\mathbf{f}) and L{(\\omega,t_{1})} = \\cos{(\\omega^{t_{1}})} and \\frac{f \\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})}}{\\cos{(\\omega^{t_{1}})}} = \\frac{f (\\frac{\\mathbf{B}}{f} - \\mathbf{f})}{\\cos{(\\omega^{t_{1}})}} and \\frac{f \\operatorname{v_{1}}{(\\mathbf{B},f,\\mathbf{f})}}{L{(\\omega,t_{1})}} = \\frac{f (\\frac{\\mathbf{B}}{f} - \\mathbf{f})}{L{(\\omega,t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "Pow(Symbol('f', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('f', commutative=True), Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('f', commutative=True), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True))))"], [["divide", 2, "cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)))"], "Equality(Mul(Symbol('f', commutative=True), Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True))), Integer(-1))), Mul(Symbol('f', commutative=True), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Pow(cos(Pow(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('f', commutative=True), Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Function('v_1')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('f', commutative=True), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))), Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}_0{(n_{1},F_{N},\\sigma_p)} = \\sigma_p (F_{N} + n_{1}), then obtain - \\sigma_p^{2} = \\sigma_p^{2} (F_{N} + n_{1})^{2} - \\sigma_p^{2} - \\sigma_p (F_{N} + n_{1}) \\hat{p}_0{(n_{1},F_{N},\\sigma_p)}", "derivation": "\\hat{p}_0{(n_{1},F_{N},\\sigma_p)} = \\sigma_p (F_{N} + n_{1}) and \\sigma_p (F_{N} + n_{1}) \\hat{p}_0{(n_{1},F_{N},\\sigma_p)} = \\sigma_p^{2} (F_{N} + n_{1})^{2} and - \\sigma_p^{2} + \\sigma_p (F_{N} + n_{1}) \\hat{p}_0{(n_{1},F_{N},\\sigma_p)} = \\sigma_p^{2} (F_{N} + n_{1})^{2} - \\sigma_p^{2} and - \\sigma_p^{2} = \\sigma_p^{2} (F_{N} + n_{1})^{2} - \\sigma_p^{2} - \\sigma_p (F_{N} + n_{1}) \\hat{p}_0{(n_{1},F_{N},\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Pow(Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Integer(2))))"], [["minus", 2, "Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Pow(Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)))))"], [["minus", 3, "Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Add(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Pow(Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Add(Symbol('F_N', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given t{(\\mathbf{A},B)} = e^{B + \\mathbf{A}}, then derive \\int t{(\\mathbf{A},B)} dB = f^{\\prime} + e^{B + \\mathbf{A}}, then obtain \\frac{\\int t{(\\mathbf{A},B)} dB}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}} = \\frac{\\int e^{B + \\mathbf{A}} dB}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}}", "derivation": "t{(\\mathbf{A},B)} = e^{B + \\mathbf{A}} and \\int t{(\\mathbf{A},B)} dB = \\int e^{B + \\mathbf{A}} dB and \\int t{(\\mathbf{A},B)} dB = f^{\\prime} + e^{B + \\mathbf{A}} and f^{\\prime} + e^{B + \\mathbf{A}} = \\int e^{B + \\mathbf{A}} dB and \\frac{\\int t{(\\mathbf{A},B)} dB}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}} = \\frac{f^{\\prime} + e^{B + \\mathbf{A}}}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}} and \\frac{\\int t{(\\mathbf{A},B)} dB}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}} = \\frac{\\int e^{B + \\mathbf{A}} dB}{\\iint t{(\\mathbf{A},B)} dB d\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Integral(exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["divide", 3, "Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Pow(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Mul(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Pow(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Pow(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Mul(Integral(exp(Add(Symbol('B', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('B', commutative=True))), Pow(Integral(Function('t')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{f}{(C_{d})} = \\int \\log{(C_{d})} dC_{d} and \\operatorname{V_{\\mathbf{B}}}{(C_{d})} = \\int \\log{(C_{d})} dC_{d}, then derive \\mathbf{f}{(C_{d})} = C_{d} \\log{(C_{d})} - C_{d} + p, then obtain (C_{d} \\log{(C_{d})} - C_{d} + p + 1)^{C_{d}} = (\\operatorname{V_{\\mathbf{B}}}{(C_{d})} + 1)^{C_{d}}", "derivation": "\\mathbf{f}{(C_{d})} = \\int \\log{(C_{d})} dC_{d} and \\mathbf{f}{(C_{d})} = C_{d} \\log{(C_{d})} - C_{d} + p and C_{d} \\log{(C_{d})} - C_{d} + p = \\int \\log{(C_{d})} dC_{d} and \\operatorname{V_{\\mathbf{B}}}{(C_{d})} = \\int \\log{(C_{d})} dC_{d} and C_{d} \\log{(C_{d})} - C_{d} + p + 1 = \\int \\log{(C_{d})} dC_{d} + 1 and C_{d} \\log{(C_{d})} - C_{d} + p + 1 = \\operatorname{V_{\\mathbf{B}}}{(C_{d})} + 1 and (C_{d} \\log{(C_{d})} - C_{d} + p + 1)^{C_{d}} = (\\operatorname{V_{\\mathbf{B}}}{(C_{d})} + 1)^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('C_d', commutative=True)), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{f}')(Symbol('C_d', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('p', commutative=True)), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C_d', commutative=True)), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('p', commutative=True), Integer(1)), Add(Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('p', commutative=True), Integer(1)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('C_d', commutative=True)), Integer(1)))"], [["power", 6, "Symbol('C_d', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('p', commutative=True), Integer(1)), Symbol('C_d', commutative=True)), Pow(Add(Function('V_{\\\\mathbf{B}}')(Symbol('C_d', commutative=True)), Integer(1)), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\log{(- \\mathbf{J}_f + i)}, then obtain \\frac{\\partial^{2}}{\\partial i\\partial \\mathbf{J}_f} \\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\frac{1}{(\\mathbf{J}_f - i)^{2}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\log{(- \\mathbf{J}_f + i)} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\log{(- \\mathbf{J}_f + i)} and \\frac{\\partial^{2}}{\\partial i\\partial \\mathbf{J}_f} \\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\frac{\\partial^{2}}{\\partial i\\partial \\mathbf{J}_f} \\log{(- \\mathbf{J}_f + i)} and \\frac{\\partial^{2}}{\\partial i\\partial \\mathbf{J}_f} \\operatorname{a^{\\dagger}}{(\\mathbf{J}_f,i)} = \\frac{1}{(\\mathbf{J}_f - i)^{2}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('i', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('i', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('i', commutative=True), Integer(1))), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given L{(\\phi_2,\\pi,\\theta)} = \\phi_2 + \\pi + \\theta, then obtain \\frac{L{(\\phi_2,\\pi,\\theta)}}{\\phi_2^{2}} = \\frac{2 \\phi_2 + 2 \\pi + 2 \\theta}{2 \\phi_2^{2}}", "derivation": "L{(\\phi_2,\\pi,\\theta)} = \\phi_2 + \\pi + \\theta and \\phi_2 + \\pi + \\theta + L{(\\phi_2,\\pi,\\theta)} = 2 \\phi_2 + 2 \\pi + 2 \\theta and 2 L{(\\phi_2,\\pi,\\theta)} = 2 \\phi_2 + 2 \\pi + 2 \\theta and \\frac{L{(\\phi_2,\\pi,\\theta)}}{\\phi_2} = \\frac{2 \\phi_2 + 2 \\pi + 2 \\theta}{2 \\phi_2} and \\frac{L{(\\phi_2,\\pi,\\theta)}}{\\phi_2} = \\frac{\\phi_2 + \\pi + \\theta + L{(\\phi_2,\\pi,\\theta)}}{2 \\phi_2} and \\frac{L{(\\phi_2,\\pi,\\theta)}}{\\phi_2^{2}} = \\frac{\\phi_2 + \\pi + \\theta + L{(\\phi_2,\\pi,\\theta)}}{2 \\phi_2^{2}} and \\frac{L{(\\phi_2,\\pi,\\theta)}}{\\phi_2^{2}} = \\frac{2 \\phi_2 + 2 \\pi + 2 \\theta}{2 \\phi_2^{2}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["times", 5, "Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Add(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Add(Mul(Integer(2), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given k{(a^{\\dagger},E_{n})} = \\frac{\\partial}{\\partial E_{n}} \\frac{a^{\\dagger}}{E_{n}}, then derive k^{2}{(a^{\\dagger},E_{n})} = - \\frac{a^{\\dagger} k{(a^{\\dagger},E_{n})}}{E_{n}^{2}}, then obtain \\frac{\\partial}{\\partial E_{n}} (- E_{n} + k^{2}{(a^{\\dagger},E_{n})}) = \\frac{\\partial}{\\partial E_{n}} (- E_{n} - \\frac{a^{\\dagger} k{(a^{\\dagger},E_{n})}}{E_{n}^{2}})", "derivation": "k{(a^{\\dagger},E_{n})} = \\frac{\\partial}{\\partial E_{n}} \\frac{a^{\\dagger}}{E_{n}} and k^{2}{(a^{\\dagger},E_{n})} = k{(a^{\\dagger},E_{n})} \\frac{\\partial}{\\partial E_{n}} \\frac{a^{\\dagger}}{E_{n}} and k^{2}{(a^{\\dagger},E_{n})} = - \\frac{a^{\\dagger} k{(a^{\\dagger},E_{n})}}{E_{n}^{2}} and - E_{n} + k^{2}{(a^{\\dagger},E_{n})} = - E_{n} - \\frac{a^{\\dagger} k{(a^{\\dagger},E_{n})}}{E_{n}^{2}} and \\frac{\\partial}{\\partial E_{n}} (- E_{n} + k^{2}{(a^{\\dagger},E_{n})}) = \\frac{\\partial}{\\partial E_{n}} (- E_{n} - \\frac{a^{\\dagger} k{(a^{\\dagger},E_{n})}}{E_{n}^{2}})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Derivative(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["times", 1, "Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True))"], "Equality(Pow(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Derivative(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('a^{\\\\dagger}', commutative=True), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True))))"], [["minus", 3, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('a^{\\\\dagger}', commutative=True), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)))))"], [["differentiate", 4, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)), Integer(2))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('a^{\\\\dagger}', commutative=True), Function('k')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(l,\\mathbf{S})} = \\frac{\\mathbf{S}}{l}, then obtain (\\mathbf{S} + \\mathbf{r}{(l,\\mathbf{S})}) (\\frac{\\mathbf{S}}{l} + \\mathbf{r}{(l,\\mathbf{S})}) = \\frac{2 \\mathbf{S} (\\mathbf{S} + \\mathbf{r}{(l,\\mathbf{S})})}{l}", "derivation": "\\mathbf{r}{(l,\\mathbf{S})} = \\frac{\\mathbf{S}}{l} and \\frac{\\mathbf{S}}{l} + \\mathbf{r}{(l,\\mathbf{S})} = \\frac{2 \\mathbf{S}}{l} and \\mathbf{S} + \\mathbf{r}{(l,\\mathbf{S})} = \\mathbf{S} + \\frac{\\mathbf{S}}{l} and (\\mathbf{S} + \\frac{\\mathbf{S}}{l}) (\\frac{\\mathbf{S}}{l} + \\mathbf{r}{(l,\\mathbf{S})}) = \\frac{2 \\mathbf{S} (\\mathbf{S} + \\frac{\\mathbf{S}}{l})}{l} and (\\mathbf{S} + \\mathbf{r}{(l,\\mathbf{S})}) (\\frac{\\mathbf{S}}{l} + \\mathbf{r}{(l,\\mathbf{S})}) = \\frac{2 \\mathbf{S} (\\mathbf{S} + \\mathbf{r}{(l,\\mathbf{S})})}{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)))))"], [["times", 2, "Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain \\frac{d}{d \\mathbf{r}} \\frac{1}{2} = \\frac{d}{d \\mathbf{r}} \\frac{\\cos{(\\mathbf{r})}}{\\operatorname{C_{2}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})}}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and 2 \\operatorname{C_{2}}{(\\mathbf{r})} = \\operatorname{C_{2}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})} and \\frac{1}{2} = \\frac{\\cos{(\\mathbf{r})}}{2 \\operatorname{C_{2}}{(\\mathbf{r})}} and \\frac{d}{d \\mathbf{r}} \\frac{1}{2} = \\frac{d}{d \\mathbf{r}} \\frac{\\cos{(\\mathbf{r})}}{2 \\operatorname{C_{2}}{(\\mathbf{r})}} and \\frac{d}{d \\mathbf{r}} \\frac{1}{2} = \\frac{d}{d \\mathbf{r}} \\frac{\\cos{(\\mathbf{r})}}{\\operatorname{C_{2}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 1, "Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Rational(1, 2), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Pow(Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Rational(1, 2), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Function('C_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(U)} = \\log{(U)} and \\operatorname{v_{y}}{(U)} = \\omega^{U}{(U)} and u{(U)} = \\omega^{U}{(U)}, then obtain u{(U)} \\operatorname{v_{y}}{(U)} = u^{2}{(U)}", "derivation": "\\omega{(U)} = \\log{(U)} and \\omega^{U}{(U)} = \\log{(U)}^{U} and \\operatorname{v_{y}}{(U)} = \\omega^{U}{(U)} and u{(U)} = \\omega^{U}{(U)} and \\operatorname{v_{y}}{(U)} \\log{(U)}^{U} = \\omega^{U}{(U)} \\log{(U)}^{U} and \\operatorname{v_{y}}{(U)} \\log{(U)}^{U} = \\log{(U)}^{2 U} and u{(U)} = \\log{(U)}^{U} and u{(U)} \\operatorname{v_{y}}{(U)} = u^{2}{(U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('U', commutative=True)), Pow(Function('\\\\omega')(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('U', commutative=True)), Pow(Function('\\\\omega')(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["times", 3, "Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Mul(Function('v_y')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Function('\\\\omega')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('v_y')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Pow(log(Symbol('U', commutative=True)), Mul(Integer(2), Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('u')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Function('u')(Symbol('U', commutative=True)), Function('v_y')(Symbol('U', commutative=True))), Pow(Function('u')(Symbol('U', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\eta{(u)} = \\log{(e^{u})}, then derive (u + \\eta{(u)}) (\\frac{d}{d u} \\eta{(u)} + 1) = 2 u + 2 \\eta{(u)}, then obtain \\int ((u + \\log{(e^{u})}) (\\frac{d}{d u} \\log{(e^{u})} + 1) + \\log{(e^{u})}) du = \\int (2 u + 3 \\log{(e^{u})}) du", "derivation": "\\eta{(u)} = \\log{(e^{u})} and u + \\eta{(u)} = u + \\log{(e^{u})} and \\frac{d}{d u} (u + \\eta{(u)}) = \\frac{d}{d u} (u + \\log{(e^{u})}) and (u + \\eta{(u)}) \\frac{d}{d u} (u + \\eta{(u)}) = (u + \\eta{(u)}) \\frac{d}{d u} (u + \\log{(e^{u})}) and (u + \\eta{(u)}) (\\frac{d}{d u} \\eta{(u)} + 1) = 2 u + 2 \\eta{(u)} and (u + \\log{(e^{u})}) (\\frac{d}{d u} \\log{(e^{u})} + 1) = 2 u + 2 \\log{(e^{u})} and (u + \\log{(e^{u})}) (\\frac{d}{d u} \\log{(e^{u})} + 1) + \\log{(e^{u})} = 2 u + 3 \\log{(e^{u})} and \\int ((u + \\log{(e^{u})}) (\\frac{d}{d u} \\log{(e^{u})} + 1) + \\log{(e^{u})}) du = \\int (2 u + 3 \\log{(e^{u})}) du", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True)), log(exp(Symbol('u', commutative=True))))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 3, "Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True)))"], "Equality(Mul(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Derivative(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Derivative(Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Symbol('u', commutative=True), Function('\\\\eta')(Symbol('u', commutative=True))), Add(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))), Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(2), Function('\\\\eta')(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))), Add(Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))), Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(2), log(exp(Symbol('u', commutative=True))))))"], [["add", 6, "log(exp(Symbol('u', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))), Add(Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))), log(exp(Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(3), log(exp(Symbol('u', commutative=True))))))"], [["integrate", 7, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Add(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))), Add(Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(1))), log(exp(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(3), log(exp(Symbol('u', commutative=True))))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then derive \\frac{d}{d \\eta^{\\prime}} \\mathbf{P}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}, then obtain \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}", "derivation": "\\mathbf{P}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\mathbf{P}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\mathbf{P}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(\\Psi)} = e^{\\Psi}, then obtain e^{(\\mathbf{f}^{\\Psi}{(\\Psi)} + (e^{\\Psi})^{\\Psi})^{\\Psi}} = e^{(2 (e^{\\Psi})^{\\Psi})^{\\Psi}}", "derivation": "\\mathbf{f}{(\\Psi)} = e^{\\Psi} and \\mathbf{f}^{\\Psi}{(\\Psi)} = (e^{\\Psi})^{\\Psi} and \\mathbf{f}^{\\Psi}{(\\Psi)} + (e^{\\Psi})^{\\Psi} = 2 (e^{\\Psi})^{\\Psi} and (\\mathbf{f}^{\\Psi}{(\\Psi)} + (e^{\\Psi})^{\\Psi})^{\\Psi} = (2 (e^{\\Psi})^{\\Psi})^{\\Psi} and e^{(\\mathbf{f}^{\\Psi}{(\\Psi)} + (e^{\\Psi})^{\\Psi})^{\\Psi}} = e^{(2 (e^{\\Psi})^{\\Psi})^{\\Psi}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["add", 2, "Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Mul(Integer(2), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Add(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))), exp(Pow(Mul(Integer(2), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(E_{n},f)} = E_{n} f, then obtain 1 = \\frac{\\mathbf{M} + f}{\\int \\frac{\\operatorname{F_{N}}{(E_{n},f)}}{E_{n} f} df}", "derivation": "\\operatorname{F_{N}}{(E_{n},f)} = E_{n} f and 1 = \\frac{E_{n} f}{\\operatorname{F_{N}}{(E_{n},f)}} and \\frac{\\operatorname{F_{N}}{(E_{n},f)}}{E_{n} f} = 1 and \\int \\frac{\\operatorname{F_{N}}{(E_{n},f)}}{E_{n} f} df = \\int 1 df and 1 = \\frac{\\int 1 df}{\\int \\frac{\\operatorname{F_{N}}{(E_{n},f)}}{E_{n} f} df} and 1 = \\frac{\\mathbf{M} + f}{\\int \\frac{\\operatorname{F_{N}}{(E_{n},f)}}{E_{n} f} df}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(1), Mul(Symbol('E_n', commutative=True), Symbol('f', commutative=True), Pow(Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"], [["divide", 2, "Mul(Symbol('E_n', commutative=True), Symbol('f', commutative=True), Pow(Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))), Integer(1))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Integer(1), Tuple(Symbol('f', commutative=True))))"], [["divide", 4, "Integral(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Integer(1), Tuple(Symbol('f', commutative=True))), Pow(Integral(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 5], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f', commutative=True)), Pow(Integral(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('F_N')(Symbol('E_n', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\phi_{2}{(f^{*},y)} = y + \\sin{(f^{*})}, then derive \\dot{x} + \\phi_{2}{(f^{*},y)} = \\lambda + \\sin{(f^{*})}, then obtain \\dot{x} + \\phi_{2}{(f^{*},y)} = \\dot{x} + y + \\sin{(f^{*})}", "derivation": "\\phi_{2}{(f^{*},y)} = y + \\sin{(f^{*})} and \\frac{\\partial}{\\partial f^{*}} \\phi_{2}{(f^{*},y)} = \\frac{\\partial}{\\partial f^{*}} (y + \\sin{(f^{*})}) and \\int \\frac{\\partial}{\\partial f^{*}} \\phi_{2}{(f^{*},y)} df^{*} = \\int \\frac{\\partial}{\\partial f^{*}} (y + \\sin{(f^{*})}) df^{*} and \\dot{x} + \\phi_{2}{(f^{*},y)} = \\lambda + \\sin{(f^{*})} and \\dot{x} + y + \\sin{(f^{*})} = \\lambda + \\sin{(f^{*})} and \\dot{x} + \\phi_{2}{(f^{*},y)} = \\dot{x} + y + \\sin{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('y', commutative=True), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))), Integral(Derivative(Add(Symbol('y', commutative=True), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\phi_2')(Symbol('f^*', commutative=True), Symbol('y', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True), sin(Symbol('f^*', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\phi_2')(Symbol('f^*', commutative=True), Symbol('y', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True), sin(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(A_{2},q)} = \\cos{(A_{2} q)} and \\mathbf{E}{(\\theta)} = \\log{(e^{\\theta})}, then obtain \\frac{\\mathbf{E}{(\\theta)} \\log{(e^{\\theta})}}{\\frac{\\partial}{\\partial A_{2}} \\cos{(A_{2} q)}} = \\frac{\\log{(e^{\\theta})}^{2}}{\\frac{\\partial}{\\partial A_{2}} \\cos{(A_{2} q)}}", "derivation": "\\operatorname{c_{0}}{(A_{2},q)} = \\cos{(A_{2} q)} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{c_{0}}{(A_{2},q)} = \\frac{\\partial}{\\partial A_{2}} \\cos{(A_{2} q)} and \\mathbf{E}{(\\theta)} = \\log{(e^{\\theta})} and \\mathbf{E}{(\\theta)} \\log{(e^{\\theta})} = \\log{(e^{\\theta})}^{2} and \\frac{\\mathbf{E}{(\\theta)} \\log{(e^{\\theta})}}{\\frac{\\partial}{\\partial A_{2}} \\operatorname{c_{0}}{(A_{2},q)}} = \\frac{\\log{(e^{\\theta})}^{2}}{\\frac{\\partial}{\\partial A_{2}} \\operatorname{c_{0}}{(A_{2},q)}} and \\frac{\\mathbf{E}{(\\theta)} \\log{(e^{\\theta})}}{\\frac{\\partial}{\\partial A_{2}} \\cos{(A_{2} q)}} = \\frac{\\log{(e^{\\theta})}^{2}}{\\frac{\\partial}{\\partial A_{2}} \\cos{(A_{2} q)}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), cos(Mul(Symbol('A_2', commutative=True), Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True))))"], [["times", 3, "log(exp(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True)))), Pow(log(exp(Symbol('\\\\theta', commutative=True))), Integer(2)))"], [["divide", 4, "Derivative(Function('c_0')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True))), Pow(Derivative(Function('c_0')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(log(exp(Symbol('\\\\theta', commutative=True))), Integer(2)), Pow(Derivative(Function('c_0')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\theta', commutative=True)), log(exp(Symbol('\\\\theta', commutative=True))), Pow(Derivative(cos(Mul(Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(log(exp(Symbol('\\\\theta', commutative=True))), Integer(2)), Pow(Derivative(cos(Mul(Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given E{(\\phi_2,M_{E},A_{1})} = \\frac{A_{1} M_{E}}{\\phi_2} and I{(\\phi_2)} = \\phi_2, then obtain 0 = \\int \\frac{A_{1} M_{E}}{\\phi_2} dI{(\\phi_2)} - \\int E{(\\phi_2,M_{E},A_{1})} dI{(\\phi_2)}", "derivation": "E{(\\phi_2,M_{E},A_{1})} = \\frac{A_{1} M_{E}}{\\phi_2} and \\int E{(\\phi_2,M_{E},A_{1})} d\\phi_2 = \\int \\frac{A_{1} M_{E}}{\\phi_2} d\\phi_2 and I{(\\phi_2)} = \\phi_2 and \\int E{(\\phi_2,M_{E},A_{1})} dI{(\\phi_2)} = \\int \\frac{A_{1} M_{E}}{\\phi_2} dI{(\\phi_2)} and - \\int E{(\\phi_2,M_{E},A_{1})} d\\phi_2 + \\int E{(\\phi_2,M_{E},A_{1})} dI{(\\phi_2)} = \\int \\frac{A_{1} M_{E}}{\\phi_2} dI{(\\phi_2)} - \\int E{(\\phi_2,M_{E},A_{1})} d\\phi_2 and 0 = \\int \\frac{A_{1} M_{E}}{\\phi_2} d\\phi_2 - \\int E{(\\phi_2,M_{E},A_{1})} d\\phi_2 and 0 = \\int \\frac{A_{1} M_{E}}{\\phi_2} dI{(\\phi_2)} - \\int E{(\\phi_2,M_{E},A_{1})} dI{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True)))), Integral(Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 4, "Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True))))), Add(Integral(Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Add(Integral(Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(0), Add(Integral(Mul(Symbol('A_1', commutative=True), Symbol('M_E', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(-1), Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('M_E', commutative=True), Symbol('A_1', commutative=True)), Tuple(Function('I')(Symbol('\\\\phi_2', commutative=True)))))))"]]}, {"prompt": "Given v{(g,\\hat{H}_{\\lambda})} = - g + e^{\\hat{H}_{\\lambda}} and \\rho{(g)} = - g, then obtain ((\\rho{(g)} + e^{\\hat{H}_{\\lambda}}) \\rho{(g)})^{g} = (- g (\\rho{(g)} + e^{\\hat{H}_{\\lambda}}))^{g}", "derivation": "v{(g,\\hat{H}_{\\lambda})} = - g + e^{\\hat{H}_{\\lambda}} and \\rho{(g)} = - g and \\rho{(g)} v{(g,\\hat{H}_{\\lambda})} = - g v{(g,\\hat{H}_{\\lambda})} and v{(g,\\hat{H}_{\\lambda})} = \\rho{(g)} + e^{\\hat{H}_{\\lambda}} and (\\rho{(g)} + e^{\\hat{H}_{\\lambda}}) \\rho{(g)} = - g (\\rho{(g)} + e^{\\hat{H}_{\\lambda}}) and ((\\rho{(g)} + e^{\\hat{H}_{\\lambda}}) \\rho{(g)})^{g} = (- g (\\rho{(g)} + e^{\\hat{H}_{\\lambda}}))^{g}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["times", 2, "Function('v')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('\\\\rho')(Symbol('g', commutative=True)), Function('v')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True), Function('v')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Function('\\\\rho')(Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Function('\\\\rho')(Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Function('\\\\rho')(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True), Add(Function('\\\\rho')(Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\rho')(Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Function('\\\\rho')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Symbol('g', commutative=True), Add(Function('\\\\rho')(Symbol('g', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given U{(M,C_{1})} = C_{1}^{M} and \\operatorname{m_{s}}{(M,C_{1})} = e^{U{(M,C_{1})}}, then obtain \\frac{(C_{1} + \\frac{\\hat{p}}{v_{2}} - \\nabla) \\operatorname{m_{s}}{(M,C_{1})}}{M^{2}} = \\frac{(C_{1} + \\frac{\\hat{p}}{v_{2}} - \\nabla) e^{C_{1}^{M}}}{M^{2}}", "derivation": "U{(M,C_{1})} = C_{1}^{M} and e^{U{(M,C_{1})}} = e^{C_{1}^{M}} and \\operatorname{m_{s}}{(M,C_{1})} = e^{U{(M,C_{1})}} and \\operatorname{m_{s}}{(M,C_{1})} = e^{C_{1}^{M}} and \\frac{\\operatorname{m_{s}}{(M,C_{1})}}{M^{2}} = \\frac{e^{C_{1}^{M}}}{M^{2}} and \\frac{(C_{1} + \\frac{\\hat{p}}{v_{2}} - \\nabla) \\operatorname{m_{s}}{(M,C_{1})}}{M^{2}} = \\frac{(C_{1} + \\frac{\\hat{p}}{v_{2}} - \\nabla) e^{C_{1}^{M}}}{M^{2}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('M', commutative=True)))"], [["exp", 1], "Equality(exp(Function('U')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))), exp(Pow(Symbol('C_1', commutative=True), Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), exp(Function('U')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('m_s')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), exp(Pow(Symbol('C_1', commutative=True), Symbol('M', commutative=True))))"], [["times", 4, "Pow(Symbol('M', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-2)), Function('m_s')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-2)), exp(Pow(Symbol('C_1', commutative=True), Symbol('M', commutative=True)))))"], [["times", 5, "Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-2)), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('m_s')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-2)), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), exp(Pow(Symbol('C_1', commutative=True), Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\varphi)} = \\sin{(\\varphi)} and \\theta{(\\varphi)} = 2 \\mathbf{v}{(\\varphi)}, then obtain \\frac{\\int 2 \\sin{(\\varphi)} d\\varphi}{\\mathbf{v}^{4}{(\\varphi)}} = \\frac{\\int (\\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)}) d\\varphi}{\\mathbf{v}^{4}{(\\varphi)}}", "derivation": "\\mathbf{v}{(\\varphi)} = \\sin{(\\varphi)} and 2 \\mathbf{v}{(\\varphi)} = \\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)} and \\theta{(\\varphi)} = 2 \\mathbf{v}{(\\varphi)} and \\theta{(\\varphi)} = \\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)} and \\theta{(\\varphi)} = 2 \\sin{(\\varphi)} and 2 \\sin{(\\varphi)} = \\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)} and \\int 2 \\sin{(\\varphi)} d\\varphi = \\int (\\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)}) d\\varphi and \\frac{\\int 2 \\sin{(\\varphi)} d\\varphi}{\\mathbf{v}^{4}{(\\varphi)}} = \\frac{\\int (\\mathbf{v}{(\\varphi)} + \\sin{(\\varphi)}) d\\varphi}{\\mathbf{v}^{4}{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))), Add(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\theta')(Symbol('\\\\varphi', commutative=True)), Add(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\theta')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))), Add(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["divide", 7, "Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Integer(4))"], "Equality(Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Integer(-4)), Integral(Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), Integer(-4)), Integral(Add(Function('\\\\mathbf{v}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\pi{(L)} = e^{L}, then derive \\frac{d}{d L} \\pi{(L)} = e^{L}, then obtain \\frac{(\\frac{d^{2}}{d L^{2}} \\pi{(L)})^{L}}{\\pi{(L)}} = \\frac{(\\frac{d}{d L} \\pi{(L)})^{L}}{\\pi{(L)}}", "derivation": "\\pi{(L)} = e^{L} and \\frac{d}{d L} \\pi{(L)} = \\frac{d}{d L} e^{L} and \\frac{d}{d L} \\pi{(L)} = e^{L} and \\frac{d}{d L} \\pi{(L)} = \\frac{d^{2}}{d L^{2}} \\pi{(L)} and (\\frac{d}{d L} \\pi{(L)})^{L} = (\\frac{d}{d L} e^{L})^{L} and (\\frac{d^{2}}{d L^{2}} \\pi{(L)})^{L} = (\\frac{d}{d L} e^{L})^{L} and (\\frac{d^{2}}{d L^{2}} \\pi{(L)})^{L} = (\\frac{d}{d L} \\pi{(L)})^{L} and \\frac{(\\frac{d^{2}}{d L^{2}} \\pi{(L)})^{L}}{\\pi{(L)}} = \\frac{(\\frac{d}{d L} \\pi{(L)})^{L}}{\\pi{(L)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), exp(Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)), Pow(Derivative(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Symbol('L', commutative=True)), Pow(Derivative(exp(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Symbol('L', commutative=True)), Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"], [["divide", 7, "Function('\\\\pi')(Symbol('L', commutative=True))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('L', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(2))), Symbol('L', commutative=True))), Mul(Pow(Function('\\\\pi')(Symbol('L', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\pi')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(F_{c},\\pi)} = F_{c} \\pi and \\theta_{2}{(F_{N})} = \\frac{d}{d F_{N}} \\log{(F_{N})}, then derive \\theta_{2}{(F_{N})} + 1 = 1 + \\frac{1}{F_{N}}, then obtain - 0^{\\pi} + \\theta_{2}{(F_{N})} + 1 = - 0^{\\pi} + 1 + \\frac{1}{F_{N}}", "derivation": "\\operatorname{A_{2}}{(F_{c},\\pi)} = F_{c} \\pi and \\theta_{2}{(F_{N})} = \\frac{d}{d F_{N}} \\log{(F_{N})} and \\theta_{2}{(F_{N})} + 1 = \\frac{d}{d F_{N}} \\log{(F_{N})} + 1 and \\theta_{2}{(F_{N})} + 1 = 1 + \\frac{1}{F_{N}} and - (- \\frac{F_{c} \\pi}{\\operatorname{A_{2}}{(F_{c},\\pi)}} + 1)^{\\pi} + \\theta_{2}{(F_{N})} + 1 = - (- \\frac{F_{c} \\pi}{\\operatorname{A_{2}}{(F_{c},\\pi)}} + 1)^{\\pi} + 1 + \\frac{1}{F_{N}} and - 0^{\\pi} + \\theta_{2}{(F_{N})} + 1 = - 0^{\\pi} + 1 + \\frac{1}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Integer(1)), Add(Derivative(log(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Integer(1)), Add(Integer(1), Pow(Symbol('F_N', commutative=True), Integer(-1))))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True), Pow(Function('A_2')(Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1)), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True), Pow(Function('A_2')(Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1)), Symbol('\\\\pi', commutative=True))), Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True), Pow(Function('A_2')(Symbol('F_c', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1)), Symbol('\\\\pi', commutative=True))), Integer(1), Pow(Symbol('F_N', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\pi', commutative=True))), Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\pi', commutative=True))), Integer(1), Pow(Symbol('F_N', commutative=True), Integer(-1))))"]]}, {"prompt": "Given x{(v_{2},\\varphi)} = \\varphi v_{2}, then obtain (\\varphi v_{2} + \\varphi) (\\varphi v_{2} + x{(v_{2},\\varphi)}) x{(v_{2},\\varphi)} = 2 \\varphi v_{2} (\\varphi v_{2} + \\varphi) x{(v_{2},\\varphi)}", "derivation": "x{(v_{2},\\varphi)} = \\varphi v_{2} and \\varphi v_{2} + x{(v_{2},\\varphi)} = 2 \\varphi v_{2} and (\\varphi v_{2} + x{(v_{2},\\varphi)}) x{(v_{2},\\varphi)} = 2 \\varphi v_{2} x{(v_{2},\\varphi)} and \\varphi + x{(v_{2},\\varphi)} = \\varphi v_{2} + \\varphi and (\\varphi + x{(v_{2},\\varphi)}) (\\varphi v_{2} + x{(v_{2},\\varphi)}) x{(v_{2},\\varphi)} = 2 \\varphi v_{2} (\\varphi + x{(v_{2},\\varphi)}) x{(v_{2},\\varphi)} and (\\varphi v_{2} + \\varphi) (\\varphi v_{2} + x{(v_{2},\\varphi)}) x{(v_{2},\\varphi)} = 2 \\varphi v_{2} (\\varphi v_{2} + \\varphi) x{(v_{2},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 2, "Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["times", 3, "Add(Symbol('\\\\varphi', commutative=True), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\varphi', commutative=True), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True), Add(Symbol('\\\\varphi', commutative=True), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\varphi', commutative=True)), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True), Add(Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_2', commutative=True)), Symbol('\\\\varphi', commutative=True)), Function('x')(Symbol('v_2', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(b)} = e^{\\cos{(b)}} and \\hat{p}{(b)} = \\cos{(b)}, then obtain (\\hat{p}{(b)} + e^{\\cos{(b)}}) e^{- \\cos{(b)}} = (e^{\\cos{(b)}} + \\cos{(b)}) e^{- \\cos{(b)}}", "derivation": "\\Psi_{nl}{(b)} = e^{\\cos{(b)}} and \\hat{p}{(b)} = \\cos{(b)} and \\hat{p}{(b)} + e^{\\cos{(b)}} = e^{\\cos{(b)}} + \\cos{(b)} and \\frac{\\hat{p}{(b)} + e^{\\cos{(b)}}}{\\Psi_{nl}{(b)}} = \\frac{e^{\\cos{(b)}} + \\cos{(b)}}{\\Psi_{nl}{(b)}} and (\\hat{p}{(b)} + e^{\\cos{(b)}}) e^{- \\cos{(b)}} = (e^{\\cos{(b)}} + \\cos{(b)}) e^{- \\cos{(b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["add", 2, "exp(cos(Symbol('b', commutative=True)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True)))), Add(exp(cos(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True))))"], [["divide", 3, "Function('\\\\Psi_{nl}')(Symbol('b', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True)))), Pow(Function('\\\\Psi_{nl}')(Symbol('b', commutative=True)), Integer(-1))), Mul(Add(exp(cos(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True))), Pow(Function('\\\\Psi_{nl}')(Symbol('b', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True)))), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True))))), Mul(Add(exp(cos(Symbol('b', commutative=True))), cos(Symbol('b', commutative=True))), exp(Mul(Integer(-1), cos(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given W{(g,J)} = J - g, then obtain 0 = \\frac{- \\frac{(J - g) \\frac{\\partial}{\\partial g} W{(g,J)}}{W^{2}{(g,J)}} - \\frac{1}{W{(g,J)}}}{\\int W{(g,J)} dJ}", "derivation": "W{(g,J)} = J - g and 1 = \\frac{J - g}{W{(g,J)}} and \\frac{d}{d g} 1 = \\frac{\\partial}{\\partial g} \\frac{J - g}{W{(g,J)}} and \\frac{\\frac{d}{d g} 1}{\\int W{(g,J)} dJ} = \\frac{\\frac{\\partial}{\\partial g} \\frac{J - g}{W{(g,J)}}}{\\int W{(g,J)} dJ} and 0 = \\frac{- \\frac{(J - g) \\frac{\\partial}{\\partial g} W{(g,J)}}{W^{2}{(g,J)}} - \\frac{1}{W{(g,J)}}}{\\int W{(g,J)} dJ}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["divide", 1, "Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))), Mul(Derivative(Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('g', commutative=True), Integer(1))), Pow(Integral(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Pow(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Integer(-2)), Derivative(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Integer(-1)))), Pow(Integral(Function('W')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given n{(\\psi)} = \\psi, then derive (\\int n{(\\psi)} d\\psi)^{\\psi} = (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi}, then obtain - \\frac{\\psi^{2}}{2} - \\theta_1 + \\frac{d}{d \\psi} (\\int \\psi d\\psi)^{\\psi} = - \\frac{\\psi^{2}}{2} - \\theta_1 + \\frac{\\partial}{\\partial \\psi} (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi}", "derivation": "n{(\\psi)} = \\psi and \\int n{(\\psi)} d\\psi = \\int \\psi d\\psi and (\\int n{(\\psi)} d\\psi)^{\\psi} = (\\int \\psi d\\psi)^{\\psi} and (\\int n{(\\psi)} d\\psi)^{\\psi} = (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi} and (\\int \\psi d\\psi)^{\\psi} = (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi} and \\frac{d}{d \\psi} (\\int \\psi d\\psi)^{\\psi} = \\frac{\\partial}{\\partial \\psi} (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi} and - \\frac{\\psi^{2}}{2} - \\theta_1 + \\frac{d}{d \\psi} (\\int \\psi d\\psi)^{\\psi} = - \\frac{\\psi^{2}}{2} - \\theta_1 + \\frac{\\partial}{\\partial \\psi} (\\frac{\\psi^{2}}{2} + \\theta_1)^{\\psi}", "srepr_derivation": [["renaming_premise", "Equality(Function('n')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Pow(Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["minus", 6, "Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Derivative(Pow(Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Derivative(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\phi)} = \\sin{(\\phi)}, then derive \\frac{d}{d \\phi} \\mathbf{p}{(\\phi)} = \\cos{(\\phi)}, then obtain \\frac{d^{2}}{d \\phi^{2}} \\mathbf{p}{(\\phi)} = \\frac{d}{d \\phi} \\cos{(\\phi)}", "derivation": "\\mathbf{p}{(\\phi)} = \\sin{(\\phi)} and \\frac{d}{d \\phi} \\mathbf{p}{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and \\frac{d}{d \\phi} \\mathbf{p}{(\\phi)} = \\cos{(\\phi)} and \\frac{d^{2}}{d \\phi^{2}} \\mathbf{p}{(\\phi)} = \\frac{d}{d \\phi} \\cos{(\\phi)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), cos(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(p,L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{p}, then obtain \\hat{H}_l + \\frac{\\int - L_{\\varepsilon}^{2} dL_{\\varepsilon} + \\int p \\operatorname{f_{E}}{(p,L_{\\varepsilon})} dL_{\\varepsilon}}{p} = A_{y} - \\frac{L_{\\varepsilon}^{3}}{3 p} + \\frac{L_{\\varepsilon}^{2}}{2 p}", "derivation": "\\operatorname{f_{E}}{(p,L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{p} and - \\frac{L_{\\varepsilon}^{2}}{p} + \\operatorname{f_{E}}{(p,L_{\\varepsilon})} = - \\frac{L_{\\varepsilon}^{2}}{p} + \\frac{L_{\\varepsilon}}{p} and \\int (- \\frac{L_{\\varepsilon}^{2}}{p} + \\operatorname{f_{E}}{(p,L_{\\varepsilon})}) dL_{\\varepsilon} = \\int (- \\frac{L_{\\varepsilon}^{2}}{p} + \\frac{L_{\\varepsilon}}{p}) dL_{\\varepsilon} and \\hat{H}_l + \\frac{\\int - L_{\\varepsilon}^{2} dL_{\\varepsilon} + \\int p \\operatorname{f_{E}}{(p,L_{\\varepsilon})} dL_{\\varepsilon}}{p} = A_{y} - \\frac{L_{\\varepsilon}^{3}}{3 p} + \\frac{L_{\\varepsilon}^{2}}{2 p}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["minus", 1, "Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1))), Function('f_E')(Symbol('p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1))), Function('f_E')(Symbol('p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('p', commutative=True), Function('f_E')(Symbol('p', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Rational(1, 3), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(3)), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Rational(1, 2), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2)), Pow(Symbol('p', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})} = \\phi_1 g_{\\varepsilon} and y{(\\phi_1)} = \\phi_1, then obtain \\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})} + y^{\\phi_1}{(\\phi_1)} = \\phi_1^{\\phi_1} + \\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})}", "derivation": "\\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})} = \\phi_1 g_{\\varepsilon} and y{(\\phi_1)} = \\phi_1 and y^{\\phi_1}{(\\phi_1)} = \\phi_1^{\\phi_1} and \\phi_1 g_{\\varepsilon} + y^{\\phi_1}{(\\phi_1)} = \\phi_1 g_{\\varepsilon} + \\phi_1^{\\phi_1} and \\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})} + y^{\\phi_1}{(\\phi_1)} = \\phi_1^{\\phi_1} + \\operatorname{F_{x}}{(\\phi_1,g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('F_x')(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('F_x')(Symbol('\\\\phi_1', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(I,n)} = \\cos{(I - n)}, then obtain 0 = - \\frac{\\operatorname{g_{\\varepsilon}}{(I,n)}}{- n + \\operatorname{g_{\\varepsilon}}{(I,n)}} + \\frac{\\cos{(I - n)}}{- n + \\operatorname{g_{\\varepsilon}}{(I,n)}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(I,n)} = \\cos{(I - n)} and - n + \\operatorname{g_{\\varepsilon}}{(I,n)} = - n + \\cos{(I - n)} and \\frac{\\operatorname{g_{\\varepsilon}}{(I,n)}}{- n + \\cos{(I - n)}} = \\frac{\\cos{(I - n)}}{- n + \\cos{(I - n)}} and 0 = - \\frac{\\operatorname{g_{\\varepsilon}}{(I,n)}}{- n + \\cos{(I - n)}} + \\frac{\\cos{(I - n)}}{- n + \\cos{(I - n)}} and 0 = - \\frac{\\operatorname{g_{\\varepsilon}}{(I,n)}}{- n + \\operatorname{g_{\\varepsilon}}{(I,n)}} + \\frac{\\cos{(I - n)}}{- n + \\operatorname{g_{\\varepsilon}}{(I,n)}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["add", 1, "Mul(Integer(-1), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(-1)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"], [["minus", 3, "Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Integer(-1)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Integer(-1)), cos(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))))"]]}, {"prompt": "Given \\chi{(W)} = \\log{(W)}, then derive (\\frac{W \\frac{d}{d W} \\chi{(W)}}{\\chi{(W)}} + \\log{(\\chi{(W)})}) \\chi^{W}{(W)} + 1 = (\\log{(\\log{(W)})} + \\frac{1}{\\log{(W)}}) \\log{(W)}^{W} + 1, then obtain (\\frac{W \\frac{d}{d W} \\log{(W)}}{\\log{(W)}} + \\log{(\\log{(W)})}) \\chi^{W}{(W)} + 1 = (\\log{(\\log{(W)})} + \\frac{1}{\\log{(W)}}) \\chi^{W}{(W)} + 1", "derivation": "\\chi{(W)} = \\log{(W)} and \\chi^{W}{(W)} = \\log{(W)}^{W} and W + \\chi^{W}{(W)} = W + \\log{(W)}^{W} and \\frac{d}{d W} (W + \\chi^{W}{(W)}) = \\frac{d}{d W} (W + \\log{(W)}^{W}) and (\\frac{W \\frac{d}{d W} \\chi{(W)}}{\\chi{(W)}} + \\log{(\\chi{(W)})}) \\chi^{W}{(W)} + 1 = (\\log{(\\log{(W)})} + \\frac{1}{\\log{(W)}}) \\log{(W)}^{W} + 1 and (\\frac{W \\frac{d}{d W} \\log{(W)}}{\\log{(W)}} + \\log{(\\log{(W)})}) \\log{(W)}^{W} + 1 = (\\log{(\\log{(W)})} + \\frac{1}{\\log{(W)}}) \\log{(W)}^{W} + 1 and (\\frac{W \\frac{d}{d W} \\log{(W)}}{\\log{(W)}} + \\log{(\\log{(W)})}) \\chi^{W}{(W)} + 1 = (\\log{(\\log{(W)})} + \\frac{1}{\\log{(W)}}) \\chi^{W}{(W)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["add", 2, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Symbol('W', commutative=True), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Mul(Symbol('W', commutative=True), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), log(Function('\\\\chi')(Symbol('W', commutative=True)))), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)), Add(Mul(Add(log(log(Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Integer(-1))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Add(Mul(Symbol('W', commutative=True), Pow(log(Symbol('W', commutative=True)), Integer(-1)), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), log(log(Symbol('W', commutative=True)))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)), Add(Mul(Add(log(log(Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Integer(-1))), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Add(Mul(Symbol('W', commutative=True), Pow(log(Symbol('W', commutative=True)), Integer(-1)), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), log(log(Symbol('W', commutative=True)))), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)), Add(Mul(Add(log(log(Symbol('W', commutative=True))), Pow(log(Symbol('W', commutative=True)), Integer(-1))), Pow(Function('\\\\chi')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\ddot{x}{(\\phi_2,F_{c})} = \\log{(F_{c})}^{\\phi_2}, then obtain - 2 \\log{(F_{c})}^{\\phi_2} \\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} = - (\\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} + 1) \\log{(F_{c})}^{\\phi_2}", "derivation": "\\ddot{x}{(\\phi_2,F_{c})} = \\log{(F_{c})}^{\\phi_2} and \\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2} = 0 and \\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} = 1 and 2 \\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} = \\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} + 1 and - 2 \\log{(F_{c})}^{\\phi_2} \\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} = - (\\cos{(\\ddot{x}{(\\phi_2,F_{c})} - \\log{(F_{c})}^{\\phi_2})} + 1) \\log{(F_{c})}^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Integer(0))"], [["cos", 2], "Equality(cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True))))), Integer(1))"], [["add", 3, "cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))"], "Equality(Mul(Integer(2), cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))), Add(cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True))))), Integer(1)))"], [["times", 4, "Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)), cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))), Mul(Integer(-1), Add(cos(Add(Function('\\\\ddot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True))))), Integer(1)), Pow(log(Symbol('F_c', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\phi{(x)} = \\sin{(x)}, then obtain (\\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)}) \\sin{(x)} + (\\phi^{2}{(x)} - \\sin^{x}{(x)}) \\sin{(x)} = 2 (\\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)}) \\sin{(x)}", "derivation": "\\phi{(x)} = \\sin{(x)} and \\phi^{x}{(x)} = \\sin^{x}{(x)} and \\phi^{2}{(x)} = \\phi{(x)} \\sin{(x)} and \\phi^{2}{(x)} - \\phi^{x}{(x)} = \\phi{(x)} \\sin{(x)} - \\phi^{x}{(x)} and \\phi^{2}{(x)} - \\sin^{x}{(x)} = \\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)} and (\\phi^{2}{(x)} - \\sin^{x}{(x)}) \\sin{(x)} = (\\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)}) \\sin{(x)} and (\\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)}) \\sin{(x)} + (\\phi^{2}{(x)} - \\sin^{x}{(x)}) \\sin{(x)} = 2 (\\phi{(x)} \\sin{(x)} - \\sin^{x}{(x)}) \\sin{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["times", 1, "Function('\\\\phi')(Symbol('x', commutative=True))"], "Equality(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Integer(2)), Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Symbol('x', commutative=True))"], "Equality(Add(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))))"], [["times", 5, "sin(Symbol('x', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True))), Mul(Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True))))"], [["add", 6, "Mul(Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True))), Mul(Add(Pow(Function('\\\\phi')(Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True)))), Mul(Integer(2), Add(Mul(Function('\\\\phi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), sin(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(C_{2})} = C_{2}, then derive m + \\frac{\\mathbf{r}^{2}{(C_{2})}}{2} = \\int C_{2} d\\mathbf{r}{(C_{2})}, then obtain (\\int \\mathbf{r}{(C_{2})} d\\mathbf{r}{(C_{2})}) \\iint C_{2} dC_{2} dC_{2} = (m + \\frac{\\mathbf{r}^{2}{(C_{2})}}{2}) \\iint C_{2} dC_{2} dC_{2}", "derivation": "\\mathbf{r}{(C_{2})} = C_{2} and \\int \\mathbf{r}{(C_{2})} dC_{2} = \\int C_{2} dC_{2} and \\int \\mathbf{r}{(C_{2})} d\\mathbf{r}{(C_{2})} = \\int C_{2} d\\mathbf{r}{(C_{2})} and m + \\frac{\\mathbf{r}^{2}{(C_{2})}}{2} = \\int C_{2} d\\mathbf{r}{(C_{2})} and \\int \\mathbf{r}{(C_{2})} d\\mathbf{r}{(C_{2})} = m + \\frac{\\mathbf{r}^{2}{(C_{2})}}{2} and (\\int \\mathbf{r}{(C_{2})} d\\mathbf{r}{(C_{2})}) \\iint C_{2} dC_{2} dC_{2} = (m + \\frac{\\mathbf{r}^{2}{(C_{2})}}{2}) \\iint C_{2} dC_{2} dC_{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Tuple(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)))), Integral(Symbol('C_2', commutative=True), Tuple(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Integer(2)))), Integral(Symbol('C_2', commutative=True), Tuple(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Tuple(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)))), Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Integer(2)))))"], [["times", 5, "Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Tuple(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)))), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Mul(Add(Symbol('m', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{r}')(Symbol('C_2', commutative=True)), Integer(2)))), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given A{(\\mu,v_{t})} = \\mu^{v_{t}}, then obtain 2 \\mu^{v_{t}} \\frac{\\partial^{2}}{\\partial \\mu^{2}} A^{v_{t}}{(\\mu,v_{t})} = 2 \\mu^{v_{t}} \\frac{\\partial^{2}}{\\partial \\mu^{2}} (\\mu^{v_{t}})^{v_{t}}", "derivation": "A{(\\mu,v_{t})} = \\mu^{v_{t}} and A^{v_{t}}{(\\mu,v_{t})} = (\\mu^{v_{t}})^{v_{t}} and \\frac{\\partial}{\\partial \\mu} A^{v_{t}}{(\\mu,v_{t})} = \\frac{\\partial}{\\partial \\mu} (\\mu^{v_{t}})^{v_{t}} and \\frac{\\partial^{2}}{\\partial \\mu^{2}} A^{v_{t}}{(\\mu,v_{t})} = \\frac{\\partial^{2}}{\\partial \\mu^{2}} (\\mu^{v_{t}})^{v_{t}} and 2 \\mu^{v_{t}} \\frac{\\partial^{2}}{\\partial \\mu^{2}} A^{v_{t}}{(\\mu,v_{t})} = 2 \\mu^{v_{t}} \\frac{\\partial^{2}}{\\partial \\mu^{2}} (\\mu^{v_{t}})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))), Derivative(Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))))"], [["times", 4, "Mul(Integer(2), Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Derivative(Pow(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2)))), Mul(Integer(2), Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Derivative(Pow(Pow(Symbol('\\\\mu', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2)))))"]]}, {"prompt": "Given V{(\\dot{\\mathbf{r}})} = \\sin{(\\log{(\\dot{\\mathbf{r}})})}, then obtain 2 (V{(\\dot{\\mathbf{r}})} + \\sin{(\\log{(\\dot{\\mathbf{r}})})}) \\sin^{3}{(\\log{(\\dot{\\mathbf{r}})})} = 4 \\sin^{4}{(\\log{(\\dot{\\mathbf{r}})})}", "derivation": "V{(\\dot{\\mathbf{r}})} = \\sin{(\\log{(\\dot{\\mathbf{r}})})} and V{(\\dot{\\mathbf{r}})} + \\sin{(\\log{(\\dot{\\mathbf{r}})})} = 2 \\sin{(\\log{(\\dot{\\mathbf{r}})})} and (V{(\\dot{\\mathbf{r}})} + \\sin{(\\log{(\\dot{\\mathbf{r}})})}) \\sin{(\\log{(\\dot{\\mathbf{r}})})} = 2 \\sin^{2}{(\\log{(\\dot{\\mathbf{r}})})} and 2 (V{(\\dot{\\mathbf{r}})} + \\sin{(\\log{(\\dot{\\mathbf{r}})})}) \\sin^{3}{(\\log{(\\dot{\\mathbf{r}})})} = 4 \\sin^{4}{(\\log{(\\dot{\\mathbf{r}})})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Add(Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Mul(Integer(2), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["times", 2, "sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Add(Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Mul(Integer(2), Pow(sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2))))"], [["times", 3, "Mul(Integer(2), Pow(sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2)))"], "Equality(Mul(Integer(2), Add(Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Pow(sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(3))), Mul(Integer(4), Pow(sin(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(4))))"]]}, {"prompt": "Given \\phi_{1}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then derive \\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then obtain - (\\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = - \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})}", "derivation": "\\phi_{1}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\phi_{1}{(V_{\\mathbf{E}})} + 1 = \\sin{(V_{\\mathbf{E}})} + 1 and \\frac{d}{d V_{\\mathbf{E}}} (\\phi_{1}{(V_{\\mathbf{E}})} + 1) = \\frac{d}{d V_{\\mathbf{E}}} (\\sin{(V_{\\mathbf{E}})} + 1) and \\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and (\\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} and - (\\frac{d}{d V_{\\mathbf{E}}} \\phi_{1}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = - \\cos^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)), Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["power", 4, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Derivative(Function('\\\\phi_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(U,v_{x})} = - U + v_{x}, then obtain \\frac{\\partial}{\\partial U} \\int (U + \\rho_{f}{(U,v_{x})}) dv_{x} = \\frac{\\partial}{\\partial U} (E + \\frac{v_{x}^{2}}{2})", "derivation": "\\rho_{f}{(U,v_{x})} = - U + v_{x} and U + \\rho_{f}{(U,v_{x})} = v_{x} and \\int (U + \\rho_{f}{(U,v_{x})}) dv_{x} = \\int v_{x} dv_{x} and \\frac{\\partial}{\\partial U} \\int (U + \\rho_{f}{(U,v_{x})}) dv_{x} = \\frac{d}{d U} \\int v_{x} dv_{x} and \\frac{\\partial}{\\partial U} \\int (U + \\rho_{f}{(U,v_{x})}) dv_{x} = \\frac{\\partial}{\\partial U} (E + \\frac{v_{x}^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('U', commutative=True), Symbol('v_x', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('v_x', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('\\\\rho_f')(Symbol('U', commutative=True), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Symbol('U', commutative=True), Function('\\\\rho_f')(Symbol('U', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Symbol('v_x', commutative=True), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('U', commutative=True), Function('\\\\rho_f')(Symbol('U', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(Symbol('v_x', commutative=True), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Add(Symbol('U', commutative=True), Function('\\\\rho_f')(Symbol('U', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Symbol('E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_x', commutative=True), Integer(2)))), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\pi)} = e^{\\pi}, then obtain - \\eta - e^{\\pi} + \\frac{d}{d \\pi} H{(\\pi)} \\int H{(\\pi)} d\\pi = - \\eta - e^{\\pi} + \\frac{\\partial}{\\partial \\pi} (\\eta + e^{\\pi}) H{(\\pi)}", "derivation": "H{(\\pi)} = e^{\\pi} and \\int H{(\\pi)} d\\pi = \\int e^{\\pi} d\\pi and H{(\\pi)} \\int H{(\\pi)} d\\pi = H{(\\pi)} \\int e^{\\pi} d\\pi and \\frac{d}{d \\pi} H{(\\pi)} \\int H{(\\pi)} d\\pi = \\frac{d}{d \\pi} H{(\\pi)} \\int e^{\\pi} d\\pi and \\frac{d}{d \\pi} H{(\\pi)} \\int H{(\\pi)} d\\pi - \\int e^{\\pi} d\\pi = \\frac{d}{d \\pi} H{(\\pi)} \\int e^{\\pi} d\\pi - \\int e^{\\pi} d\\pi and - \\eta - e^{\\pi} + \\frac{d}{d \\pi} H{(\\pi)} \\int H{(\\pi)} d\\pi = - \\eta - e^{\\pi} + \\frac{\\partial}{\\partial \\pi} (\\eta + e^{\\pi}) H{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["times", 2, "Function('H')(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(Function('H')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(Function('H')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["minus", 4, "Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Derivative(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(Function('H')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))), Add(Derivative(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Integral(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True))), Derivative(Mul(Function('H')(Symbol('\\\\pi', commutative=True)), Integral(Function('H')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True))), Derivative(Mul(Add(Symbol('\\\\eta', commutative=True), exp(Symbol('\\\\pi', commutative=True))), Function('H')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(c)} = e^{\\sin{(c)}} and \\tilde{g}{(c)} = \\int \\hat{p}_0{(c)} dc and \\mathbf{S}{(c)} = \\int \\hat{p}_0{(c)} dc, then obtain (\\frac{\\tilde{g}{(c)}}{\\hat{p}_0{(c)}})^{c} = (\\frac{\\int e^{\\sin{(c)}} dc}{\\hat{p}_0{(c)}})^{c}", "derivation": "\\hat{p}_0{(c)} = e^{\\sin{(c)}} and \\tilde{g}{(c)} = \\int \\hat{p}_0{(c)} dc and \\mathbf{S}{(c)} = \\int \\hat{p}_0{(c)} dc and \\tilde{g}{(c)} = \\mathbf{S}{(c)} and \\mathbf{S}{(c)} = \\int e^{\\sin{(c)}} dc and \\frac{\\mathbf{S}{(c)}}{\\hat{p}_0{(c)}} = \\frac{\\int e^{\\sin{(c)}} dc}{\\hat{p}_0{(c)}} and \\frac{\\tilde{g}{(c)}}{\\hat{p}_0{(c)}} = \\frac{\\int e^{\\sin{(c)}} dc}{\\hat{p}_0{(c)}} and (\\frac{\\tilde{g}{(c)}}{\\hat{p}_0{(c)}})^{c} = (\\frac{\\int e^{\\sin{(c)}} dc}{\\hat{p}_0{(c)}})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), exp(sin(Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('c', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\tilde{g}')(Symbol('c', commutative=True)), Function('\\\\mathbf{S}')(Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('c', commutative=True)), Integral(exp(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["divide", 5, "Function('\\\\hat{p}_0')(Symbol('c', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('c', commutative=True))), Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('c', commutative=True))), Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))))"], [["power", 7, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('c', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(\\phi)} = \\cos{(\\phi)}, then obtain \\frac{\\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\sigma_{p}{(\\phi)} d\\phi}{\\frac{d}{d \\phi} \\cos{(\\phi)}} = \\frac{\\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\cos{(\\phi)} d\\phi}{\\frac{d}{d \\phi} \\cos{(\\phi)}}", "derivation": "\\sigma_{p}{(\\phi)} = \\cos{(\\phi)} and \\frac{d}{d \\phi} \\sigma_{p}{(\\phi)} = \\frac{d}{d \\phi} \\cos{(\\phi)} and \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\sigma_{p}{(\\phi)} = \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\cos{(\\phi)} and \\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\sigma_{p}{(\\phi)} d\\phi = \\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\cos{(\\phi)} d\\phi and \\frac{\\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\sigma_{p}{(\\phi)} d\\phi}{\\frac{d}{d \\phi} \\cos{(\\phi)}} = \\frac{\\int \\sigma_{p}{(\\phi)} \\frac{d}{d \\phi} \\cos{(\\phi)} d\\phi}{\\frac{d}{d \\phi} \\cos{(\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["divide", 4, "Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Function('\\\\sigma_p')(Symbol('\\\\phi', commutative=True)), Derivative(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} = \\mathbf{D}^{G}, then derive \\frac{\\partial}{\\partial G} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} - 1 = \\mathbf{D}^{G} \\log{(\\mathbf{D})} - 1, then obtain \\frac{\\partial}{\\partial G} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} - 1 = \\frac{\\partial}{\\partial G} \\mathbf{D}^{G} - 1", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} = \\mathbf{D}^{G} and - G + \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} = - G + \\mathbf{D}^{G} and \\frac{\\partial}{\\partial G} (- G + \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)}) = \\frac{\\partial}{\\partial G} (- G + \\mathbf{D}^{G}) and \\frac{\\partial}{\\partial G} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} - 1 = \\mathbf{D}^{G} \\log{(\\mathbf{D})} - 1 and \\frac{\\partial}{\\partial G} \\mathbf{D}^{G} - 1 = \\mathbf{D}^{G} \\log{(\\mathbf{D})} - 1 and \\frac{\\partial}{\\partial G} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{D},G)} - 1 = \\frac{\\partial}{\\partial G} \\mathbf{D}^{G} - 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given H{(p,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - p and \\dot{x}{(p,\\hat{H}_{\\lambda})} = (\\hat{H}_{\\lambda} - p)^{2}, then obtain \\dot{x}{(p,\\hat{H}_{\\lambda})} = H^{2}{(p,\\hat{H}_{\\lambda})}", "derivation": "H{(p,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - p and (\\hat{H}_{\\lambda} - p) H{(p,\\hat{H}_{\\lambda})} = (\\hat{H}_{\\lambda} - p)^{2} and \\dot{x}{(p,\\hat{H}_{\\lambda})} = (\\hat{H}_{\\lambda} - p)^{2} and \\dot{x}{(p,\\hat{H}_{\\lambda})} = (\\hat{H}_{\\lambda} - p) H{(p,\\hat{H}_{\\lambda})} and \\dot{x}{(p,\\hat{H}_{\\lambda})} = H^{2}{(p,\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Function('H')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\dot{x}')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))), Function('H')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\dot{x}')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('H')(Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given H{(c,\\mathbf{A})} = \\log{(\\mathbf{A} c)}, then obtain - \\mathbf{A} + (\\mathbf{A} + H{(c,\\mathbf{A})})^{\\mathbf{A}} - H{(c,\\mathbf{A})} = - \\mathbf{A} + (\\mathbf{A} + \\log{(\\mathbf{A} c)})^{\\mathbf{A}} - H{(c,\\mathbf{A})}", "derivation": "H{(c,\\mathbf{A})} = \\log{(\\mathbf{A} c)} and \\mathbf{A} + H{(c,\\mathbf{A})} = \\mathbf{A} + \\log{(\\mathbf{A} c)} and (\\mathbf{A} + H{(c,\\mathbf{A})})^{\\mathbf{A}} = (\\mathbf{A} + \\log{(\\mathbf{A} c)})^{\\mathbf{A}} and - \\mathbf{A} + (\\mathbf{A} + H{(c,\\mathbf{A})})^{\\mathbf{A}} - H{(c,\\mathbf{A})} = - \\mathbf{A} + (\\mathbf{A} + \\log{(\\mathbf{A} c)})^{\\mathbf{A}} - H{(c,\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 3, "Add(Symbol('\\\\mathbf{A}', commutative=True), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(n_{1})} = e^{n_{1}} and \\operatorname{A_{x}}{(n_{1})} = \\sin{(e^{n_{1}})}, then obtain \\operatorname{A_{x}}{(n_{1})} + \\operatorname{A_{x}}^{n_{1}}{(n_{1})} = \\operatorname{A_{x}}{(n_{1})} + \\sin^{n_{1}}{(\\operatorname{n_{2}}{(n_{1})})}", "derivation": "\\operatorname{n_{2}}{(n_{1})} = e^{n_{1}} and \\sin{(\\operatorname{n_{2}}{(n_{1})})} = \\sin{(e^{n_{1}})} and \\operatorname{A_{x}}{(n_{1})} = \\sin{(e^{n_{1}})} and \\operatorname{A_{x}}{(n_{1})} = \\sin{(\\operatorname{n_{2}}{(n_{1})})} and \\operatorname{A_{x}}^{n_{1}}{(n_{1})} = \\sin^{n_{1}}{(\\operatorname{n_{2}}{(n_{1})})} and \\operatorname{A_{x}}{(n_{1})} + \\operatorname{A_{x}}^{n_{1}}{(n_{1})} = \\operatorname{A_{x}}{(n_{1})} + \\sin^{n_{1}}{(\\operatorname{n_{2}}{(n_{1})})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["sin", 1], "Equality(sin(Function('n_2')(Symbol('n_1', commutative=True))), sin(exp(Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('n_1', commutative=True)), sin(exp(Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('A_x')(Symbol('n_1', commutative=True)), sin(Function('n_2')(Symbol('n_1', commutative=True))))"], [["power", 4, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(sin(Function('n_2')(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["add", 5, "Function('A_x')(Symbol('n_1', commutative=True))"], "Equality(Add(Function('A_x')(Symbol('n_1', commutative=True)), Pow(Function('A_x')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Add(Function('A_x')(Symbol('n_1', commutative=True)), Pow(sin(Function('n_2')(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given W{(f,i,\\Psi^{\\dagger})} = i (- \\Psi^{\\dagger} + f) and \\varepsilon_{0}{(f,i,\\Psi^{\\dagger})} = (i (- \\Psi^{\\dagger} + f))^{f} and \\operatorname{A_{y}}{(\\Psi^{\\dagger})} = - \\Psi^{\\dagger}, then obtain (i (f + \\operatorname{A_{y}}{(\\Psi^{\\dagger})}))^{f} = (i (- \\Psi^{\\dagger} + f))^{f}", "derivation": "W{(f,i,\\Psi^{\\dagger})} = i (- \\Psi^{\\dagger} + f) and W^{f}{(f,i,\\Psi^{\\dagger})} = (i (- \\Psi^{\\dagger} + f))^{f} and \\varepsilon_{0}{(f,i,\\Psi^{\\dagger})} = (i (- \\Psi^{\\dagger} + f))^{f} and W^{f}{(f,i,\\Psi^{\\dagger})} = \\varepsilon_{0}{(f,i,\\Psi^{\\dagger})} and \\operatorname{A_{y}}{(\\Psi^{\\dagger})} = - \\Psi^{\\dagger} and \\varepsilon_{0}{(f,i,\\Psi^{\\dagger})} = (i (f + \\operatorname{A_{y}}{(\\Psi^{\\dagger})}))^{f} and W^{f}{(f,i,\\Psi^{\\dagger})} = (i (f + \\operatorname{A_{y}}{(\\Psi^{\\dagger})}))^{f} and (i (f + \\operatorname{A_{y}}{(\\Psi^{\\dagger})}))^{f} = (i (- \\Psi^{\\dagger} + f))^{f}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('W')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('W')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True)), Function('\\\\varepsilon_0')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\varepsilon_0')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Symbol('f', commutative=True), Function('A_y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Function('W')(Symbol('f', commutative=True), Symbol('i', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Symbol('f', commutative=True), Function('A_y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 7], "Equality(Pow(Mul(Symbol('i', commutative=True), Add(Symbol('f', commutative=True), Function('A_y')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('f', commutative=True)), Pow(Mul(Symbol('i', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)} = \\Omega + \\eta^{\\prime} and \\mathbf{v}{(g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda}^{g_{\\varepsilon}}, then obtain \\mathbf{v}{(g_{\\varepsilon},E_{\\lambda})} + \\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)} = E_{\\lambda}^{g_{\\varepsilon}} + \\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)}", "derivation": "\\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)} = \\Omega + \\eta^{\\prime} and \\mathbf{v}{(g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda}^{g_{\\varepsilon}} and \\Omega + \\eta^{\\prime} + \\mathbf{v}{(g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda}^{g_{\\varepsilon}} + \\Omega + \\eta^{\\prime} and \\mathbf{v}{(g_{\\varepsilon},E_{\\lambda})} + \\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)} = E_{\\lambda}^{g_{\\varepsilon}} + \\operatorname{m_{s}}{(\\eta^{\\prime},\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\Omega', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('m_s')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('m_s')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(A)} = e^{\\cos{(A)}} and \\theta_{1}{(A)} = \\cos{(A)}, then obtain e^{\\cos{(A)}} = e^{\\theta_{1}{(A)}}", "derivation": "\\operatorname{C_{1}}{(A)} = e^{\\cos{(A)}} and \\theta_{1}{(A)} = \\cos{(A)} and \\operatorname{C_{1}}{(A)} = e^{\\theta_{1}{(A)}} and e^{\\cos{(A)}} = e^{\\theta_{1}{(A)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_1')(Symbol('A', commutative=True)), exp(Function('\\\\theta_1')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(cos(Symbol('A', commutative=True))), exp(Function('\\\\theta_1')(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(i,p)} = i + p, then obtain i (i + p) + p (i + p) = i^{2} + 2 i p + p^{2}", "derivation": "\\operatorname{M_{E}}{(i,p)} = i + p and (i + p) \\operatorname{M_{E}}{(i,p)} = (i + p)^{2} and i \\operatorname{M_{E}}{(i,p)} + p \\operatorname{M_{E}}{(i,p)} = i^{2} + 2 i p + p^{2} and i (i + p) + p (i + p) = i^{2} + 2 i p + p^{2}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('i', commutative=True), Symbol('p', commutative=True)), Add(Symbol('i', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Add(Symbol('i', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Add(Symbol('i', commutative=True), Symbol('p', commutative=True)), Function('M_E')(Symbol('i', commutative=True), Symbol('p', commutative=True))), Pow(Add(Symbol('i', commutative=True), Symbol('p', commutative=True)), Integer(2)))"], [["expand", 2], "Equality(Add(Mul(Symbol('i', commutative=True), Function('M_E')(Symbol('i', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Function('M_E')(Symbol('i', commutative=True), Symbol('p', commutative=True)))), Add(Pow(Symbol('i', commutative=True), Integer(2)), Mul(Integer(2), Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Symbol('i', commutative=True), Add(Symbol('i', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Add(Symbol('i', commutative=True), Symbol('p', commutative=True)))), Add(Pow(Symbol('i', commutative=True), Integer(2)), Mul(Integer(2), Symbol('i', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(g,A,\\Omega)} = A + \\Omega - g, then obtain (\\int \\operatorname{F_{c}}{(g,A,\\Omega)} dA)^{A} - \\frac{A + \\Omega - g}{g} = (\\frac{A^{2}}{2} + A (\\Omega - g) + \\dot{\\mathbf{r}})^{A} - \\frac{A + \\Omega - g}{g}", "derivation": "\\operatorname{F_{c}}{(g,A,\\Omega)} = A + \\Omega - g and \\int \\operatorname{F_{c}}{(g,A,\\Omega)} dA = \\int (A + \\Omega - g) dA and (\\int \\operatorname{F_{c}}{(g,A,\\Omega)} dA)^{A} = (\\int (A + \\Omega - g) dA)^{A} and (\\int \\operatorname{F_{c}}{(g,A,\\Omega)} dA)^{A} - \\frac{A + \\Omega - g}{g} = (\\int (A + \\Omega - g) dA)^{A} - \\frac{A + \\Omega - g}{g} and (\\int \\operatorname{F_{c}}{(g,A,\\Omega)} dA)^{A} - \\frac{A + \\Omega - g}{g} = (\\frac{A^{2}}{2} + A (\\Omega - g) + \\dot{\\mathbf{r}})^{A} - \\frac{A + \\Omega - g}{g}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('g', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('g', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Function('F_c')(Symbol('g', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], "Equality(Add(Pow(Integral(Function('F_c')(Symbol('g', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))), Add(Pow(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Pow(Integral(Function('F_c')(Symbol('g', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Mul(Symbol('A', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(n)} = \\cos{(\\sin{(n)})}, then obtain \\frac{((\\int \\operatorname{g_{\\varepsilon}}{(n)} dn)^{n}) (\\int \\cos{(\\sin{(n)})} dn)^{- n}}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn} = \\frac{1}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn}", "derivation": "\\operatorname{g_{\\varepsilon}}{(n)} = \\cos{(\\sin{(n)})} and \\int \\operatorname{g_{\\varepsilon}}{(n)} dn = \\int \\cos{(\\sin{(n)})} dn and (\\int \\operatorname{g_{\\varepsilon}}{(n)} dn)^{n} = (\\int \\cos{(\\sin{(n)})} dn)^{n} and \\frac{(\\int \\operatorname{g_{\\varepsilon}}{(n)} dn)^{n}}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn} = \\frac{(\\int \\cos{(\\sin{(n)})} dn)^{n}}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn} and \\frac{((\\int \\operatorname{g_{\\varepsilon}}{(n)} dn)^{n}) (\\int \\cos{(\\sin{(n)})} dn)^{- n}}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn} = \\frac{1}{\\int \\operatorname{g_{\\varepsilon}}{(n)} dn}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), cos(sin(Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Integral(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["divide", 3, "Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True))), Mul(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Pow(Integral(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True))))"], [["divide", 4, "Pow(Integral(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True))"], "Equality(Mul(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Integral(cos(sin(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)))), Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})}, then obtain A_{2} - f^{\\prime} + (- \\Psi^{\\dagger} - f^{\\prime} + \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})})^{f^{\\prime}} = A_{2} - f^{\\prime} + (- \\Psi^{\\dagger} - f^{\\prime} + \\cos{(\\Psi^{\\dagger})})^{f^{\\prime}}", "derivation": "\\hat{\\mathbf{r}}{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})} and - f^{\\prime} + \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})} = - f^{\\prime} + \\cos{(\\Psi^{\\dagger})} and - \\Psi^{\\dagger} - f^{\\prime} + \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})} = - \\Psi^{\\dagger} - f^{\\prime} + \\cos{(\\Psi^{\\dagger})} and (- \\Psi^{\\dagger} - f^{\\prime} + \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})})^{f^{\\prime}} = (- \\Psi^{\\dagger} - f^{\\prime} + \\cos{(\\Psi^{\\dagger})})^{f^{\\prime}} and A_{2} - f^{\\prime} + (- \\Psi^{\\dagger} - f^{\\prime} + \\hat{\\mathbf{r}}{(\\Psi^{\\dagger})})^{f^{\\prime}} = A_{2} - f^{\\prime} + (- \\Psi^{\\dagger} - f^{\\prime} + \\cos{(\\Psi^{\\dagger})})^{f^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["minus", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 4, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given u{(\\mathbf{E},\\mu_0)} = \\mu_0^{\\mathbf{E}} and \\mathbf{A}{(l)} = \\cos{(l)} and \\varepsilon{(\\mathbf{E},\\mu_0,l)} = \\cos{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2}, then obtain \\frac{\\partial}{\\partial l} \\varepsilon{(\\mathbf{E},\\mu_0,l)} = \\frac{\\partial}{\\partial l} (\\mathbf{A}{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2})", "derivation": "u{(\\mathbf{E},\\mu_0)} = \\mu_0^{\\mathbf{E}} and \\mathbf{A}{(l)} = \\cos{(l)} and \\mathbf{A}{(l)} - \\frac{1}{\\mu_0^{\\mathbf{E}} + u{(\\mathbf{E},\\mu_0)}} = \\cos{(l)} - \\frac{1}{\\mu_0^{\\mathbf{E}} + u{(\\mathbf{E},\\mu_0)}} and \\mathbf{A}{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2} = \\cos{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2} and \\varepsilon{(\\mathbf{E},\\mu_0,l)} = \\cos{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2} and \\frac{\\partial}{\\partial l} \\varepsilon{(\\mathbf{E},\\mu_0,l)} = \\frac{\\partial}{\\partial l} (\\cos{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2}) and \\frac{\\partial}{\\partial l} \\varepsilon{(\\mathbf{E},\\mu_0,l)} = \\frac{\\partial}{\\partial l} (\\mathbf{A}{(l)} - \\frac{\\mu_0^{- \\mathbf{E}}}{2})", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["minus", 2, "Pow(Add(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Add(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(-1)))), Add(cos(Symbol('l', commutative=True)), Mul(Integer(-1), Pow(Add(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('l', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))), Add(cos(Symbol('l', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('l', commutative=True)), Add(cos(Symbol('l', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))))"], [["differentiate", 5, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('l', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{A}')(Symbol('l', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(\\chi)} = \\cos{(\\chi)} and \\sigma_{p}{(\\Psi)} = \\sin{(\\Psi)}, then obtain (- \\chi + \\cos{(\\chi)}) \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} = (- \\chi + \\cos{(\\chi)}) \\frac{d}{d \\Psi} \\sin{(\\Psi)}", "derivation": "v{(\\chi)} = \\cos{(\\chi)} and - \\chi + v{(\\chi)} = - \\chi + \\cos{(\\chi)} and \\sigma_{p}{(\\Psi)} = \\sin{(\\Psi)} and \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\Psi)} and (- \\chi + v{(\\chi)}) \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} = (- \\chi + v{(\\chi)}) \\frac{d}{d \\Psi} \\sin{(\\Psi)} and (- \\chi + \\cos{(\\chi)}) \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} = (- \\chi + \\cos{(\\chi)}) \\frac{d}{d \\Psi} \\sin{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('v')(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))))"], ["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('v')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('v')(Symbol('\\\\chi', commutative=True))), Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('v')(Symbol('\\\\chi', commutative=True))), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Derivative(sin(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(r,\\varphi,\\rho)} = \\rho^{r} \\varphi, then obtain \\frac{- \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} \\lambda^{\\rho}{(r,\\varphi,\\rho)}}{- \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} (\\rho^{r} \\varphi)^{\\rho}} = 1", "derivation": "\\lambda{(r,\\varphi,\\rho)} = \\rho^{r} \\varphi and \\lambda^{\\rho}{(r,\\varphi,\\rho)} = (\\rho^{r} \\varphi)^{\\rho} and \\frac{\\partial}{\\partial \\varphi} \\lambda^{\\rho}{(r,\\varphi,\\rho)} = \\frac{\\partial}{\\partial \\varphi} (\\rho^{r} \\varphi)^{\\rho} and - \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} \\lambda^{\\rho}{(r,\\varphi,\\rho)} = - \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} (\\rho^{r} \\varphi)^{\\rho} and \\frac{- \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} \\lambda^{\\rho}{(r,\\varphi,\\rho)}}{- \\rho^{r} + \\frac{\\partial}{\\partial \\varphi} (\\rho^{r} \\varphi)^{\\rho}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["minus", 3, "Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))), Derivative(Pow(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))), Derivative(Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["divide", 4, "Add(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))), Derivative(Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))), Derivative(Pow(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True))), Derivative(Pow(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Integer(1))"]]}, {"prompt": "Given H{(\\mathbf{J}_P,n)} = n^{\\mathbf{J}_P} and \\operatorname{f_{E}}{(C_{2})} = \\log{(\\sin{(C_{2})})}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_P} H{(\\mathbf{J}_P,n)} = n^{\\mathbf{J}_P} \\log{(n)}, then obtain \\operatorname{f_{E}}{(C_{2})} \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P} = \\log{(\\sin{(C_{2})})} \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P}", "derivation": "H{(\\mathbf{J}_P,n)} = n^{\\mathbf{J}_P} and \\frac{\\partial}{\\partial \\mathbf{J}_P} H{(\\mathbf{J}_P,n)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P} and \\frac{\\partial}{\\partial \\mathbf{J}_P} H{(\\mathbf{J}_P,n)} = n^{\\mathbf{J}_P} \\log{(n)} and \\operatorname{f_{E}}{(C_{2})} = \\log{(\\sin{(C_{2})})} and n^{\\mathbf{J}_P} \\operatorname{f_{E}}{(C_{2})} \\log{(n)} = n^{\\mathbf{J}_P} \\log{(n)} \\log{(\\sin{(C_{2})})} and \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P} = n^{\\mathbf{J}_P} \\log{(n)} and \\operatorname{f_{E}}{(C_{2})} \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P} = \\log{(\\sin{(C_{2})})} \\frac{\\partial}{\\partial \\mathbf{J}_P} n^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('n', commutative=True))))"], ["get_premise", "Equality(Function('f_E')(Symbol('C_2', commutative=True)), log(sin(Symbol('C_2', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('n', commutative=True)))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('f_E')(Symbol('C_2', commutative=True)), log(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('n', commutative=True)), log(sin(Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(log(sin(Symbol('C_2', commutative=True))), Derivative(Pow(Symbol('n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"]]}, {"prompt": "Given c{(\\psi^*)} = \\int \\sin{(\\psi^*)} d\\psi^*, then derive c{(\\psi^*)} = \\mathbf{B} - \\cos{(\\psi^*)}, then obtain \\sin{((\\mathbf{B} - \\cos{(\\psi^*)})^{\\psi^*})} = \\sin{((\\int \\sin{(\\psi^*)} d\\psi^*)^{\\psi^*})}", "derivation": "c{(\\psi^*)} = \\int \\sin{(\\psi^*)} d\\psi^* and c^{\\psi^*}{(\\psi^*)} = (\\int \\sin{(\\psi^*)} d\\psi^*)^{\\psi^*} and \\sin{(c^{\\psi^*}{(\\psi^*)})} = \\sin{((\\int \\sin{(\\psi^*)} d\\psi^*)^{\\psi^*})} and c{(\\psi^*)} = \\mathbf{B} - \\cos{(\\psi^*)} and \\sin{((\\mathbf{B} - \\cos{(\\psi^*)})^{\\psi^*})} = \\sin{((\\int \\sin{(\\psi^*)} d\\psi^*)^{\\psi^*})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\psi^*', commutative=True)), Integral(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Integral(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('c')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), sin(Pow(Integral(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('c')(Symbol('\\\\psi^*', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(sin(Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True))), sin(Pow(Integral(sin(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\psi{(\\lambda,a)} = \\lambda^{a}, then obtain \\frac{\\lambda^{- a} \\frac{\\partial}{\\partial \\lambda} \\int a \\psi^{2}{(\\lambda,a)} d\\lambda}{a} = \\frac{\\lambda^{- a} \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{2 a} a d\\lambda}{a}", "derivation": "\\psi{(\\lambda,a)} = \\lambda^{a} and \\lambda^{a} \\psi{(\\lambda,a)} = \\lambda^{2 a} and a \\psi{(\\lambda,a)} = \\lambda^{a} a and \\lambda^{a} a \\psi{(\\lambda,a)} = \\lambda^{2 a} a and a \\psi^{2}{(\\lambda,a)} = \\lambda^{a} a \\psi{(\\lambda,a)} and a \\psi^{2}{(\\lambda,a)} = \\lambda^{2 a} a and \\int a \\psi^{2}{(\\lambda,a)} d\\lambda = \\int \\lambda^{2 a} a d\\lambda and \\frac{\\partial}{\\partial \\lambda} \\int a \\psi^{2}{(\\lambda,a)} d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{2 a} a d\\lambda and \\frac{\\lambda^{- a} \\frac{\\partial}{\\partial \\lambda} \\int a \\psi^{2}{(\\lambda,a)} d\\lambda}{a} = \\frac{\\lambda^{- a} \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{2 a} a d\\lambda}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True))), Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], [["times", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True), Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["integrate", 6, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Symbol('a', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Integer(2))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('a', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Integer(2))), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["divide", 8, "Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Pow(Symbol('a', commutative=True), Integer(-1)), Derivative(Integral(Mul(Symbol('a', commutative=True), Pow(Function('\\\\psi')(Symbol('\\\\lambda', commutative=True), Symbol('a', commutative=True)), Integer(2))), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Pow(Symbol('a', commutative=True), Integer(-1)), Derivative(Integral(Mul(Pow(Symbol('\\\\lambda', commutative=True), Mul(Integer(2), Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\delta,m_{s})} = \\delta^{m_{s}}, then obtain \\frac{\\mathbf{D}{(\\delta,m_{s})}}{m_{s}^{3}} = \\frac{\\delta^{m_{s}}}{m_{s}^{3}}", "derivation": "\\mathbf{D}{(\\delta,m_{s})} = \\delta^{m_{s}} and \\frac{\\mathbf{D}{(\\delta,m_{s})}}{m_{s}} = \\frac{\\delta^{m_{s}}}{m_{s}} and \\frac{\\mathbf{D}{(\\delta,m_{s})}}{m_{s}^{2}} = \\frac{\\delta^{m_{s}}}{m_{s}^{2}} and \\frac{\\mathbf{D}{(\\delta,m_{s})}}{m_{s}^{3}} = \\frac{\\delta^{m_{s}}}{m_{s}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)))"], [["divide", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["times", 2, "Pow(Symbol('m_s', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-2)), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Integer(-2))))"], [["divide", 3, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-3)), Function('\\\\mathbf{D}')(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('m_s', commutative=True)), Pow(Symbol('m_s', commutative=True), Integer(-3))))"]]}, {"prompt": "Given i{(U,\\lambda)} = \\frac{\\partial}{\\partial \\lambda} U^{\\lambda}, then derive \\int U i{(U,\\lambda)} d\\lambda = U U^{\\lambda} + \\tilde{g}^*, then obtain \\iint U \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} d\\lambda d\\lambda = \\int (U U^{\\lambda} + \\tilde{g}^*) d\\lambda", "derivation": "i{(U,\\lambda)} = \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} and U i{(U,\\lambda)} = U \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} and \\int U i{(U,\\lambda)} d\\lambda = \\int U \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} d\\lambda and \\int U i{(U,\\lambda)} d\\lambda = U U^{\\lambda} + \\tilde{g}^* and \\int U \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} d\\lambda = U U^{\\lambda} + \\tilde{g}^* and \\iint U \\frac{\\partial}{\\partial \\lambda} U^{\\lambda} d\\lambda d\\lambda = \\int (U U^{\\lambda} + \\tilde{g}^*) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('U', commutative=True), Derivative(Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Symbol('U', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Derivative(Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('U', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Symbol('U', commutative=True), Derivative(Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Mul(Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Symbol('U', commutative=True), Derivative(Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Mul(Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)}, then derive \\frac{\\sigma_p \\operatorname{f_{E}}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = \\frac{\\sigma_p \\cos{(\\sigma_p)}}{\\sin{(\\sigma_p)}}, then obtain \\frac{\\sigma_p \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = \\frac{\\sigma_p \\cos{(\\sigma_p)}}{\\sin{(\\sigma_p)}}", "derivation": "\\operatorname{f_{E}}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)} and \\sigma_p \\operatorname{f_{E}}{(\\sigma_p)} = \\sigma_p \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)} and \\frac{\\sigma_p \\operatorname{f_{E}}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = \\frac{\\sigma_p \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)}}{\\sin{(\\sigma_p)}} and \\frac{\\sigma_p \\operatorname{f_{E}}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = \\frac{\\sigma_p \\cos{(\\sigma_p)}}{\\sin{(\\sigma_p)}} and \\frac{\\sigma_p \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = \\frac{\\sigma_p \\cos{(\\sigma_p)}}{\\sin{(\\sigma_p)}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\sigma_p', commutative=True)), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('f_E')(Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('\\\\sigma_p', commutative=True), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["divide", 2, "sin(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('f_E')(Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('f_E')(Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(x)} = \\sin{(x)}, then obtain \\int (x \\varepsilon{(x)} + x) dx = \\mathbf{S} + \\frac{x^{2}}{2} - x \\cos{(x)} + \\sin{(x)}", "derivation": "\\varepsilon{(x)} = \\sin{(x)} and x \\varepsilon{(x)} = x \\sin{(x)} and x \\varepsilon{(x)} + x = x \\sin{(x)} + x and \\int (x \\varepsilon{(x)} + x) dx = \\int (x \\sin{(x)} + x) dx and \\int (x \\varepsilon{(x)} + x) dx = \\mathbf{S} + \\frac{x^{2}}{2} - x \\cos{(x)} + \\sin{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('\\\\varepsilon')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), sin(Symbol('x', commutative=True))))"], [["add", 2, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Symbol('x', commutative=True), Function('\\\\varepsilon')(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Add(Mul(Symbol('x', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('x', commutative=True), Function('\\\\varepsilon')(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Mul(Symbol('x', commutative=True), sin(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Symbol('x', commutative=True), Function('\\\\varepsilon')(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('x', commutative=True), cos(Symbol('x', commutative=True))), sin(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(A_{y},v_{1})} = A_{y} + v_{1}, then obtain - A_{y} - v_{1} + e^{- A_{y} - v_{1} + \\bar{\\h}{(A_{y},v_{1})}} = - A_{y} - v_{1} + 1", "derivation": "\\bar{\\h}{(A_{y},v_{1})} = A_{y} + v_{1} and - A_{y} - v_{1} + \\bar{\\h}{(A_{y},v_{1})} = 0 and e^{- A_{y} - v_{1} + \\bar{\\h}{(A_{y},v_{1})}} = 1 and - A_{y} - v_{1} + e^{- A_{y} - v_{1} + \\bar{\\h}{(A_{y},v_{1})}} = - A_{y} - v_{1} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True)))"], [["minus", 1, "Add(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hbar')(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hbar')(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True)))), Integer(1))"], [["minus", 3, "Add(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('\\\\hbar')(Symbol('A_y', commutative=True), Symbol('v_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), Integer(1)))"]]}, {"prompt": "Given L{(P_{g})} = e^{P_{g}}, then obtain L{(P_{g})} e^{- P_{g}} = 1", "derivation": "L{(P_{g})} = e^{P_{g}} and P_{g} + L{(P_{g})} = P_{g} + e^{P_{g}} and \\frac{L{(P_{g})}}{P_{g} + e^{P_{g}}} = \\frac{e^{P_{g}}}{P_{g} + e^{P_{g}}} and \\frac{L{(P_{g})}}{P_{g} + L{(P_{g})}} = \\frac{e^{P_{g}}}{P_{g} + L{(P_{g})}} and \\frac{L^{2}{(P_{g})}}{(P_{g} + L{(P_{g})}) (P_{g} + e^{P_{g}})} = \\frac{L{(P_{g})} e^{P_{g}}}{(P_{g} + L{(P_{g})}) (P_{g} + e^{P_{g}})} and \\frac{(P_{g} + L{(P_{g})}) L{(P_{g})} e^{- P_{g}}}{P_{g} + e^{P_{g}}} = \\frac{P_{g} + L{(P_{g})}}{P_{g} + e^{P_{g}}} and L{(P_{g})} e^{- P_{g}} = 1", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], [["add", 1, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))))"], [["divide", 1, "Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True))), Mul(Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), exp(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True))), Mul(Pow(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), exp(Symbol('P_g', commutative=True))))"], [["times", 4, "Mul(Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), Pow(Function('L')(Symbol('P_g', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True))))"], [["divide", 5, "Mul(Pow(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-2)), Function('L')(Symbol('P_g', commutative=True)), exp(Symbol('P_g', commutative=True)))"], "Equality(Mul(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True)), exp(Mul(Integer(-1), Symbol('P_g', commutative=True)))), Mul(Add(Symbol('P_g', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Pow(Add(Symbol('P_g', commutative=True), exp(Symbol('P_g', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Function('L')(Symbol('P_g', commutative=True)), exp(Mul(Integer(-1), Symbol('P_g', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\hat{H}{(\\hat{x}_0,Z)} = Z + \\hat{x}_0 and \\mathbf{p}{(\\hat{x}_0,Z)} = Z + \\hat{x}_0, then obtain \\frac{d}{d \\hat{x}_0} 1 = \\frac{\\partial}{\\partial \\hat{x}_0} (\\frac{\\hat{H}{(\\hat{x}_0,Z)}}{\\mathbf{p}{(\\hat{x}_0,Z)}})^{\\hat{x}_0}", "derivation": "\\hat{H}{(\\hat{x}_0,Z)} = Z + \\hat{x}_0 and \\mathbf{p}{(\\hat{x}_0,Z)} = Z + \\hat{x}_0 and \\mathbf{p}{(\\hat{x}_0,Z)} = \\hat{H}{(\\hat{x}_0,Z)} and \\frac{\\mathbf{p}{(\\hat{x}_0,Z)}}{Z + \\hat{x}_0} = \\frac{\\hat{H}{(\\hat{x}_0,Z)}}{Z + \\hat{x}_0} and 1 = \\frac{\\hat{H}{(\\hat{x}_0,Z)}}{\\mathbf{p}{(\\hat{x}_0,Z)}} and 1 = (\\frac{\\hat{H}{(\\hat{x}_0,Z)}}{\\mathbf{p}{(\\hat{x}_0,Z)}})^{\\hat{x}_0} and \\frac{d}{d \\hat{x}_0} 1 = \\frac{\\partial}{\\partial \\hat{x}_0} (\\frac{\\hat{H}{(\\hat{x}_0,Z)}}{\\mathbf{p}{(\\hat{x}_0,Z)}})^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)))"], [["divide", 3, "Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Mul(Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integer(1), Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 6, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Function('\\\\hat{H}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(T,Z)} = T + Z, then obtain - A + T + Z + \\operatorname{z^{*}}{(T,Z)} = - A + 2 \\operatorname{z^{*}}{(T,Z)}", "derivation": "\\operatorname{z^{*}}{(T,Z)} = T + Z and T + Z + \\operatorname{z^{*}}{(T,Z)} = 2 T + 2 Z and 2 \\operatorname{z^{*}}{(T,Z)} = 2 T + 2 Z and - A + T + Z + \\operatorname{z^{*}}{(T,Z)} = - A + 2 T + 2 Z and - A + T + Z + \\operatorname{z^{*}}{(T,Z)} = - A + 2 \\operatorname{z^{*}}{(T,Z)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('T', commutative=True), Symbol('Z', commutative=True)))"], [["add", 1, "Add(Symbol('T', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Symbol('Z', commutative=True), Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["minus", 2, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('T', commutative=True), Symbol('Z', commutative=True), Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('T', commutative=True), Symbol('Z', commutative=True), Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(2), Function('z^*')(Symbol('T', commutative=True), Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(n_{1},F_{H})} = e^{F_{H} n_{1}}, then obtain n_{1} + \\frac{\\operatorname{E_{n}}{(n_{1},F_{H})}}{- F_{H} + e^{F_{H} n_{1}}} = n_{1} + \\frac{e^{F_{H} n_{1}}}{- F_{H} + e^{F_{H} n_{1}}}", "derivation": "\\operatorname{E_{n}}{(n_{1},F_{H})} = e^{F_{H} n_{1}} and - F_{H} + \\operatorname{E_{n}}{(n_{1},F_{H})} = - F_{H} + e^{F_{H} n_{1}} and \\frac{\\operatorname{E_{n}}{(n_{1},F_{H})}}{- F_{H} + \\operatorname{E_{n}}{(n_{1},F_{H})}} = \\frac{e^{F_{H} n_{1}}}{- F_{H} + \\operatorname{E_{n}}{(n_{1},F_{H})}} and n_{1} + \\frac{\\operatorname{E_{n}}{(n_{1},F_{H})}}{- F_{H} + \\operatorname{E_{n}}{(n_{1},F_{H})}} = n_{1} + \\frac{e^{F_{H} n_{1}}}{- F_{H} + \\operatorname{E_{n}}{(n_{1},F_{H})}} and n_{1} + \\frac{\\operatorname{E_{n}}{(n_{1},F_{H})}}{- F_{H} + e^{F_{H} n_{1}}} = n_{1} + \\frac{e^{F_{H} n_{1}}}{- F_{H} + e^{F_{H} n_{1}}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True))))"], [["minus", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True)))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Integer(-1)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Integer(-1)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True)))))"], [["add", 3, "Symbol('n_1', commutative=True)"], "Equality(Add(Symbol('n_1', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Integer(-1)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True)))), Add(Symbol('n_1', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True))), Integer(-1)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('n_1', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True)))), Integer(-1)), Function('E_n')(Symbol('n_1', commutative=True), Symbol('F_H', commutative=True)))), Add(Symbol('n_1', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True)))), Integer(-1)), exp(Mul(Symbol('F_H', commutative=True), Symbol('n_1', commutative=True))))))"]]}, {"prompt": "Given A{(\\chi,E_{\\lambda})} = \\cos{(E_{\\lambda} \\chi)} and \\operatorname{y^{\\prime}}{(\\chi,E_{\\lambda})} = A^{2}{(\\chi,E_{\\lambda})}, then obtain \\operatorname{y^{\\prime}}{(\\chi,E_{\\lambda})} = \\cos^{2}{(E_{\\lambda} \\chi)}", "derivation": "A{(\\chi,E_{\\lambda})} = \\cos{(E_{\\lambda} \\chi)} and A^{2}{(\\chi,E_{\\lambda})} = A{(\\chi,E_{\\lambda})} \\cos{(E_{\\lambda} \\chi)} and \\operatorname{y^{\\prime}}{(\\chi,E_{\\lambda})} = A^{2}{(\\chi,E_{\\lambda})} and \\operatorname{y^{\\prime}}{(\\chi,E_{\\lambda})} = A{(\\chi,E_{\\lambda})} \\cos{(E_{\\lambda} \\chi)} and \\operatorname{y^{\\prime}}{(\\chi,E_{\\lambda})} = \\cos^{2}{(E_{\\lambda} \\chi)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\chi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(cos(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{\\mathbf{x}},P_{e})} = P_{e} \\hat{\\mathbf{x}} and \\chi{(\\hat{x}_0,v_{2})} = \\frac{\\hat{x}_0}{v_{2}}, then obtain - (P_{e} \\hat{\\mathbf{x}})^{P_{e}} + \\chi{(\\hat{x}_0,v_{2})} = \\frac{\\hat{x}_0}{v_{2}} - (P_{e} \\hat{\\mathbf{x}})^{P_{e}}", "derivation": "\\operatorname{E_{n}}{(\\hat{\\mathbf{x}},P_{e})} = P_{e} \\hat{\\mathbf{x}} and \\operatorname{E_{n}}^{P_{e}}{(\\hat{\\mathbf{x}},P_{e})} = (P_{e} \\hat{\\mathbf{x}})^{P_{e}} and \\chi{(\\hat{x}_0,v_{2})} = \\frac{\\hat{x}_0}{v_{2}} and - \\operatorname{E_{n}}^{P_{e}}{(\\hat{\\mathbf{x}},P_{e})} + \\chi{(\\hat{x}_0,v_{2})} = \\frac{\\hat{x}_0}{v_{2}} - \\operatorname{E_{n}}^{P_{e}}{(\\hat{\\mathbf{x}},P_{e})} and - (P_{e} \\hat{\\mathbf{x}})^{P_{e}} + \\chi{(\\hat{x}_0,v_{2})} = \\frac{\\hat{x}_0}{v_{2}} - (P_{e} \\hat{\\mathbf{x}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('P_e', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(Mul(Symbol('P_e', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('P_e', commutative=True)))"], ["get_premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))"], [["minus", 3, "Pow(Function('E_n')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Function('\\\\chi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('P_e', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('P_e', commutative=True))), Function('\\\\chi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('P_e', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(f)} = \\sin{(f)}, then obtain (\\Psi_{\\lambda}{(f)} \\sin{(f)} - \\Psi_{\\lambda}{(f)})^{f} - 1 = (- \\Psi_{\\lambda}{(f)} + \\sin^{2}{(f)})^{f} - 1", "derivation": "\\Psi_{\\lambda}{(f)} = \\sin{(f)} and \\Psi_{\\lambda}{(f)} \\sin{(f)} = \\sin^{2}{(f)} and \\Psi_{\\lambda}{(f)} \\sin{(f)} - \\Psi_{\\lambda}{(f)} = - \\Psi_{\\lambda}{(f)} + \\sin^{2}{(f)} and (\\Psi_{\\lambda}{(f)} \\sin{(f)} - \\Psi_{\\lambda}{(f)})^{f} = (- \\Psi_{\\lambda}{(f)} + \\sin^{2}{(f)})^{f} and (\\Psi_{\\lambda}{(f)} \\sin{(f)} - \\Psi_{\\lambda}{(f)})^{f} - 1 = (- \\Psi_{\\lambda}{(f)} + \\sin^{2}{(f)})^{f} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["times", 1, "sin(Symbol('f', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Integer(2)))"], [["minus", 2, "Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True))"], "Equality(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Integer(2))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Integer(2))), Symbol('f', commutative=True)))"], [["add", 4, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))), Symbol('f', commutative=True)), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True))), Pow(sin(Symbol('f', commutative=True)), Integer(2))), Symbol('f', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbb{I}{(t_{2},F_{H})} = \\log{(F_{H} t_{2})}, then obtain \\int (\\mathbb{I}^{F_{H}}{(t_{2},F_{H})})^{t_{2}} dF_{H} = \\int (\\log{(F_{H} t_{2})}^{F_{H}})^{t_{2}} dF_{H}", "derivation": "\\mathbb{I}{(t_{2},F_{H})} = \\log{(F_{H} t_{2})} and \\mathbb{I}^{F_{H}}{(t_{2},F_{H})} = \\log{(F_{H} t_{2})}^{F_{H}} and (\\mathbb{I}^{F_{H}}{(t_{2},F_{H})})^{t_{2}} = (\\log{(F_{H} t_{2})}^{F_{H}})^{t_{2}} and \\int (\\mathbb{I}^{F_{H}}{(t_{2},F_{H})})^{t_{2}} dF_{H} = \\int (\\log{(F_{H} t_{2})}^{F_{H}})^{t_{2}} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('t_2', commutative=True), Symbol('F_H', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('t_2', commutative=True), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(log(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))), Symbol('F_H', commutative=True)))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbb{I}')(Symbol('t_2', commutative=True), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('t_2', commutative=True)), Pow(Pow(log(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))), Symbol('F_H', commutative=True)), Symbol('t_2', commutative=True)))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\mathbb{I}')(Symbol('t_2', commutative=True), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Pow(log(Mul(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True))), Symbol('F_H', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(m)} = \\log{(m)}, then derive \\int \\operatorname{F_{N}}{(m)} dm = F_{H} + m \\log{(m)} - m, then obtain \\frac{d}{d m} \\int \\operatorname{F_{N}}{(m)} dm = \\frac{\\partial}{\\partial m} (F_{H} + m \\operatorname{F_{N}}{(m)} - m)", "derivation": "\\operatorname{F_{N}}{(m)} = \\log{(m)} and \\int \\operatorname{F_{N}}{(m)} dm = \\int \\log{(m)} dm and \\int \\operatorname{F_{N}}{(m)} dm = F_{H} + m \\log{(m)} - m and \\int \\operatorname{F_{N}}{(m)} dm = F_{H} + m \\operatorname{F_{N}}{(m)} - m and \\frac{d}{d m} \\int \\operatorname{F_{N}}{(m)} dm = \\frac{\\partial}{\\partial m} (F_{H} + m \\operatorname{F_{N}}{(m)} - m)", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_N')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('F_N')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Symbol('m', commutative=True), Function('F_N')(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["differentiate", 4, "Symbol('m', commutative=True)"], "Equality(Derivative(Integral(Function('F_N')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Mul(Symbol('m', commutative=True), Function('F_N')(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(\\theta)} = e^{\\theta}, then derive \\int m{(\\theta)} d\\theta = H + e^{\\theta}, then obtain (\\int e^{\\theta} d\\theta)^{2} = (H + m{(\\theta)}) \\int e^{\\theta} d\\theta", "derivation": "m{(\\theta)} = e^{\\theta} and \\int m{(\\theta)} d\\theta = \\int e^{\\theta} d\\theta and \\int m{(\\theta)} d\\theta = H + e^{\\theta} and (\\int m{(\\theta)} d\\theta)^{2} = (H + e^{\\theta}) \\int m{(\\theta)} d\\theta and (\\int m{(\\theta)} d\\theta)^{2} = (H + m{(\\theta)}) \\int m{(\\theta)} d\\theta and (\\int e^{\\theta} d\\theta)^{2} = (H + m{(\\theta)}) \\int e^{\\theta} d\\theta", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('H', commutative=True), exp(Symbol('\\\\theta', commutative=True))))"], [["times", 3, "Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Pow(Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integer(2)), Mul(Add(Symbol('H', commutative=True), exp(Symbol('\\\\theta', commutative=True))), Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integer(2)), Mul(Add(Symbol('H', commutative=True), Function('m')(Symbol('\\\\theta', commutative=True))), Integral(Function('m')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integer(2)), Mul(Add(Symbol('H', commutative=True), Function('m')(Symbol('\\\\theta', commutative=True))), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given q{(W)} = \\cos{(W)} and \\lambda{(W)} = \\cos{(W)}, then obtain - F_{g} v_{y} \\cos{(W)} = - F_{g} v_{y} \\cos{(W)} + (\\lambda{(W)} - q{(W)}) (- q{(W)} + \\cos{(W)})", "derivation": "q{(W)} = \\cos{(W)} and 0 = - q{(W)} + \\cos{(W)} and \\lambda{(W)} = \\cos{(W)} and 0 = \\lambda{(W)} - q{(W)} and 0 = (\\lambda{(W)} - q{(W)}) (- q{(W)} + \\cos{(W)}) and - F_{g} v_{y} \\cos{(W)} = - F_{g} v_{y} \\cos{(W)} + (\\lambda{(W)} - q{(W)}) (- q{(W)} + \\cos{(W)})", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["minus", 1, "Function('q')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('q')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('\\\\lambda')(Symbol('W', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('W', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Function('q')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)))"], "Equality(Integer(0), Mul(Add(Function('\\\\lambda')(Symbol('W', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('q')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)))))"], [["minus", 5, "Mul(Symbol('F_g', commutative=True), Symbol('v_y', commutative=True), cos(Symbol('W', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('v_y', commutative=True), cos(Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('v_y', commutative=True), cos(Symbol('W', commutative=True))), Mul(Add(Function('\\\\lambda')(Symbol('W', commutative=True)), Mul(Integer(-1), Function('q')(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('q')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = \\log{(C_{d})} and \\mathbf{g}{(C_{d})} = \\log{(C_{d})}, then obtain 0 = - C_{d} \\mathbf{g}{(C_{d})} + C_{d} \\log{(C_{d})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = \\log{(C_{d})} and C_{d} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = C_{d} \\log{(C_{d})} and 0 = - C_{d} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} + C_{d} \\log{(C_{d})} and \\mathbf{g}{(C_{d})} = \\log{(C_{d})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = \\mathbf{g}{(C_{d})} and 0 = - C_{d} \\mathbf{g}{(C_{d})} + C_{d} \\log{(C_{d})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["times", 1, "Symbol('C_d', commutative=True)"], "Equality(Mul(Symbol('C_d', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True))), Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))))"], [["minus", 2, "Mul(Symbol('C_d', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True))), Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Function('\\\\mathbf{g}')(Symbol('C_d', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Function('\\\\mathbf{g}')(Symbol('C_d', commutative=True))), Mul(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(F_{g})} = \\sin{(F_{g})} and m{(F_{g})} = \\sin{(F_{g})}, then obtain \\iint (A \\hat{x} + Q - a) m{(F_{g})} da da = \\iint (A \\hat{x} + Q - a) \\operatorname{v_{z}}{(F_{g})} da da", "derivation": "\\operatorname{v_{z}}{(F_{g})} = \\sin{(F_{g})} and m{(F_{g})} = \\sin{(F_{g})} and m{(F_{g})} = \\operatorname{v_{z}}{(F_{g})} and (A \\hat{x} + Q - a) m{(F_{g})} = (A \\hat{x} + Q - a) \\operatorname{v_{z}}{(F_{g})} and \\int (A \\hat{x} + Q - a) m{(F_{g})} da = \\int (A \\hat{x} + Q - a) \\operatorname{v_{z}}{(F_{g})} da and \\iint (A \\hat{x} + Q - a) m{(F_{g})} da da = \\iint (A \\hat{x} + Q - a) \\operatorname{v_{z}}{(F_{g})} da da", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('F_g', commutative=True)), Function('v_z')(Symbol('F_g', commutative=True)))"], [["times", 3, "Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('m')(Symbol('F_g', commutative=True))), Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('v_z')(Symbol('F_g', commutative=True))))"], [["integrate", 4, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('m')(Symbol('F_g', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('v_z')(Symbol('F_g', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["integrate", 5, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('m')(Symbol('F_g', commutative=True))), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Function('v_z')(Symbol('F_g', commutative=True))), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\dot{y})} = e^{\\dot{y}} and \\Psi_{\\lambda}{(\\dot{y})} = \\hat{p}_0^{\\dot{y}}{(\\dot{y})}, then obtain \\frac{d}{d \\dot{y}} \\Psi_{\\lambda}{(\\dot{y})} = \\frac{d}{d \\dot{y}} (e^{\\dot{y}})^{\\dot{y}}", "derivation": "\\hat{p}_0{(\\dot{y})} = e^{\\dot{y}} and \\hat{p}_0^{\\dot{y}}{(\\dot{y})} = (e^{\\dot{y}})^{\\dot{y}} and \\frac{d}{d \\dot{y}} \\hat{p}_0^{\\dot{y}}{(\\dot{y})} = \\frac{d}{d \\dot{y}} (e^{\\dot{y}})^{\\dot{y}} and \\Psi_{\\lambda}{(\\dot{y})} = \\hat{p}_0^{\\dot{y}}{(\\dot{y})} and \\frac{d}{d \\dot{y}} \\Psi_{\\lambda}{(\\dot{y})} = \\frac{d}{d \\dot{y}} (e^{\\dot{y}})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{y}', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(a^{\\dagger},\\nabla)} = \\nabla^{a^{\\dagger}} and \\tilde{g}{(S)} = \\log{(S)}, then obtain \\mathbf{S}{(a^{\\dagger},\\nabla)} + \\log{(S)} = \\nabla^{a^{\\dagger}} + \\log{(S)}", "derivation": "\\mathbf{S}{(a^{\\dagger},\\nabla)} = \\nabla^{a^{\\dagger}} and \\tilde{g}{(S)} = \\log{(S)} and \\mathbf{S}{(a^{\\dagger},\\nabla)} + \\tilde{g}{(S)} = \\nabla^{a^{\\dagger}} + \\tilde{g}{(S)} and \\mathbf{S}{(a^{\\dagger},\\nabla)} + \\log{(S)} = \\nabla^{a^{\\dagger}} + \\log{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}')(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('\\\\tilde{g}')(Symbol('S', commutative=True))), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\tilde{g}')(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('S', commutative=True))), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('S', commutative=True))))"]]}, {"prompt": "Given t{(g)} = \\cos{(\\log{(g)})} and \\mathbf{S}{(g)} = \\cos{(\\log{(g)})}, then obtain t^{g}{(g)} = \\mathbf{S}^{g}{(g)}", "derivation": "t{(g)} = \\cos{(\\log{(g)})} and \\mathbf{S}{(g)} = \\cos{(\\log{(g)})} and t^{g}{(g)} = \\cos^{g}{(\\log{(g)})} and t^{g}{(g)} = \\mathbf{S}^{g}{(g)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('g', commutative=True)), cos(log(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('g', commutative=True)), cos(log(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('t')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(cos(log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('t')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(c)} = \\cos{(\\log{(c)})}, then obtain \\frac{d}{d c} \\operatorname{y^{\\prime}}{(c)} + \\frac{\\sin{(\\log{(c)})}}{c} = 0", "derivation": "\\operatorname{y^{\\prime}}{(c)} = \\cos{(\\log{(c)})} and \\operatorname{y^{\\prime}}{(c)} - \\cos{(\\log{(c)})} = 0 and \\frac{d}{d c} (\\operatorname{y^{\\prime}}{(c)} - \\cos{(\\log{(c)})}) = \\frac{d}{d c} 0 and \\frac{d}{d c} \\operatorname{y^{\\prime}}{(c)} + \\frac{\\sin{(\\log{(c)})}}{c} = 0", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True))))"], [["minus", 1, "cos(log(Symbol('c', commutative=True)))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('c', commutative=True)), Mul(Integer(-1), cos(log(Symbol('c', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Function('y^{\\\\prime}')(Symbol('c', commutative=True)), Mul(Integer(-1), cos(log(Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('y^{\\\\prime}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), sin(log(Symbol('c', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\hat{\\mathbf{r}}{(J_{\\varepsilon})} = J_{\\varepsilon}, then derive \\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = J_{\\varepsilon} + g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon}, then obtain \\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + \\hat{\\mathbf{r}}{(J_{\\varepsilon})}", "derivation": "\\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = J_{\\varepsilon} + g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} and \\hat{\\mathbf{r}}{(J_{\\varepsilon})} = J_{\\varepsilon} and \\operatorname{M_{E}}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} \\log{(g^{\\prime}_{\\varepsilon})} - g^{\\prime}_{\\varepsilon} + \\hat{\\mathbf{r}}{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('M_E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('M_E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given Q{(F_{H})} = \\sin{(F_{H})}, then obtain (- \\frac{d}{d F_{H}} Q{(F_{H})})^{F_{H}} = (- \\frac{d}{d F_{H}} \\sin{(F_{H})})^{F_{H}}", "derivation": "Q{(F_{H})} = \\sin{(F_{H})} and \\frac{d}{d F_{H}} Q{(F_{H})} = \\frac{d}{d F_{H}} \\sin{(F_{H})} and - \\frac{d}{d F_{H}} Q{(F_{H})} = - \\frac{d}{d F_{H}} \\sin{(F_{H})} and (- \\frac{d}{d F_{H}} Q{(F_{H})})^{F_{H}} = (- \\frac{d}{d F_{H}} \\sin{(F_{H})})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('Q')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Derivative(Function('Q')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Symbol('F_H', commutative=True)), Pow(Mul(Integer(-1), Derivative(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then derive (\\int \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (F_{x} - \\cos{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}}, then obtain (\\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (F_{x} - \\cos{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\int \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} and (\\int \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (\\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} and (\\int \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (F_{x} - \\cos{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}} and (\\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (F_{x} - \\cos{(\\hat{\\mathbf{x}})})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(h)} = \\int \\cos{(h)} dh, then obtain (\\operatorname{P_{g}}{(h)} + \\int \\operatorname{P_{g}}{(h)} dh)^{h} = (\\operatorname{P_{g}}{(h)} + \\iint \\cos{(h)} dh dh)^{h}", "derivation": "\\operatorname{P_{g}}{(h)} = \\int \\cos{(h)} dh and \\int \\operatorname{P_{g}}{(h)} dh = \\iint \\cos{(h)} dh dh and \\int \\operatorname{P_{g}}{(h)} dh + \\int \\cos{(h)} dh = \\int \\cos{(h)} dh + \\iint \\cos{(h)} dh dh and (\\int \\operatorname{P_{g}}{(h)} dh + \\int \\cos{(h)} dh)^{h} = (\\int \\cos{(h)} dh + \\iint \\cos{(h)} dh dh)^{h} and (\\operatorname{P_{g}}{(h)} + \\int \\operatorname{P_{g}}{(h)} dh)^{h} = (\\operatorname{P_{g}}{(h)} + \\iint \\cos{(h)} dh dh)^{h}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('h', commutative=True)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Integral(Function('P_g')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Integral(Function('P_g')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Add(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Add(Function('P_g')(Symbol('h', commutative=True)), Integral(Function('P_g')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Add(Function('P_g')(Symbol('h', commutative=True)), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given S{(\\mathbb{I})} = e^{\\mathbb{I}}, then derive \\frac{d}{d \\mathbb{I}} S{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain 1 = \\frac{\\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}}}{\\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}}", "derivation": "S{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} S{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} S{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} S{(\\mathbb{I})} = \\frac{d^{2}}{d \\mathbb{I}^{2}} S{(\\mathbb{I})} and e^{- \\mathbb{I}} \\frac{d}{d \\mathbb{I}} S{(\\mathbb{I})} = e^{- \\mathbb{I}} \\frac{d^{2}}{d \\mathbb{I}^{2}} S{(\\mathbb{I})} and \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} = e^{\\mathbb{I}} and 1 = e^{- \\mathbb{I}} \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and 1 = \\frac{\\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}}}{\\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"], [["divide", 4, "exp(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Derivative(Function('S')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(1), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Integer(1), Mul(Pow(Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(b)} = \\log{(b)}, then derive \\int \\operatorname{a^{\\dagger}}{(b)} db = S + b \\log{(b)} - b, then obtain \\log{(b)} (\\int \\log{(b)} db)^{b} = (S + b \\log{(b)} - b)^{b} \\log{(b)}", "derivation": "\\operatorname{a^{\\dagger}}{(b)} = \\log{(b)} and \\int \\operatorname{a^{\\dagger}}{(b)} db = \\int \\log{(b)} db and \\int \\operatorname{a^{\\dagger}}{(b)} db = S + b \\log{(b)} - b and (\\int \\operatorname{a^{\\dagger}}{(b)} db)^{b} = (S + b \\log{(b)} - b)^{b} and (\\int \\log{(b)} db)^{b} = (S + b \\log{(b)} - b)^{b} and \\log{(b)} (\\int \\log{(b)} db)^{b} = (S + b \\log{(b)} - b)^{b} \\log{(b)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Function('a^{\\\\dagger}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["times", 5, "log(Symbol('b', commutative=True))"], "Equality(Mul(log(Symbol('b', commutative=True)), Pow(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Mul(Pow(Add(Symbol('S', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Symbol('b', commutative=True)), log(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\chi)} = \\cos{(\\chi)} and Z{(M)} = \\int \\log{(M)} dM, then obtain \\int (\\mathbf{g}^{\\chi}{(\\chi)} - \\int \\log{(M)} dM) dM = \\int (\\cos^{\\chi}{(\\chi)} - \\int \\log{(M)} dM) dM", "derivation": "\\mathbf{g}{(\\chi)} = \\cos{(\\chi)} and \\mathbf{g}^{\\chi}{(\\chi)} = \\cos^{\\chi}{(\\chi)} and Z{(M)} = \\int \\log{(M)} dM and - Z{(M)} + \\mathbf{g}^{\\chi}{(\\chi)} = - Z{(M)} + \\cos^{\\chi}{(\\chi)} and \\int (- Z{(M)} + \\mathbf{g}^{\\chi}{(\\chi)}) dM = \\int (- Z{(M)} + \\cos^{\\chi}{(\\chi)}) dM and \\int (\\mathbf{g}^{\\chi}{(\\chi)} - \\int \\log{(M)} dM) dM = \\int (\\cos^{\\chi}{(\\chi)} - \\int \\log{(M)} dM) dM", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], ["get_premise", "Equality(Function('Z')(Symbol('M', commutative=True)), Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["minus", 2, "Function('Z')(Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Z')(Symbol('M', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Function('Z')(Symbol('M', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('Z')(Symbol('M', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Add(Mul(Integer(-1), Function('Z')(Symbol('M', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True))), Integral(Add(Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given W{(\\mathbf{E},\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})}, then obtain \\frac{1}{(- W{(\\mathbf{E},\\hat{H}_{\\lambda})} - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})})^{2}} = \\frac{1}{4 \\sin^{2}{(\\hat{H}_{\\lambda} - \\mathbf{E})}}", "derivation": "W{(\\mathbf{E},\\hat{H}_{\\lambda})} = \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})} and - W{(\\mathbf{E},\\hat{H}_{\\lambda})} = - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})} and - W{(\\mathbf{E},\\hat{H}_{\\lambda})} - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})} = - 2 \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})} and \\frac{1}{(- W{(\\mathbf{E},\\hat{H}_{\\lambda})} - \\sin{(\\hat{H}_{\\lambda} - \\mathbf{E})})^{2}} = \\frac{1}{4 \\sin^{2}{(\\hat{H}_{\\lambda} - \\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))))"], [["minus", 2, "sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))))), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))))"], [["power", 3, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))))), Integer(-2)), Mul(Rational(1, 4), Pow(sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(m_{s})} = m_{s}, then obtain \\frac{d}{d m_{s}} \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} - \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} = \\frac{d}{d m_{s}} 1 - \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}}", "derivation": "\\operatorname{F_{H}}{(m_{s})} = m_{s} and \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} = 1 and \\frac{d}{d m_{s}} \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} = \\frac{d}{d m_{s}} 1 and \\frac{d}{d m_{s}} \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} - \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}} = \\frac{d}{d m_{s}} 1 - \\frac{\\operatorname{F_{H}}{(m_{s})}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], [["divide", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True)))), Add(Derivative(Integer(1), Tuple(Symbol('m_s', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('F_H')(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(y^{\\prime},\\mathbf{D})} = \\cos{(\\mathbf{D} y^{\\prime})}, then obtain \\dot{y}^{4 \\mathbf{D}}{(y^{\\prime},\\mathbf{D})} = \\dot{y}^{2 \\mathbf{D}}{(y^{\\prime},\\mathbf{D})} \\cos^{2 \\mathbf{D}}{(\\mathbf{D} y^{\\prime})}", "derivation": "\\dot{y}{(y^{\\prime},\\mathbf{D})} = \\cos{(\\mathbf{D} y^{\\prime})} and \\dot{y}^{\\mathbf{D}}{(y^{\\prime},\\mathbf{D})} = \\cos^{\\mathbf{D}}{(\\mathbf{D} y^{\\prime})} and \\dot{y}^{2 \\mathbf{D}}{(y^{\\prime},\\mathbf{D})} = \\dot{y}^{\\mathbf{D}}{(y^{\\prime},\\mathbf{D})} \\cos^{\\mathbf{D}}{(\\mathbf{D} y^{\\prime})} and \\dot{y}^{4 \\mathbf{D}}{(y^{\\prime},\\mathbf{D})} = \\dot{y}^{2 \\mathbf{D}}{(y^{\\prime},\\mathbf{D})} \\cos^{2 \\mathbf{D}}{(\\mathbf{D} y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["times", 2, "Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(4), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True))), Pow(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\chi,\\mathbf{F})} = \\mathbf{F}^{\\chi}, then obtain (- \\mathbf{F} + \\frac{\\operatorname{v_{z}}{(\\chi,\\mathbf{F})}}{\\chi})^{\\mathbf{F}} = (- \\mathbf{F} + \\frac{\\mathbf{F}^{\\chi}}{\\chi})^{\\mathbf{F}}", "derivation": "\\operatorname{v_{z}}{(\\chi,\\mathbf{F})} = \\mathbf{F}^{\\chi} and \\frac{\\operatorname{v_{z}}{(\\chi,\\mathbf{F})}}{\\chi} = \\frac{\\mathbf{F}^{\\chi}}{\\chi} and - \\mathbf{F} + \\frac{\\operatorname{v_{z}}{(\\chi,\\mathbf{F})}}{\\chi} = - \\mathbf{F} + \\frac{\\mathbf{F}^{\\chi}}{\\chi} and (- \\mathbf{F} + \\frac{\\operatorname{v_{z}}{(\\chi,\\mathbf{F})}}{\\chi})^{\\mathbf{F}} = (- \\mathbf{F} + \\frac{\\mathbf{F}^{\\chi}}{\\chi})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('v_z')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\chi', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\mathbf{E},\\mathbf{F})} = \\mathbf{E} + \\cos{(\\mathbf{F})}, then obtain \\int (\\operatorname{J_{\\varepsilon}}^{\\mathbf{E}}{(\\mathbf{E},\\mathbf{F})} - 1) d\\mathbf{F} = \\int ((\\mathbf{E} + \\cos{(\\mathbf{F})})^{\\mathbf{E}} - 1) d\\mathbf{F}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\mathbf{E},\\mathbf{F})} = \\mathbf{E} + \\cos{(\\mathbf{F})} and \\operatorname{J_{\\varepsilon}}^{\\mathbf{E}}{(\\mathbf{E},\\mathbf{F})} = (\\mathbf{E} + \\cos{(\\mathbf{F})})^{\\mathbf{E}} and \\operatorname{J_{\\varepsilon}}^{\\mathbf{E}}{(\\mathbf{E},\\mathbf{F})} - 1 = (\\mathbf{E} + \\cos{(\\mathbf{F})})^{\\mathbf{E}} - 1 and \\int (\\operatorname{J_{\\varepsilon}}^{\\mathbf{E}}{(\\mathbf{E},\\mathbf{F})} - 1) d\\mathbf{F} = \\int ((\\mathbf{E} + \\cos{(\\mathbf{F})})^{\\mathbf{E}} - 1) d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Add(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Add(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), cos(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(f^{\\prime},v_{2})} = \\frac{v_{2}}{f^{\\prime}}, then derive \\int \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} = \\chi + \\log{(f^{\\prime})}, then obtain \\iint \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} dv_{2} = \\iint \\frac{1}{f^{\\prime}} df^{\\prime} dv_{2}", "derivation": "\\operatorname{A_{y}}{(f^{\\prime},v_{2})} = \\frac{v_{2}}{f^{\\prime}} and \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} = \\frac{1}{f^{\\prime}} and \\int \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} = \\int \\frac{1}{f^{\\prime}} df^{\\prime} and \\int \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} = \\chi + \\log{(f^{\\prime})} and \\iint \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} dv_{2} = \\int (\\chi + \\log{(f^{\\prime})}) dv_{2} and \\int \\frac{1}{f^{\\prime}} df^{\\prime} = \\chi + \\log{(f^{\\prime})} and \\iint \\frac{\\operatorname{A_{y}}{(f^{\\prime},v_{2})}}{v_{2}} df^{\\prime} dv_{2} = \\iint \\frac{1}{f^{\\prime}} df^{\\prime} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["divide", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True))), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Symbol('\\\\chi', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\chi', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mu_0,v_{1})} = \\frac{v_{1}}{\\mu_0}, then obtain \\frac{(\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})})^{v_{1}}}{\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})}} = \\frac{(\\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\mu_0})^{v_{1}}}{\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})}}", "derivation": "\\mathbf{A}{(\\mu_0,v_{1})} = \\frac{v_{1}}{\\mu_0} and \\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})} = \\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\mu_0} and (\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})})^{v_{1}} = (\\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\mu_0})^{v_{1}} and \\frac{(\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})})^{v_{1}}}{\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})}} = \\frac{(\\frac{\\partial}{\\partial v_{1}} \\frac{v_{1}}{\\mu_0})^{v_{1}}}{\\frac{\\partial}{\\partial v_{1}} \\mathbf{A}{(\\mu_0,v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)))"], [["divide", 3, "Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True))), Mul(Pow(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Symbol('v_1', commutative=True)), Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mu_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(g,A_{x})} = e^{\\frac{A_{x}}{g}}, then obtain g \\mathbf{J}_f^{g}{(g,A_{x})} + (e^{\\frac{A_{x}}{g}})^{g} = g (e^{\\frac{A_{x}}{g}})^{g} + (e^{\\frac{A_{x}}{g}})^{g}", "derivation": "\\mathbf{J}_f{(g,A_{x})} = e^{\\frac{A_{x}}{g}} and \\mathbf{J}_f^{g}{(g,A_{x})} = (e^{\\frac{A_{x}}{g}})^{g} and g \\mathbf{J}_f^{g}{(g,A_{x})} = g (e^{\\frac{A_{x}}{g}})^{g} and g \\mathbf{J}_f^{g}{(g,A_{x})} + (e^{\\frac{A_{x}}{g}})^{g} = g (e^{\\frac{A_{x}}{g}})^{g} + (e^{\\frac{A_{x}}{g}})^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True), Symbol('A_x', commutative=True)), exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True), Symbol('A_x', commutative=True)), Symbol('g', commutative=True)), Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True)))"], [["times", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True), Symbol('A_x', commutative=True)), Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True))))"], [["add", 3, "Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True))"], "Equality(Add(Mul(Symbol('g', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('g', commutative=True), Symbol('A_x', commutative=True)), Symbol('g', commutative=True))), Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True))), Add(Mul(Symbol('g', commutative=True), Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True))), Pow(exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(k,J)} = J + \\cos{(k)} and \\psi{(k,J)} = \\frac{\\partial}{\\partial J} (J + 2 \\cos{(k)}), then obtain \\psi{(k,J)} = \\frac{\\partial}{\\partial J} \\hat{x}{(k,J)}", "derivation": "\\hat{x}{(k,J)} = J + \\cos{(k)} and \\hat{x}{(k,J)} + \\cos{(k)} = J + 2 \\cos{(k)} and \\frac{\\partial}{\\partial J} (\\hat{x}{(k,J)} + \\cos{(k)}) = \\frac{\\partial}{\\partial J} (J + 2 \\cos{(k)}) and \\psi{(k,J)} = \\frac{\\partial}{\\partial J} (J + 2 \\cos{(k)}) and \\psi{(k,J)} = \\frac{\\partial}{\\partial J} (\\hat{x}{(k,J)} + \\cos{(k)}) and \\psi{(k,J)} = \\frac{\\partial}{\\partial J} \\hat{x}{(k,J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), cos(Symbol('k', commutative=True))))"], [["add", 1, "cos(Symbol('k', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), cos(Symbol('k', commutative=True))), Add(Symbol('J', commutative=True), Mul(Integer(2), cos(Symbol('k', commutative=True)))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), cos(Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Mul(Integer(2), cos(Symbol('k', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Derivative(Add(Symbol('J', commutative=True), Mul(Integer(2), cos(Symbol('k', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Derivative(Add(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), cos(Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\psi')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{p})} = \\log{(e^{\\mathbf{p}})} and y{(\\mathbf{p})} = \\int 0 d\\mathbf{p}, then obtain y{(\\mathbf{p})} = \\int (\\mathbf{f}{(\\mathbf{p})} - \\log{(e^{\\mathbf{p}})}) d\\mathbf{p}", "derivation": "\\mathbf{f}{(\\mathbf{p})} = \\log{(e^{\\mathbf{p}})} and \\mathbf{f}{(\\mathbf{p})} - \\log{(e^{\\mathbf{p}})} = 0 and \\int (\\mathbf{f}{(\\mathbf{p})} - \\log{(e^{\\mathbf{p}})}) d\\mathbf{p} = \\int 0 d\\mathbf{p} and y{(\\mathbf{p})} = \\int 0 d\\mathbf{p} and y{(\\mathbf{p})} = \\int (\\mathbf{f}{(\\mathbf{p})} - \\log{(e^{\\mathbf{p}})}) d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True)), log(exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 1, "log(exp(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{p}', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{p}', commutative=True))))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('y')(Symbol('\\\\mathbf{p}', commutative=True)), Integral(Add(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{p}', commutative=True))))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given f{(\\mathbf{r})} = e^{\\mathbf{r}} and \\operatorname{c_{0}}{(\\mathbf{r})} = \\frac{e^{\\mathbf{r}}}{f{(\\mathbf{r})}}, then obtain \\operatorname{c_{0}}{(\\mathbf{r})} + e^{\\mathbf{r}} = e^{\\mathbf{r}} + 1", "derivation": "f{(\\mathbf{r})} = e^{\\mathbf{r}} and \\operatorname{c_{0}}{(\\mathbf{r})} = \\frac{e^{\\mathbf{r}}}{f{(\\mathbf{r})}} and \\operatorname{c_{0}}{(\\mathbf{r})} + e^{\\mathbf{r}} = e^{\\mathbf{r}} + \\frac{e^{\\mathbf{r}}}{f{(\\mathbf{r})}} and \\operatorname{c_{0}}{(\\mathbf{r})} + e^{\\mathbf{r}} = e^{\\mathbf{r}} + 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Function('f')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True))), Add(exp(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Function('f')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), exp(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('c_0')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True))), Add(exp(Symbol('\\\\mathbf{r}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(U,m)} = U + m, then obtain - 2 U + \\operatorname{t_{1}}{(U,m)} = - U + m", "derivation": "\\operatorname{t_{1}}{(U,m)} = U + m and - U + \\operatorname{t_{1}}{(U,m)} = m and - 2 U - m + \\operatorname{t_{1}}{(U,m)} = - U and - 2 U + \\operatorname{t_{1}}{(U,m)} = - U + m", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Add(Symbol('U', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('t_1')(Symbol('U', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))"], [["minus", 2, "Add(Symbol('U', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)), Function('t_1')(Symbol('U', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)))"], [["add", 3, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Function('t_1')(Symbol('U', commutative=True), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given l{(U,\\eta^{\\prime})} = \\cos{(U - \\eta^{\\prime})} and \\operatorname{r_{0}}{(U,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} l{(U,\\eta^{\\prime})}, then obtain - \\operatorname{r_{0}}{(U,\\eta^{\\prime})} = - \\frac{\\partial}{\\partial \\eta^{\\prime}} \\cos{(U - \\eta^{\\prime})}", "derivation": "l{(U,\\eta^{\\prime})} = \\cos{(U - \\eta^{\\prime})} and \\operatorname{r_{0}}{(U,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} l{(U,\\eta^{\\prime})} and \\operatorname{r_{0}}{(U,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\cos{(U - \\eta^{\\prime})} and - \\operatorname{r_{0}}{(U,\\eta^{\\prime})} = - \\frac{\\partial}{\\partial \\eta^{\\prime}} \\cos{(U - \\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('U', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('U', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Function('l')(Symbol('U', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('r_0')(Symbol('U', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('r_0')(Symbol('U', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(-1), Derivative(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(J)} = \\log{(\\log{(J)})}, then derive \\frac{d}{d J} v{(J)} = \\frac{1}{J \\log{(J)}}, then obtain \\frac{1}{J \\log{(J)} \\frac{d}{d J} \\log{(\\log{(J)})}} = 1", "derivation": "v{(J)} = \\log{(\\log{(J)})} and \\frac{d}{d J} v{(J)} = \\frac{d}{d J} \\log{(\\log{(J)})} and \\frac{d}{d J} v{(J)} = \\frac{1}{J \\log{(J)}} and \\frac{\\frac{d}{d J} v{(J)}}{\\frac{d}{d J} \\log{(\\log{(J)})}} = 1 and \\frac{1}{J \\log{(J)} \\frac{d}{d J} \\log{(\\log{(J)})}} = 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(log(log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(log(Symbol('J', commutative=True)), Integer(-1))))"], [["divide", 2, "Derivative(log(log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('v')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Derivative(log(log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(log(Symbol('J', commutative=True)), Integer(-1)), Pow(Derivative(log(log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{c},\\delta)} = - \\delta + \\cos{(F_{c})}, then obtain \\int (\\int (\\eta^{\\prime}{(F_{c},\\delta)} - \\cos{(F_{c})}) d\\delta)^{\\delta} d\\delta = \\int (\\int - \\delta d\\delta)^{\\delta} d\\delta", "derivation": "\\eta^{\\prime}{(F_{c},\\delta)} = - \\delta + \\cos{(F_{c})} and \\eta^{\\prime}{(F_{c},\\delta)} - \\cos{(F_{c})} = - \\delta and \\int (\\eta^{\\prime}{(F_{c},\\delta)} - \\cos{(F_{c})}) d\\delta = \\int - \\delta d\\delta and (\\int (\\eta^{\\prime}{(F_{c},\\delta)} - \\cos{(F_{c})}) d\\delta)^{\\delta} = (\\int - \\delta d\\delta)^{\\delta} and \\int (\\int (\\eta^{\\prime}{(F_{c},\\delta)} - \\cos{(F_{c})}) d\\delta)^{\\delta} d\\delta = \\int (\\int - \\delta d\\delta)^{\\delta} d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), cos(Symbol('F_c', commutative=True))))"], [["minus", 1, "cos(Symbol('F_c', commutative=True))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Pow(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Integral(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(v_{1},m_{s})} = m_{s} - v_{1} and \\varepsilon{(v_{1},m_{s})} = m_{s} - v_{1}, then obtain - v_{1} (m_{s} - v_{1})^{v_{1}} - 1 = - v_{1} \\varepsilon^{v_{1}}{(v_{1},m_{s})} - 1", "derivation": "\\eta^{\\prime}{(v_{1},m_{s})} = m_{s} - v_{1} and \\eta^{\\prime}^{v_{1}}{(v_{1},m_{s})} = (m_{s} - v_{1})^{v_{1}} and - v_{1} \\eta^{\\prime}^{v_{1}}{(v_{1},m_{s})} = - v_{1} (m_{s} - v_{1})^{v_{1}} and \\varepsilon{(v_{1},m_{s})} = m_{s} - v_{1} and - v_{1} \\eta^{\\prime}^{v_{1}}{(v_{1},m_{s})} = - v_{1} \\varepsilon^{v_{1}}{(v_{1},m_{s})} and - v_{1} (m_{s} - v_{1})^{v_{1}} = - v_{1} \\varepsilon^{v_{1}}{(v_{1},m_{s})} and - v_{1} (m_{s} - v_{1})^{v_{1}} - 1 = - v_{1} \\varepsilon^{v_{1}}{(v_{1},m_{s})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))))"], [["add", 6, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_1', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(z)} = \\log{(\\cos{(z)})}, then obtain \\iint (\\operatorname{v_{t}}{(z)} \\cos{(z)} + \\operatorname{v_{t}}{(z)}) dz dz = \\iint (\\operatorname{v_{t}}{(z)} + \\log{(\\cos{(z)})} \\cos{(z)}) dz dz", "derivation": "\\operatorname{v_{t}}{(z)} = \\log{(\\cos{(z)})} and \\operatorname{v_{t}}{(z)} \\cos{(z)} = \\log{(\\cos{(z)})} \\cos{(z)} and \\operatorname{v_{t}}{(z)} \\cos{(z)} + \\operatorname{v_{t}}{(z)} = \\operatorname{v_{t}}{(z)} + \\log{(\\cos{(z)})} \\cos{(z)} and \\int (\\operatorname{v_{t}}{(z)} \\cos{(z)} + \\operatorname{v_{t}}{(z)}) dz = \\int (\\operatorname{v_{t}}{(z)} + \\log{(\\cos{(z)})} \\cos{(z)}) dz and \\iint (\\operatorname{v_{t}}{(z)} \\cos{(z)} + \\operatorname{v_{t}}{(z)}) dz dz = \\iint (\\operatorname{v_{t}}{(z)} + \\log{(\\cos{(z)})} \\cos{(z)}) dz dz", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('z', commutative=True)), log(cos(Symbol('z', commutative=True))))"], [["times", 1, "cos(Symbol('z', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Mul(log(cos(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["add", 2, "Function('v_t')(Symbol('z', commutative=True))"], "Equality(Add(Mul(Function('v_t')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Function('v_t')(Symbol('z', commutative=True))), Add(Function('v_t')(Symbol('z', commutative=True)), Mul(log(cos(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True)))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Mul(Function('v_t')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Function('v_t')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Add(Function('v_t')(Symbol('z', commutative=True)), Mul(log(cos(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))"], [["integrate", 4, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Mul(Function('v_t')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Function('v_t')(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Function('v_t')(Symbol('z', commutative=True)), Mul(log(cos(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(n,G)} = \\frac{\\partial}{\\partial G} (G + n), then derive 1 = 2 - \\operatorname{A_{1}}{(n,G)}, then obtain \\int 1 dn = \\int (2 - \\operatorname{A_{1}}{(n,G)}) dn", "derivation": "\\operatorname{A_{1}}{(n,G)} = \\frac{\\partial}{\\partial G} (G + n) and \\operatorname{A_{1}}{(n,G)} + 1 = \\frac{\\partial}{\\partial G} (G + n) + 1 and 1 = - \\operatorname{A_{1}}{(n,G)} + \\frac{\\partial}{\\partial G} (G + n) + 1 and 1 = 2 - \\operatorname{A_{1}}{(n,G)} and \\int 1 dn = \\int (2 - \\operatorname{A_{1}}{(n,G)}) dn", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Integer(1)), Add(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)))"], [["minus", 2, "Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True))"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True))), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Add(Integer(2), Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True)))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('n', commutative=True))), Integral(Add(Integer(2), Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given v{(A_{x})} = \\cos{(A_{x})}, then obtain v^{2}{(A_{x})} \\cos{(A_{x})} = \\cos^{3}{(A_{x})}", "derivation": "v{(A_{x})} = \\cos{(A_{x})} and v{(A_{x})} \\cos{(A_{x})} = \\cos^{2}{(A_{x})} and v^{2}{(A_{x})} \\cos{(A_{x})} = v{(A_{x})} \\cos^{2}{(A_{x})} and v{(A_{x})} \\cos^{2}{(A_{x})} = \\cos^{3}{(A_{x})} and v^{2}{(A_{x})} \\cos{(A_{x})} = \\cos^{3}{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["times", 1, "cos(Symbol('A_x', commutative=True))"], "Equality(Mul(Function('v')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True))), Pow(cos(Symbol('A_x', commutative=True)), Integer(2)))"], [["times", 2, "Function('v')(Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Function('v')(Symbol('A_x', commutative=True)), Integer(2)), cos(Symbol('A_x', commutative=True))), Mul(Function('v')(Symbol('A_x', commutative=True)), Pow(cos(Symbol('A_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('v')(Symbol('A_x', commutative=True)), Pow(cos(Symbol('A_x', commutative=True)), Integer(2))), Pow(cos(Symbol('A_x', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('v')(Symbol('A_x', commutative=True)), Integer(2)), cos(Symbol('A_x', commutative=True))), Pow(cos(Symbol('A_x', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(A,u)} = u^{A} and f{(A)} = - A, then obtain (f{(A)} + \\int u^{A} dA) (f{(A)} + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA) = (- A + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA) (f{(A)} + \\int u^{A} dA)", "derivation": "\\operatorname{a^{\\dagger}}{(A,u)} = u^{A} and \\int \\operatorname{a^{\\dagger}}{(A,u)} dA = \\int u^{A} dA and f{(A)} = - A and f{(A)} + \\int u^{A} dA = - A + \\int u^{A} dA and f{(A)} + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA = - A + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA and (f{(A)} + \\int u^{A} dA) (f{(A)} + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA) = (- A + \\int \\operatorname{a^{\\dagger}}{(A,u)} dA) (f{(A)} + \\int u^{A} dA)", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)))"], [["add", 3, "Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Function('f')(Symbol('A', commutative=True)), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('f')(Symbol('A', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["times", 5, "Add(Function('f')(Symbol('A', commutative=True)), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], "Equality(Mul(Add(Function('f')(Symbol('A', commutative=True)), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Function('f')(Symbol('A', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('A', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Function('f')(Symbol('A', commutative=True)), Integral(Pow(Symbol('u', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(v)} = \\sin{(v)}, then derive (\\int \\Psi_{nl}{(v)} dv)^{v} = (r_{0} - \\cos{(v)})^{v}, then obtain 2 \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv = \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv + \\int (\\int \\sin{(v)} dv)^{v} dv", "derivation": "\\Psi_{nl}{(v)} = \\sin{(v)} and \\int \\Psi_{nl}{(v)} dv = \\int \\sin{(v)} dv and (\\int \\Psi_{nl}{(v)} dv)^{v} = (\\int \\sin{(v)} dv)^{v} and (\\int \\Psi_{nl}{(v)} dv)^{v} = (r_{0} - \\cos{(v)})^{v} and \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv = \\int (r_{0} - \\cos{(v)})^{v} dv and 2 \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv = \\int (r_{0} - \\cos{(v)})^{v} dv + \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv and \\int (\\int \\sin{(v)} dv)^{v} dv = \\int (r_{0} - \\cos{(v)})^{v} dv and 2 \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv = \\int (\\int \\Psi_{nl}{(v)} dv)^{v} dv + \\int (\\int \\sin{(v)} dv)^{v} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), cos(Symbol('v', commutative=True)))), Symbol('v', commutative=True)))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), cos(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 5, "Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Integral(Pow(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), cos(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Pow(Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), cos(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Integer(2), Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Integral(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\rho)} = \\rho, then derive \\log{(e^{\\frac{d}{d \\rho} \\dot{\\mathbf{r}}{(\\rho)}})} = 1, then obtain \\log{(e^{\\frac{d}{d \\dot{\\mathbf{r}}{(\\rho)}} \\dot{\\mathbf{r}}{(\\rho)}})} = 1", "derivation": "\\dot{\\mathbf{r}}{(\\rho)} = \\rho and \\frac{d}{d \\rho} \\dot{\\mathbf{r}}{(\\rho)} = \\frac{d}{d \\rho} \\rho and e^{\\frac{d}{d \\rho} \\dot{\\mathbf{r}}{(\\rho)}} = e^{\\frac{d}{d \\rho} \\rho} and \\log{(e^{\\frac{d}{d \\rho} \\dot{\\mathbf{r}}{(\\rho)}})} = \\log{(e^{\\frac{d}{d \\rho} \\rho})} and \\log{(e^{\\frac{d}{d \\rho} \\dot{\\mathbf{r}}{(\\rho)}})} = 1 and \\log{(e^{\\frac{d}{d \\rho} \\rho})} = 1 and \\log{(e^{\\frac{d}{d \\dot{\\mathbf{r}}{(\\rho)}} \\dot{\\mathbf{r}}{(\\rho)}})} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), exp(Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["log", 3], "Equality(log(exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))), log(exp(Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(log(exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(log(exp(Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(log(exp(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Tuple(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho', commutative=True)), Integer(1))))), Integer(1))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{E},z^{*})} = \\mathbf{E} z^{*}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},z^{*})} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*} = \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{E} z^{*} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*}", "derivation": "\\mathbf{D}{(\\mathbf{E},z^{*})} = \\mathbf{E} z^{*} and \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*} = \\int \\mathbf{E} z^{*} dz^{*} and \\mathbf{D}{(\\mathbf{E},z^{*})} \\int \\mathbf{E} z^{*} dz^{*} = \\mathbf{E} z^{*} \\int \\mathbf{E} z^{*} dz^{*} and \\mathbf{D}{(\\mathbf{E},z^{*})} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*} = \\mathbf{E} z^{*} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*} and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},z^{*})} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*} = \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{E} z^{*} \\int \\mathbf{D}{(\\mathbf{E},z^{*})} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["times", 1, "Integral(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True), Integral(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(C_{2})} = \\frac{d}{d C_{2}} \\cos{(C_{2})}, then derive \\operatorname{C_{1}}{(C_{2})} = - \\sin{(C_{2})}, then obtain (C_{2} + \\int \\operatorname{C_{1}}{(C_{2})} dC_{2})^{C_{2}} = (C_{2} + \\int - \\sin{(C_{2})} dC_{2})^{C_{2}}", "derivation": "\\operatorname{C_{1}}{(C_{2})} = \\frac{d}{d C_{2}} \\cos{(C_{2})} and \\int \\operatorname{C_{1}}{(C_{2})} dC_{2} = \\int \\frac{d}{d C_{2}} \\cos{(C_{2})} dC_{2} and \\operatorname{C_{1}}{(C_{2})} = - \\sin{(C_{2})} and C_{2} + \\int \\operatorname{C_{1}}{(C_{2})} dC_{2} = C_{2} + \\int \\frac{d}{d C_{2}} \\cos{(C_{2})} dC_{2} and C_{2} + \\int - \\sin{(C_{2})} dC_{2} = C_{2} + \\int \\frac{d}{d C_{2}} \\cos{(C_{2})} dC_{2} and C_{2} + \\int \\operatorname{C_{1}}{(C_{2})} dC_{2} = C_{2} + \\int - \\sin{(C_{2})} dC_{2} and (C_{2} + \\int \\operatorname{C_{1}}{(C_{2})} dC_{2})^{C_{2}} = (C_{2} + \\int - \\sin{(C_{2})} dC_{2})^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('C_2', commutative=True)), Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('C_1')(Symbol('C_2', commutative=True)), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"], [["add", 2, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('C_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('C_2', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('C_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["power", 6, "Symbol('C_2', commutative=True)"], "Equality(Pow(Add(Symbol('C_2', commutative=True), Integral(Function('C_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Symbol('C_2', commutative=True)), Pow(Add(Symbol('C_2', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Symbol('C_2', commutative=True)))"]]}, {"prompt": "Given p{(n_{2},v_{2})} = \\frac{n_{2}}{v_{2}}, then derive \\frac{\\partial}{\\partial v_{2}} p{(n_{2},v_{2})} = - \\frac{n_{2}}{v_{2}^{2}}, then obtain - \\frac{n_{2}}{v_{2}^{2}} = \\frac{\\partial}{\\partial v_{2}} \\frac{n_{2}}{v_{2}}", "derivation": "p{(n_{2},v_{2})} = \\frac{n_{2}}{v_{2}} and \\frac{\\partial}{\\partial v_{2}} p{(n_{2},v_{2})} = \\frac{\\partial}{\\partial v_{2}} \\frac{n_{2}}{v_{2}} and \\frac{\\partial}{\\partial v_{2}} p{(n_{2},v_{2})} = - \\frac{n_{2}}{v_{2}^{2}} and - \\frac{n_{2}}{v_{2}^{2}} = \\frac{\\partial}{\\partial v_{2}} \\frac{n_{2}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('n_2', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('n_2', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('n_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('n_2', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('n_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('n_2', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('n_2', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-2))), Derivative(Mul(Symbol('n_2', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(c_{0})} = \\log{(c_{0})} and \\operatorname{v_{1}}{(c_{0})} = \\frac{1}{\\operatorname{A_{1}}{(c_{0})}}, then obtain \\operatorname{A_{1}}{(c_{0})} + \\frac{1}{\\log{(c_{0})}} = \\log{(c_{0})} + \\frac{1}{\\log{(c_{0})}}", "derivation": "\\operatorname{A_{1}}{(c_{0})} = \\log{(c_{0})} and \\operatorname{A_{1}}{(c_{0})} + \\frac{1}{\\operatorname{A_{1}}{(c_{0})}} = \\log{(c_{0})} + \\frac{1}{\\operatorname{A_{1}}{(c_{0})}} and \\operatorname{v_{1}}{(c_{0})} = \\frac{1}{\\operatorname{A_{1}}{(c_{0})}} and \\operatorname{v_{1}}{(c_{0})} = \\frac{1}{\\log{(c_{0})}} and \\frac{1}{\\operatorname{A_{1}}{(c_{0})}} = \\frac{1}{\\log{(c_{0})}} and \\operatorname{A_{1}}{(c_{0})} + \\frac{1}{\\log{(c_{0})}} = \\log{(c_{0})} + \\frac{1}{\\log{(c_{0})}}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["add", 1, "Pow(Function('A_1')(Symbol('c_0', commutative=True)), Integer(-1))"], "Equality(Add(Function('A_1')(Symbol('c_0', commutative=True)), Pow(Function('A_1')(Symbol('c_0', commutative=True)), Integer(-1))), Add(log(Symbol('c_0', commutative=True)), Pow(Function('A_1')(Symbol('c_0', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('c_0', commutative=True)), Pow(Function('A_1')(Symbol('c_0', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('v_1')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('A_1')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Function('A_1')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Add(log(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_l{(J_{\\varepsilon},y)} = \\frac{y}{J_{\\varepsilon}}, then obtain \\int 2 \\hat{H}_l{(J_{\\varepsilon},y)} dy = \\phi + \\frac{\\int y dy + \\int J_{\\varepsilon} \\hat{H}_l{(J_{\\varepsilon},y)} dy}{J_{\\varepsilon}}", "derivation": "\\hat{H}_l{(J_{\\varepsilon},y)} = \\frac{y}{J_{\\varepsilon}} and 2 \\hat{H}_l{(J_{\\varepsilon},y)} = \\hat{H}_l{(J_{\\varepsilon},y)} + \\frac{y}{J_{\\varepsilon}} and \\int 2 \\hat{H}_l{(J_{\\varepsilon},y)} dy = \\int (\\hat{H}_l{(J_{\\varepsilon},y)} + \\frac{y}{J_{\\varepsilon}}) dy and \\int 2 \\hat{H}_l{(J_{\\varepsilon},y)} dy = \\phi + \\frac{\\int y dy + \\int J_{\\varepsilon} \\hat{H}_l{(J_{\\varepsilon},y)} dy}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))), Add(Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Integral(Symbol('y', commutative=True), Tuple(Symbol('y', commutative=True))), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('\\\\hat{H}_l')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))))))"]]}, {"prompt": "Given \\phi_{1}{(b,x)} = \\sin{(b^{x})}, then obtain 2 \\phi_{1}{(b,x)} \\sin{(b^{x})} - \\phi_{1}{(b,x)} = \\phi_{1}{(b,x)} \\sin{(b^{x})} - \\phi_{1}{(b,x)} + \\sin^{2}{(b^{x})}", "derivation": "\\phi_{1}{(b,x)} = \\sin{(b^{x})} and \\phi_{1}{(b,x)} \\sin{(b^{x})} = \\sin^{2}{(b^{x})} and 2 \\phi_{1}{(b,x)} \\sin{(b^{x})} = \\phi_{1}{(b,x)} \\sin{(b^{x})} + \\sin^{2}{(b^{x})} and 2 \\phi_{1}{(b,x)} \\sin{(b^{x})} - \\phi_{1}{(b,x)} = \\phi_{1}{(b,x)} \\sin{(b^{x})} - \\phi_{1}{(b,x)} + \\sin^{2}{(b^{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True))))"], [["times", 1, "sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Pow(sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True))), Integer(2)))"], [["add", 2, "Mul(Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True))))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Add(Mul(Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Pow(sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True))), Integer(2))))"], [["minus", 3, "Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Add(Mul(Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True)), sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('b', commutative=True), Symbol('x', commutative=True))), Pow(sin(Pow(Symbol('b', commutative=True), Symbol('x', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mu_{0}{(k)} = \\sin{(k)}, then derive 0 = \\cos{(k)} - \\frac{d}{d k} \\mu_{0}{(k)}, then obtain - \\frac{- \\cos{(k)} + \\frac{d}{d k} \\sin{(k)}}{\\frac{d}{d k} \\mu_{0}{(k)}} = 0", "derivation": "\\mu_{0}{(k)} = \\sin{(k)} and \\frac{d}{d k} \\mu_{0}{(k)} = \\frac{d}{d k} \\sin{(k)} and 0 = - \\frac{d}{d k} \\mu_{0}{(k)} + \\frac{d}{d k} \\sin{(k)} and 0 = \\cos{(k)} - \\frac{d}{d k} \\mu_{0}{(k)} and - \\cos{(k)} + \\frac{d}{d k} \\sin{(k)} = - \\frac{d}{d k} \\mu_{0}{(k)} + \\frac{d}{d k} \\sin{(k)} and - \\cos{(k)} + \\frac{d}{d k} \\sin{(k)} = 0 and - \\frac{- \\cos{(k)} + \\frac{d}{d k} \\sin{(k)}}{\\frac{d}{d k} \\mu_{0}{(k)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(cos(Symbol('k', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["minus", 4, "Add(cos(Symbol('k', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(0))"], [["divide", 6, "Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), cos(Symbol('k', commutative=True))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\mu_0')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\phi{(m)} = e^{\\sin{(m)}} and \\mathbf{P}{(m)} = \\phi{(m)} \\sin{(m)}, then obtain \\mathbf{P}{(m)} = e^{\\sin{(m)}} \\sin{(m)}", "derivation": "\\phi{(m)} = e^{\\sin{(m)}} and \\phi{(m)} \\sin{(m)} = e^{\\sin{(m)}} \\sin{(m)} and \\mathbf{P}{(m)} = \\phi{(m)} \\sin{(m)} and \\mathbf{P}{(m)} = e^{\\sin{(m)}} \\sin{(m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('m', commutative=True)), exp(sin(Symbol('m', commutative=True))))"], [["times", 1, "sin(Symbol('m', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(exp(sin(Symbol('m', commutative=True))), sin(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('m', commutative=True)), Mul(Function('\\\\phi')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{P}')(Symbol('m', commutative=True)), Mul(exp(sin(Symbol('m', commutative=True))), sin(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\lambda{(h,t,\\mathbf{M})} = \\frac{h - t}{\\mathbf{M}}, then obtain - t + \\frac{\\partial}{\\partial h} \\lambda{(h,t,\\mathbf{M})} - \\int \\frac{h - t}{\\mathbf{M}} dh = - t + \\frac{\\partial}{\\partial h} \\frac{h - t}{\\mathbf{M}} - \\int \\frac{h - t}{\\mathbf{M}} dh", "derivation": "\\lambda{(h,t,\\mathbf{M})} = \\frac{h - t}{\\mathbf{M}} and \\frac{\\partial}{\\partial h} \\lambda{(h,t,\\mathbf{M})} = \\frac{\\partial}{\\partial h} \\frac{h - t}{\\mathbf{M}} and \\frac{\\partial}{\\partial h} \\lambda{(h,t,\\mathbf{M})} - \\int \\frac{h - t}{\\mathbf{M}} dh = \\frac{\\partial}{\\partial h} \\frac{h - t}{\\mathbf{M}} - \\int \\frac{h - t}{\\mathbf{M}} dh and - t + \\frac{\\partial}{\\partial h} \\lambda{(h,t,\\mathbf{M})} - \\int \\frac{h - t}{\\mathbf{M}} dh = - t + \\frac{\\partial}{\\partial h} \\frac{h - t}{\\mathbf{M}} - \\int \\frac{h - t}{\\mathbf{M}} dh", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('h', commutative=True), Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('h', commutative=True), Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 2, "Integral(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('h', commutative=True), Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True))))), Add(Derivative(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True))))))"], [["minus", 3, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('h', commutative=True), Symbol('t', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True))))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('h', commutative=True))))))"]]}, {"prompt": "Given r{(v_{y},F_{x})} = F_{x} v_{y}, then derive \\frac{\\partial}{\\partial v_{y}} r{(v_{y},F_{x})} = F_{x}, then obtain \\int\\limits^{\\frac{\\partial}{\\partial v_{y}} F_{x} v_{y}} F_{x} r{(v_{y},F_{x})} dF_{x} = \\int\\limits^{\\frac{\\partial}{\\partial v_{y}} F_{x} v_{y}} F_{x}^{2} v_{y} dF_{x}", "derivation": "r{(v_{y},F_{x})} = F_{x} v_{y} and \\frac{\\partial}{\\partial v_{y}} r{(v_{y},F_{x})} = \\frac{\\partial}{\\partial v_{y}} F_{x} v_{y} and F_{x} r{(v_{y},F_{x})} = F_{x}^{2} v_{y} and \\frac{\\partial}{\\partial v_{y}} r{(v_{y},F_{x})} = F_{x} and F_{x} = \\frac{\\partial}{\\partial v_{y}} F_{x} v_{y} and \\int F_{x} r{(v_{y},F_{x})} dF_{x} = \\int F_{x}^{2} v_{y} dF_{x} and \\int\\limits^{\\frac{\\partial}{\\partial v_{y}} F_{x} v_{y}} F_{x} r{(v_{y},F_{x})} dF_{x} = \\int\\limits^{\\frac{\\partial}{\\partial v_{y}} F_{x} v_{y}} F_{x}^{2} v_{y} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["times", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Symbol('v_y', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('F_x', commutative=True))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Symbol('F_x', commutative=True), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Symbol('F_x', commutative=True), Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Symbol('v_y', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Mul(Symbol('F_x', commutative=True), Function('r')(Symbol('v_y', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(2)), Symbol('v_y', commutative=True)), Tuple(Symbol('F_x', commutative=True), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(c,f_{E})} = \\cos{(c^{f_{E}})} and L{(y^{\\prime},\\hat{p}_0)} = \\hat{p}_0 - y^{\\prime}, then obtain 2 (L{(y^{\\prime},\\hat{p}_0)} + \\operatorname{x^{{\\}'}}{(c,f_{E})}) \\operatorname{x^{{\\}'}}{(c,f_{E})} = 2 (\\hat{p}_0 - y^{\\prime} + \\operatorname{x^{{\\}'}}{(c,f_{E})}) \\operatorname{x^{{\\}'}}{(c,f_{E})}", "derivation": "\\operatorname{x^{{\\}'}}{(c,f_{E})} = \\cos{(c^{f_{E}})} and L{(y^{\\prime},\\hat{p}_0)} = \\hat{p}_0 - y^{\\prime} and L{(y^{\\prime},\\hat{p}_0)} + 2 \\operatorname{x^{{\\}'}}{(c,f_{E})} - \\cos{(c^{f_{E}})} = \\hat{p}_0 - y^{\\prime} + 2 \\operatorname{x^{{\\}'}}{(c,f_{E})} - \\cos{(c^{f_{E}})} and L{(y^{\\prime},\\hat{p}_0)} + \\operatorname{x^{{\\}'}}{(c,f_{E})} = \\hat{p}_0 - y^{\\prime} + \\operatorname{x^{{\\}'}}{(c,f_{E})} and 2 (L{(y^{\\prime},\\hat{p}_0)} + \\operatorname{x^{{\\}'}}{(c,f_{E})}) \\operatorname{x^{{\\}'}}{(c,f_{E})} = 2 (\\hat{p}_0 - y^{\\prime} + \\operatorname{x^{{\\}'}}{(c,f_{E})}) \\operatorname{x^{{\\}'}}{(c,f_{E})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True)), cos(Pow(Symbol('c', commutative=True), Symbol('f_E', commutative=True))))"], ["get_premise", "Equality(Function('L')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 2, "Add(Mul(Integer(2), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), cos(Pow(Symbol('c', commutative=True), Symbol('f_E', commutative=True)))))"], "Equality(Add(Function('L')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), cos(Pow(Symbol('c', commutative=True), Symbol('f_E', commutative=True))))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), cos(Pow(Symbol('c', commutative=True), Symbol('f_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('L')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))))"], [["times", 4, "Mul(Integer(2), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(2), Add(Function('L')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(2), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))), Function('x^\\\\prime')(Symbol('c', commutative=True), Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\omega,\\Omega)} = \\Omega + \\omega and f{(\\phi,F_{N})} = e^{F_{N}^{\\phi}}, then obtain (\\Omega + \\omega) f{(\\phi,F_{N})} = (\\Omega + \\omega) e^{F_{N}^{\\phi}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\omega,\\Omega)} = \\Omega + \\omega and f{(\\phi,F_{N})} = e^{F_{N}^{\\phi}} and \\operatorname{E_{\\lambda}}{(\\omega,\\Omega)} f{(\\phi,F_{N})} = \\operatorname{E_{\\lambda}}{(\\omega,\\Omega)} e^{F_{N}^{\\phi}} and (\\Omega + \\omega) f{(\\phi,F_{N})} = (\\Omega + \\omega) e^{F_{N}^{\\phi}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True)))"], ["get_premise", "Equality(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('F_N', commutative=True)), exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["times", 2, "Function('E_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('f')(Symbol('\\\\phi', commutative=True), Symbol('F_N', commutative=True))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Omega', commutative=True)), exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True)), Function('f')(Symbol('\\\\phi', commutative=True), Symbol('F_N', commutative=True))), Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Pow(Symbol('F_N', commutative=True), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(M,M_{E})} = M + M_{E}, then derive \\frac{\\frac{\\partial}{\\partial M} \\operatorname{C_{1}}{(M,M_{E})}}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} = \\frac{1}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}}, then obtain \\frac{\\frac{\\partial}{\\partial M} (M + M_{E})}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} = \\frac{1}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}}", "derivation": "\\operatorname{C_{1}}{(M,M_{E})} = M + M_{E} and \\frac{\\partial}{\\partial M} \\operatorname{C_{1}}{(M,M_{E})} = \\frac{\\partial}{\\partial M} (M + M_{E}) and \\frac{\\frac{\\partial}{\\partial M} \\operatorname{C_{1}}{(M,M_{E})}}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} = \\frac{\\frac{\\partial}{\\partial M} (M + M_{E})}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} and \\frac{\\frac{\\partial}{\\partial M} \\operatorname{C_{1}}{(M,M_{E})}}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} = \\frac{1}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} and \\frac{\\frac{\\partial}{\\partial M} (M + M_{E})}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}} = \\frac{1}{(M + M_{E}) \\operatorname{C_{1}}{(M,M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Derivative(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Derivative(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Derivative(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Derivative(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('C_1')(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{s}{(H,\\Psi_{nl})} = \\log{(H \\Psi_{nl})}, then obtain H (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}^{2}{(H,\\Psi_{nl})})^{2} = H (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}{(H,\\Psi_{nl})} \\log{(H \\Psi_{nl})})^{2}", "derivation": "\\mathbf{s}{(H,\\Psi_{nl})} = \\log{(H \\Psi_{nl})} and \\mathbf{s}^{2}{(H,\\Psi_{nl})} = \\mathbf{s}{(H,\\Psi_{nl})} \\log{(H \\Psi_{nl})} and \\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}^{2}{(H,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}{(H,\\Psi_{nl})} \\log{(H \\Psi_{nl})} and (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}^{2}{(H,\\Psi_{nl})})^{2} = (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}{(H,\\Psi_{nl})} \\log{(H \\Psi_{nl})})^{2} and H (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}^{2}{(H,\\Psi_{nl})})^{2} = H (\\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{s}{(H,\\Psi_{nl})} \\log{(H \\Psi_{nl})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(2)))"], [["times", 4, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Pow(Derivative(Pow(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(2))), Mul(Symbol('H', commutative=True), Pow(Derivative(Mul(Function('\\\\mathbf{s}')(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\phi{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and \\tilde{g}^*{(\\phi)} = \\cos{(\\phi)}, then obtain \\frac{- \\phi{(\\hat{\\mathbf{x}})} + \\tilde{g}^*{(\\phi)}}{\\phi} = \\frac{- \\phi{(\\hat{\\mathbf{x}})} + \\cos{(\\phi)}}{\\phi}", "derivation": "\\phi{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and \\tilde{g}^*{(\\phi)} = \\cos{(\\phi)} and \\tilde{g}^*{(\\phi)} - e^{\\hat{\\mathbf{x}}} = - e^{\\hat{\\mathbf{x}}} + \\cos{(\\phi)} and \\frac{\\tilde{g}^*{(\\phi)} - e^{\\hat{\\mathbf{x}}}}{\\phi} = \\frac{- e^{\\hat{\\mathbf{x}}} + \\cos{(\\phi)}}{\\phi} and \\frac{- \\phi{(\\hat{\\mathbf{x}})} + \\tilde{g}^*{(\\phi)}}{\\phi} = \\frac{- \\phi{(\\hat{\\mathbf{x}})} + \\cos{(\\phi)}}{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["minus", 2, "exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\phi', commutative=True))))"], [["divide", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\hat{p},c)} = \\frac{c}{\\hat{p}}, then obtain \\operatorname{P_{g}}{(\\hat{p})} \\frac{\\partial}{\\partial c} (e^{\\delta{(\\hat{p},c)}} + \\frac{1}{\\hat{p}}) = \\operatorname{P_{g}}{(\\hat{p})} \\frac{\\partial}{\\partial c} (e^{\\frac{c}{\\hat{p}}} + \\frac{1}{\\hat{p}})", "derivation": "\\delta{(\\hat{p},c)} = \\frac{c}{\\hat{p}} and e^{\\delta{(\\hat{p},c)}} = e^{\\frac{c}{\\hat{p}}} and e^{\\delta{(\\hat{p},c)}} + \\frac{1}{\\hat{p}} = e^{\\frac{c}{\\hat{p}}} + \\frac{1}{\\hat{p}} and \\frac{\\partial}{\\partial c} (e^{\\delta{(\\hat{p},c)}} + \\frac{1}{\\hat{p}}) = \\frac{\\partial}{\\partial c} (e^{\\frac{c}{\\hat{p}}} + \\frac{1}{\\hat{p}}) and \\operatorname{P_{g}}{(\\hat{p})} \\frac{\\partial}{\\partial c} (e^{\\delta{(\\hat{p},c)}} + \\frac{1}{\\hat{p}}) = \\operatorname{P_{g}}{(\\hat{p})} \\frac{\\partial}{\\partial c} (e^{\\frac{c}{\\hat{p}}} + \\frac{1}{\\hat{p}})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('c', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\delta')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True))), exp(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))"], "Equality(Add(exp(Function('\\\\delta')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Add(exp(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(exp(Function('\\\\delta')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(exp(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["times", 4, "Function('P_g')(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('\\\\hat{p}', commutative=True)), Derivative(Add(exp(Function('\\\\delta')(Symbol('\\\\hat{p}', commutative=True), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Function('P_g')(Symbol('\\\\hat{p}', commutative=True)), Derivative(Add(exp(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(\\omega)} = \\log{(\\omega)}, then obtain \\int (f{(\\omega)} - \\log{(\\omega)}) d\\omega = \\int (- f{(\\omega)} + \\log{(\\omega)}) d\\omega", "derivation": "f{(\\omega)} = \\log{(\\omega)} and f{(\\omega)} - \\log{(\\omega)} = 0 and 0 = - f{(\\omega)} + \\log{(\\omega)} and \\int 0 d\\omega = \\int (- f{(\\omega)} + \\log{(\\omega)}) d\\omega and \\int (f{(\\omega)} - \\log{(\\omega)}) d\\omega = \\int 0 d\\omega and \\int (f{(\\omega)} - \\log{(\\omega)}) d\\omega = \\int (- f{(\\omega)} + \\log{(\\omega)}) d\\omega", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["minus", 2, "Add(Function('f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f')(Symbol('\\\\omega', commutative=True))), log(Symbol('\\\\omega', commutative=True))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\omega', commutative=True))), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Function('f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Function('f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\omega', commutative=True))), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\hat{H},c_{0})} = - \\hat{H} + c_{0}, then obtain - \\frac{- \\hat{H} + c_{0} + 2 \\hat{H}_{\\lambda}{(\\hat{H},c_{0})}}{2 \\hat{H}} = - \\frac{- 3 \\hat{H} + 3 c_{0}}{2 \\hat{H}}", "derivation": "\\hat{H}_{\\lambda}{(\\hat{H},c_{0})} = - \\hat{H} + c_{0} and - \\hat{H} + c_{0} + \\hat{H}_{\\lambda}{(\\hat{H},c_{0})} = - 2 \\hat{H} + 2 c_{0} and - 2 \\hat{H} + 2 c_{0} + \\hat{H}_{\\lambda}{(\\hat{H},c_{0})} = - 3 \\hat{H} + 3 c_{0} and - \\frac{- 2 \\hat{H} + 2 c_{0} + \\hat{H}_{\\lambda}{(\\hat{H},c_{0})}}{2 \\hat{H}} = - \\frac{- 3 \\hat{H} + 3 c_{0}}{2 \\hat{H}} and - \\frac{- \\hat{H} + c_{0} + 2 \\hat{H}_{\\lambda}{(\\hat{H},c_{0})}}{2 \\hat{H}} = - \\frac{- 3 \\hat{H} + 3 c_{0}}{2 \\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), Symbol('c_0', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), Symbol('c_0', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(3), Symbol('c_0', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(2), Symbol('c_0', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(3), Symbol('c_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True), Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True))))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(3), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(3), Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\pi,T)} = \\frac{\\partial}{\\partial \\pi} (T + \\pi), then derive J + \\int (\\mathbf{B}{(\\pi,T)} - 1) dT = \\int 0 dT, then obtain - (J + \\int (\\mathbf{B}{(\\pi,T)} - 1) dT) \\int (\\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi)) dT = - (\\int 0 dT) \\int (\\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi)) dT", "derivation": "\\mathbf{B}{(\\pi,T)} = \\frac{\\partial}{\\partial \\pi} (T + \\pi) and \\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi) = 0 and \\int (\\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi)) dT = \\int 0 dT and J + \\int (\\mathbf{B}{(\\pi,T)} - 1) dT = \\int 0 dT and - (J + \\int (\\mathbf{B}{(\\pi,T)} - 1) dT) \\int (\\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi)) dT = - (\\int 0 dT) \\int (\\mathbf{B}{(\\pi,T)} - \\frac{\\partial}{\\partial \\pi} (T + \\pi)) dT", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Integer(0))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('T', commutative=True))), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('J', commutative=True), Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Tuple(Symbol('T', commutative=True)))), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('T', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Symbol('J', commutative=True), Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Tuple(Symbol('T', commutative=True)))), Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('T', commutative=True))), Integral(Add(Function('\\\\mathbf{B}')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(I,\\hat{p}_0)} = I^{\\hat{p}_0} and c{(I,\\hat{p}_0)} = I^{\\hat{p}_0}, then obtain \\int - I^{\\hat{p}_0} dI + \\int c{(I,\\hat{p}_0)} dI = \\int 0 dI", "derivation": "\\operatorname{v_{x}}{(I,\\hat{p}_0)} = I^{\\hat{p}_0} and - I^{\\hat{p}_0} + \\operatorname{v_{x}}{(I,\\hat{p}_0)} = 0 and \\int (- I^{\\hat{p}_0} + \\operatorname{v_{x}}{(I,\\hat{p}_0)}) dI = \\int 0 dI and c{(I,\\hat{p}_0)} = I^{\\hat{p}_0} and \\operatorname{v_{x}}{(I,\\hat{p}_0)} = c{(I,\\hat{p}_0)} and \\int - I^{\\hat{p}_0} dI + \\int \\operatorname{v_{x}}{(I,\\hat{p}_0)} dI = \\int 0 dI and \\int - I^{\\hat{p}_0} dI + \\int c{(I,\\hat{p}_0)} dI = \\int 0 dI", "srepr_derivation": [["renaming_premise", "Equality(Function('v_x')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Function('v_x')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Function('v_x')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('v_x')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('c')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["expand", 3], "Equality(Add(Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Function('v_x')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Function('c')(Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given l{(h,q)} = h \\sin{(q)}, then obtain h l^{h}{(h,q)} \\sin{(q)} = h (h \\sin{(q)})^{h} \\sin{(q)}", "derivation": "l{(h,q)} = h \\sin{(q)} and l^{h}{(h,q)} = (h \\sin{(q)})^{h} and h l^{h}{(h,q)} = h (h \\sin{(q)})^{h} and h l^{h}{(h,q)} \\frac{\\partial}{\\partial h} h \\sin{(q)} = h (h \\sin{(q)})^{h} \\frac{\\partial}{\\partial h} h \\sin{(q)} and h l^{h}{(h,q)} \\sin{(q)} = h (h \\sin{(q)})^{h} \\sin{(q)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('h', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('l')(Symbol('h', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Symbol('h', commutative=True)))"], [["times", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('l')(Symbol('h', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), Pow(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Symbol('h', commutative=True))))"], [["times", 3, "Derivative(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('l')(Symbol('h', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True)), Derivative(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Pow(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Symbol('h', commutative=True)), Derivative(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('l')(Symbol('h', commutative=True), Symbol('q', commutative=True)), Symbol('h', commutative=True)), sin(Symbol('q', commutative=True))), Mul(Symbol('h', commutative=True), Pow(Mul(Symbol('h', commutative=True), sin(Symbol('q', commutative=True))), Symbol('h', commutative=True)), sin(Symbol('q', commutative=True))))"]]}, {"prompt": "Given q{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^*, then derive q{(\\varphi^*)} = r - \\cos{(\\varphi^*)}, then derive - W + r - \\cos{(\\varphi^*)} = - W + \\hat{H}_l - \\cos{(\\varphi^*)}, then obtain - W + \\hat{H}_l + g^{\\prime}_{\\varepsilon} - \\cos{(\\varphi^*)} = - W + g^{\\prime}_{\\varepsilon} + q{(\\varphi^*)}", "derivation": "q{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^* and q{(\\varphi^*)} = r - \\cos{(\\varphi^*)} and r - \\cos{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^* and - W + r - \\cos{(\\varphi^*)} = - W + \\int \\sin{(\\varphi^*)} d\\varphi^* and - W + r - \\cos{(\\varphi^*)} = - W + \\hat{H}_l - \\cos{(\\varphi^*)} and - W + \\hat{H}_l - \\cos{(\\varphi^*)} = - W + \\int \\sin{(\\varphi^*)} d\\varphi^* and - W + \\hat{H}_l + g^{\\prime}_{\\varepsilon} - \\cos{(\\varphi^*)} = - W + g^{\\prime}_{\\varepsilon} + \\int \\sin{(\\varphi^*)} d\\varphi^* and - W + \\hat{H}_l + g^{\\prime}_{\\varepsilon} - \\cos{(\\varphi^*)} = - W + g^{\\prime}_{\\varepsilon} + q{(\\varphi^*)}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('\\\\varphi^*', commutative=True)), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('q')(Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 3, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 6, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('q')(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(J,\\rho)} = - \\rho + \\cos{(J)} and \\theta{(J,\\rho)} = \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)})^{2}, then obtain \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)}) \\operatorname{v_{1}}{(J,\\rho)} = \\theta{(J,\\rho)}", "derivation": "\\operatorname{v_{1}}{(J,\\rho)} = - \\rho + \\cos{(J)} and (- \\rho + \\cos{(J)}) \\operatorname{v_{1}}{(J,\\rho)} = (- \\rho + \\cos{(J)})^{2} and \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)}) \\operatorname{v_{1}}{(J,\\rho)} = \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)})^{2} and \\theta{(J,\\rho)} = \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)})^{2} and \\frac{\\partial}{\\partial J} (- \\rho + \\cos{(J)}) \\operatorname{v_{1}}{(J,\\rho)} = \\theta{(J,\\rho)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Function('v_1')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Function('v_1')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Symbol('J', commutative=True))), Function('v_1')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(F_{c})} = \\frac{d}{d F_{c}} \\sin{(F_{c})}, then derive \\operatorname{v_{t}}{(F_{c})} = \\cos{(F_{c})}, then obtain - \\sin{(F_{c})} + \\cos{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{v_{t}}{(F_{c})} = - 2 \\sin{(F_{c})} + \\cos{(F_{c})}", "derivation": "\\operatorname{v_{t}}{(F_{c})} = \\frac{d}{d F_{c}} \\sin{(F_{c})} and \\operatorname{v_{t}}{(F_{c})} = \\cos{(F_{c})} and \\frac{d}{d F_{c}} \\operatorname{v_{t}}{(F_{c})} = \\frac{d}{d F_{c}} \\cos{(F_{c})} and - \\sin{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{v_{t}}{(F_{c})} + \\frac{d}{d F_{c}} \\sin{(F_{c})} = - \\sin{(F_{c})} + \\frac{d}{d F_{c}} \\sin{(F_{c})} + \\frac{d}{d F_{c}} \\cos{(F_{c})} and - \\sin{(F_{c})} + \\cos{(F_{c})} + \\frac{d}{d F_{c}} \\operatorname{v_{t}}{(F_{c})} = - 2 \\sin{(F_{c})} + \\cos{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('F_c', commutative=True)), Derivative(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_t')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["add", 3, "Add(Mul(Integer(-1), sin(Symbol('F_c', commutative=True))), Derivative(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('F_c', commutative=True))), Derivative(Function('v_t')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('F_c', commutative=True))), Derivative(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True)), Derivative(Function('v_t')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), sin(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(P_{e},H)} = H P_{e}, then derive \\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} = P_{e}, then obtain \\int (\\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} dP_{e} = \\int (P_{e} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} dP_{e}", "derivation": "\\mathbf{H}{(P_{e},H)} = H P_{e} and \\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} = \\frac{\\partial}{\\partial H} H P_{e} and \\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} = P_{e} and \\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} + \\frac{1}{\\mathbf{H}{(P_{e},H)}} = P_{e} + \\frac{1}{\\mathbf{H}{(P_{e},H)}} and (\\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} = (P_{e} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} and \\int (\\frac{\\partial}{\\partial H} \\mathbf{H}{(P_{e},H)} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} dP_{e} = \\int (P_{e} + \\frac{1}{\\mathbf{H}{(P_{e},H)}})^{H} dP_{e}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('P_e', commutative=True))"], [["add", 3, "Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Add(Symbol('P_e', commutative=True), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Symbol('H', commutative=True)), Pow(Add(Symbol('P_e', commutative=True), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Symbol('H', commutative=True)))"], [["integrate", 5, "Symbol('P_e', commutative=True)"], "Equality(Integral(Pow(Add(Derivative(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Symbol('H', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Pow(Add(Symbol('P_e', commutative=True), Pow(Function('\\\\mathbf{H}')(Symbol('P_e', commutative=True), Symbol('H', commutative=True)), Integer(-1))), Symbol('H', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(x,\\mathbf{J})} = \\frac{\\mathbf{J}}{x}, then obtain \\frac{\\mathbf{J}^{2}}{x} = \\frac{\\mathbf{J} (\\mathbf{J} + \\frac{\\partial}{\\partial x} \\frac{\\mathbf{J}}{x} - \\frac{\\partial}{\\partial x} \\operatorname{v_{z}}{(x,\\mathbf{J})})}{x}", "derivation": "\\operatorname{v_{z}}{(x,\\mathbf{J})} = \\frac{\\mathbf{J}}{x} and \\frac{\\partial}{\\partial x} \\operatorname{v_{z}}{(x,\\mathbf{J})} = \\frac{\\partial}{\\partial x} \\frac{\\mathbf{J}}{x} and \\mathbf{J} + \\frac{\\partial}{\\partial x} \\operatorname{v_{z}}{(x,\\mathbf{J})} = \\mathbf{J} + \\frac{\\partial}{\\partial x} \\frac{\\mathbf{J}}{x} and \\mathbf{J} = \\mathbf{J} + \\frac{\\partial}{\\partial x} \\frac{\\mathbf{J}}{x} - \\frac{\\partial}{\\partial x} \\operatorname{v_{z}}{(x,\\mathbf{J})} and \\frac{\\mathbf{J}^{2}}{x} = \\frac{\\mathbf{J} (\\mathbf{J} + \\frac{\\partial}{\\partial x} \\frac{\\mathbf{J}}{x} - \\frac{\\partial}{\\partial x} \\operatorname{v_{z}}{(x,\\mathbf{J})})}{x}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Derivative(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{J}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["minus", 3, "Derivative(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))"], [["times", 4, "Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2)), Pow(Symbol('x', commutative=True), Integer(-1))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{J}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('v_z')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given n{(V,\\theta_1)} = \\int V^{\\theta_1} d\\theta_1, then obtain 1 = \\frac{2 \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1}{\\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} n{(V,\\theta_1)} + \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1}", "derivation": "n{(V,\\theta_1)} = \\int V^{\\theta_1} d\\theta_1 and \\frac{\\partial}{\\partial V} n{(V,\\theta_1)} = \\frac{\\partial}{\\partial V} \\int V^{\\theta_1} d\\theta_1 and \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} n{(V,\\theta_1)} = \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1 and \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} n{(V,\\theta_1)} + \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1 = 2 \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1 and 1 = \\frac{2 \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1}{\\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} n{(V,\\theta_1)} + \\frac{\\partial^{2}}{\\partial \\theta_1\\partial V} \\int V^{\\theta_1} d\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["divide", 4, "Add(Derivative(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Integer(2), Pow(Add(Derivative(Function('n')(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Integer(-1)), Derivative(Integral(Pow(Symbol('V', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\lambda,W,c_{0})} = W \\lambda - c_{0}, then obtain - 2 \\lambda + 2 (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})} = - 2 \\lambda + (W \\lambda - c_{0})^{2} + (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})}", "derivation": "\\hat{H}_l{(\\lambda,W,c_{0})} = W \\lambda - c_{0} and (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})} = (W \\lambda - c_{0})^{2} and - \\lambda + (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})} = - \\lambda + (W \\lambda - c_{0})^{2} and - 2 \\lambda + 2 (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})} = - 2 \\lambda + (W \\lambda - c_{0})^{2} + (W \\lambda - c_{0}) \\hat{H}_l{(\\lambda,W,c_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["times", 1, "Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True))), Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Integer(2)))"], [["minus", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Integer(2))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Mul(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(2), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Integer(2)), Mul(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\lambda', commutative=True), Symbol('W', commutative=True), Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(P_{e})} = \\log{(\\sin{(P_{e})})} and G{(\\mathbf{r})} = \\cos{(\\sin{(\\mathbf{r})})}, then obtain G{(\\mathbf{r})} - 1 = \\cos{(\\sin{(\\mathbf{r})})} - 1", "derivation": "\\Psi{(P_{e})} = \\log{(\\sin{(P_{e})})} and G{(\\mathbf{r})} = \\cos{(\\sin{(\\mathbf{r})})} and G{(\\mathbf{r})} - \\frac{\\Psi{(P_{e})}}{\\log{(\\sin{(P_{e})})}} = - \\frac{\\Psi{(P_{e})}}{\\log{(\\sin{(P_{e})})}} + \\cos{(\\sin{(\\mathbf{r})})} and G{(\\mathbf{r})} - 1 = \\cos{(\\sin{(\\mathbf{r})})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('P_e', commutative=True)), log(sin(Symbol('P_e', commutative=True))))"], ["get_premise", "Equality(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), cos(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 2, "Mul(Function('\\\\Psi')(Symbol('P_e', commutative=True)), Pow(log(sin(Symbol('P_e', commutative=True))), Integer(-1)))"], "Equality(Add(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('P_e', commutative=True)), Pow(log(sin(Symbol('P_e', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('P_e', commutative=True)), Pow(log(sin(Symbol('P_e', commutative=True))), Integer(-1))), cos(sin(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1)), Add(cos(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\rho_{f}{(\\mu,\\rho_b)} = \\mu \\rho_b, then derive \\frac{\\partial}{\\partial \\mu} \\rho_{f}{(\\mu,\\rho_b)} = \\rho_b, then obtain \\frac{\\frac{\\partial}{\\partial \\mu} \\rho_{f}{(\\mu,\\rho_b)}}{\\mu} = \\frac{\\rho_b}{\\mu}", "derivation": "\\rho_{f}{(\\mu,\\rho_b)} = \\mu \\rho_b and \\frac{\\partial}{\\partial \\mu} \\rho_{f}{(\\mu,\\rho_b)} = \\frac{\\partial}{\\partial \\mu} \\mu \\rho_b and \\frac{\\partial}{\\partial \\mu} \\rho_{f}{(\\mu,\\rho_b)} = \\rho_b and \\frac{\\frac{\\partial}{\\partial \\mu} \\rho_{f}{(\\mu,\\rho_b)}}{\\mu} = \\frac{\\rho_b}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\rho_b', commutative=True))"], [["divide", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Derivative(Function('\\\\rho_f')(Symbol('\\\\mu', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\mu{(U,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (U - \\hbar), then derive \\mu{(U,\\hbar)} = -1, then obtain - A_{1} + \\int \\mu^{\\hbar}{(U,\\hbar)} dU = - A_{1} + \\int (-1)^{\\hbar} dU", "derivation": "\\mu{(U,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (U - \\hbar) and \\mu{(U,\\hbar)} = -1 and \\mu^{\\hbar}{(U,\\hbar)} = (-1)^{\\hbar} and \\int \\mu^{\\hbar}{(U,\\hbar)} dU = \\int (-1)^{\\hbar} dU and - A_{1} + \\int \\mu^{\\hbar}{(U,\\hbar)} dU = - A_{1} + \\int (-1)^{\\hbar} dU", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('U', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu')(Symbol('U', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('U', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mu')(Symbol('U', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Integer(-1), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["minus", 4, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Integral(Pow(Function('\\\\mu')(Symbol('U', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Integral(Pow(Integer(-1), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(r_{0})} = \\cos{(r_{0})}, then obtain (\\mathbf{E}^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} - \\cos{(r_{0})} = (\\cos^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} - \\cos{(r_{0})}", "derivation": "\\mathbf{E}{(r_{0})} = \\cos{(r_{0})} and \\mathbf{E}^{r_{0}}{(r_{0})} = \\cos^{r_{0}}{(r_{0})} and (\\mathbf{E}^{r_{0}}{(r_{0})})^{r_{0}} = (\\cos^{r_{0}}{(r_{0})})^{r_{0}} and (\\mathbf{E}^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} = (\\cos^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} and (\\mathbf{E}^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} - \\cos{(r_{0})} = (\\cos^{r_{0}}{(r_{0})})^{r_{0}} \\mathbf{E}^{- r_{0}}{(r_{0})} - \\cos{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["divide", 3, "Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)))), Mul(Pow(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["minus", 4, "cos(Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)))), Mul(Integer(-1), cos(Symbol('r_0', commutative=True)))), Add(Mul(Pow(Pow(cos(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)))), Mul(Integer(-1), cos(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given G{(v_{t})} = \\sin{(v_{t})}, then derive \\int G{(v_{t})} dv_{t} = r - \\cos{(v_{t})}, then derive r - \\cos{(v_{t})} = l - \\cos{(v_{t})}, then obtain \\int 0 dl = \\int (l - \\cos{(v_{t})} - \\int G{(v_{t})} dv_{t}) dl", "derivation": "G{(v_{t})} = \\sin{(v_{t})} and \\int G{(v_{t})} dv_{t} = \\int \\sin{(v_{t})} dv_{t} and \\int G{(v_{t})} dv_{t} = r - \\cos{(v_{t})} and r - \\cos{(v_{t})} = \\int \\sin{(v_{t})} dv_{t} and r - \\cos{(v_{t})} = l - \\cos{(v_{t})} and \\int G{(v_{t})} dv_{t} = l - \\cos{(v_{t})} and 0 = l - \\cos{(v_{t})} - \\int G{(v_{t})} dv_{t} and \\int 0 dl = \\int (l - \\cos{(v_{t})} - \\int G{(v_{t})} dv_{t}) dl", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["minus", 6, "Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Integer(0), Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))))"], [["integrate", 7, "Symbol('l', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Function('G')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J}, then derive \\operatorname{f^{*}}{(\\mathbf{J})} = \\Omega + \\sin{(\\mathbf{J})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} (\\Omega + \\sin{(\\mathbf{J})}) = \\frac{d}{d \\mathbf{J}} \\int \\cos{(\\mathbf{J})} d\\mathbf{J}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{J})} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\operatorname{f^{*}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and \\operatorname{f^{*}}{(\\mathbf{J})} = \\Omega + \\sin{(\\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} (\\Omega + \\sin{(\\mathbf{J})}) = \\frac{d}{d \\mathbf{J}} \\int \\cos{(\\mathbf{J})} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_integrals", 1], "Equality(Function('f^*')(Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(I,t)} = - I + \\sin{(t)} and \\operatorname{n_{1}}{(I,t)} = \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}), then obtain 2 I + 2 \\operatorname{n_{1}}{(I,t)} - 2 \\sin{(t)} - 1 = 2 I + \\operatorname{n_{1}}{(I,t)} - 2 \\sin{(t)} + \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) - 1", "derivation": "\\operatorname{f^{\\prime}}{(I,t)} = - I + \\sin{(t)} and \\operatorname{n_{1}}{(I,t)} = \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) and - \\operatorname{f^{\\prime}}{(I,t)} + \\operatorname{n_{1}}{(I,t)} = - \\operatorname{f^{\\prime}}{(I,t)} + \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) and I + \\operatorname{n_{1}}{(I,t)} - \\sin{(t)} = I - \\sin{(t)} + \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) and I - \\operatorname{f^{\\prime}}{(I,t)} + 2 \\operatorname{n_{1}}{(I,t)} - \\sin{(t)} - 1 = I - \\operatorname{f^{\\prime}}{(I,t)} + \\operatorname{n_{1}}{(I,t)} - \\sin{(t)} + \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) - 1 and 2 I + 2 \\operatorname{n_{1}}{(I,t)} - 2 \\sin{(t)} - 1 = 2 I + \\operatorname{n_{1}}{(I,t)} - 2 \\sin{(t)} + \\frac{\\partial}{\\partial I} (- I + \\sin{(t)}) - 1", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["minus", 2, "Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('I', commutative=True), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), sin(Symbol('t', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["add", 4, "Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Integer(-1))"], "Equality(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Mul(Integer(2), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), sin(Symbol('t', commutative=True))), Integer(-1)), Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(2), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('t', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('I', commutative=True)), Function('n_1')(Symbol('I', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('t', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('t', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given L{(\\nabla)} = \\log{(\\nabla)}, then derive J = \\int (- L{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} d\\nabla, then obtain \\log{(\\nabla)} \\frac{d}{d \\nabla} \\int 0^{\\nabla} d\\nabla = \\log{(\\nabla)} \\frac{d}{d \\nabla} J", "derivation": "L{(\\nabla)} = \\log{(\\nabla)} and 0 = - L{(\\nabla)} + \\log{(\\nabla)} and 0^{\\nabla} = (- L{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} and \\int 0^{\\nabla} d\\nabla = \\int (- L{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} d\\nabla and J = \\int (- L{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} d\\nabla and \\frac{d}{d \\nabla} \\int 0^{\\nabla} d\\nabla = \\frac{d}{d \\nabla} \\int (- L{(\\nabla)} + \\log{(\\nabla)})^{\\nabla} d\\nabla and \\frac{d}{d \\nabla} \\int 0^{\\nabla} d\\nabla = \\frac{d}{d \\nabla} J and \\log{(\\nabla)} \\frac{d}{d \\nabla} \\int 0^{\\nabla} d\\nabla = \\log{(\\nabla)} \\frac{d}{d \\nabla} J", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Function('L')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Pow(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Symbol('J', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\nabla', commutative=True))), log(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Symbol('J', commutative=True), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["times", 7, "log(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(log(Symbol('\\\\nabla', commutative=True)), Derivative(Integral(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(log(Symbol('\\\\nabla', commutative=True)), Derivative(Symbol('J', commutative=True), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(l,v_{1})} = l v_{1}, then obtain \\int \\frac{\\int e^{\\mathbf{g}{(l,v_{1})}} dl}{\\mathbf{g}{(l,v_{1})}} dv_{1} = \\int \\frac{\\int e^{l v_{1}} dl}{\\mathbf{g}{(l,v_{1})}} dv_{1}", "derivation": "\\mathbf{g}{(l,v_{1})} = l v_{1} and e^{\\mathbf{g}{(l,v_{1})}} = e^{l v_{1}} and \\int e^{\\mathbf{g}{(l,v_{1})}} dl = \\int e^{l v_{1}} dl and \\frac{\\int e^{\\mathbf{g}{(l,v_{1})}} dl}{l v_{1}} = \\frac{\\int e^{l v_{1}} dl}{l v_{1}} and \\frac{\\int e^{\\mathbf{g}{(l,v_{1})}} dl}{\\mathbf{g}{(l,v_{1})}} = \\frac{\\int e^{l v_{1}} dl}{\\mathbf{g}{(l,v_{1})}} and \\int \\frac{\\int e^{\\mathbf{g}{(l,v_{1})}} dl}{\\mathbf{g}{(l,v_{1})}} dv_{1} = \\int \\frac{\\int e^{l v_{1}} dl}{\\mathbf{g}{(l,v_{1})}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), exp(Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(exp(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(exp(Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["divide", 3, "Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Integral(exp(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Integral(exp(Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Integral(exp(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Integral(exp(Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))))"], [["integrate", 5, "Symbol('v_1', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Integral(exp(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Integral(exp(Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given g{(\\omega)} = \\cos{(\\log{(\\omega)})}, then obtain \\cos{(\\frac{d}{d \\omega} g{(\\omega)})} = \\cos{(\\frac{\\sin{(\\log{(\\omega)})}}{\\omega})}", "derivation": "g{(\\omega)} = \\cos{(\\log{(\\omega)})} and \\frac{d}{d \\omega} g{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})} and \\cos{(\\frac{d}{d \\omega} g{(\\omega)})} = \\cos{(\\frac{d}{d \\omega} \\cos{(\\log{(\\omega)})})} and \\cos{(\\frac{d}{d \\omega} g{(\\omega)})} = \\cos{(\\frac{\\sin{(\\log{(\\omega)})}}{\\omega})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\omega', commutative=True)), cos(log(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('g')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), cos(Derivative(cos(log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('g')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), sin(log(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{X})} = \\cos{(e^{\\hat{X}})}, then derive \\frac{d}{d \\hat{X}} \\operatorname{n_{2}}{(\\hat{X})} = - e^{\\hat{X}} \\sin{(e^{\\hat{X}})}, then obtain \\frac{d}{d \\hat{X}} \\operatorname{n_{2}}{(\\hat{X})} + 1 = \\frac{d}{d \\hat{X}} \\cos{(e^{\\hat{X}})} + 1", "derivation": "\\operatorname{n_{2}}{(\\hat{X})} = \\cos{(e^{\\hat{X}})} and \\frac{d}{d \\hat{X}} \\operatorname{n_{2}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\cos{(e^{\\hat{X}})} and \\frac{d}{d \\hat{X}} \\operatorname{n_{2}}{(\\hat{X})} = - e^{\\hat{X}} \\sin{(e^{\\hat{X}})} and - e^{\\hat{X}} \\sin{(e^{\\hat{X}})} = \\frac{d}{d \\hat{X}} \\cos{(e^{\\hat{X}})} and - e^{\\hat{X}} \\sin{(e^{\\hat{X}})} + 1 = \\frac{d}{d \\hat{X}} \\cos{(e^{\\hat{X}})} + 1 and \\frac{d}{d \\hat{X}} \\operatorname{n_{2}}{(\\hat{X})} + 1 = \\frac{d}{d \\hat{X}} \\cos{(e^{\\hat{X}})} + 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{X}', commutative=True)), cos(exp(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('\\\\hat{X}', commutative=True)), sin(exp(Symbol('\\\\hat{X}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(Symbol('\\\\hat{X}', commutative=True)), sin(exp(Symbol('\\\\hat{X}', commutative=True)))), Derivative(cos(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{X}', commutative=True)), sin(exp(Symbol('\\\\hat{X}', commutative=True)))), Integer(1)), Add(Derivative(cos(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Function('n_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(exp(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given f{(k)} = \\log{(k)}, then obtain (\\int f^{k}{(k)} dk)^{k} = (\\int \\log{(k)}^{k} dk)^{k}", "derivation": "f{(k)} = \\log{(k)} and f^{k}{(k)} = \\log{(k)}^{k} and \\int f^{k}{(k)} dk = \\int \\log{(k)}^{k} dk and (\\int f^{k}{(k)} dk)^{k} = (\\int \\log{(k)}^{k} dk)^{k}", "srepr_derivation": [["get_premise", "Equality(Function('f')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Integral(Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Integral(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(a^{\\dagger},s)} = a^{\\dagger} + \\sin{(s)}, then obtain (\\iint (- a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)}) da^{\\dagger} da^{\\dagger})^{s} + \\iint \\sin{(s)} da^{\\dagger} da^{\\dagger} = \\iint \\sin{(s)} da^{\\dagger} da^{\\dagger} + (\\iint \\sin{(s)} da^{\\dagger} da^{\\dagger})^{s}", "derivation": "\\sigma_{p}{(a^{\\dagger},s)} = a^{\\dagger} + \\sin{(s)} and - a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)} = \\sin{(s)} and \\int (- a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)}) da^{\\dagger} = \\int \\sin{(s)} da^{\\dagger} and \\iint (- a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)}) da^{\\dagger} da^{\\dagger} = \\iint \\sin{(s)} da^{\\dagger} da^{\\dagger} and (\\iint (- a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)}) da^{\\dagger} da^{\\dagger})^{s} = (\\iint \\sin{(s)} da^{\\dagger} da^{\\dagger})^{s} and (\\iint (- a^{\\dagger} + \\sigma_{p}{(a^{\\dagger},s)}) da^{\\dagger} da^{\\dagger})^{s} + \\iint \\sin{(s)} da^{\\dagger} da^{\\dagger} = \\iint \\sin{(s)} da^{\\dagger} da^{\\dagger} + (\\iint \\sin{(s)} da^{\\dagger} da^{\\dagger})^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('s', commutative=True))))"], [["minus", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), sin(Symbol('s', commutative=True)))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 4, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('s', commutative=True)))"], [["add", 5, "Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Pow(Integral(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\sigma_p')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then obtain 2 \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\operatorname{n_{1}}^{\\eta^{\\prime}}{(\\eta^{\\prime})} = 2 \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})}", "derivation": "\\operatorname{n_{1}}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and 2 \\operatorname{n_{1}}{(\\eta^{\\prime})} = \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\sin{(\\eta^{\\prime})} and \\operatorname{n_{1}}^{\\eta^{\\prime}}{(\\eta^{\\prime})} = \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})} and \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\operatorname{n_{1}}^{\\eta^{\\prime}}{(\\eta^{\\prime})} + \\sin{(\\eta^{\\prime})} = \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\sin{(\\eta^{\\prime})} + \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})} and 2 \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\operatorname{n_{1}}^{\\eta^{\\prime}}{(\\eta^{\\prime})} = 2 \\operatorname{n_{1}}{(\\eta^{\\prime})} + \\sin^{\\eta^{\\prime}}{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 1, "Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 3, "Add(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Pow(Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Function('n_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Pow(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(z^{*})} = \\sin{(z^{*})}, then obtain \\sigma_{p}{(z^{*})} \\sin{(z^{*})} = \\sigma_{p}{(z^{*})} \\sin{(z^{*})} - \\sigma_{p}{(z^{*})} + \\sin{(z^{*})}", "derivation": "\\sigma_{p}{(z^{*})} = \\sin{(z^{*})} and \\sigma_{p}^{2}{(z^{*})} = \\sigma_{p}{(z^{*})} \\sin{(z^{*})} and \\sigma_{p}^{2}{(z^{*})} + \\sigma_{p}{(z^{*})} = \\sigma_{p}^{2}{(z^{*})} + \\sin{(z^{*})} and \\sigma_{p}^{2}{(z^{*})} = \\sigma_{p}^{2}{(z^{*})} - \\sigma_{p}{(z^{*})} + \\sin{(z^{*})} and \\sigma_{p}{(z^{*})} \\sin{(z^{*})} = \\sigma_{p}{(z^{*})} \\sin{(z^{*})} - \\sigma_{p}{(z^{*})} + \\sin{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["times", 1, "Function('\\\\sigma_p')(Symbol('z^*', commutative=True))"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2)), Mul(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))))"], [["add", 1, "Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2)), Function('\\\\sigma_p')(Symbol('z^*', commutative=True))), Add(Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2)), sin(Symbol('z^*', commutative=True))))"], [["minus", 3, "Function('\\\\sigma_p')(Symbol('z^*', commutative=True))"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2)), Add(Pow(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('z^*', commutative=True))), sin(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Add(Mul(Function('\\\\sigma_p')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('z^*', commutative=True))), sin(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(W)} = e^{W}, then derive e^{W} + \\frac{d^{2}}{d W^{2}} \\operatorname{E_{n}}{(W)} = 2 e^{W}, then obtain e^{W} + \\frac{d^{2}}{d W^{2}} e^{W} = 2 e^{W}", "derivation": "\\operatorname{E_{n}}{(W)} = e^{W} and \\operatorname{E_{n}}{(W)} + e^{W} = 2 e^{W} and \\frac{d}{d W} (\\operatorname{E_{n}}{(W)} + e^{W}) = \\frac{d}{d W} 2 e^{W} and \\frac{d^{2}}{d W^{2}} (\\operatorname{E_{n}}{(W)} + e^{W}) = \\frac{d^{2}}{d W^{2}} 2 e^{W} and e^{W} + \\frac{d^{2}}{d W^{2}} \\operatorname{E_{n}}{(W)} = 2 e^{W} and e^{W} + \\frac{d^{2}}{d W^{2}} e^{W} = 2 e^{W}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["add", 1, "exp(Symbol('W', commutative=True))"], "Equality(Add(Function('E_n')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Mul(Integer(2), exp(Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Function('E_n')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Function('E_n')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(2))), Derivative(Mul(Integer(2), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Add(exp(Symbol('W', commutative=True)), Derivative(Function('E_n')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))), Mul(Integer(2), exp(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(exp(Symbol('W', commutative=True)), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2)))), Mul(Integer(2), exp(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})} = \\sin{(\\log{(\\Psi_{\\lambda})})}, then obtain \\frac{d}{d \\Psi_{\\lambda}} \\frac{\\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})}}{\\sin^{2}{(\\log{(\\Psi_{\\lambda})})}} = \\frac{d}{d \\Psi_{\\lambda}} \\frac{1}{\\sin{(\\log{(\\Psi_{\\lambda})})}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})} = \\sin{(\\log{(\\Psi_{\\lambda})})} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})}}{\\sin{(\\log{(\\Psi_{\\lambda})})}} = 1 and \\frac{\\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})}}{\\sin^{2}{(\\log{(\\Psi_{\\lambda})})}} = \\frac{1}{\\sin{(\\log{(\\Psi_{\\lambda})})}} and \\frac{d}{d \\Psi_{\\lambda}} \\frac{\\operatorname{J_{\\varepsilon}}{(\\Psi_{\\lambda})}}{\\sin^{2}{(\\log{(\\Psi_{\\lambda})})}} = \\frac{d}{d \\Psi_{\\lambda}} \\frac{1}{\\sin{(\\log{(\\Psi_{\\lambda})})}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 1, "sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-2))), Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-2))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Pow(sin(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(\\mathbf{M},\\mathbf{J}_P)} = \\mathbf{J}_P + \\mathbf{M}, then obtain \\mathbf{J}_P + \\mathbf{M} + \\frac{Z^{\\mathbf{M}}{(\\mathbf{M},\\mathbf{J}_P)}}{Z{(\\mathbf{M},\\mathbf{J}_P)}} = \\mathbf{J}_P + \\mathbf{M} + \\frac{(\\mathbf{J}_P + \\mathbf{M})^{\\mathbf{M}}}{Z{(\\mathbf{M},\\mathbf{J}_P)}}", "derivation": "Z{(\\mathbf{M},\\mathbf{J}_P)} = \\mathbf{J}_P + \\mathbf{M} and Z^{\\mathbf{M}}{(\\mathbf{M},\\mathbf{J}_P)} = (\\mathbf{J}_P + \\mathbf{M})^{\\mathbf{M}} and \\frac{Z^{\\mathbf{M}}{(\\mathbf{M},\\mathbf{J}_P)}}{Z{(\\mathbf{M},\\mathbf{J}_P)}} = \\frac{(\\mathbf{J}_P + \\mathbf{M})^{\\mathbf{M}}}{Z{(\\mathbf{M},\\mathbf{J}_P)}} and \\mathbf{J}_P + \\mathbf{M} + \\frac{Z^{\\mathbf{M}}{(\\mathbf{M},\\mathbf{J}_P)}}{Z{(\\mathbf{M},\\mathbf{J}_P)}} = \\mathbf{J}_P + \\mathbf{M} + \\frac{(\\mathbf{J}_P + \\mathbf{M})^{\\mathbf{M}}}{Z{(\\mathbf{M},\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 2, "Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"], [["add", 3, "Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Mul(Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('Z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(W)} = e^{W} and t{(W)} = \\hat{H}_{\\lambda}{(W)} e^{W}, then obtain - 4 W + \\hat{H}_{\\lambda}^{2}{(W)} t{(W)} = - 4 W + t^{2}{(W)}", "derivation": "\\hat{H}_{\\lambda}{(W)} = e^{W} and \\hat{H}_{\\lambda}{(W)} e^{W} = e^{2 W} and t{(W)} = \\hat{H}_{\\lambda}{(W)} e^{W} and t{(W)} = e^{2 W} and \\hat{H}_{\\lambda}^{2}{(W)} e^{2 W} = e^{4 W} and \\hat{H}_{\\lambda}^{2}{(W)} t{(W)} = t^{2}{(W)} and - 4 W + \\hat{H}_{\\lambda}^{2}{(W)} t{(W)} = - 4 W + t^{2}{(W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["times", 1, "exp(Symbol('W', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), exp(Mul(Integer(2), Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('W', commutative=True)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('t')(Symbol('W', commutative=True)), exp(Mul(Integer(2), Symbol('W', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('W', commutative=True)))), exp(Mul(Integer(4), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), Integer(2)), Function('t')(Symbol('W', commutative=True))), Pow(Function('t')(Symbol('W', commutative=True)), Integer(2)))"], [["minus", 6, "Mul(Integer(4), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(4), Symbol('W', commutative=True)), Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True)), Integer(2)), Function('t')(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Integer(4), Symbol('W', commutative=True)), Pow(Function('t')(Symbol('W', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{B}{(y^{\\prime})} = \\log{(e^{y^{\\prime}})} and \\operatorname{z^{*}}{(y^{\\prime})} = \\frac{\\mathbf{B}{(y^{\\prime})}}{\\log{(e^{y^{\\prime}})}}, then obtain \\mathbf{B}{(y^{\\prime})} + \\operatorname{z^{*}}^{y^{\\prime}}{(y^{\\prime})} = \\mathbf{B}{(y^{\\prime})} + 1", "derivation": "\\mathbf{B}{(y^{\\prime})} = \\log{(e^{y^{\\prime}})} and \\frac{\\mathbf{B}{(y^{\\prime})}}{\\log{(e^{y^{\\prime}})}} = 1 and (\\frac{\\mathbf{B}{(y^{\\prime})}}{\\log{(e^{y^{\\prime}})}})^{y^{\\prime}} = 1 and (\\frac{\\mathbf{B}{(y^{\\prime})}}{\\log{(e^{y^{\\prime}})}})^{y^{\\prime}} + \\mathbf{B}{(y^{\\prime})} = \\mathbf{B}{(y^{\\prime})} + 1 and \\operatorname{z^{*}}{(y^{\\prime})} = \\frac{\\mathbf{B}{(y^{\\prime})}}{\\log{(e^{y^{\\prime}})}} and \\mathbf{B}{(y^{\\prime})} + \\operatorname{z^{*}}^{y^{\\prime}}{(y^{\\prime})} = \\mathbf{B}{(y^{\\prime})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), log(exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "log(exp(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Symbol('y^{\\\\prime}', commutative=True)), Integer(1))"], [["add", 3, "Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True))), Add(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(Function('z^*')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Add(Function('\\\\mathbf{B}')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} = e^{- J_{\\varepsilon} + \\tilde{g}}, then derive 0 = t_{2} - e^{- J_{\\varepsilon} + \\tilde{g}} - \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon}, then obtain 0 = t_{2} - \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} - \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon}", "derivation": "\\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} = e^{- J_{\\varepsilon} + \\tilde{g}} and \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon} = \\int e^{- J_{\\varepsilon} + \\tilde{g}} dJ_{\\varepsilon} and 0 = - \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon} + \\int e^{- J_{\\varepsilon} + \\tilde{g}} dJ_{\\varepsilon} and 0 = t_{2} - e^{- J_{\\varepsilon} + \\tilde{g}} - \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon} and 0 = t_{2} - \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} - \\int \\operatorname{m_{s}}{(J_{\\varepsilon},\\tilde{g})} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Integral(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Integral(exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))), Mul(Integer(-1), Integral(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Integral(Function('m_s')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})} = \\hat{p} \\mathbf{r} and \\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})} = \\hat{p} \\mathbf{r}, then obtain \\int \\frac{\\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})}}{\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})}} d\\mathbf{r} = \\int 1 d\\mathbf{r}", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})} = \\hat{p} \\mathbf{r} and \\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})} = \\hat{p} \\mathbf{r} and \\frac{\\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})}}{\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})}} = \\frac{\\hat{p} \\mathbf{r}}{\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})}} and \\frac{\\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})}}{\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})}} = 1 and \\int \\frac{\\operatorname{E_{\\lambda}}{(\\hat{p},\\mathbf{r})}}{\\operatorname{y^{\\prime}}{(\\hat{p},\\mathbf{r})}} d\\mathbf{r} = \\int 1 d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["divide", 2, "Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(f,q)} = f q, then derive f q + (\\frac{\\frac{\\partial}{\\partial f} \\ddot{x}{(f,q)}}{f q} - \\frac{\\ddot{x}{(f,q)}}{f^{2} q})^{f} = 0^{f} + f q, then obtain (\\frac{\\frac{\\partial}{\\partial f} \\ddot{x}{(f,q)}}{\\ddot{x}{(f,q)}} - \\frac{1}{f})^{f} + \\ddot{x}{(f,q)} = 0^{f} + \\ddot{x}{(f,q)}", "derivation": "\\ddot{x}{(f,q)} = f q and \\frac{\\ddot{x}{(f,q)}}{f q} = 1 and \\frac{\\partial}{\\partial f} \\frac{\\ddot{x}{(f,q)}}{f q} = \\frac{d}{d f} 1 and (\\frac{\\partial}{\\partial f} \\frac{\\ddot{x}{(f,q)}}{f q})^{f} = (\\frac{d}{d f} 1)^{f} and f q + (\\frac{\\partial}{\\partial f} \\frac{\\ddot{x}{(f,q)}}{f q})^{f} = f q + (\\frac{d}{d f} 1)^{f} and f q + (\\frac{\\frac{\\partial}{\\partial f} \\ddot{x}{(f,q)}}{f q} - \\frac{\\ddot{x}{(f,q)}}{f^{2} q})^{f} = 0^{f} + f q and (\\frac{\\frac{\\partial}{\\partial f} \\ddot{x}{(f,q)}}{\\ddot{x}{(f,q)}} - \\frac{1}{f})^{f} + \\ddot{x}{(f,q)} = 0^{f} + \\ddot{x}{(f,q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('q', commutative=True)))"], [["divide", 1, "Mul(Symbol('f', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)))"], [["add", 4, "Mul(Symbol('f', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Mul(Symbol('f', commutative=True), Symbol('q', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True))), Add(Mul(Symbol('f', commutative=True), Symbol('q', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Symbol('f', commutative=True), Symbol('q', commutative=True)), Pow(Add(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-2)), Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True)))), Symbol('f', commutative=True))), Add(Pow(Integer(0), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Pow(Add(Mul(Pow(Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)))), Symbol('f', commutative=True)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))), Add(Pow(Integer(0), Symbol('f', commutative=True)), Function('\\\\ddot{x}')(Symbol('f', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} = \\Psi_{\\lambda} \\varepsilon_0 + \\varphi, then derive \\int \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} d\\varepsilon_0 = \\frac{\\Psi_{\\lambda} \\varepsilon_0^{2}}{2} + \\varepsilon_0 \\varphi + h, then obtain \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} + \\int \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} d\\varepsilon_0 = \\frac{\\Psi_{\\lambda} \\varepsilon_0^{2}}{2} + \\varepsilon_0 \\varphi + h + \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)}", "derivation": "\\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} = \\Psi_{\\lambda} \\varepsilon_0 + \\varphi and \\int \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} d\\varepsilon_0 = \\int (\\Psi_{\\lambda} \\varepsilon_0 + \\varphi) d\\varepsilon_0 and \\int \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} d\\varepsilon_0 = \\frac{\\Psi_{\\lambda} \\varepsilon_0^{2}}{2} + \\varepsilon_0 \\varphi + h and \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} + \\int \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)} d\\varepsilon_0 = \\frac{\\Psi_{\\lambda} \\varepsilon_0^{2}}{2} + \\varepsilon_0 \\varphi + h + \\operatorname{F_{H}}{(\\varepsilon_0,\\Psi_{\\lambda},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('h', commutative=True)))"], [["add", 3, "Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('h', commutative=True), Function('F_H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(r)} = \\log{(r)}, then obtain (- \\frac{d}{d r} \\mathbf{A}{(r)})^{r} (\\frac{r \\frac{d^{2}}{d r^{2}} \\mathbf{A}{(r)}}{\\frac{d}{d r} \\mathbf{A}{(r)}} + \\log{(- \\frac{d}{d r} \\mathbf{A}{(r)})}) = (- \\frac{1}{r})^{r} (\\log{(- \\frac{1}{r})} - 1)", "derivation": "\\mathbf{A}{(r)} = \\log{(r)} and - \\mathbf{A}{(r)} = - \\log{(r)} and \\frac{d}{d r} - \\mathbf{A}{(r)} = \\frac{d}{d r} - \\log{(r)} and (\\frac{d}{d r} - \\mathbf{A}{(r)})^{r} = (\\frac{d}{d r} - \\log{(r)})^{r} and \\frac{d}{d r} (\\frac{d}{d r} - \\mathbf{A}{(r)})^{r} = \\frac{d}{d r} (\\frac{d}{d r} - \\log{(r)})^{r} and (- \\frac{d}{d r} \\mathbf{A}{(r)})^{r} (\\frac{r \\frac{d^{2}}{d r^{2}} \\mathbf{A}{(r)}}{\\frac{d}{d r} \\mathbf{A}{(r)}} + \\log{(- \\frac{d}{d r} \\mathbf{A}{(r)})}) = (- \\frac{1}{r})^{r} (\\log{(- \\frac{1}{r})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('r', commutative=True))), Mul(Integer(-1), log(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Mul(Integer(-1), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Derivative(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Integer(-1), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Mul(Integer(-1), Derivative(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Symbol('r', commutative=True)), Add(Mul(Symbol('r', commutative=True), Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))), log(Mul(Integer(-1), Derivative(Function('\\\\mathbf{A}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))), Mul(Pow(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1))), Symbol('r', commutative=True)), Add(log(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(-1)))), Integer(-1))))"]]}, {"prompt": "Given \\varphi{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0, then derive \\varphi{(\\varepsilon_0)} = z - \\cos{(\\varepsilon_0)}, then obtain \\frac{d}{d \\varepsilon_0} e^{\\varphi{(\\varepsilon_0)}} = \\frac{\\partial}{\\partial \\varepsilon_0} e^{z^{*} - \\cos{(\\varepsilon_0)}}", "derivation": "\\varphi{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and \\varphi{(\\varepsilon_0)} = z - \\cos{(\\varepsilon_0)} and e^{\\varphi{(\\varepsilon_0)}} = e^{z - \\cos{(\\varepsilon_0)}} and z - \\cos{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and e^{\\varphi{(\\varepsilon_0)}} = e^{\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} e^{\\varphi{(\\varepsilon_0)}} = \\frac{d}{d \\varepsilon_0} e^{\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} e^{\\varphi{(\\varepsilon_0)}} = \\frac{\\partial}{\\partial \\varepsilon_0} e^{z^{*} - \\cos{(\\varepsilon_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True)), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True)), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["exp", 2], "Equality(exp(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True))), exp(Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True))), exp(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(exp(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(exp(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(exp(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then derive \\int r{(\\mathbf{F})} d\\mathbf{F} = t + \\sin{(\\mathbf{F})}, then obtain ((\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}})^{\\mathbf{F}} = ((\\int r{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}})^{\\mathbf{F}}", "derivation": "r{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\int r{(\\mathbf{F})} d\\mathbf{F} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and \\int r{(\\mathbf{F})} d\\mathbf{F} = t + \\sin{(\\mathbf{F})} and (\\int r{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}} = (t + \\sin{(\\mathbf{F})})^{\\mathbf{F}} and (\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}} = (t + \\sin{(\\mathbf{F})})^{\\mathbf{F}} and (\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}} = (\\int r{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}} and ((\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}})^{\\mathbf{F}} = ((\\int r{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('t', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Integral(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('t', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('t', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Integral(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 6, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Pow(Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Pow(Integral(Function('r')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hat{p},\\mathbf{s})} = \\hat{p} + \\mathbf{s} and \\hat{x}_0{(\\hat{p})} = \\hat{p}, then obtain \\int (\\hat{p} + \\mathbf{s})^{2} \\hat{x}_0{(\\hat{p})} d\\hat{p} = \\int \\hat{p} (\\hat{p} + \\mathbf{s})^{2} d\\hat{p}", "derivation": "\\mathbf{J}_f{(\\hat{p},\\mathbf{s})} = \\hat{p} + \\mathbf{s} and \\hat{x}_0{(\\hat{p})} = \\hat{p} and \\hat{x}_0{(\\hat{p})} \\mathbf{J}_f^{2}{(\\hat{p},\\mathbf{s})} = \\hat{p} \\mathbf{J}_f^{2}{(\\hat{p},\\mathbf{s})} and \\int \\hat{x}_0{(\\hat{p})} \\mathbf{J}_f^{2}{(\\hat{p},\\mathbf{s})} d\\hat{p} = \\int \\hat{p} \\mathbf{J}_f^{2}{(\\hat{p},\\mathbf{s})} d\\hat{p} and \\int (\\hat{p} + \\mathbf{s})^{2} \\hat{x}_0{(\\hat{p})} d\\hat{p} = \\int \\hat{p} (\\hat{p} + \\mathbf{s})^{2} d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], [["times", 2, "Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)), Function('\\\\hat{x}_0')(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\nabla,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\nabla})}, then derive \\int \\operatorname{E_{n}}{(\\nabla,\\mathbf{f})} d\\nabla = \\mathbf{f} \\operatorname{Si}{(\\frac{\\mathbf{f}}{\\nabla})} + \\nabla \\cos{(\\frac{\\mathbf{f}}{\\nabla})} + g_{\\varepsilon}, then obtain \\mathbf{f} \\operatorname{Si}{(\\frac{\\mathbf{f}}{\\nabla})} + \\nabla \\cos{(\\frac{\\mathbf{f}}{\\nabla})} + g_{\\varepsilon} = \\int \\cos{(\\frac{\\mathbf{f}}{\\nabla})} d\\nabla", "derivation": "\\operatorname{E_{n}}{(\\nabla,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\nabla})} and \\int \\operatorname{E_{n}}{(\\nabla,\\mathbf{f})} d\\nabla = \\int \\cos{(\\frac{\\mathbf{f}}{\\nabla})} d\\nabla and \\int \\operatorname{E_{n}}{(\\nabla,\\mathbf{f})} d\\nabla = \\mathbf{f} \\operatorname{Si}{(\\frac{\\mathbf{f}}{\\nabla})} + \\nabla \\cos{(\\frac{\\mathbf{f}}{\\nabla})} + g_{\\varepsilon} and \\mathbf{f} \\operatorname{Si}{(\\frac{\\mathbf{f}}{\\nabla})} + \\nabla \\cos{(\\frac{\\mathbf{f}}{\\nabla})} + g_{\\varepsilon} = \\int \\cos{(\\frac{\\mathbf{f}}{\\nabla})} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Si(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))), Mul(Symbol('\\\\nabla', commutative=True), cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Si(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))), Mul(Symbol('\\\\nabla', commutative=True), cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))), Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(C_{1},T)} = \\frac{\\partial}{\\partial T} (C_{1} + T), then derive \\mathbf{H}{(C_{1},T)} = 1, then obtain \\mathbf{H}{(C_{1},\\frac{T}{\\frac{\\partial}{\\partial T} (C_{1} + T)})} + \\cos{(\\frac{\\frac{\\partial}{\\partial T} (C_{1} + T)}{T})} = \\mathbf{H}{(C_{1},\\frac{T}{\\frac{\\partial}{\\partial T} (C_{1} + T)})} + \\cos{(\\frac{1}{T})}", "derivation": "\\mathbf{H}{(C_{1},T)} = \\frac{\\partial}{\\partial T} (C_{1} + T) and \\mathbf{H}{(C_{1},T)} = 1 and \\frac{\\partial}{\\partial T} (C_{1} + T) = 1 and \\frac{\\frac{\\partial}{\\partial T} (C_{1} + T)}{T} = \\frac{1}{T} and \\frac{\\mathbf{H}{(C_{1},T)}}{T} = \\frac{1}{T} and \\cos{(\\frac{\\mathbf{H}{(C_{1},T)}}{T})} = \\cos{(\\frac{1}{T})} and \\cos{(\\frac{\\frac{\\partial}{\\partial T} (C_{1} + T)}{T})} = \\cos{(\\frac{1}{T})} and \\mathbf{H}{(C_{1},\\frac{T}{\\frac{\\partial}{\\partial T} (C_{1} + T)})} + \\cos{(\\frac{\\frac{\\partial}{\\partial T} (C_{1} + T)}{T})} = \\mathbf{H}{(C_{1},\\frac{T}{\\frac{\\partial}{\\partial T} (C_{1} + T)})} + \\cos{(\\frac{1}{T})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Pow(Symbol('T', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Symbol('T', commutative=True))), Pow(Symbol('T', commutative=True), Integer(-1)))"], [["cos", 5], "Equality(cos(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Symbol('T', commutative=True)))), cos(Pow(Symbol('T', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(cos(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), cos(Pow(Symbol('T', commutative=True), Integer(-1))))"], [["add", 7, "Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Mul(Symbol('T', commutative=True), Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Mul(Symbol('T', commutative=True), Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)))), cos(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))), Add(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True), Mul(Symbol('T', commutative=True), Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)))), cos(Pow(Symbol('T', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} = \\hat{x} - g^{\\prime}_{\\varepsilon}, then obtain (\\hat{x} - g^{\\prime}_{\\varepsilon})^{\\hat{x}} (g^{\\prime}_{\\varepsilon} + \\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} - 1) = (\\hat{x} - 1) (\\hat{x} - g^{\\prime}_{\\varepsilon})^{\\hat{x}}", "derivation": "\\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} = \\hat{x} - g^{\\prime}_{\\varepsilon} and g^{\\prime}_{\\varepsilon} + \\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} = \\hat{x} and g^{\\prime}_{\\varepsilon} + \\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} - 1 = \\hat{x} - 1 and (\\hat{x} - g^{\\prime}_{\\varepsilon})^{\\hat{x}} (g^{\\prime}_{\\varepsilon} + \\theta_{1}{(g^{\\prime}_{\\varepsilon},\\hat{x})} - 1) = (\\hat{x} - 1) (\\hat{x} - g^{\\prime}_{\\varepsilon})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), Add(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))"], [["times", 3, "Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\theta_1')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\eta^{\\prime})} = \\int e^{\\eta^{\\prime}} d\\eta^{\\prime}, then obtain \\int \\eta^{\\prime} \\int \\operatorname{f^{*}}{(\\eta^{\\prime})} d\\eta^{\\prime} d\\eta^{\\prime} = \\int \\eta^{\\prime} \\iint e^{\\eta^{\\prime}} d\\eta^{\\prime} d\\eta^{\\prime} d\\eta^{\\prime}", "derivation": "\\operatorname{f^{*}}{(\\eta^{\\prime})} = \\int e^{\\eta^{\\prime}} d\\eta^{\\prime} and \\int \\operatorname{f^{*}}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\iint e^{\\eta^{\\prime}} d\\eta^{\\prime} d\\eta^{\\prime} and \\eta^{\\prime} \\int \\operatorname{f^{*}}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\eta^{\\prime} \\iint e^{\\eta^{\\prime}} d\\eta^{\\prime} d\\eta^{\\prime} and \\int \\eta^{\\prime} \\int \\operatorname{f^{*}}{(\\eta^{\\prime})} d\\eta^{\\prime} d\\eta^{\\prime} = \\int \\eta^{\\prime} \\iint e^{\\eta^{\\prime}} d\\eta^{\\prime} d\\eta^{\\prime} d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Function('f^*')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} = \\frac{\\theta_1}{\\eta^{\\prime}}, then obtain \\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} + \\theta_1)^{\\theta_1} = \\frac{d}{d \\eta^{\\prime}} (2 \\theta_1)^{\\theta_1}", "derivation": "\\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} = \\frac{\\theta_1}{\\eta^{\\prime}} and \\eta^{\\prime} \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} = \\theta_1 and \\eta^{\\prime} \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} + \\theta_1 = 2 \\theta_1 and (\\eta^{\\prime} \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} + \\theta_1)^{\\theta_1} = (2 \\theta_1)^{\\theta_1} and \\frac{\\partial}{\\partial \\eta^{\\prime}} (\\eta^{\\prime} \\operatorname{f^{*}}{(\\theta_1,\\eta^{\\prime})} + \\theta_1)^{\\theta_1} = \\frac{d}{d \\eta^{\\prime}} (2 \\theta_1)^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f^*')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\theta_1', commutative=True))"], [["add", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f^*')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f^*')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f^*')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(c_{0},\\Psi)} = \\Psi c_{0}, then derive \\frac{\\partial}{\\partial \\Psi} \\rho_{f}{(c_{0},\\Psi)} = c_{0}, then obtain \\frac{d}{d \\Psi} 0 = \\frac{\\partial}{\\partial \\Psi} (c_{0} - \\frac{\\partial}{\\partial \\Psi} \\Psi c_{0})", "derivation": "\\rho_{f}{(c_{0},\\Psi)} = \\Psi c_{0} and \\frac{\\partial}{\\partial \\Psi} \\rho_{f}{(c_{0},\\Psi)} = \\frac{\\partial}{\\partial \\Psi} \\Psi c_{0} and \\frac{\\partial}{\\partial \\Psi} \\rho_{f}{(c_{0},\\Psi)} = c_{0} and 0 = c_{0} - \\frac{\\partial}{\\partial \\Psi} \\rho_{f}{(c_{0},\\Psi)} and 0 = c_{0} - \\frac{\\partial}{\\partial \\Psi} \\Psi c_{0} and \\frac{d}{d \\Psi} 0 = \\frac{\\partial}{\\partial \\Psi} (c_{0} - \\frac{\\partial}{\\partial \\Psi} \\Psi c_{0})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('c_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('c_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('c_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Symbol('c_0', commutative=True))"], [["minus", 3, "Derivative(Function('\\\\rho_f')(Symbol('c_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\rho_f')(Symbol('c_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["differentiate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then obtain U{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} - \\log{(\\hat{H}_l)}^{\\hat{H}_l} = 2 \\log{(\\hat{H}_l)} - \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "derivation": "U{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and U^{\\hat{H}_l}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}^{\\hat{H}_l} and U{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} = 2 \\log{(\\hat{H}_l)} and U{(\\hat{H}_l)} - U^{\\hat{H}_l}{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} = - U^{\\hat{H}_l}{(\\hat{H}_l)} + 2 \\log{(\\hat{H}_l)} and U{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} - \\log{(\\hat{H}_l)}^{\\hat{H}_l} = 2 \\log{(\\hat{H}_l)} - \\log{(\\hat{H}_l)}^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "log(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{H}_l', commutative=True))))"], [["minus", 3, "Pow(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Pow(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), log(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('U')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))), Add(Mul(Integer(2), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\varphi^*,v_{z})} = \\cos{(\\varphi^* v_{z})}, then obtain (\\varphi^* v_{z} + \\Psi_{\\lambda}^{v_{z}}{(\\varphi^*,v_{z})}) \\frac{\\partial}{\\partial v_{z}} Z{(\\varphi^*,v_{z})} = (\\varphi^* v_{z} + \\cos^{v_{z}}{(\\varphi^* v_{z})}) \\frac{\\partial}{\\partial v_{z}} Z{(\\varphi^*,v_{z})}", "derivation": "\\Psi_{\\lambda}{(\\varphi^*,v_{z})} = \\cos{(\\varphi^* v_{z})} and \\Psi_{\\lambda}^{v_{z}}{(\\varphi^*,v_{z})} = \\cos^{v_{z}}{(\\varphi^* v_{z})} and \\varphi^* v_{z} + \\Psi_{\\lambda}^{v_{z}}{(\\varphi^*,v_{z})} = \\varphi^* v_{z} + \\cos^{v_{z}}{(\\varphi^* v_{z})} and (\\varphi^* v_{z} + \\Psi_{\\lambda}^{v_{z}}{(\\varphi^*,v_{z})}) \\frac{\\partial}{\\partial v_{z}} Z{(\\varphi^*,v_{z})} = (\\varphi^* v_{z} + \\cos^{v_{z}}{(\\varphi^* v_{z})}) \\frac{\\partial}{\\partial v_{z}} Z{(\\varphi^*,v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Pow(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))))"], [["times", 3, "Derivative(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Derivative(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Pow(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Derivative(Function('Z')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(f^{*})} = \\cos{(f^{*})}, then obtain (\\mathbf{s}{(f^{*})} - \\cos{(f^{*})})^{f^{*}} = 0^{f^{*}}", "derivation": "\\mathbf{s}{(f^{*})} = \\cos{(f^{*})} and \\mathbf{s}{(f^{*})} + \\cos{(f^{*})} = 2 \\cos{(f^{*})} and \\mathbf{s}{(f^{*})} - \\cos{(f^{*})} = 0 and (\\mathbf{s}{(f^{*})} - \\cos{(f^{*})})^{f^{*}} = 0^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["add", 1, "cos(Symbol('f^*', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Mul(Integer(2), cos(Symbol('f^*', commutative=True))))"], [["minus", 2, "Mul(Integer(2), cos(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Symbol('f^*', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('f^*', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{s}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)), Pow(Integer(0), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\hat{H}_l{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}}, then obtain \\int (\\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} + 1) d\\mathbf{H} = \\int (1 + \\frac{1}{\\mathbf{H}}) d\\mathbf{H}", "derivation": "\\hat{H}_l{(\\mathbf{H})} = \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\hat{H}_l{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\hat{H}_l{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} + 1 = 1 + \\frac{1}{\\mathbf{H}} and \\int (\\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} + 1) d\\mathbf{H} = \\int (1 + \\frac{1}{\\mathbf{H}}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Integer(1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(l,\\mathbf{H})} = - \\mathbf{H} + e^{l}, then obtain \\frac{e^{\\int \\mathbf{J}_f{(l,\\mathbf{H})} dl}}{\\int (- \\mathbf{H} + e^{l}) dl} = \\frac{e^{\\int (- \\mathbf{H} + e^{l}) dl}}{\\int (- \\mathbf{H} + e^{l}) dl}", "derivation": "\\mathbf{J}_f{(l,\\mathbf{H})} = - \\mathbf{H} + e^{l} and \\int \\mathbf{J}_f{(l,\\mathbf{H})} dl = \\int (- \\mathbf{H} + e^{l}) dl and e^{\\int \\mathbf{J}_f{(l,\\mathbf{H})} dl} = e^{\\int (- \\mathbf{H} + e^{l}) dl} and \\frac{e^{\\int \\mathbf{J}_f{(l,\\mathbf{H})} dl}}{\\int (- \\mathbf{H} + e^{l}) dl} = \\frac{e^{\\int (- \\mathbf{H} + e^{l}) dl}}{\\int (- \\mathbf{H} + e^{l}) dl}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\mathbf{J}_f')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('l', commutative=True)))), exp(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))))"], [["divide", 3, "Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(exp(Integral(Function('\\\\mathbf{J}_f')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(-1))), Mul(exp(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\nabla{(\\sigma_x)} = \\cos{(\\sigma_x)}, then obtain \\int \\frac{d}{d \\sigma_x} 0 d\\sigma_x = \\int \\frac{d}{d \\sigma_x} (-1 + \\frac{\\cos{(\\cos{(\\sigma_x)})}}{\\cos{(\\nabla{(\\sigma_x)})}}) d\\sigma_x", "derivation": "\\nabla{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\cos{(\\nabla{(\\sigma_x)})} = \\cos{(\\cos{(\\sigma_x)})} and 0 = - \\cos{(\\nabla{(\\sigma_x)})} + \\cos{(\\cos{(\\sigma_x)})} and 0 = \\frac{- \\cos{(\\nabla{(\\sigma_x)})} + \\cos{(\\cos{(\\sigma_x)})}}{\\cos{(\\nabla{(\\sigma_x)})}} and 0 = -1 + \\frac{\\cos{(\\cos{(\\sigma_x)})}}{\\cos{(\\nabla{(\\sigma_x)})}} and \\frac{d}{d \\sigma_x} 0 = \\frac{d}{d \\sigma_x} (-1 + \\frac{\\cos{(\\cos{(\\sigma_x)})}}{\\cos{(\\nabla{(\\sigma_x)})}}) and \\int \\frac{d}{d \\sigma_x} 0 d\\sigma_x = \\int \\frac{d}{d \\sigma_x} (-1 + \\frac{\\cos{(\\cos{(\\sigma_x)})}}{\\cos{(\\nabla{(\\sigma_x)})}}) d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True))), cos(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True)))), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["divide", 3, "cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True)))), cos(cos(Symbol('\\\\sigma_x', commutative=True)))), Pow(cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))))"], [["expand", 4], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), cos(cos(Symbol('\\\\sigma_x', commutative=True))))))"], [["differentiate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Pow(cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), cos(cos(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Derivative(Add(Integer(-1), Mul(Pow(cos(Function('\\\\nabla')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), cos(cos(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(W,v,\\dot{y})} = W \\dot{y} v and \\dot{x}{(W,v,\\dot{y})} = W \\dot{y} v, then obtain \\int \\frac{\\dot{x}{(W,v,\\dot{y})}}{S} d\\dot{y} = \\int \\frac{\\operatorname{F_{N}}{(W,v,\\dot{y})}}{S} d\\dot{y}", "derivation": "\\operatorname{F_{N}}{(W,v,\\dot{y})} = W \\dot{y} v and \\dot{x}{(W,v,\\dot{y})} = W \\dot{y} v and \\dot{x}{(W,v,\\dot{y})} = \\operatorname{F_{N}}{(W,v,\\dot{y})} and \\frac{\\dot{x}{(W,v,\\dot{y})}}{S} = \\frac{\\operatorname{F_{N}}{(W,v,\\dot{y})}}{S} and \\int \\frac{\\dot{x}{(W,v,\\dot{y})}}{S} d\\dot{y} = \\int \\frac{\\operatorname{F_{N}}{(W,v,\\dot{y})}}{S} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('\\\\dot{y}', commutative=True), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('F_N')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 3, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('F_N')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('F_N')(Symbol('W', commutative=True), Symbol('v', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(s)} = \\cos{(\\log{(s)})}, then obtain ((\\varepsilon_{0}{(s)} + \\log{(s)})^{s})^{s} + ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} + 1 = 2 ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} + 1", "derivation": "\\varepsilon_{0}{(s)} = \\cos{(\\log{(s)})} and \\varepsilon_{0}{(s)} + \\log{(s)} = \\log{(s)} + \\cos{(\\log{(s)})} and (\\varepsilon_{0}{(s)} + \\log{(s)})^{s} = (\\log{(s)} + \\cos{(\\log{(s)})})^{s} and ((\\varepsilon_{0}{(s)} + \\log{(s)})^{s})^{s} = ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} and ((\\varepsilon_{0}{(s)} + \\log{(s)})^{s})^{s} + 1 = ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} + 1 and ((\\varepsilon_{0}{(s)} + \\log{(s)})^{s})^{s} + ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} + 1 = 2 ((\\log{(s)} + \\cos{(\\log{(s)})})^{s})^{s} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True))))"], [["add", 1, "log(Symbol('s', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["add", 4, 1], "Equality(Add(Pow(Pow(Add(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(1)), Add(Pow(Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(1)))"], [["add", 5, "Pow(Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Pow(Pow(Add(Function('\\\\varepsilon_0')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integer(1)), Add(Mul(Integer(2), Pow(Pow(Add(log(Symbol('s', commutative=True)), cos(log(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Symbol('s', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(f^{\\prime})} = e^{f^{\\prime}}, then derive \\frac{d}{d f^{\\prime}} \\operatorname{F_{g}}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain \\frac{d^{2}}{d (f^{\\prime})^{2}} e^{f^{\\prime}} = e^{f^{\\prime}}", "derivation": "\\operatorname{F_{g}}{(f^{\\prime})} = e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\operatorname{F_{g}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\frac{d}{d f^{\\prime}} \\operatorname{F_{g}}{(f^{\\prime})} = e^{f^{\\prime}} and \\operatorname{F_{g}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\operatorname{F_{g}}{(f^{\\prime})} and \\operatorname{F_{g}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} e^{f^{\\prime}} and \\frac{d^{2}}{d (f^{\\prime})^{2}} e^{f^{\\prime}} = e^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('F_g')(Symbol('f^{\\\\prime}', commutative=True)), Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), exp(Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(E)} = \\log{(E)}, then obtain \\operatorname{F_{c}}{(E)} - 1 = \\log{(E)} - 1", "derivation": "\\operatorname{F_{c}}{(E)} = \\log{(E)} and 2 \\operatorname{F_{c}}{(E)} = \\operatorname{F_{c}}{(E)} + \\log{(E)} and - \\frac{\\operatorname{F_{c}}{(E)} + \\log{(E)}}{2 \\operatorname{F_{c}}{(E)}} + 2 \\operatorname{F_{c}}{(E)} = - \\frac{\\operatorname{F_{c}}{(E)} + \\log{(E)}}{2 \\operatorname{F_{c}}{(E)}} + \\operatorname{F_{c}}{(E)} + \\log{(E)} and - \\frac{\\operatorname{F_{c}}{(E)} + \\log{(E)}}{2 \\operatorname{F_{c}}{(E)}} + \\operatorname{F_{c}}{(E)} = - \\frac{\\operatorname{F_{c}}{(E)} + \\log{(E)}}{2 \\operatorname{F_{c}}{(E)}} + \\log{(E)} and \\operatorname{F_{c}}{(E)} - 1 = \\log{(E)} - 1", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["add", 1, "Function('F_c')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('E', commutative=True))), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))))"], [["minus", 2, "Mul(Rational(1, 2), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(Function('F_c')(Symbol('E', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(Function('F_c')(Symbol('E', commutative=True)), Integer(-1))), Mul(Integer(2), Function('F_c')(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(Function('F_c')(Symbol('E', commutative=True)), Integer(-1))), Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))))"], [["minus", 3, "Function('F_c')(Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(Function('F_c')(Symbol('E', commutative=True)), Integer(-1))), Function('F_c')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Add(Function('F_c')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True))), Pow(Function('F_c')(Symbol('E', commutative=True)), Integer(-1))), log(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('F_c')(Symbol('E', commutative=True)), Integer(-1)), Add(log(Symbol('E', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{g}{(x,W)} = x^{W}, then obtain (-1 + \\frac{(x^{W})^{x}}{x}) \\frac{\\partial}{\\partial x} \\mathbf{g}^{x}{(x,W)} = (-1 + \\frac{(x^{W})^{x}}{x}) \\frac{\\partial}{\\partial x} (x^{W})^{x}", "derivation": "\\mathbf{g}{(x,W)} = x^{W} and \\mathbf{g}^{x}{(x,W)} = (x^{W})^{x} and \\frac{\\mathbf{g}^{x}{(x,W)}}{x} = \\frac{(x^{W})^{x}}{x} and \\frac{\\partial}{\\partial x} \\mathbf{g}^{x}{(x,W)} = \\frac{\\partial}{\\partial x} (x^{W})^{x} and (-1 + \\frac{\\mathbf{g}^{x}{(x,W)}}{x}) \\frac{\\partial}{\\partial x} \\mathbf{g}^{x}{(x,W)} = (-1 + \\frac{\\mathbf{g}^{x}{(x,W)}}{x}) \\frac{\\partial}{\\partial x} (x^{W})^{x} and (-1 + \\frac{(x^{W})^{x}}{x}) \\frac{\\partial}{\\partial x} \\mathbf{g}^{x}{(x,W)} = (-1 + \\frac{(x^{W})^{x}}{x}) \\frac{\\partial}{\\partial x} (x^{W})^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 2, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 4, "Add(Integer(-1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True))))"], "Equality(Mul(Add(Integer(-1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)))), Derivative(Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Add(Integer(-1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)))), Derivative(Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Integer(-1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)))), Derivative(Pow(Function('\\\\mathbf{g}')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Add(Integer(-1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)))), Derivative(Pow(Pow(Symbol('x', commutative=True), Symbol('W', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu{(k,i,\\theta)} = (\\theta k)^{i}, then obtain \\frac{\\partial}{\\partial i} \\frac{(\\theta k)^{i} \\int \\mu{(k,i,\\theta)} dk}{\\theta k} = \\frac{\\partial}{\\partial i} \\frac{(\\theta k)^{i} \\int (\\theta k)^{i} dk}{\\theta k}", "derivation": "\\mu{(k,i,\\theta)} = (\\theta k)^{i} and \\int \\mu{(k,i,\\theta)} dk = \\int (\\theta k)^{i} dk and (\\theta k)^{i} \\int \\mu{(k,i,\\theta)} dk = (\\theta k)^{i} \\int (\\theta k)^{i} dk and \\frac{(\\theta k)^{i} \\int \\mu{(k,i,\\theta)} dk}{\\theta k} = \\frac{(\\theta k)^{i} \\int (\\theta k)^{i} dk}{\\theta k} and \\frac{\\partial}{\\partial i} \\frac{(\\theta k)^{i} \\int \\mu{(k,i,\\theta)} dk}{\\theta k} = \\frac{\\partial}{\\partial i} \\frac{(\\theta k)^{i} \\int (\\theta k)^{i} dk}{\\theta k}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["times", 2, "Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["divide", 3, "Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Function('\\\\mu')(Symbol('k', commutative=True), Symbol('i', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Integral(Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('k', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(s)} = \\sin{(s)}, then obtain ((\\int \\mathbf{J}_M^{s}{(s)} ds)^{s} + \\int \\sin^{s}{(s)} ds) \\int \\mathbf{J}_M^{s}{(s)} ds = (\\int \\sin^{s}{(s)} ds + (\\int \\sin^{s}{(s)} ds)^{s}) \\int \\mathbf{J}_M^{s}{(s)} ds", "derivation": "\\mathbf{J}_M{(s)} = \\sin{(s)} and \\mathbf{J}_M^{s}{(s)} = \\sin^{s}{(s)} and \\int \\mathbf{J}_M^{s}{(s)} ds = \\int \\sin^{s}{(s)} ds and (\\int \\mathbf{J}_M^{s}{(s)} ds)^{s} = (\\int \\sin^{s}{(s)} ds)^{s} and (\\int \\mathbf{J}_M^{s}{(s)} ds)^{s} + \\int \\sin^{s}{(s)} ds = \\int \\sin^{s}{(s)} ds + (\\int \\sin^{s}{(s)} ds)^{s} and ((\\int \\mathbf{J}_M^{s}{(s)} ds)^{s} + \\int \\sin^{s}{(s)} ds) \\int \\mathbf{J}_M^{s}{(s)} ds = (\\int \\sin^{s}{(s)} ds + (\\int \\sin^{s}{(s)} ds)^{s}) \\int \\mathbf{J}_M^{s}{(s)} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["add", 4, "Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Pow(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Pow(Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["times", 5, "Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Mul(Add(Pow(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Pow(Integral(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(l,v_{z})} = l v_{z}, then derive \\frac{\\partial}{\\partial v_{z}} \\operatorname{V_{\\mathbf{E}}}{(l,v_{z})} = l, then obtain l - v_{z} = - v_{z} + \\frac{\\partial}{\\partial v_{z}} l v_{z}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(l,v_{z})} = l v_{z} and \\frac{\\partial}{\\partial v_{z}} \\operatorname{V_{\\mathbf{E}}}{(l,v_{z})} = \\frac{\\partial}{\\partial v_{z}} l v_{z} and \\frac{\\partial}{\\partial v_{z}} \\operatorname{V_{\\mathbf{E}}}{(l,v_{z})} = l and l = \\frac{\\partial}{\\partial v_{z}} l v_{z} and l - v_{z} = - v_{z} + \\frac{\\partial}{\\partial v_{z}} l v_{z}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('l', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('l', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('l', commutative=True), Derivative(Mul(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(\\rho_b)} = \\log{(\\rho_b)}, then obtain \\log{(\\frac{d}{d \\rho_b} \\rho_b \\theta{(\\rho_b)})} = \\log{(\\frac{d}{d \\rho_b} \\rho_b \\log{(\\rho_b)})}", "derivation": "\\theta{(\\rho_b)} = \\log{(\\rho_b)} and \\rho_b \\theta{(\\rho_b)} = \\rho_b \\log{(\\rho_b)} and \\frac{d}{d \\rho_b} \\rho_b \\theta{(\\rho_b)} = \\frac{d}{d \\rho_b} \\rho_b \\log{(\\rho_b)} and \\log{(\\frac{d}{d \\rho_b} \\rho_b \\theta{(\\rho_b)})} = \\log{(\\frac{d}{d \\rho_b} \\rho_b \\log{(\\rho_b)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), log(Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), log(Derivative(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(k)} = \\log{(k)}, then derive e^{\\operatorname{A_{2}}{(k)}} \\frac{d}{d k} \\operatorname{A_{2}}{(k)} + \\frac{d}{d k} \\operatorname{A_{2}}{(k)} = \\frac{d}{d k} \\operatorname{A_{2}}{(k)} + 1, then obtain (k \\frac{d}{d k} \\log{(k)} + \\frac{d}{d k} \\log{(k)}) (\\frac{d}{d k} \\log{(k)} + 1) = (\\frac{d}{d k} \\log{(k)} + 1)^{2}", "derivation": "\\operatorname{A_{2}}{(k)} = \\log{(k)} and e^{\\operatorname{A_{2}}{(k)}} = k and \\operatorname{A_{2}}{(k)} + e^{\\operatorname{A_{2}}{(k)}} = k + \\operatorname{A_{2}}{(k)} and \\frac{d}{d k} (\\operatorname{A_{2}}{(k)} + e^{\\operatorname{A_{2}}{(k)}}) = \\frac{d}{d k} (k + \\operatorname{A_{2}}{(k)}) and e^{\\operatorname{A_{2}}{(k)}} \\frac{d}{d k} \\operatorname{A_{2}}{(k)} + \\frac{d}{d k} \\operatorname{A_{2}}{(k)} = \\frac{d}{d k} \\operatorname{A_{2}}{(k)} + 1 and k \\frac{d}{d k} \\log{(k)} + \\frac{d}{d k} \\log{(k)} = \\frac{d}{d k} \\log{(k)} + 1 and (k \\frac{d}{d k} \\log{(k)} + \\frac{d}{d k} \\log{(k)}) (\\frac{d}{d k} \\log{(k)} + 1) = (\\frac{d}{d k} \\log{(k)} + 1)^{2}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["exp", 1], "Equality(exp(Function('A_2')(Symbol('k', commutative=True))), Symbol('k', commutative=True))"], [["add", 2, "Function('A_2')(Symbol('k', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('k', commutative=True)), exp(Function('A_2')(Symbol('k', commutative=True)))), Add(Symbol('k', commutative=True), Function('A_2')(Symbol('k', commutative=True))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Function('A_2')(Symbol('k', commutative=True)), exp(Function('A_2')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), Function('A_2')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(exp(Function('A_2')(Symbol('k', commutative=True))), Derivative(Function('A_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(Function('A_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Derivative(Function('A_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('k', commutative=True), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1)))"], [["times", 6, "Add(Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Mul(Symbol('k', commutative=True), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1))), Pow(Add(Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{E}{(M_{E})} = e^{M_{E}} and \\mathbf{E}{(l)} = e^{\\cos{(l)}}, then obtain (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} + e^{M_{E}} + e^{\\cos{(l)}} - e^{- M_{E}} = e^{M_{E}} + e^{\\cos{(l)}} + 1 - e^{- M_{E}}", "derivation": "\\mathbf{E}{(M_{E})} = e^{M_{E}} and \\mathbf{E}{(M_{E})} e^{- M_{E}} = 1 and \\mathbf{E}{(l)} = e^{\\cos{(l)}} and (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} = 1 and (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} + \\mathbf{E}{(l)} = \\mathbf{E}{(l)} + 1 and (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} + e^{\\cos{(l)}} = e^{\\cos{(l)}} + 1 and (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} + e^{\\cos{(l)}} - e^{- M_{E}} = e^{\\cos{(l)}} + 1 - e^{- M_{E}} and (\\mathbf{E}{(M_{E})} e^{- M_{E}})^{M_{E}} + e^{M_{E}} + e^{\\cos{(l)}} - e^{- M_{E}} = e^{M_{E}} + e^{\\cos{(l)}} + 1 - e^{- M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["divide", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Integer(1))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('l', commutative=True)), exp(cos(Symbol('l', commutative=True))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Integer(1))"], [["add", 4, "Function('\\\\mathbf{E}')(Symbol('l', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), Function('\\\\mathbf{E}')(Symbol('l', commutative=True))), Add(Function('\\\\mathbf{E}')(Symbol('l', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), exp(cos(Symbol('l', commutative=True)))), Add(exp(cos(Symbol('l', commutative=True))), Integer(1)))"], [["minus", 6, "exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))"], "Equality(Add(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), exp(cos(Symbol('l', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('M_E', commutative=True))))), Add(exp(cos(Symbol('l', commutative=True))), Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('M_E', commutative=True))))))"], [["add", 7, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\mathbf{E}')(Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), Symbol('M_E', commutative=True)))), Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)), exp(cos(Symbol('l', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('M_E', commutative=True))))), Add(exp(Symbol('M_E', commutative=True)), exp(cos(Symbol('l', commutative=True))), Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('M_E', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(f^{*})} = \\sin{(f^{*})}, then obtain \\int \\frac{\\int \\operatorname{C_{2}}{(f^{*})} df^{*}}{\\sin{(f^{*})}} df^{*} = \\int \\frac{\\int \\sin{(f^{*})} df^{*}}{\\sin{(f^{*})}} df^{*}", "derivation": "\\operatorname{C_{2}}{(f^{*})} = \\sin{(f^{*})} and \\int \\operatorname{C_{2}}{(f^{*})} df^{*} = \\int \\sin{(f^{*})} df^{*} and \\frac{\\int \\operatorname{C_{2}}{(f^{*})} df^{*}}{\\sin{(f^{*})}} = \\frac{\\int \\sin{(f^{*})} df^{*}}{\\sin{(f^{*})}} and \\int \\frac{\\int \\operatorname{C_{2}}{(f^{*})} df^{*}}{\\sin{(f^{*})}} df^{*} = \\int \\frac{\\int \\sin{(f^{*})} df^{*}}{\\sin{(f^{*})}} df^{*}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["divide", 2, "sin(Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('C_2')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["integrate", 3, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('C_2')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Pow(sin(Symbol('f^*', commutative=True)), Integer(-1)), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given h{(\\varphi,v_{1})} = \\int (- \\varphi + v_{1}) d\\varphi, then derive h{(\\varphi,v_{1})} = \\eta - \\frac{\\varphi^{2}}{2} + \\varphi v_{1}, then obtain \\iiint (\\eta - \\frac{\\varphi^{2}}{2} + \\varphi v_{1}) dv_{1} d\\varphi d\\varphi = \\iiiint (- \\varphi + v_{1}) d\\varphi dv_{1} d\\varphi d\\varphi", "derivation": "h{(\\varphi,v_{1})} = \\int (- \\varphi + v_{1}) d\\varphi and h{(\\varphi,v_{1})} = \\eta - \\frac{\\varphi^{2}}{2} + \\varphi v_{1} and \\int h{(\\varphi,v_{1})} dv_{1} = \\iint (- \\varphi + v_{1}) d\\varphi dv_{1} and \\iint h{(\\varphi,v_{1})} dv_{1} d\\varphi = \\iiint (- \\varphi + v_{1}) d\\varphi dv_{1} d\\varphi and \\iiint h{(\\varphi,v_{1})} dv_{1} d\\varphi d\\varphi = \\iiiint (- \\varphi + v_{1}) d\\varphi dv_{1} d\\varphi d\\varphi and \\iiint (\\eta - \\frac{\\varphi^{2}}{2} + \\varphi v_{1}) dv_{1} d\\varphi d\\varphi = \\iiiint (- \\varphi + v_{1}) d\\varphi dv_{1} d\\varphi d\\varphi", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('h')(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(h,i)} = \\frac{i}{h}, then obtain \\frac{\\partial^{2}}{\\partial i^{2}} (- i + \\Psi_{\\lambda}{(h,i)})^{2 i} = \\frac{\\partial^{2}}{\\partial i^{2}} (- i + \\frac{i}{h})^{i} (- i + \\Psi_{\\lambda}{(h,i)})^{i}", "derivation": "\\Psi_{\\lambda}{(h,i)} = \\frac{i}{h} and - i + \\Psi_{\\lambda}{(h,i)} = - i + \\frac{i}{h} and (- i + \\Psi_{\\lambda}{(h,i)})^{i} = (- i + \\frac{i}{h})^{i} and (- i + \\Psi_{\\lambda}{(h,i)})^{2 i} = (- i + \\frac{i}{h})^{i} (- i + \\Psi_{\\lambda}{(h,i)})^{i} and \\frac{\\partial}{\\partial i} (- i + \\Psi_{\\lambda}{(h,i)})^{2 i} = \\frac{\\partial}{\\partial i} (- i + \\frac{i}{h})^{i} (- i + \\Psi_{\\lambda}{(h,i)})^{i} and \\frac{\\partial^{2}}{\\partial i^{2}} (- i + \\Psi_{\\lambda}{(h,i)})^{2 i} = \\frac{\\partial^{2}}{\\partial i^{2}} (- i + \\frac{i}{h})^{i} (- i + \\Psi_{\\lambda}{(h,i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True)))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["times", 3, "Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('i', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True))))"], [["differentiate", 4, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('i', commutative=True))), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain \\mathbf{p} (\\Psi^{\\dagger}^{\\mathbf{p}}{(\\mathbf{p})})^{\\mathbf{p}} = \\mathbf{p} ((e^{\\mathbf{p}})^{\\mathbf{p}})^{\\mathbf{p}}", "derivation": "\\Psi^{\\dagger}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\Psi^{\\dagger}^{\\mathbf{p}}{(\\mathbf{p})} = (e^{\\mathbf{p}})^{\\mathbf{p}} and (\\Psi^{\\dagger}^{\\mathbf{p}}{(\\mathbf{p})})^{\\mathbf{p}} = ((e^{\\mathbf{p}})^{\\mathbf{p}})^{\\mathbf{p}} and \\mathbf{p} (\\Psi^{\\dagger}^{\\mathbf{p}}{(\\mathbf{p})})^{\\mathbf{p}} = \\mathbf{p} ((e^{\\mathbf{p}})^{\\mathbf{p}})^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\varphi{(f^{\\prime})} = \\sin{(f^{\\prime})}, then obtain f^{\\prime} + 1 = f^{\\prime} + \\frac{\\varphi{(f^{\\prime})} + \\sin{(f^{\\prime})}}{2 \\varphi{(f^{\\prime})}}", "derivation": "\\varphi{(f^{\\prime})} = \\sin{(f^{\\prime})} and 2 \\varphi{(f^{\\prime})} = \\varphi{(f^{\\prime})} + \\sin{(f^{\\prime})} and 1 = \\frac{\\varphi{(f^{\\prime})} + \\sin{(f^{\\prime})}}{2 \\varphi{(f^{\\prime})}} and f^{\\prime} + 1 = f^{\\prime} + \\frac{\\varphi{(f^{\\prime})} + \\sin{(f^{\\prime})}}{2 \\varphi{(f^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True))), Add(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))))"], [["add", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Add(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Pow(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given M{(v)} = \\sin{(\\sin{(v)})}, then obtain \\frac{d}{d v} (\\frac{d}{d v} M{(v)} - \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}}) = \\frac{d}{d v} (- \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}} + \\frac{d}{d v} \\sin{(\\sin{(v)})})", "derivation": "M{(v)} = \\sin{(\\sin{(v)})} and \\frac{d}{d v} M{(v)} = \\frac{d}{d v} \\sin{(\\sin{(v)})} and \\frac{d}{d v} M{(v)} - \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}} = - \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}} + \\frac{d}{d v} \\sin{(\\sin{(v)})} and \\frac{d}{d v} (\\frac{d}{d v} M{(v)} - \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}}) = \\frac{d}{d v} (- \\frac{\\frac{d}{d v} M{(v)}}{\\frac{d}{d v} \\sin{(\\sin{(v)})}} + \\frac{d}{d v} \\sin{(\\sin{(v)})})", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v', commutative=True)), sin(sin(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Integer(-1), Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Function('M')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))), Derivative(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\mathbf{F},\\tilde{g}^*)} = \\frac{\\tilde{g}^*}{\\mathbf{F}}, then derive \\int \\log{(\\mathbf{F} n{(\\mathbf{F},\\tilde{g}^*)})} d\\tilde{g}^* = E_{x} + \\tilde{g}^* \\log{(\\tilde{g}^*)} - \\tilde{g}^*, then obtain \\int \\log{(\\tilde{g}^*)} d\\tilde{g}^* = E_{x} + \\tilde{g}^* \\log{(\\tilde{g}^*)} - \\tilde{g}^*", "derivation": "n{(\\mathbf{F},\\tilde{g}^*)} = \\frac{\\tilde{g}^*}{\\mathbf{F}} and \\mathbf{F} n{(\\mathbf{F},\\tilde{g}^*)} = \\tilde{g}^* and \\log{(\\mathbf{F} n{(\\mathbf{F},\\tilde{g}^*)})} = \\log{(\\tilde{g}^*)} and \\int \\log{(\\mathbf{F} n{(\\mathbf{F},\\tilde{g}^*)})} d\\tilde{g}^* = \\int \\log{(\\tilde{g}^*)} d\\tilde{g}^* and \\int \\log{(\\mathbf{F} n{(\\mathbf{F},\\tilde{g}^*)})} d\\tilde{g}^* = E_{x} + \\tilde{g}^* \\log{(\\tilde{g}^*)} - \\tilde{g}^* and \\int \\log{(\\tilde{g}^*)} d\\tilde{g}^* = E_{x} + \\tilde{g}^* \\log{(\\tilde{g}^*)} - \\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))"], [["log", 2], "Equality(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(log(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(log(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('n')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\tilde{g}^*', commutative=True), log(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(log(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\tilde{g}^*', commutative=True), log(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(P_{e},\\mathbf{A})} = \\frac{P_{e}}{\\mathbf{A}}, then obtain s + \\mathbf{S}{(P_{e},\\mathbf{A})} = E + \\frac{P_{e}}{\\mathbf{A}}", "derivation": "\\mathbf{S}{(P_{e},\\mathbf{A})} = \\frac{P_{e}}{\\mathbf{A}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{S}{(P_{e},\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{P_{e}}{\\mathbf{A}} and \\int \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{S}{(P_{e},\\mathbf{A})} d\\mathbf{A} = \\int \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{P_{e}}{\\mathbf{A}} d\\mathbf{A} and s + \\mathbf{S}{(P_{e},\\mathbf{A})} = E + \\frac{P_{e}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{S}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('s', commutative=True), Function('\\\\mathbf{S}')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given r{(h,\\tilde{g}^*)} = \\tilde{g}^* \\sin{(h)}, then obtain r^{2}{(h,\\tilde{g}^*)} + r{(h,\\tilde{g}^*)} + \\sin{(h)} = \\tilde{g}^* \\sin{(h)} + r^{2}{(h,\\tilde{g}^*)} + \\sin{(h)}", "derivation": "r{(h,\\tilde{g}^*)} = \\tilde{g}^* \\sin{(h)} and r^{2}{(h,\\tilde{g}^*)} = \\tilde{g}^* r{(h,\\tilde{g}^*)} \\sin{(h)} and r{(h,\\tilde{g}^*)} + \\sin{(h)} = \\tilde{g}^* \\sin{(h)} + \\sin{(h)} and \\tilde{g}^* r{(h,\\tilde{g}^*)} \\sin{(h)} + r{(h,\\tilde{g}^*)} + \\sin{(h)} = \\tilde{g}^* r{(h,\\tilde{g}^*)} \\sin{(h)} + \\tilde{g}^* \\sin{(h)} + \\sin{(h)} and r^{2}{(h,\\tilde{g}^*)} + r{(h,\\tilde{g}^*)} + \\sin{(h)} = \\tilde{g}^* \\sin{(h)} + r^{2}{(h,\\tilde{g}^*)} + \\sin{(h)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('h', commutative=True))))"], [["times", 1, "Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Pow(Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))))"], [["add", 1, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('h', commutative=True))), Pow(Function('r')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), sin(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\Psi{(c)} = \\cos{(\\sin{(c)})}, then obtain e^{- \\frac{\\Psi^{2}{(c)} \\cos^{- \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}}}{(\\sin{(c)})}}{\\cos{(\\sin{(c)})}}} = e^{- \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}}}", "derivation": "\\Psi{(c)} = \\cos{(\\sin{(c)})} and \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}} = 1 and - \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}} = -1 and e^{- \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}}} = e^{-1} and e^{- \\frac{\\Psi^{2}{(c)} \\cos^{- \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}}}{(\\sin{(c)})}}{\\cos{(\\sin{(c)})}}} = e^{- \\frac{\\Psi{(c)}}{\\cos{(\\sin{(c)})}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('c', commutative=True)), cos(sin(Symbol('c', commutative=True))))"], [["divide", 1, "cos(sin(Symbol('c', commutative=True)))"], "Equality(Mul(Function('\\\\Psi')(Symbol('c', commutative=True)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\Psi')(Symbol('c', commutative=True)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1))), Integer(-1))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), Function('\\\\Psi')(Symbol('c', commutative=True)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1)))), exp(Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(exp(Mul(Integer(-1), Pow(Function('\\\\Psi')(Symbol('c', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1)), Pow(cos(sin(Symbol('c', commutative=True))), Mul(Integer(-1), Function('\\\\Psi')(Symbol('c', commutative=True)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1)))))), exp(Mul(Integer(-1), Function('\\\\Psi')(Symbol('c', commutative=True)), Pow(cos(sin(Symbol('c', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(C_{2},\\mathbf{D})} = \\cos{(C_{2} + \\mathbf{D})}, then obtain \\sin{(C_{2} + \\int (\\mathbf{D} + \\operatorname{A_{2}}{(C_{2},\\mathbf{D})}) dC_{2})} = \\sin{(C_{2} + \\int (\\mathbf{D} + \\cos{(C_{2} + \\mathbf{D})}) dC_{2})}", "derivation": "\\operatorname{A_{2}}{(C_{2},\\mathbf{D})} = \\cos{(C_{2} + \\mathbf{D})} and \\mathbf{D} + \\operatorname{A_{2}}{(C_{2},\\mathbf{D})} = \\mathbf{D} + \\cos{(C_{2} + \\mathbf{D})} and \\int (\\mathbf{D} + \\operatorname{A_{2}}{(C_{2},\\mathbf{D})}) dC_{2} = \\int (\\mathbf{D} + \\cos{(C_{2} + \\mathbf{D})}) dC_{2} and C_{2} + \\int (\\mathbf{D} + \\operatorname{A_{2}}{(C_{2},\\mathbf{D})}) dC_{2} = C_{2} + \\int (\\mathbf{D} + \\cos{(C_{2} + \\mathbf{D})}) dC_{2} and \\sin{(C_{2} + \\int (\\mathbf{D} + \\operatorname{A_{2}}{(C_{2},\\mathbf{D})}) dC_{2})} = \\sin{(C_{2} + \\int (\\mathbf{D} + \\cos{(C_{2} + \\mathbf{D})}) dC_{2})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), cos(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('A_2')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('A_2')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))"], [["add", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('A_2')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('C_2', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Symbol('C_2', commutative=True), Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('A_2')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('C_2', commutative=True))))), sin(Add(Symbol('C_2', commutative=True), Integral(Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given T{(B)} = \\log{(e^{B})}, then obtain - \\frac{T{(B)}}{\\log{(e^{B})}} - \\frac{T{(B)}}{B} = - \\frac{T{(B)}}{\\log{(e^{B})}} - \\frac{\\log{(e^{B})}}{B}", "derivation": "T{(B)} = \\log{(e^{B})} and \\frac{T{(B)}}{B} = \\frac{\\log{(e^{B})}}{B} and \\frac{T{(B)}}{\\log{(e^{B})}} + \\frac{T{(B)}}{B} = \\frac{T{(B)}}{\\log{(e^{B})}} + \\frac{\\log{(e^{B})}}{B} and - \\frac{(\\frac{T{(B)}}{\\log{(e^{B})}} + \\frac{T{(B)}}{B}) \\log{(e^{B})}}{T{(B)}} = - \\frac{(\\frac{T{(B)}}{\\log{(e^{B})}} + \\frac{\\log{(e^{B})}}{B}) \\log{(e^{B})}}{T{(B)}} and - \\frac{T{(B)}}{\\log{(e^{B})}} - \\frac{T{(B)}}{B} = - \\frac{T{(B)}}{\\log{(e^{B})}} - \\frac{\\log{(e^{B})}}{B}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('B', commutative=True)), log(exp(Symbol('B', commutative=True))))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('T')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), log(exp(Symbol('B', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('T')(Symbol('B', commutative=True)))), Add(Mul(Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), log(exp(Symbol('B', commutative=True))))))"], [["divide", 3, "Mul(Integer(-1), Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(-1), Add(Mul(Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('T')(Symbol('B', commutative=True)))), Pow(Function('T')(Symbol('B', commutative=True)), Integer(-1)), log(exp(Symbol('B', commutative=True)))), Mul(Integer(-1), Add(Mul(Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), log(exp(Symbol('B', commutative=True))))), Pow(Function('T')(Symbol('B', commutative=True)), Integer(-1)), log(exp(Symbol('B', commutative=True)))))"], [["times", 4, "Mul(Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('T')(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Function('T')(Symbol('B', commutative=True)), Pow(log(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), log(exp(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}_0{(\\psi,\\varepsilon)} = - \\psi + \\log{(\\varepsilon)}, then obtain (\\int \\hat{x}_0^{2}{(\\psi,\\varepsilon)} d\\psi)^{2} = (\\int (- \\psi + \\log{(\\varepsilon)}) \\hat{x}_0{(\\psi,\\varepsilon)} d\\psi)^{2}", "derivation": "\\hat{x}_0{(\\psi,\\varepsilon)} = - \\psi + \\log{(\\varepsilon)} and \\hat{x}_0^{2}{(\\psi,\\varepsilon)} = (- \\psi + \\log{(\\varepsilon)}) \\hat{x}_0{(\\psi,\\varepsilon)} and \\int \\hat{x}_0^{2}{(\\psi,\\varepsilon)} d\\psi = \\int (- \\psi + \\log{(\\varepsilon)}) \\hat{x}_0{(\\psi,\\varepsilon)} d\\psi and (\\int \\hat{x}_0^{2}{(\\psi,\\varepsilon)} d\\psi)^{2} = (\\int (- \\psi + \\log{(\\varepsilon)}) \\hat{x}_0{(\\psi,\\varepsilon)} d\\psi)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)), Pow(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)))"]]}, {"prompt": "Given k{(Z)} = \\cos{(Z)} and \\operatorname{t_{1}}{(Z)} = \\cos{(Z)}, then obtain - \\operatorname{t_{1}}^{2}{(Z)} = - k{(Z)} \\operatorname{t_{1}}{(Z)}", "derivation": "k{(Z)} = \\cos{(Z)} and k^{2}{(Z)} = k{(Z)} \\cos{(Z)} and \\operatorname{t_{1}}{(Z)} = \\cos{(Z)} and k{(Z)} = \\operatorname{t_{1}}{(Z)} and \\operatorname{t_{1}}^{2}{(Z)} = \\operatorname{t_{1}}{(Z)} \\cos{(Z)} and \\operatorname{t_{1}}^{2}{(Z)} = k{(Z)} \\operatorname{t_{1}}{(Z)} and - \\operatorname{t_{1}}^{2}{(Z)} = - k{(Z)} \\operatorname{t_{1}}{(Z)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["times", 1, "Function('k')(Symbol('Z', commutative=True))"], "Equality(Pow(Function('k')(Symbol('Z', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('k')(Symbol('Z', commutative=True)), Function('t_1')(Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('t_1')(Symbol('Z', commutative=True)), Integer(2)), Mul(Function('t_1')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('t_1')(Symbol('Z', commutative=True)), Integer(2)), Mul(Function('k')(Symbol('Z', commutative=True)), Function('t_1')(Symbol('Z', commutative=True))))"], [["divide", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('t_1')(Symbol('Z', commutative=True)), Integer(2))), Mul(Integer(-1), Function('k')(Symbol('Z', commutative=True)), Function('t_1')(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given k{(I,x,F_{x})} = F_{x} + I + x, then obtain I + \\frac{d}{d I} 1 = I + \\frac{\\partial}{\\partial I} \\frac{F_{x} + I + x}{k{(I,x,F_{x})}}", "derivation": "k{(I,x,F_{x})} = F_{x} + I + x and 1 = \\frac{F_{x} + I + x}{k{(I,x,F_{x})}} and \\frac{d}{d I} 1 = \\frac{\\partial}{\\partial I} \\frac{F_{x} + I + x}{k{(I,x,F_{x})}} and I + \\frac{d}{d I} 1 = I + \\frac{\\partial}{\\partial I} \\frac{F_{x} + I + x}{k{(I,x,F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Symbol('I', commutative=True), Symbol('x', commutative=True)))"], [["divide", 1, "Function('k')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('F_x', commutative=True), Symbol('I', commutative=True), Symbol('x', commutative=True)), Pow(Function('k')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('F_x', commutative=True), Symbol('I', commutative=True), Symbol('x', commutative=True)), Pow(Function('k')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 3, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Derivative(Integer(1), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Symbol('I', commutative=True), Derivative(Mul(Add(Symbol('F_x', commutative=True), Symbol('I', commutative=True), Symbol('x', commutative=True)), Pow(Function('k')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{f},\\eta^{\\prime})} = e^{\\eta^{\\prime} \\mathbf{f}} and \\mu{(\\mathbf{f},\\eta^{\\prime})} = e^{\\eta^{\\prime} \\mathbf{f}}, then obtain \\operatorname{f_{\\mathbf{p}}}^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})} = \\mu^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{f},\\eta^{\\prime})} = e^{\\eta^{\\prime} \\mathbf{f}} and \\mu{(\\mathbf{f},\\eta^{\\prime})} = e^{\\eta^{\\prime} \\mathbf{f}} and \\operatorname{f_{\\mathbf{p}}}^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})} = (e^{\\eta^{\\prime} \\mathbf{f}})^{\\mathbf{f}} and \\mu^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})} = (e^{\\eta^{\\prime} \\mathbf{f}})^{\\mathbf{f}} and \\operatorname{f_{\\mathbf{p}}}^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})} = \\mu^{\\mathbf{f}}{(\\mathbf{f},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given x{(\\hat{p}_0,u)} = \\int (- \\hat{p}_0 + u) du, then derive \\frac{x{(\\hat{p}_0,u)}}{\\mathbf{s}{(\\hat{H}_{\\lambda})}} = \\frac{- \\hat{p}_0 u + \\mathbf{B} + \\frac{u^{2}}{2}}{\\mathbf{s}{(\\hat{H}_{\\lambda})}}, then obtain \\frac{x{(\\hat{p}_0,u)}}{\\int (- \\hat{p}_0 + u) du} = \\frac{- \\hat{p}_0 u + \\mathbf{B} + \\frac{u^{2}}{2}}{\\int (- \\hat{p}_0 + u) du}", "derivation": "x{(\\hat{p}_0,u)} = \\int (- \\hat{p}_0 + u) du and \\frac{x{(\\hat{p}_0,u)}}{\\mathbf{s}{(\\hat{H}_{\\lambda})}} = \\frac{\\int (- \\hat{p}_0 + u) du}{\\mathbf{s}{(\\hat{H}_{\\lambda})}} and \\frac{x{(\\hat{p}_0,u)}}{\\mathbf{s}{(\\hat{H}_{\\lambda})}} = \\frac{- \\hat{p}_0 u + \\mathbf{B} + \\frac{u^{2}}{2}}{\\mathbf{s}{(\\hat{H}_{\\lambda})}} and \\frac{x{(\\hat{p}_0,u)}}{\\int (- \\hat{p}_0 + u) du} = \\frac{- \\hat{p}_0 u + \\mathbf{B} + \\frac{u^{2}}{2}}{\\int (- \\hat{p}_0 + u) du}", "srepr_derivation": [["get_premise", "Equality(Function('x')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Function('x')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Function('x')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], "Equality(Mul(Function('x')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}{(C_{2},\\varphi)} = \\frac{\\partial}{\\partial C_{2}} \\varphi^{C_{2}}, then derive \\tilde{g}{(C_{2},\\varphi)} = \\varphi^{C_{2}} \\log{(\\varphi)}, then obtain 1 = \\frac{\\int \\varphi^{C_{2}} \\log{(\\varphi)} dC_{2}}{\\int \\tilde{g}{(C_{2},\\varphi)} dC_{2}}", "derivation": "\\tilde{g}{(C_{2},\\varphi)} = \\frac{\\partial}{\\partial C_{2}} \\varphi^{C_{2}} and \\int \\tilde{g}{(C_{2},\\varphi)} dC_{2} = \\int \\frac{\\partial}{\\partial C_{2}} \\varphi^{C_{2}} dC_{2} and \\tilde{g}{(C_{2},\\varphi)} = \\varphi^{C_{2}} \\log{(\\varphi)} and \\int \\varphi^{C_{2}} \\log{(\\varphi)} dC_{2} = \\int \\frac{\\partial}{\\partial C_{2}} \\varphi^{C_{2}} dC_{2} and \\int \\tilde{g}{(C_{2},\\varphi)} dC_{2} = \\int \\varphi^{C_{2}} \\log{(\\varphi)} dC_{2} and 1 = \\frac{\\int \\varphi^{C_{2}} \\log{(\\varphi)} dC_{2}}{\\int \\tilde{g}{(C_{2},\\varphi)} dC_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), log(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["divide", 5, "Integral(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_2', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('C_2', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Pow(Integral(Function('\\\\tilde{g}')(Symbol('C_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given s{(\\dot{y})} = \\sin{(\\dot{y})}, then obtain \\int 0^{\\dot{y}} d\\dot{y} = \\int (- s{(\\dot{y})} \\sin{(\\dot{y})} + \\sin^{2}{(\\dot{y})})^{\\dot{y}} d\\dot{y}", "derivation": "s{(\\dot{y})} = \\sin{(\\dot{y})} and s{(\\dot{y})} \\sin{(\\dot{y})} = \\sin^{2}{(\\dot{y})} and 0 = - s{(\\dot{y})} \\sin{(\\dot{y})} + \\sin^{2}{(\\dot{y})} and 0^{\\dot{y}} = (- s{(\\dot{y})} \\sin{(\\dot{y})} + \\sin^{2}{(\\dot{y})})^{\\dot{y}} and \\int 0^{\\dot{y}} d\\dot{y} = \\int (- s{(\\dot{y})} \\sin{(\\dot{y})} + \\sin^{2}{(\\dot{y})})^{\\dot{y}} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Integer(2)))"], [["minus", 2, "Mul(Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Integer(2))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Integer(2))), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('s')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Integer(2))), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(z^{*},\\mathbf{p})} = \\log{(\\mathbf{p} + z^{*})} and \\Psi{(z^{*},\\mathbf{p})} = \\log{(\\mathbf{p} + z^{*})}, then obtain (\\mathbf{p} + z^{*}) \\log{(\\mathbf{p} + z^{*})} = (\\mathbf{p} + z^{*}) \\Psi{(z^{*},\\mathbf{p})}", "derivation": "\\operatorname{F_{N}}{(z^{*},\\mathbf{p})} = \\log{(\\mathbf{p} + z^{*})} and (\\mathbf{p} + z^{*}) \\operatorname{F_{N}}{(z^{*},\\mathbf{p})} = (\\mathbf{p} + z^{*}) \\log{(\\mathbf{p} + z^{*})} and \\Psi{(z^{*},\\mathbf{p})} = \\log{(\\mathbf{p} + z^{*})} and (\\mathbf{p} + z^{*}) \\operatorname{F_{N}}{(z^{*},\\mathbf{p})} = (\\mathbf{p} + z^{*}) \\Psi{(z^{*},\\mathbf{p})} and (\\mathbf{p} + z^{*}) \\log{(\\mathbf{p} + z^{*})} = (\\mathbf{p} + z^{*}) \\Psi{(z^{*},\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), Function('F_N')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), log(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), Function('F_N')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\Psi')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), log(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\Psi')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(r)} = \\frac{d}{d r} e^{r}, then derive r \\phi_{2}{(r)} = r e^{r}, then obtain \\frac{\\int r \\phi_{2}{(r)} dr}{r \\frac{d}{d r} e^{r}} = \\frac{\\int r e^{r} dr}{r \\frac{d}{d r} e^{r}}", "derivation": "\\phi_{2}{(r)} = \\frac{d}{d r} e^{r} and r \\phi_{2}{(r)} = r \\frac{d}{d r} e^{r} and r \\phi_{2}{(r)} = r e^{r} and \\int r \\phi_{2}{(r)} dr = \\int r e^{r} dr and \\frac{\\int r \\phi_{2}{(r)} dr}{r \\frac{d}{d r} e^{r}} = \\frac{\\int r e^{r} dr}{r \\frac{d}{d r} e^{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('r', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('\\\\phi_2')(Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('r', commutative=True), Function('\\\\phi_2')(Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), exp(Symbol('r', commutative=True))))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Mul(Symbol('r', commutative=True), Function('\\\\phi_2')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(Mul(Symbol('r', commutative=True), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["divide", 4, "Mul(Symbol('r', commutative=True), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Symbol('r', commutative=True), Function('\\\\phi_2')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Symbol('r', commutative=True), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given r{(\\psi,\\phi)} = \\psi \\cos{(\\phi)}, then obtain - \\psi + \\frac{\\partial}{\\partial \\phi} \\int r{(\\psi,\\phi)} d\\phi + 1 = - \\psi + \\frac{\\partial}{\\partial \\phi} \\int \\psi \\cos{(\\phi)} d\\phi + 1", "derivation": "r{(\\psi,\\phi)} = \\psi \\cos{(\\phi)} and \\int r{(\\psi,\\phi)} d\\phi = \\int \\psi \\cos{(\\phi)} d\\phi and \\frac{\\partial}{\\partial \\phi} \\int r{(\\psi,\\phi)} d\\phi = \\frac{\\partial}{\\partial \\phi} \\int \\psi \\cos{(\\phi)} d\\phi and \\frac{\\partial}{\\partial \\phi} \\int r{(\\psi,\\phi)} d\\phi + 1 = \\frac{\\partial}{\\partial \\phi} \\int \\psi \\cos{(\\phi)} d\\phi + 1 and - \\psi + \\frac{\\partial}{\\partial \\phi} \\int r{(\\psi,\\phi)} d\\phi + 1 = - \\psi + \\frac{\\partial}{\\partial \\phi} \\int \\psi \\cos{(\\phi)} d\\phi + 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1)))"], [["minus", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Derivative(Integral(Function('r')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Derivative(Integral(Mul(Symbol('\\\\psi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\rho{(C_{d})} = \\frac{d}{d C_{d}} \\cos{(C_{d})}, then derive \\rho{(C_{d})} = - \\sin{(C_{d})}, then derive E + \\cos{(C_{d})} = \\int - \\sin{(C_{d})} dC_{d}, then obtain (\\hat{H}_{\\lambda} (E + \\cos{(C_{d})}))^{\\hat{H}_{\\lambda}} = (\\hat{H}_{\\lambda} \\int - \\sin{(C_{d})} dC_{d})^{\\hat{H}_{\\lambda}}", "derivation": "\\rho{(C_{d})} = \\frac{d}{d C_{d}} \\cos{(C_{d})} and \\rho{(C_{d})} = - \\sin{(C_{d})} and \\frac{d}{d C_{d}} \\cos{(C_{d})} = - \\sin{(C_{d})} and \\int \\frac{d}{d C_{d}} \\cos{(C_{d})} dC_{d} = \\int - \\sin{(C_{d})} dC_{d} and E + \\cos{(C_{d})} = \\int - \\sin{(C_{d})} dC_{d} and \\hat{H}_{\\lambda} (E + \\cos{(C_{d})}) = \\hat{H}_{\\lambda} \\int - \\sin{(C_{d})} dC_{d} and (\\hat{H}_{\\lambda} (E + \\cos{(C_{d})}))^{\\hat{H}_{\\lambda}} = (\\hat{H}_{\\lambda} \\int - \\sin{(C_{d})} dC_{d})^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('C_d', commutative=True)), Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\rho')(Symbol('C_d', commutative=True)), Mul(Integer(-1), sin(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_d', commutative=True))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Derivative(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('E', commutative=True), cos(Symbol('C_d', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["times", 5, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Symbol('E', commutative=True), cos(Symbol('C_d', commutative=True)))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))))"], [["power", 6, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Symbol('E', commutative=True), cos(Symbol('C_d', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(S,\\dot{x})} = \\log{(\\dot{x}^{S})} and \\operatorname{t_{1}}{(S,\\dot{x})} = \\operatorname{v_{z}}^{2}{(S,\\dot{x})}, then obtain - (\\operatorname{v_{z}}{(S,\\dot{x})} \\log{(\\dot{x}^{S})})^{S} + \\operatorname{t_{1}}^{S}{(S,\\dot{x})} = 0", "derivation": "\\operatorname{v_{z}}{(S,\\dot{x})} = \\log{(\\dot{x}^{S})} and \\operatorname{v_{z}}^{2}{(S,\\dot{x})} = \\operatorname{v_{z}}{(S,\\dot{x})} \\log{(\\dot{x}^{S})} and (\\operatorname{v_{z}}^{2}{(S,\\dot{x})})^{S} = (\\operatorname{v_{z}}{(S,\\dot{x})} \\log{(\\dot{x}^{S})})^{S} and - (\\operatorname{v_{z}}{(S,\\dot{x})} \\log{(\\dot{x}^{S})})^{S} + (\\operatorname{v_{z}}^{2}{(S,\\dot{x})})^{S} = 0 and \\operatorname{t_{1}}{(S,\\dot{x})} = \\operatorname{v_{z}}^{2}{(S,\\dot{x})} and - (\\operatorname{v_{z}}{(S,\\dot{x})} \\log{(\\dot{x}^{S})})^{S} + \\operatorname{t_{1}}^{S}{(S,\\dot{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True))))"], [["times", 1, "Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Pow(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Mul(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Symbol('S', commutative=True)), Pow(Mul(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"], [["minus", 3, "Pow(Mul(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)))), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)))), Symbol('S', commutative=True))), Pow(Pow(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)), Symbol('S', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Pow(Mul(Function('v_z')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('S', commutative=True)))), Symbol('S', commutative=True))), Pow(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('S', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(U,\\nabla,v_{y})} = U + \\nabla - v_{y}, then obtain \\frac{\\partial^{2}}{\\partial v_{y}^{2}} \\int \\operatorname{f^{*}}{(U,\\nabla,v_{y})} dv_{y} = \\frac{\\partial^{2}}{\\partial v_{y}^{2}} \\int (U + \\nabla - v_{y}) dv_{y}", "derivation": "\\operatorname{f^{*}}{(U,\\nabla,v_{y})} = U + \\nabla - v_{y} and \\int \\operatorname{f^{*}}{(U,\\nabla,v_{y})} dv_{y} = \\int (U + \\nabla - v_{y}) dv_{y} and \\frac{\\partial}{\\partial v_{y}} \\int \\operatorname{f^{*}}{(U,\\nabla,v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} \\int (U + \\nabla - v_{y}) dv_{y} and \\frac{\\partial^{2}}{\\partial v_{y}^{2}} \\int \\operatorname{f^{*}}{(U,\\nabla,v_{y})} dv_{y} = \\frac{\\partial^{2}}{\\partial v_{y}^{2}} \\int (U + \\nabla - v_{y}) dv_{y}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integral(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integral(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(2))), Derivative(Integral(Add(Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\rho{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})}, then obtain \\int - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\rho{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon} = \\int - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon}", "derivation": "\\rho{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\int \\rho{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\rho{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\int - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\rho{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon} = \\int - \\log{(g^{\\prime}_{\\varepsilon})} \\int \\log{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\rho')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\rho')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(-1), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}}, then derive \\frac{d}{d \\mathbf{P}} \\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}}, then derive \\frac{d^{2}}{d \\mathbf{P}^{2}} \\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}}, then obtain \\frac{d}{d \\mathbf{P}} \\mathbf{F}{(\\mathbf{P})} = \\frac{d^{2}}{d \\mathbf{P}^{2}} \\mathbf{F}{(\\mathbf{P})}", "derivation": "\\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\mathbf{F}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{d^{2}}{d \\mathbf{P}^{2}} \\mathbf{F}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\mathbf{P}} and \\frac{d^{2}}{d \\mathbf{P}^{2}} \\mathbf{F}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\frac{d}{d \\mathbf{P}} \\mathbf{F}{(\\mathbf{P})} = \\frac{d^{2}}{d \\mathbf{P}^{2}} \\mathbf{F}{(\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi{(\\psi^*,A_{1})} = - A_{1} + \\cos{(\\psi^*)}, then derive - A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} \\phi{(\\psi^*,A_{1})} + 2 = - A_{1} - \\cos{(\\psi^*)} + 1, then obtain \\log{(- A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} \\phi{(\\psi^*,A_{1})} + 2)} = \\log{(- A_{1} - \\cos{(\\psi^*)} + 1)}", "derivation": "\\phi{(\\psi^*,A_{1})} = - A_{1} + \\cos{(\\psi^*)} and 2 A_{1} + \\phi{(\\psi^*,A_{1})} = A_{1} + \\cos{(\\psi^*)} and \\frac{\\partial}{\\partial A_{1}} (2 A_{1} + \\phi{(\\psi^*,A_{1})}) = \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\cos{(\\psi^*)}) and - A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} (2 A_{1} + \\phi{(\\psi^*,A_{1})}) = - A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\cos{(\\psi^*)}) and - A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} \\phi{(\\psi^*,A_{1})} + 2 = - A_{1} - \\cos{(\\psi^*)} + 1 and \\log{(- A_{1} - \\cos{(\\psi^*)} + \\frac{\\partial}{\\partial A_{1}} \\phi{(\\psi^*,A_{1})} + 2)} = \\log{(- A_{1} - \\cos{(\\psi^*)} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Derivative(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Derivative(Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(2)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Integer(1)))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Derivative(Function('\\\\phi')(Symbol('\\\\psi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(2))), log(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\psi^*', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\Omega{(a)} = \\cos{(a)}, then derive \\int \\Omega{(a)} da = F_{x} + \\sin{(a)}, then obtain F_{x} + \\sin{(a)} = \\int \\cos{(a)} da", "derivation": "\\Omega{(a)} = \\cos{(a)} and \\int \\Omega{(a)} da = \\int \\cos{(a)} da and \\int \\Omega{(a)} da = F_{x} + \\sin{(a)} and F_{x} + \\sin{(a)} = \\int \\cos{(a)} da", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('F_x', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_x', commutative=True), sin(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(A_{y})} = \\cos{(A_{y})}, then derive \\int \\hat{x}_0{(A_{y})} dA_{y} = \\theta_2 + \\sin{(A_{y})}, then obtain - A_{y} + \\int \\cos{(A_{y})} dA_{y} = - A_{y} + \\int \\hat{x}_0{(A_{y})} dA_{y}", "derivation": "\\hat{x}_0{(A_{y})} = \\cos{(A_{y})} and \\int \\hat{x}_0{(A_{y})} dA_{y} = \\int \\cos{(A_{y})} dA_{y} and \\int \\hat{x}_0{(A_{y})} dA_{y} = \\theta_2 + \\sin{(A_{y})} and \\int \\cos{(A_{y})} dA_{y} = \\theta_2 + \\sin{(A_{y})} and - A_{y} + \\int \\cos{(A_{y})} dA_{y} = - A_{y} + \\theta_2 + \\sin{(A_{y})} and - A_{y} + \\int \\cos{(A_{y})} dA_{y} = - A_{y} + \\int \\hat{x}_0{(A_{y})} dA_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('A_y', commutative=True))))"], [["minus", 4, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\theta_2', commutative=True), sin(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Integral(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Integral(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given W{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain 0 = - 2 W{(\\sigma_p)} + 2 \\log{(\\sigma_p)}", "derivation": "W{(\\sigma_p)} = \\log{(\\sigma_p)} and W{(\\sigma_p)} - \\log{(\\sigma_p)} = 0 and W{(\\sigma_p)} - 2 \\log{(\\sigma_p)} = - \\log{(\\sigma_p)} and 0 = - W{(\\sigma_p)} + \\log{(\\sigma_p)} and - W{(\\sigma_p)} + \\log{(\\sigma_p)} = - 2 W{(\\sigma_p)} + 2 \\log{(\\sigma_p)} and 0 = - 2 W{(\\sigma_p)} + 2 \\log{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('W')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), Integer(0))"], [["minus", 2, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('W')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 3, "Add(Function('W')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\sigma_p', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('W')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Function('W')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('W')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('W')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('W')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given t{(\\dot{y})} = \\cos{(\\dot{y})}, then obtain (\\int (- \\dot{y} + t{(\\dot{y})}) d\\dot{y})^{2} = (\\int (- \\dot{y} + t{(\\dot{y})}) d\\dot{y}) \\int (- \\dot{y} + \\cos{(\\dot{y})}) d\\dot{y}", "derivation": "t{(\\dot{y})} = \\cos{(\\dot{y})} and - \\dot{y} + t{(\\dot{y})} = - \\dot{y} + \\cos{(\\dot{y})} and \\int (- \\dot{y} + t{(\\dot{y})}) d\\dot{y} = \\int (- \\dot{y} + \\cos{(\\dot{y})}) d\\dot{y} and (\\int (- \\dot{y} + t{(\\dot{y})}) d\\dot{y})^{2} = (\\int (- \\dot{y} + t{(\\dot{y})}) d\\dot{y}) \\int (- \\dot{y} + \\cos{(\\dot{y})}) d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('t')(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('t')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["times", 3, "Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('t')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('t')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integer(2)), Mul(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('t')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\rho)} = \\cos{(\\rho)}, then derive \\int \\operatorname{E_{n}}{(\\rho)} d\\rho = F_{H} + \\sin{(\\rho)}, then obtain \\int \\cos{(\\rho)} d\\rho = F_{H} + \\sin{(\\rho)}", "derivation": "\\operatorname{E_{n}}{(\\rho)} = \\cos{(\\rho)} and \\int \\operatorname{E_{n}}{(\\rho)} d\\rho = \\int \\cos{(\\rho)} d\\rho and \\int \\operatorname{E_{n}}{(\\rho)} d\\rho = F_{H} + \\sin{(\\rho)} and \\int \\cos{(\\rho)} d\\rho = F_{H} + \\sin{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('F_H', commutative=True), sin(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('F_H', commutative=True), sin(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(A_{2},v)} = A_{2} - v and \\mathbf{E}{(A_{2})} = A_{2}, then obtain \\iint A_{2} \\operatorname{F_{c}}{(A_{2},v)} dv dA_{2} = \\iint A_{2} (A_{2} - v) dv dA_{2}", "derivation": "\\operatorname{F_{c}}{(A_{2},v)} = A_{2} - v and \\mathbf{E}{(A_{2})} = A_{2} and \\operatorname{F_{c}}{(A_{2},v)} \\mathbf{E}{(A_{2})} = (A_{2} - v) \\mathbf{E}{(A_{2})} and \\int \\operatorname{F_{c}}{(A_{2},v)} \\mathbf{E}{(A_{2})} dv = \\int (A_{2} - v) \\mathbf{E}{(A_{2})} dv and \\iint \\operatorname{F_{c}}{(A_{2},v)} \\mathbf{E}{(A_{2})} dv dA_{2} = \\iint (A_{2} - v) \\mathbf{E}{(A_{2})} dv dA_{2} and \\iint \\operatorname{F_{c}}{(A_{2},v)} \\mathbf{E}{(A_{2})} dv d\\mathbf{E}{(A_{2})} = \\iint (A_{2} - v) \\mathbf{E}{(A_{2})} dv d\\mathbf{E}{(A_{2})} and \\iint A_{2} \\operatorname{F_{c}}{(A_{2},v)} dv dA_{2} = \\iint A_{2} (A_{2} - v) dv dA_{2}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], [["times", 1, "Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Mul(Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Mul(Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True)), Tuple(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True)))), Integral(Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True))), Tuple(Symbol('v', commutative=True)), Tuple(Function('\\\\mathbf{E}')(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Integral(Mul(Symbol('A_2', commutative=True), Function('F_c')(Symbol('A_2', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Mul(Symbol('A_2', commutative=True), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbf{S})} = \\sin{(\\sin{(\\mathbf{S})})}, then obtain \\int \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} p{(\\mathbf{S})} d\\mathbf{S} = \\int \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} \\sin{(\\sin{(\\mathbf{S})})} d\\mathbf{S}", "derivation": "p{(\\mathbf{S})} = \\sin{(\\sin{(\\mathbf{S})})} and \\frac{d}{d \\mathbf{S}} p{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\sin{(\\sin{(\\mathbf{S})})} and \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} p{(\\mathbf{S})} = \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} \\sin{(\\sin{(\\mathbf{S})})} and \\int \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} p{(\\mathbf{S})} d\\mathbf{S} = \\int \\sin{(\\sin{(\\mathbf{S})})} \\frac{d}{d \\mathbf{S}} \\sin{(\\sin{(\\mathbf{S})})} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["divide", 2, "Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))"], "Equality(Mul(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Derivative(Function('p')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Derivative(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Derivative(Function('p')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Derivative(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\hat{x}_0,\\phi)} = \\log{(\\hat{x}_0)}^{\\phi}, then obtain \\frac{\\partial}{\\partial \\hat{x}_0} - (\\frac{\\eta^{\\prime}{(\\hat{x}_0,\\phi)}}{\\phi})^{\\hat{x}_0} = \\frac{\\partial}{\\partial \\hat{x}_0} - (\\frac{\\log{(\\hat{x}_0)}^{\\phi}}{\\phi})^{\\hat{x}_0}", "derivation": "\\eta^{\\prime}{(\\hat{x}_0,\\phi)} = \\log{(\\hat{x}_0)}^{\\phi} and \\frac{\\eta^{\\prime}{(\\hat{x}_0,\\phi)}}{\\phi} = \\frac{\\log{(\\hat{x}_0)}^{\\phi}}{\\phi} and (\\frac{\\eta^{\\prime}{(\\hat{x}_0,\\phi)}}{\\phi})^{\\hat{x}_0} = (\\frac{\\log{(\\hat{x}_0)}^{\\phi}}{\\phi})^{\\hat{x}_0} and - (\\frac{\\eta^{\\prime}{(\\hat{x}_0,\\phi)}}{\\phi})^{\\hat{x}_0} = - (\\frac{\\log{(\\hat{x}_0)}^{\\phi}}{\\phi})^{\\hat{x}_0} and \\frac{\\partial}{\\partial \\hat{x}_0} - (\\frac{\\eta^{\\prime}{(\\hat{x}_0,\\phi)}}{\\phi})^{\\hat{x}_0} = \\frac{\\partial}{\\partial \\hat{x}_0} - (\\frac{\\log{(\\hat{x}_0)}^{\\phi}}{\\phi})^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\Omega)} = \\sin{(\\Omega)} and E{(\\Omega)} = \\int \\sin{(\\Omega)} d\\Omega, then derive \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\dot{\\mathbf{r}} - \\cos{(\\Omega)}, then obtain E{(\\Omega)} \\sin{(\\eta^{\\prime})} + \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\sin{(\\eta^{\\prime})} \\int \\sin{(\\Omega)} d\\Omega + \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega", "derivation": "\\operatorname{E_{x}}{(\\Omega)} = \\sin{(\\Omega)} and \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\dot{\\mathbf{r}} - \\cos{(\\Omega)} and E{(\\Omega)} = \\int \\sin{(\\Omega)} d\\Omega and \\int \\sin{(\\Omega)} d\\Omega = \\dot{\\mathbf{r}} - \\cos{(\\Omega)} and \\sin{(\\eta^{\\prime})} \\int \\sin{(\\Omega)} d\\Omega = (\\dot{\\mathbf{r}} - \\cos{(\\Omega)}) \\sin{(\\eta^{\\prime})} and E{(\\Omega)} \\sin{(\\eta^{\\prime})} = (\\dot{\\mathbf{r}} - \\cos{(\\Omega)}) \\sin{(\\eta^{\\prime})} and E{(\\Omega)} \\sin{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} \\int \\sin{(\\Omega)} d\\Omega and E{(\\Omega)} \\sin{(\\eta^{\\prime})} + \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\sin{(\\eta^{\\prime})} \\int \\sin{(\\Omega)} d\\Omega + \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\Omega', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], [["times", 5, "sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Function('E')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Function('E')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["add", 8, "Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Function('E')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\lambda)} = \\sin{(\\cos{(\\lambda)})}, then obtain 1 = - \\Omega{(\\lambda)} + \\sin{(\\cos{(\\lambda)})} + 1", "derivation": "\\Omega{(\\lambda)} = \\sin{(\\cos{(\\lambda)})} and \\lambda + \\Omega{(\\lambda)} = \\lambda + \\sin{(\\cos{(\\lambda)})} and 0 = - \\Omega{(\\lambda)} + \\sin{(\\cos{(\\lambda)})} and 1 = - \\Omega{(\\lambda)} + \\sin{(\\cos{(\\lambda)})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["minus", 2, "Add(Symbol('\\\\lambda', commutative=True), Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True))), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True))), sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\omega,\\theta_1)} = \\omega \\cos{(\\theta_1)}, then obtain \\frac{d}{d \\omega} 0 = \\frac{\\partial}{\\partial \\omega} (\\omega \\cos{(\\theta_1)} - \\operatorname{P_{e}}{(\\omega,\\theta_1)})", "derivation": "\\operatorname{P_{e}}{(\\omega,\\theta_1)} = \\omega \\cos{(\\theta_1)} and - \\theta_1 + \\operatorname{P_{e}}{(\\omega,\\theta_1)} = \\omega \\cos{(\\theta_1)} - \\theta_1 and 0 = \\omega \\cos{(\\theta_1)} - \\operatorname{P_{e}}{(\\omega,\\theta_1)} and \\frac{d}{d \\omega} 0 = \\frac{\\partial}{\\partial \\omega} (\\omega \\cos{(\\theta_1)} - \\operatorname{P_{e}}{(\\omega,\\theta_1)})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Function('P_e')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(\\Psi_{\\lambda})} = e^{\\sin{(\\Psi_{\\lambda})}}, then obtain 2 \\sigma_{x}{(\\Psi_{\\lambda})} - \\sin{(\\Psi_{\\lambda})} = 2 e^{\\sin{(\\Psi_{\\lambda})}} - \\sin{(\\Psi_{\\lambda})}", "derivation": "\\sigma_{x}{(\\Psi_{\\lambda})} = e^{\\sin{(\\Psi_{\\lambda})}} and \\sigma_{x}{(\\Psi_{\\lambda})} - \\sin{(\\Psi_{\\lambda})} = e^{\\sin{(\\Psi_{\\lambda})}} - \\sin{(\\Psi_{\\lambda})} and 2 \\sigma_{x}{(\\Psi_{\\lambda})} - \\sin{(\\Psi_{\\lambda})} = \\sigma_{x}{(\\Psi_{\\lambda})} + e^{\\sin{(\\Psi_{\\lambda})}} - \\sin{(\\Psi_{\\lambda})} and 2 \\sigma_{x}{(\\Psi_{\\lambda})} - \\sin{(\\Psi_{\\lambda})} = 2 e^{\\sin{(\\Psi_{\\lambda})}} - \\sin{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(exp(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["add", 2, "Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(2), exp(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(f_{\\mathbf{p}},p)} = \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}}, then derive \\sigma_{x}{(f_{\\mathbf{p}},p)} = \\hat{H}_{\\lambda} - \\frac{f_{\\mathbf{p}}^{2}}{2} + f_{\\mathbf{p}} p, then derive f_{\\mathbf{p}} = \\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}}, then obtain \\sin{(f_{\\mathbf{p}})} = \\sin{(\\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}})}", "derivation": "\\sigma_{x}{(f_{\\mathbf{p}},p)} = \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}} and \\frac{\\partial}{\\partial p} \\sigma_{x}{(f_{\\mathbf{p}},p)} = \\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}} and \\sigma_{x}{(f_{\\mathbf{p}},p)} = \\hat{H}_{\\lambda} - \\frac{f_{\\mathbf{p}}^{2}}{2} + f_{\\mathbf{p}} p and \\frac{\\partial}{\\partial p} (\\hat{H}_{\\lambda} - \\frac{f_{\\mathbf{p}}^{2}}{2} + f_{\\mathbf{p}} p) = \\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}} and f_{\\mathbf{p}} = \\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}} and \\sin{(f_{\\mathbf{p}})} = \\sin{(\\frac{\\partial}{\\partial p} \\int (- f_{\\mathbf{p}} + p) df_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Symbol('f_{\\\\mathbf{p}}', commutative=True), Derivative(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["sin", 5], "Equality(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(S,E_{\\lambda})} = e^{- E_{\\lambda} + S} and S{(i)} = \\sin{(\\log{(i)})}, then obtain \\frac{\\int (e^{- E_{\\lambda} + S} - 1) S{(i)} dS}{i} = \\frac{\\int (e^{- E_{\\lambda} + S} - 1) \\sin{(\\log{(i)})} dS}{i}", "derivation": "\\operatorname{v_{y}}{(S,E_{\\lambda})} = e^{- E_{\\lambda} + S} and S{(i)} = \\sin{(\\log{(i)})} and (e^{- E_{\\lambda} + S} - 1) S{(i)} = (e^{- E_{\\lambda} + S} - 1) \\sin{(\\log{(i)})} and (\\operatorname{v_{y}}{(S,E_{\\lambda})} - 1) S{(i)} = (\\operatorname{v_{y}}{(S,E_{\\lambda})} - 1) \\sin{(\\log{(i)})} and \\int (\\operatorname{v_{y}}{(S,E_{\\lambda})} - 1) S{(i)} dS = \\int (\\operatorname{v_{y}}{(S,E_{\\lambda})} - 1) \\sin{(\\log{(i)})} dS and \\int (e^{- E_{\\lambda} + S} - 1) S{(i)} dS = \\int (e^{- E_{\\lambda} + S} - 1) \\sin{(\\log{(i)})} dS and \\frac{\\int (e^{- E_{\\lambda} + S} - 1) S{(i)} dS}{i} = \\frac{\\int (e^{- E_{\\lambda} + S} - 1) \\sin{(\\log{(i)})} dS}{i}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('S', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))))"], ["get_premise", "Equality(Function('S')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], [["times", 2, "Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1))"], "Equality(Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), Function('S')(Symbol('i', commutative=True))), Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), sin(log(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Function('v_y')(Symbol('S', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), Function('S')(Symbol('i', commutative=True))), Mul(Add(Function('v_y')(Symbol('S', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), sin(log(Symbol('i', commutative=True)))))"], [["integrate", 4, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Add(Function('v_y')(Symbol('S', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), Function('S')(Symbol('i', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Add(Function('v_y')(Symbol('S', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), sin(log(Symbol('i', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), Function('S')(Symbol('i', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), sin(log(Symbol('i', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["divide", 6, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Integral(Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), Function('S')(Symbol('i', commutative=True))), Tuple(Symbol('S', commutative=True)))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Integral(Mul(Add(exp(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('S', commutative=True))), Integer(-1)), sin(log(Symbol('i', commutative=True)))), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\eta{(\\mathbf{v},\\hat{X})} = e^{\\hat{X} + \\mathbf{v}}, then obtain \\frac{(\\eta{(\\mathbf{v},\\hat{X})} - e^{\\hat{X} + \\mathbf{v}}) \\eta{(\\mathbf{v},\\hat{X})}}{\\hat{X} + \\mathbf{v}} = 0", "derivation": "\\eta{(\\mathbf{v},\\hat{X})} = e^{\\hat{X} + \\mathbf{v}} and \\frac{\\eta{(\\mathbf{v},\\hat{X})}}{\\hat{X} + \\mathbf{v}} = \\frac{e^{\\hat{X} + \\mathbf{v}}}{\\hat{X} + \\mathbf{v}} and \\eta{(\\mathbf{v},\\hat{X})} - e^{\\hat{X} + \\mathbf{v}} = 0 and \\frac{(\\eta{(\\mathbf{v},\\hat{X})} - e^{\\hat{X} + \\mathbf{v}}) e^{\\hat{X} + \\mathbf{v}}}{\\hat{X} + \\mathbf{v}} = 0 and \\frac{(\\eta{(\\mathbf{v},\\hat{X})} - e^{\\hat{X} + \\mathbf{v}}) \\eta{(\\mathbf{v},\\hat{X})}}{\\hat{X} + \\mathbf{v}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))"], [["minus", 1, "exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))), Integer(0))"], [["times", 3, "Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Add(Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Add(Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))), Function('\\\\eta')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mu_{0}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})}, then obtain \\iint \\mu_{0}{(g_{\\varepsilon})} \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint \\cos^{2}{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "derivation": "\\mu_{0}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\mu_{0}{(g_{\\varepsilon})} \\cos{(g_{\\varepsilon})} = \\cos^{2}{(g_{\\varepsilon})} and \\int \\mu_{0}{(g_{\\varepsilon})} \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\cos^{2}{(g_{\\varepsilon})} dg_{\\varepsilon} and \\iint \\mu_{0}{(g_{\\varepsilon})} \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint \\cos^{2}{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "cos(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Pow(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Pow(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(2)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{J},u)} = \\cos{(\\mathbf{J} u)} and \\hat{p}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\operatorname{v_{z}}{(\\mathbf{J},u)} and \\mathbf{E}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\operatorname{v_{z}}{(\\mathbf{J},u)}, then obtain \\hat{p}{(\\mathbf{J},u)} = \\mathbf{E}{(\\mathbf{J},u)}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{J},u)} = \\cos{(\\mathbf{J} u)} and \\frac{\\partial}{\\partial u} \\operatorname{v_{z}}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\cos{(\\mathbf{J} u)} and \\hat{p}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\operatorname{v_{z}}{(\\mathbf{J},u)} and \\hat{p}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\cos{(\\mathbf{J} u)} and \\mathbf{E}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\operatorname{v_{z}}{(\\mathbf{J},u)} and \\mathbf{E}{(\\mathbf{J},u)} = \\frac{\\partial}{\\partial u} \\cos{(\\mathbf{J} u)} and \\hat{p}{(\\mathbf{J},u)} = \\mathbf{E}{(\\mathbf{J},u)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Derivative(Function('v_z')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Derivative(Function('v_z')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Derivative(cos(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(v_{1})} = e^{v_{1}}, then derive 0 = e^{v_{1}} - \\frac{d}{d v_{1}} \\mathbf{v}{(v_{1})}, then obtain 0 = \\mathbf{v}{(v_{1})} - \\frac{d}{d v_{1}} \\mathbf{v}{(v_{1})}", "derivation": "\\mathbf{v}{(v_{1})} = e^{v_{1}} and 0 = - \\mathbf{v}{(v_{1})} + e^{v_{1}} and \\frac{d}{d v_{1}} 0 = \\frac{d}{d v_{1}} (- \\mathbf{v}{(v_{1})} + e^{v_{1}}) and 0 = e^{v_{1}} - \\frac{d}{d v_{1}} \\mathbf{v}{(v_{1})} and 0 = \\mathbf{v}{(v_{1})} - \\frac{d}{d v_{1}} \\mathbf{v}{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(exp(Symbol('v_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{v}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)} = - V_{\\mathbf{B}} + \\chi, then obtain \\frac{\\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}}{- V_{\\mathbf{B}} + \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}} = \\frac{- V_{\\mathbf{B}} + \\chi}{- V_{\\mathbf{B}} + \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}}", "derivation": "\\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)} = - V_{\\mathbf{B}} + \\chi and - V_{\\mathbf{B}} + \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)} = - 2 V_{\\mathbf{B}} + \\chi and \\frac{\\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}}{- 2 V_{\\mathbf{B}} + \\chi} = \\frac{- V_{\\mathbf{B}} + \\chi}{- 2 V_{\\mathbf{B}} + \\chi} and \\frac{\\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}}{- V_{\\mathbf{B}} + \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}} = \\frac{- V_{\\mathbf{B}} + \\chi}{- V_{\\mathbf{B}} + \\hat{\\mathbf{x}}{(V_{\\mathbf{B}},\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given Q{(\\Psi^{\\dagger},F_{x})} = \\sin{(F_{x} - \\Psi^{\\dagger})}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} Q{(\\Psi^{\\dagger},F_{x})} = - \\cos{(F_{x} - \\Psi^{\\dagger})}, then obtain F_{x} \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\sin{(F_{x} - \\Psi^{\\dagger})} = - F_{x} \\cos{(F_{x} - \\Psi^{\\dagger})}", "derivation": "Q{(\\Psi^{\\dagger},F_{x})} = \\sin{(F_{x} - \\Psi^{\\dagger})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} Q{(\\Psi^{\\dagger},F_{x})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\sin{(F_{x} - \\Psi^{\\dagger})} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} Q{(\\Psi^{\\dagger},F_{x})} = - \\cos{(F_{x} - \\Psi^{\\dagger})} and F_{x} \\frac{\\partial}{\\partial \\Psi^{\\dagger}} Q{(\\Psi^{\\dagger},F_{x})} = - F_{x} \\cos{(F_{x} - \\Psi^{\\dagger})} and F_{x} \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\sin{(F_{x} - \\Psi^{\\dagger})} = - F_{x} \\cos{(F_{x} - \\Psi^{\\dagger})}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('F_x', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"], [["times", 3, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Derivative(Function('Q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('F_x', commutative=True), cos(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('F_x', commutative=True), Derivative(sin(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('F_x', commutative=True), cos(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbb{I})} = \\mathbb{I} and \\rho_{b}{(\\mathbb{I},\\chi)} = \\chi + \\frac{\\mathbb{I}^{2}}{2}, then derive \\chi + \\frac{\\Psi^{\\dagger}^{2}{(\\mathbb{I})}}{2} = \\int \\mathbb{I} d\\Psi^{\\dagger}{(\\mathbb{I})}, then obtain \\rho_{b}{(\\mathbb{I},\\chi)} = \\int \\mathbb{I} d\\mathbb{I}", "derivation": "\\Psi^{\\dagger}{(\\mathbb{I})} = \\mathbb{I} and \\int \\Psi^{\\dagger}{(\\mathbb{I})} d\\mathbb{I} = \\int \\mathbb{I} d\\mathbb{I} and \\int \\Psi^{\\dagger}{(\\mathbb{I})} d\\Psi^{\\dagger}{(\\mathbb{I})} = \\int \\mathbb{I} d\\Psi^{\\dagger}{(\\mathbb{I})} and \\chi + \\frac{\\Psi^{\\dagger}^{2}{(\\mathbb{I})}}{2} = \\int \\mathbb{I} d\\Psi^{\\dagger}{(\\mathbb{I})} and \\chi + \\frac{\\mathbb{I}^{2}}{2} = \\int \\mathbb{I} d\\mathbb{I} and \\rho_{b}{(\\mathbb{I},\\chi)} = \\chi + \\frac{\\mathbb{I}^{2}}{2} and \\rho_{b}{(\\mathbb{I},\\chi)} = \\int \\mathbb{I} d\\mathbb{I}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)))), Integral(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))), Integral(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))), Integral(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\mathbb{I})} = \\sin{(\\mathbb{I})}, then obtain - 2 \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\varepsilon_{0}{(\\mathbb{I})}}{\\mathbb{I}} = - 2 \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\sin{(\\mathbb{I})}}{\\mathbb{I}}", "derivation": "\\varepsilon_{0}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\frac{\\varepsilon_{0}{(\\mathbb{I})}}{\\mathbb{I}} = \\frac{\\sin{(\\mathbb{I})}}{\\mathbb{I}} and - \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\varepsilon_{0}{(\\mathbb{I})}}{\\mathbb{I}} = - \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\sin{(\\mathbb{I})}}{\\mathbb{I}} and - 2 \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\varepsilon_{0}{(\\mathbb{I})}}{\\mathbb{I}} = - 2 \\varepsilon_{0}{(\\mathbb{I})} + \\frac{\\sin{(\\mathbb{I})}}{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["minus", 3, "Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(x,A,\\Psi)} = (A + \\Psi)^{x} and c{(x,A,\\Psi)} = (((A + \\Psi)^{x})^{x})^{x}, then obtain \\int (\\Psi + (\\rho_{b}^{x}{(x,A,\\Psi)})^{x}) d\\Psi = \\int (\\Psi + (((A + \\Psi)^{x})^{x})^{x}) d\\Psi", "derivation": "\\rho_{b}{(x,A,\\Psi)} = (A + \\Psi)^{x} and \\rho_{b}^{x}{(x,A,\\Psi)} = ((A + \\Psi)^{x})^{x} and (\\rho_{b}^{x}{(x,A,\\Psi)})^{x} = (((A + \\Psi)^{x})^{x})^{x} and c{(x,A,\\Psi)} = (((A + \\Psi)^{x})^{x})^{x} and \\Psi + c{(x,A,\\Psi)} = \\Psi + (((A + \\Psi)^{x})^{x})^{x} and (\\rho_{b}^{x}{(x,A,\\Psi)})^{x} = c{(x,A,\\Psi)} and \\Psi + (\\rho_{b}^{x}{(x,A,\\Psi)})^{x} = \\Psi + (((A + \\Psi)^{x})^{x})^{x} and \\int (\\Psi + (\\rho_{b}^{x}{(x,A,\\Psi)})^{x}) d\\Psi = \\int (\\Psi + (((A + \\Psi)^{x})^{x})^{x}) d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Pow(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('c')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Pow(Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Function('c')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('\\\\Psi', commutative=True), Pow(Pow(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Pow(Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["integrate", 7, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Psi', commutative=True), Pow(Pow(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Pow(Pow(Pow(Add(Symbol('A', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\rho,H)} = H + \\rho, then derive \\frac{\\frac{\\partial}{\\partial H} \\operatorname{t_{2}}{(\\rho,H)}}{H} - \\frac{\\operatorname{t_{2}}{(\\rho,H)}}{H^{2}} = \\frac{1}{H} - \\frac{H + \\rho}{H^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial H} (H + \\rho)}{H} - \\frac{H + \\rho}{H^{2}} = \\frac{1}{H} + \\frac{- H - \\rho}{H^{2}}", "derivation": "\\operatorname{t_{2}}{(\\rho,H)} = H + \\rho and \\frac{\\operatorname{t_{2}}{(\\rho,H)}}{H} = \\frac{H + \\rho}{H} and \\frac{\\partial}{\\partial H} \\frac{\\operatorname{t_{2}}{(\\rho,H)}}{H} = \\frac{\\partial}{\\partial H} \\frac{H + \\rho}{H} and \\frac{\\frac{\\partial}{\\partial H} \\operatorname{t_{2}}{(\\rho,H)}}{H} - \\frac{\\operatorname{t_{2}}{(\\rho,H)}}{H^{2}} = \\frac{1}{H} - \\frac{H + \\rho}{H^{2}} and \\frac{\\frac{\\partial}{\\partial H} \\operatorname{t_{2}}{(\\rho,H)}}{H} - \\frac{H + \\rho}{H^{2}} = \\frac{1}{H} + \\frac{- H - \\rho}{H^{2}} and \\frac{\\frac{\\partial}{\\partial H} (H + \\rho)}{H} - \\frac{H + \\rho}{H^{2}} = \\frac{1}{H} + \\frac{- H - \\rho}{H^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Derivative(Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-2)), Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)))), Add(Pow(Symbol('H', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-2)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Derivative(Function('t_2')(Symbol('\\\\rho', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-2)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Pow(Symbol('H', commutative=True), Integer(-1)), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-2)), Add(Symbol('H', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Pow(Symbol('H', commutative=True), Integer(-1)), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{s})} = e^{\\mathbf{s}}, then derive \\int \\mathbf{P}{(\\mathbf{s})} d\\mathbf{s} = f + e^{\\mathbf{s}}, then obtain - f + \\int \\mathbf{P}{(\\mathbf{s})} d\\mathbf{s} = e^{\\mathbf{s}}", "derivation": "\\mathbf{P}{(\\mathbf{s})} = e^{\\mathbf{s}} and \\int \\mathbf{P}{(\\mathbf{s})} d\\mathbf{s} = \\int e^{\\mathbf{s}} d\\mathbf{s} and \\int \\mathbf{P}{(\\mathbf{s})} d\\mathbf{s} = f + e^{\\mathbf{s}} and - f + \\int \\mathbf{P}{(\\mathbf{s})} d\\mathbf{s} = e^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('f', commutative=True), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 3, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), exp(Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then obtain \\mathbf{D} + \\frac{d}{d \\mathbf{D}} 1 = \\mathbf{D} + \\frac{d}{d \\mathbf{D}} \\frac{2 \\sin{(\\mathbf{D})}}{\\operatorname{v_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\operatorname{v_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})} = 2 \\sin{(\\mathbf{D})} and 1 = \\frac{2 \\sin{(\\mathbf{D})}}{\\operatorname{v_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}} and \\frac{d}{d \\mathbf{D}} 1 = \\frac{d}{d \\mathbf{D}} \\frac{2 \\sin{(\\mathbf{D})}}{\\operatorname{v_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}} and \\mathbf{D} + \\frac{d}{d \\mathbf{D}} 1 = \\mathbf{D} + \\frac{d}{d \\mathbf{D}} \\frac{2 \\sin{(\\mathbf{D})}}{\\operatorname{v_{2}}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 2, "Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Integer(1), Mul(Integer(2), Pow(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Mul(Integer(2), Pow(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(\\hat{p},\\mu)} = \\cos{(\\hat{p}^{\\mu})} and V{(\\hat{p},\\mu)} = \\cos{(\\hat{p}^{\\mu})}, then obtain \\frac{V{(\\hat{p},\\mu)}}{\\hat{p}} = \\frac{i{(\\hat{p},\\mu)}}{\\hat{p}}", "derivation": "i{(\\hat{p},\\mu)} = \\cos{(\\hat{p}^{\\mu})} and V{(\\hat{p},\\mu)} = \\cos{(\\hat{p}^{\\mu})} and i{(\\hat{p},\\mu)} = V{(\\hat{p},\\mu)} and \\frac{i{(\\hat{p},\\mu)}}{\\hat{p}} = \\frac{\\cos{(\\hat{p}^{\\mu})}}{\\hat{p}} and \\frac{V{(\\hat{p},\\mu)}}{\\hat{p}} = \\frac{\\cos{(\\hat{p}^{\\mu})}}{\\hat{p}} and \\frac{V{(\\hat{p},\\mu)}}{\\hat{p}} = \\frac{i{(\\hat{p},\\mu)}}{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)), Function('V')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('V')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)} = \\mu - \\tilde{g}^*, then obtain \\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)} = (\\frac{\\mu}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}} - \\frac{\\tilde{g}^*}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}}) \\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}", "derivation": "\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)} = \\mu - \\tilde{g}^* and 1 = \\frac{\\mu - \\tilde{g}^*}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}} and 1 = \\frac{\\mu}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}} - \\frac{\\tilde{g}^*}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}} and \\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)} = (\\frac{\\mu}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}} - \\frac{\\tilde{g}^*}{\\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}}) \\operatorname{A_{1}}{(\\mu,\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["divide", 1, "Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["expand", 2], "Equality(Integer(1), Add(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), Pow(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)))))"], [["times", 3, "Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Add(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True), Pow(Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)))), Function('A_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(t)} = e^{t}, then obtain \\operatorname{y^{\\prime}}^{2}{(t)} e^{2 t} = \\operatorname{y^{\\prime}}{(t)} e^{3 t}", "derivation": "\\operatorname{y^{\\prime}}{(t)} = e^{t} and \\operatorname{y^{\\prime}}^{2}{(t)} = \\operatorname{y^{\\prime}}{(t)} e^{t} and \\operatorname{y^{\\prime}}^{4}{(t)} = \\operatorname{y^{\\prime}}^{2}{(t)} e^{2 t} and \\operatorname{y^{\\prime}}^{2}{(t)} e^{2 t} = \\operatorname{y^{\\prime}}{(t)} e^{3 t}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('t', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), Integer(2)), Mul(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), Integer(4)), Mul(Pow(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('t', commutative=True)))), Mul(Function('y^{\\\\prime}')(Symbol('t', commutative=True)), exp(Mul(Integer(3), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(F_{g},L)} = \\frac{F_{g}}{L} and \\Psi_{nl}{(u)} = \\cos{(u)}, then obtain - L (\\frac{F_{g}}{L} - L) + \\int (- F_{g} + \\Psi_{nl}{(u)}) dF_{g} = - L (\\frac{F_{g}}{L} - L) + \\int (- F_{g} + \\cos{(u)}) dF_{g}", "derivation": "\\mathbf{F}{(F_{g},L)} = \\frac{F_{g}}{L} and - L + \\mathbf{F}{(F_{g},L)} = \\frac{F_{g}}{L} - L and L (- L + \\mathbf{F}{(F_{g},L)}) = L (\\frac{F_{g}}{L} - L) and \\Psi_{nl}{(u)} = \\cos{(u)} and - F_{g} + \\Psi_{nl}{(u)} = - F_{g} + \\cos{(u)} and \\int (- F_{g} + \\Psi_{nl}{(u)}) dF_{g} = \\int (- F_{g} + \\cos{(u)}) dF_{g} and - L (- L + \\mathbf{F}{(F_{g},L)}) + \\int (- F_{g} + \\Psi_{nl}{(u)}) dF_{g} = - L (- L + \\mathbf{F}{(F_{g},L)}) + \\int (- F_{g} + \\cos{(u)}) dF_{g} and - L (\\frac{F_{g}}{L} - L) + \\int (- F_{g} + \\Psi_{nl}{(u)}) dF_{g} = - L (\\frac{F_{g}}{L} - L) + \\int (- F_{g} + \\cos{(u)}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('L', commutative=True))))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)))), Mul(Symbol('L', commutative=True), Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('L', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["minus", 4, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))))"], [["integrate", 5, "Symbol('F_g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["minus", 6, "Mul(Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('L', commutative=True), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\mathbf{F}')(Symbol('F_g', commutative=True), Symbol('L', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True), Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('L', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('L', commutative=True), Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('L', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\dot{x})} = \\sin{(\\dot{x})} and \\dot{y}{(\\dot{x})} = \\int \\sin{(\\dot{x})} d\\dot{x}, then obtain \\frac{\\operatorname{f^{*}}{(\\dot{x})}}{\\dot{y}{(\\dot{x})}} = \\frac{\\sin{(\\dot{x})}}{\\dot{y}{(\\dot{x})}}", "derivation": "\\operatorname{f^{*}}{(\\dot{x})} = \\sin{(\\dot{x})} and \\int \\operatorname{f^{*}}{(\\dot{x})} d\\dot{x} = \\int \\sin{(\\dot{x})} d\\dot{x} and \\frac{\\operatorname{f^{*}}{(\\dot{x})}}{\\int \\operatorname{f^{*}}{(\\dot{x})} d\\dot{x}} = \\frac{\\sin{(\\dot{x})}}{\\int \\operatorname{f^{*}}{(\\dot{x})} d\\dot{x}} and \\frac{\\operatorname{f^{*}}{(\\dot{x})}}{\\int \\sin{(\\dot{x})} d\\dot{x}} = \\frac{\\sin{(\\dot{x})}}{\\int \\sin{(\\dot{x})} d\\dot{x}} and \\dot{y}{(\\dot{x})} = \\int \\sin{(\\dot{x})} d\\dot{x} and \\frac{\\operatorname{f^{*}}{(\\dot{x})}}{\\dot{y}{(\\dot{x})}} = \\frac{\\sin{(\\dot{x})}}{\\dot{y}{(\\dot{x})}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 1, "Integral(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('f^*')(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\dot{x}', commutative=True)), Integral(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), Function('f^*')(Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Function('\\\\dot{y}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(h,u)} = \\int (- h + u) dh, then derive \\mu_{0}{(h,u)} = W - \\frac{h^{2}}{2} + h u, then derive W - \\frac{h^{2}}{2} + h u = \\mathbf{g} - \\frac{h^{2}}{2} + h u, then obtain \\mathbf{g} - \\frac{h^{2}}{2} + h u - u - \\int (- h + u) dh = - u", "derivation": "\\mu_{0}{(h,u)} = \\int (- h + u) dh and - u + \\mu_{0}{(h,u)} = - u + \\int (- h + u) dh and \\mu_{0}{(h,u)} = W - \\frac{h^{2}}{2} + h u and W - \\frac{h^{2}}{2} + h u - u = - u + \\int (- h + u) dh and W - \\frac{h^{2}}{2} + h u = \\int (- h + u) dh and W - \\frac{h^{2}}{2} + h u = \\mathbf{g} - \\frac{h^{2}}{2} + h u and \\mathbf{g} - \\frac{h^{2}}{2} + h u - u = - u + \\int (- h + u) dh and \\mathbf{g} - \\frac{h^{2}}{2} + h u - u - \\int (- h + u) dh = - u", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True), Symbol('u', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["minus", 7, "Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('h', commutative=True))))), Mul(Integer(-1), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain \\frac{\\int (g_{\\varepsilon} + e^{\\tilde{g}{(x^\\prime)}}) dg_{\\varepsilon}}{x^\\prime} = \\frac{\\int (g_{\\varepsilon} + e^{\\cos{(x^\\prime)}}) dg_{\\varepsilon}}{x^\\prime}", "derivation": "\\tilde{g}{(x^\\prime)} = \\cos{(x^\\prime)} and e^{\\tilde{g}{(x^\\prime)}} = e^{\\cos{(x^\\prime)}} and g_{\\varepsilon} + e^{\\tilde{g}{(x^\\prime)}} = g_{\\varepsilon} + e^{\\cos{(x^\\prime)}} and \\int (g_{\\varepsilon} + e^{\\tilde{g}{(x^\\prime)}}) dg_{\\varepsilon} = \\int (g_{\\varepsilon} + e^{\\cos{(x^\\prime)}}) dg_{\\varepsilon} and \\frac{\\int (g_{\\varepsilon} + e^{\\tilde{g}{(x^\\prime)}}) dg_{\\varepsilon}}{x^\\prime} = \\frac{\\int (g_{\\varepsilon} + e^{\\cos{(x^\\prime)}}) dg_{\\varepsilon}}{x^\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\tilde{g}')(Symbol('x^\\\\prime', commutative=True))), exp(cos(Symbol('x^\\\\prime', commutative=True))))"], [["add", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Function('\\\\tilde{g}')(Symbol('x^\\\\prime', commutative=True)))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(cos(Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Function('\\\\tilde{g}')(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Function('\\\\tilde{g}')(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain \\frac{\\int \\frac{\\mathbb{I}{(V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}})}} dV_{\\mathbf{E}}}{\\sin{(V_{\\mathbf{E}})}} = \\frac{\\int 1 dV_{\\mathbf{E}}}{\\sin{(V_{\\mathbf{E}})}}", "derivation": "\\mathbb{I}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\frac{\\mathbb{I}{(V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}})}} = 1 and \\int \\frac{\\mathbb{I}{(V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}})}} dV_{\\mathbf{E}} = \\int 1 dV_{\\mathbf{E}} and \\frac{\\int \\frac{\\mathbb{I}{(V_{\\mathbf{E}})}}{\\sin{(V_{\\mathbf{E}})}} dV_{\\mathbf{E}}}{\\sin{(V_{\\mathbf{E}})}} = \\frac{\\int 1 dV_{\\mathbf{E}}}{\\sin{(V_{\\mathbf{E}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["divide", 1, "sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["divide", 3, "sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\mathbb{I}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(A)} = \\cos{(A)}, then obtain \\frac{2 (\\operatorname{A_{z}}{(A)} + \\cos{(A)})}{A} = \\frac{4 \\cos{(A)}}{A}", "derivation": "\\operatorname{A_{z}}{(A)} = \\cos{(A)} and \\operatorname{A_{z}}{(A)} + \\cos{(A)} = 2 \\cos{(A)} and \\frac{\\operatorname{A_{z}}{(A)} + \\cos{(A)}}{A} = \\frac{2 \\cos{(A)}}{A} and \\frac{2 (\\operatorname{A_{z}}{(A)} + \\cos{(A)})}{A} = \\frac{\\operatorname{A_{z}}{(A)} + \\cos{(A)}}{A} + \\frac{2 \\cos{(A)}}{A} and \\frac{2 \\operatorname{A_{z}}{(A)} + 2 \\cos{(A)}}{A} = \\frac{2 (\\operatorname{A_{z}}{(A)} + \\cos{(A)})}{A} and \\frac{2 \\operatorname{A_{z}}{(A)} + 2 \\cos{(A)}}{A} = \\frac{4 \\cos{(A)}}{A} and \\frac{2 (\\operatorname{A_{z}}{(A)} + \\cos{(A)})}{A} = \\frac{4 \\cos{(A)}}{A}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["add", 1, "cos(Symbol('A', commutative=True))"], "Equality(Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))), Mul(Integer(2), cos(Symbol('A', commutative=True))))"], [["divide", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('A', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))), Add(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(2), Function('A_z')(Symbol('A', commutative=True))), Mul(Integer(2), cos(Symbol('A', commutative=True))))), Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(2), Function('A_z')(Symbol('A', commutative=True))), Mul(Integer(2), cos(Symbol('A', commutative=True))))), Mul(Integer(4), Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Integer(2), Pow(Symbol('A', commutative=True), Integer(-1)), Add(Function('A_z')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))), Mul(Integer(4), Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbf{J}_f,F_{N})} = \\log{(F_{N} + \\mathbf{J}_f)}, then obtain \\rho_f + \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\mathbf{J}_P{(\\mathbf{J}_f,F_{N})}) = \\rho_f + \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\log{(F_{N} + \\mathbf{J}_f)})", "derivation": "\\mathbf{J}_P{(\\mathbf{J}_f,F_{N})} = \\log{(F_{N} + \\mathbf{J}_f)} and - \\mathbf{J}_f + \\mathbf{J}_P{(\\mathbf{J}_f,F_{N})} = - \\mathbf{J}_f + \\log{(F_{N} + \\mathbf{J}_f)} and \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\mathbf{J}_P{(\\mathbf{J}_f,F_{N})}) = \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\log{(F_{N} + \\mathbf{J}_f)}) and \\rho_f + \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\mathbf{J}_P{(\\mathbf{J}_f,F_{N})}) = \\rho_f + \\frac{\\partial}{\\partial F_{N}} (- \\mathbf{J}_f + \\log{(F_{N} + \\mathbf{J}_f)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True)), log(Add(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Symbol('\\\\rho_f', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given p{(S,i)} = S^{i}, then derive \\frac{\\partial}{\\partial S} p{(S,i)} = \\frac{S^{i} i}{S}, then obtain (\\frac{\\partial}{\\partial S} p{(S,i)})^{2} = \\frac{i p{(S,i)} \\frac{\\partial}{\\partial S} p{(S,i)}}{S}", "derivation": "p{(S,i)} = S^{i} and \\frac{\\partial}{\\partial S} p{(S,i)} = \\frac{\\partial}{\\partial S} S^{i} and \\frac{\\partial}{\\partial S} p{(S,i)} = \\frac{S^{i} i}{S} and \\frac{\\partial}{\\partial S} S^{i} \\frac{\\partial}{\\partial S} p{(S,i)} = \\frac{S^{i} i \\frac{\\partial}{\\partial S} S^{i}}{S} and (\\frac{\\partial}{\\partial S} p{(S,i)})^{2} = \\frac{i p{(S,i)} \\frac{\\partial}{\\partial S} p{(S,i)}}{S}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["times", 3, "Derivative(Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True), Derivative(Pow(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('i', commutative=True), Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Derivative(Function('p')(Symbol('S', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho{(L)} = \\sin{(L)}, then derive \\int \\rho{(L)} \\sin{(L)} dL = \\frac{L}{2} + \\mathbf{J}_f - \\frac{\\sin{(L)} \\cos{(L)}}{2}, then obtain \\frac{L}{2} + \\mathbf{J}_f - \\frac{\\sin{(L)} \\cos{(L)}}{2} = \\int \\sin^{2}{(L)} dL", "derivation": "\\rho{(L)} = \\sin{(L)} and \\rho{(L)} \\sin{(L)} = \\sin^{2}{(L)} and \\int \\rho{(L)} \\sin{(L)} dL = \\int \\sin^{2}{(L)} dL and \\int \\rho{(L)} \\sin{(L)} dL = \\frac{L}{2} + \\mathbf{J}_f - \\frac{\\sin{(L)} \\cos{(L)}}{2} and \\frac{L}{2} + \\mathbf{J}_f - \\frac{\\sin{(L)} \\cos{(L)}}{2} = \\int \\sin^{2}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["times", 1, "sin(Symbol('L', commutative=True))"], "Equality(Mul(Function('\\\\rho')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Pow(sin(Symbol('L', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Function('\\\\rho')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Pow(sin(Symbol('L', commutative=True)), Integer(2)), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\rho')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Add(Mul(Rational(1, 2), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Rational(1, 2), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))), Integral(Pow(sin(Symbol('L', commutative=True)), Integer(2)), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given t{(I,q)} = \\sin{(I q)}, then obtain \\int t{(I,q)} dI - \\frac{1}{t{(I,q)}} = \\int \\sin{(I q)} dI - \\frac{1}{t{(I,q)}}", "derivation": "t{(I,q)} = \\sin{(I q)} and \\int t{(I,q)} dI = \\int \\sin{(I q)} dI and \\frac{\\int t{(I,q)} dI}{\\sin{(I q)}} = \\frac{\\int \\sin{(I q)} dI}{\\sin{(I q)}} and \\frac{t{(I,q)} \\int t{(I,q)} dI}{\\sin{(I q)}} = \\frac{t{(I,q)} \\int \\sin{(I q)} dI}{\\sin{(I q)}} and \\frac{t{(I,q)} \\int t{(I,q)} dI}{\\sin{(I q)}} - \\frac{1}{t{(I,q)}} = \\frac{t{(I,q)} \\int \\sin{(I q)} dI}{\\sin{(I q)}} - \\frac{1}{t{(I,q)}} and \\int t{(I,q)} dI - \\frac{1}{t{(I,q)}} = \\int \\sin{(I q)} dI - \\frac{1}{t{(I,q)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["divide", 2, "sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["times", 3, "Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["minus", 4, "Pow(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Integer(-1)))), Add(Mul(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Pow(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Integral(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Integral(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Pow(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Integer(-1)))), Add(Integral(sin(Mul(Symbol('I', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('I', commutative=True))), Mul(Integer(-1), Pow(Function('t')(Symbol('I', commutative=True), Symbol('q', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\hat{H}_l{(Z)} = \\cos{(e^{Z})}, then derive \\frac{d}{d Z} \\hat{H}_l{(Z)} = - e^{Z} \\sin{(e^{Z})}, then obtain - e^{Z} \\sin{(e^{Z})} \\cos{(e^{Z})} = \\cos{(e^{Z})} \\frac{d}{d Z} \\cos{(e^{Z})}", "derivation": "\\hat{H}_l{(Z)} = \\cos{(e^{Z})} and \\frac{d}{d Z} \\hat{H}_l{(Z)} = \\frac{d}{d Z} \\cos{(e^{Z})} and \\cos{(e^{Z})} \\frac{d}{d Z} \\hat{H}_l{(Z)} = \\cos{(e^{Z})} \\frac{d}{d Z} \\cos{(e^{Z})} and \\frac{d}{d Z} \\hat{H}_l{(Z)} = - e^{Z} \\sin{(e^{Z})} and - e^{Z} \\sin{(e^{Z})} \\cos{(e^{Z})} = \\cos{(e^{Z})} \\frac{d}{d Z} \\cos{(e^{Z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(exp(Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["times", 2, "cos(exp(Symbol('Z', commutative=True)))"], "Equality(Mul(cos(exp(Symbol('Z', commutative=True))), Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(cos(exp(Symbol('Z', commutative=True))), Derivative(cos(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), exp(Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True))), cos(exp(Symbol('Z', commutative=True)))), Mul(cos(exp(Symbol('Z', commutative=True))), Derivative(cos(exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(r_{0})} = \\sin{(\\log{(r_{0})})}, then obtain \\sin{(r_{0} + \\log{(r_{0})} + \\int \\operatorname{A_{z}}{(r_{0})} dr_{0})} = \\sin{(r_{0} + \\log{(r_{0})} + \\int \\sin{(\\log{(r_{0})})} dr_{0})}", "derivation": "\\operatorname{A_{z}}{(r_{0})} = \\sin{(\\log{(r_{0})})} and \\int \\operatorname{A_{z}}{(r_{0})} dr_{0} = \\int \\sin{(\\log{(r_{0})})} dr_{0} and r_{0} + \\int \\operatorname{A_{z}}{(r_{0})} dr_{0} = r_{0} + \\int \\sin{(\\log{(r_{0})})} dr_{0} and r_{0} + \\log{(r_{0})} + \\int \\operatorname{A_{z}}{(r_{0})} dr_{0} = r_{0} + \\log{(r_{0})} + \\int \\sin{(\\log{(r_{0})})} dr_{0} and \\sin{(r_{0} + \\log{(r_{0})} + \\int \\operatorname{A_{z}}{(r_{0})} dr_{0})} = \\sin{(r_{0} + \\log{(r_{0})} + \\int \\sin{(\\log{(r_{0})})} dr_{0})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('r_0', commutative=True)), sin(log(Symbol('r_0', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(sin(log(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["add", 2, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Integral(Function('A_z')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Add(Symbol('r_0', commutative=True), Integral(sin(log(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))))"], [["add", 3, "log(Symbol('r_0', commutative=True))"], "Equality(Add(Symbol('r_0', commutative=True), log(Symbol('r_0', commutative=True)), Integral(Function('A_z')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Add(Symbol('r_0', commutative=True), log(Symbol('r_0', commutative=True)), Integral(sin(log(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Symbol('r_0', commutative=True), log(Symbol('r_0', commutative=True)), Integral(Function('A_z')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))), sin(Add(Symbol('r_0', commutative=True), log(Symbol('r_0', commutative=True)), Integral(sin(log(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(f^{*})} = \\cos{(f^{*})}, then obtain \\frac{d}{d f^{*}} \\int (\\mathbf{J}_f{(f^{*})} + \\cos{(f^{*})}) df^{*} = \\frac{d}{d f^{*}} \\int 2 \\cos{(f^{*})} df^{*}", "derivation": "\\mathbf{J}_f{(f^{*})} = \\cos{(f^{*})} and \\mathbf{J}_f{(f^{*})} + \\cos{(f^{*})} = 2 \\cos{(f^{*})} and \\int (\\mathbf{J}_f{(f^{*})} + \\cos{(f^{*})}) df^{*} = \\int 2 \\cos{(f^{*})} df^{*} and \\frac{d}{d f^{*}} \\int (\\mathbf{J}_f{(f^{*})} + \\cos{(f^{*})}) df^{*} = \\frac{d}{d f^{*}} \\int 2 \\cos{(f^{*})} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["add", 1, "cos(Symbol('f^*', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Mul(Integer(2), cos(Symbol('f^*', commutative=True))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{J}_f')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mathbf{J}_f')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(E_{x})} = \\int \\log{(E_{x})} dE_{x}, then derive M{(E_{x})} = E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}}, then obtain (E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}})^{E_{x}} M{(E_{x})} \\log{(E_{x})} = M{(E_{x})} \\log{(E_{x})} (\\int \\log{(E_{x})} dE_{x})^{E_{x}}", "derivation": "M{(E_{x})} = \\int \\log{(E_{x})} dE_{x} and M^{E_{x}}{(E_{x})} = (\\int \\log{(E_{x})} dE_{x})^{E_{x}} and M{(E_{x})} = E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}} and (E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}})^{E_{x}} = (\\int \\log{(E_{x})} dE_{x})^{E_{x}} and (E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}})^{E_{x}} \\log{(E_{x})} = \\log{(E_{x})} (\\int \\log{(E_{x})} dE_{x})^{E_{x}} and (E_{x} \\log{(E_{x})} - E_{x} + \\hat{\\mathbf{r}})^{E_{x}} M{(E_{x})} \\log{(E_{x})} = M{(E_{x})} \\log{(E_{x})} (\\int \\log{(E_{x})} dE_{x})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('E_x', commutative=True)), Integral(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('M')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Integral(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('M')(Symbol('E_x', commutative=True)), Add(Mul(Symbol('E_x', commutative=True), log(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Mul(Symbol('E_x', commutative=True), log(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('E_x', commutative=True)), Pow(Integral(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["divide", 4, "Pow(log(Symbol('E_x', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), log(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True))), Mul(log(Symbol('E_x', commutative=True)), Pow(Integral(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"], [["times", 5, "Function('M')(Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), log(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('E_x', commutative=True)), Function('M')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True))), Mul(Function('M')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)), Pow(Integral(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given V{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then obtain V^{2}{(\\mathbf{F})} \\sin^{2}{(\\mathbf{F})} - V{(\\mathbf{F})} \\sin{(\\mathbf{F})} = V^{3}{(\\mathbf{F})} \\sin{(\\mathbf{F})} - V{(\\mathbf{F})} \\sin{(\\mathbf{F})}", "derivation": "V{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and V{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\sin^{2}{(\\mathbf{F})} and V^{2}{(\\mathbf{F})} \\sin^{2}{(\\mathbf{F})} = \\sin^{4}{(\\mathbf{F})} and V^{3}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = V^{2}{(\\mathbf{F})} \\sin^{2}{(\\mathbf{F})} and V^{3}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\sin^{4}{(\\mathbf{F})} and V^{2}{(\\mathbf{F})} \\sin^{2}{(\\mathbf{F})} - V{(\\mathbf{F})} \\sin{(\\mathbf{F})} = - V{(\\mathbf{F})} \\sin{(\\mathbf{F})} + \\sin^{4}{(\\mathbf{F})} and V^{2}{(\\mathbf{F})} \\sin^{2}{(\\mathbf{F})} - V{(\\mathbf{F})} \\sin{(\\mathbf{F})} = V^{3}{(\\mathbf{F})} \\sin{(\\mathbf{F})} - V{(\\mathbf{F})} \\sin{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(4)))"], [["minus", 3, "Mul(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(4))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Pow(Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Function('V')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and \\hat{H}{(\\mathbf{f})} = \\sin{(\\mathbf{f})}, then obtain - 2 \\hat{H}{(\\mathbf{f})} + \\hat{H}^{\\mathbf{f}}{(\\mathbf{f})} = - 2 \\hat{H}{(\\mathbf{f})} + \\pi^{\\mathbf{f}}{(\\mathbf{f})}", "derivation": "\\pi{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and \\hat{H}{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and \\hat{H}{(\\mathbf{f})} = \\pi{(\\mathbf{f})} and \\hat{H}^{\\mathbf{f}}{(\\mathbf{f})} = \\pi^{\\mathbf{f}}{(\\mathbf{f})} and - 2 \\hat{H}{(\\mathbf{f})} + \\hat{H}^{\\mathbf{f}}{(\\mathbf{f})} = - 2 \\hat{H}{(\\mathbf{f})} + \\pi^{\\mathbf{f}}{(\\mathbf{f})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True)), Function('\\\\pi')(Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 4, "Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True))), Pow(Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}')(Symbol('\\\\mathbf{f}', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given B{(g)} = \\cos{(\\cos{(g)})}, then obtain \\int (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int B{(g)} dg)^{g} dg = \\int (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int \\cos{(\\cos{(g)})} dg)^{g} dg", "derivation": "B{(g)} = \\cos{(\\cos{(g)})} and \\int B{(g)} dg = \\int \\cos{(\\cos{(g)})} dg and e^{\\int B{(g)} dg} = e^{\\int \\cos{(\\cos{(g)})} dg} and e^{\\int B{(g)} dg} + \\int B{(g)} dg = e^{\\int B{(g)} dg} + \\int \\cos{(\\cos{(g)})} dg and e^{\\int \\cos{(\\cos{(g)})} dg} + \\int B{(g)} dg = e^{\\int \\cos{(\\cos{(g)})} dg} + \\int \\cos{(\\cos{(g)})} dg and (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int B{(g)} dg)^{g} = (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int \\cos{(\\cos{(g)})} dg)^{g} and \\int (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int B{(g)} dg)^{g} dg = \\int (e^{\\int \\cos{(\\cos{(g)})} dg} + \\int \\cos{(\\cos{(g)})} dg)^{g} dg", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('g', commutative=True)), cos(cos(Symbol('g', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["add", 2, "exp(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], "Equality(Add(exp(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(exp(Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["integrate", 6, "Symbol('g', commutative=True)"], "Equality(Integral(Pow(Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(Function('B')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(Add(exp(Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\delta)} = \\sin{(\\delta)} and p{(\\delta)} = \\int \\sin{(\\delta)} d\\delta, then obtain 0 = - \\sin{(\\delta)} \\int p{(\\delta)} d\\delta + \\sin{(\\delta)} \\iint \\hat{\\mathbf{x}}{(\\delta)} d\\delta d\\delta", "derivation": "\\hat{\\mathbf{x}}{(\\delta)} = \\sin{(\\delta)} and \\int \\hat{\\mathbf{x}}{(\\delta)} d\\delta = \\int \\sin{(\\delta)} d\\delta and p{(\\delta)} = \\int \\sin{(\\delta)} d\\delta and \\int \\hat{\\mathbf{x}}{(\\delta)} d\\delta = p{(\\delta)} and \\int p{(\\delta)} d\\delta = \\iint \\sin{(\\delta)} d\\delta d\\delta and \\iint \\hat{\\mathbf{x}}{(\\delta)} d\\delta d\\delta = \\iint \\sin{(\\delta)} d\\delta d\\delta and \\int p{(\\delta)} d\\delta = \\iint \\hat{\\mathbf{x}}{(\\delta)} d\\delta d\\delta and \\sin{(\\delta)} \\int p{(\\delta)} d\\delta = \\sin{(\\delta)} \\iint \\hat{\\mathbf{x}}{(\\delta)} d\\delta d\\delta and 0 = - \\sin{(\\delta)} \\int p{(\\delta)} d\\delta + \\sin{(\\delta)} \\iint \\hat{\\mathbf{x}}{(\\delta)} d\\delta d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\delta', commutative=True)), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Function('p')(Symbol('\\\\delta', commutative=True)))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 7, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\delta', commutative=True)), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(sin(Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["minus", 8, "Mul(sin(Symbol('\\\\delta', commutative=True)), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True)), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(sin(Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"]]}, {"prompt": "Given \\Psi{(A,L)} = A L, then obtain - A L + A + (A L - A)^{2} + (2 A L - A - \\Psi{(A,L)})^{2} + \\Psi{(A,L)} = - A L + A + 2 (2 A L - A - \\Psi{(A,L)})^{2} + \\Psi{(A,L)}", "derivation": "\\Psi{(A,L)} = A L and - A + \\Psi{(A,L)} = A L - A and - A = A L - A - \\Psi{(A,L)} and A L - A = 2 A L - A - \\Psi{(A,L)} and (A L - A)^{2} = (2 A L - A - \\Psi{(A,L)})^{2} and (A L - A)^{2} + (2 A L - A - \\Psi{(A,L)})^{2} = 2 (2 A L - A - \\Psi{(A,L)})^{2} and - A L + A + (A L - A)^{2} + (2 A L - A - \\Psi{(A,L)})^{2} + \\Psi{(A,L)} = - A L + A + 2 (2 A L - A - \\Psi{(A,L)})^{2} + \\Psi{(A,L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)))"], [["minus", 1, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))))"], [["minus", 2, "Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('A', commutative=True)), Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2)))"], [["add", 5, "Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2))"], "Equality(Add(Pow(Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2))), Mul(Integer(2), Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2))))"], [["minus", 6, "Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('A', commutative=True), Pow(Add(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2)), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('A', commutative=True), Mul(Integer(2), Pow(Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True)))), Integer(2))), Function('\\\\Psi')(Symbol('A', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{f})} = \\log{(\\sin{(\\mathbf{f})})}, then obtain 0 = \\frac{- \\operatorname{A_{1}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}}{\\mathbf{f} + \\log{(\\sin{(\\mathbf{f})})}}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{f})} = \\log{(\\sin{(\\mathbf{f})})} and \\mathbf{f} + \\operatorname{A_{1}}{(\\mathbf{f})} = \\mathbf{f} + \\log{(\\sin{(\\mathbf{f})})} and 0 = - \\operatorname{A_{1}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})} and 0 = \\frac{- \\operatorname{A_{1}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}}{\\mathbf{f} + \\operatorname{A_{1}}{(\\mathbf{f})}} and 0 = \\frac{- \\operatorname{A_{1}}{(\\mathbf{f})} + \\log{(\\sin{(\\mathbf{f})})}}{\\mathbf{f} + \\log{(\\sin{(\\mathbf{f})})}}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True)), log(sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), log(sin(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["minus", 1, "Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), log(sin(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), log(sin(Symbol('\\\\mathbf{f}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), log(sin(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), log(sin(Symbol('\\\\mathbf{f}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\omega,A)} = A \\sin{(\\omega)} and \\operatorname{f^{\\prime}}{(\\omega,A)} = \\frac{1}{\\operatorname{V_{\\mathbf{B}}}{(\\omega,A)}}, then obtain \\frac{A \\operatorname{V_{\\mathbf{B}}}{(\\omega,A)} \\operatorname{f^{\\prime}}^{2}{(\\omega,A)} \\sin{(\\omega)}}{\\operatorname{F_{g}}{(A_{y})}} = \\frac{A \\operatorname{f^{\\prime}}{(\\omega,A)} \\sin{(\\omega)}}{\\operatorname{F_{g}}{(A_{y})}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\omega,A)} = A \\sin{(\\omega)} and \\operatorname{f^{\\prime}}{(\\omega,A)} = \\frac{1}{\\operatorname{V_{\\mathbf{B}}}{(\\omega,A)}} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\omega,A)} \\operatorname{f^{\\prime}}{(\\omega,A)}}{\\operatorname{F_{g}}{(A_{y})}} = \\frac{1}{\\operatorname{F_{g}}{(A_{y})}} and \\frac{A \\operatorname{f^{\\prime}}{(\\omega,A)} \\sin{(\\omega)}}{\\operatorname{F_{g}}{(A_{y})}} = \\frac{1}{\\operatorname{F_{g}}{(A_{y})}} and \\frac{A \\operatorname{V_{\\mathbf{B}}}{(\\omega,A)} \\operatorname{f^{\\prime}}^{2}{(\\omega,A)} \\sin{(\\omega)}}{\\operatorname{F_{g}}{(A_{y})}} = \\frac{A \\operatorname{f^{\\prime}}{(\\omega,A)} \\sin{(\\omega)}}{\\operatorname{F_{g}}{(A_{y})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Integer(-1)))"], [["divide", 2, "Mul(Function('F_g')(Symbol('A_y', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Function('f^{\\\\prime}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True))), Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)), Function('f^{\\\\prime}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), sin(Symbol('\\\\omega', commutative=True))), Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('A', commutative=True), Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Integer(2)), sin(Symbol('\\\\omega', commutative=True))), Mul(Symbol('A', commutative=True), Pow(Function('F_g')(Symbol('A_y', commutative=True)), Integer(-1)), Function('f^{\\\\prime}')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), sin(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given s{(f_{\\mathbf{v}},T)} = T + f_{\\mathbf{v}}, then obtain f_{\\mathbf{v}} (T + s^{2}{(f_{\\mathbf{v}},T)})^{T} = f_{\\mathbf{v}} (T + (T + f_{\\mathbf{v}}) s{(f_{\\mathbf{v}},T)})^{T}", "derivation": "s{(f_{\\mathbf{v}},T)} = T + f_{\\mathbf{v}} and s^{2}{(f_{\\mathbf{v}},T)} = (T + f_{\\mathbf{v}}) s{(f_{\\mathbf{v}},T)} and T + s^{2}{(f_{\\mathbf{v}},T)} = T + (T + f_{\\mathbf{v}}) s{(f_{\\mathbf{v}},T)} and (T + s^{2}{(f_{\\mathbf{v}},T)})^{T} = (T + (T + f_{\\mathbf{v}}) s{(f_{\\mathbf{v}},T)})^{T} and f_{\\mathbf{v}} (T + s^{2}{(f_{\\mathbf{v}},T)})^{T} = f_{\\mathbf{v}} (T + (T + f_{\\mathbf{v}}) s{(f_{\\mathbf{v}},T)})^{T}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 1, "Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True))"], "Equality(Pow(Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)), Integer(2)), Mul(Add(Symbol('T', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True))))"], [["add", 2, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Pow(Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Add(Symbol('T', commutative=True), Mul(Add(Symbol('T', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)))))"], [["power", 3, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Symbol('T', commutative=True), Pow(Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Symbol('T', commutative=True)), Pow(Add(Symbol('T', commutative=True), Mul(Add(Symbol('T', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)))), Symbol('T', commutative=True)))"], [["times", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Add(Symbol('T', commutative=True), Pow(Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)), Integer(2))), Symbol('T', commutative=True))), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Add(Symbol('T', commutative=True), Mul(Add(Symbol('T', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('s')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\pi)} = e^{e^{\\pi}}, then obtain \\iiint (\\mathbf{A}^{\\pi}{(\\pi)} - e^{e^{\\pi}}) d\\pi d\\pi d\\pi = \\iiint (- e^{e^{\\pi}} + (e^{e^{\\pi}})^{\\pi}) d\\pi d\\pi d\\pi", "derivation": "\\mathbf{A}{(\\pi)} = e^{e^{\\pi}} and \\mathbf{A}^{\\pi}{(\\pi)} = (e^{e^{\\pi}})^{\\pi} and \\mathbf{A}^{\\pi}{(\\pi)} - e^{e^{\\pi}} = - e^{e^{\\pi}} + (e^{e^{\\pi}})^{\\pi} and \\int (\\mathbf{A}^{\\pi}{(\\pi)} - e^{e^{\\pi}}) d\\pi = \\int (- e^{e^{\\pi}} + (e^{e^{\\pi}})^{\\pi}) d\\pi and \\iint (\\mathbf{A}^{\\pi}{(\\pi)} - e^{e^{\\pi}}) d\\pi d\\pi = \\iint (- e^{e^{\\pi}} + (e^{e^{\\pi}})^{\\pi}) d\\pi d\\pi and \\iiint (\\mathbf{A}^{\\pi}{(\\pi)} - e^{e^{\\pi}}) d\\pi d\\pi d\\pi = \\iiint (- e^{e^{\\pi}} + (e^{e^{\\pi}})^{\\pi}) d\\pi d\\pi d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), exp(exp(Symbol('\\\\pi', commutative=True))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(exp(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "exp(exp(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True)))), Pow(exp(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True)))), Pow(exp(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True)))), Pow(exp(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), exp(exp(Symbol('\\\\pi', commutative=True)))), Pow(exp(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given p{(\\sigma_x)} = \\sin{(\\sigma_x)} and \\varepsilon_{0}{(\\sigma_x)} = (\\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}})^{\\sigma_x}, then obtain (\\varepsilon_{0}^{\\sigma_x}{(\\sigma_x)})^{\\sigma_x} = 1", "derivation": "p{(\\sigma_x)} = \\sin{(\\sigma_x)} and p^{2}{(\\sigma_x)} = p{(\\sigma_x)} \\sin{(\\sigma_x)} and \\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}} = 1 and (\\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}})^{\\sigma_x} = 1 and ((\\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}})^{\\sigma_x})^{\\sigma_x} = 1 and \\varepsilon_{0}{(\\sigma_x)} = (\\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}})^{\\sigma_x} and (((\\frac{p{(\\sigma_x)}}{\\sin{(\\sigma_x)}})^{\\sigma_x})^{\\sigma_x})^{\\sigma_x} = 1 and (\\varepsilon_{0}^{\\sigma_x}{(\\sigma_x)})^{\\sigma_x} = 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Function('p')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 2, "Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Symbol('\\\\sigma_x', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Pow(Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Function('p')(Symbol('\\\\sigma_x', commutative=True)), Pow(sin(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Integer(1))"]]}, {"prompt": "Given k{(t)} = \\sin{(t)}, then obtain \\int \\frac{1 - \\int 0 dt}{- \\rho + c_{0}} d\\rho = \\int \\frac{1 - \\int (- k{(t)} + \\sin{(t)}) dt}{- \\rho + c_{0}} d\\rho", "derivation": "k{(t)} = \\sin{(t)} and 0 = - k{(t)} + \\sin{(t)} and \\int 0 dt = \\int (- k{(t)} + \\sin{(t)}) dt and - \\int 0 dt = - \\int (- k{(t)} + \\sin{(t)}) dt and 1 - \\int 0 dt = 1 - \\int (- k{(t)} + \\sin{(t)}) dt and \\frac{1 - \\int 0 dt}{- \\rho + c_{0}} = \\frac{1 - \\int (- k{(t)} + \\sin{(t)}) dt}{- \\rho + c_{0}} and \\int \\frac{1 - \\int 0 dt}{- \\rho + c_{0}} d\\rho = \\int \\frac{1 - \\int (- k{(t)} + \\sin{(t)}) dt}{- \\rho + c_{0}} d\\rho", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["minus", 1, "Function('k')(Symbol('t', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('t', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('t', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('c_0', commutative=True))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('t', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('c_0', commutative=True)), Integer(-1))), Mul(Add(Integer(1), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('c_0', commutative=True)), Integer(-1))))"], [["integrate", 6, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Mul(Add(Integer(1), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('t', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('c_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Add(Integer(1), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Function('k')(Symbol('t', commutative=True))), sin(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('c_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given r{(A_{x},u)} = - A_{x} + u and \\Psi{(A_{x},u)} = r{(A_{x},u)} + 1, then obtain \\int \\log{(\\Psi{(A_{x},u)} + r{(A_{x},u)} + 1)} du = \\int \\log{(2 r{(A_{x},u)} + 2)} du", "derivation": "r{(A_{x},u)} = - A_{x} + u and \\Psi{(A_{x},u)} = r{(A_{x},u)} + 1 and - A_{x} + u + \\Psi{(A_{x},u)} + 1 = - A_{x} + u + r{(A_{x},u)} + 2 and \\Psi{(A_{x},u)} + r{(A_{x},u)} + 1 = 2 r{(A_{x},u)} + 2 and \\log{(\\Psi{(A_{x},u)} + r{(A_{x},u)} + 1)} = \\log{(2 r{(A_{x},u)} + 2)} and \\int \\log{(\\Psi{(A_{x},u)} + r{(A_{x},u)} + 1)} du = \\int \\log{(2 r{(A_{x},u)} + 2)} du", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Add(Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(1)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('u', commutative=True), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('u', commutative=True), Function('\\\\Psi')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('u', commutative=True), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\Psi')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(1)), Add(Mul(Integer(2), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Integer(2)))"], [["log", 4], "Equality(log(Add(Function('\\\\Psi')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(1))), log(Add(Mul(Integer(2), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Integer(2))))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(log(Add(Function('\\\\Psi')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(log(Add(Mul(Integer(2), Function('r')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Integer(2))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\sigma_x)} = e^{\\sigma_x} and \\phi{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\mathbf{M}{(\\sigma_x)}, then derive \\frac{d}{d \\sigma_x} \\mathbf{M}{(\\sigma_x)} = e^{\\sigma_x}, then obtain \\phi{(\\sigma_x)} = \\frac{d^{2}}{d \\sigma_x^{2}} \\mathbf{M}{(\\sigma_x)}", "derivation": "\\mathbf{M}{(\\sigma_x)} = e^{\\sigma_x} and \\frac{d}{d \\sigma_x} \\mathbf{M}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} e^{\\sigma_x} and \\phi{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\mathbf{M}{(\\sigma_x)} and \\phi{(\\sigma_x)} = \\frac{d}{d \\sigma_x} e^{\\sigma_x} and \\frac{d}{d \\sigma_x} \\mathbf{M}{(\\sigma_x)} = e^{\\sigma_x} and \\phi{(\\sigma_x)} = \\frac{d^{2}}{d \\sigma_x^{2}} \\mathbf{M}{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Derivative(exp(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\phi')(Symbol('\\\\sigma_x', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}^*{(r)} = \\sin{(r)}, then obtain \\frac{d}{d r} (2 (2 \\tilde{g}^*{(r)} - \\sin{(r)})^{2} + 2 \\tilde{g}^*{(r)} - \\sin{(r)}) = \\frac{d}{d r} (2 (2 \\tilde{g}^*{(r)} - \\sin{(r)})^{2} + \\sin{(r)})", "derivation": "\\tilde{g}^*{(r)} = \\sin{(r)} and 2 \\tilde{g}^*{(r)} = \\tilde{g}^*{(r)} + \\sin{(r)} and 2 \\tilde{g}^*{(r)} - \\sin{(r)} = \\tilde{g}^*{(r)} and 2 \\tilde{g}^*^{2}{(r)} + \\tilde{g}^*{(r)} = 2 \\tilde{g}^*^{2}{(r)} + \\sin{(r)} and \\frac{d}{d r} (2 \\tilde{g}^*^{2}{(r)} + \\tilde{g}^*{(r)}) = \\frac{d}{d r} (2 \\tilde{g}^*^{2}{(r)} + \\sin{(r)}) and \\frac{d}{d r} (2 (2 \\tilde{g}^*{(r)} - \\sin{(r)})^{2} + 2 \\tilde{g}^*{(r)} - \\sin{(r)}) = \\frac{d}{d r} (2 (2 \\tilde{g}^*{(r)} - \\sin{(r)})^{2} + \\sin{(r)})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))))"], [["minus", 2, "sin(Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True)))), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)))"], [["add", 1, "Mul(Integer(2), Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Integer(2))), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Add(Mul(Integer(2), Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Integer(2))), sin(Symbol('r', commutative=True))))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Integer(2))), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Pow(Function('\\\\tilde{g}^*')(Symbol('r', commutative=True)), Integer(2))), sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(2), Pow(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True)))), Integer(2))), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Pow(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('r', commutative=True))), Mul(Integer(-1), sin(Symbol('r', commutative=True)))), Integer(2))), sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{A})} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A}, then obtain \\frac{\\int \\operatorname{A_{1}}{(\\mathbf{A})} d\\mathbf{A}}{\\operatorname{A_{1}}{(\\mathbf{A})}} = \\frac{\\iint \\sin{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A}}{\\operatorname{A_{1}}{(\\mathbf{A})}}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{A})} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A} and \\int \\operatorname{A_{1}}{(\\mathbf{A})} d\\mathbf{A} = \\iint \\sin{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A} and \\frac{\\int \\operatorname{A_{1}}{(\\mathbf{A})} d\\mathbf{A}}{\\int \\sin{(\\mathbf{A})} d\\mathbf{A}} = \\frac{\\iint \\sin{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A}}{\\int \\sin{(\\mathbf{A})} d\\mathbf{A}} and \\frac{\\int \\operatorname{A_{1}}{(\\mathbf{A})} d\\mathbf{A}}{\\operatorname{A_{1}}{(\\mathbf{A})}} = \\frac{\\iint \\sin{(\\mathbf{A})} d\\mathbf{A} d\\mathbf{A}}{\\operatorname{A_{1}}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Integral(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Pow(Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Mul(Pow(Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Integral(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Pow(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(F_{g},\\hat{x})} = F_{g} - \\hat{x}, then obtain (\\frac{\\partial}{\\partial \\hat{x}} \\frac{- \\hat{x} + \\operatorname{A_{y}}{(F_{g},\\hat{x})}}{F_{g}})^{F_{g}} = (\\frac{\\partial}{\\partial \\hat{x}} \\frac{F_{g} - 2 \\hat{x}}{F_{g}})^{F_{g}}", "derivation": "\\operatorname{A_{y}}{(F_{g},\\hat{x})} = F_{g} - \\hat{x} and - \\hat{x} + \\operatorname{A_{y}}{(F_{g},\\hat{x})} = F_{g} - 2 \\hat{x} and \\frac{- \\hat{x} + \\operatorname{A_{y}}{(F_{g},\\hat{x})}}{F_{g}} = \\frac{F_{g} - 2 \\hat{x}}{F_{g}} and \\frac{\\partial}{\\partial \\hat{x}} \\frac{- \\hat{x} + \\operatorname{A_{y}}{(F_{g},\\hat{x})}}{F_{g}} = \\frac{\\partial}{\\partial \\hat{x}} \\frac{F_{g} - 2 \\hat{x}}{F_{g}} and (\\frac{\\partial}{\\partial \\hat{x}} \\frac{- \\hat{x} + \\operatorname{A_{y}}{(F_{g},\\hat{x})}}{F_{g}})^{F_{g}} = (\\frac{\\partial}{\\partial \\hat{x}} \\frac{F_{g} - 2 \\hat{x}}{F_{g}})^{F_{g}}", "srepr_derivation": [["get_premise", "Equality(Function('A_y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('A_y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('A_y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('A_y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('F_g', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('A_y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\omega)} = \\omega, then obtain - \\omega = - \\mathbf{M}{(\\omega)}", "derivation": "\\mathbf{M}{(\\omega)} = \\omega and 0 = \\omega - \\mathbf{M}{(\\omega)} and - \\omega + \\mathbf{M}{(\\omega)} = 0 and - \\omega = - \\mathbf{M}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["minus", 1, "Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)))))"], [["minus", 2, "Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))), Integer(0))"], [["add", 3, "Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given y{(\\mathbf{s})} = e^{\\mathbf{s}}, then derive \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} = - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})}", "derivation": "y{(\\mathbf{s})} = e^{\\mathbf{s}} and \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} and \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})} = e^{\\mathbf{s}} and e^{\\mathbf{s}} = \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} and - \\mathbf{s} + e^{\\mathbf{s}} = - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} and - \\mathbf{s} + e^{\\mathbf{s}} = - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})} and - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} e^{\\mathbf{s}} = - \\mathbf{s} + \\frac{d}{d \\mathbf{s}} y{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('y')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\omega)} = \\cos{(\\omega)}, then derive \\int (\\omega + \\varphi^{*}{(\\omega)}) d\\omega = \\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)}, then obtain \\int \\frac{\\partial}{\\partial \\omega} (\\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)}) d\\mathbf{P} = \\int \\frac{d}{d \\omega} \\int (\\omega + \\cos{(\\omega)}) d\\omega d\\mathbf{P}", "derivation": "\\varphi^{*}{(\\omega)} = \\cos{(\\omega)} and \\omega + \\varphi^{*}{(\\omega)} = \\omega + \\cos{(\\omega)} and \\int (\\omega + \\varphi^{*}{(\\omega)}) d\\omega = \\int (\\omega + \\cos{(\\omega)}) d\\omega and \\int (\\omega + \\varphi^{*}{(\\omega)}) d\\omega = \\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)} and \\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)} = \\int (\\omega + \\cos{(\\omega)}) d\\omega and \\frac{\\partial}{\\partial \\omega} (\\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)}) = \\frac{d}{d \\omega} \\int (\\omega + \\cos{(\\omega)}) d\\omega and \\int \\frac{\\partial}{\\partial \\omega} (\\mathbf{P} + \\frac{\\omega^{2}}{2} + \\sin{(\\omega)}) d\\mathbf{P} = \\int \\frac{d}{d \\omega} \\int (\\omega + \\cos{(\\omega)}) d\\omega d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\omega', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\omega', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), sin(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), sin(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(Integral(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} = e^{g_{\\varepsilon} + n} and \\operatorname{t_{1}}{(g_{\\varepsilon},n)} = \\log{(n^{g_{\\varepsilon}})}, then obtain \\log{((n + \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} - e^{g_{\\varepsilon} + n})^{g_{\\varepsilon}})} = \\operatorname{t_{1}}{(g_{\\varepsilon},n)}", "derivation": "\\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} = e^{g_{\\varepsilon} + n} and n + \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} - e^{g_{\\varepsilon} + n} = n and (n + \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} - e^{g_{\\varepsilon} + n})^{g_{\\varepsilon}} = n^{g_{\\varepsilon}} and \\log{((n + \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} - e^{g_{\\varepsilon} + n})^{g_{\\varepsilon}})} = \\log{(n^{g_{\\varepsilon}})} and \\operatorname{t_{1}}{(g_{\\varepsilon},n)} = \\log{(n^{g_{\\varepsilon}})} and \\log{((n + \\operatorname{x^{{\\}'}}{(g_{\\varepsilon},n)} - e^{g_{\\varepsilon} + n})^{g_{\\varepsilon}})} = \\operatorname{t_{1}}{(g_{\\varepsilon},n)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('n', commutative=True)), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))"], "Equality(Add(Symbol('n', commutative=True), Function('x^\\\\prime')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))), Symbol('n', commutative=True))"], [["power", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Symbol('n', commutative=True), Function('x^\\\\prime')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["log", 3], "Equality(log(Pow(Add(Symbol('n', commutative=True), Function('x^\\\\prime')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))), Symbol('g_{\\\\varepsilon}', commutative=True))), log(Pow(Symbol('n', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), log(Pow(Symbol('n', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Pow(Add(Symbol('n', commutative=True), Function('x^\\\\prime')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True))))), Symbol('g_{\\\\varepsilon}', commutative=True))), Function('t_1')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('n', commutative=True)))"]]}, {"prompt": "Given u{(V,A_{x})} = \\frac{V}{A_{x}}, then derive \\frac{\\partial}{\\partial A_{x}} u{(V,A_{x})} = - \\frac{V}{A_{x}^{2}}, then obtain - \\frac{V}{A_{x}^{2}} = \\frac{\\partial}{\\partial A_{x}} \\frac{V}{A_{x}}", "derivation": "u{(V,A_{x})} = \\frac{V}{A_{x}} and \\frac{\\partial}{\\partial A_{x}} u{(V,A_{x})} = \\frac{\\partial}{\\partial A_{x}} \\frac{V}{A_{x}} and \\frac{\\partial}{\\partial A_{x}} u{(V,A_{x})} = - \\frac{V}{A_{x}^{2}} and - \\frac{V}{A_{x}^{2}} = \\frac{\\partial}{\\partial A_{x}} \\frac{V}{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('V', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('V', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('V', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('V', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-2)), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('A_x', commutative=True), Integer(-2)), Symbol('V', commutative=True)), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('V', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(f^{*},\\Psi)} = e^{\\Psi f^{*}}, then derive \\frac{\\partial}{\\partial \\Psi} \\operatorname{M_{E}}{(f^{*},\\Psi)} - 1 = f^{*} e^{\\Psi f^{*}} - 1, then obtain \\frac{\\partial}{\\partial \\Psi} e^{\\Psi f^{*}} - 1 = f^{*} e^{\\Psi f^{*}} - 1", "derivation": "\\operatorname{M_{E}}{(f^{*},\\Psi)} = e^{\\Psi f^{*}} and - \\Psi + \\operatorname{M_{E}}{(f^{*},\\Psi)} = - \\Psi + e^{\\Psi f^{*}} and \\frac{\\partial}{\\partial \\Psi} (- \\Psi + \\operatorname{M_{E}}{(f^{*},\\Psi)}) = \\frac{\\partial}{\\partial \\Psi} (- \\Psi + e^{\\Psi f^{*}}) and \\frac{\\partial}{\\partial \\Psi} \\operatorname{M_{E}}{(f^{*},\\Psi)} - 1 = f^{*} e^{\\Psi f^{*}} - 1 and \\frac{\\partial}{\\partial \\Psi} e^{\\Psi f^{*}} - 1 = f^{*} e^{\\Psi f^{*}} - 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('f^*', commutative=True), Symbol('\\\\Psi', commutative=True)), exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True))))"], [["minus", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('M_E')(Symbol('f^*', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('M_E')(Symbol('f^*', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('M_E')(Symbol('f^*', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('f^*', commutative=True), exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('f^*', commutative=True), exp(Mul(Symbol('\\\\Psi', commutative=True), Symbol('f^*', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given i{(\\Psi_{nl},L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi_{nl}}, then obtain 2 \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi_{nl}} \\log{(L_{\\varepsilon})} + \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})}", "derivation": "i{(\\Psi_{nl},L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi_{nl}} and \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})} = \\frac{\\partial}{\\partial \\Psi_{nl}} L_{\\varepsilon}^{\\Psi_{nl}} and 2 \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})} = \\frac{\\partial}{\\partial \\Psi_{nl}} L_{\\varepsilon}^{\\Psi_{nl}} + \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})} and 2 \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})} = L_{\\varepsilon}^{\\Psi_{nl}} \\log{(L_{\\varepsilon})} + \\frac{\\partial}{\\partial \\Psi_{nl}} i{(\\Psi_{nl},L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Add(Derivative(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Symbol('L_{\\\\varepsilon}', commutative=True))), Derivative(Function('i')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{2})} = C_{2}, then obtain (E + \\int \\operatorname{E_{n}}{(C_{2})} dC_{2}) (\\int C_{2} dC_{2})^{C_{2}} = (E + \\int C_{2} dC_{2}) (\\int C_{2} dC_{2})^{C_{2}}", "derivation": "\\operatorname{E_{n}}{(C_{2})} = C_{2} and \\int \\operatorname{E_{n}}{(C_{2})} dC_{2} = \\int C_{2} dC_{2} and (\\int \\operatorname{E_{n}}{(C_{2})} dC_{2})^{C_{2}} = (\\int C_{2} dC_{2})^{C_{2}} and E + \\int \\operatorname{E_{n}}{(C_{2})} dC_{2} = E + \\int C_{2} dC_{2} and (E + \\int \\operatorname{E_{n}}{(C_{2})} dC_{2}) (\\int \\operatorname{E_{n}}{(C_{2})} dC_{2})^{C_{2}} = (E + \\int C_{2} dC_{2}) (\\int \\operatorname{E_{n}}{(C_{2})} dC_{2})^{C_{2}} and (E + \\int \\operatorname{E_{n}}{(C_{2})} dC_{2}) (\\int C_{2} dC_{2})^{C_{2}} = (E + \\int C_{2} dC_{2}) (\\int C_{2} dC_{2})^{C_{2}}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True))))"], [["power", 2, "Symbol('C_2', commutative=True)"], "Equality(Pow(Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)), Pow(Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True)))"], [["add", 2, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('E', commutative=True), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)))))"], [["times", 4, "Pow(Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))"], "Equality(Mul(Add(Symbol('E', commutative=True), Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Pow(Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))), Mul(Add(Symbol('E', commutative=True), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)))), Pow(Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Symbol('E', commutative=True), Integral(Function('E_n')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Pow(Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))), Mul(Add(Symbol('E', commutative=True), Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True)))), Pow(Integral(Symbol('C_2', commutative=True), Tuple(Symbol('C_2', commutative=True))), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} = \\mathbf{F}^{\\mathbb{I}}, then obtain \\cos^{\\mathbb{I}}{(\\int \\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} d\\mathbb{I})} = \\cos^{\\mathbb{I}}{(\\int \\mathbf{F}^{\\mathbb{I}} d\\mathbb{I})}", "derivation": "\\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} = \\mathbf{F}^{\\mathbb{I}} and \\int \\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} d\\mathbb{I} = \\int \\mathbf{F}^{\\mathbb{I}} d\\mathbb{I} and \\cos{(\\int \\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} d\\mathbb{I})} = \\cos{(\\int \\mathbf{F}^{\\mathbb{I}} d\\mathbb{I})} and \\cos^{\\mathbb{I}}{(\\int \\operatorname{v_{z}}{(\\mathbb{I},\\mathbf{F})} d\\mathbb{I})} = \\cos^{\\mathbb{I}}{(\\int \\mathbf{F}^{\\mathbb{I}} d\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('v_z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), cos(Integral(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(cos(Integral(Function('v_z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(cos(Integral(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(E_{x})} = \\log{(E_{x})}, then obtain E_{x} \\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})} = 1", "derivation": "\\mathbf{F}{(E_{x})} = \\log{(E_{x})} and \\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and E_{x} \\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})} = E_{x} \\frac{d}{d E_{x}} \\log{(E_{x})} and E_{x} \\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["times", 2, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Derivative(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Symbol('E_x', commutative=True), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('E_x', commutative=True), Derivative(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\phi_{1}{(i,v)} = i - v, then derive v + \\int \\phi_{1}{(i,v)} di = g + \\frac{i^{2}}{2} - i v + v, then obtain \\frac{2 (v + \\int (i - v) di)}{i^{2}} = \\frac{2 (g + \\frac{i^{2}}{2} - i v + v)}{i^{2}}", "derivation": "\\phi_{1}{(i,v)} = i - v and \\int \\phi_{1}{(i,v)} di = \\int (i - v) di and v + \\int \\phi_{1}{(i,v)} di = v + \\int (i - v) di and v + \\int \\phi_{1}{(i,v)} di = g + \\frac{i^{2}}{2} - i v + v and v + \\int (i - v) di = g + \\frac{i^{2}}{2} - i v + v and \\frac{2 (v + \\int (i - v) di)}{i^{2}} = \\frac{2 (g + \\frac{i^{2}}{2} - i v + v)}{i^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Integral(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Symbol('v', commutative=True), Integral(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('i', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v', commutative=True), Integral(Function('\\\\phi_1')(Symbol('i', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Symbol('g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('i', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('v', commutative=True), Integral(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('i', commutative=True)))), Add(Symbol('g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('i', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["divide", 5, "Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('i', commutative=True), Integer(-2)), Add(Symbol('v', commutative=True), Integral(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Tuple(Symbol('i', commutative=True))))), Mul(Integer(2), Pow(Symbol('i', commutative=True), Integer(-2)), Add(Symbol('g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('i', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\pi{(l)} = \\log{(l)}, then derive e^{\\frac{d}{d l} \\pi{(l)}} = e^{\\frac{1}{l}}, then obtain e^{- \\frac{d}{d l} \\pi{(l)}} e^{\\frac{d}{d l} \\log{(l)}} = e^{\\frac{1}{l}} e^{- \\frac{d}{d l} \\pi{(l)}}", "derivation": "\\pi{(l)} = \\log{(l)} and \\frac{d}{d l} \\pi{(l)} = \\frac{d}{d l} \\log{(l)} and e^{\\frac{d}{d l} \\pi{(l)}} = e^{\\frac{d}{d l} \\log{(l)}} and e^{\\frac{d}{d l} \\pi{(l)}} = e^{\\frac{1}{l}} and e^{\\frac{d}{d l} \\log{(l)}} = e^{\\frac{1}{l}} and e^{- \\frac{d}{d l} \\pi{(l)}} e^{\\frac{d}{d l} \\log{(l)}} = e^{\\frac{1}{l}} e^{- \\frac{d}{d l} \\pi{(l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), exp(Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(exp(Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), exp(Pow(Symbol('l', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), exp(Pow(Symbol('l', commutative=True), Integer(-1))))"], [["divide", 5, "exp(Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], "Equality(Mul(exp(Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), exp(Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Mul(exp(Pow(Symbol('l', commutative=True), Integer(-1))), exp(Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\mathbf{F}{(J)} = \\cos{(e^{J})}, then obtain 3 \\mathbf{F}{(J)} = \\mathbf{F}{(J)} + 2 \\cos{(e^{J})}", "derivation": "\\mathbf{F}{(J)} = \\cos{(e^{J})} and 2 \\mathbf{F}{(J)} = \\mathbf{F}{(J)} + \\cos{(e^{J})} and 3 \\mathbf{F}{(J)} = 2 \\mathbf{F}{(J)} + \\cos{(e^{J})} and 3 \\mathbf{F}{(J)} = \\mathbf{F}{(J)} + 2 \\cos{(e^{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('J', commutative=True)), cos(exp(Symbol('J', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{F}')(Symbol('J', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('J', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('J', commutative=True)), cos(exp(Symbol('J', commutative=True)))))"], [["add", 2, "Function('\\\\mathbf{F}')(Symbol('J', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\mathbf{F}')(Symbol('J', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('J', commutative=True))), cos(exp(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\mathbf{F}')(Symbol('J', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('J', commutative=True)), Mul(Integer(2), cos(exp(Symbol('J', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\pi)} = \\sin{(\\pi)}, then obtain \\pi \\cos{(\\frac{d}{d \\pi} \\dot{\\mathbf{r}}{(\\pi)})} = \\pi \\cos{(\\frac{d}{d \\pi} \\sin{(\\pi)})}", "derivation": "\\dot{\\mathbf{r}}{(\\pi)} = \\sin{(\\pi)} and \\frac{d}{d \\pi} \\dot{\\mathbf{r}}{(\\pi)} = \\frac{d}{d \\pi} \\sin{(\\pi)} and \\cos{(\\frac{d}{d \\pi} \\dot{\\mathbf{r}}{(\\pi)})} = \\cos{(\\frac{d}{d \\pi} \\sin{(\\pi)})} and \\pi \\cos{(\\frac{d}{d \\pi} \\dot{\\mathbf{r}}{(\\pi)})} = \\pi \\cos{(\\frac{d}{d \\pi} \\sin{(\\pi)})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), cos(Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), cos(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Mul(Symbol('\\\\pi', commutative=True), cos(Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Psi{(r)} = \\frac{d}{d r} e^{r} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\log{(\\hat{x})}, then derive \\Psi{(r)} = e^{r}, then obtain e^{r} \\log{(\\hat{x})} = \\log{(\\hat{x})} \\frac{d}{d r} e^{r}", "derivation": "\\Psi{(r)} = \\frac{d}{d r} e^{r} and \\Psi{(r)} = e^{r} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\log{(\\hat{x})} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} \\Psi{(r)} = \\operatorname{J_{\\varepsilon}}{(\\hat{x})} \\frac{d}{d r} e^{r} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} e^{r} = \\operatorname{J_{\\varepsilon}}{(\\hat{x})} \\frac{d}{d r} e^{r} and e^{r} \\log{(\\hat{x})} = \\log{(\\hat{x})} \\frac{d}{d r} e^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('r', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Psi')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], ["get_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], [["times", 1, "Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Function('\\\\Psi')(Symbol('r', commutative=True))), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('r', commutative=True))), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(exp(Symbol('r', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True))), Mul(log(Symbol('\\\\hat{x}', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(x,\\mathbf{f})} = \\mathbf{f} - x and \\Psi{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}}, then derive \\int \\delta{(x,\\mathbf{f})} dx = \\dot{\\mathbf{r}} + \\mathbf{f} x - \\frac{x^{2}}{2}, then obtain (\\mathbf{f} x + \\Psi{(\\dot{\\mathbf{r}})}) \\int \\delta{(x,\\mathbf{f})} dx = (\\mathbf{f} x + \\Psi{(\\dot{\\mathbf{r}})}) (\\mathbf{f} x - \\frac{x^{2}}{2} + \\Psi{(\\dot{\\mathbf{r}})})", "derivation": "\\delta{(x,\\mathbf{f})} = \\mathbf{f} - x and \\int \\delta{(x,\\mathbf{f})} dx = \\int (\\mathbf{f} - x) dx and \\int \\delta{(x,\\mathbf{f})} dx = \\dot{\\mathbf{r}} + \\mathbf{f} x - \\frac{x^{2}}{2} and \\Psi{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} and \\mathbf{f} x + \\Psi{(\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} + \\mathbf{f} x and \\int \\delta{(x,\\mathbf{f})} dx = \\mathbf{f} x - \\frac{x^{2}}{2} + \\Psi{(\\dot{\\mathbf{r}})} and (\\mathbf{f} x + \\Psi{(\\dot{\\mathbf{r}})}) \\int \\delta{(x,\\mathbf{f})} dx = (\\mathbf{f} x + \\Psi{(\\dot{\\mathbf{r}})}) (\\mathbf{f} x - \\frac{x^{2}}{2} + \\Psi{(\\dot{\\mathbf{r}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["add", 4, "Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('\\\\delta')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2))), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["times", 6, "Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Function('\\\\delta')(Symbol('x', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2))), Function('\\\\Psi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\psi{(x^\\prime)} = e^{x^\\prime}, then derive \\int \\psi{(x^\\prime)} dx^\\prime = h + e^{x^\\prime}, then obtain \\int e^{x^\\prime} dx^\\prime = h + e^{x^\\prime}", "derivation": "\\psi{(x^\\prime)} = e^{x^\\prime} and \\int \\psi{(x^\\prime)} dx^\\prime = \\int e^{x^\\prime} dx^\\prime and \\int \\psi{(x^\\prime)} dx^\\prime = h + e^{x^\\prime} and \\int \\psi{(x^\\prime)} dx^\\prime = h + \\psi{(x^\\prime)} and \\int e^{x^\\prime} dx^\\prime = h + e^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('h', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\psi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('h', commutative=True), Function('\\\\psi')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('h', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given U{(n_{1})} = \\cos{(n_{1})} and \\eta{(n_{1})} = ((U{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}}, then obtain \\eta{(n_{1})} = ((\\cos{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}}", "derivation": "U{(n_{1})} = \\cos{(n_{1})} and U{(n_{1})} \\cos^{- n_{1}}{(n_{1})} = \\cos{(n_{1})} \\cos^{- n_{1}}{(n_{1})} and (U{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}} = (\\cos{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}} and ((U{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}} = ((\\cos{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}} and \\eta{(n_{1})} = ((U{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}} and \\eta{(n_{1})} = ((\\cos{(n_{1})} \\cos^{- n_{1}}{(n_{1})})^{n_{1}})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["divide", 1, "Pow(cos(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], "Equality(Mul(Function('U')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Mul(cos(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Mul(Function('U')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Pow(Mul(cos(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Pow(Mul(Function('U')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Pow(Mul(cos(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('n_1', commutative=True)), Pow(Pow(Mul(Function('U')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\eta')(Symbol('n_1', commutative=True)), Pow(Pow(Mul(cos(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(E_{x})} = e^{E_{x}}, then obtain \\delta + \\mathbf{s}{(E_{x})} = f^{*} + e^{E_{x}}", "derivation": "\\mathbf{s}{(E_{x})} = e^{E_{x}} and \\frac{d}{d E_{x}} \\mathbf{s}{(E_{x})} = \\frac{d}{d E_{x}} e^{E_{x}} and \\int \\frac{d}{d E_{x}} \\mathbf{s}{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} e^{E_{x}} dE_{x} and \\delta + \\mathbf{s}{(E_{x})} = f^{*} + e^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{s}')(Symbol('E_x', commutative=True))), Add(Symbol('f^*', commutative=True), exp(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mu,p)} = \\frac{p}{\\mu}, then obtain \\int (- \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(\\mu,p)}) \\mathbf{B}{(\\mu,p)} dp = \\int (- \\mu + \\frac{\\partial}{\\partial \\mu} \\frac{p}{\\mu}) \\mathbf{B}{(\\mu,p)} dp", "derivation": "\\mathbf{B}{(\\mu,p)} = \\frac{p}{\\mu} and \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(\\mu,p)} = \\frac{\\partial}{\\partial \\mu} \\frac{p}{\\mu} and - \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(\\mu,p)} = - \\mu + \\frac{\\partial}{\\partial \\mu} \\frac{p}{\\mu} and (- \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(\\mu,p)}) \\mathbf{B}{(\\mu,p)} = (- \\mu + \\frac{\\partial}{\\partial \\mu} \\frac{p}{\\mu}) \\mathbf{B}{(\\mu,p)} and \\int (- \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(\\mu,p)}) \\mathbf{B}{(\\mu,p)} dp = \\int (- \\mu + \\frac{\\partial}{\\partial \\mu} \\frac{p}{\\mu}) \\mathbf{B}{(\\mu,p)} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["times", 3, "Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(a)} = \\log{(a)}, then obtain \\frac{1}{\\tilde{g}{(a)} - \\log{(a)}} = \\frac{(\\tilde{g}{(a)} - \\log{(a)})^{a}}{\\tilde{g}{(a)} - \\log{(a)}}", "derivation": "\\tilde{g}{(a)} = \\log{(a)} and 2 \\tilde{g}{(a)} = \\tilde{g}{(a)} + \\log{(a)} and \\tilde{g}{(a)} - \\log{(a)} = 0 and (\\tilde{g}{(a)} - \\log{(a)})^{a} = 0^{a} and \\frac{(\\tilde{g}{(a)} - \\log{(a)})^{a}}{\\tilde{g}{(a)} - \\log{(a)}} = \\frac{0^{a}}{\\tilde{g}{(a)} - \\log{(a)}} and \\frac{1}{\\tilde{g}{(a)} - \\log{(a)}} = \\frac{(\\tilde{g}{(a)} - \\log{(a)})^{a}}{\\tilde{g}{(a)} - \\log{(a)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}')(Symbol('a', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('a', commutative=True))), Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True))))"], [["minus", 2, "Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Symbol('a', commutative=True)), Pow(Integer(0), Symbol('a', commutative=True)))"], [["divide", 4, "Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Symbol('a', commutative=True))), Mul(Pow(Integer(0), Symbol('a', commutative=True)), Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Mul(Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Integer(-1)), Pow(Add(Function('\\\\tilde{g}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(T)} = \\int \\log{(T)} dT, then derive - T \\log{(T)} + T - \\mathbf{J}_M + \\operatorname{C_{2}}{(T)} = 0, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\Psi - \\mathbf{J}_M) = \\frac{d}{d \\mathbf{J}_M} 0, then obtain -1 = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- T \\log{(T)} + T - \\mathbf{J}_M + \\int \\log{(T)} dT)", "derivation": "\\operatorname{C_{2}}{(T)} = \\int \\log{(T)} dT and \\operatorname{C_{2}}{(T)} - \\int \\log{(T)} dT = 0 and - T \\log{(T)} + T - \\mathbf{J}_M + \\operatorname{C_{2}}{(T)} = 0 and - T \\log{(T)} + T - \\mathbf{J}_M + \\int \\log{(T)} dT = 0 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (- T \\log{(T)} + T - \\mathbf{J}_M + \\int \\log{(T)} dT) = \\frac{d}{d \\mathbf{J}_M} 0 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\Psi - \\mathbf{J}_M) = \\frac{d}{d \\mathbf{J}_M} 0 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\Psi - \\mathbf{J}_M) = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- T \\log{(T)} + T - \\mathbf{J}_M + \\int \\log{(T)} dT) and -1 = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- T \\log{(T)} + T - \\mathbf{J}_M + \\int \\log{(T)} dT)", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('T', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["minus", 1, "Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Function('C_2')(Symbol('T', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('C_2')(Symbol('T', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True), log(Symbol('T', commutative=True))), Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(C_{d})} = \\frac{1}{C_{d}}, then obtain C_{d} + \\frac{(C_{d} + \\operatorname{F_{H}}{(C_{d})} + \\frac{1}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} + \\frac{1}{C_{d}} = C_{d} + \\frac{(C_{d} + \\frac{2}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} + \\frac{1}{C_{d}}", "derivation": "\\operatorname{F_{H}}{(C_{d})} = \\frac{1}{C_{d}} and C_{d} + \\operatorname{F_{H}}{(C_{d})} + \\frac{1}{C_{d}} = C_{d} + \\frac{2}{C_{d}} and (C_{d} + \\operatorname{F_{H}}{(C_{d})} + \\frac{1}{C_{d}})^{2} = (C_{d} + \\frac{2}{C_{d}})^{2} and \\frac{(C_{d} + \\operatorname{F_{H}}{(C_{d})} + \\frac{1}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} = \\frac{(C_{d} + \\frac{2}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} and C_{d} + \\frac{(C_{d} + \\operatorname{F_{H}}{(C_{d})} + \\frac{1}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} + \\frac{1}{C_{d}} = C_{d} + \\frac{(C_{d} + \\frac{2}{C_{d}})^{2}}{C_{d} + \\frac{1}{C_{d}}} + \\frac{1}{C_{d}}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_H')(Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Integer(-1)))"], [["add", 1, "Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1)))"], "Equality(Add(Symbol('C_d', commutative=True), Function('F_H')(Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Integer(-1))), Add(Symbol('C_d', commutative=True), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)))))"], [["power", 2, 2], "Equality(Pow(Add(Symbol('C_d', commutative=True), Function('F_H')(Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(2)), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Integer(2)))"], [["divide", 3, "Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(-1)), Pow(Add(Symbol('C_d', commutative=True), Function('F_H')(Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(2))), Mul(Pow(Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(-1)), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Integer(2))))"], [["add", 4, "Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1)))"], "Equality(Add(Symbol('C_d', commutative=True), Mul(Pow(Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(-1)), Pow(Add(Symbol('C_d', commutative=True), Function('F_H')(Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(2))), Pow(Symbol('C_d', commutative=True), Integer(-1))), Add(Symbol('C_d', commutative=True), Mul(Pow(Add(Symbol('C_d', commutative=True), Pow(Symbol('C_d', commutative=True), Integer(-1))), Integer(-1)), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(2), Pow(Symbol('C_d', commutative=True), Integer(-1)))), Integer(2))), Pow(Symbol('C_d', commutative=True), Integer(-1))))"]]}, {"prompt": "Given A{(\\chi,\\mathbf{S})} = \\mathbf{S}^{\\chi}, then derive \\chi + a = \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi, then derive \\chi + a = \\chi + \\mathbf{r}, then obtain a \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi = a (\\chi + \\mathbf{r})", "derivation": "A{(\\chi,\\mathbf{S})} = \\mathbf{S}^{\\chi} and A^{\\mathbf{S}}{(\\chi,\\mathbf{S})} = (\\mathbf{S}^{\\chi})^{\\mathbf{S}} and 1 = (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} and \\int 1 d\\chi = \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi and \\chi + a = \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi and \\chi + a = \\int 1 d\\chi and \\chi + a = \\chi + \\mathbf{r} and \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi = \\chi + \\mathbf{r} and a \\int (\\mathbf{S}^{\\chi})^{\\mathbf{S}} A^{- \\mathbf{S}}{(\\chi,\\mathbf{S})} d\\chi = a (\\chi + \\mathbf{r})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["divide", 2, "Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\chi', commutative=True), Symbol('a', commutative=True)), Integral(Mul(Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\chi', commutative=True), Symbol('a', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\chi', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integral(Mul(Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 8, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Integral(Mul(Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Symbol('a', commutative=True), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(q)} = \\sin{(q)}, then derive \\frac{d}{d q} \\operatorname{f^{*}}{(q)} = \\cos{(q)}, then obtain \\frac{\\frac{d}{d q} \\operatorname{f^{*}}{(q)}}{\\sin{(q)}} = \\frac{\\cos{(q)}}{\\sin{(q)}}", "derivation": "\\operatorname{f^{*}}{(q)} = \\sin{(q)} and \\frac{d}{d q} \\operatorname{f^{*}}{(q)} = \\frac{d}{d q} \\sin{(q)} and \\frac{\\frac{d}{d q} \\operatorname{f^{*}}{(q)}}{\\sin{(q)}} = \\frac{\\frac{d}{d q} \\sin{(q)}}{\\sin{(q)}} and \\frac{d}{d q} \\operatorname{f^{*}}{(q)} = \\cos{(q)} and \\cos{(q)} = \\frac{d}{d q} \\sin{(q)} and \\frac{\\frac{d}{d q} \\operatorname{f^{*}}{(q)}}{\\sin{(q)}} = \\frac{\\cos{(q)}}{\\sin{(q)}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 2, "sin(Symbol('q', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Derivative(Function('f^*')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), cos(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(cos(Symbol('q', commutative=True)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Derivative(Function('f^*')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), cos(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})} = \\cos{(v_{2})}, then obtain 2 e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} - 1 = e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} + e^{v_{2} \\cos{(v_{2})}} - 1", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})} = \\cos{(v_{2})} and v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})} = v_{2} \\cos{(v_{2})} and e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} = e^{v_{2} \\cos{(v_{2})}} and 2 e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} = e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} + e^{v_{2} \\cos{(v_{2})}} and 2 e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} - 1 = e^{v_{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{2})}} + e^{v_{2} \\cos{(v_{2})}} - 1", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["times", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))))"], [["exp", 2], "Equality(exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True)))), exp(Mul(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))))"], [["add", 3, "exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True))))"], "Equality(Mul(Integer(2), exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True))))), Add(exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True)))), exp(Mul(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True))))))"], [["minus", 4, 1], "Equality(Add(Mul(Integer(2), exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True))))), Integer(-1)), Add(exp(Mul(Symbol('v_2', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_2', commutative=True)))), exp(Mul(Symbol('v_2', commutative=True), cos(Symbol('v_2', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}_0{(\\hat{H}_{\\lambda})} = \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda}, then obtain \\int (\\hat{x}_0^{2}{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} d\\hat{H}_{\\lambda} = \\int (\\hat{x}_0{(\\hat{H}_{\\lambda})} \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})^{\\hat{H}_{\\lambda}} d\\hat{H}_{\\lambda}", "derivation": "\\hat{x}_0{(\\hat{H}_{\\lambda})} = \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and \\hat{x}_0^{2}{(\\hat{H}_{\\lambda})} = \\hat{x}_0{(\\hat{H}_{\\lambda})} \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and (\\hat{x}_0^{2}{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} = (\\hat{x}_0{(\\hat{H}_{\\lambda})} \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})^{\\hat{H}_{\\lambda}} and \\int (\\hat{x}_0^{2}{(\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} d\\hat{H}_{\\lambda} = \\int (\\hat{x}_0{(\\hat{H}_{\\lambda})} \\int \\sin{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda})^{\\hat{H}_{\\lambda}} d\\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["times", 1, "Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Pow(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(z,n)} = \\sin{(n + z)}, then obtain (\\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\operatorname{C_{d}}{(z,n)}))^{z} = (\\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\sin{(n + z)}))^{z}", "derivation": "\\operatorname{C_{d}}{(z,n)} = \\sin{(n + z)} and z + \\operatorname{C_{d}}{(z,n)} = z + \\sin{(n + z)} and \\frac{\\partial}{\\partial z} (z + \\operatorname{C_{d}}{(z,n)}) = \\frac{\\partial}{\\partial z} (z + \\sin{(n + z)}) and \\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\operatorname{C_{d}}{(z,n)}) = \\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\sin{(n + z)}) and (\\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\operatorname{C_{d}}{(z,n)}))^{z} = (\\frac{\\partial^{2}}{\\partial n\\partial z} (z + \\sin{(n + z)}))^{z}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('z', commutative=True), Symbol('n', commutative=True)), sin(Add(Symbol('n', commutative=True), Symbol('z', commutative=True))))"], [["add", 1, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('z', commutative=True), Function('C_d')(Symbol('z', commutative=True), Symbol('n', commutative=True))), Add(Symbol('z', commutative=True), sin(Add(Symbol('n', commutative=True), Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Symbol('z', commutative=True), Function('C_d')(Symbol('z', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('z', commutative=True), sin(Add(Symbol('n', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('z', commutative=True), Function('C_d')(Symbol('z', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('z', commutative=True), sin(Add(Symbol('n', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('z', commutative=True), Function('C_d')(Symbol('z', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Add(Symbol('z', commutative=True), sin(Add(Symbol('n', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(b)} = \\cos{(e^{b})}, then obtain \\frac{d^{2}}{d b^{2}} 1 = \\frac{d^{2}}{d b^{2}} (0^{b})^{b}", "derivation": "\\mathbf{F}{(b)} = \\cos{(e^{b})} and \\mathbf{F}{(b)} - \\cos{(e^{b})} = 0 and (\\mathbf{F}{(b)} - \\cos{(e^{b})})^{b} = 0^{b} and ((\\mathbf{F}{(b)} - \\cos{(e^{b})})^{b})^{b} = (0^{b})^{b} and 1 = ((\\mathbf{F}{(b)} - \\cos{(e^{b})})^{b})^{b} and \\frac{d}{d b} 1 = \\frac{d}{d b} ((\\mathbf{F}{(b)} - \\cos{(e^{b})})^{b})^{b} and \\frac{d^{2}}{d b^{2}} 1 = \\frac{d^{2}}{d b^{2}} ((\\mathbf{F}{(b)} - \\cos{(e^{b})})^{b})^{b} and \\frac{d^{2}}{d b^{2}} 1 = \\frac{d^{2}}{d b^{2}} (0^{b})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), cos(exp(Symbol('b', commutative=True))))"], [["minus", 1, "cos(exp(Symbol('b', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Symbol('b', commutative=True)), Pow(Integer(0), Symbol('b', commutative=True)))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Integer(0), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["differentiate", 5, "Symbol('b', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('b', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Pow(Pow(Add(Function('\\\\mathbf{F}')(Symbol('b', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('b', commutative=True))))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Derivative(Integer(1), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Pow(Pow(Integer(0), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\lambda)} = \\lambda, then derive \\int \\operatorname{E_{x}}{(\\lambda)} d\\lambda = \\dot{z} + \\frac{\\lambda^{2}}{2}, then obtain \\lambda + (\\dot{z} + \\frac{\\lambda^{2}}{2}) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda = \\lambda + (\\int \\lambda d\\lambda) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda", "derivation": "\\operatorname{E_{x}}{(\\lambda)} = \\lambda and \\int \\operatorname{E_{x}}{(\\lambda)} d\\lambda = \\int \\lambda d\\lambda and (\\int \\operatorname{E_{x}}{(\\lambda)} d\\lambda) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda = (\\int \\lambda d\\lambda) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda and \\int \\operatorname{E_{x}}{(\\lambda)} d\\lambda = \\dot{z} + \\frac{\\lambda^{2}}{2} and (\\dot{z} + \\frac{\\lambda^{2}}{2}) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda = (\\int \\lambda d\\lambda) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda and \\lambda + (\\dot{z} + \\frac{\\lambda^{2}}{2}) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda = \\lambda + (\\int \\lambda d\\lambda) \\int b{(\\lambda,\\eta^{\\prime})} d\\lambda", "srepr_derivation": [["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Integral(Function('E_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2)))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["add", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2)))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integral(Symbol('\\\\lambda', commutative=True), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Function('b')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(Z,f_{E})} = \\frac{Z}{f_{E}}, then obtain \\iint (\\mathbf{f}{(Z,f_{E})} - 1)^{Z} dZ dZ = \\iint (\\frac{Z}{f_{E}} - 1)^{Z} dZ dZ", "derivation": "\\mathbf{f}{(Z,f_{E})} = \\frac{Z}{f_{E}} and \\mathbf{f}{(Z,f_{E})} - 1 = \\frac{Z}{f_{E}} - 1 and (\\mathbf{f}{(Z,f_{E})} - 1)^{Z} = (\\frac{Z}{f_{E}} - 1)^{Z} and \\int (\\mathbf{f}{(Z,f_{E})} - 1)^{Z} dZ = \\int (\\frac{Z}{f_{E}} - 1)^{Z} dZ and \\iint (\\mathbf{f}{(Z,f_{E})} - 1)^{Z} dZ dZ = \\iint (\\frac{Z}{f_{E}} - 1)^{Z} dZ dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('Z', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('Z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Add(Mul(Symbol('Z', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Integer(-1)))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{f}')(Symbol('Z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Symbol('Z', commutative=True)), Pow(Add(Mul(Symbol('Z', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Integer(-1)), Symbol('Z', commutative=True)))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\mathbf{f}')(Symbol('Z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Add(Mul(Symbol('Z', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Integer(-1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["integrate", 4, "Symbol('Z', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\mathbf{f}')(Symbol('Z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Add(Mul(Symbol('Z', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Integer(-1)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(F_{x})} = \\cos{(F_{x})}, then obtain (\\int \\frac{\\operatorname{F_{H}}{(F_{x})}}{F_{x}} dF_{x})^{F_{x}} = (\\int \\frac{\\cos{(F_{x})}}{F_{x}} dF_{x})^{F_{x}}", "derivation": "\\operatorname{F_{H}}{(F_{x})} = \\cos{(F_{x})} and \\frac{\\operatorname{F_{H}}{(F_{x})}}{F_{x}} = \\frac{\\cos{(F_{x})}}{F_{x}} and \\int \\frac{\\operatorname{F_{H}}{(F_{x})}}{F_{x}} dF_{x} = \\int \\frac{\\cos{(F_{x})}}{F_{x}} dF_{x} and (\\int \\frac{\\operatorname{F_{H}}{(F_{x})}}{F_{x}} dF_{x})^{F_{x}} = (\\int \\frac{\\cos{(F_{x})}}{F_{x}} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["divide", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('F_H')(Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(Symbol('F_x', commutative=True))))"], [["integrate", 2, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('F_H')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('F_H')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{r})} = \\sin{(\\cos{(\\mathbf{r})})}, then obtain \\frac{d}{d \\mathbf{r}} \\frac{\\int 2 \\hat{p}{(\\mathbf{r})} d\\mathbf{r}}{\\mathbf{r}} = \\frac{d}{d \\mathbf{r}} \\frac{\\int (\\hat{p}{(\\mathbf{r})} + \\sin{(\\cos{(\\mathbf{r})})}) d\\mathbf{r}}{\\mathbf{r}}", "derivation": "\\hat{p}{(\\mathbf{r})} = \\sin{(\\cos{(\\mathbf{r})})} and 2 \\hat{p}{(\\mathbf{r})} = \\hat{p}{(\\mathbf{r})} + \\sin{(\\cos{(\\mathbf{r})})} and \\int 2 \\hat{p}{(\\mathbf{r})} d\\mathbf{r} = \\int (\\hat{p}{(\\mathbf{r})} + \\sin{(\\cos{(\\mathbf{r})})}) d\\mathbf{r} and \\frac{\\int 2 \\hat{p}{(\\mathbf{r})} d\\mathbf{r}}{\\mathbf{r}} = \\frac{\\int (\\hat{p}{(\\mathbf{r})} + \\sin{(\\cos{(\\mathbf{r})})}) d\\mathbf{r}}{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\frac{\\int 2 \\hat{p}{(\\mathbf{r})} d\\mathbf{r}}{\\mathbf{r}} = \\frac{d}{d \\mathbf{r}} \\frac{\\int (\\hat{p}{(\\mathbf{r})} + \\sin{(\\cos{(\\mathbf{r})})}) d\\mathbf{r}}{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(cos(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Add(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(cos(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["divide", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Add(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(cos(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Integral(Add(Function('\\\\hat{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(cos(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(\\hat{H},A)} = A + \\hat{H}, then derive 2 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} = \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} + 1, then obtain 4 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} = 3 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} + 1", "derivation": "c{(\\hat{H},A)} = A + \\hat{H} and 2 c{(\\hat{H},A)} = A + \\hat{H} + c{(\\hat{H},A)} and \\frac{\\partial}{\\partial \\hat{H}} 2 c{(\\hat{H},A)} = \\frac{\\partial}{\\partial \\hat{H}} (A + \\hat{H} + c{(\\hat{H},A)}) and 2 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} = \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} + 1 and 2 \\frac{\\partial}{\\partial \\hat{H}} (A + \\hat{H}) = \\frac{\\partial}{\\partial \\hat{H}} (A + \\hat{H}) + 1 and 2 \\frac{\\partial}{\\partial \\hat{H}} (A + \\hat{H}) + 2 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} = \\frac{\\partial}{\\partial \\hat{H}} (A + \\hat{H}) + 2 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} + 1 and 4 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} = 3 \\frac{\\partial}{\\partial \\hat{H}} c{(\\hat{H},A)} + 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["add", 1, "Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True), Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integer(1)))"], [["add", 5, "Mul(Integer(2), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(2), Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))), Add(Derivative(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(2), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(4), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Mul(Integer(3), Derivative(Function('c')(Symbol('\\\\hat{H}', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\phi{(g,F_{x})} = F_{x} + g, then obtain - g (- g + \\phi{(g,F_{x})}) - (- F_{x} g)^{F_{x}} = - F_{x} g - (- F_{x} g)^{F_{x}}", "derivation": "\\phi{(g,F_{x})} = F_{x} + g and - g + \\phi{(g,F_{x})} = F_{x} and - g (- g + \\phi{(g,F_{x})}) = - F_{x} g and (- g (- g + \\phi{(g,F_{x})}))^{F_{x}} = (- F_{x} g)^{F_{x}} and - g (- g + \\phi{(g,F_{x})}) - (- g (- g + \\phi{(g,F_{x})}))^{F_{x}} = - F_{x} g - (- g (- g + \\phi{(g,F_{x})}))^{F_{x}} and - g (- g + \\phi{(g,F_{x})}) - (- F_{x} g)^{F_{x}} = - F_{x} g - (- F_{x} g)^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))"], [["times", 2, "Mul(Integer(-1), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)), Pow(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)), Symbol('F_x', commutative=True)))"], [["minus", 3, "Pow(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('F_x', commutative=True)))), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)), Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('F_x', commutative=True), Symbol('g', commutative=True)), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(u)} = \\cos{(u)}, then obtain \\cos^{u}{(u)} = \\operatorname{r_{0}}^{- u}{(u)} \\cos^{2 u}{(u)}", "derivation": "\\operatorname{r_{0}}{(u)} = \\cos{(u)} and \\operatorname{r_{0}}^{u}{(u)} = \\cos^{u}{(u)} and \\operatorname{r_{0}}^{u}{(u)} \\cos^{u}{(u)} = \\cos^{2 u}{(u)} and \\cos^{u}{(u)} = \\operatorname{r_{0}}^{- u}{(u)} \\cos^{2 u}{(u)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["times", 2, "Pow(cos(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('r_0')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))))"], [["divide", 3, "Pow(Function('r_0')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Pow(cos(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(Pow(Function('r_0')(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(cos(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given r{(f^{\\prime},Q)} = Q f^{\\prime}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} e^{Q f^{\\prime} + r{(f^{\\prime},Q)}} = \\frac{\\partial}{\\partial f^{\\prime}} e^{2 Q f^{\\prime}}", "derivation": "r{(f^{\\prime},Q)} = Q f^{\\prime} and Q f^{\\prime} + r{(f^{\\prime},Q)} = 2 Q f^{\\prime} and e^{Q f^{\\prime} + r{(f^{\\prime},Q)}} = e^{2 Q f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} e^{Q f^{\\prime} + r{(f^{\\prime},Q)}} = \\frac{\\partial}{\\partial f^{\\prime}} e^{2 Q f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Mul(Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(2), Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["exp", 2], "Equality(exp(Add(Mul(Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('Q', commutative=True)))), exp(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(exp(Add(Mul(Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\hat{p}_0,f_{E})} = \\hat{p}_0 f_{E}, then obtain \\hat{p}_0 + \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{\\mathbf{x}}^{\\hat{p}_0}{(\\hat{p}_0,f_{E})} = \\hat{p}_0 + \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 f_{E})^{\\hat{p}_0}", "derivation": "\\hat{\\mathbf{x}}{(\\hat{p}_0,f_{E})} = \\hat{p}_0 f_{E} and \\hat{\\mathbf{x}}^{\\hat{p}_0}{(\\hat{p}_0,f_{E})} = (\\hat{p}_0 f_{E})^{\\hat{p}_0} and \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{\\mathbf{x}}^{\\hat{p}_0}{(\\hat{p}_0,f_{E})} = \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 f_{E})^{\\hat{p}_0} and \\hat{p}_0 + \\frac{\\partial}{\\partial \\hat{p}_0} \\hat{\\mathbf{x}}^{\\hat{p}_0}{(\\hat{p}_0,f_{E})} = \\hat{p}_0 + \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 f_{E})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Add(Symbol('\\\\hat{p}_0', commutative=True), Derivative(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(J_{\\varepsilon})} = e^{\\cos{(J_{\\varepsilon})}} and \\varepsilon{(J_{\\varepsilon})} = - e^{\\cos{(J_{\\varepsilon})}}, then obtain \\varepsilon_{0}{(S,\\psi^*)} \\frac{d}{d J_{\\varepsilon}} \\varepsilon{(J_{\\varepsilon})} = \\varepsilon_{0}{(S,\\psi^*)} \\frac{d}{d J_{\\varepsilon}} - \\hat{x}_0{(J_{\\varepsilon})}", "derivation": "\\hat{x}_0{(J_{\\varepsilon})} = e^{\\cos{(J_{\\varepsilon})}} and \\varepsilon{(J_{\\varepsilon})} = - e^{\\cos{(J_{\\varepsilon})}} and \\varepsilon{(J_{\\varepsilon})} = - \\hat{x}_0{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} \\varepsilon{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} - \\hat{x}_0{(J_{\\varepsilon})} and \\varepsilon_{0}{(S,\\psi^*)} \\frac{d}{d J_{\\varepsilon}} \\varepsilon{(J_{\\varepsilon})} = \\varepsilon_{0}{(S,\\psi^*)} \\frac{d}{d J_{\\varepsilon}} - \\hat{x}_0{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["times", 4, "Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Function('\\\\varepsilon_0')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{f}{(z)} = \\frac{d}{d z} e^{z}, then derive \\mathbf{f}{(z)} - 1 = e^{z} - 1, then obtain 0^{z} - \\frac{d}{d z} e^{z} = (e^{z} - \\frac{d}{d z} e^{z})^{z} - \\frac{d}{d z} e^{z}", "derivation": "\\mathbf{f}{(z)} = \\frac{d}{d z} e^{z} and \\mathbf{f}{(z)} - 1 = \\frac{d}{d z} e^{z} - 1 and \\mathbf{f}{(z)} - 1 = e^{z} - 1 and 0 = - \\mathbf{f}{(z)} + e^{z} and 0^{z} = (- \\mathbf{f}{(z)} + e^{z})^{z} and 0^{z} - \\frac{d}{d z} e^{z} = (- \\mathbf{f}{(z)} + e^{z})^{z} - \\frac{d}{d z} e^{z} and 0^{z} - \\frac{d}{d z} e^{z} = (e^{z} - \\frac{d}{d z} e^{z})^{z} - \\frac{d}{d z} e^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Integer(-1)), Add(Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Integer(-1)), Add(exp(Symbol('z', commutative=True)), Integer(-1)))"], [["minus", 3, "Add(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["minus", 5, "Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Add(Pow(Integer(0), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Add(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Pow(Integer(0), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Add(Pow(Add(exp(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{x}_0{(\\phi_2)} = \\log{(\\phi_2)}, then obtain (2 \\hat{x}_0{(\\phi_2)} - \\log{(\\phi_2)})^{\\phi_2} = \\log{(\\phi_2)}^{\\phi_2}", "derivation": "\\hat{x}_0{(\\phi_2)} = \\log{(\\phi_2)} and \\hat{x}_0{(\\phi_2)} - \\log{(\\phi_2)} = 0 and 2 \\hat{x}_0{(\\phi_2)} - \\log{(\\phi_2)} = \\hat{x}_0{(\\phi_2)} and 2 \\hat{x}_0{(\\phi_2)} - \\log{(\\phi_2)} = \\log{(\\phi_2)} and (2 \\hat{x}_0{(\\phi_2)} - \\log{(\\phi_2)})^{\\phi_2} = \\log{(\\phi_2)}^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\phi_2', commutative=True)))), Integer(0))"], [["add", 2, "Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi_2', commutative=True)))), Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Add(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi_2', commutative=True)))), log(Symbol('\\\\phi_2', commutative=True)))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\nabla{(A_{y})} = e^{\\sin{(A_{y})}}, then obtain 2 \\nabla{(A_{y})} e^{- \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})}} = \\nabla{(A_{y})} e^{- \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})}} + 1", "derivation": "\\nabla{(A_{y})} = e^{\\sin{(A_{y})}} and \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} = 1 and \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})} = \\sin{(A_{y})} and \\nabla{(A_{y})} e^{- \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})}} = 1 and 2 \\nabla{(A_{y})} e^{- \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})}} = \\nabla{(A_{y})} e^{- \\nabla{(A_{y})} e^{- \\sin{(A_{y})}} \\sin{(A_{y})}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(sin(Symbol('A_y', commutative=True))))"], [["divide", 1, "exp(sin(Symbol('A_y', commutative=True)))"], "Equality(Mul(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))), Integer(1))"], [["times", 2, "sin(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))), sin(Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))))), Integer(1))"], [["add", 4, "Mul(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True)))))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))))), Add(Mul(Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))), sin(Symbol('A_y', commutative=True))))), Integer(1)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\phi,\\delta)} = \\delta^{\\phi}, then obtain \\delta^{\\phi} \\varepsilon_{0}^{5}{(\\phi,\\delta)} = \\delta^{4 \\phi} \\varepsilon_{0}^{2}{(\\phi,\\delta)}", "derivation": "\\varepsilon_{0}{(\\phi,\\delta)} = \\delta^{\\phi} and \\delta^{\\phi} \\varepsilon_{0}{(\\phi,\\delta)} = \\delta^{2 \\phi} and \\delta^{2 \\phi} \\varepsilon_{0}^{2}{(\\phi,\\delta)} = \\delta^{4 \\phi} and \\delta^{\\phi} \\varepsilon_{0}^{3}{(\\phi,\\delta)} = \\delta^{2 \\phi} \\varepsilon_{0}^{2}{(\\phi,\\delta)} and \\delta^{\\phi} \\varepsilon_{0}^{3}{(\\phi,\\delta)} = \\delta^{4 \\phi} and \\delta^{\\phi} \\varepsilon_{0}^{5}{(\\phi,\\delta)} = \\delta^{4 \\phi} \\varepsilon_{0}^{2}{(\\phi,\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True))), Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))), Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(4), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(3))), Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(3))), Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(4), Symbol('\\\\phi', commutative=True))))"], [["times", 5, "Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(5))), Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(4), Symbol('\\\\phi', commutative=True))), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(Q)} = e^{Q} and \\mathbf{g}{(\\Psi,Q)} = \\Psi + e^{Q}, then derive \\int \\operatorname{n_{1}}{(Q)} dQ = \\Psi + e^{Q}, then obtain (\\int e^{Q} dQ)^{2} = \\mathbf{g}{(\\Psi,Q)} \\int e^{Q} dQ", "derivation": "\\operatorname{n_{1}}{(Q)} = e^{Q} and \\int \\operatorname{n_{1}}{(Q)} dQ = \\int e^{Q} dQ and \\int \\operatorname{n_{1}}{(Q)} dQ = \\Psi + e^{Q} and \\mathbf{g}{(\\Psi,Q)} = \\Psi + e^{Q} and \\int \\operatorname{n_{1}}{(Q)} dQ = \\mathbf{g}{(\\Psi,Q)} and (\\int \\operatorname{n_{1}}{(Q)} dQ)^{2} = \\mathbf{g}{(\\Psi,Q)} \\int \\operatorname{n_{1}}{(Q)} dQ and (\\int e^{Q} dQ)^{2} = \\mathbf{g}{(\\Psi,Q)} \\int e^{Q} dQ", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), exp(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True)))"], [["times", 5, "Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Pow(Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(2)), Mul(Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True)), Integral(Function('n_1')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Integral(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(2)), Mul(Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True)), Integral(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\Psi_{\\lambda})} = \\cos{(e^{\\Psi_{\\lambda}})}, then obtain \\frac{2 \\hat{H}_l^{2}{(\\Psi_{\\lambda})}}{\\cos^{2}{(e^{\\Psi_{\\lambda}})}} = \\frac{(\\frac{\\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}} + 1) \\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}}", "derivation": "\\hat{H}_l{(\\Psi_{\\lambda})} = \\cos{(e^{\\Psi_{\\lambda}})} and \\frac{\\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}} = 1 and \\frac{2 \\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}} = \\frac{\\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}} + 1 and \\frac{2 \\hat{H}_l^{2}{(\\Psi_{\\lambda})}}{\\cos^{2}{(e^{\\Psi_{\\lambda}})}} = \\frac{(\\frac{\\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}} + 1) \\hat{H}_l{(\\Psi_{\\lambda})}}{\\cos{(e^{\\Psi_{\\lambda}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 1, "cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Add(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(1)))"], [["times", 3, "Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-2))), Mul(Add(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(1)), Function('\\\\hat{H}_l')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(v_{2},v)} = v_{2} + \\sin{(v)}, then obtain (- v_{2} + \\operatorname{P_{e}}{(v_{2},v)})^{v} = (v_{2} - \\operatorname{P_{e}}{(v_{2},v)} + 2 \\sin{(v)})^{v}", "derivation": "\\operatorname{P_{e}}{(v_{2},v)} = v_{2} + \\sin{(v)} and - v_{2} + \\operatorname{P_{e}}{(v_{2},v)} = \\sin{(v)} and \\operatorname{P_{e}}{(v_{2},v)} + \\sin{(v)} = v_{2} + 2 \\sin{(v)} and (- v_{2} + \\operatorname{P_{e}}{(v_{2},v)})^{v} = \\sin^{v}{(v)} and \\sin{(v)} = v_{2} - \\operatorname{P_{e}}{(v_{2},v)} + 2 \\sin{(v)} and (- v_{2} + \\operatorname{P_{e}}{(v_{2},v)})^{v} = (v_{2} - \\operatorname{P_{e}}{(v_{2},v)} + 2 \\sin{(v)})^{v}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Add(Symbol('v_2', commutative=True), sin(Symbol('v', commutative=True))))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), sin(Symbol('v', commutative=True)))"], [["add", 1, "sin(Symbol('v', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), sin(Symbol('v', commutative=True))), Add(Symbol('v_2', commutative=True), Mul(Integer(2), sin(Symbol('v', commutative=True)))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["minus", 3, "Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))"], "Equality(sin(Symbol('v', commutative=True)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), sin(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), sin(Symbol('v', commutative=True)))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\mathbf{S},v_{y})} = \\mathbf{S} + v_{y}, then obtain 1 = \\frac{\\cos{(\\mathbf{S} + v_{y})}}{\\cos{(\\hat{p}{(\\mathbf{S},v_{y})})} \\frac{\\partial}{\\partial v_{y}} \\hat{p}{(\\mathbf{S},v_{y})}}", "derivation": "\\hat{p}{(\\mathbf{S},v_{y})} = \\mathbf{S} + v_{y} and \\sin{(\\hat{p}{(\\mathbf{S},v_{y})})} = \\sin{(\\mathbf{S} + v_{y})} and \\frac{\\partial}{\\partial v_{y}} \\sin{(\\hat{p}{(\\mathbf{S},v_{y})})} = \\frac{\\partial}{\\partial v_{y}} \\sin{(\\mathbf{S} + v_{y})} and 1 = \\frac{\\frac{\\partial}{\\partial v_{y}} \\sin{(\\mathbf{S} + v_{y})}}{\\frac{\\partial}{\\partial v_{y}} \\sin{(\\hat{p}{(\\mathbf{S},v_{y})})}} and 1 = \\frac{\\cos{(\\mathbf{S} + v_{y})}}{\\cos{(\\hat{p}{(\\mathbf{S},v_{y})})} \\frac{\\partial}{\\partial v_{y}} \\hat{p}{(\\mathbf{S},v_{y})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(sin(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(sin(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Pow(Derivative(sin(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Mul(cos(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Pow(cos(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Integer(-1)), Pow(Derivative(Function('\\\\hat{p}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\nabla,\\eta)} = \\nabla + e^{\\eta}, then obtain - \\eta + \\mathbf{r}{(\\nabla,\\eta)} e^{- \\eta} + e^{\\eta} = - \\eta + (\\nabla + e^{\\eta}) e^{- \\eta} + e^{\\eta}", "derivation": "\\mathbf{r}{(\\nabla,\\eta)} = \\nabla + e^{\\eta} and \\mathbf{r}{(\\nabla,\\eta)} e^{- \\eta} = (\\nabla + e^{\\eta}) e^{- \\eta} and - \\eta + \\mathbf{r}{(\\nabla,\\eta)} e^{- \\eta} = - \\eta + (\\nabla + e^{\\eta}) e^{- \\eta} and - \\eta + \\mathbf{r}{(\\nabla,\\eta)} e^{- \\eta} + e^{\\eta} = - \\eta + (\\nabla + e^{\\eta}) e^{- \\eta} + e^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\eta', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), Mul(Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))))"], [["minus", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Function('\\\\mathbf{r}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True))))))"], [["add", 3, "exp(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Function('\\\\mathbf{r}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), exp(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\eta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))), exp(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given g{(\\psi^*,k)} = \\sin{(\\psi^* + k)}, then obtain g^{- k}{(\\psi^*,k)} \\int (g{(\\psi^*,k)} + \\sin^{k}{(\\psi^* + k)}) dk = g^{- k}{(\\psi^*,k)} \\int (\\sin{(\\psi^* + k)} + \\sin^{k}{(\\psi^* + k)}) dk", "derivation": "g{(\\psi^*,k)} = \\sin{(\\psi^* + k)} and g^{k}{(\\psi^*,k)} = \\sin^{k}{(\\psi^* + k)} and g{(\\psi^*,k)} + g^{k}{(\\psi^*,k)} = g^{k}{(\\psi^*,k)} + \\sin{(\\psi^* + k)} and g{(\\psi^*,k)} + \\sin^{k}{(\\psi^* + k)} = \\sin{(\\psi^* + k)} + \\sin^{k}{(\\psi^* + k)} and \\int (g{(\\psi^*,k)} + \\sin^{k}{(\\psi^* + k)}) dk = \\int (\\sin{(\\psi^* + k)} + \\sin^{k}{(\\psi^* + k)}) dk and g^{- k}{(\\psi^*,k)} \\int (g{(\\psi^*,k)} + \\sin^{k}{(\\psi^* + k)}) dk = g^{- k}{(\\psi^*,k)} \\int (\\sin{(\\psi^* + k)} + \\sin^{k}{(\\psi^* + k)}) dk", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["add", 1, "Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Add(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Add(Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Add(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["divide", 5, "Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(Add(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Pow(Function('g')(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(Add(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Pow(sin(Add(Symbol('\\\\psi^*', commutative=True), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})}, then obtain \\frac{d}{d E_{\\lambda}} 1 = \\frac{d}{d E_{\\lambda}} \\cos{(\\operatorname{v_{t}}{(E_{\\lambda})} - \\sin{(E_{\\lambda})})}", "derivation": "\\operatorname{v_{t}}{(E_{\\lambda})} = \\sin{(E_{\\lambda})} and \\operatorname{v_{t}}{(E_{\\lambda})} - \\sin{(E_{\\lambda})} = 0 and 0 = - \\operatorname{v_{t}}{(E_{\\lambda})} + \\sin{(E_{\\lambda})} and 1 = \\cos{(\\operatorname{v_{t}}{(E_{\\lambda})} - \\sin{(E_{\\lambda})})} and \\frac{d}{d E_{\\lambda}} 1 = \\frac{d}{d E_{\\lambda}} \\cos{(\\operatorname{v_{t}}{(E_{\\lambda})} - \\sin{(E_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 1, "sin(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('E_{\\\\lambda}', commutative=True)))), Integer(0))"], [["minus", 2, "Add(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True))), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["cos", 3], "Equality(Integer(1), cos(Add(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('E_{\\\\lambda}', commutative=True))))))"], [["differentiate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(cos(Add(Function('v_t')(Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Symbol('E_{\\\\lambda}', commutative=True))))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(L_{\\varepsilon})} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and \\operatorname{z^{*}}{(L_{\\varepsilon})} = i^{2}{(L_{\\varepsilon})}, then derive i{(L_{\\varepsilon})} = f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})}, then obtain \\operatorname{z^{*}}{(L_{\\varepsilon})} = (f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})}) i{(L_{\\varepsilon})}", "derivation": "i{(L_{\\varepsilon})} = \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and i^{2}{(L_{\\varepsilon})} = i{(L_{\\varepsilon})} \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} and i{(L_{\\varepsilon})} = f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})} and \\int \\cos{(L_{\\varepsilon})} dL_{\\varepsilon} = f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})} and i^{2}{(L_{\\varepsilon})} = (f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})}) i{(L_{\\varepsilon})} and \\operatorname{z^{*}}{(L_{\\varepsilon})} = i^{2}{(L_{\\varepsilon})} and \\operatorname{z^{*}}{(L_{\\varepsilon})} = (f_{\\mathbf{v}} + \\sin{(L_{\\varepsilon})}) i{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Pow(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2)), Mul(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2)), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('z^*')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Function('i')(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given H{(\\dot{y},E_{x})} = - \\sin{(E_{x} - \\dot{y})}, then obtain - H{(\\dot{y},E_{x})} + \\frac{\\partial}{\\partial \\dot{y}} E_{x} H{(\\dot{y},E_{x})} + 1 = - H{(\\dot{y},E_{x})} + \\frac{\\partial}{\\partial \\dot{y}} - E_{x} \\sin{(E_{x} - \\dot{y})} + 1", "derivation": "H{(\\dot{y},E_{x})} = - \\sin{(E_{x} - \\dot{y})} and E_{x} H{(\\dot{y},E_{x})} = - E_{x} \\sin{(E_{x} - \\dot{y})} and \\frac{\\partial}{\\partial \\dot{y}} E_{x} H{(\\dot{y},E_{x})} = \\frac{\\partial}{\\partial \\dot{y}} - E_{x} \\sin{(E_{x} - \\dot{y})} and - H{(\\dot{y},E_{x})} + \\frac{\\partial}{\\partial \\dot{y}} E_{x} H{(\\dot{y},E_{x})} + 1 = - H{(\\dot{y},E_{x})} + \\frac{\\partial}{\\partial \\dot{y}} - E_{x} \\sin{(E_{x} - \\dot{y})} + 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))))"], [["times", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True), sin(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Mul(Symbol('E_x', commutative=True), Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('E_x', commutative=True), sin(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["minus", 3, "Add(Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True))), Derivative(Mul(Symbol('E_x', commutative=True), Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Function('H')(Symbol('\\\\dot{y}', commutative=True), Symbol('E_x', commutative=True))), Derivative(Mul(Integer(-1), Symbol('E_x', commutative=True), sin(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\hat{p}{(T)} = \\log{(T)}, then derive (\\frac{d}{d T} \\hat{p}{(T)})^{2} = \\frac{\\frac{d}{d T} \\hat{p}{(T)}}{T}, then obtain (\\frac{d}{d T} \\hat{p}{(T)})^{2} - 1 = -1 + \\frac{\\frac{d}{d T} \\hat{p}{(T)}}{T}", "derivation": "\\hat{p}{(T)} = \\log{(T)} and \\frac{d}{d T} \\hat{p}{(T)} = \\frac{d}{d T} \\log{(T)} and (\\frac{d}{d T} \\hat{p}{(T)})^{2} = \\frac{d}{d T} \\hat{p}{(T)} \\frac{d}{d T} \\log{(T)} and (\\frac{d}{d T} \\hat{p}{(T)})^{2} = \\frac{\\frac{d}{d T} \\hat{p}{(T)}}{T} and (\\frac{d}{d T} \\log{(T)})^{2} = \\frac{\\frac{d}{d T} \\log{(T)}}{T} and (\\frac{d}{d T} \\log{(T)})^{2} - 1 = -1 + \\frac{\\frac{d}{d T} \\log{(T)}}{T} and (\\frac{d}{d T} \\hat{p}{(T)})^{2} - 1 = -1 + \\frac{\\frac{d}{d T} \\hat{p}{(T)}}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["minus", 5, 1], "Equality(Add(Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Pow(Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(2)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"]]}, {"prompt": "Given Q{(W,\\mathbf{F})} = \\log{(\\frac{W}{\\mathbf{F}})}, then derive \\frac{\\frac{\\partial}{\\partial W} Q{(W,\\mathbf{F})}}{\\mathbf{F}} = \\frac{1}{W \\mathbf{F}}, then obtain \\frac{\\frac{\\partial}{\\partial W} \\log{(\\frac{W}{\\mathbf{F}})}}{\\mathbf{F}} = \\frac{1}{W \\mathbf{F}}", "derivation": "Q{(W,\\mathbf{F})} = \\log{(\\frac{W}{\\mathbf{F}})} and \\frac{\\partial}{\\partial W} Q{(W,\\mathbf{F})} = \\frac{\\partial}{\\partial W} \\log{(\\frac{W}{\\mathbf{F}})} and \\frac{\\frac{\\partial}{\\partial W} Q{(W,\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\frac{\\partial}{\\partial W} \\log{(\\frac{W}{\\mathbf{F}})}}{\\mathbf{F}} and \\frac{\\frac{\\partial}{\\partial W} Q{(W,\\mathbf{F})}}{\\mathbf{F}} = \\frac{1}{W \\mathbf{F}} and \\frac{\\frac{\\partial}{\\partial W} \\log{(\\frac{W}{\\mathbf{F}})}}{\\mathbf{F}} = \\frac{1}{W \\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), log(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(Function('Q')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(log(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(Function('Q')(Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Derivative(log(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(F_{g},\\pi)} = F_{g} + \\pi, then derive \\frac{\\partial}{\\partial F_{g}} \\hat{x}{(F_{g},\\pi)} = 1, then obtain \\frac{\\partial}{\\partial F_{g}} \\frac{\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\pi)}{\\pi} = \\frac{d}{d F_{g}} \\frac{1}{\\pi}", "derivation": "\\hat{x}{(F_{g},\\pi)} = F_{g} + \\pi and \\frac{\\partial}{\\partial F_{g}} \\hat{x}{(F_{g},\\pi)} = \\frac{\\partial}{\\partial F_{g}} (F_{g} + \\pi) and \\frac{\\partial}{\\partial F_{g}} \\hat{x}{(F_{g},\\pi)} = 1 and \\frac{\\frac{\\partial}{\\partial F_{g}} \\hat{x}{(F_{g},\\pi)}}{\\pi} = \\frac{1}{\\pi} and \\frac{\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\pi)}{\\pi} = \\frac{1}{\\pi} and \\frac{\\partial}{\\partial F_{g}} \\frac{\\frac{\\partial}{\\partial F_{g}} (F_{g} + \\pi)}{\\pi} = \\frac{d}{d F_{g}} \\frac{1}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["differentiate", 5, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(u)} = \\log{(e^{u})}, then obtain \\eta^{3}{(u)} = \\eta^{2}{(u)} \\log{(e^{u})}", "derivation": "\\eta{(u)} = \\log{(e^{u})} and \\frac{d}{d u} \\eta{(u)} = \\frac{d}{d u} \\log{(e^{u})} and \\frac{\\eta{(u)}}{\\frac{d}{d u} \\eta{(u)}} = \\frac{\\log{(e^{u})}}{\\frac{d}{d u} \\eta{(u)}} and \\frac{\\eta^{3}{(u)}}{(\\frac{d}{d u} \\eta{(u)})^{3}} = \\frac{\\eta^{2}{(u)} \\log{(e^{u})}}{(\\frac{d}{d u} \\eta{(u)})^{3}} and \\frac{\\eta^{3}{(u)}}{(\\frac{d}{d u} \\log{(e^{u})})^{3}} = \\frac{\\eta^{2}{(u)} \\log{(e^{u})}}{(\\frac{d}{d u} \\log{(e^{u})})^{3}} and \\eta^{3}{(u)} = \\eta^{2}{(u)} \\log{(e^{u})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True)), log(exp(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\eta')(Symbol('u', commutative=True)), Pow(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1))), Mul(log(exp(Symbol('u', commutative=True))), Pow(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1))))"], [["times", 3, "Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(2)), Pow(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-2)))"], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(3)), Pow(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-3))), Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(2)), log(exp(Symbol('u', commutative=True))), Pow(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-3))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(3)), Pow(Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-3))), Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(2)), log(exp(Symbol('u', commutative=True))), Pow(Derivative(log(exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-3))))"], [["evaluate_derivatives", 5], "Equality(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\eta')(Symbol('u', commutative=True)), Integer(2)), log(exp(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given k{(\\hat{p},A)} = \\cos{(A + \\hat{p})} and r{(\\hat{p},A)} = - A + k{(\\hat{p},A)}, then obtain (\\frac{- A + \\cos{(A + \\hat{p})}}{A})^{A} = (\\frac{- A + k{(\\hat{p},A)}}{A})^{A}", "derivation": "k{(\\hat{p},A)} = \\cos{(A + \\hat{p})} and - A + k{(\\hat{p},A)} = - A + \\cos{(A + \\hat{p})} and r{(\\hat{p},A)} = - A + k{(\\hat{p},A)} and r{(\\hat{p},A)} = - A + \\cos{(A + \\hat{p})} and \\frac{r{(\\hat{p},A)}}{A} = \\frac{- A + k{(\\hat{p},A)}}{A} and (\\frac{r{(\\hat{p},A)}}{A})^{A} = (\\frac{- A + k{(\\hat{p},A)}}{A})^{A} and (\\frac{- A + \\cos{(A + \\hat{p})}}{A})^{A} = (\\frac{- A + k{(\\hat{p},A)}}{A})^{A}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["minus", 1, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["divide", 3, "Symbol('A', commutative=True)"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), cos(Add(Symbol('A', commutative=True), Symbol('\\\\hat{p}', commutative=True))))), Symbol('A', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('k')(Symbol('\\\\hat{p}', commutative=True), Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(a)} = \\frac{d}{d a} e^{a}, then derive (\\mathbf{p}{(a)} + 2 \\frac{d}{d a} \\mathbf{p}{(a)} + \\frac{d^{2}}{d a^{2}} \\mathbf{p}{(a)}) e^{a} = 4 e^{2 a}, then obtain (\\frac{d}{d a} e^{a} + 2 \\frac{d^{2}}{d a^{2}} e^{a} + \\frac{d^{3}}{d a^{3}} e^{a}) e^{a} = 4 e^{2 a}", "derivation": "\\mathbf{p}{(a)} = \\frac{d}{d a} e^{a} and \\mathbf{p}{(a)} \\frac{d}{d a} e^{a} = (\\frac{d}{d a} e^{a})^{2} and \\frac{d}{d a} \\mathbf{p}{(a)} \\frac{d}{d a} e^{a} = \\frac{d}{d a} (\\frac{d}{d a} e^{a})^{2} and \\frac{d^{2}}{d a^{2}} \\mathbf{p}{(a)} \\frac{d}{d a} e^{a} = \\frac{d^{2}}{d a^{2}} (\\frac{d}{d a} e^{a})^{2} and (\\mathbf{p}{(a)} + 2 \\frac{d}{d a} \\mathbf{p}{(a)} + \\frac{d^{2}}{d a^{2}} \\mathbf{p}{(a)}) e^{a} = 4 e^{2 a} and (\\frac{d}{d a} e^{a} + 2 \\frac{d^{2}}{d a^{2}} e^{a} + \\frac{d^{3}}{d a^{3}} e^{a}) e^{a} = 4 e^{2 a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 1, "Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Pow(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('a', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Mul(Integer(2), Derivative(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbf{p}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), exp(Symbol('a', commutative=True))), Mul(Integer(4), exp(Mul(Integer(2), Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(2), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2)))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(3)))), exp(Symbol('a', commutative=True))), Mul(Integer(4), exp(Mul(Integer(2), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then derive \\frac{d}{d J_{\\varepsilon}} \\operatorname{E_{n}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\operatorname{E_{n}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\operatorname{E_{n}}{(J_{\\varepsilon})}", "derivation": "\\operatorname{E_{n}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\frac{d}{d J_{\\varepsilon}} \\operatorname{E_{n}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and \\frac{d}{d J_{\\varepsilon}} \\operatorname{E_{n}}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\operatorname{E_{n}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\operatorname{E_{n}}{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('E_n')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('E_n')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(x,\\theta_1)} = \\theta_1 + x, then derive \\int \\operatorname{f^{*}}{(x,\\theta_1)} dx = F_{x} + \\theta_1 x + \\frac{x^{2}}{2}, then obtain \\frac{\\partial}{\\partial F_{x}} \\frac{\\int (\\theta_1 + x) dx}{2} = \\frac{\\partial}{\\partial F_{x}} (\\frac{F_{x}}{2} + \\frac{\\theta_1 x}{2} + \\frac{x^{2}}{4})", "derivation": "\\operatorname{f^{*}}{(x,\\theta_1)} = \\theta_1 + x and \\int \\operatorname{f^{*}}{(x,\\theta_1)} dx = \\int (\\theta_1 + x) dx and \\int \\operatorname{f^{*}}{(x,\\theta_1)} dx = F_{x} + \\theta_1 x + \\frac{x^{2}}{2} and \\int (\\theta_1 + x) dx = F_{x} + \\theta_1 x + \\frac{x^{2}}{2} and \\frac{\\int (\\theta_1 + x) dx}{2} = \\frac{F_{x}}{2} + \\frac{\\theta_1 x}{2} + \\frac{x^{2}}{4} and \\frac{\\partial}{\\partial F_{x}} \\frac{\\int (\\theta_1 + x) dx}{2} = \\frac{\\partial}{\\partial F_{x}} (\\frac{F_{x}}{2} + \\frac{\\theta_1 x}{2} + \\frac{x^{2}}{4})", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)))))"], [["times", 4, "Rational(1, 2)"], "Equality(Mul(Rational(1, 2), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('F_x', commutative=True)), Mul(Rational(1, 2), Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Mul(Rational(1, 4), Pow(Symbol('x', commutative=True), Integer(2)))))"], [["differentiate", 5, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Integral(Add(Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Symbol('F_x', commutative=True)), Mul(Rational(1, 2), Symbol('\\\\theta_1', commutative=True), Symbol('x', commutative=True)), Mul(Rational(1, 4), Pow(Symbol('x', commutative=True), Integer(2)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(U,v_{t})} = v_{t}^{U}, then obtain v_{t} ((T^{2 U}{(U,v_{t})})^{v_{t}})^{U} = v_{t} (((v_{t}^{U})^{U} T^{U}{(U,v_{t})})^{v_{t}})^{U}", "derivation": "T{(U,v_{t})} = v_{t}^{U} and T^{U}{(U,v_{t})} = (v_{t}^{U})^{U} and T^{2 U}{(U,v_{t})} = (v_{t}^{U})^{U} T^{U}{(U,v_{t})} and (T^{2 U}{(U,v_{t})})^{v_{t}} = ((v_{t}^{U})^{U} T^{U}{(U,v_{t})})^{v_{t}} and ((T^{2 U}{(U,v_{t})})^{v_{t}})^{U} = (((v_{t}^{U})^{U} T^{U}{(U,v_{t})})^{v_{t}})^{U} and v_{t} ((T^{2 U}{(U,v_{t})})^{v_{t}})^{U} = v_{t} (((v_{t}^{U})^{U} T^{U}{(U,v_{t})})^{v_{t}})^{U}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["times", 2, "Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))"], "Equality(Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), Symbol('U', commutative=True))), Mul(Pow(Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)), Pow(Mul(Pow(Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Pow(Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(Mul(Pow(Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)), Symbol('U', commutative=True)))"], [["divide", 5, "Pow(Symbol('v_t', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v_t', commutative=True), Pow(Pow(Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(2), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))), Mul(Symbol('v_t', commutative=True), Pow(Pow(Mul(Pow(Pow(Symbol('v_t', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('T')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))), Symbol('v_t', commutative=True)), Symbol('U', commutative=True))))"]]}, {"prompt": "Given r{(f^{\\prime},f_{E})} = \\cos{(f^{\\prime} + f_{E})}, then obtain \\int (-1 + \\frac{r{(f^{\\prime},f_{E})}}{f^{\\prime} + f_{E}}) df^{\\prime} = \\int (-1 + \\frac{\\cos{(f^{\\prime} + f_{E})}}{f^{\\prime} + f_{E}}) df^{\\prime}", "derivation": "r{(f^{\\prime},f_{E})} = \\cos{(f^{\\prime} + f_{E})} and \\frac{r{(f^{\\prime},f_{E})}}{f^{\\prime} + f_{E}} = \\frac{\\cos{(f^{\\prime} + f_{E})}}{f^{\\prime} + f_{E}} and -1 + \\frac{r{(f^{\\prime},f_{E})}}{f^{\\prime} + f_{E}} = -1 + \\frac{\\cos{(f^{\\prime} + f_{E})}}{f^{\\prime} + f_{E}} and \\int (-1 + \\frac{r{(f^{\\prime},f_{E})}}{f^{\\prime} + f_{E}}) df^{\\prime} = \\int (-1 + \\frac{\\cos{(f^{\\prime} + f_{E})}}{f^{\\prime} + f_{E}}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), cos(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))))"], [["divide", 1, "Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), cos(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)))), Add(Integer(-1), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), cos(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))))))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Integer(-1), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Function('r')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Integer(-1), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), cos(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(b,x^\\prime)} = - b + x^\\prime, then derive \\frac{\\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)}}{- b + x^\\prime} = - \\frac{\\mathbf{v}{(b,x^\\prime)}}{- b + x^\\prime}, then obtain - \\frac{\\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)}}{b} = \\frac{1}{b}", "derivation": "\\mathbf{v}{(b,x^\\prime)} = - b + x^\\prime and \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)} = \\frac{\\partial}{\\partial b} (- b + x^\\prime) and \\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)} = \\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} (- b + x^\\prime) and \\frac{\\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)}}{- b + x^\\prime} = \\frac{\\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} (- b + x^\\prime)}{- b + x^\\prime} and \\frac{\\mathbf{v}{(b,x^\\prime)} \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)}}{- b + x^\\prime} = - \\frac{\\mathbf{v}{(b,x^\\prime)}}{- b + x^\\prime} and \\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)} = -1 and - \\frac{\\frac{\\partial}{\\partial b} \\mathbf{v}{(b,x^\\prime)}}{b} = \\frac{1}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1))"], [["divide", 6, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{v}')(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Pow(Symbol('b', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\hat{p}_0{(Q)} = \\log{(Q)}, then derive \\frac{\\hat{p}_0{(Q)} \\int \\hat{p}_0{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta} = \\frac{\\log{(Q)} \\int \\hat{p}_0{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta}, then obtain \\frac{\\hat{p}_0{(Q)} \\int \\log{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta} = \\frac{\\log{(Q)} \\int \\log{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta}", "derivation": "\\hat{p}_0{(Q)} = \\log{(Q)} and \\int \\hat{p}_0{(Q)} dQ = \\int \\log{(Q)} dQ and \\hat{p}_0{(Q)} \\int \\hat{p}_0{(Q)} dQ = \\log{(Q)} \\int \\hat{p}_0{(Q)} dQ and \\frac{\\hat{p}_0{(Q)} \\int \\hat{p}_0{(Q)} dQ}{\\int \\log{(Q)} dQ} = \\frac{\\log{(Q)} \\int \\hat{p}_0{(Q)} dQ}{\\int \\log{(Q)} dQ} and \\frac{\\hat{p}_0{(Q)} \\int \\hat{p}_0{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta} = \\frac{\\log{(Q)} \\int \\hat{p}_0{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta} and \\frac{\\hat{p}_0{(Q)} \\int \\log{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta} = \\frac{\\log{(Q)} \\int \\log{(Q)} dQ}{Q \\log{(Q)} - Q + \\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["times", 1, "Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(log(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["divide", 3, "Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Pow(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))), Mul(log(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Pow(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\eta', commutative=True)), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\eta', commutative=True)), Integer(-1)), log(Symbol('Q', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\eta', commutative=True)), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('Q', commutative=True)), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\eta', commutative=True)), Integer(-1)), log(Symbol('Q', commutative=True)), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mu)} = \\log{(\\mu)}, then obtain \\frac{d}{d \\mu} \\int \\hat{p}_0{(\\mu)} d\\mu = \\frac{\\partial}{\\partial \\mu} (\\mathbf{J}_f + \\mu \\log{(\\mu)} - \\mu)", "derivation": "\\hat{p}_0{(\\mu)} = \\log{(\\mu)} and \\int \\hat{p}_0{(\\mu)} d\\mu = \\int \\log{(\\mu)} d\\mu and \\frac{d}{d \\mu} \\int \\hat{p}_0{(\\mu)} d\\mu = \\frac{d}{d \\mu} \\int \\log{(\\mu)} d\\mu and \\frac{d}{d \\mu} \\int \\hat{p}_0{(\\mu)} d\\mu = \\frac{\\partial}{\\partial \\mu} (\\mathbf{J}_f + \\mu \\log{(\\mu)} - \\mu)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Symbol('\\\\mu', commutative=True), log(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(g_{\\varepsilon},c_{0},\\ddot{x})} = g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}}, then obtain \\frac{\\ddot{x} G{(g_{\\varepsilon},c_{0},\\ddot{x})}}{c_{0} (g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}}) \\cos{(A_{y} + W)}} = \\frac{\\ddot{x}}{c_{0} \\cos{(A_{y} + W)}}", "derivation": "G{(g_{\\varepsilon},c_{0},\\ddot{x})} = g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}} and \\frac{G{(g_{\\varepsilon},c_{0},\\ddot{x})}}{g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}}} = 1 and \\frac{\\ddot{x} G{(g_{\\varepsilon},c_{0},\\ddot{x})}}{c_{0} (g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}})} = \\frac{\\ddot{x}}{c_{0}} and \\frac{\\ddot{x} G{(g_{\\varepsilon},c_{0},\\ddot{x})}}{c_{0} (g_{\\varepsilon} + \\frac{c_{0}}{\\ddot{x}}) \\cos{(A_{y} + W)}} = \\frac{\\ddot{x}}{c_{0} \\cos{(A_{y} + W)}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))))"], [["divide", 1, "Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Integer(-1)), Function('G')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(1))"], [["divide", 2, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Integer(-1)), Function('G')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["divide", 3, "cos(Add(Symbol('A_y', commutative=True), Symbol('W', commutative=True)))"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Integer(-1)), Function('G')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(cos(Add(Symbol('A_y', commutative=True), Symbol('W', commutative=True))), Integer(-1))), Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('A_y', commutative=True), Symbol('W', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given n{(S,\\rho_f,F_{x})} = (\\frac{F_{x}}{S})^{\\rho_f} and \\operatorname{C_{2}}{(S,\\rho_f,F_{x})} = (- F_{x} + n{(S,\\rho_f,F_{x})})^{F_{x}}, then obtain \\frac{(- F_{x} + (\\frac{F_{x}}{S})^{\\rho_f})^{- F_{x}} \\operatorname{C_{2}}{(S,\\rho_f,F_{x})}}{\\mathbf{v}} = \\frac{1}{\\mathbf{v}}", "derivation": "n{(S,\\rho_f,F_{x})} = (\\frac{F_{x}}{S})^{\\rho_f} and - F_{x} + n{(S,\\rho_f,F_{x})} = - F_{x} + (\\frac{F_{x}}{S})^{\\rho_f} and \\operatorname{C_{2}}{(S,\\rho_f,F_{x})} = (- F_{x} + n{(S,\\rho_f,F_{x})})^{F_{x}} and (- F_{x} + n{(S,\\rho_f,F_{x})})^{- F_{x}} \\operatorname{C_{2}}{(S,\\rho_f,F_{x})} = 1 and \\frac{(- F_{x} + n{(S,\\rho_f,F_{x})})^{- F_{x}} \\operatorname{C_{2}}{(S,\\rho_f,F_{x})}}{\\mathbf{v}} = \\frac{1}{\\mathbf{v}} and \\frac{(- F_{x} + (\\frac{F_{x}}{S})^{\\rho_f})^{- F_{x}} \\operatorname{C_{2}}{(S,\\rho_f,F_{x})}}{\\mathbf{v}} = \\frac{1}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True)), Pow(Mul(Symbol('F_x', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Mul(Symbol('F_x', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["divide", 3, "Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Function('C_2')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Integer(1))"], [["divide", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('n')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Function('C_2')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Mul(Symbol('F_x', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Function('C_2')(Symbol('S', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('F_x', commutative=True))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\hat{H})} = e^{e^{\\hat{H}}}, then obtain e^{- e^{\\hat{H}}} \\frac{d}{d \\hat{H}} (\\hat{H} + \\operatorname{C_{d}}{(\\hat{H})}) = e^{- e^{\\hat{H}}} \\frac{d}{d \\hat{H}} (\\hat{H} + e^{e^{\\hat{H}}})", "derivation": "\\operatorname{C_{d}}{(\\hat{H})} = e^{e^{\\hat{H}}} and \\hat{H} + \\operatorname{C_{d}}{(\\hat{H})} = \\hat{H} + e^{e^{\\hat{H}}} and \\frac{d}{d \\hat{H}} (\\hat{H} + \\operatorname{C_{d}}{(\\hat{H})}) = \\frac{d}{d \\hat{H}} (\\hat{H} + e^{e^{\\hat{H}}}) and e^{- e^{\\hat{H}}} \\frac{d}{d \\hat{H}} (\\hat{H} + \\operatorname{C_{d}}{(\\hat{H})}) = e^{- e^{\\hat{H}}} \\frac{d}{d \\hat{H}} (\\hat{H} + e^{e^{\\hat{H}}})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\hat{H}', commutative=True)), exp(exp(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('C_d')(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), exp(exp(Symbol('\\\\hat{H}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Function('C_d')(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), exp(exp(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["divide", 3, "exp(exp(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Function('C_d')(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), exp(exp(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given T{(\\hat{H})} = e^{\\sin{(\\hat{H})}} and \\operatorname{t_{1}}{(\\hat{H})} = \\int (T{(\\hat{H})} - \\sin{(\\hat{H})}) d\\hat{H}, then obtain \\operatorname{t_{1}}^{\\hat{H}}{(\\hat{H})} = (\\int (e^{\\sin{(\\hat{H})}} - \\sin{(\\hat{H})}) d\\hat{H})^{\\hat{H}}", "derivation": "T{(\\hat{H})} = e^{\\sin{(\\hat{H})}} and T{(\\hat{H})} - \\sin{(\\hat{H})} = e^{\\sin{(\\hat{H})}} - \\sin{(\\hat{H})} and \\int (T{(\\hat{H})} - \\sin{(\\hat{H})}) d\\hat{H} = \\int (e^{\\sin{(\\hat{H})}} - \\sin{(\\hat{H})}) d\\hat{H} and (\\int (T{(\\hat{H})} - \\sin{(\\hat{H})}) d\\hat{H})^{\\hat{H}} = (\\int (e^{\\sin{(\\hat{H})}} - \\sin{(\\hat{H})}) d\\hat{H})^{\\hat{H}} and \\operatorname{t_{1}}{(\\hat{H})} = \\int (T{(\\hat{H})} - \\sin{(\\hat{H})}) d\\hat{H} and \\operatorname{t_{1}}^{\\hat{H}}{(\\hat{H})} = (\\int (e^{\\sin{(\\hat{H})}} - \\sin{(\\hat{H})}) d\\hat{H})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\hat{H}', commutative=True)), exp(sin(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Add(exp(sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Add(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(exp(sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Integral(Add(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(Add(exp(sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\hat{H}', commutative=True)), Integral(Add(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('t_1')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(Add(exp(sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(t,u)} = \\sin{(\\frac{u}{t})}, then derive \\log{(- \\frac{\\partial}{\\partial t} \\operatorname{C_{1}}{(t,u)})} = \\log{(\\frac{u \\cos{(\\frac{u}{t})}}{t^{2}})}, then obtain \\log{(- \\frac{\\partial}{\\partial t} \\sin{(\\frac{u}{t})})} = \\log{(\\frac{u \\cos{(\\frac{u}{t})}}{t^{2}})}", "derivation": "\\operatorname{C_{1}}{(t,u)} = \\sin{(\\frac{u}{t})} and - \\operatorname{C_{1}}{(t,u)} = - \\sin{(\\frac{u}{t})} and \\frac{\\partial}{\\partial t} - \\operatorname{C_{1}}{(t,u)} = \\frac{\\partial}{\\partial t} - \\sin{(\\frac{u}{t})} and \\log{(\\frac{\\partial}{\\partial t} - \\operatorname{C_{1}}{(t,u)})} = \\log{(\\frac{\\partial}{\\partial t} - \\sin{(\\frac{u}{t})})} and \\log{(- \\frac{\\partial}{\\partial t} \\operatorname{C_{1}}{(t,u)})} = \\log{(\\frac{u \\cos{(\\frac{u}{t})}}{t^{2}})} and \\log{(- \\frac{\\partial}{\\partial t} \\sin{(\\frac{u}{t})})} = \\log{(\\frac{u \\cos{(\\frac{u}{t})}}{t^{2}})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('t', commutative=True), Symbol('u', commutative=True)), sin(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('C_1')(Symbol('t', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), sin(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('C_1')(Symbol('t', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Mul(Integer(-1), Function('C_1')(Symbol('t', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), log(Derivative(Mul(Integer(-1), sin(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(log(Mul(Integer(-1), Derivative(Function('C_1')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('u', commutative=True), cos(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(log(Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('t', commutative=True), Integer(-2)), Symbol('u', commutative=True), cos(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(i)} = \\cos{(\\sin{(i)})} and \\mu_{0}{(i)} = \\cos{(\\sin{(i)})}, then obtain \\mu_{0}{(i)} + \\operatorname{a^{\\dagger}}{(i)} = 2 \\mu_{0}{(i)}", "derivation": "\\operatorname{a^{\\dagger}}{(i)} = \\cos{(\\sin{(i)})} and \\mu_{0}{(i)} = \\cos{(\\sin{(i)})} and \\operatorname{a^{\\dagger}}{(i)} = \\mu_{0}{(i)} and \\mu_{0}{(i)} + \\operatorname{a^{\\dagger}}{(i)} = 2 \\mu_{0}{(i)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('i', commutative=True)), cos(sin(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('i', commutative=True)), cos(sin(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('a^{\\\\dagger}')(Symbol('i', commutative=True)), Function('\\\\mu_0')(Symbol('i', commutative=True)))"], [["add", 3, "Function('\\\\mu_0')(Symbol('i', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('i', commutative=True)), Function('a^{\\\\dagger}')(Symbol('i', commutative=True))), Mul(Integer(2), Function('\\\\mu_0')(Symbol('i', commutative=True))))"]]}, {"prompt": "Given Q{(\\psi^*,p)} = \\frac{p}{\\psi^*} and \\Psi_{nl}{(\\psi^*,p)} = Q{(\\psi^*,p)} + \\frac{p}{\\psi^*}, then obtain 2 \\frac{\\partial}{\\partial \\psi^*} Q{(\\psi^*,p)} = \\frac{\\partial}{\\partial \\psi^*} Q{(\\psi^*,p)} - \\frac{p}{(\\psi^*)^{2}}", "derivation": "Q{(\\psi^*,p)} = \\frac{p}{\\psi^*} and \\Psi_{nl}{(\\psi^*,p)} = Q{(\\psi^*,p)} + \\frac{p}{\\psi^*} and \\frac{\\partial}{\\partial \\psi^*} \\Psi_{nl}{(\\psi^*,p)} = \\frac{\\partial}{\\partial \\psi^*} (Q{(\\psi^*,p)} + \\frac{p}{\\psi^*}) and \\Psi_{nl}{(\\psi^*,p)} = 2 Q{(\\psi^*,p)} and \\frac{\\partial}{\\partial \\psi^*} 2 Q{(\\psi^*,p)} = \\frac{\\partial}{\\partial \\psi^*} (Q{(\\psi^*,p)} + \\frac{p}{\\psi^*}) and 2 \\frac{\\partial}{\\partial \\psi^*} Q{(\\psi^*,p)} = \\frac{\\partial}{\\partial \\psi^*} Q{(\\psi^*,p)} - \\frac{p}{(\\psi^*)^{2}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Add(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Mul(Integer(2), Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Mul(Integer(2), Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(2), Derivative(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Add(Derivative(Function('Q')(Symbol('\\\\psi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{p},u)} = \\mathbf{p} + u, then obtain \\int \\mathbf{p} du + \\int (- u + \\mathbf{v}{(\\mathbf{p},u)}) du = 2 \\int \\mathbf{p} du", "derivation": "\\mathbf{v}{(\\mathbf{p},u)} = \\mathbf{p} + u and - u + \\mathbf{v}{(\\mathbf{p},u)} = \\mathbf{p} and \\int (- u + \\mathbf{v}{(\\mathbf{p},u)}) du = \\int \\mathbf{p} du and \\int \\mathbf{p} du + \\int (- u + \\mathbf{v}{(\\mathbf{p},u)}) du = 2 \\int \\mathbf{p} du", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('u', commutative=True))))"], [["add", 3, "Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('u', commutative=True)))"], "Equality(Add(Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('u', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Integer(2), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\mathbf{P})} = \\int e^{\\mathbf{P}} d\\mathbf{P}, then derive \\frac{d}{d \\mathbf{P}} \\omega{(\\mathbf{P})} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\dot{z} + e^{\\mathbf{P}}), then obtain \\frac{d}{d \\mathbf{P}} \\omega{(\\mathbf{P})} = e^{\\mathbf{P}}", "derivation": "\\omega{(\\mathbf{P})} = \\int e^{\\mathbf{P}} d\\mathbf{P} and \\frac{d}{d \\mathbf{P}} \\omega{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\int e^{\\mathbf{P}} d\\mathbf{P} and \\frac{d}{d \\mathbf{P}} \\omega{(\\mathbf{P})} = \\frac{\\partial}{\\partial \\mathbf{P}} (\\dot{z} + e^{\\mathbf{P}}) and \\frac{d}{d \\mathbf{P}} \\omega{(\\mathbf{P})} = e^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\eta)} = \\log{(\\eta)}, then obtain 2 \\log{(\\eta)} \\frac{d}{d \\eta} (\\operatorname{v_{t}}{(\\eta)} + \\log{(\\eta)}) = 2 \\log{(\\eta)} \\frac{d}{d \\eta} 2 \\log{(\\eta)}", "derivation": "\\operatorname{v_{t}}{(\\eta)} = \\log{(\\eta)} and \\operatorname{v_{t}}{(\\eta)} + \\log{(\\eta)} = 2 \\log{(\\eta)} and \\frac{d}{d \\eta} (\\operatorname{v_{t}}{(\\eta)} + \\log{(\\eta)}) = \\frac{d}{d \\eta} 2 \\log{(\\eta)} and 2 \\log{(\\eta)} \\frac{d}{d \\eta} (\\operatorname{v_{t}}{(\\eta)} + \\log{(\\eta)}) = 2 \\log{(\\eta)} \\frac{d}{d \\eta} 2 \\log{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["add", 1, "log(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('v_t')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True))), Mul(Integer(2), log(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Function('v_t')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(2), log(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(2), log(Symbol('\\\\eta', commutative=True)), Derivative(Add(Function('v_t')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(2), log(Symbol('\\\\eta', commutative=True)), Derivative(Mul(Integer(2), log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(B)} = e^{B}, then obtain - e^{B} + \\int 2 \\delta{(B)} dB = - e^{B} + \\int (\\delta{(B)} + e^{B}) dB", "derivation": "\\delta{(B)} = e^{B} and 2 \\delta{(B)} = \\delta{(B)} + e^{B} and \\int 2 \\delta{(B)} dB = \\int (\\delta{(B)} + e^{B}) dB and - e^{B} + \\int 2 \\delta{(B)} dB = - e^{B} + \\int (\\delta{(B)} + e^{B}) dB", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["add", 1, "Function('\\\\delta')(Symbol('B', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('B', commutative=True))), Add(Function('\\\\delta')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\delta')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Add(Function('\\\\delta')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["minus", 3, "exp(Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), Integral(Mul(Integer(2), Function('\\\\delta')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('B', commutative=True))), Integral(Add(Function('\\\\delta')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given v{(\\omega,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + \\cos{(\\omega)}, then obtain 0 = 2 \\hat{\\mathbf{x}} - 2 v{(\\omega,\\hat{\\mathbf{x}})} + 2 \\cos{(\\omega)}", "derivation": "v{(\\omega,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + \\cos{(\\omega)} and \\omega + v{(\\omega,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} + \\omega + \\cos{(\\omega)} and 0 = \\hat{\\mathbf{x}} - v{(\\omega,\\hat{\\mathbf{x}})} + \\cos{(\\omega)} and \\hat{\\mathbf{x}} - v{(\\omega,\\hat{\\mathbf{x}})} + \\cos{(\\omega)} = 2 \\hat{\\mathbf{x}} - 2 v{(\\omega,\\hat{\\mathbf{x}})} + 2 \\cos{(\\omega)} and 0 = 2 \\hat{\\mathbf{x}} - 2 v{(\\omega,\\hat{\\mathbf{x}})} + 2 \\cos{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), cos(Symbol('\\\\omega', commutative=True))))"], [["add", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\omega', commutative=True), cos(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\omega', commutative=True), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Integer(2), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Integer(2), Function('v')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(y)} = \\cos{(y)}, then obtain ((\\int \\operatorname{C_{d}}{(y)} dy)^{y})^{y} = ((\\int \\cos{(y)} dy)^{y})^{y}", "derivation": "\\operatorname{C_{d}}{(y)} = \\cos{(y)} and \\int \\operatorname{C_{d}}{(y)} dy = \\int \\cos{(y)} dy and (\\int \\operatorname{C_{d}}{(y)} dy)^{y} = (\\int \\cos{(y)} dy)^{y} and ((\\int \\operatorname{C_{d}}{(y)} dy)^{y})^{y} = ((\\int \\cos{(y)} dy)^{y})^{y}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Function('C_d')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(Integral(Function('C_d')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(n_{2},A_{z})} = \\int A_{z} n_{2} dA_{z}, then obtain A_{z} + 1 + \\frac{\\mathbf{v}{(n_{2},A_{z})}}{A_{z} n_{2}} = A_{z} + 1 + \\frac{\\int A_{z} n_{2} dA_{z}}{A_{z} n_{2}}", "derivation": "\\mathbf{v}{(n_{2},A_{z})} = \\int A_{z} n_{2} dA_{z} and \\frac{\\mathbf{v}{(n_{2},A_{z})}}{A_{z} n_{2}} = \\frac{\\int A_{z} n_{2} dA_{z}}{A_{z} n_{2}} and 1 + \\frac{\\mathbf{v}{(n_{2},A_{z})}}{A_{z} n_{2}} = 1 + \\frac{\\int A_{z} n_{2} dA_{z}}{A_{z} n_{2}} and A_{z} + 1 + \\frac{\\mathbf{v}{(n_{2},A_{z})}}{A_{z} n_{2}} = A_{z} + 1 + \\frac{\\int A_{z} n_{2} dA_{z}}{A_{z} n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('n_2', commutative=True), Symbol('A_z', commutative=True)), Integral(Mul(Symbol('A_z', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["divide", 1, "Mul(Symbol('A_z', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('n_2', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('A_z', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('n_2', commutative=True), Symbol('A_z', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('A_z', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('A_z', commutative=True))))))"], [["add", 3, "Symbol('A_z', commutative=True)"], "Equality(Add(Symbol('A_z', commutative=True), Integer(1), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('n_2', commutative=True), Symbol('A_z', commutative=True)))), Add(Symbol('A_z', commutative=True), Integer(1), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Integral(Mul(Symbol('A_z', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(V,B)} = B V and \\mathbf{J}_f{(\\mathbf{s},C_{d})} = \\mathbf{s}^{C_{d}}, then obtain \\mathbf{J}_f{(\\mathbf{s},C_{d})} = \\mathbf{s}^{C_{d}} - \\int B \\operatorname{C_{2}}{(V,B)} dV + \\int B^{2} V dV", "derivation": "\\operatorname{C_{2}}{(V,B)} = B V and B \\operatorname{C_{2}}{(V,B)} = B^{2} V and \\int B \\operatorname{C_{2}}{(V,B)} dV = \\int B^{2} V dV and \\mathbf{J}_f{(\\mathbf{s},C_{d})} = \\mathbf{s}^{C_{d}} and 0 = - \\int B \\operatorname{C_{2}}{(V,B)} dV + \\int B^{2} V dV and \\mathbf{s}^{C_{d}} = \\mathbf{s}^{C_{d}} - \\int B \\operatorname{C_{2}}{(V,B)} dV + \\int B^{2} V dV and \\mathbf{J}_f{(\\mathbf{s},C_{d})} = \\mathbf{s}^{C_{d}} - \\int B \\operatorname{C_{2}}{(V,B)} dV + \\int B^{2} V dV", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('V', commutative=True)))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)))"], [["minus", 3, "Integral(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('V', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('V', commutative=True)))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["add", 5, "Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)), Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('V', commutative=True)))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)), Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('B', commutative=True), Function('C_2')(Symbol('V', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('V', commutative=True)))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then obtain \\frac{a^{\\dagger} + 2 \\cos{(a^{\\dagger})}}{\\mathbf{P}{(a^{\\dagger})}} = \\frac{a^{\\dagger} + \\mathbf{P}{(a^{\\dagger})} + \\cos{(a^{\\dagger})}}{\\mathbf{P}{(a^{\\dagger})}}", "derivation": "\\mathbf{P}{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and 2 \\mathbf{P}{(a^{\\dagger})} = \\mathbf{P}{(a^{\\dagger})} + \\cos{(a^{\\dagger})} and a^{\\dagger} + 2 \\mathbf{P}{(a^{\\dagger})} = a^{\\dagger} + \\mathbf{P}{(a^{\\dagger})} + \\cos{(a^{\\dagger})} and a^{\\dagger} + \\mathbf{P}{(a^{\\dagger})} = a^{\\dagger} + \\cos{(a^{\\dagger})} and a^{\\dagger} + 2 \\mathbf{P}{(a^{\\dagger})} = a^{\\dagger} + 2 \\cos{(a^{\\dagger})} and a^{\\dagger} + 2 \\cos{(a^{\\dagger})} = a^{\\dagger} + \\mathbf{P}{(a^{\\dagger})} + \\cos{(a^{\\dagger})} and \\frac{a^{\\dagger} + 2 \\cos{(a^{\\dagger})}}{\\mathbf{P}{(a^{\\dagger})}} = \\frac{a^{\\dagger} + \\mathbf{P}{(a^{\\dagger})} + \\cos{(a^{\\dagger})}}{\\mathbf{P}{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 6, "Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(2), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Pow(Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Mul(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\omega{(V,\\mathbf{M})} = e^{- V + \\mathbf{M}}, then derive - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} \\omega{(V,\\mathbf{M})} = - 2 e^{- V + \\mathbf{M}}, then obtain - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} e^{- V + \\mathbf{M}} = - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} \\omega{(V,\\mathbf{M})}", "derivation": "\\omega{(V,\\mathbf{M})} = e^{- V + \\mathbf{M}} and \\omega{(V,\\mathbf{M})} + e^{- V + \\mathbf{M}} = 2 e^{- V + \\mathbf{M}} and \\frac{\\partial}{\\partial V} (\\omega{(V,\\mathbf{M})} + e^{- V + \\mathbf{M}}) = \\frac{\\partial}{\\partial V} 2 e^{- V + \\mathbf{M}} and - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} \\omega{(V,\\mathbf{M})} = - 2 e^{- V + \\mathbf{M}} and - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} e^{- V + \\mathbf{M}} = - 2 e^{- V + \\mathbf{M}} and - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} e^{- V + \\mathbf{M}} = - e^{- V + \\mathbf{M}} + \\frac{\\partial}{\\partial V} \\omega{(V,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('V', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 1, "exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('V', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integer(2), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Function('\\\\omega')(Symbol('V', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Derivative(Function('\\\\omega')(Symbol('V', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))), Derivative(Function('\\\\omega')(Symbol('V', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})} = \\cos{(\\ddot{x})} and \\operatorname{v_{2}}{(\\ddot{x})} = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})} + \\cos{(\\ddot{x})}, then obtain \\frac{\\sin{(\\operatorname{v_{2}}{(\\ddot{x})})}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})}} = 0", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})} = \\cos{(\\ddot{x})} and \\operatorname{v_{2}}{(\\ddot{x})} = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})} + \\cos{(\\ddot{x})} and \\operatorname{v_{2}}{(\\ddot{x})} = 0 and \\sin{(\\operatorname{v_{2}}{(\\ddot{x})})} = 0 and \\frac{\\sin{(\\operatorname{v_{2}}{(\\ddot{x})})}}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\ddot{x})}} = 0", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\ddot{x}', commutative=True))), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_2')(Symbol('\\\\ddot{x}', commutative=True)), Integer(0))"], [["sin", 3], "Equality(sin(Function('v_2')(Symbol('\\\\ddot{x}', commutative=True))), Integer(0))"], [["divide", 4, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), sin(Function('v_2')(Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given a{(\\theta,H)} = \\frac{\\theta}{H} and \\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} = \\frac{\\theta}{H}, then obtain \\int \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} d\\theta = \\int \\frac{a{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} d\\theta", "derivation": "a{(\\theta,H)} = \\frac{\\theta}{H} and \\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} = \\frac{\\theta}{H} and \\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} = a{(\\theta,H)} and \\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} - \\frac{2 \\theta}{H} = a{(\\theta,H)} - \\frac{2 \\theta}{H} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} = \\frac{a{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} and \\int \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} d\\theta = \\int \\frac{a{(\\theta,H)} - \\frac{2 \\theta}{H}}{H} d\\theta", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Function('a')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Add(Function('a')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))))"], [["times", 4, "Pow(Symbol('H', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('a')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('a')(Symbol('\\\\theta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(J)} = \\sin{(J)}, then derive \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} = \\frac{\\cos{(J)} - 1}{J}, then obtain 1 = \\frac{n_{2} - \\log{(J)} + \\operatorname{Ci}{(J)}}{\\int \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} dJ}", "derivation": "\\operatorname{E_{n}}{(J)} = \\sin{(J)} and \\frac{d}{d J} \\operatorname{E_{n}}{(J)} = \\frac{d}{d J} \\sin{(J)} and \\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1 = \\frac{d}{d J} \\sin{(J)} - 1 and \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} = \\frac{\\frac{d}{d J} \\sin{(J)} - 1}{J} and \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} = \\frac{\\cos{(J)} - 1}{J} and \\int \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} dJ = \\int \\frac{\\cos{(J)} - 1}{J} dJ and 1 = \\frac{\\int \\frac{\\cos{(J)} - 1}{J} dJ}{\\int \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} dJ} and 1 = \\frac{n_{2} - \\log{(J)} + \\operatorname{Ci}{(J)}}{\\int \\frac{\\frac{d}{d J} \\operatorname{E_{n}}{(J)} - 1}{J} dJ}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 3, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(sin(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(cos(Symbol('J', commutative=True)), Integer(-1))))"], [["integrate", 5, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(cos(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))))"], [["divide", 6, "Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(cos(Symbol('J', commutative=True)), Integer(-1))), Tuple(Symbol('J', commutative=True))), Pow(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 7], "Equality(Integer(1), Mul(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), log(Symbol('J', commutative=True))), Ci(Symbol('J', commutative=True))), Pow(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Derivative(Function('E_n')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\Psi,F_{x})} = \\log{(\\frac{F_{x}}{\\Psi})} and \\psi{(\\Psi,F_{x})} = \\log{(\\frac{F_{x}}{\\Psi})}, then obtain \\frac{\\partial}{\\partial \\Psi} \\psi{(\\Psi,F_{x})} = \\frac{\\partial}{\\partial \\Psi} \\dot{\\mathbf{r}}{(\\Psi,F_{x})}", "derivation": "\\dot{\\mathbf{r}}{(\\Psi,F_{x})} = \\log{(\\frac{F_{x}}{\\Psi})} and \\psi{(\\Psi,F_{x})} = \\log{(\\frac{F_{x}}{\\Psi})} and \\psi{(\\Psi,F_{x})} = \\dot{\\mathbf{r}}{(\\Psi,F_{x})} and \\frac{\\partial}{\\partial \\Psi} \\psi{(\\Psi,F_{x})} = \\frac{\\partial}{\\partial \\Psi} \\dot{\\mathbf{r}}{(\\Psi,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)), log(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)), log(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\psi')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\Psi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(U)} = \\sin{(U)} and a{(U)} = \\frac{\\sin{(U)}}{q{(U)}}, then obtain (\\frac{1}{\\operatorname{F_{H}}{(U)}})^{U} = (\\frac{(\\frac{\\sin{(U)}}{q{(U)}})^{U}}{\\operatorname{F_{H}}{(U)}})^{U}", "derivation": "q{(U)} = \\sin{(U)} and 1 = \\frac{\\sin{(U)}}{q{(U)}} and 1 = (\\frac{\\sin{(U)}}{q{(U)}})^{U} and \\frac{1}{\\operatorname{F_{H}}{(U)}} = \\frac{(\\frac{\\sin{(U)}}{q{(U)}})^{U}}{\\operatorname{F_{H}}{(U)}} and a{(U)} = \\frac{\\sin{(U)}}{q{(U)}} and \\frac{1}{\\operatorname{F_{H}}{(U)}} = \\frac{a^{U}{(U)}}{\\operatorname{F_{H}}{(U)}} and (\\frac{1}{\\operatorname{F_{H}}{(U)}})^{U} = (\\frac{a^{U}{(U)}}{\\operatorname{F_{H}}{(U)}})^{U} and (\\frac{1}{\\operatorname{F_{H}}{(U)}})^{U} = (\\frac{(\\frac{\\sin{(U)}}{q{(U)}})^{U}}{\\operatorname{F_{H}}{(U)}})^{U}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["divide", 1, "Function('q')(Symbol('U', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('q')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('q')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["divide", 3, "Function('F_H')(Symbol('U', commutative=True))"], "Equality(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Mul(Pow(Mul(Pow(Function('q')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('a')(Symbol('U', commutative=True)), Mul(Pow(Function('q')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Mul(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Pow(Function('a')(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["power", 6, "Symbol('U', commutative=True)"], "Equality(Pow(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Mul(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Pow(Function('a')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Mul(Pow(Mul(Pow(Function('q')(Symbol('U', commutative=True)), Integer(-1)), sin(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Function('F_H')(Symbol('U', commutative=True)), Integer(-1))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given z{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\chi{(\\mathbf{H})} = 2 \\cos{(\\mathbf{H})} and \\mathbf{D}{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain 2 \\mathbf{D}{(\\mathbf{H})} = z{(\\mathbf{H})} + \\cos{(\\mathbf{H})}", "derivation": "z{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and z{(\\mathbf{H})} + \\cos{(\\mathbf{H})} = 2 \\cos{(\\mathbf{H})} and \\chi{(\\mathbf{H})} = 2 \\cos{(\\mathbf{H})} and \\chi{(\\mathbf{H})} = 2 z{(\\mathbf{H})} and \\chi{(\\mathbf{H})} = z{(\\mathbf{H})} + \\cos{(\\mathbf{H})} and 2 z{(\\mathbf{H})} = 2 \\cos{(\\mathbf{H})} and \\mathbf{D}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and 2 z{(\\mathbf{H})} = z{(\\mathbf{H})} + \\cos{(\\mathbf{H})} and 2 z{(\\mathbf{H})} = 2 \\mathbf{D}{(\\mathbf{H})} and 2 \\mathbf{D}{(\\mathbf{H})} = z{(\\mathbf{H})} + \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{H}', commutative=True)), Add(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Integer(2), Function('z')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 9], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Function('z')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain \\varepsilon^{3}{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\varepsilon^{2}{(\\mathbf{H})} \\cos^{2}{(\\mathbf{H})}", "derivation": "\\varepsilon{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\varepsilon{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\cos^{2}{(\\mathbf{H})} and \\varepsilon^{2}{(\\mathbf{H})} \\cos^{2}{(\\mathbf{H})} = \\cos^{4}{(\\mathbf{H})} and \\varepsilon^{3}{(\\mathbf{H})} \\cos{(\\mathbf{H})} = \\varepsilon^{2}{(\\mathbf{H})} \\cos^{2}{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(3)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{p}{(r_{0},f_{E})} = (e^{r_{0}})^{f_{E}} and t{(r_{0},f_{E})} = e^{f_{E} \\mathbf{p}{(r_{0},f_{E})}}, then obtain t{(r_{0},f_{E})} = e^{f_{E} (e^{r_{0}})^{f_{E}}}", "derivation": "\\mathbf{p}{(r_{0},f_{E})} = (e^{r_{0}})^{f_{E}} and f_{E} \\mathbf{p}{(r_{0},f_{E})} = f_{E} (e^{r_{0}})^{f_{E}} and e^{f_{E} \\mathbf{p}{(r_{0},f_{E})}} = e^{f_{E} (e^{r_{0}})^{f_{E}}} and t{(r_{0},f_{E})} = e^{f_{E} \\mathbf{p}{(r_{0},f_{E})}} and t{(r_{0},f_{E})} = e^{f_{E} (e^{r_{0}})^{f_{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True)), Pow(exp(Symbol('r_0', commutative=True)), Symbol('f_E', commutative=True)))"], [["times", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('r_0', commutative=True)), Symbol('f_E', commutative=True))))"], [["exp", 2], "Equality(exp(Mul(Symbol('f_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True)))), exp(Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('r_0', commutative=True)), Symbol('f_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('t')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True)), exp(Mul(Symbol('f_E', commutative=True), Function('\\\\mathbf{p}')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('t')(Symbol('r_0', commutative=True), Symbol('f_E', commutative=True)), exp(Mul(Symbol('f_E', commutative=True), Pow(exp(Symbol('r_0', commutative=True)), Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(f^{*})} = e^{\\sin{(f^{*})}}, then obtain (\\int \\rho_{f}{(f^{*})} df^{*}) \\iint \\rho_{f}{(f^{*})} df^{*} df^{*} = (\\int e^{\\sin{(f^{*})}} df^{*}) \\iint \\rho_{f}{(f^{*})} df^{*} df^{*}", "derivation": "\\rho_{f}{(f^{*})} = e^{\\sin{(f^{*})}} and \\int \\rho_{f}{(f^{*})} df^{*} = \\int e^{\\sin{(f^{*})}} df^{*} and \\iint \\rho_{f}{(f^{*})} df^{*} df^{*} = \\iint e^{\\sin{(f^{*})}} df^{*} df^{*} and (\\int \\rho_{f}{(f^{*})} df^{*}) \\iint e^{\\sin{(f^{*})}} df^{*} df^{*} = (\\int e^{\\sin{(f^{*})}} df^{*}) \\iint e^{\\sin{(f^{*})}} df^{*} df^{*} and (\\int \\rho_{f}{(f^{*})} df^{*}) \\iint \\rho_{f}{(f^{*})} df^{*} df^{*} = (\\int e^{\\sin{(f^{*})}} df^{*}) \\iint \\rho_{f}{(f^{*})} df^{*} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), exp(sin(Symbol('f^*', commutative=True))))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["times", 2, "Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Integral(exp(sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Function('\\\\rho_f')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given u{(v_{t})} = e^{v_{t}} and \\dot{x}{(\\mathbf{P},v_{t})} = \\mathbf{P} + e^{v_{t}}, then obtain \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} u{(v_{t})} + r = \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} e^{v_{t}} + \\phi", "derivation": "u{(v_{t})} = e^{v_{t}} and \\dot{x}{(\\mathbf{P},v_{t})} = \\mathbf{P} + e^{v_{t}} and \\dot{x}{(\\mathbf{P},v_{t})} = \\mathbf{P} + u{(v_{t})} and \\mathbf{P} + u{(v_{t})} = \\mathbf{P} + e^{v_{t}} and \\int (\\mathbf{P} + u{(v_{t})}) d\\mathbf{P} = \\int (\\mathbf{P} + e^{v_{t}}) d\\mathbf{P} and \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} u{(v_{t})} + r = \\frac{\\mathbf{P}^{2}}{2} + \\mathbf{P} e^{v_{t}} + \\phi", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Function('u')(Symbol('v_t', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('u')(Symbol('v_t', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('u')(Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('u')(Symbol('v_t', commutative=True))), Symbol('r', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{P}', commutative=True), exp(Symbol('v_t', commutative=True))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(H)} = \\int \\cos{(H)} dH and \\operatorname{A_{x}}{(H)} = \\cos{(H)}, then derive \\frac{d}{d H} \\hat{H}{(H)} = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)}), then derive \\frac{d}{d H} \\hat{H}{(H)} = \\cos{(H)}, then obtain \\frac{d}{d H} \\int \\operatorname{A_{x}}{(H)} dH = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)})", "derivation": "\\hat{H}{(H)} = \\int \\cos{(H)} dH and \\frac{d}{d H} \\hat{H}{(H)} = \\frac{d}{d H} \\int \\cos{(H)} dH and \\frac{d}{d H} \\hat{H}{(H)} = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)}) and \\frac{d}{d H} \\hat{H}{(H)} = \\cos{(H)} and \\cos{(H)} = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)}) and \\operatorname{A_{x}}{(H)} = \\cos{(H)} and \\int \\operatorname{A_{x}}{(H)} dH = \\int \\cos{(H)} dH and \\cos{(H)} = \\frac{d}{d H} \\int \\cos{(H)} dH and \\frac{d}{d H} \\int \\cos{(H)} dH = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)}) and \\frac{d}{d H} \\int \\operatorname{A_{x}}{(H)} dH = \\frac{\\partial}{\\partial H} (c_{0} + \\sin{(H)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('H', commutative=True)), Integral(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('c_0', commutative=True), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), cos(Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(cos(Symbol('H', commutative=True)), Derivative(Add(Symbol('c_0', commutative=True), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(cos(Symbol('H', commutative=True)), Derivative(Integral(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 8], "Equality(Derivative(Integral(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('c_0', commutative=True), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 9, 7], "Equality(Derivative(Integral(Function('A_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('c_0', commutative=True), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}}, then derive (\\eta^{\\prime} + v_{y})^{\\eta^{\\prime}} = (\\int \\frac{e^{e^{\\eta^{\\prime}}}}{\\operatorname{f^{\\prime}}{(\\eta^{\\prime})}} d\\eta^{\\prime})^{\\eta^{\\prime}}, then obtain (\\eta^{\\prime} + v_{y})^{\\eta^{\\prime}} = (\\int 1 d\\eta^{\\prime})^{\\eta^{\\prime}}", "derivation": "\\operatorname{f^{\\prime}}{(\\eta^{\\prime})} = e^{e^{\\eta^{\\prime}}} and 1 = \\frac{e^{e^{\\eta^{\\prime}}}}{\\operatorname{f^{\\prime}}{(\\eta^{\\prime})}} and \\int 1 d\\eta^{\\prime} = \\int \\frac{e^{e^{\\eta^{\\prime}}}}{\\operatorname{f^{\\prime}}{(\\eta^{\\prime})}} d\\eta^{\\prime} and (\\int 1 d\\eta^{\\prime})^{\\eta^{\\prime}} = (\\int \\frac{e^{e^{\\eta^{\\prime}}}}{\\operatorname{f^{\\prime}}{(\\eta^{\\prime})}} d\\eta^{\\prime})^{\\eta^{\\prime}} and (\\eta^{\\prime} + v_{y})^{\\eta^{\\prime}} = (\\int \\frac{e^{e^{\\eta^{\\prime}}}}{\\operatorname{f^{\\prime}}{(\\eta^{\\prime})}} d\\eta^{\\prime})^{\\eta^{\\prime}} and (\\eta^{\\prime} + v_{y})^{\\eta^{\\prime}} = (\\int 1 d\\eta^{\\prime})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_y', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Integral(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_y', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = \\sin{(\\frac{c}{a})}, then obtain a \\frac{\\partial}{\\partial c} \\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = \\cos{(\\frac{c}{a})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = \\sin{(\\frac{c}{a})} and \\frac{\\partial}{\\partial c} \\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = \\frac{\\partial}{\\partial c} \\sin{(\\frac{c}{a})} and a \\frac{\\partial}{\\partial c} \\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = a \\frac{\\partial}{\\partial c} \\sin{(\\frac{c}{a})} and a \\frac{\\partial}{\\partial c} \\operatorname{g^{\\prime}_{\\varepsilon}}{(a,c)} = \\cos{(\\frac{c}{a})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["times", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Symbol('a', commutative=True), Derivative(sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('a', commutative=True), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), cos(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\mathbf{J},v_{z})} = \\frac{\\mathbf{J}}{v_{z}} and \\operatorname{m_{s}}{(\\mathbf{J},v_{z})} = 1 - \\sigma_{x}{(\\mathbf{J},v_{z})}, then obtain \\operatorname{m_{s}}^{\\mathbf{J}}{(\\mathbf{J},v_{z})} = (1 - \\sigma_{x}{(\\mathbf{J},v_{z})})^{\\mathbf{J}}", "derivation": "\\sigma_{x}{(\\mathbf{J},v_{z})} = \\frac{\\mathbf{J}}{v_{z}} and - \\sigma_{x}{(\\mathbf{J},v_{z})} = - \\frac{\\mathbf{J}}{v_{z}} and 1 - \\sigma_{x}{(\\mathbf{J},v_{z})} = - \\frac{\\mathbf{J}}{v_{z}} + 1 and \\operatorname{m_{s}}{(\\mathbf{J},v_{z})} = 1 - \\sigma_{x}{(\\mathbf{J},v_{z})} and (1 - \\sigma_{x}{(\\mathbf{J},v_{z})})^{\\mathbf{J}} = (- \\frac{\\mathbf{J}}{v_{z}} + 1)^{\\mathbf{J}} and \\operatorname{m_{s}}^{\\mathbf{J}}{(\\mathbf{J},v_{z})} = (- \\frac{\\mathbf{J}}{v_{z}} + 1)^{\\mathbf{J}} and \\operatorname{m_{s}}^{\\mathbf{J}}{(\\mathbf{J},v_{z})} = (1 - \\sigma_{x}{(\\mathbf{J},v_{z})})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)), Add(Integer(1), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))), Integer(1)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(V,\\mathbf{F})} = \\frac{\\mathbf{F}}{V}, then obtain 0 = 2 \\operatorname{F_{N}}{(V,\\mathbf{F})} - \\frac{2 \\mathbf{F}}{V}", "derivation": "\\operatorname{F_{N}}{(V,\\mathbf{F})} = \\frac{\\mathbf{F}}{V} and 0 = - \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{\\mathbf{F}}{V} and - \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{\\mathbf{F}}{V} = - 2 \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{2 \\mathbf{F}}{V} and \\frac{\\mathbf{F}}{V} = - \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{2 \\mathbf{F}}{V} and 0 = - 2 \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{2 \\mathbf{F}}{V} and \\operatorname{F_{N}}{(V,\\mathbf{F})} = - \\operatorname{F_{N}}{(V,\\mathbf{F})} + \\frac{2 \\mathbf{F}}{V} and 0 = 2 \\operatorname{F_{N}}{(V,\\mathbf{F})} - \\frac{2 \\mathbf{F}}{V}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Integer(0), Add(Mul(Integer(2), Function('F_N')(Symbol('V', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(k,r_{0})} = k r_{0}, then obtain (\\frac{d}{d k} \\frac{1}{r_{0}})^{k} = (\\frac{\\partial}{\\partial k} \\frac{\\iint k r_{0} dk dk}{r_{0} \\iint \\phi_{1}{(k,r_{0})} dk dk})^{k}", "derivation": "\\phi_{1}{(k,r_{0})} = k r_{0} and \\int \\phi_{1}{(k,r_{0})} dk = \\int k r_{0} dk and \\iint \\phi_{1}{(k,r_{0})} dk dk = \\iint k r_{0} dk dk and 1 = \\frac{\\iint k r_{0} dk dk}{\\iint \\phi_{1}{(k,r_{0})} dk dk} and \\frac{1}{r_{0}} = \\frac{\\iint k r_{0} dk dk}{r_{0} \\iint \\phi_{1}{(k,r_{0})} dk dk} and \\frac{d}{d k} \\frac{1}{r_{0}} = \\frac{\\partial}{\\partial k} \\frac{\\iint k r_{0} dk dk}{r_{0} \\iint \\phi_{1}{(k,r_{0})} dk dk} and (\\frac{d}{d k} \\frac{1}{r_{0}})^{k} = (\\frac{\\partial}{\\partial k} \\frac{\\iint k r_{0} dk dk}{r_{0} \\iint \\phi_{1}{(k,r_{0})} dk dk})^{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["divide", 3, "Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Pow(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))))"], [["divide", 4, "Symbol('r_0', commutative=True)"], "Equality(Pow(Symbol('r_0', commutative=True), Integer(-1)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Pow(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))))"], [["differentiate", 5, "Symbol('k', commutative=True)"], "Equality(Derivative(Pow(Symbol('r_0', commutative=True), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Pow(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Pow(Symbol('r_0', commutative=True), Integer(-1)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Mul(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Pow(Integral(Function('\\\\phi_1')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given U{(\\mathbf{P},v_{1})} = \\mathbf{P} + v_{1}, then obtain - U{(\\mathbf{P},v_{1})} + \\int U{(\\mathbf{P},v_{1})} dv_{1} = - \\mathbf{P} - v_{1} + \\int U{(\\mathbf{P},v_{1})} dv_{1}", "derivation": "U{(\\mathbf{P},v_{1})} = \\mathbf{P} + v_{1} and \\int U{(\\mathbf{P},v_{1})} dv_{1} = \\int (\\mathbf{P} + v_{1}) dv_{1} and U{(\\mathbf{P},v_{1})} - \\int (\\mathbf{P} + v_{1}) dv_{1} = \\mathbf{P} + v_{1} - \\int (\\mathbf{P} + v_{1}) dv_{1} and U{(\\mathbf{P},v_{1})} - \\int U{(\\mathbf{P},v_{1})} dv_{1} = \\mathbf{P} + v_{1} - \\int U{(\\mathbf{P},v_{1})} dv_{1} and - U{(\\mathbf{P},v_{1})} + \\int U{(\\mathbf{P},v_{1})} dv_{1} = - \\mathbf{P} - v_{1} + \\int U{(\\mathbf{P},v_{1})} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["minus", 1, "Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True), Mul(Integer(-1), Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True))), Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Function('U')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})}, then derive 0 = V + \\sin{(f_{\\mathbf{v}})} - \\int \\operatorname{F_{H}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}}, then derive 0 = V - q, then obtain 0 = - V + q", "derivation": "\\operatorname{F_{H}}{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})} and \\int \\operatorname{F_{H}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int \\cos{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and 0 = - \\int \\operatorname{F_{H}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} + \\int \\cos{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and 0 = V + \\sin{(f_{\\mathbf{v}})} - \\int \\operatorname{F_{H}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and 0 = V + \\sin{(f_{\\mathbf{v}})} - \\int \\cos{(f_{\\mathbf{v}})} df_{\\mathbf{v}} and 0 = V - q and \\frac{d}{d q} 0 = \\frac{\\partial}{\\partial q} (V - q) and (V - q) \\frac{d}{d q} 0 = (V - q) \\frac{\\partial}{\\partial q} (V - q) and 0 = - V + q", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["minus", 2, "Integral(Function('F_H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('F_H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Integral(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Symbol('V', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Integral(Function('F_H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Symbol('V', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Integer(0), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["differentiate", 6, "Symbol('q', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["times", 7, "Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))"], "Equality(Mul(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 8], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\rho_b,E_{x})} = E_{x} + \\cos{(\\rho_b)}, then obtain \\int E_{x} \\int \\operatorname{v_{2}}{(\\rho_b,E_{x})} d\\rho_b d\\rho_b = \\int E_{x} \\int (E_{x} + \\cos{(\\rho_b)}) d\\rho_b d\\rho_b", "derivation": "\\operatorname{v_{2}}{(\\rho_b,E_{x})} = E_{x} + \\cos{(\\rho_b)} and \\int \\operatorname{v_{2}}{(\\rho_b,E_{x})} d\\rho_b = \\int (E_{x} + \\cos{(\\rho_b)}) d\\rho_b and E_{x} \\int \\operatorname{v_{2}}{(\\rho_b,E_{x})} d\\rho_b = E_{x} \\int (E_{x} + \\cos{(\\rho_b)}) d\\rho_b and \\int E_{x} \\int \\operatorname{v_{2}}{(\\rho_b,E_{x})} d\\rho_b d\\rho_b = \\int E_{x} \\int (E_{x} + \\cos{(\\rho_b)}) d\\rho_b d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\rho_b', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), cos(Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\rho_b', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Add(Symbol('E_x', commutative=True), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["times", 2, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Integral(Function('v_2')(Symbol('\\\\rho_b', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Symbol('E_x', commutative=True), Integral(Add(Symbol('E_x', commutative=True), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(Symbol('E_x', commutative=True), Integral(Function('v_2')(Symbol('\\\\rho_b', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Symbol('E_x', commutative=True), Integral(Add(Symbol('E_x', commutative=True), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(S,f^{\\prime})} = \\frac{f^{\\prime}}{S} and Q{(t_{2})} = \\cos{(\\cos{(t_{2})})}, then obtain \\operatorname{t_{2}}{(S,f^{\\prime})} \\cos{(\\cos{(t_{2})})} = \\frac{f^{\\prime} \\cos{(\\cos{(t_{2})})}}{S}", "derivation": "\\operatorname{t_{2}}{(S,f^{\\prime})} = \\frac{f^{\\prime}}{S} and Q{(t_{2})} = \\cos{(\\cos{(t_{2})})} and Q{(t_{2})} \\operatorname{t_{2}}{(S,f^{\\prime})} = \\frac{f^{\\prime} Q{(t_{2})}}{S} and \\operatorname{t_{2}}{(S,f^{\\prime})} \\cos{(\\cos{(t_{2})})} = \\frac{f^{\\prime} \\cos{(\\cos{(t_{2})})}}{S}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('S', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], ["get_premise", "Equality(Function('Q')(Symbol('t_2', commutative=True)), cos(cos(Symbol('t_2', commutative=True))))"], [["times", 1, "Function('Q')(Symbol('t_2', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('t_2', commutative=True)), Function('t_2')(Symbol('S', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), Function('Q')(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('t_2')(Symbol('S', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), cos(cos(Symbol('t_2', commutative=True)))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True), cos(cos(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given b{(\\dot{\\mathbf{r}},L)} = L \\dot{\\mathbf{r}}, then obtain 0 = L \\dot{\\mathbf{r}} (- \\iint (L + b{(\\dot{\\mathbf{r}},L)}) dL dL + \\iint (L \\dot{\\mathbf{r}} + L) dL dL)", "derivation": "b{(\\dot{\\mathbf{r}},L)} = L \\dot{\\mathbf{r}} and L + b{(\\dot{\\mathbf{r}},L)} = L \\dot{\\mathbf{r}} + L and \\int (L + b{(\\dot{\\mathbf{r}},L)}) dL = \\int (L \\dot{\\mathbf{r}} + L) dL and \\iint (L + b{(\\dot{\\mathbf{r}},L)}) dL dL = \\iint (L \\dot{\\mathbf{r}} + L) dL dL and 0 = - \\iint (L + b{(\\dot{\\mathbf{r}},L)}) dL dL + \\iint (L \\dot{\\mathbf{r}} + L) dL dL and 0 = L \\dot{\\mathbf{r}} (- \\iint (L + b{(\\dot{\\mathbf{r}},L)}) dL dL + \\iint (L \\dot{\\mathbf{r}} + L) dL dL)", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["add", 1, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('L', commutative=True)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["minus", 4, "Integral(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integral(Add(Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["times", 5, "Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(0), Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Add(Mul(Integer(-1), Integral(Add(Symbol('L', commutative=True), Function('b')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integral(Add(Mul(Symbol('L', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))))"]]}, {"prompt": "Given f{(n_{1},A_{y})} = - A_{y} + n_{1}, then derive \\frac{\\partial}{\\partial n_{1}} f{(n_{1},A_{y})} = 1, then obtain \\frac{\\partial}{\\partial n_{1}} (- A_{y} + n_{1}) = 1", "derivation": "f{(n_{1},A_{y})} = - A_{y} + n_{1} and \\frac{\\partial}{\\partial n_{1}} f{(n_{1},A_{y})} = \\frac{\\partial}{\\partial n_{1}} (- A_{y} + n_{1}) and \\frac{\\partial}{\\partial n_{1}} f{(n_{1},A_{y})} = 1 and \\frac{\\partial}{\\partial n_{1}} (- A_{y} + n_{1}) = 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('n_1', commutative=True), Symbol('A_y', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('n_1', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('n_1', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given z{(S,z^{*})} = S + z^{*} and \\hat{x}_0{(S,z^{*})} = \\frac{\\partial}{\\partial S} (S + z^{*}) \\frac{\\partial}{\\partial S} z{(S,z^{*})}, then derive \\hat{x}_0{(S,z^{*})} = 1, then obtain \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,z^{*})} = \\frac{d}{d S} 1", "derivation": "z{(S,z^{*})} = S + z^{*} and \\frac{\\partial}{\\partial S} z{(S,z^{*})} = \\frac{\\partial}{\\partial S} (S + z^{*}) and \\frac{\\partial}{\\partial S} (S + z^{*}) \\frac{\\partial}{\\partial S} z{(S,z^{*})} = (\\frac{\\partial}{\\partial S} (S + z^{*}))^{2} and \\hat{x}_0{(S,z^{*})} = \\frac{\\partial}{\\partial S} (S + z^{*}) \\frac{\\partial}{\\partial S} z{(S,z^{*})} and \\hat{x}_0{(S,z^{*})} = (\\frac{\\partial}{\\partial S} (S + z^{*}))^{2} and \\hat{x}_0{(S,z^{*})} = 1 and \\frac{\\partial}{\\partial S} \\hat{x}_0{(S,z^{*})} = \\frac{d}{d S} 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Mul(Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Pow(Derivative(Add(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Integer(1))"], [["differentiate", 6, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(r_{0},J)} = \\frac{r_{0}}{J}, then obtain r_{0} + \\hat{H}_{\\lambda}{(r_{0},J)} - \\frac{2 r_{0}}{J} = r_{0} - \\frac{r_{0}}{J}", "derivation": "\\hat{H}_{\\lambda}{(r_{0},J)} = \\frac{r_{0}}{J} and \\hat{H}_{\\lambda}{(r_{0},J)} - \\frac{r_{0}}{J} = 0 and \\hat{H}_{\\lambda}{(r_{0},J)} - \\frac{2 r_{0}}{J} = - \\frac{r_{0}}{J} and r_{0} + \\hat{H}_{\\lambda}{(r_{0},J)} - \\frac{2 r_{0}}{J} = r_{0} - \\frac{r_{0}}{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Integer(0))"], [["add", 2, "Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["add", 3, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given v{(\\mathbf{S},H)} = \\cos{(\\frac{\\mathbf{S}}{H})}, then obtain \\frac{E_{\\lambda} + (- v{(\\mathbf{S},H)})^{\\mathbf{S}}}{H} = \\frac{E_{\\lambda} + (- \\cos{(\\frac{\\mathbf{S}}{H})})^{\\mathbf{S}}}{H}", "derivation": "v{(\\mathbf{S},H)} = \\cos{(\\frac{\\mathbf{S}}{H})} and - v{(\\mathbf{S},H)} = - \\cos{(\\frac{\\mathbf{S}}{H})} and (- v{(\\mathbf{S},H)})^{\\mathbf{S}} = (- \\cos{(\\frac{\\mathbf{S}}{H})})^{\\mathbf{S}} and E_{\\lambda} + (- v{(\\mathbf{S},H)})^{\\mathbf{S}} = E_{\\lambda} + (- \\cos{(\\frac{\\mathbf{S}}{H})})^{\\mathbf{S}} and \\frac{E_{\\lambda} + (- v{(\\mathbf{S},H)})^{\\mathbf{S}}}{H} = \\frac{E_{\\lambda} + (- \\cos{(\\frac{\\mathbf{S}}{H})})^{\\mathbf{S}}}{H}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True)), cos(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), cos(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('v')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Integer(-1), cos(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Mul(Integer(-1), Function('v')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Mul(Integer(-1), cos(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 4, "Pow(Symbol('H', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Mul(Integer(-1), Function('v')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Pow(Mul(Integer(-1), cos(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\chi,\\hat{x},c_{0})} = (\\chi \\hat{x})^{c_{0}} and \\varphi{(\\mathbf{M},v_{1})} = \\mathbf{M} + v_{1}, then obtain - \\int \\rho{(\\chi,\\hat{x},c_{0})} d\\chi + \\int \\varphi{(\\mathbf{M},v_{1})} d\\mathbf{M} = \\int (\\mathbf{M} + v_{1}) d\\mathbf{M} - \\int \\rho{(\\chi,\\hat{x},c_{0})} d\\chi", "derivation": "\\rho{(\\chi,\\hat{x},c_{0})} = (\\chi \\hat{x})^{c_{0}} and \\int \\rho{(\\chi,\\hat{x},c_{0})} d\\chi = \\int (\\chi \\hat{x})^{c_{0}} d\\chi and \\varphi{(\\mathbf{M},v_{1})} = \\mathbf{M} + v_{1} and \\int \\varphi{(\\mathbf{M},v_{1})} d\\mathbf{M} = \\int (\\mathbf{M} + v_{1}) d\\mathbf{M} and - \\int (\\chi \\hat{x})^{c_{0}} d\\chi + \\int \\varphi{(\\mathbf{M},v_{1})} d\\mathbf{M} = - \\int (\\chi \\hat{x})^{c_{0}} d\\chi + \\int (\\mathbf{M} + v_{1}) d\\mathbf{M} and - \\int \\rho{(\\chi,\\hat{x},c_{0})} d\\chi + \\int \\varphi{(\\mathbf{M},v_{1})} d\\mathbf{M} = \\int (\\mathbf{M} + v_{1}) d\\mathbf{M} - \\int \\rho{(\\chi,\\hat{x},c_{0})} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('c_0', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 4, "Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Integral(Function('\\\\varphi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Integral(Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Integral(Function('\\\\varphi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{S}{(\\rho,\\hat{X})} = \\sin{(\\hat{X} - \\rho)} and i{(\\hat{X},\\rho)} = (\\mathbf{S}^{\\hat{X}}{(\\rho,\\hat{X})})^{\\hat{X}}, then obtain i{(\\hat{X},\\rho)} = (\\sin^{\\hat{X}}{(\\hat{X} - \\rho)})^{\\hat{X}}", "derivation": "\\mathbf{S}{(\\rho,\\hat{X})} = \\sin{(\\hat{X} - \\rho)} and \\mathbf{S}^{\\hat{X}}{(\\rho,\\hat{X})} = \\sin^{\\hat{X}}{(\\hat{X} - \\rho)} and (\\mathbf{S}^{\\hat{X}}{(\\rho,\\hat{X})})^{\\hat{X}} = (\\sin^{\\hat{X}}{(\\hat{X} - \\rho)})^{\\hat{X}} and i{(\\hat{X},\\rho)} = (\\mathbf{S}^{\\hat{X}}{(\\rho,\\hat{X})})^{\\hat{X}} and i{(\\hat{X},\\rho)} = (\\sin^{\\hat{X}}{(\\hat{X} - \\rho)})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('i')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Pow(sin(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\lambda{(\\hat{X})} = e^{e^{\\hat{X}}} and \\mu{(\\hat{X})} = - \\hat{X} + \\lambda{(\\hat{X})}, then obtain (\\lambda{(\\hat{X})} + e^{e^{\\hat{X}}}) \\int (- \\hat{X} + \\lambda{(\\hat{X})}) d\\hat{X} = (\\lambda{(\\hat{X})} + e^{e^{\\hat{X}}}) \\int (- \\hat{X} + e^{e^{\\hat{X}}}) d\\hat{X}", "derivation": "\\lambda{(\\hat{X})} = e^{e^{\\hat{X}}} and - \\hat{X} + \\lambda{(\\hat{X})} = - \\hat{X} + e^{e^{\\hat{X}}} and \\mu{(\\hat{X})} = - \\hat{X} + \\lambda{(\\hat{X})} and \\mu{(\\hat{X})} = - \\hat{X} + e^{e^{\\hat{X}}} and \\int \\mu{(\\hat{X})} d\\hat{X} = \\int (- \\hat{X} + e^{e^{\\hat{X}}}) d\\hat{X} and \\int (- \\hat{X} + \\lambda{(\\hat{X})}) d\\hat{X} = \\int (- \\hat{X} + e^{e^{\\hat{X}}}) d\\hat{X} and (\\lambda{(\\hat{X})} + e^{e^{\\hat{X}}}) \\int (- \\hat{X} + \\lambda{(\\hat{X})}) d\\hat{X} = (\\lambda{(\\hat{X})} + e^{e^{\\hat{X}}}) \\int (- \\hat{X} + e^{e^{\\hat{X}}}) d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mu')(Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 6, "Add(Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True))))"], "Equality(Mul(Add(Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Mul(Add(Function('\\\\lambda')(Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(exp(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\phi_2)} = \\log{(\\phi_2)}, then obtain (\\operatorname{C_{d}}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)})^{\\phi_2} = (\\log{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)})^{\\phi_2}", "derivation": "\\operatorname{C_{d}}{(\\phi_2)} = \\log{(\\phi_2)} and \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\log{(\\phi_2)} and \\operatorname{C_{d}}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\log{(\\phi_2)} = \\log{(\\phi_2)} + \\frac{d}{d \\phi_2} \\log{(\\phi_2)} and \\operatorname{C_{d}}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)} = \\log{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)} and (\\operatorname{C_{d}}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)})^{\\phi_2} = (\\log{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{C_{d}}{(\\phi_2)})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 1, "Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Add(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(log(Symbol('\\\\phi_2', commutative=True)), Derivative(log(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(log(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Symbol('\\\\phi_2', commutative=True)), Pow(Add(log(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given U{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})}, then derive \\mathbf{g} + U{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})}, then obtain \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})}", "derivation": "U{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and \\mathbf{g} + U{(\\mathbf{g})} = \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} and \\mathbf{g} + U{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})} and \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})} = \\mathbf{g} - \\sin{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{g}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Function('U')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Function('U')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(i)} = \\cos{(i)}, then derive \\cos{(i)} + \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = - \\sin{(i)} + \\cos{(i)}, then obtain \\operatorname{y^{\\prime}}{(i)} + \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = \\operatorname{y^{\\prime}}{(i)} - \\sin{(i)}", "derivation": "\\operatorname{y^{\\prime}}{(i)} = \\cos{(i)} and \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = \\frac{d}{d i} \\cos{(i)} and \\cos{(i)} + \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = \\cos{(i)} + \\frac{d}{d i} \\cos{(i)} and \\cos{(i)} + \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = - \\sin{(i)} + \\cos{(i)} and \\operatorname{y^{\\prime}}{(i)} + \\frac{d}{d i} \\operatorname{y^{\\prime}}{(i)} = \\operatorname{y^{\\prime}}{(i)} - \\sin{(i)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["add", 2, "cos(Symbol('i', commutative=True))"], "Equality(Add(cos(Symbol('i', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(cos(Symbol('i', commutative=True)), Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('i', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Function('y^{\\\\prime}')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(h)} = e^{h} and \\operatorname{F_{c}}{(h)} = h + (\\Psi_{nl}{(h)} - e^{h})^{h}, then obtain \\int (h + 1) dh = \\int (h + (\\Psi_{nl}{(h)} - e^{h})^{h}) dh", "derivation": "\\Psi_{nl}{(h)} = e^{h} and \\Psi_{nl}{(h)} - e^{h} = 0 and (\\Psi_{nl}{(h)} - e^{h})^{h} = 0^{h} and h + (\\Psi_{nl}{(h)} - e^{h})^{h} = 0^{h} + h and \\operatorname{F_{c}}{(h)} = h + (\\Psi_{nl}{(h)} - e^{h})^{h} and h + 1 = h + (\\Psi_{nl}{(h)} - e^{h})^{h} and h + 1 = \\operatorname{F_{c}}{(h)} and \\int (h + 1) dh = \\int \\operatorname{F_{c}}{(h)} dh and \\int (h + 1) dh = \\int (h + (\\Psi_{nl}{(h)} - e^{h})^{h}) dh", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["minus", 1, "exp(Symbol('h', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Integer(0), Symbol('h', commutative=True)))"], [["add", 3, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Pow(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Symbol('h', commutative=True))), Add(Pow(Integer(0), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Pow(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('h', commutative=True), Integer(1)), Add(Symbol('h', commutative=True), Pow(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('h', commutative=True), Integer(1)), Function('F_c')(Symbol('h', commutative=True)))"], [["integrate", 7, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Symbol('h', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True))), Integral(Function('F_c')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Integral(Add(Symbol('h', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True))), Integral(Add(Symbol('h', commutative=True), Pow(Add(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True)), Mul(Integer(-1), exp(Symbol('h', commutative=True)))), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(B)} = \\log{(e^{B})} and \\operatorname{P_{g}}{(B)} = e^{B}, then derive 2 = 1 + \\frac{\\frac{d}{d B} \\operatorname{P_{g}}{(B)}}{\\operatorname{P_{g}}{(B)}}, then obtain \\int (2 - B) dB = \\int (- B + 1 + \\frac{\\frac{d}{d B} \\operatorname{P_{g}}{(B)}}{\\operatorname{P_{g}}{(B)}}) dB", "derivation": "\\mathbf{S}{(B)} = \\log{(e^{B})} and \\frac{d}{d B} \\mathbf{S}{(B)} = \\frac{d}{d B} \\log{(e^{B})} and \\operatorname{P_{g}}{(B)} = e^{B} and \\frac{d}{d B} \\mathbf{S}{(B)} = \\frac{d}{d B} \\log{(\\operatorname{P_{g}}{(B)})} and \\frac{d}{d B} \\log{(e^{B})} = \\frac{d}{d B} \\log{(\\operatorname{P_{g}}{(B)})} and 2 \\frac{d}{d B} \\log{(e^{B})} = \\frac{d}{d B} \\log{(\\operatorname{P_{g}}{(B)})} + \\frac{d}{d B} \\log{(e^{B})} and 2 = 1 + \\frac{\\frac{d}{d B} \\operatorname{P_{g}}{(B)}}{\\operatorname{P_{g}}{(B)}} and 2 - B = - B + 1 + \\frac{\\frac{d}{d B} \\operatorname{P_{g}}{(B)}}{\\operatorname{P_{g}}{(B)}} and \\int (2 - B) dB = \\int (- B + 1 + \\frac{\\frac{d}{d B} \\operatorname{P_{g}}{(B)}}{\\operatorname{P_{g}}{(B)}}) dB", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('B', commutative=True)), log(exp(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(Function('P_g')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(Function('P_g')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["add", 5, "Derivative(log(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(log(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Derivative(log(Function('P_g')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(log(exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('P_g')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))))"], [["minus", 7, "Symbol('B', commutative=True)"], "Equality(Add(Integer(2), Mul(Integer(-1), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integer(1), Mul(Pow(Function('P_g')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))))"], [["integrate", 8, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Integer(2), Mul(Integer(-1), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integer(1), Mul(Pow(Function('P_g')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(F_{N})} = e^{F_{N}}, then obtain (\\frac{1}{2})^{F_{N}} = (\\frac{e^{F_{N}}}{2 \\mathbf{H}{(F_{N})}})^{F_{N}}", "derivation": "\\mathbf{H}{(F_{N})} = e^{F_{N}} and 2 \\mathbf{H}{(F_{N})} = \\mathbf{H}{(F_{N})} + e^{F_{N}} and \\frac{\\mathbf{H}{(F_{N})}}{\\mathbf{H}{(F_{N})} + e^{F_{N}}} = \\frac{e^{F_{N}}}{\\mathbf{H}{(F_{N})} + e^{F_{N}}} and \\frac{1}{2} = \\frac{e^{F_{N}}}{2 \\mathbf{H}{(F_{N})}} and (\\frac{1}{2})^{F_{N}} = (\\frac{e^{F_{N}}}{2 \\mathbf{H}{(F_{N})}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True))))"], [["divide", 1, "Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True))), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True))), Mul(Pow(Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True))), Integer(-1)), exp(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), Integer(-1)), exp(Symbol('F_N', commutative=True))))"], [["power", 4, "Symbol('F_N', commutative=True)"], "Equality(Pow(Rational(1, 2), Symbol('F_N', commutative=True)), Pow(Mul(Rational(1, 2), Pow(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), Integer(-1)), exp(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\Omega)} = e^{\\Omega} and \\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega \\hat{p}^{2}{(\\Omega)}}, then obtain \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\Omega \\hat{p}{(\\Omega)} e^{\\Omega}}", "derivation": "\\hat{p}{(\\Omega)} = e^{\\Omega} and \\hat{p}^{2}{(\\Omega)} = \\hat{p}{(\\Omega)} e^{\\Omega} and \\Omega \\hat{p}^{2}{(\\Omega)} = \\Omega \\hat{p}{(\\Omega)} e^{\\Omega} and e^{\\Omega \\hat{p}^{2}{(\\Omega)}} = e^{\\Omega \\hat{p}{(\\Omega)} e^{\\Omega}} and \\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega \\hat{p}^{2}{(\\Omega)}} and \\frac{d}{d \\Omega} e^{\\Omega \\hat{p}^{2}{(\\Omega)}} = \\frac{d}{d \\Omega} e^{\\Omega \\hat{p}{(\\Omega)} e^{\\Omega}} and \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\Omega \\hat{p}{(\\Omega)} e^{\\Omega}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True))))"], [["times", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integer(2)))), exp(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integer(2)))))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(exp(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{A},Z)} = \\cos{(Z \\mathbf{A})} and \\operatorname{f_{E}}{(Z,\\mathbf{A})} = \\int (\\mathbf{A} + \\cos{(Z \\mathbf{A})}) d\\mathbf{A}, then obtain \\int (\\mathbf{A} + \\operatorname{m_{s}}{(\\mathbf{A},Z)}) d\\mathbf{A} = \\operatorname{f_{E}}{(Z,\\mathbf{A})}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{A},Z)} = \\cos{(Z \\mathbf{A})} and \\mathbf{A} + \\operatorname{m_{s}}{(\\mathbf{A},Z)} = \\mathbf{A} + \\cos{(Z \\mathbf{A})} and \\int (\\mathbf{A} + \\operatorname{m_{s}}{(\\mathbf{A},Z)}) d\\mathbf{A} = \\int (\\mathbf{A} + \\cos{(Z \\mathbf{A})}) d\\mathbf{A} and \\operatorname{f_{E}}{(Z,\\mathbf{A})} = \\int (\\mathbf{A} + \\cos{(Z \\mathbf{A})}) d\\mathbf{A} and \\int (\\mathbf{A} + \\operatorname{m_{s}}{(\\mathbf{A},Z)}) d\\mathbf{A} = \\operatorname{f_{E}}{(Z,\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Z', commutative=True)), cos(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Mul(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Function('f_E')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\rho,p)} = \\rho \\cos{(p)}, then derive \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)} = \\cos{(p)}, then derive (\\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)})^{p} = \\cos^{p}{(p)}, then obtain (\\frac{\\partial}{\\partial \\rho} \\rho \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)})^{p} = \\cos^{p}{(p)}", "derivation": "\\mathbf{J}_f{(\\rho,p)} = \\rho \\cos{(p)} and \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)} = \\frac{\\partial}{\\partial \\rho} \\rho \\cos{(p)} and \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)} = \\cos{(p)} and (\\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)})^{p} = (\\frac{\\partial}{\\partial \\rho} \\rho \\cos{(p)})^{p} and (\\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)})^{p} = \\cos^{p}{(p)} and \\mathbf{J}_f{(\\rho,p)} = \\rho \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)} and (\\frac{\\partial}{\\partial \\rho} \\rho \\frac{\\partial}{\\partial \\rho} \\mathbf{J}_f{(\\rho,p)})^{p} = \\cos^{p}{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), cos(Symbol('p', commutative=True)))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Derivative(Mul(Symbol('\\\\rho', commutative=True), Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} = \\frac{\\mathbf{J}}{\\mathbf{A}} and \\operatorname{P_{g}}{(\\mathbf{J},\\mathbf{A})} = \\frac{\\mathbf{J}}{\\mathbf{A}}, then obtain - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}} = - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\operatorname{P_{g}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} = \\frac{\\mathbf{J}}{\\mathbf{A}} and \\frac{\\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\mathbf{J}}{\\mathbf{A}^{2}} and \\operatorname{P_{g}}{(\\mathbf{J},\\mathbf{A})} = \\frac{\\mathbf{J}}{\\mathbf{A}} and - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}} = - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\mathbf{J}}{\\mathbf{A}^{2}} and - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}} = - \\operatorname{E_{n}}{(\\mathbf{J},\\mathbf{A})} + \\frac{\\operatorname{P_{g}}{(\\mathbf{J},\\mathbf{A})}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["minus", 2, "Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('P_g')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\hat{H})} = e^{\\cos{(\\hat{H})}}, then obtain - e^{\\cos{(\\hat{H})}} + (\\int \\theta^{\\hat{H}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = - e^{\\cos{(\\hat{H})}} + (\\int (e^{\\cos{(\\hat{H})}})^{\\hat{H}} d\\hat{H})^{\\hat{H}}", "derivation": "\\theta{(\\hat{H})} = e^{\\cos{(\\hat{H})}} and \\theta^{\\hat{H}}{(\\hat{H})} = (e^{\\cos{(\\hat{H})}})^{\\hat{H}} and \\int \\theta^{\\hat{H}}{(\\hat{H})} d\\hat{H} = \\int (e^{\\cos{(\\hat{H})}})^{\\hat{H}} d\\hat{H} and (\\int \\theta^{\\hat{H}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = (\\int (e^{\\cos{(\\hat{H})}})^{\\hat{H}} d\\hat{H})^{\\hat{H}} and - e^{\\cos{(\\hat{H})}} + (\\int \\theta^{\\hat{H}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = - e^{\\cos{(\\hat{H})}} + (\\int (e^{\\cos{(\\hat{H})}})^{\\hat{H}} d\\hat{H})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True)), exp(cos(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(exp(cos(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Pow(exp(cos(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(Pow(exp(cos(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 4, "exp(cos(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(cos(Symbol('\\\\hat{H}', commutative=True)))), Pow(Integral(Pow(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), exp(cos(Symbol('\\\\hat{H}', commutative=True)))), Pow(Integral(Pow(exp(cos(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given Q{(b,\\eta^{\\prime})} = \\eta^{\\prime} + b, then obtain \\int (\\eta^{\\prime} + b)^{2} Q^{2}{(b,\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{(\\eta^{\\prime})^{5}}{5} + (\\eta^{\\prime})^{4} b + 2 (\\eta^{\\prime})^{3} b^{2} + 2 (\\eta^{\\prime})^{2} b^{3} + \\eta^{\\prime} b^{4} + \\psi", "derivation": "Q{(b,\\eta^{\\prime})} = \\eta^{\\prime} + b and (\\eta^{\\prime} + b) Q{(b,\\eta^{\\prime})} = (\\eta^{\\prime} + b)^{2} and (\\eta^{\\prime} + b)^{2} Q^{2}{(b,\\eta^{\\prime})} = (\\eta^{\\prime} + b)^{4} and \\int (\\eta^{\\prime} + b)^{2} Q^{2}{(b,\\eta^{\\prime})} d\\eta^{\\prime} = \\int (\\eta^{\\prime} + b)^{4} d\\eta^{\\prime} and \\int (\\eta^{\\prime} + b)^{2} Q^{2}{(b,\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{(\\eta^{\\prime})^{5}}{5} + (\\eta^{\\prime})^{4} b + 2 (\\eta^{\\prime})^{3} b^{2} + 2 (\\eta^{\\prime})^{2} b^{3} + \\eta^{\\prime} b^{4} + \\psi", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Function('Q')(Symbol('b', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(2)), Pow(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(4)))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(2)), Pow(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(4)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('b', commutative=True)), Integer(2)), Pow(Function('Q')(Symbol('b', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 5), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(5))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(4)), Symbol('b', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(3)), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Integer(2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2)), Pow(Symbol('b', commutative=True), Integer(3))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('b', commutative=True), Integer(4))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(\\delta,v_{t},p)} = \\frac{\\delta}{p} - v_{t}, then obtain \\frac{\\partial^{2}}{\\partial \\delta\\partial p} (- p + \\dot{x}{(\\delta,v_{t},p)}) = \\frac{\\partial^{2}}{\\partial \\delta\\partial p} (\\frac{\\delta}{p} - p - v_{t})", "derivation": "\\dot{x}{(\\delta,v_{t},p)} = \\frac{\\delta}{p} - v_{t} and - p + \\dot{x}{(\\delta,v_{t},p)} = \\frac{\\delta}{p} - p - v_{t} and \\frac{\\partial}{\\partial p} (- p + \\dot{x}{(\\delta,v_{t},p)}) = \\frac{\\partial}{\\partial p} (\\frac{\\delta}{p} - p - v_{t}) and \\frac{\\partial^{2}}{\\partial \\delta\\partial p} (- p + \\dot{x}{(\\delta,v_{t},p)}) = \\frac{\\partial^{2}}{\\partial \\delta\\partial p} (\\frac{\\delta}{p} - p - v_{t})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["minus", 1, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(H)} = \\cos{(H)} and V{(\\pi)} = \\log{(e^{\\pi})} and \\nabla{(\\pi)} = \\pi, then obtain (I{(H)} - \\cos{(H)}) I{(H)} \\log{(e^{\\nabla{(\\pi)}})} = (I{(H)} - \\cos{(H)}) \\log{(e^{\\nabla{(\\pi)}})} \\cos{(H)}", "derivation": "I{(H)} = \\cos{(H)} and V{(\\pi)} = \\log{(e^{\\pi})} and (I{(H)} - \\cos{(H)}) I{(H)} V{(\\pi)} = (I{(H)} - \\cos{(H)}) V{(\\pi)} \\cos{(H)} and \\nabla{(\\pi)} = \\pi and (I{(H)} - \\cos{(H)}) I{(H)} \\log{(e^{\\pi})} = (I{(H)} - \\cos{(H)}) \\log{(e^{\\pi})} \\cos{(H)} and (I{(H)} - \\cos{(H)}) I{(H)} \\log{(e^{\\nabla{(\\pi)}})} = (I{(H)} - \\cos{(H)}) \\log{(e^{\\nabla{(\\pi)}})} \\cos{(H)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], ["get_premise", "Equality(Function('V')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["times", 1, "Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Function('V')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Function('I')(Symbol('H', commutative=True)), Function('V')(Symbol('\\\\pi', commutative=True))), Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Function('V')(Symbol('\\\\pi', commutative=True)), cos(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Function('I')(Symbol('H', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True)))), Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), log(exp(Symbol('\\\\pi', commutative=True))), cos(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Function('I')(Symbol('H', commutative=True)), log(exp(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True))))), Mul(Add(Function('I')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), log(exp(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)))), cos(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(T)} = e^{T}, then obtain \\frac{d}{d T} \\operatorname{F_{H}}{(T)} \\int \\frac{\\operatorname{F_{H}}{(T)}}{T} dT = \\frac{d}{d T} \\operatorname{F_{H}}{(T)} \\int \\frac{e^{T}}{T} dT", "derivation": "\\operatorname{F_{H}}{(T)} = e^{T} and \\frac{\\operatorname{F_{H}}{(T)}}{T} = \\frac{e^{T}}{T} and \\int \\frac{\\operatorname{F_{H}}{(T)}}{T} dT = \\int \\frac{e^{T}}{T} dT and \\operatorname{F_{H}}{(T)} \\int \\frac{\\operatorname{F_{H}}{(T)}}{T} dT = \\operatorname{F_{H}}{(T)} \\int \\frac{e^{T}}{T} dT and \\frac{d}{d T} \\operatorname{F_{H}}{(T)} \\int \\frac{\\operatorname{F_{H}}{(T)}}{T} dT = \\frac{d}{d T} \\operatorname{F_{H}}{(T)} \\int \\frac{e^{T}}{T} dT", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["times", 3, "Function('F_H')(Symbol('T', commutative=True))"], "Equality(Mul(Function('F_H')(Symbol('T', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Mul(Function('F_H')(Symbol('T', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["differentiate", 4, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Function('F_H')(Symbol('T', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('F_H')(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Function('F_H')(Symbol('T', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mu)} = \\log{(\\mu)} and \\operatorname{A_{2}}{(\\mu)} = \\mu and \\mathbf{g}{(\\mu)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\mathbf{S}{(\\mu)}}, then obtain \\frac{\\mu}{\\mathbf{S}{(\\mu)}} + \\Psi{(\\mathbf{r},\\mathbf{p},p)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\log{(\\mu)}} + \\Psi{(\\mathbf{r},\\mathbf{p},p)}", "derivation": "\\mathbf{S}{(\\mu)} = \\log{(\\mu)} and \\operatorname{A_{2}}{(\\mu)} = \\mu and \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\mathbf{S}{(\\mu)}} = \\frac{\\mu}{\\mathbf{S}{(\\mu)}} and \\mathbf{g}{(\\mu)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\mathbf{S}{(\\mu)}} and \\mathbf{g}{(\\mu)} = \\frac{\\mu}{\\mathbf{S}{(\\mu)}} and \\mathbf{g}{(\\mu)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\log{(\\mu)}} and \\Psi{(\\mathbf{r},\\mathbf{p},p)} + \\mathbf{g}{(\\mu)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\log{(\\mu)}} + \\Psi{(\\mathbf{r},\\mathbf{p},p)} and \\frac{\\mu}{\\mathbf{S}{(\\mu)}} + \\Psi{(\\mathbf{r},\\mathbf{p},p)} = \\frac{\\operatorname{A_{2}}{(\\mu)}}{\\log{(\\mu)}} + \\Psi{(\\mathbf{r},\\mathbf{p},p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["divide", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mu', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True)), Mul(Function('A_2')(Symbol('\\\\mu', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["add", 6, "Function('\\\\Psi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Function('\\\\Psi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('p', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\mu', commutative=True))), Add(Mul(Function('A_2')(Symbol('\\\\mu', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))), Function('\\\\Psi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True)), Integer(-1))), Function('\\\\Psi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Function('A_2')(Symbol('\\\\mu', commutative=True)), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))), Function('\\\\Psi')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(f^{\\prime})} = \\log{(f^{\\prime})}, then derive \\frac{d^{2}}{d (f^{\\prime})^{2}} \\hat{\\mathbf{x}}{(f^{\\prime})} = - \\frac{1}{(f^{\\prime})^{2}}, then obtain \\frac{d}{d f^{\\prime}} \\hat{\\mathbf{x}}{(f^{\\prime})} + \\frac{d^{2}}{d (f^{\\prime})^{2}} \\log{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\hat{\\mathbf{x}}{(f^{\\prime})} - \\frac{1}{(f^{\\prime})^{2}}", "derivation": "\\hat{\\mathbf{x}}{(f^{\\prime})} = \\log{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} \\hat{\\mathbf{x}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})} and \\frac{d^{2}}{d (f^{\\prime})^{2}} \\hat{\\mathbf{x}}{(f^{\\prime})} = \\frac{d^{2}}{d (f^{\\prime})^{2}} \\log{(f^{\\prime})} and \\frac{d^{2}}{d (f^{\\prime})^{2}} \\hat{\\mathbf{x}}{(f^{\\prime})} = - \\frac{1}{(f^{\\prime})^{2}} and \\frac{d^{2}}{d (f^{\\prime})^{2}} \\log{(f^{\\prime})} = - \\frac{1}{(f^{\\prime})^{2}} and \\frac{d}{d f^{\\prime}} \\hat{\\mathbf{x}}{(f^{\\prime})} + \\frac{d^{2}}{d (f^{\\prime})^{2}} \\log{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\hat{\\mathbf{x}}{(f^{\\prime})} - \\frac{1}{(f^{\\prime})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2))))"], [["add", 5, "Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))), Add(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(I,\\phi,\\theta)} = \\frac{\\phi}{I \\theta}, then obtain 2 \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} = 2 (\\frac{\\phi}{I \\theta})^{I}", "derivation": "\\operatorname{F_{g}}{(I,\\phi,\\theta)} = \\frac{\\phi}{I \\theta} and \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} = (\\frac{\\phi}{I \\theta})^{I} and (\\frac{\\phi}{I \\theta})^{I} + \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} = 2 (\\frac{\\phi}{I \\theta})^{I} and 2 \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} = (\\frac{\\phi}{I \\theta})^{I} + \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} and 2 \\operatorname{F_{g}}^{I}{(I,\\phi,\\theta)} = 2 (\\frac{\\phi}{I \\theta})^{I}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True)))"], [["add", 2, "Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True)), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True))), Mul(Integer(2), Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True))))"], [["add", 2, "Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True))), Add(Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True)), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('F_g')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('I', commutative=True))), Mul(Integer(2), Pow(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\eta,\\Omega)} = \\log{(\\eta^{\\Omega})}, then obtain \\frac{\\hat{x}_0{(\\eta,\\Omega)}}{\\eta} = \\frac{\\log{(\\eta^{\\Omega})}}{\\eta}", "derivation": "\\hat{x}_0{(\\eta,\\Omega)} = \\log{(\\eta^{\\Omega})} and \\eta^{- \\Omega} \\hat{x}_0{(\\eta,\\Omega)} = \\eta^{- \\Omega} \\log{(\\eta^{\\Omega})} and \\frac{\\eta^{- \\Omega} \\hat{x}_0{(\\eta,\\Omega)}}{\\eta} = \\frac{\\eta^{- \\Omega} \\log{(\\eta^{\\Omega})}}{\\eta} and \\frac{\\hat{x}_0{(\\eta,\\Omega)}}{\\eta} = \\frac{\\log{(\\eta^{\\Omega})}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["divide", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["times", 3, "Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), log(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given y{(\\psi^*)} = \\log{(\\psi^*)}, then obtain y^{2}{(\\psi^*)} \\log{(\\psi^*)}^{2} + y{(\\psi^*)} = y{(\\psi^*)} + \\log{(\\psi^*)}^{4}", "derivation": "y{(\\psi^*)} = \\log{(\\psi^*)} and y{(\\psi^*)} \\log{(\\psi^*)} = \\log{(\\psi^*)}^{2} and y^{2}{(\\psi^*)} \\log{(\\psi^*)}^{2} = \\log{(\\psi^*)}^{4} and y^{2}{(\\psi^*)} \\log{(\\psi^*)}^{2} + y{(\\psi^*)} = y{(\\psi^*)} + \\log{(\\psi^*)}^{4}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "log(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('y')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True))), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('y')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(4)))"], [["add", 3, "Function('y')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Pow(Function('y')(Symbol('\\\\psi^*', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Function('y')(Symbol('\\\\psi^*', commutative=True))), Add(Function('y')(Symbol('\\\\psi^*', commutative=True)), Pow(log(Symbol('\\\\psi^*', commutative=True)), Integer(4))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\rho)} = e^{e^{\\rho}}, then derive \\frac{d}{d \\rho} \\operatorname{v_{t}}{(\\rho)} = e^{\\rho} e^{e^{\\rho}}, then obtain (e^{\\rho} e^{e^{\\rho}} + e^{\\rho}) e^{- e^{\\rho}} = (e^{\\rho} + \\frac{d}{d \\rho} e^{e^{\\rho}}) e^{- e^{\\rho}}", "derivation": "\\operatorname{v_{t}}{(\\rho)} = e^{e^{\\rho}} and \\frac{d}{d \\rho} \\operatorname{v_{t}}{(\\rho)} = \\frac{d}{d \\rho} e^{e^{\\rho}} and e^{\\rho} + \\frac{d}{d \\rho} \\operatorname{v_{t}}{(\\rho)} = e^{\\rho} + \\frac{d}{d \\rho} e^{e^{\\rho}} and \\frac{d}{d \\rho} \\operatorname{v_{t}}{(\\rho)} = e^{\\rho} e^{e^{\\rho}} and e^{\\rho} e^{e^{\\rho}} + e^{\\rho} = e^{\\rho} + \\frac{d}{d \\rho} e^{e^{\\rho}} and (e^{\\rho} e^{e^{\\rho}} + e^{\\rho}) e^{- e^{\\rho}} = (e^{\\rho} + \\frac{d}{d \\rho} e^{e^{\\rho}}) e^{- e^{\\rho}}", "srepr_derivation": [["get_premise", "Equality(Function('v_t')(Symbol('\\\\rho', commutative=True)), exp(exp(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["add", 2, "exp(Symbol('\\\\rho', commutative=True))"], "Equality(Add(exp(Symbol('\\\\rho', commutative=True)), Derivative(Function('v_t')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\rho', commutative=True)), Derivative(exp(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\rho', commutative=True)), exp(exp(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(exp(Symbol('\\\\rho', commutative=True)), exp(exp(Symbol('\\\\rho', commutative=True)))), exp(Symbol('\\\\rho', commutative=True))), Add(exp(Symbol('\\\\rho', commutative=True)), Derivative(exp(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["divide", 5, "exp(exp(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Add(Mul(exp(Symbol('\\\\rho', commutative=True)), exp(exp(Symbol('\\\\rho', commutative=True)))), exp(Symbol('\\\\rho', commutative=True))), exp(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))))), Mul(Add(exp(Symbol('\\\\rho', commutative=True)), Derivative(exp(exp(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), exp(Mul(Integer(-1), exp(Symbol('\\\\rho', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(a)} = \\sin{(a)}, then derive \\cos{(a)} + \\frac{d}{d a} \\operatorname{A_{1}}{(a)} = 2 \\cos{(a)}, then obtain - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} - \\frac{\\operatorname{A_{1}}{(a)}}{2} - \\cos{(a)} + \\frac{d}{d a} \\sin{(a)} = - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} - \\frac{\\operatorname{A_{1}}{(a)}}{2}", "derivation": "\\operatorname{A_{1}}{(a)} = \\sin{(a)} and \\operatorname{A_{1}}{(a)} + \\sin{(a)} = 2 \\sin{(a)} and \\frac{d}{d a} (\\operatorname{A_{1}}{(a)} + \\sin{(a)}) = \\frac{d}{d a} 2 \\sin{(a)} and \\cos{(a)} + \\frac{d}{d a} \\operatorname{A_{1}}{(a)} = 2 \\cos{(a)} and \\cos{(a)} + \\frac{d}{d a} \\sin{(a)} = 2 \\cos{(a)} and - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} + \\cos{(a)} + \\frac{d}{d a} \\sin{(a)} = - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} + 2 \\cos{(a)} and - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} - \\frac{\\operatorname{A_{1}}{(a)}}{2} - \\cos{(a)} + \\frac{d}{d a} \\sin{(a)} = - 2 \\operatorname{A_{1}}{(a)} \\sin^{2}{(a)} - \\frac{\\operatorname{A_{1}}{(a)}}{2}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["add", 1, "sin(Symbol('a', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Mul(Integer(2), sin(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Function('A_1')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('a', commutative=True)), Derivative(Function('A_1')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('a', commutative=True))))"], [["minus", 5, "Mul(Integer(2), Function('A_1')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('A_1')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(2))), cos(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('A_1')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(2))), Mul(Integer(2), cos(Symbol('a', commutative=True)))))"], [["minus", 6, "Add(Mul(Rational(1, 2), Function('A_1')(Symbol('a', commutative=True))), Mul(Integer(2), cos(Symbol('a', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('A_1')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Function('A_1')(Symbol('a', commutative=True))), Mul(Integer(-1), cos(Symbol('a', commutative=True))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('A_1')(Symbol('a', commutative=True)), Pow(sin(Symbol('a', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Function('A_1')(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\chi)} = \\log{(\\chi)}, then derive \\frac{d}{d \\chi} \\dot{z}{(\\chi)} = \\frac{1}{\\chi}, then obtain 1 = \\frac{1}{\\chi \\frac{d}{d \\chi} \\dot{z}{(\\chi)}}", "derivation": "\\dot{z}{(\\chi)} = \\log{(\\chi)} and \\frac{d}{d \\chi} \\dot{z}{(\\chi)} = \\frac{d}{d \\chi} \\log{(\\chi)} and \\frac{d}{d \\chi} \\dot{z}{(\\chi)} = \\frac{1}{\\chi} and \\frac{\\frac{d}{d \\chi} \\dot{z}{(\\chi)}}{\\frac{d}{d \\chi} \\log{(\\chi)}} = \\frac{1}{\\chi \\frac{d}{d \\chi} \\log{(\\chi)}} and 1 = \\frac{1}{\\chi \\frac{d}{d \\chi} \\dot{z}{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))"], [["divide", 3, "Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\dot{z}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(1), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\dot{z}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{v}{(a^{\\dagger},W)} = (a^{\\dagger})^{W}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{W - \\mathbf{v}{(a^{\\dagger},W)}}{W}} = \\frac{\\partial}{\\partial a^{\\dagger}} e^{- \\frac{- W + \\mathbf{v}{(a^{\\dagger},W)}}{W}}", "derivation": "\\mathbf{v}{(a^{\\dagger},W)} = (a^{\\dagger})^{W} and - W + \\mathbf{v}{(a^{\\dagger},W)} = - W + (a^{\\dagger})^{W} and - \\frac{- W + \\mathbf{v}{(a^{\\dagger},W)}}{W} = - \\frac{- W + (a^{\\dagger})^{W}}{W} and e^{- \\frac{- W + \\mathbf{v}{(a^{\\dagger},W)}}{W}} = e^{- \\frac{- W + (a^{\\dagger})^{W}}{W}} and e^{\\frac{W - \\mathbf{v}{(a^{\\dagger},W)}}{W}} = e^{- \\frac{- W + \\mathbf{v}{(a^{\\dagger},W)}}{W}} and \\frac{\\partial}{\\partial a^{\\dagger}} e^{\\frac{W - \\mathbf{v}{(a^{\\dagger},W)}}{W}} = \\frac{\\partial}{\\partial a^{\\dagger}} e^{- \\frac{- W + \\mathbf{v}{(a^{\\dagger},W)}}{W}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))"], [["minus", 1, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('W', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))))), exp(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(exp(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))))), exp(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))))))"], [["differentiate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(exp(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)))))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Function('\\\\mathbf{v}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v)} = e^{v} and \\operatorname{v_{x}}{(v)} = e^{v} and H{(v)} = \\operatorname{v_{x}}^{v}{(v)}, then obtain \\frac{H{(v)}}{\\operatorname{v_{x}}{(v)}} = \\frac{\\operatorname{M_{E}}^{v}{(v)}}{\\operatorname{v_{x}}{(v)}}", "derivation": "\\operatorname{M_{E}}{(v)} = e^{v} and \\operatorname{v_{x}}{(v)} = e^{v} and H{(v)} = \\operatorname{v_{x}}^{v}{(v)} and H{(v)} = (e^{v})^{v} and \\frac{H{(v)}}{\\operatorname{v_{x}}{(v)}} = \\frac{(e^{v})^{v}}{\\operatorname{v_{x}}{(v)}} and \\frac{H{(v)}}{\\operatorname{v_{x}}{(v)}} = \\frac{\\operatorname{M_{E}}^{v}{(v)}}{\\operatorname{v_{x}}{(v)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('v', commutative=True)), Pow(Function('v_x')(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('H')(Symbol('v', commutative=True)), Pow(exp(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["divide", 4, "Function('v_x')(Symbol('v', commutative=True))"], "Equality(Mul(Function('H')(Symbol('v', commutative=True)), Pow(Function('v_x')(Symbol('v', commutative=True)), Integer(-1))), Mul(Pow(Function('v_x')(Symbol('v', commutative=True)), Integer(-1)), Pow(exp(Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('H')(Symbol('v', commutative=True)), Pow(Function('v_x')(Symbol('v', commutative=True)), Integer(-1))), Mul(Pow(Function('M_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Function('v_x')(Symbol('v', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} = F_{x} - \\phi_1, then obtain \\int 2 \\phi_1 \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} d\\phi_1 = \\int \\phi_1 (F_{x} - \\phi_1 + \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})}) d\\phi_1", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} = F_{x} - \\phi_1 and 2 \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} = F_{x} - \\phi_1 + \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} and - 2 \\phi_1 \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} = - \\phi_1 (F_{x} - \\phi_1 + \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})}) and 2 \\phi_1 \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} = \\phi_1 (F_{x} - \\phi_1 + \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})}) and \\int 2 \\phi_1 \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})} d\\phi_1 = \\int \\phi_1 (F_{x} - \\phi_1 + \\operatorname{J_{\\varepsilon}}{(\\phi_1,F_{x})}) d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["add", 1, "Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\phi_1', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Symbol('\\\\phi_1', commutative=True), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given v{(\\mathbf{J}_P,B)} = \\frac{\\mathbf{J}_P}{B}, then obtain \\int (\\int v{(\\mathbf{J}_P,B)} dB)^{B} d\\mathbf{J}_P = \\int (\\int \\frac{\\mathbf{J}_P}{B} dB)^{B} d\\mathbf{J}_P", "derivation": "v{(\\mathbf{J}_P,B)} = \\frac{\\mathbf{J}_P}{B} and \\int v{(\\mathbf{J}_P,B)} dB = \\int \\frac{\\mathbf{J}_P}{B} dB and (\\int v{(\\mathbf{J}_P,B)} dB)^{B} = (\\int \\frac{\\mathbf{J}_P}{B} dB)^{B} and \\int (\\int v{(\\mathbf{J}_P,B)} dB)^{B} d\\mathbf{J}_P = \\int (\\int \\frac{\\mathbf{J}_P}{B} dB)^{B} d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Pow(Integral(Function('v')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(Integral(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(g,\\hat{p},v_{z})} = - \\hat{p} + g + v_{z} and \\phi_{2}{(\\hat{p},v_{z})} = \\hat{p} - v_{z}, then obtain \\int (g - \\sigma_{p}{(g,\\hat{p},v_{z})}) dv_{z} = \\int \\phi_{2}{(\\hat{p},v_{z})} dv_{z}", "derivation": "\\sigma_{p}{(g,\\hat{p},v_{z})} = - \\hat{p} + g + v_{z} and - g + \\sigma_{p}{(g,\\hat{p},v_{z})} = - \\hat{p} + v_{z} and g - \\sigma_{p}{(g,\\hat{p},v_{z})} = \\hat{p} - v_{z} and \\phi_{2}{(\\hat{p},v_{z})} = \\hat{p} - v_{z} and g - \\sigma_{p}{(g,\\hat{p},v_{z})} = \\phi_{2}{(\\hat{p},v_{z})} and \\int (g - \\sigma_{p}{(g,\\hat{p},v_{z})}) dv_{z} = \\int \\phi_{2}{(\\hat{p},v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('g', commutative=True), Symbol('v_z', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('v_z', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)))), Function('\\\\phi_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 5, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True))), Integral(Function('\\\\phi_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{J}_M,\\theta_2)} = - \\mathbf{J}_M + \\theta_2, then derive \\frac{\\partial}{\\partial \\theta_2} \\operatorname{v_{t}}{(\\mathbf{J}_M,\\theta_2)} = 1, then obtain \\frac{\\partial}{\\partial \\theta_2} (- \\mathbf{J}_M + \\theta_2) = 1", "derivation": "\\operatorname{v_{t}}{(\\mathbf{J}_M,\\theta_2)} = - \\mathbf{J}_M + \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{v_{t}}{(\\mathbf{J}_M,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (- \\mathbf{J}_M + \\theta_2) and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{v_{t}}{(\\mathbf{J}_M,\\theta_2)} = 1 and \\frac{\\partial}{\\partial \\theta_2} (- \\mathbf{J}_M + \\theta_2) = 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(q)} = \\sin{(q)}, then obtain (\\int 2 \\operatorname{A_{y}}{(q)} dq)^{2} = (\\int (\\operatorname{A_{y}}{(q)} + \\sin{(q)}) dq)^{2}", "derivation": "\\operatorname{A_{y}}{(q)} = \\sin{(q)} and 2 \\operatorname{A_{y}}{(q)} = \\operatorname{A_{y}}{(q)} + \\sin{(q)} and \\int 2 \\operatorname{A_{y}}{(q)} dq = \\int (\\operatorname{A_{y}}{(q)} + \\sin{(q)}) dq and (\\int 2 \\operatorname{A_{y}}{(q)} dq)^{2} = (\\int (\\operatorname{A_{y}}{(q)} + \\sin{(q)}) dq)^{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["add", 1, "Function('A_y')(Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('q', commutative=True))), Add(Function('A_y')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('A_y')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Add(Function('A_y')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Mul(Integer(2), Function('A_y')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integer(2)), Pow(Integral(Add(Function('A_y')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integer(2)))"]]}, {"prompt": "Given V{(f)} = \\log{(f)}, then obtain V^{2}{(f)} \\log{(f)} + \\frac{d}{d f} (f + \\log{(f)}) = \\log{(f)}^{3} + \\frac{d}{d f} (f + \\log{(f)})", "derivation": "V{(f)} = \\log{(f)} and V{(f)} \\log{(f)} = \\log{(f)}^{2} and f + V{(f)} = f + \\log{(f)} and \\frac{d}{d f} (f + V{(f)}) = \\frac{d}{d f} (f + \\log{(f)}) and V{(f)} \\log{(f)}^{2} = \\log{(f)}^{3} and V{(f)} \\log{(f)}^{2} + \\frac{d}{d f} (f + V{(f)}) = \\log{(f)}^{3} + \\frac{d}{d f} (f + V{(f)}) and V{(f)} \\log{(f)}^{2} + \\frac{d}{d f} (f + \\log{(f)}) = \\log{(f)}^{3} + \\frac{d}{d f} (f + \\log{(f)}) and V^{2}{(f)} \\log{(f)} + \\frac{d}{d f} (f + \\log{(f)}) = \\log{(f)}^{3} + \\frac{d}{d f} (f + \\log{(f)})", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["times", 1, "log(Symbol('f', commutative=True))"], "Equality(Mul(Function('V')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True))), Pow(log(Symbol('f', commutative=True)), Integer(2)))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('V')(Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Symbol('f', commutative=True), Function('V')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["times", 2, "log(Symbol('f', commutative=True))"], "Equality(Mul(Function('V')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(2))), Pow(log(Symbol('f', commutative=True)), Integer(3)))"], [["add", 5, "Derivative(Add(Symbol('f', commutative=True), Function('V')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('V')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(2))), Derivative(Add(Symbol('f', commutative=True), Function('V')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Pow(log(Symbol('f', commutative=True)), Integer(3)), Derivative(Add(Symbol('f', commutative=True), Function('V')(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Function('V')(Symbol('f', commutative=True)), Pow(log(Symbol('f', commutative=True)), Integer(2))), Derivative(Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Pow(log(Symbol('f', commutative=True)), Integer(3)), Derivative(Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Mul(Pow(Function('V')(Symbol('f', commutative=True)), Integer(2)), log(Symbol('f', commutative=True))), Derivative(Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Pow(log(Symbol('f', commutative=True)), Integer(3)), Derivative(Add(Symbol('f', commutative=True), log(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(A)} = e^{A}, then derive - 2 A + \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = - 2 A + e^{A}, then obtain (- 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\frac{d}{d A} \\operatorname{v_{t}}{(A)}) e^{- A} = (- 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\operatorname{v_{t}}{(A)}) e^{- A}", "derivation": "\\operatorname{v_{t}}{(A)} = e^{A} and \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = \\frac{d}{d A} e^{A} and - 2 A + \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = - 2 A + \\frac{d}{d A} e^{A} and - 2 A + \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = - 2 A + e^{A} and - 2 A + \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = - 2 A + \\operatorname{v_{t}}{(A)} and - 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\frac{d}{d A} \\operatorname{v_{t}}{(A)} = - 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\operatorname{v_{t}}{(A)} and (- 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\frac{d}{d A} \\operatorname{v_{t}}{(A)}) e^{- A} = (- 2 A - \\operatorname{v_{t}}{(A)} e^{A} + \\operatorname{v_{t}}{(A)}) e^{- A}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(2), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Derivative(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Function('v_t')(Symbol('A', commutative=True))))"], [["minus", 5, "Mul(Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('v_t')(Symbol('A', commutative=True))))"], [["divide", 6, "exp(Symbol('A', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Derivative(Function('v_t')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('A', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('A', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Function('v_t')(Symbol('A', commutative=True))), exp(Mul(Integer(-1), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(U)} = e^{U}, then obtain \\int (e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU)^{2}) dU = \\int (e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU) \\int \\frac{d}{d U} e^{U} dU) dU", "derivation": "\\operatorname{n_{1}}{(U)} = e^{U} and \\frac{d}{d U} \\operatorname{n_{1}}{(U)} = \\frac{d}{d U} e^{U} and \\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU = \\int \\frac{d}{d U} e^{U} dU and (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU)^{2} = (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU) \\int \\frac{d}{d U} e^{U} dU and e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU)^{2} = e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU) \\int \\frac{d}{d U} e^{U} dU and \\int (e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU)^{2}) dU = \\int (e^{U} + (\\int \\frac{d}{d U} \\operatorname{n_{1}}{(U)} dU) \\int \\frac{d}{d U} e^{U} dU) dU", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["times", 3, "Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))"], "Equality(Pow(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integer(2)), Mul(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))))"], [["add", 4, "exp(Symbol('U', commutative=True))"], "Equality(Add(exp(Symbol('U', commutative=True)), Pow(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integer(2))), Add(exp(Symbol('U', commutative=True)), Mul(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Add(exp(Symbol('U', commutative=True)), Pow(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integer(2))), Tuple(Symbol('U', commutative=True))), Integral(Add(exp(Symbol('U', commutative=True)), Mul(Integral(Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\ddot{x})} = \\log{(\\ddot{x})}, then derive \\int \\operatorname{n_{2}}{(\\ddot{x})} d\\ddot{x} = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\varphi^*, then obtain \\ddot{x} \\operatorname{n_{2}}{(\\ddot{x})} - \\ddot{x} + \\varphi^* = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\varphi^*", "derivation": "\\operatorname{n_{2}}{(\\ddot{x})} = \\log{(\\ddot{x})} and \\int \\operatorname{n_{2}}{(\\ddot{x})} d\\ddot{x} = \\int \\log{(\\ddot{x})} d\\ddot{x} and \\int \\operatorname{n_{2}}{(\\ddot{x})} d\\ddot{x} = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\varphi^* and \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\varphi^* = \\int \\log{(\\ddot{x})} d\\ddot{x} and \\ddot{x} \\operatorname{n_{2}}{(\\ddot{x})} - \\ddot{x} + \\varphi^* = \\int \\log{(\\ddot{x})} d\\ddot{x} and \\ddot{x} \\operatorname{n_{2}}{(\\ddot{x})} - \\ddot{x} + \\varphi^* = \\ddot{x} \\log{(\\ddot{x})} - \\ddot{x} + \\varphi^*", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True)), log(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n_2')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('n_2')(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integral(log(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('n_2')(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(g,t,\\sigma_p)} = g + t^{\\sigma_p}, then derive \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg = \\frac{g^{2}}{2} + g t^{\\sigma_p} + x^\\prime, then obtain \\hat{\\mathbf{r}} + \\frac{g^{2}}{2} + g t^{\\sigma_p} + x^\\prime = \\hat{\\mathbf{r}} + \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg", "derivation": "\\operatorname{F_{H}}{(g,t,\\sigma_p)} = g + t^{\\sigma_p} and \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg = \\int (g + t^{\\sigma_p}) dg and \\hat{\\mathbf{r}} + \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg = \\hat{\\mathbf{r}} + \\int (g + t^{\\sigma_p}) dg and \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg = \\frac{g^{2}}{2} + g t^{\\sigma_p} + x^\\prime and \\hat{\\mathbf{r}} + \\frac{g^{2}}{2} + g t^{\\sigma_p} + x^\\prime = \\hat{\\mathbf{r}} + \\int (g + t^{\\sigma_p}) dg and \\hat{\\mathbf{r}} + \\frac{g^{2}}{2} + g t^{\\sigma_p} + x^\\prime = \\hat{\\mathbf{r}} + \\int \\operatorname{F_{H}}{(g,t,\\sigma_p)} dg", "srepr_derivation": [["get_premise", "Equality(Function('F_H')(Symbol('g', commutative=True), Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('g', commutative=True), Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integral(Function('F_H')(Symbol('g', commutative=True), Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integral(Add(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('g', commutative=True), Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integral(Add(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Pow(Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integral(Function('F_H')(Symbol('g', commutative=True), Symbol('t', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(C_{2},M_{E})} = \\frac{C_{2}}{M_{E}}, then obtain - \\frac{C_{2}}{M_{E}^{2}} + \\nabla^{M_{E}}{(C_{2},M_{E})} = - \\frac{C_{2}}{M_{E}^{2}} + (\\frac{C_{2}}{M_{E}})^{M_{E}}", "derivation": "\\nabla{(C_{2},M_{E})} = \\frac{C_{2}}{M_{E}} and \\frac{\\nabla{(C_{2},M_{E})}}{M_{E}} = \\frac{C_{2}}{M_{E}^{2}} and \\nabla^{M_{E}}{(C_{2},M_{E})} = (\\frac{C_{2}}{M_{E}})^{M_{E}} and \\nabla^{M_{E}}{(C_{2},M_{E})} - \\frac{\\nabla{(C_{2},M_{E})}}{M_{E}} = (\\frac{C_{2}}{M_{E}})^{M_{E}} - \\frac{\\nabla{(C_{2},M_{E})}}{M_{E}} and - \\frac{C_{2}}{M_{E}^{2}} + \\nabla^{M_{E}}{(C_{2},M_{E})} = - \\frac{C_{2}}{M_{E}^{2}} + (\\frac{C_{2}}{M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('M_E', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-2))))"], [["power", 1, "Symbol('M_E', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Symbol('M_E', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Add(Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)))), Add(Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Symbol('M_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-2))), Pow(Function('\\\\nabla')(Symbol('C_2', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-2))), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{g})} = \\log{(\\cos{(\\mathbf{g})})}, then obtain (\\int (\\operatorname{v_{1}}{(\\mathbf{g})} - \\log{(\\cos{(\\mathbf{g})})} - 1) d\\mathbf{g})^{\\mathbf{g}} = (\\int (-1) d\\mathbf{g})^{\\mathbf{g}}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{g})} = \\log{(\\cos{(\\mathbf{g})})} and - \\mathbf{g} + \\operatorname{v_{1}}{(\\mathbf{g})} = - \\mathbf{g} + \\log{(\\cos{(\\mathbf{g})})} and - \\mathbf{g} + \\operatorname{v_{1}}{(\\mathbf{g})} - 1 = - \\mathbf{g} + \\log{(\\cos{(\\mathbf{g})})} - 1 and \\operatorname{v_{1}}{(\\mathbf{g})} - \\log{(\\cos{(\\mathbf{g})})} - 1 = -1 and \\int (\\operatorname{v_{1}}{(\\mathbf{g})} - \\log{(\\cos{(\\mathbf{g})})} - 1) d\\mathbf{g} = \\int (-1) d\\mathbf{g} and (\\int (\\operatorname{v_{1}}{(\\mathbf{g})} - \\log{(\\cos{(\\mathbf{g})})} - 1) d\\mathbf{g})^{\\mathbf{g}} = (\\int (-1) d\\mathbf{g})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), log(cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), log(cos(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), log(cos(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), log(cos(Symbol('\\\\mathbf{g}', commutative=True))))"], "Equality(Add(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{g}', commutative=True)))), Integer(-1)), Integer(-1))"], [["integrate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Add(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{g}', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Integral(Add(Function('v_1')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{g}', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integral(Integer(-1), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})} = - \\mathbf{B} + v_{x} - v_{z}, then derive \\int (- \\mathbf{B} + \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})}) dv_{x} = \\mathbf{J}_M + \\frac{v_{x}^{2}}{2} + v_{x} (- 2 \\mathbf{B} - v_{z}), then obtain \\mathbf{J}_M + \\frac{v_{x}^{2}}{2} + v_{x} (- 2 \\mathbf{B} - v_{z}) = \\int (- 2 \\mathbf{B} + v_{x} - v_{z}) dv_{x}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})} = - \\mathbf{B} + v_{x} - v_{z} and - \\mathbf{B} + \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})} = - 2 \\mathbf{B} + v_{x} - v_{z} and \\int (- \\mathbf{B} + \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})}) dv_{x} = \\int (- 2 \\mathbf{B} + v_{x} - v_{z}) dv_{x} and \\int (- \\mathbf{B} + \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{B},v_{x},v_{z})}) dv_{x} = \\mathbf{J}_M + \\frac{v_{x}^{2}}{2} + v_{x} (- 2 \\mathbf{B} - v_{z}) and \\mathbf{J}_M + \\frac{v_{x}^{2}}{2} + v_{x} (- 2 \\mathbf{B} - v_{z}) = \\int (- 2 \\mathbf{B} + v_{x} - v_{z}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_x', commutative=True), Integer(2))), Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_x', commutative=True), Integer(2))), Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('v_x', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(B)} = e^{B}, then derive \\int (- B + \\operatorname{y^{\\prime}}{(B)}) dB = - \\frac{B^{2}}{2} + \\mathbf{D} + e^{B}, then obtain \\iint (- B + e^{B}) dB dB = \\int (- \\frac{B^{2}}{2} + \\mathbf{D} + e^{B}) dB", "derivation": "\\operatorname{y^{\\prime}}{(B)} = e^{B} and - B + \\operatorname{y^{\\prime}}{(B)} = - B + e^{B} and \\int (- B + \\operatorname{y^{\\prime}}{(B)}) dB = \\int (- B + e^{B}) dB and \\int (- B + \\operatorname{y^{\\prime}}{(B)}) dB = - \\frac{B^{2}}{2} + \\mathbf{D} + e^{B} and \\int (- B + \\operatorname{y^{\\prime}}{(B)}) dB = - \\frac{B^{2}}{2} + \\mathbf{D} + \\operatorname{y^{\\prime}}{(B)} and \\int (- B + e^{B}) dB = - \\frac{B^{2}}{2} + \\mathbf{D} + e^{B} and \\iint (- B + e^{B}) dB dB = \\int (- \\frac{B^{2}}{2} + \\mathbf{D} + e^{B}) dB", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('y^{\\\\prime}')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('y^{\\\\prime}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('y^{\\\\prime}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('y^{\\\\prime}')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True), Function('y^{\\\\prime}')(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('B', commutative=True))))"], [["integrate", 6, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{x}_0,\\nabla)} = \\int \\hat{x}_0^{\\nabla} d\\nabla and \\mathbf{g}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\operatorname{F_{x}}{(\\hat{x}_0,\\nabla)}, then obtain \\mathbf{g}^{2}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\mathbf{g}{(\\hat{x}_0,\\nabla)} \\int \\hat{x}_0^{\\nabla} d\\nabla", "derivation": "\\operatorname{F_{x}}{(\\hat{x}_0,\\nabla)} = \\int \\hat{x}_0^{\\nabla} d\\nabla and \\hat{x}_0^{- \\nabla} \\operatorname{F_{x}}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\int \\hat{x}_0^{\\nabla} d\\nabla and \\mathbf{g}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\operatorname{F_{x}}{(\\hat{x}_0,\\nabla)} and \\mathbf{g}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\int \\hat{x}_0^{\\nabla} d\\nabla and \\mathbf{g}^{2}{(\\hat{x}_0,\\nabla)} = \\hat{x}_0^{- \\nabla} \\mathbf{g}{(\\hat{x}_0,\\nabla)} \\int \\hat{x}_0^{\\nabla} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('F_x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('F_x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["times", 4, "Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Function('\\\\mathbf{g}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integral(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(t_{1},\\hbar)} = \\hbar + t_{1}, then obtain 2 t_{1} (\\hbar + t_{1}) = t_{1} (2 \\hbar + 2 t_{1})", "derivation": "\\hat{H}{(t_{1},\\hbar)} = \\hbar + t_{1} and 2 \\hat{H}{(t_{1},\\hbar)} = \\hbar + t_{1} + \\hat{H}{(t_{1},\\hbar)} and 2 t_{1} \\hat{H}{(t_{1},\\hbar)} = t_{1} (\\hbar + t_{1} + \\hat{H}{(t_{1},\\hbar)}) and 2 t_{1} (\\hbar + t_{1}) = t_{1} (2 \\hbar + 2 t_{1})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('t_1', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["times", 2, "Symbol('t_1', commutative=True)"], "Equality(Mul(Integer(2), Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('t_1', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('t_1', commutative=True), Function('\\\\hat{H}')(Symbol('t_1', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Symbol('t_1', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Symbol('t_1', commutative=True))), Mul(Symbol('t_1', commutative=True), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\nabla)} = \\log{(\\nabla)} and \\mathbf{J}_P{(\\nabla)} = 2 \\log{(\\nabla)}, then obtain \\sin{(\\nabla \\mathbf{J}_P{(\\nabla)})} = \\sin{(\\nabla (\\mathbf{g}{(\\nabla)} + \\log{(\\nabla)}))}", "derivation": "\\mathbf{g}{(\\nabla)} = \\log{(\\nabla)} and \\mathbf{g}{(\\nabla)} + \\log{(\\nabla)} = 2 \\log{(\\nabla)} and \\mathbf{J}_P{(\\nabla)} = 2 \\log{(\\nabla)} and \\mathbf{J}_P{(\\nabla)} = \\mathbf{g}{(\\nabla)} + \\log{(\\nabla)} and \\nabla \\mathbf{J}_P{(\\nabla)} = \\nabla (\\mathbf{g}{(\\nabla)} + \\log{(\\nabla)}) and \\sin{(\\nabla \\mathbf{J}_P{(\\nabla)})} = \\sin{(\\nabla (\\mathbf{g}{(\\nabla)} + \\log{(\\nabla)}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "log(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(2), log(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), log(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\nabla', commutative=True)), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))"], [["times", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))))"], [["sin", 5], "Equality(sin(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\nabla', commutative=True)))), sin(Mul(Symbol('\\\\nabla', commutative=True), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\chi{(\\hat{H}_l,\\mathbf{M})} = \\hat{H}_l \\mathbf{M}, then derive \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\chi{(\\hat{H}_l,\\mathbf{M})}}{\\mathbf{M}} = 1, then obtain - \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M} - 1 + \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}}{\\mathbf{M}} = - \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}", "derivation": "\\chi{(\\hat{H}_l,\\mathbf{M})} = \\hat{H}_l \\mathbf{M} and \\frac{\\partial}{\\partial \\hat{H}_l} \\chi{(\\hat{H}_l,\\mathbf{M})} = \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M} and \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\chi{(\\hat{H}_l,\\mathbf{M})}}{\\mathbf{M}} = \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}}{\\mathbf{M}} and \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\chi{(\\hat{H}_l,\\mathbf{M})}}{\\mathbf{M}} = 1 and \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}}{\\mathbf{M}} = 1 and - \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M} - 1 + \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}}{\\mathbf{M}} = - \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l \\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Function('\\\\chi')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Integer(1))"], [["minus", 5, "Add(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(1))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(g,W)} = W g and B{(g,W)} = W g, then obtain - \\frac{\\operatorname{n_{1}}{(g,W)}}{W g} = -1", "derivation": "\\operatorname{n_{1}}{(g,W)} = W g and B{(g,W)} = W g and \\frac{\\operatorname{n_{1}}{(g,W)}}{B{(g,W)}} = \\frac{W g}{B{(g,W)}} and \\frac{\\operatorname{n_{1}}{(g,W)}}{W g} = 1 and - \\frac{\\operatorname{n_{1}}{(g,W)}}{W g} = -1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('g', commutative=True)))"], [["divide", 1, "Function('B')(Symbol('g', commutative=True), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Function('B')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Integer(-1)), Function('n_1')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Symbol('g', commutative=True), Pow(Function('B')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('n_1')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Integer(1))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('n_1')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(Q)} = e^{Q}, then obtain 2 \\operatorname{A_{y}}{(Q)} - 2 e^{Q} = 0", "derivation": "\\operatorname{A_{y}}{(Q)} = e^{Q} and 2 \\operatorname{A_{y}}{(Q)} = \\operatorname{A_{y}}{(Q)} + e^{Q} and 3 \\operatorname{A_{y}}{(Q)} = 2 \\operatorname{A_{y}}{(Q)} + e^{Q} and 3 \\operatorname{A_{y}}{(Q)} = \\operatorname{A_{y}}{(Q)} + 2 e^{Q} and 2 \\operatorname{A_{y}}{(Q)} - 2 e^{Q} = 0", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["add", 1, "Function('A_y')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('Q', commutative=True))), Add(Function('A_y')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["add", 2, "Function('A_y')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(3), Function('A_y')(Symbol('Q', commutative=True))), Add(Mul(Integer(2), Function('A_y')(Symbol('Q', commutative=True))), exp(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('A_y')(Symbol('Q', commutative=True))), Add(Function('A_y')(Symbol('Q', commutative=True)), Mul(Integer(2), exp(Symbol('Q', commutative=True)))))"], [["minus", 4, "Add(Function('A_y')(Symbol('Q', commutative=True)), Mul(Integer(2), exp(Symbol('Q', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('A_y')(Symbol('Q', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('Q', commutative=True)))), Integer(0))"]]}, {"prompt": "Given q{(\\mathbf{s},\\mathbf{F})} = \\frac{\\mathbf{s}}{\\mathbf{F}}, then obtain \\int (\\mathbf{s} + \\frac{d}{d \\mathbf{s}} 0) d\\mathbf{F} = \\int (\\mathbf{s} + \\frac{\\partial}{\\partial \\mathbf{s}} (- q{(\\mathbf{s},\\mathbf{F})} + \\frac{\\mathbf{s}}{\\mathbf{F}})) d\\mathbf{F}", "derivation": "q{(\\mathbf{s},\\mathbf{F})} = \\frac{\\mathbf{s}}{\\mathbf{F}} and q{(\\mathbf{s},\\mathbf{F})} - \\frac{\\mathbf{s}}{\\mathbf{F}} = 0 and 0 = - q{(\\mathbf{s},\\mathbf{F})} + \\frac{\\mathbf{s}}{\\mathbf{F}} and \\frac{d}{d \\mathbf{s}} 0 = \\frac{\\partial}{\\partial \\mathbf{s}} (- q{(\\mathbf{s},\\mathbf{F})} + \\frac{\\mathbf{s}}{\\mathbf{F}}) and \\mathbf{s} + \\frac{d}{d \\mathbf{s}} 0 = \\mathbf{s} + \\frac{\\partial}{\\partial \\mathbf{s}} (- q{(\\mathbf{s},\\mathbf{F})} + \\frac{\\mathbf{s}}{\\mathbf{F}}) and \\int (\\mathbf{s} + \\frac{d}{d \\mathbf{s}} 0) d\\mathbf{F} = \\int (\\mathbf{s} + \\frac{\\partial}{\\partial \\mathbf{s}} (- q{(\\mathbf{s},\\mathbf{F})} + \\frac{\\mathbf{s}}{\\mathbf{F}})) d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(0))"], [["minus", 2, "Add(Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)}, then derive \\mathbf{B} + \\mathbf{J}_P = \\int \\frac{\\sin{(\\mathbf{J}_P)}}{\\mathbf{p}{(\\mathbf{J}_P)}} d\\mathbf{J}_P, then obtain \\mathbf{B} + \\mathbf{J}_P = \\mathbf{J}_M + \\mathbf{J}_P", "derivation": "\\mathbf{p}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and 1 = \\frac{\\sin{(\\mathbf{J}_P)}}{\\mathbf{p}{(\\mathbf{J}_P)}} and \\int 1 d\\mathbf{J}_P = \\int \\frac{\\sin{(\\mathbf{J}_P)}}{\\mathbf{p}{(\\mathbf{J}_P)}} d\\mathbf{J}_P and \\mathbf{B} + \\mathbf{J}_P = \\int \\frac{\\sin{(\\mathbf{J}_P)}}{\\mathbf{p}{(\\mathbf{J}_P)}} d\\mathbf{J}_P and \\mathbf{B} + \\mathbf{J}_P = \\int 1 d\\mathbf{J}_P and \\mathbf{B} + \\mathbf{J}_P = \\mathbf{J}_M + \\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\eta{(C_{d})} = \\sin{(C_{d})}, then obtain \\int \\frac{d}{d C_{d}} \\int (\\eta{(C_{d})} + \\sin{(C_{d})}) dC_{d} dC_{d} = \\int \\frac{d}{d C_{d}} \\int 2 \\sin{(C_{d})} dC_{d} dC_{d}", "derivation": "\\eta{(C_{d})} = \\sin{(C_{d})} and \\eta{(C_{d})} + \\sin{(C_{d})} = 2 \\sin{(C_{d})} and \\int (\\eta{(C_{d})} + \\sin{(C_{d})}) dC_{d} = \\int 2 \\sin{(C_{d})} dC_{d} and \\frac{d}{d C_{d}} \\int (\\eta{(C_{d})} + \\sin{(C_{d})}) dC_{d} = \\frac{d}{d C_{d}} \\int 2 \\sin{(C_{d})} dC_{d} and \\int \\frac{d}{d C_{d}} \\int (\\eta{(C_{d})} + \\sin{(C_{d})}) dC_{d} dC_{d} = \\int \\frac{d}{d C_{d}} \\int 2 \\sin{(C_{d})} dC_{d} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["add", 1, "sin(Symbol('C_d', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Mul(Integer(2), sin(Symbol('C_d', commutative=True))))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\eta')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Derivative(Integral(Add(Function('\\\\eta')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))), Integral(Derivative(Integral(Mul(Integer(2), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(A_{x})} = \\log{(A_{x})} and \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})} = \\frac{J_{\\varepsilon} x}{I}, then obtain - (- A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})}) \\operatorname{v_{x}}{(A_{x})} = - (- A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\frac{J_{\\varepsilon} x}{I}) \\operatorname{v_{x}}{(A_{x})}", "derivation": "\\operatorname{v_{x}}{(A_{x})} = \\log{(A_{x})} and \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})} = \\frac{J_{\\varepsilon} x}{I} and - A_{x} + \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})} - \\log{(A_{x})} = - A_{x} - \\log{(A_{x})} + \\frac{J_{\\varepsilon} x}{I} and - A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})} = - A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\frac{J_{\\varepsilon} x}{I} and - (- A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\operatorname{v_{y}}{(x,I,J_{\\varepsilon})}) \\operatorname{v_{x}}{(A_{x})} = - (- A_{x} - \\operatorname{v_{x}}{(A_{x})} + \\frac{J_{\\varepsilon} x}{I}) \\operatorname{v_{x}}{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], ["get_premise", "Equality(Function('v_y')(Symbol('x', commutative=True), Symbol('I', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)))"], [["minus", 2, "Add(Symbol('A_x', commutative=True), log(Symbol('A_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('v_y')(Symbol('x', commutative=True), Symbol('I', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(Symbol('A_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('A_x', commutative=True))), Function('v_y')(Symbol('x', commutative=True), Symbol('I', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Function('v_x')(Symbol('A_x', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('A_x', commutative=True))), Function('v_y')(Symbol('x', commutative=True), Symbol('I', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Function('v_x')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Function('v_x')(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(g)} = \\sin{(g)} and \\rho_{b}{(g)} = \\frac{d}{d g} \\operatorname{f^{\\prime}}{(g)}, then derive \\rho_{b}^{g}{(g)} = \\cos^{g}{(g)}, then obtain \\rho_{b}^{g}{(g)} + \\cos{(g)} = \\cos{(g)} + \\cos^{g}{(g)}", "derivation": "\\operatorname{f^{\\prime}}{(g)} = \\sin{(g)} and \\frac{d}{d g} \\operatorname{f^{\\prime}}{(g)} = \\frac{d}{d g} \\sin{(g)} and \\rho_{b}{(g)} = \\frac{d}{d g} \\operatorname{f^{\\prime}}{(g)} and \\rho_{b}{(g)} = \\frac{d}{d g} \\sin{(g)} and \\rho_{b}^{g}{(g)} = (\\frac{d}{d g} \\sin{(g)})^{g} and \\rho_{b}^{g}{(g)} = \\cos^{g}{(g)} and \\rho_{b}^{g}{(g)} + \\cos{(g)} = \\cos{(g)} + \\cos^{g}{(g)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('g', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\rho_b')(Symbol('g', commutative=True)), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Function('\\\\rho_b')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["add", 6, "cos(Symbol('g', commutative=True))"], "Equality(Add(Pow(Function('\\\\rho_b')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Add(cos(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True))))"]]}, {"prompt": "Given p{(n_{1})} = \\int \\sin{(n_{1})} dn_{1}, then derive p{(n_{1})} = x^\\prime - \\cos{(n_{1})}, then obtain \\frac{d}{d x^\\prime} 0 = \\frac{\\partial}{\\partial x^\\prime} ((x^\\prime - \\cos{(n_{1})})^{x^\\prime} - (\\int \\sin{(n_{1})} dn_{1})^{x^\\prime})", "derivation": "p{(n_{1})} = \\int \\sin{(n_{1})} dn_{1} and p{(n_{1})} = x^\\prime - \\cos{(n_{1})} and \\int \\sin{(n_{1})} dn_{1} = x^\\prime - \\cos{(n_{1})} and (\\int \\sin{(n_{1})} dn_{1})^{x^\\prime} = (x^\\prime - \\cos{(n_{1})})^{x^\\prime} and 0 = (x^\\prime - \\cos{(n_{1})})^{x^\\prime} - (\\int \\sin{(n_{1})} dn_{1})^{x^\\prime} and \\frac{d}{d x^\\prime} 0 = \\frac{\\partial}{\\partial x^\\prime} ((x^\\prime - \\cos{(n_{1})})^{x^\\prime} - (\\int \\sin{(n_{1})} dn_{1})^{x^\\prime})", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('n_1', commutative=True)), Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('p')(Symbol('n_1', commutative=True)), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 4, "Pow(Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('x^\\\\prime', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), cos(Symbol('n_1', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Integral(sin(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{J}_f,F_{N},y)} = \\mathbf{J}_f + y^{F_{N}}, then obtain \\frac{\\partial}{\\partial y} e^{- B \\mathbf{s}^{\\mathbf{J}_f}{(\\mathbf{J}_f,F_{N},y)}} = \\frac{\\partial}{\\partial y} e^{- B (\\mathbf{J}_f + y^{F_{N}})^{\\mathbf{J}_f}}", "derivation": "\\mathbf{s}{(\\mathbf{J}_f,F_{N},y)} = \\mathbf{J}_f + y^{F_{N}} and \\mathbf{s}^{\\mathbf{J}_f}{(\\mathbf{J}_f,F_{N},y)} = (\\mathbf{J}_f + y^{F_{N}})^{\\mathbf{J}_f} and - B \\mathbf{s}^{\\mathbf{J}_f}{(\\mathbf{J}_f,F_{N},y)} = - B (\\mathbf{J}_f + y^{F_{N}})^{\\mathbf{J}_f} and e^{- B \\mathbf{s}^{\\mathbf{J}_f}{(\\mathbf{J}_f,F_{N},y)}} = e^{- B (\\mathbf{J}_f + y^{F_{N}})^{\\mathbf{J}_f}} and \\frac{\\partial}{\\partial y} e^{- B \\mathbf{s}^{\\mathbf{J}_f}{(\\mathbf{J}_f,F_{N},y)}} = \\frac{\\partial}{\\partial y} e^{- B (\\mathbf{J}_f + y^{F_{N}})^{\\mathbf{J}_f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Symbol('F_N', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Symbol('F_N', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('B', commutative=True), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Symbol('F_N', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))), exp(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Symbol('F_N', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(exp(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('F_N', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Symbol('B', commutative=True), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Symbol('F_N', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(\\hbar)} = \\int e^{\\hbar} d\\hbar, then obtain \\frac{d}{d \\hbar} (B{(\\hbar)} + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar) = \\frac{d}{d \\hbar} (\\int e^{\\hbar} d\\hbar + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar)", "derivation": "B{(\\hbar)} = \\int e^{\\hbar} d\\hbar and B^{\\hbar}{(\\hbar)} = (\\int e^{\\hbar} d\\hbar)^{\\hbar} and \\int B^{\\hbar}{(\\hbar)} d\\hbar = \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar and B{(\\hbar)} + \\int B^{\\hbar}{(\\hbar)} d\\hbar = \\int B^{\\hbar}{(\\hbar)} d\\hbar + \\int e^{\\hbar} d\\hbar and B{(\\hbar)} + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar = \\int e^{\\hbar} d\\hbar + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar and \\frac{d}{d \\hbar} (B{(\\hbar)} + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar) = \\frac{d}{d \\hbar} (\\int e^{\\hbar} d\\hbar + \\int (\\int e^{\\hbar} d\\hbar)^{\\hbar} d\\hbar)", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('B')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Function('B')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "Integral(Pow(Function('B')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('B')(Symbol('\\\\hbar', commutative=True)), Integral(Pow(Function('B')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Integral(Pow(Function('B')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('B')(Symbol('\\\\hbar', commutative=True)), Integral(Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('B')(Symbol('\\\\hbar', commutative=True)), Integral(Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\Omega)} = \\cos{(\\Omega)}, then obtain 0 = - \\frac{\\sin{(\\Omega)}}{\\operatorname{F_{c}}{(\\Omega)}} - \\frac{\\cos{(\\Omega)} \\frac{d}{d \\Omega} \\operatorname{F_{c}}{(\\Omega)}}{\\operatorname{F_{c}}^{2}{(\\Omega)}}", "derivation": "\\operatorname{F_{c}}{(\\Omega)} = \\cos{(\\Omega)} and \\Omega \\operatorname{F_{c}}{(\\Omega)} = \\Omega \\cos{(\\Omega)} and 1 = \\frac{\\cos{(\\Omega)}}{\\operatorname{F_{c}}{(\\Omega)}} and \\frac{d}{d \\Omega} 1 = \\frac{d}{d \\Omega} \\frac{\\cos{(\\Omega)}}{\\operatorname{F_{c}}{(\\Omega)}} and 0 = - \\frac{\\sin{(\\Omega)}}{\\operatorname{F_{c}}{(\\Omega)}} - \\frac{\\cos{(\\Omega)} \\frac{d}{d \\Omega} \\operatorname{F_{c}}{(\\Omega)}}{\\operatorname{F_{c}}^{2}{(\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('F_c')(Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\Omega', commutative=True), Function('F_c')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Integer(-2)), cos(Symbol('\\\\Omega', commutative=True)), Derivative(Function('F_c')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))))"]]}, {"prompt": "Given i{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}}, then obtain (i^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} e^{- \\hat{\\mathbf{r}}} = ((e^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} e^{- \\hat{\\mathbf{r}}}", "derivation": "i{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and i^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})} = (e^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} and (i^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = ((e^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} and (i^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} e^{- \\hat{\\mathbf{r}}} = ((e^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} e^{- \\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Pow(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["divide", 3, "exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Pow(Pow(Function('i')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Pow(Pow(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})} = e^{- M_{E} + Z}, then derive (\\frac{\\partial}{\\partial M_{E}} \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})})^{Z} = (- e^{- M_{E} + Z})^{Z}, then obtain ((\\frac{\\partial}{\\partial M_{E}} e^{- M_{E} + Z})^{Z})^{M_{E}} = ((- e^{- M_{E} + Z})^{Z})^{M_{E}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})} = e^{- M_{E} + Z} and \\frac{\\partial}{\\partial M_{E}} \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})} = \\frac{\\partial}{\\partial M_{E}} e^{- M_{E} + Z} and (\\frac{\\partial}{\\partial M_{E}} \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})})^{Z} = (\\frac{\\partial}{\\partial M_{E}} e^{- M_{E} + Z})^{Z} and (\\frac{\\partial}{\\partial M_{E}} \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})})^{Z} = (- e^{- M_{E} + Z})^{Z} and ((\\frac{\\partial}{\\partial M_{E}} \\operatorname{V_{\\mathbf{E}}}{(Z,M_{E})})^{Z})^{M_{E}} = ((- e^{- M_{E} + Z})^{Z})^{M_{E}} and ((\\frac{\\partial}{\\partial M_{E}} e^{- M_{E} + Z})^{Z})^{M_{E}} = ((- e^{- M_{E} + Z})^{Z})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Derivative(exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('Z', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Symbol('M_E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Derivative(exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('Z', commutative=True)))), Symbol('Z', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given C{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})}, then derive \\int C{(L_{\\varepsilon})} dL_{\\varepsilon} = L_{\\varepsilon} \\log{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta, then obtain - \\frac{L_{\\varepsilon} C{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta}{L_{\\varepsilon}} = - \\frac{\\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon}}{L_{\\varepsilon}}", "derivation": "C{(L_{\\varepsilon})} = \\log{(L_{\\varepsilon})} and \\int C{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon} and \\int C{(L_{\\varepsilon})} dL_{\\varepsilon} = L_{\\varepsilon} \\log{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta and \\int C{(L_{\\varepsilon})} dL_{\\varepsilon} = L_{\\varepsilon} C{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta and L_{\\varepsilon} C{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta = \\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon} and - \\frac{L_{\\varepsilon} C{(L_{\\varepsilon})} - L_{\\varepsilon} + \\eta}{L_{\\varepsilon}} = - \\frac{\\int \\log{(L_{\\varepsilon})} dL_{\\varepsilon}}{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), log(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\eta', commutative=True)), Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('C')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Integral(log(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbb{I})} = \\cos{(e^{\\mathbb{I}})} and \\operatorname{g_{\\varepsilon}}{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain a + \\operatorname{Ci}{(e^{\\mathbb{I}})} = \\int \\cos{(\\operatorname{g_{\\varepsilon}}{(\\mathbb{I})})} d\\mathbb{I}", "derivation": "\\operatorname{C_{d}}{(\\mathbb{I})} = \\cos{(e^{\\mathbb{I}})} and \\operatorname{g_{\\varepsilon}}{(\\mathbb{I})} = e^{\\mathbb{I}} and \\operatorname{C_{d}}{(\\mathbb{I})} = \\cos{(\\operatorname{g_{\\varepsilon}}{(\\mathbb{I})})} and \\cos{(e^{\\mathbb{I}})} = \\cos{(\\operatorname{g_{\\varepsilon}}{(\\mathbb{I})})} and \\int \\cos{(e^{\\mathbb{I}})} d\\mathbb{I} = \\int \\cos{(\\operatorname{g_{\\varepsilon}}{(\\mathbb{I})})} d\\mathbb{I} and a + \\operatorname{Ci}{(e^{\\mathbb{I}})} = \\int \\cos{(\\operatorname{g_{\\varepsilon}}{(\\mathbb{I})})} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True)), cos(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_d')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(cos(exp(Symbol('\\\\mathbb{I}', commutative=True))), cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(cos(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('a', commutative=True), Ci(exp(Symbol('\\\\mathbb{I}', commutative=True)))), Integral(cos(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(F_{x})} = \\cos{(F_{x})} and \\mathbf{B}{(F_{x})} = F_{x} \\cos{(F_{x})}, then obtain \\int F_{x} \\Psi_{nl}{(F_{x})} dF_{x} = \\int \\mathbf{B}{(F_{x})} dF_{x}", "derivation": "\\Psi_{nl}{(F_{x})} = \\cos{(F_{x})} and F_{x} \\Psi_{nl}{(F_{x})} = F_{x} \\cos{(F_{x})} and \\mathbf{B}{(F_{x})} = F_{x} \\cos{(F_{x})} and F_{x} \\Psi_{nl}{(F_{x})} = \\mathbf{B}{(F_{x})} and \\int F_{x} \\Psi_{nl}{(F_{x})} dF_{x} = \\int \\mathbf{B}{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["times", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Function('\\\\Psi_{nl}')(Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), cos(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('F_x', commutative=True), Function('\\\\Psi_{nl}')(Symbol('F_x', commutative=True))), Function('\\\\mathbf{B}')(Symbol('F_x', commutative=True)))"], [["integrate", 4, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Symbol('F_x', commutative=True), Function('\\\\Psi_{nl}')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given M{(i,\\varphi)} = - \\varphi + i, then obtain \\frac{\\partial}{\\partial i} \\frac{- \\varphi - M{(i,\\varphi)}}{\\varphi} = \\frac{\\partial}{\\partial i} - \\frac{i}{\\varphi}", "derivation": "M{(i,\\varphi)} = - \\varphi + i and \\varphi + M{(i,\\varphi)} = i and \\frac{\\varphi + M{(i,\\varphi)}}{\\varphi} = \\frac{i}{\\varphi} and - \\varphi - M{(i,\\varphi)} = - i and - \\frac{\\varphi + M{(i,\\varphi)}}{\\varphi} = - \\frac{i}{\\varphi} and \\frac{\\varphi + M{(i,\\varphi)}}{- i + M{(i,\\varphi)}} = \\frac{i}{- i + M{(i,\\varphi)}} and \\frac{\\partial}{\\partial i} - \\frac{\\varphi + M{(i,\\varphi)}}{\\varphi} = \\frac{\\partial}{\\partial i} - \\frac{i}{\\varphi} and - \\frac{\\varphi + M{(i,\\varphi)}}{\\varphi} = \\frac{- \\varphi - M{(i,\\varphi)}}{\\varphi} and \\frac{\\partial}{\\partial i} \\frac{- \\varphi - M{(i,\\varphi)}}{\\varphi} = \\frac{\\partial}{\\partial i} - \\frac{i}{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Symbol('i', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('i', commutative=True))"], [["divide", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('i', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('i', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1))), Mul(Symbol('i', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(-1))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Derivative(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('M')(Symbol('i', commutative=True), Symbol('\\\\varphi', commutative=True))))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(x,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (C_{d} + x), then derive U{(x,C_{d})} + 1 = 2, then obtain \\frac{U{(x,C_{d})} + 1}{C_{d} + x} = \\frac{2}{C_{d} + x}", "derivation": "U{(x,C_{d})} = \\frac{\\partial}{\\partial C_{d}} (C_{d} + x) and U{(x,C_{d})} + 1 = \\frac{\\partial}{\\partial C_{d}} (C_{d} + x) + 1 and \\frac{U{(x,C_{d})} + 1}{C_{d} + x} = \\frac{\\frac{\\partial}{\\partial C_{d}} (C_{d} + x) + 1}{C_{d} + x} and U{(x,C_{d})} + 1 = 2 and \\frac{\\partial}{\\partial C_{d}} (C_{d} + x) + 1 = 2 and \\frac{U{(x,C_{d})} + 1}{C_{d} + x} = \\frac{2}{C_{d} + x}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Derivative(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('U')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Integer(1)), Add(Derivative(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(1)))"], [["divide", 2, "Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Integer(-1)), Add(Function('U')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Integer(1))), Mul(Pow(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Integer(-1)), Add(Derivative(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('U')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Integer(-1)), Add(Function('U')(Symbol('x', commutative=True), Symbol('C_d', commutative=True)), Integer(1))), Mul(Integer(2), Pow(Add(Symbol('C_d', commutative=True), Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\theta)} = e^{\\theta}, then derive \\frac{d}{d \\theta} \\operatorname{f_{E}}{(\\theta)} = e^{\\theta}, then obtain \\frac{d}{d \\theta} e^{\\theta} = e^{\\theta}", "derivation": "\\operatorname{f_{E}}{(\\theta)} = e^{\\theta} and \\frac{d}{d \\theta} \\operatorname{f_{E}}{(\\theta)} = \\frac{d}{d \\theta} e^{\\theta} and \\frac{d}{d \\theta} \\operatorname{f_{E}}{(\\theta)} = e^{\\theta} and \\frac{d}{d \\theta} e^{\\theta} = e^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), exp(Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), exp(Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(F_{N},t)} = F_{N} + t, then derive \\int \\ddot{x}{(F_{N},t)} dF_{N} = \\frac{F_{N}^{2}}{2} + F_{N} t + \\Psi_{\\lambda}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{\\int \\ddot{x}{(F_{N},t)} dF_{N}}{F_{N}} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{\\frac{F_{N}^{2}}{2} + F_{N} t + \\Psi_{\\lambda}}{F_{N}}", "derivation": "\\ddot{x}{(F_{N},t)} = F_{N} + t and \\int \\ddot{x}{(F_{N},t)} dF_{N} = \\int (F_{N} + t) dF_{N} and \\int \\ddot{x}{(F_{N},t)} dF_{N} = \\frac{F_{N}^{2}}{2} + F_{N} t + \\Psi_{\\lambda} and \\frac{\\int \\ddot{x}{(F_{N},t)} dF_{N}}{F_{N}} = \\frac{\\frac{F_{N}^{2}}{2} + F_{N} t + \\Psi_{\\lambda}}{F_{N}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{\\int \\ddot{x}{(F_{N},t)} dF_{N}}{F_{N}} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{\\frac{F_{N}^{2}}{2} + F_{N} t + \\Psi_{\\lambda}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 3, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Function('\\\\ddot{x}')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Function('\\\\ddot{x}')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(Z)} = \\log{(e^{Z})} and \\operatorname{E_{\\lambda}}{(Z)} = \\log{(e^{Z})}, then obtain (\\frac{\\operatorname{E_{\\lambda}}{(Z)}}{\\Psi_{\\lambda}{(Z)}})^{Z} = 1", "derivation": "\\Psi_{\\lambda}{(Z)} = \\log{(e^{Z})} and \\frac{\\Psi_{\\lambda}{(Z)}}{\\log{(e^{Z})}} = 1 and (\\frac{\\Psi_{\\lambda}{(Z)}}{\\log{(e^{Z})}})^{Z} = 1 and \\operatorname{E_{\\lambda}}{(Z)} = \\log{(e^{Z})} and \\operatorname{E_{\\lambda}}{(Z)} = \\Psi_{\\lambda}{(Z)} and (\\frac{\\operatorname{E_{\\lambda}}{(Z)}}{\\log{(e^{Z})}})^{Z} = 1 and (\\frac{\\operatorname{E_{\\lambda}}{(Z)}}{\\Psi_{\\lambda}{(Z)}})^{Z} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True)), log(exp(Symbol('Z', commutative=True))))"], [["divide", 1, "log(exp(Symbol('Z', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True)), Pow(log(exp(Symbol('Z', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True)), Pow(log(exp(Symbol('Z', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('Z', commutative=True)), log(exp(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('E_{\\\\lambda}')(Symbol('Z', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Mul(Function('E_{\\\\lambda}')(Symbol('Z', commutative=True)), Pow(log(exp(Symbol('Z', commutative=True))), Integer(-1))), Symbol('Z', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Mul(Function('E_{\\\\lambda}')(Symbol('Z', commutative=True)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{v}{(\\mathbf{p},\\hbar)} = \\hbar - \\mathbf{p}, then obtain - \\mathbf{p} + \\frac{\\cos{(\\mathbf{v}{(\\mathbf{p},\\hbar)})}}{\\hbar^{2}} = - \\mathbf{p} + \\frac{\\cos{(\\hbar - \\mathbf{p})}}{\\hbar^{2}}", "derivation": "\\mathbf{v}{(\\mathbf{p},\\hbar)} = \\hbar - \\mathbf{p} and \\cos{(\\mathbf{v}{(\\mathbf{p},\\hbar)})} = \\cos{(\\hbar - \\mathbf{p})} and \\frac{\\cos{(\\mathbf{v}{(\\mathbf{p},\\hbar)})}}{\\hbar} = \\frac{\\cos{(\\hbar - \\mathbf{p})}}{\\hbar} and \\frac{\\cos{(\\mathbf{v}{(\\mathbf{p},\\hbar)})}}{\\hbar^{2}} = \\frac{\\cos{(\\hbar - \\mathbf{p})}}{\\hbar^{2}} and - \\mathbf{p} + \\frac{\\cos{(\\mathbf{v}{(\\mathbf{p},\\hbar)})}}{\\hbar^{2}} = - \\mathbf{p} + \\frac{\\cos{(\\hbar - \\mathbf{p})}}{\\hbar^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["divide", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))))"], [["divide", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), cos(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))))"], [["minus", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), cos(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\hbar', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), cos(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))))"]]}, {"prompt": "Given y{(g)} = e^{g}, then derive \\frac{(\\frac{d}{d g} y{(g)})^{g}}{L} = \\frac{(e^{g})^{g}}{L}, then obtain (\\frac{(\\frac{d}{d g} e^{g})^{g}}{L})^{L} = (\\frac{(e^{g})^{g}}{L})^{L}", "derivation": "y{(g)} = e^{g} and \\frac{d}{d g} y{(g)} = \\frac{d}{d g} e^{g} and (\\frac{d}{d g} y{(g)})^{g} = (\\frac{d}{d g} e^{g})^{g} and \\frac{(\\frac{d}{d g} y{(g)})^{g}}{L} = \\frac{(\\frac{d}{d g} e^{g})^{g}}{L} and \\frac{(\\frac{d}{d g} y{(g)})^{g}}{L} = \\frac{(e^{g})^{g}}{L} and \\frac{(\\frac{d}{d g} y{(g)})^{g}}{L} = \\frac{y^{g}{(g)}}{L} and \\frac{(\\frac{d}{d g} e^{g})^{g}}{L} = \\frac{(e^{g})^{g}}{L} and (\\frac{(\\frac{d}{d g} e^{g})^{g}}{L})^{L} = (\\frac{(e^{g})^{g}}{L})^{L}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Derivative(Function('y')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)), Pow(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)))"], [["divide", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(Function('y')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(Function('y')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(exp(Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(Function('y')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('y')(Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(exp(Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["power", 7, "Symbol('L', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True))), Symbol('L', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(exp(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} = f_{\\mathbf{v}} g^{Q}, then obtain - (\\int f_{\\mathbf{v}} g^{Q} dg)^{Q} + (\\int \\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} dg)^{Q} = 0", "derivation": "\\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} = f_{\\mathbf{v}} g^{Q} and \\int \\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} dg = \\int f_{\\mathbf{v}} g^{Q} dg and (\\int \\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} dg)^{Q} = (\\int f_{\\mathbf{v}} g^{Q} dg)^{Q} and - (\\int f_{\\mathbf{v}} g^{Q} dg)^{Q} + (\\int \\operatorname{v_{2}}{(g,f_{\\mathbf{v}},Q)} dg)^{Q} = 0", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('g', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('g', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Integral(Function('v_2')(Symbol('g', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('Q', commutative=True)), Pow(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('Q', commutative=True)))"], [["minus", 3, "Pow(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Symbol('g', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True))), Symbol('Q', commutative=True))), Pow(Integral(Function('v_2')(Symbol('g', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('g', commutative=True))), Symbol('Q', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{p}{(n_{2})} = n_{2}, then obtain \\iint \\mathbf{p}{(n_{2})} d\\mathbf{p}{(n_{2})} dn_{2} = \\iint n_{2} d\\mathbf{p}{(n_{2})} dn_{2}", "derivation": "\\mathbf{p}{(n_{2})} = n_{2} and \\int \\mathbf{p}{(n_{2})} dn_{2} = \\int n_{2} dn_{2} and \\int \\mathbf{p}{(n_{2})} d\\mathbf{p}{(n_{2})} = \\int n_{2} d\\mathbf{p}{(n_{2})} and \\iint \\mathbf{p}{(n_{2})} d\\mathbf{p}{(n_{2})} dn_{2} = \\iint n_{2} d\\mathbf{p}{(n_{2})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Symbol('n_2', commutative=True), Tuple(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)), Tuple(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)))), Integral(Symbol('n_2', commutative=True), Tuple(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True)), Tuple(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Symbol('n_2', commutative=True), Tuple(Function('\\\\mathbf{p}')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(i)} = \\cos{(i)}, then obtain 2 i + \\mathbf{p}{(i)} + 2 \\cos{(i)} = 2 i + 3 \\cos{(i)}", "derivation": "\\mathbf{p}{(i)} = \\cos{(i)} and i + \\mathbf{p}{(i)} = i + \\cos{(i)} and 2 i + \\mathbf{p}{(i)} + \\cos{(i)} = 2 i + 2 \\cos{(i)} and 2 i + \\mathbf{p}{(i)} + 2 \\cos{(i)} = 2 i + 3 \\cos{(i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["add", 1, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Function('\\\\mathbf{p}')(Symbol('i', commutative=True))), Add(Symbol('i', commutative=True), cos(Symbol('i', commutative=True))))"], [["add", 2, "Add(Symbol('i', commutative=True), cos(Symbol('i', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('i', commutative=True)), Function('\\\\mathbf{p}')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(2), cos(Symbol('i', commutative=True)))))"], [["add", 3, "cos(Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('i', commutative=True)), Function('\\\\mathbf{p}')(Symbol('i', commutative=True)), Mul(Integer(2), cos(Symbol('i', commutative=True)))), Add(Mul(Integer(2), Symbol('i', commutative=True)), Mul(Integer(3), cos(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\omega)} = e^{\\omega}, then derive 0 = \\frac{e^{\\omega}}{\\ddot{x}{(\\omega)}} - \\frac{e^{\\omega} \\frac{d}{d \\omega} \\ddot{x}{(\\omega)}}{\\ddot{x}^{2}{(\\omega)}}, then obtain 0 = - \\frac{(1 - e^{- \\omega} \\frac{d}{d \\omega} e^{\\omega}) e^{\\omega}}{\\frac{d}{d \\omega} e^{\\omega}}", "derivation": "\\ddot{x}{(\\omega)} = e^{\\omega} and 1 = \\frac{e^{\\omega}}{\\ddot{x}{(\\omega)}} and \\frac{d}{d \\omega} 1 = \\frac{d}{d \\omega} \\frac{e^{\\omega}}{\\ddot{x}{(\\omega)}} and 0 = \\frac{e^{\\omega}}{\\ddot{x}{(\\omega)}} - \\frac{e^{\\omega} \\frac{d}{d \\omega} \\ddot{x}{(\\omega)}}{\\ddot{x}^{2}{(\\omega)}} and 0 = 1 - e^{- \\omega} \\frac{d}{d \\omega} e^{\\omega} and 0 = 1 - \\frac{\\frac{d}{d \\omega} \\ddot{x}{(\\omega)}}{\\ddot{x}{(\\omega)}} and 0 = - \\frac{(1 - \\frac{\\frac{d}{d \\omega} \\ddot{x}{(\\omega)}}{\\ddot{x}{(\\omega)}}) e^{\\omega}}{\\frac{d}{d \\omega} e^{\\omega}} and 0 = - \\frac{(1 - e^{- \\omega} \\frac{d}{d \\omega} e^{\\omega}) e^{\\omega}}{\\frac{d}{d \\omega} e^{\\omega}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), exp(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), exp(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-2)), exp(Symbol('\\\\omega', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["divide", 6, "Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], "Equality(Integer(0), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), exp(Symbol('\\\\omega', commutative=True)), Pow(Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), exp(Symbol('\\\\omega', commutative=True)), Pow(Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} = e^{\\mathbf{H}}, then obtain \\frac{d}{d \\mathbf{H}} \\int 0 d\\mathbf{H} = \\frac{d}{d \\mathbf{H}} \\int (\\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} - e^{\\mathbf{H}}) d\\mathbf{H}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} = e^{\\mathbf{H}} and 0 = - \\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} + e^{\\mathbf{H}} and 0 = \\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} - e^{\\mathbf{H}} and \\int 0 d\\mathbf{H} = \\int (\\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} - e^{\\mathbf{H}}) d\\mathbf{H} and \\frac{d}{d \\mathbf{H}} \\int 0 d\\mathbf{H} = \\frac{d}{d \\mathbf{H}} \\int (\\operatorname{L_{\\varepsilon}}{(\\mathbf{H})} - e^{\\mathbf{H}}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True))), exp(Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Integral(Add(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(x^\\prime)} = e^{e^{x^\\prime}}, then derive (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)})^{x^\\prime} = (e^{x^\\prime} e^{e^{x^\\prime}})^{x^\\prime}, then obtain (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)})^{x^\\prime} = (\\hat{H}{(x^\\prime)} e^{x^\\prime})^{x^\\prime}", "derivation": "\\hat{H}{(x^\\prime)} = e^{e^{x^\\prime}} and \\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{e^{x^\\prime}} and (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)})^{x^\\prime} = (\\frac{d}{d x^\\prime} e^{e^{x^\\prime}})^{x^\\prime} and (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)})^{x^\\prime} = (e^{x^\\prime} e^{e^{x^\\prime}})^{x^\\prime} and (\\frac{d}{d x^\\prime} \\hat{H}{(x^\\prime)})^{x^\\prime} = (\\hat{H}{(x^\\prime)} e^{x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(exp(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(exp(Symbol('x^\\\\prime', commutative=True)), exp(exp(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Function('\\\\hat{H}')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\dot{z},\\hat{p}_0)} = \\dot{z} - \\hat{p}_0, then obtain \\ddot{x}{(\\dot{z},\\hat{p}_0)} + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)} = \\dot{z} - \\hat{p}_0 + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)}", "derivation": "\\ddot{x}{(\\dot{z},\\hat{p}_0)} = \\dot{z} - \\hat{p}_0 and \\ddot{x}{(\\dot{z},\\hat{p}_0)} + 1 = \\dot{z} - \\hat{p}_0 + 1 and \\ddot{x}{(\\dot{z},\\hat{p}_0)} + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)} + 1 = \\dot{z} - \\hat{p}_0 + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)} + 1 and \\ddot{x}{(\\dot{z},\\hat{p}_0)} + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)} = \\dot{z} - \\hat{p}_0 + \\frac{\\partial}{\\partial \\dot{z}} \\ddot{x}{(\\dot{z},\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 1, 1], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)))"], [["add", 2, "Derivative(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(1)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(t_{2})} = e^{t_{2}}, then obtain 4 \\eta^{2}{(t_{2})} + \\sin{(\\log{(e^{t_{2}})})} = (\\eta{(t_{2})} + e^{t_{2}})^{2} + \\sin{(\\log{(e^{t_{2}})})}", "derivation": "\\eta{(t_{2})} = e^{t_{2}} and 2 \\eta{(t_{2})} = \\eta{(t_{2})} + e^{t_{2}} and 4 \\eta^{2}{(t_{2})} = (\\eta{(t_{2})} + e^{t_{2}})^{2} and \\log{(\\eta{(t_{2})})} = \\log{(e^{t_{2}})} and 4 \\eta^{2}{(t_{2})} + \\sin{(\\log{(\\eta{(t_{2})})})} = (\\eta{(t_{2})} + e^{t_{2}})^{2} + \\sin{(\\log{(\\eta{(t_{2})})})} and 4 \\eta^{2}{(t_{2})} + \\sin{(\\log{(e^{t_{2}})})} = (\\eta{(t_{2})} + e^{t_{2}})^{2} + \\sin{(\\log{(e^{t_{2}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["add", 1, "Function('\\\\eta')(Symbol('t_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('t_2', commutative=True))), Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Integer(2))), Pow(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(2)))"], [["log", 1], "Equality(log(Function('\\\\eta')(Symbol('t_2', commutative=True))), log(exp(Symbol('t_2', commutative=True))))"], [["add", 3, "sin(log(Function('\\\\eta')(Symbol('t_2', commutative=True))))"], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Integer(2))), sin(log(Function('\\\\eta')(Symbol('t_2', commutative=True))))), Add(Pow(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(2)), sin(log(Function('\\\\eta')(Symbol('t_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\eta')(Symbol('t_2', commutative=True)), Integer(2))), sin(log(exp(Symbol('t_2', commutative=True))))), Add(Pow(Add(Function('\\\\eta')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Integer(2)), sin(log(exp(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(r,C)} = C + r, then derive \\frac{\\partial}{\\partial C} \\int \\operatorname{f_{E}}{(r,C)} dr = \\frac{\\partial}{\\partial C} (C r + \\mathbf{B} + \\frac{r^{2}}{2}), then obtain \\frac{\\partial}{\\partial C} \\int (C + r) dr = r", "derivation": "\\operatorname{f_{E}}{(r,C)} = C + r and \\int \\operatorname{f_{E}}{(r,C)} dr = \\int (C + r) dr and \\frac{\\partial}{\\partial C} \\int \\operatorname{f_{E}}{(r,C)} dr = \\frac{\\partial}{\\partial C} \\int (C + r) dr and \\frac{\\partial}{\\partial C} \\int \\operatorname{f_{E}}{(r,C)} dr = \\frac{\\partial}{\\partial C} (C r + \\mathbf{B} + \\frac{r^{2}}{2}) and \\frac{\\partial}{\\partial C} \\int (C + r) dr = \\frac{\\partial}{\\partial C} (C r + \\mathbf{B} + \\frac{r^{2}}{2}) and \\frac{\\partial}{\\partial C} \\int (C + r) dr = r", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Integral(Function('f_E')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('f_E')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True))"]]}, {"prompt": "Given \\mathbf{A}{(M,v_{x})} = - M + v_{x}, then obtain 3 \\mathbf{A}{(M,v_{x})} \\int (- M + v_{x} + \\mathbf{A}{(M,v_{x})}) dM = 3 \\mathbf{A}{(M,v_{x})} \\int 2 \\mathbf{A}{(M,v_{x})} dM", "derivation": "\\mathbf{A}{(M,v_{x})} = - M + v_{x} and - M + v_{x} + \\mathbf{A}{(M,v_{x})} = - 2 M + 2 v_{x} and 2 \\mathbf{A}{(M,v_{x})} = - 2 M + 2 v_{x} and - M + v_{x} + \\mathbf{A}{(M,v_{x})} = 2 \\mathbf{A}{(M,v_{x})} and \\int (- M + v_{x} + \\mathbf{A}{(M,v_{x})}) dM = \\int 2 \\mathbf{A}{(M,v_{x})} dM and 3 \\mathbf{A}{(M,v_{x})} \\int (- M + v_{x} + \\mathbf{A}{(M,v_{x})}) dM = 3 \\mathbf{A}{(M,v_{x})} \\int 2 \\mathbf{A}{(M,v_{x})} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["times", 5, "Mul(Integer(3), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_x', commutative=True), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(3), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True)), Integral(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(l,y)} = e^{- l + y}, then derive \\frac{\\partial}{\\partial y} \\mathbf{r}{(l,y)} = e^{- l} e^{y}, then derive e^{y} \\frac{\\partial}{\\partial y} \\mathbf{r}{(l,y)} = e^{- l} e^{2 y}, then obtain e^{y} \\frac{\\partial^{2}}{\\partial y^{2}} \\mathbf{r}{(l,y)} = e^{- l} e^{2 y}", "derivation": "\\mathbf{r}{(l,y)} = e^{- l + y} and \\mathbf{r}{(l,y)} = e^{- l} e^{y} and - l + \\mathbf{r}{(l,y)} = - l + e^{- l} e^{y} and \\frac{\\partial}{\\partial y} (- l + \\mathbf{r}{(l,y)}) = \\frac{\\partial}{\\partial y} (- l + e^{- l} e^{y}) and \\frac{\\partial}{\\partial y} \\mathbf{r}{(l,y)} = e^{- l} e^{y} and e^{y} \\frac{\\partial}{\\partial y} (- l + \\mathbf{r}{(l,y)}) = e^{y} \\frac{\\partial}{\\partial y} (- l + e^{- l} e^{y}) and \\mathbf{r}{(l,y)} = \\frac{\\partial}{\\partial y} \\mathbf{r}{(l,y)} and e^{y} \\frac{\\partial}{\\partial y} \\mathbf{r}{(l,y)} = e^{- l} e^{2 y} and e^{y} \\frac{\\partial^{2}}{\\partial y^{2}} \\mathbf{r}{(l,y)} = e^{- l} e^{2 y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('y', commutative=True))))"], [["expand", 1], "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Symbol('y', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Symbol('y', commutative=True)))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Symbol('y', commutative=True))))"], [["times", 4, "exp(Symbol('y', commutative=True))"], "Equality(Mul(exp(Symbol('y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(exp(Symbol('y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(exp(Symbol('y', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Mul(Integer(2), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Mul(exp(Symbol('y', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('l', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2)))), Mul(exp(Mul(Integer(-1), Symbol('l', commutative=True))), exp(Mul(Integer(2), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\omega{(y)} = \\int \\cos{(y)} dy, then derive \\frac{\\omega{(y)} - \\cos{(y)}}{y} - \\frac{F_{g} + \\sin{(y)} - \\cos{(y)}}{y} = 0, then obtain \\frac{- \\cos{(y)} + \\int \\cos{(y)} dy}{y} - \\frac{F_{g} + \\sin{(y)} - \\cos{(y)}}{y} = 0", "derivation": "\\omega{(y)} = \\int \\cos{(y)} dy and \\omega{(y)} - \\cos{(y)} = - \\cos{(y)} + \\int \\cos{(y)} dy and \\frac{\\omega{(y)} - \\cos{(y)}}{y} = \\frac{- \\cos{(y)} + \\int \\cos{(y)} dy}{y} and 0 = - \\frac{\\omega{(y)} - \\cos{(y)}}{y} + \\frac{- \\cos{(y)} + \\int \\cos{(y)} dy}{y} and \\frac{\\omega{(y)} - \\cos{(y)}}{y} - \\frac{- \\cos{(y)} + \\int \\cos{(y)} dy}{y} = 0 and \\frac{\\omega{(y)} - \\cos{(y)}}{y} - \\frac{F_{g} + \\sin{(y)} - \\cos{(y)}}{y} = 0 and \\frac{- \\cos{(y)} + \\int \\cos{(y)} dy}{y} - \\frac{F_{g} + \\sin{(y)} - \\cos{(y)}}{y} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["minus", 1, "cos(Symbol('y', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["divide", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"], [["minus", 3, "Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))))"], [["minus", 4, "Add(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"], "Equality(Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))), Integer(0))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), sin(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), sin(Symbol('y', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given s{(\\hat{H}_l)} = e^{e^{\\hat{H}_l}}, then derive \\frac{d}{d \\hat{H}_l} s{(\\hat{H}_l)} = e^{\\hat{H}_l} e^{e^{\\hat{H}_l}}, then obtain \\frac{d}{d \\hat{H}_l} s{(\\hat{H}_l)} = s{(\\hat{H}_l)} e^{\\hat{H}_l}", "derivation": "s{(\\hat{H}_l)} = e^{e^{\\hat{H}_l}} and \\frac{d}{d \\hat{H}_l} s{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{e^{\\hat{H}_l}} and \\frac{d}{d \\hat{H}_l} s{(\\hat{H}_l)} = e^{\\hat{H}_l} e^{e^{\\hat{H}_l}} and \\frac{d}{d \\hat{H}_l} s{(\\hat{H}_l)} = s{(\\hat{H}_l)} e^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{H}_l', commutative=True)), exp(exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\hat{H}_l', commutative=True)), exp(exp(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('s')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Mul(Function('s')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\hat{x}_0,P_{g})} = P_{g} + \\hat{x}_0, then derive \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} = 1, then obtain \\int \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} dP_{g} = \\int \\frac{\\partial}{\\partial \\hat{x}_0} (P_{g} + \\hat{x}_0) dP_{g}", "derivation": "\\hat{p}{(\\hat{x}_0,P_{g})} = P_{g} + \\hat{x}_0 and \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} = \\frac{\\partial}{\\partial \\hat{x}_0} (P_{g} + \\hat{x}_0) and \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} = 1 and \\int \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} dP_{g} = \\int 1 dP_{g} and \\frac{\\partial}{\\partial \\hat{x}_0} (P_{g} + \\hat{x}_0) = 1 and \\int \\frac{\\partial}{\\partial \\hat{x}_0} (P_{g} + \\hat{x}_0) dP_{g} = \\int 1 dP_{g} and \\int \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{p}{(\\hat{x}_0,P_{g})} dP_{g} = \\int \\frac{\\partial}{\\partial \\hat{x}_0} (P_{g} + \\hat{x}_0) dP_{g}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('P_g', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('P_g', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(1))"], [["integrate", 5, "Symbol('P_g', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))), Integral(Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Tuple(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given L{(\\rho)} = e^{\\rho}, then derive \\frac{d}{d \\rho} L{(\\rho)} = e^{\\rho}, then obtain \\int \\frac{d}{d \\rho} L{(\\rho)} d\\rho = \\int e^{\\rho} d\\rho", "derivation": "L{(\\rho)} = e^{\\rho} and \\frac{d}{d \\rho} L{(\\rho)} = \\frac{d}{d \\rho} e^{\\rho} and \\frac{d}{d \\rho} L{(\\rho)} = e^{\\rho} and \\int \\frac{d}{d \\rho} L{(\\rho)} d\\rho = \\int e^{\\rho} d\\rho", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), exp(Symbol('\\\\rho', commutative=True)))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Function('L')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(exp(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(h)} = \\cos{(h)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = h (- \\operatorname{x^{{\\}'}}{(h)} + \\cos{(h)}), then obtain 0^{h} \\tilde{\\infty} = \\tilde{\\infty} \\operatorname{g^{\\prime}_{\\varepsilon}}^{h}{(h)}", "derivation": "\\operatorname{x^{{\\}'}}{(h)} = \\cos{(h)} and 0 = - \\operatorname{x^{{\\}'}}{(h)} + \\cos{(h)} and 0 = h (- \\operatorname{x^{{\\}'}}{(h)} + \\cos{(h)}) and 0^{h} = (h (- \\operatorname{x^{{\\}'}}{(h)} + \\cos{(h)}))^{h} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = h (- \\operatorname{x^{{\\}'}}{(h)} + \\cos{(h)}) and 0^{h} = \\operatorname{g^{\\prime}_{\\varepsilon}}^{h}{(h)} and 0^{h} \\tilde{\\infty} = \\tilde{\\infty} \\operatorname{g^{\\prime}_{\\varepsilon}}^{h}{(h)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["minus", 1, "Function('x^\\\\prime')(Symbol('h', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"], [["times", 2, "Symbol('h', commutative=True)"], "Equality(Integer(0), Mul(Symbol('h', commutative=True), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True)))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Mul(Symbol('h', commutative=True), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Mul(Symbol('h', commutative=True), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["divide", 6, 0], "Equality(Mul(Pow(Integer(0), Symbol('h', commutative=True)), zoo), Mul(zoo, Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(E,M_{E})} = M_{E}^{E}, then obtain - M_{E}^{E} + (\\frac{\\partial}{\\partial E} \\operatorname{J_{\\varepsilon}}{(E,M_{E})})^{E} + 1 = - M_{E}^{E} + (\\frac{\\partial}{\\partial E} M_{E}^{E})^{E} + 1", "derivation": "\\operatorname{J_{\\varepsilon}}{(E,M_{E})} = M_{E}^{E} and \\frac{\\partial}{\\partial E} \\operatorname{J_{\\varepsilon}}{(E,M_{E})} = \\frac{\\partial}{\\partial E} M_{E}^{E} and (\\frac{\\partial}{\\partial E} \\operatorname{J_{\\varepsilon}}{(E,M_{E})})^{E} = (\\frac{\\partial}{\\partial E} M_{E}^{E})^{E} and - M_{E}^{E} + (\\frac{\\partial}{\\partial E} \\operatorname{J_{\\varepsilon}}{(E,M_{E})})^{E} = - M_{E}^{E} + (\\frac{\\partial}{\\partial E} M_{E}^{E})^{E} and - M_{E}^{E} + (\\frac{\\partial}{\\partial E} \\operatorname{J_{\\varepsilon}}{(E,M_{E})})^{E} + 1 = - M_{E}^{E} + (\\frac{\\partial}{\\partial E} M_{E}^{E})^{E} + 1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["minus", 3, "Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True))), Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True))), Pow(Derivative(Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["add", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True))), Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True))), Pow(Derivative(Pow(Symbol('M_E', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Integer(1)))"]]}, {"prompt": "Given x{(\\mathbf{H},L)} = L \\mathbf{H}, then derive \\frac{\\partial}{\\partial \\mathbf{H}} x{(\\mathbf{H},L)} = L, then obtain \\frac{L}{\\frac{\\partial}{\\partial \\mathbf{H}} L \\mathbf{H}} = 1", "derivation": "x{(\\mathbf{H},L)} = L \\mathbf{H} and \\frac{\\partial}{\\partial \\mathbf{H}} x{(\\mathbf{H},L)} = \\frac{\\partial}{\\partial \\mathbf{H}} L \\mathbf{H} and \\frac{\\partial}{\\partial \\mathbf{H}} x{(\\mathbf{H},L)} = L and L = \\frac{\\partial}{\\partial \\mathbf{H}} L \\mathbf{H} and \\frac{L}{\\frac{\\partial}{\\partial \\mathbf{H}} L \\mathbf{H}} = 1", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Symbol('L', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('L', commutative=True), Derivative(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('L', commutative=True), Pow(Derivative(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(x,i)} = e^{x^{i}} and \\operatorname{f_{\\mathbf{v}}}{(x,i)} = e^{x^{i}}, then obtain \\int (x^{i} + \\operatorname{f_{\\mathbf{v}}}{(x,i)}) di = \\int (x^{i} + e^{x^{i}}) di", "derivation": "\\operatorname{C_{d}}{(x,i)} = e^{x^{i}} and x^{i} + \\operatorname{C_{d}}{(x,i)} = x^{i} + e^{x^{i}} and \\operatorname{f_{\\mathbf{v}}}{(x,i)} = e^{x^{i}} and \\int (x^{i} + \\operatorname{C_{d}}{(x,i)}) di = \\int (x^{i} + e^{x^{i}}) di and \\operatorname{C_{d}}{(x,i)} = \\operatorname{f_{\\mathbf{v}}}{(x,i)} and \\int (x^{i} + \\operatorname{f_{\\mathbf{v}}}{(x,i)}) di = \\int (x^{i} + e^{x^{i}}) di", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('x', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True))))"], [["add", 1, "Pow(Symbol('x', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), Function('C_d')(Symbol('x', commutative=True), Symbol('i', commutative=True))), Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('x', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), Function('C_d')(Symbol('x', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('C_d')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('x', commutative=True), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('x', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Add(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('x', commutative=True), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(I)} = \\cos{(\\sin{(I)})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} = \\cos{(\\sin{(I)})}, then obtain - 4 \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} \\cos^{3}{(\\sin{(I)})} = - 4 \\cos^{4}{(\\sin{(I)})}", "derivation": "\\mathbf{M}{(I)} = \\cos{(\\sin{(I)})} and \\mathbf{M}{(I)} + \\cos{(\\sin{(I)})} = 2 \\cos{(\\sin{(I)})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} = \\cos{(\\sin{(I)})} and (\\mathbf{M}{(I)} + \\cos{(\\sin{(I)})}) \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} = (\\mathbf{M}{(I)} + \\cos{(\\sin{(I)})}) \\cos{(\\sin{(I)})} and - (\\mathbf{M}{(I)} + \\cos{(\\sin{(I)})})^{2} \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} \\cos{(\\sin{(I)})} = - (\\mathbf{M}{(I)} + \\cos{(\\sin{(I)})})^{2} \\cos^{2}{(\\sin{(I)})} and - 4 \\operatorname{g^{\\prime}_{\\varepsilon}}{(I)} \\cos^{3}{(\\sin{(I)})} = - 4 \\cos^{4}{(\\sin{(I)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True))))"], [["add", 1, "cos(sin(Symbol('I', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('I', commutative=True)))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True))))"], [["times", 3, "Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True))), Mul(Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), cos(sin(Symbol('I', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), cos(sin(Symbol('I', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), Integer(2)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Add(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(sin(Symbol('I', commutative=True)))), Integer(2)), Pow(cos(sin(Symbol('I', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Integer(4), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('I', commutative=True)), Pow(cos(sin(Symbol('I', commutative=True))), Integer(3))), Mul(Integer(-1), Integer(4), Pow(cos(sin(Symbol('I', commutative=True))), Integer(4))))"]]}, {"prompt": "Given \\mathbf{E}{(V,J_{\\varepsilon})} = (e^{V})^{J_{\\varepsilon}} and \\operatorname{L_{\\varepsilon}}{(F_{H})} = \\log{(F_{H})}, then obtain (\\mathbf{E}^{V}{(V,J_{\\varepsilon})})^{V} \\operatorname{L_{\\varepsilon}}{(F_{H})} = (((e^{V})^{J_{\\varepsilon}})^{V})^{V} \\operatorname{L_{\\varepsilon}}{(F_{H})}", "derivation": "\\mathbf{E}{(V,J_{\\varepsilon})} = (e^{V})^{J_{\\varepsilon}} and \\mathbf{E}^{V}{(V,J_{\\varepsilon})} = ((e^{V})^{J_{\\varepsilon}})^{V} and (\\mathbf{E}^{V}{(V,J_{\\varepsilon})})^{V} = (((e^{V})^{J_{\\varepsilon}})^{V})^{V} and \\operatorname{L_{\\varepsilon}}{(F_{H})} = \\log{(F_{H})} and (\\mathbf{E}^{V}{(V,J_{\\varepsilon})})^{V} \\log{(F_{H})} = (((e^{V})^{J_{\\varepsilon}})^{V})^{V} \\log{(F_{H})} and (\\mathbf{E}^{V}{(V,J_{\\varepsilon})})^{V} \\operatorname{L_{\\varepsilon}}{(F_{H})} = (((e^{V})^{J_{\\varepsilon}})^{V})^{V} \\operatorname{L_{\\varepsilon}}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('V', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('V', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('V', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Pow(Pow(exp(Symbol('V', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)))"], [["power", 2, "Symbol('V', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('V', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Pow(Pow(exp(Symbol('V', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], ["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["times", 3, "log(Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('V', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), log(Symbol('F_H', commutative=True))), Mul(Pow(Pow(Pow(exp(Symbol('V', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), log(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('V', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('F_H', commutative=True))), Mul(Pow(Pow(Pow(exp(Symbol('V', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given E{(\\mathbf{E},\\nabla)} = \\frac{\\mathbf{E}}{\\nabla}, then derive - \\frac{\\frac{\\partial}{\\partial \\nabla} E{(\\mathbf{E},\\nabla)}}{\\nabla} + \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla^{2}} = \\frac{2 \\mathbf{E}}{\\nabla^{3}}, then obtain \\frac{- \\frac{\\frac{\\partial}{\\partial \\nabla} E{(\\mathbf{E},\\nabla)}}{\\nabla} + \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla^{2}}}{\\mathbf{E}} = \\frac{2}{\\nabla^{3}}", "derivation": "E{(\\mathbf{E},\\nabla)} = \\frac{\\mathbf{E}}{\\nabla} and - E{(\\mathbf{E},\\nabla)} = - \\frac{\\mathbf{E}}{\\nabla} and - \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla} = - \\frac{\\mathbf{E}}{\\nabla^{2}} and \\frac{\\partial}{\\partial \\nabla} - \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla} = \\frac{\\partial}{\\partial \\nabla} - \\frac{\\mathbf{E}}{\\nabla^{2}} and - \\frac{\\frac{\\partial}{\\partial \\nabla} E{(\\mathbf{E},\\nabla)}}{\\nabla} + \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla^{2}} = \\frac{2 \\mathbf{E}}{\\nabla^{3}} and \\frac{- \\frac{\\frac{\\partial}{\\partial \\nabla} E{(\\mathbf{E},\\nabla)}}{\\nabla} + \\frac{E{(\\mathbf{E},\\nabla)}}{\\nabla^{2}}}{\\mathbf{E}} = \\frac{2}{\\nabla^{3}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["divide", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-2)), Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-3))))"], [["divide", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-2)), Function('E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\nabla', commutative=True))))), Mul(Integer(2), Pow(Symbol('\\\\nabla', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} = A_{2} - \\mathbf{P}, then obtain \\iiint \\frac{\\partial}{\\partial A_{2}} \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} dA_{2} = \\iiint \\frac{\\partial}{\\partial A_{2}} (A_{2} - \\mathbf{P}) d\\mathbf{P} d\\mathbf{P} dA_{2}", "derivation": "\\Psi^{\\dagger}{(A_{2},\\mathbf{P})} = A_{2} - \\mathbf{P} and \\frac{\\partial}{\\partial A_{2}} \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} = \\frac{\\partial}{\\partial A_{2}} (A_{2} - \\mathbf{P}) and \\int \\frac{\\partial}{\\partial A_{2}} \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} d\\mathbf{P} = \\int \\frac{\\partial}{\\partial A_{2}} (A_{2} - \\mathbf{P}) d\\mathbf{P} and \\iint \\frac{\\partial}{\\partial A_{2}} \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} = \\iint \\frac{\\partial}{\\partial A_{2}} (A_{2} - \\mathbf{P}) d\\mathbf{P} d\\mathbf{P} and \\iiint \\frac{\\partial}{\\partial A_{2}} \\Psi^{\\dagger}{(A_{2},\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} dA_{2} = \\iiint \\frac{\\partial}{\\partial A_{2}} (A_{2} - \\mathbf{P}) d\\mathbf{P} d\\mathbf{P} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Derivative(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\theta,v_{z})} = - \\theta + v_{z} and \\tilde{g}{(y)} = \\sin{(y)}, then obtain \\frac{- (- \\theta + v_{z})^{v_{z}} + \\tilde{g}{(y)} - 1}{\\Omega} = \\frac{- (- \\theta + v_{z})^{v_{z}} + \\sin{(y)} - 1}{\\Omega}", "derivation": "\\theta_{2}{(\\theta,v_{z})} = - \\theta + v_{z} and \\theta_{2}^{v_{z}}{(\\theta,v_{z})} = (- \\theta + v_{z})^{v_{z}} and \\tilde{g}{(y)} = \\sin{(y)} and - \\theta_{2}^{v_{z}}{(\\theta,v_{z})} + \\tilde{g}{(y)} = - \\theta_{2}^{v_{z}}{(\\theta,v_{z})} + \\sin{(y)} and - (- \\theta + v_{z})^{v_{z}} + \\tilde{g}{(y)} = - (- \\theta + v_{z})^{v_{z}} + \\sin{(y)} and - (- \\theta + v_{z})^{v_{z}} + \\tilde{g}{(y)} - 1 = - (- \\theta + v_{z})^{v_{z}} + \\sin{(y)} - 1 and \\frac{- (- \\theta + v_{z})^{v_{z}} + \\tilde{g}{(y)} - 1}{\\Omega} = \\frac{- (- \\theta + v_{z})^{v_{z}} + \\sin{(y)} - 1}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["minus", 3, "Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Function('\\\\tilde{g}')(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), sin(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Function('\\\\tilde{g}')(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), sin(Symbol('y', commutative=True))))"], [["minus", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Function('\\\\tilde{g}')(Symbol('y', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), sin(Symbol('y', commutative=True)), Integer(-1)))"], [["divide", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Function('\\\\tilde{g}')(Symbol('y', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), sin(Symbol('y', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\pi{(\\Omega,y,v_{z})} = \\frac{\\Omega}{y} - v_{z}, then obtain \\int (- v_{z} + \\pi{(\\Omega,y,v_{z})}) dv_{z} = \\frac{\\Omega v_{z}}{y} + \\sigma_x - v_{z}^{2}", "derivation": "\\pi{(\\Omega,y,v_{z})} = \\frac{\\Omega}{y} - v_{z} and - v_{z} + \\pi{(\\Omega,y,v_{z})} = \\frac{\\Omega}{y} - 2 v_{z} and \\int (- v_{z} + \\pi{(\\Omega,y,v_{z})}) dv_{z} = \\int (\\frac{\\Omega}{y} - 2 v_{z}) dv_{z} and \\int (- v_{z} + \\pi{(\\Omega,y,v_{z})}) dv_{z} = \\frac{\\Omega v_{z}}{y} + \\sigma_x - v_{z}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["minus", 1, "Symbol('v_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\pi')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\pi')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\pi')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{p}{(m,\\Psi)} = \\frac{\\partial}{\\partial m} (\\Psi + m), then derive \\mathbf{p}{(m,\\Psi)} = 1, then obtain - m = - m - \\frac{\\partial}{\\partial m} (\\Psi + m) + 1", "derivation": "\\mathbf{p}{(m,\\Psi)} = \\frac{\\partial}{\\partial m} (\\Psi + m) and \\mathbf{p}{(m,\\Psi)} = 1 and \\frac{\\partial}{\\partial m} (\\Psi + m) = 1 and - \\mathbf{p}{(m,\\Psi)} + \\frac{\\partial}{\\partial m} (\\Psi + m) = 1 - \\mathbf{p}{(m,\\Psi)} and 0 = 1 - \\mathbf{p}{(m,\\Psi)} and - m = - m - \\mathbf{p}{(m,\\Psi)} + 1 and - m = - m - \\frac{\\partial}{\\partial m} (\\Psi + m) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True)), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["minus", 5, "Symbol('m', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('m', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(-1), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{c})} = \\log{(F_{c})}, then derive \\int \\eta^{\\prime}{(F_{c})} dF_{c} = F_{c} \\log{(F_{c})} - F_{c} + \\mathbf{D}, then obtain \\frac{d}{d F_{c}} \\int \\log{(F_{c})} dF_{c} = \\log{(F_{c})}", "derivation": "\\eta^{\\prime}{(F_{c})} = \\log{(F_{c})} and \\int \\eta^{\\prime}{(F_{c})} dF_{c} = \\int \\log{(F_{c})} dF_{c} and \\int \\eta^{\\prime}{(F_{c})} dF_{c} = F_{c} \\log{(F_{c})} - F_{c} + \\mathbf{D} and \\int \\log{(F_{c})} dF_{c} = F_{c} \\log{(F_{c})} - F_{c} + \\mathbf{D} and \\frac{d}{d F_{c}} \\int \\log{(F_{c})} dF_{c} = \\frac{\\partial}{\\partial F_{c}} (F_{c} \\log{(F_{c})} - F_{c} + \\mathbf{D}) and \\frac{d}{d F_{c}} \\int \\log{(F_{c})} dF_{c} = \\log{(F_{c})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 4, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), log(Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})} = \\mathbf{H} \\cos{(\\Psi_{\\lambda})}, then obtain 4 = (\\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1)^{\\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1}", "derivation": "\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})} = \\mathbf{H} \\cos{(\\Psi_{\\lambda})} and 1 = \\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} and 2 = \\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1 and 4 = (\\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1)^{2} and 4 = (\\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1)^{\\frac{\\mathbf{H} \\cos{(\\Psi_{\\lambda})}}{\\varepsilon{(\\Psi_{\\lambda},\\mathbf{H})}} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)))"], [["power", 3, 2], "Equality(Integer(4), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(4), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1))))"]]}, {"prompt": "Given p{(\\mu_0,v)} = \\mu_0 v, then obtain - p{(\\mu_0,v)} + \\frac{2 \\frac{\\partial}{\\partial \\mu_0} p{(\\mu_0,v)}}{\\mu_0 v} - \\frac{2 p{(\\mu_0,v)}}{\\mu_0^{2} v} = - p{(\\mu_0,v)} + \\frac{\\frac{\\partial}{\\partial \\mu_0} p{(\\mu_0,v)}}{\\mu_0 v} - \\frac{p{(\\mu_0,v)}}{\\mu_0^{2} v}", "derivation": "p{(\\mu_0,v)} = \\mu_0 v and \\frac{p{(\\mu_0,v)}}{\\mu_0 v} = 1 and \\frac{2 p{(\\mu_0,v)}}{\\mu_0 v} = 1 + \\frac{p{(\\mu_0,v)}}{\\mu_0 v} and \\frac{\\partial}{\\partial \\mu_0} \\frac{2 p{(\\mu_0,v)}}{\\mu_0 v} = \\frac{\\partial}{\\partial \\mu_0} (1 + \\frac{p{(\\mu_0,v)}}{\\mu_0 v}) and - p{(\\mu_0,v)} + \\frac{\\partial}{\\partial \\mu_0} \\frac{2 p{(\\mu_0,v)}}{\\mu_0 v} = - p{(\\mu_0,v)} + \\frac{\\partial}{\\partial \\mu_0} (1 + \\frac{p{(\\mu_0,v)}}{\\mu_0 v}) and - p{(\\mu_0,v)} + \\frac{2 \\frac{\\partial}{\\partial \\mu_0} p{(\\mu_0,v)}}{\\mu_0 v} - \\frac{2 p{(\\mu_0,v)}}{\\mu_0^{2} v} = - p{(\\mu_0,v)} + \\frac{\\frac{\\partial}{\\partial \\mu_0} p{(\\mu_0,v)}}{\\mu_0 v} - \\frac{p{(\\mu_0,v)}}{\\mu_0^{2} v}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Integer(1))"], [["add", 2, "Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Add(Integer(1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 4, "Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Derivative(Mul(Integer(2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Derivative(Add(Integer(1), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Derivative(Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Derivative(Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-2)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given x{(\\hat{\\mathbf{r}})} = e^{\\cos{(\\hat{\\mathbf{r}})}}, then obtain 4 x{(\\hat{\\mathbf{r}})} - 2 e^{\\cos{(\\hat{\\mathbf{r}})}} = 2 x{(\\hat{\\mathbf{r}})}", "derivation": "x{(\\hat{\\mathbf{r}})} = e^{\\cos{(\\hat{\\mathbf{r}})}} and 2 x{(\\hat{\\mathbf{r}})} = x{(\\hat{\\mathbf{r}})} + e^{\\cos{(\\hat{\\mathbf{r}})}} and 2 x{(\\hat{\\mathbf{r}})} + e^{\\cos{(\\hat{\\mathbf{r}})}} = x{(\\hat{\\mathbf{r}})} + 2 e^{\\cos{(\\hat{\\mathbf{r}})}} and 2 x{(\\hat{\\mathbf{r}})} - e^{\\cos{(\\hat{\\mathbf{r}})}} = x{(\\hat{\\mathbf{r}})} and 4 x{(\\hat{\\mathbf{r}})} - 2 e^{\\cos{(\\hat{\\mathbf{r}})}} = 2 x{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["add", 1, "Add(Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Add(Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(2), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))))"], [["minus", 3, "Mul(Integer(2), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(4), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), Integer(2), exp(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Mul(Integer(2), Function('x')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\omega)} = \\log{(\\omega)}, then obtain 1 = (\\mathbf{J}_f{(\\omega)} - \\log{(\\omega)})^{\\omega}", "derivation": "\\mathbf{J}_f{(\\omega)} = \\log{(\\omega)} and \\mathbf{J}_f{(\\omega)} - \\log{(\\omega)} = 0 and (\\mathbf{J}_f{(\\omega)} - \\log{(\\omega)})^{\\omega} = 0^{\\omega} and (\\mathbf{J}_f{(\\omega)} - \\log{(\\omega)})^{\\omega} \\mathbf{J}_f^{\\omega}{(\\omega)} = 0^{\\omega} \\mathbf{J}_f^{\\omega}{(\\omega)} and 1 = (\\mathbf{J}_f{(\\omega)} - \\log{(\\omega)})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Integer(0), Symbol('\\\\omega', commutative=True)))"], [["times", 3, "Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(\\ddot{x},Z,q)} = Z + \\ddot{x} + q, then derive \\frac{\\frac{\\partial}{\\partial Z} \\phi_{1}{(\\ddot{x},Z,q)} - 1}{q} = 0, then obtain \\frac{\\partial}{\\partial q} \\frac{\\frac{\\partial}{\\partial Z} (Z + \\ddot{x} + q) - 1}{q} = \\frac{d}{d q} 0", "derivation": "\\phi_{1}{(\\ddot{x},Z,q)} = Z + \\ddot{x} + q and - Z - \\ddot{x} + \\phi_{1}{(\\ddot{x},Z,q)} = q and \\frac{\\partial}{\\partial Z} (- Z - \\ddot{x} + \\phi_{1}{(\\ddot{x},Z,q)}) = \\frac{d}{d Z} q and \\frac{\\frac{\\partial}{\\partial Z} (- Z - \\ddot{x} + \\phi_{1}{(\\ddot{x},Z,q)})}{q} = \\frac{\\frac{d}{d Z} q}{q} and \\frac{\\frac{\\partial}{\\partial Z} \\phi_{1}{(\\ddot{x},Z,q)} - 1}{q} = 0 and \\frac{\\frac{\\partial}{\\partial Z} (Z + \\ddot{x} + q) - 1}{q} = 0 and \\frac{\\partial}{\\partial q} \\frac{\\frac{\\partial}{\\partial Z} (Z + \\ddot{x} + q) - 1}{q} = \\frac{d}{d q} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\ddot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('q', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Add(Symbol('Z', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\ddot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\ddot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Symbol('q', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\ddot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Symbol('q', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\ddot{x}', commutative=True), Symbol('Z', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Derivative(Add(Symbol('Z', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["differentiate", 6, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Derivative(Add(Symbol('Z', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(\\lambda)} = \\sin{(\\cos{(\\lambda)})} and \\hat{H}_{\\lambda}{(\\lambda)} = \\cos{(\\lambda)}, then obtain \\frac{k{(\\lambda)}}{\\sin{(\\cos{(\\lambda)})}} + 1 = \\frac{k{(\\lambda)}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} + 1", "derivation": "k{(\\lambda)} = \\sin{(\\cos{(\\lambda)})} and \\hat{H}_{\\lambda}{(\\lambda)} = \\cos{(\\lambda)} and k{(\\lambda)} = \\sin{(\\hat{H}_{\\lambda}{(\\lambda)})} and \\frac{k{(\\lambda)}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} = \\frac{\\sin{(\\cos{(\\lambda)})}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} and 1 = \\frac{\\sin{(\\cos{(\\lambda)})}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} and \\frac{k{(\\lambda)}}{\\sin{(\\cos{(\\lambda)})}} = \\frac{k{(\\lambda)}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} and \\frac{k{(\\lambda)}}{\\sin{(\\cos{(\\lambda)})}} + 1 = \\frac{k{(\\lambda)}}{\\sin{(\\hat{H}_{\\lambda}{(\\lambda)})}} + 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\lambda', commutative=True)), sin(cos(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('k')(Symbol('\\\\lambda', commutative=True)), sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Mul(Pow(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(1), Mul(Pow(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), sin(cos(Symbol('\\\\lambda', commutative=True)))))"], [["times", 5, "Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(-1)))"], "Equality(Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))), Integer(-1))))"], [["add", 6, 1], "Equality(Add(Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(cos(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(1)), Add(Mul(Function('k')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(Q)} = \\sin{(Q)}, then obtain (\\frac{d}{d Q} Q \\operatorname{E_{x}}{(Q)})^{Q} = (\\frac{d}{d Q} Q \\sin{(Q)})^{Q}", "derivation": "\\operatorname{E_{x}}{(Q)} = \\sin{(Q)} and Q \\operatorname{E_{x}}{(Q)} = Q \\sin{(Q)} and \\frac{d}{d Q} Q \\operatorname{E_{x}}{(Q)} = \\frac{d}{d Q} Q \\sin{(Q)} and (\\frac{d}{d Q} Q \\operatorname{E_{x}}{(Q)})^{Q} = (\\frac{d}{d Q} Q \\sin{(Q)})^{Q}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('E_x')(Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Symbol('Q', commutative=True), Function('E_x')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('Q', commutative=True), Function('E_x')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given h{(a,J_{\\varepsilon})} = \\sin{(\\frac{a}{J_{\\varepsilon}})} and \\mathbf{J}_M{(a,J_{\\varepsilon})} = a \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})}, then derive a \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})} = \\frac{a \\cos{(\\frac{a}{J_{\\varepsilon}})}}{J_{\\varepsilon}}, then obtain \\mathbf{J}_M{(a,J_{\\varepsilon})} = \\frac{a \\cos{(\\frac{a}{J_{\\varepsilon}})}}{J_{\\varepsilon}}", "derivation": "h{(a,J_{\\varepsilon})} = \\sin{(\\frac{a}{J_{\\varepsilon}})} and \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})} = \\frac{\\partial}{\\partial a} \\sin{(\\frac{a}{J_{\\varepsilon}})} and a \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})} = a \\frac{\\partial}{\\partial a} \\sin{(\\frac{a}{J_{\\varepsilon}})} and \\mathbf{J}_M{(a,J_{\\varepsilon})} = a \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})} and a \\frac{\\partial}{\\partial a} h{(a,J_{\\varepsilon})} = \\frac{a \\cos{(\\frac{a}{J_{\\varepsilon}})}}{J_{\\varepsilon}} and \\mathbf{J}_M{(a,J_{\\varepsilon})} = \\frac{a \\cos{(\\frac{a}{J_{\\varepsilon}})}}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Derivative(Function('h')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Symbol('a', commutative=True), Derivative(sin(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('a', commutative=True), Derivative(Function('h')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('a', commutative=True), Derivative(Function('h')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True), cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mathbf{J}_M')(Symbol('a', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True), cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(k)} = \\sin{(k)} and \\operatorname{v_{2}}{(k,v_{z})} = v_{z} - \\cos{(k)}, then derive (\\int \\tilde{g}{(k)} dk)^{k} = (v_{z} - \\cos{(k)})^{k}, then obtain v_{z} + (\\int \\sin{(k)} dk)^{k} = v_{z} + \\operatorname{v_{2}}^{k}{(k,v_{z})}", "derivation": "\\tilde{g}{(k)} = \\sin{(k)} and \\int \\tilde{g}{(k)} dk = \\int \\sin{(k)} dk and (\\int \\tilde{g}{(k)} dk)^{k} = (\\int \\sin{(k)} dk)^{k} and (\\int \\tilde{g}{(k)} dk)^{k} = (v_{z} - \\cos{(k)})^{k} and (\\int \\sin{(k)} dk)^{k} = (v_{z} - \\cos{(k)})^{k} and \\operatorname{v_{2}}{(k,v_{z})} = v_{z} - \\cos{(k)} and v_{z} + (\\int \\sin{(k)} dk)^{k} = v_{z} + (v_{z} - \\cos{(k)})^{k} and v_{z} + (\\int \\sin{(k)} dk)^{k} = v_{z} + \\operatorname{v_{2}}^{k}{(k,v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Integral(Function('\\\\tilde{g}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\tilde{g}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))))"], [["add", 5, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True))), Add(Symbol('v_z', commutative=True), Pow(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Symbol('v_z', commutative=True), Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True))), Add(Symbol('v_z', commutative=True), Pow(Function('v_2')(Symbol('k', commutative=True), Symbol('v_z', commutative=True)), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\lambda,\\rho)} = \\sin{(\\lambda \\rho)}, then obtain \\int ((\\omega^{\\rho}{(\\lambda,\\rho)})^{\\lambda})^{\\rho} d\\rho = \\int ((\\sin^{\\rho}{(\\lambda \\rho)})^{\\lambda})^{\\rho} d\\rho", "derivation": "\\omega{(\\lambda,\\rho)} = \\sin{(\\lambda \\rho)} and \\omega^{\\rho}{(\\lambda,\\rho)} = \\sin^{\\rho}{(\\lambda \\rho)} and (\\omega^{\\rho}{(\\lambda,\\rho)})^{\\lambda} = (\\sin^{\\rho}{(\\lambda \\rho)})^{\\lambda} and ((\\omega^{\\rho}{(\\lambda,\\rho)})^{\\lambda})^{\\rho} = ((\\sin^{\\rho}{(\\lambda \\rho)})^{\\lambda})^{\\rho} and \\int ((\\omega^{\\rho}{(\\lambda,\\rho)})^{\\lambda})^{\\rho} d\\rho = \\int ((\\sin^{\\rho}{(\\lambda \\rho)})^{\\lambda})^{\\rho} d\\rho", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True)), sin(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(sin(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Pow(Function('\\\\omega')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(sin(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\omega')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Pow(sin(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["integrate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Pow(Pow(Pow(Function('\\\\omega')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Pow(Pow(Pow(sin(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(c)} = \\sin{(c)}, then obtain \\operatorname{f^{*}}^{c}{(c)} + \\sin{(c)} + 1 = \\sin{(c)} + \\sin^{c}{(c)} + 1", "derivation": "\\operatorname{f^{*}}{(c)} = \\sin{(c)} and \\operatorname{f^{*}}^{c}{(c)} = \\sin^{c}{(c)} and \\operatorname{f^{*}}^{c}{(c)} + 1 = \\sin^{c}{(c)} + 1 and \\operatorname{f^{*}}^{c}{(c)} + \\sin{(c)} + 1 = \\sin{(c)} + \\sin^{c}{(c)} + 1", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Pow(Function('f^*')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integer(1)), Add(Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integer(1)))"], [["add", 3, "sin(Symbol('c', commutative=True))"], "Equality(Add(Pow(Function('f^*')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)), Integer(1)), Add(sin(Symbol('c', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{P}{(U,\\omega)} = U + \\omega, then obtain (- U - \\omega + \\mathbf{P}^{\\omega}{(U,\\omega)})^{U} = (- U - \\omega + (U + \\omega)^{\\omega})^{U}", "derivation": "\\mathbf{P}{(U,\\omega)} = U + \\omega and \\mathbf{P}^{\\omega}{(U,\\omega)} = (U + \\omega)^{\\omega} and - U - \\omega + \\mathbf{P}^{\\omega}{(U,\\omega)} = - U - \\omega + (U + \\omega)^{\\omega} and (- U - \\omega + \\mathbf{P}^{\\omega}{(U,\\omega)})^{U} = (- U - \\omega + (U + \\omega)^{\\omega})^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["minus", 2, "Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('U', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('U', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f, then obtain \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} - 1 + \\frac{1}{\\mathbf{A}} + \\frac{\\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f} = \\mathbf{A} \\rho_f + \\frac{1}{\\mathbf{A}}", "derivation": "\\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f and \\frac{\\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f} = 1 and \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} + \\frac{\\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f} = \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} + 1 and \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} + \\frac{1}{\\mathbf{A}} = \\mathbf{A} \\rho_f + \\frac{1}{\\mathbf{A}} and \\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})} - 1 + \\frac{1}{\\mathbf{A}} + \\frac{\\operatorname{A_{1}}{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f} = \\mathbf{A} \\rho_f + \\frac{1}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))"], [["add", 2, "Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Add(Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)))"], [["add", 1, "Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))"], "Equality(Add(Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(v_{x},m)} = v_{x}^{m}, then obtain \\iint \\cos{(v_{x}^{m} + \\operatorname{m_{s}}{(v_{x},m)})} dm dv_{x} = \\iint \\cos{(2 v_{x}^{m})} dm dv_{x}", "derivation": "\\operatorname{m_{s}}{(v_{x},m)} = v_{x}^{m} and v_{x}^{m} + \\operatorname{m_{s}}{(v_{x},m)} = 2 v_{x}^{m} and \\cos{(v_{x}^{m} + \\operatorname{m_{s}}{(v_{x},m)})} = \\cos{(2 v_{x}^{m})} and \\int \\cos{(v_{x}^{m} + \\operatorname{m_{s}}{(v_{x},m)})} dm = \\int \\cos{(2 v_{x}^{m})} dm and \\iint \\cos{(v_{x}^{m} + \\operatorname{m_{s}}{(v_{x},m)})} dm dv_{x} = \\iint \\cos{(2 v_{x}^{m})} dm dv_{x}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))"], [["add", 1, "Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Function('m_s')(Symbol('v_x', commutative=True), Symbol('m', commutative=True))), Mul(Integer(2), Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Function('m_s')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))), cos(Mul(Integer(2), Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(cos(Add(Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Function('m_s')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(cos(Mul(Integer(2), Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(cos(Add(Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Function('m_s')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(cos(Mul(Integer(2), Pow(Symbol('v_x', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})}, then obtain \\int \\frac{\\mathbf{J}_M{(V_{\\mathbf{B}})} + \\cos{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} dV_{\\mathbf{B}} - 1 = \\int 2 dV_{\\mathbf{B}} - 1", "derivation": "\\mathbf{J}_M{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\mathbf{J}_M{(V_{\\mathbf{B}})} + \\cos{(V_{\\mathbf{B}})} = 2 \\cos{(V_{\\mathbf{B}})} and \\frac{\\mathbf{J}_M{(V_{\\mathbf{B}})} + \\cos{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} = 2 and \\int \\frac{\\mathbf{J}_M{(V_{\\mathbf{B}})} + \\cos{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} dV_{\\mathbf{B}} = \\int 2 dV_{\\mathbf{B}} and \\int \\frac{\\mathbf{J}_M{(V_{\\mathbf{B}})} + \\cos{(V_{\\mathbf{B}})}}{\\cos{(V_{\\mathbf{B}})}} dV_{\\mathbf{B}} - 1 = \\int 2 dV_{\\mathbf{B}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 1, "cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["divide", 2, "cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))), Integer(2))"], [["integrate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Integer(2), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integral(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(-1)), Add(Integral(Integer(2), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{A})} = \\log{(\\mathbf{A})}, then derive \\int \\mathbf{M}{(\\mathbf{A})} d\\mathbf{A} = J_{\\varepsilon} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A}, then obtain \\int \\mathbf{M}{(\\mathbf{A})} d\\mathbf{A} = J_{\\varepsilon} + \\mathbf{A} \\mathbf{M}{(\\mathbf{A})} - \\mathbf{A}", "derivation": "\\mathbf{M}{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\int \\mathbf{M}{(\\mathbf{A})} d\\mathbf{A} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\int \\mathbf{M}{(\\mathbf{A})} d\\mathbf{A} = J_{\\varepsilon} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A} and \\int \\mathbf{M}{(\\mathbf{A})} d\\mathbf{A} = J_{\\varepsilon} + \\mathbf{A} \\mathbf{M}{(\\mathbf{A})} - \\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(T)} = \\frac{d}{d T} \\cos{(T)}, then derive \\frac{\\operatorname{v_{1}}{(T)}}{T} = - \\frac{\\sin{(T)}}{T}, then obtain \\frac{\\operatorname{v_{1}}{(T)} \\cos{(T)}}{T} = \\frac{\\cos{(T)} \\frac{d}{d T} \\cos{(T)}}{T}", "derivation": "\\operatorname{v_{1}}{(T)} = \\frac{d}{d T} \\cos{(T)} and \\frac{\\operatorname{v_{1}}{(T)}}{T} = \\frac{\\frac{d}{d T} \\cos{(T)}}{T} and \\frac{\\operatorname{v_{1}}{(T)}}{T} = - \\frac{\\sin{(T)}}{T} and \\frac{\\operatorname{v_{1}}{(T)} \\cos{(T)}}{T} = - \\frac{\\sin{(T)} \\cos{(T)}}{T} and \\frac{\\cos{(T)} \\frac{d}{d T} \\cos{(T)}}{T} = - \\frac{\\sin{(T)} \\cos{(T)}}{T} and \\frac{\\operatorname{v_{1}}{(T)} \\cos{(T)}}{T} = \\frac{\\cos{(T)} \\frac{d}{d T} \\cos{(T)}}{T}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('v_1')(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('v_1')(Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True))))"], [["times", 3, "cos(Symbol('T', commutative=True))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('v_1')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)), sin(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('v_1')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), cos(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(\\phi,\\dot{z})} = \\log{(\\dot{z}^{\\phi})}, then obtain \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\mathbf{F}{(\\phi,\\dot{z})})} = \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\log{(\\dot{z}^{\\phi})})}", "derivation": "\\mathbf{F}{(\\phi,\\dot{z})} = \\log{(\\dot{z}^{\\phi})} and \\cos{(\\mathbf{F}{(\\phi,\\dot{z})})} = \\cos{(\\log{(\\dot{z}^{\\phi})})} and - \\dot{\\mathbf{r}} + \\lambda + \\cos{(\\mathbf{F}{(\\phi,\\dot{z})})} = - \\dot{\\mathbf{r}} + \\lambda + \\cos{(\\log{(\\dot{z}^{\\phi})})} and - \\dot{\\mathbf{r}} + \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\mathbf{F}{(\\phi,\\dot{z})})} = - \\dot{\\mathbf{r}} + \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\log{(\\dot{z}^{\\phi})})} and \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\mathbf{F}{(\\phi,\\dot{z})})} = \\lambda - \\mathbf{B}{(\\lambda,\\dot{\\mathbf{r}})} + \\cos{(\\log{(\\dot{z}^{\\phi})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{F}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{z}', commutative=True))), cos(log(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True)))))"], [["minus", 2, "Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\lambda', commutative=True), cos(Function('\\\\mathbf{F}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\lambda', commutative=True), cos(log(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))))))"], [["minus", 3, "Function('\\\\mathbf{B}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(Function('\\\\mathbf{F}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(log(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))))))"], [["add", 4, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(Function('\\\\mathbf{F}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), cos(log(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then obtain (\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}})^{\\Psi_{nl}} + \\log{(\\operatorname{a^{\\dagger}}{(\\Psi_{nl})})} = (\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}})^{\\Psi_{nl}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} = e^{\\Psi_{nl}} and \\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}} = 0 and \\log{(\\operatorname{a^{\\dagger}}{(\\Psi_{nl})})} = \\log{(e^{\\Psi_{nl}})} and (\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}})^{\\Psi_{nl}} = 0^{\\Psi_{nl}} and 0^{\\Psi_{nl}} + \\log{(\\operatorname{a^{\\dagger}}{(\\Psi_{nl})})} = 0^{\\Psi_{nl}} + \\log{(e^{\\Psi_{nl}})} and (\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}})^{\\Psi_{nl}} + \\log{(\\operatorname{a^{\\dagger}}{(\\Psi_{nl})})} = (\\operatorname{a^{\\dagger}}{(\\Psi_{nl})} - e^{\\Psi_{nl}})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(0))"], [["log", 1], "Equality(log(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True))), log(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Add(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Integer(0), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["add", 3, "Pow(Integer(0), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('\\\\Psi_{nl}', commutative=True)), log(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Pow(Integer(0), Symbol('\\\\Psi_{nl}', commutative=True)), log(exp(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Add(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)), log(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)))), Pow(Add(Function('a^{\\\\dagger}')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(M_{E})} = \\log{(M_{E})}, then derive M_{E} + \\frac{d}{d M_{E}} \\hat{H}_l{(M_{E})} + 1 = M_{E} + 1 + \\frac{1}{M_{E}}, then obtain M_{E} + \\frac{d}{d M_{E}} \\log{(M_{E})} + 1 = M_{E} + \\frac{d}{d M_{E}} \\hat{H}_l{(M_{E})} + 1", "derivation": "\\hat{H}_l{(M_{E})} = \\log{(M_{E})} and M_{E} + \\hat{H}_l{(M_{E})} = M_{E} + \\log{(M_{E})} and \\frac{d}{d M_{E}} (M_{E} + \\hat{H}_l{(M_{E})}) = \\frac{d}{d M_{E}} (M_{E} + \\log{(M_{E})}) and M_{E} + \\frac{d}{d M_{E}} (M_{E} + \\hat{H}_l{(M_{E})}) = M_{E} + \\frac{d}{d M_{E}} (M_{E} + \\log{(M_{E})}) and M_{E} + \\frac{d}{d M_{E}} \\hat{H}_l{(M_{E})} + 1 = M_{E} + 1 + \\frac{1}{M_{E}} and M_{E} + \\frac{d}{d M_{E}} \\log{(M_{E})} + 1 = M_{E} + 1 + \\frac{1}{M_{E}} and M_{E} + \\frac{d}{d M_{E}} \\log{(M_{E})} + 1 = M_{E} + \\frac{d}{d M_{E}} \\hat{H}_l{(M_{E})} + 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Symbol('M_E', commutative=True), Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["add", 3, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Derivative(Add(Symbol('M_E', commutative=True), Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Symbol('M_E', commutative=True), Derivative(Add(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('M_E', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1)), Add(Symbol('M_E', commutative=True), Integer(1), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('M_E', commutative=True), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1)), Add(Symbol('M_E', commutative=True), Integer(1), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('M_E', commutative=True), Derivative(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1)), Add(Symbol('M_E', commutative=True), Derivative(Function('\\\\hat{H}_l')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given l{(\\rho_f)} = \\log{(\\rho_f)} and \\mathbf{r}{(\\rho_f)} = \\rho_f, then obtain \\rho_f (l{(\\rho_f)} + \\log{(\\rho_f)}) + \\log{(\\rho_f)} = 2 \\rho_f \\log{(\\rho_f)} + \\log{(\\rho_f)}", "derivation": "l{(\\rho_f)} = \\log{(\\rho_f)} and l{(\\rho_f)} + \\log{(\\rho_f)} = 2 \\log{(\\rho_f)} and \\mathbf{r}{(\\rho_f)} = \\rho_f and (l{(\\rho_f)} + \\log{(\\rho_f)}) \\mathbf{r}{(\\rho_f)} = 2 \\mathbf{r}{(\\rho_f)} \\log{(\\rho_f)} and \\rho_f (l{(\\rho_f)} + \\log{(\\rho_f)}) = 2 \\rho_f \\log{(\\rho_f)} and \\rho_f (l{(\\rho_f)} + \\log{(\\rho_f)}) + \\log{(\\rho_f)} = 2 \\rho_f \\log{(\\rho_f)} + \\log{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["add", 1, "log(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Function('l')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), log(Symbol('\\\\rho_f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["times", 2, "Function('\\\\mathbf{r}')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Add(Function('l')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Add(Function('l')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))), Mul(Integer(2), Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))))"], [["add", 5, "log(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\rho_f', commutative=True), Add(Function('l')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))), log(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), log(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(t_{1},A_{z})} = \\sin{(A_{z} - t_{1})}, then obtain \\hat{X} + t_{1} + \\varepsilon_{0}{(t_{1},A_{z})} = P_{e} + t_{1} + \\sin{(A_{z} - t_{1})}", "derivation": "\\varepsilon_{0}{(t_{1},A_{z})} = \\sin{(A_{z} - t_{1})} and - A_{z} + t_{1} + \\varepsilon_{0}{(t_{1},A_{z})} = - A_{z} + t_{1} + \\sin{(A_{z} - t_{1})} and \\frac{\\partial}{\\partial t_{1}} (- A_{z} + t_{1} + \\varepsilon_{0}{(t_{1},A_{z})}) = \\frac{\\partial}{\\partial t_{1}} (- A_{z} + t_{1} + \\sin{(A_{z} - t_{1})}) and \\int \\frac{\\partial}{\\partial t_{1}} (- A_{z} + t_{1} + \\varepsilon_{0}{(t_{1},A_{z})}) dt_{1} = \\int \\frac{\\partial}{\\partial t_{1}} (- A_{z} + t_{1} + \\sin{(A_{z} - t_{1})}) dt_{1} and \\hat{X} + t_{1} + \\varepsilon_{0}{(t_{1},A_{z})} = P_{e} + t_{1} + \\sin{(A_{z} - t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('t_1', commutative=True), Symbol('A_z', commutative=True)), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))))"], [["minus", 1, "Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), Function('\\\\varepsilon_0')(Symbol('t_1', commutative=True), Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), Function('\\\\varepsilon_0')(Symbol('t_1', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('t_1', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), Function('\\\\varepsilon_0')(Symbol('t_1', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('t_1', commutative=True), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('t_1', commutative=True), Function('\\\\varepsilon_0')(Symbol('t_1', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('P_e', commutative=True), Symbol('t_1', commutative=True), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))))"]]}, {"prompt": "Given q{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})} and E{(V_{\\mathbf{B}})} = q{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} - 1, then obtain \\frac{d}{d V_{\\mathbf{B}}} (q^{2}{(V_{\\mathbf{B}})} - 1) = \\frac{d}{d V_{\\mathbf{B}}} E{(V_{\\mathbf{B}})}", "derivation": "q{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})} and q^{2}{(V_{\\mathbf{B}})} = q{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} and q^{2}{(V_{\\mathbf{B}})} - 1 = q{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} - 1 and \\frac{d}{d V_{\\mathbf{B}}} (q^{2}{(V_{\\mathbf{B}})} - 1) = \\frac{d}{d V_{\\mathbf{B}}} (q{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} - 1) and E{(V_{\\mathbf{B}})} = q{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} - 1 and \\frac{d}{d V_{\\mathbf{B}}} (q^{2}{(V_{\\mathbf{B}})} - 1) = \\frac{d}{d V_{\\mathbf{B}}} E{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 1, "Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Pow(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Integer(-1)), Add(Mul(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(-1)))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Pow(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Add(Mul(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Mul(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Pow(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Function('E')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = C + \\mathbf{J}_f, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{J}_f^{2}} \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = \\frac{d}{d \\mathbf{J}_f} 1", "derivation": "\\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = C + \\mathbf{J}_f and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (C + \\mathbf{J}_f) and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = 1 and \\frac{\\partial^{2}}{\\partial \\mathbf{J}_f^{2}} \\operatorname{v_{x}}{(\\mathbf{J}_f,C)} = \\frac{d}{d \\mathbf{J}_f} 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{d})} = C_{d}, then obtain ((- E + \\operatorname{E_{n}}{(C_{d})})^{E})^{E} - \\operatorname{E_{n}}{(C_{d})} = ((C_{d} - E)^{E})^{E} - \\operatorname{E_{n}}{(C_{d})}", "derivation": "\\operatorname{E_{n}}{(C_{d})} = C_{d} and - E + \\operatorname{E_{n}}{(C_{d})} = C_{d} - E and (- E + \\operatorname{E_{n}}{(C_{d})})^{E} = (C_{d} - E)^{E} and ((- E + \\operatorname{E_{n}}{(C_{d})})^{E})^{E} = ((C_{d} - E)^{E})^{E} and ((- E + \\operatorname{E_{n}}{(C_{d})})^{E})^{E} - \\operatorname{E_{n}}{(C_{d})} = ((C_{d} - E)^{E})^{E} - \\operatorname{E_{n}}{(C_{d})}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["minus", 1, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('E_n')(Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('E_n')(Symbol('C_d', commutative=True))), Symbol('E', commutative=True)), Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('E_n')(Symbol('C_d', commutative=True))), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["minus", 4, "Function('E_n')(Symbol('C_d', commutative=True))"], "Equality(Add(Pow(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('E_n')(Symbol('C_d', commutative=True))), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Function('E_n')(Symbol('C_d', commutative=True)))), Add(Pow(Pow(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Function('E_n')(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(x,\\delta)} = \\delta + x, then derive (\\int (\\delta + \\hat{x}{(x,\\delta)}) d\\delta)^{\\delta} = (\\delta^{2} + \\delta x + \\phi_2)^{\\delta}, then obtain (\\int (2 \\delta + x) d\\delta)^{\\delta} = (\\delta^{2} + \\delta x + \\phi_2)^{\\delta}", "derivation": "\\hat{x}{(x,\\delta)} = \\delta + x and \\delta + \\hat{x}{(x,\\delta)} = 2 \\delta + x and \\int (\\delta + \\hat{x}{(x,\\delta)}) d\\delta = \\int (2 \\delta + x) d\\delta and (\\int (\\delta + \\hat{x}{(x,\\delta)}) d\\delta)^{\\delta} = (\\int (2 \\delta + x) d\\delta)^{\\delta} and (\\int (\\delta + \\hat{x}{(x,\\delta)}) d\\delta)^{\\delta} = (\\delta^{2} + \\delta x + \\phi_2)^{\\delta} and (\\int (2 \\delta + x) d\\delta)^{\\delta} = (\\delta^{2} + \\delta x + \\phi_2)^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('x', commutative=True)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\delta', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\delta', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Add(Symbol('\\\\delta', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(L)} = \\log{(L)}, then obtain \\frac{d}{d L} (- L + \\rho_{b}{(L)} + \\log{(L)} - 1) = \\frac{d}{d L} (- L + 2 \\rho_{b}{(L)} - 1)", "derivation": "\\rho_{b}{(L)} = \\log{(L)} and 2 \\rho_{b}{(L)} = \\rho_{b}{(L)} + \\log{(L)} and - L + \\rho_{b}{(L)} = - L + \\log{(L)} and - L + \\rho_{b}{(L)} - 1 = - L + \\log{(L)} - 1 and - L + \\rho_{b}{(L)} + \\log{(L)} - 1 = - L + 2 \\log{(L)} - 1 and - L + 2 \\rho_{b}{(L)} - 1 = - L + 2 \\log{(L)} - 1 and - L + \\rho_{b}{(L)} + \\log{(L)} - 1 = - L + 2 \\rho_{b}{(L)} - 1 and \\frac{d}{d L} (- L + \\rho_{b}{(L)} + \\log{(L)} - 1) = \\frac{d}{d L} (- L + 2 \\rho_{b}{(L)} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["add", 1, "Function('\\\\rho_b')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_b')(Symbol('L', commutative=True))), Add(Function('\\\\rho_b')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\rho_b')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\rho_b')(Symbol('L', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Integer(-1)))"], [["add", 4, "log(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\rho_b')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), log(Symbol('L', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('L', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), log(Symbol('L', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\rho_b')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('L', commutative=True))), Integer(-1)))"], [["differentiate", 7, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\rho_b')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Function('\\\\rho_b')(Symbol('L', commutative=True))), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(x^\\prime)} = x^\\prime and \\operatorname{E_{n}}{(x^\\prime)} = x^\\prime, then obtain (\\frac{b^{x^\\prime}{(x^\\prime)}}{x^\\prime})^{x^\\prime} = (\\frac{(x^\\prime)^{x^\\prime}}{x^\\prime})^{x^\\prime}", "derivation": "b{(x^\\prime)} = x^\\prime and \\operatorname{E_{n}}{(x^\\prime)} = x^\\prime and b^{x^\\prime}{(x^\\prime)} = (x^\\prime)^{x^\\prime} and \\frac{b^{x^\\prime}{(x^\\prime)}}{\\operatorname{E_{n}}{(x^\\prime)}} = \\frac{(x^\\prime)^{x^\\prime}}{\\operatorname{E_{n}}{(x^\\prime)}} and \\frac{b^{x^\\prime}{(x^\\prime)}}{x^\\prime} = \\frac{(x^\\prime)^{x^\\prime}}{x^\\prime} and (\\frac{b^{x^\\prime}{(x^\\prime)}}{x^\\prime})^{x^\\prime} = (\\frac{(x^\\prime)^{x^\\prime}}{x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('b')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 3, "Function('E_n')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Function('E_n')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Pow(Function('b')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('E_n')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('b')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["power", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Function('b')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\sigma_x,x^\\prime)} = \\frac{e^{\\sigma_x}}{x^\\prime}, then obtain \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + (\\int \\varphi^{*}{(\\sigma_x,x^\\prime)} d\\sigma_x)^{x^\\prime}) = \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + (\\int \\frac{e^{\\sigma_x}}{x^\\prime} d\\sigma_x)^{x^\\prime})", "derivation": "\\varphi^{*}{(\\sigma_x,x^\\prime)} = \\frac{e^{\\sigma_x}}{x^\\prime} and \\int \\varphi^{*}{(\\sigma_x,x^\\prime)} d\\sigma_x = \\int \\frac{e^{\\sigma_x}}{x^\\prime} d\\sigma_x and (\\int \\varphi^{*}{(\\sigma_x,x^\\prime)} d\\sigma_x)^{x^\\prime} = (\\int \\frac{e^{\\sigma_x}}{x^\\prime} d\\sigma_x)^{x^\\prime} and - \\sigma_x + (\\int \\varphi^{*}{(\\sigma_x,x^\\prime)} d\\sigma_x)^{x^\\prime} = - \\sigma_x + (\\int \\frac{e^{\\sigma_x}}{x^\\prime} d\\sigma_x)^{x^\\prime} and \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + (\\int \\varphi^{*}{(\\sigma_x,x^\\prime)} d\\sigma_x)^{x^\\prime}) = \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x + (\\int \\frac{e^{\\sigma_x}}{x^\\prime} d\\sigma_x)^{x^\\prime})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\sigma_x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\sigma_x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integral(Function('\\\\varphi^*')(Symbol('\\\\sigma_x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('\\\\sigma_x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Function('\\\\varphi^*')(Symbol('\\\\sigma_x', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(G)} = \\log{(\\log{(G)})} and h{(G)} = \\log{(G)}, then obtain (\\int \\frac{h{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG)^{G} = (\\int \\frac{\\log{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG)^{G}", "derivation": "\\operatorname{f^{*}}{(G)} = \\log{(\\log{(G)})} and h{(G)} = \\log{(G)} and \\frac{h{(G)}}{\\operatorname{f^{*}}{(G)} - \\log{(G)}} = \\frac{\\log{(G)}}{\\operatorname{f^{*}}{(G)} - \\log{(G)}} and \\int \\frac{h{(G)}}{\\operatorname{f^{*}}{(G)} - \\log{(G)}} dG = \\int \\frac{\\log{(G)}}{\\operatorname{f^{*}}{(G)} - \\log{(G)}} dG and \\int \\frac{h{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG = \\int \\frac{\\log{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG and (\\int \\frac{h{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG)^{G} = (\\int \\frac{\\log{(G)}}{- \\log{(G)} + \\log{(\\log{(G)})}} dG)^{G}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('G', commutative=True)), log(log(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('h')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["divide", 2, "Add(Function('f^*')(Symbol('G', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True))))"], "Equality(Mul(Pow(Add(Function('f^*')(Symbol('G', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(-1)), Function('h')(Symbol('G', commutative=True))), Mul(Pow(Add(Function('f^*')(Symbol('G', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(-1)), log(Symbol('G', commutative=True))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Function('f^*')(Symbol('G', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(-1)), Function('h')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Add(Function('f^*')(Symbol('G', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(-1)), log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), log(Symbol('G', commutative=True))), log(log(Symbol('G', commutative=True)))), Integer(-1)), Function('h')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), log(Symbol('G', commutative=True))), log(log(Symbol('G', commutative=True)))), Integer(-1)), log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["power", 5, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Add(Mul(Integer(-1), log(Symbol('G', commutative=True))), log(log(Symbol('G', commutative=True)))), Integer(-1)), Function('h')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Integral(Mul(Pow(Add(Mul(Integer(-1), log(Symbol('G', commutative=True))), log(log(Symbol('G', commutative=True)))), Integer(-1)), log(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(F_{c})} = F_{c}, then derive P_{g} + \\operatorname{c_{0}}{(F_{c})} = F_{c} + \\rho_b, then obtain (F_{c} + P_{g})^{\\rho_b} = (F_{c} + \\rho_b)^{\\rho_b}", "derivation": "\\operatorname{c_{0}}{(F_{c})} = F_{c} and \\frac{d}{d F_{c}} \\operatorname{c_{0}}{(F_{c})} = \\frac{d}{d F_{c}} F_{c} and \\int \\frac{d}{d F_{c}} \\operatorname{c_{0}}{(F_{c})} dF_{c} = \\int \\frac{d}{d F_{c}} F_{c} dF_{c} and P_{g} + \\operatorname{c_{0}}{(F_{c})} = F_{c} + \\rho_b and F_{c} + P_{g} = F_{c} + \\rho_b and (F_{c} + P_{g})^{\\rho_b} = (F_{c} + \\rho_b)^{\\rho_b}", "srepr_derivation": [["renaming_premise", "Equality(Function('c_0')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Derivative(Function('c_0')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Tuple(Symbol('F_c', commutative=True))), Integral(Derivative(Symbol('F_c', commutative=True), Tuple(Symbol('F_c', commutative=True), Integer(1))), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('P_g', commutative=True), Function('c_0')(Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('F_c', commutative=True), Symbol('P_g', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["power", 5, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Add(Symbol('F_c', commutative=True), Symbol('P_g', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f^{\\prime},F_{N})} = (e^{f^{\\prime}})^{F_{N}} and U{(f^{\\prime},F_{N})} = (F_{N} + \\operatorname{n_{1}}{(f^{\\prime},F_{N})}) e^{f^{\\prime}}, then obtain U{(f^{\\prime},F_{N})} + 1 = (F_{N} + (e^{f^{\\prime}})^{F_{N}}) e^{f^{\\prime}} + 1", "derivation": "\\operatorname{n_{1}}{(f^{\\prime},F_{N})} = (e^{f^{\\prime}})^{F_{N}} and F_{N} + \\operatorname{n_{1}}{(f^{\\prime},F_{N})} = F_{N} + (e^{f^{\\prime}})^{F_{N}} and (F_{N} + \\operatorname{n_{1}}{(f^{\\prime},F_{N})}) e^{f^{\\prime}} = (F_{N} + (e^{f^{\\prime}})^{F_{N}}) e^{f^{\\prime}} and U{(f^{\\prime},F_{N})} = (F_{N} + \\operatorname{n_{1}}{(f^{\\prime},F_{N})}) e^{f^{\\prime}} and U{(f^{\\prime},F_{N})} = (F_{N} + (e^{f^{\\prime}})^{F_{N}}) e^{f^{\\prime}} and U{(f^{\\prime},F_{N})} + 1 = (F_{N} + (e^{f^{\\prime}})^{F_{N}}) e^{f^{\\prime}} + 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True)), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('F_N', commutative=True)))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('n_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('F_N', commutative=True))))"], [["times", 2, "exp(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Symbol('F_N', commutative=True), Function('n_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True))), exp(Symbol('f^{\\\\prime}', commutative=True))), Mul(Add(Symbol('F_N', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('F_N', commutative=True))), exp(Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True)), Mul(Add(Symbol('F_N', commutative=True), Function('n_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True))), exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('U')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True)), Mul(Add(Symbol('F_N', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('F_N', commutative=True))), exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Function('U')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_N', commutative=True)), Integer(1)), Add(Mul(Add(Symbol('F_N', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('F_N', commutative=True))), exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\eta{(\\hbar,L)} = \\frac{L}{\\hbar}, then obtain \\frac{\\hbar^{2} \\eta^{\\hbar}{(\\hbar,L)} \\frac{\\partial}{\\partial L} \\eta{(\\hbar,L)}}{L \\eta{(\\hbar,L)}} = \\frac{\\hbar^{2} (\\frac{L}{\\hbar})^{\\hbar}}{L^{2}}", "derivation": "\\eta{(\\hbar,L)} = \\frac{L}{\\hbar} and \\eta^{\\hbar}{(\\hbar,L)} = (\\frac{L}{\\hbar})^{\\hbar} and \\frac{\\partial}{\\partial L} \\eta^{\\hbar}{(\\hbar,L)} = \\frac{\\partial}{\\partial L} (\\frac{L}{\\hbar})^{\\hbar} and \\frac{\\hbar \\frac{\\partial}{\\partial L} \\eta^{\\hbar}{(\\hbar,L)}}{L} = \\frac{\\hbar \\frac{\\partial}{\\partial L} (\\frac{L}{\\hbar})^{\\hbar}}{L} and \\frac{\\hbar^{2} \\eta^{\\hbar}{(\\hbar,L)} \\frac{\\partial}{\\partial L} \\eta{(\\hbar,L)}}{L \\eta{(\\hbar,L)}} = \\frac{\\hbar^{2} (\\frac{L}{\\hbar})^{\\hbar}}{L^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True), Derivative(Pow(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True), Derivative(Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Pow(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Pow(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\hbar', commutative=True)), Derivative(Function('\\\\eta')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-2)), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Pow(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(g,B)} = \\int g^{B} dg, then obtain \\frac{\\partial}{\\partial g} (\\frac{\\varepsilon{(g,B)}}{g})^{g} - \\int g^{B} dg = \\frac{\\partial}{\\partial g} (\\frac{\\int g^{B} dg}{g})^{g} - \\int g^{B} dg", "derivation": "\\varepsilon{(g,B)} = \\int g^{B} dg and \\frac{\\varepsilon{(g,B)}}{g} = \\frac{\\int g^{B} dg}{g} and (\\frac{\\varepsilon{(g,B)}}{g})^{g} = (\\frac{\\int g^{B} dg}{g})^{g} and \\frac{\\partial}{\\partial g} (\\frac{\\varepsilon{(g,B)}}{g})^{g} = \\frac{\\partial}{\\partial g} (\\frac{\\int g^{B} dg}{g})^{g} and \\frac{\\partial}{\\partial g} (\\frac{\\varepsilon{(g,B)}}{g})^{g} - \\int g^{B} dg = \\frac{\\partial}{\\partial g} (\\frac{\\int g^{B} dg}{g})^{g} - \\int g^{B} dg", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('B', commutative=True)), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 4, "Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Derivative(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('g', commutative=True), Symbol('B', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True))))), Add(Derivative(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Pow(Symbol('g', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(s,B)} = B s, then derive \\frac{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}}{B s} - \\frac{\\operatorname{F_{N}}{(s,B)}}{B^{2} s} = 0, then obtain \\frac{\\frac{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}}{B s} - \\frac{\\operatorname{F_{N}}{(s,B)}}{B^{2} s}}{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}} = 0", "derivation": "\\operatorname{F_{N}}{(s,B)} = B s and \\frac{\\operatorname{F_{N}}{(s,B)}}{B} = s and \\frac{\\operatorname{F_{N}}{(s,B)}}{B s} = 1 and \\frac{\\partial}{\\partial B} \\frac{\\operatorname{F_{N}}{(s,B)}}{B s} = \\frac{d}{d B} 1 and \\frac{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}}{B s} - \\frac{\\operatorname{F_{N}}{(s,B)}}{B^{2} s} = 0 and \\frac{\\frac{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}}{B s} - \\frac{\\operatorname{F_{N}}{(s,B)}}{B^{2} s}}{\\frac{\\partial}{\\partial B} \\operatorname{F_{N}}{(s,B)}} = 0", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('s', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Symbol('s', commutative=True))"], [["divide", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)))), Integer(0))"], [["divide", 5, "Derivative(Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Derivative(Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)))), Pow(Derivative(Function('F_N')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given q{(L)} = \\log{(L)}, then derive (\\int (- L + q{(L)}) dL)^{L} = (- \\frac{L^{2}}{2} + L \\log{(L)} - L + f_{\\mathbf{v}})^{L}, then obtain (- \\frac{L^{2}}{2} + L \\log{(L)} - L + f_{\\mathbf{v}})^{L} = (\\int (- L + \\log{(L)}) dL)^{L}", "derivation": "q{(L)} = \\log{(L)} and - L + q{(L)} = - L + \\log{(L)} and \\int (- L + q{(L)}) dL = \\int (- L + \\log{(L)}) dL and (\\int (- L + q{(L)}) dL)^{L} = (\\int (- L + \\log{(L)}) dL)^{L} and (\\int (- L + q{(L)}) dL)^{L} = (- \\frac{L^{2}}{2} + L \\log{(L)} - L + f_{\\mathbf{v}})^{L} and (- \\frac{L^{2}}{2} + L \\log{(L)} - L + f_{\\mathbf{v}})^{L} = (\\int (- L + \\log{(L)}) dL)^{L}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["minus", 1, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('q')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('q')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('q')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('q')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('L', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2))), Mul(Symbol('L', commutative=True), log(Symbol('L', commutative=True))), Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('L', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given M{(B)} = \\frac{d}{d B} \\log{(B)}, then derive M^{B}{(B)} = (\\frac{1}{B})^{B}, then derive (\\frac{B \\frac{d}{d B} M{(B)}}{M{(B)}} + \\log{(M{(B)})}) M^{B}{(B)} = (\\log{(\\frac{1}{B})} - 1) (\\frac{1}{B})^{B}, then obtain (\\frac{B \\frac{d}{d B} M{(B)}}{M{(B)}} + \\log{(M{(B)})}) (\\frac{1}{B})^{B} - (\\frac{1}{B})^{B} = (\\log{(\\frac{1}{B})} - 1) (\\frac{1}{B})^{B} - (\\frac{1}{B})^{B}", "derivation": "M{(B)} = \\frac{d}{d B} \\log{(B)} and M^{B}{(B)} = (\\frac{d}{d B} \\log{(B)})^{B} and M^{B}{(B)} = (\\frac{1}{B})^{B} and \\frac{d}{d B} M^{B}{(B)} = \\frac{d}{d B} (\\frac{1}{B})^{B} and (\\frac{B \\frac{d}{d B} M{(B)}}{M{(B)}} + \\log{(M{(B)})}) M^{B}{(B)} = (\\log{(\\frac{1}{B})} - 1) (\\frac{1}{B})^{B} and (\\frac{B \\frac{d}{d B} M{(B)}}{M{(B)}} + \\log{(M{(B)})}) M^{B}{(B)} - M^{B}{(B)} = (\\log{(\\frac{1}{B})} - 1) (\\frac{1}{B})^{B} - M^{B}{(B)} and (\\frac{B \\frac{d}{d B} M{(B)}}{M{(B)}} + \\log{(M{(B)})}) (\\frac{1}{B})^{B} - (\\frac{1}{B})^{B} = (\\log{(\\frac{1}{B})} - 1) (\\frac{1}{B})^{B} - (\\frac{1}{B})^{B}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('B', commutative=True)), Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(log(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True)))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Mul(Symbol('B', commutative=True), Pow(Function('M')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), log(Function('M')(Symbol('B', commutative=True)))), Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Add(log(Pow(Symbol('B', commutative=True), Integer(-1))), Integer(-1)), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True))))"], [["minus", 5, "Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True))"], "Equality(Add(Mul(Add(Mul(Symbol('B', commutative=True), Pow(Function('M')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), log(Function('M')(Symbol('B', commutative=True)))), Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True)))), Add(Mul(Add(log(Pow(Symbol('B', commutative=True), Integer(-1))), Integer(-1)), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('M')(Symbol('B', commutative=True)), Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Add(Mul(Symbol('B', commutative=True), Pow(Function('M')(Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('M')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), log(Function('M')(Symbol('B', commutative=True)))), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True)))), Add(Mul(Add(log(Pow(Symbol('B', commutative=True), Integer(-1))), Integer(-1)), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\pi{(x)} = \\int \\cos{(x)} dx and \\phi_{2}{(x)} = \\frac{\\pi{(x)}}{\\int \\cos{(x)} dx}, then obtain \\phi_{2}{(x)} = 1", "derivation": "\\pi{(x)} = \\int \\cos{(x)} dx and \\frac{\\pi{(x)}}{\\int \\cos{(x)} dx} = 1 and \\phi_{2}{(x)} = \\frac{\\pi{(x)}}{\\int \\cos{(x)} dx} and \\phi_{2}{(x)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('x', commutative=True)), Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["divide", 1, "Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('x', commutative=True)), Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('x', commutative=True)), Mul(Function('\\\\pi')(Symbol('x', commutative=True)), Pow(Integral(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\phi_2')(Symbol('x', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\dot{y}{(\\mu,H)} = (e^{\\mu})^{H}, then obtain 4 (\\int \\dot{y}{(\\mu,H)} d\\mu)^{2} = (\\int \\dot{y}{(\\mu,H)} d\\mu + \\int (e^{\\mu})^{H} d\\mu)^{2}", "derivation": "\\dot{y}{(\\mu,H)} = (e^{\\mu})^{H} and \\int \\dot{y}{(\\mu,H)} d\\mu = \\int (e^{\\mu})^{H} d\\mu and 2 \\int \\dot{y}{(\\mu,H)} d\\mu = \\int \\dot{y}{(\\mu,H)} d\\mu + \\int (e^{\\mu})^{H} d\\mu and 4 (\\int \\dot{y}{(\\mu,H)} d\\mu)^{2} = (\\int \\dot{y}{(\\mu,H)} d\\mu + \\int (e^{\\mu})^{H} d\\mu)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["add", 2, "Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(2))), Pow(Add(Integral(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(exp(Symbol('\\\\mu', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(t_{1})} = e^{t_{1}}, then obtain 0 = (\\frac{e^{t_{1}}}{\\operatorname{F_{c}}{(t_{1})}} - \\frac{e^{t_{1}} \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})}}{\\operatorname{F_{c}}^{2}{(t_{1})}}) \\operatorname{F_{c}}{(t_{1})}", "derivation": "\\operatorname{F_{c}}{(t_{1})} = e^{t_{1}} and 1 = \\frac{e^{t_{1}}}{\\operatorname{F_{c}}{(t_{1})}} and \\frac{d}{d t_{1}} 1 = \\frac{d}{d t_{1}} \\frac{e^{t_{1}}}{\\operatorname{F_{c}}{(t_{1})}} and \\operatorname{F_{c}}{(t_{1})} \\frac{d}{d t_{1}} 1 = \\operatorname{F_{c}}{(t_{1})} \\frac{d}{d t_{1}} \\frac{e^{t_{1}}}{\\operatorname{F_{c}}{(t_{1})}} and 0 = (\\frac{e^{t_{1}}}{\\operatorname{F_{c}}{(t_{1})}} - \\frac{e^{t_{1}} \\frac{d}{d t_{1}} \\operatorname{F_{c}}{(t_{1})}}{\\operatorname{F_{c}}^{2}{(t_{1})}}) \\operatorname{F_{c}}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["divide", 1, "Function('F_c')(Symbol('t_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["divide", 3, "Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-1))"], "Equality(Mul(Function('F_c')(Symbol('t_1', commutative=True)), Derivative(Integer(1), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Function('F_c')(Symbol('t_1', commutative=True)), Derivative(Mul(Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Add(Mul(Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))), Mul(Integer(-1), Pow(Function('F_c')(Symbol('t_1', commutative=True)), Integer(-2)), exp(Symbol('t_1', commutative=True)), Derivative(Function('F_c')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Function('F_c')(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given r{(\\tilde{g})} = \\int \\log{(\\tilde{g})} d\\tilde{g}, then derive r{(\\tilde{g})} = \\theta_1 + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g}, then obtain \\frac{\\int \\log{(\\tilde{g})} d\\tilde{g}}{\\theta_1 + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g}} = 1", "derivation": "r{(\\tilde{g})} = \\int \\log{(\\tilde{g})} d\\tilde{g} and r{(\\tilde{g})} = \\theta_1 + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g} and \\int \\log{(\\tilde{g})} d\\tilde{g} = \\theta_1 + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g} and \\frac{\\int \\log{(\\tilde{g})} d\\tilde{g}}{\\theta_1 + \\tilde{g} \\log{(\\tilde{g})} - \\tilde{g}} = 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('r')(Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\ddot{x}{(\\mu,F_{x})} = \\int \\frac{\\mu}{F_{x}} d\\mu and \\lambda{(\\mu,F_{x})} = \\int \\frac{\\mu}{F_{x}} d\\mu, then obtain \\log{(\\lambda{(\\mu,F_{x})})} = \\log{(\\ddot{x}{(\\mu,F_{x})})}", "derivation": "\\ddot{x}{(\\mu,F_{x})} = \\int \\frac{\\mu}{F_{x}} d\\mu and \\lambda{(\\mu,F_{x})} = \\int \\frac{\\mu}{F_{x}} d\\mu and \\log{(\\lambda{(\\mu,F_{x})})} = \\log{(\\int \\frac{\\mu}{F_{x}} d\\mu)} and \\log{(\\lambda{(\\mu,F_{x})})} = \\log{(\\ddot{x}{(\\mu,F_{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True), Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True), Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["log", 2], "Equality(log(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True), Symbol('F_x', commutative=True))), log(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(log(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True), Symbol('F_x', commutative=True))), log(Function('\\\\ddot{x}')(Symbol('\\\\mu', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(k,A_{z})} = A_{z} k, then derive A_{z} k \\frac{\\partial}{\\partial k} \\mathbf{P}{(k,A_{z})} + A_{z} \\mathbf{P}{(k,A_{z})} = 2 A_{z}^{2} k, then obtain A_{z}^{2} k + A_{z} k \\frac{\\partial}{\\partial k} A_{z} k = 2 A_{z}^{2} k", "derivation": "\\mathbf{P}{(k,A_{z})} = A_{z} k and A_{z} k \\mathbf{P}{(k,A_{z})} = A_{z}^{2} k^{2} and \\frac{\\partial}{\\partial k} A_{z} k \\mathbf{P}{(k,A_{z})} = \\frac{\\partial}{\\partial k} A_{z}^{2} k^{2} and A_{z} k \\frac{\\partial}{\\partial k} \\mathbf{P}{(k,A_{z})} + A_{z} \\mathbf{P}{(k,A_{z})} = 2 A_{z}^{2} k and A_{z}^{2} k + A_{z} k \\frac{\\partial}{\\partial k} A_{z} k = 2 A_{z}^{2} k", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True)))"], [["times", 1, "Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True), Function('\\\\mathbf{P}')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True), Function('\\\\mathbf{P}')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_z', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True), Derivative(Function('\\\\mathbf{P}')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Symbol('A_z', commutative=True), Function('\\\\mathbf{P}')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)))), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Integer(2)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('A_z', commutative=True), Integer(2)), Symbol('k', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Integer(2)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{t})} = \\sin{(v_{t})}, then obtain v_{t} - \\operatorname{g^{\\prime}_{\\varepsilon}}^{v_{t}}{(v_{t})} = v_{t} - \\sin^{v_{t}}{(v_{t})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v_{t})} = \\sin{(v_{t})} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{v_{t}}{(v_{t})} = \\sin^{v_{t}}{(v_{t})} and - v_{t} + \\operatorname{g^{\\prime}_{\\varepsilon}}^{v_{t}}{(v_{t})} = - v_{t} + \\sin^{v_{t}}{(v_{t})} and v_{t} - \\operatorname{g^{\\prime}_{\\varepsilon}}^{v_{t}}{(v_{t})} = v_{t} - \\sin^{v_{t}}{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Symbol('v_t', commutative=True), Mul(Integer(-1), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))), Add(Symbol('v_t', commutative=True), Mul(Integer(-1), Pow(sin(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\chi{(c_{0},H)} = - c_{0} + \\cos{(H)}, then obtain \\frac{\\partial}{\\partial c_{0}} (- 2 c_{0} + 2 \\chi{(c_{0},H)} + 2 \\cos{(H)})^{H} = \\frac{\\partial}{\\partial c_{0}} (- 3 c_{0} + \\chi{(c_{0},H)} + 3 \\cos{(H)})^{H}", "derivation": "\\chi{(c_{0},H)} = - c_{0} + \\cos{(H)} and - c_{0} + \\chi{(c_{0},H)} + \\cos{(H)} = - 2 c_{0} + 2 \\cos{(H)} and - 2 c_{0} + 2 \\chi{(c_{0},H)} + 2 \\cos{(H)} = - 3 c_{0} + \\chi{(c_{0},H)} + 3 \\cos{(H)} and (- 2 c_{0} + 2 \\chi{(c_{0},H)} + 2 \\cos{(H)})^{H} = (- 3 c_{0} + \\chi{(c_{0},H)} + 3 \\cos{(H)})^{H} and \\frac{\\partial}{\\partial c_{0}} (- 2 c_{0} + 2 \\chi{(c_{0},H)} + 2 \\cos{(H)})^{H} = \\frac{\\partial}{\\partial c_{0}} (- 3 c_{0} + \\chi{(c_{0},H)} + 3 \\cos{(H)})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), cos(Symbol('H', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), cos(Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), cos(Symbol('H', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Integer(3), Symbol('c_0', commutative=True)), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Mul(Integer(3), cos(Symbol('H', commutative=True)))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(3), Symbol('c_0', commutative=True)), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Mul(Integer(3), cos(Symbol('H', commutative=True)))), Symbol('H', commutative=True)))"], [["differentiate", 4, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Integer(3), Symbol('c_0', commutative=True)), Function('\\\\chi')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Mul(Integer(3), cos(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\chi,\\mathbf{S})} = \\chi + \\mathbf{S}, then obtain (\\chi + \\mathbf{S}) \\frac{\\partial}{\\partial \\chi} \\mathbf{g}{(\\chi,\\mathbf{S})} + \\mathbf{g}{(\\chi,\\mathbf{S})} = 2 \\chi + 2 \\mathbf{S}", "derivation": "\\mathbf{g}{(\\chi,\\mathbf{S})} = \\chi + \\mathbf{S} and (\\chi + \\mathbf{S}) \\mathbf{g}{(\\chi,\\mathbf{S})} = (\\chi + \\mathbf{S})^{2} and (\\chi + \\mathbf{S}) \\mathbf{g}{(\\chi,\\mathbf{S})} + 1 = (\\chi + \\mathbf{S})^{2} + 1 and \\frac{\\partial}{\\partial \\chi} ((\\chi + \\mathbf{S}) \\mathbf{g}{(\\chi,\\mathbf{S})} + 1) = \\frac{\\partial}{\\partial \\chi} ((\\chi + \\mathbf{S})^{2} + 1) and (\\chi + \\mathbf{S}) \\frac{\\partial}{\\partial \\chi} \\mathbf{g}{(\\chi,\\mathbf{S})} + \\mathbf{g}{(\\chi,\\mathbf{S})} = 2 \\chi + 2 \\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(1)), Add(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Integer(1)))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Integer(1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\mathbf{D},E_{x})} = \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D}, then derive - E_{x} \\mathbf{D} + \\phi{(\\mathbf{D},E_{x})} = - E_{x} \\mathbf{D} + E_{x}, then obtain - E_{x} \\mathbf{D} + \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D} + 1 = - E_{x} \\mathbf{D} + E_{x} + 1", "derivation": "\\phi{(\\mathbf{D},E_{x})} = \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D} and - E_{x} \\mathbf{D} + \\phi{(\\mathbf{D},E_{x})} = - E_{x} \\mathbf{D} + \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D} and - E_{x} \\mathbf{D} + \\phi{(\\mathbf{D},E_{x})} = - E_{x} \\mathbf{D} + E_{x} and - E_{x} \\mathbf{D} + \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D} = - E_{x} \\mathbf{D} + E_{x} and - E_{x} \\mathbf{D} + \\frac{\\partial}{\\partial \\mathbf{D}} E_{x} \\mathbf{D} + 1 = - E_{x} \\mathbf{D} + E_{x} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["minus", 1, "Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\phi')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('E_x', commutative=True)))"], [["add", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('E_x', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(M)} = \\log{(\\cos{(M)})}, then obtain (- M + (\\operatorname{v_{y}}{(M)} \\cos{(M)})^{M})^{M} = (- M + (\\log{(\\cos{(M)})} \\cos{(M)})^{M})^{M}", "derivation": "\\operatorname{v_{y}}{(M)} = \\log{(\\cos{(M)})} and \\operatorname{v_{y}}{(M)} \\cos{(M)} = \\log{(\\cos{(M)})} \\cos{(M)} and (\\operatorname{v_{y}}{(M)} \\cos{(M)})^{M} = (\\log{(\\cos{(M)})} \\cos{(M)})^{M} and - M + (\\operatorname{v_{y}}{(M)} \\cos{(M)})^{M} = - M + (\\log{(\\cos{(M)})} \\cos{(M)})^{M} and (- M + (\\operatorname{v_{y}}{(M)} \\cos{(M)})^{M})^{M} = (- M + (\\log{(\\cos{(M)})} \\cos{(M)})^{M})^{M}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('M', commutative=True)), log(cos(Symbol('M', commutative=True))))"], [["times", 1, "cos(Symbol('M', commutative=True))"], "Equality(Mul(Function('v_y')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Mul(log(cos(Symbol('M', commutative=True))), cos(Symbol('M', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Function('v_y')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(log(cos(Symbol('M', commutative=True))), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["minus", 3, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Mul(Function('v_y')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Mul(log(cos(Symbol('M', commutative=True))), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True))))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Mul(Function('v_y')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Mul(log(cos(Symbol('M', commutative=True))), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\omega{(v_{y})} = \\sin{(v_{y})}, then obtain - v_{y} (- v_{y} + \\omega{(v_{y})} + \\sin{(\\hat{\\mathbf{r}})}) - 1 = - v_{y} (- v_{y} + \\sin{(\\hat{\\mathbf{r}})} + \\sin{(v_{y})}) - 1", "derivation": "\\omega{(v_{y})} = \\sin{(v_{y})} and \\omega{(v_{y})} + \\sin{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} + \\sin{(v_{y})} and - v_{y} + \\omega{(v_{y})} + \\sin{(\\hat{\\mathbf{r}})} = - v_{y} + \\sin{(\\hat{\\mathbf{r}})} + \\sin{(v_{y})} and - v_{y} (- v_{y} + \\omega{(v_{y})} + \\sin{(\\hat{\\mathbf{r}})}) = - v_{y} (- v_{y} + \\sin{(\\hat{\\mathbf{r}})} + \\sin{(v_{y})}) and - v_{y} (- v_{y} + \\omega{(v_{y})} + \\sin{(\\hat{\\mathbf{r}})}) - 1 = - v_{y} (- v_{y} + \\sin{(\\hat{\\mathbf{r}})} + \\sin{(v_{y})}) - 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\omega')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["minus", 2, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\omega')(Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_y', commutative=True), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\omega')(Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Integer(-1), Symbol('v_y', commutative=True), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('v_y', commutative=True)))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\omega')(Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_y', commutative=True), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('v_y', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\dot{z}{(f_{\\mathbf{p}})} = \\log{(\\log{(f_{\\mathbf{p}})})}, then obtain \\frac{\\dot{z}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} \\log{(f_{\\mathbf{p}})}}{f_{\\mathbf{p}}} = \\frac{\\log{(f_{\\mathbf{p}})} \\log{(\\log{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}}}{f_{\\mathbf{p}}}", "derivation": "\\dot{z}{(f_{\\mathbf{p}})} = \\log{(\\log{(f_{\\mathbf{p}})})} and \\dot{z}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} = \\log{(\\log{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}} and \\frac{\\dot{z}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})}}{f_{\\mathbf{p}}} = \\frac{\\log{(\\log{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}}}{f_{\\mathbf{p}}} and \\frac{\\dot{z}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} \\log{(f_{\\mathbf{p}})}}{f_{\\mathbf{p}}} = \\frac{\\log{(f_{\\mathbf{p}})} \\log{(\\log{(f_{\\mathbf{p}})})}^{f_{\\mathbf{p}}}}{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(log(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["divide", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(log(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["times", 3, "log(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), log(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(log(log(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(c,W,\\eta^{\\prime})} = \\frac{W}{c} - \\eta^{\\prime}, then obtain \\frac{\\partial}{\\partial W} \\mathbb{I}{(c,W,\\eta^{\\prime})} - \\frac{1}{c} = 0", "derivation": "\\mathbb{I}{(c,W,\\eta^{\\prime})} = \\frac{W}{c} - \\eta^{\\prime} and - \\frac{W}{c} + \\mathbb{I}{(c,W,\\eta^{\\prime})} = - \\eta^{\\prime} and - \\frac{W}{c} + \\mathbb{I}{(c,W,\\eta^{\\prime})} - 1 = - \\eta^{\\prime} - 1 and \\frac{\\partial}{\\partial W} (- \\frac{W}{c} + \\mathbb{I}{(c,W,\\eta^{\\prime})} - 1) = \\frac{d}{d W} (- \\eta^{\\prime} - 1) and \\frac{\\partial}{\\partial W} \\mathbb{I}{(c,W,\\eta^{\\prime})} - \\frac{1}{c} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('c', commutative=True), Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["minus", 1, "Mul(Symbol('W', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('c', commutative=True), Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('c', commutative=True), Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('c', commutative=True), Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\mathbb{I}')(Symbol('c', commutative=True), Symbol('W', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(F_{c},c)} = e^{F_{c} c}, then derive \\frac{\\partial}{\\partial F_{c}} \\operatorname{F_{g}}{(F_{c},c)} = c e^{F_{c} c}, then obtain \\frac{\\partial^{2}}{\\partial F_{c}^{2}} e^{F_{c} c} = \\frac{\\partial^{2}}{\\partial F_{c}^{2}} \\operatorname{F_{g}}{(F_{c},c)}", "derivation": "\\operatorname{F_{g}}{(F_{c},c)} = e^{F_{c} c} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{F_{g}}{(F_{c},c)} = \\frac{\\partial}{\\partial F_{c}} e^{F_{c} c} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{F_{g}}{(F_{c},c)} = c e^{F_{c} c} and \\frac{\\partial^{2}}{\\partial F_{c}^{2}} \\operatorname{F_{g}}{(F_{c},c)} = \\frac{\\partial}{\\partial F_{c}} c e^{F_{c} c} and \\frac{\\partial^{2}}{\\partial F_{c}^{2}} e^{F_{c} c} = \\frac{\\partial}{\\partial F_{c}} c e^{F_{c} c} and \\frac{\\partial^{2}}{\\partial F_{c}^{2}} e^{F_{c} c} = \\frac{\\partial^{2}}{\\partial F_{c}^{2}} \\operatorname{F_{g}}{(F_{c},c)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('F_c', commutative=True), Symbol('c', commutative=True)), exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('F_c', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('F_c', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Mul(Symbol('c', commutative=True), exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True)))))"], [["differentiate", 3, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('F_c', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(2))), Derivative(Mul(Symbol('c', commutative=True), exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(2))), Derivative(Mul(Symbol('c', commutative=True), exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(exp(Mul(Symbol('F_c', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(2))), Derivative(Function('F_g')(Symbol('F_c', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\theta_1)} = \\int e^{\\theta_1} d\\theta_1, then derive \\operatorname{M_{E}}^{2}{(\\theta_1)} = (\\dot{x} + e^{\\theta_1}) \\operatorname{M_{E}}{(\\theta_1)}, then obtain \\sin^{\\dot{x}}{((\\int e^{\\theta_1} d\\theta_1)^{2})} = \\sin^{\\dot{x}}{((\\dot{x} + e^{\\theta_1}) \\int e^{\\theta_1} d\\theta_1)}", "derivation": "\\operatorname{M_{E}}{(\\theta_1)} = \\int e^{\\theta_1} d\\theta_1 and \\operatorname{M_{E}}^{2}{(\\theta_1)} = \\operatorname{M_{E}}{(\\theta_1)} \\int e^{\\theta_1} d\\theta_1 and \\operatorname{M_{E}}^{2}{(\\theta_1)} = (\\dot{x} + e^{\\theta_1}) \\operatorname{M_{E}}{(\\theta_1)} and (\\int e^{\\theta_1} d\\theta_1)^{2} = (\\dot{x} + e^{\\theta_1}) \\int e^{\\theta_1} d\\theta_1 and \\sin{((\\int e^{\\theta_1} d\\theta_1)^{2})} = \\sin{((\\dot{x} + e^{\\theta_1}) \\int e^{\\theta_1} d\\theta_1)} and \\sin^{\\dot{x}}{((\\int e^{\\theta_1} d\\theta_1)^{2})} = \\sin^{\\dot{x}}{((\\dot{x} + e^{\\theta_1}) \\int e^{\\theta_1} d\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Function('M_E')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Pow(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Mul(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Pow(Function('M_E')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Function('M_E')(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["sin", 4], "Equality(sin(Pow(Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))), sin(Mul(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"], [["power", 5, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(sin(Pow(Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2))), Symbol('\\\\dot{x}', commutative=True)), Pow(sin(Mul(Add(Symbol('\\\\dot{x}', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\hat{H},\\delta,F_{H})} = \\frac{F_{H}}{\\delta \\hat{H}} and \\phi_{2}{(\\hat{H},\\delta,F_{H})} = \\frac{F_{H}^{2}}{\\delta \\hat{H}} - 1, then obtain \\frac{F_{H} \\phi_{2}{(\\hat{H},\\delta,F_{H})}}{\\delta \\hat{H}} = \\frac{F_{H} (\\frac{F_{H}^{2}}{\\delta \\hat{H}} - 1)}{\\delta \\hat{H}}", "derivation": "\\mathbf{s}{(\\hat{H},\\delta,F_{H})} = \\frac{F_{H}}{\\delta \\hat{H}} and \\phi_{2}{(\\hat{H},\\delta,F_{H})} = \\frac{F_{H}^{2}}{\\delta \\hat{H}} - 1 and \\mathbf{s}{(\\hat{H},\\delta,F_{H})} \\phi_{2}{(\\hat{H},\\delta,F_{H})} = (\\frac{F_{H}^{2}}{\\delta \\hat{H}} - 1) \\mathbf{s}{(\\hat{H},\\delta,F_{H})} and \\frac{F_{H} \\phi_{2}{(\\hat{H},\\delta,F_{H})}}{\\delta \\hat{H}} = \\frac{F_{H} (\\frac{F_{H}^{2}}{\\delta \\hat{H}} - 1)}{\\delta \\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True)), Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(2)), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Integer(-1)))"], [["times", 2, "Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))), Mul(Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(2)), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('F_H', commutative=True), Integer(2)), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A_{2},y^{\\prime})} = - A_{2} + \\cos{(y^{\\prime})}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\Psi_{\\lambda}{(A_{2},y^{\\prime})} = - \\sin{(y^{\\prime})}, then obtain \\frac{\\partial}{\\partial y^{\\prime}} (- A_{2} + \\cos{(y^{\\prime})}) = - \\sin{(y^{\\prime})}", "derivation": "\\Psi_{\\lambda}{(A_{2},y^{\\prime})} = - A_{2} + \\cos{(y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\Psi_{\\lambda}{(A_{2},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (- A_{2} + \\cos{(y^{\\prime})}) and \\frac{\\partial}{\\partial y^{\\prime}} \\Psi_{\\lambda}{(A_{2},y^{\\prime})} = - \\sin{(y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} (- A_{2} + \\cos{(y^{\\prime})}) = - \\sin{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\eta,q)} = \\int (\\eta + q) dq, then derive \\operatorname{g_{\\varepsilon}}{(\\eta,q)} = \\Psi + \\eta q + \\frac{q^{2}}{2}, then derive \\Psi + \\eta q + \\frac{q^{2}}{2} = \\eta q + \\varphi^* + \\frac{q^{2}}{2}, then obtain \\int (\\eta + q) dq = \\eta q + \\varphi^* + \\frac{q^{2}}{2}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\eta,q)} = \\int (\\eta + q) dq and \\operatorname{g_{\\varepsilon}}{(\\eta,q)} = \\Psi + \\eta q + \\frac{q^{2}}{2} and \\Psi + \\eta q + \\frac{q^{2}}{2} = \\int (\\eta + q) dq and \\Psi + \\eta q + \\frac{q^{2}}{2} = \\eta q + \\varphi^* + \\frac{q^{2}}{2} and \\int (\\eta + q) dq = \\eta q + \\varphi^* + \\frac{q^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\theta_{2}{(f_{E})} = \\cos{(f_{E})}, then obtain \\cos{(f_{E} \\theta_{2}{(f_{E})} + \\theta_{2}{(f_{E})})} = \\cos{(f_{E} \\theta_{2}{(f_{E})} + \\cos{(f_{E})})}", "derivation": "\\theta_{2}{(f_{E})} = \\cos{(f_{E})} and f_{E} \\theta_{2}{(f_{E})} = f_{E} \\cos{(f_{E})} and f_{E} \\cos{(f_{E})} + \\theta_{2}{(f_{E})} = f_{E} \\cos{(f_{E})} + \\cos{(f_{E})} and \\cos{(f_{E} \\cos{(f_{E})} + \\theta_{2}{(f_{E})})} = \\cos{(f_{E} \\cos{(f_{E})} + \\cos{(f_{E})})} and \\cos{(f_{E} \\theta_{2}{(f_{E})} + \\theta_{2}{(f_{E})})} = \\cos{(f_{E} \\theta_{2}{(f_{E})} + \\cos{(f_{E})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["times", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\theta_2')(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True))))"], [["add", 1, "Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True)))"], "Equality(Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True))), Function('\\\\theta_2')(Symbol('f_E', commutative=True))), Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True))))"], [["cos", 3], "Equality(cos(Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True))), Function('\\\\theta_2')(Symbol('f_E', commutative=True)))), cos(Add(Mul(Symbol('f_E', commutative=True), cos(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(cos(Add(Mul(Symbol('f_E', commutative=True), Function('\\\\theta_2')(Symbol('f_E', commutative=True))), Function('\\\\theta_2')(Symbol('f_E', commutative=True)))), cos(Add(Mul(Symbol('f_E', commutative=True), Function('\\\\theta_2')(Symbol('f_E', commutative=True))), cos(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given q{(\\pi,P_{e},m_{s})} = P_{e} m_{s} + \\pi, then obtain q^{\\pi}{(\\pi,P_{e},m_{s})} + \\frac{\\partial}{\\partial \\pi} q^{\\pi}{(\\pi,P_{e},m_{s})} = q^{\\pi}{(\\pi,P_{e},m_{s})} + \\frac{\\partial}{\\partial \\pi} (P_{e} m_{s} + \\pi)^{\\pi}", "derivation": "q{(\\pi,P_{e},m_{s})} = P_{e} m_{s} + \\pi and q^{\\pi}{(\\pi,P_{e},m_{s})} = (P_{e} m_{s} + \\pi)^{\\pi} and \\frac{\\partial}{\\partial \\pi} q^{\\pi}{(\\pi,P_{e},m_{s})} = \\frac{\\partial}{\\partial \\pi} (P_{e} m_{s} + \\pi)^{\\pi} and q^{\\pi}{(\\pi,P_{e},m_{s})} + \\frac{\\partial}{\\partial \\pi} q^{\\pi}{(\\pi,P_{e},m_{s})} = q^{\\pi}{(\\pi,P_{e},m_{s})} + \\frac{\\partial}{\\partial \\pi} (P_{e} m_{s} + \\pi)^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 3, "Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Derivative(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Derivative(Pow(Add(Mul(Symbol('P_e', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(f^{*})} = \\cos{(f^{*})}, then obtain (\\frac{\\sin{(\\operatorname{v_{1}}{(f^{*})})} \\cos{(f^{*})}}{\\operatorname{v_{1}}{(f^{*})}})^{f^{*}} = \\sin^{f^{*}}{(\\cos{(f^{*})})}", "derivation": "\\operatorname{v_{1}}{(f^{*})} = \\cos{(f^{*})} and 1 = \\frac{\\cos{(f^{*})}}{\\operatorname{v_{1}}{(f^{*})}} and \\sin{(\\operatorname{v_{1}}{(f^{*})})} = \\sin{(\\cos{(f^{*})})} and \\sin^{f^{*}}{(\\operatorname{v_{1}}{(f^{*})})} = \\sin^{f^{*}}{(\\cos{(f^{*})})} and \\sin{(\\cos{(f^{*})})} = \\frac{\\sin{(\\cos{(f^{*})})} \\cos{(f^{*})}}{\\operatorname{v_{1}}{(f^{*})}} and \\sin{(\\operatorname{v_{1}}{(f^{*})})} = \\frac{\\sin{(\\operatorname{v_{1}}{(f^{*})})} \\cos{(f^{*})}}{\\operatorname{v_{1}}{(f^{*})}} and (\\frac{\\sin{(\\operatorname{v_{1}}{(f^{*})})} \\cos{(f^{*})}}{\\operatorname{v_{1}}{(f^{*})}})^{f^{*}} = \\sin^{f^{*}}{(\\cos{(f^{*})})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["divide", 1, "Function('v_1')(Symbol('f^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Integer(-1)), cos(Symbol('f^*', commutative=True))))"], [["sin", 1], "Equality(sin(Function('v_1')(Symbol('f^*', commutative=True))), sin(cos(Symbol('f^*', commutative=True))))"], [["power", 3, "Symbol('f^*', commutative=True)"], "Equality(Pow(sin(Function('v_1')(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(sin(cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["times", 2, "sin(cos(Symbol('f^*', commutative=True)))"], "Equality(sin(cos(Symbol('f^*', commutative=True))), Mul(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Integer(-1)), sin(cos(Symbol('f^*', commutative=True))), cos(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(sin(Function('v_1')(Symbol('f^*', commutative=True))), Mul(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Function('v_1')(Symbol('f^*', commutative=True))), cos(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Mul(Pow(Function('v_1')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Function('v_1')(Symbol('f^*', commutative=True))), cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(sin(cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(W)} = e^{W}, then obtain \\chi + \\phi_{2}{(W)} + e^{W} = \\dot{z} + 2 e^{W}", "derivation": "\\phi_{2}{(W)} = e^{W} and \\phi_{2}{(W)} + e^{W} = 2 e^{W} and \\frac{d}{d W} (\\phi_{2}{(W)} + e^{W}) = \\frac{d}{d W} 2 e^{W} and \\int \\frac{d}{d W} (\\phi_{2}{(W)} + e^{W}) dW = \\int \\frac{d}{d W} 2 e^{W} dW and \\chi + \\phi_{2}{(W)} + e^{W} = \\dot{z} + 2 e^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["add", 1, "exp(Symbol('W', commutative=True))"], "Equality(Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Mul(Integer(2), exp(Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integral(Derivative(Mul(Integer(2), exp(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), exp(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f}, then derive \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f = E_{\\lambda} + e^{\\mathbf{J}_f}, then obtain \\log{(\\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f + \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f)} = \\log{(E_{\\lambda} + e^{\\mathbf{J}_f} + \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f)}", "derivation": "\\mathbf{B}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f and \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f = E_{\\lambda} + e^{\\mathbf{J}_f} and \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f = E_{\\lambda} + e^{\\mathbf{J}_f} and \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f + \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f = E_{\\lambda} + e^{\\mathbf{J}_f} + \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\log{(\\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f + \\int e^{\\mathbf{J}_f} d\\mathbf{J}_f)} = \\log{(E_{\\lambda} + e^{\\mathbf{J}_f} + \\int \\mathbf{B}{(\\mathbf{J}_f)} d\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 4, "Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["log", 5], "Equality(log(Add(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))), log(Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(x)} = \\log{(x)}, then obtain (\\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}})^{2} = \\frac{d^{2}}{d x^{2}} 1 \\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}}", "derivation": "\\operatorname{m_{s}}{(x)} = \\log{(x)} and \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}} = 1 and \\frac{d}{d x} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}} = \\frac{d}{d x} 1 and \\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}} = \\frac{d^{2}}{d x^{2}} 1 and (\\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}})^{2} = \\frac{d^{2}}{d x^{2}} 1 \\frac{d^{2}}{d x^{2}} \\frac{\\operatorname{m_{s}}{(x)}}{\\log{(x)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["divide", 1, "log(Symbol('x', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(2))))"], [["times", 4, "Derivative(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(2)))"], "Equality(Pow(Derivative(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(2))), Integer(2)), Mul(Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(2))), Derivative(Mul(Function('m_s')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{\\partial}{\\partial \\phi} \\frac{\\phi}{F_{H}}, then derive \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{1}{F_{H}}, then derive \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = 0, then obtain \\frac{d^{2}}{d F_{H}d \\phi} \\frac{1}{F_{H}} = \\frac{d}{d F_{H}} 0", "derivation": "\\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{\\partial}{\\partial \\phi} \\frac{\\phi}{F_{H}} and \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{1}{F_{H}} and \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{\\partial^{2}}{\\partial \\phi^{2}} \\frac{\\phi}{F_{H}} and \\frac{\\partial}{\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = 0 and \\frac{\\partial^{2}}{\\partial F_{H}\\partial \\phi} \\operatorname{y^{\\prime}}{(\\phi,F_{H})} = \\frac{d}{d F_{H}} 0 and \\frac{d^{2}}{d F_{H}d \\phi} \\frac{1}{F_{H}} = \\frac{d}{d F_{H}} 0", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True)), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(0))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\phi', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Symbol('F_H', commutative=True), Integer(-1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(S,\\hbar)} = S^{\\hbar}, then derive 2 \\hbar l{(S,\\hbar)} \\frac{\\partial}{\\partial S} l{(S,\\hbar)} = \\frac{2 S^{2 \\hbar} \\hbar^{2}}{S}, then obtain 2 \\hbar l{(S,\\hbar)} \\frac{\\partial}{\\partial S} l{(S,\\hbar)} = \\frac{2 S^{\\hbar} \\hbar^{2} l{(S,\\hbar)}}{S}", "derivation": "l{(S,\\hbar)} = S^{\\hbar} and \\hbar l{(S,\\hbar)} = S^{\\hbar} \\hbar and \\hbar l^{2}{(S,\\hbar)} = S^{\\hbar} \\hbar l{(S,\\hbar)} and S^{\\hbar} \\hbar l{(S,\\hbar)} = S^{2 \\hbar} \\hbar and \\frac{\\partial}{\\partial S} S^{\\hbar} \\hbar l{(S,\\hbar)} = \\frac{\\partial}{\\partial S} S^{2 \\hbar} \\hbar and \\frac{\\partial}{\\partial S} \\hbar l^{2}{(S,\\hbar)} = \\frac{\\partial}{\\partial S} S^{\\hbar} \\hbar l{(S,\\hbar)} and \\frac{\\partial}{\\partial S} \\hbar l^{2}{(S,\\hbar)} = \\frac{\\partial}{\\partial S} S^{2 \\hbar} \\hbar and 2 \\hbar l{(S,\\hbar)} \\frac{\\partial}{\\partial S} l{(S,\\hbar)} = \\frac{2 S^{2 \\hbar} \\hbar^{2}}{S} and 2 \\hbar l{(S,\\hbar)} \\frac{\\partial}{\\partial S} l{(S,\\hbar)} = \\frac{2 S^{\\hbar} \\hbar^{2} l{(S,\\hbar)}}{S}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 4, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Pow(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('S', commutative=True), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 8, 4], "Equality(Mul(Integer(2), Symbol('\\\\hbar', commutative=True), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Function('l')(Symbol('S', commutative=True), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(t)} = \\sin{(t)} and \\operatorname{L_{\\varepsilon}}{(c,\\chi)} = \\log{(\\chi c)}, then obtain \\operatorname{L_{\\varepsilon}}{(c,\\chi)} + \\sin{(t)} = \\log{(\\chi c)} + \\sin{(t)}", "derivation": "\\operatorname{v_{t}}{(t)} = \\sin{(t)} and \\operatorname{L_{\\varepsilon}}{(c,\\chi)} = \\log{(\\chi c)} and \\operatorname{L_{\\varepsilon}}{(c,\\chi)} + \\operatorname{v_{t}}{(t)} = \\operatorname{v_{t}}{(t)} + \\log{(\\chi c)} and \\operatorname{L_{\\varepsilon}}{(c,\\chi)} + \\sin{(t)} = \\log{(\\chi c)} + \\sin{(t)}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], ["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True)), log(Mul(Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True))))"], [["add", 2, "Function('v_t')(Symbol('t', commutative=True))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True)), Function('v_t')(Symbol('t', commutative=True))), Add(Function('v_t')(Symbol('t', commutative=True)), log(Mul(Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Symbol('t', commutative=True))), Add(log(Mul(Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True))), sin(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\dot{z},B)} = \\sin{(B + \\dot{z})} and i{(\\delta,H)} = H \\delta, then obtain i{(\\delta,H)} - \\int \\hat{x}_0{(\\dot{z},B)} dB = H \\delta - \\int \\hat{x}_0{(\\dot{z},B)} dB", "derivation": "\\hat{x}_0{(\\dot{z},B)} = \\sin{(B + \\dot{z})} and \\int \\hat{x}_0{(\\dot{z},B)} dB = \\int \\sin{(B + \\dot{z})} dB and i{(\\delta,H)} = H \\delta and i{(\\delta,H)} - \\int \\sin{(B + \\dot{z})} dB = H \\delta - \\int \\sin{(B + \\dot{z})} dB and i{(\\delta,H)} - \\int \\hat{x}_0{(\\dot{z},B)} dB = H \\delta - \\int \\hat{x}_0{(\\dot{z},B)} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(sin(Add(Symbol('B', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('B', commutative=True))))"], ["get_premise", "Equality(Function('i')(Symbol('\\\\delta', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["minus", 3, "Integral(sin(Add(Symbol('B', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('B', commutative=True)))"], "Equality(Add(Function('i')(Symbol('\\\\delta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integral(sin(Add(Symbol('B', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('B', commutative=True))))), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integral(sin(Add(Symbol('B', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('B', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('i')(Symbol('\\\\delta', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given a{(C_{d})} = \\log{(C_{d})}, then obtain 0 = (C_{d} + \\log{(C_{d})}) (- a{(C_{d})} + \\log{(C_{d})})", "derivation": "a{(C_{d})} = \\log{(C_{d})} and C_{d} + a{(C_{d})} = C_{d} + \\log{(C_{d})} and 0 = - a{(C_{d})} + \\log{(C_{d})} and 0 = (C_{d} + a{(C_{d})}) (- a{(C_{d})} + \\log{(C_{d})}) and 0 = (C_{d} + \\log{(C_{d})}) (- a{(C_{d})} + \\log{(C_{d})})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["add", 1, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Function('a')(Symbol('C_d', commutative=True))), Add(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))))"], [["minus", 2, "Add(Symbol('C_d', commutative=True), Function('a')(Symbol('C_d', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a')(Symbol('C_d', commutative=True))), log(Symbol('C_d', commutative=True))))"], [["times", 3, "Add(Symbol('C_d', commutative=True), Function('a')(Symbol('C_d', commutative=True)))"], "Equality(Integer(0), Mul(Add(Symbol('C_d', commutative=True), Function('a')(Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Function('a')(Symbol('C_d', commutative=True))), log(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Add(Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Add(Mul(Integer(-1), Function('a')(Symbol('C_d', commutative=True))), log(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\chi,\\hbar)} = - \\chi + \\hbar and T{(C_{2},\\hbar)} = \\sin{(C_{2} - \\hbar)}, then obtain T{(C_{2},\\hbar)} + \\int (- \\chi + \\hbar)^{\\hbar} d\\chi = \\sin{(C_{2} - \\hbar)} + \\int (- \\chi + \\hbar)^{\\hbar} d\\chi", "derivation": "\\varphi{(\\chi,\\hbar)} = - \\chi + \\hbar and \\varphi^{\\hbar}{(\\chi,\\hbar)} = (- \\chi + \\hbar)^{\\hbar} and \\int \\varphi^{\\hbar}{(\\chi,\\hbar)} d\\chi = \\int (- \\chi + \\hbar)^{\\hbar} d\\chi and T{(C_{2},\\hbar)} = \\sin{(C_{2} - \\hbar)} and T{(C_{2},\\hbar)} + \\int \\varphi^{\\hbar}{(\\chi,\\hbar)} d\\chi = \\sin{(C_{2} - \\hbar)} + \\int \\varphi^{\\hbar}{(\\chi,\\hbar)} d\\chi and T{(C_{2},\\hbar)} + \\int (- \\chi + \\hbar)^{\\hbar} d\\chi = \\sin{(C_{2} - \\hbar)} + \\int (- \\chi + \\hbar)^{\\hbar} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], ["get_premise", "Equality(Function('T')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], [["add", 4, "Integral(Pow(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Function('T')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Pow(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(sin(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Integral(Pow(Function('\\\\varphi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('T')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(sin(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given A{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})}, then obtain \\int \\frac{\\int \\frac{A{(\\rho_f)}}{\\rho_f} d\\rho_f}{\\rho_f} d\\rho_f = \\int \\frac{\\int \\frac{\\sin{(\\sin{(\\rho_f)})}}{\\rho_f} d\\rho_f}{\\rho_f} d\\rho_f", "derivation": "A{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})} and \\frac{A{(\\rho_f)}}{\\rho_f} = \\frac{\\sin{(\\sin{(\\rho_f)})}}{\\rho_f} and \\int \\frac{A{(\\rho_f)}}{\\rho_f} d\\rho_f = \\int \\frac{\\sin{(\\sin{(\\rho_f)})}}{\\rho_f} d\\rho_f and \\frac{\\int \\frac{A{(\\rho_f)}}{\\rho_f} d\\rho_f}{\\rho_f} = \\frac{\\int \\frac{\\sin{(\\sin{(\\rho_f)})}}{\\rho_f} d\\rho_f}{\\rho_f} and \\int \\frac{\\int \\frac{A{(\\rho_f)}}{\\rho_f} d\\rho_f}{\\rho_f} d\\rho_f = \\int \\frac{\\int \\frac{\\sin{(\\sin{(\\rho_f)})}}{\\rho_f} d\\rho_f}{\\rho_f} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(v,E_{x})} = E_{x} e^{v}, then obtain \\frac{\\frac{\\partial}{\\partial v} \\int \\tilde{g}{(v,E_{x})} dE_{x}}{E_{x} e^{v} + 1} = \\frac{\\frac{\\partial}{\\partial v} \\int E_{x} e^{v} dE_{x}}{E_{x} e^{v} + 1}", "derivation": "\\tilde{g}{(v,E_{x})} = E_{x} e^{v} and \\tilde{g}{(v,E_{x})} + 1 = E_{x} e^{v} + 1 and \\int \\tilde{g}{(v,E_{x})} dE_{x} = \\int E_{x} e^{v} dE_{x} and \\frac{\\partial}{\\partial v} \\int \\tilde{g}{(v,E_{x})} dE_{x} = \\frac{\\partial}{\\partial v} \\int E_{x} e^{v} dE_{x} and \\frac{\\frac{\\partial}{\\partial v} \\int \\tilde{g}{(v,E_{x})} dE_{x}}{\\tilde{g}{(v,E_{x})} + 1} = \\frac{\\frac{\\partial}{\\partial v} \\int E_{x} e^{v} dE_{x}}{\\tilde{g}{(v,E_{x})} + 1} and \\frac{\\frac{\\partial}{\\partial v} \\int \\tilde{g}{(v,E_{x})} dE_{x}}{E_{x} e^{v} + 1} = \\frac{\\frac{\\partial}{\\partial v} \\int E_{x} e^{v} dE_{x}}{E_{x} e^{v} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Integer(1)), Add(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Integer(1)))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["divide", 4, "Add(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Integer(1))"], "Equality(Mul(Pow(Add(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Integer(1)), Integer(-1)), Derivative(Integral(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Add(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Integer(1)), Integer(-1)), Derivative(Integral(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Integer(1)), Integer(-1)), Derivative(Integral(Function('\\\\tilde{g}')(Symbol('v', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Integer(1)), Integer(-1)), Derivative(Integral(Mul(Symbol('E_x', commutative=True), exp(Symbol('v', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(T,\\mathbf{F})} = e^{T - \\mathbf{F}}, then derive \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(T,\\mathbf{F})} = - e^{T - \\mathbf{F}}, then obtain \\cos{(\\mathbf{F} \\cos{(e^{T - \\mathbf{F}})})} = \\cos{(\\mathbf{F} \\cos{(\\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}})})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(T,\\mathbf{F})} = e^{T - \\mathbf{F}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(T,\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(T,\\mathbf{F})} = - e^{T - \\mathbf{F}} and - e^{T - \\mathbf{F}} = \\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}} and \\cos{(e^{T - \\mathbf{F}})} = \\cos{(\\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}})} and \\mathbf{F} \\cos{(e^{T - \\mathbf{F}})} = \\mathbf{F} \\cos{(\\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}})} and \\cos{(\\mathbf{F} \\cos{(e^{T - \\mathbf{F}})})} = \\cos{(\\mathbf{F} \\cos{(\\frac{\\partial}{\\partial \\mathbf{F}} e^{T - \\mathbf{F}})})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('T', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))), Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))), cos(Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["divide", 5, "Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))))), Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))))"], [["cos", 6], "Equality(cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))))))), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), cos(Derivative(exp(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(y^{\\prime},E_{\\lambda})} = \\frac{\\cos{(E_{\\lambda})}}{y^{\\prime}} and \\operatorname{r_{0}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})}, then obtain \\int \\operatorname{r_{0}}{(E_{\\lambda})} dy^{\\prime} + \\frac{1}{y^{\\prime}} = \\int \\cos{(E_{\\lambda})} dy^{\\prime} + \\frac{1}{y^{\\prime}}", "derivation": "\\operatorname{P_{e}}{(y^{\\prime},E_{\\lambda})} = \\frac{\\cos{(E_{\\lambda})}}{y^{\\prime}} and y^{\\prime} \\operatorname{P_{e}}{(y^{\\prime},E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\int y^{\\prime} \\operatorname{P_{e}}{(y^{\\prime},E_{\\lambda})} dy^{\\prime} = \\int \\cos{(E_{\\lambda})} dy^{\\prime} and \\operatorname{r_{0}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\operatorname{P_{e}}{(y^{\\prime},E_{\\lambda})} = \\frac{\\operatorname{r_{0}}{(E_{\\lambda})}}{y^{\\prime}} and \\int \\operatorname{r_{0}}{(E_{\\lambda})} dy^{\\prime} = \\int \\cos{(E_{\\lambda})} dy^{\\prime} and \\int \\operatorname{r_{0}}{(E_{\\lambda})} dy^{\\prime} + \\frac{1}{y^{\\prime}} = \\int \\cos{(E_{\\lambda})} dy^{\\prime} + \\frac{1}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), cos(Symbol('E_{\\\\lambda}', commutative=True))))"], [["times", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('P_e')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('P_e')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('P_e')(Symbol('y^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('r_0')(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Function('r_0')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 6, "Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('r_0')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))), Add(Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(S,u)} = S + u, then obtain (\\frac{\\int \\Psi_{nl}{(S,u)} du}{\\int (S + u) du})^{u} = 1", "derivation": "\\Psi_{nl}{(S,u)} = S + u and \\int \\Psi_{nl}{(S,u)} du = \\int (S + u) du and \\frac{\\int \\Psi_{nl}{(S,u)} du}{\\int (S + u) du} = 1 and (\\frac{\\int \\Psi_{nl}{(S,u)} du}{\\int (S + u) du})^{u} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('u', commutative=True)), Add(Symbol('S', commutative=True), Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["divide", 2, "Integral(Add(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Pow(Integral(Add(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Symbol('u', commutative=True)), Integer(1))"]]}, {"prompt": "Given f{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\mathbf{r}{(\\mathbf{H})} = - \\cos{(\\mathbf{H})}, then obtain - (f{(\\mathbf{H})} + \\cos{(\\mathbf{H})})^{2} + \\mathbf{r}{(\\mathbf{H})} = - (f{(\\mathbf{H})} + \\cos{(\\mathbf{H})})^{2} - \\cos{(\\mathbf{H})}", "derivation": "f{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and 2 f{(\\mathbf{H})} = f{(\\mathbf{H})} + \\cos{(\\mathbf{H})} and 4 f^{2}{(\\mathbf{H})} = (f{(\\mathbf{H})} + \\cos{(\\mathbf{H})})^{2} and \\mathbf{r}{(\\mathbf{H})} = - \\cos{(\\mathbf{H})} and \\mathbf{r}{(\\mathbf{H})} - 4 f^{2}{(\\mathbf{H})} = - 4 f^{2}{(\\mathbf{H})} - \\cos{(\\mathbf{H})} and - (f{(\\mathbf{H})} + \\cos{(\\mathbf{H})})^{2} + \\mathbf{r}{(\\mathbf{H})} = - (f{(\\mathbf{H})} + \\cos{(\\mathbf{H})})^{2} - \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Function('f')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(2), Function('f')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Pow(Add(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 4, "Mul(Integer(4), Pow(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integer(4), Pow(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Integer(4), Pow(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Pow(Add(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(z)} = \\sin{(z)}, then obtain (2 \\sin{(z)})^{z} (\\hat{p}{(z)} + \\sin{(z)})^{z} \\hat{p}{(z)} = (2 \\sin{(z)})^{2 z} \\hat{p}{(z)}", "derivation": "\\hat{p}{(z)} = \\sin{(z)} and \\hat{p}{(z)} + \\sin{(z)} = 2 \\sin{(z)} and (\\hat{p}{(z)} + \\sin{(z)})^{z} = (2 \\sin{(z)})^{z} and (\\hat{p}{(z)} + \\sin{(z)})^{z} \\hat{p}{(z)} = (2 \\sin{(z)})^{z} \\hat{p}{(z)} and (2 \\sin{(z)})^{z} (\\hat{p}{(z)} + \\sin{(z)})^{z} \\hat{p}{(z)} = (2 \\sin{(z)})^{2 z} \\hat{p}{(z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["add", 1, "sin(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Mul(Integer(2), sin(Symbol('z', commutative=True))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{p}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Mul(Integer(2), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["times", 3, "Function('\\\\hat{p}')(Symbol('z', commutative=True))"], "Equality(Mul(Pow(Add(Function('\\\\hat{p}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Function('\\\\hat{p}')(Symbol('z', commutative=True))), Mul(Pow(Mul(Integer(2), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Function('\\\\hat{p}')(Symbol('z', commutative=True))))"], [["times", 4, "Pow(Mul(Integer(2), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Function('\\\\hat{p}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Function('\\\\hat{p}')(Symbol('z', commutative=True))), Mul(Pow(Mul(Integer(2), sin(Symbol('z', commutative=True))), Mul(Integer(2), Symbol('z', commutative=True))), Function('\\\\hat{p}')(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(s,B)} = B - s and \\operatorname{A_{z}}{(s,B)} = B - s, then obtain \\frac{\\mathbf{P}{(s,B)}}{B s + \\tilde{g} - \\frac{s^{2}}{2} - s} = \\frac{\\operatorname{A_{z}}{(s,B)}}{B s + \\tilde{g} - \\frac{s^{2}}{2} - s}", "derivation": "\\mathbf{P}{(s,B)} = B - s and \\operatorname{A_{z}}{(s,B)} = B - s and \\frac{\\operatorname{A_{z}}{(s,B)}}{- s + \\int (B - s) ds} = \\frac{B - s}{- s + \\int (B - s) ds} and \\operatorname{A_{z}}{(s,B)} = \\mathbf{P}{(s,B)} and \\frac{\\mathbf{P}{(s,B)}}{- s + \\int (B - s) ds} = \\frac{B - s}{- s + \\int (B - s) ds} and \\frac{\\mathbf{P}{(s,B)}}{- s + \\int (B - s) ds} = \\frac{\\operatorname{A_{z}}{(s,B)}}{- s + \\int (B - s) ds} and \\frac{\\mathbf{P}{(s,B)}}{B s + \\tilde{g} - \\frac{s^{2}}{2} - s} = \\frac{\\operatorname{A_{z}}{(s,B)}}{B s + \\tilde{g} - \\frac{s^{2}}{2} - s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1)), Function('A_z')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_z')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Function('\\\\mathbf{P}')(Symbol('s', commutative=True), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Integer(-1)), Function('A_z')(Symbol('s', commutative=True), Symbol('B', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('s', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('s', commutative=True))), Integer(-1)), Function('A_z')(Symbol('s', commutative=True), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hbar)} = \\log{(\\log{(\\hbar)})}, then derive \\frac{d}{d \\hbar} \\operatorname{P_{g}}{(\\hbar)} = \\frac{1}{\\hbar \\log{(\\hbar)}}, then obtain - \\frac{1}{\\log{(\\hbar)}} + \\frac{1}{\\hbar \\log{(\\hbar)}} = \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} - \\frac{1}{\\log{(\\hbar)}}", "derivation": "\\operatorname{P_{g}}{(\\hbar)} = \\log{(\\log{(\\hbar)})} and \\frac{d}{d \\hbar} \\operatorname{P_{g}}{(\\hbar)} = \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} and \\frac{d}{d \\hbar} \\operatorname{P_{g}}{(\\hbar)} = \\frac{1}{\\hbar \\log{(\\hbar)}} and \\frac{1}{\\hbar \\log{(\\hbar)}} = \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} and - \\frac{1}{\\log{(\\hbar)}} + \\frac{1}{\\hbar \\log{(\\hbar)}} = \\frac{d}{d \\hbar} \\log{(\\log{(\\hbar)})} - \\frac{1}{\\log{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hbar', commutative=True)), log(log(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Derivative(log(log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["minus", 4, "Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1)))), Add(Derivative(log(log(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(log(Symbol('\\\\hbar', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given r{(\\hat{\\mathbf{x}},\\tilde{g})} = \\hat{\\mathbf{x}} + \\log{(\\tilde{g})} and \\mathbf{r}{(\\tilde{g})} = \\log{(\\tilde{g})}, then obtain \\frac{\\partial}{\\partial \\tilde{g}} r{(\\hat{\\mathbf{x}},\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} (\\hat{\\mathbf{x}} + \\mathbf{r}{(\\tilde{g})})", "derivation": "r{(\\hat{\\mathbf{x}},\\tilde{g})} = \\hat{\\mathbf{x}} + \\log{(\\tilde{g})} and \\mathbf{r}{(\\tilde{g})} = \\log{(\\tilde{g})} and \\frac{\\partial}{\\partial \\tilde{g}} r{(\\hat{\\mathbf{x}},\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} (\\hat{\\mathbf{x}} + \\log{(\\tilde{g})}) and \\frac{\\partial}{\\partial \\tilde{g}} r{(\\hat{\\mathbf{x}},\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} (\\hat{\\mathbf{x}} + \\mathbf{r}{(\\tilde{g})})", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('r')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\psi,f)} = \\sin{(\\frac{\\psi}{f})}, then obtain \\frac{h^{2}{(\\psi,f)}}{f} + \\frac{h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} - \\frac{1}{f} = \\frac{2 h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} - \\frac{1}{f}", "derivation": "h{(\\psi,f)} = \\sin{(\\frac{\\psi}{f})} and \\frac{h^{2}{(\\psi,f)}}{f} = \\frac{h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} and \\frac{h^{2}{(\\psi,f)}}{f} + \\frac{h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} = \\frac{2 h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} and \\frac{h^{2}{(\\psi,f)}}{f} + \\frac{h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} - \\frac{1}{f} = \\frac{2 h{(\\psi,f)} \\sin{(\\frac{\\psi}{f})}}{f} - \\frac{1}{f}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))))"], [["times", 1, "Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))))"], [["add", 2, "Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))))"], "Equality(Add(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))))), Mul(Integer(2), Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))))"], [["minus", 3, "Pow(Symbol('f', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)))), Add(Mul(Integer(2), Pow(Symbol('f', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True)), sin(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('f', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given L{(A_{y})} = \\sin{(A_{y})} and \\eta^{\\prime}{(A_{y})} = A_{y}, then obtain \\eta^{\\prime}{(A_{y})} - \\frac{d}{d A_{y}} (A_{y} + \\sin{(A_{y})}) = A_{y} - \\frac{d}{d A_{y}} (A_{y} + \\sin{(A_{y})})", "derivation": "L{(A_{y})} = \\sin{(A_{y})} and \\eta^{\\prime}{(A_{y})} = A_{y} and A_{y} + L{(A_{y})} = A_{y} + \\sin{(A_{y})} and \\frac{d}{d A_{y}} (A_{y} + L{(A_{y})}) = \\frac{d}{d A_{y}} (A_{y} + \\sin{(A_{y})}) and \\eta^{\\prime}{(A_{y})} - \\frac{d}{d A_{y}} (A_{y} + L{(A_{y})}) = A_{y} - \\frac{d}{d A_{y}} (A_{y} + L{(A_{y})}) and \\eta^{\\prime}{(A_{y})} - \\frac{d}{d A_{y}} (A_{y} + \\sin{(A_{y})}) = A_{y} - \\frac{d}{d A_{y}} (A_{y} + \\sin{(A_{y})})", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["add", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), Function('L')(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True))))"], [["differentiate", 3, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Add(Symbol('A_y', commutative=True), Function('L')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('A_y', commutative=True), Function('L')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), Function('L')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), Function('L')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('A_y', commutative=True), sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(k)} = \\cos{(e^{k})} and J{(\\theta_2,\\mathbf{J}_M)} = (e^{\\mathbf{J}_M})^{\\theta_2}, then obtain \\int J{(\\theta_2,\\mathbf{J}_M)} d\\theta_2 - \\frac{\\operatorname{P_{e}}^{k}{(k)}}{k} = \\int (e^{\\mathbf{J}_M})^{\\theta_2} d\\theta_2 - \\frac{\\operatorname{P_{e}}^{k}{(k)}}{k}", "derivation": "\\operatorname{P_{e}}{(k)} = \\cos{(e^{k})} and J{(\\theta_2,\\mathbf{J}_M)} = (e^{\\mathbf{J}_M})^{\\theta_2} and \\int J{(\\theta_2,\\mathbf{J}_M)} d\\theta_2 = \\int (e^{\\mathbf{J}_M})^{\\theta_2} d\\theta_2 and \\int J{(\\theta_2,\\mathbf{J}_M)} d\\theta_2 - \\frac{\\cos^{k}{(e^{k})}}{k} = \\int (e^{\\mathbf{J}_M})^{\\theta_2} d\\theta_2 - \\frac{\\cos^{k}{(e^{k})}}{k} and \\int J{(\\theta_2,\\mathbf{J}_M)} d\\theta_2 - \\frac{\\operatorname{P_{e}}^{k}{(k)}}{k} = \\int (e^{\\mathbf{J}_M})^{\\theta_2} d\\theta_2 - \\frac{\\operatorname{P_{e}}^{k}{(k)}}{k}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('k', commutative=True)), cos(exp(Symbol('k', commutative=True))))"], ["get_premise", "Equality(Function('J')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], "Equality(Add(Integral(Function('J')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))), Add(Integral(Pow(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Function('J')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('P_e')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Add(Integral(Pow(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('P_e')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(\\dot{\\mathbf{r}},\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} (\\dot{\\mathbf{r}} + \\varepsilon), then derive \\mathbf{F}{(\\dot{\\mathbf{r}},\\varepsilon)} = 1, then obtain \\varepsilon \\mathbf{F}^{\\varepsilon}{(\\dot{\\mathbf{r}},\\varepsilon)} = \\varepsilon", "derivation": "\\mathbf{F}{(\\dot{\\mathbf{r}},\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} (\\dot{\\mathbf{r}} + \\varepsilon) and \\mathbf{F}{(\\dot{\\mathbf{r}},\\varepsilon)} = 1 and \\mathbf{F}^{\\varepsilon}{(\\dot{\\mathbf{r}},\\varepsilon)} = 1 and \\varepsilon \\mathbf{F}^{\\varepsilon}{(\\dot{\\mathbf{r}},\\varepsilon)} = \\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(1))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Integer(1))"], [["times", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))"]]}, {"prompt": "Given p{(g,\\mathbf{P},\\tilde{g}^*)} = \\mathbf{P} g + \\tilde{g}^*, then derive \\frac{\\partial}{\\partial g} p{(g,\\mathbf{P},\\tilde{g}^*)} = \\mathbf{P}, then obtain - (m_{s} \\cos{(m_{s})})^{m_{s}} - \\frac{\\partial}{\\partial g} (\\mathbf{P} g + \\tilde{g}^*) = - \\mathbf{P} - (m_{s} \\cos{(m_{s})})^{m_{s}}", "derivation": "p{(g,\\mathbf{P},\\tilde{g}^*)} = \\mathbf{P} g + \\tilde{g}^* and \\frac{\\partial}{\\partial g} p{(g,\\mathbf{P},\\tilde{g}^*)} = \\frac{\\partial}{\\partial g} (\\mathbf{P} g + \\tilde{g}^*) and \\frac{\\partial}{\\partial g} p{(g,\\mathbf{P},\\tilde{g}^*)} = \\mathbf{P} and - \\frac{\\partial}{\\partial g} p{(g,\\mathbf{P},\\tilde{g}^*)} = - \\mathbf{P} and - \\frac{\\partial}{\\partial g} (\\mathbf{P} g + \\tilde{g}^*) = - \\mathbf{P} and - (m_{s} \\cos{(m_{s})})^{m_{s}} - \\frac{\\partial}{\\partial g} (\\mathbf{P} g + \\tilde{g}^*) = - \\mathbf{P} - (m_{s} \\cos{(m_{s})})^{m_{s}}", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('g', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('g', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('g', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('p')(Symbol('g', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 5, "Pow(Mul(Symbol('m_s', commutative=True), cos(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('m_s', commutative=True), cos(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('g', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('m_s', commutative=True), cos(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(t_{2})} = - t_{2}, then derive n_{2} - t_{2} + \\operatorname{E_{n}}{(t_{2})} = M - 2 t_{2}, then obtain n_{2} - 2 t_{2} = M - 2 t_{2}", "derivation": "\\operatorname{E_{n}}{(t_{2})} = - t_{2} and - t_{2} + \\operatorname{E_{n}}{(t_{2})} = - 2 t_{2} and \\frac{d}{d t_{2}} (- t_{2} + \\operatorname{E_{n}}{(t_{2})}) = \\frac{d}{d t_{2}} - 2 t_{2} and \\int \\frac{d}{d t_{2}} (- t_{2} + \\operatorname{E_{n}}{(t_{2})}) dt_{2} = \\int \\frac{d}{d t_{2}} - 2 t_{2} dt_{2} and n_{2} - t_{2} + \\operatorname{E_{n}}{(t_{2})} = M - 2 t_{2} and n_{2} - 2 t_{2} = M - 2 t_{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True)))"], [["minus", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('E_n')(Symbol('t_2', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('E_n')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('E_n')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('t_2', commutative=True))), Integral(Derivative(Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('E_n')(Symbol('t_2', commutative=True))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True))), Add(Symbol('M', commutative=True), Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain (\\theta_{1}{(\\mathbb{I})} + 2 \\cos{(\\mathbb{I})}) \\cos{(\\mathbb{I})} = 3 \\cos^{2}{(\\mathbb{I})}", "derivation": "\\theta_{1}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\theta_{1}{(\\mathbb{I})} + \\cos{(\\mathbb{I})} = 2 \\cos{(\\mathbb{I})} and \\theta_{1}{(\\mathbb{I})} + 2 \\cos{(\\mathbb{I})} = 3 \\cos{(\\mathbb{I})} and (\\theta_{1}{(\\mathbb{I})} + 2 \\cos{(\\mathbb{I})}) \\cos{(\\mathbb{I})} = 3 \\cos^{2}{(\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Integer(3), cos(Symbol('\\\\mathbb{I}', commutative=True))))"], [["times", 3, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Function('\\\\theta_1')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbb{I}', commutative=True)))), cos(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(3), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)} = \\rho_b \\log{(\\mathbf{J}_f)} and \\mathbf{M}{(\\rho_b,\\mathbf{J}_f)} = \\cos{(\\rho_b \\log{(\\mathbf{J}_f)})}, then obtain (\\mathbf{M}^{\\rho_b}{(\\rho_b,\\mathbf{J}_f)})^{\\mathbf{J}_f} + \\cos{(\\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)})} = (\\mathbf{M}^{\\rho_b}{(\\rho_b,\\mathbf{J}_f)})^{\\mathbf{J}_f} + \\mathbf{M}{(\\rho_b,\\mathbf{J}_f)}", "derivation": "\\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)} = \\rho_b \\log{(\\mathbf{J}_f)} and \\cos{(\\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)})} = \\cos{(\\rho_b \\log{(\\mathbf{J}_f)})} and \\mathbf{M}{(\\rho_b,\\mathbf{J}_f)} = \\cos{(\\rho_b \\log{(\\mathbf{J}_f)})} and \\cos{(\\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)})} = \\mathbf{M}{(\\rho_b,\\mathbf{J}_f)} and (\\mathbf{M}^{\\rho_b}{(\\rho_b,\\mathbf{J}_f)})^{\\mathbf{J}_f} + \\cos{(\\mathbf{J}_P{(\\rho_b,\\mathbf{J}_f)})} = (\\mathbf{M}^{\\rho_b}{(\\rho_b,\\mathbf{J}_f)})^{\\mathbf{J}_f} + \\mathbf{M}{(\\rho_b,\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), cos(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Mul(Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 4, "Pow(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Pow(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Function('\\\\mathbf{M}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\omega)} = \\log{(\\omega)}, then obtain \\omega \\log{(\\omega)} = \\frac{\\omega \\log{(\\omega)}^{2}}{\\operatorname{g_{\\varepsilon}}{(\\omega)}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\omega)} = \\log{(\\omega)} and \\omega \\operatorname{g_{\\varepsilon}}{(\\omega)} = \\omega \\log{(\\omega)} and \\operatorname{g_{\\varepsilon}}{(\\omega)} \\log{(\\omega)} = \\log{(\\omega)}^{2} and \\log{(\\omega)} = \\frac{\\log{(\\omega)}^{2}}{\\operatorname{g_{\\varepsilon}}{(\\omega)}} and \\omega \\operatorname{g_{\\varepsilon}}{(\\omega)} = \\frac{\\omega \\log{(\\omega)}^{2}}{\\operatorname{g_{\\varepsilon}}{(\\omega)}} and \\omega \\log{(\\omega)} = \\frac{\\omega \\log{(\\omega)}^{2}}{\\operatorname{g_{\\varepsilon}}{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True))), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2)))"], [["divide", 3, "Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))"], "Equality(log(Symbol('\\\\omega', commutative=True)), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(2))))"]]}, {"prompt": "Given U{(k)} = \\cos{(\\sin{(k)})}, then obtain - \\frac{- k - \\int U{(k)} dk}{k U{(k)}} = - \\frac{- k - \\int \\cos{(\\sin{(k)})} dk}{k U{(k)}}", "derivation": "U{(k)} = \\cos{(\\sin{(k)})} and \\int U{(k)} dk = \\int \\cos{(\\sin{(k)})} dk and k + \\int U{(k)} dk = k + \\int \\cos{(\\sin{(k)})} dk and \\frac{k + \\int U{(k)} dk}{\\cos{(\\sin{(k)})}} = \\frac{k + \\int \\cos{(\\sin{(k)})} dk}{\\cos{(\\sin{(k)})}} and \\frac{k + \\int U{(k)} dk}{U{(k)}} = \\frac{k + \\int \\cos{(\\sin{(k)})} dk}{U{(k)}} and - \\frac{k + \\int U{(k)} dk}{U{(k)}} = - \\frac{k + \\int \\cos{(\\sin{(k)})} dk}{U{(k)}} and \\frac{- k - \\int U{(k)} dk}{U{(k)}} = \\frac{- k - \\int \\cos{(\\sin{(k)})} dk}{U{(k)}} and - \\frac{- k - \\int U{(k)} dk}{k U{(k)}} = - \\frac{- k - \\int \\cos{(\\sin{(k)})} dk}{k U{(k)}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('k', commutative=True)), cos(sin(Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["add", 2, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(Symbol('k', commutative=True), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["divide", 3, "cos(sin(Symbol('k', commutative=True)))"], "Equality(Mul(Add(Symbol('k', commutative=True), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Pow(cos(sin(Symbol('k', commutative=True))), Integer(-1))), Mul(Add(Symbol('k', commutative=True), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Pow(cos(sin(Symbol('k', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Symbol('k', commutative=True), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))), Mul(Add(Symbol('k', commutative=True), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Symbol('k', commutative=True), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('k', commutative=True), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))))"], [["divide", 7, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(Function('U')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Integral(cos(sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))), Pow(Function('U')(Symbol('k', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{p}{(t)} = e^{e^{t}} and a{(t)} = \\frac{\\mathbf{p}{(t)}}{t}, then obtain \\frac{a{(t)} - \\frac{\\mathbf{p}{(t)}}{t}}{\\log{(\\dot{z})}} = \\frac{- \\frac{\\mathbf{p}{(t)}}{t} + \\frac{e^{e^{t}}}{t}}{\\log{(\\dot{z})}}", "derivation": "\\mathbf{p}{(t)} = e^{e^{t}} and a{(t)} = \\frac{\\mathbf{p}{(t)}}{t} and a{(t)} = \\frac{e^{e^{t}}}{t} and a{(t)} - \\frac{\\mathbf{p}{(t)}}{t} = - \\frac{\\mathbf{p}{(t)}}{t} + \\frac{e^{e^{t}}}{t} and \\frac{a{(t)} - \\frac{\\mathbf{p}{(t)}}{t}}{\\log{(\\dot{z})}} = \\frac{- \\frac{\\mathbf{p}{(t)}}{t} + \\frac{e^{e^{t}}}{t}}{\\log{(\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('t', commutative=True)), exp(exp(Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('a')(Symbol('t', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('a')(Symbol('t', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(exp(Symbol('t', commutative=True)))))"], [["minus", 3, "Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True)))"], "Equality(Add(Function('a')(Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(exp(Symbol('t', commutative=True))))))"], [["divide", 4, "log(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Add(Function('a')(Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True)))), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), exp(exp(Symbol('t', commutative=True))))), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_l{(\\varepsilon_0,n_{2})} = - n_{2} + e^{\\varepsilon_0}, then obtain - \\frac{\\partial}{\\partial \\varepsilon_0} \\hat{H}_l{(\\varepsilon_0,n_{2})} = - e^{\\varepsilon_0}", "derivation": "\\hat{H}_l{(\\varepsilon_0,n_{2})} = - n_{2} + e^{\\varepsilon_0} and - \\hat{H}_l{(\\varepsilon_0,n_{2})} = n_{2} - e^{\\varepsilon_0} and \\frac{\\partial}{\\partial \\varepsilon_0} - \\hat{H}_l{(\\varepsilon_0,n_{2})} = \\frac{\\partial}{\\partial \\varepsilon_0} (n_{2} - e^{\\varepsilon_0}) and - \\frac{\\partial}{\\partial \\varepsilon_0} \\hat{H}_l{(\\varepsilon_0,n_{2})} = - e^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\sigma_x)} = \\int \\log{(\\sigma_x)} d\\sigma_x, then derive \\rho_{f}^{\\sigma_x}{(\\sigma_x)} = (\\sigma_x \\log{(\\sigma_x)} - \\sigma_x + n_{1})^{\\sigma_x}, then obtain (\\int \\log{(\\sigma_x)} d\\sigma_x)^{\\sigma_x} = (\\sigma_x \\log{(\\sigma_x)} - \\sigma_x + n_{1})^{\\sigma_x}", "derivation": "\\rho_{f}{(\\sigma_x)} = \\int \\log{(\\sigma_x)} d\\sigma_x and \\rho_{f}^{\\sigma_x}{(\\sigma_x)} = (\\int \\log{(\\sigma_x)} d\\sigma_x)^{\\sigma_x} and \\rho_{f}^{\\sigma_x}{(\\sigma_x)} = (\\sigma_x \\log{(\\sigma_x)} - \\sigma_x + n_{1})^{\\sigma_x} and (\\int \\log{(\\sigma_x)} d\\sigma_x)^{\\sigma_x} = (\\sigma_x \\log{(\\sigma_x)} - \\sigma_x + n_{1})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\sigma_x', commutative=True)), Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('n_1', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Integral(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('n_1', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(n_{1},\\hat{H}_l)} = \\log{(- \\hat{H}_l + n_{1})} and \\omega{(n_{1},\\hat{H}_l)} = \\log{(- \\hat{H}_l + n_{1})}, then obtain \\hat{H}_l + \\mathbf{P}{(n_{1},\\hat{H}_l)} = \\hat{H}_l + \\log{(- \\hat{H}_l + n_{1})}", "derivation": "\\mathbf{P}{(n_{1},\\hat{H}_l)} = \\log{(- \\hat{H}_l + n_{1})} and \\omega{(n_{1},\\hat{H}_l)} = \\log{(- \\hat{H}_l + n_{1})} and \\mathbf{P}{(n_{1},\\hat{H}_l)} = \\omega{(n_{1},\\hat{H}_l)} and \\hat{H}_l + \\mathbf{P}{(n_{1},\\hat{H}_l)} = \\hat{H}_l + \\omega{(n_{1},\\hat{H}_l)} and \\hat{H}_l + \\mathbf{P}{(n_{1},\\hat{H}_l)} = \\hat{H}_l + \\log{(- \\hat{H}_l + n_{1})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{P}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\omega')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbf{P}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\omega')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\mathbf{P}')(Symbol('n_1', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given U{(\\chi,l,\\mathbf{H})} = (- \\mathbf{H} + l)^{\\chi}, then obtain l + \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} U{(\\chi,l,\\mathbf{H})} = l + \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} (- \\mathbf{H} + l)^{\\chi}", "derivation": "U{(\\chi,l,\\mathbf{H})} = (- \\mathbf{H} + l)^{\\chi} and - \\mathbf{H} U{(\\chi,l,\\mathbf{H})} = - \\mathbf{H} (- \\mathbf{H} + l)^{\\chi} and \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} U{(\\chi,l,\\mathbf{H})} = \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} (- \\mathbf{H} + l)^{\\chi} and l + \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} U{(\\chi,l,\\mathbf{H})} = l + \\frac{\\partial}{\\partial \\chi} - \\mathbf{H} (- \\mathbf{H} + l)^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('l', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Function('U')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('l', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Function('U')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('l', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 3, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Function('U')(Symbol('\\\\chi', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('l', commutative=True), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('l', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})}, then obtain \\frac{\\iint \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda d\\lambda}{\\int \\cos{(\\cos{(\\lambda)})} d\\lambda} = \\frac{\\iint \\cos{(\\cos{(\\lambda)})} d\\lambda d\\lambda}{\\int \\cos{(\\cos{(\\lambda)})} d\\lambda}", "derivation": "\\hat{\\mathbf{x}}{(\\lambda)} = \\cos{(\\cos{(\\lambda)})} and \\int \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda = \\int \\cos{(\\cos{(\\lambda)})} d\\lambda and \\iint \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda d\\lambda = \\iint \\cos{(\\cos{(\\lambda)})} d\\lambda d\\lambda and \\frac{\\iint \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda d\\lambda}{\\int \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda} = \\frac{\\iint \\cos{(\\cos{(\\lambda)})} d\\lambda d\\lambda}{\\int \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda} and \\frac{\\iint \\hat{\\mathbf{x}}{(\\lambda)} d\\lambda d\\lambda}{\\int \\cos{(\\cos{(\\lambda)})} d\\lambda} = \\frac{\\iint \\cos{(\\cos{(\\lambda)})} d\\lambda d\\lambda}{\\int \\cos{(\\cos{(\\lambda)})} d\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), cos(cos(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 3, "Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Pow(Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Integral(cos(cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\hat{x}_0 + f^{\\prime})}, then obtain (\\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})})^{2} = 0", "derivation": "\\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} = \\sin{(\\hat{x}_0 + f^{\\prime})} and \\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})} = 0 and \\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})} + \\frac{d}{d f^{\\prime}} 0 = \\frac{d}{d f^{\\prime}} 0 and (\\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})} + \\frac{d}{d f^{\\prime}} 0)^{2} = (\\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})} + \\frac{d}{d f^{\\prime}} 0) \\frac{d}{d f^{\\prime}} 0 and (\\operatorname{v_{t}}{(\\hat{x}_0,f^{\\prime})} - \\sin{(\\hat{x}_0 + f^{\\prime})})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))), Integer(0))"], [["add", 2, "Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["times", 3, "Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], "Equality(Pow(Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Integer(2)), Mul(Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Function('v_t')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))), Integer(2)), Integer(0))"]]}, {"prompt": "Given \\hat{H}{(v_{t},u)} = v_{t}^{u} and \\hat{H}_{\\lambda}{(v_{t},u)} = \\int (- v_{t} + v_{t}^{u} - \\hat{H}{(v_{t},u)}) dv_{t}, then obtain \\hat{H}_{\\lambda}^{v_{t}}{(v_{t},u)} = (\\int - v_{t} dv_{t})^{v_{t}}", "derivation": "\\hat{H}{(v_{t},u)} = v_{t}^{u} and 0 = v_{t}^{u} - \\hat{H}{(v_{t},u)} and - v_{t} = - v_{t} + v_{t}^{u} - \\hat{H}{(v_{t},u)} and \\int - v_{t} dv_{t} = \\int (- v_{t} + v_{t}^{u} - \\hat{H}{(v_{t},u)}) dv_{t} and \\hat{H}_{\\lambda}{(v_{t},u)} = \\int (- v_{t} + v_{t}^{u} - \\hat{H}{(v_{t},u)}) dv_{t} and \\hat{H}_{\\lambda}^{v_{t}}{(v_{t},u)} = (\\int (- v_{t} + v_{t}^{u} - \\hat{H}{(v_{t},u)}) dv_{t})^{v_{t}} and \\hat{H}_{\\lambda}^{v_{t}}{(v_{t},u)} = (\\int - v_{t} dv_{t})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))))"], [["minus", 2, "Symbol('v_t', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))))"], [["integrate", 3, "Symbol('v_t', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('v_t', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('v_t', commutative=True))))"], [["power", 5, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Symbol('v_t', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Pow(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True), Symbol('u', commutative=True)), Symbol('v_t', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(S,V)} = S - V and g{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})}, then obtain S \\sin{(\\hat{\\mathbf{r}})} + \\int \\mathbf{J}_M^{V}{(S,V)} dS = S \\sin{(\\hat{\\mathbf{r}})} + \\int (S - V)^{V} dS", "derivation": "\\mathbf{J}_M{(S,V)} = S - V and \\mathbf{J}_M^{V}{(S,V)} = (S - V)^{V} and g{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and S g{(\\hat{\\mathbf{r}})} = S \\sin{(\\hat{\\mathbf{r}})} and \\int \\mathbf{J}_M^{V}{(S,V)} dS = \\int (S - V)^{V} dS and S g{(\\hat{\\mathbf{r}})} + \\int \\mathbf{J}_M^{V}{(S,V)} dS = S g{(\\hat{\\mathbf{r}})} + \\int (S - V)^{V} dS and S \\sin{(\\hat{\\mathbf{r}})} + \\int \\mathbf{J}_M^{V}{(S,V)} dS = S \\sin{(\\hat{\\mathbf{r}})} + \\int (S - V)^{V} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('V', commutative=True)), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], ["get_premise", "Equality(Function('g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["times", 3, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('S', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["add", 5, "Mul(Symbol('S', commutative=True), Function('g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Add(Mul(Symbol('S', commutative=True), Function('g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), Function('g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Symbol('S', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('S', commutative=True), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Pow(Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} = \\frac{c_{0}}{\\mathbf{v}}, then obtain (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1)^{2} + 1 - \\frac{c_{0}}{\\mathbf{v}} = (-1 + \\frac{c_{0}}{\\mathbf{v}}) (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1) + 1 - \\frac{c_{0}}{\\mathbf{v}}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} = \\frac{c_{0}}{\\mathbf{v}} and \\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1 = -1 + \\frac{c_{0}}{\\mathbf{v}} and (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1)^{2} = (-1 + \\frac{c_{0}}{\\mathbf{v}}) (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1) and (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1)^{2} + 1 - \\frac{c_{0}}{\\mathbf{v}} = (-1 + \\frac{c_{0}}{\\mathbf{v}}) (\\hat{\\mathbf{r}}{(\\mathbf{v},c_{0})} - 1) + 1 - \\frac{c_{0}}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))))"], [["times", 2, "Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1))"], "Equality(Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), Integer(2)), Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1))))"], [["minus", 3, "Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True)))"], "Equality(Add(Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), Integer(2)), Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Add(Mul(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1))), Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given f{(\\mathbf{B},t_{1})} = \\mathbf{B} - t_{1}, then obtain \\int (- \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int f{(\\mathbf{B},t_{1})} d\\mathbf{B}) d\\varphi^* = \\int (- \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int (\\mathbf{B} - t_{1}) d\\mathbf{B}) d\\varphi^*", "derivation": "f{(\\mathbf{B},t_{1})} = \\mathbf{B} - t_{1} and \\int f{(\\mathbf{B},t_{1})} d\\mathbf{B} = \\int (\\mathbf{B} - t_{1}) d\\mathbf{B} and - \\int f{(\\mathbf{B},t_{1})} d\\mathbf{B} = - \\int (\\mathbf{B} - t_{1}) d\\mathbf{B} and - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int f{(\\mathbf{B},t_{1})} d\\mathbf{B} = - \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int (\\mathbf{B} - t_{1}) d\\mathbf{B} and \\int (- \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int f{(\\mathbf{B},t_{1})} d\\mathbf{B}) d\\varphi^* = \\int (- \\frac{\\mathbf{B}^{2}}{2} + \\mathbf{B} t_{1} - \\varphi^* - \\int (\\mathbf{B} - t_{1}) d\\mathbf{B}) d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["minus", 3, "Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integral(Function('f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integral(Function('f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given I{(\\chi)} = \\log{(\\chi)} and \\varphi^{*}{(\\chi)} = \\log{(\\chi)}^{\\chi}, then obtain \\frac{I{(\\chi)} + \\varphi^{*}{(\\chi)}}{\\chi} = \\frac{I{(\\chi)} + I^{\\chi}{(\\chi)}}{\\chi}", "derivation": "I{(\\chi)} = \\log{(\\chi)} and I^{\\chi}{(\\chi)} = \\log{(\\chi)}^{\\chi} and \\varphi^{*}{(\\chi)} = \\log{(\\chi)}^{\\chi} and \\varphi^{*}{(\\chi)} = I^{\\chi}{(\\chi)} and \\varphi^{*}{(\\chi)} + \\log{(\\chi)} = I^{\\chi}{(\\chi)} + \\log{(\\chi)} and \\frac{\\varphi^{*}{(\\chi)} + \\log{(\\chi)}}{\\chi} = \\frac{I^{\\chi}{(\\chi)} + \\log{(\\chi)}}{\\chi} and \\frac{I{(\\chi)} + \\varphi^{*}{(\\chi)}}{\\chi} = \\frac{I{(\\chi)} + I^{\\chi}{(\\chi)}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)), Pow(Function('I')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["add", 4, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Add(Pow(Function('I')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["divide", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Pow(Function('I')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Function('I')(Symbol('\\\\chi', commutative=True)), Function('\\\\varphi^*')(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Function('I')(Symbol('\\\\chi', commutative=True)), Pow(Function('I')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(x^\\prime,p)} = \\sin{(p x^\\prime)}, then obtain (\\operatorname{P_{e}}^{p}{(x^\\prime,p)} (\\int 0 dp)^{p})^{p} = (\\sin^{p}{(p x^\\prime)} (\\int 0 dp)^{p})^{p}", "derivation": "\\operatorname{P_{e}}{(x^\\prime,p)} = \\sin{(p x^\\prime)} and 0 = - \\operatorname{P_{e}}{(x^\\prime,p)} + \\sin{(p x^\\prime)} and \\operatorname{P_{e}}^{p}{(x^\\prime,p)} = \\sin^{p}{(p x^\\prime)} and \\int 0 dp = \\int (- \\operatorname{P_{e}}{(x^\\prime,p)} + \\sin{(p x^\\prime)}) dp and \\operatorname{P_{e}}^{p}{(x^\\prime,p)} (\\int (- \\operatorname{P_{e}}{(x^\\prime,p)} + \\sin{(p x^\\prime)}) dp)^{p} = \\sin^{p}{(p x^\\prime)} (\\int (- \\operatorname{P_{e}}{(x^\\prime,p)} + \\sin{(p x^\\prime)}) dp)^{p} and \\operatorname{P_{e}}^{p}{(x^\\prime,p)} (\\int 0 dp)^{p} = \\sin^{p}{(p x^\\prime)} (\\int 0 dp)^{p} and (\\operatorname{P_{e}}^{p}{(x^\\prime,p)} (\\int 0 dp)^{p})^{p} = (\\sin^{p}{(p x^\\prime)} (\\int 0 dp)^{p})^{p}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True)), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 1, "Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('p', commutative=True)))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('p', commutative=True))), Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('p', commutative=True))))"], [["times", 3, "Pow(Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))), Mul(Pow(sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True))), sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))), Mul(Pow(sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))))"], [["power", 6, "Symbol('p', commutative=True)"], "Equality(Pow(Mul(Pow(Function('P_e')(Symbol('x^\\\\prime', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Mul(Pow(sin(Mul(Symbol('p', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(b)} = \\log{(b)}, then derive I + \\psi + \\mathbf{J}_M{(b)} + \\log{(b)} = 2 \\psi + 2 \\log{(b)}, then obtain \\frac{\\partial}{\\partial \\psi} (I + \\psi + 2 \\mathbf{J}_M{(b)} + \\log{(b)}) = \\frac{\\partial}{\\partial \\psi} (2 \\psi + \\mathbf{J}_M{(b)} + 2 \\log{(b)})", "derivation": "\\mathbf{J}_M{(b)} = \\log{(b)} and \\frac{d}{d b} \\mathbf{J}_M{(b)} = \\frac{d}{d b} \\log{(b)} and \\int \\frac{d}{d b} \\mathbf{J}_M{(b)} db = \\int \\frac{d}{d b} \\log{(b)} db and \\int \\frac{d}{d b} \\mathbf{J}_M{(b)} db + \\int \\frac{d}{d b} \\log{(b)} db = 2 \\int \\frac{d}{d b} \\log{(b)} db and I + \\psi + \\mathbf{J}_M{(b)} + \\log{(b)} = 2 \\psi + 2 \\log{(b)} and I + \\psi + 2 \\mathbf{J}_M{(b)} + \\log{(b)} = 2 \\psi + \\mathbf{J}_M{(b)} + 2 \\log{(b)} and \\frac{\\partial}{\\partial \\psi} (I + \\psi + 2 \\mathbf{J}_M{(b)} + \\log{(b)}) = \\frac{\\partial}{\\partial \\psi} (2 \\psi + \\mathbf{J}_M{(b)} + 2 \\log{(b)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], [["add", 3, "Integral(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Integral(Derivative(Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))), Mul(Integer(2), Integral(Derivative(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), log(Symbol('b', commutative=True)))))"], [["add", 5, "Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True))"], "Equality(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), Mul(Integer(2), log(Symbol('b', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\psi', commutative=True), Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True))), log(Symbol('b', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('b', commutative=True)), Mul(Integer(2), log(Symbol('b', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(g,J)} = \\cos{(\\frac{J}{g})}, then obtain (J + Q{(g,J)}) \\frac{\\partial}{\\partial g} (- Q{(g,J)})^{J} = (J + Q{(g,J)}) \\frac{\\partial}{\\partial g} (- \\cos{(\\frac{J}{g})})^{J}", "derivation": "Q{(g,J)} = \\cos{(\\frac{J}{g})} and J + Q{(g,J)} = J + \\cos{(\\frac{J}{g})} and - Q{(g,J)} = - \\cos{(\\frac{J}{g})} and (- Q{(g,J)})^{J} = (- \\cos{(\\frac{J}{g})})^{J} and \\frac{\\partial}{\\partial g} (- Q{(g,J)})^{J} = \\frac{\\partial}{\\partial g} (- \\cos{(\\frac{J}{g})})^{J} and (J + \\cos{(\\frac{J}{g})}) \\frac{\\partial}{\\partial g} (- Q{(g,J)})^{J} = (J + \\cos{(\\frac{J}{g})}) \\frac{\\partial}{\\partial g} (- \\cos{(\\frac{J}{g})})^{J} and (J + Q{(g,J)}) \\frac{\\partial}{\\partial g} (- Q{(g,J)})^{J} = (J + Q{(g,J)}) \\frac{\\partial}{\\partial g} (- \\cos{(\\frac{J}{g})})^{J}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True)), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Mul(Integer(-1), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('J', commutative=True)))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 5, "Add(Symbol('J', commutative=True), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))))"], "Equality(Mul(Add(Symbol('J', commutative=True), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Derivative(Pow(Mul(Integer(-1), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Add(Symbol('J', commutative=True), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Derivative(Pow(Mul(Integer(-1), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Add(Symbol('J', commutative=True), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Derivative(Pow(Mul(Integer(-1), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Add(Symbol('J', commutative=True), Function('Q')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Derivative(Pow(Mul(Integer(-1), cos(Mul(Symbol('J', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))), Symbol('J', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbb{I}{(W,q)} = W + q, then derive - (\\frac{\\frac{\\partial}{\\partial q} \\mathbb{I}{(W,q)}}{q} - \\frac{\\mathbb{I}{(W,q)}}{q^{2}}) \\sin{(\\frac{\\mathbb{I}{(W,q)}}{q})} = - (\\frac{1}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})}, then obtain - (\\frac{\\frac{\\partial}{\\partial q} (W + q)}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})} = - (\\frac{1}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})}", "derivation": "\\mathbb{I}{(W,q)} = W + q and \\frac{\\mathbb{I}{(W,q)}}{q} = \\frac{W + q}{q} and \\cos{(\\frac{\\mathbb{I}{(W,q)}}{q})} = \\cos{(\\frac{W + q}{q})} and \\frac{\\partial}{\\partial q} \\cos{(\\frac{\\mathbb{I}{(W,q)}}{q})} = \\frac{\\partial}{\\partial q} \\cos{(\\frac{W + q}{q})} and - (\\frac{\\frac{\\partial}{\\partial q} \\mathbb{I}{(W,q)}}{q} - \\frac{\\mathbb{I}{(W,q)}}{q^{2}}) \\sin{(\\frac{\\mathbb{I}{(W,q)}}{q})} = - (\\frac{1}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})} and - (\\frac{\\frac{\\partial}{\\partial q} (W + q)}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})} = - (\\frac{1}{q} - \\frac{W + q}{q^{2}}) \\sin{(\\frac{W + q}{q})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))"], [["divide", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True)))), cos(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(cos(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Add(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-2)), Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True)))), sin(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('W', commutative=True), Symbol('q', commutative=True))))), Mul(Integer(-1), Add(Pow(Symbol('q', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-2)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))), sin(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Add(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Derivative(Add(Symbol('W', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-2)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))), sin(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True))))), Mul(Integer(-1), Add(Pow(Symbol('q', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-2)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True)))), sin(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('q', commutative=True))))))"]]}, {"prompt": "Given G{(z)} = e^{z}, then derive 1 - z = - z + e^{z} - \\frac{d}{d z} G{(z)} + 1, then obtain 1 - z = - z + G{(z)} + e^{z} - 2 \\frac{d}{d z} G{(z)} + 1", "derivation": "G{(z)} = e^{z} and 0 = - G{(z)} + e^{z} and z = z - G{(z)} + e^{z} and \\frac{d}{d z} z = \\frac{d}{d z} (z - G{(z)} + e^{z}) and - z + \\frac{d}{d z} z = - z + \\frac{d}{d z} (z - G{(z)} + e^{z}) and 1 - z = - z + e^{z} - \\frac{d}{d z} G{(z)} + 1 and 1 - z = - z + G{(z)} - \\frac{d}{d z} G{(z)} + 1 and - z + e^{z} - \\frac{d}{d z} G{(z)} + 1 = - z + G{(z)} + e^{z} - 2 \\frac{d}{d z} G{(z)} + 1 and 1 - z = - z + G{(z)} + e^{z} - 2 \\frac{d}{d z} G{(z)} + 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["minus", 1, "Function('G')(Symbol('z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('G')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))))"], [["add", 2, "Symbol('z', commutative=True)"], "Equality(Symbol('z', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Function('G')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Symbol('z', commutative=True), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Function('G')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Symbol('z', commutative=True), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Function('G')(Symbol('z', commutative=True))), exp(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('G')(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('G')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(Function('G')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(1)))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('G')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(Function('G')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given u{(\\mathbb{I},\\mathbf{D})} = \\log{(- \\mathbb{I} + \\mathbf{D})}, then obtain \\int u{(\\mathbb{I},\\mathbf{D})} d\\mathbb{I} + 1 = \\mathbb{I} \\log{(- \\mathbb{I} + \\mathbf{D})} - \\mathbb{I} - \\mathbf{D} \\log{(\\mathbb{I} - \\mathbf{D})} + f_{\\mathbf{v}} + 1", "derivation": "u{(\\mathbb{I},\\mathbf{D})} = \\log{(- \\mathbb{I} + \\mathbf{D})} and \\int u{(\\mathbb{I},\\mathbf{D})} d\\mathbb{I} = \\int \\log{(- \\mathbb{I} + \\mathbf{D})} d\\mathbb{I} and \\int u{(\\mathbb{I},\\mathbf{D})} d\\mathbb{I} + 1 = \\int \\log{(- \\mathbb{I} + \\mathbf{D})} d\\mathbb{I} + 1 and \\int u{(\\mathbb{I},\\mathbf{D})} d\\mathbb{I} + 1 = \\mathbb{I} \\log{(- \\mathbb{I} + \\mathbf{D})} - \\mathbb{I} - \\mathbf{D} \\log{(\\mathbb{I} - \\mathbf{D})} + f_{\\mathbf{v}} + 1", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('u')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1)), Add(Integral(log(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1)))"], [["evaluate_integrals", 3], "Equality(Add(Integral(Function('u')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))), Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(p)} = \\sin{(p)} and C{(p)} = \\sin^{p}{(p)}, then obtain C{(p)} - \\operatorname{v_{y}}{(p)} = - \\operatorname{v_{y}}{(p)} + \\sin^{p}{(p)}", "derivation": "\\operatorname{v_{y}}{(p)} = \\sin{(p)} and \\operatorname{v_{y}}^{p}{(p)} = \\sin^{p}{(p)} and C{(p)} = \\sin^{p}{(p)} and C{(p)} - \\sin{(p)} = - \\sin{(p)} + \\sin^{p}{(p)} and C{(p)} - \\operatorname{v_{y}}{(p)} = - \\operatorname{v_{y}}{(p)} + \\operatorname{v_{y}}^{p}{(p)} and C{(p)} - \\operatorname{v_{y}}{(p)} = - \\operatorname{v_{y}}{(p)} + \\sin^{p}{(p)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["minus", 3, "sin(Symbol('p', commutative=True))"], "Equality(Add(Function('C')(Symbol('p', commutative=True)), Mul(Integer(-1), sin(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('C')(Symbol('p', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('v_y')(Symbol('p', commutative=True))), Pow(Function('v_y')(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('C')(Symbol('p', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('v_y')(Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"]]}, {"prompt": "Given Z{(\\varphi,\\tilde{g})} = \\varphi \\cos{(\\tilde{g})} and \\operatorname{P_{g}}{(\\varphi,\\tilde{g})} = \\varphi Z{(\\varphi,\\tilde{g})}, then obtain \\frac{\\cos{(\\tilde{g})}}{\\varphi} = \\frac{Z{(\\varphi,\\tilde{g})}}{\\varphi^{2}}", "derivation": "Z{(\\varphi,\\tilde{g})} = \\varphi \\cos{(\\tilde{g})} and \\varphi Z{(\\varphi,\\tilde{g})} \\cos{(\\tilde{g})} = \\varphi^{2} \\cos^{2}{(\\tilde{g})} and \\varphi Z{(\\varphi,\\tilde{g})} = \\varphi^{2} \\cos{(\\tilde{g})} and \\operatorname{P_{g}}{(\\varphi,\\tilde{g})} = \\varphi Z{(\\varphi,\\tilde{g})} and \\frac{\\operatorname{P_{g}}{(\\varphi,\\tilde{g})}}{\\varphi^{3}} = \\frac{Z{(\\varphi,\\tilde{g})}}{\\varphi^{2}} and \\operatorname{P_{g}}{(\\varphi,\\tilde{g})} = \\varphi^{2} \\cos{(\\tilde{g})} and \\frac{\\cos{(\\tilde{g})}}{\\varphi} = \\frac{Z{(\\varphi,\\tilde{g})}}{\\varphi^{2}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(2)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(2))))"], [["divide", 2, "cos(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(2)), cos(Symbol('\\\\tilde{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\varphi', commutative=True), Integer(3))"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-3)), Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-2)), Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('P_g')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(2)), cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), cos(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-2)), Function('Z')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\phi_2)} = e^{\\phi_2}, then obtain \\int (\\nabla{(\\phi_2)} \\int \\nabla^{2}{(\\phi_2)} d\\phi_2)^{\\phi_2} d\\phi_2 = \\int (\\nabla{(\\phi_2)} \\int \\nabla{(\\phi_2)} e^{\\phi_2} d\\phi_2)^{\\phi_2} d\\phi_2", "derivation": "\\nabla{(\\phi_2)} = e^{\\phi_2} and \\nabla^{2}{(\\phi_2)} = \\nabla{(\\phi_2)} e^{\\phi_2} and \\int \\nabla^{2}{(\\phi_2)} d\\phi_2 = \\int \\nabla{(\\phi_2)} e^{\\phi_2} d\\phi_2 and \\nabla{(\\phi_2)} \\int \\nabla^{2}{(\\phi_2)} d\\phi_2 = \\nabla{(\\phi_2)} \\int \\nabla{(\\phi_2)} e^{\\phi_2} d\\phi_2 and (\\nabla{(\\phi_2)} \\int \\nabla^{2}{(\\phi_2)} d\\phi_2)^{\\phi_2} = (\\nabla{(\\phi_2)} \\int \\nabla{(\\phi_2)} e^{\\phi_2} d\\phi_2)^{\\phi_2} and \\int (\\nabla{(\\phi_2)} \\int \\nabla^{2}{(\\phi_2)} d\\phi_2)^{\\phi_2} d\\phi_2 = \\int (\\nabla{(\\phi_2)} \\int \\nabla{(\\phi_2)} e^{\\phi_2} d\\phi_2)^{\\phi_2} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["times", 3, "Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Pow(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Pow(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Pow(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Pow(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), Integral(Mul(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(E_{x})} = \\cos{(e^{E_{x}})} and \\operatorname{C_{1}}{(E_{x})} = \\cos{(e^{E_{x}})}, then obtain (- e^{E_{x}} + \\cos{(e^{E_{x}})})^{E_{x}} = (\\operatorname{C_{1}}{(E_{x})} - e^{E_{x}})^{E_{x}}", "derivation": "\\mathbf{P}{(E_{x})} = \\cos{(e^{E_{x}})} and \\mathbf{P}{(E_{x})} - e^{E_{x}} = - e^{E_{x}} + \\cos{(e^{E_{x}})} and \\operatorname{C_{1}}{(E_{x})} = \\cos{(e^{E_{x}})} and \\mathbf{P}{(E_{x})} - e^{E_{x}} = \\operatorname{C_{1}}{(E_{x})} - e^{E_{x}} and - e^{E_{x}} + \\cos{(e^{E_{x}})} = \\operatorname{C_{1}}{(E_{x})} - e^{E_{x}} and (- e^{E_{x}} + \\cos{(e^{E_{x}})})^{E_{x}} = (\\operatorname{C_{1}}{(E_{x})} - e^{E_{x}})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), cos(exp(Symbol('E_x', commutative=True))))"], [["minus", 1, "exp(Symbol('E_x', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('E_x', commutative=True))), cos(exp(Symbol('E_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('E_x', commutative=True)), cos(exp(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Add(Function('C_1')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('E_x', commutative=True))), cos(exp(Symbol('E_x', commutative=True)))), Add(Function('C_1')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))))"], [["power", 5, "Symbol('E_x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), exp(Symbol('E_x', commutative=True))), cos(exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Pow(Add(Function('C_1')(Symbol('E_x', commutative=True)), Mul(Integer(-1), exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given u{(\\phi_2,\\mathbf{s})} = \\mathbf{s} \\phi_2, then obtain - \\mathbf{s} + u{(\\phi_2,\\mathbf{s})} + \\int (- \\mathbf{s} + u{(\\phi_2,\\mathbf{s})}) d\\phi_2 = - \\mathbf{s} + u{(\\phi_2,\\mathbf{s})} + \\int (\\mathbf{s} \\phi_2 - \\mathbf{s}) d\\phi_2", "derivation": "u{(\\phi_2,\\mathbf{s})} = \\mathbf{s} \\phi_2 and - \\mathbf{s} + u{(\\phi_2,\\mathbf{s})} = \\mathbf{s} \\phi_2 - \\mathbf{s} and \\int (- \\mathbf{s} + u{(\\phi_2,\\mathbf{s})}) d\\phi_2 = \\int (\\mathbf{s} \\phi_2 - \\mathbf{s}) d\\phi_2 and - \\mathbf{s} + u{(\\phi_2,\\mathbf{s})} + \\int (- \\mathbf{s} + u{(\\phi_2,\\mathbf{s})}) d\\phi_2 = - \\mathbf{s} + u{(\\phi_2,\\mathbf{s})} + \\int (\\mathbf{s} \\phi_2 - \\mathbf{s}) d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Function('u')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integral(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(n_{2})} = e^{n_{2}}, then derive \\mathbf{P} + \\operatorname{C_{d}}{(n_{2})} = \\varepsilon + e^{n_{2}}, then obtain \\frac{\\mathbf{P} + \\operatorname{C_{d}}{(n_{2})}}{\\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})}} = \\frac{\\varepsilon + e^{n_{2}}}{\\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})}}", "derivation": "\\operatorname{C_{d}}{(n_{2})} = e^{n_{2}} and \\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\int \\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})} dn_{2} = \\int \\frac{d}{d n_{2}} e^{n_{2}} dn_{2} and \\mathbf{P} + \\operatorname{C_{d}}{(n_{2})} = \\varepsilon + e^{n_{2}} and \\frac{\\mathbf{P} + \\operatorname{C_{d}}{(n_{2})}}{\\frac{d}{d n_{2}} e^{n_{2}}} = \\frac{\\varepsilon + e^{n_{2}}}{\\frac{d}{d n_{2}} e^{n_{2}}} and \\frac{\\mathbf{P} + \\operatorname{C_{d}}{(n_{2})}}{\\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})}} = \\frac{\\varepsilon + e^{n_{2}}}{\\frac{d}{d n_{2}} \\operatorname{C_{d}}{(n_{2})}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Derivative(Function('C_d')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))), Integral(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('C_d')(Symbol('n_2', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), exp(Symbol('n_2', commutative=True))))"], [["divide", 4, "Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('C_d')(Symbol('n_2', commutative=True))), Pow(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), exp(Symbol('n_2', commutative=True))), Pow(Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('C_d')(Symbol('n_2', commutative=True))), Pow(Derivative(Function('C_d')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), exp(Symbol('n_2', commutative=True))), Pow(Derivative(Function('C_d')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given G{(V,E_{\\lambda})} = \\sin{(E_{\\lambda}^{V})}, then derive \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} G{(V,E_{\\lambda})}}{\\sin{(E_{\\lambda}^{V})}} - \\frac{E_{\\lambda}^{V} V G{(V,E_{\\lambda})} \\cos{(E_{\\lambda}^{V})}}{E_{\\lambda} \\sin^{2}{(E_{\\lambda}^{V})}} = 0, then obtain \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda}^{V})}}{\\sin{(E_{\\lambda}^{V})}} - \\frac{E_{\\lambda}^{V} V \\cos{(E_{\\lambda}^{V})}}{E_{\\lambda} \\sin{(E_{\\lambda}^{V})}} = 0", "derivation": "G{(V,E_{\\lambda})} = \\sin{(E_{\\lambda}^{V})} and \\frac{G{(V,E_{\\lambda})}}{\\sin{(E_{\\lambda}^{V})}} = 1 and \\frac{\\partial}{\\partial E_{\\lambda}} \\frac{G{(V,E_{\\lambda})}}{\\sin{(E_{\\lambda}^{V})}} = \\frac{d}{d E_{\\lambda}} 1 and \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} G{(V,E_{\\lambda})}}{\\sin{(E_{\\lambda}^{V})}} - \\frac{E_{\\lambda}^{V} V G{(V,E_{\\lambda})} \\cos{(E_{\\lambda}^{V})}}{E_{\\lambda} \\sin^{2}{(E_{\\lambda}^{V})}} = 0 and \\frac{\\frac{\\partial}{\\partial E_{\\lambda}} \\sin{(E_{\\lambda}^{V})}}{\\sin{(E_{\\lambda}^{V})}} - \\frac{E_{\\lambda}^{V} V \\cos{(E_{\\lambda}^{V})}}{E_{\\lambda} \\sin{(E_{\\lambda}^{V})}} = 0", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('V', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))))"], [["divide", 1, "sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)))"], "Equality(Mul(Function('G')(Symbol('V', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Function('G')(Symbol('V', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-1))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Derivative(Function('G')(Symbol('V', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True), Function('G')(Symbol('V', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-2)), cos(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-1)), Derivative(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True), Pow(sin(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))), Integer(-1)), cos(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('V', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\varphi)} = \\sin{(\\log{(\\varphi)})}, then derive \\int \\operatorname{f^{*}}{(\\varphi)} d\\varphi = \\frac{\\varphi \\sin{(\\log{(\\varphi)})}}{2} - \\frac{\\varphi \\cos{(\\log{(\\varphi)})}}{2} + \\varphi^*, then obtain \\iint \\operatorname{f^{*}}{(\\varphi)} d\\varphi d\\varphi = \\int (\\frac{\\varphi \\sin{(\\log{(\\varphi)})}}{2} - \\frac{\\varphi \\cos{(\\log{(\\varphi)})}}{2} + \\varphi^*) d\\varphi", "derivation": "\\operatorname{f^{*}}{(\\varphi)} = \\sin{(\\log{(\\varphi)})} and \\int \\operatorname{f^{*}}{(\\varphi)} d\\varphi = \\int \\sin{(\\log{(\\varphi)})} d\\varphi and \\int \\operatorname{f^{*}}{(\\varphi)} d\\varphi = \\frac{\\varphi \\sin{(\\log{(\\varphi)})}}{2} - \\frac{\\varphi \\cos{(\\log{(\\varphi)})}}{2} + \\varphi^* and \\iint \\operatorname{f^{*}}{(\\varphi)} d\\varphi d\\varphi = \\int (\\frac{\\varphi \\sin{(\\log{(\\varphi)})}}{2} - \\frac{\\varphi \\cos{(\\log{(\\varphi)})}}{2} + \\varphi^*) d\\varphi", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\varphi', commutative=True)), sin(log(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\varphi', commutative=True), sin(log(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\varphi', commutative=True), cos(log(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Rational(1, 2), Symbol('\\\\varphi', commutative=True), sin(log(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\varphi', commutative=True), cos(log(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given Z{(C_{1})} = e^{C_{1}}, then obtain Z{(C_{1})} \\int (-1) dC_{1} = Z{(C_{1})} \\int - \\frac{e^{C_{1}}}{Z{(C_{1})}} dC_{1}", "derivation": "Z{(C_{1})} = e^{C_{1}} and -1 = - \\frac{e^{C_{1}}}{Z{(C_{1})}} and \\int (-1) dC_{1} = \\int - \\frac{e^{C_{1}}}{Z{(C_{1})}} dC_{1} and Z{(C_{1})} \\int (-1) dC_{1} = Z{(C_{1})} \\int - \\frac{e^{C_{1}}}{Z{(C_{1})}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Function('Z')(Symbol('C_1', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('Z')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), Pow(Function('Z')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["divide", 3, "Pow(Function('Z')(Symbol('C_1', commutative=True)), Integer(-1))"], "Equality(Mul(Function('Z')(Symbol('C_1', commutative=True)), Integral(Integer(-1), Tuple(Symbol('C_1', commutative=True)))), Mul(Function('Z')(Symbol('C_1', commutative=True)), Integral(Mul(Integer(-1), Pow(Function('Z')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(k,n,q)} = \\frac{n - q}{k}, then obtain \\int (n - q + ((\\frac{\\nabla{(k,n,q)}}{q})^{q})^{q})^{k} dq = \\int (n - q + ((\\frac{n - q}{k q})^{q})^{q})^{k} dq", "derivation": "\\nabla{(k,n,q)} = \\frac{n - q}{k} and \\frac{\\nabla{(k,n,q)}}{q} = \\frac{n - q}{k q} and (\\frac{\\nabla{(k,n,q)}}{q})^{q} = (\\frac{n - q}{k q})^{q} and ((\\frac{\\nabla{(k,n,q)}}{q})^{q})^{q} = ((\\frac{n - q}{k q})^{q})^{q} and n - q + ((\\frac{\\nabla{(k,n,q)}}{q})^{q})^{q} = n - q + ((\\frac{n - q}{k q})^{q})^{q} and (n - q + ((\\frac{\\nabla{(k,n,q)}}{q})^{q})^{q})^{k} = (n - q + ((\\frac{n - q}{k q})^{q})^{q})^{k} and \\int (n - q + ((\\frac{\\nabla{(k,n,q)}}{q})^{q})^{q})^{k} dq = \\int (n - q + ((\\frac{n - q}{k q})^{q})^{q})^{k} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["divide", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Symbol('q', commutative=True)))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('n', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["power", 5, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Symbol('k', commutative=True)))"], [["integrate", 6, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('k', commutative=True), Symbol('n', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Pow(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given p{(n,\\hat{H},\\Psi)} = \\Psi + \\hat{H} + n, then derive \\int p{(n,\\hat{H},\\Psi)} d\\hat{H} = \\frac{\\hat{H}^{2}}{2} + \\hat{H} (\\Psi + n) + v_{2}, then obtain \\iint p{(n,\\hat{H},\\Psi)} d\\hat{H} dv_{2} = \\iint (\\Psi + \\hat{H} + n) d\\hat{H} dv_{2}", "derivation": "p{(n,\\hat{H},\\Psi)} = \\Psi + \\hat{H} + n and \\int p{(n,\\hat{H},\\Psi)} d\\hat{H} = \\int (\\Psi + \\hat{H} + n) d\\hat{H} and \\int p{(n,\\hat{H},\\Psi)} d\\hat{H} = \\frac{\\hat{H}^{2}}{2} + \\hat{H} (\\Psi + n) + v_{2} and \\iint p{(n,\\hat{H},\\Psi)} d\\hat{H} dv_{2} = \\int (\\frac{\\hat{H}^{2}}{2} + \\hat{H} (\\Psi + n) + v_{2}) dv_{2} and \\iint (\\Psi + \\hat{H} + n) d\\hat{H} dv_{2} = \\int (\\frac{\\hat{H}^{2}}{2} + \\hat{H} (\\Psi + n) + v_{2}) dv_{2} and \\iint p{(n,\\hat{H},\\Psi)} d\\hat{H} dv_{2} = \\iint (\\Psi + \\hat{H} + n) d\\hat{H} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('p')(Symbol('n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True))), Symbol('v_2', commutative=True)))"], [["integrate", 3, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('p')(Symbol('n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Symbol('n', commutative=True))), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('p')(Symbol('n', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{\\hat{p}_0}{t}, then obtain 2 \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{2 \\hat{p}_0}{t}", "derivation": "\\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{\\hat{p}_0}{t} and \\frac{\\hat{p}_0}{t} + \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{2 \\hat{p}_0}{t} and 2 \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{\\hat{p}_0}{t} + \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} and 2 \\hat{\\mathbf{r}}{(t,\\hat{p}_0)} = \\frac{2 \\hat{p}_0}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"]]}, {"prompt": "Given z{(A_{z})} = \\int \\log{(A_{z})} dA_{z}, then derive A_{z} z{(A_{z})} = A_{z} (A_{z} \\log{(A_{z})} - A_{z} + \\rho), then obtain ((A_{z} z{(A_{z})})^{A_{z}})^{A_{z}} = ((A_{z} (A_{z} \\log{(A_{z})} - A_{z} + \\rho))^{A_{z}})^{A_{z}}", "derivation": "z{(A_{z})} = \\int \\log{(A_{z})} dA_{z} and A_{z} z{(A_{z})} = A_{z} \\int \\log{(A_{z})} dA_{z} and A_{z} z{(A_{z})} = A_{z} (A_{z} \\log{(A_{z})} - A_{z} + \\rho) and A_{z} (A_{z} \\log{(A_{z})} - A_{z} + \\rho) = A_{z} \\int \\log{(A_{z})} dA_{z} and (A_{z} z{(A_{z})})^{A_{z}} = (A_{z} \\int \\log{(A_{z})} dA_{z})^{A_{z}} and ((A_{z} z{(A_{z})})^{A_{z}})^{A_{z}} = ((A_{z} \\int \\log{(A_{z})} dA_{z})^{A_{z}})^{A_{z}} and ((A_{z} z{(A_{z})})^{A_{z}})^{A_{z}} = ((A_{z} (A_{z} \\log{(A_{z})} - A_{z} + \\rho))^{A_{z}})^{A_{z}}", "srepr_derivation": [["get_premise", "Equality(Function('z')(Symbol('A_z', commutative=True)), Integral(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('z')(Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), Integral(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('A_z', commutative=True), Function('z')(Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), Add(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('A_z', commutative=True), Add(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\rho', commutative=True))), Mul(Symbol('A_z', commutative=True), Integral(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Symbol('A_z', commutative=True), Function('z')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Integral(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('A_z', commutative=True), Function('z')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(Mul(Symbol('A_z', commutative=True), Integral(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Pow(Mul(Symbol('A_z', commutative=True), Function('z')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(Mul(Symbol('A_z', commutative=True), Add(Mul(Symbol('A_z', commutative=True), log(Symbol('A_z', commutative=True))), Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\rho', commutative=True))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(\\dot{y})} = \\sin{(\\log{(\\dot{y})})} and i{(\\dot{y})} = \\sin{(\\log{(\\dot{y})})} and x{(\\dot{y})} = \\log{(\\dot{y})}, then obtain x{(\\dot{y})} \\sin{(\\log{(\\dot{y})})} = \\mathbf{E}{(\\dot{y})} x{(\\dot{y})}", "derivation": "\\mathbf{E}{(\\dot{y})} = \\sin{(\\log{(\\dot{y})})} and i{(\\dot{y})} = \\sin{(\\log{(\\dot{y})})} and i{(\\dot{y})} = \\mathbf{E}{(\\dot{y})} and x{(\\dot{y})} = \\log{(\\dot{y})} and i{(\\dot{y})} = \\sin{(x{(\\dot{y})})} and \\sin{(x{(\\dot{y})})} = \\mathbf{E}{(\\dot{y})} and x{(\\dot{y})} \\sin{(x{(\\dot{y})})} = \\mathbf{E}{(\\dot{y})} x{(\\dot{y})} and i{(\\dot{y})} x{(\\dot{y})} = \\mathbf{E}{(\\dot{y})} x{(\\dot{y})} and x{(\\dot{y})} \\sin{(\\log{(\\dot{y})})} = \\mathbf{E}{(\\dot{y})} x{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)), sin(log(Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\dot{y}', commutative=True)), sin(log(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('i')(Symbol('\\\\dot{y}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('i')(Symbol('\\\\dot{y}', commutative=True)), sin(Function('x')(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(sin(Function('x')(Symbol('\\\\dot{y}', commutative=True))), Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 6, "Function('x')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Function('x')(Symbol('\\\\dot{y}', commutative=True)), sin(Function('x')(Symbol('\\\\dot{y}', commutative=True)))), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)), Function('x')(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Function('i')(Symbol('\\\\dot{y}', commutative=True)), Function('x')(Symbol('\\\\dot{y}', commutative=True))), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)), Function('x')(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Mul(Function('x')(Symbol('\\\\dot{y}', commutative=True)), sin(log(Symbol('\\\\dot{y}', commutative=True)))), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\dot{y}', commutative=True)), Function('x')(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\sigma_x)} = \\cos{(\\sigma_x)}, then obtain - \\sigma_x + 2 \\rho{(\\sigma_x)} - \\cos{(\\sigma_x)} = - \\sigma_x + \\cos{(\\sigma_x)}", "derivation": "\\rho{(\\sigma_x)} = \\cos{(\\sigma_x)} and - \\sigma_x + \\rho{(\\sigma_x)} = - \\sigma_x + \\cos{(\\sigma_x)} and - \\sigma_x + \\rho{(\\sigma_x)} - \\cos{(\\sigma_x)} = - \\sigma_x and - \\sigma_x + 2 \\rho{(\\sigma_x)} - \\cos{(\\sigma_x)} = - \\sigma_x + \\rho{(\\sigma_x)} and - \\sigma_x + 2 \\rho{(\\sigma_x)} - \\cos{(\\sigma_x)} = - \\sigma_x + \\cos{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\chi,\\hbar,u)} = \\chi u - \\hbar, then obtain \\int \\operatorname{c_{0}}{(\\chi,\\hbar,u)} d\\chi + 1 = \\frac{\\chi^{2} u}{2} - \\chi \\hbar + \\ddot{x} + 1", "derivation": "\\operatorname{c_{0}}{(\\chi,\\hbar,u)} = \\chi u - \\hbar and \\int \\operatorname{c_{0}}{(\\chi,\\hbar,u)} d\\chi = \\int (\\chi u - \\hbar) d\\chi and \\int \\operatorname{c_{0}}{(\\chi,\\hbar,u)} d\\chi + 1 = \\int (\\chi u - \\hbar) d\\chi + 1 and \\int \\operatorname{c_{0}}{(\\chi,\\hbar,u)} d\\chi + 1 = \\frac{\\chi^{2} u}{2} - \\chi \\hbar + \\ddot{x} + 1", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('c_0')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)), Add(Integral(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)))"], [["evaluate_integrals", 3], "Equality(Add(Integral(Function('c_0')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(V,\\psi^*)} = V \\psi^*, then obtain \\hat{H}_{\\lambda}^{2}{(V,\\psi^*)} - \\hat{H}_{\\lambda}{(V,\\psi^*)} = V^{2} (\\psi^*)^{2} - \\hat{H}_{\\lambda}{(V,\\psi^*)}", "derivation": "\\hat{H}_{\\lambda}{(V,\\psi^*)} = V \\psi^* and \\hat{H}_{\\lambda}^{2}{(V,\\psi^*)} = V \\psi^* \\hat{H}_{\\lambda}{(V,\\psi^*)} and \\psi^* \\hat{H}_{\\lambda}{(V,\\psi^*)} = V (\\psi^*)^{2} and \\hat{H}_{\\lambda}^{2}{(V,\\psi^*)} - \\hat{H}_{\\lambda}{(V,\\psi^*)} = V \\psi^* \\hat{H}_{\\lambda}{(V,\\psi^*)} - \\hat{H}_{\\lambda}{(V,\\psi^*)} and \\hat{H}_{\\lambda}^{2}{(V,\\psi^*)} - \\hat{H}_{\\lambda}{(V,\\psi^*)} = V^{2} (\\psi^*)^{2} - \\hat{H}_{\\lambda}{(V,\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["times", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))))"], [["minus", 2, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Pow(Symbol('V', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{v},\\pi)} = \\mathbf{v} \\pi and \\operatorname{v_{z}}{(\\mathbf{v},\\pi)} = \\mathbf{v} \\pi, then obtain \\log{(2 \\operatorname{v_{z}}{(\\mathbf{v},\\pi)})} = \\log{(\\mathbf{v} \\pi + \\operatorname{v_{z}}{(\\mathbf{v},\\pi)})}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{v},\\pi)} = \\mathbf{v} \\pi and 2 \\operatorname{F_{x}}{(\\mathbf{v},\\pi)} = \\mathbf{v} \\pi + \\operatorname{F_{x}}{(\\mathbf{v},\\pi)} and \\log{(2 \\operatorname{F_{x}}{(\\mathbf{v},\\pi)})} = \\log{(\\mathbf{v} \\pi + \\operatorname{F_{x}}{(\\mathbf{v},\\pi)})} and \\operatorname{v_{z}}{(\\mathbf{v},\\pi)} = \\mathbf{v} \\pi and \\operatorname{F_{x}}{(\\mathbf{v},\\pi)} = \\operatorname{v_{z}}{(\\mathbf{v},\\pi)} and \\log{(2 \\operatorname{v_{z}}{(\\mathbf{v},\\pi)})} = \\log{(\\mathbf{v} \\pi + \\operatorname{v_{z}}{(\\mathbf{v},\\pi)})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(2), Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["log", 2], "Equality(log(Mul(Integer(2), Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))), log(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('F_x')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('v_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(log(Mul(Integer(2), Function('v_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))), log(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)), Function('v_z')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(H,\\varepsilon)} = H - \\varepsilon, then obtain - \\dot{z} + (\\hat{\\mathbf{r}}^{H}{(H,\\varepsilon)})^{\\varepsilon} + \\operatorname{r_{0}}{(\\phi_2,\\dot{z})} = - \\dot{z} + ((H - \\varepsilon)^{H})^{\\varepsilon} + \\operatorname{r_{0}}{(\\phi_2,\\dot{z})}", "derivation": "\\hat{\\mathbf{r}}{(H,\\varepsilon)} = H - \\varepsilon and \\hat{\\mathbf{r}}^{H}{(H,\\varepsilon)} = (H - \\varepsilon)^{H} and (\\hat{\\mathbf{r}}^{H}{(H,\\varepsilon)})^{\\varepsilon} = ((H - \\varepsilon)^{H})^{\\varepsilon} and - \\dot{z} + (\\hat{\\mathbf{r}}^{H}{(H,\\varepsilon)})^{\\varepsilon} = - \\dot{z} + ((H - \\varepsilon)^{H})^{\\varepsilon} and - \\dot{z} + (\\hat{\\mathbf{r}}^{H}{(H,\\varepsilon)})^{\\varepsilon} + \\operatorname{r_{0}}{(\\phi_2,\\dot{z})} = - \\dot{z} + ((H - \\varepsilon)^{H})^{\\varepsilon} + \\operatorname{r_{0}}{(\\phi_2,\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"], [["add", 4, "Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('H', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Pow(Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Symbol('H', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Function('r_0')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\sigma_x,f)} = \\log{(- \\sigma_x + f)} and \\operatorname{A_{z}}{(\\sigma_x,f)} = \\cos{(\\log{(- \\sigma_x + f)})}, then obtain \\sigma_x - f + \\cos{(\\log{(- \\sigma_x + f)})} = \\sigma_x - f + \\cos{(\\omega{(\\sigma_x,f)})}", "derivation": "\\omega{(\\sigma_x,f)} = \\log{(- \\sigma_x + f)} and \\cos{(\\omega{(\\sigma_x,f)})} = \\cos{(\\log{(- \\sigma_x + f)})} and \\operatorname{A_{z}}{(\\sigma_x,f)} = \\cos{(\\log{(- \\sigma_x + f)})} and \\sigma_x - f + \\operatorname{A_{z}}{(\\sigma_x,f)} = \\sigma_x - f + \\cos{(\\log{(- \\sigma_x + f)})} and \\sigma_x - f + \\operatorname{A_{z}}{(\\sigma_x,f)} = \\sigma_x - f + \\cos{(\\omega{(\\sigma_x,f)})} and \\sigma_x - f + \\cos{(\\log{(- \\sigma_x + f)})} = \\sigma_x - f + \\cos{(\\omega{(\\sigma_x,f)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))), cos(log(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), cos(log(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), Function('A_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), cos(log(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), Function('A_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), cos(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), cos(log(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True))))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True)), cos(Function('\\\\omega')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(n_{2})} = \\cos{(n_{2})}, then obtain - \\sin{(n_{2})} + \\frac{d}{d n_{2}} \\lambda{(n_{2})} + 1 = 1 - 2 \\sin{(n_{2})}", "derivation": "\\lambda{(n_{2})} = \\cos{(n_{2})} and \\frac{d}{d n_{2}} \\lambda{(n_{2})} = \\frac{d}{d n_{2}} \\cos{(n_{2})} and \\frac{d}{d n_{2}} \\lambda{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})} = 2 \\frac{d}{d n_{2}} \\cos{(n_{2})} and \\frac{d}{d n_{2}} \\lambda{(n_{2})} + \\frac{d}{d n_{2}} \\cos{(n_{2})} + 1 = 2 \\frac{d}{d n_{2}} \\cos{(n_{2})} + 1 and - \\sin{(n_{2})} + \\frac{d}{d n_{2}} \\lambda{(n_{2})} + 1 = 1 - 2 \\sin{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(2), Derivative(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('n_2', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Integer(2), sin(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\pi,v_{1})} = \\log{(- \\pi + v_{1})}, then derive (- \\pi + v_{1}) \\frac{\\partial}{\\partial \\pi} \\operatorname{v_{x}}{(\\pi,v_{1})} - \\operatorname{v_{x}}{(\\pi,v_{1})} = - \\log{(- \\pi + v_{1})} - 1, then obtain (- \\pi + v_{1}) \\frac{\\partial}{\\partial \\pi} \\operatorname{v_{x}}{(\\pi,v_{1})} - \\operatorname{v_{x}}{(\\pi,v_{1})} = - \\operatorname{v_{x}}{(\\pi,v_{1})} - 1", "derivation": "\\operatorname{v_{x}}{(\\pi,v_{1})} = \\log{(- \\pi + v_{1})} and (- \\pi + v_{1}) \\operatorname{v_{x}}{(\\pi,v_{1})} = (- \\pi + v_{1}) \\log{(- \\pi + v_{1})} and \\frac{\\partial}{\\partial \\pi} (- \\pi + v_{1}) \\operatorname{v_{x}}{(\\pi,v_{1})} = \\frac{\\partial}{\\partial \\pi} (- \\pi + v_{1}) \\log{(- \\pi + v_{1})} and (- \\pi + v_{1}) \\frac{\\partial}{\\partial \\pi} \\operatorname{v_{x}}{(\\pi,v_{1})} - \\operatorname{v_{x}}{(\\pi,v_{1})} = - \\log{(- \\pi + v_{1})} - 1 and (- \\pi + v_{1}) \\frac{\\partial}{\\partial \\pi} \\operatorname{v_{x}}{(\\pi,v_{1})} - \\operatorname{v_{x}}{(\\pi,v_{1})} = - \\operatorname{v_{x}}{(\\pi,v_{1})} - 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Derivative(Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('v_1', commutative=True)), Derivative(Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True)))), Add(Mul(Integer(-1), Function('v_x')(Symbol('\\\\pi', commutative=True), Symbol('v_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(E_{n},c)} = \\sin{(E_{n} c)} and \\operatorname{F_{H}}{(E_{n})} = E_{n}, then obtain - E_{n} \\cos{(E_{n} c)} + c + \\operatorname{F_{H}}{(E_{n})} = - E_{n} \\cos{(E_{n} c)} + E_{n} + c", "derivation": "\\operatorname{C_{d}}{(E_{n},c)} = \\sin{(E_{n} c)} and \\operatorname{F_{H}}{(E_{n})} = E_{n} and \\operatorname{F_{H}}{(E_{n})} - \\frac{\\partial}{\\partial c} \\operatorname{C_{d}}{(E_{n},c)} = E_{n} - \\frac{\\partial}{\\partial c} \\operatorname{C_{d}}{(E_{n},c)} and \\operatorname{F_{H}}{(E_{n})} - \\frac{\\partial}{\\partial c} \\sin{(E_{n} c)} = E_{n} - \\frac{\\partial}{\\partial c} \\sin{(E_{n} c)} and c + \\operatorname{F_{H}}{(E_{n})} - \\frac{\\partial}{\\partial c} \\sin{(E_{n} c)} = E_{n} + c - \\frac{\\partial}{\\partial c} \\sin{(E_{n} c)} and - E_{n} \\cos{(E_{n} c)} + c + \\operatorname{F_{H}}{(E_{n})} = - E_{n} \\cos{(E_{n} c)} + E_{n} + c", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('E_n', commutative=True), Symbol('c', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], [["minus", 2, "Derivative(Function('C_d')(Symbol('E_n', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Add(Function('F_H')(Symbol('E_n', commutative=True)), Mul(Integer(-1), Derivative(Function('C_d')(Symbol('E_n', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Derivative(Function('C_d')(Symbol('E_n', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('F_H')(Symbol('E_n', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Derivative(sin(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["add", 4, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('F_H')(Symbol('E_n', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))), Add(Symbol('E_n', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Derivative(sin(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True), cos(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True)))), Symbol('c', commutative=True), Function('F_H')(Symbol('E_n', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True), cos(Mul(Symbol('E_n', commutative=True), Symbol('c', commutative=True)))), Symbol('E_n', commutative=True), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(b,\\varepsilon)} = \\varepsilon + b, then obtain \\varepsilon \\operatorname{F_{g}}^{2}{(b,\\varepsilon)} = \\varepsilon (\\varepsilon + b)^{2}", "derivation": "\\operatorname{F_{g}}{(b,\\varepsilon)} = \\varepsilon + b and \\varepsilon \\operatorname{F_{g}}{(b,\\varepsilon)} = \\varepsilon (\\varepsilon + b) and \\varepsilon \\operatorname{F_{g}}^{2}{(b,\\varepsilon)} = \\varepsilon (\\varepsilon + b) \\operatorname{F_{g}}{(b,\\varepsilon)} and \\varepsilon (\\varepsilon + b) \\operatorname{F_{g}}{(b,\\varepsilon)} = \\varepsilon (\\varepsilon + b)^{2} and \\varepsilon \\operatorname{F_{g}}^{2}{(b,\\varepsilon)} = \\varepsilon (\\varepsilon + b)^{2}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True))))"], [["times", 2, "Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Mul(Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True)), Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True)), Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('F_g')(Symbol('b', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('b', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(T,q)} = T + q, then obtain \\operatorname{P_{g}}^{T}{(T,q)} - 2 \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q} = (T + q)^{T} - 2 \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q}", "derivation": "\\operatorname{P_{g}}{(T,q)} = T + q and \\operatorname{P_{g}}^{T}{(T,q)} = (T + q)^{T} and (\\operatorname{P_{g}}^{T}{(T,q)})^{q} = ((T + q)^{T})^{q} and \\frac{\\partial}{\\partial T} (\\operatorname{P_{g}}^{T}{(T,q)})^{q} = \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q} and \\operatorname{P_{g}}^{T}{(T,q)} - \\frac{\\partial}{\\partial T} (\\operatorname{P_{g}}^{T}{(T,q)})^{q} = (T + q)^{T} - \\frac{\\partial}{\\partial T} (\\operatorname{P_{g}}^{T}{(T,q)})^{q} and \\operatorname{P_{g}}^{T}{(T,q)} - \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q} = (T + q)^{T} - \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q} and \\operatorname{P_{g}}^{T}{(T,q)} - 2 \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q} = (T + q)^{T} - 2 \\frac{\\partial}{\\partial T} ((T + q)^{T})^{q}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Add(Symbol('T', commutative=True), Symbol('q', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"], [["minus", 6, "Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('P_g')(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(Pow(Pow(Add(Symbol('T', commutative=True), Symbol('q', commutative=True)), Symbol('T', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{A})} = \\sin{(\\sin{(\\mathbf{A})})}, then derive \\int \\frac{\\operatorname{v_{z}}{(\\mathbf{A})}}{\\sin{(\\sin{(\\mathbf{A})})}} d\\mathbf{A} = \\ddot{x} + \\mathbf{A}, then derive \\ddot{x} + \\mathbf{A} = C_{d} + \\mathbf{A}, then obtain \\int 1 d\\mathbf{A} = C_{d} + \\mathbf{A}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{A})} = \\sin{(\\sin{(\\mathbf{A})})} and \\frac{\\operatorname{v_{z}}{(\\mathbf{A})}}{\\sin{(\\sin{(\\mathbf{A})})}} = 1 and \\int \\frac{\\operatorname{v_{z}}{(\\mathbf{A})}}{\\sin{(\\sin{(\\mathbf{A})})}} d\\mathbf{A} = \\int 1 d\\mathbf{A} and \\int \\frac{\\operatorname{v_{z}}{(\\mathbf{A})}}{\\sin{(\\sin{(\\mathbf{A})})}} d\\mathbf{A} = \\ddot{x} + \\mathbf{A} and \\ddot{x} + \\mathbf{A} = \\int 1 d\\mathbf{A} and \\ddot{x} + \\mathbf{A} = C_{d} + \\mathbf{A} and \\int 1 d\\mathbf{A} = C_{d} + \\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), sin(sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Mul(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('v_z')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('C_d', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(J,\\tilde{g})} = (e^{\\tilde{g}})^{J}, then obtain (0^{J})^{\\tilde{g}} = ((- \\mathbf{r}{(J,\\tilde{g})} + (e^{\\tilde{g}})^{J})^{J})^{\\tilde{g}}", "derivation": "\\mathbf{r}{(J,\\tilde{g})} = (e^{\\tilde{g}})^{J} and 0 = - \\mathbf{r}{(J,\\tilde{g})} + (e^{\\tilde{g}})^{J} and 0^{J} = (- \\mathbf{r}{(J,\\tilde{g})} + (e^{\\tilde{g}})^{J})^{J} and (0^{J})^{\\tilde{g}} = ((- \\mathbf{r}{(J,\\tilde{g})} + (e^{\\tilde{g}})^{J})^{J})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(exp(Symbol('\\\\tilde{g}', commutative=True)), Symbol('J', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(exp(Symbol('\\\\tilde{g}', commutative=True)), Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Integer(0), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(exp(Symbol('\\\\tilde{g}', commutative=True)), Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["power", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('J', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(exp(Symbol('\\\\tilde{g}', commutative=True)), Symbol('J', commutative=True))), Symbol('J', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\theta)} = \\cos{(\\theta)}, then derive \\int \\operatorname{v_{y}}{(\\theta)} d\\theta = Q + \\sin{(\\theta)}, then derive Q + \\sin{(\\theta)} = \\delta + \\sin{(\\theta)}, then obtain \\delta + \\sin{(\\theta)} = \\int \\cos{(\\theta)} d\\theta", "derivation": "\\operatorname{v_{y}}{(\\theta)} = \\cos{(\\theta)} and \\int \\operatorname{v_{y}}{(\\theta)} d\\theta = \\int \\cos{(\\theta)} d\\theta and \\int \\operatorname{v_{y}}{(\\theta)} d\\theta = Q + \\sin{(\\theta)} and Q + \\sin{(\\theta)} = \\int \\cos{(\\theta)} d\\theta and Q + \\sin{(\\theta)} = \\delta + \\sin{(\\theta)} and \\delta + \\sin{(\\theta)} = \\int \\cos{(\\theta)} d\\theta", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_y')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Add(Symbol('Q', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('Q', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('Q', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), sin(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\tilde{g})} = e^{\\tilde{g}}, then obtain (\\int (\\operatorname{c_{0}}{(\\tilde{g})} - 2 e^{\\tilde{g}}) d\\tilde{g})^{\\tilde{g}} = (\\int - e^{\\tilde{g}} d\\tilde{g})^{\\tilde{g}}", "derivation": "\\operatorname{c_{0}}{(\\tilde{g})} = e^{\\tilde{g}} and \\operatorname{c_{0}}{(\\tilde{g})} - e^{\\tilde{g}} = 0 and \\operatorname{c_{0}}{(\\tilde{g})} - 2 e^{\\tilde{g}} = - e^{\\tilde{g}} and \\int (\\operatorname{c_{0}}{(\\tilde{g})} - 2 e^{\\tilde{g}}) d\\tilde{g} = \\int - e^{\\tilde{g}} d\\tilde{g} and (\\int (\\operatorname{c_{0}}{(\\tilde{g})} - 2 e^{\\tilde{g}}) d\\tilde{g})^{\\tilde{g}} = (\\int - e^{\\tilde{g}} d\\tilde{g})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True)))), Integer(0))"], [["minus", 2, "exp(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Integral(Add(Function('c_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Integral(Mul(Integer(-1), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} = \\mathbf{A} \\cos{(\\Psi_{\\lambda})}, then obtain \\mathbf{A} \\cos{(\\Psi_{\\lambda})} \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} + \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} \\cos{(\\Psi_{\\lambda})} = 2 \\mathbf{A} \\cos^{2}{(\\Psi_{\\lambda})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} = \\mathbf{A} \\cos{(\\Psi_{\\lambda})} and \\mathbf{A} \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} \\cos{(\\Psi_{\\lambda})} = \\mathbf{A}^{2} \\cos^{2}{(\\Psi_{\\lambda})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A} \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} \\cos{(\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{2} \\cos^{2}{(\\Psi_{\\lambda})} and \\mathbf{A} \\cos{(\\Psi_{\\lambda})} \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} + \\operatorname{V_{\\mathbf{B}}}{(\\Psi_{\\lambda},\\mathbf{A})} \\cos{(\\Psi_{\\lambda})} = 2 \\mathbf{A} \\cos^{2}{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given u{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}, then derive u{(\\sigma_p)} = \\frac{1}{\\sigma_p}, then obtain \\frac{d}{d \\sigma_p} 1 = \\frac{d}{d \\sigma_p} \\frac{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}}{u{(\\frac{1}{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}})}}", "derivation": "u{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and u{(\\sigma_p)} = \\frac{1}{\\sigma_p} and \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} = \\frac{1}{\\sigma_p} and u{(\\frac{1}{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}})} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and 1 = \\frac{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}}{u{(\\frac{1}{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}})}} and \\frac{d}{d \\sigma_p} 1 = \\frac{d}{d \\sigma_p} \\frac{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}}{u{(\\frac{1}{\\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}})}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('u')(Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["divide", 4, "Function('u')(Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Function('u')(Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))), Integer(-1)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('u')(Pow(Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Integer(-1))), Integer(-1)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(k,E_{x})} = E_{x} k and \\operatorname{C_{1}}{(k,E_{x})} = k \\operatorname{v_{1}}{(k,E_{x})}, then obtain (E_{x}^{- E_{x}} \\phi{(E_{x})})^{E_{x}} \\operatorname{C_{1}}{(k,E_{x})} \\operatorname{v_{1}}{(k,E_{x})} = E_{x}^{2} k^{3} (E_{x}^{- E_{x}} \\phi{(E_{x})})^{E_{x}}", "derivation": "\\operatorname{v_{1}}{(k,E_{x})} = E_{x} k and k \\operatorname{v_{1}}{(k,E_{x})} = E_{x} k^{2} and k \\operatorname{v_{1}}^{2}{(k,E_{x})} = E_{x} k^{2} \\operatorname{v_{1}}{(k,E_{x})} and E_{x} k^{2} \\operatorname{v_{1}}{(k,E_{x})} = E_{x}^{2} k^{3} and k \\operatorname{v_{1}}^{2}{(k,E_{x})} = E_{x}^{2} k^{3} and \\operatorname{C_{1}}{(k,E_{x})} = k \\operatorname{v_{1}}{(k,E_{x})} and \\operatorname{C_{1}}{(k,E_{x})} \\operatorname{v_{1}}{(k,E_{x})} = E_{x}^{2} k^{3} and (E_{x}^{- E_{x}} \\phi{(E_{x})})^{E_{x}} \\operatorname{C_{1}}{(k,E_{x})} \\operatorname{v_{1}}{(k,E_{x})} = E_{x}^{2} k^{3} (E_{x}^{- E_{x}} \\phi{(E_{x})})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('k', commutative=True)))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))), Mul(Symbol('E_x', commutative=True), Pow(Symbol('k', commutative=True), Integer(2))))"], [["times", 2, "Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Symbol('k', commutative=True), Pow(Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Integer(2))), Mul(Symbol('E_x', commutative=True), Pow(Symbol('k', commutative=True), Integer(2)), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('E_x', commutative=True), Pow(Symbol('k', commutative=True), Integer(2)), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('k', commutative=True), Pow(Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Integer(2))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(3))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('k', commutative=True), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Function('C_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(3))))"], [["times", 7, "Pow(Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Function('\\\\phi')(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Function('\\\\phi')(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Function('C_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True)), Function('v_1')(Symbol('k', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(3)), Pow(Mul(Pow(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('E_x', commutative=True))), Function('\\\\phi')(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(c)} = \\log{(c)}, then obtain \\frac{2 \\mathbf{f}{(c)}}{\\mathbf{f}^{2}{(c)} + 2 \\mathbf{f}{(c)} \\log{(c)}} + \\frac{\\log{(c)}}{\\mathbf{f}^{2}{(c)} + 2 \\mathbf{f}{(c)} \\log{(c)}} = \\frac{1}{\\mathbf{f}{(c)}}", "derivation": "\\mathbf{f}{(c)} = \\log{(c)} and 2 \\mathbf{f}{(c)} + \\log{(c)} = \\mathbf{f}{(c)} + 2 \\log{(c)} and \\frac{2 \\mathbf{f}{(c)} + \\log{(c)}}{\\mathbf{f}{(c)}} = \\frac{\\mathbf{f}{(c)} + 2 \\log{(c)}}{\\mathbf{f}{(c)}} and \\frac{2 \\mathbf{f}{(c)} + \\log{(c)}}{(\\mathbf{f}{(c)} + 2 \\log{(c)}) \\mathbf{f}{(c)}} = \\frac{1}{\\mathbf{f}{(c)}} and \\frac{2 \\mathbf{f}{(c)}}{\\mathbf{f}^{2}{(c)} + 2 \\mathbf{f}{(c)} \\log{(c)}} + \\frac{\\log{(c)}}{\\mathbf{f}^{2}{(c)} + 2 \\mathbf{f}{(c)} \\log{(c)}} = \\frac{1}{\\mathbf{f}{(c)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["add", 1, "Add(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('c', commutative=True))), log(Symbol('c', commutative=True))), Add(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))))"], [["divide", 2, "Function('\\\\mathbf{f}')(Symbol('c', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('c', commutative=True))), log(Symbol('c', commutative=True))), Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))), Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(-1))))"], [["divide", 3, "Add(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Mul(Integer(2), log(Symbol('c', commutative=True)))), Integer(-1)), Add(Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('c', commutative=True))), log(Symbol('c', commutative=True))), Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(-1))), Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(-1)))"], [["expand", 4], "Equality(Add(Mul(Integer(2), Pow(Add(Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(2)), Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('c', commutative=True))), Mul(Pow(Add(Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(2)), Mul(Integer(2), Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))), Integer(-1)), log(Symbol('c', commutative=True)))), Pow(Function('\\\\mathbf{f}')(Symbol('c', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given y{(u)} = \\log{(e^{u})}, then obtain (\\frac{d}{d u} \\frac{y{(u)}}{\\log{(e^{u})}})^{u} = (\\frac{d}{d u} 1)^{u}", "derivation": "y{(u)} = \\log{(e^{u})} and u y{(u)} = u \\log{(e^{u})} and \\frac{y{(u)}}{\\log{(e^{u})}} = 1 and \\frac{d}{d u} \\frac{y{(u)}}{\\log{(e^{u})}} = \\frac{d}{d u} 1 and (\\frac{d}{d u} \\frac{y{(u)}}{\\log{(e^{u})}})^{u} = (\\frac{d}{d u} 1)^{u}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('u', commutative=True)), log(exp(Symbol('u', commutative=True))))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('y')(Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True)))))"], [["divide", 2, "Mul(Symbol('u', commutative=True), log(exp(Symbol('u', commutative=True))))"], "Equality(Mul(Function('y')(Symbol('u', commutative=True)), Pow(log(exp(Symbol('u', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Function('y')(Symbol('u', commutative=True)), Pow(log(exp(Symbol('u', commutative=True))), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('y')(Symbol('u', commutative=True)), Pow(log(exp(Symbol('u', commutative=True))), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(c)} = \\cos{(c)}, then derive \\int \\operatorname{F_{H}}{(c)} dc = \\hat{x} + \\sin{(c)}, then obtain \\hat{x} + \\sin{(c)} + \\int \\cos{(c)} dc = 2 \\int \\cos{(c)} dc", "derivation": "\\operatorname{F_{H}}{(c)} = \\cos{(c)} and \\int \\operatorname{F_{H}}{(c)} dc = \\int \\cos{(c)} dc and \\int \\operatorname{F_{H}}{(c)} dc = \\hat{x} + \\sin{(c)} and \\hat{x} + \\sin{(c)} = \\int \\cos{(c)} dc and \\hat{x} + \\sin{(c)} + \\int \\operatorname{F_{H}}{(c)} dc = \\int \\operatorname{F_{H}}{(c)} dc + \\int \\cos{(c)} dc and \\hat{x} + \\sin{(c)} + \\int \\cos{(c)} dc = 2 \\int \\cos{(c)} dc", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('c', commutative=True))), Integral(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 4, "Integral(Function('F_H')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('c', commutative=True)), Integral(Function('F_H')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Integral(Function('F_H')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), sin(Symbol('c', commutative=True)), Integral(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)} = \\int \\frac{Q}{\\mathbb{I}} dQ, then obtain (\\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)})^{Q} = (\\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\int \\frac{Q}{\\mathbb{I}} dQ)^{Q}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)} = \\int \\frac{Q}{\\mathbb{I}} dQ and \\frac{\\partial}{\\partial Q} \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)} = \\frac{\\partial}{\\partial Q} \\int \\frac{Q}{\\mathbb{I}} dQ and \\frac{\\partial^{2}}{\\partial \\mathbb{I}\\partial Q} \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)} = \\frac{\\partial^{2}}{\\partial \\mathbb{I}\\partial Q} \\int \\frac{Q}{\\mathbb{I}} dQ and \\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)} = \\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\int \\frac{Q}{\\mathbb{I}} dQ and (\\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\operatorname{E_{\\lambda}}{(\\mathbb{I},Q)})^{Q} = (\\frac{\\partial^{3}}{\\partial Q\\partial \\mathbb{I}\\partial Q} \\int \\frac{Q}{\\mathbb{I}} dQ)^{Q}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given v{(\\hat{p})} = \\log{(\\hat{p})} and \\hat{p}_0{(\\hat{p})} = \\int v{(\\hat{p})} d\\hat{p}, then derive \\hat{p}_0{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} - \\hat{p} + \\mathbf{A}, then obtain \\hat{p}_0{(\\hat{p})} - v{(\\hat{p})} + \\log{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} - \\hat{p} + \\mathbf{A} - v{(\\hat{p})} + \\log{(\\hat{p})}", "derivation": "v{(\\hat{p})} = \\log{(\\hat{p})} and \\int v{(\\hat{p})} d\\hat{p} = \\int \\log{(\\hat{p})} d\\hat{p} and \\hat{p}_0{(\\hat{p})} = \\int v{(\\hat{p})} d\\hat{p} and \\hat{p}_0{(\\hat{p})} = \\int \\log{(\\hat{p})} d\\hat{p} and \\hat{p}_0{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} - \\hat{p} + \\mathbf{A} and \\hat{p}_0{(\\hat{p})} - v{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} - \\hat{p} + \\mathbf{A} - v{(\\hat{p})} and \\hat{p}_0{(\\hat{p})} - v{(\\hat{p})} + \\log{(\\hat{p})} = \\hat{p} \\log{(\\hat{p})} - \\hat{p} + \\mathbf{A} - v{(\\hat{p})} + \\log{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{p}', commutative=True)), log(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(log(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\hat{p}', commutative=True)), Integral(Function('v')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\hat{p}', commutative=True)), Integral(log(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\hat{p}', commutative=True)), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 5, "Function('v')(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Function('v')(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\hat{p}', commutative=True)))))"], [["add", 6, "log(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Function('v')(Symbol('\\\\hat{p}', commutative=True))), log(Symbol('\\\\hat{p}', commutative=True))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), log(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('v')(Symbol('\\\\hat{p}', commutative=True))), log(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(f_{E})} = \\sin{(f_{E})}, then derive \\int \\mathbf{H}{(f_{E})} df_{E} = I - \\cos{(f_{E})}, then derive \\hat{H}_l - \\cos{(f_{E})} = I - \\cos{(f_{E})}, then obtain \\int \\sin{(f_{E})} df_{E} = \\hat{H}_l - \\cos{(f_{E})}", "derivation": "\\mathbf{H}{(f_{E})} = \\sin{(f_{E})} and \\int \\mathbf{H}{(f_{E})} df_{E} = \\int \\sin{(f_{E})} df_{E} and \\int \\mathbf{H}{(f_{E})} df_{E} = I - \\cos{(f_{E})} and \\int \\sin{(f_{E})} df_{E} = I - \\cos{(f_{E})} and \\hat{H}_l - \\cos{(f_{E})} = I - \\cos{(f_{E})} and \\int \\sin{(f_{E})} df_{E} = \\hat{H}_l - \\cos{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\hat{X})} = \\hat{X}, then obtain (\\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} - 1)^{\\hat{X}} + \\dot{y}{(\\hat{X})} = \\dot{y}{(\\hat{X})} + 1", "derivation": "\\dot{y}{(\\hat{X})} = \\hat{X} and 1 = \\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} and 0 = \\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} - 1 and 0^{\\hat{X}} = (\\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} - 1)^{\\hat{X}} and 0^{\\hat{X}} + \\dot{y}{(\\hat{X})} = (\\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} - 1)^{\\hat{X}} + \\dot{y}{(\\hat{X})} and (\\frac{\\hat{X}}{\\dot{y}{(\\hat{X})}} - 1)^{\\hat{X}} + \\dot{y}{(\\hat{X})} = \\dot{y}{(\\hat{X})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], [["divide", 1, "Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Integer(-1)))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 4, "Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True))), Add(Pow(Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Add(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True))), Add(Function('\\\\dot{y}')(Symbol('\\\\hat{X}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\theta)} = \\frac{d}{d \\theta} \\log{(\\theta)}, then derive \\int \\mathbf{J}_M{(\\theta)} d\\theta - \\frac{1}{\\theta} = \\int \\frac{d}{d \\theta} \\log{(\\theta)} d\\theta - \\frac{1}{\\theta}, then obtain \\int \\mathbf{J}_M{(\\theta)} d\\theta - \\frac{1}{\\theta} = U + \\log{(\\theta)} - \\frac{1}{\\theta}", "derivation": "\\mathbf{J}_M{(\\theta)} = \\frac{d}{d \\theta} \\log{(\\theta)} and \\int \\mathbf{J}_M{(\\theta)} d\\theta = \\int \\frac{d}{d \\theta} \\log{(\\theta)} d\\theta and - \\frac{d}{d \\theta} \\log{(\\theta)} + \\int \\mathbf{J}_M{(\\theta)} d\\theta = - \\frac{d}{d \\theta} \\log{(\\theta)} + \\int \\frac{d}{d \\theta} \\log{(\\theta)} d\\theta and \\int \\mathbf{J}_M{(\\theta)} d\\theta - \\frac{1}{\\theta} = \\int \\frac{d}{d \\theta} \\log{(\\theta)} d\\theta - \\frac{1}{\\theta} and \\int \\mathbf{J}_M{(\\theta)} d\\theta - \\frac{1}{\\theta} = U + \\log{(\\theta)} - \\frac{1}{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["minus", 2, "Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(Mul(Integer(-1), Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Integral(Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Add(Integral(Derivative(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))))"], [["evaluate_integrals", 4], "Equality(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))), Add(Symbol('U', commutative=True), log(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given y{(y^{\\prime},s)} = y^{\\prime} + e^{s}, then obtain s (y^{\\prime} + e^{s}) y{(y^{\\prime},s)} = s (y^{\\prime} + e^{s})^{2}", "derivation": "y{(y^{\\prime},s)} = y^{\\prime} + e^{s} and s y{(y^{\\prime},s)} = s (y^{\\prime} + e^{s}) and s y^{2}{(y^{\\prime},s)} = s (y^{\\prime} + e^{s}) y{(y^{\\prime},s)} and s (y^{\\prime} + e^{s}) y{(y^{\\prime},s)} = s (y^{\\prime} + e^{s})^{2}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))))"], [["times", 1, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('s', commutative=True), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True)))))"], [["times", 1, "Mul(Symbol('s', commutative=True), Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Symbol('s', commutative=True), Pow(Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Symbol('s', commutative=True), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))), Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('s', commutative=True), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))), Function('y')(Symbol('y^{\\\\prime}', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('s', commutative=True), Pow(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('s', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\omega{(f_{E})} = \\log{(f_{E})}, then obtain \\log{(f_{E})} \\int \\frac{\\omega{(f_{E})}}{f_{E}} df_{E} = \\log{(f_{E})} \\int \\frac{\\log{(f_{E})}}{f_{E}} df_{E}", "derivation": "\\omega{(f_{E})} = \\log{(f_{E})} and \\frac{\\omega{(f_{E})}}{f_{E}} = \\frac{\\log{(f_{E})}}{f_{E}} and \\int \\frac{\\omega{(f_{E})}}{f_{E}} df_{E} = \\int \\frac{\\log{(f_{E})}}{f_{E}} df_{E} and \\log{(f_{E})} \\int \\frac{\\omega{(f_{E})}}{f_{E}} df_{E} = \\log{(f_{E})} \\int \\frac{\\log{(f_{E})}}{f_{E}} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["divide", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["times", 3, "log(Symbol('f_E', commutative=True))"], "Equality(Mul(log(Symbol('f_E', commutative=True)), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True)))), Mul(log(Symbol('f_E', commutative=True)), Integral(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(k,m)} = \\int k m dm and \\mathbf{J}_P{(k,m)} = \\operatorname{F_{x}}^{4}{(k,m)}, then obtain \\operatorname{F_{x}}^{2}{(k,m)} + \\mathbf{J}_P{(k,m)} = \\operatorname{F_{x}}^{2}{(k,m)} (\\int k m dm)^{2} + \\operatorname{F_{x}}^{2}{(k,m)}", "derivation": "\\operatorname{F_{x}}{(k,m)} = \\int k m dm and \\operatorname{F_{x}}^{2}{(k,m)} = \\operatorname{F_{x}}{(k,m)} \\int k m dm and \\operatorname{F_{x}}^{4}{(k,m)} = \\operatorname{F_{x}}^{2}{(k,m)} (\\int k m dm)^{2} and \\mathbf{J}_P{(k,m)} = \\operatorname{F_{x}}^{4}{(k,m)} and \\operatorname{F_{x}}^{2}{(k,m)} + \\mathbf{J}_P{(k,m)} = \\operatorname{F_{x}}^{4}{(k,m)} + \\operatorname{F_{x}}^{2}{(k,m)} and \\operatorname{F_{x}}^{2}{(k,m)} + \\mathbf{J}_P{(k,m)} = \\operatorname{F_{x}}^{2}{(k,m)} (\\int k m dm)^{2} + \\operatorname{F_{x}}^{2}{(k,m)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integral(Mul(Symbol('k', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["times", 1, "Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2)), Mul(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integral(Mul(Symbol('k', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(4)), Mul(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2)), Pow(Integral(Mul(Symbol('k', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(4)))"], [["add", 4, "Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2)), Function('\\\\mathbf{J}_P')(Symbol('k', commutative=True), Symbol('m', commutative=True))), Add(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(4)), Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2)), Function('\\\\mathbf{J}_P')(Symbol('k', commutative=True), Symbol('m', commutative=True))), Add(Mul(Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2)), Pow(Integral(Mul(Symbol('k', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2))), Pow(Function('F_x')(Symbol('k', commutative=True), Symbol('m', commutative=True)), Integer(2))))"]]}, {"prompt": "Given U{(x^\\prime)} = \\cos{(x^\\prime)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(x^\\prime)} = U^{2}{(x^\\prime)}, then obtain \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(x^\\prime)}}{(x^\\prime)^{2}} = \\frac{U{(x^\\prime)} \\cos{(x^\\prime)}}{(x^\\prime)^{2}}", "derivation": "U{(x^\\prime)} = \\cos{(x^\\prime)} and \\frac{U{(x^\\prime)}}{x^\\prime} = \\frac{\\cos{(x^\\prime)}}{x^\\prime} and \\frac{U^{2}{(x^\\prime)}}{(x^\\prime)^{2}} = \\frac{U{(x^\\prime)} \\cos{(x^\\prime)}}{(x^\\prime)^{2}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(x^\\prime)} = U^{2}{(x^\\prime)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(x^\\prime)}}{(x^\\prime)^{2}} = \\frac{U{(x^\\prime)} \\cos{(x^\\prime)}}{(x^\\prime)^{2}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('U')(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), cos(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('U')(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2)), Pow(Function('U')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2)), Function('U')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('x^\\\\prime', commutative=True)), Pow(Function('U')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2)), Function('U')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\varphi{(n_{2},r)} = n_{2} + r, then obtain ((\\frac{\\partial}{\\partial r} 2 \\varphi{(n_{2},r)})^{n_{2}})^{n_{2}} = ((\\frac{\\partial}{\\partial r} (n_{2} + r + \\varphi{(n_{2},r)}))^{n_{2}})^{n_{2}}", "derivation": "\\varphi{(n_{2},r)} = n_{2} + r and 2 \\varphi{(n_{2},r)} = n_{2} + r + \\varphi{(n_{2},r)} and \\frac{\\partial}{\\partial r} 2 \\varphi{(n_{2},r)} = \\frac{\\partial}{\\partial r} (n_{2} + r + \\varphi{(n_{2},r)}) and (\\frac{\\partial}{\\partial r} 2 \\varphi{(n_{2},r)})^{n_{2}} = (\\frac{\\partial}{\\partial r} (n_{2} + r + \\varphi{(n_{2},r)}))^{n_{2}} and ((\\frac{\\partial}{\\partial r} 2 \\varphi{(n_{2},r)})^{n_{2}})^{n_{2}} = ((\\frac{\\partial}{\\partial r} (n_{2} + r + \\varphi{(n_{2},r)}))^{n_{2}})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True)), Add(Symbol('n_2', commutative=True), Symbol('r', commutative=True)))"], [["add", 1, "Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Add(Symbol('n_2', commutative=True), Symbol('r', commutative=True), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('n_2', commutative=True), Symbol('r', commutative=True), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(2), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Add(Symbol('n_2', commutative=True), Symbol('r', commutative=True), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Pow(Derivative(Mul(Integer(2), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Pow(Derivative(Add(Symbol('n_2', commutative=True), Symbol('r', commutative=True), Function('\\\\varphi')(Symbol('n_2', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(J)} = \\cos{(J)}, then obtain - J (n_{1} + \\sin{(J)}) + \\frac{d}{d J} \\int \\phi_{2}{(J)} dJ = - J (n_{1} + \\sin{(J)}) + \\frac{d}{d J} \\int \\cos{(J)} dJ", "derivation": "\\phi_{2}{(J)} = \\cos{(J)} and \\int \\phi_{2}{(J)} dJ = \\int \\cos{(J)} dJ and - \\phi_{2}{(J)} + \\cos{(J)} + \\int \\phi_{2}{(J)} dJ = - \\phi_{2}{(J)} + \\cos{(J)} + \\int \\cos{(J)} dJ and \\frac{d}{d J} (- \\phi_{2}{(J)} + \\cos{(J)} + \\int \\phi_{2}{(J)} dJ) = \\frac{d}{d J} (- \\phi_{2}{(J)} + \\cos{(J)} + \\int \\cos{(J)} dJ) and \\frac{d}{d J} \\int \\phi_{2}{(J)} dJ = \\frac{d}{d J} \\int \\cos{(J)} dJ and - J (n_{1} + \\sin{(J)}) + \\frac{d}{d J} \\int \\phi_{2}{(J)} dJ = - J (n_{1} + \\sin{(J)}) + \\frac{d}{d J} \\int \\cos{(J)} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["minus", 2, "Add(Function('\\\\phi_2')(Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)), Integral(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)), Integral(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('J', commutative=True))), cos(Symbol('J', commutative=True)), Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Symbol('J', commutative=True), Add(Symbol('n_1', commutative=True), sin(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Add(Symbol('n_1', commutative=True), sin(Symbol('J', commutative=True)))), Derivative(Integral(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('J', commutative=True), Add(Symbol('n_1', commutative=True), sin(Symbol('J', commutative=True)))), Derivative(Integral(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given p{(l)} = \\cos{(l)}, then obtain \\int p^{9}{(l)} dl = \\int p^{3}{(l)} \\cos^{6}{(l)} dl", "derivation": "p{(l)} = \\cos{(l)} and p^{2}{(l)} = p{(l)} \\cos{(l)} and p^{2}{(l)} \\cos{(l)} = p{(l)} \\cos^{2}{(l)} and p^{3}{(l)} = p^{2}{(l)} \\cos{(l)} and p^{3}{(l)} = p{(l)} \\cos^{2}{(l)} and p^{9}{(l)} = p^{3}{(l)} \\cos^{6}{(l)} and \\int p^{9}{(l)} dl = \\int p^{3}{(l)} \\cos^{6}{(l)} dl", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["times", 1, "Function('p')(Symbol('l', commutative=True))"], "Equality(Pow(Function('p')(Symbol('l', commutative=True)), Integer(2)), Mul(Function('p')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))))"], [["times", 2, "cos(Symbol('l', commutative=True))"], "Equality(Mul(Pow(Function('p')(Symbol('l', commutative=True)), Integer(2)), cos(Symbol('l', commutative=True))), Mul(Function('p')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('p')(Symbol('l', commutative=True)), Integer(3)), Mul(Pow(Function('p')(Symbol('l', commutative=True)), Integer(2)), cos(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('p')(Symbol('l', commutative=True)), Integer(3)), Mul(Function('p')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(2))))"], [["power", 5, 3], "Equality(Pow(Function('p')(Symbol('l', commutative=True)), Integer(9)), Mul(Pow(Function('p')(Symbol('l', commutative=True)), Integer(3)), Pow(cos(Symbol('l', commutative=True)), Integer(6))))"], [["integrate", 6, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Function('p')(Symbol('l', commutative=True)), Integer(9)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Pow(Function('p')(Symbol('l', commutative=True)), Integer(3)), Pow(cos(Symbol('l', commutative=True)), Integer(6))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{J},\\chi)} = - \\chi + e^{\\mathbf{J}}, then derive \\frac{\\partial}{\\partial \\mathbf{J}} \\hat{H}{(\\mathbf{J},\\chi)} = e^{\\mathbf{J}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + e^{\\mathbf{J}})) = e^{\\mathbf{J}}", "derivation": "\\hat{H}{(\\mathbf{J},\\chi)} = - \\chi + e^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mathbf{J}} \\hat{H}{(\\mathbf{J},\\chi)} = \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + e^{\\mathbf{J}}) and \\frac{\\partial}{\\partial \\mathbf{J}} \\hat{H}{(\\mathbf{J},\\chi)} = e^{\\mathbf{J}} and \\hat{H}{(\\mathbf{J},\\chi)} = - \\chi + \\frac{\\partial}{\\partial \\mathbf{J}} \\hat{H}{(\\mathbf{J},\\chi)} and - \\chi + \\frac{\\partial}{\\partial \\mathbf{J}} \\hat{H}{(\\mathbf{J},\\chi)} = - \\chi + e^{\\mathbf{J}} and - \\chi + \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + e^{\\mathbf{J}}) = - \\chi + e^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + e^{\\mathbf{J}}) = e^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + \\frac{\\partial}{\\partial \\mathbf{J}} (- \\chi + e^{\\mathbf{J}})) = e^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\pi)} = \\cos{(\\pi)}, then obtain 0 = - \\dot{\\mathbf{r}} + \\hat{\\mathbf{r}} - \\hat{p}{(\\pi)} + \\cos{(\\pi)}", "derivation": "\\hat{p}{(\\pi)} = \\cos{(\\pi)} and \\frac{d}{d \\pi} \\hat{p}{(\\pi)} = \\frac{d}{d \\pi} \\cos{(\\pi)} and \\int \\frac{d}{d \\pi} \\hat{p}{(\\pi)} d\\pi = \\int \\frac{d}{d \\pi} \\cos{(\\pi)} d\\pi and 0 = - \\int \\frac{d}{d \\pi} \\hat{p}{(\\pi)} d\\pi + \\int \\frac{d}{d \\pi} \\cos{(\\pi)} d\\pi and 0 = - \\dot{\\mathbf{r}} + \\hat{\\mathbf{r}} - \\hat{p}{(\\pi)} + \\cos{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["minus", 3, "Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Derivative(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True)))), Integral(Derivative(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\Psi)} = e^{\\Psi}, then obtain \\int (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) \\varepsilon{(\\Psi)} d\\Psi = \\int (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) e^{\\Psi} d\\Psi", "derivation": "\\varepsilon{(\\Psi)} = e^{\\Psi} and \\varepsilon^{\\Psi}{(\\Psi)} = (e^{\\Psi})^{\\Psi} and (\\varepsilon{(\\Psi)} + (e^{\\Psi})^{\\Psi}) \\varepsilon{(\\Psi)} = (\\varepsilon{(\\Psi)} + (e^{\\Psi})^{\\Psi}) e^{\\Psi} and (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) \\varepsilon{(\\Psi)} = (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) e^{\\Psi} and \\int (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) \\varepsilon{(\\Psi)} d\\Psi = \\int (\\varepsilon{(\\Psi)} + \\varepsilon^{\\Psi}{(\\Psi)}) e^{\\Psi} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["times", 1, "Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True))), Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(exp(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True))), Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given E{(L)} = \\log{(L)} and \\varepsilon_{0}{(L)} = E^{2}{(L)}, then obtain - E{(L)} + \\log{(L)}^{2} = E{(L)} \\log{(L)} - E{(L)}", "derivation": "E{(L)} = \\log{(L)} and E^{2}{(L)} = E{(L)} \\log{(L)} and \\varepsilon_{0}{(L)} = E^{2}{(L)} and \\varepsilon_{0}{(L)} = E{(L)} \\log{(L)} and - E{(L)} + \\varepsilon_{0}{(L)} = E{(L)} \\log{(L)} - E{(L)} and \\varepsilon_{0}{(L)} = \\log{(L)}^{2} and - E{(L)} + \\log{(L)}^{2} = E{(L)} \\log{(L)} - E{(L)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["times", 1, "Function('E')(Symbol('L', commutative=True))"], "Equality(Pow(Function('E')(Symbol('L', commutative=True)), Integer(2)), Mul(Function('E')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('L', commutative=True)), Pow(Function('E')(Symbol('L', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varepsilon_0')(Symbol('L', commutative=True)), Mul(Function('E')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))"], [["minus", 4, "Function('E')(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('L', commutative=True))), Function('\\\\varepsilon_0')(Symbol('L', commutative=True))), Add(Mul(Function('E')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Function('E')(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\varepsilon_0')(Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(2))), Add(Mul(Function('E')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(-1), Function('E')(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\dot{z})} = \\log{(e^{\\dot{z}})} and f{(\\dot{z})} = \\dot{z}, then derive \\frac{d}{d \\dot{z}} f{(\\dot{z})} = 1, then obtain \\frac{\\dot{z} \\frac{d}{d \\dot{z}} f{(\\dot{z})}}{\\log{(e^{\\dot{z}})}} = \\frac{\\dot{z}}{\\log{(e^{\\dot{z}})}}", "derivation": "\\operatorname{f_{E}}{(\\dot{z})} = \\log{(e^{\\dot{z}})} and f{(\\dot{z})} = \\dot{z} and \\frac{d}{d \\dot{z}} f{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\dot{z} and \\frac{d}{d \\dot{z}} f{(\\dot{z})} = 1 and \\frac{\\dot{z} \\frac{d}{d \\dot{z}} f{(\\dot{z})}}{\\operatorname{f_{E}}{(\\dot{z})}} = \\frac{\\dot{z}}{\\operatorname{f_{E}}{(\\dot{z})}} and \\frac{\\dot{z} \\frac{d}{d \\dot{z}} f{(\\dot{z})}}{\\log{(e^{\\dot{z}})}} = \\frac{\\dot{z}}{\\log{(e^{\\dot{z}})}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\dot{z}', commutative=True)), log(exp(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Symbol('\\\\dot{z}', commutative=True), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Function('f_E')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Function('f_E')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(log(exp(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Derivative(Function('f')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(log(exp(Symbol('\\\\dot{z}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\Omega)} = \\cos{(\\Omega)} and A{(\\Omega)} = \\cos{(\\Omega)}, then obtain (\\Omega + \\operatorname{M_{E}}{(\\Omega)} - u{(\\Omega)})^{\\Omega} = (\\Omega + A{(\\Omega)} - u{(\\Omega)})^{\\Omega}", "derivation": "\\operatorname{M_{E}}{(\\Omega)} = \\cos{(\\Omega)} and \\Omega + \\operatorname{M_{E}}{(\\Omega)} = \\Omega + \\cos{(\\Omega)} and A{(\\Omega)} = \\cos{(\\Omega)} and \\Omega + \\operatorname{M_{E}}{(\\Omega)} - u{(\\Omega)} = \\Omega - u{(\\Omega)} + \\cos{(\\Omega)} and \\Omega + \\operatorname{M_{E}}{(\\Omega)} - u{(\\Omega)} = \\Omega + A{(\\Omega)} - u{(\\Omega)} and (\\Omega + \\operatorname{M_{E}}{(\\Omega)} - u{(\\Omega)})^{\\Omega} = (\\Omega + A{(\\Omega)} - u{(\\Omega)})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('M_E')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["minus", 2, "Function('u')(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('M_E')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('M_E')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Function('A')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True)))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), Function('M_E')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), Function('A')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(F_{g})} = e^{e^{F_{g}}}, then obtain 2 \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}} - e^{e^{F_{g}}} = \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}}", "derivation": "\\operatorname{n_{1}}{(F_{g})} = e^{e^{F_{g}}} and \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}} = - e^{F_{g}} + e^{e^{F_{g}}} and \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}} - e^{e^{F_{g}}} = - e^{F_{g}} and 2 \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}} - e^{e^{F_{g}}} = \\operatorname{n_{1}}{(F_{g})} - e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('F_g', commutative=True)), exp(exp(Symbol('F_g', commutative=True))))"], [["minus", 1, "exp(Symbol('F_g', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('F_g', commutative=True))), exp(exp(Symbol('F_g', commutative=True)))))"], [["minus", 2, "exp(exp(Symbol('F_g', commutative=True)))"], "Equality(Add(Function('n_1')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('F_g', commutative=True))))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Function('n_1')(Symbol('F_g', commutative=True))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('F_g', commutative=True))))), Add(Function('n_1')(Symbol('F_g', commutative=True)), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(A_{y},E_{x})} = E_{x}^{A_{y}} and B{(A_{y})} = A_{y}, then obtain \\frac{\\operatorname{E_{\\lambda}}{(A_{y},E_{x})}}{A_{y} + E_{x}^{A_{y}}} = \\frac{E_{x}^{A_{y}}}{A_{y} + E_{x}^{A_{y}}}", "derivation": "\\operatorname{E_{\\lambda}}{(A_{y},E_{x})} = E_{x}^{A_{y}} and B{(A_{y})} = A_{y} and B{(A_{y})} + \\operatorname{E_{\\lambda}}{(A_{y},E_{x})} = E_{x}^{A_{y}} + B{(A_{y})} and A_{y} + \\operatorname{E_{\\lambda}}{(A_{y},E_{x})} = A_{y} + E_{x}^{A_{y}} and \\frac{\\operatorname{E_{\\lambda}}{(A_{y},E_{x})}}{A_{y} + \\operatorname{E_{\\lambda}}{(A_{y},E_{x})}} = \\frac{E_{x}^{A_{y}}}{A_{y} + \\operatorname{E_{\\lambda}}{(A_{y},E_{x})}} and \\frac{\\operatorname{E_{\\lambda}}{(A_{y},E_{x})}}{A_{y} + E_{x}^{A_{y}}} = \\frac{E_{x}^{A_{y}}}{A_{y} + E_{x}^{A_{y}}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["add", 1, "Function('B')(Symbol('A_y', commutative=True))"], "Equality(Add(Function('B')(Symbol('A_y', commutative=True)), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Add(Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True)), Function('B')(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('A_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Add(Symbol('A_y', commutative=True), Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True))))"], [["divide", 1, "Add(Symbol('A_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True)), Pow(Add(Symbol('A_y', commutative=True), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('A_y', commutative=True), Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True))), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('A_y', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True)), Pow(Add(Symbol('A_y', commutative=True), Pow(Symbol('E_x', commutative=True), Symbol('A_y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} = H - \\pi, then derive \\int \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} d\\pi = H \\pi + W - \\frac{\\pi^{2}}{2}, then obtain W - \\frac{\\pi^{2}}{2} + \\pi (\\pi + \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)}) = \\int \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} d\\pi", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} = H - \\pi and \\pi + \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} = H and \\int \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} d\\pi = \\int (H - \\pi) d\\pi and \\int \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} d\\pi = H \\pi + W - \\frac{\\pi^{2}}{2} and H \\pi + W - \\frac{\\pi^{2}}{2} = \\int (H - \\pi) d\\pi and W - \\frac{\\pi^{2}}{2} + \\pi (\\pi + \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)}) = \\int \\operatorname{V_{\\mathbf{B}}}{(\\pi,H)} d\\pi", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2)))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('W', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Mul(Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\pi', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given g{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} and \\phi_{1}{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} - 1 and \\operatorname{A_{x}}{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} - 1, then obtain (g{(U,\\tilde{g}^*)} - 1)^{U} = \\operatorname{A_{x}}^{U}{(U,\\tilde{g}^*)}", "derivation": "g{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} and g{(U,\\tilde{g}^*)} - 1 = \\log{(U - \\tilde{g}^*)} - 1 and \\phi_{1}{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} - 1 and g{(U,\\tilde{g}^*)} - 1 = \\phi_{1}{(U,\\tilde{g}^*)} and (g{(U,\\tilde{g}^*)} - 1)^{U} = \\phi_{1}^{U}{(U,\\tilde{g}^*)} and (g{(U,\\tilde{g}^*)} - 1)^{U} = (\\log{(U - \\tilde{g}^*)} - 1)^{U} and \\operatorname{A_{x}}{(U,\\tilde{g}^*)} = \\log{(U - \\tilde{g}^*)} - 1 and (g{(U,\\tilde{g}^*)} - 1)^{U} = \\operatorname{A_{x}}^{U}{(U,\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), log(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Add(log(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(log(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Function('\\\\phi_1')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Add(log(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)), Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(log(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(Add(Function('g')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Symbol('U', commutative=True)), Pow(Function('A_x')(Symbol('U', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(E)} = \\sin{(E)}, then obtain (E + \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE)^{2} = (E + \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE) (E + \\int (E + \\sin{(E)}) \\ddot{x}^{- E}{(E)} dE)", "derivation": "\\ddot{x}{(E)} = \\sin{(E)} and E + \\ddot{x}{(E)} = E + \\sin{(E)} and (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} = (E + \\sin{(E)}) \\ddot{x}^{- E}{(E)} and \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE = \\int (E + \\sin{(E)}) \\ddot{x}^{- E}{(E)} dE and E + \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE = E + \\int (E + \\sin{(E)}) \\ddot{x}^{- E}{(E)} dE and (E + \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE)^{2} = (E + \\int (E + \\ddot{x}{(E)}) \\ddot{x}^{- E}{(E)} dE) (E + \\int (E + \\sin{(E)}) \\ddot{x}^{- E}{(E)} dE)", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Symbol('E', commutative=True))"], "Equality(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Mul(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["add", 4, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))))"], [["times", 5, "Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], "Equality(Pow(Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Integer(2)), Mul(Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), Function('\\\\ddot{x}')(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Integral(Mul(Add(Symbol('E', commutative=True), sin(Symbol('E', commutative=True))), Pow(Function('\\\\ddot{x}')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(F_{g},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*}, then derive F_{g} + \\varepsilon{(F_{g},\\tilde{g}^*)} = F_{g} - \\frac{F_{g}}{(\\tilde{g}^*)^{2}}, then obtain F_{g} - \\frac{F_{g}}{(\\tilde{g}^*)^{2}} = F_{g} + \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*}", "derivation": "\\varepsilon{(F_{g},\\tilde{g}^*)} = \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*} and F_{g} + \\varepsilon{(F_{g},\\tilde{g}^*)} = F_{g} + \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*} and F_{g} + \\varepsilon{(F_{g},\\tilde{g}^*)} = F_{g} - \\frac{F_{g}}{(\\tilde{g}^*)^{2}} and F_{g} - \\frac{F_{g}}{(\\tilde{g}^*)^{2}} = F_{g} + \\frac{\\partial}{\\partial \\tilde{g}^*} \\frac{F_{g}}{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["add", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Symbol('F_g', commutative=True), Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('F_g', commutative=True), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('F_g', commutative=True), Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-2)))), Add(Symbol('F_g', commutative=True), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(P_{g})} = \\cos{(\\cos{(P_{g})})} and h{(a,\\mathbf{D})} = \\sin{(\\mathbf{D} - a)}, then obtain \\int (2 \\mathbf{J}{(P_{g})} - \\sin{(\\mathbf{D} - a)}) d\\mathbf{D} = \\int (\\mathbf{J}{(P_{g})} - \\sin{(\\mathbf{D} - a)} + \\cos{(\\cos{(P_{g})})}) d\\mathbf{D}", "derivation": "\\mathbf{J}{(P_{g})} = \\cos{(\\cos{(P_{g})})} and 2 \\mathbf{J}{(P_{g})} = \\mathbf{J}{(P_{g})} + \\cos{(\\cos{(P_{g})})} and h{(a,\\mathbf{D})} = \\sin{(\\mathbf{D} - a)} and 2 \\mathbf{J}{(P_{g})} - h{(a,\\mathbf{D})} = \\mathbf{J}{(P_{g})} - h{(a,\\mathbf{D})} + \\cos{(\\cos{(P_{g})})} and \\int (2 \\mathbf{J}{(P_{g})} - h{(a,\\mathbf{D})}) d\\mathbf{D} = \\int (\\mathbf{J}{(P_{g})} - h{(a,\\mathbf{D})} + \\cos{(\\cos{(P_{g})})}) d\\mathbf{D} and \\int (2 \\mathbf{J}{(P_{g})} - \\sin{(\\mathbf{D} - a)}) d\\mathbf{D} = \\int (\\mathbf{J}{(P_{g})} - \\sin{(\\mathbf{D} - a)} + \\cos{(\\cos{(P_{g})})}) d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True)), cos(cos(Symbol('P_g', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True))), Add(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True)), cos(cos(Symbol('P_g', commutative=True)))))"], ["get_premise", "Equality(Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["minus", 2, "Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True))), Mul(Integer(-1), Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Add(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True)), Mul(Integer(-1), Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), cos(cos(Symbol('P_g', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True))), Mul(Integer(-1), Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Add(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True)), Mul(Integer(-1), Function('h')(Symbol('a', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), cos(cos(Symbol('P_g', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Add(Function('\\\\mathbf{J}')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))), cos(cos(Symbol('P_g', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(Q)} = \\log{(\\log{(Q)})}, then obtain \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{d}{d Q} \\operatorname{E_{\\lambda}}{(Q)} + \\frac{1}{Q} = \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{1}{Q} + \\frac{1}{Q \\log{(Q)}}", "derivation": "\\operatorname{E_{\\lambda}}{(Q)} = \\log{(\\log{(Q)})} and \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} = \\log{(Q)} + \\log{(\\log{(Q)})} and \\frac{d}{d Q} (\\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)}) = \\frac{d}{d Q} (\\log{(Q)} + \\log{(\\log{(Q)})}) and \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{d}{d Q} (\\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)}) = \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{d}{d Q} (\\log{(Q)} + \\log{(\\log{(Q)})}) and \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{d}{d Q} \\operatorname{E_{\\lambda}}{(Q)} + \\frac{1}{Q} = \\operatorname{E_{\\lambda}}{(Q)} + \\log{(Q)} + \\frac{1}{Q} + \\frac{1}{Q \\log{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))"], [["add", 1, "log(Symbol('Q', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True)))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 3, "Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)), Derivative(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)), Derivative(Add(log(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Pow(Symbol('Q', commutative=True), Integer(-1))), Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(log(Symbol('Q', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(E_{\\lambda})} = e^{E_{\\lambda}}, then derive \\frac{d}{d E_{\\lambda}} \\operatorname{n_{1}}{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain e^{E_{\\lambda}} = \\frac{d}{d E_{\\lambda}} e^{E_{\\lambda}}", "derivation": "\\operatorname{n_{1}}{(E_{\\lambda})} = e^{E_{\\lambda}} and \\frac{d}{d E_{\\lambda}} \\operatorname{n_{1}}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} e^{E_{\\lambda}} and \\frac{d}{d E_{\\lambda}} \\operatorname{n_{1}}{(E_{\\lambda})} = e^{E_{\\lambda}} and e^{E_{\\lambda}} = \\frac{d}{d E_{\\lambda}} e^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(exp(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('E_{\\\\lambda}', commutative=True)), Derivative(exp(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(n)} = e^{n}, then obtain (\\int \\frac{\\dot{z}{(n)}}{n} dn)^{n} = (h + \\operatorname{Ei}{(n)})^{n}", "derivation": "\\dot{z}{(n)} = e^{n} and \\frac{\\dot{z}{(n)}}{n} = \\frac{e^{n}}{n} and \\int \\frac{\\dot{z}{(n)}}{n} dn = \\int \\frac{e^{n}}{n} dn and (\\int \\frac{\\dot{z}{(n)}}{n} dn)^{n} = (\\int \\frac{e^{n}}{n} dn)^{n} and (\\int \\frac{\\dot{z}{(n)}}{n} dn)^{n} = (h + \\operatorname{Ei}{(n)})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["divide", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), exp(Symbol('n', commutative=True))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Add(Symbol('h', commutative=True), Ei(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\delta{(I)} = I, then obtain \\frac{d}{d I} \\delta{(I)} + 1 = 2", "derivation": "\\delta{(I)} = I and \\frac{d}{d I} \\delta{(I)} = \\frac{d}{d I} I and \\frac{d}{d I} I + \\frac{d}{d I} \\delta{(I)} = 2 \\frac{d}{d I} I and \\frac{d}{d I} \\delta{(I)} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Function('\\\\delta')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\delta')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1)), Integer(2))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(x)} = \\cos{(x)}, then obtain 0 = - \\frac{\\sin{(x)}}{\\operatorname{E_{x}}{(x)}} - \\frac{\\cos{(x)} \\frac{d}{d x} \\operatorname{E_{x}}{(x)}}{\\operatorname{E_{x}}^{2}{(x)}}", "derivation": "\\operatorname{E_{x}}{(x)} = \\cos{(x)} and 1 = \\frac{\\cos{(x)}}{\\operatorname{E_{x}}{(x)}} and \\frac{d}{d x} 1 = \\frac{d}{d x} \\frac{\\cos{(x)}}{\\operatorname{E_{x}}{(x)}} and 0 = - \\frac{\\sin{(x)}}{\\operatorname{E_{x}}{(x)}} - \\frac{\\cos{(x)} \\frac{d}{d x} \\operatorname{E_{x}}{(x)}}{\\operatorname{E_{x}}^{2}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["divide", 1, "Function('E_x')(Symbol('x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(-1)), cos(Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(-1)), cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(-1)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Pow(Function('E_x')(Symbol('x', commutative=True)), Integer(-2)), cos(Symbol('x', commutative=True)), Derivative(Function('E_x')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(h,U)} = h \\cos{(U)}, then derive \\frac{\\partial}{\\partial h} \\operatorname{x^{{\\}'}}{(h,U)} = \\cos{(U)}, then obtain \\int \\frac{\\partial}{\\partial h} h \\cos{(U)} dU = \\int \\cos{(U)} dU", "derivation": "\\operatorname{x^{{\\}'}}{(h,U)} = h \\cos{(U)} and \\frac{\\partial}{\\partial h} \\operatorname{x^{{\\}'}}{(h,U)} = \\frac{\\partial}{\\partial h} h \\cos{(U)} and \\frac{\\partial}{\\partial h} \\operatorname{x^{{\\}'}}{(h,U)} = \\cos{(U)} and \\int \\frac{\\partial}{\\partial h} \\operatorname{x^{{\\}'}}{(h,U)} dU = \\int \\cos{(U)} dU and \\int \\frac{\\partial}{\\partial h} h \\cos{(U)} dU = \\int \\cos{(U)} dU", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('h', commutative=True), cos(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Symbol('h', commutative=True), cos(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), cos(Symbol('U', commutative=True)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Mul(Symbol('h', commutative=True), cos(Symbol('U', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{p},G)} = G \\mathbf{p}, then derive \\frac{\\partial}{\\partial G} \\mathbf{H}{(\\mathbf{p},G)} = \\mathbf{p}, then obtain \\frac{\\partial}{\\partial G} G \\mathbf{p} = \\mathbf{p}", "derivation": "\\mathbf{H}{(\\mathbf{p},G)} = G \\mathbf{p} and \\frac{\\partial}{\\partial G} \\mathbf{H}{(\\mathbf{p},G)} = \\frac{\\partial}{\\partial G} G \\mathbf{p} and \\frac{\\partial}{\\partial G} \\mathbf{H}{(\\mathbf{p},G)} = \\mathbf{p} and \\frac{\\partial}{\\partial G} G \\mathbf{p} = \\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\mathbf{p}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\mathbf{p}', commutative=True))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{J}_M,n)} = \\int (\\mathbf{J}_M + n) dn, then derive \\operatorname{v_{1}}{(\\mathbf{J}_M,n)} = V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2}, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2}) = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{v_{1}}{(\\mathbf{J}_M,n)}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{J}_M,n)} = \\int (\\mathbf{J}_M + n) dn and \\operatorname{v_{1}}{(\\mathbf{J}_M,n)} = V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2} and V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2} = \\int (\\mathbf{J}_M + n) dn and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2}) = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\int (\\mathbf{J}_M + n) dn and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + \\mathbf{J}_M n + \\frac{n^{2}}{2}) = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{v_{1}}{(\\mathbf{J}_M,n)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('v_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Function('v_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} = F_{H} + g_{\\varepsilon}, then derive \\int \\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} g_{\\varepsilon} + \\chi, then obtain \\int (F_{H} + g_{\\varepsilon}) dF_{H} + 1 = \\frac{F_{H}^{2}}{2} + F_{H} g_{\\varepsilon} + \\chi + 1", "derivation": "\\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} = F_{H} + g_{\\varepsilon} and \\int \\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} dF_{H} = \\int (F_{H} + g_{\\varepsilon}) dF_{H} and \\int \\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} g_{\\varepsilon} + \\chi and \\int \\operatorname{M_{E}}{(g_{\\varepsilon},F_{H})} dF_{H} + 1 = \\frac{F_{H}^{2}}{2} + F_{H} g_{\\varepsilon} + \\chi + 1 and \\int (F_{H} + g_{\\varepsilon}) dF_{H} + 1 = \\frac{F_{H}^{2}}{2} + F_{H} g_{\\varepsilon} + \\chi + 1", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Function('M_E')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integer(1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\chi', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integral(Add(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integer(1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\chi', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\hat{p}_0{(n_{2})} = \\sin{(n_{2})}, then derive (\\frac{d}{d n_{2}} \\int \\hat{p}_0{(n_{2})} dn_{2})^{n_{2}} = (\\frac{\\partial}{\\partial n_{2}} (\\sigma_x - \\cos{(n_{2})}))^{n_{2}}, then obtain (\\frac{d}{d n_{2}} \\int \\sin{(n_{2})} dn_{2})^{n_{2}} = (\\frac{\\partial}{\\partial n_{2}} (\\sigma_x - \\cos{(n_{2})}))^{n_{2}}", "derivation": "\\hat{p}_0{(n_{2})} = \\sin{(n_{2})} and \\int \\hat{p}_0{(n_{2})} dn_{2} = \\int \\sin{(n_{2})} dn_{2} and \\frac{d}{d n_{2}} \\int \\hat{p}_0{(n_{2})} dn_{2} = \\frac{d}{d n_{2}} \\int \\sin{(n_{2})} dn_{2} and (\\frac{d}{d n_{2}} \\int \\hat{p}_0{(n_{2})} dn_{2})^{n_{2}} = (\\frac{d}{d n_{2}} \\int \\sin{(n_{2})} dn_{2})^{n_{2}} and (\\frac{d}{d n_{2}} \\int \\hat{p}_0{(n_{2})} dn_{2})^{n_{2}} = (\\frac{\\partial}{\\partial n_{2}} (\\sigma_x - \\cos{(n_{2})}))^{n_{2}} and (\\frac{d}{d n_{2}} \\int \\sin{(n_{2})} dn_{2})^{n_{2}} = (\\frac{\\partial}{\\partial n_{2}} (\\sigma_x - \\cos{(n_{2})}))^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Derivative(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)), Pow(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(b,P_{g},\\dot{z})} = (P_{g} - \\dot{z})^{b}, then obtain \\frac{\\partial}{\\partial b} (- P_{g} + \\frac{\\operatorname{f^{\\prime}}^{\\dot{z}}{(b,P_{g},\\dot{z})}}{b}) = \\frac{\\partial}{\\partial b} (- P_{g} + \\frac{((P_{g} - \\dot{z})^{b})^{\\dot{z}}}{b})", "derivation": "\\operatorname{f^{\\prime}}{(b,P_{g},\\dot{z})} = (P_{g} - \\dot{z})^{b} and \\operatorname{f^{\\prime}}^{\\dot{z}}{(b,P_{g},\\dot{z})} = ((P_{g} - \\dot{z})^{b})^{\\dot{z}} and \\frac{\\operatorname{f^{\\prime}}^{\\dot{z}}{(b,P_{g},\\dot{z})}}{b} = \\frac{((P_{g} - \\dot{z})^{b})^{\\dot{z}}}{b} and - P_{g} + \\frac{\\operatorname{f^{\\prime}}^{\\dot{z}}{(b,P_{g},\\dot{z})}}{b} = - P_{g} + \\frac{((P_{g} - \\dot{z})^{b})^{\\dot{z}}}{b} and \\frac{\\partial}{\\partial b} (- P_{g} + \\frac{\\operatorname{f^{\\prime}}^{\\dot{z}}{(b,P_{g},\\dot{z})}}{b}) = \\frac{\\partial}{\\partial b} (- P_{g} + \\frac{((P_{g} - \\dot{z})^{b})^{\\dot{z}}}{b})", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Pow(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('b', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True))), Symbol('b', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(C_{d},\\hat{X})} = \\frac{\\hat{X}}{C_{d}} and \\operatorname{m_{s}}{(C_{d},\\hat{X})} = \\frac{\\hat{X}}{C_{d}}, then obtain \\frac{\\hat{X} \\operatorname{m_{s}}{(C_{d},\\hat{X})}}{C_{d}} = \\frac{\\hat{X}^{2}}{C_{d}^{2}}", "derivation": "\\tilde{g}^*{(C_{d},\\hat{X})} = \\frac{\\hat{X}}{C_{d}} and \\operatorname{m_{s}}{(C_{d},\\hat{X})} = \\frac{\\hat{X}}{C_{d}} and \\tilde{g}^*{(C_{d},\\hat{X})} \\operatorname{m_{s}}{(C_{d},\\hat{X})} = \\frac{\\hat{X} \\operatorname{m_{s}}{(C_{d},\\hat{X})}}{C_{d}} and \\frac{\\hat{X} \\tilde{g}^*{(C_{d},\\hat{X})}}{C_{d}} = \\frac{\\hat{X}^{2}}{C_{d}^{2}} and \\operatorname{m_{s}}{(C_{d},\\hat{X})} = \\tilde{g}^*{(C_{d},\\hat{X})} and \\frac{\\hat{X} \\operatorname{m_{s}}{(C_{d},\\hat{X})}}{C_{d}} = \\frac{\\hat{X}^{2}}{C_{d}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 1, "Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True), Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True), Function('m_s')(Symbol('C_d', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\sigma_{p}{(\\dot{\\mathbf{r}},u)} = u + \\log{(\\dot{\\mathbf{r}})}, then obtain \\frac{\\partial}{\\partial u} e^{4 (u + \\log{(\\dot{\\mathbf{r}})})^{2}} = \\frac{\\partial}{\\partial u} e^{(2 u + 2 \\log{(\\dot{\\mathbf{r}})})^{2}}", "derivation": "\\sigma_{p}{(\\dot{\\mathbf{r}},u)} = u + \\log{(\\dot{\\mathbf{r}})} and 2 \\sigma_{p}{(\\dot{\\mathbf{r}},u)} = u + \\sigma_{p}{(\\dot{\\mathbf{r}},u)} + \\log{(\\dot{\\mathbf{r}})} and 4 \\sigma_{p}^{2}{(\\dot{\\mathbf{r}},u)} = (u + \\sigma_{p}{(\\dot{\\mathbf{r}},u)} + \\log{(\\dot{\\mathbf{r}})})^{2} and 4 (u + \\log{(\\dot{\\mathbf{r}})})^{2} = (2 u + 2 \\log{(\\dot{\\mathbf{r}})})^{2} and e^{4 (u + \\log{(\\dot{\\mathbf{r}})})^{2}} = e^{(2 u + 2 \\log{(\\dot{\\mathbf{r}})})^{2}} and \\frac{\\partial}{\\partial u} e^{4 (u + \\log{(\\dot{\\mathbf{r}})})^{2}} = \\frac{\\partial}{\\partial u} e^{(2 u + 2 \\log{(\\dot{\\mathbf{r}})})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), Integer(2))), Pow(Add(Symbol('u', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('u', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('u', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(2), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integer(2)))"], [["exp", 4], "Equality(exp(Mul(Integer(4), Pow(Add(Symbol('u', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2)))), exp(Pow(Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(2), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integer(2))))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(exp(Mul(Integer(4), Pow(Add(Symbol('u', commutative=True), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integer(2)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(exp(Pow(Add(Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(2), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(C_{1},q)} = \\log{(C_{1} q)}, then obtain \\frac{\\partial}{\\partial q} (q + Z{(C_{1},q)} + \\frac{\\partial}{\\partial C_{1}} (q + \\log{(C_{1} q)})) = \\frac{\\partial}{\\partial q} (q + \\log{(C_{1} q)} + \\frac{\\partial}{\\partial C_{1}} (q + \\log{(C_{1} q)}))", "derivation": "Z{(C_{1},q)} = \\log{(C_{1} q)} and q + Z{(C_{1},q)} = q + \\log{(C_{1} q)} and \\frac{\\partial}{\\partial C_{1}} (q + Z{(C_{1},q)}) = \\frac{\\partial}{\\partial C_{1}} (q + \\log{(C_{1} q)}) and q + Z{(C_{1},q)} + \\frac{\\partial}{\\partial C_{1}} (q + Z{(C_{1},q)}) = q + \\log{(C_{1} q)} + \\frac{\\partial}{\\partial C_{1}} (q + Z{(C_{1},q)}) and \\frac{\\partial}{\\partial q} (q + Z{(C_{1},q)} + \\frac{\\partial}{\\partial C_{1}} (q + Z{(C_{1},q)})) = \\frac{\\partial}{\\partial q} (q + \\log{(C_{1} q)} + \\frac{\\partial}{\\partial C_{1}} (q + Z{(C_{1},q)})) and \\frac{\\partial}{\\partial q} (q + Z{(C_{1},q)} + \\frac{\\partial}{\\partial C_{1}} (q + \\log{(C_{1} q)})) = \\frac{\\partial}{\\partial q} (q + \\log{(C_{1} q)} + \\frac{\\partial}{\\partial C_{1}} (q + \\log{(C_{1} q)}))", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Symbol('q', commutative=True), Function('Z')(Symbol('C_1', commutative=True), Symbol('q', commutative=True)), Derivative(Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True))), Derivative(Add(Symbol('q', commutative=True), log(Mul(Symbol('C_1', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbb{I},\\theta_2,A)} = \\frac{A + \\theta_2}{\\mathbb{I}}, then obtain (\\int \\frac{A + \\theta_2}{\\mathbb{I}} d\\theta_2)^{A} = (\\int \\frac{A}{\\mathbb{I}} d\\theta_2 + \\int \\frac{\\theta_2}{\\mathbb{I}} d\\theta_2)^{A}", "derivation": "\\operatorname{v_{t}}{(\\mathbb{I},\\theta_2,A)} = \\frac{A + \\theta_2}{\\mathbb{I}} and \\int \\operatorname{v_{t}}{(\\mathbb{I},\\theta_2,A)} d\\theta_2 = \\int \\frac{A + \\theta_2}{\\mathbb{I}} d\\theta_2 and (\\int \\operatorname{v_{t}}{(\\mathbb{I},\\theta_2,A)} d\\theta_2)^{A} = (\\int \\frac{A + \\theta_2}{\\mathbb{I}} d\\theta_2)^{A} and (\\int \\operatorname{v_{t}}{(\\mathbb{I},\\theta_2,A)} d\\theta_2)^{A} = (\\int \\frac{A}{\\mathbb{I}} d\\theta_2 + \\int \\frac{\\theta_2}{\\mathbb{I}} d\\theta_2)^{A} and (\\int \\frac{A + \\theta_2}{\\mathbb{I}} d\\theta_2)^{A} = (\\int \\frac{A}{\\mathbb{I}} d\\theta_2 + \\int \\frac{\\theta_2}{\\mathbb{I}} d\\theta_2)^{A}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('A', commutative=True)))"], [["expand", 3], "Equality(Pow(Integral(Function('v_t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('A', commutative=True)), Pow(Add(Integral(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('A', commutative=True)), Pow(Add(Integral(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given z{(t)} = \\log{(t)} and \\mathbf{J}_M{(t)} = \\frac{d}{d t} z{(t)}, then derive \\frac{d}{d t} z{(t)} = \\frac{1}{t}, then obtain \\frac{d}{d t} \\log{(t)} = \\frac{1}{t}", "derivation": "z{(t)} = \\log{(t)} and \\frac{d}{d t} z{(t)} = \\frac{d}{d t} \\log{(t)} and \\mathbf{J}_M{(t)} = \\frac{d}{d t} z{(t)} and \\mathbf{J}_M{(t)} = \\frac{d}{d t} \\log{(t)} and \\frac{d}{d t} z{(t)} = \\frac{1}{t} and \\mathbf{J}_M{(t)} = \\frac{1}{t} and \\frac{d}{d t} \\log{(t)} = \\frac{1}{t}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('t', commutative=True)), Derivative(Function('z')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}_M')(Symbol('t', commutative=True)), Derivative(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Pow(Symbol('t', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Function('\\\\mathbf{J}_M')(Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Pow(Symbol('t', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mu_0,Q)} = Q^{\\mu_0} and \\mathbf{F}{(\\mu_0,Q)} = (Q^{\\mu_0})^{\\mu_0}, then obtain 2 \\mathbf{F}{(\\mu_0,Q)} = \\mathbf{F}{(\\mu_0,Q)} + \\operatorname{t_{1}}^{\\mu_0}{(\\mu_0,Q)}", "derivation": "\\operatorname{t_{1}}{(\\mu_0,Q)} = Q^{\\mu_0} and \\mathbf{F}{(\\mu_0,Q)} = (Q^{\\mu_0})^{\\mu_0} and \\mathbf{F}{(\\mu_0,Q)} = \\operatorname{t_{1}}^{\\mu_0}{(\\mu_0,Q)} and 2 \\mathbf{F}{(\\mu_0,Q)} = \\mathbf{F}{(\\mu_0,Q)} + \\operatorname{t_{1}}^{\\mu_0}{(\\mu_0,Q)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Pow(Symbol('Q', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["add", 3, "Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given Q{(V_{\\mathbf{B}},z)} = \\frac{V_{\\mathbf{B}}}{z}, then obtain Q^{2}{(V_{\\mathbf{B}},z)} + 1 - \\frac{1}{z} = \\frac{V_{\\mathbf{B}} Q{(V_{\\mathbf{B}},z)}}{z} + 1 - \\frac{1}{z}", "derivation": "Q{(V_{\\mathbf{B}},z)} = \\frac{V_{\\mathbf{B}}}{z} and Q^{2}{(V_{\\mathbf{B}},z)} = \\frac{V_{\\mathbf{B}} Q{(V_{\\mathbf{B}},z)}}{z} and Q^{2}{(V_{\\mathbf{B}},z)} + 1 = \\frac{V_{\\mathbf{B}} Q{(V_{\\mathbf{B}},z)}}{z} + 1 and Q^{2}{(V_{\\mathbf{B}},z)} + 1 - \\frac{1}{z} = \\frac{V_{\\mathbf{B}} Q{(V_{\\mathbf{B}},z)}}{z} + 1 - \\frac{1}{z}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["times", 1, "Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))"], "Equality(Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Integer(1)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))), Integer(1)))"], [["minus", 3, "Pow(Symbol('z', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)), Integer(2)), Integer(1), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\chi{(\\mathbf{J}_M,F_{N},B)} = B + F_{N} - \\mathbf{J}_M, then obtain \\frac{\\partial}{\\partial B} \\frac{\\chi{(\\mathbf{J}_M,F_{N},B)} - 1}{B + F_{N} - \\mathbf{J}_M} = \\frac{\\partial}{\\partial B} \\frac{B + F_{N} - \\mathbf{J}_M - 1}{B + F_{N} - \\mathbf{J}_M}", "derivation": "\\chi{(\\mathbf{J}_M,F_{N},B)} = B + F_{N} - \\mathbf{J}_M and \\chi{(\\mathbf{J}_M,F_{N},B)} - 1 = B + F_{N} - \\mathbf{J}_M - 1 and \\frac{\\chi{(\\mathbf{J}_M,F_{N},B)} - 1}{B + F_{N} - \\mathbf{J}_M} = \\frac{B + F_{N} - \\mathbf{J}_M - 1}{B + F_{N} - \\mathbf{J}_M} and \\frac{\\partial}{\\partial B} \\frac{\\chi{(\\mathbf{J}_M,F_{N},B)} - 1}{B + F_{N} - \\mathbf{J}_M} = \\frac{\\partial}{\\partial B} \\frac{B + F_{N} - \\mathbf{J}_M - 1}{B + F_{N} - \\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_N', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_N', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))"], [["divide", 2, "Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Mul(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_N', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\chi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('F_N', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(v_{2})} = e^{v_{2}}, then obtain \\int \\log{((\\mathbf{p}{(v_{2})} + e^{v_{2}}) \\mathbf{p}{(v_{2})})} dv_{2} = \\int \\log{(2 \\mathbf{p}{(v_{2})} e^{v_{2}})} dv_{2}", "derivation": "\\mathbf{p}{(v_{2})} = e^{v_{2}} and \\mathbf{p}{(v_{2})} + e^{v_{2}} = 2 e^{v_{2}} and (\\mathbf{p}{(v_{2})} + e^{v_{2}}) \\mathbf{p}{(v_{2})} = 2 \\mathbf{p}{(v_{2})} e^{v_{2}} and \\log{((\\mathbf{p}{(v_{2})} + e^{v_{2}}) \\mathbf{p}{(v_{2})})} = \\log{(2 \\mathbf{p}{(v_{2})} e^{v_{2}})} and \\int \\log{((\\mathbf{p}{(v_{2})} + e^{v_{2}}) \\mathbf{p}{(v_{2})})} dv_{2} = \\int \\log{(2 \\mathbf{p}{(v_{2})} e^{v_{2}})} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), exp(Symbol('v_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Mul(Integer(2), exp(Symbol('v_2', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))))"], [["log", 3], "Equality(log(Mul(Add(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)))), log(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(log(Mul(Add(Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True))), Integral(log(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(t_{2},\\ddot{x})} = \\ddot{x} t_{2}, then obtain (\\int (- \\ddot{x}^{2} t_{2}^{2} + \\ddot{x} t_{2} \\mathbf{J}_f{(t_{2},\\ddot{x})}) dt_{2})^{\\ddot{x}} = (\\int 0 dt_{2})^{\\ddot{x}}", "derivation": "\\mathbf{J}_f{(t_{2},\\ddot{x})} = \\ddot{x} t_{2} and \\ddot{x} t_{2} \\mathbf{J}_f{(t_{2},\\ddot{x})} = \\ddot{x}^{2} t_{2}^{2} and - \\ddot{x}^{2} t_{2}^{2} + \\ddot{x} t_{2} \\mathbf{J}_f{(t_{2},\\ddot{x})} = 0 and \\int (- \\ddot{x}^{2} t_{2}^{2} + \\ddot{x} t_{2} \\mathbf{J}_f{(t_{2},\\ddot{x})}) dt_{2} = \\int 0 dt_{2} and (\\int (- \\ddot{x}^{2} t_{2}^{2} + \\ddot{x} t_{2} \\mathbf{J}_f{(t_{2},\\ddot{x})}) dt_{2})^{\\ddot{x}} = (\\int 0 dt_{2})^{\\ddot{x}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Pow(Symbol('t_2', commutative=True), Integer(2))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Pow(Symbol('t_2', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Pow(Symbol('t_2', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Pow(Symbol('t_2', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('t_2', commutative=True))))"], [["power", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Pow(Symbol('t_2', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('t_2', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('t_2', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(t_{1},\\dot{x})} = \\dot{x} t_{1}, then obtain t_{1} + \\dot{\\mathbf{r}}{(t_{1},\\dot{x})} + \\dot{\\mathbf{r}}^{t_{1}}{(t_{1},\\dot{x})} = t_{1} + (\\dot{x} t_{1})^{t_{1}} + \\dot{\\mathbf{r}}{(t_{1},\\dot{x})}", "derivation": "\\dot{\\mathbf{r}}{(t_{1},\\dot{x})} = \\dot{x} t_{1} and t_{1} + \\dot{\\mathbf{r}}{(t_{1},\\dot{x})} = \\dot{x} t_{1} + t_{1} and \\dot{\\mathbf{r}}^{t_{1}}{(t_{1},\\dot{x})} = (\\dot{x} t_{1})^{t_{1}} and \\dot{x} t_{1} + t_{1} + \\dot{\\mathbf{r}}^{t_{1}}{(t_{1},\\dot{x})} = \\dot{x} t_{1} + t_{1} + (\\dot{x} t_{1})^{t_{1}} and t_{1} + \\dot{\\mathbf{r}}{(t_{1},\\dot{x})} + \\dot{\\mathbf{r}}^{t_{1}}{(t_{1},\\dot{x})} = t_{1} + (\\dot{x} t_{1})^{t_{1}} + \\dot{\\mathbf{r}}{(t_{1},\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)))"], [["add", 1, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('t_1', commutative=True)), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["add", 3, "Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('t_1', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('t_1', commutative=True))), Add(Symbol('t_1', commutative=True), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_1', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given k{(\\nabla,\\Psi_{nl})} = \\log{(\\nabla^{\\Psi_{nl}})} and \\mathbf{v}{(\\nabla,\\Psi_{nl})} = k^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})}, then obtain (k^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})})^{\\Psi_{nl}} = (\\log{(\\nabla^{\\Psi_{nl}})}^{\\Psi_{nl}})^{\\Psi_{nl}}", "derivation": "k{(\\nabla,\\Psi_{nl})} = \\log{(\\nabla^{\\Psi_{nl}})} and k^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})} = \\log{(\\nabla^{\\Psi_{nl}})}^{\\Psi_{nl}} and \\mathbf{v}{(\\nabla,\\Psi_{nl})} = k^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})} and \\mathbf{v}{(\\nabla,\\Psi_{nl})} = \\log{(\\nabla^{\\Psi_{nl}})}^{\\Psi_{nl}} and \\mathbf{v}^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})} = (\\log{(\\nabla^{\\Psi_{nl}})}^{\\Psi_{nl}})^{\\Psi_{nl}} and (k^{\\Psi_{nl}}{(\\nabla,\\Psi_{nl})})^{\\Psi_{nl}} = (\\log{(\\nabla^{\\Psi_{nl}})}^{\\Psi_{nl}})^{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(log(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Function('k')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(log(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["power", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Pow(log(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Pow(Function('k')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Pow(log(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given B{(\\theta,y)} = \\log{(- \\theta + y)}, then obtain (\\int (1 - y) dy) \\int (- y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}}) dy = (\\int (- y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}}) dy)^{2}", "derivation": "B{(\\theta,y)} = \\log{(- \\theta + y)} and 1 = \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}} and 1 - y = - y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}} and \\int (1 - y) dy = \\int (- y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}}) dy and (\\int (1 - y) dy) \\int (- y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}}) dy = (\\int (- y + \\frac{\\log{(- \\theta + y)}}{B{(\\theta,y)}}) dy)^{2}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))"], [["divide", 1, "Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True)))))"], [["minus", 2, "Symbol('y', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True))))"], [["times", 4, "Integral(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)))), Pow(Integral(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Function('B')(Symbol('\\\\theta', commutative=True), Symbol('y', commutative=True)), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\theta{(C_{d},\\mu)} = - C_{d} + \\mu, then derive \\frac{\\partial}{\\partial C_{d}} \\theta{(C_{d},\\mu)} = -1, then obtain \\frac{\\partial^{- \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)}}{\\partial C_{d}^{- \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)}} \\theta{(C_{d},\\mu)} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)", "derivation": "\\theta{(C_{d},\\mu)} = - C_{d} + \\mu and \\frac{\\partial}{\\partial C_{d}} \\theta{(C_{d},\\mu)} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu) and \\frac{\\partial}{\\partial C_{d}} \\theta{(C_{d},\\mu)} = -1 and -1 = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu) and \\frac{\\partial^{- \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)}}{\\partial C_{d}^{- \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)}} \\theta{(C_{d},\\mu)} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + \\mu)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\theta')(Symbol('C_d', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(x)} = \\cos{(\\cos{(x)})}, then obtain \\frac{- x - \\phi_{1}{(x)}}{x^{2} \\phi_{1}{(x)}} = \\frac{- x - \\cos{(\\cos{(x)})}}{x^{2} \\phi_{1}{(x)}}", "derivation": "\\phi_{1}{(x)} = \\cos{(\\cos{(x)})} and x + \\phi_{1}{(x)} = x + \\cos{(\\cos{(x)})} and \\frac{x + \\phi_{1}{(x)}}{\\phi_{1}{(x)}} = \\frac{x + \\cos{(\\cos{(x)})}}{\\phi_{1}{(x)}} and \\frac{x + \\phi_{1}{(x)}}{x \\phi_{1}{(x)}} = \\frac{x + \\cos{(\\cos{(x)})}}{x \\phi_{1}{(x)}} and \\frac{x + \\phi_{1}{(x)}}{x^{2} \\phi_{1}{(x)}} = \\frac{x + \\cos{(\\cos{(x)})}}{x^{2} \\phi_{1}{(x)}} and - \\frac{x + \\phi_{1}{(x)}}{x^{2} \\phi_{1}{(x)}} = - \\frac{x + \\cos{(\\cos{(x)})}}{x^{2} \\phi_{1}{(x)}} and \\frac{- x - \\phi_{1}{(x)}}{x^{2} \\phi_{1}{(x)}} = \\frac{- x - \\cos{(\\cos{(x)})}}{x^{2} \\phi_{1}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True))))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('\\\\phi_1')(Symbol('x', commutative=True))), Add(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))))"], [["divide", 2, "Function('\\\\phi_1')(Symbol('x', commutative=True))"], "Equality(Mul(Add(Symbol('x', commutative=True), Function('\\\\phi_1')(Symbol('x', commutative=True))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))), Mul(Add(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))))"], [["divide", 3, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), Function('\\\\phi_1')(Symbol('x', commutative=True))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))))"], [["times", 4, "Pow(Symbol('x', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-2)), Add(Symbol('x', commutative=True), Function('\\\\phi_1')(Symbol('x', commutative=True))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))), Mul(Pow(Symbol('x', commutative=True), Integer(-2)), Add(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-2)), Add(Symbol('x', commutative=True), Function('\\\\phi_1')(Symbol('x', commutative=True))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-2)), Add(Symbol('x', commutative=True), cos(cos(Symbol('x', commutative=True)))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('x', commutative=True)))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))), Mul(Pow(Symbol('x', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('x', commutative=True))))), Pow(Function('\\\\phi_1')(Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} = \\mathbf{v} y^{\\prime}, then obtain (\\mathbf{v} + \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})}) \\int \\mathbf{v} y^{\\prime} dy^{\\prime} = (\\mathbf{v} y^{\\prime} + \\mathbf{v}) \\int \\mathbf{v} y^{\\prime} dy^{\\prime}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} = \\mathbf{v} y^{\\prime} and \\mathbf{v} + \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} = \\mathbf{v} y^{\\prime} + \\mathbf{v} and \\int \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} dy^{\\prime} = \\int \\mathbf{v} y^{\\prime} dy^{\\prime} and (\\mathbf{v} + \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})}) \\int \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} dy^{\\prime} = (\\mathbf{v} y^{\\prime} + \\mathbf{v}) \\int \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})} dy^{\\prime} and (\\mathbf{v} + \\operatorname{n_{2}}{(\\mathbf{v},y^{\\prime})}) \\int \\mathbf{v} y^{\\prime} dy^{\\prime} = (\\mathbf{v} y^{\\prime} + \\mathbf{v}) \\int \\mathbf{v} y^{\\prime} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 2, "Integral(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('n_2')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{v}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(b,C)} = \\frac{\\partial}{\\partial b} C b, then obtain \\frac{\\partial^{3}}{\\partial b^{3}} \\operatorname{z^{*}}{(b,C)} = \\frac{\\partial^{4}}{\\partial b^{4}} C b", "derivation": "\\operatorname{z^{*}}{(b,C)} = \\frac{\\partial}{\\partial b} C b and \\frac{\\partial}{\\partial b} \\operatorname{z^{*}}{(b,C)} = \\frac{\\partial^{2}}{\\partial b^{2}} C b and \\frac{\\partial^{2}}{\\partial b^{2}} \\operatorname{z^{*}}{(b,C)} = \\frac{\\partial^{3}}{\\partial b^{3}} C b and \\frac{\\partial^{3}}{\\partial b^{3}} \\operatorname{z^{*}}{(b,C)} = \\frac{\\partial^{4}}{\\partial b^{4}} C b", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('b', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('b', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('b', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Mul(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(3))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('b', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(3))), Derivative(Mul(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(4))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(W)} = \\sin{(W)}, then derive - \\frac{\\operatorname{f_{E}}{(W)} \\cos{(W)}}{2 \\sin^{2}{(W)}} + \\frac{\\frac{d}{d W} \\operatorname{f_{E}}{(W)}}{2 \\sin{(W)}} = 0, then obtain (- \\frac{\\cos{(W)}}{2 \\sin{(W)}} + \\frac{\\frac{d}{d W} \\sin{(W)}}{2 \\sin{(W)}})^{W} = 0^{W}", "derivation": "\\operatorname{f_{E}}{(W)} = \\sin{(W)} and \\operatorname{f_{E}}{(W)} \\sin{(W)} = \\sin^{2}{(W)} and \\frac{\\operatorname{f_{E}}{(W)}}{2 \\sin{(W)}} = \\frac{1}{2} and \\frac{d}{d W} \\frac{\\operatorname{f_{E}}{(W)}}{2 \\sin{(W)}} = \\frac{d}{d W} \\frac{1}{2} and - \\frac{\\operatorname{f_{E}}{(W)} \\cos{(W)}}{2 \\sin^{2}{(W)}} + \\frac{\\frac{d}{d W} \\operatorname{f_{E}}{(W)}}{2 \\sin{(W)}} = 0 and - \\frac{\\cos{(W)}}{2 \\sin{(W)}} + \\frac{\\frac{d}{d W} \\sin{(W)}}{2 \\sin{(W)}} = 0 and (- \\frac{\\cos{(W)}}{2 \\sin{(W)}} + \\frac{\\frac{d}{d W} \\sin{(W)}}{2 \\sin{(W)}})^{W} = 0^{W}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["times", 1, "sin(Symbol('W', commutative=True))"], "Equality(Mul(Function('f_E')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Pow(sin(Symbol('W', commutative=True)), Integer(2)))"], [["divide", 2, "Mul(Integer(2), Pow(sin(Symbol('W', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 2), Function('f_E')(Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Integer(-1))), Rational(1, 2))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Function('f_E')(Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Rational(1, 2), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Function('f_E')(Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Integer(-2)), cos(Symbol('W', commutative=True))), Mul(Rational(1, 2), Pow(sin(Symbol('W', commutative=True)), Integer(-1)), Derivative(Function('f_E')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(sin(Symbol('W', commutative=True)), Integer(-1)), cos(Symbol('W', commutative=True))), Mul(Rational(1, 2), Pow(sin(Symbol('W', commutative=True)), Integer(-1)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))), Integer(0))"], [["power", 6, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(sin(Symbol('W', commutative=True)), Integer(-1)), cos(Symbol('W', commutative=True))), Mul(Rational(1, 2), Pow(sin(Symbol('W', commutative=True)), Integer(-1)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))), Symbol('W', commutative=True)), Pow(Integer(0), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(M_{E})} = \\sin{(M_{E})}, then derive \\int \\hat{x}{(M_{E})} dM_{E} = J_{\\varepsilon} - \\cos{(M_{E})}, then obtain \\frac{d}{d M_{E}} \\int \\sin{(M_{E})} dM_{E} = \\frac{\\partial}{\\partial M_{E}} (J_{\\varepsilon} - \\cos{(M_{E})})", "derivation": "\\hat{x}{(M_{E})} = \\sin{(M_{E})} and \\int \\hat{x}{(M_{E})} dM_{E} = \\int \\sin{(M_{E})} dM_{E} and \\int \\hat{x}{(M_{E})} dM_{E} = J_{\\varepsilon} - \\cos{(M_{E})} and \\int \\sin{(M_{E})} dM_{E} = J_{\\varepsilon} - \\cos{(M_{E})} and \\frac{d}{d M_{E}} \\int \\sin{(M_{E})} dM_{E} = \\frac{\\partial}{\\partial M_{E}} (J_{\\varepsilon} - \\cos{(M_{E})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"], [["differentiate", 4, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(c_{0},t)} = c_{0} + t, then obtain \\int \\frac{f{(c_{0},t)}}{t} dt - \\frac{f{(c_{0},t)}}{t} = \\int \\frac{c_{0} + t}{t} dt - \\frac{f{(c_{0},t)}}{t}", "derivation": "f{(c_{0},t)} = c_{0} + t and \\frac{f{(c_{0},t)}}{t} = \\frac{c_{0} + t}{t} and \\int \\frac{f{(c_{0},t)}}{t} dt = \\int \\frac{c_{0} + t}{t} dt and \\int \\frac{f{(c_{0},t)}}{t} dt - \\frac{f{(c_{0},t)}}{t} = \\int \\frac{c_{0} + t}{t} dt - \\frac{f{(c_{0},t)}}{t}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True)), Add(Symbol('c_0', commutative=True), Symbol('t', commutative=True)))"], [["divide", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('c_0', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('c_0', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True)))), Add(Integral(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('c_0', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('f')(Symbol('c_0', commutative=True), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})} = P_{e} \\mathbf{J} - \\rho_f, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} e^{\\rho_f (- \\mathbf{J} + \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})})} = \\frac{\\partial}{\\partial \\mathbf{J}} e^{\\rho_f (P_{e} \\mathbf{J} - \\mathbf{J} - \\rho_f)}", "derivation": "\\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})} = P_{e} \\mathbf{J} - \\rho_f and - \\mathbf{J} + \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})} = P_{e} \\mathbf{J} - \\mathbf{J} - \\rho_f and \\rho_f (- \\mathbf{J} + \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})}) = \\rho_f (P_{e} \\mathbf{J} - \\mathbf{J} - \\rho_f) and e^{\\rho_f (- \\mathbf{J} + \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})})} = e^{\\rho_f (P_{e} \\mathbf{J} - \\mathbf{J} - \\rho_f)} and \\frac{\\partial}{\\partial \\mathbf{J}} e^{\\rho_f (- \\mathbf{J} + \\operatorname{A_{z}}{(\\rho_f,P_{e},\\mathbf{J})})} = \\frac{\\partial}{\\partial \\mathbf{J}} e^{\\rho_f (P_{e} \\mathbf{J} - \\mathbf{J} - \\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\rho_f', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('A_z')(Symbol('\\\\rho_f', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('A_z')(Symbol('\\\\rho_f', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))"], [["exp", 3], "Equality(exp(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('A_z')(Symbol('\\\\rho_f', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), exp(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(exp(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('A_z')(Symbol('\\\\rho_f', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('\\\\rho_f', commutative=True), Add(Mul(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} = e^{\\rho_b}, then obtain (\\operatorname{f_{\\mathbf{v}}}^{\\rho_b}{(\\rho_b)})^{\\rho_b} + \\iint \\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} d\\rho_b d\\rho_b = (\\operatorname{f_{\\mathbf{v}}}^{\\rho_b}{(\\rho_b)})^{\\rho_b} + \\iint e^{\\rho_b} d\\rho_b d\\rho_b", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} = e^{\\rho_b} and \\int \\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} d\\rho_b = \\int e^{\\rho_b} d\\rho_b and \\iint \\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} d\\rho_b d\\rho_b = \\iint e^{\\rho_b} d\\rho_b d\\rho_b and (\\operatorname{f_{\\mathbf{v}}}^{\\rho_b}{(\\rho_b)})^{\\rho_b} + \\iint \\operatorname{f_{\\mathbf{v}}}{(\\rho_b)} d\\rho_b d\\rho_b = (\\operatorname{f_{\\mathbf{v}}}^{\\rho_b}{(\\rho_b)})^{\\rho_b} + \\iint e^{\\rho_b} d\\rho_b d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["add", 3, "Pow(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Pow(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Add(Pow(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given k{(F_{x})} = e^{F_{x}}, then derive \\int k{(F_{x})} dF_{x} = \\eta + e^{F_{x}}, then obtain e^{F_{x}} \\frac{d}{d F_{x}} \\int k{(F_{x})} dF_{x} = e^{2 F_{x}}", "derivation": "k{(F_{x})} = e^{F_{x}} and \\int k{(F_{x})} dF_{x} = \\int e^{F_{x}} dF_{x} and \\int k{(F_{x})} dF_{x} = \\eta + e^{F_{x}} and \\frac{d}{d F_{x}} \\int k{(F_{x})} dF_{x} = \\frac{\\partial}{\\partial F_{x}} (\\eta + e^{F_{x}}) and e^{F_{x}} \\frac{d}{d F_{x}} \\int k{(F_{x})} dF_{x} = e^{F_{x}} \\frac{\\partial}{\\partial F_{x}} (\\eta + e^{F_{x}}) and e^{F_{x}} \\frac{d}{d F_{x}} \\int k{(F_{x})} dF_{x} = e^{2 F_{x}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('k')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\eta', commutative=True), exp(Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integral(Function('k')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 4, "exp(Symbol('F_x', commutative=True))"], "Equality(Mul(exp(Symbol('F_x', commutative=True)), Derivative(Integral(Function('k')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(exp(Symbol('F_x', commutative=True)), Derivative(Add(Symbol('\\\\eta', commutative=True), exp(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(exp(Symbol('F_x', commutative=True)), Derivative(Integral(Function('k')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given J{(M,C_{1})} = \\frac{e^{C_{1}}}{M} and \\dot{\\mathbf{r}}{(M,C_{1})} = - \\frac{e^{C_{1}}}{M}, then obtain \\cos^{C_{1}}{(0^{C_{1}})} = \\cos^{C_{1}}{((\\dot{\\mathbf{r}}{(M,C_{1})} + \\frac{e^{C_{1}}}{M})^{C_{1}})}", "derivation": "J{(M,C_{1})} = \\frac{e^{C_{1}}}{M} and 0 = - J{(M,C_{1})} + \\frac{e^{C_{1}}}{M} and \\dot{\\mathbf{r}}{(M,C_{1})} = - \\frac{e^{C_{1}}}{M} and \\dot{\\mathbf{r}}{(M,C_{1})} = - J{(M,C_{1})} and 0 = \\dot{\\mathbf{r}}{(M,C_{1})} + \\frac{e^{C_{1}}}{M} and 0^{C_{1}} = (\\dot{\\mathbf{r}}{(M,C_{1})} + \\frac{e^{C_{1}}}{M})^{C_{1}} and \\cos{(0^{C_{1}})} = \\cos{((\\dot{\\mathbf{r}}{(M,C_{1})} + \\frac{e^{C_{1}}}{M})^{C_{1}})} and \\cos^{C_{1}}{(0^{C_{1}})} = \\cos^{C_{1}}{((\\dot{\\mathbf{r}}{(M,C_{1})} + \\frac{e^{C_{1}}}{M})^{C_{1}})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True))))"], [["minus", 1, "Function('J')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('J')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('J')(Symbol('M', commutative=True), Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(0), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True)))))"], [["power", 5, "Symbol('C_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('C_1', commutative=True)), Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)))"], [["cos", 6], "Equality(cos(Pow(Integer(0), Symbol('C_1', commutative=True))), cos(Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))))"], [["power", 7, "Symbol('C_1', commutative=True)"], "Equality(Pow(cos(Pow(Integer(0), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(cos(Pow(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('C_1', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), exp(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(V)} = \\sin{(e^{V})}, then derive \\frac{d}{d V} \\mathbf{F}{(V)} = e^{V} \\cos{(e^{V})}, then obtain (e^{V} \\cos{(e^{V})})^{V} - (\\mathbf{F}^{2}{(V)})^{V} + \\mathbf{F}{(V)} = - (\\mathbf{F}^{2}{(V)})^{V} + \\mathbf{F}{(V)} + (\\frac{d}{d V} \\sin{(e^{V})})^{V}", "derivation": "\\mathbf{F}{(V)} = \\sin{(e^{V})} and \\frac{d}{d V} \\mathbf{F}{(V)} = \\frac{d}{d V} \\sin{(e^{V})} and \\frac{d}{d V} \\mathbf{F}{(V)} = e^{V} \\cos{(e^{V})} and e^{V} \\cos{(e^{V})} = \\frac{d}{d V} \\sin{(e^{V})} and (e^{V} \\cos{(e^{V})})^{V} = (\\frac{d}{d V} \\sin{(e^{V})})^{V} and (e^{V} \\cos{(e^{V})})^{V} - (\\mathbf{F}^{2}{(V)})^{V} = - (\\mathbf{F}^{2}{(V)})^{V} + (\\frac{d}{d V} \\sin{(e^{V})})^{V} and (e^{V} \\cos{(e^{V})})^{V} - (\\mathbf{F}^{2}{(V)})^{V} + \\mathbf{F}{(V)} = - (\\mathbf{F}^{2}{(V)})^{V} + \\mathbf{F}{(V)} + (\\frac{d}{d V} \\sin{(e^{V})})^{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), sin(exp(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Mul(exp(Symbol('V', commutative=True)), cos(exp(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(exp(Symbol('V', commutative=True)), cos(exp(Symbol('V', commutative=True)))), Derivative(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Mul(exp(Symbol('V', commutative=True)), cos(exp(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Derivative(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["minus", 5, "Pow(Pow(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Integer(2)), Symbol('V', commutative=True))"], "Equality(Add(Pow(Mul(exp(Symbol('V', commutative=True)), cos(exp(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Integer(2)), Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Integer(2)), Symbol('V', commutative=True))), Pow(Derivative(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True))))"], [["add", 6, "Function('\\\\mathbf{F}')(Symbol('V', commutative=True))"], "Equality(Add(Pow(Mul(exp(Symbol('V', commutative=True)), cos(exp(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Integer(2)), Symbol('V', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Integer(2)), Symbol('V', commutative=True))), Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Pow(Derivative(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(\\pi)} = \\cos{(\\pi)}, then obtain \\pi \\dot{z}{(\\pi)} - \\frac{\\pi}{\\cos{(\\pi)}} = \\pi \\cos{(\\pi)} - \\frac{\\pi}{\\cos{(\\pi)}}", "derivation": "\\dot{z}{(\\pi)} = \\cos{(\\pi)} and \\pi \\dot{z}{(\\pi)} = \\pi \\cos{(\\pi)} and \\pi = \\frac{\\pi \\cos{(\\pi)}}{\\dot{z}{(\\pi)}} and \\frac{\\pi}{\\cos{(\\pi)}} = \\frac{\\pi}{\\dot{z}{(\\pi)}} and \\pi \\dot{z}{(\\pi)} - \\frac{\\pi}{\\dot{z}{(\\pi)}} = \\pi \\cos{(\\pi)} - \\frac{\\pi}{\\dot{z}{(\\pi)}} and \\pi \\dot{z}{(\\pi)} - \\frac{\\pi}{\\cos{(\\pi)}} = \\pi \\cos{(\\pi)} - \\frac{\\pi}{\\cos{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True))"], "Equality(Symbol('\\\\pi', commutative=True), Mul(Symbol('\\\\pi', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))))"], [["divide", 3, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Symbol('\\\\pi', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Symbol('\\\\pi', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\pi', commutative=True), cos(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given c{(t_{2},U)} = U + t_{2} and I{(t_{2},U)} = U + t_{2}, then obtain U I{(t_{2},U)} + c{(t_{2},U)} - \\log{(I{(t_{2},U)})} = U I{(t_{2},U)} + I{(t_{2},U)} - \\log{(I{(t_{2},U)})}", "derivation": "c{(t_{2},U)} = U + t_{2} and \\log{(c{(t_{2},U)})} = \\log{(U + t_{2})} and c{(t_{2},U)} - \\log{(c{(t_{2},U)})} = U + t_{2} - \\log{(c{(t_{2},U)})} and c{(t_{2},U)} - \\log{(U + t_{2})} = U + t_{2} - \\log{(U + t_{2})} and I{(t_{2},U)} = U + t_{2} and c{(t_{2},U)} - \\log{(I{(t_{2},U)})} = I{(t_{2},U)} - \\log{(I{(t_{2},U)})} and U (U + t_{2}) + c{(t_{2},U)} - \\log{(I{(t_{2},U)})} = U (U + t_{2}) + I{(t_{2},U)} - \\log{(I{(t_{2},U)})} and U I{(t_{2},U)} + c{(t_{2},U)} - \\log{(I{(t_{2},U)})} = U I{(t_{2},U)} + I{(t_{2},U)} - \\log{(I{(t_{2},U)})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True)))"], [["log", 1], "Equality(log(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))), log(Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True))))"], [["minus", 1, "log(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)))"], "Equality(Add(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), log(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True))))), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), log(Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True))))))"], ["renaming_premise", "Equality(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))), Add(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))))"], [["add", 6, "Mul(Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True))), Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))), Add(Mul(Symbol('U', commutative=True), Add(Symbol('U', commutative=True), Symbol('t_2', commutative=True))), Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Mul(Symbol('U', commutative=True), Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))), Function('c')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))), Add(Mul(Symbol('U', commutative=True), Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))), Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), log(Function('I')(Symbol('t_2', commutative=True), Symbol('U', commutative=True))))))"]]}, {"prompt": "Given \\varphi{(a,\\mathbf{J}_P)} = \\mathbf{J}_P + \\sin{(a)} and I{(a)} = \\sin{(a)}, then obtain \\int (\\varphi{(a,\\mathbf{J}_P)} - \\sin{(a)}) da = \\int \\mathbf{J}_P da", "derivation": "\\varphi{(a,\\mathbf{J}_P)} = \\mathbf{J}_P + \\sin{(a)} and I{(a)} = \\sin{(a)} and \\varphi{(a,\\mathbf{J}_P)} = \\mathbf{J}_P + I{(a)} and \\varphi{(a,\\mathbf{J}_P)} - \\sin{(a)} = \\mathbf{J}_P + I{(a)} - \\sin{(a)} and \\mathbf{J}_P + \\sin{(a)} = \\mathbf{J}_P + I{(a)} and \\varphi{(a,\\mathbf{J}_P)} - \\sin{(a)} = \\mathbf{J}_P and \\int (\\varphi{(a,\\mathbf{J}_P)} - \\sin{(a)}) da = \\int \\mathbf{J}_P da", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varphi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('a', commutative=True))))"], [["minus", 3, "sin(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Function('I')(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\varphi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["integrate", 6, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))), Integral(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given G{(p,B)} = B + p, then obtain \\frac{B + p}{B} + \\frac{\\frac{\\partial}{\\partial p} G{(p,B)}}{B} = \\frac{B + p}{B} + \\frac{1}{B}", "derivation": "G{(p,B)} = B + p and \\frac{G{(p,B)}}{B} = \\frac{B + p}{B} and \\frac{\\partial}{\\partial p} \\frac{G{(p,B)}}{B} = \\frac{\\partial}{\\partial p} \\frac{B + p}{B} and \\frac{\\partial}{\\partial p} \\frac{G{(p,B)}}{B} + \\frac{B + p}{B} = \\frac{\\partial}{\\partial p} \\frac{B + p}{B} + \\frac{B + p}{B} and \\frac{B + p}{B} + \\frac{\\frac{\\partial}{\\partial p} G{(p,B)}}{B} = \\frac{B + p}{B} + \\frac{1}{B}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('p', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('G')(Symbol('p', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('G')(Symbol('p', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 3, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('G')(Symbol('p', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True)))), Add(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Derivative(Function('G')(Symbol('p', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('p', commutative=True))), Pow(Symbol('B', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{x}{(v_{t},Z)} = \\frac{\\partial}{\\partial v_{t}} (Z + v_{t}), then derive \\sigma_{x}^{Z}{(v_{t},Z)} = 1, then obtain \\frac{\\partial}{\\partial Z} (\\frac{\\partial}{\\partial v_{t}} (Z + v_{t}))^{Z} = \\frac{d}{d Z} 1", "derivation": "\\sigma_{x}{(v_{t},Z)} = \\frac{\\partial}{\\partial v_{t}} (Z + v_{t}) and \\sigma_{x}^{Z}{(v_{t},Z)} = (\\frac{\\partial}{\\partial v_{t}} (Z + v_{t}))^{Z} and \\sigma_{x}^{Z}{(v_{t},Z)} = 1 and (\\frac{\\partial}{\\partial v_{t}} (Z + v_{t}))^{Z} = 1 and \\frac{\\partial}{\\partial Z} (\\frac{\\partial}{\\partial v_{t}} (Z + v_{t}))^{Z} = \\frac{d}{d Z} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Derivative(Add(Symbol('Z', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\sigma_x')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(Add(Symbol('Z', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Integer(1))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Symbol('Z', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(M,c)} = M + \\cos{(c)}, then obtain \\int (\\hat{p}{(M,c)} \\cos{(c)} + \\int \\hat{p}{(M,c)} dc) dM = \\int (\\hat{p}{(M,c)} \\cos{(c)} + \\int (M + \\cos{(c)}) dc) dM", "derivation": "\\hat{p}{(M,c)} = M + \\cos{(c)} and \\hat{p}{(M,c)} \\cos{(c)} = (M + \\cos{(c)}) \\cos{(c)} and \\int \\hat{p}{(M,c)} dc = \\int (M + \\cos{(c)}) dc and (M + \\cos{(c)}) \\cos{(c)} + \\int \\hat{p}{(M,c)} dc = (M + \\cos{(c)}) \\cos{(c)} + \\int (M + \\cos{(c)}) dc and \\hat{p}{(M,c)} \\cos{(c)} + \\int \\hat{p}{(M,c)} dc = \\hat{p}{(M,c)} \\cos{(c)} + \\int (M + \\cos{(c)}) dc and \\int (\\hat{p}{(M,c)} \\cos{(c)} + \\int \\hat{p}{(M,c)} dc) dM = \\int (\\hat{p}{(M,c)} \\cos{(c)} + \\int (M + \\cos{(c)}) dc) dM", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))))"], [["times", 1, "cos(Symbol('c', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Mul(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["add", 3, "Mul(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True))), Integral(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integral(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))))"], [["integrate", 5, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Mul(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integral(Add(Mul(Function('\\\\hat{p}')(Symbol('M', commutative=True), Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), Integral(Add(Symbol('M', commutative=True), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and \\operatorname{n_{2}}{(t)} = \\log{(t)}, then obtain (\\operatorname{n_{2}}{(t)} + e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}})^{t} = (e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}} + \\log{(t)})^{t}", "derivation": "\\varphi^{*}{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and \\operatorname{n_{2}}{(t)} = \\log{(t)} and \\operatorname{n_{2}}{(t)} + e^{\\hat{H}_{\\lambda}} = e^{\\hat{H}_{\\lambda}} + \\log{(t)} and \\operatorname{n_{2}}{(t)} + e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}} = e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}} + \\log{(t)} and (\\operatorname{n_{2}}{(t)} + e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}})^{t} = (e^{\\varphi^{*}{(\\hat{H}_{\\lambda})}} + \\log{(t)})^{t}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], ["get_premise", "Equality(Function('n_2')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["add", 2, "exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('n_2')(Symbol('t', commutative=True)), exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(exp(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('n_2')(Symbol('t', commutative=True)), exp(Function('\\\\varphi^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(exp(Function('\\\\varphi^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(Symbol('t', commutative=True))))"], [["power", 4, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Function('n_2')(Symbol('t', commutative=True)), exp(Function('\\\\varphi^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Symbol('t', commutative=True)), Pow(Add(exp(Function('\\\\varphi^*')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(H)} = \\log{(H)}, then derive \\int \\dot{x}{(H)} dH = H \\log{(H)} - H + t_{1}, then obtain \\frac{\\partial^{2}}{\\partial H^{2}} (H \\log{(H)} - H + t_{1}) = \\frac{d^{2}}{d H^{2}} \\int \\dot{x}{(H)} dH", "derivation": "\\dot{x}{(H)} = \\log{(H)} and \\int \\dot{x}{(H)} dH = \\int \\log{(H)} dH and \\int \\dot{x}{(H)} dH = H \\log{(H)} - H + t_{1} and \\int \\dot{x}{(H)} dH = H \\dot{x}{(H)} - H + t_{1} and H \\log{(H)} - H + t_{1} = H \\dot{x}{(H)} - H + t_{1} and \\frac{\\partial}{\\partial H} (H \\log{(H)} - H + t_{1}) = \\frac{\\partial}{\\partial H} (H \\dot{x}{(H)} - H + t_{1}) and \\frac{\\partial^{2}}{\\partial H^{2}} (H \\log{(H)} - H + t_{1}) = \\frac{\\partial^{2}}{\\partial H^{2}} (H \\dot{x}{(H)} - H + t_{1}) and \\frac{\\partial^{2}}{\\partial H^{2}} (H \\log{(H)} - H + t_{1}) = \\frac{d^{2}}{d H^{2}} \\int \\dot{x}{(H)} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), Function('\\\\dot{x}')(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Add(Mul(Symbol('H', commutative=True), Function('\\\\dot{x}')(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('H', commutative=True), Function('\\\\dot{x}')(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('H', commutative=True), Function('\\\\dot{x}')(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Derivative(Add(Mul(Symbol('H', commutative=True), log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Derivative(Integral(Function('\\\\dot{x}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(2))))"]]}, {"prompt": "Given v{(x,\\mathbf{H})} = - \\mathbf{H} + x, then obtain \\frac{(- \\mathbf{H} + x) v{(x,\\mathbf{H})} - ((- \\mathbf{H} + x)^{2})^{x}}{\\mathbf{H}} = \\frac{(- \\mathbf{H} + x)^{2} - ((- \\mathbf{H} + x)^{2})^{x}}{\\mathbf{H}}", "derivation": "v{(x,\\mathbf{H})} = - \\mathbf{H} + x and (- \\mathbf{H} + x) v{(x,\\mathbf{H})} = (- \\mathbf{H} + x)^{2} and ((- \\mathbf{H} + x) v{(x,\\mathbf{H})})^{x} = ((- \\mathbf{H} + x)^{2})^{x} and - ((- \\mathbf{H} + x) v{(x,\\mathbf{H})})^{x} + (- \\mathbf{H} + x) v{(x,\\mathbf{H})} = - ((- \\mathbf{H} + x) v{(x,\\mathbf{H})})^{x} + (- \\mathbf{H} + x)^{2} and (- \\mathbf{H} + x) v{(x,\\mathbf{H})} - ((- \\mathbf{H} + x)^{2})^{x} = (- \\mathbf{H} + x)^{2} - ((- \\mathbf{H} + x)^{2})^{x} and \\frac{(- \\mathbf{H} + x) v{(x,\\mathbf{H})} - ((- \\mathbf{H} + x)^{2})^{x}}{\\mathbf{H}} = \\frac{(- \\mathbf{H} + x)^{2} - ((- \\mathbf{H} + x)^{2})^{x}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('x', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Symbol('x', commutative=True)))"], [["minus", 2, "Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('x', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('x', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Symbol('x', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Symbol('x', commutative=True)))))"], [["divide", 5, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Function('v')(Symbol('x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Symbol('x', commutative=True))))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x', commutative=True)), Integer(2)), Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\mathbf{E} + \\rho, then obtain \\Psi^{v_{x}} + 2 \\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\Psi^{v_{x}} + 2 \\mathbf{E} + 2 \\rho", "derivation": "\\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\mathbf{E} + \\rho and \\Psi^{v_{x}} + \\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\Psi^{v_{x}} + \\mathbf{E} + \\rho and \\Psi^{v_{x}} + \\mathbf{E} + \\rho + \\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\Psi^{v_{x}} + 2 \\mathbf{E} + 2 \\rho and \\Psi^{v_{x}} + 2 \\Psi_{\\lambda}{(\\mathbf{E},\\rho)} = \\Psi^{v_{x}} + 2 \\mathbf{E} + 2 \\rho", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(V_{\\mathbf{B}},B)} = \\sin{(B + V_{\\mathbf{B}})}, then derive \\int \\dot{y}{(V_{\\mathbf{B}},B)} dV_{\\mathbf{B}} = m_{s} - \\cos{(B + V_{\\mathbf{B}})}, then obtain \\int (- B - V_{\\mathbf{B}} + \\int \\sin{(B + V_{\\mathbf{B}})} dV_{\\mathbf{B}}) dV_{\\mathbf{B}} = \\int (- B - V_{\\mathbf{B}} + m_{s} - \\cos{(B + V_{\\mathbf{B}})}) dV_{\\mathbf{B}}", "derivation": "\\dot{y}{(V_{\\mathbf{B}},B)} = \\sin{(B + V_{\\mathbf{B}})} and \\int \\dot{y}{(V_{\\mathbf{B}},B)} dV_{\\mathbf{B}} = \\int \\sin{(B + V_{\\mathbf{B}})} dV_{\\mathbf{B}} and \\int \\dot{y}{(V_{\\mathbf{B}},B)} dV_{\\mathbf{B}} = m_{s} - \\cos{(B + V_{\\mathbf{B}})} and \\int \\sin{(B + V_{\\mathbf{B}})} dV_{\\mathbf{B}} = m_{s} - \\cos{(B + V_{\\mathbf{B}})} and - B - V_{\\mathbf{B}} + \\int \\sin{(B + V_{\\mathbf{B}})} dV_{\\mathbf{B}} = - B - V_{\\mathbf{B}} + m_{s} - \\cos{(B + V_{\\mathbf{B}})} and \\int (- B - V_{\\mathbf{B}} + \\int \\sin{(B + V_{\\mathbf{B}})} dV_{\\mathbf{B}}) dV_{\\mathbf{B}} = \\int (- B - V_{\\mathbf{B}} + m_{s} - \\cos{(B + V_{\\mathbf{B}})}) dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(sin(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))))"], [["minus", 4, "Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(sin(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))))"], [["integrate", 5, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(sin(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Add(Symbol('B', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(z^{*})} = \\int \\log{(z^{*})} dz^{*}, then derive \\frac{d}{d z^{*}} \\hat{\\mathbf{x}}{(z^{*})} - 1 = \\frac{d}{d z^{*}} (- z^{*} + \\int \\log{(z^{*})} dz^{*}), then obtain \\frac{d}{d z^{*}} \\int \\log{(z^{*})} dz^{*} - 1 = \\frac{d}{d z^{*}} (- z^{*} + \\int \\log{(z^{*})} dz^{*})", "derivation": "\\hat{\\mathbf{x}}{(z^{*})} = \\int \\log{(z^{*})} dz^{*} and - z^{*} + \\hat{\\mathbf{x}}{(z^{*})} = - z^{*} + \\int \\log{(z^{*})} dz^{*} and \\frac{d}{d z^{*}} (- z^{*} + \\hat{\\mathbf{x}}{(z^{*})}) = \\frac{d}{d z^{*}} (- z^{*} + \\int \\log{(z^{*})} dz^{*}) and \\frac{d}{d z^{*}} \\hat{\\mathbf{x}}{(z^{*})} - 1 = \\frac{d}{d z^{*}} (- z^{*} + \\int \\log{(z^{*})} dz^{*}) and \\frac{d}{d z^{*}} \\int \\log{(z^{*})} dz^{*} - 1 = \\frac{d}{d z^{*}} (- z^{*} + \\int \\log{(z^{*})} dz^{*})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('z^*', commutative=True)), Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["minus", 1, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(i)} = \\int \\log{(i)} di and \\operatorname{t_{1}}{(i)} = \\int \\log{(i)} di, then derive z{(i)} = \\mathbf{F} + i \\log{(i)} - i, then obtain \\frac{\\partial}{\\partial i} \\frac{\\mathbf{F} + i \\log{(i)} - i}{\\log{(i)}} = \\frac{d}{d i} \\frac{\\operatorname{t_{1}}{(i)}}{\\log{(i)}}", "derivation": "z{(i)} = \\int \\log{(i)} di and z{(i)} = \\mathbf{F} + i \\log{(i)} - i and \\operatorname{t_{1}}{(i)} = \\int \\log{(i)} di and z{(i)} = \\operatorname{t_{1}}{(i)} and \\frac{z{(i)}}{\\log{(i)}} = \\frac{\\operatorname{t_{1}}{(i)}}{\\log{(i)}} and \\frac{\\mathbf{F} + i \\log{(i)} - i}{\\log{(i)}} = \\frac{\\operatorname{t_{1}}{(i)}}{\\log{(i)}} and \\frac{\\partial}{\\partial i} \\frac{\\mathbf{F} + i \\log{(i)} - i}{\\log{(i)}} = \\frac{d}{d i} \\frac{\\operatorname{t_{1}}{(i)}}{\\log{(i)}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('i', commutative=True)), Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('z')(Symbol('i', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('i', commutative=True), log(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('i', commutative=True)), Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('z')(Symbol('i', commutative=True)), Function('t_1')(Symbol('i', commutative=True)))"], [["divide", 4, "log(Symbol('i', commutative=True))"], "Equality(Mul(Function('z')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Integer(-1))), Mul(Function('t_1')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('i', commutative=True), log(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1))), Mul(Function('t_1')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Integer(-1))))"], [["differentiate", 6, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Symbol('i', commutative=True), log(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Pow(log(Symbol('i', commutative=True)), Integer(-1))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Function('t_1')(Symbol('i', commutative=True)), Pow(log(Symbol('i', commutative=True)), Integer(-1))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{1}{(E,a^{\\dagger})} = \\frac{E}{a^{\\dagger}}, then obtain \\frac{\\partial}{\\partial E} (\\theta_{1}{(E,a^{\\dagger})} + \\int \\theta_{1}{(E,a^{\\dagger})} dE) = \\frac{\\partial}{\\partial E} (\\theta_{1}{(E,a^{\\dagger})} + \\int \\frac{E}{a^{\\dagger}} dE)", "derivation": "\\theta_{1}{(E,a^{\\dagger})} = \\frac{E}{a^{\\dagger}} and \\int \\theta_{1}{(E,a^{\\dagger})} dE = \\int \\frac{E}{a^{\\dagger}} dE and \\frac{E}{a^{\\dagger}} + \\int \\theta_{1}{(E,a^{\\dagger})} dE = \\frac{E}{a^{\\dagger}} + \\int \\frac{E}{a^{\\dagger}} dE and \\theta_{1}{(E,a^{\\dagger})} + \\int \\theta_{1}{(E,a^{\\dagger})} dE = \\theta_{1}{(E,a^{\\dagger})} + \\int \\frac{E}{a^{\\dagger}} dE and \\frac{\\partial}{\\partial E} (\\theta_{1}{(E,a^{\\dagger})} + \\int \\theta_{1}{(E,a^{\\dagger})} dE) = \\frac{\\partial}{\\partial E} (\\theta_{1}{(E,a^{\\dagger})} + \\int \\frac{E}{a^{\\dagger}} dE)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Tuple(Symbol('E', commutative=True))))"], [["add", 2, "Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Integral(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Tuple(Symbol('E', commutative=True)))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Function('\\\\theta_1')(Symbol('E', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Tuple(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(I,y)} = \\log{(I - y)}, then obtain - \\frac{- \\operatorname{A_{y}}{(I,y)} + \\operatorname{A_{y}}^{I}{(I,y)}}{y} = - \\frac{\\operatorname{A_{y}}^{I}{(I,y)} - \\log{(I - y)}}{y}", "derivation": "\\operatorname{A_{y}}{(I,y)} = \\log{(I - y)} and \\operatorname{A_{y}}^{I}{(I,y)} = \\log{(I - y)}^{I} and - \\operatorname{A_{y}}{(I,y)} = - \\log{(I - y)} and - \\operatorname{A_{y}}{(I,y)} + \\log{(I - y)}^{I} = - \\log{(I - y)} + \\log{(I - y)}^{I} and - \\frac{- \\operatorname{A_{y}}{(I,y)} + \\log{(I - y)}^{I}}{y} = - \\frac{- \\log{(I - y)} + \\log{(I - y)}^{I}}{y} and - \\frac{- \\operatorname{A_{y}}{(I,y)} + \\operatorname{A_{y}}^{I}{(I,y)}}{y} = - \\frac{\\operatorname{A_{y}}^{I}{(I,y)} - \\log{(I - y)}}{y}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True)), log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True)), Symbol('I', commutative=True)), Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["add", 3, "Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True))), Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True))), Add(Mul(Integer(-1), log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True))), Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Pow(log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True))), Pow(Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True)), Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Pow(Function('A_y')(Symbol('I', commutative=True), Symbol('y', commutative=True)), Symbol('I', commutative=True)), Mul(Integer(-1), log(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = - \\mathbf{J}_M + \\log{(I)}, then derive - \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = 1, then obtain \\frac{\\partial}{\\partial I} - \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\log{(I)}) = \\frac{d}{d I} 1", "derivation": "\\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = - \\mathbf{J}_M + \\log{(I)} and - \\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = \\mathbf{J}_M - \\log{(I)} and \\frac{\\partial}{\\partial \\mathbf{J}_M} - \\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M - \\log{(I)}) and - \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{c_{0}}{(I,\\mathbf{J}_M)} = 1 and - \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\log{(I)}) = 1 and \\frac{\\partial}{\\partial I} - \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\log{(I)}) = \\frac{d}{d I} 1", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('I', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), log(Symbol('I', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), log(Symbol('I', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Integer(1))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('I', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(s)} = \\sin{(s)} and \\Psi{(s)} = \\sin^{s}{(s)} and \\varphi^{*}{(s)} = \\sin^{s}{(s)}, then obtain \\Psi{(s)} = \\eta^{s}{(s)}", "derivation": "\\eta{(s)} = \\sin{(s)} and \\eta^{s}{(s)} = \\sin^{s}{(s)} and \\Psi{(s)} = \\sin^{s}{(s)} and \\varphi^{*}{(s)} = \\sin^{s}{(s)} and \\varphi^{*}{(s)} = \\eta^{s}{(s)} and \\Psi{(s)} = \\varphi^{*}{(s)} and \\Psi{(s)} = \\eta^{s}{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('s', commutative=True)), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\varphi^*')(Symbol('s', commutative=True)), Pow(Function('\\\\eta')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\Psi')(Symbol('s', commutative=True)), Function('\\\\varphi^*')(Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\Psi')(Symbol('s', commutative=True)), Pow(Function('\\\\eta')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\eta{(n_{1},z)} = n_{1} z and \\theta{(n_{1},z)} = \\frac{\\frac{\\partial}{\\partial n_{1}} n_{1} z}{\\eta{(n_{1},z)}}, then derive \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = - \\frac{z \\frac{\\partial}{\\partial n_{1}} \\eta{(n_{1},z)}}{\\eta^{2}{(n_{1},z)}}, then obtain - \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = \\frac{\\frac{\\partial}{\\partial n_{1}} n_{1} z}{n_{1}^{2} z}", "derivation": "\\eta{(n_{1},z)} = n_{1} z and \\theta{(n_{1},z)} = \\frac{\\frac{\\partial}{\\partial n_{1}} n_{1} z}{\\eta{(n_{1},z)}} and \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = \\frac{\\partial}{\\partial n_{1}} \\frac{\\frac{\\partial}{\\partial n_{1}} n_{1} z}{\\eta{(n_{1},z)}} and \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = - \\frac{z \\frac{\\partial}{\\partial n_{1}} \\eta{(n_{1},z)}}{\\eta^{2}{(n_{1},z)}} and - \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = \\frac{z \\frac{\\partial}{\\partial n_{1}} \\eta{(n_{1},z)}}{\\eta^{2}{(n_{1},z)}} and - \\frac{\\partial}{\\partial n_{1}} \\theta{(n_{1},z)} = \\frac{\\frac{\\partial}{\\partial n_{1}} n_{1} z}{n_{1}^{2} z}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('n_1', commutative=True), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('z', commutative=True), Pow(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(-2)), Derivative(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Symbol('z', commutative=True), Pow(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(-2)), Derivative(Function('\\\\eta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Derivative(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Mul(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(U,m)} = e^{m^{U}} and \\mathbf{J}_P{(U,m)} = 4 e^{2 m^{U}}, then obtain \\mathbf{J}_P{(U,m)} = 4 \\mathbf{J}_M{(U,m)} e^{m^{U}}", "derivation": "\\mathbf{J}_M{(U,m)} = e^{m^{U}} and \\mathbf{J}_M{(U,m)} + e^{m^{U}} = 2 e^{m^{U}} and (\\mathbf{J}_M{(U,m)} + e^{m^{U}})^{2} = 4 e^{2 m^{U}} and \\mathbf{J}_M{(U,m)} e^{m^{U}} = e^{2 m^{U}} and \\mathbf{J}_P{(U,m)} = 4 e^{2 m^{U}} and (\\mathbf{J}_M{(U,m)} + e^{m^{U}})^{2} = 4 \\mathbf{J}_M{(U,m)} e^{m^{U}} and 4 e^{2 m^{U}} = 4 \\mathbf{J}_M{(U,m)} e^{m^{U}} and \\mathbf{J}_P{(U,m)} = 4 \\mathbf{J}_M{(U,m)} e^{m^{U}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True))))"], [["add", 1, "exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))), Mul(Integer(2), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('U', commutative=True))))))"], [["times", 1, "exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))), exp(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Mul(Integer(4), exp(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('U', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))), Integer(2)), Mul(Integer(4), Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(4), exp(Mul(Integer(2), Pow(Symbol('m', commutative=True), Symbol('U', commutative=True))))), Mul(Integer(4), Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Function('\\\\mathbf{J}_P')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Mul(Integer(4), Function('\\\\mathbf{J}_M')(Symbol('U', commutative=True), Symbol('m', commutative=True)), exp(Pow(Symbol('m', commutative=True), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r}, then derive \\frac{\\partial}{\\partial r} u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r}, then obtain e^{\\mathbf{J} + r} \\frac{\\partial}{\\partial \\mathbf{J}} u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r} \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial r} u{(\\mathbf{J},r)}", "derivation": "u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r} and \\frac{\\partial}{\\partial r} u{(\\mathbf{J},r)} = \\frac{\\partial}{\\partial r} e^{\\mathbf{J} + r} and \\frac{\\partial}{\\partial r} u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r} and e^{\\mathbf{J} + r} = \\frac{\\partial}{\\partial r} e^{\\mathbf{J} + r} and \\frac{\\partial}{\\partial \\mathbf{J}} e^{\\mathbf{J} + r} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial r} e^{\\mathbf{J} + r} and \\frac{\\partial}{\\partial \\mathbf{J}} u{(\\mathbf{J},r)} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial r} u{(\\mathbf{J},r)} and e^{\\mathbf{J} + r} \\frac{\\partial}{\\partial \\mathbf{J}} u{(\\mathbf{J},r)} = e^{\\mathbf{J} + r} \\frac{\\partial^{2}}{\\partial \\mathbf{J}\\partial r} u{(\\mathbf{J},r)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Derivative(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["times", 6, "exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)))"], "Equality(Mul(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(exp(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True))), Derivative(Function('u')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\varepsilon_0,F_{H})} = F_{H} \\varepsilon_0, then derive \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{n_{1}}{(\\varepsilon_0,F_{H})} = F_{H}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} F_{H} \\varepsilon_0 = F_{H}", "derivation": "\\operatorname{n_{1}}{(\\varepsilon_0,F_{H})} = F_{H} \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{n_{1}}{(\\varepsilon_0,F_{H})} = \\frac{\\partial}{\\partial \\varepsilon_0} F_{H} \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{n_{1}}{(\\varepsilon_0,F_{H})} = F_{H} and \\frac{\\partial}{\\partial \\varepsilon_0} F_{H} \\varepsilon_0 = F_{H}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('F_H', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Symbol('F_H', commutative=True))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\theta_2,v_{t})} = - \\theta_2 + v_{t} and \\phi_{1}{(\\theta_2)} = - \\theta_2, then obtain \\cos{(\\theta_2 + \\phi_{1}{(\\theta_2)})} = 1", "derivation": "\\operatorname{P_{g}}{(\\theta_2,v_{t})} = - \\theta_2 + v_{t} and \\theta_2 - v_{t} + \\operatorname{P_{g}}{(\\theta_2,v_{t})} = 0 and \\cos{(\\theta_2 - v_{t} + \\operatorname{P_{g}}{(\\theta_2,v_{t})})} = 1 and \\phi_{1}{(\\theta_2)} = - \\theta_2 and \\operatorname{P_{g}}{(\\theta_2,v_{t})} = v_{t} + \\phi_{1}{(\\theta_2)} and \\cos{(\\theta_2 + \\phi_{1}{(\\theta_2)})} = 1", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_t', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))), Integer(0))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('v_t', commutative=True), Function('\\\\phi_1')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(cos(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\theta_2', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\dot{\\mathbf{r}},m_{s})} = \\int (\\dot{\\mathbf{r}} + m_{s}) d\\dot{\\mathbf{r}}, then derive \\operatorname{A_{1}}{(\\dot{\\mathbf{r}},m_{s})} = F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s}, then derive \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} + f = F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s}, then obtain (\\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} + f)^{2} = (F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s})^{2}", "derivation": "\\operatorname{A_{1}}{(\\dot{\\mathbf{r}},m_{s})} = \\int (\\dot{\\mathbf{r}} + m_{s}) d\\dot{\\mathbf{r}} and \\operatorname{A_{1}}{(\\dot{\\mathbf{r}},m_{s})} = F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} and \\int (\\dot{\\mathbf{r}} + m_{s}) d\\dot{\\mathbf{r}} = F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} and \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} + f = F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} and (\\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s} + f)^{2} = (F_{x} + \\frac{\\dot{\\mathbf{r}}^{2}}{2} + \\dot{\\mathbf{r}} m_{s})^{2}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('A_1')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Symbol('f', commutative=True)), Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True)), Symbol('f', commutative=True)), Integer(2)), Pow(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('m_s', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\dot{\\mathbf{r}},v_{x})} = \\dot{\\mathbf{r}} v_{x}, then obtain \\int 0 dv_{x} = \\int (\\frac{(\\dot{\\mathbf{r}} v_{x})^{v_{x}}}{v_{x}} - \\frac{\\varepsilon_{0}^{v_{x}}{(\\dot{\\mathbf{r}},v_{x})}}{v_{x}}) dv_{x}", "derivation": "\\varepsilon_{0}{(\\dot{\\mathbf{r}},v_{x})} = \\dot{\\mathbf{r}} v_{x} and \\varepsilon_{0}^{v_{x}}{(\\dot{\\mathbf{r}},v_{x})} = (\\dot{\\mathbf{r}} v_{x})^{v_{x}} and \\frac{\\varepsilon_{0}^{v_{x}}{(\\dot{\\mathbf{r}},v_{x})}}{v_{x}} = \\frac{(\\dot{\\mathbf{r}} v_{x})^{v_{x}}}{v_{x}} and 0 = \\frac{(\\dot{\\mathbf{r}} v_{x})^{v_{x}}}{v_{x}} - \\frac{\\varepsilon_{0}^{v_{x}}{(\\dot{\\mathbf{r}},v_{x})}}{v_{x}} and \\int 0 dv_{x} = \\int (\\frac{(\\dot{\\mathbf{r}} v_{x})^{v_{x}}}{v_{x}} - \\frac{\\varepsilon_{0}^{v_{x}}{(\\dot{\\mathbf{r}},v_{x})}}{v_{x}}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["divide", 2, "Symbol('v_x', commutative=True)"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given b{(a^{\\dagger})} = e^{a^{\\dagger}}, then derive \\frac{d}{d a^{\\dagger}} b{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain b{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} b{(a^{\\dagger})}", "derivation": "b{(a^{\\dagger})} = e^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} b{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} b{(a^{\\dagger})} = e^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} b{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} b{(a^{\\dagger})} and \\frac{d^{2}}{d (a^{\\dagger})^{2}} b{(a^{\\dagger})} = e^{a^{\\dagger}} and b{(a^{\\dagger})} = \\frac{d^{2}}{d (a^{\\dagger})^{2}} b{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('b')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(F_{g})} = F_{g} and \\mu{(F_{g})} = F_{g} \\operatorname{n_{2}}{(F_{g})}, then obtain F_{g}^{2} = F_{g} \\operatorname{n_{2}}{(F_{g})}", "derivation": "\\operatorname{n_{2}}{(F_{g})} = F_{g} and \\mu{(F_{g})} = F_{g} \\operatorname{n_{2}}{(F_{g})} and \\mu{(F_{g})} = F_{g}^{2} and F_{g}^{2} = F_{g} \\operatorname{n_{2}}{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('F_g', commutative=True)), Mul(Symbol('F_g', commutative=True), Function('n_2')(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mu')(Symbol('F_g', commutative=True)), Pow(Symbol('F_g', commutative=True), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('F_g', commutative=True), Integer(2)), Mul(Symbol('F_g', commutative=True), Function('n_2')(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{2},\\mathbb{I})} = C_{2} \\mathbb{I}, then obtain (e^{\\int (- C_{2} \\mathbb{I} + \\operatorname{E_{n}}{(C_{2},\\mathbb{I})}) dC_{2}})^{C_{2}} = 1", "derivation": "\\operatorname{E_{n}}{(C_{2},\\mathbb{I})} = C_{2} \\mathbb{I} and 0 = C_{2} \\mathbb{I} - \\operatorname{E_{n}}{(C_{2},\\mathbb{I})} and - C_{2} \\mathbb{I} + \\operatorname{E_{n}}{(C_{2},\\mathbb{I})} = 0 and \\int (- C_{2} \\mathbb{I} + \\operatorname{E_{n}}{(C_{2},\\mathbb{I})}) dC_{2} = \\int 0 dC_{2} and e^{\\int (- C_{2} \\mathbb{I} + \\operatorname{E_{n}}{(C_{2},\\mathbb{I})}) dC_{2}} = 1 and (e^{\\int (- C_{2} \\mathbb{I} + \\operatorname{E_{n}}{(C_{2},\\mathbb{I})}) dC_{2}})^{C_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], [["minus", 2, "Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('C_2', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Integer(1))"], [["power", 5, "Symbol('C_2', commutative=True)"], "Equality(Pow(exp(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('E_n')(Symbol('C_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('C_2', commutative=True)))), Symbol('C_2', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{B}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}, then derive \\sigma_x \\mathbf{B}{(\\sigma_x)} = 1, then obtain (- \\sigma_x \\mathbf{B}{(\\sigma_x)} + 1) \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} = (- 2 \\sigma_x \\mathbf{B}{(\\sigma_x)} + 2) \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}", "derivation": "\\mathbf{B}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} and \\frac{\\mathbf{B}{(\\sigma_x)}}{\\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}} = 1 and \\sigma_x \\mathbf{B}{(\\sigma_x)} = 1 and \\sigma_x \\mathbf{B}{(\\sigma_x)} - 1 = 0 and 2 \\sigma_x \\mathbf{B}{(\\sigma_x)} - 2 = \\sigma_x \\mathbf{B}{(\\sigma_x)} - 1 and - \\sigma_x \\mathbf{B}{(\\sigma_x)} + 1 = - 2 \\sigma_x \\mathbf{B}{(\\sigma_x)} + 2 and (- \\sigma_x \\mathbf{B}{(\\sigma_x)} + 1) \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} = (- 2 \\sigma_x \\mathbf{B}{(\\sigma_x)} + 2) \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True)), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True)), Pow(Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(1))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Integer(0))"], [["add", 4, "Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(-2)), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)))"], [["minus", 5, "Add(Mul(Integer(3), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(-3))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(2)))"], [["divide", 6, "Pow(Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(1)), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\sigma_x', commutative=True))), Integer(2)), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{H})} = \\sin{(\\sin{(\\mathbf{H})})} and q{(\\mathbf{H})} = \\rho_{f}{(\\mathbf{H})} + \\sin{(\\sin{(\\mathbf{H})})}, then obtain \\log{(\\rho_{f}{(\\mathbf{H})} + \\sin{(\\sin{(\\mathbf{H})})})} = \\log{(q{(\\mathbf{H})})}", "derivation": "\\rho_{f}{(\\mathbf{H})} = \\sin{(\\sin{(\\mathbf{H})})} and 2 \\rho_{f}{(\\mathbf{H})} = \\rho_{f}{(\\mathbf{H})} + \\sin{(\\sin{(\\mathbf{H})})} and q{(\\mathbf{H})} = \\rho_{f}{(\\mathbf{H})} + \\sin{(\\sin{(\\mathbf{H})})} and 2 \\rho_{f}{(\\mathbf{H})} = q{(\\mathbf{H})} and \\log{(2 \\rho_{f}{(\\mathbf{H})})} = \\log{(q{(\\mathbf{H})})} and \\log{(\\rho_{f}{(\\mathbf{H})} + \\sin{(\\sin{(\\mathbf{H})})})} = \\log{(q{(\\mathbf{H})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)), sin(sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)), sin(sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\mathbf{H}', commutative=True)), Add(Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)), sin(sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True))), Function('q')(Symbol('\\\\mathbf{H}', commutative=True)))"], [["log", 4], "Equality(log(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)))), log(Function('q')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(log(Add(Function('\\\\rho_f')(Symbol('\\\\mathbf{H}', commutative=True)), sin(sin(Symbol('\\\\mathbf{H}', commutative=True))))), log(Function('q')(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{M})} = \\sin{(\\sin{(\\mathbf{M})})} and \\bar{\\h}{(\\mathbf{M})} = \\sin{(\\sin{(\\mathbf{M})})}, then obtain \\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{M})} = \\sin{(\\sin{(\\mathbf{M})})} and \\bar{\\h}{(\\mathbf{M})} = \\sin{(\\sin{(\\mathbf{M})})} and \\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\sin{(\\mathbf{M})})} and \\frac{d}{d \\mathbf{M}} \\operatorname{A_{z}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\bar{\\h}{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), sin(sin(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), sin(sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('A_z')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(u)} = \\log{(u)} and \\operatorname{E_{n}}{(u)} = \\frac{d}{d u} \\operatorname{C_{d}}{(u)} \\log{(u)}, then obtain \\operatorname{E_{n}}{(u)} e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} = e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} \\frac{d}{d u} \\log{(u)}^{2}", "derivation": "\\operatorname{C_{d}}{(u)} = \\log{(u)} and \\operatorname{C_{d}}{(u)} \\log{(u)} = \\log{(u)}^{2} and \\frac{d}{d u} \\operatorname{C_{d}}{(u)} \\log{(u)} = \\frac{d}{d u} \\log{(u)}^{2} and \\operatorname{E_{n}}{(u)} = \\frac{d}{d u} \\operatorname{C_{d}}{(u)} \\log{(u)} and \\operatorname{E_{n}}{(u)} e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} = e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} \\frac{d}{d u} \\operatorname{C_{d}}{(u)} \\log{(u)} and \\operatorname{E_{n}}{(u)} e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} = e^{- \\operatorname{C_{d}}{(u)} \\log{(u)} - \\log{(u)}^{2}} \\frac{d}{d u} \\log{(u)}^{2}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["times", 1, "log(Symbol('u', commutative=True))"], "Equality(Mul(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(log(Symbol('u', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('u', commutative=True)), Integer(2)), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('u', commutative=True)), Derivative(Mul(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 4, "exp(Add(Mul(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Pow(log(Symbol('u', commutative=True)), Integer(2))))"], "Equality(Mul(Function('E_n')(Symbol('u', commutative=True)), exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2)))))), Mul(exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2))))), Derivative(Mul(Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Function('E_n')(Symbol('u', commutative=True)), exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2)))))), Mul(exp(Add(Mul(Integer(-1), Function('C_d')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('u', commutative=True)), Integer(2))))), Derivative(Pow(log(Symbol('u', commutative=True)), Integer(2)), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(x^\\prime)} = \\sin{(x^\\prime)}, then obtain 0^{x^\\prime} + (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime - 1 = (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime", "derivation": "\\dot{\\mathbf{r}}{(x^\\prime)} = \\sin{(x^\\prime)} and 0 = - \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)} and 0^{x^\\prime} = (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} and 0^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime = (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime and (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime = \\int 0^{x^\\prime} dx^\\prime + 1 and 0^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime = \\int 0^{x^\\prime} dx^\\prime + 1 and 0^{x^\\prime} + (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime - 1 = (- \\dot{\\mathbf{r}}{(x^\\prime)} + \\sin{(x^\\prime)})^{x^\\prime} + \\int 0^{x^\\prime} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 3, "Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Integral(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given B{(Z)} = \\log{(Z)}, then derive 0 = - \\frac{d}{d Z} B{(Z)} + \\frac{1}{Z}, then obtain \\frac{d}{d Z} \\int 0 dZ = \\frac{d}{d Z} \\int (- \\frac{d}{d Z} B{(Z)} + \\frac{1}{Z}) dZ", "derivation": "B{(Z)} = \\log{(Z)} and B{(Z)} + \\log{(Z)} = 2 \\log{(Z)} and 0 = - B{(Z)} + \\log{(Z)} and \\frac{d}{d Z} 0 = \\frac{d}{d Z} (- B{(Z)} + \\log{(Z)}) and 0 = - \\frac{d}{d Z} B{(Z)} + \\frac{1}{Z} and \\int 0 dZ = \\int (- \\frac{d}{d Z} B{(Z)} + \\frac{1}{Z}) dZ and \\frac{d}{d Z} \\int 0 dZ = \\frac{d}{d Z} \\int (- \\frac{d}{d Z} B{(Z)} + \\frac{1}{Z}) dZ", "srepr_derivation": [["get_premise", "Equality(Function('B')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["add", 1, "log(Symbol('Z', commutative=True))"], "Equality(Add(Function('B')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Mul(Integer(2), log(Symbol('Z', commutative=True))))"], [["minus", 2, "Add(Function('B')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('B')(Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('B')(Symbol('Z', commutative=True))), log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('B')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Function('B')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Pow(Symbol('Z', commutative=True), Integer(-1))), Tuple(Symbol('Z', commutative=True))))"], [["differentiate", 6, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Derivative(Function('B')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Pow(Symbol('Z', commutative=True), Integer(-1))), Tuple(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\nabla,\\dot{z})} = \\dot{z} + \\nabla, then derive 2 \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} + 1, then obtain 2 \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) + 1", "derivation": "\\mathbf{E}{(\\nabla,\\dot{z})} = \\dot{z} + \\nabla and \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) and 2 \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) + \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} and 2 \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{E}{(\\nabla,\\dot{z})} + 1 and 2 \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) = \\frac{\\partial}{\\partial \\dot{z}} (\\dot{z} + \\nabla) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\dot{y}{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain (\\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\dot{y}{(\\sigma_p)}))^{\\sigma_p} = (\\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}))^{\\sigma_p}", "derivation": "\\dot{y}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} \\dot{y}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\dot{y}{(\\sigma_p)} = \\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\dot{y}{(\\sigma_p)}) = \\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}) and (\\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\dot{y}{(\\sigma_p)}))^{\\sigma_p} = (\\frac{d}{d \\sigma_p} (\\dot{y}{(\\sigma_p)} + \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)}))^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Add(Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{E})} = \\mathbf{E}, then derive \\int \\operatorname{M_{E}}{(\\mathbf{E})} d\\mathbf{E} = \\frac{\\mathbf{E}^{2}}{2} + n_{2}, then obtain \\frac{5 \\mathbf{E}^{2}}{2} + \\int \\mathbf{E} d\\mathbf{E} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})} = 3 \\mathbf{E}^{2} + n_{2} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{E})} = \\mathbf{E} and \\int \\operatorname{M_{E}}{(\\mathbf{E})} d\\mathbf{E} = \\int \\mathbf{E} d\\mathbf{E} and \\int \\operatorname{M_{E}}{(\\mathbf{E})} d\\mathbf{E} = \\frac{\\mathbf{E}^{2}}{2} + n_{2} and \\int \\mathbf{E} d\\mathbf{E} = \\frac{\\mathbf{E}^{2}}{2} + n_{2} and \\mathbf{E}^{2} + \\int \\mathbf{E} d\\mathbf{E} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})} = \\frac{3 \\mathbf{E}^{2}}{2} + n_{2} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})} and \\frac{5 \\mathbf{E}^{2}}{2} + \\int \\mathbf{E} d\\mathbf{E} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})} = 3 \\mathbf{E}^{2} + n_{2} + \\int \\mathbf{E} d\\operatorname{M_{E}}{(\\mathbf{E})}", "srepr_derivation": [["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('n_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('n_2', commutative=True)))"], [["add", 4, "Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True)))))"], "Equality(Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True))))), Add(Mul(Rational(3, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('n_2', commutative=True), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["add", 5, "Mul(Rational(3, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)))"], "Equality(Add(Mul(Rational(5, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True))))), Add(Mul(Integer(3), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('n_2', commutative=True), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True))))))"]]}, {"prompt": "Given x{(x^\\prime)} = \\log{(x^\\prime)}, then derive \\frac{d}{d x^\\prime} x{(x^\\prime)} = \\frac{1}{x^\\prime}, then obtain \\frac{d^{2}}{d (x^\\prime)^{2}} x{(x^\\prime)} + \\frac{1}{x^\\prime} = \\frac{d}{d x^\\prime} \\frac{1}{x^\\prime} + \\frac{1}{x^\\prime}", "derivation": "x{(x^\\prime)} = \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} x{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} x{(x^\\prime)} = \\frac{1}{x^\\prime} and \\frac{d^{2}}{d (x^\\prime)^{2}} x{(x^\\prime)} = \\frac{d^{2}}{d (x^\\prime)^{2}} \\log{(x^\\prime)} and \\frac{d^{2}}{d (x^\\prime)^{2}} x{(x^\\prime)} + \\frac{d}{d x^\\prime} \\log{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} + \\frac{d^{2}}{d (x^\\prime)^{2}} \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} \\log{(x^\\prime)} = \\frac{1}{x^\\prime} and \\frac{d^{2}}{d (x^\\prime)^{2}} x{(x^\\prime)} + \\frac{1}{x^\\prime} = \\frac{d}{d x^\\prime} \\frac{1}{x^\\prime} + \\frac{1}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["add", 4, "Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Derivative(Function('x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Add(Derivative(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\rho{(c,F_{N},z^{*})} = F_{N} (- c + z^{*}), then obtain (- c + z^{*}) e^{\\int (z^{*} + \\rho{(c,F_{N},z^{*})}) dc} = (- c + z^{*}) e^{\\int (F_{N} (- c + z^{*}) + z^{*}) dc}", "derivation": "\\rho{(c,F_{N},z^{*})} = F_{N} (- c + z^{*}) and z^{*} + \\rho{(c,F_{N},z^{*})} = F_{N} (- c + z^{*}) + z^{*} and \\int (z^{*} + \\rho{(c,F_{N},z^{*})}) dc = \\int (F_{N} (- c + z^{*}) + z^{*}) dc and e^{\\int (z^{*} + \\rho{(c,F_{N},z^{*})}) dc} = e^{\\int (F_{N} (- c + z^{*}) + z^{*}) dc} and (- c + z^{*}) e^{\\int (z^{*} + \\rho{(c,F_{N},z^{*})}) dc} = (- c + z^{*}) e^{\\int (F_{N} (- c + z^{*}) + z^{*}) dc}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('c', commutative=True), Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))))"], [["add", 1, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Function('\\\\rho')(Symbol('c', commutative=True), Symbol('F_N', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Symbol('z^*', commutative=True), Function('\\\\rho')(Symbol('c', commutative=True), Symbol('F_N', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Add(Symbol('z^*', commutative=True), Function('\\\\rho')(Symbol('c', commutative=True), Symbol('F_N', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('c', commutative=True)))), exp(Integral(Add(Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True)), exp(Integral(Add(Symbol('z^*', commutative=True), Function('\\\\rho')(Symbol('c', commutative=True), Symbol('F_N', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('c', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True)), exp(Integral(Add(Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('z^*', commutative=True))), Symbol('z^*', commutative=True)), Tuple(Symbol('c', commutative=True))))))"]]}, {"prompt": "Given Q{(v_{1},M_{E})} = \\sin{(v_{1}^{M_{E}})} and \\operatorname{v_{z}}{(v_{1},M_{E})} = Q{(v_{1},M_{E})} - \\sin{(v_{1}^{M_{E}})}, then obtain \\operatorname{v_{z}}^{v_{1}}{(v_{1},M_{E})} = 0^{v_{1}}", "derivation": "Q{(v_{1},M_{E})} = \\sin{(v_{1}^{M_{E}})} and Q{(v_{1},M_{E})} - \\sin{(v_{1}^{M_{E}})} = 0 and (Q{(v_{1},M_{E})} - \\sin{(v_{1}^{M_{E}})})^{v_{1}} = 0^{v_{1}} and \\operatorname{v_{z}}{(v_{1},M_{E})} = Q{(v_{1},M_{E})} - \\sin{(v_{1}^{M_{E}})} and \\operatorname{v_{z}}^{v_{1}}{(v_{1},M_{E})} = 0^{v_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), sin(Pow(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True))))"], [["minus", 1, "sin(Pow(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Add(Function('Q')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Function('Q')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True))))), Symbol('v_1', commutative=True)), Pow(Integer(0), Symbol('v_1', commutative=True)))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), Add(Function('Q')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Pow(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('v_z')(Symbol('v_1', commutative=True), Symbol('M_E', commutative=True)), Symbol('v_1', commutative=True)), Pow(Integer(0), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\lambda,A_{1})} = \\lambda^{A_{1}} and \\mathbb{I}{(\\lambda,A_{1})} = \\lambda^{A_{1}} + \\varepsilon_{0}{(\\lambda,A_{1})}, then obtain \\mathbb{I}{(\\lambda,A_{1})} = 2 \\lambda^{A_{1}}", "derivation": "\\varepsilon_{0}{(\\lambda,A_{1})} = \\lambda^{A_{1}} and \\lambda^{A_{1}} + \\varepsilon_{0}{(\\lambda,A_{1})} = 2 \\lambda^{A_{1}} and \\mathbb{I}{(\\lambda,A_{1})} = \\lambda^{A_{1}} + \\varepsilon_{0}{(\\lambda,A_{1})} and \\mathbb{I}{(\\lambda,A_{1})} = 2 \\lambda^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)), Add(Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\lambda', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(L)} = e^{L} and \\varepsilon{(L)} = \\frac{e^{L}}{\\operatorname{C_{d}}{(L)}}, then obtain \\cos{(1)} = \\cos{(\\varepsilon{(L)})}", "derivation": "\\operatorname{C_{d}}{(L)} = e^{L} and 1 = \\frac{e^{L}}{\\operatorname{C_{d}}{(L)}} and \\varepsilon{(L)} = \\frac{e^{L}}{\\operatorname{C_{d}}{(L)}} and 1 = \\varepsilon{(L)} and \\cos{(1)} = \\cos{(\\varepsilon{(L)})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["divide", 1, "Function('C_d')(Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_d')(Symbol('L', commutative=True)), Integer(-1)), exp(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('L', commutative=True)), Mul(Pow(Function('C_d')(Symbol('L', commutative=True)), Integer(-1)), exp(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Function('\\\\varepsilon')(Symbol('L', commutative=True)))"], [["cos", 4], "Equality(cos(Integer(1)), cos(Function('\\\\varepsilon')(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\lambda,\\dot{y})} = \\frac{\\log{(\\lambda)}}{\\dot{y}}, then obtain 1 = (((\\int \\frac{\\log{(\\lambda)}}{\\dot{y}} d\\dot{y})^{\\dot{y}}) (\\int \\theta{(\\lambda,\\dot{y})} d\\dot{y})^{- \\dot{y}})^{\\lambda}", "derivation": "\\theta{(\\lambda,\\dot{y})} = \\frac{\\log{(\\lambda)}}{\\dot{y}} and \\int \\theta{(\\lambda,\\dot{y})} d\\dot{y} = \\int \\frac{\\log{(\\lambda)}}{\\dot{y}} d\\dot{y} and (\\int \\theta{(\\lambda,\\dot{y})} d\\dot{y})^{\\dot{y}} = (\\int \\frac{\\log{(\\lambda)}}{\\dot{y}} d\\dot{y})^{\\dot{y}} and 1 = ((\\int \\frac{\\log{(\\lambda)}}{\\dot{y}} d\\dot{y})^{\\dot{y}}) (\\int \\theta{(\\lambda,\\dot{y})} d\\dot{y})^{- \\dot{y}} and 1 = (((\\int \\frac{\\log{(\\lambda)}}{\\dot{y}} d\\dot{y})^{\\dot{y}}) (\\int \\theta{(\\lambda,\\dot{y})} d\\dot{y})^{- \\dot{y}})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 3, "Pow(Integral(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integral(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))))"], [["power", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integral(Function('\\\\theta')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given t{(V_{\\mathbf{E}},B)} = e^{\\frac{B}{V_{\\mathbf{E}}}}, then obtain \\int \\frac{- B + t{(V_{\\mathbf{E}},B)}}{V_{\\mathbf{E}}} dB = \\int \\frac{- B + e^{\\frac{B}{V_{\\mathbf{E}}}}}{V_{\\mathbf{E}}} dB", "derivation": "t{(V_{\\mathbf{E}},B)} = e^{\\frac{B}{V_{\\mathbf{E}}}} and - B + t{(V_{\\mathbf{E}},B)} = - B + e^{\\frac{B}{V_{\\mathbf{E}}}} and \\frac{- B + t{(V_{\\mathbf{E}},B)}}{V_{\\mathbf{E}}} = \\frac{- B + e^{\\frac{B}{V_{\\mathbf{E}}}}}{V_{\\mathbf{E}}} and \\int \\frac{- B + t{(V_{\\mathbf{E}},B)}}{V_{\\mathbf{E}}} dB = \\int \\frac{- B + e^{\\frac{B}{V_{\\mathbf{E}}}}}{V_{\\mathbf{E}}} dB", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('t')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1))))))"], [["divide", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('t')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)))))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('t')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True))), Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)))))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\pi,\\eta^{\\prime})} = \\frac{\\pi}{\\eta^{\\prime}}, then obtain \\int \\log{(\\pi \\phi_{2}{(\\pi,\\eta^{\\prime})})} d\\eta^{\\prime} = \\int \\log{(\\frac{\\pi^{2}}{\\eta^{\\prime}})} d\\eta^{\\prime}", "derivation": "\\phi_{2}{(\\pi,\\eta^{\\prime})} = \\frac{\\pi}{\\eta^{\\prime}} and \\pi \\phi_{2}{(\\pi,\\eta^{\\prime})} = \\frac{\\pi^{2}}{\\eta^{\\prime}} and \\log{(\\pi \\phi_{2}{(\\pi,\\eta^{\\prime})})} = \\log{(\\frac{\\pi^{2}}{\\eta^{\\prime}})} and \\int \\log{(\\pi \\phi_{2}{(\\pi,\\eta^{\\prime})})} d\\eta^{\\prime} = \\int \\log{(\\frac{\\pi^{2}}{\\eta^{\\prime}})} d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)))"], [["times", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(2))))"], [["log", 2], "Equality(log(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), log(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(log(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(2)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)} = \\log{(\\eta - \\sigma_p)}, then obtain \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)}}{\\eta} d\\eta = \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\log{(\\eta - \\sigma_p)}}{\\eta} d\\eta", "derivation": "\\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)} = \\log{(\\eta - \\sigma_p)} and \\frac{\\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)}}{\\eta} = \\frac{\\log{(\\eta - \\sigma_p)}}{\\eta} and \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)}}{\\eta} = \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\log{(\\eta - \\sigma_p)}}{\\eta} and \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\operatorname{y^{\\prime}}{(\\eta,\\sigma_p)}}{\\eta} d\\eta = \\int \\frac{\\partial}{\\partial \\sigma_p} \\frac{\\log{(\\eta - \\sigma_p)}}{\\eta} d\\eta", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))))"], [["divide", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mu{(n_{2})} = \\cos{(\\cos{(n_{2})})} and \\mathbf{s}{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})}, then obtain \\mathbf{s}^{2}{(\\mathbb{I})} + \\mathbf{s}{(\\mathbb{I})} + \\mu{(n_{2})} = \\mathbf{s}^{2}{(\\mathbb{I})} + \\mathbf{s}{(\\mathbb{I})} + \\cos{(\\cos{(n_{2})})}", "derivation": "\\mu{(n_{2})} = \\cos{(\\cos{(n_{2})})} and \\mathbf{s}{(\\mathbb{I})} = \\log{(\\log{(\\mathbb{I})})} and \\mu{(n_{2})} + \\log{(\\log{(\\mathbb{I})})} = \\log{(\\log{(\\mathbb{I})})} + \\cos{(\\cos{(n_{2})})} and \\mathbf{s}^{2}{(\\mathbb{I})} + \\mu{(n_{2})} + \\log{(\\log{(\\mathbb{I})})} = \\mathbf{s}^{2}{(\\mathbb{I})} + \\log{(\\log{(\\mathbb{I})})} + \\cos{(\\cos{(n_{2})})} and \\mathbf{s}^{2}{(\\mathbb{I})} + \\mathbf{s}{(\\mathbb{I})} + \\mu{(n_{2})} = \\mathbf{s}^{2}{(\\mathbb{I})} + \\mathbf{s}{(\\mathbb{I})} + \\cos{(\\cos{(n_{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('n_2', commutative=True)), cos(cos(Symbol('n_2', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), log(log(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 1, "log(log(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Function('\\\\mu')(Symbol('n_2', commutative=True)), log(log(Symbol('\\\\mathbb{I}', commutative=True)))), Add(log(log(Symbol('\\\\mathbb{I}', commutative=True))), cos(cos(Symbol('n_2', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))"], "Equality(Add(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Function('\\\\mu')(Symbol('n_2', commutative=True)), log(log(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), log(log(Symbol('\\\\mathbb{I}', commutative=True))), cos(cos(Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mu')(Symbol('n_2', commutative=True))), Add(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(cos(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(c_{0})} = \\cos{(c_{0})}, then obtain \\sigma_{x}{(c_{0})} = \\frac{\\cos^{3}{(c_{0})}}{\\sigma_{x}^{2}{(c_{0})}}", "derivation": "\\sigma_{x}{(c_{0})} = \\cos{(c_{0})} and \\sigma_{x}{(c_{0})} \\cos{(c_{0})} = \\cos^{2}{(c_{0})} and \\cos{(c_{0})} = \\frac{\\cos^{2}{(c_{0})}}{\\sigma_{x}{(c_{0})}} and \\sigma_{x}{(c_{0})} = \\frac{\\cos^{2}{(c_{0})}}{\\sigma_{x}{(c_{0})}} and \\frac{\\cos^{3}{(c_{0})}}{\\sigma_{x}{(c_{0})}} = \\cos^{2}{(c_{0})} and \\cos{(c_{0})} = \\frac{\\cos^{3}{(c_{0})}}{\\sigma_{x}^{2}{(c_{0})}} and \\sigma_{x}{(c_{0})} = \\frac{\\cos^{3}{(c_{0})}}{\\sigma_{x}^{2}{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["times", 1, "cos(Symbol('c_0', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True))), Pow(cos(Symbol('c_0', commutative=True)), Integer(2)))"], [["divide", 2, "Function('\\\\sigma_x')(Symbol('c_0', commutative=True))"], "Equality(cos(Symbol('c_0', commutative=True)), Mul(Pow(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(cos(Symbol('c_0', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Pow(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(cos(Symbol('c_0', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(cos(Symbol('c_0', commutative=True)), Integer(3))), Pow(cos(Symbol('c_0', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(cos(Symbol('c_0', commutative=True)), Mul(Pow(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Integer(-2)), Pow(cos(Symbol('c_0', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 1, 6], "Equality(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Pow(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Integer(-2)), Pow(cos(Symbol('c_0', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(r)} = e^{r} and L{(r)} = e^{r}, then obtain e^{- r} \\int \\operatorname{g_{\\varepsilon}}{(r)} dr = e^{- r} \\int e^{r} dr", "derivation": "\\operatorname{g_{\\varepsilon}}{(r)} = e^{r} and L{(r)} = e^{r} and L{(r)} = \\operatorname{g_{\\varepsilon}}{(r)} and \\int L{(r)} dr = \\int e^{r} dr and \\int \\operatorname{g_{\\varepsilon}}{(r)} dr = \\int e^{r} dr and e^{- r} \\int \\operatorname{g_{\\varepsilon}}{(r)} dr = e^{- r} \\int e^{r} dr", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('L')(Symbol('r', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Function('L')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["divide", 5, "exp(Symbol('r', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('r', commutative=True))), Integral(Function('g_{\\\\varepsilon}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('r', commutative=True))), Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{S})} = \\sin{(\\sin{(\\mathbf{S})})}, then obtain 16 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})})^{5} \\sin^{4}{(\\sin{(\\mathbf{S})})} = 256 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})}) \\sin^{8}{(\\sin{(\\mathbf{S})})}", "derivation": "\\Omega{(\\mathbf{S})} = \\sin{(\\sin{(\\mathbf{S})})} and \\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})} = 2 \\sin{(\\sin{(\\mathbf{S})})} and 2 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})}) \\sin{(\\sin{(\\mathbf{S})})} = 4 \\sin^{2}{(\\sin{(\\mathbf{S})})} and 16 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})})^{4} \\sin^{4}{(\\sin{(\\mathbf{S})})} = 256 \\sin^{8}{(\\sin{(\\mathbf{S})})} and 16 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})})^{5} \\sin^{4}{(\\sin{(\\mathbf{S})})} = 256 (\\Omega{(\\mathbf{S})} + \\sin{(\\sin{(\\mathbf{S})})}) \\sin^{8}{(\\sin{(\\mathbf{S})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(2), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["times", 2, "Mul(Integer(2), sin(sin(Symbol('\\\\mathbf{S}', commutative=True))))"], "Equality(Mul(Integer(2), Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Integer(4), Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(2))))"], [["power", 3, 4], "Equality(Mul(Integer(16), Pow(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(4)), Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(4))), Mul(Integer(256), Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(8))))"], [["times", 4, "Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True))))"], "Equality(Mul(Integer(16), Pow(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(5)), Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(4))), Mul(Integer(256), Add(Function('\\\\Omega')(Symbol('\\\\mathbf{S}', commutative=True)), sin(sin(Symbol('\\\\mathbf{S}', commutative=True)))), Pow(sin(sin(Symbol('\\\\mathbf{S}', commutative=True))), Integer(8))))"]]}, {"prompt": "Given \\mathbf{s}{(\\chi,J_{\\varepsilon})} = \\chi^{J_{\\varepsilon}}, then obtain (\\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}} + \\frac{\\partial}{\\partial \\chi} \\mathbf{s}{(\\chi,J_{\\varepsilon})})^{2} = 4 (\\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}})^{2}", "derivation": "\\mathbf{s}{(\\chi,J_{\\varepsilon})} = \\chi^{J_{\\varepsilon}} and \\frac{\\partial}{\\partial \\chi} \\mathbf{s}{(\\chi,J_{\\varepsilon})} = \\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}} and \\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}} + \\frac{\\partial}{\\partial \\chi} \\mathbf{s}{(\\chi,J_{\\varepsilon})} = 2 \\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}} and (\\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}} + \\frac{\\partial}{\\partial \\chi} \\mathbf{s}{(\\chi,J_{\\varepsilon})})^{2} = 4 (\\frac{\\partial}{\\partial \\chi} \\chi^{J_{\\varepsilon}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["power", 3, 2], "Equality(Pow(Add(Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(4), Pow(Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(t)} = \\log{(\\cos{(t)})} and \\Psi_{\\lambda}{(\\theta_1)} = \\log{(e^{\\theta_1})}, then obtain \\operatorname{A_{1}}{(t)} + \\Psi_{\\lambda}{(\\theta_1)} = \\operatorname{A_{1}}{(t)} + \\log{(e^{\\theta_1})}", "derivation": "\\operatorname{A_{1}}{(t)} = \\log{(\\cos{(t)})} and \\Psi_{\\lambda}{(\\theta_1)} = \\log{(e^{\\theta_1})} and \\Psi_{\\lambda}{(\\theta_1)} + \\log{(\\cos{(t)})} = \\log{(e^{\\theta_1})} + \\log{(\\cos{(t)})} and \\operatorname{A_{1}}{(t)} + \\Psi_{\\lambda}{(\\theta_1)} = \\operatorname{A_{1}}{(t)} + \\log{(e^{\\theta_1})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('t', commutative=True)), log(cos(Symbol('t', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True))))"], [["add", 2, "log(cos(Symbol('t', commutative=True)))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), log(cos(Symbol('t', commutative=True)))), Add(log(exp(Symbol('\\\\theta_1', commutative=True))), log(cos(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A_1')(Symbol('t', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True))), Add(Function('A_1')(Symbol('t', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{p},L)} = \\mathbf{p}^{L}, then derive \\frac{\\partial}{\\partial L} \\hat{H}{(\\mathbf{p},L)} = \\mathbf{p}^{L} \\log{(\\mathbf{p})}, then obtain \\frac{\\partial}{\\partial L} \\hat{H}{(\\mathbf{p},L)} = \\hat{H}{(\\mathbf{p},L)} \\log{(\\mathbf{p})}", "derivation": "\\hat{H}{(\\mathbf{p},L)} = \\mathbf{p}^{L} and \\frac{\\partial}{\\partial L} \\hat{H}{(\\mathbf{p},L)} = \\frac{\\partial}{\\partial L} \\mathbf{p}^{L} and \\frac{\\partial}{\\partial L} \\hat{H}{(\\mathbf{p},L)} = \\mathbf{p}^{L} \\log{(\\mathbf{p})} and \\frac{\\partial}{\\partial L} \\hat{H}{(\\mathbf{p},L)} = \\hat{H}{(\\mathbf{p},L)} \\log{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Function('\\\\hat{H}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(z)} = e^{z} and S{(z)} = \\frac{z + \\mathbf{p}{(z)}}{z + e^{z}}, then obtain S{(z)} = 1", "derivation": "\\mathbf{p}{(z)} = e^{z} and z + \\mathbf{p}{(z)} = z + e^{z} and \\frac{z + \\mathbf{p}{(z)}}{z + e^{z}} = 1 and S{(z)} = \\frac{z + \\mathbf{p}{(z)}}{z + e^{z}} and S{(z)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["add", 1, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('z', commutative=True))), Add(Symbol('z', commutative=True), exp(Symbol('z', commutative=True))))"], [["divide", 2, "Add(Symbol('z', commutative=True), exp(Symbol('z', commutative=True)))"], "Equality(Mul(Add(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('z', commutative=True))), Pow(Add(Symbol('z', commutative=True), exp(Symbol('z', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('S')(Symbol('z', commutative=True)), Mul(Add(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('z', commutative=True))), Pow(Add(Symbol('z', commutative=True), exp(Symbol('z', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('S')(Symbol('z', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(C_{2})} = \\sin{(C_{2})}, then derive \\frac{d}{d C_{2}} \\operatorname{f^{*}}{(C_{2})} = \\cos{(C_{2})}, then obtain E_{x} + \\sin{(C_{2})} + \\operatorname{Si}{(C_{2})} = x + \\sin{(C_{2})} + \\operatorname{Si}{(C_{2})}", "derivation": "\\operatorname{f^{*}}{(C_{2})} = \\sin{(C_{2})} and \\frac{d}{d C_{2}} \\operatorname{f^{*}}{(C_{2})} = \\frac{d}{d C_{2}} \\sin{(C_{2})} and \\frac{d}{d C_{2}} \\operatorname{f^{*}}{(C_{2})} = \\cos{(C_{2})} and \\cos{(C_{2})} = \\frac{d}{d C_{2}} \\sin{(C_{2})} and \\cos{(C_{2})} + \\frac{\\sin{(C_{2})}}{C_{2}} = \\frac{d}{d C_{2}} \\sin{(C_{2})} + \\frac{\\sin{(C_{2})}}{C_{2}} and \\int (\\cos{(C_{2})} + \\frac{\\sin{(C_{2})}}{C_{2}}) dC_{2} = \\int (\\frac{d}{d C_{2}} \\sin{(C_{2})} + \\frac{\\sin{(C_{2})}}{C_{2}}) dC_{2} and E_{x} + \\sin{(C_{2})} + \\operatorname{Si}{(C_{2})} = x + \\sin{(C_{2})} + \\operatorname{Si}{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), cos(Symbol('C_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('C_2', commutative=True)), Derivative(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["add", 4, "Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(Symbol('C_2', commutative=True)))"], "Equality(Add(cos(Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(Symbol('C_2', commutative=True)))), Add(Derivative(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(Symbol('C_2', commutative=True)))))"], [["integrate", 5, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(cos(Symbol('C_2', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Derivative(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('E_x', commutative=True), sin(Symbol('C_2', commutative=True)), Si(Symbol('C_2', commutative=True))), Add(Symbol('x', commutative=True), sin(Symbol('C_2', commutative=True)), Si(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = - \\mathbf{g} + e^{\\varepsilon_0}, then derive \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = e^{\\varepsilon_0}, then obtain \\frac{\\partial^{3}}{\\partial \\varepsilon_0^{3}} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0}", "derivation": "\\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = - \\mathbf{g} + e^{\\varepsilon_0} and \\frac{\\partial}{\\partial \\varepsilon_0} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = \\frac{\\partial}{\\partial \\varepsilon_0} (- \\mathbf{g} + e^{\\varepsilon_0}) and \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} (- \\mathbf{g} + e^{\\varepsilon_0}) and \\frac{\\partial^{2}}{\\partial \\varepsilon_0^{2}} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = e^{\\varepsilon_0} and \\frac{\\partial^{3}}{\\partial \\varepsilon_0^{3}} \\hat{H}_{\\lambda}{(\\varepsilon_0,\\mathbf{g})} = \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(2))), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(3))), Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\dot{y})} = e^{\\dot{y}}, then obtain - \\dot{y} = \\chi - \\dot{y} + \\frac{e^{2 \\dot{y}}}{2} - \\int V{(\\dot{y})} e^{\\dot{y}} d\\dot{y}", "derivation": "V{(\\dot{y})} = e^{\\dot{y}} and V{(\\dot{y})} e^{\\dot{y}} = e^{2 \\dot{y}} and \\int V{(\\dot{y})} e^{\\dot{y}} d\\dot{y} = \\int e^{2 \\dot{y}} d\\dot{y} and - \\dot{y} = - \\dot{y} - \\int V{(\\dot{y})} e^{\\dot{y}} d\\dot{y} + \\int e^{2 \\dot{y}} d\\dot{y} and - \\dot{y} = \\chi - \\dot{y} + \\frac{e^{2 \\dot{y}}}{2} - \\int V{(\\dot{y})} e^{\\dot{y}} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Mul(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\dot{y}', commutative=True), Integral(Mul(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Integral(Mul(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Integral(exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Rational(1, 2), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Function('V')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\varphi)} = \\cos{(\\varphi)}, then obtain (((\\mathbf{A}^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi})^{\\varphi} = (((\\cos^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi})^{\\varphi}", "derivation": "\\mathbf{A}{(\\varphi)} = \\cos{(\\varphi)} and \\mathbf{A}^{\\varphi}{(\\varphi)} = \\cos^{\\varphi}{(\\varphi)} and (\\mathbf{A}^{\\varphi}{(\\varphi)})^{\\varphi} = (\\cos^{\\varphi}{(\\varphi)})^{\\varphi} and ((\\mathbf{A}^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi} = ((\\cos^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi} and (((\\mathbf{A}^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi})^{\\varphi} = (((\\cos^{\\varphi}{(\\varphi)})^{\\varphi})^{\\varphi})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["power", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Pow(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Pow(Pow(cos(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(m_{s})} = \\sin{(m_{s})}, then obtain \\frac{d}{d m_{s}} m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{3}{(m_{s})} \\sin{(m_{s})} = \\frac{d}{d m_{s}} m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{2}{(m_{s})} \\sin^{2}{(m_{s})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(m_{s})} = \\sin{(m_{s})} and m_{s} \\operatorname{V_{\\mathbf{B}}}{(m_{s})} = m_{s} \\sin{(m_{s})} and m_{s} \\operatorname{V_{\\mathbf{B}}}^{2}{(m_{s})} = m_{s} \\operatorname{V_{\\mathbf{B}}}{(m_{s})} \\sin{(m_{s})} and m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{3}{(m_{s})} \\sin{(m_{s})} = m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{2}{(m_{s})} \\sin^{2}{(m_{s})} and \\frac{d}{d m_{s}} m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{3}{(m_{s})} \\sin{(m_{s})} = \\frac{d}{d m_{s}} m_{s}^{2} \\operatorname{V_{\\mathbf{B}}}^{2}{(m_{s})} \\sin^{2}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["times", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Symbol('m_s', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), sin(Symbol('m_s', commutative=True))))"], [["times", 2, "Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Symbol('m_s', commutative=True), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), Integer(2))), Mul(Symbol('m_s', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["times", 3, "Mul(Symbol('m_s', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), Integer(3)), sin(Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2))))"], [["differentiate", 4, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), Integer(3)), sin(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('m_s', commutative=True)), Integer(2)), Pow(sin(Symbol('m_s', commutative=True)), Integer(2))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain \\mathbb{I}{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} (\\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}}) = e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} (\\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}})", "derivation": "\\mathbb{I}{(a^{\\dagger})} = e^{a^{\\dagger}} and \\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}} = 0 and \\frac{d}{d a^{\\dagger}} (\\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}}) = \\frac{d}{d a^{\\dagger}} 0 and \\mathbb{I}{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} 0 = e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} 0 and \\mathbb{I}{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} (\\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}}) = e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} (\\mathbb{I}{(a^{\\dagger})} - e^{a^{\\dagger}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 1, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(exp(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(exp(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Add(Function('\\\\mathbb{I}')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\varphi)} = \\sin{(\\varphi)}, then derive \\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi = \\Psi - \\cos{(\\varphi)}, then obtain (\\int \\sin{(\\varphi)} d\\varphi)^{\\varphi} = (\\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi)^{\\varphi}", "derivation": "\\operatorname{P_{e}}{(\\varphi)} = \\sin{(\\varphi)} and \\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi = \\int \\sin{(\\varphi)} d\\varphi and \\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi = \\Psi - \\cos{(\\varphi)} and (\\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi)^{\\varphi} = (\\Psi - \\cos{(\\varphi)})^{\\varphi} and (\\int \\sin{(\\varphi)} d\\varphi)^{\\varphi} = (\\Psi - \\cos{(\\varphi)})^{\\varphi} and (\\int \\sin{(\\varphi)} d\\varphi)^{\\varphi} = (\\int \\operatorname{P_{e}}{(\\varphi)} d\\varphi)^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True)))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integral(Function('P_e')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Function('P_e')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mu)} = - \\mu, then derive - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{\\operatorname{F_{g}}{(\\mu)}} d\\mu = c_{0} - e^{\\operatorname{F_{g}}{(\\mu)}} - e^{- \\mu}, then obtain - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{- \\mu} d\\mu = c_{0} - e^{\\operatorname{F_{g}}{(\\mu)}} - e^{- \\mu}", "derivation": "\\operatorname{F_{g}}{(\\mu)} = - \\mu and e^{\\operatorname{F_{g}}{(\\mu)}} = e^{- \\mu} and \\int e^{\\operatorname{F_{g}}{(\\mu)}} d\\mu = \\int e^{- \\mu} d\\mu and - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{\\operatorname{F_{g}}{(\\mu)}} d\\mu = - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{- \\mu} d\\mu and - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{\\operatorname{F_{g}}{(\\mu)}} d\\mu = c_{0} - e^{\\operatorname{F_{g}}{(\\mu)}} - e^{- \\mu} and - e^{\\operatorname{F_{g}}{(\\mu)}} + \\int e^{- \\mu} d\\mu = c_{0} - e^{\\operatorname{F_{g}}{(\\mu)}} - e^{- \\mu}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))"], [["exp", 1], "Equality(exp(Function('F_g')(Symbol('\\\\mu', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(exp(Function('F_g')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 3, "exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Integral(exp(Function('F_g')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Integral(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Integral(exp(Function('F_g')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Integral(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), exp(Function('F_g')(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{S})} = \\log{(\\mathbf{S})}, then derive \\int \\operatorname{z^{*}}{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{B} + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S}, then obtain - \\mathbf{B} + \\int \\log{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{S})} = \\log{(\\mathbf{S})} and \\int \\operatorname{z^{*}}{(\\mathbf{S})} d\\mathbf{S} = \\int \\log{(\\mathbf{S})} d\\mathbf{S} and \\int \\operatorname{z^{*}}{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{B} + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} and \\int \\log{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{B} + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} and - \\mathbf{B} + \\int \\log{(\\mathbf{S})} d\\mathbf{S} = \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given B{(S,\\mu,\\hat{H}_l)} = \\frac{- S + \\mu}{\\hat{H}_l}, then obtain S + (S + B{(S,\\mu,\\hat{H}_l)})^{\\mu} = S + (S + B{(S,\\mu,\\hat{H}_l)})^{\\mu} - B{(S,\\mu,\\hat{H}_l)} + \\frac{- S + \\mu}{\\hat{H}_l}", "derivation": "B{(S,\\mu,\\hat{H}_l)} = \\frac{- S + \\mu}{\\hat{H}_l} and S + B{(S,\\mu,\\hat{H}_l)} = S + \\frac{- S + \\mu}{\\hat{H}_l} and S = S - B{(S,\\mu,\\hat{H}_l)} + \\frac{- S + \\mu}{\\hat{H}_l} and (S + B{(S,\\mu,\\hat{H}_l)})^{\\mu} = (S + \\frac{- S + \\mu}{\\hat{H}_l})^{\\mu} and S + (S + \\frac{- S + \\mu}{\\hat{H}_l})^{\\mu} = S + (S + \\frac{- S + \\mu}{\\hat{H}_l})^{\\mu} - B{(S,\\mu,\\hat{H}_l)} + \\frac{- S + \\mu}{\\hat{H}_l} and S + (S + B{(S,\\mu,\\hat{H}_l)})^{\\mu} = S + (S + B{(S,\\mu,\\hat{H}_l)})^{\\mu} - B{(S,\\mu,\\hat{H}_l)} + \\frac{- S + \\mu}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('S', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["minus", 2, "Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Symbol('S', commutative=True), Add(Symbol('S', commutative=True), Mul(Integer(-1), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('S', commutative=True), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], [["add", 3, "Pow(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Symbol('S', commutative=True), Pow(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True))), Add(Symbol('S', commutative=True), Pow(Add(Symbol('S', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('S', commutative=True), Pow(Add(Symbol('S', commutative=True), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\mu', commutative=True))), Add(Symbol('S', commutative=True), Pow(Add(Symbol('S', commutative=True), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('S', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(P_{g})} = e^{e^{P_{g}}}, then obtain \\operatorname{E_{\\lambda}}{(P_{g})} e^{- e^{P_{g}}} - e^{e^{P_{g}}} = 1 - e^{e^{P_{g}}}", "derivation": "\\operatorname{E_{\\lambda}}{(P_{g})} = e^{e^{P_{g}}} and P_{g} \\operatorname{E_{\\lambda}}{(P_{g})} = P_{g} e^{e^{P_{g}}} and \\operatorname{E_{\\lambda}}{(P_{g})} e^{- e^{P_{g}}} = 1 and \\operatorname{E_{\\lambda}}{(P_{g})} e^{- e^{P_{g}}} - e^{e^{P_{g}}} = 1 - e^{e^{P_{g}}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True))))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('E_{\\\\lambda}')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), exp(exp(Symbol('P_g', commutative=True)))))"], [["divide", 2, "Mul(Symbol('P_g', commutative=True), exp(exp(Symbol('P_g', commutative=True))))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('P_g', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('P_g', commutative=True))))), Integer(1))"], [["minus", 3, "exp(exp(Symbol('P_g', commutative=True)))"], "Equality(Add(Mul(Function('E_{\\\\lambda}')(Symbol('P_g', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('P_g', commutative=True))))), Mul(Integer(-1), exp(exp(Symbol('P_g', commutative=True))))), Add(Integer(1), Mul(Integer(-1), exp(exp(Symbol('P_g', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\varphi,Z)} = Z - \\varphi, then obtain \\operatorname{v_{x}}{(\\varphi,Z)} = \\operatorname{v_{x}}{(\\varphi,Z)} + \\int (Z - \\varphi) d\\varphi - \\int \\operatorname{v_{x}}{(\\varphi,Z)} d\\varphi", "derivation": "\\operatorname{v_{x}}{(\\varphi,Z)} = Z - \\varphi and \\int \\operatorname{v_{x}}{(\\varphi,Z)} d\\varphi = \\int (Z - \\varphi) d\\varphi and \\operatorname{v_{x}}{(\\varphi,Z)} + \\int \\operatorname{v_{x}}{(\\varphi,Z)} d\\varphi = \\operatorname{v_{x}}{(\\varphi,Z)} + \\int (Z - \\varphi) d\\varphi and \\operatorname{v_{x}}{(\\varphi,Z)} = \\operatorname{v_{x}}{(\\varphi,Z)} + \\int (Z - \\varphi) d\\varphi - \\int \\operatorname{v_{x}}{(\\varphi,Z)} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Integral(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["minus", 3, "Integral(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Add(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Integral(Function('v_x')(Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{E})} = \\sin{(\\sin{(\\mathbf{E})})}, then derive \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} \\cos{(\\sin{(\\mathbf{E})})}, then obtain \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} + 1 = \\frac{d}{d \\mathbf{E}} \\sin{(\\sin{(\\mathbf{E})})} + 1", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{E})} = \\sin{(\\sin{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\sin{(\\sin{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} = \\cos{(\\mathbf{E})} \\cos{(\\sin{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\sin{(\\sin{(\\mathbf{E})})} = \\cos{(\\mathbf{E})} \\cos{(\\sin{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} + 1 = \\cos{(\\mathbf{E})} \\cos{(\\sin{(\\mathbf{E})})} + 1 and \\frac{d}{d \\mathbf{E}} \\operatorname{a^{\\dagger}}{(\\mathbf{E})} + 1 = \\frac{d}{d \\mathbf{E}} \\sin{(\\sin{(\\mathbf{E})})} + 1", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{E}', commutative=True)), sin(sin(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{E}', commutative=True)), cos(sin(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mathbf{E}', commutative=True)), cos(sin(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1)), Add(Mul(cos(Symbol('\\\\mathbf{E}', commutative=True)), cos(sin(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(sin(sin(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given B{(\\omega,\\tilde{g})} = \\int \\omega \\tilde{g} d\\omega and m{(\\omega,\\tilde{g})} = \\int \\omega \\tilde{g} d\\omega, then obtain (B{(\\omega,\\tilde{g})} m{(\\omega,\\tilde{g})})^{\\omega} = (m{(\\omega,\\tilde{g})} \\int \\omega \\tilde{g} d\\omega)^{\\omega}", "derivation": "B{(\\omega,\\tilde{g})} = \\int \\omega \\tilde{g} d\\omega and B{(\\omega,\\tilde{g})} \\int \\omega \\tilde{g} d\\omega = (\\int \\omega \\tilde{g} d\\omega)^{2} and m{(\\omega,\\tilde{g})} = \\int \\omega \\tilde{g} d\\omega and B{(\\omega,\\tilde{g})} m{(\\omega,\\tilde{g})} = m^{2}{(\\omega,\\tilde{g})} and m{(\\omega,\\tilde{g})} \\int \\omega \\tilde{g} d\\omega = m^{2}{(\\omega,\\tilde{g})} and (B{(\\omega,\\tilde{g})} m{(\\omega,\\tilde{g})})^{\\omega} = (m^{2}{(\\omega,\\tilde{g})})^{\\omega} and (B{(\\omega,\\tilde{g})} m{(\\omega,\\tilde{g})})^{\\omega} = (m{(\\omega,\\tilde{g})} \\int \\omega \\tilde{g} d\\omega)^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('B')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Pow(Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('B')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Pow(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Pow(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(2)))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Function('B')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Pow(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Mul(Function('B')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Function('m')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given M{(\\rho)} = \\cos{(\\rho)}, then derive \\int \\rho M{(\\rho)} d\\rho = \\rho \\sin{(\\rho)} + \\varphi + \\cos{(\\rho)}, then obtain \\int \\rho M{(\\rho)} d\\rho = \\rho \\sin{(\\rho)} + \\varphi + M{(\\rho)}", "derivation": "M{(\\rho)} = \\cos{(\\rho)} and \\rho M{(\\rho)} = \\rho \\cos{(\\rho)} and \\int \\rho M{(\\rho)} d\\rho = \\int \\rho \\cos{(\\rho)} d\\rho and \\int \\rho M{(\\rho)} d\\rho = \\rho \\sin{(\\rho)} + \\varphi + \\cos{(\\rho)} and \\int \\rho M{(\\rho)} d\\rho = \\rho \\sin{(\\rho)} + \\varphi + M{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["times", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Function('M')(Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('\\\\rho', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\rho', commutative=True), Function('M')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\rho', commutative=True), Function('M')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\rho', commutative=True))), Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Symbol('\\\\rho', commutative=True), Function('M')(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\rho', commutative=True))), Symbol('\\\\varphi', commutative=True), Function('M')(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given T{(\\mathbf{J}_f,\\hat{X})} = \\mathbf{J}_f^{\\hat{X}}, then obtain - \\mathbf{J}_f^{2 \\hat{X}} + T{(\\mathbf{J}_f,\\hat{X})} = - \\mathbf{J}_f^{2 \\hat{X}} + \\mathbf{J}_f^{\\hat{X}}", "derivation": "T{(\\mathbf{J}_f,\\hat{X})} = \\mathbf{J}_f^{\\hat{X}} and \\mathbf{J}_f^{\\hat{X}} T{(\\mathbf{J}_f,\\hat{X})} = \\mathbf{J}_f^{2 \\hat{X}} and - \\mathbf{J}_f^{\\hat{X}} T{(\\mathbf{J}_f,\\hat{X})} + T{(\\mathbf{J}_f,\\hat{X})} = - \\mathbf{J}_f^{\\hat{X}} T{(\\mathbf{J}_f,\\hat{X})} + \\mathbf{J}_f^{\\hat{X}} and - \\mathbf{J}_f^{2 \\hat{X}} + T{(\\mathbf{J}_f,\\hat{X})} = - \\mathbf{J}_f^{2 \\hat{X}} + \\mathbf{J}_f^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True)))), Function('T')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(2), Symbol('\\\\hat{X}', commutative=True)))), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = \\Psi + \\mathbf{J}_M - r, then obtain 4 \\Psi \\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = 2 \\Psi (2 \\Psi + 2 \\mathbf{J}_M - 2 r)", "derivation": "\\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = \\Psi + \\mathbf{J}_M - r and \\Psi + \\mathbf{J}_M - r + \\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = 2 \\Psi + 2 \\mathbf{J}_M - 2 r and 2 \\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = 2 \\Psi + 2 \\mathbf{J}_M - 2 r and 4 \\Psi \\operatorname{y^{\\prime}}{(\\mathbf{J}_M,\\Psi,r)} = 2 \\Psi (2 \\Psi + 2 \\mathbf{J}_M - 2 r)", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["times", 3, "Mul(Integer(2), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Integer(4), Symbol('\\\\Psi', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('r', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given y{(t,s)} = s t and L{(t,s)} = \\int y{(t,s)} ds, then obtain \\frac{\\partial}{\\partial t} L{(t,s)} = \\frac{\\partial}{\\partial t} \\int s t ds", "derivation": "y{(t,s)} = s t and \\int y{(t,s)} ds = \\int s t ds and L{(t,s)} = \\int y{(t,s)} ds and \\frac{\\partial}{\\partial t} L{(t,s)} = \\frac{\\partial}{\\partial t} \\int y{(t,s)} ds and \\frac{\\partial}{\\partial t} L{(t,s)} = \\frac{\\partial}{\\partial t} \\int s t ds", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('y')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Mul(Symbol('s', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Integral(Function('y')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Function('y')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('L')(Symbol('t', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('s', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)} = \\chi \\phi_2 - \\mu, then derive 0 = - \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)}, then obtain \\frac{d}{d \\phi_2} 0 = \\frac{\\partial}{\\partial \\phi_2} - \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)}", "derivation": "\\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)} = \\chi \\phi_2 - \\mu and 0 = \\chi \\phi_2 - \\mu - \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)} and \\frac{d}{d \\phi_2} 0 = \\frac{\\partial}{\\partial \\phi_2} (\\chi \\phi_2 - \\mu - \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)}) and \\frac{d^{2}}{d \\phi_2^{2}} 0 = \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} (\\chi \\phi_2 - \\mu - \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)}) and 0 = - \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)} and \\frac{d}{d \\phi_2} 0 = \\frac{\\partial}{\\partial \\phi_2} - \\frac{\\partial^{2}}{\\partial \\phi_2^{2}} \\dot{\\mathbf{r}}{(\\phi_2,\\mu,\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2)))))"], [["differentiate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(2)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(E_{n},n_{1})} = \\int E_{n} n_{1} dn_{1}, then obtain \\cos^{E_{n}}{(\\int \\operatorname{A_{1}}{(E_{n},n_{1})} dn_{1})} = \\cos^{E_{n}}{(\\iint E_{n} n_{1} dn_{1} dn_{1})}", "derivation": "\\operatorname{A_{1}}{(E_{n},n_{1})} = \\int E_{n} n_{1} dn_{1} and \\int \\operatorname{A_{1}}{(E_{n},n_{1})} dn_{1} = \\iint E_{n} n_{1} dn_{1} dn_{1} and \\cos{(\\int \\operatorname{A_{1}}{(E_{n},n_{1})} dn_{1})} = \\cos{(\\iint E_{n} n_{1} dn_{1} dn_{1})} and \\cos^{E_{n}}{(\\int \\operatorname{A_{1}}{(E_{n},n_{1})} dn_{1})} = \\cos^{E_{n}}{(\\iint E_{n} n_{1} dn_{1} dn_{1})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Integral(Mul(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('A_1')(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), cos(Integral(Mul(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(cos(Integral(Function('A_1')(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Symbol('E_n', commutative=True)), Pow(cos(Integral(Mul(Symbol('E_n', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(\\Psi^{\\dagger},\\tilde{g}^*)} = \\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}} and f{(\\Psi^{\\dagger},\\tilde{g}^*)} = (\\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}})^{\\tilde{g}^*} \\phi_{2}^{- \\tilde{g}^*}{(\\Psi^{\\dagger},\\tilde{g}^*)}, then obtain 1 = f{(\\Psi^{\\dagger},\\tilde{g}^*)}", "derivation": "\\phi_{2}{(\\Psi^{\\dagger},\\tilde{g}^*)} = \\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}} and \\phi_{2}^{\\tilde{g}^*}{(\\Psi^{\\dagger},\\tilde{g}^*)} = (\\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}})^{\\tilde{g}^*} and 1 = (\\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}})^{\\tilde{g}^*} \\phi_{2}^{- \\tilde{g}^*}{(\\Psi^{\\dagger},\\tilde{g}^*)} and f{(\\Psi^{\\dagger},\\tilde{g}^*)} = (\\log{(\\tilde{g}^*)}^{\\Psi^{\\dagger}})^{\\tilde{g}^*} \\phi_{2}^{- \\tilde{g}^*}{(\\Psi^{\\dagger},\\tilde{g}^*)} and 1 = f{(\\Psi^{\\dagger},\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\phi_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(a,m)} = a^{m}, then obtain \\frac{\\partial^{2}}{\\partial a^{2}} (- m)^{a} = \\frac{\\partial^{2}}{\\partial a^{2}} (a^{m} - m - \\operatorname{c_{0}}{(a,m)})^{a}", "derivation": "\\operatorname{c_{0}}{(a,m)} = a^{m} and - m = a^{m} - m - \\operatorname{c_{0}}{(a,m)} and (- m)^{a} = (a^{m} - m - \\operatorname{c_{0}}{(a,m)})^{a} and \\frac{\\partial}{\\partial a} (- m)^{a} = \\frac{\\partial}{\\partial a} (a^{m} - m - \\operatorname{c_{0}}{(a,m)})^{a} and \\frac{\\partial^{2}}{\\partial a^{2}} (- m)^{a} = \\frac{\\partial^{2}}{\\partial a^{2}} (a^{m} - m - \\operatorname{c_{0}}{(a,m)})^{a}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Add(Symbol('m', commutative=True), Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('m', commutative=True)), Add(Pow(Symbol('a', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('a', commutative=True)), Pow(Add(Pow(Symbol('a', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Symbol('a', commutative=True)))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Add(Pow(Symbol('a', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Pow(Add(Pow(Symbol('a', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)), Mul(Integer(-1), Function('c_0')(Symbol('a', commutative=True), Symbol('m', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(G)} = \\cos{(G)} and z{(G)} = \\cos{(G)}, then obtain \\operatorname{M_{E}}^{G}{(G)} = \\cos^{G}{(G)}", "derivation": "\\operatorname{M_{E}}{(G)} = \\cos{(G)} and z{(G)} = \\cos{(G)} and z{(G)} = \\operatorname{M_{E}}{(G)} and z^{G}{(G)} = \\cos^{G}{(G)} and \\operatorname{M_{E}}^{G}{(G)} = \\cos^{G}{(G)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('z')(Symbol('G', commutative=True)), Function('M_E')(Symbol('G', commutative=True)))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Function('z')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('M_E')(Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(cos(Symbol('G', commutative=True)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(J,\\mathbf{v})} = \\int (J + \\mathbf{v}) dJ, then obtain ((- J + \\mathbf{g}{(J,\\mathbf{v})})^{2})^{J} = ((- J + \\mathbf{g}{(J,\\mathbf{v})}) (- J + \\int (J + \\mathbf{v}) dJ))^{J}", "derivation": "\\mathbf{g}{(J,\\mathbf{v})} = \\int (J + \\mathbf{v}) dJ and - J + \\mathbf{g}{(J,\\mathbf{v})} = - J + \\int (J + \\mathbf{v}) dJ and (- J + \\mathbf{g}{(J,\\mathbf{v})})^{2} = (- J + \\mathbf{g}{(J,\\mathbf{v})}) (- J + \\int (J + \\mathbf{v}) dJ) and ((- J + \\mathbf{g}{(J,\\mathbf{v})})^{2})^{J} = ((- J + \\mathbf{g}{(J,\\mathbf{v})}) (- J + \\int (J + \\mathbf{v}) dJ))^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["minus", 1, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('J', commutative=True))))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(2)), Symbol('J', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Function('\\\\mathbf{g}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('J', commutative=True))))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given t{(u)} = \\frac{d}{d u} e^{u}, then derive 2 t{(u)} = t{(u)} + e^{u}, then derive e^{u} + \\frac{d}{d u} t{(u)} = 2 \\frac{d}{d u} t{(u)}, then obtain 2 e^{u} + \\frac{d}{d u} t{(u)} = e^{u} + 2 \\frac{d}{d u} t{(u)}", "derivation": "t{(u)} = \\frac{d}{d u} e^{u} and 2 t{(u)} = t{(u)} + \\frac{d}{d u} e^{u} and \\frac{d}{d u} 2 t{(u)} = \\frac{d}{d u} (t{(u)} + \\frac{d}{d u} e^{u}) and 2 t{(u)} = t{(u)} + e^{u} and \\frac{d}{d u} (t{(u)} + e^{u}) = \\frac{d}{d u} (t{(u)} + \\frac{d}{d u} e^{u}) and \\frac{d}{d u} (t{(u)} + e^{u}) = \\frac{d}{d u} 2 t{(u)} and e^{u} + \\frac{d}{d u} t{(u)} = 2 \\frac{d}{d u} t{(u)} and 2 e^{u} + \\frac{d}{d u} t{(u)} = e^{u} + 2 \\frac{d}{d u} t{(u)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["add", 1, "Function('t')(Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Function('t')(Symbol('u', commutative=True))), Add(Function('t')(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('t')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Function('t')(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(2), Function('t')(Symbol('u', commutative=True))), Add(Function('t')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Function('t')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Function('t')(Symbol('u', commutative=True)), Derivative(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Add(Function('t')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('t')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(exp(Symbol('u', commutative=True)), Derivative(Function('t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["add", 7, "exp(Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(2), exp(Symbol('u', commutative=True))), Derivative(Function('t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(exp(Symbol('u', commutative=True)), Mul(Integer(2), Derivative(Function('t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\hat{p}_0 and \\lambda{(\\hat{p},t_{1})} = \\int \\hat{p} t_{1} dt_{1}, then obtain \\frac{\\lambda{(\\hat{p},t_{1})}}{- 2 \\hat{H}_{\\lambda} + \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})}} = \\frac{\\int \\hat{p} t_{1} dt_{1}}{- 2 \\hat{H}_{\\lambda} + \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})}}", "derivation": "\\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\hat{p}_0 and - 2 \\hat{H}_{\\lambda} + \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + \\hat{p}_0 and \\lambda{(\\hat{p},t_{1})} = \\int \\hat{p} t_{1} dt_{1} and \\frac{\\lambda{(\\hat{p},t_{1})}}{- \\hat{H}_{\\lambda} + \\hat{p}_0} = \\frac{\\int \\hat{p} t_{1} dt_{1}}{- \\hat{H}_{\\lambda} + \\hat{p}_0} and \\frac{\\lambda{(\\hat{p},t_{1})}}{- 2 \\hat{H}_{\\lambda} + \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})}} = \\frac{\\int \\hat{p} t_{1} dt_{1}}{- 2 \\hat{H}_{\\lambda} + \\operatorname{m_{s}}{(\\hat{p}_0,\\hat{H}_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], ["get_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Function('\\\\lambda')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1)), Function('\\\\lambda')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1)), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given J{(\\mu)} = \\cos{(\\mu)}, then obtain \\mu \\cos{(\\mu)} + \\tilde{\\infty} (J^{\\mu}{(\\mu)})^{\\mu} = \\mu \\cos{(\\mu)} + \\tilde{\\infty} (\\cos^{\\mu}{(\\mu)})^{\\mu}", "derivation": "J{(\\mu)} = \\cos{(\\mu)} and \\mu J{(\\mu)} = \\mu \\cos{(\\mu)} and J^{\\mu}{(\\mu)} = \\cos^{\\mu}{(\\mu)} and (J^{\\mu}{(\\mu)})^{\\mu} = (\\cos^{\\mu}{(\\mu)})^{\\mu} and \\tilde{\\infty} (J^{\\mu}{(\\mu)})^{\\mu} = \\tilde{\\infty} (\\cos^{\\mu}{(\\mu)})^{\\mu} and \\mu J{(\\mu)} + \\tilde{\\infty} (J^{\\mu}{(\\mu)})^{\\mu} = \\mu J{(\\mu)} + \\tilde{\\infty} (\\cos^{\\mu}{(\\mu)})^{\\mu} and \\mu \\cos{(\\mu)} + \\tilde{\\infty} (J^{\\mu}{(\\mu)})^{\\mu} = \\mu \\cos{(\\mu)} + \\tilde{\\infty} (\\cos^{\\mu}{(\\mu)})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('J')(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(cos(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["divide", 4, 0], "Equality(Mul(zoo, Pow(Pow(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Mul(zoo, Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Function('J')(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Function('J')(Symbol('\\\\mu', commutative=True))), Mul(zoo, Pow(Pow(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\mu', commutative=True), Function('J')(Symbol('\\\\mu', commutative=True))), Mul(zoo, Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))), Mul(zoo, Pow(Pow(Function('J')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))), Mul(zoo, Pow(Pow(cos(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(F_{N})} = \\log{(F_{N})}, then obtain (\\frac{d}{d F_{N}} (- F_{N} + \\operatorname{f_{\\mathbf{v}}}{(F_{N})}))^{F_{N}} = (\\frac{d}{d F_{N}} (- F_{N} + \\log{(F_{N})}))^{F_{N}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(F_{N})} = \\log{(F_{N})} and - F_{N} + \\operatorname{f_{\\mathbf{v}}}{(F_{N})} = - F_{N} + \\log{(F_{N})} and \\frac{d}{d F_{N}} (- F_{N} + \\operatorname{f_{\\mathbf{v}}}{(F_{N})}) = \\frac{d}{d F_{N}} (- F_{N} + \\log{(F_{N})}) and (\\frac{d}{d F_{N}} (- F_{N} + \\operatorname{f_{\\mathbf{v}}}{(F_{N})}))^{F_{N}} = (\\frac{d}{d F_{N}} (- F_{N} + \\log{(F_{N})}))^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["minus", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\mu{(E,y^{\\prime})} = \\cos{(E + y^{\\prime})} and U{(f^{*},\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\log{(f^{*})}, then obtain \\int \\frac{\\mu{(E,y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} dy^{\\prime} = \\int \\frac{\\cos{(E + y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} dy^{\\prime}", "derivation": "\\mu{(E,y^{\\prime})} = \\cos{(E + y^{\\prime})} and U{(f^{*},\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\log{(f^{*})} and \\frac{\\mu{(E,y^{\\prime})}}{\\dot{\\mathbf{r}} \\log{(f^{*})}} = \\frac{\\cos{(E + y^{\\prime})}}{\\dot{\\mathbf{r}} \\log{(f^{*})}} and \\frac{\\mu{(E,y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} = \\frac{\\cos{(E + y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} and \\int \\frac{\\mu{(E,y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} dy^{\\prime} = \\int \\frac{\\cos{(E + y^{\\prime})}}{U{(f^{*},\\dot{\\mathbf{r}})}} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], ["get_premise", "Equality(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), log(Symbol('f^*', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), log(Symbol('f^*', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(log(Symbol('f^*', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(log(Symbol('f^*', commutative=True)), Integer(-1)), cos(Add(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Function('\\\\mu')(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), cos(Add(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["integrate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Function('\\\\mu')(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), cos(Add(Symbol('E', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(n_{1},\\mathbf{J})} = \\frac{\\sin{(\\mathbf{J})}}{n_{1}}, then obtain \\frac{\\varepsilon_{0}^{2}{(n_{1},\\mathbf{J})}}{n_{1}} = \\frac{\\sin^{2}{(\\mathbf{J})}}{n_{1}^{3}}", "derivation": "\\varepsilon_{0}{(n_{1},\\mathbf{J})} = \\frac{\\sin{(\\mathbf{J})}}{n_{1}} and \\frac{\\varepsilon_{0}{(n_{1},\\mathbf{J})}}{n_{1}} = \\frac{\\sin{(\\mathbf{J})}}{n_{1}^{2}} and \\frac{\\varepsilon_{0}^{2}{(n_{1},\\mathbf{J})}}{n_{1}} = \\frac{\\varepsilon_{0}{(n_{1},\\mathbf{J})} \\sin{(\\mathbf{J})}}{n_{1}^{2}} and \\frac{\\varepsilon_{0}{(n_{1},\\mathbf{J})} \\sin{(\\mathbf{J})}}{n_{1}^{2}} = \\frac{\\sin^{2}{(\\mathbf{J})}}{n_{1}^{3}} and \\frac{\\varepsilon_{0}^{2}{(n_{1},\\mathbf{J})}}{n_{1}} = \\frac{\\sin^{2}{(\\mathbf{J})}}{n_{1}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 2, "Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-3)), Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-3)), Pow(sin(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(v)} = \\log{(v)}, then derive \\frac{d}{d v} \\operatorname{F_{g}}{(v)} = \\frac{1}{v}, then obtain \\frac{d^{2}}{d v^{2}} \\operatorname{F_{g}}{(v)} = \\frac{d}{d v} \\frac{1}{v}", "derivation": "\\operatorname{F_{g}}{(v)} = \\log{(v)} and \\frac{d}{d v} \\operatorname{F_{g}}{(v)} = \\frac{d}{d v} \\log{(v)} and \\frac{d}{d v} \\operatorname{F_{g}}{(v)} = \\frac{1}{v} and \\frac{d^{2}}{d v^{2}} \\operatorname{F_{g}}{(v)} = \\frac{d}{d v} \\frac{1}{v}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Symbol('v', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(2))), Derivative(Pow(Symbol('v', commutative=True), Integer(-1)), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(A_{2},z)} = A_{2}^{z}, then derive \\frac{\\frac{\\partial}{\\partial z} \\operatorname{v_{z}}{(A_{2},z)}}{z} = \\frac{A_{2}^{z} \\log{(A_{2})}}{z}, then obtain \\frac{\\frac{\\partial}{\\partial z} A_{2}^{z}}{z} = \\frac{A_{2}^{z} \\log{(A_{2})}}{z}", "derivation": "\\operatorname{v_{z}}{(A_{2},z)} = A_{2}^{z} and \\frac{\\partial}{\\partial z} \\operatorname{v_{z}}{(A_{2},z)} = \\frac{\\partial}{\\partial z} A_{2}^{z} and \\frac{\\frac{\\partial}{\\partial z} \\operatorname{v_{z}}{(A_{2},z)}}{z} = \\frac{\\frac{\\partial}{\\partial z} A_{2}^{z}}{z} and \\frac{\\frac{\\partial}{\\partial z} \\operatorname{v_{z}}{(A_{2},z)}}{z} = \\frac{A_{2}^{z} \\log{(A_{2})}}{z} and \\frac{\\frac{\\partial}{\\partial z} A_{2}^{z}}{z} = \\frac{A_{2}^{z} \\log{(A_{2})}}{z}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Function('v_z')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Function('v_z')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given k{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\operatorname{f^{*}}{(\\varphi^*)} = \\cos^{\\varphi^*}{(\\varphi^*)}, then obtain k^{- \\varphi^*}{(\\varphi^*)} \\cos^{\\varphi^*}{(\\varphi^*)} = 1", "derivation": "k{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\operatorname{f^{*}}{(\\varphi^*)} = \\cos^{\\varphi^*}{(\\varphi^*)} and \\operatorname{f^{*}}{(\\varphi^*)} \\cos^{- \\varphi^*}{(\\varphi^*)} = 1 and \\operatorname{f^{*}}{(\\varphi^*)} k^{- \\varphi^*}{(\\varphi^*)} = 1 and k^{- \\varphi^*}{(\\varphi^*)} \\cos^{\\varphi^*}{(\\varphi^*)} = 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 2, "Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Function('f^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('f^*')(Symbol('\\\\varphi^*', commutative=True)), Pow(Function('k')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('k')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})} = (\\frac{\\nabla}{A_{x}})^{z^{*}}, then obtain (\\frac{d}{d A_{x}} 0)^{z^{*}} = (\\frac{\\partial}{\\partial A_{x}} ((\\frac{\\nabla}{A_{x}})^{z^{*}} - \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})}))^{z^{*}}", "derivation": "\\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})} = (\\frac{\\nabla}{A_{x}})^{z^{*}} and \\nabla + \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})} = \\nabla + (\\frac{\\nabla}{A_{x}})^{z^{*}} and 0 = (\\frac{\\nabla}{A_{x}})^{z^{*}} - \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})} and \\frac{d}{d A_{x}} 0 = \\frac{\\partial}{\\partial A_{x}} ((\\frac{\\nabla}{A_{x}})^{z^{*}} - \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})}) and (\\frac{d}{d A_{x}} 0)^{z^{*}} = (\\frac{\\partial}{\\partial A_{x}} ((\\frac{\\nabla}{A_{x}})^{z^{*}} - \\operatorname{v_{y}}{(A_{x},\\nabla,z^{*})}))^{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Symbol('z^*', commutative=True)))"], [["add", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Symbol('z^*', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True))))"], "Equality(Integer(0), Add(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(Add(Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\nabla', commutative=True)), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('A_x', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given J{(\\hat{H},\\Omega)} = \\log{(\\frac{\\Omega}{\\hat{H}})}, then obtain - \\hat{H} + \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + J{(\\hat{H},\\Omega)}) d\\hat{H} = - \\hat{H} + \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + \\log{(\\frac{\\Omega}{\\hat{H}})}) d\\hat{H}", "derivation": "J{(\\hat{H},\\Omega)} = \\log{(\\frac{\\Omega}{\\hat{H}})} and \\frac{\\Omega}{\\hat{H}} + J{(\\hat{H},\\Omega)} = \\frac{\\Omega}{\\hat{H}} + \\log{(\\frac{\\Omega}{\\hat{H}})} and \\int (\\frac{\\Omega}{\\hat{H}} + J{(\\hat{H},\\Omega)}) d\\hat{H} = \\int (\\frac{\\Omega}{\\hat{H}} + \\log{(\\frac{\\Omega}{\\hat{H}})}) d\\hat{H} and \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + J{(\\hat{H},\\Omega)}) d\\hat{H} = \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + \\log{(\\frac{\\Omega}{\\hat{H}})}) d\\hat{H} and - \\hat{H} + \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + J{(\\hat{H},\\Omega)}) d\\hat{H} = - \\hat{H} + \\frac{\\partial}{\\partial \\Omega} \\int (\\frac{\\Omega}{\\hat{H}} + \\log{(\\frac{\\Omega}{\\hat{H}})}) d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))))"], [["add", 1, "Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Function('J')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Function('J')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Function('J')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Function('J')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Derivative(Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), log(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\ddot{x}{(k,A)} = A k, then obtain (0^{A})^{A} + 1 = 2", "derivation": "\\ddot{x}{(k,A)} = A k and \\int \\ddot{x}{(k,A)} dk = \\int A k dk and 0 = \\int A k dk - \\int \\ddot{x}{(k,A)} dk and 0^{A} = (\\int A k dk - \\int \\ddot{x}{(k,A)} dk)^{A} and (0^{A})^{A} = ((\\int A k dk - \\int \\ddot{x}{(k,A)} dk)^{A})^{A} and ((\\int A k dk - \\int \\ddot{x}{(k,A)} dk)^{A})^{A} = 1 and (0^{A})^{A} = 1 and (0^{A})^{A} + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True))))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A', commutative=True)), Pow(Add(Integral(Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True))))), Symbol('A', commutative=True)))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Pow(Add(Integral(Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True))))), Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Add(Integral(Mul(Symbol('A', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\ddot{x}')(Symbol('k', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('k', commutative=True))))), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Pow(Integer(0), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integer(1))"], [["add", 7, 1], "Equality(Add(Pow(Pow(Integer(0), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integer(1)), Integer(2))"]]}, {"prompt": "Given y{(F_{g},\\hat{p})} = F_{g} + e^{\\hat{p}}, then derive \\frac{\\partial}{\\partial \\hat{p}} y{(F_{g},\\hat{p})} = e^{\\hat{p}}, then obtain y{(F_{g},\\hat{p})} = F_{g} + \\frac{\\partial}{\\partial \\hat{p}} (F_{g} + e^{\\hat{p}})", "derivation": "y{(F_{g},\\hat{p})} = F_{g} + e^{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} y{(F_{g},\\hat{p})} = \\frac{\\partial}{\\partial \\hat{p}} (F_{g} + e^{\\hat{p}}) and \\frac{\\partial}{\\partial \\hat{p}} y{(F_{g},\\hat{p})} = e^{\\hat{p}} and e^{\\hat{p}} = \\frac{\\partial}{\\partial \\hat{p}} (F_{g} + e^{\\hat{p}}) and y{(F_{g},\\hat{p})} = F_{g} + \\frac{\\partial}{\\partial \\hat{p}} (F_{g} + e^{\\hat{p}})", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('F_g', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\hat{p}', commutative=True)), Derivative(Add(Symbol('F_g', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('y')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Add(Symbol('F_g', commutative=True), Derivative(Add(Symbol('F_g', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi{(f^{\\prime},\\chi)} = e^{\\chi f^{\\prime}} and \\operatorname{P_{e}}{(f^{\\prime})} = f^{\\prime}, then obtain e^{\\frac{e^{\\chi f^{\\prime}} \\sin{(\\operatorname{P_{e}}{(f^{\\prime})})}}{\\sin{(f^{\\prime})}}} = e^{e^{\\chi f^{\\prime}}}", "derivation": "\\psi{(f^{\\prime},\\chi)} = e^{\\chi f^{\\prime}} and \\operatorname{P_{e}}{(f^{\\prime})} = f^{\\prime} and \\sin{(\\operatorname{P_{e}}{(f^{\\prime})})} = \\sin{(f^{\\prime})} and \\frac{\\psi{(f^{\\prime},\\chi)} \\sin{(\\operatorname{P_{e}}{(f^{\\prime})})}}{\\sin{(f^{\\prime})}} = \\psi{(f^{\\prime},\\chi)} and \\frac{e^{\\chi f^{\\prime}} \\sin{(\\operatorname{P_{e}}{(f^{\\prime})})}}{\\sin{(f^{\\prime})}} = e^{\\chi f^{\\prime}} and e^{\\frac{e^{\\chi f^{\\prime}} \\sin{(\\operatorname{P_{e}}{(f^{\\prime})})}}{\\sin{(f^{\\prime})}}} = e^{e^{\\chi f^{\\prime}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["sin", 2], "Equality(sin(Function('P_e')(Symbol('f^{\\\\prime}', commutative=True))), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 3, "Mul(Pow(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), sin(Function('P_e')(Symbol('f^{\\\\prime}', commutative=True)))), Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), sin(Function('P_e')(Symbol('f^{\\\\prime}', commutative=True)))), exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), sin(Function('P_e')(Symbol('f^{\\\\prime}', commutative=True))))), exp(exp(Mul(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(C)} = \\log{(C)} and J{(C)} = \\log{(C)}, then obtain J{(C)} + \\varphi^{*}{(C)} = 2 \\varphi^{*}{(C)}", "derivation": "\\varphi^{*}{(C)} = \\log{(C)} and J{(C)} = \\log{(C)} and J{(C)} = \\varphi^{*}{(C)} and J{(C)} + \\varphi^{*}{(C)} = 2 \\varphi^{*}{(C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('J')(Symbol('C', commutative=True)), Function('\\\\varphi^*')(Symbol('C', commutative=True)))"], [["add", 3, "Function('\\\\varphi^*')(Symbol('C', commutative=True))"], "Equality(Add(Function('J')(Symbol('C', commutative=True)), Function('\\\\varphi^*')(Symbol('C', commutative=True))), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain (\\mathbf{B}{(\\sigma_p)} + \\cos{(\\sigma_p)}) \\cos{(\\sigma_p)} + 2 \\cos{(\\sigma_p)} = 2 \\cos^{2}{(\\sigma_p)} + 2 \\cos{(\\sigma_p)}", "derivation": "\\mathbf{B}{(\\sigma_p)} = \\cos{(\\sigma_p)} and \\mathbf{B}{(\\sigma_p)} + \\cos{(\\sigma_p)} = 2 \\cos{(\\sigma_p)} and (\\mathbf{B}{(\\sigma_p)} + \\cos{(\\sigma_p)}) \\cos{(\\sigma_p)} = 2 \\cos^{2}{(\\sigma_p)} and (\\mathbf{B}{(\\sigma_p)} + \\cos{(\\sigma_p)}) \\cos{(\\sigma_p)} + 2 \\cos{(\\sigma_p)} = 2 \\cos^{2}{(\\sigma_p)} + 2 \\cos{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 2, "cos(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), Pow(cos(Symbol('\\\\sigma_p', commutative=True)), Integer(2))))"], [["add", 3, "Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Add(Function('\\\\mathbf{B}')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(2), Pow(cos(Symbol('\\\\sigma_p', commutative=True)), Integer(2))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True)))))"]]}, {"prompt": "Given U{(h,v)} = h^{v}, then obtain \\frac{\\partial^{2}}{\\partial h\\partial v} 4 (h^{v} + U{(h,v)})^{2 v} = \\frac{\\partial^{2}}{\\partial h\\partial v} ((2 h^{v})^{2 v} + 3 (h^{v} + U{(h,v)})^{2 v})", "derivation": "U{(h,v)} = h^{v} and h^{v} + U{(h,v)} = 2 h^{v} and (h^{v} + U{(h,v)})^{v} = (2 h^{v})^{v} and (h^{v} + U{(h,v)})^{2 v} = (2 h^{v})^{2 v} and 2 (h^{v} + U{(h,v)})^{2 v} = (2 h^{v})^{2 v} + (h^{v} + U{(h,v)})^{2 v} and 4 (h^{v} + U{(h,v)})^{2 v} = (2 h^{v})^{2 v} + 3 (h^{v} + U{(h,v)})^{2 v} and \\frac{\\partial}{\\partial v} 4 (h^{v} + U{(h,v)})^{2 v} = \\frac{\\partial}{\\partial v} ((2 h^{v})^{2 v} + 3 (h^{v} + U{(h,v)})^{2 v}) and \\frac{\\partial^{2}}{\\partial h\\partial v} 4 (h^{v} + U{(h,v)})^{2 v} = \\frac{\\partial^{2}}{\\partial h\\partial v} ((2 h^{v})^{2 v} + 3 (h^{v} + U{(h,v)})^{2 v})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)))"], [["add", 1, "Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))), Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))))"], [["add", 4, "Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))), Add(Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))))"], [["add", 5, "Mul(Integer(2), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))))"], "Equality(Mul(Integer(4), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))), Add(Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))), Mul(Integer(3), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))))))"], [["differentiate", 6, "Symbol('v', commutative=True)"], "Equality(Derivative(Mul(Integer(4), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))), Mul(Integer(3), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Integer(4), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))), Mul(Integer(3), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('h', commutative=True), Symbol('v', commutative=True))), Mul(Integer(2), Symbol('v', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1)), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(i)} = \\frac{d}{d i} e^{i}, then derive \\sin{(i \\operatorname{v_{y}}{(i)})} = \\sin{(i e^{i})}, then obtain \\int \\sin{(i \\frac{d}{d i} e^{i})} di = \\int \\sin{(i e^{i})} di", "derivation": "\\operatorname{v_{y}}{(i)} = \\frac{d}{d i} e^{i} and i \\operatorname{v_{y}}{(i)} = i \\frac{d}{d i} e^{i} and \\sin{(i \\operatorname{v_{y}}{(i)})} = \\sin{(i \\frac{d}{d i} e^{i})} and \\sin{(i \\operatorname{v_{y}}{(i)})} = \\sin{(i e^{i})} and \\sin{(i \\frac{d}{d i} e^{i})} = \\sin{(i e^{i})} and \\int \\sin{(i \\frac{d}{d i} e^{i})} di = \\int \\sin{(i e^{i})} di", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('i', commutative=True)), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('v_y')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["sin", 2], "Equality(sin(Mul(Symbol('i', commutative=True), Function('v_y')(Symbol('i', commutative=True)))), sin(Mul(Symbol('i', commutative=True), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(sin(Mul(Symbol('i', commutative=True), Function('v_y')(Symbol('i', commutative=True)))), sin(Mul(Symbol('i', commutative=True), exp(Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(sin(Mul(Symbol('i', commutative=True), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), sin(Mul(Symbol('i', commutative=True), exp(Symbol('i', commutative=True)))))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(sin(Mul(Symbol('i', commutative=True), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), Tuple(Symbol('i', commutative=True))), Integral(sin(Mul(Symbol('i', commutative=True), exp(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given W{(t_{1},\\lambda)} = \\cos{(\\frac{t_{1}}{\\lambda})}, then obtain - \\frac{- W{(t_{1},\\lambda)} - 1}{\\cos{(\\frac{t_{1}}{\\lambda})}} = - \\frac{- \\cos{(\\frac{t_{1}}{\\lambda})} - 1}{\\cos{(\\frac{t_{1}}{\\lambda})}}", "derivation": "W{(t_{1},\\lambda)} = \\cos{(\\frac{t_{1}}{\\lambda})} and W{(t_{1},\\lambda)} + 1 = \\cos{(\\frac{t_{1}}{\\lambda})} + 1 and - W{(t_{1},\\lambda)} - 1 = - \\cos{(\\frac{t_{1}}{\\lambda})} - 1 and - \\frac{- W{(t_{1},\\lambda)} - 1}{\\cos{(\\frac{t_{1}}{\\lambda})}} = - \\frac{- \\cos{(\\frac{t_{1}}{\\lambda})} - 1}{\\cos{(\\frac{t_{1}}{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)), cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('W')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(1)), Add(cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Integer(1)))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('W')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))), Integer(-1)))"], [["divide", 3, "Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('W')(Symbol('t_1', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(-1)), Pow(cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))), Integer(-1)), Pow(cos(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\eta{(\\psi^*)} = \\log{(\\psi^*)}, then derive \\int \\eta{(\\psi^*)} d\\psi^* = \\lambda + \\psi^* \\log{(\\psi^*)} - \\psi^*, then obtain - \\hbar + \\lambda + \\psi^* \\log{(\\psi^*)} - \\psi^* = - \\hbar + \\int \\log{(\\psi^*)} d\\psi^*", "derivation": "\\eta{(\\psi^*)} = \\log{(\\psi^*)} and \\int \\eta{(\\psi^*)} d\\psi^* = \\int \\log{(\\psi^*)} d\\psi^* and \\int \\eta{(\\psi^*)} d\\psi^* = \\lambda + \\psi^* \\log{(\\psi^*)} - \\psi^* and \\lambda + \\psi^* \\log{(\\psi^*)} - \\psi^* = \\int \\log{(\\psi^*)} d\\psi^* and - \\hbar + \\lambda + \\psi^* \\log{(\\psi^*)} - \\psi^* = - \\hbar + \\int \\log{(\\psi^*)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(b,F_{c})} = F_{c} b and \\hat{\\mathbf{x}}{(b,F_{c})} = 2 F_{c} b, then obtain 2 F_{c} b + \\tilde{g}{(b,F_{c})} = 3 F_{c} b", "derivation": "\\tilde{g}{(b,F_{c})} = F_{c} b and \\hat{\\mathbf{x}}{(b,F_{c})} = 2 F_{c} b and \\hat{\\mathbf{x}}{(b,F_{c})} + \\tilde{g}{(b,F_{c})} = F_{c} b + \\hat{\\mathbf{x}}{(b,F_{c})} and 2 F_{c} b + \\tilde{g}{(b,F_{c})} = 3 F_{c} b", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(2), Symbol('F_c', commutative=True), Symbol('b', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True)), Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), Symbol('b', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Symbol('F_c', commutative=True), Symbol('b', commutative=True)), Function('\\\\tilde{g}')(Symbol('b', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(3), Symbol('F_c', commutative=True), Symbol('b', commutative=True)))"]]}, {"prompt": "Given x{(\\dot{x},v_{2})} = \\dot{x} v_{2}, then obtain \\frac{d}{d v_{2}} 1 \\frac{\\partial}{\\partial v_{2}} x{(\\dot{x},v_{2})} = \\frac{\\partial}{\\partial v_{2}} \\frac{\\dot{x} v_{2}}{x{(\\dot{x},v_{2})}} \\frac{\\partial}{\\partial v_{2}} x{(\\dot{x},v_{2})}", "derivation": "x{(\\dot{x},v_{2})} = \\dot{x} v_{2} and 1 = \\frac{\\dot{x} v_{2}}{x{(\\dot{x},v_{2})}} and \\frac{d}{d v_{2}} 1 = \\frac{\\partial}{\\partial v_{2}} \\frac{\\dot{x} v_{2}}{x{(\\dot{x},v_{2})}} and \\frac{d}{d v_{2}} 1 \\frac{\\partial}{\\partial v_{2}} x{(\\dot{x},v_{2})} = \\frac{\\partial}{\\partial v_{2}} \\frac{\\dot{x} v_{2}}{x{(\\dot{x},v_{2})}} \\frac{\\partial}{\\partial v_{2}} x{(\\dot{x},v_{2})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)))"], [["divide", 1, "Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True), Pow(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True), Pow(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Integer(1), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True), Pow(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Function('x')(Symbol('\\\\dot{x}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then obtain \\frac{\\int \\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} dg_{\\varepsilon}}{F_{H} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}} = 1", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\int \\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon} and \\frac{\\int \\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} dg_{\\varepsilon}}{\\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon}} = 1 and \\frac{\\int \\operatorname{f_{\\mathbf{p}}}{(g_{\\varepsilon})} dg_{\\varepsilon}}{F_{H} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}} = 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('F_H', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given h{(c,M_{E})} = \\sin{(c^{M_{E}})}, then obtain \\frac{h{(c,M_{E})} \\iint (- M_{E} + h{(c,M_{E})}) dc dM_{E}}{M_{E}} = \\frac{h{(c,M_{E})} \\iint (- M_{E} + \\sin{(c^{M_{E}})}) dc dM_{E}}{M_{E}}", "derivation": "h{(c,M_{E})} = \\sin{(c^{M_{E}})} and - M_{E} + h{(c,M_{E})} = - M_{E} + \\sin{(c^{M_{E}})} and \\int (- M_{E} + h{(c,M_{E})}) dc = \\int (- M_{E} + \\sin{(c^{M_{E}})}) dc and \\iint (- M_{E} + h{(c,M_{E})}) dc dM_{E} = \\iint (- M_{E} + \\sin{(c^{M_{E}})}) dc dM_{E} and \\frac{\\iint (- M_{E} + h{(c,M_{E})}) dc dM_{E}}{M_{E}} = \\frac{\\iint (- M_{E} + \\sin{(c^{M_{E}})}) dc dM_{E}}{M_{E}} and \\frac{h{(c,M_{E})} \\iint (- M_{E} + h{(c,M_{E})}) dc dM_{E}}{M_{E}} = \\frac{h{(c,M_{E})} \\iint (- M_{E} + \\sin{(c^{M_{E}})}) dc dM_{E}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True))))"], [["minus", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True)))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('c', commutative=True))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["divide", 4, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["times", 5, "Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('h')(Symbol('c', commutative=True), Symbol('M_E', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Pow(Symbol('c', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi,\\eta)} = \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi}, then derive \\int \\dot{\\mathbf{r}}{(\\phi,\\eta)} d\\eta = \\frac{\\eta^{2}}{4} + \\mathbf{S}, then obtain \\frac{\\eta^{2}}{16} + \\frac{\\mathbf{S}}{4} - \\int \\log{(\\cos{(t)})} dt = \\frac{\\int \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} d\\eta}{4} - \\int \\log{(\\cos{(t)})} dt", "derivation": "\\dot{\\mathbf{r}}{(\\phi,\\eta)} = \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} and \\int \\dot{\\mathbf{r}}{(\\phi,\\eta)} d\\eta = \\int \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} d\\eta and \\int \\dot{\\mathbf{r}}{(\\phi,\\eta)} d\\eta = \\frac{\\eta^{2}}{4} + \\mathbf{S} and \\frac{\\eta^{2}}{4} + \\mathbf{S} = \\int \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} d\\eta and \\frac{\\eta^{2}}{16} + \\frac{\\mathbf{S}}{4} = \\frac{\\int \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} d\\eta}{4} and \\frac{\\eta^{2}}{16} + \\frac{\\mathbf{S}}{4} - \\int \\log{(\\cos{(t)})} dt = \\frac{\\int \\frac{\\int \\eta \\phi d\\eta}{\\eta \\phi} d\\eta}{4} - \\int \\log{(\\cos{(t)})} dt", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Rational(1, 4), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 4), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["times", 4, "Rational(1, 4)"], "Equality(Add(Mul(Rational(1, 16), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Rational(1, 4), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Rational(1, 4), Integral(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["minus", 5, "Integral(log(cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Rational(1, 16), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Mul(Rational(1, 4), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integral(log(cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))), Add(Mul(Rational(1, 4), Integral(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(Integer(-1), Integral(log(cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(I,F_{N},\\hat{p}_0)} = \\hat{p}_0 (F_{N} - I), then obtain (\\hat{p}_0 (F_{N} - I) - \\mathbf{P} + v_{2}^{B})^{\\hat{p}_0} = (F_{N} \\hat{p}_0 - I \\hat{p}_0 - \\mathbf{P} + v_{2}^{B})^{\\hat{p}_0}", "derivation": "\\hat{\\mathbf{x}}{(I,F_{N},\\hat{p}_0)} = \\hat{p}_0 (F_{N} - I) and \\hat{\\mathbf{x}}{(I,F_{N},\\hat{p}_0)} = F_{N} \\hat{p}_0 - I \\hat{p}_0 and - \\mathbf{P} + v_{2}^{B} + \\hat{\\mathbf{x}}{(I,F_{N},\\hat{p}_0)} = F_{N} \\hat{p}_0 - I \\hat{p}_0 - \\mathbf{P} + v_{2}^{B} and \\hat{p}_0 (F_{N} - I) - \\mathbf{P} + v_{2}^{B} = F_{N} \\hat{p}_0 - I \\hat{p}_0 - \\mathbf{P} + v_{2}^{B} and (\\hat{p}_0 (F_{N} - I) - \\mathbf{P} + v_{2}^{B})^{\\hat{p}_0} = (F_{N} \\hat{p}_0 - I \\hat{p}_0 - \\mathbf{P} + v_{2}^{B})^{\\hat{p}_0}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["expand", 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('I', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Mul(Symbol('F_N', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('B', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(E_{\\lambda})} = E_{\\lambda}, then obtain 1 + \\frac{E_{\\lambda}^{E_{\\lambda}} \\operatorname{F_{c}}^{E_{\\lambda}}{(E_{\\lambda})}}{E_{\\lambda}} = 1 + \\frac{E_{\\lambda}^{2 E_{\\lambda}}}{E_{\\lambda}}", "derivation": "\\operatorname{F_{c}}{(E_{\\lambda})} = E_{\\lambda} and \\operatorname{F_{c}}^{E_{\\lambda}}{(E_{\\lambda})} = E_{\\lambda}^{E_{\\lambda}} and E_{\\lambda}^{E_{\\lambda}} \\operatorname{F_{c}}^{E_{\\lambda}}{(E_{\\lambda})} = E_{\\lambda}^{2 E_{\\lambda}} and \\frac{E_{\\lambda}^{E_{\\lambda}} \\operatorname{F_{c}}^{E_{\\lambda}}{(E_{\\lambda})}}{E_{\\lambda}} = \\frac{E_{\\lambda}^{2 E_{\\lambda}}}{E_{\\lambda}} and 1 + \\frac{E_{\\lambda}^{E_{\\lambda}} \\operatorname{F_{c}}^{E_{\\lambda}}{(E_{\\lambda})}}{E_{\\lambda}} = 1 + \\frac{E_{\\lambda}^{2 E_{\\lambda}}}{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["times", 2, "Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('F_c')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Pow(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('F_c')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Function('F_c')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given f{(s,Z)} = \\int Z s ds and \\operatorname{y^{\\prime}}{(s,Z)} = (\\frac{\\partial}{\\partial s} f{(s,Z)})^{s}, then obtain \\cos{((\\frac{\\partial}{\\partial s} f{(s,Z)})^{s})} = \\cos{(\\operatorname{y^{\\prime}}{(s,Z)})}", "derivation": "f{(s,Z)} = \\int Z s ds and \\frac{\\partial}{\\partial s} f{(s,Z)} = \\frac{\\partial}{\\partial s} \\int Z s ds and (\\frac{\\partial}{\\partial s} f{(s,Z)})^{s} = (\\frac{\\partial}{\\partial s} \\int Z s ds)^{s} and \\cos{((\\frac{\\partial}{\\partial s} f{(s,Z)})^{s})} = \\cos{((\\frac{\\partial}{\\partial s} \\int Z s ds)^{s})} and \\operatorname{y^{\\prime}}{(s,Z)} = (\\frac{\\partial}{\\partial s} f{(s,Z)})^{s} and \\operatorname{y^{\\prime}}{(s,Z)} = (\\frac{\\partial}{\\partial s} \\int Z s ds)^{s} and \\cos{((\\frac{\\partial}{\\partial s} f{(s,Z)})^{s})} = \\cos{(\\operatorname{y^{\\prime}}{(s,Z)})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Integral(Mul(Symbol('Z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Derivative(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('Z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Derivative(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))), cos(Pow(Derivative(Integral(Mul(Symbol('Z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Pow(Derivative(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('y^{\\\\prime}')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('Z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(cos(Pow(Derivative(Function('f')(Symbol('s', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))), cos(Function('y^{\\\\prime}')(Symbol('s', commutative=True), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\phi_2)} = e^{\\phi_2} and \\operatorname{g_{\\varepsilon}}{(\\phi_2)} = \\mathbf{P}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)}, then obtain \\int \\operatorname{g_{\\varepsilon}}{(\\phi_2)} d\\phi_2 = \\int (e^{\\phi_2} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)}) d\\phi_2", "derivation": "\\mathbf{P}{(\\phi_2)} = e^{\\phi_2} and \\mathbf{P}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)} = e^{\\phi_2} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)} and \\operatorname{g_{\\varepsilon}}{(\\phi_2)} = \\mathbf{P}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)} and \\operatorname{g_{\\varepsilon}}{(\\phi_2)} = e^{\\phi_2} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)} and \\int \\operatorname{g_{\\varepsilon}}{(\\phi_2)} d\\phi_2 = \\int (e^{\\phi_2} + \\frac{d}{d \\phi_2} \\mathbf{P}{(\\phi_2)}) d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)), Add(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)), Add(exp(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(exp(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} = \\cos{(\\mathbf{D} v_{t})}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{J}_P + \\sin{(\\int \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} dv_{t})}) = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{J}_P + \\sin{(\\int \\cos{(\\mathbf{D} v_{t})} dv_{t})})", "derivation": "\\operatorname{t_{1}}{(\\mathbf{D},v_{t})} = \\cos{(\\mathbf{D} v_{t})} and \\int \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} dv_{t} = \\int \\cos{(\\mathbf{D} v_{t})} dv_{t} and \\sin{(\\int \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} dv_{t})} = \\sin{(\\int \\cos{(\\mathbf{D} v_{t})} dv_{t})} and \\mathbf{J}_P + \\sin{(\\int \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} dv_{t})} = \\mathbf{J}_P + \\sin{(\\int \\cos{(\\mathbf{D} v_{t})} dv_{t})} and \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{J}_P + \\sin{(\\int \\operatorname{t_{1}}{(\\mathbf{D},v_{t})} dv_{t})}) = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{J}_P + \\sin{(\\int \\cos{(\\mathbf{D} v_{t})} dv_{t})})", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True)), cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('t_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), sin(Integral(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Integral(Function('t_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Integral(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Integral(Function('t_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Integral(cos(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(F_{N})} = \\cos{(F_{N})}, then obtain \\frac{F_{N} \\mathbf{B}{(F_{N})}}{\\cos{(F_{N})}} + F_{N} \\cos{(F_{N})} = F_{N} \\cos{(F_{N})} + F_{N}", "derivation": "\\mathbf{B}{(F_{N})} = \\cos{(F_{N})} and \\frac{\\mathbf{B}{(F_{N})}}{\\cos{(F_{N})}} = 1 and \\frac{F_{N} \\mathbf{B}{(F_{N})}}{\\cos{(F_{N})}} = F_{N} and \\frac{F_{N} \\mathbf{B}{(F_{N})}}{\\cos{(F_{N})}} + F_{N} \\cos{(F_{N})} = F_{N} \\cos{(F_{N})} + F_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["divide", 1, "cos(Symbol('F_N', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Pow(cos(Symbol('F_N', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Pow(cos(Symbol('F_N', commutative=True)), Integer(-1))), Symbol('F_N', commutative=True))"], [["add", 3, "Mul(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Pow(cos(Symbol('F_N', commutative=True)), Integer(-1))), Mul(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True)))), Add(Mul(Symbol('F_N', commutative=True), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given h{(a,T)} = T a, then derive (\\frac{\\partial}{\\partial T} h{(a,T)})^{T} = a^{T}, then obtain (\\frac{\\partial}{\\partial T} T a)^{T} - \\frac{1}{\\int h{(a,T)} da} = a^{T} - \\frac{1}{\\int h{(a,T)} da}", "derivation": "h{(a,T)} = T a and \\frac{\\partial}{\\partial T} h{(a,T)} = \\frac{\\partial}{\\partial T} T a and (\\frac{\\partial}{\\partial T} h{(a,T)})^{T} = (\\frac{\\partial}{\\partial T} T a)^{T} and (\\frac{\\partial}{\\partial T} h{(a,T)})^{T} = a^{T} and (\\frac{\\partial}{\\partial T} T a)^{T} = a^{T} and (\\frac{\\partial}{\\partial T} T a)^{T} - \\frac{1}{\\int h{(a,T)} da} = a^{T} - \\frac{1}{\\int h{(a,T)} da}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Derivative(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Pow(Derivative(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('T', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('T', commutative=True)))"], [["minus", 5, "Pow(Integral(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))"], "Equality(Add(Pow(Derivative(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('T', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1)))), Add(Pow(Symbol('a', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('h')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(v_{x},\\theta)} = e^{\\theta + v_{x}}, then obtain \\frac{\\partial}{\\partial \\theta} \\operatorname{E_{\\lambda}}^{v_{x}}{(v_{x},\\theta)} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)} = \\frac{\\partial}{\\partial \\theta} (e^{\\theta + v_{x}})^{v_{x}} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)}", "derivation": "\\operatorname{E_{\\lambda}}{(v_{x},\\theta)} = e^{\\theta + v_{x}} and \\operatorname{E_{\\lambda}}^{v_{x}}{(v_{x},\\theta)} = (e^{\\theta + v_{x}})^{v_{x}} and \\operatorname{E_{\\lambda}}^{v_{x}}{(v_{x},\\theta)} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)} = (e^{\\theta + v_{x}})^{v_{x}} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)} and \\frac{\\partial}{\\partial \\theta} \\operatorname{E_{\\lambda}}^{v_{x}}{(v_{x},\\theta)} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)} = \\frac{\\partial}{\\partial \\theta} (e^{\\theta + v_{x}})^{v_{x}} \\frac{\\partial}{\\partial v_{x}} \\operatorname{E_{\\lambda}}{(v_{x},\\theta)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), exp(Add(Symbol('\\\\theta', commutative=True), Symbol('v_x', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('v_x', commutative=True)), Pow(exp(Add(Symbol('\\\\theta', commutative=True), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["times", 2, "Derivative(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('v_x', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Pow(exp(Add(Symbol('\\\\theta', commutative=True), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('v_x', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Pow(exp(Add(Symbol('\\\\theta', commutative=True), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('v_x', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} = \\mathbb{I} + \\sin{(\\hat{H})}, then obtain (- \\mathbb{I} + \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} - \\sin{(\\hat{H})}) \\sin{(\\hat{H})} - \\sin{(\\hat{H})} = - \\sin{(\\hat{H})}", "derivation": "\\mathbf{J}_f{(\\hat{H},\\mathbb{I})} = \\mathbb{I} + \\sin{(\\hat{H})} and \\hat{H} + \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} = \\hat{H} + \\mathbb{I} + \\sin{(\\hat{H})} and - \\mathbb{I} + \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} - \\sin{(\\hat{H})} = 0 and (- \\mathbb{I} + \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} - \\sin{(\\hat{H})}) \\sin{(\\hat{H})} = 0 and (- \\mathbb{I} + \\mathbf{J}_f{(\\hat{H},\\mathbb{I})} - \\sin{(\\hat{H})}) \\sin{(\\hat{H})} - \\sin{(\\hat{H})} = - \\sin{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Integer(0))"], [["times", 3, "sin(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), sin(Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["add", 4, "Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), sin(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain \\hat{x}^{3}{(\\rho_f)} = \\hat{x}^{2}{(\\rho_f)} \\sin{(\\rho_f)}", "derivation": "\\hat{x}{(\\rho_f)} = \\sin{(\\rho_f)} and \\hat{x}^{2}{(\\rho_f)} = \\hat{x}{(\\rho_f)} \\sin{(\\rho_f)} and \\hat{x}^{2}{(\\rho_f)} \\sin{(\\rho_f)} = \\hat{x}{(\\rho_f)} \\sin^{2}{(\\rho_f)} and \\hat{x}^{3}{(\\rho_f)} = \\hat{x}^{2}{(\\rho_f)} \\sin{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Mul(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "Mul(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), sin(Symbol('\\\\rho_f', commutative=True))), Mul(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), Pow(sin(Symbol('\\\\rho_f', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\hat{x}')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), sin(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hbar)} = \\hbar, then derive (- U - \\frac{\\hbar^{2}}{2} + \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar)^{\\hbar} = 0^{\\hbar}, then obtain \\int 0^{\\hbar} d\\hbar + \\int (- U - \\frac{\\hbar^{2}}{2} + \\int \\hbar d\\hbar)^{\\hbar} d\\hbar = 2 \\int 0^{\\hbar} d\\hbar", "derivation": "\\operatorname{F_{c}}{(\\hbar)} = \\hbar and \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar = \\int \\hbar d\\hbar and - \\int \\hbar d\\hbar + \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar = 0 and (- \\int \\hbar d\\hbar + \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar)^{\\hbar} = 0^{\\hbar} and (- U - \\frac{\\hbar^{2}}{2} + \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar)^{\\hbar} = 0^{\\hbar} and \\int (- U - \\frac{\\hbar^{2}}{2} + \\int \\operatorname{F_{c}}{(\\hbar)} d\\hbar)^{\\hbar} d\\hbar = \\int 0^{\\hbar} d\\hbar and \\int (- U - \\frac{\\hbar^{2}}{2} + \\int \\hbar d\\hbar)^{\\hbar} d\\hbar = \\int 0^{\\hbar} d\\hbar and \\int 0^{\\hbar} d\\hbar + \\int (- U - \\frac{\\hbar^{2}}{2} + \\int \\hbar d\\hbar)^{\\hbar} d\\hbar = 2 \\int 0^{\\hbar} d\\hbar", "srepr_derivation": [["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True)))), Integral(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True)))), Integral(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Integral(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Integral(Function('F_c')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 7, "Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Integral(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(2), Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given J{(F_{N})} = \\cos{(\\cos{(F_{N})})}, then obtain \\frac{d^{2}}{d F_{N}^{2}} (F_{N} J{(F_{N})} - \\cos{(F_{N})}) = \\frac{d^{2}}{d F_{N}^{2}} (F_{N} \\cos{(\\cos{(F_{N})})} - \\cos{(F_{N})})", "derivation": "J{(F_{N})} = \\cos{(\\cos{(F_{N})})} and F_{N} J{(F_{N})} = F_{N} \\cos{(\\cos{(F_{N})})} and F_{N} J{(F_{N})} - \\cos{(F_{N})} = F_{N} \\cos{(\\cos{(F_{N})})} - \\cos{(F_{N})} and \\frac{d}{d F_{N}} (F_{N} J{(F_{N})} - \\cos{(F_{N})}) = \\frac{d}{d F_{N}} (F_{N} \\cos{(\\cos{(F_{N})})} - \\cos{(F_{N})}) and \\frac{d^{2}}{d F_{N}^{2}} (F_{N} J{(F_{N})} - \\cos{(F_{N})}) = \\frac{d^{2}}{d F_{N}^{2}} (F_{N} \\cos{(\\cos{(F_{N})})} - \\cos{(F_{N})})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('F_N', commutative=True)), cos(cos(Symbol('F_N', commutative=True))))"], [["times", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Function('J')(Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), cos(cos(Symbol('F_N', commutative=True)))))"], [["minus", 2, "cos(Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Symbol('F_N', commutative=True), Function('J')(Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Add(Mul(Symbol('F_N', commutative=True), cos(cos(Symbol('F_N', commutative=True)))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))))"], [["differentiate", 3, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_N', commutative=True), Function('J')(Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_N', commutative=True), cos(cos(Symbol('F_N', commutative=True)))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_N', commutative=True), Function('J')(Symbol('F_N', commutative=True))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('F_N', commutative=True), cos(cos(Symbol('F_N', commutative=True)))), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(2))))"]]}, {"prompt": "Given i{(v_{t},\\rho)} = \\rho^{v_{t}}, then derive \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)} = \\frac{\\rho^{v_{t}} v_{t}}{\\rho}, then obtain (- v_{t} - i{(v_{t},\\rho)} + \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)})^{\\rho} = (- v_{t} - i{(v_{t},\\rho)} + \\frac{\\rho^{v_{t}} v_{t}}{\\rho})^{\\rho}", "derivation": "i{(v_{t},\\rho)} = \\rho^{v_{t}} and \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)} = \\frac{\\partial}{\\partial \\rho} \\rho^{v_{t}} and \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)} = \\frac{\\rho^{v_{t}} v_{t}}{\\rho} and - v_{t} - i{(v_{t},\\rho)} + \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)} = - v_{t} - i{(v_{t},\\rho)} + \\frac{\\rho^{v_{t}} v_{t}}{\\rho} and (- v_{t} - i{(v_{t},\\rho)} + \\frac{\\partial}{\\partial \\rho} i{(v_{t},\\rho)})^{\\rho} = (- v_{t} - i{(v_{t},\\rho)} + \\frac{\\rho^{v_{t}} v_{t}}{\\rho})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('v_t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["minus", 3, "Add(Symbol('v_t', commutative=True), Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Derivative(Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Derivative(Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('v_t', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\rho_b,t)} = \\rho_b \\cos{(t)}, then obtain (\\rho_b^{2} + \\rho_b \\mathbf{A}{(\\rho_b,t)}) \\mathbf{A}{(\\rho_b,t)} \\int \\mathbf{A}{(\\rho_b,t)} d\\rho_b = (\\rho_b^{2} \\cos{(t)} + \\rho_b^{2}) \\mathbf{A}{(\\rho_b,t)} \\int \\mathbf{A}{(\\rho_b,t)} d\\rho_b", "derivation": "\\mathbf{A}{(\\rho_b,t)} = \\rho_b \\cos{(t)} and \\rho_b \\mathbf{A}{(\\rho_b,t)} = \\rho_b^{2} \\cos{(t)} and \\rho_b^{2} + \\rho_b \\mathbf{A}{(\\rho_b,t)} = \\rho_b^{2} \\cos{(t)} + \\rho_b^{2} and (\\rho_b^{2} + \\rho_b \\mathbf{A}{(\\rho_b,t)}) \\mathbf{A}{(\\rho_b,t)} = (\\rho_b^{2} \\cos{(t)} + \\rho_b^{2}) \\mathbf{A}{(\\rho_b,t)} and (\\rho_b^{2} + \\rho_b \\mathbf{A}{(\\rho_b,t)}) \\mathbf{A}{(\\rho_b,t)} \\int \\mathbf{A}{(\\rho_b,t)} d\\rho_b = (\\rho_b^{2} \\cos{(t)} + \\rho_b^{2}) \\mathbf{A}{(\\rho_b,t)} \\int \\mathbf{A}{(\\rho_b,t)} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), cos(Symbol('t', commutative=True))))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), cos(Symbol('t', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)))), Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), cos(Symbol('t', commutative=True))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))))"], [["times", 3, "Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Add(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)))), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True))), Mul(Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), cos(Symbol('t', commutative=True))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True))))"], [["times", 4, "Integral(Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Add(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)))), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Add(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(2)), cos(Symbol('t', commutative=True))), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('\\\\rho_b', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given v{(\\phi_1,n)} = n + \\log{(\\phi_1)}, then obtain v^{2}{(\\phi_1,n)} = (n + \\log{(\\phi_1)})^{2}", "derivation": "v{(\\phi_1,n)} = n + \\log{(\\phi_1)} and (n + \\log{(\\phi_1)}) v{(\\phi_1,n)} = (n + \\log{(\\phi_1)})^{2} and v^{2}{(\\phi_1,n)} = (n + \\log{(\\phi_1)}) v{(\\phi_1,n)} and v^{2}{(\\phi_1,n)} = (n + \\log{(\\phi_1)})^{2}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True))), Pow(Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Integer(2)))"], [["times", 1, "Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True))"], "Equality(Pow(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Mul(Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('v')(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Integer(2)), Pow(Add(Symbol('n', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\hat{H}{(S,A)} = A \\cos{(S)}, then obtain - \\cos{(S)} = A S \\cos{(S)} - S \\hat{H}{(S,A)} - \\cos{(S)}", "derivation": "\\hat{H}{(S,A)} = A \\cos{(S)} and S \\hat{H}{(S,A)} = A S \\cos{(S)} and 0 = A S \\cos{(S)} - S \\hat{H}{(S,A)} and - \\cos{(S)} = A S \\cos{(S)} - S \\hat{H}{(S,A)} - \\cos{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), cos(Symbol('S', commutative=True))))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('S', commutative=True), cos(Symbol('S', commutative=True))))"], [["minus", 2, "Mul(Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('A', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('A', commutative=True), Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('A', commutative=True)))))"], [["minus", 3, "cos(Symbol('S', commutative=True))"], "Equality(Mul(Integer(-1), cos(Symbol('S', commutative=True))), Add(Mul(Symbol('A', commutative=True), Symbol('S', commutative=True), cos(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), cos(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given v{(v_{1})} = \\sin{(\\log{(v_{1})})}, then derive \\int v{(v_{1})} dv_{1} = \\hat{p} + \\frac{v_{1} \\sin{(\\log{(v_{1})})}}{2} - \\frac{v_{1} \\cos{(\\log{(v_{1})})}}{2}, then obtain \\iint v{(v_{1})} dv_{1} d\\hat{p} = \\int (\\hat{p} + \\frac{v_{1} v{(v_{1})}}{2} - \\frac{v_{1} \\cos{(\\log{(v_{1})})}}{2}) d\\hat{p}", "derivation": "v{(v_{1})} = \\sin{(\\log{(v_{1})})} and \\int v{(v_{1})} dv_{1} = \\int \\sin{(\\log{(v_{1})})} dv_{1} and \\int v{(v_{1})} dv_{1} = \\hat{p} + \\frac{v_{1} \\sin{(\\log{(v_{1})})}}{2} - \\frac{v_{1} \\cos{(\\log{(v_{1})})}}{2} and \\int v{(v_{1})} dv_{1} = \\hat{p} + \\frac{v_{1} v{(v_{1})}}{2} - \\frac{v_{1} \\cos{(\\log{(v_{1})})}}{2} and \\iint v{(v_{1})} dv_{1} d\\hat{p} = \\int (\\hat{p} + \\frac{v_{1} v{(v_{1})}}{2} - \\frac{v_{1} \\cos{(\\log{(v_{1})})}}{2}) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('v_1', commutative=True)), sin(log(Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('v')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('v_1', commutative=True), sin(log(Symbol('v_1', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('v_1', commutative=True), cos(log(Symbol('v_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('v_1', commutative=True), Function('v')(Symbol('v_1', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('v_1', commutative=True), cos(log(Symbol('v_1', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('v_1', commutative=True), Function('v')(Symbol('v_1', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('v_1', commutative=True), cos(log(Symbol('v_1', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\varphi^*)} = \\log{(\\varphi^*)}, then obtain (\\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\psi^{*}{(\\varphi^*)})})^{\\varphi^*} = (\\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\log{(\\varphi^*)})})^{\\varphi^*}", "derivation": "\\psi^{*}{(\\varphi^*)} = \\log{(\\varphi^*)} and - \\psi^{*}{(\\varphi^*)} = - \\log{(\\varphi^*)} and - \\sin{(\\psi^{*}{(\\varphi^*)})} = - \\sin{(\\log{(\\varphi^*)})} and \\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\psi^{*}{(\\varphi^*)})} = \\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\log{(\\varphi^*)})} and (\\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\psi^{*}{(\\varphi^*)})})^{\\varphi^*} = (\\psi^{*}{(\\varphi^*)} - \\log{(\\varphi^*)} - \\sin{(\\log{(\\varphi^*)})})^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), sin(log(Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 3, "Add(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))))"], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), sin(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True))))), Add(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\varphi^*', commutative=True))))))"], [["power", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Add(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), sin(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True))))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Function('\\\\psi^*')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\varphi^*', commutative=True))))), Symbol('\\\\varphi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(h)} = \\log{(h)} and \\mathbf{S}{(h)} = \\frac{d}{d h} (\\operatorname{f_{E}}{(h)} + 2 \\log{(h)}), then derive \\mathbf{S}{(h)} = 3 \\frac{d}{d h} \\operatorname{f_{E}}{(h)}, then obtain \\mathbf{S}^{h}{(h)} = (3 \\frac{d}{d h} \\operatorname{f_{E}}{(h)})^{h}", "derivation": "\\operatorname{f_{E}}{(h)} = \\log{(h)} and 2 \\operatorname{f_{E}}{(h)} + \\log{(h)} = \\operatorname{f_{E}}{(h)} + 2 \\log{(h)} and \\frac{d}{d h} (2 \\operatorname{f_{E}}{(h)} + \\log{(h)}) = \\frac{d}{d h} (\\operatorname{f_{E}}{(h)} + 2 \\log{(h)}) and \\mathbf{S}{(h)} = \\frac{d}{d h} (\\operatorname{f_{E}}{(h)} + 2 \\log{(h)}) and \\mathbf{S}{(h)} = \\frac{d}{d h} (2 \\operatorname{f_{E}}{(h)} + \\log{(h)}) and \\mathbf{S}{(h)} = \\frac{d}{d h} 3 \\operatorname{f_{E}}{(h)} and \\mathbf{S}{(h)} = 3 \\frac{d}{d h} \\operatorname{f_{E}}{(h)} and \\mathbf{S}^{h}{(h)} = (3 \\frac{d}{d h} \\operatorname{f_{E}}{(h)})^{h}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["add", 1, "Add(Function('f_E')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('f_E')(Symbol('h', commutative=True))), log(Symbol('h', commutative=True))), Add(Function('f_E')(Symbol('h', commutative=True)), Mul(Integer(2), log(Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('f_E')(Symbol('h', commutative=True))), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Function('f_E')(Symbol('h', commutative=True)), Mul(Integer(2), log(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Derivative(Add(Function('f_E')(Symbol('h', commutative=True)), Mul(Integer(2), log(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Derivative(Add(Mul(Integer(2), Function('f_E')(Symbol('h', commutative=True))), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Derivative(Mul(Integer(3), Function('f_E')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Mul(Integer(3), Derivative(Function('f_E')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["power", 7, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Integer(3), Derivative(Function('f_E')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(i)} = \\sin{(i)} and \\sigma_{p}{(i)} = \\sin{(i)}, then obtain \\hat{x}_0{(i)} \\sin^{2}{(i)} = \\sin^{3}{(i)}", "derivation": "\\hat{x}_0{(i)} = \\sin{(i)} and \\sigma_{p}{(i)} = \\sin{(i)} and \\hat{x}_0{(i)} = \\sigma_{p}{(i)} and \\sigma_{p}{(i)} \\sin{(i)} = \\sin^{2}{(i)} and \\hat{x}_0{(i)} \\sin{(i)} = \\sin^{2}{(i)} and \\hat{x}_0{(i)} \\sin^{2}{(i)} = \\sin^{3}{(i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{x}_0')(Symbol('i', commutative=True)), Function('\\\\sigma_p')(Symbol('i', commutative=True)))"], [["times", 2, "sin(Symbol('i', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Pow(sin(Symbol('i', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True))), Pow(sin(Symbol('i', commutative=True)), Integer(2)))"], [["times", 5, "sin(Symbol('i', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('i', commutative=True)), Pow(sin(Symbol('i', commutative=True)), Integer(2))), Pow(sin(Symbol('i', commutative=True)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(W,L)} = L W and m{(W,L)} = \\int W dL, then obtain L W + \\cos{(\\frac{m{(W,L)}}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL = L W + \\cos{(\\frac{\\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL", "derivation": "\\operatorname{x^{{\\}'}}{(W,L)} = L W and \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} = W and \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL = \\int W dL and m{(W,L)} = \\int W dL and \\frac{m{(W,L)}}{L} = \\frac{\\int W dL}{L} and \\cos{(\\frac{m{(W,L)}}{L})} = \\cos{(\\frac{\\int W dL}{L})} and L W + \\cos{(\\frac{m{(W,L)}}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL = L W + \\cos{(\\frac{\\int W dL}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL and L W + \\cos{(\\frac{m{(W,L)}}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL = L W + \\cos{(\\frac{\\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL}{L})} + \\int \\frac{\\operatorname{x^{{\\}'}}{(W,L)}}{L} dL", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)))"], [["divide", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Symbol('W', commutative=True))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Symbol('W', commutative=True), Tuple(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('W', commutative=True), Symbol('L', commutative=True)), Integral(Symbol('W', commutative=True), Tuple(Symbol('L', commutative=True))))"], [["times", 4, "Pow(Symbol('L', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('m')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Symbol('W', commutative=True), Tuple(Symbol('L', commutative=True)))))"], [["cos", 5], "Equality(cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('m')(Symbol('W', commutative=True), Symbol('L', commutative=True)))), cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Symbol('W', commutative=True), Tuple(Symbol('L', commutative=True))))))"], [["add", 6, "Add(Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], "Equality(Add(Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)), cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('m')(Symbol('W', commutative=True), Symbol('L', commutative=True)))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))), Add(Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)), cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Symbol('W', commutative=True), Tuple(Symbol('L', commutative=True))))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)), cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('m')(Symbol('W', commutative=True), Symbol('L', commutative=True)))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))), Add(Mul(Symbol('L', commutative=True), Symbol('W', commutative=True)), cos(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))), Integral(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('W', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\omega,H)} = H + \\omega and \\mathbf{J}{(\\omega)} = - \\omega, then obtain (\\frac{\\partial}{\\partial \\omega} (\\mathbf{J}{(\\omega)} - \\frac{1}{H + \\omega}))^{H} = (\\frac{\\partial}{\\partial \\omega} (- \\omega - \\frac{1}{H + \\omega}))^{H}", "derivation": "\\Psi_{\\lambda}{(\\omega,H)} = H + \\omega and \\mathbf{J}{(\\omega)} = - \\omega and \\mathbf{J}{(\\omega)} - \\frac{1}{\\Psi_{\\lambda}{(\\omega,H)}} = - \\omega - \\frac{1}{\\Psi_{\\lambda}{(\\omega,H)}} and \\mathbf{J}{(\\omega)} - \\frac{1}{H + \\omega} = - \\omega - \\frac{1}{H + \\omega} and \\frac{\\partial}{\\partial \\omega} (\\mathbf{J}{(\\omega)} - \\frac{1}{H + \\omega}) = \\frac{\\partial}{\\partial \\omega} (- \\omega - \\frac{1}{H + \\omega}) and (\\frac{\\partial}{\\partial \\omega} (\\mathbf{J}{(\\omega)} - \\frac{1}{H + \\omega}))^{H} = (\\frac{\\partial}{\\partial \\omega} (- \\omega - \\frac{1}{H + \\omega}))^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('H', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('H', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\omega', commutative=True), Symbol('H', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given v{(\\delta,\\hat{H}_{\\lambda})} = \\delta \\hat{H}_{\\lambda}, then derive \\frac{\\partial}{\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}, then obtain \\hat{H}_{\\lambda}^{2} \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}^{2} \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda}", "derivation": "v{(\\delta,\\hat{H}_{\\lambda})} = \\delta \\hat{H}_{\\lambda} and \\frac{\\partial}{\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\delta} \\delta \\hat{H}_{\\lambda} and \\frac{\\partial}{\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} and \\hat{H}_{\\lambda}^{2} \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}\\partial \\delta} v{(\\delta,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}^{2} \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["differentiate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["times", 4, "Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2)), Derivative(Function('v')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2)), Derivative(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(\\chi)} = \\cos{(\\sin{(\\chi)})}, then obtain \\frac{1}{2 \\hat{H}{(\\chi)}} = \\frac{\\frac{d}{d \\chi} \\int (\\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})}) d\\chi}{2 \\hat{H}{(\\chi)} \\frac{d}{d \\chi} \\int 2 \\hat{H}{(\\chi)} d\\chi}", "derivation": "\\hat{H}{(\\chi)} = \\cos{(\\sin{(\\chi)})} and 2 \\hat{H}{(\\chi)} = \\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})} and \\int 2 \\hat{H}{(\\chi)} d\\chi = \\int (\\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})}) d\\chi and \\frac{d}{d \\chi} \\int 2 \\hat{H}{(\\chi)} d\\chi = \\frac{d}{d \\chi} \\int (\\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})}) d\\chi and \\frac{\\frac{d}{d \\chi} \\int 2 \\hat{H}{(\\chi)} d\\chi}{2 \\hat{H}{(\\chi)}} = \\frac{\\frac{d}{d \\chi} \\int (\\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})}) d\\chi}{2 \\hat{H}{(\\chi)}} and \\frac{1}{2 \\hat{H}{(\\chi)}} = \\frac{\\frac{d}{d \\chi} \\int (\\hat{H}{(\\chi)} + \\cos{(\\sin{(\\chi)})}) d\\chi}{2 \\hat{H}{(\\chi)} \\frac{d}{d \\chi} \\int 2 \\hat{H}{(\\chi)} d\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Add(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Derivative(Integral(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Derivative(Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["divide", 5, "Derivative(Integral(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Mul(Rational(1, 2), Pow(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), Integer(-1)), Derivative(Integral(Add(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Derivative(Integral(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\theta{(T,m)} = T + \\sin{(m)}, then obtain (\\frac{T + \\sin{(m)}}{T})^{T} (\\frac{\\theta{(T,m)}}{T})^{T} = (\\frac{T + \\sin{(m)}}{T})^{2 T}", "derivation": "\\theta{(T,m)} = T + \\sin{(m)} and \\frac{\\theta{(T,m)}}{T} = \\frac{T + \\sin{(m)}}{T} and (\\frac{\\theta{(T,m)}}{T})^{T} = (\\frac{T + \\sin{(m)}}{T})^{T} and (\\frac{T + \\sin{(m)}}{T})^{T} (\\frac{\\theta{(T,m)}}{T})^{T} = (\\frac{T + \\sin{(m)}}{T})^{2 T}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True))))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True)))))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Symbol('T', commutative=True)), Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True)))), Symbol('T', commutative=True)))"], [["times", 3, "Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True)))), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True)))), Symbol('T', commutative=True)), Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Symbol('T', commutative=True))), Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), sin(Symbol('m', commutative=True)))), Mul(Integer(2), Symbol('T', commutative=True))))"]]}, {"prompt": "Given W{(z)} = \\sin{(z)}, then derive \\int W{(z)} dz = \\mu - \\cos{(z)}, then obtain \\frac{\\int \\sin{(z)} dz}{\\int W{(z)} dz} = 1", "derivation": "W{(z)} = \\sin{(z)} and \\int W{(z)} dz = \\int \\sin{(z)} dz and \\int W{(z)} dz = \\mu - \\cos{(z)} and (\\int W{(z)} dz) \\int \\sin{(z)} dz = (\\mu - \\cos{(z)}) \\int \\sin{(z)} dz and (\\int W{(z)} dz)^{2} = (\\mu - \\cos{(z)}) \\int W{(z)} dz and (\\int \\sin{(z)} dz)^{2} = (\\mu - \\cos{(z)}) \\int \\sin{(z)} dz and (\\int \\sin{(z)} dz)^{2} = (\\int W{(z)} dz) \\int \\sin{(z)} dz and \\frac{\\int \\sin{(z)} dz}{\\int W{(z)} dz} = 1", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["times", 3, "Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Mul(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(2)), Mul(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["divide", 7, "Mul(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], "Equality(Mul(Pow(Integral(Function('W')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(-1)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(z)} = \\cos{(\\log{(z)})} and \\rho_{b}{(z)} = (- z + \\operatorname{C_{2}}{(z)})^{z}, then obtain (z - \\operatorname{C_{2}}{(z)} + \\rho_{b}{(z)})^{z} = (z + (- z + \\cos{(\\log{(z)})})^{z} - \\operatorname{C_{2}}{(z)})^{z}", "derivation": "\\operatorname{C_{2}}{(z)} = \\cos{(\\log{(z)})} and - z + \\operatorname{C_{2}}{(z)} = - z + \\cos{(\\log{(z)})} and (- z + \\operatorname{C_{2}}{(z)})^{z} = (- z + \\cos{(\\log{(z)})})^{z} and z + (- z + \\operatorname{C_{2}}{(z)})^{z} - \\operatorname{C_{2}}{(z)} = z + (- z + \\cos{(\\log{(z)})})^{z} - \\operatorname{C_{2}}{(z)} and (z + (- z + \\operatorname{C_{2}}{(z)})^{z} - \\operatorname{C_{2}}{(z)})^{z} = (z + (- z + \\cos{(\\log{(z)})})^{z} - \\operatorname{C_{2}}{(z)})^{z} and \\rho_{b}{(z)} = (- z + \\operatorname{C_{2}}{(z)})^{z} and (z - \\operatorname{C_{2}}{(z)} + \\rho_{b}{(z)})^{z} = (z + (- z + \\cos{(\\log{(z)})})^{z} - \\operatorname{C_{2}}{(z)})^{z}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True))))"], [["minus", 1, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True)))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True)))), Symbol('z', commutative=True)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True)))"], "Equality(Add(Symbol('z', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True)))), Add(Symbol('z', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True)))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Symbol('z', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Add(Symbol('z', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True)))), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('C_2')(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Add(Symbol('z', commutative=True), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True))), Function('\\\\rho_b')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Add(Symbol('z', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(log(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), Function('C_2')(Symbol('z', commutative=True)))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(F_{H})} = \\sin{(F_{H})}, then obtain \\int - \\frac{\\frac{\\mathbf{v}{(F_{H})}}{\\sin{(F_{H})}} - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} dF_{H} = \\int - \\frac{1 - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} dF_{H}", "derivation": "\\mathbf{v}{(F_{H})} = \\sin{(F_{H})} and \\frac{\\mathbf{v}{(F_{H})}}{\\sin{(F_{H})}} = 1 and \\frac{\\mathbf{v}{(F_{H})}}{\\sin{(F_{H})}} - \\frac{1}{\\sin{(F_{H})}} = 1 - \\frac{1}{\\sin{(F_{H})}} and - \\frac{\\frac{\\mathbf{v}{(F_{H})}}{\\sin{(F_{H})}} - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} = - \\frac{1 - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} and \\int - \\frac{\\frac{\\mathbf{v}{(F_{H})}}{\\sin{(F_{H})}} - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} dF_{H} = \\int - \\frac{1 - \\frac{1}{\\sin{(F_{H})}}}{\\sin{(F_{H})}} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["divide", 1, "sin(Symbol('F_H', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('F_H', commutative=True)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('\\\\mathbf{v}')(Symbol('F_H', commutative=True)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))))"], [["times", 3, "Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Add(Mul(Function('\\\\mathbf{v}')(Symbol('F_H', commutative=True)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Add(Mul(Function('\\\\mathbf{v}')(Symbol('F_H', commutative=True)), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)))), Pow(sin(Symbol('F_H', commutative=True)), Integer(-1))), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} = (f^{\\prime})^{\\mathbf{P}} + t, then obtain \\frac{d}{d v_{2}} r^{v_{2}}{(v_{2})} + \\frac{\\partial}{\\partial \\mathbf{P}} \\int \\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} df^{\\prime} = \\frac{d}{d v_{2}} r^{v_{2}}{(v_{2})} + \\frac{\\partial}{\\partial \\mathbf{P}} \\int ((f^{\\prime})^{\\mathbf{P}} + t) df^{\\prime}", "derivation": "\\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} = (f^{\\prime})^{\\mathbf{P}} + t and \\int \\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} df^{\\prime} = \\int ((f^{\\prime})^{\\mathbf{P}} + t) df^{\\prime} and \\frac{\\partial}{\\partial \\mathbf{P}} \\int \\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} df^{\\prime} = \\frac{\\partial}{\\partial \\mathbf{P}} \\int ((f^{\\prime})^{\\mathbf{P}} + t) df^{\\prime} and \\frac{d}{d v_{2}} r^{v_{2}}{(v_{2})} + \\frac{\\partial}{\\partial \\mathbf{P}} \\int \\mathbb{I}{(f^{\\prime},\\mathbf{P},t)} df^{\\prime} = \\frac{d}{d v_{2}} r^{v_{2}}{(v_{2})} + \\frac{\\partial}{\\partial \\mathbf{P}} \\int ((f^{\\prime})^{\\mathbf{P}} + t) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Pow(Function('r')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Function('r')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Add(Derivative(Pow(Function('r')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(r_{0})} = \\log{(r_{0})} and \\operatorname{A_{2}}{(r_{0})} = - \\operatorname{v_{y}}{(r_{0})} + \\log{(r_{0})}, then obtain \\operatorname{A_{2}}{(r_{0})} = - \\operatorname{A_{2}}{(r_{0})} - \\operatorname{v_{y}}{(r_{0})} + \\log{(r_{0})}", "derivation": "\\operatorname{v_{y}}{(r_{0})} = \\log{(r_{0})} and \\operatorname{A_{2}}{(r_{0})} = - \\operatorname{v_{y}}{(r_{0})} + \\log{(r_{0})} and \\operatorname{A_{2}}{(r_{0})} = 0 and \\operatorname{A_{2}}{(r_{0})} + \\operatorname{v_{y}}{(r_{0})} = \\operatorname{v_{y}}{(r_{0})} and \\operatorname{A_{2}}{(r_{0})} = - \\operatorname{A_{2}}{(r_{0})} - \\operatorname{v_{y}}{(r_{0})} + \\log{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Function('v_y')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_2')(Symbol('r_0', commutative=True)), Integer(0))"], [["minus", 3, "Mul(Integer(-1), Function('v_y')(Symbol('r_0', commutative=True)))"], "Equality(Add(Function('A_2')(Symbol('r_0', commutative=True)), Function('v_y')(Symbol('r_0', commutative=True))), Function('v_y')(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('A_2')(Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Function('A_2')(Symbol('r_0', commutative=True))), Mul(Integer(-1), Function('v_y')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given m{(\\varepsilon)} = \\varepsilon, then obtain - \\frac{\\varepsilon}{4 m{(\\varepsilon)}} = - \\frac{\\varepsilon^{\\frac{\\varepsilon + m{(\\varepsilon)}}{m{(\\varepsilon)}}}}{2 (\\varepsilon + m{(\\varepsilon)}) m{(\\varepsilon)}}", "derivation": "m{(\\varepsilon)} = \\varepsilon and 2 m{(\\varepsilon)} = \\varepsilon + m{(\\varepsilon)} and \\frac{1}{2} = \\frac{\\varepsilon}{2 m{(\\varepsilon)}} and \\frac{1}{2} = \\frac{\\varepsilon}{\\varepsilon + m{(\\varepsilon)}} and 2 = \\frac{\\varepsilon + m{(\\varepsilon)}}{m{(\\varepsilon)}} and - \\frac{\\varepsilon}{2} = - \\frac{\\varepsilon^{2}}{\\varepsilon + m{(\\varepsilon)}} and - \\frac{\\varepsilon}{4 m{(\\varepsilon)}} = - \\frac{\\varepsilon^{2}}{2 (\\varepsilon + m{(\\varepsilon)}) m{(\\varepsilon)}} and - \\frac{\\varepsilon}{4 m{(\\varepsilon)}} = - \\frac{\\varepsilon^{\\frac{\\varepsilon + m{(\\varepsilon)}}{m{(\\varepsilon)}}}}{2 (\\varepsilon + m{(\\varepsilon)}) m{(\\varepsilon)}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["add", 1, "Function('m')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(2), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('m')(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Symbol('\\\\varepsilon', commutative=True), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))))"], [["divide", 2, "Function('m')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(2), Mul(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1))))"], [["divide", 6, "Mul(Integer(2), Function('m')(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(-1), Rational(1, 4), Symbol('\\\\varepsilon', commutative=True), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Integer(-1), Rational(1, 4), Symbol('\\\\varepsilon', commutative=True), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Mul(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))), Pow(Add(Symbol('\\\\varepsilon', commutative=True), Function('m')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(l)} = \\cos{(l)}, then obtain \\frac{0^{l}}{l} = \\frac{(\\frac{- \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}}{\\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}})^{l}}{l}", "derivation": "\\operatorname{y^{\\prime}}{(l)} = \\cos{(l)} and 0 = - \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)} and \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)} = 2 \\cos{(l)} and 0 = \\frac{- \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}}{2 \\cos{(l)}} and 0^{l} = (\\frac{- \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}}{2 \\cos{(l)}})^{l} and \\frac{0^{l}}{l} = \\frac{(\\frac{- \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}}{2 \\cos{(l)}})^{l}}{l} and \\frac{0^{l}}{l} = \\frac{(\\frac{- \\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}}{\\operatorname{y^{\\prime}}{(l)} + \\cos{(l)}})^{l}}{l}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["minus", 1, "Function('y^{\\\\prime}')(Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))))"], [["add", 1, "cos(Symbol('l', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Mul(Integer(2), cos(Symbol('l', commutative=True))))"], [["divide", 2, "Mul(Integer(2), cos(Symbol('l', commutative=True)))"], "Equality(Integer(0), Mul(Rational(1, 2), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Integer(0), Symbol('l', commutative=True)), Pow(Mul(Rational(1, 2), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Symbol('l', commutative=True)))"], [["divide", 5, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Rational(1, 2), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Pow(cos(Symbol('l', commutative=True)), Integer(-1))), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Integer(0), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('l', commutative=True))), cos(Symbol('l', commutative=True))), Pow(Add(Function('y^{\\\\prime}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Integer(-1))), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}}, then derive \\operatorname{P_{g}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain \\frac{d}{d L_{\\varepsilon}} \\operatorname{P_{g}}{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{P_{g}}{(L_{\\varepsilon})}", "derivation": "\\operatorname{P_{g}}{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\operatorname{P_{g}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\operatorname{P_{g}}{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\operatorname{P_{g}}{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\operatorname{P_{g}}{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('P_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('P_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Function('P_g')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(\\hat{p}_0)} = \\sin{(\\cos{(\\hat{p}_0)})} and \\mathbf{r}{(F_{c},C_{d})} = C_{d} - F_{c}, then obtain \\frac{\\mathbf{r}{(F_{c},C_{d})}}{F_{c} (- F_{c} + \\mathbf{J}{(\\hat{p}_0)})} = \\frac{C_{d} - F_{c}}{F_{c} (- F_{c} + \\mathbf{J}{(\\hat{p}_0)})}", "derivation": "\\mathbf{J}{(\\hat{p}_0)} = \\sin{(\\cos{(\\hat{p}_0)})} and \\mathbf{r}{(F_{c},C_{d})} = C_{d} - F_{c} and - F_{c} + \\mathbf{J}{(\\hat{p}_0)} = - F_{c} + \\sin{(\\cos{(\\hat{p}_0)})} and \\frac{\\mathbf{r}{(F_{c},C_{d})}}{F_{c}} = \\frac{C_{d} - F_{c}}{F_{c}} and \\frac{\\mathbf{r}{(F_{c},C_{d})}}{F_{c} (- F_{c} + \\sin{(\\cos{(\\hat{p}_0)})})} = \\frac{C_{d} - F_{c}}{F_{c} (- F_{c} + \\sin{(\\cos{(\\hat{p}_0)})})} and \\frac{\\mathbf{r}{(F_{c},C_{d})}}{F_{c} (- F_{c} + \\mathbf{J}{(\\hat{p}_0)})} = \\frac{C_{d} - F_{c}}{F_{c} (- F_{c} + \\mathbf{J}{(\\hat{p}_0)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}_0', commutative=True)), sin(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_c', commutative=True))))"], [["minus", 1, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), sin(cos(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["divide", 2, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('C_d', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_c', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), sin(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), sin(cos(Symbol('\\\\hat{p}_0', commutative=True)))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('C_d', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), sin(cos(Symbol('\\\\hat{p}_0', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('F_c', commutative=True), Symbol('C_d', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(i,r)} = i + r and \\operatorname{E_{\\lambda}}{(r,i)} = r + \\operatorname{P_{g}}{(i,r)}, then obtain \\frac{2 r \\operatorname{E_{\\lambda}}{(r,i)}}{(i + r) \\sin{(r + \\operatorname{P_{g}}{(i,r)})}} = \\frac{2 r (i + 2 r)}{(i + r) \\sin{(r + \\operatorname{P_{g}}{(i,r)})}}", "derivation": "\\operatorname{P_{g}}{(i,r)} = i + r and r + \\operatorname{P_{g}}{(i,r)} = i + 2 r and \\operatorname{E_{\\lambda}}{(r,i)} = r + \\operatorname{P_{g}}{(i,r)} and 2 r \\operatorname{E_{\\lambda}}{(r,i)} = 2 r (r + \\operatorname{P_{g}}{(i,r)}) and 2 r \\operatorname{E_{\\lambda}}{(r,i)} = 2 r (i + 2 r) and \\frac{2 r \\operatorname{E_{\\lambda}}{(r,i)}}{i + r} = \\frac{2 r (i + 2 r)}{i + r} and \\frac{2 r \\operatorname{E_{\\lambda}}{(r,i)}}{(i + r) \\sin{(r + \\operatorname{P_{g}}{(i,r)})}} = \\frac{2 r (i + 2 r)}{(i + r) \\sin{(r + \\operatorname{P_{g}}{(i,r)})}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))"], [["add", 1, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True))), Add(Symbol('i', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('i', commutative=True)), Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], [["times", 3, "Mul(Integer(2), Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Symbol('r', commutative=True), Function('E_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True), Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Symbol('r', commutative=True), Function('E_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True), Add(Symbol('i', commutative=True), Mul(Integer(2), Symbol('r', commutative=True)))))"], [["divide", 5, "Add(Symbol('i', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Symbol('r', commutative=True), Pow(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), Symbol('r', commutative=True), Pow(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Add(Symbol('i', commutative=True), Mul(Integer(2), Symbol('r', commutative=True)))))"], [["divide", 6, "sin(Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], "Equality(Mul(Integer(2), Symbol('r', commutative=True), Pow(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('r', commutative=True), Symbol('i', commutative=True)), Pow(sin(Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True)))), Integer(-1))), Mul(Integer(2), Symbol('r', commutative=True), Pow(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Add(Symbol('i', commutative=True), Mul(Integer(2), Symbol('r', commutative=True))), Pow(sin(Add(Symbol('r', commutative=True), Function('P_g')(Symbol('i', commutative=True), Symbol('r', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} = V_{\\mathbf{B}} \\eta n_{1}, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\int \\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} dn_{1}}{\\int V_{\\mathbf{B}} \\eta n_{1} dn_{1}} = \\frac{d}{d V_{\\mathbf{B}}} 1", "derivation": "\\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} = V_{\\mathbf{B}} \\eta n_{1} and \\int \\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} dn_{1} = \\int V_{\\mathbf{B}} \\eta n_{1} dn_{1} and \\frac{\\int \\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} dn_{1}}{\\int V_{\\mathbf{B}} \\eta n_{1} dn_{1}} = 1 and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\int \\operatorname{f^{\\prime}}{(V_{\\mathbf{B}},n_{1},\\eta)} dn_{1}}{\\int V_{\\mathbf{B}} \\eta n_{1} dn_{1}} = \\frac{d}{d V_{\\mathbf{B}}} 1", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["divide", 2, "Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integer(-1)), Integral(Function('f^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Integer(1))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Mul(Pow(Integral(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integer(-1)), Integral(Function('f^{\\\\prime}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(v_{z})} = e^{v_{z}} and \\bar{\\h}{(v_{z})} = \\operatorname{A_{1}}{(v_{z})} - e^{v_{z}}, then obtain \\bar{\\h}{(v_{z})} = 0", "derivation": "\\operatorname{A_{1}}{(v_{z})} = e^{v_{z}} and \\operatorname{A_{1}}{(v_{z})} - e^{v_{z}} = 0 and \\bar{\\h}{(v_{z})} = \\operatorname{A_{1}}{(v_{z})} - e^{v_{z}} and \\bar{\\h}{(v_{z})} = 0", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["minus", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('v_z', commutative=True)), Add(Function('A_1')(Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hbar')(Symbol('v_z', commutative=True)), Integer(0))"]]}, {"prompt": "Given r{(\\Psi,v_{1})} = \\frac{\\Psi}{v_{1}}, then obtain \\Psi + \\frac{v_{1}^{2} r^{2}{(\\Psi,v_{1})}}{\\Psi^{2}} = \\Psi + 1", "derivation": "r{(\\Psi,v_{1})} = \\frac{\\Psi}{v_{1}} and \\frac{v_{1} r{(\\Psi,v_{1})}}{\\Psi} = 1 and \\frac{v_{1} r^{2}{(\\Psi,v_{1})}}{\\Psi} = r{(\\Psi,v_{1})} and \\frac{v_{1}^{2} r^{2}{(\\Psi,v_{1})}}{\\Psi^{2}} = 1 and \\Psi + \\frac{v_{1}^{2} r^{2}{(\\Psi,v_{1})}}{\\Psi^{2}} = \\Psi + 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v_1', commutative=True), Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True))), Integer(1))"], [["times", 2, "Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('v_1', commutative=True), Pow(Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True)), Integer(2))), Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Pow(Symbol('v_1', commutative=True), Integer(2)), Pow(Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True)), Integer(2))), Integer(1))"], [["add", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Pow(Symbol('v_1', commutative=True), Integer(2)), Pow(Function('r')(Symbol('\\\\Psi', commutative=True), Symbol('v_1', commutative=True)), Integer(2)))), Add(Symbol('\\\\Psi', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\lambda{(r_{0},Q)} = \\int r_{0}^{Q} dQ, then obtain \\frac{\\int (- g + \\lambda{(r_{0},Q)}) dg}{- g + \\int r_{0}^{Q} dQ} = \\frac{\\int (- g + \\int r_{0}^{Q} dQ) dg}{- g + \\int r_{0}^{Q} dQ}", "derivation": "\\lambda{(r_{0},Q)} = \\int r_{0}^{Q} dQ and - g + \\lambda{(r_{0},Q)} = - g + \\int r_{0}^{Q} dQ and \\int (- g + \\lambda{(r_{0},Q)}) dg = \\int (- g + \\int r_{0}^{Q} dQ) dg and \\frac{\\int (- g + \\lambda{(r_{0},Q)}) dg}{- g + \\int r_{0}^{Q} dQ} = \\frac{\\int (- g + \\int r_{0}^{Q} dQ) dg}{- g + \\int r_{0}^{Q} dQ}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\lambda')(Symbol('r_0', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\lambda')(Symbol('r_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\lambda')(Symbol('r_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given s{(V)} = \\cos{(V)} and \\operatorname{f_{E}}{(V)} = \\int \\cos{(V)} dV, then obtain \\operatorname{f_{E}}{(V)} = \\int s{(V)} dV", "derivation": "s{(V)} = \\cos{(V)} and \\int s{(V)} dV = \\int \\cos{(V)} dV and \\operatorname{f_{E}}{(V)} = \\int \\cos{(V)} dV and \\operatorname{f_{E}}{(V)} = \\int s{(V)} dV", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('V', commutative=True)), Integral(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('f_E')(Symbol('V', commutative=True)), Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(a,\\tilde{g})} = a + e^{\\tilde{g}}, then obtain a + (a + e^{\\tilde{g}}) e^{\\tilde{g}} = a + \\frac{(a + e^{\\tilde{g}})^{2} e^{\\tilde{g}}}{\\rho_{f}{(a,\\tilde{g})}}", "derivation": "\\rho_{f}{(a,\\tilde{g})} = a + e^{\\tilde{g}} and 1 = \\frac{a + e^{\\tilde{g}}}{\\rho_{f}{(a,\\tilde{g})}} and (a + e^{\\tilde{g}}) e^{\\tilde{g}} = \\frac{(a + e^{\\tilde{g}})^{2} e^{\\tilde{g}}}{\\rho_{f}{(a,\\tilde{g})}} and a + (a + e^{\\tilde{g}}) e^{\\tilde{g}} = a + \\frac{(a + e^{\\tilde{g}})^{2} e^{\\tilde{g}}}{\\rho_{f}{(a,\\tilde{g})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 1, "Function('\\\\rho_f')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Pow(Function('\\\\rho_f')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))))"], [["times", 2, "Mul(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Integer(2)), Pow(Function('\\\\rho_f')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["add", 3, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Mul(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Pow(Add(Symbol('a', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Integer(2)), Pow(Function('\\\\rho_f')(Symbol('a', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\hbar)} = \\log{(\\sin{(\\hbar)})}, then obtain \\frac{\\hat{p}{(\\hbar)} - \\int \\hat{p}{(\\hbar)} d\\hbar}{\\log{(\\sin{(\\hbar)})} \\sin{(\\hbar)}} = \\frac{\\log{(\\sin{(\\hbar)})} - \\int \\hat{p}{(\\hbar)} d\\hbar}{\\log{(\\sin{(\\hbar)})} \\sin{(\\hbar)}}", "derivation": "\\hat{p}{(\\hbar)} = \\log{(\\sin{(\\hbar)})} and \\int \\hat{p}{(\\hbar)} d\\hbar = \\int \\log{(\\sin{(\\hbar)})} d\\hbar and \\hat{p}{(\\hbar)} - \\int \\log{(\\sin{(\\hbar)})} d\\hbar = \\log{(\\sin{(\\hbar)})} - \\int \\log{(\\sin{(\\hbar)})} d\\hbar and \\hat{p}{(\\hbar)} - \\int \\hat{p}{(\\hbar)} d\\hbar = \\log{(\\sin{(\\hbar)})} - \\int \\hat{p}{(\\hbar)} d\\hbar and \\frac{\\hat{p}{(\\hbar)} - \\int \\hat{p}{(\\hbar)} d\\hbar}{\\log{(\\sin{(\\hbar)})} \\sin{(\\hbar)}} = \\frac{\\log{(\\sin{(\\hbar)})} - \\int \\hat{p}{(\\hbar)} d\\hbar}{\\log{(\\sin{(\\hbar)})} \\sin{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), log(sin(Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Integral(log(sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(log(sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))), Add(log(sin(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integral(log(sin(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Add(log(sin(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))))"], [["divide", 4, "Mul(log(sin(Symbol('\\\\hbar', commutative=True))), sin(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Pow(log(sin(Symbol('\\\\hbar', commutative=True))), Integer(-1)), Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(-1))), Mul(Add(log(sin(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{p}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))), Pow(log(sin(Symbol('\\\\hbar', commutative=True))), Integer(-1)), Pow(sin(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given A{(c)} = \\cos{(\\cos{(c)})}, then obtain (\\int A{(c)} dc + 1) ((\\int \\cos{(\\cos{(c)})} dc)^{c})^{c} = (\\int \\cos{(\\cos{(c)})} dc + 1) ((\\int \\cos{(\\cos{(c)})} dc)^{c})^{c}", "derivation": "A{(c)} = \\cos{(\\cos{(c)})} and \\int A{(c)} dc = \\int \\cos{(\\cos{(c)})} dc and (\\int A{(c)} dc)^{c} = (\\int \\cos{(\\cos{(c)})} dc)^{c} and ((\\int A{(c)} dc)^{c})^{c} = ((\\int \\cos{(\\cos{(c)})} dc)^{c})^{c} and \\int A{(c)} dc + 1 = \\int \\cos{(\\cos{(c)})} dc + 1 and (\\int A{(c)} dc + 1) ((\\int A{(c)} dc)^{c})^{c} = (\\int \\cos{(\\cos{(c)})} dc + 1) ((\\int A{(c)} dc)^{c})^{c} and (\\int A{(c)} dc + 1) ((\\int \\cos{(\\cos{(c)})} dc)^{c})^{c} = (\\int \\cos{(\\cos{(c)})} dc + 1) ((\\int \\cos{(\\cos{(c)})} dc)^{c})^{c}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('c', commutative=True)), cos(cos(Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Pow(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["add", 2, 1], "Equality(Add(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(1)), Add(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(1)))"], [["times", 5, "Pow(Pow(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True))"], "Equality(Mul(Add(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(1)), Pow(Pow(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Add(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(1)), Pow(Pow(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Integral(Function('A')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integer(1)), Pow(Pow(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Add(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(1)), Pow(Pow(Integral(cos(cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Symbol('c', commutative=True)), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(Z)} = \\sin{(\\log{(Z)})}, then derive \\frac{d}{d Z} \\varepsilon{(Z)} - 1 = -1 + \\frac{\\cos{(\\log{(Z)})}}{Z}, then obtain \\frac{\\frac{d}{d Z} \\sin{(\\log{(Z)})} - 1}{\\varepsilon{(Z)}} = \\frac{-1 + \\frac{\\cos{(\\log{(Z)})}}{Z}}{\\varepsilon{(Z)}}", "derivation": "\\varepsilon{(Z)} = \\sin{(\\log{(Z)})} and \\frac{d}{d Z} \\varepsilon{(Z)} = \\frac{d}{d Z} \\sin{(\\log{(Z)})} and \\frac{d}{d Z} \\varepsilon{(Z)} - 1 = \\frac{d}{d Z} \\sin{(\\log{(Z)})} - 1 and \\frac{d}{d Z} \\varepsilon{(Z)} - 1 = -1 + \\frac{\\cos{(\\log{(Z)})}}{Z} and \\frac{\\frac{d}{d Z} \\varepsilon{(Z)} - 1}{\\varepsilon{(Z)}} = \\frac{-1 + \\frac{\\cos{(\\log{(Z)})}}{Z}}{\\varepsilon{(Z)}} and \\frac{\\frac{d}{d Z} \\sin{(\\log{(Z)})} - 1}{\\varepsilon{(Z)}} = \\frac{-1 + \\frac{\\cos{(\\log{(Z)})}}{Z}}{\\varepsilon{(Z)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), sin(log(Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(sin(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), cos(log(Symbol('Z', commutative=True))))))"], [["divide", 4, "Function('\\\\varepsilon')(Symbol('Z', commutative=True))"], "Equality(Mul(Add(Derivative(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1))), Mul(Add(Integer(-1), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), cos(log(Symbol('Z', commutative=True))))), Pow(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Derivative(sin(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1)), Pow(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1))), Mul(Add(Integer(-1), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), cos(log(Symbol('Z', commutative=True))))), Pow(Function('\\\\varepsilon')(Symbol('Z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(r_{0},\\hat{H})} = \\hat{H} + r_{0}, then obtain 2 \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\operatorname{P_{g}}{(r_{0},\\hat{H})}}{r_{0}} = \\hat{H} + r_{0} + \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\operatorname{P_{g}}{(r_{0},\\hat{H})}}{r_{0}}", "derivation": "\\operatorname{P_{g}}{(r_{0},\\hat{H})} = \\hat{H} + r_{0} and 2 \\operatorname{P_{g}}{(r_{0},\\hat{H})} = \\hat{H} + r_{0} + \\operatorname{P_{g}}{(r_{0},\\hat{H})} and \\frac{\\operatorname{P_{g}}{(r_{0},\\hat{H})}}{r_{0}} = \\frac{\\hat{H} + r_{0}}{r_{0}} and 2 \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\hat{H} + r_{0}}{r_{0}} = \\hat{H} + r_{0} + \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\hat{H} + r_{0}}{r_{0}} and 2 \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\operatorname{P_{g}}{(r_{0},\\hat{H})}}{r_{0}} = \\hat{H} + r_{0} + \\operatorname{P_{g}}{(r_{0},\\hat{H})} + \\frac{\\operatorname{P_{g}}{(r_{0},\\hat{H})}}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 1, "Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Integer(2), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('r_0', commutative=True), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('P_g')(Symbol('r_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(z^{*})} = \\sin{(z^{*})}, then derive a^{\\dagger} + \\operatorname{A_{2}}{(z^{*})} = f_{\\mathbf{v}} + \\sin{(z^{*})}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (a^{\\dagger} + \\operatorname{A_{2}}{(z^{*})}) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + \\operatorname{A_{2}}{(z^{*})})", "derivation": "\\operatorname{A_{2}}{(z^{*})} = \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{A_{2}}{(z^{*})} = \\frac{d}{d z^{*}} \\sin{(z^{*})} and \\int \\frac{d}{d z^{*}} \\operatorname{A_{2}}{(z^{*})} dz^{*} = \\int \\frac{d}{d z^{*}} \\sin{(z^{*})} dz^{*} and a^{\\dagger} + \\operatorname{A_{2}}{(z^{*})} = f_{\\mathbf{v}} + \\sin{(z^{*})} and a^{\\dagger} + \\operatorname{A_{2}}{(z^{*})} = f_{\\mathbf{v}} + \\operatorname{A_{2}}{(z^{*})} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (a^{\\dagger} + \\operatorname{A_{2}}{(z^{*})}) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + \\operatorname{A_{2}}{(z^{*})})", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Derivative(Function('A_2')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Tuple(Symbol('z^*', commutative=True))), Integral(Derivative(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('A_2')(Symbol('z^*', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('A_2')(Symbol('z^*', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_2')(Symbol('z^*', commutative=True))))"], [["differentiate", 5, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('A_2')(Symbol('z^*', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_2')(Symbol('z^*', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(v)} = e^{\\cos{(v)}}, then obtain \\int \\frac{d}{d v} 2 \\phi_{2}^{v}{(v)} dv = \\int \\frac{d}{d v} (\\phi_{2}^{v}{(v)} + (e^{\\cos{(v)}})^{v}) dv", "derivation": "\\phi_{2}{(v)} = e^{\\cos{(v)}} and \\phi_{2}^{v}{(v)} = (e^{\\cos{(v)}})^{v} and 2 \\phi_{2}^{v}{(v)} = \\phi_{2}^{v}{(v)} + (e^{\\cos{(v)}})^{v} and \\frac{d}{d v} 2 \\phi_{2}^{v}{(v)} = \\frac{d}{d v} (\\phi_{2}^{v}{(v)} + (e^{\\cos{(v)}})^{v}) and \\int \\frac{d}{d v} 2 \\phi_{2}^{v}{(v)} dv = \\int \\frac{d}{d v} (\\phi_{2}^{v}{(v)} + (e^{\\cos{(v)}})^{v}) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('v', commutative=True)), exp(cos(Symbol('v', commutative=True))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(exp(cos(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["add", 2, "Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Add(Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(exp(cos(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(exp(cos(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Integral(Derivative(Add(Pow(Function('\\\\phi_2')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(exp(cos(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given m{(\\psi^*,r)} = \\psi^* + \\sin{(r)}, then obtain \\int \\frac{\\cos{(m{(\\psi^*,r)} \\sin{(r)})}}{\\cos{((\\psi^* + \\sin{(r)}) \\sin{(r)})}} d\\psi^* = \\int 1 d\\psi^*", "derivation": "m{(\\psi^*,r)} = \\psi^* + \\sin{(r)} and m{(\\psi^*,r)} \\sin{(r)} = (\\psi^* + \\sin{(r)}) \\sin{(r)} and \\cos{(m{(\\psi^*,r)} \\sin{(r)})} = \\cos{((\\psi^* + \\sin{(r)}) \\sin{(r)})} and \\frac{\\cos{(m{(\\psi^*,r)} \\sin{(r)})}}{\\cos{((\\psi^* + \\sin{(r)}) \\sin{(r)})}} = 1 and \\int \\frac{\\cos{(m{(\\psi^*,r)} \\sin{(r)})}}{\\cos{((\\psi^* + \\sin{(r)}) \\sin{(r)})}} d\\psi^* = \\int 1 d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))))"], [["times", 1, "sin(Symbol('r', commutative=True))"], "Equality(Mul(Function('m')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Mul(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Function('m')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))), cos(Mul(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True)))))"], [["divide", 3, "cos(Mul(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True))))"], "Equality(Mul(Pow(cos(Mul(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True)))), Integer(-1)), cos(Mul(Function('m')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))))), Integer(1))"], [["integrate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Mul(Pow(cos(Mul(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('r', commutative=True))), sin(Symbol('r', commutative=True)))), Integer(-1)), cos(Mul(Function('m')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given B{(\\mathbf{J})} = \\cos{(\\mathbf{J})}, then derive \\int B{(\\mathbf{J})} d\\mathbf{J} = t_{2} + \\sin{(\\mathbf{J})}, then obtain 1 = \\frac{t_{2} + \\sin{(\\mathbf{J})}}{\\int B{(\\mathbf{J})} d\\mathbf{J}}", "derivation": "B{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\int B{(\\mathbf{J})} d\\mathbf{J} = \\int \\cos{(\\mathbf{J})} d\\mathbf{J} and 1 = \\frac{\\int \\cos{(\\mathbf{J})} d\\mathbf{J}}{\\int B{(\\mathbf{J})} d\\mathbf{J}} and \\int B{(\\mathbf{J})} d\\mathbf{J} = t_{2} + \\sin{(\\mathbf{J})} and \\int \\cos{(\\mathbf{J})} d\\mathbf{J} = t_{2} + \\sin{(\\mathbf{J})} and 1 = \\frac{t_{2} + \\sin{(\\mathbf{J})}}{\\int B{(\\mathbf{J})} d\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 2, "Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(1), Mul(Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\mathbf{J}', commutative=True))), Pow(Integral(Function('B')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given B{(h,\\nabla)} = \\log{(\\frac{\\nabla}{h})}, then obtain \\cos{(\\iint \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} d\\nabla d\\nabla)} + \\frac{1}{h} = \\cos{(\\iint 1 d\\nabla d\\nabla)} + \\frac{1}{h}", "derivation": "B{(h,\\nabla)} = \\log{(\\frac{\\nabla}{h})} and \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} = 1 and \\int \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} d\\nabla = \\int 1 d\\nabla and \\iint \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} d\\nabla d\\nabla = \\iint 1 d\\nabla d\\nabla and \\cos{(\\iint \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} d\\nabla d\\nabla)} = \\cos{(\\iint 1 d\\nabla d\\nabla)} and \\cos{(\\iint \\frac{B{(h,\\nabla)}}{\\log{(\\frac{\\nabla}{h})}} d\\nabla d\\nabla)} + \\frac{1}{h} = \\cos{(\\iint 1 d\\nabla d\\nabla)} + \\frac{1}{h}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["divide", 1, "log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], "Equality(Mul(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Mul(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Mul(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["cos", 4], "Equality(cos(Integral(Mul(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), cos(Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["add", 5, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Add(cos(Integral(Mul(Function('B')(Symbol('h', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Pow(Symbol('h', commutative=True), Integer(-1))), Add(cos(Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Pow(Symbol('h', commutative=True), Integer(-1))))"]]}, {"prompt": "Given J{(n_{1},l)} = l n_{1}, then obtain \\frac{\\partial}{\\partial l} (- n_{1} + \\frac{\\partial}{\\partial n_{1}} J{(n_{1},l)}) = \\frac{\\partial}{\\partial l} (- n_{1} + \\frac{\\partial}{\\partial n_{1}} l n_{1})", "derivation": "J{(n_{1},l)} = l n_{1} and \\frac{\\partial}{\\partial n_{1}} J{(n_{1},l)} = \\frac{\\partial}{\\partial n_{1}} l n_{1} and - n_{1} + \\frac{\\partial}{\\partial n_{1}} J{(n_{1},l)} = - n_{1} + \\frac{\\partial}{\\partial n_{1}} l n_{1} and \\frac{\\partial}{\\partial l} (- n_{1} + \\frac{\\partial}{\\partial n_{1}} J{(n_{1},l)}) = \\frac{\\partial}{\\partial l} (- n_{1} + \\frac{\\partial}{\\partial n_{1}} l n_{1})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('n_1', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('n_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Function('J')(Symbol('n_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Function('J')(Symbol('n_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Mul(Symbol('l', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(h,n)} = \\frac{n}{h}, then obtain 2 \\operatorname{F_{N}}{(h,n)} \\frac{\\partial}{\\partial h} \\operatorname{F_{N}}{(h,n)} = \\frac{n \\frac{\\partial}{\\partial h} \\operatorname{F_{N}}{(h,n)}}{h} - \\frac{n \\operatorname{F_{N}}{(h,n)}}{h^{2}}", "derivation": "\\operatorname{F_{N}}{(h,n)} = \\frac{n}{h} and \\operatorname{F_{N}}^{2}{(h,n)} = \\frac{n \\operatorname{F_{N}}{(h,n)}}{h} and \\frac{\\partial}{\\partial h} \\operatorname{F_{N}}^{2}{(h,n)} = \\frac{\\partial}{\\partial h} \\frac{n \\operatorname{F_{N}}{(h,n)}}{h} and 2 \\operatorname{F_{N}}{(h,n)} \\frac{\\partial}{\\partial h} \\operatorname{F_{N}}{(h,n)} = \\frac{n \\frac{\\partial}{\\partial h} \\operatorname{F_{N}}{(h,n)}}{h} - \\frac{n \\operatorname{F_{N}}{(h,n)}}{h^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["times", 1, "Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Integer(2)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n', commutative=True), Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n', commutative=True), Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Derivative(Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('n', commutative=True), Derivative(Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-2)), Symbol('n', commutative=True), Function('F_N')(Symbol('h', commutative=True), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\eta,\\mathbf{r})} = \\mathbf{r} + e^{\\eta}, then obtain \\int (\\operatorname{F_{N}}{(\\eta,\\mathbf{r})} + \\sin{(\\mathbf{r} + e^{\\eta})}) d\\eta = \\int (\\mathbf{r} + e^{\\eta} + \\sin{(\\mathbf{r} + e^{\\eta})}) d\\eta", "derivation": "\\operatorname{F_{N}}{(\\eta,\\mathbf{r})} = \\mathbf{r} + e^{\\eta} and \\sin{(\\operatorname{F_{N}}{(\\eta,\\mathbf{r})})} = \\sin{(\\mathbf{r} + e^{\\eta})} and \\operatorname{F_{N}}{(\\eta,\\mathbf{r})} + \\sin{(\\operatorname{F_{N}}{(\\eta,\\mathbf{r})})} = \\mathbf{r} + e^{\\eta} + \\sin{(\\operatorname{F_{N}}{(\\eta,\\mathbf{r})})} and \\operatorname{F_{N}}{(\\eta,\\mathbf{r})} + \\sin{(\\mathbf{r} + e^{\\eta})} = \\mathbf{r} + e^{\\eta} + \\sin{(\\mathbf{r} + e^{\\eta})} and \\int (\\operatorname{F_{N}}{(\\eta,\\mathbf{r})} + \\sin{(\\mathbf{r} + e^{\\eta})}) d\\eta = \\int (\\mathbf{r} + e^{\\eta} + \\sin{(\\mathbf{r} + e^{\\eta})}) d\\eta", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))"], [["sin", 1], "Equality(sin(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), sin(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True)))))"], [["add", 1, "sin(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True)), sin(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))), Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True)), sin(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Add(Function('F_N')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), sin(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True)), sin(Add(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\eta', commutative=True))))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\pi)} = \\sin{(\\pi)} and \\lambda{(\\pi)} = \\sin{(\\pi)}, then obtain \\int \\frac{d}{d \\pi} \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\pi)})} d\\pi = \\int \\frac{d}{d \\pi} \\sin{(\\lambda{(\\pi)})} d\\pi", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\pi)} = \\sin{(\\pi)} and \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\pi)})} = \\sin{(\\sin{(\\pi)})} and \\frac{d}{d \\pi} \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\pi)})} = \\frac{d}{d \\pi} \\sin{(\\sin{(\\pi)})} and \\lambda{(\\pi)} = \\sin{(\\pi)} and \\frac{d}{d \\pi} \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\pi)})} = \\frac{d}{d \\pi} \\sin{(\\lambda{(\\pi)})} and \\int \\frac{d}{d \\pi} \\sin{(\\operatorname{f_{\\mathbf{v}}}{(\\pi)})} d\\pi = \\int \\frac{d}{d \\pi} \\sin{(\\lambda{(\\pi)})} d\\pi", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["sin", 1], "Equality(sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\pi', commutative=True))), sin(sin(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(Function('\\\\lambda')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(sin(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Derivative(sin(Function('\\\\lambda')(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given V{(\\rho_f)} = e^{\\rho_f}, then obtain \\int \\frac{d^{2}}{d \\rho_f^{2}} V{(\\rho_f)} d\\rho_f = \\int \\frac{d^{2}}{d \\rho_f^{2}} e^{\\rho_f} d\\rho_f", "derivation": "V{(\\rho_f)} = e^{\\rho_f} and \\frac{d}{d \\rho_f} V{(\\rho_f)} = \\frac{d}{d \\rho_f} e^{\\rho_f} and \\frac{d^{2}}{d \\rho_f^{2}} V{(\\rho_f)} = \\frac{d^{2}}{d \\rho_f^{2}} e^{\\rho_f} and \\int \\frac{d^{2}}{d \\rho_f^{2}} V{(\\rho_f)} d\\rho_f = \\int \\frac{d^{2}}{d \\rho_f^{2}} e^{\\rho_f} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\rho_f', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Derivative(Function('V')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Derivative(exp(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(2))), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given W{(C,\\eta^{\\prime})} = - C + \\eta^{\\prime}, then obtain - \\frac{\\cos{(\\frac{\\partial}{\\partial C} W{(C,\\eta^{\\prime})} - 1)}}{\\eta^{\\prime}} = - \\frac{\\cos{(\\frac{\\partial}{\\partial C} (- C + \\eta^{\\prime}) - 1)}}{\\eta^{\\prime}}", "derivation": "W{(C,\\eta^{\\prime})} = - C + \\eta^{\\prime} and \\frac{\\partial}{\\partial C} W{(C,\\eta^{\\prime})} = \\frac{\\partial}{\\partial C} (- C + \\eta^{\\prime}) and \\frac{\\partial}{\\partial C} W{(C,\\eta^{\\prime})} - 1 = \\frac{\\partial}{\\partial C} (- C + \\eta^{\\prime}) - 1 and \\cos{(\\frac{\\partial}{\\partial C} W{(C,\\eta^{\\prime})} - 1)} = \\cos{(\\frac{\\partial}{\\partial C} (- C + \\eta^{\\prime}) - 1)} and - \\frac{\\cos{(\\frac{\\partial}{\\partial C} W{(C,\\eta^{\\prime})} - 1)}}{\\eta^{\\prime}} = - \\frac{\\cos{(\\frac{\\partial}{\\partial C} (- C + \\eta^{\\prime}) - 1)}}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('W')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)))"], [["cos", 3], "Equality(cos(Add(Derivative(Function('W')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))), cos(Add(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), cos(Add(Derivative(Function('W')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), cos(Add(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given M{(v,\\eta^{\\prime})} = \\eta^{\\prime} + v, then derive v + \\int M{(v,\\eta^{\\prime})} dv = \\eta^{\\prime} v + \\hat{H}_{\\lambda} + \\frac{v^{2}}{2} + v, then obtain v + \\int (\\eta^{\\prime} + v) dv = \\eta^{\\prime} v + \\hat{H}_{\\lambda} + \\frac{v^{2}}{2} + v", "derivation": "M{(v,\\eta^{\\prime})} = \\eta^{\\prime} + v and \\int M{(v,\\eta^{\\prime})} dv = \\int (\\eta^{\\prime} + v) dv and v + \\int M{(v,\\eta^{\\prime})} dv = v + \\int (\\eta^{\\prime} + v) dv and v + \\int M{(v,\\eta^{\\prime})} dv = \\eta^{\\prime} v + \\hat{H}_{\\lambda} + \\frac{v^{2}}{2} + v and v + \\int (\\eta^{\\prime} + v) dv = \\eta^{\\prime} v + \\hat{H}_{\\lambda} + \\frac{v^{2}}{2} + v", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('M')(Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Integral(Function('M')(Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Symbol('v', commutative=True), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v', commutative=True), Integral(Function('M')(Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('v', commutative=True), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\varepsilon f_{E}, then obtain \\frac{\\partial}{\\partial f_{E}} (f_{E} + \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)}) \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\frac{\\partial}{\\partial f_{E}} \\varepsilon f_{E} (f_{E} + \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)})", "derivation": "\\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\varepsilon f_{E} and f_{E} + \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\varepsilon f_{E} + f_{E} and (\\varepsilon f_{E} + f_{E}) \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\varepsilon f_{E} (\\varepsilon f_{E} + f_{E}) and \\frac{\\partial}{\\partial f_{E}} (\\varepsilon f_{E} + f_{E}) \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\frac{\\partial}{\\partial f_{E}} \\varepsilon f_{E} (\\varepsilon f_{E} + f_{E}) and \\frac{\\partial}{\\partial f_{E}} (f_{E} + \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)}) \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)} = \\frac{\\partial}{\\partial f_{E}} \\varepsilon f_{E} (f_{E} + \\operatorname{y^{\\prime}}{(f_{E},\\varepsilon)})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)))"], [["add", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["times", 1, "Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Add(Symbol('f_E', commutative=True), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f_E', commutative=True), Add(Symbol('f_E', commutative=True), Function('y^{\\\\prime}')(Symbol('f_E', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(F_{c})} = \\cos{(F_{c})}, then obtain \\int 0^{F_{c}} dF_{c} = \\int (- (- \\hat{x}_0{(F_{c})} + \\cos{(F_{c})}) \\hat{x}_0{(F_{c})})^{F_{c}} dF_{c}", "derivation": "\\hat{x}_0{(F_{c})} = \\cos{(F_{c})} and 0 = - \\hat{x}_0{(F_{c})} + \\cos{(F_{c})} and 0 = - (- \\hat{x}_0{(F_{c})} + \\cos{(F_{c})}) \\hat{x}_0{(F_{c})} and 0^{F_{c}} = (- (- \\hat{x}_0{(F_{c})} + \\cos{(F_{c})}) \\hat{x}_0{(F_{c})})^{F_{c}} and \\int 0^{F_{c}} dF_{c} = \\int (- (- \\hat{x}_0{(F_{c})} + \\cos{(F_{c})}) \\hat{x}_0{(F_{c})})^{F_{c}} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["minus", 1, "Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True))), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integer(0), Symbol('F_c', commutative=True)), Pow(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True))), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["integrate", 4, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Pow(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), cos(Symbol('F_c', commutative=True))), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(W,\\ddot{x})} = \\ddot{x} \\cos{(W)}, then obtain \\frac{(\\int \\sigma_{p}{(W,\\ddot{x})} dW)^{W}}{\\sigma_{p}{(W,\\ddot{x})}} = \\frac{(\\int \\ddot{x} \\cos{(W)} dW)^{W}}{\\sigma_{p}{(W,\\ddot{x})}}", "derivation": "\\sigma_{p}{(W,\\ddot{x})} = \\ddot{x} \\cos{(W)} and \\int \\sigma_{p}{(W,\\ddot{x})} dW = \\int \\ddot{x} \\cos{(W)} dW and (\\int \\sigma_{p}{(W,\\ddot{x})} dW)^{W} = (\\int \\ddot{x} \\cos{(W)} dW)^{W} and \\frac{(\\int \\sigma_{p}{(W,\\ddot{x})} dW)^{W}}{\\sigma_{p}{(W,\\ddot{x})}} = \\frac{(\\int \\ddot{x} \\cos{(W)} dW)^{W}}{\\sigma_{p}{(W,\\ddot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["divide", 3, "Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), Pow(Integral(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Mul(Pow(Function('\\\\sigma_p')(Symbol('W', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(p)} = \\log{(p)}, then derive \\frac{d}{d p} \\operatorname{E_{x}}{(p)} = \\frac{1}{p}, then obtain \\frac{d}{d p} \\log{(p)} = \\frac{1}{p}", "derivation": "\\operatorname{E_{x}}{(p)} = \\log{(p)} and \\frac{d}{d p} \\operatorname{E_{x}}{(p)} = \\frac{d}{d p} \\log{(p)} and \\frac{d}{d p} \\operatorname{E_{x}}{(p)} = \\frac{1}{p} and \\frac{d}{d p} \\log{(p)} = \\frac{1}{p}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Pow(Symbol('p', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Pow(Symbol('p', commutative=True), Integer(-1)))"]]}, {"prompt": "Given B{(\\mathbf{B})} = \\cos{(e^{\\mathbf{B}})} and \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} = \\cos{(e^{\\mathbf{B}})}, then obtain \\frac{\\cos^{\\mathbf{B}}{(e^{\\mathbf{B}})}}{\\cos{(e^{\\mathbf{B}})}} = \\frac{B^{\\mathbf{B}}{(\\mathbf{B})}}{\\cos{(e^{\\mathbf{B}})}}", "derivation": "B{(\\mathbf{B})} = \\cos{(e^{\\mathbf{B}})} and \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} = \\cos{(e^{\\mathbf{B}})} and \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{B}}{(\\mathbf{B})} = \\cos^{\\mathbf{B}}{(e^{\\mathbf{B}})} and \\operatorname{f_{\\mathbf{v}}}^{\\mathbf{B}}{(\\mathbf{B})} = B^{\\mathbf{B}}{(\\mathbf{B})} and \\cos^{\\mathbf{B}}{(e^{\\mathbf{B}})} = B^{\\mathbf{B}}{(\\mathbf{B})} and \\frac{\\cos^{\\mathbf{B}}{(e^{\\mathbf{B}})}}{\\cos{(e^{\\mathbf{B}})}} = \\frac{B^{\\mathbf{B}}{(\\mathbf{B})}}{\\cos{(e^{\\mathbf{B}})}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), cos(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), cos(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(cos(exp(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 5, "cos(exp(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Pow(cos(exp(Symbol('\\\\mathbf{B}', commutative=True))), Integer(-1)), Pow(cos(exp(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{B}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\pi{(p)} = e^{e^{p}} and W{(p)} = e^{2 e^{p}}, then obtain \\frac{W{(p)} \\pi^{2}{(p)} e^{- p}}{3} = \\frac{\\pi{(p)} e^{- p} e^{3 e^{p}}}{3}", "derivation": "\\pi{(p)} = e^{e^{p}} and \\pi^{2}{(p)} = \\pi{(p)} e^{e^{p}} and \\pi^{4}{(p)} = \\pi^{2}{(p)} e^{2 e^{p}} and \\pi^{2}{(p)} e^{2 e^{p}} = \\pi{(p)} e^{3 e^{p}} and \\frac{\\pi^{2}{(p)} e^{- p} e^{2 e^{p}}}{3} = \\frac{\\pi{(p)} e^{- p} e^{3 e^{p}}}{3} and W{(p)} = e^{2 e^{p}} and \\frac{W{(p)} \\pi^{2}{(p)} e^{- p}}{3} = \\frac{\\pi{(p)} e^{- p} e^{3 e^{p}}}{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('p', commutative=True)), exp(exp(Symbol('p', commutative=True))))"], [["times", 1, "Function('\\\\pi')(Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(2)), Mul(Function('\\\\pi')(Symbol('p', commutative=True)), exp(exp(Symbol('p', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(2)), exp(Mul(Integer(2), exp(Symbol('p', commutative=True))))), Mul(Function('\\\\pi')(Symbol('p', commutative=True)), exp(Mul(Integer(3), exp(Symbol('p', commutative=True))))))"], [["divide", 4, "Mul(Integer(3), exp(Symbol('p', commutative=True)))"], "Equality(Mul(Rational(1, 3), Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Symbol('p', commutative=True))), exp(Mul(Integer(2), exp(Symbol('p', commutative=True))))), Mul(Rational(1, 3), Function('\\\\pi')(Symbol('p', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True))), exp(Mul(Integer(3), exp(Symbol('p', commutative=True))))))"], ["renaming_premise", "Equality(Function('W')(Symbol('p', commutative=True)), exp(Mul(Integer(2), exp(Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Rational(1, 3), Function('W')(Symbol('p', commutative=True)), Pow(Function('\\\\pi')(Symbol('p', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Rational(1, 3), Function('\\\\pi')(Symbol('p', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True))), exp(Mul(Integer(3), exp(Symbol('p', commutative=True))))))"]]}, {"prompt": "Given z{(f^{\\prime},L)} = \\frac{L}{f^{\\prime}} and \\operatorname{v_{2}}{(f^{\\prime},L)} = \\frac{L z{(f^{\\prime},L)}}{f^{\\prime}}, then obtain \\operatorname{v_{2}}{(f^{\\prime},L)} + z^{2}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}} = 2 z^{2}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}}", "derivation": "z{(f^{\\prime},L)} = \\frac{L}{f^{\\prime}} and z^{2}{(f^{\\prime},L)} = \\frac{L z{(f^{\\prime},L)}}{f^{\\prime}} and \\operatorname{v_{2}}{(f^{\\prime},L)} = \\frac{L z{(f^{\\prime},L)}}{f^{\\prime}} and \\operatorname{v_{2}}{(f^{\\prime},L)} = z^{2}{(f^{\\prime},L)} and \\operatorname{v_{2}}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}} = z^{2}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}} and \\operatorname{v_{2}}{(f^{\\prime},L)} + z^{2}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}} = 2 z^{2}{(f^{\\prime},L)} - \\frac{1}{(f^{\\prime})^{2}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["times", 1, "Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True))"], "Equality(Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Symbol('L', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('v_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2)))"], [["minus", 4, "Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2))"], "Equality(Add(Function('v_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)))), Add(Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)))))"], [["add", 5, "Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2))"], "Equality(Add(Function('v_2')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)))), Add(Mul(Integer(2), Pow(Function('z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\mathbf{H}{(a^{\\dagger},\\mathbf{f})} = \\sin{(\\mathbf{f} a^{\\dagger})}, then obtain \\frac{\\iint \\mathbf{H}{(a^{\\dagger},\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}}{\\sin{(\\mathbf{f} a^{\\dagger})}} = \\frac{\\iint \\sin{(\\mathbf{f} a^{\\dagger})} d\\mathbf{f} d\\mathbf{f}}{\\sin{(\\mathbf{f} a^{\\dagger})}}", "derivation": "\\mathbf{H}{(a^{\\dagger},\\mathbf{f})} = \\sin{(\\mathbf{f} a^{\\dagger})} and \\int \\mathbf{H}{(a^{\\dagger},\\mathbf{f})} d\\mathbf{f} = \\int \\sin{(\\mathbf{f} a^{\\dagger})} d\\mathbf{f} and \\iint \\mathbf{H}{(a^{\\dagger},\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} = \\iint \\sin{(\\mathbf{f} a^{\\dagger})} d\\mathbf{f} d\\mathbf{f} and \\frac{\\iint \\mathbf{H}{(a^{\\dagger},\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}}{\\sin{(\\mathbf{f} a^{\\dagger})}} = \\frac{\\iint \\sin{(\\mathbf{f} a^{\\dagger})} d\\mathbf{f} d\\mathbf{f}}{\\sin{(\\mathbf{f} a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 3, "sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1)), Integral(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(E_{n},p)} = \\frac{E_{n}}{p}, then obtain \\frac{d^{2}}{d p^{2}} \\iint 1 dE_{n} dE_{n} = \\frac{\\partial^{2}}{\\partial p^{2}} \\iint \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} dE_{n} dE_{n}", "derivation": "\\operatorname{t_{2}}{(E_{n},p)} = \\frac{E_{n}}{p} and 1 = \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} and \\int 1 dE_{n} = \\int \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} dE_{n} and \\iint 1 dE_{n} dE_{n} = \\iint \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} dE_{n} dE_{n} and \\frac{d}{d p} \\iint 1 dE_{n} dE_{n} = \\frac{\\partial}{\\partial p} \\iint \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} dE_{n} dE_{n} and \\frac{d^{2}}{d p^{2}} \\iint 1 dE_{n} dE_{n} = \\frac{\\partial^{2}}{\\partial p^{2}} \\iint \\frac{E_{n}}{p \\operatorname{t_{2}}{(E_{n},p)}} dE_{n} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["divide", 1, "Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True))"], "Equality(Integer(1), Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["differentiate", 4, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(2))), Derivative(Integral(Mul(Symbol('E_n', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Function('t_2')(Symbol('E_n', commutative=True), Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(G)} = \\sin{(G)} and U{(G)} = - \\sin{(\\sin{(G)})}, then obtain (U{(G)} - \\sin{(\\operatorname{E_{n}}{(G)})})^{G} = (- 2 \\sin{(\\operatorname{E_{n}}{(G)})})^{G}", "derivation": "\\operatorname{E_{n}}{(G)} = \\sin{(G)} and \\sin{(\\operatorname{E_{n}}{(G)})} = \\sin{(\\sin{(G)})} and U{(G)} = - \\sin{(\\sin{(G)})} and U{(G)} - \\sin{(\\operatorname{E_{n}}{(G)})} = - \\sin{(\\operatorname{E_{n}}{(G)})} - \\sin{(\\sin{(G)})} and U{(G)} - \\sin{(\\sin{(G)})} = - 2 \\sin{(\\sin{(G)})} and U{(G)} - \\sin{(\\operatorname{E_{n}}{(G)})} = - 2 \\sin{(\\operatorname{E_{n}}{(G)})} and (U{(G)} - \\sin{(\\operatorname{E_{n}}{(G)})})^{G} = (- 2 \\sin{(\\operatorname{E_{n}}{(G)})})^{G}", "srepr_derivation": [["get_premise", "Equality(Function('E_n')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True)))"], [["sin", 1], "Equality(sin(Function('E_n')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('G', commutative=True)))))"], [["minus", 3, "sin(Function('E_n')(Symbol('G', commutative=True)))"], "Equality(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Function('E_n')(Symbol('G', commutative=True))))), Add(Mul(Integer(-1), sin(Function('E_n')(Symbol('G', commutative=True)))), Mul(Integer(-1), sin(sin(Symbol('G', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('G', commutative=True))))), Mul(Integer(-1), Integer(2), sin(sin(Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Function('E_n')(Symbol('G', commutative=True))))), Mul(Integer(-1), Integer(2), sin(Function('E_n')(Symbol('G', commutative=True)))))"], [["power", 6, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Function('E_n')(Symbol('G', commutative=True))))), Symbol('G', commutative=True)), Pow(Mul(Integer(-1), Integer(2), sin(Function('E_n')(Symbol('G', commutative=True)))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(Q)} = \\cos{(Q)}, then obtain \\int \\mathbf{J}_M{(Q)} \\cos{(Q)} \\cos^{2 Q}{(Q)} dQ = \\int \\cos^{2}{(Q)} \\cos^{2 Q}{(Q)} dQ", "derivation": "\\mathbf{J}_M{(Q)} = \\cos{(Q)} and \\mathbf{J}_M^{Q}{(Q)} = \\cos^{Q}{(Q)} and \\mathbf{J}_M{(Q)} \\mathbf{J}_M^{Q}{(Q)} = \\mathbf{J}_M^{Q}{(Q)} \\cos{(Q)} and \\mathbf{J}_M^{Q}{(Q)} \\cos{(Q)} = \\cos{(Q)} \\cos^{Q}{(Q)} and \\mathbf{J}_M{(Q)} \\mathbf{J}_M^{Q}{(Q)} = \\cos{(Q)} \\cos^{Q}{(Q)} and \\mathbf{J}_M{(Q)} \\mathbf{J}_M^{Q}{(Q)} \\cos{(Q)} \\cos^{Q}{(Q)} = \\cos^{2}{(Q)} \\cos^{2 Q}{(Q)} and \\mathbf{J}_M{(Q)} \\cos{(Q)} \\cos^{2 Q}{(Q)} = \\cos^{2}{(Q)} \\cos^{2 Q}{(Q)} and \\int \\mathbf{J}_M{(Q)} \\cos{(Q)} \\cos^{2 Q}{(Q)} dQ = \\int \\cos^{2}{(Q)} \\cos^{2 Q}{(Q)} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["times", 1, "Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))))"], [["times", 2, "cos(Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Mul(cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))))"], [["times", 5, "Mul(cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Pow(cos(Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True)))), Mul(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Pow(cos(Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True)))))"], [["integrate", 7, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)), Pow(cos(Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Pow(cos(Symbol('Q', commutative=True)), Integer(2)), Pow(cos(Symbol('Q', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given p{(a)} = \\sin{(a)}, then obtain \\frac{d}{d a} p{(a)} + 1 = \\cos{(a)} + 1", "derivation": "p{(a)} = \\sin{(a)} and a + p{(a)} = a + \\sin{(a)} and \\frac{d}{d a} (a + p{(a)}) = \\frac{d}{d a} (a + \\sin{(a)}) and \\frac{d}{d a} p{(a)} + 1 = \\cos{(a)} + 1", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('p')(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Symbol('a', commutative=True), Function('p')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('p')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('a', commutative=True)), Integer(1)))"]]}, {"prompt": "Given A{(E_{x})} = e^{E_{x}}, then obtain 0 = - \\frac{(- 2 A{(E_{x})} + 2 e^{E_{x}}) q{(A_{y},\\dot{z})}}{A{(E_{x})}}", "derivation": "A{(E_{x})} = e^{E_{x}} and 0 = - A{(E_{x})} + e^{E_{x}} and e^{E_{x}} = - A{(E_{x})} + 2 e^{E_{x}} and 0 = - 2 A{(E_{x})} + 2 e^{E_{x}} and 0 = (- 2 A{(E_{x})} + 2 e^{E_{x}}) q{(A_{y},\\dot{z})} and 0 = - \\frac{(- 2 A{(E_{x})} + 2 e^{E_{x}}) q{(A_{y},\\dot{z})}}{A{(E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], [["minus", 1, "Function('A')(Symbol('E_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A')(Symbol('E_x', commutative=True))), exp(Symbol('E_x', commutative=True))))"], [["add", 2, "exp(Symbol('E_x', commutative=True))"], "Equality(exp(Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Function('A')(Symbol('E_x', commutative=True))), Mul(Integer(2), exp(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('A')(Symbol('E_x', commutative=True))), Mul(Integer(2), exp(Symbol('E_x', commutative=True)))))"], [["times", 4, "Function('q')(Symbol('A_y', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integer(2), Function('A')(Symbol('E_x', commutative=True))), Mul(Integer(2), exp(Symbol('E_x', commutative=True)))), Function('q')(Symbol('A_y', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Function('A')(Symbol('E_x', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Integer(2), Function('A')(Symbol('E_x', commutative=True))), Mul(Integer(2), exp(Symbol('E_x', commutative=True)))), Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(-1)), Function('q')(Symbol('A_y', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given E{(S,A)} = - A + S and r{(S,A)} = - A + S - E{(S,A)}, then obtain \\int \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial A} r^{2}{(S,A)})^{S} dS = \\int \\frac{d}{d S} (\\frac{d}{d A} 0)^{S} dS", "derivation": "E{(S,A)} = - A + S and r{(S,A)} = - A + S - E{(S,A)} and r{(S,A)} = 0 and (- A + S - E{(S,A)}) r{(S,A)} = 0 and \\frac{\\partial}{\\partial A} (- A + S - E{(S,A)}) r{(S,A)} = \\frac{d}{d A} 0 and (\\frac{\\partial}{\\partial A} (- A + S - E{(S,A)}) r{(S,A)})^{S} = (\\frac{d}{d A} 0)^{S} and (\\frac{\\partial}{\\partial A} r^{2}{(S,A)})^{S} = (\\frac{d}{d A} 0)^{S} and \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial A} r^{2}{(S,A)})^{S} = \\frac{d}{d S} (\\frac{d}{d A} 0)^{S} and \\int \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial A} r^{2}{(S,A)})^{S} dS = \\int \\frac{d}{d S} (\\frac{d}{d A} 0)^{S} dS", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Integer(0))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True)))), Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True)))), Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 5, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('S', commutative=True), Mul(Integer(-1), Function('E')(Symbol('S', commutative=True), Symbol('A', commutative=True)))), Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Derivative(Pow(Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)))"], [["differentiate", 7, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Derivative(Pow(Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Pow(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["integrate", 8, "Symbol('S', commutative=True)"], "Equality(Integral(Derivative(Pow(Derivative(Pow(Function('r')(Symbol('S', commutative=True), Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))), Integral(Derivative(Pow(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given G{(f,\\sigma_x)} = \\frac{\\sigma_x}{f} and M{(r_{0})} = \\log{(\\cos{(r_{0})})}, then obtain (G{(f,\\sigma_x)} - 1)^{2 f} M{(r_{0})} \\log{(\\cos{(r_{0})})} = (G{(f,\\sigma_x)} - 1)^{2 f} \\log{(\\cos{(r_{0})})}^{2}", "derivation": "G{(f,\\sigma_x)} = \\frac{\\sigma_x}{f} and M{(r_{0})} = \\log{(\\cos{(r_{0})})} and - M{(r_{0})} = - \\log{(\\cos{(r_{0})})} and - (\\frac{\\sigma_x}{f} - 1)^{f} M{(r_{0})} = - (\\frac{\\sigma_x}{f} - 1)^{f} \\log{(\\cos{(r_{0})})} and (\\frac{\\sigma_x}{f} - 1)^{2 f} M{(r_{0})} \\log{(\\cos{(r_{0})})} = (\\frac{\\sigma_x}{f} - 1)^{2 f} \\log{(\\cos{(r_{0})})}^{2} and (G{(f,\\sigma_x)} - 1)^{2 f} M{(r_{0})} \\log{(\\cos{(r_{0})})} = (G{(f,\\sigma_x)} - 1)^{2 f} \\log{(\\cos{(r_{0})})}^{2}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('M')(Symbol('r_0', commutative=True)), log(cos(Symbol('r_0', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('M')(Symbol('r_0', commutative=True))), Mul(Integer(-1), log(cos(Symbol('r_0', commutative=True)))))"], [["times", 3, "Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Symbol('f', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Symbol('f', commutative=True)), Function('M')(Symbol('r_0', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Symbol('f', commutative=True)), log(cos(Symbol('r_0', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Symbol('f', commutative=True)), log(cos(Symbol('r_0', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Mul(Integer(2), Symbol('f', commutative=True))), Function('M')(Symbol('r_0', commutative=True)), log(cos(Symbol('r_0', commutative=True)))), Mul(Pow(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integer(-1)), Mul(Integer(2), Symbol('f', commutative=True))), Pow(log(cos(Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Add(Function('G')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Mul(Integer(2), Symbol('f', commutative=True))), Function('M')(Symbol('r_0', commutative=True)), log(cos(Symbol('r_0', commutative=True)))), Mul(Pow(Add(Function('G')(Symbol('f', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Mul(Integer(2), Symbol('f', commutative=True))), Pow(log(cos(Symbol('r_0', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(U)} = \\cos{(U)}, then obtain \\operatorname{v_{x}}{(U)} e^{2 \\operatorname{v_{x}}{(U)} + \\cos{(U)}} = \\operatorname{v_{x}}{(U)} e^{\\operatorname{v_{x}}{(U)} + 2 \\cos{(U)}}", "derivation": "\\operatorname{v_{x}}{(U)} = \\cos{(U)} and 2 \\operatorname{v_{x}}{(U)} + \\cos{(U)} = \\operatorname{v_{x}}{(U)} + 2 \\cos{(U)} and e^{2 \\operatorname{v_{x}}{(U)} + \\cos{(U)}} = e^{\\operatorname{v_{x}}{(U)} + 2 \\cos{(U)}} and \\operatorname{v_{x}}{(U)} e^{2 \\operatorname{v_{x}}{(U)} + \\cos{(U)}} = \\operatorname{v_{x}}{(U)} e^{\\operatorname{v_{x}}{(U)} + 2 \\cos{(U)}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["add", 1, "Add(Function('v_x')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v_x')(Symbol('U', commutative=True))), cos(Symbol('U', commutative=True))), Add(Function('v_x')(Symbol('U', commutative=True)), Mul(Integer(2), cos(Symbol('U', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(2), Function('v_x')(Symbol('U', commutative=True))), cos(Symbol('U', commutative=True)))), exp(Add(Function('v_x')(Symbol('U', commutative=True)), Mul(Integer(2), cos(Symbol('U', commutative=True))))))"], [["times", 3, "Function('v_x')(Symbol('U', commutative=True))"], "Equality(Mul(Function('v_x')(Symbol('U', commutative=True)), exp(Add(Mul(Integer(2), Function('v_x')(Symbol('U', commutative=True))), cos(Symbol('U', commutative=True))))), Mul(Function('v_x')(Symbol('U', commutative=True)), exp(Add(Function('v_x')(Symbol('U', commutative=True)), Mul(Integer(2), cos(Symbol('U', commutative=True)))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\operatorname{v_{t}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})}, then obtain 0 = 2 (- \\Psi^{\\dagger}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}) \\Psi^{\\dagger}{(\\mathbf{D})}", "derivation": "\\Psi^{\\dagger}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and 0 = - \\Psi^{\\dagger}{(\\mathbf{D})} + \\sin{(\\mathbf{D})} and \\operatorname{v_{t}}{(\\mathbf{D})} = \\sin{(\\mathbf{D})} and \\Psi^{\\dagger}{(\\mathbf{D})} = \\operatorname{v_{t}}{(\\mathbf{D})} and 2 \\Psi^{\\dagger}{(\\mathbf{D})} = \\Psi^{\\dagger}{(\\mathbf{D})} + \\operatorname{v_{t}}{(\\mathbf{D})} and 0 = (- \\Psi^{\\dagger}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}) (\\Psi^{\\dagger}{(\\mathbf{D})} + \\operatorname{v_{t}}{(\\mathbf{D})}) and 0 = 2 (- \\Psi^{\\dagger}{(\\mathbf{D})} + \\sin{(\\mathbf{D})}) \\Psi^{\\dagger}{(\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{D}', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))), sin(Symbol('\\\\mathbf{D}', commutative=True))), Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(0), Mul(Integer(2), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))), sin(Symbol('\\\\mathbf{D}', commutative=True))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(H)} = \\sin{(H)}, then derive \\cos{(H \\frac{d}{d H} \\operatorname{t_{2}}{(H)})} = \\cos{(H \\cos{(H)})}, then obtain \\frac{d}{d H} \\cos{(H \\frac{d}{d H} \\sin{(H)})} = \\frac{d}{d H} \\cos{(H \\cos{(H)})}", "derivation": "\\operatorname{t_{2}}{(H)} = \\sin{(H)} and \\frac{d}{d H} \\operatorname{t_{2}}{(H)} = \\frac{d}{d H} \\sin{(H)} and H \\frac{d}{d H} \\operatorname{t_{2}}{(H)} = H \\frac{d}{d H} \\sin{(H)} and \\cos{(H \\frac{d}{d H} \\operatorname{t_{2}}{(H)})} = \\cos{(H \\frac{d}{d H} \\sin{(H)})} and \\cos{(H \\frac{d}{d H} \\operatorname{t_{2}}{(H)})} = \\cos{(H \\cos{(H)})} and \\cos{(H \\frac{d}{d H} \\sin{(H)})} = \\cos{(H \\cos{(H)})} and \\frac{d}{d H} \\cos{(H \\frac{d}{d H} \\sin{(H)})} = \\frac{d}{d H} \\cos{(H \\cos{(H)})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["times", 2, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Derivative(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('H', commutative=True), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Mul(Symbol('H', commutative=True), Derivative(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), cos(Mul(Symbol('H', commutative=True), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(cos(Mul(Symbol('H', commutative=True), Derivative(Function('t_2')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), cos(Mul(Symbol('H', commutative=True), cos(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(cos(Mul(Symbol('H', commutative=True), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), cos(Mul(Symbol('H', commutative=True), cos(Symbol('H', commutative=True)))))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(cos(Mul(Symbol('H', commutative=True), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('H', commutative=True), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(A_{y})} = A_{y}, then obtain \\psi{(A_{y})} = (\\frac{A_{y}}{\\phi{(r_{0},A_{y})}})^{r_{0}} - (\\frac{\\psi{(A_{y})}}{\\phi{(r_{0},A_{y})}})^{r_{0}} + \\psi{(A_{y})}", "derivation": "\\psi{(A_{y})} = A_{y} and \\frac{\\psi{(A_{y})}}{\\phi{(r_{0},A_{y})}} = \\frac{A_{y}}{\\phi{(r_{0},A_{y})}} and (\\frac{\\psi{(A_{y})}}{\\phi{(r_{0},A_{y})}})^{r_{0}} = (\\frac{A_{y}}{\\phi{(r_{0},A_{y})}})^{r_{0}} and 0 = (\\frac{A_{y}}{\\phi{(r_{0},A_{y})}})^{r_{0}} - (\\frac{\\psi{(A_{y})}}{\\phi{(r_{0},A_{y})}})^{r_{0}} and \\psi{(A_{y})} = (\\frac{A_{y}}{\\phi{(r_{0},A_{y})}})^{r_{0}} - (\\frac{\\psi{(A_{y})}}{\\phi{(r_{0},A_{y})}})^{r_{0}} + \\psi{(A_{y})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\psi')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], [["divide", 1, "Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True))"], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('A_y', commutative=True))), Mul(Symbol('A_y', commutative=True), Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('A_y', commutative=True))), Symbol('r_0', commutative=True)), Pow(Mul(Symbol('A_y', commutative=True), Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)))"], [["minus", 3, "Pow(Mul(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('A_y', commutative=True))), Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Symbol('A_y', commutative=True), Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('A_y', commutative=True))), Symbol('r_0', commutative=True)))))"], [["add", 4, "Function('\\\\psi')(Symbol('A_y', commutative=True))"], "Equality(Function('\\\\psi')(Symbol('A_y', commutative=True)), Add(Pow(Mul(Symbol('A_y', commutative=True), Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Mul(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\psi')(Symbol('A_y', commutative=True))), Symbol('r_0', commutative=True))), Function('\\\\psi')(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{J},\\dot{z})} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J}^{\\dot{z}}, then derive \\operatorname{f_{E}}^{\\mathbf{J}}{(\\mathbf{J},\\dot{z})} = (\\frac{\\dot{z} \\mathbf{J}^{\\dot{z}}}{\\mathbf{J}})^{\\mathbf{J}}, then obtain (\\frac{\\dot{z} \\mathbf{J}^{\\dot{z}}}{\\mathbf{J}})^{\\mathbf{J}} = (\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J}^{\\dot{z}})^{\\mathbf{J}}", "derivation": "\\operatorname{f_{E}}{(\\mathbf{J},\\dot{z})} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J}^{\\dot{z}} and \\operatorname{f_{E}}^{\\mathbf{J}}{(\\mathbf{J},\\dot{z})} = (\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J}^{\\dot{z}})^{\\mathbf{J}} and \\operatorname{f_{E}}^{\\mathbf{J}}{(\\mathbf{J},\\dot{z})} = (\\frac{\\dot{z} \\mathbf{J}^{\\dot{z}}}{\\mathbf{J}})^{\\mathbf{J}} and (\\frac{\\dot{z} \\mathbf{J}^{\\dot{z}}}{\\mathbf{J}})^{\\mathbf{J}} = (\\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J}^{\\dot{z}})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('f_E')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(m_{s},v_{z})} = \\cos{(m_{s} + v_{z})}, then obtain - v_{z} + \\frac{\\int \\operatorname{A_{1}}^{v_{z}}{(m_{s},v_{z})} dv_{z}}{\\int \\cos^{v_{z}}{(m_{s} + v_{z})} dv_{z}} = 1 - v_{z}", "derivation": "\\operatorname{A_{1}}{(m_{s},v_{z})} = \\cos{(m_{s} + v_{z})} and \\operatorname{A_{1}}^{v_{z}}{(m_{s},v_{z})} = \\cos^{v_{z}}{(m_{s} + v_{z})} and \\int \\operatorname{A_{1}}^{v_{z}}{(m_{s},v_{z})} dv_{z} = \\int \\cos^{v_{z}}{(m_{s} + v_{z})} dv_{z} and \\frac{\\int \\operatorname{A_{1}}^{v_{z}}{(m_{s},v_{z})} dv_{z}}{\\int \\cos^{v_{z}}{(m_{s} + v_{z})} dv_{z}} = 1 and - v_{z} + \\frac{\\int \\operatorname{A_{1}}^{v_{z}}{(m_{s},v_{z})} dv_{z}}{\\int \\cos^{v_{z}}{(m_{s} + v_{z})} dv_{z}} = 1 - v_{z}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Pow(Function('A_1')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Pow(cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["divide", 3, "Integral(Pow(cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Mul(Integral(Pow(Function('A_1')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Pow(Integral(Pow(cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 4, "Symbol('v_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Mul(Integral(Pow(Function('A_1')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Pow(Integral(Pow(cos(Add(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\chi{(J)} = \\sin{(J)} and k{(J)} = \\chi^{2}{(J)}, then obtain (k^{2}{(J)})^{J} = (\\sin^{4}{(J)})^{J}", "derivation": "\\chi{(J)} = \\sin{(J)} and k{(J)} = \\chi^{2}{(J)} and k^{2}{(J)} = \\chi^{4}{(J)} and k^{2}{(J)} = \\sin^{4}{(J)} and \\chi^{4}{(J)} = \\sin^{4}{(J)} and (\\chi^{4}{(J)})^{J} = (\\sin^{4}{(J)})^{J} and (k^{2}{(J)})^{J} = (\\sin^{4}{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('J', commutative=True)), Pow(Function('\\\\chi')(Symbol('J', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Pow(Function('k')(Symbol('J', commutative=True)), Integer(2)), Pow(Function('\\\\chi')(Symbol('J', commutative=True)), Integer(4)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('k')(Symbol('J', commutative=True)), Integer(2)), Pow(sin(Symbol('J', commutative=True)), Integer(4)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('\\\\chi')(Symbol('J', commutative=True)), Integer(4)), Pow(sin(Symbol('J', commutative=True)), Integer(4)))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Function('\\\\chi')(Symbol('J', commutative=True)), Integer(4)), Symbol('J', commutative=True)), Pow(Pow(sin(Symbol('J', commutative=True)), Integer(4)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Pow(Function('k')(Symbol('J', commutative=True)), Integer(2)), Symbol('J', commutative=True)), Pow(Pow(sin(Symbol('J', commutative=True)), Integer(4)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(v)} = \\sin{(v)}, then derive \\frac{\\frac{d}{d v} \\operatorname{E_{x}}{(v)}}{\\operatorname{E_{x}}{(v)}} = \\frac{\\cos{(v)}}{\\sin{(v)}}, then obtain \\int \\frac{\\frac{d}{d v} \\sin{(v)}}{\\sin{(v)}} dv = \\int \\frac{\\frac{d}{d v} \\operatorname{E_{x}}{(v)}}{\\operatorname{E_{x}}{(v)}} dv", "derivation": "\\operatorname{E_{x}}{(v)} = \\sin{(v)} and \\log{(\\operatorname{E_{x}}{(v)})} = \\log{(\\sin{(v)})} and \\frac{d}{d v} \\log{(\\operatorname{E_{x}}{(v)})} = \\frac{d}{d v} \\log{(\\sin{(v)})} and \\frac{\\frac{d}{d v} \\operatorname{E_{x}}{(v)}}{\\operatorname{E_{x}}{(v)}} = \\frac{\\cos{(v)}}{\\sin{(v)}} and \\int \\frac{\\frac{d}{d v} \\operatorname{E_{x}}{(v)}}{\\operatorname{E_{x}}{(v)}} dv = \\int \\frac{\\cos{(v)}}{\\sin{(v)}} dv and \\int \\frac{\\frac{d}{d v} \\sin{(v)}}{\\sin{(v)}} dv = \\int \\frac{\\cos{(v)}}{\\sin{(v)}} dv and \\int \\frac{\\frac{d}{d v} \\sin{(v)}}{\\sin{(v)}} dv = \\int \\frac{\\frac{d}{d v} \\operatorname{E_{x}}{(v)}}{\\operatorname{E_{x}}{(v)}} dv", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["log", 1], "Equality(log(Function('E_x')(Symbol('v', commutative=True))), log(sin(Symbol('v', commutative=True))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(log(Function('E_x')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('E_x')(Symbol('v', commutative=True)), Integer(-1)), Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Pow(Function('E_x')(Symbol('v', commutative=True)), Integer(-1)), Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(sin(Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Pow(sin(Symbol('v', commutative=True)), Integer(-1)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(sin(Symbol('v', commutative=True)), Integer(-1)), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integral(Mul(Pow(sin(Symbol('v', commutative=True)), Integer(-1)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(Function('E_x')(Symbol('v', commutative=True)), Integer(-1)), Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\phi_1,C_{d})} = C_{d} - \\phi_1 and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = \\mathbf{J}_P^{\\tilde{g}}, then obtain C_{d} - 2 \\phi_1 + \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = C_{d} + \\mathbf{J}_P^{\\tilde{g}} - 2 \\phi_1", "derivation": "\\mathbf{S}{(\\phi_1,C_{d})} = C_{d} - \\phi_1 and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = \\mathbf{J}_P^{\\tilde{g}} and \\mathbf{S}{(\\phi_1,C_{d})} + \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = \\mathbf{J}_P^{\\tilde{g}} + \\mathbf{S}{(\\phi_1,C_{d})} and C_{d} - \\phi_1 + \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = C_{d} + \\mathbf{J}_P^{\\tilde{g}} - \\phi_1 and C_{d} - 2 \\phi_1 + \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_P,\\tilde{g})} = C_{d} + \\mathbf{J}_P^{\\tilde{g}} - 2 \\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], ["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\phi_1', commutative=True), Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('C_d', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\phi_1', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('C_d', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(t,H)} = t^{H}, then obtain (\\iint \\frac{\\partial}{\\partial H} \\operatorname{v_{z}}{(t,H)} dt dt)^{t} = (\\iint \\frac{\\partial}{\\partial H} t^{H} dt dt)^{t}", "derivation": "\\operatorname{v_{z}}{(t,H)} = t^{H} and \\frac{\\partial}{\\partial H} \\operatorname{v_{z}}{(t,H)} = \\frac{\\partial}{\\partial H} t^{H} and \\int \\frac{\\partial}{\\partial H} \\operatorname{v_{z}}{(t,H)} dt = \\int \\frac{\\partial}{\\partial H} t^{H} dt and \\iint \\frac{\\partial}{\\partial H} \\operatorname{v_{z}}{(t,H)} dt dt = \\iint \\frac{\\partial}{\\partial H} t^{H} dt dt and (\\iint \\frac{\\partial}{\\partial H} \\operatorname{v_{z}}{(t,H)} dt dt)^{t} = (\\iint \\frac{\\partial}{\\partial H} t^{H} dt dt)^{t}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('t', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('v_z')(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Pow(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('v_z')(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Pow(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["power", 4, "Symbol('t', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('v_z')(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Integral(Derivative(Pow(Symbol('t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(M,\\mathbf{J}_M)} = M + \\mathbf{J}_M, then obtain \\mathbf{J}_M \\phi_{2}^{2}{(M,\\mathbf{J}_M)} = \\mathbf{J}_M (M + \\mathbf{J}_M)^{2}", "derivation": "\\phi_{2}{(M,\\mathbf{J}_M)} = M + \\mathbf{J}_M and \\mathbf{J}_M \\phi_{2}{(M,\\mathbf{J}_M)} = \\mathbf{J}_M (M + \\mathbf{J}_M) and \\mathbf{J}_M \\phi_{2}^{2}{(M,\\mathbf{J}_M)} = \\mathbf{J}_M (M + \\mathbf{J}_M) \\phi_{2}{(M,\\mathbf{J}_M)} and \\mathbf{J}_M (M + \\mathbf{J}_M) \\phi_{2}{(M,\\mathbf{J}_M)} = \\mathbf{J}_M (M + \\mathbf{J}_M)^{2} and \\mathbf{J}_M \\phi_{2}^{2}{(M,\\mathbf{J}_M)} = \\mathbf{J}_M (M + \\mathbf{J}_M)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 2, "Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\phi_2')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{v})} = e^{\\mathbf{v}}, then derive \\int \\dot{y}{(\\mathbf{v})} d\\mathbf{v} = v_{x} + e^{\\mathbf{v}}, then obtain \\iint \\dot{y}{(\\mathbf{v})} d\\mathbf{v} d\\mathbf{v} = \\int (v_{x} + \\dot{y}{(\\mathbf{v})}) d\\mathbf{v}", "derivation": "\\dot{y}{(\\mathbf{v})} = e^{\\mathbf{v}} and \\int \\dot{y}{(\\mathbf{v})} d\\mathbf{v} = \\int e^{\\mathbf{v}} d\\mathbf{v} and \\int \\dot{y}{(\\mathbf{v})} d\\mathbf{v} = v_{x} + e^{\\mathbf{v}} and \\int \\dot{y}{(\\mathbf{v})} d\\mathbf{v} = v_{x} + \\dot{y}{(\\mathbf{v})} and \\iint \\dot{y}{(\\mathbf{v})} d\\mathbf{v} d\\mathbf{v} = \\int (v_{x} + \\dot{y}{(\\mathbf{v})}) d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('v_x', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('v_x', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(A_{y})} = \\cos{(A_{y})}, then obtain \\frac{\\int (\\varepsilon{(A_{y})} + \\cos{(A_{y})}) dA_{y}}{\\int 2 \\cos{(A_{y})} dA_{y}} + \\int \\varepsilon{(A_{y})} dA_{y} = \\int \\varepsilon{(A_{y})} dA_{y} + 1", "derivation": "\\varepsilon{(A_{y})} = \\cos{(A_{y})} and \\varepsilon{(A_{y})} + \\cos{(A_{y})} = 2 \\cos{(A_{y})} and \\int (\\varepsilon{(A_{y})} + \\cos{(A_{y})}) dA_{y} = \\int 2 \\cos{(A_{y})} dA_{y} and \\frac{\\int (\\varepsilon{(A_{y})} + \\cos{(A_{y})}) dA_{y}}{\\int 2 \\cos{(A_{y})} dA_{y}} = 1 and \\frac{\\int (\\varepsilon{(A_{y})} + \\cos{(A_{y})}) dA_{y}}{\\int 2 \\cos{(A_{y})} dA_{y}} + \\int \\varepsilon{(A_{y})} dA_{y} = \\int \\varepsilon{(A_{y})} dA_{y} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["add", 1, "cos(Symbol('A_y', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Mul(Integer(2), cos(Symbol('A_y', commutative=True))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))))"], [["divide", 3, "Integral(Mul(Integer(2), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True)))"], "Equality(Mul(Integral(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Pow(Integral(Mul(Integer(2), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))), Integer(1))"], [["add", 4, "Integral(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))"], "Equality(Add(Mul(Integral(Add(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Pow(Integral(Mul(Integer(2), cos(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True))), Integer(-1))), Integral(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Integral(Function('\\\\varepsilon')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\sigma_{x}{(\\tilde{g})} = \\tilde{g}, then obtain \\operatorname{A_{2}}^{- \\varphi^*}{(\\varphi^*)} \\sigma_{x}{(\\tilde{g})} = \\tilde{g} \\operatorname{A_{2}}^{- \\varphi^*}{(\\varphi^*)}", "derivation": "\\operatorname{A_{2}}{(\\varphi^*)} = \\log{(\\varphi^*)} and \\sigma_{x}{(\\tilde{g})} = \\tilde{g} and \\sigma_{x}{(\\tilde{g})} \\log{(\\varphi^*)}^{- \\varphi^*} = \\tilde{g} \\log{(\\varphi^*)}^{- \\varphi^*} and \\operatorname{A_{2}}^{- \\varphi^*}{(\\varphi^*)} \\sigma_{x}{(\\tilde{g})} = \\tilde{g} \\operatorname{A_{2}}^{- \\varphi^*}{(\\varphi^*)}", "srepr_derivation": [["get_premise", "Equality(Function('A_2')(Symbol('\\\\varphi^*', commutative=True)), log(Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))"], [["divide", 2, "Pow(log(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\tilde{g}', commutative=True)), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(log(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Function('\\\\sigma_x')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('A_2')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given q{(\\hat{x}_0,\\tilde{g})} = \\hat{x}_0 \\tilde{g} and \\operatorname{t_{2}}{(\\hat{x}_0,\\tilde{g})} = \\int q{(\\hat{x}_0,\\tilde{g})} d\\hat{x}_0, then obtain \\operatorname{t_{2}}{(\\hat{x}_0,\\tilde{g})} = \\int \\hat{x}_0 \\tilde{g} d\\hat{x}_0", "derivation": "q{(\\hat{x}_0,\\tilde{g})} = \\hat{x}_0 \\tilde{g} and \\int q{(\\hat{x}_0,\\tilde{g})} d\\hat{x}_0 = \\int \\hat{x}_0 \\tilde{g} d\\hat{x}_0 and \\operatorname{t_{2}}{(\\hat{x}_0,\\tilde{g})} = \\int q{(\\hat{x}_0,\\tilde{g})} d\\hat{x}_0 and \\operatorname{t_{2}}{(\\hat{x}_0,\\tilde{g})} = \\int \\hat{x}_0 \\tilde{g} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('q')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('t_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(s,\\hbar)} = \\sin{(\\hbar s)} and m{(s,\\hbar)} = \\sin{(\\hbar s)}, then obtain \\frac{\\operatorname{F_{x}}{(s,\\hbar)}}{E} = \\frac{\\sin{(\\hbar s)}}{E}", "derivation": "\\operatorname{F_{x}}{(s,\\hbar)} = \\sin{(\\hbar s)} and m{(s,\\hbar)} = \\sin{(\\hbar s)} and \\frac{m{(s,\\hbar)}}{E} = \\frac{\\sin{(\\hbar s)}}{E} and m{(s,\\hbar)} = \\operatorname{F_{x}}{(s,\\hbar)} and \\frac{\\operatorname{F_{x}}{(s,\\hbar)}}{E} = \\frac{\\sin{(\\hbar s)}}{E}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True))))"], [["divide", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('F_x')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('F_x')(Symbol('s', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given q{(u)} = e^{u}, then obtain 1 - e^{2 u} = - e^{2 u} + (q{(u)} e^{3 u})^{- u} (q^{2}{(u)} e^{2 u})^{u}", "derivation": "q{(u)} = e^{u} and q^{2}{(u)} = q{(u)} e^{u} and q^{3}{(u)} e^{u} = q^{2}{(u)} e^{2 u} and q^{3}{(u)} e^{u} = q{(u)} e^{3 u} and q{(u)} e^{3 u} = q^{2}{(u)} e^{2 u} and (q{(u)} e^{3 u})^{u} = (q^{2}{(u)} e^{2 u})^{u} and 2 u (q{(u)} e^{3 u})^{u} = 2 u (q^{2}{(u)} e^{2 u})^{u} and 1 = (q{(u)} e^{3 u})^{- u} (q^{2}{(u)} e^{2 u})^{u} and 1 - e^{2 u} = - e^{2 u} + (q{(u)} e^{3 u})^{- u} (q^{2}{(u)} e^{2 u})^{u}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["times", 1, "Function('q')(Symbol('u', commutative=True))"], "Equality(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('q')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True))))"], [["times", 2, "Mul(Function('q')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(3)), exp(Symbol('u', commutative=True))), Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(3)), exp(Symbol('u', commutative=True))), Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))))"], [["power", 5, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Symbol('u', commutative=True)), Pow(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], [["times", 6, "Mul(Integer(2), Symbol('u', commutative=True))"], "Equality(Mul(Integer(2), Symbol('u', commutative=True), Pow(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('u', commutative=True), Pow(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))), Symbol('u', commutative=True))))"], [["divide", 7, "Mul(Integer(2), Symbol('u', commutative=True), Pow(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))), Symbol('u', commutative=True))))"], [["minus", 8, "exp(Mul(Integer(2), Symbol('u', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('u', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('u', commutative=True)))), Mul(Pow(Mul(Function('q')(Symbol('u', commutative=True)), exp(Mul(Integer(3), Symbol('u', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Mul(Pow(Function('q')(Symbol('u', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\mu_0,Q)} = Q + \\log{(\\mu_0)}, then obtain \\iint \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} \\ddot{x}{(\\mu_0,Q)} dQ dQ = \\iint \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} (Q + \\log{(\\mu_0)}) dQ dQ", "derivation": "\\ddot{x}{(\\mu_0,Q)} = Q + \\log{(\\mu_0)} and \\frac{\\partial}{\\partial \\mu_0} \\ddot{x}{(\\mu_0,Q)} = \\frac{\\partial}{\\partial \\mu_0} (Q + \\log{(\\mu_0)}) and \\frac{\\partial^{2}}{\\partial \\mu_0^{2}} \\ddot{x}{(\\mu_0,Q)} = \\frac{\\partial^{2}}{\\partial \\mu_0^{2}} (Q + \\log{(\\mu_0)}) and \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} \\ddot{x}{(\\mu_0,Q)} = \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} (Q + \\log{(\\mu_0)}) and \\int \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} \\ddot{x}{(\\mu_0,Q)} dQ = \\int \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} (Q + \\log{(\\mu_0)}) dQ and \\iint \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} \\ddot{x}{(\\mu_0,Q)} dQ dQ = \\iint \\frac{\\partial^{3}}{\\partial \\mu_0^{3}} (Q + \\log{(\\mu_0)}) dQ dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Derivative(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))), Derivative(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))), Tuple(Symbol('Q', commutative=True))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Add(Symbol('Q', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(3))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given n{(h)} = \\sin{(h)}, then obtain \\frac{d}{d h} - n{(h)} = - n^{h}{(h)} + \\sin^{h}{(h)} + \\frac{d}{d h} - n{(h)}", "derivation": "n{(h)} = \\sin{(h)} and n^{h}{(h)} = \\sin^{h}{(h)} and 0 = - n^{h}{(h)} + \\sin^{h}{(h)} and - n{(h)} = - \\sin{(h)} and \\frac{d}{d h} - n{(h)} = \\frac{d}{d h} - \\sin{(h)} and \\frac{d}{d h} - \\sin{(h)} = - n^{h}{(h)} + \\sin^{h}{(h)} + \\frac{d}{d h} - \\sin{(h)} and \\frac{d}{d h} - n{(h)} = - n^{h}{(h)} + \\sin^{h}{(h)} + \\frac{d}{d h} - n{(h)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('n')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 2, "Pow(Function('n')(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('n')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('n')(Symbol('h', commutative=True))), Mul(Integer(-1), sin(Symbol('h', commutative=True))))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('n')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Add(Mul(Integer(-1), Pow(Function('n')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Derivative(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Mul(Integer(-1), Function('n')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Add(Mul(Integer(-1), Pow(Function('n')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Derivative(Mul(Integer(-1), Function('n')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(V,v_{t})} = - V + e^{v_{t}}, then derive \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = e^{v_{t}}, then obtain \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = \\frac{\\partial^{2}}{\\partial v_{t}^{2}} \\delta{(V,v_{t})}", "derivation": "\\delta{(V,v_{t})} = - V + e^{v_{t}} and \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = \\frac{\\partial}{\\partial v_{t}} (- V + e^{v_{t}}) and \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = e^{v_{t}} and \\delta{(V,v_{t})} = - V + \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} and \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = \\frac{\\partial}{\\partial v_{t}} (- V + \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})}) and \\frac{\\partial}{\\partial v_{t}} \\delta{(V,v_{t})} = \\frac{\\partial^{2}}{\\partial v_{t}^{2}} \\delta{(V,v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), exp(Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), exp(Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Function('\\\\delta')(Symbol('V', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(2))))"]]}, {"prompt": "Given A{(\\psi,z^{*})} = \\psi z^{*}, then obtain \\frac{\\partial}{\\partial z^{*}} (\\psi + \\int \\frac{A{(\\psi,z^{*})}}{z^{*}} dz^{*}) = \\frac{\\partial}{\\partial z^{*}} (\\psi + \\int \\psi dz^{*})", "derivation": "A{(\\psi,z^{*})} = \\psi z^{*} and \\frac{A{(\\psi,z^{*})}}{z^{*}} = \\psi and \\int \\frac{A{(\\psi,z^{*})}}{z^{*}} dz^{*} = \\int \\psi dz^{*} and \\int \\frac{A{(\\psi,z^{*})}}{z^{*}} dz^{*} + \\frac{A{(\\psi,z^{*})}}{z^{*}} = \\int \\psi dz^{*} + \\frac{A{(\\psi,z^{*})}}{z^{*}} and \\psi + \\int \\frac{A{(\\psi,z^{*})}}{z^{*}} dz^{*} = \\psi + \\int \\psi dz^{*} and \\frac{\\partial}{\\partial z^{*}} (\\psi + \\int \\frac{A{(\\psi,z^{*})}}{z^{*}} dz^{*}) = \\frac{\\partial}{\\partial z^{*}} (\\psi + \\int \\psi dz^{*})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)))"], [["divide", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Symbol('\\\\psi', commutative=True))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('z^*', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)))), Add(Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('z^*', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\psi', commutative=True), Integral(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))), Add(Symbol('\\\\psi', commutative=True), Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('z^*', commutative=True)))))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\psi', commutative=True), Integral(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\psi', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\psi', commutative=True), Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(E,\\dot{x})} = - E + \\dot{x}, then obtain - \\frac{\\frac{\\partial}{\\partial \\dot{x}} \\hat{p}_0{(E,\\dot{x})}}{2 E} = - \\frac{1}{2 E}", "derivation": "\\hat{p}_0{(E,\\dot{x})} = - E + \\dot{x} and - E + \\hat{p}_0{(E,\\dot{x})} = - 2 E + \\dot{x} and - \\frac{- E + \\hat{p}_0{(E,\\dot{x})}}{2 E} = - \\frac{- 2 E + \\dot{x}}{2 E} and \\frac{\\partial}{\\partial \\dot{x}} - \\frac{- E + \\hat{p}_0{(E,\\dot{x})}}{2 E} = \\frac{\\partial}{\\partial \\dot{x}} - \\frac{- 2 E + \\dot{x}}{2 E} and - \\frac{\\frac{\\partial}{\\partial \\dot{x}} \\hat{p}_0{(E,\\dot{x})}}{2 E} = - \\frac{1}{2 E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 1, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Integer(2), Symbol('E', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(P_{g},Z)} = Z^{P_{g}}, then obtain (\\int (P_{g} + \\operatorname{a^{\\dagger}}{(P_{g},Z)}) dZ)^{P_{g}} = (\\int (P_{g} + Z^{P_{g}}) dZ)^{P_{g}}", "derivation": "\\operatorname{a^{\\dagger}}{(P_{g},Z)} = Z^{P_{g}} and P_{g} + \\operatorname{a^{\\dagger}}{(P_{g},Z)} = P_{g} + Z^{P_{g}} and \\int (P_{g} + \\operatorname{a^{\\dagger}}{(P_{g},Z)}) dZ = \\int (P_{g} + Z^{P_{g}}) dZ and (\\int (P_{g} + \\operatorname{a^{\\dagger}}{(P_{g},Z)}) dZ)^{P_{g}} = (\\int (P_{g} + Z^{P_{g}}) dZ)^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('P_g', commutative=True)))"], [["add", 1, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Function('a^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('P_g', commutative=True), Pow(Symbol('Z', commutative=True), Symbol('P_g', commutative=True))))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Symbol('P_g', commutative=True), Function('a^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Pow(Symbol('Z', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('P_g', commutative=True), Function('a^{\\\\dagger}')(Symbol('P_g', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('P_g', commutative=True)), Pow(Integral(Add(Symbol('P_g', commutative=True), Pow(Symbol('Z', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('Z', commutative=True))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\delta)} = \\delta, then obtain \\log{(\\operatorname{F_{H}}{(\\delta)})} \\int \\operatorname{F_{H}}{(\\delta)} d\\delta = \\log{(\\operatorname{F_{H}}{(\\delta)})} \\int \\delta d\\delta", "derivation": "\\operatorname{F_{H}}{(\\delta)} = \\delta and \\log{(\\operatorname{F_{H}}{(\\delta)})} = \\log{(\\delta)} and \\int \\operatorname{F_{H}}{(\\delta)} d\\delta = \\int \\delta d\\delta and \\log{(\\delta)} \\int \\operatorname{F_{H}}{(\\delta)} d\\delta = \\log{(\\delta)} \\int \\delta d\\delta and \\log{(\\operatorname{F_{H}}{(\\delta)})} \\int \\operatorname{F_{H}}{(\\delta)} d\\delta = \\log{(\\operatorname{F_{H}}{(\\delta)})} \\int \\delta d\\delta", "srepr_derivation": [["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["log", 1], "Equality(log(Function('F_H')(Symbol('\\\\delta', commutative=True))), log(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 3, "log(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(log(Symbol('\\\\delta', commutative=True)), Integral(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(log(Symbol('\\\\delta', commutative=True)), Integral(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(log(Function('F_H')(Symbol('\\\\delta', commutative=True))), Integral(Function('F_H')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(log(Function('F_H')(Symbol('\\\\delta', commutative=True))), Integral(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} = i^{\\hbar}, then derive \\frac{\\partial}{\\partial \\hbar} \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} - \\frac{1}{\\hbar^{2}} = i^{\\hbar} \\log{(i)} - \\frac{1}{\\hbar^{2}}, then obtain \\frac{\\partial}{\\partial \\hbar} \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} - \\frac{1}{\\hbar^{2}} = \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} \\log{(i)} - \\frac{1}{\\hbar^{2}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(i,\\hbar)} = i^{\\hbar} and \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} + \\frac{1}{\\hbar} = i^{\\hbar} + \\frac{1}{\\hbar} and \\frac{\\partial}{\\partial \\hbar} (\\operatorname{L_{\\varepsilon}}{(i,\\hbar)} + \\frac{1}{\\hbar}) = \\frac{\\partial}{\\partial \\hbar} (i^{\\hbar} + \\frac{1}{\\hbar}) and \\frac{\\partial}{\\partial \\hbar} \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} - \\frac{1}{\\hbar^{2}} = i^{\\hbar} \\log{(i)} - \\frac{1}{\\hbar^{2}} and \\frac{\\partial}{\\partial \\hbar} i^{\\hbar} - \\frac{1}{\\hbar^{2}} = i^{\\hbar} \\log{(i)} - \\frac{1}{\\hbar^{2}} and \\frac{\\partial}{\\partial \\hbar} \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} - \\frac{1}{\\hbar^{2}} = \\operatorname{L_{\\varepsilon}}{(i,\\hbar)} \\log{(i)} - \\frac{1}{\\hbar^{2}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Add(Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))), Add(Mul(Function('L_{\\\\varepsilon}')(Symbol('i', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(t_{2})} = e^{t_{2}}, then obtain \\operatorname{y^{\\prime}}^{2 t_{2}}{(t_{2})} e^{t_{2}} (e^{t_{2}})^{2 t_{2}} = e^{t_{2}} (e^{t_{2}})^{4 t_{2}}", "derivation": "\\operatorname{y^{\\prime}}{(t_{2})} = e^{t_{2}} and \\operatorname{y^{\\prime}}^{t_{2}}{(t_{2})} = (e^{t_{2}})^{t_{2}} and \\operatorname{y^{\\prime}}^{t_{2}}{(t_{2})} (e^{t_{2}})^{t_{2}} = (e^{t_{2}})^{2 t_{2}} and \\operatorname{y^{\\prime}}^{2 t_{2}}{(t_{2})} (e^{t_{2}})^{2 t_{2}} = (e^{t_{2}})^{4 t_{2}} and \\operatorname{y^{\\prime}}^{2 t_{2}}{(t_{2})} e^{t_{2}} (e^{t_{2}})^{2 t_{2}} = e^{t_{2}} (e^{t_{2}})^{4 t_{2}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Pow(exp(Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))), Pow(exp(Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True)))), Pow(exp(Symbol('t_2', commutative=True)), Mul(Integer(4), Symbol('t_2', commutative=True))))"], [["times", 4, "exp(Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))), exp(Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True)))), Mul(exp(Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Mul(Integer(4), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given y{(A_{x})} = \\cos{(A_{x})}, then derive \\frac{\\cos{(A_{x})} \\int y{(A_{x})} dA_{x}}{y{(A_{x})}} = \\frac{(s + \\sin{(A_{x})}) \\cos{(A_{x})}}{y{(A_{x})}}, then obtain \\int \\cos{(A_{x})} dA_{x} = s + \\sin{(A_{x})}", "derivation": "y{(A_{x})} = \\cos{(A_{x})} and \\int y{(A_{x})} dA_{x} = \\int \\cos{(A_{x})} dA_{x} and \\frac{\\cos{(A_{x})} \\int y{(A_{x})} dA_{x}}{y{(A_{x})}} = \\frac{\\cos{(A_{x})} \\int \\cos{(A_{x})} dA_{x}}{y{(A_{x})}} and \\frac{\\cos{(A_{x})} \\int y{(A_{x})} dA_{x}}{y{(A_{x})}} = \\frac{(s + \\sin{(A_{x})}) \\cos{(A_{x})}}{y{(A_{x})}} and \\int \\cos{(A_{x})} dA_{x} = s + \\sin{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["times", 2, "Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True)))"], "Equality(Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True)), Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True)), Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True)), Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Mul(Add(Symbol('s', commutative=True), sin(Symbol('A_x', commutative=True))), Pow(Function('y')(Symbol('A_x', commutative=True)), Integer(-1)), cos(Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('s', commutative=True), sin(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given W{(\\mathbf{F})} = e^{\\mathbf{F}}, then obtain \\int - W^{\\mathbf{F}}{(\\mathbf{F})} e^{- \\mathbf{F}} d\\mathbf{F} = \\int - e^{- \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}} d\\mathbf{F}", "derivation": "W{(\\mathbf{F})} = e^{\\mathbf{F}} and W^{\\mathbf{F}}{(\\mathbf{F})} = (e^{\\mathbf{F}})^{\\mathbf{F}} and - W^{\\mathbf{F}}{(\\mathbf{F})} e^{- \\mathbf{F}} = - e^{- \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}} and \\int - W^{\\mathbf{F}}{(\\mathbf{F})} e^{- \\mathbf{F}} d\\mathbf{F} = \\int - e^{- \\mathbf{F}} (e^{\\mathbf{F}})^{\\mathbf{F}} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('W')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Function('W')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} = \\mathbf{J}_M e^{H}, then obtain \\frac{e^{- H}}{\\operatorname{E_{x}}{(\\mathbf{J}_M,H)}} = \\frac{e^{- H} \\int \\mathbf{J}_M e^{H} dH}{\\operatorname{E_{x}}{(\\mathbf{J}_M,H)} \\int \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} dH}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{J}_M,H)} = \\mathbf{J}_M e^{H} and \\int \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} dH = \\int \\mathbf{J}_M e^{H} dH and e^{- H} = \\frac{e^{- H} \\int \\mathbf{J}_M e^{H} dH}{\\int \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} dH} and \\frac{e^{- 2 H}}{\\mathbf{J}_M} = \\frac{e^{- 2 H} \\int \\mathbf{J}_M e^{H} dH}{\\mathbf{J}_M \\int \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} dH} and \\frac{e^{- H}}{\\operatorname{E_{x}}{(\\mathbf{J}_M,H)}} = \\frac{e^{- H} \\int \\mathbf{J}_M e^{H} dH}{\\operatorname{E_{x}}{(\\mathbf{J}_M,H)} \\int \\operatorname{E_{x}}{(\\mathbf{J}_M,H)} dH}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["divide", 2, "Mul(exp(Symbol('H', commutative=True)), Integral(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], "Equality(exp(Mul(Integer(-1), Symbol('H', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('H', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Pow(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Integer(2), Symbol('H', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Pow(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('H', commutative=True)))), Mul(Pow(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('H', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Pow(Integral(Function('E_x')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}} n_{2})}, then derive n_{2} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} - \\frac{1}{\\hat{\\mathbf{x}}} = n_{2}, then obtain n_{2} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\log{(\\hat{\\mathbf{x}} n_{2})} - \\frac{1}{\\hat{\\mathbf{x}}} = n_{2}", "derivation": "\\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}} n_{2})} and \\hat{\\mathbf{x}} n_{2} + \\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}} n_{2})} = \\hat{\\mathbf{x}} n_{2} and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} (\\hat{\\mathbf{x}} n_{2} + \\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} - \\log{(\\hat{\\mathbf{x}} n_{2})}) = \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\hat{\\mathbf{x}} n_{2} and n_{2} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\mathbf{B}{(n_{2},\\hat{\\mathbf{x}})} - \\frac{1}{\\hat{\\mathbf{x}}} = n_{2} and n_{2} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\log{(\\hat{\\mathbf{x}} n_{2})} - \\frac{1}{\\hat{\\mathbf{x}}} = n_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('n_2', commutative=True), Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)))), Symbol('n_2', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('n_2', commutative=True), Derivative(log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)))), Symbol('n_2', commutative=True))"]]}, {"prompt": "Given \\varepsilon{(\\omega,M_{E})} = \\frac{\\partial}{\\partial M_{E}} M_{E} \\omega, then derive 0 = \\frac{\\omega - \\varepsilon{(\\omega,M_{E})}}{\\theta}, then obtain (\\frac{\\omega - \\varepsilon{(\\omega,M_{E})}}{\\theta})^{\\omega} = 1", "derivation": "\\varepsilon{(\\omega,M_{E})} = \\frac{\\partial}{\\partial M_{E}} M_{E} \\omega and 0 = - \\varepsilon{(\\omega,M_{E})} + \\frac{\\partial}{\\partial M_{E}} M_{E} \\omega and 0 = \\frac{- \\varepsilon{(\\omega,M_{E})} + \\frac{\\partial}{\\partial M_{E}} M_{E} \\omega}{\\theta} and 0 = \\frac{\\omega - \\varepsilon{(\\omega,M_{E})}}{\\theta} and 0^{\\omega} = (\\frac{\\omega - \\varepsilon{(\\omega,M_{E})}}{\\theta})^{\\omega} and 0^{\\omega} = (\\frac{\\omega - \\frac{\\partial}{\\partial M_{E}} M_{E} \\omega}{\\theta})^{\\omega} and (\\frac{\\omega - \\varepsilon{(\\omega,M_{E})}}{\\theta})^{\\omega} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True)), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["divide", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))))))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\omega', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integer(0), Symbol('\\\\omega', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('M_E', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\omega', commutative=True), Symbol('M_E', commutative=True))))), Symbol('\\\\omega', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mu_{0}{(\\omega,\\theta_1)} = \\omega \\theta_1, then obtain \\mu_{0}{(\\omega,\\theta_1)} - \\frac{\\partial}{\\partial \\omega} \\omega \\theta_1 = \\omega \\theta_1 - \\frac{\\partial}{\\partial \\omega} \\omega \\theta_1", "derivation": "\\mu_{0}{(\\omega,\\theta_1)} = \\omega \\theta_1 and \\frac{\\partial}{\\partial \\omega} \\mu_{0}{(\\omega,\\theta_1)} = \\frac{\\partial}{\\partial \\omega} \\omega \\theta_1 and \\mu_{0}{(\\omega,\\theta_1)} - \\frac{\\partial}{\\partial \\omega} \\mu_{0}{(\\omega,\\theta_1)} = \\omega \\theta_1 - \\frac{\\partial}{\\partial \\omega} \\mu_{0}{(\\omega,\\theta_1)} and \\mu_{0}{(\\omega,\\theta_1)} - \\frac{\\partial}{\\partial \\omega} \\omega \\theta_1 = \\omega \\theta_1 - \\frac{\\partial}{\\partial \\omega} \\omega \\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mu_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}, then obtain \\int \\frac{d}{d \\hat{x}} (- \\hat{x} + \\mathbf{H}{(\\hat{x})} - \\cos{(\\hat{x})}) d\\hat{x} = \\int \\frac{d}{d \\hat{x}} (- \\hat{x} + \\log{(\\sin{(\\hat{x})})} - \\cos{(\\hat{x})}) d\\hat{x}", "derivation": "\\mathbf{H}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})} and - \\hat{x} + \\mathbf{H}{(\\hat{x})} = - \\hat{x} + \\log{(\\sin{(\\hat{x})})} and - \\hat{x} + \\mathbf{H}{(\\hat{x})} - \\cos{(\\hat{x})} = - \\hat{x} + \\log{(\\sin{(\\hat{x})})} - \\cos{(\\hat{x})} and \\frac{d}{d \\hat{x}} (- \\hat{x} + \\mathbf{H}{(\\hat{x})} - \\cos{(\\hat{x})}) = \\frac{d}{d \\hat{x}} (- \\hat{x} + \\log{(\\sin{(\\hat{x})})} - \\cos{(\\hat{x})}) and \\int \\frac{d}{d \\hat{x}} (- \\hat{x} + \\mathbf{H}{(\\hat{x})} - \\cos{(\\hat{x})}) d\\hat{x} = \\int \\frac{d}{d \\hat{x}} (- \\hat{x} + \\log{(\\sin{(\\hat{x})})} - \\cos{(\\hat{x})}) d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True)))))"], [["minus", 2, "cos(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(t_{2})} = t_{2}, then obtain - \\log{(2 t_{2})} + 2 \\log{(t_{2} + \\mathbf{B}{(t_{2})})} = t_{2} - \\mathbf{B}{(t_{2})} - \\log{(2 t_{2})} + 2 \\log{(t_{2} + \\mathbf{B}{(t_{2})})}", "derivation": "\\mathbf{B}{(t_{2})} = t_{2} and t_{2} + \\mathbf{B}{(t_{2})} = 2 t_{2} and \\mathbf{B}{(t_{2})} - \\log{(t_{2} + \\mathbf{B}{(t_{2})})} = t_{2} - \\log{(t_{2} + \\mathbf{B}{(t_{2})})} and \\mathbf{B}{(t_{2})} - \\log{(2 t_{2})} = t_{2} - \\log{(2 t_{2})} and \\mathbf{B}{(t_{2})} - \\log{(2 t_{2})} + \\log{(t_{2} + \\mathbf{B}{(t_{2})})} = t_{2} - \\log{(2 t_{2})} + \\log{(t_{2} + \\mathbf{B}{(t_{2})})} and - \\log{(2 t_{2})} + 2 \\log{(t_{2} + \\mathbf{B}{(t_{2})})} = t_{2} - \\mathbf{B}{(t_{2})} - \\log{(2 t_{2})} + 2 \\log{(t_{2} + \\mathbf{B}{(t_{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["add", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))), Mul(Integer(2), Symbol('t_2', commutative=True)))"], [["minus", 1, "log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)))))), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True))))), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True))))))"], [["minus", 4, "Mul(Integer(-1), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)))))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True)))), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))))), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True)))), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))))))"], [["minus", 5, "Add(Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)), Mul(Integer(-1), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))))))"], "Equality(Add(Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True)))), Mul(Integer(2), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)))))), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True))), Mul(Integer(-1), log(Mul(Integer(2), Symbol('t_2', commutative=True)))), Mul(Integer(2), log(Add(Symbol('t_2', commutative=True), Function('\\\\mathbf{B}')(Symbol('t_2', commutative=True)))))))"]]}, {"prompt": "Given \\phi_{1}{(x)} = \\log{(\\cos{(x)})}, then derive \\sin{(x)} - \\frac{d}{d x} \\phi_{1}{(x)} = \\sin{(x)} + \\frac{\\sin{(x)}}{\\cos{(x)}}, then obtain \\frac{\\sin{(x)} - \\frac{d}{d x} \\phi_{1}{(x)}}{\\cos{(x)}} = \\frac{\\sin{(x)} + \\frac{\\sin{(x)}}{\\cos{(x)}}}{\\cos{(x)}}", "derivation": "\\phi_{1}{(x)} = \\log{(\\cos{(x)})} and \\phi_{1}{(x)} + \\cos{(x)} = \\log{(\\cos{(x)})} + \\cos{(x)} and - \\phi_{1}{(x)} - \\cos{(x)} = - \\log{(\\cos{(x)})} - \\cos{(x)} and \\frac{d}{d x} (- \\phi_{1}{(x)} - \\cos{(x)}) = \\frac{d}{d x} (- \\log{(\\cos{(x)})} - \\cos{(x)}) and \\sin{(x)} - \\frac{d}{d x} \\phi_{1}{(x)} = \\sin{(x)} + \\frac{\\sin{(x)}}{\\cos{(x)}} and \\frac{\\sin{(x)} - \\frac{d}{d x} \\phi_{1}{(x)}}{\\cos{(x)}} = \\frac{\\sin{(x)} + \\frac{\\sin{(x)}}{\\cos{(x)}}}{\\cos{(x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True))))"], [["add", 1, "cos(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Add(log(cos(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), log(cos(Symbol('x', commutative=True)))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\phi_1')(Symbol('x', commutative=True))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(cos(Symbol('x', commutative=True)))), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Add(sin(Symbol('x', commutative=True)), Mul(sin(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1)))))"], [["times", 5, "Pow(cos(Symbol('x', commutative=True)), Integer(-1))"], "Equality(Mul(Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\phi_1')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Mul(Add(sin(Symbol('x', commutative=True)), Mul(sin(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1)))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given i{(\\phi_1,\\omega)} = \\omega^{\\phi_1}, then obtain \\omega + (\\frac{i{(\\phi_1,\\omega)}}{\\omega})^{\\phi_1} = \\omega + (\\frac{\\omega^{\\phi_1}}{\\omega})^{\\phi_1}", "derivation": "i{(\\phi_1,\\omega)} = \\omega^{\\phi_1} and \\frac{i{(\\phi_1,\\omega)}}{\\omega} = \\frac{\\omega^{\\phi_1}}{\\omega} and (\\frac{i{(\\phi_1,\\omega)}}{\\omega})^{\\phi_1} = (\\frac{\\omega^{\\phi_1}}{\\omega})^{\\phi_1} and \\omega + (\\frac{i{(\\phi_1,\\omega)}}{\\omega})^{\\phi_1} = \\omega + (\\frac{\\omega^{\\phi_1}}{\\omega})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["add", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Pow(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Pow(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(F_{c},C)} = \\log{(C + F_{c})} and \\delta{(F_{c},C)} = C \\log{(C + F_{c})}, then obtain - C + (C \\hat{x}{(F_{c},C)})^{F_{c}} = - C + \\delta^{F_{c}}{(F_{c},C)}", "derivation": "\\hat{x}{(F_{c},C)} = \\log{(C + F_{c})} and C \\hat{x}{(F_{c},C)} = C \\log{(C + F_{c})} and (C \\hat{x}{(F_{c},C)})^{F_{c}} = (C \\log{(C + F_{c})})^{F_{c}} and - C + (C \\hat{x}{(F_{c},C)})^{F_{c}} = - C + (C \\log{(C + F_{c})})^{F_{c}} and \\delta{(F_{c},C)} = C \\log{(C + F_{c})} and - C + (C \\hat{x}{(F_{c},C)})^{F_{c}} = - C + \\delta^{F_{c}}{(F_{c},C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True))))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\hat{x}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Function('\\\\hat{x}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Symbol('F_c', commutative=True)), Pow(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)))"], [["minus", 3, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Function('\\\\hat{x}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), log(Add(Symbol('C', commutative=True), Symbol('F_c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Function('\\\\hat{x}')(Symbol('F_c', commutative=True), Symbol('C', commutative=True))), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('C', commutative=True)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a,\\chi)} = - a + \\log{(\\chi)} and n{(a,\\chi)} = - a + \\log{(\\chi)}, then obtain \\frac{d}{d \\chi} \\log{(a)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\frac{a n{(a,\\chi)}}{\\operatorname{M_{E}}{(a,\\chi)}})}", "derivation": "\\operatorname{M_{E}}{(a,\\chi)} = - a + \\log{(\\chi)} and a \\operatorname{M_{E}}{(a,\\chi)} = a (- a + \\log{(\\chi)}) and a = \\frac{a (- a + \\log{(\\chi)})}{\\operatorname{M_{E}}{(a,\\chi)}} and \\log{(a)} = \\log{(\\frac{a (- a + \\log{(\\chi)})}{\\operatorname{M_{E}}{(a,\\chi)}})} and \\frac{d}{d \\chi} \\log{(a)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\frac{a (- a + \\log{(\\chi)})}{\\operatorname{M_{E}}{(a,\\chi)}})} and n{(a,\\chi)} = - a + \\log{(\\chi)} and \\frac{d}{d \\chi} \\log{(a)} = \\frac{\\partial}{\\partial \\chi} \\log{(\\frac{a n{(a,\\chi)}}{\\operatorname{M_{E}}{(a,\\chi)}})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Symbol('a', commutative=True), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True)))))"], [["divide", 2, "Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Symbol('a', commutative=True), Mul(Symbol('a', commutative=True), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Pow(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["log", 3], "Equality(log(Symbol('a', commutative=True)), log(Mul(Symbol('a', commutative=True), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Pow(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(log(Symbol('a', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('a', commutative=True), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Pow(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(log(Symbol('a', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('a', commutative=True), Pow(Function('M_E')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)), Function('n')(Symbol('a', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(b,\\ddot{x})} = \\ddot{x} b and \\operatorname{t_{1}}{(b,\\ddot{x})} = (\\ddot{x} b)^{- \\ddot{x}} \\mathbf{M}{(b,\\ddot{x})}, then obtain \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}{(b,\\ddot{x})} = \\frac{\\partial}{\\partial b} \\mathbf{M}{(b,\\ddot{x})} \\mathbf{M}^{- \\ddot{x}}{(b,\\ddot{x})}", "derivation": "\\mathbf{M}{(b,\\ddot{x})} = \\ddot{x} b and \\operatorname{t_{1}}{(b,\\ddot{x})} = (\\ddot{x} b)^{- \\ddot{x}} \\mathbf{M}{(b,\\ddot{x})} and \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}{(b,\\ddot{x})} = \\frac{\\partial}{\\partial b} (\\ddot{x} b)^{- \\ddot{x}} \\mathbf{M}{(b,\\ddot{x})} and \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}{(b,\\ddot{x})} = \\frac{\\partial}{\\partial b} \\mathbf{M}{(b,\\ddot{x})} \\mathbf{M}^{- \\ddot{x}}{(b,\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))), Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True))), Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('t_1')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Function('\\\\mathbf{M}')(Symbol('b', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})}, then obtain \\cos{(\\frac{\\frac{d}{d x^\\prime} \\operatorname{r_{0}}{(x^\\prime)}}{x^\\prime})} = \\cos{(\\frac{\\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})}}{x^\\prime})}", "derivation": "\\operatorname{r_{0}}{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\operatorname{r_{0}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} and \\frac{\\frac{d}{d x^\\prime} \\operatorname{r_{0}}{(x^\\prime)}}{x^\\prime} = \\frac{\\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})}}{x^\\prime} and \\cos{(\\frac{\\frac{d}{d x^\\prime} \\operatorname{r_{0}}{(x^\\prime)}}{x^\\prime})} = \\cos{(\\frac{\\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})}}{x^\\prime})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('x^\\\\prime', commutative=True)), log(sin(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Function('r_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(log(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Function('r_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))), cos(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(log(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))))"]]}, {"prompt": "Given u{(\\mathbf{E},t)} = \\frac{\\mathbf{E}}{t}, then obtain \\frac{\\mathbf{E} (- u{(\\mathbf{E},t)})^{\\mathbf{E}}}{t} = \\frac{\\mathbf{E} (- \\frac{\\mathbf{E}}{t})^{\\mathbf{E}}}{t}", "derivation": "u{(\\mathbf{E},t)} = \\frac{\\mathbf{E}}{t} and - u{(\\mathbf{E},t)} = - \\frac{\\mathbf{E}}{t} and (- u{(\\mathbf{E},t)})^{\\mathbf{E}} = (- \\frac{\\mathbf{E}}{t})^{\\mathbf{E}} and \\frac{\\mathbf{E} (- u{(\\mathbf{E},t)})^{\\mathbf{E}}}{t} = \\frac{\\mathbf{E} (- \\frac{\\mathbf{E}}{t})^{\\mathbf{E}}}{t}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 3, "Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(h,u)} = u \\cos{(h)}, then obtain \\frac{\\frac{\\partial}{\\partial h} (- u - \\sigma_{x}{(h,u)})}{\\sigma_{x}{(h,u)}} = \\frac{\\frac{\\partial}{\\partial h} (- u \\cos{(h)} - u)}{\\sigma_{x}{(h,u)}}", "derivation": "\\sigma_{x}{(h,u)} = u \\cos{(h)} and u + \\sigma_{x}{(h,u)} = u \\cos{(h)} + u and - u - \\sigma_{x}{(h,u)} = - u \\cos{(h)} - u and \\frac{\\partial}{\\partial h} (- u - \\sigma_{x}{(h,u)}) = \\frac{\\partial}{\\partial h} (- u \\cos{(h)} - u) and \\frac{\\frac{\\partial}{\\partial h} (- u - \\sigma_{x}{(h,u)})}{\\sigma_{x}{(h,u)}} = \\frac{\\frac{\\partial}{\\partial h} (- u \\cos{(h)} - u)}{\\sigma_{x}{(h,u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('u', commutative=True), cos(Symbol('h', commutative=True))))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('u', commutative=True), cos(Symbol('h', commutative=True))), Symbol('u', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Symbol('u', commutative=True), cos(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True), cos(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 4, "Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\sigma_x')(Symbol('h', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True), cos(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given c{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}}, then derive \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}}, then obtain 0 = - c{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})}", "derivation": "c{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and 0 = - c{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and 0 = - c{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} \\Psi_{\\lambda}{(\\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["minus", 1, "Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(b,\\hat{H})} = - \\hat{H} + b, then obtain b + \\frac{b \\operatorname{C_{d}}{(b,\\hat{H})}}{- \\hat{H} + b} = 2 b", "derivation": "\\operatorname{C_{d}}{(b,\\hat{H})} = - \\hat{H} + b and \\frac{\\operatorname{C_{d}}{(b,\\hat{H})}}{- \\hat{H} + b} = 1 and \\frac{b \\operatorname{C_{d}}{(b,\\hat{H})}}{- \\hat{H} + b} = b and b + \\frac{b \\operatorname{C_{d}}{(b,\\hat{H})}}{- \\hat{H} + b} = 2 b", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('b', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('b', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('b', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('b', commutative=True)), Integer(-1)), Function('C_d')(Symbol('b', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(1))"], [["times", 2, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('b', commutative=True)), Integer(-1)), Function('C_d')(Symbol('b', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('b', commutative=True))"], [["add", 3, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Mul(Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('b', commutative=True)), Integer(-1)), Function('C_d')(Symbol('b', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Mul(Integer(2), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{s},\\mathbf{E})} = - \\mathbf{E} + \\mathbf{s} and \\bar{\\h}{(\\mathbf{E})} = - \\mathbf{E}, then obtain 1 = \\frac{\\mathbf{s} + \\bar{\\h}{(\\mathbf{E})}}{2 \\mathbf{D}{(\\mathbf{s},\\mathbf{E})}} + \\frac{1}{2}", "derivation": "\\mathbf{D}{(\\mathbf{s},\\mathbf{E})} = - \\mathbf{E} + \\mathbf{s} and \\frac{1}{2} = \\frac{- \\mathbf{E} + \\mathbf{s}}{2 \\mathbf{D}{(\\mathbf{s},\\mathbf{E})}} and \\bar{\\h}{(\\mathbf{E})} = - \\mathbf{E} and \\frac{1}{2} = \\frac{\\mathbf{s} + \\bar{\\h}{(\\mathbf{E})}}{2 \\mathbf{D}{(\\mathbf{s},\\mathbf{E})}} and 0 = \\frac{\\mathbf{s} + \\bar{\\h}{(\\mathbf{E})}}{2 \\mathbf{D}{(\\mathbf{s},\\mathbf{E})}} - \\frac{1}{2} and 1 = \\frac{\\mathbf{s} + \\bar{\\h}{(\\mathbf{E})}}{2 \\mathbf{D}{(\\mathbf{s},\\mathbf{E})}} + \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Rational(1, 2), Mul(Rational(1, 2), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{E}', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))))"], [["minus", 4, "Rational(1, 2)"], "Equality(Integer(0), Add(Mul(Rational(1, 2), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{E}', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Rational(-1, 2)))"], [["minus", 5, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Rational(1, 2), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{E}', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Rational(1, 2)))"]]}, {"prompt": "Given \\hat{p}{(n_{2})} = e^{n_{2}}, then obtain \\int (\\hat{p}{(n_{2})} e^{- n_{2}})^{n_{2}} dn_{2} = n_{2} + p", "derivation": "\\hat{p}{(n_{2})} = e^{n_{2}} and \\hat{p}{(n_{2})} e^{- n_{2}} = 1 and (\\hat{p}{(n_{2})} e^{- n_{2}})^{n_{2}} = 1 and \\int (\\hat{p}{(n_{2})} e^{- n_{2}})^{n_{2}} dn_{2} = \\int 1 dn_{2} and \\int (\\hat{p}{(n_{2})} e^{- n_{2}})^{n_{2}} dn_{2} = n_{2} + p", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["divide", 1, "exp(Symbol('n_2', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Integer(1))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{p}')(Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\hat{p}')(Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Mul(Function('\\\\hat{p}')(Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('n_2', commutative=True), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\hbar)} = \\int e^{\\hbar} d\\hbar, then obtain \\int \\frac{d}{d \\hbar} 0 d\\hbar = \\int \\frac{d}{d \\hbar} (- \\Omega{(\\hbar)} + \\int e^{\\hbar} d\\hbar) d\\hbar", "derivation": "\\Omega{(\\hbar)} = \\int e^{\\hbar} d\\hbar and 0 = - \\Omega{(\\hbar)} + \\int e^{\\hbar} d\\hbar and \\frac{d}{d \\hbar} 0 = \\frac{d}{d \\hbar} (- \\Omega{(\\hbar)} + \\int e^{\\hbar} d\\hbar) and \\int \\frac{d}{d \\hbar} 0 d\\hbar = \\int \\frac{d}{d \\hbar} (- \\Omega{(\\hbar)} + \\int e^{\\hbar} d\\hbar) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Function('\\\\Omega')(Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{p},B)} = \\log{(\\mathbf{p}^{B})} and n{(\\mathbf{p},B)} = \\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p}, then obtain 1 = \\frac{n{(\\mathbf{p},B)}}{\\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p}}", "derivation": "\\mathbf{J}{(\\mathbf{p},B)} = \\log{(\\mathbf{p}^{B})} and \\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p} = \\int \\log{(\\mathbf{p}^{B})} d\\mathbf{p} and 1 = \\frac{\\int \\log{(\\mathbf{p}^{B})} d\\mathbf{p}}{\\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p}} and n{(\\mathbf{p},B)} = \\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p} and n{(\\mathbf{p},B)} = \\int \\log{(\\mathbf{p}^{B})} d\\mathbf{p} and 1 = \\frac{n{(\\mathbf{p},B)}}{\\int \\mathbf{J}{(\\mathbf{p},B)} d\\mathbf{p}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), log(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(log(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integer(-1)), Integral(log(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Integral(log(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(1), Mul(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Pow(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then derive \\int k{(\\mathbf{A})} d\\mathbf{A} = \\hat{H}_l - \\cos{(\\mathbf{A})}, then derive \\hat{H}_l - \\cos{(\\mathbf{A})} = \\varepsilon_0 - \\cos{(\\mathbf{A})}, then obtain \\int k{(\\mathbf{A})} d\\mathbf{A} = \\varepsilon_0 - \\cos{(\\mathbf{A})}", "derivation": "k{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\int k{(\\mathbf{A})} d\\mathbf{A} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A} and \\int k{(\\mathbf{A})} d\\mathbf{A} = \\hat{H}_l - \\cos{(\\mathbf{A})} and \\hat{H}_l - \\cos{(\\mathbf{A})} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A} and \\hat{H}_l - \\cos{(\\mathbf{A})} = \\varepsilon_0 - \\cos{(\\mathbf{A})} and \\int k{(\\mathbf{A})} d\\mathbf{A} = \\varepsilon_0 - \\cos{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Function('k')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given S{(t_{2},P_{e})} = P_{e}^{t_{2}}, then derive \\int 0 dP_{e} = n + \\int \\frac{P_{e}^{t_{2}} - S{(t_{2},P_{e})}}{S{(t_{2},P_{e})}} dP_{e}, then obtain 2 n + \\int 0 dP_{e} = \\int (\\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} - 1) dP_{e}", "derivation": "S{(t_{2},P_{e})} = P_{e}^{t_{2}} and 1 = \\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} and 0 = \\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} - 1 and \\int 0 dP_{e} = \\int (\\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} - 1) dP_{e} and \\int 0 dP_{e} = n + \\int \\frac{P_{e}^{t_{2}} - S{(t_{2},P_{e})}}{S{(t_{2},P_{e})}} dP_{e} and \\int 0 dP_{e} = n + \\int 0 dP_{e} and n + \\int 0 dP_{e} = \\int (\\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} - 1) dP_{e} and 2 n + \\int 0 dP_{e} = \\int (\\frac{P_{e}^{t_{2}}}{S{(t_{2},P_{e})}} - 1) dP_{e}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)))"], [["divide", 1, "Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))))"], [["minus", 2, 1], "Equality(Integer(0), Add(Mul(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Integer(-1)))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Mul(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Integer(0), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('n', commutative=True), Integral(Mul(Add(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)))), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Integer(0), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('n', commutative=True), Integral(Integer(0), Tuple(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('n', commutative=True), Integral(Integer(0), Tuple(Symbol('P_e', commutative=True)))), Integral(Add(Mul(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Mul(Integer(2), Symbol('n', commutative=True)), Integral(Integer(0), Tuple(Symbol('P_e', commutative=True)))), Integral(Add(Mul(Pow(Symbol('P_e', commutative=True), Symbol('t_2', commutative=True)), Pow(Function('S')(Symbol('t_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(A_{x},\\hat{X})} = \\hat{X}^{A_{x}}, then obtain A_{x} + (- A_{x} + \\operatorname{n_{2}}{(A_{x},\\hat{X})})^{A_{x}} - \\operatorname{n_{2}}{(A_{x},\\hat{X})} = A_{x} + (- A_{x} + \\hat{X}^{A_{x}})^{A_{x}} - \\operatorname{n_{2}}{(A_{x},\\hat{X})}", "derivation": "\\operatorname{n_{2}}{(A_{x},\\hat{X})} = \\hat{X}^{A_{x}} and - A_{x} + \\operatorname{n_{2}}{(A_{x},\\hat{X})} = - A_{x} + \\hat{X}^{A_{x}} and (- A_{x} + \\operatorname{n_{2}}{(A_{x},\\hat{X})})^{A_{x}} = (- A_{x} + \\hat{X}^{A_{x}})^{A_{x}} and A_{x} + (- A_{x} + \\operatorname{n_{2}}{(A_{x},\\hat{X})})^{A_{x}} - \\operatorname{n_{2}}{(A_{x},\\hat{X})} = A_{x} + (- A_{x} + \\hat{X}^{A_{x}})^{A_{x}} - \\operatorname{n_{2}}{(A_{x},\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('A_x', commutative=True)))"], [["minus", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('A_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Symbol('A_x', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Add(Symbol('A_x', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('A_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(h)} = \\cos{(h)} and \\mu_{0}{(h)} = \\frac{\\theta_{2}{(h)}}{\\int \\cos{(h)} dh}, then derive \\mu_{0}{(h)} = \\frac{\\theta_{2}{(h)}}{\\pi + \\sin{(h)}}, then obtain - \\int \\theta_{2}{(h)} dh + \\frac{\\theta_{2}{(h)}}{\\pi + \\sin{(h)}} = \\frac{\\theta_{2}{(h)}}{\\int \\theta_{2}{(h)} dh} - \\int \\theta_{2}{(h)} dh", "derivation": "\\theta_{2}{(h)} = \\cos{(h)} and \\int \\theta_{2}{(h)} dh = \\int \\cos{(h)} dh and \\mu_{0}{(h)} = \\frac{\\theta_{2}{(h)}}{\\int \\cos{(h)} dh} and \\mu_{0}{(h)} = \\frac{\\theta_{2}{(h)}}{\\pi + \\sin{(h)}} and \\mu_{0}{(h)} = \\frac{\\theta_{2}{(h)}}{\\int \\theta_{2}{(h)} dh} and \\mu_{0}{(h)} - \\int \\theta_{2}{(h)} dh = \\frac{\\theta_{2}{(h)}}{\\int \\theta_{2}{(h)} dh} - \\int \\theta_{2}{(h)} dh and - \\int \\theta_{2}{(h)} dh + \\frac{\\theta_{2}{(h)}}{\\pi + \\sin{(h)}} = \\frac{\\theta_{2}{(h)}}{\\int \\theta_{2}{(h)} dh} - \\int \\theta_{2}{(h)} dh", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True)), Mul(Function('\\\\theta_2')(Symbol('h', commutative=True)), Pow(Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 3], "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True)), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('h', commutative=True))), Integer(-1)), Function('\\\\theta_2')(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mu_0')(Symbol('h', commutative=True)), Mul(Function('\\\\theta_2')(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))))"], [["minus", 5, "Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Add(Function('\\\\mu_0')(Symbol('h', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))), Add(Mul(Function('\\\\theta_2')(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Add(Symbol('\\\\pi', commutative=True), sin(Symbol('h', commutative=True))), Integer(-1)), Function('\\\\theta_2')(Symbol('h', commutative=True)))), Add(Mul(Function('\\\\theta_2')(Symbol('h', commutative=True)), Pow(Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1))), Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\eta{(u)} = \\log{(u)}, then obtain \\int \\frac{d^{2}}{d u^{2}} \\eta{(u)} \\frac{d}{d u} \\log{(u)} du = V_{\\mathbf{B}} + \\frac{1}{2 u^{2}}", "derivation": "\\eta{(u)} = \\log{(u)} and \\frac{d}{d u} \\eta{(u)} = \\frac{d}{d u} \\log{(u)} and \\frac{d^{2}}{d u^{2}} \\eta{(u)} = \\frac{d^{2}}{d u^{2}} \\log{(u)} and \\frac{d^{2}}{d u^{2}} \\eta{(u)} \\frac{d}{d u} \\log{(u)} = \\frac{d}{d u} \\log{(u)} \\frac{d^{2}}{d u^{2}} \\log{(u)} and \\int \\frac{d^{2}}{d u^{2}} \\eta{(u)} \\frac{d}{d u} \\log{(u)} du = \\int \\frac{d}{d u} \\log{(u)} \\frac{d^{2}}{d u^{2}} \\log{(u)} du and \\int \\frac{d^{2}}{d u^{2}} \\eta{(u)} \\frac{d}{d u} \\log{(u)} du = V_{\\mathbf{B}} + \\frac{1}{2 u^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))))"], [["times", 3, "Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2)))))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Derivative(Function('\\\\eta')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(2))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\varphi^*)} = \\int e^{\\varphi^*} d\\varphi^*, then obtain \\frac{d}{d \\varphi^*} (3 \\hat{\\mathbf{x}}{(\\varphi^*)} - 4 \\int e^{\\varphi^*} d\\varphi^*) = \\frac{d}{d \\varphi^*} (\\hat{\\mathbf{x}}{(\\varphi^*)} - 2 \\int e^{\\varphi^*} d\\varphi^*)", "derivation": "\\hat{\\mathbf{x}}{(\\varphi^*)} = \\int e^{\\varphi^*} d\\varphi^* and \\hat{\\mathbf{x}}{(\\varphi^*)} - \\int e^{\\varphi^*} d\\varphi^* = 0 and \\hat{\\mathbf{x}}{(\\varphi^*)} - 2 \\int e^{\\varphi^*} d\\varphi^* = - \\int e^{\\varphi^*} d\\varphi^* and \\frac{d}{d \\varphi^*} (\\hat{\\mathbf{x}}{(\\varphi^*)} - 2 \\int e^{\\varphi^*} d\\varphi^*) = \\frac{d}{d \\varphi^*} - \\int e^{\\varphi^*} d\\varphi^* and \\frac{d}{d \\varphi^*} (3 \\hat{\\mathbf{x}}{(\\varphi^*)} - 4 \\int e^{\\varphi^*} d\\varphi^*) = \\frac{d}{d \\varphi^*} (\\hat{\\mathbf{x}}{(\\varphi^*)} - 2 \\int e^{\\varphi^*} d\\varphi^*)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True)), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 1, "Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Integer(0))"], [["minus", 2, "Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integer(2), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Mul(Integer(-1), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integer(2), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(3), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Integer(4), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Integer(2), Integral(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\theta_2)} = \\theta_2, then obtain ((\\theta_2 + \\eta^{\\prime}{(\\theta_2)}) \\eta^{\\prime}{(\\theta_2)})^{\\theta_2} = (\\theta_2 (\\theta_2 + \\eta^{\\prime}{(\\theta_2)}))^{\\theta_2}", "derivation": "\\eta^{\\prime}{(\\theta_2)} = \\theta_2 and 2 \\eta^{\\prime}{(\\theta_2)} = \\theta_2 + \\eta^{\\prime}{(\\theta_2)} and 2 \\eta^{\\prime}^{2}{(\\theta_2)} = 2 \\theta_2 \\eta^{\\prime}{(\\theta_2)} and (\\theta_2 + \\eta^{\\prime}{(\\theta_2)}) \\eta^{\\prime}{(\\theta_2)} = \\theta_2 (\\theta_2 + \\eta^{\\prime}{(\\theta_2)}) and ((\\theta_2 + \\eta^{\\prime}{(\\theta_2)}) \\eta^{\\prime}{(\\theta_2)})^{\\theta_2} = (\\theta_2 (\\theta_2 + \\eta^{\\prime}{(\\theta_2)}))^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], [["add", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given h{(\\sigma_x,\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + \\cos{(\\sigma_x)}, then derive \\frac{\\partial}{\\partial \\Psi^{\\dagger}} h{(\\sigma_x,\\Psi^{\\dagger})} - 1 = -2, then obtain \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} + \\cos{(\\sigma_x)}) - 1 = -2", "derivation": "h{(\\sigma_x,\\Psi^{\\dagger})} = - \\Psi^{\\dagger} + \\cos{(\\sigma_x)} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} h{(\\sigma_x,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} + \\cos{(\\sigma_x)}) and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} h{(\\sigma_x,\\Psi^{\\dagger})} - 1 = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} + \\cos{(\\sigma_x)}) - 1 and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} h{(\\sigma_x,\\Psi^{\\dagger})} - 1 = -2 and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} + \\cos{(\\sigma_x)}) - 1 = -2", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('h')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('h')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(M,B)} = \\int (B + M) dM and \\mathbf{J}{(J_{\\varepsilon},M_{E})} = J_{\\varepsilon}^{M_{E}} and G{(M,M_{E},J_{\\varepsilon},B)} = \\operatorname{F_{g}}^{M}{(M,B)} + \\mathbf{J}{(J_{\\varepsilon},M_{E})}, then derive \\operatorname{F_{g}}{(M,B)} = B M + \\frac{M^{2}}{2} + \\mathbf{g}, then obtain G{(M,M_{E},J_{\\varepsilon},B)} = J_{\\varepsilon}^{M_{E}} + (B M + \\frac{M^{2}}{2} + \\mathbf{g})^{M}", "derivation": "\\operatorname{F_{g}}{(M,B)} = \\int (B + M) dM and \\operatorname{F_{g}}{(M,B)} = B M + \\frac{M^{2}}{2} + \\mathbf{g} and \\operatorname{F_{g}}^{M}{(M,B)} = (\\int (B + M) dM)^{M} and (B M + \\frac{M^{2}}{2} + \\mathbf{g})^{M} = (\\int (B + M) dM)^{M} and \\mathbf{J}{(J_{\\varepsilon},M_{E})} = J_{\\varepsilon}^{M_{E}} and (B M + \\frac{M^{2}}{2} + \\mathbf{g})^{M} = \\operatorname{F_{g}}^{M}{(M,B)} and G{(M,M_{E},J_{\\varepsilon},B)} = \\operatorname{F_{g}}^{M}{(M,B)} + \\mathbf{J}{(J_{\\varepsilon},M_{E})} and G{(M,M_{E},J_{\\varepsilon},B)} = J_{\\varepsilon}^{M_{E}} + \\operatorname{F_{g}}^{M}{(M,B)} and G{(M,M_{E},J_{\\varepsilon},B)} = J_{\\varepsilon}^{M_{E}} + (B M + \\frac{M^{2}}{2} + \\mathbf{g})^{M}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Add(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Symbol('M', commutative=True)), Pow(Integral(Add(Symbol('B', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Pow(Integral(Add(Symbol('B', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('G')(Symbol('M', commutative=True), Symbol('M_E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Add(Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Symbol('M', commutative=True)), Function('\\\\mathbf{J}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Function('G')(Symbol('M', commutative=True), Symbol('M_E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Add(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Pow(Function('F_g')(Symbol('M', commutative=True), Symbol('B', commutative=True)), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Function('G')(Symbol('M', commutative=True), Symbol('M_E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('B', commutative=True)), Add(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('M_E', commutative=True)), Pow(Add(Mul(Symbol('B', commutative=True), Symbol('M', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('M', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}} and \\nabla{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - r + \\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})}, then obtain \\nabla{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - r + \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}}", "derivation": "\\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}} and \\hat{H}_{\\lambda} + \\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}} and \\hat{H}_{\\lambda} - r + \\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - r + \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}} and \\nabla{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - r + \\mathbf{r}{(r,\\mathbf{M},\\hat{H}_{\\lambda})} and \\nabla{(r,\\mathbf{M},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - r + \\frac{\\mathbf{M} r}{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('r', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('r', commutative=True))))"], [["minus", 2, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\mathbf{r}')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\nabla')(Symbol('r', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True), Symbol('r', commutative=True))))"]]}, {"prompt": "Given z{(T)} = \\cos{(\\cos{(T)})}, then obtain \\int 0 dT - \\int z{(T)} dT = \\int (- z{(T)} + \\cos{(\\cos{(T)})}) dT - \\int z{(T)} dT", "derivation": "z{(T)} = \\cos{(\\cos{(T)})} and 0 = - z{(T)} + \\cos{(\\cos{(T)})} and \\int 0 dT = \\int (- z{(T)} + \\cos{(\\cos{(T)})}) dT and \\int 0 dT - \\int z{(T)} dT = \\int (- z{(T)} + \\cos{(\\cos{(T)})}) dT - \\int z{(T)} dT", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('T', commutative=True)), cos(cos(Symbol('T', commutative=True))))"], [["minus", 1, "Function('z')(Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z')(Symbol('T', commutative=True))), cos(cos(Symbol('T', commutative=True)))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Function('z')(Symbol('T', commutative=True))), cos(cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], [["minus", 3, "Integral(Function('z')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('T', commutative=True))), Mul(Integer(-1), Integral(Function('z')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))), Add(Integral(Add(Mul(Integer(-1), Function('z')(Symbol('T', commutative=True))), cos(cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))), Mul(Integer(-1), Integral(Function('z')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))))"]]}, {"prompt": "Given \\eta{(m,v)} = m v, then obtain m - \\frac{\\frac{\\partial}{\\partial v} (m v)^{v}}{\\eta{(m,v)}} + \\frac{\\frac{\\partial}{\\partial v} \\eta^{v}{(m,v)}}{\\eta{(m,v)}} = m", "derivation": "\\eta{(m,v)} = m v and \\eta^{v}{(m,v)} = (m v)^{v} and \\frac{\\partial}{\\partial v} \\eta^{v}{(m,v)} = \\frac{\\partial}{\\partial v} (m v)^{v} and \\frac{\\frac{\\partial}{\\partial v} \\eta^{v}{(m,v)}}{\\eta{(m,v)}} = \\frac{\\frac{\\partial}{\\partial v} (m v)^{v}}{\\eta{(m,v)}} and - \\frac{\\frac{\\partial}{\\partial v} (m v)^{v}}{\\eta{(m,v)}} + \\frac{\\frac{\\partial}{\\partial v} \\eta^{v}{(m,v)}}{\\eta{(m,v)}} = 0 and m - \\frac{\\frac{\\partial}{\\partial v} (m v)^{v}}{\\eta{(m,v)}} + \\frac{\\frac{\\partial}{\\partial v} \\eta^{v}{(m,v)}}{\\eta{(m,v)}} = m", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["divide", 3, "Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["minus", 4, "Mul(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))), Integer(0))"], [["add", 5, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Mul(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\eta')(Symbol('m', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))), Symbol('m', commutative=True))"]]}, {"prompt": "Given \\varphi{(\\hat{p}_0,G)} = \\frac{\\log{(\\hat{p}_0)}}{G} and \\operatorname{C_{2}}{(A_{y})} = e^{\\sin{(A_{y})}}, then obtain \\operatorname{C_{2}}{(A_{y})} - (\\int \\varphi{(\\hat{p}_0,G)} dG)^{G} = e^{\\sin{(A_{y})}} - (\\int \\varphi{(\\hat{p}_0,G)} dG)^{G}", "derivation": "\\varphi{(\\hat{p}_0,G)} = \\frac{\\log{(\\hat{p}_0)}}{G} and \\int \\varphi{(\\hat{p}_0,G)} dG = \\int \\frac{\\log{(\\hat{p}_0)}}{G} dG and (\\int \\varphi{(\\hat{p}_0,G)} dG)^{G} = (\\int \\frac{\\log{(\\hat{p}_0)}}{G} dG)^{G} and \\operatorname{C_{2}}{(A_{y})} = e^{\\sin{(A_{y})}} and \\operatorname{C_{2}}{(A_{y})} - (\\int \\frac{\\log{(\\hat{p}_0)}}{G} dG)^{G} = e^{\\sin{(A_{y})}} - (\\int \\frac{\\log{(\\hat{p}_0)}}{G} dG)^{G} and \\operatorname{C_{2}}{(A_{y})} - (\\int \\varphi{(\\hat{p}_0,G)} dG)^{G} = e^{\\sin{(A_{y})}} - (\\int \\varphi{(\\hat{p}_0,G)} dG)^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Function('\\\\varphi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], ["get_premise", "Equality(Function('C_2')(Symbol('A_y', commutative=True)), exp(sin(Symbol('A_y', commutative=True))))"], [["minus", 4, "Pow(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True))"], "Equality(Add(Function('C_2')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Add(exp(sin(Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), log(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('C_2')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('\\\\varphi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Add(exp(sin(Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Integral(Function('\\\\varphi')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(k,\\hat{X},\\mathbf{F})} = \\frac{\\mathbf{F} k}{\\hat{X}} and \\dot{x}{(k,\\hat{X},\\mathbf{F})} = \\frac{\\mathbf{F} k}{\\hat{X}}, then obtain 0 = \\dot{x}{(k,\\hat{X},\\mathbf{F})} - \\varphi{(k,\\hat{X},\\mathbf{F})}", "derivation": "\\varphi{(k,\\hat{X},\\mathbf{F})} = \\frac{\\mathbf{F} k}{\\hat{X}} and \\varphi{(k,\\hat{X},\\mathbf{F})} - 1 = -1 + \\frac{\\mathbf{F} k}{\\hat{X}} and \\varphi{(k,\\hat{X},\\mathbf{F})} - 1 + \\frac{\\mathbf{F} k}{\\hat{X}} = -1 + \\frac{2 \\mathbf{F} k}{\\hat{X}} and 0 = - \\varphi{(k,\\hat{X},\\mathbf{F})} + \\frac{\\mathbf{F} k}{\\hat{X}} and \\dot{x}{(k,\\hat{X},\\mathbf{F})} = \\frac{\\mathbf{F} k}{\\hat{X}} and 0 = \\dot{x}{(k,\\hat{X},\\mathbf{F})} - \\varphi{(k,\\hat{X},\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True))), Add(Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True))))"], [["minus", 3, "Add(Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Add(Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Function('\\\\varphi')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then derive \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\Psi_{\\lambda}{(V_{\\mathbf{B}})})} = \\sin{(\\frac{1}{V_{\\mathbf{B}}})}, then obtain - \\sin{(\\frac{1}{V_{\\mathbf{B}}})} = - \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})})}", "derivation": "\\Psi_{\\lambda}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\frac{d}{d V_{\\mathbf{B}}} \\Psi_{\\lambda}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})} and \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\Psi_{\\lambda}{(V_{\\mathbf{B}})})} = \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})})} and \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\Psi_{\\lambda}{(V_{\\mathbf{B}})})} = \\sin{(\\frac{1}{V_{\\mathbf{B}}})} and \\sin{(\\frac{1}{V_{\\mathbf{B}}})} = \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})})} and - \\sin{(\\frac{1}{V_{\\mathbf{B}}})} = - \\sin{(\\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), sin(Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), sin(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(sin(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1))), sin(Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))), Mul(Integer(-1), sin(Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta{(P_{e})} = \\cos{(P_{e})}, then derive \\frac{d}{d P_{e}} \\eta{(P_{e})} = - \\sin{(P_{e})}, then obtain \\frac{d^{2}}{d P_{e}^{2}} \\eta{(P_{e})} = \\frac{d^{2}}{d P_{e}^{2}} \\cos{(P_{e})}", "derivation": "\\eta{(P_{e})} = \\cos{(P_{e})} and \\frac{d}{d P_{e}} \\eta{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(P_{e})} and \\frac{d}{d P_{e}} \\eta{(P_{e})} = - \\sin{(P_{e})} and \\frac{d^{2}}{d P_{e}^{2}} \\eta{(P_{e})} = \\frac{d}{d P_{e}} - \\sin{(P_{e})} and \\frac{d}{d P_{e}} \\cos{(P_{e})} = - \\sin{(P_{e})} and \\frac{d^{2}}{d P_{e}^{2}} \\eta{(P_{e})} = \\frac{d^{2}}{d P_{e}^{2}} \\cos{(P_{e})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('P_e', commutative=True))))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('\\\\eta')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(2))), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\chi{(\\psi,B)} = \\sin{(B - \\psi)}, then derive \\int \\chi{(\\psi,B)} d\\psi = \\mathbf{F} + \\cos{(B - \\psi)}, then obtain \\frac{\\partial}{\\partial \\psi} \\int \\chi{(\\psi,B)} d\\psi = \\frac{\\partial}{\\partial \\psi} (\\mathbf{F} + \\cos{(B - \\psi)})", "derivation": "\\chi{(\\psi,B)} = \\sin{(B - \\psi)} and \\int \\chi{(\\psi,B)} d\\psi = \\int \\sin{(B - \\psi)} d\\psi and \\int \\chi{(\\psi,B)} d\\psi = \\mathbf{F} + \\cos{(B - \\psi)} and \\frac{\\partial}{\\partial \\psi} \\int \\chi{(\\psi,B)} d\\psi = \\frac{\\partial}{\\partial \\psi} (\\mathbf{F} + \\cos{(B - \\psi)})", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(sin(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), cos(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(A)} = e^{A}, then derive \\int \\Psi_{nl}{(A)} dA = \\dot{y} + e^{A}, then obtain \\dot{y} + e^{A} = \\dot{y} + \\Psi_{nl}{(A)}", "derivation": "\\Psi_{nl}{(A)} = e^{A} and \\int \\Psi_{nl}{(A)} dA = \\int e^{A} dA and \\int \\Psi_{nl}{(A)} dA = \\dot{y} + e^{A} and \\int \\Psi_{nl}{(A)} dA = \\dot{y} + \\Psi_{nl}{(A)} and \\dot{y} + e^{A} = \\dot{y} + \\Psi_{nl}{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), exp(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), exp(Symbol('A', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(E)} = \\sin{(\\log{(E)})} and \\mu{(E)} = \\sin{(\\log{(E)})}, then obtain \\int (- E + \\mu^{E}{(E)}) dE = \\int (- E + \\sin^{E}{(\\log{(E)})}) dE", "derivation": "\\operatorname{z^{*}}{(E)} = \\sin{(\\log{(E)})} and \\mu{(E)} = \\sin{(\\log{(E)})} and \\operatorname{z^{*}}^{E}{(E)} = \\sin^{E}{(\\log{(E)})} and - E + \\operatorname{z^{*}}^{E}{(E)} = - E + \\sin^{E}{(\\log{(E)})} and - E + \\operatorname{z^{*}}^{E}{(E)} = - E + \\mu^{E}{(E)} and - E + \\mu^{E}{(E)} = - E + \\sin^{E}{(\\log{(E)})} and \\int (- E + \\mu^{E}{(E)}) dE = \\int (- E + \\sin^{E}{(\\log{(E)})}) dE", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('E', commutative=True)), sin(log(Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True)), sin(log(Symbol('E', commutative=True))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(sin(log(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["minus", 3, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(Function('z^*')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(sin(log(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(Function('z^*')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(Function('\\\\mu')(Symbol('E', commutative=True)), Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(Function('\\\\mu')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(sin(log(Symbol('E', commutative=True))), Symbol('E', commutative=True))))"], [["integrate", 6, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(Function('\\\\mu')(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Pow(sin(log(Symbol('E', commutative=True))), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{v},Z)} = \\frac{e^{Z}}{\\mathbf{v}} and \\dot{z}{(\\mathbf{v},Z)} = \\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v}, then obtain e^{\\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v}} = e^{\\int \\operatorname{z^{*}}{(\\mathbf{v},Z)} d\\mathbf{v}}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{v},Z)} = \\frac{e^{Z}}{\\mathbf{v}} and \\int \\operatorname{z^{*}}{(\\mathbf{v},Z)} d\\mathbf{v} = \\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v} and \\dot{z}{(\\mathbf{v},Z)} = \\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v} and e^{\\dot{z}{(\\mathbf{v},Z)}} = e^{\\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v}} and e^{\\dot{z}{(\\mathbf{v},Z)}} = e^{\\int \\operatorname{z^{*}}{(\\mathbf{v},Z)} d\\mathbf{v}} and e^{\\int \\frac{e^{Z}}{\\mathbf{v}} d\\mathbf{v}} = e^{\\int \\operatorname{z^{*}}{(\\mathbf{v},Z)} d\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Symbol('Z', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["exp", 3], "Equality(exp(Function('\\\\dot{z}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True))), exp(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(exp(Function('\\\\dot{z}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True))), exp(Integral(Function('z^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(exp(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), exp(Integral(Function('z^*')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given c{(m)} = e^{\\sin{(m)}}, then obtain (m c^{2}{(m)})^{m} = (m e^{2 \\sin{(m)}})^{m}", "derivation": "c{(m)} = e^{\\sin{(m)}} and m c{(m)} = m e^{\\sin{(m)}} and m c^{2}{(m)} = m c{(m)} e^{\\sin{(m)}} and (m c^{2}{(m)})^{m} = (m c{(m)} e^{\\sin{(m)}})^{m} and (m c{(m)} e^{\\sin{(m)}})^{m} = (m e^{2 \\sin{(m)}})^{m} and (m c^{2}{(m)})^{m} = (m e^{2 \\sin{(m)}})^{m}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('m', commutative=True)), exp(sin(Symbol('m', commutative=True))))"], [["times", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Function('c')(Symbol('m', commutative=True))), Mul(Symbol('m', commutative=True), exp(sin(Symbol('m', commutative=True)))))"], [["times", 1, "Mul(Symbol('m', commutative=True), Function('c')(Symbol('m', commutative=True)))"], "Equality(Mul(Symbol('m', commutative=True), Pow(Function('c')(Symbol('m', commutative=True)), Integer(2))), Mul(Symbol('m', commutative=True), Function('c')(Symbol('m', commutative=True)), exp(sin(Symbol('m', commutative=True)))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Symbol('m', commutative=True), Pow(Function('c')(Symbol('m', commutative=True)), Integer(2))), Symbol('m', commutative=True)), Pow(Mul(Symbol('m', commutative=True), Function('c')(Symbol('m', commutative=True)), exp(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Symbol('m', commutative=True), Function('c')(Symbol('m', commutative=True)), exp(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(Mul(Symbol('m', commutative=True), exp(Mul(Integer(2), sin(Symbol('m', commutative=True))))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Mul(Symbol('m', commutative=True), Pow(Function('c')(Symbol('m', commutative=True)), Integer(2))), Symbol('m', commutative=True)), Pow(Mul(Symbol('m', commutative=True), exp(Mul(Integer(2), sin(Symbol('m', commutative=True))))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given r{(u)} = \\cos{(\\sin{(u)})}, then obtain \\frac{r{(u)}}{r{(u)} + \\cos{(\\sin{(u)})}} = \\frac{1}{2}", "derivation": "r{(u)} = \\cos{(\\sin{(u)})} and r{(u)} + \\cos{(\\sin{(u)})} = 2 \\cos{(\\sin{(u)})} and \\frac{r{(u)}}{2 \\cos{(\\sin{(u)})}} = \\frac{1}{2} and \\frac{r{(u)}}{r{(u)} + \\cos{(\\sin{(u)})}} = \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True))))"], [["add", 1, "cos(sin(Symbol('u', commutative=True)))"], "Equality(Add(Function('r')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Mul(Integer(2), cos(sin(Symbol('u', commutative=True)))))"], [["divide", 1, "Mul(Integer(2), cos(sin(Symbol('u', commutative=True))))"], "Equality(Mul(Rational(1, 2), Function('r')(Symbol('u', commutative=True)), Pow(cos(sin(Symbol('u', commutative=True))), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Function('r')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Integer(-1)), Function('r')(Symbol('u', commutative=True))), Rational(1, 2))"]]}, {"prompt": "Given \\hat{H}{(A_{z})} = \\log{(A_{z})}, then derive \\frac{d}{d A_{z}} \\hat{H}{(A_{z})} = \\frac{1}{A_{z}}, then obtain A_{z}^{2} (\\frac{d}{d A_{z}} \\hat{H}{(A_{z})})^{2} = A_{z} \\frac{d}{d A_{z}} \\hat{H}{(A_{z})}", "derivation": "\\hat{H}{(A_{z})} = \\log{(A_{z})} and \\frac{d}{d A_{z}} \\hat{H}{(A_{z})} = \\frac{d}{d A_{z}} \\log{(A_{z})} and \\frac{d}{d A_{z}} \\hat{H}{(A_{z})} = \\frac{1}{A_{z}} and 1 = \\frac{1}{A_{z} \\frac{d}{d A_{z}} \\hat{H}{(A_{z})}} and A_{z} \\frac{d}{d A_{z}} \\hat{H}{(A_{z})} = 1 and A_{z}^{2} (\\frac{d}{d A_{z}} \\hat{H}{(A_{z})})^{2} = A_{z} \\frac{d}{d A_{z}} \\hat{H}{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), log(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(log(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Pow(Symbol('A_z', commutative=True), Integer(-1)))"], [["divide", 3, "Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 4, "Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Mul(Symbol('A_z', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Integer(1))"], [["times", 5, "Mul(Symbol('A_z', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(2)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(2))), Mul(Symbol('A_z', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} = (h^{c})^{\\varepsilon_0}, then obtain \\frac{\\partial}{\\partial h} \\int \\frac{\\partial}{\\partial c} h^{c} \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} dc = \\frac{\\partial}{\\partial h} \\int \\frac{\\partial}{\\partial c} h^{c} (h^{c})^{\\varepsilon_0} dc", "derivation": "\\operatorname{m_{s}}{(h,c,\\varepsilon_0)} = (h^{c})^{\\varepsilon_0} and h^{c} \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} = h^{c} (h^{c})^{\\varepsilon_0} and \\frac{\\partial}{\\partial c} h^{c} \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} = \\frac{\\partial}{\\partial c} h^{c} (h^{c})^{\\varepsilon_0} and \\int \\frac{\\partial}{\\partial c} h^{c} \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} dc = \\int \\frac{\\partial}{\\partial c} h^{c} (h^{c})^{\\varepsilon_0} dc and \\frac{\\partial}{\\partial h} \\int \\frac{\\partial}{\\partial c} h^{c} \\operatorname{m_{s}}{(h,c,\\varepsilon_0)} dc = \\frac{\\partial}{\\partial h} \\int \\frac{\\partial}{\\partial c} h^{c} (h^{c})^{\\varepsilon_0} dc", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('h', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 1, "Pow(Symbol('h', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Function('m_s')(Symbol('h', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Function('m_s')(Symbol('h', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('c', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Function('m_s')(Symbol('h', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Integral(Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Function('m_s')(Symbol('h', commutative=True), Symbol('c', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Pow(Pow(Symbol('h', commutative=True), Symbol('c', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(L,M_{E})} = \\frac{M_{E}}{L}, then obtain \\frac{\\partial}{\\partial M_{E}} \\int (L G{(L,M_{E})} - G{(L,M_{E})}) dL = \\frac{\\partial}{\\partial M_{E}} \\int (M_{E} - G{(L,M_{E})}) dL", "derivation": "G{(L,M_{E})} = \\frac{M_{E}}{L} and L G{(L,M_{E})} = M_{E} and L G{(L,M_{E})} - \\frac{M_{E}}{L} = M_{E} - \\frac{M_{E}}{L} and L G{(L,M_{E})} - G{(L,M_{E})} = M_{E} - G{(L,M_{E})} and \\int (L G{(L,M_{E})} - G{(L,M_{E})}) dL = \\int (M_{E} - G{(L,M_{E})}) dL and \\frac{\\partial}{\\partial M_{E}} \\int (L G{(L,M_{E})} - G{(L,M_{E})}) dL = \\frac{\\partial}{\\partial M_{E}} \\int (M_{E} - G{(L,M_{E})}) dL", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)))"], [["divide", 1, "Pow(Symbol('L', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('L', commutative=True), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))"], [["minus", 2, "Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Symbol('L', commutative=True), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Symbol('L', commutative=True), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('L', commutative=True), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('L', commutative=True))), Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('L', commutative=True))))"], [["differentiate", 5, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Symbol('L', commutative=True), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Function('G')(Symbol('L', commutative=True), Symbol('M_E', commutative=True)))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\tilde{g})} = \\log{(\\tilde{g})}, then derive \\frac{d}{d \\tilde{g}} \\operatorname{v_{1}}{(\\tilde{g})} = \\frac{1}{\\tilde{g}}, then obtain \\log{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} = \\log{(\\tilde{g})} + \\frac{1}{\\tilde{g}}", "derivation": "\\operatorname{v_{1}}{(\\tilde{g})} = \\log{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\operatorname{v_{1}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\operatorname{v_{1}}{(\\tilde{g})} = \\frac{1}{\\tilde{g}} and \\operatorname{v_{1}}{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\operatorname{v_{1}}{(\\tilde{g})} = \\operatorname{v_{1}}{(\\tilde{g})} + \\frac{1}{\\tilde{g}} and \\log{(\\tilde{g})} + \\frac{d}{d \\tilde{g}} \\log{(\\tilde{g})} = \\log{(\\tilde{g})} + \\frac{1}{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))"], [["add", 3, "Function('v_1')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(Function('v_1')(Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('\\\\tilde{g}', commutative=True)), Derivative(log(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given v{(M,H)} = \\log{(H - M)}, then obtain \\log{(H - M)} + \\sin{(v{(M,H)} - 1)} = \\log{(H - M)} + \\sin{(\\log{(H - M)} - 1)}", "derivation": "v{(M,H)} = \\log{(H - M)} and v{(M,H)} - 1 = \\log{(H - M)} - 1 and \\sin{(v{(M,H)} - 1)} = \\sin{(\\log{(H - M)} - 1)} and \\log{(H - M)} + \\sin{(v{(M,H)} - 1)} = \\log{(H - M)} + \\sin{(\\log{(H - M)} - 1)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('M', commutative=True), Symbol('H', commutative=True)), log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v')(Symbol('M', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Add(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(-1)))"], [["sin", 2], "Equality(sin(Add(Function('v')(Symbol('M', commutative=True), Symbol('H', commutative=True)), Integer(-1))), sin(Add(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(-1))))"], [["add", 3, "log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))"], "Equality(Add(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))), sin(Add(Function('v')(Symbol('M', commutative=True), Symbol('H', commutative=True)), Integer(-1)))), Add(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))), sin(Add(log(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(-1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(x)} = \\sin{(x)}, then obtain - \\sin{(x)} + \\int \\sin{(x)} dx = - \\Psi^{\\dagger}{(x)} + \\int \\sin{(x)} dx", "derivation": "\\Psi^{\\dagger}{(x)} = \\sin{(x)} and \\int \\Psi^{\\dagger}{(x)} dx = \\int \\sin{(x)} dx and \\Psi^{\\dagger}{(x)} - \\int \\sin{(x)} dx = \\sin{(x)} - \\int \\sin{(x)} dx and \\Psi^{\\dagger}{(x)} - \\int \\Psi^{\\dagger}{(x)} dx = \\sin{(x)} - \\int \\Psi^{\\dagger}{(x)} dx and \\Psi^{\\dagger}{(x)} - \\sin{(x)} - \\int \\Psi^{\\dagger}{(x)} dx + \\int \\sin{(x)} dx = - \\int \\Psi^{\\dagger}{(x)} dx + \\int \\sin{(x)} dx and - \\sin{(x)} - \\int \\Psi^{\\dagger}{(x)} dx + \\int \\sin{(x)} dx = - \\Psi^{\\dagger}{(x)} - \\int \\Psi^{\\dagger}{(x)} dx + \\int \\sin{(x)} dx and - \\sin{(x)} + \\int \\sin{(x)} dx = - \\Psi^{\\dagger}{(x)} + \\int \\sin{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["minus", 1, "Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"], [["minus", 4, "Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["minus", 5, "Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["minus", 6, "Mul(Integer(-1), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbb{I},A_{1})} = - A_{1} + \\mathbb{I} and I{(\\mathbb{I},A_{1})} = - 2 A_{1} + \\mathbb{I}, then obtain (- A_{1} + \\mathbb{I}) (- A_{1} + \\operatorname{P_{e}}{(\\mathbb{I},A_{1})}) = (- A_{1} + \\mathbb{I}) I{(\\mathbb{I},A_{1})}", "derivation": "\\operatorname{P_{e}}{(\\mathbb{I},A_{1})} = - A_{1} + \\mathbb{I} and - A_{1} + \\operatorname{P_{e}}{(\\mathbb{I},A_{1})} = - 2 A_{1} + \\mathbb{I} and (- A_{1} + \\mathbb{I}) (- A_{1} + \\operatorname{P_{e}}{(\\mathbb{I},A_{1})}) = (- 2 A_{1} + \\mathbb{I}) (- A_{1} + \\mathbb{I}) and I{(\\mathbb{I},A_{1})} = - 2 A_{1} + \\mathbb{I} and (- A_{1} + \\mathbb{I}) (- A_{1} + \\operatorname{P_{e}}{(\\mathbb{I},A_{1})}) = (- A_{1} + \\mathbb{I}) I{(\\mathbb{I},A_{1})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('P_e')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('P_e')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('P_e')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Function('I')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} = \\Psi^{F_{N}} - \\mathbf{A}, then obtain (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi)^{2} + (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi) \\int \\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} d\\Psi = 2 (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi)^{2}", "derivation": "\\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} = \\Psi^{F_{N}} - \\mathbf{A} and \\int \\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} d\\Psi = \\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi and (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi) \\int \\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} d\\Psi = (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi)^{2} and (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi)^{2} + (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi) \\int \\operatorname{F_{x}}{(F_{N},\\Psi,\\mathbf{A})} d\\Psi = 2 (\\int (\\Psi^{F_{N}} - \\mathbf{A}) d\\Psi)^{2}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('F_N', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('F_N', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["times", 2, "Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Function('F_x')(Symbol('F_N', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Pow(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(2)))"], [["add", 3, "Pow(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(2))"], "Equality(Add(Pow(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(2)), Mul(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Function('F_x')(Symbol('F_N', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))), Mul(Integer(2), Pow(Integral(Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\rho)} = \\cos{(\\log{(\\rho)})} and h{(\\rho)} = \\log{(\\rho)}, then obtain h{(\\rho)} \\frac{d}{d \\rho} \\cos^{2}{(\\log{(\\rho)})} = \\log{(\\rho)} \\frac{d}{d \\rho} \\cos^{2}{(\\log{(\\rho)})}", "derivation": "\\operatorname{F_{c}}{(\\rho)} = \\cos{(\\log{(\\rho)})} and \\operatorname{F_{c}}^{2}{(\\rho)} = \\operatorname{F_{c}}{(\\rho)} \\cos{(\\log{(\\rho)})} and h{(\\rho)} = \\log{(\\rho)} and h{(\\rho)} \\frac{d}{d \\rho} \\operatorname{F_{c}}{(\\rho)} \\cos{(\\log{(\\rho)})} = \\log{(\\rho)} \\frac{d}{d \\rho} \\operatorname{F_{c}}{(\\rho)} \\cos{(\\log{(\\rho)})} and h{(\\rho)} \\frac{d}{d \\rho} \\operatorname{F_{c}}^{2}{(\\rho)} = \\log{(\\rho)} \\frac{d}{d \\rho} \\operatorname{F_{c}}^{2}{(\\rho)} and h{(\\rho)} \\frac{d}{d \\rho} \\cos^{2}{(\\log{(\\rho)})} = \\log{(\\rho)} \\frac{d}{d \\rho} \\cos^{2}{(\\log{(\\rho)})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True))))"], [["times", 1, "Function('F_c')(Symbol('\\\\rho', commutative=True))"], "Equality(Pow(Function('F_c')(Symbol('\\\\rho', commutative=True)), Integer(2)), Mul(Function('F_c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True)))))"], ["renaming_premise", "Equality(Function('h')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["times", 3, "Derivative(Mul(Function('F_c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))"], "Equality(Mul(Function('h')(Symbol('\\\\rho', commutative=True)), Derivative(Mul(Function('F_c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(log(Symbol('\\\\rho', commutative=True)), Derivative(Mul(Function('F_c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('h')(Symbol('\\\\rho', commutative=True)), Derivative(Pow(Function('F_c')(Symbol('\\\\rho', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(log(Symbol('\\\\rho', commutative=True)), Derivative(Pow(Function('F_c')(Symbol('\\\\rho', commutative=True)), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('h')(Symbol('\\\\rho', commutative=True)), Derivative(Pow(cos(log(Symbol('\\\\rho', commutative=True))), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(log(Symbol('\\\\rho', commutative=True)), Derivative(Pow(cos(log(Symbol('\\\\rho', commutative=True))), Integer(2)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{v}{(G,u)} = - u + \\log{(G)}, then obtain \\sin^{2}{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} + \\log{(G)})} = \\sin^{2}{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{4} + \\log{(G)})}", "derivation": "\\mathbf{v}{(G,u)} = - u + \\log{(G)} and u + \\mathbf{v}{(G,u)} - \\log{(G)} = 0 and (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} = 0 and - u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} = - u and \\mathbf{v}{(G,u)} = - u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} + \\log{(G)} and \\sin{(\\mathbf{v}{(G,u)})} = \\sin{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} + \\log{(G)})} and \\sin^{2}{(\\mathbf{v}{(G,u)})} = \\sin^{2}{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} + \\log{(G)})} and \\sin^{2}{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{2} + \\log{(G)})} = \\sin^{2}{(- u + (u + \\mathbf{v}{(G,u)} - \\log{(G)})^{4} + \\log{(G)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('G', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Symbol('G', commutative=True)))"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(0))"], [["times", 2, "Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True))))"], "Equality(Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2)), Integer(0))"], [["add", 3, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2))), Mul(Integer(-1), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2)), log(Symbol('G', commutative=True))))"], [["sin", 5], "Equality(sin(Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True))), sin(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2)), log(Symbol('G', commutative=True)))))"], [["power", 6, 2], "Equality(Pow(sin(Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True))), Integer(2)), Pow(sin(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2)), log(Symbol('G', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(sin(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(2)), log(Symbol('G', commutative=True)))), Integer(2)), Pow(sin(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), log(Symbol('G', commutative=True)))), Integer(4)), log(Symbol('G', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mu_0,\\Omega)} = \\int (\\Omega + \\mu_0) d\\Omega, then derive \\Omega \\operatorname{F_{g}}{(\\mu_0,\\Omega)} = \\Omega (S + \\frac{\\Omega^{2}}{2} + \\Omega \\mu_0), then obtain \\Omega \\int (\\Omega + \\mu_0) d\\Omega = \\Omega (S + \\frac{\\Omega^{2}}{2} + \\Omega \\mu_0)", "derivation": "\\operatorname{F_{g}}{(\\mu_0,\\Omega)} = \\int (\\Omega + \\mu_0) d\\Omega and \\Omega \\operatorname{F_{g}}{(\\mu_0,\\Omega)} = \\Omega \\int (\\Omega + \\mu_0) d\\Omega and \\Omega \\operatorname{F_{g}}{(\\mu_0,\\Omega)} = \\Omega (S + \\frac{\\Omega^{2}}{2} + \\Omega \\mu_0) and \\Omega \\int (\\Omega + \\mu_0) d\\Omega = \\Omega (S + \\frac{\\Omega^{2}}{2} + \\Omega \\mu_0)", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["times", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('F_g')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('F_g')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('S', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('\\\\Omega', commutative=True), Add(Symbol('S', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\hat{p},v_{2},v_{z})} = \\hat{p} v_{2} v_{z}, then obtain \\frac{- \\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})}}{2 \\hat{H}_l{(\\hat{p},v_{2},v_{z})}} = 0", "derivation": "\\hat{H}_l{(\\hat{p},v_{2},v_{z})} = \\hat{p} v_{2} v_{z} and - \\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})} = 0 and 2 \\hat{H}_l{(\\hat{p},v_{2},v_{z})} = \\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})} and \\frac{- \\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})}}{\\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})}} = 0 and \\frac{- \\hat{p} v_{2} v_{z} + \\hat{H}_l{(\\hat{p},v_{2},v_{z})}}{2 \\hat{H}_l{(\\hat{p},v_{2},v_{z})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))), Integer(0))"], [["add", 1, "Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 2, "Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))), Pow(Add(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(V_{\\mathbf{E}},\\dot{y})} = V_{\\mathbf{E}} - \\dot{y} and \\hat{H}{(V_{\\mathbf{E}},\\dot{y})} = V_{\\mathbf{E}} - \\dot{y}, then obtain \\frac{\\hat{H}{(V_{\\mathbf{E}},\\dot{y})}}{V_{\\mathbf{E}}} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(V_{\\mathbf{E}},\\dot{y})}}{V_{\\mathbf{E}}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(V_{\\mathbf{E}},\\dot{y})} = V_{\\mathbf{E}} - \\dot{y} and \\hat{H}{(V_{\\mathbf{E}},\\dot{y})} = V_{\\mathbf{E}} - \\dot{y} and \\frac{\\hat{H}{(V_{\\mathbf{E}},\\dot{y})}}{V_{\\mathbf{E}}} = \\frac{V_{\\mathbf{E}} - \\dot{y}}{V_{\\mathbf{E}}} and \\frac{\\hat{H}{(V_{\\mathbf{E}},\\dot{y})}}{V_{\\mathbf{E}}} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(V_{\\mathbf{E}},\\dot{y})}}{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))))"], [["divide", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given T{(\\hat{x})} = \\cos{(\\log{(\\hat{x})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} = T^{\\hat{x}}{(\\hat{x})}, then obtain \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} d\\hat{x} = \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} \\int \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} d\\hat{x}", "derivation": "T{(\\hat{x})} = \\cos{(\\log{(\\hat{x})})} and T^{\\hat{x}}{(\\hat{x})} = \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} = T^{\\hat{x}}{(\\hat{x})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} = \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} d\\hat{x} = \\int \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} d\\hat{x} and \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{x})} d\\hat{x} = \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} \\int \\cos^{\\hat{x}}{(\\log{(\\hat{x})})} d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\hat{x}', commutative=True)), cos(log(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('T')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Pow(Function('T')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 5, "Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Integral(Pow(cos(log(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})}, then derive \\frac{d}{d \\hat{\\mathbf{r}}} \\sigma_{p}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})}, then obtain \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})} = \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\sin{(\\hat{\\mathbf{r}})}", "derivation": "\\sigma_{p}{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\sigma_{p}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\sin{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\sigma_{p}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\cos{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\sin{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})} = \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\sin{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(z^{*},\\varepsilon)} = \\frac{\\varepsilon}{z^{*}}, then obtain \\int (- \\operatorname{n_{2}}{(z^{*},\\varepsilon)} + \\frac{1}{(z^{*})^{2}}) d\\varepsilon = \\int (- \\frac{\\varepsilon}{z^{*}} + \\frac{1}{(z^{*})^{2}}) d\\varepsilon", "derivation": "\\operatorname{n_{2}}{(z^{*},\\varepsilon)} = \\frac{\\varepsilon}{z^{*}} and \\operatorname{n_{2}}{(z^{*},\\varepsilon)} - \\frac{1}{(z^{*})^{2}} = \\frac{\\varepsilon}{z^{*}} - \\frac{1}{(z^{*})^{2}} and - \\operatorname{n_{2}}{(z^{*},\\varepsilon)} + \\frac{1}{(z^{*})^{2}} = - \\frac{\\varepsilon}{z^{*}} + \\frac{1}{(z^{*})^{2}} and \\int (- \\operatorname{n_{2}}{(z^{*},\\varepsilon)} + \\frac{1}{(z^{*})^{2}}) d\\varepsilon = \\int (- \\frac{\\varepsilon}{z^{*}} + \\frac{1}{(z^{*})^{2}}) d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('z^*', commutative=True), Integer(-2))"], "Equality(Add(Function('n_2')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-2)))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-2)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('n_2')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('z^*', commutative=True), Integer(-2))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Pow(Symbol('z^*', commutative=True), Integer(-2))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('n_2')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('z^*', commutative=True), Integer(-2))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))), Pow(Symbol('z^*', commutative=True), Integer(-2))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)} = \\frac{L}{g^{\\prime}_{\\varepsilon}}, then obtain \\int (\\phi_2 + (g^{\\prime}_{\\varepsilon} \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)})^{g^{\\prime}_{\\varepsilon}}) dL = \\int (L^{g^{\\prime}_{\\varepsilon}} + \\phi_2) dL", "derivation": "\\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)} = \\frac{L}{g^{\\prime}_{\\varepsilon}} and g^{\\prime}_{\\varepsilon} \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)} = L and (g^{\\prime}_{\\varepsilon} \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)})^{g^{\\prime}_{\\varepsilon}} = L^{g^{\\prime}_{\\varepsilon}} and \\phi_2 + (g^{\\prime}_{\\varepsilon} \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)})^{g^{\\prime}_{\\varepsilon}} = L^{g^{\\prime}_{\\varepsilon}} + \\phi_2 and \\int (\\phi_2 + (g^{\\prime}_{\\varepsilon} \\mathbf{H}{(g^{\\prime}_{\\varepsilon},L)})^{g^{\\prime}_{\\varepsilon}}) dL = \\int (L^{g^{\\prime}_{\\varepsilon}} + \\phi_2) dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{H}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Symbol('L', commutative=True))"], [["power", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{H}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["add", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Pow(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{H}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\phi_2', commutative=True), Pow(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{H}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Pow(Symbol('L', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\theta)} = \\log{(\\cos{(\\theta)})} and \\operatorname{A_{z}}{(\\theta)} = 2 \\log{(\\cos{(\\theta)})}, then obtain 2 \\operatorname{x^{{\\}'}}{(\\theta)} \\operatorname{x^{{\\}'}}^{\\theta}{(\\theta)} = \\operatorname{A_{z}}{(\\theta)} \\operatorname{x^{{\\}'}}^{\\theta}{(\\theta)}", "derivation": "\\operatorname{x^{{\\}'}}{(\\theta)} = \\log{(\\cos{(\\theta)})} and \\operatorname{x^{{\\}'}}{(\\theta)} + \\log{(\\cos{(\\theta)})} = 2 \\log{(\\cos{(\\theta)})} and \\operatorname{A_{z}}{(\\theta)} = 2 \\log{(\\cos{(\\theta)})} and \\operatorname{x^{{\\}'}}{(\\theta)} + \\log{(\\cos{(\\theta)})} = \\operatorname{A_{z}}{(\\theta)} and (\\operatorname{x^{{\\}'}}{(\\theta)} + \\log{(\\cos{(\\theta)})}) \\log{(\\cos{(\\theta)})}^{\\theta} = \\operatorname{A_{z}}{(\\theta)} \\log{(\\cos{(\\theta)})}^{\\theta} and 2 \\operatorname{x^{{\\}'}}{(\\theta)} \\operatorname{x^{{\\}'}}^{\\theta}{(\\theta)} = \\operatorname{A_{z}}{(\\theta)} \\operatorname{x^{{\\}'}}^{\\theta}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), log(cos(Symbol('\\\\theta', commutative=True))))"], [["add", 1, "log(cos(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), log(cos(Symbol('\\\\theta', commutative=True)))), Mul(Integer(2), log(cos(Symbol('\\\\theta', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\theta', commutative=True)), Mul(Integer(2), log(cos(Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), log(cos(Symbol('\\\\theta', commutative=True)))), Function('A_z')(Symbol('\\\\theta', commutative=True)))"], [["times", 4, "Pow(log(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), log(cos(Symbol('\\\\theta', commutative=True)))), Pow(log(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))), Mul(Function('A_z')(Symbol('\\\\theta', commutative=True)), Pow(log(cos(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(2), Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Function('A_z')(Symbol('\\\\theta', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\eta^{\\prime},M_{E})} = - M_{E} + \\sin{(\\eta^{\\prime})}, then obtain \\sin{((\\Omega{(\\eta^{\\prime},M_{E})} - 1)^{M_{E}})} = \\sin{((- M_{E} + \\sin{(\\eta^{\\prime})} - 1)^{M_{E}})}", "derivation": "\\Omega{(\\eta^{\\prime},M_{E})} = - M_{E} + \\sin{(\\eta^{\\prime})} and \\Omega{(\\eta^{\\prime},M_{E})} - 1 = - M_{E} + \\sin{(\\eta^{\\prime})} - 1 and (\\Omega{(\\eta^{\\prime},M_{E})} - 1)^{M_{E}} = (- M_{E} + \\sin{(\\eta^{\\prime})} - 1)^{M_{E}} and \\sin{((\\Omega{(\\eta^{\\prime},M_{E})} - 1)^{M_{E}})} = \\sin{((- M_{E} + \\sin{(\\eta^{\\prime})} - 1)^{M_{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Function('\\\\Omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Add(Function('\\\\Omega')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then derive \\int \\pi{(\\mathbf{p})} d\\mathbf{p} = \\varepsilon + \\sin{(\\mathbf{p})}, then obtain \\int \\cos{(\\mathbf{p})} d\\mathbf{p} + \\iint \\cos{(\\mathbf{p})} d\\mathbf{p} d\\mathbf{p} = \\int (\\varepsilon + \\sin{(\\mathbf{p})}) d\\mathbf{p} + \\int \\cos{(\\mathbf{p})} d\\mathbf{p}", "derivation": "\\pi{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\int \\pi{(\\mathbf{p})} d\\mathbf{p} = \\int \\cos{(\\mathbf{p})} d\\mathbf{p} and \\int \\pi{(\\mathbf{p})} d\\mathbf{p} = \\varepsilon + \\sin{(\\mathbf{p})} and \\int \\cos{(\\mathbf{p})} d\\mathbf{p} = \\varepsilon + \\sin{(\\mathbf{p})} and \\iint \\cos{(\\mathbf{p})} d\\mathbf{p} d\\mathbf{p} = \\int (\\varepsilon + \\sin{(\\mathbf{p})}) d\\mathbf{p} and \\int \\cos{(\\mathbf{p})} d\\mathbf{p} + \\iint \\cos{(\\mathbf{p})} d\\mathbf{p} d\\mathbf{p} = \\int (\\varepsilon + \\sin{(\\mathbf{p})}) d\\mathbf{p} + \\int \\cos{(\\mathbf{p})} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 5, "Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Integral(Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given n{(\\psi)} = \\log{(\\psi)}, then obtain \\frac{d}{d \\psi} n{(\\psi)} - \\frac{2}{\\psi} = - \\frac{1}{\\psi}", "derivation": "n{(\\psi)} = \\log{(\\psi)} and n{(\\psi)} - \\log{(\\psi)} = 0 and \\frac{d}{d \\psi} (n{(\\psi)} - \\log{(\\psi)}) = \\frac{d}{d \\psi} 0 and \\frac{d}{d \\psi} (n{(\\psi)} - \\log{(\\psi)}) - \\frac{1}{\\psi} = \\frac{d}{d \\psi} 0 - \\frac{1}{\\psi} and \\frac{d}{d \\psi} n{(\\psi)} - \\frac{2}{\\psi} = - \\frac{1}{\\psi}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('n')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Add(Function('n')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))"], "Equality(Add(Derivative(Add(Function('n')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))), Add(Derivative(Integer(0), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(q)} = \\frac{d}{d q} \\sin{(q)}, then derive q + r = \\int \\frac{\\frac{d}{d q} \\sin{(q)}}{\\operatorname{t_{1}}{(q)}} dq, then obtain - \\int 0^{q} dq + \\iint 1 dq dr = - \\int 0^{q} dq + \\int (q + r) dr", "derivation": "\\operatorname{t_{1}}{(q)} = \\frac{d}{d q} \\sin{(q)} and 1 = \\frac{\\frac{d}{d q} \\sin{(q)}}{\\operatorname{t_{1}}{(q)}} and \\int 1 dq = \\int \\frac{\\frac{d}{d q} \\sin{(q)}}{\\operatorname{t_{1}}{(q)}} dq and q + r = \\int \\frac{\\frac{d}{d q} \\sin{(q)}}{\\operatorname{t_{1}}{(q)}} dq and \\int 1 dq = q + r and \\iint 1 dq dr = \\int (q + r) dr and - \\int (\\frac{d}{d q} 1)^{q} dq + \\iint 1 dq dr = \\int (q + r) dr - \\int (\\frac{d}{d q} 1)^{q} dq and - \\int 0^{q} dq + \\iint 1 dq dr = - \\int 0^{q} dq + \\int (q + r) dr", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('q', commutative=True)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 1, "Function('t_1')(Symbol('q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t_1')(Symbol('q', commutative=True)), Integer(-1)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('q', commutative=True))), Integral(Mul(Pow(Function('t_1')(Symbol('q', commutative=True)), Integer(-1)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('q', commutative=True), Symbol('r', commutative=True)), Integral(Mul(Pow(Function('t_1')(Symbol('q', commutative=True)), Integer(-1)), Derivative(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('q', commutative=True))), Add(Symbol('q', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 5, "Symbol('r', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('q', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["minus", 6, "Integral(Pow(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Integer(1), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Integral(Add(Symbol('q', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Mul(Integer(-1), Integral(Pow(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))))"], [["evaluate_derivatives", 7], "Equality(Add(Mul(Integer(-1), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Integer(1), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Integral(Pow(Integer(0), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Integral(Add(Symbol('q', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(k,y^{\\prime})} = \\int (k + y^{\\prime}) dy^{\\prime}, then obtain - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{2}{(k,y^{\\prime})} - (\\int (k + y^{\\prime}) dy^{\\prime})^{y^{\\prime}} = - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\int (k + y^{\\prime}) dy^{\\prime} - (\\int (k + y^{\\prime}) dy^{\\prime})^{y^{\\prime}}", "derivation": "\\phi_{2}{(k,y^{\\prime})} = \\int (k + y^{\\prime}) dy^{\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{2}{(k,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} \\int (k + y^{\\prime}) dy^{\\prime} and - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{2}{(k,y^{\\prime})} = - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\int (k + y^{\\prime}) dy^{\\prime} and - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{2}{(k,y^{\\prime})} - (\\int (k + y^{\\prime}) dy^{\\prime})^{y^{\\prime}} = - y^{\\prime} + \\frac{\\partial}{\\partial y^{\\prime}} \\int (k + y^{\\prime}) dy^{\\prime} - (\\int (k + y^{\\prime}) dy^{\\prime})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["minus", 3, "Pow(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Integral(Add(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})} = - \\pi + f^{\\prime} and \\mathbf{J}_f{(f_{E},\\Psi_{nl})} = \\Psi_{nl} f_{E}, then obtain \\mathbf{J}_f{(f_{E},\\Psi_{nl})} - \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})} = \\Psi_{nl} f_{E} - \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})} = - \\pi + f^{\\prime} and - \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})} = \\pi - f^{\\prime} and \\mathbf{J}_f{(f_{E},\\Psi_{nl})} = \\Psi_{nl} f_{E} and \\pi - f^{\\prime} + \\mathbf{J}_f{(f_{E},\\Psi_{nl})} = \\Psi_{nl} f_{E} + \\pi - f^{\\prime} and \\mathbf{J}_f{(f_{E},\\Psi_{nl})} - \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})} = \\Psi_{nl} f_{E} - \\operatorname{f_{\\mathbf{p}}}{(\\pi,f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\pi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\pi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f_E', commutative=True)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\pi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\pi', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given Z{(\\hat{X},F_{N},\\ddot{x})} = (\\ddot{x} + \\hat{X})^{F_{N}}, then obtain \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} Z{(\\hat{X},F_{N},\\ddot{x})} = \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}}", "derivation": "Z{(\\hat{X},F_{N},\\ddot{x})} = (\\ddot{x} + \\hat{X})^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} Z{(\\hat{X},F_{N},\\ddot{x})} = \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} and (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} Z{(\\hat{X},F_{N},\\ddot{x})} = (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} Z{(\\hat{X},F_{N},\\ddot{x})} = \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} (\\ddot{x} + \\hat{X})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["times", 2, "Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Derivative(Function('Z')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Derivative(Function('Z')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Derivative(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(x^\\prime)} = \\log{(\\log{(x^\\prime)})}, then derive - \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\varphi^{*}{(x^\\prime)} = - \\log{(x^\\prime)} + \\frac{1}{x^\\prime \\log{(x^\\prime)}}, then obtain (- \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})})^{x^\\prime} = (- \\log{(x^\\prime)} + \\frac{1}{x^\\prime \\log{(x^\\prime)}})^{x^\\prime}", "derivation": "\\varphi^{*}{(x^\\prime)} = \\log{(\\log{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\varphi^{*}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})} and - \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\varphi^{*}{(x^\\prime)} = - \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})} and - \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\varphi^{*}{(x^\\prime)} = - \\log{(x^\\prime)} + \\frac{1}{x^\\prime \\log{(x^\\prime)}} and (- \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\varphi^{*}{(x^\\prime)})^{x^\\prime} = (- \\log{(x^\\prime)} + \\frac{1}{x^\\prime \\log{(x^\\prime)}})^{x^\\prime} and (- \\log{(x^\\prime)} + \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})})^{x^\\prime} = (- \\log{(x^\\prime)} + \\frac{1}{x^\\prime \\log{(x^\\prime)}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), log(log(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 2, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Derivative(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Derivative(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Derivative(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Integer(-1)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\theta{(E_{x})} = \\sin{(e^{E_{x}})}, then derive (\\int \\theta{(E_{x})} dE_{x})^{E_{x}} = (\\tilde{g}^* + \\operatorname{Si}{(e^{E_{x}})})^{E_{x}}, then obtain ((\\int \\theta{(E_{x})} dE_{x})^{E_{x}})^{\\tilde{g}^*} = ((\\tilde{g}^* + \\operatorname{Si}{(e^{E_{x}})})^{E_{x}})^{\\tilde{g}^*}", "derivation": "\\theta{(E_{x})} = \\sin{(e^{E_{x}})} and \\int \\theta{(E_{x})} dE_{x} = \\int \\sin{(e^{E_{x}})} dE_{x} and (\\int \\theta{(E_{x})} dE_{x})^{E_{x}} = (\\int \\sin{(e^{E_{x}})} dE_{x})^{E_{x}} and (\\int \\theta{(E_{x})} dE_{x})^{E_{x}} = (\\tilde{g}^* + \\operatorname{Si}{(e^{E_{x}})})^{E_{x}} and ((\\int \\theta{(E_{x})} dE_{x})^{E_{x}})^{\\tilde{g}^*} = ((\\tilde{g}^* + \\operatorname{Si}{(e^{E_{x}})})^{E_{x}})^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('E_x', commutative=True)), sin(exp(Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(sin(exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\theta')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Pow(Integral(sin(exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\theta')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Si(exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)))"], [["power", 4, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\theta')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Si(exp(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi)} = e^{\\phi}, then obtain \\frac{\\operatorname{t_{2}}{(\\phi)} e^{\\phi} + \\operatorname{t_{2}}{(\\phi)}}{\\frac{d}{d \\phi} \\operatorname{t_{2}}{(\\phi)} e^{\\phi}} = \\frac{\\operatorname{t_{2}}{(\\phi)} e^{\\phi} + e^{\\phi}}{\\frac{d}{d \\phi} \\operatorname{t_{2}}{(\\phi)} e^{\\phi}}", "derivation": "\\operatorname{t_{2}}{(\\phi)} = e^{\\phi} and \\operatorname{t_{2}}{(\\phi)} e^{\\phi} = e^{2 \\phi} and \\operatorname{t_{2}}{(\\phi)} e^{\\phi} + \\operatorname{t_{2}}{(\\phi)} = \\operatorname{t_{2}}{(\\phi)} e^{\\phi} + e^{\\phi} and \\operatorname{t_{2}}{(\\phi)} + e^{2 \\phi} = e^{2 \\phi} + e^{\\phi} and \\frac{\\operatorname{t_{2}}{(\\phi)} + e^{2 \\phi}}{\\frac{d}{d \\phi} e^{2 \\phi}} = \\frac{e^{2 \\phi} + e^{\\phi}}{\\frac{d}{d \\phi} e^{2 \\phi}} and \\frac{\\operatorname{t_{2}}{(\\phi)} e^{\\phi} + \\operatorname{t_{2}}{(\\phi)}}{\\frac{d}{d \\phi} \\operatorname{t_{2}}{(\\phi)} e^{\\phi}} = \\frac{\\operatorname{t_{2}}{(\\phi)} e^{\\phi} + e^{\\phi}}{\\frac{d}{d \\phi} \\operatorname{t_{2}}{(\\phi)} e^{\\phi}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), Function('t_2')(Symbol('\\\\phi', commutative=True))), Add(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Add(exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))))"], [["divide", 4, "Derivative(exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Pow(Derivative(exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))), Pow(Derivative(exp(Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), Function('t_2')(Symbol('\\\\phi', commutative=True))), Pow(Derivative(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))), Pow(Derivative(Mul(Function('t_2')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(k)} = \\sin{(k)} and \\hat{H}_l{(k)} = \\sin{(k)}, then obtain \\int \\frac{d}{d k} \\dot{z}{(k)} dk = \\int \\frac{d}{d k} \\sin{(k)} dk", "derivation": "\\dot{z}{(k)} = \\sin{(k)} and \\hat{H}_l{(k)} = \\sin{(k)} and \\dot{z}{(k)} = \\hat{H}_l{(k)} and \\frac{d}{d k} \\dot{z}{(k)} = \\frac{d}{d k} \\hat{H}_l{(k)} and \\frac{d}{d k} \\dot{z}{(k)} = \\frac{d}{d k} \\sin{(k)} and \\int \\frac{d}{d k} \\dot{z}{(k)} dk = \\int \\frac{d}{d k} \\sin{(k)} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Function('\\\\hat{H}_l')(Symbol('k', commutative=True)))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('\\\\hat{H}_l')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))), Integral(Derivative(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\mathbf{p} e^{\\hat{p}}, then derive \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\mathbf{p} e^{\\hat{p}}, then obtain \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\frac{\\partial^{3}}{\\partial \\hat{p}^{3}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})}", "derivation": "\\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\mathbf{p} e^{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\frac{\\partial}{\\partial \\hat{p}} \\mathbf{p} e^{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\mathbf{p} e^{\\hat{p}} and \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} \\mathbf{p} e^{\\hat{p}} and \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})} = \\frac{\\partial^{3}}{\\partial \\hat{p}^{3}} \\operatorname{A_{1}}{(\\hat{p},\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Derivative(Function('A_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(m_{s})} = \\sin{(m_{s})}, then derive \\int \\operatorname{C_{2}}{(m_{s})} dm_{s} = f^{*} - \\cos{(m_{s})}, then obtain v - \\cos{(m_{s})} = f^{*} - \\cos{(m_{s})}", "derivation": "\\operatorname{C_{2}}{(m_{s})} = \\sin{(m_{s})} and \\int \\operatorname{C_{2}}{(m_{s})} dm_{s} = \\int \\sin{(m_{s})} dm_{s} and \\int \\operatorname{C_{2}}{(m_{s})} dm_{s} = f^{*} - \\cos{(m_{s})} and \\int \\sin{(m_{s})} dm_{s} = f^{*} - \\cos{(m_{s})} and v - \\cos{(m_{s})} = f^{*} - \\cos{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), cos(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given h{(F_{g},\\omega)} = F_{g} \\omega, then obtain \\int h{(F_{g},\\omega)} \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} d\\omega = \\int F_{g} \\omega \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} d\\omega", "derivation": "h{(F_{g},\\omega)} = F_{g} \\omega and F_{g} \\omega + h{(F_{g},\\omega)} = 2 F_{g} \\omega and h{(F_{g},\\omega)} \\log{(2 F_{g} \\omega)} = F_{g} \\omega \\log{(2 F_{g} \\omega)} and h{(F_{g},\\omega)} \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} = F_{g} \\omega \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} and \\int h{(F_{g},\\omega)} \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} d\\omega = \\int F_{g} \\omega \\log{(F_{g} \\omega + h{(F_{g},\\omega)})} d\\omega", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["add", 1, "Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["times", 1, "log(Mul(Integer(2), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), log(Mul(Integer(2), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))), Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True), log(Mul(Integer(2), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), log(Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))))), Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True), log(Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), log(Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True), log(Add(Mul(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Function('h')(Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain 2 \\operatorname{V_{\\mathbf{B}}}^{2}{(\\dot{x})} = 2 \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} \\sin{(\\dot{x})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} = \\sin{(\\dot{x})} and 2 \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} = \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} + \\sin{(\\dot{x})} and (\\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} + \\sin{(\\dot{x})}) \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} = (\\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} + \\sin{(\\dot{x})}) \\sin{(\\dot{x})} and 2 \\operatorname{V_{\\mathbf{B}}}^{2}{(\\dot{x})} = 2 \\operatorname{V_{\\mathbf{B}}}{(\\dot{x})} \\sin{(\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["add", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 1, "Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True))), Mul(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True))), sin(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), Integer(2))), Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given c{(F_{g},\\delta,l)} = F_{g} - \\delta + l, then obtain \\int (- \\frac{c{(F_{g},\\delta,l)}}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} dF_{g} = \\int (- \\frac{F_{g} - \\delta + l}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} dF_{g}", "derivation": "c{(F_{g},\\delta,l)} = F_{g} - \\delta + l and - \\frac{c{(F_{g},\\delta,l)}}{\\delta} = - \\frac{F_{g} - \\delta + l}{\\delta} and - \\frac{c{(F_{g},\\delta,l)}}{\\delta \\sin{(e^{\\sigma_p})}} = - \\frac{F_{g} - \\delta + l}{\\delta \\sin{(e^{\\sigma_p})}} and (- \\frac{c{(F_{g},\\delta,l)}}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} = (- \\frac{F_{g} - \\delta + l}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} and \\int (- \\frac{c{(F_{g},\\delta,l)}}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} dF_{g} = \\int (- \\frac{F_{g} - \\delta + l}{\\delta \\sin{(e^{\\sigma_p})}})^{\\delta} dF_{g}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('F_g', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('c')(Symbol('F_g', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('l', commutative=True))))"], [["divide", 2, "sin(exp(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('c')(Symbol('F_g', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('c')(Symbol('F_g', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\delta', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\delta', commutative=True)))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('c')(Symbol('F_g', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(F_{g},r)} = \\cos{(F_{g} r)}, then derive \\frac{\\partial}{\\partial r} \\tilde{g}{(F_{g},r)} = - F_{g} \\sin{(F_{g} r)}, then obtain - F_{g} + \\frac{\\partial}{\\partial r} \\cos{(F_{g} r)} = - F_{g} \\sin{(F_{g} r)} - F_{g}", "derivation": "\\tilde{g}{(F_{g},r)} = \\cos{(F_{g} r)} and \\frac{\\partial}{\\partial r} \\tilde{g}{(F_{g},r)} = \\frac{\\partial}{\\partial r} \\cos{(F_{g} r)} and - F_{g} + \\frac{\\partial}{\\partial r} \\tilde{g}{(F_{g},r)} = - F_{g} + \\frac{\\partial}{\\partial r} \\cos{(F_{g} r)} and \\frac{\\partial}{\\partial r} \\tilde{g}{(F_{g},r)} = - F_{g} \\sin{(F_{g} r)} and - F_{g} \\sin{(F_{g} r)} = \\frac{\\partial}{\\partial r} \\cos{(F_{g} r)} and - F_{g} + \\frac{\\partial}{\\partial r} \\tilde{g}{(F_{g},r)} = - F_{g} \\sin{(F_{g} r)} - F_{g} and - F_{g} + \\frac{\\partial}{\\partial r} \\cos{(F_{g} r)} = - F_{g} \\sin{(F_{g} r)} - F_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('r', commutative=True)), cos(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('F_g', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Symbol('F_g', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True)))), Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True)))), Mul(Integer(-1), Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(cos(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), sin(Mul(Symbol('F_g', commutative=True), Symbol('r', commutative=True)))), Mul(Integer(-1), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given p{(c_{0})} = \\log{(\\cos{(c_{0})})}, then derive \\frac{d}{d c_{0}} p{(c_{0})} = - \\frac{\\sin{(c_{0})}}{\\cos{(c_{0})}}, then obtain \\frac{d}{d c_{0}} \\log{(\\cos{(c_{0})})} = - \\frac{\\sin{(c_{0})}}{\\cos{(c_{0})}}", "derivation": "p{(c_{0})} = \\log{(\\cos{(c_{0})})} and \\frac{d}{d c_{0}} p{(c_{0})} = \\frac{d}{d c_{0}} \\log{(\\cos{(c_{0})})} and \\frac{d}{d c_{0}} p{(c_{0})} = - \\frac{\\sin{(c_{0})}}{\\cos{(c_{0})}} and \\frac{d}{d c_{0}} \\log{(\\cos{(c_{0})})} = - \\frac{\\sin{(c_{0})}}{\\cos{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('c_0', commutative=True)), log(cos(Symbol('c_0', commutative=True))))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(log(cos(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(cos(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given G{(\\eta)} = \\sin{(\\eta)} and \\operatorname{n_{1}}{(\\eta)} = \\sin{(\\eta)}, then obtain \\frac{G{(\\eta)}}{\\eta} = \\frac{\\operatorname{n_{1}}{(\\eta)}}{\\eta}", "derivation": "G{(\\eta)} = \\sin{(\\eta)} and \\frac{G{(\\eta)}}{\\eta} = \\frac{\\sin{(\\eta)}}{\\eta} and \\operatorname{n_{1}}{(\\eta)} = \\sin{(\\eta)} and \\frac{G{(\\eta)}}{\\eta} = \\frac{\\operatorname{n_{1}}{(\\eta)}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('G')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), sin(Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('G')(Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('n_1')(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(h,\\mu)} = \\mu h, then obtain \\int (- 2 \\mu h + 2 \\mathbf{v}{(h,\\mu)}) dh = \\int 0 dh", "derivation": "\\mathbf{v}{(h,\\mu)} = \\mu h and \\mu + \\mathbf{v}{(h,\\mu)} = \\mu h + \\mu and - \\mu h + \\mathbf{v}{(h,\\mu)} = 0 and \\mu h = 2 \\mu h - \\mathbf{v}{(h,\\mu)} and - 2 \\mu h + 2 \\mathbf{v}{(h,\\mu)} = 0 and \\int (- 2 \\mu h + 2 \\mathbf{v}{(h,\\mu)}) dh = \\int 0 dh", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 2, "Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(0))"], [["integrate", 5, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('h', commutative=True))), Integral(Integer(0), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(E,v)} = - v + \\cos{(E)}, then obtain - \\frac{\\partial}{\\partial E} (- v + \\cos{(E)}) + 2 \\int \\hat{H}_l{(E,v)} dE = - \\frac{\\partial}{\\partial E} (- v + \\cos{(E)}) + \\int (- v + \\cos{(E)}) dE + \\int \\hat{H}_l{(E,v)} dE", "derivation": "\\hat{H}_l{(E,v)} = - v + \\cos{(E)} and \\int \\hat{H}_l{(E,v)} dE = \\int (- v + \\cos{(E)}) dE and \\frac{\\partial}{\\partial E} \\hat{H}_l{(E,v)} = \\frac{\\partial}{\\partial E} (- v + \\cos{(E)}) and - \\frac{\\partial}{\\partial E} \\hat{H}_l{(E,v)} + 2 \\int \\hat{H}_l{(E,v)} dE = - \\frac{\\partial}{\\partial E} \\hat{H}_l{(E,v)} + \\int (- v + \\cos{(E)}) dE + \\int \\hat{H}_l{(E,v)} dE and - \\frac{\\partial}{\\partial E} (- v + \\cos{(E)}) + 2 \\int \\hat{H}_l{(E,v)} dE = - \\frac{\\partial}{\\partial E} (- v + \\cos{(E)}) + \\int (- v + \\cos{(E)}) dE + \\int \\hat{H}_l{(E,v)} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["add", 2, "Add(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(2), Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True))))), Add(Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(2), Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True))))), Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Integral(Add(Mul(Integer(-1), Symbol('v', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\chi{(V)} = \\log{(V)}, then obtain (\\log{(V)}^{V} + \\frac{\\chi{(V)}}{V})^{V} - \\log{(V)} = (\\log{(V)}^{V} + \\frac{\\log{(V)}}{V})^{V} - \\log{(V)}", "derivation": "\\chi{(V)} = \\log{(V)} and \\frac{\\chi{(V)}}{V} = \\frac{\\log{(V)}}{V} and \\chi^{V}{(V)} = \\log{(V)}^{V} and \\chi^{V}{(V)} + \\frac{\\chi{(V)}}{V} = \\chi^{V}{(V)} + \\frac{\\log{(V)}}{V} and (\\chi^{V}{(V)} + \\frac{\\chi{(V)}}{V})^{V} = (\\chi^{V}{(V)} + \\frac{\\log{(V)}}{V})^{V} and (\\chi^{V}{(V)} + \\frac{\\chi{(V)}}{V})^{V} - \\log{(V)} = (\\chi^{V}{(V)} + \\frac{\\log{(V)}}{V})^{V} - \\log{(V)} and (\\log{(V)}^{V} + \\frac{\\chi{(V)}}{V})^{V} - \\log{(V)} = (\\log{(V)}^{V} + \\frac{\\log{(V)}}{V})^{V} - \\log{(V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["add", 2, "Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], "Equality(Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('V', commutative=True)))), Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))), Symbol('V', commutative=True)))"], [["minus", 5, "log(Symbol('V', commutative=True))"], "Equality(Add(Pow(Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))), Add(Pow(Add(Pow(Function('\\\\chi')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Add(Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))), Add(Pow(Add(Pow(log(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), log(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Mul(Integer(-1), log(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given x{(b,v_{1})} = b + v_{1} and \\hat{x}{(v_{1})} = v_{1}^{2}, then derive \\int (x{(b,v_{1})} + 1) dv_{1} = B + \\frac{v_{1}^{2}}{2} + v_{1} (b + 1), then obtain (B + v_{1} (b + 1) + \\frac{\\hat{x}{(v_{1})}}{2}) h{(b,v_{1})} = (B + \\frac{v_{1}^{2}}{2} + v_{1} (b + 1)) h{(b,v_{1})}", "derivation": "x{(b,v_{1})} = b + v_{1} and x{(b,v_{1})} + 1 = b + v_{1} + 1 and \\int (x{(b,v_{1})} + 1) dv_{1} = \\int (b + v_{1} + 1) dv_{1} and \\int (x{(b,v_{1})} + 1) dv_{1} = B + \\frac{v_{1}^{2}}{2} + v_{1} (b + 1) and \\hat{x}{(v_{1})} = v_{1}^{2} and \\int (x{(b,v_{1})} + 1) dv_{1} = B + v_{1} (b + 1) + \\frac{\\hat{x}{(v_{1})}}{2} and B + v_{1} (b + 1) + \\frac{\\hat{x}{(v_{1})}}{2} = B + \\frac{v_{1}^{2}}{2} + v_{1} (b + 1) and (B + v_{1} (b + 1) + \\frac{\\hat{x}{(v_{1})}}{2}) h{(b,v_{1})} = (B + \\frac{v_{1}^{2}}{2} + v_{1} (b + 1)) h{(b,v_{1})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('x')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(1)), Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('v_1', commutative=True)"], "Equality(Integral(Add(Function('x')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(1)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True), Integer(1)), Tuple(Symbol('v_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('x')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(1)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Function('x')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(1)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1))), Mul(Rational(1, 2), Function('\\\\hat{x}')(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('B', commutative=True), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1))), Mul(Rational(1, 2), Function('\\\\hat{x}')(Symbol('v_1', commutative=True)))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1)))))"], [["times", 7, "Function('h')(Symbol('b', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Add(Symbol('B', commutative=True), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1))), Mul(Rational(1, 2), Function('\\\\hat{x}')(Symbol('v_1', commutative=True)))), Function('h')(Symbol('b', commutative=True), Symbol('v_1', commutative=True))), Mul(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Integer(1)))), Function('h')(Symbol('b', commutative=True), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\omega,\\varphi)} = \\omega - \\varphi, then obtain \\frac{\\partial}{\\partial \\omega} \\int \\Omega{(\\omega,\\varphi)} d\\omega = \\frac{\\partial}{\\partial \\omega} (\\frac{\\omega^{2}}{2} - \\omega \\varphi + m)", "derivation": "\\Omega{(\\omega,\\varphi)} = \\omega - \\varphi and \\int \\Omega{(\\omega,\\varphi)} d\\omega = \\int (\\omega - \\varphi) d\\omega and \\frac{\\partial}{\\partial \\omega} \\int \\Omega{(\\omega,\\varphi)} d\\omega = \\frac{\\partial}{\\partial \\omega} \\int (\\omega - \\varphi) d\\omega and \\frac{\\partial}{\\partial \\omega} \\int \\Omega{(\\omega,\\varphi)} d\\omega = \\frac{\\partial}{\\partial \\omega} (\\frac{\\omega^{2}}{2} - \\omega \\varphi + m)", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\Omega')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} = F_{c} + x^\\prime, then derive \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2}, then obtain - \\frac{\\partial}{\\partial x^\\prime} (F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2}) + \\frac{\\partial}{\\partial x^\\prime} \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = 0", "derivation": "\\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} = F_{c} + x^\\prime and \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = \\int (F_{c} + x^\\prime) dx^\\prime and \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2} and \\frac{\\partial}{\\partial x^\\prime} \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} (F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2}) and \\frac{\\partial}{\\partial x^\\prime} \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime + 1 = \\frac{\\partial}{\\partial x^\\prime} (F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2}) + 1 and - \\frac{\\partial}{\\partial x^\\prime} (F_{c} x^\\prime + h + \\frac{(x^\\prime)^{2}}{2}) + \\frac{\\partial}{\\partial x^\\prime} \\int \\operatorname{a^{\\dagger}}{(F_{c},x^\\prime)} dx^\\prime = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)))"], [["minus", 5, "Add(Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Derivative(Integral(Function('a^{\\\\dagger}')(Symbol('F_c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(v_{1})} = \\sin{(v_{1})}, then obtain \\int \\frac{\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} dv_{1} = \\int \\frac{\\sin^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} dv_{1}", "derivation": "\\operatorname{L_{\\varepsilon}}{(v_{1})} = \\sin{(v_{1})} and \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} = \\sin^{v_{1}}{(v_{1})} and \\frac{\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} = \\frac{\\sin^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} and \\int \\frac{\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} dv_{1} = \\int \\frac{\\sin^{v_{1}}{(v_{1})}}{\\operatorname{L_{\\varepsilon}}{(v_{1})}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["divide", 2, "Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["integrate", 3, "Symbol('v_1', commutative=True)"], "Equality(Integral(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and C{(g_{\\varepsilon})} = \\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}}, then obtain \\frac{d}{d g_{\\varepsilon}} C{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} (\\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}})", "derivation": "\\operatorname{A_{x}}{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and 2 \\operatorname{A_{x}}{(g_{\\varepsilon})} = \\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} and C{(g_{\\varepsilon})} = \\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}} and C{(g_{\\varepsilon})} = 2 \\operatorname{A_{x}}{(g_{\\varepsilon})} and \\frac{d}{d g_{\\varepsilon}} 2 \\operatorname{A_{x}}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} (\\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}}) and \\frac{d}{d g_{\\varepsilon}} C{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} (\\operatorname{A_{x}}{(g_{\\varepsilon})} + e^{g_{\\varepsilon}})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(F_{g})} = \\log{(F_{g})}, then obtain (\\mathbf{J}_M{(F_{g})} + \\log{(F_{g})})^{2} + \\mathbf{J}_M{(F_{g})} - \\log{(F_{g})} = 4 \\log{(F_{g})}^{2}", "derivation": "\\mathbf{J}_M{(F_{g})} = \\log{(F_{g})} and \\mathbf{J}_M{(F_{g})} + \\log{(F_{g})} = 2 \\log{(F_{g})} and (\\mathbf{J}_M{(F_{g})} + \\log{(F_{g})})^{2} = 4 \\log{(F_{g})}^{2} and \\mathbf{J}_M{(F_{g})} - \\log{(F_{g})} = 0 and (\\mathbf{J}_M{(F_{g})} + \\log{(F_{g})})^{2} + \\mathbf{J}_M{(F_{g})} - \\log{(F_{g})} = (\\mathbf{J}_M{(F_{g})} + \\log{(F_{g})})^{2} and (\\mathbf{J}_M{(F_{g})} + \\log{(F_{g})})^{2} + \\mathbf{J}_M{(F_{g})} - \\log{(F_{g})} = 4 \\log{(F_{g})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))"], [["add", 1, "log(Symbol('F_g', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Mul(Integer(2), log(Symbol('F_g', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Integer(2)), Mul(Integer(4), Pow(log(Symbol('F_g', commutative=True)), Integer(2))))"], [["minus", 2, "Mul(Integer(2), log(Symbol('F_g', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Mul(Integer(-1), log(Symbol('F_g', commutative=True)))), Integer(0))"], [["add", 4, "Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Integer(2))"], "Equality(Add(Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Mul(Integer(-1), log(Symbol('F_g', commutative=True)))), Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Pow(Add(Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Integer(2)), Function('\\\\mathbf{J}_M')(Symbol('F_g', commutative=True)), Mul(Integer(-1), log(Symbol('F_g', commutative=True)))), Mul(Integer(4), Pow(log(Symbol('F_g', commutative=True)), Integer(2))))"]]}, {"prompt": "Given v{(\\pi,y^{\\prime})} = - \\pi + y^{\\prime}, then obtain - \\frac{\\frac{\\partial}{\\partial y^{\\prime}} v{(\\pi,y^{\\prime})}}{\\pi} = - \\frac{1}{\\pi}", "derivation": "v{(\\pi,y^{\\prime})} = - \\pi + y^{\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} v{(\\pi,y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (- \\pi + y^{\\prime}) and - \\frac{\\frac{\\partial}{\\partial y^{\\prime}} v{(\\pi,y^{\\prime})}}{\\pi} = - \\frac{\\frac{\\partial}{\\partial y^{\\prime}} (- \\pi + y^{\\prime})}{\\pi} and - \\frac{\\frac{\\partial}{\\partial y^{\\prime}} v{(\\pi,y^{\\prime})}}{\\pi} = - \\frac{1}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Function('v')(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Derivative(Function('v')(Symbol('\\\\pi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\psi^{*}{(y^{\\prime})} = \\log{(e^{y^{\\prime}})}, then derive \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime} = C + \\frac{(y^{\\prime})^{2}}{2}, then obtain \\frac{(C + \\frac{(y^{\\prime})^{2}}{2}) e^{y^{\\prime}}}{\\log{(e^{y^{\\prime}})}} = \\frac{e^{y^{\\prime}} \\int \\log{(e^{y^{\\prime}})} dy^{\\prime}}{\\log{(e^{y^{\\prime}})}}", "derivation": "\\psi^{*}{(y^{\\prime})} = \\log{(e^{y^{\\prime}})} and \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime} = \\int \\log{(e^{y^{\\prime}})} dy^{\\prime} and \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime} = C + \\frac{(y^{\\prime})^{2}}{2} and C + \\frac{(y^{\\prime})^{2}}{2} = \\int \\log{(e^{y^{\\prime}})} dy^{\\prime} and \\frac{C + \\frac{(y^{\\prime})^{2}}{2}}{\\log{(e^{y^{\\prime}})}} = \\frac{\\int \\log{(e^{y^{\\prime}})} dy^{\\prime}}{\\log{(e^{y^{\\prime}})}} and \\frac{(C + \\frac{(y^{\\prime})^{2}}{2}) e^{y^{\\prime}}}{\\log{(e^{y^{\\prime}})}} = \\frac{e^{y^{\\prime}} \\int \\log{(e^{y^{\\prime}})} dy^{\\prime}}{\\log{(e^{y^{\\prime}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), log(exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)))), Integral(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 4, "log(exp(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)))), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Integral(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["times", 5, "exp(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Symbol('C', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)))), exp(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1))), Mul(exp(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)), Integral(log(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(\\Psi)} = \\log{(\\Psi)}, then derive \\int \\psi^{*}{(\\Psi)} d\\Psi = \\Psi \\log{(\\Psi)} - \\Psi + \\Psi^{\\dagger}, then obtain \\frac{d}{d \\Psi} \\int \\psi^{*}{(\\Psi)} d\\Psi = \\frac{\\partial}{\\partial \\Psi} (\\Psi \\psi^{*}{(\\Psi)} - \\Psi + \\Psi^{\\dagger})", "derivation": "\\psi^{*}{(\\Psi)} = \\log{(\\Psi)} and \\int \\psi^{*}{(\\Psi)} d\\Psi = \\int \\log{(\\Psi)} d\\Psi and \\int \\psi^{*}{(\\Psi)} d\\Psi = \\Psi \\log{(\\Psi)} - \\Psi + \\Psi^{\\dagger} and \\frac{d}{d \\Psi} \\int \\psi^{*}{(\\Psi)} d\\Psi = \\frac{\\partial}{\\partial \\Psi} (\\Psi \\log{(\\Psi)} - \\Psi + \\Psi^{\\dagger}) and \\frac{d}{d \\Psi} \\int \\psi^{*}{(\\Psi)} d\\Psi = \\frac{\\partial}{\\partial \\Psi} (\\Psi \\psi^{*}{(\\Psi)} - \\Psi + \\Psi^{\\dagger})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integral(Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\psi^*')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(\\rho,\\mathbf{g})} = \\log{(\\mathbf{g} \\rho)} and Q{(\\mathbf{g},\\rho)} = S^{\\rho}{(\\rho,\\mathbf{g})}, then obtain Q{(\\mathbf{g},\\rho)} = \\log{(\\mathbf{g} \\rho)}^{\\rho}", "derivation": "S{(\\rho,\\mathbf{g})} = \\log{(\\mathbf{g} \\rho)} and S^{\\rho}{(\\rho,\\mathbf{g})} = \\log{(\\mathbf{g} \\rho)}^{\\rho} and Q{(\\mathbf{g},\\rho)} = S^{\\rho}{(\\rho,\\mathbf{g})} and Q{(\\mathbf{g},\\rho)} = \\log{(\\mathbf{g} \\rho)}^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Function('S')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('Q')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)} = (P_{g} + \\eta)^{f^{*}}, then derive \\frac{\\partial}{\\partial P_{g}} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)} = \\frac{f^{*} (P_{g} + \\eta)^{f^{*}}}{P_{g} + \\eta}, then obtain \\frac{f^{*} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)}}{P_{g} + \\eta} = \\frac{\\partial}{\\partial P_{g}} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)}", "derivation": "\\operatorname{v_{x}}{(f^{*},P_{g},\\eta)} = (P_{g} + \\eta)^{f^{*}} and \\frac{\\partial}{\\partial P_{g}} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)} = \\frac{\\partial}{\\partial P_{g}} (P_{g} + \\eta)^{f^{*}} and \\frac{\\partial}{\\partial P_{g}} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)} = \\frac{f^{*} (P_{g} + \\eta)^{f^{*}}}{P_{g} + \\eta} and \\frac{f^{*} (P_{g} + \\eta)^{f^{*}}}{P_{g} + \\eta} = \\frac{\\partial}{\\partial P_{g}} (P_{g} + \\eta)^{f^{*}} and \\frac{f^{*} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)}}{P_{g} + \\eta} = \\frac{\\partial}{\\partial P_{g}} \\operatorname{v_{x}}{(f^{*},P_{g},\\eta)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('f^*', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Mul(Symbol('f^*', commutative=True), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('f^*', commutative=True), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('f^*', commutative=True))), Derivative(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('f^*', commutative=True), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Function('v_x')(Symbol('f^*', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True))), Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('P_g', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(x,v_{x})} = \\frac{\\partial}{\\partial v_{x}} v_{x} x, then derive \\pi{(x,v_{x})} = x, then obtain \\int \\frac{d}{d x} x dv_{x} = \\int \\frac{\\partial^{2}}{\\partial x\\partial v_{x}} v_{x} x dv_{x}", "derivation": "\\pi{(x,v_{x})} = \\frac{\\partial}{\\partial v_{x}} v_{x} x and \\pi{(x,v_{x})} = x and \\frac{\\partial}{\\partial x} \\pi{(x,v_{x})} = \\frac{\\partial^{2}}{\\partial x\\partial v_{x}} v_{x} x and \\frac{d}{d x} x = \\frac{\\partial^{2}}{\\partial x\\partial v_{x}} v_{x} x and \\int \\frac{d}{d x} x dv_{x} = \\int \\frac{\\partial^{2}}{\\partial x\\partial v_{x}} v_{x} x dv_{x}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)), Derivative(Mul(Symbol('v_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\pi')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)), Symbol('x', commutative=True))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Derivative(Symbol('x', commutative=True), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))), Integral(Derivative(Mul(Symbol('v_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\varphi)} = e^{\\varphi}, then derive \\frac{d}{d \\varphi} \\operatorname{v_{1}}{(\\varphi)} = e^{\\varphi}, then obtain e^{\\operatorname{v_{1}}{(\\varphi)}} = e^{\\frac{d}{d \\varphi} \\operatorname{v_{1}}{(\\varphi)}}", "derivation": "\\operatorname{v_{1}}{(\\varphi)} = e^{\\varphi} and e^{\\operatorname{v_{1}}{(\\varphi)}} = e^{e^{\\varphi}} and \\frac{d}{d \\varphi} \\operatorname{v_{1}}{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and \\frac{d}{d \\varphi} \\operatorname{v_{1}}{(\\varphi)} = e^{\\varphi} and e^{\\operatorname{v_{1}}{(\\varphi)}} = e^{\\frac{d}{d \\varphi} \\operatorname{v_{1}}{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["exp", 1], "Equality(exp(Function('v_1')(Symbol('\\\\varphi', commutative=True))), exp(exp(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_1')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), exp(Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(exp(Function('v_1')(Symbol('\\\\varphi', commutative=True))), exp(Derivative(Function('v_1')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(A_{x},\\chi)} = \\sin{(A_{x} \\chi)}, then obtain \\frac{\\partial}{\\partial A_{x}} (A_{x} \\operatorname{F_{c}}^{A_{x}}{(A_{x},\\chi)} + \\sin{(A_{x} \\chi)}) = \\frac{\\partial}{\\partial A_{x}} (A_{x} \\sin^{A_{x}}{(A_{x} \\chi)} + \\sin{(A_{x} \\chi)})", "derivation": "\\operatorname{F_{c}}{(A_{x},\\chi)} = \\sin{(A_{x} \\chi)} and \\operatorname{F_{c}}^{A_{x}}{(A_{x},\\chi)} = \\sin^{A_{x}}{(A_{x} \\chi)} and A_{x} \\operatorname{F_{c}}^{A_{x}}{(A_{x},\\chi)} = A_{x} \\sin^{A_{x}}{(A_{x} \\chi)} and A_{x} \\operatorname{F_{c}}^{A_{x}}{(A_{x},\\chi)} + \\sin{(A_{x} \\chi)} = A_{x} \\sin^{A_{x}}{(A_{x} \\chi)} + \\sin{(A_{x} \\chi)} and \\frac{\\partial}{\\partial A_{x}} (A_{x} \\operatorname{F_{c}}^{A_{x}}{(A_{x},\\chi)} + \\sin{(A_{x} \\chi)}) = \\frac{\\partial}{\\partial A_{x}} (A_{x} \\sin^{A_{x}}{(A_{x} \\chi)} + \\sin{(A_{x} \\chi)})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('A_x', commutative=True)), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('A_x', commutative=True)))"], [["times", 2, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Pow(Function('F_c')(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('A_x', commutative=True))), Mul(Symbol('A_x', commutative=True), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('A_x', commutative=True))))"], [["add", 3, "sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Symbol('A_x', commutative=True), Pow(Function('F_c')(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('A_x', commutative=True))), sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Symbol('A_x', commutative=True), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('A_x', commutative=True))), sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 4, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('A_x', commutative=True), Pow(Function('F_c')(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('A_x', commutative=True))), sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_x', commutative=True), Pow(sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('A_x', commutative=True))), sin(Mul(Symbol('A_x', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{B},A_{1})} = A_{1}^{\\mathbf{B}} and \\operatorname{x^{{\\}'}}{(\\mathbf{B},A_{1})} = \\frac{\\mathbf{M}{(\\mathbf{B},A_{1})}}{A_{1}}, then obtain \\operatorname{x^{{\\}'}}{(\\mathbf{B},A_{1})} = \\frac{A_{1}^{\\mathbf{B}}}{A_{1}}", "derivation": "\\mathbf{M}{(\\mathbf{B},A_{1})} = A_{1}^{\\mathbf{B}} and \\frac{\\mathbf{M}{(\\mathbf{B},A_{1})}}{A_{1}} = \\frac{A_{1}^{\\mathbf{B}}}{A_{1}} and \\operatorname{x^{{\\}'}}{(\\mathbf{B},A_{1})} = \\frac{\\mathbf{M}{(\\mathbf{B},A_{1})}}{A_{1}} and \\operatorname{x^{{\\}'}}{(\\mathbf{B},A_{1})} = \\frac{A_{1}^{\\mathbf{B}}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(r,F_{H})} = \\log{(F_{H} r)} and \\operatorname{A_{z}}{(r)} = 0^{r}, then obtain 1 = \\frac{1}{\\operatorname{A_{z}}{(r)}}", "derivation": "\\Psi^{\\dagger}{(r,F_{H})} = \\log{(F_{H} r)} and 0 = - \\Psi^{\\dagger}{(r,F_{H})} + \\log{(F_{H} r)} and 0^{r} = (- \\Psi^{\\dagger}{(r,F_{H})} + \\log{(F_{H} r)})^{r} and \\operatorname{A_{z}}{(r)} = 0^{r} and \\operatorname{A_{z}}{(r)} = 1 and 1 = \\frac{1}{\\operatorname{A_{z}}{(r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True))))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), log(Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True)))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), log(Mul(Symbol('F_H', commutative=True), Symbol('r', commutative=True)))), Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('r', commutative=True)), Pow(Integer(0), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('A_z')(Symbol('r', commutative=True)), Integer(1))"], [["divide", 5, "Function('A_z')(Symbol('r', commutative=True))"], "Equality(Integer(1), Pow(Function('A_z')(Symbol('r', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} = - y^{\\prime} + \\sin{(\\dot{y})} and \\varepsilon_{0}{(y^{\\prime},\\dot{y})} = \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} = -1, then obtain \\varepsilon_{0}{(y^{\\prime},\\dot{y})} = -1", "derivation": "\\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} = - y^{\\prime} + \\sin{(\\dot{y})} and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} = \\frac{\\partial}{\\partial y^{\\prime}} (- y^{\\prime} + \\sin{(\\dot{y})}) and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} = -1 and -1 = \\frac{\\partial}{\\partial y^{\\prime}} (- y^{\\prime} + \\sin{(\\dot{y})}) and \\varepsilon_{0}{(y^{\\prime},\\dot{y})} = \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\dot{y})} and \\varepsilon_{0}{(y^{\\prime},\\dot{y})} = \\frac{\\partial}{\\partial y^{\\prime}} (- y^{\\prime} + \\sin{(\\dot{y})}) and \\varepsilon_{0}{(y^{\\prime},\\dot{y})} = -1", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))"]]}, {"prompt": "Given \\dot{z}{(U,C_{1},l)} = \\frac{C_{1} + l}{U}, then derive 0 = \\frac{- \\frac{\\partial}{\\partial l} \\dot{z}{(U,C_{1},l)} + \\frac{1}{U}}{U}, then obtain \\int 0 dC_{1} = \\int - \\frac{- \\frac{\\partial}{\\partial l} \\dot{z}{(U,C_{1},l)} + \\frac{1}{U}}{U \\frac{\\partial}{\\partial C_{1}} \\dot{z}{(U,C_{1},l)}} dC_{1}", "derivation": "\\dot{z}{(U,C_{1},l)} = \\frac{C_{1} + l}{U} and 0 = - \\dot{z}{(U,C_{1},l)} + \\frac{C_{1} + l}{U} and 0 = \\frac{- \\dot{z}{(U,C_{1},l)} + \\frac{C_{1} + l}{U}}{U} and \\frac{d}{d l} 0 = \\frac{\\partial}{\\partial l} \\frac{- \\dot{z}{(U,C_{1},l)} + \\frac{C_{1} + l}{U}}{U} and 0 = \\frac{- \\frac{\\partial}{\\partial l} \\dot{z}{(U,C_{1},l)} + \\frac{1}{U}}{U} and 0 = - \\frac{- \\frac{\\partial}{\\partial l} \\dot{z}{(U,C_{1},l)} + \\frac{1}{U}}{U \\frac{\\partial}{\\partial C_{1}} \\dot{z}{(U,C_{1},l)}} and \\int 0 dC_{1} = \\int - \\frac{- \\frac{\\partial}{\\partial l} \\dot{z}{(U,C_{1},l)} + \\frac{1}{U}}{U \\frac{\\partial}{\\partial C_{1}} \\dot{z}{(U,C_{1},l)}} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Symbol('l', commutative=True))))"], [["minus", 1, "Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Symbol('l', commutative=True)))))"], [["times", 2, "Pow(Symbol('U', commutative=True), Integer(-1))"], "Equality(Integer(0), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Symbol('l', commutative=True))))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('C_1', commutative=True), Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('U', commutative=True), Integer(-1)))))"], [["divide", 5, "Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('U', commutative=True), Integer(-1))), Pow(Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))))"], [["integrate", 6, "Symbol('C_1', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Symbol('U', commutative=True), Integer(-1))), Pow(Derivative(Function('\\\\dot{z}')(Symbol('U', commutative=True), Symbol('C_1', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{E},\\rho)} = \\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho), then derive \\mathbf{H}{(\\mathbf{E},\\rho)} = -1, then obtain \\frac{\\frac{d}{d \\mathbf{E}} (-1)}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)} = \\frac{\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial \\rho} (\\mathbf{E} - \\rho)}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)}", "derivation": "\\mathbf{H}{(\\mathbf{E},\\rho)} = \\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho) and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{H}{(\\mathbf{E},\\rho)} = \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial \\rho} (\\mathbf{E} - \\rho) and \\frac{\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{H}{(\\mathbf{E},\\rho)}}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)} = \\frac{\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial \\rho} (\\mathbf{E} - \\rho)}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)} and \\mathbf{H}{(\\mathbf{E},\\rho)} = -1 and \\frac{\\frac{d}{d \\mathbf{E}} (-1)}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)} = \\frac{\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial \\rho} (\\mathbf{E} - \\rho)}{\\frac{\\partial}{\\partial \\rho} (\\mathbf{E} - \\rho)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Mul(Pow(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}{(\\mu_0,Z)} = Z - \\mu_0 and S{(f_{\\mathbf{p}})} = \\cos{(e^{f_{\\mathbf{p}}})}, then obtain - \\mu_0 + S{(f_{\\mathbf{p}})} - \\hat{x}{(\\mu_0,Z)} = - \\mu_0 - \\hat{x}{(\\mu_0,Z)} + \\cos{(e^{f_{\\mathbf{p}}})}", "derivation": "\\hat{x}{(\\mu_0,Z)} = Z - \\mu_0 and \\mu_0 + \\hat{x}{(\\mu_0,Z)} = Z and S{(f_{\\mathbf{p}})} = \\cos{(e^{f_{\\mathbf{p}}})} and - Z + S{(f_{\\mathbf{p}})} = - Z + \\cos{(e^{f_{\\mathbf{p}}})} and - \\mu_0 + S{(f_{\\mathbf{p}})} - \\hat{x}{(\\mu_0,Z)} = - \\mu_0 - \\hat{x}{(\\mu_0,Z)} + \\cos{(e^{f_{\\mathbf{p}}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))"], ["get_premise", "Equality(Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 3, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), cos(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('S')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('Z', commutative=True))), cos(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(V_{\\mathbf{B}},\\mathbf{v})} = V_{\\mathbf{B}} + \\mathbf{v}, then derive \\int \\hat{H}_l{(V_{\\mathbf{B}},\\mathbf{v})} d\\mathbf{v} = V_{\\mathbf{B}} \\mathbf{v} + \\mathbf{D} + \\frac{\\mathbf{v}^{2}}{2}, then obtain \\int (V_{\\mathbf{B}} + \\mathbf{v}) d\\mathbf{v} = V_{\\mathbf{B}} \\mathbf{v} + \\mathbf{D} + \\frac{\\mathbf{v}^{2}}{2}", "derivation": "\\hat{H}_l{(V_{\\mathbf{B}},\\mathbf{v})} = V_{\\mathbf{B}} + \\mathbf{v} and \\int \\hat{H}_l{(V_{\\mathbf{B}},\\mathbf{v})} d\\mathbf{v} = \\int (V_{\\mathbf{B}} + \\mathbf{v}) d\\mathbf{v} and \\int \\hat{H}_l{(V_{\\mathbf{B}},\\mathbf{v})} d\\mathbf{v} = V_{\\mathbf{B}} \\mathbf{v} + \\mathbf{D} + \\frac{\\mathbf{v}^{2}}{2} and \\int (V_{\\mathbf{B}} + \\mathbf{v}) d\\mathbf{v} = V_{\\mathbf{B}} \\mathbf{v} + \\mathbf{D} + \\frac{\\mathbf{v}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(k,\\rho)} = k \\cos{(\\rho)}, then obtain \\int (\\cos{(\\rho)} + \\int \\operatorname{z^{*}}{(k,\\rho)} dk) d\\rho = \\int (\\cos{(\\rho)} + \\int k \\cos{(\\rho)} dk) d\\rho", "derivation": "\\operatorname{z^{*}}{(k,\\rho)} = k \\cos{(\\rho)} and \\int \\operatorname{z^{*}}{(k,\\rho)} dk = \\int k \\cos{(\\rho)} dk and \\cos{(\\rho)} + \\int \\operatorname{z^{*}}{(k,\\rho)} dk = \\cos{(\\rho)} + \\int k \\cos{(\\rho)} dk and \\int (\\cos{(\\rho)} + \\int \\operatorname{z^{*}}{(k,\\rho)} dk) d\\rho = \\int (\\cos{(\\rho)} + \\int k \\cos{(\\rho)} dk) d\\rho", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('k', commutative=True), cos(Symbol('\\\\rho', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('k', commutative=True), cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\rho', commutative=True))"], "Equality(Add(cos(Symbol('\\\\rho', commutative=True)), Integral(Function('z^*')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True)))), Add(cos(Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('k', commutative=True), cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Add(cos(Symbol('\\\\rho', commutative=True)), Integral(Function('z^*')(Symbol('k', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Add(cos(Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('k', commutative=True), cos(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given L{(v_{1},v_{z})} = v_{1} - v_{z}, then obtain e^{(- v_{1} + L{(v_{1},v_{z})})^{v_{1}}} = e^{(- v_{z})^{v_{1}}}", "derivation": "L{(v_{1},v_{z})} = v_{1} - v_{z} and - v_{1} + L{(v_{1},v_{z})} = - v_{z} and (- v_{1} + L{(v_{1},v_{z})})^{v_{1}} = (- v_{z})^{v_{1}} and e^{(- v_{1} + L{(v_{1},v_{z})})^{v_{1}}} = e^{(- v_{z})^{v_{1}}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v_1', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('v_1', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["minus", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('L')(Symbol('v_1', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True)))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('L')(Symbol('v_1', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_1', commutative=True)), Pow(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('v_1', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('L')(Symbol('v_1', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_1', commutative=True))), exp(Pow(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(F_{x})} = e^{F_{x}}, then obtain \\frac{d}{d F_{x}} (- F_{x} + \\dot{y}^{F_{x}}{(F_{x})}) - \\int (e^{F_{x}})^{F_{x}} dF_{x} = \\frac{d}{d F_{x}} (- F_{x} + (e^{F_{x}})^{F_{x}}) - \\int (e^{F_{x}})^{F_{x}} dF_{x}", "derivation": "\\dot{y}{(F_{x})} = e^{F_{x}} and \\dot{y}^{F_{x}}{(F_{x})} = (e^{F_{x}})^{F_{x}} and - F_{x} + \\dot{y}^{F_{x}}{(F_{x})} = - F_{x} + (e^{F_{x}})^{F_{x}} and \\frac{d}{d F_{x}} (- F_{x} + \\dot{y}^{F_{x}}{(F_{x})}) = \\frac{d}{d F_{x}} (- F_{x} + (e^{F_{x}})^{F_{x}}) and \\frac{d}{d F_{x}} (- F_{x} + \\dot{y}^{F_{x}}{(F_{x})}) - \\int (e^{F_{x}})^{F_{x}} dF_{x} = \\frac{d}{d F_{x}} (- F_{x} + (e^{F_{x}})^{F_{x}}) - \\int (e^{F_{x}})^{F_{x}} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["minus", 2, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["minus", 4, "Integral(Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))), Add(Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))))"]]}, {"prompt": "Given L{(\\hat{p},\\omega)} = \\hat{p} \\omega, then obtain \\frac{\\omega L{(\\hat{p},\\omega)}}{\\hat{p}} = \\omega^{2}", "derivation": "L{(\\hat{p},\\omega)} = \\hat{p} \\omega and \\frac{L{(\\hat{p},\\omega)}}{\\hat{p}} = \\omega and \\omega L{(\\hat{p},\\omega)} = \\hat{p} \\omega^{2} and \\frac{\\omega L{(\\hat{p},\\omega)}}{\\hat{p}} = \\omega^{2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))"], [["times", 2, "Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["divide", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Symbol('\\\\omega', commutative=True), Integer(2)))"]]}, {"prompt": "Given x{(\\chi,u)} = \\chi \\sin{(u)} and \\operatorname{y^{\\prime}}{(u)} = \\sin{(u)}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\frac{\\chi \\operatorname{y^{\\prime}}{(u)}}{\\sin{(u)}} + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi = \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\chi + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi", "derivation": "x{(\\chi,u)} = \\chi \\sin{(u)} and \\frac{x{(\\chi,u)}}{\\sin{(u)}} = \\chi and \\operatorname{y^{\\prime}}{(u)} = \\sin{(u)} and x{(\\chi,u)} = \\chi \\operatorname{y^{\\prime}}{(u)} and \\frac{\\chi \\operatorname{y^{\\prime}}{(u)}}{\\sin{(u)}} = \\chi and \\frac{\\chi \\operatorname{y^{\\prime}}{(u)}}{\\sin{(u)}} + \\cos^{v_{t}}{(\\mathbb{I})} = \\chi + \\cos^{v_{t}}{(\\mathbb{I})} and \\int (\\frac{\\chi \\operatorname{y^{\\prime}}{(u)}}{\\sin{(u)}} + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi = \\int (\\chi + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi and \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\frac{\\chi \\operatorname{y^{\\prime}}{(u)}}{\\sin{(u)}} + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi = \\frac{\\partial}{\\partial \\mathbb{I}} \\int (\\chi + \\cos^{v_{t}}{(\\mathbb{I})}) d\\chi", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), sin(Symbol('u', commutative=True))))"], [["divide", 1, "sin(Symbol('u', commutative=True))"], "Equality(Mul(Function('x')(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('x')(Symbol('\\\\chi', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Function('y^{\\\\prime}')(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Symbol('\\\\chi', commutative=True))"], [["add", 5, "Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))))"], [["integrate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\chi', commutative=True), Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Symbol('\\\\chi', commutative=True), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Symbol('\\\\chi', commutative=True), Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\chi', commutative=True), Pow(cos(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(t,G)} = \\frac{G}{t} and m{(t,G)} = \\log{(\\mu_{0}{(t,G)})}, then obtain 0 = - m{(t,G)} + \\log{(\\frac{G}{t})}", "derivation": "\\mu_{0}{(t,G)} = \\frac{G}{t} and \\log{(\\mu_{0}{(t,G)})} = \\log{(\\frac{G}{t})} and 0 = \\log{(\\frac{G}{t})} - \\log{(\\mu_{0}{(t,G)})} and m{(t,G)} = \\log{(\\mu_{0}{(t,G)})} and 0 = - m{(t,G)} + \\log{(\\frac{G}{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["log", 1], "Equality(log(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('G', commutative=True))), log(Mul(Symbol('G', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))))"], [["minus", 2, "log(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('G', commutative=True)))"], "Equality(Integer(0), Add(log(Mul(Symbol('G', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))), Mul(Integer(-1), log(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('G', commutative=True))))))"], ["renaming_premise", "Equality(Function('m')(Symbol('t', commutative=True), Symbol('G', commutative=True)), log(Function('\\\\mu_0')(Symbol('t', commutative=True), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m')(Symbol('t', commutative=True), Symbol('G', commutative=True))), log(Mul(Symbol('G', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\hat{p}{(n_{2},\\varphi)} = n_{2}^{\\varphi}, then derive 0 = \\frac{\\varphi n_{2}^{\\varphi}}{n_{2}} - \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)}, then obtain 0 = \\frac{\\varphi \\hat{p}{(n_{2},\\varphi)}}{n_{2}} - \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)}", "derivation": "\\hat{p}{(n_{2},\\varphi)} = n_{2}^{\\varphi} and \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)} = \\frac{\\partial}{\\partial n_{2}} n_{2}^{\\varphi} and 0 = \\frac{\\partial}{\\partial n_{2}} n_{2}^{\\varphi} - \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)} and 0 = \\frac{\\varphi n_{2}^{\\varphi}}{n_{2}} - \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)} and 0 = \\frac{\\varphi n_{2}^{\\varphi}}{n_{2}} - \\frac{\\partial}{\\partial n_{2}} n_{2}^{\\varphi} and 0 = \\frac{\\varphi \\hat{p}{(n_{2},\\varphi)}}{n_{2}} - \\frac{\\partial}{\\partial n_{2}} \\hat{p}{(n_{2},\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hat{p}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\rho_f,F_{H})} = - F_{H} + e^{\\rho_f}, then obtain F_{H} - e^{\\rho_f} + \\frac{1}{\\hat{H}_{\\lambda}{(\\rho_f,F_{H})}} = F_{H} + \\frac{- F_{H} + e^{\\rho_f}}{\\hat{H}_{\\lambda}^{2}{(\\rho_f,F_{H})}} - e^{\\rho_f}", "derivation": "\\hat{H}_{\\lambda}{(\\rho_f,F_{H})} = - F_{H} + e^{\\rho_f} and 1 = \\frac{- F_{H} + e^{\\rho_f}}{\\hat{H}_{\\lambda}{(\\rho_f,F_{H})}} and \\frac{1}{\\hat{H}_{\\lambda}{(\\rho_f,F_{H})}} = \\frac{- F_{H} + e^{\\rho_f}}{\\hat{H}_{\\lambda}^{2}{(\\rho_f,F_{H})}} and F_{H} - e^{\\rho_f} + \\frac{1}{\\hat{H}_{\\lambda}{(\\rho_f,F_{H})}} = F_{H} + \\frac{- F_{H} + e^{\\rho_f}}{\\hat{H}_{\\lambda}^{2}{(\\rho_f,F_{H})}} - e^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Integer(-1))))"], [["divide", 2, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Mul(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Integer(-2))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), exp(Symbol('\\\\rho_f', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Integer(-1))), Add(Symbol('F_H', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), exp(Symbol('\\\\rho_f', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Integer(-2))), Mul(Integer(-1), exp(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given I{(n_{1},\\phi,u)} = u (\\phi + n_{1}) and \\operatorname{V_{\\mathbf{B}}}{(C_{2})} = \\cos{(C_{2})}, then obtain - u (\\phi + n_{1}) + \\operatorname{V_{\\mathbf{B}}}{(C_{2})} - \\cos{(C_{2})} = - u (\\phi + n_{1})", "derivation": "I{(n_{1},\\phi,u)} = u (\\phi + n_{1}) and \\operatorname{V_{\\mathbf{B}}}{(C_{2})} = \\cos{(C_{2})} and - I{(n_{1},\\phi,u)} + \\operatorname{V_{\\mathbf{B}}}{(C_{2})} = - I{(n_{1},\\phi,u)} + \\cos{(C_{2})} and - I{(n_{1},\\phi,u)} + \\operatorname{V_{\\mathbf{B}}}{(C_{2})} - \\cos{(C_{2})} = - I{(n_{1},\\phi,u)} and - u (\\phi + n_{1}) + \\operatorname{V_{\\mathbf{B}}}{(C_{2})} - \\cos{(C_{2})} = - u (\\phi + n_{1})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('u', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('n_1', commutative=True))))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["minus", 2, "Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('C_2', commutative=True))), Add(Mul(Integer(-1), Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))), cos(Symbol('C_2', commutative=True))))"], [["minus", 3, "cos(Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Function('I')(Symbol('n_1', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('n_1', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('C_2', commutative=True)), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given y{(t_{2},y^{\\prime},\\tilde{g}^*)} = \\tilde{g}^* y^{\\prime} + t_{2}, then obtain (\\iint y{(t_{2},y^{\\prime},\\tilde{g}^*)} d\\tilde{g}^* dt_{2})^{\\tilde{g}^*} = (\\iint (\\tilde{g}^* y^{\\prime} + t_{2}) d\\tilde{g}^* dt_{2})^{\\tilde{g}^*}", "derivation": "y{(t_{2},y^{\\prime},\\tilde{g}^*)} = \\tilde{g}^* y^{\\prime} + t_{2} and \\int y{(t_{2},y^{\\prime},\\tilde{g}^*)} d\\tilde{g}^* = \\int (\\tilde{g}^* y^{\\prime} + t_{2}) d\\tilde{g}^* and \\iint y{(t_{2},y^{\\prime},\\tilde{g}^*)} d\\tilde{g}^* dt_{2} = \\iint (\\tilde{g}^* y^{\\prime} + t_{2}) d\\tilde{g}^* dt_{2} and (\\iint y{(t_{2},y^{\\prime},\\tilde{g}^*)} d\\tilde{g}^* dt_{2})^{\\tilde{g}^*} = (\\iint (\\tilde{g}^* y^{\\prime} + t_{2}) d\\tilde{g}^* dt_{2})^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('y')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('y')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["power", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Integral(Function('y')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(L)} = \\log{(L)}, then obtain \\sin{(\\int \\operatorname{r_{0}}^{L}{(L)} dL)} + \\int \\log{(L)}^{L} dL = \\sin{(\\int \\log{(L)}^{L} dL)} + \\int \\log{(L)}^{L} dL", "derivation": "\\operatorname{r_{0}}{(L)} = \\log{(L)} and \\operatorname{r_{0}}^{L}{(L)} = \\log{(L)}^{L} and \\int \\operatorname{r_{0}}^{L}{(L)} dL = \\int \\log{(L)}^{L} dL and \\sin{(\\int \\operatorname{r_{0}}^{L}{(L)} dL)} = \\sin{(\\int \\log{(L)}^{L} dL)} and \\sin{(\\int \\operatorname{r_{0}}^{L}{(L)} dL)} + \\int \\log{(L)}^{L} dL = \\sin{(\\int \\log{(L)}^{L} dL)} + \\int \\log{(L)}^{L} dL", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Pow(Function('r_0')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Pow(Function('r_0')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), sin(Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["add", 4, "Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))"], "Equality(Add(sin(Integral(Pow(Function('r_0')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(sin(Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integral(Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(Q)} = e^{Q} and \\Psi_{\\lambda}{(Q)} = \\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)}), then derive \\Psi_{\\lambda}{(Q)} = e^{Q} + 1, then obtain \\frac{\\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)})}{e^{Q} + 1} = \\frac{\\frac{d}{d Q} (Q + e^{Q})}{e^{Q} + 1}", "derivation": "\\operatorname{v_{2}}{(Q)} = e^{Q} and Q + \\operatorname{v_{2}}{(Q)} = Q + e^{Q} and \\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)}) = \\frac{d}{d Q} (Q + e^{Q}) and \\Psi_{\\lambda}{(Q)} = \\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)}) and \\Psi_{\\lambda}{(Q)} = \\frac{d}{d Q} (Q + e^{Q}) and \\Psi_{\\lambda}{(Q)} = e^{Q} + 1 and \\frac{\\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)})}{\\Psi_{\\lambda}{(Q)}} = \\frac{\\frac{d}{d Q} (Q + e^{Q})}{\\Psi_{\\lambda}{(Q)}} and \\frac{\\frac{d}{d Q} (Q + \\operatorname{v_{2}}{(Q)})}{e^{Q} + 1} = \\frac{\\frac{d}{d Q} (Q + e^{Q})}{e^{Q} + 1}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["add", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Function('v_2')(Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), exp(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Symbol('Q', commutative=True), Function('v_2')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True)), Derivative(Add(Symbol('Q', commutative=True), Function('v_2')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True)), Derivative(Add(Symbol('Q', commutative=True), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True)), Add(exp(Symbol('Q', commutative=True)), Integer(1)))"], [["divide", 3, "Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Add(Symbol('Q', commutative=True), Function('v_2')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Add(Symbol('Q', commutative=True), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Pow(Add(exp(Symbol('Q', commutative=True)), Integer(1)), Integer(-1)), Derivative(Add(Symbol('Q', commutative=True), Function('v_2')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Add(exp(Symbol('Q', commutative=True)), Integer(1)), Integer(-1)), Derivative(Add(Symbol('Q', commutative=True), exp(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\phi)} = \\cos{(\\phi)}, then derive \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi - 1 = \\frac{\\partial}{\\partial \\phi} (\\mathbf{J} + \\sin{(\\phi)}) - 1, then obtain - E_{n} + \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi - 1 = - E_{n} + \\frac{\\partial}{\\partial \\phi} (\\mathbf{J} + \\sin{(\\phi)}) - 1", "derivation": "\\mathbf{J}_P{(\\phi)} = \\cos{(\\phi)} and \\int \\mathbf{J}_P{(\\phi)} d\\phi = \\int \\cos{(\\phi)} d\\phi and \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi = \\frac{d}{d \\phi} \\int \\cos{(\\phi)} d\\phi and \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi - 1 = \\frac{d}{d \\phi} \\int \\cos{(\\phi)} d\\phi - 1 and \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi - 1 = \\frac{\\partial}{\\partial \\phi} (\\mathbf{J} + \\sin{(\\phi)}) - 1 and - E_{n} + \\frac{d}{d \\phi} \\int \\mathbf{J}_P{(\\phi)} d\\phi - 1 = - E_{n} + \\frac{\\partial}{\\partial \\phi} (\\mathbf{J} + \\sin{(\\phi)}) - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Derivative(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 5, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Derivative(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}{(\\omega,l)} = \\omega - l, then derive \\frac{\\int \\hat{x}{(\\omega,l)} d\\omega}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + x}{B}, then derive \\frac{\\frac{\\omega^{2}}{2} - \\omega l + i}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + x}{B}, then obtain \\frac{\\int (\\omega - l) d\\omega}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + i}{B}", "derivation": "\\hat{x}{(\\omega,l)} = \\omega - l and \\int \\hat{x}{(\\omega,l)} d\\omega = \\int (\\omega - l) d\\omega and \\frac{\\int \\hat{x}{(\\omega,l)} d\\omega}{B} = \\frac{\\int (\\omega - l) d\\omega}{B} and \\frac{\\int \\hat{x}{(\\omega,l)} d\\omega}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + x}{B} and \\frac{\\int (\\omega - l) d\\omega}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + x}{B} and \\frac{\\frac{\\omega^{2}}{2} - \\omega l + i}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + x}{B} and \\frac{\\int (\\omega - l) d\\omega}{B} = \\frac{\\frac{\\omega^{2}}{2} - \\omega l + i}{B}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Function('\\\\hat{x}')(Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Function('\\\\hat{x}')(Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Symbol('x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Symbol('i', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\omega', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('l', commutative=True)), Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mu{(I)} = \\sin{(I)}, then obtain \\mu{(I)} + \\int 0^{I} dI = \\mu{(I)} + \\int (- \\mu{(I)} + \\sin{(I)})^{I} dI", "derivation": "\\mu{(I)} = \\sin{(I)} and 0 = - \\mu{(I)} + \\sin{(I)} and 0^{I} = (- \\mu{(I)} + \\sin{(I)})^{I} and \\int 0^{I} dI = \\int (- \\mu{(I)} + \\sin{(I)})^{I} dI and \\mu{(I)} + \\int 0^{I} dI = \\mu{(I)} + \\int (- \\mu{(I)} + \\sin{(I)})^{I} dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["minus", 1, "Function('\\\\mu')(Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Integer(0), Symbol('I', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True)))"], "Equality(Add(Function('\\\\mu')(Symbol('I', commutative=True)), Integral(Pow(Integer(0), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(Function('\\\\mu')(Symbol('I', commutative=True)), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\rho_f,G)} = G \\rho_f, then derive \\frac{\\partial}{\\partial G} \\delta{(\\rho_f,G)} = \\rho_f, then obtain \\frac{\\partial}{\\partial G} G \\rho_f = \\rho_f", "derivation": "\\delta{(\\rho_f,G)} = G \\rho_f and \\rho_f + \\delta{(\\rho_f,G)} = G \\rho_f + \\rho_f and \\frac{\\partial}{\\partial G} (\\rho_f + \\delta{(\\rho_f,G)}) = \\frac{\\partial}{\\partial G} (G \\rho_f + \\rho_f) and \\frac{\\partial}{\\partial G} \\delta{(\\rho_f,G)} = \\rho_f and \\frac{\\partial}{\\partial G} G \\rho_f = \\rho_f", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["add", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('G', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\rho_f', commutative=True))"]]}, {"prompt": "Given \\rho{(\\rho_b,W,\\mathbf{g})} = W^{\\mathbf{g}} + \\rho_b, then obtain \\frac{\\partial^{2}}{\\partial \\rho_b^{2}} W^{- \\mathbf{g}} \\rho{(\\rho_b,W,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\rho_b^{2}} W^{- \\mathbf{g}} (W^{\\mathbf{g}} + \\rho_b)", "derivation": "\\rho{(\\rho_b,W,\\mathbf{g})} = W^{\\mathbf{g}} + \\rho_b and W^{- \\mathbf{g}} \\rho{(\\rho_b,W,\\mathbf{g})} = W^{- \\mathbf{g}} (W^{\\mathbf{g}} + \\rho_b) and \\frac{\\partial}{\\partial \\rho_b} W^{- \\mathbf{g}} \\rho{(\\rho_b,W,\\mathbf{g})} = \\frac{\\partial}{\\partial \\rho_b} W^{- \\mathbf{g}} (W^{\\mathbf{g}} + \\rho_b) and \\frac{\\partial^{2}}{\\partial \\rho_b^{2}} W^{- \\mathbf{g}} \\rho{(\\rho_b,W,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\rho_b^{2}} W^{- \\mathbf{g}} (W^{\\mathbf{g}} + \\rho_b)", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\rho_b', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 1, "Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\rho')(Symbol('\\\\rho_b', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Add(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\rho')(Symbol('\\\\rho_b', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Add(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\rho')(Symbol('\\\\rho_b', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Add(Pow(Symbol('W', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(J,f)} = J f and \\mathbf{r}{(J,f)} = J f, then derive \\int \\frac{\\partial}{\\partial f} \\operatorname{v_{2}}{(J,f)} dJ = \\frac{J^{2}}{2} + m_{s}, then obtain \\int \\frac{\\partial}{\\partial f} \\mathbf{r}{(J,f)} dJ = \\frac{J^{2}}{2} + m_{s}", "derivation": "\\operatorname{v_{2}}{(J,f)} = J f and \\frac{\\partial}{\\partial f} \\operatorname{v_{2}}{(J,f)} = \\frac{\\partial}{\\partial f} J f and \\int \\frac{\\partial}{\\partial f} \\operatorname{v_{2}}{(J,f)} dJ = \\int \\frac{\\partial}{\\partial f} J f dJ and \\int \\frac{\\partial}{\\partial f} \\operatorname{v_{2}}{(J,f)} dJ = \\frac{J^{2}}{2} + m_{s} and \\mathbf{r}{(J,f)} = J f and \\operatorname{v_{2}}{(J,f)} = \\mathbf{r}{(J,f)} and \\int \\frac{\\partial}{\\partial f} \\mathbf{r}{(J,f)} dJ = \\frac{J^{2}}{2} + m_{s}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('v_2')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Derivative(Mul(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('v_2')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('m_s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('v_2')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Integral(Derivative(Function('\\\\mathbf{r}')(Symbol('J', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(s,\\ddot{x})} = \\ddot{x} + s, then obtain \\frac{\\int \\operatorname{a^{\\dagger}}^{\\ddot{x}}{(s,\\ddot{x})} d\\ddot{x}}{\\ddot{x}} = \\frac{\\int (\\ddot{x} + s)^{\\ddot{x}} d\\ddot{x}}{\\ddot{x}}", "derivation": "\\operatorname{a^{\\dagger}}{(s,\\ddot{x})} = \\ddot{x} + s and \\operatorname{a^{\\dagger}}^{\\ddot{x}}{(s,\\ddot{x})} = (\\ddot{x} + s)^{\\ddot{x}} and \\int \\operatorname{a^{\\dagger}}^{\\ddot{x}}{(s,\\ddot{x})} d\\ddot{x} = \\int (\\ddot{x} + s)^{\\ddot{x}} d\\ddot{x} and \\frac{\\int \\operatorname{a^{\\dagger}}^{\\ddot{x}}{(s,\\ddot{x})} d\\ddot{x}}{\\ddot{x}} = \\frac{\\int (\\ddot{x} + s)^{\\ddot{x}} d\\ddot{x}}{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)))"], [["power", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Pow(Function('a^{\\\\dagger}')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Integral(Pow(Function('a^{\\\\dagger}')(Symbol('s', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Integral(Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(t_{1},c)} = \\frac{c}{t_{1}} and \\operatorname{M_{E}}{(\\mathbf{H},C)} = \\mathbf{H} \\cos{(C)}, then obtain (\\operatorname{M_{E}}{(\\mathbf{H},C)} + 1) (\\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc - 1) = (\\mathbf{H} \\cos{(C)} + 1) (\\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc - 1)", "derivation": "\\hat{H}_l{(t_{1},c)} = \\frac{c}{t_{1}} and \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} = 1 and \\operatorname{M_{E}}{(\\mathbf{H},C)} = \\mathbf{H} \\cos{(C)} and \\operatorname{M_{E}}{(\\mathbf{H},C)} + 1 = \\mathbf{H} \\cos{(C)} + 1 and \\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc = \\int 1 dc and \\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc - 1 = \\int 1 dc - 1 and (\\operatorname{M_{E}}{(\\mathbf{H},C)} + 1) (\\int 1 dc - 1) = (\\mathbf{H} \\cos{(C)} + 1) (\\int 1 dc - 1) and (\\operatorname{M_{E}}{(\\mathbf{H},C)} + 1) (\\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc - 1) = (\\mathbf{H} \\cos{(C)} + 1) (\\int \\frac{t_{1} \\hat{H}_l{(t_{1},c)}}{c} dc - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True)), Mul(Symbol('c', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('c', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True))), Integer(1))"], ["get_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('C', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('C', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('C', commutative=True))), Integer(1)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Integer(1), Tuple(Symbol('c', commutative=True))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Integral(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1)))"], [["times", 4, "Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1))"], "Equality(Mul(Add(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('C', commutative=True)), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('C', commutative=True))), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Add(Function('M_E')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('C', commutative=True)), Integer(1)), Add(Integral(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('C', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Symbol('t_1', commutative=True), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given Z{(\\phi_2,v)} = \\phi_2 - v and \\mathbf{f}{(\\phi_2,v)} = Z{(\\phi_2,v)} - 1, then obtain - v + \\mathbf{f}{(\\phi_2,v)} = \\phi_2 - 2 v - 1", "derivation": "Z{(\\phi_2,v)} = \\phi_2 - v and Z{(\\phi_2,v)} - 1 = \\phi_2 - v - 1 and \\mathbf{f}{(\\phi_2,v)} = Z{(\\phi_2,v)} - 1 and \\mathbf{f}{(\\phi_2,v)} = \\phi_2 - v - 1 and - v + \\mathbf{f}{(\\phi_2,v)} = \\phi_2 - 2 v - 1", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True)), Add(Function('Z')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(-1)))"], [["minus", 4, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\mathbf{f}')(Symbol('\\\\phi_2', commutative=True), Symbol('v', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(t)} = \\cos{(e^{t})}, then obtain e^{(\\operatorname{P_{e}}{(t)} + \\cos{(e^{t})})^{4}} = e^{16 \\cos^{4}{(e^{t})}}", "derivation": "\\operatorname{P_{e}}{(t)} = \\cos{(e^{t})} and \\operatorname{P_{e}}{(t)} + \\cos{(e^{t})} = 2 \\cos{(e^{t})} and (\\operatorname{P_{e}}{(t)} + \\cos{(e^{t})})^{2} = 4 \\cos^{2}{(e^{t})} and (\\operatorname{P_{e}}{(t)} + \\cos{(e^{t})})^{4} = 16 \\cos^{4}{(e^{t})} and e^{(\\operatorname{P_{e}}{(t)} + \\cos{(e^{t})})^{4}} = e^{16 \\cos^{4}{(e^{t})}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('t', commutative=True)), cos(exp(Symbol('t', commutative=True))))"], [["add", 1, "cos(exp(Symbol('t', commutative=True)))"], "Equality(Add(Function('P_e')(Symbol('t', commutative=True)), cos(exp(Symbol('t', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('t', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Add(Function('P_e')(Symbol('t', commutative=True)), cos(exp(Symbol('t', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(cos(exp(Symbol('t', commutative=True))), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Add(Function('P_e')(Symbol('t', commutative=True)), cos(exp(Symbol('t', commutative=True)))), Integer(4)), Mul(Integer(16), Pow(cos(exp(Symbol('t', commutative=True))), Integer(4))))"], [["exp", 4], "Equality(exp(Pow(Add(Function('P_e')(Symbol('t', commutative=True)), cos(exp(Symbol('t', commutative=True)))), Integer(4))), exp(Mul(Integer(16), Pow(cos(exp(Symbol('t', commutative=True))), Integer(4)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},\\delta)} = e^{- \\delta + a^{\\dagger}} and \\operatorname{r_{0}}{(\\delta)} = - \\delta, then obtain - 2 \\delta + a^{\\dagger} + e^{- \\delta + a^{\\dagger}} = - 2 \\delta + a^{\\dagger} + e^{a^{\\dagger} + \\operatorname{r_{0}}{(\\delta)}}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},\\delta)} = e^{- \\delta + a^{\\dagger}} and \\operatorname{r_{0}}{(\\delta)} = - \\delta and \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger},\\delta)} = e^{a^{\\dagger} + \\operatorname{r_{0}}{(\\delta)}} and e^{- \\delta + a^{\\dagger}} = e^{a^{\\dagger} + \\operatorname{r_{0}}{(\\delta)}} and - \\delta + e^{- \\delta + a^{\\dagger}} = - \\delta + e^{a^{\\dagger} + \\operatorname{r_{0}}{(\\delta)}} and - 2 \\delta + a^{\\dagger} + e^{- \\delta + a^{\\dagger}} = - 2 \\delta + a^{\\dagger} + e^{a^{\\dagger} + \\operatorname{r_{0}}{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), exp(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('r_0')(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), exp(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('r_0')(Symbol('\\\\delta', commutative=True)))))"], [["minus", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('r_0')(Symbol('\\\\delta', commutative=True))))))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True), exp(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('r_0')(Symbol('\\\\delta', commutative=True))))))"]]}, {"prompt": "Given \\pi{(E,G)} = \\frac{G}{E}, then derive E \\frac{\\partial}{\\partial E} \\pi{(E,G)} + \\pi{(E,G)} = 0, then obtain E \\frac{\\partial}{\\partial E} \\frac{G}{E} + \\frac{G}{E} = 0", "derivation": "\\pi{(E,G)} = \\frac{G}{E} and E \\pi{(E,G)} = G and \\frac{\\partial}{\\partial E} E \\pi{(E,G)} = \\frac{d}{d E} G and E \\frac{\\partial}{\\partial E} \\pi{(E,G)} + \\pi{(E,G)} = 0 and E \\frac{\\partial}{\\partial E} \\frac{G}{E} + \\frac{G}{E} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('E', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('G', commutative=True)))"], [["divide", 1, "Pow(Symbol('E', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('E', commutative=True), Function('\\\\pi')(Symbol('E', commutative=True), Symbol('G', commutative=True))), Symbol('G', commutative=True))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Symbol('E', commutative=True), Function('\\\\pi')(Symbol('E', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Symbol('G', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('E', commutative=True), Derivative(Function('\\\\pi')(Symbol('E', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Function('\\\\pi')(Symbol('E', commutative=True), Symbol('G', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('E', commutative=True), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('G', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('G', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(J,T)} = T^{J}, then obtain - J - \\operatorname{A_{y}}{(J,T)} + \\frac{\\operatorname{A_{y}}{(J,T)}}{T} = - J - \\operatorname{A_{y}}{(J,T)} + \\frac{T^{J}}{T}", "derivation": "\\operatorname{A_{y}}{(J,T)} = T^{J} and J + \\operatorname{A_{y}}{(J,T)} = J + T^{J} and \\frac{\\operatorname{A_{y}}{(J,T)}}{T} = \\frac{T^{J}}{T} and - J - T^{J} + \\frac{\\operatorname{A_{y}}{(J,T)}}{T} = - J - T^{J} + \\frac{T^{J}}{T} and - J - \\operatorname{A_{y}}{(J,T)} + \\frac{\\operatorname{A_{y}}{(J,T)}}{T} = - J - \\operatorname{A_{y}}{(J,T)} + \\frac{T^{J}}{T}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True)))"], [["add", 1, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True))), Add(Symbol('J', commutative=True), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True))))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True))))"], [["minus", 3, "Add(Symbol('J', commutative=True), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('J', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(P_{e})} = \\sin{(\\cos{(P_{e})})}, then derive \\frac{d}{d P_{e}} \\varepsilon{(P_{e})} = - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})}, then obtain - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} = \\frac{d}{d P_{e}} \\sin{(\\cos{(P_{e})})}", "derivation": "\\varepsilon{(P_{e})} = \\sin{(\\cos{(P_{e})})} and \\frac{d}{d P_{e}} \\varepsilon{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(\\cos{(P_{e})})} and \\frac{d}{d P_{e}} \\varepsilon{(P_{e})} = - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} and - \\sin{(P_{e})} \\cos{(\\cos{(P_{e})})} = \\frac{d}{d P_{e}} \\sin{(\\cos{(P_{e})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('P_e', commutative=True)), sin(cos(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), cos(cos(Symbol('P_e', commutative=True)))), Derivative(sin(cos(Symbol('P_e', commutative=True))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{B},H,M_{E})} = (- H + M_{E})^{\\mathbf{B}}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{H}{(\\mathbf{B},H,M_{E})} - 1 = (- H + M_{E})^{\\mathbf{B}} \\log{(- H + M_{E})} - 1, then obtain \\frac{\\partial}{\\partial \\mathbf{B}} (- H + M_{E})^{\\mathbf{B}} - 1 = (- H + M_{E})^{\\mathbf{B}} \\log{(- H + M_{E})} - 1", "derivation": "\\mathbf{H}{(\\mathbf{B},H,M_{E})} = (- H + M_{E})^{\\mathbf{B}} and - \\mathbf{B} + \\mathbf{H}{(\\mathbf{B},H,M_{E})} = - \\mathbf{B} + (- H + M_{E})^{\\mathbf{B}} and \\frac{\\partial}{\\partial \\mathbf{B}} (- \\mathbf{B} + \\mathbf{H}{(\\mathbf{B},H,M_{E})}) = \\frac{\\partial}{\\partial \\mathbf{B}} (- \\mathbf{B} + (- H + M_{E})^{\\mathbf{B}}) and \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{H}{(\\mathbf{B},H,M_{E})} - 1 = (- H + M_{E})^{\\mathbf{B}} \\log{(- H + M_{E})} - 1 and \\frac{\\partial}{\\partial \\mathbf{B}} (- H + M_{E})^{\\mathbf{B}} - 1 = (- H + M_{E})^{\\mathbf{B}} \\log{(- H + M_{E})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('H', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('M_E', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given z{(h)} = \\sin{(h)}, then obtain (2 z{(h)} + 1)^{h} = (2 \\sin{(h)} + 1)^{h}", "derivation": "z{(h)} = \\sin{(h)} and 2 z{(h)} = z{(h)} + \\sin{(h)} and z{(h)} + 1 = \\sin{(h)} + 1 and z{(h)} + \\sin{(h)} + 1 = 2 \\sin{(h)} + 1 and (z{(h)} + \\sin{(h)} + 1)^{h} = (2 \\sin{(h)} + 1)^{h} and (2 z{(h)} + 1)^{h} = (2 \\sin{(h)} + 1)^{h}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, "Function('z')(Symbol('h', commutative=True))"], "Equality(Mul(Integer(2), Function('z')(Symbol('h', commutative=True))), Add(Function('z')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('z')(Symbol('h', commutative=True)), Integer(1)), Add(sin(Symbol('h', commutative=True)), Integer(1)))"], [["add", 3, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('z')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)), Integer(1)), Add(Mul(Integer(2), sin(Symbol('h', commutative=True))), Integer(1)))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Function('z')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)), Integer(1)), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(2), sin(Symbol('h', commutative=True))), Integer(1)), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Add(Mul(Integer(2), Function('z')(Symbol('h', commutative=True))), Integer(1)), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(2), sin(Symbol('h', commutative=True))), Integer(1)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and G{(\\Psi)} = \\Psi, then derive \\int \\hat{x}{(\\mathbf{M})} d\\mathbf{M} = \\Psi + \\sin{(\\mathbf{M})}, then obtain \\iint \\cos{(\\mathbf{M})} d\\mathbf{M} dG{(\\Psi)} = \\int (\\Psi + \\sin{(\\mathbf{M})}) dG{(\\Psi)}", "derivation": "\\hat{x}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\int \\hat{x}{(\\mathbf{M})} d\\mathbf{M} = \\int \\cos{(\\mathbf{M})} d\\mathbf{M} and \\int \\hat{x}{(\\mathbf{M})} d\\mathbf{M} = \\Psi + \\sin{(\\mathbf{M})} and G{(\\Psi)} = \\Psi and \\iint \\hat{x}{(\\mathbf{M})} d\\mathbf{M} d\\Psi = \\int (\\Psi + \\sin{(\\mathbf{M})}) d\\Psi and \\iint \\cos{(\\mathbf{M})} d\\mathbf{M} d\\Psi = \\int (\\Psi + \\sin{(\\mathbf{M})}) d\\Psi and \\iint \\cos{(\\mathbf{M})} d\\mathbf{M} dG{(\\Psi)} = \\int (\\Psi + \\sin{(\\mathbf{M})}) dG{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integral(cos(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Function('G')(Symbol('\\\\Psi', commutative=True)))), Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Function('G')(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\operatorname{v_{t}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then derive \\int \\frac{\\operatorname{A_{y}}{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} = \\sigma_x + \\operatorname{Si}{(\\mathbf{v})}, then obtain \\int \\frac{\\operatorname{v_{t}}{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} = \\sigma_x + \\operatorname{Si}{(\\mathbf{v})}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\frac{\\operatorname{A_{y}}{(\\mathbf{v})}}{\\mathbf{v}} = \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}} and \\operatorname{v_{t}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\operatorname{A_{y}}{(\\mathbf{v})} = \\operatorname{v_{t}}{(\\mathbf{v})} and \\int \\frac{\\operatorname{A_{y}}{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} = \\int \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} and \\int \\frac{\\operatorname{A_{y}}{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} = \\sigma_x + \\operatorname{Si}{(\\mathbf{v})} and \\int \\frac{\\operatorname{v_{t}}{(\\mathbf{v})}}{\\mathbf{v}} d\\mathbf{v} = \\sigma_x + \\operatorname{Si}{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A_y')(Symbol('\\\\mathbf{v}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Si(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('v_t')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Si(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given I{(Q)} = \\frac{d}{d Q} \\log{(Q)}, then derive Q I{(Q)} + 1 = 2, then obtain Q \\frac{d}{d Q} \\log{(Q)} + \\log{(Q)} - \\int I{(Q)} dQ + 1 = \\log{(Q)} - \\int I{(Q)} dQ + 2", "derivation": "I{(Q)} = \\frac{d}{d Q} \\log{(Q)} and Q I{(Q)} = Q \\frac{d}{d Q} \\log{(Q)} and Q I{(Q)} + 1 = Q \\frac{d}{d Q} \\log{(Q)} + 1 and Q I{(Q)} + 1 = 2 and Q I{(Q)} + \\log{(Q)} + 1 = \\log{(Q)} + 2 and Q \\frac{d}{d Q} \\log{(Q)} + \\log{(Q)} + 1 = \\log{(Q)} + 2 and Q \\frac{d}{d Q} \\log{(Q)} + \\log{(Q)} - \\int I{(Q)} dQ + 1 = \\log{(Q)} - \\int I{(Q)} dQ + 2", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('Q', commutative=True)), Derivative(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('I')(Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), Derivative(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('I')(Symbol('Q', commutative=True))), Integer(1)), Add(Mul(Symbol('Q', commutative=True), Derivative(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('I')(Symbol('Q', commutative=True))), Integer(1)), Integer(2))"], [["add", 4, "log(Symbol('Q', commutative=True))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Function('I')(Symbol('Q', commutative=True))), log(Symbol('Q', commutative=True)), Integer(1)), Add(log(Symbol('Q', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('Q', commutative=True), Derivative(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), log(Symbol('Q', commutative=True)), Integer(1)), Add(log(Symbol('Q', commutative=True)), Integer(2)))"], [["minus", 6, "Integral(Function('I')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Derivative(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), log(Symbol('Q', commutative=True)), Mul(Integer(-1), Integral(Function('I')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(1)), Add(log(Symbol('Q', commutative=True)), Mul(Integer(-1), Integral(Function('I')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given Z{(n_{1},G)} = G^{n_{1}}, then obtain 3 G^{n_{1}} + Z{(n_{1},G)} = 2 G^{n_{1}} + 2 Z{(n_{1},G)}", "derivation": "Z{(n_{1},G)} = G^{n_{1}} and 2 Z{(n_{1},G)} = G^{n_{1}} + Z{(n_{1},G)} and G^{n_{1}} + 3 Z{(n_{1},G)} = 2 G^{n_{1}} + 2 Z{(n_{1},G)} and G^{n_{1}} + 3 Z{(n_{1},G)} = 3 G^{n_{1}} + Z{(n_{1},G)} and 3 G^{n_{1}} + Z{(n_{1},G)} = 2 G^{n_{1}} + 2 Z{(n_{1},G)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True)), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True))))"], [["add", 2, "Add(Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True)), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)))"], "Equality(Add(Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(3), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(2), Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True))), Mul(Integer(2), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(3), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(3), Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True))), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(3), Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True))), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('G', commutative=True), Symbol('n_1', commutative=True))), Mul(Integer(2), Function('Z')(Symbol('n_1', commutative=True), Symbol('G', commutative=True)))))"]]}, {"prompt": "Given U{(G)} = \\int \\sin{(G)} dG, then obtain (U{(G)} - \\sin{(G)}) (- \\sin{(G)} + \\int \\sin{(G)} dG) \\sin^{2}{(G)} = ((- \\sin{(G)} + \\int \\sin{(G)} dG)^{2}) \\sin^{2}{(G)}", "derivation": "U{(G)} = \\int \\sin{(G)} dG and U{(G)} - \\sin{(G)} = - \\sin{(G)} + \\int \\sin{(G)} dG and (U{(G)} - \\sin{(G)}) \\sin{(G)} = (- \\sin{(G)} + \\int \\sin{(G)} dG) \\sin{(G)} and (U{(G)} - \\sin{(G)}) (- \\sin{(G)} + \\int \\sin{(G)} dG) \\sin^{2}{(G)} = ((- \\sin{(G)} + \\int \\sin{(G)} dG)^{2}) \\sin^{2}{(G)}", "srepr_derivation": [["renaming_premise", "Equality(Function('U')(Symbol('G', commutative=True)), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["minus", 1, "sin(Symbol('G', commutative=True))"], "Equality(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["times", 2, "sin(Symbol('G', commutative=True))"], "Equality(Mul(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True))), Mul(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True))))"], [["times", 3, "Mul(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True)))"], "Equality(Mul(Add(Function('U')(Symbol('G', commutative=True)), Mul(Integer(-1), sin(Symbol('G', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Pow(sin(Symbol('G', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Integer(2)), Pow(sin(Symbol('G', commutative=True)), Integer(2))))"]]}, {"prompt": "Given r{(F_{c},\\dot{x})} = \\log{(\\frac{F_{c}}{\\dot{x}})}, then derive \\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})} = - \\frac{1}{\\dot{x}}, then obtain \\frac{1}{\\dot{x}^{2}} = - \\frac{\\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})}}{\\dot{x}}", "derivation": "r{(F_{c},\\dot{x})} = \\log{(\\frac{F_{c}}{\\dot{x}})} and \\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} \\log{(\\frac{F_{c}}{\\dot{x}})} and (\\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})})^{2} = \\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})} \\frac{\\partial}{\\partial \\dot{x}} \\log{(\\frac{F_{c}}{\\dot{x}})} and \\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})} = - \\frac{1}{\\dot{x}} and \\frac{1}{\\dot{x}^{2}} = - \\frac{\\frac{\\partial}{\\partial \\dot{x}} \\log{(\\frac{F_{c}}{\\dot{x}})}}{\\dot{x}} and \\frac{1}{\\dot{x}^{2}} = - \\frac{\\frac{\\partial}{\\partial \\dot{x}} r{(F_{c},\\dot{x})}}{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-2)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Derivative(log(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-2)), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Derivative(Function('r')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(Q,\\Psi_{nl})} = \\frac{Q}{\\Psi_{nl}} and \\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} = e^{\\mathbf{J}_M{(Q,\\Psi_{nl})}}, then obtain (\\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} e^{- \\frac{Q}{\\Psi_{nl}}})^{Q} = 1", "derivation": "\\mathbf{J}_M{(Q,\\Psi_{nl})} = \\frac{Q}{\\Psi_{nl}} and e^{\\mathbf{J}_M{(Q,\\Psi_{nl})}} = e^{\\frac{Q}{\\Psi_{nl}}} and \\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} = e^{\\mathbf{J}_M{(Q,\\Psi_{nl})}} and \\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} = e^{\\frac{Q}{\\Psi_{nl}}} and \\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} e^{- \\mathbf{J}_M{(Q,\\Psi_{nl})}} = e^{\\frac{Q}{\\Psi_{nl}}} e^{- \\mathbf{J}_M{(Q,\\Psi_{nl})}} and \\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} e^{- \\frac{Q}{\\Psi_{nl}}} = 1 and (\\operatorname{J_{\\varepsilon}}{(Q,\\Psi_{nl})} e^{- \\frac{Q}{\\Psi_{nl}}})^{Q} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), exp(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('J_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))))"], [["divide", 4, "exp(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))), Mul(exp(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), exp(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))), Integer(1))"], [["power", 6, "Symbol('Q', commutative=True)"], "Equality(Pow(Mul(Function('J_{\\\\varepsilon}')(Symbol('Q', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), exp(Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))), Symbol('Q', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\delta{(\\rho_f,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\rho_f and \\dot{z}{(\\rho_f)} = \\rho_f, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\dot{z}{(\\rho_f)}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\rho_f)", "derivation": "\\delta{(\\rho_f,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\rho_f and \\dot{z}{(\\rho_f)} = \\rho_f and - \\rho_f + \\delta{(\\rho_f,\\Psi_{\\lambda})} + \\dot{z}{(\\rho_f)} = \\delta{(\\rho_f,\\Psi_{\\lambda})} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- \\rho_f + \\delta{(\\rho_f,\\Psi_{\\lambda})} + \\dot{z}{(\\rho_f)}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\delta{(\\rho_f,\\Psi_{\\lambda})} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\dot{z}{(\\rho_f)}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\rho_f)", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\rho_f', commutative=True))), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\theta_2,\\chi)} = \\chi + \\theta_2, then obtain (\\chi + \\theta_2) \\operatorname{V_{\\mathbf{B}}}^{3}{(\\theta_2,\\chi)} = (\\chi + \\theta_2)^{4}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\theta_2,\\chi)} = \\chi + \\theta_2 and (\\chi + \\theta_2) \\operatorname{V_{\\mathbf{B}}}{(\\theta_2,\\chi)} = (\\chi + \\theta_2)^{2} and (\\chi + \\theta_2)^{2} \\operatorname{V_{\\mathbf{B}}}^{2}{(\\theta_2,\\chi)} = (\\chi + \\theta_2)^{4} and (\\chi + \\theta_2) \\operatorname{V_{\\mathbf{B}}}^{3}{(\\theta_2,\\chi)} = (\\chi + \\theta_2)^{2} \\operatorname{V_{\\mathbf{B}}}^{2}{(\\theta_2,\\chi)} and (\\chi + \\theta_2) \\operatorname{V_{\\mathbf{B}}}^{3}{(\\theta_2,\\chi)} = (\\chi + \\theta_2)^{4}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(3))), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(3))), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(4)))"]]}, {"prompt": "Given t{(r_{0},P_{g})} = (e^{P_{g}})^{r_{0}}, then obtain \\frac{\\partial^{2}}{\\partial r_{0}\\partial P_{g}} (t^{2}{(r_{0},P_{g})})^{P_{g}} = \\frac{\\partial^{2}}{\\partial r_{0}\\partial P_{g}} (t{(r_{0},P_{g})} (e^{P_{g}})^{r_{0}})^{P_{g}}", "derivation": "t{(r_{0},P_{g})} = (e^{P_{g}})^{r_{0}} and t^{2}{(r_{0},P_{g})} = t{(r_{0},P_{g})} (e^{P_{g}})^{r_{0}} and (t^{2}{(r_{0},P_{g})})^{P_{g}} = (t{(r_{0},P_{g})} (e^{P_{g}})^{r_{0}})^{P_{g}} and \\frac{\\partial}{\\partial P_{g}} (t^{2}{(r_{0},P_{g})})^{P_{g}} = \\frac{\\partial}{\\partial P_{g}} (t{(r_{0},P_{g})} (e^{P_{g}})^{r_{0}})^{P_{g}} and \\frac{\\partial^{2}}{\\partial r_{0}\\partial P_{g}} (t^{2}{(r_{0},P_{g})})^{P_{g}} = \\frac{\\partial^{2}}{\\partial r_{0}\\partial P_{g}} (t{(r_{0},P_{g})} (e^{P_{g}})^{r_{0}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('r_0', commutative=True)))"], [["times", 1, "Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True))"], "Equality(Pow(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Integer(2)), Mul(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('r_0', commutative=True))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Pow(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Integer(2)), Symbol('P_g', commutative=True)), Pow(Mul(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('r_0', commutative=True))), Symbol('P_g', commutative=True)))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Integer(2)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Pow(Mul(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('r_0', commutative=True))), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Integer(2)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Function('t')(Symbol('r_0', commutative=True), Symbol('P_g', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('r_0', commutative=True))), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\delta, then derive \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\delta + 1, then obtain \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta) + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} - 1 = \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta)", "derivation": "\\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\delta and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta) and \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta) and \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} = \\delta + 1 and \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta) = \\delta + 1 and \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta) + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mu_{0}{(\\delta,\\Psi_{\\lambda})} - 1 = \\delta + \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\delta)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(M,f,A_{1})} = (f^{A_{1}})^{M}, then obtain \\frac{\\partial}{\\partial A_{1}} (- 2 g_{\\varepsilon} + \\frac{\\varepsilon_{0}{(M,f,A_{1})}}{f}) = \\frac{\\partial}{\\partial A_{1}} (- 2 g_{\\varepsilon} + \\frac{(f^{A_{1}})^{M}}{f})", "derivation": "\\varepsilon_{0}{(M,f,A_{1})} = (f^{A_{1}})^{M} and \\frac{\\varepsilon_{0}{(M,f,A_{1})}}{f} = \\frac{(f^{A_{1}})^{M}}{f} and - g_{\\varepsilon} + \\frac{\\varepsilon_{0}{(M,f,A_{1})}}{f} = - g_{\\varepsilon} + \\frac{(f^{A_{1}})^{M}}{f} and - 2 g_{\\varepsilon} + \\frac{\\varepsilon_{0}{(M,f,A_{1})}}{f} = - 2 g_{\\varepsilon} + \\frac{(f^{A_{1}})^{M}}{f} and \\frac{\\partial}{\\partial A_{1}} (- 2 g_{\\varepsilon} + \\frac{\\varepsilon_{0}{(M,f,A_{1})}}{f}) = \\frac{\\partial}{\\partial A_{1}} (- 2 g_{\\varepsilon} + \\frac{(f^{A_{1}})^{M}}{f})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('M', commutative=True), Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Symbol('M', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('M', commutative=True), Symbol('f', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Symbol('M', commutative=True))))"], [["minus", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('M', commutative=True), Symbol('f', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Symbol('M', commutative=True)))))"], [["minus", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('M', commutative=True), Symbol('f', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Symbol('M', commutative=True)))))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('M', commutative=True), Symbol('f', commutative=True), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Pow(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Symbol('M', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(n_{1})} = \\cos{(n_{1})}, then derive \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} \\frac{z{(n_{1})}}{\\cos{(n_{1})}} dn_{1} = \\int 0 dn_{1}, then obtain \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} 1 dn_{1} = \\int 0 dn_{1}", "derivation": "z{(n_{1})} = \\cos{(n_{1})} and \\frac{z{(n_{1})}}{\\cos{(n_{1})}} = 1 and \\frac{d}{d n_{1}} \\frac{z{(n_{1})}}{\\cos{(n_{1})}} = \\frac{d}{d n_{1}} 1 and \\cos{(n_{1})} \\frac{d}{d n_{1}} \\frac{z{(n_{1})}}{\\cos{(n_{1})}} = \\cos{(n_{1})} \\frac{d}{d n_{1}} 1 and \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} \\frac{z{(n_{1})}}{\\cos{(n_{1})}} dn_{1} = \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} 1 dn_{1} and \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} \\frac{z{(n_{1})}}{\\cos{(n_{1})}} dn_{1} = \\int 0 dn_{1} and \\int \\cos{(n_{1})} \\frac{d}{d n_{1}} 1 dn_{1} = \\int 0 dn_{1}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["divide", 1, "cos(Symbol('n_1', commutative=True))"], "Equality(Mul(Function('z')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Mul(Function('z')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["divide", 3, "Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))"], "Equality(Mul(cos(Symbol('n_1', commutative=True)), Derivative(Mul(Function('z')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(cos(Symbol('n_1', commutative=True)), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('n_1', commutative=True)"], "Equality(Integral(Mul(cos(Symbol('n_1', commutative=True)), Derivative(Mul(Function('z')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(cos(Symbol('n_1', commutative=True)), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integral(Mul(cos(Symbol('n_1', commutative=True)), Derivative(Mul(Function('z')(Symbol('n_1', commutative=True)), Pow(cos(Symbol('n_1', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))), Integral(Integer(0), Tuple(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Mul(cos(Symbol('n_1', commutative=True)), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True))), Integral(Integer(0), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(g,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{g}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{A_{1}}{(g,\\mathbf{J}_M)} = \\frac{1}{g}, then obtain \\mathbf{r} (- \\mathbf{r} + \\cos{(\\log{(\\mathbf{r})})}) + (\\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{g})^{\\mathbf{J}_M} = \\mathbf{r} (- \\mathbf{r} + \\cos{(\\log{(\\mathbf{r})})}) + (\\frac{1}{g})^{\\mathbf{J}_M}", "derivation": "\\operatorname{A_{1}}{(g,\\mathbf{J}_M)} = \\frac{\\mathbf{J}_M}{g} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{A_{1}}{(g,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{g} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\operatorname{A_{1}}{(g,\\mathbf{J}_M)} = \\frac{1}{g} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{g} = \\frac{1}{g} and (\\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{g})^{\\mathbf{J}_M} = (\\frac{1}{g})^{\\mathbf{J}_M} and \\mathbf{r} (- \\mathbf{r} + \\cos{(\\log{(\\mathbf{r})})}) + (\\frac{\\partial}{\\partial \\mathbf{J}_M} \\frac{\\mathbf{J}_M}{g})^{\\mathbf{J}_M} = \\mathbf{r} (- \\mathbf{r} + \\cos{(\\log{(\\mathbf{r})})}) + (\\frac{1}{g})^{\\mathbf{J}_M}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Pow(Symbol('g', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Pow(Symbol('g', commutative=True), Integer(-1)))"], [["power", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 5, "Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), cos(log(Symbol('\\\\mathbf{r}', commutative=True)))))"], "Equality(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), cos(log(Symbol('\\\\mathbf{r}', commutative=True))))), Pow(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), cos(log(Symbol('\\\\mathbf{r}', commutative=True))))), Pow(Pow(Symbol('g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(a)} = \\cos{(a)}, then derive \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} = - \\sin{(a)}, then obtain (\\int \\frac{d}{d a} \\cos{(a)} \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} da)^{a} = (\\int \\frac{d}{d a} - \\sin{(a)} \\cos{(a)} da)^{a}", "derivation": "\\operatorname{y^{\\prime}}{(a)} = \\cos{(a)} and \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} = \\frac{d}{d a} \\cos{(a)} and \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} = - \\sin{(a)} and \\cos{(a)} \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} = - \\sin{(a)} \\cos{(a)} and \\frac{d}{d a} \\cos{(a)} \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} = \\frac{d}{d a} - \\sin{(a)} \\cos{(a)} and \\int \\frac{d}{d a} \\cos{(a)} \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} da = \\int \\frac{d}{d a} - \\sin{(a)} \\cos{(a)} da and (\\int \\frac{d}{d a} \\cos{(a)} \\frac{d}{d a} \\operatorname{y^{\\prime}}{(a)} da)^{a} = (\\int \\frac{d}{d a} - \\sin{(a)} \\cos{(a)} da)^{a}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('a', commutative=True))))"], [["times", 3, "cos(Symbol('a', commutative=True))"], "Equality(Mul(cos(Symbol('a', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(cos(Symbol('a', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('a', commutative=True)"], "Equality(Integral(Derivative(Mul(cos(Symbol('a', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Integral(Derivative(Mul(Integer(-1), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(Derivative(Mul(cos(Symbol('a', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(Derivative(Mul(Integer(-1), sin(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(V_{\\mathbf{B}},\\Psi)} = e^{V_{\\mathbf{B}} + \\Psi}, then obtain \\frac{\\partial}{\\partial \\Psi} \\frac{\\int \\tilde{g}^*^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\Psi)} d\\Psi}{\\int (e^{V_{\\mathbf{B}} + \\Psi})^{V_{\\mathbf{B}}} d\\Psi} = \\frac{d}{d \\Psi} 1", "derivation": "\\tilde{g}^*{(V_{\\mathbf{B}},\\Psi)} = e^{V_{\\mathbf{B}} + \\Psi} and \\tilde{g}^*^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\Psi)} = (e^{V_{\\mathbf{B}} + \\Psi})^{V_{\\mathbf{B}}} and \\int \\tilde{g}^*^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\Psi)} d\\Psi = \\int (e^{V_{\\mathbf{B}} + \\Psi})^{V_{\\mathbf{B}}} d\\Psi and \\frac{\\int \\tilde{g}^*^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\Psi)} d\\Psi}{\\int (e^{V_{\\mathbf{B}} + \\Psi})^{V_{\\mathbf{B}}} d\\Psi} = 1 and \\frac{\\partial}{\\partial \\Psi} \\frac{\\int \\tilde{g}^*^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\Psi)} d\\Psi}{\\int (e^{V_{\\mathbf{B}} + \\Psi})^{V_{\\mathbf{B}}} d\\Psi} = \\frac{d}{d \\Psi} 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True)), exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 3, "Integral(Pow(exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Pow(Integral(Pow(exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Pow(Integral(Pow(exp(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(t_{2})} = e^{t_{2}} and \\phi_{2}{(t_{2})} = e^{t_{2}}, then obtain \\int J{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2}", "derivation": "J{(t_{2})} = e^{t_{2}} and \\phi_{2}{(t_{2})} = e^{t_{2}} and \\phi_{2}{(t_{2})} = J{(t_{2})} and \\int \\phi_{2}{(t_{2})} dt_{2} = \\int J{(t_{2})} dt_{2} and \\int \\phi_{2}{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2} and \\int J{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\phi_2')(Symbol('t_2', commutative=True)), Function('J')(Symbol('t_2', commutative=True)))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Function('J')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Function('\\\\phi_2')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Function('J')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given H{(v_{x},\\ddot{x})} = \\int \\ddot{x} v_{x} dv_{x}, then obtain \\sin{(\\int H{(v_{x},\\ddot{x})} d\\ddot{x} - 1)} = \\sin{(\\iint \\ddot{x} v_{x} dv_{x} d\\ddot{x} - 1)}", "derivation": "H{(v_{x},\\ddot{x})} = \\int \\ddot{x} v_{x} dv_{x} and \\int H{(v_{x},\\ddot{x})} d\\ddot{x} = \\iint \\ddot{x} v_{x} dv_{x} d\\ddot{x} and \\int H{(v_{x},\\ddot{x})} d\\ddot{x} - 1 = \\iint \\ddot{x} v_{x} dv_{x} d\\ddot{x} - 1 and \\sin{(\\int H{(v_{x},\\ddot{x})} d\\ddot{x} - 1)} = \\sin{(\\iint \\ddot{x} v_{x} dv_{x} d\\ddot{x} - 1)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('H')(Symbol('v_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integral(Function('H')(Symbol('v_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-1)), Add(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-1)))"], [["sin", 3], "Equality(sin(Add(Integral(Function('H')(Symbol('v_x', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))), sin(Add(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\pi{(J_{\\varepsilon})} = J_{\\varepsilon}, then derive \\int \\pi{(J_{\\varepsilon})} dJ_{\\varepsilon} = E_{n} + \\frac{J_{\\varepsilon}^{2}}{2}, then obtain \\int \\pi{(J_{\\varepsilon})} d\\pi{(J_{\\varepsilon})} = E_{n} + \\frac{\\pi^{2}{(J_{\\varepsilon})}}{2}", "derivation": "\\pi{(J_{\\varepsilon})} = J_{\\varepsilon} and \\int \\pi{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int J_{\\varepsilon} dJ_{\\varepsilon} and \\int \\pi{(J_{\\varepsilon})} dJ_{\\varepsilon} = E_{n} + \\frac{J_{\\varepsilon}^{2}}{2} and \\int \\pi{(J_{\\varepsilon})} d\\pi{(J_{\\varepsilon})} = E_{n} + \\frac{\\pi^{2}{(J_{\\varepsilon})}}{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Symbol('J_{\\\\varepsilon}', commutative=True), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\tilde{g}{(P_{g},\\hat{H}_l)} = P_{g} + \\hat{H}_l and \\mathbf{s}{(P_{g})} = - P_{g}, then obtain P_{g} + \\mathbf{s}{(P_{g})} = 0", "derivation": "\\tilde{g}{(P_{g},\\hat{H}_l)} = P_{g} + \\hat{H}_l and - P_{g} - \\hat{H}_l + \\tilde{g}{(P_{g},\\hat{H}_l)} = 0 and \\mathbf{s}{(P_{g})} = - P_{g} and - \\hat{H}_l + \\mathbf{s}{(P_{g})} + \\tilde{g}{(P_{g},\\hat{H}_l)} = 0 and P_{g} + \\mathbf{s}{(P_{g})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Add(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\tilde{g}')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{s}')(Symbol('P_g', commutative=True)), Function('\\\\tilde{g}')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('P_g', commutative=True), Function('\\\\mathbf{s}')(Symbol('P_g', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(m,\\hat{x})} = \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m}, then derive \\operatorname{A_{1}}{(m,\\hat{x})} = - \\frac{\\hat{x}}{m^{2}}, then obtain \\frac{\\partial}{\\partial m} \\operatorname{A_{1}}{(m,\\hat{x})} = \\frac{\\partial}{\\partial m} (2 \\operatorname{A_{1}}{(m,\\hat{x})} - \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m})", "derivation": "\\operatorname{A_{1}}{(m,\\hat{x})} = \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m} and \\operatorname{A_{1}}{(m,\\hat{x})} = - \\frac{\\hat{x}}{m^{2}} and \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m} = - \\frac{\\hat{x}}{m^{2}} and \\operatorname{A_{1}}{(m,\\hat{x})} = - \\frac{\\hat{x}}{m^{2}} + \\operatorname{A_{1}}{(m,\\hat{x})} - \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m} and \\frac{\\partial}{\\partial m} \\operatorname{A_{1}}{(m,\\hat{x})} = \\frac{\\partial}{\\partial m} (- \\frac{\\hat{x}}{m^{2}} + \\operatorname{A_{1}}{(m,\\hat{x})} - \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m}) and \\frac{\\partial}{\\partial m} \\operatorname{A_{1}}{(m,\\hat{x})} = \\frac{\\partial}{\\partial m} (2 \\operatorname{A_{1}}{(m,\\hat{x})} - \\frac{\\partial}{\\partial m} \\frac{\\hat{x}}{m})", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-2))))"], [["minus", 3, "Add(Mul(Integer(-1), Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], "Equality(Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-2))), Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))))"], [["differentiate", 4, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-2))), Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Function('A_1')(Symbol('m', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(P_{e})} = \\log{(P_{e})}, then derive \\frac{d}{d P_{e}} \\mathbf{g}{(P_{e})} = \\frac{1}{P_{e}}, then obtain \\frac{d}{d P_{e}} \\log{(P_{e})} = \\frac{1}{P_{e}}", "derivation": "\\mathbf{g}{(P_{e})} = \\log{(P_{e})} and \\frac{d}{d P_{e}} \\mathbf{g}{(P_{e})} = \\frac{d}{d P_{e}} \\log{(P_{e})} and \\frac{d}{d P_{e}} \\mathbf{g}{(P_{e})} = \\frac{1}{P_{e}} and \\frac{d}{d P_{e}} \\log{(P_{e})} = \\frac{1}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Pow(Symbol('P_e', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Pow(Symbol('P_e', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{p}{(\\nabla)} = e^{\\nabla}, then derive \\int \\mathbf{p}{(\\nabla)} d\\nabla = g + e^{\\nabla}, then obtain g \\mathbf{p}{(\\nabla)} = (- \\mathbf{p}{(\\nabla)} + \\int e^{\\nabla} d\\nabla) \\mathbf{p}{(\\nabla)}", "derivation": "\\mathbf{p}{(\\nabla)} = e^{\\nabla} and \\int \\mathbf{p}{(\\nabla)} d\\nabla = \\int e^{\\nabla} d\\nabla and \\int \\mathbf{p}{(\\nabla)} d\\nabla = g + e^{\\nabla} and g + e^{\\nabla} = \\int e^{\\nabla} d\\nabla and g - \\mathbf{p}{(\\nabla)} + e^{\\nabla} = - \\mathbf{p}{(\\nabla)} + \\int e^{\\nabla} d\\nabla and (g - \\mathbf{p}{(\\nabla)} + e^{\\nabla}) e^{\\nabla} = (- \\mathbf{p}{(\\nabla)} + \\int e^{\\nabla} d\\nabla) e^{\\nabla} and g \\mathbf{p}{(\\nabla)} = (- \\mathbf{p}{(\\nabla)} + \\int e^{\\nabla} d\\nabla) \\mathbf{p}{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('g', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('g', commutative=True), exp(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["minus", 4, "Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["times", 5, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))), exp(Symbol('\\\\nabla', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Function('\\\\mathbf{p}')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda} - g)}, then obtain ((\\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - \\cos{(\\hat{H}_{\\lambda} - g)}) \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - 1)^{\\hat{H}_{\\lambda}} = (-1)^{\\hat{H}_{\\lambda}}", "derivation": "\\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda} - g)} and \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - \\cos{(\\hat{H}_{\\lambda} - g)} = 0 and (\\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - \\cos{(\\hat{H}_{\\lambda} - g)}) \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} = 0 and (\\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - \\cos{(\\hat{H}_{\\lambda} - g)}) \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - 1 = -1 and ((\\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - \\cos{(\\hat{H}_{\\lambda} - g)}) \\operatorname{a^{\\dagger}}{(g,\\hat{H}_{\\lambda})} - 1)^{\\hat{H}_{\\lambda}} = (-1)^{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))"], [["minus", 1, "cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))), Integer(0))"], [["times", 2, "Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))), Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(0))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))), Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1)), Integer(-1))"], [["power", 4, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Mul(Add(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))), Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given U{(\\dot{x})} = \\sin{(\\sin{(\\dot{x})})}, then obtain U{(\\dot{x})} \\sin{(\\dot{x})} \\int U{(\\dot{x})} d\\dot{x} = \\sin{(\\dot{x})} \\sin{(\\sin{(\\dot{x})})} \\int U{(\\dot{x})} d\\dot{x}", "derivation": "U{(\\dot{x})} = \\sin{(\\sin{(\\dot{x})})} and U{(\\dot{x})} \\sin{(\\dot{x})} = \\sin{(\\dot{x})} \\sin{(\\sin{(\\dot{x})})} and \\int U{(\\dot{x})} d\\dot{x} = \\int \\sin{(\\sin{(\\dot{x})})} d\\dot{x} and U{(\\dot{x})} \\sin{(\\dot{x})} \\int \\sin{(\\sin{(\\dot{x})})} d\\dot{x} = \\sin{(\\dot{x})} \\sin{(\\sin{(\\dot{x})})} \\int \\sin{(\\sin{(\\dot{x})})} d\\dot{x} and U{(\\dot{x})} \\sin{(\\dot{x})} \\int U{(\\dot{x})} d\\dot{x} = \\sin{(\\dot{x})} \\sin{(\\sin{(\\dot{x})})} \\int U{(\\dot{x})} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\dot{x}', commutative=True)), sin(sin(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('U')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), sin(sin(Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 2, "Integral(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Function('U')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)), Integral(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), sin(sin(Symbol('\\\\dot{x}', commutative=True))), Integral(sin(sin(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('U')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)), Integral(Function('U')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(sin(Symbol('\\\\dot{x}', commutative=True)), sin(sin(Symbol('\\\\dot{x}', commutative=True))), Integral(Function('U')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)} = \\dot{z} (C_{d} - \\Omega), then derive \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}} = \\frac{C_{d} - \\Omega}{C_{d}}, then obtain (\\frac{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}})^{\\Omega} = (\\frac{C_{d} - \\Omega}{C_{d}})^{\\Omega}", "derivation": "\\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)} = \\dot{z} (C_{d} - \\Omega) and \\frac{\\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}} = \\frac{\\dot{z} (C_{d} - \\Omega)}{C_{d}} and \\frac{\\partial}{\\partial \\dot{z}} \\frac{\\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}} = \\frac{\\partial}{\\partial \\dot{z}} \\frac{\\dot{z} (C_{d} - \\Omega)}{C_{d}} and \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}} = \\frac{C_{d} - \\Omega}{C_{d}} and (\\frac{\\frac{\\partial}{\\partial \\dot{z}} \\operatorname{F_{g}}{(C_{d},\\dot{z},\\Omega)}}{C_{d}})^{\\Omega} = (\\frac{C_{d} - \\Omega}{C_{d}})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('C_d', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))))"], [["divide", 1, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('F_g')(Symbol('C_d', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('F_g')(Symbol('C_d', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\dot{z}', commutative=True), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Function('F_g')(Symbol('C_d', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Derivative(Function('F_g')(Symbol('C_d', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given T{(\\theta)} = \\cos{(e^{\\theta})}, then obtain \\int \\frac{d}{d \\theta} - \\cos{(e^{\\theta})} d\\theta = \\int \\frac{d}{d \\theta} (- 2 T{(\\theta)} + \\cos{(e^{\\theta})}) d\\theta", "derivation": "T{(\\theta)} = \\cos{(e^{\\theta})} and 0 = - T{(\\theta)} + \\cos{(e^{\\theta})} and - \\cos{(e^{\\theta})} = - T{(\\theta)} and - T{(\\theta)} = - 2 T{(\\theta)} + \\cos{(e^{\\theta})} and - \\cos{(e^{\\theta})} = - 2 T{(\\theta)} + \\cos{(e^{\\theta})} and \\frac{d}{d \\theta} - \\cos{(e^{\\theta})} = \\frac{d}{d \\theta} (- 2 T{(\\theta)} + \\cos{(e^{\\theta})}) and \\int \\frac{d}{d \\theta} - \\cos{(e^{\\theta})} d\\theta = \\int \\frac{d}{d \\theta} (- 2 T{(\\theta)} + \\cos{(e^{\\theta})}) d\\theta", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\theta', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True))))"], [["minus", 1, "Function('T')(Symbol('\\\\theta', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["minus", 2, "cos(exp(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Integer(-1), cos(exp(Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Function('T')(Symbol('\\\\theta', commutative=True))))"], [["minus", 2, "Function('T')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(-1), Function('T')(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), cos(exp(Symbol('\\\\theta', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), cos(exp(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), cos(exp(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varepsilon_0)} = \\cos{(\\cos{(\\varepsilon_0)})}, then obtain (\\operatorname{A_{x}}^{\\varepsilon_0}{(\\varepsilon_0)})^{\\varepsilon_0} - \\cos{(\\cos{(\\varepsilon_0)})} = (\\cos^{\\varepsilon_0}{(\\cos{(\\varepsilon_0)})})^{\\varepsilon_0} - \\cos{(\\cos{(\\varepsilon_0)})}", "derivation": "\\operatorname{A_{x}}{(\\varepsilon_0)} = \\cos{(\\cos{(\\varepsilon_0)})} and \\operatorname{A_{x}}^{\\varepsilon_0}{(\\varepsilon_0)} = \\cos^{\\varepsilon_0}{(\\cos{(\\varepsilon_0)})} and (\\operatorname{A_{x}}^{\\varepsilon_0}{(\\varepsilon_0)})^{\\varepsilon_0} = (\\cos^{\\varepsilon_0}{(\\cos{(\\varepsilon_0)})})^{\\varepsilon_0} and (\\operatorname{A_{x}}^{\\varepsilon_0}{(\\varepsilon_0)})^{\\varepsilon_0} - \\cos{(\\cos{(\\varepsilon_0)})} = (\\cos^{\\varepsilon_0}{(\\cos{(\\varepsilon_0)})})^{\\varepsilon_0} - \\cos{(\\cos{(\\varepsilon_0)})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), cos(cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Pow(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(cos(cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 3, "cos(cos(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Add(Pow(Pow(Function('A_x')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\varepsilon_0', commutative=True))))), Add(Pow(Pow(cos(cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('\\\\varepsilon_0', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(\\hat{H},i)} = \\hat{H} - i and Q{(\\hat{H},i)} = \\int (\\hat{H} - i) d\\hat{H}, then obtain Q^{2}{(\\hat{H},i)} = Q{(\\hat{H},i)} \\int (\\hat{H} - i) d\\hat{H}", "derivation": "\\hat{x}{(\\hat{H},i)} = \\hat{H} - i and \\int \\hat{x}{(\\hat{H},i)} d\\hat{H} = \\int (\\hat{H} - i) d\\hat{H} and Q{(\\hat{H},i)} = \\int (\\hat{H} - i) d\\hat{H} and Q{(\\hat{H},i)} = \\int \\hat{x}{(\\hat{H},i)} d\\hat{H} and Q{(\\hat{H},i)} \\int \\hat{x}{(\\hat{H},i)} d\\hat{H} = Q{(\\hat{H},i)} \\int (\\hat{H} - i) d\\hat{H} and Q^{2}{(\\hat{H},i)} = Q{(\\hat{H},i)} \\int (\\hat{H} - i) d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 2, "Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integer(2)), Mul(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('i', commutative=True)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(f_{\\mathbf{v}},C_{d})} = \\sin{(C_{d} + f_{\\mathbf{v}})} and y{(k)} = \\cos{(e^{k})}, then obtain \\phi_{1}{(f_{\\mathbf{v}},C_{d})} + \\sin{(\\int (y{(k)} + 1)^{k} dk)} = \\phi_{1}{(f_{\\mathbf{v}},C_{d})} + \\sin{(\\int (\\cos{(e^{k})} + 1)^{k} dk)}", "derivation": "\\phi_{1}{(f_{\\mathbf{v}},C_{d})} = \\sin{(C_{d} + f_{\\mathbf{v}})} and y{(k)} = \\cos{(e^{k})} and y{(k)} + 1 = \\cos{(e^{k})} + 1 and (y{(k)} + 1)^{k} = (\\cos{(e^{k})} + 1)^{k} and \\int (y{(k)} + 1)^{k} dk = \\int (\\cos{(e^{k})} + 1)^{k} dk and \\sin{(\\int (y{(k)} + 1)^{k} dk)} = \\sin{(\\int (\\cos{(e^{k})} + 1)^{k} dk)} and \\sin{(C_{d} + f_{\\mathbf{v}})} + \\sin{(\\int (y{(k)} + 1)^{k} dk)} = \\sin{(C_{d} + f_{\\mathbf{v}})} + \\sin{(\\int (\\cos{(e^{k})} + 1)^{k} dk)} and \\phi_{1}{(f_{\\mathbf{v}},C_{d})} + \\sin{(\\int (y{(k)} + 1)^{k} dk)} = \\phi_{1}{(f_{\\mathbf{v}},C_{d})} + \\sin{(\\int (\\cos{(e^{k})} + 1)^{k} dk)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C_d', commutative=True)), sin(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], ["get_premise", "Equality(Function('y')(Symbol('k', commutative=True)), cos(exp(Symbol('k', commutative=True))))"], [["add", 2, 1], "Equality(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Add(cos(exp(Symbol('k', commutative=True))), Integer(1)))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Pow(Add(cos(exp(Symbol('k', commutative=True))), Integer(1)), Symbol('k', commutative=True)))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Pow(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Pow(Add(cos(exp(Symbol('k', commutative=True))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["sin", 5], "Equality(sin(Integral(Pow(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), sin(Integral(Pow(Add(cos(exp(Symbol('k', commutative=True))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))))"], [["add", 6, "sin(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(sin(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), sin(Integral(Pow(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Add(sin(Add(Symbol('C_d', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), sin(Integral(Pow(Add(cos(exp(Symbol('k', commutative=True))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Function('\\\\phi_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C_d', commutative=True)), sin(Integral(Pow(Add(Function('y')(Symbol('k', commutative=True)), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Add(Function('\\\\phi_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C_d', commutative=True)), sin(Integral(Pow(Add(cos(exp(Symbol('k', commutative=True))), Integer(1)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))))"]]}, {"prompt": "Given r{(L)} = \\log{(L)}, then obtain \\int L r{(L)} \\log{(L)} dL = \\int L \\log{(L)}^{2} dL", "derivation": "r{(L)} = \\log{(L)} and r{(L)} \\log{(L)} = \\log{(L)}^{2} and L r{(L)} \\log{(L)} = L \\log{(L)}^{2} and \\int L r{(L)} \\log{(L)} dL = \\int L \\log{(L)}^{2} dL", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["times", 1, "log(Symbol('L', commutative=True))"], "Equality(Mul(Function('r')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('r')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Pow(log(Symbol('L', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Symbol('L', commutative=True), Function('r')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Symbol('L', commutative=True), Pow(log(Symbol('L', commutative=True)), Integer(2))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(c_{0},\\varphi^*,\\rho)} = \\rho c_{0} + \\varphi^*, then obtain \\int e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} dc_{0} + \\int e^{\\operatorname{n_{2}}^{\\rho}{(c_{0},\\varphi^*,\\rho)}} dc_{0} = 2 \\int e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} dc_{0}", "derivation": "\\operatorname{n_{2}}{(c_{0},\\varphi^*,\\rho)} = \\rho c_{0} + \\varphi^* and \\operatorname{n_{2}}^{\\rho}{(c_{0},\\varphi^*,\\rho)} = (\\rho c_{0} + \\varphi^*)^{\\rho} and e^{\\operatorname{n_{2}}^{\\rho}{(c_{0},\\varphi^*,\\rho)}} = e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} and \\int e^{\\operatorname{n_{2}}^{\\rho}{(c_{0},\\varphi^*,\\rho)}} dc_{0} = \\int e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} dc_{0} and \\int e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} dc_{0} + \\int e^{\\operatorname{n_{2}}^{\\rho}{(c_{0},\\varphi^*,\\rho)}} dc_{0} = 2 \\int e^{(\\rho c_{0} + \\varphi^*)^{\\rho}} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), exp(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True))))"], [["integrate", 3, "Symbol('c_0', commutative=True)"], "Equality(Integral(exp(Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integral(exp(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True))))"], [["add", 4, "Integral(exp(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True)))"], "Equality(Add(Integral(exp(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True))), Integral(exp(Pow(Function('n_2')(Symbol('c_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True)))), Mul(Integer(2), Integral(exp(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(k)} = \\log{(k)}, then obtain 0^{k} \\log{(k)} - (- \\mathbf{D}{(k)} + \\log{(k)})^{k} = (- \\mathbf{D}{(k)} + \\log{(k)})^{k} \\log{(k)} - (- \\mathbf{D}{(k)} + \\log{(k)})^{k}", "derivation": "\\mathbf{D}{(k)} = \\log{(k)} and 0 = - \\mathbf{D}{(k)} + \\log{(k)} and 0^{k} = (- \\mathbf{D}{(k)} + \\log{(k)})^{k} and 0^{k} \\log{(k)} = (- \\mathbf{D}{(k)} + \\log{(k)})^{k} \\log{(k)} and 0^{k} \\log{(k)} - (- \\mathbf{D}{(k)} + \\log{(k)})^{k} = (- \\mathbf{D}{(k)} + \\log{(k)})^{k} \\log{(k)} - (- \\mathbf{D}{(k)} + \\log{(k)})^{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{D}')(Symbol('k', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Integer(0), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["times", 3, "log(Symbol('k', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True))"], "Equality(Add(Mul(Pow(Integer(0), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('k', commutative=True))), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given U{(\\mu_0,u)} = \\mu_0 u, then obtain \\frac{- \\mu_0 u + U{(\\mu_0,u)}}{2 U{(\\mu_0,u)}} = 0", "derivation": "U{(\\mu_0,u)} = \\mu_0 u and 0 = \\mu_0 u - U{(\\mu_0,u)} and -1 = \\mu_0 u - U{(\\mu_0,u)} - 1 and \\mu_0 u - U{(\\mu_0,u)} - 1 = 2 \\mu_0 u - 2 U{(\\mu_0,u)} - 1 and -1 = 2 \\mu_0 u - 2 U{(\\mu_0,u)} - 1 and - \\mu_0 u + U{(\\mu_0,u)} = 0 and - \\frac{- \\mu_0 u + U{(\\mu_0,u)}}{2 U{(\\mu_0,u)}} = 0 and - \\frac{(- \\mu_0 u + U{(\\mu_0,u)}) (2 \\mu_0 u - 2 U{(\\mu_0,u)} - 1)}{2 U{(\\mu_0,u)}} = 0 and \\frac{- \\mu_0 u + U{(\\mu_0,u)}}{2 U{(\\mu_0,u)}} = 0", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1)))"], [["add", 2, "Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(-1), Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1)))"], [["minus", 1, "Mul(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(0))"], [["divide", 6, "Mul(Integer(-1), Integer(2), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Pow(Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Integer(0))"], [["times", 7, "Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1))"], "Equality(Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Integer(-1)), Pow(Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True))), Pow(Function('U')(Symbol('\\\\mu_0', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} = - \\chi + \\omega g_{\\varepsilon}, then derive \\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} = g_{\\varepsilon}, then obtain \\frac{\\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} + 1}{\\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)}} = \\frac{g_{\\varepsilon} + 1}{\\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)}}", "derivation": "\\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} = - \\chi + \\omega g_{\\varepsilon} and \\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} = \\frac{\\partial}{\\partial \\omega} (- \\chi + \\omega g_{\\varepsilon}) and \\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} = g_{\\varepsilon} and \\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} + 1 = g_{\\varepsilon} + 1 and \\frac{\\frac{\\partial}{\\partial \\omega} \\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)} + 1}{\\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)}} = \\frac{g_{\\varepsilon} + 1}{\\phi_{1}{(\\omega,g_{\\varepsilon},\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))"], [["divide", 4, "Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Derivative(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)), Pow(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)), Pow(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(p)} = \\sin{(p)} and \\operatorname{L_{\\varepsilon}}{(p)} = \\sin{(p)}, then obtain 0 = \\frac{(- \\operatorname{L_{\\varepsilon}}{(p)} + \\sin{(p)}) \\operatorname{L_{\\varepsilon}}{(p)}}{- \\operatorname{F_{x}}{(p)} + \\operatorname{L_{\\varepsilon}}{(p)}}", "derivation": "\\operatorname{F_{x}}{(p)} = \\sin{(p)} and \\operatorname{L_{\\varepsilon}}{(p)} = \\sin{(p)} and - \\operatorname{F_{x}}{(p)} + \\operatorname{L_{\\varepsilon}}{(p)} = - \\operatorname{F_{x}}{(p)} + \\sin{(p)} and \\operatorname{F_{x}}{(p)} = \\operatorname{L_{\\varepsilon}}{(p)} and (- \\operatorname{F_{x}}{(p)} + \\operatorname{L_{\\varepsilon}}{(p)}) \\operatorname{L_{\\varepsilon}}{(p)} = (- \\operatorname{F_{x}}{(p)} + \\sin{(p)}) \\operatorname{L_{\\varepsilon}}{(p)} and 0 = (- \\operatorname{L_{\\varepsilon}}{(p)} + \\sin{(p)}) \\operatorname{L_{\\varepsilon}}{(p)} and 0 = \\frac{(- \\operatorname{L_{\\varepsilon}}{(p)} + \\sin{(p)}) \\operatorname{L_{\\varepsilon}}{(p)}}{- \\operatorname{F_{x}}{(p)} + \\operatorname{L_{\\varepsilon}}{(p)}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["minus", 2, "Function('F_x')(Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_x')(Symbol('p', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True)))"], [["times", 3, "Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), Mul(Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))))"], [["divide", 6, "Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))), sin(Symbol('p', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\psi,\\phi_1)} = - \\phi_1 + \\psi and \\operatorname{F_{H}}{(\\phi_1)} = - \\frac{\\phi_1^{2}}{2}, then derive \\int \\mathbf{E}{(\\psi,\\phi_1)} d\\phi_1 = - \\frac{\\phi_1^{2}}{2} + \\phi_1 \\psi + k, then obtain \\sin{(\\int (- \\phi_1 + \\psi) d\\phi_1)} = \\sin{(\\phi_1 \\psi + k + \\operatorname{F_{H}}{(\\phi_1)})}", "derivation": "\\mathbf{E}{(\\psi,\\phi_1)} = - \\phi_1 + \\psi and \\int \\mathbf{E}{(\\psi,\\phi_1)} d\\phi_1 = \\int (- \\phi_1 + \\psi) d\\phi_1 and \\int \\mathbf{E}{(\\psi,\\phi_1)} d\\phi_1 = - \\frac{\\phi_1^{2}}{2} + \\phi_1 \\psi + k and \\int (- \\phi_1 + \\psi) d\\phi_1 = - \\frac{\\phi_1^{2}}{2} + \\phi_1 \\psi + k and \\operatorname{F_{H}}{(\\phi_1)} = - \\frac{\\phi_1^{2}}{2} and \\int (- \\phi_1 + \\psi) d\\phi_1 = \\phi_1 \\psi + k + \\operatorname{F_{H}}{(\\phi_1)} and \\sin{(\\int (- \\phi_1 + \\psi) d\\phi_1)} = \\sin{(\\phi_1 \\psi + k + \\operatorname{F_{H}}{(\\phi_1)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_1', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('k', commutative=True), Function('F_H')(Symbol('\\\\phi_1', commutative=True))))"], [["sin", 6], "Equality(sin(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), sin(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('k', commutative=True), Function('F_H')(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\hat{X},q)} = \\frac{q}{\\hat{X}}, then obtain - \\int q \\int \\hat{\\mathbf{x}}{(\\hat{X},q)} d\\hat{X} dq = - \\int q \\int \\frac{q}{\\hat{X}} d\\hat{X} dq", "derivation": "\\hat{\\mathbf{x}}{(\\hat{X},q)} = \\frac{q}{\\hat{X}} and \\int \\hat{\\mathbf{x}}{(\\hat{X},q)} d\\hat{X} = \\int \\frac{q}{\\hat{X}} d\\hat{X} and q \\int \\hat{\\mathbf{x}}{(\\hat{X},q)} d\\hat{X} = q \\int \\frac{q}{\\hat{X}} d\\hat{X} and \\int q \\int \\hat{\\mathbf{x}}{(\\hat{X},q)} d\\hat{X} dq = \\int q \\int \\frac{q}{\\hat{X}} d\\hat{X} dq and - \\int q \\int \\hat{\\mathbf{x}}{(\\hat{X},q)} d\\hat{X} dq = - \\int q \\int \\frac{q}{\\hat{X}} d\\hat{X} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Mul(Symbol('q', commutative=True), Integral(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Symbol('q', commutative=True), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(Mul(Symbol('q', commutative=True), Integral(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('q', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Symbol('q', commutative=True), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('q', commutative=True), Integral(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(U,C)} = - U + \\cos{(C)} and \\operatorname{v_{t}}{(t_{1},\\mathbf{B})} = \\cos{(t_{1}^{\\mathbf{B}})}, then obtain \\frac{\\operatorname{v_{t}}{(t_{1},\\mathbf{B})}}{- U + \\cos{(C)}} = \\frac{\\cos{(t_{1}^{\\mathbf{B}})}}{- U + \\cos{(C)}}", "derivation": "\\ddot{x}{(U,C)} = - U + \\cos{(C)} and \\operatorname{v_{t}}{(t_{1},\\mathbf{B})} = \\cos{(t_{1}^{\\mathbf{B}})} and \\frac{\\operatorname{v_{t}}{(t_{1},\\mathbf{B})}}{\\ddot{x}{(U,C)}} = \\frac{\\cos{(t_{1}^{\\mathbf{B}})}}{\\ddot{x}{(U,C)}} and \\frac{\\operatorname{v_{t}}{(t_{1},\\mathbf{B})}}{- U + \\cos{(C)}} = \\frac{\\cos{(t_{1}^{\\mathbf{B}})}}{- U + \\cos{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('U', commutative=True), Symbol('C', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('C', commutative=True))))"], ["get_premise", "Equality(Function('v_t')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), cos(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Function('\\\\ddot{x}')(Symbol('U', commutative=True), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Function('\\\\ddot{x}')(Symbol('U', commutative=True), Symbol('C', commutative=True)), Integer(-1)), cos(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('C', commutative=True))), Integer(-1)), Function('v_t')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('C', commutative=True))), Integer(-1)), cos(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\lambda)} = \\int e^{\\lambda} d\\lambda, then derive \\mathbb{I}{(\\lambda)} e^{\\lambda} = (r + e^{\\lambda}) e^{\\lambda}, then derive \\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda} = \\frac{\\partial}{\\partial \\lambda} (I + e^{\\lambda}) e^{\\lambda}, then obtain (\\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda})^{2} = \\frac{\\partial}{\\partial \\lambda} (I + e^{\\lambda}) e^{\\lambda} \\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda}", "derivation": "\\mathbb{I}{(\\lambda)} = \\int e^{\\lambda} d\\lambda and \\mathbb{I}{(\\lambda)} e^{\\lambda} = e^{\\lambda} \\int e^{\\lambda} d\\lambda and \\frac{d}{d \\lambda} \\mathbb{I}{(\\lambda)} e^{\\lambda} = \\frac{d}{d \\lambda} e^{\\lambda} \\int e^{\\lambda} d\\lambda and \\mathbb{I}{(\\lambda)} e^{\\lambda} = (r + e^{\\lambda}) e^{\\lambda} and \\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda} = \\frac{d}{d \\lambda} e^{\\lambda} \\int e^{\\lambda} d\\lambda and \\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda} = \\frac{\\partial}{\\partial \\lambda} (I + e^{\\lambda}) e^{\\lambda} and (\\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda})^{2} = \\frac{\\partial}{\\partial \\lambda} (I + e^{\\lambda}) e^{\\lambda} \\frac{\\partial}{\\partial \\lambda} (r + e^{\\lambda}) e^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Mul(exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('I', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["times", 6, "Derivative(Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Add(Symbol('I', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('r', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(n_{1},F_{c})} = n_{1}^{F_{c}}, then obtain n_{1}^{F_{c}} = (n_{1} + n_{1}^{F_{c}} - \\hat{\\mathbf{x}}{(n_{1},F_{c})})^{F_{c}}", "derivation": "\\hat{\\mathbf{x}}{(n_{1},F_{c})} = n_{1}^{F_{c}} and 0 = n_{1}^{F_{c}} - \\hat{\\mathbf{x}}{(n_{1},F_{c})} and n_{1} = n_{1} + n_{1}^{F_{c}} - \\hat{\\mathbf{x}}{(n_{1},F_{c})} and n_{1}^{F_{c}} = (n_{1} + n_{1}^{F_{c}} - \\hat{\\mathbf{x}}{(n_{1},F_{c})})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)))"], [["minus", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)))))"], [["add", 2, "Symbol('n_1', commutative=True)"], "Equality(Symbol('n_1', commutative=True), Add(Symbol('n_1', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)))))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Add(Symbol('n_1', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(x^\\prime,F_{x})} = - F_{x} + x^\\prime and \\dot{x}{(x^\\prime,F_{x})} = - F_{x} + \\operatorname{f_{\\mathbf{p}}}{(x^\\prime,F_{x})}, then obtain \\frac{\\partial^{2}}{\\partial V\\partial x^\\prime} (- V + \\dot{x}{(x^\\prime,F_{x})}) = \\frac{\\partial^{2}}{\\partial V\\partial x^\\prime} (- 2 F_{x} - V + x^\\prime)", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(x^\\prime,F_{x})} = - F_{x} + x^\\prime and - F_{x} + \\operatorname{f_{\\mathbf{p}}}{(x^\\prime,F_{x})} = - 2 F_{x} + x^\\prime and \\dot{x}{(x^\\prime,F_{x})} = - F_{x} + \\operatorname{f_{\\mathbf{p}}}{(x^\\prime,F_{x})} and \\dot{x}{(x^\\prime,F_{x})} = - 2 F_{x} + x^\\prime and - V + \\dot{x}{(x^\\prime,F_{x})} = - 2 F_{x} - V + x^\\prime and \\frac{\\partial}{\\partial x^\\prime} (- V + \\dot{x}{(x^\\prime,F_{x})}) = \\frac{\\partial}{\\partial x^\\prime} (- 2 F_{x} - V + x^\\prime) and \\frac{\\partial^{2}}{\\partial V\\partial x^\\prime} (- V + \\dot{x}{(x^\\prime,F_{x})}) = \\frac{\\partial^{2}}{\\partial V\\partial x^\\prime} (- 2 F_{x} - V + x^\\prime)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 4, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(x^\\prime,\\varphi^*)} = \\varphi^* x^\\prime, then obtain (\\varphi^*)^{2} - \\mathbf{J}_f{(x^\\prime,\\varphi^*)} = (\\varphi^*)^{2} + \\varphi^* x^\\prime - 2 \\mathbf{J}_f{(x^\\prime,\\varphi^*)}", "derivation": "\\mathbf{J}_f{(x^\\prime,\\varphi^*)} = \\varphi^* x^\\prime and - (\\varphi^*)^{2} + \\mathbf{J}_f{(x^\\prime,\\varphi^*)} = - (\\varphi^*)^{2} + \\varphi^* x^\\prime and 0 = \\varphi^* x^\\prime - \\mathbf{J}_f{(x^\\prime,\\varphi^*)} and (\\varphi^*)^{2} - \\mathbf{J}_f{(x^\\prime,\\varphi^*)} = (\\varphi^*)^{2} + \\varphi^* x^\\prime - 2 \\mathbf{J}_f{(x^\\prime,\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2))), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} = \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})} and \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})}, then obtain - \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = - \\sin^{2}{(\\cos{(g^{\\prime}_{\\varepsilon})})}", "derivation": "\\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} = \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})} and \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})} = \\sin^{2}{(\\cos{(g^{\\prime}_{\\varepsilon})})} and \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})} and - \\mathbf{J}_f{(g^{\\prime}_{\\varepsilon})} \\sin{(\\cos{(g^{\\prime}_{\\varepsilon})})} = - \\sin^{2}{(\\cos{(g^{\\prime}_{\\varepsilon})})} and - \\eta^{\\prime}{(g^{\\prime}_{\\varepsilon})} = - \\sin^{2}{(\\cos{(g^{\\prime}_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Pow(sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Pow(sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(sin(cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\sigma_p,\\rho)} = \\frac{\\rho}{\\sigma_p}, then obtain (- \\operatorname{F_{x}}{(\\sigma_p,\\rho)} + \\frac{1}{\\sigma_p}) \\operatorname{F_{x}}{(\\sigma_p,\\rho)} = (- \\frac{\\rho}{\\sigma_p} + \\frac{1}{\\sigma_p}) \\operatorname{F_{x}}{(\\sigma_p,\\rho)}", "derivation": "\\operatorname{F_{x}}{(\\sigma_p,\\rho)} = \\frac{\\rho}{\\sigma_p} and \\operatorname{F_{x}}{(\\sigma_p,\\rho)} - \\frac{1}{\\sigma_p} = \\frac{\\rho}{\\sigma_p} - \\frac{1}{\\sigma_p} and - \\operatorname{F_{x}}{(\\sigma_p,\\rho)} + \\frac{1}{\\sigma_p} = - \\frac{\\rho}{\\sigma_p} + \\frac{1}{\\sigma_p} and (- \\operatorname{F_{x}}{(\\sigma_p,\\rho)} + \\frac{1}{\\sigma_p}) \\operatorname{F_{x}}{(\\sigma_p,\\rho)} = (- \\frac{\\rho}{\\sigma_p} + \\frac{1}{\\sigma_p}) \\operatorname{F_{x}}{(\\sigma_p,\\rho)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))"], "Equality(Add(Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))), Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], [["times", 3, "Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Function('F_x')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(n_{2})} = \\sin{(\\sin{(n_{2})})}, then obtain \\frac{d}{d n_{2}} 1 = \\frac{d}{d n_{2}} 0^{n_{2}}", "derivation": "\\operatorname{x^{{\\}'}}{(n_{2})} = \\sin{(\\sin{(n_{2})})} and \\operatorname{x^{{\\}'}}{(n_{2})} - \\sin{(\\sin{(n_{2})})} = 0 and (\\operatorname{x^{{\\}'}}{(n_{2})} - \\sin{(\\sin{(n_{2})})})^{n_{2}} = 0^{n_{2}} and \\frac{d}{d n_{2}} (\\operatorname{x^{{\\}'}}{(n_{2})} - \\sin{(\\sin{(n_{2})})})^{n_{2}} = \\frac{d}{d n_{2}} 0^{n_{2}} and \\frac{d}{d n_{2}} 1 = \\frac{d}{d n_{2}} (\\operatorname{x^{{\\}'}}{(n_{2})} - \\sin{(\\sin{(n_{2})})})^{n_{2}} and \\frac{d}{d n_{2}} 1 = \\frac{d}{d n_{2}} 0^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('n_2', commutative=True)), sin(sin(Symbol('n_2', commutative=True))))"], [["minus", 1, "sin(sin(Symbol('n_2', commutative=True)))"], "Equality(Add(Function('x^\\\\prime')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('n_2', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Add(Function('x^\\\\prime')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('n_2', commutative=True))))), Symbol('n_2', commutative=True)), Pow(Integer(0), Symbol('n_2', commutative=True)))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Pow(Add(Function('x^\\\\prime')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('n_2', commutative=True))))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Pow(Add(Function('x^\\\\prime')(Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('n_2', commutative=True))))), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integer(1), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(F_{g},L_{\\varepsilon})} = \\log{(- F_{g} + L_{\\varepsilon})}, then obtain \\int (- F_{g} + L_{\\varepsilon}) \\int M{(F_{g},L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon} = \\int (- F_{g} + L_{\\varepsilon}) \\int \\log{(- F_{g} + L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon}", "derivation": "M{(F_{g},L_{\\varepsilon})} = \\log{(- F_{g} + L_{\\varepsilon})} and \\int M{(F_{g},L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\log{(- F_{g} + L_{\\varepsilon})} dL_{\\varepsilon} and (- F_{g} + L_{\\varepsilon}) \\int M{(F_{g},L_{\\varepsilon})} dL_{\\varepsilon} = (- F_{g} + L_{\\varepsilon}) \\int \\log{(- F_{g} + L_{\\varepsilon})} dL_{\\varepsilon} and \\int (- F_{g} + L_{\\varepsilon}) \\int M{(F_{g},L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon} = \\int (- F_{g} + L_{\\varepsilon}) \\int \\log{(- F_{g} + L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('F_g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('M')(Symbol('F_g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(log(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('M')(Symbol('F_g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(log(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} = \\cos{(A_{y} - y^{\\prime})} and \\mathbf{J}_M{(y^{\\prime},A_{y})} = \\cos{(A_{y} - y^{\\prime})} + 1, then obtain 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 2 = \\mathbf{J}_M{(y^{\\prime},A_{y})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 1", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} = \\cos{(A_{y} - y^{\\prime})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 1 = \\cos{(A_{y} - y^{\\prime})} + 1 and \\mathbf{J}_M{(y^{\\prime},A_{y})} = \\cos{(A_{y} - y^{\\prime})} + 1 and \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 1 = \\mathbf{J}_M{(y^{\\prime},A_{y})} and 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 2 = \\mathbf{J}_M{(y^{\\prime},A_{y})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(y^{\\prime},A_{y})} + 1", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), cos(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 1, 1], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integer(1)), Add(cos(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Add(cos(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integer(1)), Function('\\\\mathbf{J}_M')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)))"], [["add", 4, "Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True))), Integer(2)), Add(Function('\\\\mathbf{J}_M')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\omega{(\\mathbf{v},\\hat{X})} = \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} and \\dot{\\mathbf{r}}{(\\mathbf{v},\\hat{X})} = \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{\\mathbf{r}}{(\\mathbf{v},\\hat{X})}", "derivation": "\\omega{(\\mathbf{v},\\hat{X})} = \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} and \\dot{\\mathbf{r}}{(\\mathbf{v},\\hat{X})} = \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} and \\frac{\\partial}{\\partial \\mathbf{v}} \\omega{(\\mathbf{v},\\hat{X})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} and \\frac{\\partial}{\\partial \\mathbf{v}} \\omega{(\\mathbf{v},\\hat{X})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{\\mathbf{r}}{(\\mathbf{v},\\hat{X})} and \\frac{\\partial}{\\partial \\mathbf{v}} \\sin{(\\frac{\\hat{X}}{\\mathbf{v}})} = \\frac{\\partial}{\\partial \\mathbf{v}} \\dot{\\mathbf{r}}{(\\mathbf{v},\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(c)} = \\cos{(\\log{(c)})} and \\nabla{(c)} = \\log{(c)}, then obtain (\\Psi_{nl}{(c)} + \\cos{(\\nabla{(c)})})^{2} - 2 \\cos{(\\nabla{(c)})} = 4 \\cos^{2}{(\\nabla{(c)})} - 2 \\cos{(\\nabla{(c)})}", "derivation": "\\Psi_{nl}{(c)} = \\cos{(\\log{(c)})} and \\Psi_{nl}{(c)} + \\cos{(\\log{(c)})} = 2 \\cos{(\\log{(c)})} and \\nabla{(c)} = \\log{(c)} and \\Psi_{nl}{(c)} + \\cos{(\\nabla{(c)})} = 2 \\cos{(\\nabla{(c)})} and (\\Psi_{nl}{(c)} + \\cos{(\\nabla{(c)})})^{2} = 4 \\cos^{2}{(\\nabla{(c)})} and (\\Psi_{nl}{(c)} + \\cos{(\\nabla{(c)})})^{2} - 2 \\cos{(\\nabla{(c)})} = 4 \\cos^{2}{(\\nabla{(c)})} - 2 \\cos{(\\nabla{(c)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True))))"], [["add", 1, "cos(log(Symbol('c', commutative=True)))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True)), cos(log(Symbol('c', commutative=True)))), Mul(Integer(2), cos(log(Symbol('c', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True)), cos(Function('\\\\nabla')(Symbol('c', commutative=True)))), Mul(Integer(2), cos(Function('\\\\nabla')(Symbol('c', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True)), cos(Function('\\\\nabla')(Symbol('c', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(cos(Function('\\\\nabla')(Symbol('c', commutative=True))), Integer(2))))"], [["minus", 5, "Mul(Integer(2), cos(Function('\\\\nabla')(Symbol('c', commutative=True))))"], "Equality(Add(Pow(Add(Function('\\\\Psi_{nl}')(Symbol('c', commutative=True)), cos(Function('\\\\nabla')(Symbol('c', commutative=True)))), Integer(2)), Mul(Integer(-1), Integer(2), cos(Function('\\\\nabla')(Symbol('c', commutative=True))))), Add(Mul(Integer(4), Pow(cos(Function('\\\\nabla')(Symbol('c', commutative=True))), Integer(2))), Mul(Integer(-1), Integer(2), cos(Function('\\\\nabla')(Symbol('c', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(V,q)} = V + q, then obtain (\\int (2 V + \\frac{\\partial}{\\partial q} (V + q) + \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)}) dq)^{2} = (\\int (2 V + 2 \\frac{\\partial}{\\partial q} (V + q)) dq)^{2}", "derivation": "\\hat{\\mathbf{r}}{(V,q)} = V + q and \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)} = \\frac{\\partial}{\\partial q} (V + q) and V + \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)} = V + \\frac{\\partial}{\\partial q} (V + q) and 2 V + \\frac{\\partial}{\\partial q} (V + q) + \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)} = 2 V + 2 \\frac{\\partial}{\\partial q} (V + q) and \\int (2 V + \\frac{\\partial}{\\partial q} (V + q) + \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)}) dq = \\int (2 V + 2 \\frac{\\partial}{\\partial q} (V + q)) dq and (\\int (2 V + \\frac{\\partial}{\\partial q} (V + q) + \\frac{\\partial}{\\partial q} \\hat{\\mathbf{r}}{(V,q)}) dq)^{2} = (\\int (2 V + 2 \\frac{\\partial}{\\partial q} (V + q)) dq)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Add(Symbol('V', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 2, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Symbol('V', commutative=True), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["add", 3, "Add(Symbol('V', commutative=True), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(2), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Integer(2), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Integer(2), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True))))"], [["power", 5, 2], "Equality(Pow(Integral(Add(Mul(Integer(2), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True))), Integer(2)), Pow(Integral(Add(Mul(Integer(2), Symbol('V', commutative=True)), Mul(Integer(2), Derivative(Add(Symbol('V', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(l)} = \\log{(\\log{(l)})} and \\operatorname{E_{\\lambda}}{(l)} = \\log{(\\log{(l)})}, then derive \\frac{\\partial}{\\partial l} (v_{2} + \\operatorname{E_{\\lambda}}{(l)}) = \\frac{\\partial}{\\partial l} (u + \\log{(\\log{(l)})}), then derive \\frac{d}{d l} \\operatorname{E_{\\lambda}}{(l)} = \\frac{1}{l \\log{(l)}}, then obtain \\frac{d}{d l} \\operatorname{m_{s}}{(l)} = \\frac{1}{l \\log{(l)}}", "derivation": "\\operatorname{m_{s}}{(l)} = \\log{(\\log{(l)})} and \\operatorname{E_{\\lambda}}{(l)} = \\log{(\\log{(l)})} and \\operatorname{m_{s}}{(l)} = \\operatorname{E_{\\lambda}}{(l)} and \\frac{d}{d l} \\operatorname{E_{\\lambda}}{(l)} = \\frac{d}{d l} \\log{(\\log{(l)})} and \\int \\frac{d}{d l} \\operatorname{E_{\\lambda}}{(l)} dl = \\int \\frac{d}{d l} \\log{(\\log{(l)})} dl and \\frac{d}{d l} \\int \\frac{d}{d l} \\operatorname{E_{\\lambda}}{(l)} dl = \\frac{d}{d l} \\int \\frac{d}{d l} \\log{(\\log{(l)})} dl and \\frac{\\partial}{\\partial l} (v_{2} + \\operatorname{E_{\\lambda}}{(l)}) = \\frac{\\partial}{\\partial l} (u + \\log{(\\log{(l)})}) and \\frac{d}{d l} \\operatorname{E_{\\lambda}}{(l)} = \\frac{1}{l \\log{(l)}} and \\frac{d}{d l} \\operatorname{m_{s}}{(l)} = \\frac{1}{l \\log{(l)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('m_s')(Symbol('l', commutative=True)), Function('E_{\\\\lambda}')(Symbol('l', commutative=True)))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Function('E_{\\\\lambda}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(log(log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))))"], [["differentiate", 5, "Symbol('l', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('E_{\\\\lambda}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(Derivative(log(log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Add(Symbol('v_2', commutative=True), Function('E_{\\\\lambda}')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Symbol('u', commutative=True), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(log(Symbol('l', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Derivative(Function('m_s')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(log(Symbol('l', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{p})} = \\sin{(\\mathbf{p})}, then obtain \\sin{((\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p})^{2})} = \\sin{((\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p}) (\\mathbf{p} \\sin{(\\mathbf{p})} + \\mathbf{p}))}", "derivation": "\\tilde{g}^*{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} = \\mathbf{p} \\sin{(\\mathbf{p})} and \\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p} = \\mathbf{p} \\sin{(\\mathbf{p})} + \\mathbf{p} and (\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p})^{2} = (\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p}) (\\mathbf{p} \\sin{(\\mathbf{p})} + \\mathbf{p}) and \\sin{((\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p})^{2})} = \\sin{((\\mathbf{p} \\tilde{g}^*{(\\mathbf{p})} + \\mathbf{p}) (\\mathbf{p} \\sin{(\\mathbf{p})} + \\mathbf{p}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 3, "Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Mul(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True))))"], [["sin", 4], "Equality(sin(Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), sin(Mul(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), sin(Symbol('\\\\mathbf{p}', commutative=True))), Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{s})} = \\log{(\\mathbf{s})} and \\operatorname{m_{s}}{(\\mathbf{s})} = \\mathbf{s} \\log{(\\mathbf{s})}, then obtain \\frac{\\operatorname{m_{s}}{(\\mathbf{s})}}{\\log{(\\mathbf{s})}} = \\mathbf{s}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{s})} = \\log{(\\mathbf{s})} and \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s})} = \\mathbf{s} \\log{(\\mathbf{s})} and \\operatorname{m_{s}}{(\\mathbf{s})} = \\mathbf{s} \\log{(\\mathbf{s})} and \\operatorname{m_{s}}{(\\mathbf{s})} = \\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s})} and \\frac{\\operatorname{m_{s}}{(\\mathbf{s})}}{\\log{(\\mathbf{s})}} = \\frac{\\mathbf{s} \\operatorname{x^{{\\}'}}{(\\mathbf{s})}}{\\log{(\\mathbf{s})}} and \\frac{\\operatorname{m_{s}}{(\\mathbf{s})}}{\\log{(\\mathbf{s})}} = \\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True)), log(Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('m_s')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 4, "log(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('x^\\\\prime')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('m_s')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True))"]]}, {"prompt": "Given q{(C_{1})} = \\log{(\\log{(C_{1})})} and \\operatorname{g_{\\varepsilon}}{(C_{1})} = \\log{(\\log{(C_{1})})}, then obtain (- \\log{(C_{1})} + \\log{(\\log{(C_{1})})} + 1)^{C_{1}} = (\\operatorname{g_{\\varepsilon}}{(C_{1})} - \\log{(C_{1})} + 1)^{C_{1}}", "derivation": "q{(C_{1})} = \\log{(\\log{(C_{1})})} and q{(C_{1})} - \\log{(C_{1})} = - \\log{(C_{1})} + \\log{(\\log{(C_{1})})} and q{(C_{1})} - \\log{(C_{1})} + 1 = - \\log{(C_{1})} + \\log{(\\log{(C_{1})})} + 1 and \\operatorname{g_{\\varepsilon}}{(C_{1})} = \\log{(\\log{(C_{1})})} and q{(C_{1})} - \\log{(C_{1})} + 1 = \\operatorname{g_{\\varepsilon}}{(C_{1})} - \\log{(C_{1})} + 1 and - \\log{(C_{1})} + \\log{(\\log{(C_{1})})} + 1 = \\operatorname{g_{\\varepsilon}}{(C_{1})} - \\log{(C_{1})} + 1 and (- \\log{(C_{1})} + \\log{(\\log{(C_{1})})} + 1)^{C_{1}} = (\\operatorname{g_{\\varepsilon}}{(C_{1})} - \\log{(C_{1})} + 1)^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('C_1', commutative=True)), log(log(Symbol('C_1', commutative=True))))"], [["minus", 1, "log(Symbol('C_1', commutative=True))"], "Equality(Add(Function('q')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('C_1', commutative=True))), log(log(Symbol('C_1', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('q')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Integer(1)), Add(Mul(Integer(-1), log(Symbol('C_1', commutative=True))), log(log(Symbol('C_1', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True)), log(log(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('q')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Integer(1)), Add(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), log(Symbol('C_1', commutative=True))), log(log(Symbol('C_1', commutative=True))), Integer(1)), Add(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Integer(1)))"], [["power", 6, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('C_1', commutative=True))), log(log(Symbol('C_1', commutative=True))), Integer(1)), Symbol('C_1', commutative=True)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Symbol('C_1', commutative=True))), Integer(1)), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(A_{y})} = \\sin{(\\sin{(A_{y})})}, then obtain A_{y} (- \\cos{(A_{y})} \\cos{(\\sin{(A_{y})})} + \\frac{d}{d A_{y}} \\hat{x}_0{(A_{y})}) + \\hat{x}_0{(A_{y})} - \\sin{(\\sin{(A_{y})})} = 0", "derivation": "\\hat{x}_0{(A_{y})} = \\sin{(\\sin{(A_{y})})} and \\hat{x}_0{(A_{y})} - \\sin{(\\sin{(A_{y})})} = 0 and A_{y} (\\hat{x}_0{(A_{y})} - \\sin{(\\sin{(A_{y})})}) = 0 and \\frac{d}{d A_{y}} A_{y} (\\hat{x}_0{(A_{y})} - \\sin{(\\sin{(A_{y})})}) = \\frac{d}{d A_{y}} 0 and A_{y} (- \\cos{(A_{y})} \\cos{(\\sin{(A_{y})})} + \\frac{d}{d A_{y}} \\hat{x}_0{(A_{y})}) + \\hat{x}_0{(A_{y})} - \\sin{(\\sin{(A_{y})})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), sin(sin(Symbol('A_y', commutative=True))))"], [["minus", 1, "sin(sin(Symbol('A_y', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('A_y', commutative=True))))), Integer(0))"], [["times", 2, "Symbol('A_y', commutative=True)"], "Equality(Mul(Symbol('A_y', commutative=True), Add(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('A_y', commutative=True)))))), Integer(0))"], [["differentiate", 3, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Symbol('A_y', commutative=True), Add(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('A_y', commutative=True)))))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('A_y', commutative=True), Add(Mul(Integer(-1), cos(Symbol('A_y', commutative=True)), cos(sin(Symbol('A_y', commutative=True)))), Derivative(Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))), Function('\\\\hat{x}_0')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('A_y', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(r_{0},\\rho)} = \\log{(\\rho^{r_{0}})}, then obtain (\\int \\rho^{r_{0}} \\operatorname{f_{E}}{(r_{0},\\rho)} dr_{0})^{\\rho} = (\\int \\rho^{r_{0}} \\log{(\\rho^{r_{0}})} dr_{0})^{\\rho}", "derivation": "\\operatorname{f_{E}}{(r_{0},\\rho)} = \\log{(\\rho^{r_{0}})} and \\rho^{r_{0}} \\operatorname{f_{E}}{(r_{0},\\rho)} = \\rho^{r_{0}} \\log{(\\rho^{r_{0}})} and \\int \\rho^{r_{0}} \\operatorname{f_{E}}{(r_{0},\\rho)} dr_{0} = \\int \\rho^{r_{0}} \\log{(\\rho^{r_{0}})} dr_{0} and (\\int \\rho^{r_{0}} \\operatorname{f_{E}}{(r_{0},\\rho)} dr_{0})^{\\rho} = (\\int \\rho^{r_{0}} \\log{(\\rho^{r_{0}})} dr_{0})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True)), log(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Function('f_E')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), log(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Function('f_E')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), log(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True))))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Function('f_E')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), log(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{P},H)} = H + \\cos{(\\mathbf{P})} and \\operatorname{A_{2}}{(\\mathbf{P},H)} = (- \\mathbf{P} + \\mu_{0}{(\\mathbf{P},H)})^{\\mathbf{P}}, then obtain \\operatorname{A_{2}}{(\\mathbf{P},H)} = (H - \\mathbf{P} + \\cos{(\\mathbf{P})})^{\\mathbf{P}}", "derivation": "\\mu_{0}{(\\mathbf{P},H)} = H + \\cos{(\\mathbf{P})} and - \\mathbf{P} + \\mu_{0}{(\\mathbf{P},H)} = H - \\mathbf{P} + \\cos{(\\mathbf{P})} and (- \\mathbf{P} + \\mu_{0}{(\\mathbf{P},H)})^{\\mathbf{P}} = (H - \\mathbf{P} + \\cos{(\\mathbf{P})})^{\\mathbf{P}} and \\operatorname{A_{2}}{(\\mathbf{P},H)} = (- \\mathbf{P} + \\mu_{0}{(\\mathbf{P},H)})^{\\mathbf{P}} and \\operatorname{A_{2}}{(\\mathbf{P},H)} = (H - \\mathbf{P} + \\cos{(\\mathbf{P})})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('A_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\theta_1,Q)} = \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q} and \\Psi{(\\theta_1,Q)} = \\frac{\\operatorname{E_{x}}{(\\theta_1,Q)}}{Q}, then derive \\Psi{(\\theta_1,Q)} = - \\frac{\\theta_1}{Q^{3}}, then obtain (- \\frac{\\theta_1}{Q^{3}})^{\\theta_1} = (\\frac{\\operatorname{E_{x}}{(\\theta_1,Q)}}{Q})^{\\theta_1}", "derivation": "\\operatorname{E_{x}}{(\\theta_1,Q)} = \\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q} and \\frac{\\operatorname{E_{x}}{(\\theta_1,Q)}}{Q} = \\frac{\\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q}}{Q} and \\Psi{(\\theta_1,Q)} = \\frac{\\operatorname{E_{x}}{(\\theta_1,Q)}}{Q} and \\Psi{(\\theta_1,Q)} = \\frac{\\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q}}{Q} and \\Psi{(\\theta_1,Q)} = - \\frac{\\theta_1}{Q^{3}} and - \\frac{\\theta_1}{Q^{3}} = \\frac{\\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q}}{Q} and (- \\frac{\\theta_1}{Q^{3}})^{\\theta_1} = (\\frac{\\frac{\\partial}{\\partial Q} \\frac{\\theta_1}{Q}}{Q})^{\\theta_1} and (- \\frac{\\theta_1}{Q^{3}})^{\\theta_1} = (\\frac{\\operatorname{E_{x}}{(\\theta_1,Q)}}{Q})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Psi')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Function('\\\\Psi')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-3)), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-3)), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-3)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-3)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(b,s)} = \\log{(\\frac{s}{b})}, then obtain 0 = - \\frac{\\int (- s + \\operatorname{v_{z}}{(b,s)}) db}{b} + \\frac{\\int (- s + \\log{(\\frac{s}{b})}) db}{b}", "derivation": "\\operatorname{v_{z}}{(b,s)} = \\log{(\\frac{s}{b})} and - s + \\operatorname{v_{z}}{(b,s)} = - s + \\log{(\\frac{s}{b})} and \\int (- s + \\operatorname{v_{z}}{(b,s)}) db = \\int (- s + \\log{(\\frac{s}{b})}) db and \\frac{\\int (- s + \\operatorname{v_{z}}{(b,s)}) db}{b} = \\frac{\\int (- s + \\log{(\\frac{s}{b})}) db}{b} and 0 = - \\frac{\\int (- s + \\operatorname{v_{z}}{(b,s)}) db}{b} + \\frac{\\int (- s + \\log{(\\frac{s}{b})}) db}{b}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["minus", 1, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('s', commutative=True)))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Tuple(Symbol('b', commutative=True))))"], [["times", 3, "Pow(Symbol('b', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Tuple(Symbol('b', commutative=True)))))"], [["minus", 4, "Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('v_z')(Symbol('b', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('b', commutative=True)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), log(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('s', commutative=True)))), Tuple(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\omega)} = \\log{(\\cos{(\\omega)})} and \\dot{z}{(\\omega)} = \\log{(\\cos{(\\omega)})}, then obtain \\log{(\\cos{(\\omega)})}^{\\omega} = \\dot{z}^{\\omega}{(\\omega)}", "derivation": "\\tilde{g}^*{(\\omega)} = \\log{(\\cos{(\\omega)})} and \\tilde{g}^*^{\\omega}{(\\omega)} = \\log{(\\cos{(\\omega)})}^{\\omega} and \\dot{z}{(\\omega)} = \\log{(\\cos{(\\omega)})} and \\tilde{g}^*^{\\omega}{(\\omega)} = \\dot{z}^{\\omega}{(\\omega)} and \\log{(\\cos{(\\omega)})}^{\\omega} = \\dot{z}^{\\omega}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), log(cos(Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(log(cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), log(cos(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(log(cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} = \\frac{H}{\\dot{\\mathbf{r}}} and t{(\\omega,A)} = A \\omega, then obtain - \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} + \\int t{(\\omega,A)} d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}} = - \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} + \\int A \\omega d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}}", "derivation": "\\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} = \\frac{H}{\\dot{\\mathbf{r}}} and t{(\\omega,A)} = A \\omega and \\int t{(\\omega,A)} d\\omega = \\int A \\omega d\\omega and - \\frac{H}{\\dot{\\mathbf{r}}} + \\int t{(\\omega,A)} d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}} = - \\frac{H}{\\dot{\\mathbf{r}}} + \\int A \\omega d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}} and - \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} + \\int t{(\\omega,A)} d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}} = - \\operatorname{t_{2}}{(\\dot{\\mathbf{r}},H)} + \\int A \\omega d\\omega - 1 + \\frac{2}{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('t')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Add(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Integer(1), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Integral(Function('t')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('H', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True))), Integral(Function('t')(Symbol('\\\\omega', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('H', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{r})} = \\sin{(\\mathbf{r})}, then obtain \\frac{\\mathbf{p}{(\\mathbf{r})} + \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})}}{\\mathbf{p}{(\\mathbf{r})}} = \\frac{2 \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})}}{\\mathbf{p}{(\\mathbf{r})}}", "derivation": "\\mathbf{p}{(\\mathbf{r})} = \\sin{(\\mathbf{r})} and \\mathbf{p}{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})} = \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})} and \\mathbf{p}{(\\mathbf{r})} + \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})} = 2 \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})} and \\frac{\\mathbf{p}{(\\mathbf{r})} + \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})}}{\\mathbf{p}{(\\mathbf{r})}} = \\frac{2 \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\mathbf{p}{(\\mathbf{r})}}{\\mathbf{p}{(\\mathbf{r})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 1, "Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["add", 2, "sin(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Add(Mul(Integer(2), sin(Symbol('\\\\mathbf{r}', commutative=True))), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["divide", 3, "Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(2), sin(Symbol('\\\\mathbf{r}', commutative=True))), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\omega{(g_{\\varepsilon},W)} = e^{W - g_{\\varepsilon}}, then obtain (\\frac{d}{d W} 1)^{W} = (\\frac{\\partial}{\\partial W} e^{- W + g_{\\varepsilon}} e^{W - g_{\\varepsilon}})^{W}", "derivation": "\\omega{(g_{\\varepsilon},W)} = e^{W - g_{\\varepsilon}} and 1 = \\frac{e^{W - g_{\\varepsilon}}}{\\omega{(g_{\\varepsilon},W)}} and \\frac{d}{d W} 1 = \\frac{\\partial}{\\partial W} \\frac{e^{W - g_{\\varepsilon}}}{\\omega{(g_{\\varepsilon},W)}} and (\\frac{d}{d W} 1)^{W} = (\\frac{\\partial}{\\partial W} \\frac{e^{W - g_{\\varepsilon}}}{\\omega{(g_{\\varepsilon},W)}})^{W} and (\\frac{d}{d W} 1)^{W} = (\\frac{\\partial}{\\partial W} e^{- W + g_{\\varepsilon}} e^{W - g_{\\varepsilon}})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["divide", 1, "Function('\\\\omega')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\omega')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\omega')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Mul(Pow(Function('\\\\omega')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(n)} = \\sin{(n)} and \\operatorname{t_{1}}{(\\mathbf{J},y)} = \\int \\frac{\\mathbf{J}}{y} d\\mathbf{J}, then obtain \\frac{\\sin{(\\frac{d}{d n} \\varphi^{*}{(n)})}}{\\operatorname{t_{1}}{(\\mathbf{J},y)}} = \\frac{\\sin{(\\frac{d}{d n} \\sin{(n)})}}{\\operatorname{t_{1}}{(\\mathbf{J},y)}}", "derivation": "\\varphi^{*}{(n)} = \\sin{(n)} and \\operatorname{t_{1}}{(\\mathbf{J},y)} = \\int \\frac{\\mathbf{J}}{y} d\\mathbf{J} and \\frac{d}{d n} \\varphi^{*}{(n)} = \\frac{d}{d n} \\sin{(n)} and \\sin{(\\frac{d}{d n} \\varphi^{*}{(n)})} = \\sin{(\\frac{d}{d n} \\sin{(n)})} and \\frac{\\sin{(\\frac{d}{d n} \\varphi^{*}{(n)})}}{\\int \\frac{\\mathbf{J}}{y} d\\mathbf{J}} = \\frac{\\sin{(\\frac{d}{d n} \\sin{(n)})}}{\\int \\frac{\\mathbf{J}}{y} d\\mathbf{J}} and \\frac{\\sin{(\\frac{d}{d n} \\varphi^{*}{(n)})}}{\\operatorname{t_{1}}{(\\mathbf{J},y)}} = \\frac{\\sin{(\\frac{d}{d n} \\sin{(n)})}}{\\operatorname{t_{1}}{(\\mathbf{J},y)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], ["get_premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Function('\\\\varphi^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), sin(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["divide", 4, "Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(sin(Derivative(Function('\\\\varphi^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Mul(sin(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Integral(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), sin(Derivative(Function('\\\\varphi^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Mul(Pow(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), sin(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})} = - \\mathbf{J}_f + \\log{(f^{\\prime})} and \\hat{x}_0{(\\mathbf{J}_f,f^{\\prime})} = - \\mathbf{J}_f - f^{\\prime} + \\log{(f^{\\prime})}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} (- f^{\\prime} + \\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})}) = \\frac{\\partial}{\\partial f^{\\prime}} \\hat{x}_0{(\\mathbf{J}_f,f^{\\prime})}", "derivation": "\\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})} = - \\mathbf{J}_f + \\log{(f^{\\prime})} and - f^{\\prime} + \\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})} = - \\mathbf{J}_f - f^{\\prime} + \\log{(f^{\\prime})} and \\frac{\\partial}{\\partial f^{\\prime}} (- f^{\\prime} + \\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})}) = \\frac{\\partial}{\\partial f^{\\prime}} (- \\mathbf{J}_f - f^{\\prime} + \\log{(f^{\\prime})}) and \\hat{x}_0{(\\mathbf{J}_f,f^{\\prime})} = - \\mathbf{J}_f - f^{\\prime} + \\log{(f^{\\prime})} and \\frac{\\partial}{\\partial f^{\\prime}} (- f^{\\prime} + \\mathbf{s}{(\\mathbf{J}_f,f^{\\prime})}) = \\frac{\\partial}{\\partial f^{\\prime}} \\hat{x}_0{(\\mathbf{J}_f,f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(l,\\mathbf{H})} = \\log{(l)}^{\\mathbf{H}}, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} (v{(l,\\mathbf{H})} \\log{(l)})^{l} = \\frac{\\partial}{\\partial \\mathbf{H}} (\\log{(l)} \\log{(l)}^{\\mathbf{H}})^{l}", "derivation": "v{(l,\\mathbf{H})} = \\log{(l)}^{\\mathbf{H}} and v{(l,\\mathbf{H})} \\log{(l)} = \\log{(l)} \\log{(l)}^{\\mathbf{H}} and (v{(l,\\mathbf{H})} \\log{(l)})^{l} = (\\log{(l)} \\log{(l)}^{\\mathbf{H}})^{l} and \\frac{\\partial}{\\partial \\mathbf{H}} (v{(l,\\mathbf{H})} \\log{(l)})^{l} = \\frac{\\partial}{\\partial \\mathbf{H}} (\\log{(l)} \\log{(l)}^{\\mathbf{H}})^{l}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "log(Symbol('l', commutative=True))"], "Equality(Mul(Function('v')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('l', commutative=True))), Mul(log(Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(Function('v')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Mul(log(Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('l', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('v')(Symbol('l', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Pow(Mul(log(Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(z)} = \\cos{(z)} and \\operatorname{v_{2}}{(z)} = \\cos{(z)}, then obtain (u + (\\mathbf{A} + u) q{(z)}) \\bar{\\h}{(\\mathbf{A},u)} = (u + (\\mathbf{A} + u) \\cos{(z)}) \\bar{\\h}{(\\mathbf{A},u)}", "derivation": "q{(z)} = \\cos{(z)} and \\operatorname{v_{2}}{(z)} = \\cos{(z)} and (\\mathbf{A} + u) \\operatorname{v_{2}}{(z)} = (\\mathbf{A} + u) \\cos{(z)} and (\\mathbf{A} + u) \\operatorname{v_{2}}{(z)} = (\\mathbf{A} + u) q{(z)} and (\\mathbf{A} + u) q{(z)} = (\\mathbf{A} + u) \\cos{(z)} and u + (\\mathbf{A} + u) q{(z)} = u + (\\mathbf{A} + u) \\cos{(z)} and (u + (\\mathbf{A} + u) q{(z)}) \\bar{\\h}{(\\mathbf{A},u)} = (u + (\\mathbf{A} + u) \\cos{(z)}) \\bar{\\h}{(\\mathbf{A},u)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('v_2')(Symbol('z', commutative=True))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), cos(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('v_2')(Symbol('z', commutative=True))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('q')(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('q')(Symbol('z', commutative=True))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), cos(Symbol('z', commutative=True))))"], [["add", 5, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('q')(Symbol('z', commutative=True)))), Add(Symbol('u', commutative=True), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), cos(Symbol('z', commutative=True)))))"], [["times", 6, "Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Add(Symbol('u', commutative=True), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), Function('q')(Symbol('z', commutative=True)))), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True))), Mul(Add(Symbol('u', commutative=True), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True)), cos(Symbol('z', commutative=True)))), Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given L{(\\varepsilon_0)} = \\log{(\\varepsilon_0)}, then obtain \\frac{(L{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} + L{(\\varepsilon_0)} = \\frac{(\\log{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} + L{(\\varepsilon_0)}", "derivation": "L{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and L{(\\varepsilon_0)} + 1 = \\log{(\\varepsilon_0)} + 1 and (L{(\\varepsilon_0)} + 1) L{(\\varepsilon_0)} = (\\log{(\\varepsilon_0)} + 1) L{(\\varepsilon_0)} and \\frac{(L{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} = \\frac{(\\log{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} and \\frac{(L{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} + L{(\\varepsilon_0)} = \\frac{(\\log{(\\varepsilon_0)} + 1) L^{2}{(\\varepsilon_0)}}{\\log{(\\varepsilon_0)}} + L{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Add(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)))"], [["divide", 2, "Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Function('L')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Add(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Function('L')(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 3, "Mul(Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Add(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Mul(Add(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))))"], [["add", 4, "Function('L')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Mul(Add(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Function('L')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Add(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Pow(Function('L')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Function('L')(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and c{(\\mathbf{F})} = \\sin^{2}{(\\mathbf{F})}, then obtain \\frac{d}{d \\mathbf{F}} 1 = \\frac{d}{d \\mathbf{F}} \\frac{c{(\\mathbf{F})}}{\\sin^{2}{(\\mathbf{F})}}", "derivation": "\\tilde{g}^*{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and 1 = \\frac{\\sin{(\\mathbf{F})}}{\\tilde{g}^*{(\\mathbf{F})}} and \\sin{(\\mathbf{F})} = \\frac{\\sin^{2}{(\\mathbf{F})}}{\\tilde{g}^*{(\\mathbf{F})}} and 1 = \\frac{\\sin^{2}{(\\mathbf{F})}}{\\tilde{g}^*^{2}{(\\mathbf{F})}} and c{(\\mathbf{F})} = \\sin^{2}{(\\mathbf{F})} and 1 = \\frac{c{(\\mathbf{F})}}{\\tilde{g}^*^{2}{(\\mathbf{F})}} and 1 = \\frac{c{(\\mathbf{F})}}{\\sin^{2}{(\\mathbf{F})}} and \\frac{d}{d \\mathbf{F}} 1 = \\frac{d}{d \\mathbf{F}} \\frac{c{(\\mathbf{F})}}{\\sin^{2}{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 2, "sin(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(sin(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(1), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2)), Function('c')(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(1), Mul(Function('c')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2))))"], [["differentiate", 7, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Function('c')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega}, then derive \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega}, then derive \\int \\operatorname{n_{1}}{(\\Omega)} d\\Omega = L + \\operatorname{n_{1}}{(\\Omega)}, then obtain L + \\operatorname{n_{1}}{(\\Omega)} = \\mathbf{F} + \\operatorname{n_{1}}{(\\Omega)}", "derivation": "\\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega} and \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\Omega} and \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} = e^{\\Omega} and \\operatorname{n_{1}}{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} and \\int \\operatorname{n_{1}}{(\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} d\\Omega and \\int \\operatorname{n_{1}}{(\\Omega)} d\\Omega = L + \\operatorname{n_{1}}{(\\Omega)} and L + \\operatorname{n_{1}}{(\\Omega)} = \\int \\frac{d}{d \\Omega} \\operatorname{n_{1}}{(\\Omega)} d\\Omega and L + \\operatorname{n_{1}}{(\\Omega)} = \\mathbf{F} + \\operatorname{n_{1}}{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), exp(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('L', commutative=True), Function('n_1')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('L', commutative=True), Function('n_1')(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Function('n_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('L', commutative=True), Function('n_1')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Function('n_1')(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given l{(a)} = \\cos{(e^{a})}, then obtain \\frac{d^{2}}{d a^{2}} (l{(a)} - l^{a}{(a)}) = \\frac{d^{2}}{d a^{2}} (- l^{a}{(a)} + \\cos{(e^{a})})", "derivation": "l{(a)} = \\cos{(e^{a})} and l^{a}{(a)} = \\cos^{a}{(e^{a})} and l{(a)} - l^{a}{(a)} = - l^{a}{(a)} + \\cos{(e^{a})} and l{(a)} - \\cos^{a}{(e^{a})} = \\cos{(e^{a})} - \\cos^{a}{(e^{a})} and \\frac{d}{d a} (l{(a)} - \\cos^{a}{(e^{a})}) = \\frac{d}{d a} (\\cos{(e^{a})} - \\cos^{a}{(e^{a})}) and \\frac{d}{d a} (l{(a)} - l^{a}{(a)}) = \\frac{d}{d a} (- l^{a}{(a)} + \\cos{(e^{a})}) and \\frac{d^{2}}{d a^{2}} (l{(a)} - l^{a}{(a)}) = \\frac{d^{2}}{d a^{2}} (- l^{a}{(a)} + \\cos{(e^{a})})", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('a', commutative=True)), cos(exp(Symbol('a', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(cos(exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["minus", 1, "Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))"], "Equality(Add(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), cos(exp(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(cos(exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))), Add(cos(exp(Symbol('a', commutative=True))), Mul(Integer(-1), Pow(cos(exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(cos(exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(cos(exp(Symbol('a', commutative=True))), Mul(Integer(-1), Pow(cos(exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Add(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), cos(exp(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), cos(exp(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\hat{H}{(\\lambda)} = \\log{(\\lambda)} and \\mathbf{P}{(\\lambda)} = \\lambda, then obtain \\log{(\\lambda)} + \\frac{\\mathbf{P}{(\\lambda)}}{\\lambda + \\log{(\\lambda)}} = \\frac{\\lambda}{\\lambda + \\log{(\\lambda)}} + \\log{(\\lambda)}", "derivation": "\\hat{H}{(\\lambda)} = \\log{(\\lambda)} and \\mathbf{P}{(\\lambda)} = \\lambda and \\frac{\\mathbf{P}{(\\lambda)}}{\\lambda + \\hat{H}{(\\lambda)}} = \\frac{\\lambda}{\\lambda + \\hat{H}{(\\lambda)}} and \\hat{H}{(\\lambda)} + \\frac{\\mathbf{P}{(\\lambda)}}{\\lambda + \\hat{H}{(\\lambda)}} = \\frac{\\lambda}{\\lambda + \\hat{H}{(\\lambda)}} + \\hat{H}{(\\lambda)} and \\log{(\\lambda)} + \\frac{\\mathbf{P}{(\\lambda)}}{\\lambda + \\log{(\\lambda)}} = \\frac{\\lambda}{\\lambda + \\log{(\\lambda)}} + \\log{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["divide", 2, "Add(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True))), Integer(-1))))"], [["minus", 3, "Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True))), Integer(-1))), Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Integer(-1))), log(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(S)} = \\log{(S)} and \\mathbf{D}{(S)} = \\log{(S)} and \\delta{(S)} = \\log{(S)}, then obtain \\delta^{2}{(S)} + \\mathbf{D}{(S)} \\psi^{*}{(S)} = \\delta{(S)} \\log{(S)} + \\mathbf{D}{(S)} \\psi^{*}{(S)}", "derivation": "\\psi^{*}{(S)} = \\log{(S)} and \\mathbf{D}{(S)} = \\log{(S)} and \\mathbf{D}{(S)} = \\psi^{*}{(S)} and \\mathbf{D}{(S)} \\psi^{*}{(S)} = \\psi^{*}{(S)} \\log{(S)} and \\psi^{*}^{2}{(S)} = \\psi^{*}{(S)} \\log{(S)} and \\delta{(S)} = \\log{(S)} and \\psi^{*}{(S)} = \\delta{(S)} and \\delta^{2}{(S)} = \\delta{(S)} \\log{(S)} and \\delta^{2}{(S)} + \\mathbf{D}{(S)} \\psi^{*}{(S)} = \\delta{(S)} \\log{(S)} + \\mathbf{D}{(S)} \\psi^{*}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), Function('\\\\psi^*')(Symbol('S', commutative=True)))"], [["times", 2, "Function('\\\\psi^*')(Symbol('S', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), Function('\\\\psi^*')(Symbol('S', commutative=True))), Mul(Function('\\\\psi^*')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\psi^*')(Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\psi^*')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 6], "Equality(Function('\\\\psi^*')(Symbol('S', commutative=True)), Function('\\\\delta')(Symbol('S', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Pow(Function('\\\\delta')(Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\delta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))))"], [["add", 8, "Mul(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), Function('\\\\psi^*')(Symbol('S', commutative=True)))"], "Equality(Add(Pow(Function('\\\\delta')(Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), Function('\\\\psi^*')(Symbol('S', commutative=True)))), Add(Mul(Function('\\\\delta')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Mul(Function('\\\\mathbf{D}')(Symbol('S', commutative=True)), Function('\\\\psi^*')(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given B{(\\theta_1,\\mu_0)} = \\log{(\\mu_0 + \\theta_1)}, then obtain \\int \\frac{\\int (- \\mu_0 - \\theta_1 + B{(\\theta_1,\\mu_0)}) d\\mu_0}{q} d\\mu_0 = \\int \\frac{\\int (- \\mu_0 - \\theta_1 + \\log{(\\mu_0 + \\theta_1)}) d\\mu_0}{q} d\\mu_0", "derivation": "B{(\\theta_1,\\mu_0)} = \\log{(\\mu_0 + \\theta_1)} and - \\mu_0 - \\theta_1 + B{(\\theta_1,\\mu_0)} = - \\mu_0 - \\theta_1 + \\log{(\\mu_0 + \\theta_1)} and \\int (- \\mu_0 - \\theta_1 + B{(\\theta_1,\\mu_0)}) d\\mu_0 = \\int (- \\mu_0 - \\theta_1 + \\log{(\\mu_0 + \\theta_1)}) d\\mu_0 and \\frac{\\int (- \\mu_0 - \\theta_1 + B{(\\theta_1,\\mu_0)}) d\\mu_0}{q} = \\frac{\\int (- \\mu_0 - \\theta_1 + \\log{(\\mu_0 + \\theta_1)}) d\\mu_0}{q} and \\int \\frac{\\int (- \\mu_0 - \\theta_1 + B{(\\theta_1,\\mu_0)}) d\\mu_0}{q} d\\mu_0 = \\int \\frac{\\int (- \\mu_0 - \\theta_1 + \\log{(\\mu_0 + \\theta_1)}) d\\mu_0}{q} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('B')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('B')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 3, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('B')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('B')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), log(Add(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(m,G)} = G + m and H{(m)} = \\int m dm, then obtain H^{2}{(m)} - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm = H{(m)} \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm", "derivation": "\\operatorname{v_{z}}{(m,G)} = G + m and - G + \\operatorname{v_{z}}{(m,G)} = m and \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm = \\int m dm and H{(m)} = \\int m dm and H^{2}{(m)} = H{(m)} \\int m dm and H^{2}{(m)} - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm = H{(m)} \\int m dm - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm and H^{2}{(m)} - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm = H{(m)} \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm - \\int (- G + \\operatorname{v_{z}}{(m,G)}) dm", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Symbol('m', commutative=True))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('m', commutative=True)), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True))))"], [["times", 4, "Function('H')(Symbol('m', commutative=True))"], "Equality(Pow(Function('H')(Symbol('m', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('m', commutative=True)), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True)))))"], [["minus", 5, "Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))"], "Equality(Add(Pow(Function('H')(Symbol('m', commutative=True)), Integer(2)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))), Add(Mul(Function('H')(Symbol('m', commutative=True)), Integral(Symbol('m', commutative=True), Tuple(Symbol('m', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Pow(Function('H')(Symbol('m', commutative=True)), Integer(2)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))), Add(Mul(Function('H')(Symbol('m', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('v_z')(Symbol('m', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(g)} = \\cos{(g)}, then obtain \\frac{\\operatorname{C_{d}}{(g)}}{\\int \\operatorname{C_{d}}^{g}{(g)} dg} = \\frac{\\cos{(g)}}{\\int \\operatorname{C_{d}}^{g}{(g)} dg}", "derivation": "\\operatorname{C_{d}}{(g)} = \\cos{(g)} and \\operatorname{C_{d}}^{g}{(g)} = \\cos^{g}{(g)} and \\int \\operatorname{C_{d}}^{g}{(g)} dg = \\int \\cos^{g}{(g)} dg and \\frac{\\operatorname{C_{d}}{(g)}}{\\int \\cos^{g}{(g)} dg} = \\frac{\\cos{(g)}}{\\int \\cos^{g}{(g)} dg} and \\frac{\\operatorname{C_{d}}{(g)}}{\\int \\operatorname{C_{d}}^{g}{(g)} dg} = \\frac{\\cos{(g)}}{\\int \\operatorname{C_{d}}^{g}{(g)} dg}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Pow(Function('C_d')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["divide", 1, "Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Function('C_d')(Symbol('g', commutative=True)), Pow(Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(cos(Symbol('g', commutative=True)), Pow(Integral(Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('C_d')(Symbol('g', commutative=True)), Pow(Integral(Pow(Function('C_d')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))), Mul(cos(Symbol('g', commutative=True)), Pow(Integral(Pow(Function('C_d')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(\\dot{z})} = \\log{(\\log{(\\dot{z})})}, then derive \\frac{d}{d \\dot{z}} k{(\\dot{z})} = \\frac{1}{\\dot{z} \\log{(\\dot{z})}}, then obtain \\frac{d}{d \\dot{z}} (\\frac{d}{d \\dot{z}} \\log{(\\log{(\\dot{z})})})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (\\frac{1}{\\dot{z} \\log{(\\dot{z})}})^{\\dot{z}}", "derivation": "k{(\\dot{z})} = \\log{(\\log{(\\dot{z})})} and \\frac{d}{d \\dot{z}} k{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\log{(\\log{(\\dot{z})})} and \\frac{d}{d \\dot{z}} k{(\\dot{z})} = \\frac{1}{\\dot{z} \\log{(\\dot{z})}} and \\frac{d}{d \\dot{z}} \\log{(\\log{(\\dot{z})})} = \\frac{1}{\\dot{z} \\log{(\\dot{z})}} and (\\frac{d}{d \\dot{z}} \\log{(\\log{(\\dot{z})})})^{\\dot{z}} = (\\frac{1}{\\dot{z} \\log{(\\dot{z})}})^{\\dot{z}} and \\frac{d}{d \\dot{z}} (\\frac{d}{d \\dot{z}} \\log{(\\log{(\\dot{z})})})^{\\dot{z}} = \\frac{d}{d \\dot{z}} (\\frac{1}{\\dot{z} \\log{(\\dot{z})}})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\dot{z}', commutative=True)), log(log(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(log(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Derivative(log(log(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('\\\\dot{z}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Pow(Derivative(log(log(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\phi,G)} = G + \\phi, then derive \\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)} = 1, then derive \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G (G + \\mathbf{p})} = \\frac{1}{G (G + \\mathbf{p})}, then obtain \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G (G + \\mathbf{p})} = \\frac{\\partial}{\\partial G} \\frac{1}{G (G + \\mathbf{p})}", "derivation": "\\operatorname{E_{n}}{(\\phi,G)} = G + \\phi and G + \\operatorname{E_{n}}{(\\phi,G)} = 2 G + \\phi and \\frac{\\partial}{\\partial \\phi} (G + \\operatorname{E_{n}}{(\\phi,G)}) = \\frac{\\partial}{\\partial \\phi} (2 G + \\phi) and \\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)} = 1 and \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G} = \\frac{1}{G} and \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G \\int \\frac{\\partial}{\\partial \\phi} (2 G + \\phi) dG} = \\frac{1}{G \\int \\frac{\\partial}{\\partial \\phi} (2 G + \\phi) dG} and \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G (G + \\mathbf{p})} = \\frac{1}{G (G + \\mathbf{p})} and \\frac{\\partial}{\\partial G} \\frac{\\frac{\\partial}{\\partial \\phi} \\operatorname{E_{n}}{(\\phi,G)}}{G (G + \\mathbf{p})} = \\frac{\\partial}{\\partial G} \\frac{1}{G (G + \\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('G', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Pow(Symbol('G', commutative=True), Integer(-1)))"], [["divide", 5, "Integral(Derivative(Add(Mul(Integer(2), Symbol('G', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Pow(Integral(Derivative(Add(Mul(Integer(2), Symbol('G', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integer(-1))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Integral(Derivative(Add(Mul(Integer(2), Symbol('G', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Derivative(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))))"], [["differentiate", 7, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Derivative(Function('E_n')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(\\mu_0)} = e^{\\mu_0}, then obtain \\int \\frac{- \\mu_0 + \\theta_{2}{(\\mu_0)}}{\\theta_{2}{(\\mu_0)}} d\\mu_0 = \\int \\frac{- \\mu_0 + e^{\\mu_0}}{\\theta_{2}{(\\mu_0)}} d\\mu_0", "derivation": "\\theta_{2}{(\\mu_0)} = e^{\\mu_0} and - \\mu_0 + \\theta_{2}{(\\mu_0)} = - \\mu_0 + e^{\\mu_0} and \\frac{- \\mu_0 + \\theta_{2}{(\\mu_0)}}{\\theta_{2}{(\\mu_0)}} = \\frac{- \\mu_0 + e^{\\mu_0}}{\\theta_{2}{(\\mu_0)}} and \\int \\frac{- \\mu_0 + \\theta_{2}{(\\mu_0)}}{\\theta_{2}{(\\mu_0)}} d\\mu_0 = \\int \\frac{- \\mu_0 + e^{\\mu_0}}{\\theta_{2}{(\\mu_0)}} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 2, "Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},r)} = \\log{(f_{\\mathbf{v}} r)}^{r} and \\operatorname{y^{\\prime}}{(\\mathbf{v})} = e^{\\mathbf{v}}, then obtain \\operatorname{y^{\\prime}}{(\\mathbf{v})} - \\log{(f_{\\mathbf{v}} r)}^{r} = e^{\\mathbf{v}} - \\log{(f_{\\mathbf{v}} r)}^{r}", "derivation": "\\hat{\\mathbf{r}}{(f_{\\mathbf{v}},r)} = \\log{(f_{\\mathbf{v}} r)}^{r} and \\operatorname{y^{\\prime}}{(\\mathbf{v})} = e^{\\mathbf{v}} and - \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},r)} + \\operatorname{y^{\\prime}}{(\\mathbf{v})} = - \\hat{\\mathbf{r}}{(f_{\\mathbf{v}},r)} + e^{\\mathbf{v}} and \\operatorname{y^{\\prime}}{(\\mathbf{v})} - \\log{(f_{\\mathbf{v}} r)}^{r} = e^{\\mathbf{v}} - \\log{(f_{\\mathbf{v}} r)}^{r}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True)), Pow(log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], ["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["minus", 2, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))), exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))), Add(exp(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(log(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given y{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial \\mathbf{p}} y{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}} \\log{(t_{1})}, then obtain \\frac{\\partial}{\\partial \\mathbf{p}} y{(t_{1},\\mathbf{p})} = y{(t_{1},\\mathbf{p})} \\log{(t_{1})}", "derivation": "y{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{p}} y{(t_{1},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} t_{1}^{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{p}} y{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}} \\log{(t_{1})} and \\frac{\\partial}{\\partial \\mathbf{p}} y{(t_{1},\\mathbf{p})} = y{(t_{1},\\mathbf{p})} \\log{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Function('y')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\theta)} = \\log{(\\log{(\\theta)})}, then obtain - \\frac{((- \\operatorname{c_{0}}{(\\theta)} + \\log{(\\log{(\\theta)})})^{\\theta})^{\\theta}}{\\log{(\\log{(\\theta)})}} = - \\frac{(0^{\\theta})^{\\theta}}{\\log{(\\log{(\\theta)})}}", "derivation": "\\operatorname{c_{0}}{(\\theta)} = \\log{(\\log{(\\theta)})} and - \\operatorname{c_{0}}{(\\theta)} = - \\log{(\\log{(\\theta)})} and - \\operatorname{c_{0}}{(\\theta)} + \\log{(\\log{(\\theta)})} = 0 and (- \\operatorname{c_{0}}{(\\theta)} + \\log{(\\log{(\\theta)})})^{\\theta} = 0^{\\theta} and ((- \\operatorname{c_{0}}{(\\theta)} + \\log{(\\log{(\\theta)})})^{\\theta})^{\\theta} = (0^{\\theta})^{\\theta} and - \\frac{((- \\operatorname{c_{0}}{(\\theta)} + \\log{(\\log{(\\theta)})})^{\\theta})^{\\theta}}{\\log{(\\log{(\\theta)})}} = - \\frac{(0^{\\theta})^{\\theta}}{\\log{(\\log{(\\theta)})}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\theta', commutative=True)), log(log(Symbol('\\\\theta', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c_0')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), log(log(Symbol('\\\\theta', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), log(log(Symbol('\\\\theta', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\theta', commutative=True))), log(log(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\theta', commutative=True))), log(log(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(Integer(0), Symbol('\\\\theta', commutative=True)))"], [["power", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\theta', commutative=True))), log(log(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["divide", 5, "Mul(Integer(-1), log(log(Symbol('\\\\theta', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Pow(Add(Mul(Integer(-1), Function('c_0')(Symbol('\\\\theta', commutative=True))), log(log(Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(log(log(Symbol('\\\\theta', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Pow(Integer(0), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(log(log(Symbol('\\\\theta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then obtain \\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\mathbf{J}_f{(a^{\\dagger})}", "derivation": "\\mathbf{J}_f{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\mathbf{J}_f{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\operatorname{x^{{\\}'}}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\mathbf{J}_f{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\sin{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\mathbf{J}_f{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\pi)} = \\cos{(\\pi)}, then derive \\int \\operatorname{z^{*}}{(\\pi)} d\\pi = \\sigma_x + \\sin{(\\pi)}, then obtain \\pi + \\sin{(\\pi)} (\\int \\operatorname{z^{*}}{(\\pi)} d\\pi)^{\\pi} = \\pi + (\\sigma_x + \\sin{(\\pi)})^{\\pi} \\sin{(\\pi)}", "derivation": "\\operatorname{z^{*}}{(\\pi)} = \\cos{(\\pi)} and \\int \\operatorname{z^{*}}{(\\pi)} d\\pi = \\int \\cos{(\\pi)} d\\pi and \\int \\operatorname{z^{*}}{(\\pi)} d\\pi = \\sigma_x + \\sin{(\\pi)} and (\\int \\operatorname{z^{*}}{(\\pi)} d\\pi)^{\\pi} = (\\sigma_x + \\sin{(\\pi)})^{\\pi} and \\sin{(\\pi)} (\\int \\operatorname{z^{*}}{(\\pi)} d\\pi)^{\\pi} = (\\sigma_x + \\sin{(\\pi)})^{\\pi} \\sin{(\\pi)} and \\pi + \\sin{(\\pi)} (\\int \\operatorname{z^{*}}{(\\pi)} d\\pi)^{\\pi} = \\pi + (\\sigma_x + \\sin{(\\pi)})^{\\pi} \\sin{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\pi', commutative=True))))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Function('z^*')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["times", 4, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\pi', commutative=True)), Pow(Integral(Function('z^*')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Pow(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True))))"], [["add", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(sin(Symbol('\\\\pi', commutative=True)), Pow(Integral(Function('z^*')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Mul(Pow(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(S,b)} = S \\log{(b)}, then derive \\frac{\\partial}{\\partial b} \\operatorname{A_{x}}{(S,b)} = \\frac{S}{b}, then obtain \\frac{\\partial^{3}}{\\partial b^{3}} \\operatorname{A_{x}}{(S,b)} = \\frac{2 S}{b^{3}}", "derivation": "\\operatorname{A_{x}}{(S,b)} = S \\log{(b)} and \\frac{\\partial}{\\partial b} \\operatorname{A_{x}}{(S,b)} = \\frac{\\partial}{\\partial b} S \\log{(b)} and \\frac{\\partial}{\\partial b} \\operatorname{A_{x}}{(S,b)} = \\frac{S}{b} and \\frac{\\partial^{2}}{\\partial b^{2}} \\operatorname{A_{x}}{(S,b)} = \\frac{\\partial}{\\partial b} \\frac{S}{b} and \\frac{\\partial^{3}}{\\partial b^{3}} \\operatorname{A_{x}}{(S,b)} = \\frac{\\partial^{2}}{\\partial b^{2}} \\frac{S}{b} and \\frac{\\partial^{3}}{\\partial b^{3}} \\operatorname{A_{x}}{(S,b)} = \\frac{2 S}{b^{3}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('S', commutative=True), log(Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('S', commutative=True), log(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Mul(Symbol('S', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(3))), Derivative(Mul(Symbol('S', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('A_x')(Symbol('S', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(3))), Mul(Integer(2), Symbol('S', commutative=True), Pow(Symbol('b', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mu)} = e^{\\mu}, then obtain \\cos{(\\mu + \\hat{p}_0{(\\mu)} + 1)} (\\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu)^{\\mu} = \\cos{(\\mu + e^{\\mu} + 1)} (\\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu)^{\\mu}", "derivation": "\\hat{p}_0{(\\mu)} = e^{\\mu} and \\mu + \\hat{p}_0{(\\mu)} = \\mu + e^{\\mu} and \\mu + \\hat{p}_0{(\\mu)} + 1 = \\mu + e^{\\mu} + 1 and \\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu = \\int (\\mu + e^{\\mu} + 1) d\\mu and (\\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu)^{\\mu} = (\\int (\\mu + e^{\\mu} + 1) d\\mu)^{\\mu} and \\cos{(\\mu + \\hat{p}_0{(\\mu)} + 1)} = \\cos{(\\mu + e^{\\mu} + 1)} and \\cos{(\\mu + \\hat{p}_0{(\\mu)} + 1)} (\\int (\\mu + e^{\\mu} + 1) d\\mu)^{\\mu} = \\cos{(\\mu + e^{\\mu} + 1)} (\\int (\\mu + e^{\\mu} + 1) d\\mu)^{\\mu} and \\cos{(\\mu + \\hat{p}_0{(\\mu)} + 1)} (\\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu)^{\\mu} = \\cos{(\\mu + e^{\\mu} + 1)} (\\int (\\mu + \\hat{p}_0{(\\mu)} + 1) d\\mu)^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1)), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["cos", 3], "Equality(cos(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1))), cos(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1))))"], [["times", 6, "Pow(Integral(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(cos(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1))), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Mul(cos(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1))), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(cos(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1))), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Mul(cos(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('\\\\mu', commutative=True)), Integer(1))), Pow(Integral(Add(Symbol('\\\\mu', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given A{(h,f)} = \\int (f + h) dh, then derive A^{h}{(h,f)} = (\\mathbf{D} + f h + \\frac{h^{2}}{2})^{h}, then obtain ((\\mathbf{D} + f h + \\frac{h^{2}}{2}) A^{h}{(h,f)})^{\\mathbf{D}} = ((\\mathbf{D} + f h + \\frac{h^{2}}{2}) (\\mathbf{D} + f h + \\frac{h^{2}}{2})^{h})^{\\mathbf{D}}", "derivation": "A{(h,f)} = \\int (f + h) dh and A^{h}{(h,f)} = (\\int (f + h) dh)^{h} and A^{h}{(h,f)} = (\\mathbf{D} + f h + \\frac{h^{2}}{2})^{h} and (\\mathbf{D} + f h + \\frac{h^{2}}{2}) A^{h}{(h,f)} = (\\mathbf{D} + f h + \\frac{h^{2}}{2}) (\\mathbf{D} + f h + \\frac{h^{2}}{2})^{h} and ((\\mathbf{D} + f h + \\frac{h^{2}}{2}) A^{h}{(h,f)})^{\\mathbf{D}} = ((\\mathbf{D} + f h + \\frac{h^{2}}{2}) (\\mathbf{D} + f h + \\frac{h^{2}}{2})^{h})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Integral(Add(Symbol('f', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('A')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Symbol('h', commutative=True)), Pow(Integral(Add(Symbol('f', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('A')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Symbol('h', commutative=True)))"], [["times", 3, "Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Pow(Function('A')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Symbol('h', commutative=True))), Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Symbol('h', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Pow(Function('A')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Symbol('h', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Symbol('f', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Symbol('h', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given s{(n)} = \\log{(n)}, then derive (\\frac{d}{d n} s{(n)} - \\frac{1}{n})^{n} = 0^{n}, then obtain \\iint (\\frac{d}{d n} s{(n)} - \\frac{1}{n})^{n} dn dn = \\iint 0^{n} dn dn", "derivation": "s{(n)} = \\log{(n)} and \\frac{d}{d n} s{(n)} = \\frac{d}{d n} \\log{(n)} and \\frac{d}{d n} s{(n)} - \\frac{d}{d n} \\log{(n)} = 0 and (\\frac{d}{d n} s{(n)} - \\frac{d}{d n} \\log{(n)})^{n} = 0^{n} and (\\frac{d}{d n} s{(n)} - \\frac{1}{n})^{n} = 0^{n} and \\int (\\frac{d}{d n} s{(n)} - \\frac{1}{n})^{n} dn = \\int 0^{n} dn and \\iint (\\frac{d}{d n} s{(n)} - \\frac{1}{n})^{n} dn dn = \\iint 0^{n} dn dn", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Integer(0))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Symbol('n', commutative=True)), Pow(Integer(0), Symbol('n', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('n', commutative=True)), Pow(Integer(0), Symbol('n', commutative=True)))"], [["integrate", 5, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Add(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Integer(0), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["integrate", 6, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Add(Derivative(Function('s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Integer(0), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(a)} = \\sin{(a)} and \\mathbf{J}_P{(a)} = a + \\hat{x}_0{(a)}, then obtain \\int (\\mathbf{J}_P{(a)} - \\int (a + \\sin{(a)}) da) da = \\int (a + \\hat{x}_0{(a)} - \\int (a + \\sin{(a)}) da) da", "derivation": "\\hat{x}_0{(a)} = \\sin{(a)} and a + \\hat{x}_0{(a)} = a + \\sin{(a)} and \\mathbf{J}_P{(a)} = a + \\hat{x}_0{(a)} and \\int (a + \\hat{x}_0{(a)}) da = \\int (a + \\sin{(a)}) da and \\mathbf{J}_P{(a)} = a + \\sin{(a)} and \\mathbf{J}_P{(a)} - \\int (a + \\hat{x}_0{(a)}) da = a + \\sin{(a)} - \\int (a + \\hat{x}_0{(a)}) da and \\mathbf{J}_P{(a)} - \\int (a + \\sin{(a)}) da = a + \\sin{(a)} - \\int (a + \\sin{(a)}) da and \\mathbf{J}_P{(a)} - \\int (a + \\sin{(a)}) da = a + \\hat{x}_0{(a)} - \\int (a + \\sin{(a)}) da and \\int (\\mathbf{J}_P{(a)} - \\int (a + \\sin{(a)}) da) da = \\int (a + \\hat{x}_0{(a)} - \\int (a + \\sin{(a)}) da) da", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))), Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))))"], [["minus", 5, "Integral(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))), Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))), Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))), Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))))"], [["integrate", 8, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{J}_P')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('a', commutative=True), Function('\\\\hat{x}_0')(Symbol('a', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('a', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hbar)} = \\frac{1}{\\hbar^{2}} and i{(C,\\hbar)} = \\frac{\\hbar}{C}, then obtain 0 = ((\\int (\\frac{\\hbar}{C})^{\\hbar} dC)^{C} - (\\int i^{\\hbar}{(C,\\hbar)} dC)^{C}) (\\frac{1}{\\hbar^{2}})^{\\hbar}", "derivation": "\\operatorname{F_{N}}{(\\hbar)} = \\frac{1}{\\hbar^{2}} and i{(C,\\hbar)} = \\frac{\\hbar}{C} and i^{\\hbar}{(C,\\hbar)} = (\\frac{\\hbar}{C})^{\\hbar} and \\int i^{\\hbar}{(C,\\hbar)} dC = \\int (\\frac{\\hbar}{C})^{\\hbar} dC and (\\int i^{\\hbar}{(C,\\hbar)} dC)^{C} = (\\int (\\frac{\\hbar}{C})^{\\hbar} dC)^{C} and 0 = (\\int (\\frac{\\hbar}{C})^{\\hbar} dC)^{C} - (\\int i^{\\hbar}{(C,\\hbar)} dC)^{C} and 0 = ((\\int (\\frac{\\hbar}{C})^{\\hbar} dC)^{C} - (\\int i^{\\hbar}{(C,\\hbar)} dC)^{C}) \\operatorname{F_{N}}^{\\hbar}{(\\hbar)} and 0 = ((\\int (\\frac{\\hbar}{C})^{\\hbar} dC)^{C} - (\\int i^{\\hbar}{(C,\\hbar)} dC)^{C}) (\\frac{1}{\\hbar^{2}})^{\\hbar}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\hbar', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)))"], ["get_premise", "Equality(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["power", 4, "Symbol('C', commutative=True)"], "Equality(Pow(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Integral(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["minus", 5, "Pow(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True))"], "Equality(Integer(0), Add(Pow(Integral(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))))"], [["times", 6, "Pow(Function('F_N')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(0), Mul(Add(Pow(Integral(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))), Pow(Function('F_N')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Mul(Add(Pow(Integral(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Integral(Pow(Function('i')(Symbol('C', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('C', commutative=True))), Symbol('C', commutative=True)))), Pow(Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\pi)} = \\sin{(\\pi)}, then derive \\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1 = \\cos{(\\pi)} + 1, then obtain 0 = \\frac{\\cos{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}} - \\frac{\\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}}", "derivation": "\\operatorname{t_{1}}{(\\pi)} = \\sin{(\\pi)} and \\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} = \\frac{d}{d \\pi} \\sin{(\\pi)} and \\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1 = \\frac{d}{d \\pi} \\sin{(\\pi)} + 1 and \\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1 = \\cos{(\\pi)} + 1 and \\frac{\\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1}{\\sin{(\\pi)}} = \\frac{\\cos{(\\pi)} + 1}{\\sin{(\\pi)}} and \\frac{\\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}} = \\frac{\\cos{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}} and 0 = \\frac{\\cos{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}} - \\frac{\\frac{d}{d \\pi} \\operatorname{t_{1}}{(\\pi)} + 1}{\\operatorname{t_{1}}{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('\\\\pi', commutative=True)), Integer(1)))"], [["divide", 4, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('\\\\pi', commutative=True)), Integer(1)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Pow(Function('t_1')(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Add(cos(Symbol('\\\\pi', commutative=True)), Integer(1)), Pow(Function('t_1')(Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["minus", 6, "Mul(Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Pow(Function('t_1')(Symbol('\\\\pi', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Add(cos(Symbol('\\\\pi', commutative=True)), Integer(1)), Pow(Function('t_1')(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Derivative(Function('t_1')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Pow(Function('t_1')(Symbol('\\\\pi', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given I{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})}, then derive - \\cos{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} I{(g^{\\prime}_{\\varepsilon})} = 0, then obtain - 2 \\cos{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\sin{(g^{\\prime}_{\\varepsilon})} = - \\cos{(g^{\\prime}_{\\varepsilon})}", "derivation": "I{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and I{(g^{\\prime}_{\\varepsilon})} - \\sin{(g^{\\prime}_{\\varepsilon})} = 0 and \\frac{d}{d g^{\\prime}_{\\varepsilon}} (I{(g^{\\prime}_{\\varepsilon})} - \\sin{(g^{\\prime}_{\\varepsilon})}) = \\frac{d}{d g^{\\prime}_{\\varepsilon}} 0 and - \\cos{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} I{(g^{\\prime}_{\\varepsilon})} = 0 and - \\cos{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\sin{(g^{\\prime}_{\\varepsilon})} = 0 and - 2 \\cos{(g^{\\prime}_{\\varepsilon})} + \\frac{d}{d g^{\\prime}_{\\varepsilon}} \\sin{(g^{\\prime}_{\\varepsilon})} = - \\cos{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('I')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Function('I')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Derivative(Function('I')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(0))"], [["minus", 5, "cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Derivative(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(x)} = \\log{(\\log{(x)})}, then obtain (\\mathbf{r}{(x)} \\sin{(\\log{(\\log{(x)})})})^{x} = (\\log{(\\log{(x)})} \\sin{(\\log{(\\log{(x)})})})^{x}", "derivation": "\\mathbf{r}{(x)} = \\log{(\\log{(x)})} and \\sin{(\\mathbf{r}{(x)})} = \\sin{(\\log{(\\log{(x)})})} and \\mathbf{r}{(x)} \\sin{(\\mathbf{r}{(x)})} = \\log{(\\log{(x)})} \\sin{(\\mathbf{r}{(x)})} and (\\mathbf{r}{(x)} \\sin{(\\mathbf{r}{(x)})})^{x} = (\\log{(\\log{(x)})} \\sin{(\\mathbf{r}{(x)})})^{x} and (\\mathbf{r}{(x)} \\sin{(\\log{(\\log{(x)})})})^{x} = (\\log{(\\log{(x)})} \\sin{(\\log{(\\log{(x)})})})^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)), log(log(Symbol('x', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True))), sin(log(log(Symbol('x', commutative=True)))))"], [["times", 1, "sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)), sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)))), Mul(log(log(Symbol('x', commutative=True))), sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)))))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)), sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), Pow(Mul(log(log(Symbol('x', commutative=True))), sin(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)))), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Function('\\\\mathbf{r}')(Symbol('x', commutative=True)), sin(log(log(Symbol('x', commutative=True))))), Symbol('x', commutative=True)), Pow(Mul(log(log(Symbol('x', commutative=True))), sin(log(log(Symbol('x', commutative=True))))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given r{(U)} = \\sin{(U)}, then derive \\int r{(U)} dU = \\mathbf{J}_M - \\cos{(U)}, then obtain \\sin{(U)} + \\int r{(U)} dU + 1 = \\mathbf{J}_M + \\sin{(U)} - \\cos{(U)} + 1", "derivation": "r{(U)} = \\sin{(U)} and \\int r{(U)} dU = \\int \\sin{(U)} dU and \\sin{(U)} + \\int r{(U)} dU = \\sin{(U)} + \\int \\sin{(U)} dU and \\int r{(U)} dU = \\mathbf{J}_M - \\cos{(U)} and \\sin{(U)} + \\int r{(U)} dU + 1 = \\sin{(U)} + \\int \\sin{(U)} dU + 1 and \\int \\sin{(U)} dU = \\mathbf{J}_M - \\cos{(U)} and \\sin{(U)} + \\int r{(U)} dU + 1 = \\mathbf{J}_M + \\sin{(U)} - \\cos{(U)} + 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["add", 2, "sin(Symbol('U', commutative=True))"], "Equality(Add(sin(Symbol('U', commutative=True)), Integral(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(sin(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(sin(Symbol('U', commutative=True)), Integral(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(1)), Add(sin(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(sin(Symbol('U', commutative=True)), Integral(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integer(1)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('U', commutative=True)), Mul(Integer(-1), cos(Symbol('U', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} = \\mathbf{J} \\theta_2, then obtain \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} + \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} d\\theta_2 = \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} + \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\theta_2 d\\theta_2", "derivation": "\\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} = \\mathbf{J} \\theta_2 and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\theta_2 and \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} d\\theta_2 = \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\theta_2 d\\theta_2 and \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} + \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} d\\theta_2 = \\operatorname{m_{s}}{(\\mathbf{J},\\theta_2)} + \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} \\theta_2 d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["add", 3, "Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Derivative(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Function('m_s')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(t,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + t and B{(t,\\Psi^{\\dagger})} = \\cos{(t (\\Psi^{\\dagger} + t) + t)}, then obtain t B{(t,\\Psi^{\\dagger})} = t \\cos{(t (\\Psi^{\\dagger} + t) + t)}", "derivation": "\\nabla{(t,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + t and t \\nabla{(t,\\Psi^{\\dagger})} = t (\\Psi^{\\dagger} + t) and t \\nabla{(t,\\Psi^{\\dagger})} + t = t (\\Psi^{\\dagger} + t) + t and \\cos{(t \\nabla{(t,\\Psi^{\\dagger})} + t)} = \\cos{(t (\\Psi^{\\dagger} + t) + t)} and B{(t,\\Psi^{\\dagger})} = \\cos{(t (\\Psi^{\\dagger} + t) + t)} and B{(t,\\Psi^{\\dagger})} = \\cos{(t \\nabla{(t,\\Psi^{\\dagger})} + t)} and - B{(t,\\Psi^{\\dagger})} = - \\cos{(t \\nabla{(t,\\Psi^{\\dagger})} + t)} and - B{(t,\\Psi^{\\dagger})} = - \\cos{(t (\\Psi^{\\dagger} + t) + t)} and t B{(t,\\Psi^{\\dagger})} = t \\cos{(t (\\Psi^{\\dagger} + t) + t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)))"], [["times", 1, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))))"], [["add", 2, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Symbol('t', commutative=True), Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('t', commutative=True)), Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["cos", 3], "Equality(cos(Add(Mul(Symbol('t', commutative=True), Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('t', commutative=True))), cos(Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('B')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Add(Mul(Symbol('t', commutative=True), Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('t', commutative=True))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('B')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), cos(Add(Mul(Symbol('t', commutative=True), Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Integer(-1), Function('B')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), cos(Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)))))"], [["times", 8, "Mul(Integer(-1), Symbol('t', commutative=True))"], "Equality(Mul(Symbol('t', commutative=True), Function('B')(Symbol('t', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Symbol('t', commutative=True), cos(Add(Mul(Symbol('t', commutative=True), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\lambda,y)} = \\lambda^{y}, then derive \\sin{(\\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)})} = \\sin{(\\lambda^{y} \\log{(\\lambda)})}, then obtain \\sin{(\\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)})} = \\sin{(\\operatorname{t_{1}}{(\\lambda,y)} \\log{(\\lambda)})}", "derivation": "\\operatorname{t_{1}}{(\\lambda,y)} = \\lambda^{y} and \\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)} = \\frac{\\partial}{\\partial y} \\lambda^{y} and \\sin{(\\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)})} = \\sin{(\\frac{\\partial}{\\partial y} \\lambda^{y})} and \\sin{(\\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)})} = \\sin{(\\lambda^{y} \\log{(\\lambda)})} and \\sin{(\\frac{\\partial}{\\partial y} \\operatorname{t_{1}}{(\\lambda,y)})} = \\sin{(\\operatorname{t_{1}}{(\\lambda,y)} \\log{(\\lambda)})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), sin(Derivative(Pow(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), sin(Mul(Pow(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(sin(Derivative(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), sin(Mul(Function('t_1')(Symbol('\\\\lambda', commutative=True), Symbol('y', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\delta)} = \\log{(e^{\\delta})}, then derive (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta} = (A_{x} + \\frac{\\delta^{2}}{2})^{\\delta}, then obtain \\hat{H}{(\\delta)} + (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta} = \\log{(e^{\\delta})} + (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta}", "derivation": "\\hat{H}{(\\delta)} = \\log{(e^{\\delta})} and \\int \\hat{H}{(\\delta)} d\\delta = \\int \\log{(e^{\\delta})} d\\delta and (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta} = (\\int \\log{(e^{\\delta})} d\\delta)^{\\delta} and (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta} = (A_{x} + \\frac{\\delta^{2}}{2})^{\\delta} and (A_{x} + \\frac{\\delta^{2}}{2})^{\\delta} + \\hat{H}{(\\delta)} = (A_{x} + \\frac{\\delta^{2}}{2})^{\\delta} + \\log{(e^{\\delta})} and \\hat{H}{(\\delta)} + (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta} = \\log{(e^{\\delta})} + (\\int \\hat{H}{(\\delta)} d\\delta)^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), log(exp(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(log(exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(log(exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2)))), Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Pow(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2)))), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Pow(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2)))), Symbol('\\\\delta', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True))), Add(Pow(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\delta', commutative=True), Integer(2)))), Symbol('\\\\delta', commutative=True)), log(exp(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Pow(Integral(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Add(log(exp(Symbol('\\\\delta', commutative=True))), Pow(Integral(Function('\\\\hat{H}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given S{(\\phi,A_{y})} = e^{\\frac{\\phi}{A_{y}}}, then derive \\int S^{A_{y}}{(\\phi,A_{y})} d\\phi = n_{1} + (e^{\\frac{\\phi}{A_{y}}})^{A_{y}}, then obtain n_{1} + (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} = \\int (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} d\\phi", "derivation": "S{(\\phi,A_{y})} = e^{\\frac{\\phi}{A_{y}}} and S^{A_{y}}{(\\phi,A_{y})} = (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} and \\int S^{A_{y}}{(\\phi,A_{y})} d\\phi = \\int (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} d\\phi and \\int S^{A_{y}}{(\\phi,A_{y})} d\\phi = n_{1} + (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} and n_{1} + (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} = \\int (e^{\\frac{\\phi}{A_{y}}})^{A_{y}} d\\phi", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\phi', commutative=True), Symbol('A_y', commutative=True)), exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\phi', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('A_y', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Function('S')(Symbol('\\\\phi', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Pow(Function('S')(Symbol('\\\\phi', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('n_1', commutative=True), Pow(exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('n_1', commutative=True), Pow(exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('A_y', commutative=True))), Integral(Pow(exp(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Symbol('A_y', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(x,y)} = x + y, then derive \\int \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy = J_{\\varepsilon} + y, then obtain \\frac{y \\tilde{g}^*{(x,y)}}{x + y} + \\iint \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy dx = \\frac{y \\tilde{g}^*{(x,y)}}{x + y} + \\int (J_{\\varepsilon} + y) dx", "derivation": "\\tilde{g}^*{(x,y)} = x + y and y \\tilde{g}^*{(x,y)} = y (x + y) and \\frac{y \\tilde{g}^*{(x,y)}}{x + y} = y and \\frac{\\tilde{g}^*{(x,y)}}{x + y} = 1 and \\int \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy = \\int 1 dy and \\int \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy = J_{\\varepsilon} + y and \\iint \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy dx = \\int (J_{\\varepsilon} + y) dx and \\frac{y \\tilde{g}^*{(x,y)}}{x + y} + \\iint \\frac{\\tilde{g}^*{(x,y)}}{x + y} dy dx = \\frac{y \\tilde{g}^*{(x,y)}}{x + y} + \\int (J_{\\varepsilon} + y) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True)), Add(Symbol('x', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Symbol('y', commutative=True), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('y', commutative=True), Add(Symbol('x', commutative=True), Symbol('y', commutative=True))))"], [["divide", 2, "Add(Symbol('x', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Symbol('y', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], [["divide", 3, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Integer(1))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Integer(1), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 6, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["add", 7, "Mul(Symbol('y', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Symbol('y', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Integral(Mul(Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Symbol('y', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given S{(g)} = e^{\\cos{(g)}}, then obtain (g (g S{(g)} + g) S{(g)})^{g} = (g (g e^{\\cos{(g)}} + g) S{(g)})^{g}", "derivation": "S{(g)} = e^{\\cos{(g)}} and g S{(g)} = g e^{\\cos{(g)}} and g S{(g)} + g = g e^{\\cos{(g)}} + g and g (g S{(g)} + g) S{(g)} = g (g e^{\\cos{(g)}} + g) S{(g)} and (g (g S{(g)} + g) S{(g)})^{g} = (g (g e^{\\cos{(g)}} + g) S{(g)})^{g}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('g', commutative=True)), exp(cos(Symbol('g', commutative=True))))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('S')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), exp(cos(Symbol('g', commutative=True)))))"], [["add", 2, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Symbol('g', commutative=True), Function('S')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Add(Mul(Symbol('g', commutative=True), exp(cos(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["times", 3, "Mul(Symbol('g', commutative=True), Function('S')(Symbol('g', commutative=True)))"], "Equality(Mul(Symbol('g', commutative=True), Add(Mul(Symbol('g', commutative=True), Function('S')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Function('S')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), Add(Mul(Symbol('g', commutative=True), exp(cos(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Function('S')(Symbol('g', commutative=True))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Symbol('g', commutative=True), Add(Mul(Symbol('g', commutative=True), Function('S')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Function('S')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Symbol('g', commutative=True), Add(Mul(Symbol('g', commutative=True), exp(cos(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Function('S')(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given v{(J,z)} = \\frac{z}{J}, then obtain v{(J,z)} \\int v{(J,z)} dJ - \\frac{z \\int v{(J,z)} dJ}{J} = 0", "derivation": "v{(J,z)} = \\frac{z}{J} and \\int v{(J,z)} dJ = \\int \\frac{z}{J} dJ and v{(J,z)} \\int \\frac{z}{J} dJ = \\frac{z \\int \\frac{z}{J} dJ}{J} and v{(J,z)} \\int v{(J,z)} dJ = \\frac{z \\int v{(J,z)} dJ}{J} and v{(J,z)} \\int v{(J,z)} dJ - \\frac{z \\int v{(J,z)} dJ}{J} = 0", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["times", 1, "Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["minus", 4, "Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('z', commutative=True), Integral(Function('v')(Symbol('J', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('J', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\theta_{1}{(\\lambda)} = \\sin{(\\lambda)}, then obtain \\cos{(2 \\theta_{1}{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}})} = \\cos{(\\theta_{1}{(\\lambda)} + \\sin{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}})}", "derivation": "\\theta_{1}{(\\lambda)} = \\sin{(\\lambda)} and 2 \\theta_{1}{(\\lambda)} = \\theta_{1}{(\\lambda)} + \\sin{(\\lambda)} and 2 \\theta_{1}{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}} = \\theta_{1}{(\\lambda)} + \\sin{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}} and \\cos{(2 \\theta_{1}{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}})} = \\cos{(\\theta_{1}{(\\lambda)} + \\sin{(\\lambda)} - \\frac{1}{\\sin{(\\lambda)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True))), Add(Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))))"], [["minus", 2, "Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)))), Add(Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1)))))"], [["cos", 3], "Equality(cos(Add(Mul(Integer(2), Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1))))), cos(Add(Function('\\\\theta_1')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\lambda', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(A_{y},Q)} = - A_{y} + \\log{(Q)}, then obtain - A_{y} + 2 Q (- A_{y} + \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)} = - A_{y} + Q (- 2 A_{y} + 2 \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)}", "derivation": "\\eta^{\\prime}{(A_{y},Q)} = - A_{y} + \\log{(Q)} and 2 \\eta^{\\prime}{(A_{y},Q)} = - A_{y} + \\eta^{\\prime}{(A_{y},Q)} + \\log{(Q)} and 2 Q \\eta^{\\prime}{(A_{y},Q)} = Q (- A_{y} + \\eta^{\\prime}{(A_{y},Q)} + \\log{(Q)}) and 2 Q (- A_{y} + \\log{(Q)}) = Q (- 2 A_{y} + 2 \\log{(Q)}) and 2 Q (- A_{y} + \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)} = Q (- 2 A_{y} + 2 \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)} and - A_{y} + 2 Q (- A_{y} + \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)} = - A_{y} + Q (- 2 A_{y} + 2 \\log{(Q)}) \\eta^{\\prime}^{- Q}{(A_{y},Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('Q', commutative=True))))"], [["add", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))))"], [["times", 2, "Symbol('Q', commutative=True)"], "Equality(Mul(Integer(2), Symbol('Q', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Symbol('Q', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('Q', commutative=True)))), Mul(Symbol('Q', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), log(Symbol('Q', commutative=True))))))"], [["divide", 4, "Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Symbol('Q', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('Q', commutative=True))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)))), Mul(Symbol('Q', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), log(Symbol('Q', commutative=True)))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)))))"], [["add", 5, "Mul(Integer(-1), Symbol('A_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), log(Symbol('Q', commutative=True))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Mul(Symbol('Q', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), log(Symbol('Q', commutative=True)))), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\omega)} = \\omega, then derive \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} = \\operatorname{C_{d}}{(\\omega)}, then obtain \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} + \\operatorname{C_{d}}{(\\omega)} = 2 \\operatorname{C_{d}}{(\\omega)}", "derivation": "\\operatorname{C_{d}}{(\\omega)} = \\omega and \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} = \\frac{d}{d \\omega} \\omega and \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} = \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\omega and \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} = \\operatorname{C_{d}}{(\\omega)} and \\operatorname{C_{d}}{(\\omega)} \\frac{d}{d \\omega} \\operatorname{C_{d}}{(\\omega)} + \\operatorname{C_{d}}{(\\omega)} = 2 \\operatorname{C_{d}}{(\\omega)}", "srepr_derivation": [["renaming_premise", "Equality(Function('C_d')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["times", 2, "Function('C_d')(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('C_d')(Symbol('\\\\omega', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Mul(Function('C_d')(Symbol('\\\\omega', commutative=True)), Derivative(Symbol('\\\\omega', commutative=True), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('C_d')(Symbol('\\\\omega', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Function('C_d')(Symbol('\\\\omega', commutative=True)))"], [["add", 4, "Function('C_d')(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Function('C_d')(Symbol('\\\\omega', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Function('C_d')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Function('C_d')(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(S)} = \\cos{(S)}, then obtain \\frac{\\operatorname{f^{\\prime}}{(S)} \\sin{(S)}}{\\cos^{2}{(S)}} + \\frac{\\frac{d}{d S} \\operatorname{f^{\\prime}}{(S)}}{\\cos{(S)}} = 0", "derivation": "\\operatorname{f^{\\prime}}{(S)} = \\cos{(S)} and \\frac{\\operatorname{f^{\\prime}}{(S)}}{\\cos{(S)}} = 1 and \\frac{d}{d S} \\frac{\\operatorname{f^{\\prime}}{(S)}}{\\cos{(S)}} = \\frac{d}{d S} 1 and \\frac{\\operatorname{f^{\\prime}}{(S)} \\sin{(S)}}{\\cos^{2}{(S)}} + \\frac{\\frac{d}{d S} \\operatorname{f^{\\prime}}{(S)}}{\\cos{(S)}} = 0", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["divide", 1, "cos(Symbol('S', commutative=True))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Function('f^{\\\\prime}')(Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Integer(-1))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('f^{\\\\prime}')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Integer(-2))), Mul(Pow(cos(Symbol('S', commutative=True)), Integer(-1)), Derivative(Function('f^{\\\\prime}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given i{(b)} = e^{b} and \\mathbf{S}{(b)} = b^{2} i{(b)} e^{b}, then obtain \\cos{(b^{2} i{(b)} e^{b} + \\mathbf{S}{(b)})} = \\cos{(b^{2} i^{2}{(b)} + b^{2} i{(b)} e^{b})}", "derivation": "i{(b)} = e^{b} and b i{(b)} = b e^{b} and b^{2} i{(b)} e^{b} = b^{2} e^{2 b} and \\mathbf{S}{(b)} = b^{2} i{(b)} e^{b} and \\mathbf{S}{(b)} = b^{2} i^{2}{(b)} and b^{2} e^{2 b} + \\mathbf{S}{(b)} = b^{2} i^{2}{(b)} + b^{2} e^{2 b} and \\cos{(b^{2} e^{2 b} + \\mathbf{S}{(b)})} = \\cos{(b^{2} i^{2}{(b)} + b^{2} e^{2 b})} and \\cos{(b^{2} i{(b)} e^{b} + \\mathbf{S}{(b)})} = \\cos{(b^{2} i^{2}{(b)} + b^{2} i{(b)} e^{b})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('i')(Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))))"], [["times", 2, "Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('i')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('b', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('i')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('b', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('i')(Symbol('b', commutative=True)), Integer(2))))"], [["add", 5, "Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('b', commutative=True))), Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('i')(Symbol('b', commutative=True)), Integer(2))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True))))))"], [["cos", 6], "Equality(cos(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('b', commutative=True)))), cos(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('i')(Symbol('b', commutative=True)), Integer(2))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('b', commutative=True)))))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(cos(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('i')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Function('\\\\mathbf{S}')(Symbol('b', commutative=True)))), cos(Add(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('i')(Symbol('b', commutative=True)), Integer(2))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('i')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(\\hat{p}_0,u)} = e^{\\hat{p}_0 + u}, then derive \\int \\hat{x}{(\\hat{p}_0,u)} du = z^{*} + e^{\\hat{p}_0 + u}, then obtain \\frac{\\partial}{\\partial z^{*}} (z^{*} + e^{\\hat{p}_0 + u}) = \\frac{\\partial}{\\partial z^{*}} \\int e^{\\hat{p}_0 + u} du", "derivation": "\\hat{x}{(\\hat{p}_0,u)} = e^{\\hat{p}_0 + u} and \\int \\hat{x}{(\\hat{p}_0,u)} du = \\int e^{\\hat{p}_0 + u} du and \\int \\hat{x}{(\\hat{p}_0,u)} du = z^{*} + e^{\\hat{p}_0 + u} and z^{*} + e^{\\hat{p}_0 + u} = \\int e^{\\hat{p}_0 + u} du and \\frac{\\partial}{\\partial z^{*}} (z^{*} + e^{\\hat{p}_0 + u}) = \\frac{\\partial}{\\partial z^{*}} \\int e^{\\hat{p}_0 + u} du", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('z^*', commutative=True), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('z^*', commutative=True), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)))), Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["differentiate", 4, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Add(Symbol('z^*', commutative=True), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(F_{x})} = \\frac{d}{d F_{x}} \\sin{(F_{x})}, then derive f^{2}{(F_{x})} = f{(F_{x})} \\cos{(F_{x})}, then obtain f{(F_{x})} \\cos{(F_{x})} = f{(F_{x})} \\frac{d}{d F_{x}} \\sin{(F_{x})}", "derivation": "f{(F_{x})} = \\frac{d}{d F_{x}} \\sin{(F_{x})} and f^{2}{(F_{x})} = f{(F_{x})} \\frac{d}{d F_{x}} \\sin{(F_{x})} and f^{2}{(F_{x})} = f{(F_{x})} \\cos{(F_{x})} and f{(F_{x})} \\cos{(F_{x})} = f{(F_{x})} \\frac{d}{d F_{x}} \\sin{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('F_x', commutative=True)), Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 1, "Function('f')(Symbol('F_x', commutative=True))"], "Equality(Pow(Function('f')(Symbol('F_x', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('F_x', commutative=True)), Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('f')(Symbol('F_x', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Function('f')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Mul(Function('f')(Symbol('F_x', commutative=True)), Derivative(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(E_{\\lambda},\\tilde{g})} = E_{\\lambda} + \\tilde{g}, then obtain \\int - \\frac{(- \\tilde{g} + u{(E_{\\lambda},\\tilde{g})})^{2}}{E_{\\lambda}} dE_{\\lambda} = \\int (\\tilde{g} - u{(E_{\\lambda},\\tilde{g})}) dE_{\\lambda}", "derivation": "u{(E_{\\lambda},\\tilde{g})} = E_{\\lambda} + \\tilde{g} and - \\tilde{g} + u{(E_{\\lambda},\\tilde{g})} = E_{\\lambda} and (- \\tilde{g} + u{(E_{\\lambda},\\tilde{g})})^{2} = E_{\\lambda} (- \\tilde{g} + u{(E_{\\lambda},\\tilde{g})}) and - \\frac{(- \\tilde{g} + u{(E_{\\lambda},\\tilde{g})})^{2}}{E_{\\lambda}} = \\tilde{g} - u{(E_{\\lambda},\\tilde{g})} and \\int - \\frac{(- \\tilde{g} + u{(E_{\\lambda},\\tilde{g})})^{2}}{E_{\\lambda}} dE_{\\lambda} = \\int (\\tilde{g} - u{(E_{\\lambda},\\tilde{g})}) dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integer(2)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integer(2))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["integrate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integer(2))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Function('u')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given A{(h,\\mathbf{S})} = \\mathbf{S}^{h}, then obtain \\frac{\\partial}{\\partial h} (\\mathbf{S} + \\mathbf{S}^{h} + A^{h}{(h,\\mathbf{S})}) = \\frac{\\partial}{\\partial h} (\\mathbf{S} + \\mathbf{S}^{h} + (\\mathbf{S}^{h})^{h})", "derivation": "A{(h,\\mathbf{S})} = \\mathbf{S}^{h} and \\mathbf{S} + A{(h,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{h} and A^{h}{(h,\\mathbf{S})} = (\\mathbf{S}^{h})^{h} and \\mathbf{S} + A{(h,\\mathbf{S})} + A^{h}{(h,\\mathbf{S})} = \\mathbf{S} + (\\mathbf{S}^{h})^{h} + A{(h,\\mathbf{S})} and \\mathbf{S} + \\mathbf{S}^{h} + A^{h}{(h,\\mathbf{S})} = \\mathbf{S} + \\mathbf{S}^{h} + (\\mathbf{S}^{h})^{h} and \\frac{\\partial}{\\partial h} (\\mathbf{S} + \\mathbf{S}^{h} + A^{h}{(h,\\mathbf{S})}) = \\frac{\\partial}{\\partial h} (\\mathbf{S} + \\mathbf{S}^{h} + (\\mathbf{S}^{h})^{h})", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('h', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["add", 3, "Add(Symbol('\\\\mathbf{S}', commutative=True), Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('h', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Pow(Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('h', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["differentiate", 5, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Pow(Function('A')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(g,n_{2},f^{\\prime})} = - f^{\\prime} + g - n_{2}, then derive \\int \\hat{H}{(g,n_{2},f^{\\prime})} dg = \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) + t, then obtain \\theta_2 + \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) = \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) + t", "derivation": "\\hat{H}{(g,n_{2},f^{\\prime})} = - f^{\\prime} + g - n_{2} and \\int \\hat{H}{(g,n_{2},f^{\\prime})} dg = \\int (- f^{\\prime} + g - n_{2}) dg and \\int \\hat{H}{(g,n_{2},f^{\\prime})} dg = \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) + t and \\int (- f^{\\prime} + g - n_{2}) dg = \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) + t and \\theta_2 + \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) = \\frac{g^{2}}{2} + g (- f^{\\prime} - n_{2}) + t", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('g', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('g', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('t', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))))), Add(Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given T{(g,\\mathbf{D})} = \\frac{\\partial}{\\partial g} (\\mathbf{D} + g), then derive T{(g,\\mathbf{D})} = 1, then derive \\int T{(g,\\mathbf{D})} d\\mathbf{D} = \\dot{z} + \\mathbf{D}, then obtain \\frac{\\partial}{\\partial g} \\int \\frac{\\partial}{\\partial g} (\\mathbf{D} + g) d\\mathbf{D} = \\frac{\\partial}{\\partial g} \\int T{(g,\\mathbf{D})} d\\mathbf{D}", "derivation": "T{(g,\\mathbf{D})} = \\frac{\\partial}{\\partial g} (\\mathbf{D} + g) and T{(g,\\mathbf{D})} = 1 and \\int T{(g,\\mathbf{D})} d\\mathbf{D} = \\int 1 d\\mathbf{D} and \\int T{(g,\\mathbf{D})} d\\mathbf{D} = \\dot{z} + \\mathbf{D} and \\frac{\\partial}{\\partial g} \\int T{(g,\\mathbf{D})} d\\mathbf{D} = \\frac{\\partial}{\\partial g} (\\dot{z} + \\mathbf{D}) and \\frac{\\partial}{\\partial g} \\int \\frac{\\partial}{\\partial g} (\\mathbf{D} + g) d\\mathbf{D} = \\frac{\\partial}{\\partial g} (\\dot{z} + \\mathbf{D}) and \\frac{\\partial}{\\partial g} \\int \\frac{\\partial}{\\partial g} (\\mathbf{D} + g) d\\mathbf{D} = \\frac{\\partial}{\\partial g} \\int T{(g,\\mathbf{D})} d\\mathbf{D}", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Integral(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(Function('T')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(t)} = \\log{(t)}, then obtain (\\frac{t \\frac{d}{d t} \\mathbf{v}{(t)}}{\\mathbf{v}{(t)}} + \\log{(\\mathbf{v}{(t)})}) \\mathbf{v}^{t}{(t)} = (\\log{(\\log{(t)})} + \\frac{1}{\\log{(t)}}) \\log{(t)}^{t}", "derivation": "\\mathbf{v}{(t)} = \\log{(t)} and \\mathbf{v}^{t}{(t)} = \\log{(t)}^{t} and \\frac{d}{d t} \\mathbf{v}^{t}{(t)} = \\frac{d}{d t} \\log{(t)}^{t} and (\\frac{t \\frac{d}{d t} \\mathbf{v}{(t)}}{\\mathbf{v}{(t)}} + \\log{(\\mathbf{v}{(t)})}) \\mathbf{v}^{t}{(t)} = (\\log{(\\log{(t)})} + \\frac{1}{\\log{(t)}}) \\log{(t)}^{t}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(log(Symbol('t', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('t', commutative=True), Pow(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), log(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)))), Pow(Function('\\\\mathbf{v}')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Add(log(log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Integer(-1))), Pow(log(Symbol('t', commutative=True)), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then obtain \\frac{d}{d \\mathbf{B}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} \\int \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{d}{d \\mathbf{B}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} \\int \\log{(\\mathbf{B})} d\\mathbf{B}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\int \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} d\\mathbf{B} = \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\frac{d}{d \\mathbf{B}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} \\int \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\frac{d}{d \\mathbf{B}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} \\int \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} d\\mathbf{B} = \\frac{d}{d \\mathbf{B}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{B})} \\int \\log{(\\mathbf{B})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} = C_{1} - V_{\\mathbf{E}}, then obtain (- (C_{1} - V_{\\mathbf{E}}) \\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} + \\operatorname{A_{y}}^{2}{(C_{1},V_{\\mathbf{E}})})^{C_{1}} = 0^{C_{1}}", "derivation": "\\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} = C_{1} - V_{\\mathbf{E}} and \\operatorname{A_{y}}^{2}{(C_{1},V_{\\mathbf{E}})} = (C_{1} - V_{\\mathbf{E}}) \\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} and - (C_{1} - V_{\\mathbf{E}}) \\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} + \\operatorname{A_{y}}^{2}{(C_{1},V_{\\mathbf{E}})} = 0 and (- (C_{1} - V_{\\mathbf{E}}) \\operatorname{A_{y}}{(C_{1},V_{\\mathbf{E}})} + \\operatorname{A_{y}}^{2}{(C_{1},V_{\\mathbf{E}})})^{C_{1}} = 0^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 1, "Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Pow(Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2)), Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["minus", 2, "Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2))), Integer(0))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Function('A_y')(Symbol('C_1', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2))), Symbol('C_1', commutative=True)), Pow(Integer(0), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(q)} = \\cos{(\\log{(q)})}, then obtain (\\frac{d}{d q} 2 \\operatorname{F_{x}}^{3}{(q)})^{q} = (\\frac{d}{d q} 2 \\operatorname{F_{x}}^{2}{(q)} \\cos{(\\log{(q)})})^{q}", "derivation": "\\operatorname{F_{x}}{(q)} = \\cos{(\\log{(q)})} and 2 \\operatorname{F_{x}}{(q)} = \\operatorname{F_{x}}{(q)} + \\cos{(\\log{(q)})} and \\operatorname{F_{x}}^{2}{(q)} = \\operatorname{F_{x}}{(q)} \\cos{(\\log{(q)})} and (\\operatorname{F_{x}}{(q)} + \\cos{(\\log{(q)})}) \\operatorname{F_{x}}^{2}{(q)} = (\\operatorname{F_{x}}{(q)} + \\cos{(\\log{(q)})}) \\operatorname{F_{x}}{(q)} \\cos{(\\log{(q)})} and 2 \\operatorname{F_{x}}^{3}{(q)} = 2 \\operatorname{F_{x}}^{2}{(q)} \\cos{(\\log{(q)})} and \\frac{d}{d q} 2 \\operatorname{F_{x}}^{3}{(q)} = \\frac{d}{d q} 2 \\operatorname{F_{x}}^{2}{(q)} \\cos{(\\log{(q)})} and (\\frac{d}{d q} 2 \\operatorname{F_{x}}^{3}{(q)})^{q} = (\\frac{d}{d q} 2 \\operatorname{F_{x}}^{2}{(q)} \\cos{(\\log{(q)})})^{q}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True))))"], [["add", 1, "Function('F_x')(Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Function('F_x')(Symbol('q', commutative=True))), Add(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))))"], [["times", 1, "Function('F_x')(Symbol('q', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(2)), Mul(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))))"], [["times", 3, "Add(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True))))"], "Equality(Mul(Add(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(2))), Mul(Add(Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))), Function('F_x')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(3))), Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(2)), cos(log(Symbol('q', commutative=True)))))"], [["differentiate", 5, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(3))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(2)), cos(log(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["power", 6, "Symbol('q', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(3))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Pow(Derivative(Mul(Integer(2), Pow(Function('F_x')(Symbol('q', commutative=True)), Integer(2)), cos(log(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given r{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})}, then derive r{(\\hat{\\mathbf{r}})} + \\log{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}}, then obtain \\log{(\\hat{\\mathbf{r}})} + \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}}", "derivation": "r{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} and r{(\\hat{\\mathbf{r}})} + \\log{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} + \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} and r{(\\hat{\\mathbf{r}})} + \\log{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}} and \\log{(\\hat{\\mathbf{r}})} + \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["add", 1, "log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})} = e^{J_{\\varepsilon} + P_{g}}, then derive \\int \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})} dP_{g} = x + e^{J_{\\varepsilon} + P_{g}}, then obtain \\int e^{J_{\\varepsilon} + P_{g}} dP_{g} = x + \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})}", "derivation": "\\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})} = e^{J_{\\varepsilon} + P_{g}} and \\int \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})} dP_{g} = \\int e^{J_{\\varepsilon} + P_{g}} dP_{g} and \\int \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})} dP_{g} = x + e^{J_{\\varepsilon} + P_{g}} and \\int e^{J_{\\varepsilon} + P_{g}} dP_{g} = x + e^{J_{\\varepsilon} + P_{g}} and \\int e^{J_{\\varepsilon} + P_{g}} dP_{g} = x + \\operatorname{M_{E}}{(P_{g},J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('P_g', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True))))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('P_g', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('P_g', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('x', commutative=True), exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('x', commutative=True), exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(exp(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('x', commutative=True), Function('M_E')(Symbol('P_g', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\chi)} = e^{\\chi}, then derive \\frac{d}{d \\chi} \\bar{\\h}{(\\chi)} = e^{\\chi}, then obtain \\frac{d}{d \\chi} e^{\\chi} = e^{\\chi}", "derivation": "\\bar{\\h}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\bar{\\h}{(\\chi)} = \\frac{d}{d \\chi} e^{\\chi} and \\frac{d}{d \\chi} \\bar{\\h}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} e^{\\chi} = e^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), exp(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), exp(Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\eta{(h)} = \\cos{(h)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = \\eta^{h}{(h)} and \\operatorname{c_{0}}{(t_{1})} = \\log{(t_{1})}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} - \\log{(t_{1})} = - \\log{(t_{1})} + \\cos^{h}{(h)}", "derivation": "\\eta{(h)} = \\cos{(h)} and \\eta^{h}{(h)} = \\cos^{h}{(h)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = \\eta^{h}{(h)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = \\cos^{h}{(h)} and \\operatorname{c_{0}}{(t_{1})} = \\log{(t_{1})} and - \\operatorname{c_{0}}{(t_{1})} + \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} = - \\operatorname{c_{0}}{(t_{1})} + \\cos^{h}{(h)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(h)} - \\log{(t_{1})} = - \\log{(t_{1})} + \\cos^{h}{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Pow(Function('\\\\eta')(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], ["get_premise", "Equality(Function('c_0')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["minus", 4, "Function('c_0')(Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('c_0')(Symbol('t_1', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True))), Add(Mul(Integer(-1), Function('c_0')(Symbol('t_1', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('h', commutative=True)), Mul(Integer(-1), log(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('t_1', commutative=True))), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given E{(g^{\\prime}_{\\varepsilon},y)} = g^{\\prime}_{\\varepsilon} y, then obtain - \\sin{(\\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} E{(g^{\\prime}_{\\varepsilon},y)})} + \\frac{g^{\\prime}_{\\varepsilon} y + E{(g^{\\prime}_{\\varepsilon},y)}}{y} = 2 g^{\\prime}_{\\varepsilon} - \\sin{(\\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} E{(g^{\\prime}_{\\varepsilon},y)})}", "derivation": "E{(g^{\\prime}_{\\varepsilon},y)} = g^{\\prime}_{\\varepsilon} y and g^{\\prime}_{\\varepsilon} y + E{(g^{\\prime}_{\\varepsilon},y)} = 2 g^{\\prime}_{\\varepsilon} y and \\frac{g^{\\prime}_{\\varepsilon} y + E{(g^{\\prime}_{\\varepsilon},y)}}{y} = 2 g^{\\prime}_{\\varepsilon} and - \\sin{(\\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} E{(g^{\\prime}_{\\varepsilon},y)})} + \\frac{g^{\\prime}_{\\varepsilon} y + E{(g^{\\prime}_{\\varepsilon},y)}}{y} = 2 g^{\\prime}_{\\varepsilon} - \\sin{(\\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} E{(g^{\\prime}_{\\varepsilon},y)})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))), Mul(Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))"], [["divide", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 3, "sin(Derivative(Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), sin(Derivative(Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True))))), Add(Mul(Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), sin(Derivative(Function('E')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given m{(\\delta)} = \\log{(\\delta)}, then obtain \\frac{m^{4}{(\\delta)}}{\\log{(\\delta)}^{2}} = m{(\\delta)} \\log{(\\delta)}", "derivation": "m{(\\delta)} = \\log{(\\delta)} and m^{2}{(\\delta)} = m{(\\delta)} \\log{(\\delta)} and \\frac{m^{2}{(\\delta)}}{\\log{(\\delta)}} = m{(\\delta)} and \\frac{m^{4}{(\\delta)}}{\\log{(\\delta)}^{2}} = m^{2}{(\\delta)} and \\frac{m^{4}{(\\delta)}}{\\log{(\\delta)}^{2}} = m{(\\delta)} \\log{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Function('m')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('m')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Function('m')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True))))"], [["divide", 2, "log(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Pow(Function('m')(Symbol('\\\\delta', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1))), Function('m')(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('m')(Symbol('\\\\delta', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-2))), Pow(Function('m')(Symbol('\\\\delta', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Function('m')(Symbol('\\\\delta', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-2))), Mul(Function('m')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given n{(a,C_{2})} = C_{2} a, then derive \\int (a + n{(a,C_{2})}) dC_{2} = \\frac{C_{2}^{2} a}{2} + C_{2} a + \\hat{H}, then obtain \\frac{C_{2}^{2} a}{2} + C_{2} a + \\lambda = \\frac{C_{2}^{2} a}{2} + C_{2} a + \\hat{H}", "derivation": "n{(a,C_{2})} = C_{2} a and a + n{(a,C_{2})} = C_{2} a + a and \\int (a + n{(a,C_{2})}) dC_{2} = \\int (C_{2} a + a) dC_{2} and \\int (a + n{(a,C_{2})}) dC_{2} = \\frac{C_{2}^{2} a}{2} + C_{2} a + \\hat{H} and \\int (C_{2} a + a) dC_{2} = \\frac{C_{2}^{2} a}{2} + C_{2} a + \\hat{H} and \\frac{C_{2}^{2} a}{2} + C_{2} a + \\lambda = \\frac{C_{2}^{2} a}{2} + C_{2} a + \\hat{H}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('a', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)))"], [["add", 1, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), Function('n')(Symbol('a', commutative=True), Symbol('C_2', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Symbol('a', commutative=True), Function('n')(Symbol('a', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Add(Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('a', commutative=True), Function('n')(Symbol('a', commutative=True), Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('a', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('a', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('a', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\lambda', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('a', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('a', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(n)} = \\log{(n)}, then obtain i{(\\mathbf{J},F_{x})} + \\frac{d}{d n} \\operatorname{E_{x}}^{n}{(n)} = i{(\\mathbf{J},F_{x})} + \\frac{d}{d n} \\log{(n)}^{n}", "derivation": "\\operatorname{E_{x}}{(n)} = \\log{(n)} and \\operatorname{E_{x}}^{n}{(n)} = \\log{(n)}^{n} and \\frac{d}{d n} \\operatorname{E_{x}}^{n}{(n)} = \\frac{d}{d n} \\log{(n)}^{n} and i{(\\mathbf{J},F_{x})} + \\frac{d}{d n} \\operatorname{E_{x}}^{n}{(n)} = i{(\\mathbf{J},F_{x})} + \\frac{d}{d n} \\log{(n)}^{n}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(log(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Function('E_x')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 3, "Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Add(Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True)), Derivative(Pow(Function('E_x')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('i')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('F_x', commutative=True)), Derivative(Pow(log(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(p)} = e^{p} and \\theta_{2}{(v_{x},\\mu_0)} = \\cos{(\\mu_0 v_{x})}, then obtain \\frac{\\theta_{2}{(v_{x},\\mu_0)} e^{- p}}{2} = \\frac{e^{- p} \\cos{(\\mu_0 v_{x})}}{2}", "derivation": "\\operatorname{m_{s}}{(p)} = e^{p} and \\theta_{2}{(v_{x},\\mu_0)} = \\cos{(\\mu_0 v_{x})} and \\frac{\\theta_{2}{(v_{x},\\mu_0)}}{2 \\operatorname{m_{s}}{(p)}} = \\frac{\\cos{(\\mu_0 v_{x})}}{2 \\operatorname{m_{s}}{(p)}} and \\frac{\\theta_{2}{(v_{x},\\mu_0)} e^{- p}}{2} = \\frac{e^{- p} \\cos{(\\mu_0 v_{x})}}{2}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_x', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('m_s')(Symbol('p', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Function('m_s')(Symbol('p', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Function('m_s')(Symbol('p', commutative=True)), Integer(-1)), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Rational(1, 2), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), exp(Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('p', commutative=True))), cos(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\lambda,A_{x})} = e^{A_{x} + \\lambda}, then derive \\int \\operatorname{f^{*}}{(\\lambda,A_{x})} dA_{x} = f^{\\prime} + e^{A_{x} + \\lambda}, then obtain (A_{x} + \\lambda + \\int e^{A_{x} + \\lambda} dA_{x})^{f^{\\prime}} = (A_{x} + \\lambda + f^{\\prime} + e^{A_{x} + \\lambda})^{f^{\\prime}}", "derivation": "\\operatorname{f^{*}}{(\\lambda,A_{x})} = e^{A_{x} + \\lambda} and \\int \\operatorname{f^{*}}{(\\lambda,A_{x})} dA_{x} = \\int e^{A_{x} + \\lambda} dA_{x} and \\int \\operatorname{f^{*}}{(\\lambda,A_{x})} dA_{x} = f^{\\prime} + e^{A_{x} + \\lambda} and A_{x} + \\lambda + \\int \\operatorname{f^{*}}{(\\lambda,A_{x})} dA_{x} = A_{x} + \\lambda + f^{\\prime} + e^{A_{x} + \\lambda} and (A_{x} + \\lambda + \\int \\operatorname{f^{*}}{(\\lambda,A_{x})} dA_{x})^{f^{\\prime}} = (A_{x} + \\lambda + f^{\\prime} + e^{A_{x} + \\lambda})^{f^{\\prime}} and (A_{x} + \\lambda + \\int e^{A_{x} + \\lambda} dA_{x})^{f^{\\prime}} = (A_{x} + \\lambda + f^{\\prime} + e^{A_{x} + \\lambda})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["add", 3, "Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["power", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Integral(Function('f^*')(Symbol('\\\\lambda', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('A_x', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('f^{\\\\prime}', commutative=True), exp(Add(Symbol('A_x', commutative=True), Symbol('\\\\lambda', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})} = F_{x} \\lambda, then obtain F_{x} \\lambda - \\lambda + 1 = F_{x} \\lambda + \\frac{F_{x} \\lambda}{\\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})}} - \\lambda", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})} = F_{x} \\lambda and 1 = \\frac{F_{x} \\lambda}{\\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})}} and 1 - \\lambda = \\frac{F_{x} \\lambda}{\\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})}} - \\lambda and F_{x} \\lambda - \\lambda + 1 = F_{x} \\lambda + \\frac{F_{x} \\lambda}{\\operatorname{g_{\\varepsilon}}{(\\lambda,F_{x})}} - \\lambda", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(1), Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))))"], [["minus", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["add", 3, "Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Integer(1)), Add(Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\lambda', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given G{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\rho_{b}{(f_{\\mathbf{p}})} = G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})}, then obtain 2 \\cos^{2}{(f_{\\mathbf{p}})} = (G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})}) \\cos{(f_{\\mathbf{p}})}", "derivation": "G{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})} = 2 \\cos{(f_{\\mathbf{p}})} and \\rho_{b}{(f_{\\mathbf{p}})} = G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})} and \\rho_{b}{(f_{\\mathbf{p}})} = 2 \\cos{(f_{\\mathbf{p}})} and \\rho_{b}{(f_{\\mathbf{p}})} \\cos{(f_{\\mathbf{p}})} = (G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})}) \\cos{(f_{\\mathbf{p}})} and 2 \\cos^{2}{(f_{\\mathbf{p}})} = (G{(f_{\\mathbf{p}})} + \\cos{(f_{\\mathbf{p}})}) \\cos{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(2), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(2), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["times", 3, "cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Add(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Pow(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(2))), Mul(Add(Function('G')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(r_{0},C_{2})} = r_{0}^{C_{2}} and Q{(r_{0},C_{2})} = \\operatorname{m_{s}}^{2}{(r_{0},C_{2})}, then obtain Q^{2}{(r_{0},C_{2})} = r_{0}^{2 C_{2}} Q{(r_{0},C_{2})}", "derivation": "\\operatorname{m_{s}}{(r_{0},C_{2})} = r_{0}^{C_{2}} and \\operatorname{m_{s}}^{2}{(r_{0},C_{2})} = r_{0}^{C_{2}} \\operatorname{m_{s}}{(r_{0},C_{2})} and Q{(r_{0},C_{2})} = \\operatorname{m_{s}}^{2}{(r_{0},C_{2})} and \\operatorname{m_{s}}^{4}{(r_{0},C_{2})} = r_{0}^{2 C_{2}} \\operatorname{m_{s}}^{2}{(r_{0},C_{2})} and Q^{2}{(r_{0},C_{2})} = r_{0}^{2 C_{2}} Q{(r_{0},C_{2})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)))"], [["times", 1, "Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Integer(2)), Mul(Pow(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Integer(4)), Mul(Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('C_2', commutative=True))), Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('Q')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True)), Integer(2)), Mul(Pow(Symbol('r_0', commutative=True), Mul(Integer(2), Symbol('C_2', commutative=True))), Function('Q')(Symbol('r_0', commutative=True), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given L{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)}, then derive \\mu + L{(\\mu)} = \\mu + \\frac{1}{\\mu}, then obtain \\log{(\\mu)} \\int (\\mu + L{(\\mu)}) d\\mu = \\log{(\\mu)} \\int (\\mu + \\frac{d}{d \\mu} \\log{(\\mu)}) d\\mu", "derivation": "L{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)} and \\mu + L{(\\mu)} = \\mu + \\frac{d}{d \\mu} \\log{(\\mu)} and \\mu + L{(\\mu)} = \\mu + \\frac{1}{\\mu} and \\mu + \\frac{1}{\\mu} = \\mu + \\frac{d}{d \\mu} \\log{(\\mu)} and \\int (\\mu + \\frac{1}{\\mu}) d\\mu = \\int (\\mu + \\frac{d}{d \\mu} \\log{(\\mu)}) d\\mu and \\int (\\mu + L{(\\mu)}) d\\mu = \\int (\\mu + \\frac{1}{\\mu}) d\\mu and \\log{(\\mu)} \\int (\\mu + \\frac{1}{\\mu}) d\\mu = \\log{(\\mu)} \\int (\\mu + \\frac{d}{d \\mu} \\log{(\\mu)}) d\\mu and \\log{(\\mu)} \\int (\\mu + L{(\\mu)}) d\\mu = \\log{(\\mu)} \\int (\\mu + \\frac{d}{d \\mu} \\log{(\\mu)}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mu', commutative=True)), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('L')(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('L')(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Add(Symbol('\\\\mu', commutative=True), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\mu', commutative=True), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mu', commutative=True), Function('L')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["divide", 5, "Pow(log(Symbol('\\\\mu', commutative=True)), Integer(-1))"], "Equality(Mul(log(Symbol('\\\\mu', commutative=True)), Integral(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(log(Symbol('\\\\mu', commutative=True)), Integral(Add(Symbol('\\\\mu', commutative=True), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(log(Symbol('\\\\mu', commutative=True)), Integral(Add(Symbol('\\\\mu', commutative=True), Function('L')(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(log(Symbol('\\\\mu', commutative=True)), Integral(Add(Symbol('\\\\mu', commutative=True), Derivative(log(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given H{(n)} = \\sin{(n)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)} = H^{n}{(n)}, then obtain \\frac{d}{d n} n \\sin^{n}{(n)} = \\frac{d}{d n} n \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)}", "derivation": "H{(n)} = \\sin{(n)} and H^{n}{(n)} = \\sin^{n}{(n)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)} = H^{n}{(n)} and n H^{n}{(n)} = n \\sin^{n}{(n)} and \\frac{d}{d n} n H^{n}{(n)} = \\frac{d}{d n} n \\sin^{n}{(n)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)} = \\sin^{n}{(n)} and \\frac{d}{d n} n H^{n}{(n)} = \\frac{d}{d n} n \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)} and \\frac{d}{d n} n \\sin^{n}{(n)} = \\frac{d}{d n} n \\operatorname{g^{\\prime}_{\\varepsilon}}{(n)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('H')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('n', commutative=True)), Pow(Function('H')(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["times", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Pow(Function('H')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Mul(Symbol('n', commutative=True), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True))))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Symbol('n', commutative=True), Pow(Function('H')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('n', commutative=True), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Mul(Symbol('n', commutative=True), Pow(Function('H')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('n', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Derivative(Mul(Symbol('n', commutative=True), Pow(sin(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Symbol('n', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(g,W)} = g^{W}, then derive \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} = \\frac{W g^{W}}{g}, then obtain \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} = \\frac{\\partial}{\\partial g} g^{W} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}}", "derivation": "\\hat{p}{(g,W)} = g^{W} and \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} = \\frac{\\partial}{\\partial g} g^{W} and \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} = \\frac{W g^{W}}{g} and \\frac{\\partial}{\\partial g} g^{W} = \\frac{W g^{W}}{g} and \\frac{\\partial}{\\partial g} g^{W} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} = \\frac{W g^{W}}{g} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} and \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} = \\frac{W g^{W}}{g} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} and \\frac{\\partial}{\\partial g} \\hat{p}{(g,W)} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}} = \\frac{\\partial}{\\partial g} g^{W} - \\frac{1}{\\frac{\\partial}{\\partial g} \\hat{p}{(g,W)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('W', commutative=True))))"], [["minus", 4, "Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Pow(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Symbol('W', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))), Add(Derivative(Pow(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('\\\\hat{p}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\Psi_{nl})} = \\log{(\\cos{(\\Psi_{nl})})}, then obtain \\frac{d}{d \\Psi_{nl}} \\operatorname{F_{c}}{(\\Psi_{nl})} + 1 = - \\frac{\\sin{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} + 1", "derivation": "\\operatorname{F_{c}}{(\\Psi_{nl})} = \\log{(\\cos{(\\Psi_{nl})})} and \\Psi_{nl} + \\operatorname{F_{c}}{(\\Psi_{nl})} = \\Psi_{nl} + \\log{(\\cos{(\\Psi_{nl})})} and \\frac{d}{d \\Psi_{nl}} (\\Psi_{nl} + \\operatorname{F_{c}}{(\\Psi_{nl})}) = \\frac{d}{d \\Psi_{nl}} (\\Psi_{nl} + \\log{(\\cos{(\\Psi_{nl})})}) and \\frac{d}{d \\Psi_{nl}} \\operatorname{F_{c}}{(\\Psi_{nl})} + 1 = - \\frac{\\sin{(\\Psi_{nl})}}{\\cos{(\\Psi_{nl})}} + 1", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\Psi_{nl}', commutative=True)), log(cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('F_c')(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), log(cos(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('F_c')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{nl}', commutative=True), log(cos(Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('F_c')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given \\phi_{2}{(a)} = \\sin{(a)}, then obtain \\frac{d}{d a} (1 + \\frac{- a + \\phi_{2}{(a)}}{a}) = \\frac{d}{d a} (1 + \\frac{- a + \\sin{(a)}}{a})", "derivation": "\\phi_{2}{(a)} = \\sin{(a)} and - a + \\phi_{2}{(a)} = - a + \\sin{(a)} and - \\frac{- a + \\phi_{2}{(a)}}{a} = - \\frac{- a + \\sin{(a)}}{a} and \\frac{- a + \\phi_{2}{(a)}}{a} = \\frac{- a + \\sin{(a)}}{a} and 1 + \\frac{- a + \\phi_{2}{(a)}}{a} = 1 + \\frac{- a + \\sin{(a)}}{a} and \\frac{d}{d a} (1 + \\frac{- a + \\phi_{2}{(a)}}{a}) = \\frac{d}{d a} (1 + \\frac{- a + \\sin{(a)}}{a})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["minus", 1, "Symbol('a', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\phi_2')(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\phi_2')(Symbol('a', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\phi_2')(Symbol('a', commutative=True)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\phi_2')(Symbol('a', commutative=True))))), Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))))"], [["differentiate", 5, "Symbol('a', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Function('\\\\phi_2')(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(k)} = \\cos{(k)}, then obtain \\frac{\\operatorname{f_{\\mathbf{p}}}{(k)}}{\\cos{(k)}} - \\frac{d}{d k} \\operatorname{f_{\\mathbf{p}}}{(k)} = 1 - \\frac{d}{d k} \\operatorname{f_{\\mathbf{p}}}{(k)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(k)} = \\cos{(k)} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(k)}}{\\cos{(k)}} = 1 and \\frac{d}{d k} \\operatorname{f_{\\mathbf{p}}}{(k)} = \\frac{d}{d k} \\cos{(k)} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(k)}}{\\cos{(k)}} - \\frac{d}{d k} \\cos{(k)} = 1 - \\frac{d}{d k} \\cos{(k)} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(k)}}{\\cos{(k)}} - \\frac{d}{d k} \\operatorname{f_{\\mathbf{p}}}{(k)} = 1 - \\frac{d}{d k} \\operatorname{f_{\\mathbf{p}}}{(k)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["divide", 1, "cos(Symbol('k', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Add(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(-1))), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"]]}, {"prompt": "Given M{(\\omega,k)} = \\frac{\\partial}{\\partial \\omega} (\\omega - k), then derive M^{\\omega}{(\\omega,k)} - 1 = 0, then obtain \\sin{(M^{\\omega}{(\\omega,k)})} = \\sin{(1)}", "derivation": "M{(\\omega,k)} = \\frac{\\partial}{\\partial \\omega} (\\omega - k) and M^{\\omega}{(\\omega,k)} = (\\frac{\\partial}{\\partial \\omega} (\\omega - k))^{\\omega} and M^{\\omega}{(\\omega,k)} (\\frac{\\partial}{\\partial \\omega} (\\omega - k))^{\\omega} = (\\frac{\\partial}{\\partial \\omega} (\\omega - k))^{2 \\omega} and M^{\\omega}{(\\omega,k)} (\\frac{\\partial}{\\partial \\omega} (\\omega - k))^{\\omega} - 1 = (\\frac{\\partial}{\\partial \\omega} (\\omega - k))^{2 \\omega} - 1 and M^{\\omega}{(\\omega,k)} - 1 = 0 and M^{\\omega}{(\\omega,k)} = 1 and \\sin{(M^{\\omega}{(\\omega,k)})} = \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["times", 2, "Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True))), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Integer(2), Symbol('\\\\omega', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Mul(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True))), Integer(-1)), Add(Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Integer(2), Symbol('\\\\omega', commutative=True))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True)), Integer(-1)), Integer(0))"], [["minus", 5, "Integer(-1)"], "Equality(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True)), Integer(1))"], [["sin", 6], "Equality(sin(Pow(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('\\\\omega', commutative=True))), sin(Integer(1)))"]]}, {"prompt": "Given z{(\\phi_2,\\phi)} = \\phi + \\phi_2, then derive \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1) = \\frac{\\partial}{\\partial \\phi_2} (\\delta + \\frac{\\phi \\phi_2^{2}}{2} + \\frac{\\phi_2^{3}}{3} + 1), then obtain \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1) = \\phi \\phi_2 + \\phi_2^{2}", "derivation": "z{(\\phi_2,\\phi)} = \\phi + \\phi_2 and \\phi_2 z{(\\phi_2,\\phi)} = \\phi_2 (\\phi + \\phi_2) and \\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 = \\int \\phi_2 (\\phi + \\phi_2) d\\phi_2 and \\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1 = \\int \\phi_2 (\\phi + \\phi_2) d\\phi_2 + 1 and \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1) = \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 (\\phi + \\phi_2) d\\phi_2 + 1) and \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1) = \\frac{\\partial}{\\partial \\phi_2} (\\delta + \\frac{\\phi \\phi_2^{2}}{2} + \\frac{\\phi_2^{3}}{3} + 1) and \\frac{\\partial}{\\partial \\phi_2} (\\int \\phi_2 z{(\\phi_2,\\phi)} d\\phi_2 + 1) = \\phi \\phi_2 + \\phi_2^{2}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)), Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)))"], [["differentiate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Rational(1, 3), Pow(Symbol('\\\\phi_2', commutative=True), Integer(3))), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Add(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Function('z')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\mathbf{r} + \\log{(\\mathbf{J}_f)}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\frac{1}{\\mathbf{J}_f}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{J}_f^{2}} \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\frac{d}{d \\mathbf{J}_f} \\frac{1}{\\mathbf{J}_f}", "derivation": "\\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\mathbf{r} + \\log{(\\mathbf{J}_f)} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{r} + \\log{(\\mathbf{J}_f)}) and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\frac{1}{\\mathbf{J}_f} and \\frac{\\partial^{2}}{\\partial \\mathbf{J}_f^{2}} \\operatorname{A_{2}}{(\\mathbf{J}_f,\\mathbf{r})} = \\frac{d}{d \\mathbf{J}_f} \\frac{1}{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(A_{y})} = \\log{(\\sin{(A_{y})})} and \\operatorname{A_{2}}{(Q)} = \\cos{(Q)}, then obtain \\frac{(\\operatorname{A_{2}}{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})}) \\mathbf{r}{(A_{y})}}{\\log{(\\sin{(A_{y})})}} = \\operatorname{A_{2}}{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})}", "derivation": "\\mathbf{r}{(A_{y})} = \\log{(\\sin{(A_{y})})} and \\frac{\\mathbf{r}{(A_{y})}}{\\log{(\\sin{(A_{y})})}} = 1 and \\operatorname{A_{2}}{(Q)} = \\cos{(Q)} and \\operatorname{A_{2}}{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})} = \\cos{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})} and \\frac{(\\cos{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})}) \\mathbf{r}{(A_{y})}}{\\log{(\\sin{(A_{y})})}} = \\cos{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})} and \\frac{(\\operatorname{A_{2}}{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})}) \\mathbf{r}{(A_{y})}}{\\log{(\\sin{(A_{y})})}} = \\operatorname{A_{2}}{(Q)} + \\frac{d}{d A_{y}} \\mathbf{r}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), log(sin(Symbol('A_y', commutative=True))))"], [["divide", 1, "log(sin(Symbol('A_y', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Pow(log(sin(Symbol('A_y', commutative=True))), Integer(-1))), Integer(1))"], ["get_premise", "Equality(Function('A_2')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["add", 3, "Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Add(Function('A_2')(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Add(cos(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["times", 2, "Add(cos(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], "Equality(Mul(Add(cos(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Pow(log(sin(Symbol('A_y', commutative=True))), Integer(-1))), Add(cos(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Function('A_2')(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Pow(log(sin(Symbol('A_y', commutative=True))), Integer(-1))), Add(Function('A_2')(Symbol('Q', commutative=True)), Derivative(Function('\\\\mathbf{r}')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} = \\mathbf{E} + \\phi_2 and \\varepsilon{(\\phi_2,\\mathbf{E})} = \\int (\\mathbf{E} + \\phi_2) d\\phi_2, then derive \\int \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} d\\phi_2 = \\mathbf{E} \\phi_2 + \\frac{\\phi_2^{2}}{2} + a, then obtain \\sin{(\\varepsilon{(\\phi_2,\\mathbf{E})})} = \\sin{(\\mathbf{E} \\phi_2 + \\frac{\\phi_2^{2}}{2} + a)}", "derivation": "\\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} = \\mathbf{E} + \\phi_2 and \\int \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} d\\phi_2 = \\int (\\mathbf{E} + \\phi_2) d\\phi_2 and \\int \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} d\\phi_2 = \\mathbf{E} \\phi_2 + \\frac{\\phi_2^{2}}{2} + a and \\sin{(\\int \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} d\\phi_2)} = \\sin{(\\mathbf{E} \\phi_2 + \\frac{\\phi_2^{2}}{2} + a)} and \\varepsilon{(\\phi_2,\\mathbf{E})} = \\int (\\mathbf{E} + \\phi_2) d\\phi_2 and \\int \\operatorname{v_{x}}{(\\phi_2,\\mathbf{E})} d\\phi_2 = \\varepsilon{(\\phi_2,\\mathbf{E})} and \\sin{(\\varepsilon{(\\phi_2,\\mathbf{E})})} = \\sin{(\\mathbf{E} \\phi_2 + \\frac{\\phi_2^{2}}{2} + a)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('a', commutative=True)))"], [["sin", 3], "Equality(sin(Integral(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), sin(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Integral(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(sin(Function('\\\\varepsilon')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), sin(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(n,B)} = \\frac{B}{n} and \\theta_{1}{(n,B)} = B + \\operatorname{v_{1}}{(n,B)} + 1, then obtain \\theta_{1}^{B}{(n,B)} \\operatorname{v_{1}}{(n,B)} = (B + \\frac{B}{n} + 1)^{B} \\operatorname{v_{1}}{(n,B)}", "derivation": "\\operatorname{v_{1}}{(n,B)} = \\frac{B}{n} and \\operatorname{v_{1}}{(n,B)} + 1 = \\frac{B}{n} + 1 and B + \\operatorname{v_{1}}{(n,B)} + 1 = B + \\frac{B}{n} + 1 and (B + \\operatorname{v_{1}}{(n,B)} + 1)^{B} = (B + \\frac{B}{n} + 1)^{B} and (B + \\operatorname{v_{1}}{(n,B)} + 1)^{B} \\operatorname{v_{1}}{(n,B)} = (B + \\frac{B}{n} + 1)^{B} \\operatorname{v_{1}}{(n,B)} and \\theta_{1}{(n,B)} = B + \\operatorname{v_{1}}{(n,B)} + 1 and \\theta_{1}^{B}{(n,B)} \\operatorname{v_{1}}{(n,B)} = (B + \\frac{B}{n} + 1)^{B} \\operatorname{v_{1}}{(n,B)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Integer(1)), Add(Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], [["add", 2, "Symbol('B', commutative=True)"], "Equality(Add(Symbol('B', commutative=True), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Integer(1)), Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Symbol('B', commutative=True), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Integer(1)), Symbol('B', commutative=True)), Pow(Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Symbol('B', commutative=True)))"], [["times", 4, "Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Integer(1)), Symbol('B', commutative=True)), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Symbol('B', commutative=True)), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('n', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Add(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Symbol('B', commutative=True)), Function('v_1')(Symbol('n', commutative=True), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\varphi{(n_{2})} = \\cos{(n_{2})}, then obtain \\frac{d}{d n_{2}} (\\varphi{(n_{2})} - \\cos{(n_{2})})^{2} \\int (\\varphi{(n_{2})} - \\cos{(n_{2})}) dn_{2} = \\frac{d}{d n_{2}} 0 \\int (\\varphi{(n_{2})} - \\cos{(n_{2})}) dn_{2}", "derivation": "\\varphi{(n_{2})} = \\cos{(n_{2})} and \\varphi{(n_{2})} - \\cos{(n_{2})} = 0 and (\\varphi{(n_{2})} - \\cos{(n_{2})})^{2} = 0 and \\frac{d}{d n_{2}} (\\varphi{(n_{2})} - \\cos{(n_{2})})^{2} = \\frac{d}{d n_{2}} 0 and \\int (\\varphi{(n_{2})} - \\cos{(n_{2})}) dn_{2} = \\int 0 dn_{2} and \\frac{d}{d n_{2}} (\\varphi{(n_{2})} - \\cos{(n_{2})})^{2} \\int 0 dn_{2} = \\frac{d}{d n_{2}} 0 \\int 0 dn_{2} and \\frac{d}{d n_{2}} (\\varphi{(n_{2})} - \\cos{(n_{2})})^{2} \\int (\\varphi{(n_{2})} - \\cos{(n_{2})}) dn_{2} = \\frac{d}{d n_{2}} 0 \\int (\\varphi{(n_{2})} - \\cos{(n_{2})}) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('n_2', commutative=True)), cos(Symbol('n_2', commutative=True)))"], [["minus", 1, "cos(Symbol('n_2', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(0))"], [["times", 2, "Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True))))"], "Equality(Pow(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)), Integer(0))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('n_2', commutative=True))))"], [["times", 4, "Integral(Integer(0), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Derivative(Pow(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Integer(0), Tuple(Symbol('n_2', commutative=True)))), Mul(Derivative(Integer(0), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Integer(0), Tuple(Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Derivative(Pow(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integer(2)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True)))), Mul(Derivative(Integer(0), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integral(Add(Function('\\\\varphi')(Symbol('n_2', commutative=True)), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{f})} = - \\mathbf{f}, then obtain \\mathbf{f} + (\\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})})^{2} + \\operatorname{F_{N}}{(\\mathbf{f})} = 0", "derivation": "\\operatorname{F_{N}}{(\\mathbf{f})} = - \\mathbf{f} and \\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})} = 0 and (\\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})})^{2} = 0 and - \\mathbf{f} + (\\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})})^{2} = - \\mathbf{f} and (\\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})})^{2} + \\operatorname{F_{N}}{(\\mathbf{f})} = \\operatorname{F_{N}}{(\\mathbf{f})} and \\mathbf{f} + (\\mathbf{f} + \\operatorname{F_{N}}{(\\mathbf{f})})^{2} + \\operatorname{F_{N}}{(\\mathbf{f})} = 0", "srepr_derivation": [["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"], [["times", 2, "Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(2)), Integer(0))"], [["minus", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(2))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(2)), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(2)), Function('F_N')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"]]}, {"prompt": "Given B{(V)} = \\log{(V)}, then derive (B{(V)} + \\int B{(V)} dV) \\int B{(V)} dV = (B{(V)} + \\int B{(V)} dV) (V \\log{(V)} - V + \\Psi^{\\dagger}), then obtain (\\log{(V)} + \\int \\log{(V)} dV) \\int \\log{(V)} dV = (\\log{(V)} + \\int \\log{(V)} dV) (V \\log{(V)} - V + \\Psi^{\\dagger})", "derivation": "B{(V)} = \\log{(V)} and \\int B{(V)} dV = \\int \\log{(V)} dV and (B{(V)} + \\int B{(V)} dV) \\int B{(V)} dV = (B{(V)} + \\int B{(V)} dV) \\int \\log{(V)} dV and (B{(V)} + \\int B{(V)} dV) \\int B{(V)} dV = (B{(V)} + \\int B{(V)} dV) (V \\log{(V)} - V + \\Psi^{\\dagger}) and (\\log{(V)} + \\int \\log{(V)} dV) \\int \\log{(V)} dV = (\\log{(V)} + \\int \\log{(V)} dV) (V \\log{(V)} - V + \\Psi^{\\dagger})", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["times", 2, "Add(Function('B')(Symbol('V', commutative=True)), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], "Equality(Mul(Add(Function('B')(Symbol('V', commutative=True)), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Add(Function('B')(Symbol('V', commutative=True)), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Function('B')(Symbol('V', commutative=True)), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Add(Function('B')(Symbol('V', commutative=True)), Integral(Function('B')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(log(Symbol('V', commutative=True)), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Add(log(Symbol('V', commutative=True)), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given s{(l)} = e^{l} and \\dot{x}{(l)} = e^{l}, then derive l e^{l} = l \\frac{d}{d l} \\dot{x}{(l)}, then obtain l e^{l} - l \\frac{d}{d l} e^{l} = l \\frac{d}{d l} \\dot{x}{(l)} - l \\frac{d}{d l} e^{l}", "derivation": "s{(l)} = e^{l} and \\frac{d}{d l} s{(l)} = \\frac{d}{d l} e^{l} and \\dot{x}{(l)} = e^{l} and \\frac{d}{d l} s{(l)} = \\frac{d}{d l} \\dot{x}{(l)} and \\frac{d}{d l} e^{l} = \\frac{d}{d l} \\dot{x}{(l)} and l \\frac{d}{d l} e^{l} = l \\frac{d}{d l} \\dot{x}{(l)} and l e^{l} = l \\frac{d}{d l} \\dot{x}{(l)} and l e^{l} - l \\frac{d}{d l} e^{l} = l \\frac{d}{d l} \\dot{x}{(l)} - l \\frac{d}{d l} e^{l}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('s')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 5, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Symbol('l', commutative=True), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Symbol('l', commutative=True), exp(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["minus", 7, "Mul(Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], "Equality(Add(Mul(Symbol('l', commutative=True), exp(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Mul(Symbol('l', commutative=True), Derivative(Function('\\\\dot{x}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"]]}, {"prompt": "Given V{(U,\\sigma_p)} = U + \\sigma_p, then obtain \\frac{- U + V{(U,\\sigma_p)}}{(U + V{(U,\\sigma_p)})^{2}} = \\frac{\\sigma_p}{(U + V{(U,\\sigma_p)})^{2}}", "derivation": "V{(U,\\sigma_p)} = U + \\sigma_p and U + V{(U,\\sigma_p)} = 2 U + \\sigma_p and - U + V{(U,\\sigma_p)} = \\sigma_p and \\frac{- U + V{(U,\\sigma_p)}}{U + V{(U,\\sigma_p)}} = \\frac{\\sigma_p}{U + V{(U,\\sigma_p)}} and \\frac{- U + V{(U,\\sigma_p)}}{2 U + \\sigma_p} = \\frac{\\sigma_p}{2 U + \\sigma_p} and \\frac{- U + V{(U,\\sigma_p)}}{(2 U + \\sigma_p)^{2}} = \\frac{\\sigma_p}{(2 U + \\sigma_p)^{2}} and \\frac{- U + V{(U,\\sigma_p)}}{(U + V{(U,\\sigma_p)})^{2}} = \\frac{\\sigma_p}{(U + V{(U,\\sigma_p)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))"], [["divide", 3, "Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"], [["times", 5, "Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-2))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Integer(2), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-2))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-2))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} = \\frac{\\varphi^*}{\\mathbf{J}_P} and q{(k)} = - 2 k, then obtain 2 \\mathbf{J}_P \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} - k q{(k)} = 2 \\mathbf{J}_P \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} + 2 k^{2}", "derivation": "\\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} = \\frac{\\varphi^*}{\\mathbf{J}_P} and \\mathbf{J}_P \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} = \\varphi^* and q{(k)} = - 2 k and - k q{(k)} = 2 k^{2} and 2 \\varphi^* - k q{(k)} = 2 \\varphi^* + 2 k^{2} and 2 \\mathbf{J}_P \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} - k q{(k)} = 2 \\mathbf{J}_P \\bar{\\h}{(\\mathbf{J}_P,\\varphi^*)} + 2 k^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))"], ["renaming_premise", "Equality(Function('q')(Symbol('k', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('k', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('k', commutative=True), Function('q')(Symbol('k', commutative=True))), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(2))))"], [["add", 4, "Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True), Function('q')(Symbol('k', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True), Function('q')(Symbol('k', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(n_{1},\\Psi_{nl})} = \\Psi_{nl} n_{1} and \\operatorname{g_{\\varepsilon}}{(n_{1},\\Psi_{nl})} = \\operatorname{v_{2}}^{\\Psi_{nl}}{(n_{1},\\Psi_{nl})}, then obtain 1 = \\frac{(\\Psi_{nl} n_{1})^{\\Psi_{nl}}}{\\operatorname{g_{\\varepsilon}}{(n_{1},\\Psi_{nl})}}", "derivation": "\\operatorname{v_{2}}{(n_{1},\\Psi_{nl})} = \\Psi_{nl} n_{1} and \\operatorname{v_{2}}^{\\Psi_{nl}}{(n_{1},\\Psi_{nl})} = (\\Psi_{nl} n_{1})^{\\Psi_{nl}} and \\operatorname{g_{\\varepsilon}}{(n_{1},\\Psi_{nl})} = \\operatorname{v_{2}}^{\\Psi_{nl}}{(n_{1},\\Psi_{nl})} and \\operatorname{g_{\\varepsilon}}{(n_{1},\\Psi_{nl})} = (\\Psi_{nl} n_{1})^{\\Psi_{nl}} and 1 = \\frac{(\\Psi_{nl} n_{1})^{\\Psi_{nl}}}{\\operatorname{g_{\\varepsilon}}{(n_{1},\\Psi_{nl})}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Function('v_2')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["divide", 4, "Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given u{(\\mathbf{J}_P,H,\\Psi_{nl})} = \\Psi_{nl}^{H} - \\mathbf{J}_P, then derive \\frac{\\partial}{\\partial H} u{(\\mathbf{J}_P,H,\\Psi_{nl})} = \\Psi_{nl}^{H} \\log{(\\Psi_{nl})}, then obtain \\int H \\frac{\\partial}{\\partial H} (\\Psi_{nl}^{H} - \\mathbf{J}_P) d\\mathbf{J}_P = \\int H \\Psi_{nl}^{H} \\log{(\\Psi_{nl})} d\\mathbf{J}_P", "derivation": "u{(\\mathbf{J}_P,H,\\Psi_{nl})} = \\Psi_{nl}^{H} - \\mathbf{J}_P and \\frac{\\partial}{\\partial H} u{(\\mathbf{J}_P,H,\\Psi_{nl})} = \\frac{\\partial}{\\partial H} (\\Psi_{nl}^{H} - \\mathbf{J}_P) and \\frac{\\partial}{\\partial H} u{(\\mathbf{J}_P,H,\\Psi_{nl})} = \\Psi_{nl}^{H} \\log{(\\Psi_{nl})} and H \\frac{\\partial}{\\partial H} u{(\\mathbf{J}_P,H,\\Psi_{nl})} = H \\Psi_{nl}^{H} \\log{(\\Psi_{nl})} and H \\frac{\\partial}{\\partial H} (\\Psi_{nl}^{H} - \\mathbf{J}_P) = H \\Psi_{nl}^{H} \\log{(\\Psi_{nl})} and \\int H \\frac{\\partial}{\\partial H} (\\Psi_{nl}^{H} - \\mathbf{J}_P) d\\mathbf{J}_P = \\int H \\Psi_{nl}^{H} \\log{(\\Psi_{nl})} d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), log(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["times", 3, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Derivative(Function('u')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('H', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), log(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('H', commutative=True), Derivative(Add(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), log(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Mul(Symbol('H', commutative=True), Derivative(Add(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('H', commutative=True)), log(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\chi,\\eta)} = \\chi + \\eta and \\mathbf{M}{(\\chi,\\eta)} = \\chi + \\eta, then obtain \\frac{\\int \\mathbf{M}{(\\chi,\\eta)} d\\chi}{\\eta} = \\frac{\\int (\\chi + \\eta) d\\chi}{\\eta}", "derivation": "\\tilde{g}{(\\chi,\\eta)} = \\chi + \\eta and \\mathbf{M}{(\\chi,\\eta)} = \\chi + \\eta and \\mathbf{M}{(\\chi,\\eta)} = \\tilde{g}{(\\chi,\\eta)} and \\int \\tilde{g}{(\\chi,\\eta)} d\\chi = \\int (\\chi + \\eta) d\\chi and \\frac{\\int \\tilde{g}{(\\chi,\\eta)} d\\chi}{\\eta} = \\frac{\\int (\\chi + \\eta) d\\chi}{\\eta} and \\frac{\\int \\mathbf{M}{(\\chi,\\eta)} d\\chi}{\\eta} = \\frac{\\int (\\chi + \\eta) d\\chi}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Function('\\\\tilde{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(t_{2},\\delta)} = \\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}), then derive \\theta_{2}^{\\delta}{(t_{2},\\delta)} = 1, then obtain ((\\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}))^{\\delta})^{- \\delta} = ((\\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}))^{\\delta})^{- 2 \\delta}", "derivation": "\\theta_{2}{(t_{2},\\delta)} = \\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}) and \\theta_{2}^{\\delta}{(t_{2},\\delta)} = (\\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}))^{\\delta} and \\theta_{2}^{\\delta}{(t_{2},\\delta)} = 1 and (\\theta_{2}^{\\delta}{(t_{2},\\delta)})^{\\delta} = 1 and 1 = (\\theta_{2}^{\\delta}{(t_{2},\\delta)})^{- \\delta} and (\\theta_{2}^{\\delta}{(t_{2},\\delta)})^{- \\delta} = (\\theta_{2}^{\\delta}{(t_{2},\\delta)})^{- 2 \\delta} and ((\\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}))^{\\delta})^{- \\delta} = ((\\frac{\\partial}{\\partial \\delta} (\\delta + t_{2}))^{\\delta})^{- 2 \\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(1))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(1))"], [["divide", 4, "Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Integer(1), Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))"], [["times", 5, "Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))"], "Equality(Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Pow(Function('\\\\theta_2')(Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Pow(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Pow(Derivative(Add(Symbol('\\\\delta', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\phi,\\mathbf{g})} = \\frac{\\cos{(\\phi)}}{\\mathbf{g}}, then obtain \\iint \\varphi^{*}{(\\phi,\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} - 1 = \\iint \\frac{\\cos{(\\phi)}}{\\mathbf{g}} d\\mathbf{g} d\\mathbf{g} - 1", "derivation": "\\varphi^{*}{(\\phi,\\mathbf{g})} = \\frac{\\cos{(\\phi)}}{\\mathbf{g}} and \\int \\varphi^{*}{(\\phi,\\mathbf{g})} d\\mathbf{g} = \\int \\frac{\\cos{(\\phi)}}{\\mathbf{g}} d\\mathbf{g} and \\iint \\varphi^{*}{(\\phi,\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} = \\iint \\frac{\\cos{(\\phi)}}{\\mathbf{g}} d\\mathbf{g} d\\mathbf{g} and \\iint \\varphi^{*}{(\\phi,\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} - 1 = \\iint \\frac{\\cos{(\\phi)}}{\\mathbf{g}} d\\mathbf{g} d\\mathbf{g} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)), Add(Integral(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain (\\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbb{I}}{(\\mathbb{I})} + (e^{\\mathbb{I}})^{\\mathbb{I}})^{2} = 4 (e^{\\mathbb{I}})^{2 \\mathbb{I}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} = e^{\\mathbb{I}} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbb{I}}{(\\mathbb{I})} = (e^{\\mathbb{I}})^{\\mathbb{I}} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbb{I}}{(\\mathbb{I})} + (e^{\\mathbb{I}})^{\\mathbb{I}} = 2 (e^{\\mathbb{I}})^{\\mathbb{I}} and (\\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbb{I}}{(\\mathbb{I})} + (e^{\\mathbb{I}})^{\\mathbb{I}})^{2} = 4 (e^{\\mathbb{I}})^{2 \\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 2, "Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Integer(2)), Mul(Integer(4), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\rho_b,y)} = - \\rho_b + y and G{(y)} = \\frac{1}{y}, then obtain G{(y)} + \\log{(\\rho_b \\operatorname{E_{n}}{(\\rho_b,y)})} = \\log{(\\rho_b \\operatorname{E_{n}}{(\\rho_b,y)})} + \\frac{1}{y}", "derivation": "\\operatorname{E_{n}}{(\\rho_b,y)} = - \\rho_b + y and \\rho_b \\operatorname{E_{n}}{(\\rho_b,y)} = \\rho_b (- \\rho_b + y) and \\log{(\\rho_b \\operatorname{E_{n}}{(\\rho_b,y)})} = \\log{(\\rho_b (- \\rho_b + y))} and G{(y)} = \\frac{1}{y} and G{(y)} + \\log{(\\rho_b (- \\rho_b + y))} = \\log{(\\rho_b (- \\rho_b + y))} + \\frac{1}{y} and G{(y)} + \\log{(\\rho_b \\operatorname{E_{n}}{(\\rho_b,y)})} = \\log{(\\rho_b \\operatorname{E_{n}}{(\\rho_b,y)})} + \\frac{1}{y}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\rho_b', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('E_n')(Symbol('\\\\rho_b', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True))))"], [["log", 2], "Equality(log(Mul(Symbol('\\\\rho_b', commutative=True), Function('E_n')(Symbol('\\\\rho_b', commutative=True), Symbol('y', commutative=True)))), log(Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True)))))"], ["renaming_premise", "Equality(Function('G')(Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1)))"], [["add", 4, "log(Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True))))"], "Equality(Add(Function('G')(Symbol('y', commutative=True)), log(Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True))))), Add(log(Mul(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('y', commutative=True)))), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('G')(Symbol('y', commutative=True)), log(Mul(Symbol('\\\\rho_b', commutative=True), Function('E_n')(Symbol('\\\\rho_b', commutative=True), Symbol('y', commutative=True))))), Add(log(Mul(Symbol('\\\\rho_b', commutative=True), Function('E_n')(Symbol('\\\\rho_b', commutative=True), Symbol('y', commutative=True)))), Pow(Symbol('y', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\rho{(\\theta)} = \\log{(\\theta)}, then obtain (\\frac{d}{d \\theta} \\int 0 d\\theta) \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta = (\\frac{d}{d \\theta} \\int 0 d\\theta) \\int 0 d\\theta", "derivation": "\\rho{(\\theta)} = \\log{(\\theta)} and \\rho{(\\theta)} - \\log{(\\theta)} = 0 and \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta = \\int 0 d\\theta and \\frac{d}{d \\theta} \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta = \\frac{d}{d \\theta} \\int 0 d\\theta and (\\frac{d}{d \\theta} \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta) \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta = (\\frac{d}{d \\theta} \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta) \\int 0 d\\theta and (\\frac{d}{d \\theta} \\int 0 d\\theta) \\int (\\rho{(\\theta)} - \\log{(\\theta)}) d\\theta = (\\frac{d}{d \\theta} \\int 0 d\\theta) \\int 0 d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Derivative(Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integral(Add(Function('\\\\rho')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integral(Integer(0), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(E)} = \\cos{(\\cos{(E)})}, then derive \\log{(E + \\mathbf{r})} = \\log{(\\int \\frac{\\cos{(\\cos{(E)})}}{\\hat{H}{(E)}} dE)}, then obtain \\log{(E + \\mathbf{r})} = \\log{(\\int 1 dE)}", "derivation": "\\hat{H}{(E)} = \\cos{(\\cos{(E)})} and 1 = \\frac{\\cos{(\\cos{(E)})}}{\\hat{H}{(E)}} and \\int 1 dE = \\int \\frac{\\cos{(\\cos{(E)})}}{\\hat{H}{(E)}} dE and \\log{(\\int 1 dE)} = \\log{(\\int \\frac{\\cos{(\\cos{(E)})}}{\\hat{H}{(E)}} dE)} and \\log{(E + \\mathbf{r})} = \\log{(\\int \\frac{\\cos{(\\cos{(E)})}}{\\hat{H}{(E)}} dE)} and \\log{(E + \\mathbf{r})} = \\log{(\\int 1 dE)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('E', commutative=True)), cos(cos(Symbol('E', commutative=True))))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}')(Symbol('E', commutative=True)), Integer(-1)), cos(cos(Symbol('E', commutative=True)))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E', commutative=True))), Integral(Mul(Pow(Function('\\\\hat{H}')(Symbol('E', commutative=True)), Integer(-1)), cos(cos(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["log", 3], "Equality(log(Integral(Integer(1), Tuple(Symbol('E', commutative=True)))), log(Integral(Mul(Pow(Function('\\\\hat{H}')(Symbol('E', commutative=True)), Integer(-1)), cos(cos(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(log(Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), log(Integral(Mul(Pow(Function('\\\\hat{H}')(Symbol('E', commutative=True)), Integer(-1)), cos(cos(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(log(Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), log(Integral(Integer(1), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(\\pi,\\mathbf{J}_f)} = \\mathbf{J}_f + \\pi, then obtain \\dot{x}{(\\pi,\\mathbf{J}_f)} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\dot{x}{(\\pi,\\mathbf{J}_f)} = \\dot{x}{(\\pi,\\mathbf{J}_f)}", "derivation": "\\dot{x}{(\\pi,\\mathbf{J}_f)} = \\mathbf{J}_f + \\pi and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\dot{x}{(\\pi,\\mathbf{J}_f)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f + \\pi) and \\dot{x}{(\\pi,\\mathbf{J}_f)} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\dot{x}{(\\pi,\\mathbf{J}_f)} = \\dot{x}{(\\pi,\\mathbf{J}_f)} \\frac{\\partial}{\\partial \\mathbf{J}_f} (\\mathbf{J}_f + \\pi) and \\dot{x}{(\\pi,\\mathbf{J}_f)} \\frac{\\partial}{\\partial \\mathbf{J}_f} \\dot{x}{(\\pi,\\mathbf{J}_f)} = \\dot{x}{(\\pi,\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Mul(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Function('\\\\dot{x}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given x{(\\mathbf{B},E_{n},M_{E})} = E_{n} M_{E} - \\mathbf{B}, then derive \\frac{\\partial}{\\partial M_{E}} x{(\\mathbf{B},E_{n},M_{E})} = E_{n}, then obtain \\frac{E_{n}^{2}}{2} + k = \\frac{E_{n}^{2}}{2} + \\hat{H}_l", "derivation": "x{(\\mathbf{B},E_{n},M_{E})} = E_{n} M_{E} - \\mathbf{B} and \\frac{\\partial}{\\partial M_{E}} x{(\\mathbf{B},E_{n},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (E_{n} M_{E} - \\mathbf{B}) and \\frac{\\partial}{\\partial M_{E}} x{(\\mathbf{B},E_{n},M_{E})} = E_{n} and E_{n} = \\frac{\\partial}{\\partial M_{E}} (E_{n} M_{E} - \\mathbf{B}) and \\int E_{n} dE_{n} = \\int \\frac{\\partial}{\\partial M_{E}} (E_{n} M_{E} - \\mathbf{B}) dE_{n} and \\frac{E_{n}^{2}}{2} + k = \\frac{E_{n}^{2}}{2} + \\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Symbol('E_n', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('E_n', commutative=True), Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('E_n', commutative=True)"], "Equality(Integral(Symbol('E_n', commutative=True), Tuple(Symbol('E_n', commutative=True))), Integral(Derivative(Add(Mul(Symbol('E_n', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('k', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('E_n', commutative=True), Integer(2))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given A{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain \\frac{d}{d E_{\\lambda}} \\frac{1}{(- (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} - 1)^{2}} = \\frac{d}{d E_{\\lambda}} \\frac{1}{4}", "derivation": "A{(E_{\\lambda})} = e^{E_{\\lambda}} and 0 = - A{(E_{\\lambda})} + e^{E_{\\lambda}} and 0^{E_{\\lambda}} = (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} and - 0^{E_{\\lambda}} = - (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} and - 0^{E_{\\lambda}} - e^{E_{\\lambda}} = - (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} - e^{E_{\\lambda}} and - (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} - 1 = -2 and \\frac{1}{(- (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} - 1)^{2}} = \\frac{1}{4} and \\frac{d}{d E_{\\lambda}} \\frac{1}{(- (- A{(E_{\\lambda})} + e^{E_{\\lambda}})^{E_{\\lambda}} - 1)^{2}} = \\frac{d}{d E_{\\lambda}} \\frac{1}{4}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 1, "Function('A')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 4, "exp(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1)), Integer(-2))"], [["power", 6, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1)), Integer(-2)), Rational(1, 4))"], [["differentiate", 7, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1)), Integer(-2)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Rational(1, 4), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(U,c)} = \\sin{(U^{c})} and I{(U,c)} = U^{c} and \\chi{(U,c)} = \\mathbf{f}^{U}{(U,c)}, then obtain \\frac{\\chi{(U,c)}}{\\frac{\\partial}{\\partial c} \\sin{(U^{c})}} = \\frac{\\sin^{U}{(I{(U,c)})}}{\\frac{\\partial}{\\partial c} \\sin{(U^{c})}}", "derivation": "\\mathbf{f}{(U,c)} = \\sin{(U^{c})} and I{(U,c)} = U^{c} and \\mathbf{f}{(U,c)} = \\sin{(I{(U,c)})} and \\mathbf{f}^{U}{(U,c)} = \\sin^{U}{(I{(U,c)})} and \\chi{(U,c)} = \\mathbf{f}^{U}{(U,c)} and \\chi{(U,c)} = \\sin^{U}{(I{(U,c)})} and \\frac{\\chi{(U,c)}}{\\frac{\\partial}{\\partial c} \\sin{(U^{c})}} = \\frac{\\sin^{U}{(I{(U,c)})}}{\\frac{\\partial}{\\partial c} \\sin{(U^{c})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('U', commutative=True), Symbol('c', commutative=True)), sin(Pow(Symbol('U', commutative=True), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('U', commutative=True), Symbol('c', commutative=True)), sin(Function('I')(Symbol('U', commutative=True), Symbol('c', commutative=True))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Symbol('U', commutative=True)), Pow(sin(Function('I')(Symbol('U', commutative=True), Symbol('c', commutative=True))), Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Pow(Function('\\\\mathbf{f}')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Pow(sin(Function('I')(Symbol('U', commutative=True), Symbol('c', commutative=True))), Symbol('U', commutative=True)))"], [["divide", 6, "Derivative(sin(Pow(Symbol('U', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\chi')(Symbol('U', commutative=True), Symbol('c', commutative=True)), Pow(Derivative(sin(Pow(Symbol('U', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(sin(Function('I')(Symbol('U', commutative=True), Symbol('c', commutative=True))), Symbol('U', commutative=True)), Pow(Derivative(sin(Pow(Symbol('U', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(W)} = W, then derive \\int \\phi_{1}{(W)} dW = \\frac{W^{2}}{2} + f_{\\mathbf{p}}, then obtain \\frac{W^{2}}{2} + \\lambda - \\frac{1}{\\frac{W^{2}}{2} + \\lambda} = \\frac{W^{2}}{2} + f_{\\mathbf{p}} - \\frac{1}{\\frac{W^{2}}{2} + \\lambda}", "derivation": "\\phi_{1}{(W)} = W and \\int \\phi_{1}{(W)} dW = \\int W dW and \\int \\phi_{1}{(W)} dW = \\frac{W^{2}}{2} + f_{\\mathbf{p}} and \\int \\phi_{1}{(W)} dW - \\frac{1}{\\int W dW} = \\frac{W^{2}}{2} + f_{\\mathbf{p}} - \\frac{1}{\\int W dW} and \\int W dW - \\frac{1}{\\int W dW} = \\frac{W^{2}}{2} + f_{\\mathbf{p}} - \\frac{1}{\\int W dW} and \\frac{W^{2}}{2} + \\lambda - \\frac{1}{\\frac{W^{2}}{2} + \\lambda} = \\frac{W^{2}}{2} + f_{\\mathbf{p}} - \\frac{1}{\\frac{W^{2}}{2} + \\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["minus", 3, "Pow(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Integer(-1))"], "Equality(Add(Integral(Function('\\\\phi_1')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Integer(-1)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Pow(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Integer(-1)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Pow(Integral(Symbol('W', commutative=True), Tuple(Symbol('W', commutative=True))), Integer(-1)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)), Integer(-1)))), Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('W', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given C{(g,C_{d})} = - C_{d} + g and \\operatorname{A_{2}}{(g,C_{d})} = \\frac{\\partial}{\\partial g} C{(g,C_{d})}, then derive (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} = 1, then obtain 2 (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} = (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} + 1", "derivation": "C{(g,C_{d})} = - C_{d} + g and \\frac{\\partial}{\\partial g} C{(g,C_{d})} = \\frac{\\partial}{\\partial g} (- C_{d} + g) and (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} = (\\frac{\\partial}{\\partial g} (- C_{d} + g))^{C_{d}} and (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} = 1 and \\operatorname{A_{2}}{(g,C_{d})} = \\frac{\\partial}{\\partial g} C{(g,C_{d})} and \\operatorname{A_{2}}^{C_{d}}{(g,C_{d})} = 1 and 2 \\operatorname{A_{2}}^{C_{d}}{(g,C_{d})} = \\operatorname{A_{2}}^{C_{d}}{(g,C_{d})} + 1 and 2 (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} = (\\frac{\\partial}{\\partial g} C{(g,C_{d})})^{C_{d}} + 1", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('C_d', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Integer(1))"], [["add", 6, "Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Add(Pow(Function('A_2')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Integer(2), Pow(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('C_d', commutative=True))), Add(Pow(Derivative(Function('C')(Symbol('g', commutative=True), Symbol('C_d', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Integer(1)))"]]}, {"prompt": "Given C{(Z,F_{H})} = F_{H} + Z, then derive (\\frac{F_{H}^{2}}{2} + F_{H} Z + V) \\int C{(Z,F_{H})} dF_{H} = (\\frac{F_{H}^{2}}{2} + F_{H} Z + V)^{2}, then obtain (\\frac{F_{H}^{2}}{2} + F_{H} Z + V) \\int (F_{H} + Z) dF_{H} + \\int (F_{H} + Z) dF_{H} = (\\frac{F_{H}^{2}}{2} + F_{H} Z + V)^{2} + \\int (F_{H} + Z) dF_{H}", "derivation": "C{(Z,F_{H})} = F_{H} + Z and \\int C{(Z,F_{H})} dF_{H} = \\int (F_{H} + Z) dF_{H} and (\\int (F_{H} + Z) dF_{H}) \\int C{(Z,F_{H})} dF_{H} = (\\int (F_{H} + Z) dF_{H})^{2} and (\\frac{F_{H}^{2}}{2} + F_{H} Z + V) \\int C{(Z,F_{H})} dF_{H} = (\\frac{F_{H}^{2}}{2} + F_{H} Z + V)^{2} and (\\frac{F_{H}^{2}}{2} + F_{H} Z + V) \\int C{(Z,F_{H})} dF_{H} + \\int (F_{H} + Z) dF_{H} = (\\frac{F_{H}^{2}}{2} + F_{H} Z + V)^{2} + \\int (F_{H} + Z) dF_{H} and (\\frac{F_{H}^{2}}{2} + F_{H} Z + V) \\int (F_{H} + Z) dF_{H} + \\int (F_{H} + Z) dF_{H} = (\\frac{F_{H}^{2}}{2} + F_{H} Z + V)^{2} + \\int (F_{H} + Z) dF_{H}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('C')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["times", 2, "Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Function('C')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Pow(Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integral(Function('C')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integer(2)))"], [["add", 4, "Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integral(Function('C')(Symbol('Z', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integer(2)), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Symbol('V', commutative=True)), Integer(2)), Integral(Add(Symbol('F_H', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(a^{\\dagger},\\mathbf{J}_P)} = (a^{\\dagger})^{\\mathbf{J}_P} and \\operatorname{E_{\\lambda}}{(A)} = e^{A}, then obtain (\\frac{\\hat{H}_l^{2}{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int \\operatorname{E_{\\lambda}}{(A)} dA = (\\frac{\\hat{H}_l^{2}{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int e^{A} dA", "derivation": "\\hat{H}_l{(a^{\\dagger},\\mathbf{J}_P)} = (a^{\\dagger})^{\\mathbf{J}_P} and \\operatorname{E_{\\lambda}}{(A)} = e^{A} and \\int \\operatorname{E_{\\lambda}}{(A)} dA = \\int e^{A} dA and (\\frac{(a^{\\dagger})^{\\mathbf{J}_P} \\hat{H}_l{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int \\operatorname{E_{\\lambda}}{(A)} dA = (\\frac{(a^{\\dagger})^{\\mathbf{J}_P} \\hat{H}_l{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int e^{A} dA and (\\frac{\\hat{H}_l^{2}{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int \\operatorname{E_{\\lambda}}{(A)} dA = (\\frac{\\hat{H}_l^{2}{(a^{\\dagger},\\mathbf{J}_P)}}{a^{\\dagger}})^{\\mathbf{J}_P} \\int e^{A} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 3, "Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Function('E_{\\\\lambda}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Function('E_{\\\\lambda}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_l')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(2))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given J{(U)} = \\log{(U)}, then derive (U + J{(U)}) \\frac{d}{d U} J{(U)} = \\frac{U + J{(U)}}{U}, then obtain (U + J{(U)}) \\frac{d}{d U} J{(U)} - J{(U)} = - J{(U)} + \\frac{U + J{(U)}}{U}", "derivation": "J{(U)} = \\log{(U)} and \\frac{d}{d U} J{(U)} = \\frac{d}{d U} \\log{(U)} and (U + J{(U)}) \\frac{d}{d U} J{(U)} = (U + J{(U)}) \\frac{d}{d U} \\log{(U)} and (U + J{(U)}) \\frac{d}{d U} J{(U)} = \\frac{U + J{(U)}}{U} and (U + \\log{(U)}) \\frac{d}{d U} \\log{(U)} = \\frac{U + \\log{(U)}}{U} and (U + \\log{(U)}) \\frac{d}{d U} J{(U)} = \\frac{U + \\log{(U)}}{U} and (U + \\log{(U)}) \\frac{d}{d U} J{(U)} - J{(U)} = - J{(U)} + \\frac{U + \\log{(U)}}{U} and (U + J{(U)}) \\frac{d}{d U} J{(U)} - J{(U)} = - J{(U)} + \\frac{U + J{(U)}}{U}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["times", 2, "Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True)))"], "Equality(Mul(Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True))), Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True))), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True))), Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Derivative(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True)))))"], [["minus", 6, "Function('J')(Symbol('U', commutative=True))"], "Equality(Add(Mul(Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Function('J')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Function('J')(Symbol('U', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), log(Symbol('U', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Mul(Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True))), Derivative(Function('J')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Function('J')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Function('J')(Symbol('U', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Function('J')(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{E}{(U)} = e^{U}, then obtain \\frac{(U + \\mathbf{E}{(U)}) (- e^{U} - 1)}{(U + e^{U})^{2}} + \\frac{\\frac{d}{d U} \\mathbf{E}{(U)} + 1}{U + e^{U}} = 0", "derivation": "\\mathbf{E}{(U)} = e^{U} and U + \\mathbf{E}{(U)} = U + e^{U} and \\frac{U + \\mathbf{E}{(U)}}{U + e^{U}} = 1 and \\frac{d}{d U} \\frac{U + \\mathbf{E}{(U)}}{U + e^{U}} = \\frac{d}{d U} 1 and \\frac{(U + \\mathbf{E}{(U)}) (- e^{U} - 1)}{(U + e^{U})^{2}} + \\frac{\\frac{d}{d U} \\mathbf{E}{(U)} + 1}{U + e^{U}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('\\\\mathbf{E}')(Symbol('U', commutative=True))), Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True))))"], [["divide", 2, "Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True)))"], "Equality(Mul(Add(Symbol('U', commutative=True), Function('\\\\mathbf{E}')(Symbol('U', commutative=True))), Pow(Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('U', commutative=True), Function('\\\\mathbf{E}')(Symbol('U', commutative=True))), Pow(Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True))), Integer(-1))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Symbol('U', commutative=True), Function('\\\\mathbf{E}')(Symbol('U', commutative=True))), Pow(Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), exp(Symbol('U', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('U', commutative=True), exp(Symbol('U', commutative=True))), Integer(-1)), Add(Derivative(Function('\\\\mathbf{E}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(U,m_{s})} = - U + m_{s}, then obtain - \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1)^{2}}{U} + \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1) (- U + m_{s} - 1)}{U} = 0", "derivation": "\\hat{H}_{\\lambda}{(U,m_{s})} = - U + m_{s} and \\hat{H}_{\\lambda}{(U,m_{s})} - 1 = - U + m_{s} - 1 and - \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1)^{2}}{U} = - \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1) (- U + m_{s} - 1)}{U} and - \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1)^{2}}{U} + \\frac{(\\hat{H}_{\\lambda}{(U,m_{s})} - 1) (- U + m_{s} - 1)}{U} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m_s', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)))"], [["times", 2, "Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Integer(2))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m_s', commutative=True), Integer(-1))))"], [["minus", 3, "Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Integer(2))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{r},A_{z})} = \\sin{(A_{z}^{\\mathbf{r}})}, then derive \\frac{\\partial}{\\partial A_{z}} \\mathbf{B}{(\\mathbf{r},A_{z})} = \\frac{A_{z}^{\\mathbf{r}} \\mathbf{r} \\cos{(A_{z}^{\\mathbf{r}})}}{A_{z}}, then obtain \\frac{\\partial}{\\partial A_{z}} \\sin{(A_{z}^{\\mathbf{r}})} = \\frac{A_{z}^{\\mathbf{r}} \\mathbf{r} \\cos{(A_{z}^{\\mathbf{r}})}}{A_{z}}", "derivation": "\\mathbf{B}{(\\mathbf{r},A_{z})} = \\sin{(A_{z}^{\\mathbf{r}})} and \\frac{\\partial}{\\partial A_{z}} \\mathbf{B}{(\\mathbf{r},A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\sin{(A_{z}^{\\mathbf{r}})} and \\frac{\\partial}{\\partial A_{z}} \\mathbf{B}{(\\mathbf{r},A_{z})} = \\frac{A_{z}^{\\mathbf{r}} \\mathbf{r} \\cos{(A_{z}^{\\mathbf{r}})}}{A_{z}} and \\frac{\\partial}{\\partial A_{z}} \\sin{(A_{z}^{\\mathbf{r}})} = \\frac{A_{z}^{\\mathbf{r}} \\mathbf{r} \\cos{(A_{z}^{\\mathbf{r}})}}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True), cos(Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True), cos(Pow(Symbol('A_z', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given H{(A)} = \\log{(A)}, then derive \\phi + H{(A)} = y + \\log{(A)}, then obtain \\frac{\\partial}{\\partial y} (y + H{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA) = \\frac{\\partial}{\\partial y} (y + \\log{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA)", "derivation": "H{(A)} = \\log{(A)} and \\frac{d}{d A} H{(A)} = \\frac{d}{d A} \\log{(A)} and \\int \\frac{d}{d A} H{(A)} dA = \\int \\frac{d}{d A} \\log{(A)} dA and \\phi + H{(A)} = y + \\log{(A)} and \\phi + H{(A)} = y + H{(A)} and \\phi + H{(A)} - \\int \\frac{d}{d A} H{(A)} dA = y + \\log{(A)} - \\int \\frac{d}{d A} H{(A)} dA and \\phi + H{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA = y + \\log{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA and \\frac{\\partial}{\\partial y} (\\phi + H{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA) = \\frac{\\partial}{\\partial y} (y + \\log{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA) and \\frac{\\partial}{\\partial y} (y + H{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA) = \\frac{\\partial}{\\partial y} (y + \\log{(A)} - \\int \\frac{d}{d A} \\log{(A)} dA)", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Derivative(Function('H')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('H')(Symbol('A', commutative=True))), Add(Symbol('y', commutative=True), log(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('H')(Symbol('A', commutative=True))), Add(Symbol('y', commutative=True), Function('H')(Symbol('A', commutative=True))))"], [["minus", 4, "Integral(Derivative(Function('H')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('H')(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(Function('H')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Add(Symbol('y', commutative=True), log(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(Function('H')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('H')(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Add(Symbol('y', commutative=True), log(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))))"], [["differentiate", 7, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\phi', commutative=True), Function('H')(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('y', commutative=True), log(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Derivative(Add(Symbol('y', commutative=True), Function('H')(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('y', commutative=True), log(Symbol('A', commutative=True)), Mul(Integer(-1), Integral(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(h,z)} = e^{h z}, then obtain \\frac{(- z + \\int \\frac{\\mathbf{F}{(h,z)}}{h} dh) e^{h z}}{h^{2}} = \\frac{(- z + \\int \\frac{e^{h z}}{h} dh) e^{h z}}{h^{2}}", "derivation": "\\mathbf{F}{(h,z)} = e^{h z} and \\frac{\\mathbf{F}{(h,z)}}{h} = \\frac{e^{h z}}{h} and \\int \\frac{\\mathbf{F}{(h,z)}}{h} dh = \\int \\frac{e^{h z}}{h} dh and - z + \\int \\frac{\\mathbf{F}{(h,z)}}{h} dh = - z + \\int \\frac{e^{h z}}{h} dh and \\frac{- z + \\int \\frac{\\mathbf{F}{(h,z)}}{h} dh}{h} = \\frac{- z + \\int \\frac{e^{h z}}{h} dh}{h} and \\frac{(- z + \\int \\frac{\\mathbf{F}{(h,z)}}{h} dh) e^{h z}}{h^{2}} = \\frac{(- z + \\int \\frac{e^{h z}}{h} dh) e^{h z}}{h^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True))))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('h', commutative=True))))"], [["minus", 3, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('h', commutative=True)))))"], [["times", 4, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('h', commutative=True))))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('h', commutative=True))))))"], [["times", 5, "Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True))))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('h', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('h', commutative=True)))), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))), Mul(Pow(Symbol('h', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('h', commutative=True)))), exp(Mul(Symbol('h', commutative=True), Symbol('z', commutative=True)))))"]]}, {"prompt": "Given q{(x^\\prime)} = \\sin{(\\log{(x^\\prime)})}, then obtain \\frac{d}{d x^\\prime} \\int q{(x^\\prime)} \\sin^{2}{(\\log{(x^\\prime)})} dx^\\prime = \\frac{d}{d x^\\prime} \\int \\sin^{3}{(\\log{(x^\\prime)})} dx^\\prime", "derivation": "q{(x^\\prime)} = \\sin{(\\log{(x^\\prime)})} and q{(x^\\prime)} \\sin{(\\log{(x^\\prime)})} = \\sin^{2}{(\\log{(x^\\prime)})} and q{(x^\\prime)} \\sin^{2}{(\\log{(x^\\prime)})} = \\sin^{3}{(\\log{(x^\\prime)})} and \\int q{(x^\\prime)} \\sin^{2}{(\\log{(x^\\prime)})} dx^\\prime = \\int \\sin^{3}{(\\log{(x^\\prime)})} dx^\\prime and \\frac{d}{d x^\\prime} \\int q{(x^\\prime)} \\sin^{2}{(\\log{(x^\\prime)})} dx^\\prime = \\frac{d}{d x^\\prime} \\int \\sin^{3}{(\\log{(x^\\prime)})} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('x^\\\\prime', commutative=True)), sin(log(Symbol('x^\\\\prime', commutative=True))))"], [["times", 1, "sin(log(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('x^\\\\prime', commutative=True)), sin(log(Symbol('x^\\\\prime', commutative=True)))), Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(2)))"], [["times", 2, "sin(log(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('q')(Symbol('x^\\\\prime', commutative=True)), Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(2))), Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(3)))"], [["integrate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Function('q')(Symbol('x^\\\\prime', commutative=True)), Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(2))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(3)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('q')(Symbol('x^\\\\prime', commutative=True)), Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(2))), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(log(Symbol('x^\\\\prime', commutative=True))), Integer(3)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(\\hat{p}_0)} = \\log{(\\hat{p}_0)}, then obtain \\int \\frac{d^{2}}{d \\hat{p}_0^{2}} m{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\frac{d^{2}}{d \\hat{p}_0^{2}} \\log{(\\hat{p}_0)} d\\hat{p}_0", "derivation": "m{(\\hat{p}_0)} = \\log{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} m{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\log{(\\hat{p}_0)} and \\frac{d^{2}}{d \\hat{p}_0^{2}} m{(\\hat{p}_0)} = \\frac{d^{2}}{d \\hat{p}_0^{2}} \\log{(\\hat{p}_0)} and \\int \\frac{d^{2}}{d \\hat{p}_0^{2}} m{(\\hat{p}_0)} d\\hat{p}_0 = \\int \\frac{d^{2}}{d \\hat{p}_0^{2}} \\log{(\\hat{p}_0)} d\\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\hat{p}_0', commutative=True)), log(Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Derivative(log(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Derivative(Function('m')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Derivative(log(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})} and \\operatorname{C_{2}}{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{2}, then obtain \\operatorname{C_{2}}{(\\mathbf{M})} = \\Omega{(\\mathbf{M})} \\log{(\\cos{(\\mathbf{M})})}", "derivation": "\\Omega{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})} and \\Omega{(\\mathbf{M})} \\log{(\\cos{(\\mathbf{M})})} = \\log{(\\cos{(\\mathbf{M})})}^{2} and \\operatorname{C_{2}}{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{2} and \\operatorname{C_{2}}{(\\mathbf{M})} = \\Omega{(\\mathbf{M})} \\log{(\\cos{(\\mathbf{M})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{M}', commutative=True)), log(cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 1, "log(cos(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{M}', commutative=True)), log(cos(Symbol('\\\\mathbf{M}', commutative=True)))), Pow(log(cos(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{M}', commutative=True)), log(cos(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\rho_f,n,y)} = \\frac{y}{\\rho_f n} and u{(\\rho_f,n,y)} = y + \\delta{(\\rho_f,n,y)}, then obtain u{(\\rho_f,n,y)} = y + \\frac{y}{\\rho_f n}", "derivation": "\\delta{(\\rho_f,n,y)} = \\frac{y}{\\rho_f n} and y + \\delta{(\\rho_f,n,y)} = y + \\frac{y}{\\rho_f n} and u{(\\rho_f,n,y)} = y + \\delta{(\\rho_f,n,y)} and u{(\\rho_f,n,y)} = y + \\frac{y}{\\rho_f n}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Add(Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\rho_f', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Function('\\\\delta')(Symbol('\\\\rho_f', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('u')(Symbol('\\\\rho_f', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Add(Symbol('y', commutative=True), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given M{(f,\\mathbf{J}_f)} = \\mathbf{J}_f f and J{(f,\\mathbf{J}_f)} = \\int \\mathbf{J}_f f df, then obtain (\\frac{\\partial}{\\partial f} \\int M{(f,\\mathbf{J}_f)} df)^{\\mathbf{J}_f} = (\\frac{\\partial}{\\partial f} J{(f,\\mathbf{J}_f)})^{\\mathbf{J}_f}", "derivation": "M{(f,\\mathbf{J}_f)} = \\mathbf{J}_f f and \\int M{(f,\\mathbf{J}_f)} df = \\int \\mathbf{J}_f f df and J{(f,\\mathbf{J}_f)} = \\int \\mathbf{J}_f f df and \\int M{(f,\\mathbf{J}_f)} df = J{(f,\\mathbf{J}_f)} and \\frac{\\partial}{\\partial f} J{(f,\\mathbf{J}_f)} = \\frac{\\partial}{\\partial f} \\int \\mathbf{J}_f f df and (\\frac{\\partial}{\\partial f} J{(f,\\mathbf{J}_f)})^{\\mathbf{J}_f} = (\\frac{\\partial}{\\partial f} \\int \\mathbf{J}_f f df)^{\\mathbf{J}_f} and (\\frac{\\partial}{\\partial f} \\int M{(f,\\mathbf{J}_f)} df)^{\\mathbf{J}_f} = (\\frac{\\partial}{\\partial f} \\int \\mathbf{J}_f f df)^{\\mathbf{J}_f} and (\\frac{\\partial}{\\partial f} \\int M{(f,\\mathbf{J}_f)} df)^{\\mathbf{J}_f} = (\\frac{\\partial}{\\partial f} J{(f,\\mathbf{J}_f)})^{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('M')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('M')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True))), Function('J')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Derivative(Function('J')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Derivative(Integral(Function('M')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Pow(Derivative(Integral(Function('M')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(Function('J')(Symbol('f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(g,W)} = W g and \\operatorname{L_{\\varepsilon}}{(g,W)} = (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W}, then obtain (- W g + \\frac{\\partial}{\\partial g} \\operatorname{L_{\\varepsilon}}{(g,W)})^{W} = (- W g + \\frac{d}{d g} 1)^{W}", "derivation": "\\mathbf{g}{(g,W)} = W g and \\frac{\\mathbf{g}{(g,W)}}{W g} = 1 and (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W} = 1 and \\operatorname{L_{\\varepsilon}}{(g,W)} = (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W} and \\frac{\\partial}{\\partial g} (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W} = \\frac{d}{d g} 1 and - W g + \\frac{\\partial}{\\partial g} (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W} = - W g + \\frac{d}{d g} 1 and (- W g + \\frac{\\partial}{\\partial g} (\\frac{\\mathbf{g}{(g,W)}}{W g})^{W})^{W} = (- W g + \\frac{d}{d g} 1)^{W} and (- W g + \\frac{\\partial}{\\partial g} \\operatorname{L_{\\varepsilon}}{(g,W)})^{W} = (- W g + \\frac{d}{d g} 1)^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('g', commutative=True)))"], [["divide", 1, "Mul(Symbol('W', commutative=True), Symbol('g', commutative=True))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Integer(1))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Symbol('W', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Function('L_{\\\\varepsilon}')(Symbol('g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('W', commutative=True), Symbol('g', commutative=True)), Derivative(Integer(1), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(t_{1},\\mathbf{P})} = \\mathbf{P} + t_{1}, then derive \\frac{\\partial}{\\partial t_{1}} \\mathbf{r}{(t_{1},\\mathbf{P})} = 1, then obtain \\mathbf{r}^{\\mathbf{P}}{(t_{1},\\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}))} = (\\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) + t_{1})^{\\mathbf{P}}", "derivation": "\\mathbf{r}{(t_{1},\\mathbf{P})} = \\mathbf{P} + t_{1} and \\frac{\\partial}{\\partial t_{1}} \\mathbf{r}{(t_{1},\\mathbf{P})} = \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) and \\frac{\\partial}{\\partial t_{1}} \\mathbf{r}{(t_{1},\\mathbf{P})} = 1 and \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) = 1 and \\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) = \\mathbf{P} and \\mathbf{r}{(t_{1},\\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}))} = \\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) + t_{1} and \\mathbf{r}^{\\mathbf{P}}{(t_{1},\\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}))} = (\\mathbf{P} \\frac{\\partial}{\\partial t_{1}} (\\mathbf{P} + t_{1}) + t_{1})^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Symbol('\\\\mathbf{P}', commutative=True))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True), Mul(Symbol('\\\\mathbf{P}', commutative=True), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Symbol('t_1', commutative=True)))"], [["power", 6, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True), Mul(Symbol('\\\\mathbf{P}', commutative=True), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given I{(\\sigma_p)} = \\log{(\\sigma_p)} and \\hat{x}{(\\sigma_p)} = \\frac{1}{I{(\\sigma_p)}}, then obtain \\frac{\\frac{d}{d \\sigma_p} \\hat{x}{(\\sigma_p)}}{\\log{(\\sigma_p)}} = - \\frac{1}{\\sigma_p \\log{(\\sigma_p)}^{3}}", "derivation": "I{(\\sigma_p)} = \\log{(\\sigma_p)} and \\hat{x}{(\\sigma_p)} = \\frac{1}{I{(\\sigma_p)}} and \\frac{d}{d \\sigma_p} \\hat{x}{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\frac{1}{I{(\\sigma_p)}} and \\frac{\\frac{d}{d \\sigma_p} \\hat{x}{(\\sigma_p)}}{I{(\\sigma_p)}} = \\frac{\\frac{d}{d \\sigma_p} \\frac{1}{I{(\\sigma_p)}}}{I{(\\sigma_p)}} and \\frac{\\frac{d}{d \\sigma_p} \\hat{x}{(\\sigma_p)}}{\\log{(\\sigma_p)}} = \\frac{\\frac{d}{d \\sigma_p} \\frac{1}{\\log{(\\sigma_p)}}}{\\log{(\\sigma_p)}} and \\frac{\\frac{d}{d \\sigma_p} \\hat{x}{(\\sigma_p)}}{\\log{(\\sigma_p)}} = - \\frac{1}{\\sigma_p \\log{(\\sigma_p)}^{3}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["times", 3, "Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(Pow(Function('I')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Integer(-3))))"]]}, {"prompt": "Given \\theta{(V)} = \\log{(V)}, then obtain \\frac{d}{d V} \\int \\frac{\\theta{(V)}}{\\log{(V)}} dV = \\frac{d}{d V} \\int 1 dV", "derivation": "\\theta{(V)} = \\log{(V)} and \\frac{\\theta{(V)}}{\\log{(V)}} = 1 and \\int \\frac{\\theta{(V)}}{\\log{(V)}} dV = \\int 1 dV and \\frac{d}{d V} \\int \\frac{\\theta{(V)}}{\\log{(V)}} dV = \\frac{d}{d V} \\int 1 dV", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["divide", 1, "log(Symbol('V', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Function('\\\\theta')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integral(Integer(1), Tuple(Symbol('V', commutative=True))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\theta')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(J)} = e^{J} and \\mathbf{f}{(J)} = - e^{J}, then obtain (\\mathbf{f}{(J)} - 1)^{J} = (- \\operatorname{E_{n}}{(J)} - 1)^{J}", "derivation": "\\operatorname{E_{n}}{(J)} = e^{J} and \\mathbf{f}{(J)} = - e^{J} and \\mathbf{f}{(J)} = - \\operatorname{E_{n}}{(J)} and \\mathbf{f}{(J)} - 1 = - e^{J} - 1 and - \\operatorname{E_{n}}{(J)} - 1 = - e^{J} - 1 and (\\mathbf{f}{(J)} - 1)^{J} = (- e^{J} - 1)^{J} and (\\mathbf{f}{(J)} - 1)^{J} = (- \\operatorname{E_{n}}{(J)} - 1)^{J}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('J', commutative=True)), Mul(Integer(-1), exp(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{f}')(Symbol('J', commutative=True)), Mul(Integer(-1), Function('E_n')(Symbol('J', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('J', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('J', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('E_n')(Symbol('J', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('J', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{f}')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), exp(Symbol('J', commutative=True))), Integer(-1)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Add(Function('\\\\mathbf{f}')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Function('E_n')(Symbol('J', commutative=True))), Integer(-1)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(g_{\\varepsilon},s)} = \\frac{s}{g_{\\varepsilon}} and y{(g_{\\varepsilon},s)} = 2 \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{2 s}{g_{\\varepsilon}}, then obtain y{(g_{\\varepsilon},s)} + \\frac{s}{g_{\\varepsilon}} = \\operatorname{m_{s}}{(g_{\\varepsilon},s)}", "derivation": "\\operatorname{m_{s}}{(g_{\\varepsilon},s)} = \\frac{s}{g_{\\varepsilon}} and \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{s}{g_{\\varepsilon}} = 0 and 2 \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{2 s}{g_{\\varepsilon}} = \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{s}{g_{\\varepsilon}} and y{(g_{\\varepsilon},s)} = 2 \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{2 s}{g_{\\varepsilon}} and y{(g_{\\varepsilon},s)} = \\operatorname{m_{s}}{(g_{\\varepsilon},s)} - \\frac{s}{g_{\\varepsilon}} and y{(g_{\\varepsilon},s)} + \\frac{s}{g_{\\varepsilon}} = \\operatorname{m_{s}}{(g_{\\varepsilon},s)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))"], "Equality(Add(Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Integer(0))"], [["add", 2, "Add(Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Add(Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(2), Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Add(Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))"], "Equality(Add(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Function('m_s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\phi{(S,\\hat{x})} = S + \\hat{x}, then obtain \\hat{x} + (\\int \\phi{(S,\\hat{x})} dS)^{S} = \\hat{x} + (\\int (S + \\hat{x}) dS)^{S}", "derivation": "\\phi{(S,\\hat{x})} = S + \\hat{x} and \\int \\phi{(S,\\hat{x})} dS = \\int (S + \\hat{x}) dS and (\\int \\phi{(S,\\hat{x})} dS)^{S} = (\\int (S + \\hat{x}) dS)^{S} and \\hat{x} + (\\int \\phi{(S,\\hat{x})} dS)^{S} = \\hat{x} + (\\int (S + \\hat{x}) dS)^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Integral(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Integral(Add(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["add", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Pow(Integral(Function('\\\\phi')(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Pow(Integral(Add(Symbol('S', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"]]}, {"prompt": "Given B{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain \\mathbf{H} \\cos{(B{(\\mathbf{H})} - 1)} = \\mathbf{H} \\cos{(\\cos{(\\mathbf{H})} - 1)}", "derivation": "B{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and B{(\\mathbf{H})} - 1 = \\cos{(\\mathbf{H})} - 1 and \\cos{(B{(\\mathbf{H})} - 1)} = \\cos{(\\cos{(\\mathbf{H})} - 1)} and \\mathbf{H} \\cos{(B{(\\mathbf{H})} - 1)} = \\mathbf{H} \\cos{(\\cos{(\\mathbf{H})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('B')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], [["cos", 2], "Equality(cos(Add(Function('B')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), cos(Add(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"], [["divide", 3, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Add(Function('B')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))), Mul(Symbol('\\\\mathbf{H}', commutative=True), cos(Add(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given l{(\\theta)} = \\frac{d}{d \\theta} e^{\\theta}, then derive l{(\\theta)} = e^{\\theta}, then obtain ((l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta)^{2}) (\\int e^{\\theta} d\\theta)^{2} = ((l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta)^{2}) (\\int e^{\\theta} d\\theta) \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta", "derivation": "l{(\\theta)} = \\frac{d}{d \\theta} e^{\\theta} and \\int l{(\\theta)} d\\theta = \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta and l{(\\theta)} = e^{\\theta} and \\int e^{\\theta} d\\theta = \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta and (l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta) \\int e^{\\theta} d\\theta = (l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta) \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta and ((l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta)^{2}) (\\int e^{\\theta} d\\theta)^{2} = ((l{(\\theta)} - \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta)^{2}) (\\int e^{\\theta} d\\theta) \\int \\frac{d}{d \\theta} e^{\\theta} d\\theta", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\theta', commutative=True)), Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('l')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 4, "Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)))))"], "Equality(Mul(Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["times", 5, "Mul(Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], "Equality(Mul(Pow(Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))), Integer(2)), Pow(Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integer(2))), Mul(Pow(Add(Function('l')(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))), Integer(2)), Integral(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(a)} = \\sin{(a)} and l{(a)} = - \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)} + \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)})})}, then obtain 1 - l{(a)} = - l{(a)} + l^{a}{(a)}", "derivation": "\\operatorname{A_{x}}{(a)} = \\sin{(a)} and l{(a)} = - \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)} + \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)})})} and l^{a}{(a)} = (- \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)} + \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)})})})^{a} and l^{a}{(a)} = 0^{a} and - l{(a)} + l^{a}{(a)} = 0^{a} - l{(a)} and (- \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)} + \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)})})})^{a} = 0^{a} and 1 - l{(a)} = (- \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)} + \\sin{(\\operatorname{A_{x}}{(a)} - \\sin{(a)})})})^{a} - l{(a)} and 1 - l{(a)} = - l{(a)} + l^{a}{(a)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True))), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True))), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))))), Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Integer(0), Symbol('a', commutative=True)))"], [["minus", 4, "Function('l')(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('a', commutative=True))), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Add(Pow(Integer(0), Symbol('a', commutative=True)), Mul(Integer(-1), Function('l')(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Integer(-1), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True))), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))))), Symbol('a', commutative=True)), Pow(Integer(0), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Integer(1), Mul(Integer(-1), Function('l')(Symbol('a', commutative=True)))), Add(Pow(Mul(Integer(-1), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True))), sin(Add(Function('A_x')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))))))), Symbol('a', commutative=True)), Mul(Integer(-1), Function('l')(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Function('l')(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Function('l')(Symbol('a', commutative=True))), Pow(Function('l')(Symbol('a', commutative=True)), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(V_{\\mathbf{B}},\\mathbf{v},\\rho_b)} = - \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}}, then derive \\frac{\\partial}{\\partial \\mathbf{v}} \\hat{p}{(V_{\\mathbf{B}},\\mathbf{v},\\rho_b)} = \\frac{1}{V_{\\mathbf{B}}}, then obtain \\int \\frac{1}{V_{\\mathbf{B}}} dV_{\\mathbf{B}} = \\int \\frac{\\partial}{\\partial \\mathbf{v}} (- \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}}) dV_{\\mathbf{B}}", "derivation": "\\hat{p}{(V_{\\mathbf{B}},\\mathbf{v},\\rho_b)} = - \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}} and \\frac{\\partial}{\\partial \\mathbf{v}} \\hat{p}{(V_{\\mathbf{B}},\\mathbf{v},\\rho_b)} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}}) and \\frac{\\partial}{\\partial \\mathbf{v}} \\hat{p}{(V_{\\mathbf{B}},\\mathbf{v},\\rho_b)} = \\frac{1}{V_{\\mathbf{B}}} and \\frac{1}{V_{\\mathbf{B}}} = \\frac{\\partial}{\\partial \\mathbf{v}} (- \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}}) and \\int \\frac{1}{V_{\\mathbf{B}}} dV_{\\mathbf{B}} = \\int \\frac{\\partial}{\\partial \\mathbf{v}} (- \\rho_b + \\frac{\\mathbf{v}}{V_{\\mathbf{B}}}) dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(t_{1})} = \\cos{(t_{1})}, then derive \\int \\mathbf{A}{(t_{1})} dt_{1} = E + \\sin{(t_{1})}, then obtain (\\int (\\int \\mathbf{A}{(t_{1})} dt_{1})^{t_{1}} dE)^{E} = (\\int (E + \\sin{(t_{1})})^{t_{1}} dE)^{E}", "derivation": "\\mathbf{A}{(t_{1})} = \\cos{(t_{1})} and \\int \\mathbf{A}{(t_{1})} dt_{1} = \\int \\cos{(t_{1})} dt_{1} and \\int \\mathbf{A}{(t_{1})} dt_{1} = E + \\sin{(t_{1})} and (\\int \\mathbf{A}{(t_{1})} dt_{1})^{t_{1}} = (E + \\sin{(t_{1})})^{t_{1}} and \\int (\\int \\mathbf{A}{(t_{1})} dt_{1})^{t_{1}} dE = \\int (E + \\sin{(t_{1})})^{t_{1}} dE and (\\int (\\int \\mathbf{A}{(t_{1})} dt_{1})^{t_{1}} dE)^{E} = (\\int (E + \\sin{(t_{1})})^{t_{1}} dE)^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('E', commutative=True), sin(Symbol('t_1', commutative=True))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Pow(Add(Symbol('E', commutative=True), sin(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Add(Symbol('E', commutative=True), sin(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["power", 5, "Symbol('E', commutative=True)"], "Equality(Pow(Integral(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(Pow(Add(Symbol('E', commutative=True), sin(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(A_{2},A_{z})} = - A_{2} + A_{z}, then derive \\frac{\\partial}{\\partial A_{2}} \\operatorname{C_{1}}{(A_{2},A_{z})} = -1, then obtain \\operatorname{C_{1}}{(A_{2},A_{z})} + \\cos{(\\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}))} = \\operatorname{C_{1}}{(A_{2},A_{z})} + \\cos{(1)}", "derivation": "\\operatorname{C_{1}}{(A_{2},A_{z})} = - A_{2} + A_{z} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{C_{1}}{(A_{2},A_{z})} = \\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}) and \\cos{(\\frac{\\partial}{\\partial A_{2}} \\operatorname{C_{1}}{(A_{2},A_{z})})} = \\cos{(\\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}))} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{C_{1}}{(A_{2},A_{z})} = -1 and \\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}) = -1 and \\cos{(\\frac{\\partial}{\\partial A_{2}} \\operatorname{C_{1}}{(A_{2},A_{z})})} = \\cos{(1)} and \\cos{(\\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}))} = \\cos{(1)} and \\operatorname{C_{1}}{(A_{2},A_{z})} + \\cos{(\\frac{\\partial}{\\partial A_{2}} (- A_{2} + A_{z}))} = \\operatorname{C_{1}}{(A_{2},A_{z})} + \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), cos(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(cos(Derivative(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), cos(Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(cos(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), cos(Integer(1)))"], [["add", 7, "Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), cos(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))), Add(Function('C_1')(Symbol('A_2', commutative=True), Symbol('A_z', commutative=True)), cos(Integer(1))))"]]}, {"prompt": "Given A{(\\mathbf{J}_f,\\tilde{g})} = \\mathbf{J}_f \\tilde{g} and \\mathbf{S}{(\\mathbf{J}_f,\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} A{(\\mathbf{J}_f,\\tilde{g})}, then obtain \\mathbf{S}{(\\mathbf{J}_f,\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f \\tilde{g}", "derivation": "A{(\\mathbf{J}_f,\\tilde{g})} = \\mathbf{J}_f \\tilde{g} and \\frac{\\partial}{\\partial \\mathbf{J}_f} A{(\\mathbf{J}_f,\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f \\tilde{g} and \\mathbf{S}{(\\mathbf{J}_f,\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} A{(\\mathbf{J}_f,\\tilde{g})} and \\mathbf{S}{(\\mathbf{J}_f,\\tilde{g})} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{J}_f \\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('A')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\phi)} = \\sin{(\\phi)}, then derive (\\int \\operatorname{f^{\\prime}}{(\\phi)} d\\phi)^{\\phi} = (M - \\cos{(\\phi)})^{\\phi}, then derive (\\theta_1 - \\cos{(\\phi)})^{\\phi} = (M - \\cos{(\\phi)})^{\\phi}, then obtain (M - \\cos{(\\phi)})^{\\phi} + h{(\\phi)} = h{(\\phi)} + (\\int \\sin{(\\phi)} d\\phi)^{\\phi}", "derivation": "\\operatorname{f^{\\prime}}{(\\phi)} = \\sin{(\\phi)} and \\int \\operatorname{f^{\\prime}}{(\\phi)} d\\phi = \\int \\sin{(\\phi)} d\\phi and (\\int \\operatorname{f^{\\prime}}{(\\phi)} d\\phi)^{\\phi} = (\\int \\sin{(\\phi)} d\\phi)^{\\phi} and (\\int \\operatorname{f^{\\prime}}{(\\phi)} d\\phi)^{\\phi} = (M - \\cos{(\\phi)})^{\\phi} and (\\int \\sin{(\\phi)} d\\phi)^{\\phi} = (M - \\cos{(\\phi)})^{\\phi} and (\\theta_1 - \\cos{(\\phi)})^{\\phi} = (M - \\cos{(\\phi)})^{\\phi} and (\\theta_1 - \\cos{(\\phi)})^{\\phi} + h{(\\phi)} = (M - \\cos{(\\phi)})^{\\phi} + h{(\\phi)} and (\\theta_1 - \\cos{(\\phi)})^{\\phi} + h{(\\phi)} = h{(\\phi)} + (\\int \\sin{(\\phi)} d\\phi)^{\\phi} and (M - \\cos{(\\phi)})^{\\phi} + h{(\\phi)} = h{(\\phi)} + (\\int \\sin{(\\phi)} d\\phi)^{\\phi}", "srepr_derivation": [["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Integral(Function('f^{\\\\prime}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('f^{\\\\prime}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["add", 6, "Function('h')(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('\\\\phi', commutative=True))), Add(Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Pow(Add(Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('\\\\phi', commutative=True))), Add(Function('h')(Symbol('\\\\phi', commutative=True)), Pow(Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Add(Pow(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('\\\\phi', commutative=True))), Add(Function('h')(Symbol('\\\\phi', commutative=True)), Pow(Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(x,Q)} = - x + \\sin{(Q)} and \\varepsilon_{0}{(x,Q)} = - \\mathbf{v}{(x,Q)}, then derive - x \\frac{\\partial}{\\partial Q} \\varepsilon_{0}{(x,Q)} = x \\frac{\\partial}{\\partial Q} \\mathbf{v}{(x,Q)}, then obtain - x \\frac{\\partial}{\\partial Q} (x - \\sin{(Q)}) = x \\frac{\\partial}{\\partial Q} \\mathbf{v}{(x,Q)}", "derivation": "\\mathbf{v}{(x,Q)} = - x + \\sin{(Q)} and \\varepsilon_{0}{(x,Q)} = - \\mathbf{v}{(x,Q)} and \\varepsilon_{0}{(x,Q)} = x - \\sin{(Q)} and - x \\varepsilon_{0}{(x,Q)} = x \\mathbf{v}{(x,Q)} and \\frac{\\partial}{\\partial Q} - x \\varepsilon_{0}{(x,Q)} = \\frac{\\partial}{\\partial Q} x \\mathbf{v}{(x,Q)} and - x \\frac{\\partial}{\\partial Q} \\varepsilon_{0}{(x,Q)} = x \\frac{\\partial}{\\partial Q} \\mathbf{v}{(x,Q)} and - x \\frac{\\partial}{\\partial Q} (x - \\sin{(Q)}) = x \\frac{\\partial}{\\partial Q} \\mathbf{v}{(x,Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), sin(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\varepsilon_0')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('x', commutative=True), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\varepsilon_0')(Symbol('x', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('x', commutative=True), Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\varepsilon_0')(Symbol('x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Symbol('x', commutative=True), Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Derivative(Function('\\\\varepsilon_0')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('x', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Derivative(Add(Symbol('x', commutative=True), Mul(Integer(-1), sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('x', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(\\varphi)} = \\sin{(\\varphi)}, then derive \\int M{(\\varphi)} d\\varphi = \\sigma_x - \\cos{(\\varphi)}, then obtain 8 (\\int M{(\\varphi)} d\\varphi)^{2} = (\\sigma_x - \\cos{(\\varphi)} + \\int M{(\\varphi)} d\\varphi)^{2} + 4 (\\int M{(\\varphi)} d\\varphi)^{2}", "derivation": "M{(\\varphi)} = \\sin{(\\varphi)} and \\int M{(\\varphi)} d\\varphi = \\int \\sin{(\\varphi)} d\\varphi and \\int M{(\\varphi)} d\\varphi = \\sigma_x - \\cos{(\\varphi)} and \\sigma_x - \\cos{(\\varphi)} + \\int M{(\\varphi)} d\\varphi = 2 \\sigma_x - 2 \\cos{(\\varphi)} and 2 \\int M{(\\varphi)} d\\varphi = 2 \\sigma_x - 2 \\cos{(\\varphi)} and 4 (\\int M{(\\varphi)} d\\varphi)^{2} = (2 \\sigma_x - 2 \\cos{(\\varphi)})^{2} and 8 (\\int M{(\\varphi)} d\\varphi)^{2} = (2 \\sigma_x - 2 \\cos{(\\varphi)})^{2} + 4 (\\int M{(\\varphi)} d\\varphi)^{2} and 8 (\\int M{(\\varphi)} d\\varphi)^{2} = (\\sigma_x - \\cos{(\\varphi)} + \\int M{(\\varphi)} d\\varphi)^{2} + 4 (\\int M{(\\varphi)} d\\varphi)^{2}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True)))))"], [["add", 3, "Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True))))"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True))), Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\varphi', commutative=True)))))"], [["power", 5, 2], "Equality(Mul(Integer(4), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\varphi', commutative=True)))), Integer(2)))"], [["add", 6, "Mul(Integer(4), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2)))"], "Equality(Mul(Integer(8), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2))), Add(Pow(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\varphi', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(8), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2))), Add(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi', commutative=True))), Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(Integral(Function('M')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given A{(Z)} = \\int \\log{(Z)} dZ, then derive A{(Z)} = Z \\log{(Z)} - Z + \\rho_f, then obtain \\frac{\\int A{(Z)} dZ}{\\log{(Z)}} = \\frac{\\int (Z \\log{(Z)} - Z + \\rho_f) dZ}{\\log{(Z)}}", "derivation": "A{(Z)} = \\int \\log{(Z)} dZ and \\int A{(Z)} dZ = \\iint \\log{(Z)} dZ dZ and \\frac{\\int A{(Z)} dZ}{\\log{(Z)}} = \\frac{\\iint \\log{(Z)} dZ dZ}{\\log{(Z)}} and A{(Z)} = Z \\log{(Z)} - Z + \\rho_f and \\int (Z \\log{(Z)} - Z + \\rho_f) dZ = \\iint \\log{(Z)} dZ dZ and \\frac{\\int A{(Z)} dZ}{\\log{(Z)}} = \\frac{\\int (Z \\log{(Z)} - Z + \\rho_f) dZ}{\\log{(Z)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('Z', commutative=True)), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('A')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["divide", 2, "log(Symbol('Z', commutative=True))"], "Equality(Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), Integral(Function('A')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('A')(Symbol('Z', commutative=True)), Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), Integral(Function('A')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Mul(Pow(log(Symbol('Z', commutative=True)), Integer(-1)), Integral(Add(Mul(Symbol('Z', commutative=True), log(Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given u{(f^{\\prime},v_{2})} = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + v_{2}), then derive \\int u{(f^{\\prime},v_{2})} dv_{2} = B + v_{2}, then derive u{(f^{\\prime},v_{2})} = 1, then obtain \\cos{(\\frac{\\int 1 dv_{2}}{B + v_{2}})} = \\cos{(1)}", "derivation": "u{(f^{\\prime},v_{2})} = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + v_{2}) and \\int u{(f^{\\prime},v_{2})} dv_{2} = \\int \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + v_{2}) dv_{2} and \\int u{(f^{\\prime},v_{2})} dv_{2} = B + v_{2} and u{(f^{\\prime},v_{2})} = 1 and \\int u{(f^{\\prime},v_{2})} dv_{2} = \\int 1 dv_{2} and \\frac{\\int u{(f^{\\prime},v_{2})} dv_{2}}{B + v_{2}} = 1 and \\cos{(\\frac{\\int u{(f^{\\prime},v_{2})} dv_{2}}{B + v_{2}})} = \\cos{(1)} and \\cos{(\\frac{\\int 1 dv_{2}}{B + v_{2}})} = \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('B', commutative=True), Symbol('v_2', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))))"], [["divide", 3, "Add(Symbol('B', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Integral(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Integer(1))"], [["cos", 6], "Equality(cos(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Integral(Function('u')(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))), cos(Integer(1)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(cos(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('v_2', commutative=True))))), cos(Integer(1)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)} = \\mathbf{J}_M + \\sin{(\\theta_1)}, then obtain (\\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}})^{\\mathbf{J}_M} = (\\frac{\\mathbf{J}_M + \\sin{(\\theta_1)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}})^{\\mathbf{J}_M}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)} = \\mathbf{J}_M + \\sin{(\\theta_1)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)}}{\\sin{(\\theta_1)}} = \\frac{\\mathbf{J}_M + \\sin{(\\theta_1)}}{\\sin{(\\theta_1)}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}} = \\frac{\\mathbf{J}_M + \\sin{(\\theta_1)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}} and (\\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\theta_1,\\mathbf{J}_M)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}})^{\\mathbf{J}_M} = (\\frac{\\mathbf{J}_M + \\sin{(\\theta_1)}}{\\sin{(\\theta_1)}} + \\frac{1}{\\sin{(\\theta_1)}})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"], [["add", 2, "Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Add(Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Add(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(E_{n})} = E_{n}, then obtain \\int \\frac{\\partial}{\\partial l} (- l + \\operatorname{F_{c}}{(E_{n})}) dE_{n} = \\int \\frac{\\partial}{\\partial l} (E_{n} - l) dE_{n}", "derivation": "\\operatorname{F_{c}}{(E_{n})} = E_{n} and - l + \\operatorname{F_{c}}{(E_{n})} = E_{n} - l and \\frac{\\partial}{\\partial l} (- l + \\operatorname{F_{c}}{(E_{n})}) = \\frac{\\partial}{\\partial l} (E_{n} - l) and \\int \\frac{\\partial}{\\partial l} (- l + \\operatorname{F_{c}}{(E_{n})}) dE_{n} = \\int \\frac{\\partial}{\\partial l} (E_{n} - l) dE_{n}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_c')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], [["minus", 1, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('F_c')(Symbol('E_n', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('F_c')(Symbol('E_n', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('F_c')(Symbol('E_n', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('E_n', commutative=True))), Integral(Derivative(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{x})} = \\int e^{v_{x}} dv_{x}, then derive 0 = \\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}}, then obtain \\int (\\mathbf{s} - m_{s} + \\int 0 d\\mathbf{s}) dm_{s} = \\int (\\mathbf{s} - m_{s} + \\int (\\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}}) d\\mathbf{s}) dm_{s}", "derivation": "\\mathbf{f}{(v_{x})} = \\int e^{v_{x}} dv_{x} and 0 = - \\mathbf{f}{(v_{x})} + \\int e^{v_{x}} dv_{x} and 0 = \\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}} and \\int 0 d\\mathbf{s} = \\int (\\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}}) d\\mathbf{s} and \\mathbf{s} - m_{s} + \\int 0 d\\mathbf{s} = \\mathbf{s} - m_{s} + \\int (\\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}}) d\\mathbf{s} and \\int (\\mathbf{s} - m_{s} + \\int 0 d\\mathbf{s}) dm_{s} = \\int (\\mathbf{s} - m_{s} + \\int (\\mathbf{s} - \\mathbf{f}{(v_{x})} + e^{v_{x}}) d\\mathbf{s}) dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True)), Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))), Integral(exp(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(0), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 4, "Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["integrate", 5, "Symbol('m_s', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('v_x', commutative=True))), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\lambda,g_{\\varepsilon},C)} = \\frac{\\lambda + g_{\\varepsilon}}{C}, then obtain \\int (\\tilde{g}{(\\lambda,g_{\\varepsilon},C)} + \\tilde{g}^{g_{\\varepsilon}}{(\\lambda,g_{\\varepsilon},C)} - 1) dC = \\int ((\\frac{\\lambda + g_{\\varepsilon}}{C})^{g_{\\varepsilon}} + \\tilde{g}{(\\lambda,g_{\\varepsilon},C)} - 1) dC", "derivation": "\\tilde{g}{(\\lambda,g_{\\varepsilon},C)} = \\frac{\\lambda + g_{\\varepsilon}}{C} and \\tilde{g}^{g_{\\varepsilon}}{(\\lambda,g_{\\varepsilon},C)} = (\\frac{\\lambda + g_{\\varepsilon}}{C})^{g_{\\varepsilon}} and \\tilde{g}{(\\lambda,g_{\\varepsilon},C)} + \\tilde{g}^{g_{\\varepsilon}}{(\\lambda,g_{\\varepsilon},C)} - 1 = (\\frac{\\lambda + g_{\\varepsilon}}{C})^{g_{\\varepsilon}} + \\tilde{g}{(\\lambda,g_{\\varepsilon},C)} - 1 and \\int (\\tilde{g}{(\\lambda,g_{\\varepsilon},C)} + \\tilde{g}^{g_{\\varepsilon}}{(\\lambda,g_{\\varepsilon},C)} - 1) dC = \\int ((\\frac{\\lambda + g_{\\varepsilon}}{C})^{g_{\\varepsilon}} + \\tilde{g}{(\\lambda,g_{\\varepsilon},C)} - 1) dC", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Add(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))), Integral(Add(Pow(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(x,\\mathbf{v})} = \\mathbf{v} + x, then derive \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\mathbf{v})} = 1, then derive (A_{x} + x)^{x} = (\\mathbb{I} + x)^{x}, then obtain (A_{x} + x)^{x} (\\frac{x}{A_{x} + x} + \\log{(A_{x} + x)}) = (\\mathbb{I} + x)^{x} (\\frac{x}{\\mathbb{I} + x} + \\log{(\\mathbb{I} + x)})", "derivation": "\\dot{x}{(x,\\mathbf{v})} = \\mathbf{v} + x and \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\mathbf{v})} = \\frac{\\partial}{\\partial x} (\\mathbf{v} + x) and \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\mathbf{v})} = 1 and \\frac{\\partial}{\\partial x} (\\mathbf{v} + x) = 1 and \\int \\frac{\\partial}{\\partial x} (\\mathbf{v} + x) dx = \\int 1 dx and (\\int \\frac{\\partial}{\\partial x} (\\mathbf{v} + x) dx)^{x} = (\\int 1 dx)^{x} and (A_{x} + x)^{x} = (\\mathbb{I} + x)^{x} and \\frac{\\partial}{\\partial x} (A_{x} + x)^{x} = \\frac{\\partial}{\\partial x} (\\mathbb{I} + x)^{x} and (A_{x} + x)^{x} (\\frac{x}{A_{x} + x} + \\log{(A_{x} + x)}) = (\\mathbb{I} + x)^{x} (\\frac{x}{\\mathbb{I} + x} + \\log{(\\mathbb{I} + x)})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Integer(1), Tuple(Symbol('x', commutative=True))))"], [["power", 5, "Symbol('x', commutative=True)"], "Equality(Pow(Integral(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('A_x', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 7, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('A_x', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 8], "Equality(Mul(Pow(Add(Symbol('A_x', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Add(Mul(Symbol('x', commutative=True), Pow(Add(Symbol('A_x', commutative=True), Symbol('x', commutative=True)), Integer(-1))), log(Add(Symbol('A_x', commutative=True), Symbol('x', commutative=True))))), Mul(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Add(Mul(Symbol('x', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x', commutative=True)), Integer(-1))), log(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\chi{(t_{1})} = \\log{(\\sin{(t_{1})})} and H{(t_{1})} = \\log{(\\sin{(t_{1})})}, then obtain 0 = H{(t_{1})} - \\chi{(t_{1})}", "derivation": "\\chi{(t_{1})} = \\log{(\\sin{(t_{1})})} and 0 = - \\chi{(t_{1})} + \\log{(\\sin{(t_{1})})} and H{(t_{1})} = \\log{(\\sin{(t_{1})})} and 0 = H{(t_{1})} - \\chi{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('t_1', commutative=True)), log(sin(Symbol('t_1', commutative=True))))"], [["minus", 1, "Function('\\\\chi')(Symbol('t_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('t_1', commutative=True))), log(sin(Symbol('t_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('H')(Symbol('t_1', commutative=True)), log(sin(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('H')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given b{(a)} = \\sin{(\\sin{(a)})} and \\operatorname{r_{0}}{(a)} = \\log{(\\sin{(\\sin{(a)})})}, then obtain - b{(a)} + 2 \\operatorname{r_{0}}{(a)} = - b{(a)} + \\operatorname{r_{0}}{(a)} + \\log{(\\sin{(\\sin{(a)})})}", "derivation": "b{(a)} = \\sin{(\\sin{(a)})} and \\log{(b{(a)})} = \\log{(\\sin{(\\sin{(a)})})} and \\operatorname{r_{0}}{(a)} = \\log{(\\sin{(\\sin{(a)})})} and \\log{(b{(a)})} = \\operatorname{r_{0}}{(a)} and - b{(a)} + \\log{(b{(a)})} = - b{(a)} + \\log{(\\sin{(\\sin{(a)})})} and - b{(a)} + 2 \\log{(b{(a)})} = - b{(a)} + \\log{(b{(a)})} + \\log{(\\sin{(\\sin{(a)})})} and - b{(a)} + 2 \\operatorname{r_{0}}{(a)} = - b{(a)} + \\operatorname{r_{0}}{(a)} + \\log{(\\sin{(\\sin{(a)})})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a', commutative=True)), sin(sin(Symbol('a', commutative=True))))"], [["log", 1], "Equality(log(Function('b')(Symbol('a', commutative=True))), log(sin(sin(Symbol('a', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('a', commutative=True)), log(sin(sin(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(log(Function('b')(Symbol('a', commutative=True))), Function('r_0')(Symbol('a', commutative=True)))"], [["minus", 2, "Function('b')(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), log(Function('b')(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), log(sin(sin(Symbol('a', commutative=True))))))"], [["add", 5, "log(Function('b')(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), Mul(Integer(2), log(Function('b')(Symbol('a', commutative=True))))), Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), log(Function('b')(Symbol('a', commutative=True))), log(sin(sin(Symbol('a', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), Mul(Integer(2), Function('r_0')(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('a', commutative=True))), Function('r_0')(Symbol('a', commutative=True)), log(sin(sin(Symbol('a', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(r_{0})} = \\log{(\\cos{(r_{0})})}, then obtain \\frac{\\frac{\\operatorname{F_{g}}{(r_{0})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}}}{\\cos{(r_{0})}} = \\frac{\\frac{\\log{(\\cos{(r_{0})})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}}}{\\cos{(r_{0})}}", "derivation": "\\operatorname{F_{g}}{(r_{0})} = \\log{(\\cos{(r_{0})})} and \\frac{\\operatorname{F_{g}}{(r_{0})}}{\\cos{(r_{0})}} = \\frac{\\log{(\\cos{(r_{0})})}}{\\cos{(r_{0})}} and \\frac{\\operatorname{F_{g}}{(r_{0})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}} = \\frac{\\log{(\\cos{(r_{0})})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}} and \\frac{\\frac{\\operatorname{F_{g}}{(r_{0})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}}}{\\cos{(r_{0})}} = \\frac{\\frac{\\log{(\\cos{(r_{0})})}}{\\cos{(r_{0})}} + \\frac{1}{\\cos{(r_{0})}}}{\\cos{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('r_0', commutative=True)), log(cos(Symbol('r_0', commutative=True))))"], [["divide", 1, "cos(Symbol('r_0', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Mul(log(cos(Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))))"], [["add", 2, "Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('F_g')(Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Add(Mul(log(cos(Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))))"], [["times", 3, "Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Function('F_g')(Symbol('r_0', commutative=True)), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Mul(Add(Mul(log(cos(Symbol('r_0', commutative=True))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))), Pow(cos(Symbol('r_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0}, then derive \\phi_{1}{(\\mu_0)} = e^{\\mu_0}, then obtain \\log{(\\phi_{1}^{\\mu_0}{(\\mu_0)})} = \\log{((\\frac{d}{d \\mu_0} e^{\\mu_0})^{\\mu_0})}", "derivation": "\\phi_{1}{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\phi_{1}{(\\mu_0)} = e^{\\mu_0} and \\phi_{1}^{\\mu_0}{(\\mu_0)} = (e^{\\mu_0})^{\\mu_0} and \\log{(\\phi_{1}^{\\mu_0}{(\\mu_0)})} = \\log{((e^{\\mu_0})^{\\mu_0})} and \\log{((\\frac{d}{d \\mu_0} e^{\\mu_0})^{\\mu_0})} = \\log{((e^{\\mu_0})^{\\mu_0})} and \\log{(\\phi_{1}^{\\mu_0}{(\\mu_0)})} = \\log{((\\frac{d}{d \\mu_0} e^{\\mu_0})^{\\mu_0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True)), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["log", 3], "Equality(log(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), log(Pow(exp(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(log(Pow(Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True))), log(Pow(exp(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Pow(Function('\\\\phi_1')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), log(Pow(Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\lambda{(U)} = e^{\\cos{(U)}}, then obtain (\\int \\lambda{(U)} dU) \\iiint \\lambda{(U)} dU dU dU = (\\int \\lambda{(U)} dU) \\iiint e^{\\cos{(U)}} dU dU dU", "derivation": "\\lambda{(U)} = e^{\\cos{(U)}} and \\int \\lambda{(U)} dU = \\int e^{\\cos{(U)}} dU and \\iint \\lambda{(U)} dU dU = \\iint e^{\\cos{(U)}} dU dU and \\iiint \\lambda{(U)} dU dU dU = \\iiint e^{\\cos{(U)}} dU dU dU and (\\int e^{\\cos{(U)}} dU) \\iiint \\lambda{(U)} dU dU dU = (\\int e^{\\cos{(U)}} dU) \\iiint e^{\\cos{(U)}} dU dU dU and (\\int \\lambda{(U)} dU) \\iiint \\lambda{(U)} dU dU dU = (\\int \\lambda{(U)} dU) \\iiint e^{\\cos{(U)}} dU dU dU", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('U', commutative=True)), exp(cos(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["times", 4, "Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integral(Function('\\\\lambda')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(c_{0})} = c_{0}, then obtain \\mathbf{F}{(c_{0})} + \\int c_{0} d\\mathbf{F}{(c_{0})} = c_{0} + \\int c_{0} d\\mathbf{F}{(c_{0})}", "derivation": "\\mathbf{F}{(c_{0})} = c_{0} and \\int \\mathbf{F}{(c_{0})} dc_{0} = \\int c_{0} dc_{0} and \\int \\mathbf{F}{(c_{0})} d\\mathbf{F}{(c_{0})} = \\int c_{0} d\\mathbf{F}{(c_{0})} and \\mathbf{F}{(c_{0})} + \\int \\mathbf{F}{(c_{0})} d\\mathbf{F}{(c_{0})} = c_{0} + \\int \\mathbf{F}{(c_{0})} d\\mathbf{F}{(c_{0})} and \\mathbf{F}{(c_{0})} + \\int c_{0} d\\mathbf{F}{(c_{0})} = c_{0} + \\int c_{0} d\\mathbf{F}{(c_{0})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)))), Integral(Symbol('c_0', commutative=True), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)))))"], [["add", 1, "Integral(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Integral(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True))))), Add(Symbol('c_0', commutative=True), Integral(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True)), Integral(Symbol('c_0', commutative=True), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True))))), Add(Symbol('c_0', commutative=True), Integral(Symbol('c_0', commutative=True), Tuple(Function('\\\\mathbf{F}')(Symbol('c_0', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(\\mu,\\dot{x},\\hbar)} = (- \\dot{x} + \\hbar)^{\\mu}, then obtain (- \\dot{x} + \\hbar)^{\\mu} + \\int (\\phi_{2}{(\\mu,\\dot{x},\\hbar)} + 1) d\\mu = (- \\dot{x} + \\hbar)^{\\mu} + \\int ((- \\dot{x} + \\hbar)^{\\mu} + 1) d\\mu", "derivation": "\\phi_{2}{(\\mu,\\dot{x},\\hbar)} = (- \\dot{x} + \\hbar)^{\\mu} and \\phi_{2}{(\\mu,\\dot{x},\\hbar)} + 1 = (- \\dot{x} + \\hbar)^{\\mu} + 1 and \\int (\\phi_{2}{(\\mu,\\dot{x},\\hbar)} + 1) d\\mu = \\int ((- \\dot{x} + \\hbar)^{\\mu} + 1) d\\mu and (- \\dot{x} + \\hbar)^{\\mu} + \\int (\\phi_{2}{(\\mu,\\dot{x},\\hbar)} + 1) d\\mu = (- \\dot{x} + \\hbar)^{\\mu} + \\int ((- \\dot{x} + \\hbar)^{\\mu} + 1) d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)), Integral(Add(Function('\\\\phi_2')(Symbol('\\\\mu', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)), Integral(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mu', commutative=True)), Integer(1)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(Z)} = \\log{(Z)}, then obtain \\int \\frac{d}{d Z} Z (\\frac{\\mathbf{E}{(Z)}}{Z} + \\frac{1}{Z}) dZ = \\int \\frac{d}{d Z} Z (\\frac{\\log{(Z)}}{Z} + \\frac{1}{Z}) dZ", "derivation": "\\mathbf{E}{(Z)} = \\log{(Z)} and \\frac{\\mathbf{E}{(Z)}}{Z} = \\frac{\\log{(Z)}}{Z} and \\frac{\\mathbf{E}{(Z)}}{Z} + \\frac{1}{Z} = \\frac{\\log{(Z)}}{Z} + \\frac{1}{Z} and Z (\\frac{\\mathbf{E}{(Z)}}{Z} + \\frac{1}{Z}) = Z (\\frac{\\log{(Z)}}{Z} + \\frac{1}{Z}) and \\frac{d}{d Z} Z (\\frac{\\mathbf{E}{(Z)}}{Z} + \\frac{1}{Z}) = \\frac{d}{d Z} Z (\\frac{\\log{(Z)}}{Z} + \\frac{1}{Z}) and \\int \\frac{d}{d Z} Z (\\frac{\\mathbf{E}{(Z)}}{Z} + \\frac{1}{Z}) dZ = \\int \\frac{d}{d Z} Z (\\frac{\\log{(Z)}}{Z} + \\frac{1}{Z}) dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["divide", 1, "Symbol('Z', commutative=True)"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('Z', commutative=True))))"], [["add", 2, "Pow(Symbol('Z', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1))))"], [["times", 3, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))), Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Mul(Symbol('Z', commutative=True), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('Z', commutative=True))), Pow(Symbol('Z', commutative=True), Integer(-1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} = \\sin{(\\theta_1)} and I{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain 1 + \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} \\sin{(\\theta_1)}}{\\theta_1 I{(\\theta_1)}} = 1 + \\frac{\\sin{(\\theta_1)}}{\\theta_1}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} = \\sin{(\\theta_1)} and I{(\\theta_1)} = \\sin{(\\theta_1)} and \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} = I{(\\theta_1)} and \\frac{\\theta_1 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)}}{\\sin{(\\theta_1)}} = \\frac{\\theta_1 I{(\\theta_1)}}{\\sin{(\\theta_1)}} and \\frac{\\theta_1 \\operatorname{f_{\\mathbf{v}}}{(\\theta_1)}}{I{(\\theta_1)}} = \\theta_1 and \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)}}{I{(\\theta_1)}} = 1 and \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} \\sin{(\\theta_1)}}{\\theta_1 I{(\\theta_1)}} = \\frac{\\sin{(\\theta_1)}}{\\theta_1} and 1 + \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\theta_1)} \\sin{(\\theta_1)}}{\\theta_1 I{(\\theta_1)}} = 1 + \\frac{\\sin{(\\theta_1)}}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), Function('I')(Symbol('\\\\theta_1', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Mul(Symbol('\\\\theta_1', commutative=True), Function('I')(Symbol('\\\\theta_1', commutative=True)), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('I')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True))"], [["divide", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Function('I')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True))), Integer(1))"], [["times", 6, "Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["add", 7, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given t{(v,\\varphi^*)} = \\int \\varphi^* v d\\varphi^*, then derive \\int \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} d\\varphi^* = \\Psi_{\\lambda} + \\frac{(\\varphi^*)^{3}}{6}, then obtain - \\operatorname{F_{N}}{(\\lambda,\\nabla,c_{0})} + \\int \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} d\\varphi^* = \\Psi_{\\lambda} + \\frac{(\\varphi^*)^{3}}{6} - \\operatorname{F_{N}}{(\\lambda,\\nabla,c_{0})}", "derivation": "t{(v,\\varphi^*)} = \\int \\varphi^* v d\\varphi^* and \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} = \\frac{\\partial}{\\partial v} \\int \\varphi^* v d\\varphi^* and \\int \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} d\\varphi^* = \\int \\frac{\\partial}{\\partial v} \\int \\varphi^* v d\\varphi^* d\\varphi^* and \\int \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} d\\varphi^* = \\Psi_{\\lambda} + \\frac{(\\varphi^*)^{3}}{6} and - \\operatorname{F_{N}}{(\\lambda,\\nabla,c_{0})} + \\int \\frac{\\partial}{\\partial v} t{(v,\\varphi^*)} d\\varphi^* = \\Psi_{\\lambda} + \\frac{(\\varphi^*)^{3}}{6} - \\operatorname{F_{N}}{(\\lambda,\\nabla,c_{0})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Derivative(Function('t')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Derivative(Integral(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('t')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Rational(1, 6), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(3)))))"], [["minus", 4, "Function('F_N')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('c_0', commutative=True))), Integral(Derivative(Function('t')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Rational(1, 6), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(3))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(h)} = \\sin{(h)}, then obtain \\int (- h - \\lambda{(h)} \\sin{(h)} + \\lambda{(h)} - \\sin{(h)}) dh = \\int (- h + \\lambda{(h)} - \\sin^{2}{(h)} - \\sin{(h)}) dh", "derivation": "\\lambda{(h)} = \\sin{(h)} and \\lambda{(h)} \\sin{(h)} = \\sin^{2}{(h)} and h + \\lambda{(h)} \\sin{(h)} = h + \\sin^{2}{(h)} and h + \\lambda{(h)} \\sin{(h)} - \\lambda{(h)} = h - \\lambda{(h)} + \\sin^{2}{(h)} and h + \\lambda{(h)} \\sin{(h)} - \\lambda{(h)} + \\sin{(h)} = h - \\lambda{(h)} + \\sin^{2}{(h)} + \\sin{(h)} and - h - \\lambda{(h)} \\sin{(h)} + \\lambda{(h)} - \\sin{(h)} = - h + \\lambda{(h)} - \\sin^{2}{(h)} - \\sin{(h)} and \\int (- h - \\lambda{(h)} \\sin{(h)} + \\lambda{(h)} - \\sin{(h)}) dh = \\int (- h + \\lambda{(h)} - \\sin^{2}{(h)} - \\sin{(h)}) dh", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["times", 1, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(2)))"], [["add", 2, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Mul(Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))), Add(Symbol('h', commutative=True), Pow(sin(Symbol('h', commutative=True)), Integer(2))))"], [["minus", 3, "Function('\\\\lambda')(Symbol('h', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Mul(Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True)))), Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(2))))"], [["add", 4, "sin(Symbol('h', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Mul(Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True))), sin(Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(2)), sin(Symbol('h', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Function('\\\\lambda')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\lambda')(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('h', commutative=True)))))"], [["integrate", 6, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Function('\\\\lambda')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\lambda')(Symbol('h', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given q{(r)} = \\sin{(e^{r})} and \\phi{(r)} = e^{r}, then obtain 1 = \\frac{\\sin{(e^{r})}}{\\sin{(\\phi{(r)})}}", "derivation": "q{(r)} = \\sin{(e^{r})} and \\phi{(r)} = e^{r} and q{(r)} = \\sin{(\\phi{(r)})} and 1 = \\frac{\\sin{(e^{r})}}{q{(r)}} and 1 = \\frac{\\sin{(e^{r})}}{\\sin{(\\phi{(r)})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('r', commutative=True)), sin(exp(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('q')(Symbol('r', commutative=True)), sin(Function('\\\\phi')(Symbol('r', commutative=True))))"], [["divide", 1, "Function('q')(Symbol('r', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('q')(Symbol('r', commutative=True)), Integer(-1)), sin(exp(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(1), Mul(Pow(sin(Function('\\\\phi')(Symbol('r', commutative=True))), Integer(-1)), sin(exp(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given x{(\\hat{H}_l,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + e^{\\hat{H}_l} and L{(\\mathbf{p},F_{H})} = F_{H} - \\mathbf{p}, then obtain L{(\\mathbf{p},F_{H})} \\int (V_{\\mathbf{E}} + e^{\\hat{H}_l}) dV_{\\mathbf{E}} = (F_{H} - \\mathbf{p}) \\int (V_{\\mathbf{E}} + e^{\\hat{H}_l}) dV_{\\mathbf{E}}", "derivation": "x{(\\hat{H}_l,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + e^{\\hat{H}_l} and \\int x{(\\hat{H}_l,V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int (V_{\\mathbf{E}} + e^{\\hat{H}_l}) dV_{\\mathbf{E}} and L{(\\mathbf{p},F_{H})} = F_{H} - \\mathbf{p} and L{(\\mathbf{p},F_{H})} \\int x{(\\hat{H}_l,V_{\\mathbf{E}})} dV_{\\mathbf{E}} = (F_{H} - \\mathbf{p}) \\int x{(\\hat{H}_l,V_{\\mathbf{E}})} dV_{\\mathbf{E}} and L{(\\mathbf{p},F_{H})} \\int (V_{\\mathbf{E}} + e^{\\hat{H}_l}) dV_{\\mathbf{E}} = (F_{H} - \\mathbf{p}) \\int (V_{\\mathbf{E}} + e^{\\hat{H}_l}) dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["times", 3, "Integral(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Mul(Function('L')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_H', commutative=True)), Integral(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Function('x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('L')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_H', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(g,L_{\\varepsilon})} = (e^{g})^{L_{\\varepsilon}}, then obtain \\int (\\int \\mathbf{H}{(g,L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} dL_{\\varepsilon} = \\int (\\int (e^{g})^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} dL_{\\varepsilon}", "derivation": "\\mathbf{H}{(g,L_{\\varepsilon})} = (e^{g})^{L_{\\varepsilon}} and \\int \\mathbf{H}{(g,L_{\\varepsilon})} dL_{\\varepsilon} = \\int (e^{g})^{L_{\\varepsilon}} dL_{\\varepsilon} and (\\int \\mathbf{H}{(g,L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} = (\\int (e^{g})^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} and \\int (\\int \\mathbf{H}{(g,L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} dL_{\\varepsilon} = \\int (\\int (e^{g})^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(exp(Symbol('g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Pow(exp(Symbol('g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integral(Pow(exp(Symbol('g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mathbf{H}')(Symbol('g', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Pow(Integral(Pow(exp(Symbol('g', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(f_{E})} = \\log{(f_{E})}, then obtain f_{E}^{2} \\hat{x}^{2}{(f_{E})} = f_{E}^{2} \\hat{x}{(f_{E})} \\log{(f_{E})}", "derivation": "\\hat{x}{(f_{E})} = \\log{(f_{E})} and f_{E} \\hat{x}{(f_{E})} = f_{E} \\log{(f_{E})} and f_{E}^{2} \\hat{x}{(f_{E})} = f_{E}^{2} \\log{(f_{E})} and f_{E}^{2} \\hat{x}^{2}{(f_{E})} = f_{E}^{2} \\hat{x}{(f_{E})} \\log{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["times", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\hat{x}')(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True))))"], [["times", 2, "Symbol('f_E', commutative=True)"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(2)), Function('\\\\hat{x}')(Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Integer(2)), log(Symbol('f_E', commutative=True))))"], [["times", 3, "Function('\\\\hat{x}')(Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(2)), Pow(Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), Integer(2))), Mul(Pow(Symbol('f_E', commutative=True), Integer(2)), Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(E,\\mathbf{S})} = \\mathbf{S}^{E}, then obtain \\frac{\\int \\operatorname{E_{x}}{(E,\\mathbf{S})} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\mathbf{S}^{E}} = \\frac{\\int \\mathbf{S}^{E} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\mathbf{S}^{E}}", "derivation": "\\operatorname{E_{x}}{(E,\\mathbf{S})} = \\mathbf{S}^{E} and \\frac{\\partial}{\\partial E} \\operatorname{E_{x}}{(E,\\mathbf{S})} = \\frac{\\partial}{\\partial E} \\mathbf{S}^{E} and \\int \\operatorname{E_{x}}{(E,\\mathbf{S})} d\\mathbf{S} = \\int \\mathbf{S}^{E} d\\mathbf{S} and \\frac{\\int \\operatorname{E_{x}}{(E,\\mathbf{S})} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\operatorname{E_{x}}{(E,\\mathbf{S})}} = \\frac{\\int \\mathbf{S}^{E} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\operatorname{E_{x}}{(E,\\mathbf{S})}} and \\frac{\\int \\operatorname{E_{x}}{(E,\\mathbf{S})} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\mathbf{S}^{E}} = \\frac{\\int \\mathbf{S}^{E} d\\mathbf{S}}{\\frac{\\partial}{\\partial E} \\mathbf{S}^{E}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 3, "Derivative(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(-1)), Integral(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Pow(Derivative(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Derivative(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(-1)), Integral(Function('E_x')(Symbol('E', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Pow(Derivative(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(I,\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} I^{\\mathbf{s}} and v{(I,\\mathbf{s})} = I^{\\mathbf{s}} + \\frac{\\partial}{\\partial \\mathbf{s}} I^{\\mathbf{s}}, then obtain I^{\\mathbf{s}} + \\mathbf{H}{(I,\\mathbf{s})} = v{(I,\\mathbf{s})}", "derivation": "\\mathbf{H}{(I,\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} I^{\\mathbf{s}} and I^{\\mathbf{s}} + \\mathbf{H}{(I,\\mathbf{s})} = I^{\\mathbf{s}} + \\frac{\\partial}{\\partial \\mathbf{s}} I^{\\mathbf{s}} and v{(I,\\mathbf{s})} = I^{\\mathbf{s}} + \\frac{\\partial}{\\partial \\mathbf{s}} I^{\\mathbf{s}} and I^{\\mathbf{s}} + \\mathbf{H}{(I,\\mathbf{s})} = v{(I,\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["add", 1, "Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Function('v')(Symbol('I', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(S,A_{2})} = S^{A_{2}}, then obtain \\frac{d}{d S} 0 + \\frac{\\partial}{\\partial A_{2}} \\operatorname{v_{t}}{(S,A_{2})} = \\frac{\\partial}{\\partial S} (S^{A_{2}} - \\operatorname{v_{t}}{(S,A_{2})}) + \\frac{\\partial}{\\partial A_{2}} \\operatorname{v_{t}}{(S,A_{2})}", "derivation": "\\operatorname{v_{t}}{(S,A_{2})} = S^{A_{2}} and 0 = S^{A_{2}} - \\operatorname{v_{t}}{(S,A_{2})} and \\frac{d}{d S} 0 = \\frac{\\partial}{\\partial S} (S^{A_{2}} - \\operatorname{v_{t}}{(S,A_{2})}) and \\frac{d}{d S} 0 + \\frac{\\partial}{\\partial A_{2}} \\operatorname{v_{t}}{(S,A_{2})} = \\frac{\\partial}{\\partial S} (S^{A_{2}} - \\operatorname{v_{t}}{(S,A_{2})}) + \\frac{\\partial}{\\partial A_{2}} \\operatorname{v_{t}}{(S,A_{2})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('A_2', commutative=True)))"], [["minus", 1, "Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Derivative(Add(Pow(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Function('v_t')(Symbol('S', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(\\hat{\\mathbf{r}},W)} = \\hat{\\mathbf{r}}^{W}, then obtain (\\int \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} d\\hat{\\mathbf{r}})^{W} = \\frac{\\hat{\\mathbf{r}}^{W} (\\int \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} d\\hat{\\mathbf{r}})^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}}", "derivation": "\\theta{(\\hat{\\mathbf{r}},W)} = \\hat{\\mathbf{r}}^{W} and 1 = \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} and \\int 1 d\\hat{\\mathbf{r}} = \\int \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} d\\hat{\\mathbf{r}} and (\\int 1 d\\hat{\\mathbf{r}})^{W} = \\frac{\\hat{\\mathbf{r}}^{W} (\\int 1 d\\hat{\\mathbf{r}})^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} and (\\int \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} d\\hat{\\mathbf{r}})^{W} = \\frac{\\hat{\\mathbf{r}}^{W} (\\int \\frac{\\hat{\\mathbf{r}}^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}} d\\hat{\\mathbf{r}})^{W}}{\\theta{(\\hat{\\mathbf{r}},W)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)))"], [["divide", 1, "Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 2, "Pow(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('W', commutative=True))"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('W', commutative=True)), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('W', commutative=True)), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1)), Pow(Integral(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('W', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(a)} = \\frac{d}{d a} \\sin{(a)}, then obtain (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)})^{2} \\operatorname{a^{\\dagger}}^{a}{(a)} = (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)})^{2} (\\frac{d}{d a} \\sin{(a)})^{a}", "derivation": "\\operatorname{a^{\\dagger}}{(a)} = \\frac{d}{d a} \\sin{(a)} and \\operatorname{a^{\\dagger}}^{a}{(a)} = (\\frac{d}{d a} \\sin{(a)})^{a} and (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)}) \\operatorname{a^{\\dagger}}^{a}{(a)} = (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)}) (\\frac{d}{d a} \\sin{(a)})^{a} and (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)})^{2} \\operatorname{a^{\\dagger}}^{a}{(a)} = (\\operatorname{a^{\\dagger}}{(a)} + \\frac{d}{d a} \\sin{(a)})^{2} (\\frac{d}{d a} \\sin{(a)})^{a}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)))"], [["times", 2, "Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Mul(Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Pow(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True))))"], [["times", 3, "Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Pow(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Pow(Add(Function('a^{\\\\dagger}')(Symbol('a', commutative=True)), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Integer(2)), Pow(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given n{(\\mathbf{P})} = \\sin{(\\cos{(\\mathbf{P})})} and \\operatorname{m_{s}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then obtain \\sin{(\\operatorname{m_{s}}{(\\mathbf{P})})} \\sin{(\\cos{(\\mathbf{P})})} = \\sin^{2}{(\\cos{(\\mathbf{P})})}", "derivation": "n{(\\mathbf{P})} = \\sin{(\\cos{(\\mathbf{P})})} and n{(\\mathbf{P})} \\sin{(\\cos{(\\mathbf{P})})} = \\sin^{2}{(\\cos{(\\mathbf{P})})} and \\operatorname{m_{s}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and n{(\\mathbf{P})} = \\sin{(\\operatorname{m_{s}}{(\\mathbf{P})})} and \\sin{(\\operatorname{m_{s}}{(\\mathbf{P})})} \\sin{(\\cos{(\\mathbf{P})})} = \\sin^{2}{(\\cos{(\\mathbf{P})})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{P}', commutative=True)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["times", 1, "sin(cos(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Function('n')(Symbol('\\\\mathbf{P}', commutative=True)), sin(cos(Symbol('\\\\mathbf{P}', commutative=True)))), Pow(sin(cos(Symbol('\\\\mathbf{P}', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('n')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(sin(Function('m_s')(Symbol('\\\\mathbf{P}', commutative=True))), sin(cos(Symbol('\\\\mathbf{P}', commutative=True)))), Pow(sin(cos(Symbol('\\\\mathbf{P}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{g}{(\\nabla)} = \\frac{d}{d \\nabla} \\log{(\\nabla)}, then derive \\mathbf{g}{(\\nabla)} = \\frac{1}{\\nabla}, then obtain \\frac{d}{d \\nabla} (\\mathbf{g}{(\\nabla)} + \\frac{d}{d \\nabla} \\log{(\\nabla)}) = \\frac{d}{d \\nabla} (\\mathbf{g}{(\\nabla)} + \\frac{1}{\\nabla})", "derivation": "\\mathbf{g}{(\\nabla)} = \\frac{d}{d \\nabla} \\log{(\\nabla)} and \\mathbf{g}{(\\nabla)} = \\frac{1}{\\nabla} and \\mathbf{g}{(\\nabla)} + \\frac{1}{\\nabla} = \\frac{d}{d \\nabla} \\log{(\\nabla)} + \\frac{1}{\\nabla} and \\mathbf{g}{(\\nabla)} + \\frac{d}{d \\nabla} \\log{(\\nabla)} = \\frac{d}{d \\nabla} \\log{(\\nabla)} + \\frac{1}{\\nabla} and \\mathbf{g}{(\\nabla)} + \\frac{d}{d \\nabla} \\log{(\\nabla)} = \\mathbf{g}{(\\nabla)} + \\frac{1}{\\nabla} and \\frac{d}{d \\nabla} (\\mathbf{g}{(\\nabla)} + \\frac{d}{d \\nabla} \\log{(\\nabla)}) = \\frac{d}{d \\nabla} (\\mathbf{g}{(\\nabla)} + \\frac{1}{\\nabla})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))"], [["add", 1, "Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Add(Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["add", 2, "Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Derivative(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{g}')(Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(M)} = e^{M}, then obtain (\\frac{d}{d M} \\int \\Omega{(M)} dM)^{M} = (\\frac{d}{d M} \\int e^{M} dM)^{M}", "derivation": "\\Omega{(M)} = e^{M} and \\int \\Omega{(M)} dM = \\int e^{M} dM and \\frac{d}{d M} \\int \\Omega{(M)} dM = \\frac{d}{d M} \\int e^{M} dM and (\\frac{d}{d M} \\int \\Omega{(M)} dM)^{M} = (\\frac{d}{d M} \\int e^{M} dM)^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Omega')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\Omega')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = - L_{\\varepsilon} + v_{1}, then derive L_{\\varepsilon} \\frac{\\partial}{\\partial v_{1}} \\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = L_{\\varepsilon}, then obtain (L_{\\varepsilon} \\frac{\\partial}{\\partial v_{1}} (- L_{\\varepsilon} + v_{1}))^{L_{\\varepsilon}} = L_{\\varepsilon}^{L_{\\varepsilon}}", "derivation": "\\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = - L_{\\varepsilon} + v_{1} and L_{\\varepsilon} \\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = L_{\\varepsilon} (- L_{\\varepsilon} + v_{1}) and \\frac{\\partial}{\\partial v_{1}} L_{\\varepsilon} \\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = \\frac{\\partial}{\\partial v_{1}} L_{\\varepsilon} (- L_{\\varepsilon} + v_{1}) and L_{\\varepsilon} \\frac{\\partial}{\\partial v_{1}} \\operatorname{n_{1}}{(v_{1},L_{\\varepsilon})} = L_{\\varepsilon} and L_{\\varepsilon} \\frac{\\partial}{\\partial v_{1}} (- L_{\\varepsilon} + v_{1}) = L_{\\varepsilon} and (L_{\\varepsilon} \\frac{\\partial}{\\partial v_{1}} (- L_{\\varepsilon} + v_{1}))^{L_{\\varepsilon}} = L_{\\varepsilon}^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('v_1', commutative=True)))"], [["times", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('n_1')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('v_1', commutative=True))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Function('n_1')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(Function('n_1')(Symbol('v_1', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('L_{\\\\varepsilon}', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('L_{\\\\varepsilon}', commutative=True))"], [["power", 5, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mu{(T,H)} = \\sin{(H - T)}, then obtain - H + T + \\frac{T \\mu{(T,H)}}{- T \\mu{(T,H)} + \\sin{(H - T)}} = - H + T + \\frac{T \\sin{(H - T)}}{- T \\mu{(T,H)} + \\sin{(H - T)}}", "derivation": "\\mu{(T,H)} = \\sin{(H - T)} and T \\mu{(T,H)} = T \\sin{(H - T)} and - T \\mu{(T,H)} + \\mu{(T,H)} = - T \\mu{(T,H)} + \\sin{(H - T)} and \\frac{T \\mu{(T,H)}}{- T \\mu{(T,H)} + \\mu{(T,H)}} = \\frac{T \\sin{(H - T)}}{- T \\mu{(T,H)} + \\mu{(T,H)}} and - H + T + \\frac{T \\mu{(T,H)}}{- T \\mu{(T,H)} + \\mu{(T,H)}} = - H + T + \\frac{T \\sin{(H - T)}}{- T \\mu{(T,H)} + \\mu{(T,H)}} and - H + T + \\frac{T \\mu{(T,H)}}{- T \\mu{(T,H)} + \\sin{(H - T)}} = - H + T + \\frac{T \\sin{(H - T)}}{- T \\mu{(T,H)} + \\sin{(H - T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True)), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))))"], [["times", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Mul(Symbol('T', commutative=True), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))))"], [["minus", 1, "Mul(Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True)))"], "Equality(Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(-1)), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))))"], [["minus", 4, "Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('T', commutative=True), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('T', commutative=True), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(-1)), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('T', commutative=True), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Integer(-1)), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('T', commutative=True), Mul(Symbol('T', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('\\\\mu')(Symbol('T', commutative=True), Symbol('H', commutative=True))), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True))))), Integer(-1)), sin(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('T', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{x},y^{\\prime})} = \\frac{y^{\\prime}}{v_{x}} and \\theta_{1}{(v_{x},y^{\\prime})} = \\frac{\\operatorname{n_{1}}{(v_{x},y^{\\prime})}}{v_{x}} - \\frac{1}{v_{x}}, then obtain \\theta_{1}{(v_{x},y^{\\prime})} - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} = - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} + \\frac{\\operatorname{n_{1}}{(v_{x},y^{\\prime})}}{v_{x}} - \\frac{1}{v_{x}}", "derivation": "\\operatorname{n_{1}}{(v_{x},y^{\\prime})} = \\frac{y^{\\prime}}{v_{x}} and \\theta_{1}{(v_{x},y^{\\prime})} = \\frac{\\operatorname{n_{1}}{(v_{x},y^{\\prime})}}{v_{x}} - \\frac{1}{v_{x}} and \\theta_{1}{(v_{x},y^{\\prime})} = - \\frac{1}{v_{x}} + \\frac{y^{\\prime}}{v_{x}^{2}} and \\theta_{1}{(v_{x},y^{\\prime})} - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} = - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} - \\frac{1}{v_{x}} + \\frac{y^{\\prime}}{v_{x}^{2}} and \\theta_{1}{(v_{x},y^{\\prime})} - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} = - \\operatorname{n_{1}}{(v_{x},y^{\\prime})} + \\frac{\\operatorname{n_{1}}{(v_{x},y^{\\prime})}}{v_{x}} - \\frac{1}{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\theta_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 3, "Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\theta_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\theta_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('n_1')(Symbol('v_x', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given f{(y^{\\prime})} = e^{e^{y^{\\prime}}}, then obtain \\int \\cos{(f{(y^{\\prime})} e^{e^{y^{\\prime}}})} dy^{\\prime} = \\int \\cos{(e^{2 e^{y^{\\prime}}})} dy^{\\prime}", "derivation": "f{(y^{\\prime})} = e^{e^{y^{\\prime}}} and f{(y^{\\prime})} e^{e^{y^{\\prime}}} = e^{2 e^{y^{\\prime}}} and \\cos{(f{(y^{\\prime})} e^{e^{y^{\\prime}}})} = \\cos{(e^{2 e^{y^{\\prime}}})} and \\int \\cos{(f{(y^{\\prime})} e^{e^{y^{\\prime}}})} dy^{\\prime} = \\int \\cos{(e^{2 e^{y^{\\prime}}})} dy^{\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('f')(Symbol('y^{\\\\prime}', commutative=True)), exp(exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 1, "exp(exp(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('f')(Symbol('y^{\\\\prime}', commutative=True)), exp(exp(Symbol('y^{\\\\prime}', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["cos", 2], "Equality(cos(Mul(Function('f')(Symbol('y^{\\\\prime}', commutative=True)), exp(exp(Symbol('y^{\\\\prime}', commutative=True))))), cos(exp(Mul(Integer(2), exp(Symbol('y^{\\\\prime}', commutative=True))))))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(cos(Mul(Function('f')(Symbol('y^{\\\\prime}', commutative=True)), exp(exp(Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(exp(Mul(Integer(2), exp(Symbol('y^{\\\\prime}', commutative=True))))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{f})} = \\sin{(\\log{(\\mathbf{f})})}, then obtain 0 = (\\mathbf{f} + \\sin{(\\log{(\\mathbf{f})})}) (- \\operatorname{t_{2}}{(\\mathbf{f})} + \\sin{(\\log{(\\mathbf{f})})})", "derivation": "\\operatorname{t_{2}}{(\\mathbf{f})} = \\sin{(\\log{(\\mathbf{f})})} and \\mathbf{f} + \\operatorname{t_{2}}{(\\mathbf{f})} = \\mathbf{f} + \\sin{(\\log{(\\mathbf{f})})} and 0 = - \\operatorname{t_{2}}{(\\mathbf{f})} + \\sin{(\\log{(\\mathbf{f})})} and 0 = (\\mathbf{f} + \\operatorname{t_{2}}{(\\mathbf{f})}) (- \\operatorname{t_{2}}{(\\mathbf{f})} + \\sin{(\\log{(\\mathbf{f})})}) and 0 = (\\mathbf{f} + \\sin{(\\log{(\\mathbf{f})})}) (- \\operatorname{t_{2}}{(\\mathbf{f})} + \\sin{(\\log{(\\mathbf{f})})})", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True)), sin(log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), sin(log(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["minus", 1, "Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))), sin(log(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["times", 3, "Add(Symbol('\\\\mathbf{f}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Integer(0), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))), sin(log(Symbol('\\\\mathbf{f}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(log(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{f}', commutative=True))), sin(log(Symbol('\\\\mathbf{f}', commutative=True))))))"]]}, {"prompt": "Given \\phi{(\\nabla)} = \\sin{(\\nabla)} and \\mathbf{M}{(\\chi,\\nabla,r)} = - \\nabla \\phi{(\\nabla)} + \\mathbf{B}{(\\chi,r)}, then obtain \\int \\frac{\\partial}{\\partial \\chi} \\mathbf{M}{(\\chi,\\nabla,r)} dr = \\int \\frac{\\partial}{\\partial \\chi} (- \\nabla \\sin{(\\nabla)} + \\mathbf{B}{(\\chi,r)}) dr", "derivation": "\\phi{(\\nabla)} = \\sin{(\\nabla)} and \\nabla \\phi{(\\nabla)} = \\nabla \\sin{(\\nabla)} and \\mathbf{M}{(\\chi,\\nabla,r)} = - \\nabla \\phi{(\\nabla)} + \\mathbf{B}{(\\chi,r)} and \\frac{\\partial}{\\partial \\chi} \\mathbf{M}{(\\chi,\\nabla,r)} = \\frac{\\partial}{\\partial \\chi} (- \\nabla \\phi{(\\nabla)} + \\mathbf{B}{(\\chi,r)}) and \\int \\frac{\\partial}{\\partial \\chi} \\mathbf{M}{(\\chi,\\nabla,r)} dr = \\int \\frac{\\partial}{\\partial \\chi} (- \\nabla \\phi{(\\nabla)} + \\mathbf{B}{(\\chi,r)}) dr and \\int \\frac{\\partial}{\\partial \\chi} \\mathbf{M}{(\\chi,\\nabla,r)} dr = \\int \\frac{\\partial}{\\partial \\chi} (- \\nabla \\sin{(\\nabla)} + \\mathbf{B}{(\\chi,r)}) dr", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\chi', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\chi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\chi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\chi', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} = \\frac{\\rho_b}{V_{\\mathbf{E}}}, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (\\int \\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (\\int \\frac{\\rho_b}{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}}", "derivation": "\\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} = \\frac{\\rho_b}{V_{\\mathbf{E}}} and \\int \\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\int \\frac{\\rho_b}{V_{\\mathbf{E}}} dV_{\\mathbf{E}} and (\\int \\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} = (\\int \\frac{\\rho_b}{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (\\int \\mathbf{r}{(\\rho_b,V_{\\mathbf{E}})} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (\\int \\frac{\\rho_b}{V_{\\mathbf{E}}} dV_{\\mathbf{E}})^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["power", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\rho_b', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(n_{1})} = e^{n_{1}} and H{(n_{1})} = - (e^{n_{1}})^{n_{1}}, then obtain (\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}})^{2} = \\frac{d}{d n_{1}} - \\mu_{0}^{n_{1}}{(n_{1})} \\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}", "derivation": "\\mu_{0}{(n_{1})} = e^{n_{1}} and \\mu_{0}^{n_{1}}{(n_{1})} = (e^{n_{1}})^{n_{1}} and H{(n_{1})} = - (e^{n_{1}})^{n_{1}} and \\frac{d}{d n_{1}} H{(n_{1})} = \\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}} and \\frac{d}{d n_{1}} H{(n_{1})} = \\frac{d}{d n_{1}} - \\mu_{0}^{n_{1}}{(n_{1})} and \\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}} = \\frac{d}{d n_{1}} - \\mu_{0}^{n_{1}}{(n_{1})} and (\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}})^{2} = \\frac{d}{d n_{1}} - \\mu_{0}^{n_{1}}{(n_{1})} \\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('n_1', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('H')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 6, "Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Integer(-1), Pow(Function('\\\\mu_0')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(\\nabla,\\ddot{x})} = \\frac{\\sin{(\\nabla)}}{\\ddot{x}}, then obtain \\frac{\\nabla + \\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}}{\\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}} = \\frac{\\nabla - \\sin{(\\nabla)} + \\frac{\\sin{(\\nabla)}}{\\ddot{x}}}{\\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}}", "derivation": "\\varepsilon{(\\nabla,\\ddot{x})} = \\frac{\\sin{(\\nabla)}}{\\ddot{x}} and \\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)} = - \\sin{(\\nabla)} + \\frac{\\sin{(\\nabla)}}{\\ddot{x}} and \\nabla + \\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)} = \\nabla - \\sin{(\\nabla)} + \\frac{\\sin{(\\nabla)}}{\\ddot{x}} and \\frac{\\nabla + \\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}}{\\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}} = \\frac{\\nabla - \\sin{(\\nabla)} + \\frac{\\sin{(\\nabla)}}{\\ddot{x}}}{\\varepsilon{(\\nabla,\\ddot{x})} - \\sin{(\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True)))))"], [["add", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 3, "Add(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))), Mul(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(g)} = \\cos{(g)} and l{(g)} = g \\operatorname{f_{\\mathbf{v}}}{(g)}, then obtain - g \\cos{(g)} + l{(g)} - \\cos{(g)} = - \\cos{(g)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(g)} = \\cos{(g)} and l{(g)} = g \\operatorname{f_{\\mathbf{v}}}{(g)} and l{(g)} = g \\cos{(g)} and - g \\cos{(g)} + l{(g)} - \\cos{(g)} = - \\cos{(g)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('l')(Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))))"], [["minus", 3, "Add(Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), cos(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), cos(Symbol('g', commutative=True))), Function('l')(Symbol('g', commutative=True)), Mul(Integer(-1), cos(Symbol('g', commutative=True)))), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"]]}, {"prompt": "Given T{(\\phi,H)} = H + \\phi, then obtain (H + 2 \\phi)^{- H} (- 2 \\phi + (H + 2 \\phi)^{H}) = (H + 2 \\phi)^{- H} (- 2 \\phi + 2 (H + 2 \\phi)^{H} - (\\phi + T{(\\phi,H)})^{H})", "derivation": "T{(\\phi,H)} = H + \\phi and \\phi + T{(\\phi,H)} = H + 2 \\phi and (\\phi + T{(\\phi,H)})^{H} = (H + 2 \\phi)^{H} and - 2 \\phi + (\\phi + T{(\\phi,H)})^{H} = - 2 \\phi + (H + 2 \\phi)^{H} and - 2 \\phi = - 2 \\phi + (H + 2 \\phi)^{H} - (\\phi + T{(\\phi,H)})^{H} and (H + 2 \\phi)^{- H} (- 2 \\phi + (\\phi + T{(\\phi,H)})^{H}) = (H + 2 \\phi)^{- H} (- 2 \\phi + (H + 2 \\phi)^{H}) and (H + 2 \\phi)^{- H} (- 2 \\phi + (H + 2 \\phi)^{H}) = (H + 2 \\phi)^{- H} (- 2 \\phi + 2 (H + 2 \\phi)^{H} - (\\phi + T{(\\phi,H)})^{H})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True)))"], [["minus", 3, "Mul(Integer(2), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True))))"], [["minus", 4, "Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)))))"], [["divide", 4, "Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)))), Mul(Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True)))), Mul(Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\phi', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('H', commutative=True), Mul(Integer(2), Symbol('\\\\phi', commutative=True))), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\phi', commutative=True), Function('T')(Symbol('\\\\phi', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(i,G)} = G i, then obtain \\frac{2 (G + \\operatorname{E_{n}}{(i,G)}) \\int \\frac{2 G + 2 \\operatorname{E_{n}}{(i,G)}}{G + \\operatorname{E_{n}}{(i,G)}} di}{G i + G} = \\frac{2 (G + \\operatorname{E_{n}}{(i,G)}) \\int 2 di}{G i + G}", "derivation": "\\operatorname{E_{n}}{(i,G)} = G i and G + \\operatorname{E_{n}}{(i,G)} = G i + G and \\frac{G + \\operatorname{E_{n}}{(i,G)}}{G i + G} = 1 and \\mathbf{s} + \\frac{2 (G + \\operatorname{E_{n}}{(i,G)})}{G i + G} = \\mathbf{s} + \\frac{G + \\operatorname{E_{n}}{(i,G)}}{G i + G} + 1 and \\frac{2 (G + \\operatorname{E_{n}}{(i,G)})}{G i + G} = \\frac{G + \\operatorname{E_{n}}{(i,G)}}{G i + G} + 1 and \\frac{2 G + 2 \\operatorname{E_{n}}{(i,G)}}{G + \\operatorname{E_{n}}{(i,G)}} = 2 and \\int \\frac{2 G + 2 \\operatorname{E_{n}}{(i,G)}}{G + \\operatorname{E_{n}}{(i,G)}} di = \\int 2 di and \\frac{2 (G + \\operatorname{E_{n}}{(i,G)}) \\int \\frac{2 G + 2 \\operatorname{E_{n}}{(i,G)}}{G + \\operatorname{E_{n}}{(i,G)}} di}{G i + G} = \\frac{2 (G + \\operatorname{E_{n}}{(i,G)}) \\int 2 di}{G i + G}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)))"], [["divide", 2, "Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True))"], "Equality(Mul(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1))), Integer(1))"], [["add", 3, "Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1))))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1)))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1))), Integer(1)))"], [["minus", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Integer(2), Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1))), Add(Mul(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))))), Integer(2))"], [["integrate", 6, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))))), Tuple(Symbol('i', commutative=True))), Integral(Integer(2), Tuple(Symbol('i', commutative=True))))"], [["times", 7, "Mul(Integer(2), Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1)), Integral(Mul(Pow(Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))))), Tuple(Symbol('i', commutative=True)))), Mul(Integer(2), Add(Symbol('G', commutative=True), Function('E_n')(Symbol('i', commutative=True), Symbol('G', commutative=True))), Pow(Add(Mul(Symbol('G', commutative=True), Symbol('i', commutative=True)), Symbol('G', commutative=True)), Integer(-1)), Integral(Integer(2), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(m)} = \\log{(e^{m})} and \\mathbf{f}{(m)} = \\log{(e^{m})}, then obtain \\operatorname{E_{x}}^{m}{(m)} - \\frac{d}{d m} \\log{(e^{m})} = \\log{(e^{m})}^{m} - \\frac{d}{d m} \\log{(e^{m})}", "derivation": "\\operatorname{E_{x}}{(m)} = \\log{(e^{m})} and \\mathbf{f}{(m)} = \\log{(e^{m})} and \\operatorname{E_{x}}{(m)} = \\mathbf{f}{(m)} and \\mathbf{f}^{m}{(m)} = \\log{(e^{m})}^{m} and \\mathbf{f}^{m}{(m)} - \\frac{d}{d m} \\log{(e^{m})} = \\log{(e^{m})}^{m} - \\frac{d}{d m} \\log{(e^{m})} and \\operatorname{E_{x}}^{m}{(m)} - \\frac{d}{d m} \\log{(e^{m})} = \\log{(e^{m})}^{m} - \\frac{d}{d m} \\log{(e^{m})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E_x')(Symbol('m', commutative=True)), Function('\\\\mathbf{f}')(Symbol('m', commutative=True)))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["minus", 4, "Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))), Add(Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Function('E_x')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))), Add(Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Mul(Integer(-1), Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))))"]]}, {"prompt": "Given U{(\\mathbf{A})} = e^{\\mathbf{A}}, then derive 0 = e^{\\mathbf{A}} - \\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}, then obtain 0 = - \\frac{(U{(\\mathbf{A})} - \\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}) U{(\\mathbf{A})}}{\\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}}", "derivation": "U{(\\mathbf{A})} = e^{\\mathbf{A}} and 0 = - U{(\\mathbf{A})} + e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} 0 = \\frac{d}{d \\mathbf{A}} (- U{(\\mathbf{A})} + e^{\\mathbf{A}}) and 0 = e^{\\mathbf{A}} - \\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})} and 0 = e^{\\mathbf{A}} - \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and 0 = - (e^{\\mathbf{A}} - \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}) U{(\\mathbf{A})} and 0 = - (U{(\\mathbf{A})} - \\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}) U{(\\mathbf{A})} and 0 = - \\frac{(U{(\\mathbf{A})} - \\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}) U{(\\mathbf{A})}}{\\frac{d}{d \\mathbf{A}} U{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Function('U')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{A}', commutative=True))), exp(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{A}', commutative=True))), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(exp(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(exp(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))))"], [["times", 5, "Mul(Integer(-1), Function('U')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(exp(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))), Function('U')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Mul(Integer(-1), Add(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))), Function('U')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 7, "Derivative(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))), Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Pow(Derivative(Function('U')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given g{(k,\\hat{X})} = \\sin{(\\hat{X} k)}, then derive \\frac{\\hat{X} g^{\\hat{X}}{(k,\\hat{X})} \\frac{\\partial}{\\partial k} g{(k,\\hat{X})}}{g{(k,\\hat{X})}} = \\frac{\\hat{X}^{2} \\sin^{\\hat{X}}{(\\hat{X} k)} \\cos{(\\hat{X} k)}}{\\sin{(\\hat{X} k)}}, then obtain \\frac{\\hat{X} \\sin^{\\hat{X}}{(\\hat{X} k)} \\frac{\\partial}{\\partial k} g{(k,\\hat{X})}}{g{(k,\\hat{X})}} = \\frac{\\hat{X}^{2} \\sin^{\\hat{X}}{(\\hat{X} k)} \\cos{(\\hat{X} k)}}{\\sin{(\\hat{X} k)}}", "derivation": "g{(k,\\hat{X})} = \\sin{(\\hat{X} k)} and g^{\\hat{X}}{(k,\\hat{X})} = \\sin^{\\hat{X}}{(\\hat{X} k)} and \\frac{\\partial}{\\partial k} g^{\\hat{X}}{(k,\\hat{X})} = \\frac{\\partial}{\\partial k} \\sin^{\\hat{X}}{(\\hat{X} k)} and \\frac{\\hat{X} g^{\\hat{X}}{(k,\\hat{X})} \\frac{\\partial}{\\partial k} g{(k,\\hat{X})}}{g{(k,\\hat{X})}} = \\frac{\\hat{X}^{2} \\sin^{\\hat{X}}{(\\hat{X} k)} \\cos{(\\hat{X} k)}}{\\sin{(\\hat{X} k)}} and \\frac{\\hat{X} \\sin^{\\hat{X}}{(\\hat{X} k)} \\frac{\\partial}{\\partial k} g{(k,\\hat{X})}}{g{(k,\\hat{X})}} = \\frac{\\hat{X}^{2} \\sin^{\\hat{X}}{(\\hat{X} k)} \\cos{(\\hat{X} k)}}{\\sin{(\\hat{X} k)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Pow(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Derivative(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Derivative(Function('g')(Symbol('k', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Pow(sin(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\theta,v_{z})} = \\theta v_{z}, then obtain \\int \\theta \\operatorname{F_{g}}^{v_{z}}{(\\theta,v_{z})} d\\theta = \\int \\theta (\\theta v_{z})^{v_{z}} d\\theta", "derivation": "\\operatorname{F_{g}}{(\\theta,v_{z})} = \\theta v_{z} and \\operatorname{F_{g}}^{v_{z}}{(\\theta,v_{z})} = (\\theta v_{z})^{v_{z}} and \\theta \\operatorname{F_{g}}^{v_{z}}{(\\theta,v_{z})} = \\theta (\\theta v_{z})^{v_{z}} and \\int \\theta \\operatorname{F_{g}}^{v_{z}}{(\\theta,v_{z})} d\\theta = \\int \\theta (\\theta v_{z})^{v_{z}} d\\theta", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["times", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('F_g')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('F_g')(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Symbol('\\\\theta', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mu)} = e^{\\mu}, then obtain \\int (1 - \\frac{e^{\\mu}}{\\operatorname{v_{x}}{(\\mu)}}) d\\mu = \\int 0 d\\mu", "derivation": "\\operatorname{v_{x}}{(\\mu)} = e^{\\mu} and 1 = \\frac{e^{\\mu}}{\\operatorname{v_{x}}{(\\mu)}} and 1 - \\frac{e^{\\mu}}{\\operatorname{v_{x}}{(\\mu)}} = 0 and \\int (1 - \\frac{e^{\\mu}}{\\operatorname{v_{x}}{(\\mu)}}) d\\mu = \\int 0 d\\mu", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Function('v_x')(Symbol('\\\\mu', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_x')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('v_x')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('v_x')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Pow(Function('v_x')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given L{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain - \\cos{(\\hat{H}_l + \\mathbf{M})} + e^{- A_{z}} \\int L{(\\mathbf{p})} d\\mathbf{p} = - \\cos{(\\hat{H}_l + \\mathbf{M})} + e^{- A_{z}} \\int \\cos{(\\mathbf{p})} d\\mathbf{p}", "derivation": "L{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\int L{(\\mathbf{p})} d\\mathbf{p} = \\int \\cos{(\\mathbf{p})} d\\mathbf{p} and e^{- A_{z}} \\int L{(\\mathbf{p})} d\\mathbf{p} = e^{- A_{z}} \\int \\cos{(\\mathbf{p})} d\\mathbf{p} and - \\cos{(\\hat{H}_l + \\mathbf{M})} + e^{- A_{z}} \\int L{(\\mathbf{p})} d\\mathbf{p} = - \\cos{(\\hat{H}_l + \\mathbf{M})} + e^{- A_{z}} \\int \\cos{(\\mathbf{p})} d\\mathbf{p}", "srepr_derivation": [["get_premise", "Equality(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["divide", 2, "exp(Symbol('A_z', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('A_z', commutative=True))), Integral(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('A_z', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 3, "cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('A_z', commutative=True))), Integral(Function('L')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))), Add(Mul(Integer(-1), cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('A_z', commutative=True))), Integral(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(S,J)} = S^{J}, then obtain (J (- \\sin{(S^{J})} + \\sin{(\\Psi_{nl}{(S,J)})}))^{S} + \\frac{1}{\\sin{(\\Psi_{nl}{(S,J)})}} = 0^{S} + \\frac{1}{\\sin{(\\Psi_{nl}{(S,J)})}}", "derivation": "\\Psi_{nl}{(S,J)} = S^{J} and \\sin{(\\Psi_{nl}{(S,J)})} = \\sin{(S^{J})} and - \\sin{(S^{J})} + \\sin{(\\Psi_{nl}{(S,J)})} = 0 and J (- \\sin{(S^{J})} + \\sin{(\\Psi_{nl}{(S,J)})}) = 0 and (J (- \\sin{(S^{J})} + \\sin{(\\Psi_{nl}{(S,J)})}))^{S} = 0^{S} and (J (- \\sin{(S^{J})} + \\sin{(\\Psi_{nl}{(S,J)})}))^{S} + \\frac{1}{\\sin{(\\Psi_{nl}{(S,J)})}} = 0^{S} + \\frac{1}{\\sin{(\\Psi_{nl}{(S,J)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))), sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True))))"], [["minus", 2, "sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))), sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True)))), Integer(0))"], [["times", 3, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))), sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))), sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))))), Symbol('S', commutative=True)), Pow(Integer(0), Symbol('S', commutative=True)))"], [["add", 5, "Pow(sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))), Integer(-1))"], "Equality(Add(Pow(Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), sin(Pow(Symbol('S', commutative=True), Symbol('J', commutative=True)))), sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))))), Symbol('S', commutative=True)), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))), Integer(-1))), Add(Pow(Integer(0), Symbol('S', commutative=True)), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('S', commutative=True), Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} = \\frac{n}{g^{\\prime}_{\\varepsilon} x} and \\chi{(g^{\\prime}_{\\varepsilon},x,n)} = \\frac{n}{g^{\\prime}_{\\varepsilon} x}, then obtain n (\\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} + 1) = n (\\chi{(g^{\\prime}_{\\varepsilon},x,n)} + 1)", "derivation": "\\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} = \\frac{n}{g^{\\prime}_{\\varepsilon} x} and \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} + 1 = 1 + \\frac{n}{g^{\\prime}_{\\varepsilon} x} and n (\\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} + 1) = n (1 + \\frac{n}{g^{\\prime}_{\\varepsilon} x}) and \\chi{(g^{\\prime}_{\\varepsilon},x,n)} = \\frac{n}{g^{\\prime}_{\\varepsilon} x} and n (\\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},x,n)} + 1) = n (\\chi{(g^{\\prime}_{\\varepsilon},x,n)} + 1)", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))))"], [["times", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Add(Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Integer(1))), Mul(Symbol('n', commutative=True), Add(Integer(1), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('n', commutative=True), Add(Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Integer(1))), Mul(Symbol('n', commutative=True), Add(Function('\\\\chi')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True), Symbol('n', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(y)} = \\log{(y)}, then obtain (\\log{(y)} + \\frac{d}{d y} (\\int (\\mathbf{p}{(y)} - \\log{(y)}) dy + 1))^{y} = (\\log{(y)} + \\frac{d}{d y} (\\int 0 dy + 1))^{y}", "derivation": "\\mathbf{p}{(y)} = \\log{(y)} and \\mathbf{p}{(y)} - \\log{(y)} = 0 and \\int (\\mathbf{p}{(y)} - \\log{(y)}) dy = \\int 0 dy and \\int (\\mathbf{p}{(y)} - \\log{(y)}) dy + 1 = \\int 0 dy + 1 and \\frac{d}{d y} (\\int (\\mathbf{p}{(y)} - \\log{(y)}) dy + 1) = \\frac{d}{d y} (\\int 0 dy + 1) and \\log{(y)} + \\frac{d}{d y} (\\int (\\mathbf{p}{(y)} - \\log{(y)}) dy + 1) = \\log{(y)} + \\frac{d}{d y} (\\int 0 dy + 1) and (\\log{(y)} + \\frac{d}{d y} (\\int (\\mathbf{p}{(y)} - \\log{(y)}) dy + 1))^{y} = (\\log{(y)} + \\frac{d}{d y} (\\int 0 dy + 1))^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["minus", 1, "log(Symbol('y', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Integer(0), Tuple(Symbol('y', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integer(1)), Add(Integral(Integer(0), Tuple(Symbol('y', commutative=True))), Integer(1)))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Integral(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Integral(Integer(0), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Integer(-1), log(Symbol('y', commutative=True)))"], "Equality(Add(log(Symbol('y', commutative=True)), Derivative(Add(Integral(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(log(Symbol('y', commutative=True)), Derivative(Add(Integral(Integer(0), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('y', commutative=True)"], "Equality(Pow(Add(log(Symbol('y', commutative=True)), Derivative(Add(Integral(Add(Function('\\\\mathbf{p}')(Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))), Symbol('y', commutative=True)), Pow(Add(log(Symbol('y', commutative=True)), Derivative(Add(Integral(Integer(0), Tuple(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(F_{N},\\Omega)} = \\sin{(F_{N} + \\Omega)}, then derive \\int \\mathbf{B}{(F_{N},\\Omega)} d\\Omega = \\nabla - \\cos{(F_{N} + \\Omega)}, then obtain 0^{\\nabla} = (\\nabla - \\cos{(F_{N} + \\Omega)} - \\int \\sin{(F_{N} + \\Omega)} d\\Omega)^{\\nabla}", "derivation": "\\mathbf{B}{(F_{N},\\Omega)} = \\sin{(F_{N} + \\Omega)} and \\int \\mathbf{B}{(F_{N},\\Omega)} d\\Omega = \\int \\sin{(F_{N} + \\Omega)} d\\Omega and \\int \\mathbf{B}{(F_{N},\\Omega)} d\\Omega = \\nabla - \\cos{(F_{N} + \\Omega)} and \\int \\sin{(F_{N} + \\Omega)} d\\Omega = \\nabla - \\cos{(F_{N} + \\Omega)} and 0 = \\nabla - \\cos{(F_{N} + \\Omega)} - \\int \\sin{(F_{N} + \\Omega)} d\\Omega and 0^{\\nabla} = (\\nabla - \\cos{(F_{N} + \\Omega)} - \\int \\sin{(F_{N} + \\Omega)} d\\Omega)^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["minus", 4, "Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["power", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), cos(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), Integral(sin(Add(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(F_{N},\\sigma_p)} = e^{F_{N} - \\sigma_p} and z{(F_{N},\\sigma_p)} = \\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)}, then obtain \\frac{z^{2}{(F_{N},\\sigma_p)}}{\\dot{y}{(F_{N},\\sigma_p)}} = \\frac{\\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)} z{(F_{N},\\sigma_p)}}{\\dot{y}{(F_{N},\\sigma_p)}}", "derivation": "\\dot{y}{(F_{N},\\sigma_p)} = e^{F_{N} - \\sigma_p} and \\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)} = (e^{F_{N} - \\sigma_p})^{\\sigma_p} and z{(F_{N},\\sigma_p)} = \\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)} and z{(F_{N},\\sigma_p)} = (e^{F_{N} - \\sigma_p})^{\\sigma_p} and z^{2}{(F_{N},\\sigma_p)} = z{(F_{N},\\sigma_p)} (e^{F_{N} - \\sigma_p})^{\\sigma_p} and z^{2}{(F_{N},\\sigma_p)} = \\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)} z{(F_{N},\\sigma_p)} and \\frac{z^{2}{(F_{N},\\sigma_p)}}{\\dot{y}{(F_{N},\\sigma_p)}} = \\frac{\\dot{y}^{\\sigma_p}{(F_{N},\\sigma_p)} z{(F_{N},\\sigma_p)}}{\\dot{y}{(F_{N},\\sigma_p)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"], [["times", 4, "Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Pow(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2)), Mul(Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 6, "Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Pow(Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Pow(Function('\\\\dot{y}')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Function('z')(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\varepsilon)} = \\sin{(\\log{(\\varepsilon)})}, then derive \\frac{d}{d \\varepsilon} \\mathbf{g}{(\\varepsilon)} = \\frac{\\cos{(\\log{(\\varepsilon)})}}{\\varepsilon}, then obtain - \\log{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\sin{(\\log{(\\varepsilon)})} = - \\log{(\\varepsilon)} + \\frac{\\cos{(\\log{(\\varepsilon)})}}{\\varepsilon}", "derivation": "\\mathbf{g}{(\\varepsilon)} = \\sin{(\\log{(\\varepsilon)})} and \\frac{d}{d \\varepsilon} \\mathbf{g}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\sin{(\\log{(\\varepsilon)})} and \\frac{d}{d \\varepsilon} \\mathbf{g}{(\\varepsilon)} = \\frac{\\cos{(\\log{(\\varepsilon)})}}{\\varepsilon} and \\frac{d}{d \\varepsilon} \\sin{(\\log{(\\varepsilon)})} = \\frac{\\cos{(\\log{(\\varepsilon)})}}{\\varepsilon} and - \\log{(\\varepsilon)} + \\frac{d}{d \\varepsilon} \\sin{(\\log{(\\varepsilon)})} = - \\log{(\\varepsilon)} + \\frac{\\cos{(\\log{(\\varepsilon)})}}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\varepsilon', commutative=True)), sin(log(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(sin(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), cos(log(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), cos(log(Symbol('\\\\varepsilon', commutative=True)))))"], [["minus", 4, "log(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\varepsilon', commutative=True))), Derivative(sin(log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), cos(log(Symbol('\\\\varepsilon', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{F_{x}}^{4}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{F_{x}}^{3}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})}", "derivation": "\\operatorname{F_{x}}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\operatorname{F_{x}}^{2}{(V_{\\mathbf{E}})} = \\operatorname{F_{x}}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})} and \\operatorname{F_{x}}^{4}{(V_{\\mathbf{E}})} = \\operatorname{F_{x}}^{3}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})} and \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{F_{x}}^{4}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} \\operatorname{F_{x}}^{3}{(V_{\\mathbf{E}})} \\sin{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["times", 1, "Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2)), Mul(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 2, "Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2))"], "Equality(Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(4)), Mul(Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(3)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["differentiate", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(4)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('F_x')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(3)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(\\tilde{g},A_{z})} = \\log{(A_{z} \\tilde{g})}, then obtain 0 = \\frac{\\partial}{\\partial \\tilde{g}} A_{z} \\tilde{g} e^{A_{z}} - \\frac{\\partial}{\\partial \\tilde{g}} e^{A_{z} + \\omega{(\\tilde{g},A_{z})}}", "derivation": "\\omega{(\\tilde{g},A_{z})} = \\log{(A_{z} \\tilde{g})} and A_{z} + \\omega{(\\tilde{g},A_{z})} = A_{z} + \\log{(A_{z} \\tilde{g})} and e^{A_{z} + \\omega{(\\tilde{g},A_{z})}} = A_{z} \\tilde{g} e^{A_{z}} and \\frac{\\partial}{\\partial \\tilde{g}} e^{A_{z} + \\omega{(\\tilde{g},A_{z})}} = \\frac{\\partial}{\\partial \\tilde{g}} A_{z} \\tilde{g} e^{A_{z}} and 0 = \\frac{\\partial}{\\partial \\tilde{g}} A_{z} \\tilde{g} e^{A_{z}} - \\frac{\\partial}{\\partial \\tilde{g}} e^{A_{z} + \\omega{(\\tilde{g},A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)), log(Mul(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["add", 1, "Symbol('A_z', commutative=True)"], "Equality(Add(Symbol('A_z', commutative=True), Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('A_z', commutative=True), log(Mul(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Symbol('A_z', commutative=True), Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)))), Mul(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('A_z', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(exp(Add(Symbol('A_z', commutative=True), Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(exp(Add(Symbol('A_z', commutative=True), Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Add(Symbol('A_z', commutative=True), Function('\\\\omega')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{x}{(r_{0})} = \\cos{(\\sin{(r_{0})})}, then obtain ((\\int \\hat{x}{(r_{0})} dr_{0})^{r_{0}})^{r_{0}} = ((\\int \\cos{(\\sin{(r_{0})})} dr_{0})^{r_{0}})^{r_{0}}", "derivation": "\\hat{x}{(r_{0})} = \\cos{(\\sin{(r_{0})})} and \\int \\hat{x}{(r_{0})} dr_{0} = \\int \\cos{(\\sin{(r_{0})})} dr_{0} and (\\int \\hat{x}{(r_{0})} dr_{0})^{r_{0}} = (\\int \\cos{(\\sin{(r_{0})})} dr_{0})^{r_{0}} and ((\\int \\hat{x}{(r_{0})} dr_{0})^{r_{0}})^{r_{0}} = ((\\int \\cos{(\\sin{(r_{0})})} dr_{0})^{r_{0}})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True))))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(cos(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Pow(Integral(cos(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\hat{x}')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Integral(cos(sin(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(y,\\mathbf{F})} = \\sin^{y}{(\\mathbf{F})} and \\dot{x}{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then obtain \\int (\\dot{x}^{y}{(\\mathbf{F})} + \\tilde{g}^*{(y,\\mathbf{F})}) \\tilde{g}^*{(y,\\mathbf{F})} dy = \\int 2 \\dot{x}^{y}{(\\mathbf{F})} \\tilde{g}^*{(y,\\mathbf{F})} dy", "derivation": "\\tilde{g}^*{(y,\\mathbf{F})} = \\sin^{y}{(\\mathbf{F})} and \\tilde{g}^*{(y,\\mathbf{F})} + \\sin^{y}{(\\mathbf{F})} = 2 \\sin^{y}{(\\mathbf{F})} and \\dot{x}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\dot{x}^{y}{(\\mathbf{F})} + \\tilde{g}^*{(y,\\mathbf{F})} = 2 \\dot{x}^{y}{(\\mathbf{F})} and (\\dot{x}^{y}{(\\mathbf{F})} + \\tilde{g}^*{(y,\\mathbf{F})}) \\tilde{g}^*{(y,\\mathbf{F})} = 2 \\dot{x}^{y}{(\\mathbf{F})} \\tilde{g}^*{(y,\\mathbf{F})} and \\int (\\dot{x}^{y}{(\\mathbf{F})} + \\tilde{g}^*{(y,\\mathbf{F})}) \\tilde{g}^*{(y,\\mathbf{F})} dy = \\int 2 \\dot{x}^{y}{(\\mathbf{F})} \\tilde{g}^*{(y,\\mathbf{F})} dy", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True))))"], [["times", 4, "Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Add(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Integer(2), Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('y', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('y', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\pi)} = \\log{(e^{\\pi})}, then derive \\int \\operatorname{E_{x}}{(\\pi)} d\\pi = \\frac{\\pi^{2}}{2} + \\theta_1, then obtain (\\frac{d}{d \\pi} \\int \\operatorname{E_{x}}{(\\pi)} d\\pi)^{\\theta_1} = (\\frac{\\partial}{\\partial \\pi} (\\frac{\\pi^{2}}{2} + \\theta_1))^{\\theta_1}", "derivation": "\\operatorname{E_{x}}{(\\pi)} = \\log{(e^{\\pi})} and \\int \\operatorname{E_{x}}{(\\pi)} d\\pi = \\int \\log{(e^{\\pi})} d\\pi and \\int \\operatorname{E_{x}}{(\\pi)} d\\pi = \\frac{\\pi^{2}}{2} + \\theta_1 and \\frac{d}{d \\pi} \\int \\operatorname{E_{x}}{(\\pi)} d\\pi = \\frac{\\partial}{\\partial \\pi} (\\frac{\\pi^{2}}{2} + \\theta_1) and (\\frac{d}{d \\pi} \\int \\operatorname{E_{x}}{(\\pi)} d\\pi)^{\\theta_1} = (\\frac{\\partial}{\\partial \\pi} (\\frac{\\pi^{2}}{2} + \\theta_1))^{\\theta_1}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(log(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Integral(Function('E_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('E_x')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\pi', commutative=True), Integer(2))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(M)} = \\log{(M)}, then obtain ((\\frac{\\frac{d}{d M} \\operatorname{E_{\\lambda}}^{M}{(M)}}{\\frac{d}{d M} \\log{(M)}^{M}})^{M})^{M} = 1", "derivation": "\\operatorname{E_{\\lambda}}{(M)} = \\log{(M)} and \\operatorname{E_{\\lambda}}^{M}{(M)} = \\log{(M)}^{M} and \\frac{d}{d M} \\operatorname{E_{\\lambda}}^{M}{(M)} = \\frac{d}{d M} \\log{(M)}^{M} and \\frac{\\frac{d}{d M} \\operatorname{E_{\\lambda}}^{M}{(M)}}{\\frac{d}{d M} \\log{(M)}^{M}} = 1 and (\\frac{\\frac{d}{d M} \\operatorname{E_{\\lambda}}^{M}{(M)}}{\\frac{d}{d M} \\log{(M)}^{M}})^{M} = 1 and ((\\frac{\\frac{d}{d M} \\operatorname{E_{\\lambda}}^{M}{(M)}}{\\frac{d}{d M} \\log{(M)}^{M}})^{M})^{M} = 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Pow(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Derivative(Pow(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))), Symbol('M', commutative=True)), Integer(1))"], [["power", 5, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Mul(Derivative(Pow(Function('E_{\\\\lambda}')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Pow(Derivative(Pow(log(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\theta_{2}{(r_{0})} = \\log{(r_{0})}, then obtain - r_{0} + \\log{(r_{0})}^{r_{0}} = - r_{0} - \\theta_{2}^{r_{0}}{(r_{0})} + 2 \\log{(r_{0})}^{r_{0}}", "derivation": "\\theta_{2}{(r_{0})} = \\log{(r_{0})} and \\theta_{2}^{r_{0}}{(r_{0})} = \\log{(r_{0})}^{r_{0}} and - r_{0} + \\theta_{2}^{r_{0}}{(r_{0})} = - r_{0} + \\log{(r_{0})}^{r_{0}} and - r_{0} = - r_{0} - \\theta_{2}^{r_{0}}{(r_{0})} + \\log{(r_{0})}^{r_{0}} and - r_{0} + \\log{(r_{0})}^{r_{0}} = - r_{0} - \\theta_{2}^{r_{0}}{(r_{0})} + 2 \\log{(r_{0})}^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["minus", 2, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Mul(Integer(2), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given H{(F_{N},v_{y})} = F_{N} + v_{y}, then derive \\frac{F_{N} H^{F_{N}}{(F_{N},v_{y})} \\frac{\\partial}{\\partial v_{y}} H{(F_{N},v_{y})}}{H{(F_{N},v_{y})}} = \\frac{F_{N} (F_{N} + v_{y})^{F_{N}}}{F_{N} + v_{y}}, then obtain \\frac{F_{N} (F_{N} + v_{y})^{F_{N}} \\frac{\\partial}{\\partial v_{y}} H{(F_{N},v_{y})}}{H{(F_{N},v_{y})}} = \\frac{F_{N} (F_{N} + v_{y})^{F_{N}}}{F_{N} + v_{y}}", "derivation": "H{(F_{N},v_{y})} = F_{N} + v_{y} and H^{F_{N}}{(F_{N},v_{y})} = (F_{N} + v_{y})^{F_{N}} and \\frac{\\partial}{\\partial v_{y}} H^{F_{N}}{(F_{N},v_{y})} = \\frac{\\partial}{\\partial v_{y}} (F_{N} + v_{y})^{F_{N}} and \\frac{F_{N} H^{F_{N}}{(F_{N},v_{y})} \\frac{\\partial}{\\partial v_{y}} H{(F_{N},v_{y})}}{H{(F_{N},v_{y})}} = \\frac{F_{N} (F_{N} + v_{y})^{F_{N}}}{F_{N} + v_{y}} and \\frac{F_{N} (F_{N} + v_{y})^{F_{N}} \\frac{\\partial}{\\partial v_{y}} H{(F_{N},v_{y})}}{H{(F_{N},v_{y})}} = \\frac{F_{N} (F_{N} + v_{y})^{F_{N}}}{F_{N} + v_{y}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Pow(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('F_N', commutative=True), Pow(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Pow(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)), Derivative(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Symbol('F_N', commutative=True), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('F_N', commutative=True), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True)), Pow(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Derivative(Function('H')(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Symbol('F_N', commutative=True), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Pow(Add(Symbol('F_N', commutative=True), Symbol('v_y', commutative=True)), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given s{(V)} = \\sin{(e^{V})}, then obtain (\\iint s{(V)} dV dV - 1)^{V} = (\\iint \\sin{(e^{V})} dV dV - 1)^{V}", "derivation": "s{(V)} = \\sin{(e^{V})} and \\int s{(V)} dV = \\int \\sin{(e^{V})} dV and \\iint s{(V)} dV dV = \\iint \\sin{(e^{V})} dV dV and \\iint s{(V)} dV dV - 1 = \\iint \\sin{(e^{V})} dV dV - 1 and (\\iint s{(V)} dV dV - 1)^{V} = (\\iint \\sin{(e^{V})} dV dV - 1)^{V}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('V', commutative=True)), sin(exp(Symbol('V', commutative=True))))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1)), Add(Integral(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Integral(Function('s')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1)), Symbol('V', commutative=True)), Pow(Add(Integral(sin(exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1)), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(V,\\mathbf{p})} = \\mathbf{p}^{V}, then obtain \\mathbf{p}^{2 V} \\varphi^{*}^{2}{(V,\\mathbf{p})} = \\mathbf{p}^{4 V}", "derivation": "\\varphi^{*}{(V,\\mathbf{p})} = \\mathbf{p}^{V} and \\mathbf{p}^{V} \\varphi^{*}{(V,\\mathbf{p})} = \\mathbf{p}^{2 V} and \\mathbf{p}^{2 V} \\varphi^{*}^{2}{(V,\\mathbf{p})} = \\mathbf{p}^{3 V} \\varphi^{*}{(V,\\mathbf{p})} and \\mathbf{p}^{V} \\varphi^{*}^{3}{(V,\\mathbf{p})} = \\mathbf{p}^{3 V} \\varphi^{*}{(V,\\mathbf{p})} and \\mathbf{p}^{2 V} \\varphi^{*}^{2}{(V,\\mathbf{p})} = \\mathbf{p}^{4 V}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)), Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)), Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(3), Symbol('V', commutative=True))), Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(3))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(3), Symbol('V', commutative=True))), Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(4), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(x)} = e^{x}, then derive \\frac{d}{d x} \\operatorname{A_{1}}{(x)} = e^{x}, then obtain \\frac{d}{d x} \\cos{(e^{x})} = \\frac{d}{d x} \\cos{(\\frac{d}{d x} e^{x})}", "derivation": "\\operatorname{A_{1}}{(x)} = e^{x} and \\cos{(\\operatorname{A_{1}}{(x)})} = \\cos{(e^{x})} and \\frac{d}{d x} \\operatorname{A_{1}}{(x)} = \\frac{d}{d x} e^{x} and \\frac{d}{d x} \\cos{(\\operatorname{A_{1}}{(x)})} = \\frac{d}{d x} \\cos{(e^{x})} and \\frac{d}{d x} \\operatorname{A_{1}}{(x)} = e^{x} and \\frac{d}{d x} \\cos{(\\operatorname{A_{1}}{(x)})} = \\frac{d}{d x} \\cos{(\\frac{d}{d x} \\operatorname{A_{1}}{(x)})} and \\frac{d}{d x} \\cos{(e^{x})} = \\frac{d}{d x} \\cos{(\\frac{d}{d x} e^{x})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["cos", 1], "Equality(cos(Function('A_1')(Symbol('x', commutative=True))), cos(exp(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(cos(Function('A_1')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_1')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), exp(Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(cos(Function('A_1')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(Derivative(Function('A_1')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(cos(exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(Derivative(exp(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} = e^{\\hat{p}_0 - \\pi}, then derive \\frac{\\partial}{\\partial \\pi} \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + 1 = 1 - e^{\\hat{p}_0 - \\pi}, then obtain e^{\\hat{p}_0 - \\pi} + \\frac{\\partial}{\\partial \\pi} \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + 1 = - \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + e^{\\hat{p}_0 - \\pi} + 1", "derivation": "\\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} = e^{\\hat{p}_0 - \\pi} and \\pi + \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} = \\pi + e^{\\hat{p}_0 - \\pi} and \\frac{\\partial}{\\partial \\pi} (\\pi + \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)}) = \\frac{\\partial}{\\partial \\pi} (\\pi + e^{\\hat{p}_0 - \\pi}) and \\frac{\\partial}{\\partial \\pi} \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + 1 = 1 - e^{\\hat{p}_0 - \\pi} and \\frac{\\partial}{\\partial \\pi} \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + 1 = 1 - \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} and e^{\\hat{p}_0 - \\pi} + \\frac{\\partial}{\\partial \\pi} \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + 1 = - \\operatorname{C_{d}}{(\\pi,\\hat{p}_0)} + e^{\\hat{p}_0 - \\pi} + 1", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\pi', commutative=True), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\pi', commutative=True), Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\pi', commutative=True), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], "Equality(Add(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Derivative(Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Function('C_d')(Symbol('\\\\pi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{S}{(H)} = \\cos{(e^{H})} and \\dot{\\mathbf{r}}{(H)} = \\mathbf{S}^{H}{(H)}, then obtain e^{H} + \\frac{d}{d H} \\dot{\\mathbf{r}}{(H)} = (- \\frac{H e^{H} \\sin{(e^{H})}}{\\cos{(e^{H})}} + \\log{(\\cos{(e^{H})})}) \\cos^{H}{(e^{H})} + e^{H}", "derivation": "\\mathbf{S}{(H)} = \\cos{(e^{H})} and \\dot{\\mathbf{r}}{(H)} = \\mathbf{S}^{H}{(H)} and \\dot{\\mathbf{r}}{(H)} = \\cos^{H}{(e^{H})} and \\dot{\\mathbf{r}}{(H)} + 1 = \\cos^{H}{(e^{H})} + 1 and \\dot{\\mathbf{r}}{(H)} + e^{H} + 1 = e^{H} + \\cos^{H}{(e^{H})} + 1 and \\frac{d}{d H} (\\dot{\\mathbf{r}}{(H)} + e^{H} + 1) = \\frac{d}{d H} (e^{H} + \\cos^{H}{(e^{H})} + 1) and e^{H} + \\frac{d}{d H} \\dot{\\mathbf{r}}{(H)} = (- \\frac{H e^{H} \\sin{(e^{H})}}{\\cos{(e^{H})}} + \\log{(\\cos{(e^{H})})}) \\cos^{H}{(e^{H})} + e^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('H', commutative=True)), Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), Integer(1)), Add(Pow(cos(exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(1)))"], [["add", 4, "exp(Symbol('H', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Integer(1)), Add(exp(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(1)))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('H', commutative=True)), Pow(cos(exp(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(exp(Symbol('H', commutative=True)), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True), exp(Symbol('H', commutative=True)), sin(exp(Symbol('H', commutative=True))), Pow(cos(exp(Symbol('H', commutative=True))), Integer(-1))), log(cos(exp(Symbol('H', commutative=True))))), Pow(cos(exp(Symbol('H', commutative=True))), Symbol('H', commutative=True))), exp(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(s,\\tilde{g}^*)} = \\tilde{g}^* s, then obtain \\frac{d}{d \\tilde{g}^*} 0 = \\frac{d}{d \\tilde{g}^*} (1 - 0^{\\tilde{g}^*})", "derivation": "\\operatorname{C_{1}}{(s,\\tilde{g}^*)} = \\tilde{g}^* s and 0 = \\tilde{g}^* s - \\operatorname{C_{1}}{(s,\\tilde{g}^*)} and 0^{\\tilde{g}^*} = (\\tilde{g}^* s - \\operatorname{C_{1}}{(s,\\tilde{g}^*)})^{\\tilde{g}^*} and 0 = - 0^{\\tilde{g}^*} + (\\tilde{g}^* s - \\operatorname{C_{1}}{(s,\\tilde{g}^*)})^{\\tilde{g}^*} and 0 = 1 - (\\tilde{g}^* s - \\operatorname{C_{1}}{(s,\\tilde{g}^*)})^{\\tilde{g}^*} and 0 = 1 - 0^{\\tilde{g}^*} and \\frac{d}{d \\tilde{g}^*} 0 = \\frac{d}{d \\tilde{g}^*} (1 - 0^{\\tilde{g}^*})", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["power", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 3, "Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(\\psi^*,J,\\eta^{\\prime})} = \\frac{\\eta^{\\prime} \\psi^*}{J}, then obtain J \\pi{(\\psi^*,J,\\eta^{\\prime})} + J + \\eta^{\\prime} = J + \\eta^{\\prime} \\psi^* + \\eta^{\\prime}", "derivation": "\\pi{(\\psi^*,J,\\eta^{\\prime})} = \\frac{\\eta^{\\prime} \\psi^*}{J} and J \\pi{(\\psi^*,J,\\eta^{\\prime})} = \\eta^{\\prime} \\psi^* and J \\pi{(\\psi^*,J,\\eta^{\\prime})} + J = J + \\eta^{\\prime} \\psi^* and J \\pi{(\\psi^*,J,\\eta^{\\prime})} + J + \\eta^{\\prime} = J + \\eta^{\\prime} \\psi^* + \\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\psi^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\pi')(Symbol('\\\\psi^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["add", 2, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Symbol('J', commutative=True), Function('\\\\pi')(Symbol('\\\\psi^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["add", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Symbol('J', commutative=True), Function('\\\\pi')(Symbol('\\\\psi^*', commutative=True), Symbol('J', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('J', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('J', commutative=True), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(\\mu_0,\\eta)} = \\mu_0 + \\sin{(\\eta)}, then obtain \\frac{- \\eta + \\int \\mathbf{f}{(\\mu_0,\\eta)} d\\mu_0}{\\mu_0} = \\frac{S - \\eta + \\frac{\\mu_0^{2}}{2} + \\mu_0 \\sin{(\\eta)}}{\\mu_0}", "derivation": "\\mathbf{f}{(\\mu_0,\\eta)} = \\mu_0 + \\sin{(\\eta)} and \\int \\mathbf{f}{(\\mu_0,\\eta)} d\\mu_0 = \\int (\\mu_0 + \\sin{(\\eta)}) d\\mu_0 and - \\eta + \\int \\mathbf{f}{(\\mu_0,\\eta)} d\\mu_0 = - \\eta + \\int (\\mu_0 + \\sin{(\\eta)}) d\\mu_0 and \\frac{- \\eta + \\int \\mathbf{f}{(\\mu_0,\\eta)} d\\mu_0}{\\mu_0} = \\frac{- \\eta + \\int (\\mu_0 + \\sin{(\\eta)}) d\\mu_0}{\\mu_0} and \\frac{- \\eta + \\int \\mathbf{f}{(\\mu_0,\\eta)} d\\mu_0}{\\mu_0} = \\frac{S - \\eta + \\frac{\\mu_0^{2}}{2} + \\mu_0 \\sin{(\\eta)}}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\eta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Add(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["divide", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Add(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('S', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), Mul(Symbol('\\\\mu_0', commutative=True), sin(Symbol('\\\\eta', commutative=True))))))"]]}, {"prompt": "Given M{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})}, then obtain \\frac{d}{d \\hat{\\mathbf{r}}} (M^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = \\frac{d}{d \\hat{\\mathbf{r}}} (\\log{(\\hat{\\mathbf{r}})}^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}}", "derivation": "M{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} and M^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})}^{\\hat{\\mathbf{r}}} and (M^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = (\\log{(\\hat{\\mathbf{r}})}^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}} and \\frac{d}{d \\hat{\\mathbf{r}}} (M^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = \\frac{d}{d \\hat{\\mathbf{r}}} (\\log{(\\hat{\\mathbf{r}})}^{\\hat{\\mathbf{r}}})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Function('M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Pow(Function('M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Pow(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(Pow(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(F_{g},A)} = \\frac{F_{g}}{A}, then obtain \\int \\frac{\\partial^{2}}{\\partial A^{2}} \\phi_{1}{(F_{g},A)} dF_{g} = \\int \\frac{\\partial^{2}}{\\partial A^{2}} \\frac{F_{g}}{A} dF_{g}", "derivation": "\\phi_{1}{(F_{g},A)} = \\frac{F_{g}}{A} and \\frac{\\partial}{\\partial A} \\phi_{1}{(F_{g},A)} = \\frac{\\partial}{\\partial A} \\frac{F_{g}}{A} and \\frac{\\partial^{2}}{\\partial A^{2}} \\phi_{1}{(F_{g},A)} = \\frac{\\partial^{2}}{\\partial A^{2}} \\frac{F_{g}}{A} and \\int \\frac{\\partial^{2}}{\\partial A^{2}} \\phi_{1}{(F_{g},A)} dF_{g} = \\int \\frac{\\partial^{2}}{\\partial A^{2}} \\frac{F_{g}}{A} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('F_g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('F_g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Tuple(Symbol('F_g', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Symbol('F_g', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(y^{\\prime})} = \\cos{(\\sin{(y^{\\prime})})}, then obtain ((\\varepsilon_{0}{(y^{\\prime})} - \\sin{(y^{\\prime})}) \\cos{(\\sin{(y^{\\prime})})})^{y^{\\prime}} = ((- \\sin{(y^{\\prime})} + \\cos{(\\sin{(y^{\\prime})})}) \\cos{(\\sin{(y^{\\prime})})})^{y^{\\prime}}", "derivation": "\\varepsilon_{0}{(y^{\\prime})} = \\cos{(\\sin{(y^{\\prime})})} and \\varepsilon_{0}{(y^{\\prime})} - \\sin{(y^{\\prime})} = - \\sin{(y^{\\prime})} + \\cos{(\\sin{(y^{\\prime})})} and (\\varepsilon_{0}{(y^{\\prime})} - \\sin{(y^{\\prime})}) \\cos{(\\sin{(y^{\\prime})})} = (- \\sin{(y^{\\prime})} + \\cos{(\\sin{(y^{\\prime})})}) \\cos{(\\sin{(y^{\\prime})})} and ((\\varepsilon_{0}{(y^{\\prime})} - \\sin{(y^{\\prime})}) \\cos{(\\sin{(y^{\\prime})})})^{y^{\\prime}} = ((- \\sin{(y^{\\prime})} + \\cos{(\\sin{(y^{\\prime})})}) \\cos{(\\sin{(y^{\\prime})})})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True)), cos(sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "sin(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["times", 2, "cos(sin(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\varepsilon_0')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True)))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), sin(Symbol('y^{\\\\prime}', commutative=True))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\omega{(\\theta_2)} = \\log{(\\theta_2)}, then obtain (\\frac{2 \\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}})^{\\theta_2} = (\\frac{\\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} + 1)^{\\theta_2}", "derivation": "\\omega{(\\theta_2)} = \\log{(\\theta_2)} and \\frac{d}{d \\theta_2} \\omega{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and \\frac{\\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} = 1 and \\frac{2 \\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} = \\frac{\\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} + 1 and (\\frac{2 \\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}})^{\\theta_2} = (\\frac{\\frac{d}{d \\theta_2} \\omega{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} + 1)^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["add", 3, "Mul(Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Add(Mul(Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Integer(2), Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Mul(Derivative(Function('\\\\omega')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Integer(1)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(q)} = \\log{(\\cos{(q)})}, then obtain \\frac{d}{d q} (\\frac{d}{d q} \\operatorname{g^{\\prime}_{\\varepsilon}}{(q)})^{q} = \\frac{d}{d q} (\\frac{d}{d q} \\log{(\\cos{(q)})})^{q}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(q)} = \\log{(\\cos{(q)})} and \\frac{d}{d q} \\operatorname{g^{\\prime}_{\\varepsilon}}{(q)} = \\frac{d}{d q} \\log{(\\cos{(q)})} and (\\frac{d}{d q} \\operatorname{g^{\\prime}_{\\varepsilon}}{(q)})^{q} = (\\frac{d}{d q} \\log{(\\cos{(q)})})^{q} and \\frac{d}{d q} (\\frac{d}{d q} \\operatorname{g^{\\prime}_{\\varepsilon}}{(q)})^{q} = \\frac{d}{d q} (\\frac{d}{d q} \\log{(\\cos{(q)})})^{q}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('q', commutative=True)), log(cos(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(log(cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Pow(Derivative(log(cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Derivative(log(cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(\\pi)} = \\sin{(\\pi)}, then derive \\frac{\\int \\mathbf{M}{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} = \\frac{A - \\cos{(\\pi)}}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}}, then obtain \\frac{\\int \\sin{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} = \\frac{A - \\cos{(\\pi)}}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}}", "derivation": "\\mathbf{M}{(\\pi)} = \\sin{(\\pi)} and \\int \\mathbf{M}{(\\pi)} d\\pi = \\int \\sin{(\\pi)} d\\pi and \\frac{\\int \\mathbf{M}{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} = \\frac{\\int \\sin{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} and \\frac{\\int \\mathbf{M}{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} = \\frac{A - \\cos{(\\pi)}}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} and \\frac{\\int \\sin{(\\pi)} d\\pi}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}} = \\frac{A - \\cos{(\\pi)}}{\\operatorname{n_{1}}{(Z)} \\sin{(e^{Z})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "Mul(Function('n_1')(Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True))))"], "Equality(Mul(Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1)), Integral(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Pow(Function('n_1')(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(exp(Symbol('Z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}_0{(n)} = e^{n}, then derive \\int (\\hat{x}_0{(n)} - \\int e^{n} dn) dn = \\rho, then obtain \\frac{\\int (e^{n} - \\int e^{n} dn) dn}{\\int \\hat{x}_0{(n)} dn} = \\frac{\\rho}{\\int \\hat{x}_0{(n)} dn}", "derivation": "\\hat{x}_0{(n)} = e^{n} and \\int \\hat{x}_0{(n)} dn = \\int e^{n} dn and \\hat{x}_0{(n)} - \\int e^{n} dn = e^{n} - \\int e^{n} dn and \\int (\\hat{x}_0{(n)} - \\int e^{n} dn) dn = \\int (e^{n} - \\int e^{n} dn) dn and \\int (\\hat{x}_0{(n)} - \\int e^{n} dn) dn = \\rho and \\int (e^{n} - \\int e^{n} dn) dn = \\rho and \\frac{\\int (e^{n} - \\int e^{n} dn) dn}{\\int e^{n} dn} = \\frac{\\rho}{\\int e^{n} dn} and \\frac{\\int (e^{n} - \\int e^{n} dn) dn}{\\int \\hat{x}_0{(n)} dn} = \\frac{\\rho}{\\int \\hat{x}_0{(n)} dn}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["minus", 1, "Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Add(exp(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Integral(Add(exp(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Symbol('\\\\rho', commutative=True))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(exp(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Symbol('\\\\rho', commutative=True))"], [["divide", 6, "Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Mul(Integral(Add(exp(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Pow(Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))), Mul(Symbol('\\\\rho', commutative=True), Pow(Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Mul(Integral(Add(exp(Symbol('n', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), Tuple(Symbol('n', commutative=True))), Pow(Integral(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))), Mul(Symbol('\\\\rho', commutative=True), Pow(Integral(Function('\\\\hat{x}_0')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(\\hat{H})} = e^{\\hat{H}}, then obtain - \\theta_{2}{(\\hat{H})} \\frac{d}{d \\hat{H}} ((\\theta_{2}{(\\hat{H})} - e^{\\hat{H}})^{\\hat{H}})^{\\hat{H}} = - \\theta_{2}{(\\hat{H})} \\frac{d}{d \\hat{H}} (0^{\\hat{H}})^{\\hat{H}}", "derivation": "\\theta_{2}{(\\hat{H})} = e^{\\hat{H}} and \\theta_{2}{(\\hat{H})} - e^{\\hat{H}} = 0 and (\\theta_{2}{(\\hat{H})} - e^{\\hat{H}})^{\\hat{H}} = 0^{\\hat{H}} and ((\\theta_{2}{(\\hat{H})} - e^{\\hat{H}})^{\\hat{H}})^{\\hat{H}} = (0^{\\hat{H}})^{\\hat{H}} and \\frac{d}{d \\hat{H}} ((\\theta_{2}{(\\hat{H})} - e^{\\hat{H}})^{\\hat{H}})^{\\hat{H}} = \\frac{d}{d \\hat{H}} (0^{\\hat{H}})^{\\hat{H}} and - \\theta_{2}{(\\hat{H})} \\frac{d}{d \\hat{H}} ((\\theta_{2}{(\\hat{H})} - e^{\\hat{H}})^{\\hat{H}})^{\\hat{H}} = - \\theta_{2}{(\\hat{H})} \\frac{d}{d \\hat{H}} (0^{\\hat{H}})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["add", 1, "Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Add(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Pow(Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["times", 5, "Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(Pow(Add(Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(Pow(Integer(0), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\delta)} = e^{\\cos{(\\delta)}} and g{(\\delta)} = \\delta, then obtain (\\int \\mathbf{r}^{\\delta}{(\\delta)} dg{(\\delta)})^{g{(\\delta)}} = (\\int (e^{\\cos{(\\delta)}})^{\\delta} dg{(\\delta)})^{g{(\\delta)}}", "derivation": "\\mathbf{r}{(\\delta)} = e^{\\cos{(\\delta)}} and \\mathbf{r}^{\\delta}{(\\delta)} = (e^{\\cos{(\\delta)}})^{\\delta} and \\int \\mathbf{r}^{\\delta}{(\\delta)} d\\delta = \\int (e^{\\cos{(\\delta)}})^{\\delta} d\\delta and (\\int \\mathbf{r}^{\\delta}{(\\delta)} d\\delta)^{\\delta} = (\\int (e^{\\cos{(\\delta)}})^{\\delta} d\\delta)^{\\delta} and g{(\\delta)} = \\delta and (\\int \\mathbf{r}^{\\delta}{(\\delta)} dg{(\\delta)})^{g{(\\delta)}} = (\\int (e^{\\cos{(\\delta)}})^{\\delta} dg{(\\delta)})^{g{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Function('g')(Symbol('\\\\delta', commutative=True)))), Function('g')(Symbol('\\\\delta', commutative=True))), Pow(Integral(Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Function('g')(Symbol('\\\\delta', commutative=True)))), Function('g')(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(f_{\\mathbf{v}},E_{n})} = \\log{(f_{\\mathbf{v}}^{E_{n}})}, then obtain - \\eta^{\\prime} + \\int \\operatorname{A_{x}}^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},E_{n})} dE_{n} = - \\eta^{\\prime} + \\int \\log{(f_{\\mathbf{v}}^{E_{n}})}^{f_{\\mathbf{v}}} dE_{n}", "derivation": "\\operatorname{A_{x}}{(f_{\\mathbf{v}},E_{n})} = \\log{(f_{\\mathbf{v}}^{E_{n}})} and \\operatorname{A_{x}}^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},E_{n})} = \\log{(f_{\\mathbf{v}}^{E_{n}})}^{f_{\\mathbf{v}}} and \\int \\operatorname{A_{x}}^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},E_{n})} dE_{n} = \\int \\log{(f_{\\mathbf{v}}^{E_{n}})}^{f_{\\mathbf{v}}} dE_{n} and - \\eta^{\\prime} + \\int \\operatorname{A_{x}}^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}},E_{n})} dE_{n} = - \\eta^{\\prime} + \\int \\log{(f_{\\mathbf{v}}^{E_{n}})}^{f_{\\mathbf{v}}} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True)), log(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True))))"], [["power", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(log(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Pow(Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Pow(log(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["minus", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Pow(Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Pow(log(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('E_n', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given n{(V_{\\mathbf{E}},\\dot{y})} = - V_{\\mathbf{E}} + \\dot{y}, then derive \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} = \\frac{\\dot{y}^{2}}{2} + \\mathbf{F}, then obtain -1 = \\frac{\\dot{y}^{2}}{2} + \\mathbf{F} - \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} - 1", "derivation": "n{(V_{\\mathbf{E}},\\dot{y})} = - V_{\\mathbf{E}} + \\dot{y} and V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})} = \\dot{y} and \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} = \\int \\dot{y} d\\dot{y} and \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} = \\frac{\\dot{y}^{2}}{2} + \\mathbf{F} and 0 = \\frac{\\dot{y}^{2}}{2} + \\mathbf{F} - \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} and -1 = \\frac{\\dot{y}^{2}}{2} + \\mathbf{F} - \\int (V_{\\mathbf{E}} + n{(V_{\\mathbf{E}},\\dot{y})}) d\\dot{y} - 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Symbol('\\\\dot{y}', commutative=True), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))))"], [["add", 5, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integral(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('n')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then derive \\frac{d}{d J_{\\varepsilon}} \\hat{x}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\frac{d}{d J_{\\varepsilon}} \\hat{x}{(J_{\\varepsilon})} - 1 = \\frac{d^{2}}{d J_{\\varepsilon}^{2}} \\hat{x}{(J_{\\varepsilon})} - 1", "derivation": "\\hat{x}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\frac{d}{d J_{\\varepsilon}} \\hat{x}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and \\frac{d}{d J_{\\varepsilon}} \\hat{x}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and e^{J_{\\varepsilon}} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and e^{J_{\\varepsilon}} - 1 = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} - 1 and \\frac{d}{d J_{\\varepsilon}} \\hat{x}{(J_{\\varepsilon})} - 1 = \\frac{d^{2}}{d J_{\\varepsilon}^{2}} \\hat{x}{(J_{\\varepsilon})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2))), Integer(-1)))"]]}, {"prompt": "Given G{(\\mathbf{r})} = \\log{(\\sin{(\\mathbf{r})})}, then obtain - 2 G^{2}{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})}^{3} = - 2 G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})}^{4}", "derivation": "G{(\\mathbf{r})} = \\log{(\\sin{(\\mathbf{r})})} and G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})} = \\log{(\\sin{(\\mathbf{r})})}^{2} and G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})} + \\log{(\\sin{(\\mathbf{r})})}^{2} = 2 \\log{(\\sin{(\\mathbf{r})})}^{2} and (G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})} + \\log{(\\sin{(\\mathbf{r})})}^{2}) \\log{(\\sin{(\\mathbf{r})})} = 2 \\log{(\\sin{(\\mathbf{r})})}^{3} and 2 G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})}^{2} = 2 \\log{(\\sin{(\\mathbf{r})})}^{3} and - 2 G^{2}{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})}^{3} = - 2 G{(\\mathbf{r})} \\log{(\\sin{(\\mathbf{r})})}^{4}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 1, "log(sin(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True)))), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2)))"], [["add", 2, "Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2))"], "Equality(Add(Mul(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True)))), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2))), Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2))))"], [["times", 3, "log(sin(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Add(Mul(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True)))), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2))), log(sin(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(2))), Mul(Integer(2), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(3))))"], [["times", 5, "Mul(Integer(-1), Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), log(sin(Symbol('\\\\mathbf{r}', commutative=True))))"], "Equality(Mul(Integer(-1), Integer(2), Pow(Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(3))), Mul(Integer(-1), Integer(2), Function('G')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{r}', commutative=True))), Integer(4))))"]]}, {"prompt": "Given v{(z,s)} = \\cos^{z}{(s)}, then derive \\frac{z \\sin{(s)} \\cos^{z}{(s)}}{\\cos{(s)}} + \\frac{\\partial}{\\partial s} v{(z,s)} = 0, then obtain \\frac{z v{(z,s)} \\sin{(s)}}{\\cos{(s)}} + \\frac{\\partial}{\\partial s} v{(z,s)} = 0", "derivation": "v{(z,s)} = \\cos^{z}{(s)} and \\frac{\\partial}{\\partial s} v{(z,s)} = \\frac{\\partial}{\\partial s} \\cos^{z}{(s)} and \\frac{\\partial}{\\partial s} v{(z,s)} - \\frac{\\partial}{\\partial s} \\cos^{z}{(s)} = 0 and \\frac{z \\sin{(s)} \\cos^{z}{(s)}}{\\cos{(s)}} + \\frac{\\partial}{\\partial s} v{(z,s)} = 0 and \\frac{z v{(z,s)} \\sin{(s)}}{\\cos{(s)}} + \\frac{\\partial}{\\partial s} v{(z,s)} = 0", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('s', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(cos(Symbol('s', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(cos(Symbol('s', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('z', commutative=True), sin(Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Integer(-1)), Pow(cos(Symbol('s', commutative=True)), Symbol('z', commutative=True))), Derivative(Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('z', commutative=True), Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Integer(-1))), Derivative(Function('v')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = \\mathbf{D} + f^{\\prime}, then derive \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = 1, then obtain \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = 0", "derivation": "\\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = \\mathbf{D} + f^{\\prime} and \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = \\frac{\\partial}{\\partial f^{\\prime}} (\\mathbf{D} + f^{\\prime}) and \\frac{\\partial}{\\partial f^{\\prime}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = 1 and \\frac{\\partial}{\\partial f^{\\prime}} (\\mathbf{D} + f^{\\prime}) = 1 and \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} (\\mathbf{D} + f^{\\prime}) = \\frac{d}{d f^{\\prime}} 1 and \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = \\frac{d}{d f^{\\prime}} 1 and \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} \\operatorname{f^{*}}{(f^{\\prime},\\mathbf{D})} = 0", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('f^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\lambda{(\\varepsilon_0,P_{g})} = P_{g} + e^{\\varepsilon_0}, then derive \\frac{\\partial}{\\partial \\varepsilon_0} \\lambda{(\\varepsilon_0,P_{g})} + 1 = e^{\\varepsilon_0} + 1, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} \\lambda{(\\varepsilon_0,P_{g})} + 1 = \\frac{\\partial}{\\partial \\varepsilon_0} (P_{g} + e^{\\varepsilon_0}) + 1", "derivation": "\\lambda{(\\varepsilon_0,P_{g})} = P_{g} + e^{\\varepsilon_0} and \\varepsilon_0 + \\lambda{(\\varepsilon_0,P_{g})} = P_{g} + \\varepsilon_0 + e^{\\varepsilon_0} and \\frac{\\partial}{\\partial \\varepsilon_0} (\\varepsilon_0 + \\lambda{(\\varepsilon_0,P_{g})}) = \\frac{\\partial}{\\partial \\varepsilon_0} (P_{g} + \\varepsilon_0 + e^{\\varepsilon_0}) and \\frac{\\partial}{\\partial \\varepsilon_0} \\lambda{(\\varepsilon_0,P_{g})} + 1 = e^{\\varepsilon_0} + 1 and \\frac{\\partial}{\\partial \\varepsilon_0} (P_{g} + e^{\\varepsilon_0}) + 1 = e^{\\varepsilon_0} + 1 and \\frac{\\partial}{\\partial \\varepsilon_0} \\lambda{(\\varepsilon_0,P_{g})} + 1 = \\frac{\\partial}{\\partial \\varepsilon_0} (P_{g} + e^{\\varepsilon_0}) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_g', commutative=True)), Add(Symbol('P_g', commutative=True), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_g', commutative=True))), Add(Symbol('P_g', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('\\\\varepsilon_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Add(Symbol('P_g', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Symbol('P_g', commutative=True), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('\\\\lambda')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Symbol('P_g', commutative=True), exp(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\rho_{f}{(F_{c},\\dot{x})} = F_{c} \\dot{x}, then derive (\\frac{\\partial}{\\partial F_{c}} \\rho_{f}{(F_{c},\\dot{x})})^{F_{c}} = \\dot{x}^{F_{c}}, then obtain F_{c} + (\\frac{\\partial}{\\partial F_{c}} F_{c} \\dot{x})^{F_{c}} = F_{c} + \\dot{x}^{F_{c}}", "derivation": "\\rho_{f}{(F_{c},\\dot{x})} = F_{c} \\dot{x} and \\frac{\\partial}{\\partial F_{c}} \\rho_{f}{(F_{c},\\dot{x})} = \\frac{\\partial}{\\partial F_{c}} F_{c} \\dot{x} and (\\frac{\\partial}{\\partial F_{c}} \\rho_{f}{(F_{c},\\dot{x})})^{F_{c}} = (\\frac{\\partial}{\\partial F_{c}} F_{c} \\dot{x})^{F_{c}} and (\\frac{\\partial}{\\partial F_{c}} \\rho_{f}{(F_{c},\\dot{x})})^{F_{c}} = \\dot{x}^{F_{c}} and (\\frac{\\partial}{\\partial F_{c}} F_{c} \\dot{x})^{F_{c}} = \\dot{x}^{F_{c}} and F_{c} + (\\frac{\\partial}{\\partial F_{c}} F_{c} \\dot{x})^{F_{c}} = F_{c} + \\dot{x}^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho_f')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(Derivative(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\rho_f')(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('F_c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('F_c', commutative=True)))"], [["add", 5, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Pow(Derivative(Mul(Symbol('F_c', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given B{(m,\\mathbf{S})} = \\mathbf{S} + m and \\rho_{f}{(u)} = \\int e^{u} du, then obtain \\int (B^{\\mathbf{S}}{(m,\\mathbf{S})} - \\rho_{f}{(u)}) dm = \\int ((\\mathbf{S} + m)^{\\mathbf{S}} - \\rho_{f}{(u)}) dm", "derivation": "B{(m,\\mathbf{S})} = \\mathbf{S} + m and B^{\\mathbf{S}}{(m,\\mathbf{S})} = (\\mathbf{S} + m)^{\\mathbf{S}} and \\rho_{f}{(u)} = \\int e^{u} du and B^{\\mathbf{S}}{(m,\\mathbf{S})} - \\int e^{u} du = (\\mathbf{S} + m)^{\\mathbf{S}} - \\int e^{u} du and B^{\\mathbf{S}}{(m,\\mathbf{S})} - \\rho_{f}{(u)} = (\\mathbf{S} + m)^{\\mathbf{S}} - \\rho_{f}{(u)} and \\int (B^{\\mathbf{S}}{(m,\\mathbf{S})} - \\rho_{f}{(u)}) dm = \\int ((\\mathbf{S} + m)^{\\mathbf{S}} - \\rho_{f}{(u)}) dm", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('B')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('u', commutative=True)), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["minus", 2, "Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Add(Pow(Function('B')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))), Add(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('B')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('u', commutative=True)))), Add(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('u', commutative=True)))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Pow(Function('B')(Symbol('m', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('u', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(Add(Pow(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('u', commutative=True)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(n_{1},T)} = - n_{1} + \\log{(T)}, then derive \\omega + n_{1} = \\int \\frac{- n_{1} + \\log{(T)}}{\\operatorname{v_{y}}{(n_{1},T)}} dn_{1}, then derive \\omega + n_{1} = \\Psi_{nl} + n_{1}, then obtain \\int (\\omega + n_{1})^{\\omega} d\\omega = \\int (\\Psi_{nl} + n_{1})^{\\omega} d\\omega", "derivation": "\\operatorname{v_{y}}{(n_{1},T)} = - n_{1} + \\log{(T)} and 1 = \\frac{- n_{1} + \\log{(T)}}{\\operatorname{v_{y}}{(n_{1},T)}} and \\int 1 dn_{1} = \\int \\frac{- n_{1} + \\log{(T)}}{\\operatorname{v_{y}}{(n_{1},T)}} dn_{1} and \\omega + n_{1} = \\int \\frac{- n_{1} + \\log{(T)}}{\\operatorname{v_{y}}{(n_{1},T)}} dn_{1} and \\omega + n_{1} = \\int 1 dn_{1} and \\omega + n_{1} = \\Psi_{nl} + n_{1} and (\\omega + n_{1})^{\\omega} = (\\Psi_{nl} + n_{1})^{\\omega} and \\int (\\omega + n_{1})^{\\omega} d\\omega = \\int (\\Psi_{nl} + n_{1})^{\\omega} d\\omega", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('n_1', commutative=True), Symbol('T', commutative=True)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), log(Symbol('T', commutative=True))))"], [["divide", 1, "Function('v_y')(Symbol('n_1', commutative=True), Symbol('T', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), log(Symbol('T', commutative=True))), Pow(Function('v_y')(Symbol('n_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('n_1', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), log(Symbol('T', commutative=True))), Pow(Function('v_y')(Symbol('n_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True)), Integral(Mul(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), log(Symbol('T', commutative=True))), Pow(Function('v_y')(Symbol('n_1', commutative=True), Symbol('T', commutative=True)), Integer(-1))), Tuple(Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('n_1', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)))"], [["power", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["integrate", 7, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given Q{(n,E_{x})} = \\frac{E_{x}}{n} and \\operatorname{t_{2}}{(E_{x})} = E_{x}, then obtain 2 Q{(n,E_{x})} + 1 = 1 + \\frac{E_{x} + \\operatorname{t_{2}}{(E_{x})}}{n}", "derivation": "Q{(n,E_{x})} = \\frac{E_{x}}{n} and Q{(n,E_{x})} + 1 = \\frac{E_{x}}{n} + 1 and 2 Q{(n,E_{x})} + 1 = \\frac{E_{x}}{n} + Q{(n,E_{x})} + 1 and \\operatorname{t_{2}}{(E_{x})} = E_{x} and E_{x} + \\operatorname{t_{2}}{(E_{x})} = 2 E_{x} and 2 Q{(n,E_{x})} + 1 = \\frac{2 E_{x}}{n} + 1 and 2 Q{(n,E_{x})} + 1 = 1 + \\frac{E_{x} + \\operatorname{t_{2}}{(E_{x})}}{n}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)), Integer(1)), Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], [["add", 2, "Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(1)), Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], [["add", 4, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Function('t_2')(Symbol('E_x', commutative=True))), Mul(Integer(2), Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('E_x', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(2), Function('Q')(Symbol('n', commutative=True), Symbol('E_x', commutative=True))), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Function('t_2')(Symbol('E_x', commutative=True))))))"]]}, {"prompt": "Given G{(r_{0})} = \\log{(r_{0})} and B{(r_{0})} = \\log{(r_{0})}^{r_{0}}, then obtain G^{r_{0}}{(r_{0})} + \\log{(r_{0})} = B{(r_{0})} + \\log{(r_{0})}", "derivation": "G{(r_{0})} = \\log{(r_{0})} and G^{r_{0}}{(r_{0})} = \\log{(r_{0})}^{r_{0}} and G^{r_{0}}{(r_{0})} + \\log{(r_{0})} = \\log{(r_{0})} + \\log{(r_{0})}^{r_{0}} and B{(r_{0})} = \\log{(r_{0})}^{r_{0}} and G^{r_{0}}{(r_{0})} + \\log{(r_{0})} = B{(r_{0})} + \\log{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('G')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["add", 2, "log(Symbol('r_0', commutative=True))"], "Equality(Add(Pow(Function('G')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True))), Add(log(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], ["renaming_premise", "Equality(Function('B')(Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('G')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True))), Add(Function('B')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given L{(h)} = \\log{(h)}, then derive \\int L{(h)} dh = h \\log{(h)} - h + x, then derive \\eta + h \\log{(h)} - h = h \\log{(h)} - h + x, then obtain \\eta + h \\log{(h)} - h = h L{(h)} - h + x", "derivation": "L{(h)} = \\log{(h)} and \\int L{(h)} dh = \\int \\log{(h)} dh and \\int L{(h)} dh = h \\log{(h)} - h + x and \\int L{(h)} dh = h L{(h)} - h + x and \\int \\log{(h)} dh = h L{(h)} - h + x and \\int \\log{(h)} dh = h \\log{(h)} - h + x and \\eta + h \\log{(h)} - h = h \\log{(h)} - h + x and \\int \\log{(h)} dh = \\eta + h \\log{(h)} - h and \\eta + h \\log{(h)} - h = h L{(h)} - h + x", "srepr_derivation": [["get_premise", "Equality(Function('L')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('L')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('L')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Function('L')(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Function('L')(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\eta', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\eta', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 8], "Equality(Add(Symbol('\\\\eta', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Function('L')(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(y^{\\prime},\\varphi^*)} = - \\varphi^* + y^{\\prime} and \\hat{H}_{\\lambda}{(\\phi_2)} = e^{\\phi_2}, then obtain - \\varphi^* \\hat{H}_{\\lambda}{(\\phi_2)} + \\frac{\\partial}{\\partial \\hat{p}} \\int e^{\\hat{p} h} d\\hat{p} = - \\varphi^* e^{\\phi_2} + \\frac{\\partial}{\\partial \\hat{p}} \\int e^{\\hat{p} h} d\\hat{p}", "derivation": "\\dot{y}{(y^{\\prime},\\varphi^*)} = - \\varphi^* + y^{\\prime} and - y^{\\prime} + \\dot{y}{(y^{\\prime},\\varphi^*)} = - \\varphi^* and \\hat{H}_{\\lambda}{(\\phi_2)} = e^{\\phi_2} and (- y^{\\prime} + \\dot{y}{(y^{\\prime},\\varphi^*)}) \\hat{H}_{\\lambda}{(\\phi_2)} = (- y^{\\prime} + \\dot{y}{(y^{\\prime},\\varphi^*)}) e^{\\phi_2} and - \\varphi^* \\hat{H}_{\\lambda}{(\\phi_2)} = - \\varphi^* e^{\\phi_2} and - \\varphi^* \\hat{H}_{\\lambda}{(\\phi_2)} + \\frac{\\partial}{\\partial \\hat{p}} \\int e^{\\hat{p} h} d\\hat{p} = - \\varphi^* e^{\\phi_2} + \\frac{\\partial}{\\partial \\hat{p}} \\int e^{\\hat{p} h} d\\hat{p}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))"], [["add", 5, "Derivative(Integral(exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Derivative(Integral(exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Derivative(Integral(exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi{(\\mathbf{p},A_{1})} = \\frac{A_{1}}{\\mathbf{p}}, then obtain \\int (- \\mathbf{p} + \\psi{(\\mathbf{p},A_{1})}) d\\mathbf{p} = A_{1} \\log{(\\mathbf{p})} - \\frac{\\mathbf{p}^{2}}{2} + \\varepsilon_0", "derivation": "\\psi{(\\mathbf{p},A_{1})} = \\frac{A_{1}}{\\mathbf{p}} and - \\mathbf{p} + \\psi{(\\mathbf{p},A_{1})} = \\frac{A_{1}}{\\mathbf{p}} - \\mathbf{p} and \\int (- \\mathbf{p} + \\psi{(\\mathbf{p},A_{1})}) d\\mathbf{p} = \\int (\\frac{A_{1}}{\\mathbf{p}} - \\mathbf{p}) d\\mathbf{p} and \\int (- \\mathbf{p} + \\psi{(\\mathbf{p},A_{1})}) d\\mathbf{p} = A_{1} \\log{(\\mathbf{p})} - \\frac{\\mathbf{p}^{2}}{2} + \\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(2))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given u{(\\mathbf{B},\\nabla)} = \\mathbf{B} \\nabla, then obtain \\frac{e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\nabla u{(\\mathbf{B},\\nabla)}}}{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}} = \\frac{e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}}}{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}}", "derivation": "u{(\\mathbf{B},\\nabla)} = \\mathbf{B} \\nabla and \\nabla u{(\\mathbf{B},\\nabla)} = \\mathbf{B} \\nabla^{2} and \\frac{\\partial}{\\partial \\mathbf{B}} \\nabla u{(\\mathbf{B},\\nabla)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2} and e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\nabla u{(\\mathbf{B},\\nabla)}} = e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}} and \\frac{e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\nabla u{(\\mathbf{B},\\nabla)}}}{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}} = \\frac{e^{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}}}{\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\nabla^{2}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), exp(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["divide", 4, "Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))"], "Equality(Mul(exp(Derivative(Mul(Symbol('\\\\nabla', commutative=True), Function('u')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))), Mul(exp(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}{(c,\\mathbf{r})} = \\mathbf{r} c and \\sigma_{p}{(\\sigma_x)} = e^{\\sigma_x}, then obtain \\sigma_{p}{(\\sigma_x)} - \\frac{\\int (- c + \\hat{x}{(c,\\mathbf{r})}) dc}{c} = e^{\\sigma_x} - \\frac{\\int (- c + \\hat{x}{(c,\\mathbf{r})}) dc}{c}", "derivation": "\\hat{x}{(c,\\mathbf{r})} = \\mathbf{r} c and - c + \\hat{x}{(c,\\mathbf{r})} = \\mathbf{r} c - c and \\int (- c + \\hat{x}{(c,\\mathbf{r})}) dc = \\int (\\mathbf{r} c - c) dc and \\sigma_{p}{(\\sigma_x)} = e^{\\sigma_x} and \\sigma_{p}{(\\sigma_x)} - \\frac{\\int (\\mathbf{r} c - c) dc}{c} = e^{\\sigma_x} - \\frac{\\int (\\mathbf{r} c - c) dc}{c} and \\sigma_{p}{(\\sigma_x)} - \\frac{\\int (- c + \\hat{x}{(c,\\mathbf{r})}) dc}{c} = e^{\\sigma_x} - \\frac{\\int (- c + \\hat{x}{(c,\\mathbf{r})}) dc}{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)))"], [["minus", 1, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{x}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{x}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], ["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 4, "Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))), Add(exp(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{x}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('c', commutative=True))))), Add(exp(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\hat{x}')(Symbol('c', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('c', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(M,z^{*})} = (z^{*})^{M}, then derive (z^{*})^{M} \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = (z^{*})^{2 M} \\log{(z^{*})}, then obtain \\Psi_{nl}{(M,z^{*})} \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = \\Psi_{nl}^{2}{(M,z^{*})} \\log{(z^{*})}", "derivation": "\\Psi_{nl}{(M,z^{*})} = (z^{*})^{M} and \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = \\frac{\\partial}{\\partial M} (z^{*})^{M} and (z^{*})^{M} \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = (z^{*})^{M} \\frac{\\partial}{\\partial M} (z^{*})^{M} and (z^{*})^{M} \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = (z^{*})^{2 M} \\log{(z^{*})} and \\Psi_{nl}{(M,z^{*})} \\frac{\\partial}{\\partial M} \\Psi_{nl}{(M,z^{*})} = \\Psi_{nl}^{2}{(M,z^{*})} \\log{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 2, "Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('z^*', commutative=True), Mul(Integer(2), Symbol('M', commutative=True))), log(Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(2)), log(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} = \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} (\\int \\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\int \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}}", "derivation": "\\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} = \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}} and \\int \\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} d\\hat{\\mathbf{x}} = \\int \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} and (\\int \\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = (\\int \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} and \\frac{\\partial}{\\partial \\mathbf{D}} (\\int \\operatorname{f^{*}}{(\\hat{\\mathbf{x}},\\mathbf{D},Q)} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\int \\frac{Q^{\\mathbf{D}}}{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Integral(Function('f^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('f^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(x,E_{n})} = \\sin{(x^{E_{n}})}, then obtain E_{n} x^{- 3 E_{n}} \\dot{\\mathbf{r}}^{2}{(x,E_{n})} = E_{n} x^{- 3 E_{n}} \\dot{\\mathbf{r}}{(x,E_{n})} \\sin{(x^{E_{n}})}", "derivation": "\\dot{\\mathbf{r}}{(x,E_{n})} = \\sin{(x^{E_{n}})} and x^{- E_{n}} \\dot{\\mathbf{r}}{(x,E_{n})} = x^{- E_{n}} \\sin{(x^{E_{n}})} and E_{n} x^{- E_{n}} \\dot{\\mathbf{r}}{(x,E_{n})} = E_{n} x^{- E_{n}} \\sin{(x^{E_{n}})} and E_{n} x^{- 2 E_{n}} \\dot{\\mathbf{r}}^{2}{(x,E_{n})} = E_{n} x^{- 2 E_{n}} \\dot{\\mathbf{r}}{(x,E_{n})} \\sin{(x^{E_{n}})} and E_{n} x^{- 3 E_{n}} \\dot{\\mathbf{r}}^{2}{(x,E_{n})} = E_{n} x^{- 3 E_{n}} \\dot{\\mathbf{r}}{(x,E_{n})} \\sin{(x^{E_{n}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)), sin(Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True))))"], [["divide", 1, "Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), sin(Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True)))))"], [["times", 2, "Symbol('E_n', commutative=True)"], "Equality(Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True))), Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), sin(Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)))"], "Equality(Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Integer(2), Symbol('E_n', commutative=True))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)), Integer(2))), Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Integer(2), Symbol('E_n', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)), sin(Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True)))))"], [["times", 4, "Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True)))"], "Equality(Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('E_n', commutative=True))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)), Integer(2))), Mul(Symbol('E_n', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Integer(3), Symbol('E_n', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('E_n', commutative=True)), sin(Pow(Symbol('x', commutative=True), Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given E{(C_{1})} = e^{\\sin{(C_{1})}}, then obtain (\\iint E{(C_{1})} dC_{1} dC_{1})^{C_{1}} = (\\iint e^{\\sin{(C_{1})}} dC_{1} dC_{1})^{C_{1}}", "derivation": "E{(C_{1})} = e^{\\sin{(C_{1})}} and \\int E{(C_{1})} dC_{1} = \\int e^{\\sin{(C_{1})}} dC_{1} and \\iint E{(C_{1})} dC_{1} dC_{1} = \\iint e^{\\sin{(C_{1})}} dC_{1} dC_{1} and (\\iint E{(C_{1})} dC_{1} dC_{1})^{C_{1}} = (\\iint e^{\\sin{(C_{1})}} dC_{1} dC_{1})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('C_1', commutative=True)), exp(sin(Symbol('C_1', commutative=True))))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('E')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(exp(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('E')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(exp(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Integral(Function('E')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Integral(exp(sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(g,v_{1})} = \\sin{(g v_{1})}, then obtain \\frac{\\mathbf{J}{(g,v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})}} = \\frac{\\sin{(g v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})}}", "derivation": "\\mathbf{J}{(g,v_{1})} = \\sin{(g v_{1})} and \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})} = \\frac{\\partial}{\\partial v_{1}} \\sin{(g v_{1})} and - \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})} = - \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\sin{(g v_{1})} and \\frac{\\mathbf{J}{(g,v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\sin{(g v_{1})}} = \\frac{\\sin{(g v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\sin{(g v_{1})}} and \\frac{\\mathbf{J}{(g,v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})}} = \\frac{\\sin{(g v_{1})}}{- \\sin{(g v_{1})} + \\frac{\\partial}{\\partial v_{1}} \\mathbf{J}{(g,v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["divide", 1, "Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Integer(-1)), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))), Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Integer(-1)), sin(Mul(Symbol('g', commutative=True), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given f{(\\theta)} = \\int \\log{(\\theta)} d\\theta, then obtain (\\sin{(f{(\\theta)})} - \\int \\log{(\\theta)} d\\theta)^{\\theta} = (\\sin{(\\int \\log{(\\theta)} d\\theta)} - \\int \\log{(\\theta)} d\\theta)^{\\theta}", "derivation": "f{(\\theta)} = \\int \\log{(\\theta)} d\\theta and \\sin{(f{(\\theta)})} = \\sin{(\\int \\log{(\\theta)} d\\theta)} and \\sin{(f{(\\theta)})} - \\int \\log{(\\theta)} d\\theta = \\sin{(\\int \\log{(\\theta)} d\\theta)} - \\int \\log{(\\theta)} d\\theta and (\\sin{(f{(\\theta)})} - \\int \\log{(\\theta)} d\\theta)^{\\theta} = (\\sin{(\\int \\log{(\\theta)} d\\theta)} - \\int \\log{(\\theta)} d\\theta)^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\theta', commutative=True)), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["sin", 1], "Equality(sin(Function('f')(Symbol('\\\\theta', commutative=True))), sin(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["minus", 2, "Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(sin(Function('f')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))), Add(sin(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(sin(Function('f')(Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True)), Pow(Add(sin(Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Integral(log(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\theta{(q)} = \\sin{(q)}, then obtain \\frac{\\int \\frac{\\theta{(q)}}{\\sin{(q)}} dq}{\\theta{(q)}} = \\frac{\\int 1 dq}{\\theta{(q)}}", "derivation": "\\theta{(q)} = \\sin{(q)} and \\frac{\\theta{(q)}}{\\sin{(q)}} = 1 and \\int \\frac{\\theta{(q)}}{\\sin{(q)}} dq = \\int 1 dq and \\frac{\\int \\frac{\\theta{(q)}}{\\sin{(q)}} dq}{\\sin{(q)}} = \\frac{\\int 1 dq}{\\sin{(q)}} and \\frac{\\int \\frac{\\theta{(q)}}{\\sin{(q)}} dq}{\\theta{(q)}} = \\frac{\\int 1 dq}{\\theta{(q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["divide", 1, "sin(Symbol('q', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Mul(Function('\\\\theta')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True))), Integral(Integer(1), Tuple(Symbol('q', commutative=True))))"], [["times", 3, "Pow(sin(Symbol('q', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\theta')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\theta')(Symbol('q', commutative=True)), Integer(-1)), Integral(Mul(Function('\\\\theta')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Mul(Pow(Function('\\\\theta')(Symbol('q', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given V{(\\Psi_{\\lambda})} = \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}, then derive \\Psi_{\\lambda} + V{(\\Psi_{\\lambda})} = E_{\\lambda} + \\Psi_{\\lambda} - \\cos{(\\Psi_{\\lambda})}, then obtain \\Psi_{\\lambda} + \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = E_{\\lambda} + \\Psi_{\\lambda} - \\cos{(\\Psi_{\\lambda})}", "derivation": "V{(\\Psi_{\\lambda})} = \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} and \\Psi_{\\lambda} + V{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} and \\Psi_{\\lambda} + V{(\\Psi_{\\lambda})} = E_{\\lambda} + \\Psi_{\\lambda} - \\cos{(\\Psi_{\\lambda})} and \\Psi_{\\lambda} + \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = E_{\\lambda} + \\Psi_{\\lambda} - \\cos{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integral(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('V')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('V')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given m{(U)} = \\sin{(\\log{(U)})}, then obtain (U + m^{4}{(U)} \\sin^{2}{(\\log{(U)})}) m^{2}{(U)} \\sin{(\\log{(U)})} = (U + m^{3}{(U)} \\sin^{3}{(\\log{(U)})}) m^{2}{(U)} \\sin{(\\log{(U)})}", "derivation": "m{(U)} = \\sin{(\\log{(U)})} and m^{2}{(U)} = m{(U)} \\sin{(\\log{(U)})} and m^{3}{(U)} = m^{2}{(U)} \\sin{(\\log{(U)})} and m^{3}{(U)} = m{(U)} \\sin^{2}{(\\log{(U)})} and m^{2}{(U)} \\sin{(\\log{(U)})} = m{(U)} \\sin^{2}{(\\log{(U)})} and m^{3}{(U)} \\sin^{3}{(\\log{(U)})} = m^{2}{(U)} \\sin^{4}{(\\log{(U)})} and U + m^{3}{(U)} \\sin^{3}{(\\log{(U)})} = U + m^{2}{(U)} \\sin^{4}{(\\log{(U)})} and U + m^{4}{(U)} \\sin^{2}{(\\log{(U)})} = U + m^{3}{(U)} \\sin^{3}{(\\log{(U)})} and (U + m^{4}{(U)} \\sin^{2}{(\\log{(U)})}) m^{2}{(U)} \\sin{(\\log{(U)})} = (U + m^{3}{(U)} \\sin^{3}{(\\log{(U)})}) m^{2}{(U)} \\sin{(\\log{(U)})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('U', commutative=True)), sin(log(Symbol('U', commutative=True))))"], [["times", 1, "Function('m')(Symbol('U', commutative=True))"], "Equality(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), Mul(Function('m')(Symbol('U', commutative=True)), sin(log(Symbol('U', commutative=True)))))"], [["times", 1, "Pow(Function('m')(Symbol('U', commutative=True)), Integer(2))"], "Equality(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), sin(log(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Mul(Function('m')(Symbol('U', commutative=True)), Pow(sin(log(Symbol('U', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), sin(log(Symbol('U', commutative=True)))), Mul(Function('m')(Symbol('U', commutative=True)), Pow(sin(log(Symbol('U', commutative=True))), Integer(2))))"], [["times", 5, "Mul(Function('m')(Symbol('U', commutative=True)), Pow(sin(log(Symbol('U', commutative=True))), Integer(2)))"], "Equality(Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Pow(sin(log(Symbol('U', commutative=True))), Integer(3))), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), Pow(sin(log(Symbol('U', commutative=True))), Integer(4))))"], [["add", 6, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Pow(sin(log(Symbol('U', commutative=True))), Integer(3)))), Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), Pow(sin(log(Symbol('U', commutative=True))), Integer(4)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(4)), Pow(sin(log(Symbol('U', commutative=True))), Integer(2)))), Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Pow(sin(log(Symbol('U', commutative=True))), Integer(3)))))"], [["times", 8, "Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), sin(log(Symbol('U', commutative=True))))"], "Equality(Mul(Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(4)), Pow(sin(log(Symbol('U', commutative=True))), Integer(2)))), Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), sin(log(Symbol('U', commutative=True)))), Mul(Add(Symbol('U', commutative=True), Mul(Pow(Function('m')(Symbol('U', commutative=True)), Integer(3)), Pow(sin(log(Symbol('U', commutative=True))), Integer(3)))), Pow(Function('m')(Symbol('U', commutative=True)), Integer(2)), sin(log(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(g,s)} = - s + \\cos{(g)} and \\mathbf{P}{(g,\\lambda,A_{y},s)} = (- A_{y} + \\lambda) \\sin^{s}{(\\dot{x}^{g}{(g,s)})}, then obtain \\mathbf{P}{(g,\\lambda,A_{y},s)} = (- A_{y} + \\lambda) \\sin^{s}{((- s + \\cos{(g)})^{g})}", "derivation": "\\dot{x}{(g,s)} = - s + \\cos{(g)} and \\dot{x}^{g}{(g,s)} = (- s + \\cos{(g)})^{g} and \\sin{(\\dot{x}^{g}{(g,s)})} = \\sin{((- s + \\cos{(g)})^{g})} and \\sin^{s}{(\\dot{x}^{g}{(g,s)})} = \\sin^{s}{((- s + \\cos{(g)})^{g})} and (- A_{y} + \\lambda) \\sin^{s}{(\\dot{x}^{g}{(g,s)})} = (- A_{y} + \\lambda) \\sin^{s}{((- s + \\cos{(g)})^{g})} and \\mathbf{P}{(g,\\lambda,A_{y},s)} = (- A_{y} + \\lambda) \\sin^{s}{(\\dot{x}^{g}{(g,s)})} and \\mathbf{P}{(g,\\lambda,A_{y},s)} = (- A_{y} + \\lambda) \\sin^{s}{((- s + \\cos{(g)})^{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Symbol('g', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(sin(Pow(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Symbol('g', commutative=True))), Symbol('s', commutative=True)), Pow(sin(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True))), Symbol('s', commutative=True)))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Pow(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Symbol('g', commutative=True))), Symbol('s', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True))), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('A_y', commutative=True), Symbol('s', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Pow(Function('\\\\dot{x}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Symbol('g', commutative=True))), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('A_y', commutative=True), Symbol('s', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Pow(Add(Mul(Integer(-1), Symbol('s', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given r{(g)} = \\log{(g)}, then obtain \\frac{2 g (g + r{(g)})^{g}}{2 g + 2 \\log{(g)}} = \\frac{2 g (g + \\log{(g)})^{g}}{2 g + 2 \\log{(g)}}", "derivation": "r{(g)} = \\log{(g)} and g + r{(g)} = g + \\log{(g)} and (g + r{(g)})^{g} = (g + \\log{(g)})^{g} and 2 g + r{(g)} + \\log{(g)} = 2 g + 2 \\log{(g)} and \\frac{(g + r{(g)})^{g}}{2 g + r{(g)} + \\log{(g)}} = \\frac{(g + \\log{(g)})^{g}}{2 g + r{(g)} + \\log{(g)}} and \\frac{2 g (g + r{(g)})^{g}}{2 g + r{(g)} + \\log{(g)}} = \\frac{2 g (g + \\log{(g)})^{g}}{2 g + r{(g)} + \\log{(g)}} and \\frac{2 g (g + r{(g)})^{g}}{2 g + 2 \\log{(g)}} = \\frac{2 g (g + \\log{(g)})^{g}}{2 g + 2 \\log{(g)}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["add", 1, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Function('r')(Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Symbol('g', commutative=True), Function('r')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["add", 2, "Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(2), log(Symbol('g', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Function('r')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Integer(-1))))"], [["times", 5, "Mul(Integer(2), Symbol('g', commutative=True))"], "Equality(Mul(Integer(2), Symbol('g', commutative=True), Pow(Add(Symbol('g', commutative=True), Function('r')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Integer(-1))), Mul(Integer(2), Symbol('g', commutative=True), Pow(Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Function('r')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(2), Symbol('g', commutative=True), Pow(Add(Symbol('g', commutative=True), Function('r')(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(2), log(Symbol('g', commutative=True)))), Integer(-1))), Mul(Integer(2), Symbol('g', commutative=True), Pow(Add(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Integer(2), log(Symbol('g', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\Omega)} = \\cos{(\\Omega)} and \\mathbf{p}{(\\Omega)} = (\\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega)^{\\Omega}, then derive (\\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega)^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega}, then obtain \\mathbf{p}^{\\Omega}{(\\Omega)} = ((k + \\sin{(\\Omega)})^{\\Omega})^{\\Omega}", "derivation": "\\operatorname{v_{x}}{(\\Omega)} = \\cos{(\\Omega)} and \\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega = \\int \\cos{(\\Omega)} d\\Omega and (\\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega)^{\\Omega} = (\\int \\cos{(\\Omega)} d\\Omega)^{\\Omega} and (\\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega)^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega} and \\mathbf{p}{(\\Omega)} = (\\int \\operatorname{v_{x}}{(\\Omega)} d\\Omega)^{\\Omega} and \\mathbf{p}{(\\Omega)} = (k + \\sin{(\\Omega)})^{\\Omega} and \\mathbf{p}^{\\Omega}{(\\Omega)} = ((k + \\sin{(\\Omega)})^{\\Omega})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integral(Function('v_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('v_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\Omega', commutative=True)), Pow(Integral(Function('v_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["power", 6, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(C,p)} = \\frac{C}{p}, then obtain (\\theta_{1}{(C,p)} + 1 - \\frac{1}{p})^{p} = (\\frac{C}{p} + 1 - \\frac{1}{p})^{p}", "derivation": "\\theta_{1}{(C,p)} = \\frac{C}{p} and \\theta_{1}{(C,p)} + 1 = \\frac{C}{p} + 1 and \\theta_{1}{(C,p)} + 1 - \\frac{1}{p} = \\frac{C}{p} + 1 - \\frac{1}{p} and (\\theta_{1}{(C,p)} + 1 - \\frac{1}{p})^{p} = (\\frac{C}{p} + 1 - \\frac{1}{p})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta_1')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Integer(1)), Add(Mul(Symbol('C', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Integer(1)))"], [["minus", 2, "Pow(Symbol('p', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\theta_1')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)))), Add(Mul(Symbol('C', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Integer(1), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Function('\\\\theta_1')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('p', commutative=True)), Pow(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Integer(1), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{M})} = e^{\\mathbf{M}}, then derive \\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain \\frac{\\frac{d^{2}}{d \\mathbf{M}^{2}} \\dot{y}{(\\mathbf{M})}}{\\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})}} = 1", "derivation": "\\dot{y}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} and \\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})} = e^{\\mathbf{M}} and - \\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}} = - \\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})} and \\frac{\\frac{d}{d \\mathbf{M}} e^{\\mathbf{M}}}{\\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})}} = 1 and \\frac{\\frac{d^{2}}{d \\mathbf{M}^{2}} \\dot{y}{(\\mathbf{M})}}{\\frac{d}{d \\mathbf{M}} \\dot{y}{(\\mathbf{M})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 2, "Add(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))"], [["divide", 4, "Mul(Integer(-1), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))), Integer(1))"]]}, {"prompt": "Given t{(\\mathbb{I},\\mathbf{A},v_{y})} = \\mathbb{I} (\\mathbf{A} + v_{y}), then obtain \\mathbb{I} t{(\\mathbb{I},\\mathbf{A},v_{y})} - \\frac{\\partial}{\\partial v_{y}} (\\mathbf{A} + v_{y}) - \\frac{1}{\\mathbb{I}} = \\mathbb{I}^{2} (\\mathbf{A} + v_{y}) - \\frac{\\partial}{\\partial v_{y}} (\\mathbf{A} + v_{y}) - \\frac{1}{\\mathbb{I}}", "derivation": "t{(\\mathbb{I},\\mathbf{A},v_{y})} = \\mathbb{I} (\\mathbf{A} + v_{y}) and \\mathbb{I} t{(\\mathbb{I},\\mathbf{A},v_{y})} = \\mathbb{I}^{2} (\\mathbf{A} + v_{y}) and \\mathbb{I} t{(\\mathbb{I},\\mathbf{A},v_{y})} - \\frac{1}{\\mathbb{I}} = \\mathbb{I}^{2} (\\mathbf{A} + v_{y}) - \\frac{1}{\\mathbb{I}} and \\mathbb{I} t{(\\mathbb{I},\\mathbf{A},v_{y})} - \\frac{\\partial}{\\partial v_{y}} (\\mathbf{A} + v_{y}) - \\frac{1}{\\mathbb{I}} = \\mathbb{I}^{2} (\\mathbf{A} + v_{y}) - \\frac{\\partial}{\\partial v_{y}} (\\mathbf{A} + v_{y}) - \\frac{1}{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], [["minus", 3, "Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('t')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given h{(y)} = e^{y}, then derive \\frac{d}{d y} h{(y)} = e^{y}, then obtain (\\frac{d}{d y} h{(y)})^{y} = h^{y}{(y)}", "derivation": "h{(y)} = e^{y} and \\frac{d}{d y} h{(y)} = \\frac{d}{d y} e^{y} and \\frac{d}{d y} h{(y)} = e^{y} and \\frac{d}{d y} h{(y)} = h{(y)} and (\\frac{d}{d y} h{(y)})^{y} = h^{y}{(y)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), exp(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('h')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Function('h')(Symbol('y', commutative=True)))"], [["power", 4, "Symbol('y', commutative=True)"], "Equality(Pow(Derivative(Function('h')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('y', commutative=True)), Pow(Function('h')(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)} = \\omega + \\tilde{g}^*, then obtain - \\omega \\tilde{g}^* (- \\omega - \\tilde{g}^* + \\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)}) = 0", "derivation": "\\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)} = \\omega + \\tilde{g}^* and - \\omega - \\tilde{g}^* + \\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)} = 0 and \\tilde{g}^* (- \\omega - \\tilde{g}^* + \\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)}) = 0 and - \\omega \\tilde{g}^* (- \\omega - \\tilde{g}^* + \\operatorname{F_{H}}{(\\omega,\\tilde{g}^*)}) = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(0))"], [["times", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)} = (\\mathbf{S}^{Z})^{g_{\\varepsilon}}, then obtain (\\mathbf{S} \\mathbf{S}^{- Z} \\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)})^{Z} = (\\mathbf{S} \\mathbf{S}^{- Z} (\\mathbf{S}^{Z})^{g_{\\varepsilon}})^{Z}", "derivation": "\\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)} = (\\mathbf{S}^{Z})^{g_{\\varepsilon}} and \\mathbf{S} \\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)} = \\mathbf{S} (\\mathbf{S}^{Z})^{g_{\\varepsilon}} and \\mathbf{S} \\mathbf{S}^{- Z} \\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)} = \\mathbf{S} \\mathbf{S}^{- Z} (\\mathbf{S}^{Z})^{g_{\\varepsilon}} and (\\mathbf{S} \\mathbf{S}^{- Z} \\operatorname{t_{2}}{(g_{\\varepsilon},\\mathbf{S},Z)})^{Z} = (\\mathbf{S} \\mathbf{S}^{- Z} (\\mathbf{S}^{Z})^{g_{\\varepsilon}})^{Z}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('Z', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given T{(P_{g},\\dot{y})} = (e^{P_{g}})^{\\dot{y}} and \\operatorname{v_{x}}{(s)} = e^{s}, then obtain \\operatorname{v_{x}}{(s)} (e^{P_{g}})^{- \\dot{y}} = (e^{P_{g}})^{- \\dot{y}} e^{s}", "derivation": "T{(P_{g},\\dot{y})} = (e^{P_{g}})^{\\dot{y}} and \\operatorname{v_{x}}{(s)} = e^{s} and \\frac{\\operatorname{v_{x}}{(s)}}{T{(P_{g},\\dot{y})}} = \\frac{e^{s}}{T{(P_{g},\\dot{y})}} and \\operatorname{v_{x}}{(s)} (e^{P_{g}})^{- \\dot{y}} = (e^{P_{g}})^{- \\dot{y}} e^{s}", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('P_g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('v_x')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["divide", 2, "Function('T')(Symbol('P_g', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Function('T')(Symbol('P_g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Function('v_x')(Symbol('s', commutative=True))), Mul(Pow(Function('T')(Symbol('P_g', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), exp(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('v_x')(Symbol('s', commutative=True)), Pow(exp(Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)))), Mul(Pow(exp(Symbol('P_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('s', commutative=True))))"]]}, {"prompt": "Given B{(E,M_{E})} = E - M_{E}, then obtain (2 B{(E,M_{E})} - 1)^{M_{E}} = (E - M_{E} + B{(E,M_{E})} - 1)^{M_{E}}", "derivation": "B{(E,M_{E})} = E - M_{E} and E - M_{E} + B{(E,M_{E})} - 1 = 2 E - 2 M_{E} - 1 and 2 B{(E,M_{E})} - 1 = 2 E - 2 M_{E} - 1 and (2 B{(E,M_{E})} - 1)^{M_{E}} = (2 E - 2 M_{E} - 1)^{M_{E}} and (2 B{(E,M_{E})} - 1)^{M_{E}} = (E - M_{E} + B{(E,M_{E})} - 1)^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["add", 1, "Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Integer(-1))"], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('M_E', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Add(Mul(Integer(2), Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('M_E', commutative=True)), Integer(-1)))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), Symbol('M_E', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('M_E', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Mul(Integer(2), Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), Symbol('M_E', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Function('B')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given E{(\\theta_2,\\mathbf{f})} = \\int (\\mathbf{f} - \\theta_2) d\\theta_2, then obtain - 2 \\theta_2^{2} - \\frac{E{(\\theta_2,\\mathbf{f})}}{2} = - \\frac{J_{\\varepsilon}}{2} - \\frac{\\mathbf{f} \\theta_2}{2} - \\frac{7 \\theta_2^{2}}{4}", "derivation": "E{(\\theta_2,\\mathbf{f})} = \\int (\\mathbf{f} - \\theta_2) d\\theta_2 and - \\frac{E{(\\theta_2,\\mathbf{f})}}{2} = - \\frac{\\int (\\mathbf{f} - \\theta_2) d\\theta_2}{2} and - \\theta_2^{2} - \\frac{E{(\\theta_2,\\mathbf{f})}}{2} = - \\theta_2^{2} - \\frac{\\int (\\mathbf{f} - \\theta_2) d\\theta_2}{2} and - 2 \\theta_2^{2} - \\frac{E{(\\theta_2,\\mathbf{f})}}{2} = - 2 \\theta_2^{2} - \\frac{\\int (\\mathbf{f} - \\theta_2) d\\theta_2}{2} and - 2 \\theta_2^{2} - \\frac{E{(\\theta_2,\\mathbf{f})}}{2} = - \\frac{J_{\\varepsilon}}{2} - \\frac{\\mathbf{f} \\theta_2}{2} - \\frac{7 \\theta_2^{2}}{4}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Rational(-1, 2)"], "Equality(Mul(Integer(-1), Rational(1, 2), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Rational(1, 2), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 2, "Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))))"], [["minus", 3, "Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Integer(-1), Rational(1, 2), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Rational(7, 4), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{F}{(C_{2},\\hat{p}_0)} = C_{2} \\hat{p}_0, then derive \\sin{(\\frac{\\partial}{\\partial C_{2}} \\mathbf{F}{(C_{2},\\hat{p}_0)})} = \\sin{(\\hat{p}_0)}, then obtain \\frac{\\sin{(\\frac{\\partial}{\\partial C_{2}} C_{2} \\hat{p}_0)}}{\\hat{p}_0} = \\frac{\\sin{(\\hat{p}_0)}}{\\hat{p}_0}", "derivation": "\\mathbf{F}{(C_{2},\\hat{p}_0)} = C_{2} \\hat{p}_0 and \\frac{\\partial}{\\partial C_{2}} \\mathbf{F}{(C_{2},\\hat{p}_0)} = \\frac{\\partial}{\\partial C_{2}} C_{2} \\hat{p}_0 and \\sin{(\\frac{\\partial}{\\partial C_{2}} \\mathbf{F}{(C_{2},\\hat{p}_0)})} = \\sin{(\\frac{\\partial}{\\partial C_{2}} C_{2} \\hat{p}_0)} and \\sin{(\\frac{\\partial}{\\partial C_{2}} \\mathbf{F}{(C_{2},\\hat{p}_0)})} = \\sin{(\\hat{p}_0)} and \\sin{(\\frac{\\partial}{\\partial C_{2}} C_{2} \\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\frac{\\sin{(\\frac{\\partial}{\\partial C_{2}} C_{2} \\hat{p}_0)}}{\\hat{p}_0} = \\frac{\\sin{(\\hat{p}_0)}}{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), sin(Derivative(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('\\\\mathbf{F}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(sin(Derivative(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 5, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), sin(Derivative(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\phi_2,\\mathbf{J})} = \\log{(\\mathbf{J} + \\phi_2)} and \\eta^{\\prime}{(\\phi_2,\\mathbf{J})} = \\log{(\\mathbf{J} + \\phi_2)} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2}, then obtain \\eta^{\\prime}{(\\phi_2,\\mathbf{J})} = \\mathbf{F}{(\\phi_2,\\mathbf{J})} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2}", "derivation": "\\mathbf{F}{(\\phi_2,\\mathbf{J})} = \\log{(\\mathbf{J} + \\phi_2)} and \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2} = \\frac{\\log{(\\mathbf{J} + \\phi_2)}}{\\phi_2} and \\mathbf{F}{(\\phi_2,\\mathbf{J})} + \\frac{\\log{(\\mathbf{J} + \\phi_2)}}{\\phi_2} = \\log{(\\mathbf{J} + \\phi_2)} + \\frac{\\log{(\\mathbf{J} + \\phi_2)}}{\\phi_2} and \\mathbf{F}{(\\phi_2,\\mathbf{J})} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2} = \\log{(\\mathbf{J} + \\phi_2)} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2} and \\eta^{\\prime}{(\\phi_2,\\mathbf{J})} = \\log{(\\mathbf{J} + \\phi_2)} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2} and \\eta^{\\prime}{(\\phi_2,\\mathbf{J})} = \\mathbf{F}{(\\phi_2,\\mathbf{J})} + \\frac{\\mathbf{F}{(\\phi_2,\\mathbf{J})}}{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["add", 1, "Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))))), Add(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Add(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(log(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given E{(f,m_{s})} = \\log{(f m_{s})}, then obtain m_{s} + (m_{s} E{(f,m_{s})})^{f} = m_{s} + (m_{s} \\log{(f m_{s})})^{f}", "derivation": "E{(f,m_{s})} = \\log{(f m_{s})} and m_{s} E{(f,m_{s})} = m_{s} \\log{(f m_{s})} and (m_{s} E{(f,m_{s})})^{f} = (m_{s} \\log{(f m_{s})})^{f} and m_{s} + (m_{s} E{(f,m_{s})})^{f} = m_{s} + (m_{s} \\log{(f m_{s})})^{f}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f', commutative=True), Symbol('m_s', commutative=True)), log(Mul(Symbol('f', commutative=True), Symbol('m_s', commutative=True))))"], [["times", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Symbol('m_s', commutative=True), Function('E')(Symbol('f', commutative=True), Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), log(Mul(Symbol('f', commutative=True), Symbol('m_s', commutative=True)))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Symbol('m_s', commutative=True), Function('E')(Symbol('f', commutative=True), Symbol('m_s', commutative=True))), Symbol('f', commutative=True)), Pow(Mul(Symbol('m_s', commutative=True), log(Mul(Symbol('f', commutative=True), Symbol('m_s', commutative=True)))), Symbol('f', commutative=True)))"], [["add", 3, "Symbol('m_s', commutative=True)"], "Equality(Add(Symbol('m_s', commutative=True), Pow(Mul(Symbol('m_s', commutative=True), Function('E')(Symbol('f', commutative=True), Symbol('m_s', commutative=True))), Symbol('f', commutative=True))), Add(Symbol('m_s', commutative=True), Pow(Mul(Symbol('m_s', commutative=True), log(Mul(Symbol('f', commutative=True), Symbol('m_s', commutative=True)))), Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(F_{H})} = e^{F_{H}}, then obtain \\int \\frac{d}{d F_{H}} 0 dF_{H} = f^{\\prime} - \\operatorname{g_{\\varepsilon}}{(F_{H})} + e^{F_{H}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(F_{H})} = e^{F_{H}} and 0 = - \\operatorname{g_{\\varepsilon}}{(F_{H})} + e^{F_{H}} and \\frac{d}{d F_{H}} 0 = \\frac{d}{d F_{H}} (- \\operatorname{g_{\\varepsilon}}{(F_{H})} + e^{F_{H}}) and \\int \\frac{d}{d F_{H}} 0 dF_{H} = \\int \\frac{d}{d F_{H}} (- \\operatorname{g_{\\varepsilon}}{(F_{H})} + e^{F_{H}}) dF_{H} and \\int \\frac{d}{d F_{H}} 0 dF_{H} = f^{\\prime} - \\operatorname{g_{\\varepsilon}}{(F_{H})} + e^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["minus", 1, "Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True))), exp(Symbol('F_H', commutative=True))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True))), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True))), exp(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('F_H', commutative=True))), exp(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)} = \\mathbf{J}_M + r_{0} and Q{(r_{0},\\mathbf{J}_M)} = - r_{0} + \\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)}, then obtain \\int Q{(r_{0},\\mathbf{J}_M)} d\\mathbf{J}_M = \\int \\mathbf{J}_M d\\mathbf{J}_M", "derivation": "\\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)} = \\mathbf{J}_M + r_{0} and \\mathbf{J}_M + \\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)} = 2 \\mathbf{J}_M + r_{0} and - r_{0} + \\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)} = \\mathbf{J}_M and Q{(r_{0},\\mathbf{J}_M)} = - r_{0} + \\operatorname{y^{\\prime}}{(r_{0},\\mathbf{J}_M)} and Q{(r_{0},\\mathbf{J}_M)} = \\mathbf{J}_M and \\int Q{(r_{0},\\mathbf{J}_M)} d\\mathbf{J}_M = \\int \\mathbf{J}_M d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('y^{\\\\prime}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('r_0', commutative=True)))"], [["minus", 2, "Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('y^{\\\\prime}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))"], ["renaming_premise", "Equality(Function('Q')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('y^{\\\\prime}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('Q')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))"], [["integrate", 5, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Symbol('\\\\mathbf{J}_M', commutative=True), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\nabla{(n_{2},l)} = n_{2} \\sin{(l)}, then derive \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)} = n_{2} \\cos{(l)}, then obtain (n_{2} \\cos{(l)} - \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)})^{n_{2}} = 0^{n_{2}}", "derivation": "\\nabla{(n_{2},l)} = n_{2} \\sin{(l)} and \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)} = \\frac{\\partial}{\\partial l} n_{2} \\sin{(l)} and \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)} = n_{2} \\cos{(l)} and n_{2} \\cos{(l)} = \\frac{\\partial}{\\partial l} n_{2} \\sin{(l)} and n_{2} \\cos{(l)} - \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)} = \\frac{\\partial}{\\partial l} n_{2} \\sin{(l)} - \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)} and n_{2} \\cos{(l)} - \\frac{\\partial}{\\partial l} n_{2} \\sin{(l)} = 0 and (n_{2} \\cos{(l)} - \\frac{\\partial}{\\partial l} n_{2} \\sin{(l)})^{n_{2}} = 0^{n_{2}} and (n_{2} \\cos{(l)} - \\frac{\\partial}{\\partial l} \\nabla{(n_{2},l)})^{n_{2}} = 0^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))), Derivative(Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Derivative(Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))), Integer(0))"], [["power", 6, "Symbol('n_2', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('n_2', commutative=True), sin(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))), Symbol('n_2', commutative=True)), Pow(Integer(0), Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Pow(Add(Mul(Symbol('n_2', commutative=True), cos(Symbol('l', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\nabla')(Symbol('n_2', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Symbol('n_2', commutative=True)), Pow(Integer(0), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given V{(A_{y})} = \\cos{(A_{y})}, then derive \\frac{d}{d A_{y}} V{(A_{y})} = - \\sin{(A_{y})}, then obtain \\frac{d}{d A_{y}} V{(A_{y})} (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} = (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} \\frac{d}{d A_{y}} \\cos{(A_{y})}", "derivation": "V{(A_{y})} = \\cos{(A_{y})} and \\frac{d}{d A_{y}} V{(A_{y})} = \\frac{d}{d A_{y}} \\cos{(A_{y})} and \\frac{d}{d A_{y}} V{(A_{y})} = - \\sin{(A_{y})} and \\frac{d}{d A_{y}} \\cos{(A_{y})} = - \\sin{(A_{y})} and \\frac{d}{d A_{y}} V{(A_{y})} (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} = - \\sin{(A_{y})} (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} and \\frac{d}{d A_{y}} V{(A_{y})} (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} = (\\frac{d}{d A_{y}} V{(A_{y})})^{- A_{y}} \\frac{d}{d A_{y}} \\cos{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], [["divide", 3, "Pow(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Symbol('A_y', commutative=True))"], "Equality(Mul(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Pow(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_y', commutative=True)))), Mul(Integer(-1), sin(Symbol('A_y', commutative=True)), Pow(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Pow(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_y', commutative=True)))), Mul(Pow(Derivative(Function('V')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A_y', commutative=True))), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(E,E_{\\lambda})} = \\sin{(E E_{\\lambda})} and \\operatorname{n_{2}}{(E_{\\lambda})} = 0^{E_{\\lambda}}, then obtain 1 = (\\sin^{E_{\\lambda}}{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})})^{E}", "derivation": "L{(E,E_{\\lambda})} = \\sin{(E E_{\\lambda})} and L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})} = 0 and \\sin{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})} = 0 and \\sin^{E_{\\lambda}}{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})} = 0^{E_{\\lambda}} and \\operatorname{n_{2}}{(E_{\\lambda})} = 0^{E_{\\lambda}} and 1 = \\operatorname{n_{2}}{(E_{\\lambda})} and \\operatorname{n_{2}}{(E_{\\lambda})} = \\sin^{E_{\\lambda}}{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})} and 1 = \\sin^{E_{\\lambda}}{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})} and 1 = (\\sin^{E_{\\lambda}}{(L{(E,E_{\\lambda})} - \\sin{(E E_{\\lambda})})})^{E}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 1, "sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))), Integer(0))"], [["sin", 2], "Equality(sin(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))), Integer(0))"], [["power", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(sin(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(1), Function('n_2')(Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('n_2')(Symbol('E_{\\\\lambda}', commutative=True)), Pow(sin(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Integer(1), Pow(sin(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 8, "Symbol('E', commutative=True)"], "Equality(Integer(1), Pow(Pow(sin(Add(Function('L')(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('E', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(n_{1})} = \\cos{(e^{n_{1}})} and C{(n_{1})} = e^{1 - \\frac{\\cos{(e^{n_{1}})}}{\\eta^{\\prime}{(n_{1})}}}, then obtain \\frac{d}{d n_{1}} e^{1 - \\frac{\\cos{(e^{n_{1}})}}{\\eta^{\\prime}{(n_{1})}}} = \\frac{d}{d n_{1}} 1", "derivation": "\\eta^{\\prime}{(n_{1})} = \\cos{(e^{n_{1}})} and C{(n_{1})} = e^{1 - \\frac{\\cos{(e^{n_{1}})}}{\\eta^{\\prime}{(n_{1})}}} and C{(n_{1})} = 1 and \\frac{d}{d n_{1}} C{(n_{1})} = \\frac{d}{d n_{1}} 1 and \\frac{d}{d n_{1}} e^{1 - \\frac{\\cos{(e^{n_{1}})}}{\\eta^{\\prime}{(n_{1})}}} = \\frac{d}{d n_{1}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('n_1', commutative=True)), exp(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('n_1', commutative=True)), Integer(-1)), cos(exp(Symbol('n_1', commutative=True)))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C')(Symbol('n_1', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('n_1', commutative=True)), Integer(-1)), cos(exp(Symbol('n_1', commutative=True)))))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(r,v_{t})} = r + v_{t}, then obtain (\\frac{1}{\\operatorname{f_{\\mathbf{p}}}{(r,v_{t})}})^{r} = (\\frac{r + v_{t}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(r,v_{t})}})^{r}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(r,v_{t})} = r + v_{t} and 1 = \\frac{r + v_{t}}{\\operatorname{f_{\\mathbf{p}}}{(r,v_{t})}} and \\frac{1}{\\operatorname{f_{\\mathbf{p}}}{(r,v_{t})}} = \\frac{r + v_{t}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(r,v_{t})}} and (\\frac{1}{\\operatorname{f_{\\mathbf{p}}}{(r,v_{t})}})^{r} = (\\frac{r + v_{t}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(r,v_{t})}})^{r}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True)))"], [["divide", 1, "Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Integer(-1))))"], [["divide", 2, "Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Integer(-1)), Mul(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Integer(-2))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Integer(-1)), Symbol('r', commutative=True)), Pow(Mul(Add(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True), Symbol('v_t', commutative=True)), Integer(-2))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given v{(u)} = \\log{(u)} and \\operatorname{c_{0}}{(u)} = \\frac{\\log{(u)}}{u}, then obtain (\\frac{v{(u)}}{u})^{u} = \\operatorname{c_{0}}^{u}{(u)}", "derivation": "v{(u)} = \\log{(u)} and \\frac{v{(u)}}{u} = \\frac{\\log{(u)}}{u} and (\\frac{v{(u)}}{u})^{u} = (\\frac{\\log{(u)}}{u})^{u} and \\operatorname{c_{0}}{(u)} = \\frac{\\log{(u)}}{u} and (\\frac{v{(u)}}{u})^{u} = \\operatorname{c_{0}}^{u}{(u)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["divide", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('v')(Symbol('u', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('v')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('u', commutative=True)), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), log(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('v')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Function('c_0')(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\Psi{(v_{x},\\mu)} = - \\sin{(\\mu - v_{x})}, then obtain 0 = (- v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})}) \\frac{d}{d \\mu} 0", "derivation": "\\Psi{(v_{x},\\mu)} = - \\sin{(\\mu - v_{x})} and v_{x} \\Psi{(v_{x},\\mu)} = - v_{x} \\sin{(\\mu - v_{x})} and 0 = - v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})} and \\frac{d}{d \\mu} 0 = \\frac{\\partial}{\\partial \\mu} (- v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})}) and 0 = (- v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})}) \\frac{\\partial}{\\partial \\mu} (- v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})}) and 0 = (- v_{x} \\Psi{(v_{x},\\mu)} - v_{x} \\sin{(\\mu - v_{x})}) \\frac{d}{d \\mu} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))))"], [["times", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))))"], [["minus", 2, "Mul(Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Function('\\\\Psi')(Symbol('v_x', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), sin(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))), Derivative(Integer(0), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})}, then obtain 0 = - \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{n_{2}}{(\\hat{\\mathbf{r}})})} + \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})})}", "derivation": "\\operatorname{n_{2}}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{n_{2}}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})} and \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{n_{2}}{(\\hat{\\mathbf{r}})})} = \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})})} and 0 = - \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{n_{2}}{(\\hat{\\mathbf{r}})})} + \\log{(\\frac{d}{d \\hat{\\mathbf{r}}} \\cos{(\\hat{\\mathbf{r}})})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), log(Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))))"], [["minus", 3, "log(Derivative(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), log(Derivative(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))), log(Derivative(cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(x)} = \\sin{(x)}, then obtain - 16 (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{4} \\operatorname{A_{2}}^{4}{(x)} = - (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{8}", "derivation": "\\operatorname{A_{2}}{(x)} = \\sin{(x)} and 2 \\operatorname{A_{2}}{(x)} = \\operatorname{A_{2}}{(x)} + \\sin{(x)} and 4 \\operatorname{A_{2}}^{2}{(x)} = (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{2} and 4 (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{2} \\operatorname{A_{2}}^{2}{(x)} = (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{4} and 16 (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{4} \\operatorname{A_{2}}^{4}{(x)} = (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{8} and - 16 (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{4} \\operatorname{A_{2}}^{4}{(x)} = - (\\operatorname{A_{2}}{(x)} + \\sin{(x)})^{8}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["add", 1, "Function('A_2')(Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('A_2')(Symbol('x', commutative=True))), Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('A_2')(Symbol('x', commutative=True)), Integer(2))), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(2)))"], [["times", 3, "Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(2))"], "Equality(Mul(Integer(4), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(2)), Pow(Function('A_2')(Symbol('x', commutative=True)), Integer(2))), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(4)))"], [["power", 4, 2], "Equality(Mul(Integer(16), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(4)), Pow(Function('A_2')(Symbol('x', commutative=True)), Integer(4))), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(8)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integer(16), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(4)), Pow(Function('A_2')(Symbol('x', commutative=True)), Integer(4))), Mul(Integer(-1), Pow(Add(Function('A_2')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True))), Integer(8))))"]]}, {"prompt": "Given p{(\\Psi^{\\dagger},L)} = \\Psi^{\\dagger} e^{L} and q{(\\Psi^{\\dagger},L)} = \\int p{(\\Psi^{\\dagger},L)} dL, then obtain q{(\\Psi^{\\dagger},L)} = \\int \\Psi^{\\dagger} e^{L} dL", "derivation": "p{(\\Psi^{\\dagger},L)} = \\Psi^{\\dagger} e^{L} and \\int p{(\\Psi^{\\dagger},L)} dL = \\int \\Psi^{\\dagger} e^{L} dL and q{(\\Psi^{\\dagger},L)} = \\int p{(\\Psi^{\\dagger},L)} dL and q{(\\Psi^{\\dagger},L)} = \\int \\Psi^{\\dagger} e^{L} dL", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Symbol('L', commutative=True))))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Integral(Function('p')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('q')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('L', commutative=True)), Integral(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(F_{c})} = \\cos{(F_{c})}, then obtain (\\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} + 1)^{F_{c}} - \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} = (\\frac{d}{d F_{c}} \\cos{(F_{c})} + 1)^{F_{c}} - \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})}", "derivation": "\\psi^{*}{(F_{c})} = \\cos{(F_{c})} and \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} = \\frac{d}{d F_{c}} \\cos{(F_{c})} and \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} + 1 = \\frac{d}{d F_{c}} \\cos{(F_{c})} + 1 and (\\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} + 1)^{F_{c}} = (\\frac{d}{d F_{c}} \\cos{(F_{c})} + 1)^{F_{c}} and (\\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} + 1)^{F_{c}} - \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})} = (\\frac{d}{d F_{c}} \\cos{(F_{c})} + 1)^{F_{c}} - \\frac{d}{d F_{c}} \\psi^{*}{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)), Symbol('F_c', commutative=True)), Pow(Add(Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)), Symbol('F_c', commutative=True)))"], [["minus", 4, "Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Add(Pow(Add(Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))), Add(Pow(Add(Derivative(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1)), Symbol('F_c', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\psi^*')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\eta)} = \\cos{(\\eta)}, then obtain - \\eta + \\frac{d}{d \\eta} \\operatorname{L_{\\varepsilon}}{(\\eta)} - 1 = - \\eta - \\sin{(\\eta)} - 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\eta)} = \\cos{(\\eta)} and - \\eta + \\operatorname{L_{\\varepsilon}}{(\\eta)} = - \\eta + \\cos{(\\eta)} and \\frac{d}{d \\eta} (- \\eta + \\operatorname{L_{\\varepsilon}}{(\\eta)}) = \\frac{d}{d \\eta} (- \\eta + \\cos{(\\eta)}) and - \\eta + \\frac{d}{d \\eta} (- \\eta + \\operatorname{L_{\\varepsilon}}{(\\eta)}) = - \\eta + \\frac{d}{d \\eta} (- \\eta + \\cos{(\\eta)}) and - \\eta + \\frac{d}{d \\eta} \\operatorname{L_{\\varepsilon}}{(\\eta)} - 1 = - \\eta - \\sin{(\\eta)} - 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given t{(g,\\pi)} = \\int \\pi^{g} dg, then obtain 2 g \\int \\pi^{g} dg - t{(g,\\pi)} - \\int \\pi^{g} dg = 2 g \\int \\pi^{g} dg - 2 \\int \\pi^{g} dg", "derivation": "t{(g,\\pi)} = \\int \\pi^{g} dg and - g \\int \\pi^{g} dg + t{(g,\\pi)} = - g \\int \\pi^{g} dg + \\int \\pi^{g} dg and - 2 g \\int \\pi^{g} dg + t{(g,\\pi)} + \\int \\pi^{g} dg = - 2 g \\int \\pi^{g} dg + 2 \\int \\pi^{g} dg and 2 g \\int \\pi^{g} dg - t{(g,\\pi)} - \\int \\pi^{g} dg = 2 g \\int \\pi^{g} dg - 2 \\int \\pi^{g} dg", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["minus", 1, "Mul(Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Function('t')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Function('t')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(2), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Function('t')(Symbol('g', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))), Add(Mul(Integer(2), Symbol('g', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Integer(2), Integral(Pow(Symbol('\\\\pi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given i{(s,\\Omega)} = (e^{s})^{\\Omega}, then obtain \\iint \\frac{i{(s,\\Omega)}}{\\Omega} d\\Omega d\\Omega = \\iint \\frac{(e^{s})^{\\Omega}}{\\Omega} d\\Omega d\\Omega", "derivation": "i{(s,\\Omega)} = (e^{s})^{\\Omega} and \\frac{i{(s,\\Omega)}}{\\Omega} = \\frac{(e^{s})^{\\Omega}}{\\Omega} and \\int \\frac{i{(s,\\Omega)}}{\\Omega} d\\Omega = \\int \\frac{(e^{s})^{\\Omega}}{\\Omega} d\\Omega and \\iint \\frac{i{(s,\\Omega)}}{\\Omega} d\\Omega d\\Omega = \\iint \\frac{(e^{s})^{\\Omega}}{\\Omega} d\\Omega d\\Omega", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('i')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(exp(Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('i')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(exp(Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('i')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(exp(Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given p{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain 0 = - \\frac{p^{3}{(V_{\\mathbf{E}})}}{\\sin^{4}{(V_{\\mathbf{E}})}} + \\frac{p{(V_{\\mathbf{E}})}}{\\sin^{2}{(V_{\\mathbf{E}})}}", "derivation": "p{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\frac{p{(V_{\\mathbf{E}})}}{\\sin^{2}{(V_{\\mathbf{E}})}} = \\frac{1}{\\sin{(V_{\\mathbf{E}})}} and 0 = - \\frac{p{(V_{\\mathbf{E}})}}{\\sin^{2}{(V_{\\mathbf{E}})}} + \\frac{1}{\\sin{(V_{\\mathbf{E}})}} and 0 = - \\frac{p^{3}{(V_{\\mathbf{E}})}}{\\sin^{4}{(V_{\\mathbf{E}})}} + \\frac{p{(V_{\\mathbf{E}})}}{\\sin^{2}{(V_{\\mathbf{E}})}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["divide", 1, "Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(2))"], "Equality(Mul(Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2))), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)))"], [["minus", 2, "Mul(Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2))), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(3)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-4))), Mul(Function('p')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} = (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} and \\theta{(\\mathbf{r},\\mu,\\mathbf{f})} = \\mu \\int \\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} d\\mu, then obtain \\theta{(\\mathbf{r},\\mu,\\mathbf{f})} = \\mu \\int (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} d\\mu", "derivation": "\\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} = (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} and \\int \\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} d\\mu = \\int (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} d\\mu and \\mu \\int \\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} d\\mu = \\mu \\int (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} d\\mu and \\theta{(\\mathbf{r},\\mu,\\mathbf{f})} = \\mu \\int \\tilde{g}{(\\mathbf{r},\\mu,\\mathbf{f})} d\\mu and \\theta{(\\mathbf{r},\\mu,\\mathbf{f})} = \\mu \\int (\\frac{\\mu}{\\mathbf{f}})^{\\mathbf{r}} d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["times", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Integral(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then derive \\frac{d}{d \\mathbf{B}} \\theta{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}}, then obtain - \\log{(\\mathbf{B})} + \\frac{1}{\\mathbf{B}} = - \\log{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})}", "derivation": "\\theta{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\theta{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\theta{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}} and - \\log{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\theta{(\\mathbf{B})} = - \\log{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}} and - \\log{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\theta{(\\mathbf{B})} = - \\log{(\\mathbf{B})} + \\frac{1}{\\mathbf{B}} and - \\log{(\\mathbf{B})} + \\frac{1}{\\mathbf{B}} = - \\log{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))"], [["minus", 2, "log(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(b,E_{\\lambda})} = \\log{(\\frac{b}{E_{\\lambda}})}, then obtain \\frac{1 - \\frac{1}{E_{\\lambda}}}{\\frac{\\log{(\\frac{b}{E_{\\lambda}})}}{\\mathbf{F}{(b,E_{\\lambda})}} - \\frac{1}{E_{\\lambda}}} = 1", "derivation": "\\mathbf{F}{(b,E_{\\lambda})} = \\log{(\\frac{b}{E_{\\lambda}})} and 1 = \\frac{\\log{(\\frac{b}{E_{\\lambda}})}}{\\mathbf{F}{(b,E_{\\lambda})}} and 1 - \\frac{1}{E_{\\lambda}} = \\frac{\\log{(\\frac{b}{E_{\\lambda}})}}{\\mathbf{F}{(b,E_{\\lambda})}} - \\frac{1}{E_{\\lambda}} and \\frac{1 - \\frac{1}{E_{\\lambda}}}{\\frac{\\log{(\\frac{b}{E_{\\lambda}})}}{\\mathbf{F}{(b,E_{\\lambda})}} - \\frac{1}{E_{\\lambda}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), log(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('b', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))))"], [["minus", 2, "Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Add(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))))"], [["divide", 3, "Add(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1))))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Pow(Add(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('b', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('b', commutative=True)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{f},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\mathbf{f}} (- \\hat{H}_{\\lambda} + \\mathbf{f}), then derive \\operatorname{P_{g}}{(\\mathbf{f},\\hat{H}_{\\lambda})} = 1, then obtain A + \\hat{H}_{\\lambda} = \\hat{H}_{\\lambda} + v_{z}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{f},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\mathbf{f}} (- \\hat{H}_{\\lambda} + \\mathbf{f}) and \\operatorname{P_{g}}{(\\mathbf{f},\\hat{H}_{\\lambda})} = 1 and \\frac{\\partial}{\\partial \\mathbf{f}} (- \\hat{H}_{\\lambda} + \\mathbf{f}) = 1 and \\int \\frac{\\partial}{\\partial \\mathbf{f}} (- \\hat{H}_{\\lambda} + \\mathbf{f}) d\\hat{H}_{\\lambda} = \\int 1 d\\hat{H}_{\\lambda} and A + \\hat{H}_{\\lambda} = \\hat{H}_{\\lambda} + v_{z}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('P_g')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(r,C)} = r^{C}, then obtain \\frac{\\varepsilon{(r,C)}}{\\int r^{C} dr} = \\frac{r^{C}}{\\int r^{C} dr}", "derivation": "\\varepsilon{(r,C)} = r^{C} and \\int \\varepsilon{(r,C)} dr = \\int r^{C} dr and \\frac{\\varepsilon{(r,C)}}{\\int \\varepsilon{(r,C)} dr} = \\frac{r^{C}}{\\int \\varepsilon{(r,C)} dr} and \\frac{\\varepsilon{(r,C)}}{\\int r^{C} dr} = \\frac{r^{C}}{\\int r^{C} dr}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Pow(Integral(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))), Mul(Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)), Pow(Integral(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\varepsilon')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Pow(Integral(Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))), Mul(Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)), Pow(Integral(Pow(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(v_{z})} = e^{v_{z}}, then derive \\frac{d}{d v_{z}} \\hat{H}_{\\lambda}{(v_{z})} = e^{v_{z}}, then obtain 2 \\hat{H}_{\\lambda}{(v_{z})} = \\hat{H}_{\\lambda}{(v_{z})} + \\frac{d}{d v_{z}} \\hat{H}_{\\lambda}{(v_{z})}", "derivation": "\\hat{H}_{\\lambda}{(v_{z})} = e^{v_{z}} and 2 \\hat{H}_{\\lambda}{(v_{z})} = \\hat{H}_{\\lambda}{(v_{z})} + e^{v_{z}} and \\frac{d}{d v_{z}} \\hat{H}_{\\lambda}{(v_{z})} = \\frac{d}{d v_{z}} e^{v_{z}} and \\frac{d}{d v_{z}} \\hat{H}_{\\lambda}{(v_{z})} = e^{v_{z}} and 2 \\hat{H}_{\\lambda}{(v_{z})} = \\hat{H}_{\\lambda}{(v_{z})} + \\frac{d}{d v_{z}} \\hat{H}_{\\lambda}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), exp(Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(F_{c},f,y^{\\prime})} = \\frac{F_{c} y^{\\prime}}{f}, then obtain \\sin{((M{(F_{c},f,y^{\\prime})} - 1)^{F_{c}})} = \\sin{((\\frac{F_{c} y^{\\prime}}{f} - 1)^{F_{c}})}", "derivation": "M{(F_{c},f,y^{\\prime})} = \\frac{F_{c} y^{\\prime}}{f} and M{(F_{c},f,y^{\\prime})} - 1 = \\frac{F_{c} y^{\\prime}}{f} - 1 and (M{(F_{c},f,y^{\\prime})} - 1)^{F_{c}} = (\\frac{F_{c} y^{\\prime}}{f} - 1)^{F_{c}} and \\sin{((M{(F_{c},f,y^{\\prime})} - 1)^{F_{c}})} = \\sin{((\\frac{F_{c} y^{\\prime}}{f} - 1)^{F_{c}})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_c', commutative=True), Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('M')(Symbol('F_c', commutative=True), Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Function('M')(Symbol('F_c', commutative=True), Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('F_c', commutative=True)), Pow(Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('F_c', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Add(Function('M')(Symbol('F_c', commutative=True), Symbol('f', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('F_c', commutative=True))), sin(Pow(Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{E},l,f^{\\prime})} = \\mathbf{E} + f^{\\prime} + l and \\operatorname{M_{E}}{(\\mathbf{E},l,f^{\\prime})} = \\mathbf{S}^{f^{\\prime}}{(\\mathbf{E},l,f^{\\prime})}, then obtain \\operatorname{M_{E}}{(\\mathbf{E},l,f^{\\prime})} = (\\mathbf{E} + f^{\\prime} + l)^{f^{\\prime}}", "derivation": "\\mathbf{S}{(\\mathbf{E},l,f^{\\prime})} = \\mathbf{E} + f^{\\prime} + l and \\mathbf{S}^{f^{\\prime}}{(\\mathbf{E},l,f^{\\prime})} = (\\mathbf{E} + f^{\\prime} + l)^{f^{\\prime}} and \\operatorname{M_{E}}{(\\mathbf{E},l,f^{\\prime})} = \\mathbf{S}^{f^{\\prime}}{(\\mathbf{E},l,f^{\\prime})} and \\operatorname{M_{E}}{(\\mathbf{E},l,f^{\\prime})} = (\\mathbf{E} + f^{\\prime} + l)^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('M_E')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('l', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(T)} = \\log{(T)}, then derive - \\mathbf{f}{(T)} + \\log{(T)} + \\frac{d}{d T} \\mathbf{f}{(T)} - \\frac{1}{T} = - \\mathbf{f}{(T)} + \\log{(T)}, then obtain \\frac{d^{2}}{d T^{2}} 0 + \\frac{d}{d T} \\log{(T)} - \\frac{1}{T} = \\frac{d^{2}}{d T^{2}} 0", "derivation": "\\mathbf{f}{(T)} = \\log{(T)} and \\mathbf{f}{(T)} - \\log{(T)} = 0 and \\frac{d}{d T} (\\mathbf{f}{(T)} - \\log{(T)}) = \\frac{d}{d T} 0 and - \\mathbf{f}{(T)} + \\log{(T)} + \\frac{d}{d T} (\\mathbf{f}{(T)} - \\log{(T)}) = - \\mathbf{f}{(T)} + \\log{(T)} + \\frac{d}{d T} 0 and - \\mathbf{f}{(T)} + \\log{(T)} + \\frac{d}{d T} \\mathbf{f}{(T)} - \\frac{1}{T} = - \\mathbf{f}{(T)} + \\log{(T)} and \\frac{d}{d T} \\mathbf{f}{(T)} - \\frac{1}{T} = 0 and \\frac{d^{2}}{d T^{2}} 0 + \\frac{d}{d T} \\mathbf{f}{(T)} - \\frac{1}{T} = \\frac{d^{2}}{d T^{2}} 0 and \\frac{d^{2}}{d T^{2}} 0 + \\frac{d}{d T} \\log{(T)} - \\frac{1}{T} = \\frac{d^{2}}{d T^{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["minus", 1, "log(Symbol('T', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 3, "Add(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True)), Derivative(Add(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True)), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))), Integer(0))"], [["add", 6, "Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(2))), Derivative(Function('\\\\mathbf{f}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(2))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\psi{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{P_{e}}{(\\mathbf{P})} = \\psi{(\\mathbf{P})} + \\log{(\\mathbf{P})}, then obtain \\frac{\\cos{(\\operatorname{P_{e}}{(\\mathbf{P})})}}{\\mathbf{P}} = \\frac{\\cos{(2 \\psi{(\\mathbf{P})})}}{\\mathbf{P}}", "derivation": "\\psi{(\\mathbf{P})} = \\log{(\\mathbf{P})} and 2 \\psi{(\\mathbf{P})} = \\psi{(\\mathbf{P})} + \\log{(\\mathbf{P})} and \\operatorname{P_{e}}{(\\mathbf{P})} = \\psi{(\\mathbf{P})} + \\log{(\\mathbf{P})} and \\cos{(\\operatorname{P_{e}}{(\\mathbf{P})})} = \\cos{(\\psi{(\\mathbf{P})} + \\log{(\\mathbf{P})})} and \\cos{(\\operatorname{P_{e}}{(\\mathbf{P})})} = \\cos{(2 \\psi{(\\mathbf{P})})} and \\frac{\\cos{(\\operatorname{P_{e}}{(\\mathbf{P})})}}{\\mathbf{P}} = \\frac{\\cos{(2 \\psi{(\\mathbf{P})})}}{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{P}', commutative=True)), Add(Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["cos", 3], "Equality(cos(Function('P_e')(Symbol('\\\\mathbf{P}', commutative=True))), cos(Add(Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(cos(Function('P_e')(Symbol('\\\\mathbf{P}', commutative=True))), cos(Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), cos(Function('P_e')(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), cos(Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given M{(\\delta)} = \\cos{(\\delta)}, then derive \\frac{d}{d \\delta} M{(\\delta)} = - \\sin{(\\delta)}, then obtain \\frac{d}{d \\delta} - \\frac{M{(\\delta)}}{\\sin^{2}{(\\delta)}} = \\frac{d}{d \\delta} - \\frac{\\cos{(\\delta)}}{\\sin^{2}{(\\delta)}}", "derivation": "M{(\\delta)} = \\cos{(\\delta)} and \\frac{d}{d \\delta} M{(\\delta)} = \\frac{d}{d \\delta} \\cos{(\\delta)} and \\frac{d}{d \\delta} M{(\\delta)} = - \\sin{(\\delta)} and \\frac{M{(\\delta)}}{\\sin{(\\delta)}} = \\frac{\\cos{(\\delta)}}{\\sin{(\\delta)}} and \\frac{M{(\\delta)}}{\\sin{(\\delta)} \\frac{d}{d \\delta} M{(\\delta)}} = \\frac{\\cos{(\\delta)}}{\\sin{(\\delta)} \\frac{d}{d \\delta} M{(\\delta)}} and - \\frac{M{(\\delta)}}{\\sin^{2}{(\\delta)}} = - \\frac{\\cos{(\\delta)}}{\\sin^{2}{(\\delta)}} and \\frac{d}{d \\delta} - \\frac{M{(\\delta)}}{\\sin^{2}{(\\delta)}} = \\frac{d}{d \\delta} - \\frac{\\cos{(\\delta)}}{\\sin^{2}{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "sin(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('M')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1)), cos(Symbol('\\\\delta', commutative=True))))"], [["divide", 4, "Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Mul(Function('M')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-1)), cos(Symbol('\\\\delta', commutative=True)), Pow(Derivative(Function('M')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-2)), cos(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(sin(Symbol('\\\\delta', commutative=True)), Integer(-2)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(U,s)} = U - s, then obtain \\frac{\\partial}{\\partial U} (U - s) \\frac{\\partial}{\\partial U} \\operatorname{n_{1}}^{s}{(U,s)} = \\frac{\\partial}{\\partial U} (U - s) \\frac{\\partial}{\\partial U} (U - s)^{s}", "derivation": "\\operatorname{n_{1}}{(U,s)} = U - s and \\operatorname{n_{1}}^{s}{(U,s)} = (U - s)^{s} and \\frac{\\partial}{\\partial U} \\operatorname{n_{1}}^{s}{(U,s)} = \\frac{\\partial}{\\partial U} (U - s)^{s} and \\frac{\\partial}{\\partial U} (U - s) \\frac{\\partial}{\\partial U} \\operatorname{n_{1}}^{s}{(U,s)} = \\frac{\\partial}{\\partial U} (U - s) \\frac{\\partial}{\\partial U} (U - s)^{s}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('U', commutative=True), Symbol('s', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('U', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Pow(Function('n_1')(Symbol('U', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Function('n_1')(Symbol('U', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain \\frac{d}{d \\phi_2} W^{\\phi_2}{(\\phi_2)} + \\int W{(\\phi_2)} d\\phi_2 = \\frac{d}{d \\phi_2} W^{\\phi_2}{(\\phi_2)} + \\int \\cos{(\\phi_2)} d\\phi_2", "derivation": "W{(\\phi_2)} = \\cos{(\\phi_2)} and \\int W{(\\phi_2)} d\\phi_2 = \\int \\cos{(\\phi_2)} d\\phi_2 and W^{\\phi_2}{(\\phi_2)} = \\cos^{\\phi_2}{(\\phi_2)} and \\frac{d}{d \\phi_2} W^{\\phi_2}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)} and \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)} + \\int W{(\\phi_2)} d\\phi_2 = \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)} + \\int \\cos{(\\phi_2)} d\\phi_2 and \\frac{d}{d \\phi_2} W^{\\phi_2}{(\\phi_2)} + \\int W{(\\phi_2)} d\\phi_2 = \\frac{d}{d \\phi_2} W^{\\phi_2}{(\\phi_2)} + \\int \\cos{(\\phi_2)} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Pow(Function('W')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integral(Function('W')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integral(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Pow(Function('W')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integral(Function('W')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Derivative(Pow(Function('W')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integral(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\theta)} = e^{e^{\\theta}}, then obtain \\iint \\frac{d}{d \\theta} \\operatorname{A_{x}}{(\\theta)} d\\theta d\\theta = \\iint \\frac{d}{d \\theta} e^{e^{\\theta}} d\\theta d\\theta", "derivation": "\\operatorname{A_{x}}{(\\theta)} = e^{e^{\\theta}} and \\frac{d}{d \\theta} \\operatorname{A_{x}}{(\\theta)} = \\frac{d}{d \\theta} e^{e^{\\theta}} and \\int \\frac{d}{d \\theta} \\operatorname{A_{x}}{(\\theta)} d\\theta = \\int \\frac{d}{d \\theta} e^{e^{\\theta}} d\\theta and \\iint \\frac{d}{d \\theta} \\operatorname{A_{x}}{(\\theta)} d\\theta d\\theta = \\iint \\frac{d}{d \\theta} e^{e^{\\theta}} d\\theta d\\theta", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\theta', commutative=True)), exp(exp(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Function('A_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(exp(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Derivative(Function('A_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Derivative(exp(exp(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})} = \\log{(\\frac{a^{\\dagger}}{\\mathbf{E}})}, then obtain \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})}}{\\mathbf{E}} = \\frac{1}{\\mathbf{E} a^{\\dagger}}", "derivation": "\\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})} = \\log{(\\frac{a^{\\dagger}}{\\mathbf{E}})} and \\frac{\\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})}}{\\mathbf{E}} = \\frac{\\log{(\\frac{a^{\\dagger}}{\\mathbf{E}})}}{\\mathbf{E}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\frac{\\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})}}{\\mathbf{E}} = \\frac{\\partial}{\\partial a^{\\dagger}} \\frac{\\log{(\\frac{a^{\\dagger}}{\\mathbf{E}})}}{\\mathbf{E}} and \\frac{\\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},\\mathbf{E})}}{\\mathbf{E}} = \\frac{1}{\\mathbf{E} a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), log(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), log(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Derivative(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\phi{(\\hat{H},\\Psi)} = \\log{(\\Psi + \\hat{H})}, then obtain \\int 0 d\\hat{H} = \\int (- \\phi{(\\hat{H},\\Psi)} + \\log{(\\Psi + \\hat{H})}) \\log{(\\Psi + \\hat{H})} d\\hat{H}", "derivation": "\\phi{(\\hat{H},\\Psi)} = \\log{(\\Psi + \\hat{H})} and 0 = - \\phi{(\\hat{H},\\Psi)} + \\log{(\\Psi + \\hat{H})} and 0 = (- \\phi{(\\hat{H},\\Psi)} + \\log{(\\Psi + \\hat{H})}) \\log{(\\Psi + \\hat{H})} and \\int 0 d\\hat{H} = \\int (- \\phi{(\\hat{H},\\Psi)} + \\log{(\\Psi + \\hat{H})}) \\log{(\\Psi + \\hat{H})} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True)), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True))), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["times", 2, "log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True))), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\Psi', commutative=True))), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), log(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given n{(v_{2})} = e^{v_{2}}, then derive \\frac{d}{d v_{2}} n{(v_{2})} = e^{v_{2}}, then obtain n{(v_{2})} = \\frac{d}{d v_{2}} n{(v_{2})}", "derivation": "n{(v_{2})} = e^{v_{2}} and \\frac{d}{d v_{2}} n{(v_{2})} = \\frac{d}{d v_{2}} e^{v_{2}} and \\frac{d}{d v_{2}} n{(v_{2})} = e^{v_{2}} and n{(v_{2})} = \\frac{d}{d v_{2}} n{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(exp(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), exp(Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('n')(Symbol('v_2', commutative=True)), Derivative(Function('n')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{M},y)} = \\mathbf{M} y, then obtain (\\int \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} dy)^{2} = (\\int \\mathbf{M} y \\operatorname{F_{N}}{(\\mathbf{M},y)} dy) \\int \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} dy", "derivation": "\\operatorname{F_{N}}{(\\mathbf{M},y)} = \\mathbf{M} y and \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} = \\mathbf{M} y \\operatorname{F_{N}}{(\\mathbf{M},y)} and \\int \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} dy = \\int \\mathbf{M} y \\operatorname{F_{N}}{(\\mathbf{M},y)} dy and (\\int \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} dy)^{2} = (\\int \\mathbf{M} y \\operatorname{F_{N}}{(\\mathbf{M},y)} dy) \\int \\operatorname{F_{N}}^{2}{(\\mathbf{M},y)} dy", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True), Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Tuple(Symbol('y', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True), Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["times", 3, "Integral(Pow(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Tuple(Symbol('y', commutative=True)))"], "Equality(Pow(Integral(Pow(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Tuple(Symbol('y', commutative=True))), Integer(2)), Mul(Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True), Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Pow(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\eta{(l)} = \\log{(l)} and \\operatorname{A_{1}}{(l)} = \\log{(l)}^{2}, then obtain (\\eta{(l)} \\log{(l)})^{2 l} (\\int \\log{(l)}^{2} dl)^{2} = (\\log{(l)}^{2})^{2 l} (\\int \\log{(l)}^{2} dl)^{2}", "derivation": "\\eta{(l)} = \\log{(l)} and \\eta{(l)} \\log{(l)} = \\log{(l)}^{2} and (\\eta{(l)} \\log{(l)})^{l} = (\\log{(l)}^{2})^{l} and \\operatorname{A_{1}}{(l)} = \\log{(l)}^{2} and \\int \\operatorname{A_{1}}{(l)} dl = \\int \\log{(l)}^{2} dl and (\\eta{(l)} \\log{(l)})^{l} \\int \\operatorname{A_{1}}{(l)} dl = (\\log{(l)}^{2})^{l} \\int \\operatorname{A_{1}}{(l)} dl and (\\eta{(l)} \\log{(l)})^{l} \\int \\log{(l)}^{2} dl = (\\log{(l)}^{2})^{l} \\int \\log{(l)}^{2} dl and (\\eta{(l)} \\log{(l)})^{2 l} (\\int \\log{(l)}^{2} dl)^{2} = (\\log{(l)}^{2})^{2 l} (\\int \\log{(l)}^{2} dl)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["times", 1, "log(Symbol('l', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Pow(log(Symbol('l', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(Pow(log(Symbol('l', commutative=True)), Integer(2)), Symbol('l', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Integer(2)))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))))"], [["times", 3, "Integral(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Mul(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integral(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Pow(log(Symbol('l', commutative=True)), Integer(2)), Symbol('l', commutative=True)), Integral(Function('A_1')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Mul(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Pow(log(Symbol('l', commutative=True)), Integer(2)), Symbol('l', commutative=True)), Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True)))))"], [["power", 7, 2], "Equality(Mul(Pow(Mul(Function('\\\\eta')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Mul(Integer(2), Symbol('l', commutative=True))), Pow(Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integer(2))), Mul(Pow(Pow(log(Symbol('l', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('l', commutative=True))), Pow(Integral(Pow(log(Symbol('l', commutative=True)), Integer(2)), Tuple(Symbol('l', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\lambda{(V_{\\mathbf{B}},\\rho_f)} = V_{\\mathbf{B}} \\cos{(\\rho_f)}, then obtain \\lambda{(V_{\\mathbf{B}},\\rho_f)} \\cos{(\\rho_f)} + 1 = V_{\\mathbf{B}} \\cos^{2}{(\\rho_f)} + 1", "derivation": "\\lambda{(V_{\\mathbf{B}},\\rho_f)} = V_{\\mathbf{B}} \\cos{(\\rho_f)} and \\frac{\\lambda{(V_{\\mathbf{B}},\\rho_f)}}{V_{\\mathbf{B}}} = \\cos{(\\rho_f)} and \\lambda{(V_{\\mathbf{B}},\\rho_f)} \\cos{(\\rho_f)} = V_{\\mathbf{B}} \\cos^{2}{(\\rho_f)} and \\lambda{(V_{\\mathbf{B}},\\rho_f)} \\cos{(\\rho_f)} + 1 = V_{\\mathbf{B}} \\cos^{2}{(\\rho_f)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True)))"], [["times", 2, "Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), cos(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(2))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\lambda')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Integer(1)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Integer(1)))"]]}, {"prompt": "Given \\varphi^{*}{(M,z,\\mathbf{f})} = - M + \\mathbf{f} + z, then derive \\int \\varphi^{*}{(M,z,\\mathbf{f})} dz = \\mathbf{D} + \\frac{z^{2}}{2} + z (- M + \\mathbf{f}), then obtain (\\int (- M + \\mathbf{f} + z) dz)^{\\mathbf{f}} = (\\mathbf{D} + \\frac{z^{2}}{2} + z (- M + \\mathbf{f}))^{\\mathbf{f}}", "derivation": "\\varphi^{*}{(M,z,\\mathbf{f})} = - M + \\mathbf{f} + z and \\int \\varphi^{*}{(M,z,\\mathbf{f})} dz = \\int (- M + \\mathbf{f} + z) dz and \\int \\varphi^{*}{(M,z,\\mathbf{f})} dz = \\mathbf{D} + \\frac{z^{2}}{2} + z (- M + \\mathbf{f}) and (\\int \\varphi^{*}{(M,z,\\mathbf{f})} dz)^{\\mathbf{f}} = (\\mathbf{D} + \\frac{z^{2}}{2} + z (- M + \\mathbf{f}))^{\\mathbf{f}} and (\\int (- M + \\mathbf{f} + z) dz)^{\\mathbf{f}} = (\\mathbf{D} + \\frac{z^{2}}{2} + z (- M + \\mathbf{f}))^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('M', commutative=True), Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('M', commutative=True), Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('M', commutative=True), Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\varphi^*')(Symbol('M', commutative=True), Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('z', commutative=True), Integer(2))), Mul(Symbol('z', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given U{(h)} = \\cos{(\\sin{(h)})}, then derive \\cos{(h)} + \\frac{d}{d h} U{(h)} = - \\sin{(\\sin{(h)})} \\cos{(h)} + \\cos{(h)}, then obtain \\cos{(h)} + \\frac{d}{d h} \\cos{(\\sin{(h)})} = - \\sin{(\\sin{(h)})} \\cos{(h)} + \\cos{(h)}", "derivation": "U{(h)} = \\cos{(\\sin{(h)})} and U{(h)} + \\sin{(h)} = \\sin{(h)} + \\cos{(\\sin{(h)})} and \\frac{d}{d h} (U{(h)} + \\sin{(h)}) = \\frac{d}{d h} (\\sin{(h)} + \\cos{(\\sin{(h)})}) and \\cos{(h)} + \\frac{d}{d h} U{(h)} = - \\sin{(\\sin{(h)})} \\cos{(h)} + \\cos{(h)} and \\cos{(h)} + \\frac{d}{d h} \\cos{(\\sin{(h)})} = - \\sin{(\\sin{(h)})} \\cos{(h)} + \\cos{(h)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('h', commutative=True)), cos(sin(Symbol('h', commutative=True))))"], [["add", 1, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('U')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Add(sin(Symbol('h', commutative=True)), cos(sin(Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Function('U')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('h', commutative=True)), cos(sin(Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('h', commutative=True)), Derivative(Function('U')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('h', commutative=True)), Derivative(cos(sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(sin(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))), cos(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\lambda{(Q)} = \\int \\sin{(Q)} dQ, then derive \\lambda{(Q)} = a - \\cos{(Q)}, then obtain 0 = a - \\cos{(Q)} - \\int \\sin{(Q)} dQ", "derivation": "\\lambda{(Q)} = \\int \\sin{(Q)} dQ and \\lambda{(Q)} = a - \\cos{(Q)} and \\int \\sin{(Q)} dQ = a - \\cos{(Q)} and 0 = a - \\cos{(Q)} - \\int \\sin{(Q)} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('Q', commutative=True)), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\lambda')(Symbol('Q', commutative=True)), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"], [["minus", 3, "Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Integer(0), Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{v}{(g)} = \\cos{(\\sin{(g)})}, then obtain \\iint (\\mathbf{v}{(g)} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} dM_{E} = \\iint (\\cos{(\\sin{(g)})} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} dM_{E}", "derivation": "\\mathbf{v}{(g)} = \\cos{(\\sin{(g)})} and \\frac{\\mathbf{v}{(g)}}{M_{E}} = \\frac{\\cos{(\\sin{(g)})}}{M_{E}} and \\mathbf{v}{(g)} - \\frac{\\mathbf{v}{(g)}}{M_{E}} = \\cos{(\\sin{(g)})} - \\frac{\\mathbf{v}{(g)}}{M_{E}} and \\mathbf{v}{(g)} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}} = \\cos{(\\sin{(g)})} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}} and \\int (\\mathbf{v}{(g)} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} = \\int (\\cos{(\\sin{(g)})} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} and \\iint (\\mathbf{v}{(g)} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} dM_{E} = \\iint (\\cos{(\\sin{(g)})} - \\frac{\\cos{(\\sin{(g)})}}{M_{E}}) dM_{E} dM_{E}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), cos(sin(Symbol('g', commutative=True))))"], [["divide", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True)))))"], [["minus", 1, "Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)))), Add(cos(sin(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))), Add(cos(sin(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))), Tuple(Symbol('M_E', commutative=True))), Integral(Add(cos(sin(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 5, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(cos(sin(Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), cos(sin(Symbol('g', commutative=True))))), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(H,A_{1})} = \\frac{A_{1}}{H}, then derive 0 = - \\frac{A_{1} \\frac{\\partial}{\\partial H} \\operatorname{F_{c}}{(H,A_{1})}}{H \\operatorname{F_{c}}^{2}{(H,A_{1})}} - \\frac{A_{1}}{H^{2} \\operatorname{F_{c}}{(H,A_{1})}}, then obtain 0 = - \\frac{1}{H} - \\frac{H \\frac{\\partial}{\\partial H} \\operatorname{F_{c}}{(H,A_{1})}}{A_{1}}", "derivation": "\\operatorname{F_{c}}{(H,A_{1})} = \\frac{A_{1}}{H} and 1 = \\frac{A_{1}}{H \\operatorname{F_{c}}{(H,A_{1})}} and \\frac{d}{d H} 1 = \\frac{\\partial}{\\partial H} \\frac{A_{1}}{H \\operatorname{F_{c}}{(H,A_{1})}} and 0 = - \\frac{A_{1} \\frac{\\partial}{\\partial H} \\operatorname{F_{c}}{(H,A_{1})}}{H \\operatorname{F_{c}}^{2}{(H,A_{1})}} - \\frac{A_{1}}{H^{2} \\operatorname{F_{c}}{(H,A_{1})}} and 0 = - \\frac{1}{H} - \\frac{H \\frac{\\partial}{\\partial H} \\frac{A_{1}}{H}}{A_{1}} and 0 = - \\frac{1}{H} - \\frac{H \\frac{\\partial}{\\partial H} \\operatorname{F_{c}}{(H,A_{1})}}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1))))"], [["divide", 1, "Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Integer(1), Mul(Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Integer(-2)), Derivative(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('H', commutative=True), Derivative(Mul(Symbol('A_1', commutative=True), Pow(Symbol('H', commutative=True), Integer(-1))), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('H', commutative=True), Derivative(Function('F_c')(Symbol('H', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)}, then obtain (\\mathbf{F}{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)} = (2 \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)}", "derivation": "\\mathbf{F}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\mathbf{F}{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)} = 2 \\cos{(\\hat{H}_l)} and (\\mathbf{F}{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} = (2 \\cos{(\\hat{H}_l)})^{\\hat{H}_l} and (\\mathbf{F}{(\\hat{H}_l)} + \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)} = (2 \\cos{(\\hat{H}_l)})^{\\hat{H}_l} - \\frac{d}{d \\hat{H}_l} \\cos{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{F}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Mul(Integer(2), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 3, "Derivative(cos(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))"], "Equality(Add(Pow(Add(Function('\\\\mathbf{F}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Add(Pow(Mul(Integer(2), cos(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})} = \\sin{(A_{2}^{F_{H}})}, then obtain \\sin{((F_{H} \\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})})^{A_{2}})} = \\sin{((F_{H} \\sin{(A_{2}^{F_{H}})})^{A_{2}})}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})} = \\sin{(A_{2}^{F_{H}})} and F_{H} \\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})} = F_{H} \\sin{(A_{2}^{F_{H}})} and (F_{H} \\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})})^{A_{2}} = (F_{H} \\sin{(A_{2}^{F_{H}})})^{A_{2}} and \\sin{((F_{H} \\operatorname{V_{\\mathbf{E}}}{(A_{2},F_{H})})^{A_{2}})} = \\sin{((F_{H} \\sin{(A_{2}^{F_{H}})})^{A_{2}})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True)), sin(Pow(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True))))"], [["times", 1, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), sin(Pow(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True)))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Mul(Symbol('F_H', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True))), Symbol('A_2', commutative=True)), Pow(Mul(Symbol('F_H', commutative=True), sin(Pow(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True)))), Symbol('A_2', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Mul(Symbol('F_H', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True))), Symbol('A_2', commutative=True))), sin(Pow(Mul(Symbol('F_H', commutative=True), sin(Pow(Symbol('A_2', commutative=True), Symbol('F_H', commutative=True)))), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given l{(\\phi_2,\\eta)} = \\eta \\phi_2 and \\operatorname{v_{x}}{(\\mathbf{B},i)} = e^{i^{\\mathbf{B}}}, then obtain (\\eta + l{(\\phi_2,\\eta)}) \\operatorname{v_{x}}{(\\mathbf{B},i)} = (\\eta \\phi_2 + \\eta) \\operatorname{v_{x}}{(\\mathbf{B},i)}", "derivation": "l{(\\phi_2,\\eta)} = \\eta \\phi_2 and \\eta + l{(\\phi_2,\\eta)} = \\eta \\phi_2 + \\eta and \\operatorname{v_{x}}{(\\mathbf{B},i)} = e^{i^{\\mathbf{B}}} and (\\eta + l{(\\phi_2,\\eta)}) e^{i^{\\mathbf{B}}} = (\\eta \\phi_2 + \\eta) e^{i^{\\mathbf{B}}} and (\\eta + l{(\\phi_2,\\eta)}) \\operatorname{v_{x}}{(\\mathbf{B},i)} = (\\eta \\phi_2 + \\eta) \\operatorname{v_{x}}{(\\mathbf{B},i)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('l')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\eta', commutative=True)))"], ["get_premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True)), exp(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 2, "exp(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\eta', commutative=True), Function('l')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta', commutative=True))), exp(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\eta', commutative=True)), exp(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\eta', commutative=True), Function('l')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\eta', commutative=True))), Function('v_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))), Mul(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Function('v_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(A_{z})} = \\cos{(A_{z})}, then obtain \\frac{\\frac{d}{d A_{z}} \\operatorname{n_{2}}{(A_{z})}}{\\frac{d}{d A_{z}} \\cos{(A_{z})}} = 1", "derivation": "\\operatorname{n_{2}}{(A_{z})} = \\cos{(A_{z})} and \\frac{d}{d A_{z}} \\operatorname{n_{2}}{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})} and \\frac{\\frac{d}{d A_{z}} \\operatorname{n_{2}}{(A_{z})}}{\\cos{(A_{z})}} = \\frac{\\frac{d}{d A_{z}} \\cos{(A_{z})}}{\\cos{(A_{z})}} and \\frac{\\frac{d}{d A_{z}} \\operatorname{n_{2}}{(A_{z})}}{\\frac{d}{d A_{z}} \\cos{(A_{z})}} = 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["divide", 2, "cos(Symbol('A_z', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)), Derivative(Function('n_2')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Pow(cos(Symbol('A_z', commutative=True)), Integer(-1)), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], "Equality(Mul(Derivative(Function('n_2')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{M}{(W)} = e^{W}, then derive \\int \\mathbf{M}{(W)} dW = G + e^{W}, then derive G + \\mathbf{M}{(W)} = y^{\\prime} + e^{W}, then obtain G + e^{W} = y^{\\prime} + e^{W}", "derivation": "\\mathbf{M}{(W)} = e^{W} and \\int \\mathbf{M}{(W)} dW = \\int e^{W} dW and \\int \\mathbf{M}{(W)} dW = G + e^{W} and \\int \\mathbf{M}{(W)} dW = G + \\mathbf{M}{(W)} and G + \\mathbf{M}{(W)} = \\int e^{W} dW and G + \\mathbf{M}{(W)} = y^{\\prime} + e^{W} and \\int \\mathbf{M}{(W)} dW = y^{\\prime} + e^{W} and G + e^{W} = y^{\\prime} + e^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('G', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('G', commutative=True), Function('\\\\mathbf{M}')(Symbol('W', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Symbol('G', commutative=True), exp(Symbol('W', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('W', commutative=True))))"]]}, {"prompt": "Given L{(p)} = p, then derive \\frac{d}{d p} L{(p)} = 1, then obtain \\frac{1}{\\frac{d}{d p} p} = \\frac{\\int 1 dp}{\\frac{d}{d p} p \\int \\frac{d}{d p} p dp}", "derivation": "L{(p)} = p and \\frac{d}{d p} L{(p)} = \\frac{d}{d p} p and \\frac{d}{d p} L{(p)} = 1 and \\frac{d}{d p} p = 1 and \\int \\frac{d}{d p} p dp = \\int 1 dp and \\frac{\\int \\frac{d}{d p} p dp}{\\frac{d}{d L{(p)}} L{(p)}} = \\frac{\\int 1 dp}{\\frac{d}{d L{(p)}} L{(p)}} and \\frac{\\int \\frac{d}{d p} p dp}{\\frac{d}{d p} p} = \\frac{\\int 1 dp}{\\frac{d}{d p} p} and \\frac{1}{\\frac{d}{d p} p} = \\frac{\\int 1 dp}{\\frac{d}{d p} p \\int \\frac{d}{d p} p dp}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(Integer(1), Tuple(Symbol('p', commutative=True))))"], [["divide", 5, "Derivative(Function('L')(Symbol('p', commutative=True)), Tuple(Function('L')(Symbol('p', commutative=True)), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('L')(Symbol('p', commutative=True)), Tuple(Function('L')(Symbol('p', commutative=True)), Integer(1))), Integer(-1)), Integral(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Derivative(Function('L')(Symbol('p', commutative=True)), Tuple(Function('L')(Symbol('p', commutative=True)), Integer(1))), Integer(-1)), Integral(Integer(1), Tuple(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Integral(Integer(1), Tuple(Symbol('p', commutative=True)))))"], [["divide", 7, "Integral(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))"], "Equality(Pow(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Mul(Pow(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Integral(Integer(1), Tuple(Symbol('p', commutative=True))), Pow(Integral(Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(V)} = \\sin{(\\log{(V)})}, then obtain \\frac{1}{7 (- V + 11 \\mathbb{I}{(V)})^{9} \\mathbb{I}{(V)}} = \\frac{1}{7 (- V + 10 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})})^{9} \\mathbb{I}{(V)}}", "derivation": "\\mathbb{I}{(V)} = \\sin{(\\log{(V)})} and 2 \\mathbb{I}{(V)} = \\mathbb{I}{(V)} + \\sin{(\\log{(V)})} and - V + 4 \\mathbb{I}{(V)} = - V + 3 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})} and - V + 11 \\mathbb{I}{(V)} = - V + 10 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})} and (- V + 11 \\mathbb{I}{(V)})^{10} = (- V + 10 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})})^{10} and \\frac{1}{7 (- V + 11 \\mathbb{I}{(V)})^{9} \\mathbb{I}{(V)}} = \\frac{- V + 10 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})}}{7 (- V + 11 \\mathbb{I}{(V)})^{10} \\mathbb{I}{(V)}} and \\frac{1}{7 (- V + 11 \\mathbb{I}{(V)})^{9} \\mathbb{I}{(V)}} = \\frac{1}{7 (- V + 10 \\mathbb{I}{(V)} + \\sin{(\\log{(V)})})^{9} \\mathbb{I}{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True))))"], [["add", 1, "Function('\\\\mathbb{I}')(Symbol('V', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), Add(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), sin(log(Symbol('V', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(4), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(3), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))))"], [["add", 3, "Mul(Integer(7), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(10), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))))"], [["power", 4, "Integer(10)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Integer(10)), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(10), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))), Integer(10)))"], [["divide", 4, "Mul(Integer(7), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Integer(10)), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))"], "Equality(Mul(Rational(1, 7), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Integer(-9)), Pow(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), Integer(-1))), Mul(Rational(1, 7), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Integer(-10)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(10), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))), Pow(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Rational(1, 7), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(11), Function('\\\\mathbb{I}')(Symbol('V', commutative=True)))), Integer(-9)), Pow(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), Integer(-1))), Mul(Rational(1, 7), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Mul(Integer(10), Function('\\\\mathbb{I}')(Symbol('V', commutative=True))), sin(log(Symbol('V', commutative=True)))), Integer(-9)), Pow(Function('\\\\mathbb{I}')(Symbol('V', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(H,B)} = - B + H, then obtain \\iint 0 dH dB = \\iint (- B + H - \\varepsilon{(H,B)}) dH dB", "derivation": "\\varepsilon{(H,B)} = - B + H and 0 = - B + H - \\varepsilon{(H,B)} and \\int 0 dH = \\int (- B + H - \\varepsilon{(H,B)}) dH and \\iint 0 dH dB = \\iint (- B + H - \\varepsilon{(H,B)}) dH dB", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('H', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('H', commutative=True)))"], [["minus", 1, "Function('\\\\varepsilon')(Symbol('H', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('H', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('H', commutative=True), Symbol('B', commutative=True)))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('H', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('H', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('H', commutative=True), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('H', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('H', commutative=True)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} = \\frac{\\nabla}{\\theta_1}, then obtain \\frac{\\partial}{\\partial \\nabla} (- \\int \\frac{\\nabla}{\\theta_1} d\\nabla + \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla) - \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla = \\frac{d}{d \\nabla} 0 - \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla", "derivation": "\\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} = \\frac{\\nabla}{\\theta_1} and \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla = \\int \\frac{\\nabla}{\\theta_1} d\\nabla and - \\int \\frac{\\nabla}{\\theta_1} d\\nabla + \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla = 0 and \\frac{\\partial}{\\partial \\nabla} (- \\int \\frac{\\nabla}{\\theta_1} d\\nabla + \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla) = \\frac{d}{d \\nabla} 0 and \\frac{\\partial}{\\partial \\nabla} (- \\int \\frac{\\nabla}{\\theta_1} d\\nabla + \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla) - \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla = \\frac{d}{d \\nabla} 0 - \\int \\operatorname{x^{{\\}'}}{(\\theta_1,\\nabla)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))), Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))), Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["minus", 4, "Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))), Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))), Add(Derivative(Integer(0), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\varphi^{*}{(\\mu_0,A_{z})} = \\log{(\\mu_0^{A_{z}})}, then obtain \\mu_0 + \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\varphi^{*}{(\\mu_0,A_{z})} + 1 = \\mu_0 + \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\log{(\\mu_0^{A_{z}})} + 1", "derivation": "\\varphi^{*}{(\\mu_0,A_{z})} = \\log{(\\mu_0^{A_{z}})} and \\frac{\\partial}{\\partial A_{z}} \\varphi^{*}{(\\mu_0,A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\log{(\\mu_0^{A_{z}})} and \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\varphi^{*}{(\\mu_0,A_{z})} = \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\log{(\\mu_0^{A_{z}})} and \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\varphi^{*}{(\\mu_0,A_{z})} + 1 = \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\log{(\\mu_0^{A_{z}})} + 1 and \\mu_0 + \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\varphi^{*}{(\\mu_0,A_{z})} + 1 = \\mu_0 + \\frac{\\partial^{2}}{\\partial A_{z}^{2}} \\log{(\\mu_0^{A_{z}})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Derivative(log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(2))))"], [["add", 3, 1], "Equality(Add(Derivative(Function('\\\\varphi^*')(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(1)), Add(Derivative(log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(1)))"], [["add", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Derivative(Function('\\\\varphi^*')(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(1)), Add(Symbol('\\\\mu_0', commutative=True), Derivative(log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(1)))"]]}, {"prompt": "Given Z{(\\Omega)} = e^{\\cos{(\\Omega)}} and \\hat{\\mathbf{x}}{(\\Omega)} = \\frac{Z{(\\Omega)} - e^{\\cos{(\\Omega)}}}{\\Omega}, then obtain \\hat{\\mathbf{x}}{(\\Omega)} e^{- \\cos{(\\Omega)}} = 0", "derivation": "Z{(\\Omega)} = e^{\\cos{(\\Omega)}} and \\hat{\\mathbf{x}}{(\\Omega)} = \\frac{Z{(\\Omega)} - e^{\\cos{(\\Omega)}}}{\\Omega} and \\hat{\\mathbf{x}}{(\\Omega)} = 0 and \\hat{\\mathbf{x}}{(\\Omega)} e^{- \\cos{(\\Omega)}} = 0", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\Omega', commutative=True)), exp(cos(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Add(Function('Z')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('\\\\Omega', commutative=True)))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Omega', commutative=True)), Integer(0))"], [["divide", 3, "exp(cos(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))))), Integer(0))"]]}, {"prompt": "Given Q{(z)} = \\cos{(z)}, then obtain \\frac{d}{d z} ((- z + \\cos{(z)}) Q{(z)} + \\int (- z + \\cos{(z)}) dz) = \\frac{d}{d z} ((- z + \\cos{(z)}) \\cos{(z)} + \\int (- z + \\cos{(z)}) dz)", "derivation": "Q{(z)} = \\cos{(z)} and - z + Q{(z)} = - z + \\cos{(z)} and (- z + Q{(z)}) Q{(z)} = (- z + Q{(z)}) \\cos{(z)} and (- z + Q{(z)}) Q{(z)} + \\int (- z + \\cos{(z)}) dz = (- z + Q{(z)}) \\cos{(z)} + \\int (- z + \\cos{(z)}) dz and \\frac{d}{d z} ((- z + Q{(z)}) Q{(z)} + \\int (- z + \\cos{(z)}) dz) = \\frac{d}{d z} ((- z + Q{(z)}) \\cos{(z)} + \\int (- z + \\cos{(z)}) dz) and \\frac{d}{d z} ((- z + \\cos{(z)}) Q{(z)} + \\int (- z + \\cos{(z)}) dz) = \\frac{d}{d z} ((- z + \\cos{(z)}) \\cos{(z)} + \\int (- z + \\cos{(z)}) dz)", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["minus", 1, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), Function('Q')(Symbol('z', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), Function('Q')(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), Function('Q')(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('Q')(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Function('Q')(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), cos(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{y},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + v_{y}, then derive \\frac{\\partial}{\\partial v_{y}} \\operatorname{x^{{\\}'}}{(v_{y},\\hat{H}_{\\lambda})} = 1, then obtain 0 = -1 + \\frac{1}{\\frac{\\partial}{\\partial v_{y}} (\\hat{H}_{\\lambda} + v_{y})}", "derivation": "\\operatorname{x^{{\\}'}}{(v_{y},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} + v_{y} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{x^{{\\}'}}{(v_{y},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial v_{y}} (\\hat{H}_{\\lambda} + v_{y}) and \\frac{\\partial}{\\partial v_{y}} \\operatorname{x^{{\\}'}}{(v_{y},\\hat{H}_{\\lambda})} = 1 and \\frac{\\partial}{\\partial v_{y}} (\\hat{H}_{\\lambda} + v_{y}) = 1 and 1 = \\frac{1}{\\frac{\\partial}{\\partial v_{y}} (\\hat{H}_{\\lambda} + v_{y})} and 0 = -1 + \\frac{1}{\\frac{\\partial}{\\partial v_{y}} (\\hat{H}_{\\lambda} + v_{y})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('v_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('v_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Integer(1), Pow(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1)))"], [["add", 5, "Integer(-1)"], "Equality(Integer(0), Add(Integer(-1), Pow(Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(J,C)} = C + J, then obtain \\frac{\\frac{\\partial}{\\partial J} \\operatorname{P_{e}}{(J,C)} - 1}{\\frac{\\partial}{\\partial J} (C + J)} = \\frac{\\frac{\\partial}{\\partial J} (C + J) - 1}{\\frac{\\partial}{\\partial J} (C + J)}", "derivation": "\\operatorname{P_{e}}{(J,C)} = C + J and \\frac{\\partial}{\\partial J} \\operatorname{P_{e}}{(J,C)} = \\frac{\\partial}{\\partial J} (C + J) and \\frac{\\partial}{\\partial J} \\operatorname{P_{e}}{(J,C)} - 1 = \\frac{\\partial}{\\partial J} (C + J) - 1 and \\frac{\\frac{\\partial}{\\partial J} \\operatorname{P_{e}}{(J,C)} - 1}{\\frac{\\partial}{\\partial J} (C + J)} = \\frac{\\frac{\\partial}{\\partial J} (C + J) - 1}{\\frac{\\partial}{\\partial J} (C + J)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('P_e')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 3, "Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('P_e')(Symbol('J', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Add(Symbol('C', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(F_{x})} = \\int \\sin{(F_{x})} dF_{x}, then derive \\dot{z}{(F_{x})} = g^{\\prime}_{\\varepsilon} - \\cos{(F_{x})}, then derive \\dot{z}{(F_{x})} \\sin{(F_{x})} = (A - \\cos{(F_{x})}) \\sin{(F_{x})}, then obtain (g^{\\prime}_{\\varepsilon} - \\cos{(F_{x})}) \\sin{(F_{x})} = \\sin{(F_{x})} \\int \\sin{(F_{x})} dF_{x}", "derivation": "\\dot{z}{(F_{x})} = \\int \\sin{(F_{x})} dF_{x} and \\dot{z}{(F_{x})} \\sin{(F_{x})} = \\sin{(F_{x})} \\int \\sin{(F_{x})} dF_{x} and \\dot{z}{(F_{x})} = g^{\\prime}_{\\varepsilon} - \\cos{(F_{x})} and \\dot{z}{(F_{x})} \\sin{(F_{x})} = (A - \\cos{(F_{x})}) \\sin{(F_{x})} and (A - \\cos{(F_{x})}) \\sin{(F_{x})} = \\sin{(F_{x})} \\int \\sin{(F_{x})} dF_{x} and (g^{\\prime}_{\\varepsilon} - \\cos{(F_{x})}) \\sin{(F_{x})} = (A - \\cos{(F_{x})}) \\sin{(F_{x})} and (g^{\\prime}_{\\varepsilon} - \\cos{(F_{x})}) \\sin{(F_{x})} = \\sin{(F_{x})} \\int \\sin{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('F_x', commutative=True)), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["times", 1, "sin(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True))), Mul(sin(Symbol('F_x', commutative=True)), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{z}')(Symbol('F_x', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('\\\\dot{z}')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True))), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), sin(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), sin(Symbol('F_x', commutative=True))), Mul(sin(Symbol('F_x', commutative=True)), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), sin(Symbol('F_x', commutative=True))), Mul(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), sin(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), sin(Symbol('F_x', commutative=True))), Mul(sin(Symbol('F_x', commutative=True)), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given b{(\\mathbf{H},z)} = \\cos{(\\mathbf{H}^{z})} and \\operatorname{A_{1}}{(\\mathbf{H})} = \\mathbf{H}, then obtain - b^{z}{(\\mathbf{H},z)} + \\sin{(\\operatorname{A_{1}}{(\\mathbf{H})})} = - b^{z}{(\\mathbf{H},z)} + \\sin{(\\mathbf{H})}", "derivation": "b{(\\mathbf{H},z)} = \\cos{(\\mathbf{H}^{z})} and \\operatorname{A_{1}}{(\\mathbf{H})} = \\mathbf{H} and \\sin{(\\operatorname{A_{1}}{(\\mathbf{H})})} = \\sin{(\\mathbf{H})} and \\sin{(\\operatorname{A_{1}}{(\\mathbf{H})})} - \\cos^{z}{(\\mathbf{H}^{z})} = \\sin{(\\mathbf{H})} - \\cos^{z}{(\\mathbf{H}^{z})} and - b^{z}{(\\mathbf{H},z)} + \\sin{(\\operatorname{A_{1}}{(\\mathbf{H})})} = - b^{z}{(\\mathbf{H},z)} + \\sin{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["sin", 2], "Equality(sin(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True))), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 3, "Pow(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Add(sin(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True)))), Add(sin(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(cos(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True))), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True))), sin(Function('A_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True))), sin(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} = \\frac{I}{\\dot{z} t_{2}}, then obtain t_{2} \\iint I \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} dt_{2} d\\dot{z} = t_{2} \\iint \\frac{I^{2}}{\\dot{z} t_{2}} dt_{2} d\\dot{z}", "derivation": "\\operatorname{v_{1}}{(t_{2},I,\\dot{z})} = \\frac{I}{\\dot{z} t_{2}} and I \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} = \\frac{I^{2}}{\\dot{z} t_{2}} and \\int I \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} dt_{2} = \\int \\frac{I^{2}}{\\dot{z} t_{2}} dt_{2} and \\iint I \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} dt_{2} d\\dot{z} = \\iint \\frac{I^{2}}{\\dot{z} t_{2}} dt_{2} d\\dot{z} and t_{2} \\iint I \\operatorname{v_{1}}{(t_{2},I,\\dot{z})} dt_{2} d\\dot{z} = t_{2} \\iint \\frac{I^{2}}{\\dot{z} t_{2}} dt_{2} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('I', commutative=True), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('v_1')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Symbol('I', commutative=True), Function('v_1')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Symbol('I', commutative=True), Function('v_1')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 4, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Integral(Mul(Symbol('I', commutative=True), Function('v_1')(Symbol('t_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Symbol('t_2', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given t{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\operatorname{C_{2}}{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then derive \\int t{(\\mathbf{F})} d\\mathbf{F} = \\pi - \\cos{(\\mathbf{F})}, then obtain \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = \\pi - \\cos{(\\mathbf{F})}", "derivation": "t{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\int t{(\\mathbf{F})} d\\mathbf{F} = \\int \\sin{(\\mathbf{F})} d\\mathbf{F} and \\operatorname{C_{2}}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\int t{(\\mathbf{F})} d\\mathbf{F} = \\pi - \\cos{(\\mathbf{F})} and t{(\\mathbf{F})} = \\operatorname{C_{2}}{(\\mathbf{F})} and \\int \\sin{(\\mathbf{F})} d\\mathbf{F} = \\pi - \\cos{(\\mathbf{F})} and \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = \\int \\sin{(\\mathbf{F})} d\\mathbf{F} and \\int \\operatorname{C_{2}}{(\\mathbf{F})} d\\mathbf{F} = \\pi - \\cos{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('t')(Symbol('\\\\mathbf{F}', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integral(Function('C_2')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}, then obtain - \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} \\log{(\\mathbf{J}_f)} + 2 \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = - \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} \\log{(\\mathbf{J}_f)} + \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} + \\log{(\\mathbf{J}_f)}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and \\hat{\\mathbf{x}}^{2}{(\\mathbf{J}_f)} = \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} \\log{(\\mathbf{J}_f)} and 2 \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} + \\log{(\\mathbf{J}_f)} and - \\hat{\\mathbf{x}}^{2}{(\\mathbf{J}_f)} + 2 \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = - \\hat{\\mathbf{x}}^{2}{(\\mathbf{J}_f)} + \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} + \\log{(\\mathbf{J}_f)} and - \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} \\log{(\\mathbf{J}_f)} + 2 \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} = - \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} \\log{(\\mathbf{J}_f)} + \\hat{\\mathbf{x}}{(\\mathbf{J}_f)} + \\log{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given Q{(\\dot{z})} = \\cos{(\\dot{z})} and S{(\\dot{z})} = \\int Q{(\\dot{z})} d\\dot{z}, then derive \\int Q{(\\dot{z})} d\\dot{z} = y^{\\prime} + \\sin{(\\dot{z})}, then obtain y^{\\prime} + \\sin{(\\dot{z})} = \\int \\cos{(\\dot{z})} d\\dot{z}", "derivation": "Q{(\\dot{z})} = \\cos{(\\dot{z})} and \\int Q{(\\dot{z})} d\\dot{z} = \\int \\cos{(\\dot{z})} d\\dot{z} and \\int Q{(\\dot{z})} d\\dot{z} = y^{\\prime} + \\sin{(\\dot{z})} and S{(\\dot{z})} = \\int Q{(\\dot{z})} d\\dot{z} and S{(\\dot{z})} = \\int \\cos{(\\dot{z})} d\\dot{z} and S{(\\dot{z})} = y^{\\prime} + \\sin{(\\dot{z})} and y^{\\prime} + \\sin{(\\dot{z})} = \\int \\cos{(\\dot{z})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('\\\\dot{z}', commutative=True)), Integral(Function('Q')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('S')(Symbol('\\\\dot{z}', commutative=True)), Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('S')(Symbol('\\\\dot{z}', commutative=True)), Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), sin(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\psi{(V,p)} = \\frac{\\sin{(p)}}{V}, then derive - p + \\frac{\\partial}{\\partial p} \\psi{(V,p)} = - p + \\frac{\\cos{(p)}}{V}, then obtain (- p + \\frac{\\partial}{\\partial p} \\psi{(V,p)})^{p} = (- p + \\frac{\\cos{(p)}}{V})^{p}", "derivation": "\\psi{(V,p)} = \\frac{\\sin{(p)}}{V} and \\frac{\\partial}{\\partial p} \\psi{(V,p)} = \\frac{\\partial}{\\partial p} \\frac{\\sin{(p)}}{V} and - p + \\frac{\\partial}{\\partial p} \\psi{(V,p)} = - p + \\frac{\\partial}{\\partial p} \\frac{\\sin{(p)}}{V} and - p + \\frac{\\partial}{\\partial p} \\psi{(V,p)} = - p + \\frac{\\cos{(p)}}{V} and (- p + \\frac{\\partial}{\\partial p} \\psi{(V,p)})^{p} = (- p + \\frac{\\cos{(p)}}{V})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('p', commutative=True)))))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('p', commutative=True)))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given l{(I,\\mathbf{J}_f,A_{1})} = - A_{1} + I \\mathbf{J}_f, then derive \\int - l{(I,\\mathbf{J}_f,A_{1})} d\\mathbf{J}_f = A_{1} \\mathbf{J}_f - \\frac{I \\mathbf{J}_f^{2}}{2} + J, then obtain \\frac{\\partial}{\\partial J} \\int - l{(I,\\mathbf{J}_f,A_{1})} d\\mathbf{J}_f = \\frac{\\partial}{\\partial J} (A_{1} \\mathbf{J}_f - \\frac{I \\mathbf{J}_f^{2}}{2} + J)", "derivation": "l{(I,\\mathbf{J}_f,A_{1})} = - A_{1} + I \\mathbf{J}_f and - l{(I,\\mathbf{J}_f,A_{1})} = A_{1} - I \\mathbf{J}_f and \\int - l{(I,\\mathbf{J}_f,A_{1})} d\\mathbf{J}_f = \\int (A_{1} - I \\mathbf{J}_f) d\\mathbf{J}_f and \\int - l{(I,\\mathbf{J}_f,A_{1})} d\\mathbf{J}_f = A_{1} \\mathbf{J}_f - \\frac{I \\mathbf{J}_f^{2}}{2} + J and \\frac{\\partial}{\\partial J} \\int - l{(I,\\mathbf{J}_f,A_{1})} d\\mathbf{J}_f = \\frac{\\partial}{\\partial J} (A_{1} \\mathbf{J}_f - \\frac{I \\mathbf{J}_f^{2}}{2} + J)", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('l')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('l')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Function('l')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Symbol('J', commutative=True)))"], [["differentiate", 4, "Symbol('J', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), Function('l')(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('I', commutative=True), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(2))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(J,c_{0})} = J - c_{0}, then derive \\int \\dot{\\mathbf{r}}{(J,c_{0})} dJ = F_{c} + \\frac{J^{2}}{2} - J c_{0}, then obtain \\int (F_{c} + \\frac{J^{2}}{2} - J c_{0} + J)^{F_{c}} dJ = \\int (J + \\int (J - c_{0}) dJ)^{F_{c}} dJ", "derivation": "\\dot{\\mathbf{r}}{(J,c_{0})} = J - c_{0} and \\int \\dot{\\mathbf{r}}{(J,c_{0})} dJ = \\int (J - c_{0}) dJ and J + \\int \\dot{\\mathbf{r}}{(J,c_{0})} dJ = J + \\int (J - c_{0}) dJ and \\int \\dot{\\mathbf{r}}{(J,c_{0})} dJ = F_{c} + \\frac{J^{2}}{2} - J c_{0} and F_{c} + \\frac{J^{2}}{2} - J c_{0} + J = J + \\int (J - c_{0}) dJ and (F_{c} + \\frac{J^{2}}{2} - J c_{0} + J)^{F_{c}} = (J + \\int (J - c_{0}) dJ)^{F_{c}} and \\int (F_{c} + \\frac{J^{2}}{2} - J c_{0} + J)^{F_{c}} dJ = \\int (J + \\int (J - c_{0}) dJ)^{F_{c}} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["add", 2, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Symbol('J', commutative=True), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))))"], [["power", 5, "Symbol('F_c', commutative=True)"], "Equality(Pow(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Symbol('J', commutative=True)), Symbol('F_c', commutative=True)), Pow(Add(Symbol('J', commutative=True), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))), Symbol('F_c', commutative=True)))"], [["integrate", 6, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('c_0', commutative=True)), Symbol('J', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Add(Symbol('J', commutative=True), Integral(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))), Symbol('F_c', commutative=True)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(\\sigma_x,q)} = \\sin{(\\sigma_x q)}, then obtain \\frac{(\\int \\ddot{x}{(\\sigma_x,q)} dq)^{q}}{\\ddot{x}{(\\sigma_x,q)}} = \\frac{(\\int \\sin{(\\sigma_x q)} dq)^{q}}{\\ddot{x}{(\\sigma_x,q)}}", "derivation": "\\ddot{x}{(\\sigma_x,q)} = \\sin{(\\sigma_x q)} and \\int \\ddot{x}{(\\sigma_x,q)} dq = \\int \\sin{(\\sigma_x q)} dq and (\\int \\ddot{x}{(\\sigma_x,q)} dq)^{q} = (\\int \\sin{(\\sigma_x q)} dq)^{q} and \\frac{(\\int \\ddot{x}{(\\sigma_x,q)} dq)^{q}}{\\ddot{x}{(\\sigma_x,q)}} = \\frac{(\\int \\sin{(\\sigma_x q)} dq)^{q}}{\\ddot{x}{(\\sigma_x,q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), sin(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(sin(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Integral(sin(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["divide", 3, "Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True))), Mul(Pow(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(sin(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given H{(A_{x},\\mathbf{H})} = A_{x} \\mathbf{H}, then derive \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} = \\mathbf{H}, then obtain \\int \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} d\\mathbf{H} = \\Omega + \\frac{\\mathbf{H}^{2}}{2}", "derivation": "H{(A_{x},\\mathbf{H})} = A_{x} \\mathbf{H} and \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\mathbf{H} and \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} = \\mathbf{H} and \\int \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H} d\\mathbf{H} and \\int \\frac{\\partial}{\\partial A_{x}} H{(A_{x},\\mathbf{H})} d\\mathbf{H} = \\Omega + \\frac{\\mathbf{H}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('H')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Derivative(Function('H')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Function('H')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(h)} = \\sin{(h)}, then obtain (\\dot{\\mathbf{r}}{(h)} - \\sin{(h)}) \\int \\sin{(h)} dh = 0", "derivation": "\\dot{\\mathbf{r}}{(h)} = \\sin{(h)} and \\dot{\\mathbf{r}}{(h)} - \\sin{(h)} = 0 and \\int \\dot{\\mathbf{r}}{(h)} dh = \\int \\sin{(h)} dh and (\\dot{\\mathbf{r}}{(h)} - \\sin{(h)}) \\int \\dot{\\mathbf{r}}{(h)} dh = 0 and (\\dot{\\mathbf{r}}{(h)} - \\sin{(h)}) \\int \\sin{(h)} dh = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["minus", 1, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Integer(0))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["times", 2, "Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))"], "Equality(Mul(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('h', commutative=True)), Mul(Integer(-1), sin(Symbol('h', commutative=True)))), Integral(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{A},\\varepsilon)} = \\frac{\\varepsilon}{\\mathbf{A}}, then obtain - \\varepsilon + \\int - \\mathbf{S}{(\\mathbf{A},\\varepsilon)} d\\varepsilon = - \\varepsilon + \\int - \\frac{\\varepsilon}{\\mathbf{A}} d\\varepsilon", "derivation": "\\mathbf{S}{(\\mathbf{A},\\varepsilon)} = \\frac{\\varepsilon}{\\mathbf{A}} and - \\mathbf{S}{(\\mathbf{A},\\varepsilon)} = - \\frac{\\varepsilon}{\\mathbf{A}} and \\int - \\mathbf{S}{(\\mathbf{A},\\varepsilon)} d\\varepsilon = \\int - \\frac{\\varepsilon}{\\mathbf{A}} d\\varepsilon and - \\varepsilon + \\int - \\mathbf{S}{(\\mathbf{A},\\varepsilon)} d\\varepsilon = - \\varepsilon + \\int - \\frac{\\varepsilon}{\\mathbf{A}} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Integral(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Integral(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given W{(T,A_{1})} = A_{1} T, then derive \\frac{\\partial}{\\partial T} W{(T,A_{1})} = A_{1}, then obtain 2 A_{1} = A_{1} + \\frac{\\partial}{\\partial T} A_{1} T", "derivation": "W{(T,A_{1})} = A_{1} T and \\frac{\\partial}{\\partial T} W{(T,A_{1})} = \\frac{\\partial}{\\partial T} A_{1} T and 2 \\frac{\\partial}{\\partial T} W{(T,A_{1})} = \\frac{\\partial}{\\partial T} A_{1} T + \\frac{\\partial}{\\partial T} W{(T,A_{1})} and \\frac{\\partial}{\\partial T} W{(T,A_{1})} = A_{1} and 2 A_{1} = A_{1} + \\frac{\\partial}{\\partial T} A_{1} T", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('A_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('T', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Symbol('A_1', commutative=True))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\chi,f_{E})} = \\frac{\\sin{(\\chi)}}{f_{E}} and \\mathbf{v}{(\\chi,f_{E})} = - \\frac{\\partial}{\\partial \\chi} \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})}, then obtain \\mathbf{v}{(\\chi,f_{E})} + 1 = 1 - \\frac{\\partial}{\\partial \\chi} \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})}", "derivation": "\\operatorname{C_{2}}{(\\chi,f_{E})} = \\frac{\\sin{(\\chi)}}{f_{E}} and \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})} = \\sin{(\\frac{\\sin{(\\chi)}}{f_{E}})} and \\frac{\\partial}{\\partial \\chi} \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})} = \\frac{\\partial}{\\partial \\chi} \\sin{(\\frac{\\sin{(\\chi)}}{f_{E}})} and \\mathbf{v}{(\\chi,f_{E})} = - \\frac{\\partial}{\\partial \\chi} \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})} and \\mathbf{v}{(\\chi,f_{E})} = - \\frac{\\partial}{\\partial \\chi} \\sin{(\\frac{\\sin{(\\chi)}}{f_{E}})} and \\mathbf{v}{(\\chi,f_{E})} + 1 = 1 - \\frac{\\partial}{\\partial \\chi} \\sin{(\\frac{\\sin{(\\chi)}}{f_{E}})} and \\mathbf{v}{(\\chi,f_{E})} + 1 = 1 - \\frac{\\partial}{\\partial \\chi} \\sin{(\\operatorname{C_{2}}{(\\chi,f_{E})})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True))))"], [["sin", 1], "Equality(sin(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True))), sin(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(sin(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(sin(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["add", 5, 1], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Add(Integer(1), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), sin(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True)), Integer(1)), Add(Integer(1), Mul(Integer(-1), Derivative(sin(Function('C_2')(Symbol('\\\\chi', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"]]}, {"prompt": "Given y{(S,E_{x})} = E_{x}^{S}, then obtain \\int (- g^{p} + \\int \\frac{\\partial}{\\partial S} p y{(S,E_{x})} dp) dg = \\int (- g^{p} + \\int \\frac{\\partial}{\\partial S} E_{x}^{S} p dp) dg", "derivation": "y{(S,E_{x})} = E_{x}^{S} and p y{(S,E_{x})} = E_{x}^{S} p and \\frac{\\partial}{\\partial S} p y{(S,E_{x})} = \\frac{\\partial}{\\partial S} E_{x}^{S} p and \\int \\frac{\\partial}{\\partial S} p y{(S,E_{x})} dp = \\int \\frac{\\partial}{\\partial S} E_{x}^{S} p dp and - g^{p} + \\int \\frac{\\partial}{\\partial S} p y{(S,E_{x})} dp = - g^{p} + \\int \\frac{\\partial}{\\partial S} E_{x}^{S} p dp and \\int (- g^{p} + \\int \\frac{\\partial}{\\partial S} p y{(S,E_{x})} dp) dg = \\int (- g^{p} + \\int \\frac{\\partial}{\\partial S} E_{x}^{S} p dp) dg", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Symbol('p', commutative=True), Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('p', commutative=True), Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))))"], [["minus", 4, "Pow(Symbol('g', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('p', commutative=True))), Integral(Derivative(Mul(Symbol('p', commutative=True), Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('p', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))))"], [["integrate", 5, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('p', commutative=True))), Integral(Derivative(Mul(Symbol('p', commutative=True), Function('y')(Symbol('S', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('p', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\omega,\\mathbf{p})} = \\mathbf{p} - \\omega, then obtain - \\int (\\mathbf{p} - \\omega) d\\mathbf{p} + \\int \\varphi^{*}{(\\omega,\\mathbf{p})} d\\mathbf{p} = 0", "derivation": "\\varphi^{*}{(\\omega,\\mathbf{p})} = \\mathbf{p} - \\omega and \\int \\varphi^{*}{(\\omega,\\mathbf{p})} d\\mathbf{p} = \\int (\\mathbf{p} - \\omega) d\\mathbf{p} and - \\mathbf{p} + \\omega + \\int \\varphi^{*}{(\\omega,\\mathbf{p})} d\\mathbf{p} = - \\mathbf{p} + \\omega + \\int (\\mathbf{p} - \\omega) d\\mathbf{p} and - \\varphi^{*}{(\\omega,\\mathbf{p})} + \\int \\varphi^{*}{(\\omega,\\mathbf{p})} d\\mathbf{p} = - \\varphi^{*}{(\\omega,\\mathbf{p})} + \\int (\\mathbf{p} - \\omega) d\\mathbf{p} and - \\int (\\mathbf{p} - \\omega) d\\mathbf{p} + \\int \\varphi^{*}{(\\omega,\\mathbf{p})} d\\mathbf{p} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\omega', commutative=True), Integral(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\omega', commutative=True), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Integral(Function('\\\\varphi^*')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\Omega{(z)} = e^{z} and \\operatorname{A_{2}}{(f^{*})} = e^{f^{*}}, then obtain \\operatorname{A_{2}}{(f^{*})} \\Omega{(z)} = \\Omega{(z)} e^{f^{*}}", "derivation": "\\Omega{(z)} = e^{z} and \\operatorname{A_{2}}{(f^{*})} = e^{f^{*}} and \\operatorname{A_{2}}{(f^{*})} e^{z} = e^{f^{*}} e^{z} and \\operatorname{A_{2}}{(f^{*})} \\Omega{(z)} = \\Omega{(z)} e^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], ["get_premise", "Equality(Function('A_2')(Symbol('f^*', commutative=True)), exp(Symbol('f^*', commutative=True)))"], [["times", 2, "exp(Symbol('z', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('f^*', commutative=True)), exp(Symbol('z', commutative=True))), Mul(exp(Symbol('f^*', commutative=True)), exp(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('A_2')(Symbol('f^*', commutative=True)), Function('\\\\Omega')(Symbol('z', commutative=True))), Mul(Function('\\\\Omega')(Symbol('z', commutative=True)), exp(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(h,F_{c})} = F_{c} + h, then derive \\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})} = 1, then obtain \\cos^{h}{(\\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})})} = \\cos^{h}{(1)}", "derivation": "\\mathbf{P}{(h,F_{c})} = F_{c} + h and \\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})} = \\frac{\\partial}{\\partial F_{c}} (F_{c} + h) and \\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})} = 1 and \\cos{(\\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})})} = \\cos{(1)} and \\cos^{h}{(\\frac{\\partial}{\\partial F_{c}} \\mathbf{P}{(h,F_{c})})} = \\cos^{h}{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('h', commutative=True), Symbol('F_c', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('h', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Symbol('F_c', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('h', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1))"], [["cos", 3], "Equality(cos(Derivative(Function('\\\\mathbf{P}')(Symbol('h', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), cos(Integer(1)))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(cos(Derivative(Function('\\\\mathbf{P}')(Symbol('h', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Symbol('h', commutative=True)), Pow(cos(Integer(1)), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(t)} = e^{t}, then derive \\frac{d}{d t} \\sigma_{x}{(t)} = e^{t}, then obtain \\Psi_{nl} + \\sigma_{x}{(t)} = H + e^{t}", "derivation": "\\sigma_{x}{(t)} = e^{t} and \\frac{d}{d t} \\sigma_{x}{(t)} = \\frac{d}{d t} e^{t} and \\frac{d}{d t} \\sigma_{x}{(t)} = e^{t} and \\int \\frac{d}{d t} \\sigma_{x}{(t)} dt = \\int e^{t} dt and \\Psi_{nl} + \\sigma_{x}{(t)} = H + e^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), exp(Symbol('t', commutative=True)))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\sigma_x')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\sigma_x')(Symbol('t', commutative=True))), Add(Symbol('H', commutative=True), exp(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(\\varepsilon_0,a^{\\dagger})} = a^{\\dagger} \\log{(\\varepsilon_0)} and c{(\\varepsilon_0,a^{\\dagger})} = \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} \\log{(\\varepsilon_0)}, then derive c{(\\varepsilon_0,a^{\\dagger})} = \\log{(\\varepsilon_0)}, then obtain \\phi_{1}{(\\varepsilon_0,a^{\\dagger})} = a^{\\dagger} \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} \\log{(\\varepsilon_0)}", "derivation": "\\phi_{1}{(\\varepsilon_0,a^{\\dagger})} = a^{\\dagger} \\log{(\\varepsilon_0)} and c{(\\varepsilon_0,a^{\\dagger})} = \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} \\log{(\\varepsilon_0)} and c{(\\varepsilon_0,a^{\\dagger})} = \\log{(\\varepsilon_0)} and \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} \\log{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and \\phi_{1}{(\\varepsilon_0,a^{\\dagger})} = a^{\\dagger} \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} \\log{(\\varepsilon_0)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Function('c')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\phi_1')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(y^{\\prime},L)} = \\log{((y^{\\prime})^{L})}, then derive \\int \\operatorname{v_{x}}{(y^{\\prime},L)} dy^{\\prime} = - L y^{\\prime} + t_{2} + y^{\\prime} \\log{((y^{\\prime})^{L})}, then obtain L y^{\\prime} - t_{2} - y^{\\prime} \\log{((y^{\\prime})^{L})} = - \\int \\log{((y^{\\prime})^{L})} dy^{\\prime}", "derivation": "\\operatorname{v_{x}}{(y^{\\prime},L)} = \\log{((y^{\\prime})^{L})} and \\int \\operatorname{v_{x}}{(y^{\\prime},L)} dy^{\\prime} = \\int \\log{((y^{\\prime})^{L})} dy^{\\prime} and \\int \\operatorname{v_{x}}{(y^{\\prime},L)} dy^{\\prime} = - L y^{\\prime} + t_{2} + y^{\\prime} \\log{((y^{\\prime})^{L})} and - \\int \\operatorname{v_{x}}{(y^{\\prime},L)} dy^{\\prime} = - \\int \\log{((y^{\\prime})^{L})} dy^{\\prime} and L y^{\\prime} - t_{2} - y^{\\prime} \\log{((y^{\\prime})^{L})} = - \\int \\log{((y^{\\prime})^{L})} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('t_2', commutative=True), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('v_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Integral(log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True), log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))), Mul(Integer(-1), Integral(log(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given q{(m,v_{2})} = m v_{2}, then obtain \\cos{(\\frac{1}{1 + \\frac{1}{q{(m,v_{2})}}})} = \\cos{(\\frac{m v_{2}}{(1 + \\frac{1}{q{(m,v_{2})}}) q{(m,v_{2})}})}", "derivation": "q{(m,v_{2})} = m v_{2} and 1 = \\frac{m v_{2}}{q{(m,v_{2})}} and 1 + \\frac{1}{q{(m,v_{2})}} = \\frac{m v_{2}}{q{(m,v_{2})}} + \\frac{1}{q{(m,v_{2})}} and \\frac{1}{\\frac{m v_{2}}{q{(m,v_{2})}} + \\frac{1}{q{(m,v_{2})}}} = \\frac{m v_{2}}{(\\frac{m v_{2}}{q{(m,v_{2})}} + \\frac{1}{q{(m,v_{2})}}) q{(m,v_{2})}} and \\frac{1}{1 + \\frac{1}{q{(m,v_{2})}}} = \\frac{m v_{2}}{(1 + \\frac{1}{q{(m,v_{2})}}) q{(m,v_{2})}} and \\cos{(\\frac{1}{1 + \\frac{1}{q{(m,v_{2})}}})} = \\cos{(\\frac{m v_{2}}{(1 + \\frac{1}{q{(m,v_{2})}}) q{(m,v_{2})}})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True)))"], [["divide", 1, "Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Integer(1), Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["add", 2, "Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Add(Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["divide", 2, "Add(Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))"], "Equality(Pow(Add(Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1)), Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1)), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Integer(1), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1)), Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Integer(1), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1)), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))))"], [["cos", 5], "Equality(cos(Pow(Add(Integer(1), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1))), cos(Mul(Symbol('m', commutative=True), Symbol('v_2', commutative=True), Pow(Add(Integer(1), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1))), Integer(-1)), Pow(Function('q')(Symbol('m', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(W,G)} = G W, then derive \\frac{\\partial}{\\partial W} \\mathbf{J}_f{(W,G)} = G, then obtain G = \\frac{\\partial}{\\partial W} G W", "derivation": "\\mathbf{J}_f{(W,G)} = G W and \\frac{\\partial}{\\partial W} \\mathbf{J}_f{(W,G)} = \\frac{\\partial}{\\partial W} G W and \\frac{\\partial}{\\partial W} \\mathbf{J}_f{(W,G)} = G and G = \\frac{\\partial}{\\partial W} G W", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('W', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('G', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('G', commutative=True), Derivative(Mul(Symbol('G', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain M_{E}^{\\Psi_{nl}} + \\frac{\\phi \\operatorname{A_{1}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = M_{E}^{\\Psi_{nl}} + \\frac{\\phi \\sin^{\\dot{z}}{(\\dot{z})}}{\\dot{z}}", "derivation": "\\operatorname{A_{1}}{(\\dot{z})} = \\sin{(\\dot{z})} and \\operatorname{A_{1}}^{\\dot{z}}{(\\dot{z})} = \\sin^{\\dot{z}}{(\\dot{z})} and \\frac{\\operatorname{A_{1}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = \\frac{\\sin^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} and \\frac{\\phi \\operatorname{A_{1}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = \\frac{\\phi \\sin^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} and M_{E}^{\\Psi_{nl}} + \\frac{\\phi \\operatorname{A_{1}}^{\\dot{z}}{(\\dot{z})}}{\\dot{z}} = M_{E}^{\\Psi_{nl}} + \\frac{\\phi \\sin^{\\dot{z}}{(\\dot{z})}}{\\dot{z}}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(Function('A_1')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["times", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Function('A_1')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["add", 4, "Pow(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Pow(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(Function('A_1')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))), Add(Pow(Symbol('M_E', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)} and \\Psi{(\\mathbf{J}_f)} = \\mathbf{J}_f, then obtain \\int \\frac{\\operatorname{M_{E}}{(\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} d\\Psi{(\\mathbf{J}_f)} = \\int 1 d\\Psi{(\\mathbf{J}_f)}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)} and \\frac{\\operatorname{M_{E}}{(\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} = 1 and \\int \\frac{\\operatorname{M_{E}}{(\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} d\\mathbf{J}_f = \\int 1 d\\mathbf{J}_f and \\Psi{(\\mathbf{J}_f)} = \\mathbf{J}_f and \\int \\frac{\\operatorname{M_{E}}{(\\mathbf{J}_f)}}{\\sin{(\\mathbf{J}_f)}} d\\Psi{(\\mathbf{J}_f)} = \\int 1 d\\Psi{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(Function('M_E')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Mul(Function('M_E')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Function('\\\\Psi')(Symbol('\\\\mathbf{J}_f', commutative=True)))), Integral(Integer(1), Tuple(Function('\\\\Psi')(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\nabla)} = \\frac{d}{d \\nabla} \\sin{(\\nabla)}, then obtain \\frac{1}{\\cos{(\\nabla)}} - \\frac{1}{\\mathbf{s}{(\\nabla)}} = 0", "derivation": "\\mathbf{s}{(\\nabla)} = \\frac{d}{d \\nabla} \\sin{(\\nabla)} and 1 = \\frac{\\frac{d}{d \\nabla} \\sin{(\\nabla)}}{\\mathbf{s}{(\\nabla)}} and \\frac{1}{\\frac{d}{d \\nabla} \\sin{(\\nabla)}} = \\frac{1}{\\mathbf{s}{(\\nabla)}} and - 2 \\mathbf{s}{(\\nabla)} + \\frac{1}{\\frac{d}{d \\nabla} \\sin{(\\nabla)}} = - 2 \\mathbf{s}{(\\nabla)} + \\frac{1}{\\mathbf{s}{(\\nabla)}} and \\frac{1}{\\frac{d}{d \\nabla} \\sin{(\\nabla)}} - \\frac{1}{\\mathbf{s}{(\\nabla)}} = 0 and \\frac{1}{\\cos{(\\nabla)}} - \\frac{1}{\\mathbf{s}{(\\nabla)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["divide", 1, "Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["divide", 2, "Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)))"], [["minus", 3, "Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True))), Pow(Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["minus", 4, "Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)))"], "Equality(Add(Pow(Derivative(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)))), Integer(0))"], [["evaluate_derivatives", 5], "Equality(Add(Pow(cos(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Mul(Integer(-1), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given y{(\\phi_1)} = \\cos{(\\phi_1)} and \\operatorname{C_{2}}{(\\phi_1)} = \\cos{(\\phi_1)}, then obtain \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)} + y{(\\phi_1)} \\cos{(\\phi_1)} = 2 \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)}", "derivation": "y{(\\phi_1)} = \\cos{(\\phi_1)} and y^{2}{(\\phi_1)} = y{(\\phi_1)} \\cos{(\\phi_1)} and y^{2}{(\\phi_1)} + y{(\\phi_1)} \\cos{(\\phi_1)} = 2 y{(\\phi_1)} \\cos{(\\phi_1)} and \\operatorname{C_{2}}{(\\phi_1)} = \\cos{(\\phi_1)} and \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)} + y^{2}{(\\phi_1)} = 2 \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)} and \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)} + y{(\\phi_1)} \\cos{(\\phi_1)} = 2 \\operatorname{C_{2}}{(\\phi_1)} y{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Function('y')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Mul(Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"], [["add", 2, "Mul(Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Mul(Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Function('C_2')(Symbol('\\\\phi_1', commutative=True)), Function('y')(Symbol('\\\\phi_1', commutative=True))), Pow(Function('y')(Symbol('\\\\phi_1', commutative=True)), Integer(2))), Mul(Integer(2), Function('C_2')(Symbol('\\\\phi_1', commutative=True)), Function('y')(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Function('C_2')(Symbol('\\\\phi_1', commutative=True)), Function('y')(Symbol('\\\\phi_1', commutative=True))), Mul(Function('y')(Symbol('\\\\phi_1', commutative=True)), cos(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), Function('C_2')(Symbol('\\\\phi_1', commutative=True)), Function('y')(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(C_{1})} = \\sin{(C_{1})}, then derive \\int \\theta_{1}{(C_{1})} dC_{1} = \\mathbb{I} - \\cos{(C_{1})}, then obtain - \\cos{(C_{1})} + \\int \\theta_{1}{(C_{1})} dC_{1} - \\int \\sin{(C_{1})} dC_{1} = \\mathbb{I} - 2 \\cos{(C_{1})} - \\int \\sin{(C_{1})} dC_{1}", "derivation": "\\theta_{1}{(C_{1})} = \\sin{(C_{1})} and \\int \\theta_{1}{(C_{1})} dC_{1} = \\int \\sin{(C_{1})} dC_{1} and \\int \\theta_{1}{(C_{1})} dC_{1} = \\mathbb{I} - \\cos{(C_{1})} and - \\mathbb{I} + \\cos{(C_{1})} + \\int \\theta_{1}{(C_{1})} dC_{1} = 0 and \\int \\theta_{1}{(C_{1})} dC_{1} - \\int \\sin{(C_{1})} dC_{1} = \\mathbb{I} - \\cos{(C_{1})} - \\int \\sin{(C_{1})} dC_{1} and - \\cos{(C_{1})} + \\int \\theta_{1}{(C_{1})} dC_{1} - \\int \\sin{(C_{1})} dC_{1} = \\mathbb{I} - 2 \\cos{(C_{1})} - \\int \\sin{(C_{1})} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))))"], [["minus", 3, "Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('C_1', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Integer(0))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('C_1', commutative=True)), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], "Equality(Add(Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('C_1', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))))"], [["add", 5, "Mul(Integer(-1), cos(Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('C_1', commutative=True))), Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Integer(2), cos(Symbol('C_1', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{v}{(J,\\dot{y})} = J \\dot{y}, then obtain \\frac{\\int J \\dot{y} dJ + \\int \\mathbf{v}{(J,\\dot{y})} dJ}{\\mathbf{v}{(J,\\dot{y})}} = \\frac{2 \\int J \\dot{y} dJ}{\\mathbf{v}{(J,\\dot{y})}}", "derivation": "\\mathbf{v}{(J,\\dot{y})} = J \\dot{y} and \\int \\mathbf{v}{(J,\\dot{y})} dJ = \\int J \\dot{y} dJ and \\int J \\dot{y} dJ + \\int \\mathbf{v}{(J,\\dot{y})} dJ = 2 \\int J \\dot{y} dJ and \\frac{\\int J \\dot{y} dJ + \\int \\mathbf{v}{(J,\\dot{y})} dJ}{J \\dot{y}} = \\frac{2 \\int J \\dot{y} dJ}{J \\dot{y}} and \\frac{\\int J \\dot{y} dJ + \\int \\mathbf{v}{(J,\\dot{y})} dJ}{\\mathbf{v}{(J,\\dot{y})}} = \\frac{2 \\int J \\dot{y} dJ}{\\mathbf{v}{(J,\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["add", 2, "Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["divide", 3, "Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Add(Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))))), Mul(Integer(2), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))), Pow(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(n_{2})} = \\log{(n_{2})}, then obtain (- \\operatorname{L_{\\varepsilon}}{(n_{2})} + \\log{(n_{2})}) (\\operatorname{L_{\\varepsilon}}{(n_{2})} - \\log{(n_{2})}) = 0", "derivation": "\\operatorname{L_{\\varepsilon}}{(n_{2})} = \\log{(n_{2})} and 0 = - \\operatorname{L_{\\varepsilon}}{(n_{2})} + \\log{(n_{2})} and \\operatorname{L_{\\varepsilon}}{(n_{2})} - \\log{(n_{2})} = 0 and (- \\operatorname{L_{\\varepsilon}}{(n_{2})} + \\log{(n_{2})}) (\\operatorname{L_{\\varepsilon}}{(n_{2})} - \\log{(n_{2})}) = 0", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True))), log(Symbol('n_2', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True))), log(Symbol('n_2', commutative=True)))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True)))), Integer(0))"], [["times", 3, "Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True))), log(Symbol('n_2', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True))), log(Symbol('n_2', commutative=True))), Add(Function('L_{\\\\varepsilon}')(Symbol('n_2', commutative=True)), Mul(Integer(-1), log(Symbol('n_2', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\sigma_{p}{(b,\\mathbf{r})} = \\mathbf{r}^{b}, then obtain (\\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} \\sigma_{p}^{\\mathbf{r}}{(b,\\mathbf{r})})^{\\mathbf{r}} = (\\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} (\\mathbf{r}^{b})^{\\mathbf{r}})^{\\mathbf{r}}", "derivation": "\\sigma_{p}{(b,\\mathbf{r})} = \\mathbf{r}^{b} and \\sigma_{p}^{\\mathbf{r}}{(b,\\mathbf{r})} = (\\mathbf{r}^{b})^{\\mathbf{r}} and \\frac{\\partial}{\\partial \\mathbf{r}} \\sigma_{p}^{\\mathbf{r}}{(b,\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} (\\mathbf{r}^{b})^{\\mathbf{r}} and \\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} \\sigma_{p}^{\\mathbf{r}}{(b,\\mathbf{r})} = \\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} (\\mathbf{r}^{b})^{\\mathbf{r}} and (\\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} \\sigma_{p}^{\\mathbf{r}}{(b,\\mathbf{r})})^{\\mathbf{r}} = (\\frac{\\partial^{2}}{\\partial b\\partial \\mathbf{r}} (\\mathbf{r}^{b})^{\\mathbf{r}})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('b', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('b', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('b', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('b', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\sigma_p')(Symbol('b', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('b', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\sigma_p')(Symbol('b', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('b', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\sigma_p')(Symbol('b', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Derivative(Pow(Pow(Symbol('\\\\mathbf{r}', commutative=True), Symbol('b', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(S)} = \\log{(S)}, then derive \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{1}{S}, then obtain 2 \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{d}{d S} \\operatorname{E_{x}}{(S)} + \\frac{1}{S}", "derivation": "\\operatorname{E_{x}}{(S)} = \\log{(S)} and \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{d}{d S} \\log{(S)} and \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{1}{S} and 2 \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{d}{d S} \\operatorname{E_{x}}{(S)} + \\frac{d}{d S} \\log{(S)} and \\frac{d}{d S} \\log{(S)} = \\frac{1}{S} and 2 \\frac{d}{d S} \\operatorname{E_{x}}{(S)} = \\frac{d}{d S} \\operatorname{E_{x}}{(S)} + \\frac{1}{S}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('S', commutative=True), Integer(-1)))"], [["add", 2, "Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('S', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Derivative(Function('E_x')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Pow(Symbol('S', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\omega{(a)} = \\sin{(\\log{(a)})}, then derive \\frac{d}{d a} \\omega{(a)} = \\frac{\\cos{(\\log{(a)})}}{a}, then obtain M \\frac{d}{d a} \\omega{(a)} + \\frac{d}{d a} \\sin{(\\log{(a)})} = M \\frac{d}{d a} \\omega{(a)} + \\frac{\\cos{(\\log{(a)})}}{a}", "derivation": "\\omega{(a)} = \\sin{(\\log{(a)})} and \\frac{d}{d a} \\omega{(a)} = \\frac{d}{d a} \\sin{(\\log{(a)})} and \\frac{d}{d a} \\omega{(a)} = \\frac{\\cos{(\\log{(a)})}}{a} and \\frac{d}{d a} \\sin{(\\log{(a)})} = \\frac{\\cos{(\\log{(a)})}}{a} and M \\frac{d}{d a} \\omega{(a)} + \\frac{d}{d a} \\sin{(\\log{(a)})} = M \\frac{d}{d a} \\omega{(a)} + \\frac{\\cos{(\\log{(a)})}}{a}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\omega')(Symbol('a', commutative=True)), sin(log(Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(log(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(log(Symbol('a', commutative=True)))))"], [["add", 4, "Mul(Symbol('M', commutative=True), Derivative(Function('\\\\omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], "Equality(Add(Mul(Symbol('M', commutative=True), Derivative(Function('\\\\omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Derivative(sin(log(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Symbol('M', commutative=True), Derivative(Function('\\\\omega')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), cos(log(Symbol('a', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\rho_f)} = \\log{(\\cos{(\\rho_f)})}, then obtain (- 2 \\rho_f + \\operatorname{v_{y}}{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} = (- 2 \\rho_f + 2 \\log{(\\cos{(\\rho_f)})})^{\\rho_f}", "derivation": "\\operatorname{v_{y}}{(\\rho_f)} = \\log{(\\cos{(\\rho_f)})} and - \\rho_f + \\operatorname{v_{y}}{(\\rho_f)} = - \\rho_f + \\log{(\\cos{(\\rho_f)})} and - 2 \\rho_f + \\operatorname{v_{y}}{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})} = - 2 \\rho_f + 2 \\log{(\\cos{(\\rho_f)})} and (- 2 \\rho_f + \\operatorname{v_{y}}{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} = (- 2 \\rho_f + 2 \\log{(\\cos{(\\rho_f)})})^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('v_y')(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_f', commutative=True)), Function('v_y')(Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), log(cos(Symbol('\\\\rho_f', commutative=True))))))"], [["power", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_f', commutative=True)), Function('v_y')(Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), log(cos(Symbol('\\\\rho_f', commutative=True))))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(f,g_{\\varepsilon})} = - f + g_{\\varepsilon}, then obtain - 2 f + g_{\\varepsilon} + \\mathbf{F}{(f,g_{\\varepsilon})} = - f + 2 \\mathbf{F}{(f,g_{\\varepsilon})}", "derivation": "\\mathbf{F}{(f,g_{\\varepsilon})} = - f + g_{\\varepsilon} and - f + \\mathbf{F}{(f,g_{\\varepsilon})} = - 2 f + g_{\\varepsilon} and - 2 f + g_{\\varepsilon} + \\mathbf{F}{(f,g_{\\varepsilon})} = - 3 f + 2 g_{\\varepsilon} and - f + 2 \\mathbf{F}{(f,g_{\\varepsilon})} = - 3 f + 2 g_{\\varepsilon} and - 2 f + g_{\\varepsilon} + \\mathbf{F}{(f,g_{\\varepsilon})} = - f + 2 \\mathbf{F}{(f,g_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(3), Symbol('f', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Integer(3), Symbol('f', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then obtain - \\operatorname{v_{2}}{(\\mathbf{g})} + \\int \\operatorname{v_{2}}{(\\mathbf{g})} d\\mathbf{g} = - \\operatorname{v_{2}}{(\\mathbf{g})} + \\int \\sin{(\\mathbf{g})} d\\mathbf{g}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\int \\operatorname{v_{2}}{(\\mathbf{g})} d\\mathbf{g} = \\int \\sin{(\\mathbf{g})} d\\mathbf{g} and - \\sin{(\\mathbf{g})} + \\int \\operatorname{v_{2}}{(\\mathbf{g})} d\\mathbf{g} = - \\sin{(\\mathbf{g})} + \\int \\sin{(\\mathbf{g})} d\\mathbf{g} and - \\operatorname{v_{2}}{(\\mathbf{g})} + \\int \\operatorname{v_{2}}{(\\mathbf{g})} d\\mathbf{g} = - \\operatorname{v_{2}}{(\\mathbf{g})} + \\int \\sin{(\\mathbf{g})} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 2, "sin(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Integer(-1), Function('v_2')(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(v_{1},\\Psi_{nl})} = \\Psi_{nl} v_{1}, then obtain 0 = \\frac{\\lambda^{2}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl} v_{1}} - \\frac{\\lambda^{4}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl}^{3} v_{1}^{3}}", "derivation": "\\lambda{(v_{1},\\Psi_{nl})} = \\Psi_{nl} v_{1} and \\lambda^{2}{(v_{1},\\Psi_{nl})} = \\Psi_{nl} v_{1} \\lambda{(v_{1},\\Psi_{nl})} and \\frac{\\lambda^{2}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl} v_{1}} = \\lambda{(v_{1},\\Psi_{nl})} and 0 = \\lambda{(v_{1},\\Psi_{nl})} - \\frac{\\lambda^{2}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl} v_{1}} and 0 = \\frac{\\lambda^{2}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl} v_{1}} - \\frac{\\lambda^{4}{(v_{1},\\Psi_{nl})}}{\\Psi_{nl}^{3} v_{1}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_1', commutative=True)))"], [["times", 1, "Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_1', commutative=True), Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2))), Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)))"], "Equality(Integer(0), Add(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-3)), Pow(Symbol('v_1', commutative=True), Integer(-3)), Pow(Function('\\\\lambda')(Symbol('v_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(4)))))"]]}, {"prompt": "Given \\mathbf{A}{(y)} = e^{y}, then derive \\int \\mathbf{A}{(y)} dy = F_{H} + e^{y}, then obtain \\frac{d}{d y} \\int e^{y} dy = \\frac{\\partial}{\\partial y} (F_{H} + \\mathbf{A}{(y)})", "derivation": "\\mathbf{A}{(y)} = e^{y} and \\int \\mathbf{A}{(y)} dy = \\int e^{y} dy and \\int \\mathbf{A}{(y)} dy = F_{H} + e^{y} and \\int \\mathbf{A}{(y)} dy = F_{H} + \\mathbf{A}{(y)} and \\int e^{y} dy = F_{H} + \\mathbf{A}{(y)} and \\frac{d}{d y} \\int e^{y} dy = \\frac{\\partial}{\\partial y} (F_{H} + \\mathbf{A}{(y)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('F_H', commutative=True), exp(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{A}')(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{A}')(Symbol('y', commutative=True))))"], [["differentiate", 5, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{A}')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(C,\\sigma_p)} = \\log{(C + \\sigma_p)} and \\eta^{\\prime}{(h,\\tilde{g}^*)} = \\sin{(\\tilde{g}^* + h)}, then obtain \\eta^{\\prime}{(h,\\tilde{g}^*)} + \\operatorname{x^{{\\}'}}{(C,\\sigma_p)} = \\operatorname{x^{{\\}'}}{(C,\\sigma_p)} + \\sin{(\\tilde{g}^* + h)}", "derivation": "\\operatorname{x^{{\\}'}}{(C,\\sigma_p)} = \\log{(C + \\sigma_p)} and \\eta^{\\prime}{(h,\\tilde{g}^*)} = \\sin{(\\tilde{g}^* + h)} and \\eta^{\\prime}{(h,\\tilde{g}^*)} + \\log{(C + \\sigma_p)} = \\log{(C + \\sigma_p)} + \\sin{(\\tilde{g}^* + h)} and \\eta^{\\prime}{(h,\\tilde{g}^*)} + \\operatorname{x^{{\\}'}}{(C,\\sigma_p)} = \\operatorname{x^{{\\}'}}{(C,\\sigma_p)} + \\sin{(\\tilde{g}^* + h)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], ["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True))))"], [["add", 2, "log(Add(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(log(Add(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True))), sin(Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('x^\\\\prime')(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Function('x^\\\\prime')(Symbol('C', commutative=True), Symbol('\\\\sigma_p', commutative=True)), sin(Add(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given I{(\\delta)} = \\log{(\\delta)}, then obtain \\delta + I{(\\delta)} + \\int \\frac{d}{d \\delta} (\\delta + I{(\\delta)} - \\log{(\\delta)}) d\\delta = \\delta + I{(\\delta)} + \\int \\frac{d}{d \\delta} \\delta d\\delta", "derivation": "I{(\\delta)} = \\log{(\\delta)} and \\delta + I{(\\delta)} = \\delta + \\log{(\\delta)} and \\delta + I{(\\delta)} - \\log{(\\delta)} = \\delta and \\frac{d}{d \\delta} (\\delta + I{(\\delta)} - \\log{(\\delta)}) = \\frac{d}{d \\delta} \\delta and \\int \\frac{d}{d \\delta} (\\delta + I{(\\delta)} - \\log{(\\delta)}) d\\delta = \\int \\frac{d}{d \\delta} \\delta d\\delta and \\delta + I{(\\delta)} + \\int \\frac{d}{d \\delta} (\\delta + I{(\\delta)} - \\log{(\\delta)}) d\\delta = \\delta + I{(\\delta)} + \\int \\frac{d}{d \\delta} \\delta d\\delta", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "log(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["add", 5, "Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Integral(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Symbol('\\\\delta', commutative=True), Function('I')(Symbol('\\\\delta', commutative=True)), Integral(Derivative(Symbol('\\\\delta', commutative=True), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(F_{g},\\tilde{g})} = \\frac{\\tilde{g}}{F_{g}}, then obtain - \\tilde{g} + 2 \\phi_{1}{(F_{g},\\tilde{g})} - \\frac{1}{F_{g}} = - \\tilde{g} + \\frac{2 \\tilde{g}}{F_{g}} - \\frac{1}{F_{g}}", "derivation": "\\phi_{1}{(F_{g},\\tilde{g})} = \\frac{\\tilde{g}}{F_{g}} and \\phi_{1}{(F_{g},\\tilde{g})} - \\frac{1}{F_{g}} = \\frac{\\tilde{g}}{F_{g}} - \\frac{1}{F_{g}} and - \\tilde{g} + \\phi_{1}{(F_{g},\\tilde{g})} - \\frac{1}{F_{g}} = - \\tilde{g} + \\frac{\\tilde{g}}{F_{g}} - \\frac{1}{F_{g}} and - \\tilde{g} + \\phi_{1}{(F_{g},\\tilde{g})} + \\frac{\\tilde{g}}{F_{g}} - \\frac{1}{F_{g}} = - \\tilde{g} + \\frac{2 \\tilde{g}}{F_{g}} - \\frac{1}{F_{g}} and - \\tilde{g} + 2 \\phi_{1}{(F_{g},\\tilde{g})} - \\frac{1}{F_{g}} = - \\tilde{g} + \\frac{2 \\tilde{g}}{F_{g}} - \\frac{1}{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)))"], [["minus", 1, "Pow(Symbol('F_g', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))))"], [["minus", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))))"], [["add", 3, "Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Function('\\\\phi_1')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{s},q)} = \\mathbf{s} q and H{(\\mathbf{s},q)} = \\mathbf{s} q, then obtain \\mathbf{s} q + \\mathbf{s} + \\frac{\\partial}{\\partial q} \\mathbf{S}{(\\mathbf{s},q)} = \\mathbf{s} q + \\mathbf{s} + \\frac{\\partial}{\\partial q} H{(\\mathbf{s},q)}", "derivation": "\\mathbf{S}{(\\mathbf{s},q)} = \\mathbf{s} q and \\frac{\\partial}{\\partial q} \\mathbf{S}{(\\mathbf{s},q)} = \\frac{\\partial}{\\partial q} \\mathbf{s} q and H{(\\mathbf{s},q)} = \\mathbf{s} q and \\frac{\\partial}{\\partial q} \\mathbf{S}{(\\mathbf{s},q)} = \\frac{\\partial}{\\partial q} H{(\\mathbf{s},q)} and \\mathbf{s} q + \\mathbf{s} + \\frac{\\partial}{\\partial q} \\mathbf{S}{(\\mathbf{s},q)} = \\mathbf{s} q + \\mathbf{s} + \\frac{\\partial}{\\partial q} H{(\\mathbf{s},q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 4, "Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('H')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(\\theta,\\tilde{g})} = \\tilde{g} e^{\\theta} and \\operatorname{C_{1}}{(\\theta,\\tilde{g})} = \\int G^{\\theta}{(\\theta,\\tilde{g})} d\\theta, then obtain \\frac{\\operatorname{C_{1}}{(\\theta,\\tilde{g})} G^{- \\theta}{(\\theta,\\tilde{g})} e^{- \\theta}}{\\tilde{g}} = \\frac{G^{- \\theta}{(\\theta,\\tilde{g})} e^{- \\theta} \\int (\\tilde{g} e^{\\theta})^{\\theta} d\\theta}{\\tilde{g}}", "derivation": "G{(\\theta,\\tilde{g})} = \\tilde{g} e^{\\theta} and G^{\\theta}{(\\theta,\\tilde{g})} = (\\tilde{g} e^{\\theta})^{\\theta} and \\int G^{\\theta}{(\\theta,\\tilde{g})} d\\theta = \\int (\\tilde{g} e^{\\theta})^{\\theta} d\\theta and \\operatorname{C_{1}}{(\\theta,\\tilde{g})} = \\int G^{\\theta}{(\\theta,\\tilde{g})} d\\theta and \\operatorname{C_{1}}{(\\theta,\\tilde{g})} = \\int (\\tilde{g} e^{\\theta})^{\\theta} d\\theta and \\frac{\\operatorname{C_{1}}{(\\theta,\\tilde{g})} G^{- \\theta}{(\\theta,\\tilde{g})} e^{- \\theta}}{\\tilde{g}} = \\frac{G^{- \\theta}{(\\theta,\\tilde{g})} e^{- \\theta} \\int (\\tilde{g} e^{\\theta})^{\\theta} d\\theta}{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\theta', commutative=True))))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Pow(Mul(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('C_1')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Mul(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["divide", 5, "Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('C_1')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Pow(Function('G')(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(Pow(Mul(Symbol('\\\\tilde{g}', commutative=True), exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{P},p)} = p^{\\mathbf{P}}, then obtain - \\frac{- p + \\operatorname{A_{y}}{(\\mathbf{P},p)}}{p^{2}} = - \\frac{- p + p^{\\mathbf{P}}}{p^{2}}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{P},p)} = p^{\\mathbf{P}} and - p + \\operatorname{A_{y}}{(\\mathbf{P},p)} = - p + p^{\\mathbf{P}} and \\frac{- p + \\operatorname{A_{y}}{(\\mathbf{P},p)}}{p} = \\frac{- p + p^{\\mathbf{P}}}{p} and - \\frac{- p + \\operatorname{A_{y}}{(\\mathbf{P},p)}}{p^{2}} = - \\frac{- p + p^{\\mathbf{P}}}{p^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 1, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('A_y')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 2, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('A_y')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('p', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('A_y')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('p', commutative=True)))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and \\operatorname{A_{1}}{(\\Psi_{nl})} = \\Psi_{nl}, then obtain \\int \\mathbf{v}{(\\Psi_{nl})} d\\operatorname{A_{1}}{(\\Psi_{nl})} = \\int \\cos{(\\Psi_{nl})} d\\operatorname{A_{1}}{(\\Psi_{nl})}", "derivation": "\\mathbf{v}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and \\int \\mathbf{v}{(\\Psi_{nl})} d\\Psi_{nl} = \\int \\cos{(\\Psi_{nl})} d\\Psi_{nl} and \\operatorname{A_{1}}{(\\Psi_{nl})} = \\Psi_{nl} and \\int \\mathbf{v}{(\\Psi_{nl})} d\\operatorname{A_{1}}{(\\Psi_{nl})} = \\int \\cos{(\\Psi_{nl})} d\\operatorname{A_{1}}{(\\Psi_{nl})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Function('A_1')(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Function('A_1')(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(l,\\varphi^*,H)} = \\frac{H + \\varphi^*}{l}, then obtain \\int \\frac{\\partial}{\\partial l} (1 - \\frac{H + \\varphi^*}{l}) dH = \\int \\frac{\\partial}{\\partial l} (1 + \\frac{- H - \\varphi^*}{l}) dH", "derivation": "\\mathbf{B}{(l,\\varphi^*,H)} = \\frac{H + \\varphi^*}{l} and \\mathbf{B}{(l,\\varphi^*,H)} - 1 = -1 + \\frac{H + \\varphi^*}{l} and 1 - \\mathbf{B}{(l,\\varphi^*,H)} = 1 - \\frac{H + \\varphi^*}{l} and 1 - \\mathbf{B}{(l,\\varphi^*,H)} = 1 + \\frac{- H - \\varphi^*}{l} and \\frac{\\partial}{\\partial l} (1 - \\mathbf{B}{(l,\\varphi^*,H)}) = \\frac{\\partial}{\\partial l} (1 + \\frac{- H - \\varphi^*}{l}) and \\frac{\\partial}{\\partial l} (1 - \\frac{H + \\varphi^*}{l}) = \\frac{\\partial}{\\partial l} (1 + \\frac{- H - \\varphi^*}{l}) and \\int \\frac{\\partial}{\\partial l} (1 - \\frac{H + \\varphi^*}{l}) dH = \\int \\frac{\\partial}{\\partial l} (1 + \\frac{- H - \\varphi^*}{l}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('l', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('l', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{v},x^\\prime)} = \\frac{\\mathbf{v}}{x^\\prime} and I{(\\mathbf{v})} = - \\mathbf{v}, then obtain \\frac{d}{d \\mathbf{v}} e^{I{(\\mathbf{v})}} = \\frac{\\partial}{\\partial \\mathbf{v}} e^{- x^\\prime \\hat{H}{(\\mathbf{v},x^\\prime)}}", "derivation": "\\hat{H}{(\\mathbf{v},x^\\prime)} = \\frac{\\mathbf{v}}{x^\\prime} and - \\hat{H}{(\\mathbf{v},x^\\prime)} = - \\frac{\\mathbf{v}}{x^\\prime} and - x^\\prime \\hat{H}{(\\mathbf{v},x^\\prime)} = - \\mathbf{v} and e^{- x^\\prime \\hat{H}{(\\mathbf{v},x^\\prime)}} = e^{- \\mathbf{v}} and I{(\\mathbf{v})} = - \\mathbf{v} and e^{I{(\\mathbf{v})}} = e^{- \\mathbf{v}} and \\frac{d}{d \\mathbf{v}} e^{I{(\\mathbf{v})}} = \\frac{d}{d \\mathbf{v}} e^{- \\mathbf{v}} and \\frac{d}{d \\mathbf{v}} e^{I{(\\mathbf{v})}} = \\frac{\\partial}{\\partial \\mathbf{v}} e^{- x^\\prime \\hat{H}{(\\mathbf{v},x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["divide", 2, "Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))"], [["exp", 3], "Equality(exp(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))"], [["exp", 5], "Equality(exp(Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(exp(Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Derivative(exp(Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(C_{1},A_{1})} = - C_{1} + \\sin{(A_{1})}, then obtain - 2 C_{1} + n{(C_{1},A_{1})} + \\sin{(A_{1})} - 1 = - 3 C_{1} + 2 \\sin{(A_{1})} - 1", "derivation": "n{(C_{1},A_{1})} = - C_{1} + \\sin{(A_{1})} and - C_{1} + n{(C_{1},A_{1})} = - 2 C_{1} + \\sin{(A_{1})} and - 2 C_{1} + n{(C_{1},A_{1})} + \\sin{(A_{1})} = - 3 C_{1} + 2 \\sin{(A_{1})} and - 2 C_{1} + n{(C_{1},A_{1})} + \\sin{(A_{1})} - 1 = - 3 C_{1} + 2 \\sin{(A_{1})} - 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Integer(3), Symbol('C_1', commutative=True)), Mul(Integer(2), sin(Symbol('A_1', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('C_1', commutative=True)), Function('n')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(3), Symbol('C_1', commutative=True)), Mul(Integer(2), sin(Symbol('A_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(f_{\\mathbf{v}})} = f_{\\mathbf{v}}, then derive \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} = 1, then obtain \\iint \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} df_{\\mathbf{v}} = \\iint 1 df_{\\mathbf{v}} df_{\\mathbf{v}}", "derivation": "\\operatorname{m_{s}}{(f_{\\mathbf{v}})} = f_{\\mathbf{v}} and \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} f_{\\mathbf{v}} and \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} = 1 and \\int \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int 1 df_{\\mathbf{v}} and \\iint \\frac{d}{d f_{\\mathbf{v}}} \\operatorname{m_{s}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} df_{\\mathbf{v}} = \\iint 1 df_{\\mathbf{v}} df_{\\mathbf{v}}", "srepr_derivation": [["renaming_premise", "Equality(Function('m_s')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Symbol('f_{\\\\mathbf{v}}', commutative=True), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Derivative(Function('m_s')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["integrate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Derivative(Function('m_s')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\psi{(f^{\\prime},\\hat{x})} = f^{\\prime} + \\cos{(\\hat{x})}, then obtain (\\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} \\psi^{f^{\\prime}}{(f^{\\prime},\\hat{x})})^{\\hat{x}} = (\\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\cos{(\\hat{x})})^{f^{\\prime}})^{\\hat{x}}", "derivation": "\\psi{(f^{\\prime},\\hat{x})} = f^{\\prime} + \\cos{(\\hat{x})} and \\psi^{f^{\\prime}}{(f^{\\prime},\\hat{x})} = (f^{\\prime} + \\cos{(\\hat{x})})^{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} \\psi^{f^{\\prime}}{(f^{\\prime},\\hat{x})} = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\cos{(\\hat{x})})^{f^{\\prime}} and \\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} \\psi^{f^{\\prime}}{(f^{\\prime},\\hat{x})} = \\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\cos{(\\hat{x})})^{f^{\\prime}} and (\\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} \\psi^{f^{\\prime}}{(f^{\\prime},\\hat{x})})^{\\hat{x}} = (\\mathbf{D} + \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\cos{(\\hat{x})})^{f^{\\prime}})^{\\hat{x}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Pow(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Pow(Function('\\\\psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(\\eta,M_{E})} = M_{E} + \\eta, then obtain \\frac{M_{E} \\mathbf{H}^{M_{E}}{(\\eta,M_{E})} \\frac{\\partial}{\\partial \\eta} \\mathbf{H}{(\\eta,M_{E})}}{\\mathbf{H}{(\\eta,M_{E})}} = \\frac{M_{E} (M_{E} + \\eta)^{M_{E}}}{M_{E} + \\eta}", "derivation": "\\mathbf{H}{(\\eta,M_{E})} = M_{E} + \\eta and \\mathbf{H}^{M_{E}}{(\\eta,M_{E})} = (M_{E} + \\eta)^{M_{E}} and \\frac{\\partial}{\\partial \\eta} \\mathbf{H}^{M_{E}}{(\\eta,M_{E})} = \\frac{\\partial}{\\partial \\eta} (M_{E} + \\eta)^{M_{E}} and \\frac{M_{E} \\mathbf{H}^{M_{E}}{(\\eta,M_{E})} \\frac{\\partial}{\\partial \\eta} \\mathbf{H}{(\\eta,M_{E})}}{\\mathbf{H}{(\\eta,M_{E})}} = \\frac{M_{E} (M_{E} + \\eta)^{M_{E}}}{M_{E} + \\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["power", 1, "Symbol('M_E', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('M_E', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('M_E', commutative=True), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Symbol('M_E', commutative=True), Pow(Add(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), Pow(Add(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(c,\\rho_b)} = \\rho_b e^{c} and \\mathbf{B}{(c,\\rho_b)} = 2 \\int \\rho_b e^{c} d\\rho_b, then obtain \\frac{\\partial}{\\partial c} (\\int \\rho_b e^{c} d\\rho_b + \\int \\operatorname{E_{n}}{(c,\\rho_b)} d\\rho_b) = \\frac{\\partial}{\\partial c} \\mathbf{B}{(c,\\rho_b)}", "derivation": "\\operatorname{E_{n}}{(c,\\rho_b)} = \\rho_b e^{c} and \\int \\operatorname{E_{n}}{(c,\\rho_b)} d\\rho_b = \\int \\rho_b e^{c} d\\rho_b and \\int \\rho_b e^{c} d\\rho_b + \\int \\operatorname{E_{n}}{(c,\\rho_b)} d\\rho_b = 2 \\int \\rho_b e^{c} d\\rho_b and \\mathbf{B}{(c,\\rho_b)} = 2 \\int \\rho_b e^{c} d\\rho_b and \\int \\rho_b e^{c} d\\rho_b + \\int \\operatorname{E_{n}}{(c,\\rho_b)} d\\rho_b = \\mathbf{B}{(c,\\rho_b)} and \\frac{\\partial}{\\partial c} (\\int \\rho_b e^{c} d\\rho_b + \\int \\operatorname{E_{n}}{(c,\\rho_b)} d\\rho_b) = \\frac{\\partial}{\\partial c} \\mathbf{B}{(c,\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["add", 2, "Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Function('E_n')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Function('E_n')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Function('\\\\mathbf{B}')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 5, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Integral(Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('c', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Function('E_n')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{B}')(Symbol('c', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(x,k)} = - k + x and \\varphi{(x)} = 2 x, then obtain \\frac{\\frac{d}{d x} \\varphi{(x)}}{x + y{(x,k)}} = \\frac{\\frac{d}{d x} 2 x}{x + y{(x,k)}}", "derivation": "y{(x,k)} = - k + x and x + y{(x,k)} = - k + 2 x and \\varphi{(x)} = 2 x and \\frac{d}{d x} \\varphi{(x)} = \\frac{d}{d x} 2 x and \\frac{\\frac{d}{d x} \\varphi{(x)}}{- k + 2 x} = \\frac{\\frac{d}{d x} 2 x}{- k + 2 x} and \\frac{\\frac{d}{d x} \\varphi{(x)}}{x + y{(x,k)}} = \\frac{\\frac{d}{d x} 2 x}{x + y{(x,k)}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('x', commutative=True), Symbol('k', commutative=True)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('y')(Symbol('x', commutative=True), Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('x', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True))), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True))), Integer(-1)), Derivative(Mul(Integer(2), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Add(Symbol('x', commutative=True), Function('y')(Symbol('x', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('x', commutative=True), Function('y')(Symbol('x', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Derivative(Mul(Integer(2), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - \\dot{y} + \\eta \\mathbb{I}, then obtain - \\dot{y} + 2 \\eta \\mathbb{I} + \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - 2 \\dot{y} + 3 \\eta \\mathbb{I}", "derivation": "\\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - \\dot{y} + \\eta \\mathbb{I} and - \\dot{y} + \\eta \\mathbb{I} + \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - 2 \\dot{y} + 2 \\eta \\mathbb{I} and 2 \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - 2 \\dot{y} + 2 \\eta \\mathbb{I} and \\eta \\mathbb{I} + 2 \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - 2 \\dot{y} + 3 \\eta \\mathbb{I} and 2 \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - \\dot{y} + \\eta \\mathbb{I} + \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} and - \\dot{y} + 2 \\eta \\mathbb{I} + \\operatorname{f^{*}}{(\\mathbb{I},\\eta,\\dot{y})} = - 2 \\dot{y} + 3 \\eta \\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(3), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('f^*')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(3), Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\psi^*,\\tilde{g})} = \\psi^* \\tilde{g}, then obtain \\frac{\\mathbf{F}{(\\psi^*,\\tilde{g})}}{\\psi^* \\tilde{g} + \\frac{1}{\\tilde{g}}} = \\frac{\\psi^* \\tilde{g}}{\\psi^* \\tilde{g} + \\frac{1}{\\tilde{g}}}", "derivation": "\\mathbf{F}{(\\psi^*,\\tilde{g})} = \\psi^* \\tilde{g} and \\mathbf{F}{(\\psi^*,\\tilde{g})} + \\frac{1}{\\tilde{g}} = \\psi^* \\tilde{g} + \\frac{1}{\\tilde{g}} and \\frac{\\mathbf{F}{(\\psi^*,\\tilde{g})}}{\\mathbf{F}{(\\psi^*,\\tilde{g})} + \\frac{1}{\\tilde{g}}} = \\frac{\\psi^* \\tilde{g}}{\\mathbf{F}{(\\psi^*,\\tilde{g})} + \\frac{1}{\\tilde{g}}} and \\frac{\\mathbf{F}{(\\psi^*,\\tilde{g})}}{\\psi^* \\tilde{g} + \\frac{1}{\\tilde{g}}} = \\frac{\\psi^* \\tilde{g}}{\\psi^* \\tilde{g} + \\frac{1}{\\tilde{g}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["divide", 1, "Add(Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Add(Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1))))"]]}, {"prompt": "Given \\eta{(\\rho_f,c_{0},Z)} = (\\rho_f^{Z})^{c_{0}}, then obtain \\int (\\eta^{2}{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} - 1 = \\int ((\\rho_f^{Z})^{c_{0}} \\eta{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} - 1", "derivation": "\\eta{(\\rho_f,c_{0},Z)} = (\\rho_f^{Z})^{c_{0}} and \\eta^{2}{(\\rho_f,c_{0},Z)} = (\\rho_f^{Z})^{c_{0}} \\eta{(\\rho_f,c_{0},Z)} and (\\eta^{2}{(\\rho_f,c_{0},Z)})^{\\rho_f} = ((\\rho_f^{Z})^{c_{0}} \\eta{(\\rho_f,c_{0},Z)})^{\\rho_f} and \\int (\\eta^{2}{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} = \\int ((\\rho_f^{Z})^{c_{0}} \\eta{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} and \\int (\\eta^{2}{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} - 1 = \\int ((\\rho_f^{Z})^{c_{0}} \\eta{(\\rho_f,c_{0},Z)})^{\\rho_f} dc_{0} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Pow(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Symbol('c_0', commutative=True)))"], [["times", 1, "Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True))"], "Equality(Pow(Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Integer(2)), Mul(Pow(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Symbol('c_0', commutative=True)), Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True))))"], [["power", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Pow(Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Integer(2)), Symbol('\\\\rho_f', commutative=True)), Pow(Mul(Pow(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Symbol('c_0', commutative=True)), Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 3, "Symbol('c_0', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Integer(2)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Pow(Mul(Pow(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Symbol('c_0', commutative=True)), Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["minus", 4, 1], "Equality(Add(Integral(Pow(Pow(Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True)), Integer(2)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integer(-1)), Add(Integral(Pow(Mul(Pow(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Symbol('c_0', commutative=True)), Function('\\\\eta')(Symbol('\\\\rho_f', commutative=True), Symbol('c_0', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{B}{(p)} = \\sin{(\\log{(p)})}, then obtain \\frac{\\frac{d}{d p} (\\int \\mathbf{B}{(p)} dp + \\int \\sin{(\\log{(p)})} dp)}{\\frac{d}{d p} 2 \\int \\sin{(\\log{(p)})} dp} = 1", "derivation": "\\mathbf{B}{(p)} = \\sin{(\\log{(p)})} and \\int \\mathbf{B}{(p)} dp = \\int \\sin{(\\log{(p)})} dp and \\int \\mathbf{B}{(p)} dp + \\int \\sin{(\\log{(p)})} dp = 2 \\int \\sin{(\\log{(p)})} dp and \\frac{d}{d p} (\\int \\mathbf{B}{(p)} dp + \\int \\sin{(\\log{(p)})} dp) = \\frac{d}{d p} 2 \\int \\sin{(\\log{(p)})} dp and \\frac{\\frac{d}{d p} (\\int \\mathbf{B}{(p)} dp + \\int \\sin{(\\log{(p)})} dp)}{\\frac{d}{d p} 2 \\int \\sin{(\\log{(p)})} dp} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), sin(log(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Integral(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Mul(Integer(2), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Integral(Function('\\\\mathbf{B}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Pow(Derivative(Mul(Integer(2), Integral(sin(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(F_{c},z)} = \\int F_{c} z dz and \\theta{(F_{c},z)} = z \\int F_{c} z dz, then obtain \\theta{(F_{c},z)} = z \\operatorname{t_{2}}{(F_{c},z)}", "derivation": "\\operatorname{t_{2}}{(F_{c},z)} = \\int F_{c} z dz and z \\operatorname{t_{2}}{(F_{c},z)} = z \\int F_{c} z dz and \\theta{(F_{c},z)} = z \\int F_{c} z dz and \\theta{(F_{c},z)} = z \\operatorname{t_{2}}{(F_{c},z)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('t_2')(Symbol('F_c', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), Integral(Mul(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), Integral(Mul(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\theta')(Symbol('F_c', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), Function('t_2')(Symbol('F_c', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(c)} = \\log{(c)}, then derive - \\frac{1}{c \\log{(c)}^{2}} = - \\frac{\\frac{d}{d c} \\mathbb{I}{(c)}}{\\mathbb{I}^{2}{(c)}}, then obtain - \\frac{1}{c \\log{(c)}^{2}} = - \\frac{\\frac{d}{d c} \\log{(c)}}{\\log{(c)}^{2}}", "derivation": "\\mathbb{I}{(c)} = \\log{(c)} and \\frac{\\mathbb{I}{(c)}}{\\log{(c)}} = 1 and \\frac{1}{\\log{(c)}} = \\frac{1}{\\mathbb{I}{(c)}} and \\frac{d}{d c} \\frac{1}{\\log{(c)}} = \\frac{d}{d c} \\frac{1}{\\mathbb{I}{(c)}} and - \\frac{1}{c \\log{(c)}^{2}} = - \\frac{\\frac{d}{d c} \\mathbb{I}{(c)}}{\\mathbb{I}^{2}{(c)}} and - \\frac{1}{c \\mathbb{I}^{2}{(c)}} = - \\frac{\\frac{d}{d c} \\mathbb{I}{(c)}}{\\mathbb{I}^{2}{(c)}} and - \\frac{1}{c \\log{(c)}^{2}} = - \\frac{\\frac{d}{d c} \\log{(c)}}{\\log{(c)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), log(Symbol('c', commutative=True)))"], [["divide", 1, "log(Symbol('c', commutative=True))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Pow(log(Symbol('c', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Function('\\\\mathbb{I}')(Symbol('c', commutative=True))"], "Equality(Pow(log(Symbol('c', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Pow(log(Symbol('c', commutative=True)), Integer(-1)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Integer(-1)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Pow(log(Symbol('c', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Integer(-2)), Derivative(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Pow(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Integer(-2)), Derivative(Function('\\\\mathbb{I}')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Pow(log(Symbol('c', commutative=True)), Integer(-2))), Mul(Integer(-1), Pow(log(Symbol('c', commutative=True)), Integer(-2)), Derivative(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} and y{(\\hat{p})} = \\int (- \\sin{(\\hat{p})})^{\\hat{p}} d\\hat{p}, then derive \\operatorname{A_{2}}{(\\hat{p})} = - \\sin{(\\hat{p})}, then obtain \\int (\\frac{d}{d \\hat{p}} \\cos{(\\hat{p})})^{\\hat{p}} d\\hat{p} = y{(\\hat{p})}", "derivation": "\\operatorname{A_{2}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\cos{(\\hat{p})} and \\operatorname{A_{2}}{(\\hat{p})} = - \\sin{(\\hat{p})} and \\operatorname{A_{2}}^{\\hat{p}}{(\\hat{p})} = (- \\sin{(\\hat{p})})^{\\hat{p}} and (\\frac{d}{d \\hat{p}} \\cos{(\\hat{p})})^{\\hat{p}} = (- \\sin{(\\hat{p})})^{\\hat{p}} and \\int (\\frac{d}{d \\hat{p}} \\cos{(\\hat{p})})^{\\hat{p}} d\\hat{p} = \\int (- \\sin{(\\hat{p})})^{\\hat{p}} d\\hat{p} and y{(\\hat{p})} = \\int (- \\sin{(\\hat{p})})^{\\hat{p}} d\\hat{p} and \\int (\\frac{d}{d \\hat{p}} \\cos{(\\hat{p})})^{\\hat{p}} d\\hat{p} = y{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{p}', commutative=True)), Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_2')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Pow(Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\hat{p}', commutative=True)), Integral(Pow(Mul(Integer(-1), sin(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Pow(Derivative(cos(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Function('y')(Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\Psi_{nl})} = e^{\\Psi_{nl}} and v{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then obtain \\int \\sin{(v{(\\Psi_{nl})})} d\\Psi_{nl} = \\int \\sin{(e^{\\Psi_{nl}})} d\\Psi_{nl}", "derivation": "\\eta^{\\prime}{(\\Psi_{nl})} = e^{\\Psi_{nl}} and v{(\\Psi_{nl})} = e^{\\Psi_{nl}} and v{(\\Psi_{nl})} = \\eta^{\\prime}{(\\Psi_{nl})} and \\sin{(v{(\\Psi_{nl})})} = \\sin{(\\eta^{\\prime}{(\\Psi_{nl})})} and \\int \\sin{(v{(\\Psi_{nl})})} d\\Psi_{nl} = \\int \\sin{(\\eta^{\\prime}{(\\Psi_{nl})})} d\\Psi_{nl} and \\int \\sin{(v{(\\Psi_{nl})})} d\\Psi_{nl} = \\int \\sin{(e^{\\Psi_{nl}})} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v')(Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["sin", 3], "Equality(sin(Function('v')(Symbol('\\\\Psi_{nl}', commutative=True))), sin(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(sin(Function('v')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(sin(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(sin(Function('v')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(sin(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\hat{x}_0,Q)} = \\cos{(Q \\hat{x}_0)} and \\hat{H}_l{(\\hat{x}_0,Q)} = \\int \\cos{(Q \\hat{x}_0)} dQ, then obtain \\hat{x}_0 + \\int \\hat{p}{(\\hat{x}_0,Q)} dQ = \\hat{x}_0 + \\hat{H}_l{(\\hat{x}_0,Q)}", "derivation": "\\hat{p}{(\\hat{x}_0,Q)} = \\cos{(Q \\hat{x}_0)} and \\int \\hat{p}{(\\hat{x}_0,Q)} dQ = \\int \\cos{(Q \\hat{x}_0)} dQ and \\hat{H}_l{(\\hat{x}_0,Q)} = \\int \\cos{(Q \\hat{x}_0)} dQ and \\hat{x}_0 + \\int \\hat{p}{(\\hat{x}_0,Q)} dQ = \\hat{x}_0 + \\int \\cos{(Q \\hat{x}_0)} dQ and \\hat{x}_0 + \\int \\hat{p}{(\\hat{x}_0,Q)} dQ = \\hat{x}_0 + \\hat{H}_l{(\\hat{x}_0,Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True)), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True)), Integral(cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["add", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Integral(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('\\\\hat{x}_0', commutative=True), Integral(cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Integral(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(t_{2})} = \\log{(t_{2})}, then obtain 0 = - \\mathbf{p}{(t_{2})} + \\log{(t_{2})}", "derivation": "\\mathbf{p}{(t_{2})} = \\log{(t_{2})} and - t_{2} + \\mathbf{p}{(t_{2})} = - t_{2} + \\log{(t_{2})} and - 2 t_{2} + 2 \\mathbf{p}{(t_{2})} = - 2 t_{2} + \\mathbf{p}{(t_{2})} + \\log{(t_{2})} and 0 = - \\mathbf{p}{(t_{2})} + \\log{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True)))"], [["minus", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('t_2', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('t_2', commutative=True))), log(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)} = C_{d} \\rho_b, then obtain e^{\\rho_b + \\frac{\\partial}{\\partial C_{d}} \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)}} = e^{2 \\rho_b}", "derivation": "\\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)} = C_{d} \\rho_b and \\frac{\\partial}{\\partial C_{d}} \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)} = \\frac{\\partial}{\\partial C_{d}} C_{d} \\rho_b and \\rho_b + \\frac{\\partial}{\\partial C_{d}} \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)} = \\rho_b + \\frac{\\partial}{\\partial C_{d}} C_{d} \\rho_b and e^{\\rho_b + \\frac{\\partial}{\\partial C_{d}} \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)}} = e^{\\rho_b + \\frac{\\partial}{\\partial C_{d}} C_{d} \\rho_b} and e^{\\rho_b + \\frac{\\partial}{\\partial C_{d}} \\operatorname{g_{\\varepsilon}}{(C_{d},\\rho_b)}} = e^{2 \\rho_b}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Derivative(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Symbol('\\\\rho_b', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\rho_b', commutative=True), Derivative(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), exp(Add(Symbol('\\\\rho_b', commutative=True), Derivative(Mul(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(exp(Add(Symbol('\\\\rho_b', commutative=True), Derivative(Function('g_{\\\\varepsilon}')(Symbol('C_d', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), exp(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(f^{*})} = \\log{(f^{*})}, then obtain \\dot{y}^{3}{(f^{*})} = \\dot{y}{(f^{*})} \\log{(f^{*})}^{2}", "derivation": "\\dot{y}{(f^{*})} = \\log{(f^{*})} and \\dot{y}^{2}{(f^{*})} = \\dot{y}{(f^{*})} \\log{(f^{*})} and \\dot{y}^{3}{(f^{*})} = \\dot{y}^{2}{(f^{*})} \\log{(f^{*})} and \\dot{y}^{3}{(f^{*})} = \\dot{y}{(f^{*})} \\log{(f^{*})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('f^*', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Integer(2)), Mul(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True))))"], [["times", 1, "Pow(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Integer(2)), log(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Integer(3)), Mul(Function('\\\\dot{y}')(Symbol('f^*', commutative=True)), Pow(log(Symbol('f^*', commutative=True)), Integer(2))))"]]}, {"prompt": "Given t{(\\sigma_x)} = \\cos{(\\sigma_x)}, then obtain t{(\\sigma_x)} \\cos{(\\sigma_x)} + t{(\\sigma_x)} - 2 \\sin{(\\sigma_x)} \\cos{(\\sigma_x)} = t{(\\sigma_x)} \\cos{(\\sigma_x)} - 2 \\sin{(\\sigma_x)} \\cos{(\\sigma_x)} + \\cos{(\\sigma_x)}", "derivation": "t{(\\sigma_x)} = \\cos{(\\sigma_x)} and t{(\\sigma_x)} \\cos{(\\sigma_x)} = \\cos^{2}{(\\sigma_x)} and t{(\\sigma_x)} + \\cos^{2}{(\\sigma_x)} = \\cos^{2}{(\\sigma_x)} + \\cos{(\\sigma_x)} and t{(\\sigma_x)} \\cos{(\\sigma_x)} + t{(\\sigma_x)} = t{(\\sigma_x)} \\cos{(\\sigma_x)} + \\cos{(\\sigma_x)} and t{(\\sigma_x)} \\cos{(\\sigma_x)} + t{(\\sigma_x)} - 2 \\sin{(\\sigma_x)} \\cos{(\\sigma_x)} = t{(\\sigma_x)} \\cos{(\\sigma_x)} - 2 \\sin{(\\sigma_x)} \\cos{(\\sigma_x)} + \\cos{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(2)))"], [["add", 1, "Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(2))"], "Equality(Add(Function('t')(Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(2))), Add(Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Function('t')(Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Integer(2), sin(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Mul(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Function('t')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Function('t')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(a^{\\dagger},k)} = \\cos{(a^{\\dagger} k)}, then obtain (\\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)})^{a^{\\dagger}} + (\\tilde{g}^{2}{(a^{\\dagger},k)})^{a^{\\dagger}} = 2 (\\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)})^{a^{\\dagger}}", "derivation": "\\tilde{g}{(a^{\\dagger},k)} = \\cos{(a^{\\dagger} k)} and \\tilde{g}^{2}{(a^{\\dagger},k)} = \\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)} and (\\tilde{g}^{2}{(a^{\\dagger},k)})^{a^{\\dagger}} = (\\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)})^{a^{\\dagger}} and (\\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)})^{a^{\\dagger}} + (\\tilde{g}^{2}{(a^{\\dagger},k)})^{a^{\\dagger}} = 2 (\\tilde{g}{(a^{\\dagger},k)} \\cos{(a^{\\dagger} k)})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True))))"], [["times", 1, "Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), Integer(2)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 3, "Pow(Mul(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Pow(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), Integer(2)), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), Pow(Mul(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('k', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\pi{(v_{z})} = v_{z} and \\mathbf{P}{(v_{z})} = v_{z} \\pi{(v_{z})} - v_{z}, then obtain \\frac{d}{d v_{z}} \\mathbf{P}^{2}{(v_{z})} = \\frac{d}{d v_{z}} (v_{z} \\pi{(v_{z})} - v_{z}) \\mathbf{P}{(v_{z})}", "derivation": "\\pi{(v_{z})} = v_{z} and \\mathbf{P}{(v_{z})} = v_{z} \\pi{(v_{z})} - v_{z} and \\mathbf{P}{(v_{z})} = v_{z}^{2} - v_{z} and (v_{z}^{2} - v_{z}) \\mathbf{P}{(v_{z})} = (v_{z}^{2} - v_{z}) (v_{z} \\pi{(v_{z})} - v_{z}) and \\frac{d}{d v_{z}} (v_{z}^{2} - v_{z}) \\mathbf{P}{(v_{z})} = \\frac{d}{d v_{z}} (v_{z}^{2} - v_{z}) (v_{z} \\pi{(v_{z})} - v_{z}) and \\frac{d}{d v_{z}} \\mathbf{P}^{2}{(v_{z})} = \\frac{d}{d v_{z}} (v_{z} \\pi{(v_{z})} - v_{z}) \\mathbf{P}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True)), Add(Mul(Symbol('v_z', commutative=True), Function('\\\\pi')(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True)), Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["times", 2, "Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True)))"], "Equality(Mul(Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True))), Mul(Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Add(Mul(Symbol('v_z', commutative=True), Function('\\\\pi')(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True)))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Add(Pow(Symbol('v_z', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Add(Mul(Symbol('v_z', commutative=True), Function('\\\\pi')(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Pow(Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True)), Integer(2)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Symbol('v_z', commutative=True), Function('\\\\pi')(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{P}')(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(A_{2},\\chi)} = e^{- A_{2} + \\chi} and \\hat{x}{(A_{2},\\chi)} = - A_{2} + \\chi, then obtain - A_{2} + \\chi + e^{\\hat{x}{(A_{2},\\chi)}} = - A_{2} + \\chi + e^{- A_{2} + \\chi}", "derivation": "\\operatorname{F_{c}}{(A_{2},\\chi)} = e^{- A_{2} + \\chi} and - A_{2} + \\operatorname{F_{c}}{(A_{2},\\chi)} = - A_{2} + e^{- A_{2} + \\chi} and - A_{2} + \\chi + \\operatorname{F_{c}}{(A_{2},\\chi)} = - A_{2} + \\chi + e^{- A_{2} + \\chi} and \\hat{x}{(A_{2},\\chi)} = - A_{2} + \\chi and \\operatorname{F_{c}}{(A_{2},\\chi)} = e^{\\hat{x}{(A_{2},\\chi)}} and - A_{2} + \\chi + e^{\\hat{x}{(A_{2},\\chi)}} = - A_{2} + \\chi + e^{- A_{2} + \\chi}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('F_c')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)))))"], [["add", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True), Function('F_c')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('F_c')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Function('\\\\hat{x}')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True), exp(Function('\\\\hat{x}')(Symbol('A_2', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\varepsilon)} = \\cos{(\\varepsilon)} and r{(\\varepsilon)} = \\varepsilon, then obtain - \\varepsilon \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} + 1 = - \\varepsilon \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} + \\frac{\\varepsilon}{r{(\\varepsilon)}}", "derivation": "\\nabla{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\nabla{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and r{(\\varepsilon)} = \\varepsilon and 1 = \\frac{\\varepsilon}{r{(\\varepsilon)}} and - \\varepsilon \\frac{d}{d \\varepsilon} \\nabla{(\\varepsilon)} + 1 = - \\varepsilon \\frac{d}{d \\varepsilon} \\nabla{(\\varepsilon)} + \\frac{\\varepsilon}{r{(\\varepsilon)}} and - \\varepsilon \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} + 1 = - \\varepsilon \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} + \\frac{\\varepsilon}{r{(\\varepsilon)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["divide", 3, "Function('r')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('r')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["minus", 4, "Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('r')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Mul(Symbol('\\\\varepsilon', commutative=True), Pow(Function('r')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\nabla)} = \\sin{(\\nabla)}, then obtain - \\nabla + e^{\\nabla} e^{\\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} = - \\nabla + e^{\\nabla}", "derivation": "\\mathbf{D}{(\\nabla)} = \\sin{(\\nabla)} and \\nabla + \\mathbf{D}{(\\nabla)} = \\nabla + \\sin{(\\nabla)} and \\nabla + \\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)} = \\nabla and e^{\\nabla + \\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} = e^{\\nabla} and - \\nabla + e^{\\nabla + \\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} = - \\nabla + e^{\\nabla} and - \\nabla + e^{\\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} e^{\\nabla + \\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} = - \\nabla + e^{\\nabla + \\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} and - \\nabla + e^{\\nabla} e^{\\mathbf{D}{(\\nabla)} - \\sin{(\\nabla)}} = - \\nabla + e^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "sin(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Symbol('\\\\nabla', commutative=True))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))), exp(Symbol('\\\\nabla', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), exp(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(exp(Add(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))), exp(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), exp(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(exp(Symbol('\\\\nabla', commutative=True)), exp(Add(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True))))))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\psi{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})}, then derive \\int \\psi{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\theta_2 + \\sin{(f_{\\mathbf{p}})}, then obtain \\theta_2 + \\sin{(f_{\\mathbf{p}})} = P_{e} + \\sin{(f_{\\mathbf{p}})}", "derivation": "\\psi{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\int \\psi{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}} and \\int \\psi{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\theta_2 + \\sin{(f_{\\mathbf{p}})} and \\theta_2 + \\sin{(f_{\\mathbf{p}})} = \\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}} and \\theta_2 + \\sin{(f_{\\mathbf{p}})} = P_{e} + \\sin{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('P_e', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given p{(\\phi_1)} = \\log{(\\phi_1)}, then obtain \\frac{d}{d \\phi_1} (p{(\\phi_1)} - \\log{(\\phi_1)}^{2}) = \\frac{d}{d \\phi_1} (- \\log{(\\phi_1)}^{2} + \\log{(\\phi_1)})", "derivation": "p{(\\phi_1)} = \\log{(\\phi_1)} and p{(\\phi_1)} \\log{(\\phi_1)} = \\log{(\\phi_1)}^{2} and - p{(\\phi_1)} \\log{(\\phi_1)} + p{(\\phi_1)} = - p{(\\phi_1)} \\log{(\\phi_1)} + \\log{(\\phi_1)} and p{(\\phi_1)} - \\log{(\\phi_1)}^{2} = - \\log{(\\phi_1)}^{2} + \\log{(\\phi_1)} and \\frac{d}{d \\phi_1} (p{(\\phi_1)} - \\log{(\\phi_1)}^{2}) = \\frac{d}{d \\phi_1} (- \\log{(\\phi_1)}^{2} + \\log{(\\phi_1)})", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "log(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('p')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))), Pow(log(Symbol('\\\\phi_1', commutative=True)), Integer(2)))"], [["minus", 1, "Mul(Function('p')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('p')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))), Function('p')(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Function('p')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True))), log(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('p')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\phi_1', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(log(Symbol('\\\\phi_1', commutative=True)), Integer(2))), log(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Add(Function('p')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('\\\\phi_1', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(log(Symbol('\\\\phi_1', commutative=True)), Integer(2))), log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(\\delta)} = e^{\\delta}, then derive \\frac{d}{d \\delta} \\mathbf{F}{(\\delta)} = e^{\\delta}, then obtain \\frac{d}{d \\delta} e^{\\delta} + \\frac{\\frac{d^{2}}{d \\delta^{2}} \\mathbf{F}{(\\delta)}}{\\delta} = \\frac{d}{d \\delta} e^{\\delta} + \\frac{\\mathbf{F}{(\\delta)}}{\\delta}", "derivation": "\\mathbf{F}{(\\delta)} = e^{\\delta} and \\frac{d}{d \\delta} \\mathbf{F}{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta} and \\frac{d}{d \\delta} \\mathbf{F}{(\\delta)} = e^{\\delta} and \\frac{d}{d \\delta} \\mathbf{F}{(\\delta)} = \\frac{d^{2}}{d \\delta^{2}} \\mathbf{F}{(\\delta)} and \\frac{\\frac{d}{d \\delta} \\mathbf{F}{(\\delta)}}{\\delta} = \\frac{e^{\\delta}}{\\delta} and \\frac{\\frac{d}{d \\delta} \\mathbf{F}{(\\delta)}}{\\delta} = \\frac{\\mathbf{F}{(\\delta)}}{\\delta} and \\frac{\\frac{d^{2}}{d \\delta^{2}} \\mathbf{F}{(\\delta)}}{\\delta} = \\frac{\\mathbf{F}{(\\delta)}}{\\delta} and \\frac{d}{d \\delta} e^{\\delta} + \\frac{\\frac{d^{2}}{d \\delta^{2}} \\mathbf{F}{(\\delta)}}{\\delta} = \\frac{d}{d \\delta} e^{\\delta} + \\frac{\\mathbf{F}{(\\delta)}}{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), exp(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))"], [["divide", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True))))"], [["add", 7, "Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Add(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2))))), Add(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\omega,v)} = - \\omega + v, then obtain - \\sin{(\\frac{\\operatorname{x^{{\\}'}}{(\\omega,v)}}{\\omega^{2}})} = \\sin{(\\frac{\\omega - v}{\\omega^{2}})}", "derivation": "\\operatorname{x^{{\\}'}}{(\\omega,v)} = - \\omega + v and \\frac{\\operatorname{x^{{\\}'}}{(\\omega,v)}}{\\omega} = \\frac{- \\omega + v}{\\omega} and - \\frac{\\operatorname{x^{{\\}'}}{(\\omega,v)}}{\\omega^{2}} = - \\frac{- \\omega + v}{\\omega^{2}} and - \\frac{\\operatorname{x^{{\\}'}}{(\\omega,v)}}{\\omega^{2}} = \\frac{\\omega - v}{\\omega^{2}} and - \\sin{(\\frac{\\operatorname{x^{{\\}'}}{(\\omega,v)}}{\\omega^{2}})} = \\sin{(\\frac{\\omega - v}{\\omega^{2}})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\omega', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v', commutative=True)))"], [["divide", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\omega', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('x^\\\\prime')(Symbol('\\\\omega', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('x^\\\\prime')(Symbol('\\\\omega', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('x^\\\\prime')(Symbol('\\\\omega', commutative=True), Symbol('v', commutative=True))))), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{p})} = \\log{(\\mathbf{p})}, then obtain \\cos{(\\frac{d}{d \\mathbf{p}} \\int 0 d\\mathbf{p})} = \\cos{(\\frac{d}{d \\mathbf{p}} \\int (- \\mathbf{J}{(\\mathbf{p})} + \\log{(\\mathbf{p})}) d\\mathbf{p})}", "derivation": "\\mathbf{J}{(\\mathbf{p})} = \\log{(\\mathbf{p})} and 0 = - \\mathbf{J}{(\\mathbf{p})} + \\log{(\\mathbf{p})} and \\int 0 d\\mathbf{p} = \\int (- \\mathbf{J}{(\\mathbf{p})} + \\log{(\\mathbf{p})}) d\\mathbf{p} and \\frac{d}{d \\mathbf{p}} \\int 0 d\\mathbf{p} = \\frac{d}{d \\mathbf{p}} \\int (- \\mathbf{J}{(\\mathbf{p})} + \\log{(\\mathbf{p})}) d\\mathbf{p} and \\cos{(\\frac{d}{d \\mathbf{p}} \\int 0 d\\mathbf{p})} = \\cos{(\\frac{d}{d \\mathbf{p}} \\int (- \\mathbf{J}{(\\mathbf{p})} + \\log{(\\mathbf{p})}) d\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True)), log(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), cos(Derivative(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True))), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain - \\mathbf{A} + \\int G{(\\mathbf{A})} d\\mathbf{A} - 1 = - \\mathbf{A} + \\int \\cos{(\\mathbf{A})} d\\mathbf{A} - 1", "derivation": "G{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\int G{(\\mathbf{A})} d\\mathbf{A} = \\int \\cos{(\\mathbf{A})} d\\mathbf{A} and - \\mathbf{A} + \\int G{(\\mathbf{A})} d\\mathbf{A} = - \\mathbf{A} + \\int \\cos{(\\mathbf{A})} d\\mathbf{A} and - \\mathbf{A} + \\int G{(\\mathbf{A})} d\\mathbf{A} - 1 = - \\mathbf{A} + \\int \\cos{(\\mathbf{A})} d\\mathbf{A} - 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('G')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('G')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\theta_{2}{(r_{0},s)} = r_{0} s and U{(r_{0},s)} = \\frac{1}{\\theta_{2}{(r_{0},s)}}, then obtain - \\theta_{2}{(r_{0},s)} + \\frac{1}{\\theta_{2}{(r_{0},s)}} = U{(r_{0},s)} - \\theta_{2}{(r_{0},s)}", "derivation": "\\theta_{2}{(r_{0},s)} = r_{0} s and U{(r_{0},s)} = \\frac{1}{\\theta_{2}{(r_{0},s)}} and U{(r_{0},s)} = \\frac{1}{r_{0} s} and \\frac{1}{r_{0} s} = \\frac{1}{\\theta_{2}{(r_{0},s)}} and - \\theta_{2}{(r_{0},s)} + \\frac{1}{r_{0} s} = - \\theta_{2}{(r_{0},s)} + \\frac{1}{\\theta_{2}{(r_{0},s)}} and - \\theta_{2}{(r_{0},s)} + \\frac{1}{r_{0} s} = U{(r_{0},s)} - \\theta_{2}{(r_{0},s)} and - \\theta_{2}{(r_{0},s)} + \\frac{1}{\\theta_{2}{(r_{0},s)}} = U{(r_{0},s)} - \\theta_{2}{(r_{0},s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('r_0', commutative=True), Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('U')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('U')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1))), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Integer(-1)))"], [["minus", 4, "Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)))), Add(Function('U')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True))), Pow(Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Integer(-1))), Add(Function('U')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(F_{H})} = \\log{(F_{H})}, then derive \\frac{\\int \\operatorname{F_{c}}{(F_{H})} dF_{H}}{\\log{(F_{H})}} = \\frac{F_{H} \\log{(F_{H})} - F_{H} + \\psi^*}{\\log{(F_{H})}}, then obtain \\int \\frac{\\int \\operatorname{F_{c}}{(F_{H})} dF_{H}}{\\log{(F_{H})}} dF_{H} = \\int \\frac{F_{H} \\log{(F_{H})} - F_{H} + \\psi^*}{\\log{(F_{H})}} dF_{H}", "derivation": "\\operatorname{F_{c}}{(F_{H})} = \\log{(F_{H})} and \\int \\operatorname{F_{c}}{(F_{H})} dF_{H} = \\int \\log{(F_{H})} dF_{H} and \\frac{\\int \\operatorname{F_{c}}{(F_{H})} dF_{H}}{\\log{(F_{H})}} = \\frac{\\int \\log{(F_{H})} dF_{H}}{\\log{(F_{H})}} and \\frac{\\int \\operatorname{F_{c}}{(F_{H})} dF_{H}}{\\log{(F_{H})}} = \\frac{F_{H} \\log{(F_{H})} - F_{H} + \\psi^*}{\\log{(F_{H})}} and \\int \\frac{\\int \\operatorname{F_{c}}{(F_{H})} dF_{H}}{\\log{(F_{H})}} dF_{H} = \\int \\frac{F_{H} \\log{(F_{H})} - F_{H} + \\psi^*}{\\log{(F_{H})}} dF_{H}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["divide", 2, "log(Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(log(Symbol('F_H', commutative=True)), Integer(-1)), Integral(Function('F_c')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Pow(log(Symbol('F_H', commutative=True)), Integer(-1)), Integral(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(log(Symbol('F_H', commutative=True)), Integer(-1)), Integral(Function('F_c')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Add(Mul(Symbol('F_H', commutative=True), log(Symbol('F_H', commutative=True))), Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(log(Symbol('F_H', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Pow(log(Symbol('F_H', commutative=True)), Integer(-1)), Integral(Function('F_c')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Add(Mul(Symbol('F_H', commutative=True), log(Symbol('F_H', commutative=True))), Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(log(Symbol('F_H', commutative=True)), Integer(-1))), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(m)} = \\log{(m)}, then derive \\frac{d}{d m} \\mathbf{M}{(m)} = \\frac{1}{m}, then obtain \\frac{d}{d m} \\mathbf{M}{(m)} + \\frac{1}{m} = \\frac{2}{m}", "derivation": "\\mathbf{M}{(m)} = \\log{(m)} and \\frac{d}{d m} \\mathbf{M}{(m)} = \\frac{d}{d m} \\log{(m)} and \\frac{d}{d m} \\mathbf{M}{(m)} + \\frac{d}{d m} \\log{(m)} = 2 \\frac{d}{d m} \\log{(m)} and \\frac{d}{d m} \\mathbf{M}{(m)} = \\frac{1}{m} and \\frac{d}{d m} \\log{(m)} = \\frac{1}{m} and \\frac{d}{d m} \\mathbf{M}{(m)} + \\frac{1}{m} = \\frac{2}{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["add", 2, "Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Derivative(Function('\\\\mathbf{M}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('m', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{2})} = \\cos{(v_{2})} and V{(g,\\mathbf{J}_f,v_{2})} = - \\frac{\\cos{(v_{2})}}{(\\mathbf{J}_f - g)^{2}}, then derive - \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} = - \\frac{\\cos{(v_{2})}}{(\\mathbf{J}_f - g)^{2}}, then obtain - \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} = V{(g,\\mathbf{J}_f,v_{2})}", "derivation": "\\operatorname{x^{{\\}'}}{(v_{2})} = \\cos{(v_{2})} and \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{\\mathbf{J}_f - g} = \\frac{\\cos{(v_{2})}}{\\mathbf{J}_f - g} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{\\mathbf{J}_f - g} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\cos{(v_{2})}}{\\mathbf{J}_f - g} and - \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} = - \\frac{\\cos{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} and V{(g,\\mathbf{J}_f,v_{2})} = - \\frac{\\cos{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} and - \\frac{\\operatorname{x^{{\\}'}}{(v_{2})}}{(\\mathbf{J}_f - g)^{2}} = V{(g,\\mathbf{J}_f,v_{2})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-1)), Function('x^\\\\prime')(Symbol('v_2', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-1)), cos(Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-1)), Function('x^\\\\prime')(Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-1)), cos(Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-2)), Function('x^\\\\prime')(Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-2)), cos(Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-2)), cos(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Integer(-2)), Function('x^\\\\prime')(Symbol('v_2', commutative=True))), Function('V')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(J,E_{x})} = - E_{x} + J, then obtain - \\frac{E_{x} ((\\operatorname{F_{x}}^{E_{x}}{(J,E_{x})})^{E_{x}} + \\frac{1}{E_{x}})}{\\operatorname{F_{x}}{(J,E_{x})}} = - \\frac{E_{x} (((- E_{x} + J)^{E_{x}})^{E_{x}} + \\frac{1}{E_{x}})}{\\operatorname{F_{x}}{(J,E_{x})}}", "derivation": "\\operatorname{F_{x}}{(J,E_{x})} = - E_{x} + J and \\operatorname{F_{x}}^{E_{x}}{(J,E_{x})} = (- E_{x} + J)^{E_{x}} and (\\operatorname{F_{x}}^{E_{x}}{(J,E_{x})})^{E_{x}} = ((- E_{x} + J)^{E_{x}})^{E_{x}} and (\\operatorname{F_{x}}^{E_{x}}{(J,E_{x})})^{E_{x}} + \\frac{1}{E_{x}} = ((- E_{x} + J)^{E_{x}})^{E_{x}} + \\frac{1}{E_{x}} and - \\frac{E_{x} ((\\operatorname{F_{x}}^{E_{x}}{(J,E_{x})})^{E_{x}} + \\frac{1}{E_{x}})}{\\operatorname{F_{x}}{(J,E_{x})}} = - \\frac{E_{x} (((- E_{x} + J)^{E_{x}})^{E_{x}} + \\frac{1}{E_{x}})}{\\operatorname{F_{x}}{(J,E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True)), Symbol('E_x', commutative=True)))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["add", 3, "Pow(Symbol('E_x', commutative=True), Integer(-1))"], "Equality(Add(Pow(Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1))), Add(Pow(Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('E_x', commutative=True), Add(Pow(Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1))), Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('E_x', commutative=True), Add(Pow(Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('J', commutative=True)), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1))), Pow(Function('F_x')(Symbol('J', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\nabla{(G)} = e^{G}, then derive \\int \\nabla{(G)} dG = L_{\\varepsilon} + e^{G}, then obtain \\int e^{G} dG = L_{\\varepsilon} + e^{G}", "derivation": "\\nabla{(G)} = e^{G} and \\int \\nabla{(G)} dG = \\int e^{G} dG and \\int \\nabla{(G)} dG = L_{\\varepsilon} + e^{G} and \\int e^{G} dG = L_{\\varepsilon} + e^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), exp(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(V,G)} = G V, then obtain \\frac{\\partial}{\\partial V} V \\sin{(\\operatorname{J_{\\varepsilon}}{(V,G)})} = \\frac{\\partial}{\\partial V} V \\sin{(G V)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(V,G)} = G V and \\sin{(\\operatorname{J_{\\varepsilon}}{(V,G)})} = \\sin{(G V)} and V \\sin{(\\operatorname{J_{\\varepsilon}}{(V,G)})} = V \\sin{(G V)} and \\frac{\\partial}{\\partial V} V \\sin{(\\operatorname{J_{\\varepsilon}}{(V,G)})} = \\frac{\\partial}{\\partial V} V \\sin{(G V)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('V', commutative=True)))"], [["sin", 1], "Equality(sin(Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('G', commutative=True))), sin(Mul(Symbol('G', commutative=True), Symbol('V', commutative=True))))"], [["times", 2, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), sin(Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('G', commutative=True)))), Mul(Symbol('V', commutative=True), sin(Mul(Symbol('G', commutative=True), Symbol('V', commutative=True)))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Symbol('V', commutative=True), sin(Function('J_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), sin(Mul(Symbol('G', commutative=True), Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(S,\\Psi)} = \\frac{S}{\\Psi}, then obtain - \\operatorname{t_{1}}^{\\Psi}{(S,\\Psi)} + e^{(\\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S \\Psi})^{S}} = - \\operatorname{t_{1}}^{\\Psi}{(S,\\Psi)} + e^{(\\frac{1}{\\Psi^{2}})^{S}}", "derivation": "\\operatorname{t_{1}}{(S,\\Psi)} = \\frac{S}{\\Psi} and \\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S} = \\frac{1}{\\Psi} and \\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S \\Psi} = \\frac{1}{\\Psi^{2}} and (\\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S \\Psi})^{S} = (\\frac{1}{\\Psi^{2}})^{S} and e^{(\\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S \\Psi})^{S}} = e^{(\\frac{1}{\\Psi^{2}})^{S}} and - \\operatorname{t_{1}}^{\\Psi}{(S,\\Psi)} + e^{(\\frac{\\operatorname{t_{1}}{(S,\\Psi)}}{S \\Psi})^{S}} = - \\operatorname{t_{1}}^{\\Psi}{(S,\\Psi)} + e^{(\\frac{1}{\\Psi^{2}})^{S}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True))), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)))"], [["times", 2, "Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True))), Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('S', commutative=True)), Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Symbol('S', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('S', commutative=True))), exp(Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Symbol('S', commutative=True))))"], [["minus", 5, "Pow(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), exp(Pow(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('t_1')(Symbol('S', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), exp(Pow(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\omega)} = \\cos{(\\sin{(\\omega)})}, then obtain - \\sin{(\\omega)} + \\cos{(\\sin{(\\omega)})} = \\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)}", "derivation": "\\operatorname{E_{n}}{(\\omega)} = \\cos{(\\sin{(\\omega)})} and \\int \\operatorname{E_{n}}{(\\omega)} d\\omega = \\int \\cos{(\\sin{(\\omega)})} d\\omega and 0 = - \\int \\operatorname{E_{n}}{(\\omega)} d\\omega + \\int \\cos{(\\sin{(\\omega)})} d\\omega and \\cos{(\\sin{(\\omega)})} = \\cos{(\\sin{(\\omega)})} - \\int \\operatorname{E_{n}}{(\\omega)} d\\omega + \\int \\cos{(\\sin{(\\omega)})} d\\omega and - \\sin{(\\omega)} + \\cos{(\\sin{(\\omega)})} = - \\sin{(\\omega)} + \\cos{(\\sin{(\\omega)})} - \\int \\operatorname{E_{n}}{(\\omega)} d\\omega + \\int \\cos{(\\sin{(\\omega)})} d\\omega and \\operatorname{E_{n}}{(\\omega)} = \\cos{(\\sin{(\\omega)})} - \\int \\operatorname{E_{n}}{(\\omega)} d\\omega + \\int \\cos{(\\sin{(\\omega)})} d\\omega and - \\sin{(\\omega)} + \\cos{(\\sin{(\\omega)})} = \\operatorname{E_{n}}{(\\omega)} - \\sin{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\omega', commutative=True)), cos(sin(Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(cos(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(cos(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["add", 3, "cos(sin(Symbol('\\\\omega', commutative=True)))"], "Equality(cos(sin(Symbol('\\\\omega', commutative=True))), Add(cos(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(cos(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["minus", 4, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), cos(sin(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), cos(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(cos(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('E_n')(Symbol('\\\\omega', commutative=True)), Add(cos(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Integral(Function('E_n')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(cos(sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), cos(sin(Symbol('\\\\omega', commutative=True)))), Add(Function('E_n')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given B{(\\hat{p}_0,\\varphi^*,s)} = - \\hat{p}_0 + \\varphi^* - s, then obtain (B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{B{(\\hat{p}_0,\\varphi^*,s)}}{\\varphi^*})^{\\hat{p}_0} = (B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{- \\hat{p}_0 + \\varphi^* - s}{\\varphi^*})^{\\hat{p}_0}", "derivation": "B{(\\hat{p}_0,\\varphi^*,s)} = - \\hat{p}_0 + \\varphi^* - s and \\frac{B{(\\hat{p}_0,\\varphi^*,s)}}{\\varphi^*} = \\frac{- \\hat{p}_0 + \\varphi^* - s}{\\varphi^*} and B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{B{(\\hat{p}_0,\\varphi^*,s)}}{\\varphi^*} = B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{- \\hat{p}_0 + \\varphi^* - s}{\\varphi^*} and (B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{B{(\\hat{p}_0,\\varphi^*,s)}}{\\varphi^*})^{\\hat{p}_0} = (B{(\\hat{p}_0,\\varphi^*,s)} + \\frac{- \\hat{p}_0 + \\varphi^* - s}{\\varphi^*})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["divide", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["add", 2, "Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))), Add(Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))))"], [["power", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Add(Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Function('B')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\ddot{x})} = \\sin{(\\ddot{x})}, then obtain \\int (\\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} \\frac{\\eta^{\\prime}{(\\ddot{x})}}{\\sin{(\\ddot{x})}}) d\\ddot{x} = \\int (\\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} 1) d\\ddot{x}", "derivation": "\\eta^{\\prime}{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\frac{\\eta^{\\prime}{(\\ddot{x})}}{\\sin{(\\ddot{x})}} = 1 and \\frac{d}{d \\ddot{x}} \\frac{\\eta^{\\prime}{(\\ddot{x})}}{\\sin{(\\ddot{x})}} = \\frac{d}{d \\ddot{x}} 1 and \\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} \\frac{\\eta^{\\prime}{(\\ddot{x})}}{\\sin{(\\ddot{x})}} = \\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} 1 and \\int (\\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} \\frac{\\eta^{\\prime}{(\\ddot{x})}}{\\sin{(\\ddot{x})}}) d\\ddot{x} = \\int (\\sin{(\\ddot{x})} + \\frac{d}{d \\ddot{x}} 1) d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["add", 3, "sin(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(sin(Symbol('\\\\ddot{x}', commutative=True)), Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\ddot{x}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Add(sin(Symbol('\\\\ddot{x}', commutative=True)), Derivative(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(sin(Symbol('\\\\ddot{x}', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\hbar)} = \\cos{(\\hbar)}, then obtain \\mathbf{p}^{2}{(\\hbar)} \\cos{(\\hbar)} + \\cos{(\\hbar)} = \\mathbf{p}{(\\hbar)} \\cos^{2}{(\\hbar)} + \\cos{(\\hbar)}", "derivation": "\\mathbf{p}{(\\hbar)} = \\cos{(\\hbar)} and \\mathbf{p}^{2}{(\\hbar)} = \\mathbf{p}{(\\hbar)} \\cos{(\\hbar)} and \\mathbf{p}^{3}{(\\hbar)} = \\mathbf{p}^{2}{(\\hbar)} \\cos{(\\hbar)} and \\mathbf{p}^{3}{(\\hbar)} = \\mathbf{p}{(\\hbar)} \\cos^{2}{(\\hbar)} and \\mathbf{p}^{2}{(\\hbar)} \\cos{(\\hbar)} = \\mathbf{p}{(\\hbar)} \\cos^{2}{(\\hbar)} and \\mathbf{p}^{2}{(\\hbar)} \\cos{(\\hbar)} + \\cos{(\\hbar)} = \\mathbf{p}{(\\hbar)} \\cos^{2}{(\\hbar)} + \\cos{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(2)), cos(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(3)), Mul(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(2)), cos(Symbol('\\\\hbar', commutative=True))), Mul(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2))))"], [["add", 5, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Integer(2)), cos(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Add(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\hbar', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(2))), cos(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\rho_f)} = \\cos{(\\rho_f)}, then obtain \\cos{(\\rho_f)} = \\operatorname{E_{x}}^{- \\rho_f}{(\\rho_f)} \\cos{(\\rho_f)} \\cos^{\\rho_f}{(\\rho_f)}", "derivation": "\\operatorname{E_{x}}{(\\rho_f)} = \\cos{(\\rho_f)} and \\operatorname{E_{x}}^{\\rho_f}{(\\rho_f)} = \\cos^{\\rho_f}{(\\rho_f)} and \\operatorname{E_{x}}^{\\rho_f}{(\\rho_f)} \\cos{(\\rho_f)} = \\cos{(\\rho_f)} \\cos^{\\rho_f}{(\\rho_f)} and \\cos{(\\rho_f)} = \\operatorname{E_{x}}^{- \\rho_f}{(\\rho_f)} \\cos{(\\rho_f)} \\cos^{\\rho_f}{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["times", 2, "cos(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Mul(cos(Symbol('\\\\rho_f', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))))"], [["divide", 3, "Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], "Equality(cos(Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{J}_M)} = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M, then obtain \\mathbf{J}_M \\operatorname{v_{2}}^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M (\\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M)^{2}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{J}_M)} = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\mathbf{J}_M \\operatorname{v_{2}}{(\\mathbf{J}_M)} = \\mathbf{J}_M \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\mathbf{J}_M \\operatorname{v_{2}}^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M \\operatorname{v_{2}}{(\\mathbf{J}_M)} \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\mathbf{J}_M \\operatorname{v_{2}}{(\\mathbf{J}_M)} \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\mathbf{J}_M (\\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M)^{2} and \\mathbf{J}_M \\operatorname{v_{2}}^{2}{(\\mathbf{J}_M)} = \\mathbf{J}_M (\\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M)^{2}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["times", 2, "Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{H}{(Z)} = \\log{(Z)} and \\operatorname{C_{2}}{(Z)} = \\log{(Z)}, then obtain \\log{(Z)}^{Z} = \\mathbf{H}^{Z}{(Z)}", "derivation": "\\mathbf{H}{(Z)} = \\log{(Z)} and \\mathbf{H}{(Z)} \\log{(Z)} = \\log{(Z)}^{2} and \\operatorname{C_{2}}{(Z)} = \\log{(Z)} and \\log{(Z)} = \\frac{\\log{(Z)}^{2}}{\\mathbf{H}{(Z)}} and \\operatorname{C_{2}}{(Z)} = \\mathbf{H}{(Z)} and \\operatorname{C_{2}}^{Z}{(Z)} = \\mathbf{H}^{Z}{(Z)} and \\operatorname{C_{2}}{(Z)} = \\frac{\\log{(Z)}^{2}}{\\mathbf{H}{(Z)}} and (\\frac{\\log{(Z)}^{2}}{\\mathbf{H}{(Z)}})^{Z} = \\mathbf{H}^{Z}{(Z)} and \\log{(Z)}^{Z} = \\mathbf{H}^{Z}{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["times", 1, "log(Symbol('Z', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["divide", 2, "Function('\\\\mathbf{H}')(Symbol('Z', commutative=True))"], "Equality(log(Symbol('Z', commutative=True)), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), Pow(log(Symbol('Z', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('C_2')(Symbol('Z', commutative=True)), Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)))"], [["power", 5, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Function('C_2')(Symbol('Z', commutative=True)), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), Pow(log(Symbol('Z', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Pow(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 2], "Equality(Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(Q)} = \\int \\log{(Q)} dQ, then derive \\mathbf{F}{(Q)} = Q \\log{(Q)} - Q + n_{1}, then obtain - 2 Q + \\log{(Q \\log{(Q)} - Q + n_{1})}^{Q} \\log{(\\int \\log{(Q)} dQ)}^{Q} = - 2 Q + \\log{(\\int \\log{(Q)} dQ)}^{2 Q}", "derivation": "\\mathbf{F}{(Q)} = \\int \\log{(Q)} dQ and \\log{(\\mathbf{F}{(Q)})} = \\log{(\\int \\log{(Q)} dQ)} and \\log{(\\mathbf{F}{(Q)})}^{Q} = \\log{(\\int \\log{(Q)} dQ)}^{Q} and \\log{(\\mathbf{F}{(Q)})}^{Q} \\log{(\\int \\log{(Q)} dQ)}^{Q} = \\log{(\\int \\log{(Q)} dQ)}^{2 Q} and \\mathbf{F}{(Q)} = Q \\log{(Q)} - Q + n_{1} and - 2 Q + \\log{(\\mathbf{F}{(Q)})}^{Q} \\log{(\\int \\log{(Q)} dQ)}^{Q} = - 2 Q + \\log{(\\int \\log{(Q)} dQ)}^{2 Q} and - 2 Q + \\log{(Q \\log{(Q)} - Q + n_{1})}^{Q} \\log{(\\int \\log{(Q)} dQ)}^{Q} = - 2 Q + \\log{(\\int \\log{(Q)} dQ)}^{2 Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True)), Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True))), log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(log(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["times", 3, "Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(log(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True))), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Integer(2), Symbol('Q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True)), Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('n_1', commutative=True)))"], [["minus", 4, "Mul(Integer(2), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('Q', commutative=True)), Mul(Pow(log(Function('\\\\mathbf{F}')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Integer(2), Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('Q', commutative=True)), Mul(Pow(log(Add(Mul(Symbol('Q', commutative=True), log(Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('n_1', commutative=True))), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('Q', commutative=True)), Pow(log(Integral(log(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Integer(2), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(I)} = \\cos{(I)}, then derive \\int \\mathbf{M}{(I)} dI = C + \\sin{(I)}, then obtain - \\int \\cos{(I)} dI + \\frac{\\frac{\\partial}{\\partial I} (C + \\sin{(I)} - \\int \\cos{(I)} dI)}{\\cos{(I)}} = - \\int \\cos{(I)} dI + \\frac{\\frac{d}{d I} 0}{\\cos{(I)}}", "derivation": "\\mathbf{M}{(I)} = \\cos{(I)} and \\int \\mathbf{M}{(I)} dI = \\int \\cos{(I)} dI and \\int \\mathbf{M}{(I)} dI = C + \\sin{(I)} and C + \\sin{(I)} = \\int \\cos{(I)} dI and C + \\sin{(I)} - \\int \\cos{(I)} dI = 0 and \\frac{\\partial}{\\partial I} (C + \\sin{(I)} - \\int \\cos{(I)} dI) = \\frac{d}{d I} 0 and \\frac{\\frac{\\partial}{\\partial I} (C + \\sin{(I)} - \\int \\cos{(I)} dI)}{\\cos{(I)}} = \\frac{\\frac{d}{d I} 0}{\\cos{(I)}} and - \\int \\cos{(I)} dI + \\frac{\\frac{\\partial}{\\partial I} (C + \\sin{(I)} - \\int \\cos{(I)} dI)}{\\cos{(I)}} = - \\int \\cos{(I)} dI + \\frac{\\frac{d}{d I} 0}{\\cos{(I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True))), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["minus", 4, "Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Integer(0))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["divide", 6, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["minus", 7, "Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Add(Symbol('C', commutative=True), sin(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Tuple(Symbol('I', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Integral(cos(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(cos(Symbol('I', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('I', commutative=True), Integer(1))))))"]]}, {"prompt": "Given f{(\\mu)} = \\sin{(\\mu)}, then obtain (\\log{(f{(\\mu)})} \\log{(\\sin{(\\mu)})})^{\\mu} = (\\log{(\\sin{(\\mu)})}^{2})^{\\mu}", "derivation": "f{(\\mu)} = \\sin{(\\mu)} and \\log{(f{(\\mu)})} = \\log{(\\sin{(\\mu)})} and \\log{(f{(\\mu)})} \\log{(\\sin{(\\mu)})} = \\log{(\\sin{(\\mu)})}^{2} and (\\log{(f{(\\mu)})} \\log{(\\sin{(\\mu)})})^{\\mu} = (\\log{(\\sin{(\\mu)})}^{2})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["log", 1], "Equality(log(Function('f')(Symbol('\\\\mu', commutative=True))), log(sin(Symbol('\\\\mu', commutative=True))))"], [["times", 2, "log(sin(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(log(Function('f')(Symbol('\\\\mu', commutative=True))), log(sin(Symbol('\\\\mu', commutative=True)))), Pow(log(sin(Symbol('\\\\mu', commutative=True))), Integer(2)))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(log(Function('f')(Symbol('\\\\mu', commutative=True))), log(sin(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Pow(Pow(log(sin(Symbol('\\\\mu', commutative=True))), Integer(2)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given l{(f)} = f, then obtain (l{(f)} \\int l{(f)} dl{(f)})^{f} = (l{(f)} \\int f dl{(f)})^{f}", "derivation": "l{(f)} = f and \\int l{(f)} df = \\int f df and \\int l{(f)} dl{(f)} = \\int f dl{(f)} and l{(f)} \\int l{(f)} dl{(f)} = l{(f)} \\int f dl{(f)} and (l{(f)} \\int l{(f)} dl{(f)})^{f} = (l{(f)} \\int f dl{(f)})^{f}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('f', commutative=True)), Symbol('f', commutative=True))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Function('l')(Symbol('f', commutative=True)))), Integral(Symbol('f', commutative=True), Tuple(Function('l')(Symbol('f', commutative=True)))))"], [["times", 3, "Function('l')(Symbol('f', commutative=True))"], "Equality(Mul(Function('l')(Symbol('f', commutative=True)), Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Function('l')(Symbol('f', commutative=True))))), Mul(Function('l')(Symbol('f', commutative=True)), Integral(Symbol('f', commutative=True), Tuple(Function('l')(Symbol('f', commutative=True))))))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Mul(Function('l')(Symbol('f', commutative=True)), Integral(Function('l')(Symbol('f', commutative=True)), Tuple(Function('l')(Symbol('f', commutative=True))))), Symbol('f', commutative=True)), Pow(Mul(Function('l')(Symbol('f', commutative=True)), Integral(Symbol('f', commutative=True), Tuple(Function('l')(Symbol('f', commutative=True))))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given B{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} = 2 V_{\\mathbf{E}}, then obtain B{(V_{\\mathbf{E}})} + \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} = \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} + \\cos{(V_{\\mathbf{E}})}", "derivation": "B{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and V_{\\mathbf{E}} + B{(V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})} and 2 V_{\\mathbf{E}} + B{(V_{\\mathbf{E}})} = 2 V_{\\mathbf{E}} + \\cos{(V_{\\mathbf{E}})} and \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} = 2 V_{\\mathbf{E}} and B{(V_{\\mathbf{E}})} + \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} = \\operatorname{a^{\\dagger}}{(V_{\\mathbf{E}})} + \\cos{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('B')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["add", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('B')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(2), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('B')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('a^{\\\\dagger}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('a^{\\\\dagger}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\dot{y})} = e^{\\dot{y}}, then obtain (\\nabla{(\\dot{y})} + 2 e^{\\dot{y}}) \\nabla{(\\dot{y})} = (\\nabla{(\\dot{y})} + 2 e^{\\dot{y}}) e^{\\dot{y}}", "derivation": "\\nabla{(\\dot{y})} = e^{\\dot{y}} and \\nabla{(\\dot{y})} + e^{\\dot{y}} = 2 e^{\\dot{y}} and 2 \\nabla{(\\dot{y})} + e^{\\dot{y}} = \\nabla{(\\dot{y})} + 2 e^{\\dot{y}} and (2 \\nabla{(\\dot{y})} + e^{\\dot{y}}) \\nabla{(\\dot{y})} = (2 \\nabla{(\\dot{y})} + e^{\\dot{y}}) e^{\\dot{y}} and (\\nabla{(\\dot{y})} + 2 e^{\\dot{y}}) \\nabla{(\\dot{y})} = (\\nabla{(\\dot{y})} + 2 e^{\\dot{y}}) e^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 2, "Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('\\\\dot{y}', commutative=True))), Add(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["times", 1, "Add(Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('\\\\dot{y}', commutative=True))), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), Mul(Add(Mul(Integer(2), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('\\\\dot{y}', commutative=True))), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\dot{y}', commutative=True)))), Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True))), Mul(Add(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\dot{y}', commutative=True)))), exp(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(b)} = \\log{(b)} and \\hat{x}_0{(b)} = \\log{(b)}, then obtain - 2 \\operatorname{A_{1}}{(b)} \\hat{x}_0{(b)} \\log{(\\operatorname{A_{1}}{(b)})} = - 2 \\operatorname{A_{1}}^{2}{(b)} \\log{(\\operatorname{A_{1}}{(b)})}", "derivation": "\\operatorname{A_{1}}{(b)} = \\log{(b)} and \\log{(\\operatorname{A_{1}}{(b)})} = \\log{(\\log{(b)})} and 2 \\log{(\\operatorname{A_{1}}{(b)})} = \\log{(\\operatorname{A_{1}}{(b)})} + \\log{(\\log{(b)})} and \\hat{x}_0{(b)} = \\log{(b)} and \\hat{x}_0{(b)} = \\operatorname{A_{1}}{(b)} and - \\operatorname{A_{1}}{(b)} \\hat{x}_0{(b)} = - \\operatorname{A_{1}}^{2}{(b)} and - (\\log{(\\operatorname{A_{1}}{(b)})} + \\log{(\\log{(b)})}) \\operatorname{A_{1}}{(b)} \\hat{x}_0{(b)} = - (\\log{(\\operatorname{A_{1}}{(b)})} + \\log{(\\log{(b)})}) \\operatorname{A_{1}}^{2}{(b)} and - 2 \\operatorname{A_{1}}{(b)} \\hat{x}_0{(b)} \\log{(\\operatorname{A_{1}}{(b)})} = - 2 \\operatorname{A_{1}}^{2}{(b)} \\log{(\\operatorname{A_{1}}{(b)})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["log", 1], "Equality(log(Function('A_1')(Symbol('b', commutative=True))), log(log(Symbol('b', commutative=True))))"], [["add", 2, "log(Function('A_1')(Symbol('b', commutative=True)))"], "Equality(Mul(Integer(2), log(Function('A_1')(Symbol('b', commutative=True)))), Add(log(Function('A_1')(Symbol('b', commutative=True))), log(log(Symbol('b', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), Function('A_1')(Symbol('b', commutative=True)))"], [["times", 5, "Mul(Integer(-1), Function('A_1')(Symbol('b', commutative=True)))"], "Equality(Mul(Integer(-1), Function('A_1')(Symbol('b', commutative=True)), Function('\\\\hat{x}_0')(Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Function('A_1')(Symbol('b', commutative=True)), Integer(2))))"], [["times", 6, "Add(log(Function('A_1')(Symbol('b', commutative=True))), log(log(Symbol('b', commutative=True))))"], "Equality(Mul(Integer(-1), Add(log(Function('A_1')(Symbol('b', commutative=True))), log(log(Symbol('b', commutative=True)))), Function('A_1')(Symbol('b', commutative=True)), Function('\\\\hat{x}_0')(Symbol('b', commutative=True))), Mul(Integer(-1), Add(log(Function('A_1')(Symbol('b', commutative=True))), log(log(Symbol('b', commutative=True)))), Pow(Function('A_1')(Symbol('b', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Integer(-1), Integer(2), Function('A_1')(Symbol('b', commutative=True)), Function('\\\\hat{x}_0')(Symbol('b', commutative=True)), log(Function('A_1')(Symbol('b', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Function('A_1')(Symbol('b', commutative=True)), Integer(2)), log(Function('A_1')(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(a,i)} = a + i and p{(a,i)} = a + i + \\bar{\\h}{(a,i)}, then obtain \\int 2 \\bar{\\h}{(a,i)} di = \\int (a + i + \\bar{\\h}{(a,i)}) di", "derivation": "\\bar{\\h}{(a,i)} = a + i and 2 \\bar{\\h}{(a,i)} = a + i + \\bar{\\h}{(a,i)} and p{(a,i)} = a + i + \\bar{\\h}{(a,i)} and p{(a,i)} = 2 \\bar{\\h}{(a,i)} and \\int p{(a,i)} di = \\int (a + i + \\bar{\\h}{(a,i)}) di and \\int 2 \\bar{\\h}{(a,i)} di = \\int (a + i + \\bar{\\h}{(a,i)}) di", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Add(Symbol('a', commutative=True), Symbol('i', commutative=True)))"], [["add", 1, "Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Add(Symbol('a', commutative=True), Symbol('i', commutative=True), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Add(Symbol('a', commutative=True), Symbol('i', commutative=True), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('p')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Mul(Integer(2), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Function('p')(Symbol('a', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('a', commutative=True), Symbol('i', commutative=True), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Mul(Integer(2), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('a', commutative=True), Symbol('i', commutative=True), Function('\\\\hbar')(Symbol('a', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(I)} = e^{I}, then obtain \\frac{(\\psi^{*}^{2}{(I)})^{I}}{I} = \\frac{(\\psi^{*}{(I)} e^{I})^{I}}{I}", "derivation": "\\psi^{*}{(I)} = e^{I} and \\psi^{*}^{2}{(I)} = \\psi^{*}{(I)} e^{I} and (\\psi^{*}^{2}{(I)})^{I} = (\\psi^{*}{(I)} e^{I})^{I} and \\frac{(\\psi^{*}^{2}{(I)})^{I}}{I} = \\frac{(\\psi^{*}{(I)} e^{I})^{I}}{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["times", 1, "Function('\\\\psi^*')(Symbol('I', commutative=True))"], "Equality(Pow(Function('\\\\psi^*')(Symbol('I', commutative=True)), Integer(2)), Mul(Function('\\\\psi^*')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('I', commutative=True)), Integer(2)), Symbol('I', commutative=True)), Pow(Mul(Function('\\\\psi^*')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["divide", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\psi^*')(Symbol('I', commutative=True)), Integer(2)), Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Mul(Function('\\\\psi^*')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given E{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)}, then obtain 1 = \\frac{\\frac{d}{d x^\\prime} \\log{(x^\\prime)}}{E{(x^\\prime)}}", "derivation": "E{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and E^{x^\\prime}{(x^\\prime)} = (\\frac{d}{d x^\\prime} \\log{(x^\\prime)})^{x^\\prime} and E{(x^\\prime)} E^{x^\\prime}{(x^\\prime)} = E^{x^\\prime}{(x^\\prime)} \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and E{(x^\\prime)} (\\frac{d}{d x^\\prime} \\log{(x^\\prime)})^{x^\\prime} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} (\\frac{d}{d x^\\prime} \\log{(x^\\prime)})^{x^\\prime} and 1 = \\frac{\\frac{d}{d x^\\prime} \\log{(x^\\prime)}}{E{(x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('x^\\\\prime', commutative=True)), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('E')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Pow(Function('E')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Function('E')(Symbol('x^\\\\prime', commutative=True)), Pow(Function('E')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Function('E')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('E')(Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))), Mul(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True))))"], [["divide", 4, "Mul(Function('E')(Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('E')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(c)} = \\cos{(c)}, then derive \\int (c + \\varepsilon_{0}{(c)}) dc = \\frac{c^{2}}{2} + v_{x} + \\sin{(c)}, then obtain \\mathbf{J} + \\frac{\\int c^{2} \\int c dc dc}{2} + \\frac{\\int c^{2} \\int \\varepsilon_{0}{(c)} dc dc}{2} = \\int \\frac{c^{2} (\\frac{c^{2}}{2} + v_{x} + \\sin{(c)})}{2} dc", "derivation": "\\varepsilon_{0}{(c)} = \\cos{(c)} and c + \\varepsilon_{0}{(c)} = c + \\cos{(c)} and \\int (c + \\varepsilon_{0}{(c)}) dc = \\int (c + \\cos{(c)}) dc and \\int (c + \\varepsilon_{0}{(c)}) dc = \\frac{c^{2}}{2} + v_{x} + \\sin{(c)} and \\frac{c^{2} \\int (c + \\varepsilon_{0}{(c)}) dc}{2} = \\frac{c^{2} (\\frac{c^{2}}{2} + v_{x} + \\sin{(c)})}{2} and \\int \\frac{c^{2} \\int (c + \\varepsilon_{0}{(c)}) dc}{2} dc = \\int \\frac{c^{2} (\\frac{c^{2}}{2} + v_{x} + \\sin{(c)})}{2} dc and \\mathbf{J} + \\frac{\\int c^{2} \\int c dc dc}{2} + \\frac{\\int c^{2} \\int \\varepsilon_{0}{(c)} dc dc}{2} = \\int \\frac{c^{2} (\\frac{c^{2}}{2} + v_{x} + \\sin{(c)})}{2} dc", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\varepsilon_0')(Symbol('c', commutative=True))), Add(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Symbol('c', commutative=True), Function('\\\\varepsilon_0')(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('c', commutative=True), Function('\\\\varepsilon_0')(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2))), Symbol('v_x', commutative=True), sin(Symbol('c', commutative=True))))"], [["times", 4, "Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Integral(Add(Symbol('c', commutative=True), Function('\\\\varepsilon_0')(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Add(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2))), Symbol('v_x', commutative=True), sin(Symbol('c', commutative=True)))))"], [["integrate", 5, "Symbol('c', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Integral(Add(Symbol('c', commutative=True), Function('\\\\varepsilon_0')(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Integral(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Add(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2))), Symbol('v_x', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Rational(1, 2), Add(Integral(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Integral(Symbol('c', commutative=True), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Integral(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Integral(Function('\\\\varepsilon_0')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True)))))), Integral(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2)), Add(Mul(Rational(1, 2), Pow(Symbol('c', commutative=True), Integer(2))), Symbol('v_x', commutative=True), sin(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(b,E_{n})} = E_{n}^{b} and \\operatorname{f_{E}}{(a,k)} = a + k, then obtain (- E_{n}^{b} + \\operatorname{J_{\\varepsilon}}{(b,E_{n})}) \\operatorname{f_{E}}{(a,k)} = 0", "derivation": "\\operatorname{J_{\\varepsilon}}{(b,E_{n})} = E_{n}^{b} and - E_{n}^{b} + \\operatorname{J_{\\varepsilon}}{(b,E_{n})} = 0 and \\operatorname{f_{E}}{(a,k)} = a + k and (- E_{n}^{b} + \\operatorname{J_{\\varepsilon}}{(b,E_{n})}) (a + k) = 0 and (- E_{n}^{b} + \\operatorname{J_{\\varepsilon}}{(b,E_{n})}) \\operatorname{f_{E}}{(a,k)} = 0", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('b', commutative=True)))"], [["minus", 1, "Pow(Symbol('E_n', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Symbol('b', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('E_n', commutative=True))), Integer(0))"], ["get_premise", "Equality(Function('f_E')(Symbol('a', commutative=True), Symbol('k', commutative=True)), Add(Symbol('a', commutative=True), Symbol('k', commutative=True)))"], [["times", 2, "Add(Symbol('a', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Symbol('b', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('E_n', commutative=True))), Add(Symbol('a', commutative=True), Symbol('k', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Symbol('b', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('E_n', commutative=True))), Function('f_E')(Symbol('a', commutative=True), Symbol('k', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\dot{x}{(b,x)} = x e^{b}, then derive \\int (b + \\dot{x}{(b,x)}) dx = b x + h + \\frac{x^{2} e^{b}}{2}, then obtain \\frac{\\partial}{\\partial b} \\int (b + \\dot{x}{(b,x)}) dx = \\frac{\\partial}{\\partial b} (b x + h + \\frac{x^{2} e^{b}}{2})", "derivation": "\\dot{x}{(b,x)} = x e^{b} and b + \\dot{x}{(b,x)} = b + x e^{b} and \\int (b + \\dot{x}{(b,x)}) dx = \\int (b + x e^{b}) dx and \\int (b + \\dot{x}{(b,x)}) dx = b x + h + \\frac{x^{2} e^{b}}{2} and \\frac{\\partial}{\\partial b} \\int (b + \\dot{x}{(b,x)}) dx = \\frac{\\partial}{\\partial b} (b x + h + \\frac{x^{2} e^{b}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('x', commutative=True), exp(Symbol('b', commutative=True))))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('x', commutative=True))), Add(Symbol('b', commutative=True), Mul(Symbol('x', commutative=True), exp(Symbol('b', commutative=True)))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Symbol('b', commutative=True), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Add(Symbol('b', commutative=True), Mul(Symbol('x', commutative=True), exp(Symbol('b', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('b', commutative=True), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Add(Mul(Symbol('b', commutative=True), Symbol('x', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)), exp(Symbol('b', commutative=True)))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('b', commutative=True), Function('\\\\dot{x}')(Symbol('b', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('b', commutative=True), Symbol('x', commutative=True)), Symbol('h', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x', commutative=True), Integer(2)), exp(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(A)} = \\frac{d}{d A} \\cos{(A)}, then derive \\delta{(A)} = - \\sin{(A)}, then obtain (\\delta^{A}{(A)})^{A} + \\delta{(A)} = ((- \\sin{(A)})^{A})^{A} + \\delta{(A)}", "derivation": "\\delta{(A)} = \\frac{d}{d A} \\cos{(A)} and \\delta{(A)} = - \\sin{(A)} and \\delta^{A}{(A)} = (- \\sin{(A)})^{A} and (\\delta^{A}{(A)})^{A} = ((- \\sin{(A)})^{A})^{A} and (\\delta^{A}{(A)})^{A} + \\delta{(A)} = ((- \\sin{(A)})^{A})^{A} + \\delta{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('A', commutative=True)), Derivative(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\delta')(Symbol('A', commutative=True)), Mul(Integer(-1), sin(Symbol('A', commutative=True))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Pow(Function('\\\\delta')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["add", 4, "Function('\\\\delta')(Symbol('A', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\delta')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Function('\\\\delta')(Symbol('A', commutative=True))), Add(Pow(Pow(Mul(Integer(-1), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Function('\\\\delta')(Symbol('A', commutative=True))))"]]}, {"prompt": "Given y{(t_{1},H)} = \\log{(H^{t_{1}})}, then obtain \\frac{\\partial}{\\partial H} \\log{(H^{t_{1}})} \\int \\frac{\\partial}{\\partial t_{1}} y{(t_{1},H)} dH = \\frac{\\partial}{\\partial H} \\log{(H^{t_{1}})} \\int \\frac{\\partial}{\\partial t_{1}} \\log{(H^{t_{1}})} dH", "derivation": "y{(t_{1},H)} = \\log{(H^{t_{1}})} and \\frac{\\partial}{\\partial t_{1}} y{(t_{1},H)} = \\frac{\\partial}{\\partial t_{1}} \\log{(H^{t_{1}})} and \\int \\frac{\\partial}{\\partial t_{1}} y{(t_{1},H)} dH = \\int \\frac{\\partial}{\\partial t_{1}} \\log{(H^{t_{1}})} dH and \\frac{\\partial}{\\partial H} \\log{(H^{t_{1}})} \\int \\frac{\\partial}{\\partial t_{1}} y{(t_{1},H)} dH = \\frac{\\partial}{\\partial H} \\log{(H^{t_{1}})} \\int \\frac{\\partial}{\\partial t_{1}} \\log{(H^{t_{1}})} dH", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t_1', commutative=True), Symbol('H', commutative=True)), log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"], [["times", 3, "Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(Function('y')(Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)))), Mul(Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integral(Derivative(log(Pow(Symbol('H', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\sigma_{x}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} - 1 and k{(\\mathbb{I})} = \\mathbf{S}{(\\mathbb{I})} - 1, then obtain \\sigma_{x}{(\\mathbb{I})} = k{(\\mathbb{I})}", "derivation": "\\mathbf{S}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\mathbf{S}{(\\mathbb{I})} - 1 = \\cos{(\\mathbb{I})} - 1 and \\sigma_{x}{(\\mathbb{I})} = \\cos{(\\mathbb{I})} - 1 and k{(\\mathbb{I})} = \\mathbf{S}{(\\mathbb{I})} - 1 and k{(\\mathbb{I})} = \\cos{(\\mathbb{I})} - 1 and \\sigma_{x}{(\\mathbb{I})} = k{(\\mathbb{I})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbb{I}', commutative=True)), Add(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), Add(Function('\\\\mathbf{S}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('k')(Symbol('\\\\mathbb{I}', commutative=True)), Add(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbb{I}', commutative=True)), Function('k')(Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(A)} = A, then obtain (\\frac{A \\frac{d}{d A} \\mathbf{A}{(A)}}{\\mathbf{A}{(A)}} + \\log{(\\mathbf{A}{(A)})}) \\mathbf{A}^{A}{(A)} = A^{A} (\\log{(A)} + 1)", "derivation": "\\mathbf{A}{(A)} = A and \\mathbf{A}^{A}{(A)} = A^{A} and \\frac{d}{d A} \\mathbf{A}^{A}{(A)} = \\frac{d}{d A} A^{A} and (\\frac{A \\frac{d}{d A} \\mathbf{A}{(A)}}{\\mathbf{A}{(A)}} + \\log{(\\mathbf{A}{(A)})}) \\mathbf{A}^{A}{(A)} = A^{A} (\\log{(A)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Symbol('A', commutative=True))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('A', commutative=True)))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Symbol('A', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('A', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), log(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)))), Pow(Function('\\\\mathbf{A}')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Symbol('A', commutative=True)), Add(log(Symbol('A', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} = J_{\\varepsilon} - \\sigma_x, then derive \\sigma_x \\int \\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} dJ_{\\varepsilon} = \\sigma_x (\\frac{J_{\\varepsilon}^{2}}{2} - J_{\\varepsilon} \\sigma_x + \\Omega), then obtain \\sigma_x \\int (J_{\\varepsilon} - \\sigma_x) dJ_{\\varepsilon} = \\sigma_x (\\frac{J_{\\varepsilon}^{2}}{2} - J_{\\varepsilon} \\sigma_x + \\Omega)", "derivation": "\\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} = J_{\\varepsilon} - \\sigma_x and \\int \\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (J_{\\varepsilon} - \\sigma_x) dJ_{\\varepsilon} and \\sigma_x \\int \\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} dJ_{\\varepsilon} = \\sigma_x \\int (J_{\\varepsilon} - \\sigma_x) dJ_{\\varepsilon} and \\sigma_x \\int \\mathbf{J}{(\\sigma_x,J_{\\varepsilon})} dJ_{\\varepsilon} = \\sigma_x (\\frac{J_{\\varepsilon}^{2}}{2} - J_{\\varepsilon} \\sigma_x + \\Omega) and \\sigma_x \\int (J_{\\varepsilon} - \\sigma_x) dJ_{\\varepsilon} = \\sigma_x (\\frac{J_{\\varepsilon}^{2}}{2} - J_{\\varepsilon} \\sigma_x + \\Omega)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_x', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Integral(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(b,\\hat{x})} = \\hat{x} - b and \\mathbf{E}{(b,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{E}}{(b,\\hat{x})}, then derive 0 = 1 - \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{E}}{(b,\\hat{x})}, then obtain 0^{\\hat{x}} = (1 - \\mathbf{E}{(b,\\hat{x})})^{\\hat{x}}", "derivation": "\\operatorname{f_{E}}{(b,\\hat{x})} = \\hat{x} - b and 0 = \\hat{x} - b - \\operatorname{f_{E}}{(b,\\hat{x})} and \\frac{d}{d \\hat{x}} 0 = \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} - b - \\operatorname{f_{E}}{(b,\\hat{x})}) and 0 = 1 - \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{E}}{(b,\\hat{x})} and \\mathbf{E}{(b,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{E}}{(b,\\hat{x})} and 0 = 1 - \\mathbf{E}{(b,\\hat{x})} and 0^{\\hat{x}} = (1 - \\mathbf{E}{(b,\\hat{x})})^{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["minus", 1, "Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Derivative(Function('f_E')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"], [["power", 6, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given u{(\\mathbf{p})} = \\sin{(\\mathbf{p})}, then obtain \\frac{d}{d \\mathbf{p}} (\\int u{(\\mathbf{p})} d\\mathbf{p}) \\int \\sin{(\\mathbf{p})} d\\mathbf{p} = \\frac{d}{d \\mathbf{p}} (\\int \\sin{(\\mathbf{p})} d\\mathbf{p})^{2}", "derivation": "u{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\int u{(\\mathbf{p})} d\\mathbf{p} = \\int \\sin{(\\mathbf{p})} d\\mathbf{p} and (\\int u{(\\mathbf{p})} d\\mathbf{p}) \\int \\sin{(\\mathbf{p})} d\\mathbf{p} = (\\int \\sin{(\\mathbf{p})} d\\mathbf{p})^{2} and \\frac{d}{d \\mathbf{p}} (\\int u{(\\mathbf{p})} d\\mathbf{p}) \\int \\sin{(\\mathbf{p})} d\\mathbf{p} = \\frac{d}{d \\mathbf{p}} (\\int \\sin{(\\mathbf{p})} d\\mathbf{p})^{2}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Integral(Function('u')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Pow(Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('u')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integer(2)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(\\lambda,\\theta_2)} = \\frac{\\log{(\\lambda)}}{\\theta_2} and \\operatorname{A_{x}}{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain \\operatorname{A_{x}}{(\\hat{H})} + u{(\\lambda,\\theta_2)} = u{(\\lambda,\\theta_2)} + \\cos{(\\hat{H})}", "derivation": "u{(\\lambda,\\theta_2)} = \\frac{\\log{(\\lambda)}}{\\theta_2} and \\operatorname{A_{x}}{(\\hat{H})} = \\cos{(\\hat{H})} and \\operatorname{A_{x}}{(\\hat{H})} + \\frac{\\log{(\\lambda)}}{\\theta_2} = \\cos{(\\hat{H})} + \\frac{\\log{(\\lambda)}}{\\theta_2} and \\operatorname{A_{x}}{(\\hat{H})} + u{(\\lambda,\\theta_2)} = u{(\\lambda,\\theta_2)} + \\cos{(\\hat{H})}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))))"], ["get_premise", "Equality(Function('A_x')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('A_x')(Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True)))), Add(cos(Symbol('\\\\hat{H}', commutative=True)), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('A_x')(Symbol('\\\\hat{H}', commutative=True)), Function('u')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Function('u')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(J)} = J and n{(J)} = J^{J} \\operatorname{A_{y}}^{J}{(J)}, then obtain n{(J)} = J^{2 J}", "derivation": "\\operatorname{A_{y}}{(J)} = J and \\operatorname{A_{y}}^{J}{(J)} = J^{J} and J^{J} \\operatorname{A_{y}}^{J}{(J)} = J^{2 J} and n{(J)} = J^{J} \\operatorname{A_{y}}^{J}{(J)} and n{(J)} = J^{2 J}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('J', commutative=True)))"], [["times", 2, "Pow(Symbol('J', commutative=True), Symbol('J', commutative=True))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Symbol('J', commutative=True)), Pow(Function('A_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Pow(Symbol('J', commutative=True), Mul(Integer(2), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Symbol('J', commutative=True)), Pow(Function('A_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('n')(Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Mul(Integer(2), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(t_{2},y)} = e^{t_{2} + y}, then obtain \\frac{\\hat{p}_0{(t_{2},y)} e^{- t_{2} - y}}{\\int e^{t_{2} + y} dy} = \\frac{e^{- t_{2} - y} e^{t_{2} + y}}{\\int e^{t_{2} + y} dy}", "derivation": "\\hat{p}_0{(t_{2},y)} = e^{t_{2} + y} and \\int \\hat{p}_0{(t_{2},y)} dy = \\int e^{t_{2} + y} dy and \\hat{p}_0{(t_{2},y)} e^{- t_{2} - y} = e^{- t_{2} - y} e^{t_{2} + y} and \\frac{\\hat{p}_0{(t_{2},y)} e^{- t_{2} - y}}{\\int \\hat{p}_0{(t_{2},y)} dy} = \\frac{e^{- t_{2} - y} e^{t_{2} + y}}{\\int \\hat{p}_0{(t_{2},y)} dy} and \\frac{\\hat{p}_0{(t_{2},y)} e^{- t_{2} - y}}{\\int e^{t_{2} + y} dy} = \\frac{e^{- t_{2} - y} e^{t_{2} + y}}{\\int e^{t_{2} + y} dy}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["divide", 1, "exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))))), Mul(exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))), exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))), Pow(Integral(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))), exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))), Pow(Integral(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('t_2', commutative=True), Symbol('y', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))), Pow(Integral(exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(exp(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True)))), exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))), Pow(Integral(exp(Add(Symbol('t_2', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mu)} = e^{\\mu}, then obtain \\frac{d}{d \\mu} \\int \\frac{\\mathbf{r}{(\\mu)} e^{\\mu} - e^{2 \\mu}}{2 \\mu} d\\mu = \\frac{d}{d \\mu} \\int 0 d\\mu", "derivation": "\\mathbf{r}{(\\mu)} = e^{\\mu} and \\mathbf{r}{(\\mu)} e^{\\mu} = e^{2 \\mu} and \\mathbf{r}{(\\mu)} e^{\\mu} - e^{2 \\mu} = 0 and \\frac{\\mathbf{r}{(\\mu)} e^{\\mu} - e^{2 \\mu}}{2 \\mu} = 0 and \\int \\frac{\\mathbf{r}{(\\mu)} e^{\\mu} - e^{2 \\mu}}{2 \\mu} d\\mu = \\int 0 d\\mu and \\frac{d}{d \\mu} \\int \\frac{\\mathbf{r}{(\\mu)} e^{\\mu} - e^{2 \\mu}}{2 \\mu} d\\mu = \\frac{d}{d \\mu} \\int 0 d\\mu", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True))))), Integer(0))"], [["divide", 3, "Mul(Integer(2), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))), Integer(0))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Integral(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Function('\\\\mathbf{r}')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\mu', commutative=True)))))), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(a^{\\dagger},U,s)} = \\frac{a^{\\dagger} s}{U}, then obtain L_{\\varepsilon} (U + s + \\operatorname{A_{z}}{(a^{\\dagger},U,s)}) - \\frac{1}{U} = L_{\\varepsilon} (U + s + \\frac{a^{\\dagger} s}{U}) - \\frac{1}{U}", "derivation": "\\operatorname{A_{z}}{(a^{\\dagger},U,s)} = \\frac{a^{\\dagger} s}{U} and U + \\operatorname{A_{z}}{(a^{\\dagger},U,s)} = U + \\frac{a^{\\dagger} s}{U} and U + s + \\operatorname{A_{z}}{(a^{\\dagger},U,s)} = U + s + \\frac{a^{\\dagger} s}{U} and L_{\\varepsilon} (U + s + \\operatorname{A_{z}}{(a^{\\dagger},U,s)}) = L_{\\varepsilon} (U + s + \\frac{a^{\\dagger} s}{U}) and L_{\\varepsilon} (U + s + \\operatorname{A_{z}}{(a^{\\dagger},U,s)}) - \\frac{1}{U} = L_{\\varepsilon} (U + s + \\frac{a^{\\dagger} s}{U}) - \\frac{1}{U}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True), Symbol('s', commutative=True))), Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True), Symbol('s', commutative=True))), Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True))))"], [["times", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True), Symbol('s', commutative=True)))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)))))"], [["minus", 4, "Pow(Symbol('U', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('U', commutative=True), Symbol('s', commutative=True)))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('U', commutative=True), Symbol('s', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('s', commutative=True)))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given m{(S,v_{z})} = \\frac{v_{z}}{S}, then obtain \\frac{\\partial}{\\partial v_{z}} \\int\\limits^{S m{(S,v_{z})}} S m{(S,v_{z})} dv_{z} = \\frac{\\partial}{\\partial v_{z}} \\int\\limits^{S m{(S,v_{z})}} v_{z} dv_{z}", "derivation": "m{(S,v_{z})} = \\frac{v_{z}}{S} and S m{(S,v_{z})} = v_{z} and \\int S m{(S,v_{z})} dv_{z} = \\int v_{z} dv_{z} and \\int\\limits^{S m{(S,v_{z})}} S m{(S,v_{z})} dv_{z} = \\int\\limits^{S m{(S,v_{z})}} v_{z} dv_{z} and \\frac{\\partial}{\\partial v_{z}} \\int\\limits^{S m{(S,v_{z})}} S m{(S,v_{z})} dv_{z} = \\frac{\\partial}{\\partial v_{z}} \\int\\limits^{S m{(S,v_{z})}} v_{z} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["divide", 1, "Pow(Symbol('S', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))))), Integral(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Mul(Symbol('S', commutative=True), Function('m')(Symbol('S', commutative=True), Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}}, then obtain 1 = \\frac{\\int \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}} d\\mathbf{D}}{\\int \\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} d\\mathbf{D}}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} = \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}} and \\int \\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} d\\mathbf{D} = \\int \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}} d\\mathbf{D} and L_{\\varepsilon} \\int \\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} d\\mathbf{D} = L_{\\varepsilon} \\int \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}} d\\mathbf{D} and 1 = \\frac{\\int \\frac{L_{\\varepsilon}}{\\mathbf{D} \\mathbf{E}} d\\mathbf{D}}{\\int \\operatorname{m_{s}}{(\\mathbf{E},\\mathbf{D},L_{\\varepsilon})} d\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Function('m_s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["divide", 3, "Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Integral(Function('m_s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], "Equality(Integer(1), Mul(Integral(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Pow(Integral(Function('m_s')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given S{(\\delta,\\mu)} = \\sin{(\\mu^{\\delta})} and \\ddot{x}{(f^{*})} = \\int e^{f^{*}} df^{*}, then derive \\ddot{x}{(f^{*})} = s + e^{f^{*}}, then obtain s (\\ddot{x}{(f^{*})} \\sin{(\\mu^{\\delta})})^{f^{*}} = s ((s + e^{f^{*}}) \\sin{(\\mu^{\\delta})})^{f^{*}}", "derivation": "S{(\\delta,\\mu)} = \\sin{(\\mu^{\\delta})} and \\ddot{x}{(f^{*})} = \\int e^{f^{*}} df^{*} and \\ddot{x}{(f^{*})} = s + e^{f^{*}} and S{(\\delta,\\mu)} \\ddot{x}{(f^{*})} = (s + e^{f^{*}}) S{(\\delta,\\mu)} and (S{(\\delta,\\mu)} \\ddot{x}{(f^{*})})^{f^{*}} = ((s + e^{f^{*}}) S{(\\delta,\\mu)})^{f^{*}} and s (S{(\\delta,\\mu)} \\ddot{x}{(f^{*})})^{f^{*}} = s ((s + e^{f^{*}}) S{(\\delta,\\mu)})^{f^{*}} and s (\\ddot{x}{(f^{*})} \\sin{(\\mu^{\\delta})})^{f^{*}} = s ((s + e^{f^{*}}) \\sin{(\\mu^{\\delta})})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('f^*', commutative=True)), Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Function('\\\\ddot{x}')(Symbol('f^*', commutative=True)), Add(Symbol('s', commutative=True), exp(Symbol('f^*', commutative=True))))"], [["times", 3, "Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\ddot{x}')(Symbol('f^*', commutative=True))), Mul(Add(Symbol('s', commutative=True), exp(Symbol('f^*', commutative=True))), Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["power", 4, "Symbol('f^*', commutative=True)"], "Equality(Pow(Mul(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\ddot{x}')(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Mul(Add(Symbol('s', commutative=True), exp(Symbol('f^*', commutative=True))), Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('f^*', commutative=True)))"], [["times", 5, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Pow(Mul(Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\ddot{x}')(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True))), Mul(Symbol('s', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), exp(Symbol('f^*', commutative=True))), Function('S')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('s', commutative=True), Pow(Mul(Function('\\\\ddot{x}')(Symbol('f^*', commutative=True)), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True)))), Symbol('f^*', commutative=True))), Mul(Symbol('s', commutative=True), Pow(Mul(Add(Symbol('s', commutative=True), exp(Symbol('f^*', commutative=True))), sin(Pow(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True)))), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(U,u)} = - U + u and \\theta_{1}{(U,u)} = - U + u, then obtain \\frac{\\partial^{2}}{\\partial U\\partial u} \\mathbf{r}{(U,u)} = \\frac{\\partial^{2}}{\\partial U\\partial u} \\theta_{1}{(U,u)}", "derivation": "\\mathbf{r}{(U,u)} = - U + u and \\theta_{1}{(U,u)} = - U + u and \\frac{\\partial}{\\partial u} \\theta_{1}{(U,u)} = \\frac{\\partial}{\\partial u} (- U + u) and \\frac{\\partial}{\\partial u} \\theta_{1}{(U,u)} = \\frac{\\partial}{\\partial u} \\mathbf{r}{(U,u)} and \\frac{\\partial}{\\partial u} \\mathbf{r}{(U,u)} = \\frac{\\partial}{\\partial u} (- U + u) and \\frac{\\partial^{2}}{\\partial U\\partial u} \\mathbf{r}{(U,u)} = \\frac{\\partial^{2}}{\\partial U\\partial u} (- U + u) and \\frac{\\partial^{2}}{\\partial U\\partial u} \\mathbf{r}{(U,u)} = \\frac{\\partial^{2}}{\\partial U\\partial u} \\theta_{1}{(U,u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('u', commutative=True)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{r}')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(i,\\Omega)} = \\frac{\\Omega}{i}, then obtain \\frac{\\int \\varphi{(i,\\Omega)} d\\Omega}{\\frac{2 \\Omega}{i} + 1} = \\frac{\\int \\frac{\\Omega}{i} d\\Omega}{\\frac{2 \\Omega}{i} + 1}", "derivation": "\\varphi{(i,\\Omega)} = \\frac{\\Omega}{i} and \\frac{\\Omega}{i} + \\varphi{(i,\\Omega)} = \\frac{2 \\Omega}{i} and \\frac{\\Omega}{i} + \\varphi{(i,\\Omega)} + 1 = \\frac{2 \\Omega}{i} + 1 and \\int \\varphi{(i,\\Omega)} d\\Omega = \\int \\frac{\\Omega}{i} d\\Omega and \\frac{\\int \\varphi{(i,\\Omega)} d\\Omega}{\\frac{\\Omega}{i} + \\varphi{(i,\\Omega)} + 1} = \\frac{\\int \\frac{\\Omega}{i} d\\Omega}{\\frac{\\Omega}{i} + \\varphi{(i,\\Omega)} + 1} and \\frac{\\int \\varphi{(i,\\Omega)} d\\Omega}{\\frac{2 \\Omega}{i} + 1} = \\frac{\\int \\frac{\\Omega}{i} d\\Omega}{\\frac{2 \\Omega}{i} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1)), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Integer(1)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1)), Integer(-1)), Integral(Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Add(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(1)), Integer(-1)), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Integer(1)), Integer(-1)), Integral(Function('\\\\varphi')(Symbol('i', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Integer(1)), Integer(-1)), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given G{(\\mu)} = e^{\\mu}, then derive \\int G{(\\mu)} d\\mu = I + e^{\\mu}, then obtain \\mu + \\iint (\\int G{(\\mu)} d\\mu)^{I} d\\mu d\\mu = \\mu + \\iint (I + e^{\\mu})^{I} d\\mu d\\mu", "derivation": "G{(\\mu)} = e^{\\mu} and \\int G{(\\mu)} d\\mu = \\int e^{\\mu} d\\mu and \\int G{(\\mu)} d\\mu = I + e^{\\mu} and (\\int G{(\\mu)} d\\mu)^{I} = (I + e^{\\mu})^{I} and \\int (\\int G{(\\mu)} d\\mu)^{I} d\\mu = \\int (I + e^{\\mu})^{I} d\\mu and \\iint (\\int G{(\\mu)} d\\mu)^{I} d\\mu d\\mu = \\iint (I + e^{\\mu})^{I} d\\mu d\\mu and \\mu + \\iint (\\int G{(\\mu)} d\\mu)^{I} d\\mu d\\mu = \\mu + \\iint (I + e^{\\mu})^{I} d\\mu d\\mu", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('I', commutative=True), exp(Symbol('\\\\mu', commutative=True))))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Symbol('I', commutative=True), exp(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Pow(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(Add(Symbol('I', commutative=True), exp(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Pow(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(Add(Symbol('I', commutative=True), exp(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["add", 6, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Integral(Pow(Integral(Function('G')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Integral(Pow(Add(Symbol('I', commutative=True), exp(Symbol('\\\\mu', commutative=True))), Symbol('I', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\theta_2)} = \\log{(\\log{(\\theta_2)})}, then obtain \\frac{(\\varepsilon{(\\theta_2)} + \\log{(\\theta_2)})^{\\theta_2}}{\\log{(\\log{(\\theta_2)})}} = \\frac{(\\log{(\\theta_2)} + \\log{(\\log{(\\theta_2)})})^{\\theta_2}}{\\log{(\\log{(\\theta_2)})}}", "derivation": "\\varepsilon{(\\theta_2)} = \\log{(\\log{(\\theta_2)})} and \\varepsilon{(\\theta_2)} + \\log{(\\theta_2)} = \\log{(\\theta_2)} + \\log{(\\log{(\\theta_2)})} and (\\varepsilon{(\\theta_2)} + \\log{(\\theta_2)})^{\\theta_2} = (\\log{(\\theta_2)} + \\log{(\\log{(\\theta_2)})})^{\\theta_2} and \\frac{(\\varepsilon{(\\theta_2)} + \\log{(\\theta_2)})^{\\theta_2}}{\\log{(\\log{(\\theta_2)})}} = \\frac{(\\log{(\\theta_2)} + \\log{(\\log{(\\theta_2)})})^{\\theta_2}}{\\log{(\\log{(\\theta_2)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\theta_2', commutative=True)), log(log(Symbol('\\\\theta_2', commutative=True))))"], [["add", 1, "log(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Add(log(Symbol('\\\\theta_2', commutative=True)), log(log(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(log(Symbol('\\\\theta_2', commutative=True)), log(log(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 3, "log(log(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(log(log(Symbol('\\\\theta_2', commutative=True))), Integer(-1))), Mul(Pow(Add(log(Symbol('\\\\theta_2', commutative=True)), log(log(Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Pow(log(log(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(J)} = \\log{(\\log{(J)})}, then obtain - \\mathbb{I}{(J)} + \\frac{d}{d J} \\iiint \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ dJ dJ = - \\mathbb{I}{(J)} + \\frac{d}{d J} \\iiint \\log{(\\log{(J)})}^{2} dJ dJ dJ", "derivation": "\\mathbb{I}{(J)} = \\log{(\\log{(J)})} and \\mathbb{I}{(J)} \\log{(\\log{(J)})} = \\log{(\\log{(J)})}^{2} and \\int \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ = \\int \\log{(\\log{(J)})}^{2} dJ and \\iint \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ dJ = \\iint \\log{(\\log{(J)})}^{2} dJ dJ and \\iiint \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ dJ dJ = \\iiint \\log{(\\log{(J)})}^{2} dJ dJ dJ and \\frac{d}{d J} \\iiint \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ dJ dJ = \\frac{d}{d J} \\iiint \\log{(\\log{(J)})}^{2} dJ dJ dJ and - \\mathbb{I}{(J)} + \\frac{d}{d J} \\iiint \\mathbb{I}{(J)} \\log{(\\log{(J)})} dJ dJ dJ = - \\mathbb{I}{(J)} + \\frac{d}{d J} \\iiint \\log{(\\log{(J)})}^{2} dJ dJ dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True))))"], [["times", 1, "log(log(Symbol('J', commutative=True)))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Pow(log(log(Symbol('J', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))), Integral(Pow(log(log(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(log(log(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(log(log(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Integral(Pow(log(log(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 6, "Function('\\\\mathbb{I}')(Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('J', commutative=True))), Derivative(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('J', commutative=True))), Derivative(Integral(Pow(log(log(Symbol('J', commutative=True))), Integer(2)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(\\hat{H})} = \\cos{(\\hat{H})}, then derive \\int \\lambda{(\\hat{H})} d\\hat{H} = \\mathbf{J}_M + \\sin{(\\hat{H})}, then obtain (2 \\int \\cos{(\\hat{H})} d\\hat{H})^{\\hat{H}} = (\\int \\lambda{(\\hat{H})} d\\hat{H} + \\int \\cos{(\\hat{H})} d\\hat{H})^{\\hat{H}}", "derivation": "\\lambda{(\\hat{H})} = \\cos{(\\hat{H})} and \\int \\lambda{(\\hat{H})} d\\hat{H} = \\int \\cos{(\\hat{H})} d\\hat{H} and \\int \\lambda{(\\hat{H})} d\\hat{H} = \\mathbf{J}_M + \\sin{(\\hat{H})} and \\int \\cos{(\\hat{H})} d\\hat{H} = \\mathbf{J}_M + \\sin{(\\hat{H})} and 2 \\int \\cos{(\\hat{H})} d\\hat{H} = \\mathbf{J}_M + \\sin{(\\hat{H})} + \\int \\cos{(\\hat{H})} d\\hat{H} and 2 \\int \\cos{(\\hat{H})} d\\hat{H} = \\int \\lambda{(\\hat{H})} d\\hat{H} + \\int \\cos{(\\hat{H})} d\\hat{H} and (2 \\int \\cos{(\\hat{H})} d\\hat{H})^{\\hat{H}} = (\\int \\lambda{(\\hat{H})} d\\hat{H} + \\int \\cos{(\\hat{H})} d\\hat{H})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 4, "Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Add(Integral(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["power", 6, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Integral(Function('\\\\lambda')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given q{(x^\\prime,r)} = - r + x^\\prime and \\ddot{x}{(x^\\prime)} = x^\\prime, then obtain x^\\prime (x^\\prime + \\frac{q{(x^\\prime,r)}}{- r + x^\\prime}) - (x^\\prime)^{x^\\prime} = x^\\prime (x^\\prime + 1) - (x^\\prime)^{x^\\prime}", "derivation": "q{(x^\\prime,r)} = - r + x^\\prime and \\frac{q{(x^\\prime,r)}}{- r + x^\\prime} = 1 and \\ddot{x}{(x^\\prime)} = x^\\prime and \\ddot{x}{(x^\\prime)} + \\frac{q{(x^\\prime,r)}}{- r + x^\\prime} = \\ddot{x}{(x^\\prime)} + 1 and x^\\prime (\\ddot{x}{(x^\\prime)} + \\frac{q{(x^\\prime,r)}}{- r + x^\\prime}) = x^\\prime (\\ddot{x}{(x^\\prime)} + 1) and x^\\prime (\\ddot{x}{(x^\\prime)} + \\frac{q{(x^\\prime,r)}}{- r + x^\\prime}) - (x^\\prime)^{x^\\prime} = x^\\prime (\\ddot{x}{(x^\\prime)} + 1) - (x^\\prime)^{x^\\prime} and x^\\prime (x^\\prime + \\frac{q{(x^\\prime,r)}}{- r + x^\\prime}) - (x^\\prime)^{x^\\prime} = x^\\prime (x^\\prime + 1) - (x^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["add", 2, "Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True)))), Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Integer(1)))"], [["times", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True))))), Mul(Symbol('x^\\\\prime', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Integer(1))))"], [["minus", 5, "Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True))))), Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('x^\\\\prime', commutative=True)), Integer(1))), Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('x^\\\\prime', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('r', commutative=True))))), Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\hat{x}_0)} = \\sin{(\\log{(\\hat{x}_0)})}, then obtain 0 = - \\nabla{(\\hat{x}_0)} + \\sin{(\\frac{\\log{(\\hat{x}_0)} \\sin{(\\log{(\\hat{x}_0)})}}{\\nabla{(\\hat{x}_0)}})}", "derivation": "\\nabla{(\\hat{x}_0)} = \\sin{(\\log{(\\hat{x}_0)})} and \\nabla{(\\hat{x}_0)} \\log{(\\hat{x}_0)} = \\log{(\\hat{x}_0)} \\sin{(\\log{(\\hat{x}_0)})} and \\log{(\\hat{x}_0)} = \\frac{\\log{(\\hat{x}_0)} \\sin{(\\log{(\\hat{x}_0)})}}{\\nabla{(\\hat{x}_0)}} and \\nabla{(\\hat{x}_0)} = \\sin{(\\frac{\\log{(\\hat{x}_0)} \\sin{(\\log{(\\hat{x}_0)})}}{\\nabla{(\\hat{x}_0)}})} and 0 = - \\nabla{(\\hat{x}_0)} + \\sin{(\\frac{\\log{(\\hat{x}_0)} \\sin{(\\log{(\\hat{x}_0)})}}{\\nabla{(\\hat{x}_0)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), sin(log(Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 1, "log(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Mul(log(Symbol('\\\\hat{x}_0', commutative=True)), sin(log(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["divide", 2, "Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(log(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Pow(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{x}_0', commutative=True)), sin(log(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{x}_0', commutative=True)), sin(log(Symbol('\\\\hat{x}_0', commutative=True))))))"], [["minus", 4, "Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True))), sin(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{x}_0', commutative=True)), sin(log(Symbol('\\\\hat{x}_0', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} = \\frac{A_{2}}{\\hat{H}_{\\lambda}}, then obtain \\frac{\\partial}{\\partial A_{2}} \\int (\\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2} = \\frac{\\partial}{\\partial A_{2}} \\int (\\frac{A_{2}}{\\hat{H}_{\\lambda}} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2}", "derivation": "\\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} = \\frac{A_{2}}{\\hat{H}_{\\lambda}} and \\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}} = \\frac{A_{2}}{\\hat{H}_{\\lambda}} + \\frac{1}{\\hat{H}_{\\lambda}} and \\int (\\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2} = \\int (\\frac{A_{2}}{\\hat{H}_{\\lambda}} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2} and \\frac{\\partial}{\\partial A_{2}} \\int (\\mathbf{H}{(A_{2},\\hat{H}_{\\lambda})} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2} = \\frac{\\partial}{\\partial A_{2}} \\int (\\frac{A_{2}}{\\hat{H}_{\\lambda}} + \\frac{1}{\\hat{H}_{\\lambda}}) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))))"], [["add", 1, "Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(g,m)} = \\sin{(g^{m})} and \\varphi{(g,m)} = - g^{m} + \\dot{y}{(g,m)}, then obtain \\varphi{(g,m)} = - g^{m} + \\sin{(g^{m})}", "derivation": "\\dot{y}{(g,m)} = \\sin{(g^{m})} and - g^{m} + \\dot{y}{(g,m)} = - g^{m} + \\sin{(g^{m})} and \\varphi{(g,m)} = - g^{m} + \\dot{y}{(g,m)} and \\varphi{(g,m)} = - g^{m} + \\sin{(g^{m})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('g', commutative=True), Symbol('m', commutative=True)), sin(Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))))"], [["minus", 1, "Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))), Function('\\\\dot{y}')(Symbol('g', commutative=True), Symbol('m', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))), sin(Pow(Symbol('g', commutative=True), Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('g', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))), Function('\\\\dot{y}')(Symbol('g', commutative=True), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varphi')(Symbol('g', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('m', commutative=True))), sin(Pow(Symbol('g', commutative=True), Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\rho,v_{z})} = - \\rho + v_{z} and \\Psi^{\\dagger}{(\\rho,v_{z})} = - \\sigma_{x}{(\\rho,v_{z})}, then obtain \\sin{(\\Psi^{\\dagger}{(\\rho,v_{z})})} = - \\sin{(\\sigma_{x}{(\\rho,v_{z})})}", "derivation": "\\sigma_{x}{(\\rho,v_{z})} = - \\rho + v_{z} and - \\sigma_{x}{(\\rho,v_{z})} = \\rho - v_{z} and \\Psi^{\\dagger}{(\\rho,v_{z})} = - \\sigma_{x}{(\\rho,v_{z})} and \\Psi^{\\dagger}{(\\rho,v_{z})} = \\rho - v_{z} and \\sin{(\\Psi^{\\dagger}{(\\rho,v_{z})})} = \\sin{(\\rho - v_{z})} and \\sin{(\\Psi^{\\dagger}{(\\rho,v_{z})})} = - \\sin{(\\sigma_{x}{(\\rho,v_{z})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('v_z', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["sin", 4], "Equality(sin(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True))), sin(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(sin(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(-1), sin(Function('\\\\sigma_x')(Symbol('\\\\rho', commutative=True), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\phi_1)} = \\log{(\\cos{(\\phi_1)})}, then obtain \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\nabla{(\\phi_1)} d\\phi_1 = \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\log{(\\cos{(\\phi_1)})} d\\phi_1", "derivation": "\\nabla{(\\phi_1)} = \\log{(\\cos{(\\phi_1)})} and \\int \\nabla{(\\phi_1)} d\\phi_1 = \\int \\log{(\\cos{(\\phi_1)})} d\\phi_1 and \\phi_1 \\int \\nabla{(\\phi_1)} d\\phi_1 = \\phi_1 \\int \\log{(\\cos{(\\phi_1)})} d\\phi_1 and \\phi_1 \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\nabla{(\\phi_1)} d\\phi_1 = \\phi_1 \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\log{(\\cos{(\\phi_1)})} d\\phi_1 and \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\nabla{(\\phi_1)} d\\phi_1 = \\frac{d}{d \\hat{p}_0} \\hat{x}{(\\hat{p}_0)} \\int \\log{(\\cos{(\\phi_1)})} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True)), log(cos(Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(log(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Integral(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Integral(log(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["times", 3, "Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integral(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integral(log(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["divide", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integral(Function('\\\\nabla')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Derivative(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integral(log(cos(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\psi^*,z)} = \\sin{(\\psi^* z)}, then obtain ((\\frac{\\partial}{\\partial z} \\hat{H}^{\\psi^*}{(\\psi^*,z)})^{z})^{z} = ((\\frac{\\partial}{\\partial z} \\sin^{\\psi^*}{(\\psi^* z)})^{z})^{z}", "derivation": "\\hat{H}{(\\psi^*,z)} = \\sin{(\\psi^* z)} and \\hat{H}^{\\psi^*}{(\\psi^*,z)} = \\sin^{\\psi^*}{(\\psi^* z)} and \\frac{\\partial}{\\partial z} \\hat{H}^{\\psi^*}{(\\psi^*,z)} = \\frac{\\partial}{\\partial z} \\sin^{\\psi^*}{(\\psi^* z)} and (\\frac{\\partial}{\\partial z} \\hat{H}^{\\psi^*}{(\\psi^*,z)})^{z} = (\\frac{\\partial}{\\partial z} \\sin^{\\psi^*}{(\\psi^* z)})^{z} and ((\\frac{\\partial}{\\partial z} \\hat{H}^{\\psi^*}{(\\psi^*,z)})^{z})^{z} = ((\\frac{\\partial}{\\partial z} \\sin^{\\psi^*}{(\\psi^* z)})^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{H}')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\hat{H}')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Pow(sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Derivative(Pow(Function('\\\\hat{H}')(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Derivative(Pow(sin(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{J},f)} = \\mathbf{J} f and \\operatorname{g_{\\varepsilon}}{(\\mathbf{J},f)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} f, then derive \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{E_{n}}{(\\mathbf{J},f)} = f, then obtain \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J},f)} = f \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{J},f)} = \\mathbf{J} f and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{E_{n}}{(\\mathbf{J},f)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} f and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{E_{n}}{(\\mathbf{J},f)} = f and \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} f = f and \\operatorname{g_{\\varepsilon}}{(\\mathbf{J},f)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} f and \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J},f)} = \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)} \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{J} f and \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J},f)} = f \\operatorname{E_{n}}^{- \\mathbf{J}}{(\\mathbf{J},f)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('f', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('f', commutative=True))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 5, "Pow(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Derivative(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('f', commutative=True), Pow(Function('E_n')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\psi^*)} = \\cos{(e^{\\psi^*})}, then obtain \\frac{\\phi_{2}^{2}{(\\psi^*)}}{\\psi^*} + \\frac{\\phi_{2}{(\\psi^*)} \\cos{(e^{\\psi^*})}}{\\psi^*} = \\frac{2 \\phi_{2}{(\\psi^*)} \\cos{(e^{\\psi^*})}}{\\psi^*}", "derivation": "\\phi_{2}{(\\psi^*)} = \\cos{(e^{\\psi^*})} and \\frac{\\phi_{2}{(\\psi^*)}}{\\psi^*} = \\frac{\\cos{(e^{\\psi^*})}}{\\psi^*} and \\frac{\\phi_{2}^{2}{(\\psi^*)}}{\\psi^*} = \\frac{\\phi_{2}{(\\psi^*)} \\cos{(e^{\\psi^*})}}{\\psi^*} and \\frac{\\phi_{2}^{2}{(\\psi^*)}}{\\psi^*} + \\frac{\\phi_{2}{(\\psi^*)} \\cos{(e^{\\psi^*})}}{\\psi^*} = \\frac{2 \\phi_{2}{(\\psi^*)} \\cos{(e^{\\psi^*})}}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), cos(exp(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["times", 2, "Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Pow(Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), cos(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), cos(exp(Symbol('\\\\psi^*', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Pow(Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), cos(exp(Symbol('\\\\psi^*', commutative=True))))), Mul(Integer(2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('\\\\psi^*', commutative=True)), cos(exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} = - \\Omega + \\dot{y}, then obtain \\Omega - \\dot{y} + \\int (\\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})}) \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega = \\Omega - \\dot{y} + \\int \\dot{y} \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega", "derivation": "\\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} = - \\Omega + \\dot{y} and \\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} = \\dot{y} and (- \\Omega + \\dot{y}) (\\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})}) = \\dot{y} (- \\Omega + \\dot{y}) and (\\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})}) \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} = \\dot{y} \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} and \\int (\\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})}) \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega = \\int \\dot{y} \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega and \\Omega - \\dot{y} + \\int (\\Omega + \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})}) \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega = \\Omega - \\dot{y} + \\int \\dot{y} \\operatorname{f^{\\prime}}{(\\Omega,\\dot{y})} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Mul(Symbol('\\\\dot{y}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\Omega', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Integral(Mul(Add(Symbol('\\\\Omega', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} = - \\eta + \\sin{(\\mathbf{g})}, then obtain (2 \\eta + \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} - \\sin{(\\mathbf{g})})^{2 \\mathbf{g}} = \\eta^{2 \\mathbf{g}}", "derivation": "\\operatorname{A_{x}}{(\\eta,\\mathbf{g})} = - \\eta + \\sin{(\\mathbf{g})} and \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} - \\sin{(\\mathbf{g})} = - \\eta and 2 \\eta + \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} - \\sin{(\\mathbf{g})} = \\eta and (2 \\eta + \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} - \\sin{(\\mathbf{g})})^{\\mathbf{g}} = \\eta^{\\mathbf{g}} and (2 \\eta + \\operatorname{A_{x}}{(\\eta,\\mathbf{g})} - \\sin{(\\mathbf{g})})^{2 \\mathbf{g}} = \\eta^{2 \\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\eta', commutative=True)), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\eta', commutative=True))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\eta', commutative=True)), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\eta', commutative=True)), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Symbol('\\\\eta', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(k,v)} = k + v, then derive \\int \\hat{H}_{\\lambda}{(k,v)} dv = E_{\\lambda} + k v + \\frac{v^{2}}{2}, then obtain - E_{\\lambda} - k v - \\frac{v^{2}}{2} = - \\int (k + v) dv", "derivation": "\\hat{H}_{\\lambda}{(k,v)} = k + v and \\int \\hat{H}_{\\lambda}{(k,v)} dv = \\int (k + v) dv and \\int \\hat{H}_{\\lambda}{(k,v)} dv = E_{\\lambda} + k v + \\frac{v^{2}}{2} and E_{\\lambda} + k v + \\frac{v^{2}}{2} = \\int (k + v) dv and - E_{\\lambda} - k v - \\frac{v^{2}}{2} = - \\int (k + v) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Add(Symbol('k', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('k', commutative=True), Symbol('v', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('k', commutative=True), Symbol('v', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2)))), Integral(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2)))), Mul(Integer(-1), Integral(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} = \\cos^{\\sigma_x}{(b)}, then obtain - \\cos{(b)} + \\cos^{\\sigma_x}{(b)} = (\\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} + \\cos{(b)} - \\cos^{\\sigma_x}{(b)})^{\\sigma_x} - \\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} - \\cos{(b)} + \\cos^{\\sigma_x}{(b)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} = \\cos^{\\sigma_x}{(b)} and \\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} - \\cos{(b)} = - \\cos{(b)} + \\cos^{\\sigma_x}{(b)} and \\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} + \\cos{(b)} - \\cos^{\\sigma_x}{(b)} = \\cos{(b)} and - \\cos{(b)} + \\cos^{\\sigma_x}{(b)} = (\\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} + \\cos{(b)} - \\cos^{\\sigma_x}{(b)})^{\\sigma_x} - \\operatorname{J_{\\varepsilon}}{(b,\\sigma_x)} - \\cos{(b)} + \\cos^{\\sigma_x}{(b)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "cos(Symbol('b', commutative=True))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('b', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), cos(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('b', commutative=True))), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Add(Pow(Add(Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('b', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('b', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), cos(Symbol('b', commutative=True))), Pow(cos(Symbol('b', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\phi{(C_{1})} = e^{C_{1}} and \\operatorname{f_{E}}{(C_{1})} = \\int e^{C_{1}} dC_{1}, then derive \\int \\phi{(C_{1})} dC_{1} = V + e^{C_{1}}, then obtain \\int \\operatorname{f_{E}}{(C_{1})} dC_{1} = \\int (V + e^{C_{1}}) dC_{1}", "derivation": "\\phi{(C_{1})} = e^{C_{1}} and \\int \\phi{(C_{1})} dC_{1} = \\int e^{C_{1}} dC_{1} and \\operatorname{f_{E}}{(C_{1})} = \\int e^{C_{1}} dC_{1} and \\int \\phi{(C_{1})} dC_{1} = V + e^{C_{1}} and \\int e^{C_{1}} dC_{1} = V + e^{C_{1}} and \\operatorname{f_{E}}{(C_{1})} = V + e^{C_{1}} and \\int \\operatorname{f_{E}}{(C_{1})} dC_{1} = \\int (V + e^{C_{1}}) dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('C_1', commutative=True)), Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('V', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('V', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Function('f_E')(Symbol('C_1', commutative=True)), Add(Symbol('V', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["integrate", 6, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Symbol('V', commutative=True), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = \\mathbf{A} + \\mathbf{g}, then obtain \\mathbf{A} + \\mathbf{g} + \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = 2 \\mathbf{M}{(\\mathbf{g},\\mathbf{A})}", "derivation": "\\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = \\mathbf{A} + \\mathbf{g} and \\mathbf{g} + \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = \\mathbf{A} + 2 \\mathbf{g} and \\mathbf{A} + \\mathbf{g} + \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = 2 \\mathbf{A} + 2 \\mathbf{g} and 2 \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = 2 \\mathbf{A} + 2 \\mathbf{g} and \\mathbf{A} + \\mathbf{g} + \\mathbf{M}{(\\mathbf{g},\\mathbf{A})} = 2 \\mathbf{M}{(\\mathbf{g},\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given G{(M,v_{2})} = - M + v_{2}, then obtain \\int \\frac{G{(M,v_{2})}}{v_{2} (- M + v_{2})^{2}} dv_{2} = P_{g} + \\frac{- \\log{(v_{2})} + \\log{(- M + v_{2})}}{M}", "derivation": "G{(M,v_{2})} = - M + v_{2} and \\frac{G{(M,v_{2})}}{- M + v_{2}} = 1 and \\frac{G{(M,v_{2})}}{v_{2} (- M + v_{2})} = \\frac{1}{v_{2}} and \\frac{G{(M,v_{2})}}{v_{2} (- M + v_{2})^{2}} = \\frac{1}{v_{2} (- M + v_{2})} and \\int \\frac{G{(M,v_{2})}}{v_{2} (- M + v_{2})^{2}} dv_{2} = \\int \\frac{1}{v_{2} (- M + v_{2})} dv_{2} and \\int \\frac{G{(M,v_{2})}}{v_{2} (- M + v_{2})^{2}} dv_{2} = P_{g} + \\frac{- \\log{(v_{2})} + \\log{(- M + v_{2})}}{M}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-1)), Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True))), Integer(1))"], [["divide", 2, "Symbol('v_2', commutative=True)"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-1)), Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True))), Pow(Symbol('v_2', commutative=True), Integer(-1)))"], [["times", 3, "Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-2)), Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-2)), Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-1))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)), Integer(-2)), Function('G')(Symbol('M', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('P_g', commutative=True), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('v_2', commutative=True))), log(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('v_2', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(r,Z)} = \\frac{\\sin{(r)}}{Z}, then obtain r \\operatorname{A_{y}}{(r,Z)} + \\frac{r \\operatorname{A_{y}}{(r,Z)}}{\\sin{(r)}} + \\operatorname{A_{y}}{(r,Z)} - \\frac{\\sin{(r)}}{Z} = r \\operatorname{A_{y}}{(r,Z)} + \\operatorname{A_{y}}{(r,Z)} + \\frac{r}{Z} - \\frac{\\sin{(r)}}{Z}", "derivation": "\\operatorname{A_{y}}{(r,Z)} = \\frac{\\sin{(r)}}{Z} and r \\operatorname{A_{y}}{(r,Z)} = \\frac{r \\sin{(r)}}{Z} and \\frac{r \\operatorname{A_{y}}{(r,Z)}}{\\sin{(r)}} = \\frac{r}{Z} and \\frac{r \\operatorname{A_{y}}{(r,Z)}}{\\sin{(r)}} + \\operatorname{A_{y}}{(r,Z)} = \\operatorname{A_{y}}{(r,Z)} + \\frac{r}{Z} and r \\operatorname{A_{y}}{(r,Z)} + \\frac{r \\operatorname{A_{y}}{(r,Z)}}{\\sin{(r)}} + \\operatorname{A_{y}}{(r,Z)} - \\frac{\\sin{(r)}}{Z} = r \\operatorname{A_{y}}{(r,Z)} + \\operatorname{A_{y}}{(r,Z)} + \\frac{r}{Z} - \\frac{\\sin{(r)}}{Z}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(Symbol('r', commutative=True))))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('r', commutative=True), sin(Symbol('r', commutative=True))))"], [["divide", 2, "sin(Symbol('r', commutative=True))"], "Equality(Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Integer(-1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('r', commutative=True)))"], [["add", 3, "Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Integer(-1))), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))), Add(Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('r', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(Symbol('r', commutative=True))))"], "Equality(Add(Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Integer(-1))), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)), sin(Symbol('r', commutative=True)))), Add(Mul(Symbol('r', commutative=True), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True))), Function('A_y')(Symbol('r', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)), sin(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given u{(B)} = B and \\rho_{b}{(B)} = (B^{2})^{B}, then obtain - \\frac{e^{(B u{(B)})^{B}}}{u^{2}{(B)}} + \\frac{e^{(B^{2})^{B}}}{B^{2}} = - \\frac{e^{(B u{(B)})^{B}}}{u^{2}{(B)}} + \\frac{e^{\\rho_{b}{(B)}}}{B^{2}}", "derivation": "u{(B)} = B and B u{(B)} = B^{2} and (B u{(B)})^{B} = (B^{2})^{B} and \\rho_{b}{(B)} = (B^{2})^{B} and (B u{(B)})^{B} = \\rho_{b}{(B)} and e^{(B u{(B)})^{B}} = e^{\\rho_{b}{(B)}} and \\frac{e^{(B u{(B)})^{B}}}{u^{2}{(B)}} = \\frac{e^{\\rho_{b}{(B)}}}{u^{2}{(B)}} and \\frac{e^{(B^{2})^{B}}}{B^{2}} = \\frac{e^{\\rho_{b}{(B)}}}{B^{2}} and - \\frac{e^{(B u{(B)})^{B}}}{u^{2}{(B)}} + \\frac{e^{(B^{2})^{B}}}{B^{2}} = - \\frac{e^{(B u{(B)})^{B}}}{u^{2}{(B)}} + \\frac{e^{\\rho_{b}{(B)}}}{B^{2}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('B', commutative=True)), Symbol('B', commutative=True))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Pow(Symbol('B', commutative=True), Integer(2)))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('B', commutative=True)), Pow(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Function('\\\\rho_b')(Symbol('B', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True))), exp(Function('\\\\rho_b')(Symbol('B', commutative=True))))"], [["divide", 6, "Pow(Function('u')(Symbol('B', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('u')(Symbol('B', commutative=True)), Integer(-2)), exp(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Mul(Pow(Function('u')(Symbol('B', commutative=True)), Integer(-2)), exp(Function('\\\\rho_b')(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-2)), exp(Pow(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), exp(Function('\\\\rho_b')(Symbol('B', commutative=True)))))"], [["minus", 8, "Mul(Pow(Function('u')(Symbol('B', commutative=True)), Integer(-2)), exp(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Function('u')(Symbol('B', commutative=True)), Integer(-2)), exp(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), exp(Pow(Pow(Symbol('B', commutative=True), Integer(2)), Symbol('B', commutative=True))))), Add(Mul(Integer(-1), Pow(Function('u')(Symbol('B', commutative=True)), Integer(-2)), exp(Pow(Mul(Symbol('B', commutative=True), Function('u')(Symbol('B', commutative=True))), Symbol('B', commutative=True)))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), exp(Function('\\\\rho_b')(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given S{(n_{1},\\sigma_x)} = \\frac{\\cos{(\\sigma_x)}}{n_{1}}, then obtain \\int S{(n_{1},\\sigma_x)} d\\sigma_x + (\\int S{(n_{1},\\sigma_x)} dn_{1})^{n_{1}} = (\\int \\frac{\\cos{(\\sigma_x)}}{n_{1}} dn_{1})^{n_{1}} + \\int S{(n_{1},\\sigma_x)} d\\sigma_x", "derivation": "S{(n_{1},\\sigma_x)} = \\frac{\\cos{(\\sigma_x)}}{n_{1}} and \\int S{(n_{1},\\sigma_x)} dn_{1} = \\int \\frac{\\cos{(\\sigma_x)}}{n_{1}} dn_{1} and (\\int S{(n_{1},\\sigma_x)} dn_{1})^{n_{1}} = (\\int \\frac{\\cos{(\\sigma_x)}}{n_{1}} dn_{1})^{n_{1}} and \\int S{(n_{1},\\sigma_x)} d\\sigma_x + (\\int S{(n_{1},\\sigma_x)} dn_{1})^{n_{1}} = (\\int \\frac{\\cos{(\\sigma_x)}}{n_{1}} dn_{1})^{n_{1}} + \\int S{(n_{1},\\sigma_x)} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('n_1', commutative=True))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Pow(Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["add", 3, "Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Pow(Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True))), Add(Pow(Integral(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Integral(Function('S')(Symbol('n_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(S,A_{y})} = A_{y} + S and \\hat{H}_l{(S,A_{y})} = \\frac{\\partial}{\\partial S} \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})}, then derive \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})} = \\rho_{f}{(S,A_{y})}, then obtain \\hat{H}_l{(S,A_{y})} = \\frac{\\partial}{\\partial S} \\rho_{f}{(S,A_{y})}", "derivation": "\\rho_{f}{(S,A_{y})} = A_{y} + S and \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})} = \\frac{\\partial}{\\partial A_{y}} (A_{y} + S) and \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})} = \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} (A_{y} + S) and \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})} = \\rho_{f}{(S,A_{y})} and \\hat{H}_l{(S,A_{y})} = \\frac{\\partial}{\\partial S} \\rho_{f}{(S,A_{y})} \\frac{\\partial}{\\partial A_{y}} \\rho_{f}{(S,A_{y})} and \\hat{H}_l{(S,A_{y})} = \\frac{\\partial}{\\partial S} \\rho_{f}{(S,A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_y', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True))"], "Equality(Mul(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Add(Symbol('A_y', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Mul(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\hat{H}_l')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('S', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})} = \\frac{T}{V_{\\mathbf{E}}} and \\mathbf{D}{(T)} = \\int 0 dT, then obtain \\int (-1 + \\frac{V_{\\mathbf{E}} \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})}}{T}) dT = \\mathbf{D}{(T)}", "derivation": "\\operatorname{F_{H}}{(T,V_{\\mathbf{E}})} = \\frac{T}{V_{\\mathbf{E}}} and \\frac{V_{\\mathbf{E}} \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})}}{T} = 1 and -1 + \\frac{V_{\\mathbf{E}} \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})}}{T} = 0 and \\int (-1 + \\frac{V_{\\mathbf{E}} \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})}}{T}) dT = \\int 0 dT and \\mathbf{D}{(T)} = \\int 0 dT and \\int (-1 + \\frac{V_{\\mathbf{E}} \\operatorname{F_{H}}{(T,V_{\\mathbf{E}})}}{T}) dT = \\mathbf{D}{(T)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('T', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('T', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('T', commutative=True), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_H')(Symbol('T', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_H')(Symbol('T', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Integer(-1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_H')(Symbol('T', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('T', commutative=True))), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Add(Integer(-1), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('F_H')(Symbol('T', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('T', commutative=True))), Function('\\\\mathbf{D}')(Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\nabla)} = \\sin{(\\log{(\\nabla)})} and \\operatorname{A_{x}}{(M,c_{0})} = M^{c_{0}}, then obtain \\nabla (\\nabla \\sin{(\\log{(\\nabla)})} + \\operatorname{A_{x}}{(M,c_{0})}) \\sin{(\\log{(\\nabla)})} = \\nabla (M^{c_{0}} + \\nabla \\sin{(\\log{(\\nabla)})}) \\sin{(\\log{(\\nabla)})}", "derivation": "\\phi{(\\nabla)} = \\sin{(\\log{(\\nabla)})} and \\nabla \\phi{(\\nabla)} = \\nabla \\sin{(\\log{(\\nabla)})} and \\operatorname{A_{x}}{(M,c_{0})} = M^{c_{0}} and \\nabla \\phi{(\\nabla)} + \\operatorname{A_{x}}{(M,c_{0})} = M^{c_{0}} + \\nabla \\phi{(\\nabla)} and \\nabla \\sin{(\\log{(\\nabla)})} + \\operatorname{A_{x}}{(M,c_{0})} = M^{c_{0}} + \\nabla \\sin{(\\log{(\\nabla)})} and \\nabla (\\nabla \\sin{(\\log{(\\nabla)})} + \\operatorname{A_{x}}{(M,c_{0})}) \\sin{(\\log{(\\nabla)})} = \\nabla (M^{c_{0}} + \\nabla \\sin{(\\log{(\\nabla)})}) \\sin{(\\log{(\\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\nabla', commutative=True)), sin(log(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('M', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('c_0', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True))), Function('A_x')(Symbol('M', commutative=True), Symbol('c_0', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\phi')(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True)))), Function('A_x')(Symbol('M', commutative=True), Symbol('c_0', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True))))))"], [["times", 5, "Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True)))), Function('A_x')(Symbol('M', commutative=True), Symbol('c_0', commutative=True))), sin(log(Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), Add(Pow(Symbol('M', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), sin(log(Symbol('\\\\nabla', commutative=True))))), sin(log(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given a{(\\mathbf{s},m)} = \\cos{(\\mathbf{s} + m)}, then obtain \\frac{\\frac{\\partial}{\\partial \\mathbf{s}} a{(\\mathbf{s},m)}}{a{(\\mathbf{s},m)}} = - \\frac{\\sin{(\\mathbf{s} + m)}}{\\cos{(\\mathbf{s} + m)}}", "derivation": "a{(\\mathbf{s},m)} = \\cos{(\\mathbf{s} + m)} and \\log{(a{(\\mathbf{s},m)})} = \\log{(\\cos{(\\mathbf{s} + m)})} and \\frac{\\partial}{\\partial \\mathbf{s}} \\log{(a{(\\mathbf{s},m)})} = \\frac{\\partial}{\\partial \\mathbf{s}} \\log{(\\cos{(\\mathbf{s} + m)})} and \\frac{\\frac{\\partial}{\\partial \\mathbf{s}} a{(\\mathbf{s},m)}}{a{(\\mathbf{s},m)}} = - \\frac{\\sin{(\\mathbf{s} + m)}}{\\cos{(\\mathbf{s} + m)}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))))"], [["log", 1], "Equality(log(Function('a')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), log(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(log(Function('a')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(log(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('a')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True)), Integer(-1)), Derivative(Function('a')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Pow(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(y,E_{x})} = \\log{(- E_{x} + y)}, then obtain \\frac{\\operatorname{L_{\\varepsilon}}{(y,E_{x})}}{\\int \\log{(- E_{x} + y)} dE_{x}} = \\frac{\\log{(- E_{x} + y)}}{\\int \\log{(- E_{x} + y)} dE_{x}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(y,E_{x})} = \\log{(- E_{x} + y)} and \\int \\operatorname{L_{\\varepsilon}}{(y,E_{x})} dE_{x} = \\int \\log{(- E_{x} + y)} dE_{x} and \\frac{\\operatorname{L_{\\varepsilon}}{(y,E_{x})}}{\\int \\operatorname{L_{\\varepsilon}}{(y,E_{x})} dE_{x}} = \\frac{\\log{(- E_{x} + y)}}{\\int \\operatorname{L_{\\varepsilon}}{(y,E_{x})} dE_{x}} and \\frac{\\operatorname{L_{\\varepsilon}}{(y,E_{x})}}{\\int \\log{(- E_{x} + y)} dE_{x}} = \\frac{\\log{(- E_{x} + y)}}{\\int \\log{(- E_{x} + y)} dE_{x}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["divide", 1, "Integral(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integer(-1))), Mul(log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))), Pow(Integral(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('y', commutative=True), Symbol('E_x', commutative=True)), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integer(-1))), Mul(log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('E_x', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\operatorname{t_{1}}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})}, then obtain 1 = \\frac{\\operatorname{n_{2}}{(J_{\\varepsilon})}}{\\operatorname{t_{1}}{(J_{\\varepsilon})}}", "derivation": "\\operatorname{n_{2}}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\operatorname{t_{1}}{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and 1 = \\frac{\\sin{(J_{\\varepsilon})}}{\\operatorname{t_{1}}{(J_{\\varepsilon})}} and 1 = \\frac{\\operatorname{n_{2}}{(J_{\\varepsilon})}}{\\operatorname{t_{1}}{(J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 2, "Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), sin(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Function('n_2')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(E,\\rho_f)} = \\cos{(E - \\rho_f)}, then obtain 0 = - \\log{(\\rho_f + \\delta{(E,\\rho_f)})}^{\\rho_f} + \\log{(\\rho_f + \\cos{(E - \\rho_f)})}^{\\rho_f}", "derivation": "\\delta{(E,\\rho_f)} = \\cos{(E - \\rho_f)} and \\rho_f + \\delta{(E,\\rho_f)} = \\rho_f + \\cos{(E - \\rho_f)} and \\log{(\\rho_f + \\delta{(E,\\rho_f)})} = \\log{(\\rho_f + \\cos{(E - \\rho_f)})} and \\log{(\\rho_f + \\delta{(E,\\rho_f)})}^{\\rho_f} = \\log{(\\rho_f + \\cos{(E - \\rho_f)})}^{\\rho_f} and 0 = - \\log{(\\rho_f + \\delta{(E,\\rho_f)})}^{\\rho_f} + \\log{(\\rho_f + \\cos{(E - \\rho_f)})}^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), cos(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), cos(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["log", 2], "Equality(log(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))), log(Add(Symbol('\\\\rho_f', commutative=True), cos(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))))"], [["power", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(log(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), Pow(log(Add(Symbol('\\\\rho_f', commutative=True), cos(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 4, "Pow(log(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(log(Add(Symbol('\\\\rho_f', commutative=True), Function('\\\\delta')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True))), Pow(log(Add(Symbol('\\\\rho_f', commutative=True), cos(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(v_{z})} = \\sin{(v_{z})}, then derive \\int \\hat{p}_0{(v_{z})} dv_{z} = t_{2} - \\cos{(v_{z})}, then obtain v_{z} (t_{2} - \\cos{(v_{z})}) = v_{z} \\int \\sin{(v_{z})} dv_{z}", "derivation": "\\hat{p}_0{(v_{z})} = \\sin{(v_{z})} and \\int \\hat{p}_0{(v_{z})} dv_{z} = \\int \\sin{(v_{z})} dv_{z} and v_{z} \\int \\hat{p}_0{(v_{z})} dv_{z} = v_{z} \\int \\sin{(v_{z})} dv_{z} and \\int \\hat{p}_0{(v_{z})} dv_{z} = t_{2} - \\cos{(v_{z})} and v_{z} (t_{2} - \\cos{(v_{z})}) = v_{z} \\int \\sin{(v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["times", 2, "Symbol('v_z', commutative=True)"], "Equality(Mul(Symbol('v_z', commutative=True), Integral(Function('\\\\hat{p}_0')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Symbol('v_z', commutative=True), Integral(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('v_z', commutative=True), Add(Symbol('t_2', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True))))), Mul(Symbol('v_z', commutative=True), Integral(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(L)} = \\cos{(L)}, then obtain - (\\frac{\\cos{(L)}}{L})^{L} + \\frac{\\varphi{(L)}}{L} = - (\\frac{\\cos{(L)}}{L})^{L} + \\frac{\\cos{(L)}}{L}", "derivation": "\\varphi{(L)} = \\cos{(L)} and \\frac{\\varphi{(L)}}{L} = \\frac{\\cos{(L)}}{L} and (\\frac{\\varphi{(L)}}{L})^{L} = (\\frac{\\cos{(L)}}{L})^{L} and - (\\frac{\\varphi{(L)}}{L})^{L} + \\frac{\\varphi{(L)}}{L} = - (\\frac{\\varphi{(L)}}{L})^{L} + \\frac{\\cos{(L)}}{L} and - (\\frac{\\cos{(L)}}{L})^{L} + \\frac{\\varphi{(L)}}{L} = - (\\frac{\\cos{(L)}}{L})^{L} + \\frac{\\cos{(L)}}{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["divide", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["minus", 2, "Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True))), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\nabla,\\Psi)} = \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla, then derive \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = \\Psi \\nabla + \\mathbf{D}, then obtain - \\Psi^{\\dagger}{(\\nabla,\\Psi)} + \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = - \\Psi^{\\dagger}{(\\nabla,\\Psi)} + \\int \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla d\\nabla", "derivation": "\\Psi^{\\dagger}{(\\nabla,\\Psi)} = \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla and \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = \\int \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla d\\nabla and \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = \\Psi \\nabla + \\mathbf{D} and \\int \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla d\\nabla = \\Psi \\nabla + \\mathbf{D} and - \\Psi^{\\dagger}{(\\nabla,\\Psi)} + \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = \\Psi \\nabla + \\mathbf{D} - \\Psi^{\\dagger}{(\\nabla,\\Psi)} and - \\Psi^{\\dagger}{(\\nabla,\\Psi)} + \\int \\Psi^{\\dagger}{(\\nabla,\\Psi)} d\\nabla = - \\Psi^{\\dagger}{(\\nabla,\\Psi)} + \\int \\frac{\\partial}{\\partial \\nabla} \\Psi \\nabla d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 3, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\pi{(f_{\\mathbf{v}},\\sigma_x)} = \\sigma_x^{f_{\\mathbf{v}}}, then obtain \\frac{\\sigma_x^{f_{\\mathbf{v}}} (\\pi^{2}{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}}}{\\pi{(f_{\\mathbf{v}},\\sigma_x)}} = \\frac{\\sigma_x^{f_{\\mathbf{v}}} (\\sigma_x^{f_{\\mathbf{v}}} \\pi{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}}}{\\pi{(f_{\\mathbf{v}},\\sigma_x)}}", "derivation": "\\pi{(f_{\\mathbf{v}},\\sigma_x)} = \\sigma_x^{f_{\\mathbf{v}}} and \\pi^{2}{(f_{\\mathbf{v}},\\sigma_x)} = \\sigma_x^{f_{\\mathbf{v}}} \\pi{(f_{\\mathbf{v}},\\sigma_x)} and (\\pi^{2}{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}} = (\\sigma_x^{f_{\\mathbf{v}}} \\pi{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}} and \\frac{\\sigma_x^{f_{\\mathbf{v}}} (\\pi^{2}{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}}}{\\pi{(f_{\\mathbf{v}},\\sigma_x)}} = \\frac{\\sigma_x^{f_{\\mathbf{v}}} (\\sigma_x^{f_{\\mathbf{v}}} \\pi{(f_{\\mathbf{v}},\\sigma_x)})^{f_{\\mathbf{v}}}}{\\pi{(f_{\\mathbf{v}},\\sigma_x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 1, "Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{B})} = e^{e^{\\mathbf{B}}}, then derive \\frac{d}{d \\mathbf{B}} \\dot{z}{(\\mathbf{B})} = e^{\\mathbf{B}} e^{e^{\\mathbf{B}}}, then obtain 1 = \\frac{e^{- \\mathbf{B}} \\frac{d}{d \\mathbf{B}} e^{e^{\\mathbf{B}}}}{\\dot{z}{(\\mathbf{B})}}", "derivation": "\\dot{z}{(\\mathbf{B})} = e^{e^{\\mathbf{B}}} and \\dot{z}{(\\mathbf{B})} e^{\\mathbf{B}} = e^{\\mathbf{B}} e^{e^{\\mathbf{B}}} and \\frac{d}{d \\mathbf{B}} \\dot{z}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} e^{e^{\\mathbf{B}}} and \\frac{d}{d \\mathbf{B}} \\dot{z}{(\\mathbf{B})} = e^{\\mathbf{B}} e^{e^{\\mathbf{B}}} and e^{\\mathbf{B}} e^{e^{\\mathbf{B}}} = \\frac{d}{d \\mathbf{B}} e^{e^{\\mathbf{B}}} and \\dot{z}{(\\mathbf{B})} e^{\\mathbf{B}} = \\frac{d}{d \\mathbf{B}} e^{e^{\\mathbf{B}}} and 1 = \\frac{e^{- \\mathbf{B}} \\frac{d}{d \\mathbf{B}} e^{e^{\\mathbf{B}}}}{\\dot{z}{(\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), exp(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), exp(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), exp(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Derivative(exp(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(exp(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["divide", 6, "Mul(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{z}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Derivative(exp(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(c,\\mathbf{A})} = \\mathbf{A} + c, then derive - \\mathbf{A} - c + (\\mathbf{A} + c) (\\frac{\\partial}{\\partial \\mathbf{A}} \\varepsilon{(c,\\mathbf{A})} - 1) + \\varepsilon{(c,\\mathbf{A})} = 0, then obtain (\\frac{\\partial}{\\partial \\mathbf{A}} \\varepsilon{(c,\\mathbf{A})} - 1) \\varepsilon{(c,\\mathbf{A})} = 0", "derivation": "\\varepsilon{(c,\\mathbf{A})} = \\mathbf{A} + c and - \\mathbf{A} - c + \\varepsilon{(c,\\mathbf{A})} = 0 and (\\mathbf{A} + c) (- \\mathbf{A} - c + \\varepsilon{(c,\\mathbf{A})}) = 0 and \\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{A} + c) (- \\mathbf{A} - c + \\varepsilon{(c,\\mathbf{A})}) = \\frac{d}{d \\mathbf{A}} 0 and - \\mathbf{A} - c + (\\mathbf{A} + c) (\\frac{\\partial}{\\partial \\mathbf{A}} \\varepsilon{(c,\\mathbf{A})} - 1) + \\varepsilon{(c,\\mathbf{A})} = 0 and (\\frac{\\partial}{\\partial \\mathbf{A}} \\varepsilon{(c,\\mathbf{A})} - 1) \\varepsilon{(c,\\mathbf{A})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(0))"], [["times", 2, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('c', commutative=True)), Add(Derivative(Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1))), Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Derivative(Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(-1)), Function('\\\\varepsilon')(Symbol('c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\pi{(\\eta)} = \\cos{(\\eta)}, then obtain \\frac{\\Psi_{\\lambda} + \\int \\pi{(\\eta)} d\\eta}{\\tilde{g}{(P_{e})} \\cos{(P_{e})}} = \\frac{\\Psi_{\\lambda} + \\int \\cos{(\\eta)} d\\eta}{\\tilde{g}{(P_{e})} \\cos{(P_{e})}}", "derivation": "\\pi{(\\eta)} = \\cos{(\\eta)} and \\int \\pi{(\\eta)} d\\eta = \\int \\cos{(\\eta)} d\\eta and \\Psi_{\\lambda} + \\int \\pi{(\\eta)} d\\eta = \\Psi_{\\lambda} + \\int \\cos{(\\eta)} d\\eta and \\frac{\\Psi_{\\lambda} + \\int \\pi{(\\eta)} d\\eta}{\\tilde{g}{(P_{e})} \\cos{(P_{e})}} = \\frac{\\Psi_{\\lambda} + \\int \\cos{(\\eta)} d\\eta}{\\tilde{g}{(P_{e})} \\cos{(P_{e})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["add", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["divide", 3, "Mul(Function('\\\\tilde{g}')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('\\\\pi')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('P_e', commutative=True)), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('P_e', commutative=True)), Integer(-1)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given u{(B)} = \\cos{(e^{B})}, then obtain (\\frac{- B + u{(B)}}{\\cos{(e^{B})}})^{B} = (\\frac{- B + \\cos{(e^{B})}}{\\cos{(e^{B})}})^{B}", "derivation": "u{(B)} = \\cos{(e^{B})} and - B + u{(B)} = - B + \\cos{(e^{B})} and \\frac{- B + u{(B)}}{\\cos{(e^{B})}} = \\frac{- B + \\cos{(e^{B})}}{\\cos{(e^{B})}} and (\\frac{- B + u{(B)}}{\\cos{(e^{B})}})^{B} = (\\frac{- B + \\cos{(e^{B})}}{\\cos{(e^{B})}})^{B}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('B', commutative=True)), cos(exp(Symbol('B', commutative=True))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('u')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(exp(Symbol('B', commutative=True)))))"], [["divide", 2, "cos(exp(Symbol('B', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('u')(Symbol('B', commutative=True))), Pow(cos(exp(Symbol('B', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(exp(Symbol('B', commutative=True)))), Pow(cos(exp(Symbol('B', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('u')(Symbol('B', commutative=True))), Pow(cos(exp(Symbol('B', commutative=True))), Integer(-1))), Symbol('B', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('B', commutative=True)), cos(exp(Symbol('B', commutative=True)))), Pow(cos(exp(Symbol('B', commutative=True))), Integer(-1))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\mu{(u)} = \\sin{(\\log{(u)})}, then obtain \\int \\frac{d}{d u} \\mu^{2}{(u)} du = \\int \\frac{d}{d u} \\mu{(u)} \\sin{(\\log{(u)})} du", "derivation": "\\mu{(u)} = \\sin{(\\log{(u)})} and \\mu^{2}{(u)} = \\mu{(u)} \\sin{(\\log{(u)})} and \\frac{d}{d u} \\mu^{2}{(u)} = \\frac{d}{d u} \\mu{(u)} \\sin{(\\log{(u)})} and \\int \\frac{d}{d u} \\mu^{2}{(u)} du = \\int \\frac{d}{d u} \\mu{(u)} \\sin{(\\log{(u)})} du", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('u', commutative=True)), sin(log(Symbol('u', commutative=True))))"], [["times", 1, "Function('\\\\mu')(Symbol('u', commutative=True))"], "Equality(Pow(Function('\\\\mu')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('\\\\mu')(Symbol('u', commutative=True)), sin(log(Symbol('u', commutative=True)))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu')(Symbol('u', commutative=True)), Integer(2)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\mu')(Symbol('u', commutative=True)), sin(log(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\mu')(Symbol('u', commutative=True)), Integer(2)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Mul(Function('\\\\mu')(Symbol('u', commutative=True)), sin(log(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given V{(v_{2})} = v_{2}, then derive \\frac{d}{d v_{2}} V{(v_{2})} = 1, then obtain \\frac{d^{2}}{d v_{2}d V{(v_{2})}} V{(v_{2})} = \\frac{d}{d v_{2}} 1", "derivation": "V{(v_{2})} = v_{2} and \\frac{d}{d v_{2}} V{(v_{2})} = \\frac{d}{d v_{2}} v_{2} and \\frac{d}{d v_{2}} V{(v_{2})} = 1 and \\frac{d}{d v_{2}} v_{2} = 1 and \\frac{d}{d V{(v_{2})}} V{(v_{2})} = 1 and \\frac{d^{2}}{d v_{2}d V{(v_{2})}} V{(v_{2})} = \\frac{d}{d v_{2}} 1", "srepr_derivation": [["renaming_premise", "Equality(Function('V')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('V')(Symbol('v_2', commutative=True)), Tuple(Function('V')(Symbol('v_2', commutative=True)), Integer(1))), Integer(1))"], [["differentiate", 5, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('v_2', commutative=True)), Tuple(Function('V')(Symbol('v_2', commutative=True)), Integer(1)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\Omega)} = \\log{(\\Omega)} and q{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{t_{1}}^{2}{(\\Omega)}, then obtain q{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{t_{1}}{(\\Omega)} \\log{(\\Omega)}", "derivation": "\\operatorname{t_{1}}{(\\Omega)} = \\log{(\\Omega)} and \\operatorname{t_{1}}^{2}{(\\Omega)} = \\operatorname{t_{1}}{(\\Omega)} \\log{(\\Omega)} and \\frac{d}{d \\Omega} \\operatorname{t_{1}}^{2}{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{t_{1}}{(\\Omega)} \\log{(\\Omega)} and q{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{t_{1}}^{2}{(\\Omega)} and q{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{t_{1}}{(\\Omega)} \\log{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Function('t_1')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('t_1')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('t_1')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Pow(Function('t_1')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Function('t_1')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\Omega', commutative=True)), Derivative(Pow(Function('t_1')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('q')(Symbol('\\\\Omega', commutative=True)), Derivative(Mul(Function('t_1')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(A_{y})} = \\frac{d}{d A_{y}} \\cos{(A_{y})} and \\varepsilon_{0}{(A_{y})} = - \\sin{(A_{y})}, then derive E{(A_{y})} = - \\sin{(A_{y})}, then obtain - \\frac{E{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})}}{\\sin{(A_{y})}} = - \\frac{\\varepsilon_{0}{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})}}{\\sin{(A_{y})}}", "derivation": "E{(A_{y})} = \\frac{d}{d A_{y}} \\cos{(A_{y})} and E{(A_{y})} = - \\sin{(A_{y})} and \\varepsilon_{0}{(A_{y})} = - \\sin{(A_{y})} and E{(A_{y})} = \\varepsilon_{0}{(A_{y})} and E{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})} = \\varepsilon_{0}{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})} and - \\frac{E{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})}}{\\sin{(A_{y})}} = - \\frac{\\varepsilon_{0}{(A_{y})} \\frac{d}{d A_{y}} E{(A_{y})}}{\\sin{(A_{y})}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('A_y', commutative=True)), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E')(Symbol('A_y', commutative=True)), Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)))"], [["times", 4, "Derivative(Function('E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Mul(Function('E')(Symbol('A_y', commutative=True)), Derivative(Function('E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Derivative(Function('E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["divide", 5, "Mul(Integer(-1), sin(Symbol('A_y', commutative=True)))"], "Equality(Mul(Integer(-1), Function('E')(Symbol('A_y', commutative=True)), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(Function('E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('A_y', commutative=True)), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(Function('E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(\\theta_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}, then derive \\chi{(\\theta_1,\\theta_2)} = \\frac{1}{\\theta_1}, then obtain \\chi^{\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}}}{(\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1})^{\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}}}", "derivation": "\\chi{(\\theta_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1} and \\chi{(\\theta_1,\\theta_2)} = \\frac{1}{\\theta_1} and \\frac{1}{\\theta_1} = \\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1} and \\chi^{\\theta_1}{(\\theta_1,\\theta_2)} = (\\frac{1}{\\theta_1})^{\\theta_1} and \\chi^{\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}}}{(\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1})^{\\frac{1}{\\frac{\\partial}{\\partial \\theta_2} \\frac{\\theta_2}{\\theta_1}}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\chi')(Pow(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(\\psi^*,\\dot{z})} = - \\dot{z} + \\psi^*, then derive \\frac{\\partial}{\\partial \\psi^*} \\rho_{f}{(\\psi^*,\\dot{z})} = 1, then obtain \\dot{z} - \\psi^* - \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) = \\dot{z} - \\psi^* - 1", "derivation": "\\rho_{f}{(\\psi^*,\\dot{z})} = - \\dot{z} + \\psi^* and \\frac{\\partial}{\\partial \\psi^*} \\rho_{f}{(\\psi^*,\\dot{z})} = \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) and \\frac{\\partial}{\\partial \\psi^*} \\rho_{f}{(\\psi^*,\\dot{z})} = 1 and \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) = 1 and \\rho_{f}{(\\psi^*,\\dot{z})} + \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) = \\rho_{f}{(\\psi^*,\\dot{z})} + 1 and - \\dot{z} + \\psi^* + \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) = - \\dot{z} + \\psi^* + 1 and \\dot{z} - \\psi^* - \\frac{\\partial}{\\partial \\psi^*} (- \\dot{z} + \\psi^*) = \\dot{z} - \\psi^* - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Add(Function('\\\\rho_f')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True), Integer(1)))"], [["divide", 6, "Integer(-1)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{S}{(\\omega)} = \\log{(\\omega)} and \\mathbf{A}{(M,A_{z})} = \\frac{\\cos{(A_{z})}}{M}, then obtain (\\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}} - \\frac{\\cos{(A_{z})}}{M})^{\\omega} = (1 - \\frac{\\cos{(A_{z})}}{M})^{\\omega}", "derivation": "\\mathbf{S}{(\\omega)} = \\log{(\\omega)} and \\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}} = 1 and \\mathbf{A}{(M,A_{z})} = \\frac{\\cos{(A_{z})}}{M} and \\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}} - \\frac{\\cos{(A_{z})}}{M} = 1 - \\frac{\\cos{(A_{z})}}{M} and - \\mathbf{A}{(M,A_{z})} + \\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}} = 1 - \\mathbf{A}{(M,A_{z})} and (- \\mathbf{A}{(M,A_{z})} + \\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}})^{\\omega} = (1 - \\mathbf{A}{(M,A_{z})})^{\\omega} and (\\frac{\\mathbf{S}{(\\omega)}}{\\log{(\\omega)}} - \\frac{\\cos{(A_{z})}}{M})^{\\omega} = (1 - \\frac{\\cos{(A_{z})}}{M})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1))), Integer(1))"], ["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('A_z', commutative=True))), Mul(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)))))"], [["power", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('A_z', commutative=True))), Mul(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1)))), Symbol('\\\\omega', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('M', commutative=True), Symbol('A_z', commutative=True)))), Symbol('\\\\omega', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Add(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('A_z', commutative=True)))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given h{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\operatorname{z^{*}}{(\\hat{H}_l)} = \\hat{H}_l, then obtain \\frac{d}{d \\hat{H}_l} \\int 0 d\\hat{H}_l = \\frac{d}{d \\hat{H}_l} \\int \\hat{H}_l (- h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}) d\\hat{H}_l", "derivation": "h{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and 0 = - h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)} and \\operatorname{z^{*}}{(\\hat{H}_l)} = \\hat{H}_l and 0 = (- h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}) \\operatorname{z^{*}}{(\\hat{H}_l)} and \\int 0 d\\hat{H}_l = \\int (- h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}) \\operatorname{z^{*}}{(\\hat{H}_l)} d\\hat{H}_l and \\frac{d}{d \\hat{H}_l} \\int 0 d\\hat{H}_l = \\frac{d}{d \\hat{H}_l} \\int (- h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}) \\operatorname{z^{*}}{(\\hat{H}_l)} d\\hat{H}_l and \\frac{d}{d \\hat{H}_l} \\int 0 d\\hat{H}_l = \\frac{d}{d \\hat{H}_l} \\int \\hat{H}_l (- h{(\\hat{H}_l)} + \\log{(\\hat{H}_l)}) d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Function('h')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('h')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], [["times", 2, "Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True))), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True))), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integral(Mul(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True))), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\hat{H}_l', commutative=True), Add(Mul(Integer(-1), Function('h')(Symbol('\\\\hat{H}_l', commutative=True))), log(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(\\sigma_p)} = \\int \\sin{(\\sigma_p)} d\\sigma_p, then derive Q{(\\sigma_p)} = \\mathbf{F} - \\cos{(\\sigma_p)}, then obtain \\frac{\\int \\sin{(\\sigma_p)} d\\sigma_p}{\\mathbf{F}} = \\frac{\\mathbf{F} - \\cos{(\\sigma_p)}}{\\mathbf{F}}", "derivation": "Q{(\\sigma_p)} = \\int \\sin{(\\sigma_p)} d\\sigma_p and Q{(\\sigma_p)} = \\mathbf{F} - \\cos{(\\sigma_p)} and \\frac{Q{(\\sigma_p)}}{\\mathbf{F}} = \\frac{\\mathbf{F} - \\cos{(\\sigma_p)}}{\\mathbf{F}} and \\frac{\\int \\sin{(\\sigma_p)} d\\sigma_p}{\\mathbf{F}} = \\frac{\\mathbf{F} - \\cos{(\\sigma_p)}}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\sigma_p', commutative=True)), Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('Q')(Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["divide", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('Q')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given A{(\\pi,\\varphi)} = \\log{(\\pi \\varphi)} and \\operatorname{A_{z}}{(\\pi,\\varphi)} = e^{\\varphi A^{\\pi}{(\\pi,\\varphi)}}, then obtain \\int \\frac{\\operatorname{A_{z}}{(\\pi,\\varphi)}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} d\\pi = \\int \\frac{e^{\\varphi \\log{(\\pi \\varphi)}^{\\pi}}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} d\\pi", "derivation": "A{(\\pi,\\varphi)} = \\log{(\\pi \\varphi)} and A^{\\pi}{(\\pi,\\varphi)} = \\log{(\\pi \\varphi)}^{\\pi} and \\varphi A^{\\pi}{(\\pi,\\varphi)} = \\varphi \\log{(\\pi \\varphi)}^{\\pi} and e^{\\varphi A^{\\pi}{(\\pi,\\varphi)}} = e^{\\varphi \\log{(\\pi \\varphi)}^{\\pi}} and \\operatorname{A_{z}}{(\\pi,\\varphi)} = e^{\\varphi A^{\\pi}{(\\pi,\\varphi)}} and \\operatorname{A_{z}}{(\\pi,\\varphi)} = e^{\\varphi \\log{(\\pi \\varphi)}^{\\pi}} and \\frac{\\operatorname{A_{z}}{(\\pi,\\varphi)}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} = \\frac{e^{\\varphi \\log{(\\pi \\varphi)}^{\\pi}}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} and \\int \\frac{\\operatorname{A_{z}}{(\\pi,\\varphi)}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} d\\pi = \\int \\frac{e^{\\varphi \\log{(\\pi \\varphi)}^{\\pi}}}{- \\varphi + A^{\\pi}{(\\pi,\\varphi)}} d\\pi", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["times", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True)))), exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True)))))"], [["divide", 6, "Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True))), Integer(-1)), exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True))))))"], [["integrate", 7, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('A_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Pow(Function('A')(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\pi', commutative=True))), Integer(-1)), exp(Mul(Symbol('\\\\varphi', commutative=True), Pow(log(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\pi', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(C_{1},A_{1})} = - C_{1} + \\log{(A_{1})}, then obtain C_{1} (A_{1} + 2 \\operatorname{f^{\\prime}}{(C_{1},A_{1})} - \\log{(A_{1})})^{2} = C_{1} (A_{1} - C_{1} + \\operatorname{f^{\\prime}}{(C_{1},A_{1})})^{2}", "derivation": "\\operatorname{f^{\\prime}}{(C_{1},A_{1})} = - C_{1} + \\log{(A_{1})} and A_{1} + 2 \\operatorname{f^{\\prime}}{(C_{1},A_{1})} = A_{1} - C_{1} + \\operatorname{f^{\\prime}}{(C_{1},A_{1})} + \\log{(A_{1})} and A_{1} + 2 \\operatorname{f^{\\prime}}{(C_{1},A_{1})} - \\log{(A_{1})} = A_{1} - C_{1} + \\operatorname{f^{\\prime}}{(C_{1},A_{1})} and (A_{1} + 2 \\operatorname{f^{\\prime}}{(C_{1},A_{1})} - \\log{(A_{1})})^{2} = (A_{1} - C_{1} + \\operatorname{f^{\\prime}}{(C_{1},A_{1})})^{2} and C_{1} (A_{1} + 2 \\operatorname{f^{\\prime}}{(C_{1},A_{1})} - \\log{(A_{1})})^{2} = C_{1} (A_{1} - C_{1} + \\operatorname{f^{\\prime}}{(C_{1},A_{1})})^{2}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), log(Symbol('A_1', commutative=True))))"], [["add", 1, "Add(Symbol('A_1', commutative=True), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True))))"], [["minus", 2, "log(Symbol('A_1', commutative=True))"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), log(Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), log(Symbol('A_1', commutative=True)))), Integer(2)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Integer(2)))"], [["times", 4, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), log(Symbol('A_1', commutative=True)))), Integer(2))), Mul(Symbol('C_1', commutative=True), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('f^{\\\\prime}')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\phi_1)} = e^{\\sin{(\\phi_1)}}, then derive \\frac{d}{d \\phi_1} \\operatorname{F_{N}}{(\\phi_1)} = e^{\\sin{(\\phi_1)}} \\cos{(\\phi_1)}, then obtain e^{\\sin{(\\phi_1)}} \\cos{(\\phi_1)} = \\frac{d}{d \\phi_1} e^{\\sin{(\\phi_1)}}", "derivation": "\\operatorname{F_{N}}{(\\phi_1)} = e^{\\sin{(\\phi_1)}} and \\frac{d}{d \\phi_1} \\operatorname{F_{N}}{(\\phi_1)} = \\frac{d}{d \\phi_1} e^{\\sin{(\\phi_1)}} and \\frac{d}{d \\phi_1} \\operatorname{F_{N}}{(\\phi_1)} = e^{\\sin{(\\phi_1)}} \\cos{(\\phi_1)} and e^{\\sin{(\\phi_1)}} \\cos{(\\phi_1)} = \\frac{d}{d \\phi_1} e^{\\sin{(\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\phi_1', commutative=True)), exp(sin(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(exp(sin(Symbol('\\\\phi_1', commutative=True))), cos(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(exp(sin(Symbol('\\\\phi_1', commutative=True))), cos(Symbol('\\\\phi_1', commutative=True))), Derivative(exp(sin(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(n_{2},P_{e})} = P_{e}^{n_{2}} and y{(n_{2},P_{e})} = P_{e}^{n_{2}}, then obtain - \\frac{P_{e}^{n_{2}}}{2 y{(n_{2},P_{e})}} + y{(n_{2},P_{e})} = P_{e}^{n_{2}} - \\frac{P_{e}^{n_{2}}}{2 y{(n_{2},P_{e})}}", "derivation": "I{(n_{2},P_{e})} = P_{e}^{n_{2}} and y{(n_{2},P_{e})} = P_{e}^{n_{2}} and \\frac{1}{2} = \\frac{P_{e}^{n_{2}}}{2 I{(n_{2},P_{e})}} and y{(n_{2},P_{e})} - \\frac{1}{2} = P_{e}^{n_{2}} - \\frac{1}{2} and y{(n_{2},P_{e})} = I{(n_{2},P_{e})} and - \\frac{P_{e}^{n_{2}}}{2 I{(n_{2},P_{e})}} + y{(n_{2},P_{e})} = P_{e}^{n_{2}} - \\frac{P_{e}^{n_{2}}}{2 I{(n_{2},P_{e})}} and - \\frac{P_{e}^{n_{2}}}{2 y{(n_{2},P_{e})}} + y{(n_{2},P_{e})} = P_{e}^{n_{2}} - \\frac{P_{e}^{n_{2}}}{2 y{(n_{2},P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 1, "Mul(Integer(2), Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))))"], [["minus", 2, "Rational(1, 2)"], "Equality(Add(Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Rational(-1, 2)), Add(Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Rational(-1, 2)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True))), Add(Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('I')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True))), Add(Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_e', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('y')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{S})} = \\log{(e^{\\mathbf{S}})}, then obtain - \\int 0 d\\mathbf{S} + \\int (\\operatorname{F_{x}}{(\\mathbf{S})} - \\log{(e^{\\mathbf{S}})}) d\\mathbf{S} = 0", "derivation": "\\operatorname{F_{x}}{(\\mathbf{S})} = \\log{(e^{\\mathbf{S}})} and \\operatorname{F_{x}}{(\\mathbf{S})} - \\log{(e^{\\mathbf{S}})} = 0 and \\int (\\operatorname{F_{x}}{(\\mathbf{S})} - \\log{(e^{\\mathbf{S}})}) d\\mathbf{S} = \\int 0 d\\mathbf{S} and - \\int 0 d\\mathbf{S} + \\int (\\operatorname{F_{x}}{(\\mathbf{S})} - \\log{(e^{\\mathbf{S}})}) d\\mathbf{S} = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{S}', commutative=True)), log(exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 1, "log(exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Function('F_x')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{S}', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Add(Function('F_x')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{S}', commutative=True))))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 3, "Integral(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integral(Add(Function('F_x')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\mathbf{S}', commutative=True))))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\ddot{x}{(m)} = \\sin{(m)} and \\sigma_{p}{(m)} = \\log{(\\sin^{m}{(m)})}, then obtain \\frac{d}{d m} (\\ddot{x}^{m}{(m)} + \\log{(\\ddot{x}^{m}{(m)})} - \\sin^{m}{(m)}) = \\frac{d}{d m} (\\ddot{x}^{m}{(m)} + \\sigma_{p}{(m)} - \\sin^{m}{(m)})", "derivation": "\\ddot{x}{(m)} = \\sin{(m)} and \\ddot{x}^{m}{(m)} = \\sin^{m}{(m)} and \\log{(\\ddot{x}^{m}{(m)})} = \\log{(\\sin^{m}{(m)})} and \\sigma_{p}{(m)} = \\log{(\\sin^{m}{(m)})} and \\log{(\\ddot{x}^{m}{(m)})} = \\sigma_{p}{(m)} and \\ddot{x}^{m}{(m)} + \\log{(\\ddot{x}^{m}{(m)})} - \\sin^{m}{(m)} = \\ddot{x}^{m}{(m)} + \\sigma_{p}{(m)} - \\sin^{m}{(m)} and \\frac{d}{d m} (\\ddot{x}^{m}{(m)} + \\log{(\\ddot{x}^{m}{(m)})} - \\sin^{m}{(m)}) = \\frac{d}{d m} (\\ddot{x}^{m}{(m)} + \\sigma_{p}{(m)} - \\sin^{m}{(m)})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), log(Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('m', commutative=True)), log(Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Function('\\\\sigma_p')(Symbol('m', commutative=True)))"], [["add", 5, "Add(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True))))"], "Equality(Add(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), log(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Add(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\sigma_p')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["differentiate", 6, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), log(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Pow(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\sigma_p')(Symbol('m', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(c_{0})} = c_{0} and \\hat{\\mathbf{x}}{(E_{\\lambda})} = E_{\\lambda}, then derive E_{\\lambda} + \\frac{V^{2}{(c_{0})}}{2} = \\int c_{0} dV{(c_{0})}, then obtain \\int V{(c_{0})} dc_{0} = \\frac{c_{0}^{2}}{2} + \\hat{\\mathbf{x}}{(E_{\\lambda})}", "derivation": "V{(c_{0})} = c_{0} and \\int V{(c_{0})} dc_{0} = \\int c_{0} dc_{0} and \\int V{(c_{0})} dV{(c_{0})} = \\int c_{0} dV{(c_{0})} and E_{\\lambda} + \\frac{V^{2}{(c_{0})}}{2} = \\int c_{0} dV{(c_{0})} and E_{\\lambda} + \\frac{c_{0}^{2}}{2} = \\int c_{0} dc_{0} and \\hat{\\mathbf{x}}{(E_{\\lambda})} = E_{\\lambda} and \\frac{c_{0}^{2}}{2} + \\hat{\\mathbf{x}}{(E_{\\lambda})} = \\int c_{0} dc_{0} and \\int V{(c_{0})} dc_{0} = \\frac{c_{0}^{2}}{2} + \\hat{\\mathbf{x}}{(E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('V')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('V')(Symbol('c_0', commutative=True)), Tuple(Function('V')(Symbol('c_0', commutative=True)))), Integral(Symbol('c_0', commutative=True), Tuple(Function('V')(Symbol('c_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Function('V')(Symbol('c_0', commutative=True)), Integer(2)))), Integral(Symbol('c_0', commutative=True), Tuple(Function('V')(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Symbol('c_0', commutative=True), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 7], "Equality(Integral(Function('V')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{S})} = e^{\\mathbf{S}}, then derive \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S} = u + e^{\\mathbf{S}}, then obtain u \\int e^{\\mathbf{S}} d\\mathbf{S} = u \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S}", "derivation": "\\mathbf{H}{(\\mathbf{S})} = e^{\\mathbf{S}} and \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S} = \\int e^{\\mathbf{S}} d\\mathbf{S} and \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S} = u + e^{\\mathbf{S}} and \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S} = u + \\mathbf{H}{(\\mathbf{S})} and \\int e^{\\mathbf{S}} d\\mathbf{S} = u + \\mathbf{H}{(\\mathbf{S})} and u \\int e^{\\mathbf{S}} d\\mathbf{S} = u (u + \\mathbf{H}{(\\mathbf{S})}) and u \\int e^{\\mathbf{S}} d\\mathbf{S} = u \\int \\mathbf{H}{(\\mathbf{S})} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('u', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('u', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('u', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 5, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Symbol('u', commutative=True), Add(Symbol('u', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Symbol('u', commutative=True), Integral(exp(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(Symbol('u', commutative=True), Integral(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\sigma_x)} = e^{\\sigma_x} and \\mu{(\\sigma_x)} = \\sigma_x, then obtain \\int (\\sigma_x + \\operatorname{F_{x}}{(\\sigma_x)}) d\\sigma_x = \\int (\\sigma_x + e^{\\sigma_x}) d\\sigma_x", "derivation": "\\operatorname{F_{x}}{(\\sigma_x)} = e^{\\sigma_x} and \\mu{(\\sigma_x)} = \\sigma_x and \\operatorname{F_{x}}{(\\sigma_x)} + \\mu{(\\sigma_x)} = \\mu{(\\sigma_x)} + e^{\\sigma_x} and \\int (\\operatorname{F_{x}}{(\\sigma_x)} + \\mu{(\\sigma_x)}) d\\sigma_x = \\int (\\mu{(\\sigma_x)} + e^{\\sigma_x}) d\\sigma_x and \\int (\\sigma_x + \\operatorname{F_{x}}{(\\sigma_x)}) d\\sigma_x = \\int (\\sigma_x + e^{\\sigma_x}) d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["add", 1, "Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True))), Add(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Function('F_x')(Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Add(Symbol('\\\\sigma_x', commutative=True), Function('F_x')(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(I,\\dot{z})} = \\frac{\\log{(I)}}{\\dot{z}}, then obtain \\dot{z} (\\mathbf{A}{(I,\\dot{z})} + \\log{(I)} + \\frac{\\log{(I)}}{\\dot{z}}) = \\dot{z} (\\log{(I)} + \\frac{2 \\log{(I)}}{\\dot{z}})", "derivation": "\\mathbf{A}{(I,\\dot{z})} = \\frac{\\log{(I)}}{\\dot{z}} and \\mathbf{A}{(I,\\dot{z})} + \\log{(I)} = \\log{(I)} + \\frac{\\log{(I)}}{\\dot{z}} and \\mathbf{A}{(I,\\dot{z})} + \\log{(I)} + \\frac{\\log{(I)}}{\\dot{z}} = \\log{(I)} + \\frac{2 \\log{(I)}}{\\dot{z}} and \\dot{z} (\\mathbf{A}{(I,\\dot{z})} + \\log{(I)} + \\frac{\\log{(I)}}{\\dot{z}}) = \\dot{z} (\\log{(I)} + \\frac{2 \\log{(I)}}{\\dot{z}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True))))"], [["add", 1, "log(Symbol('I', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('I', commutative=True))), Add(log(Symbol('I', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('I', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True)))), Add(log(Symbol('I', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True)))))"], [["times", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{z}', commutative=True), Add(Function('\\\\mathbf{A}')(Symbol('I', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('I', commutative=True)), Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True))))), Mul(Symbol('\\\\dot{z}', commutative=True), Add(log(Symbol('I', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), log(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\chi{(H,\\hat{H}_{\\lambda})} = \\cos{(H + \\hat{H}_{\\lambda})}, then obtain \\frac{\\partial}{\\partial H} (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} + (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} = \\frac{\\partial}{\\partial H} (\\int \\cos{(H + \\hat{H}_{\\lambda})} dH)^{H} + (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H}", "derivation": "\\chi{(H,\\hat{H}_{\\lambda})} = \\cos{(H + \\hat{H}_{\\lambda})} and \\int \\chi{(H,\\hat{H}_{\\lambda})} dH = \\int \\cos{(H + \\hat{H}_{\\lambda})} dH and (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} = (\\int \\cos{(H + \\hat{H}_{\\lambda})} dH)^{H} and \\frac{\\partial}{\\partial H} (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} = \\frac{\\partial}{\\partial H} (\\int \\cos{(H + \\hat{H}_{\\lambda})} dH)^{H} and \\frac{\\partial}{\\partial H} (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} + (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H} = \\frac{\\partial}{\\partial H} (\\int \\cos{(H + \\hat{H}_{\\lambda})} dH)^{H} + (\\int \\chi{(H,\\hat{H}_{\\lambda})} dH)^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Add(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(cos(Add(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Integral(cos(Add(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Integral(cos(Add(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 4, "Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))"], "Equality(Add(Derivative(Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))), Add(Derivative(Pow(Integral(cos(Add(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Integral(Function('\\\\chi')(Symbol('H', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('H', commutative=True))), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\chi{(v,p)} = v^{p}, then obtain v + \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} \\chi{(v,p)} - 1) = v + \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} v^{p} - 1)", "derivation": "\\chi{(v,p)} = v^{p} and \\frac{\\partial}{\\partial p} \\chi{(v,p)} = \\frac{\\partial}{\\partial p} v^{p} and \\frac{\\partial}{\\partial p} \\chi{(v,p)} - 1 = \\frac{\\partial}{\\partial p} v^{p} - 1 and \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} \\chi{(v,p)} - 1) = \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} v^{p} - 1) and v + \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} \\chi{(v,p)} - 1) = v + \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial p} v^{p} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Derivative(Pow(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 4, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Derivative(Add(Derivative(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Symbol('v', commutative=True), Derivative(Add(Derivative(Pow(Symbol('v', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(\\hat{p},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + 2 \\hat{p}, then obtain (- \\mathbf{B}{(\\delta,\\mathbf{D})} + v^{2}{(\\hat{p},\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = ((\\hat{\\mathbf{r}} + 2 \\hat{p})^{2} - \\mathbf{B}{(\\delta,\\mathbf{D})})^{\\hat{\\mathbf{r}}}", "derivation": "v{(\\hat{p},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} + 2 \\hat{p} and v^{2}{(\\hat{p},\\hat{\\mathbf{r}})} = (\\hat{\\mathbf{r}} + 2 \\hat{p})^{2} and - \\mathbf{B}{(\\delta,\\mathbf{D})} + v^{2}{(\\hat{p},\\hat{\\mathbf{r}})} = (\\hat{\\mathbf{r}} + 2 \\hat{p})^{2} - \\mathbf{B}{(\\delta,\\mathbf{D})} and (- \\mathbf{B}{(\\delta,\\mathbf{D})} + v^{2}{(\\hat{p},\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} = ((\\hat{\\mathbf{r}} + 2 \\hat{p})^{2} - \\mathbf{B}{(\\delta,\\mathbf{D})})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))))"], [["power", 1, 2], "Equality(Pow(Function('v')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))), Integer(2)))"], [["minus", 2, "Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Function('v')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Add(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Function('v')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True))), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(M,\\varepsilon)} = \\frac{M}{\\varepsilon} and Z{(M,\\varepsilon)} = \\ddot{x}^{2}{(M,\\varepsilon)}, then obtain \\frac{\\varepsilon Z{(M,\\varepsilon)}}{M} = \\ddot{x}{(M,\\varepsilon)}", "derivation": "\\ddot{x}{(M,\\varepsilon)} = \\frac{M}{\\varepsilon} and \\ddot{x}^{2}{(M,\\varepsilon)} = \\frac{M \\ddot{x}{(M,\\varepsilon)}}{\\varepsilon} and Z{(M,\\varepsilon)} = \\ddot{x}^{2}{(M,\\varepsilon)} and Z{(M,\\varepsilon)} = \\frac{M \\ddot{x}{(M,\\varepsilon)}}{\\varepsilon} and \\frac{\\varepsilon Z{(M,\\varepsilon)}}{M} = \\ddot{x}{(M,\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["times", 1, "Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)), Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('Z')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 4, "Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True), Function('Z')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('\\\\ddot{x}')(Symbol('M', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(g,\\mathbf{F})} = \\mathbf{F} - g, then obtain - 2 g \\frac{\\partial}{\\partial g} (- g + \\mathbf{J}{(g,\\mathbf{F})}) = - 2 g \\frac{\\partial}{\\partial g} (\\mathbf{F} - 2 g)", "derivation": "\\mathbf{J}{(g,\\mathbf{F})} = \\mathbf{F} - g and - g + \\mathbf{J}{(g,\\mathbf{F})} = \\mathbf{F} - 2 g and \\frac{\\partial}{\\partial g} (- g + \\mathbf{J}{(g,\\mathbf{F})}) = \\frac{\\partial}{\\partial g} (\\mathbf{F} - 2 g) and - 2 g \\frac{\\partial}{\\partial g} (- g + \\mathbf{J}{(g,\\mathbf{F})}) = - 2 g \\frac{\\partial}{\\partial g} (\\mathbf{F} - 2 g)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('g', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(c,s)} = \\sin{(\\frac{c}{s})} and \\lambda{(c,s)} = \\sin^{2}{(\\frac{c}{s})}, then obtain \\frac{\\lambda^{4}{(c,s)}}{s^{4}} = \\frac{\\sin^{8}{(\\frac{c}{s})}}{s^{4}}", "derivation": "\\operatorname{E_{x}}{(c,s)} = \\sin{(\\frac{c}{s})} and \\lambda{(c,s)} = \\sin^{2}{(\\frac{c}{s})} and \\lambda{(c,s)} = \\operatorname{E_{x}}^{2}{(c,s)} and \\frac{\\lambda{(c,s)}}{s} = \\frac{\\operatorname{E_{x}}^{2}{(c,s)}}{s} and \\frac{\\lambda{(c,s)}}{s} = \\frac{\\sin^{2}{(\\frac{c}{s})}}{s} and \\frac{\\lambda^{2}{(c,s)}}{s^{2}} = \\frac{\\sin^{4}{(\\frac{c}{s})}}{s^{2}} and \\frac{\\lambda^{4}{(c,s)}}{s^{4}} = \\frac{\\sin^{8}{(\\frac{c}{s})}}{s^{4}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('c', commutative=True), Symbol('s', commutative=True)), sin(Mul(Symbol('c', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Pow(sin(Mul(Symbol('c', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Pow(Function('E_x')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Integer(2)))"], [["times", 3, "Pow(Symbol('s', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(Function('E_x')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Pow(sin(Mul(Symbol('c', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(2))))"], [["power", 5, 2], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-2)), Pow(Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Symbol('s', commutative=True), Integer(-2)), Pow(sin(Mul(Symbol('c', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(4))))"], [["power", 6, 2], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-4)), Pow(Function('\\\\lambda')(Symbol('c', commutative=True), Symbol('s', commutative=True)), Integer(4))), Mul(Pow(Symbol('s', commutative=True), Integer(-4)), Pow(sin(Mul(Symbol('c', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Integer(8))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\sigma_x)} = \\sin{(\\sigma_x)}, then derive \\int \\operatorname{P_{e}}{(\\sigma_x)} d\\sigma_x = g_{\\varepsilon} - \\cos{(\\sigma_x)}, then obtain \\int \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} \\int \\sin{(\\sigma_x)} d\\sigma_x dg_{\\varepsilon} = \\int (g_{\\varepsilon} - \\cos{(\\sigma_x)}) \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} dg_{\\varepsilon}", "derivation": "\\operatorname{P_{e}}{(\\sigma_x)} = \\sin{(\\sigma_x)} and \\int \\operatorname{P_{e}}{(\\sigma_x)} d\\sigma_x = \\int \\sin{(\\sigma_x)} d\\sigma_x and \\int \\operatorname{P_{e}}{(\\sigma_x)} d\\sigma_x = g_{\\varepsilon} - \\cos{(\\sigma_x)} and \\int \\sin{(\\sigma_x)} d\\sigma_x = g_{\\varepsilon} - \\cos{(\\sigma_x)} and \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} \\int \\sin{(\\sigma_x)} d\\sigma_x = (g_{\\varepsilon} - \\cos{(\\sigma_x)}) \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} and \\int \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} \\int \\sin{(\\sigma_x)} d\\sigma_x dg_{\\varepsilon} = \\int (g_{\\varepsilon} - \\cos{(\\sigma_x)}) \\cos{(\\sin{(\\operatorname{P_{e}}{(\\sigma_x)})})} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["times", 4, "cos(sin(Function('P_e')(Symbol('\\\\sigma_x', commutative=True))))"], "Equality(Mul(cos(sin(Function('P_e')(Symbol('\\\\sigma_x', commutative=True)))), Integral(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), cos(sin(Function('P_e')(Symbol('\\\\sigma_x', commutative=True))))))"], [["integrate", 5, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(cos(sin(Function('P_e')(Symbol('\\\\sigma_x', commutative=True)))), Integral(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), cos(sin(Function('P_e')(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(B,\\varphi^*,L)} = \\frac{L \\varphi^*}{B}, then obtain \\frac{\\partial}{\\partial \\varphi^*} - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{(\\operatorname{A_{z}}^{\\varphi^*}{(B,\\varphi^*,L)})} = \\frac{\\partial}{\\partial \\varphi^*} - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{((\\frac{L \\varphi^*}{B})^{\\varphi^*})}", "derivation": "\\operatorname{A_{z}}{(B,\\varphi^*,L)} = \\frac{L \\varphi^*}{B} and \\operatorname{A_{z}}^{\\varphi^*}{(B,\\varphi^*,L)} = (\\frac{L \\varphi^*}{B})^{\\varphi^*} and \\sin{(\\operatorname{A_{z}}^{\\varphi^*}{(B,\\varphi^*,L)})} = \\sin{((\\frac{L \\varphi^*}{B})^{\\varphi^*})} and - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{(\\operatorname{A_{z}}^{\\varphi^*}{(B,\\varphi^*,L)})} = - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{((\\frac{L \\varphi^*}{B})^{\\varphi^*})} and \\frac{\\partial}{\\partial \\varphi^*} - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{(\\operatorname{A_{z}}^{\\varphi^*}{(B,\\varphi^*,L)})} = \\frac{\\partial}{\\partial \\varphi^*} - \\operatorname{A_{z}}{(B,\\varphi^*,L)} \\sin{((\\frac{L \\varphi^*}{B})^{\\varphi^*})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('L', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('L', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi^*', commutative=True))), sin(Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('L', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(Integer(-1), Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), sin(Pow(Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), sin(Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('L', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), sin(Pow(Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('A_z')(Symbol('B', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('L', commutative=True)), sin(Pow(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('L', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{r})} = \\sin{(\\log{(\\mathbf{r})})} and \\operatorname{F_{c}}{(\\mathbf{r})} = \\int 2 \\mathbf{D}{(\\mathbf{r})} d\\mathbf{r}, then obtain \\operatorname{F_{c}}{(\\mathbf{r})} = \\int (\\mathbf{D}{(\\mathbf{r})} + \\sin{(\\log{(\\mathbf{r})})}) d\\mathbf{r}", "derivation": "\\mathbf{D}{(\\mathbf{r})} = \\sin{(\\log{(\\mathbf{r})})} and 2 \\mathbf{D}{(\\mathbf{r})} = \\mathbf{D}{(\\mathbf{r})} + \\sin{(\\log{(\\mathbf{r})})} and \\int 2 \\mathbf{D}{(\\mathbf{r})} d\\mathbf{r} = \\int (\\mathbf{D}{(\\mathbf{r})} + \\sin{(\\log{(\\mathbf{r})})}) d\\mathbf{r} and \\operatorname{F_{c}}{(\\mathbf{r})} = \\int 2 \\mathbf{D}{(\\mathbf{r})} d\\mathbf{r} and \\operatorname{F_{c}}{(\\mathbf{r})} = \\int (\\mathbf{D}{(\\mathbf{r})} + \\sin{(\\log{(\\mathbf{r})})}) d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(log(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(log(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('F_c')(Symbol('\\\\mathbf{r}', commutative=True)), Integral(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(log(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given q{(\\mu_0,\\mathbf{P})} = \\int \\mathbf{P} \\mu_0 d\\mu_0 and \\hat{X}{(\\mu_0,\\mathbf{P})} = - \\mathbf{P} \\mu_0, then obtain (\\mu_0 + q{(\\mu_0,\\mathbf{P})})^{\\mu_0} + \\hat{X}{(\\mu_0,\\mathbf{P})} = - \\mathbf{P} \\mu_0 + (\\mu_0 + q{(\\mu_0,\\mathbf{P})})^{\\mu_0}", "derivation": "q{(\\mu_0,\\mathbf{P})} = \\int \\mathbf{P} \\mu_0 d\\mu_0 and \\mu_0 + q{(\\mu_0,\\mathbf{P})} = \\mu_0 + \\int \\mathbf{P} \\mu_0 d\\mu_0 and (\\mu_0 + q{(\\mu_0,\\mathbf{P})})^{\\mu_0} = (\\mu_0 + \\int \\mathbf{P} \\mu_0 d\\mu_0)^{\\mu_0} and \\hat{X}{(\\mu_0,\\mathbf{P})} = - \\mathbf{P} \\mu_0 and (\\mu_0 + \\int \\mathbf{P} \\mu_0 d\\mu_0)^{\\mu_0} + \\hat{X}{(\\mu_0,\\mathbf{P})} = - \\mathbf{P} \\mu_0 + (\\mu_0 + \\int \\mathbf{P} \\mu_0 d\\mu_0)^{\\mu_0} and (\\mu_0 + q{(\\mu_0,\\mathbf{P})})^{\\mu_0} + \\hat{X}{(\\mu_0,\\mathbf{P})} = - \\mathbf{P} \\mu_0 + (\\mu_0 + q{(\\mu_0,\\mathbf{P})})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('q')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('q')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["add", 4, "Pow(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('q')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), Function('q')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(n_{2},y^{\\prime})} = \\frac{y^{\\prime}}{n_{2}}, then obtain (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}})^{2} e^{- \\operatorname{A_{x}}{(n_{2},y^{\\prime})}} = \\frac{2 y^{\\prime} (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}}) e^{- \\operatorname{A_{x}}{(n_{2},y^{\\prime})}}}{n_{2}}", "derivation": "\\operatorname{A_{x}}{(n_{2},y^{\\prime})} = \\frac{y^{\\prime}}{n_{2}} and \\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}} = \\frac{2 y^{\\prime}}{n_{2}} and (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}})^{2} = \\frac{2 y^{\\prime} (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}})}{n_{2}} and (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}})^{2} e^{- \\operatorname{A_{x}}{(n_{2},y^{\\prime})}} = \\frac{2 y^{\\prime} (\\operatorname{A_{x}}{(n_{2},y^{\\prime})} + \\frac{y^{\\prime}}{n_{2}}) e^{- \\operatorname{A_{x}}{(n_{2},y^{\\prime})}}}{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(2), Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 2, "Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Pow(Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Integer(2)), Mul(Integer(2), Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True), Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))))"], [["divide", 3, "exp(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Pow(Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Integer(2)), exp(Mul(Integer(-1), Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))), Mul(Integer(2), Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True), Add(Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), exp(Mul(Integer(-1), Function('A_x')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\dot{z}{(\\theta_2,\\eta,M_{E})} = \\frac{\\eta - \\theta_2}{M_{E}}, then derive \\frac{\\partial}{\\partial \\eta} \\dot{z}{(\\theta_2,\\eta,M_{E})} = \\frac{1}{M_{E}}, then obtain \\frac{\\partial}{\\partial \\eta} \\frac{\\eta - \\theta_2}{M_{E}} = \\frac{1}{M_{E}}", "derivation": "\\dot{z}{(\\theta_2,\\eta,M_{E})} = \\frac{\\eta - \\theta_2}{M_{E}} and \\frac{\\partial}{\\partial \\eta} \\dot{z}{(\\theta_2,\\eta,M_{E})} = \\frac{\\partial}{\\partial \\eta} \\frac{\\eta - \\theta_2}{M_{E}} and \\frac{\\partial}{\\partial \\eta} \\dot{z}{(\\theta_2,\\eta,M_{E})} = \\frac{1}{M_{E}} and \\frac{\\partial}{\\partial \\eta} \\frac{\\eta - \\theta_2}{M_{E}} = \\frac{1}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\hat{H}_l{(U)} = \\sin{(U)}, then derive \\sin{(\\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)})} = - \\sin{(\\sin{(U)})}, then obtain \\sin{(\\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)})} = - \\sin{(\\hat{H}_l{(U)})}", "derivation": "\\hat{H}_l{(U)} = \\sin{(U)} and \\frac{d}{d U} \\hat{H}_l{(U)} = \\frac{d}{d U} \\sin{(U)} and \\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)} = \\frac{d^{2}}{d U^{2}} \\sin{(U)} and \\sin{(\\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)})} = \\sin{(\\frac{d^{2}}{d U^{2}} \\sin{(U)})} and \\sin{(\\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)})} = - \\sin{(\\sin{(U)})} and \\sin{(\\frac{d^{2}}{d U^{2}} \\sin{(U)})} = - \\sin{(\\sin{(U)})} and \\sin{(\\frac{d^{2}}{d U^{2}} \\hat{H}_l{(U)})} = - \\sin{(\\hat{H}_l{(U)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2))))"], [["sin", 3], "Equality(sin(Derivative(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))), sin(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(sin(Derivative(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))), Mul(Integer(-1), sin(sin(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(sin(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))), Mul(Integer(-1), sin(sin(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(sin(Derivative(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))), Mul(Integer(-1), sin(Function('\\\\hat{H}_l')(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(l,\\theta)} = - l + \\log{(\\theta)}, then obtain \\int (l - \\log{(\\theta)} + \\int \\frac{\\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)}}{l} dl) d\\theta = \\int (l - \\log{(\\theta)} + \\int (-1) dl) d\\theta", "derivation": "\\operatorname{L_{\\varepsilon}}{(l,\\theta)} = - l + \\log{(\\theta)} and \\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)} = - l and \\frac{\\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)}}{l} = -1 and \\int \\frac{\\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)}}{l} dl = \\int (-1) dl and l - \\log{(\\theta)} + \\int \\frac{\\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)}}{l} dl = l - \\log{(\\theta)} + \\int (-1) dl and \\int (l - \\log{(\\theta)} + \\int \\frac{\\operatorname{L_{\\varepsilon}}{(l,\\theta)} - \\log{(\\theta)}}{l} dl) d\\theta = \\int (l - \\log{(\\theta)} + \\int (-1) dl) d\\theta", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), log(Symbol('\\\\theta', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\theta', commutative=True))"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Symbol('l', commutative=True)))"], [["divide", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))))), Integer(-1))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))))), Tuple(Symbol('l', commutative=True))), Integral(Integer(-1), Tuple(Symbol('l', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('l', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))))), Tuple(Symbol('l', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Integral(Integer(-1), Tuple(Symbol('l', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))))), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('l', commutative=True), Mul(Integer(-1), log(Symbol('\\\\theta', commutative=True))), Integral(Integer(-1), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})} = (e^{m_{s}})^{v_{1}}, then obtain 2 (- m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})}) (e^{m_{s}})^{v_{1}} = (- m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})}) (e^{m_{s}})^{v_{1}} + (- m_{s} + (e^{m_{s}})^{v_{1}}) (e^{m_{s}})^{v_{1}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})} = (e^{m_{s}})^{v_{1}} and - m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})} = - m_{s} + (e^{m_{s}})^{v_{1}} and (- m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})}) (e^{m_{s}})^{v_{1}} = (- m_{s} + (e^{m_{s}})^{v_{1}}) (e^{m_{s}})^{v_{1}} and 2 (- m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})}) (e^{m_{s}})^{v_{1}} = (- m_{s} + \\operatorname{J_{\\varepsilon}}{(v_{1},m_{s})}) (e^{m_{s}})^{v_{1}} + (- m_{s} + (e^{m_{s}})^{v_{1}}) (e^{m_{s}})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True)), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True)))"], [["minus", 1, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))))"], [["times", 2, "Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))))"], [["add", 3, "Mul(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True)))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v_1', commutative=True), Symbol('m_s', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True))), Pow(exp(Symbol('m_s', commutative=True)), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\tilde{g},p)} = - \\tilde{g} + p, then obtain \\int \\frac{(\\mathbf{J}^{p}{(\\tilde{g},p)})^{\\tilde{g}}}{p} d\\tilde{g} = \\int \\frac{((- \\tilde{g} + p)^{p})^{\\tilde{g}}}{p} d\\tilde{g}", "derivation": "\\mathbf{J}{(\\tilde{g},p)} = - \\tilde{g} + p and \\mathbf{J}^{p}{(\\tilde{g},p)} = (- \\tilde{g} + p)^{p} and (\\mathbf{J}^{p}{(\\tilde{g},p)})^{\\tilde{g}} = ((- \\tilde{g} + p)^{p})^{\\tilde{g}} and \\frac{(\\mathbf{J}^{p}{(\\tilde{g},p)})^{\\tilde{g}}}{p} = \\frac{((- \\tilde{g} + p)^{p})^{\\tilde{g}}}{p} and \\int \\frac{(\\mathbf{J}^{p}{(\\tilde{g},p)})^{\\tilde{g}}}{p} d\\tilde{g} = \\int \\frac{((- \\tilde{g} + p)^{p})^{\\tilde{g}}}{p} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 3, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given h{(r)} = e^{r}, then obtain (h{(r)} - \\sin{(h{(r)})} \\frac{d}{d r} h{(r)} + 1) M{(r)} = (e^{r} - \\sin{(h{(r)})} \\frac{d}{d r} h{(r)} + 1) M{(r)}", "derivation": "h{(r)} = e^{r} and \\cos{(h{(r)})} = \\cos{(e^{r})} and h{(r)} + \\frac{d}{d r} \\cos{(e^{r})} = e^{r} + \\frac{d}{d r} \\cos{(e^{r})} and h{(r)} + \\frac{d}{d r} \\cos{(e^{r})} + 1 = e^{r} + \\frac{d}{d r} \\cos{(e^{r})} + 1 and h{(r)} + \\frac{d}{d r} \\cos{(h{(r)})} + 1 = e^{r} + \\frac{d}{d r} \\cos{(h{(r)})} + 1 and (h{(r)} + \\frac{d}{d r} \\cos{(h{(r)})} + 1) M{(r)} = (e^{r} + \\frac{d}{d r} \\cos{(h{(r)})} + 1) M{(r)} and (h{(r)} - \\sin{(h{(r)})} \\frac{d}{d r} h{(r)} + 1) M{(r)} = (e^{r} - \\sin{(h{(r)})} \\frac{d}{d r} h{(r)} + 1) M{(r)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["cos", 1], "Equality(cos(Function('h')(Symbol('r', commutative=True))), cos(exp(Symbol('r', commutative=True))))"], [["add", 1, "Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Add(Function('h')(Symbol('r', commutative=True)), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(exp(Symbol('r', commutative=True)), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('h')(Symbol('r', commutative=True)), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('r', commutative=True)), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('h')(Symbol('r', commutative=True)), Derivative(cos(Function('h')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('r', commutative=True)), Derivative(cos(Function('h')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)))"], [["times", 5, "Function('M')(Symbol('r', commutative=True))"], "Equality(Mul(Add(Function('h')(Symbol('r', commutative=True)), Derivative(cos(Function('h')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Function('M')(Symbol('r', commutative=True))), Mul(Add(exp(Symbol('r', commutative=True)), Derivative(cos(Function('h')(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Function('M')(Symbol('r', commutative=True))))"], [["evaluate_derivatives", 6], "Equality(Mul(Add(Function('h')(Symbol('r', commutative=True)), Mul(Integer(-1), sin(Function('h')(Symbol('r', commutative=True))), Derivative(Function('h')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(1)), Function('M')(Symbol('r', commutative=True))), Mul(Add(exp(Symbol('r', commutative=True)), Mul(Integer(-1), sin(Function('h')(Symbol('r', commutative=True))), Derivative(Function('h')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(1)), Function('M')(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(G)} = \\frac{d}{d G} \\log{(G)} and \\psi^{*}{(\\theta_1,s)} = \\theta_1^{s}, then derive \\psi^{*}{(\\theta_1,s)} - \\frac{1}{G} = \\theta_1^{s} - \\frac{1}{G}, then obtain G + \\operatorname{C_{d}}{(G)} + \\psi^{*}{(\\theta_1,s)} - \\frac{1}{G} = G + \\theta_1^{s} + \\operatorname{C_{d}}{(G)} - \\frac{1}{G}", "derivation": "\\operatorname{C_{d}}{(G)} = \\frac{d}{d G} \\log{(G)} and \\psi^{*}{(\\theta_1,s)} = \\theta_1^{s} and - \\operatorname{C_{d}}{(G)} + \\psi^{*}{(\\theta_1,s)} = \\theta_1^{s} - \\operatorname{C_{d}}{(G)} and \\psi^{*}{(\\theta_1,s)} - \\frac{d}{d G} \\log{(G)} = \\theta_1^{s} - \\frac{d}{d G} \\log{(G)} and \\psi^{*}{(\\theta_1,s)} - \\frac{1}{G} = \\theta_1^{s} - \\frac{1}{G} and G + \\psi^{*}{(\\theta_1,s)} + \\frac{d}{d G} \\log{(G)} - \\frac{1}{G} = G + \\theta_1^{s} + \\frac{d}{d G} \\log{(G)} - \\frac{1}{G} and G + \\operatorname{C_{d}}{(G)} + \\psi^{*}{(\\theta_1,s)} - \\frac{1}{G} = G + \\theta_1^{s} + \\operatorname{C_{d}}{(G)} - \\frac{1}{G}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('G', commutative=True)), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)))"], [["minus", 2, "Function('C_d')(Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('C_d')(Symbol('G', commutative=True))), Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('C_d')(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))), Add(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))))"], [["add", 5, "Add(Symbol('G', commutative=True), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))), Add(Symbol('G', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('G', commutative=True), Function('C_d')(Symbol('G', commutative=True)), Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))), Add(Symbol('G', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('s', commutative=True)), Function('C_d')(Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Symbol('G', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(h)} = \\sin{(h)}, then obtain ((\\operatorname{f_{\\mathbf{p}}}{(h)} + \\sin{(h)}) \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)})^{h} = (2 \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)} \\sin{(h)})^{h}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(h)} = \\sin{(h)} and \\operatorname{f_{\\mathbf{p}}}{(h)} + \\sin{(h)} = 2 \\sin{(h)} and (\\operatorname{f_{\\mathbf{p}}}{(h)} + \\sin{(h)}) \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)} = 2 \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)} \\sin{(h)} and ((\\operatorname{f_{\\mathbf{p}}}{(h)} + \\sin{(h)}) \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)})^{h} = (2 \\operatorname{f_{\\mathbf{p}}}^{- h}{(h)} \\sin{(h)})^{h}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(2), sin(Symbol('h', commutative=True))))"], [["divide", 2, "Pow(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Mul(Add(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(Integer(2), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), sin(Symbol('h', commutative=True))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Add(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Mul(Integer(2), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('h', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), sin(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(x^\\prime)} = \\cos{(x^\\prime)} and n{(x^\\prime)} = \\frac{d}{d x^\\prime} 2 \\mu_{0}{(x^\\prime)}, then obtain \\int n{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} (\\mu_{0}{(x^\\prime)} + \\cos{(x^\\prime)}) dx^\\prime", "derivation": "\\mu_{0}{(x^\\prime)} = \\cos{(x^\\prime)} and 2 \\mu_{0}{(x^\\prime)} = \\mu_{0}{(x^\\prime)} + \\cos{(x^\\prime)} and \\frac{d}{d x^\\prime} 2 \\mu_{0}{(x^\\prime)} = \\frac{d}{d x^\\prime} (\\mu_{0}{(x^\\prime)} + \\cos{(x^\\prime)}) and n{(x^\\prime)} = \\frac{d}{d x^\\prime} 2 \\mu_{0}{(x^\\prime)} and n{(x^\\prime)} = \\frac{d}{d x^\\prime} (\\mu_{0}{(x^\\prime)} + \\cos{(x^\\prime)}) and \\int n{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} (\\mu_{0}{(x^\\prime)} + \\cos{(x^\\prime)}) dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True))), Add(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('x^\\\\prime', commutative=True)), Derivative(Mul(Integer(2), Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('n')(Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('n')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(Add(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\chi{(A_{x},C)} = \\sin{(A_{x} + C)}, then obtain e^{\\chi^{C}{(A_{x},C)}} + \\sin{(A_{x} + C)} = e^{\\sin^{C}{(A_{x} + C)}} + \\sin{(A_{x} + C)}", "derivation": "\\chi{(A_{x},C)} = \\sin{(A_{x} + C)} and \\chi^{C}{(A_{x},C)} = \\sin^{C}{(A_{x} + C)} and e^{\\chi^{C}{(A_{x},C)}} = e^{\\sin^{C}{(A_{x} + C)}} and e^{\\chi^{C}{(A_{x},C)}} + \\sin{(A_{x} + C)} = e^{\\sin^{C}{(A_{x} + C)}} + \\sin{(A_{x} + C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('A_x', commutative=True), Symbol('C', commutative=True)), sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('A_x', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\chi')(Symbol('A_x', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True))), exp(Pow(sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True))), Symbol('C', commutative=True))))"], [["add", 3, "sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True)))"], "Equality(Add(exp(Pow(Function('\\\\chi')(Symbol('A_x', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True))), sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True)))), Add(exp(Pow(sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True))), Symbol('C', commutative=True))), sin(Add(Symbol('A_x', commutative=True), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\dot{z})} = e^{\\dot{z}}, then obtain (\\tilde{g}{(\\dot{z})} + 1) \\cos{(\\tilde{g}{(\\dot{z})} - e^{\\dot{z}})} = e^{\\dot{z}} + 1", "derivation": "\\tilde{g}{(\\dot{z})} = e^{\\dot{z}} and 0 = - \\tilde{g}{(\\dot{z})} + e^{\\dot{z}} and \\tilde{g}{(\\dot{z})} + 1 = e^{\\dot{z}} + 1 and 1 = \\cos{(\\tilde{g}{(\\dot{z})} - e^{\\dot{z}})} and \\tilde{g}{(\\dot{z})} + 1 = (\\tilde{g}{(\\dot{z})} + 1) \\cos{(\\tilde{g}{(\\dot{z})} - e^{\\dot{z}})} and (\\tilde{g}{(\\dot{z})} + 1) \\cos{(\\tilde{g}{(\\dot{z})} - e^{\\dot{z}})} = e^{\\dot{z}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True))), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Integer(1)), Add(exp(Symbol('\\\\dot{z}', commutative=True)), Integer(1)))"], [["cos", 2], "Equality(Integer(1), cos(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True))))))"], [["times", 4, "Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Integer(1))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Integer(1)), Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Integer(1)), cos(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Integer(1)), cos(Add(Function('\\\\tilde{g}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True)))))), Add(exp(Symbol('\\\\dot{z}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(a)} = \\cos{(a)}, then derive \\int \\dot{\\mathbf{r}}{(a)} da = r + \\sin{(a)}, then derive \\mathbf{J} + \\sin{(a)} = r + \\sin{(a)}, then obtain \\int \\dot{\\mathbf{r}}{(a)} da = \\mathbf{J} + \\sin{(a)}", "derivation": "\\dot{\\mathbf{r}}{(a)} = \\cos{(a)} and \\int \\dot{\\mathbf{r}}{(a)} da = \\int \\cos{(a)} da and \\int \\dot{\\mathbf{r}}{(a)} da = r + \\sin{(a)} and \\int \\cos{(a)} da = r + \\sin{(a)} and \\mathbf{J} + \\sin{(a)} = r + \\sin{(a)} and \\int \\dot{\\mathbf{r}}{(a)} da = \\mathbf{J} + \\sin{(a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('r', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('r', commutative=True), sin(Symbol('a', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('a', commutative=True))), Add(Symbol('r', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\eta{(\\mathbf{s})} = \\cos{(e^{\\mathbf{s}})}, then obtain - \\eta{(\\mathbf{s})} - e^{\\mathbf{s}} + \\int \\frac{\\eta{(\\mathbf{s})}}{\\mathbf{s}} d\\mathbf{s} = - \\eta{(\\mathbf{s})} - e^{\\mathbf{s}} + \\int \\frac{\\cos{(e^{\\mathbf{s}})}}{\\mathbf{s}} d\\mathbf{s}", "derivation": "\\eta{(\\mathbf{s})} = \\cos{(e^{\\mathbf{s}})} and \\frac{\\eta{(\\mathbf{s})}}{\\mathbf{s}} = \\frac{\\cos{(e^{\\mathbf{s}})}}{\\mathbf{s}} and \\int \\frac{\\eta{(\\mathbf{s})}}{\\mathbf{s}} d\\mathbf{s} = \\int \\frac{\\cos{(e^{\\mathbf{s}})}}{\\mathbf{s}} d\\mathbf{s} and - e^{\\mathbf{s}} + \\int \\frac{\\eta{(\\mathbf{s})}}{\\mathbf{s}} d\\mathbf{s} = - e^{\\mathbf{s}} + \\int \\frac{\\cos{(e^{\\mathbf{s}})}}{\\mathbf{s}} d\\mathbf{s} and - \\eta{(\\mathbf{s})} - e^{\\mathbf{s}} + \\int \\frac{\\eta{(\\mathbf{s})}}{\\mathbf{s}} d\\mathbf{s} = - \\eta{(\\mathbf{s})} - e^{\\mathbf{s}} + \\int \\frac{\\cos{(e^{\\mathbf{s}})}}{\\mathbf{s}} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True)), cos(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 3, "exp(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["minus", 4, "Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(F_{c},\\Psi^{\\dagger})} = F_{c} + \\Psi^{\\dagger}, then obtain \\iiint \\mathbf{F}^{F_{c}}{(F_{c},\\Psi^{\\dagger})} dF_{c} d\\Psi^{\\dagger} d\\Psi^{\\dagger} = \\iiint (F_{c} + \\Psi^{\\dagger})^{F_{c}} dF_{c} d\\Psi^{\\dagger} d\\Psi^{\\dagger}", "derivation": "\\mathbf{F}{(F_{c},\\Psi^{\\dagger})} = F_{c} + \\Psi^{\\dagger} and \\mathbf{F}^{F_{c}}{(F_{c},\\Psi^{\\dagger})} = (F_{c} + \\Psi^{\\dagger})^{F_{c}} and \\int \\mathbf{F}^{F_{c}}{(F_{c},\\Psi^{\\dagger})} dF_{c} = \\int (F_{c} + \\Psi^{\\dagger})^{F_{c}} dF_{c} and \\iint \\mathbf{F}^{F_{c}}{(F_{c},\\Psi^{\\dagger})} dF_{c} d\\Psi^{\\dagger} = \\iint (F_{c} + \\Psi^{\\dagger})^{F_{c}} dF_{c} d\\Psi^{\\dagger} and \\iiint \\mathbf{F}^{F_{c}}{(F_{c},\\Psi^{\\dagger})} dF_{c} d\\Psi^{\\dagger} d\\Psi^{\\dagger} = \\iiint (F_{c} + \\Psi^{\\dagger})^{F_{c}} dF_{c} d\\Psi^{\\dagger} d\\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Pow(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{F}')(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Add(Symbol('F_c', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\varphi{(A_{z})} = \\sin{(A_{z})}, then derive E + \\varphi{(A_{z})} = A_{1} + \\sin{(A_{z})}, then obtain 2 A_{1} + \\varphi{(A_{z})} = 2 A_{1} + \\sin{(A_{z})}", "derivation": "\\varphi{(A_{z})} = \\sin{(A_{z})} and \\frac{d}{d A_{z}} \\varphi{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})} and \\int \\frac{d}{d A_{z}} \\varphi{(A_{z})} dA_{z} = \\int \\frac{d}{d A_{z}} \\sin{(A_{z})} dA_{z} and E + \\varphi{(A_{z})} = A_{1} + \\sin{(A_{z})} and E + \\varphi{(A_{z})} = A_{1} + \\varphi{(A_{z})} and A_{1} + \\varphi{(A_{z})} = A_{1} + \\sin{(A_{z})} and 2 A_{1} + \\varphi{(A_{z})} = 2 A_{1} + \\sin{(A_{z})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\varphi')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))), Integral(Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E', commutative=True), Function('\\\\varphi')(Symbol('A_z', commutative=True))), Add(Symbol('A_1', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('E', commutative=True), Function('\\\\varphi')(Symbol('A_z', commutative=True))), Add(Symbol('A_1', commutative=True), Function('\\\\varphi')(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('A_1', commutative=True), Function('\\\\varphi')(Symbol('A_z', commutative=True))), Add(Symbol('A_1', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["add", 6, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('A_1', commutative=True)), Function('\\\\varphi')(Symbol('A_z', commutative=True))), Add(Mul(Integer(2), Symbol('A_1', commutative=True)), sin(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(P_{e},v_{1})} = P_{e} - v_{1}, then obtain - P_{e} v_{1} - q + \\frac{v_{1}^{2}}{2} + \\int \\operatorname{E_{n}}{(P_{e},v_{1})} dv_{1} = 0", "derivation": "\\operatorname{E_{n}}{(P_{e},v_{1})} = P_{e} - v_{1} and \\int \\operatorname{E_{n}}{(P_{e},v_{1})} dv_{1} = \\int (P_{e} - v_{1}) dv_{1} and - \\int (P_{e} - v_{1}) dv_{1} + \\int \\operatorname{E_{n}}{(P_{e},v_{1})} dv_{1} = 0 and - P_{e} v_{1} - q + \\frac{v_{1}^{2}}{2} + \\int \\operatorname{E_{n}}{(P_{e},v_{1})} dv_{1} = 0", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('P_e', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('P_e', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))), Integral(Function('E_n')(Symbol('P_e', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Integer(0))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('v_1', commutative=True), Integer(2))), Integral(Function('E_n')(Symbol('P_e', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{g}{(\\theta_1,C_{1})} = - C_{1} + \\theta_1 and \\operatorname{C_{d}}{(\\theta_1,C_{1},A_{1})} = A_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})}, then obtain A_{1} C_{1} (- C_{1} + \\theta_1)^{\\theta_1} = A_{1} C_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})}", "derivation": "\\mathbf{g}{(\\theta_1,C_{1})} = - C_{1} + \\theta_1 and \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})} = (- C_{1} + \\theta_1)^{\\theta_1} and A_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})} = A_{1} (- C_{1} + \\theta_1)^{\\theta_1} and \\operatorname{C_{d}}{(\\theta_1,C_{1},A_{1})} = A_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})} and \\operatorname{C_{d}}{(\\theta_1,C_{1},A_{1})} = A_{1} (- C_{1} + \\theta_1)^{\\theta_1} and C_{1} \\operatorname{C_{d}}{(\\theta_1,C_{1},A_{1})} = A_{1} C_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})} and A_{1} C_{1} (- C_{1} + \\theta_1)^{\\theta_1} = A_{1} C_{1} \\mathbf{g}^{\\theta_1}{(\\theta_1,C_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 2, "Pow(Symbol('A_1', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["times", 4, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\theta_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(v_{1})} = \\log{(e^{v_{1}})}, then obtain v_{1} \\mathbb{I}{(v_{1})} \\log{(e^{v_{1}})} = v_{1} \\log{(e^{v_{1}})}^{2}", "derivation": "\\mathbb{I}{(v_{1})} = \\log{(e^{v_{1}})} and v_{1} \\mathbb{I}{(v_{1})} = v_{1} \\log{(e^{v_{1}})} and v_{1} \\mathbb{I}^{2}{(v_{1})} = v_{1} \\mathbb{I}{(v_{1})} \\log{(e^{v_{1}})} and v_{1} \\mathbb{I}{(v_{1})} \\log{(e^{v_{1}})} = v_{1} \\log{(e^{v_{1}})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True)), log(exp(Symbol('v_1', commutative=True))))"], [["times", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), log(exp(Symbol('v_1', commutative=True)))))"], [["times", 2, "Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True))"], "Equality(Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True)), Integer(2))), Mul(Symbol('v_1', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True)), log(exp(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True)), log(exp(Symbol('v_1', commutative=True)))), Mul(Symbol('v_1', commutative=True), Pow(log(exp(Symbol('v_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{P}{(g)} = e^{g}, then derive \\int (- g + \\mathbf{P}{(g)}) dg = \\mathbf{A} - \\frac{g^{2}}{2} + e^{g}, then obtain - \\sin{(g - \\mathbf{P}{(g)})} + \\int (- g + \\mathbf{P}{(g)}) dg = \\mathbf{A} - \\frac{g^{2}}{2} + e^{g} - \\sin{(g - \\mathbf{P}{(g)})}", "derivation": "\\mathbf{P}{(g)} = e^{g} and - g + \\mathbf{P}{(g)} = - g + e^{g} and - \\sin{(g - \\mathbf{P}{(g)})} = - \\sin{(g - e^{g})} and \\int (- g + \\mathbf{P}{(g)}) dg = \\int (- g + e^{g}) dg and \\int (- g + \\mathbf{P}{(g)}) dg = \\mathbf{A} - \\frac{g^{2}}{2} + e^{g} and - \\sin{(g - e^{g})} + \\int (- g + \\mathbf{P}{(g)}) dg = \\mathbf{A} - \\frac{g^{2}}{2} + e^{g} - \\sin{(g - e^{g})} and - \\sin{(g - \\mathbf{P}{(g)})} + \\int (- g + \\mathbf{P}{(g)}) dg = \\mathbf{A} - \\frac{g^{2}}{2} + e^{g} - \\sin{(g - \\mathbf{P}{(g)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('g', commutative=True)))))), Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), exp(Symbol('g', commutative=True)))))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), exp(Symbol('g', commutative=True))))"], [["add", 5, "Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), exp(Symbol('g', commutative=True))))))"], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), exp(Symbol('g', commutative=True)))))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), exp(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), exp(Symbol('g', commutative=True))))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('g', commutative=True)))))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), exp(Symbol('g', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('g', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(E,Q)} = \\cos{(E Q)}, then derive \\frac{\\partial}{\\partial E} \\operatorname{n_{1}}{(E,Q)} = - Q \\sin{(E Q)}, then obtain \\frac{\\partial^{2}}{\\partial Q\\partial E} \\cos{(E Q)} = \\frac{\\partial}{\\partial Q} - Q \\sin{(E Q)}", "derivation": "\\operatorname{n_{1}}{(E,Q)} = \\cos{(E Q)} and \\frac{\\partial}{\\partial E} \\operatorname{n_{1}}{(E,Q)} = \\frac{\\partial}{\\partial E} \\cos{(E Q)} and \\frac{\\partial}{\\partial E} \\operatorname{n_{1}}{(E,Q)} = - Q \\sin{(E Q)} and \\frac{\\partial}{\\partial E} \\cos{(E Q)} = - Q \\sin{(E Q)} and \\frac{\\partial^{2}}{\\partial Q\\partial E} \\cos{(E Q)} = \\frac{\\partial}{\\partial Q} - Q \\sin{(E Q)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('E', commutative=True), Symbol('Q', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('Q', commutative=True), sin(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('Q', commutative=True), sin(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True)))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(cos(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('Q', commutative=True), sin(Mul(Symbol('E', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(h)} = e^{h}, then obtain (\\mathbf{g}{(h)} \\int \\mathbf{g}{(h)} dh)^{h} = (\\mathbf{g}{(h)} \\int e^{h} dh)^{h}", "derivation": "\\mathbf{g}{(h)} = e^{h} and \\int \\mathbf{g}{(h)} dh = \\int e^{h} dh and e^{h} \\int \\mathbf{g}{(h)} dh = e^{h} \\int e^{h} dh and \\mathbf{g}{(h)} \\int \\mathbf{g}{(h)} dh = \\mathbf{g}{(h)} \\int e^{h} dh and (\\mathbf{g}{(h)} \\int \\mathbf{g}{(h)} dh)^{h} = (\\mathbf{g}{(h)} \\int e^{h} dh)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["times", 2, "exp(Symbol('h', commutative=True))"], "Equality(Mul(exp(Symbol('h', commutative=True)), Integral(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(exp(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Integral(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Integral(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(Mul(Function('\\\\mathbf{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\Omega)} = \\cos{(\\Omega)}, then obtain - \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)} + \\frac{d}{d \\Omega} (- \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)}) = \\frac{d}{d \\Omega} 0", "derivation": "\\mathbf{A}{(\\Omega)} = \\cos{(\\Omega)} and \\mathbf{A}{(\\Omega)} - \\cos{(\\Omega)} = 0 and - \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)} = 0 and \\frac{d}{d \\Omega} (- \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)}) = \\frac{d}{d \\Omega} 0 and - \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)} + \\frac{d}{d \\Omega} (- \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)}) = \\frac{d}{d \\Omega} (- \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)}) and - \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)} + \\frac{d}{d \\Omega} (- \\mathbf{A}{(\\Omega)} + \\cos{(\\Omega)}) = \\frac{d}{d \\Omega} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Integer(0))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\Omega', commutative=True))), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(b)} = e^{b}, then obtain e^{(u{(b)} e^{- b})^{- 2 b} (2 + e^{- b}) (u{(b)} e^{- b} + 1 + e^{- b})} = e^{(u{(b)} e^{- b})^{- 2 b} (2 + e^{- b})^{2}}", "derivation": "u{(b)} = e^{b} and u{(b)} e^{- b} = 1 and u{(b)} e^{- b} + 1 = 2 and u{(b)} e^{- b} + 1 + e^{- b} = 2 + e^{- b} and (u{(b)} e^{- b})^{- b} (u{(b)} e^{- b} + 1 + e^{- b}) = (u{(b)} e^{- b})^{- b} (2 + e^{- b}) and (u{(b)} e^{- b})^{- 2 b} (2 + e^{- b}) (u{(b)} e^{- b} + 1 + e^{- b}) = (u{(b)} e^{- b})^{- 2 b} (2 + e^{- b})^{2} and e^{(u{(b)} e^{- b})^{- 2 b} (2 + e^{- b}) (u{(b)} e^{- b} + 1 + e^{- b})} = e^{(u{(b)} e^{- b})^{- 2 b} (2 + e^{- b})^{2}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["divide", 1, "exp(Symbol('b', commutative=True))"], "Equality(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1))"], [["add", 2, 1], "Equality(Add(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1)), Integer(2))"], [["add", 3, "exp(Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Add(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["divide", 4, "Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Symbol('b', commutative=True))"], "Equality(Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Symbol('b', commutative=True))), Add(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1), exp(Mul(Integer(-1), Symbol('b', commutative=True))))), Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Symbol('b', commutative=True))), Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True))))))"], [["times", 5, "Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Symbol('b', commutative=True))), Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))))"], "Equality(Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))), Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Add(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1), exp(Mul(Integer(-1), Symbol('b', commutative=True))))), Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))), Pow(Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(2))))"], [["exp", 6], "Equality(exp(Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))), Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Add(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(1), exp(Mul(Integer(-1), Symbol('b', commutative=True)))))), exp(Mul(Pow(Mul(Function('u')(Symbol('b', commutative=True)), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('b', commutative=True))), Pow(Add(Integer(2), exp(Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\phi_1,\\Psi)} = \\Psi + \\phi_1, then obtain \\int e^{\\operatorname{M_{E}}^{\\phi_1}{(\\phi_1,\\Psi)}} d\\Psi = \\int e^{(\\Psi + \\phi_1)^{\\phi_1}} d\\Psi", "derivation": "\\operatorname{M_{E}}{(\\phi_1,\\Psi)} = \\Psi + \\phi_1 and \\operatorname{M_{E}}^{\\phi_1}{(\\phi_1,\\Psi)} = (\\Psi + \\phi_1)^{\\phi_1} and e^{\\operatorname{M_{E}}^{\\phi_1}{(\\phi_1,\\Psi)}} = e^{(\\Psi + \\phi_1)^{\\phi_1}} and \\int e^{\\operatorname{M_{E}}^{\\phi_1}{(\\phi_1,\\Psi)}} d\\Psi = \\int e^{(\\Psi + \\phi_1)^{\\phi_1}} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('M_E')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\phi_1', commutative=True))), exp(Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(exp(Pow(Function('M_E')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(exp(Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(V,E_{x})} = - E_{x} + V, then obtain - E_{x} + 2 V - \\operatorname{P_{e}}{(V,E_{x})} \\operatorname{P_{e}}^{- V}{(V,E_{x})} = - E_{x} + 2 V + (E_{x} - V) \\operatorname{P_{e}}^{- V}{(V,E_{x})}", "derivation": "\\operatorname{P_{e}}{(V,E_{x})} = - E_{x} + V and V + \\operatorname{P_{e}}{(V,E_{x})} = - E_{x} + 2 V and - \\operatorname{P_{e}}{(V,E_{x})} = E_{x} - V and - \\operatorname{P_{e}}{(V,E_{x})} \\operatorname{P_{e}}^{- V}{(V,E_{x})} = (E_{x} - V) \\operatorname{P_{e}}^{- V}{(V,E_{x})} and V + \\operatorname{P_{e}}{(V,E_{x})} - \\operatorname{P_{e}}{(V,E_{x})} \\operatorname{P_{e}}^{- V}{(V,E_{x})} = V + (E_{x} - V) \\operatorname{P_{e}}^{- V}{(V,E_{x})} + \\operatorname{P_{e}}{(V,E_{x})} and - E_{x} + 2 V - \\operatorname{P_{e}}{(V,E_{x})} \\operatorname{P_{e}}^{- V}{(V,E_{x})} = - E_{x} + 2 V + (E_{x} - V) \\operatorname{P_{e}}^{- V}{(V,E_{x})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('V', commutative=True)))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('V', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))))"], [["divide", 3, "Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Symbol('V', commutative=True))"], "Equality(Mul(Integer(-1), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), Mul(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))))"], [["add", 4, "Add(Symbol('V', commutative=True), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Add(Symbol('V', commutative=True), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))))), Add(Symbol('V', commutative=True), Mul(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True)))), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('V', commutative=True)), Mul(Integer(-1), Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('V', commutative=True)), Mul(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Pow(Function('P_e')(Symbol('V', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('V', commutative=True))))))"]]}, {"prompt": "Given Q{(\\hat{H}_l,Z)} = Z \\log{(\\hat{H}_l)}, then derive \\int Q{(\\hat{H}_l,Z)} d\\hat{H}_l = M_{E} + Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l, then obtain M_{E} + Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l = Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l + u", "derivation": "Q{(\\hat{H}_l,Z)} = Z \\log{(\\hat{H}_l)} and \\int Q{(\\hat{H}_l,Z)} d\\hat{H}_l = \\int Z \\log{(\\hat{H}_l)} d\\hat{H}_l and \\int Q{(\\hat{H}_l,Z)} d\\hat{H}_l = M_{E} + Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l and M_{E} + Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l = \\int Z \\log{(\\hat{H}_l)} d\\hat{H}_l and M_{E} + Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l = Z \\hat{H}_l \\log{(\\hat{H}_l)} - Z \\hat{H}_l + u", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('M_E', commutative=True), Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('M_E', commutative=True), Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('Z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(b)} = \\sin{(b)} and \\hat{H}_l{(b)} = (- b + \\sin{(b)})^{b}, then obtain \\frac{d}{d b} \\hat{H}_l{(b)} = \\frac{d}{d b} (- b + \\Psi^{\\dagger}{(b)})^{b}", "derivation": "\\Psi^{\\dagger}{(b)} = \\sin{(b)} and - b + \\Psi^{\\dagger}{(b)} = - b + \\sin{(b)} and (- b + \\Psi^{\\dagger}{(b)})^{b} = (- b + \\sin{(b)})^{b} and \\hat{H}_l{(b)} = (- b + \\sin{(b)})^{b} and \\hat{H}_l{(b)} = (- b + \\Psi^{\\dagger}{(b)})^{b} and \\frac{d}{d b} \\hat{H}_l{(b)} = \\frac{d}{d b} (- b + \\Psi^{\\dagger}{(b)})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('b', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\hat{H}_l')(Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["differentiate", 5, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(v_{2})} = \\cos{(v_{2})}, then obtain \\int \\sin^{v}{(v - v_{2} + \\dot{y}{(v_{2})})} dv_{2} = \\int \\sin^{v}{(v - v_{2} + \\cos{(v_{2})})} dv_{2}", "derivation": "\\dot{y}{(v_{2})} = \\cos{(v_{2})} and - v_{2} + \\dot{y}{(v_{2})} = - v_{2} + \\cos{(v_{2})} and v - v_{2} + \\dot{y}{(v_{2})} = v - v_{2} + \\cos{(v_{2})} and \\sin{(v - v_{2} + \\dot{y}{(v_{2})})} = \\sin{(v - v_{2} + \\cos{(v_{2})})} and \\sin^{v}{(v - v_{2} + \\dot{y}{(v_{2})})} = \\sin^{v}{(v - v_{2} + \\cos{(v_{2})})} and \\int \\sin^{v}{(v - v_{2} + \\dot{y}{(v_{2})})} dv_{2} = \\int \\sin^{v}{(v - v_{2} + \\cos{(v_{2})})} dv_{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\dot{y}')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["add", 2, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\dot{y}')(Symbol('v_2', commutative=True))), Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["sin", 3], "Equality(sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\dot{y}')(Symbol('v_2', commutative=True)))), sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\dot{y}')(Symbol('v_2', commutative=True)))), Symbol('v', commutative=True)), Pow(sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))), Symbol('v', commutative=True)))"], [["integrate", 5, "Symbol('v_2', commutative=True)"], "Equality(Integral(Pow(sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\dot{y}')(Symbol('v_2', commutative=True)))), Symbol('v', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Pow(sin(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))), Symbol('v', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given l{(\\phi,v_{2})} = \\frac{v_{2}}{\\phi}, then obtain - l{(\\phi,v_{2})} e^{\\int - l{(\\phi,v_{2})} d\\phi} + l{(\\phi,v_{2})} = l{(\\phi,v_{2})} - \\frac{v_{2} e^{\\int - l{(\\phi,v_{2})} d\\phi}}{\\phi}", "derivation": "l{(\\phi,v_{2})} = \\frac{v_{2}}{\\phi} and - l{(\\phi,v_{2})} = - \\frac{v_{2}}{\\phi} and \\int - l{(\\phi,v_{2})} d\\phi = \\int - \\frac{v_{2}}{\\phi} d\\phi and - l{(\\phi,v_{2})} e^{\\int - \\frac{v_{2}}{\\phi} d\\phi} = - \\frac{v_{2} e^{\\int - \\frac{v_{2}}{\\phi} d\\phi}}{\\phi} and - l{(\\phi,v_{2})} e^{\\int - \\frac{v_{2}}{\\phi} d\\phi} + l{(\\phi,v_{2})} = l{(\\phi,v_{2})} - \\frac{v_{2} e^{\\int - \\frac{v_{2}}{\\phi} d\\phi}}{\\phi} and - l{(\\phi,v_{2})} e^{\\int - l{(\\phi,v_{2})} d\\phi} + l{(\\phi,v_{2})} = l{(\\phi,v_{2})} - \\frac{v_{2} e^{\\int - l{(\\phi,v_{2})} d\\phi}}{\\phi}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["times", 2, "exp(Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], "Equality(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), exp(Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True), exp(Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))))"], [["add", 4, "Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), exp(Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Add(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True), exp(Integral(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), exp(Integral(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Add(Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('v_2', commutative=True), exp(Integral(Mul(Integer(-1), Function('l')(Symbol('\\\\phi', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(v_{y},\\theta,u)} = \\frac{u v_{y}}{\\theta}, then obtain \\int \\frac{u v_{y} (\\operatorname{L_{\\varepsilon}}^{u}{(v_{y},\\theta,u)})^{u}}{\\theta} d\\theta = \\int \\frac{u v_{y} ((\\frac{u v_{y}}{\\theta})^{u})^{u}}{\\theta} d\\theta", "derivation": "\\operatorname{L_{\\varepsilon}}{(v_{y},\\theta,u)} = \\frac{u v_{y}}{\\theta} and \\operatorname{L_{\\varepsilon}}^{u}{(v_{y},\\theta,u)} = (\\frac{u v_{y}}{\\theta})^{u} and (\\operatorname{L_{\\varepsilon}}^{u}{(v_{y},\\theta,u)})^{u} = ((\\frac{u v_{y}}{\\theta})^{u})^{u} and \\frac{u v_{y} (\\operatorname{L_{\\varepsilon}}^{u}{(v_{y},\\theta,u)})^{u}}{\\theta} = \\frac{u v_{y} ((\\frac{u v_{y}}{\\theta})^{u})^{u}}{\\theta} and \\int \\frac{u v_{y} (\\operatorname{L_{\\varepsilon}}^{u}{(v_{y},\\theta,u)})^{u}}{\\theta} d\\theta = \\int \\frac{u v_{y} ((\\frac{u v_{y}}{\\theta})^{u})^{u}}{\\theta} d\\theta", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_y', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_y', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True)), Symbol('u', commutative=True)))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_y', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True), Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_y', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True), Pow(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True), Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_y', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True), Pow(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('u', commutative=True), Symbol('v_y', commutative=True)), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given i{(\\phi)} = e^{\\cos{(\\phi)}}, then derive \\frac{d}{d \\phi} i{(\\phi)} = - e^{\\cos{(\\phi)}} \\sin{(\\phi)}, then obtain - e^{\\cos{(\\phi)}} \\sin{(\\phi)} = - i{(\\phi)} \\sin{(\\phi)}", "derivation": "i{(\\phi)} = e^{\\cos{(\\phi)}} and \\frac{d}{d \\phi} i{(\\phi)} = \\frac{d}{d \\phi} e^{\\cos{(\\phi)}} and \\frac{d}{d \\phi} i{(\\phi)} = - e^{\\cos{(\\phi)}} \\sin{(\\phi)} and \\frac{d}{d \\phi} i{(\\phi)} = - i{(\\phi)} \\sin{(\\phi)} and - e^{\\cos{(\\phi)}} \\sin{(\\phi)} = - i{(\\phi)} \\sin{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\phi', commutative=True)), exp(cos(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\phi', commutative=True))), sin(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Function('i')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('\\\\phi', commutative=True))), sin(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Function('i')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\phi{(u)} = \\sin{(\\log{(u)})} and \\operatorname{C_{1}}{(u)} = \\log{(u)}, then obtain (\\int \\sin^{u}{(\\operatorname{C_{1}}{(u)})} du)^{u} = (\\int \\sin^{u}{(\\log{(u)})} du)^{u}", "derivation": "\\phi{(u)} = \\sin{(\\log{(u)})} and \\phi^{u}{(u)} = \\sin^{u}{(\\log{(u)})} and \\operatorname{C_{1}}{(u)} = \\log{(u)} and \\phi^{u}{(u)} = \\sin^{u}{(\\operatorname{C_{1}}{(u)})} and \\sin^{u}{(\\operatorname{C_{1}}{(u)})} = \\sin^{u}{(\\log{(u)})} and \\int \\sin^{u}{(\\operatorname{C_{1}}{(u)})} du = \\int \\sin^{u}{(\\log{(u)})} du and (\\int \\sin^{u}{(\\operatorname{C_{1}}{(u)})} du)^{u} = (\\int \\sin^{u}{(\\log{(u)})} du)^{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('u', commutative=True)), sin(log(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\phi')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(sin(Function('C_1')(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(sin(Function('C_1')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(sin(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(sin(Function('C_1')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(sin(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Pow(sin(Function('C_1')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Pow(sin(log(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(U,\\ddot{x})} = \\log{(U + \\ddot{x})}, then derive (\\int \\frac{\\Psi_{\\lambda}{(U,\\ddot{x})}}{\\log{(U + \\ddot{x})}} d\\ddot{x})^{\\ddot{x}} = (\\ddot{x} + \\mathbf{f})^{\\ddot{x}}, then obtain (\\int 1 d\\ddot{x})^{\\ddot{x}} = (\\ddot{x} + \\mathbf{f})^{\\ddot{x}}", "derivation": "\\Psi_{\\lambda}{(U,\\ddot{x})} = \\log{(U + \\ddot{x})} and \\frac{\\Psi_{\\lambda}{(U,\\ddot{x})}}{\\log{(U + \\ddot{x})}} = 1 and \\int \\frac{\\Psi_{\\lambda}{(U,\\ddot{x})}}{\\log{(U + \\ddot{x})}} d\\ddot{x} = \\int 1 d\\ddot{x} and (\\int \\frac{\\Psi_{\\lambda}{(U,\\ddot{x})}}{\\log{(U + \\ddot{x})}} d\\ddot{x})^{\\ddot{x}} = (\\int 1 d\\ddot{x})^{\\ddot{x}} and (\\int \\frac{\\Psi_{\\lambda}{(U,\\ddot{x})}}{\\log{(U + \\ddot{x})}} d\\ddot{x})^{\\ddot{x}} = (\\ddot{x} + \\mathbf{f})^{\\ddot{x}} and (\\int 1 d\\ddot{x})^{\\ddot{x}} = (\\ddot{x} + \\mathbf{f})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 1, "log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["power", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Integral(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(Add(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(C_{d})} = e^{\\cos{(C_{d})}}, then derive \\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})} = - e^{\\cos{(C_{d})}} \\sin{(C_{d})}, then obtain (\\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})})^{C_{d}} = (- e^{\\cos{(C_{d})}} \\sin{(C_{d})})^{C_{d}}", "derivation": "\\mathbf{s}{(C_{d})} = e^{\\cos{(C_{d})}} and \\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})} = \\frac{d}{d C_{d}} e^{\\cos{(C_{d})}} and \\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})} = - e^{\\cos{(C_{d})}} \\sin{(C_{d})} and (\\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})})^{C_{d}} = (\\frac{d}{d C_{d}} e^{\\cos{(C_{d})}})^{C_{d}} and \\frac{d}{d C_{d}} e^{\\cos{(C_{d})}} = - e^{\\cos{(C_{d})}} \\sin{(C_{d})} and (\\frac{d}{d C_{d}} \\mathbf{s}{(C_{d})})^{C_{d}} = (- e^{\\cos{(C_{d})}} \\sin{(C_{d})})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('C_d', commutative=True)), exp(cos(Symbol('C_d', commutative=True))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('C_d', commutative=True))), sin(Symbol('C_d', commutative=True))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{s}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Pow(Derivative(exp(cos(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(cos(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('C_d', commutative=True))), sin(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\mathbf{s}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('C_d', commutative=True)), Pow(Mul(Integer(-1), exp(cos(Symbol('C_d', commutative=True))), sin(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given f{(g,r)} = g r, then obtain f{(g,r)} \\frac{\\partial}{\\partial r} g r + f{(g,r)} = g r + f{(g,r)} \\frac{\\partial}{\\partial r} g r", "derivation": "f{(g,r)} = g r and \\frac{\\partial}{\\partial r} f{(g,r)} = \\frac{\\partial}{\\partial r} g r and f{(g,r)} \\frac{\\partial}{\\partial r} f{(g,r)} = f{(g,r)} \\frac{\\partial}{\\partial r} g r and f{(g,r)} \\frac{\\partial}{\\partial r} f{(g,r)} + f{(g,r)} = g r + f{(g,r)} \\frac{\\partial}{\\partial r} f{(g,r)} and f{(g,r)} \\frac{\\partial}{\\partial r} g r + f{(g,r)} = g r + f{(g,r)} \\frac{\\partial}{\\partial r} g r", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Integer(-1))"], "Equality(Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["add", 1, "Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], "Equality(Add(Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True))), Add(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True))), Add(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Mul(Function('f')(Symbol('g', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\rho{(V_{\\mathbf{B}},t_{1})} = \\sin{(V_{\\mathbf{B}} - t_{1})}, then derive \\frac{\\partial^{2}}{\\partial t_{1}\\partial V_{\\mathbf{B}}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\sin{(V_{\\mathbf{B}} - t_{1})}, then obtain \\frac{\\partial^{3}}{\\partial t_{1}^{2}\\partial V_{\\mathbf{B}}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\frac{\\partial}{\\partial t_{1}} \\sin{(V_{\\mathbf{B}} - t_{1})}", "derivation": "\\rho{(V_{\\mathbf{B}},t_{1})} = \\sin{(V_{\\mathbf{B}} - t_{1})} and \\frac{\\partial}{\\partial t_{1}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\frac{\\partial}{\\partial t_{1}} \\sin{(V_{\\mathbf{B}} - t_{1})} and \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}\\partial t_{1}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\frac{\\partial^{2}}{\\partial V_{\\mathbf{B}}\\partial t_{1}} \\sin{(V_{\\mathbf{B}} - t_{1})} and \\frac{\\partial^{2}}{\\partial t_{1}\\partial V_{\\mathbf{B}}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\sin{(V_{\\mathbf{B}} - t_{1})} and \\frac{\\partial^{3}}{\\partial t_{1}^{2}\\partial V_{\\mathbf{B}}} \\rho{(V_{\\mathbf{B}},t_{1})} = \\frac{\\partial}{\\partial t_{1}} \\sin{(V_{\\mathbf{B}} - t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\rho')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))))"], [["differentiate", 4, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(2))), Derivative(sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(V,v_{z})} = V v_{z}, then derive \\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})} = V, then obtain V^{- V} (\\frac{\\partial}{\\partial v_{z}} V v_{z})^{V} = V^{- V} (\\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})})^{V}", "derivation": "\\mu_{0}{(V,v_{z})} = V v_{z} and \\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})} = \\frac{\\partial}{\\partial v_{z}} V v_{z} and \\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})} = V and (\\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})})^{V} = V^{V} and (\\frac{\\partial}{\\partial v_{z}} V v_{z})^{V} = V^{V} and (\\frac{\\partial}{\\partial v_{z}} V v_{z})^{V} = (\\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})})^{V} and V^{- V} (\\frac{\\partial}{\\partial v_{z}} V v_{z})^{V} = V^{- V} (\\frac{\\partial}{\\partial v_{z}} \\mu_{0}{(V,v_{z})})^{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Symbol('V', commutative=True), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Symbol('V', commutative=True), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Derivative(Mul(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True)))"], [["divide", 6, "Pow(Symbol('V', commutative=True), Symbol('V', commutative=True))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Pow(Derivative(Mul(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True))), Pow(Derivative(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{P},\\mathbf{S})} = \\mathbf{S}^{\\mathbf{P}} and \\operatorname{v_{2}}{(\\mathbf{P},\\mathbf{S})} = \\int \\varepsilon{(\\mathbf{P},\\mathbf{S})} d\\mathbf{S}, then obtain \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\operatorname{v_{2}}{(\\mathbf{P},\\mathbf{S})} = \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\int \\mathbf{S}^{\\mathbf{P}} d\\mathbf{S}", "derivation": "\\varepsilon{(\\mathbf{P},\\mathbf{S})} = \\mathbf{S}^{\\mathbf{P}} and \\int \\varepsilon{(\\mathbf{P},\\mathbf{S})} d\\mathbf{S} = \\int \\mathbf{S}^{\\mathbf{P}} d\\mathbf{S} and \\operatorname{v_{2}}{(\\mathbf{P},\\mathbf{S})} = \\int \\varepsilon{(\\mathbf{P},\\mathbf{S})} d\\mathbf{S} and \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\operatorname{v_{2}}{(\\mathbf{P},\\mathbf{S})} = \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\int \\varepsilon{(\\mathbf{P},\\mathbf{S})} d\\mathbf{S} and \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\operatorname{v_{2}}{(\\mathbf{P},\\mathbf{S})} = \\varepsilon{(\\mathbf{P},\\mathbf{S})} \\int \\mathbf{S}^{\\mathbf{P}} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 3, "Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('v_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given y{(M,\\omega)} = M + \\omega, then derive \\frac{\\partial}{\\partial M} y{(M,\\omega)} = 1, then obtain (\\frac{\\partial}{\\partial M} (M + \\omega))^{\\omega} = 1", "derivation": "y{(M,\\omega)} = M + \\omega and \\frac{\\partial}{\\partial M} y{(M,\\omega)} = \\frac{\\partial}{\\partial M} (M + \\omega) and \\frac{\\partial}{\\partial M} y{(M,\\omega)} = 1 and \\frac{\\partial}{\\partial M} (M + \\omega) = 1 and (\\frac{\\partial}{\\partial M} (M + \\omega))^{\\omega} = 1", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\psi{(i,c_{0})} = - c_{0} + i, then derive n_{1} - \\int c_{0} \\frac{\\partial}{\\partial i} \\psi{(i,c_{0})} dc_{0} = A_{z} - \\frac{c_{0}^{2}}{2}, then obtain n_{1} - \\int c_{0} \\frac{\\partial}{\\partial i} (- c_{0} + i) dc_{0} = A_{z} - \\frac{c_{0}^{2}}{2}", "derivation": "\\psi{(i,c_{0})} = - c_{0} + i and - c_{0} \\psi{(i,c_{0})} = - c_{0} (- c_{0} + i) and \\frac{\\partial}{\\partial i} - c_{0} \\psi{(i,c_{0})} = \\frac{\\partial}{\\partial i} - c_{0} (- c_{0} + i) and \\int \\frac{\\partial}{\\partial i} - c_{0} \\psi{(i,c_{0})} dc_{0} = \\int \\frac{\\partial}{\\partial i} - c_{0} (- c_{0} + i) dc_{0} and n_{1} - \\int c_{0} \\frac{\\partial}{\\partial i} \\psi{(i,c_{0})} dc_{0} = A_{z} - \\frac{c_{0}^{2}}{2} and n_{1} - \\int c_{0} \\frac{\\partial}{\\partial i} (- c_{0} + i) dc_{0} = A_{z} - \\frac{c_{0}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('i', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('i', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('c_0', commutative=True), Function('\\\\psi')(Symbol('i', commutative=True), Symbol('c_0', commutative=True))), Mul(Integer(-1), Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('c_0', commutative=True), Function('\\\\psi')(Symbol('i', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('c_0', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), Symbol('c_0', commutative=True), Function('\\\\psi')(Symbol('i', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('c_0', commutative=True))), Integral(Derivative(Mul(Integer(-1), Symbol('c_0', commutative=True), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('n_1', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('c_0', commutative=True), Derivative(Function('\\\\psi')(Symbol('i', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('c_0', commutative=True))))), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('n_1', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('c_0', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Tuple(Symbol('c_0', commutative=True))))), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))))"]]}, {"prompt": "Given h{(U)} = \\log{(U)}, then derive \\int h{(U)} dU = U \\log{(U)} - U + c, then obtain ((h{(U)} - \\log{(U)})^{- U} \\int h{(U)} dU)^{c} = ((h{(U)} - \\log{(U)})^{- U} (U \\log{(U)} - U + c))^{c}", "derivation": "h{(U)} = \\log{(U)} and \\int h{(U)} dU = \\int \\log{(U)} dU and \\int h{(U)} dU = U \\log{(U)} - U + c and (h{(U)} - \\log{(U)})^{- U} \\int h{(U)} dU = (h{(U)} - \\log{(U)})^{- U} (U \\log{(U)} - U + c) and ((h{(U)} - \\log{(U)})^{- U} \\int h{(U)} dU)^{c} = ((h{(U)} - \\log{(U)})^{- U} (U \\log{(U)} - U + c))^{c}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('h')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('c', commutative=True)))"], [["divide", 3, "Pow(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Function('h')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Pow(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Mul(Integer(-1), Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('c', commutative=True))))"], [["power", 4, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Function('h')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Pow(Add(Function('h')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Mul(Integer(-1), Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('c', commutative=True))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(\\Psi)} = e^{\\Psi}, then derive \\int \\tilde{g}{(\\Psi)} d\\Psi = m_{s} + e^{\\Psi}, then obtain \\int (m_{s} + \\tilde{g}{(\\Psi)} - 1)^{\\Psi} dm_{s} = \\int (m_{s} + e^{\\Psi} - 1)^{\\Psi} dm_{s}", "derivation": "\\tilde{g}{(\\Psi)} = e^{\\Psi} and \\int \\tilde{g}{(\\Psi)} d\\Psi = \\int e^{\\Psi} d\\Psi and \\int \\tilde{g}{(\\Psi)} d\\Psi = m_{s} + e^{\\Psi} and \\int \\tilde{g}{(\\Psi)} d\\Psi - 1 = m_{s} + e^{\\Psi} - 1 and \\int \\tilde{g}{(\\Psi)} d\\Psi - 1 = m_{s} + \\tilde{g}{(\\Psi)} - 1 and (\\int \\tilde{g}{(\\Psi)} d\\Psi - 1)^{\\Psi} = (m_{s} + e^{\\Psi} - 1)^{\\Psi} and \\int (\\int \\tilde{g}{(\\Psi)} d\\Psi - 1)^{\\Psi} dm_{s} = \\int (m_{s} + e^{\\Psi} - 1)^{\\Psi} dm_{s} and \\int (m_{s} + \\tilde{g}{(\\Psi)} - 1)^{\\Psi} dm_{s} = \\int (m_{s} + e^{\\Psi} - 1)^{\\Psi} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('m_s', commutative=True), exp(Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Add(Symbol('m_s', commutative=True), exp(Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Add(Symbol('m_s', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["power", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Add(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), exp(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Symbol('\\\\Psi', commutative=True)))"], [["integrate", 6, "Symbol('m_s', commutative=True)"], "Equality(Integral(Pow(Add(Integral(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Pow(Add(Symbol('m_s', commutative=True), exp(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integral(Pow(Add(Symbol('m_s', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Pow(Add(Symbol('m_s', commutative=True), exp(Symbol('\\\\Psi', commutative=True)), Integer(-1)), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\rho_f,\\rho)} = \\frac{\\partial}{\\partial \\rho_f} (\\rho + \\rho_f) and \\theta_{2}{(\\rho_f)} = \\rho_f, then derive \\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)} = \\rho + 1, then obtain \\int (\\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)}) d\\theta_{2}{(\\rho_f)} = \\int (\\rho + 1) d\\theta_{2}{(\\rho_f)}", "derivation": "\\operatorname{t_{1}}{(\\rho_f,\\rho)} = \\frac{\\partial}{\\partial \\rho_f} (\\rho + \\rho_f) and \\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)} = \\rho + \\frac{\\partial}{\\partial \\rho_f} (\\rho + \\rho_f) and \\theta_{2}{(\\rho_f)} = \\rho_f and \\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)} = \\rho + 1 and \\int (\\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)}) d\\rho_f = \\int (\\rho + 1) d\\rho_f and \\int (\\rho + \\operatorname{t_{1}}{(\\rho_f,\\rho)}) d\\theta_{2}{(\\rho_f)} = \\int (\\rho + 1) d\\theta_{2}{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Function('t_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Derivative(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('\\\\rho', commutative=True), Function('t_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Integer(1)))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\rho', commutative=True), Function('t_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Symbol('\\\\rho', commutative=True), Integer(1)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Add(Symbol('\\\\rho', commutative=True), Function('t_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)))), Integral(Add(Symbol('\\\\rho', commutative=True), Integer(1)), Tuple(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\phi)} = \\sin{(\\phi)} and \\Psi_{nl}{(\\phi)} = - \\hat{H}{(\\phi)}, then obtain \\phi + \\int - \\hat{H}{(\\phi)} d\\phi + 1 = \\phi + \\int - \\sin{(\\phi)} d\\phi + 1", "derivation": "\\hat{H}{(\\phi)} = \\sin{(\\phi)} and \\Psi_{nl}{(\\phi)} = - \\hat{H}{(\\phi)} and \\int \\Psi_{nl}{(\\phi)} d\\phi = \\int - \\hat{H}{(\\phi)} d\\phi and \\int \\Psi_{nl}{(\\phi)} d\\phi = \\int - \\sin{(\\phi)} d\\phi and \\int \\Psi_{nl}{(\\phi)} d\\phi + 1 = \\int - \\sin{(\\phi)} d\\phi + 1 and \\phi + \\int \\Psi_{nl}{(\\phi)} d\\phi + 1 = \\phi + \\int - \\sin{(\\phi)} d\\phi + 1 and \\phi + \\int - \\hat{H}{(\\phi)} d\\phi + 1 = \\phi + \\int - \\sin{(\\phi)} d\\phi + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Integral(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)))"], [["add", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\phi', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('\\\\phi', commutative=True), Integral(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Symbol('\\\\phi', commutative=True), Integral(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mu{(\\varphi^*,v_{t})} = e^{- \\varphi^* + v_{t}}, then derive \\frac{\\partial^{2}}{\\partial v_{t}\\partial \\varphi^*} \\mu{(\\varphi^*,v_{t})} = - e^{- \\varphi^* + v_{t}}, then obtain \\frac{\\partial^{2}}{\\partial v_{t}\\partial \\varphi^*} \\mu{(\\varphi^*,v_{t})} = - \\mu{(\\varphi^*,v_{t})}", "derivation": "\\mu{(\\varphi^*,v_{t})} = e^{- \\varphi^* + v_{t}} and \\frac{\\partial}{\\partial v_{t}} \\mu{(\\varphi^*,v_{t})} = \\frac{\\partial}{\\partial v_{t}} e^{- \\varphi^* + v_{t}} and \\frac{\\partial^{2}}{\\partial \\varphi^*\\partial v_{t}} \\mu{(\\varphi^*,v_{t})} = \\frac{\\partial^{2}}{\\partial \\varphi^*\\partial v_{t}} e^{- \\varphi^* + v_{t}} and \\frac{\\partial^{2}}{\\partial v_{t}\\partial \\varphi^*} \\mu{(\\varphi^*,v_{t})} = - e^{- \\varphi^* + v_{t}} and \\frac{\\partial^{2}}{\\partial v_{t}\\partial \\varphi^*} \\mu{(\\varphi^*,v_{t})} = - \\mu{(\\varphi^*,v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_t', commutative=True))))"], [["differentiate", 1, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\varphi^*', commutative=True), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(F_{H})} = F_{H}, then obtain \\frac{d^{2}}{d F_{H}^{2}} \\mathbf{s}{(F_{H})} = 0", "derivation": "\\mathbf{s}{(F_{H})} = F_{H} and \\frac{d}{d F_{H}} \\mathbf{s}{(F_{H})} = \\frac{d}{d F_{H}} F_{H} and \\frac{d^{2}}{d F_{H}^{2}} \\mathbf{s}{(F_{H})} = \\frac{d^{2}}{d F_{H}^{2}} F_{H} and \\frac{d^{2}}{d F_{H}^{2}} \\mathbf{s}{(F_{H})} = 0", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(2))), Derivative(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\sigma_{x}{(M,p)} = M^{p} and L{(M,p)} = (M^{p})^{M}, then obtain \\cos{(\\sigma_{x}^{M}{(M,p)})} = \\cos{(L{(M,p)})}", "derivation": "\\sigma_{x}{(M,p)} = M^{p} and \\sigma_{x}^{M}{(M,p)} = (M^{p})^{M} and L{(M,p)} = (M^{p})^{M} and \\cos{(\\sigma_{x}^{M}{(M,p)})} = \\cos{((M^{p})^{M})} and \\cos{(\\sigma_{x}^{M}{(M,p)})} = \\cos{(L{(M,p)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('M', commutative=True), Symbol('p', commutative=True)), Pow(Pow(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True))), cos(Pow(Pow(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(cos(Pow(Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('p', commutative=True)), Symbol('M', commutative=True))), cos(Function('L')(Symbol('M', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given p{(c_{0},n)} = n^{c_{0}} and \\theta_{1}{(i)} = \\log{(i)}, then obtain ((\\theta_{1}{(i)} + 1) \\int n^{c_{0}} dc_{0})^{c_{0}} = ((\\log{(i)} + 1) \\int n^{c_{0}} dc_{0})^{c_{0}}", "derivation": "p{(c_{0},n)} = n^{c_{0}} and \\int p{(c_{0},n)} dc_{0} = \\int n^{c_{0}} dc_{0} and \\theta_{1}{(i)} = \\log{(i)} and \\theta_{1}{(i)} + 1 = \\log{(i)} + 1 and (\\theta_{1}{(i)} + 1) \\int n^{c_{0}} dc_{0} = (\\log{(i)} + 1) \\int n^{c_{0}} dc_{0} and (\\theta_{1}{(i)} + 1) \\int p{(c_{0},n)} dc_{0} = (\\log{(i)} + 1) \\int p{(c_{0},n)} dc_{0} and ((\\theta_{1}{(i)} + 1) \\int p{(c_{0},n)} dc_{0})^{c_{0}} = ((\\log{(i)} + 1) \\int p{(c_{0},n)} dc_{0})^{c_{0}} and ((\\theta_{1}{(i)} + 1) \\int n^{c_{0}} dc_{0})^{c_{0}} = ((\\log{(i)} + 1) \\int n^{c_{0}} dc_{0})^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["add", 3, 1], "Equality(Add(Function('\\\\theta_1')(Symbol('i', commutative=True)), Integer(1)), Add(log(Symbol('i', commutative=True)), Integer(1)))"], [["times", 4, "Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))"], "Equality(Mul(Add(Function('\\\\theta_1')(Symbol('i', commutative=True)), Integer(1)), Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Add(log(Symbol('i', commutative=True)), Integer(1)), Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Add(Function('\\\\theta_1')(Symbol('i', commutative=True)), Integer(1)), Integral(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Add(log(Symbol('i', commutative=True)), Integer(1)), Integral(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["power", 6, "Symbol('c_0', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\theta_1')(Symbol('i', commutative=True)), Integer(1)), Integral(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)), Pow(Mul(Add(log(Symbol('i', commutative=True)), Integer(1)), Integral(Function('p')(Symbol('c_0', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Pow(Mul(Add(Function('\\\\theta_1')(Symbol('i', commutative=True)), Integer(1)), Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)), Pow(Mul(Add(log(Symbol('i', commutative=True)), Integer(1)), Integral(Pow(Symbol('n', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{y})} = e^{v_{y}} and \\operatorname{f_{E}}{(v_{y})} = \\frac{d}{d v_{y}} e^{v_{y}}, then obtain \\operatorname{f_{E}}{(v_{y})} = \\frac{d}{d v_{y}} \\operatorname{f_{\\mathbf{p}}}{(v_{y})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{y})} = e^{v_{y}} and \\frac{d}{d v_{y}} \\operatorname{f_{\\mathbf{p}}}{(v_{y})} = \\frac{d}{d v_{y}} e^{v_{y}} and \\operatorname{f_{E}}{(v_{y})} = \\frac{d}{d v_{y}} e^{v_{y}} and \\operatorname{f_{E}}{(v_{y})} = \\frac{d}{d v_{y}} \\operatorname{f_{\\mathbf{p}}}{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('v_y', commutative=True)), Derivative(exp(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('f_E')(Symbol('v_y', commutative=True)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{D},i)} = \\frac{\\sin{(\\mathbf{D})}}{i}, then obtain \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\operatorname{t_{2}}{(\\mathbf{D},i)})^{i} = \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\frac{\\sin{(\\mathbf{D})}}{i})^{i}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{D},i)} = \\frac{\\sin{(\\mathbf{D})}}{i} and \\frac{\\partial}{\\partial i} \\operatorname{t_{2}}{(\\mathbf{D},i)} = \\frac{\\partial}{\\partial i} \\frac{\\sin{(\\mathbf{D})}}{i} and (\\frac{\\partial}{\\partial i} \\operatorname{t_{2}}{(\\mathbf{D},i)})^{i} = (\\frac{\\partial}{\\partial i} \\frac{\\sin{(\\mathbf{D})}}{i})^{i} and \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\operatorname{t_{2}}{(\\mathbf{D},i)})^{i} = \\frac{\\partial}{\\partial i} (\\frac{\\partial}{\\partial i} \\frac{\\sin{(\\mathbf{D})}}{i})^{i}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('t_2')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(S)} = \\sin{(S)} and \\operatorname{v_{2}}{(S)} = \\frac{H{(S)}}{- S + \\sin{(S)}}, then obtain (\\frac{H^{2}{(S)}}{(- S + \\sin{(S)})^{2}})^{S} = (\\frac{H{(S)} \\sin{(S)}}{(- S + \\sin{(S)})^{2}})^{S}", "derivation": "H{(S)} = \\sin{(S)} and - S + H{(S)} = - S + \\sin{(S)} and \\frac{H{(S)}}{- S + H{(S)}} = \\frac{\\sin{(S)}}{- S + H{(S)}} and \\operatorname{v_{2}}{(S)} = \\frac{H{(S)}}{- S + \\sin{(S)}} and \\frac{H{(S)} \\operatorname{v_{2}}{(S)}}{- S + H{(S)}} = \\frac{\\operatorname{v_{2}}{(S)} \\sin{(S)}}{- S + H{(S)}} and \\frac{H{(S)} \\operatorname{v_{2}}{(S)}}{- S + \\sin{(S)}} = \\frac{\\operatorname{v_{2}}{(S)} \\sin{(S)}}{- S + \\sin{(S)}} and (\\frac{H{(S)} \\operatorname{v_{2}}{(S)}}{- S + \\sin{(S)}})^{S} = (\\frac{\\operatorname{v_{2}}{(S)} \\sin{(S)}}{- S + \\sin{(S)}})^{S} and (\\frac{H^{2}{(S)}}{(- S + \\sin{(S)})^{2}})^{S} = (\\frac{H{(S)} \\sin{(S)}}{(- S + \\sin{(S)})^{2}})^{S}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["minus", 1, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True))), Integer(-1)), Function('H')(Symbol('S', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True))), Integer(-1)), sin(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('S', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-1)), Function('H')(Symbol('S', commutative=True))))"], [["times", 3, "Function('v_2')(Symbol('S', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True))), Integer(-1)), Function('H')(Symbol('S', commutative=True)), Function('v_2')(Symbol('S', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('H')(Symbol('S', commutative=True))), Integer(-1)), Function('v_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-1)), Function('H')(Symbol('S', commutative=True)), Function('v_2')(Symbol('S', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-1)), Function('v_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))))"], [["power", 6, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-1)), Function('H')(Symbol('S', commutative=True)), Function('v_2')(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-1)), Function('v_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-2)), Pow(Function('H')(Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Integer(-2)), Function('H')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(n)} = \\sin{(n)} and \\dot{\\mathbf{r}}{(n)} = \\log{(\\sin{(n)})}^{n}, then obtain \\log{(\\mathbf{M}{(n)})}^{n} = \\dot{\\mathbf{r}}{(n)}", "derivation": "\\mathbf{M}{(n)} = \\sin{(n)} and \\log{(\\mathbf{M}{(n)})} = \\log{(\\sin{(n)})} and \\log{(\\mathbf{M}{(n)})}^{n} = \\log{(\\sin{(n)})}^{n} and \\dot{\\mathbf{r}}{(n)} = \\log{(\\sin{(n)})}^{n} and \\log{(\\mathbf{M}{(n)})}^{n} = \\dot{\\mathbf{r}}{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), log(sin(Symbol('n', commutative=True))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(log(Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(log(sin(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)), Pow(log(sin(Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(log(Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(P_{g},F_{x})} = \\frac{F_{x}}{P_{g}}, then obtain \\int - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int 0 dP_{g} dP_{g} = \\int - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int (\\frac{F_{x}}{P_{g}} - \\operatorname{t_{1}}{(P_{g},F_{x})}) dP_{g} dP_{g}", "derivation": "\\operatorname{t_{1}}{(P_{g},F_{x})} = \\frac{F_{x}}{P_{g}} and 0 = \\frac{F_{x}}{P_{g}} - \\operatorname{t_{1}}{(P_{g},F_{x})} and \\int 0 dP_{g} = \\int (\\frac{F_{x}}{P_{g}} - \\operatorname{t_{1}}{(P_{g},F_{x})}) dP_{g} and - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int 0 dP_{g} = - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int (\\frac{F_{x}}{P_{g}} - \\operatorname{t_{1}}{(P_{g},F_{x})}) dP_{g} and \\int - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int 0 dP_{g} dP_{g} = \\int - \\operatorname{t_{1}}{(P_{g},F_{x})} \\int (\\frac{F_{x}}{P_{g}} - \\operatorname{t_{1}}{(P_{g},F_{x})}) dP_{g} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))))"], [["minus", 1, "Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('P_g', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Integral(Integer(0), Tuple(Symbol('P_g', commutative=True)))), Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Integral(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('P_g', commutative=True)))))"], [["integrate", 4, "Symbol('P_g', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Integral(Integer(0), Tuple(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True))), Integral(Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Integral(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given L{(P_{g})} = P_{g}, then obtain \\frac{L{(P_{g})}}{(P_{g} + \\Psi_{\\lambda}) \\log{(\\rho_f)}} = \\frac{P_{g}}{(P_{g} + \\Psi_{\\lambda}) \\log{(\\rho_f)}}", "derivation": "L{(P_{g})} = P_{g} and \\Psi_{\\lambda} + L{(P_{g})} = P_{g} + \\Psi_{\\lambda} and \\frac{L{(P_{g})}}{P_{g} + \\Psi_{\\lambda}} = \\frac{P_{g}}{P_{g} + \\Psi_{\\lambda}} and \\frac{L{(P_{g})}}{\\Psi_{\\lambda} + L{(P_{g})}} = \\frac{P_{g}}{\\Psi_{\\lambda} + L{(P_{g})}} and \\frac{L{(P_{g})}}{(\\Psi_{\\lambda} + L{(P_{g})}) \\log{(\\rho_f)}} = \\frac{P_{g}}{(\\Psi_{\\lambda} + L{(P_{g})}) \\log{(\\rho_f)}} and \\frac{L{(P_{g})}}{(P_{g} + \\Psi_{\\lambda}) \\log{(\\rho_f)}} = \\frac{P_{g}}{(P_{g} + \\Psi_{\\lambda}) \\log{(\\rho_f)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('L')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["add", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Function('L')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1))))"], [["divide", 4, "log(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Function('L')(Symbol('P_g', commutative=True)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('L')(Symbol('P_g', commutative=True))), Integer(-1)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Function('L')(Symbol('P_g', commutative=True)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Mul(Symbol('P_g', commutative=True), Pow(Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given V{(S,f_{\\mathbf{v}})} = S + f_{\\mathbf{v}}, then derive \\frac{\\partial}{\\partial S} V{(S,f_{\\mathbf{v}})} - 1 = 0, then obtain (\\frac{\\partial}{\\partial S} (S + f_{\\mathbf{v}}) - 1)^{S} = 0^{S}", "derivation": "V{(S,f_{\\mathbf{v}})} = S + f_{\\mathbf{v}} and - S - f_{\\mathbf{v}} + V{(S,f_{\\mathbf{v}})} = 0 and \\frac{\\partial}{\\partial S} (- S - f_{\\mathbf{v}} + V{(S,f_{\\mathbf{v}})}) = \\frac{d}{d S} 0 and \\frac{\\partial}{\\partial S} V{(S,f_{\\mathbf{v}})} - 1 = 0 and \\frac{\\partial}{\\partial S} (S + f_{\\mathbf{v}}) - 1 = 0 and (\\frac{\\partial}{\\partial S} (S + f_{\\mathbf{v}}) - 1)^{S} = 0^{S}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Add(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('V')(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('V')(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V')(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["power", 5, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Derivative(Add(Symbol('S', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)), Pow(Integer(0), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(g)} = \\sin{(g)}, then obtain \\frac{d^{2}}{d g^{2}} \\int \\mathbf{F}{(g)} dg = \\frac{d^{2}}{d g^{2}} \\int \\sin{(g)} dg", "derivation": "\\mathbf{F}{(g)} = \\sin{(g)} and \\int \\mathbf{F}{(g)} dg = \\int \\sin{(g)} dg and \\frac{d}{d g} \\int \\mathbf{F}{(g)} dg = \\frac{d}{d g} \\int \\sin{(g)} dg and \\frac{d^{2}}{d g^{2}} \\int \\mathbf{F}{(g)} dg = \\frac{d^{2}}{d g^{2}} \\int \\sin{(g)} dg", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(2))), Derivative(Integral(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(2))))"]]}, {"prompt": "Given c{(A_{1},E_{x})} = E_{x}^{A_{1}}, then obtain (E_{x}^{A_{1}})^{A_{1}} + 2 c^{A_{1}}{(A_{1},E_{x})} = 3 (E_{x}^{A_{1}})^{A_{1}}", "derivation": "c{(A_{1},E_{x})} = E_{x}^{A_{1}} and c^{A_{1}}{(A_{1},E_{x})} = (E_{x}^{A_{1}})^{A_{1}} and (E_{x}^{A_{1}})^{A_{1}} + c^{A_{1}}{(A_{1},E_{x})} = 2 (E_{x}^{A_{1}})^{A_{1}} and 2 (E_{x}^{A_{1}})^{A_{1}} + c^{A_{1}}{(A_{1},E_{x})} = 3 (E_{x}^{A_{1}})^{A_{1}} and (E_{x}^{A_{1}})^{A_{1}} + 2 c^{A_{1}}{(A_{1},E_{x})} = 3 (E_{x}^{A_{1}})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('A_1', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('c')(Symbol('A_1', commutative=True), Symbol('E_x', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["add", 2, "Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Function('c')(Symbol('A_1', commutative=True), Symbol('E_x', commutative=True)), Symbol('A_1', commutative=True))), Mul(Integer(2), Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"], [["add", 3, "Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Pow(Function('c')(Symbol('A_1', commutative=True), Symbol('E_x', commutative=True)), Symbol('A_1', commutative=True))), Mul(Integer(3), Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Mul(Integer(2), Pow(Function('c')(Symbol('A_1', commutative=True), Symbol('E_x', commutative=True)), Symbol('A_1', commutative=True)))), Mul(Integer(3), Pow(Pow(Symbol('E_x', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(y,v_{1})} = \\log{(v_{1} y)}, then obtain \\int (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\hat{p}_0{(y,v_{1})})^{v_{1}} dy = \\int (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\log{(v_{1} y)})^{v_{1}} dy", "derivation": "\\hat{p}_0{(y,v_{1})} = \\log{(v_{1} y)} and \\frac{\\partial}{\\partial v_{1}} \\hat{p}_0{(y,v_{1})} = \\frac{\\partial}{\\partial v_{1}} \\log{(v_{1} y)} and - \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\hat{p}_0{(y,v_{1})} = - \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\log{(v_{1} y)} and (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\hat{p}_0{(y,v_{1})})^{v_{1}} = (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\log{(v_{1} y)})^{v_{1}} and \\int (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\hat{p}_0{(y,v_{1})})^{v_{1}} dy = \\int (- \\log{(v_{1} y)} + \\frac{\\partial}{\\partial v_{1}} \\log{(v_{1} y)})^{v_{1}} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(Function('\\\\hat{p}_0')(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(Function('\\\\hat{p}_0')(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)), Pow(Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(Function('\\\\hat{p}_0')(Symbol('y', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True)))), Derivative(log(Mul(Symbol('v_1', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Symbol('v_1', commutative=True)), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\dot{z},t)} = \\dot{z} + t, then obtain \\frac{\\int \\mathbf{M}^{\\dot{z}}{(\\dot{z},t)} d\\dot{z}}{t} = \\frac{\\int (\\dot{z} + t)^{\\dot{z}} d\\dot{z}}{t}", "derivation": "\\mathbf{M}{(\\dot{z},t)} = \\dot{z} + t and t + \\mathbf{M}{(\\dot{z},t)} = \\dot{z} + 2 t and \\mathbf{M}^{\\dot{z}}{(\\dot{z},t)} = (\\dot{z} + t)^{\\dot{z}} and \\int \\mathbf{M}^{\\dot{z}}{(\\dot{z},t)} d\\dot{z} = \\int (\\dot{z} + t)^{\\dot{z}} d\\dot{z} and \\frac{\\int \\mathbf{M}^{\\dot{z}}{(\\dot{z},t)} d\\dot{z}}{\\dot{z} + 2 t - \\mathbf{M}{(\\dot{z},t)}} = \\frac{\\int (\\dot{z} + t)^{\\dot{z}} d\\dot{z}}{\\dot{z} + 2 t - \\mathbf{M}{(\\dot{z},t)}} and \\frac{\\int \\mathbf{M}^{\\dot{z}}{(\\dot{z},t)} d\\dot{z}}{t} = \\frac{\\int (\\dot{z} + t)^{\\dot{z}} d\\dot{z}}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('t', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)))), Integer(-1)), Integral(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('t', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)))), Integer(-1)), Integral(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Integral(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})} = \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\mathbf{P}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}, then obtain \\frac{d}{d \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})}) = \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{P}{(\\Psi^{\\dagger})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})} = \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\Psi^{\\dagger} + \\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\frac{d}{d \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})}) = \\frac{d}{d \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger}) and \\mathbf{P}{(\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\int \\sin{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and \\frac{d}{d \\Psi^{\\dagger}} (\\Psi^{\\dagger} + \\operatorname{a^{\\dagger}}{(\\Psi^{\\dagger})}) = \\frac{d}{d \\Psi^{\\dagger}} \\mathbf{P}{(\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{P}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{p}_0,\\Psi_{nl})} = - \\Psi_{nl} + \\cos{(\\hat{p}_0)}, then obtain - \\operatorname{E_{n}}^{\\Psi_{nl}}{(\\hat{p}_0,\\Psi_{nl})} - \\cos{(\\hat{p}_0)} = - (- \\Psi_{nl} + \\cos{(\\hat{p}_0)})^{\\Psi_{nl}} - \\cos{(\\hat{p}_0)}", "derivation": "\\operatorname{E_{n}}{(\\hat{p}_0,\\Psi_{nl})} = - \\Psi_{nl} + \\cos{(\\hat{p}_0)} and \\operatorname{E_{n}}^{\\Psi_{nl}}{(\\hat{p}_0,\\Psi_{nl})} = (- \\Psi_{nl} + \\cos{(\\hat{p}_0)})^{\\Psi_{nl}} and - \\operatorname{E_{n}}^{\\Psi_{nl}}{(\\hat{p}_0,\\Psi_{nl})} = - (- \\Psi_{nl} + \\cos{(\\hat{p}_0)})^{\\Psi_{nl}} and - \\operatorname{E_{n}}^{\\Psi_{nl}}{(\\hat{p}_0,\\Psi_{nl})} - \\cos{(\\hat{p}_0)} = - (- \\Psi_{nl} + \\cos{(\\hat{p}_0)})^{\\Psi_{nl}} - \\cos{(\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 3, "cos(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('E_n')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(f^{*},\\phi)} = \\phi f^{*}, then obtain 0 = \\frac{((\\frac{\\phi f^{*}}{\\operatorname{A_{2}}{(f^{*},\\phi)}})^{f^{*}} - 1) \\operatorname{A_{2}}{(f^{*},\\phi)}}{\\phi f^{*}}", "derivation": "\\operatorname{A_{2}}{(f^{*},\\phi)} = \\phi f^{*} and 1 = \\frac{\\phi f^{*}}{\\operatorname{A_{2}}{(f^{*},\\phi)}} and 1 = (\\frac{\\phi f^{*}}{\\operatorname{A_{2}}{(f^{*},\\phi)}})^{f^{*}} and 0 = (\\frac{\\phi f^{*}}{\\operatorname{A_{2}}{(f^{*},\\phi)}})^{f^{*}} - 1 and 0 = \\frac{((\\frac{\\phi f^{*}}{\\operatorname{A_{2}}{(f^{*},\\phi)}})^{f^{*}} - 1) \\operatorname{A_{2}}{(f^{*},\\phi)}}{\\phi f^{*}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)))"], [["divide", 1, "Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True), Pow(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True), Pow(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('f^*', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Integer(0), Add(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True), Pow(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('f^*', commutative=True)), Integer(-1)))"], [["divide", 4, "Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True), Pow(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Integer(-1)), Add(Pow(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True), Pow(Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('f^*', commutative=True)), Integer(-1)), Function('A_2')(Symbol('f^*', commutative=True), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\phi{(a,\\rho_f)} = - \\rho_f + e^{a} and \\Omega{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a}, then obtain \\phi^{2 a}{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a} \\phi^{a}{(a,\\rho_f)}", "derivation": "\\phi{(a,\\rho_f)} = - \\rho_f + e^{a} and \\phi^{a}{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a} and \\Omega{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a} and \\Omega{(a,\\rho_f)} \\phi^{a}{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a} \\Omega{(a,\\rho_f)} and \\phi^{a}{(a,\\rho_f)} = \\Omega{(a,\\rho_f)} and \\phi^{2 a}{(a,\\rho_f)} = (- \\rho_f + e^{a})^{a} \\phi^{a}{(a,\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["times", 2, "Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('a', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('a', commutative=True)), Function('\\\\Omega')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(2), Symbol('a', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Function('\\\\phi')(Symbol('a', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('a', commutative=True))))"]]}, {"prompt": "Given f{(\\theta_2,v_{t})} = \\sin{(\\theta_2 + v_{t})}, then derive \\int f{(\\theta_2,v_{t})} dv_{t} = p - \\cos{(\\theta_2 + v_{t})}, then derive \\hat{x} - \\cos{(\\theta_2 + v_{t})} = p - \\cos{(\\theta_2 + v_{t})}, then obtain \\hat{x} - \\cos{(\\theta_2 + v_{t})} = \\int f{(\\theta_2,v_{t})} dv_{t}", "derivation": "f{(\\theta_2,v_{t})} = \\sin{(\\theta_2 + v_{t})} and \\int f{(\\theta_2,v_{t})} dv_{t} = \\int \\sin{(\\theta_2 + v_{t})} dv_{t} and \\int f{(\\theta_2,v_{t})} dv_{t} = p - \\cos{(\\theta_2 + v_{t})} and \\int \\sin{(\\theta_2 + v_{t})} dv_{t} = p - \\cos{(\\theta_2 + v_{t})} and \\hat{x} - \\cos{(\\theta_2 + v_{t})} = p - \\cos{(\\theta_2 + v_{t})} and \\hat{x} - \\cos{(\\theta_2 + v_{t})} = \\int f{(\\theta_2,v_{t})} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), sin(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(sin(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('p', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('p', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))), Add(Symbol('p', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), cos(Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True))))), Integral(Function('f')(Symbol('\\\\theta_2', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given V{(U,P_{g})} = \\frac{\\log{(P_{g})}}{U}, then obtain \\int (P_{g} + ((U + V{(U,P_{g})})^{P_{g}})^{U}) dU = \\int (P_{g} + ((U + \\frac{\\log{(P_{g})}}{U})^{P_{g}})^{U}) dU", "derivation": "V{(U,P_{g})} = \\frac{\\log{(P_{g})}}{U} and U + V{(U,P_{g})} = U + \\frac{\\log{(P_{g})}}{U} and (U + V{(U,P_{g})})^{P_{g}} = (U + \\frac{\\log{(P_{g})}}{U})^{P_{g}} and ((U + V{(U,P_{g})})^{P_{g}})^{U} = ((U + \\frac{\\log{(P_{g})}}{U})^{P_{g}})^{U} and P_{g} + ((U + V{(U,P_{g})})^{P_{g}})^{U} = P_{g} + ((U + \\frac{\\log{(P_{g})}}{U})^{P_{g}})^{U} and \\int (P_{g} + ((U + V{(U,P_{g})})^{P_{g}})^{U}) dU = \\int (P_{g} + ((U + \\frac{\\log{(P_{g})}}{U})^{P_{g}})^{U}) dU", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True))))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True))), Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True)))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True)))"], [["add", 4, "Symbol('P_g', commutative=True)"], "Equality(Add(Symbol('P_g', commutative=True), Pow(Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True))), Add(Symbol('P_g', commutative=True), Pow(Pow(Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Symbol('P_g', commutative=True), Pow(Pow(Add(Symbol('U', commutative=True), Function('V')(Symbol('U', commutative=True), Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Pow(Pow(Add(Symbol('U', commutative=True), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('P_g', commutative=True)))), Symbol('P_g', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given C{(h)} = \\log{(h)}, then obtain 0 = - h C{(h)} + h \\log{(h)}", "derivation": "C{(h)} = \\log{(h)} and h C{(h)} = h \\log{(h)} and 2 h C{(h)} = h C{(h)} + h \\log{(h)} and 0 = - h C{(h)} + h \\log{(h)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))))"], [["add", 2, "Mul(Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('h', commutative=True), Function('C')(Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(A_{x},\\varepsilon)} = A_{x}^{\\varepsilon}, then obtain \\varepsilon + \\frac{\\partial}{\\partial A_{x}} \\mathbf{v}{(A_{x},\\varepsilon)} = \\varepsilon + \\frac{A_{x}^{\\varepsilon} \\varepsilon}{A_{x}}", "derivation": "\\mathbf{v}{(A_{x},\\varepsilon)} = A_{x}^{\\varepsilon} and \\frac{\\partial}{\\partial A_{x}} \\mathbf{v}{(A_{x},\\varepsilon)} = \\frac{\\partial}{\\partial A_{x}} A_{x}^{\\varepsilon} and \\varepsilon + \\frac{\\partial}{\\partial A_{x}} \\mathbf{v}{(A_{x},\\varepsilon)} = \\varepsilon + \\frac{\\partial}{\\partial A_{x}} A_{x}^{\\varepsilon} and \\varepsilon + \\frac{\\partial}{\\partial A_{x}} \\mathbf{v}{(A_{x},\\varepsilon)} = \\varepsilon + \\frac{A_{x}^{\\varepsilon} \\varepsilon}{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Symbol('\\\\varepsilon', commutative=True), Derivative(Pow(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(n_{2},\\rho_f)} = \\rho_f - n_{2}, then obtain \\frac{\\partial}{\\partial \\rho_f} \\psi^{*}{(n_{2},\\rho_f)} = 1", "derivation": "\\psi^{*}{(n_{2},\\rho_f)} = \\rho_f - n_{2} and n_{2} + \\psi^{*}{(n_{2},\\rho_f)} = \\rho_f and \\frac{\\partial}{\\partial \\rho_f} (n_{2} + \\psi^{*}{(n_{2},\\rho_f)}) = \\frac{d}{d \\rho_f} \\rho_f and \\frac{\\partial}{\\partial \\rho_f} \\psi^{*}{(n_{2},\\rho_f)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('n_2', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Add(Symbol('n_2', commutative=True), Function('\\\\psi^*')(Symbol('n_2', commutative=True), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Symbol('n_2', commutative=True), Function('\\\\psi^*')(Symbol('n_2', commutative=True), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\psi^*')(Symbol('n_2', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given p{(S,r)} = e^{S^{r}}, then obtain (\\frac{\\partial}{\\partial r} (S + p{(S,r)}) \\int p{(S,r)} dS)^{S} = (\\frac{\\partial}{\\partial r} (S + p{(S,r)}) \\int e^{S^{r}} dS)^{S}", "derivation": "p{(S,r)} = e^{S^{r}} and S + p{(S,r)} = S + e^{S^{r}} and \\int p{(S,r)} dS = \\int e^{S^{r}} dS and (S + e^{S^{r}}) \\int p{(S,r)} dS = (S + e^{S^{r}}) \\int e^{S^{r}} dS and \\frac{\\partial}{\\partial r} (S + e^{S^{r}}) \\int p{(S,r)} dS = \\frac{\\partial}{\\partial r} (S + e^{S^{r}}) \\int e^{S^{r}} dS and (\\frac{\\partial}{\\partial r} (S + e^{S^{r}}) \\int p{(S,r)} dS)^{S} = (\\frac{\\partial}{\\partial r} (S + e^{S^{r}}) \\int e^{S^{r}} dS)^{S} and (\\frac{\\partial}{\\partial r} (S + p{(S,r)}) \\int p{(S,r)} dS)^{S} = (\\frac{\\partial}{\\partial r} (S + p{(S,r)}) \\int e^{S^{r}} dS)^{S}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))))"], [["add", 1, "Symbol('S', commutative=True)"], "Equality(Add(Symbol('S', commutative=True), Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True))), Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["times", 3, "Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))))"], "Equality(Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 5, "Symbol('S', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Mul(Add(Symbol('S', commutative=True), exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True)))), Integral(exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Derivative(Mul(Add(Symbol('S', commutative=True), Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True))), Integral(Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('S', commutative=True)), Pow(Derivative(Mul(Add(Symbol('S', commutative=True), Function('p')(Symbol('S', commutative=True), Symbol('r', commutative=True))), Integral(exp(Pow(Symbol('S', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(f_{E})} = \\frac{d}{d f_{E}} \\sin{(f_{E})}, then derive \\Psi_{nl}{(f_{E})} = \\cos{(f_{E})}, then obtain (\\Psi_{nl}^{2}{(f_{E})})^{f_{E}} = (\\Psi_{nl}{(f_{E})} \\frac{d}{d f_{E}} \\sin{(f_{E})})^{f_{E}}", "derivation": "\\Psi_{nl}{(f_{E})} = \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\Psi_{nl}{(f_{E})} = \\cos{(f_{E})} and \\cos{(f_{E})} = \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\Psi_{nl}^{2}{(f_{E})} = \\Psi_{nl}{(f_{E})} \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\Psi_{nl}^{2}{(f_{E})} = \\Psi_{nl}{(f_{E})} \\cos{(f_{E})} and (\\Psi_{nl}^{2}{(f_{E})})^{f_{E}} = (\\Psi_{nl}{(f_{E})} \\cos{(f_{E})})^{f_{E}} and (\\Psi_{nl}^{2}{(f_{E})})^{f_{E}} = (\\Psi_{nl}{(f_{E})} \\frac{d}{d f_{E}} \\sin{(f_{E})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('f_E', commutative=True)), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True))"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Integer(2)), Mul(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Integer(2)), Mul(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))))"], [["power", 5, "Symbol('f_E', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Integer(2)), Symbol('f_E', commutative=True)), Pow(Mul(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Pow(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Integer(2)), Symbol('f_E', commutative=True)), Pow(Mul(Function('\\\\Psi_{nl}')(Symbol('f_E', commutative=True)), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given r{(b)} = \\sin{(b)}, then obtain r^{3}{(b)} - r^{2}{(b)} \\sin{(b)} + \\cos{(b \\sin{(b)})} = \\cos{(b \\sin{(b)})}", "derivation": "r{(b)} = \\sin{(b)} and r^{3}{(b)} = r^{2}{(b)} \\sin{(b)} and r^{3}{(b)} - r^{2}{(b)} \\sin{(b)} = 0 and r^{3}{(b)} - r^{2}{(b)} \\sin{(b)} + \\cos{(b \\sin{(b)})} = \\cos{(b \\sin{(b)})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["times", 1, "Pow(Function('r')(Symbol('b', commutative=True)), Integer(2))"], "Equality(Pow(Function('r')(Symbol('b', commutative=True)), Integer(3)), Mul(Pow(Function('r')(Symbol('b', commutative=True)), Integer(2)), sin(Symbol('b', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('r')(Symbol('b', commutative=True)), Integer(2)), sin(Symbol('b', commutative=True)))"], "Equality(Add(Pow(Function('r')(Symbol('b', commutative=True)), Integer(3)), Mul(Integer(-1), Pow(Function('r')(Symbol('b', commutative=True)), Integer(2)), sin(Symbol('b', commutative=True)))), Integer(0))"], [["add", 3, "cos(Mul(Symbol('b', commutative=True), sin(Symbol('b', commutative=True))))"], "Equality(Add(Pow(Function('r')(Symbol('b', commutative=True)), Integer(3)), Mul(Integer(-1), Pow(Function('r')(Symbol('b', commutative=True)), Integer(2)), sin(Symbol('b', commutative=True))), cos(Mul(Symbol('b', commutative=True), sin(Symbol('b', commutative=True))))), cos(Mul(Symbol('b', commutative=True), sin(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given c{(f^{\\prime},u)} = \\frac{e^{u}}{f^{\\prime}}, then derive \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} c{(f^{\\prime},u)} = - \\frac{e^{u}}{(f^{\\prime})^{2}}, then obtain - \\frac{e^{u} \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}}}{(f^{\\prime})^{2}} = (\\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}})^{2}", "derivation": "c{(f^{\\prime},u)} = \\frac{e^{u}}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} c{(f^{\\prime},u)} = \\frac{\\partial}{\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}} and \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} c{(f^{\\prime},u)} = \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}} and \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} c{(f^{\\prime},u)} = - \\frac{e^{u}}{(f^{\\prime})^{2}} and - \\frac{e^{u}}{(f^{\\prime})^{2}} = \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}} and - \\frac{e^{u} \\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}}}{(f^{\\prime})^{2}} = (\\frac{\\partial^{2}}{\\partial u\\partial f^{\\prime}} \\frac{e^{u}}{f^{\\prime}})^{2}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('f^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('f^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('f^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c')(Symbol('f^{\\\\prime}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), exp(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), exp(Symbol('u', commutative=True))), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 5, "Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), exp(Symbol('u', commutative=True)), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1)))), Pow(Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), exp(Symbol('u', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\chi{(\\phi,G)} = G^{\\phi}, then obtain \\frac{\\partial^{2}}{\\partial \\phi^{2}} \\int G \\int \\chi{(\\phi,G)} dG d\\phi = \\frac{\\partial^{2}}{\\partial \\phi^{2}} \\int G \\int G^{\\phi} dG d\\phi", "derivation": "\\chi{(\\phi,G)} = G^{\\phi} and \\int \\chi{(\\phi,G)} dG = \\int G^{\\phi} dG and G \\int \\chi{(\\phi,G)} dG = G \\int G^{\\phi} dG and \\int G \\int \\chi{(\\phi,G)} dG d\\phi = \\int G \\int G^{\\phi} dG d\\phi and \\frac{\\partial}{\\partial \\phi} \\int G \\int \\chi{(\\phi,G)} dG d\\phi = \\frac{\\partial}{\\partial \\phi} \\int G \\int G^{\\phi} dG d\\phi and \\frac{\\partial^{2}}{\\partial \\phi^{2}} \\int G \\int \\chi{(\\phi,G)} dG d\\phi = \\frac{\\partial^{2}}{\\partial \\phi^{2}} \\int G \\int G^{\\phi} dG d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["times", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Integral(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('G', commutative=True), Integral(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('G', commutative=True), Integral(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('G', commutative=True), Integral(Function('\\\\chi')(Symbol('\\\\phi', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(2))), Derivative(Integral(Mul(Symbol('G', commutative=True), Integral(Pow(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{f},\\theta)} = \\theta e^{\\mathbf{f}}, then derive e^{\\frac{\\partial}{\\partial \\theta} \\operatorname{v_{1}}{(\\mathbf{f},\\theta)}} = e^{e^{\\mathbf{f}}}, then obtain - \\theta e^{\\mathbf{f}} + e^{\\frac{\\partial}{\\partial \\theta} \\theta e^{\\mathbf{f}}} = - \\theta e^{\\mathbf{f}} + e^{e^{\\mathbf{f}}}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{f},\\theta)} = \\theta e^{\\mathbf{f}} and \\frac{\\partial}{\\partial \\theta} \\operatorname{v_{1}}{(\\mathbf{f},\\theta)} = \\frac{\\partial}{\\partial \\theta} \\theta e^{\\mathbf{f}} and e^{\\frac{\\partial}{\\partial \\theta} \\operatorname{v_{1}}{(\\mathbf{f},\\theta)}} = e^{\\frac{\\partial}{\\partial \\theta} \\theta e^{\\mathbf{f}}} and e^{\\frac{\\partial}{\\partial \\theta} \\operatorname{v_{1}}{(\\mathbf{f},\\theta)}} = e^{e^{\\mathbf{f}}} and e^{\\frac{\\partial}{\\partial \\theta} \\theta e^{\\mathbf{f}}} = e^{e^{\\mathbf{f}}} and - \\theta e^{\\mathbf{f}} + e^{\\frac{\\partial}{\\partial \\theta} \\theta e^{\\mathbf{f}}} = - \\theta e^{\\mathbf{f}} + e^{e^{\\mathbf{f}}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), exp(Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(exp(Derivative(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 5, "Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), exp(Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), exp(Symbol('\\\\mathbf{f}', commutative=True))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\phi{(\\delta,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\delta, then obtain \\Psi^{\\dagger} + \\delta - \\phi{(\\delta,\\Psi^{\\dagger})} + \\iint (- \\Psi^{\\dagger} - \\delta + \\phi{(\\delta,\\Psi^{\\dagger})}) d\\delta d\\delta = \\Psi^{\\dagger} + \\delta - \\phi{(\\delta,\\Psi^{\\dagger})} + \\iint 0 d\\delta d\\delta", "derivation": "\\phi{(\\delta,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + \\delta and - \\Psi^{\\dagger} - \\delta + \\phi{(\\delta,\\Psi^{\\dagger})} = 0 and \\int (- \\Psi^{\\dagger} - \\delta + \\phi{(\\delta,\\Psi^{\\dagger})}) d\\delta = \\int 0 d\\delta and \\iint (- \\Psi^{\\dagger} - \\delta + \\phi{(\\delta,\\Psi^{\\dagger})}) d\\delta d\\delta = \\iint 0 d\\delta d\\delta and \\Psi^{\\dagger} + \\delta - \\phi{(\\delta,\\Psi^{\\dagger})} + \\iint (- \\Psi^{\\dagger} - \\delta + \\phi{(\\delta,\\Psi^{\\dagger})}) d\\delta d\\delta = \\Psi^{\\dagger} + \\delta - \\phi{(\\delta,\\Psi^{\\dagger})} + \\iint 0 d\\delta d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(i,\\mathbf{f})} = i^{\\mathbf{f}}, then obtain \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{C_{2}}{(i,\\mathbf{f})} \\int \\operatorname{C_{2}}{(i,\\mathbf{f})} d\\mathbf{f} = \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{C_{2}}{(i,\\mathbf{f})} \\int i^{\\mathbf{f}} d\\mathbf{f}", "derivation": "\\operatorname{C_{2}}{(i,\\mathbf{f})} = i^{\\mathbf{f}} and \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{C_{2}}{(i,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} i^{\\mathbf{f}} and \\int \\operatorname{C_{2}}{(i,\\mathbf{f})} d\\mathbf{f} = \\int i^{\\mathbf{f}} d\\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} i^{\\mathbf{f}} \\int \\operatorname{C_{2}}{(i,\\mathbf{f})} d\\mathbf{f} = \\frac{\\partial}{\\partial \\mathbf{f}} i^{\\mathbf{f}} \\int i^{\\mathbf{f}} d\\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{C_{2}}{(i,\\mathbf{f})} \\int \\operatorname{C_{2}}{(i,\\mathbf{f})} d\\mathbf{f} = \\frac{\\partial}{\\partial \\mathbf{f}} \\operatorname{C_{2}}{(i,\\mathbf{f})} \\int i^{\\mathbf{f}} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 3, "Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integral(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Derivative(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Derivative(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integral(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Derivative(Function('C_2')(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integral(Pow(Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(\\dot{z})} = e^{\\dot{z}}, then obtain \\int (\\hat{X}{(\\dot{z})} - e^{\\dot{z}} + 1) d\\dot{z} = \\int 1 d\\dot{z}", "derivation": "\\hat{X}{(\\dot{z})} = e^{\\dot{z}} and - \\dot{z} + \\hat{X}{(\\dot{z})} = - \\dot{z} + e^{\\dot{z}} and \\hat{X}{(\\dot{z})} - e^{\\dot{z}} = 0 and \\hat{X}{(\\dot{z})} - e^{\\dot{z}} + 1 = 1 and \\int (\\hat{X}{(\\dot{z})} - e^{\\dot{z}} + 1) d\\dot{z} = \\int 1 d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True)))), Integer(0))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Integer(1))"], [["integrate", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{X}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\dot{z}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given Z{(\\varepsilon)} = \\log{(\\varepsilon)} and \\phi_{1}{(\\varepsilon)} = \\int \\log{(\\varepsilon)}^{\\varepsilon} d\\varepsilon, then obtain Z{(\\varepsilon)} \\phi_{1}{(\\varepsilon)} \\log{(k)} = Z{(\\varepsilon)} \\log{(k)} \\int Z^{\\varepsilon}{(\\varepsilon)} d\\varepsilon", "derivation": "Z{(\\varepsilon)} = \\log{(\\varepsilon)} and Z^{\\varepsilon}{(\\varepsilon)} = \\log{(\\varepsilon)}^{\\varepsilon} and \\int Z^{\\varepsilon}{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\varepsilon)}^{\\varepsilon} d\\varepsilon and \\phi_{1}{(\\varepsilon)} = \\int \\log{(\\varepsilon)}^{\\varepsilon} d\\varepsilon and \\phi_{1}{(\\varepsilon)} \\log{(k)} = \\log{(k)} \\int \\log{(\\varepsilon)}^{\\varepsilon} d\\varepsilon and Z{(\\varepsilon)} \\phi_{1}{(\\varepsilon)} \\log{(k)} = Z{(\\varepsilon)} \\log{(k)} \\int \\log{(\\varepsilon)}^{\\varepsilon} d\\varepsilon and Z{(\\varepsilon)} \\phi_{1}{(\\varepsilon)} \\log{(k)} = Z{(\\varepsilon)} \\log{(k)} \\int Z^{\\varepsilon}{(\\varepsilon)} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(log(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Pow(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)), Integral(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 4, "log(Symbol('k', commutative=True))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('k', commutative=True))), Mul(log(Symbol('k', commutative=True)), Integral(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["divide", 5, "Pow(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))"], "Equality(Mul(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('k', commutative=True))), Mul(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('k', commutative=True)), Integral(Pow(log(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('k', commutative=True))), Mul(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('k', commutative=True)), Integral(Pow(Function('Z')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given z{(\\pi,\\eta^{\\prime})} = \\eta^{\\prime} - \\pi, then obtain \\pi + 2 z{(\\pi,\\eta^{\\prime})} = 2 \\eta^{\\prime} - \\pi", "derivation": "z{(\\pi,\\eta^{\\prime})} = \\eta^{\\prime} - \\pi and \\eta^{\\prime} - \\pi + z{(\\pi,\\eta^{\\prime})} = 2 \\eta^{\\prime} - 2 \\pi and 2 z{(\\pi,\\eta^{\\prime})} = 2 \\eta^{\\prime} - 2 \\pi and \\pi + 2 z{(\\pi,\\eta^{\\prime})} = 2 \\eta^{\\prime} - \\pi", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\pi', commutative=True))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Function('z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\phi_2,n_{2})} = \\sin{(\\frac{\\phi_2}{n_{2}})}, then obtain \\cos{((\\frac{\\operatorname{v_{t}}{(\\phi_2,n_{2})}}{\\sin{(\\frac{\\phi_2}{n_{2}})}})^{n_{2}} - \\sin{(\\frac{\\phi_2}{n_{2}})})} = \\cos{(\\sin{(\\frac{\\phi_2}{n_{2}})} - 1)}", "derivation": "\\operatorname{v_{t}}{(\\phi_2,n_{2})} = \\sin{(\\frac{\\phi_2}{n_{2}})} and \\frac{\\operatorname{v_{t}}{(\\phi_2,n_{2})}}{\\sin{(\\frac{\\phi_2}{n_{2}})}} = 1 and (\\frac{\\operatorname{v_{t}}{(\\phi_2,n_{2})}}{\\sin{(\\frac{\\phi_2}{n_{2}})}})^{n_{2}} = 1 and (\\frac{\\operatorname{v_{t}}{(\\phi_2,n_{2})}}{\\sin{(\\frac{\\phi_2}{n_{2}})}})^{n_{2}} - \\sin{(\\frac{\\phi_2}{n_{2}})} = 1 - \\sin{(\\frac{\\phi_2}{n_{2}})} and \\cos{((\\frac{\\operatorname{v_{t}}{(\\phi_2,n_{2})}}{\\sin{(\\frac{\\phi_2}{n_{2}})}})^{n_{2}} - \\sin{(\\frac{\\phi_2}{n_{2}})})} = \\cos{(\\sin{(\\frac{\\phi_2}{n_{2}})} - 1)}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\phi_2', commutative=True), Symbol('n_2', commutative=True)), sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))))"], [["divide", 1, "sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], "Equality(Mul(Function('v_t')(Symbol('\\\\phi_2', commutative=True), Symbol('n_2', commutative=True)), Pow(sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Function('v_t')(Symbol('\\\\phi_2', commutative=True), Symbol('n_2', commutative=True)), Pow(sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))), Integer(-1))), Symbol('n_2', commutative=True)), Integer(1))"], [["add", 3, "Mul(Integer(-1), sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))))"], "Equality(Add(Pow(Mul(Function('v_t')(Symbol('\\\\phi_2', commutative=True), Symbol('n_2', commutative=True)), Pow(sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))), Integer(-1))), Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))))), Add(Integer(1), Mul(Integer(-1), sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))))))"], [["cos", 4], "Equality(cos(Add(Pow(Mul(Function('v_t')(Symbol('\\\\phi_2', commutative=True), Symbol('n_2', commutative=True)), Pow(sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))), Integer(-1))), Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))))), cos(Add(sin(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(V,T,v_{y})} = T^{V} v_{y}, then obtain (\\frac{\\partial}{\\partial V} T^{V} v_{y} \\operatorname{F_{H}}{(V,T,v_{y})})^{V} = (\\frac{\\partial}{\\partial V} T^{2 V} v_{y}^{2})^{V}", "derivation": "\\operatorname{F_{H}}{(V,T,v_{y})} = T^{V} v_{y} and T^{V} v_{y} \\operatorname{F_{H}}{(V,T,v_{y})} = T^{2 V} v_{y}^{2} and \\frac{\\partial}{\\partial V} T^{V} v_{y} \\operatorname{F_{H}}{(V,T,v_{y})} = \\frac{\\partial}{\\partial V} T^{2 V} v_{y}^{2} and (\\frac{\\partial}{\\partial V} T^{V} v_{y} \\operatorname{F_{H}}{(V,T,v_{y})})^{V} = (\\frac{\\partial}{\\partial V} T^{2 V} v_{y}^{2})^{V}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Symbol('V', commutative=True)), Symbol('v_y', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('T', commutative=True), Symbol('V', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Symbol('V', commutative=True)), Symbol('v_y', commutative=True), Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True), Symbol('v_y', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Pow(Symbol('v_y', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('T', commutative=True), Symbol('V', commutative=True)), Symbol('v_y', commutative=True), Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('T', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Pow(Symbol('v_y', commutative=True), Integer(2))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('T', commutative=True), Symbol('V', commutative=True)), Symbol('v_y', commutative=True), Function('F_H')(Symbol('V', commutative=True), Symbol('T', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('T', commutative=True), Mul(Integer(2), Symbol('V', commutative=True))), Pow(Symbol('v_y', commutative=True), Integer(2))), Tuple(Symbol('V', commutative=True), Integer(1))), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\eta^{\\prime},\\sigma_x,G)} = (G - \\sigma_x)^{\\eta^{\\prime}}, then obtain (\\frac{\\hat{\\mathbf{x}}^{G}{(\\eta^{\\prime},\\sigma_x,G)}}{f_{E}})^{\\eta^{\\prime}} = (\\frac{((G - \\sigma_x)^{\\eta^{\\prime}})^{G}}{f_{E}})^{\\eta^{\\prime}}", "derivation": "\\hat{\\mathbf{x}}{(\\eta^{\\prime},\\sigma_x,G)} = (G - \\sigma_x)^{\\eta^{\\prime}} and \\hat{\\mathbf{x}}^{G}{(\\eta^{\\prime},\\sigma_x,G)} = ((G - \\sigma_x)^{\\eta^{\\prime}})^{G} and \\frac{\\hat{\\mathbf{x}}^{G}{(\\eta^{\\prime},\\sigma_x,G)}}{f_{E}} = \\frac{((G - \\sigma_x)^{\\eta^{\\prime}})^{G}}{f_{E}} and (\\frac{\\hat{\\mathbf{x}}^{G}{(\\eta^{\\prime},\\sigma_x,G)}}{f_{E}})^{\\eta^{\\prime}} = (\\frac{((G - \\sigma_x)^{\\eta^{\\prime}})^{G}}{f_{E}})^{\\eta^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('G', commutative=True)), Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True)), Pow(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('G', commutative=True)))"], [["divide", 2, "Symbol('f_E', commutative=True)"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('G', commutative=True))))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('G', commutative=True)), Symbol('G', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('G', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(F_{x})} = \\cos{(\\log{(F_{x})})}, then obtain F_{x} \\hat{p}{(F_{x})} + \\frac{d^{2}}{d F_{x}^{2}} \\hat{p}{(F_{x})} \\cos{(\\log{(F_{x})})} = F_{x} \\hat{p}{(F_{x})} + \\frac{d^{2}}{d F_{x}^{2}} \\cos^{2}{(\\log{(F_{x})})}", "derivation": "\\hat{p}{(F_{x})} = \\cos{(\\log{(F_{x})})} and \\hat{p}{(F_{x})} \\cos{(\\log{(F_{x})})} = \\cos^{2}{(\\log{(F_{x})})} and \\frac{d}{d F_{x}} \\hat{p}{(F_{x})} \\cos{(\\log{(F_{x})})} = \\frac{d}{d F_{x}} \\cos^{2}{(\\log{(F_{x})})} and \\frac{d^{2}}{d F_{x}^{2}} \\hat{p}{(F_{x})} \\cos{(\\log{(F_{x})})} = \\frac{d^{2}}{d F_{x}^{2}} \\cos^{2}{(\\log{(F_{x})})} and F_{x} \\hat{p}{(F_{x})} + \\frac{d^{2}}{d F_{x}^{2}} \\hat{p}{(F_{x})} \\cos{(\\log{(F_{x})})} = F_{x} \\hat{p}{(F_{x})} + \\frac{d^{2}}{d F_{x}^{2}} \\cos^{2}{(\\log{(F_{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('F_x', commutative=True)), cos(log(Symbol('F_x', commutative=True))))"], [["times", 1, "cos(log(Symbol('F_x', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('F_x', commutative=True)), cos(log(Symbol('F_x', commutative=True)))), Pow(cos(log(Symbol('F_x', commutative=True))), Integer(2)))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{p}')(Symbol('F_x', commutative=True)), cos(log(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Pow(cos(log(Symbol('F_x', commutative=True))), Integer(2)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{p}')(Symbol('F_x', commutative=True)), cos(log(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2))), Derivative(Pow(cos(log(Symbol('F_x', commutative=True))), Integer(2)), Tuple(Symbol('F_x', commutative=True), Integer(2))))"], [["add", 4, "Mul(Symbol('F_x', commutative=True), Function('\\\\hat{p}')(Symbol('F_x', commutative=True)))"], "Equality(Add(Mul(Symbol('F_x', commutative=True), Function('\\\\hat{p}')(Symbol('F_x', commutative=True))), Derivative(Mul(Function('\\\\hat{p}')(Symbol('F_x', commutative=True)), cos(log(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(2)))), Add(Mul(Symbol('F_x', commutative=True), Function('\\\\hat{p}')(Symbol('F_x', commutative=True))), Derivative(Pow(cos(log(Symbol('F_x', commutative=True))), Integer(2)), Tuple(Symbol('F_x', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(F_{c})} = \\log{(F_{c})}, then obtain (\\operatorname{f^{*}}^{F_{c}}{(F_{c})})^{F_{c}} \\operatorname{f^{*}}^{- F_{c}}{(F_{c})} = (\\log{(F_{c})}^{F_{c}})^{F_{c}} \\operatorname{f^{*}}^{- F_{c}}{(F_{c})}", "derivation": "\\operatorname{f^{*}}{(F_{c})} = \\log{(F_{c})} and \\operatorname{f^{*}}^{F_{c}}{(F_{c})} = \\log{(F_{c})}^{F_{c}} and (\\operatorname{f^{*}}^{F_{c}}{(F_{c})})^{F_{c}} = (\\log{(F_{c})}^{F_{c}})^{F_{c}} and (\\operatorname{f^{*}}^{F_{c}}{(F_{c})})^{F_{c}} \\operatorname{f^{*}}^{- F_{c}}{(F_{c})} = (\\log{(F_{c})}^{F_{c}})^{F_{c}} \\operatorname{f^{*}}^{- F_{c}}{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Pow(Function('f^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Pow(log(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["divide", 3, "Pow(Function('f^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], "Equality(Mul(Pow(Pow(Function('f^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Function('f^*')(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True)))), Mul(Pow(Pow(log(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(Function('f^*')(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\psi^*,r,F_{H})} = \\frac{(\\psi^*)^{r}}{F_{H}} and \\Psi_{\\lambda}{(\\psi^*)} = \\psi^*, then obtain \\Psi_{\\lambda}{(\\psi^*)} + \\frac{(\\psi^*)^{r}}{F_{H}} = \\psi^* + \\frac{(\\psi^*)^{r}}{F_{H}}", "derivation": "\\rho{(\\psi^*,r,F_{H})} = \\frac{(\\psi^*)^{r}}{F_{H}} and \\Psi_{\\lambda}{(\\psi^*)} = \\psi^* and \\Psi_{\\lambda}{(\\psi^*)} + \\rho{(\\psi^*,r,F_{H})} = \\psi^* + \\rho{(\\psi^*,r,F_{H})} and \\Psi_{\\lambda}{(\\psi^*)} + \\frac{(\\psi^*)^{r}}{F_{H}} = \\psi^* + \\frac{(\\psi^*)^{r}}{F_{H}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["add", 2, "Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi^*', commutative=True)), Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\rho')(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True), Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi^*', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{H})} = e^{\\hat{H}}, then obtain (\\operatorname{A_{2}}{(\\hat{H})} + \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})}) \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})} = (\\operatorname{A_{2}}{(\\hat{H})} + \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})}) (e^{\\hat{H}})^{\\hat{H}}", "derivation": "\\operatorname{A_{2}}{(\\hat{H})} = e^{\\hat{H}} and \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})} = (e^{\\hat{H}})^{\\hat{H}} and \\operatorname{A_{2}}{(\\hat{H})} + \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})} = \\operatorname{A_{2}}{(\\hat{H})} + (e^{\\hat{H}})^{\\hat{H}} and (\\operatorname{A_{2}}{(\\hat{H})} + (e^{\\hat{H}})^{\\hat{H}}) \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})} = (\\operatorname{A_{2}}{(\\hat{H})} + (e^{\\hat{H}})^{\\hat{H}}) (e^{\\hat{H}})^{\\hat{H}} and (\\operatorname{A_{2}}{(\\hat{H})} + \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})}) \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})} = (\\operatorname{A_{2}}{(\\hat{H})} + \\operatorname{A_{2}}^{\\hat{H}}{(\\hat{H})}) (e^{\\hat{H}})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["add", 2, "Function('A_2')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))))"], [["times", 2, "Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Mul(Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Mul(Add(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('A_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\dot{y})} = \\log{(\\dot{y})}, then obtain 2 (e^{\\tilde{g}{(\\dot{y})}})^{\\dot{y}} = \\dot{y}^{\\dot{y}} + (e^{\\tilde{g}{(\\dot{y})}})^{\\dot{y}}", "derivation": "\\tilde{g}{(\\dot{y})} = \\log{(\\dot{y})} and e^{\\tilde{g}{(\\dot{y})}} = \\dot{y} and (e^{\\tilde{g}{(\\dot{y})}})^{\\dot{y}} = \\dot{y}^{\\dot{y}} and 2 (e^{\\tilde{g}{(\\dot{y})}})^{\\dot{y}} = \\dot{y}^{\\dot{y}} + (e^{\\tilde{g}{(\\dot{y})}})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["add", 3, "Pow(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Integer(2), Pow(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))), Add(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Function('\\\\tilde{g}')(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\mathbf{B},\\dot{y})} = \\dot{y}^{\\mathbf{B}} and g{(\\mathbf{B},\\dot{y})} = \\theta^{\\dot{y}}{(\\mathbf{B},\\dot{y})}, then obtain g{(\\mathbf{B},\\dot{y})} = (\\dot{y}^{\\mathbf{B}})^{\\dot{y}}", "derivation": "\\theta{(\\mathbf{B},\\dot{y})} = \\dot{y}^{\\mathbf{B}} and \\theta^{\\dot{y}}{(\\mathbf{B},\\dot{y})} = (\\dot{y}^{\\mathbf{B}})^{\\dot{y}} and g{(\\mathbf{B},\\dot{y})} = \\theta^{\\dot{y}}{(\\mathbf{B},\\dot{y})} and g{(\\mathbf{B},\\dot{y})} = (\\dot{y}^{\\mathbf{B}})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(m_{s})} = \\cos{(m_{s})} and G{(m_{s})} = \\cos{(m_{s})}, then obtain \\frac{d}{d m_{s}} - \\sin{(\\varepsilon_{0}{(m_{s})})} = \\frac{d}{d m_{s}} - \\sin{(\\cos{(m_{s})})}", "derivation": "\\varepsilon_{0}{(m_{s})} = \\cos{(m_{s})} and G{(m_{s})} = \\cos{(m_{s})} and G{(m_{s})} = \\varepsilon_{0}{(m_{s})} and \\sin{(G{(m_{s})})} = \\sin{(\\cos{(m_{s})})} and \\sin{(\\varepsilon_{0}{(m_{s})})} = \\sin{(\\cos{(m_{s})})} and \\sin{(\\varepsilon_{0}{(m_{s})})} = \\sin{(G{(m_{s})})} and - \\sin{(\\varepsilon_{0}{(m_{s})})} = - \\sin{(G{(m_{s})})} and \\frac{d}{d m_{s}} - \\sin{(\\varepsilon_{0}{(m_{s})})} = \\frac{d}{d m_{s}} - \\sin{(G{(m_{s})})} and \\frac{d}{d m_{s}} - \\sin{(\\varepsilon_{0}{(m_{s})})} = \\frac{d}{d m_{s}} - \\sin{(\\cos{(m_{s})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)), cos(Symbol('m_s', commutative=True)))"], ["renaming_premise", "Equality(Function('G')(Symbol('m_s', commutative=True)), cos(Symbol('m_s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('G')(Symbol('m_s', commutative=True)), Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)))"], [["sin", 2], "Equality(sin(Function('G')(Symbol('m_s', commutative=True))), sin(cos(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(sin(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True))), sin(cos(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(sin(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True))), sin(Function('G')(Symbol('m_s', commutative=True))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), sin(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)))), Mul(Integer(-1), sin(Function('G')(Symbol('m_s', commutative=True)))))"], [["differentiate", 7, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Function('G')(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Derivative(Mul(Integer(-1), sin(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(cos(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(C_{1},n)} = C_{1}^{n}, then obtain \\frac{\\partial}{\\partial C_{1}} E{(C_{1},n)} - \\frac{\\partial}{\\partial C_{1}} E^{C_{1}}{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} C_{1}^{n} - \\frac{\\partial}{\\partial C_{1}} E^{C_{1}}{(C_{1},n)}", "derivation": "E{(C_{1},n)} = C_{1}^{n} and \\frac{\\partial}{\\partial C_{1}} E{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} C_{1}^{n} and E^{C_{1}}{(C_{1},n)} = (C_{1}^{n})^{C_{1}} and \\frac{\\partial}{\\partial C_{1}} E^{C_{1}}{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} (C_{1}^{n})^{C_{1}} and - \\frac{\\partial}{\\partial C_{1}} (C_{1}^{n})^{C_{1}} + \\frac{\\partial}{\\partial C_{1}} E{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} C_{1}^{n} - \\frac{\\partial}{\\partial C_{1}} (C_{1}^{n})^{C_{1}} and \\frac{\\partial}{\\partial C_{1}} E{(C_{1},n)} - \\frac{\\partial}{\\partial C_{1}} E^{C_{1}}{(C_{1},n)} = \\frac{\\partial}{\\partial C_{1}} C_{1}^{n} - \\frac{\\partial}{\\partial C_{1}} E^{C_{1}}{(C_{1},n)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Pow(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Add(Derivative(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))), Add(Derivative(Pow(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Pow(Function('E')(Symbol('C_1', commutative=True), Symbol('n', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain \\dot{\\mathbf{r}}^{2}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} - \\sin^{2}{(\\hat{H}_l)} = \\sin^{3}{(\\hat{H}_l)} - \\sin^{2}{(\\hat{H}_l)}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\dot{\\mathbf{r}}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} = \\sin^{2}{(\\hat{H}_l)} and \\dot{\\mathbf{r}}{(\\hat{H}_l)} \\sin^{2}{(\\hat{H}_l)} = \\sin^{3}{(\\hat{H}_l)} and \\dot{\\mathbf{r}}^{2}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} = \\sin^{3}{(\\hat{H}_l)} and \\dot{\\mathbf{r}}^{2}{(\\hat{H}_l)} \\sin{(\\hat{H}_l)} - \\sin^{2}{(\\hat{H}_l)} = \\sin^{3}{(\\hat{H}_l)} - \\sin^{2}{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))"], [["times", 2, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(3)))"], [["minus", 4, "Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))"], "Equality(Add(Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))), Add(Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(3)), Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\rho{(C_{d},\\ddot{x})} = C_{d} \\ddot{x}, then obtain \\ddot{x}^{2} = C_{d} \\ddot{x}^{2} + \\ddot{x}^{2} - \\ddot{x} \\rho{(C_{d},\\ddot{x})}", "derivation": "\\rho{(C_{d},\\ddot{x})} = C_{d} \\ddot{x} and \\ddot{x} \\rho{(C_{d},\\ddot{x})} = C_{d} \\ddot{x}^{2} and \\ddot{x}^{2} + \\ddot{x} \\rho{(C_{d},\\ddot{x})} = C_{d} \\ddot{x}^{2} + \\ddot{x}^{2} and \\ddot{x}^{2} = C_{d} \\ddot{x}^{2} + \\ddot{x}^{2} - \\ddot{x} \\rho{(C_{d},\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))))"], [["add", 2, "Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Mul(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))))"], [["minus", 3, "Mul(Symbol('\\\\ddot{x}', commutative=True), Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given z{(i,r)} = e^{i + r} and x{(i,\\varphi^*,r)} = \\varphi^* + z{(i,r)}, then obtain \\frac{- \\lambda \\mu + \\varphi^* + e^{i + r}}{i} = \\frac{- \\lambda \\mu + x{(i,\\varphi^*,r)}}{i}", "derivation": "z{(i,r)} = e^{i + r} and \\varphi^* + z{(i,r)} = \\varphi^* + e^{i + r} and - \\lambda \\mu + \\varphi^* + z{(i,r)} = - \\lambda \\mu + \\varphi^* + e^{i + r} and x{(i,\\varphi^*,r)} = \\varphi^* + z{(i,r)} and x{(i,\\varphi^*,r)} = \\varphi^* + e^{i + r} and - \\lambda \\mu + \\varphi^* + z{(i,r)} = - \\lambda \\mu + x{(i,\\varphi^*,r)} and - \\lambda \\mu + \\varphi^* + e^{i + r} = - \\lambda \\mu + x{(i,\\varphi^*,r)} and \\frac{- \\lambda \\mu + \\varphi^* + e^{i + r}}{i} = \\frac{- \\lambda \\mu + x{(i,\\varphi^*,r)}}{i}", "srepr_derivation": [["get_premise", "Equality(Function('z')(Symbol('i', commutative=True), Symbol('r', commutative=True)), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], [["add", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('z')(Symbol('i', commutative=True), Symbol('r', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))))"], [["minus", 2, "Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('z')(Symbol('i', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))))"], ["renaming_premise", "Equality(Function('x')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('r', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), Function('z')(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('x')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('r', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('z')(Symbol('i', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Function('x')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Function('x')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('r', commutative=True))))"], [["divide", 7, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True), exp(Add(Symbol('i', commutative=True), Symbol('r', commutative=True))))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mu', commutative=True)), Function('x')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbb{I})} = \\sin{(\\sin{(\\mathbb{I})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} = \\cos{(\\sin{(\\sin{(\\mathbb{I})})})}, then obtain \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} d\\mathbb{I} = \\int \\cos{(\\hat{\\mathbf{r}}{(\\mathbb{I})})} d\\mathbb{I}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbb{I})} = \\sin{(\\sin{(\\mathbb{I})})} and \\cos{(\\hat{\\mathbf{r}}{(\\mathbb{I})})} = \\cos{(\\sin{(\\sin{(\\mathbb{I})})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} = \\cos{(\\sin{(\\sin{(\\mathbb{I})})})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} = \\cos{(\\hat{\\mathbf{r}}{(\\mathbb{I})})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbb{I})} d\\mathbb{I} = \\int \\cos{(\\hat{\\mathbf{r}}{(\\mathbb{I})})} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True)), sin(sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True))), cos(sin(sin(Symbol('\\\\mathbb{I}', commutative=True)))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(sin(sin(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(cos(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(v_{z},f_{E})} = \\frac{f_{E}}{v_{z}}, then obtain \\int 0 dv_{z} = \\int (((\\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}})^{v_{z}})^{v_{z}} - 1) dv_{z}", "derivation": "\\operatorname{A_{y}}{(v_{z},f_{E})} = \\frac{f_{E}}{v_{z}} and 1 = \\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}} and 1 = (\\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}})^{v_{z}} and 1 = ((\\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}})^{v_{z}})^{v_{z}} and 0 = ((\\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}})^{v_{z}})^{v_{z}} - 1 and \\int 0 dv_{z} = \\int (((\\frac{f_{E}}{v_{z} \\operatorname{A_{y}}{(v_{z},f_{E})}})^{v_{z}})^{v_{z}} - 1) dv_{z}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1))))"], [["divide", 1, "Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(1), Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Symbol('v_z', commutative=True)))"], [["power", 3, "Symbol('v_z', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["add", 4, "Integer(-1)"], "Equality(Integer(0), Add(Pow(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('v_z', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Pow(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v_z', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Tuple(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(U,\\varphi)} = - U + \\varphi, then derive \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} \\operatorname{f^{\\prime}}{(U,\\varphi)} = \\operatorname{f^{\\prime}}{(U,\\varphi)} + 1, then obtain \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} (- U + \\varphi) = \\operatorname{f^{\\prime}}{(U,\\varphi)} + 1", "derivation": "\\operatorname{f^{\\prime}}{(U,\\varphi)} = - U + \\varphi and \\frac{\\partial}{\\partial \\varphi} \\operatorname{f^{\\prime}}{(U,\\varphi)} = \\frac{\\partial}{\\partial \\varphi} (- U + \\varphi) and \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} \\operatorname{f^{\\prime}}{(U,\\varphi)} = \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} (- U + \\varphi) and \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} \\operatorname{f^{\\prime}}{(U,\\varphi)} = \\operatorname{f^{\\prime}}{(U,\\varphi)} + 1 and \\operatorname{f^{\\prime}}{(U,\\varphi)} + \\frac{\\partial}{\\partial \\varphi} (- U + \\varphi) = \\operatorname{f^{\\prime}}{(U,\\varphi)} + 1", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["add", 2, "Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Function('f^{\\\\prime}')(Symbol('U', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}}, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = F_{x} + e^{\\Psi^{\\dagger}}, then obtain \\int e^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} = F_{x} + e^{\\Psi^{\\dagger}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int e^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = F_{x} + e^{\\Psi^{\\dagger}} and \\int e^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} = F_{x} + e^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('F_x', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('F_x', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\theta_1,B)} = e^{\\theta_1^{B}} and E{(\\mathbb{I},\\theta_2)} = \\sin{(\\frac{\\theta_2}{\\mathbb{I}})}, then derive \\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)} = \\frac{B \\theta_1^{B} e^{\\theta_1^{B}}}{\\theta_1}, then obtain \\frac{\\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)}}{\\sin{(\\frac{\\theta_2}{\\mathbb{I}})}} = \\frac{B \\theta_1^{B} e^{\\theta_1^{B}}}{\\theta_1 \\sin{(\\frac{\\theta_2}{\\mathbb{I}})}}", "derivation": "\\rho{(\\theta_1,B)} = e^{\\theta_1^{B}} and \\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)} = \\frac{\\partial}{\\partial \\theta_1} e^{\\theta_1^{B}} and \\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)} = \\frac{B \\theta_1^{B} e^{\\theta_1^{B}}}{\\theta_1} and E{(\\mathbb{I},\\theta_2)} = \\sin{(\\frac{\\theta_2}{\\mathbb{I}})} and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)}}{E{(\\mathbb{I},\\theta_2)}} = \\frac{B \\theta_1^{B} e^{\\theta_1^{B}}}{\\theta_1 E{(\\mathbb{I},\\theta_2)}} and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\rho{(\\theta_1,B)}}{\\sin{(\\frac{\\theta_2}{\\mathbb{I}})}} = \\frac{B \\theta_1^{B} e^{\\theta_1^{B}}}{\\theta_1 \\sin{(\\frac{\\theta_2}{\\mathbb{I}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), exp(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), exp(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)))))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 3, "Function('E')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Function('E')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Derivative(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Pow(Function('E')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), exp(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Derivative(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True)), exp(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('B', commutative=True))), Pow(sin(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given m{(v)} = \\sin{(e^{v})} and \\operatorname{J_{\\varepsilon}}{(v)} = \\sin^{v}{(e^{v})}, then obtain (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} m^{v}{(v)})^{\\mathbf{g}} = (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} \\operatorname{J_{\\varepsilon}}{(v)})^{\\mathbf{g}}", "derivation": "m{(v)} = \\sin{(e^{v})} and m^{v}{(v)} = \\sin^{v}{(e^{v})} and \\operatorname{J_{\\varepsilon}}{(v)} = \\sin^{v}{(e^{v})} and \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} m^{v}{(v)} = \\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} \\sin^{v}{(e^{v})} and (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} m^{v}{(v)})^{\\mathbf{g}} = (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} \\sin^{v}{(e^{v})})^{\\mathbf{g}} and (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} m^{v}{(v)})^{\\mathbf{g}} = (\\operatorname{J_{\\varepsilon}}{(\\mathbf{g})} \\operatorname{J_{\\varepsilon}}{(v)})^{\\mathbf{g}}", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('v', commutative=True)), sin(exp(Symbol('v', commutative=True))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('m')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(sin(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('v', commutative=True)), Pow(sin(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["times", 2, "Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('m')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('m')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('m')(Symbol('v', commutative=True)), Symbol('v', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\mathbf{g}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('v', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(\\dot{z})} = \\sin{(\\dot{z})}, then obtain \\int (\\frac{\\mathbf{H}{(\\dot{z})}}{\\sin{(\\dot{z})}})^{\\dot{z}} d\\dot{z} = \\dot{z} + \\psi", "derivation": "\\mathbf{H}{(\\dot{z})} = \\sin{(\\dot{z})} and \\frac{\\mathbf{H}{(\\dot{z})}}{\\sin{(\\dot{z})}} = 1 and (\\frac{\\mathbf{H}{(\\dot{z})}}{\\sin{(\\dot{z})}})^{\\dot{z}} = 1 and \\int (\\frac{\\mathbf{H}{(\\dot{z})}}{\\sin{(\\dot{z})}})^{\\dot{z}} d\\dot{z} = \\int 1 d\\dot{z} and \\int (\\frac{\\mathbf{H}{(\\dot{z})}}{\\sin{(\\dot{z})}})^{\\dot{z}} d\\dot{z} = \\dot{z} + \\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Mul(Function('\\\\mathbf{H}')(Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given c{(\\dot{z})} = \\log{(\\cos{(\\dot{z})})} and \\mathbf{g}{(\\dot{z})} = \\int 0 d\\dot{z}, then obtain \\frac{\\mathbf{g}{(\\dot{z})}}{c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}} = \\frac{\\int (c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}) d\\dot{z}}{c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}}", "derivation": "c{(\\dot{z})} = \\log{(\\cos{(\\dot{z})})} and c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})} = 0 and \\int (c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}) d\\dot{z} = \\int 0 d\\dot{z} and \\mathbf{g}{(\\dot{z})} = \\int 0 d\\dot{z} and \\mathbf{g}{(\\dot{z})} = \\int (c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}) d\\dot{z} and \\frac{\\mathbf{g}{(\\dot{z})}}{c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}} = \\frac{\\int (c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}) d\\dot{z}}{c{(\\dot{z})} - \\log{(\\cos{(\\dot{z})})}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\dot{z}', commutative=True)), log(cos(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 1, "log(cos(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), Integral(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["divide", 5, "Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True)))))"], "Equality(Mul(Pow(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Integer(-1)), Integral(Add(Function('c')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\dot{z}', commutative=True))))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mu_0,M_{E})} = M_{E} - \\mu_0 and \\sigma_{x}{(M_{E})} = M_{E}, then obtain (3 \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})})^{M_{E}} = (M_{E} + 2 \\mu_0)^{M_{E}}", "derivation": "\\operatorname{v_{t}}{(\\mu_0,M_{E})} = M_{E} - \\mu_0 and - M_{E} + \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})} = 0 and \\sigma_{x}{(M_{E})} = M_{E} and - M_{E} + \\mu_0 - \\sigma_{x}{(M_{E})} + \\operatorname{v_{t}}{(\\mu_0,M_{E})} = - \\sigma_{x}{(M_{E})} and - 2 M_{E} + \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})} = - M_{E} and \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})} = M_{E} and 3 \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})} = M_{E} + 2 \\mu_0 and (3 \\mu_0 + \\operatorname{v_{t}}{(\\mu_0,M_{E})})^{M_{E}} = (M_{E} + 2 \\mu_0)^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))))"], [["minus", 1, "Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mu_0', commutative=True), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["minus", 2, "Function('\\\\sigma_x')(Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('M_E', commutative=True))), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('M_E', commutative=True)), Symbol('\\\\mu_0', commutative=True), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True)))"], [["minus", 5, "Mul(Integer(-1), Integer(2), Symbol('M_E', commutative=True))"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))"], [["add", 6, "Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\mu_0', commutative=True)), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))))"], [["power", 7, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(3), Symbol('\\\\mu_0', commutative=True)), Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M_E', commutative=True), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\dot{x}{(z^{*})} = \\cos{(\\sin{(z^{*})})} and \\mathbb{I}{(z^{*})} = \\dot{x}{(z^{*})} + \\sin{(z^{*})}, then obtain 2 \\mathbb{I}{(z^{*})} = \\dot{x}{(z^{*})} + \\mathbb{I}{(z^{*})} + \\sin{(z^{*})}", "derivation": "\\dot{x}{(z^{*})} = \\cos{(\\sin{(z^{*})})} and \\dot{x}{(z^{*})} + \\sin{(z^{*})} = \\sin{(z^{*})} + \\cos{(\\sin{(z^{*})})} and \\mathbb{I}{(z^{*})} = \\dot{x}{(z^{*})} + \\sin{(z^{*})} and \\mathbb{I}{(z^{*})} = \\sin{(z^{*})} + \\cos{(\\sin{(z^{*})})} and 2 \\mathbb{I}{(z^{*})} = \\mathbb{I}{(z^{*})} + \\sin{(z^{*})} + \\cos{(\\sin{(z^{*})})} and 2 \\mathbb{I}{(z^{*})} = \\dot{x}{(z^{*})} + \\mathbb{I}{(z^{*})} + \\sin{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True))))"], [["add", 1, "sin(Symbol('z^*', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))), Add(sin(Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True)), Add(Function('\\\\dot{x}')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True)), Add(sin(Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True)))))"], [["add", 4, "Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True))), Add(Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)), cos(sin(Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True))), Add(Function('\\\\dot{x}')(Symbol('z^*', commutative=True)), Function('\\\\mathbb{I}')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given z{(\\sigma_p,y,u)} = \\sigma_p u - y, then derive (\\int z{(\\sigma_p,y,u)} d\\sigma_p)^{\\sigma_p} = (\\psi + \\frac{\\sigma_p^{2} u}{2} - \\sigma_p y)^{\\sigma_p}, then obtain ((\\int (\\sigma_p u - y) d\\sigma_p)^{\\sigma_p})^{\\psi} = ((\\psi + \\frac{\\sigma_p^{2} u}{2} - \\sigma_p y)^{\\sigma_p})^{\\psi}", "derivation": "z{(\\sigma_p,y,u)} = \\sigma_p u - y and \\int z{(\\sigma_p,y,u)} d\\sigma_p = \\int (\\sigma_p u - y) d\\sigma_p and (\\int z{(\\sigma_p,y,u)} d\\sigma_p)^{\\sigma_p} = (\\int (\\sigma_p u - y) d\\sigma_p)^{\\sigma_p} and (\\int z{(\\sigma_p,y,u)} d\\sigma_p)^{\\sigma_p} = (\\psi + \\frac{\\sigma_p^{2} u}{2} - \\sigma_p y)^{\\sigma_p} and (\\int (\\sigma_p u - y) d\\sigma_p)^{\\sigma_p} = (\\psi + \\frac{\\sigma_p^{2} u}{2} - \\sigma_p y)^{\\sigma_p} and ((\\int (\\sigma_p u - y) d\\sigma_p)^{\\sigma_p})^{\\psi} = ((\\psi + \\frac{\\sigma_p^{2} u}{2} - \\sigma_p y)^{\\sigma_p})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)), Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integral(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["power", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Pow(Integral(Add(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('y', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(2)), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True), Symbol('y', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(k)} = e^{k}, then obtain (- (\\operatorname{A_{2}}{(k)} - e^{k}) e^{k})^{k} = 0^{k}", "derivation": "\\operatorname{A_{2}}{(k)} = e^{k} and \\operatorname{A_{2}}{(k)} + e^{k} = 2 e^{k} and \\operatorname{A_{2}}{(k)} - e^{k} = 0 and - (\\operatorname{A_{2}}{(k)} - e^{k}) e^{k} = 0 and (- (\\operatorname{A_{2}}{(k)} - e^{k}) e^{k})^{k} = 0^{k}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], [["add", 1, "exp(Symbol('k', commutative=True))"], "Equality(Add(Function('A_2')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True))), Mul(Integer(2), exp(Symbol('k', commutative=True))))"], [["minus", 2, "Mul(Integer(2), exp(Symbol('k', commutative=True)))"], "Equality(Add(Function('A_2')(Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('k', commutative=True)))), Integer(0))"], [["times", 3, "Mul(Integer(-1), exp(Symbol('k', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('A_2')(Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('k', commutative=True)))), exp(Symbol('k', commutative=True))), Integer(0))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Add(Function('A_2')(Symbol('k', commutative=True)), Mul(Integer(-1), exp(Symbol('k', commutative=True)))), exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Integer(0), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon} - \\hat{H}_l)}, then obtain - \\frac{\\sin{(J_{\\varepsilon} - \\hat{H}_l)}}{\\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})}} = - \\frac{\\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon} - \\hat{H}_l)}}", "derivation": "\\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})} = - \\sin{(J_{\\varepsilon} - \\hat{H}_l)} and \\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})} \\sin{(J_{\\varepsilon} - \\hat{H}_l)} = - \\sin^{2}{(J_{\\varepsilon} - \\hat{H}_l)} and \\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})} \\sin{(J_{\\varepsilon} - \\hat{H}_l)} = - \\operatorname{A_{z}}^{2}{(\\hat{H}_l,J_{\\varepsilon})} and - \\sin^{2}{(J_{\\varepsilon} - \\hat{H}_l)} = - \\operatorname{A_{z}}^{2}{(\\hat{H}_l,J_{\\varepsilon})} and - \\frac{\\sin{(J_{\\varepsilon} - \\hat{H}_l)}}{\\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})}} = - \\frac{\\operatorname{A_{z}}{(\\hat{H}_l,J_{\\varepsilon})}}{\\sin{(J_{\\varepsilon} - \\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))))"], [["times", 1, "sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], "Equality(Mul(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))), Mul(Integer(-1), Pow(sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))), Mul(Integer(-1), Pow(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Integer(2))), Mul(Integer(-1), Pow(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))))"], [["divide", 4, "Mul(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))))"], "Equality(Mul(Integer(-1), Pow(Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))), Mul(Integer(-1), Function('A_z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}{(Z,\\mathbf{A})} = \\mathbf{A} + e^{Z}, then obtain (2 \\hat{H}^{2}{(Z,\\mathbf{A})})^{Z} = ((2 \\mathbf{A} + 2 e^{Z}) \\hat{H}{(Z,\\mathbf{A})})^{Z}", "derivation": "\\hat{H}{(Z,\\mathbf{A})} = \\mathbf{A} + e^{Z} and 2 \\hat{H}{(Z,\\mathbf{A})} = \\mathbf{A} + \\hat{H}{(Z,\\mathbf{A})} + e^{Z} and 2 \\hat{H}^{2}{(Z,\\mathbf{A})} = (\\mathbf{A} + \\hat{H}{(Z,\\mathbf{A})} + e^{Z}) \\hat{H}{(Z,\\mathbf{A})} and 2 (\\mathbf{A} + e^{Z})^{2} = (\\mathbf{A} + e^{Z}) (2 \\mathbf{A} + 2 e^{Z}) and 2 \\hat{H}^{2}{(Z,\\mathbf{A})} = (2 \\mathbf{A} + 2 e^{Z}) \\hat{H}{(Z,\\mathbf{A})} and (2 \\hat{H}^{2}{(Z,\\mathbf{A})})^{Z} = ((2 \\mathbf{A} + 2 e^{Z}) \\hat{H}{(Z,\\mathbf{A})})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('Z', commutative=True))))"], [["add", 1, "Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('Z', commutative=True))))"], [["times", 2, "Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('Z', commutative=True))), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('Z', commutative=True))), Integer(2))), Mul(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('Z', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), exp(Symbol('Z', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Mul(Add(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), exp(Symbol('Z', commutative=True)))), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 5, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Symbol('Z', commutative=True)), Pow(Mul(Add(Mul(Integer(2), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), exp(Symbol('Z', commutative=True)))), Function('\\\\hat{H}')(Symbol('Z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(n_{1},\\mathbf{s})} = \\frac{n_{1}}{\\mathbf{s}}, then obtain (\\frac{d}{d n_{1}} 0)^{\\mathbf{s}} = (\\frac{\\partial}{\\partial n_{1}} (- \\mathbf{p}{(n_{1},\\mathbf{s})} + \\frac{n_{1}}{\\mathbf{s}}))^{\\mathbf{s}}", "derivation": "\\mathbf{p}{(n_{1},\\mathbf{s})} = \\frac{n_{1}}{\\mathbf{s}} and 0 = - \\mathbf{p}{(n_{1},\\mathbf{s})} + \\frac{n_{1}}{\\mathbf{s}} and \\frac{d}{d n_{1}} 0 = \\frac{\\partial}{\\partial n_{1}} (- \\mathbf{p}{(n_{1},\\mathbf{s})} + \\frac{n_{1}}{\\mathbf{s}}) and (\\frac{d}{d n_{1}} 0)^{\\mathbf{s}} = (\\frac{\\partial}{\\partial n_{1}} (- \\mathbf{p}{(n_{1},\\mathbf{s})} + \\frac{n_{1}}{\\mathbf{s}}))^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n_1', commutative=True))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('n_1', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given f{(\\Psi_{\\lambda},Q)} = \\frac{Q}{\\Psi_{\\lambda}}, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},Q)} = - \\frac{Q}{\\Psi_{\\lambda}^{2}}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},Q)} = - \\frac{f{(\\Psi_{\\lambda},Q)}}{\\Psi_{\\lambda}}", "derivation": "f{(\\Psi_{\\lambda},Q)} = \\frac{Q}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},Q)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\frac{Q}{\\Psi_{\\lambda}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},Q)} = - \\frac{Q}{\\Psi_{\\lambda}^{2}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} f{(\\Psi_{\\lambda},Q)} = - \\frac{f{(\\Psi_{\\lambda},Q)}}{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('Q', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(k,\\rho_f)} = \\log{(- \\rho_f + k)}, then derive \\int \\dot{z}{(k,\\rho_f)} dk = \\hat{x} - \\rho_f \\log{(- \\rho_f + k)} + k \\log{(- \\rho_f + k)} - k, then obtain \\int \\dot{z}{(k,\\rho_f)} dk = \\hat{x} - \\rho_f \\dot{z}{(k,\\rho_f)} + k \\dot{z}{(k,\\rho_f)} - k", "derivation": "\\dot{z}{(k,\\rho_f)} = \\log{(- \\rho_f + k)} and \\int \\dot{z}{(k,\\rho_f)} dk = \\int \\log{(- \\rho_f + k)} dk and \\int \\dot{z}{(k,\\rho_f)} dk = \\hat{x} - \\rho_f \\log{(- \\rho_f + k)} + k \\log{(- \\rho_f + k)} - k and \\int \\dot{z}{(k,\\rho_f)} dk = \\hat{x} - \\rho_f \\dot{z}{(k,\\rho_f)} + k \\dot{z}{(k,\\rho_f)} - k", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('k', commutative=True)))), Mul(Symbol('k', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('k', commutative=True)))), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('k', commutative=True), Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(c_{0},H)} = H c_{0}, then obtain e^{H c_{0} + \\mathbf{A}{(c_{0},H)}} + \\iint \\mathbf{A}{(c_{0},H)} dc_{0} dc_{0} = e^{H c_{0} + \\mathbf{A}{(c_{0},H)}} + \\iint H c_{0} dc_{0} dc_{0}", "derivation": "\\mathbf{A}{(c_{0},H)} = H c_{0} and 2 \\mathbf{A}{(c_{0},H)} = H c_{0} + \\mathbf{A}{(c_{0},H)} and \\int \\mathbf{A}{(c_{0},H)} dc_{0} = \\int H c_{0} dc_{0} and \\iint \\mathbf{A}{(c_{0},H)} dc_{0} dc_{0} = \\iint H c_{0} dc_{0} dc_{0} and e^{2 \\mathbf{A}{(c_{0},H)}} + \\iint \\mathbf{A}{(c_{0},H)} dc_{0} dc_{0} = e^{2 \\mathbf{A}{(c_{0},H)}} + \\iint H c_{0} dc_{0} dc_{0} and e^{H c_{0} + \\mathbf{A}{(c_{0},H)}} + \\iint \\mathbf{A}{(c_{0},H)} dc_{0} dc_{0} = e^{H c_{0} + \\mathbf{A}{(c_{0},H)}} + \\iint H c_{0} dc_{0} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Add(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["integrate", 3, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["add", 4, "exp(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))))"], "Equality(Add(exp(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)))), Integral(Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Add(exp(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)))), Integral(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(exp(Add(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)))), Integral(Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Add(exp(Add(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)))), Integral(Mul(Symbol('H', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(\\pi)} = \\sin{(\\pi)}, then obtain \\iint \\frac{\\hat{X}{(\\pi)}}{\\sin{(\\pi)}} d\\pi d\\pi = \\iint 1 d\\pi d\\pi", "derivation": "\\hat{X}{(\\pi)} = \\sin{(\\pi)} and \\frac{\\hat{X}{(\\pi)}}{\\sin{(\\pi)}} = 1 and \\int \\frac{\\hat{X}{(\\pi)}}{\\sin{(\\pi)}} d\\pi = \\int 1 d\\pi and \\iint \\frac{\\hat{X}{(\\pi)}}{\\sin{(\\pi)}} d\\pi d\\pi = \\iint 1 d\\pi d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Function('\\\\hat{X}')(Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{X}')(Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{X}')(Symbol('\\\\pi', commutative=True)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\chi,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M), then derive \\hat{\\mathbf{x}}{(\\chi,\\mathbf{J}_M)} = 1, then obtain - \\int 1 d\\mathbf{J}_M + \\int \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M) d\\mathbf{J}_M = 0", "derivation": "\\hat{\\mathbf{x}}{(\\chi,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M) and \\hat{\\mathbf{x}}{(\\chi,\\mathbf{J}_M)} = 1 and \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M) = 1 and \\int \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M) d\\mathbf{J}_M = \\int 1 d\\mathbf{J}_M and - \\int 1 d\\mathbf{J}_M + \\int \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\chi + \\mathbf{J}_M) d\\mathbf{J}_M = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 4, "Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(W)} = e^{\\sin{(W)}} and \\sigma_{p}{(W)} = \\sin{(W)} and \\mathbf{S}{(W)} = e^{\\sin{(W)}}, then obtain \\sigma_{p}{(W)} - e^{\\sigma_{p}{(W)}} = - e^{\\sigma_{p}{(W)}} + \\sin{(W)}", "derivation": "\\operatorname{y^{\\prime}}{(W)} = e^{\\sin{(W)}} and \\sigma_{p}{(W)} = \\sin{(W)} and \\sigma_{p}{(W)} - \\operatorname{y^{\\prime}}{(W)} = - \\operatorname{y^{\\prime}}{(W)} + \\sin{(W)} and \\mathbf{S}{(W)} = e^{\\sin{(W)}} and \\sigma_{p}{(W)} - e^{\\sin{(W)}} = - e^{\\sin{(W)}} + \\sin{(W)} and \\mathbf{S}{(W)} = e^{\\sigma_{p}{(W)}} and e^{\\sin{(W)}} = e^{\\sigma_{p}{(W)}} and \\sigma_{p}{(W)} - e^{\\sigma_{p}{(W)}} = - e^{\\sigma_{p}{(W)}} + \\sin{(W)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('W', commutative=True)), exp(sin(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["minus", 2, "Function('y^{\\\\prime}')(Symbol('W', commutative=True))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('W', commutative=True)), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('W', commutative=True))), sin(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), exp(sin(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\sigma_p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('W', commutative=True))))), Add(Mul(Integer(-1), exp(sin(Symbol('W', commutative=True)))), sin(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\mathbf{S}')(Symbol('W', commutative=True)), exp(Function('\\\\sigma_p')(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(exp(sin(Symbol('W', commutative=True))), exp(Function('\\\\sigma_p')(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Add(Function('\\\\sigma_p')(Symbol('W', commutative=True)), Mul(Integer(-1), exp(Function('\\\\sigma_p')(Symbol('W', commutative=True))))), Add(Mul(Integer(-1), exp(Function('\\\\sigma_p')(Symbol('W', commutative=True)))), sin(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(H)} = \\log{(\\sin{(H)})}, then obtain ((H + \\operatorname{M_{E}}{(H)}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}))^{H} = ((H + \\log{(\\sin{(H)})}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}))^{H}", "derivation": "\\operatorname{M_{E}}{(H)} = \\log{(\\sin{(H)})} and - H + \\operatorname{M_{E}}{(H)} = - H + \\log{(\\sin{(H)})} and \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}) = \\frac{d}{d H} (- H + \\log{(\\sin{(H)})}) and H + \\operatorname{M_{E}}{(H)} = H + \\log{(\\sin{(H)})} and (H + \\operatorname{M_{E}}{(H)}) \\frac{d}{d H} (- H + \\log{(\\sin{(H)})}) = (H + \\log{(\\sin{(H)})}) \\frac{d}{d H} (- H + \\log{(\\sin{(H)})}) and (H + \\operatorname{M_{E}}{(H)}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}) = (H + \\log{(\\sin{(H)})}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}) and ((H + \\operatorname{M_{E}}{(H)}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}))^{H} = ((H + \\log{(\\sin{(H)})}) \\frac{d}{d H} (- H + \\operatorname{M_{E}}{(H)}))^{H}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True)))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('M_E')(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), log(sin(Symbol('H', commutative=True)))))"], [["times", 4, "Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('H', commutative=True), Function('M_E')(Symbol('H', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Symbol('H', commutative=True), log(sin(Symbol('H', commutative=True)))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(sin(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Symbol('H', commutative=True), Function('M_E')(Symbol('H', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Add(Symbol('H', commutative=True), log(sin(Symbol('H', commutative=True)))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('H', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('H', commutative=True), Function('M_E')(Symbol('H', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('H', commutative=True)), Pow(Mul(Add(Symbol('H', commutative=True), log(sin(Symbol('H', commutative=True)))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('M_E')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given z{(\\mathbf{J}_M,E_{\\lambda})} = E_{\\lambda} \\mathbf{J}_M, then obtain E_{\\lambda} (\\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} z{(\\mathbf{J}_M,E_{\\lambda})}) = E_{\\lambda} (\\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} E_{\\lambda} \\mathbf{J}_M)", "derivation": "z{(\\mathbf{J}_M,E_{\\lambda})} = E_{\\lambda} \\mathbf{J}_M and \\frac{\\partial}{\\partial \\mathbf{J}_M} z{(\\mathbf{J}_M,E_{\\lambda})} = \\frac{\\partial}{\\partial \\mathbf{J}_M} E_{\\lambda} \\mathbf{J}_M and \\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} z{(\\mathbf{J}_M,E_{\\lambda})} = \\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} E_{\\lambda} \\mathbf{J}_M and E_{\\lambda} (\\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} z{(\\mathbf{J}_M,E_{\\lambda})}) = E_{\\lambda} (\\operatorname{g_{\\varepsilon}}{(u)} + \\frac{\\partial}{\\partial \\mathbf{J}_M} E_{\\lambda} \\mathbf{J}_M)", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["add", 2, "Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True)), Derivative(Function('z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Add(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True)), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True)), Derivative(Function('z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True)), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\phi_{2}{(S,r)} = S + r and \\theta{(r)} = r^{2}, then derive \\int \\phi_{2}{(S,r)} dr = A_{2} + S r + \\frac{r^{2}}{2}, then derive P_{e} + S r + \\frac{r^{2}}{2} = A_{2} + S r + \\frac{r^{2}}{2}, then obtain \\frac{P_{e} + S r - S - r + \\frac{\\theta{(r)}}{2}}{S r + \\mathbf{J}_M + \\frac{r^{2}}{2}} = \\frac{A_{2} + S r - S - r + \\frac{\\theta{(r)}}{2}}{S r + \\mathbf{J}_M + \\frac{r^{2}}{2}}", "derivation": "\\phi_{2}{(S,r)} = S + r and \\int \\phi_{2}{(S,r)} dr = \\int (S + r) dr and \\int \\phi_{2}{(S,r)} dr = A_{2} + S r + \\frac{r^{2}}{2} and \\int (S + r) dr = A_{2} + S r + \\frac{r^{2}}{2} and P_{e} + S r + \\frac{r^{2}}{2} = A_{2} + S r + \\frac{r^{2}}{2} and P_{e} + S r - S + \\frac{r^{2}}{2} - r = A_{2} + S r - S + \\frac{r^{2}}{2} - r and \\theta{(r)} = r^{2} and P_{e} + S r - S - r + \\frac{\\theta{(r)}}{2} = A_{2} + S r - S - r + \\frac{\\theta{(r)}}{2} and \\frac{P_{e} + S r - S - r + \\frac{\\theta{(r)}}{2}}{S r + \\mathbf{J}_M + \\frac{r^{2}}{2}} = \\frac{A_{2} + S r - S - r + \\frac{\\theta{(r)}}{2}}{S r + \\mathbf{J}_M + \\frac{r^{2}}{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Add(Symbol('S', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Add(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('S', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('P_e', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))))"], [["minus", 5, "Add(Symbol('S', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Symbol('P_e', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('r', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('r', commutative=True)), Pow(Symbol('r', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Symbol('P_e', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Rational(1, 2), Function('\\\\theta')(Symbol('r', commutative=True)))), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Rational(1, 2), Function('\\\\theta')(Symbol('r', commutative=True)))))"], [["divide", 8, "Add(Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2))))"], "Equality(Mul(Pow(Add(Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Integer(-1)), Add(Symbol('P_e', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Rational(1, 2), Function('\\\\theta')(Symbol('r', commutative=True))))), Mul(Pow(Add(Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r', commutative=True), Integer(2)))), Integer(-1)), Add(Symbol('A_2', commutative=True), Mul(Symbol('S', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Rational(1, 2), Function('\\\\theta')(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{B}{(F_{N})} = F_{N}, then obtain \\frac{2 \\lambda}{f} - 2 f + 2 \\int \\mathbf{B}{(F_{N})} d\\mathbf{B}{(F_{N})} = \\frac{2 \\lambda}{f} - 2 f + \\int F_{N} d\\mathbf{B}{(F_{N})} + \\int \\mathbf{B}{(F_{N})} d\\mathbf{B}{(F_{N})}", "derivation": "\\mathbf{B}{(F_{N})} = F_{N} and \\int \\mathbf{B}{(F_{N})} dF_{N} = \\int F_{N} dF_{N} and \\frac{\\lambda}{f} + \\int \\mathbf{B}{(F_{N})} dF_{N} = \\frac{\\lambda}{f} + \\int F_{N} dF_{N} and \\frac{\\lambda}{f} - f + \\int \\mathbf{B}{(F_{N})} dF_{N} = \\frac{\\lambda}{f} - f + \\int F_{N} dF_{N} and \\frac{2 \\lambda}{f} - 2 f + 2 \\int \\mathbf{B}{(F_{N})} dF_{N} = \\frac{2 \\lambda}{f} - 2 f + \\int F_{N} dF_{N} + \\int \\mathbf{B}{(F_{N})} dF_{N} and \\frac{2 \\lambda}{f} - 2 f + 2 \\int \\mathbf{B}{(F_{N})} d\\mathbf{B}{(F_{N})} = \\frac{2 \\lambda}{f} - 2 f + \\int F_{N} d\\mathbf{B}{(F_{N})} + \\int \\mathbf{B}{(F_{N})} d\\mathbf{B}{(F_{N})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True)))))"], [["minus", 3, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('f', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('f', commutative=True)), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True)))))"], [["add", 4, "Add(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('f', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(2), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True))), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(2), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)))))), Add(Mul(Integer(2), Symbol('\\\\lambda', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Integral(Symbol('F_N', commutative=True), Tuple(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)))), Integral(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True)), Tuple(Function('\\\\mathbf{B}')(Symbol('F_N', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{p}{(I,v_{z})} = v_{z}^{I}, then obtain \\frac{\\partial}{\\partial v_{z}} ((- I + v_{z}^{I})^{v_{z}} + (- I + \\mathbf{p}{(I,v_{z})})^{v_{z}}) = \\frac{\\partial}{\\partial v_{z}} 2 (- I + v_{z}^{I})^{v_{z}}", "derivation": "\\mathbf{p}{(I,v_{z})} = v_{z}^{I} and - I + \\mathbf{p}{(I,v_{z})} = - I + v_{z}^{I} and (- I + \\mathbf{p}{(I,v_{z})})^{v_{z}} = (- I + v_{z}^{I})^{v_{z}} and (- I + v_{z}^{I})^{v_{z}} + (- I + \\mathbf{p}{(I,v_{z})})^{v_{z}} = 2 (- I + v_{z}^{I})^{v_{z}} and \\frac{\\partial}{\\partial v_{z}} ((- I + v_{z}^{I})^{v_{z}} + (- I + \\mathbf{p}{(I,v_{z})})^{v_{z}}) = \\frac{\\partial}{\\partial v_{z}} 2 (- I + v_{z}^{I})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('I', commutative=True))), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then obtain (Q^{3}{(\\mathbf{B})} \\sin{(\\mathbf{B})})^{\\mathbf{B}} = (Q^{2}{(\\mathbf{B})} \\sin^{2}{(\\mathbf{B})})^{\\mathbf{B}}", "derivation": "Q{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and Q{(\\mathbf{B})} \\sin{(\\mathbf{B})} = \\sin^{2}{(\\mathbf{B})} and Q^{2}{(\\mathbf{B})} \\sin^{2}{(\\mathbf{B})} = \\sin^{4}{(\\mathbf{B})} and Q^{3}{(\\mathbf{B})} \\sin{(\\mathbf{B})} = Q^{2}{(\\mathbf{B})} \\sin^{2}{(\\mathbf{B})} and (Q^{3}{(\\mathbf{B})} \\sin{(\\mathbf{B})})^{\\mathbf{B}} = (Q^{2}{(\\mathbf{B})} \\sin^{2}{(\\mathbf{B})})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\mathbf{B}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))))"], [["power", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Pow(Function('Q')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})} = \\frac{P_{g} \\mathbf{v}}{A_{1}} and p{(\\mathbf{v},A_{1},P_{g})} = \\frac{P_{g} \\mathbf{v}}{A_{1} \\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})}}, then obtain -1 = - p{(\\mathbf{v},A_{1},P_{g})}", "derivation": "\\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})} = \\frac{P_{g} \\mathbf{v}}{A_{1}} and 1 = \\frac{P_{g} \\mathbf{v}}{A_{1} \\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})}} and -1 = - \\frac{P_{g} \\mathbf{v}}{A_{1} \\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})}} and p{(\\mathbf{v},A_{1},P_{g})} = \\frac{P_{g} \\mathbf{v}}{A_{1} \\operatorname{t_{1}}{(P_{g},\\mathbf{v},A_{1})}} and -1 = - p{(\\mathbf{v},A_{1},P_{g})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "Function('t_1')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('t_1')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('t_1')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True), Symbol('P_g', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Pow(Function('t_1')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(-1), Mul(Integer(-1), Function('p')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('A_1', commutative=True), Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(g)} = \\log{(\\sin{(g)})}, then obtain - (\\operatorname{V_{\\mathbf{E}}}^{g}{(g)})^{g} + \\operatorname{V_{\\mathbf{E}}}^{g}{(g)} = - (\\operatorname{V_{\\mathbf{E}}}^{g}{(g)})^{g} + \\log{(\\sin{(g)})}^{g}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(g)} = \\log{(\\sin{(g)})} and \\operatorname{V_{\\mathbf{E}}}^{g}{(g)} = \\log{(\\sin{(g)})}^{g} and (\\operatorname{V_{\\mathbf{E}}}^{g}{(g)})^{g} = (\\log{(\\sin{(g)})}^{g})^{g} and - (\\log{(\\sin{(g)})}^{g})^{g} + \\operatorname{V_{\\mathbf{E}}}^{g}{(g)} = - (\\log{(\\sin{(g)})}^{g})^{g} + \\log{(\\sin{(g)})}^{g} and - (\\operatorname{V_{\\mathbf{E}}}^{g}{(g)})^{g} + \\operatorname{V_{\\mathbf{E}}}^{g}{(g)} = - (\\operatorname{V_{\\mathbf{E}}}^{g}{(g)})^{g} + \\log{(\\sin{(g)})}^{g}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), log(sin(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["minus", 2, "Pow(Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Pow(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Mul(Integer(-1), Pow(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Pow(log(sin(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\chi,\\varphi)} = \\frac{\\chi}{\\varphi}, then derive \\frac{\\partial}{\\partial \\chi} \\operatorname{t_{1}}{(\\chi,\\varphi)} = \\frac{1}{\\varphi}, then obtain \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\varphi} = \\frac{1}{\\varphi}", "derivation": "\\operatorname{t_{1}}{(\\chi,\\varphi)} = \\frac{\\chi}{\\varphi} and \\frac{\\partial}{\\partial \\chi} \\operatorname{t_{1}}{(\\chi,\\varphi)} = \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\varphi} and \\frac{\\partial}{\\partial \\chi} \\operatorname{t_{1}}{(\\chi,\\varphi)} = \\frac{1}{\\varphi} and \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\varphi} = \\frac{1}{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\omega,I)} = \\omega^{I}, then obtain \\operatorname{C_{1}}^{3}{(\\omega,I)} = \\omega^{I} \\operatorname{C_{1}}^{2}{(\\omega,I)}", "derivation": "\\operatorname{C_{1}}{(\\omega,I)} = \\omega^{I} and \\operatorname{C_{1}}^{2}{(\\omega,I)} = \\omega^{I} \\operatorname{C_{1}}{(\\omega,I)} and \\omega^{I} \\operatorname{C_{1}}^{2}{(\\omega,I)} = \\omega^{2 I} \\operatorname{C_{1}}{(\\omega,I)} and \\operatorname{C_{1}}^{3}{(\\omega,I)} = \\omega^{I} \\operatorname{C_{1}}^{2}{(\\omega,I)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)))"], [["times", 1, "Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True))"], "Equality(Pow(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True))))"], [["times", 2, "Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Pow(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Integer(3)), Mul(Pow(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Pow(Function('C_1')(Symbol('\\\\omega', commutative=True), Symbol('I', commutative=True)), Integer(2))))"]]}, {"prompt": "Given t{(V_{\\mathbf{B}},\\nabla)} = \\log{(V_{\\mathbf{B}} - \\nabla)}, then derive \\int t{(V_{\\mathbf{B}},\\nabla)} d\\nabla = A_{y} - V_{\\mathbf{B}} \\log{(- V_{\\mathbf{B}} + \\nabla)} + \\nabla \\log{(V_{\\mathbf{B}} - \\nabla)} - \\nabla, then obtain \\int t{(V_{\\mathbf{B}},\\nabla)} d\\nabla = A_{y} - V_{\\mathbf{B}} \\log{(- V_{\\mathbf{B}} + \\nabla)} + \\nabla t{(V_{\\mathbf{B}},\\nabla)} - \\nabla", "derivation": "t{(V_{\\mathbf{B}},\\nabla)} = \\log{(V_{\\mathbf{B}} - \\nabla)} and \\int t{(V_{\\mathbf{B}},\\nabla)} d\\nabla = \\int \\log{(V_{\\mathbf{B}} - \\nabla)} d\\nabla and \\int t{(V_{\\mathbf{B}},\\nabla)} d\\nabla = A_{y} - V_{\\mathbf{B}} \\log{(- V_{\\mathbf{B}} + \\nabla)} + \\nabla \\log{(V_{\\mathbf{B}} - \\nabla)} - \\nabla and \\int t{(V_{\\mathbf{B}},\\nabla)} d\\nabla = A_{y} - V_{\\mathbf{B}} \\log{(- V_{\\mathbf{B}} + \\nabla)} + \\nabla t{(V_{\\mathbf{B}},\\nabla)} - \\nabla", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('t')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('t')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), Function('t')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(k,F_{H})} = F_{H} k, then derive (\\frac{k \\frac{\\partial}{\\partial k} \\operatorname{v_{y}}{(k,F_{H})}}{\\operatorname{v_{y}}{(k,F_{H})}} + \\log{(\\operatorname{v_{y}}{(k,F_{H})})}) \\operatorname{v_{y}}^{k}{(k,F_{H})} = (F_{H} k)^{k} (\\log{(F_{H} k)} + 1), then obtain (F_{H} k)^{k} (\\frac{k \\frac{\\partial}{\\partial k} \\operatorname{v_{y}}{(k,F_{H})}}{\\operatorname{v_{y}}{(k,F_{H})}} + \\log{(\\operatorname{v_{y}}{(k,F_{H})})}) = (F_{H} k)^{k} (\\log{(F_{H} k)} + 1)", "derivation": "\\operatorname{v_{y}}{(k,F_{H})} = F_{H} k and \\operatorname{v_{y}}^{k}{(k,F_{H})} = (F_{H} k)^{k} and \\frac{\\partial}{\\partial k} \\operatorname{v_{y}}^{k}{(k,F_{H})} = \\frac{\\partial}{\\partial k} (F_{H} k)^{k} and (\\frac{k \\frac{\\partial}{\\partial k} \\operatorname{v_{y}}{(k,F_{H})}}{\\operatorname{v_{y}}{(k,F_{H})}} + \\log{(\\operatorname{v_{y}}{(k,F_{H})})}) \\operatorname{v_{y}}^{k}{(k,F_{H})} = (F_{H} k)^{k} (\\log{(F_{H} k)} + 1) and (F_{H} k)^{k} (\\frac{k \\frac{\\partial}{\\partial k} \\operatorname{v_{y}}{(k,F_{H})}}{\\operatorname{v_{y}}{(k,F_{H})}} + \\log{(\\operatorname{v_{y}}{(k,F_{H})})}) = (F_{H} k)^{k} (\\log{(F_{H} k)} + 1)", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Symbol('k', commutative=True)), Pow(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Pow(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('k', commutative=True), Pow(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Derivative(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), log(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)))), Pow(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Symbol('k', commutative=True))), Mul(Pow(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Add(log(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True))), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Add(Mul(Symbol('k', commutative=True), Pow(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Integer(-1)), Derivative(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), log(Function('v_y')(Symbol('k', commutative=True), Symbol('F_H', commutative=True))))), Mul(Pow(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Add(log(Mul(Symbol('F_H', commutative=True), Symbol('k', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(k)} = \\cos{(k)} and \\eta^{\\prime}{(c_{0})} = \\cos{(\\sin{(c_{0})})}, then obtain (- k + \\cos{(k)}) \\eta^{\\prime}{(c_{0})} = (- k + \\cos{(k)}) \\cos{(\\sin{(c_{0})})}", "derivation": "\\operatorname{P_{g}}{(k)} = \\cos{(k)} and - k + \\operatorname{P_{g}}{(k)} = - k + \\cos{(k)} and - k + \\operatorname{P_{g}}{(k)} - \\cos{(k)} = - k and \\eta^{\\prime}{(c_{0})} = \\cos{(\\sin{(c_{0})})} and - k + 2 \\operatorname{P_{g}}{(k)} - \\cos{(k)} = - k + \\operatorname{P_{g}}{(k)} and - k + 2 \\operatorname{P_{g}}{(k)} - \\cos{(k)} = - k + \\cos{(k)} and (- k + 2 \\operatorname{P_{g}}{(k)} - \\cos{(k)}) \\eta^{\\prime}{(c_{0})} = (- k + 2 \\operatorname{P_{g}}{(k)} - \\cos{(k)}) \\cos{(\\sin{(c_{0})})} and (- k + \\cos{(k)}) \\eta^{\\prime}{(c_{0})} = (- k + \\cos{(k)}) \\cos{(\\sin{(c_{0})})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["minus", 1, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('P_g')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], [["minus", 2, "cos(Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('P_g')(Symbol('k', commutative=True)), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Mul(Integer(-1), Symbol('k', commutative=True)))"], ["get_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('c_0', commutative=True)), cos(sin(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('P_g')(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Function('\\\\eta^{\\\\prime}')(Symbol('c_0', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('k', commutative=True))), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), cos(sin(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('c_0', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), cos(sin(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(A_{z})} = e^{A_{z}}, then derive \\int \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} = \\mathbf{A} + e^{A_{z}}, then obtain A_{z} \\iint \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} d\\mathbf{A} = A_{z} \\int (\\mathbf{A} + e^{A_{z}}) d\\mathbf{A}", "derivation": "\\operatorname{y^{\\prime}}{(A_{z})} = e^{A_{z}} and \\int \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} = \\int e^{A_{z}} dA_{z} and \\int \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} = \\mathbf{A} + e^{A_{z}} and \\iint \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} d\\mathbf{A} = \\int (\\mathbf{A} + e^{A_{z}}) d\\mathbf{A} and A_{z} \\iint \\operatorname{y^{\\prime}}{(A_{z})} dA_{z} d\\mathbf{A} = A_{z} \\int (\\mathbf{A} + e^{A_{z}}) d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_z', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 4, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Symbol('A_z', commutative=True), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), exp(Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\mu_0)} = \\sin{(\\mu_0)}, then derive \\frac{(\\frac{d^{2}}{d \\mu_0^{2}} \\hat{H}{(\\mu_0)})^{2}}{\\sin^{2}{(\\mu_0)}} = 1, then obtain \\frac{(\\frac{d^{2}}{d \\mu_0^{2}} \\sin{(\\mu_0)})^{2}}{\\sin^{2}{(\\mu_0)}} = 1", "derivation": "\\hat{H}{(\\mu_0)} = \\sin{(\\mu_0)} and \\frac{d}{d \\mu_0} \\hat{H}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\sin{(\\mu_0)} and \\frac{d^{2}}{d \\mu_0^{2}} \\hat{H}{(\\mu_0)} = \\frac{d^{2}}{d \\mu_0^{2}} \\sin{(\\mu_0)} and \\frac{\\frac{d^{2}}{d \\mu_0^{2}} \\hat{H}{(\\mu_0)}}{\\frac{d^{2}}{d \\mu_0^{2}} \\sin{(\\mu_0)}} = 1 and \\frac{(\\frac{d^{2}}{d \\mu_0^{2}} \\hat{H}{(\\mu_0)})^{2}}{(\\frac{d^{2}}{d \\mu_0^{2}} \\sin{(\\mu_0)})^{2}} = 1 and \\frac{(\\frac{d^{2}}{d \\mu_0^{2}} \\hat{H}{(\\mu_0)})^{2}}{\\sin^{2}{(\\mu_0)}} = 1 and \\frac{(\\frac{d^{2}}{d \\mu_0^{2}} \\sin{(\\mu_0)})^{2}}{\\sin^{2}{(\\mu_0)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))))"], [["divide", 3, "Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Pow(Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Integer(-1))), Integer(1))"], [["power", 4, 2], "Equality(Mul(Pow(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Integer(-2))), Integer(1))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-2)), Pow(Derivative(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Integer(2))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-2)), Pow(Derivative(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\rho_{b}{(\\Psi)} = \\log{(\\Psi)}, then obtain \\log{(\\frac{\\rho_{b}^{4}{(\\Psi)}}{\\log{(\\Psi)}^{3}})} = \\log{(\\rho_{b}{(\\Psi)})}", "derivation": "\\rho_{b}{(\\Psi)} = \\log{(\\Psi)} and \\rho_{b}^{2}{(\\Psi)} = \\rho_{b}{(\\Psi)} \\log{(\\Psi)} and \\frac{\\rho_{b}^{2}{(\\Psi)}}{\\log{(\\Psi)}} = \\rho_{b}{(\\Psi)} and \\log{(\\frac{\\rho_{b}^{2}{(\\Psi)}}{\\log{(\\Psi)}})} = \\log{(\\rho_{b}{(\\Psi)})} and \\log{(\\frac{\\rho_{b}^{4}{(\\Psi)}}{\\log{(\\Psi)}^{3}})} = \\log{(\\frac{\\rho_{b}^{2}{(\\Psi)}}{\\log{(\\Psi)}})} and \\log{(\\frac{\\rho_{b}^{4}{(\\Psi)}}{\\log{(\\Psi)}^{3}})} = \\log{(\\rho_{b}{(\\Psi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["times", 1, "Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True))"], "Equality(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Mul(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "log(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)))"], [["log", 3], "Equality(log(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\Psi', commutative=True)), Integer(-1)))), log(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(log(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\Psi', commutative=True)), Integer(-3)))), log(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\Psi', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(log(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\Psi', commutative=True)), Integer(-3)))), log(Function('\\\\rho_b')(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given u{(t_{2},A_{y},v_{t})} = (A_{y} t_{2})^{v_{t}}, then obtain (- 2 A_{y} t_{2} + e^{u{(t_{2},A_{y},v_{t})}})^{A_{y}} = (- 2 A_{y} t_{2} + e^{(A_{y} t_{2})^{v_{t}}})^{A_{y}}", "derivation": "u{(t_{2},A_{y},v_{t})} = (A_{y} t_{2})^{v_{t}} and e^{u{(t_{2},A_{y},v_{t})}} = e^{(A_{y} t_{2})^{v_{t}}} and - A_{y} t_{2} + e^{u{(t_{2},A_{y},v_{t})}} = - A_{y} t_{2} + e^{(A_{y} t_{2})^{v_{t}}} and - 2 A_{y} t_{2} + e^{u{(t_{2},A_{y},v_{t})}} = - 2 A_{y} t_{2} + e^{(A_{y} t_{2})^{v_{t}}} and (- 2 A_{y} t_{2} + e^{u{(t_{2},A_{y},v_{t})}})^{A_{y}} = (- 2 A_{y} t_{2} + e^{(A_{y} t_{2})^{v_{t}}})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('t_2', commutative=True), Symbol('A_y', commutative=True), Symbol('v_t', commutative=True)), Pow(Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), Symbol('v_t', commutative=True)))"], [["exp", 1], "Equality(exp(Function('u')(Symbol('t_2', commutative=True), Symbol('A_y', commutative=True), Symbol('v_t', commutative=True))), exp(Pow(Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), Symbol('v_t', commutative=True))))"], [["minus", 2, "Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Function('u')(Symbol('t_2', commutative=True), Symbol('A_y', commutative=True), Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Pow(Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), Symbol('v_t', commutative=True)))))"], [["minus", 3, "Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Function('u')(Symbol('t_2', commutative=True), Symbol('A_y', commutative=True), Symbol('v_t', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Pow(Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), Symbol('v_t', commutative=True)))))"], [["power", 4, "Symbol('A_y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Function('u')(Symbol('t_2', commutative=True), Symbol('A_y', commutative=True), Symbol('v_t', commutative=True)))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), exp(Pow(Mul(Symbol('A_y', commutative=True), Symbol('t_2', commutative=True)), Symbol('v_t', commutative=True)))), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(U,T)} = \\frac{\\log{(T)}}{U}, then derive \\frac{\\partial}{\\partial U} \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = \\frac{\\partial}{\\partial U} (\\frac{T \\log{(T)}}{U^{2}} - \\frac{T}{U^{2}} + v_{y}), then derive \\frac{\\partial}{\\partial U} \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = - \\frac{2 T \\log{(T)}}{U^{3}} + \\frac{2 T}{U^{3}}, then obtain \\frac{\\partial}{\\partial U} \\int \\frac{\\log{(T)}}{U^{2}} dT = - \\frac{2 T \\log{(T)}}{U^{3}} + \\frac{2 T}{U^{3}}", "derivation": "\\operatorname{A_{y}}{(U,T)} = \\frac{\\log{(T)}}{U} and \\frac{\\operatorname{A_{y}}{(U,T)}}{U} = \\frac{\\log{(T)}}{U^{2}} and \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = \\int \\frac{\\log{(T)}}{U^{2}} dT and \\frac{\\partial}{\\partial U} \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = \\frac{\\partial}{\\partial U} \\int \\frac{\\log{(T)}}{U^{2}} dT and \\frac{\\partial}{\\partial U} \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = \\frac{\\partial}{\\partial U} (\\frac{T \\log{(T)}}{U^{2}} - \\frac{T}{U^{2}} + v_{y}) and \\frac{\\partial}{\\partial U} \\int \\frac{\\operatorname{A_{y}}{(U,T)}}{U} dT = - \\frac{2 T \\log{(T)}}{U^{3}} + \\frac{2 T}{U^{3}} and \\frac{\\partial}{\\partial U} \\int \\frac{\\log{(T)}}{U^{2}} dT = - \\frac{2 T \\log{(T)}}{U^{3}} + \\frac{2 T}{U^{3}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), log(Symbol('T', commutative=True))))"], [["divide", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-2)), log(Symbol('T', commutative=True))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-2)), log(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-2)), log(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-2)), log(Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-2))), Symbol('v_y', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('A_y')(Symbol('U', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-3)), log(Symbol('T', commutative=True))), Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-3)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-2)), log(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-3)), log(Symbol('T', commutative=True))), Mul(Integer(2), Symbol('T', commutative=True), Pow(Symbol('U', commutative=True), Integer(-3)))))"]]}, {"prompt": "Given \\mathbf{B}{(m_{s},B)} = m_{s}^{B}, then obtain \\frac{\\partial}{\\partial m_{s}} \\int (\\int \\mathbf{B}{(m_{s},B)} dB)^{m_{s}} dB = \\frac{\\partial}{\\partial m_{s}} \\int (\\int m_{s}^{B} dB)^{m_{s}} dB", "derivation": "\\mathbf{B}{(m_{s},B)} = m_{s}^{B} and \\int \\mathbf{B}{(m_{s},B)} dB = \\int m_{s}^{B} dB and (\\int \\mathbf{B}{(m_{s},B)} dB)^{m_{s}} = (\\int m_{s}^{B} dB)^{m_{s}} and \\int (\\int \\mathbf{B}{(m_{s},B)} dB)^{m_{s}} dB = \\int (\\int m_{s}^{B} dB)^{m_{s}} dB and \\frac{\\partial}{\\partial m_{s}} \\int (\\int \\mathbf{B}{(m_{s},B)} dB)^{m_{s}} dB = \\frac{\\partial}{\\partial m_{s}} \\int (\\int m_{s}^{B} dB)^{m_{s}} dB", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)), Pow(Integral(Pow(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Integral(Pow(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 4, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Integral(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Integral(Pow(Integral(Pow(Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('m_s', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(v,q)} = \\frac{q}{v} and \\delta{(v,q)} = \\frac{1}{\\frac{q}{v} - \\frac{1}{v}}, then obtain \\frac{\\partial}{\\partial q} \\delta{(v,q)} = - \\frac{1}{v (\\frac{q}{v} - \\frac{1}{v})^{2}}", "derivation": "\\operatorname{E_{x}}{(v,q)} = \\frac{q}{v} and \\operatorname{E_{x}}{(v,q)} - \\frac{1}{v} = \\frac{q}{v} - \\frac{1}{v} and \\delta{(v,q)} = \\frac{1}{\\frac{q}{v} - \\frac{1}{v}} and \\delta{(v,q)} = \\frac{1}{\\operatorname{E_{x}}{(v,q)} - \\frac{1}{v}} and \\frac{1}{\\operatorname{E_{x}}{(v,q)} - \\frac{1}{v}} = \\frac{1}{\\frac{q}{v} - \\frac{1}{v}} and \\frac{\\partial}{\\partial q} \\delta{(v,q)} = \\frac{\\partial}{\\partial q} \\frac{1}{\\operatorname{E_{x}}{(v,q)} - \\frac{1}{v}} and \\frac{\\partial}{\\partial q} \\delta{(v,q)} = \\frac{\\partial}{\\partial q} \\frac{1}{\\frac{q}{v} - \\frac{1}{v}} and \\frac{\\partial}{\\partial q} \\delta{(v,q)} = - \\frac{1}{v (\\frac{q}{v} - \\frac{1}{v})^{2}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('v', commutative=True), Integer(-1))"], "Equality(Add(Function('E_x')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Add(Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Pow(Add(Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\delta')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Pow(Add(Function('E_x')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Function('E_x')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)), Pow(Add(Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Add(Function('E_x')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Function('\\\\delta')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-1)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('\\\\delta')(Symbol('v', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('q', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)))), Integer(-2))))"]]}, {"prompt": "Given \\mathbf{M}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\mathbf{S} + \\sigma_x + \\varphi and \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\mathbf{S} + \\sigma_x + \\varphi, then derive \\frac{\\partial}{\\partial \\varphi} \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} = 1, then obtain \\frac{\\partial}{\\partial \\varphi} (\\mathbf{S} + \\sigma_x + \\varphi) = 1", "derivation": "\\mathbf{M}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\mathbf{S} + \\sigma_x + \\varphi and \\frac{\\partial}{\\partial \\varphi} \\mathbf{M}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{S} + \\sigma_x + \\varphi) and \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\mathbf{S} + \\sigma_x + \\varphi and \\frac{\\partial}{\\partial \\varphi} \\mathbf{M}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\frac{\\partial}{\\partial \\varphi} \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{S} + \\sigma_x + \\varphi) and \\frac{\\partial}{\\partial \\varphi} \\operatorname{n_{1}}{(\\varphi,\\sigma_x,\\mathbf{S})} = 1 and \\frac{\\partial}{\\partial \\varphi} (\\mathbf{S} + \\sigma_x + \\varphi) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Function('n_1')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('n_1')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('n_1')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(c,B)} = c e^{B}, then derive \\frac{\\partial}{\\partial c} \\operatorname{v_{1}}{(c,B)} = e^{B}, then obtain 1 = \\frac{c \\frac{\\partial}{\\partial c} \\operatorname{v_{1}}{(c,B)}}{\\operatorname{v_{1}}{(c,B)}}", "derivation": "\\operatorname{v_{1}}{(c,B)} = c e^{B} and 1 = \\frac{c e^{B}}{\\operatorname{v_{1}}{(c,B)}} and \\frac{\\partial}{\\partial c} \\operatorname{v_{1}}{(c,B)} = \\frac{\\partial}{\\partial c} c e^{B} and \\frac{\\partial}{\\partial c} \\operatorname{v_{1}}{(c,B)} = e^{B} and 1 = \\frac{c \\frac{\\partial}{\\partial c} \\operatorname{v_{1}}{(c,B)}}{\\operatorname{v_{1}}{(c,B)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('c', commutative=True), exp(Symbol('B', commutative=True))))"], [["divide", 1, "Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(1), Mul(Symbol('c', commutative=True), Pow(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Symbol('c', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), exp(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(1), Mul(Symbol('c', commutative=True), Pow(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Derivative(Function('v_1')(Symbol('c', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\phi_2,E)} = \\frac{\\cos{(E)}}{\\phi_2}, then obtain \\frac{\\operatorname{x^{{\\}'}}^{E}{(\\phi_2,E)}}{\\int \\frac{\\cos{(E)}}{\\phi_2} d\\phi_2} = \\frac{(\\frac{\\cos{(E)}}{\\phi_2})^{E}}{\\int \\frac{\\cos{(E)}}{\\phi_2} d\\phi_2}", "derivation": "\\operatorname{x^{{\\}'}}{(\\phi_2,E)} = \\frac{\\cos{(E)}}{\\phi_2} and \\operatorname{x^{{\\}'}}^{E}{(\\phi_2,E)} = (\\frac{\\cos{(E)}}{\\phi_2})^{E} and \\int \\operatorname{x^{{\\}'}}{(\\phi_2,E)} d\\phi_2 = \\int \\frac{\\cos{(E)}}{\\phi_2} d\\phi_2 and \\frac{\\operatorname{x^{{\\}'}}^{E}{(\\phi_2,E)}}{\\int \\operatorname{x^{{\\}'}}{(\\phi_2,E)} d\\phi_2} = \\frac{(\\frac{\\cos{(E)}}{\\phi_2})^{E}}{\\int \\operatorname{x^{{\\}'}}{(\\phi_2,E)} d\\phi_2} and \\frac{\\operatorname{x^{{\\}'}}^{E}{(\\phi_2,E)}}{\\int \\frac{\\cos{(E)}}{\\phi_2} d\\phi_2} = \\frac{(\\frac{\\cos{(E)}}{\\phi_2})^{E}}{\\int \\frac{\\cos{(E)}}{\\phi_2} d\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 2, "Integral(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Integral(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\phi_2', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), cos(Symbol('E', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{J}_f)} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f, then derive \\mathbb{I}{(\\mathbf{J}_f)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f, then obtain M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f + \\mathbb{I}{(\\mathbf{J}_f)} + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f", "derivation": "\\mathbb{I}{(\\mathbf{J}_f)} = \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\mathbf{J}_f + \\mathbb{I}{(\\mathbf{J}_f)} = \\mathbf{J}_f + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and \\mathbb{I}{(\\mathbf{J}_f)} = M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} - \\mathbf{J}_f and M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} = \\mathbf{J}_f + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f and M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} = \\mathbf{J}_f + \\mathbb{I}{(\\mathbf{J}_f)} and M + \\mathbf{J}_f \\log{(\\mathbf{J}_f)} + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\mathbf{J}_f + \\mathbb{I}{(\\mathbf{J}_f)} + \\int \\log{(\\mathbf{J}_f)} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 5, "Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), log(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(c,T,\\mathbf{H})} = T - \\mathbf{H} - c, then obtain \\mathbf{H} \\frac{\\partial}{\\partial c} - \\frac{\\sigma_{p}{(c,T,\\mathbf{H})}}{\\mathbf{H}} = \\mathbf{H} \\frac{\\partial}{\\partial c} \\frac{- T + \\mathbf{H} + c}{\\mathbf{H}}", "derivation": "\\sigma_{p}{(c,T,\\mathbf{H})} = T - \\mathbf{H} - c and - \\frac{\\sigma_{p}{(c,T,\\mathbf{H})}}{\\mathbf{H}} = - \\frac{T - \\mathbf{H} - c}{\\mathbf{H}} and \\frac{\\partial}{\\partial c} - \\frac{\\sigma_{p}{(c,T,\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\partial}{\\partial c} - \\frac{T - \\mathbf{H} - c}{\\mathbf{H}} and \\frac{\\partial}{\\partial c} - \\frac{T - \\mathbf{H} - c}{\\mathbf{H}} = \\frac{\\partial}{\\partial c} \\frac{- T + \\mathbf{H} + c}{\\mathbf{H}} and \\frac{\\partial}{\\partial c} - \\frac{\\sigma_{p}{(c,T,\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\partial}{\\partial c} \\frac{- T + \\mathbf{H} + c}{\\mathbf{H}} and \\mathbf{H} \\frac{\\partial}{\\partial c} - \\frac{\\sigma_{p}{(c,T,\\mathbf{H})}}{\\mathbf{H}} = \\mathbf{H} \\frac{\\partial}{\\partial c} \\frac{- T + \\mathbf{H} + c}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('c', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('c', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('c', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('c', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["times", 5, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\sigma_p')(Symbol('c', commutative=True), Symbol('T', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(x)} = \\log{(x)}, then obtain g^{x}{(x)} \\frac{d}{d x} \\int \\frac{d}{d x} g{(x)} dx = g^{x}{(x)} \\frac{d}{d x} \\int \\frac{d}{d x} \\log{(x)} dx", "derivation": "g{(x)} = \\log{(x)} and \\frac{d}{d x} g{(x)} = \\frac{d}{d x} \\log{(x)} and \\int \\frac{d}{d x} g{(x)} dx = \\int \\frac{d}{d x} \\log{(x)} dx and \\frac{d}{d x} \\int \\frac{d}{d x} g{(x)} dx = \\frac{d}{d x} \\int \\frac{d}{d x} \\log{(x)} dx and \\log{(x)}^{x} \\frac{d}{d x} \\int \\frac{d}{d x} g{(x)} dx = \\log{(x)}^{x} \\frac{d}{d x} \\int \\frac{d}{d x} \\log{(x)} dx and g^{x}{(x)} \\frac{d}{d x} \\int \\frac{d}{d x} g{(x)} dx = g^{x}{(x)} \\frac{d}{d x} \\int \\frac{d}{d x} \\log{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Function('g')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('g')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 4, "Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True))"], "Equality(Mul(Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(Integral(Derivative(Function('g')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(Integral(Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('g')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(Integral(Derivative(Function('g')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Function('g')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(Integral(Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})}, then derive V{(E_{x})} = \\frac{1}{E_{x}}, then obtain E_{x} V{(E_{x})} - 1 = 0", "derivation": "V{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and V{(E_{x})} = \\frac{1}{E_{x}} and \\frac{V{(E_{x})}}{\\frac{d}{d E_{x}} \\log{(E_{x})}} = \\frac{1}{E_{x} \\frac{d}{d E_{x}} \\log{(E_{x})}} and - E_{x} \\frac{d}{d E_{x}} \\log{(E_{x})} + \\frac{V{(E_{x})}}{\\frac{d}{d E_{x}} \\log{(E_{x})}} = - E_{x} \\frac{d}{d E_{x}} \\log{(E_{x})} + \\frac{1}{E_{x} \\frac{d}{d E_{x}} \\log{(E_{x})}} and E_{x} V{(E_{x})} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('E_x', commutative=True)), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('V')(Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Integer(-1)))"], [["divide", 2, "Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))"], "Equality(Mul(Function('V')(Symbol('E_x', commutative=True)), Pow(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 3, "Mul(Symbol('E_x', commutative=True), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Function('V')(Symbol('E_x', commutative=True)), Pow(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('E_x', commutative=True), Function('V')(Symbol('E_x', commutative=True))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{f}{(v,\\Psi_{nl})} = \\frac{\\Psi_{nl}}{v}, then derive \\frac{v \\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}} - \\frac{v \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}^{2}} = 0, then obtain \\frac{v \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{\\Psi_{nl}}{v}}{\\Psi_{nl}} - \\frac{1}{\\Psi_{nl}} = 0", "derivation": "\\mathbf{f}{(v,\\Psi_{nl})} = \\frac{\\Psi_{nl}}{v} and \\frac{v \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}} = 1 and \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{v \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}} = \\frac{d}{d \\Psi_{nl}} 1 and \\frac{v \\frac{\\partial}{\\partial \\Psi_{nl}} \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}} - \\frac{v \\mathbf{f}{(v,\\Psi_{nl})}}{\\Psi_{nl}^{2}} = 0 and \\frac{v \\frac{\\partial}{\\partial \\Psi_{nl}} \\frac{\\Psi_{nl}}{v}}{\\Psi_{nl}} - \\frac{1}{\\Psi_{nl}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v', commutative=True), Function('\\\\mathbf{f}')(Symbol('v', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v', commutative=True), Function('\\\\mathbf{f}')(Symbol('v', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v', commutative=True), Derivative(Function('\\\\mathbf{f}')(Symbol('v', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-2)), Symbol('v', commutative=True), Function('\\\\mathbf{f}')(Symbol('v', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Symbol('v', commutative=True), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\bar{\\h}{(M)} = - M and \\dot{y}{(f_{\\mathbf{v}},M)} = f_{\\mathbf{v}} \\bar{\\h}^{M}{(M)}, then obtain \\dot{y}^{M}{(f_{\\mathbf{v}},M)} - (e^{M})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} \\bar{\\h}^{M}{(M)})^{M} - (e^{M})^{f_{\\mathbf{v}}}", "derivation": "\\bar{\\h}{(M)} = - M and \\bar{\\h}^{M}{(M)} = (- M)^{M} and f_{\\mathbf{v}} \\bar{\\h}^{M}{(M)} = f_{\\mathbf{v}} (- M)^{M} and \\dot{y}{(f_{\\mathbf{v}},M)} = f_{\\mathbf{v}} \\bar{\\h}^{M}{(M)} and \\dot{y}{(f_{\\mathbf{v}},M)} = f_{\\mathbf{v}} (- M)^{M} and \\dot{y}^{M}{(f_{\\mathbf{v}},M)} = (f_{\\mathbf{v}} (- M)^{M})^{M} and \\dot{y}^{M}{(f_{\\mathbf{v}},M)} - (e^{M})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} (- M)^{M})^{M} - (e^{M})^{f_{\\mathbf{v}}} and \\dot{y}^{M}{(f_{\\mathbf{v}},M)} - (e^{M})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} \\bar{\\h}^{M}{(M)})^{M} - (e^{M})^{f_{\\mathbf{v}}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["times", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hbar')(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hbar')(Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], [["power", 5, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["minus", 6, "Pow(exp(Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Pow(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Pow(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hbar')(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\mu{(f_{\\mathbf{p}},r)} = f_{\\mathbf{p}} + r and \\phi_{2}{(f_{\\mathbf{p}},r)} = - \\frac{- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}}{f_{\\mathbf{p}}}, then obtain - r^{2} + \\phi_{2}{(f_{\\mathbf{p}},r)} = - r^{2} + \\frac{f_{\\mathbf{p}} - \\mu{(f_{\\mathbf{p}},r)}}{f_{\\mathbf{p}}}", "derivation": "\\mu{(f_{\\mathbf{p}},r)} = f_{\\mathbf{p}} + r and - f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)} = r and r (- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}) = r^{2} and \\phi_{2}{(f_{\\mathbf{p}},r)} = - \\frac{- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}}{f_{\\mathbf{p}}} and - r (- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}) + \\phi_{2}{(f_{\\mathbf{p}},r)} = - r (- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}) - \\frac{- f_{\\mathbf{p}} + \\mu{(f_{\\mathbf{p}},r)}}{f_{\\mathbf{p}}} and - r^{2} + \\phi_{2}{(f_{\\mathbf{p}},r)} = - r^{2} + \\frac{f_{\\mathbf{p}} - \\mu{(f_{\\mathbf{p}},r)}}{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))"], [["minus", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True))"], [["times", 2, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))), Pow(Symbol('r', commutative=True), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))))"], [["minus", 4, "Mul(Symbol('r', commutative=True), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))), Function('\\\\phi_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('r', commutative=True), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(2))), Function('\\\\phi_2')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('r', commutative=True)))))))"]]}, {"prompt": "Given U{(M)} = \\cos{(M)} and \\mathbf{J}{(M)} = \\cos{(M)} \\frac{d}{d M} \\cos{(M)} and m{(M)} = \\frac{d}{d M} U{(M)}, then derive \\frac{d}{d M} U{(M)} = - \\sin{(M)}, then obtain \\frac{d}{d M} - m{(M)} \\sin{(M)} \\cos{(M)} = \\frac{d}{d M} - \\sin{(M)} \\cos{(M)} \\frac{d}{d M} U{(M)}", "derivation": "U{(M)} = \\cos{(M)} and \\frac{d}{d M} U{(M)} = \\frac{d}{d M} \\cos{(M)} and \\frac{d}{d M} U{(M)} = - \\sin{(M)} and \\frac{d}{d M} \\cos{(M)} = - \\sin{(M)} and \\mathbf{J}{(M)} = \\cos{(M)} \\frac{d}{d M} \\cos{(M)} and m{(M)} = \\frac{d}{d M} U{(M)} and \\mathbf{J}{(M)} m{(M)} = \\mathbf{J}{(M)} \\frac{d}{d M} U{(M)} and m{(M)} \\cos{(M)} \\frac{d}{d M} \\cos{(M)} = \\cos{(M)} \\frac{d}{d M} U{(M)} \\frac{d}{d M} \\cos{(M)} and \\frac{d}{d M} m{(M)} \\cos{(M)} \\frac{d}{d M} \\cos{(M)} = \\frac{d}{d M} \\cos{(M)} \\frac{d}{d M} U{(M)} \\frac{d}{d M} \\cos{(M)} and \\frac{d}{d M} - m{(M)} \\sin{(M)} \\cos{(M)} = \\frac{d}{d M} - \\sin{(M)} \\cos{(M)} \\frac{d}{d M} U{(M)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('M', commutative=True)), Mul(cos(Symbol('M', commutative=True)), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('M', commutative=True)), Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 6, "Function('\\\\mathbf{J}')(Symbol('M', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('M', commutative=True)), Function('m')(Symbol('M', commutative=True))), Mul(Function('\\\\mathbf{J}')(Symbol('M', commutative=True)), Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Function('m')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(cos(Symbol('M', commutative=True)), Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["differentiate", 8, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Function('m')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('M', commutative=True)), Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 9, 4], "Equality(Derivative(Mul(Integer(-1), Function('m')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)), Derivative(Function('U')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)}, then obtain \\hat{H}_l \\int \\frac{\\operatorname{z^{*}}{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l = \\hat{H}_l \\int \\frac{\\cos{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l", "derivation": "\\operatorname{z^{*}}{(\\hat{H}_l)} = \\cos{(\\hat{H}_l)} and \\frac{\\operatorname{z^{*}}{(\\hat{H}_l)}}{\\hat{H}_l} = \\frac{\\cos{(\\hat{H}_l)}}{\\hat{H}_l} and \\int \\frac{\\operatorname{z^{*}}{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l = \\int \\frac{\\cos{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l and \\hat{H}_l \\int \\frac{\\operatorname{z^{*}}{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l = \\hat{H}_l \\int \\frac{\\cos{(\\hat{H}_l)}}{\\hat{H}_l} d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\hat{H}_l', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}_l', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('z^*')(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(t)} = \\cos{(t)} and Q{(t)} = - t, then obtain \\frac{Q{(t)} + \\operatorname{n_{2}}{(t)}}{\\operatorname{n_{2}}{(t)}} - \\operatorname{n_{2}}{(t)} = \\frac{Q{(t)} + \\cos{(t)}}{\\operatorname{n_{2}}{(t)}} - \\operatorname{n_{2}}{(t)}", "derivation": "\\operatorname{n_{2}}{(t)} = \\cos{(t)} and - t + \\operatorname{n_{2}}{(t)} = - t + \\cos{(t)} and \\frac{- t + \\operatorname{n_{2}}{(t)}}{\\operatorname{n_{2}}{(t)}} = \\frac{- t + \\cos{(t)}}{\\operatorname{n_{2}}{(t)}} and Q{(t)} = - t and \\frac{Q{(t)} + \\operatorname{n_{2}}{(t)}}{\\operatorname{n_{2}}{(t)}} = \\frac{Q{(t)} + \\cos{(t)}}{\\operatorname{n_{2}}{(t)}} and \\frac{Q{(t)} + \\operatorname{n_{2}}{(t)}}{\\operatorname{n_{2}}{(t)}} - \\operatorname{n_{2}}{(t)} = \\frac{Q{(t)} + \\cos{(t)}}{\\operatorname{n_{2}}{(t)}} - \\operatorname{n_{2}}{(t)}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["minus", 1, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('n_2')(Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))))"], [["divide", 2, "Function('n_2')(Symbol('t', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('n_2')(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Function('Q')(Symbol('t', commutative=True)), Function('n_2')(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))), Mul(Add(Function('Q')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))))"], [["minus", 5, "Function('n_2')(Symbol('t', commutative=True))"], "Equality(Add(Mul(Add(Function('Q')(Symbol('t', commutative=True)), Function('n_2')(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('n_2')(Symbol('t', commutative=True)))), Add(Mul(Add(Function('Q')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Pow(Function('n_2')(Symbol('t', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('n_2')(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(A_{z})} = A_{z}, then obtain - \\frac{\\int \\mathbf{H}{(A_{z})} dA_{z}}{\\mathbf{H}{(A_{z})}} = - \\frac{\\frac{A_{z}^{2}}{2} + P_{g}}{\\mathbf{H}{(A_{z})}}", "derivation": "\\mathbf{H}{(A_{z})} = A_{z} and \\int \\mathbf{H}{(A_{z})} dA_{z} = \\int A_{z} dA_{z} and - \\int \\mathbf{H}{(A_{z})} dA_{z} = - \\int A_{z} dA_{z} and - \\frac{\\int \\mathbf{H}{(A_{z})} dA_{z}}{\\mathbf{H}{(A_{z})}} = - \\frac{\\int A_{z} dA_{z}}{\\mathbf{H}{(A_{z})}} and - \\frac{\\int \\mathbf{H}{(A_{z})} dA_{z}}{\\mathbf{H}{(A_{z})}} = - \\frac{\\frac{A_{z}^{2}}{2} + P_{g}}{\\mathbf{H}{(A_{z})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Symbol('A_z', commutative=True), Tuple(Symbol('A_z', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Integer(-1), Integral(Symbol('A_z', commutative=True), Tuple(Symbol('A_z', commutative=True)))))"], [["divide", 3, "Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Integer(-1)), Integral(Symbol('A_z', commutative=True), Tuple(Symbol('A_z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Integer(-1), Add(Mul(Rational(1, 2), Pow(Symbol('A_z', commutative=True), Integer(2))), Symbol('P_g', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given V{(V_{\\mathbf{B}},\\mathbf{g})} = e^{V_{\\mathbf{B}} \\mathbf{g}}, then obtain \\frac{\\partial}{\\partial \\mathbf{g}} (V^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\mathbf{g})})^{V_{\\mathbf{B}}} = \\frac{\\partial}{\\partial \\mathbf{g}} ((e^{V_{\\mathbf{B}} \\mathbf{g}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "derivation": "V{(V_{\\mathbf{B}},\\mathbf{g})} = e^{V_{\\mathbf{B}} \\mathbf{g}} and V^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\mathbf{g})} = (e^{V_{\\mathbf{B}} \\mathbf{g}})^{V_{\\mathbf{B}}} and (V^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\mathbf{g})})^{V_{\\mathbf{B}}} = ((e^{V_{\\mathbf{B}} \\mathbf{g}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} and \\frac{\\partial}{\\partial \\mathbf{g}} (V^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},\\mathbf{g})})^{V_{\\mathbf{B}}} = \\frac{\\partial}{\\partial \\mathbf{g}} ((e^{V_{\\mathbf{B}} \\mathbf{g}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('V')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(exp(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Pow(Function('V')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Pow(exp(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('V')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Pow(Pow(exp(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(P_{e},c_{0})} = P_{e}^{c_{0}}, then obtain P_{e} + P_{e}^{c_{0}} + \\int (P_{e} + G{(P_{e},c_{0})}) dP_{e} = P_{e} + P_{e}^{c_{0}} + \\int (P_{e} + P_{e}^{c_{0}}) dP_{e}", "derivation": "G{(P_{e},c_{0})} = P_{e}^{c_{0}} and P_{e} + G{(P_{e},c_{0})} = P_{e} + P_{e}^{c_{0}} and \\int (P_{e} + G{(P_{e},c_{0})}) dP_{e} = \\int (P_{e} + P_{e}^{c_{0}}) dP_{e} and P_{e} + G{(P_{e},c_{0})} + \\int (P_{e} + G{(P_{e},c_{0})}) dP_{e} = P_{e} + G{(P_{e},c_{0})} + \\int (P_{e} + P_{e}^{c_{0}}) dP_{e} and P_{e} + P_{e}^{c_{0}} + \\int (P_{e} + G{(P_{e},c_{0})}) dP_{e} = P_{e} + P_{e}^{c_{0}} + \\int (P_{e} + P_{e}^{c_{0}}) dP_{e}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["add", 3, "Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)))"], "Equality(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), Function('G')(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True)))), Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), Pow(Symbol('P_e', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbb{I},C)} = \\sin{(\\frac{C}{\\mathbb{I}})} and \\eta^{\\prime}{(\\mathbb{I},C)} = \\sin{(\\frac{C}{\\mathbb{I}})} and \\mathbf{B}{(\\mathbb{I})} = \\frac{1}{\\mathbb{I}}, then obtain - \\hat{x}_0{(\\mathbb{I},C)} + \\frac{\\mathbf{B}{(\\mathbb{I})}}{C} = - \\hat{x}_0{(\\mathbb{I},C)} + \\frac{1}{C \\mathbb{I}}", "derivation": "\\hat{x}_0{(\\mathbb{I},C)} = \\sin{(\\frac{C}{\\mathbb{I}})} and \\eta^{\\prime}{(\\mathbb{I},C)} = \\sin{(\\frac{C}{\\mathbb{I}})} and \\eta^{\\prime}{(\\mathbb{I},C)} = \\hat{x}_0{(\\mathbb{I},C)} and \\mathbf{B}{(\\mathbb{I})} = \\frac{1}{\\mathbb{I}} and \\frac{\\mathbf{B}{(\\mathbb{I})}}{C} = \\frac{1}{C \\mathbb{I}} and - \\eta^{\\prime}{(\\mathbb{I},C)} + \\frac{\\mathbf{B}{(\\mathbb{I})}}{C} = - \\eta^{\\prime}{(\\mathbb{I},C)} + \\frac{1}{C \\mathbb{I}} and - \\hat{x}_0{(\\mathbb{I},C)} + \\frac{\\mathbf{B}{(\\mathbb{I})}}{C} = - \\hat{x}_0{(\\mathbb{I},C)} + \\frac{1}{C \\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True)), sin(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True)), sin(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))"], [["divide", 4, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["minus", 5, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given m{(P_{g},G)} = P_{g}^{G}, then obtain \\frac{\\partial}{\\partial P_{g}} m{(P_{g},G)} (\\int m{(P_{g},G)} dP_{g})^{G} = \\frac{\\partial}{\\partial P_{g}} m{(P_{g},G)} (\\int P_{g}^{G} dP_{g})^{G}", "derivation": "m{(P_{g},G)} = P_{g}^{G} and \\int m{(P_{g},G)} dP_{g} = \\int P_{g}^{G} dP_{g} and (\\int m{(P_{g},G)} dP_{g})^{G} = (\\int P_{g}^{G} dP_{g})^{G} and P_{g}^{G} (\\int m{(P_{g},G)} dP_{g})^{G} = P_{g}^{G} (\\int P_{g}^{G} dP_{g})^{G} and \\frac{\\partial}{\\partial P_{g}} P_{g}^{G} (\\int m{(P_{g},G)} dP_{g})^{G} = \\frac{\\partial}{\\partial P_{g}} P_{g}^{G} (\\int P_{g}^{G} dP_{g})^{G} and \\frac{\\partial}{\\partial P_{g}} m{(P_{g},G)} (\\int m{(P_{g},G)} dP_{g})^{G} = \\frac{\\partial}{\\partial P_{g}} m{(P_{g},G)} (\\int P_{g}^{G} dP_{g})^{G}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True)), Pow(Integral(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True)))"], [["times", 3, "Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))))"], [["differentiate", 4, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Mul(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Function('m')(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Pow(Integral(Pow(Symbol('P_g', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('G', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(t_{1},f^{\\prime},\\mathbb{I})} = (f^{\\prime})^{\\mathbb{I}} t_{1} and E{(\\phi_2,\\hat{X})} = \\frac{\\phi_2}{\\hat{X}}, then obtain \\frac{(f^{\\prime})^{- \\mathbb{I}} E{(\\phi_2,\\hat{X})}}{t_{1}} = \\frac{\\phi_2 (f^{\\prime})^{- \\mathbb{I}}}{\\hat{X} t_{1}}", "derivation": "H{(t_{1},f^{\\prime},\\mathbb{I})} = (f^{\\prime})^{\\mathbb{I}} t_{1} and E{(\\phi_2,\\hat{X})} = \\frac{\\phi_2}{\\hat{X}} and \\frac{E{(\\phi_2,\\hat{X})}}{H{(t_{1},f^{\\prime},\\mathbb{I})}} = \\frac{\\phi_2}{\\hat{X} H{(t_{1},f^{\\prime},\\mathbb{I})}} and \\frac{(f^{\\prime})^{- \\mathbb{I}} E{(\\phi_2,\\hat{X})}}{t_{1}} = \\frac{\\phi_2 (f^{\\prime})^{- \\mathbb{I}}}{\\hat{X} t_{1}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('t_1', commutative=True)))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 2, "Function('H')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Function('H')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Function('H')(Symbol('t_1', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\phi_2', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Pow(Symbol('t_1', commutative=True), Integer(-1))))"]]}, {"prompt": "Given i{(C_{1})} = \\cos{(e^{C_{1}})} and \\hat{X}{(C_{1})} = C_{1} + i{(C_{1})}, then obtain \\frac{d^{2}}{d C_{1}^{2}} \\hat{X}{(C_{1})} = \\frac{d^{2}}{d C_{1}^{2}} (C_{1} + \\cos{(e^{C_{1}})})", "derivation": "i{(C_{1})} = \\cos{(e^{C_{1}})} and C_{1} + i{(C_{1})} = C_{1} + \\cos{(e^{C_{1}})} and \\hat{X}{(C_{1})} = C_{1} + i{(C_{1})} and \\hat{X}{(C_{1})} = C_{1} + \\cos{(e^{C_{1}})} and \\frac{d}{d C_{1}} \\hat{X}{(C_{1})} = \\frac{d}{d C_{1}} (C_{1} + \\cos{(e^{C_{1}})}) and \\frac{d^{2}}{d C_{1}^{2}} \\hat{X}{(C_{1})} = \\frac{d^{2}}{d C_{1}^{2}} (C_{1} + \\cos{(e^{C_{1}})})", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('C_1', commutative=True)), cos(exp(Symbol('C_1', commutative=True))))"], [["add", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Function('i')(Symbol('C_1', commutative=True))), Add(Symbol('C_1', commutative=True), cos(exp(Symbol('C_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('C_1', commutative=True)), Add(Symbol('C_1', commutative=True), Function('i')(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{X}')(Symbol('C_1', commutative=True)), Add(Symbol('C_1', commutative=True), cos(exp(Symbol('C_1', commutative=True)))))"], [["differentiate", 4, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('C_1', commutative=True), cos(exp(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2))), Derivative(Add(Symbol('C_1', commutative=True), cos(exp(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(u)} = \\cos{(u)} and \\mathbf{g}{(u)} = \\cos{(u)}, then obtain \\frac{\\mathbf{g}{(u)}}{u \\operatorname{A_{y}}{(u)}} = \\frac{1}{u}", "derivation": "\\operatorname{A_{y}}{(u)} = \\cos{(u)} and \\mathbf{g}{(u)} = \\cos{(u)} and \\operatorname{A_{y}}{(u)} = \\mathbf{g}{(u)} and \\frac{\\operatorname{A_{y}}{(u)}}{u \\cos{(u)}} = \\frac{1}{u} and \\frac{\\mathbf{g}{(u)}}{u \\cos{(u)}} = \\frac{1}{u} and \\frac{\\mathbf{g}{(u)}}{u \\operatorname{A_{y}}{(u)}} = \\frac{1}{u}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_y')(Symbol('u', commutative=True)), Function('\\\\mathbf{g}')(Symbol('u', commutative=True)))"], [["divide", 1, "Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A_y')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Pow(Symbol('u', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Pow(Symbol('u', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Function('A_y')(Symbol('u', commutative=True)), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('u', commutative=True))), Pow(Symbol('u', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\sigma_{x}{(v_{2})} = \\cos{(v_{2})}, then obtain (\\frac{d}{d v_{2}} \\frac{- v_{2} + \\sigma_{x}{(v_{2})}}{- v_{2} + \\cos{(v_{2})}})^{v_{2}} = (\\frac{d}{d v_{2}} 1)^{v_{2}}", "derivation": "\\sigma_{x}{(v_{2})} = \\cos{(v_{2})} and - v_{2} + \\sigma_{x}{(v_{2})} = - v_{2} + \\cos{(v_{2})} and \\frac{- v_{2} + \\sigma_{x}{(v_{2})}}{- v_{2} + \\cos{(v_{2})}} = 1 and \\frac{d}{d v_{2}} \\frac{- v_{2} + \\sigma_{x}{(v_{2})}}{- v_{2} + \\cos{(v_{2})}} = \\frac{d}{d v_{2}} 1 and (\\frac{d}{d v_{2}} \\frac{- v_{2} + \\sigma_{x}{(v_{2})}}{- v_{2} + \\cos{(v_{2})}})^{v_{2}} = (\\frac{d}{d v_{2}} 1)^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\sigma_x')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\sigma_x')(Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\sigma_x')(Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\sigma_x')(Symbol('v_2', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True))), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Symbol('v_2', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('v_2', commutative=True), Integer(1))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(t_{1},\\phi)} = t_{1} \\cos{(\\phi)} and \\tilde{g}^*{(f^{\\prime},\\chi)} = e^{\\chi + f^{\\prime}}, then obtain \\tilde{g}^*^{f^{\\prime}}{(f^{\\prime},\\chi)} + \\int \\mathbf{A}{(t_{1},\\phi)} dt_{1} = (e^{\\chi + f^{\\prime}})^{f^{\\prime}} + \\int \\mathbf{A}{(t_{1},\\phi)} dt_{1}", "derivation": "\\mathbf{A}{(t_{1},\\phi)} = t_{1} \\cos{(\\phi)} and \\int \\mathbf{A}{(t_{1},\\phi)} dt_{1} = \\int t_{1} \\cos{(\\phi)} dt_{1} and \\tilde{g}^*{(f^{\\prime},\\chi)} = e^{\\chi + f^{\\prime}} and \\tilde{g}^*^{f^{\\prime}}{(f^{\\prime},\\chi)} = (e^{\\chi + f^{\\prime}})^{f^{\\prime}} and \\tilde{g}^*^{f^{\\prime}}{(f^{\\prime},\\chi)} + \\int t_{1} \\cos{(\\phi)} dt_{1} = (e^{\\chi + f^{\\prime}})^{f^{\\prime}} + \\int t_{1} \\cos{(\\phi)} dt_{1} and \\tilde{g}^*^{f^{\\prime}}{(f^{\\prime},\\chi)} + \\int \\mathbf{A}{(t_{1},\\phi)} dt_{1} = (e^{\\chi + f^{\\prime}})^{f^{\\prime}} + \\int \\mathbf{A}{(t_{1},\\phi)} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('t_1', commutative=True), cos(Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Symbol('t_1', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), exp(Add(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(exp(Add(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 4, "Integral(Mul(Symbol('t_1', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Add(Pow(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Symbol('t_1', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('t_1', commutative=True)))), Add(Pow(exp(Add(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Symbol('t_1', commutative=True), cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Add(Pow(exp(Add(Symbol('\\\\chi', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Function('\\\\mathbf{A}')(Symbol('t_1', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\omega{(z^{*})} = \\sin{(z^{*})}, then obtain \\frac{d}{d z^{*}} \\int \\omega{(z^{*})} dz^{*} = \\frac{\\partial}{\\partial z^{*}} (\\tilde{g} - \\cos{(z^{*})})", "derivation": "\\omega{(z^{*})} = \\sin{(z^{*})} and \\int \\omega{(z^{*})} dz^{*} = \\int \\sin{(z^{*})} dz^{*} and \\frac{d}{d z^{*}} \\int \\omega{(z^{*})} dz^{*} = \\frac{d}{d z^{*}} \\int \\sin{(z^{*})} dz^{*} and \\frac{d}{d z^{*}} \\int \\omega{(z^{*})} dz^{*} = \\frac{\\partial}{\\partial z^{*}} (\\tilde{g} - \\cos{(z^{*})})", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\omega')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\omega')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})}, then obtain \\frac{d}{d \\dot{\\mathbf{r}}} \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\frac{d}{d \\dot{\\mathbf{r}}} \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}", "derivation": "\\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})} and \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})} and \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\frac{d}{d \\dot{\\mathbf{r}}} \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{x}}{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\frac{d}{d \\dot{\\mathbf{r}}} \\int \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Function('A_x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('A_x')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Integral(Derivative(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\pi)} = \\log{(e^{\\pi})}, then derive \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} = 1, then obtain - 2 \\log{(e^{\\pi})} + \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} - \\frac{d^{2}}{d \\pi^{2}} \\operatorname{A_{z}}{(\\pi)} = - 2 \\log{(e^{\\pi})} - \\frac{d^{2}}{d \\pi^{2}} \\operatorname{A_{z}}{(\\pi)} + 1", "derivation": "\\operatorname{A_{z}}{(\\pi)} = \\log{(e^{\\pi})} and \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} = \\frac{d}{d \\pi} \\log{(e^{\\pi})} and \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} = 1 and - 2 \\log{(e^{\\pi})} + \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} = 1 - 2 \\log{(e^{\\pi})} and - 2 \\log{(e^{\\pi})} + \\frac{d}{d \\pi} \\operatorname{A_{z}}{(\\pi)} - \\frac{d^{2}}{d \\pi^{2}} \\operatorname{A_{z}}{(\\pi)} = - 2 \\log{(e^{\\pi})} - \\frac{d^{2}}{d \\pi^{2}} \\operatorname{A_{z}}{(\\pi)} + 1", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Mul(Integer(2), log(exp(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), log(exp(Symbol('\\\\pi', commutative=True)))), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Integer(2), log(exp(Symbol('\\\\pi', commutative=True))))))"], [["minus", 4, "Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Integer(2), log(exp(Symbol('\\\\pi', commutative=True)))), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2))))), Add(Mul(Integer(-1), Integer(2), log(exp(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Derivative(Function('A_z')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(2)))), Integer(1)))"]]}, {"prompt": "Given \\hat{x}{(k,J)} = J k and \\varepsilon{(k,J)} = J k, then obtain \\int \\frac{\\partial}{\\partial k} J k \\varepsilon{(k,J)} dk = \\int \\frac{\\partial}{\\partial k} J^{2} k^{2} dk", "derivation": "\\hat{x}{(k,J)} = J k and \\varepsilon{(k,J)} = J k and \\hat{x}{(k,J)} \\varepsilon{(k,J)} = J k \\varepsilon{(k,J)} and J k \\hat{x}{(k,J)} = J^{2} k^{2} and \\frac{\\partial}{\\partial k} J k \\hat{x}{(k,J)} = \\frac{\\partial}{\\partial k} J^{2} k^{2} and \\int \\frac{\\partial}{\\partial k} J k \\hat{x}{(k,J)} dk = \\int \\frac{\\partial}{\\partial k} J^{2} k^{2} dk and \\varepsilon{(k,J)} = \\hat{x}{(k,J)} and \\int \\frac{\\partial}{\\partial k} J k \\varepsilon{(k,J)} dk = \\int \\frac{\\partial}{\\partial k} J^{2} k^{2} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('k', commutative=True)))"], [["times", 1, "Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), Symbol('k', commutative=True), Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True), Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True), Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True), Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Function('\\\\hat{x}')(Symbol('k', commutative=True), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Derivative(Mul(Symbol('J', commutative=True), Symbol('k', commutative=True), Function('\\\\varepsilon')(Symbol('k', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('k', commutative=True), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given f{(\\dot{z},u)} = e^{u^{\\dot{z}}} and \\varepsilon_{0}{(\\dot{z},u)} = (e^{u^{\\dot{z}}})^{u}, then obtain f^{u}{(\\dot{z},u)} + (e^{u^{\\dot{z}}})^{u} = 2 f^{u}{(\\dot{z},u)}", "derivation": "f{(\\dot{z},u)} = e^{u^{\\dot{z}}} and f^{u}{(\\dot{z},u)} = (e^{u^{\\dot{z}}})^{u} and \\varepsilon_{0}{(\\dot{z},u)} = (e^{u^{\\dot{z}}})^{u} and \\varepsilon_{0}{(\\dot{z},u)} + f^{u}{(\\dot{z},u)} = \\varepsilon_{0}{(\\dot{z},u)} + (e^{u^{\\dot{z}}})^{u} and f^{u}{(\\dot{z},u)} = \\varepsilon_{0}{(\\dot{z},u)} and \\varepsilon_{0}{(\\dot{z},u)} + f^{u}{(\\dot{z},u)} = 2 \\varepsilon_{0}{(\\dot{z},u)} and \\varepsilon_{0}{(\\dot{z},u)} + (e^{u^{\\dot{z}}})^{u} = 2 \\varepsilon_{0}{(\\dot{z},u)} and f^{u}{(\\dot{z},u)} + (e^{u^{\\dot{z}}})^{u} = 2 f^{u}{(\\dot{z},u)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Pow(exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('u', commutative=True)))"], [["add", 2, "Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Add(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Pow(exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Pow(exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('u', commutative=True))), Mul(Integer(2), Function('\\\\varepsilon_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Pow(Symbol('u', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('u', commutative=True))), Mul(Integer(2), Pow(Function('f')(Symbol('\\\\dot{z}', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(G,B)} = G^{B}, then obtain G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 2 \\Psi_{nl}{(G,B)} = 2 G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 3 \\Psi_{nl}{(G,B)}", "derivation": "\\Psi_{nl}{(G,B)} = G^{B} and 2 \\Psi_{nl}{(G,B)} = G^{B} + \\Psi_{nl}{(G,B)} and - \\Psi_{nl}{(G,B)} = G^{B} - 2 \\Psi_{nl}{(G,B)} and 4 \\Psi_{nl}{(G,B)} = G^{B} + 3 \\Psi_{nl}{(G,B)} and - \\Psi_{nl}^{2}{(G,B)} - \\Psi_{nl}{(G,B)} = G^{B} - \\Psi_{nl}^{2}{(G,B)} - 2 \\Psi_{nl}{(G,B)} and G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 2 \\Psi_{nl}{(G,B)} = 3 G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 4 \\Psi_{nl}{(G,B)} and G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 2 \\Psi_{nl}{(G,B)} = 2 G^{B} - (- G^{B} + 2 \\Psi_{nl}{(G,B)})^{2} - 3 \\Psi_{nl}{(G,B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True))))"], [["minus", 1, "Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Integer(3), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))))"], [["minus", 3, "Pow(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Integer(2))), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(3), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Integer(2))), Mul(Integer(-1), Integer(4), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(Symbol('G', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Integer(2))), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(2), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('B', commutative=True))), Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))), Integer(2))), Mul(Integer(-1), Integer(3), Function('\\\\Psi_{nl}')(Symbol('G', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)} = \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})}, then obtain \\cos{((\\Omega + \\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)})^{\\Omega})} = \\cos{((\\Omega + \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})})^{\\Omega})}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)} = \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})} and \\Omega + \\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)} = \\Omega + \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})} and (\\Omega + \\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)})^{\\Omega} = (\\Omega + \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})})^{\\Omega} and \\cos{((\\Omega + \\operatorname{n_{2}}{(\\mathbf{J}_f,\\Omega)})^{\\Omega})} = \\cos{((\\Omega + \\sin{(\\frac{\\mathbf{J}_f}{\\Omega})})^{\\Omega})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\Omega', commutative=True)), sin(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), sin(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('\\\\Omega', commutative=True), sin(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Add(Symbol('\\\\Omega', commutative=True), Function('n_2')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))), cos(Pow(Add(Symbol('\\\\Omega', commutative=True), sin(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(A)} = \\cos{(\\cos{(A)})} and \\theta_{2}{(A)} = \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA, then obtain \\frac{d}{d A} \\int 0 dA = \\frac{d}{d A} \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA", "derivation": "\\hat{H}_l{(A)} = \\cos{(\\cos{(A)})} and - A + \\hat{H}_l{(A)} = - A + \\cos{(\\cos{(A)})} and \\hat{H}_l{(A)} - \\cos{(\\cos{(A)})} = 0 and \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA = \\int 0 dA and \\theta_{2}{(A)} = \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA and \\theta_{2}{(A)} = \\int 0 dA and \\frac{d}{d A} \\theta_{2}{(A)} = \\frac{d}{d A} \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA and \\frac{d}{d A} \\int 0 dA = \\frac{d}{d A} \\int (\\hat{H}_l{(A)} - \\cos{(\\cos{(A)})}) dA", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), cos(cos(Symbol('A', commutative=True))))"], [["minus", 1, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Function('\\\\hat{H}_l')(Symbol('A', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), cos(cos(Symbol('A', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('A', commutative=True)), cos(cos(Symbol('A', commutative=True))))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('A', commutative=True))))), Integer(0))"], [["integrate", 3, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('A', commutative=True))))), Tuple(Symbol('A', commutative=True))), Integral(Integer(0), Tuple(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('A', commutative=True)), Integral(Add(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('A', commutative=True))))), Tuple(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\theta_2')(Symbol('A', commutative=True)), Integral(Integer(0), Tuple(Symbol('A', commutative=True))))"], [["differentiate", 5, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('A', commutative=True))))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\hat{H}_l')(Symbol('A', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('A', commutative=True))))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\dot{y})} = \\cos{(e^{\\dot{y}})}, then derive \\int \\Psi^{\\dagger}{(\\dot{y})} d\\dot{y} = \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})}, then derive a + \\operatorname{Ci}{(e^{\\dot{y}})} = \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})}, then obtain \\frac{\\partial}{\\partial \\dot{y}} (- \\dot{y} + a + \\operatorname{Ci}{(e^{\\dot{y}})}) = \\frac{\\partial}{\\partial \\dot{y}} (- \\dot{y} + \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})})", "derivation": "\\Psi^{\\dagger}{(\\dot{y})} = \\cos{(e^{\\dot{y}})} and \\int \\Psi^{\\dagger}{(\\dot{y})} d\\dot{y} = \\int \\cos{(e^{\\dot{y}})} d\\dot{y} and \\int \\Psi^{\\dagger}{(\\dot{y})} d\\dot{y} = \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})} and \\int \\cos{(e^{\\dot{y}})} d\\dot{y} = \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})} and a + \\operatorname{Ci}{(e^{\\dot{y}})} = \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})} and - \\dot{y} + a + \\operatorname{Ci}{(e^{\\dot{y}})} = - \\dot{y} + \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})} and \\frac{\\partial}{\\partial \\dot{y}} (- \\dot{y} + a + \\operatorname{Ci}{(e^{\\dot{y}})}) = \\frac{\\partial}{\\partial \\dot{y}} (- \\dot{y} + \\dot{z} + \\operatorname{Ci}{(e^{\\dot{y}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)), cos(exp(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(exp(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('a', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Add(Symbol('\\\\dot{z}', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["minus", 5, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('a', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('a', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{z}', commutative=True), Ci(exp(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{1}{(C)} = \\sin{(C)}, then obtain \\frac{d}{d C} (- \\theta_{1}{(C)} + \\sin{(C)}) = \\frac{d}{d C} - (- \\theta_{1}{(C)} + \\sin{(C)}) \\theta_{1}{(C)}", "derivation": "\\theta_{1}{(C)} = \\sin{(C)} and 0 = - \\theta_{1}{(C)} + \\sin{(C)} and \\frac{d}{d C} 0 = \\frac{d}{d C} (- \\theta_{1}{(C)} + \\sin{(C)}) and 0 = - (- \\theta_{1}{(C)} + \\sin{(C)}) \\theta_{1}{(C)} and \\frac{d}{d C} 0 = \\frac{d}{d C} - (- \\theta_{1}{(C)} + \\sin{(C)}) \\theta_{1}{(C)} and \\frac{d}{d C} (- \\theta_{1}{(C)} + \\sin{(C)}) = \\frac{d}{d C} - (- \\theta_{1}{(C)} + \\sin{(C)}) \\theta_{1}{(C)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["minus", 1, "Function('\\\\theta_1')(Symbol('C', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Function('\\\\theta_1')(Symbol('C', commutative=True))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Function('\\\\theta_1')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('C', commutative=True))), sin(Symbol('C', commutative=True))), Function('\\\\theta_1')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{A},F_{H})} = \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A}, then derive \\operatorname{A_{1}}{(\\mathbf{A},F_{H})} = \\mathbf{A}, then obtain - F_{H} + \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} - (\\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A})^{F_{H}} = - F_{H} + \\mathbf{A} - (\\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A})^{F_{H}}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{A},F_{H})} = \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} and \\operatorname{A_{1}}{(\\mathbf{A},F_{H})} = \\mathbf{A} and \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} = \\mathbf{A} and (\\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A})^{F_{H}} = \\mathbf{A}^{F_{H}} and - \\mathbf{A}^{F_{H}} + \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} = \\mathbf{A} - \\mathbf{A}^{F_{H}} and - F_{H} - \\mathbf{A}^{F_{H}} + \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} = - F_{H} + \\mathbf{A} - \\mathbf{A}^{F_{H}} and - F_{H} + \\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A} - (\\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A})^{F_{H}} = - F_{H} + \\mathbf{A} - (\\frac{\\partial}{\\partial F_{H}} F_{H} \\mathbf{A})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True)), Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_1')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('\\\\mathbf{A}', commutative=True))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True)))"], [["minus", 3, "Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True))), Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True)))))"], [["minus", 5, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True))), Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('F_H', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Pow(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(Q)} = \\sin{(Q)}, then obtain \\int (\\tilde{g}^{Q}{(Q)} - \\sin^{Q}{(Q)}) dQ = \\int 0 dQ", "derivation": "\\tilde{g}{(Q)} = \\sin{(Q)} and \\frac{\\tilde{g}{(Q)}}{Q} = \\frac{\\sin{(Q)}}{Q} and \\tilde{g}^{Q}{(Q)} = \\sin^{Q}{(Q)} and \\tilde{g}^{Q}{(Q)} - \\frac{\\tilde{g}{(Q)}}{Q} = \\sin^{Q}{(Q)} - \\frac{\\tilde{g}{(Q)}}{Q} and \\tilde{g}^{Q}{(Q)} - \\frac{\\sin{(Q)}}{Q} = \\sin^{Q}{(Q)} - \\frac{\\sin{(Q)}}{Q} and \\tilde{g}^{Q}{(Q)} - \\sin^{Q}{(Q)} = 0 and \\int (\\tilde{g}^{Q}{(Q)} - \\sin^{Q}{(Q)}) dQ = \\int 0 dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["divide", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), sin(Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('Q', commutative=True)))"], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('Q', commutative=True)))), Add(Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), sin(Symbol('Q', commutative=True)))), Add(Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), sin(Symbol('Q', commutative=True)))))"], [["minus", 5, "Add(Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), sin(Symbol('Q', commutative=True))))"], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))), Integer(0))"], [["integrate", 6, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Integer(0), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(q)} = \\cos{(q)}, then obtain q (\\operatorname{v_{t}}^{q}{(q)} \\int \\cos{(q)} dq)^{q} = q (\\cos^{q}{(q)} \\int \\cos{(q)} dq)^{q}", "derivation": "\\operatorname{v_{t}}{(q)} = \\cos{(q)} and \\operatorname{v_{t}}^{q}{(q)} = \\cos^{q}{(q)} and \\int \\operatorname{v_{t}}{(q)} dq = \\int \\cos{(q)} dq and \\operatorname{v_{t}}^{q}{(q)} \\int \\cos{(q)} dq = \\cos^{q}{(q)} \\int \\cos{(q)} dq and \\operatorname{v_{t}}^{q}{(q)} \\int \\operatorname{v_{t}}{(q)} dq = \\cos^{q}{(q)} \\int \\operatorname{v_{t}}{(q)} dq and (\\operatorname{v_{t}}^{q}{(q)} \\int \\operatorname{v_{t}}{(q)} dq)^{q} = (\\cos^{q}{(q)} \\int \\operatorname{v_{t}}{(q)} dq)^{q} and (\\operatorname{v_{t}}^{q}{(q)} \\int \\cos{(q)} dq)^{q} = (\\cos^{q}{(q)} \\int \\cos{(q)} dq)^{q} and q (\\operatorname{v_{t}}^{q}{(q)} \\int \\cos{(q)} dq)^{q} = q (\\cos^{q}{(q)} \\int \\cos{(q)} dq)^{q}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["times", 2, "Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["power", 5, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Pow(Mul(Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Function('v_t')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True)), Pow(Mul(Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True)))"], [["times", 7, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Pow(Mul(Pow(Function('v_t')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), Pow(Mul(Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(Z,A_{z})} = A_{z} Z and \\delta{(Z,A_{z})} = \\int (- A_{z} Z + \\mathbf{J}_M{(Z,A_{z})}) dZ, then obtain \\delta{(Z,A_{z})} = \\int 0 dZ", "derivation": "\\mathbf{J}_M{(Z,A_{z})} = A_{z} Z and - A_{z} Z + \\mathbf{J}_M{(Z,A_{z})} = 0 and \\int (- A_{z} Z + \\mathbf{J}_M{(Z,A_{z})}) dZ = \\int 0 dZ and \\delta{(Z,A_{z})} = \\int (- A_{z} Z + \\mathbf{J}_M{(Z,A_{z})}) dZ and \\delta{(Z,A_{z})} = \\int 0 dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('Z', commutative=True)))"], [["minus", 1, "Mul(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True), Symbol('Z', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True))), Integer(0))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True), Symbol('Z', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Integer(0), Tuple(Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('A_z', commutative=True), Symbol('Z', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\delta')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Integral(Integer(0), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given x{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain V_{\\mathbf{B}} (x^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = V_{\\mathbf{B}} (\\sin^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}}", "derivation": "x{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and x^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} = \\sin^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} and (x^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = (\\sin^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} and V_{\\mathbf{B}} (x^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = V_{\\mathbf{B}} (\\sin^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Pow(Function('x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Pow(Function('x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Pow(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(l,v_{1})} = v_{1}^{l}, then obtain (\\log{(\\operatorname{E_{x}}{(l,v_{1})})}^{v_{1}})^{l} = (\\log{(v_{1}^{l})}^{v_{1}})^{l}", "derivation": "\\operatorname{E_{x}}{(l,v_{1})} = v_{1}^{l} and \\log{(\\operatorname{E_{x}}{(l,v_{1})})} = \\log{(v_{1}^{l})} and \\log{(\\operatorname{E_{x}}{(l,v_{1})})}^{v_{1}} = \\log{(v_{1}^{l})}^{v_{1}} and (\\log{(\\operatorname{E_{x}}{(l,v_{1})})}^{v_{1}})^{l} = (\\log{(v_{1}^{l})}^{v_{1}})^{l}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Symbol('l', commutative=True)))"], [["log", 1], "Equality(log(Function('E_x')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), log(Pow(Symbol('v_1', commutative=True), Symbol('l', commutative=True))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(log(Function('E_x')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(log(Pow(Symbol('v_1', commutative=True), Symbol('l', commutative=True))), Symbol('v_1', commutative=True)))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(log(Function('E_x')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(log(Pow(Symbol('v_1', commutative=True), Symbol('l', commutative=True))), Symbol('v_1', commutative=True)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given i{(\\omega)} = \\sin{(\\omega)}, then obtain 2 - \\sin^{\\omega}{(\\omega)} = (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} + 1", "derivation": "i{(\\omega)} = \\sin{(\\omega)} and i^{\\omega}{(\\omega)} = \\sin^{\\omega}{(\\omega)} and 1 = \\frac{\\sin{(\\omega)}}{i{(\\omega)}} and 1 = (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} and 1 - i^{\\omega}{(\\omega)} = (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - i^{\\omega}{(\\omega)} and 1 - \\sin^{\\omega}{(\\omega)} = (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} and (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} + 1 = 2 (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} and 2 - \\sin^{\\omega}{(\\omega)} = 2 (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} and 2 - \\sin^{\\omega}{(\\omega)} = (\\frac{\\sin{(\\omega)}}{i{(\\omega)}})^{\\omega} - \\sin^{\\omega}{(\\omega)} + 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Function('i')(Symbol('\\\\omega', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["minus", 4, "Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["add", 4, "Add(Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], "Equality(Add(Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Integer(1)), Add(Mul(Integer(2), Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Integer(2), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(2), Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Add(Integer(2), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Pow(Mul(Pow(Function('i')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\pi{(\\ddot{x})} = e^{e^{\\ddot{x}}}, then obtain 0 = \\frac{- \\pi^{\\ddot{x}}{(\\ddot{x})} + (e^{e^{\\ddot{x}}})^{\\ddot{x}}}{\\pi{(\\ddot{x})}}", "derivation": "\\pi{(\\ddot{x})} = e^{e^{\\ddot{x}}} and \\pi^{\\ddot{x}}{(\\ddot{x})} = (e^{e^{\\ddot{x}}})^{\\ddot{x}} and 0 = - \\pi^{\\ddot{x}}{(\\ddot{x})} + (e^{e^{\\ddot{x}}})^{\\ddot{x}} and 0 = \\frac{- \\pi^{\\ddot{x}}{(\\ddot{x})} + (e^{e^{\\ddot{x}}})^{\\ddot{x}}}{\\pi{(\\ddot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), exp(exp(Symbol('\\\\ddot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(exp(exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))), Pow(exp(exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 3, "Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))), Pow(exp(exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{H},Z)} = Z + \\sin{(\\mathbf{H})} and C{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\frac{2 \\mathbf{f}{(\\mathbf{H},Z)}}{Z} = \\frac{Z + C{(\\mathbf{H})}}{Z} + \\frac{\\mathbf{f}{(\\mathbf{H},Z)}}{Z}", "derivation": "\\mathbf{f}{(\\mathbf{H},Z)} = Z + \\sin{(\\mathbf{H})} and C{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\mathbf{f}{(\\mathbf{H},Z)} = Z + C{(\\mathbf{H})} and \\frac{\\mathbf{f}{(\\mathbf{H},Z)}}{Z} = \\frac{Z + C{(\\mathbf{H})}}{Z} and \\frac{\\mathbf{f}{(\\mathbf{H},Z)}}{Z} = \\frac{Z + \\sin{(\\mathbf{H})}}{Z} and \\frac{Z + \\sin{(\\mathbf{H})}}{Z} + \\frac{\\mathbf{f}{(\\mathbf{H},Z)}}{Z} = \\frac{Z + C{(\\mathbf{H})}}{Z} + \\frac{Z + \\sin{(\\mathbf{H})}}{Z} and \\frac{2 \\mathbf{f}{(\\mathbf{H},Z)}}{Z} = \\frac{Z + C{(\\mathbf{H})}}{Z} + \\frac{\\mathbf{f}{(\\mathbf{H},Z)}}{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Function('C')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 3, "Symbol('Z', commutative=True)"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), Function('C')(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["add", 4, "Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)))), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), Function('C')(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(2), Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Add(Symbol('Z', commutative=True), Function('C')(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given U{(I,\\mu_0)} = \\frac{\\log{(\\mu_0)}}{I}, then obtain \\frac{\\partial}{\\partial I} e^{U{(I,\\mu_0)} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}} = \\frac{\\partial}{\\partial I} e^{\\frac{\\log{(\\mu_0)}}{I} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}}", "derivation": "U{(I,\\mu_0)} = \\frac{\\log{(\\mu_0)}}{I} and \\frac{U{(I,\\mu_0)}}{\\mu_0} = \\frac{\\log{(\\mu_0)}}{I \\mu_0} and U{(I,\\mu_0)} + \\frac{U{(I,\\mu_0)}}{\\mu_0} = \\frac{U{(I,\\mu_0)}}{\\mu_0} + \\frac{\\log{(\\mu_0)}}{I} and e^{U{(I,\\mu_0)} + \\frac{U{(I,\\mu_0)}}{\\mu_0}} = e^{\\frac{U{(I,\\mu_0)}}{\\mu_0} + \\frac{\\log{(\\mu_0)}}{I}} and e^{U{(I,\\mu_0)} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}} = e^{\\frac{\\log{(\\mu_0)}}{I} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}} and \\frac{\\partial}{\\partial I} e^{U{(I,\\mu_0)} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}} = \\frac{\\partial}{\\partial I} e^{\\frac{\\log{(\\mu_0)}}{I} + \\frac{\\log{(\\mu_0)}}{I \\mu_0}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True))))), exp(Add(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Add(Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))), exp(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(exp(Add(Function('U')(Symbol('I', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\omega)} = \\omega, then obtain \\frac{d}{d \\omega} e^{\\operatorname{J_{\\varepsilon}}{(\\omega)} + 1} = \\frac{d}{d \\omega} e^{\\omega + 1}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\omega)} = \\omega and \\operatorname{J_{\\varepsilon}}{(\\omega)} + 1 = \\omega + 1 and e^{\\operatorname{J_{\\varepsilon}}{(\\omega)} + 1} = e^{\\omega + 1} and \\frac{d}{d \\omega} e^{\\operatorname{J_{\\varepsilon}}{(\\omega)} + 1} = \\frac{d}{d \\omega} e^{\\omega + 1}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["add", 1, 1], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Integer(1)))"], [["exp", 2], "Equality(exp(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(1))), exp(Add(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(exp(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True)), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(I,B)} = I e^{B}, then obtain \\int (\\operatorname{v_{z}}{(I,B)} + \\frac{\\partial}{\\partial I} I e^{B}) dB = h + (I + 1) e^{B}", "derivation": "\\operatorname{v_{z}}{(I,B)} = I e^{B} and \\operatorname{v_{z}}{(I,B)} + \\frac{\\partial}{\\partial I} I e^{B} = I e^{B} + \\frac{\\partial}{\\partial I} I e^{B} and \\int (\\operatorname{v_{z}}{(I,B)} + \\frac{\\partial}{\\partial I} I e^{B}) dB = \\int (I e^{B} + \\frac{\\partial}{\\partial I} I e^{B}) dB and \\int (\\operatorname{v_{z}}{(I,B)} + \\frac{\\partial}{\\partial I} I e^{B}) dB = h + (I + 1) e^{B}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))))"], [["add", 1, "Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Add(Function('v_z')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Add(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Function('v_z')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('v_z')(Symbol('I', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('I', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('B', commutative=True))), Add(Symbol('h', commutative=True), Mul(Add(Symbol('I', commutative=True), Integer(1)), exp(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(\\psi)} = \\cos{(\\log{(\\psi)})}, then obtain \\psi + \\int (e^{\\mu_{0}{(\\psi)}} - \\log{(\\psi)}) d\\psi = \\psi + \\int (e^{\\cos{(\\log{(\\psi)})}} - \\log{(\\psi)}) d\\psi", "derivation": "\\mu_{0}{(\\psi)} = \\cos{(\\log{(\\psi)})} and e^{\\mu_{0}{(\\psi)}} = e^{\\cos{(\\log{(\\psi)})}} and e^{\\mu_{0}{(\\psi)}} - \\log{(\\psi)} = e^{\\cos{(\\log{(\\psi)})}} - \\log{(\\psi)} and \\int (e^{\\mu_{0}{(\\psi)}} - \\log{(\\psi)}) d\\psi = \\int (e^{\\cos{(\\log{(\\psi)})}} - \\log{(\\psi)}) d\\psi and \\psi + \\int (e^{\\mu_{0}{(\\psi)}} - \\log{(\\psi)}) d\\psi = \\psi + \\int (e^{\\cos{(\\log{(\\psi)})}} - \\log{(\\psi)}) d\\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True)), cos(log(Symbol('\\\\psi', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True))), exp(cos(log(Symbol('\\\\psi', commutative=True)))))"], [["minus", 2, "log(Symbol('\\\\psi', commutative=True))"], "Equality(Add(exp(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Add(exp(cos(log(Symbol('\\\\psi', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(exp(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(exp(cos(log(Symbol('\\\\psi', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["add", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Symbol('\\\\psi', commutative=True), Integral(Add(exp(Function('\\\\mu_0')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Symbol('\\\\psi', commutative=True), Integral(Add(exp(cos(log(Symbol('\\\\psi', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\phi{(A_{2},F_{c},\\delta)} = A_{2}^{\\delta} + F_{c}, then obtain \\frac{A_{2}^{- \\delta} (- A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)})}{- A_{2} + F_{c}} = \\frac{A_{2}^{- \\delta} F_{c}}{- A_{2} + F_{c}}", "derivation": "\\phi{(A_{2},F_{c},\\delta)} = A_{2}^{\\delta} + F_{c} and - A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)} = F_{c} and A_{2}^{- \\delta} (- A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)}) = A_{2}^{- \\delta} F_{c} and - A_{2} - A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)} = - A_{2} + F_{c} and \\frac{A_{2}^{- \\delta} (- A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)})}{- A_{2} - A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)}} = \\frac{A_{2}^{- \\delta} F_{c}}{- A_{2} - A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)}} and \\frac{A_{2}^{- \\delta} (- A_{2}^{\\delta} + \\phi{(A_{2},F_{c},\\delta)})}{- A_{2} + F_{c}} = \\frac{A_{2}^{- \\delta} F_{c}}{- A_{2} + F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('F_c', commutative=True)))"], [["minus", 1, "Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('F_c', commutative=True))"], [["divide", 2, "Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Symbol('F_c', commutative=True)))"], [["minus", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('F_c', commutative=True)))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))), Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Symbol('F_c', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('F_c', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Function('\\\\phi')(Symbol('A_2', commutative=True), Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)))), Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Symbol('F_c', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('F_c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{E},p)} = \\log{(p^{\\mathbf{E}})}, then derive \\int \\operatorname{F_{N}}{(\\mathbf{E},p)} d\\mathbf{E} = F_{c} + \\frac{\\mathbf{E}^{2} \\log{(p)}}{2}, then obtain F_{c} + \\frac{\\mathbf{E}^{2} \\log{(p)}}{2} + \\int \\log{(p^{\\mathbf{E}})} d\\mathbf{E} = 2 F_{c} + \\mathbf{E}^{2} \\log{(p)}", "derivation": "\\operatorname{F_{N}}{(\\mathbf{E},p)} = \\log{(p^{\\mathbf{E}})} and \\int \\operatorname{F_{N}}{(\\mathbf{E},p)} d\\mathbf{E} = \\int \\log{(p^{\\mathbf{E}})} d\\mathbf{E} and \\int \\operatorname{F_{N}}{(\\mathbf{E},p)} d\\mathbf{E} = F_{c} + \\frac{\\mathbf{E}^{2} \\log{(p)}}{2} and \\int \\log{(p^{\\mathbf{E}})} d\\mathbf{E} = F_{c} + \\frac{\\mathbf{E}^{2} \\log{(p)}}{2} and F_{c} + \\frac{\\mathbf{E}^{2} \\log{(p)}}{2} + \\int \\log{(p^{\\mathbf{E}})} d\\mathbf{E} = 2 F_{c} + \\mathbf{E}^{2} \\log{(p)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('p', commutative=True)), log(Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(log(Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_N')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), log(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), log(Symbol('p', commutative=True)))))"], [["add", 4, "Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), log(Symbol('p', commutative=True))))"], "Equality(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), log(Symbol('p', commutative=True))), Integral(log(Pow(Symbol('p', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2)), log(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given Z{(A,\\hat{\\mathbf{x}})} = \\cos{(A^{\\hat{\\mathbf{x}}})}, then obtain (\\iint Z{(A,\\hat{\\mathbf{x}})} dA dA)^{A} = (\\iint \\cos{(A^{\\hat{\\mathbf{x}}})} dA dA)^{A}", "derivation": "Z{(A,\\hat{\\mathbf{x}})} = \\cos{(A^{\\hat{\\mathbf{x}}})} and \\int Z{(A,\\hat{\\mathbf{x}})} dA = \\int \\cos{(A^{\\hat{\\mathbf{x}}})} dA and \\iint Z{(A,\\hat{\\mathbf{x}})} dA dA = \\iint \\cos{(A^{\\hat{\\mathbf{x}}})} dA dA and (\\iint Z{(A,\\hat{\\mathbf{x}})} dA dA)^{A} = (\\iint \\cos{(A^{\\hat{\\mathbf{x}}})} dA dA)^{A}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(Pow(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Pow(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Pow(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Integral(Function('Z')(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Integral(cos(Pow(Symbol('A', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain \\frac{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})} - \\mathbf{P}}{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})}} = \\frac{\\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P}}{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})}}", "derivation": "\\mathbf{J}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\mathbf{P} \\mathbf{J}{(\\mathbf{P})} = \\mathbf{P} \\log{(\\mathbf{P})} and \\mathbf{P} \\mathbf{J}{(\\mathbf{P})} - \\mathbf{P} = \\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P} and \\frac{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})} - \\mathbf{P}}{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})}} = \\frac{\\mathbf{P} \\log{(\\mathbf{P})} - \\mathbf{P}}{\\mathbf{P} \\mathbf{J}{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain \\hat{H}_{\\lambda}{(\\mathbf{p})} \\cos{(\\mathbf{p})} + \\hat{H}_{\\lambda}{(\\mathbf{p})} = \\hat{H}_{\\lambda}{(\\mathbf{p})} \\cos{(\\mathbf{p})} + \\cos{(\\mathbf{p})}", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\hat{H}_{\\lambda}^{2}{(\\mathbf{p})} = \\hat{H}_{\\lambda}{(\\mathbf{p})} \\cos{(\\mathbf{p})} and \\hat{H}_{\\lambda}^{2}{(\\mathbf{p})} + \\hat{H}_{\\lambda}{(\\mathbf{p})} = \\hat{H}_{\\lambda}^{2}{(\\mathbf{p})} + \\cos{(\\mathbf{p})} and \\hat{H}_{\\lambda}{(\\mathbf{p})} \\cos{(\\mathbf{p})} + \\hat{H}_{\\lambda}{(\\mathbf{p})} = \\hat{H}_{\\lambda}{(\\mathbf{p})} \\cos{(\\mathbf{p})} + \\cos{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 1, "Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True))), cos(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(k)} = \\log{(k)} and f{(k)} = \\operatorname{f^{\\prime}}{(k)} + \\log{(k)}, then obtain f^{- k}{(k)} \\log{(k)} \\sin{((\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k})} = f^{- k}{(k)} \\log{(k)} \\sin{((2 \\operatorname{f^{\\prime}}{(k)})^{k})}", "derivation": "\\operatorname{f^{\\prime}}{(k)} = \\log{(k)} and f{(k)} = \\operatorname{f^{\\prime}}{(k)} + \\log{(k)} and f^{k}{(k)} = (\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k} and f^{k}{(k)} = (2 \\operatorname{f^{\\prime}}{(k)})^{k} and (\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k} = (2 \\operatorname{f^{\\prime}}{(k)})^{k} and \\sin{((\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k})} = \\sin{((2 \\operatorname{f^{\\prime}}{(k)})^{k})} and \\log{(k)} \\sin{((\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k})} = \\log{(k)} \\sin{((2 \\operatorname{f^{\\prime}}{(k)})^{k})} and f^{- k}{(k)} \\log{(k)} \\sin{((\\operatorname{f^{\\prime}}{(k)} + \\log{(k)})^{k})} = f^{- k}{(k)} \\log{(k)} \\sin{((2 \\operatorname{f^{\\prime}}{(k)})^{k})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('k', commutative=True)), Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Pow(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["sin", 5], "Equality(sin(Pow(Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Symbol('k', commutative=True))), sin(Pow(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('k', commutative=True))), Symbol('k', commutative=True))))"], [["times", 6, "log(Symbol('k', commutative=True))"], "Equality(Mul(log(Symbol('k', commutative=True)), sin(Pow(Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))), Mul(log(Symbol('k', commutative=True)), sin(Pow(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"], [["divide", 7, "Pow(Function('f')(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Function('f')(Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), log(Symbol('k', commutative=True)), sin(Pow(Add(Function('f^{\\\\prime}')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True))), Symbol('k', commutative=True)))), Mul(Pow(Function('f')(Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), log(Symbol('k', commutative=True)), sin(Pow(Mul(Integer(2), Function('f^{\\\\prime}')(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given t{(\\mathbf{J}_M,\\Omega)} = \\frac{\\mathbf{J}_M}{\\Omega}, then derive \\frac{\\partial}{\\partial \\Omega} t{(\\mathbf{J}_M,\\Omega)} = - \\frac{\\mathbf{J}_M}{\\Omega^{2}}, then obtain \\frac{\\partial}{\\partial \\Omega} \\frac{\\mathbf{J}_M}{\\Omega} = - \\frac{\\mathbf{J}_M}{\\Omega^{2}}", "derivation": "t{(\\mathbf{J}_M,\\Omega)} = \\frac{\\mathbf{J}_M}{\\Omega} and \\frac{\\partial}{\\partial \\Omega} t{(\\mathbf{J}_M,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\frac{\\mathbf{J}_M}{\\Omega} and \\frac{\\partial}{\\partial \\Omega} t{(\\mathbf{J}_M,\\Omega)} = - \\frac{\\mathbf{J}_M}{\\Omega^{2}} and \\frac{\\partial}{\\partial \\Omega} \\frac{\\mathbf{J}_M}{\\Omega} = - \\frac{\\mathbf{J}_M}{\\Omega^{2}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\delta)} = \\int \\log{(\\delta)} d\\delta and \\operatorname{r_{0}}{(\\delta)} = \\delta \\int \\log{(\\delta)} d\\delta + \\delta, then obtain \\operatorname{r_{0}}^{\\delta}{(\\delta)} = (\\delta \\operatorname{z^{*}}{(\\delta)} + \\delta)^{\\delta}", "derivation": "\\operatorname{z^{*}}{(\\delta)} = \\int \\log{(\\delta)} d\\delta and \\operatorname{r_{0}}{(\\delta)} = \\delta \\int \\log{(\\delta)} d\\delta + \\delta and \\operatorname{r_{0}}{(\\delta)} = \\delta \\operatorname{z^{*}}{(\\delta)} + \\delta and \\operatorname{r_{0}}^{\\delta}{(\\delta)} = (\\delta \\operatorname{z^{*}}{(\\delta)} + \\delta)^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True)), Integral(log(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\delta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Integral(log(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('r_0')(Symbol('\\\\delta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Function('z^*')(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Function('z^*')(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\dot{x})} = \\log{(e^{\\dot{x}})}, then obtain (\\frac{- \\dot{x} + \\mu{(\\dot{x})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} e^{\\dot{x}} = (\\frac{- \\dot{x} + \\log{(e^{\\dot{x}})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} e^{\\dot{x}}", "derivation": "\\mu{(\\dot{x})} = \\log{(e^{\\dot{x}})} and - \\dot{x} + \\mu{(\\dot{x})} = - \\dot{x} + \\log{(e^{\\dot{x}})} and \\frac{- \\dot{x} + \\mu{(\\dot{x})}}{\\log{(e^{\\dot{x}})}} = \\frac{- \\dot{x} + \\log{(e^{\\dot{x}})}}{\\log{(e^{\\dot{x}})}} and (\\frac{- \\dot{x} + \\mu{(\\dot{x})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} = (\\frac{- \\dot{x} + \\log{(e^{\\dot{x}})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} and (\\frac{- \\dot{x} + \\mu{(\\dot{x})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} e^{\\dot{x}} = (\\frac{- \\dot{x} + \\log{(e^{\\dot{x}})}}{\\log{(e^{\\dot{x}})}})^{\\dot{x}} e^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))))"], [["divide", 2, "log(exp(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 4, "exp(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))), Pow(log(exp(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(n)} = \\sin{(n)}, then derive \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\cos{(n)}, then obtain n \\int \\frac{d}{d n} \\sin{(n)} dn = n \\int \\cos{(n)} dn", "derivation": "\\operatorname{m_{s}}{(n)} = \\sin{(n)} and \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\frac{d}{d n} \\sin{(n)} and \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\cos{(n)} and \\int \\frac{d}{d n} \\operatorname{m_{s}}{(n)} dn = \\int \\cos{(n)} dn and n \\int \\frac{d}{d n} \\operatorname{m_{s}}{(n)} dn = n \\int \\cos{(n)} dn and \\int \\frac{d}{d n} \\sin{(n)} dn = \\int \\cos{(n)} dn and n \\int \\frac{d}{d n} \\operatorname{m_{s}}{(n)} dn = n \\int \\frac{d}{d n} \\sin{(n)} dn and n \\int \\frac{d}{d n} \\sin{(n)} dn = n \\int \\cos{(n)} dn", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), cos(Symbol('n', commutative=True)))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["times", 4, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Integral(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Mul(Symbol('n', commutative=True), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Symbol('n', commutative=True), Integral(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Mul(Symbol('n', commutative=True), Integral(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Mul(Symbol('n', commutative=True), Integral(Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))), Mul(Symbol('n', commutative=True), Integral(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(m,\\tilde{g})} = e^{\\frac{\\tilde{g}}{m}}, then obtain 1 - \\frac{d}{d m} 1 = - \\frac{d}{d m} 1 + \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}}", "derivation": "\\mathbf{J}_f{(m,\\tilde{g})} = e^{\\frac{\\tilde{g}}{m}} and 1 = \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}} and \\frac{d}{d m} 1 = \\frac{\\partial}{\\partial m} \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}} and 1 - \\frac{\\partial}{\\partial m} \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}} = - \\frac{\\partial}{\\partial m} \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}} + \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}} and 1 - \\frac{d}{d m} 1 = - \\frac{d}{d m} 1 + \\frac{e^{\\frac{\\tilde{g}}{m}}}{\\mathbf{J}_f{(m,\\tilde{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('m', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), exp(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given h{(\\varphi)} = e^{e^{\\varphi}}, then obtain \\varphi h{(\\varphi)} + h^{\\varphi}{(\\varphi)} + e^{\\varphi} = \\varphi h{(\\varphi)} + e^{\\varphi} + (e^{- \\varphi h{(\\varphi)} + \\varphi e^{e^{\\varphi}} + e^{\\varphi}})^{\\varphi}", "derivation": "h{(\\varphi)} = e^{e^{\\varphi}} and \\varphi h{(\\varphi)} = \\varphi e^{e^{\\varphi}} and \\varphi h{(\\varphi)} + e^{\\varphi} = \\varphi e^{e^{\\varphi}} + e^{\\varphi} and e^{\\varphi} = - \\varphi h{(\\varphi)} + \\varphi e^{e^{\\varphi}} + e^{\\varphi} and h^{\\varphi}{(\\varphi)} = (e^{e^{\\varphi}})^{\\varphi} and h^{\\varphi}{(\\varphi)} = (e^{- \\varphi h{(\\varphi)} + \\varphi e^{e^{\\varphi}} + e^{\\varphi}})^{\\varphi} and \\varphi h{(\\varphi)} + h^{\\varphi}{(\\varphi)} + e^{\\varphi} = \\varphi h{(\\varphi)} + e^{\\varphi} + (e^{- \\varphi h{(\\varphi)} + \\varphi e^{e^{\\varphi}} + e^{\\varphi}})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\varphi', commutative=True)), exp(exp(Symbol('\\\\varphi', commutative=True))))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(exp(Symbol('\\\\varphi', commutative=True)))))"], [["add", 2, "exp(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), exp(Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\varphi', commutative=True), exp(exp(Symbol('\\\\varphi', commutative=True)))), exp(Symbol('\\\\varphi', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True)))"], "Equality(exp(Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(exp(Symbol('\\\\varphi', commutative=True)))), exp(Symbol('\\\\varphi', commutative=True))))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(exp(exp(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Function('h')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(exp(Symbol('\\\\varphi', commutative=True)))), exp(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["add", 6, "Add(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), exp(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), Pow(Function('h')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), exp(Symbol('\\\\varphi', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(exp(Symbol('\\\\varphi', commutative=True)))), exp(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(I,\\rho)} = \\frac{e^{I}}{\\rho}, then obtain (\\hat{H}_l{(I,\\rho)} + \\frac{\\hat{H}_l{(I,\\rho)}}{I})^{I} = (\\frac{e^{I}}{\\rho} + \\frac{\\hat{H}_l{(I,\\rho)}}{I})^{I}", "derivation": "\\hat{H}_l{(I,\\rho)} = \\frac{e^{I}}{\\rho} and \\frac{\\hat{H}_l{(I,\\rho)}}{I} = \\frac{e^{I}}{I \\rho} and \\hat{H}_l{(I,\\rho)} + \\frac{e^{I}}{I \\rho} = \\frac{e^{I}}{\\rho} + \\frac{e^{I}}{I \\rho} and \\hat{H}_l{(I,\\rho)} + \\frac{\\hat{H}_l{(I,\\rho)}}{I} = \\frac{e^{I}}{\\rho} + \\frac{\\hat{H}_l{(I,\\rho)}}{I} and (\\hat{H}_l{(I,\\rho)} + \\frac{\\hat{H}_l{(I,\\rho)}}{I})^{I} = (\\frac{e^{I}}{\\rho} + \\frac{\\hat{H}_l{(I,\\rho)}}{I})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True)))), Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), exp(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('I', commutative=True), Symbol('\\\\rho', commutative=True)))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given S{(\\psi,x)} = - x + \\log{(\\psi)}, then derive \\frac{x (- x S{(\\psi,x)})^{x} \\frac{\\partial}{\\partial \\psi} S{(\\psi,x)}}{S{(\\psi,x)}} = \\frac{x (- x (- x + \\log{(\\psi)}))^{x}}{\\psi (- x + \\log{(\\psi)})}, then obtain \\frac{x (- x (- x + \\log{(\\psi)}))^{x} \\frac{\\partial}{\\partial \\psi} S{(\\psi,x)}}{S{(\\psi,x)}} = \\frac{x (- x (- x + \\log{(\\psi)}))^{x}}{\\psi (- x + \\log{(\\psi)})}", "derivation": "S{(\\psi,x)} = - x + \\log{(\\psi)} and - x S{(\\psi,x)} = - x (- x + \\log{(\\psi)}) and (- x S{(\\psi,x)})^{x} = (- x (- x + \\log{(\\psi)}))^{x} and \\frac{\\partial}{\\partial \\psi} (- x S{(\\psi,x)})^{x} = \\frac{\\partial}{\\partial \\psi} (- x (- x + \\log{(\\psi)}))^{x} and \\frac{x (- x S{(\\psi,x)})^{x} \\frac{\\partial}{\\partial \\psi} S{(\\psi,x)}}{S{(\\psi,x)}} = \\frac{x (- x (- x + \\log{(\\psi)}))^{x}}{\\psi (- x + \\log{(\\psi)})} and \\frac{x (- x (- x + \\log{(\\psi)}))^{x} \\frac{\\partial}{\\partial \\psi} S{(\\psi,x)}}{S{(\\psi,x)}} = \\frac{x (- x (- x + \\log{(\\psi)}))^{x}}{\\psi (- x + \\log{(\\psi)})}", "srepr_derivation": [["renaming_premise", "Equality(Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('x', commutative=True), Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('x', commutative=True), Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Symbol('x', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Symbol('x', commutative=True), Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Symbol('x', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('x', commutative=True), Pow(Mul(Integer(-1), Symbol('x', commutative=True), Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True)), Integer(-1)), Derivative(Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x', commutative=True), Pow(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('x', commutative=True), Pow(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Symbol('x', commutative=True)), Pow(Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True)), Integer(-1)), Derivative(Function('S')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Symbol('x', commutative=True), Pow(Mul(Integer(-1), Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True)))), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\psi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given y{(s)} = \\cos{(s)} and \\dot{y}{(s)} = \\cos{(s)}, then obtain (y^{s}{(s)})^{s} = (\\dot{y}^{s}{(s)})^{s}", "derivation": "y{(s)} = \\cos{(s)} and \\dot{y}{(s)} = \\cos{(s)} and y^{s}{(s)} = \\cos^{s}{(s)} and \\dot{y}{(s)} = y{(s)} and (y^{s}{(s)})^{s} = (\\cos^{s}{(s)})^{s} and \\dot{y}^{s}{(s)} = \\cos^{s}{(s)} and (y^{s}{(s)})^{s} = (\\dot{y}^{s}{(s)})^{s}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('y')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\dot{y}')(Symbol('s', commutative=True)), Function('y')(Symbol('s', commutative=True)))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Pow(Function('y')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\dot{y}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Pow(Function('y')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(Function('\\\\dot{y}')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given k{(m_{s},v_{t})} = - m_{s} + v_{t} and \\mathbf{r}{(m_{s},v_{t})} = - \\int (- m_{s} + v_{t}) dm_{s}, then obtain \\mathbf{r}^{v_{t}}{(m_{s},v_{t})} = (- \\int k{(m_{s},v_{t})} dm_{s})^{v_{t}}", "derivation": "k{(m_{s},v_{t})} = - m_{s} + v_{t} and \\int k{(m_{s},v_{t})} dm_{s} = \\int (- m_{s} + v_{t}) dm_{s} and \\mathbf{r}{(m_{s},v_{t})} = - \\int (- m_{s} + v_{t}) dm_{s} and \\mathbf{r}{(m_{s},v_{t})} = - \\int k{(m_{s},v_{t})} dm_{s} and \\mathbf{r}^{v_{t}}{(m_{s},v_{t})} = (- \\int k{(m_{s},v_{t})} dm_{s})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('k')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Symbol('v_t', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{r}')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Function('k')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["power", 4, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(Mul(Integer(-1), Integral(Function('k')(Symbol('m_s', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given E{(\\chi,s)} = \\chi s, then obtain \\frac{d}{d s} 0 = \\frac{\\partial}{\\partial s} (2 \\chi s - 2 E{(\\chi,s)})", "derivation": "E{(\\chi,s)} = \\chi s and \\chi s + E{(\\chi,s)} = 2 \\chi s and 0 = \\chi s - E{(\\chi,s)} and \\chi s - E{(\\chi,s)} = 2 \\chi s - 2 E{(\\chi,s)} and 0 = 2 \\chi s - 2 E{(\\chi,s)} and \\frac{d}{d s} 0 = \\frac{\\partial}{\\partial s} (2 \\chi s - 2 E{(\\chi,s)})", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True))), Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))"], [["minus", 2, "Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))))"], [["differentiate", 5, "Symbol('s', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Integer(2), Function('E')(Symbol('\\\\chi', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(n_{2})} = \\sin{(n_{2})}, then obtain (\\frac{n_{2} \\frac{d}{d n_{2}} L{(n_{2})}}{L{(n_{2})}} + \\log{(L{(n_{2})})}) L^{n_{2}}{(n_{2})} = (\\frac{n_{2} \\cos{(n_{2})}}{\\sin{(n_{2})}} + \\log{(\\sin{(n_{2})})}) \\sin^{n_{2}}{(n_{2})}", "derivation": "L{(n_{2})} = \\sin{(n_{2})} and L^{n_{2}}{(n_{2})} = \\sin^{n_{2}}{(n_{2})} and \\frac{d}{d n_{2}} L^{n_{2}}{(n_{2})} = \\frac{d}{d n_{2}} \\sin^{n_{2}}{(n_{2})} and (\\frac{n_{2} \\frac{d}{d n_{2}} L{(n_{2})}}{L{(n_{2})}} + \\log{(L{(n_{2})})}) L^{n_{2}}{(n_{2})} = (\\frac{n_{2} \\cos{(n_{2})}}{\\sin{(n_{2})}} + \\log{(\\sin{(n_{2})})}) \\sin^{n_{2}}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('L')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Pow(Function('L')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('n_2', commutative=True), Pow(Function('L')(Symbol('n_2', commutative=True)), Integer(-1)), Derivative(Function('L')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), log(Function('L')(Symbol('n_2', commutative=True)))), Pow(Function('L')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Mul(Add(Mul(Symbol('n_2', commutative=True), Pow(sin(Symbol('n_2', commutative=True)), Integer(-1)), cos(Symbol('n_2', commutative=True))), log(sin(Symbol('n_2', commutative=True)))), Pow(sin(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given p{(E_{n},\\hat{H}_l)} = E_{n} - \\hat{H}_l, then obtain E_{n} p{(E_{n},\\hat{H}_l)} = E_{n} (E_{n} - \\hat{H}_l)", "derivation": "p{(E_{n},\\hat{H}_l)} = E_{n} - \\hat{H}_l and - p{(E_{n},\\hat{H}_l)} = - E_{n} + \\hat{H}_l and E_{n} p{(E_{n},\\hat{H}_l)} = - E_{n} (- E_{n} + \\hat{H}_l) and E_{n} (E_{n} - \\hat{H}_l) = - E_{n} (- E_{n} + \\hat{H}_l) and E_{n} p{(E_{n},\\hat{H}_l)} = E_{n} (E_{n} - \\hat{H}_l)", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('p')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('E_n', commutative=True))"], "Equality(Mul(Symbol('E_n', commutative=True), Function('p')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('E_n', commutative=True), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Integer(-1), Symbol('E_n', commutative=True), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('E_n', commutative=True), Function('p')(Symbol('E_n', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\hat{X})} = e^{\\hat{X}}, then derive \\frac{d}{d \\hat{X}} \\operatorname{C_{d}}{(\\hat{X})} = e^{\\hat{X}}, then obtain \\frac{d}{d \\hat{X}} \\operatorname{C_{d}}{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\operatorname{C_{d}}{(\\hat{X})}", "derivation": "\\operatorname{C_{d}}{(\\hat{X})} = e^{\\hat{X}} and \\frac{d}{d \\hat{X}} \\operatorname{C_{d}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} e^{\\hat{X}} and \\frac{d}{d \\hat{X}} \\operatorname{C_{d}}{(\\hat{X})} = e^{\\hat{X}} and \\frac{d}{d \\hat{X}} \\operatorname{C_{d}}{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\operatorname{C_{d}}{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('C_d')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Function('C_d')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(S,A)} = \\log{(A - S)}, then obtain - f^{\\prime} + \\frac{\\operatorname{M_{E}}{(S,A)} \\log{(A - S)}}{(f^{\\prime})^{2}} = - f^{\\prime} + \\frac{\\log{(A - S)}^{2}}{(f^{\\prime})^{2}}", "derivation": "\\operatorname{M_{E}}{(S,A)} = \\log{(A - S)} and \\frac{\\operatorname{M_{E}}{(S,A)}}{f^{\\prime}} = \\frac{\\log{(A - S)}}{f^{\\prime}} and \\frac{\\operatorname{M_{E}}{(S,A)} \\log{(A - S)}}{(f^{\\prime})^{2}} = \\frac{\\log{(A - S)}^{2}}{(f^{\\prime})^{2}} and - f^{\\prime} + \\frac{\\operatorname{M_{E}}{(S,A)} \\log{(A - S)}}{(f^{\\prime})^{2}} = - f^{\\prime} + \\frac{\\log{(A - S)}^{2}}{(f^{\\prime})^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('M_E')(Symbol('S', commutative=True), Symbol('A', commutative=True)), log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))))"], [["divide", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('M_E')(Symbol('S', commutative=True), Symbol('A', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))))))"], [["times", 2, "Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))))"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), Function('M_E')(Symbol('S', commutative=True), Symbol('A', commutative=True)), log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), Pow(log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Integer(2))))"], [["minus", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), Function('M_E')(Symbol('S', commutative=True), Symbol('A', commutative=True)), log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-2)), Pow(log(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given \\omega{(E_{x})} = \\sin{(E_{x})} and z{(E_{x})} = \\sin{(E_{x})} and M{(E_{x})} = z{(E_{x})} + \\sin{(E_{x})}, then obtain M^{2}{(E_{x})} = 4 \\omega^{2}{(E_{x})}", "derivation": "\\omega{(E_{x})} = \\sin{(E_{x})} and z{(E_{x})} = \\sin{(E_{x})} and z{(E_{x})} + \\sin{(E_{x})} = 2 \\sin{(E_{x})} and M{(E_{x})} = z{(E_{x})} + \\sin{(E_{x})} and M{(E_{x})} = 2 \\sin{(E_{x})} and M^{2}{(E_{x})} = 4 \\sin^{2}{(E_{x})} and M^{2}{(E_{x})} = 4 \\omega^{2}{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], [["add", 2, "sin(Symbol('E_x', commutative=True))"], "Equality(Add(Function('z')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Mul(Integer(2), sin(Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('E_x', commutative=True)), Add(Function('z')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('M')(Symbol('E_x', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True))))"], [["power", 5, 2], "Equality(Pow(Function('M')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Integer(4), Pow(sin(Symbol('E_x', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Function('M')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Integer(4), Pow(Function('\\\\omega')(Symbol('E_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\rho_{b}{(f)} = e^{\\cos{(f)}}, then obtain f (\\rho_{b}^{f}{(f)} - 3 e^{\\cos{(f)}}) = f (- 3 e^{\\cos{(f)}} + (e^{\\cos{(f)}})^{f})", "derivation": "\\rho_{b}{(f)} = e^{\\cos{(f)}} and \\rho_{b}^{f}{(f)} = (e^{\\cos{(f)}})^{f} and \\rho_{b}^{f}{(f)} - e^{\\cos{(f)}} = - e^{\\cos{(f)}} + (e^{\\cos{(f)}})^{f} and \\rho_{b}^{f}{(f)} - 3 e^{\\cos{(f)}} = - 3 e^{\\cos{(f)}} + (e^{\\cos{(f)}})^{f} and f (\\rho_{b}^{f}{(f)} - 3 e^{\\cos{(f)}}) = f (- 3 e^{\\cos{(f)}} + (e^{\\cos{(f)}})^{f})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('f', commutative=True)), exp(cos(Symbol('f', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(exp(cos(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["minus", 2, "exp(cos(Symbol('f', commutative=True)))"], "Equality(Add(Pow(Function('\\\\rho_b')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Mul(Integer(-1), exp(cos(Symbol('f', commutative=True))))), Add(Mul(Integer(-1), exp(cos(Symbol('f', commutative=True)))), Pow(exp(cos(Symbol('f', commutative=True))), Symbol('f', commutative=True))))"], [["minus", 3, "Mul(Integer(2), exp(cos(Symbol('f', commutative=True))))"], "Equality(Add(Pow(Function('\\\\rho_b')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(3), exp(cos(Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Integer(3), exp(cos(Symbol('f', commutative=True)))), Pow(exp(cos(Symbol('f', commutative=True))), Symbol('f', commutative=True))))"], [["times", 4, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Add(Pow(Function('\\\\rho_b')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(3), exp(cos(Symbol('f', commutative=True)))))), Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Integer(3), exp(cos(Symbol('f', commutative=True)))), Pow(exp(cos(Symbol('f', commutative=True))), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\psi{(b,y)} = b y and \\mu{(r,v_{1})} = r^{v_{1}}, then obtain \\frac{\\partial}{\\partial b} (r + \\iint \\psi{(b,y)} db db) = \\frac{\\partial}{\\partial b} (r + \\iint b y db db)", "derivation": "\\psi{(b,y)} = b y and \\int \\psi{(b,y)} db = \\int b y db and \\mu{(r,v_{1})} = r^{v_{1}} and \\iint \\psi{(b,y)} db db = \\iint b y db db and r - \\mu{(r,v_{1})} + \\iint \\psi{(b,y)} db db = r - \\mu{(r,v_{1})} + \\iint b y db db and r - r^{v_{1}} - \\mu{(r,v_{1})} + \\iint \\psi{(b,y)} db db = r - r^{v_{1}} - \\mu{(r,v_{1})} + \\iint b y db db and r - 2 r^{v_{1}} + \\iint \\psi{(b,y)} db db = r - 2 r^{v_{1}} + \\iint b y db db and r + \\iint \\psi{(b,y)} db db = r + \\iint b y db db and \\frac{\\partial}{\\partial b} (r + \\iint \\psi{(b,y)} db db) = \\frac{\\partial}{\\partial b} (r + \\iint b y db db)", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["minus", 5, "Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Integer(-1), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Function('\\\\mu')(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('r', commutative=True), Mul(Integer(-1), Integer(2), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Integer(-1), Integer(2), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["minus", 7, "Mul(Integer(-1), Integer(2), Pow(Symbol('r', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Add(Symbol('r', commutative=True), Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Symbol('r', commutative=True), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["differentiate", 8, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Symbol('r', commutative=True), Integral(Function('\\\\psi')(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('r', commutative=True), Integral(Mul(Symbol('b', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(a^{\\dagger})} = a^{\\dagger}, then derive \\dot{z} + \\frac{G^{2}{(a^{\\dagger})}}{2} = \\int a^{\\dagger} dG{(a^{\\dagger})}, then obtain \\dot{z} + \\frac{(a^{\\dagger})^{2}}{2} = \\int G{(a^{\\dagger})} da^{\\dagger}", "derivation": "G{(a^{\\dagger})} = a^{\\dagger} and \\int G{(a^{\\dagger})} da^{\\dagger} = \\int a^{\\dagger} da^{\\dagger} and \\int G{(a^{\\dagger})} dG{(a^{\\dagger})} = \\int a^{\\dagger} dG{(a^{\\dagger})} and \\dot{z} + \\frac{G^{2}{(a^{\\dagger})}}{2} = \\int a^{\\dagger} dG{(a^{\\dagger})} and \\dot{z} + \\frac{(a^{\\dagger})^{2}}{2} = \\int a^{\\dagger} da^{\\dagger} and \\dot{z} + \\frac{(a^{\\dagger})^{2}}{2} = \\int G{(a^{\\dagger})} da^{\\dagger}", "srepr_derivation": [["renaming_premise", "Equality(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Symbol('a^{\\\\dagger}', commutative=True), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)))), Integral(Symbol('a^{\\\\dagger}', commutative=True), Tuple(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)))), Integral(Symbol('a^{\\\\dagger}', commutative=True), Tuple(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)))), Integral(Symbol('a^{\\\\dagger}', commutative=True), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(2)))), Integral(Function('G')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(x)} = \\log{(\\cos{(x)})}, then obtain \\frac{d}{d x} \\int x \\operatorname{C_{2}}{(x)} dx = \\frac{d}{d x} \\int x \\log{(\\cos{(x)})} dx", "derivation": "\\operatorname{C_{2}}{(x)} = \\log{(\\cos{(x)})} and x \\operatorname{C_{2}}{(x)} = x \\log{(\\cos{(x)})} and \\int x \\operatorname{C_{2}}{(x)} dx = \\int x \\log{(\\cos{(x)})} dx and \\frac{d}{d x} \\int x \\operatorname{C_{2}}{(x)} dx = \\frac{d}{d x} \\int x \\log{(\\cos{(x)})} dx", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('C_2')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), log(cos(Symbol('x', commutative=True)))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('C_2')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('x', commutative=True), Function('C_2')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('x', commutative=True), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(M,\\mathbf{f})} = \\mathbf{f} \\sin{(M)} and \\operatorname{A_{z}}{(M,\\mathbf{f})} = 2 \\psi^{*}{(M,\\mathbf{f})}, then obtain (2 \\psi^{*}{(M,\\mathbf{f})})^{M} = (2 \\mathbf{f} \\sin{(M)})^{M}", "derivation": "\\psi^{*}{(M,\\mathbf{f})} = \\mathbf{f} \\sin{(M)} and 2 \\psi^{*}{(M,\\mathbf{f})} = \\mathbf{f} \\sin{(M)} + \\psi^{*}{(M,\\mathbf{f})} and \\operatorname{A_{z}}{(M,\\mathbf{f})} = 2 \\psi^{*}{(M,\\mathbf{f})} and (2 \\psi^{*}{(M,\\mathbf{f})})^{M} = (\\mathbf{f} \\sin{(M)} + \\psi^{*}{(M,\\mathbf{f})})^{M} and \\operatorname{A_{z}}^{M}{(M,\\mathbf{f})} = (\\mathbf{f} \\sin{(M)} + \\psi^{*}{(M,\\mathbf{f})})^{M} and \\operatorname{A_{z}}^{M}{(M,\\mathbf{f})} = (2 \\mathbf{f} \\sin{(M)})^{M} and \\operatorname{A_{z}}^{M}{(M,\\mathbf{f})} = (2 \\psi^{*}{(M,\\mathbf{f})})^{M} and (2 \\psi^{*}{(M,\\mathbf{f})})^{M} = (2 \\mathbf{f} \\sin{(M)})^{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))))"], [["add", 1, "Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('M', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('A_z')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('M', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Function('A_z')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('A_z')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Integer(2), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Pow(Mul(Integer(2), Function('\\\\psi^*')(Symbol('M', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(V)} = \\sin{(V)}, then derive \\int V \\mathbf{P}{(V)} dV = P_{e} - V \\cos{(V)} + \\sin{(V)}, then obtain P_{e} - V \\cos{(V)} + \\sin{(V)} + 1 = \\int V \\sin{(V)} dV + 1", "derivation": "\\mathbf{P}{(V)} = \\sin{(V)} and V \\mathbf{P}{(V)} = V \\sin{(V)} and \\int V \\mathbf{P}{(V)} dV = \\int V \\sin{(V)} dV and \\int V \\mathbf{P}{(V)} dV = P_{e} - V \\cos{(V)} + \\sin{(V)} and P_{e} - V \\cos{(V)} + \\sin{(V)} = \\int V \\sin{(V)} dV and P_{e} - V \\cos{(V)} + \\sin{(V)} + 1 = \\int V \\sin{(V)} dV + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{P}')(Symbol('V', commutative=True))), Mul(Symbol('V', commutative=True), sin(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{P}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Symbol('V', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('V', commutative=True), Function('\\\\mathbf{P}')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True), cos(Symbol('V', commutative=True))), sin(Symbol('V', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True), cos(Symbol('V', commutative=True))), sin(Symbol('V', commutative=True))), Integral(Mul(Symbol('V', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('V', commutative=True), cos(Symbol('V', commutative=True))), sin(Symbol('V', commutative=True)), Integer(1)), Add(Integral(Mul(Symbol('V', commutative=True), sin(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\rho{(\\mu_0,\\dot{y})} = \\dot{y}^{\\mu_0} and \\phi{(\\dot{\\mathbf{r}},\\varphi^*)} = \\dot{\\mathbf{r}} - \\varphi^*, then obtain \\int (\\varphi^* + \\phi{(\\dot{\\mathbf{r}},\\varphi^*)}) \\rho{(\\mu_0,\\dot{y})} d\\dot{y} = \\int \\dot{y}^{\\mu_0} (\\varphi^* + \\phi{(\\dot{\\mathbf{r}},\\varphi^*)}) d\\dot{y}", "derivation": "\\rho{(\\mu_0,\\dot{y})} = \\dot{y}^{\\mu_0} and \\phi{(\\dot{\\mathbf{r}},\\varphi^*)} = \\dot{\\mathbf{r}} - \\varphi^* and \\dot{\\mathbf{r}} \\rho{(\\mu_0,\\dot{y})} = \\dot{\\mathbf{r}} \\dot{y}^{\\mu_0} and \\int \\dot{\\mathbf{r}} \\rho{(\\mu_0,\\dot{y})} d\\dot{y} = \\int \\dot{\\mathbf{r}} \\dot{y}^{\\mu_0} d\\dot{y} and \\varphi^* + \\phi{(\\dot{\\mathbf{r}},\\varphi^*)} = \\dot{\\mathbf{r}} and \\int (\\varphi^* + \\phi{(\\dot{\\mathbf{r}},\\varphi^*)}) \\rho{(\\mu_0,\\dot{y})} d\\dot{y} = \\int \\dot{y}^{\\mu_0} (\\varphi^* + \\phi{(\\dot{\\mathbf{r}},\\varphi^*)}) d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], ["get_premise", "Equality(Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))"], [["times", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\rho')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('\\\\rho')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Function('\\\\rho')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\phi')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given g{(E)} = e^{E} and \\operatorname{v_{y}}{(E)} = g{(E)} e^{E}, then obtain \\int g^{2}{(E)} dE = \\int e^{2 E} dE", "derivation": "g{(E)} = e^{E} and g{(E)} e^{E} = e^{2 E} and \\operatorname{v_{y}}{(E)} = g{(E)} e^{E} and \\operatorname{v_{y}}{(E)} = g^{2}{(E)} and E g{(E)} e^{E} = E e^{2 E} and g^{2}{(E)} = g{(E)} e^{E} and E g^{2}{(E)} = E e^{2 E} and g^{2}{(E)} = e^{2 E} and \\int g^{2}{(E)} dE = \\int e^{2 E} dE", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["times", 1, "exp(Symbol('E', commutative=True))"], "Equality(Mul(Function('g')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), exp(Mul(Integer(2), Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('E', commutative=True)), Mul(Function('g')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('v_y')(Symbol('E', commutative=True)), Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('g')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Mul(Symbol('E', commutative=True), exp(Mul(Integer(2), Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)), Mul(Function('g')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Symbol('E', commutative=True), Pow(Function('g')(Symbol('E', commutative=True)), Integer(2))), Mul(Symbol('E', commutative=True), exp(Mul(Integer(2), Symbol('E', commutative=True)))))"], [["divide", 7, "Symbol('E', commutative=True)"], "Equality(Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('E', commutative=True))))"], [["integrate", 8, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)), Tuple(Symbol('E', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given A{(\\mathbf{S})} = \\log{(\\mathbf{S})}, then obtain 2 A{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} - e^{C_{1}} = - e^{C_{1}} + \\log{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} + \\frac{A{(\\mathbf{S})}}{\\mathbf{S}}", "derivation": "A{(\\mathbf{S})} = \\log{(\\mathbf{S})} and A^{2}{(\\mathbf{S})} = A{(\\mathbf{S})} \\log{(\\mathbf{S})} and \\frac{d}{d \\mathbf{S}} A^{2}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} \\log{(\\mathbf{S})} and - e^{C_{1}} + \\frac{d}{d \\mathbf{S}} A^{2}{(\\mathbf{S})} = - e^{C_{1}} + \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} \\log{(\\mathbf{S})} and 2 A{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} - e^{C_{1}} = - e^{C_{1}} + \\log{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} A{(\\mathbf{S})} + \\frac{A{(\\mathbf{S})}}{\\mathbf{S}}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Function('A')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Pow(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Mul(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["minus", 3, "exp(Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('C_1', commutative=True))), Derivative(Pow(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('C_1', commutative=True))), Derivative(Mul(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Integer(-1), exp(Symbol('C_1', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('C_1', commutative=True))), Mul(log(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('A')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{p})} = \\sin{(e^{\\hat{p}})} and f{(\\hat{p})} = e^{\\hat{p}}, then obtain \\frac{d}{d \\hat{p}} \\operatorname{F_{x}}{(\\hat{p})} = \\cos{(f{(\\hat{p})})} \\frac{d}{d \\hat{p}} f{(\\hat{p})}", "derivation": "\\operatorname{F_{x}}{(\\hat{p})} = \\sin{(e^{\\hat{p}})} and f{(\\hat{p})} = e^{\\hat{p}} and \\frac{d}{d \\hat{p}} \\operatorname{F_{x}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\sin{(e^{\\hat{p}})} and \\frac{d}{d \\hat{p}} \\operatorname{F_{x}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\sin{(f{(\\hat{p})})} and \\frac{d}{d \\hat{p}} \\operatorname{F_{x}}{(\\hat{p})} = \\cos{(f{(\\hat{p})})} \\frac{d}{d \\hat{p}} f{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{p}', commutative=True)), sin(exp(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Derivative(Function('F_x')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(sin(Function('f')(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('F_x')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Mul(cos(Function('f')(Symbol('\\\\hat{p}', commutative=True))), Derivative(Function('f')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(\\varepsilon)} = \\varepsilon, then obtain \\frac{0^{\\varepsilon}}{g{(\\varepsilon)}} = \\frac{1}{g{(\\varepsilon)}}", "derivation": "g{(\\varepsilon)} = \\varepsilon and 0 = \\varepsilon - g{(\\varepsilon)} and 0 = - (\\varepsilon - g{(\\varepsilon)}) g{(\\varepsilon)} and 0^{\\varepsilon} = (- (\\varepsilon - g{(\\varepsilon)}) g{(\\varepsilon)})^{\\varepsilon} and \\frac{0^{\\varepsilon}}{g{(\\varepsilon)}} = \\frac{(- (\\varepsilon - g{(\\varepsilon)}) g{(\\varepsilon)})^{\\varepsilon}}{g{(\\varepsilon)}} and \\frac{(- (\\varepsilon - g{(\\varepsilon)}) g{(\\varepsilon)})^{\\varepsilon}}{g{(\\varepsilon)}} = \\frac{1}{g{(\\varepsilon)}} and \\frac{0^{\\varepsilon}}{g{(\\varepsilon)}} = \\frac{1}{g{(\\varepsilon)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["minus", 1, "Function('g')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))), Function('g')(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Integer(-1), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))), Function('g')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 4, "Function('g')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Mul(Pow(Mul(Integer(-1), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))), Function('g')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Mul(Integer(-1), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Function('g')(Symbol('\\\\varepsilon', commutative=True)))), Function('g')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Pow(Function('g')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\rho_f)} = \\rho_f, then derive H + \\frac{d}{d \\rho_f} \\operatorname{A_{y}}{(\\rho_f)} = H + 1, then obtain \\iint (H + \\frac{d}{d \\rho_f} \\rho_f) d\\rho_f dH = \\iint (H + 1) d\\rho_f dH", "derivation": "\\operatorname{A_{y}}{(\\rho_f)} = \\rho_f and \\frac{d}{d \\rho_f} \\operatorname{A_{y}}{(\\rho_f)} = \\frac{d}{d \\rho_f} \\rho_f and H + \\frac{d}{d \\rho_f} \\operatorname{A_{y}}{(\\rho_f)} = H + \\frac{d}{d \\rho_f} \\rho_f and H + \\frac{d}{d \\rho_f} \\operatorname{A_{y}}{(\\rho_f)} = H + 1 and H + \\frac{d}{d \\rho_f} \\rho_f = H + 1 and \\int (H + \\frac{d}{d \\rho_f} \\rho_f) d\\rho_f = \\int (H + 1) d\\rho_f and \\iint (H + \\frac{d}{d \\rho_f} \\rho_f) d\\rho_f dH = \\iint (H + 1) d\\rho_f dH", "srepr_derivation": [["renaming_premise", "Equality(Function('A_y')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["add", 2, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Derivative(Function('A_y')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Add(Symbol('H', commutative=True), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('H', commutative=True), Derivative(Function('A_y')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Add(Symbol('H', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('H', commutative=True), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Add(Symbol('H', commutative=True), Integer(1)))"], [["integrate", 5, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Symbol('H', commutative=True), Integer(1)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Integer(1)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(f)} = e^{f} and \\tilde{g}^*{(f)} = \\operatorname{y^{\\prime}}^{f}{(f)} and \\chi{(f)} = \\frac{\\operatorname{y^{\\prime}}^{f}{(f)} e^{- f}}{f}, then obtain 1 = \\frac{\\tilde{g}^*{(f)} e^{- f}}{f \\chi{(f)}}", "derivation": "\\operatorname{y^{\\prime}}{(f)} = e^{f} and \\operatorname{y^{\\prime}}^{f}{(f)} = (e^{f})^{f} and \\frac{\\operatorname{y^{\\prime}}^{f}{(f)} e^{- f}}{f} = \\frac{e^{- f} (e^{f})^{f}}{f} and \\tilde{g}^*{(f)} = \\operatorname{y^{\\prime}}^{f}{(f)} and \\tilde{g}^*{(f)} = (e^{f})^{f} and \\chi{(f)} = \\frac{\\operatorname{y^{\\prime}}^{f}{(f)} e^{- f}}{f} and \\frac{\\operatorname{y^{\\prime}}^{f}{(f)} e^{- f}}{f} = \\frac{\\tilde{g}^*{(f)} e^{- f}}{f} and \\chi{(f)} = \\frac{\\tilde{g}^*{(f)} e^{- f}}{f} and 1 = \\frac{\\tilde{g}^*{(f)} e^{- f}}{f \\chi{(f)}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True)))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(exp(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["divide", 2, "Mul(Symbol('f', commutative=True), exp(Symbol('f', commutative=True)))"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('f', commutative=True))), Pow(exp(Symbol('f', commutative=True)), Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), Pow(exp(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('f', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('y^{\\\\prime}')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Function('\\\\chi')(Symbol('f', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))))"], [["divide", 8, "Function('\\\\chi')(Symbol('f', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('f', commutative=True)), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('f', commutative=True)), exp(Mul(Integer(-1), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\hat{x}_0,\\mathbf{f},v_{1})} = (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0} and E{(\\mathbf{f},v_{1})} = \\frac{\\mathbf{f}}{v_{1}}, then obtain (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0} + E{(\\mathbf{f},v_{1})} = \\frac{\\mathbf{f}}{v_{1}} + (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0}", "derivation": "\\Psi_{\\lambda}{(\\hat{x}_0,\\mathbf{f},v_{1})} = (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0} and E{(\\mathbf{f},v_{1})} = \\frac{\\mathbf{f}}{v_{1}} and \\Psi_{\\lambda}{(\\hat{x}_0,\\mathbf{f},v_{1})} = E^{\\hat{x}_0}{(\\mathbf{f},v_{1})} and E{(\\mathbf{f},v_{1})} + E^{\\hat{x}_0}{(\\mathbf{f},v_{1})} = \\frac{\\mathbf{f}}{v_{1}} + E^{\\hat{x}_0}{(\\mathbf{f},v_{1})} and (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0} = E^{\\hat{x}_0}{(\\mathbf{f},v_{1})} and (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0} + E{(\\mathbf{f},v_{1})} = \\frac{\\mathbf{f}}{v_{1}} + (\\frac{\\mathbf{f}}{v_{1}})^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 2, "Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Function('E')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_1', commutative=True))), Add(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('v_1', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(E_{n},\\eta^{\\prime})} = E_{n} \\eta^{\\prime} and \\mu_{0}{(E_{n},\\eta^{\\prime})} = E_{n} \\eta^{\\prime}, then obtain (\\operatorname{f_{E}}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})})^{\\eta^{\\prime}} = ((E_{n} \\eta^{\\prime})^{\\eta^{\\prime}})^{\\eta^{\\prime}}", "derivation": "\\operatorname{f_{E}}{(E_{n},\\eta^{\\prime})} = E_{n} \\eta^{\\prime} and \\mu_{0}{(E_{n},\\eta^{\\prime})} = E_{n} \\eta^{\\prime} and \\operatorname{f_{E}}{(E_{n},\\eta^{\\prime})} = \\mu_{0}{(E_{n},\\eta^{\\prime})} and \\operatorname{f_{E}}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})} = \\mu_{0}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})} and (\\operatorname{f_{E}}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})})^{\\eta^{\\prime}} = (\\mu_{0}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})})^{\\eta^{\\prime}} and (\\operatorname{f_{E}}^{\\eta^{\\prime}}{(E_{n},\\eta^{\\prime})})^{\\eta^{\\prime}} = ((E_{n} \\eta^{\\prime})^{\\eta^{\\prime}})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_E')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\mu_0')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Pow(Function('f_E')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Pow(Function('\\\\mu_0')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Function('f_E')(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Pow(Mul(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\rho_{b}{(H)} = e^{\\sin{(H)}}, then derive (\\frac{H \\frac{d}{d H} \\rho_{b}{(H)}}{\\rho_{b}{(H)}} + \\log{(\\rho_{b}{(H)})}) \\rho_{b}^{H}{(H)} = (H \\cos{(H)} + \\log{(e^{\\sin{(H)}})}) (e^{\\sin{(H)}})^{H}, then obtain (\\frac{H \\frac{d}{d H} \\rho_{b}{(H)}}{\\rho_{b}{(H)}} + \\log{(\\rho_{b}{(H)})}) \\rho_{b}^{H}{(H)} + \\cos{(H)} = (H \\cos{(H)} + \\log{(e^{\\sin{(H)}})}) (e^{\\sin{(H)}})^{H} + \\cos{(H)}", "derivation": "\\rho_{b}{(H)} = e^{\\sin{(H)}} and \\rho_{b}^{H}{(H)} = (e^{\\sin{(H)}})^{H} and \\frac{d}{d H} \\rho_{b}^{H}{(H)} = \\frac{d}{d H} (e^{\\sin{(H)}})^{H} and (\\frac{H \\frac{d}{d H} \\rho_{b}{(H)}}{\\rho_{b}{(H)}} + \\log{(\\rho_{b}{(H)})}) \\rho_{b}^{H}{(H)} = (H \\cos{(H)} + \\log{(e^{\\sin{(H)}})}) (e^{\\sin{(H)}})^{H} and (\\frac{H \\frac{d}{d H} \\rho_{b}{(H)}}{\\rho_{b}{(H)}} + \\log{(\\rho_{b}{(H)})}) \\rho_{b}^{H}{(H)} + \\cos{(H)} = (H \\cos{(H)} + \\log{(e^{\\sin{(H)}})}) (e^{\\sin{(H)}})^{H} + \\cos{(H)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('H', commutative=True)), exp(sin(Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(exp(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(exp(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('H', commutative=True), Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Integer(-1)), Derivative(Function('\\\\rho_b')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), log(Function('\\\\rho_b')(Symbol('H', commutative=True)))), Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Mul(Add(Mul(Symbol('H', commutative=True), cos(Symbol('H', commutative=True))), log(exp(sin(Symbol('H', commutative=True))))), Pow(exp(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True))))"], [["add", 4, "cos(Symbol('H', commutative=True))"], "Equality(Add(Mul(Add(Mul(Symbol('H', commutative=True), Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Integer(-1)), Derivative(Function('\\\\rho_b')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), log(Function('\\\\rho_b')(Symbol('H', commutative=True)))), Pow(Function('\\\\rho_b')(Symbol('H', commutative=True)), Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))), Add(Mul(Add(Mul(Symbol('H', commutative=True), cos(Symbol('H', commutative=True))), log(exp(sin(Symbol('H', commutative=True))))), Pow(exp(sin(Symbol('H', commutative=True))), Symbol('H', commutative=True))), cos(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(F_{N})} = \\sin{(F_{N})}, then obtain \\frac{d}{d F_{N}} 0 - \\int (\\hat{H} - \\cos{(F_{N})}) d\\hat{H} = \\frac{d}{d F_{N}} (- \\operatorname{C_{d}}{(F_{N})} + \\sin{(F_{N})})^{2} \\operatorname{C_{d}}{(F_{N})} - \\int (\\hat{H} - \\cos{(F_{N})}) d\\hat{H}", "derivation": "\\operatorname{C_{d}}{(F_{N})} = \\sin{(F_{N})} and 0 = - \\operatorname{C_{d}}{(F_{N})} + \\sin{(F_{N})} and 0 = (- \\operatorname{C_{d}}{(F_{N})} + \\sin{(F_{N})})^{2} \\operatorname{C_{d}}{(F_{N})} and \\frac{d}{d F_{N}} 0 = \\frac{d}{d F_{N}} (- \\operatorname{C_{d}}{(F_{N})} + \\sin{(F_{N})})^{2} \\operatorname{C_{d}}{(F_{N})} and \\frac{d}{d F_{N}} 0 - \\int (\\hat{H} - \\cos{(F_{N})}) d\\hat{H} = \\frac{d}{d F_{N}} (- \\operatorname{C_{d}}{(F_{N})} + \\sin{(F_{N})})^{2} \\operatorname{C_{d}}{(F_{N})} - \\int (\\hat{H} - \\cos{(F_{N})}) d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["minus", 1, "Function('C_d')(Symbol('F_N', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))))"], [["times", 2, "Mul(Add(Mul(Integer(-1), Function('C_d')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))), Function('C_d')(Symbol('F_N', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))), Integer(2)), Function('C_d')(Symbol('F_N', commutative=True))))"], [["differentiate", 3, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))), Integer(2)), Function('C_d')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["minus", 4, "Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))), Add(Derivative(Mul(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('F_N', commutative=True))), sin(Symbol('F_N', commutative=True))), Integer(2)), Function('C_d')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(W)} = \\log{(W)}, then obtain - \\operatorname{z^{*}}{(W)} + \\iint \\operatorname{z^{*}}^{W}{(W)} dW dW - \\iint \\log{(W)}^{W} dW dW = - \\operatorname{z^{*}}{(W)}", "derivation": "\\operatorname{z^{*}}{(W)} = \\log{(W)} and \\operatorname{z^{*}}^{W}{(W)} = \\log{(W)}^{W} and \\int \\operatorname{z^{*}}^{W}{(W)} dW = \\int \\log{(W)}^{W} dW and \\iint \\operatorname{z^{*}}^{W}{(W)} dW dW = \\iint \\log{(W)}^{W} dW dW and \\iint \\operatorname{z^{*}}^{W}{(W)} dW dW - \\iint \\log{(W)}^{W} dW dW = 0 and - \\operatorname{z^{*}}{(W)} + \\iint \\operatorname{z^{*}}^{W}{(W)} dW dW - \\iint \\log{(W)}^{W} dW dW = - \\operatorname{z^{*}}{(W)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('z^*')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('z^*')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["minus", 4, "Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Integral(Pow(Function('z^*')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Integer(0))"], [["minus", 5, "Function('z^*')(Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('z^*')(Symbol('W', commutative=True))), Integral(Pow(Function('z^*')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Mul(Integer(-1), Function('z^*')(Symbol('W', commutative=True))))"]]}, {"prompt": "Given f{(E_{x})} = \\cos{(E_{x})}, then obtain 1 = (0^{E_{x}})^{E_{x}}", "derivation": "f{(E_{x})} = \\cos{(E_{x})} and f{(E_{x})} - \\cos{(E_{x})} = 0 and (f{(E_{x})} - \\cos{(E_{x})})^{E_{x}} = 0^{E_{x}} and ((f{(E_{x})} - \\cos{(E_{x})})^{E_{x}})^{E_{x}} = (0^{E_{x}})^{E_{x}} and 1 = ((f{(E_{x})} - \\cos{(E_{x})})^{E_{x}})^{E_{x}} and 1 = (0^{E_{x}})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["minus", 1, "cos(Symbol('E_x', commutative=True))"], "Equality(Add(Function('f')(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Add(Function('f')(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Pow(Integer(0), Symbol('E_x', commutative=True)))"], [["power", 3, "Symbol('E_x', commutative=True)"], "Equality(Pow(Pow(Add(Function('f')(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Pow(Integer(0), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Function('f')(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given t{(A_{z})} = \\sin{(A_{z})}, then obtain ((t^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}} = ((\\sin^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}}", "derivation": "t{(A_{z})} = \\sin{(A_{z})} and t^{A_{z}}{(A_{z})} = \\sin^{A_{z}}{(A_{z})} and (t^{A_{z}}{(A_{z})})^{A_{z}} = (\\sin^{A_{z}}{(A_{z})})^{A_{z}} and ((t^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}} = ((\\sin^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('t')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Function('t')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(sin(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Pow(Function('t')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(Pow(sin(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(\\chi)} = \\chi, then obtain \\chi^{\\chi} (\\log{(\\chi)} + 1) \\varepsilon{(\\chi)} + \\chi^{\\chi} \\frac{d}{d \\chi} \\varepsilon{(\\chi)} = \\chi \\chi^{\\chi} (\\log{(\\chi)} + 1) + \\chi^{\\chi}", "derivation": "\\varepsilon{(\\chi)} = \\chi and \\varepsilon^{\\chi}{(\\chi)} = \\chi^{\\chi} and \\varepsilon{(\\chi)} \\varepsilon^{\\chi}{(\\chi)} = \\chi \\varepsilon^{\\chi}{(\\chi)} and \\chi^{\\chi} \\varepsilon{(\\chi)} = \\chi \\chi^{\\chi} and \\frac{d}{d \\chi} \\chi^{\\chi} \\varepsilon{(\\chi)} = \\frac{d}{d \\chi} \\chi \\chi^{\\chi} and \\chi^{\\chi} (\\log{(\\chi)} + 1) \\varepsilon{(\\chi)} + \\chi^{\\chi} \\frac{d}{d \\chi} \\varepsilon{(\\chi)} = \\chi \\chi^{\\chi} (\\log{(\\chi)} + 1) + \\chi^{\\chi}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Pow(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(log(Symbol('\\\\chi', commutative=True)), Integer(1)), Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(log(Symbol('\\\\chi', commutative=True)), Integer(1))), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(E_{x})} = \\cos{(\\cos{(E_{x})})} and \\phi_{2}{(E_{x})} = \\int \\operatorname{y^{\\prime}}{(E_{x})} dE_{x}, then obtain - E_{x} + \\iint \\phi_{2}{(E_{x})} dE_{x} dE_{x} = - E_{x} + \\iiint \\cos{(\\cos{(E_{x})})} dE_{x} dE_{x} dE_{x}", "derivation": "\\operatorname{y^{\\prime}}{(E_{x})} = \\cos{(\\cos{(E_{x})})} and \\int \\operatorname{y^{\\prime}}{(E_{x})} dE_{x} = \\int \\cos{(\\cos{(E_{x})})} dE_{x} and \\phi_{2}{(E_{x})} = \\int \\operatorname{y^{\\prime}}{(E_{x})} dE_{x} and \\phi_{2}{(E_{x})} = \\int \\cos{(\\cos{(E_{x})})} dE_{x} and \\int \\phi_{2}{(E_{x})} dE_{x} = \\iint \\cos{(\\cos{(E_{x})})} dE_{x} dE_{x} and \\iint \\phi_{2}{(E_{x})} dE_{x} dE_{x} = \\iiint \\cos{(\\cos{(E_{x})})} dE_{x} dE_{x} dE_{x} and - E_{x} + \\iint \\phi_{2}{(E_{x})} dE_{x} dE_{x} = - E_{x} + \\iiint \\cos{(\\cos{(E_{x})})} dE_{x} dE_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('E_x', commutative=True)), cos(cos(Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(cos(cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('E_x', commutative=True)), Integral(Function('y^{\\\\prime}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\phi_2')(Symbol('E_x', commutative=True)), Integral(cos(cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 4, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(cos(cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 5, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(cos(cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["minus", 6, "Symbol('E_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Integral(Function('\\\\phi_2')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Integral(cos(cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given n{(\\chi)} = \\cos{(e^{\\chi})}, then obtain \\int (n{(\\chi)} + \\cos{(e^{\\chi})}) e^{- \\chi} d\\chi = \\int 2 e^{- \\chi} \\cos{(e^{\\chi})} d\\chi", "derivation": "n{(\\chi)} = \\cos{(e^{\\chi})} and n{(\\chi)} + \\cos{(e^{\\chi})} = 2 \\cos{(e^{\\chi})} and (n{(\\chi)} + \\cos{(e^{\\chi})}) e^{- \\chi} = 2 e^{- \\chi} \\cos{(e^{\\chi})} and \\int (n{(\\chi)} + \\cos{(e^{\\chi})}) e^{- \\chi} d\\chi = \\int 2 e^{- \\chi} \\cos{(e^{\\chi})} d\\chi", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True))))"], [["add", 1, "cos(exp(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Function('n')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('\\\\chi', commutative=True)))))"], [["divide", 2, "exp(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Add(Function('n')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Mul(Integer(2), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), cos(exp(Symbol('\\\\chi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Add(Function('n')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Integer(2), exp(Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), cos(exp(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\omega,\\mathbf{r})} = \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega), then derive \\operatorname{g_{\\varepsilon}}{(\\omega,\\mathbf{r})} = 1, then obtain 2 \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) = \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) + 1", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\omega,\\mathbf{r})} = \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) and \\operatorname{g_{\\varepsilon}}{(\\omega,\\mathbf{r})} = 1 and \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) = 1 and 2 \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) = \\frac{\\partial}{\\partial \\omega} (\\mathbf{r} + \\omega) + 1", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1))"], [["add", 3, "Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} = \\Psi_{nl} + \\phi + a, then obtain \\frac{\\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} \\int (\\Psi_{nl} + \\phi + a) d\\phi}{\\int \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} d\\phi} = \\Psi_{nl} + \\phi + a", "derivation": "\\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} = \\Psi_{nl} + \\phi + a and \\int \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} d\\phi = \\int (\\Psi_{nl} + \\phi + a) d\\phi and 1 = \\frac{\\int (\\Psi_{nl} + \\phi + a) d\\phi}{\\int \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} d\\phi} and \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} = \\frac{\\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} \\int (\\Psi_{nl} + \\phi + a) d\\phi}{\\int \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} d\\phi} and \\frac{\\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} \\int (\\Psi_{nl} + \\phi + a) d\\phi}{\\int \\operatorname{t_{2}}{(a,\\phi,\\Psi_{nl})} d\\phi} = \\Psi_{nl} + \\phi + a", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["divide", 2, "Integral(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))))"], [["times", 3, "Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Mul(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Function('t_2')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi', commutative=True), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\Psi_{nl})} = e^{\\sin{(\\Psi_{nl})}}, then obtain 2 (\\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl}) \\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} = (\\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl})^{2}", "derivation": "\\operatorname{v_{z}}{(\\Psi_{nl})} = e^{\\sin{(\\Psi_{nl})}} and \\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} = \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl} and 2 \\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} = \\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl} and 2 (\\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl}) \\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} = (\\int \\operatorname{v_{z}}{(\\Psi_{nl})} d\\Psi_{nl} + \\int e^{\\sin{(\\Psi_{nl})}} d\\Psi_{nl})^{2}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 2, "Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["times", 3, "Add(Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], "Equality(Mul(Integer(2), Add(Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Pow(Add(Integral(Function('v_z')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given n{(r_{0})} = \\sin{(r_{0})}, then obtain 1 - r_{0} = - r_{0} + (n{(r_{0})} - \\sin{(r_{0})})^{r_{0}}", "derivation": "n{(r_{0})} = \\sin{(r_{0})} and n{(r_{0})} - \\sin{(r_{0})} = 0 and (n{(r_{0})} - \\sin{(r_{0})})^{r_{0}} = 0^{r_{0}} and - r_{0} + (n{(r_{0})} - \\sin{(r_{0})})^{r_{0}} = 0^{r_{0}} - r_{0} and 1 - r_{0} = - r_{0} + (n{(r_{0})} - \\sin{(r_{0})})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], [["minus", 1, "sin(Symbol('r_0', commutative=True))"], "Equality(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)), Pow(Integer(0), Symbol('r_0', commutative=True)))"], [["minus", 3, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True))), Add(Pow(Integer(0), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Pow(Add(Function('n')(Symbol('r_0', commutative=True)), Mul(Integer(-1), sin(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(i)} = \\sin{(i)}, then obtain (i + \\sin{(i)} + 1) \\sin{(i)} = (i + \\sin{(i)} + \\frac{i + \\sin{(i)}}{i + \\operatorname{z^{*}}{(i)}}) \\sin{(i)}", "derivation": "\\operatorname{z^{*}}{(i)} = \\sin{(i)} and i + \\operatorname{z^{*}}{(i)} = i + \\sin{(i)} and 1 = \\frac{i + \\sin{(i)}}{i + \\operatorname{z^{*}}{(i)}} and i + \\sin{(i)} + 1 = i + \\sin{(i)} + \\frac{i + \\sin{(i)}}{i + \\operatorname{z^{*}}{(i)}} and (i + \\sin{(i)} + 1) \\sin{(i)} = (i + \\sin{(i)} + \\frac{i + \\sin{(i)}}{i + \\operatorname{z^{*}}{(i)}}) \\sin{(i)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["add", 1, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Function('z^*')(Symbol('i', commutative=True))), Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], [["divide", 2, "Add(Symbol('i', commutative=True), Function('z^*')(Symbol('i', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('i', commutative=True), Function('z^*')(Symbol('i', commutative=True))), Integer(-1)), Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))))"], [["add", 3, "Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))"], "Equality(Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)), Integer(1)), Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)), Mul(Pow(Add(Symbol('i', commutative=True), Function('z^*')(Symbol('i', commutative=True))), Integer(-1)), Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))))"], [["times", 4, "sin(Symbol('i', commutative=True))"], "Equality(Mul(Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)), Integer(1)), sin(Symbol('i', commutative=True))), Mul(Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)), Mul(Pow(Add(Symbol('i', commutative=True), Function('z^*')(Symbol('i', commutative=True))), Integer(-1)), Add(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))), sin(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(J)} = e^{J}, then obtain ((e^{J} - 1)^{J} + \\operatorname{M_{E}}{(J)} - 1)^{J} = ((e^{J} - 1)^{J} + e^{J} - 1)^{J}", "derivation": "\\operatorname{M_{E}}{(J)} = e^{J} and \\operatorname{M_{E}}{(J)} - 1 = e^{J} - 1 and (\\operatorname{M_{E}}{(J)} - 1)^{J} = (e^{J} - 1)^{J} and (\\operatorname{M_{E}}{(J)} - 1)^{J} + \\operatorname{M_{E}}{(J)} - 1 = (\\operatorname{M_{E}}{(J)} - 1)^{J} + e^{J} - 1 and ((\\operatorname{M_{E}}{(J)} - 1)^{J} + \\operatorname{M_{E}}{(J)} - 1)^{J} = ((\\operatorname{M_{E}}{(J)} - 1)^{J} + e^{J} - 1)^{J} and ((e^{J} - 1)^{J} + \\operatorname{M_{E}}{(J)} - 1)^{J} = ((e^{J} - 1)^{J} + e^{J} - 1)^{J}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Add(exp(Symbol('J', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(exp(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)))"], [["add", 2, "Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True))"], "Equality(Add(Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Add(Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)), Integer(-1)))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(Pow(Add(Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Pow(Add(exp(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Function('M_E')(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), Pow(Add(Pow(Add(exp(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)), Integer(-1)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given u{(\\dot{y})} = \\cos{(\\dot{y})} and \\mathbf{H}{(\\mathbf{P},A,q)} = (\\mathbf{P} + q)^{A}, then obtain \\int (\\mathbf{H}{(\\mathbf{P},A,q)} - u{(\\dot{y})}) dq = \\int ((\\mathbf{P} + q)^{A} - u{(\\dot{y})}) dq", "derivation": "u{(\\dot{y})} = \\cos{(\\dot{y})} and \\mathbf{H}{(\\mathbf{P},A,q)} = (\\mathbf{P} + q)^{A} and \\mathbf{H}{(\\mathbf{P},A,q)} - \\cos{(\\dot{y})} = (\\mathbf{P} + q)^{A} - \\cos{(\\dot{y})} and \\mathbf{H}{(\\mathbf{P},A,q)} - u{(\\dot{y})} = (\\mathbf{P} + q)^{A} - u{(\\dot{y})} and \\int (\\mathbf{H}{(\\mathbf{P},A,q)} - u{(\\dot{y})}) dq = \\int ((\\mathbf{P} + q)^{A} - u{(\\dot{y})}) dq", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A', commutative=True), Symbol('q', commutative=True)), Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('q', commutative=True)), Symbol('A', commutative=True)))"], [["minus", 2, "cos(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))), Add(Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('q', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\dot{y}', commutative=True)))), Add(Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('q', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('A', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(Add(Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('q', commutative=True)), Symbol('A', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(z)} = \\sin{(z)}, then derive \\int \\phi_{1}{(z)} dz = C_{d} - \\cos{(z)}, then obtain C_{d} - \\cos{(z)} = \\int \\sin{(z)} dz", "derivation": "\\phi_{1}{(z)} = \\sin{(z)} and \\int \\phi_{1}{(z)} dz = \\int \\sin{(z)} dz and \\int \\phi_{1}{(z)} dz = C_{d} - \\cos{(z)} and C_{d} - \\cos{(z)} = \\int \\sin{(z)} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(f^{\\prime})} = \\log{(f^{\\prime})}, then derive \\frac{\\frac{d}{d f^{\\prime}} \\mathbf{J}_f{(f^{\\prime})}}{\\log{(f^{\\prime})}} - \\frac{\\mathbf{J}_f{(f^{\\prime})}}{f^{\\prime} \\log{(f^{\\prime})}^{2}} = 0, then obtain (\\frac{\\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})}}{\\log{(f^{\\prime})}} - \\frac{1}{f^{\\prime} \\log{(f^{\\prime})}})^{f^{\\prime}} = 0^{f^{\\prime}}", "derivation": "\\mathbf{J}_f{(f^{\\prime})} = \\log{(f^{\\prime})} and \\frac{\\mathbf{J}_f{(f^{\\prime})}}{\\log{(f^{\\prime})}} = 1 and \\frac{d}{d f^{\\prime}} \\frac{\\mathbf{J}_f{(f^{\\prime})}}{\\log{(f^{\\prime})}} = \\frac{d}{d f^{\\prime}} 1 and \\frac{\\frac{d}{d f^{\\prime}} \\mathbf{J}_f{(f^{\\prime})}}{\\log{(f^{\\prime})}} - \\frac{\\mathbf{J}_f{(f^{\\prime})}}{f^{\\prime} \\log{(f^{\\prime})}^{2}} = 0 and \\frac{\\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})}}{\\log{(f^{\\prime})}} - \\frac{1}{f^{\\prime} \\log{(f^{\\prime})}} = 0 and (\\frac{\\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})}}{\\log{(f^{\\prime})}} - \\frac{1}{f^{\\prime} \\log{(f^{\\prime})}})^{f^{\\prime}} = 0^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "log(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-2)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)))), Integer(0))"], [["power", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Mul(Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given E{(c,\\hbar)} = \\cos{(\\frac{\\hbar}{c})}, then obtain \\frac{\\partial}{\\partial \\hbar} \\frac{c E{(c,\\hbar)}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})} = \\frac{\\partial}{\\partial \\hbar} \\frac{c \\cos{(\\frac{\\hbar}{c})}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})}", "derivation": "E{(c,\\hbar)} = \\cos{(\\frac{\\hbar}{c})} and c E{(c,\\hbar)} = c \\cos{(\\frac{\\hbar}{c})} and \\frac{c E{(c,\\hbar)}}{\\hbar} = \\frac{c \\cos{(\\frac{\\hbar}{c})}}{\\hbar} and \\frac{c E{(c,\\hbar)}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})} = \\frac{c \\cos{(\\frac{\\hbar}{c})}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})} and \\frac{\\partial}{\\partial \\hbar} \\frac{c E{(c,\\hbar)}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})} = \\frac{\\partial}{\\partial \\hbar} \\frac{c \\cos{(\\frac{\\hbar}{c})}}{\\hbar (c \\cos{(\\frac{\\hbar}{c})} - \\frac{1}{c})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1)))))"], [["divide", 1, "Pow(Symbol('c', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('c', commutative=True), Function('E')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))))"], [["divide", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), Function('E')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))))"], [["divide", 3, "Add(Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1))))"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), Pow(Add(Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Integer(-1)), Function('E')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), Pow(Add(Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Integer(-1)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))))"], [["differentiate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), Pow(Add(Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Integer(-1)), Function('E')(Symbol('c', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('c', commutative=True), Pow(Add(Mul(Symbol('c', commutative=True), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Integer(-1)), cos(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})} = \\hat{H}_l \\varepsilon + v_{z}, then obtain \\sin{(\\frac{\\partial}{\\partial \\hat{H}_l} \\frac{\\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})}}{\\hat{H}_l \\varepsilon + v_{z}})} = \\sin{(\\frac{d}{d \\hat{H}_l} 1)}", "derivation": "\\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})} = \\hat{H}_l \\varepsilon + v_{z} and \\frac{\\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})}}{\\hat{H}_l \\varepsilon + v_{z}} = 1 and \\frac{\\partial}{\\partial \\hat{H}_l} \\frac{\\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})}}{\\hat{H}_l \\varepsilon + v_{z}} = \\frac{d}{d \\hat{H}_l} 1 and \\sin{(\\frac{\\partial}{\\partial \\hat{H}_l} \\frac{\\hat{\\mathbf{r}}{(\\varepsilon,\\hat{H}_l,v_{z})}}{\\hat{H}_l \\varepsilon + v_{z}})} = \\sin{(\\frac{d}{d \\hat{H}_l} 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('v_z', commutative=True)))"], [["divide", 1, "Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Mul(Pow(Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), sin(Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\dot{x})} = \\log{(e^{\\dot{x}})} and L{(\\dot{x})} = e^{\\dot{x}} and \\operatorname{C_{d}}{(\\dot{x})} = e^{\\dot{x}}, then obtain \\operatorname{C_{d}}{(\\dot{x})} - L{(\\dot{x})} \\log{(L{(\\dot{x})})} = - L{(\\dot{x})} \\log{(L{(\\dot{x})})} + e^{\\dot{x}}", "derivation": "\\operatorname{f^{\\prime}}{(\\dot{x})} = \\log{(e^{\\dot{x}})} and L{(\\dot{x})} = e^{\\dot{x}} and \\operatorname{f^{\\prime}}{(\\dot{x})} = \\log{(L{(\\dot{x})})} and L{(\\dot{x})} \\operatorname{f^{\\prime}}{(\\dot{x})} = L{(\\dot{x})} \\log{(L{(\\dot{x})})} and L{(\\dot{x})} \\log{(e^{\\dot{x}})} = L{(\\dot{x})} \\log{(L{(\\dot{x})})} and \\operatorname{C_{d}}{(\\dot{x})} = e^{\\dot{x}} and \\operatorname{C_{d}}{(\\dot{x})} - L{(\\dot{x})} \\log{(e^{\\dot{x}})} = - L{(\\dot{x})} \\log{(e^{\\dot{x}})} + e^{\\dot{x}} and \\operatorname{C_{d}}{(\\dot{x})} - L{(\\dot{x})} \\log{(L{(\\dot{x})})} = - L{(\\dot{x})} \\log{(L{(\\dot{x})})} + e^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True)), log(Function('L')(Symbol('\\\\dot{x}', commutative=True))))"], [["times", 3, "Function('L')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('L')(Symbol('\\\\dot{x}', commutative=True)), Function('f^{\\\\prime}')(Symbol('\\\\dot{x}', commutative=True))), Mul(Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(Function('L')(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))), Mul(Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(Function('L')(Symbol('\\\\dot{x}', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 6, "Mul(Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True))))"], "Equality(Add(Function('C_d')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True))))), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(exp(Symbol('\\\\dot{x}', commutative=True)))), exp(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Function('C_d')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(Function('L')(Symbol('\\\\dot{x}', commutative=True))))), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\dot{x}', commutative=True)), log(Function('L')(Symbol('\\\\dot{x}', commutative=True)))), exp(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(f_{E})} = e^{f_{E}}, then obtain \\frac{d}{d f_{E}} 0 + \\frac{\\operatorname{P_{e}}{(f_{E})}}{f_{E}} = \\frac{d}{d f_{E}} (- \\operatorname{P_{e}}{(f_{E})} + e^{f_{E}}) + \\frac{\\operatorname{P_{e}}{(f_{E})}}{f_{E}}", "derivation": "\\operatorname{P_{e}}{(f_{E})} = e^{f_{E}} and 0 = - \\operatorname{P_{e}}{(f_{E})} + e^{f_{E}} and \\frac{d}{d f_{E}} 0 = \\frac{d}{d f_{E}} (- \\operatorname{P_{e}}{(f_{E})} + e^{f_{E}}) and \\frac{\\operatorname{P_{e}}{(f_{E})}}{f_{E}} = \\frac{e^{f_{E}}}{f_{E}} and \\frac{d}{d f_{E}} 0 + \\frac{e^{f_{E}}}{f_{E}} = \\frac{d}{d f_{E}} (- \\operatorname{P_{e}}{(f_{E})} + e^{f_{E}}) + \\frac{e^{f_{E}}}{f_{E}} and \\frac{d}{d f_{E}} 0 + \\frac{\\operatorname{P_{e}}{(f_{E})}}{f_{E}} = \\frac{d}{d f_{E}} (- \\operatorname{P_{e}}{(f_{E})} + e^{f_{E}}) + \\frac{\\operatorname{P_{e}}{(f_{E})}}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], [["minus", 1, "Function('P_e')(Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('P_e')(Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), exp(Symbol('f_E', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), exp(Symbol('f_E', commutative=True)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), exp(Symbol('f_E', commutative=True)))), Add(Derivative(Add(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), exp(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('P_e')(Symbol('f_E', commutative=True)))), Add(Derivative(Add(Mul(Integer(-1), Function('P_e')(Symbol('f_E', commutative=True))), exp(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('P_e')(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(f,s)} = \\frac{\\partial}{\\partial s} f s, then obtain \\iint (\\mathbf{f}{(f,s)} + \\frac{\\partial}{\\partial s} f s + 1) df ds = \\iint (2 \\frac{\\partial}{\\partial s} f s + 1) df ds", "derivation": "\\mathbf{f}{(f,s)} = \\frac{\\partial}{\\partial s} f s and \\mathbf{f}{(f,s)} + \\frac{\\partial}{\\partial s} f s = 2 \\frac{\\partial}{\\partial s} f s and \\mathbf{f}{(f,s)} + \\frac{\\partial}{\\partial s} f s + 1 = 2 \\frac{\\partial}{\\partial s} f s + 1 and \\int (\\mathbf{f}{(f,s)} + \\frac{\\partial}{\\partial s} f s + 1) df = \\int (2 \\frac{\\partial}{\\partial s} f s + 1) df and \\iint (\\mathbf{f}{(f,s)} + \\frac{\\partial}{\\partial s} f s + 1) df ds = \\iint (2 \\frac{\\partial}{\\partial s} f s + 1) df ds", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(2), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(1)))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(2), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('f', commutative=True))))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{f}')(Symbol('f', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(2), Derivative(Mul(Symbol('f', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Integer(1)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = \\hat{p}_0 q - \\phi, then obtain - \\hat{p}_0 q + \\phi + (\\hat{p}_0 q - \\phi) \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} + \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = (\\hat{p}_0 q - \\phi) \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)}", "derivation": "\\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = \\hat{p}_0 q - \\phi and \\dot{\\mathbf{r}}^{2}{(\\phi,\\hat{p}_0,q)} = (\\hat{p}_0 q - \\phi) \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} and - \\hat{p}_0 q + \\phi + \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = 0 and - \\hat{p}_0 q + \\phi + \\dot{\\mathbf{r}}^{2}{(\\phi,\\hat{p}_0,q)} + \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = \\dot{\\mathbf{r}}^{2}{(\\phi,\\hat{p}_0,q)} and - \\hat{p}_0 q + \\phi + (\\hat{p}_0 q - \\phi) \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} + \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)} = (\\hat{p}_0 q - \\phi) \\dot{\\mathbf{r}}{(\\phi,\\hat{p}_0,q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["times", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Integer(2)), Mul(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))))"], [["minus", 1, "Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\phi', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))), Integer(0))"], [["add", 3, "Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\phi', commutative=True), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Integer(2)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))), Mul(Add(Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{D},F_{x})} = - F_{x} + \\sin{(\\mathbf{D})}, then obtain - \\frac{\\sigma_{p}^{4}{(\\mathbf{D},F_{x})}}{(- F_{x} + \\sin{(\\mathbf{D})})^{2}} = - \\sigma_{p}^{2}{(\\mathbf{D},F_{x})}", "derivation": "\\sigma_{p}{(\\mathbf{D},F_{x})} = - F_{x} + \\sin{(\\mathbf{D})} and \\sigma_{p}^{2}{(\\mathbf{D},F_{x})} = (- F_{x} + \\sin{(\\mathbf{D})}) \\sigma_{p}{(\\mathbf{D},F_{x})} and - \\sigma_{p}^{2}{(\\mathbf{D},F_{x})} = - (- F_{x} + \\sin{(\\mathbf{D})}) \\sigma_{p}{(\\mathbf{D},F_{x})} and - \\frac{\\sigma_{p}^{2}{(\\mathbf{D},F_{x})}}{- F_{x} + \\sin{(\\mathbf{D})}} = - \\sigma_{p}{(\\mathbf{D},F_{x})} and - \\frac{\\sigma_{p}^{4}{(\\mathbf{D},F_{x})}}{(- F_{x} + \\sin{(\\mathbf{D})})^{2}} = - \\sigma_{p}^{2}{(\\mathbf{D},F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 1, "Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Integer(2))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)), Pow(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-2)), Pow(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Integer(4))), Mul(Integer(-1), Pow(Function('\\\\sigma_p')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('F_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\omega{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})}, then obtain \\int (\\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1)^{2} d\\Psi^{\\dagger} = V_{\\mathbf{E}} + \\Psi^{\\dagger}", "derivation": "\\omega{(\\Psi^{\\dagger})} = \\sin{(\\Psi^{\\dagger})} and \\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} = 0 and \\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1 = 1 and (\\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1)^{2} = \\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1 and (\\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1)^{2} = 1 and \\int (\\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1)^{2} d\\Psi^{\\dagger} = \\int 1 d\\Psi^{\\dagger} and \\int (\\omega{(\\Psi^{\\dagger})} - \\sin{(\\Psi^{\\dagger})} + 1)^{2} d\\Psi^{\\dagger} = V_{\\mathbf{E}} + \\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Integer(0))"], [["add", 2, 1], "Equality(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Integer(1))"], [["times", 3, "Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1))"], "Equality(Pow(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Integer(2)), Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Integer(2)), Integer(1))"], [["integrate", 5, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Integer(2)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Pow(Add(Function('\\\\omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Integer(2)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given C{(\\chi,\\dot{z})} = \\dot{z}^{\\chi}, then obtain - \\frac{\\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})}}{\\chi} = \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\dot{z}^{\\chi} - 2 \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})}}{\\chi}", "derivation": "C{(\\chi,\\dot{z})} = \\dot{z}^{\\chi} and \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\dot{z}^{\\chi} and 0 = \\frac{\\partial}{\\partial \\dot{z}} \\dot{z}^{\\chi} - \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})} and - \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})} = \\frac{\\partial}{\\partial \\dot{z}} \\dot{z}^{\\chi} - 2 \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})} and - \\frac{\\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})}}{\\chi} = \\frac{\\frac{\\partial}{\\partial \\dot{z}} \\dot{z}^{\\chi} - 2 \\frac{\\partial}{\\partial \\dot{z}} C{(\\chi,\\dot{z})}}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))))"], [["minus", 3, "Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(Derivative(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))))"], [["divide", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Derivative(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Derivative(Function('C')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(g,\\mathbf{D})} = - \\mathbf{D} + g and \\sigma_{x}{(g,\\mathbf{D})} = - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}}, then obtain - \\sigma_{x}{(g,\\mathbf{D})} - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}} = \\frac{(- \\mathbf{D} + g)^{g}}{\\mathbf{D}} - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}}", "derivation": "\\dot{\\mathbf{r}}{(g,\\mathbf{D})} = - \\mathbf{D} + g and \\sigma_{x}{(g,\\mathbf{D})} = - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}} and \\sigma_{x}{(g,\\mathbf{D})} = - \\frac{(- \\mathbf{D} + g)^{g}}{\\mathbf{D}} and - \\sigma_{x}{(g,\\mathbf{D})} = \\frac{(- \\mathbf{D} + g)^{g}}{\\mathbf{D}} and - \\sigma_{x}{(g,\\mathbf{D})} - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}} = \\frac{(- \\mathbf{D} + g)^{g}}{\\mathbf{D}} - \\frac{\\dot{\\mathbf{r}}^{g}{(g,\\mathbf{D})}}{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\sigma_x')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["minus", 4, "Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(\\Psi)} = \\log{(\\Psi)}, then obtain 1 = - \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} + 1 + \\frac{1}{\\Psi}", "derivation": "\\sigma_{p}{(\\Psi)} = \\log{(\\Psi)} and 0 = - \\sigma_{p}{(\\Psi)} + \\log{(\\Psi)} and \\Psi = \\Psi - \\sigma_{p}{(\\Psi)} + \\log{(\\Psi)} and \\frac{d}{d \\Psi} \\Psi = \\frac{d}{d \\Psi} (\\Psi - \\sigma_{p}{(\\Psi)} + \\log{(\\Psi)}) and 1 = - \\frac{d}{d \\Psi} \\sigma_{p}{(\\Psi)} + 1 + \\frac{1}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Symbol('\\\\Psi', commutative=True), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Add(Mul(Integer(-1), Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Integer(1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given g{(E)} = \\sin{(E)}, then obtain 1 = e^{\\frac{d}{d E} (g{(E)} + \\sin{(E)}) - \\frac{d}{d E} 2 g{(E)}}", "derivation": "g{(E)} = \\sin{(E)} and 2 g{(E)} = g{(E)} + \\sin{(E)} and \\frac{d}{d E} 2 g{(E)} = \\frac{d}{d E} (g{(E)} + \\sin{(E)}) and 0 = \\frac{d}{d E} (g{(E)} + \\sin{(E)}) - \\frac{d}{d E} 2 g{(E)} and 1 = e^{\\frac{d}{d E} (g{(E)} + \\sin{(E)}) - \\frac{d}{d E} 2 g{(E)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["add", 1, "Function('g')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(2), Function('g')(Symbol('E', commutative=True))), Add(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('g')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Mul(Integer(2), Function('g')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Add(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Integer(2), Function('g')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["exp", 4], "Equality(Integer(1), exp(Add(Derivative(Add(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Integer(2), Function('g')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\eta)} = \\log{(\\eta)}, then obtain 3 \\operatorname{J_{\\varepsilon}}{(\\eta)} = \\operatorname{J_{\\varepsilon}}{(\\eta)} + 2 \\log{(\\eta)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\eta)} = \\log{(\\eta)} and 2 \\operatorname{J_{\\varepsilon}}{(\\eta)} = \\operatorname{J_{\\varepsilon}}{(\\eta)} + \\log{(\\eta)} and 3 \\operatorname{J_{\\varepsilon}}{(\\eta)} = 2 \\operatorname{J_{\\varepsilon}}{(\\eta)} + \\log{(\\eta)} and 3 \\operatorname{J_{\\varepsilon}}{(\\eta)} = \\operatorname{J_{\\varepsilon}}{(\\eta)} + 2 \\log{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], [["add", 1, "Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(3), Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(2), Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), log(Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True))), Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(2), log(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given s{(\\hat{x})} = \\log{(\\hat{x})} and \\lambda{(\\hat{x})} = \\log{(\\hat{x})}, then obtain g_{\\varepsilon} - z^{*} + \\int s{(\\hat{x})} d\\hat{x} = g_{\\varepsilon} - z^{*} + \\int \\log{(\\hat{x})} d\\hat{x}", "derivation": "s{(\\hat{x})} = \\log{(\\hat{x})} and \\lambda{(\\hat{x})} = \\log{(\\hat{x})} and \\int \\lambda{(\\hat{x})} d\\hat{x} = \\int \\log{(\\hat{x})} d\\hat{x} and \\lambda{(\\hat{x})} = s{(\\hat{x})} and \\int s{(\\hat{x})} d\\hat{x} = \\int \\log{(\\hat{x})} d\\hat{x} and g_{\\varepsilon} - z^{*} + \\int s{(\\hat{x})} d\\hat{x} = g_{\\varepsilon} - z^{*} + \\int \\log{(\\hat{x})} d\\hat{x}", "srepr_derivation": [["get_premise", "Equality(Function('s')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\hat{x}', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\lambda')(Symbol('\\\\hat{x}', commutative=True)), Function('s')(Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["add", 5, "Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(Function('s')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)), Integral(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(W,z)} = \\cos{(W z)} and \\hat{H}{(W,z)} = \\cos^{W}{(W z)}, then obtain (\\frac{\\partial^{2}}{\\partial z\\partial W} \\cos^{W}{(W z)})^{z} = (\\frac{\\partial^{2}}{\\partial z\\partial W} \\tilde{g}^{W}{(W,z)})^{z}", "derivation": "\\tilde{g}{(W,z)} = \\cos{(W z)} and \\tilde{g}^{W}{(W,z)} = \\cos^{W}{(W z)} and \\hat{H}{(W,z)} = \\cos^{W}{(W z)} and \\frac{\\partial}{\\partial W} \\hat{H}{(W,z)} = \\frac{\\partial}{\\partial W} \\cos^{W}{(W z)} and \\frac{\\partial}{\\partial W} \\hat{H}{(W,z)} = \\frac{\\partial}{\\partial W} \\tilde{g}^{W}{(W,z)} and \\frac{\\partial^{2}}{\\partial z\\partial W} \\hat{H}{(W,z)} = \\frac{\\partial^{2}}{\\partial z\\partial W} \\tilde{g}^{W}{(W,z)} and \\frac{\\partial^{2}}{\\partial z\\partial W} \\cos^{W}{(W z)} = \\frac{\\partial^{2}}{\\partial z\\partial W} \\tilde{g}^{W}{(W,z)} and (\\frac{\\partial^{2}}{\\partial z\\partial W} \\cos^{W}{(W z)})^{z} = (\\frac{\\partial^{2}}{\\partial z\\partial W} \\tilde{g}^{W}{(W,z)})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Symbol('W', commutative=True)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))), Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Pow(cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))), Symbol('W', commutative=True)))"], [["differentiate", 3, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Pow(cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 7, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Pow(cos(Mul(Symbol('W', commutative=True), Symbol('z', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Pow(Function('\\\\tilde{g}')(Symbol('W', commutative=True), Symbol('z', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} = g_{\\varepsilon}^{\\Psi}, then obtain - g_{\\varepsilon} + \\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} + e^{\\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)}} = - g_{\\varepsilon} + \\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} + e^{g_{\\varepsilon}^{\\Psi}}", "derivation": "\\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} = g_{\\varepsilon}^{\\Psi} and e^{\\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)}} = e^{g_{\\varepsilon}^{\\Psi}} and - g_{\\varepsilon} + e^{\\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)}} = - g_{\\varepsilon} + e^{g_{\\varepsilon}^{\\Psi}} and - g_{\\varepsilon} + \\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} + e^{\\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)}} = - g_{\\varepsilon} + \\operatorname{E_{n}}{(g_{\\varepsilon},\\Psi)} + e^{g_{\\varepsilon}^{\\Psi}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["exp", 1], "Equality(exp(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True))), exp(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["minus", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), exp(Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('E_n')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)), exp(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{\\mathbf{r}},r)} = \\cos^{\\hat{\\mathbf{r}}}{(r)}, then obtain \\phi_{1}{(\\hat{\\mathbf{r}},r)} \\int \\cos^{\\hat{\\mathbf{r}}}{(r)} dr = \\cos^{\\hat{\\mathbf{r}}}{(r)} \\int \\cos^{\\hat{\\mathbf{r}}}{(r)} dr", "derivation": "\\phi_{1}{(\\hat{\\mathbf{r}},r)} = \\cos^{\\hat{\\mathbf{r}}}{(r)} and \\int \\phi_{1}{(\\hat{\\mathbf{r}},r)} dr = \\int \\cos^{\\hat{\\mathbf{r}}}{(r)} dr and \\phi_{1}{(\\hat{\\mathbf{r}},r)} \\int \\phi_{1}{(\\hat{\\mathbf{r}},r)} dr = \\cos^{\\hat{\\mathbf{r}}}{(r)} \\int \\phi_{1}{(\\hat{\\mathbf{r}},r)} dr and \\phi_{1}{(\\hat{\\mathbf{r}},r)} \\int \\cos^{\\hat{\\mathbf{r}}}{(r)} dr = \\cos^{\\hat{\\mathbf{r}}}{(r)} \\int \\cos^{\\hat{\\mathbf{r}}}{(r)} dr", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["times", 1, "Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('r', commutative=True)), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(Pow(cos(Symbol('r', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(z,u)} = \\int (u - z) du and \\mathbf{H}{(v_{x},q)} = q + v_{x}, then obtain \\mathbf{H}{(v_{x},q)} + \\operatorname{t_{1}}{(z,u)} + \\frac{- z - \\operatorname{t_{1}}{(z,u)}}{z} = q + v_{x} + \\operatorname{t_{1}}{(z,u)} + \\frac{- z - \\operatorname{t_{1}}{(z,u)}}{z}", "derivation": "\\operatorname{t_{1}}{(z,u)} = \\int (u - z) du and z + \\operatorname{t_{1}}{(z,u)} = z + \\int (u - z) du and \\frac{z + \\operatorname{t_{1}}{(z,u)}}{z} = \\frac{z + \\int (u - z) du}{z} and \\mathbf{H}{(v_{x},q)} = q + v_{x} and \\mathbf{H}{(v_{x},q)} + \\int (u - z) du - \\frac{z + \\int (u - z) du}{z} = q + v_{x} + \\int (u - z) du - \\frac{z + \\int (u - z) du}{z} and \\mathbf{H}{(v_{x},q)} + \\int (u - z) du + \\frac{- z - \\int (u - z) du}{z} = q + v_{x} + \\int (u - z) du + \\frac{- z - \\int (u - z) du}{z} and \\mathbf{H}{(v_{x},q)} + \\operatorname{t_{1}}{(z,u)} + \\frac{- z - \\operatorname{t_{1}}{(z,u)}}{z} = q + v_{x} + \\operatorname{t_{1}}{(z,u)} + \\frac{- z - \\operatorname{t_{1}}{(z,u)}}{z}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('z', commutative=True))"], "Equality(Add(Symbol('z', commutative=True), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True))), Add(Symbol('z', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))))))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('v_x', commutative=True), Symbol('q', commutative=True)), Add(Symbol('q', commutative=True), Symbol('v_x', commutative=True)))"], [["minus", 4, "Add(Mul(Integer(-1), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))))))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('v_x', commutative=True), Symbol('q', commutative=True)), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True)))))), Add(Symbol('q', commutative=True), Symbol('v_x', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('v_x', commutative=True), Symbol('q', commutative=True)), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))))))), Add(Symbol('q', commutative=True), Symbol('v_x', commutative=True), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('u', commutative=True))))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('v_x', commutative=True), Symbol('q', commutative=True)), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)))))), Add(Symbol('q', commutative=True), Symbol('v_x', commutative=True), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('z', commutative=True), Symbol('u', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{D}{(T)} = \\log{(T)}, then obtain \\mathbf{D}^{3 T}{(T)} = \\mathbf{D}^{T}{(T)} \\log{(T)}^{2 T}", "derivation": "\\mathbf{D}{(T)} = \\log{(T)} and \\mathbf{D}^{T}{(T)} = \\log{(T)}^{T} and \\mathbf{D}^{2 T}{(T)} = \\mathbf{D}^{T}{(T)} \\log{(T)}^{T} and \\mathbf{D}^{3 T}{(T)} = \\mathbf{D}^{2 T}{(T)} \\log{(T)}^{T} and \\mathbf{D}^{3 T}{(T)} = \\mathbf{D}^{T}{(T)} \\log{(T)}^{2 T}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["times", 2, "Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True)))"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Mul(Integer(3), Symbol('T', commutative=True))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Mul(Integer(3), Symbol('T', commutative=True))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Mul(Integer(2), Symbol('T', commutative=True)))))"]]}, {"prompt": "Given s{(\\lambda,\\theta_1,f_{\\mathbf{v}})} = \\frac{\\lambda}{\\theta_1 f_{\\mathbf{v}}}, then derive \\frac{\\partial}{\\partial \\lambda} s{(\\lambda,\\theta_1,f_{\\mathbf{v}})} = \\frac{1}{\\theta_1 f_{\\mathbf{v}}}, then obtain \\frac{1}{\\theta_1 f_{\\mathbf{v}}} = \\frac{\\partial}{\\partial \\lambda} \\frac{\\lambda}{\\theta_1 f_{\\mathbf{v}}}", "derivation": "s{(\\lambda,\\theta_1,f_{\\mathbf{v}})} = \\frac{\\lambda}{\\theta_1 f_{\\mathbf{v}}} and \\frac{\\partial}{\\partial \\lambda} s{(\\lambda,\\theta_1,f_{\\mathbf{v}})} = \\frac{\\partial}{\\partial \\lambda} \\frac{\\lambda}{\\theta_1 f_{\\mathbf{v}}} and \\frac{\\partial}{\\partial \\lambda} s{(\\lambda,\\theta_1,f_{\\mathbf{v}})} = \\frac{1}{\\theta_1 f_{\\mathbf{v}}} and \\frac{1}{\\theta_1 f_{\\mathbf{v}}} = \\frac{\\partial}{\\partial \\lambda} \\frac{\\lambda}{\\theta_1 f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Derivative(Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(t_{1},\\mu_0)} = - \\mu_0 + \\cos{(t_{1})}, then derive \\frac{\\partial}{\\partial t_{1}} b{(t_{1},\\mu_0)} = - \\sin{(t_{1})}, then obtain \\frac{\\partial^{2}}{\\partial \\mu_0\\partial t_{1}} (- \\mu_0 + \\cos{(t_{1})}) = \\frac{d}{d \\mu_0} - \\sin{(t_{1})}", "derivation": "b{(t_{1},\\mu_0)} = - \\mu_0 + \\cos{(t_{1})} and \\frac{\\partial}{\\partial t_{1}} b{(t_{1},\\mu_0)} = \\frac{\\partial}{\\partial t_{1}} (- \\mu_0 + \\cos{(t_{1})}) and \\frac{\\partial}{\\partial t_{1}} b{(t_{1},\\mu_0)} = - \\sin{(t_{1})} and \\frac{\\partial}{\\partial t_{1}} (- \\mu_0 + \\cos{(t_{1})}) = - \\sin{(t_{1})} and \\frac{\\partial^{2}}{\\partial \\mu_0\\partial t_{1}} (- \\mu_0 + \\cos{(t_{1})}) = \\frac{d}{d \\mu_0} - \\sin{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('t_1', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(t_{1})} = \\cos{(e^{t_{1}})} and \\operatorname{F_{c}}{(t_{1})} = t_{1}, then obtain - t_{1} + \\frac{d}{d t_{1}} (\\theta_{2}{(t_{1})} - \\cos{(e^{t_{1}})}) = - t_{1} + \\frac{d}{d t_{1}} 0", "derivation": "\\theta_{2}{(t_{1})} = \\cos{(e^{t_{1}})} and \\theta_{2}{(t_{1})} - \\cos{(e^{t_{1}})} = 0 and \\operatorname{F_{c}}{(t_{1})} = t_{1} and \\frac{d}{d t_{1}} (\\theta_{2}{(t_{1})} - \\cos{(e^{t_{1}})}) = \\frac{d}{d t_{1}} 0 and - \\operatorname{F_{c}}{(t_{1})} + \\frac{d}{d t_{1}} (\\theta_{2}{(t_{1})} - \\cos{(e^{t_{1}})}) = - \\operatorname{F_{c}}{(t_{1})} + \\frac{d}{d t_{1}} 0 and - t_{1} + \\frac{d}{d t_{1}} (\\theta_{2}{(t_{1})} - \\cos{(e^{t_{1}})}) = - t_{1} + \\frac{d}{d t_{1}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('t_1', commutative=True)), cos(exp(Symbol('t_1', commutative=True))))"], [["minus", 1, "cos(exp(Symbol('t_1', commutative=True)))"], "Equality(Add(Function('\\\\theta_2')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('t_1', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta_2')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('t_1', commutative=True))))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["minus", 4, "Function('F_c')(Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('F_c')(Symbol('t_1', commutative=True))), Derivative(Add(Function('\\\\theta_2')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('t_1', commutative=True))))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('F_c')(Symbol('t_1', commutative=True))), Derivative(Integer(0), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(Add(Function('\\\\theta_2')(Symbol('t_1', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('t_1', commutative=True))))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,v)} = \\phi^{v}, then obtain (\\phi^{v})^{\\phi} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,v)} = \\phi^{v} (\\phi^{v})^{\\phi}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,v)} = \\phi^{v} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\phi}{(\\phi,v)} = (\\phi^{v})^{\\phi} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,v)} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\phi}{(\\phi,v)} = \\phi^{v} \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\phi}{(\\phi,v)} and (\\phi^{v})^{\\phi} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi,v)} = \\phi^{v} (\\phi^{v})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Pow(Pow(Symbol('\\\\phi', commutative=True), Symbol('v', commutative=True)), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given I{(\\lambda,G)} = G - \\lambda, then obtain (G + I{(\\lambda,G)}) (G - \\lambda + I{(\\lambda,G)}) = (G + I{(\\lambda,G)}) (2 G - 2 \\lambda)", "derivation": "I{(\\lambda,G)} = G - \\lambda and G - \\lambda + I{(\\lambda,G)} = 2 G - 2 \\lambda and G + I{(\\lambda,G)} = 2 G - \\lambda and (2 G - \\lambda) (G - \\lambda + I{(\\lambda,G)}) = (2 G - 2 \\lambda) (2 G - \\lambda) and (G + I{(\\lambda,G)}) (G - \\lambda + I{(\\lambda,G)}) = (G + I{(\\lambda,G)}) (2 G - 2 \\lambda)", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True)))), Mul(Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('G', commutative=True), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True)))), Mul(Add(Symbol('G', commutative=True), Function('I')(Symbol('\\\\lambda', commutative=True), Symbol('G', commutative=True))), Add(Mul(Integer(2), Symbol('G', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\nabla,M_{E})} = \\nabla \\cos{(M_{E})} and \\dot{z}{(B)} = \\sin{(B)}, then obtain \\nabla \\dot{z}{(B)} \\cos{(M_{E})} - \\frac{1}{\\nabla \\cos{(M_{E})}} = \\nabla \\sin{(B)} \\cos{(M_{E})} - \\frac{1}{\\nabla \\cos{(M_{E})}}", "derivation": "\\mathbf{v}{(\\nabla,M_{E})} = \\nabla \\cos{(M_{E})} and \\dot{z}{(B)} = \\sin{(B)} and \\nabla \\dot{z}{(B)} \\cos{(M_{E})} = \\nabla \\sin{(B)} \\cos{(M_{E})} and \\nabla \\dot{z}{(B)} \\cos{(M_{E})} - \\frac{1}{\\mathbf{v}{(\\nabla,M_{E})}} = \\nabla \\sin{(B)} \\cos{(M_{E})} - \\frac{1}{\\mathbf{v}{(\\nabla,M_{E})}} and \\nabla \\dot{z}{(B)} \\cos{(M_{E})} - \\frac{1}{\\nabla \\cos{(M_{E})}} = \\nabla \\sin{(B)} \\cos{(M_{E})} - \\frac{1}{\\nabla \\cos{(M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('\\\\nabla', commutative=True), cos(Symbol('M_E', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["times", 2, "Mul(Symbol('\\\\nabla', commutative=True), cos(Symbol('M_E', commutative=True)))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\dot{z}')(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(cos(Symbol('M_E', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('B', commutative=True)), cos(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(cos(Symbol('M_E', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given M{(\\delta,A)} = \\frac{\\partial}{\\partial A} A \\delta, then derive 0 = \\delta - M{(\\delta,A)}, then derive \\delta - M{(\\delta,A)} = 2 \\delta - 2 M{(\\delta,A)}, then obtain 1 = \\cos{(2 (2 \\delta - 2 M{(\\delta,A)}) M{(\\delta,A)})}", "derivation": "M{(\\delta,A)} = \\frac{\\partial}{\\partial A} A \\delta and 0 = - M{(\\delta,A)} + \\frac{\\partial}{\\partial A} A \\delta and 0 = \\delta - M{(\\delta,A)} and - M{(\\delta,A)} + \\frac{\\partial}{\\partial A} A \\delta = \\delta - 2 M{(\\delta,A)} + \\frac{\\partial}{\\partial A} A \\delta and \\delta - M{(\\delta,A)} = 2 \\delta - 2 M{(\\delta,A)} and 0 = \\delta - \\frac{\\partial}{\\partial A} A \\delta and 0 = - 2 (\\delta - \\frac{\\partial}{\\partial A} A \\delta) M{(\\delta,A)} and 0 = - 2 (\\delta - M{(\\delta,A)}) M{(\\delta,A)} and 1 = \\cos{(2 (\\delta - M{(\\delta,A)}) M{(\\delta,A)})} and 1 = \\cos{(2 (2 \\delta - 2 M{(\\delta,A)}) M{(\\delta,A)})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["minus", 1, "Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Integer(0), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Integer(2), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(0), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"], [["times", 6, "Mul(Integer(-1), Integer(2), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Integer(2), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('A', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Integer(0), Mul(Integer(-1), Integer(2), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True))))"], [["cos", 8], "Equality(Integer(1), cos(Mul(Integer(2), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 9, 5], "Equality(Integer(1), cos(Mul(Integer(2), Add(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Integer(2), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))), Function('M')(Symbol('\\\\delta', commutative=True), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\psi)} = \\cos{(\\cos{(\\psi)})} and \\operatorname{x^{{\\}'}}{(\\psi)} = \\int (\\int \\cos{(\\cos{(\\psi)})} d\\psi)^{2} d\\psi, then obtain \\int (\\int \\hat{p}_0{(\\psi)} d\\psi) \\int \\cos{(\\cos{(\\psi)})} d\\psi d\\psi = \\operatorname{x^{{\\}'}}{(\\psi)}", "derivation": "\\hat{p}_0{(\\psi)} = \\cos{(\\cos{(\\psi)})} and \\int \\hat{p}_0{(\\psi)} d\\psi = \\int \\cos{(\\cos{(\\psi)})} d\\psi and (\\int \\hat{p}_0{(\\psi)} d\\psi) \\int \\cos{(\\cos{(\\psi)})} d\\psi = (\\int \\cos{(\\cos{(\\psi)})} d\\psi)^{2} and \\int (\\int \\hat{p}_0{(\\psi)} d\\psi) \\int \\cos{(\\cos{(\\psi)})} d\\psi d\\psi = \\int (\\int \\cos{(\\cos{(\\psi)})} d\\psi)^{2} d\\psi and \\operatorname{x^{{\\}'}}{(\\psi)} = \\int (\\int \\cos{(\\cos{(\\psi)})} d\\psi)^{2} d\\psi and \\int (\\int \\hat{p}_0{(\\psi)} d\\psi) \\int \\cos{(\\cos{(\\psi)})} d\\psi d\\psi = \\operatorname{x^{{\\}'}}{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), cos(cos(Symbol('\\\\psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Pow(Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Mul(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True)), Integral(Pow(Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Mul(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given G{(P_{e})} = \\log{(P_{e})}, then obtain \\frac{(G{(P_{e})} + \\log{(P_{e})})^{2}}{P_{e}} = \\frac{2 (G{(P_{e})} + \\log{(P_{e})}) \\log{(P_{e})}}{P_{e}}", "derivation": "G{(P_{e})} = \\log{(P_{e})} and G{(P_{e})} + \\log{(P_{e})} = 2 \\log{(P_{e})} and (G{(P_{e})} + \\log{(P_{e})})^{2} = 2 (G{(P_{e})} + \\log{(P_{e})}) \\log{(P_{e})} and \\frac{(G{(P_{e})} + \\log{(P_{e})})^{2}}{P_{e}} = \\frac{2 (G{(P_{e})} + \\log{(P_{e})}) \\log{(P_{e})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["add", 1, "log(Symbol('P_e', commutative=True))"], "Equality(Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Mul(Integer(2), log(Symbol('P_e', commutative=True))))"], [["times", 2, "Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], "Equality(Pow(Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Integer(2)), Mul(Integer(2), Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), log(Symbol('P_e', commutative=True))))"], [["divide", 3, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Integer(2))), Mul(Integer(2), Pow(Symbol('P_e', commutative=True), Integer(-1)), Add(Function('G')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), log(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})}, then obtain (- g_{\\varepsilon} + \\operatorname{F_{H}}{(g_{\\varepsilon})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} - (- g_{\\varepsilon} + \\log{(e^{g_{\\varepsilon}})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} = 0", "derivation": "\\operatorname{F_{H}}{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})} and - g_{\\varepsilon} + \\operatorname{F_{H}}{(g_{\\varepsilon})} = - g_{\\varepsilon} + \\log{(e^{g_{\\varepsilon}})} and (- g_{\\varepsilon} + \\operatorname{F_{H}}{(g_{\\varepsilon})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} = (- g_{\\varepsilon} + \\log{(e^{g_{\\varepsilon}})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} and (- g_{\\varepsilon} + \\operatorname{F_{H}}{(g_{\\varepsilon})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} - (- g_{\\varepsilon} + \\log{(e^{g_{\\varepsilon}})}) \\operatorname{F_{H}}{(g_{\\varepsilon})} = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["times", 2, "Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))), Function('F_H')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given B{(\\mathbf{p},\\sigma_p)} = \\log{(\\frac{\\sigma_p}{\\mathbf{p}})}, then derive \\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p = \\dot{\\mathbf{r}} + \\sigma_p \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} - \\sigma_p, then obtain \\varphi{(\\mathbf{p})} + (\\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p)^{\\dot{\\mathbf{r}}} = (\\dot{\\mathbf{r}} + \\sigma_p \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} - \\sigma_p)^{\\dot{\\mathbf{r}}} + \\varphi{(\\mathbf{p})}", "derivation": "B{(\\mathbf{p},\\sigma_p)} = \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} and \\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p = \\int \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} d\\sigma_p and \\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p = \\dot{\\mathbf{r}} + \\sigma_p \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} - \\sigma_p and (\\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p)^{\\dot{\\mathbf{r}}} = (\\dot{\\mathbf{r}} + \\sigma_p \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} - \\sigma_p)^{\\dot{\\mathbf{r}}} and \\varphi{(\\mathbf{p})} + (\\int B{(\\mathbf{p},\\sigma_p)} d\\sigma_p)^{\\dot{\\mathbf{r}}} = (\\dot{\\mathbf{r}} + \\sigma_p \\log{(\\frac{\\sigma_p}{\\mathbf{p}})} - \\sigma_p)^{\\dot{\\mathbf{r}}} + \\varphi{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), log(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('\\\\sigma_p', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Integral(Function('B')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('\\\\sigma_p', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["add", 4, "Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(Integral(Function('B')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('\\\\sigma_p', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)} = (\\tilde{g}^*)^{C} - u, then obtain (\\tilde{g}^*)^{- C} \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)} = (\\tilde{g}^*)^{- C} ((\\tilde{g}^*)^{C} - u)", "derivation": "\\operatorname{n_{2}}{(u,C,\\tilde{g}^*)} = (\\tilde{g}^*)^{C} - u and u + \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)} = (\\tilde{g}^*)^{C} and \\frac{\\operatorname{n_{2}}{(u,C,\\tilde{g}^*)}}{u + \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)}} = \\frac{(\\tilde{g}^*)^{C} - u}{u + \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)}} and (\\tilde{g}^*)^{- C} \\operatorname{n_{2}}{(u,C,\\tilde{g}^*)} = (\\tilde{g}^*)^{- C} ((\\tilde{g}^*)^{C} - u)", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Add(Symbol('u', commutative=True), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('C', commutative=True)))"], [["divide", 1, "Add(Symbol('u', commutative=True), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('u', commutative=True), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Add(Symbol('u', commutative=True), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('C', commutative=True))), Function('n_2')(Symbol('u', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Symbol('C', commutative=True))), Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(U,x)} = \\int U x dx, then derive \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU = \\frac{U x^{2}}{2}, then obtain \\int (\\mathbf{M} + \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU) dU = \\int (\\frac{U x^{2}}{2} + \\mathbf{M}) dU", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(U,x)} = \\int U x dx and \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU = \\iint U x dx dU and \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU = \\frac{\\partial}{\\partial U} \\iint U x dx dU and \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU = \\frac{U x^{2}}{2} and \\frac{\\partial}{\\partial U} \\iint U x dx dU = \\frac{U x^{2}}{2} and \\mathbf{M} + \\frac{\\partial}{\\partial U} \\iint U x dx dU = \\frac{U x^{2}}{2} + \\mathbf{M} and \\mathbf{M} + \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU = \\frac{U x^{2}}{2} + \\mathbf{M} and \\int (\\mathbf{M} + \\frac{\\partial}{\\partial U} \\int \\operatorname{V_{\\mathbf{B}}}{(U,x)} dU) dU = \\int (\\frac{U x^{2}}{2} + \\mathbf{M}) dU", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Integral(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))))"], [["add", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Derivative(Integral(Mul(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 7, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('U', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True))), Integral(Add(Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(U)} = \\cos{(U)}, then obtain \\cos{(U)} = \\frac{\\cos^{3}{(U)}}{\\operatorname{m_{s}}^{2}{(U)}}", "derivation": "\\operatorname{m_{s}}{(U)} = \\cos{(U)} and \\operatorname{m_{s}}{(U)} \\cos{(U)} = \\cos^{2}{(U)} and \\cos{(U)} = \\frac{\\cos^{2}{(U)}}{\\operatorname{m_{s}}{(U)}} and \\cos^{2}{(U)} = \\frac{\\cos^{3}{(U)}}{\\operatorname{m_{s}}{(U)}} and \\cos{(U)} = \\frac{\\cos^{3}{(U)}}{\\operatorname{m_{s}}^{2}{(U)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["times", 1, "cos(Symbol('U', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Pow(cos(Symbol('U', commutative=True)), Integer(2)))"], [["divide", 2, "Function('m_s')(Symbol('U', commutative=True))"], "Equality(cos(Symbol('U', commutative=True)), Mul(Pow(Function('m_s')(Symbol('U', commutative=True)), Integer(-1)), Pow(cos(Symbol('U', commutative=True)), Integer(2))))"], [["times", 1, "Mul(Pow(Function('m_s')(Symbol('U', commutative=True)), Integer(-1)), Pow(cos(Symbol('U', commutative=True)), Integer(2)))"], "Equality(Pow(cos(Symbol('U', commutative=True)), Integer(2)), Mul(Pow(Function('m_s')(Symbol('U', commutative=True)), Integer(-1)), Pow(cos(Symbol('U', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(cos(Symbol('U', commutative=True)), Mul(Pow(Function('m_s')(Symbol('U', commutative=True)), Integer(-2)), Pow(cos(Symbol('U', commutative=True)), Integer(3))))"]]}, {"prompt": "Given G{(r)} = \\cos{(\\log{(r)})} and \\Omega{(r)} = \\log{(r)}, then obtain r G{(r)} + G{(r)} = r \\cos{(\\Omega{(r)})} + G{(r)}", "derivation": "G{(r)} = \\cos{(\\log{(r)})} and r G{(r)} = r \\cos{(\\log{(r)})} and r G{(r)} + G{(r)} = r \\cos{(\\log{(r)})} + G{(r)} and \\Omega{(r)} = \\log{(r)} and r G{(r)} + G{(r)} = r \\cos{(\\Omega{(r)})} + G{(r)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('r', commutative=True)), cos(log(Symbol('r', commutative=True))))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('G')(Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), cos(log(Symbol('r', commutative=True)))))"], [["add", 2, "Function('G')(Symbol('r', commutative=True))"], "Equality(Add(Mul(Symbol('r', commutative=True), Function('G')(Symbol('r', commutative=True))), Function('G')(Symbol('r', commutative=True))), Add(Mul(Symbol('r', commutative=True), cos(log(Symbol('r', commutative=True)))), Function('G')(Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('r', commutative=True), Function('G')(Symbol('r', commutative=True))), Function('G')(Symbol('r', commutative=True))), Add(Mul(Symbol('r', commutative=True), cos(Function('\\\\Omega')(Symbol('r', commutative=True)))), Function('G')(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{M})} = e^{\\mathbf{M}}, then obtain (\\mathbf{p}{(\\mathbf{M})} e^{- \\mathbf{M}})^{\\mathbf{M}} - \\frac{d}{d \\mathbf{M}} (e^{2 \\mathbf{M}})^{\\mathbf{M}} = 1 - \\frac{d}{d \\mathbf{M}} (e^{2 \\mathbf{M}})^{\\mathbf{M}}", "derivation": "\\mathbf{p}{(\\mathbf{M})} = e^{\\mathbf{M}} and \\mathbf{p}{(\\mathbf{M})} e^{\\mathbf{M}} = e^{2 \\mathbf{M}} and \\mathbf{p}{(\\mathbf{M})} e^{- \\mathbf{M}} = 1 and (\\mathbf{p}{(\\mathbf{M})} e^{- \\mathbf{M}})^{\\mathbf{M}} = 1 and (\\mathbf{p}{(\\mathbf{M})} e^{- \\mathbf{M}})^{\\mathbf{M}} - \\frac{d}{d \\mathbf{M}} (e^{2 \\mathbf{M}})^{\\mathbf{M}} = 1 - \\frac{d}{d \\mathbf{M}} (e^{2 \\mathbf{M}})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True))))"], [["divide", 2, "exp(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Integer(1))"], [["minus", 4, "Derivative(Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))"], "Equality(Add(Pow(Mul(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Derivative(Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), Derivative(Pow(exp(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\mathbf{r})} = \\cos{(e^{\\mathbf{r}})}, then obtain 2 \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} + 2 \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})} = \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} + 3 \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})}", "derivation": "\\operatorname{E_{n}}{(\\mathbf{r})} = \\cos{(e^{\\mathbf{r}})} and \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} = \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})} and \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} + \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})} = 2 \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})} and 2 \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} + 2 \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})} = \\operatorname{E_{n}}^{\\mathbf{r}}{(\\mathbf{r})} + 3 \\cos^{\\mathbf{r}}{(e^{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), cos(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 2, "Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Pow(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(2), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 3, "Add(Pow(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(2), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))), Add(Pow(Function('E_n')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(3), Pow(cos(exp(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(I)} = \\sin{(I)}, then derive (\\frac{d}{d I} \\operatorname{f^{*}}{(I)})^{I} = \\cos^{I}{(I)}, then obtain \\int \\cos{(\\cos^{I}{(I)})} dI = \\int \\cos{((\\frac{d}{d I} \\sin{(I)})^{I})} dI", "derivation": "\\operatorname{f^{*}}{(I)} = \\sin{(I)} and \\frac{d}{d I} \\operatorname{f^{*}}{(I)} = \\frac{d}{d I} \\sin{(I)} and (\\frac{d}{d I} \\operatorname{f^{*}}{(I)})^{I} = (\\frac{d}{d I} \\sin{(I)})^{I} and (\\frac{d}{d I} \\operatorname{f^{*}}{(I)})^{I} = \\cos^{I}{(I)} and \\cos{((\\frac{d}{d I} \\operatorname{f^{*}}{(I)})^{I})} = \\cos{((\\frac{d}{d I} \\sin{(I)})^{I})} and \\cos{(\\cos^{I}{(I)})} = \\cos{((\\frac{d}{d I} \\sin{(I)})^{I})} and \\int \\cos{(\\cos^{I}{(I)})} dI = \\int \\cos{((\\frac{d}{d I} \\sin{(I)})^{I})} dI", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Derivative(Function('f^*')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)), Pow(Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('f^*')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Derivative(Function('f^*')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))), cos(Pow(Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(cos(Pow(cos(Symbol('I', commutative=True)), Symbol('I', commutative=True))), cos(Pow(Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))))"], [["integrate", 6, "Symbol('I', commutative=True)"], "Equality(Integral(cos(Pow(cos(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(cos(Pow(Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} = \\frac{\\mathbf{g}}{n} + f^{*}, then obtain \\frac{\\int \\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} df^{*} + \\frac{1}{n}}{\\mathbf{g}} = \\frac{\\int (\\frac{\\mathbf{g}}{n} + f^{*}) df^{*} + \\frac{1}{n}}{\\mathbf{g}}", "derivation": "\\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} = \\frac{\\mathbf{g}}{n} + f^{*} and \\int \\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} df^{*} = \\int (\\frac{\\mathbf{g}}{n} + f^{*}) df^{*} and \\int \\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} df^{*} + \\frac{1}{n} = \\int (\\frac{\\mathbf{g}}{n} + f^{*}) df^{*} + \\frac{1}{n} and \\frac{\\int \\operatorname{f_{E}}{(f^{*},n,\\mathbf{g})} df^{*} + \\frac{1}{n}}{\\mathbf{g}} = \\frac{\\int (\\frac{\\mathbf{g}}{n} + f^{*}) df^{*} + \\frac{1}{n}}{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('f^*', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('f^*', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["add", 2, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('f_E')(Symbol('f^*', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Integral(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["divide", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Integral(Function('f_E')(Symbol('f^*', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1)))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Integral(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{f}{(A_{z})} = \\log{(\\sin{(A_{z})})} and u{(A_{z})} = \\log{(\\sin{(A_{z})})}^{A_{z}}, then obtain \\int ((\\mathbf{f}^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}} dA_{z} = \\int (u^{A_{z}}{(A_{z})})^{A_{z}} dA_{z}", "derivation": "\\mathbf{f}{(A_{z})} = \\log{(\\sin{(A_{z})})} and \\mathbf{f}^{A_{z}}{(A_{z})} = \\log{(\\sin{(A_{z})})}^{A_{z}} and u{(A_{z})} = \\log{(\\sin{(A_{z})})}^{A_{z}} and \\mathbf{f}^{A_{z}}{(A_{z})} = u{(A_{z})} and (\\mathbf{f}^{A_{z}}{(A_{z})})^{A_{z}} = u^{A_{z}}{(A_{z})} and ((\\mathbf{f}^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}} = (u^{A_{z}}{(A_{z})})^{A_{z}} and \\int ((\\mathbf{f}^{A_{z}}{(A_{z})})^{A_{z}})^{A_{z}} dA_{z} = \\int (u^{A_{z}}{(A_{z})})^{A_{z}} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), log(sin(Symbol('A_z', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(log(sin(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('A_z', commutative=True)), Pow(log(sin(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Function('u')(Symbol('A_z', commutative=True)))"], [["power", 4, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Function('u')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(Function('u')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["integrate", 6, "Symbol('A_z', commutative=True)"], "Equality(Integral(Pow(Pow(Pow(Function('\\\\mathbf{f}')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Pow(Pow(Function('u')(Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\chi)} = \\int \\sin{(\\chi)} d\\chi, then derive \\operatorname{t_{1}}{(\\chi)} = \\mathbf{J}_M - \\cos{(\\chi)}, then obtain \\frac{\\int \\sin{(\\chi)} d\\chi}{\\sin{(\\chi)}} = \\frac{\\mathbf{J}_M - \\cos{(\\chi)}}{\\sin{(\\chi)}}", "derivation": "\\operatorname{t_{1}}{(\\chi)} = \\int \\sin{(\\chi)} d\\chi and \\operatorname{t_{1}}{(\\chi)} = \\mathbf{J}_M - \\cos{(\\chi)} and \\frac{\\operatorname{t_{1}}{(\\chi)}}{\\sin{(\\chi)}} = \\frac{\\mathbf{J}_M - \\cos{(\\chi)}}{\\sin{(\\chi)}} and \\frac{\\int \\sin{(\\chi)} d\\chi}{\\sin{(\\chi)}} = \\frac{\\mathbf{J}_M - \\cos{(\\chi)}}{\\sin{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\chi', commutative=True)), Integral(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('t_1')(Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))))"], [["divide", 2, "sin(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('t_1')(Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(-1)), Integral(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Pow(sin(Symbol('\\\\chi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given r{(f^{*})} = \\log{(e^{f^{*}})}, then derive f^{*} + \\int r{(f^{*})} df^{*} = F_{c} + \\frac{(f^{*})^{2}}{2} + f^{*}, then derive Z + \\frac{(f^{*})^{2}}{2} + f^{*} = F_{c} + \\frac{(f^{*})^{2}}{2} + f^{*}, then obtain f^{*} + \\int r{(f^{*})} df^{*} = Z + \\frac{(f^{*})^{2}}{2} + f^{*}", "derivation": "r{(f^{*})} = \\log{(e^{f^{*}})} and \\int r{(f^{*})} df^{*} = \\int \\log{(e^{f^{*}})} df^{*} and f^{*} + \\int r{(f^{*})} df^{*} = f^{*} + \\int \\log{(e^{f^{*}})} df^{*} and f^{*} + \\int r{(f^{*})} df^{*} = F_{c} + \\frac{(f^{*})^{2}}{2} + f^{*} and f^{*} + \\int \\log{(e^{f^{*}})} df^{*} = F_{c} + \\frac{(f^{*})^{2}}{2} + f^{*} and Z + \\frac{(f^{*})^{2}}{2} + f^{*} = F_{c} + \\frac{(f^{*})^{2}}{2} + f^{*} and f^{*} + \\int \\log{(e^{f^{*}})} df^{*} = Z + \\frac{(f^{*})^{2}}{2} + f^{*} and f^{*} + \\int r{(f^{*})} df^{*} = Z + \\frac{(f^{*})^{2}}{2} + f^{*}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('f^*', commutative=True)), log(exp(Symbol('f^*', commutative=True))))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('r')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(log(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["add", 2, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Integral(Function('r')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('f^*', commutative=True), Integral(log(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('f^*', commutative=True), Integral(Function('r')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('f^*', commutative=True), Integral(log(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Symbol('f^*', commutative=True), Integral(log(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Add(Symbol('f^*', commutative=True), Integral(Function('r')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given s{(\\mu_0,\\nabla)} = - \\mu_0 + \\nabla, then derive \\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = \\frac{\\partial}{\\partial \\mu_0} (\\frac{\\nabla^{2}}{2} + \\nabla (- \\mu_0 - 1) + t), then derive \\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = - \\nabla, then obtain (\\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla) \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = - \\nabla \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla", "derivation": "s{(\\mu_0,\\nabla)} = - \\mu_0 + \\nabla and s{(\\mu_0,\\nabla)} - 1 = - \\mu_0 + \\nabla - 1 and \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = \\int (- \\mu_0 + \\nabla - 1) d\\nabla and \\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = \\frac{\\partial}{\\partial \\mu_0} \\int (- \\mu_0 + \\nabla - 1) d\\nabla and \\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = \\frac{\\partial}{\\partial \\mu_0} (\\frac{\\nabla^{2}}{2} + \\nabla (- \\mu_0 - 1) + t) and \\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = - \\nabla and (\\frac{\\partial}{\\partial \\mu_0} \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla) \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla = - \\nabla \\int (s{(\\mu_0,\\nabla)} - 1) d\\nabla", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\nabla', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\nabla', commutative=True), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\nabla', commutative=True), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Symbol('t', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))"], [["times", 6, "Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Derivative(Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True), Integral(Add(Function('s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(A_{2},x^\\prime)} = A_{2} x^\\prime, then derive \\frac{\\partial}{\\partial x^\\prime} \\psi^{*}{(A_{2},x^\\prime)} = A_{2}, then obtain \\frac{\\partial}{\\partial x^\\prime} A_{2} x^\\prime = A_{2}", "derivation": "\\psi^{*}{(A_{2},x^\\prime)} = A_{2} x^\\prime and \\frac{\\partial}{\\partial x^\\prime} \\psi^{*}{(A_{2},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} A_{2} x^\\prime and \\frac{\\partial}{\\partial x^\\prime} \\psi^{*}{(A_{2},x^\\prime)} = A_{2} and \\frac{\\partial}{\\partial x^\\prime} A_{2} x^\\prime = A_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi^*')(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('A_2', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_2', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('A_2', commutative=True))"]]}, {"prompt": "Given \\mathbf{J}{(\\rho_f,v_{y})} = \\rho_f - v_{y}, then derive \\int \\mathbf{J}{(\\rho_f,v_{y})} dv_{y} = \\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2}, then derive \\rho_f v_{y} + a - \\frac{v_{y}^{2}}{2} = \\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2}, then obtain \\int (\\rho_f v_{y} + a - \\frac{v_{y}^{2}}{2}) dv_{y} = \\int (\\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2}) dv_{y}", "derivation": "\\mathbf{J}{(\\rho_f,v_{y})} = \\rho_f - v_{y} and \\int \\mathbf{J}{(\\rho_f,v_{y})} dv_{y} = \\int (\\rho_f - v_{y}) dv_{y} and \\int \\mathbf{J}{(\\rho_f,v_{y})} dv_{y} = \\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2} and \\int (\\rho_f - v_{y}) dv_{y} = \\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2} and \\rho_f v_{y} + a - \\frac{v_{y}^{2}}{2} = \\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2} and \\int (\\rho_f v_{y} + a - \\frac{v_{y}^{2}}{2}) dv_{y} = \\int (\\rho_f v_{y} + v_{1} - \\frac{v_{y}^{2}}{2}) dv_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))))"], [["integrate", 5, "Symbol('v_y', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('a', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(t,B)} = B - t and \\operatorname{F_{c}}{(B)} = - B, then obtain \\frac{d}{d B} 2 \\operatorname{F_{c}}{(B)} = \\frac{\\partial}{\\partial B} (- t + \\operatorname{F_{c}}{(B)} - \\hat{H}{(t,B)})", "derivation": "\\hat{H}{(t,B)} = B - t and - B + \\hat{H}{(t,B)} = - t and - B = - t - \\hat{H}{(t,B)} and \\operatorname{F_{c}}{(B)} = - B and 2 \\operatorname{F_{c}}{(B)} = - B + \\operatorname{F_{c}}{(B)} and \\frac{d}{d B} 2 \\operatorname{F_{c}}{(B)} = \\frac{d}{d B} (- B + \\operatorname{F_{c}}{(B)}) and \\frac{d}{d B} 2 \\operatorname{F_{c}}{(B)} = \\frac{\\partial}{\\partial B} (- t + \\operatorname{F_{c}}{(B)} - \\hat{H}{(t,B)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('t', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\hat{H}')(Symbol('t', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True)))"], [["minus", 2, "Function('\\\\hat{H}')(Symbol('t', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('t', commutative=True), Symbol('B', commutative=True)))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True)))"], [["add", 4, "Function('F_c')(Symbol('B', commutative=True))"], "Equality(Mul(Integer(2), Function('F_c')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('F_c')(Symbol('B', commutative=True))))"], [["differentiate", 5, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('F_c')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('F_c')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Mul(Integer(2), Function('F_c')(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('F_c')(Symbol('B', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('t', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(n_{2})} = \\sin{(n_{2})}, then derive \\int \\mu{(n_{2})} dn_{2} = V - \\cos{(n_{2})}, then obtain e^{\\sin{(V - \\cos{(n_{2})})}} = e^{\\sin{(\\int \\sin{(n_{2})} dn_{2})}}", "derivation": "\\mu{(n_{2})} = \\sin{(n_{2})} and \\int \\mu{(n_{2})} dn_{2} = \\int \\sin{(n_{2})} dn_{2} and \\sin{(\\int \\mu{(n_{2})} dn_{2})} = \\sin{(\\int \\sin{(n_{2})} dn_{2})} and \\int \\mu{(n_{2})} dn_{2} = V - \\cos{(n_{2})} and \\int \\sin{(n_{2})} dn_{2} = V - \\cos{(n_{2})} and \\sin{(\\int \\mu{(n_{2})} dn_{2})} = \\sin{(V - \\cos{(n_{2})})} and \\sin{(V - \\cos{(n_{2})})} = \\sin{(\\int \\sin{(n_{2})} dn_{2})} and e^{\\sin{(V - \\cos{(n_{2})})}} = e^{\\sin{(\\int \\sin{(n_{2})} dn_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\mu')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(sin(Integral(Function('\\\\mu')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), sin(Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(sin(Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True))))), sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["exp", 7], "Equality(exp(sin(Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))))), exp(sin(Integral(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{A},\\Omega)} = \\int \\Omega \\mathbf{A} d\\Omega, then obtain \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi_{2}{(\\mathbf{A},\\Omega)})} + \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega = \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega)} + \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega", "derivation": "\\phi_{2}{(\\mathbf{A},\\Omega)} = \\int \\Omega \\mathbf{A} d\\Omega and \\frac{\\partial}{\\partial \\mathbf{A}} \\phi_{2}{(\\mathbf{A},\\Omega)} = \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega and \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi_{2}{(\\mathbf{A},\\Omega)})} = \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega)} and \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi_{2}{(\\mathbf{A},\\Omega)})} + \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega = \\log{(\\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega)} + \\frac{\\partial}{\\partial \\mathbf{A}} \\int \\Omega \\mathbf{A} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), log(Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["add", 3, "Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))"], "Equality(Add(log(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Add(log(Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Derivative(Integral(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{p}{(V,\\mathbb{I})} = V + \\mathbb{I}, then obtain \\frac{\\mathbf{p}{(V,\\mathbb{I})}}{V \\int (V + \\mathbb{I}) d\\mathbb{I}} = \\frac{V + \\mathbb{I}}{V \\int (V + \\mathbb{I}) d\\mathbb{I}}", "derivation": "\\mathbf{p}{(V,\\mathbb{I})} = V + \\mathbb{I} and \\frac{\\mathbf{p}{(V,\\mathbb{I})}}{V} = \\frac{V + \\mathbb{I}}{V} and \\int \\mathbf{p}{(V,\\mathbb{I})} d\\mathbb{I} = \\int (V + \\mathbb{I}) d\\mathbb{I} and \\frac{\\mathbf{p}{(V,\\mathbb{I})}}{V \\int \\mathbf{p}{(V,\\mathbb{I})} d\\mathbb{I}} = \\frac{V + \\mathbb{I}}{V \\int \\mathbf{p}{(V,\\mathbb{I})} d\\mathbb{I}} and \\frac{\\mathbf{p}{(V,\\mathbb{I})}}{V \\int (V + \\mathbb{I}) d\\mathbb{I}} = \\frac{V + \\mathbb{I}}{V \\int (V + \\mathbb{I}) d\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Add(Symbol('V', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(M)} = e^{\\cos{(M)}}, then obtain \\sin{(\\operatorname{J_{\\varepsilon}}{(M)} \\frac{d}{d M} 1)} = \\sin{(e^{\\cos{(M)}} \\frac{d}{d M} 1)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(M)} = e^{\\cos{(M)}} and \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}} = 1 and \\frac{d}{d M} \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}} = \\frac{d}{d M} 1 and \\operatorname{J_{\\varepsilon}}{(M)} \\frac{d}{d M} \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}} = e^{\\cos{(M)}} \\frac{d}{d M} \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}} and \\sin{(\\operatorname{J_{\\varepsilon}}{(M)} \\frac{d}{d M} \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}})} = \\sin{(e^{\\cos{(M)}} \\frac{d}{d M} \\operatorname{J_{\\varepsilon}}{(M)} e^{- \\cos{(M)}})} and \\sin{(\\operatorname{J_{\\varepsilon}}{(M)} \\frac{d}{d M} 1)} = \\sin{(e^{\\cos{(M)}} \\frac{d}{d M} 1)}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(cos(Symbol('M', commutative=True))))"], [["divide", 1, "exp(cos(Symbol('M', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Integer(1))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(exp(cos(Symbol('M', commutative=True))), Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["sin", 4], "Equality(sin(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))))), sin(Mul(exp(cos(Symbol('M', commutative=True))), Derivative(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), exp(Mul(Integer(-1), cos(Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(sin(Mul(Function('J_{\\\\varepsilon}')(Symbol('M', commutative=True)), Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))))), sin(Mul(exp(cos(Symbol('M', commutative=True))), Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(v_{x})} = v_{x}, then obtain ((\\frac{\\mathbf{s} + \\operatorname{v_{z}}{(v_{x})}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}})^{v_{x}} = ((\\frac{\\mathbf{s} + v_{x}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}})^{v_{x}}", "derivation": "\\operatorname{v_{z}}{(v_{x})} = v_{x} and \\mathbf{s} + \\operatorname{v_{z}}{(v_{x})} = \\mathbf{s} + v_{x} and \\frac{\\mathbf{s} + \\operatorname{v_{z}}{(v_{x})}}{\\operatorname{v_{z}}{(v_{x})}} = \\frac{\\mathbf{s} + v_{x}}{\\operatorname{v_{z}}{(v_{x})}} and (\\frac{\\mathbf{s} + \\operatorname{v_{z}}{(v_{x})}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}} = (\\frac{\\mathbf{s} + v_{x}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}} and ((\\frac{\\mathbf{s} + \\operatorname{v_{z}}{(v_{x})}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}})^{v_{x}} = ((\\frac{\\mathbf{s} + v_{x}}{\\operatorname{v_{z}}{(v_{x})}})^{v_{x}})^{v_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('v_z')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], [["add", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('v_z')(Symbol('v_x', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)))"], [["divide", 2, "Function('v_z')(Symbol('v_x', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('v_z')(Symbol('v_x', commutative=True))), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('v_z')(Symbol('v_x', commutative=True))), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))), Symbol('v_x', commutative=True)), Pow(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))), Symbol('v_x', commutative=True)))"], [["power", 4, "Symbol('v_x', commutative=True)"], "Equality(Pow(Pow(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('v_z')(Symbol('v_x', commutative=True))), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(Pow(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)), Pow(Function('v_z')(Symbol('v_x', commutative=True)), Integer(-1))), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{J}_P,B)} = B \\mathbf{J}_P, then obtain \\int \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\hat{x}_0{(\\mathbf{J}_P,B)} dB = \\mathbf{B}", "derivation": "\\hat{x}_0{(\\mathbf{J}_P,B)} = B \\mathbf{J}_P and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\hat{x}_0{(\\mathbf{J}_P,B)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} B \\mathbf{J}_P and \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\hat{x}_0{(\\mathbf{J}_P,B)} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} B \\mathbf{J}_P and \\int \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\hat{x}_0{(\\mathbf{J}_P,B)} dB = \\int \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} B \\mathbf{J}_P dB and \\int \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\hat{x}_0{(\\mathbf{J}_P,B)} dB = \\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Tuple(Symbol('B', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True))"]]}, {"prompt": "Given \\hat{x}_0{(T,n,y)} = \\frac{n + y}{T}, then obtain - y + \\hat{x}_0{(T,n,y)} - \\frac{\\partial}{\\partial T} (- y + \\frac{n + y}{T}) = - y - \\frac{\\partial}{\\partial T} (- y + \\frac{n + y}{T}) + \\frac{n + y}{T}", "derivation": "\\hat{x}_0{(T,n,y)} = \\frac{n + y}{T} and - y + \\hat{x}_0{(T,n,y)} = - y + \\frac{n + y}{T} and \\frac{\\partial}{\\partial T} (- y + \\hat{x}_0{(T,n,y)}) = \\frac{\\partial}{\\partial T} (- y + \\frac{n + y}{T}) and - y + \\hat{x}_0{(T,n,y)} - \\frac{\\partial}{\\partial T} (- y + \\hat{x}_0{(T,n,y)}) = - y - \\frac{\\partial}{\\partial T} (- y + \\hat{x}_0{(T,n,y)}) + \\frac{n + y}{T} and - y + \\hat{x}_0{(T,n,y)} - \\frac{\\partial}{\\partial T} (- y + \\frac{n + y}{T}) = - y - \\frac{\\partial}{\\partial T} (- y + \\frac{n + y}{T}) + \\frac{n + y}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{x}_0')(Symbol('T', commutative=True), Symbol('n', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{J}_P)} = \\mathbf{J}_P, then derive \\frac{d}{d \\mathbf{J}_P} \\rho_{f}{(\\mathbf{J}_P)} = 1, then obtain 1 = \\frac{1}{\\frac{d}{d \\mathbf{J}_P} \\rho_{f}{(\\mathbf{J}_P)}}", "derivation": "\\rho_{f}{(\\mathbf{J}_P)} = \\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} \\rho_{f}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\mathbf{J}_P and \\frac{d}{d \\mathbf{J}_P} \\rho_{f}{(\\mathbf{J}_P)} = 1 and 1 = \\frac{1}{\\frac{d}{d \\mathbf{J}_P} \\rho_{f}{(\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{J}_P', commutative=True), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))"], "Equality(Integer(1), Pow(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} = J + \\mathbf{J}_M, then obtain \\iiint \\frac{\\partial}{\\partial J} \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} dJ d\\mathbf{J}_M d\\mathbf{J}_M = \\iiint \\frac{\\partial}{\\partial J} (J + \\mathbf{J}_M) dJ d\\mathbf{J}_M d\\mathbf{J}_M", "derivation": "\\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} = J + \\mathbf{J}_M and \\frac{\\partial}{\\partial J} \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial J} (J + \\mathbf{J}_M) and \\int \\frac{\\partial}{\\partial J} \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} dJ = \\int \\frac{\\partial}{\\partial J} (J + \\mathbf{J}_M) dJ and \\iint \\frac{\\partial}{\\partial J} \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} dJ d\\mathbf{J}_M = \\iint \\frac{\\partial}{\\partial J} (J + \\mathbf{J}_M) dJ d\\mathbf{J}_M and \\iiint \\frac{\\partial}{\\partial J} \\operatorname{f^{\\prime}}{(J,\\mathbf{J}_M)} dJ d\\mathbf{J}_M d\\mathbf{J}_M = \\iiint \\frac{\\partial}{\\partial J} (J + \\mathbf{J}_M) dJ d\\mathbf{J}_M d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('f^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Derivative(Function('f^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Derivative(Function('f^{\\\\prime}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})}, then derive v_{x} + \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\ddot{x} + \\log{(\\hat{\\mathbf{r}})}, then obtain v_{x} + \\log{(\\hat{\\mathbf{r}})} = \\ddot{x} + \\log{(\\hat{\\mathbf{r}})}", "derivation": "\\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\log{(\\hat{\\mathbf{r}})} and \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} and \\int \\frac{d}{d \\hat{\\mathbf{r}}} \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} = \\int \\frac{d}{d \\hat{\\mathbf{r}}} \\log{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} and v_{x} + \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\ddot{x} + \\log{(\\hat{\\mathbf{r}})} and v_{x} + \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} = \\ddot{x} + \\operatorname{A_{z}}{(\\hat{\\mathbf{r}})} and v_{x} + \\log{(\\hat{\\mathbf{r}})} = \\ddot{x} + \\log{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v_x', commutative=True), Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('v_x', commutative=True), Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), Function('A_z')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('v_x', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\ddot{x}', commutative=True), log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\omega{(t,\\hat{\\mathbf{r}},\\theta)} = - \\hat{\\mathbf{r}} + \\frac{t}{\\theta}, then obtain \\int (2 \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1) dt = \\int (- \\hat{\\mathbf{r}} + \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1 + \\frac{t}{\\theta}) dt", "derivation": "\\omega{(t,\\hat{\\mathbf{r}},\\theta)} = - \\hat{\\mathbf{r}} + \\frac{t}{\\theta} and 2 \\omega{(t,\\hat{\\mathbf{r}},\\theta)} = - \\hat{\\mathbf{r}} + \\omega{(t,\\hat{\\mathbf{r}},\\theta)} + \\frac{t}{\\theta} and 2 \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1 = - \\hat{\\mathbf{r}} + \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1 + \\frac{t}{\\theta} and \\int (2 \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1) dt = \\int (- \\hat{\\mathbf{r}} + \\omega{(t,\\hat{\\mathbf{r}},\\theta)} - 1 + \\frac{t}{\\theta}) dt", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["add", 1, "Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["minus", 2, 1], "Equality(Add(Mul(Integer(2), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-1)), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Function('\\\\omega')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})} and r{(\\mathbb{I})} = \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} - 1, then derive \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} = J_{\\varepsilon} + \\frac{\\mathbb{I}^{2}}{2}, then obtain J_{\\varepsilon} + \\frac{\\mathbb{I}^{2}}{2} - 1 = \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} - 1", "derivation": "\\mathbf{P}{(\\mathbb{I})} = \\log{(e^{\\mathbb{I}})} and \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} = \\int \\log{(e^{\\mathbb{I}})} d\\mathbb{I} and \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} = J_{\\varepsilon} + \\frac{\\mathbb{I}^{2}}{2} and r{(\\mathbb{I})} = \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} - 1 and r{(\\mathbb{I})} = J_{\\varepsilon} + \\frac{\\mathbb{I}^{2}}{2} - 1 and J_{\\varepsilon} + \\frac{\\mathbb{I}^{2}}{2} - 1 = \\int \\mathbf{P}{(\\mathbb{I})} d\\mathbb{I} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True)), log(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(log(exp(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2)))))"], ["renaming_premise", "Equality(Function('r')(Symbol('\\\\mathbb{I}', commutative=True)), Add(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('r')(Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(-1)), Add(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\phi{(v)} = \\cos{(v)}, then obtain \\frac{d}{d v} \\int 2 \\phi{(v)} dv = \\frac{d}{d v} \\int (\\phi{(v)} + \\cos{(v)}) dv", "derivation": "\\phi{(v)} = \\cos{(v)} and 2 \\phi{(v)} = \\phi{(v)} + \\cos{(v)} and \\int 2 \\phi{(v)} dv = \\int (\\phi{(v)} + \\cos{(v)}) dv and \\frac{d}{d v} \\int 2 \\phi{(v)} dv = \\frac{d}{d v} \\int (\\phi{(v)} + \\cos{(v)}) dv", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('v', commutative=True))), Add(Function('\\\\phi')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\phi')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Add(Function('\\\\phi')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(2), Function('\\\\phi')(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\phi')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}}, then obtain (e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} \\hat{H}_{\\lambda}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}}", "derivation": "\\hat{H}_{\\lambda}{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}} and \\frac{d}{d V_{\\mathbf{E}}} \\hat{H}_{\\lambda}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} and e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} \\hat{H}_{\\lambda}{(V_{\\mathbf{E}})} = e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} and (e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} \\hat{H}_{\\lambda}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (e^{- V_{\\mathbf{E}}} \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["divide", 2, "exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Mul(exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Mul(exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(A_{y})} = (e^{A_{y}})^{A_{y}}, then obtain \\int (A_{y} + e^{\\hat{p}_0{(A_{y})}}) dA_{y} = \\int (A_{y} + e^{(e^{A_{y}})^{A_{y}}}) dA_{y}", "derivation": "\\hat{p}_0{(A_{y})} = (e^{A_{y}})^{A_{y}} and e^{\\hat{p}_0{(A_{y})}} = e^{(e^{A_{y}})^{A_{y}}} and A_{y} + e^{\\hat{p}_0{(A_{y})}} = A_{y} + e^{(e^{A_{y}})^{A_{y}}} and \\int (A_{y} + e^{\\hat{p}_0{(A_{y})}}) dA_{y} = \\int (A_{y} + e^{(e^{A_{y}})^{A_{y}}}) dA_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True))), exp(Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))))"], [["add", 2, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), exp(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)))), Add(Symbol('A_y', commutative=True), exp(Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))))"], [["integrate", 3, "Symbol('A_y', commutative=True)"], "Equality(Integral(Add(Symbol('A_y', commutative=True), exp(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))), Integral(Add(Symbol('A_y', commutative=True), exp(Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(I)} = e^{I}, then derive \\int (- I + \\mathbf{p}{(I)}) dI = - \\frac{I^{2}}{2} + \\mathbf{g} + e^{I}, then obtain - I^{2} + I + 2 \\mathbf{g} + 2 e^{I} = - \\frac{I^{2}}{2} + I + \\mathbf{g} + e^{I} + \\int (- I + e^{I}) dI", "derivation": "\\mathbf{p}{(I)} = e^{I} and - I + \\mathbf{p}{(I)} = - I + e^{I} and \\int (- I + \\mathbf{p}{(I)}) dI = \\int (- I + e^{I}) dI and I + \\int (- I + \\mathbf{p}{(I)}) dI = I + \\int (- I + e^{I}) dI and \\int (- I + \\mathbf{p}{(I)}) dI = - \\frac{I^{2}}{2} + \\mathbf{g} + e^{I} and - \\frac{I^{2}}{2} + I + \\mathbf{g} + e^{I} + \\int (- I + \\mathbf{p}{(I)}) dI = - \\frac{I^{2}}{2} + I + \\mathbf{g} + e^{I} + \\int (- I + e^{I}) dI and - I^{2} + I + 2 \\mathbf{g} + 2 e^{I} = - \\frac{I^{2}}{2} + I + \\mathbf{g} + e^{I} + \\int (- I + e^{I}) dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["add", 3, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Add(Symbol('I', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('I', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('I', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\mathbf{p}')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('I', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('I', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), exp(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), exp(Symbol('I', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given i{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\operatorname{A_{2}}{(y^{\\prime})} = \\log{(y^{\\prime})}, then obtain - i{(f_{\\mathbf{v}})} + \\int \\operatorname{A_{2}}{(y^{\\prime})} dy^{\\prime} = - i{(f_{\\mathbf{v}})} + \\int \\log{(y^{\\prime})} dy^{\\prime}", "derivation": "i{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\operatorname{A_{2}}{(y^{\\prime})} = \\log{(y^{\\prime})} and \\int \\operatorname{A_{2}}{(y^{\\prime})} dy^{\\prime} = \\int \\log{(y^{\\prime})} dy^{\\prime} and - \\log{(f_{\\mathbf{v}})} + \\int \\operatorname{A_{2}}{(y^{\\prime})} dy^{\\prime} = - \\log{(f_{\\mathbf{v}})} + \\int \\log{(y^{\\prime})} dy^{\\prime} and - i{(f_{\\mathbf{v}})} + \\int \\operatorname{A_{2}}{(y^{\\prime})} dy^{\\prime} = - i{(f_{\\mathbf{v}})} + \\int \\log{(y^{\\prime})} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], ["get_premise", "Equality(Function('A_2')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 3, "log(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Function('A_2')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Function('A_2')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{F})} = e^{\\mathbf{F}}, then obtain \\sin{(\\varepsilon^{\\mathbf{F}}{(\\mathbf{F})} - 1)} = \\sin{((e^{\\mathbf{F}})^{\\mathbf{F}} - 1)}", "derivation": "\\varepsilon{(\\mathbf{F})} = e^{\\mathbf{F}} and \\varepsilon^{\\mathbf{F}}{(\\mathbf{F})} = (e^{\\mathbf{F}})^{\\mathbf{F}} and \\varepsilon^{\\mathbf{F}}{(\\mathbf{F})} - 1 = (e^{\\mathbf{F}})^{\\mathbf{F}} - 1 and \\sin{(\\varepsilon^{\\mathbf{F}}{(\\mathbf{F})} - 1)} = \\sin{((e^{\\mathbf{F}})^{\\mathbf{F}} - 1)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 2, 1], "Equality(Add(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)), Add(Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1)))"], [["sin", 3], "Equality(sin(Add(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), sin(Add(Pow(exp(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{M})} = \\cos{(\\mathbf{M})}, then obtain \\frac{(\\cos{(\\operatorname{t_{2}}{(\\mathbf{M})})} - \\cos{(\\cos{(\\mathbf{M})})})^{\\mathbf{M}}}{\\mathbf{M} + \\operatorname{t_{2}}{(\\mathbf{M})}} = \\frac{0^{\\mathbf{M}}}{\\mathbf{M} + \\operatorname{t_{2}}{(\\mathbf{M})}}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\cos{(\\operatorname{t_{2}}{(\\mathbf{M})})} = \\cos{(\\cos{(\\mathbf{M})})} and \\cos{(\\operatorname{t_{2}}{(\\mathbf{M})})} - \\cos{(\\cos{(\\mathbf{M})})} = 0 and (\\cos{(\\operatorname{t_{2}}{(\\mathbf{M})})} - \\cos{(\\cos{(\\mathbf{M})})})^{\\mathbf{M}} = 0^{\\mathbf{M}} and \\frac{(\\cos{(\\operatorname{t_{2}}{(\\mathbf{M})})} - \\cos{(\\cos{(\\mathbf{M})})})^{\\mathbf{M}}}{\\mathbf{M} + \\operatorname{t_{2}}{(\\mathbf{M})}} = \\frac{0^{\\mathbf{M}}}{\\mathbf{M} + \\operatorname{t_{2}}{(\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), cos(cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 2, "cos(cos(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(cos(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), cos(cos(Symbol('\\\\mathbf{M}', commutative=True))))), Integer(0))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Add(cos(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), cos(cos(Symbol('\\\\mathbf{M}', commutative=True))))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 4, "Add(Symbol('\\\\mathbf{M}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1)), Pow(Add(cos(Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), cos(cos(Symbol('\\\\mathbf{M}', commutative=True))))), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('t_2')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\psi^*,\\dot{x})} = \\frac{\\dot{x}}{\\psi^*}, then obtain \\dot{x} + \\frac{\\dot{x}}{\\psi^*} + \\int \\operatorname{r_{0}}{(\\psi^*,\\dot{x})} d\\psi^* = \\dot{x} + \\frac{\\dot{x}}{\\psi^*} + \\int \\frac{\\dot{x}}{\\psi^*} d\\psi^*", "derivation": "\\operatorname{r_{0}}{(\\psi^*,\\dot{x})} = \\frac{\\dot{x}}{\\psi^*} and \\int \\operatorname{r_{0}}{(\\psi^*,\\dot{x})} d\\psi^* = \\int \\frac{\\dot{x}}{\\psi^*} d\\psi^* and \\dot{x} + \\int \\operatorname{r_{0}}{(\\psi^*,\\dot{x})} d\\psi^* = \\dot{x} + \\int \\frac{\\dot{x}}{\\psi^*} d\\psi^* and \\dot{x} + \\frac{\\dot{x}}{\\psi^*} + \\int \\operatorname{r_{0}}{(\\psi^*,\\dot{x})} d\\psi^* = \\dot{x} + \\frac{\\dot{x}}{\\psi^*} + \\int \\frac{\\dot{x}}{\\psi^*} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Symbol('\\\\dot{x}', commutative=True), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["add", 3, "Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Integral(Function('r_0')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Integral(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given Z{(x,\\omega)} = \\frac{\\partial}{\\partial x} (- \\omega + x) and \\operatorname{v_{2}}{(x,\\omega)} = - Z{(x,\\omega)}, then obtain \\operatorname{v_{2}}{(x,\\omega)} = - \\frac{\\partial}{\\partial x} (- \\omega + x)", "derivation": "Z{(x,\\omega)} = \\frac{\\partial}{\\partial x} (- \\omega + x) and - Z{(x,\\omega)} = - \\frac{\\partial}{\\partial x} (- \\omega + x) and \\operatorname{v_{2}}{(x,\\omega)} = - Z{(x,\\omega)} and \\operatorname{v_{2}}{(x,\\omega)} = - \\frac{\\partial}{\\partial x} (- \\omega + x)", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('x', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('x', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('Z')(Symbol('x', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('v_2')(Symbol('x', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{f}{(r)} = \\cos{(r)}, then obtain \\frac{(\\rho_{f}^{r}{(r)})^{r}}{\\cos{(r)}} = \\frac{(\\cos^{r}{(r)})^{r}}{\\cos{(r)}}", "derivation": "\\rho_{f}{(r)} = \\cos{(r)} and \\rho_{f}^{r}{(r)} = \\cos^{r}{(r)} and (\\rho_{f}^{r}{(r)})^{r} = (\\cos^{r}{(r)})^{r} and \\frac{(\\rho_{f}^{r}{(r)})^{r}}{\\cos{(r)}} = \\frac{(\\cos^{r}{(r)})^{r}}{\\cos{(r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('\\\\rho_f')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["divide", 3, "cos(Symbol('r', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\rho_f')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Integer(-1))), Mul(Pow(Pow(cos(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(cos(Symbol('r', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(z)} = \\sin{(z)} and \\operatorname{a^{\\dagger}}{(z)} = \\sin{(z)}, then obtain - \\operatorname{a^{\\dagger}}{(z)} + \\int \\hat{H}_{\\lambda}{(z)} dz = - \\sin{(z)} + \\int \\hat{H}_{\\lambda}{(z)} dz", "derivation": "\\hat{H}_{\\lambda}{(z)} = \\sin{(z)} and \\int \\hat{H}_{\\lambda}{(z)} dz = \\int \\sin{(z)} dz and \\operatorname{a^{\\dagger}}{(z)} = \\sin{(z)} and \\operatorname{a^{\\dagger}}{(z)} - \\int \\sin{(z)} dz = \\sin{(z)} - \\int \\sin{(z)} dz and - \\operatorname{a^{\\dagger}}{(z)} + \\int \\sin{(z)} dz = - \\sin{(z)} + \\int \\sin{(z)} dz and - \\operatorname{a^{\\dagger}}{(z)} + \\int \\hat{H}_{\\lambda}{(z)} dz = - \\sin{(z)} + \\int \\hat{H}_{\\lambda}{(z)} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["minus", 3, "Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('z', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(sin(Symbol('z', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('z', commutative=True))), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given U{(i,Q)} = Q i, then obtain i + \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int U{(i,Q)} dQ = i + \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int Q i dQ", "derivation": "U{(i,Q)} = Q i and \\int U{(i,Q)} dQ = \\int Q i dQ and \\frac{\\partial}{\\partial Q} \\int U{(i,Q)} dQ = \\frac{\\partial}{\\partial Q} \\int Q i dQ and \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int U{(i,Q)} dQ = \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int Q i dQ and \\frac{\\partial}{\\partial Q} \\int U{(i,Q)} dQ + \\int U{(i,Q)} dQ = \\frac{\\partial}{\\partial Q} \\int U{(i,Q)} dQ + \\int Q i dQ and i + \\frac{\\partial}{\\partial Q} \\int U{(i,Q)} dQ + \\int U{(i,Q)} dQ = i + \\frac{\\partial}{\\partial Q} \\int U{(i,Q)} dQ + \\int Q i dQ and i + \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int U{(i,Q)} dQ = i + \\frac{\\partial}{\\partial Q} \\int Q i dQ + \\int Q i dQ", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Derivative(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Derivative(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["add", 5, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Derivative(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('i', commutative=True), Derivative(Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('i', commutative=True), Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Function('U')(Symbol('i', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('i', commutative=True), Derivative(Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integral(Mul(Symbol('Q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(F_{H},F_{x})} = \\log{(\\frac{F_{x}}{F_{H}})}, then obtain (\\int (\\operatorname{M_{E}}{(F_{H},F_{x})} - \\frac{1}{F_{H}}) dF_{x} - \\int (\\log{(\\frac{F_{x}}{F_{H}})} - \\frac{1}{F_{H}}) dF_{x})^{F_{x}} = 0^{F_{x}}", "derivation": "\\operatorname{M_{E}}{(F_{H},F_{x})} = \\log{(\\frac{F_{x}}{F_{H}})} and \\operatorname{M_{E}}{(F_{H},F_{x})} - \\frac{1}{F_{H}} = \\log{(\\frac{F_{x}}{F_{H}})} - \\frac{1}{F_{H}} and \\int (\\operatorname{M_{E}}{(F_{H},F_{x})} - \\frac{1}{F_{H}}) dF_{x} = \\int (\\log{(\\frac{F_{x}}{F_{H}})} - \\frac{1}{F_{H}}) dF_{x} and \\int (\\operatorname{M_{E}}{(F_{H},F_{x})} - \\frac{1}{F_{H}}) dF_{x} - \\int (\\log{(\\frac{F_{x}}{F_{H}})} - \\frac{1}{F_{H}}) dF_{x} = 0 and (\\int (\\operatorname{M_{E}}{(F_{H},F_{x})} - \\frac{1}{F_{H}}) dF_{x} - \\int (\\log{(\\frac{F_{x}}{F_{H}})} - \\frac{1}{F_{H}}) dF_{x})^{F_{x}} = 0^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('F_H', commutative=True), Symbol('F_x', commutative=True)), log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))))"], [["minus", 1, "Pow(Symbol('F_H', commutative=True), Integer(-1))"], "Equality(Add(Function('M_E')(Symbol('F_H', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Add(log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))))"], [["integrate", 2, "Symbol('F_x', commutative=True)"], "Equality(Integral(Add(Function('M_E')(Symbol('F_H', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))), Integral(Add(log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))))"], [["minus", 3, "Integral(Add(log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Add(Integral(Add(Function('M_E')(Symbol('F_H', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))), Mul(Integer(-1), Integral(Add(log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('F_x', commutative=True)"], "Equality(Pow(Add(Integral(Add(Function('M_E')(Symbol('F_H', commutative=True), Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))), Mul(Integer(-1), Integral(Add(log(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)))), Tuple(Symbol('F_x', commutative=True))))), Symbol('F_x', commutative=True)), Pow(Integer(0), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given p{(A_{z})} = \\int e^{A_{z}} dA_{z}, then derive p{(A_{z})} = A_{1} + e^{A_{z}}, then obtain A_{1} (\\mathbf{f} + e^{A_{z}}) = A_{1} (A_{1} + e^{A_{z}})", "derivation": "p{(A_{z})} = \\int e^{A_{z}} dA_{z} and p{(A_{z})} = A_{1} + e^{A_{z}} and A_{1} p{(A_{z})} = A_{1} (A_{1} + e^{A_{z}}) and A_{1} \\int e^{A_{z}} dA_{z} = A_{1} (A_{1} + e^{A_{z}}) and A_{1} (\\mathbf{f} + e^{A_{z}}) = A_{1} (A_{1} + e^{A_{z}})", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('A_z', commutative=True)), Integral(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('p')(Symbol('A_z', commutative=True)), Add(Symbol('A_1', commutative=True), exp(Symbol('A_z', commutative=True))))"], [["times", 2, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('p')(Symbol('A_z', commutative=True))), Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), exp(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A_1', commutative=True), Integral(exp(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), exp(Symbol('A_z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('A_1', commutative=True), Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('A_z', commutative=True)))), Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), exp(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\omega,E_{\\lambda})} = \\frac{E_{\\lambda}}{\\omega}, then obtain \\frac{\\int (- \\omega + \\mathbf{s}{(\\omega,E_{\\lambda})}) d\\omega}{E_{\\lambda}} = \\frac{\\int (\\frac{E_{\\lambda}}{\\omega} - \\omega) d\\omega}{E_{\\lambda}}", "derivation": "\\mathbf{s}{(\\omega,E_{\\lambda})} = \\frac{E_{\\lambda}}{\\omega} and - \\omega + \\mathbf{s}{(\\omega,E_{\\lambda})} = \\frac{E_{\\lambda}}{\\omega} - \\omega and \\int (- \\omega + \\mathbf{s}{(\\omega,E_{\\lambda})}) d\\omega = \\int (\\frac{E_{\\lambda}}{\\omega} - \\omega) d\\omega and \\frac{\\int (- \\omega + \\mathbf{s}{(\\omega,E_{\\lambda})}) d\\omega}{E_{\\lambda}} = \\frac{\\int (\\frac{E_{\\lambda}}{\\omega} - \\omega) d\\omega}{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given i{(\\mathbf{p},\\dot{y})} = \\frac{\\dot{y}}{\\mathbf{p}}, then obtain \\frac{\\dot{y}}{\\mathbf{p}} + \\frac{\\partial}{\\partial \\mathbf{p}} i{(\\mathbf{p},\\dot{y})} - 1 = \\frac{\\dot{y}}{\\mathbf{p}} - \\frac{\\dot{y}}{\\mathbf{p}^{2}} - 1", "derivation": "i{(\\mathbf{p},\\dot{y})} = \\frac{\\dot{y}}{\\mathbf{p}} and - \\mathbf{p} + i{(\\mathbf{p},\\dot{y})} = \\frac{\\dot{y}}{\\mathbf{p}} - \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} (- \\mathbf{p} + i{(\\mathbf{p},\\dot{y})}) = \\frac{\\partial}{\\partial \\mathbf{p}} (\\frac{\\dot{y}}{\\mathbf{p}} - \\mathbf{p}) and \\frac{\\dot{y}}{\\mathbf{p}} + \\frac{\\partial}{\\partial \\mathbf{p}} (- \\mathbf{p} + i{(\\mathbf{p},\\dot{y})}) = \\frac{\\dot{y}}{\\mathbf{p}} + \\frac{\\partial}{\\partial \\mathbf{p}} (\\frac{\\dot{y}}{\\mathbf{p}} - \\mathbf{p}) and \\frac{\\dot{y}}{\\mathbf{p}} + \\frac{\\partial}{\\partial \\mathbf{p}} i{(\\mathbf{p},\\dot{y})} - 1 = \\frac{\\dot{y}}{\\mathbf{p}} - \\frac{\\dot{y}}{\\mathbf{p}^{2}} - 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('i')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('i')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('i')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Derivative(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Derivative(Function('i')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-2))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{B}{(\\delta)} = \\sin{(\\cos{(\\delta)})}, then obtain ((\\int \\mathbf{B}{(\\delta)} d\\delta)^{\\delta})^{\\delta} + \\int \\sin{(\\cos{(\\delta)})} d\\delta + 1 = ((\\int \\sin{(\\cos{(\\delta)})} d\\delta)^{\\delta})^{\\delta} + \\int \\sin{(\\cos{(\\delta)})} d\\delta + 1", "derivation": "\\mathbf{B}{(\\delta)} = \\sin{(\\cos{(\\delta)})} and \\int \\mathbf{B}{(\\delta)} d\\delta = \\int \\sin{(\\cos{(\\delta)})} d\\delta and \\int \\mathbf{B}{(\\delta)} d\\delta + 1 = \\int \\sin{(\\cos{(\\delta)})} d\\delta + 1 and (\\int \\mathbf{B}{(\\delta)} d\\delta)^{\\delta} = (\\int \\sin{(\\cos{(\\delta)})} d\\delta)^{\\delta} and ((\\int \\mathbf{B}{(\\delta)} d\\delta)^{\\delta})^{\\delta} = ((\\int \\sin{(\\cos{(\\delta)})} d\\delta)^{\\delta})^{\\delta} and ((\\int \\mathbf{B}{(\\delta)} d\\delta)^{\\delta})^{\\delta} + \\int \\mathbf{B}{(\\delta)} d\\delta + 1 = ((\\int \\sin{(\\cos{(\\delta)})} d\\delta)^{\\delta})^{\\delta} + \\int \\mathbf{B}{(\\delta)} d\\delta + 1 and ((\\int \\mathbf{B}{(\\delta)} d\\delta)^{\\delta})^{\\delta} + \\int \\sin{(\\cos{(\\delta)})} d\\delta + 1 = ((\\int \\sin{(\\cos{(\\delta)})} d\\delta)^{\\delta})^{\\delta} + \\int \\sin{(\\cos{(\\delta)})} d\\delta + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), sin(cos(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["power", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["add", 5, "Add(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1))"], "Equality(Add(Pow(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Pow(Pow(Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Pow(Pow(Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(sin(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)))"]]}, {"prompt": "Given M{(S,A_{x},A_{y})} = A_{x}^{S} - A_{y}, then obtain (A_{x}^{S} - 2 A_{y} + 2 M{(S,A_{x},A_{y})} + 1)^{A_{x}} = (3 A_{x}^{S} - 4 A_{y} + 1)^{A_{x}}", "derivation": "M{(S,A_{x},A_{y})} = A_{x}^{S} - A_{y} and M{(S,A_{x},A_{y})} + 1 = A_{x}^{S} - A_{y} + 1 and A_{x}^{S} - A_{y} + M{(S,A_{x},A_{y})} + 1 = 2 A_{x}^{S} - 2 A_{y} + 1 and A_{x}^{S} - 2 A_{y} + 2 M{(S,A_{x},A_{y})} + 1 = 2 A_{x}^{S} - 3 A_{y} + M{(S,A_{x},A_{y})} + 1 and A_{x}^{S} - 2 A_{y} + 2 M{(S,A_{x},A_{y})} + 1 = 3 A_{x}^{S} - 4 A_{y} + 1 and (A_{x}^{S} - 2 A_{y} + 2 M{(S,A_{x},A_{y})} + 1)^{A_{x}} = (3 A_{x}^{S} - 4 A_{y} + 1)^{A_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True)), Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True))))"], [["add", 1, 1], "Equality(Add(Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True)), Integer(1)), Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)), Integer(1)))"], [["add", 2, "Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))"], "Equality(Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True)), Integer(1)), Add(Mul(Integer(2), Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Integer(1)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True)))"], "Equality(Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True))), Integer(1)), Add(Mul(Integer(2), Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True))), Mul(Integer(-1), Integer(3), Symbol('A_y', commutative=True)), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True))), Integer(1)), Add(Mul(Integer(3), Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True))), Mul(Integer(-1), Integer(4), Symbol('A_y', commutative=True)), Integer(1)))"], [["power", 5, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Function('M')(Symbol('S', commutative=True), Symbol('A_x', commutative=True), Symbol('A_y', commutative=True))), Integer(1)), Symbol('A_x', commutative=True)), Pow(Add(Mul(Integer(3), Pow(Symbol('A_x', commutative=True), Symbol('S', commutative=True))), Mul(Integer(-1), Integer(4), Symbol('A_y', commutative=True)), Integer(1)), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\dot{z})} = \\log{(\\dot{z})}, then derive \\int \\dot{y}{(\\dot{z})} d\\dot{z} = \\dot{z} \\log{(\\dot{z})} - \\dot{z} + \\mathbf{P}, then obtain \\int \\dot{y}{(\\dot{z})} d\\dot{z} = \\dot{z} \\dot{y}{(\\dot{z})} - \\dot{z} + \\mathbf{P}", "derivation": "\\dot{y}{(\\dot{z})} = \\log{(\\dot{z})} and \\int \\dot{y}{(\\dot{z})} d\\dot{z} = \\int \\log{(\\dot{z})} d\\dot{z} and \\int \\dot{y}{(\\dot{z})} d\\dot{z} = \\dot{z} \\log{(\\dot{z})} - \\dot{z} + \\mathbf{P} and \\int \\dot{y}{(\\dot{z})} d\\dot{z} = \\dot{z} \\dot{y}{(\\dot{z})} - \\dot{z} + \\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(log(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Symbol('\\\\dot{z}', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given s{(\\nabla)} = \\sin{(\\nabla)}, then obtain \\nabla (\\nabla s{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)}) s{(\\nabla)} = \\nabla (\\nabla \\sin{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)}) s{(\\nabla)}", "derivation": "s{(\\nabla)} = \\sin{(\\nabla)} and 2 s{(\\nabla)} = s{(\\nabla)} + \\sin{(\\nabla)} and \\nabla s{(\\nabla)} = \\nabla \\sin{(\\nabla)} and \\nabla s{(\\nabla)} - 2 s{(\\nabla)} = \\nabla \\sin{(\\nabla)} - 2 s{(\\nabla)} and \\nabla s{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)} = \\nabla \\sin{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)} and \\nabla (\\nabla s{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)}) s{(\\nabla)} = \\nabla (\\nabla \\sin{(\\nabla)} - s{(\\nabla)} - \\sin{(\\nabla)}) s{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Function('s')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('s')(Symbol('\\\\nabla', commutative=True))), Add(Function('s')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Function('s')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integer(2), Function('s')(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Integer(2), Function('s')(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\nabla', commutative=True), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))))"], [["times", 5, "Mul(Symbol('\\\\nabla', commutative=True), Function('s')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Add(Mul(Symbol('\\\\nabla', commutative=True), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Add(Mul(Symbol('\\\\nabla', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Function('s')(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\nabla', commutative=True)))), Function('s')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(n_{1},\\hbar)} = \\frac{\\hbar}{n_{1}}, then derive \\frac{\\partial}{\\partial n_{1}} \\operatorname{C_{1}}{(n_{1},\\hbar)} = - \\frac{\\hbar}{n_{1}^{2}}, then obtain \\frac{\\partial}{\\partial n_{1}} - \\frac{\\operatorname{C_{1}}{(n_{1},\\hbar)}}{n_{1}} = \\frac{\\partial}{\\partial n_{1}} - \\frac{\\hbar}{n_{1}^{2}}", "derivation": "\\operatorname{C_{1}}{(n_{1},\\hbar)} = \\frac{\\hbar}{n_{1}} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{C_{1}}{(n_{1},\\hbar)} = \\frac{\\partial}{\\partial n_{1}} \\frac{\\hbar}{n_{1}} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{C_{1}}{(n_{1},\\hbar)} = - \\frac{\\hbar}{n_{1}^{2}} and \\frac{\\partial^{2}}{\\partial n_{1}^{2}} \\operatorname{C_{1}}{(n_{1},\\hbar)} = \\frac{\\partial}{\\partial n_{1}} - \\frac{\\hbar}{n_{1}^{2}} and \\frac{\\partial^{2}}{\\partial n_{1}^{2}} \\operatorname{C_{1}}{(n_{1},\\hbar)} = \\frac{\\partial}{\\partial n_{1}} - \\frac{\\operatorname{C_{1}}{(n_{1},\\hbar)}}{n_{1}} and \\frac{\\partial}{\\partial n_{1}} - \\frac{\\operatorname{C_{1}}{(n_{1},\\hbar)}}{n_{1}} = \\frac{\\partial}{\\partial n_{1}} - \\frac{\\hbar}{n_{1}^{2}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-2))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-2))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('C_1')(Symbol('n_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-2))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(J,c_{0},H)} = - H + J c_{0}, then obtain \\frac{(H + \\int \\operatorname{F_{c}}{(J,c_{0},H)} dJ)^{H}}{c_{0}} = \\frac{(H + \\int (- H + J c_{0}) dJ)^{H}}{c_{0}}", "derivation": "\\operatorname{F_{c}}{(J,c_{0},H)} = - H + J c_{0} and \\int \\operatorname{F_{c}}{(J,c_{0},H)} dJ = \\int (- H + J c_{0}) dJ and H + \\int \\operatorname{F_{c}}{(J,c_{0},H)} dJ = H + \\int (- H + J c_{0}) dJ and (H + \\int \\operatorname{F_{c}}{(J,c_{0},H)} dJ)^{H} = (H + \\int (- H + J c_{0}) dJ)^{H} and \\frac{(H + \\int \\operatorname{F_{c}}{(J,c_{0},H)} dJ)^{H}}{c_{0}} = \\frac{(H + \\int (- H + J c_{0}) dJ)^{H}}{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('J', commutative=True), Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('J', commutative=True), Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Add(Symbol('H', commutative=True), Integral(Function('F_c')(Symbol('J', commutative=True), Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Symbol('H', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Symbol('H', commutative=True), Integral(Function('F_c')(Symbol('J', commutative=True), Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))), Symbol('H', commutative=True)))"], [["divide", 4, "Symbol('c_0', commutative=True)"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Add(Symbol('H', commutative=True), Integral(Function('F_c')(Symbol('J', commutative=True), Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('H', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Integer(-1)), Pow(Add(Symbol('H', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('J', commutative=True)))), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(h,\\Psi_{nl})} = h \\cos{(\\Psi_{nl})}, then obtain h (\\operatorname{r_{0}}{(h,\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})}) = h (h \\cos{(\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})})", "derivation": "\\operatorname{r_{0}}{(h,\\Psi_{nl})} = h \\cos{(\\Psi_{nl})} and \\operatorname{r_{0}}{(h,\\Psi_{nl})} + \\cos{(\\Psi_{nl})} = h \\cos{(\\Psi_{nl})} + \\cos{(\\Psi_{nl})} and \\operatorname{r_{0}}{(h,\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})} = h \\cos{(\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})} and h (\\operatorname{r_{0}}{(h,\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})}) = h (h \\cos{(\\Psi_{nl})} + 2 \\cos{(\\Psi_{nl})})", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('h', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('h', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 1, "cos(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('h', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Symbol('h', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('h', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Mul(Symbol('h', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["times", 3, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Add(Function('r_0')(Symbol('h', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\Psi_{nl}', commutative=True))))), Mul(Symbol('h', commutative=True), Add(Mul(Symbol('h', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi_{nl}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = e^{\\mathbf{D} - m_{s}}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = e^{\\mathbf{D} - m_{s}}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} e^{\\mathbf{D} - m_{s}} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}^{2}} e^{\\mathbf{D} - m_{s}}", "derivation": "\\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = e^{\\mathbf{D} - m_{s}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = \\frac{\\partial}{\\partial \\mathbf{D}} e^{\\mathbf{D} - m_{s}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = e^{\\mathbf{D} - m_{s}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}^{2}} \\operatorname{F_{g}}{(\\mathbf{D},m_{s})} and \\frac{\\partial}{\\partial \\mathbf{D}} e^{\\mathbf{D} - m_{s}} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}^{2}} e^{\\mathbf{D} - m_{s}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('m_s', commutative=True)), exp(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('F_g')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Function('F_g')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(2))))"]]}, {"prompt": "Given s{(\\mathbf{s},\\nabla,\\varepsilon)} = (\\mathbf{s} + \\varepsilon)^{\\nabla}, then obtain \\frac{\\partial}{\\partial \\mathbf{s}} (- s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int s{(\\mathbf{s},\\nabla,\\varepsilon)} d\\nabla) = \\frac{\\partial}{\\partial \\mathbf{s}} (- s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int (\\mathbf{s} + \\varepsilon)^{\\nabla} d\\nabla)", "derivation": "s{(\\mathbf{s},\\nabla,\\varepsilon)} = (\\mathbf{s} + \\varepsilon)^{\\nabla} and \\int s{(\\mathbf{s},\\nabla,\\varepsilon)} d\\nabla = \\int (\\mathbf{s} + \\varepsilon)^{\\nabla} d\\nabla and - s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int s{(\\mathbf{s},\\nabla,\\varepsilon)} d\\nabla = - s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int (\\mathbf{s} + \\varepsilon)^{\\nabla} d\\nabla and \\frac{\\partial}{\\partial \\mathbf{s}} (- s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int s{(\\mathbf{s},\\nabla,\\varepsilon)} d\\nabla) = \\frac{\\partial}{\\partial \\mathbf{s}} (- s{(\\mathbf{s},\\nabla,\\varepsilon)} + \\int (\\mathbf{s} + \\varepsilon)^{\\nabla} d\\nabla)", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integral(Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integral(Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('s')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\sin{(M + \\hat{x}_0)}, then derive \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\cos{(M + \\hat{x}_0)}, then obtain \\operatorname{z^{*}}{(\\hat{x}_0,M)} \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\operatorname{z^{*}}{(\\hat{x}_0,M)} \\cos{(M + \\hat{x}_0)}", "derivation": "\\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\sin{(M + \\hat{x}_0)} and \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\frac{\\partial}{\\partial M} \\sin{(M + \\hat{x}_0)} and \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\cos{(M + \\hat{x}_0)} and \\sin{(M + \\hat{x}_0)} \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\sin{(M + \\hat{x}_0)} \\cos{(M + \\hat{x}_0)} and \\operatorname{z^{*}}{(\\hat{x}_0,M)} \\frac{\\partial}{\\partial M} \\operatorname{z^{*}}{(\\hat{x}_0,M)} = \\operatorname{z^{*}}{(\\hat{x}_0,M)} \\cos{(M + \\hat{x}_0)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), sin(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), cos(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 3, "sin(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Mul(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Derivative(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(sin(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), cos(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), Derivative(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Function('z^*')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('M', commutative=True)), cos(Add(Symbol('M', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(G)} = \\sin{(\\sin{(G)})}, then derive 1 = \\cos{(\\cos{(G)} \\cos{(\\sin{(G)})} - \\frac{d}{d G} \\mathbf{p}{(G)})}, then obtain 1 = \\cos{(\\cos{(G)} \\cos{(\\sin{(G)})} - \\frac{d}{d G} \\sin{(\\sin{(G)})})}", "derivation": "\\mathbf{p}{(G)} = \\sin{(\\sin{(G)})} and 0 = - \\mathbf{p}{(G)} + \\sin{(\\sin{(G)})} and \\frac{d}{d G} 0 = \\frac{d}{d G} (- \\mathbf{p}{(G)} + \\sin{(\\sin{(G)})}) and \\cos{(\\frac{d}{d G} 0)} = \\cos{(\\frac{d}{d G} (- \\mathbf{p}{(G)} + \\sin{(\\sin{(G)})}))} and 1 = \\cos{(\\cos{(G)} \\cos{(\\sin{(G)})} - \\frac{d}{d G} \\mathbf{p}{(G)})} and 1 = \\cos{(\\cos{(G)} \\cos{(\\sin{(G)})} - \\frac{d}{d G} \\sin{(\\sin{(G)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{p}')(Symbol('G', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True)))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Integer(0), Tuple(Symbol('G', commutative=True), Integer(1)))), cos(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('G', commutative=True))), sin(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), cos(Add(Mul(cos(Symbol('G', commutative=True)), cos(sin(Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{p}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), cos(Add(Mul(cos(Symbol('G', commutative=True)), cos(sin(Symbol('G', commutative=True)))), Mul(Integer(-1), Derivative(sin(sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given u{(W)} = e^{W}, then derive \\frac{d}{d W} u{(W)} = e^{W}, then obtain \\frac{d}{d W} (\\frac{d^{2}}{d W^{2}} u{(W)} + \\int u{(W)} dW) = \\frac{d}{d W} (u{(W)} + \\int u{(W)} dW)", "derivation": "u{(W)} = e^{W} and \\frac{d}{d W} u{(W)} = \\frac{d}{d W} e^{W} and \\frac{d}{d W} u{(W)} = e^{W} and \\frac{d}{d W} u{(W)} = u{(W)} and \\frac{d^{2}}{d W^{2}} u{(W)} = \\frac{d}{d W} u{(W)} and \\frac{d}{d W} u{(W)} + \\int u{(W)} dW = u{(W)} + \\int u{(W)} dW and \\frac{d}{d W} (\\frac{d}{d W} u{(W)} + \\int u{(W)} dW) = \\frac{d}{d W} (u{(W)} + \\int u{(W)} dW) and \\frac{d}{d W} (\\frac{d^{2}}{d W^{2}} u{(W)} + \\int u{(W)} dW) = \\frac{d}{d W} (u{(W)} + \\int u{(W)} dW)", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), exp(Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Function('u')(Symbol('W', commutative=True)))"], [["differentiate", 4, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["add", 4, "Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Function('u')(Symbol('W', commutative=True)), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Function('u')(Symbol('W', commutative=True)), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Derivative(Add(Derivative(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(2))), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Function('u')(Symbol('W', commutative=True)), Integral(Function('u')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\psi^*,\\rho_b)} = \\psi^* + \\rho_b, then obtain \\psi^* (\\frac{- \\psi^* - \\rho_b}{\\psi^*} + \\frac{\\psi^* + \\rho_b}{\\psi^*}) = 0", "derivation": "\\mathbf{S}{(\\psi^*,\\rho_b)} = \\psi^* + \\rho_b and \\frac{\\mathbf{S}{(\\psi^*,\\rho_b)}}{\\psi^*} = \\frac{\\psi^* + \\rho_b}{\\psi^*} and - \\frac{\\psi^* + \\rho_b}{\\psi^*} + \\frac{\\mathbf{S}{(\\psi^*,\\rho_b)}}{\\psi^*} = 0 and \\frac{- \\psi^* - \\rho_b}{\\psi^*} + \\frac{\\psi^* + \\rho_b}{\\psi^*} = 0 and \\psi^* (\\frac{- \\psi^* - \\rho_b}{\\psi^*} + \\frac{\\psi^* + \\rho_b}{\\psi^*}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(0))"], [["divide", 4, "Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\rho_b', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(V)} = \\log{(\\sin{(V)})} and \\eta{(V)} = \\sin{(V)}, then obtain 1 = \\frac{\\log{(\\eta{(V)})}}{\\operatorname{F_{g}}{(V)}}", "derivation": "\\operatorname{F_{g}}{(V)} = \\log{(\\sin{(V)})} and \\eta{(V)} = \\sin{(V)} and \\operatorname{F_{g}}{(V)} = \\log{(\\eta{(V)})} and 1 = \\frac{\\log{(\\eta{(V)})}}{\\operatorname{F_{g}}{(V)}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('V', commutative=True)), log(sin(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_g')(Symbol('V', commutative=True)), log(Function('\\\\eta')(Symbol('V', commutative=True))))"], [["divide", 3, "Function('F_g')(Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_g')(Symbol('V', commutative=True)), Integer(-1)), log(Function('\\\\eta')(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})} and \\hat{p}_0{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})}, then obtain \\delta{(\\mathbf{P})} + \\hat{p}_0{(\\mathbf{P})} = 2 \\hat{p}_0{(\\mathbf{P})}", "derivation": "\\delta{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})} and \\delta{(\\mathbf{P})} + \\cos{(\\log{(\\mathbf{P})})} = 2 \\cos{(\\log{(\\mathbf{P})})} and \\hat{p}_0{(\\mathbf{P})} = \\cos{(\\log{(\\mathbf{P})})} and \\delta{(\\mathbf{P})} + \\hat{p}_0{(\\mathbf{P})} = 2 \\hat{p}_0{(\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), cos(log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 1, "cos(log(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), cos(log(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Integer(2), cos(log(Symbol('\\\\mathbf{P}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), cos(log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given I{(t_{2},m_{s})} = m_{s} t_{2}, then obtain \\frac{t_{2}^{2} \\frac{\\partial}{\\partial t_{2}} I{(t_{2},m_{s})}}{\\frac{\\partial}{\\partial t_{2}} m_{s} t_{2}} = t_{2}^{2}", "derivation": "I{(t_{2},m_{s})} = m_{s} t_{2} and \\frac{\\partial}{\\partial t_{2}} I{(t_{2},m_{s})} = \\frac{\\partial}{\\partial t_{2}} m_{s} t_{2} and t_{2} \\frac{\\partial}{\\partial t_{2}} I{(t_{2},m_{s})} = t_{2} \\frac{\\partial}{\\partial t_{2}} m_{s} t_{2} and \\frac{t_{2}^{2} \\frac{\\partial}{\\partial t_{2}} I{(t_{2},m_{s})}}{\\frac{\\partial}{\\partial t_{2}} m_{s} t_{2}} = t_{2}^{2}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t_2', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('m_s', commutative=True), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('m_s', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Symbol('t_2', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('t_2', commutative=True), Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(Symbol('t_2', commutative=True), Derivative(Mul(Symbol('m_s', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Mul(Symbol('m_s', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(2)), Pow(Derivative(Mul(Symbol('m_s', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('I')(Symbol('t_2', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Pow(Symbol('t_2', commutative=True), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\varepsilon)} = \\sin{(\\varepsilon)}, then obtain - \\varepsilon - \\sin{(\\varepsilon)} + \\frac{d}{d \\varepsilon} (\\varepsilon + \\mathbf{J}_P{(\\varepsilon)}) = - \\varepsilon - \\sin{(\\varepsilon)} + \\frac{d}{d \\varepsilon} (\\varepsilon + \\sin{(\\varepsilon)})", "derivation": "\\mathbf{J}_P{(\\varepsilon)} = \\sin{(\\varepsilon)} and \\varepsilon + \\mathbf{J}_P{(\\varepsilon)} = \\varepsilon + \\sin{(\\varepsilon)} and \\frac{d}{d \\varepsilon} (\\varepsilon + \\mathbf{J}_P{(\\varepsilon)}) = \\frac{d}{d \\varepsilon} (\\varepsilon + \\sin{(\\varepsilon)}) and - \\varepsilon - \\sin{(\\varepsilon)} + \\frac{d}{d \\varepsilon} (\\varepsilon + \\mathbf{J}_P{(\\varepsilon)}) = - \\varepsilon - \\sin{(\\varepsilon)} + \\frac{d}{d \\varepsilon} (\\varepsilon + \\sin{(\\varepsilon)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('\\\\varepsilon', commutative=True)))"], [["add", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Derivative(Add(Symbol('\\\\varepsilon', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(v)} = \\cos{(v)}, then derive - \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\tilde{g}^*{(v)} = - \\operatorname{L_{\\varepsilon}}{(C_{1})} - \\sin{(v)}, then obtain (- \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\tilde{g}^*{(v)}) \\sin{(v)} = (- \\operatorname{L_{\\varepsilon}}{(C_{1})} - \\sin{(v)}) \\sin{(v)}", "derivation": "\\tilde{g}^*{(v)} = \\cos{(v)} and \\frac{d}{d v} \\tilde{g}^*{(v)} = \\frac{d}{d v} \\cos{(v)} and - \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\tilde{g}^*{(v)} = - \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\cos{(v)} and - \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\tilde{g}^*{(v)} = - \\operatorname{L_{\\varepsilon}}{(C_{1})} - \\sin{(v)} and (- \\operatorname{L_{\\varepsilon}}{(C_{1})} + \\frac{d}{d v} \\tilde{g}^*{(v)}) \\sin{(v)} = (- \\operatorname{L_{\\varepsilon}}{(C_{1})} - \\sin{(v)}) \\sin{(v)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["minus", 2, "Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Mul(Integer(-1), sin(Symbol('v', commutative=True)))))"], [["times", 4, "sin(Symbol('v', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), sin(Symbol('v', commutative=True))), Mul(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('C_1', commutative=True))), Mul(Integer(-1), sin(Symbol('v', commutative=True)))), sin(Symbol('v', commutative=True))))"]]}, {"prompt": "Given H{(u)} = \\sin{(u)}, then obtain k + e^{\\frac{d}{d u} H{(u)} \\sin^{3}{(u)}} = k + e^{\\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)}}", "derivation": "H{(u)} = \\sin{(u)} and H^{2}{(u)} = H{(u)} \\sin{(u)} and H^{4}{(u)} = H^{2}{(u)} \\sin^{2}{(u)} and \\frac{d}{d u} H^{4}{(u)} = \\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)} and e^{\\frac{d}{d u} H^{4}{(u)}} = e^{\\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)}} and e^{\\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)}} = e^{\\frac{d}{d u} H{(u)} \\sin^{3}{(u)}} and k + e^{\\frac{d}{d u} H^{4}{(u)}} = k + e^{\\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)}} and e^{\\frac{d}{d u} H^{4}{(u)}} = e^{\\frac{d}{d u} H{(u)} \\sin^{3}{(u)}} and k + e^{\\frac{d}{d u} H{(u)} \\sin^{3}{(u)}} = k + e^{\\frac{d}{d u} H^{2}{(u)} \\sin^{2}{(u)}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["times", 1, "Function('H')(Symbol('u', commutative=True))"], "Equality(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('H')(Symbol('u', commutative=True)), Integer(4)), Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('H')(Symbol('u', commutative=True)), Integer(4)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(Derivative(Pow(Function('H')(Symbol('u', commutative=True)), Integer(4)), Tuple(Symbol('u', commutative=True), Integer(1)))), exp(Derivative(Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(exp(Derivative(Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1)))), exp(Derivative(Mul(Function('H')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(3))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["add", 5, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), exp(Derivative(Pow(Function('H')(Symbol('u', commutative=True)), Integer(4)), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Symbol('k', commutative=True), exp(Derivative(Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(exp(Derivative(Pow(Function('H')(Symbol('u', commutative=True)), Integer(4)), Tuple(Symbol('u', commutative=True), Integer(1)))), exp(Derivative(Mul(Function('H')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(3))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Add(Symbol('k', commutative=True), exp(Derivative(Mul(Function('H')(Symbol('u', commutative=True)), Pow(sin(Symbol('u', commutative=True)), Integer(3))), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Symbol('k', commutative=True), exp(Derivative(Mul(Pow(Function('H')(Symbol('u', commutative=True)), Integer(2)), Pow(sin(Symbol('u', commutative=True)), Integer(2))), Tuple(Symbol('u', commutative=True), Integer(1))))))"]]}, {"prompt": "Given q{(\\hat{H},\\nabla)} = \\hat{H} e^{\\nabla}, then obtain e^{- \\nabla} \\int \\frac{q{(\\hat{H},\\nabla)} e^{- \\nabla}}{\\hat{H}} d\\nabla = e^{- \\nabla} \\int 1 d\\nabla", "derivation": "q{(\\hat{H},\\nabla)} = \\hat{H} e^{\\nabla} and \\frac{q{(\\hat{H},\\nabla)} e^{- \\nabla}}{\\hat{H}} = 1 and \\int \\frac{q{(\\hat{H},\\nabla)} e^{- \\nabla}}{\\hat{H}} d\\nabla = \\int 1 d\\nabla and e^{- \\nabla} \\int \\frac{q{(\\hat{H},\\nabla)} e^{- \\nabla}}{\\hat{H}} d\\nabla = e^{- \\nabla} \\int 1 d\\nabla", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 3, "exp(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\mathbf{s})} = \\sin{(\\mathbf{s})}, then obtain (\\frac{1}{2} - \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}}) C{(x,z^{*})} = 0", "derivation": "\\theta{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and 2 \\theta{(\\mathbf{s})} = \\theta{(\\mathbf{s})} + \\sin{(\\mathbf{s})} and \\frac{1}{2} = \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}} and \\frac{1}{2} = \\frac{\\sin{(\\mathbf{s})}}{\\theta{(\\mathbf{s})} + \\sin{(\\mathbf{s})}} and \\frac{1}{2} - \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}} = - \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}} + \\frac{\\sin{(\\mathbf{s})}}{\\theta{(\\mathbf{s})} + \\sin{(\\mathbf{s})}} and \\frac{1}{2} - \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}} = 0 and (\\frac{1}{2} - \\frac{\\sin{(\\mathbf{s})}}{2 \\theta{(\\mathbf{s})}}) C{(x,z^{*})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 1, "Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Pow(Add(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 4, "Mul(Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Add(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True)))), Integer(0))"], [["divide", 6, "Pow(Function('C')(Symbol('x', commutative=True), Symbol('z^*', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), Pow(Function('\\\\theta')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{s}', commutative=True)))), Function('C')(Symbol('x', commutative=True), Symbol('z^*', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(\\theta_1,\\theta_2)} = \\theta_2 + \\sin{(\\theta_1)} and \\operatorname{x^{{\\}'}}{(\\theta_1,\\theta_2)} = \\theta_2 + \\phi_{1}{(\\theta_1,\\theta_2)} + \\sin{(\\theta_1)}, then obtain 2 \\phi_{1}{(\\theta_1,\\theta_2)} = 2 \\theta_2 + 2 \\sin{(\\theta_1)}", "derivation": "\\phi_{1}{(\\theta_1,\\theta_2)} = \\theta_2 + \\sin{(\\theta_1)} and 2 \\phi_{1}{(\\theta_1,\\theta_2)} = \\theta_2 + \\phi_{1}{(\\theta_1,\\theta_2)} + \\sin{(\\theta_1)} and \\operatorname{x^{{\\}'}}{(\\theta_1,\\theta_2)} = \\theta_2 + \\phi_{1}{(\\theta_1,\\theta_2)} + \\sin{(\\theta_1)} and \\operatorname{x^{{\\}'}}{(\\theta_1,\\theta_2)} = 2 \\phi_{1}{(\\theta_1,\\theta_2)} and \\operatorname{x^{{\\}'}}{(\\theta_1,\\theta_2)} = 2 \\theta_2 + 2 \\sin{(\\theta_1)} and 2 \\phi_{1}{(\\theta_1,\\theta_2)} = 2 \\theta_2 + 2 \\sin{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["add", 1, "Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{2})} = \\frac{d}{d A_{2}} \\sin{(A_{2})}, then derive A_{2} \\operatorname{P_{g}}{(A_{2})} = A_{2} \\cos{(A_{2})}, then obtain A_{2} \\frac{d}{d A_{2}} \\sin{(A_{2})} - A_{2} = A_{2} \\cos{(A_{2})} - A_{2}", "derivation": "\\operatorname{P_{g}}{(A_{2})} = \\frac{d}{d A_{2}} \\sin{(A_{2})} and A_{2} \\operatorname{P_{g}}{(A_{2})} = A_{2} \\frac{d}{d A_{2}} \\sin{(A_{2})} and A_{2} \\operatorname{P_{g}}{(A_{2})} = A_{2} \\cos{(A_{2})} and A_{2} \\operatorname{P_{g}}{(A_{2})} - A_{2} = A_{2} \\cos{(A_{2})} - A_{2} and A_{2} \\frac{d}{d A_{2}} \\sin{(A_{2})} - A_{2} = A_{2} \\cos{(A_{2})} - A_{2}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_2', commutative=True)), Derivative(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["times", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Function('P_g')(Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), Derivative(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('A_2', commutative=True), Function('P_g')(Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), cos(Symbol('A_2', commutative=True))))"], [["minus", 3, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Symbol('A_2', commutative=True), Function('P_g')(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), cos(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('A_2', commutative=True), Derivative(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), cos(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given y{(f,\\hat{X})} = - \\hat{X} + f, then derive \\int y{(f,\\hat{X})} df = - \\hat{X} f + \\mathbf{J}_P + \\frac{f^{2}}{2}, then obtain \\sin{(\\int y{(f,\\hat{X})} df)} = \\sin{(- \\hat{X} f + \\mathbf{J}_P + \\frac{f^{2}}{2})}", "derivation": "y{(f,\\hat{X})} = - \\hat{X} + f and \\int y{(f,\\hat{X})} df = \\int (- \\hat{X} + f) df and \\sin{(\\int y{(f,\\hat{X})} df)} = \\sin{(\\int (- \\hat{X} + f) df)} and \\int y{(f,\\hat{X})} df = - \\hat{X} f + \\mathbf{J}_P + \\frac{f^{2}}{2} and \\int (- \\hat{X} + f) df = - \\hat{X} f + \\mathbf{J}_P + \\frac{f^{2}}{2} and \\sin{(\\int y{(f,\\hat{X})} df)} = \\sin{(- \\hat{X} f + \\mathbf{J}_P + \\frac{f^{2}}{2})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('y')(Symbol('f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('y')(Symbol('f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('f', commutative=True)))), sin(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(sin(Integral(Function('y')(Symbol('f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('f', commutative=True)))), sin(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Symbol('f', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('f', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\mathbf{P}{(n_{2})} = \\sin{(n_{2})} and \\operatorname{f_{\\mathbf{p}}}{(n_{2})} = \\sin{(n_{2})} and \\operatorname{E_{\\lambda}}{(n_{2})} = \\sin{(n_{2})}, then obtain 2 \\mathbf{P}{(n_{2})} = \\operatorname{E_{\\lambda}}{(n_{2})} + \\mathbf{P}{(n_{2})}", "derivation": "\\mathbf{P}{(n_{2})} = \\sin{(n_{2})} and \\operatorname{f_{\\mathbf{p}}}{(n_{2})} = \\sin{(n_{2})} and \\mathbf{P}{(n_{2})} = \\operatorname{f_{\\mathbf{p}}}{(n_{2})} and 2 \\operatorname{f_{\\mathbf{p}}}{(n_{2})} = \\operatorname{f_{\\mathbf{p}}}{(n_{2})} + \\sin{(n_{2})} and \\operatorname{E_{\\lambda}}{(n_{2})} = \\sin{(n_{2})} and 2 \\mathbf{P}{(n_{2})} = \\mathbf{P}{(n_{2})} + \\sin{(n_{2})} and 2 \\mathbf{P}{(n_{2})} = \\operatorname{E_{\\lambda}}{(n_{2})} + \\mathbf{P}{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)))"], [["add", 2, "Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True))"], "Equality(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True))), Add(Function('f_{\\\\mathbf{p}}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True))), Add(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True)), Function('\\\\mathbf{P}')(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(A_{z})} = \\sin{(A_{z})}, then obtain (\\frac{2 \\operatorname{A_{1}}{(A_{z})}}{\\operatorname{A_{1}}{(A_{z})} + \\sin{(A_{z})}})^{A_{z}} = 1", "derivation": "\\operatorname{A_{1}}{(A_{z})} = \\sin{(A_{z})} and 2 \\operatorname{A_{1}}{(A_{z})} = \\operatorname{A_{1}}{(A_{z})} + \\sin{(A_{z})} and \\frac{2 \\operatorname{A_{1}}{(A_{z})}}{\\operatorname{A_{1}}{(A_{z})} + \\sin{(A_{z})}} = 1 and (\\frac{2 \\operatorname{A_{1}}{(A_{z})}}{\\operatorname{A_{1}}{(A_{z})} + \\sin{(A_{z})}})^{A_{z}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["add", 1, "Function('A_1')(Symbol('A_z', commutative=True))"], "Equality(Mul(Integer(2), Function('A_1')(Symbol('A_z', commutative=True))), Add(Function('A_1')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True))))"], [["divide", 2, "Add(Function('A_1')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Function('A_1')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True))), Integer(-1)), Function('A_1')(Symbol('A_z', commutative=True))), Integer(1))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Add(Function('A_1')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True))), Integer(-1)), Function('A_1')(Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\rho_{f}{(\\hbar)} = \\cos{(\\hbar)}, then obtain (\\int (- \\rho_{f}{(\\hbar)} + \\cos{(\\hbar)})^{\\hbar} d\\hbar)^{\\hbar} = (\\int 0^{\\hbar} d\\hbar)^{\\hbar}", "derivation": "\\rho_{f}{(\\hbar)} = \\cos{(\\hbar)} and - \\rho_{f}{(\\hbar)} = - \\cos{(\\hbar)} and - \\rho_{f}{(\\hbar)} + \\cos{(\\hbar)} = 0 and (- \\rho_{f}{(\\hbar)} + \\cos{(\\hbar)})^{\\hbar} = 0^{\\hbar} and \\int (- \\rho_{f}{(\\hbar)} + \\cos{(\\hbar)})^{\\hbar} d\\hbar = \\int 0^{\\hbar} d\\hbar and (\\int (- \\rho_{f}{(\\hbar)} + \\cos{(\\hbar)})^{\\hbar} d\\hbar)^{\\hbar} = (\\int 0^{\\hbar} d\\hbar)^{\\hbar}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Integer(0))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integer(0), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["power", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Pow(Integer(0), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\rho_b)} = e^{e^{\\rho_b}}, then derive \\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)} = e^{\\rho_b} e^{e^{\\rho_b}}, then obtain 1 + \\frac{\\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)}}{\\rho_b} = 1 + \\frac{\\mathbf{J}_P{(\\rho_b)} e^{\\rho_b}}{\\rho_b}", "derivation": "\\mathbf{J}_P{(\\rho_b)} = e^{e^{\\rho_b}} and \\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)} = \\frac{d}{d \\rho_b} e^{e^{\\rho_b}} and \\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)} = e^{\\rho_b} e^{e^{\\rho_b}} and \\frac{\\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)}}{\\rho_b} = \\frac{e^{\\rho_b} e^{e^{\\rho_b}}}{\\rho_b} and 1 + \\frac{\\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)}}{\\rho_b} = 1 + \\frac{e^{\\rho_b} e^{e^{\\rho_b}}}{\\rho_b} and 1 + \\frac{\\frac{d}{d \\rho_b} \\mathbf{J}_P{(\\rho_b)}}{\\rho_b} = 1 + \\frac{\\mathbf{J}_P{(\\rho_b)} e^{\\rho_b}}{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True)))))"], [["divide", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))), Add(Integer(1), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))), Add(Integer(1), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\phi_2)} = \\log{(\\log{(\\phi_2)})}, then derive \\frac{d}{d \\phi_2} \\operatorname{f_{E}}{(\\phi_2)} = \\frac{1}{\\phi_2 \\log{(\\phi_2)}}, then obtain \\int \\frac{d}{d \\phi_2} \\log{(\\log{(\\phi_2)})} d\\phi_2 = \\int \\frac{1}{\\phi_2 \\log{(\\phi_2)}} d\\phi_2", "derivation": "\\operatorname{f_{E}}{(\\phi_2)} = \\log{(\\log{(\\phi_2)})} and \\frac{d}{d \\phi_2} \\operatorname{f_{E}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\log{(\\log{(\\phi_2)})} and \\frac{d}{d \\phi_2} \\operatorname{f_{E}}{(\\phi_2)} = \\frac{1}{\\phi_2 \\log{(\\phi_2)}} and \\frac{d}{d \\phi_2} \\log{(\\log{(\\phi_2)})} = \\frac{1}{\\phi_2 \\log{(\\phi_2)}} and \\int \\frac{d}{d \\phi_2} \\log{(\\log{(\\phi_2)})} d\\phi_2 = \\int \\frac{1}{\\phi_2 \\log{(\\phi_2)}} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\phi_2', commutative=True)), log(log(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Derivative(log(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given p{(\\rho_f,m_{s})} = - \\rho_f + m_{s}, then derive \\int p{(\\rho_f,m_{s})} dm_{s} = \\mathbf{J} - \\rho_f m_{s} + \\frac{m_{s}^{2}}{2}, then obtain (\\frac{\\int p{(\\rho_f,m_{s})} dm_{s}}{2})^{\\rho_f} = (\\frac{\\mathbf{J}}{2} - \\frac{\\rho_f m_{s}}{2} + \\frac{m_{s}^{2}}{4})^{\\rho_f}", "derivation": "p{(\\rho_f,m_{s})} = - \\rho_f + m_{s} and \\int p{(\\rho_f,m_{s})} dm_{s} = \\int (- \\rho_f + m_{s}) dm_{s} and \\int p{(\\rho_f,m_{s})} dm_{s} = \\mathbf{J} - \\rho_f m_{s} + \\frac{m_{s}^{2}}{2} and \\frac{\\int p{(\\rho_f,m_{s})} dm_{s}}{2} = \\frac{\\mathbf{J}}{2} - \\frac{\\rho_f m_{s}}{2} + \\frac{m_{s}^{2}}{4} and (\\frac{\\int p{(\\rho_f,m_{s})} dm_{s}}{2})^{\\rho_f} = (\\frac{\\mathbf{J}}{2} - \\frac{\\rho_f m_{s}}{2} + \\frac{m_{s}^{2}}{4})^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('m_s', commutative=True), Integer(2)))))"], [["times", 3, "Rational(1, 2)"], "Equality(Mul(Rational(1, 2), Integral(Function('p')(Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Mul(Rational(1, 4), Pow(Symbol('m_s', commutative=True), Integer(2)))))"], [["power", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Integral(Function('p')(Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\rho_f', commutative=True), Symbol('m_s', commutative=True)), Mul(Rational(1, 4), Pow(Symbol('m_s', commutative=True), Integer(2)))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(V,r)} = V - r, then derive \\frac{\\partial}{\\partial V} \\Psi_{nl}{(V,r)} = 1, then obtain \\int 1 dV = \\int \\frac{1}{\\frac{\\partial}{\\partial V} (V - r)} dV", "derivation": "\\Psi_{nl}{(V,r)} = V - r and \\frac{\\partial}{\\partial V} \\Psi_{nl}{(V,r)} = \\frac{\\partial}{\\partial V} (V - r) and \\frac{\\partial}{\\partial V} \\Psi_{nl}{(V,r)} = 1 and \\frac{\\partial}{\\partial V} (V - r) = 1 and \\frac{\\frac{\\partial}{\\partial V} (V - r)}{\\frac{\\partial}{\\partial V} \\Psi_{nl}{(V,r)}} = \\frac{1}{\\frac{\\partial}{\\partial V} \\Psi_{nl}{(V,r)}} and 1 = \\frac{1}{\\frac{\\partial}{\\partial V} (V - r)} and \\int 1 dV = \\int \\frac{1}{\\frac{\\partial}{\\partial V} (V - r)} dV", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Derivative(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 6, "Symbol('V', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Pow(Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given E{(\\hat{H})} = \\cos{(\\hat{H})} and \\delta{(\\hat{H})} = E{(\\hat{H})} - \\cos{(\\hat{H})}, then obtain E{(\\hat{H})} \\delta{(\\hat{H})} = 0", "derivation": "E{(\\hat{H})} = \\cos{(\\hat{H})} and \\delta{(\\hat{H})} = E{(\\hat{H})} - \\cos{(\\hat{H})} and \\delta{(\\hat{H})} = 0 and E{(\\hat{H})} \\delta{(\\hat{H})} = 0", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True)), Add(Function('E')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True)), Integer(0))"], [["times", 3, "Function('E')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Function('E')(Symbol('\\\\hat{H}', commutative=True)), Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\phi_{2}{(G)} = \\frac{d}{d G} e^{G}, then derive \\frac{d}{d G} \\int \\phi_{2}{(G)} dG = \\frac{\\partial}{\\partial G} (\\varphi^* + e^{G}), then obtain \\frac{d}{d G} \\int \\frac{d}{d G} e^{G} dG = \\frac{\\partial}{\\partial G} (\\varphi^* + e^{G})", "derivation": "\\phi_{2}{(G)} = \\frac{d}{d G} e^{G} and \\int \\phi_{2}{(G)} dG = \\int \\frac{d}{d G} e^{G} dG and \\frac{d}{d G} \\int \\phi_{2}{(G)} dG = \\frac{d}{d G} \\int \\frac{d}{d G} e^{G} dG and \\frac{d}{d G} \\int \\phi_{2}{(G)} dG = \\frac{\\partial}{\\partial G} (\\varphi^* + e^{G}) and \\frac{d}{d G} \\int \\frac{d}{d G} e^{G} dG = \\frac{\\partial}{\\partial G} (\\varphi^* + e^{G})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('G', commutative=True)), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\phi_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(C_{1},F_{c},L)} = C_{1} F_{c} L and \\mathbf{g}{(C_{1},F_{c},L)} = \\sin{(C_{1} F_{c} L)}, then obtain - \\mu_0 + \\frac{\\sin{(\\dot{z}{(C_{1},F_{c},L)})}}{\\mathbf{J}_P + \\mu_0} = - \\mu_0 + \\frac{\\sin{(C_{1} F_{c} L)}}{\\mathbf{J}_P + \\mu_0}", "derivation": "\\dot{z}{(C_{1},F_{c},L)} = C_{1} F_{c} L and \\sin{(\\dot{z}{(C_{1},F_{c},L)})} = \\sin{(C_{1} F_{c} L)} and \\mathbf{g}{(C_{1},F_{c},L)} = \\sin{(C_{1} F_{c} L)} and \\mathbf{g}{(C_{1},F_{c},L)} = \\sin{(\\dot{z}{(C_{1},F_{c},L)})} and \\frac{\\mathbf{g}{(C_{1},F_{c},L)}}{\\mathbf{J}_P + \\mu_0} = \\frac{\\sin{(C_{1} F_{c} L)}}{\\mathbf{J}_P + \\mu_0} and \\frac{\\sin{(\\dot{z}{(C_{1},F_{c},L)})}}{\\mathbf{J}_P + \\mu_0} = \\frac{\\sin{(C_{1} F_{c} L)}}{\\mathbf{J}_P + \\mu_0} and - \\mu_0 + \\frac{\\sin{(\\dot{z}{(C_{1},F_{c},L)})}}{\\mathbf{J}_P + \\mu_0} = - \\mu_0 + \\frac{\\sin{(C_{1} F_{c} L)}}{\\mathbf{J}_P + \\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\dot{z}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))), sin(Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)), sin(Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)), sin(Function('\\\\dot{z}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Function('\\\\mathbf{g}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), sin(Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), sin(Function('\\\\dot{z}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), sin(Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True)))))"], [["minus", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), sin(Function('\\\\dot{z}')(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), sin(Mul(Symbol('C_1', commutative=True), Symbol('F_c', commutative=True), Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(f^{*},\\ddot{x})} = \\ddot{x} f^{*}, then derive - f^{*} + \\frac{\\partial}{\\partial \\ddot{x}} \\Psi_{\\lambda}{(f^{*},\\ddot{x})} = 0, then obtain \\frac{- f^{*} + \\frac{\\partial}{\\partial \\ddot{x}} \\ddot{x} f^{*}}{\\frac{d}{d \\ddot{x}} 0} = 0", "derivation": "\\Psi_{\\lambda}{(f^{*},\\ddot{x})} = \\ddot{x} f^{*} and - \\ddot{x} f^{*} + \\Psi_{\\lambda}{(f^{*},\\ddot{x})} = 0 and \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} f^{*} + \\Psi_{\\lambda}{(f^{*},\\ddot{x})}) = \\frac{d}{d \\ddot{x}} 0 and - f^{*} + \\frac{\\partial}{\\partial \\ddot{x}} \\Psi_{\\lambda}{(f^{*},\\ddot{x})} = 0 and \\frac{- f^{*} + \\frac{\\partial}{\\partial \\ddot{x}} \\Psi_{\\lambda}{(f^{*},\\ddot{x})}}{\\frac{d}{d \\ddot{x}} 0} = 0 and \\frac{- f^{*} + \\frac{\\partial}{\\partial \\ddot{x}} \\ddot{x} f^{*}}{\\frac{d}{d \\ddot{x}} 0} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('f^*', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('f^*', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('f^*', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('f^*', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('f^*', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Integer(0))"], [["divide", 4, "Derivative(Integer(0), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('f^*', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Derivative(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(m)} = \\frac{d}{d m} e^{m}, then derive \\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm = \\mathbf{v}, then obtain \\sin{(\\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm)} = \\sin{(\\int (- e^{m} + \\frac{d}{d m} e^{m}) dm)}", "derivation": "\\operatorname{A_{x}}{(m)} = \\frac{d}{d m} e^{m} and \\operatorname{A_{x}}{(m)} - e^{m} = - e^{m} + \\frac{d}{d m} e^{m} and \\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm = \\int (- e^{m} + \\frac{d}{d m} e^{m}) dm and \\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm = \\mathbf{v} and \\sin{(\\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm)} = \\sin{(\\mathbf{v})} and \\sin{(\\int (- e^{m} + \\frac{d}{d m} e^{m}) dm)} = \\sin{(\\mathbf{v})} and \\sin{(\\int (\\operatorname{A_{x}}{(m)} - e^{m}) dm)} = \\sin{(\\int (- e^{m} + \\frac{d}{d m} e^{m}) dm)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('m', commutative=True)), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 1, "exp(Symbol('m', commutative=True))"], "Equality(Add(Function('A_x')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Function('A_x')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('A_x')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True))"], [["sin", 4], "Equality(sin(Integral(Add(Function('A_x')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)))), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(sin(Integral(Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True)))), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(sin(Integral(Add(Function('A_x')(Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)))), sin(Integral(Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} = A_{y} V, then obtain (\\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} + \\int \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} dV)^{V} = (A_{y} V + \\int \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} dV)^{V}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} = A_{y} V and \\int \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} dV = \\int A_{y} V dV and \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} + \\int A_{y} V dV = A_{y} V + \\int A_{y} V dV and (\\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} + \\int A_{y} V dV)^{V} = (A_{y} V + \\int A_{y} V dV)^{V} and (\\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} + \\int \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} dV)^{V} = (A_{y} V + \\int \\operatorname{V_{\\mathbf{B}}}{(V,A_{y})} dV)^{V}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 1, "Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Add(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Integral(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('V', commutative=True)))), Symbol('V', commutative=True)), Pow(Add(Mul(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('V', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('V', commutative=True)))), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\hat{x}_0,C_{1})} = C_{1} + \\hat{x}_0, then obtain - 4 \\phi{(\\hat{x}_0,C_{1})} = - \\frac{(2 C_{1} + 2 \\hat{x}_0)^{2}}{\\phi{(\\hat{x}_0,C_{1})}}", "derivation": "\\phi{(\\hat{x}_0,C_{1})} = C_{1} + \\hat{x}_0 and C_{1} + \\hat{x}_0 + \\phi{(\\hat{x}_0,C_{1})} = 2 C_{1} + 2 \\hat{x}_0 and (C_{1} + \\hat{x}_0 + \\phi{(\\hat{x}_0,C_{1})})^{2} = (2 C_{1} + 2 \\hat{x}_0)^{2} and 2 \\phi{(\\hat{x}_0,C_{1})} = 2 C_{1} + 2 \\hat{x}_0 and 2 \\phi{(\\hat{x}_0,C_{1})} = C_{1} + \\hat{x}_0 + \\phi{(\\hat{x}_0,C_{1})} and 4 \\phi^{2}{(\\hat{x}_0,C_{1})} = (2 C_{1} + 2 \\hat{x}_0)^{2} and - 4 \\phi{(\\hat{x}_0,C_{1})} = - \\frac{(2 C_{1} + 2 \\hat{x}_0)^{2}}{\\phi{(\\hat{x}_0,C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "Add(Symbol('C_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Add(Symbol('C_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(4), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True))), Integer(2)))"], [["divide", 6, "Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(4), Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}_0', commutative=True))), Integer(2)), Pow(Function('\\\\phi')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Omega{(\\eta,F_{x})} = \\frac{\\partial}{\\partial F_{x}} (- F_{x} + \\eta), then derive \\Omega{(\\eta,F_{x})} = -1, then obtain 1 - \\frac{\\partial}{\\partial F_{x}} (- F_{x} + \\eta) = 2", "derivation": "\\Omega{(\\eta,F_{x})} = \\frac{\\partial}{\\partial F_{x}} (- F_{x} + \\eta) and \\Omega{(\\eta,F_{x})} = -1 and \\Omega{(\\eta,F_{x})} - 1 = -2 and \\frac{\\partial}{\\partial F_{x}} (- F_{x} + \\eta) - 1 = -2 and 1 - \\frac{\\partial}{\\partial F_{x}} (- F_{x} + \\eta) = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))"], [["minus", 2, 1], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\eta', commutative=True), Symbol('F_x', commutative=True)), Integer(-1)), Integer(-2))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Integer(-2))"], [["times", 4, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Integer(2))"]]}, {"prompt": "Given \\mathbf{D}{(Q)} = \\cos{(Q)}, then obtain 2 \\frac{d}{d Q} \\mathbf{D}{(Q)} = - \\sin{(Q)} + \\frac{d}{d Q} \\mathbf{D}{(Q)}", "derivation": "\\mathbf{D}{(Q)} = \\cos{(Q)} and 2 \\mathbf{D}{(Q)} = \\mathbf{D}{(Q)} + \\cos{(Q)} and \\frac{d}{d Q} 2 \\mathbf{D}{(Q)} = \\frac{d}{d Q} (\\mathbf{D}{(Q)} + \\cos{(Q)}) and 2 \\frac{d}{d Q} \\mathbf{D}{(Q)} = - \\sin{(Q)} + \\frac{d}{d Q} \\mathbf{D}{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{D}')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('Q', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{D}')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{D}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{D}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Derivative(Function('\\\\mathbf{D}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(\\dot{x})} = \\dot{x}, then derive \\dot{x} \\frac{d}{d \\dot{x}} \\theta{(\\dot{x})} = \\dot{x}, then obtain \\dot{x} \\frac{d}{d \\dot{x}} \\dot{x} - \\mathbf{J}_M = \\dot{x} - \\mathbf{J}_M", "derivation": "\\theta{(\\dot{x})} = \\dot{x} and \\frac{d}{d \\dot{x}} \\theta{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\dot{x} and \\dot{x} \\frac{d}{d \\dot{x}} \\theta{(\\dot{x})} = \\dot{x} \\frac{d}{d \\dot{x}} \\dot{x} and \\dot{x} \\frac{d}{d \\dot{x}} \\theta{(\\dot{x})} = \\dot{x} and \\dot{x} \\frac{d}{d \\dot{x}} \\theta{(\\dot{x})} - \\mathbf{J}_M = \\dot{x} - \\mathbf{J}_M and \\dot{x} \\frac{d}{d \\dot{x}} \\dot{x} - \\mathbf{J}_M = \\dot{x} - \\mathbf{J}_M", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Symbol('\\\\dot{x}', commutative=True), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Symbol('\\\\dot{x}', commutative=True), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Symbol('\\\\dot{x}', commutative=True))"], [["minus", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Function('\\\\theta')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Derivative(Symbol('\\\\dot{x}', commutative=True), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(r_{0},q)} = q + e^{r_{0}}, then obtain \\frac{d^{2}}{d r_{0}d q} 0 = \\frac{\\partial^{2}}{\\partial r_{0}\\partial q} (q - \\operatorname{n_{1}}{(r_{0},q)} + e^{r_{0}}) e^{- r_{0}}", "derivation": "\\operatorname{n_{1}}{(r_{0},q)} = q + e^{r_{0}} and 0 = q - \\operatorname{n_{1}}{(r_{0},q)} + e^{r_{0}} and 0 = (q - \\operatorname{n_{1}}{(r_{0},q)} + e^{r_{0}}) e^{- r_{0}} and \\frac{d}{d q} 0 = \\frac{\\partial}{\\partial q} (q - \\operatorname{n_{1}}{(r_{0},q)} + e^{r_{0}}) e^{- r_{0}} and \\frac{d^{2}}{d r_{0}d q} 0 = \\frac{\\partial^{2}}{\\partial r_{0}\\partial q} (q - \\operatorname{n_{1}}{(r_{0},q)} + e^{r_{0}}) e^{- r_{0}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Add(Symbol('q', commutative=True), exp(Symbol('r_0', commutative=True))))"], [["minus", 1, "Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True))), exp(Symbol('r_0', commutative=True))))"], [["divide", 2, "exp(Symbol('r_0', commutative=True))"], "Equality(Integer(0), Mul(Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True))), exp(Symbol('r_0', commutative=True))), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True))), exp(Symbol('r_0', commutative=True))), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('q', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('r_0', commutative=True), Symbol('q', commutative=True))), exp(Symbol('r_0', commutative=True))), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{s})} = \\sin{(\\sin{(\\mathbf{s})})}, then obtain \\int (\\lambda^{\\mathbf{s}}{(\\mathbf{s})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s}) d\\mathbf{s} = \\int (\\sin^{\\mathbf{s}}{(\\sin{(\\mathbf{s})})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s}) d\\mathbf{s}", "derivation": "\\lambda{(\\mathbf{s})} = \\sin{(\\sin{(\\mathbf{s})})} and \\lambda^{\\mathbf{s}}{(\\mathbf{s})} = \\sin^{\\mathbf{s}}{(\\sin{(\\mathbf{s})})} and \\lambda^{\\mathbf{s}}{(\\mathbf{s})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s} = \\sin^{\\mathbf{s}}{(\\sin{(\\mathbf{s})})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s} and \\int (\\lambda^{\\mathbf{s}}{(\\mathbf{s})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s}) d\\mathbf{s} = \\int (\\sin^{\\mathbf{s}}{(\\sin{(\\mathbf{s})})} - \\int \\lambda{(\\mathbf{s})} d\\mathbf{s}) d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), sin(sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 2, "Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Add(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(Pow(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"], [["integrate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Pow(sin(sin(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\pi,\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} - \\pi)}, then obtain \\iint \\Psi_{\\lambda} \\mathbf{F}^{\\pi}{(\\pi,\\Psi_{\\lambda})} d\\pi d\\Psi_{\\lambda} = \\iint \\Psi_{\\lambda} \\sin^{\\pi}{(\\Psi_{\\lambda} - \\pi)} d\\pi d\\Psi_{\\lambda}", "derivation": "\\mathbf{F}{(\\pi,\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda} - \\pi)} and \\mathbf{F}^{\\pi}{(\\pi,\\Psi_{\\lambda})} = \\sin^{\\pi}{(\\Psi_{\\lambda} - \\pi)} and \\Psi_{\\lambda} \\mathbf{F}^{\\pi}{(\\pi,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\sin^{\\pi}{(\\Psi_{\\lambda} - \\pi)} and \\int \\Psi_{\\lambda} \\mathbf{F}^{\\pi}{(\\pi,\\Psi_{\\lambda})} d\\pi = \\int \\Psi_{\\lambda} \\sin^{\\pi}{(\\Psi_{\\lambda} - \\pi)} d\\pi and \\iint \\Psi_{\\lambda} \\mathbf{F}^{\\pi}{(\\pi,\\Psi_{\\lambda})} d\\pi d\\Psi_{\\lambda} = \\iint \\Psi_{\\lambda} \\sin^{\\pi}{(\\Psi_{\\lambda} - \\pi)} d\\pi d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)))"], [["times", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(sin(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given B{(W)} = \\log{(W)}, then derive \\frac{d}{d W} B{(W)} = \\frac{1}{W}, then obtain B{(W)} \\frac{d}{d W} B{(W)} = \\frac{\\log{(W)}}{W}", "derivation": "B{(W)} = \\log{(W)} and \\frac{B{(W)}}{W} = \\frac{\\log{(W)}}{W} and \\frac{d}{d W} B{(W)} = \\frac{d}{d W} \\log{(W)} and \\frac{d}{d W} B{(W)} = \\frac{1}{W} and B{(W)} \\frac{d}{d W} B{(W)} = \\frac{B{(W)}}{W} and B{(W)} \\frac{d}{d W} B{(W)} = \\frac{\\log{(W)}}{W}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["divide", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('B')(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('B')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Pow(Symbol('W', commutative=True), Integer(-1)))"], [["times", 4, "Function('B')(Symbol('W', commutative=True))"], "Equality(Mul(Function('B')(Symbol('W', commutative=True)), Derivative(Function('B')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('B')(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('B')(Symbol('W', commutative=True)), Derivative(Function('B')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), log(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(A)} = e^{\\cos{(A)}}, then obtain - 2 \\theta_{2}{(A)} + \\frac{d}{d A} \\theta_{2}{(A)} + 1 = - 2 \\theta_{2}{(A)} + \\frac{d}{d A} e^{\\cos{(A)}} + 1", "derivation": "\\theta_{2}{(A)} = e^{\\cos{(A)}} and \\frac{d}{d A} \\theta_{2}{(A)} = \\frac{d}{d A} e^{\\cos{(A)}} and - 2 \\theta_{2}{(A)} + \\frac{d}{d A} \\theta_{2}{(A)} = - 2 \\theta_{2}{(A)} + \\frac{d}{d A} e^{\\cos{(A)}} and - 2 \\theta_{2}{(A)} + \\frac{d}{d A} \\theta_{2}{(A)} + 1 = - 2 \\theta_{2}{(A)} + \\frac{d}{d A} e^{\\cos{(A)}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('A', commutative=True)), exp(cos(Symbol('A', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(2), Function('\\\\theta_2')(Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\theta_2')(Symbol('A', commutative=True))), Derivative(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\theta_2')(Symbol('A', commutative=True))), Derivative(exp(cos(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["add", 3, 1], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\theta_2')(Symbol('A', commutative=True))), Derivative(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Integer(2), Function('\\\\theta_2')(Symbol('A', commutative=True))), Derivative(exp(cos(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} = \\frac{F_{x}}{\\dot{\\mathbf{r}}} - \\mathbf{S}, then obtain \\int (- \\mathbf{S} + \\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} + \\frac{1}{\\dot{\\mathbf{r}}}) dF_{x} = \\int (\\frac{F_{x}}{\\dot{\\mathbf{r}}} - 2 \\mathbf{S} + \\frac{1}{\\dot{\\mathbf{r}}}) dF_{x}", "derivation": "\\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} = \\frac{F_{x}}{\\dot{\\mathbf{r}}} - \\mathbf{S} and \\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} + \\frac{1}{\\dot{\\mathbf{r}}} = \\frac{F_{x}}{\\dot{\\mathbf{r}}} - \\mathbf{S} + \\frac{1}{\\dot{\\mathbf{r}}} and - \\mathbf{S} + \\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} + \\frac{1}{\\dot{\\mathbf{r}}} = \\frac{F_{x}}{\\dot{\\mathbf{r}}} - 2 \\mathbf{S} + \\frac{1}{\\dot{\\mathbf{r}}} and \\int (- \\mathbf{S} + \\hat{X}{(\\dot{\\mathbf{r}},\\mathbf{S},F_{x})} + \\frac{1}{\\dot{\\mathbf{r}}}) dF_{x} = \\int (\\frac{F_{x}}{\\dot{\\mathbf{r}}} - 2 \\mathbf{S} + \\frac{1}{\\dot{\\mathbf{r}}}) dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["minus", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Tuple(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(P_{e})} = e^{P_{e}}, then obtain - \\sigma_{p}{(P_{e})} - 1 = - 4 \\sigma_{p}{(P_{e})} + 3 e^{P_{e}} - 1", "derivation": "\\sigma_{p}{(P_{e})} = e^{P_{e}} and 0 = - \\sigma_{p}{(P_{e})} + e^{P_{e}} and - \\sigma_{p}{(P_{e})} = - 2 \\sigma_{p}{(P_{e})} + e^{P_{e}} and - \\sigma_{p}{(P_{e})} - 1 = - 2 \\sigma_{p}{(P_{e})} + e^{P_{e}} - 1 and - 2 \\sigma_{p}{(P_{e})} + e^{P_{e}} - 1 = - 4 \\sigma_{p}{(P_{e})} + 3 e^{P_{e}} - 1 and - \\sigma_{p}{(P_{e})} - 1 = - 4 \\sigma_{p}{(P_{e})} + 3 e^{P_{e}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["minus", 1, "Function('\\\\sigma_p')(Symbol('P_e', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))))"], [["minus", 2, "Function('\\\\sigma_p')(Symbol('P_e', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(4), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), Mul(Integer(3), exp(Symbol('P_e', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(4), Function('\\\\sigma_p')(Symbol('P_e', commutative=True))), Mul(Integer(3), exp(Symbol('P_e', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\rho{(B)} = \\cos{(B)}, then obtain B \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)} - \\cos{(B)} = \\frac{B \\cos{(B)} \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)}}{\\rho{(B)}} - \\cos{(B)}", "derivation": "\\rho{(B)} = \\cos{(B)} and B \\rho{(B)} = B \\cos{(B)} and B \\rho{(B)} \\cos{(B)} = B \\cos^{2}{(B)} and B \\rho^{2}{(B)} = B \\rho{(B)} \\cos{(B)} and B = \\frac{B \\cos{(B)}}{\\rho{(B)}} and B \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)} = \\frac{B \\cos{(B)} \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)}}{\\rho{(B)}} and B \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)} - \\cos{(B)} = \\frac{B \\cos{(B)} \\frac{\\partial}{\\partial k} \\mathbf{p}{(k,\\varphi^*)}}{\\rho{(B)}} - \\cos{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["times", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Function('\\\\rho')(Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), cos(Symbol('B', commutative=True))))"], [["times", 2, "cos(Symbol('B', commutative=True))"], "Equality(Mul(Symbol('B', commutative=True), Function('\\\\rho')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Pow(cos(Symbol('B', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('\\\\rho')(Symbol('B', commutative=True)), Integer(2))), Mul(Symbol('B', commutative=True), Function('\\\\rho')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))))"], [["divide", 4, "Pow(Function('\\\\rho')(Symbol('B', commutative=True)), Integer(2))"], "Equality(Symbol('B', commutative=True), Mul(Symbol('B', commutative=True), Pow(Function('\\\\rho')(Symbol('B', commutative=True)), Integer(-1)), cos(Symbol('B', commutative=True))))"], [["times", 5, "Derivative(Function('\\\\mathbf{p}')(Symbol('k', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('B', commutative=True), Derivative(Function('\\\\mathbf{p}')(Symbol('k', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Pow(Function('\\\\rho')(Symbol('B', commutative=True)), Integer(-1)), cos(Symbol('B', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('k', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["minus", 6, "cos(Symbol('B', commutative=True))"], "Equality(Add(Mul(Symbol('B', commutative=True), Derivative(Function('\\\\mathbf{p}')(Symbol('k', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Add(Mul(Symbol('B', commutative=True), Pow(Function('\\\\rho')(Symbol('B', commutative=True)), Integer(-1)), cos(Symbol('B', commutative=True)), Derivative(Function('\\\\mathbf{p}')(Symbol('k', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(v_{2},\\mathbf{M})} = \\frac{e^{\\mathbf{M}}}{v_{2}} and \\mathbf{F}{(v_{2},\\mathbf{M})} = \\frac{\\tilde{g}{(v_{2},\\mathbf{M})}}{v_{2}}, then obtain (\\frac{e^{\\mathbf{M}}}{v_{2}^{2}})^{\\mathbf{M}} = (\\frac{\\tilde{g}{(v_{2},\\mathbf{M})}}{v_{2}})^{\\mathbf{M}}", "derivation": "\\tilde{g}{(v_{2},\\mathbf{M})} = \\frac{e^{\\mathbf{M}}}{v_{2}} and \\mathbf{F}{(v_{2},\\mathbf{M})} = \\frac{\\tilde{g}{(v_{2},\\mathbf{M})}}{v_{2}} and \\mathbf{F}^{\\mathbf{M}}{(v_{2},\\mathbf{M})} = (\\frac{\\tilde{g}{(v_{2},\\mathbf{M})}}{v_{2}})^{\\mathbf{M}} and \\mathbf{F}^{\\mathbf{M}}{(v_{2},\\mathbf{M})} = (\\frac{e^{\\mathbf{M}}}{v_{2}^{2}})^{\\mathbf{M}} and (\\frac{e^{\\mathbf{M}}}{v_{2}^{2}})^{\\mathbf{M}} = (\\frac{\\tilde{g}{(v_{2},\\mathbf{M})}}{v_{2}})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('v_2', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\hat{H}_l,A_{z})} = \\cos{(A_{z} + \\hat{H}_l)}, then obtain - A_{z} - \\cos{(A_{z} + \\hat{H}_l)} + \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\hat{p}{(\\hat{H}_l,A_{z})}) = - A_{z} - \\cos{(A_{z} + \\hat{H}_l)} + \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\cos{(A_{z} + \\hat{H}_l)})", "derivation": "\\hat{p}{(\\hat{H}_l,A_{z})} = \\cos{(A_{z} + \\hat{H}_l)} and A_{z} + \\hat{p}{(\\hat{H}_l,A_{z})} = A_{z} + \\cos{(A_{z} + \\hat{H}_l)} and \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\hat{p}{(\\hat{H}_l,A_{z})}) = \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\cos{(A_{z} + \\hat{H}_l)}) and - A_{z} - \\cos{(A_{z} + \\hat{H}_l)} + \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\hat{p}{(\\hat{H}_l,A_{z})}) = - A_{z} - \\cos{(A_{z} + \\hat{H}_l)} + \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\cos{(A_{z} + \\hat{H}_l)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_z', commutative=True)), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 1, "Symbol('A_z', commutative=True)"], "Equality(Add(Symbol('A_z', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_z', commutative=True))), Add(Symbol('A_z', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["differentiate", 2, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('A_z', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Symbol('A_z', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["minus", 3, "Add(Symbol('A_z', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Derivative(Add(Symbol('A_z', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Derivative(Add(Symbol('A_z', commutative=True), cos(Add(Symbol('A_z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given t{(\\sigma_x,z^{*},\\hbar)} = \\hbar (\\sigma_x + z^{*}), then obtain (\\frac{t{(\\sigma_x,z^{*},\\hbar)}}{z^{*}})^{\\sigma_x} - 1 = (\\frac{\\hbar (\\sigma_x + z^{*})}{z^{*}})^{\\sigma_x} - 1", "derivation": "t{(\\sigma_x,z^{*},\\hbar)} = \\hbar (\\sigma_x + z^{*}) and \\frac{t{(\\sigma_x,z^{*},\\hbar)}}{z^{*}} = \\frac{\\hbar (\\sigma_x + z^{*})}{z^{*}} and (\\frac{t{(\\sigma_x,z^{*},\\hbar)}}{z^{*}})^{\\sigma_x} = (\\frac{\\hbar (\\sigma_x + z^{*})}{z^{*}})^{\\sigma_x} and (\\frac{t{(\\sigma_x,z^{*},\\hbar)}}{z^{*}})^{\\sigma_x} - 1 = (\\frac{\\hbar (\\sigma_x + z^{*})}{z^{*}})^{\\sigma_x} - 1", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True))))"], [["divide", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 3, 1], "Equality(Add(Pow(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Add(Pow(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('z^*', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(L)} = \\log{(L)} and \\operatorname{C_{d}}{(\\phi_1,\\varepsilon)} = e^{\\phi_1 - \\varepsilon}, then obtain \\operatorname{C_{d}}{(\\phi_1,\\varepsilon)} + \\log{(\\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L})} = e^{\\phi_1 - \\varepsilon} + \\log{(\\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L})}", "derivation": "\\Psi_{\\lambda}{(L)} = \\log{(L)} and \\Psi_{\\lambda}{(L)} + \\log{(L)} = 2 \\log{(L)} and \\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L} = \\frac{2 \\log{(L)}}{L} and \\log{(\\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L})} = \\log{(\\frac{2 \\log{(L)}}{L})} and \\operatorname{C_{d}}{(\\phi_1,\\varepsilon)} = e^{\\phi_1 - \\varepsilon} and \\operatorname{C_{d}}{(\\phi_1,\\varepsilon)} + \\log{(\\frac{2 \\log{(L)}}{L})} = e^{\\phi_1 - \\varepsilon} + \\log{(\\frac{2 \\log{(L)}}{L})} and \\operatorname{C_{d}}{(\\phi_1,\\varepsilon)} + \\log{(\\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L})} = e^{\\phi_1 - \\varepsilon} + \\log{(\\frac{\\Psi_{\\lambda}{(L)} + \\log{(L)}}{L})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["add", 1, "log(Symbol('L', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Mul(Integer(2), log(Symbol('L', commutative=True))))"], [["divide", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))), Mul(Integer(2), Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))))"], [["log", 3], "Equality(log(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))))), log(Mul(Integer(2), Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True)))))"], ["get_premise", "Equality(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), exp(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 5, "log(Mul(Integer(2), Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))))"], "Equality(Add(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), log(Mul(Integer(2), Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))))), Add(exp(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), log(Mul(Integer(2), Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('C_d')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), log(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))))), Add(exp(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), log(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})} = \\cos{(\\dot{y} + \\tilde{g}^*)}, then obtain - \\cos{(\\dot{y} + \\tilde{g}^*)} = - \\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})}", "derivation": "\\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})} = \\cos{(\\dot{y} + \\tilde{g}^*)} and \\tilde{g}^* + \\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})} = \\tilde{g}^* + \\cos{(\\dot{y} + \\tilde{g}^*)} and \\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})} - \\cos{(\\dot{y} + \\tilde{g}^*)} = 0 and - \\cos{(\\dot{y} + \\tilde{g}^*)} = - \\operatorname{n_{2}}{(\\tilde{g}^*,\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), cos(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), cos(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["minus", 2, "Add(Symbol('\\\\tilde{g}^*', commutative=True), cos(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], "Equality(Add(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))), Integer(0))"], [["minus", 3, "Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Integer(-1), cos(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Integer(-1), Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given i{(C_{d})} = \\sin{(\\sin{(C_{d})})} and S{(C_{d})} = \\int C_{d} \\sin{(\\sin{(C_{d})})} dC_{d}, then obtain - S{(C_{d})} + \\int C_{d} i{(C_{d})} dC_{d} = 0", "derivation": "i{(C_{d})} = \\sin{(\\sin{(C_{d})})} and C_{d} i{(C_{d})} = C_{d} \\sin{(\\sin{(C_{d})})} and \\int C_{d} i{(C_{d})} dC_{d} = \\int C_{d} \\sin{(\\sin{(C_{d})})} dC_{d} and S{(C_{d})} = \\int C_{d} \\sin{(\\sin{(C_{d})})} dC_{d} and \\int C_{d} i{(C_{d})} dC_{d} = S{(C_{d})} and - S{(C_{d})} + \\int C_{d} i{(C_{d})} dC_{d} = 0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('C_d', commutative=True)), sin(sin(Symbol('C_d', commutative=True))))"], [["times", 1, "Symbol('C_d', commutative=True)"], "Equality(Mul(Symbol('C_d', commutative=True), Function('i')(Symbol('C_d', commutative=True))), Mul(Symbol('C_d', commutative=True), sin(sin(Symbol('C_d', commutative=True)))))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Symbol('C_d', commutative=True), Function('i')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Symbol('C_d', commutative=True), sin(sin(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('C_d', commutative=True)), Integral(Mul(Symbol('C_d', commutative=True), sin(sin(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Mul(Symbol('C_d', commutative=True), Function('i')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Function('S')(Symbol('C_d', commutative=True)))"], [["minus", 5, "Function('S')(Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('C_d', commutative=True))), Integral(Mul(Symbol('C_d', commutative=True), Function('i')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{z}{(g,\\mathbf{E},\\hat{X})} = \\hat{X} + \\mathbf{E} + g and a{(g,\\mathbf{E},\\hat{X})} = \\hat{X} + \\mathbf{E} + g, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} (- \\hat{X} + \\dot{z}{(g,\\mathbf{E},\\hat{X})}) = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + g)", "derivation": "\\dot{z}{(g,\\mathbf{E},\\hat{X})} = \\hat{X} + \\mathbf{E} + g and a{(g,\\mathbf{E},\\hat{X})} = \\hat{X} + \\mathbf{E} + g and \\dot{z}{(g,\\mathbf{E},\\hat{X})} = a{(g,\\mathbf{E},\\hat{X})} and - \\hat{X} + a{(g,\\mathbf{E},\\hat{X})} = \\mathbf{E} + g and \\frac{\\partial}{\\partial \\mathbf{E}} (- \\hat{X} + a{(g,\\mathbf{E},\\hat{X})}) = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + g) and \\frac{\\partial}{\\partial \\mathbf{E}} (- \\hat{X} + \\dot{z}{(g,\\mathbf{E},\\hat{X})}) = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + g)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('a')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["minus", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('a')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('a')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\dot{z}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(v)} = e^{v}, then obtain \\log{(\\frac{d}{d v} (- M{(v)} + \\log{(M^{2}{(v)})}))} = \\log{(\\frac{d}{d v} (- M{(v)} + \\log{(M{(v)} e^{v})}))}", "derivation": "M{(v)} = e^{v} and M^{2}{(v)} = M{(v)} e^{v} and \\log{(M^{2}{(v)})} = \\log{(M{(v)} e^{v})} and - M{(v)} + \\log{(M^{2}{(v)})} = - M{(v)} + \\log{(M{(v)} e^{v})} and \\frac{d}{d v} (- M{(v)} + \\log{(M^{2}{(v)})}) = \\frac{d}{d v} (- M{(v)} + \\log{(M{(v)} e^{v})}) and \\log{(\\frac{d}{d v} (- M{(v)} + \\log{(M^{2}{(v)})}))} = \\log{(\\frac{d}{d v} (- M{(v)} + \\log{(M{(v)} e^{v})}))}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["times", 1, "Function('M')(Symbol('v', commutative=True))"], "Equality(Pow(Function('M')(Symbol('v', commutative=True)), Integer(2)), Mul(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))"], [["log", 2], "Equality(log(Pow(Function('M')(Symbol('v', commutative=True)), Integer(2))), log(Mul(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))))"], [["minus", 3, "Function('M')(Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Pow(Function('M')(Symbol('v', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Mul(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))))"], [["differentiate", 4, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Pow(Function('M')(Symbol('v', commutative=True)), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Mul(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["log", 5], "Equality(log(Derivative(Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Pow(Function('M')(Symbol('v', commutative=True)), Integer(2)))), Tuple(Symbol('v', commutative=True), Integer(1)))), log(Derivative(Add(Mul(Integer(-1), Function('M')(Symbol('v', commutative=True))), log(Mul(Function('M')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} = - f + g^{\\prime}_{\\varepsilon} and \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} = - f + g^{\\prime}_{\\varepsilon} + \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)}, then obtain - \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} - \\int 2 \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} dg^{\\prime}_{\\varepsilon} = 2 f - 2 g^{\\prime}_{\\varepsilon} - \\int 2 \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} dg^{\\prime}_{\\varepsilon}", "derivation": "\\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} = - f + g^{\\prime}_{\\varepsilon} and \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} = - f + g^{\\prime}_{\\varepsilon} + \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} and - \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} = f - g^{\\prime}_{\\varepsilon} - \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} and - \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} = 2 f - 2 g^{\\prime}_{\\varepsilon} and - \\bar{\\h}{(g^{\\prime}_{\\varepsilon},f)} - \\int 2 \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} dg^{\\prime}_{\\varepsilon} = 2 f - 2 g^{\\prime}_{\\varepsilon} - \\int 2 \\operatorname{v_{t}}{(g^{\\prime}_{\\varepsilon},f)} dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hbar')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Function('\\\\hbar')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["minus", 4, "Integral(Mul(Integer(2), Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), Function('v_t')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbb{I})} = \\mathbb{I} and v{(\\mathbb{I})} = \\frac{1}{\\mathbb{I}}, then obtain \\frac{d}{d \\mathbb{I}} \\frac{\\operatorname{v_{x}}{(\\mathbb{I})}}{\\mathbb{I} v{(\\mathbb{I})}} = \\frac{d}{d \\mathbb{I}} \\mathbb{I}", "derivation": "\\operatorname{v_{x}}{(\\mathbb{I})} = \\mathbb{I} and v{(\\mathbb{I})} = \\frac{1}{\\mathbb{I}} and 1 = \\frac{1}{\\mathbb{I} v{(\\mathbb{I})}} and \\operatorname{v_{x}}{(\\mathbb{I})} = \\frac{\\operatorname{v_{x}}{(\\mathbb{I})}}{\\mathbb{I} v{(\\mathbb{I})}} and \\frac{\\operatorname{v_{x}}{(\\mathbb{I})}}{\\mathbb{I} v{(\\mathbb{I})}} = \\mathbb{I} and \\frac{d}{d \\mathbb{I}} \\frac{\\operatorname{v_{x}}{(\\mathbb{I})}}{\\mathbb{I} v{(\\mathbb{I})}} = \\frac{d}{d \\mathbb{I}} \\mathbb{I}", "srepr_derivation": [["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))"], [["divide", 2, "Function('v')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Function('v')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1))))"], [["times", 3, "Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Function('v')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Function('v')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))"], [["differentiate", 5, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Function('v')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(m_{s},\\Omega)} = m_{s}^{\\Omega}, then obtain (\\operatorname{F_{g}}^{\\Omega}{(m_{s},\\Omega)})^{m_{s}} - (\\operatorname{F_{g}}^{m_{s}}{(m_{s},\\Omega)})^{m_{s}} = ((m_{s}^{\\Omega})^{\\Omega})^{m_{s}} - (\\operatorname{F_{g}}^{m_{s}}{(m_{s},\\Omega)})^{m_{s}}", "derivation": "\\operatorname{F_{g}}{(m_{s},\\Omega)} = m_{s}^{\\Omega} and \\operatorname{F_{g}}^{\\Omega}{(m_{s},\\Omega)} = (m_{s}^{\\Omega})^{\\Omega} and (\\operatorname{F_{g}}^{\\Omega}{(m_{s},\\Omega)})^{m_{s}} = ((m_{s}^{\\Omega})^{\\Omega})^{m_{s}} and (\\operatorname{F_{g}}^{\\Omega}{(m_{s},\\Omega)})^{m_{s}} - (\\operatorname{F_{g}}^{m_{s}}{(m_{s},\\Omega)})^{m_{s}} = ((m_{s}^{\\Omega})^{\\Omega})^{m_{s}} - (\\operatorname{F_{g}}^{m_{s}}{(m_{s},\\Omega)})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)))"], [["minus", 3, "Pow(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], "Equality(Add(Pow(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))), Add(Pow(Pow(Pow(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('F_g')(Symbol('m_s', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(L)} = \\log{(L)} and \\theta_{1}{(L)} = \\frac{d}{d L} \\log{(L)}, then obtain \\frac{d}{d L} (\\theta_{1}{(L)} - 1) = \\frac{d}{d L} (\\frac{d}{d L} \\log{(L)} - 1)", "derivation": "\\mathbf{J}_M{(L)} = \\log{(L)} and \\frac{d}{d L} \\mathbf{J}_M{(L)} = \\frac{d}{d L} \\log{(L)} and \\theta_{1}{(L)} = \\frac{d}{d L} \\log{(L)} and \\theta_{1}{(L)} = \\frac{d}{d L} \\mathbf{J}_M{(L)} and \\theta_{1}{(L)} - 1 = \\frac{d}{d L} \\mathbf{J}_M{(L)} - 1 and \\theta_{1}{(L)} - 1 = \\frac{d}{d L} \\log{(L)} - 1 and \\frac{d}{d L} (\\theta_{1}{(L)} - 1) = \\frac{d}{d L} (\\frac{d}{d L} \\log{(L)} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\theta_1')(Symbol('L', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(Function('\\\\theta_1')(Symbol('L', commutative=True)), Integer(-1)), Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\theta_1')(Symbol('L', commutative=True)), Integer(-1)), Add(Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta_1')(Symbol('L', commutative=True)), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(\\pi,C_{d},v)} = (- C_{d} + \\pi)^{v}, then obtain \\frac{\\partial}{\\partial v} u{(\\pi,C_{d},v)} = (- C_{d} + \\pi)^{v} \\log{(- C_{d} + \\pi)}", "derivation": "u{(\\pi,C_{d},v)} = (- C_{d} + \\pi)^{v} and u{(\\pi,C_{d},v)} - 1 = (- C_{d} + \\pi)^{v} - 1 and \\frac{\\partial}{\\partial v} (u{(\\pi,C_{d},v)} - 1) = \\frac{\\partial}{\\partial v} ((- C_{d} + \\pi)^{v} - 1) and \\frac{\\partial}{\\partial v} u{(\\pi,C_{d},v)} = (- C_{d} + \\pi)^{v} \\log{(- C_{d} + \\pi)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\pi', commutative=True), Symbol('C_d', commutative=True), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('v', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('u')(Symbol('\\\\pi', commutative=True), Symbol('C_d', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('v', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('u')(Symbol('\\\\pi', commutative=True), Symbol('C_d', commutative=True), Symbol('v', commutative=True)), Integer(-1)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('v', commutative=True)), Integer(-1)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('u')(Symbol('\\\\pi', commutative=True), Symbol('C_d', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('v', commutative=True)), log(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\omega{(E)} = E \\cos{(E)} and \\operatorname{M_{E}}{(E)} = (E \\cos{(E)})^{E} + \\cos{(E)}, then obtain - E \\cos{(E)} + \\operatorname{M_{E}}{(E)} = - E \\cos{(E)} + (E \\cos{(E)})^{E} + \\cos{(E)}", "derivation": "\\omega{(E)} = E \\cos{(E)} and \\omega^{E}{(E)} = (E \\cos{(E)})^{E} and \\omega^{E}{(E)} + \\cos{(E)} = (E \\cos{(E)})^{E} + \\cos{(E)} and - E \\cos{(E)} + \\omega^{E}{(E)} + \\cos{(E)} = - E \\cos{(E)} + (E \\cos{(E)})^{E} + \\cos{(E)} and \\operatorname{M_{E}}{(E)} = (E \\cos{(E)})^{E} + \\cos{(E)} and \\operatorname{M_{E}}{(E)} = \\omega^{E}{(E)} + \\cos{(E)} and - E \\cos{(E)} + \\operatorname{M_{E}}{(E)} = - E \\cos{(E)} + (E \\cos{(E)})^{E} + \\cos{(E)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["add", 2, "cos(Symbol('E', commutative=True))"], "Equality(Add(Pow(Function('\\\\omega')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Add(Pow(Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"], [["minus", 3, "Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Pow(Function('\\\\omega')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Pow(Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('E', commutative=True)), Add(Pow(Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Function('M_E')(Symbol('E', commutative=True)), Add(Pow(Function('\\\\omega')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Function('M_E')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Pow(Mul(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(C_{d})} = \\sin{(C_{d})}, then derive \\int \\bar{\\h}{(C_{d})} dC_{d} = \\varphi - \\cos{(C_{d})}, then obtain \\int (2 \\varphi - 2 \\cos{(C_{d})}) d\\varphi = \\int 2 \\int \\sin{(C_{d})} dC_{d} d\\varphi", "derivation": "\\bar{\\h}{(C_{d})} = \\sin{(C_{d})} and \\int \\bar{\\h}{(C_{d})} dC_{d} = \\int \\sin{(C_{d})} dC_{d} and \\int \\bar{\\h}{(C_{d})} dC_{d} = \\varphi - \\cos{(C_{d})} and \\varphi - \\cos{(C_{d})} = \\int \\sin{(C_{d})} dC_{d} and 2 \\varphi - 2 \\cos{(C_{d})} = 2 \\int \\sin{(C_{d})} dC_{d} and \\int (2 \\varphi - 2 \\cos{(C_{d})}) d\\varphi = \\int 2 \\int \\sin{(C_{d})} dC_{d} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hbar')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Integral(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 4, "Rational(1, 2)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('C_d', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('C_d', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Integer(2), Integral(sin(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\Omega{(C_{d},s)} = C_{d} + \\log{(s)} and \\operatorname{v_{t}}{(C_{d},s)} = C_{d} + \\log{(s)}, then obtain \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{t}}{(C_{d},s)} = \\frac{\\partial}{\\partial C_{d}} \\Omega{(C_{d},s)}", "derivation": "\\Omega{(C_{d},s)} = C_{d} + \\log{(s)} and \\operatorname{v_{t}}{(C_{d},s)} = C_{d} + \\log{(s)} and \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{t}}{(C_{d},s)} = \\frac{\\partial}{\\partial C_{d}} (C_{d} + \\log{(s)}) and \\frac{\\partial}{\\partial C_{d}} \\operatorname{v_{t}}{(C_{d},s)} = \\frac{\\partial}{\\partial C_{d}} \\Omega{(C_{d},s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Add(Symbol('C_d', commutative=True), log(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Add(Symbol('C_d', commutative=True), log(Symbol('s', commutative=True))))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Symbol('C_d', commutative=True), log(Symbol('s', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('v_t')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Function('\\\\Omega')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)}, then derive (G{(\\mu_0)} \\log{(\\mu_0)})^{\\mu_0} = (\\frac{\\log{(\\mu_0)}}{\\mu_0})^{\\mu_0}, then obtain \\int (\\log{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)})^{\\mu_0} d\\mu_0 = \\int (\\frac{\\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} d\\mu_0", "derivation": "G{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and G{(\\mu_0)} \\log{(\\mu_0)} = \\log{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and (G{(\\mu_0)} \\log{(\\mu_0)})^{\\mu_0} = (\\log{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)})^{\\mu_0} and (G{(\\mu_0)} \\log{(\\mu_0)})^{\\mu_0} = (\\frac{\\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} and (\\log{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)})^{\\mu_0} = (\\frac{\\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} and \\int (\\log{(\\mu_0)} \\frac{d}{d \\mu_0} \\log{(\\mu_0)})^{\\mu_0} d\\mu_0 = \\int (\\frac{\\log{(\\mu_0)}}{\\mu_0})^{\\mu_0} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["times", 1, "log(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True))), Mul(log(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(log(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(log(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Pow(Mul(log(Symbol('\\\\mu_0', commutative=True)), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), log(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\chi)} = \\cos{(\\chi)}, then obtain - \\log{(\\chi + \\mathbf{s}{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\mathbf{s}{(\\chi)}) = - \\log{(\\chi + \\mathbf{s}{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\cos{(\\chi)})", "derivation": "\\mathbf{s}{(\\chi)} = \\cos{(\\chi)} and \\chi + \\mathbf{s}{(\\chi)} = \\chi + \\cos{(\\chi)} and \\frac{d}{d \\chi} (\\chi + \\mathbf{s}{(\\chi)}) = \\frac{d}{d \\chi} (\\chi + \\cos{(\\chi)}) and \\log{(\\chi + \\mathbf{s}{(\\chi)})} = \\log{(\\chi + \\cos{(\\chi)})} and - \\log{(\\chi + \\cos{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\mathbf{s}{(\\chi)}) = - \\log{(\\chi + \\cos{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\cos{(\\chi)}) and - \\log{(\\chi + \\mathbf{s}{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\mathbf{s}{(\\chi)}) = - \\log{(\\chi + \\mathbf{s}{(\\chi)})} + \\frac{d}{d \\chi} (\\chi + \\cos{(\\chi)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True)))), log(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True)))))"], [["minus", 3, "log(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))), Derivative(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))))), Derivative(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), log(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))))), Derivative(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Add(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\chi', commutative=True))))), Derivative(Add(Symbol('\\\\chi', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(v_{x},v_{t})} = v_{x}^{v_{t}}, then obtain 2 (v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})})^{v_{x}} = v_{x}^{v_{x}} + (v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})})^{v_{x}}", "derivation": "\\pi{(v_{x},v_{t})} = v_{x}^{v_{t}} and v_{x} + \\pi{(v_{x},v_{t})} = v_{x} + v_{x}^{v_{t}} and v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})} = v_{x} and (v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})})^{v_{x}} = v_{x}^{v_{x}} and 2 (v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})})^{v_{x}} = v_{x}^{v_{x}} + (v_{x} - v_{x}^{v_{t}} + \\pi{(v_{x},v_{t})})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)))"], [["add", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Add(Symbol('v_x', commutative=True), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))))"], [["minus", 2, "Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_x', commutative=True))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_x', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 4, "Pow(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Pow(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_x', commutative=True))), Add(Pow(Symbol('v_x', commutative=True), Symbol('v_x', commutative=True)), Pow(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Function('\\\\pi')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\Psi_{nl})} = e^{e^{\\Psi_{nl}}}, then obtain - 2 e^{e^{\\Psi_{nl}}} + \\int \\mathbf{M}{(\\Psi_{nl})} d\\Psi_{nl} = - 2 e^{e^{\\Psi_{nl}}} + \\int e^{e^{\\Psi_{nl}}} d\\Psi_{nl}", "derivation": "\\mathbf{M}{(\\Psi_{nl})} = e^{e^{\\Psi_{nl}}} and \\int \\mathbf{M}{(\\Psi_{nl})} d\\Psi_{nl} = \\int e^{e^{\\Psi_{nl}}} d\\Psi_{nl} and - e^{e^{\\Psi_{nl}}} + \\int \\mathbf{M}{(\\Psi_{nl})} d\\Psi_{nl} = - e^{e^{\\Psi_{nl}}} + \\int e^{e^{\\Psi_{nl}}} d\\Psi_{nl} and - 2 e^{e^{\\Psi_{nl}}} + \\int \\mathbf{M}{(\\Psi_{nl})} d\\Psi_{nl} = - 2 e^{e^{\\Psi_{nl}}} + \\int e^{e^{\\Psi_{nl}}} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(exp(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["minus", 2, "exp(exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Mul(Integer(-1), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(exp(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["add", 3, "Mul(Integer(-1), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), exp(exp(Symbol('\\\\Psi_{nl}', commutative=True)))), Integral(exp(exp(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\pi{(C_{d},s)} = \\frac{\\partial}{\\partial s} (C_{d} + s), then derive \\pi{(C_{d},s)} = 1, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\frac{\\partial}{\\partial s} (C_{d} + s)}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})}} = \\frac{d}{d \\mathbf{A}} \\frac{1}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\pi{(C_{d},s)} = \\frac{\\partial}{\\partial s} (C_{d} + s) and \\pi{(C_{d},s)} = 1 and \\frac{\\pi{(C_{d},s)}}{\\cos{(\\mathbf{A})}} = \\frac{1}{\\cos{(\\mathbf{A})}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\pi{(C_{d},s)}}{\\cos{(\\mathbf{A})}} = \\frac{d}{d \\mathbf{A}} \\frac{1}{\\cos{(\\mathbf{A})}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\frac{\\partial}{\\partial s} (C_{d} + s)}{\\cos{(\\mathbf{A})}} = \\frac{d}{d \\mathbf{A}} \\frac{1}{\\cos{(\\mathbf{A})}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\frac{\\partial}{\\partial s} (C_{d} + s)}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})}} = \\frac{d}{d \\mathbf{A}} \\frac{1}{\\operatorname{V_{\\mathbf{E}}}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\pi')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Add(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Function('\\\\pi')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Integer(1))"], [["divide", 3, "cos(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('\\\\pi')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\pi')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Mul(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Derivative(Add(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(E_{n},I)} = E_{n} + I, then obtain (\\int (I + \\mu{(E_{n},I)})^{E_{n}} dE_{n})^{2} = (\\int (E_{n} + 2 I)^{E_{n}} dE_{n})^{2}", "derivation": "\\mu{(E_{n},I)} = E_{n} + I and I + \\mu{(E_{n},I)} = E_{n} + 2 I and (I + \\mu{(E_{n},I)})^{E_{n}} = (E_{n} + 2 I)^{E_{n}} and \\int (I + \\mu{(E_{n},I)})^{E_{n}} dE_{n} = \\int (E_{n} + 2 I)^{E_{n}} dE_{n} and (\\int (I + \\mu{(E_{n},I)})^{E_{n}} dE_{n})^{2} = (\\int (E_{n} + 2 I)^{E_{n}} dE_{n})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('I', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('I', commutative=True)))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('I', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))))"], [["power", 2, "Symbol('E_n', commutative=True)"], "Equality(Pow(Add(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)), Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Pow(Add(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(2)), Pow(Integral(Pow(Add(Symbol('E_n', commutative=True), Mul(Integer(2), Symbol('I', commutative=True))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integer(2)))"]]}, {"prompt": "Given Q{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then obtain - \\mathbf{S} (- \\mathbf{S} + \\frac{d}{d \\mathbf{S}} Q{(\\mathbf{S})}) = - \\mathbf{S} (- \\mathbf{S} - \\sin{(\\mathbf{S})})", "derivation": "Q{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\frac{d}{d \\mathbf{S}} Q{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})} and - \\mathbf{S} + \\frac{d}{d \\mathbf{S}} Q{(\\mathbf{S})} = - \\mathbf{S} + \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})} and - \\mathbf{S} (- \\mathbf{S} + \\frac{d}{d \\mathbf{S}} Q{(\\mathbf{S})}) = - \\mathbf{S} (- \\mathbf{S} + \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})}) and - \\mathbf{S} (- \\mathbf{S} + \\frac{d}{d \\mathbf{S}} Q{(\\mathbf{S})}) = - \\mathbf{S} (- \\mathbf{S} - \\sin{(\\mathbf{S})})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('Q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('Q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('Q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given z{(f_{E})} = \\sin{(f_{E})} and \\operatorname{v_{z}}{(f_{E})} = - \\frac{d}{d f_{E}} z{(f_{E})}, then derive \\operatorname{v_{z}}{(f_{E})} = - \\cos{(f_{E})}, then obtain 0 = - \\cos{(f_{E})} + \\frac{d}{d f_{E}} z{(f_{E})}", "derivation": "z{(f_{E})} = \\sin{(f_{E})} and \\frac{d}{d f_{E}} z{(f_{E})} = \\frac{d}{d f_{E}} \\sin{(f_{E})} and 0 = - \\frac{d}{d f_{E}} z{(f_{E})} + \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\operatorname{v_{z}}{(f_{E})} = - \\frac{d}{d f_{E}} z{(f_{E})} and 0 = \\operatorname{v_{z}}{(f_{E})} + \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\operatorname{v_{z}}{(f_{E})} = - \\frac{d}{d f_{E}} \\sin{(f_{E})} and \\operatorname{v_{z}}{(f_{E})} = - \\cos{(f_{E})} and 0 = \\operatorname{v_{z}}{(f_{E})} + \\frac{d}{d f_{E}} z{(f_{E})} and 0 = - \\cos{(f_{E})} + \\frac{d}{d f_{E}} z{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Function('v_z')(Symbol('f_E', commutative=True)), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('v_z')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Function('v_z')(Symbol('f_E', commutative=True)), Mul(Integer(-1), cos(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(0), Add(Function('v_z')(Symbol('f_E', commutative=True)), Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Integer(0), Add(Mul(Integer(-1), cos(Symbol('f_E', commutative=True))), Derivative(Function('z')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(x^\\prime,u)} = e^{u + x^\\prime}, then derive 1 = \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{F_{N}}{(x^\\prime,u)}}, then obtain (e^{u + x^\\prime})^{u} + 1 = \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime}} + (e^{u + x^\\prime})^{u}", "derivation": "\\operatorname{F_{N}}{(x^\\prime,u)} = e^{u + x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\operatorname{F_{N}}{(x^\\prime,u)} = \\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime} and 1 = \\frac{\\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{F_{N}}{(x^\\prime,u)}} and 1 = \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} \\operatorname{F_{N}}{(x^\\prime,u)}} and 1 = \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime}} and \\operatorname{F_{N}}^{u}{(x^\\prime,u)} + 1 = \\operatorname{F_{N}}^{u}{(x^\\prime,u)} + \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime}} and (e^{u + x^\\prime})^{u} + 1 = \\frac{e^{u + x^\\prime}}{\\frac{\\partial}{\\partial x^\\prime} e^{u + x^\\prime}} + (e^{u + x^\\prime})^{u}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)), Derivative(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Pow(Derivative(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Pow(Derivative(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))))"], [["add", 5, "Pow(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Add(Pow(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(1)), Add(Pow(Function('F_N')(Symbol('x^\\\\prime', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Mul(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Pow(Derivative(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Pow(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('u', commutative=True)), Integer(1)), Add(Mul(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Pow(Derivative(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Pow(exp(Add(Symbol('u', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mu{(f^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} - f^{\\prime}), then derive f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1 = f^{\\prime} + 2, then obtain \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + 2 \\mu{(f^{\\prime},E_{\\lambda})}) = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1)", "derivation": "\\mu{(f^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} - f^{\\prime}) and f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} = f^{\\prime} + \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} - f^{\\prime}) and f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1 = f^{\\prime} + \\frac{\\partial}{\\partial E_{\\lambda}} (E_{\\lambda} - f^{\\prime}) + 1 and f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1 = f^{\\prime} + 2 and \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1) = \\frac{d}{d f^{\\prime}} (f^{\\prime} + 2) and \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + 2 \\mu{(f^{\\prime},E_{\\lambda})}) = \\frac{\\partial}{\\partial f^{\\prime}} (f^{\\prime} + \\mu{(f^{\\prime},E_{\\lambda})} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["minus", 1, "Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Add(Symbol('f^{\\\\prime}', commutative=True), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Add(Symbol('f^{\\\\prime}', commutative=True), Integer(2)))"], [["differentiate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Integer(2)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(2), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(W)} = \\sin{(\\log{(W)})}, then derive \\frac{d}{d W} \\delta{(W)} = \\frac{\\cos{(\\log{(W)})}}{W}, then obtain \\frac{1}{\\cos{(\\log{(W)})}} = \\frac{1}{W \\frac{d}{d W} \\delta{(W)}}", "derivation": "\\delta{(W)} = \\sin{(\\log{(W)})} and \\frac{d}{d W} \\delta{(W)} = \\frac{d}{d W} \\sin{(\\log{(W)})} and \\frac{d}{d W} \\delta{(W)} = \\frac{\\cos{(\\log{(W)})}}{W} and \\frac{\\cos{(\\log{(W)})}}{W} = \\frac{d}{d W} \\sin{(\\log{(W)})} and 1 = \\frac{\\frac{d}{d W} \\sin{(\\log{(W)})}}{\\frac{d}{d W} \\delta{(W)}} and \\frac{1}{\\cos{(\\log{(W)})}} = \\frac{\\frac{d}{d W} \\sin{(\\log{(W)})}}{\\cos{(\\log{(W)})} \\frac{d}{d W} \\delta{(W)}} and \\frac{1}{\\cos{(\\log{(W)})}} = \\frac{1}{W \\frac{d}{d W} \\delta{(W)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\delta')(Symbol('W', commutative=True)), sin(log(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(log(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(log(Symbol('W', commutative=True)))), Derivative(sin(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["divide", 5, "cos(log(Symbol('W', commutative=True)))"], "Equality(Pow(cos(log(Symbol('W', commutative=True))), Integer(-1)), Mul(Pow(cos(log(Symbol('W', commutative=True))), Integer(-1)), Pow(Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(cos(log(Symbol('W', commutative=True))), Integer(-1)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\delta')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given s{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain \\log{(((\\frac{s{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}})} = \\log{(((\\frac{\\log{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}})}", "derivation": "s{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\frac{s{(\\mathbf{P})}}{\\mathbf{P}} = \\frac{\\log{(\\mathbf{P})}}{\\mathbf{P}} and (\\frac{s{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}} = (\\frac{\\log{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}} and ((\\frac{s{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}} = ((\\frac{\\log{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}} and \\log{(((\\frac{s{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}})} = \\log{(((\\frac{\\log{(\\mathbf{P})}}{\\mathbf{P}})^{\\mathbf{P}})^{\\mathbf{P}})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["log", 4], "Equality(log(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), log(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given v{(F_{N})} = \\cos{(\\cos{(F_{N})})}, then derive \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})} + \\frac{d}{d F_{N}} v{(F_{N})} = 2 \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})}, then obtain \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})} + \\frac{d}{d F_{N}} \\cos{(\\cos{(F_{N})})} = 2 \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})}", "derivation": "v{(F_{N})} = \\cos{(\\cos{(F_{N})})} and v{(F_{N})} + \\cos{(\\cos{(F_{N})})} = 2 \\cos{(\\cos{(F_{N})})} and \\frac{d}{d F_{N}} (v{(F_{N})} + \\cos{(\\cos{(F_{N})})}) = \\frac{d}{d F_{N}} 2 \\cos{(\\cos{(F_{N})})} and \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})} + \\frac{d}{d F_{N}} v{(F_{N})} = 2 \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})} and \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})} + \\frac{d}{d F_{N}} \\cos{(\\cos{(F_{N})})} = 2 \\sin{(F_{N})} \\sin{(\\cos{(F_{N})})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('F_N', commutative=True)), cos(cos(Symbol('F_N', commutative=True))))"], [["add", 1, "cos(cos(Symbol('F_N', commutative=True)))"], "Equality(Add(Function('v')(Symbol('F_N', commutative=True)), cos(cos(Symbol('F_N', commutative=True)))), Mul(Integer(2), cos(cos(Symbol('F_N', commutative=True)))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Function('v')(Symbol('F_N', commutative=True)), cos(cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(cos(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(sin(Symbol('F_N', commutative=True)), sin(cos(Symbol('F_N', commutative=True)))), Derivative(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Integer(2), sin(Symbol('F_N', commutative=True)), sin(cos(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(sin(Symbol('F_N', commutative=True)), sin(cos(Symbol('F_N', commutative=True)))), Derivative(cos(cos(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Integer(2), sin(Symbol('F_N', commutative=True)), sin(cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(C_{d},v_{z})} = - C_{d} + v_{z}, then obtain (\\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} \\mu_{0}{(C_{d},v_{z})}) dv_{z})^{v_{z}} = (\\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} (- C_{d} + v_{z})) dv_{z})^{v_{z}}", "derivation": "\\mu_{0}{(C_{d},v_{z})} = - C_{d} + v_{z} and \\frac{\\partial}{\\partial C_{d}} \\mu_{0}{(C_{d},v_{z})} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + v_{z}) and C_{d} + \\frac{\\partial}{\\partial C_{d}} \\mu_{0}{(C_{d},v_{z})} = C_{d} + \\frac{\\partial}{\\partial C_{d}} (- C_{d} + v_{z}) and \\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} \\mu_{0}{(C_{d},v_{z})}) dv_{z} = \\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} (- C_{d} + v_{z})) dv_{z} and (\\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} \\mu_{0}{(C_{d},v_{z})}) dv_{z})^{v_{z}} = (\\int (C_{d} + \\frac{\\partial}{\\partial C_{d}} (- C_{d} + v_{z})) dv_{z})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('C_d', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('C_d', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["add", 2, "Symbol('C_d', commutative=True)"], "Equality(Add(Symbol('C_d', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('C_d', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Add(Symbol('C_d', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Symbol('C_d', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('C_d', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('C_d', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))))"], [["power", 4, "Symbol('v_z', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('C_d', commutative=True), Derivative(Function('\\\\mu_0')(Symbol('C_d', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Pow(Integral(Add(Symbol('C_d', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\operatorname{F_{x}}{(y^{\\prime})} = \\cos^{y^{\\prime}}{(y^{\\prime})}, then obtain e^{(y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{M_{E}}^{y^{\\prime}}{(y^{\\prime})}} = e^{(y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{F_{x}}{(y^{\\prime})}}", "derivation": "\\operatorname{M_{E}}{(y^{\\prime})} = \\cos{(y^{\\prime})} and \\operatorname{M_{E}}^{y^{\\prime}}{(y^{\\prime})} = \\cos^{y^{\\prime}}{(y^{\\prime})} and (y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{M_{E}}^{y^{\\prime}}{(y^{\\prime})} = (y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\cos^{y^{\\prime}}{(y^{\\prime})} and \\operatorname{F_{x}}{(y^{\\prime})} = \\cos^{y^{\\prime}}{(y^{\\prime})} and (y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{M_{E}}^{y^{\\prime}}{(y^{\\prime})} = (y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{F_{x}}{(y^{\\prime})} and e^{(y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{M_{E}}^{y^{\\prime}}{(y^{\\prime})}} = e^{(y^{\\prime} + \\operatorname{M_{E}}{(y^{\\prime})}) \\operatorname{F_{x}}{(y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 2, "Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('y^{\\\\prime}', commutative=True)), Pow(cos(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Function('F_x')(Symbol('y^{\\\\prime}', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('M_E')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))), exp(Mul(Add(Symbol('y^{\\\\prime}', commutative=True), Function('M_E')(Symbol('y^{\\\\prime}', commutative=True))), Function('F_x')(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given V{(\\phi_1)} = \\log{(\\phi_1)}, then derive \\int V{(\\phi_1)} d\\phi_1 = \\phi + \\phi_1 \\log{(\\phi_1)} - \\phi_1, then obtain \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi + \\phi_1 V{(\\phi_1)} - \\phi_1}{\\phi_1} = \\frac{d}{d \\phi_1} \\frac{\\int \\log{(\\phi_1)} d\\phi_1}{\\phi_1}", "derivation": "V{(\\phi_1)} = \\log{(\\phi_1)} and \\int V{(\\phi_1)} d\\phi_1 = \\int \\log{(\\phi_1)} d\\phi_1 and \\frac{\\int V{(\\phi_1)} d\\phi_1}{\\phi_1} = \\frac{\\int \\log{(\\phi_1)} d\\phi_1}{\\phi_1} and \\frac{d}{d \\phi_1} \\frac{\\int V{(\\phi_1)} d\\phi_1}{\\phi_1} = \\frac{d}{d \\phi_1} \\frac{\\int \\log{(\\phi_1)} d\\phi_1}{\\phi_1} and \\int V{(\\phi_1)} d\\phi_1 = \\phi + \\phi_1 \\log{(\\phi_1)} - \\phi_1 and \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi + \\phi_1 \\log{(\\phi_1)} - \\phi_1}{\\phi_1} = \\frac{d}{d \\phi_1} \\frac{\\int \\log{(\\phi_1)} d\\phi_1}{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi + \\phi_1 V{(\\phi_1)} - \\phi_1}{\\phi_1} = \\frac{d}{d \\phi_1} \\frac{\\int \\log{(\\phi_1)} d\\phi_1}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('V')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["divide", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(Function('V')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(Function('V')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Mul(Symbol('\\\\phi_1', commutative=True), Function('V')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})}, then obtain \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\frac{\\mathbf{E}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\frac{\\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}}", "derivation": "\\mathbf{E}{(\\hat{\\mathbf{r}})} = \\cos{(\\hat{\\mathbf{r}})} and \\frac{\\mathbf{E}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{\\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} and \\frac{d}{d \\hat{\\mathbf{r}}} \\frac{\\mathbf{E}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{d}{d \\hat{\\mathbf{r}}} \\frac{\\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} and \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\frac{\\mathbf{E}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{d^{2}}{d \\hat{\\mathbf{r}}^{2}} \\frac{\\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{v}{(G)} = \\frac{d}{d G} \\cos{(G)}, then derive A \\sin{(G)} + (A \\mathbf{v}{(G)})^{G} = A \\sin{(G)} + (- A \\sin{(G)})^{G}, then obtain (A \\sin{(G)} + (A \\mathbf{v}{(G)})^{G}) ((A \\frac{d}{d G} \\cos{(G)})^{G} - (- A \\sin{(G)})^{G}) = 0", "derivation": "\\mathbf{v}{(G)} = \\frac{d}{d G} \\cos{(G)} and A \\mathbf{v}{(G)} = A \\frac{d}{d G} \\cos{(G)} and (A \\mathbf{v}{(G)})^{G} = (A \\frac{d}{d G} \\cos{(G)})^{G} and - A \\frac{d}{d G} \\cos{(G)} + (A \\mathbf{v}{(G)})^{G} = - A \\frac{d}{d G} \\cos{(G)} + (A \\frac{d}{d G} \\cos{(G)})^{G} and A \\sin{(G)} + (A \\mathbf{v}{(G)})^{G} = A \\sin{(G)} + (- A \\sin{(G)})^{G} and (A \\mathbf{v}{(G)})^{G} - (- A \\sin{(G)})^{G} = 0 and (A \\frac{d}{d G} \\cos{(G)})^{G} - (- A \\sin{(G)})^{G} = 0 and (A \\sin{(G)} + (A \\mathbf{v}{(G)})^{G}) ((A \\frac{d}{d G} \\cos{(G)})^{G} - (- A \\sin{(G)})^{G}) = 0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('G', commutative=True)), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["times", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True)))"], [["minus", 3, "Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Pow(Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True))), Add(Mul(Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Pow(Mul(Integer(-1), Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True))))"], [["minus", 5, "Add(Mul(Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Pow(Mul(Integer(-1), Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], "Equality(Add(Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True)))), Integer(0))"], [["times", 7, "Add(Mul(Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Pow(Mul(Symbol('A', commutative=True), Function('\\\\mathbf{v}')(Symbol('G', commutative=True))), Symbol('G', commutative=True))), Add(Pow(Mul(Symbol('A', commutative=True), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Symbol('G', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(-1), Symbol('A', commutative=True), sin(Symbol('G', commutative=True))), Symbol('G', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\varphi{(\\mathbf{S},v)} = v^{\\mathbf{S}}, then derive \\frac{\\partial}{\\partial \\mathbf{S}} \\varphi{(\\mathbf{S},v)} = v^{\\mathbf{S}} \\log{(v)}, then obtain \\mathbf{S} + \\varphi{(\\mathbf{S},v)} \\log{(v)} = \\mathbf{S} + \\frac{\\partial}{\\partial \\mathbf{S}} \\varphi{(\\mathbf{S},v)}", "derivation": "\\varphi{(\\mathbf{S},v)} = v^{\\mathbf{S}} and v + \\varphi{(\\mathbf{S},v)} = v + v^{\\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} (v + \\varphi{(\\mathbf{S},v)}) = \\frac{\\partial}{\\partial \\mathbf{S}} (v + v^{\\mathbf{S}}) and \\frac{\\partial}{\\partial \\mathbf{S}} \\varphi{(\\mathbf{S},v)} = v^{\\mathbf{S}} \\log{(v)} and \\frac{\\partial}{\\partial \\mathbf{S}} \\varphi{(\\mathbf{S},v)} = \\varphi{(\\mathbf{S},v)} \\log{(v)} and \\varphi{(\\mathbf{S},v)} \\log{(v)} = v^{\\mathbf{S}} \\log{(v)} and \\mathbf{S} + \\varphi{(\\mathbf{S},v)} \\log{(v)} = \\mathbf{S} + v^{\\mathbf{S}} \\log{(v)} and \\mathbf{S} + \\varphi{(\\mathbf{S},v)} \\log{(v)} = \\mathbf{S} + \\frac{\\partial}{\\partial \\mathbf{S}} \\varphi{(\\mathbf{S},v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Add(Symbol('v', commutative=True), Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True))))"], [["add", 6, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Pow(Symbol('v', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(x,F_{N})} = F_{N}^{x}, then obtain \\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})} (\\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})})^{F_{N}} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{x} (\\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})})^{F_{N}}", "derivation": "\\Psi_{nl}{(x,F_{N})} = F_{N}^{x} and \\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{x} and (\\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})})^{F_{N}} = (\\frac{\\partial}{\\partial F_{N}} F_{N}^{x})^{F_{N}} and (\\frac{\\partial}{\\partial F_{N}} F_{N}^{x})^{F_{N}} \\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{x} (\\frac{\\partial}{\\partial F_{N}} F_{N}^{x})^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})} (\\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})})^{F_{N}} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{x} (\\frac{\\partial}{\\partial F_{N}} \\Psi_{nl}{(x,F_{N})})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["power", 2, "Symbol('F_N', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Pow(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)))"], [["times", 2, "Pow(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))), Mul(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('x', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given l{(V_{\\mathbf{E}},L)} = V_{\\mathbf{E}}^{L} and \\eta^{\\prime}{(V_{\\mathbf{E}},L)} = V_{\\mathbf{E}}^{L}, then obtain \\int 1 dL = \\int V_{\\mathbf{E}}^{- L} \\eta^{\\prime}{(V_{\\mathbf{E}},L)} dL", "derivation": "l{(V_{\\mathbf{E}},L)} = V_{\\mathbf{E}}^{L} and 1 = \\frac{V_{\\mathbf{E}}^{L}}{l{(V_{\\mathbf{E}},L)}} and \\eta^{\\prime}{(V_{\\mathbf{E}},L)} = V_{\\mathbf{E}}^{L} and 1 = \\frac{\\eta^{\\prime}{(V_{\\mathbf{E}},L)}}{l{(V_{\\mathbf{E}},L)}} and 1 = V_{\\mathbf{E}}^{- L} \\eta^{\\prime}{(V_{\\mathbf{E}},L)} and \\int 1 dL = \\int V_{\\mathbf{E}}^{- L} \\eta^{\\prime}{(V_{\\mathbf{E}},L)} dL", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)))"], [["divide", 1, "Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Pow(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Pow(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('L', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True))))"], [["integrate", 5, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('L', commutative=True))), Integral(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), Symbol('L', commutative=True))), Function('\\\\eta^{\\\\prime}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given a{(\\rho_f)} = \\log{(\\cos{(\\rho_f)})} and H{(\\mathbf{J},\\mathbf{A})} = \\mathbf{A} + \\log{(\\mathbf{J})}, then obtain 0^{\\rho_f} + H{(\\mathbf{J},\\mathbf{A})} = (- a{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} + H{(\\mathbf{J},\\mathbf{A})}", "derivation": "a{(\\rho_f)} = \\log{(\\cos{(\\rho_f)})} and 0 = - a{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})} and 0^{\\rho_f} = (- a{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} and H{(\\mathbf{J},\\mathbf{A})} = \\mathbf{A} + \\log{(\\mathbf{J})} and 0^{\\rho_f} + \\mathbf{A} + \\log{(\\mathbf{J})} = \\mathbf{A} + (- a{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} + \\log{(\\mathbf{J})} and 0^{\\rho_f} + H{(\\mathbf{J},\\mathbf{A})} = (- a{(\\rho_f)} + \\log{(\\cos{(\\rho_f)})})^{\\rho_f} + H{(\\mathbf{J},\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\rho_f', commutative=True)), log(cos(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Function('a')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('a')(Symbol('\\\\rho_f', commutative=True))), log(cos(Symbol('\\\\rho_f', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\rho_f', commutative=True))), log(cos(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)))"], ["get_premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Pow(Integer(0), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\rho_f', commutative=True))), log(cos(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Integer(0), Symbol('\\\\rho_f', commutative=True)), Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Function('a')(Symbol('\\\\rho_f', commutative=True))), log(cos(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given x{(a,\\hbar)} = \\log{(\\frac{a}{\\hbar})}, then derive \\frac{\\partial}{\\partial a} x{(a,\\hbar)} + 1 = 1 + \\frac{1}{a}, then obtain \\frac{\\partial}{\\partial a} \\log{(\\frac{a}{\\hbar})} + 1 = 1 + \\frac{1}{a}", "derivation": "x{(a,\\hbar)} = \\log{(\\frac{a}{\\hbar})} and \\frac{\\partial}{\\partial a} x{(a,\\hbar)} = \\frac{\\partial}{\\partial a} \\log{(\\frac{a}{\\hbar})} and \\frac{\\partial}{\\partial a} x{(a,\\hbar)} + 1 = \\frac{\\partial}{\\partial a} \\log{(\\frac{a}{\\hbar})} + 1 and \\frac{\\partial}{\\partial a} x{(a,\\hbar)} + 1 = 1 + \\frac{1}{a} and \\frac{\\partial}{\\partial a} \\log{(\\frac{a}{\\hbar})} + 1 = 1 + \\frac{1}{a}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('a', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('a', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('x')(Symbol('a', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1)), Add(Derivative(log(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('x')(Symbol('a', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(log(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('a', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{X}{(y)} = \\frac{d}{d y} \\log{(y)}, then derive \\int \\hat{X}{(y)} dy = \\tilde{g} + \\log{(y)}, then obtain \\frac{\\int (3 \\log{(y)} + \\int \\hat{X}{(y)} dy) dy}{y} = \\frac{\\int (\\tilde{g} + 4 \\log{(y)}) dy}{y}", "derivation": "\\hat{X}{(y)} = \\frac{d}{d y} \\log{(y)} and \\int \\hat{X}{(y)} dy = \\int \\frac{d}{d y} \\log{(y)} dy and \\int \\hat{X}{(y)} dy = \\tilde{g} + \\log{(y)} and \\log{(y)} + \\int \\hat{X}{(y)} dy = \\tilde{g} + 2 \\log{(y)} and 3 \\log{(y)} + \\int \\hat{X}{(y)} dy = \\tilde{g} + 4 \\log{(y)} and \\int (3 \\log{(y)} + \\int \\hat{X}{(y)} dy) dy = \\int (\\tilde{g} + 4 \\log{(y)}) dy and \\frac{\\int (3 \\log{(y)} + \\int \\hat{X}{(y)} dy) dy}{y} = \\frac{\\int (\\tilde{g} + 4 \\log{(y)}) dy}{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Add(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('y', commutative=True))))"], [["add", 3, "log(Symbol('y', commutative=True))"], "Equality(Add(log(Symbol('y', commutative=True)), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(2), log(Symbol('y', commutative=True)))))"], [["add", 4, "Mul(Integer(2), log(Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(3), log(Symbol('y', commutative=True))), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(4), log(Symbol('y', commutative=True)))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(3), log(Symbol('y', commutative=True))), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(4), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["divide", 6, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(3), log(Symbol('y', commutative=True))), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(4), log(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\phi_2)} = \\sin{(\\phi_2)}, then derive \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 = \\mathbf{F} - \\cos{(\\phi_2)}, then obtain - \\hat{H}_l + \\cos{(\\phi_2)} + \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2} = - \\hat{H}_l + \\mathbf{F} + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2}", "derivation": "\\mathbf{s}{(\\phi_2)} = \\sin{(\\phi_2)} and \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 = \\int \\sin{(\\phi_2)} d\\phi_2 and \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 = \\mathbf{F} - \\cos{(\\phi_2)} and \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 - \\int \\sin{(\\phi_2)} d\\phi_2 = \\mathbf{F} - \\cos{(\\phi_2)} - \\int \\sin{(\\phi_2)} d\\phi_2 and \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 - \\int \\sin{(\\phi_2)} d\\phi_2 + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2} = \\mathbf{F} - \\cos{(\\phi_2)} - \\int \\sin{(\\phi_2)} d\\phi_2 + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2} and - \\hat{H}_l + \\cos{(\\phi_2)} + \\int \\mathbf{s}{(\\phi_2)} d\\phi_2 + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2} = - \\hat{H}_l + \\mathbf{F} + \\frac{\\mathbf{s}{(\\phi_2)}}{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 3, "Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"], [["add", 4, "Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)))), Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given k{(C,\\Psi_{nl})} = \\sin{(C + \\Psi_{nl})} and \\lambda{(\\sigma_x,\\mathbf{M})} = - \\mathbf{M} + \\sigma_x, then obtain 0^{C} + \\mathbf{M} - \\sigma_x + \\frac{d}{d \\Psi_{nl}} 0 = \\mathbf{M} - \\sigma_x + (- k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})})^{C} + \\frac{d}{d \\Psi_{nl}} 0", "derivation": "k{(C,\\Psi_{nl})} = \\sin{(C + \\Psi_{nl})} and \\lambda{(\\sigma_x,\\mathbf{M})} = - \\mathbf{M} + \\sigma_x and 0 = - k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})} and 0^{C} = (- k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})})^{C} and 0^{C} - \\lambda{(\\sigma_x,\\mathbf{M})} = (- k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})})^{C} - \\lambda{(\\sigma_x,\\mathbf{M})} and 0^{C} + \\mathbf{M} - \\sigma_x = \\mathbf{M} - \\sigma_x + (- k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})})^{C} and 0^{C} + \\mathbf{M} - \\sigma_x + \\frac{d}{d \\Psi_{nl}} 0 = \\mathbf{M} - \\sigma_x + (- k{(C,\\Psi_{nl})} + \\sin{(C + \\Psi_{nl})})^{C} + \\frac{d}{d \\Psi_{nl}} 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Integer(0), Symbol('C', commutative=True)), Pow(Add(Mul(Integer(-1), Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('C', commutative=True)))"], [["minus", 4, "Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('C', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('C', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Integer(0), Symbol('C', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('C', commutative=True))))"], [["add", 6, "Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))"], "Equality(Add(Pow(Integer(0), Symbol('C', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('k')(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), sin(Add(Symbol('C', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Symbol('C', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain \\lambda{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\lambda{(\\mathbf{r})} = \\cos{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\lambda{(\\mathbf{r})}", "derivation": "\\lambda{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\frac{d}{d \\mathbf{r}} \\lambda{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} and \\lambda{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} = \\cos{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} and \\lambda{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\lambda{(\\mathbf{r})} = \\cos{(\\mathbf{r})} \\frac{d}{d \\mathbf{r}} \\lambda{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\varphi)} = \\int \\log{(\\varphi)} d\\varphi, then obtain \\frac{d}{d \\varphi} \\iint \\hat{x}_0^{\\varphi}{(\\varphi)} d\\varphi d\\varphi = \\frac{d}{d \\varphi} \\iint (\\int \\log{(\\varphi)} d\\varphi)^{\\varphi} d\\varphi d\\varphi", "derivation": "\\hat{x}_0{(\\varphi)} = \\int \\log{(\\varphi)} d\\varphi and \\hat{x}_0^{\\varphi}{(\\varphi)} = (\\int \\log{(\\varphi)} d\\varphi)^{\\varphi} and \\int \\hat{x}_0^{\\varphi}{(\\varphi)} d\\varphi = \\int (\\int \\log{(\\varphi)} d\\varphi)^{\\varphi} d\\varphi and \\iint \\hat{x}_0^{\\varphi}{(\\varphi)} d\\varphi d\\varphi = \\iint (\\int \\log{(\\varphi)} d\\varphi)^{\\varphi} d\\varphi d\\varphi and \\frac{d}{d \\varphi} \\iint \\hat{x}_0^{\\varphi}{(\\varphi)} d\\varphi d\\varphi = \\frac{d}{d \\varphi} \\iint (\\int \\log{(\\varphi)} d\\varphi)^{\\varphi} d\\varphi d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varphi', commutative=True)), Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Integral(Pow(Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(n_{2},F_{H})} = F_{H}^{n_{2}}, then derive 0 = - \\frac{\\partial}{\\partial F_{H}} \\operatorname{v_{z}}{(n_{2},F_{H})} + \\frac{F_{H}^{n_{2}} n_{2}}{F_{H}}, then obtain 0 = - \\frac{\\partial}{\\partial F_{H}} \\operatorname{v_{z}}{(n_{2},F_{H})} + \\frac{n_{2} \\operatorname{v_{z}}{(n_{2},F_{H})}}{F_{H}}", "derivation": "\\operatorname{v_{z}}{(n_{2},F_{H})} = F_{H}^{n_{2}} and 0 = F_{H}^{n_{2}} - \\operatorname{v_{z}}{(n_{2},F_{H})} and \\frac{d}{d F_{H}} 0 = \\frac{\\partial}{\\partial F_{H}} (F_{H}^{n_{2}} - \\operatorname{v_{z}}{(n_{2},F_{H})}) and 0 = - \\frac{\\partial}{\\partial F_{H}} \\operatorname{v_{z}}{(n_{2},F_{H})} + \\frac{F_{H}^{n_{2}} n_{2}}{F_{H}} and 0 = - \\frac{\\partial}{\\partial F_{H}} \\operatorname{v_{z}}{(n_{2},F_{H})} + \\frac{n_{2} \\operatorname{v_{z}}{(n_{2},F_{H})}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 1, "Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('F_H', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('F_H', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('F_H', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), Function('v_z')(Symbol('n_2', commutative=True), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given f{(y)} = \\sin{(y)}, then obtain (\\frac{y \\frac{d}{d y} f{(y)}}{f{(y)}} + \\log{(f{(y)})}) f^{y}{(y)} = (\\frac{y \\cos{(y)}}{\\sin{(y)}} + \\log{(\\sin{(y)})}) \\sin^{y}{(y)}", "derivation": "f{(y)} = \\sin{(y)} and f^{y}{(y)} = \\sin^{y}{(y)} and \\frac{d}{d y} f^{y}{(y)} = \\frac{d}{d y} \\sin^{y}{(y)} and (\\frac{y \\frac{d}{d y} f{(y)}}{f{(y)}} + \\log{(f{(y)})}) f^{y}{(y)} = (\\frac{y \\cos{(y)}}{\\sin{(y)}} + \\log{(\\sin{(y)})}) \\sin^{y}{(y)}", "srepr_derivation": [["get_premise", "Equality(Function('f')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('f')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Function('f')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('y', commutative=True), Pow(Function('f')(Symbol('y', commutative=True)), Integer(-1)), Derivative(Function('f')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), log(Function('f')(Symbol('y', commutative=True)))), Pow(Function('f')(Symbol('y', commutative=True)), Symbol('y', commutative=True))), Mul(Add(Mul(Symbol('y', commutative=True), Pow(sin(Symbol('y', commutative=True)), Integer(-1)), cos(Symbol('y', commutative=True))), log(sin(Symbol('y', commutative=True)))), Pow(sin(Symbol('y', commutative=True)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(r,s)} = e^{\\frac{r}{s}} and \\operatorname{v_{t}}{(r,s)} = - e^{\\frac{r}{s}}, then obtain \\operatorname{v_{t}}{(r,s)} = - \\theta_{1}{(r,s)}", "derivation": "\\theta_{1}{(r,s)} = e^{\\frac{r}{s}} and \\operatorname{v_{t}}{(r,s)} = - e^{\\frac{r}{s}} and s \\operatorname{v_{t}}{(r,s)} = - s e^{\\frac{r}{s}} and s \\operatorname{v_{t}}{(r,s)} = - s \\theta_{1}{(r,s)} and \\operatorname{v_{t}}{(r,s)} = - \\theta_{1}{(r,s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('r', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('r', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('r', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], [["times", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Function('v_t')(Symbol('r', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True), exp(Mul(Symbol('r', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Symbol('s', commutative=True), Function('v_t')(Symbol('r', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('s', commutative=True), Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('s', commutative=True))))"], [["times", 4, "Pow(Symbol('s', commutative=True), Integer(-1))"], "Equality(Function('v_t')(Symbol('r', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} = (y^{\\prime})^{\\theta}, then obtain \\int \\frac{\\partial}{\\partial \\theta} y^{\\prime} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} dy^{\\prime} = \\int \\frac{\\partial}{\\partial \\theta} y^{\\prime} (y^{\\prime})^{\\theta} dy^{\\prime}", "derivation": "\\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} = (y^{\\prime})^{\\theta} and y^{\\prime} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} = y^{\\prime} (y^{\\prime})^{\\theta} and \\frac{\\partial}{\\partial \\theta} y^{\\prime} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} = \\frac{\\partial}{\\partial \\theta} y^{\\prime} (y^{\\prime})^{\\theta} and \\int \\frac{\\partial}{\\partial \\theta} y^{\\prime} \\operatorname{a^{\\dagger}}{(y^{\\prime},\\theta)} dy^{\\prime} = \\int \\frac{\\partial}{\\partial \\theta} y^{\\prime} (y^{\\prime})^{\\theta} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('a^{\\\\dagger}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Mul(Symbol('y^{\\\\prime}', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(A_{1})} = \\sin{(A_{1})} and Z{(A_{1})} = \\mu_{0}{(A_{1})} \\sin{(A_{1})}, then obtain (\\int Z{(A_{1})} dA_{1})^{A_{1}} = (\\frac{A_{1}}{2} + \\ddot{x} - \\frac{\\sin{(A_{1})} \\cos{(A_{1})}}{2})^{A_{1}}", "derivation": "\\mu_{0}{(A_{1})} = \\sin{(A_{1})} and Z{(A_{1})} = \\mu_{0}{(A_{1})} \\sin{(A_{1})} and Z{(A_{1})} = \\mu_{0}^{2}{(A_{1})} and \\int Z{(A_{1})} dA_{1} = \\int \\mu_{0}^{2}{(A_{1})} dA_{1} and (\\int Z{(A_{1})} dA_{1})^{A_{1}} = (\\int \\mu_{0}^{2}{(A_{1})} dA_{1})^{A_{1}} and (\\int Z{(A_{1})} dA_{1})^{A_{1}} = (\\int \\sin^{2}{(A_{1})} dA_{1})^{A_{1}} and (\\int Z{(A_{1})} dA_{1})^{A_{1}} = (\\frac{A_{1}}{2} + \\ddot{x} - \\frac{\\sin{(A_{1})} \\cos{(A_{1})}}{2})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], ["renaming_premise", "Equality(Function('Z')(Symbol('A_1', commutative=True)), Mul(Function('\\\\mu_0')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Z')(Symbol('A_1', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('A_1', commutative=True)), Integer(2)))"], [["integrate", 3, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Pow(Function('\\\\mu_0')(Symbol('A_1', commutative=True)), Integer(2)), Tuple(Symbol('A_1', commutative=True))))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Pow(Integral(Pow(Function('\\\\mu_0')(Symbol('A_1', commutative=True)), Integer(2)), Tuple(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Pow(Integral(Pow(sin(Symbol('A_1', commutative=True)), Integer(2)), Tuple(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Mul(Rational(1, 2), Symbol('A_1', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given v{(\\psi^*)} = e^{\\psi^*}, then obtain 0^{\\psi^*} e^{\\psi^*} + 2 \\psi^* = 2 \\psi^* + (1 - v{(\\psi^*)})^{2 \\psi^*} e^{\\psi^*}", "derivation": "v{(\\psi^*)} = e^{\\psi^*} and 0 = - v{(\\psi^*)} + e^{\\psi^*} and 0^{\\psi^*} = (- v{(\\psi^*)} + e^{\\psi^*})^{\\psi^*} and 0^{\\psi^*} (- v{(\\psi^*)} + e^{\\psi^*})^{\\psi^*} = (- v{(\\psi^*)} + e^{\\psi^*})^{2 \\psi^*} and (- v{(\\psi^*)} + e^{\\psi^*})^{\\psi^*} = (1 - v{(\\psi^*)})^{2 \\psi^*} and 0^{\\psi^*} = (1 - v{(\\psi^*)})^{2 \\psi^*} and 0^{\\psi^*} e^{\\psi^*} = (1 - v{(\\psi^*)})^{2 \\psi^*} e^{\\psi^*} and 0^{\\psi^*} e^{\\psi^*} + 2 \\psi^* = 2 \\psi^* + (1 - v{(\\psi^*)})^{2 \\psi^*} e^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "Function('v')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"], [["times", 3, "Pow(Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True))), Pow(Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["times", 6, "exp(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Add(Integer(1), Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))))"], [["add", 7, "Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Add(Integer(1), Mul(Integer(-1), Function('v')(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\theta{(Q)} = \\cos{(Q)}, then derive - \\sin{(Q)} + \\frac{d}{d Q} \\theta{(Q)} + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q} = - 2 \\sin{(Q)} + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q}, then obtain \\frac{d}{d Q} (- \\sin{(Q)} + \\frac{d}{d Q} \\cos{(Q)}) = \\frac{d}{d Q} - 2 \\sin{(Q)}", "derivation": "\\theta{(Q)} = \\cos{(Q)} and \\theta{(Q)} + \\cos{(Q)} = 2 \\cos{(Q)} and \\frac{d}{d Q} (\\theta{(Q)} + \\cos{(Q)}) = \\frac{d}{d Q} 2 \\cos{(Q)} and \\frac{d}{d Q} (\\theta{(Q)} + \\cos{(Q)}) + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q} = \\frac{d}{d Q} 2 \\cos{(Q)} + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q} and - \\sin{(Q)} + \\frac{d}{d Q} \\theta{(Q)} + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q} = - 2 \\sin{(Q)} + \\frac{\\theta{(Q)} - \\cos{(Q)}}{Q} and - \\sin{(Q)} + \\frac{d}{d Q} \\theta{(Q)} = - 2 \\sin{(Q)} and - \\sin{(Q)} + \\frac{d}{d Q} \\cos{(Q)} = - 2 \\sin{(Q)} and \\frac{d}{d Q} (- \\sin{(Q)} + \\frac{d}{d Q} \\cos{(Q)}) = \\frac{d}{d Q} - 2 \\sin{(Q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["add", 1, "cos(Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Mul(Integer(2), cos(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 3, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"], "Equality(Add(Derivative(Add(Function('\\\\theta')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))), Add(Derivative(Mul(Integer(2), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Derivative(Function('\\\\theta')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))), Add(Mul(Integer(-1), Integer(2), sin(Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Function('\\\\theta')(Symbol('Q', commutative=True)), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Derivative(Function('\\\\theta')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('Q', commutative=True))))"], [["differentiate", 7, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}}, then derive \\frac{d}{d g_{\\varepsilon}} \\dot{y}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}} \\cos{(g_{\\varepsilon})}, then obtain \\frac{\\frac{d}{d g_{\\varepsilon}} e^{\\sin{(g_{\\varepsilon})}}}{\\sin{(g_{\\varepsilon})}} = \\frac{e^{\\sin{(g_{\\varepsilon})}} \\cos{(g_{\\varepsilon})}}{\\sin{(g_{\\varepsilon})}}", "derivation": "\\dot{y}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}} and \\frac{d}{d g_{\\varepsilon}} \\dot{y}{(g_{\\varepsilon})} = \\frac{d}{d g_{\\varepsilon}} e^{\\sin{(g_{\\varepsilon})}} and \\frac{d}{d g_{\\varepsilon}} \\dot{y}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}} \\cos{(g_{\\varepsilon})} and \\frac{d}{d g_{\\varepsilon}} e^{\\sin{(g_{\\varepsilon})}} = e^{\\sin{(g_{\\varepsilon})}} \\cos{(g_{\\varepsilon})} and \\frac{\\frac{d}{d g_{\\varepsilon}} e^{\\sin{(g_{\\varepsilon})}}}{\\sin{(g_{\\varepsilon})}} = \\frac{e^{\\sin{(g_{\\varepsilon})}} \\cos{(g_{\\varepsilon})}}{\\sin{(g_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 4, "sin(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Derivative(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), cos(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given M{(\\delta)} = e^{\\delta}, then obtain (\\frac{d}{d \\delta} (M{(\\delta)} + e^{\\delta}))^{\\delta} = (\\frac{d}{d \\delta} 2 e^{\\delta})^{\\delta}", "derivation": "M{(\\delta)} = e^{\\delta} and M{(\\delta)} + e^{\\delta} = 2 e^{\\delta} and \\frac{d}{d \\delta} (M{(\\delta)} + e^{\\delta}) = \\frac{d}{d \\delta} 2 e^{\\delta} and (\\frac{d}{d \\delta} (M{(\\delta)} + e^{\\delta}))^{\\delta} = (\\frac{d}{d \\delta} 2 e^{\\delta})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('M')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Function('M')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Derivative(Add(Function('M')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)), Pow(Derivative(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(H,f_{E})} = - H + f_{E}, then obtain - E H \\tilde{g}{(H,f_{E})} + \\operatorname{P_{e}}^{2}{(E)} = - E H (- H + f_{E}) + \\operatorname{P_{e}}^{2}{(E)}", "derivation": "\\tilde{g}{(H,f_{E})} = - H + f_{E} and E \\tilde{g}{(H,f_{E})} = E (- H + f_{E}) and - E H \\tilde{g}{(H,f_{E})} = - E H (- H + f_{E}) and - E H \\tilde{g}{(H,f_{E})} + \\operatorname{P_{e}}^{2}{(E)} = - E H (- H + f_{E}) + \\operatorname{P_{e}}^{2}{(E)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f_E', commutative=True)))"], [["times", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('E', commutative=True), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f_E', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('H', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('H', commutative=True), Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True), Symbol('H', commutative=True), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f_E', commutative=True))))"], [["add", 3, "Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('H', commutative=True), Function('\\\\tilde{g}')(Symbol('H', commutative=True), Symbol('f_E', commutative=True))), Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('H', commutative=True), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('f_E', commutative=True))), Pow(Function('P_e')(Symbol('E', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\varepsilon_0,F_{x},p)} = \\frac{F_{x}}{\\varepsilon_0} - p, then obtain 1 - F_{x} = - F_{x} + (- \\frac{F_{x}}{\\varepsilon_0} + p + \\tilde{g}^*{(\\varepsilon_0,F_{x},p)})^{p}", "derivation": "\\tilde{g}^*{(\\varepsilon_0,F_{x},p)} = \\frac{F_{x}}{\\varepsilon_0} - p and - \\frac{F_{x}}{\\varepsilon_0} + p + \\tilde{g}^*{(\\varepsilon_0,F_{x},p)} = 0 and (- \\frac{F_{x}}{\\varepsilon_0} + p + \\tilde{g}^*{(\\varepsilon_0,F_{x},p)})^{p} = 0^{p} and - F_{x} + (- \\frac{F_{x}}{\\varepsilon_0} + p + \\tilde{g}^*{(\\varepsilon_0,F_{x},p)})^{p} = 0^{p} - F_{x} and 1 - F_{x} = - F_{x} + (- \\frac{F_{x}}{\\varepsilon_0} + p + \\tilde{g}^*{(\\varepsilon_0,F_{x},p)})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["minus", 1, "Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Symbol('p', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_x', commutative=True), Symbol('p', commutative=True))), Integer(0))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Symbol('p', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_x', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Integer(0), Symbol('p', commutative=True)))"], [["minus", 3, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Symbol('p', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_x', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Add(Pow(Integer(0), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Symbol('p', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_x', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(C_{1})} = e^{\\sin{(C_{1})}} and I{(C_{1})} = - \\operatorname{A_{x}}{(C_{1})} + e^{\\sin{(C_{1})}}, then obtain (- C_{1} + 2 I{(C_{1})})^{C_{1}} = (- C_{1} + I{(C_{1})})^{C_{1}}", "derivation": "\\operatorname{A_{x}}{(C_{1})} = e^{\\sin{(C_{1})}} and I{(C_{1})} = - \\operatorname{A_{x}}{(C_{1})} + e^{\\sin{(C_{1})}} and I{(C_{1})} = 0 and - C_{1} + I{(C_{1})} = - C_{1} and (- C_{1} + I{(C_{1})})^{C_{1}} = (- C_{1})^{C_{1}} and (- C_{1} + 2 I{(C_{1})})^{C_{1}} = (- C_{1} + I{(C_{1})})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('C_1', commutative=True)), exp(sin(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Function('A_x')(Symbol('C_1', commutative=True))), exp(sin(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('I')(Symbol('C_1', commutative=True)), Integer(0))"], [["minus", 3, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True))), Mul(Integer(-1), Symbol('C_1', commutative=True)))"], [["power", 4, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(2), Function('I')(Symbol('C_1', commutative=True)))), Symbol('C_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('I')(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(v_{1})} = e^{v_{1}}, then obtain \\frac{\\partial}{\\partial k} (\\frac{d}{d v_{1}} \\sigma_{x}{(v_{1})} - \\int \\operatorname{F_{c}}{(k)} dk) = \\frac{\\partial}{\\partial k} (\\frac{d}{d v_{1}} e^{v_{1}} - \\int \\operatorname{F_{c}}{(k)} dk)", "derivation": "\\sigma_{x}{(v_{1})} = e^{v_{1}} and \\frac{d}{d v_{1}} \\sigma_{x}{(v_{1})} = \\frac{d}{d v_{1}} e^{v_{1}} and \\frac{d}{d v_{1}} \\sigma_{x}{(v_{1})} - \\int \\operatorname{F_{c}}{(k)} dk = \\frac{d}{d v_{1}} e^{v_{1}} - \\int \\operatorname{F_{c}}{(k)} dk and \\frac{\\partial}{\\partial k} (\\frac{d}{d v_{1}} \\sigma_{x}{(v_{1})} - \\int \\operatorname{F_{c}}{(k)} dk) = \\frac{\\partial}{\\partial k} (\\frac{d}{d v_{1}} e^{v_{1}} - \\int \\operatorname{F_{c}}{(k)} dk)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "Integral(Function('F_c')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\sigma_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('F_c')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Add(Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('F_c')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\sigma_x')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('F_c')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Derivative(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('F_c')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(E)} = \\frac{d}{d E} e^{E} and H{(E)} = e^{E}, then obtain \\dot{z}{(E)} + \\frac{d}{d E} H{(E)} = \\dot{z}{(E)} + \\frac{d}{d E} e^{E}", "derivation": "\\dot{z}{(E)} = \\frac{d}{d E} e^{E} and H{(E)} = e^{E} and 2 \\dot{z}{(E)} = \\dot{z}{(E)} + \\frac{d}{d E} e^{E} and 2 \\dot{z}{(E)} = \\dot{z}{(E)} + \\frac{d}{d E} H{(E)} and \\dot{z}{(E)} + \\frac{d}{d E} H{(E)} = \\dot{z}{(E)} + \\frac{d}{d E} e^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('H')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["add", 1, "Function('\\\\dot{z}')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('E', commutative=True))), Add(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('E', commutative=True))), Add(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(Function('H')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(Function('H')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Function('\\\\dot{z}')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{1}{(L)} = \\sin{(L)}, then derive L + \\frac{d}{d L} \\theta_{1}{(L)} = L + \\cos{(L)}, then obtain \\int (L + \\frac{d}{d L} \\theta_{1}{(L)}) \\theta_{1}{(L)} dL = \\int (L + \\frac{d}{d L} \\sin{(L)}) \\theta_{1}{(L)} dL", "derivation": "\\theta_{1}{(L)} = \\sin{(L)} and \\frac{d}{d L} \\theta_{1}{(L)} = \\frac{d}{d L} \\sin{(L)} and L + \\frac{d}{d L} \\theta_{1}{(L)} = L + \\frac{d}{d L} \\sin{(L)} and L + \\frac{d}{d L} \\theta_{1}{(L)} = L + \\cos{(L)} and L + \\cos{(L)} = L + \\frac{d}{d L} \\sin{(L)} and (L + \\cos{(L)}) \\theta_{1}{(L)} = (L + \\frac{d}{d L} \\sin{(L)}) \\theta_{1}{(L)} and (L + \\frac{d}{d L} \\theta_{1}{(L)}) \\theta_{1}{(L)} = (L + \\frac{d}{d L} \\sin{(L)}) \\theta_{1}{(L)} and \\int (L + \\frac{d}{d L} \\theta_{1}{(L)}) \\theta_{1}{(L)} dL = \\int (L + \\frac{d}{d L} \\sin{(L)}) \\theta_{1}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["add", 2, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Derivative(Function('\\\\theta_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Symbol('L', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('L', commutative=True), Derivative(Function('\\\\theta_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["times", 5, "Function('\\\\theta_1')(Symbol('L', commutative=True))"], "Equality(Mul(Add(Symbol('L', commutative=True), cos(Symbol('L', commutative=True))), Function('\\\\theta_1')(Symbol('L', commutative=True))), Mul(Add(Symbol('L', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Function('\\\\theta_1')(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Add(Symbol('L', commutative=True), Derivative(Function('\\\\theta_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Function('\\\\theta_1')(Symbol('L', commutative=True))), Mul(Add(Symbol('L', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Function('\\\\theta_1')(Symbol('L', commutative=True))))"], [["integrate", 7, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('L', commutative=True), Derivative(Function('\\\\theta_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Function('\\\\theta_1')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Add(Symbol('L', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Function('\\\\theta_1')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g})} = \\cos{(\\log{(\\tilde{g})})}, then obtain \\operatorname{C_{d}}{(\\sigma_x)} - \\cos{(\\log{(\\tilde{g})})} = \\cos{(\\sigma_x)} - \\cos{(\\log{(\\tilde{g})})}", "derivation": "\\operatorname{C_{d}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g})} = \\cos{(\\log{(\\tilde{g})})} and \\operatorname{C_{d}}{(\\sigma_x)} - \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g})} = - \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g})} + \\cos{(\\sigma_x)} and \\operatorname{C_{d}}{(\\sigma_x)} - \\cos{(\\log{(\\tilde{g})})} = \\cos{(\\sigma_x)} - \\cos{(\\log{(\\tilde{g})})}", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}', commutative=True)), cos(log(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('C_d')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}', commutative=True)))), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}', commutative=True))), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('C_d')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\tilde{g}', commutative=True))))), Add(cos(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), cos(log(Symbol('\\\\tilde{g}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{y})} = \\sin{(\\dot{y})} and \\ddot{x}{(\\mathbf{J},\\eta)} = \\eta^{\\mathbf{J}}, then obtain \\frac{\\partial}{\\partial \\eta} \\operatorname{A_{y}}^{\\dot{y}}{(\\dot{y})} \\ddot{x}{(\\mathbf{J},\\eta)} = \\frac{\\partial}{\\partial \\eta} \\eta^{\\mathbf{J}} \\operatorname{A_{y}}^{\\dot{y}}{(\\dot{y})}", "derivation": "\\operatorname{A_{y}}{(\\dot{y})} = \\sin{(\\dot{y})} and \\operatorname{A_{y}}^{\\dot{y}}{(\\dot{y})} = \\sin^{\\dot{y}}{(\\dot{y})} and \\ddot{x}{(\\mathbf{J},\\eta)} = \\eta^{\\mathbf{J}} and \\ddot{x}{(\\mathbf{J},\\eta)} \\sin^{\\dot{y}}{(\\dot{y})} = \\eta^{\\mathbf{J}} \\sin^{\\dot{y}}{(\\dot{y})} and \\frac{\\partial}{\\partial \\eta} \\ddot{x}{(\\mathbf{J},\\eta)} \\sin^{\\dot{y}}{(\\dot{y})} = \\frac{\\partial}{\\partial \\eta} \\eta^{\\mathbf{J}} \\sin^{\\dot{y}}{(\\dot{y})} and \\frac{\\partial}{\\partial \\eta} \\operatorname{A_{y}}^{\\dot{y}}{(\\dot{y})} \\ddot{x}{(\\mathbf{J},\\eta)} = \\frac{\\partial}{\\partial \\eta} \\eta^{\\mathbf{J}} \\operatorname{A_{y}}^{\\dot{y}}{(\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{y}', commutative=True)), sin(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 3, "Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Mul(Pow(Function('A_y')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Function('A_y')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\varepsilon_0,a,r)} = \\varepsilon_0 (a + r), then obtain - \\varepsilon_0 (a + r) \\dot{y}{(\\varepsilon_0,a,r)} + \\dot{y}{(\\varepsilon_0,a,r)} = - \\varepsilon_0 (a + r) \\dot{y}{(\\varepsilon_0,a,r)} + \\varepsilon_0 (a + r)", "derivation": "\\dot{y}{(\\varepsilon_0,a,r)} = \\varepsilon_0 (a + r) and \\varepsilon_0 (a + r) \\dot{y}{(\\varepsilon_0,a,r)} = \\varepsilon_0^{2} (a + r)^{2} and - \\varepsilon_0^{2} (a + r)^{2} + \\dot{y}{(\\varepsilon_0,a,r)} = - \\varepsilon_0^{2} (a + r)^{2} + \\varepsilon_0 (a + r) and - \\varepsilon_0 (a + r) \\dot{y}{(\\varepsilon_0,a,r)} + \\dot{y}{(\\varepsilon_0,a,r)} = - \\varepsilon_0 (a + r) \\dot{y}{(\\varepsilon_0,a,r)} + \\varepsilon_0 (a + r)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)))"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)), Pow(Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Integer(2))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)), Pow(Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)), Pow(Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Integer(2))), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)), Pow(Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Integer(2))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True))), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('a', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Add(Symbol('a', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(F_{N},\\theta)} = \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta), then obtain - \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta) + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} - 1 = - \\hat{x}{(F_{N},\\theta)} + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} - 1", "derivation": "\\hat{x}{(F_{N},\\theta)} = \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta) and \\hat{x}{(F_{N},\\theta)} + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} = \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta) + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} and \\hat{x}{(F_{N},\\theta)} - \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta) + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} = (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} and - \\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta) + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} - 1 = - \\hat{x}{(F_{N},\\theta)} + (\\frac{\\partial}{\\partial F_{N}} (F_{N} + \\theta))^{F_{N}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 1, "Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))), Add(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))))"], [["minus", 2, "Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)))"], [["minus", 3, "Add(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Derivative(Add(Symbol('F_N', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}_0{(H)} = \\log{(H)}, then obtain \\log{(H)}^{- 2 H} = \\frac{\\log{(H)} \\log{(H)}^{- 2 H}}{\\hat{x}_0{(H)}}", "derivation": "\\hat{x}_0{(H)} = \\log{(H)} and 1 = \\frac{\\log{(H)}}{\\hat{x}_0{(H)}} and \\log{(H)}^{- H} = \\frac{\\log{(H)} \\log{(H)}^{- H}}{\\hat{x}_0{(H)}} and \\log{(H)}^{- 2 H} = \\frac{\\log{(H)} \\log{(H)}^{- 2 H}}{\\hat{x}_0{(H)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["divide", 1, "Function('\\\\hat{x}_0')(Symbol('H', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), Integer(-1)), log(Symbol('H', commutative=True))))"], [["divide", 2, "Pow(log(Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Pow(log(Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), Integer(-1)), log(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)))))"], [["times", 3, "Pow(log(Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)))"], "Equality(Pow(log(Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('H', commutative=True))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('H', commutative=True)), Integer(-1)), log(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\hat{X})} = \\log{(\\log{(\\hat{X})})}, then obtain \\log{(\\log{(\\hat{X})})}^{\\hat{X}} + (\\frac{d^{2}}{d \\hat{X}^{2}} \\theta_{2}{(\\hat{X})})^{2} = \\log{(\\log{(\\hat{X})})}^{\\hat{X}} + (\\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\log{(\\hat{X})})})^{2}", "derivation": "\\theta_{2}{(\\hat{X})} = \\log{(\\log{(\\hat{X})})} and \\frac{d}{d \\hat{X}} \\theta_{2}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\log{(\\log{(\\hat{X})})} and \\frac{d^{2}}{d \\hat{X}^{2}} \\theta_{2}{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\log{(\\hat{X})})} and (\\frac{d^{2}}{d \\hat{X}^{2}} \\theta_{2}{(\\hat{X})})^{2} = (\\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\log{(\\hat{X})})})^{2} and \\log{(\\log{(\\hat{X})})}^{\\hat{X}} + (\\frac{d^{2}}{d \\hat{X}^{2}} \\theta_{2}{(\\hat{X})})^{2} = \\log{(\\log{(\\hat{X})})}^{\\hat{X}} + (\\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\log{(\\hat{X})})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), log(log(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Derivative(log(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(log(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Integer(2)))"], [["add", 4, "Pow(log(log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(log(log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Integer(2))), Add(Pow(log(log(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(log(log(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Integer(2))))"]]}, {"prompt": "Given \\omega{(v_{x},\\mathbf{F})} = - \\mathbf{F} + v_{x}, then obtain \\omega{(v_{x},\\mathbf{F})} \\int e^{\\omega^{v_{x}}{(v_{x},\\mathbf{F})}} d\\mathbf{F} = \\omega{(v_{x},\\mathbf{F})} \\int e^{(- \\mathbf{F} + v_{x})^{v_{x}}} d\\mathbf{F}", "derivation": "\\omega{(v_{x},\\mathbf{F})} = - \\mathbf{F} + v_{x} and \\omega^{v_{x}}{(v_{x},\\mathbf{F})} = (- \\mathbf{F} + v_{x})^{v_{x}} and e^{\\omega^{v_{x}}{(v_{x},\\mathbf{F})}} = e^{(- \\mathbf{F} + v_{x})^{v_{x}}} and \\int e^{\\omega^{v_{x}}{(v_{x},\\mathbf{F})}} d\\mathbf{F} = \\int e^{(- \\mathbf{F} + v_{x})^{v_{x}}} d\\mathbf{F} and \\omega{(v_{x},\\mathbf{F})} \\int e^{\\omega^{v_{x}}{(v_{x},\\mathbf{F})}} d\\mathbf{F} = \\omega{(v_{x},\\mathbf{F})} \\int e^{(- \\mathbf{F} + v_{x})^{v_{x}}} d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(exp(Pow(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(exp(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 4, "Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integral(exp(Pow(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Function('\\\\omega')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integral(exp(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}}, then obtain \\int (- \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}} = \\int (- \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}}) d\\hat{\\mathbf{x}}", "derivation": "\\psi{(\\hat{\\mathbf{x}})} = e^{\\hat{\\mathbf{x}}} and \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})} = \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and - \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})} = - \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}} and \\int (- \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} \\psi{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}} = \\int (- \\psi{(\\hat{\\mathbf{x}})} + \\frac{d}{d \\hat{\\mathbf{x}}} e^{\\hat{\\mathbf{x}}}) d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Derivative(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A_{x})} = \\log{(A_{x})}, then obtain (A_{x} + \\operatorname{v_{y}}^{A_{x}}{(A_{x})}) \\operatorname{v_{y}}{(A_{x})} = (A_{x} + \\operatorname{v_{y}}^{A_{x}}{(A_{x})}) \\log{(A_{x})}", "derivation": "\\operatorname{v_{y}}{(A_{x})} = \\log{(A_{x})} and \\operatorname{v_{y}}^{A_{x}}{(A_{x})} = \\log{(A_{x})}^{A_{x}} and A_{x} + \\operatorname{v_{y}}^{A_{x}}{(A_{x})} = A_{x} + \\log{(A_{x})}^{A_{x}} and (A_{x} + \\log{(A_{x})}^{A_{x}}) \\operatorname{v_{y}}{(A_{x})} = (A_{x} + \\log{(A_{x})}^{A_{x}}) \\log{(A_{x})} and (A_{x} + \\operatorname{v_{y}}^{A_{x}}{(A_{x})}) \\operatorname{v_{y}}{(A_{x})} = (A_{x} + \\operatorname{v_{y}}^{A_{x}}{(A_{x})}) \\log{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["power", 1, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["add", 2, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Pow(Function('v_y')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))))"], [["times", 1, "Add(Symbol('A_x', commutative=True), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Function('v_y')(Symbol('A_x', commutative=True))), Mul(Add(Symbol('A_x', commutative=True), Pow(log(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('A_x', commutative=True), Pow(Function('v_y')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), Function('v_y')(Symbol('A_x', commutative=True))), Mul(Add(Symbol('A_x', commutative=True), Pow(Function('v_y')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(h)} = \\log{(h)}, then derive \\sin{(\\int \\sigma_{x}{(h)} dh)} = \\sin{(\\mathbf{E} + h \\log{(h)} - h)}, then obtain \\sin^{h}{(\\mathbf{E} + h \\log{(h)} - h)} = \\sin^{h}{(\\mathbf{E} + h \\sigma_{x}{(h)} - h)}", "derivation": "\\sigma_{x}{(h)} = \\log{(h)} and \\int \\sigma_{x}{(h)} dh = \\int \\log{(h)} dh and \\sin{(\\int \\sigma_{x}{(h)} dh)} = \\sin{(\\int \\log{(h)} dh)} and \\sin{(\\int \\sigma_{x}{(h)} dh)} = \\sin{(\\mathbf{E} + h \\log{(h)} - h)} and \\sin^{h}{(\\int \\sigma_{x}{(h)} dh)} = \\sin^{h}{(\\mathbf{E} + h \\log{(h)} - h)} and \\sin^{h}{(\\int \\log{(h)} dh)} = \\sin^{h}{(\\mathbf{E} + h \\log{(h)} - h)} and \\sin^{h}{(\\int \\log{(h)} dh)} = \\sin^{h}{(\\mathbf{E} + h \\sigma_{x}{(h)} - h)} and \\sin^{h}{(\\mathbf{E} + h \\log{(h)} - h)} = \\sin^{h}{(\\mathbf{E} + h \\sigma_{x}{(h)} - h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\sigma_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), sin(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(sin(Integral(Function('\\\\sigma_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(sin(Integral(Function('\\\\sigma_x')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(sin(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(sin(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), Function('\\\\sigma_x')(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Symbol('h', commutative=True), Function('\\\\sigma_x')(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given T{(h,\\hbar)} = \\cos{(h^{\\hbar})}, then obtain \\frac{\\partial}{\\partial \\hbar} (T^{2}{(h,\\hbar)})^{h} = \\frac{\\partial}{\\partial \\hbar} (T{(h,\\hbar)} \\cos{(h^{\\hbar})})^{h}", "derivation": "T{(h,\\hbar)} = \\cos{(h^{\\hbar})} and T^{2}{(h,\\hbar)} = T{(h,\\hbar)} \\cos{(h^{\\hbar})} and (T^{2}{(h,\\hbar)})^{h} = (T{(h,\\hbar)} \\cos{(h^{\\hbar})})^{h} and \\frac{\\partial}{\\partial \\hbar} (T^{2}{(h,\\hbar)})^{h} = \\frac{\\partial}{\\partial \\hbar} (T{(h,\\hbar)} \\cos{(h^{\\hbar})})^{h}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Pow(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["times", 1, "Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)), Mul(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Pow(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)), Symbol('h', commutative=True)), Pow(Mul(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Pow(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('h', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(2)), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Pow(Mul(Function('T')(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Pow(Symbol('h', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(\\hat{x}_0)} = \\hat{x}_0, then derive \\int \\rho_{f}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2}, then obtain 0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2} - \\int \\hat{x}_0 d\\hat{x}_0", "derivation": "\\rho_{f}{(\\hat{x}_0)} = \\hat{x}_0 and \\int \\rho_{f}{(\\hat{x}_0)} d\\hat{x}_0 = \\int \\hat{x}_0 d\\hat{x}_0 and \\int \\rho_{f}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2} and \\rho_{f}^{\\hat{x}_0}{(\\hat{x}_0)} + \\int \\rho_{f}{(\\hat{x}_0)} d\\hat{x}_0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2} + \\rho_{f}^{\\hat{x}_0}{(\\hat{x}_0)} and 0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2} - \\int \\rho_{f}{(\\hat{x}_0)} d\\hat{x}_0 and 0 = \\frac{\\hat{x}_0^{2}}{2} + t_{2} - \\int \\hat{x}_0 d\\hat{x}_0", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Symbol('\\\\hat{x}_0', commutative=True), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))), Symbol('t_2', commutative=True)))"], [["add", 3, "Pow(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Pow(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))), Symbol('t_2', commutative=True), Pow(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 4, "Add(Pow(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))), Symbol('t_2', commutative=True), Mul(Integer(-1), Integral(Function('\\\\rho_f')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2))), Symbol('t_2', commutative=True), Mul(Integer(-1), Integral(Symbol('\\\\hat{x}_0', commutative=True), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain ((\\int \\Psi_{\\lambda}{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}})^{\\mathbf{A}} = ((\\int \\cos{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}})^{\\mathbf{A}}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\int \\Psi_{\\lambda}{(\\mathbf{A})} d\\mathbf{A} = \\int \\cos{(\\mathbf{A})} d\\mathbf{A} and (\\int \\Psi_{\\lambda}{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} = (\\int \\cos{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} and ((\\int \\Psi_{\\lambda}{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}})^{\\mathbf{A}} = ((\\int \\cos{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Pow(Integral(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(y)} = \\sin{(y)}, then derive \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)}) \\int \\hat{X}{(y)} dy = \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)})^{2}, then obtain \\int \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)}) \\int \\hat{X}{(y)} dy dy = \\int \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)})^{2} dy", "derivation": "\\hat{X}{(y)} = \\sin{(y)} and \\int \\hat{X}{(y)} dy = \\int \\sin{(y)} dy and (\\int \\hat{X}{(y)} dy) \\int \\sin{(y)} dy = (\\int \\sin{(y)} dy)^{2} and \\frac{d}{d y} (\\int \\hat{X}{(y)} dy) \\int \\sin{(y)} dy = \\frac{d}{d y} (\\int \\sin{(y)} dy)^{2} and \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)}) \\int \\hat{X}{(y)} dy = \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)})^{2} and \\int \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)}) \\int \\hat{X}{(y)} dy dy = \\int \\frac{\\partial}{\\partial y} (\\theta - \\cos{(y)})^{2} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('y', commutative=True)), sin(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["times", 2, "Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Pow(Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(2)))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(2)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Mul(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integer(2)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Mul(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integral(Function('\\\\hat{X}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('y', commutative=True)))), Integer(2)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(L_{\\varepsilon})} = L_{\\varepsilon}, then derive \\int (\\frac{\\operatorname{A_{1}}{(L_{\\varepsilon})}}{L_{\\varepsilon}})^{L_{\\varepsilon}} dL_{\\varepsilon} = L_{\\varepsilon} + v_{x}, then obtain L_{\\varepsilon} + v_{x} = \\int 1 dL_{\\varepsilon}", "derivation": "\\operatorname{A_{1}}{(L_{\\varepsilon})} = L_{\\varepsilon} and \\frac{\\operatorname{A_{1}}{(L_{\\varepsilon})}}{L_{\\varepsilon}} = 1 and (\\frac{\\operatorname{A_{1}}{(L_{\\varepsilon})}}{L_{\\varepsilon}})^{L_{\\varepsilon}} = 1 and \\int (\\frac{\\operatorname{A_{1}}{(L_{\\varepsilon})}}{L_{\\varepsilon}})^{L_{\\varepsilon}} dL_{\\varepsilon} = \\int 1 dL_{\\varepsilon} and \\int (\\frac{\\operatorname{A_{1}}{(L_{\\varepsilon})}}{L_{\\varepsilon}})^{L_{\\varepsilon}} dL_{\\varepsilon} = L_{\\varepsilon} + v_{x} and L_{\\varepsilon} + v_{x} = \\int 1 dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))"], [["divide", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('A_1')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(1))"], [["power", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('A_1')(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('A_1')(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('A_1')(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_x', commutative=True)), Integral(Integer(1), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(C)} = e^{C}, then derive \\frac{d}{d C} \\dot{y}{(C)} = e^{C}, then obtain \\frac{d}{d C} \\dot{y}{(C)} = \\frac{d^{2}}{d C^{2}} e^{C}", "derivation": "\\dot{y}{(C)} = e^{C} and \\frac{d}{d C} \\dot{y}{(C)} = \\frac{d}{d C} e^{C} and \\frac{d}{d C} \\dot{y}{(C)} = e^{C} and \\frac{d}{d C} \\dot{y}{(C)} = \\frac{d^{2}}{d C^{2}} \\dot{y}{(C)} and \\frac{d}{d C} e^{C} = \\frac{d^{2}}{d C^{2}} e^{C} and \\frac{d}{d C} \\dot{y}{(C)} = \\frac{d^{2}}{d C^{2}} e^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), exp(Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('\\\\dot{y}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"]]}, {"prompt": "Given E{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})}, then obtain \\iint 2 E{(J_{\\varepsilon})} dJ_{\\varepsilon} dJ_{\\varepsilon} = \\iint (E{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})}) dJ_{\\varepsilon} dJ_{\\varepsilon}", "derivation": "E{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})} and 2 E{(J_{\\varepsilon})} = E{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})} and \\int 2 E{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (E{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})}) dJ_{\\varepsilon} and \\iint 2 E{(J_{\\varepsilon})} dJ_{\\varepsilon} dJ_{\\varepsilon} = \\iint (E{(J_{\\varepsilon})} + \\frac{d}{d J_{\\varepsilon}} \\log{(J_{\\varepsilon})}) dJ_{\\varepsilon} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["add", 1, "Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Add(Function('E')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(T,H)} = \\frac{H}{T}, then obtain \\sin^{2}{(\\Omega{(T,H)})} (\\int \\Omega{(T,H)} dT)^{2} = \\sin{(\\frac{H}{T})} \\sin{(\\Omega{(T,H)})} (\\int \\Omega{(T,H)} dT)^{2}", "derivation": "\\Omega{(T,H)} = \\frac{H}{T} and \\int \\Omega{(T,H)} dT = \\int \\frac{H}{T} dT and \\sin{(\\Omega{(T,H)})} = \\sin{(\\frac{H}{T})} and \\sin{(\\Omega{(T,H)})} \\int \\Omega{(T,H)} dT = \\sin{(\\frac{H}{T})} \\int \\Omega{(T,H)} dT and \\sin{(\\Omega{(T,H)})} \\int \\frac{H}{T} dT = \\sin{(\\frac{H}{T})} \\int \\frac{H}{T} dT and \\sin^{2}{(\\Omega{(T,H)})} (\\int \\frac{H}{T} dT) \\int \\Omega{(T,H)} dT = \\sin{(\\frac{H}{T})} \\sin{(\\Omega{(T,H)})} (\\int \\frac{H}{T} dT) \\int \\Omega{(T,H)} dT and \\sin^{2}{(\\Omega{(T,H)})} (\\int \\Omega{(T,H)} dT)^{2} = \\sin{(\\frac{H}{T})} \\sin{(\\Omega{(T,H)})} (\\int \\Omega{(T,H)} dT)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), sin(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)))))"], [["times", 3, "Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(sin(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)))), Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True)))), Mul(sin(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)))), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True)))))"], [["times", 5, "Mul(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True))))"], "Equality(Mul(Pow(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(2)), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))), Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(sin(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)))), sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True))), Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Integer(2)), Pow(Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(2))), Mul(sin(Mul(Symbol('H', commutative=True), Pow(Symbol('T', commutative=True), Integer(-1)))), sin(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True))), Pow(Integral(Function('\\\\Omega')(Symbol('T', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('T', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\phi{(I)} = \\sin{(I)} and \\Omega{(I)} = I \\sin{(I)}, then obtain \\frac{\\int \\Omega{(I)} dI}{\\Omega{(I)}} = \\frac{\\int I \\sin{(I)} dI}{\\Omega{(I)}}", "derivation": "\\phi{(I)} = \\sin{(I)} and I \\phi{(I)} = I \\sin{(I)} and \\Omega{(I)} = I \\sin{(I)} and \\int \\Omega{(I)} dI = \\int I \\sin{(I)} dI and \\frac{I \\sin{(I)} \\int \\Omega{(I)} dI}{\\phi{(I)}} = \\frac{I \\sin{(I)} \\int I \\sin{(I)} dI}{\\phi{(I)}} and I \\int \\Omega{(I)} dI = I \\int I \\sin{(I)} dI and \\frac{\\int \\Omega{(I)} dI}{I \\phi{(I)}} = \\frac{\\int I \\sin{(I)} dI}{I \\phi{(I)}} and \\Omega{(I)} = I \\phi{(I)} and \\frac{\\int \\Omega{(I)} dI}{\\Omega{(I)}} = \\frac{\\int I \\sin{(I)} dI}{\\Omega{(I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\phi')(Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Integer(-1)))"], "Equality(Mul(Symbol('I', commutative=True), Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True)), Integral(Function('\\\\Omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('I', commutative=True)), Integral(Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('I', commutative=True), Integral(Function('\\\\Omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Integral(Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["divide", 6, "Mul(Pow(Symbol('I', commutative=True), Integer(2)), Function('\\\\phi')(Symbol('I', commutative=True)))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\phi')(Symbol('I', commutative=True)), Integer(-1)), Integral(Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\Omega')(Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), Function('\\\\phi')(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Mul(Pow(Function('\\\\Omega')(Symbol('I', commutative=True)), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Pow(Function('\\\\Omega')(Symbol('I', commutative=True)), Integer(-1)), Integral(Mul(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(T,\\tilde{g}^*)} = e^{T + \\tilde{g}^*}, then derive \\int \\operatorname{A_{2}}{(T,\\tilde{g}^*)} dT = J + e^{T + \\tilde{g}^*}, then obtain J + \\operatorname{A_{2}}{(T,\\tilde{g}^*)} = \\int e^{T + \\tilde{g}^*} dT", "derivation": "\\operatorname{A_{2}}{(T,\\tilde{g}^*)} = e^{T + \\tilde{g}^*} and \\int \\operatorname{A_{2}}{(T,\\tilde{g}^*)} dT = \\int e^{T + \\tilde{g}^*} dT and \\int \\operatorname{A_{2}}{(T,\\tilde{g}^*)} dT = J + e^{T + \\tilde{g}^*} and J + e^{T + \\tilde{g}^*} = \\int e^{T + \\tilde{g}^*} dT and J + \\operatorname{A_{2}}{(T,\\tilde{g}^*)} = \\int e^{T + \\tilde{g}^*} dT", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_2')(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('J', commutative=True), exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('J', commutative=True), exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integral(exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('J', commutative=True), Function('A_2')(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Integral(exp(Add(Symbol('T', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(F_{c})} = \\frac{d}{d F_{c}} e^{F_{c}} and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} = \\operatorname{L_{\\varepsilon}}{(F_{c})} - e^{F_{c}}, then derive \\operatorname{L_{\\varepsilon}}{(F_{c})} = e^{F_{c}}, then obtain \\operatorname{V_{\\mathbf{B}}}{(F_{c})} + 1 = 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(F_{c})} = \\frac{d}{d F_{c}} e^{F_{c}} and \\operatorname{L_{\\varepsilon}}{(F_{c})} = e^{F_{c}} and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} = \\operatorname{L_{\\varepsilon}}{(F_{c})} - e^{F_{c}} and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} = 0 and \\operatorname{V_{\\mathbf{B}}}{(F_{c})} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Derivative(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True)), Add(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Mul(Integer(-1), exp(Symbol('F_c', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True)), Integer(0))"], [["add", 4, 1], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('F_c', commutative=True)), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(t_{2})} = \\sin{(t_{2})}, then obtain \\operatorname{V_{\\mathbf{B}}}{(t_{2})} \\int 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} dt_{2} = \\sin{(t_{2})} \\int 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} dt_{2}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(t_{2})} = \\sin{(t_{2})} and 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} = \\operatorname{V_{\\mathbf{B}}}{(t_{2})} + \\sin{(t_{2})} and \\int 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} dt_{2} = \\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + \\sin{(t_{2})}) dt_{2} and \\operatorname{V_{\\mathbf{B}}}{(t_{2})} \\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + \\sin{(t_{2})}) dt_{2} = \\sin{(t_{2})} \\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + \\sin{(t_{2})}) dt_{2} and \\operatorname{V_{\\mathbf{B}}}{(t_{2})} \\int 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} dt_{2} = \\sin{(t_{2})} \\int 2 \\operatorname{V_{\\mathbf{B}}}{(t_{2})} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["add", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True))"], "Equality(Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["times", 1, "Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))), Mul(sin(Symbol('t_2', commutative=True)), Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), Integral(Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))), Mul(sin(Symbol('t_2', commutative=True)), Integral(Mul(Integer(2), Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given V{(\\dot{\\mathbf{r}},i,\\tilde{g})} = \\dot{\\mathbf{r}}^{\\tilde{g}} - i and z{(\\dot{\\mathbf{r}},i,\\tilde{g})} = \\dot{\\mathbf{r}}^{\\tilde{g}} + \\tilde{g} - i, then obtain \\tilde{g} = - V{(\\dot{\\mathbf{r}},i,\\tilde{g})} + z{(\\dot{\\mathbf{r}},i,\\tilde{g})}", "derivation": "V{(\\dot{\\mathbf{r}},i,\\tilde{g})} = \\dot{\\mathbf{r}}^{\\tilde{g}} - i and \\tilde{g} + V{(\\dot{\\mathbf{r}},i,\\tilde{g})} = \\dot{\\mathbf{r}}^{\\tilde{g}} + \\tilde{g} - i and z{(\\dot{\\mathbf{r}},i,\\tilde{g})} = \\dot{\\mathbf{r}}^{\\tilde{g}} + \\tilde{g} - i and \\tilde{g} = \\dot{\\mathbf{r}}^{\\tilde{g}} + \\tilde{g} - i - V{(\\dot{\\mathbf{r}},i,\\tilde{g})} and \\tilde{g} = - V{(\\dot{\\mathbf{r}},i,\\tilde{g})} + z{(\\dot{\\mathbf{r}},i,\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["add", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["minus", 2, "Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Symbol('\\\\tilde{g}', commutative=True), Add(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Symbol('\\\\tilde{g}', commutative=True), Add(Mul(Integer(-1), Function('V')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Function('z')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('i', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(W,\\mu)} = \\frac{\\partial}{\\partial W} (W + \\mu), then derive \\psi^{*}{(W,\\mu)} = 1, then derive \\int \\frac{\\partial}{\\partial W} (W + \\mu) dW + 1 = \\int 1 dW + 1, then obtain - \\frac{\\int \\frac{\\partial}{\\partial W} (W + \\mu) dW + 1}{\\psi^{*}{(W,\\mu)}} = - \\frac{\\int 1 dW + 1}{\\psi^{*}{(W,\\mu)}}", "derivation": "\\psi^{*}{(W,\\mu)} = \\frac{\\partial}{\\partial W} (W + \\mu) and \\psi^{*}{(W,\\mu)} = 1 and \\frac{\\partial}{\\partial W} (W + \\mu) = 1 and \\int \\frac{\\partial}{\\partial W} (W + \\mu) dW = \\int 1 dW and ((\\frac{\\partial}{\\partial W} (W + \\mu))^{2})^{W} + \\int \\frac{\\partial}{\\partial W} (W + \\mu) dW = ((\\frac{\\partial}{\\partial W} (W + \\mu))^{2})^{W} + \\int 1 dW and \\int \\frac{\\partial}{\\partial W} (W + \\mu) dW + 1 = \\int 1 dW + 1 and - \\frac{\\int \\frac{\\partial}{\\partial W} (W + \\mu) dW + 1}{\\psi^{*}{(W,\\mu)}} = - \\frac{\\int 1 dW + 1}{\\psi^{*}{(W,\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\psi^*')(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integral(Integer(1), Tuple(Symbol('W', commutative=True))))"], [["add", 4, "Pow(Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Symbol('W', commutative=True))"], "Equality(Add(Pow(Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Symbol('W', commutative=True)), Integral(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True)))), Add(Pow(Pow(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Symbol('W', commutative=True)), Integral(Integer(1), Tuple(Symbol('W', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Add(Integral(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('W', commutative=True))), Integer(1)))"], [["divide", 6, "Mul(Integer(-1), Function('\\\\psi^*')(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Integral(Derivative(Add(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integer(1)), Pow(Function('\\\\psi^*')(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Integral(Integer(1), Tuple(Symbol('W', commutative=True))), Integer(1)), Pow(Function('\\\\psi^*')(Symbol('W', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{M},q)} = \\mathbf{M} q, then obtain \\operatorname{v_{y}}{(\\mathbf{M},q)} + \\frac{\\partial}{\\partial q} \\operatorname{v_{y}}{(\\mathbf{M},q)} = \\mathbf{M} + \\operatorname{v_{y}}{(\\mathbf{M},q)}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{M},q)} = \\mathbf{M} q and \\frac{\\partial}{\\partial q} \\operatorname{v_{y}}{(\\mathbf{M},q)} = \\frac{\\partial}{\\partial q} \\mathbf{M} q and \\operatorname{v_{y}}{(\\mathbf{M},q)} + \\frac{\\partial}{\\partial q} \\operatorname{v_{y}}{(\\mathbf{M},q)} = \\operatorname{v_{y}}{(\\mathbf{M},q)} + \\frac{\\partial}{\\partial q} \\mathbf{M} q and \\operatorname{v_{y}}{(\\mathbf{M},q)} + \\frac{\\partial}{\\partial q} \\operatorname{v_{y}}{(\\mathbf{M},q)} = \\mathbf{M} + \\operatorname{v_{y}}{(\\mathbf{M},q)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 2, "Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Derivative(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Derivative(Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{M}', commutative=True), Function('v_y')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(b)} = e^{b}, then obtain (b \\frac{d}{d b} b \\mathbf{H}{(b)})^{b} \\mathbf{H}{(b)} = (b \\frac{d}{d b} b e^{b})^{b} \\mathbf{H}{(b)}", "derivation": "\\mathbf{H}{(b)} = e^{b} and b \\mathbf{H}{(b)} = b e^{b} and \\frac{d}{d b} b \\mathbf{H}{(b)} = \\frac{d}{d b} b e^{b} and b \\frac{d}{d b} b \\mathbf{H}{(b)} = b \\frac{d}{d b} b e^{b} and (b \\frac{d}{d b} b \\mathbf{H}{(b)})^{b} = (b \\frac{d}{d b} b e^{b})^{b} and (b \\frac{d}{d b} b \\mathbf{H}{(b)})^{b} \\mathbf{H}{(b)} = (b \\frac{d}{d b} b e^{b})^{b} \\mathbf{H}{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Symbol('b', commutative=True), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 3, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Symbol('b', commutative=True)), Pow(Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Symbol('b', commutative=True)))"], [["times", 5, "Function('\\\\mathbf{H}')(Symbol('b', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Symbol('b', commutative=True)), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))), Mul(Pow(Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('b', commutative=True), exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Symbol('b', commutative=True)), Function('\\\\mathbf{H}')(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}}, then obtain \\mathbf{p} t_{1}^{\\mathbf{p}} + \\mathbf{p} \\psi^{*}{(t_{1},\\mathbf{p})} + (\\int t_{1}^{\\mathbf{p}} dt_{1})^{2} = 2 \\mathbf{p} t_{1}^{\\mathbf{p}} + (\\int t_{1}^{\\mathbf{p}} dt_{1})^{2}", "derivation": "\\psi^{*}{(t_{1},\\mathbf{p})} = t_{1}^{\\mathbf{p}} and \\mathbf{p} \\psi^{*}{(t_{1},\\mathbf{p})} = \\mathbf{p} t_{1}^{\\mathbf{p}} and \\mathbf{p} t_{1}^{\\mathbf{p}} + \\mathbf{p} \\psi^{*}{(t_{1},\\mathbf{p})} = 2 \\mathbf{p} t_{1}^{\\mathbf{p}} and \\mathbf{p} t_{1}^{\\mathbf{p}} + \\mathbf{p} \\psi^{*}{(t_{1},\\mathbf{p})} + (\\int t_{1}^{\\mathbf{p}} dt_{1})^{2} = 2 \\mathbf{p} t_{1}^{\\mathbf{p}} + (\\int t_{1}^{\\mathbf{p}} dt_{1})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["add", 3, "Pow(Integral(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2))"], "Equality(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Pow(Integral(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2))), Add(Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Pow(Integral(Pow(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(n_{1},\\varphi^*)} = e^{\\frac{\\varphi^*}{n_{1}}}, then obtain (- n_{1} + \\frac{\\hat{\\mathbf{x}}{(n_{1},\\varphi^*)}}{n_{1}})^{n_{1}} = (- n_{1} + \\frac{e^{\\frac{\\varphi^*}{n_{1}}}}{n_{1}})^{n_{1}}", "derivation": "\\hat{\\mathbf{x}}{(n_{1},\\varphi^*)} = e^{\\frac{\\varphi^*}{n_{1}}} and \\frac{\\hat{\\mathbf{x}}{(n_{1},\\varphi^*)}}{n_{1}} = \\frac{e^{\\frac{\\varphi^*}{n_{1}}}}{n_{1}} and - n_{1} + \\frac{\\hat{\\mathbf{x}}{(n_{1},\\varphi^*)}}{n_{1}} = - n_{1} + \\frac{e^{\\frac{\\varphi^*}{n_{1}}}}{n_{1}} and (- n_{1} + \\frac{\\hat{\\mathbf{x}}{(n_{1},\\varphi^*)}}{n_{1}})^{n_{1}} = (- n_{1} + \\frac{e^{\\frac{\\varphi^*}{n_{1}}}}{n_{1}})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"], [["divide", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))))"], [["minus", 2, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(a)} = \\log{(a)}, then obtain \\int (\\operatorname{V_{\\mathbf{B}}}{(a)} + \\operatorname{V_{\\mathbf{B}}}^{a}{(a)}) da = \\int (\\operatorname{V_{\\mathbf{B}}}{(a)} + \\log{(a)}^{a}) da", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(a)} = \\log{(a)} and \\operatorname{V_{\\mathbf{B}}}^{a}{(a)} = \\log{(a)}^{a} and \\operatorname{V_{\\mathbf{B}}}{(a)} + \\operatorname{V_{\\mathbf{B}}}^{a}{(a)} = \\operatorname{V_{\\mathbf{B}}}{(a)} + \\log{(a)}^{a} and \\int (\\operatorname{V_{\\mathbf{B}}}{(a)} + \\operatorname{V_{\\mathbf{B}}}^{a}{(a)}) da = \\int (\\operatorname{V_{\\mathbf{B}}}{(a)} + \\log{(a)}^{a}) da", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('a', commutative=True)))"], [["add", 2, "Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))), Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('a', commutative=True)), Pow(log(Symbol('a', commutative=True)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given u{(V_{\\mathbf{B}},s)} = V_{\\mathbf{B}} + s and \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)} = V_{\\mathbf{B}} + s, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (\\frac{V_{\\mathbf{B}} + s}{\\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)}})^{s} = \\frac{d}{d V_{\\mathbf{B}}} 1", "derivation": "u{(V_{\\mathbf{B}},s)} = V_{\\mathbf{B}} + s and \\frac{u{(V_{\\mathbf{B}},s)}}{V_{\\mathbf{B}} + s} = 1 and (\\frac{u{(V_{\\mathbf{B}},s)}}{V_{\\mathbf{B}} + s})^{s} = 1 and \\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)} = V_{\\mathbf{B}} + s and (\\frac{u{(V_{\\mathbf{B}},s)}}{\\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)}})^{s} = 1 and (\\frac{V_{\\mathbf{B}} + s}{\\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)}})^{s} = 1 and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (\\frac{V_{\\mathbf{B}} + s}{\\operatorname{L_{\\varepsilon}}{(V_{\\mathbf{B}},s)}})^{s} = \\frac{d}{d V_{\\mathbf{B}}} 1", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)))"], [["divide", 1, "Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True))), Integer(1))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Integer(-1))), Symbol('s', commutative=True)), Integer(1))"], [["differentiate", 6, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Pow(Mul(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('s', commutative=True)), Integer(-1))), Symbol('s', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(h,\\dot{z})} = \\frac{\\dot{z}}{h}, then derive (\\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})})^{h} = (- \\frac{\\dot{z}}{h^{2}})^{h}, then obtain (\\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})})^{h} = (- \\frac{\\Psi_{\\lambda}{(h,\\dot{z})}}{h})^{h}", "derivation": "\\Psi_{\\lambda}{(h,\\dot{z})} = \\frac{\\dot{z}}{h} and \\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})} = \\frac{\\partial}{\\partial h} \\frac{\\dot{z}}{h} and (\\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})})^{h} = (\\frac{\\partial}{\\partial h} \\frac{\\dot{z}}{h})^{h} and (\\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})})^{h} = (- \\frac{\\dot{z}}{h^{2}})^{h} and (\\frac{\\partial}{\\partial h} \\Psi_{\\lambda}{(h,\\dot{z})})^{h} = (- \\frac{\\Psi_{\\lambda}{(h,\\dot{z})}}{h})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-2))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('h', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(E,v_{2})} = - v_{2} + e^{E} and \\phi_{2}{(E,v_{2})} = - v_{2} - \\Psi^{\\dagger}{(E,v_{2})} + e^{E}, then obtain \\frac{d}{d v_{2}} 0 = \\frac{d}{d v_{2}} (-1)", "derivation": "\\Psi^{\\dagger}{(E,v_{2})} = - v_{2} + e^{E} and 0 = - v_{2} - \\Psi^{\\dagger}{(E,v_{2})} + e^{E} and \\phi_{2}{(E,v_{2})} = - v_{2} - \\Psi^{\\dagger}{(E,v_{2})} + e^{E} and 0 = \\phi_{2}{(E,v_{2})} and 0 = \\frac{\\phi_{2}{(E,v_{2})}}{v_{2} + \\Psi^{\\dagger}{(E,v_{2})} - e^{E}} and 0 = \\frac{- v_{2} - \\Psi^{\\dagger}{(E,v_{2})} + e^{E}}{v_{2} + \\Psi^{\\dagger}{(E,v_{2})} - e^{E}} and \\frac{d}{d v_{2}} 0 = \\frac{\\partial}{\\partial v_{2}} \\frac{- v_{2} - \\Psi^{\\dagger}{(E,v_{2})} + e^{E}}{v_{2} + \\Psi^{\\dagger}{(E,v_{2})} - e^{E}} and \\frac{d}{d v_{2}} 0 = \\frac{d}{d v_{2}} (-1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), exp(Symbol('E', commutative=True))))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))), exp(Symbol('E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))), exp(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)))"], [["divide", 4, "Add(Symbol('v_2', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Symbol('E', commutative=True))))"], "Equality(Integer(0), Mul(Pow(Add(Symbol('v_2', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Symbol('E', commutative=True)))), Integer(-1)), Function('\\\\phi_2')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))), exp(Symbol('E', commutative=True))), Pow(Add(Symbol('v_2', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Symbol('E', commutative=True)))), Integer(-1))))"], [["differentiate", 6, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True))), exp(Symbol('E', commutative=True))), Pow(Add(Symbol('v_2', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Symbol('E', commutative=True)))), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(G)} = \\log{(G)}, then obtain \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\mathbf{M}{(G)}} + \\frac{d}{d G} \\mathbf{M}{(G)} = \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\mathbf{M}{(G)}} + \\frac{d}{d G} \\log{(G)}", "derivation": "\\mathbf{M}{(G)} = \\log{(G)} and \\frac{d}{d G} \\mathbf{M}{(G)} = \\frac{d}{d G} \\log{(G)} and e^{\\frac{d}{d G} \\mathbf{M}{(G)}} = e^{\\frac{d}{d G} \\log{(G)}} and \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\log{(G)}} + \\frac{d}{d G} \\mathbf{M}{(G)} = \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\log{(G)}} + \\frac{d}{d G} \\log{(G)} and \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\mathbf{M}{(G)}} + \\frac{d}{d G} \\mathbf{M}{(G)} = \\mathbf{M}{(G)} - e^{\\frac{d}{d G} \\mathbf{M}{(G)}} + \\frac{d}{d G} \\log{(G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), exp(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('G', commutative=True))), exp(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Mul(Integer(-1), exp(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Mul(Integer(-1), exp(Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Mul(Integer(-1), exp(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Mul(Integer(-1), exp(Derivative(Function('\\\\mathbf{M}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))), Derivative(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(S)} = e^{S}, then obtain - (\\varepsilon{(S)} + 3 e^{S})^{4} - (\\varepsilon{(S)} + 3 e^{S})^{S} = - (\\varepsilon{(S)} + 3 e^{S})^{S} - 256 e^{4 S}", "derivation": "\\varepsilon{(S)} = e^{S} and \\varepsilon{(S)} + e^{S} = 2 e^{S} and \\varepsilon{(S)} + 3 e^{S} = 4 e^{S} and (\\varepsilon{(S)} + 3 e^{S})^{4} = 256 e^{4 S} and - (\\varepsilon{(S)} + 3 e^{S})^{4} = - 256 e^{4 S} and - (\\varepsilon{(S)} + 3 e^{S})^{4} - (\\varepsilon{(S)} + 3 e^{S})^{S} = - (\\varepsilon{(S)} + 3 e^{S})^{S} - 256 e^{4 S}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["add", 1, "exp(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Mul(Integer(2), exp(Symbol('S', commutative=True))))"], [["add", 2, "Mul(Integer(2), exp(Symbol('S', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Mul(Integer(4), exp(Symbol('S', commutative=True))))"], [["power", 3, 4], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Integer(4)), Mul(Integer(256), exp(Mul(Integer(4), Symbol('S', commutative=True)))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Integer(4))), Mul(Integer(-1), Integer(256), exp(Mul(Integer(4), Symbol('S', commutative=True)))))"], [["minus", 5, "Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Integer(4))), Mul(Integer(-1), Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Function('\\\\varepsilon')(Symbol('S', commutative=True)), Mul(Integer(3), exp(Symbol('S', commutative=True)))), Symbol('S', commutative=True))), Mul(Integer(-1), Integer(256), exp(Mul(Integer(4), Symbol('S', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(i)} = \\cos{(i)}, then obtain \\frac{1}{((- \\int 0 di)^{i} + \\cos{(i)})^{2}} = \\frac{1}{((- 2 \\operatorname{C_{2}}{(i)} + 2 \\cos{(i)} - \\int 0 di)^{i} + \\cos{(i)})^{2}}", "derivation": "\\operatorname{C_{2}}{(i)} = \\cos{(i)} and 0 = - \\operatorname{C_{2}}{(i)} + \\cos{(i)} and - \\int 0 di = - \\operatorname{C_{2}}{(i)} + \\cos{(i)} - \\int 0 di and (- \\int 0 di)^{i} = (- \\operatorname{C_{2}}{(i)} + \\cos{(i)} - \\int 0 di)^{i} and (- \\operatorname{C_{2}}{(i)} + \\cos{(i)} - \\int 0 di)^{i} = (- 2 \\operatorname{C_{2}}{(i)} + 2 \\cos{(i)} - \\int 0 di)^{i} and (- \\int 0 di)^{i} = (- 2 \\operatorname{C_{2}}{(i)} + 2 \\cos{(i)} - \\int 0 di)^{i} and (- \\int 0 di)^{i} + \\cos{(i)} = (- 2 \\operatorname{C_{2}}{(i)} + 2 \\cos{(i)} - \\int 0 di)^{i} + \\cos{(i)} and \\frac{1}{((- \\int 0 di)^{i} + \\cos{(i)})^{2}} = \\frac{1}{((- 2 \\operatorname{C_{2}}{(i)} + 2 \\cos{(i)} - \\int 0 di)^{i} + \\cos{(i)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["minus", 1, "Function('C_2')(Symbol('i', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_2')(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True))))"], [["minus", 2, "Integral(Integer(0), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Function('C_2')(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))))"], [["power", 3, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Function('C_2')(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('C_2')(Symbol('i', commutative=True))), cos(Symbol('i', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Function('C_2')(Symbol('i', commutative=True))), Mul(Integer(2), cos(Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Function('C_2')(Symbol('i', commutative=True))), Mul(Integer(2), cos(Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)))"], [["minus", 6, "Mul(Integer(-1), cos(Symbol('i', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Integer(2), Function('C_2')(Symbol('i', commutative=True))), Mul(Integer(2), cos(Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))))"], [["power", 7, "Integer(-2)"], "Equality(Pow(Add(Pow(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Integer(-2)), Pow(Add(Pow(Add(Mul(Integer(-1), Integer(2), Function('C_2')(Symbol('i', commutative=True))), Mul(Integer(2), cos(Symbol('i', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\theta_1,\\rho)} = \\log{(\\rho^{\\theta_1})}, then obtain \\frac{\\int \\operatorname{P_{g}}{(\\theta_1,\\rho)} d\\theta_1}{\\theta_1} = \\frac{\\frac{\\theta_1^{2} \\log{(\\rho)}}{2} + a^{\\dagger}}{\\theta_1}", "derivation": "\\operatorname{P_{g}}{(\\theta_1,\\rho)} = \\log{(\\rho^{\\theta_1})} and \\int \\operatorname{P_{g}}{(\\theta_1,\\rho)} d\\theta_1 = \\int \\log{(\\rho^{\\theta_1})} d\\theta_1 and \\frac{\\int \\operatorname{P_{g}}{(\\theta_1,\\rho)} d\\theta_1}{\\theta_1} = \\frac{\\int \\log{(\\rho^{\\theta_1})} d\\theta_1}{\\theta_1} and \\frac{\\int \\operatorname{P_{g}}{(\\theta_1,\\rho)} d\\theta_1}{\\theta_1} = \\frac{\\frac{\\theta_1^{2} \\log{(\\rho)}}{2} + a^{\\dagger}}{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\rho', commutative=True)), log(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(log(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Integral(Function('P_g')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Integral(log(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Integral(Function('P_g')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)), log(Symbol('\\\\rho', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\chi{(t_{1},p)} = \\log{(p - t_{1})}, then obtain \\int \\frac{\\partial}{\\partial p} \\int \\chi{(t_{1},p)} dt_{1} dt_{1} = \\int \\frac{\\partial}{\\partial p} \\int \\log{(p - t_{1})} dt_{1} dt_{1}", "derivation": "\\chi{(t_{1},p)} = \\log{(p - t_{1})} and \\int \\chi{(t_{1},p)} dt_{1} = \\int \\log{(p - t_{1})} dt_{1} and \\frac{\\partial}{\\partial p} \\int \\chi{(t_{1},p)} dt_{1} = \\frac{\\partial}{\\partial p} \\int \\log{(p - t_{1})} dt_{1} and \\int \\frac{\\partial}{\\partial p} \\int \\chi{(t_{1},p)} dt_{1} dt_{1} = \\int \\frac{\\partial}{\\partial p} \\int \\log{(p - t_{1})} dt_{1} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('t_1', commutative=True), Symbol('p', commutative=True)), log(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('t_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(log(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\chi')(Symbol('t_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(log(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('t_1', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\chi')(Symbol('t_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))), Integral(Derivative(Integral(log(Add(Symbol('p', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given H{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then obtain ((- \\varepsilon_0 + H{(\\varepsilon_0)} + 1)^{\\varepsilon_0})^{\\varepsilon_0} = ((- \\varepsilon_0 + \\sin{(\\varepsilon_0)} + 1)^{\\varepsilon_0})^{\\varepsilon_0}", "derivation": "H{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and - \\varepsilon_0 + H{(\\varepsilon_0)} = - \\varepsilon_0 + \\sin{(\\varepsilon_0)} and - \\varepsilon_0 + H{(\\varepsilon_0)} + 1 = - \\varepsilon_0 + \\sin{(\\varepsilon_0)} + 1 and (- \\varepsilon_0 + H{(\\varepsilon_0)} + 1)^{\\varepsilon_0} = (- \\varepsilon_0 + \\sin{(\\varepsilon_0)} + 1)^{\\varepsilon_0} and ((- \\varepsilon_0 + H{(\\varepsilon_0)} + 1)^{\\varepsilon_0})^{\\varepsilon_0} = ((- \\varepsilon_0 + \\sin{(\\varepsilon_0)} + 1)^{\\varepsilon_0})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('H')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('H')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('H')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 4, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('H')(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(p)} = \\cos{(e^{p})}, then derive \\frac{d}{d p} \\int \\operatorname{f_{\\mathbf{v}}}{(p)} dp = \\frac{\\partial}{\\partial p} (\\mathbf{J}_f + \\operatorname{Ci}{(e^{p})}), then obtain \\frac{d}{d p} \\int \\cos{(e^{p})} dp = \\frac{\\partial}{\\partial p} (\\mathbf{J}_f + \\operatorname{Ci}{(e^{p})})", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(p)} = \\cos{(e^{p})} and \\int \\operatorname{f_{\\mathbf{v}}}{(p)} dp = \\int \\cos{(e^{p})} dp and \\frac{d}{d p} \\int \\operatorname{f_{\\mathbf{v}}}{(p)} dp = \\frac{d}{d p} \\int \\cos{(e^{p})} dp and \\frac{d}{d p} \\int \\operatorname{f_{\\mathbf{v}}}{(p)} dp = \\frac{\\partial}{\\partial p} (\\mathbf{J}_f + \\operatorname{Ci}{(e^{p})}) and \\frac{d}{d p} \\int \\cos{(e^{p})} dp = \\frac{\\partial}{\\partial p} (\\mathbf{J}_f + \\operatorname{Ci}{(e^{p})})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Ci(exp(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integral(cos(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Ci(exp(Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = A_{x} + u, then derive \\frac{\\partial}{\\partial A_{x}} \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = 1, then obtain 0 = 1 - \\frac{\\partial}{\\partial A_{x}} (A_{x} + u)", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = A_{x} + u and \\frac{\\partial}{\\partial A_{x}} \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + u) and \\frac{\\partial}{\\partial A_{x}} \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = 1 and - \\frac{\\partial}{\\partial A_{x}} (A_{x} + u) + \\frac{\\partial}{\\partial A_{x}} \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} = 1 - \\frac{\\partial}{\\partial A_{x}} (A_{x} + u) and 0 = 1 - \\frac{\\partial}{\\partial A_{x}} \\operatorname{f_{\\mathbf{p}}}{(A_{x},u)} and 0 = 1 - \\frac{\\partial}{\\partial A_{x}} (A_{x} + u)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Derivative(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(T,F_{N})} = \\int F_{N} T dT, then obtain \\operatorname{v_{x}}{(T,F_{N})} + \\operatorname{v_{x}}^{F_{N}}{(T,F_{N})} = \\operatorname{v_{x}}^{F_{N}}{(T,F_{N})} + \\int F_{N} T dT", "derivation": "\\operatorname{v_{x}}{(T,F_{N})} = \\int F_{N} T dT and \\operatorname{v_{x}}^{F_{N}}{(T,F_{N})} = (\\int F_{N} T dT)^{F_{N}} and \\operatorname{v_{x}}{(T,F_{N})} + (\\int F_{N} T dT)^{F_{N}} = \\int F_{N} T dT + (\\int F_{N} T dT)^{F_{N}} and \\operatorname{v_{x}}{(T,F_{N})} + \\operatorname{v_{x}}^{F_{N}}{(T,F_{N})} = \\operatorname{v_{x}}^{F_{N}}{(T,F_{N})} + \\int F_{N} T dT", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('F_N', commutative=True)))"], [["add", 1, "Pow(Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('F_N', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Pow(Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('F_N', commutative=True))), Add(Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Pow(Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Pow(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Pow(Function('v_x')(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Integral(Mul(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(F_{c},\\hat{p}_0)} = F_{c} + \\hat{p}_0, then obtain \\int\\limits^{- \\hat{p}_0 + \\dot{x}{(F_{c},\\hat{p}_0)}} \\cos{(\\dot{x}{(F_{c},\\hat{p}_0)})} dF_{c} = \\int\\limits^{- \\hat{p}_0 + \\dot{x}{(F_{c},\\hat{p}_0)}} \\cos{(F_{c} + \\hat{p}_0)} dF_{c}", "derivation": "\\dot{x}{(F_{c},\\hat{p}_0)} = F_{c} + \\hat{p}_0 and - \\hat{p}_0 + \\dot{x}{(F_{c},\\hat{p}_0)} = F_{c} and \\cos{(\\dot{x}{(F_{c},\\hat{p}_0)})} = \\cos{(F_{c} + \\hat{p}_0)} and \\int \\cos{(\\dot{x}{(F_{c},\\hat{p}_0)})} dF_{c} = \\int \\cos{(F_{c} + \\hat{p}_0)} dF_{c} and \\int\\limits^{- \\hat{p}_0 + \\dot{x}{(F_{c},\\hat{p}_0)}} \\cos{(\\dot{x}{(F_{c},\\hat{p}_0)})} dF_{c} = \\int\\limits^{- \\hat{p}_0 + \\dot{x}{(F_{c},\\hat{p}_0)}} \\cos{(F_{c} + \\hat{p}_0)} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('F_c', commutative=True))"], [["cos", 1], "Equality(cos(Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(cos(Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(cos(Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('F_c', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))), Integral(cos(Add(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('F_c', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('\\\\dot{x}')(Symbol('F_c', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(\\hat{x})} = e^{\\hat{x}}, then obtain \\int \\frac{\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}}{\\hat{x}} d\\hat{x} = \\int (\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}) d\\hat{x}", "derivation": "\\mathbf{s}{(\\hat{x})} = e^{\\hat{x}} and \\mathbf{s}{(\\hat{x})} - e^{\\hat{x}} = 0 and \\int (\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}) d\\hat{x} = \\int 0 d\\hat{x} and \\frac{\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}}{\\hat{x}} = 0 and \\int \\frac{\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}}{\\hat{x}} d\\hat{x} = \\int 0 d\\hat{x} and \\int \\frac{\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}}{\\hat{x}} d\\hat{x} = \\int (\\mathbf{s}{(\\hat{x})} - e^{\\hat{x}}) d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given c{(a^{\\dagger},V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}} a^{\\dagger})}, then obtain c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})} + \\frac{c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})}}{V_{\\mathbf{E}}} = c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})}", "derivation": "c{(a^{\\dagger},V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}} a^{\\dagger})} and c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})} = 0 and \\frac{c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})}}{V_{\\mathbf{E}}} = 0 and c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})} + \\frac{c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})}}{V_{\\mathbf{E}}} = c{(a^{\\dagger},V_{\\mathbf{E}})} - \\log{(V_{\\mathbf{E}} a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 1, "log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))), Integer(0))"], [["divide", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))), Integer(0))"], [["add", 3, "Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"], "Equality(Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))))), Add(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given i{(\\phi_2,\\Psi_{nl})} = \\phi_2 + \\sin{(\\Psi_{nl})}, then obtain e^{i{(\\phi_2,\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2}} = e^{\\phi_2 + \\sin{(\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2}}", "derivation": "i{(\\phi_2,\\Psi_{nl})} = \\phi_2 + \\sin{(\\Psi_{nl})} and \\frac{i{(\\phi_2,\\Psi_{nl})}}{\\phi_2} = \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2} and i{(\\phi_2,\\Psi_{nl})} + \\frac{i{(\\phi_2,\\Psi_{nl})}}{\\phi_2} = \\phi_2 + \\sin{(\\Psi_{nl})} + \\frac{i{(\\phi_2,\\Psi_{nl})}}{\\phi_2} and i{(\\phi_2,\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2} = \\phi_2 + \\sin{(\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2} and e^{i{(\\phi_2,\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2}} = e^{\\phi_2 + \\sin{(\\Psi_{nl})} + \\frac{\\phi_2 + \\sin{(\\Psi_{nl})}}{\\phi_2}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["add", 1, "Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Add(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))))"], [["exp", 4], "Equality(exp(Add(Function('i')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)))))), exp(Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\theta_2)} = \\sin{(\\sin{(\\theta_2)})} and C{(\\theta_2)} = \\frac{\\sin{(\\sin{(\\theta_2)})}}{\\operatorname{E_{\\lambda}}{(\\theta_2)}}, then obtain C{(\\theta_2)} - \\sin^{\\theta_2}{(\\sin{(\\theta_2)})} - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}} = - \\sin^{\\theta_2}{(\\sin{(\\theta_2)})} + 1 - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\theta_2)} = \\sin{(\\sin{(\\theta_2)})} and C{(\\theta_2)} = \\frac{\\sin{(\\sin{(\\theta_2)})}}{\\operatorname{E_{\\lambda}}{(\\theta_2)}} and C{(\\theta_2)} = 1 and C{(\\theta_2)} - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}} = 1 - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}} and C{(\\theta_2)} - \\sin^{\\theta_2}{(\\sin{(\\theta_2)})} - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}} = - \\sin^{\\theta_2}{(\\sin{(\\theta_2)})} + 1 - \\frac{1}{\\operatorname{E_{\\lambda}}{(\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), sin(sin(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C')(Symbol('\\\\theta_2', commutative=True)), Integer(1))"], [["minus", 3, "Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1))"], "Equality(Add(Function('C')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))))"], [["minus", 4, "Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('C')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(f^{*})} = \\int \\cos{(f^{*})} df^{*}, then derive \\operatorname{c_{0}}{(f^{*})} = S + \\sin{(f^{*})}, then derive S + \\sin{(f^{*})} = g_{\\varepsilon} + \\sin{(f^{*})}, then obtain (g_{\\varepsilon} + \\sin{(f^{*})}) \\log{(\\theta)} = (\\mathbf{D} + \\sin{(f^{*})}) \\log{(\\theta)}", "derivation": "\\operatorname{c_{0}}{(f^{*})} = \\int \\cos{(f^{*})} df^{*} and \\operatorname{c_{0}}{(f^{*})} = S + \\sin{(f^{*})} and S + \\sin{(f^{*})} = \\int \\cos{(f^{*})} df^{*} and S + \\sin{(f^{*})} = g_{\\varepsilon} + \\sin{(f^{*})} and \\operatorname{c_{0}}{(f^{*})} = g_{\\varepsilon} + \\sin{(f^{*})} and g_{\\varepsilon} + \\sin{(f^{*})} = \\int \\cos{(f^{*})} df^{*} and (g_{\\varepsilon} + \\sin{(f^{*})}) \\log{(\\theta)} = \\log{(\\theta)} \\int \\cos{(f^{*})} df^{*} and (g_{\\varepsilon} + \\sin{(f^{*})}) \\log{(\\theta)} = (\\mathbf{D} + \\sin{(f^{*})}) \\log{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('f^*', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('c_0')(Symbol('f^*', commutative=True)), Add(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('S', commutative=True), sin(Symbol('f^*', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Function('c_0')(Symbol('f^*', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('f^*', commutative=True))), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["times", 6, "log(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('f^*', commutative=True))), log(Symbol('\\\\theta', commutative=True))), Mul(log(Symbol('\\\\theta', commutative=True)), Integral(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Mul(Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('f^*', commutative=True))), log(Symbol('\\\\theta', commutative=True))), Mul(Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('f^*', commutative=True))), log(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\phi{(k,u)} = k + u, then obtain 2 k + 2 u - 2 \\phi{(k,u)} = 0", "derivation": "\\phi{(k,u)} = k + u and k + u + \\phi{(k,u)} = 2 k + 2 u and 2 \\phi{(k,u)} = 2 k + 2 u and k + u - \\phi{(k,u)} = 2 k + 2 u - 2 \\phi{(k,u)} and k + u - \\phi{(k,u)} = 0 and 2 k + 2 u - 2 \\phi{(k,u)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)), Add(Symbol('k', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Add(Symbol('k', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Symbol('u', commutative=True), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)))"], "Equality(Add(Symbol('k', commutative=True), Symbol('u', commutative=True), Mul(Integer(-1), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('k', commutative=True), Symbol('u', commutative=True), Mul(Integer(-1), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\phi')(Symbol('k', commutative=True), Symbol('u', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{z})} = \\int \\sin{(A_{z})} dA_{z}, then derive \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})}, then derive c_{0} - \\cos{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})}, then derive c_{0} - \\cos{(A_{z})} = A_{1} - \\cos{(A_{z})}, then obtain A_{1} - \\cos{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{z})} = \\int \\sin{(A_{z})} dA_{z} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})} and \\int \\sin{(A_{z})} dA_{z} = \\mathbf{P} - \\cos{(A_{z})} and c_{0} - \\cos{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})} and c_{0} - \\cos{(A_{z})} = \\int \\sin{(A_{z})} dA_{z} and c_{0} - \\cos{(A_{z})} = A_{1} - \\cos{(A_{z})} and A_{1} - \\cos{(A_{z})} = \\mathbf{P} - \\cos{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_z', commutative=True)), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('A_z', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given H{(S)} = \\cos{(S)} and \\mathbf{r}{(S)} = H{(S)} + \\cos{(S)}, then obtain \\log{(2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + \\mathbf{r}{(S)}) \\cos{(S)})} = \\log{(2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + 2 \\cos{(S)}) \\cos{(S)})}", "derivation": "H{(S)} = \\cos{(S)} and \\mathbf{r}{(S)} = H{(S)} + \\cos{(S)} and \\mathbf{r}{(S)} = 2 \\cos{(S)} and - (H{(S)} + \\cos{(S)}) \\cos{(S)} + \\mathbf{r}{(S)} = - (H{(S)} + \\cos{(S)}) \\cos{(S)} + 2 \\cos{(S)} and 2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + \\mathbf{r}{(S)}) \\cos{(S)} = 2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + 2 \\cos{(S)}) \\cos{(S)} and \\log{(2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + \\mathbf{r}{(S)}) \\cos{(S)})} = \\log{(2 (- (H{(S)} + \\cos{(S)}) \\cos{(S)} + 2 \\cos{(S)}) \\cos{(S)})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('S', commutative=True)), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{r}')(Symbol('S', commutative=True)), Mul(Integer(2), cos(Symbol('S', commutative=True))))"], [["minus", 3, "Mul(Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Function('\\\\mathbf{r}')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Mul(Integer(2), cos(Symbol('S', commutative=True)))))"], [["times", 4, "Mul(Integer(2), cos(Symbol('S', commutative=True)))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Function('\\\\mathbf{r}')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Mul(Integer(2), Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Mul(Integer(2), cos(Symbol('S', commutative=True)))), cos(Symbol('S', commutative=True))))"], [["log", 5], "Equality(log(Mul(Integer(2), Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Function('\\\\mathbf{r}')(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True)))), log(Mul(Integer(2), Add(Mul(Integer(-1), Add(Function('H')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), cos(Symbol('S', commutative=True))), Mul(Integer(2), cos(Symbol('S', commutative=True)))), cos(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\psi)} = \\psi and C{(\\hat{x}_0,p)} = \\frac{p}{\\hat{x}_0}, then obtain - \\hat{x}_0 + \\int (C{(\\hat{x}_0,p)} + \\frac{\\psi}{\\nabla}) d\\psi = - \\hat{x}_0 + \\int (\\frac{\\psi}{\\nabla} + \\frac{p}{\\hat{x}_0}) d\\psi", "derivation": "\\operatorname{r_{0}}{(\\psi)} = \\psi and C{(\\hat{x}_0,p)} = \\frac{p}{\\hat{x}_0} and C{(\\hat{x}_0,p)} + \\frac{\\operatorname{r_{0}}{(\\psi)}}{\\nabla} = \\frac{\\operatorname{r_{0}}{(\\psi)}}{\\nabla} + \\frac{p}{\\hat{x}_0} and C{(\\hat{x}_0,p)} + \\frac{\\psi}{\\nabla} = \\frac{\\psi}{\\nabla} + \\frac{p}{\\hat{x}_0} and \\int (C{(\\hat{x}_0,p)} + \\frac{\\psi}{\\nabla}) d\\psi = \\int (\\frac{\\psi}{\\nabla} + \\frac{p}{\\hat{x}_0}) d\\psi and - \\hat{x}_0 + \\int (C{(\\hat{x}_0,p)} + \\frac{\\psi}{\\nabla}) d\\psi = - \\hat{x}_0 + \\int (\\frac{\\psi}{\\nabla} + \\frac{p}{\\hat{x}_0}) d\\psi", "srepr_derivation": [["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))"], ["get_premise", "Equality(Function('C')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Function('C')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\psi', commutative=True)))), Add(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('r_0')(Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('C')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Add(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Function('C')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["minus", 5, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Add(Function('C')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Add(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\psi', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given A{(v_{z},\\rho,f_{E})} = - \\rho + f_{E} + v_{z}, then obtain ((\\frac{A{(v_{z},\\rho,f_{E})}}{f_{E}})^{f_{E}})^{\\rho} = ((\\frac{- \\rho + f_{E} + v_{z}}{f_{E}})^{f_{E}})^{\\rho}", "derivation": "A{(v_{z},\\rho,f_{E})} = - \\rho + f_{E} + v_{z} and \\frac{A{(v_{z},\\rho,f_{E})}}{f_{E}} = \\frac{- \\rho + f_{E} + v_{z}}{f_{E}} and (\\frac{A{(v_{z},\\rho,f_{E})}}{f_{E}})^{f_{E}} = (\\frac{- \\rho + f_{E} + v_{z}}{f_{E}})^{f_{E}} and ((\\frac{A{(v_{z},\\rho,f_{E})}}{f_{E}})^{f_{E}})^{\\rho} = ((\\frac{- \\rho + f_{E} + v_{z}}{f_{E}})^{f_{E}})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('v_z', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('f_E', commutative=True), Symbol('v_z', commutative=True)))"], [["divide", 1, "Symbol('f_E', commutative=True)"], "Equality(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('A')(Symbol('v_z', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('f_E', commutative=True), Symbol('v_z', commutative=True))))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('A')(Symbol('v_z', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)), Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('f_E', commutative=True), Symbol('v_z', commutative=True))), Symbol('f_E', commutative=True)))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('A')(Symbol('v_z', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Mul(Pow(Symbol('f_E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('f_E', commutative=True), Symbol('v_z', commutative=True))), Symbol('f_E', commutative=True)), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\nabla{(x^\\prime)} = e^{x^\\prime}, then derive \\int \\nabla{(x^\\prime)} dx^\\prime = C_{d} + e^{x^\\prime}, then obtain \\int \\nabla{(x^\\prime)} dx^\\prime = C_{d} + \\nabla{(x^\\prime)}", "derivation": "\\nabla{(x^\\prime)} = e^{x^\\prime} and \\int \\nabla{(x^\\prime)} dx^\\prime = \\int e^{x^\\prime} dx^\\prime and \\int \\nabla{(x^\\prime)} dx^\\prime = C_{d} + e^{x^\\prime} and \\int \\nabla{(x^\\prime)} dx^\\prime = C_{d} + \\nabla{(x^\\prime)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('C_d', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('C_d', commutative=True), Function('\\\\nabla')(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})} = ((a^{\\dagger})^{f_{E}})^{v_{z}}, then obtain \\frac{-2 + ((a^{\\dagger})^{f_{E}})^{- v_{z}} \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}}{\\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}} = - \\frac{1}{\\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}}", "derivation": "\\mathbf{E}{(a^{\\dagger},v_{z},f_{E})} = ((a^{\\dagger})^{f_{E}})^{v_{z}} and ((a^{\\dagger})^{f_{E}})^{- v_{z}} \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})} = 1 and -1 + ((a^{\\dagger})^{f_{E}})^{- v_{z}} \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})} = 0 and -2 + ((a^{\\dagger})^{f_{E}})^{- v_{z}} \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})} = -1 and \\frac{-2 + ((a^{\\dagger})^{f_{E}})^{- v_{z}} \\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}}{\\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}} = - \\frac{1}{\\mathbf{E}{(a^{\\dagger},v_{z},f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Symbol('v_z', commutative=True)))"], [["divide", 1, "Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)))), Integer(0))"], [["minus", 3, 1], "Equality(Add(Integer(-2), Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)))), Integer(-1))"], [["divide", 4, "Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Add(Integer(-2), Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True))), Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)))), Pow(Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{E}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(\\mu,\\mathbf{r})} = \\frac{\\mu}{\\mathbf{r}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mu,\\mathbf{r})} = - \\dot{y}{(\\mu,\\mathbf{r})}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mu,\\mathbf{r})} = - \\frac{\\mu}{\\mathbf{r}}", "derivation": "\\dot{y}{(\\mu,\\mathbf{r})} = \\frac{\\mu}{\\mathbf{r}} and - \\dot{y}{(\\mu,\\mathbf{r})} = - \\frac{\\mu}{\\mathbf{r}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mu,\\mathbf{r})} = - \\dot{y}{(\\mu,\\mathbf{r})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mu,\\mathbf{r})} = - \\frac{\\mu}{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given c{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} e^{V_{\\mathbf{E}}} and Z{(\\mathbb{I},V_{\\mathbf{E}})} = \\int \\mathbb{I} e^{V_{\\mathbf{E}}} d\\mathbb{I}, then obtain (\\int c{(\\mathbb{I},V_{\\mathbf{E}})} d\\mathbb{I})^{\\mathbb{I}} = Z^{\\mathbb{I}}{(\\mathbb{I},V_{\\mathbf{E}})}", "derivation": "c{(\\mathbb{I},V_{\\mathbf{E}})} = \\mathbb{I} e^{V_{\\mathbf{E}}} and \\int c{(\\mathbb{I},V_{\\mathbf{E}})} d\\mathbb{I} = \\int \\mathbb{I} e^{V_{\\mathbf{E}}} d\\mathbb{I} and Z{(\\mathbb{I},V_{\\mathbf{E}})} = \\int \\mathbb{I} e^{V_{\\mathbf{E}}} d\\mathbb{I} and (\\int c{(\\mathbb{I},V_{\\mathbf{E}})} d\\mathbb{I})^{\\mathbb{I}} = (\\int \\mathbb{I} e^{V_{\\mathbf{E}}} d\\mathbb{I})^{\\mathbb{I}} and (\\int c{(\\mathbb{I},V_{\\mathbf{E}})} d\\mathbb{I})^{\\mathbb{I}} = Z^{\\mathbb{I}}{(\\mathbb{I},V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('Z')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(v,\\tilde{g})} = \\tilde{g} + v, then derive \\int \\frac{\\partial}{\\partial v} \\operatorname{A_{x}}{(v,\\tilde{g})} d\\tilde{g} = \\mathbf{H} + \\tilde{g}, then derive \\mathbf{J}_M + \\tilde{g} = \\mathbf{H} + \\tilde{g}, then obtain \\tilde{g} + f_{\\mathbf{p}} = \\mathbf{J}_M + \\tilde{g}", "derivation": "\\operatorname{A_{x}}{(v,\\tilde{g})} = \\tilde{g} + v and \\frac{\\partial}{\\partial v} \\operatorname{A_{x}}{(v,\\tilde{g})} = \\frac{\\partial}{\\partial v} (\\tilde{g} + v) and \\int \\frac{\\partial}{\\partial v} \\operatorname{A_{x}}{(v,\\tilde{g})} d\\tilde{g} = \\int \\frac{\\partial}{\\partial v} (\\tilde{g} + v) d\\tilde{g} and \\int \\frac{\\partial}{\\partial v} \\operatorname{A_{x}}{(v,\\tilde{g})} d\\tilde{g} = \\mathbf{H} + \\tilde{g} and \\int \\frac{\\partial}{\\partial v} (\\tilde{g} + v) d\\tilde{g} = \\mathbf{H} + \\tilde{g} and \\mathbf{J}_M + \\tilde{g} = \\mathbf{H} + \\tilde{g} and \\int \\frac{\\partial}{\\partial v} (\\tilde{g} + v) d\\tilde{g} = \\mathbf{J}_M + \\tilde{g} and \\tilde{g} + f_{\\mathbf{p}} = \\mathbf{J}_M + \\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Derivative(Function('A_x')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('A_x')(Symbol('v', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\Psi_{nl},L)} = L \\cos{(\\Psi_{nl})}, then obtain \\int (\\Psi_{nl} + \\phi{(\\Psi_{nl},L)}) dL = A_{y} + \\frac{L^{2} \\cos{(\\Psi_{nl})}}{2} + L \\Psi_{nl}", "derivation": "\\phi{(\\Psi_{nl},L)} = L \\cos{(\\Psi_{nl})} and \\Psi_{nl} + \\phi{(\\Psi_{nl},L)} = L \\cos{(\\Psi_{nl})} + \\Psi_{nl} and \\int (\\Psi_{nl} + \\phi{(\\Psi_{nl},L)}) dL = \\int (L \\cos{(\\Psi_{nl})} + \\Psi_{nl}) dL and \\int (\\Psi_{nl} + \\phi{(\\Psi_{nl},L)}) dL = A_{y} + \\frac{L^{2} \\cos{(\\Psi_{nl})}}{2} + L \\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Mul(Symbol('L', commutative=True), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('L', commutative=True), Integer(2)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('L', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} = - \\mathbf{F} + \\psi^* c_{0}, then derive \\int \\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} d\\psi^* = - \\mathbf{F} \\psi^* + \\frac{(\\psi^*)^{2} c_{0}}{2} + r_{0}, then obtain \\iint (- \\mathbf{F} + \\psi^* c_{0}) d\\psi^* dr_{0} = \\int (- \\mathbf{F} \\psi^* + \\frac{(\\psi^*)^{2} c_{0}}{2} + r_{0}) dr_{0}", "derivation": "\\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} = - \\mathbf{F} + \\psi^* c_{0} and \\int \\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} d\\psi^* = \\int (- \\mathbf{F} + \\psi^* c_{0}) d\\psi^* and \\int \\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} d\\psi^* = - \\mathbf{F} \\psi^* + \\frac{(\\psi^*)^{2} c_{0}}{2} + r_{0} and \\iint \\hat{x}{(\\psi^*,c_{0},\\mathbf{F})} d\\psi^* dr_{0} = \\int (- \\mathbf{F} \\psi^* + \\frac{(\\psi^*)^{2} c_{0}}{2} + r_{0}) dr_{0} and \\iint (- \\mathbf{F} + \\psi^* c_{0}) d\\psi^* dr_{0} = \\int (- \\mathbf{F} \\psi^* + \\frac{(\\psi^*)^{2} c_{0}}{2} + r_{0}) dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Symbol('c_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Symbol('c_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), Symbol('c_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given H{(\\phi,\\mathbf{P})} = \\log{(\\mathbf{P} \\phi)}, then obtain \\int (- H{(\\phi,\\mathbf{P})} + H^{\\phi}{(\\phi,\\mathbf{P})}) d\\phi = \\int (- H{(\\phi,\\mathbf{P})} + \\log{(\\mathbf{P} \\phi)}^{\\phi}) d\\phi", "derivation": "H{(\\phi,\\mathbf{P})} = \\log{(\\mathbf{P} \\phi)} and H^{\\phi}{(\\phi,\\mathbf{P})} = \\log{(\\mathbf{P} \\phi)}^{\\phi} and - H{(\\phi,\\mathbf{P})} + H^{\\phi}{(\\phi,\\mathbf{P})} = - H{(\\phi,\\mathbf{P})} + \\log{(\\mathbf{P} \\phi)}^{\\phi} and \\int (- H{(\\phi,\\mathbf{P})} + H^{\\phi}{(\\phi,\\mathbf{P})}) d\\phi = \\int (- H{(\\phi,\\mathbf{P})} + \\log{(\\mathbf{P} \\phi)}^{\\phi}) d\\phi", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["minus", 2, "Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\omega{(g,E)} = E g, then derive \\frac{\\partial}{\\partial g} \\omega{(g,E)} = E, then obtain \\frac{g - \\frac{\\partial^{2}}{\\partial g\\partial E} \\omega{(g,E)}}{g - 1} = 1", "derivation": "\\omega{(g,E)} = E g and \\frac{\\partial}{\\partial g} \\omega{(g,E)} = \\frac{\\partial}{\\partial g} E g and \\frac{\\partial}{\\partial g} \\omega{(g,E)} = E and \\frac{\\partial^{2}}{\\partial E\\partial g} \\omega{(g,E)} = \\frac{d}{d E} E and - g + \\frac{\\partial^{2}}{\\partial E\\partial g} \\omega{(g,E)} = - g + \\frac{d}{d E} E and g - \\frac{\\partial^{2}}{\\partial E\\partial g} \\omega{(g,E)} = g - \\frac{d}{d E} E and \\frac{g - \\frac{\\partial^{2}}{\\partial E\\partial g} \\omega{(g,E)}}{g - \\frac{d}{d E} E} = 1 and \\frac{g - \\frac{\\partial^{2}}{\\partial g\\partial E} \\omega{(g,E)}}{g - 1} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('E', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('E', commutative=True))"], [["differentiate", 3, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))))), Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["divide", 6, "Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)))))"], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))), Integer(-1)), Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1)))))), Integer(1))"], [["evaluate_derivatives", 7], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Integer(-1)), Integer(-1)), Add(Symbol('g', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\omega')(Symbol('g', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1)))))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(f^{*},\\mathbf{A})} = \\mathbf{A} - f^{*} and \\mathbf{S}{(f^{*},\\mathbf{A})} = (- \\frac{(\\mathbf{A} - f^{*})^{\\mathbf{A}}}{f^{*}})^{\\mathbf{A}}, then obtain - f^{*} \\mathbf{S}{(f^{*},\\mathbf{A})} = - f^{*} (- \\frac{\\operatorname{A_{x}}^{\\mathbf{A}}{(f^{*},\\mathbf{A})}}{f^{*}})^{\\mathbf{A}}", "derivation": "\\operatorname{A_{x}}{(f^{*},\\mathbf{A})} = \\mathbf{A} - f^{*} and \\operatorname{A_{x}}^{\\mathbf{A}}{(f^{*},\\mathbf{A})} = (\\mathbf{A} - f^{*})^{\\mathbf{A}} and - \\frac{\\operatorname{A_{x}}^{\\mathbf{A}}{(f^{*},\\mathbf{A})}}{f^{*}} = - \\frac{(\\mathbf{A} - f^{*})^{\\mathbf{A}}}{f^{*}} and \\mathbf{S}{(f^{*},\\mathbf{A})} = (- \\frac{(\\mathbf{A} - f^{*})^{\\mathbf{A}}}{f^{*}})^{\\mathbf{A}} and \\mathbf{S}{(f^{*},\\mathbf{A})} = (- \\frac{\\operatorname{A_{x}}^{\\mathbf{A}}{(f^{*},\\mathbf{A})}}{f^{*}})^{\\mathbf{A}} and - f^{*} \\mathbf{S}{(f^{*},\\mathbf{A})} = - f^{*} (- \\frac{\\operatorname{A_{x}}^{\\mathbf{A}}{(f^{*},\\mathbf{A})}}{f^{*}})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A_x')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{S}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A_x')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 5, "Mul(Integer(-1), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('f^*', commutative=True), Function('\\\\mathbf{S}')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True), Pow(Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('A_x')(Symbol('f^*', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{1})} = \\sin{(A_{1})}, then obtain \\operatorname{P_{g}}{(A_{1})} - \\cos{(\\sin{(A_{1})} - 1)} - 1 = \\sin{(A_{1})} - \\cos{(\\sin{(A_{1})} - 1)} - 1", "derivation": "\\operatorname{P_{g}}{(A_{1})} = \\sin{(A_{1})} and \\operatorname{P_{g}}{(A_{1})} - 1 = \\sin{(A_{1})} - 1 and \\cos{(\\operatorname{P_{g}}{(A_{1})} - 1)} = \\cos{(\\sin{(A_{1})} - 1)} and \\operatorname{P_{g}}{(A_{1})} - \\cos{(\\operatorname{P_{g}}{(A_{1})} - 1)} - 1 = \\sin{(A_{1})} - \\cos{(\\operatorname{P_{g}}{(A_{1})} - 1)} - 1 and \\operatorname{P_{g}}{(A_{1})} - \\cos{(\\sin{(A_{1})} - 1)} - 1 = \\sin{(A_{1})} - \\cos{(\\sin{(A_{1})} - 1)} - 1", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_g')(Symbol('A_1', commutative=True)), Integer(-1)), Add(sin(Symbol('A_1', commutative=True)), Integer(-1)))"], [["cos", 2], "Equality(cos(Add(Function('P_g')(Symbol('A_1', commutative=True)), Integer(-1))), cos(Add(sin(Symbol('A_1', commutative=True)), Integer(-1))))"], [["minus", 2, "cos(Add(Function('P_g')(Symbol('A_1', commutative=True)), Integer(-1)))"], "Equality(Add(Function('P_g')(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Add(Function('P_g')(Symbol('A_1', commutative=True)), Integer(-1)))), Integer(-1)), Add(sin(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Add(Function('P_g')(Symbol('A_1', commutative=True)), Integer(-1)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('P_g')(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Add(sin(Symbol('A_1', commutative=True)), Integer(-1)))), Integer(-1)), Add(sin(Symbol('A_1', commutative=True)), Mul(Integer(-1), cos(Add(sin(Symbol('A_1', commutative=True)), Integer(-1)))), Integer(-1)))"]]}, {"prompt": "Given T{(\\pi)} = \\int \\log{(\\pi)} d\\pi, then derive - \\pi \\log{(\\pi)} - v + T{(\\pi)} = - \\pi, then obtain \\frac{- \\pi \\log{(\\pi)} - v + 2 T{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} = \\frac{- \\pi \\log{(\\pi)} - v + T{(\\pi)} + \\int \\log{(\\pi)} d\\pi}{\\int \\log{(\\pi)} d\\pi}", "derivation": "T{(\\pi)} = \\int \\log{(\\pi)} d\\pi and - \\pi + T{(\\pi)} = - \\pi + \\int \\log{(\\pi)} d\\pi and \\frac{- \\pi + T{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} = \\frac{- \\pi + \\int \\log{(\\pi)} d\\pi}{\\int \\log{(\\pi)} d\\pi} and - \\pi + T{(\\pi)} - \\int \\log{(\\pi)} d\\pi = - \\pi and - \\pi \\log{(\\pi)} - v + T{(\\pi)} = - \\pi and \\frac{- \\pi \\log{(\\pi)} - v + 2 T{(\\pi)}}{\\int \\log{(\\pi)} d\\pi} = \\frac{- \\pi \\log{(\\pi)} - v + T{(\\pi)} + \\int \\log{(\\pi)} d\\pi}{\\int \\log{(\\pi)} d\\pi}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["minus", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["divide", 2, "Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('\\\\pi', commutative=True))), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))))"], [["minus", 2, "Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('T')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True)), Function('T')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Function('T')(Symbol('\\\\pi', commutative=True)))), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), log(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True)), Function('T')(Symbol('\\\\pi', commutative=True)), Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Pow(Integral(log(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(f,Q)} = - Q + f, then obtain \\frac{\\partial}{\\partial f} (- Q + f) = \\frac{\\partial}{\\partial f} \\frac{(- Q + f)^{2}}{\\operatorname{x^{{\\}'}}{(f,Q)}}", "derivation": "\\operatorname{x^{{\\}'}}{(f,Q)} = - Q + f and (- Q + f) \\operatorname{x^{{\\}'}}{(f,Q)} = (- Q + f)^{2} and - Q + f = \\frac{(- Q + f)^{2}}{\\operatorname{x^{{\\}'}}{(f,Q)}} and \\frac{\\partial}{\\partial f} (- Q + f) = \\frac{\\partial}{\\partial f} \\frac{(- Q + f)^{2}}{\\operatorname{x^{{\\}'}}{(f,Q)}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Function('x^\\\\prime')(Symbol('f', commutative=True), Symbol('Q', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Integer(2)))"], [["divide", 2, "Function('x^\\\\prime')(Symbol('f', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Integer(2)), Pow(Function('x^\\\\prime')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('f', commutative=True)), Integer(2)), Pow(Function('x^\\\\prime')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(H)} = \\cos{(H)}, then derive 2 \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} = - \\tilde{g}{(H)} \\sin{(H)} + \\cos{(H)} \\frac{d}{d H} \\tilde{g}{(H)}, then obtain - \\tilde{g}{(H)} \\sin{(H)} + \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} = - \\tilde{g}{(H)} \\sin{(H)} + \\cos{(H)} \\frac{d}{d H} \\tilde{g}{(H)}", "derivation": "\\tilde{g}{(H)} = \\cos{(H)} and \\tilde{g}^{2}{(H)} = \\tilde{g}{(H)} \\cos{(H)} and \\frac{d}{d H} \\tilde{g}^{2}{(H)} = \\frac{d}{d H} \\tilde{g}{(H)} \\cos{(H)} and 2 \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} = - \\tilde{g}{(H)} \\sin{(H)} + \\cos{(H)} \\frac{d}{d H} \\tilde{g}{(H)} and 2 \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} = - \\tilde{g}{(H)} \\sin{(H)} + \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} and - \\tilde{g}{(H)} \\sin{(H)} + \\tilde{g}{(H)} \\frac{d}{d H} \\tilde{g}{(H)} = - \\tilde{g}{(H)} \\sin{(H)} + \\cos{(H)} \\frac{d}{d H} \\tilde{g}{(H)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["times", 1, "Function('\\\\tilde{g}')(Symbol('H', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Integer(2)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Mul(cos(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Mul(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Mul(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True))), Mul(cos(Symbol('H', commutative=True)), Derivative(Function('\\\\tilde{g}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))))"]]}, {"prompt": "Given m{(\\phi_1,p)} = \\frac{e^{p}}{\\phi_1}, then obtain \\frac{\\partial}{\\partial p} (- p (m^{p}{(\\phi_1,p)} + \\frac{1}{\\phi_1}))^{p} = \\frac{\\partial}{\\partial p} (- p ((\\frac{e^{p}}{\\phi_1})^{p} + \\frac{1}{\\phi_1}))^{p}", "derivation": "m{(\\phi_1,p)} = \\frac{e^{p}}{\\phi_1} and m^{p}{(\\phi_1,p)} = (\\frac{e^{p}}{\\phi_1})^{p} and m^{p}{(\\phi_1,p)} + \\frac{1}{\\phi_1} = (\\frac{e^{p}}{\\phi_1})^{p} + \\frac{1}{\\phi_1} and - p (m^{p}{(\\phi_1,p)} + \\frac{1}{\\phi_1}) = - p ((\\frac{e^{p}}{\\phi_1})^{p} + \\frac{1}{\\phi_1}) and (- p (m^{p}{(\\phi_1,p)} + \\frac{1}{\\phi_1}))^{p} = (- p ((\\frac{e^{p}}{\\phi_1})^{p} + \\frac{1}{\\phi_1}))^{p} and \\frac{\\partial}{\\partial p} (- p (m^{p}{(\\phi_1,p)} + \\frac{1}{\\phi_1}))^{p} = \\frac{\\partial}{\\partial p} (- p ((\\frac{e^{p}}{\\phi_1})^{p} + \\frac{1}{\\phi_1}))^{p}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Add(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))))"], [["times", 3, "Mul(Integer(-1), Symbol('p', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Symbol('p', commutative=True)), Pow(Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Symbol('p', commutative=True)))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Function('m')(Symbol('\\\\phi_1', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Symbol('p', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), exp(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(P_{e})} = \\cos{(e^{P_{e}})}, then obtain (e^{\\cos{(J{(P_{e})})}})^{P_{e}} = (e^{\\cos{(\\cos{(e^{P_{e}})})}})^{P_{e}}", "derivation": "J{(P_{e})} = \\cos{(e^{P_{e}})} and \\cos{(J{(P_{e})})} = \\cos{(\\cos{(e^{P_{e}})})} and e^{\\cos{(J{(P_{e})})}} = e^{\\cos{(\\cos{(e^{P_{e}})})}} and (e^{\\cos{(J{(P_{e})})}})^{P_{e}} = (e^{\\cos{(\\cos{(e^{P_{e}})})}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('P_e', commutative=True)), cos(exp(Symbol('P_e', commutative=True))))"], [["cos", 1], "Equality(cos(Function('J')(Symbol('P_e', commutative=True))), cos(cos(exp(Symbol('P_e', commutative=True)))))"], [["exp", 2], "Equality(exp(cos(Function('J')(Symbol('P_e', commutative=True)))), exp(cos(cos(exp(Symbol('P_e', commutative=True))))))"], [["power", 3, "Symbol('P_e', commutative=True)"], "Equality(Pow(exp(cos(Function('J')(Symbol('P_e', commutative=True)))), Symbol('P_e', commutative=True)), Pow(exp(cos(cos(exp(Symbol('P_e', commutative=True))))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given x{(t_{2},p,\\hat{H}_{\\lambda})} = \\frac{\\hat{H}_{\\lambda} t_{2}}{p}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\frac{\\hat{H}_{\\lambda} t_{2}}{p})^{- p} x^{p}{(t_{2},p,\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} 1", "derivation": "x{(t_{2},p,\\hat{H}_{\\lambda})} = \\frac{\\hat{H}_{\\lambda} t_{2}}{p} and x^{p}{(t_{2},p,\\hat{H}_{\\lambda})} = (\\frac{\\hat{H}_{\\lambda} t_{2}}{p})^{p} and (\\frac{\\hat{H}_{\\lambda} t_{2}}{p})^{- p} x^{p}{(t_{2},p,\\hat{H}_{\\lambda})} = 1 and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\frac{\\hat{H}_{\\lambda} t_{2}}{p})^{- p} x^{p}{(t_{2},p,\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} 1", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('t_2', commutative=True), Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('x')(Symbol('t_2', commutative=True), Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('p', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('p', commutative=True)))"], [["divide", 2, "Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True))), Pow(Function('x')(Symbol('t_2', commutative=True), Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('p', commutative=True))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Pow(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True))), Pow(Function('x')(Symbol('t_2', commutative=True), Symbol('p', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(f_{E})} = \\log{(f_{E})} and \\mathbf{p}{(C_{2},Q,\\tilde{g}^*)} = C_{2} + Q + \\tilde{g}^*, then obtain (C_{2} + Q + \\tilde{g}^*) \\frac{d}{d f_{E}} \\lambda{(f_{E})} = (C_{2} + Q + \\tilde{g}^*) \\frac{d}{d f_{E}} \\log{(f_{E})}", "derivation": "\\lambda{(f_{E})} = \\log{(f_{E})} and \\frac{d}{d f_{E}} \\lambda{(f_{E})} = \\frac{d}{d f_{E}} \\log{(f_{E})} and \\mathbf{p}{(C_{2},Q,\\tilde{g}^*)} = C_{2} + Q + \\tilde{g}^* and \\mathbf{p}{(C_{2},Q,\\tilde{g}^*)} \\frac{d}{d f_{E}} \\lambda{(f_{E})} = \\mathbf{p}{(C_{2},Q,\\tilde{g}^*)} \\frac{d}{d f_{E}} \\log{(f_{E})} and (C_{2} + Q + \\tilde{g}^*) \\frac{d}{d f_{E}} \\lambda{(f_{E})} = (C_{2} + Q + \\tilde{g}^*) \\frac{d}{d f_{E}} \\log{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{p}')(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Function('\\\\mathbf{p}')(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Mul(Add(Symbol('C_2', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\hat{X},m_{s})} = - \\hat{X} + m_{s}, then obtain (- \\hat{X} + m_{s}) \\frac{\\partial}{\\partial \\hat{X}} \\sin{(\\operatorname{v_{x}}^{m_{s}}{(\\hat{X},m_{s})})} = (- \\hat{X} + m_{s}) \\frac{\\partial}{\\partial \\hat{X}} \\sin{((- \\hat{X} + m_{s})^{m_{s}})}", "derivation": "\\operatorname{v_{x}}{(\\hat{X},m_{s})} = - \\hat{X} + m_{s} and \\operatorname{v_{x}}^{m_{s}}{(\\hat{X},m_{s})} = (- \\hat{X} + m_{s})^{m_{s}} and \\sin{(\\operatorname{v_{x}}^{m_{s}}{(\\hat{X},m_{s})})} = \\sin{((- \\hat{X} + m_{s})^{m_{s}})} and \\frac{\\partial}{\\partial \\hat{X}} \\sin{(\\operatorname{v_{x}}^{m_{s}}{(\\hat{X},m_{s})})} = \\frac{\\partial}{\\partial \\hat{X}} \\sin{((- \\hat{X} + m_{s})^{m_{s}})} and (- \\hat{X} + m_{s}) \\frac{\\partial}{\\partial \\hat{X}} \\sin{(\\operatorname{v_{x}}^{m_{s}}{(\\hat{X},m_{s})})} = (- \\hat{X} + m_{s}) \\frac{\\partial}{\\partial \\hat{X}} \\sin{((- \\hat{X} + m_{s})^{m_{s}})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('v_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(sin(Pow(Function('v_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["divide", 4, "Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Derivative(sin(Pow(Function('v_x')(Symbol('\\\\hat{X}', commutative=True), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Derivative(sin(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(C_{d})} = \\log{(C_{d})}, then derive \\int \\frac{\\Psi_{nl}{(C_{d})}}{C_{d}} dC_{d} = g_{\\varepsilon} + \\frac{\\log{(C_{d})}^{2}}{2}, then derive J + \\frac{\\log{(C_{d})}^{2}}{2} = g_{\\varepsilon} + \\frac{\\log{(C_{d})}^{2}}{2}, then obtain J + \\frac{\\log{(C_{d})}^{2}}{2} = \\int \\frac{\\log{(C_{d})}}{C_{d}} dC_{d}", "derivation": "\\Psi_{nl}{(C_{d})} = \\log{(C_{d})} and \\frac{\\Psi_{nl}{(C_{d})}}{C_{d}} = \\frac{\\log{(C_{d})}}{C_{d}} and \\int \\frac{\\Psi_{nl}{(C_{d})}}{C_{d}} dC_{d} = \\int \\frac{\\log{(C_{d})}}{C_{d}} dC_{d} and \\int \\frac{\\Psi_{nl}{(C_{d})}}{C_{d}} dC_{d} = g_{\\varepsilon} + \\frac{\\log{(C_{d})}^{2}}{2} and \\int \\frac{\\log{(C_{d})}}{C_{d}} dC_{d} = g_{\\varepsilon} + \\frac{\\log{(C_{d})}^{2}}{2} and J + \\frac{\\log{(C_{d})}^{2}}{2} = g_{\\varepsilon} + \\frac{\\log{(C_{d})}^{2}}{2} and J + \\frac{\\log{(C_{d})}^{2}}{2} = \\int \\frac{\\log{(C_{d})}}{C_{d}} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["divide", 1, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), log(Symbol('C_d', commutative=True))))"], [["integrate", 2, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), log(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), log(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('J', commutative=True), Mul(Rational(1, 2), Pow(log(Symbol('C_d', commutative=True)), Integer(2)))), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), log(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(t_{1},M_{E})} = - M_{E} + t_{1}, then derive \\int \\theta_{1}{(t_{1},M_{E})} dt_{1} = - M_{E} t_{1} + \\theta_1 + \\frac{t_{1}^{2}}{2}, then obtain - M_{E} + \\int \\theta_{1}{(t_{1},M_{E})} dt_{1} = - M_{E} t_{1} - M_{E} + \\theta_1 + \\frac{t_{1}^{2}}{2}", "derivation": "\\theta_{1}{(t_{1},M_{E})} = - M_{E} + t_{1} and \\int \\theta_{1}{(t_{1},M_{E})} dt_{1} = \\int (- M_{E} + t_{1}) dt_{1} and \\int \\theta_{1}{(t_{1},M_{E})} dt_{1} = - M_{E} t_{1} + \\theta_1 + \\frac{t_{1}^{2}}{2} and - M_{E} + \\int \\theta_{1}{(t_{1},M_{E})} dt_{1} = - M_{E} t_{1} - M_{E} + \\theta_1 + \\frac{t_{1}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('t_1', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["minus", 3, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Integral(Function('\\\\theta_1')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})} and Q{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})}, then obtain \\sin^{4}{(\\sin{(V_{\\mathbf{B}})})} = \\operatorname{c_{0}}^{2}{(V_{\\mathbf{B}})} \\sin^{2}{(\\sin{(V_{\\mathbf{B}})})}", "derivation": "\\operatorname{c_{0}}{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})} and Q{(V_{\\mathbf{B}})} = \\sin{(\\sin{(V_{\\mathbf{B}})})} and Q{(V_{\\mathbf{B}})} = \\operatorname{c_{0}}{(V_{\\mathbf{B}})} and Q^{2}{(V_{\\mathbf{B}})} = Q{(V_{\\mathbf{B}})} \\operatorname{c_{0}}{(V_{\\mathbf{B}})} and \\sin^{2}{(\\sin{(V_{\\mathbf{B}})})} = \\operatorname{c_{0}}{(V_{\\mathbf{B}})} \\sin{(\\sin{(V_{\\mathbf{B}})})} and \\sin^{4}{(\\sin{(V_{\\mathbf{B}})})} = \\operatorname{c_{0}}^{2}{(V_{\\mathbf{B}})} \\sin^{2}{(\\sin{(V_{\\mathbf{B}})})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('c_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 3, "Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Pow(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Function('Q')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('c_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(2)), Mul(Function('c_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["power", 5, 2], "Equality(Pow(sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(4)), Mul(Pow(Function('c_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Pow(sin(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\psi{(Z)} = \\cos{(Z)}, then obtain - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}} = \\frac{\\sin{(Z)}}{\\psi{(Z)}} + \\frac{\\cos{(Z)} \\frac{d}{d Z} \\psi{(Z)}}{\\psi^{2}{(Z)}} - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}}", "derivation": "\\psi{(Z)} = \\cos{(Z)} and 1 = \\frac{\\cos{(Z)}}{\\psi{(Z)}} and \\frac{d}{d Z} 1 = \\frac{d}{d Z} \\frac{\\cos{(Z)}}{\\psi{(Z)}} and - \\frac{d}{d Z} 1 = - \\frac{d}{d Z} \\frac{\\cos{(Z)}}{\\psi{(Z)}} and - \\frac{d}{d Z} 1 - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}} = - \\frac{d}{d Z} \\frac{\\cos{(Z)}}{\\psi{(Z)}} - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}} and - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}} = \\frac{\\sin{(Z)}}{\\psi{(Z)}} + \\frac{\\cos{(Z)} \\frac{d}{d Z} \\psi{(Z)}}{\\psi^{2}{(Z)}} - \\frac{\\cos{(Z)}}{\\psi^{2}{(Z)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('Z', commutative=True)), cos(Symbol('Z', commutative=True)))"], [["divide", 1, "Function('\\\\psi')(Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-1)), cos(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-1)), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-1)), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["minus", 4, "Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-1)), cos(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True))), Add(Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-1)), sin(Symbol('Z', commutative=True))), Mul(Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True)), Derivative(Function('\\\\psi')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('Z', commutative=True)), Integer(-2)), cos(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} = \\frac{\\ddot{x} \\mathbf{r}}{z^{*}}, then obtain \\iint z^{*} \\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} d\\mathbf{r} d\\mathbf{r} = \\iint \\ddot{x} \\mathbf{r} d\\mathbf{r} d\\mathbf{r}", "derivation": "\\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} = \\frac{\\ddot{x} \\mathbf{r}}{z^{*}} and z^{*} \\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} = \\ddot{x} \\mathbf{r} and \\int z^{*} \\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} d\\mathbf{r} = \\int \\ddot{x} \\mathbf{r} d\\mathbf{r} and \\iint z^{*} \\Psi_{nl}{(z^{*},\\mathbf{r},\\ddot{x})} d\\mathbf{r} d\\mathbf{r} = \\iint \\ddot{x} \\mathbf{r} d\\mathbf{r} d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Symbol('z^*', commutative=True), Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Symbol('z^*', commutative=True), Function('\\\\Psi_{nl}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given u{(\\hat{x}_0,\\mathbf{H},q)} = \\hat{x}_0^{q} + \\mathbf{H}, then obtain \\cos{(\\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) - \\frac{\\partial}{\\partial \\hat{x}_0} u{(\\hat{x}_0,\\mathbf{H},q)})} = \\cos{(\\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) - \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0^{q} + \\mathbf{H}))}", "derivation": "u{(\\hat{x}_0,\\mathbf{H},q)} = \\hat{x}_0^{q} + \\mathbf{H} and \\frac{\\partial}{\\partial \\hat{x}_0} u{(\\hat{x}_0,\\mathbf{H},q)} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0^{q} + \\mathbf{H}) and - \\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) + \\frac{\\partial}{\\partial \\hat{x}_0} u{(\\hat{x}_0,\\mathbf{H},q)} = - \\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) + \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0^{q} + \\mathbf{H}) and \\cos{(\\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) - \\frac{\\partial}{\\partial \\hat{x}_0} u{(\\hat{x}_0,\\mathbf{H},q)})} = \\cos{(\\hat{x}_0 (\\hat{x}_0^{q} + \\mathbf{H}) - \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0^{q} + \\mathbf{H}))}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('q', commutative=True)), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Derivative(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Derivative(Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Derivative(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))), cos(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Derivative(Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\psi^{*}{(\\mu_0)} = \\cos{(\\mu_0)} and \\theta_{2}{(\\mu_0)} = \\frac{1}{\\mu_0}, then obtain - \\frac{\\theta_{2}^{\\mu_0}{(\\mu_0)}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\cos{(\\mu_0)}} = - \\frac{(\\frac{1}{\\mu_0})^{\\mu_0}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\cos{(\\mu_0)}}", "derivation": "\\psi^{*}{(\\mu_0)} = \\cos{(\\mu_0)} and \\frac{d}{d \\mu_0} \\psi^{*}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\cos{(\\mu_0)} and \\theta_{2}{(\\mu_0)} = \\frac{1}{\\mu_0} and \\theta_{2}^{\\mu_0}{(\\mu_0)} = (\\frac{1}{\\mu_0})^{\\mu_0} and - \\frac{\\theta_{2}^{\\mu_0}{(\\mu_0)}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\psi^{*}{(\\mu_0)}} = - \\frac{(\\frac{1}{\\mu_0})^{\\mu_0}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\psi^{*}{(\\mu_0)}} and - \\frac{\\theta_{2}^{\\mu_0}{(\\mu_0)}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\cos{(\\mu_0)}} = - \\frac{(\\frac{1}{\\mu_0})^{\\mu_0}}{\\sin{(\\mu_0)} \\frac{d}{d \\mu_0} \\cos{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), sin(Symbol('\\\\mu_0', commutative=True)), Derivative(Function('\\\\psi^*')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\psi^*')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Derivative(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Pow(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Pow(Derivative(cos(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\theta_1,P_{e})} = \\frac{P_{e}}{\\theta_1}, then derive \\frac{\\partial}{\\partial P_{e}} \\operatorname{t_{2}}{(\\theta_1,P_{e})} = \\frac{1}{\\theta_1}, then obtain \\theta_1 \\int \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e}}{\\theta_1} d\\theta_1 = \\theta_1 \\int \\frac{1}{\\theta_1} d\\theta_1", "derivation": "\\operatorname{t_{2}}{(\\theta_1,P_{e})} = \\frac{P_{e}}{\\theta_1} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{t_{2}}{(\\theta_1,P_{e})} = \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e}}{\\theta_1} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{t_{2}}{(\\theta_1,P_{e})} = \\frac{1}{\\theta_1} and \\int \\frac{\\partial}{\\partial P_{e}} \\operatorname{t_{2}}{(\\theta_1,P_{e})} d\\theta_1 = \\int \\frac{1}{\\theta_1} d\\theta_1 and \\theta_1 \\int \\frac{\\partial}{\\partial P_{e}} \\operatorname{t_{2}}{(\\theta_1,P_{e})} d\\theta_1 = \\theta_1 \\int \\frac{1}{\\theta_1} d\\theta_1 and \\theta_1 \\int \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e}}{\\theta_1} d\\theta_1 = \\theta_1 \\int \\frac{1}{\\theta_1} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Function('t_2')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Integral(Derivative(Function('t_2')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Integral(Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(v_{1})} = e^{v_{1}} and \\omega{(v_{1})} = - 2 \\mathbf{M}{(v_{1})} + 2 e^{v_{1}}, then obtain \\frac{d}{d v_{1}} 0 = \\frac{\\partial}{\\partial v_{1}} \\frac{\\omega{(v_{1})}}{v_{1} \\operatorname{C_{1}}{(\\rho_b)}}", "derivation": "\\mathbf{M}{(v_{1})} = e^{v_{1}} and 0 = - \\mathbf{M}{(v_{1})} + e^{v_{1}} and 0 = \\frac{- \\mathbf{M}{(v_{1})} + e^{v_{1}}}{v_{1}} and - \\mathbf{M}{(v_{1})} + e^{v_{1}} = - 2 \\mathbf{M}{(v_{1})} + 2 e^{v_{1}} and 0 = \\frac{- 2 \\mathbf{M}{(v_{1})} + 2 e^{v_{1}}}{v_{1}} and \\omega{(v_{1})} = - 2 \\mathbf{M}{(v_{1})} + 2 e^{v_{1}} and 0 = \\frac{\\omega{(v_{1})}}{v_{1}} and 0 = \\frac{\\omega{(v_{1})}}{v_{1} \\operatorname{C_{1}}{(\\rho_b)}} and \\frac{d}{d v_{1}} 0 = \\frac{\\partial}{\\partial v_{1}} \\frac{\\omega{(v_{1})}}{v_{1} \\operatorname{C_{1}}{(\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True))))"], [["divide", 2, "Symbol('v_1', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), exp(Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), Mul(Integer(2), exp(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), Mul(Integer(2), exp(Symbol('v_1', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True))), Mul(Integer(2), exp(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integer(0), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Function('\\\\omega')(Symbol('v_1', commutative=True))))"], [["divide", 7, "Function('C_1')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Integer(0), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('C_1')(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Function('\\\\omega')(Symbol('v_1', commutative=True))))"], [["differentiate", 8, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('C_1')(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), Function('\\\\omega')(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(t,\\hat{x})} = \\hat{x} t, then derive \\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})} = t, then obtain - t + (\\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})})^{t} + 1 = - t + t^{t} + 1", "derivation": "\\varphi^{*}{(t,\\hat{x})} = \\hat{x} t and \\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} t and \\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})} = t and (\\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})})^{t} = (\\frac{\\partial}{\\partial \\hat{x}} \\hat{x} t)^{t} and \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} t = t and (\\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})})^{t} = t^{t} and (\\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})})^{t} + 1 = t^{t} + 1 and - t + (\\frac{\\partial}{\\partial \\hat{x}} \\varphi^{*}{(t,\\hat{x})})^{t} + 1 = - t + t^{t} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('t', commutative=True)))"], [["add", 6, 1], "Equality(Add(Pow(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True)), Integer(1)), Add(Pow(Symbol('t', commutative=True), Symbol('t', commutative=True)), Integer(1)))"], [["minus", 7, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Pow(Derivative(Function('\\\\varphi^*')(Symbol('t', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Symbol('t', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('t', commutative=True)), Integer(1)))"]]}, {"prompt": "Given M{(\\mathbf{g})} = \\sin{(\\mathbf{g})}, then derive \\frac{d}{d \\mathbf{g}} M{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then obtain \\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})}", "derivation": "M{(\\mathbf{g})} = \\sin{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} M{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\sin{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} M{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\frac{d}{d \\mathbf{g}} \\sin{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\cos{(\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{g}', commutative=True)), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(sin(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} = \\sin{(\\log{(\\sigma_p)})}, then obtain \\frac{d}{d \\sigma_p} (\\sigma_p + \\cos{(\\int \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} d\\sigma_p)}) = \\frac{d}{d \\sigma_p} (\\sigma_p + \\cos{(\\int \\sin{(\\log{(\\sigma_p)})} d\\sigma_p)})", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\sigma_p)} = \\sin{(\\log{(\\sigma_p)})} and \\int \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} d\\sigma_p = \\int \\sin{(\\log{(\\sigma_p)})} d\\sigma_p and \\cos{(\\int \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} d\\sigma_p)} = \\cos{(\\int \\sin{(\\log{(\\sigma_p)})} d\\sigma_p)} and \\sigma_p + \\cos{(\\int \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} d\\sigma_p)} = \\sigma_p + \\cos{(\\int \\sin{(\\log{(\\sigma_p)})} d\\sigma_p)} and \\frac{d}{d \\sigma_p} (\\sigma_p + \\cos{(\\int \\operatorname{J_{\\varepsilon}}{(\\sigma_p)} d\\sigma_p)}) = \\frac{d}{d \\sigma_p} (\\sigma_p + \\cos{(\\int \\sin{(\\log{(\\sigma_p)})} d\\sigma_p)})", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\sigma_p', commutative=True)), sin(log(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(sin(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), cos(Integral(sin(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"], [["add", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), cos(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))), Add(Symbol('\\\\sigma_p', commutative=True), cos(Integral(sin(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\sigma_p', commutative=True), cos(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\sigma_p', commutative=True), cos(Integral(sin(log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(Q)} = \\log{(Q)}, then obtain (\\int \\mathbf{p}^{Q}{(Q)} \\int \\mathbf{p}^{Q}{(Q)} dQ dQ)^{Q} = (\\int \\mathbf{p}^{Q}{(Q)} \\int \\log{(Q)}^{Q} dQ dQ)^{Q}", "derivation": "\\mathbf{p}{(Q)} = \\log{(Q)} and \\mathbf{p}^{Q}{(Q)} = \\log{(Q)}^{Q} and \\int \\mathbf{p}^{Q}{(Q)} dQ = \\int \\log{(Q)}^{Q} dQ and \\log{(Q)}^{Q} \\int \\mathbf{p}^{Q}{(Q)} dQ = \\log{(Q)}^{Q} \\int \\log{(Q)}^{Q} dQ and \\mathbf{p}^{Q}{(Q)} \\int \\mathbf{p}^{Q}{(Q)} dQ = \\mathbf{p}^{Q}{(Q)} \\int \\log{(Q)}^{Q} dQ and \\int \\mathbf{p}^{Q}{(Q)} \\int \\mathbf{p}^{Q}{(Q)} dQ dQ = \\int \\mathbf{p}^{Q}{(Q)} \\int \\log{(Q)}^{Q} dQ dQ and (\\int \\mathbf{p}^{Q}{(Q)} \\int \\mathbf{p}^{Q}{(Q)} dQ dQ)^{Q} = (\\int \\mathbf{p}^{Q}{(Q)} \\int \\log{(Q)}^{Q} dQ dQ)^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["times", 3, "Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["power", 6, "Symbol('Q', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Integral(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Integral(Pow(log(Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given T{(\\varepsilon_0,t_{2})} = \\log{(\\varepsilon_0)}^{t_{2}}, then obtain \\frac{T{(\\varepsilon_0,t_{2})}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}^{t_{2}}} = \\frac{\\log{(\\varepsilon_0)}^{t_{2}}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}^{t_{2}}}", "derivation": "T{(\\varepsilon_0,t_{2})} = \\log{(\\varepsilon_0)}^{t_{2}} and \\varepsilon_0 + T{(\\varepsilon_0,t_{2})} = \\varepsilon_0 + \\log{(\\varepsilon_0)}^{t_{2}} and \\frac{T{(\\varepsilon_0,t_{2})}}{\\varepsilon_0 + T{(\\varepsilon_0,t_{2})}} = \\frac{\\log{(\\varepsilon_0)}^{t_{2}}}{\\varepsilon_0 + T{(\\varepsilon_0,t_{2})}} and \\frac{T{(\\varepsilon_0,t_{2})}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}^{t_{2}}} = \\frac{\\log{(\\varepsilon_0)}^{t_{2}}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}^{t_{2}}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\varepsilon_0', commutative=True), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True))), Integer(-1)), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True))), Integer(-1)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True))), Integer(-1)), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True))), Integer(-1)), Pow(log(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\theta_1,A)} = \\frac{\\cos{(\\theta_1)}}{A}, then obtain \\log{(\\theta_1 + \\int \\operatorname{C_{d}}^{\\theta_1}{(\\theta_1,A)} d\\theta_1)} = \\log{(\\theta_1 + \\int (\\frac{\\cos{(\\theta_1)}}{A})^{\\theta_1} d\\theta_1)}", "derivation": "\\operatorname{C_{d}}{(\\theta_1,A)} = \\frac{\\cos{(\\theta_1)}}{A} and \\operatorname{C_{d}}^{\\theta_1}{(\\theta_1,A)} = (\\frac{\\cos{(\\theta_1)}}{A})^{\\theta_1} and \\int \\operatorname{C_{d}}^{\\theta_1}{(\\theta_1,A)} d\\theta_1 = \\int (\\frac{\\cos{(\\theta_1)}}{A})^{\\theta_1} d\\theta_1 and \\theta_1 + \\int \\operatorname{C_{d}}^{\\theta_1}{(\\theta_1,A)} d\\theta_1 = \\theta_1 + \\int (\\frac{\\cos{(\\theta_1)}}{A})^{\\theta_1} d\\theta_1 and \\log{(\\theta_1 + \\int \\operatorname{C_{d}}^{\\theta_1}{(\\theta_1,A)} d\\theta_1)} = \\log{(\\theta_1 + \\int (\\frac{\\cos{(\\theta_1)}}{A})^{\\theta_1} d\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_1', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('A', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Pow(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('A', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["add", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('A', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Add(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["log", 4], "Equality(log(Add(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Function('C_d')(Symbol('\\\\theta_1', commutative=True), Symbol('A', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), log(Add(Symbol('\\\\theta_1', commutative=True), Integral(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), cos(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mu,\\theta)} = \\mu + \\theta, then obtain - \\mu + (\\mu + \\theta)^{2} \\operatorname{v_{z}}^{2}{(\\mu,\\theta)} = - \\mu + (\\mu + \\theta)^{3} \\operatorname{v_{z}}{(\\mu,\\theta)}", "derivation": "\\operatorname{v_{z}}{(\\mu,\\theta)} = \\mu + \\theta and \\operatorname{v_{z}}^{2}{(\\mu,\\theta)} = (\\mu + \\theta) \\operatorname{v_{z}}{(\\mu,\\theta)} and \\operatorname{v_{z}}^{4}{(\\mu,\\theta)} = (\\mu + \\theta)^{2} \\operatorname{v_{z}}^{2}{(\\mu,\\theta)} and (\\mu + \\theta)^{2} \\operatorname{v_{z}}^{2}{(\\mu,\\theta)} = (\\mu + \\theta)^{3} \\operatorname{v_{z}}{(\\mu,\\theta)} and - \\mu + (\\mu + \\theta)^{2} \\operatorname{v_{z}}^{2}{(\\mu,\\theta)} = - \\mu + (\\mu + \\theta)^{3} \\operatorname{v_{z}}{(\\mu,\\theta)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["divide", 1, "Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))"], "Equality(Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(4)), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2))), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(3)), Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["minus", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)), Pow(Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(3)), Function('v_z')(Symbol('\\\\mu', commutative=True), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(F_{N})} = \\log{(F_{N})}, then obtain \\tilde{g}^*{(F_{N})} - \\log{(F_{N})}^{2} + \\log{(F_{N})} = - \\log{(F_{N})}^{2} + 2 \\log{(F_{N})}", "derivation": "\\tilde{g}^*{(F_{N})} = \\log{(F_{N})} and \\tilde{g}^*{(F_{N})} \\log{(F_{N})} = \\log{(F_{N})}^{2} and \\tilde{g}^*{(F_{N})} + \\log{(F_{N})} = 2 \\log{(F_{N})} and - \\tilde{g}^*{(F_{N})} \\log{(F_{N})} + \\tilde{g}^*{(F_{N})} + \\log{(F_{N})} = - \\tilde{g}^*{(F_{N})} \\log{(F_{N})} + 2 \\log{(F_{N})} and \\tilde{g}^*{(F_{N})} - \\log{(F_{N})}^{2} + \\log{(F_{N})} = - \\log{(F_{N})}^{2} + 2 \\log{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["times", 1, "log(Symbol('F_N', commutative=True))"], "Equality(Mul(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Pow(log(Symbol('F_N', commutative=True)), Integer(2)))"], [["add", 1, "log(Symbol('F_N', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Mul(Integer(2), log(Symbol('F_N', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Mul(Integer(2), log(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('F_N', commutative=True)), Integer(2))), log(Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Pow(log(Symbol('F_N', commutative=True)), Integer(2))), Mul(Integer(2), log(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain \\int (\\hat{p}_0{(\\mathbf{J}_P)} - e^{\\mathbf{J}_P} - 1) d\\mathbf{J}_P = \\int (-1) d\\mathbf{J}_P", "derivation": "\\hat{p}_0{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\hat{p}_0{(\\mathbf{J}_P)} - e^{\\mathbf{J}_P} = 0 and \\hat{p}_0{(\\mathbf{J}_P)} - e^{\\mathbf{J}_P} - 1 = -1 and \\int (\\hat{p}_0{(\\mathbf{J}_P)} - e^{\\mathbf{J}_P} - 1) d\\mathbf{J}_P = \\int (-1) d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1)), Integer(-1))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Integer(-1), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\psi{(\\chi)} = \\sin{(e^{\\chi})} and \\phi_{2}{(\\chi)} = \\cos{(e^{\\chi})}, then derive \\frac{d}{d \\chi} \\psi{(\\chi)} = e^{\\chi} \\cos{(e^{\\chi})}, then obtain (- \\psi{(\\chi)} + \\frac{d}{d \\chi} \\psi{(\\chi)})^{\\chi} = (\\phi_{2}{(\\chi)} e^{\\chi} - \\psi{(\\chi)})^{\\chi}", "derivation": "\\psi{(\\chi)} = \\sin{(e^{\\chi})} and \\frac{d}{d \\chi} \\psi{(\\chi)} = \\frac{d}{d \\chi} \\sin{(e^{\\chi})} and \\frac{d}{d \\chi} \\psi{(\\chi)} = e^{\\chi} \\cos{(e^{\\chi})} and \\phi_{2}{(\\chi)} = \\cos{(e^{\\chi})} and \\frac{d}{d \\chi} \\psi{(\\chi)} = \\phi_{2}{(\\chi)} e^{\\chi} and - \\psi{(\\chi)} + \\frac{d}{d \\chi} \\psi{(\\chi)} = \\phi_{2}{(\\chi)} e^{\\chi} - \\psi{(\\chi)} and (- \\psi{(\\chi)} + \\frac{d}{d \\chi} \\psi{(\\chi)})^{\\chi} = (\\phi_{2}{(\\chi)} e^{\\chi} - \\psi{(\\chi)})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), sin(exp(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(exp(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Function('\\\\phi_2')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))))"], [["minus", 5, "Function('\\\\psi')(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\chi', commutative=True))), Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Function('\\\\phi_2')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\chi', commutative=True)))))"], [["power", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\chi', commutative=True))), Derivative(Function('\\\\psi')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Mul(Function('\\\\phi_2')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Function('\\\\psi')(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\rho{(v_{1})} = \\sin{(v_{1})}, then derive (\\int \\rho{(v_{1})} dv_{1})^{v_{1}} = (\\varphi - \\cos{(v_{1})})^{v_{1}}, then obtain \\sin{(v_{1})} = (\\varphi - \\cos{(v_{1})})^{v_{1}} + \\sin{(v_{1})} - (\\int \\sin{(v_{1})} dv_{1})^{v_{1}}", "derivation": "\\rho{(v_{1})} = \\sin{(v_{1})} and \\int \\rho{(v_{1})} dv_{1} = \\int \\sin{(v_{1})} dv_{1} and (\\int \\rho{(v_{1})} dv_{1})^{v_{1}} = (\\int \\sin{(v_{1})} dv_{1})^{v_{1}} and (\\int \\rho{(v_{1})} dv_{1})^{v_{1}} = (\\varphi - \\cos{(v_{1})})^{v_{1}} and \\rho{(v_{1})} + (\\int \\rho{(v_{1})} dv_{1})^{v_{1}} - (\\int \\sin{(v_{1})} dv_{1})^{v_{1}} = (\\varphi - \\cos{(v_{1})})^{v_{1}} + \\rho{(v_{1})} - (\\int \\sin{(v_{1})} dv_{1})^{v_{1}} and \\sin{(v_{1})} = (\\varphi - \\cos{(v_{1})})^{v_{1}} + \\sin{(v_{1})} - (\\int \\sin{(v_{1})} dv_{1})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\rho')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)))"], [["minus", 4, "Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('v_1', commutative=True))), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))"], "Equality(Add(Function('\\\\rho')(Symbol('v_1', commutative=True)), Pow(Integral(Function('\\\\rho')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))), Add(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), Function('\\\\rho')(Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(sin(Symbol('v_1', commutative=True)), Add(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))), Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Integral(sin(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{x})} = \\cos{(\\hat{x})}, then derive \\int \\mathbf{p}{(\\hat{x})} d\\hat{x} = z^{*} + \\sin{(\\hat{x})}, then obtain \\sin{(\\hat{x})} \\int \\mathbf{p}{(\\hat{x})} d\\hat{x} = (z^{*} + \\sin{(\\hat{x})}) \\sin{(\\hat{x})}", "derivation": "\\mathbf{p}{(\\hat{x})} = \\cos{(\\hat{x})} and \\int \\mathbf{p}{(\\hat{x})} d\\hat{x} = \\int \\cos{(\\hat{x})} d\\hat{x} and \\int \\mathbf{p}{(\\hat{x})} d\\hat{x} = z^{*} + \\sin{(\\hat{x})} and \\sin{(\\hat{x})} \\int \\mathbf{p}{(\\hat{x})} d\\hat{x} = (z^{*} + \\sin{(\\hat{x})}) \\sin{(\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(cos(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 3, "sin(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\hat{x}', commutative=True)), Integral(Function('\\\\mathbf{p}')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), sin(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then derive \\hat{\\mathbf{x}} + \\phi_1 = \\int \\frac{\\log{(\\hat{\\mathbf{x}})}}{\\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})}} d\\hat{\\mathbf{x}}, then obtain \\int (\\int 1 d\\hat{\\mathbf{x}})^{\\phi_1} d\\hat{\\mathbf{x}} = \\int (\\hat{\\mathbf{x}} + \\phi_1)^{\\phi_1} d\\hat{\\mathbf{x}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and 1 = \\frac{\\log{(\\hat{\\mathbf{x}})}}{\\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})}} and \\int 1 d\\hat{\\mathbf{x}} = \\int \\frac{\\log{(\\hat{\\mathbf{x}})}}{\\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})}} d\\hat{\\mathbf{x}} and \\hat{\\mathbf{x}} + \\phi_1 = \\int \\frac{\\log{(\\hat{\\mathbf{x}})}}{\\operatorname{x^{{\\}'}}{(\\hat{\\mathbf{x}})}} d\\hat{\\mathbf{x}} and \\int 1 d\\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} + \\phi_1 and (\\int 1 d\\hat{\\mathbf{x}})^{\\phi_1} = (\\hat{\\mathbf{x}} + \\phi_1)^{\\phi_1} and \\int (\\int 1 d\\hat{\\mathbf{x}})^{\\phi_1} d\\hat{\\mathbf{x}} = \\int (\\hat{\\mathbf{x}} + \\phi_1)^{\\phi_1} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["divide", 1, "Function('x^\\\\prime')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["power", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 6, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Pow(Integral(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = \\cos{(\\log{(\\mathbf{J}_M)})} and v{(a,\\mathbf{J}_M)} = \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} d\\mathbf{J}_M, then obtain v{(a,\\mathbf{J}_M)} = \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\cos{(\\log{(\\mathbf{J}_M)})} d\\mathbf{J}_M", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = \\cos{(\\log{(\\mathbf{J}_M)})} and (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\cos{(\\log{(\\mathbf{J}_M)})} and \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\cos{(\\log{(\\mathbf{J}_M)})} d\\mathbf{J}_M and v{(a,\\mathbf{J}_M)} = \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} d\\mathbf{J}_M and v{(a,\\mathbf{J}_M)} = \\int (a + \\cos{(\\log{(\\mathbf{J}_M)})}) \\cos{(\\log{(\\mathbf{J}_M)})} d\\mathbf{J}_M", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 1, "Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('v')(Symbol('a', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Mul(Add(Symbol('a', commutative=True), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), cos(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(n_{2})} = \\sin{(n_{2})}, then obtain 2 \\operatorname{A_{2}}{(n_{2})} + \\frac{d}{d n_{2}} \\operatorname{A_{2}}{(n_{2})} = 2 \\operatorname{A_{2}}{(n_{2})} + \\frac{d}{d n_{2}} \\sin{(n_{2})}", "derivation": "\\operatorname{A_{2}}{(n_{2})} = \\sin{(n_{2})} and 2 \\operatorname{A_{2}}{(n_{2})} = \\operatorname{A_{2}}{(n_{2})} + \\sin{(n_{2})} and \\frac{d}{d n_{2}} \\operatorname{A_{2}}{(n_{2})} = \\frac{d}{d n_{2}} \\sin{(n_{2})} and \\operatorname{A_{2}}{(n_{2})} + \\sin{(n_{2})} + \\frac{d}{d n_{2}} \\operatorname{A_{2}}{(n_{2})} = \\operatorname{A_{2}}{(n_{2})} + \\sin{(n_{2})} + \\frac{d}{d n_{2}} \\sin{(n_{2})} and 2 \\operatorname{A_{2}}{(n_{2})} + \\frac{d}{d n_{2}} \\operatorname{A_{2}}{(n_{2})} = 2 \\operatorname{A_{2}}{(n_{2})} + \\frac{d}{d n_{2}} \\sin{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["add", 1, "Function('A_2')(Symbol('n_2', commutative=True))"], "Equality(Mul(Integer(2), Function('A_2')(Symbol('n_2', commutative=True))), Add(Function('A_2')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["add", 3, "Add(Function('A_2')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], "Equality(Add(Function('A_2')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)), Derivative(Function('A_2')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Function('A_2')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)), Derivative(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(2), Function('A_2')(Symbol('n_2', commutative=True))), Derivative(Function('A_2')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(2), Function('A_2')(Symbol('n_2', commutative=True))), Derivative(sin(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(\\mathbf{J}_M,q)} = \\mathbf{J}_M q, then obtain ((q + \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M})^{4})^{\\mathbf{J}_M} = (16 q^{4})^{\\mathbf{J}_M}", "derivation": "s{(\\mathbf{J}_M,q)} = \\mathbf{J}_M q and \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M} = q and q + \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M} = 2 q and (q + \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M})^{2} = 4 q^{2} and (q + \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M})^{4} = 16 q^{4} and ((q + \\frac{s{(\\mathbf{J}_M,q)}}{\\mathbf{J}_M})^{4})^{\\mathbf{J}_M} = (16 q^{4})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))"], [["add", 2, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)))), Mul(Integer(2), Symbol('q', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(Symbol('q', commutative=True), Integer(2))))"], [["power", 4, 2], "Equality(Pow(Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)))), Integer(4)), Mul(Integer(16), Pow(Symbol('q', commutative=True), Integer(4))))"], [["power", 5, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('q', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('q', commutative=True)))), Integer(4)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Integer(16), Pow(Symbol('q', commutative=True), Integer(4))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given M{(\\eta^{\\prime})} = e^{\\eta^{\\prime}}, then derive \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} = \\mathbf{E} + e^{\\eta^{\\prime}}, then obtain \\eta^{\\prime} \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} - \\eta^{\\prime} = \\eta^{\\prime} (\\mathbf{E} + e^{\\eta^{\\prime}}) - \\eta^{\\prime}", "derivation": "M{(\\eta^{\\prime})} = e^{\\eta^{\\prime}} and \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} = \\int e^{\\eta^{\\prime}} d\\eta^{\\prime} and \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} = \\mathbf{E} + e^{\\eta^{\\prime}} and \\eta^{\\prime} \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} = \\eta^{\\prime} (\\mathbf{E} + e^{\\eta^{\\prime}}) and \\eta^{\\prime} \\int M{(\\eta^{\\prime})} d\\eta^{\\prime} - \\eta^{\\prime} = \\eta^{\\prime} (\\mathbf{E} + e^{\\eta^{\\prime}}) - \\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Function('M')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["minus", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Function('M')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Add(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(T)} = \\log{(T)}, then derive \\int \\frac{d}{d T} \\frac{\\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)}}{\\frac{d}{d T} \\log{(T)}} dT = \\int 0 dT, then obtain (\\int \\frac{d}{d T} 1 dT)^{T} = (\\int 0 dT)^{T}", "derivation": "\\operatorname{L_{\\varepsilon}}{(T)} = \\log{(T)} and \\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)} = \\frac{d}{d T} \\log{(T)} and \\frac{\\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)}}{\\frac{d}{d T} \\log{(T)}} = 1 and \\frac{d}{d T} \\frac{\\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)}}{\\frac{d}{d T} \\log{(T)}} = \\frac{d}{d T} 1 and \\int \\frac{d}{d T} \\frac{\\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)}}{\\frac{d}{d T} \\log{(T)}} dT = \\int \\frac{d}{d T} 1 dT and \\int \\frac{d}{d T} \\frac{\\frac{d}{d T} \\operatorname{L_{\\varepsilon}}{(T)}}{\\frac{d}{d T} \\log{(T)}} dT = \\int 0 dT and \\int \\frac{d}{d T} 1 dT = \\int 0 dT and (\\int \\frac{d}{d T} 1 dT)^{T} = (\\int 0 dT)^{T}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Mul(Derivative(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Integral(Derivative(Mul(Derivative(Function('L_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Derivative(Integer(1), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Integer(0), Tuple(Symbol('T', commutative=True))))"], [["power", 7, "Symbol('T', commutative=True)"], "Equality(Pow(Integral(Derivative(Integer(1), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('T', commutative=True))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(a,n)} = a^{n}, then obtain a^{n} + \\mathbf{F}{(a,n)} + 2 = 2 a^{n} + 2", "derivation": "\\mathbf{F}{(a,n)} = a^{n} and - a^{n} + \\mathbf{F}{(a,n)} = 0 and - a^{n} + \\mathbf{F}{(a,n)} + 1 = 1 and \\mathbf{F}{(a,n)} + 1 = a^{n} + 1 and a^{n} + \\mathbf{F}{(a,n)} + 2 = 2 a^{n} + 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)))"], [["minus", 1, "Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(1)), Integer(1))"], [["minus", 3, "Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(1)))"], [["add", 4, "Add(Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(1))"], "Equality(Add(Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(2)), Add(Mul(Integer(2), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(r_{0},\\rho)} = e^{\\rho^{r_{0}}}, then obtain - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho + \\operatorname{z^{*}}{(r_{0},\\rho)}) = - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho + e^{\\rho^{r_{0}}})", "derivation": "\\operatorname{z^{*}}{(r_{0},\\rho)} = e^{\\rho^{r_{0}}} and \\rho + \\operatorname{z^{*}}{(r_{0},\\rho)} = \\rho + e^{\\rho^{r_{0}}} and \\frac{\\partial}{\\partial r_{0}} (\\rho + \\operatorname{z^{*}}{(r_{0},\\rho)}) = \\frac{\\partial}{\\partial r_{0}} (\\rho + e^{\\rho^{r_{0}}}) and - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho + \\operatorname{z^{*}}{(r_{0},\\rho)}) = - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho + e^{\\rho^{r_{0}}})", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True)), exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Function('z^*')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\rho', commutative=True), exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\rho', commutative=True), Function('z^*')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho', commutative=True), exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho', commutative=True), Function('z^*')(Symbol('r_0', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho', commutative=True), exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(P_{g})} = \\cos{(P_{g})}, then obtain \\cos{(P_{g})} \\int z^{P_{g}}{(P_{g})} dP_{g} = \\cos{(P_{g})} \\int \\cos^{P_{g}}{(P_{g})} dP_{g}", "derivation": "z{(P_{g})} = \\cos{(P_{g})} and z^{P_{g}}{(P_{g})} = \\cos^{P_{g}}{(P_{g})} and \\int z^{P_{g}}{(P_{g})} dP_{g} = \\int \\cos^{P_{g}}{(P_{g})} dP_{g} and \\cos{(P_{g})} \\int z^{P_{g}}{(P_{g})} dP_{g} = \\cos{(P_{g})} \\int \\cos^{P_{g}}{(P_{g})} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True)))"], [["power", 1, "Symbol('P_g', commutative=True)"], "Equality(Pow(Function('z')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Pow(Function('z')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["times", 3, "cos(Symbol('P_g', commutative=True))"], "Equality(Mul(cos(Symbol('P_g', commutative=True)), Integral(Pow(Function('z')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Mul(cos(Symbol('P_g', commutative=True)), Integral(Pow(cos(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\delta)} = e^{e^{\\delta}} and a{(\\delta)} = \\int e^{e^{\\delta}} d\\delta, then obtain \\int \\operatorname{f^{\\prime}}{(\\delta)} d\\delta + 1 = a{(\\delta)} + 1", "derivation": "\\operatorname{f^{\\prime}}{(\\delta)} = e^{e^{\\delta}} and \\int \\operatorname{f^{\\prime}}{(\\delta)} d\\delta = \\int e^{e^{\\delta}} d\\delta and a{(\\delta)} = \\int e^{e^{\\delta}} d\\delta and \\int \\operatorname{f^{\\prime}}{(\\delta)} d\\delta + 1 = \\int e^{e^{\\delta}} d\\delta + 1 and \\int \\operatorname{f^{\\prime}}{(\\delta)} d\\delta + 1 = a{(\\delta)} + 1", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\delta', commutative=True)), exp(exp(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(exp(exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('a')(Symbol('\\\\delta', commutative=True)), Integral(exp(exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["add", 2, 1], "Equality(Add(Integral(Function('f^{\\\\prime}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Integral(exp(exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integral(Function('f^{\\\\prime}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Function('a')(Symbol('\\\\delta', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(E,x^\\prime,y)} = - E - x^\\prime + y, then obtain \\frac{\\partial}{\\partial x^\\prime} 2 \\operatorname{v_{t}}{(E,x^\\prime,y)} = \\frac{\\partial}{\\partial x^\\prime} (- 2 E - 2 x^\\prime + 2 y)", "derivation": "\\operatorname{v_{t}}{(E,x^\\prime,y)} = - E - x^\\prime + y and - E - x^\\prime + y + \\operatorname{v_{t}}{(E,x^\\prime,y)} = - 2 E - 2 x^\\prime + 2 y and \\frac{\\partial}{\\partial x^\\prime} (- E - x^\\prime + y + \\operatorname{v_{t}}{(E,x^\\prime,y)}) = \\frac{\\partial}{\\partial x^\\prime} (- 2 E - 2 x^\\prime + 2 y) and \\frac{\\partial}{\\partial x^\\prime} 2 \\operatorname{v_{t}}{(E,x^\\prime,y)} = \\frac{\\partial}{\\partial x^\\prime} (- 2 E - 2 x^\\prime + 2 y)", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('E', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Symbol('y', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Symbol('y', commutative=True), Function('v_t')(Symbol('E', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Symbol('y', commutative=True), Function('v_t')(Symbol('E', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Mul(Integer(2), Function('v_t')(Symbol('E', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('E', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(I)} = \\sin{(I)} and W{(\\mathbf{J}_P)} = \\log{(\\sin{(\\mathbf{J}_P)})}, then obtain W{(\\mathbf{J}_P)} + \\frac{- I + W{(I)}}{\\mathbf{J}_P} = W{(\\mathbf{J}_P)} + \\frac{- I + \\sin{(I)}}{\\mathbf{J}_P}", "derivation": "W{(I)} = \\sin{(I)} and - I + W{(I)} = - I + \\sin{(I)} and W{(\\mathbf{J}_P)} = \\log{(\\sin{(\\mathbf{J}_P)})} and \\frac{- I + W{(I)}}{\\mathbf{J}_P} = \\frac{- I + \\sin{(I)}}{\\mathbf{J}_P} and \\log{(\\sin{(\\mathbf{J}_P)})} + \\frac{- I + W{(I)}}{\\mathbf{J}_P} = \\log{(\\sin{(\\mathbf{J}_P)})} + \\frac{- I + \\sin{(I)}}{\\mathbf{J}_P} and W{(\\mathbf{J}_P)} + \\frac{- I + W{(I)}}{\\mathbf{J}_P} = W{(\\mathbf{J}_P)} + \\frac{- I + \\sin{(I)}}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('W')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))))"], ["get_premise", "Equality(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('W')(Symbol('I', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))))"], [["add", 4, "log(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(log(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('W')(Symbol('I', commutative=True))))), Add(log(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('W')(Symbol('I', commutative=True))))), Add(Function('W')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), sin(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given z{(g,\\mathbf{M})} = \\frac{\\mathbf{M}}{g} and s{(g,\\mathbf{M})} = \\frac{\\mathbf{M}}{g}, then obtain z^{\\mathbf{M}}{(g,\\mathbf{M})} = s^{\\mathbf{M}}{(g,\\mathbf{M})}", "derivation": "z{(g,\\mathbf{M})} = \\frac{\\mathbf{M}}{g} and s{(g,\\mathbf{M})} = \\frac{\\mathbf{M}}{g} and s{(g,\\mathbf{M})} = z{(g,\\mathbf{M})} and s^{\\mathbf{M}}{(g,\\mathbf{M})} = (\\frac{\\mathbf{M}}{g})^{\\mathbf{M}} and z^{\\mathbf{M}}{(g,\\mathbf{M})} = (\\frac{\\mathbf{M}}{g})^{\\mathbf{M}} and z^{\\mathbf{M}}{(g,\\mathbf{M})} = s^{\\mathbf{M}}{(g,\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Function('z')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('z')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('z')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given Z{(g)} = \\cos{(g)} and \\mathbf{P}{(g,\\delta)} = \\frac{\\delta}{Z{(g)}} - 1, then obtain - \\operatorname{n_{1}}{(\\delta)} - \\int (\\frac{\\delta}{\\cos{(g)}} - 1) d\\delta + \\int \\mathbf{P}{(g,\\delta)} d\\delta = - \\operatorname{n_{1}}{(\\delta)}", "derivation": "Z{(g)} = \\cos{(g)} and \\mathbf{P}{(g,\\delta)} = \\frac{\\delta}{Z{(g)}} - 1 and \\int \\mathbf{P}{(g,\\delta)} d\\delta = \\int (\\frac{\\delta}{Z{(g)}} - 1) d\\delta and - \\operatorname{n_{1}}{(\\delta)} + \\int \\mathbf{P}{(g,\\delta)} d\\delta = - \\operatorname{n_{1}}{(\\delta)} + \\int (\\frac{\\delta}{Z{(g)}} - 1) d\\delta and - \\int (\\frac{\\delta}{Z{(g)}} - 1) d\\delta + \\int \\mathbf{P}{(g,\\delta)} d\\delta = 0 and - \\operatorname{n_{1}}{(\\delta)} - \\int (\\frac{\\delta}{Z{(g)}} - 1) d\\delta + \\int \\mathbf{P}{(g,\\delta)} d\\delta = - \\operatorname{n_{1}}{(\\delta)} and - \\operatorname{n_{1}}{(\\delta)} - \\int (\\frac{\\delta}{\\cos{(g)}} - 1) d\\delta + \\int \\mathbf{P}{(g,\\delta)} d\\delta = - \\operatorname{n_{1}}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 3, "Function('n_1')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))), Integral(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))), Integral(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Integer(0))"], [["add", 5, "Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('Z')(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))), Integral(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Add(Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\delta', commutative=True)))), Integral(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integer(-1), Function('n_1')(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given Z{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)}, then derive Z{(\\sigma_x)} = - \\sin{(\\sigma_x)}, then obtain \\frac{Z{(\\sigma_x)}}{\\sigma_x} = - \\frac{\\sin{(\\sigma_x)}}{\\sigma_x}", "derivation": "Z{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)} and Z{(\\sigma_x)} = - \\sin{(\\sigma_x)} and \\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)} = - \\sin{(\\sigma_x)} and \\frac{\\frac{d}{d \\sigma_x} \\cos{(\\sigma_x)}}{\\sigma_x} = - \\frac{\\sin{(\\sigma_x)}}{\\sigma_x} and \\frac{Z{(\\sigma_x)}}{\\sigma_x} = - \\frac{\\sin{(\\sigma_x)}}{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\sigma_x', commutative=True)), Derivative(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Z')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('Z')(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\rho_b,A_{z})} = A_{z} + e^{\\rho_b}, then obtain \\log{((A_{z} - \\operatorname{m_{s}}{(\\rho_b,A_{z})} + e^{\\rho_b})^{A_{z}})} = 0", "derivation": "\\operatorname{m_{s}}{(\\rho_b,A_{z})} = A_{z} + e^{\\rho_b} and 0 = A_{z} - \\operatorname{m_{s}}{(\\rho_b,A_{z})} + e^{\\rho_b} and 0^{A_{z}} = (A_{z} - \\operatorname{m_{s}}{(\\rho_b,A_{z})} + e^{\\rho_b})^{A_{z}} and \\log{(0^{A_{z}})} = \\log{((A_{z} - \\operatorname{m_{s}}{(\\rho_b,A_{z})} + e^{\\rho_b})^{A_{z}})} and \\log{((A_{z} - \\operatorname{m_{s}}{(\\rho_b,A_{z})} + e^{\\rho_b})^{A_{z}})} = 0", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 1, "Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Integer(0), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True))), exp(Symbol('\\\\rho_b', commutative=True))))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_z', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True))), exp(Symbol('\\\\rho_b', commutative=True))), Symbol('A_z', commutative=True)))"], [["log", 3], "Equality(log(Pow(Integer(0), Symbol('A_z', commutative=True))), log(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True))), exp(Symbol('\\\\rho_b', commutative=True))), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(log(Pow(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Function('m_s')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True))), exp(Symbol('\\\\rho_b', commutative=True))), Symbol('A_z', commutative=True))), Integer(0))"]]}, {"prompt": "Given c{(i)} = \\sin{(i)}, then obtain c{(i)} \\frac{d^{2}}{d i^{2}} c{(i)} = c{(i)} \\frac{d^{2}}{d i^{2}} \\sin{(i)}", "derivation": "c{(i)} = \\sin{(i)} and \\frac{d}{d i} c{(i)} = \\frac{d}{d i} \\sin{(i)} and \\frac{d^{2}}{d i^{2}} c{(i)} = \\frac{d^{2}}{d i^{2}} \\sin{(i)} and c{(i)} \\frac{d^{2}}{d i^{2}} c{(i)} = c{(i)} \\frac{d^{2}}{d i^{2}} \\sin{(i)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))), Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2))))"], [["times", 3, "Function('c')(Symbol('i', commutative=True))"], "Equality(Mul(Function('c')(Symbol('i', commutative=True)), Derivative(Function('c')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2)))), Mul(Function('c')(Symbol('i', commutative=True)), Derivative(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})} = C_{1} + \\sin{(\\Psi_{nl})}, then derive \\frac{\\partial}{\\partial C_{1}} \\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})} = 1, then obtain (\\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}))^{\\Psi_{nl}} \\int \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}) dC_{1} = \\int \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}) dC_{1}", "derivation": "\\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})} = C_{1} + \\sin{(\\Psi_{nl})} and \\frac{\\partial}{\\partial C_{1}} \\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})} = \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}) and \\frac{\\partial}{\\partial C_{1}} \\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})} = 1 and (\\frac{\\partial}{\\partial C_{1}} \\operatorname{E_{\\lambda}}{(C_{1},\\Psi_{nl})})^{\\Psi_{nl}} = 1 and (\\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}))^{\\Psi_{nl}} = 1 and (\\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}))^{\\Psi_{nl}} \\int \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}) dC_{1} = \\int \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\sin{(\\Psi_{nl})}) dC_{1}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Derivative(Function('E_{\\\\lambda}')(Symbol('C_1', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1))"], [["times", 5, "Integral(Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('\\\\Psi_{nl}', commutative=True)), Integral(Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True)))), Integral(Derivative(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(v_{y})} = e^{v_{y}} and \\dot{\\mathbf{r}}{(\\mathbf{B},a^{\\dagger},v_{y})} = - \\bar{\\h}{(v_{y})} + \\mathbf{v}{(\\mathbf{B},a^{\\dagger})}, then obtain - \\bar{\\h}{(v_{y})} + \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} = \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} - e^{v_{y}}", "derivation": "\\bar{\\h}{(v_{y})} = e^{v_{y}} and \\dot{\\mathbf{r}}{(\\mathbf{B},a^{\\dagger},v_{y})} = - \\bar{\\h}{(v_{y})} + \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} and \\dot{\\mathbf{r}}{(\\mathbf{B},a^{\\dagger},v_{y})} = \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} - e^{v_{y}} and - \\bar{\\h}{(v_{y})} + \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} = \\mathbf{v}{(\\mathbf{B},a^{\\dagger})} - e^{v_{y}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hbar')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Add(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('v_y', commutative=True))), Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('\\\\mathbf{v}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\omega,\\varepsilon)} = \\omega - \\varepsilon, then derive \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon = \\dot{z} + \\omega \\varepsilon - \\frac{\\varepsilon^{2}}{2}, then obtain \\int (- \\dot{z} + \\frac{\\varepsilon^{2}}{2} + \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon) d\\dot{z} = \\int \\omega \\varepsilon d\\dot{z}", "derivation": "\\operatorname{c_{0}}{(\\omega,\\varepsilon)} = \\omega - \\varepsilon and \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon = \\int (\\omega - \\varepsilon) d\\varepsilon and \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon = \\dot{z} + \\omega \\varepsilon - \\frac{\\varepsilon^{2}}{2} and \\omega \\varepsilon + \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon = \\dot{z} + 2 \\omega \\varepsilon - \\frac{\\varepsilon^{2}}{2} and - \\dot{z} + \\frac{\\varepsilon^{2}}{2} + \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon = \\omega \\varepsilon and \\int (- \\dot{z} + \\frac{\\varepsilon^{2}}{2} + \\int \\operatorname{c_{0}}{(\\omega,\\varepsilon)} d\\varepsilon) d\\dot{z} = \\int \\omega \\varepsilon d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)))))"], [["add", 3, "Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integral(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)))))"], [["minus", 4, "Add(Symbol('\\\\dot{z}', commutative=True), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Integral(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 5, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Integral(Function('c_0')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\theta_1)} = \\log{(\\theta_1)}, then derive \\int \\operatorname{m_{s}}{(\\theta_1)} d\\theta_1 = \\eta^{\\prime} + \\theta_1 \\log{(\\theta_1)} - \\theta_1, then obtain \\int \\operatorname{m_{s}}{(\\theta_1)} d\\theta_1 = \\eta^{\\prime} + \\theta_1 \\operatorname{m_{s}}{(\\theta_1)} - \\theta_1", "derivation": "\\operatorname{m_{s}}{(\\theta_1)} = \\log{(\\theta_1)} and \\int \\operatorname{m_{s}}{(\\theta_1)} d\\theta_1 = \\int \\log{(\\theta_1)} d\\theta_1 and \\int \\operatorname{m_{s}}{(\\theta_1)} d\\theta_1 = \\eta^{\\prime} + \\theta_1 \\log{(\\theta_1)} - \\theta_1 and \\int \\operatorname{m_{s}}{(\\theta_1)} d\\theta_1 = \\eta^{\\prime} + \\theta_1 \\operatorname{m_{s}}{(\\theta_1)} - \\theta_1", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\theta_1', commutative=True)), log(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m_s')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('m_s')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('m_s')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(a)} = \\sin{(\\log{(a)})}, then obtain (\\int \\cos{(1)} da)^{a} = (\\int \\cos{(\\frac{\\sin{(\\log{(a)})}}{\\operatorname{M_{E}}{(a)}})} da)^{a}", "derivation": "\\operatorname{M_{E}}{(a)} = \\sin{(\\log{(a)})} and 1 = \\frac{\\sin{(\\log{(a)})}}{\\operatorname{M_{E}}{(a)}} and \\cos{(1)} = \\cos{(\\frac{\\sin{(\\log{(a)})}}{\\operatorname{M_{E}}{(a)}})} and \\int \\cos{(1)} da = \\int \\cos{(\\frac{\\sin{(\\log{(a)})}}{\\operatorname{M_{E}}{(a)}})} da and (\\int \\cos{(1)} da)^{a} = (\\int \\cos{(\\frac{\\sin{(\\log{(a)})}}{\\operatorname{M_{E}}{(a)}})} da)^{a}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('a', commutative=True)), sin(log(Symbol('a', commutative=True))))"], [["divide", 1, "Function('M_E')(Symbol('a', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('M_E')(Symbol('a', commutative=True)), Integer(-1)), sin(log(Symbol('a', commutative=True)))))"], [["cos", 2], "Equality(cos(Integer(1)), cos(Mul(Pow(Function('M_E')(Symbol('a', commutative=True)), Integer(-1)), sin(log(Symbol('a', commutative=True))))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(cos(Integer(1)), Tuple(Symbol('a', commutative=True))), Integral(cos(Mul(Pow(Function('M_E')(Symbol('a', commutative=True)), Integer(-1)), sin(log(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True))))"], [["power", 4, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(cos(Integer(1)), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(cos(Mul(Pow(Function('M_E')(Symbol('a', commutative=True)), Integer(-1)), sin(log(Symbol('a', commutative=True))))), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(m_{s})} = \\sin{(m_{s})} and \\operatorname{f^{*}}{(m_{s})} = \\sin{(m_{s})}, then obtain \\log{(\\phi_{1}{(m_{s})} \\int \\operatorname{f^{*}}{(m_{s})} dm_{s})} = \\log{(\\operatorname{f^{*}}{(m_{s})} \\int \\operatorname{f^{*}}{(m_{s})} dm_{s})}", "derivation": "\\phi_{1}{(m_{s})} = \\sin{(m_{s})} and \\operatorname{f^{*}}{(m_{s})} = \\sin{(m_{s})} and \\int \\operatorname{f^{*}}{(m_{s})} dm_{s} = \\int \\sin{(m_{s})} dm_{s} and \\phi_{1}{(m_{s})} = \\operatorname{f^{*}}{(m_{s})} and \\phi_{1}{(m_{s})} \\int \\sin{(m_{s})} dm_{s} = \\operatorname{f^{*}}{(m_{s})} \\int \\sin{(m_{s})} dm_{s} and \\log{(\\phi_{1}{(m_{s})} \\int \\sin{(m_{s})} dm_{s})} = \\log{(\\operatorname{f^{*}}{(m_{s})} \\int \\sin{(m_{s})} dm_{s})} and \\log{(\\phi_{1}{(m_{s})} \\int \\operatorname{f^{*}}{(m_{s})} dm_{s})} = \\log{(\\operatorname{f^{*}}{(m_{s})} \\int \\operatorname{f^{*}}{(m_{s})} dm_{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Function('f^*')(Symbol('m_s', commutative=True)))"], [["times", 4, "Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Function('f^*')(Symbol('m_s', commutative=True)), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["log", 5], "Equality(log(Mul(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))), log(Mul(Function('f^*')(Symbol('m_s', commutative=True)), Integral(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(log(Mul(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Integral(Function('f^*')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))), log(Mul(Function('f^*')(Symbol('m_s', commutative=True)), Integral(Function('f^*')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(S,U)} = \\frac{U}{S}, then obtain \\mathbf{g} \\int (- \\int \\operatorname{A_{1}}{(S,U)} dU + (\\int \\operatorname{A_{1}}{(S,U)} dU)^{U}) dS = \\mathbf{g} \\int ((\\int \\frac{U}{S} dU)^{U} - \\int \\operatorname{A_{1}}{(S,U)} dU) dS", "derivation": "\\operatorname{A_{1}}{(S,U)} = \\frac{U}{S} and \\int \\operatorname{A_{1}}{(S,U)} dU = \\int \\frac{U}{S} dU and (\\int \\operatorname{A_{1}}{(S,U)} dU)^{U} = (\\int \\frac{U}{S} dU)^{U} and - \\int \\operatorname{A_{1}}{(S,U)} dU + (\\int \\operatorname{A_{1}}{(S,U)} dU)^{U} = (\\int \\frac{U}{S} dU)^{U} - \\int \\operatorname{A_{1}}{(S,U)} dU and \\int (- \\int \\operatorname{A_{1}}{(S,U)} dU + (\\int \\operatorname{A_{1}}{(S,U)} dU)^{U}) dS = \\int ((\\int \\frac{U}{S} dU)^{U} - \\int \\operatorname{A_{1}}{(S,U)} dU) dS and \\mathbf{g} \\int (- \\int \\operatorname{A_{1}}{(S,U)} dU + (\\int \\operatorname{A_{1}}{(S,U)} dU)^{U}) dS = \\mathbf{g} \\int ((\\int \\frac{U}{S} dU)^{U} - \\int \\operatorname{A_{1}}{(S,U)} dU) dS", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["minus", 3, "Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Pow(Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Add(Pow(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["integrate", 4, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Pow(Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Add(Pow(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('S', commutative=True))))"], [["times", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Integral(Add(Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Pow(Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Tuple(Symbol('S', commutative=True)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Integral(Add(Pow(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('A_1')(Symbol('S', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given W{(\\pi,f_{E})} = e^{\\pi f_{E}}, then obtain - \\pi f_{E} + f_{E} W^{\\pi}{(\\pi,f_{E})} = - \\pi f_{E} + f_{E} (e^{\\pi f_{E}})^{\\pi}", "derivation": "W{(\\pi,f_{E})} = e^{\\pi f_{E}} and W^{\\pi}{(\\pi,f_{E})} = (e^{\\pi f_{E}})^{\\pi} and f_{E} W^{\\pi}{(\\pi,f_{E})} = f_{E} (e^{\\pi f_{E}})^{\\pi} and - \\pi f_{E} + f_{E} W^{\\pi}{(\\pi,f_{E})} = - \\pi f_{E} + f_{E} (e^{\\pi f_{E}})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["times", 2, "Symbol('f_E', commutative=True)"], "Equality(Mul(Symbol('f_E', commutative=True), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True))), Symbol('\\\\pi', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(Function('W')(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('f_E', commutative=True))), Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(V,\\mathbf{J}_P)} = e^{V \\mathbf{J}_P}, then obtain \\frac{e^{- V \\mathbf{J}_P} (\\int (\\operatorname{f_{E}}{(V,\\mathbf{J}_P)} + e^{V \\mathbf{J}_P}) dV)^{2}}{3} = \\frac{e^{- V \\mathbf{J}_P} (\\int 2 e^{V \\mathbf{J}_P} dV)^{2}}{3}", "derivation": "\\operatorname{f_{E}}{(V,\\mathbf{J}_P)} = e^{V \\mathbf{J}_P} and \\operatorname{f_{E}}{(V,\\mathbf{J}_P)} + e^{V \\mathbf{J}_P} = 2 e^{V \\mathbf{J}_P} and \\int (\\operatorname{f_{E}}{(V,\\mathbf{J}_P)} + e^{V \\mathbf{J}_P}) dV = \\int 2 e^{V \\mathbf{J}_P} dV and (\\int (\\operatorname{f_{E}}{(V,\\mathbf{J}_P)} + e^{V \\mathbf{J}_P}) dV)^{2} = (\\int 2 e^{V \\mathbf{J}_P} dV)^{2} and \\frac{e^{- V \\mathbf{J}_P} (\\int (\\operatorname{f_{E}}{(V,\\mathbf{J}_P)} + e^{V \\mathbf{J}_P}) dV)^{2}}{3} = \\frac{e^{- V \\mathbf{J}_P} (\\int 2 e^{V \\mathbf{J}_P} dV)^{2}}{3}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 1, "exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Function('f_E')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(2), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('f_E')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Integer(2), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Add(Function('f_E')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integer(2)))"], [["divide", 4, "Mul(Integer(3), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], "Equality(Mul(Rational(1, 3), exp(Mul(Integer(-1), Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Integral(Add(Function('f_E')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integer(2))), Mul(Rational(1, 3), exp(Mul(Integer(-1), Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Integral(Mul(Integer(2), exp(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('V', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\delta{(\\rho)} = \\log{(\\rho)} and \\varphi{(\\dot{x})} = \\dot{x}, then derive \\frac{d}{d \\rho} \\delta{(\\rho)} = \\frac{1}{\\rho}, then obtain e^{\\dot{x}} + \\frac{d}{d \\rho} \\log{(\\rho)} = e^{\\dot{x}} + \\frac{1}{\\rho}", "derivation": "\\delta{(\\rho)} = \\log{(\\rho)} and \\frac{d}{d \\rho} \\delta{(\\rho)} = \\frac{d}{d \\rho} \\log{(\\rho)} and \\varphi{(\\dot{x})} = \\dot{x} and \\frac{d}{d \\rho} \\delta{(\\rho)} = \\frac{1}{\\rho} and \\frac{d}{d \\rho} \\log{(\\rho)} = \\frac{1}{\\rho} and e^{\\varphi{(\\dot{x})}} + \\frac{d}{d \\rho} \\log{(\\rho)} = e^{\\varphi{(\\dot{x})}} + \\frac{1}{\\rho} and e^{\\dot{x}} + \\frac{d}{d \\rho} \\log{(\\rho)} = e^{\\dot{x}} + \\frac{1}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))"], [["add", 5, "exp(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(exp(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True))), Derivative(log(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(exp(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True))), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(exp(Symbol('\\\\dot{x}', commutative=True)), Derivative(log(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\dot{x}', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"]]}, {"prompt": "Given M{(A)} = \\frac{d}{d A} \\sin{(A)}, then derive M{(A)} = \\cos{(A)}, then obtain A \\cos{(A)} + \\frac{d}{d A} \\frac{1}{A} = A \\cos{(A)} + \\frac{d}{d A} \\frac{\\frac{d}{d A} \\sin{(A)}}{A M{(A)}}", "derivation": "M{(A)} = \\frac{d}{d A} \\sin{(A)} and M{(A)} = \\cos{(A)} and A M{(A)} = A \\cos{(A)} and \\frac{M{(A)}}{A \\cos{(A)}} = \\frac{\\frac{d}{d A} \\sin{(A)}}{A \\cos{(A)}} and \\frac{1}{A} = \\frac{\\frac{d}{d A} \\sin{(A)}}{A M{(A)}} and \\frac{d}{d A} \\frac{1}{A} = \\frac{d}{d A} \\frac{\\frac{d}{d A} \\sin{(A)}}{A M{(A)}} and A \\cos{(A)} + \\frac{d}{d A} \\frac{1}{A} = A \\cos{(A)} + \\frac{d}{d A} \\frac{\\frac{d}{d A} \\sin{(A)}}{A M{(A)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('M')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["times", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('M')(Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True))))"], [["divide", 1, "Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True)))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Function('M')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(cos(Symbol('A', commutative=True)), Integer(-1)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Symbol('A', commutative=True), Integer(-1)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('M')(Symbol('A', commutative=True)), Integer(-1)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Symbol('A', commutative=True), Integer(-1)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('M')(Symbol('A', commutative=True)), Integer(-1)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["add", 6, "Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True)))"], "Equality(Add(Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True))), Derivative(Pow(Symbol('A', commutative=True), Integer(-1)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Mul(Symbol('A', commutative=True), cos(Symbol('A', commutative=True))), Derivative(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Function('M')(Symbol('A', commutative=True)), Integer(-1)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{r}{(a)} = \\int \\cos{(a)} da, then derive \\mathbf{r}{(a)} = V + \\sin{(a)}, then derive V + v_{y} = \\int \\frac{\\partial}{\\partial V} (a^{\\dagger} + \\sin{(a)}) dV, then obtain (V + v_{y})^{a^{\\dagger}} = (\\int \\frac{\\partial}{\\partial V} (a^{\\dagger} + \\sin{(a)}) dV)^{a^{\\dagger}}", "derivation": "\\mathbf{r}{(a)} = \\int \\cos{(a)} da and \\mathbf{r}{(a)} = V + \\sin{(a)} and V + \\sin{(a)} = \\int \\cos{(a)} da and \\frac{\\partial}{\\partial V} (V + \\sin{(a)}) = \\frac{d}{d V} \\int \\cos{(a)} da and \\int \\frac{\\partial}{\\partial V} (V + \\sin{(a)}) dV = \\int \\frac{d}{d V} \\int \\cos{(a)} da dV and V + v_{y} = \\int \\frac{\\partial}{\\partial V} (a^{\\dagger} + \\sin{(a)}) dV and (V + v_{y})^{a^{\\dagger}} = (\\int \\frac{\\partial}{\\partial V} (a^{\\dagger} + \\sin{(a)}) dV)^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('a', commutative=True)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{r}')(Symbol('a', commutative=True)), Add(Symbol('V', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('V', commutative=True), sin(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('V', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integral(Derivative(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integral(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["power", 6, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), sin(Symbol('a', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain (2 \\operatorname{t_{2}}{(\\mathbf{P})} - \\log{(\\mathbf{P})})^{\\mathbf{P}} = \\operatorname{t_{2}}^{\\mathbf{P}}{(\\mathbf{P})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and 2 \\operatorname{t_{2}}{(\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{P})} + \\log{(\\mathbf{P})} and 2 \\operatorname{t_{2}}{(\\mathbf{P})} - \\log{(\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{P})} and (2 \\operatorname{t_{2}}{(\\mathbf{P})} - \\log{(\\mathbf{P})})^{\\mathbf{P}} = \\operatorname{t_{2}}^{\\mathbf{P}}{(\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Integer(2), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 2, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True)))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given t{(h,f)} = f \\cos{(h)}, then obtain h + \\frac{(- h + (\\int t{(h,f)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h} = h + \\frac{(- h + (\\int f \\cos{(h)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h}", "derivation": "t{(h,f)} = f \\cos{(h)} and \\int t{(h,f)} df = \\int f \\cos{(h)} df and (\\int t{(h,f)} df)^{h} = (\\int f \\cos{(h)} df)^{h} and - h + (\\int t{(h,f)} df)^{h} = - h + (\\int f \\cos{(h)} df)^{h} and \\frac{(- h + (\\int t{(h,f)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h} = \\frac{(- h + (\\int f \\cos{(h)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h} and h + \\frac{(- h + (\\int t{(h,f)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h} = h + \\frac{(- h + (\\int f \\cos{(h)} df)^{h}) (\\int t{(h,f)} df)^{- h}}{h}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))))"], [["divide", 4, "Mul(Symbol('h', commutative=True), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True)))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Add(Symbol('h', commutative=True), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))))), Add(Symbol('h', commutative=True), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Mul(Symbol('f', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('h', commutative=True))), Pow(Integral(Function('t')(Symbol('h', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))))))"]]}, {"prompt": "Given a{(\\theta)} = e^{\\theta}, then obtain (- \\theta + a{(\\theta)} + 1)^{\\theta} = (- \\theta + e^{\\theta} + 1)^{\\theta}", "derivation": "a{(\\theta)} = e^{\\theta} and - \\theta + a{(\\theta)} = - \\theta + e^{\\theta} and - \\theta + a{(\\theta)} + 1 = - \\theta + e^{\\theta} + 1 and (- \\theta + a{(\\theta)} + 1)^{\\theta} = (- \\theta + e^{\\theta} + 1)^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True))))"], [["add", 2, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)), Integer(1)))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('\\\\theta', commutative=True)), Integer(1)), Symbol('\\\\theta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), exp(Symbol('\\\\theta', commutative=True)), Integer(1)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given c{(f_{E})} = \\mathbf{E}^{f_{E}}{(f_{E})}, then obtain \\int \\frac{d}{d f_{E}} (- f_{E} + c{(f_{E})}) df_{E} = \\int \\frac{d}{d f_{E}} (- f_{E} + \\mathbf{E}^{f_{E}}{(f_{E})}) df_{E}", "derivation": "c{(f_{E})} = \\mathbf{E}^{f_{E}}{(f_{E})} and - f_{E} + c{(f_{E})} = - f_{E} + \\mathbf{E}^{f_{E}}{(f_{E})} and \\frac{d}{d f_{E}} (- f_{E} + c{(f_{E})}) = \\frac{d}{d f_{E}} (- f_{E} + \\mathbf{E}^{f_{E}}{(f_{E})}) and \\int \\frac{d}{d f_{E}} (- f_{E} + c{(f_{E})}) df_{E} = \\int \\frac{d}{d f_{E}} (- f_{E} + \\mathbf{E}^{f_{E}}{(f_{E})}) df_{E}", "srepr_derivation": [["renaming_premise", "Equality(Function('c')(Symbol('f_E', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True)))"], [["minus", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('f_E', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('c')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Tuple(Symbol('f_E', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(m)} = \\int \\log{(m)} dm, then derive \\operatorname{f^{\\prime}}^{m}{(m)} = (m \\log{(m)} - m + v_{t})^{m}, then derive \\operatorname{f^{\\prime}}{(m)} = J + m \\log{(m)} - m, then obtain (J + m \\log{(m)} - m)^{m} = (m \\log{(m)} - m + v_{t})^{m}", "derivation": "\\operatorname{f^{\\prime}}{(m)} = \\int \\log{(m)} dm and \\operatorname{f^{\\prime}}^{m}{(m)} = (\\int \\log{(m)} dm)^{m} and \\operatorname{f^{\\prime}}^{m}{(m)} = (m \\log{(m)} - m + v_{t})^{m} and \\operatorname{f^{\\prime}}{(m)} = J + m \\log{(m)} - m and (J + m \\log{(m)} - m)^{m} = (m \\log{(m)} - m + v_{t})^{m}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('f^{\\\\prime}')(Symbol('m', commutative=True)), Add(Symbol('J', commutative=True), Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('J', commutative=True), Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Add(Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)), Symbol('v_t', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(c,\\mathbf{D})} = e^{\\frac{c}{\\mathbf{D}}}, then obtain (\\frac{\\partial}{\\partial c} \\operatorname{A_{z}}{(c,\\mathbf{D})})^{\\mathbf{D}} = (\\frac{e^{\\frac{c}{\\mathbf{D}}}}{\\mathbf{D}})^{\\mathbf{D}}", "derivation": "\\operatorname{A_{z}}{(c,\\mathbf{D})} = e^{\\frac{c}{\\mathbf{D}}} and \\frac{\\partial}{\\partial c} \\operatorname{A_{z}}{(c,\\mathbf{D})} = \\frac{\\partial}{\\partial c} e^{\\frac{c}{\\mathbf{D}}} and (\\frac{\\partial}{\\partial c} \\operatorname{A_{z}}{(c,\\mathbf{D})})^{\\mathbf{D}} = (\\frac{\\partial}{\\partial c} e^{\\frac{c}{\\mathbf{D}}})^{\\mathbf{D}} and (\\frac{\\partial}{\\partial c} \\operatorname{A_{z}}{(c,\\mathbf{D})})^{\\mathbf{D}} = (\\frac{e^{\\frac{c}{\\mathbf{D}}}}{\\mathbf{D}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('c', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('c', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Derivative(Function('A_z')(Symbol('c', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('A_z')(Symbol('c', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('c', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(Q)} = \\log{(Q)}, then obtain (\\operatorname{M_{E}}{(Q)} - 1)^{2} = (\\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} - 1)^{2}", "derivation": "\\operatorname{M_{E}}{(Q)} = \\log{(Q)} and \\frac{\\operatorname{M_{E}}{(Q)}}{\\log{(Q)}} = 1 and \\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} = \\operatorname{M_{E}}{(Q)} and \\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} - 1 = \\operatorname{M_{E}}{(Q)} - 1 and \\frac{\\operatorname{M_{E}}^{4}{(Q)}}{\\log{(Q)}^{3}} - 1 = \\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} - 1 and \\frac{\\operatorname{M_{E}}^{4}{(Q)}}{\\log{(Q)}^{3}} - 1 = \\operatorname{M_{E}}{(Q)} - 1 and (\\frac{\\operatorname{M_{E}}^{4}{(Q)}}{\\log{(Q)}^{3}} - 1)^{2} = (\\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} - 1)^{2} and (\\operatorname{M_{E}}{(Q)} - 1)^{2} = (\\frac{\\operatorname{M_{E}}^{2}{(Q)}}{\\log{(Q)}} - 1)^{2}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["divide", 1, "log(Symbol('Q', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Function('M_E')(Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(2)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Function('M_E')(Symbol('Q', commutative=True)))"], [["minus", 3, 1], "Equality(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(2)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(-1)), Add(Function('M_E')(Symbol('Q', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(4)), Pow(log(Symbol('Q', commutative=True)), Integer(-3))), Integer(-1)), Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(2)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(4)), Pow(log(Symbol('Q', commutative=True)), Integer(-3))), Integer(-1)), Add(Function('M_E')(Symbol('Q', commutative=True)), Integer(-1)))"], [["power", 5, 2], "Equality(Pow(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(4)), Pow(log(Symbol('Q', commutative=True)), Integer(-3))), Integer(-1)), Integer(2)), Pow(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(2)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(-1)), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Add(Function('M_E')(Symbol('Q', commutative=True)), Integer(-1)), Integer(2)), Pow(Add(Mul(Pow(Function('M_E')(Symbol('Q', commutative=True)), Integer(2)), Pow(log(Symbol('Q', commutative=True)), Integer(-1))), Integer(-1)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(k,\\mathbf{J})} = \\mathbf{J} k and \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = \\mathbf{J} k - \\operatorname{v_{z}}{(k,\\mathbf{J})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} - \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} 0", "derivation": "\\operatorname{v_{z}}{(k,\\mathbf{J})} = \\mathbf{J} k and \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = \\mathbf{J} k - \\operatorname{v_{z}}{(k,\\mathbf{J})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = 0 and - \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = 0 and \\frac{\\partial}{\\partial \\mathbf{J}} - \\operatorname{g^{\\prime}_{\\varepsilon}}{(k,\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} 0", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(0))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(A_{x},\\Omega)} = - A_{x} + \\log{(\\Omega)} and \\operatorname{P_{e}}{(\\Omega)} = \\log{(\\Omega)}, then obtain e^{\\cos{(B{(A_{x},\\Omega)})}} = e^{\\cos{(A_{x} - \\operatorname{P_{e}}{(\\Omega)})}}", "derivation": "B{(A_{x},\\Omega)} = - A_{x} + \\log{(\\Omega)} and \\cos{(B{(A_{x},\\Omega)})} = \\cos{(A_{x} - \\log{(\\Omega)})} and e^{\\cos{(B{(A_{x},\\Omega)})}} = e^{\\cos{(A_{x} - \\log{(\\Omega)})}} and \\operatorname{P_{e}}{(\\Omega)} = \\log{(\\Omega)} and e^{\\cos{(B{(A_{x},\\Omega)})}} = e^{\\cos{(A_{x} - \\operatorname{P_{e}}{(\\Omega)})}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('A_x', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), log(Symbol('\\\\Omega', commutative=True))))"], [["cos", 1], "Equality(cos(Function('B')(Symbol('A_x', commutative=True), Symbol('\\\\Omega', commutative=True))), cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\Omega', commutative=True))))))"], [["exp", 2], "Equality(exp(cos(Function('B')(Symbol('A_x', commutative=True), Symbol('\\\\Omega', commutative=True)))), exp(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\Omega', commutative=True)))))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(exp(cos(Function('B')(Symbol('A_x', commutative=True), Symbol('\\\\Omega', commutative=True)))), exp(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('\\\\Omega', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(b)} = \\cos{(b)}, then obtain \\cos^{- \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} - 1}{(b)} + \\cos{(\\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}})} = \\cos^{- \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} - 1}{(b)} + \\cos{(1)}", "derivation": "\\operatorname{v_{1}}{(b)} = \\cos{(b)} and \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} = 1 and \\frac{\\operatorname{v_{1}}^{2}{(b)}}{\\cos^{2}{(b)}} = \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} and \\frac{\\operatorname{v_{1}}^{2}{(b)}}{\\cos^{2}{(b)}} = 1 and \\cos{(\\frac{\\operatorname{v_{1}}^{2}{(b)}}{\\cos^{2}{(b)}})} = \\cos{(1)} and \\cos{(\\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}})} = \\cos{(1)} and \\cos^{- \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} - 1}{(b)} + \\cos{(\\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}})} = \\cos^{- \\frac{\\operatorname{v_{1}}{(b)}}{\\cos{(b)}} - 1}{(b)} + \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["divide", 1, "cos(Symbol('b', commutative=True))"], "Equality(Mul(Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Mul(Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('v_1')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(Symbol('b', commutative=True)), Integer(-2))), Mul(Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('v_1')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(Symbol('b', commutative=True)), Integer(-2))), Integer(1))"], [["cos", 4], "Equality(cos(Mul(Pow(Function('v_1')(Symbol('b', commutative=True)), Integer(2)), Pow(cos(Symbol('b', commutative=True)), Integer(-2)))), cos(Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(cos(Mul(Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1)))), cos(Integer(1)))"], [["add", 6, "Pow(cos(Symbol('b', commutative=True)), Add(Mul(Integer(-1), Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Integer(-1)))"], "Equality(Add(Pow(cos(Symbol('b', commutative=True)), Add(Mul(Integer(-1), Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Integer(-1))), cos(Mul(Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))))), Add(Pow(cos(Symbol('b', commutative=True)), Add(Mul(Integer(-1), Function('v_1')(Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Integer(-1))), Integer(-1))), cos(Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = \\ddot{x}^{c} - \\hat{H}_{\\lambda}, then obtain \\ddot{x}^{c} \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = \\ddot{x}^{c} \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} + \\ddot{x}^{c} - 2 \\hat{H}_{\\lambda}", "derivation": "\\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = \\ddot{x}^{c} - \\hat{H}_{\\lambda} and \\ddot{x}^{c} - \\hat{H}_{\\lambda} + \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = 2 \\ddot{x}^{c} - 2 \\hat{H}_{\\lambda} and - \\hat{H}_{\\lambda} + \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = \\ddot{x}^{c} - 2 \\hat{H}_{\\lambda} and \\ddot{x}^{c} \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} - \\hat{H}_{\\lambda} + \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} = \\ddot{x}^{c} \\sigma_{x}{(c,\\ddot{x},\\hat{H}_{\\lambda})} + \\ddot{x}^{c} - 2 \\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["add", 1, "Add(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Add(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Function('\\\\sigma_x')(Symbol('c', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(B)} = \\sin{(e^{B})}, then obtain \\frac{\\operatorname{c_{0}}{(B)} \\sin^{B}{(e^{B})}}{B} = \\frac{\\sin{(e^{B})} \\sin^{B}{(e^{B})}}{B}", "derivation": "\\operatorname{c_{0}}{(B)} = \\sin{(e^{B})} and \\operatorname{c_{0}}^{B}{(B)} = \\sin^{B}{(e^{B})} and \\frac{\\operatorname{c_{0}}^{B}{(B)}}{B} = \\frac{\\sin^{B}{(e^{B})}}{B} and \\frac{\\operatorname{c_{0}}{(B)} \\operatorname{c_{0}}^{B}{(B)}}{B} = \\frac{\\operatorname{c_{0}}^{B}{(B)} \\sin{(e^{B})}}{B} and \\frac{\\operatorname{c_{0}}{(B)} \\sin^{B}{(e^{B})}}{B} = \\frac{\\sin{(e^{B})} \\sin^{B}{(e^{B})}}{B}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('B', commutative=True)), sin(exp(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(sin(exp(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["divide", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Function('c_0')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(sin(exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Function('c_0')(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('c_0')(Symbol('B', commutative=True)), Pow(Function('c_0')(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Function('c_0')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), sin(exp(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('c_0')(Symbol('B', commutative=True)), Pow(sin(exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(exp(Symbol('B', commutative=True))), Pow(sin(exp(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(P_{e},r,H)} = H + P_{e} + r and \\operatorname{v_{2}}{(p)} = \\sin{(p)} and \\hat{x}{(P_{e},r,H)} = (H + P_{e} + r)^{P_{e}}, then obtain \\hat{x}{(P_{e},r,H)} \\sin{(p)} = \\dot{\\mathbf{r}}^{P_{e}}{(P_{e},r,H)} \\sin{(p)}", "derivation": "\\dot{\\mathbf{r}}{(P_{e},r,H)} = H + P_{e} + r and \\dot{\\mathbf{r}}^{P_{e}}{(P_{e},r,H)} = (H + P_{e} + r)^{P_{e}} and \\operatorname{v_{2}}{(p)} = \\sin{(p)} and \\hat{x}{(P_{e},r,H)} = (H + P_{e} + r)^{P_{e}} and \\hat{x}{(P_{e},r,H)} = \\dot{\\mathbf{r}}^{P_{e}}{(P_{e},r,H)} and \\hat{x}{(P_{e},r,H)} \\operatorname{v_{2}}{(p)} = \\dot{\\mathbf{r}}^{P_{e}}{(P_{e},r,H)} \\operatorname{v_{2}}{(p)} and \\hat{x}{(P_{e},r,H)} \\sin{(p)} = \\dot{\\mathbf{r}}^{P_{e}}{(P_{e},r,H)} \\sin{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('P_e', commutative=True), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Symbol('P_e', commutative=True)), Pow(Add(Symbol('H', commutative=True), Symbol('P_e', commutative=True), Symbol('r', commutative=True)), Symbol('P_e', commutative=True)))"], ["get_premise", "Equality(Function('v_2')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Pow(Add(Symbol('H', commutative=True), Symbol('P_e', commutative=True), Symbol('r', commutative=True)), Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\hat{x}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Symbol('P_e', commutative=True)))"], [["times", 5, "Function('v_2')(Symbol('p', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Function('v_2')(Symbol('p', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Symbol('P_e', commutative=True)), Function('v_2')(Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Function('\\\\hat{x}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('P_e', commutative=True), Symbol('r', commutative=True), Symbol('H', commutative=True)), Symbol('P_e', commutative=True)), sin(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(P_{g},\\theta_2)} = \\cos{(P_{g} - \\theta_2)} and \\rho_{b}{(P_{g},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\operatorname{f^{*}}{(P_{g},\\theta_2)})^{\\theta_2}, then obtain \\rho_{b}{(P_{g},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\cos{(P_{g} - \\theta_2)})^{\\theta_2}", "derivation": "\\operatorname{f^{*}}{(P_{g},\\theta_2)} = \\cos{(P_{g} - \\theta_2)} and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{f^{*}}{(P_{g},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\cos{(P_{g} - \\theta_2)} and \\rho_{b}{(P_{g},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\operatorname{f^{*}}{(P_{g},\\theta_2)})^{\\theta_2} and \\rho_{b}{(P_{g},\\theta_2)} = (\\frac{\\partial}{\\partial \\theta_2} \\cos{(P_{g} - \\theta_2)})^{\\theta_2}", "srepr_derivation": [["get_premise", "Equality(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('P_g', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Function('f^*')(Symbol('P_g', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\rho_b')(Symbol('P_g', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(cos(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = \\hat{x}_0 + \\log{(\\Psi)}, then derive \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial \\Psi} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = 0, then obtain \\frac{\\partial^{3}}{\\partial \\hat{x}_0\\partial \\Psi^{2}} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = 0", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = \\hat{x}_0 + \\log{(\\Psi)} and \\frac{\\partial}{\\partial \\Psi} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = \\frac{\\partial}{\\partial \\Psi} (\\hat{x}_0 + \\log{(\\Psi)}) and \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial \\Psi} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial \\Psi} (\\hat{x}_0 + \\log{(\\Psi)}) and \\frac{\\partial^{2}}{\\partial \\hat{x}_0\\partial \\Psi} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = 0 and \\frac{\\partial^{3}}{\\partial \\Psi\\partial \\hat{x}_0\\partial \\Psi} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = \\frac{d}{d \\Psi} 0 and \\frac{\\partial^{3}}{\\partial \\hat{x}_0\\partial \\Psi^{2}} \\operatorname{V_{\\mathbf{E}}}{(\\hat{x}_0,\\Psi)} = 0", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)} = \\mathbf{J}_f (\\omega - \\phi) and \\nabla{(\\omega,\\phi)} = \\frac{1}{\\omega - \\phi}, then obtain \\frac{\\mathbf{s}^{2}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f^{2} (\\omega - \\phi)} = \\frac{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f}", "derivation": "\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)} = \\mathbf{J}_f (\\omega - \\phi) and \\frac{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f (\\omega - \\phi)} = 1 and \\nabla{(\\omega,\\phi)} = \\frac{1}{\\omega - \\phi} and \\frac{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f (\\omega - \\phi) \\nabla{(\\omega,\\phi)}} = \\frac{1}{\\nabla{(\\omega,\\phi)}} and \\frac{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f} = \\omega - \\phi and \\nabla{(\\omega,\\phi)} = \\frac{\\mathbf{J}_f}{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}} and \\frac{\\mathbf{s}^{2}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f^{2} (\\omega - \\phi)} = \\frac{\\mathbf{s}{(\\omega,\\mathbf{J}_f,\\phi)}}{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Integer(-1)))"], [["divide", 2, "Function('\\\\nabla')(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\nabla')(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Pow(Function('\\\\nabla')(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\nabla')(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Integer(-1)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(F_{N},\\mathbf{A})} = \\frac{e^{\\mathbf{A}}}{F_{N}} and \\mathbf{S}{(F_{N},\\mathbf{A})} = \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N}, then obtain \\mathbf{S}{(F_{N},\\mathbf{A})} + \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} - 1 = \\int \\frac{e^{\\mathbf{A}}}{F_{N}} dF_{N} + \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} - 1", "derivation": "\\Psi_{nl}{(F_{N},\\mathbf{A})} = \\frac{e^{\\mathbf{A}}}{F_{N}} and \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} = \\int \\frac{e^{\\mathbf{A}}}{F_{N}} dF_{N} and \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} - 1 = \\int \\frac{e^{\\mathbf{A}}}{F_{N}} dF_{N} - 1 and \\mathbf{S}{(F_{N},\\mathbf{A})} = \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} and \\mathbf{S}{(F_{N},\\mathbf{A})} - 1 = \\int \\frac{e^{\\mathbf{A}}}{F_{N}} dF_{N} - 1 and \\mathbf{S}{(F_{N},\\mathbf{A})} + \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} - 1 = \\int \\frac{e^{\\mathbf{A}}}{F_{N}} dF_{N} + \\int \\Psi_{nl}{(F_{N},\\mathbf{A})} dF_{N} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)), Add(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Add(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(-1)))"], [["add", 5, "Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)), Add(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(a^{\\dagger},\\mathbb{I})} = \\mathbb{I} + a^{\\dagger}, then obtain (\\mathbb{I} + a^{\\dagger})^{2} = - (- \\mathbb{I} - a^{\\dagger}) (\\mathbb{I} + a^{\\dagger})", "derivation": "\\operatorname{A_{z}}{(a^{\\dagger},\\mathbb{I})} = \\mathbb{I} + a^{\\dagger} and - \\operatorname{A_{z}}{(a^{\\dagger},\\mathbb{I})} = - \\mathbb{I} - a^{\\dagger} and \\operatorname{A_{z}}^{2}{(a^{\\dagger},\\mathbb{I})} = - (- \\mathbb{I} - a^{\\dagger}) \\operatorname{A_{z}}{(a^{\\dagger},\\mathbb{I})} and (\\mathbb{I} + a^{\\dagger})^{2} = - (- \\mathbb{I} - a^{\\dagger}) (\\mathbb{I} + a^{\\dagger})", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Pow(Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('A_z')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given k{(E,\\sigma_p)} = \\frac{\\sin{(\\sigma_p)}}{E} and V{(E,\\sigma_p)} = \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})}, then obtain \\frac{(- L + \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})})^{\\sigma_p}}{\\sin{(\\frac{\\sin{(\\sigma_p)}}{E})}} = \\frac{(- L + \\sin{(k{(E,\\sigma_p)})})^{\\sigma_p}}{\\sin{(\\frac{\\sin{(\\sigma_p)}}{E})}}", "derivation": "k{(E,\\sigma_p)} = \\frac{\\sin{(\\sigma_p)}}{E} and V{(E,\\sigma_p)} = \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})} and V{(E,\\sigma_p)} = \\sin{(k{(E,\\sigma_p)})} and - L + V{(E,\\sigma_p)} = - L + \\sin{(k{(E,\\sigma_p)})} and - L + \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})} = - L + \\sin{(k{(E,\\sigma_p)})} and (- L + \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})})^{\\sigma_p} = (- L + \\sin{(k{(E,\\sigma_p)})})^{\\sigma_p} and \\frac{(- L + \\sin{(\\frac{\\sin{(\\sigma_p)}}{E})})^{\\sigma_p}}{\\sin{(\\frac{\\sin{(\\sigma_p)}}{E})}} = \\frac{(- L + \\sin{(k{(E,\\sigma_p)})})^{\\sigma_p}}{\\sin{(\\frac{\\sin{(\\sigma_p)}}{E})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('V')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)), sin(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 3, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('V')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["power", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 6, "sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True))))), Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True)))), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('L', commutative=True)), sin(Function('k')(Symbol('E', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('\\\\sigma_p', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\eta{(f_{\\mathbf{v}})} = \\cos{(\\sin{(f_{\\mathbf{v}})})}, then obtain - \\frac{\\eta^{2}{(f_{\\mathbf{v}})}}{\\cos{(\\sin{(f_{\\mathbf{v}})})}} = - \\cos{(\\sin{(f_{\\mathbf{v}})})}", "derivation": "\\eta{(f_{\\mathbf{v}})} = \\cos{(\\sin{(f_{\\mathbf{v}})})} and - \\eta{(f_{\\mathbf{v}})} = - \\cos{(\\sin{(f_{\\mathbf{v}})})} and - \\frac{\\eta^{2}{(f_{\\mathbf{v}})}}{\\cos{(\\sin{(f_{\\mathbf{v}})})}} = - \\eta{(f_{\\mathbf{v}})} and - \\frac{\\eta^{2}{(f_{\\mathbf{v}})}}{\\cos{(\\sin{(f_{\\mathbf{v}})})}} = - \\cos{(\\sin{(f_{\\mathbf{v}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(-1), cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["divide", 2, "Mul(Pow(Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1))), Mul(Integer(-1), Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Pow(cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(g_{\\varepsilon},c)} = \\frac{\\partial}{\\partial g_{\\varepsilon}} c^{g_{\\varepsilon}}, then derive \\mathbf{P}{(g_{\\varepsilon},c)} = c^{g_{\\varepsilon}} \\log{(c)}, then obtain c^{g_{\\varepsilon}} + \\mathbf{P}^{c}{(g_{\\varepsilon},c)} = c^{g_{\\varepsilon}} + (c^{g_{\\varepsilon}} \\log{(c)})^{c}", "derivation": "\\mathbf{P}{(g_{\\varepsilon},c)} = \\frac{\\partial}{\\partial g_{\\varepsilon}} c^{g_{\\varepsilon}} and \\mathbf{P}{(g_{\\varepsilon},c)} = c^{g_{\\varepsilon}} \\log{(c)} and \\mathbf{P}^{c}{(g_{\\varepsilon},c)} = (c^{g_{\\varepsilon}} \\log{(c)})^{c} and c^{g_{\\varepsilon}} + \\mathbf{P}^{c}{(g_{\\varepsilon},c)} = c^{g_{\\varepsilon}} + (c^{g_{\\varepsilon}} \\log{(c)})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c', commutative=True)), Derivative(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('c', commutative=True))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Mul(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["add", 3, "Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Mul(Pow(Symbol('c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('c', commutative=True))), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} = \\sin{(\\chi \\phi_1)}, then obtain 0 = - \\frac{2 (- 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + 2 \\sin{(\\chi \\phi_1)}) \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)}}{\\chi \\phi_1}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} = \\sin{(\\chi \\phi_1)} and 0 = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + \\sin{(\\chi \\phi_1)} and - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + \\sin{(\\chi \\phi_1)} = - 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + 2 \\sin{(\\chi \\phi_1)} and 0 = - 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + 2 \\sin{(\\chi \\phi_1)} and 0 = - 2 (- 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + 2 \\sin{(\\chi \\phi_1)}) \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} and 0 = - \\frac{2 (- 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)} + 2 \\sin{(\\chi \\phi_1)}) \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_1,\\chi)}}{\\chi \\phi_1}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))))"], [["times", 4, "Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["divide", 5, "Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi_1', commutative=True))))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})}, then obtain \\cos^{f_{\\mathbf{p}}}{(\\int \\Psi_{nl}{(f_{\\mathbf{p}})} df_{\\mathbf{p}})} = \\cos^{f_{\\mathbf{p}}}{(\\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}})}", "derivation": "\\Psi_{nl}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\int \\Psi_{nl}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}} and \\cos{(\\int \\Psi_{nl}{(f_{\\mathbf{p}})} df_{\\mathbf{p}})} = \\cos{(\\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}})} and \\cos^{f_{\\mathbf{p}}}{(\\int \\Psi_{nl}{(f_{\\mathbf{p}})} df_{\\mathbf{p}})} = \\cos^{f_{\\mathbf{p}}}{(\\int \\cos{(f_{\\mathbf{p}})} df_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), cos(Integral(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["power", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(cos(Integral(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(cos(Integral(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(t_{2},\\varphi)} = \\log{(\\varphi + t_{2})}, then obtain \\frac{d}{d t_{2}} 1 = \\frac{\\partial}{\\partial t_{2}} (- \\mathbf{J}_f{(t_{2},\\varphi)} + \\log{(\\varphi + t_{2})} + 1)", "derivation": "\\mathbf{J}_f{(t_{2},\\varphi)} = \\log{(\\varphi + t_{2})} and \\mathbf{J}_f{(t_{2},\\varphi)} - \\log{(\\varphi + t_{2})} = 0 and \\mathbf{J}_f{(t_{2},\\varphi)} - \\log{(\\varphi + t_{2})} + 1 = 1 and 1 = - \\mathbf{J}_f{(t_{2},\\varphi)} + \\log{(\\varphi + t_{2})} + 1 and \\frac{d}{d t_{2}} 1 = \\frac{\\partial}{\\partial t_{2}} (- \\mathbf{J}_f{(t_{2},\\varphi)} + \\log{(\\varphi + t_{2})} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True))))"], [["minus", 1, "log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True))))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)))), Integer(1)), Integer(1))"], [["minus", 3, "Add(Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True)))))"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True))), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True))), Integer(1)))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('t_2', commutative=True), Symbol('\\\\varphi', commutative=True))), log(Add(Symbol('\\\\varphi', commutative=True), Symbol('t_2', commutative=True))), Integer(1)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(k,F_{N})} = F_{N} + \\log{(k)} and \\operatorname{m_{s}}{(k,F_{N})} = \\frac{\\rho_{b}{(k,F_{N})}}{F_{N} + \\log{(k)}}, then obtain - \\int \\operatorname{m_{s}}{(k,F_{N})} dk - 1 = - \\int 1 dk - 1", "derivation": "\\rho_{b}{(k,F_{N})} = F_{N} + \\log{(k)} and \\frac{\\rho_{b}{(k,F_{N})}}{F_{N} + \\log{(k)}} = 1 and \\operatorname{m_{s}}{(k,F_{N})} = \\frac{\\rho_{b}{(k,F_{N})}}{F_{N} + \\log{(k)}} and \\operatorname{m_{s}}{(k,F_{N})} = 1 and \\int \\operatorname{m_{s}}{(k,F_{N})} dk = \\int 1 dk and - \\int \\operatorname{m_{s}}{(k,F_{N})} dk = - \\int 1 dk and - \\int \\operatorname{m_{s}}{(k,F_{N})} dk - 1 = - \\int 1 dk - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), log(Symbol('k', commutative=True))))"], [["divide", 1, "Add(Symbol('F_N', commutative=True), log(Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('F_N', commutative=True), log(Symbol('k', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('k', commutative=True), Symbol('F_N', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Mul(Pow(Add(Symbol('F_N', commutative=True), log(Symbol('k', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('k', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('m_s')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Integer(1), Tuple(Symbol('k', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('m_s')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('k', commutative=True)))))"], [["minus", 6, 1], "Equality(Add(Mul(Integer(-1), Integral(Function('m_s')(Symbol('k', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('k', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('k', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\varphi^{*}{(c_{0})} = \\log{(c_{0})}, then obtain \\frac{1}{\\varphi^{*}{(c_{0})} \\log{(c_{0})}^{2}} = \\frac{1}{\\varphi^{*}^{3}{(c_{0})}}", "derivation": "\\varphi^{*}{(c_{0})} = \\log{(c_{0})} and \\varphi^{*}^{2}{(c_{0})} = \\varphi^{*}{(c_{0})} \\log{(c_{0})} and 1 = \\frac{\\log{(c_{0})}}{\\varphi^{*}{(c_{0})}} and \\frac{1}{\\varphi^{*}{(c_{0})} \\log{(c_{0})}} = \\frac{1}{\\varphi^{*}^{2}{(c_{0})}} and \\frac{1}{\\varphi^{*}^{2}{(c_{0})} \\log{(c_{0})}^{2}} = \\frac{1}{\\varphi^{*}^{3}{(c_{0})} \\log{(c_{0})}} and \\frac{1}{\\varphi^{*}^{2}{(c_{0})} \\log{(c_{0})}} = \\frac{1}{\\varphi^{*}^{3}{(c_{0})}} and \\frac{1}{\\varphi^{*}{(c_{0})} \\log{(c_{0})}^{2}} = \\frac{1}{\\varphi^{*}^{3}{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["times", 1, "Function('\\\\varphi^*')(Symbol('c_0', commutative=True))"], "Equality(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(2)), Mul(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["divide", 1, "Function('\\\\varphi^*')(Symbol('c_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-1)), log(Symbol('c_0', commutative=True))))"], [["divide", 3, "Mul(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-2)))"], [["divide", 4, "Mul(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-2)), Pow(log(Symbol('c_0', commutative=True)), Integer(-2))), Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-3)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))))"], [["divide", 5, "Pow(log(Symbol('c_0', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-2)), Pow(log(Symbol('c_0', commutative=True)), Integer(-1))), Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-3)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-1)), Pow(log(Symbol('c_0', commutative=True)), Integer(-2))), Pow(Function('\\\\varphi^*')(Symbol('c_0', commutative=True)), Integer(-3)))"]]}, {"prompt": "Given \\mathbf{J}_f{(m,u)} = \\cos^{m}{(u)}, then obtain - \\mathbf{J}_f{(m,u)} + (\\int \\mathbf{J}_f{(m,u)} dm) \\int \\cos^{m}{(u)} dm = - \\cos^{m}{(u)} + (\\int \\mathbf{J}_f{(m,u)} dm) \\int \\cos^{m}{(u)} dm", "derivation": "\\mathbf{J}_f{(m,u)} = \\cos^{m}{(u)} and \\int \\mathbf{J}_f{(m,u)} dm = \\int \\cos^{m}{(u)} dm and (\\int \\mathbf{J}_f{(m,u)} dm)^{2} = (\\int \\mathbf{J}_f{(m,u)} dm) \\int \\cos^{m}{(u)} dm and \\mathbf{J}_f{(m,u)} - (\\int \\mathbf{J}_f{(m,u)} dm)^{2} = \\cos^{m}{(u)} - (\\int \\mathbf{J}_f{(m,u)} dm)^{2} and - \\mathbf{J}_f{(m,u)} + (\\int \\mathbf{J}_f{(m,u)} dm)^{2} = - \\cos^{m}{(u)} + (\\int \\mathbf{J}_f{(m,u)} dm)^{2} and - \\mathbf{J}_f{(m,u)} + (\\int \\mathbf{J}_f{(m,u)} dm) \\int \\cos^{m}{(u)} dm = - \\cos^{m}{(u)} + (\\int \\mathbf{J}_f{(m,u)} dm) \\int \\cos^{m}{(u)} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["times", 2, "Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2)), Mul(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["minus", 1, "Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2)))), Add(Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2)))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True))), Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2))), Add(Mul(Integer(-1), Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True))), Pow(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True))), Mul(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True))), Mul(Integral(Function('\\\\mathbf{J}_f')(Symbol('m', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(cos(Symbol('u', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))))"]]}, {"prompt": "Given h{(M_{E})} = e^{M_{E}} and \\Psi{(M_{E})} = \\frac{d}{d M_{E}} (h{(M_{E})} e^{M_{E}})^{M_{E}}, then obtain \\Psi{(M_{E})} + e^{2 M_{E}} (e^{2 M_{E}})^{M_{E}} = e^{2 M_{E}} (e^{2 M_{E}})^{M_{E}} + \\frac{d}{d M_{E}} (e^{2 M_{E}})^{M_{E}}", "derivation": "h{(M_{E})} = e^{M_{E}} and h{(M_{E})} e^{M_{E}} = e^{2 M_{E}} and (h{(M_{E})} e^{M_{E}})^{M_{E}} = (e^{2 M_{E}})^{M_{E}} and \\Psi{(M_{E})} = \\frac{d}{d M_{E}} (h{(M_{E})} e^{M_{E}})^{M_{E}} and (h{(M_{E})} e^{M_{E}})^{M_{E}} e^{2 M_{E}} + \\Psi{(M_{E})} = (h{(M_{E})} e^{M_{E}})^{M_{E}} e^{2 M_{E}} + \\frac{d}{d M_{E}} (h{(M_{E})} e^{M_{E}})^{M_{E}} and \\Psi{(M_{E})} + e^{2 M_{E}} (e^{2 M_{E}})^{M_{E}} = e^{2 M_{E}} (e^{2 M_{E}})^{M_{E}} + \\frac{d}{d M_{E}} (e^{2 M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["times", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), exp(Mul(Integer(2), Symbol('M_E', commutative=True))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('M_E', commutative=True)), Derivative(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["add", 4, "Mul(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), exp(Mul(Integer(2), Symbol('M_E', commutative=True))))"], "Equality(Add(Mul(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), exp(Mul(Integer(2), Symbol('M_E', commutative=True)))), Function('\\\\Psi')(Symbol('M_E', commutative=True))), Add(Mul(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), exp(Mul(Integer(2), Symbol('M_E', commutative=True)))), Derivative(Pow(Mul(Function('h')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('\\\\Psi')(Symbol('M_E', commutative=True)), Mul(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Pow(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))), Add(Mul(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Pow(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))), Derivative(Pow(exp(Mul(Integer(2), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(x,A)} = - A + x, then derive (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = (-1)^{x}, then obtain - (-1)^{x} + \\operatorname{F_{c}}{(x,A)} + (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = \\operatorname{F_{c}}{(x,A)}", "derivation": "G{(x,A)} = - A + x and \\frac{\\partial}{\\partial A} G{(x,A)} = \\frac{\\partial}{\\partial A} (- A + x) and (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = (\\frac{\\partial}{\\partial A} (- A + x))^{x} and (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = (-1)^{x} and (-1)^{x} = (\\frac{\\partial}{\\partial A} (- A + x))^{x} and (-1)^{x} - (\\frac{\\partial}{\\partial A} (- A + x))^{x} = 0 and - (\\frac{\\partial}{\\partial A} (- A + x))^{x} + (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = 0 and \\operatorname{F_{c}}{(x,A)} - (\\frac{\\partial}{\\partial A} (- A + x))^{x} + (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = \\operatorname{F_{c}}{(x,A)} and - (-1)^{x} + \\operatorname{F_{c}}{(x,A)} + (\\frac{\\partial}{\\partial A} G{(x,A)})^{x} = \\operatorname{F_{c}}{(x,A)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Integer(-1), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Integer(-1), Symbol('x', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["minus", 5, "Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))"], "Equality(Add(Pow(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))), Pow(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))), Integer(0))"], [["add", 7, "Function('F_c')(Symbol('x', commutative=True), Symbol('A', commutative=True))"], "Equality(Add(Function('F_c')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))), Pow(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))), Function('F_c')(Symbol('x', commutative=True), Symbol('A', commutative=True)))"], [["evaluate_derivatives", 8], "Equality(Add(Mul(Integer(-1), Pow(Integer(-1), Symbol('x', commutative=True))), Function('F_c')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Pow(Derivative(Function('G')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('x', commutative=True))), Function('F_c')(Symbol('x', commutative=True), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f,Q)} = \\int f^{Q} df and \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} = \\ddot{x}^{V_{\\mathbf{E}}}, then obtain \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} \\int \\hat{\\mathbf{r}}{(f,Q)} dQ = \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} \\iint f^{Q} df dQ", "derivation": "\\hat{\\mathbf{r}}{(f,Q)} = \\int f^{Q} df and \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} = \\ddot{x}^{V_{\\mathbf{E}}} and \\int \\hat{\\mathbf{r}}{(f,Q)} dQ = \\iint f^{Q} df dQ and \\ddot{x}^{V_{\\mathbf{E}}} \\int \\hat{\\mathbf{r}}{(f,Q)} dQ = \\ddot{x}^{V_{\\mathbf{E}}} \\iint f^{Q} df dQ and \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} \\int \\hat{\\mathbf{r}}{(f,Q)} dQ = \\rho_{f}{(\\ddot{x},V_{\\mathbf{E}})} \\iint f^{Q} df dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Integral(Pow(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('f', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["times", 3, "Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Pow(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('\\\\rho_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Mul(Function('\\\\rho_f')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Pow(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given p{(c,\\chi,H)} = H + \\chi + c, then obtain H + (p^{c}{(c,\\chi,H)})^{c} + \\int p^{c}{(c,\\chi,H)} dH = H + ((H + \\chi + c)^{c})^{c} + \\int p^{c}{(c,\\chi,H)} dH", "derivation": "p{(c,\\chi,H)} = H + \\chi + c and p^{c}{(c,\\chi,H)} = (H + \\chi + c)^{c} and (p^{c}{(c,\\chi,H)})^{c} = ((H + \\chi + c)^{c})^{c} and \\int p^{c}{(c,\\chi,H)} dH = \\int (H + \\chi + c)^{c} dH and H + (p^{c}{(c,\\chi,H)})^{c} = H + ((H + \\chi + c)^{c})^{c} and H + (p^{c}{(c,\\chi,H)})^{c} + \\int (H + \\chi + c)^{c} dH = H + ((H + \\chi + c)^{c})^{c} + \\int (H + \\chi + c)^{c} dH and H + (p^{c}{(c,\\chi,H)})^{c} + \\int p^{c}{(c,\\chi,H)} dH = H + ((H + \\chi + c)^{c})^{c} + \\int p^{c}{(c,\\chi,H)} dH", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["add", 3, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Pow(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Add(Symbol('H', commutative=True), Pow(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], [["add", 5, "Integral(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Symbol('H', commutative=True), Pow(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integral(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), Pow(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integral(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('H', commutative=True), Pow(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integral(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), Pow(Pow(Add(Symbol('H', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Integral(Pow(Function('p')(Symbol('c', commutative=True), Symbol('\\\\chi', commutative=True), Symbol('H', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(t)} = \\cos{(t)}, then derive (\\Psi + \\sin{(t)}) \\int \\dot{y}{(t)} dt = (\\Psi + \\sin{(t)})^{2}, then obtain (\\Psi + \\sin{(t)}) \\int \\cos{(t)} dt = (\\Psi + \\sin{(t)})^{2}", "derivation": "\\dot{y}{(t)} = \\cos{(t)} and \\int \\dot{y}{(t)} dt = \\int \\cos{(t)} dt and (\\int \\dot{y}{(t)} dt) \\int \\cos{(t)} dt = (\\int \\cos{(t)} dt)^{2} and (\\Psi + \\sin{(t)}) \\int \\dot{y}{(t)} dt = (\\Psi + \\sin{(t)})^{2} and (\\Psi + \\sin{(t)}) \\int \\cos{(t)} dt = (\\Psi + \\sin{(t)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 2, "Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\dot{y}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Pow(Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('t', commutative=True))), Integral(Function('\\\\dot{y}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Pow(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('t', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('t', commutative=True))), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Pow(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('t', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\dot{x})} = e^{\\dot{x}} and \\operatorname{F_{c}}{(\\dot{x},P_{e})} = \\iint e^{\\dot{x}} d\\dot{x} dP_{e}, then derive \\int \\operatorname{P_{e}}{(\\dot{x})} d\\dot{x} = P_{e} + e^{\\dot{x}}, then obtain \\operatorname{F_{c}}{(\\dot{x},P_{e})} = \\int (P_{e} + e^{\\dot{x}}) dP_{e}", "derivation": "\\operatorname{P_{e}}{(\\dot{x})} = e^{\\dot{x}} and \\int \\operatorname{P_{e}}{(\\dot{x})} d\\dot{x} = \\int e^{\\dot{x}} d\\dot{x} and \\int \\operatorname{P_{e}}{(\\dot{x})} d\\dot{x} = P_{e} + e^{\\dot{x}} and \\iint \\operatorname{P_{e}}{(\\dot{x})} d\\dot{x} dP_{e} = \\int (P_{e} + e^{\\dot{x}}) dP_{e} and \\iint e^{\\dot{x}} d\\dot{x} dP_{e} = \\int (P_{e} + e^{\\dot{x}}) dP_{e} and \\operatorname{F_{c}}{(\\dot{x},P_{e})} = \\iint e^{\\dot{x}} d\\dot{x} dP_{e} and \\operatorname{F_{c}}{(\\dot{x},P_{e})} = \\int (P_{e} + e^{\\dot{x}}) dP_{e}", "srepr_derivation": [["get_premise", "Equality(Function('P_e')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Symbol('P_e', commutative=True), exp(Symbol('\\\\dot{x}', commutative=True))))"], [["integrate", 3, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\dot{x}', commutative=True), Symbol('P_e', commutative=True)), Integral(exp(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('F_c')(Symbol('\\\\dot{x}', commutative=True), Symbol('P_e', commutative=True)), Integral(Add(Symbol('P_e', commutative=True), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given H{(S,a,m_{s})} = a + m_{s}^{S}, then obtain - \\frac{a m_{s}^{- S} + 1}{S} = \\frac{m_{s}^{- S} (- a - m_{s}^{S})}{S}", "derivation": "H{(S,a,m_{s})} = a + m_{s}^{S} and m_{s}^{- S} H{(S,a,m_{s})} = m_{s}^{- S} (a + m_{s}^{S}) and - \\frac{m_{s}^{- S} H{(S,a,m_{s})}}{S} = - \\frac{m_{s}^{- S} (a + m_{s}^{S})}{S} and m_{s}^{- S} H{(S,a,m_{s})} = a m_{s}^{- S} + 1 and - \\frac{m_{s}^{- S} H{(S,a,m_{s})}}{S} = \\frac{m_{s}^{- S} (- a - m_{s}^{S})}{S} and - \\frac{a m_{s}^{- S} + 1}{S} = \\frac{m_{s}^{- S} (- a - m_{s}^{S})}{S}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('S', commutative=True), Symbol('a', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('a', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True))))"], [["divide", 1, "Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True))"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('H')(Symbol('S', commutative=True), Symbol('a', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Add(Symbol('a', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('H')(Symbol('S', commutative=True), Symbol('a', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Add(Symbol('a', commutative=True), Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True)))))"], [["expand", 2], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('H')(Symbol('S', commutative=True), Symbol('a', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Symbol('a', commutative=True), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Function('H')(Symbol('S', commutative=True), Symbol('a', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Symbol('a', commutative=True), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True)))), Integer(1))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Mul(Integer(-1), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Symbol('S', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} = e^{\\hat{x} - \\mathbf{D}}, then derive \\int \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} d\\mathbf{D} = F_{H} - e^{\\hat{x} - \\mathbf{D}}, then obtain \\int \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} d\\mathbf{D} = F_{H} - \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} = e^{\\hat{x} - \\mathbf{D}} and \\int \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} d\\mathbf{D} = \\int e^{\\hat{x} - \\mathbf{D}} d\\mathbf{D} and \\int \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} d\\mathbf{D} = F_{H} - e^{\\hat{x} - \\mathbf{D}} and \\int \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})} d\\mathbf{D} = F_{H} - \\operatorname{F_{c}}{(\\mathbf{D},\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), exp(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_c')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), exp(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('F_c')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given V{(y^{\\prime})} = \\cos{(e^{y^{\\prime}})}, then derive \\frac{d}{d y^{\\prime}} V{(y^{\\prime})} = - e^{y^{\\prime}} \\sin{(e^{y^{\\prime}})}, then obtain \\int \\frac{d}{d y^{\\prime}} \\cos{(e^{y^{\\prime}})} dy^{\\prime} = \\int - e^{y^{\\prime}} \\sin{(e^{y^{\\prime}})} dy^{\\prime}", "derivation": "V{(y^{\\prime})} = \\cos{(e^{y^{\\prime}})} and \\frac{d}{d y^{\\prime}} V{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\cos{(e^{y^{\\prime}})} and \\frac{d}{d y^{\\prime}} V{(y^{\\prime})} = - e^{y^{\\prime}} \\sin{(e^{y^{\\prime}})} and \\frac{d}{d y^{\\prime}} \\cos{(e^{y^{\\prime}})} = - e^{y^{\\prime}} \\sin{(e^{y^{\\prime}})} and \\int \\frac{d}{d y^{\\prime}} \\cos{(e^{y^{\\prime}})} dy^{\\prime} = \\int - e^{y^{\\prime}} \\sin{(e^{y^{\\prime}})} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('y^{\\\\prime}', commutative=True)), cos(exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('y^{\\\\prime}', commutative=True)), sin(exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('y^{\\\\prime}', commutative=True)), sin(exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["integrate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('y^{\\\\prime}', commutative=True)), sin(exp(Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(E_{x},L)} = \\frac{L}{E_{x}} and \\operatorname{t_{2}}{(t_{1})} = \\cos{(t_{1})}, then obtain \\frac{\\sin{(\\operatorname{F_{N}}{(E_{x},L)})} \\cos{(t_{1})}}{\\operatorname{F_{N}}{(E_{x},L)}} = \\frac{\\sin{(\\frac{L}{E_{x}})} \\cos{(t_{1})}}{\\operatorname{F_{N}}{(E_{x},L)}}", "derivation": "\\operatorname{F_{N}}{(E_{x},L)} = \\frac{L}{E_{x}} and \\sin{(\\operatorname{F_{N}}{(E_{x},L)})} = \\sin{(\\frac{L}{E_{x}})} and \\operatorname{t_{2}}{(t_{1})} = \\cos{(t_{1})} and \\frac{\\operatorname{t_{2}}{(t_{1})} \\sin{(\\operatorname{F_{N}}{(E_{x},L)})}}{\\operatorname{F_{N}}{(E_{x},L)}} = \\frac{\\operatorname{t_{2}}{(t_{1})} \\sin{(\\frac{L}{E_{x}})}}{\\operatorname{F_{N}}{(E_{x},L)}} and \\frac{\\sin{(\\operatorname{F_{N}}{(E_{x},L)})} \\cos{(t_{1})}}{\\operatorname{F_{N}}{(E_{x},L)}} = \\frac{\\sin{(\\frac{L}{E_{x}})} \\cos{(t_{1})}}{\\operatorname{F_{N}}{(E_{x},L)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('L', commutative=True)))"], [["sin", 1], "Equality(sin(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True))), sin(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('L', commutative=True))))"], ["get_premise", "Equality(Function('t_2')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["times", 2, "Mul(Pow(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Function('t_2')(Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Function('t_2')(Symbol('t_1', commutative=True)), sin(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)))), Mul(Pow(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Function('t_2')(Symbol('t_1', commutative=True)), sin(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Integer(-1)), sin(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True))), cos(Symbol('t_1', commutative=True))), Mul(Pow(Function('F_N')(Symbol('E_x', commutative=True), Symbol('L', commutative=True)), Integer(-1)), sin(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Symbol('L', commutative=True))), cos(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\phi{(V,\\nabla)} = \\log{(V + \\nabla)}, then obtain (V + \\nabla) \\phi^{2}{(V,\\nabla)} = (V + \\nabla) \\log{(V + \\nabla)}^{2}", "derivation": "\\phi{(V,\\nabla)} = \\log{(V + \\nabla)} and (V + \\nabla) \\phi{(V,\\nabla)} = (V + \\nabla) \\log{(V + \\nabla)} and (V + \\nabla) \\phi{(V,\\nabla)} \\log{(V + \\nabla)} = (V + \\nabla) \\log{(V + \\nabla)}^{2} and (V + \\nabla) \\phi^{2}{(V,\\nabla)} = (V + \\nabla) \\phi{(V,\\nabla)} \\log{(V + \\nabla)} and (V + \\nabla) \\phi^{2}{(V,\\nabla)} = (V + \\nabla) \\log{(V + \\nabla)}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["times", 2, "log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)))), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(2))), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Function('\\\\phi')(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(2))), Mul(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(log(Add(Symbol('V', commutative=True), Symbol('\\\\nabla', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(C_{1},\\mathbf{P})} = \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}), then derive \\theta_{1}{(C_{1},\\mathbf{P})} = 1, then obtain \\int \\theta_{1}^{2}{(C_{1},\\mathbf{P})} d\\mathbf{P} = \\int \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) d\\mathbf{P}", "derivation": "\\theta_{1}{(C_{1},\\mathbf{P})} = \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) and \\int \\theta_{1}{(C_{1},\\mathbf{P})} d\\mathbf{P} = \\int \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) d\\mathbf{P} and \\theta_{1}{(C_{1},\\mathbf{P})} = 1 and \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) = 1 and (\\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}))^{2} = \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) and \\theta_{1}^{2}{(C_{1},\\mathbf{P})} = \\theta_{1}{(C_{1},\\mathbf{P})} and \\int \\theta_{1}^{2}{(C_{1},\\mathbf{P})} d\\mathbf{P} = \\int \\frac{\\partial}{\\partial \\mathbf{P}} (C_{1} + \\mathbf{P}) d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1))"], [["times", 4, "Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(2)), Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Integral(Pow(Function('\\\\theta_1')(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(L)} = e^{e^{L}}, then obtain 2 (\\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}))^{L} = (\\frac{d}{d L} 0)^{L} + (\\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}))^{L}", "derivation": "\\mathbf{J}_M{(L)} = e^{e^{L}} and \\mathbf{J}_M{(L)} - e^{e^{L}} = 0 and \\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}) = \\frac{d}{d L} 0 and (\\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}))^{L} = (\\frac{d}{d L} 0)^{L} and 2 (\\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}))^{L} = (\\frac{d}{d L} 0)^{L} + (\\frac{d}{d L} (\\mathbf{J}_M{(L)} - e^{e^{L}}))^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), exp(exp(Symbol('L', commutative=True))))"], [["minus", 1, "exp(exp(Symbol('L', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"], [["add", 4, "Pow(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Pow(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True))), Add(Pow(Derivative(Integer(0), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)), Pow(Derivative(Add(Function('\\\\mathbf{J}_M')(Symbol('L', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('L', commutative=True))))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(Q,\\hbar)} = Q^{\\hbar}, then obtain \\sigma_{p}{(Q,\\hbar)} \\frac{\\partial}{\\partial Q} \\frac{\\int \\sigma_{p}{(Q,\\hbar)} d\\hbar}{\\hbar} = \\sigma_{p}{(Q,\\hbar)} \\frac{\\partial}{\\partial Q} \\frac{\\int Q^{\\hbar} d\\hbar}{\\hbar}", "derivation": "\\sigma_{p}{(Q,\\hbar)} = Q^{\\hbar} and \\int \\sigma_{p}{(Q,\\hbar)} d\\hbar = \\int Q^{\\hbar} d\\hbar and \\frac{\\int \\sigma_{p}{(Q,\\hbar)} d\\hbar}{\\hbar} = \\frac{\\int Q^{\\hbar} d\\hbar}{\\hbar} and \\frac{\\partial}{\\partial Q} \\frac{\\int \\sigma_{p}{(Q,\\hbar)} d\\hbar}{\\hbar} = \\frac{\\partial}{\\partial Q} \\frac{\\int Q^{\\hbar} d\\hbar}{\\hbar} and Q^{\\hbar} \\frac{\\partial}{\\partial Q} \\frac{\\int \\sigma_{p}{(Q,\\hbar)} d\\hbar}{\\hbar} = Q^{\\hbar} \\frac{\\partial}{\\partial Q} \\frac{\\int Q^{\\hbar} d\\hbar}{\\hbar} and \\sigma_{p}{(Q,\\hbar)} \\frac{\\partial}{\\partial Q} \\frac{\\int \\sigma_{p}{(Q,\\hbar)} d\\hbar}{\\hbar} = \\sigma_{p}{(Q,\\hbar)} \\frac{\\partial}{\\partial Q} \\frac{\\int Q^{\\hbar} d\\hbar}{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["divide", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 4, "Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Function('\\\\sigma_p')(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Integral(Pow(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu{(J,E_{\\lambda})} = \\frac{\\sin{(E_{\\lambda})}}{J} and \\sigma_{p}{(J,E_{\\lambda})} = \\frac{\\sin{(E_{\\lambda})}}{J}, then obtain -1 = - (\\frac{\\sin{(E_{\\lambda})}}{J})^{- J} \\sigma_{p}^{J}{(J,E_{\\lambda})}", "derivation": "\\mu{(J,E_{\\lambda})} = \\frac{\\sin{(E_{\\lambda})}}{J} and \\sigma_{p}{(J,E_{\\lambda})} = \\frac{\\sin{(E_{\\lambda})}}{J} and \\sigma_{p}{(J,E_{\\lambda})} = \\mu{(J,E_{\\lambda})} and \\sigma_{p}^{J}{(J,E_{\\lambda})} = \\mu^{J}{(J,E_{\\lambda})} and (\\frac{\\sin{(E_{\\lambda})}}{J})^{J} = \\mu^{J}{(J,E_{\\lambda})} and (\\frac{\\sin{(E_{\\lambda})}}{J})^{J} = \\sigma_{p}^{J}{(J,E_{\\lambda})} and -1 = - (\\frac{\\sin{(E_{\\lambda})}}{J})^{- J} \\sigma_{p}^{J}{(J,E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\sigma_p')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mu')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J', commutative=True)), Pow(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('J', commutative=True)), Pow(Function('\\\\mu')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('J', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J', commutative=True)))"], [["divide", 6, "Mul(Integer(-1), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('J', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('J', commutative=True))), Pow(Function('\\\\sigma_p')(Symbol('J', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\psi{(z,\\tilde{g})} = \\tilde{g} z, then derive \\frac{\\partial^{2}}{\\partial z\\partial \\tilde{g}} \\psi{(z,\\tilde{g})} = 1, then obtain \\frac{\\partial^{3}}{\\partial \\tilde{g}\\partial z\\partial \\tilde{g}} \\tilde{g} z = \\frac{d}{d \\tilde{g}} 1", "derivation": "\\psi{(z,\\tilde{g})} = \\tilde{g} z and \\frac{\\partial}{\\partial \\tilde{g}} \\psi{(z,\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} \\tilde{g} z and \\frac{\\partial}{\\partial \\tilde{g}} \\psi{(z,\\tilde{g})} + 1 = \\frac{\\partial}{\\partial \\tilde{g}} \\tilde{g} z + 1 and \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\tilde{g}} \\psi{(z,\\tilde{g})} + 1) = \\frac{\\partial}{\\partial z} (\\frac{\\partial}{\\partial \\tilde{g}} \\tilde{g} z + 1) and \\frac{\\partial^{2}}{\\partial z\\partial \\tilde{g}} \\psi{(z,\\tilde{g})} = 1 and \\frac{\\partial^{2}}{\\partial z\\partial \\tilde{g}} \\tilde{g} z = 1 and \\frac{\\partial^{3}}{\\partial \\tilde{g}\\partial z\\partial \\tilde{g}} \\tilde{g} z = \\frac{d}{d \\tilde{g}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 6, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)), Tuple(Symbol('z', commutative=True), Integer(1)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(a)} = e^{\\cos{(a)}} and v{(a)} = q{(a)} + 1, then obtain \\frac{d^{2}}{d a^{2}} v{(a)} = \\frac{d^{2}}{d a^{2}} (e^{\\cos{(a)}} + 1)", "derivation": "q{(a)} = e^{\\cos{(a)}} and q{(a)} + 1 = e^{\\cos{(a)}} + 1 and v{(a)} = q{(a)} + 1 and \\frac{d}{d a} v{(a)} = \\frac{d}{d a} (q{(a)} + 1) and \\frac{d^{2}}{d a^{2}} v{(a)} = \\frac{d^{2}}{d a^{2}} (q{(a)} + 1) and \\frac{d^{2}}{d a^{2}} v{(a)} = \\frac{d^{2}}{d a^{2}} (e^{\\cos{(a)}} + 1)", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('a', commutative=True)), exp(cos(Symbol('a', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('q')(Symbol('a', commutative=True)), Integer(1)), Add(exp(cos(Symbol('a', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('v')(Symbol('a', commutative=True)), Add(Function('q')(Symbol('a', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Function('q')(Symbol('a', commutative=True)), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Add(Function('q')(Symbol('a', commutative=True)), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Function('v')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(Add(exp(cos(Symbol('a', commutative=True))), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(2))))"]]}, {"prompt": "Given Z{(\\mathbf{s})} = e^{\\mathbf{s}} and \\mathbf{F}{(\\mathbf{s})} = e^{\\mathbf{s}}, then obtain 1 = \\frac{Z{(\\mathbf{s})} + \\mathbf{F}{(\\mathbf{s})}}{2 Z{(\\mathbf{s})}}", "derivation": "Z{(\\mathbf{s})} = e^{\\mathbf{s}} and 2 Z{(\\mathbf{s})} = Z{(\\mathbf{s})} + e^{\\mathbf{s}} and 1 = \\frac{Z{(\\mathbf{s})} + e^{\\mathbf{s}}}{2 Z{(\\mathbf{s})}} and \\mathbf{F}{(\\mathbf{s})} = e^{\\mathbf{s}} and 1 = \\frac{Z{(\\mathbf{s})} + \\mathbf{F}{(\\mathbf{s})}}{2 Z{(\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 1, "Function('Z')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True))), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{s}', commutative=True))), Pow(Function('Z')(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given p{(x^\\prime)} = e^{x^\\prime} and \\operatorname{f_{\\mathbf{p}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} p{(x^\\prime)}, then obtain \\operatorname{f_{\\mathbf{p}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime}", "derivation": "p{(x^\\prime)} = e^{x^\\prime} and \\frac{d}{d x^\\prime} p{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} and \\operatorname{f_{\\mathbf{p}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} p{(x^\\prime)} and \\operatorname{f_{\\mathbf{p}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('p')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('x^\\\\prime', commutative=True)), Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(v_{y})} = \\frac{d}{d v_{y}} \\log{(v_{y})}, then derive \\mathbf{J}_f^{v_{y}}{(v_{y})} = (\\frac{1}{v_{y}})^{v_{y}}, then obtain (\\frac{1}{v_{y}})^{v_{y}} = (\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}}", "derivation": "\\mathbf{J}_f{(v_{y})} = \\frac{d}{d v_{y}} \\log{(v_{y})} and \\mathbf{J}_f^{v_{y}}{(v_{y})} = (\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}} and \\mathbf{J}_f^{v_{y}}{(v_{y})} = (\\frac{1}{v_{y}})^{v_{y}} and (\\frac{1}{v_{y}})^{v_{y}} = (\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('v_y', commutative=True)), Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)), Pow(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\phi{(H)} = \\sin{(H)}, then derive \\int \\phi{(H)} dH = \\varphi^* - \\cos{(H)}, then obtain \\varphi^* - \\cos{(H)} = \\int \\sin{(H)} dH", "derivation": "\\phi{(H)} = \\sin{(H)} and \\int \\phi{(H)} dH = \\int \\sin{(H)} dH and \\int \\phi{(H)} dH = \\varphi^* - \\cos{(H)} and \\varphi^* - \\cos{(H)} = \\int \\sin{(H)} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(P_{g},\\mathbf{H},\\hbar)} = P_{g} \\hbar \\mathbf{H} and \\operatorname{f_{E}}{(P_{g},\\mathbf{H},\\hbar)} = (\\operatorname{f^{\\prime}}^{\\mathbf{H}}{(P_{g},\\mathbf{H},\\hbar)})^{\\hbar}, then obtain \\cos{(\\operatorname{f_{E}}{(P_{g},\\mathbf{H},\\hbar)})} = \\cos{(((P_{g} \\hbar \\mathbf{H})^{\\mathbf{H}})^{\\hbar})}", "derivation": "\\operatorname{f^{\\prime}}{(P_{g},\\mathbf{H},\\hbar)} = P_{g} \\hbar \\mathbf{H} and \\operatorname{f^{\\prime}}^{\\mathbf{H}}{(P_{g},\\mathbf{H},\\hbar)} = (P_{g} \\hbar \\mathbf{H})^{\\mathbf{H}} and (\\operatorname{f^{\\prime}}^{\\mathbf{H}}{(P_{g},\\mathbf{H},\\hbar)})^{\\hbar} = ((P_{g} \\hbar \\mathbf{H})^{\\mathbf{H}})^{\\hbar} and \\operatorname{f_{E}}{(P_{g},\\mathbf{H},\\hbar)} = (\\operatorname{f^{\\prime}}^{\\mathbf{H}}{(P_{g},\\mathbf{H},\\hbar)})^{\\hbar} and \\operatorname{f_{E}}{(P_{g},\\mathbf{H},\\hbar)} = ((P_{g} \\hbar \\mathbf{H})^{\\mathbf{H}})^{\\hbar} and \\cos{(\\operatorname{f_{E}}{(P_{g},\\mathbf{H},\\hbar)})} = \\cos{(((P_{g} \\hbar \\mathbf{H})^{\\mathbf{H}})^{\\hbar})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('P_g', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Symbol('P_g', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Pow(Function('f^{\\\\prime}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Pow(Mul(Symbol('P_g', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Pow(Function('f^{\\\\prime}')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('f_E')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Pow(Mul(Symbol('P_g', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["cos", 5], "Equality(cos(Function('f_E')(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True))), cos(Pow(Pow(Mul(Symbol('P_g', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} = n_{2} (P_{e} + \\hbar), then derive \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = J_{\\varepsilon} + P_{e} \\hbar n_{2} + \\frac{\\hbar^{2} n_{2}}{2}, then derive 2 \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} + 2 m_{s}, then obtain 2 J_{\\varepsilon} + 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} = 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} + 2 m_{s}", "derivation": "\\eta^{\\prime}{(n_{2},P_{e},\\hbar)} = n_{2} (P_{e} + \\hbar) and \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = \\int n_{2} (P_{e} + \\hbar) d\\hbar and \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = J_{\\varepsilon} + P_{e} \\hbar n_{2} + \\frac{\\hbar^{2} n_{2}}{2} and 2 \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = 2 \\int n_{2} (P_{e} + \\hbar) d\\hbar and 2 \\int \\eta^{\\prime}{(n_{2},P_{e},\\hbar)} d\\hbar = 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} + 2 m_{s} and 2 J_{\\varepsilon} + 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} = 2 P_{e} \\hbar n_{2} + \\hbar^{2} n_{2} + 2 m_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('n_2', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Symbol('n_2', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('n_2', commutative=True))))"], [["divide", 2, "Rational(1, 2)"], "Equality(Mul(Integer(2), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('n_2', commutative=True), Add(Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(2), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('n_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('n_2', commutative=True))), Add(Mul(Integer(2), Symbol('P_e', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('n_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(Z)} = e^{Z} and l{(Z)} = \\frac{e^{Z}}{\\mathbf{H}{(Z)}}, then obtain \\int - \\frac{d^{2}}{d Z^{2}} l{(Z)} dZ = \\int - \\frac{d^{2}}{d Z^{2}} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} dZ", "derivation": "\\mathbf{H}{(Z)} = e^{Z} and 1 = \\frac{e^{Z}}{\\mathbf{H}{(Z)}} and \\frac{d}{d Z} 1 = \\frac{d}{d Z} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} and \\frac{d^{2}}{d Z^{2}} 1 = \\frac{d^{2}}{d Z^{2}} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} and - \\frac{d^{2}}{d Z^{2}} 1 = - \\frac{d^{2}}{d Z^{2}} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} and \\int - \\frac{d^{2}}{d Z^{2}} 1 dZ = \\int - \\frac{d^{2}}{d Z^{2}} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} dZ and l{(Z)} = \\frac{e^{Z}}{\\mathbf{H}{(Z)}} and - \\frac{d^{2}}{d Z^{2}} 1 = - \\frac{d^{2}}{d Z^{2}} l{(Z)} and \\int - \\frac{d^{2}}{d Z^{2}} l{(Z)} dZ = \\int - \\frac{d^{2}}{d Z^{2}} \\frac{e^{Z}}{\\mathbf{H}{(Z)}} dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{H}')(Symbol('Z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2))), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2)))), Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2)))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2)))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2)))), Tuple(Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('l')(Symbol('Z', commutative=True)), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2)))), Mul(Integer(-1), Derivative(Function('l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 8], "Equality(Integral(Mul(Integer(-1), Derivative(Function('l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(2)))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Integer(-1), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('Z', commutative=True)), Integer(-1)), exp(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(2)))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given S{(t_{1},E)} = \\frac{t_{1}}{E} and \\operatorname{A_{y}}{(t_{1},E)} = \\frac{t_{1}}{E}, then obtain E + \\int \\frac{S{(t_{1},E)}}{E} dE - \\frac{2 t_{1}}{E} = E + \\int \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} dE - \\frac{2 t_{1}}{E}", "derivation": "S{(t_{1},E)} = \\frac{t_{1}}{E} and \\frac{S{(t_{1},E)}}{E} = \\frac{t_{1}}{E^{2}} and \\operatorname{A_{y}}{(t_{1},E)} = \\frac{t_{1}}{E} and \\frac{S{(t_{1},E)}}{E} = \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} and \\int \\frac{S{(t_{1},E)}}{E} dE = \\int \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} dE and E + \\int \\frac{S{(t_{1},E)}}{E} dE = E + \\int \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} dE and E + \\int \\frac{S{(t_{1},E)}}{E} dE - \\frac{t_{1}}{E} = E + \\int \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} dE - \\frac{t_{1}}{E} and E + \\int \\frac{S{(t_{1},E)}}{E} dE - \\frac{2 t_{1}}{E} = E + \\int \\frac{\\operatorname{A_{y}}{(t_{1},E)}}{E} dE - \\frac{2 t_{1}}{E}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["divide", 1, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-2)), Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))))"], [["integrate", 4, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["add", 5, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["minus", 6, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))"], "Equality(Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Mul(Integer(-1), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], [["minus", 7, "Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))"], "Equality(Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('S')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Add(Symbol('E', commutative=True), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_1', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(p)} = p and \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} = (\\mathbf{D} + \\mathbf{M})^{\\phi_1}, then obtain \\mathbf{M} \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} + \\frac{d}{d \\operatorname{f^{*}}{(p)}} \\operatorname{f^{*}}{(p)} = \\mathbf{M} (\\mathbf{D} + \\mathbf{M})^{\\phi_1} + \\frac{d}{d \\operatorname{f^{*}}{(p)}} \\operatorname{f^{*}}{(p)}", "derivation": "\\operatorname{f^{*}}{(p)} = p and \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} = (\\mathbf{D} + \\mathbf{M})^{\\phi_1} and \\mathbf{M} \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} = \\mathbf{M} (\\mathbf{D} + \\mathbf{M})^{\\phi_1} and \\mathbf{M} \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} + \\frac{d}{d p} p = \\mathbf{M} (\\mathbf{D} + \\mathbf{M})^{\\phi_1} + \\frac{d}{d p} p and \\mathbf{M} \\rho_{b}{(\\phi_1,\\mathbf{D},\\mathbf{M})} + \\frac{d}{d \\operatorname{f^{*}}{(p)}} \\operatorname{f^{*}}{(p)} = \\mathbf{M} (\\mathbf{D} + \\mathbf{M})^{\\phi_1} + \\frac{d}{d \\operatorname{f^{*}}{(p)}} \\operatorname{f^{*}}{(p)}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"], [["add", 3, "Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Derivative(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Function('f^*')(Symbol('p', commutative=True)), Tuple(Function('f^*')(Symbol('p', commutative=True)), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Derivative(Function('f^*')(Symbol('p', commutative=True)), Tuple(Function('f^*')(Symbol('p', commutative=True)), Integer(1)))))"]]}, {"prompt": "Given E{(n_{2},c)} = \\int (c + n_{2}) dn_{2}, then obtain 0 = - E{(n_{2},c)} + \\int (c + n_{2}) dn_{2}", "derivation": "E{(n_{2},c)} = \\int (c + n_{2}) dn_{2} and c + E{(n_{2},c)} = c + \\int (c + n_{2}) dn_{2} and 2 c + E{(n_{2},c)} = 2 c + \\int (c + n_{2}) dn_{2} and 0 = - E{(n_{2},c)} + \\int (c + n_{2}) dn_{2}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('n_2', commutative=True), Symbol('c', commutative=True)), Integral(Add(Symbol('c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('E')(Symbol('n_2', commutative=True), Symbol('c', commutative=True))), Add(Symbol('c', commutative=True), Integral(Add(Symbol('c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["add", 2, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('c', commutative=True)), Function('E')(Symbol('n_2', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(2), Symbol('c', commutative=True)), Integral(Add(Symbol('c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(2), Symbol('c', commutative=True)), Function('E')(Symbol('n_2', commutative=True), Symbol('c', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E')(Symbol('n_2', commutative=True), Symbol('c', commutative=True))), Integral(Add(Symbol('c', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given f{(p)} = \\cos{(p)} and \\dot{\\mathbf{r}}{(p)} = - (\\cos^{p}{(p)})^{p}, then obtain \\frac{d}{d p} (f^{p}{(p)} + 1) \\dot{\\mathbf{r}}{(p)} = \\frac{d}{d p} - (f^{p}{(p)} + 1) (f^{p}{(p)})^{p}", "derivation": "f{(p)} = \\cos{(p)} and f^{p}{(p)} = \\cos^{p}{(p)} and \\dot{\\mathbf{r}}{(p)} = - (\\cos^{p}{(p)})^{p} and \\dot{\\mathbf{r}}{(p)} = - (f^{p}{(p)})^{p} and (\\cos^{p}{(p)} + 1) \\dot{\\mathbf{r}}{(p)} = - (\\cos^{p}{(p)} + 1) (f^{p}{(p)})^{p} and \\frac{d}{d p} (\\cos^{p}{(p)} + 1) \\dot{\\mathbf{r}}{(p)} = \\frac{d}{d p} - (\\cos^{p}{(p)} + 1) (f^{p}{(p)})^{p} and \\frac{d}{d p} (f^{p}{(p)} + 1) \\dot{\\mathbf{r}}{(p)} = \\frac{d}{d p} - (f^{p}{(p)} + 1) (f^{p}{(p)})^{p}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["times", 4, "Add(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1))"], "Equality(Mul(Add(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('p', commutative=True))), Mul(Integer(-1), Add(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Pow(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Add(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Pow(cos(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Pow(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Mul(Add(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Add(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Integer(1)), Pow(Pow(Function('f')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(z)} = e^{z}, then obtain \\frac{e^{z} (\\frac{d}{d z} \\varepsilon_{0}{(z)})^{2}}{\\varepsilon_{0}^{2}{(z)}} = \\frac{e^{z} \\frac{d}{d z} \\varepsilon_{0}{(z)} \\frac{d}{d z} e^{z}}{\\varepsilon_{0}^{2}{(z)}}", "derivation": "\\varepsilon_{0}{(z)} = e^{z} and \\frac{d}{d z} \\varepsilon_{0}{(z)} = \\frac{d}{d z} e^{z} and \\frac{\\frac{d}{d z} \\varepsilon_{0}{(z)}}{\\varepsilon_{0}{(z)}} = \\frac{\\frac{d}{d z} e^{z}}{\\varepsilon_{0}{(z)}} and \\frac{(\\frac{d}{d z} \\varepsilon_{0}{(z)})^{2}}{\\varepsilon_{0}^{2}{(z)}} = \\frac{\\frac{d}{d z} \\varepsilon_{0}{(z)} \\frac{d}{d z} e^{z}}{\\varepsilon_{0}^{2}{(z)}} and \\frac{e^{z} (\\frac{d}{d z} \\varepsilon_{0}{(z)})^{2}}{\\varepsilon_{0}^{2}{(z)}} = \\frac{e^{z} \\frac{d}{d z} \\varepsilon_{0}{(z)} \\frac{d}{d z} e^{z}}{\\varepsilon_{0}^{2}{(z)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\varepsilon_0')(Symbol('z', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-1)), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["times", 3, "Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-2)), Pow(Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-2)), Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["times", 4, "exp(Symbol('z', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-2)), exp(Symbol('z', commutative=True)), Pow(Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(2))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Integer(-2)), exp(Symbol('z', commutative=True)), Derivative(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given m{(C_{d})} = \\log{(C_{d})}, then derive \\frac{\\partial}{\\partial C_{d}} (- A_{2} - C_{d} \\log{(C_{d})} + C_{d} + \\int m{(C_{d})} dC_{d}) = \\frac{d}{d C_{d}} 0, then obtain \\frac{\\partial}{\\partial C_{d}} (- A_{2} - C_{d} \\log{(C_{d})} + C_{d} + \\int \\log{(C_{d})} dC_{d}) = \\frac{d}{d C_{d}} 0", "derivation": "m{(C_{d})} = \\log{(C_{d})} and \\int m{(C_{d})} dC_{d} = \\int \\log{(C_{d})} dC_{d} and \\int m{(C_{d})} dC_{d} - \\int \\log{(C_{d})} dC_{d} = 0 and \\frac{d}{d C_{d}} (\\int m{(C_{d})} dC_{d} - \\int \\log{(C_{d})} dC_{d}) = \\frac{d}{d C_{d}} 0 and \\frac{\\partial}{\\partial C_{d}} (- A_{2} - C_{d} \\log{(C_{d})} + C_{d} + \\int m{(C_{d})} dC_{d}) = \\frac{d}{d C_{d}} 0 and \\frac{\\partial}{\\partial C_{d}} (- A_{2} - C_{d} \\log{(C_{d})} + C_{d} + \\int \\log{(C_{d})} dC_{d}) = \\frac{d}{d C_{d}} 0", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('m')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["minus", 2, "Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))"], "Equality(Add(Integral(Function('m')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Add(Integral(Function('m')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True), Integral(Function('m')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True), log(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True), Integral(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})}, then derive \\frac{d}{d \\tilde{g}} \\operatorname{n_{2}}{(\\tilde{g})} = - \\frac{\\sin{(\\tilde{g})}}{\\cos{(\\tilde{g})}}, then obtain \\frac{d^{2}}{d \\tilde{g}^{2}} \\log{(\\cos{(\\tilde{g})})} = \\frac{d}{d \\tilde{g}} - \\frac{\\sin{(\\tilde{g})}}{\\cos{(\\tilde{g})}}", "derivation": "\\operatorname{n_{2}}{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})} and \\frac{d}{d \\tilde{g}} \\operatorname{n_{2}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\log{(\\cos{(\\tilde{g})})} and \\frac{d}{d \\tilde{g}} \\operatorname{n_{2}}{(\\tilde{g})} = - \\frac{\\sin{(\\tilde{g})}}{\\cos{(\\tilde{g})}} and \\frac{d^{2}}{d \\tilde{g}^{2}} \\operatorname{n_{2}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} - \\frac{\\sin{(\\tilde{g})}}{\\cos{(\\tilde{g})}} and \\frac{d^{2}}{d \\tilde{g}^{2}} \\log{(\\cos{(\\tilde{g})})} = \\frac{d}{d \\tilde{g}} - \\frac{\\sin{(\\tilde{g})}}{\\cos{(\\tilde{g})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), log(cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(log(cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(cos(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(u)} = e^{u}, then obtain (\\frac{\\iint b{(u)} du du}{u})^{u} = (\\frac{\\iint e^{u} du du}{u})^{u}", "derivation": "b{(u)} = e^{u} and \\int b{(u)} du = \\int e^{u} du and \\iint b{(u)} du du = \\iint e^{u} du du and \\frac{\\iint b{(u)} du du}{u} = \\frac{\\iint e^{u} du du}{u} and (\\frac{\\iint b{(u)} du du}{u})^{u} = (\\frac{\\iint e^{u} du du}{u})^{u}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["divide", 3, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Symbol('u', commutative=True)), Pow(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(exp(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given L{(f^{\\prime})} = \\log{(f^{\\prime})} and \\operatorname{A_{x}}{(\\nabla,v,M_{E})} = \\frac{v^{\\nabla}}{M_{E}}, then obtain \\operatorname{A_{x}}{(\\nabla,v,M_{E})} - L{(f^{\\prime})} - \\log{(f^{\\prime})} = - L{(f^{\\prime})} - \\log{(f^{\\prime})} + \\frac{v^{\\nabla}}{M_{E}}", "derivation": "L{(f^{\\prime})} = \\log{(f^{\\prime})} and 2 L{(f^{\\prime})} = L{(f^{\\prime})} + \\log{(f^{\\prime})} and \\operatorname{A_{x}}{(\\nabla,v,M_{E})} = \\frac{v^{\\nabla}}{M_{E}} and \\operatorname{A_{x}}{(\\nabla,v,M_{E})} - 2 L{(f^{\\prime})} = - 2 L{(f^{\\prime})} + \\frac{v^{\\nabla}}{M_{E}} and \\operatorname{A_{x}}{(\\nabla,v,M_{E})} - L{(f^{\\prime})} - \\log{(f^{\\prime})} = - L{(f^{\\prime})} - \\log{(f^{\\prime})} + \\frac{v^{\\nabla}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Function('L')(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(2), Function('L')(Symbol('f^{\\\\prime}', commutative=True))), Add(Function('L')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], ["get_premise", "Equality(Function('A_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Function('L')(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Function('A_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Integer(2), Function('L')(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('L')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('A_x')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Function('L')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given n{(F_{x})} = \\int \\log{(F_{x})} dF_{x}, then derive n^{F_{x}}{(F_{x})} = (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}}, then obtain 2 (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}} = (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}} + (\\int \\log{(F_{x})} dF_{x})^{F_{x}}", "derivation": "n{(F_{x})} = \\int \\log{(F_{x})} dF_{x} and n^{F_{x}}{(F_{x})} = (\\int \\log{(F_{x})} dF_{x})^{F_{x}} and n^{F_{x}}{(F_{x})} = (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}} and 2 n^{F_{x}}{(F_{x})} = n^{F_{x}}{(F_{x})} + (\\int \\log{(F_{x})} dF_{x})^{F_{x}} and 2 (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}} = (F_{x} \\log{(F_{x})} - F_{x} + \\mathbf{A})^{F_{x}} + (\\int \\log{(F_{x})} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('F_x', commutative=True)), Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('n')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('n')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Add(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('F_x', commutative=True)))"], [["add", 2, "Pow(Function('n')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('n')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Add(Pow(Function('n')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Add(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('F_x', commutative=True))), Add(Pow(Add(Mul(Symbol('F_x', commutative=True), log(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('F_x', commutative=True)), Pow(Integral(log(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(A_{2},\\delta)} = A_{2} + \\delta, then obtain \\frac{1}{4} = \\frac{A_{2} + \\delta}{4 \\hat{x}_0{(A_{2},\\delta)}}", "derivation": "\\hat{x}_0{(A_{2},\\delta)} = A_{2} + \\delta and 2 \\hat{x}_0{(A_{2},\\delta)} = A_{2} + \\delta + \\hat{x}_0{(A_{2},\\delta)} and 4 \\hat{x}_0{(A_{2},\\delta)} = A_{2} + \\delta + 3 \\hat{x}_0{(A_{2},\\delta)} and \\frac{\\hat{x}_0{(A_{2},\\delta)}}{A_{2} + \\delta + 3 \\hat{x}_0{(A_{2},\\delta)}} = \\frac{A_{2} + \\delta}{A_{2} + \\delta + 3 \\hat{x}_0{(A_{2},\\delta)}} and \\frac{1}{4} = \\frac{A_{2} + \\delta}{4 \\hat{x}_0{(A_{2},\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(3), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)))))"], [["divide", 1, "Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(3), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(3), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)))), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True), Mul(Integer(3), Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Rational(1, 4), Mul(Rational(1, 4), Add(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('A_2', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given z{(g)} = \\cos{(\\cos{(g)})}, then derive \\int 0 dg = A_{2} + 2 \\int - z{(g)} dg + 2 \\int \\cos{(\\cos{(g)})} dg, then obtain (\\int 0 dg)^{2} = (A_{2} + 2 \\int - z{(g)} dg + 2 \\int \\cos{(\\cos{(g)})} dg) \\int 0 dg", "derivation": "z{(g)} = \\cos{(\\cos{(g)})} and - z{(g)} = - 2 z{(g)} + \\cos{(\\cos{(g)})} and 0 = - z{(g)} + \\cos{(\\cos{(g)})} and 0 = - 2 z{(g)} + 2 \\cos{(\\cos{(g)})} and \\int 0 dg = \\int (- 2 z{(g)} + 2 \\cos{(\\cos{(g)})}) dg and \\int 0 dg = A_{2} + 2 \\int - z{(g)} dg + 2 \\int \\cos{(\\cos{(g)})} dg and (\\int 0 dg)^{2} = (A_{2} + 2 \\int - z{(g)} dg + 2 \\int \\cos{(\\cos{(g)})} dg) \\int 0 dg", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('g', commutative=True)), cos(cos(Symbol('g', commutative=True))))"], [["minus", 1, "Mul(Integer(2), Function('z')(Symbol('g', commutative=True)))"], "Equality(Mul(Integer(-1), Function('z')(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('z')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Function('z')(Symbol('g', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('z')(Symbol('g', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g', commutative=True))))))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Function('z')(Symbol('g', commutative=True))), Mul(Integer(2), cos(cos(Symbol('g', commutative=True))))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(2), Add(Integral(Mul(Integer(-1), Function('z')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))))"], [["times", 6, "Integral(Integer(0), Tuple(Symbol('g', commutative=True)))"], "Equality(Pow(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integer(2)), Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(2), Add(Integral(Mul(Integer(-1), Function('z')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(cos(cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\omega)} = \\sin{(\\sin{(\\omega)})} and \\tilde{g}^*{(\\omega)} = \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}}, then obtain e^{\\tilde{g}^*{(\\omega)}} - \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}} = e - \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}}", "derivation": "\\dot{z}{(\\omega)} = \\sin{(\\sin{(\\omega)})} and \\tilde{g}^*{(\\omega)} = \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}} and e^{\\tilde{g}^*{(\\omega)}} = e^{\\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}}} and e^{\\tilde{g}^*{(\\omega)}} = e and e^{\\tilde{g}^*{(\\omega)}} - \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}} = e - \\frac{\\sin{(\\sin{(\\omega)})}}{\\dot{z}{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), sin(sin(Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Mul(Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\omega', commutative=True)))))"], [["exp", 2], "Equality(exp(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True))), exp(Mul(Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True))), E)"], [["minus", 4, "Mul(Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\omega', commutative=True))))"], "Equality(Add(exp(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\omega', commutative=True))))), Add(E, Mul(Integer(-1), Pow(Function('\\\\dot{z}')(Symbol('\\\\omega', commutative=True)), Integer(-1)), sin(sin(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{B}{(n_{2},\\mu)} = \\mu e^{n_{2}}, then obtain \\mu \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(n_{2},\\mu)} + \\mathbf{B}{(n_{2},\\mu)} = 2 \\mu e^{n_{2}}", "derivation": "\\mathbf{B}{(n_{2},\\mu)} = \\mu e^{n_{2}} and \\mu \\mathbf{B}{(n_{2},\\mu)} = \\mu^{2} e^{n_{2}} and \\frac{\\partial}{\\partial \\mu} \\mu \\mathbf{B}{(n_{2},\\mu)} = \\frac{\\partial}{\\partial \\mu} \\mu^{2} e^{n_{2}} and \\mu \\frac{\\partial}{\\partial \\mu} \\mathbf{B}{(n_{2},\\mu)} + \\mathbf{B}{(n_{2},\\mu)} = 2 \\mu e^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('n_2', commutative=True))))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(2)), exp(Symbol('n_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(2)), exp(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Derivative(Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Function('\\\\mathbf{B}')(Symbol('n_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Symbol('\\\\mu', commutative=True), exp(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given C{(A_{x})} = e^{A_{x}}, then obtain C{(A_{x})} - e^{A_{x}} = 4 C{(A_{x})} - 4 e^{A_{x}}", "derivation": "C{(A_{x})} = e^{A_{x}} and 0 = - C{(A_{x})} + e^{A_{x}} and - C{(A_{x})} = - 2 C{(A_{x})} + e^{A_{x}} and - C{(A_{x})} + e^{A_{x}} = - 2 C{(A_{x})} + 2 e^{A_{x}} and - 2 C{(A_{x})} + 2 e^{A_{x}} = - 4 C{(A_{x})} + 4 e^{A_{x}} and - C{(A_{x})} + e^{A_{x}} = - 4 C{(A_{x})} + 4 e^{A_{x}} and C{(A_{x})} - e^{A_{x}} = 4 C{(A_{x})} - 4 e^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["minus", 1, "Function('C')(Symbol('A_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C')(Symbol('A_x', commutative=True))), exp(Symbol('A_x', commutative=True))))"], [["minus", 2, "Function('C')(Symbol('A_x', commutative=True))"], "Equality(Mul(Integer(-1), Function('C')(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('C')(Symbol('A_x', commutative=True))), exp(Symbol('A_x', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('C')(Symbol('A_x', commutative=True))), exp(Symbol('A_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('C')(Symbol('A_x', commutative=True))), exp(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('C')(Symbol('A_x', commutative=True))), Mul(Integer(2), exp(Symbol('A_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Function('C')(Symbol('A_x', commutative=True))), Mul(Integer(2), exp(Symbol('A_x', commutative=True)))), Add(Mul(Integer(-1), Integer(4), Function('C')(Symbol('A_x', commutative=True))), Mul(Integer(4), exp(Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('C')(Symbol('A_x', commutative=True))), exp(Symbol('A_x', commutative=True))), Add(Mul(Integer(-1), Integer(4), Function('C')(Symbol('A_x', commutative=True))), Mul(Integer(4), exp(Symbol('A_x', commutative=True)))))"], [["divide", 6, "Integer(-1)"], "Equality(Add(Function('C')(Symbol('A_x', commutative=True)), Mul(Integer(-1), exp(Symbol('A_x', commutative=True)))), Add(Mul(Integer(4), Function('C')(Symbol('A_x', commutative=True))), Mul(Integer(-1), Integer(4), exp(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given k{(c,x^\\prime)} = - c + x^\\prime, then obtain (x^\\prime)^{x^\\prime} + ((c + k{(c,x^\\prime)})^{x^\\prime})^{c} = (x^\\prime)^{x^\\prime} + ((x^\\prime)^{x^\\prime})^{c}", "derivation": "k{(c,x^\\prime)} = - c + x^\\prime and c + k{(c,x^\\prime)} = x^\\prime and (c + k{(c,x^\\prime)})^{x^\\prime} = (x^\\prime)^{x^\\prime} and ((c + k{(c,x^\\prime)})^{x^\\prime})^{c} = ((x^\\prime)^{x^\\prime})^{c} and (x^\\prime)^{x^\\prime} + ((c + k{(c,x^\\prime)})^{x^\\prime})^{c} = (x^\\prime)^{x^\\prime} + ((x^\\prime)^{x^\\prime})^{c}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('c', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), Function('k')(Symbol('c', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Symbol('c', commutative=True), Function('k')(Symbol('c', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('c', commutative=True), Function('k')(Symbol('c', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('c', commutative=True)))"], [["add", 4, "Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Add(Symbol('c', commutative=True), Function('k')(Symbol('c', commutative=True), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Symbol('c', commutative=True))), Add(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(c_{0})} = \\log{(c_{0})}, then obtain \\frac{2 \\frac{d}{d c_{0}} \\Psi_{nl}{(c_{0})}}{\\frac{d}{d c_{0}} \\Psi_{nl}{(c_{0})} + \\frac{1}{c_{0}}} = 1", "derivation": "\\Psi_{nl}{(c_{0})} = \\log{(c_{0})} and 2 \\Psi_{nl}{(c_{0})} = \\Psi_{nl}{(c_{0})} + \\log{(c_{0})} and \\frac{d}{d c_{0}} 2 \\Psi_{nl}{(c_{0})} = \\frac{d}{d c_{0}} (\\Psi_{nl}{(c_{0})} + \\log{(c_{0})}) and \\frac{\\frac{d}{d c_{0}} 2 \\Psi_{nl}{(c_{0})}}{\\frac{d}{d c_{0}} (\\Psi_{nl}{(c_{0})} + \\log{(c_{0})})} = 1 and \\frac{2 \\frac{d}{d c_{0}} \\Psi_{nl}{(c_{0})}}{\\frac{d}{d c_{0}} \\Psi_{nl}{(c_{0})} + \\frac{1}{c_{0}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True)))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Pow(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Pow(Symbol('c_0', commutative=True), Integer(-1))), Integer(-1)), Derivative(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\Omega{(f^{*},v_{z})} = f^{*} - v_{z}, then obtain \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\Omega{(f^{*},v_{z})} = 0", "derivation": "\\Omega{(f^{*},v_{z})} = f^{*} - v_{z} and \\frac{\\partial}{\\partial v_{z}} \\Omega{(f^{*},v_{z})} = \\frac{\\partial}{\\partial v_{z}} (f^{*} - v_{z}) and \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\Omega{(f^{*},v_{z})} = \\frac{\\partial^{2}}{\\partial v_{z}^{2}} (f^{*} - v_{z}) and \\frac{\\partial^{2}}{\\partial v_{z}^{2}} \\Omega{(f^{*},v_{z})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('f^*', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('f^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('f^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2))), Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Omega')(Symbol('f^*', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given C{(\\omega,C_{2})} = C_{2} \\omega, then obtain \\frac{\\frac{\\partial}{\\partial C_{2}} (C_{2} \\omega + 2 C_{2} C{(\\omega,C_{2})})}{\\frac{\\partial}{\\partial \\omega} C_{2}^{2} \\omega} = \\frac{\\frac{\\partial}{\\partial C_{2}} (C_{2}^{2} \\omega + C_{2} \\omega + C_{2} C{(\\omega,C_{2})})}{\\frac{\\partial}{\\partial \\omega} C_{2}^{2} \\omega}", "derivation": "C{(\\omega,C_{2})} = C_{2} \\omega and C_{2} C{(\\omega,C_{2})} = C_{2}^{2} \\omega and 2 C_{2} C{(\\omega,C_{2})} = C_{2}^{2} \\omega + C_{2} C{(\\omega,C_{2})} and C_{2} \\omega + 2 C_{2} C{(\\omega,C_{2})} = C_{2}^{2} \\omega + C_{2} \\omega + C_{2} C{(\\omega,C_{2})} and \\frac{\\partial}{\\partial C_{2}} (C_{2} \\omega + 2 C_{2} C{(\\omega,C_{2})}) = \\frac{\\partial}{\\partial C_{2}} (C_{2}^{2} \\omega + C_{2} \\omega + C_{2} C{(\\omega,C_{2})}) and \\frac{\\frac{\\partial}{\\partial C_{2}} (C_{2} \\omega + 2 C_{2} C{(\\omega,C_{2})})}{\\frac{\\partial}{\\partial \\omega} C_{2}^{2} \\omega} = \\frac{\\frac{\\partial}{\\partial C_{2}} (C_{2}^{2} \\omega + C_{2} \\omega + C_{2} C{(\\omega,C_{2})})}{\\frac{\\partial}{\\partial \\omega} C_{2}^{2} \\omega}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)))"], [["add", 2, "Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))))"], [["add", 3, "Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))), Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))))"], [["differentiate", 4, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Mul(Pow(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Mul(Pow(Symbol('C_2', commutative=True), Integer(2)), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('C_2', commutative=True), Function('C')(Symbol('\\\\omega', commutative=True), Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\pi,E_{x})} = \\sin{(E_{x} + \\pi)}, then obtain \\frac{d}{d E_{x}} (-1) = \\frac{\\partial}{\\partial E_{x}} - \\frac{\\sin{(E_{x} + \\pi)}}{\\operatorname{E_{\\lambda}}{(\\pi,E_{x})}}", "derivation": "\\operatorname{E_{\\lambda}}{(\\pi,E_{x})} = \\sin{(E_{x} + \\pi)} and 1 = \\frac{\\sin{(E_{x} + \\pi)}}{\\operatorname{E_{\\lambda}}{(\\pi,E_{x})}} and -1 = - \\frac{\\sin{(E_{x} + \\pi)}}{\\operatorname{E_{\\lambda}}{(\\pi,E_{x})}} and \\frac{d}{d E_{x}} (-1) = \\frac{\\partial}{\\partial E_{x}} - \\frac{\\sin{(E_{x} + \\pi)}}{\\operatorname{E_{\\lambda}}{(\\pi,E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), sin(Add(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), sin(Add(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), sin(Add(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integer(-1), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), sin(Add(Symbol('E_x', commutative=True), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(F_{g})} = F_{g}, then derive 2 \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} = \\frac{F_{g}^{2}}{2} + \\theta + \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g}, then obtain 2 \\int F_{g} dF_{g} = \\frac{F_{g}^{2}}{2} + \\theta + \\int F_{g} dF_{g}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(F_{g})} = F_{g} and \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} = \\int F_{g} dF_{g} and 2 \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} = \\int F_{g} dF_{g} + \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} and 2 \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} = \\frac{F_{g}^{2}}{2} + \\theta + \\int \\operatorname{f_{\\mathbf{v}}}{(F_{g})} dF_{g} and 2 \\int F_{g} dF_{g} = \\frac{F_{g}^{2}}{2} + \\theta + \\int F_{g} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Symbol('F_g', commutative=True))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True))))"], [["add", 2, "Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(Integral(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True))), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_g', commutative=True), Integer(2))), Symbol('\\\\theta', commutative=True), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Integral(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_g', commutative=True), Integer(2))), Symbol('\\\\theta', commutative=True), Integral(Symbol('F_g', commutative=True), Tuple(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(n_{1},E_{\\lambda})} = \\sin{(E_{\\lambda} n_{1})}, then obtain - \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} + \\int \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} dn_{1} = - \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} + \\int \\sin^{n_{1}}{(E_{\\lambda} n_{1})} dn_{1}", "derivation": "\\operatorname{g_{\\varepsilon}}{(n_{1},E_{\\lambda})} = \\sin{(E_{\\lambda} n_{1})} and \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} = \\sin^{n_{1}}{(E_{\\lambda} n_{1})} and \\int \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} dn_{1} = \\int \\sin^{n_{1}}{(E_{\\lambda} n_{1})} dn_{1} and - \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} + \\int \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} dn_{1} = - \\operatorname{g_{\\varepsilon}}^{n_{1}}{(n_{1},E_{\\lambda})} + \\int \\sin^{n_{1}}{(E_{\\lambda} n_{1})} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n_1', commutative=True))))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Pow(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["integrate", 2, "Symbol('n_1', commutative=True)"], "Equality(Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Pow(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["minus", 3, "Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True))), Integral(Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('g_{\\\\varepsilon}')(Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True))), Integral(Pow(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(f^{\\prime})} = \\cos{(f^{\\prime})}, then obtain (\\cos^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\ddot{x}{(f^{\\prime})} = (\\cos^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\cos{(f^{\\prime})}", "derivation": "\\ddot{x}{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\ddot{x}^{f^{\\prime}}{(f^{\\prime})} = \\cos^{f^{\\prime}}{(f^{\\prime})} and (\\ddot{x}^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\ddot{x}{(f^{\\prime})} = (\\ddot{x}^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\cos{(f^{\\prime})} and (\\cos^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\ddot{x}{(f^{\\prime})} = (\\cos^{f^{\\prime}}{(f^{\\prime})})^{- f^{\\prime}} \\cos{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 1, "Pow(Pow(Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Pow(Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Function('\\\\ddot{x}')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), cos(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\theta)} = \\cos{(\\theta)} and b{(\\theta)} = \\cos{(\\theta)}, then obtain \\theta^{2} \\operatorname{v_{z}}{(\\theta)} \\cos{(\\theta)} = \\theta^{2} b{(\\theta)} \\operatorname{v_{z}}{(\\theta)}", "derivation": "\\operatorname{v_{z}}{(\\theta)} = \\cos{(\\theta)} and b{(\\theta)} = \\cos{(\\theta)} and \\operatorname{v_{z}}{(\\theta)} = b{(\\theta)} and \\theta \\operatorname{v_{z}}{(\\theta)} = \\theta b{(\\theta)} and \\theta \\cos{(\\theta)} = \\theta b{(\\theta)} and \\theta^{2} \\operatorname{v_{z}}{(\\theta)} \\cos{(\\theta)} = \\theta^{2} b{(\\theta)} \\operatorname{v_{z}}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('b')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v_z')(Symbol('\\\\theta', commutative=True)), Function('b')(Symbol('\\\\theta', commutative=True)))"], [["times", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('v_z')(Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Function('b')(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\theta', commutative=True), cos(Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Function('b')(Symbol('\\\\theta', commutative=True))))"], [["times", 5, "Mul(Symbol('\\\\theta', commutative=True), Function('v_z')(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('v_z')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('b')(Symbol('\\\\theta', commutative=True)), Function('v_z')(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(m)} = \\sin{(\\log{(m)})}, then obtain (m + \\dot{x}{(m)}) \\int \\frac{m + \\dot{x}{(m)}}{\\log{(m)}} dm = (m + \\dot{x}{(m)}) \\int \\frac{m + \\sin{(\\log{(m)})}}{\\log{(m)}} dm", "derivation": "\\dot{x}{(m)} = \\sin{(\\log{(m)})} and m + \\dot{x}{(m)} = m + \\sin{(\\log{(m)})} and \\frac{m + \\dot{x}{(m)}}{\\log{(m)}} = \\frac{m + \\sin{(\\log{(m)})}}{\\log{(m)}} and \\int \\frac{m + \\dot{x}{(m)}}{\\log{(m)}} dm = \\int \\frac{m + \\sin{(\\log{(m)})}}{\\log{(m)}} dm and (m + \\dot{x}{(m)}) \\int \\frac{m + \\dot{x}{(m)}}{\\log{(m)}} dm = (m + \\dot{x}{(m)}) \\int \\frac{m + \\sin{(\\log{(m)})}}{\\log{(m)}} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('m', commutative=True)), sin(log(Symbol('m', commutative=True))))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Add(Symbol('m', commutative=True), sin(log(Symbol('m', commutative=True)))))"], [["divide", 2, "log(Symbol('m', commutative=True))"], "Equality(Mul(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Mul(Add(Symbol('m', commutative=True), sin(log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Add(Symbol('m', commutative=True), sin(log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('m', commutative=True))))"], [["times", 4, "Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True)))"], "Equality(Mul(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Integral(Mul(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('m', commutative=True)))), Mul(Add(Symbol('m', commutative=True), Function('\\\\dot{x}')(Symbol('m', commutative=True))), Integral(Mul(Add(Symbol('m', commutative=True), sin(log(Symbol('m', commutative=True)))), Pow(log(Symbol('m', commutative=True)), Integer(-1))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given B{(P_{g},u)} = \\cos{(u^{P_{g}})} and \\rho_{f}{(P_{g},u)} = u^{P_{g}}, then obtain \\frac{\\partial}{\\partial u} \\frac{\\cos{(u^{P_{g}})}}{P_{g}} = \\frac{\\partial}{\\partial u} \\frac{\\cos{(\\rho_{f}{(P_{g},u)})}}{P_{g}}", "derivation": "B{(P_{g},u)} = \\cos{(u^{P_{g}})} and \\frac{B{(P_{g},u)}}{P_{g}} = \\frac{\\cos{(u^{P_{g}})}}{P_{g}} and \\rho_{f}{(P_{g},u)} = u^{P_{g}} and \\frac{B{(P_{g},u)}}{P_{g}} = \\frac{\\cos{(\\rho_{f}{(P_{g},u)})}}{P_{g}} and \\frac{\\cos{(u^{P_{g}})}}{P_{g}} = \\frac{\\cos{(\\rho_{f}{(P_{g},u)})}}{P_{g}} and \\frac{\\partial}{\\partial u} \\frac{\\cos{(u^{P_{g}})}}{P_{g}} = \\frac{\\partial}{\\partial u} \\frac{\\cos{(\\rho_{f}{(P_{g},u)})}}{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('P_g', commutative=True), Symbol('u', commutative=True)), cos(Pow(Symbol('u', commutative=True), Symbol('P_g', commutative=True))))"], [["divide", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('B')(Symbol('P_g', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Pow(Symbol('u', commutative=True), Symbol('P_g', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('P_g', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('P_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('B')(Symbol('P_g', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Function('\\\\rho_f')(Symbol('P_g', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Pow(Symbol('u', commutative=True), Symbol('P_g', commutative=True)))), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Function('\\\\rho_f')(Symbol('P_g', commutative=True), Symbol('u', commutative=True)))))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Pow(Symbol('u', commutative=True), Symbol('P_g', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), cos(Function('\\\\rho_f')(Symbol('P_g', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(s,\\tilde{g}^*)} = \\cos{((\\tilde{g}^*)^{s})}, then obtain (B^{\\tilde{g}^*}{(s,\\tilde{g}^*)} - \\int B{(s,\\tilde{g}^*)} ds)^{s} = (\\cos^{\\tilde{g}^*}{((\\tilde{g}^*)^{s})} - \\int B{(s,\\tilde{g}^*)} ds)^{s}", "derivation": "B{(s,\\tilde{g}^*)} = \\cos{((\\tilde{g}^*)^{s})} and B^{\\tilde{g}^*}{(s,\\tilde{g}^*)} = \\cos^{\\tilde{g}^*}{((\\tilde{g}^*)^{s})} and B^{\\tilde{g}^*}{(s,\\tilde{g}^*)} - \\int B{(s,\\tilde{g}^*)} ds = \\cos^{\\tilde{g}^*}{((\\tilde{g}^*)^{s})} - \\int B{(s,\\tilde{g}^*)} ds and (B^{\\tilde{g}^*}{(s,\\tilde{g}^*)} - \\int B{(s,\\tilde{g}^*)} ds)^{s} = (\\cos^{\\tilde{g}^*}{((\\tilde{g}^*)^{s})} - \\int B{(s,\\tilde{g}^*)} ds)^{s}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), cos(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True))))"], [["power", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(cos(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 2, "Integral(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Pow(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integral(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('s', commutative=True))))), Add(Pow(cos(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integral(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('s', commutative=True))))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Add(Pow(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integral(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('s', commutative=True))))), Symbol('s', commutative=True)), Pow(Add(Pow(cos(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('s', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Integral(Function('B')(Symbol('s', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('s', commutative=True))))), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(t,G,\\hbar)} = G (\\hbar - t), then obtain \\frac{\\partial}{\\partial G} (\\hbar + \\operatorname{E_{n}}{(t,G,\\hbar)})^{t} = \\frac{\\partial}{\\partial G} (G (\\hbar - t) + \\hbar)^{t}", "derivation": "\\operatorname{E_{n}}{(t,G,\\hbar)} = G (\\hbar - t) and \\hbar + \\operatorname{E_{n}}{(t,G,\\hbar)} = G (\\hbar - t) + \\hbar and (\\hbar + \\operatorname{E_{n}}{(t,G,\\hbar)})^{t} = (G (\\hbar - t) + \\hbar)^{t} and \\frac{\\partial}{\\partial G} (\\hbar + \\operatorname{E_{n}}{(t,G,\\hbar)})^{t} = \\frac{\\partial}{\\partial G} (G (\\hbar - t) + \\hbar)^{t}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('t', commutative=True), Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('G', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["add", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('E_n')(Symbol('t', commutative=True), Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('G', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Symbol('\\\\hbar', commutative=True)))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hbar', commutative=True), Function('E_n')(Symbol('t', commutative=True), Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Mul(Symbol('G', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('\\\\hbar', commutative=True), Function('E_n')(Symbol('t', commutative=True), Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('t', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('G', commutative=True), Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(C,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{C}, then obtain \\frac{\\sin^{2}{(M{(C,\\mathbf{J}_P)})}}{\\sin^{2}{(\\frac{\\mathbf{J}_P}{C})}} = 1", "derivation": "M{(C,\\mathbf{J}_P)} = \\frac{\\mathbf{J}_P}{C} and \\sin{(M{(C,\\mathbf{J}_P)})} = \\sin{(\\frac{\\mathbf{J}_P}{C})} and \\frac{\\sin{(M{(C,\\mathbf{J}_P)})}}{\\sin{(\\frac{\\mathbf{J}_P}{C})}} = 1 and \\frac{\\sin^{2}{(M{(C,\\mathbf{J}_P)})}}{\\sin{(\\frac{\\mathbf{J}_P}{C})}} = \\sin{(M{(C,\\mathbf{J}_P)})} and \\frac{\\sin^{2}{(M{(C,\\mathbf{J}_P)})}}{\\sin^{2}{(\\frac{\\mathbf{J}_P}{C})}} = 1", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["sin", 1], "Equality(sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 2, "sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1)), sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(1))"], [["times", 3, "sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1)), Pow(sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(2))), sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(sin(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-2)), Pow(sin(Function('M')(Symbol('C', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(2))), Integer(1))"]]}, {"prompt": "Given a{(M,v_{1})} = v_{1}^{M}, then derive - v_{1} + \\frac{\\partial}{\\partial M} a{(M,v_{1})} = - v_{1} + v_{1}^{M} \\log{(v_{1})}, then obtain \\frac{\\partial^{2}}{\\partial M^{2}} a{(M,v_{1})} = v_{1}^{M} \\log{(v_{1})}^{2}", "derivation": "a{(M,v_{1})} = v_{1}^{M} and \\frac{\\partial}{\\partial M} a{(M,v_{1})} = \\frac{\\partial}{\\partial M} v_{1}^{M} and - v_{1} + \\frac{\\partial}{\\partial M} a{(M,v_{1})} = - v_{1} + \\frac{\\partial}{\\partial M} v_{1}^{M} and - v_{1} + \\frac{\\partial}{\\partial M} a{(M,v_{1})} = - v_{1} + v_{1}^{M} \\log{(v_{1})} and \\frac{\\partial}{\\partial M} (- v_{1} + \\frac{\\partial}{\\partial M} a{(M,v_{1})}) = \\frac{\\partial}{\\partial M} (- v_{1} + v_{1}^{M} \\log{(v_{1})}) and \\frac{\\partial^{2}}{\\partial M^{2}} a{(M,v_{1})} = v_{1}^{M} \\log{(v_{1})}^{2}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)), log(Symbol('v_1', commutative=True)))))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)), log(Symbol('v_1', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('a')(Symbol('M', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(2))), Mul(Pow(Symbol('v_1', commutative=True), Symbol('M', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{f}{(y)} = \\cos{(y)}, then obtain \\int \\frac{d}{d y} \\mathbf{f}^{4}{(y)} dy = \\int \\frac{d}{d y} \\mathbf{f}^{3}{(y)} \\cos{(y)} dy", "derivation": "\\mathbf{f}{(y)} = \\cos{(y)} and \\mathbf{f}^{2}{(y)} = \\mathbf{f}{(y)} \\cos{(y)} and \\mathbf{f}^{4}{(y)} = \\mathbf{f}^{3}{(y)} \\cos{(y)} and \\frac{d}{d y} \\mathbf{f}^{4}{(y)} = \\frac{d}{d y} \\mathbf{f}^{3}{(y)} \\cos{(y)} and \\int \\frac{d}{d y} \\mathbf{f}^{4}{(y)} dy = \\int \\frac{d}{d y} \\mathbf{f}^{3}{(y)} \\cos{(y)} dy", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{f}')(Symbol('y', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(3)), cos(Symbol('y', commutative=True))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(4)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(3)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(4)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Mul(Pow(Function('\\\\mathbf{f}')(Symbol('y', commutative=True)), Integer(3)), cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(A,\\mathbf{F})} = A \\mathbf{F}, then obtain \\frac{d}{d A} 0^{\\mathbf{F}} = \\frac{\\partial}{\\partial A} (\\frac{2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})}}{A})^{\\mathbf{F}}", "derivation": "\\varepsilon{(A,\\mathbf{F})} = A \\mathbf{F} and 0 = A \\mathbf{F} - \\varepsilon{(A,\\mathbf{F})} and A \\mathbf{F} - \\varepsilon{(A,\\mathbf{F})} = 2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})} and 0 = 2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})} and 0 = \\frac{2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})}}{A} and 0^{\\mathbf{F}} = (\\frac{2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})}}{A})^{\\mathbf{F}} and \\frac{d}{d A} 0^{\\mathbf{F}} = \\frac{\\partial}{\\partial A} (\\frac{2 A \\mathbf{F} - 2 \\varepsilon{(A,\\mathbf{F})}}{A})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["add", 2, "Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 4, "Symbol('A', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))))"], [["power", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 6, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\varepsilon')(Symbol('A', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(W,r)} = W + r, then obtain \\frac{r^{3} ((\\int (W + r) dW)^{4}) (\\int \\varepsilon_{0}{(W,r)} dW)^{2}}{W + r} = \\frac{r^{3} ((\\int (W + r) dW)^{5}) \\int \\varepsilon_{0}{(W,r)} dW}{W + r}", "derivation": "\\varepsilon_{0}{(W,r)} = W + r and \\int \\varepsilon_{0}{(W,r)} dW = \\int (W + r) dW and (\\int (W + r) dW) \\int \\varepsilon_{0}{(W,r)} dW = (\\int (W + r) dW)^{2} and r (\\int (W + r) dW) \\int \\varepsilon_{0}{(W,r)} dW = r (\\int (W + r) dW)^{2} and r^{2} ((\\int (W + r) dW)^{3}) \\int \\varepsilon_{0}{(W,r)} dW = r^{2} (\\int (W + r) dW)^{4} and \\frac{r^{2} ((\\int (W + r) dW)^{3}) \\int \\varepsilon_{0}{(W,r)} dW}{W + r} = \\frac{r^{2} (\\int (W + r) dW)^{4}}{W + r} and \\frac{r^{3} ((\\int (W + r) dW)^{4}) (\\int \\varepsilon_{0}{(W,r)} dW)^{2}}{W + r} = \\frac{r^{3} ((\\int (W + r) dW)^{5}) \\int \\varepsilon_{0}{(W,r)} dW}{W + r}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Add(Symbol('W', commutative=True), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["times", 2, "Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)))"], [["times", 3, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Symbol('r', commutative=True), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2))))"], [["times", 4, "Mul(Symbol('r', commutative=True), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(2)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(3)), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Pow(Symbol('r', commutative=True), Integer(2)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(4))))"], [["divide", 5, "Add(Symbol('W', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(2)), Pow(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(3)), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Pow(Symbol('r', commutative=True), Integer(2)), Pow(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(4))))"], [["times", 6, "Mul(Symbol('r', commutative=True), Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(3)), Pow(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(4)), Pow(Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2))), Mul(Pow(Symbol('r', commutative=True), Integer(3)), Pow(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Pow(Integral(Add(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(5)), Integral(Function('\\\\varepsilon_0')(Symbol('W', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given s{(t_{1},\\varepsilon)} = t_{1} + \\sin{(\\varepsilon)} and \\psi^{*}{(t_{1},\\varepsilon)} = \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}}, then obtain 2 \\psi^{*}{(t_{1},\\varepsilon)} = \\psi^{*}{(t_{1},\\varepsilon)} + \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}}", "derivation": "s{(t_{1},\\varepsilon)} = t_{1} + \\sin{(\\varepsilon)} and \\frac{s{(t_{1},\\varepsilon)}}{t_{1}} = \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}} and \\psi^{*}{(t_{1},\\varepsilon)} = \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}} and \\frac{2 s{(t_{1},\\varepsilon)}}{t_{1}} = \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}} + \\frac{s{(t_{1},\\varepsilon)}}{t_{1}} and \\frac{s{(t_{1},\\varepsilon)}}{t_{1}} = \\psi^{*}{(t_{1},\\varepsilon)} and 2 \\psi^{*}{(t_{1},\\varepsilon)} = \\psi^{*}{(t_{1},\\varepsilon)} + \\frac{t_{1} + \\sin{(\\varepsilon)}}{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Symbol('t_1', commutative=True)"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 2, "Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True)))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('s')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Function('\\\\psi^*')(Symbol('t_1', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\varepsilon', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(f_{\\mathbf{p}})} = \\sin{(f_{\\mathbf{p}})}, then derive \\int \\operatorname{P_{g}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = f_{E} - \\cos{(f_{\\mathbf{p}})}, then obtain \\frac{d}{d f_{\\mathbf{p}}} \\int \\sin{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\sin{(f_{\\mathbf{p}})}", "derivation": "\\operatorname{P_{g}}{(f_{\\mathbf{p}})} = \\sin{(f_{\\mathbf{p}})} and \\int \\operatorname{P_{g}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\int \\sin{(f_{\\mathbf{p}})} df_{\\mathbf{p}} and \\int \\operatorname{P_{g}}{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = f_{E} - \\cos{(f_{\\mathbf{p}})} and \\int \\sin{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = f_{E} - \\cos{(f_{\\mathbf{p}})} and \\frac{d}{d f_{\\mathbf{p}}} \\int \\sin{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (f_{E} - \\cos{(f_{\\mathbf{p}})}) and \\frac{d}{d f_{\\mathbf{p}}} \\int \\sin{(f_{\\mathbf{p}})} df_{\\mathbf{p}} = \\sin{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(u)} = \\cos{(u)}, then obtain u (u + \\mathbb{I}{(u)})^{2 u} = u (u + \\mathbb{I}{(u)})^{u} (u + \\cos{(u)})^{u}", "derivation": "\\mathbb{I}{(u)} = \\cos{(u)} and u + \\mathbb{I}{(u)} = u + \\cos{(u)} and (u + \\mathbb{I}{(u)})^{u} = (u + \\cos{(u)})^{u} and u (u + \\mathbb{I}{(u)})^{u} = u (u + \\cos{(u)})^{u} and u (u + \\mathbb{I}{(u)})^{u} (u + \\cos{(u)})^{u} = u (u + \\cos{(u)})^{2 u} and u (u + \\mathbb{I}{(u)})^{2 u} = u (u + \\mathbb{I}{(u)})^{u} (u + \\cos{(u)})^{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["times", 3, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["times", 4, "Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))"], "Equality(Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Mul(Integer(2), Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Mul(Integer(2), Symbol('u', commutative=True)))), Mul(Symbol('u', commutative=True), Pow(Add(Symbol('u', commutative=True), Function('\\\\mathbb{I}')(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\theta_1)} = \\cos{(\\theta_1)}, then derive e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} = e^{f^{\\prime} + \\sin{(\\theta_1)}}, then obtain e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} - \\sin{(\\theta_1)} = e^{\\int \\cos{(\\theta_1)} d\\theta_1} - \\sin{(\\theta_1)}", "derivation": "\\operatorname{t_{1}}{(\\theta_1)} = \\cos{(\\theta_1)} and \\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1 = \\int \\cos{(\\theta_1)} d\\theta_1 and e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} = e^{\\int \\cos{(\\theta_1)} d\\theta_1} and e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} = e^{f^{\\prime} + \\sin{(\\theta_1)}} and e^{\\int \\cos{(\\theta_1)} d\\theta_1} = e^{f^{\\prime} + \\sin{(\\theta_1)}} and e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} - \\sin{(\\theta_1)} = e^{f^{\\prime} + \\sin{(\\theta_1)}} - \\sin{(\\theta_1)} and e^{\\int \\operatorname{t_{1}}{(\\theta_1)} d\\theta_1} - \\sin{(\\theta_1)} = e^{\\int \\cos{(\\theta_1)} d\\theta_1} - \\sin{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Integral(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Integral(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), exp(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 4, "sin(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(exp(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Add(exp(Add(Symbol('f^{\\\\prime}', commutative=True), sin(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(exp(Integral(Function('t_1')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))), Add(exp(Integral(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(c,T)} = \\cos{(T c)}, then obtain (\\frac{\\frac{\\partial}{\\partial T} \\operatorname{J_{\\varepsilon}}^{c}{(c,T)}}{\\cos{(T c)}})^{T} = (\\frac{\\frac{\\partial}{\\partial T} \\cos^{c}{(T c)}}{\\cos{(T c)}})^{T}", "derivation": "\\operatorname{J_{\\varepsilon}}{(c,T)} = \\cos{(T c)} and \\operatorname{J_{\\varepsilon}}^{c}{(c,T)} = \\cos^{c}{(T c)} and \\frac{\\partial}{\\partial T} \\operatorname{J_{\\varepsilon}}^{c}{(c,T)} = \\frac{\\partial}{\\partial T} \\cos^{c}{(T c)} and \\frac{\\frac{\\partial}{\\partial T} \\operatorname{J_{\\varepsilon}}^{c}{(c,T)}}{\\cos{(T c)}} = \\frac{\\frac{\\partial}{\\partial T} \\cos^{c}{(T c)}}{\\cos{(T c)}} and (\\frac{\\frac{\\partial}{\\partial T} \\operatorname{J_{\\varepsilon}}^{c}{(c,T)}}{\\cos{(T c)}})^{T} = (\\frac{\\frac{\\partial}{\\partial T} \\cos^{c}{(T c)}}{\\cos{(T c)}})^{T}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('T', commutative=True)), cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Pow(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 3, "cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True)))"], "Equality(Mul(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Integer(-1)), Derivative(Pow(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Integer(-1)), Derivative(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('T', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Integer(-1)), Derivative(Pow(Function('J_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)), Pow(Mul(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Integer(-1)), Derivative(Pow(cos(Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Symbol('T', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(J,M_{E})} = \\frac{J}{M_{E}}, then derive \\frac{\\partial}{\\partial J} \\mathbf{P}{(J,M_{E})} = \\frac{1}{M_{E}}, then obtain 2 \\frac{\\partial}{\\partial J} \\frac{J}{M_{E}} = \\frac{\\partial}{\\partial J} \\frac{J}{M_{E}} + \\frac{1}{M_{E}}", "derivation": "\\mathbf{P}{(J,M_{E})} = \\frac{J}{M_{E}} and \\frac{\\partial}{\\partial J} \\mathbf{P}{(J,M_{E})} = \\frac{\\partial}{\\partial J} \\frac{J}{M_{E}} and \\frac{\\partial}{\\partial J} \\mathbf{P}{(J,M_{E})} = \\frac{1}{M_{E}} and 2 \\frac{\\partial}{\\partial J} \\mathbf{P}{(J,M_{E})} = \\frac{\\partial}{\\partial J} \\mathbf{P}{(J,M_{E})} + \\frac{1}{M_{E}} and 2 \\frac{\\partial}{\\partial J} \\frac{J}{M_{E}} = \\frac{\\partial}{\\partial J} \\frac{J}{M_{E}} + \\frac{1}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Symbol('J', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], [["add", 3, "Derivative(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\mathbf{P}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Derivative(Mul(Symbol('J', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('J', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Symbol('M_E', commutative=True), Integer(-1))))"]]}, {"prompt": "Given p{(\\theta_2,\\theta_1,l)} = - \\theta_1 - \\theta_2 + l, then obtain (\\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{\\theta_2 + p{(\\theta_2,\\theta_1,l)}}{l})^{\\theta_1} = (\\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{- \\theta_1 + l}{l})^{\\theta_1}", "derivation": "p{(\\theta_2,\\theta_1,l)} = - \\theta_1 - \\theta_2 + l and \\theta_2 + p{(\\theta_2,\\theta_1,l)} = - \\theta_1 + l and \\frac{\\theta_2 + p{(\\theta_2,\\theta_1,l)}}{l} = \\frac{- \\theta_1 + l}{l} and \\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{\\theta_2 + p{(\\theta_2,\\theta_1,l)}}{l} = \\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{- \\theta_1 + l}{l} and (\\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{\\theta_2 + p{(\\theta_2,\\theta_1,l)}}{l})^{\\theta_1} = (\\theta_2 + p{(\\theta_2,\\theta_1,l)} + \\frac{- \\theta_1 + l}{l})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Symbol('l', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('l', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True))))), Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('l', commutative=True)))))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True))))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('\\\\theta_2', commutative=True), Function('p')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('l', commutative=True)))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(I,M_{E})} = \\sin{(M_{E}^{I})}, then obtain M_{E}^{I} \\log{(M_{E})} \\cos{(M_{E}^{I})} + \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,M_{E})} = 2 M_{E}^{I} \\log{(M_{E})} \\cos{(M_{E}^{I})}", "derivation": "\\operatorname{P_{e}}{(I,M_{E})} = \\sin{(M_{E}^{I})} and \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,M_{E})} = \\frac{\\partial}{\\partial I} \\sin{(M_{E}^{I})} and \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,M_{E})} + \\frac{\\partial}{\\partial I} \\sin{(M_{E}^{I})} = 2 \\frac{\\partial}{\\partial I} \\sin{(M_{E}^{I})} and M_{E}^{I} \\log{(M_{E})} \\cos{(M_{E}^{I})} + \\frac{\\partial}{\\partial I} \\operatorname{P_{e}}{(I,M_{E})} = 2 M_{E}^{I} \\log{(M_{E})} \\cos{(M_{E}^{I})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('I', commutative=True), Symbol('M_E', commutative=True)), sin(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["add", 2, "Derivative(sin(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(sin(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), log(Symbol('M_E', commutative=True)), cos(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True)))), Derivative(Function('P_e')(Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True)), log(Symbol('M_E', commutative=True)), cos(Pow(Symbol('M_E', commutative=True), Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)} = - \\varepsilon_0 + x^\\prime, then obtain \\frac{\\int - \\frac{\\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)}}{\\varepsilon_0} dx^\\prime}{x^\\prime} = \\frac{\\int - \\frac{- \\varepsilon_0 + x^\\prime}{\\varepsilon_0} dx^\\prime}{x^\\prime}", "derivation": "\\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)} = - \\varepsilon_0 + x^\\prime and - \\frac{\\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)}}{\\varepsilon_0} = - \\frac{- \\varepsilon_0 + x^\\prime}{\\varepsilon_0} and \\int - \\frac{\\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)}}{\\varepsilon_0} dx^\\prime = \\int - \\frac{- \\varepsilon_0 + x^\\prime}{\\varepsilon_0} dx^\\prime and \\frac{\\int - \\frac{\\operatorname{t_{2}}{(\\varepsilon_0,x^\\prime)}}{\\varepsilon_0} dx^\\prime}{x^\\prime} = \\frac{\\int - \\frac{- \\varepsilon_0 + x^\\prime}{\\varepsilon_0} dx^\\prime}{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given E{(\\mathbf{S})} = \\log{(\\mathbf{S})}, then derive \\int E{(\\mathbf{S})} d\\mathbf{S} = W + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S}, then obtain - W - \\mathbf{S} \\log{(\\mathbf{S})} + \\mathbf{S} + \\int E{(\\mathbf{S})} d\\mathbf{S} - 1 = -1", "derivation": "E{(\\mathbf{S})} = \\log{(\\mathbf{S})} and \\int E{(\\mathbf{S})} d\\mathbf{S} = \\int \\log{(\\mathbf{S})} d\\mathbf{S} and \\int E{(\\mathbf{S})} d\\mathbf{S} = W + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} and - W - \\mathbf{S} \\log{(\\mathbf{S})} + \\mathbf{S} + \\int E{(\\mathbf{S})} d\\mathbf{S} = - W - \\mathbf{S} \\log{(\\mathbf{S})} + \\mathbf{S} + \\int \\log{(\\mathbf{S})} d\\mathbf{S} and \\int \\log{(\\mathbf{S})} d\\mathbf{S} = W + \\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} and - W - \\mathbf{S} \\log{(\\mathbf{S})} + \\mathbf{S} + \\int E{(\\mathbf{S})} d\\mathbf{S} = 0 and - W - \\mathbf{S} \\log{(\\mathbf{S})} + \\mathbf{S} + \\int E{(\\mathbf{S})} d\\mathbf{S} - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('W', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 2, "Add(Symbol('W', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True), Integral(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('W', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True), Integral(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(0))"], [["add", 6, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True), Integral(Function('E')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given V{(x,n_{1})} = \\log{(\\frac{x}{n_{1}})}, then derive \\frac{\\partial}{\\partial x} V{(x,n_{1})} = \\frac{1}{x}, then obtain \\frac{\\frac{\\partial}{\\partial x} V{(x,n_{1})}}{\\log{(\\frac{x}{n_{1}})}} = \\frac{1}{x \\log{(\\frac{x}{n_{1}})}}", "derivation": "V{(x,n_{1})} = \\log{(\\frac{x}{n_{1}})} and \\frac{\\partial}{\\partial x} V{(x,n_{1})} = \\frac{\\partial}{\\partial x} \\log{(\\frac{x}{n_{1}})} and \\frac{\\partial}{\\partial x} V{(x,n_{1})} = \\frac{1}{x} and \\frac{\\partial}{\\partial x} \\log{(\\frac{x}{n_{1}})} = \\frac{1}{x} and \\frac{\\frac{\\partial}{\\partial x} \\log{(\\frac{x}{n_{1}})}}{V{(x,n_{1})}} = \\frac{1}{x V{(x,n_{1})}} and \\frac{\\frac{\\partial}{\\partial x} \\log{(\\frac{x}{n_{1}})}}{\\log{(\\frac{x}{n_{1}})}} = \\frac{1}{x \\log{(\\frac{x}{n_{1}})}} and \\frac{\\frac{\\partial}{\\partial x} V{(x,n_{1})}}{\\log{(\\frac{x}{n_{1}})}} = \\frac{1}{x \\log{(\\frac{x}{n_{1}})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["divide", 4, "Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Derivative(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Integer(-1)), Derivative(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Integer(-1)), Derivative(Function('V')(Symbol('x', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(log(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(A,\\mathbf{J}_P)} = \\int (A + \\mathbf{J}_P) d\\mathbf{J}_P, then derive \\operatorname{n_{1}}{(A,\\mathbf{J}_P)} = A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x}, then obtain \\frac{\\operatorname{n_{1}}^{2 \\mathbf{J}_P}{(A,\\mathbf{J}_P)}}{v_{x}^{2}} = \\frac{(A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x})^{2 \\mathbf{J}_P}}{v_{x}^{2}}", "derivation": "\\operatorname{n_{1}}{(A,\\mathbf{J}_P)} = \\int (A + \\mathbf{J}_P) d\\mathbf{J}_P and \\operatorname{n_{1}}{(A,\\mathbf{J}_P)} = A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x} and \\operatorname{n_{1}}^{\\mathbf{J}_P}{(A,\\mathbf{J}_P)} = (A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x})^{\\mathbf{J}_P} and \\frac{\\operatorname{n_{1}}^{\\mathbf{J}_P}{(A,\\mathbf{J}_P)}}{v_{x}} = \\frac{(A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x})^{\\mathbf{J}_P}}{v_{x}} and \\frac{\\operatorname{n_{1}}^{2 \\mathbf{J}_P}{(A,\\mathbf{J}_P)}}{v_{x}^{2}} = \\frac{(A \\mathbf{J}_P + \\frac{\\mathbf{J}_P^{2}}{2} + v_{x})^{2 \\mathbf{J}_P}}{v_{x}^{2}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Symbol('v_x', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Symbol('v_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 3, "Symbol('v_x', commutative=True)"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Symbol('v_x', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Pow(Function('n_1')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Pow(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} = \\frac{C_{2} \\mathbf{H}}{m}, then obtain \\pi^{- \\rho} (- \\pi + \\frac{\\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} - \\frac{1}{m}}{m}) = \\pi^{- \\rho} (- \\pi + \\frac{\\frac{C_{2} \\mathbf{H}}{m} - \\frac{1}{m}}{m})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} = \\frac{C_{2} \\mathbf{H}}{m} and \\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} - \\frac{1}{m} = \\frac{C_{2} \\mathbf{H}}{m} - \\frac{1}{m} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} - \\frac{1}{m}}{m} = \\frac{\\frac{C_{2} \\mathbf{H}}{m} - \\frac{1}{m}}{m} and - \\pi + \\frac{\\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} - \\frac{1}{m}}{m} = - \\pi + \\frac{\\frac{C_{2} \\mathbf{H}}{m} - \\frac{1}{m}}{m} and \\pi^{- \\rho} (- \\pi + \\frac{\\operatorname{V_{\\mathbf{B}}}{(m,C_{2},\\mathbf{H})} - \\frac{1}{m}}{m}) = \\pi^{- \\rho} (- \\pi + \\frac{\\frac{C_{2} \\mathbf{H}}{m} - \\frac{1}{m}}{m})", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('m', commutative=True), Integer(-1))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["divide", 2, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))))"], [["minus", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)))))))"], [["divide", 4, "Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('m', commutative=True), Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))))), Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1))))))))"]]}, {"prompt": "Given \\mathbf{S}{(T,z^{*})} = - T + e^{z^{*}}, then derive \\int (T + \\mathbf{S}{(T,z^{*})}) dz^{*} = \\delta + e^{z^{*}}, then obtain T + \\delta + \\mathbf{S}{(T,z^{*})} = \\int e^{z^{*}} dz^{*}", "derivation": "\\mathbf{S}{(T,z^{*})} = - T + e^{z^{*}} and T + \\mathbf{S}{(T,z^{*})} = e^{z^{*}} and \\int (T + \\mathbf{S}{(T,z^{*})}) dz^{*} = \\int e^{z^{*}} dz^{*} and \\int (T + \\mathbf{S}{(T,z^{*})}) dz^{*} = \\delta + e^{z^{*}} and \\int (T + \\mathbf{S}{(T,z^{*})}) dz^{*} = T + \\delta + \\mathbf{S}{(T,z^{*})} and T + \\delta + \\mathbf{S}{(T,z^{*})} = \\int e^{z^{*}} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), exp(Symbol('z^*', commutative=True))))"], [["add", 1, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))), exp(Symbol('z^*', commutative=True)))"], [["integrate", 2, "Symbol('z^*', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Integral(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('T', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('\\\\delta', commutative=True), exp(Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Add(Symbol('T', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('T', commutative=True), Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('T', commutative=True), Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('z^*', commutative=True))), Integral(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(l,J_{\\varepsilon})} = \\frac{l}{J_{\\varepsilon}}, then obtain l \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{H}{(l,J_{\\varepsilon})} = - \\frac{l^{2}}{J_{\\varepsilon}^{2}}", "derivation": "\\mathbf{H}{(l,J_{\\varepsilon})} = \\frac{l}{J_{\\varepsilon}} and l \\mathbf{H}{(l,J_{\\varepsilon})} = \\frac{l^{2}}{J_{\\varepsilon}} and \\frac{\\partial}{\\partial J_{\\varepsilon}} l \\mathbf{H}{(l,J_{\\varepsilon})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} \\frac{l^{2}}{J_{\\varepsilon}} and l \\frac{\\partial}{\\partial J_{\\varepsilon}} \\mathbf{H}{(l,J_{\\varepsilon})} = - \\frac{l^{2}}{J_{\\varepsilon}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Symbol('l', commutative=True), Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(2))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('l', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('l', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\eta,\\hat{X})} = \\hat{X}^{\\eta}, then obtain \\frac{(e^{\\hat{X} + \\operatorname{A_{x}}{(\\eta,\\hat{X})}})^{\\hat{X}}}{\\operatorname{A_{x}}{(\\eta,\\hat{X})}} = \\frac{(e^{\\hat{X} + \\hat{X}^{\\eta}})^{\\hat{X}}}{\\operatorname{A_{x}}{(\\eta,\\hat{X})}}", "derivation": "\\operatorname{A_{x}}{(\\eta,\\hat{X})} = \\hat{X}^{\\eta} and \\hat{X} + \\operatorname{A_{x}}{(\\eta,\\hat{X})} = \\hat{X} + \\hat{X}^{\\eta} and e^{\\hat{X} + \\operatorname{A_{x}}{(\\eta,\\hat{X})}} = e^{\\hat{X} + \\hat{X}^{\\eta}} and (e^{\\hat{X} + \\operatorname{A_{x}}{(\\eta,\\hat{X})}})^{\\hat{X}} = (e^{\\hat{X} + \\hat{X}^{\\eta}})^{\\hat{X}} and \\frac{(e^{\\hat{X} + \\operatorname{A_{x}}{(\\eta,\\hat{X})}})^{\\hat{X}}}{\\operatorname{A_{x}}{(\\eta,\\hat{X})}} = \\frac{(e^{\\hat{X} + \\hat{X}^{\\eta}})^{\\hat{X}}}{\\operatorname{A_{x}}{(\\eta,\\hat{X})}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["exp", 2], "Equality(exp(Add(Symbol('\\\\hat{X}', commutative=True), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), exp(Add(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(exp(Add(Symbol('\\\\hat{X}', commutative=True), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)), Pow(exp(Add(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\eta', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 4, "Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Pow(Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(exp(Add(Symbol('\\\\hat{X}', commutative=True), Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Function('A_x')(Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(exp(Add(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\eta', commutative=True)))), Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given h{(\\hat{H}_l,\\hat{X})} = \\hat{H}_l \\hat{X}, then obtain \\int \\frac{h{(\\hat{H}_l,\\hat{X})}}{\\hat{H}_l^{2}} d\\hat{H}_l + \\frac{1}{\\hat{H}_l} = \\int \\frac{\\hat{X}}{\\hat{H}_l} d\\hat{H}_l + \\frac{1}{\\hat{H}_l}", "derivation": "h{(\\hat{H}_l,\\hat{X})} = \\hat{H}_l \\hat{X} and \\frac{h{(\\hat{H}_l,\\hat{X})}}{\\hat{H}_l} = \\hat{X} and \\frac{h{(\\hat{H}_l,\\hat{X})}}{\\hat{H}_l^{2}} = \\frac{\\hat{X}}{\\hat{H}_l} and \\int \\frac{h{(\\hat{H}_l,\\hat{X})}}{\\hat{H}_l^{2}} d\\hat{H}_l = \\int \\frac{\\hat{X}}{\\hat{H}_l} d\\hat{H}_l and \\int \\frac{h{(\\hat{H}_l,\\hat{X})}}{\\hat{H}_l^{2}} d\\hat{H}_l + \\frac{1}{\\hat{H}_l} = \\int \\frac{\\hat{X}}{\\hat{H}_l} d\\hat{H}_l + \\frac{1}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))"], [["divide", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 4, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))), Add(Integral(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1))))"]]}, {"prompt": "Given b{(\\theta_2,\\hat{x})} = \\hat{x} + \\theta_2, then derive \\int b{(\\theta_2,\\hat{x})} d\\theta_2 = \\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1}, then obtain \\frac{\\partial}{\\partial v_{1}} \\int (\\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1}) d\\theta_2 = \\frac{\\partial}{\\partial v_{1}} \\iint (\\hat{x} + \\theta_2) d\\theta_2 d\\theta_2", "derivation": "b{(\\theta_2,\\hat{x})} = \\hat{x} + \\theta_2 and \\int b{(\\theta_2,\\hat{x})} d\\theta_2 = \\int (\\hat{x} + \\theta_2) d\\theta_2 and \\int b{(\\theta_2,\\hat{x})} d\\theta_2 = \\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1} and \\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1} = \\int (\\hat{x} + \\theta_2) d\\theta_2 and \\int (\\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1}) d\\theta_2 = \\iint (\\hat{x} + \\theta_2) d\\theta_2 d\\theta_2 and \\frac{\\partial}{\\partial v_{1}} \\int (\\hat{x} \\theta_2 + \\frac{\\theta_2^{2}}{2} + v_{1}) d\\theta_2 = \\frac{\\partial}{\\partial v_{1}} \\iint (\\hat{x} + \\theta_2) d\\theta_2 d\\theta_2", "srepr_derivation": [["renaming_premise", "Equality(Function('b')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(J)} = J, then derive \\frac{\\int G{(J)} dJ}{\\frac{J^{2}}{2} + f^{\\prime}} = 1, then obtain (\\frac{J^{2}}{2} + f^{\\prime}) \\int \\frac{\\int J dJ}{\\frac{J^{2}}{2} + f^{\\prime}} df^{\\prime} = (\\frac{J^{2}}{2} + f^{\\prime}) \\int 1 df^{\\prime}", "derivation": "G{(J)} = J and \\int G{(J)} dJ = \\int J dJ and \\frac{\\int G{(J)} dJ}{\\int J dJ} = 1 and \\frac{\\int G{(J)} dJ}{\\frac{J^{2}}{2} + f^{\\prime}} = 1 and \\int \\frac{\\int G{(J)} dJ}{\\frac{J^{2}}{2} + f^{\\prime}} df^{\\prime} = \\int 1 df^{\\prime} and (\\frac{J^{2}}{2} + f^{\\prime}) \\int \\frac{\\int G{(J)} dJ}{\\frac{J^{2}}{2} + f^{\\prime}} df^{\\prime} = (\\frac{J^{2}}{2} + f^{\\prime}) \\int 1 df^{\\prime} and (\\frac{J^{2}}{2} + f^{\\prime}) \\int \\frac{\\int J dJ}{\\frac{J^{2}}{2} + f^{\\prime}} df^{\\prime} = (\\frac{J^{2}}{2} + f^{\\prime}) \\int 1 df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('G')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True))))"], [["divide", 2, "Integral(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Integral(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True))), Integer(-1)), Integral(Function('G')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Function('G')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Integer(1))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Function('G')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 5, "Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Function('G')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Integral(Symbol('J', commutative=True), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Symbol('f^{\\\\prime}', commutative=True)), Integral(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given S{(\\Omega)} = e^{\\Omega} and \\operatorname{v_{z}}{(\\Omega)} = e^{\\Omega}, then obtain S^{3}{(\\Omega)} \\operatorname{v_{z}}^{3}{(\\Omega)} = S^{4}{(\\Omega)} \\operatorname{v_{z}}^{2}{(\\Omega)}", "derivation": "S{(\\Omega)} = e^{\\Omega} and \\operatorname{v_{z}}{(\\Omega)} = e^{\\Omega} and \\operatorname{v_{z}}^{2}{(\\Omega)} e^{\\Omega} = \\operatorname{v_{z}}{(\\Omega)} e^{2 \\Omega} and S{(\\Omega)} \\operatorname{v_{z}}^{2}{(\\Omega)} = S^{2}{(\\Omega)} \\operatorname{v_{z}}{(\\Omega)} and S^{3}{(\\Omega)} \\operatorname{v_{z}}^{3}{(\\Omega)} = S^{4}{(\\Omega)} \\operatorname{v_{z}}^{2}{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Mul(Function('v_z')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Pow(Function('v_z')(Symbol('\\\\Omega', commutative=True)), Integer(2)), exp(Symbol('\\\\Omega', commutative=True))), Mul(Function('v_z')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('S')(Symbol('\\\\Omega', commutative=True)), Pow(Function('v_z')(Symbol('\\\\Omega', commutative=True)), Integer(2))), Mul(Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Function('v_z')(Symbol('\\\\Omega', commutative=True))))"], [["times", 4, "Mul(Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Function('v_z')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(3)), Pow(Function('v_z')(Symbol('\\\\Omega', commutative=True)), Integer(3))), Mul(Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(4)), Pow(Function('v_z')(Symbol('\\\\Omega', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(i)} = \\cos{(i)} and f{(i)} = \\int \\operatorname{C_{d}}{(i)} di, then obtain \\frac{f^{i}{(i)}}{f{(i)}} = \\frac{(\\dot{x} + \\sin{(i)})^{i}}{f{(i)}}", "derivation": "\\operatorname{C_{d}}{(i)} = \\cos{(i)} and \\int \\operatorname{C_{d}}{(i)} di = \\int \\cos{(i)} di and f{(i)} = \\int \\operatorname{C_{d}}{(i)} di and f{(i)} = \\int \\cos{(i)} di and f^{i}{(i)} = (\\int \\cos{(i)} di)^{i} and \\frac{f^{i}{(i)}}{f{(i)}} = \\frac{(\\int \\cos{(i)} di)^{i}}{f{(i)}} and \\frac{f^{i}{(i)}}{f{(i)}} = \\frac{(\\dot{x} + \\sin{(i)})^{i}}{f{(i)}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('f')(Symbol('i', commutative=True)), Integral(Function('C_d')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('f')(Symbol('i', commutative=True)), Integral(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Function('f')(Symbol('i', commutative=True)), Symbol('i', commutative=True)), Pow(Integral(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["times", 5, "Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1)), Pow(Function('f')(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Mul(Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1)), Pow(Integral(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1)), Pow(Function('f')(Symbol('i', commutative=True)), Symbol('i', commutative=True))), Mul(Pow(Add(Symbol('\\\\dot{x}', commutative=True), sin(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Function('f')(Symbol('i', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{E})} = \\log{(\\cos{(\\mathbf{E})})}, then derive \\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})} = - \\frac{\\sin{(\\mathbf{E})}}{\\cos{(\\mathbf{E})}}, then obtain - \\frac{\\sin{(\\mathbf{E})}}{\\cos{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})}} = \\frac{\\frac{d}{d \\mathbf{E}} \\log{(\\cos{(\\mathbf{E})})}}{\\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})}}", "derivation": "\\phi_{1}{(\\mathbf{E})} = \\log{(\\cos{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\log{(\\cos{(\\mathbf{E})})} and \\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})} = - \\frac{\\sin{(\\mathbf{E})}}{\\cos{(\\mathbf{E})}} and - \\frac{\\sin{(\\mathbf{E})}}{\\cos{(\\mathbf{E})}} = \\frac{d}{d \\mathbf{E}} \\log{(\\cos{(\\mathbf{E})})} and - \\frac{\\sin{(\\mathbf{E})}}{\\cos{(\\mathbf{E})} \\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})}} = \\frac{\\frac{d}{d \\mathbf{E}} \\log{(\\cos{(\\mathbf{E})})}}{\\frac{d}{d \\mathbf{E}} \\phi_{1}{(\\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), log(cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(log(cos(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Derivative(log(cos(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Derivative(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(-1)), Derivative(log(cos(Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(i)} = \\log{(i)}, then derive z^{*} + \\dot{x}{(i)} = v_{t} + \\log{(i)}, then obtain \\sin{(z^{*} + \\log{(i)})} = \\sin{(v_{t} + \\log{(i)})}", "derivation": "\\dot{x}{(i)} = \\log{(i)} and \\frac{d}{d i} \\dot{x}{(i)} = \\frac{d}{d i} \\log{(i)} and \\int \\frac{d}{d i} \\dot{x}{(i)} di = \\int \\frac{d}{d i} \\log{(i)} di and z^{*} + \\dot{x}{(i)} = v_{t} + \\log{(i)} and z^{*} + \\log{(i)} = v_{t} + \\log{(i)} and \\sin{(z^{*} + \\log{(i)})} = \\sin{(v_{t} + \\log{(i)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{x}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Derivative(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('z^*', commutative=True), Function('\\\\dot{x}')(Symbol('i', commutative=True))), Add(Symbol('v_t', commutative=True), log(Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('z^*', commutative=True), log(Symbol('i', commutative=True))), Add(Symbol('v_t', commutative=True), log(Symbol('i', commutative=True))))"], [["sin", 5], "Equality(sin(Add(Symbol('z^*', commutative=True), log(Symbol('i', commutative=True)))), sin(Add(Symbol('v_t', commutative=True), log(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\lambda,F_{c})} = F_{c} - \\lambda, then derive \\frac{\\partial}{\\partial F_{c}} \\operatorname{E_{n}}{(\\lambda,F_{c})} = 1, then obtain F_{c} (\\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda))^{2} = F_{c}", "derivation": "\\operatorname{E_{n}}{(\\lambda,F_{c})} = F_{c} - \\lambda and \\frac{\\partial}{\\partial F_{c}} \\operatorname{E_{n}}{(\\lambda,F_{c})} = \\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda) and \\frac{\\partial}{\\partial F_{c}} \\operatorname{E_{n}}{(\\lambda,F_{c})} = 1 and \\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda) = 1 and F_{c} \\frac{\\partial}{\\partial F_{c}} \\operatorname{E_{n}}{(\\lambda,F_{c})} = F_{c} and F_{c} \\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda) = F_{c} and (\\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda))^{2} = \\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda) and F_{c} (\\frac{\\partial}{\\partial F_{c}} (F_{c} - \\lambda))^{2} = F_{c}", "srepr_derivation": [["get_premise", "Equality(Function('E_n')(Symbol('\\\\lambda', commutative=True), Symbol('F_c', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\lambda', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\lambda', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Derivative(Function('E_n')(Symbol('\\\\lambda', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Symbol('F_c', commutative=True))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('F_c', commutative=True), Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Symbol('F_c', commutative=True))"], [["times", 4, "Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(2)), Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Symbol('F_c', commutative=True), Pow(Derivative(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(2))), Symbol('F_c', commutative=True))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\omega,\\mathbf{r})} = \\mathbf{r} + \\omega and \\mathbf{B}{(\\omega,\\mathbf{r})} = \\mathbf{r} + \\omega, then obtain 0^{\\omega} = (- \\mathbf{r} - \\omega + \\operatorname{F_{c}}{(\\omega,\\mathbf{r})})^{\\omega}", "derivation": "\\operatorname{F_{c}}{(\\omega,\\mathbf{r})} = \\mathbf{r} + \\omega and \\mathbf{B}{(\\omega,\\mathbf{r})} = \\mathbf{r} + \\omega and \\operatorname{F_{c}}{(\\omega,\\mathbf{r})} = \\mathbf{B}{(\\omega,\\mathbf{r})} and - \\operatorname{F_{c}}{(\\omega,\\mathbf{r})} = - \\mathbf{B}{(\\omega,\\mathbf{r})} and - \\operatorname{F_{c}}{(\\omega,\\mathbf{r})} = - \\mathbf{r} - \\omega and 0 = - \\mathbf{r} - \\omega + \\operatorname{F_{c}}{(\\omega,\\mathbf{r})} and 0^{\\omega} = (- \\mathbf{r} - \\omega + \\operatorname{F_{c}}{(\\omega,\\mathbf{r})})^{\\omega}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given p{(h,U)} = U^{h}, then obtain - 2 U U^{h} p{(h,U)} + U^{h} + p{(h,U)} = - 2 U U^{h} p{(h,U)} + 2 U^{h}", "derivation": "p{(h,U)} = U^{h} and U^{h} + p{(h,U)} = 2 U^{h} and (U^{h} + p{(h,U)}) p{(h,U)} = 2 U^{h} p{(h,U)} and U (U^{h} + p{(h,U)}) p{(h,U)} = 2 U U^{h} p{(h,U)} and - U (U^{h} + p{(h,U)}) p{(h,U)} + U^{h} + p{(h,U)} = - U (U^{h} + p{(h,U)}) p{(h,U)} + 2 U^{h} and - 2 U U^{h} p{(h,U)} + U^{h} + p{(h,U)} = - 2 U U^{h} p{(h,U)} + 2 U^{h}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Pow(Symbol('U', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True))))"], [["times", 2, "Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))))"], [["times", 3, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))))"], [["minus", 2, "Mul(Symbol('U', commutative=True), Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True), Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True), Add(Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)), Function('p')(Symbol('h', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{1})} = \\log{(\\log{(C_{1})})}, then obtain \\frac{e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\Psi^{\\dagger}{(C_{1})}}}{\\Psi^{\\dagger}{(C_{1})}} = \\frac{e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\log{(\\log{(C_{1})})}}}{\\Psi^{\\dagger}{(C_{1})}}", "derivation": "\\Psi^{\\dagger}{(C_{1})} = \\log{(\\log{(C_{1})})} and (C_{1} + \\log{(\\log{(C_{1})})}) \\Psi^{\\dagger}{(C_{1})} = (C_{1} + \\log{(\\log{(C_{1})})}) \\log{(\\log{(C_{1})})} and e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\Psi^{\\dagger}{(C_{1})}} = e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\log{(\\log{(C_{1})})}} and \\frac{e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\Psi^{\\dagger}{(C_{1})}}}{\\Psi^{\\dagger}{(C_{1})}} = \\frac{e^{(C_{1} + \\log{(\\log{(C_{1})})}) \\log{(\\log{(C_{1})})}}}{\\Psi^{\\dagger}{(C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True)), log(log(Symbol('C_1', commutative=True))))"], [["times", 1, "Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True))))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True))), Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), log(log(Symbol('C_1', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True)))), exp(Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), log(log(Symbol('C_1', commutative=True))))))"], [["divide", 3, "Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True))))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Mul(Add(Symbol('C_1', commutative=True), log(log(Symbol('C_1', commutative=True)))), log(log(Symbol('C_1', commutative=True)))))))"]]}, {"prompt": "Given v{(\\varepsilon_0,J)} = J^{\\varepsilon_0} and \\theta_{2}{(\\varepsilon_0,J)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\int J^{\\varepsilon_0} dJ, then obtain J^{\\varepsilon_0} + \\frac{\\partial}{\\partial \\varepsilon_0} \\int v{(\\varepsilon_0,J)} dJ = J^{\\varepsilon_0} + \\theta_{2}{(\\varepsilon_0,J)}", "derivation": "v{(\\varepsilon_0,J)} = J^{\\varepsilon_0} and \\int v{(\\varepsilon_0,J)} dJ = \\int J^{\\varepsilon_0} dJ and \\frac{\\partial}{\\partial \\varepsilon_0} \\int v{(\\varepsilon_0,J)} dJ = \\frac{\\partial}{\\partial \\varepsilon_0} \\int J^{\\varepsilon_0} dJ and \\theta_{2}{(\\varepsilon_0,J)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\int J^{\\varepsilon_0} dJ and \\frac{\\partial}{\\partial \\varepsilon_0} \\int v{(\\varepsilon_0,J)} dJ = \\theta_{2}{(\\varepsilon_0,J)} and J^{\\varepsilon_0} + \\frac{\\partial}{\\partial \\varepsilon_0} \\int v{(\\varepsilon_0,J)} dJ = J^{\\varepsilon_0} + \\theta_{2}{(\\varepsilon_0,J)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Integral(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Derivative(Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integral(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)))"], [["add", 5, "Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Integral(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Add(Pow(Symbol('J', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(n_{1})} = \\sin{(e^{n_{1}})}, then derive \\frac{d}{d n_{1}} \\operatorname{A_{y}}{(n_{1})} = e^{n_{1}} \\cos{(e^{n_{1}})}, then obtain \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} = e^{n_{1}} \\cos{(e^{n_{1}})}", "derivation": "\\operatorname{A_{y}}{(n_{1})} = \\sin{(e^{n_{1}})} and \\frac{d}{d n_{1}} \\operatorname{A_{y}}{(n_{1})} = \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} and \\frac{d}{d n_{1}} \\operatorname{A_{y}}{(n_{1})} = e^{n_{1}} \\cos{(e^{n_{1}})} and \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} = e^{n_{1}} \\cos{(e^{n_{1}})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('n_1', commutative=True)), sin(exp(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(exp(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(exp(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(t_{2},r)} = \\frac{t_{2}}{r} and \\operatorname{E_{x}}{(t_{2})} = t_{2}, then obtain (r + t_{2} + \\operatorname{r_{0}}{(t_{2},r)})^{r} = (r + t_{2} + \\frac{t_{2}}{r})^{r}", "derivation": "\\operatorname{r_{0}}{(t_{2},r)} = \\frac{t_{2}}{r} and \\operatorname{E_{x}}{(t_{2})} = t_{2} and r + \\operatorname{r_{0}}{(t_{2},r)} = r + \\frac{t_{2}}{r} and r + \\operatorname{E_{x}}{(t_{2})} + \\operatorname{r_{0}}{(t_{2},r)} = r + \\operatorname{E_{x}}{(t_{2})} + \\frac{t_{2}}{r} and r + t_{2} + \\operatorname{r_{0}}{(t_{2},r)} = r + t_{2} + \\frac{t_{2}}{r} and (r + t_{2} + \\operatorname{r_{0}}{(t_{2},r)})^{r} = (r + t_{2} + \\frac{t_{2}}{r})^{r}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('t_2', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], [["add", 1, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Function('r_0')(Symbol('t_2', commutative=True), Symbol('r', commutative=True))), Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["add", 3, "Function('E_x')(Symbol('t_2', commutative=True))"], "Equality(Add(Symbol('r', commutative=True), Function('E_x')(Symbol('t_2', commutative=True)), Function('r_0')(Symbol('t_2', commutative=True), Symbol('r', commutative=True))), Add(Symbol('r', commutative=True), Function('E_x')(Symbol('t_2', commutative=True)), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('r', commutative=True), Symbol('t_2', commutative=True), Function('r_0')(Symbol('t_2', commutative=True), Symbol('r', commutative=True))), Add(Symbol('r', commutative=True), Symbol('t_2', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["power", 5, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Symbol('r', commutative=True), Symbol('t_2', commutative=True), Function('r_0')(Symbol('t_2', commutative=True), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Add(Symbol('r', commutative=True), Symbol('t_2', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\psi{(\\mathbf{J}_f,\\varphi^*)} = \\frac{\\varphi^*}{\\mathbf{J}_f}, then derive 1 = \\frac{1}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}}, then obtain 1 + \\frac{1}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}} = \\frac{2}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}}", "derivation": "\\psi{(\\mathbf{J}_f,\\varphi^*)} = \\frac{\\varphi^*}{\\mathbf{J}_f} and \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\varphi^*}{\\mathbf{J}_f} and 1 = \\frac{\\frac{\\partial}{\\partial \\varphi^*} \\frac{\\varphi^*}{\\mathbf{J}_f}}{\\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}} and 1 = \\frac{1}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}} and 1 + \\frac{1}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}} = \\frac{2}{\\mathbf{J}_f \\frac{\\partial}{\\partial \\varphi^*} \\psi{(\\mathbf{J}_f,\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(-1))))"], [["add", 4, "Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(-1)))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})}, then obtain \\operatorname{a^{\\dagger}}{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\sin^{\\mathbf{p}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\sin^{\\mathbf{p}}{(\\mathbf{p})}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\operatorname{a^{\\dagger}}^{\\mathbf{p}}{(\\mathbf{p})} = \\sin^{\\mathbf{p}}{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\operatorname{a^{\\dagger}}^{\\mathbf{p}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\sin^{\\mathbf{p}}{(\\mathbf{p})} and \\operatorname{a^{\\dagger}}{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\operatorname{a^{\\dagger}}^{\\mathbf{p}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\operatorname{a^{\\dagger}}^{\\mathbf{p}}{(\\mathbf{p})} and \\operatorname{a^{\\dagger}}{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\sin^{\\mathbf{p}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} \\frac{d}{d \\mathbf{p}} \\sin^{\\mathbf{p}}{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(sin(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(sin(Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\ddot{x}{(b,\\chi)} = \\chi + b and \\mathbf{J}_P{(b,\\chi)} = \\ddot{x}{(b,\\chi)} - 1, then obtain - \\chi + \\mathbf{J}_P{(b,\\chi)} = b - 1", "derivation": "\\ddot{x}{(b,\\chi)} = \\chi + b and \\ddot{x}{(b,\\chi)} - 1 = \\chi + b - 1 and \\mathbf{J}_P{(b,\\chi)} = \\ddot{x}{(b,\\chi)} - 1 and \\mathbf{J}_P{(b,\\chi)} = \\chi + b - 1 and - \\chi + \\mathbf{J}_P{(b,\\chi)} = b - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('b', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Symbol('b', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Function('\\\\ddot{x}')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('b', commutative=True), Integer(-1)))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('b', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Symbol('b', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\phi_1)} = \\sin{(\\phi_1)}, then derive \\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)} = \\cos{(\\phi_1)}, then obtain \\frac{\\frac{d}{d \\phi_1} \\sin{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)}} = \\frac{\\cos{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)}}", "derivation": "\\operatorname{f^{\\prime}}{(\\phi_1)} = \\sin{(\\phi_1)} and \\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\sin{(\\phi_1)} and \\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)} = \\cos{(\\phi_1)} and \\frac{d}{d \\phi_1} \\sin{(\\phi_1)} = \\cos{(\\phi_1)} and \\frac{\\frac{d}{d \\phi_1} \\sin{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)}} = \\frac{\\cos{(\\phi_1)}}{\\frac{d}{d \\phi_1} \\operatorname{f^{\\prime}}{(\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), cos(Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), cos(Symbol('\\\\phi_1', commutative=True)))"], [["divide", 4, "Derivative(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1)), Derivative(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\lambda{(n_{1},L)} = \\log{(L n_{1})}, then obtain (L + \\lambda{(n_{1},L)} - \\log{(L n_{1})})^{L} = L^{L}", "derivation": "\\lambda{(n_{1},L)} = \\log{(L n_{1})} and \\lambda{(n_{1},L)} - \\log{(L n_{1})} = 0 and L + \\lambda{(n_{1},L)} - \\log{(L n_{1})} = L and (L + \\lambda{(n_{1},L)} - \\log{(L n_{1})})^{L} = L^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), log(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))))"], [["minus", 1, "log(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))))), Integer(0))"], [["add", 2, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('\\\\lambda')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))))), Symbol('L', commutative=True))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Symbol('L', commutative=True), Function('\\\\lambda')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))))), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('L', commutative=True)))"]]}, {"prompt": "Given Q{(C,G)} = - C + G, then derive \\frac{\\partial}{\\partial G} Q{(C,G)} = 1, then obtain - E_{n} \\frac{\\partial^{2}}{\\partial G^{2}} (- C + G) = - E_{n} \\frac{d}{d G} 1", "derivation": "Q{(C,G)} = - C + G and \\frac{\\partial}{\\partial G} Q{(C,G)} = \\frac{\\partial}{\\partial G} (- C + G) and \\frac{\\partial}{\\partial G} Q{(C,G)} = 1 and \\frac{\\partial}{\\partial G} (- C + G) = 1 and \\frac{\\partial^{2}}{\\partial G^{2}} (- C + G) = \\frac{d}{d G} 1 and - E_{n} \\frac{\\partial^{2}}{\\partial G^{2}} (- C + G) = - E_{n} \\frac{d}{d G} 1", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('C', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Q')(Symbol('C', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["times", 5, "Mul(Integer(-1), Symbol('E_n', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E_n', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))), Mul(Integer(-1), Symbol('E_n', commutative=True), Derivative(Integer(1), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mu,f^{\\prime})} = \\cos{(\\mu + f^{\\prime})}, then obtain - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} + \\frac{\\operatorname{E_{x}}{(\\mu,f^{\\prime})}}{\\cos{(\\mu + f^{\\prime})}} - \\frac{1}{\\cos{(\\mu + f^{\\prime})}} = - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} + 1 - \\frac{1}{\\cos{(\\mu + f^{\\prime})}}", "derivation": "\\operatorname{E_{x}}{(\\mu,f^{\\prime})} = \\cos{(\\mu + f^{\\prime})} and \\operatorname{E_{x}}{(\\mu,f^{\\prime})} \\cos{(\\mu + f^{\\prime})} = \\cos^{2}{(\\mu + f^{\\prime})} and \\frac{\\operatorname{E_{x}}{(\\mu,f^{\\prime})}}{\\cos{(\\mu + f^{\\prime})}} = 1 and - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} + \\frac{\\operatorname{E_{x}}{(\\mu,f^{\\prime})}}{\\cos{(\\mu + f^{\\prime})}} = 1 - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} and - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} + \\frac{\\operatorname{E_{x}}{(\\mu,f^{\\prime})}}{\\cos{(\\mu + f^{\\prime})}} - \\frac{1}{\\cos{(\\mu + f^{\\prime})}} = - \\operatorname{E_{x}}{(\\mu,f^{\\prime})} + 1 - \\frac{1}{\\cos{(\\mu + f^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))"], [["divide", 2, "Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(2))"], "Equality(Mul(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 4, "Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(1), Mul(Integer(-1), Pow(cos(Add(Symbol('\\\\mu', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given S{(I)} = e^{I} and Z{(I)} = e^{I}, then obtain ((- I + S{(I)})^{2} - 1) Z{(I)} = ((- I + S{(I)})^{2} - 1) e^{I}", "derivation": "S{(I)} = e^{I} and - I + S{(I)} = - I + e^{I} and Z{(I)} = e^{I} and ((- I + e^{I})^{2} - 1) Z{(I)} = ((- I + e^{I})^{2} - 1) e^{I} and ((- I + S{(I)})^{2} - 1) Z{(I)} = ((- I + S{(I)})^{2} - 1) e^{I}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["times", 3, "Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2)), Integer(-1))"], "Equality(Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2)), Integer(-1)), Function('Z')(Symbol('I', commutative=True))), Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True))), Integer(2)), Integer(-1)), exp(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True))), Integer(2)), Integer(-1)), Function('Z')(Symbol('I', commutative=True))), Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True))), Integer(2)), Integer(-1)), exp(Symbol('I', commutative=True))))"]]}, {"prompt": "Given q{(V_{\\mathbf{B}},\\nabla)} = \\frac{V_{\\mathbf{B}}}{\\nabla}, then obtain q{(V_{\\mathbf{B}},\\nabla)} - 1 + \\frac{q{(V_{\\mathbf{B}},\\nabla)}}{\\nabla} = \\frac{V_{\\mathbf{B}}}{\\nabla^{2}} + q{(V_{\\mathbf{B}},\\nabla)} - 1", "derivation": "q{(V_{\\mathbf{B}},\\nabla)} = \\frac{V_{\\mathbf{B}}}{\\nabla} and q{(V_{\\mathbf{B}},\\nabla)} - 1 = \\frac{V_{\\mathbf{B}}}{\\nabla} - 1 and \\frac{q{(V_{\\mathbf{B}},\\nabla)}}{\\nabla} = \\frac{V_{\\mathbf{B}}}{\\nabla^{2}} and \\frac{V_{\\mathbf{B}}}{\\nabla} - 1 + \\frac{q{(V_{\\mathbf{B}},\\nabla)}}{\\nabla} = \\frac{V_{\\mathbf{B}}}{\\nabla} + \\frac{V_{\\mathbf{B}}}{\\nabla^{2}} - 1 and q{(V_{\\mathbf{B}},\\nabla)} - 1 + \\frac{q{(V_{\\mathbf{B}},\\nabla)}}{\\nabla} = \\frac{V_{\\mathbf{B}}}{\\nabla^{2}} + q{(V_{\\mathbf{B}},\\nabla)} - 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Integer(-1)))"], [["times", 1, "Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))))"], [["add", 3, "Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Integer(-1))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Integer(-1), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)))), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Function('q')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)}, then derive \\int \\mathbf{J}_f{(\\varepsilon_0)} d\\varepsilon_0 = m - \\cos{(\\varepsilon_0)}, then obtain \\cos{(\\dot{x} - \\cos{(\\varepsilon_0)})} = \\cos{(m - \\cos{(\\varepsilon_0)})}", "derivation": "\\mathbf{J}_f{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\int \\mathbf{J}_f{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and \\int \\mathbf{J}_f{(\\varepsilon_0)} d\\varepsilon_0 = m - \\cos{(\\varepsilon_0)} and \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 = m - \\cos{(\\varepsilon_0)} and \\cos{(\\int \\sin{(\\varepsilon_0)} d\\varepsilon_0)} = \\cos{(m - \\cos{(\\varepsilon_0)})} and \\cos{(\\dot{x} - \\cos{(\\varepsilon_0)})} = \\cos{(m - \\cos{(\\varepsilon_0)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('m', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('m', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True)))))"], [["cos", 4], "Equality(cos(Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), cos(Add(Symbol('m', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(cos(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))), cos(Add(Symbol('m', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varepsilon_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(H)} = \\sin{(H)}, then derive H \\int \\operatorname{F_{x}}{(H)} dH = H (v_{y} - \\cos{(H)}), then obtain \\int (H \\int \\operatorname{F_{x}}{(H)} dH)^{v_{y}} dv_{y} = \\int (H (v_{y} - \\cos{(H)}))^{v_{y}} dv_{y}", "derivation": "\\operatorname{F_{x}}{(H)} = \\sin{(H)} and \\int \\operatorname{F_{x}}{(H)} dH = \\int \\sin{(H)} dH and H \\int \\operatorname{F_{x}}{(H)} dH = H \\int \\sin{(H)} dH and H \\int \\operatorname{F_{x}}{(H)} dH = H (v_{y} - \\cos{(H)}) and (H \\int \\operatorname{F_{x}}{(H)} dH)^{v_{y}} = (H (v_{y} - \\cos{(H)}))^{v_{y}} and \\int (H \\int \\operatorname{F_{x}}{(H)} dH)^{v_{y}} dv_{y} = \\int (H (v_{y} - \\cos{(H)}))^{v_{y}} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["times", 2, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Integral(Function('F_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Symbol('H', commutative=True), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('H', commutative=True), Integral(Function('F_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Mul(Symbol('H', commutative=True), Add(Symbol('v_y', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True))))))"], [["power", 4, "Symbol('v_y', commutative=True)"], "Equality(Pow(Mul(Symbol('H', commutative=True), Integral(Function('F_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Symbol('v_y', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Add(Symbol('v_y', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True))))), Symbol('v_y', commutative=True)))"], [["integrate", 5, "Symbol('v_y', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('H', commutative=True), Integral(Function('F_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Pow(Mul(Symbol('H', commutative=True), Add(Symbol('v_y', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True))))), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\omega)} = \\log{(\\omega)}, then derive \\int (\\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)}) d\\omega = 0, then obtain \\int - \\log{(\\omega)} d\\omega + \\int \\log{(\\omega)} d\\omega = 0", "derivation": "\\operatorname{C_{1}}{(\\omega)} = \\log{(\\omega)} and \\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)} = 0 and \\int (\\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)}) d\\omega = \\int 0 d\\omega and \\iint (\\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)}) d\\omega d\\omega = \\iint 0 d\\omega d\\omega and \\frac{d}{d \\omega} \\iint (\\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)}) d\\omega d\\omega = \\frac{d}{d \\omega} \\iint 0 d\\omega d\\omega and \\int (\\operatorname{C_{1}}{(\\omega)} - \\log{(\\omega)}) d\\omega = 0 and \\int \\operatorname{C_{1}}{(\\omega)} d\\omega + \\int - \\log{(\\omega)} d\\omega = 0 and \\int - \\log{(\\omega)} d\\omega + \\int \\log{(\\omega)} d\\omega = 0", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Function('C_1')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Function('C_1')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Integral(Add(Function('C_1')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integral(Add(Function('C_1')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integer(0))"], [["expand", 6], "Equality(Add(Integral(Function('C_1')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Integral(Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(0))"]]}, {"prompt": "Given Q{(\\lambda)} = e^{\\lambda}, then derive \\mathbb{I} - \\int Q{(\\lambda)} e^{- \\int Q{(\\lambda)} d\\lambda} d\\lambda = q + e^{- e^{\\lambda}}, then obtain \\int (\\mathbb{I} - \\int Q{(\\lambda)} e^{- \\int Q{(\\lambda)} d\\lambda} d\\lambda) d\\lambda = \\int (q + e^{- e^{\\lambda}}) d\\lambda", "derivation": "Q{(\\lambda)} = e^{\\lambda} and - Q{(\\lambda)} = - e^{\\lambda} and \\int - Q{(\\lambda)} d\\lambda = \\int - e^{\\lambda} d\\lambda and e^{\\int - Q{(\\lambda)} d\\lambda} = e^{\\int - e^{\\lambda} d\\lambda} and \\frac{d}{d \\lambda} e^{\\int - Q{(\\lambda)} d\\lambda} = \\frac{d}{d \\lambda} e^{\\int - e^{\\lambda} d\\lambda} and \\int \\frac{d}{d \\lambda} e^{\\int - Q{(\\lambda)} d\\lambda} d\\lambda = \\int \\frac{d}{d \\lambda} e^{\\int - e^{\\lambda} d\\lambda} d\\lambda and \\mathbb{I} - \\int Q{(\\lambda)} e^{- \\int Q{(\\lambda)} d\\lambda} d\\lambda = q + e^{- e^{\\lambda}} and \\int (\\mathbb{I} - \\int Q{(\\lambda)} e^{- \\int Q{(\\lambda)} d\\lambda} d\\lambda) d\\lambda = \\int (q + e^{- e^{\\lambda}}) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('Q')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('Q')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Mul(Integer(-1), Function('Q')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), exp(Integral(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(exp(Integral(Mul(Integer(-1), Function('Q')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Integral(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Derivative(exp(Integral(Mul(Integer(-1), Function('Q')(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Derivative(exp(Integral(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Integral(Mul(Function('Q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Symbol('q', commutative=True), exp(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))))))"], [["integrate", 7, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Integral(Mul(Function('Q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Integral(Function('Q')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))), Tuple(Symbol('\\\\lambda', commutative=True))))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Symbol('q', commutative=True), exp(Mul(Integer(-1), exp(Symbol('\\\\lambda', commutative=True))))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then derive \\cos{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\operatorname{P_{g}}{(\\mathbf{v})} = 2 \\cos{(\\mathbf{v})}, then obtain \\cos{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\sin{(\\mathbf{v})} = 2 \\cos{(\\mathbf{v})}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\operatorname{P_{g}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})} = 2 \\sin{(\\mathbf{v})} and \\frac{d}{d \\mathbf{v}} (\\operatorname{P_{g}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})}) = \\frac{d}{d \\mathbf{v}} 2 \\sin{(\\mathbf{v})} and \\cos{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\operatorname{P_{g}}{(\\mathbf{v})} = 2 \\cos{(\\mathbf{v})} and \\cos{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\sin{(\\mathbf{v})} = 2 \\cos{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Add(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given g{(G)} = \\frac{d}{d G} \\cos{(G)}, then derive \\int g{(G)} dG = \\mathbb{I} + \\cos{(G)}, then obtain (\\int g{(G)} dG)^{G} = (\\mathbb{I} + \\cos{(G)})^{G}", "derivation": "g{(G)} = \\frac{d}{d G} \\cos{(G)} and \\int g{(G)} dG = \\int \\frac{d}{d G} \\cos{(G)} dG and (\\int g{(G)} dG)^{G} = (\\int \\frac{d}{d G} \\cos{(G)} dG)^{G} and \\int g{(G)} dG = \\mathbb{I} + \\cos{(G)} and \\int \\frac{d}{d G} \\cos{(G)} dG = \\mathbb{I} + \\cos{(G)} and (\\int g{(G)} dG)^{G} = (\\mathbb{I} + \\cos{(G)})^{G}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('G', commutative=True)), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('g')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Function('g')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Integral(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Integral(Function('g')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"]]}, {"prompt": "Given V{(\\mathbf{E})} = \\sin{(\\mathbf{E})}, then derive \\int (V{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}}) d\\mathbf{E} = v_{x} - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})}, then obtain v_{x} - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})} = C - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})}", "derivation": "V{(\\mathbf{E})} = \\sin{(\\mathbf{E})} and V{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}} = \\sin{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}} and \\int (V{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}}) d\\mathbf{E} = \\int (\\sin{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}}) d\\mathbf{E} and \\int (V{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}}) d\\mathbf{E} = v_{x} - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})} and v_{x} - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})} = \\int (\\sin{(\\mathbf{E})} - \\frac{1}{\\mathbf{E}}) d\\mathbf{E} and v_{x} - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})} = C - \\log{(\\mathbf{E})} - \\cos{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))"], "Equality(Add(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))), Add(sin(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(sin(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('V')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('v_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Integral(Add(sin(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('v_x', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Symbol('C', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{D},E_{x})} = - E_{x} + \\log{(\\mathbf{D})}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D},E_{x})} + \\frac{1}{\\mathbf{D}} = \\frac{2}{\\mathbf{D}}, then obtain \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} + \\log{(\\mathbf{D})}) + \\frac{1}{\\mathbf{D}} = \\frac{2}{\\mathbf{D}}", "derivation": "\\Psi_{nl}{(\\mathbf{D},E_{x})} = - E_{x} + \\log{(\\mathbf{D})} and \\Psi_{nl}{(\\mathbf{D},E_{x})} + \\log{(\\mathbf{D})} = - E_{x} + 2 \\log{(\\mathbf{D})} and \\frac{\\partial}{\\partial \\mathbf{D}} (\\Psi_{nl}{(\\mathbf{D},E_{x})} + \\log{(\\mathbf{D})}) = \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} + 2 \\log{(\\mathbf{D})}) and \\frac{\\partial}{\\partial \\mathbf{D}} \\Psi_{nl}{(\\mathbf{D},E_{x})} + \\frac{1}{\\mathbf{D}} = \\frac{2}{\\mathbf{D}} and \\frac{\\partial}{\\partial \\mathbf{D}} (- E_{x} + \\log{(\\mathbf{D})}) + \\frac{1}{\\mathbf{D}} = \\frac{2}{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 1, "log(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), log(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbb{I},Q)} = Q + \\mathbb{I}, then obtain - \\frac{Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{\\partial}{\\partial \\mathbb{I}} (- Q + \\operatorname{A_{2}}{(\\mathbb{I},Q)})}{\\frac{d}{d \\mathbb{I}} \\mathbb{I}} = - \\frac{Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{d}{d \\mathbb{I}} \\mathbb{I}}{\\frac{d}{d \\mathbb{I}} \\mathbb{I}}", "derivation": "\\operatorname{A_{2}}{(\\mathbb{I},Q)} = Q + \\mathbb{I} and - Q + \\operatorname{A_{2}}{(\\mathbb{I},Q)} = \\mathbb{I} and \\frac{\\partial}{\\partial \\mathbb{I}} (- Q + \\operatorname{A_{2}}{(\\mathbb{I},Q)}) = \\frac{d}{d \\mathbb{I}} \\mathbb{I} and Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{\\partial}{\\partial \\mathbb{I}} (- Q + \\operatorname{A_{2}}{(\\mathbb{I},Q)}) = Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{d}{d \\mathbb{I}} \\mathbb{I} and - \\frac{Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{\\partial}{\\partial \\mathbb{I}} (- Q + \\operatorname{A_{2}}{(\\mathbb{I},Q)})}{\\frac{d}{d \\mathbb{I}} \\mathbb{I}} = - \\frac{Q - \\operatorname{A_{2}}{(\\mathbb{I},Q)} + \\frac{d}{d \\mathbb{I}} \\mathbb{I}}{\\frac{d}{d \\mathbb{I}} \\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"], [["divide", 4, "Mul(Integer(-1), Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Pow(Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Function('A_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True))), Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Pow(Derivative(Symbol('\\\\mathbb{I}', commutative=True), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\psi)} = \\cos{(\\psi)}, then obtain 1 - \\int \\cos{(\\psi)} d\\psi = - \\int \\cos{(\\psi)} d\\psi + \\frac{\\cos{(\\psi)}}{\\tilde{g}^*{(\\psi)}}", "derivation": "\\tilde{g}^*{(\\psi)} = \\cos{(\\psi)} and \\int \\tilde{g}^*{(\\psi)} d\\psi = \\int \\cos{(\\psi)} d\\psi and 1 = \\frac{\\cos{(\\psi)}}{\\tilde{g}^*{(\\psi)}} and 1 - \\int \\tilde{g}^*{(\\psi)} d\\psi = - \\int \\tilde{g}^*{(\\psi)} d\\psi + \\frac{\\cos{(\\psi)}}{\\tilde{g}^*{(\\psi)}} and 1 - \\int \\cos{(\\psi)} d\\psi = - \\int \\cos{(\\psi)} d\\psi + \\frac{\\cos{(\\psi)}}{\\tilde{g}^*{(\\psi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))))"], [["minus", 3, "Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))), Add(Mul(Integer(-1), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))), Add(Mul(Integer(-1), Integral(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\psi', commutative=True)), Integer(-1)), cos(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given i{(p)} = \\sin{(e^{p})}, then obtain e^{p} + \\frac{d}{d p} i{(p)} = e^{p} \\cos{(e^{p})} + e^{p}", "derivation": "i{(p)} = \\sin{(e^{p})} and \\frac{d}{d p} i{(p)} = \\frac{d}{d p} \\sin{(e^{p})} and e^{p} + \\frac{d}{d p} i{(p)} = e^{p} + \\frac{d}{d p} \\sin{(e^{p})} and e^{p} + \\frac{d}{d p} i{(p)} = e^{p} \\cos{(e^{p})} + e^{p}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('p', commutative=True)), sin(exp(Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 2, "exp(Symbol('p', commutative=True))"], "Equality(Add(exp(Symbol('p', commutative=True)), Derivative(Function('i')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(exp(Symbol('p', commutative=True)), Derivative(sin(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('p', commutative=True)), Derivative(Function('i')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Mul(exp(Symbol('p', commutative=True)), cos(exp(Symbol('p', commutative=True)))), exp(Symbol('p', commutative=True))))"]]}, {"prompt": "Given V{(\\nabla)} = \\sin{(\\nabla)}, then obtain \\frac{1}{2} = \\frac{\\sin^{2}{(\\nabla)}}{2 V^{2}{(\\nabla)}}", "derivation": "V{(\\nabla)} = \\sin{(\\nabla)} and 2 V{(\\nabla)} = V{(\\nabla)} + \\sin{(\\nabla)} and \\frac{1}{2} = \\frac{\\sin{(\\nabla)}}{2 V{(\\nabla)}} and \\frac{1}{2} = \\frac{\\sin{(\\nabla)}}{V{(\\nabla)} + \\sin{(\\nabla)}} and \\frac{\\sin{(\\nabla)}}{V{(\\nabla)} + \\sin{(\\nabla)}} = \\frac{\\sin^{2}{(\\nabla)}}{(V{(\\nabla)} + \\sin{(\\nabla)}) V{(\\nabla)}} and \\frac{\\sin{(\\nabla)}}{2 V{(\\nabla)}} = \\frac{\\sin^{2}{(\\nabla)}}{2 V^{2}{(\\nabla)}} and \\frac{1}{2} = \\frac{\\sin^{2}{(\\nabla)}}{2 V^{2}{(\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Function('V')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('V')(Symbol('\\\\nabla', commutative=True))), Add(Function('V')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))))"], [["divide", 1, "Mul(Integer(2), Function('V')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('V')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Pow(Add(Function('V')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Add(Function('V')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))), Mul(Pow(Add(Function('V')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True))), Integer(-1)), Pow(Function('V')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\nabla', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Rational(1, 2), Pow(Function('V')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), sin(Symbol('\\\\nabla', commutative=True))), Mul(Rational(1, 2), Pow(Function('V')(Symbol('\\\\nabla', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\nabla', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('V')(Symbol('\\\\nabla', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\nabla', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(F_{c})} = \\sin{(F_{c})}, then derive \\frac{d}{d F_{c}} \\theta_{1}{(F_{c})} = \\cos{(F_{c})}, then obtain \\int \\frac{d}{d F_{c}} \\theta_{1}{(F_{c})} dF_{c} = \\int \\cos{(F_{c})} dF_{c}", "derivation": "\\theta_{1}{(F_{c})} = \\sin{(F_{c})} and \\frac{d}{d F_{c}} \\theta_{1}{(F_{c})} = \\frac{d}{d F_{c}} \\sin{(F_{c})} and \\frac{d}{d F_{c}} \\theta_{1}{(F_{c})} = \\cos{(F_{c})} and \\int \\frac{d}{d F_{c}} \\theta_{1}{(F_{c})} dF_{c} = \\int \\cos{(F_{c})} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(sin(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), cos(Symbol('F_c', commutative=True)))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta_1')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Tuple(Symbol('F_c', commutative=True))), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given f{(\\pi,A_{1})} = A_{1}^{\\pi}, then obtain \\int \\frac{\\partial}{\\partial A_{1}} f^{A_{1}}{(\\pi,A_{1})} d\\pi = \\int \\frac{\\partial}{\\partial A_{1}} (A_{1}^{\\pi})^{A_{1}} d\\pi", "derivation": "f{(\\pi,A_{1})} = A_{1}^{\\pi} and f^{A_{1}}{(\\pi,A_{1})} = (A_{1}^{\\pi})^{A_{1}} and \\frac{\\partial}{\\partial A_{1}} f^{A_{1}}{(\\pi,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1}^{\\pi})^{A_{1}} and \\int \\frac{\\partial}{\\partial A_{1}} f^{A_{1}}{(\\pi,A_{1})} d\\pi = \\int \\frac{\\partial}{\\partial A_{1}} (A_{1}^{\\pi})^{A_{1}} d\\pi", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('f')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Pow(Function('f')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('f')(Symbol('\\\\pi', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Derivative(Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(h,\\sigma_x)} = \\sigma_x h, then obtain \\log{(\\sigma_x + \\operatorname{v_{x}}{(h,\\sigma_x)} - \\int \\sigma_x h d\\sigma_x)} = \\log{(\\sigma_x h + \\sigma_x - \\int \\sigma_x h d\\sigma_x)}", "derivation": "\\operatorname{v_{x}}{(h,\\sigma_x)} = \\sigma_x h and \\sigma_x + \\operatorname{v_{x}}{(h,\\sigma_x)} = \\sigma_x h + \\sigma_x and \\int \\operatorname{v_{x}}{(h,\\sigma_x)} d\\sigma_x = \\int \\sigma_x h d\\sigma_x and \\sigma_x + \\operatorname{v_{x}}{(h,\\sigma_x)} - \\int \\operatorname{v_{x}}{(h,\\sigma_x)} d\\sigma_x = \\sigma_x h + \\sigma_x - \\int \\operatorname{v_{x}}{(h,\\sigma_x)} d\\sigma_x and \\log{(\\sigma_x + \\operatorname{v_{x}}{(h,\\sigma_x)} - \\int \\operatorname{v_{x}}{(h,\\sigma_x)} d\\sigma_x)} = \\log{(\\sigma_x h + \\sigma_x - \\int \\operatorname{v_{x}}{(h,\\sigma_x)} d\\sigma_x)} and \\log{(\\sigma_x + \\operatorname{v_{x}}{(h,\\sigma_x)} - \\int \\sigma_x h d\\sigma_x)} = \\log{(\\sigma_x h + \\sigma_x - \\int \\sigma_x h d\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))))"], [["log", 4], "Equality(log(Add(Symbol('\\\\sigma_x', commutative=True), Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))), log(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Integral(Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(log(Add(Symbol('\\\\sigma_x', commutative=True), Function('v_x')(Symbol('h', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))), log(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(Z)} = \\int e^{Z} dZ, then derive \\operatorname{y^{\\prime}}{(Z)} - e^{Z} = \\sigma_x, then obtain - \\frac{\\sigma_x e^{Z}}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}} = - \\frac{(- e^{Z} + \\int e^{Z} dZ) e^{Z}}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}}", "derivation": "\\operatorname{y^{\\prime}}{(Z)} = \\int e^{Z} dZ and \\operatorname{y^{\\prime}}{(Z)} - e^{Z} = - e^{Z} + \\int e^{Z} dZ and \\operatorname{y^{\\prime}}{(Z)} - e^{Z} = \\sigma_x and \\sigma_x = - e^{Z} + \\int e^{Z} dZ and \\frac{\\sigma_x}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}} = \\frac{- e^{Z} + \\int e^{Z} dZ}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}} and - \\frac{\\sigma_x e^{Z}}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}} = - \\frac{(- e^{Z} + \\int e^{Z} dZ) e^{Z}}{(\\operatorname{y^{\\prime}}{(Z)} - e^{Z})^{2}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["minus", 1, "exp(Symbol('Z', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Symbol('\\\\sigma_x', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Integer(-1), exp(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["divide", 4, "Pow(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Integer(2))"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Integer(-2))), Mul(Pow(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Integer(-2)), Add(Mul(Integer(-1), exp(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))))"], [["times", 5, "Mul(Integer(-1), exp(Symbol('Z', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Pow(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Integer(-2)), exp(Symbol('Z', commutative=True))), Mul(Integer(-1), Pow(Add(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Mul(Integer(-1), exp(Symbol('Z', commutative=True)))), Integer(-2)), Add(Mul(Integer(-1), exp(Symbol('Z', commutative=True))), Integral(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), exp(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\mathbf{r})} = \\sin{(\\mathbf{r})}, then derive \\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r} = F_{c} - \\cos{(\\mathbf{r})}, then obtain (\\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = (\\int \\sin{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "derivation": "\\operatorname{M_{E}}{(\\mathbf{r})} = \\sin{(\\mathbf{r})} and \\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r} = \\int \\sin{(\\mathbf{r})} d\\mathbf{r} and \\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r} = F_{c} - \\cos{(\\mathbf{r})} and \\int \\sin{(\\mathbf{r})} d\\mathbf{r} = F_{c} - \\cos{(\\mathbf{r})} and (\\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = (F_{c} - \\cos{(\\mathbf{r})})^{\\mathbf{r}} and (\\int \\operatorname{M_{E}}{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = (\\int \\sin{(\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M_E')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Integral(Function('M_E')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(Function('M_E')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)} = \\sigma_x, then obtain \\int \\operatorname{f^{\\prime}}{(\\sigma_x)} d\\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)} = \\int \\cos{(\\sigma_x)} d\\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)}", "derivation": "\\operatorname{f^{\\prime}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)} = \\sigma_x and \\int \\operatorname{f^{\\prime}}{(\\sigma_x)} d\\sigma_x = \\int \\cos{(\\sigma_x)} d\\sigma_x and \\int \\operatorname{f^{\\prime}}{(\\sigma_x)} d\\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)} = \\int \\cos{(\\sigma_x)} d\\operatorname{V_{\\mathbf{B}}}{(\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\sigma_x', commutative=True)))), Integral(cos(Symbol('\\\\sigma_x', commutative=True)), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(n_{1})} = \\log{(e^{n_{1}})}, then obtain \\frac{\\rho_{f}{(n_{1})} - \\log{(e^{n_{1}})}}{n_{1} \\log{(e^{n_{1}})}} = 0", "derivation": "\\rho_{f}{(n_{1})} = \\log{(e^{n_{1}})} and \\rho_{f}{(n_{1})} - \\log{(e^{n_{1}})} = 0 and \\frac{\\rho_{f}{(n_{1})} - \\log{(e^{n_{1}})}}{n_{1}} = 0 and \\frac{\\rho_{f}{(n_{1})} - \\log{(e^{n_{1}})}}{n_{1} \\log{(e^{n_{1}})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), log(exp(Symbol('n_1', commutative=True))))"], [["minus", 1, "log(exp(Symbol('n_1', commutative=True)))"], "Equality(Add(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), Mul(Integer(-1), log(exp(Symbol('n_1', commutative=True))))), Integer(0))"], [["divide", 2, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Add(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), Mul(Integer(-1), log(exp(Symbol('n_1', commutative=True)))))), Integer(0))"], [["divide", 3, "log(exp(Symbol('n_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Add(Function('\\\\rho_f')(Symbol('n_1', commutative=True)), Mul(Integer(-1), log(exp(Symbol('n_1', commutative=True))))), Pow(log(exp(Symbol('n_1', commutative=True))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given m{(\\sigma_x,B)} = B + \\sigma_x, then obtain - \\frac{1}{B} + \\frac{m{(\\sigma_x,B)}}{B^{2}} = - \\frac{1}{B} + \\frac{B + \\sigma_x}{B^{2}}", "derivation": "m{(\\sigma_x,B)} = B + \\sigma_x and \\frac{m{(\\sigma_x,B)}}{B} = \\frac{B + \\sigma_x}{B} and \\frac{m{(\\sigma_x,B)}}{B^{2}} = \\frac{B + \\sigma_x}{B^{2}} and - \\frac{1}{B} + \\frac{m{(\\sigma_x,B)}}{B^{2}} = - \\frac{1}{B} + \\frac{B + \\sigma_x}{B^{2}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\sigma_x', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('m')(Symbol('\\\\sigma_x', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 2, "Pow(Symbol('B', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Function('m')(Symbol('\\\\sigma_x', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 3, "Pow(Symbol('B', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Function('m')(Symbol('\\\\sigma_x', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1))), Mul(Pow(Symbol('B', commutative=True), Integer(-2)), Add(Symbol('B', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(z)} = \\frac{d}{d z} \\cos{(z)}, then derive \\varepsilon_{0}{(z)} = - \\sin{(z)}, then obtain \\int (\\frac{d}{d z} \\cos{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)}) dz = \\int (- \\sin{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)}) dz", "derivation": "\\varepsilon_{0}{(z)} = \\frac{d}{d z} \\cos{(z)} and \\varepsilon_{0}{(z)} = - \\sin{(z)} and \\frac{d}{d z} \\cos{(z)} = - \\sin{(z)} and \\frac{d}{d z} \\cos{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)} = - \\sin{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)} and \\int (\\frac{d}{d z} \\cos{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)}) dz = \\int (- \\sin{(z)} - \\frac{d^{2}}{d z^{2}} \\cos{(z)}) dz", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varepsilon_0')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('z', commutative=True))))"], [["minus", 3, "Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(2)))"], "Equality(Add(Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(2))))), Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(2))))))"], [["integrate", 4, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(2))))), Tuple(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(2))))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\omega{(h)} = \\log{(h)}, then derive (\\int \\omega{(h)} dh)^{h} = (\\dot{\\mathbf{r}} + h \\log{(h)} - h)^{h}, then obtain (\\int \\log{(h)} dh)^{h} = (\\dot{\\mathbf{r}} + h \\log{(h)} - h)^{h}", "derivation": "\\omega{(h)} = \\log{(h)} and \\int \\omega{(h)} dh = \\int \\log{(h)} dh and (\\int \\omega{(h)} dh)^{h} = (\\int \\log{(h)} dh)^{h} and (\\int \\omega{(h)} dh)^{h} = (\\dot{\\mathbf{r}} + h \\log{(h)} - h)^{h} and (\\int \\log{(h)} dh)^{h} = (\\dot{\\mathbf{r}} + h \\log{(h)} - h)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('\\\\omega')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\omega')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Symbol('h', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(r_{0},V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial r_{0}} (- V_{\\mathbf{E}} + r_{0}), then derive \\hat{X}{(r_{0},V_{\\mathbf{E}})} = 1, then obtain \\hat{X}{(r_{0},V_{\\mathbf{E}})} - 1 = 0", "derivation": "\\hat{X}{(r_{0},V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial r_{0}} (- V_{\\mathbf{E}} + r_{0}) and \\hat{X}{(r_{0},V_{\\mathbf{E}})} = 1 and \\frac{\\partial}{\\partial r_{0}} (- V_{\\mathbf{E}} + r_{0}) = 1 and \\frac{\\partial}{\\partial r_{0}} (- V_{\\mathbf{E}} + r_{0}) - 1 = 0 and \\hat{X}{(r_{0},V_{\\mathbf{E}})} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('r_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{X}')(Symbol('r_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{X}')(Symbol('r_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\hat{x}{(I,\\rho_b,a)} = \\frac{I \\rho_b}{a}, then obtain \\hat{x}^{a}{(I,\\rho_b,a)} \\cos{((\\frac{I \\rho_b}{a})^{a})} \\cos{(\\hat{x}^{a}{(I,\\rho_b,a)})} = \\hat{x}^{a}{(I,\\rho_b,a)} \\cos^{2}{((\\frac{I \\rho_b}{a})^{a})}", "derivation": "\\hat{x}{(I,\\rho_b,a)} = \\frac{I \\rho_b}{a} and \\hat{x}^{a}{(I,\\rho_b,a)} = (\\frac{I \\rho_b}{a})^{a} and \\cos{(\\hat{x}^{a}{(I,\\rho_b,a)})} = \\cos{((\\frac{I \\rho_b}{a})^{a})} and \\cos{((\\frac{I \\rho_b}{a})^{a})} \\cos{(\\hat{x}^{a}{(I,\\rho_b,a)})} = \\cos^{2}{((\\frac{I \\rho_b}{a})^{a})} and \\hat{x}^{a}{(I,\\rho_b,a)} \\cos{((\\frac{I \\rho_b}{a})^{a})} \\cos{(\\hat{x}^{a}{(I,\\rho_b,a)})} = \\hat{x}^{a}{(I,\\rho_b,a)} \\cos^{2}{((\\frac{I \\rho_b}{a})^{a})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))))"], [["times", 3, "cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True)))"], "Equality(Mul(cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))), cos(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Pow(cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))), Integer(2)))"], [["times", 4, "Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))), cos(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))), Mul(Pow(Function('\\\\hat{x}')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(cos(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Symbol('a', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\phi_{2}{(T)} = \\log{(\\cos{(T)})} and f{(A_{z},v_{1})} = v_{1} + \\log{(A_{z})}, then obtain (-1 - \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}}) \\cos{(T)} + f{(A_{z},v_{1})} = v_{1} + (-1 - \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}}) \\cos{(T)} + \\log{(A_{z})}", "derivation": "\\phi_{2}{(T)} = \\log{(\\cos{(T)})} and 1 = \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}} and 2 = 1 + \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}} and f{(A_{z},v_{1})} = v_{1} + \\log{(A_{z})} and f{(A_{z},v_{1})} - \\cos{(T)} = v_{1} + \\log{(A_{z})} - \\cos{(T)} and f{(A_{z},v_{1})} - 2 \\cos{(T)} = v_{1} + \\log{(A_{z})} - 2 \\cos{(T)} and (-1 - \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}}) \\cos{(T)} + f{(A_{z},v_{1})} = v_{1} + (-1 - \\frac{\\log{(\\cos{(T)})}}{\\phi_{2}{(T)}}) \\cos{(T)} + \\log{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('T', commutative=True)), log(cos(Symbol('T', commutative=True))))"], [["divide", 1, "Function('\\\\phi_2')(Symbol('T', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\phi_2')(Symbol('T', commutative=True)), Integer(-1)), log(cos(Symbol('T', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('\\\\phi_2')(Symbol('T', commutative=True)), Integer(-1)), log(cos(Symbol('T', commutative=True))))))"], ["get_premise", "Equality(Function('f')(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Add(Symbol('v_1', commutative=True), log(Symbol('A_z', commutative=True))))"], [["minus", 4, "cos(Symbol('T', commutative=True))"], "Equality(Add(Function('f')(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), Add(Symbol('v_1', commutative=True), log(Symbol('A_z', commutative=True)), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["minus", 5, "cos(Symbol('T', commutative=True))"], "Equality(Add(Function('f')(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('T', commutative=True)))), Add(Symbol('v_1', commutative=True), log(Symbol('A_z', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Add(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\phi_2')(Symbol('T', commutative=True)), Integer(-1)), log(cos(Symbol('T', commutative=True))))), cos(Symbol('T', commutative=True))), Function('f')(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), Mul(Add(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\phi_2')(Symbol('T', commutative=True)), Integer(-1)), log(cos(Symbol('T', commutative=True))))), cos(Symbol('T', commutative=True))), log(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given A{(\\nabla)} = \\cos{(\\nabla)}, then obtain A^{2}{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(\\cos{(\\nabla)})} = A{(\\nabla)} \\cos{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(\\cos{(\\nabla)})}", "derivation": "A{(\\nabla)} = \\cos{(\\nabla)} and \\log{(A{(\\nabla)})} = \\log{(\\cos{(\\nabla)})} and A^{2}{(\\nabla)} = A{(\\nabla)} \\cos{(\\nabla)} and A^{2}{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(A{(\\nabla)})} = A{(\\nabla)} \\cos{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(A{(\\nabla)})} and A^{2}{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(\\cos{(\\nabla)})} = A{(\\nabla)} \\cos{(\\nabla)} - \\frac{d}{d \\nabla} \\log{(\\cos{(\\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["log", 1], "Equality(log(Function('A')(Symbol('\\\\nabla', commutative=True))), log(cos(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Function('A')(Symbol('\\\\nabla', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True))))"], [["minus", 3, "Derivative(log(Function('A')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('A')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(log(Function('A')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))), Add(Mul(Function('A')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Derivative(log(Function('A')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('A')(Symbol('\\\\nabla', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(log(cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))), Add(Mul(Function('A')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Derivative(log(cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{S}{(n_{1},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}), then derive \\mathbf{S}{(n_{1},\\mathbf{s})} = 1, then obtain \\mathbf{J}{(\\hat{p},C_{1})} + \\cos{(\\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}))} = \\mathbf{J}{(\\hat{p},C_{1})} + \\cos{(1)}", "derivation": "\\mathbf{S}{(n_{1},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}) and \\mathbf{S}{(n_{1},\\mathbf{s})} = 1 and \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}) = 1 and \\cos{(\\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}))} = \\cos{(1)} and \\mathbf{J}{(\\hat{p},C_{1})} + \\cos{(\\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + n_{1}))} = \\mathbf{J}{(\\hat{p},C_{1})} + \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Integer(1))"], [["cos", 3], "Equality(cos(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), cos(Integer(1)))"], [["add", 4, "Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_1', commutative=True)), cos(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Add(Function('\\\\mathbf{J}')(Symbol('\\\\hat{p}', commutative=True), Symbol('C_1', commutative=True)), cos(Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(\\rho_f)} = \\sin{(\\rho_f)}, then obtain \\int \\frac{- \\rho_f + \\theta_{2}{(\\rho_f)}}{- \\rho_f + \\sin{(\\rho_f)}} d\\rho_f = \\int 1 d\\rho_f", "derivation": "\\theta_{2}{(\\rho_f)} = \\sin{(\\rho_f)} and - \\rho_f + \\theta_{2}{(\\rho_f)} = - \\rho_f + \\sin{(\\rho_f)} and \\frac{- \\rho_f + \\theta_{2}{(\\rho_f)}}{- \\rho_f + \\sin{(\\rho_f)}} = 1 and \\int \\frac{- \\rho_f + \\theta_{2}{(\\rho_f)}}{- \\rho_f + \\sin{(\\rho_f)}} d\\rho_f = \\int 1 d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\rho_f', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\mu{(c)} = \\cos{(c)}, then obtain \\mu^{c}{(c)} + \\frac{d}{d c} \\mu{(c)} = \\cos^{c}{(c)} + \\frac{d}{d c} \\mu{(c)}", "derivation": "\\mu{(c)} = \\cos{(c)} and \\frac{d}{d c} \\mu{(c)} = \\frac{d}{d c} \\cos{(c)} and \\mu^{c}{(c)} = \\cos^{c}{(c)} and \\mu^{c}{(c)} + \\frac{d}{d c} \\cos{(c)} = \\cos^{c}{(c)} + \\frac{d}{d c} \\cos{(c)} and \\mu^{c}{(c)} + \\frac{d}{d c} \\mu{(c)} = \\cos^{c}{(c)} + \\frac{d}{d c} \\mu{(c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["add", 3, "Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(cos(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('\\\\mu')(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('\\\\mu')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('\\\\mu')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then derive \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = F_{x} - \\cos{(\\varphi^*)}, then obtain - \\cos{(\\varphi^*)} + \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = F_{x} - 2 \\cos{(\\varphi^*)}", "derivation": "\\mathbf{P}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = \\int \\sin{(\\varphi^*)} d\\varphi^* and \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = F_{x} - \\cos{(\\varphi^*)} and F_{x} - \\cos{(\\varphi^*)} = \\int \\sin{(\\varphi^*)} d\\varphi^* and - \\cos{(\\varphi^*)} + \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = - \\cos{(\\varphi^*)} + \\int \\sin{(\\varphi^*)} d\\varphi^* and - \\cos{(\\varphi^*)} + \\int \\mathbf{P}{(\\varphi^*)} d\\varphi^* = F_{x} - 2 \\cos{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 2, "Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True))), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True))), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(V,s)} = \\cos{(V + s)}, then derive \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,s)} = - \\sin{(V + s)}, then obtain (\\frac{\\partial^{2}}{\\partial V^{2}} \\operatorname{L_{\\varepsilon}}{(V,s)})^{2} = \\cos^{2}{(V + s)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(V,s)} = \\cos{(V + s)} and \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,s)} = \\frac{\\partial}{\\partial V} \\cos{(V + s)} and \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,s)} = - \\sin{(V + s)} and \\frac{\\partial}{\\partial V} \\cos{(V + s)} = - \\sin{(V + s)} and \\frac{\\partial^{2}}{\\partial V^{2}} \\cos{(V + s)} = \\frac{\\partial}{\\partial V} - \\sin{(V + s)} and \\frac{\\partial^{2}}{\\partial V^{2}} \\operatorname{L_{\\varepsilon}}{(V,s)} = \\frac{\\partial}{\\partial V} - \\sin{(V + s)} and (\\frac{\\partial^{2}}{\\partial V^{2}} \\operatorname{L_{\\varepsilon}}{(V,s)})^{2} = (\\frac{\\partial}{\\partial V} - \\sin{(V + s)})^{2} and (\\frac{\\partial^{2}}{\\partial V^{2}} \\operatorname{L_{\\varepsilon}}{(V,s)})^{2} = \\cos^{2}{(V + s)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), cos(Add(Symbol('V', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('V', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Add(Symbol('V', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(cos(Add(Symbol('V', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 6, 2], "Equality(Pow(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Mul(Integer(-1), sin(Add(Symbol('V', commutative=True), Symbol('s', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 7], "Equality(Pow(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))), Integer(2)), Pow(cos(Add(Symbol('V', commutative=True), Symbol('s', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\mathbf{J},\\mathbf{J}_M)} = - \\mathbf{J}_M + e^{\\mathbf{J}} and \\mathbb{I}{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain \\operatorname{t_{1}}^{\\mathbf{J}}{(\\mathbf{J},\\mathbf{J}_M)} = (- \\mathbf{J}_M + \\mathbb{I}{(\\mathbf{J})})^{\\mathbf{J}}", "derivation": "\\operatorname{t_{1}}{(\\mathbf{J},\\mathbf{J}_M)} = - \\mathbf{J}_M + e^{\\mathbf{J}} and \\operatorname{t_{1}}^{\\mathbf{J}}{(\\mathbf{J},\\mathbf{J}_M)} = (- \\mathbf{J}_M + e^{\\mathbf{J}})^{\\mathbf{J}} and \\mathbb{I}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\operatorname{t_{1}}^{\\mathbf{J}}{(\\mathbf{J},\\mathbf{J}_M)} = (- \\mathbf{J}_M + \\mathbb{I}{(\\mathbf{J})})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('t_1')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\mu{(x,\\mu_0)} = \\mu_0 - x, then obtain (\\frac{\\mu_0 \\frac{\\partial}{\\partial \\mu_0} \\mu{(x,\\mu_0)}}{\\mu{(x,\\mu_0)}} + \\log{(\\mu{(x,\\mu_0)})}) \\mu^{\\mu_0}{(x,\\mu_0)} = (\\mu_0 - x)^{\\mu_0} (\\frac{\\mu_0}{\\mu_0 - x} + \\log{(\\mu_0 - x)})", "derivation": "\\mu{(x,\\mu_0)} = \\mu_0 - x and \\mu^{\\mu_0}{(x,\\mu_0)} = (\\mu_0 - x)^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} \\mu^{\\mu_0}{(x,\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (\\mu_0 - x)^{\\mu_0} and (\\frac{\\mu_0 \\frac{\\partial}{\\partial \\mu_0} \\mu{(x,\\mu_0)}}{\\mu{(x,\\mu_0)}} + \\log{(\\mu{(x,\\mu_0)})}) \\mu^{\\mu_0}{(x,\\mu_0)} = (\\mu_0 - x)^{\\mu_0} (\\frac{\\mu_0}{\\mu_0 - x} + \\log{(\\mu_0 - x)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), log(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Pow(Function('\\\\mu')(Symbol('x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Add(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))), log(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))))"]]}, {"prompt": "Given T{(v_{t})} = \\sin{(v_{t})}, then obtain (\\frac{d}{d v_{t}} 0^{v_{t}})^{v_{t}} = (\\frac{d}{d v_{t}} (- T{(v_{t})} + \\sin{(v_{t})})^{v_{t}})^{v_{t}}", "derivation": "T{(v_{t})} = \\sin{(v_{t})} and 0 = - T{(v_{t})} + \\sin{(v_{t})} and 0^{v_{t}} = (- T{(v_{t})} + \\sin{(v_{t})})^{v_{t}} and \\frac{d}{d v_{t}} 0^{v_{t}} = \\frac{d}{d v_{t}} (- T{(v_{t})} + \\sin{(v_{t})})^{v_{t}} and (\\frac{d}{d v_{t}} 0^{v_{t}})^{v_{t}} = (\\frac{d}{d v_{t}} (- T{(v_{t})} + \\sin{(v_{t})})^{v_{t}})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["minus", 1, "Function('T')(Symbol('v_t', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('T')(Symbol('v_t', commutative=True))), sin(Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_t', commutative=True)), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('v_t', commutative=True))), sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('T')(Symbol('v_t', commutative=True))), sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["power", 4, "Symbol('v_t', commutative=True)"], "Equality(Pow(Derivative(Pow(Integer(0), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)), Pow(Derivative(Pow(Add(Mul(Integer(-1), Function('T')(Symbol('v_t', commutative=True))), sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} = \\dot{y} + \\mathbf{E}, then obtain - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} + 2 \\int 0 d\\mathbf{E} - \\int (\\dot{y} + \\mathbf{E} - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})}) d\\mathbf{E} = - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} + \\int 0 d\\mathbf{E}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} = \\dot{y} + \\mathbf{E} and 0 = \\dot{y} + \\mathbf{E} - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} and \\int 0 d\\mathbf{E} = \\int (\\dot{y} + \\mathbf{E} - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})}) d\\mathbf{E} and - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} + \\int 0 d\\mathbf{E} - \\int (\\dot{y} + \\mathbf{E} - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})}) d\\mathbf{E} = - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} and - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} + 2 \\int 0 d\\mathbf{E} - \\int (\\dot{y} + \\mathbf{E} - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})}) d\\mathbf{E} = - \\operatorname{f^{\\prime}}{(\\mathbf{E},\\dot{y})} + \\int 0 d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 3, "Add(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["add", 4, "Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)} = \\sin{(Q)}, then obtain - Q + \\int (Q + \\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)}) dQ = - Q + \\int (Q + \\sin{(Q)}) dQ", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)} = \\sin{(Q)} and Q + \\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)} = Q + \\sin{(Q)} and \\int (Q + \\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)}) dQ = \\int (Q + \\sin{(Q)}) dQ and - Q + \\int (Q + \\operatorname{g^{\\prime}_{\\varepsilon}}{(Q)}) dQ = - Q + \\int (Q + \\sin{(Q)}) dQ", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["add", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["minus", 3, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Integral(Add(Symbol('Q', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Integral(Add(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\phi_2,f)} = \\log{(- \\phi_2 + f)}, then derive (\\frac{\\partial}{\\partial f} \\lambda{(\\phi_2,f)})^{\\phi_2} = (\\frac{1}{- \\phi_2 + f})^{\\phi_2}, then obtain \\sin{(f (\\frac{1}{- \\phi_2 + f})^{\\phi_2})} = \\sin{(f (\\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)})^{\\phi_2})}", "derivation": "\\lambda{(\\phi_2,f)} = \\log{(- \\phi_2 + f)} and \\frac{\\partial}{\\partial f} \\lambda{(\\phi_2,f)} = \\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)} and (\\frac{\\partial}{\\partial f} \\lambda{(\\phi_2,f)})^{\\phi_2} = (\\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)})^{\\phi_2} and (\\frac{\\partial}{\\partial f} \\lambda{(\\phi_2,f)})^{\\phi_2} = (\\frac{1}{- \\phi_2 + f})^{\\phi_2} and (\\frac{1}{- \\phi_2 + f})^{\\phi_2} = (\\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)})^{\\phi_2} and f (\\frac{1}{- \\phi_2 + f})^{\\phi_2} = f (\\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)})^{\\phi_2} and \\sin{(f (\\frac{1}{- \\phi_2 + f})^{\\phi_2})} = \\sin{(f (\\frac{\\partial}{\\partial f} \\log{(- \\phi_2 + f)})^{\\phi_2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\lambda')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\lambda')(Symbol('\\\\phi_2', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Symbol('\\\\phi_2', commutative=True)), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))"], [["times", 5, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('f', commutative=True), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True))))"], [["sin", 6], "Equality(sin(Mul(Symbol('f', commutative=True), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Symbol('\\\\phi_2', commutative=True)))), sin(Mul(Symbol('f', commutative=True), Pow(Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(V)} = \\cos{(V)}, then obtain e^{\\int (\\mathbf{s}{(V)} - \\frac{d}{d V} \\cos{(V)}) dV} = e^{f_{\\mathbf{p}} + \\sin{(V)} - \\cos{(V)}}", "derivation": "\\mathbf{s}{(V)} = \\cos{(V)} and \\mathbf{s}{(V)} - \\frac{d}{d V} \\cos{(V)} = \\cos{(V)} - \\frac{d}{d V} \\cos{(V)} and \\int (\\mathbf{s}{(V)} - \\frac{d}{d V} \\cos{(V)}) dV = \\int (\\cos{(V)} - \\frac{d}{d V} \\cos{(V)}) dV and e^{\\int (\\mathbf{s}{(V)} - \\frac{d}{d V} \\cos{(V)}) dV} = e^{\\int (\\cos{(V)} - \\frac{d}{d V} \\cos{(V)}) dV} and e^{\\int (\\mathbf{s}{(V)} - \\frac{d}{d V} \\cos{(V)}) dV} = e^{f_{\\mathbf{p}} + \\sin{(V)} - \\cos{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["minus", 1, "Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Add(cos(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Tuple(Symbol('V', commutative=True))), Integral(Add(cos(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Tuple(Symbol('V', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Tuple(Symbol('V', commutative=True)))), exp(Integral(Add(cos(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Tuple(Symbol('V', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(exp(Integral(Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))), Tuple(Symbol('V', commutative=True)))), exp(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('V', commutative=True)), Mul(Integer(-1), cos(Symbol('V', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = \\frac{\\psi^*}{\\mathbf{F}}, then obtain \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = - \\frac{\\psi^*}{\\mathbf{F}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = \\frac{\\psi^*}{\\mathbf{F}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\psi^*}{\\mathbf{F}} and \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\psi^*}{\\mathbf{F}} and \\mathbf{F} \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{f_{\\mathbf{p}}}{(\\psi^*,\\mathbf{F})} = - \\frac{\\psi^*}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given c{(\\varphi)} = e^{\\varphi}, then obtain \\cos{(\\int c^{\\varphi}{(\\varphi)} d\\varphi)} = \\cos{(\\int (e^{\\varphi})^{\\varphi} d\\varphi)}", "derivation": "c{(\\varphi)} = e^{\\varphi} and c^{\\varphi}{(\\varphi)} = (e^{\\varphi})^{\\varphi} and \\int c^{\\varphi}{(\\varphi)} d\\varphi = \\int (e^{\\varphi})^{\\varphi} d\\varphi and \\cos{(\\int c^{\\varphi}{(\\varphi)} d\\varphi)} = \\cos{(\\int (e^{\\varphi})^{\\varphi} d\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Function('c')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Pow(Function('c')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), cos(Integral(Pow(exp(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\mathbf{g},q)} = \\mathbf{g} q and \\operatorname{t_{1}}{(\\mathbf{g},q)} = \\mathbf{g} q, then obtain (\\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_P^{q}{(\\mathbf{g},q)})^{q} = (\\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} q)^{q})^{q}", "derivation": "\\mathbf{J}_P{(\\mathbf{g},q)} = \\mathbf{g} q and \\operatorname{t_{1}}{(\\mathbf{g},q)} = \\mathbf{g} q and \\mathbf{J}_P{(\\mathbf{g},q)} = \\operatorname{t_{1}}{(\\mathbf{g},q)} and \\mathbf{J}_P^{q}{(\\mathbf{g},q)} = \\operatorname{t_{1}}^{q}{(\\mathbf{g},q)} and \\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_P^{q}{(\\mathbf{g},q)} = \\frac{\\partial}{\\partial \\mathbf{g}} \\operatorname{t_{1}}^{q}{(\\mathbf{g},q)} and \\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_P^{q}{(\\mathbf{g},q)} = \\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} q)^{q} and (\\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_P^{q}{(\\mathbf{g},q)})^{q} = (\\frac{\\partial}{\\partial \\mathbf{g}} (\\mathbf{g} q)^{q})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Function('t_1')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Function('t_1')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Pow(Function('t_1')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["power", 6, "Symbol('q', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Symbol('q', commutative=True)), Pow(Derivative(Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given z{(H)} = \\frac{d}{d H} e^{H}, then derive z{(H)} = e^{H}, then obtain \\frac{d^{2}}{d H^{2}} e^{H} = e^{H}", "derivation": "z{(H)} = \\frac{d}{d H} e^{H} and z{(H)} = e^{H} and z{(H)} = \\frac{d}{d H} z{(H)} and \\frac{d}{d H} e^{H} = \\frac{d^{2}}{d H^{2}} e^{H} and z{(H)} = \\frac{d^{2}}{d H^{2}} e^{H} and \\frac{d^{2}}{d H^{2}} e^{H} = e^{H}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('H', commutative=True)), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('z')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('z')(Symbol('H', commutative=True)), Derivative(Function('z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('z')(Symbol('H', commutative=True)), Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Derivative(exp(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), exp(Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(z,V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}} + z}, then obtain H + \\varepsilon_{0}{(z,V_{\\mathbf{E}})} = H + e^{V_{\\mathbf{E}} + z}", "derivation": "\\varepsilon_{0}{(z,V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}} + z} and V_{\\mathbf{E}} + \\varepsilon_{0}{(z,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + e^{V_{\\mathbf{E}} + z} and H + V_{\\mathbf{E}} + \\varepsilon_{0}{(z,V_{\\mathbf{E}})} = H + V_{\\mathbf{E}} + e^{V_{\\mathbf{E}} + z} and H + \\varepsilon_{0}{(z,V_{\\mathbf{E}})} = H + e^{V_{\\mathbf{E}} + z}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True))))"], [["add", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\varepsilon_0')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True)))))"], [["add", 2, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\varepsilon_0')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('H', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True)))))"], [["minus", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('\\\\varepsilon_0')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('H', commutative=True), exp(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{1})} = \\log{(v_{1})}, then obtain (\\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} + 1)^{v_{1}} = (\\frac{d}{d v_{1}} \\log{(v_{1})} + 1)^{v_{1}}", "derivation": "\\operatorname{n_{1}}{(v_{1})} = \\log{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} = \\frac{d}{d v_{1}} \\log{(v_{1})} and \\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} + 1 = \\frac{d}{d v_{1}} \\log{(v_{1})} + 1 and (\\frac{d}{d v_{1}} \\operatorname{n_{1}}{(v_{1})} + 1)^{v_{1}} = (\\frac{d}{d v_{1}} \\log{(v_{1})} + 1)^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Add(Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Derivative(Function('n_1')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Symbol('v_1', commutative=True)), Pow(Add(Derivative(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Integer(1)), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(\\phi_1)} = \\log{(\\sin{(\\phi_1)})}, then obtain \\sin{((\\dot{z}{(\\phi_1)} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})})} = \\sin{((\\log{(\\sin{(\\phi_1)})} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})})}", "derivation": "\\dot{z}{(\\phi_1)} = \\log{(\\sin{(\\phi_1)})} and \\dot{z}{(\\phi_1)} + \\sin{(\\phi_1)} = \\log{(\\sin{(\\phi_1)})} + \\sin{(\\phi_1)} and (\\dot{z}{(\\phi_1)} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})} = (\\log{(\\sin{(\\phi_1)})} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})} and \\sin{((\\dot{z}{(\\phi_1)} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})})} = \\sin{((\\log{(\\sin{(\\phi_1)})} + \\sin{(\\phi_1)}) \\log{(\\sin{(\\phi_1)})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\phi_1', commutative=True)), log(sin(Symbol('\\\\phi_1', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Add(log(sin(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "log(sin(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Add(Function('\\\\dot{z}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), log(sin(Symbol('\\\\phi_1', commutative=True)))), Mul(Add(log(sin(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))), log(sin(Symbol('\\\\phi_1', commutative=True)))))"], [["sin", 3], "Equality(sin(Mul(Add(Function('\\\\dot{z}')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), log(sin(Symbol('\\\\phi_1', commutative=True))))), sin(Mul(Add(log(sin(Symbol('\\\\phi_1', commutative=True))), sin(Symbol('\\\\phi_1', commutative=True))), log(sin(Symbol('\\\\phi_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{E},\\hat{x})} = \\mathbf{E} e^{\\hat{x}} and \\mu_{0}{(V,\\mathbf{B})} = \\sin{(V + \\mathbf{B})}, then obtain \\mu_{0}{(V,\\mathbf{B})} e^{\\hat{x}} = e^{\\hat{x}} \\sin{(V + \\mathbf{B})}", "derivation": "\\mathbf{f}{(\\mathbf{E},\\hat{x})} = \\mathbf{E} e^{\\hat{x}} and \\mu_{0}{(V,\\mathbf{B})} = \\sin{(V + \\mathbf{B})} and \\frac{\\mu_{0}{(V,\\mathbf{B})}}{\\mathbf{E}} = \\frac{\\sin{(V + \\mathbf{B})}}{\\mathbf{E}} and \\frac{\\mathbf{f}{(\\mathbf{E},\\hat{x})} \\mu_{0}{(V,\\mathbf{B})}}{\\mathbf{E}} = \\frac{\\mathbf{f}{(\\mathbf{E},\\hat{x})} \\sin{(V + \\mathbf{B})}}{\\mathbf{E}} and \\mu_{0}{(V,\\mathbf{B})} e^{\\hat{x}} = e^{\\hat{x}} \\sin{(V + \\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('\\\\hat{x}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), sin(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), sin(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 3, "Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), sin(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('\\\\mu_0')(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True))), Mul(exp(Symbol('\\\\hat{x}', commutative=True)), sin(Add(Symbol('V', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\rho{(a,\\pi,f)} = \\pi + a^{f}, then obtain \\int \\cos{((- a^{f} + \\rho{(a,\\pi,f)})^{\\pi})} d\\pi = \\int \\cos{(\\pi^{\\pi})} d\\pi", "derivation": "\\rho{(a,\\pi,f)} = \\pi + a^{f} and - a^{f} + \\rho{(a,\\pi,f)} = \\pi and (- a^{f} + \\rho{(a,\\pi,f)})^{\\pi} = \\pi^{\\pi} and \\cos{((- a^{f} + \\rho{(a,\\pi,f)})^{\\pi})} = \\cos{(\\pi^{\\pi})} and \\int \\cos{((- a^{f} + \\rho{(a,\\pi,f)})^{\\pi})} d\\pi = \\int \\cos{(\\pi^{\\pi})} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('f', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))))"], [["minus", 1, "Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))), Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\pi', commutative=True))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))), Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))), Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\pi', commutative=True))), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(cos(Pow(Add(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Symbol('f', commutative=True))), Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(A_{y})} = \\frac{d}{d A_{y}} \\cos{(A_{y})}, then derive - \\frac{\\operatorname{n_{2}}{(A_{y})}}{\\sin{(A_{y})}} = 1, then obtain \\frac{d}{d A_{y}} - \\frac{\\operatorname{n_{2}}{(A_{y})}}{\\sin{(A_{y})}} = \\frac{d}{d A_{y}} 1", "derivation": "\\operatorname{n_{2}}{(A_{y})} = \\frac{d}{d A_{y}} \\cos{(A_{y})} and \\frac{\\operatorname{n_{2}}{(A_{y})}}{\\frac{d}{d A_{y}} \\cos{(A_{y})}} = 1 and - \\frac{\\operatorname{n_{2}}{(A_{y})}}{\\sin{(A_{y})}} = 1 and - \\frac{\\frac{d}{d A_{y}} \\cos{(A_{y})}}{\\sin{(A_{y})}} = 1 and \\frac{d}{d A_{y}} - \\frac{\\frac{d}{d A_{y}} \\cos{(A_{y})}}{\\sin{(A_{y})}} = \\frac{d}{d A_{y}} 1 and \\frac{d}{d A_{y}} - \\frac{\\operatorname{n_{2}}{(A_{y})}}{\\sin{(A_{y})}} = \\frac{d}{d A_{y}} 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('A_y', commutative=True)), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Mul(Function('n_2')(Symbol('A_y', commutative=True)), Pow(Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(-1), Function('n_2')(Symbol('A_y', commutative=True)), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Integer(1))"], [["differentiate", 4, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Mul(Integer(-1), Function('n_2')(Symbol('A_y', commutative=True)), Pow(sin(Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(A_{2},P_{e})} = \\int (A_{2} + P_{e}) dA_{2}, then derive \\operatorname{f_{E}}^{2}{(A_{2},P_{e})} = (\\frac{A_{2}^{2}}{2} + A_{2} P_{e} + A_{z}) \\operatorname{f_{E}}{(A_{2},P_{e})}, then obtain (\\frac{A_{2}^{2}}{2} + A_{2} P_{e} + A_{z}) \\operatorname{f_{E}}{(A_{2},P_{e})} = \\operatorname{f_{E}}{(A_{2},P_{e})} \\int (A_{2} + P_{e}) dA_{2}", "derivation": "\\operatorname{f_{E}}{(A_{2},P_{e})} = \\int (A_{2} + P_{e}) dA_{2} and \\operatorname{f_{E}}^{2}{(A_{2},P_{e})} = \\operatorname{f_{E}}{(A_{2},P_{e})} \\int (A_{2} + P_{e}) dA_{2} and \\operatorname{f_{E}}^{2}{(A_{2},P_{e})} = (\\frac{A_{2}^{2}}{2} + A_{2} P_{e} + A_{z}) \\operatorname{f_{E}}{(A_{2},P_{e})} and (\\frac{A_{2}^{2}}{2} + A_{2} P_{e} + A_{z}) \\operatorname{f_{E}}{(A_{2},P_{e})} = \\operatorname{f_{E}}{(A_{2},P_{e})} \\int (A_{2} + P_{e}) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["times", 1, "Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True))"], "Equality(Pow(Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Pow(Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Integer(2)), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Symbol('A_z', commutative=True)), Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Mul(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Symbol('A_z', commutative=True)), Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True))), Mul(Function('f_E')(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(c,n)} = \\frac{\\partial}{\\partial n} c n, then obtain (\\frac{\\partial}{\\partial n} c n)^{2} (\\iint \\operatorname{t_{1}}^{n}{(c,n)} dc dn) \\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn = (\\frac{\\partial}{\\partial n} c n)^{2} (\\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn)^{2}", "derivation": "\\operatorname{t_{1}}{(c,n)} = \\frac{\\partial}{\\partial n} c n and \\operatorname{t_{1}}^{n}{(c,n)} = (\\frac{\\partial}{\\partial n} c n)^{n} and \\int \\operatorname{t_{1}}^{n}{(c,n)} dc = \\int (\\frac{\\partial}{\\partial n} c n)^{n} dc and \\iint \\operatorname{t_{1}}^{n}{(c,n)} dc dn = \\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn and \\frac{\\partial}{\\partial n} c n \\iint \\operatorname{t_{1}}^{n}{(c,n)} dc dn = \\frac{\\partial}{\\partial n} c n \\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn and (\\frac{\\partial}{\\partial n} c n)^{2} (\\iint \\operatorname{t_{1}}^{n}{(c,n)} dc dn) \\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn = (\\frac{\\partial}{\\partial n} c n)^{2} (\\iint (\\frac{\\partial}{\\partial n} c n)^{n} dc dn)^{2}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Pow(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["times", 4, "Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integral(Pow(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["times", 5, "Mul(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True))))"], "Equality(Mul(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(2)), Integral(Pow(Function('t_1')(Symbol('c', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(2)), Pow(Integral(Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(2))))"]]}, {"prompt": "Given b{(\\phi_1)} = \\sin{(\\phi_1)}, then derive \\int b{(\\phi_1)} d\\phi_1 = \\sigma_p - \\cos{(\\phi_1)}, then obtain (\\frac{\\partial}{\\partial \\phi_1} (\\sigma_p - \\cos{(\\phi_1)}))^{\\sigma_p} = (\\frac{d}{d \\phi_1} \\int \\sin{(\\phi_1)} d\\phi_1)^{\\sigma_p}", "derivation": "b{(\\phi_1)} = \\sin{(\\phi_1)} and \\int b{(\\phi_1)} d\\phi_1 = \\int \\sin{(\\phi_1)} d\\phi_1 and \\frac{d}{d \\phi_1} \\int b{(\\phi_1)} d\\phi_1 = \\frac{d}{d \\phi_1} \\int \\sin{(\\phi_1)} d\\phi_1 and \\int b{(\\phi_1)} d\\phi_1 = \\sigma_p - \\cos{(\\phi_1)} and \\frac{\\partial}{\\partial \\phi_1} (\\sigma_p - \\cos{(\\phi_1)}) = \\frac{d}{d \\phi_1} \\int \\sin{(\\phi_1)} d\\phi_1 and (\\frac{\\partial}{\\partial \\phi_1} (\\sigma_p - \\cos{(\\phi_1)}))^{\\sigma_p} = (\\frac{d}{d \\phi_1} \\int \\sin{(\\phi_1)} d\\phi_1)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Integral(Function('b')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integral(sin(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given I{(\\lambda)} = \\cos{(\\lambda)}, then obtain (1 + \\frac{I^{\\lambda}{(\\lambda)}}{I{(\\lambda)}}) (1 + \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}}) = (1 + \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}})^{2}", "derivation": "I{(\\lambda)} = \\cos{(\\lambda)} and I^{\\lambda}{(\\lambda)} = \\cos^{\\lambda}{(\\lambda)} and \\frac{I^{\\lambda}{(\\lambda)}}{I{(\\lambda)}} = \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}} and 1 + \\frac{I^{\\lambda}{(\\lambda)}}{I{(\\lambda)}} = 1 + \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}} and (1 + \\frac{I^{\\lambda}{(\\lambda)}}{I{(\\lambda)}}) (1 + \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}}) = (1 + \\frac{\\cos^{\\lambda}{(\\lambda)}}{I{(\\lambda)}})^{2}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["divide", 2, "Function('I')(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))))"], [["times", 4, "Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], "Equality(Mul(Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))), Pow(Add(Integer(1), Mul(Pow(Function('I')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given E{(\\pi,T,f_{E})} = \\frac{\\pi}{T f_{E}}, then derive \\frac{\\partial}{\\partial \\pi} E{(\\pi,T,f_{E})} = \\frac{1}{T f_{E}}, then obtain (\\frac{\\partial}{\\partial \\pi} \\frac{\\pi}{T f_{E}})^{f_{E}} = (\\frac{1}{T f_{E}})^{f_{E}}", "derivation": "E{(\\pi,T,f_{E})} = \\frac{\\pi}{T f_{E}} and \\frac{\\partial}{\\partial \\pi} E{(\\pi,T,f_{E})} = \\frac{\\partial}{\\partial \\pi} \\frac{\\pi}{T f_{E}} and \\frac{\\partial}{\\partial \\pi} E{(\\pi,T,f_{E})} = \\frac{1}{T f_{E}} and \\frac{\\partial}{\\partial \\pi} \\frac{\\pi}{T f_{E}} = \\frac{1}{T f_{E}} and (\\frac{\\partial}{\\partial \\pi} \\frac{\\pi}{T f_{E}})^{f_{E}} = (\\frac{1}{T f_{E}})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('\\\\pi', commutative=True), Symbol('T', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('f_E', commutative=True)), Pow(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('f_E', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given k{(\\phi_2,m_{s},B)} = B + \\phi_2 - m_{s}, then derive \\frac{\\partial}{\\partial B} k{(\\phi_2,m_{s},B)} = 1, then obtain \\frac{\\partial}{\\partial B} (B + \\phi_2 - m_{s}) = 1", "derivation": "k{(\\phi_2,m_{s},B)} = B + \\phi_2 - m_{s} and \\frac{\\partial}{\\partial B} k{(\\phi_2,m_{s},B)} = \\frac{\\partial}{\\partial B} (B + \\phi_2 - m_{s}) and \\frac{\\partial}{\\partial B} k{(\\phi_2,m_{s},B)} = 1 and \\frac{\\partial}{\\partial B} (B + \\phi_2 - m_{s}) = 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\phi_2', commutative=True), Symbol('m_s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('B', commutative=True), Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} = \\cos^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})}", "derivation": "\\operatorname{C_{2}}{(\\hat{H})} = \\cos{(\\hat{H})} and \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} = \\cos^{\\hat{H}}{(\\hat{H})} and \\frac{d}{d \\hat{H}} \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} = \\frac{d}{d \\hat{H}} \\cos^{\\hat{H}}{(\\hat{H})} and \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\cos^{\\hat{H}}{(\\hat{H})} = \\cos^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\cos^{\\hat{H}}{(\\hat{H})} and \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})} = \\cos^{\\hat{H}}{(\\hat{H})} + \\frac{d}{d \\hat{H}} \\operatorname{C_{2}}^{\\hat{H}}{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Derivative(Pow(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(\\hat{p},\\Psi_{nl})} = \\frac{\\cos{(\\Psi_{nl})}}{\\hat{p}} and \\tilde{g}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})}, then obtain - \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\tilde{g}{(\\Psi_{nl})}}{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} a{(\\hat{p},\\Psi_{nl})} = 0", "derivation": "a{(\\hat{p},\\Psi_{nl})} = \\frac{\\cos{(\\Psi_{nl})}}{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} a{(\\hat{p},\\Psi_{nl})} = \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\cos{(\\Psi_{nl})}}{\\hat{p}} and \\tilde{g}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and - \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\cos{(\\Psi_{nl})}}{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} a{(\\hat{p},\\Psi_{nl})} = 0 and - \\frac{\\partial}{\\partial \\hat{p}} \\frac{\\tilde{g}{(\\Psi_{nl})}}{\\hat{p}} + \\frac{\\partial}{\\partial \\hat{p}} a{(\\hat{p},\\Psi_{nl})} = 0", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 2, "Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Derivative(Function('a')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Derivative(Function('a')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\rho_{b}{(z,A_{1})} = A_{1} + z and \\mathbf{S}{(E)} = \\cos{(E)}, then obtain - (A_{1} + z) \\cos{(E)} + (\\frac{d}{d E} \\mathbf{S}{(E)})^{E} = - (A_{1} + z) \\cos{(E)} + (\\frac{d}{d E} \\cos{(E)})^{E}", "derivation": "\\rho_{b}{(z,A_{1})} = A_{1} + z and \\mathbf{S}{(E)} = \\cos{(E)} and \\rho_{b}{(z,A_{1})} \\cos{(E)} = (A_{1} + z) \\cos{(E)} and \\frac{d}{d E} \\mathbf{S}{(E)} = \\frac{d}{d E} \\cos{(E)} and (\\frac{d}{d E} \\mathbf{S}{(E)})^{E} = (\\frac{d}{d E} \\cos{(E)})^{E} and - \\rho_{b}{(z,A_{1})} \\cos{(E)} + (\\frac{d}{d E} \\mathbf{S}{(E)})^{E} = - \\rho_{b}{(z,A_{1})} \\cos{(E)} + (\\frac{d}{d E} \\cos{(E)})^{E} and - (A_{1} + z) \\cos{(E)} + (\\frac{d}{d E} \\mathbf{S}{(E)})^{E} = - (A_{1} + z) \\cos{(E)} + (\\frac{d}{d E} \\cos{(E)})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('z', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('z', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["times", 1, "cos(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\rho_b')(Symbol('z', commutative=True), Symbol('A_1', commutative=True)), cos(Symbol('E', commutative=True))), Mul(Add(Symbol('A_1', commutative=True), Symbol('z', commutative=True)), cos(Symbol('E', commutative=True))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 4, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{S}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["minus", 5, "Mul(Function('\\\\rho_b')(Symbol('z', commutative=True), Symbol('A_1', commutative=True)), cos(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('z', commutative=True), Symbol('A_1', commutative=True)), cos(Symbol('E', commutative=True))), Pow(Derivative(Function('\\\\mathbf{S}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('z', commutative=True), Symbol('A_1', commutative=True)), cos(Symbol('E', commutative=True))), Pow(Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Add(Symbol('A_1', commutative=True), Symbol('z', commutative=True)), cos(Symbol('E', commutative=True))), Pow(Derivative(Function('\\\\mathbf{S}')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Add(Mul(Integer(-1), Add(Symbol('A_1', commutative=True), Symbol('z', commutative=True)), cos(Symbol('E', commutative=True))), Pow(Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"]]}, {"prompt": "Given Q{(f_{E})} = e^{f_{E}} and \\mathbf{J}_P{(f_{E})} = f_{E}, then obtain f_{E} - Q{(f_{E})} + \\mathbf{J}_P{(f_{E})} - 2 e^{f_{E}} - 2 = 2 f_{E} - Q{(f_{E})} - 2 e^{f_{E}} - 2", "derivation": "Q{(f_{E})} = e^{f_{E}} and \\mathbf{J}_P{(f_{E})} = f_{E} and - Q{(f_{E})} + \\mathbf{J}_P{(f_{E})} = f_{E} - Q{(f_{E})} and \\mathbf{J}_P{(f_{E})} - e^{f_{E}} = f_{E} - e^{f_{E}} and \\mathbf{J}_P{(f_{E})} - e^{f_{E}} - 1 = f_{E} - e^{f_{E}} - 1 and - Q{(f_{E})} + \\mathbf{J}_P{(f_{E})} - e^{f_{E}} - 1 = f_{E} - Q{(f_{E})} - e^{f_{E}} - 1 and f_{E} - Q{(f_{E})} + \\mathbf{J}_P{(f_{E})} - 2 e^{f_{E}} - 2 = 2 f_{E} - Q{(f_{E})} - 2 e^{f_{E}} - 2", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('f_E', commutative=True)), exp(Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True)), Symbol('f_E', commutative=True))"], [["minus", 2, "Function('Q')(Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True)), Mul(Integer(-1), exp(Symbol('f_E', commutative=True)))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), exp(Symbol('f_E', commutative=True)))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True)), Mul(Integer(-1), exp(Symbol('f_E', commutative=True))), Integer(-1)), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), exp(Symbol('f_E', commutative=True))), Integer(-1)))"], [["minus", 5, "Function('Q')(Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True)), Mul(Integer(-1), exp(Symbol('f_E', commutative=True))), Integer(-1)), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True))), Mul(Integer(-1), exp(Symbol('f_E', commutative=True))), Integer(-1)))"], [["add", 6, "Add(Symbol('f_E', commutative=True), Mul(Integer(-1), exp(Symbol('f_E', commutative=True))), Integer(-1))"], "Equality(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('f_E', commutative=True))), Integer(-2)), Add(Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('f_E', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('f_E', commutative=True))), Integer(-2)))"]]}, {"prompt": "Given \\mathbf{E}{(v_{x},m)} = m - v_{x} and \\rho_{f}{(\\nabla)} = \\cos{(\\nabla)}, then obtain \\frac{\\mathbf{E}^{v_{x}}{(v_{x},m)} - \\cos{(\\nabla)}}{(m - v_{x})^{v_{x}} - \\rho_{f}{(\\nabla)}} = \\frac{(m - v_{x})^{v_{x}} - \\cos{(\\nabla)}}{(m - v_{x})^{v_{x}} - \\rho_{f}{(\\nabla)}}", "derivation": "\\mathbf{E}{(v_{x},m)} = m - v_{x} and \\rho_{f}{(\\nabla)} = \\cos{(\\nabla)} and \\mathbf{E}^{v_{x}}{(v_{x},m)} = (m - v_{x})^{v_{x}} and \\mathbf{E}^{v_{x}}{(v_{x},m)} - \\rho_{f}{(\\nabla)} = (m - v_{x})^{v_{x}} - \\rho_{f}{(\\nabla)} and \\mathbf{E}^{v_{x}}{(v_{x},m)} - \\cos{(\\nabla)} = (m - v_{x})^{v_{x}} - \\cos{(\\nabla)} and \\frac{\\mathbf{E}^{v_{x}}{(v_{x},m)} - \\cos{(\\nabla)}}{(m - v_{x})^{v_{x}} - \\rho_{f}{(\\nabla)}} = \\frac{(m - v_{x})^{v_{x}} - \\cos{(\\nabla)}}{(m - v_{x})^{v_{x}} - \\rho_{f}{(\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Symbol('v_x', commutative=True)), Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["minus", 3, "Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True)))), Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True)))), Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 5, "Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True))))"], "Equality(Mul(Pow(Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(Pow(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('m', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True))))), Mul(Pow(Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\rho_f')(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(Pow(Add(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given k{(b,l)} = b \\operatorname{v_{z}}{(b,l)} and a{(b,l)} = b \\operatorname{v_{z}}{(b,l)}, then obtain ((b \\operatorname{v_{z}}{(b,l)})^{b})^{b} k^{b}{(b,l)} = (a^{b}{(b,l)})^{b} k^{b}{(b,l)}", "derivation": "k{(b,l)} = b \\operatorname{v_{z}}{(b,l)} and k^{b}{(b,l)} = (b \\operatorname{v_{z}}{(b,l)})^{b} and a{(b,l)} = b \\operatorname{v_{z}}{(b,l)} and k^{b}{(b,l)} = a^{b}{(b,l)} and (b \\operatorname{v_{z}}{(b,l)})^{b} = a^{b}{(b,l)} and ((b \\operatorname{v_{z}}{(b,l)})^{b})^{b} = (a^{b}{(b,l)})^{b} and ((b \\operatorname{v_{z}}{(b,l)})^{b})^{b} k^{b}{(b,l)} = (a^{b}{(b,l)})^{b} k^{b}{(b,l)}", "srepr_derivation": [["renaming_premise", "Equality(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), Pow(Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), Pow(Function('a')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True)), Pow(Function('a')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Pow(Function('a')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["times", 6, "Pow(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))"], "Equality(Mul(Pow(Pow(Mul(Symbol('b', commutative=True), Function('v_z')(Symbol('b', commutative=True), Symbol('l', commutative=True))), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))), Mul(Pow(Pow(Function('a')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(Function('k')(Symbol('b', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))))"]]}, {"prompt": "Given L{(v_{2})} = e^{\\cos{(v_{2})}}, then obtain 2 \\frac{d}{d v_{2}} \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}} = \\frac{d}{d v_{2}} \\frac{e^{2 \\cos{(v_{2})}}}{L^{2}{(v_{2})}} + \\frac{d}{d v_{2}} \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}}", "derivation": "L{(v_{2})} = e^{\\cos{(v_{2})}} and 1 = \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}} and \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}} = \\frac{e^{2 \\cos{(v_{2})}}}{L^{2}{(v_{2})}} and \\frac{d}{d v_{2}} \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}} = \\frac{d}{d v_{2}} \\frac{e^{2 \\cos{(v_{2})}}}{L^{2}{(v_{2})}} and 2 \\frac{d}{d v_{2}} \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}} = \\frac{d}{d v_{2}} \\frac{e^{2 \\cos{(v_{2})}}}{L^{2}{(v_{2})}} + \\frac{d}{d v_{2}} \\frac{e^{\\cos{(v_{2})}}}{L{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v_2', commutative=True)), exp(cos(Symbol('v_2', commutative=True))))"], [["divide", 1, "Function('L')(Symbol('v_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))))"], [["times", 2, "Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True))))"], "Equality(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))), Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-2)), exp(Mul(Integer(2), cos(Symbol('v_2', commutative=True))))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-2)), exp(Mul(Integer(2), cos(Symbol('v_2', commutative=True))))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-2)), exp(Mul(Integer(2), cos(Symbol('v_2', commutative=True))))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('L')(Symbol('v_2', commutative=True)), Integer(-1)), exp(cos(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = \\Psi_{\\lambda} e^{V_{\\mathbf{B}}}, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = e^{V_{\\mathbf{B}}}, then obtain \\Psi_{\\lambda} e^{V_{\\mathbf{B}}} = \\Psi_{\\lambda} \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} e^{V_{\\mathbf{B}}}", "derivation": "\\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = \\Psi_{\\lambda} e^{V_{\\mathbf{B}}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} e^{V_{\\mathbf{B}}} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = e^{V_{\\mathbf{B}}} and \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{f^{*}}{(V_{\\mathbf{B}},\\Psi_{\\lambda})} and \\Psi_{\\lambda} e^{V_{\\mathbf{B}}} = \\Psi_{\\lambda} \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} e^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('f^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Derivative(Function('f^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\delta{(A_{x})} = \\log{(A_{x})}, then obtain (0^{A_{x}})^{A_{x}} - \\delta{(A_{x})} = 1 - \\delta{(A_{x})}", "derivation": "\\delta{(A_{x})} = \\log{(A_{x})} and 0 = - \\delta{(A_{x})} + \\log{(A_{x})} and 0^{A_{x}} = (- \\delta{(A_{x})} + \\log{(A_{x})})^{A_{x}} and (0^{A_{x}})^{A_{x}} = ((- \\delta{(A_{x})} + \\log{(A_{x})})^{A_{x}})^{A_{x}} and (0^{A_{x}})^{A_{x}} - \\delta{(A_{x})} = ((- \\delta{(A_{x})} + \\log{(A_{x})})^{A_{x}})^{A_{x}} - \\delta{(A_{x})} and ((- \\delta{(A_{x})} + \\log{(A_{x})})^{A_{x}})^{A_{x}} - \\delta{(A_{x})} = 1 - \\delta{(A_{x})} and (0^{A_{x}})^{A_{x}} - \\delta{(A_{x})} = 1 - \\delta{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["minus", 1, "Function('\\\\delta')(Symbol('A_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)))"], [["minus", 4, "Function('\\\\delta')(Symbol('A_x', commutative=True))"], "Equality(Add(Pow(Pow(Integer(0), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))), Add(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True))), log(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Pow(Pow(Integer(0), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\delta')(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\chi)} = \\chi and \\operatorname{E_{x}}{(\\chi)} = - \\chi + \\operatorname{c_{0}}{(\\chi)}, then obtain (\\frac{- \\chi + \\operatorname{c_{0}}{(\\chi)}}{2 \\chi})^{\\chi} - (\\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi})^{\\chi} = 0^{\\chi} - (\\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi})^{\\chi}", "derivation": "\\operatorname{c_{0}}{(\\chi)} = \\chi and \\operatorname{E_{x}}{(\\chi)} = - \\chi + \\operatorname{c_{0}}{(\\chi)} and \\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi} = \\frac{- \\chi + \\operatorname{c_{0}}{(\\chi)}}{2 \\chi} and \\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi} = 0 and (\\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi})^{\\chi} = 0^{\\chi} and (\\frac{- \\chi + \\operatorname{c_{0}}{(\\chi)}}{2 \\chi})^{\\chi} = 0^{\\chi} and (\\frac{- \\chi + \\operatorname{c_{0}}{(\\chi)}}{2 \\chi})^{\\chi} - (\\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi})^{\\chi} = 0^{\\chi} - (\\frac{\\operatorname{E_{x}}{(\\chi)}}{2 \\chi})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\chi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('c_0')(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('c_0')(Symbol('\\\\chi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Integer(0))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integer(0), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('c_0')(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Integer(0), Symbol('\\\\chi', commutative=True)))"], [["minus", 6, "Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('c_0')(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))), Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Mul(Rational(1, 2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('E_x')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(t)} = \\sin{(t)}, then obtain \\operatorname{v_{2}}{(t)} \\sin{(t)} - 2 \\operatorname{v_{2}}{(t)} + \\sin^{2}{(t)} = - 2 \\operatorname{v_{2}}{(t)} + 2 \\sin^{2}{(t)}", "derivation": "\\operatorname{v_{2}}{(t)} = \\sin{(t)} and \\operatorname{v_{2}}{(t)} \\sin{(t)} = \\sin^{2}{(t)} and \\operatorname{v_{2}}{(t)} \\sin{(t)} - \\operatorname{v_{2}}{(t)} = - \\operatorname{v_{2}}{(t)} + \\sin^{2}{(t)} and \\operatorname{v_{2}}{(t)} \\sin{(t)} - 2 \\operatorname{v_{2}}{(t)} + \\sin^{2}{(t)} = - 2 \\operatorname{v_{2}}{(t)} + 2 \\sin^{2}{(t)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["times", 1, "sin(Symbol('t', commutative=True))"], "Equality(Mul(Function('v_2')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Pow(sin(Symbol('t', commutative=True)), Integer(2)))"], [["minus", 2, "Function('v_2')(Symbol('t', commutative=True))"], "Equality(Add(Mul(Function('v_2')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(sin(Symbol('t', commutative=True)), Integer(2))))"], [["add", 3, "Add(Mul(Integer(-1), Function('v_2')(Symbol('t', commutative=True))), Pow(sin(Symbol('t', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Function('v_2')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Mul(Integer(-1), Integer(2), Function('v_2')(Symbol('t', commutative=True))), Pow(sin(Symbol('t', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Integer(2), Function('v_2')(Symbol('t', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('t', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(Z,v_{z})} = Z + v_{z}, then derive \\int \\operatorname{v_{t}}{(Z,v_{z})} dv_{z} = Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2}, then obtain \\int \\frac{\\partial}{\\partial v_{z}} (Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2}) d\\ddot{x} = \\int \\frac{\\partial}{\\partial v_{z}} \\int (Z + v_{z}) dv_{z} d\\ddot{x}", "derivation": "\\operatorname{v_{t}}{(Z,v_{z})} = Z + v_{z} and \\int \\operatorname{v_{t}}{(Z,v_{z})} dv_{z} = \\int (Z + v_{z}) dv_{z} and \\int \\operatorname{v_{t}}{(Z,v_{z})} dv_{z} = Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2} and Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2} = \\int (Z + v_{z}) dv_{z} and \\frac{\\partial}{\\partial v_{z}} (Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2}) = \\frac{\\partial}{\\partial v_{z}} \\int (Z + v_{z}) dv_{z} and \\int \\frac{\\partial}{\\partial v_{z}} (Z v_{z} + \\ddot{x} + \\frac{v_{z}^{2}}{2}) d\\ddot{x} = \\int \\frac{\\partial}{\\partial v_{z}} \\int (Z + v_{z}) dv_{z} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Mul(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))), Integral(Add(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\ddot{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Derivative(Integral(Add(Symbol('Z', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbf{J}_M,v_{z})} = e^{\\mathbf{J}_M - v_{z}}, then obtain (\\operatorname{r_{0}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,v_{z})})^{v_{z}} - ((e^{\\mathbf{J}_M - v_{z}})^{\\mathbf{J}_M})^{v_{z}} = 0", "derivation": "\\operatorname{r_{0}}{(\\mathbf{J}_M,v_{z})} = e^{\\mathbf{J}_M - v_{z}} and \\operatorname{r_{0}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,v_{z})} = (e^{\\mathbf{J}_M - v_{z}})^{\\mathbf{J}_M} and (\\operatorname{r_{0}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,v_{z})})^{v_{z}} = ((e^{\\mathbf{J}_M - v_{z}})^{\\mathbf{J}_M})^{v_{z}} and (\\operatorname{r_{0}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,v_{z})})^{v_{z}} - ((e^{\\mathbf{J}_M - v_{z}})^{\\mathbf{J}_M})^{v_{z}} = 0", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), exp(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(exp(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Pow(Function('r_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('v_z', commutative=True)), Pow(Pow(exp(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('v_z', commutative=True)))"], [["minus", 3, "Pow(Pow(exp(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Pow(Pow(Function('r_0')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Pow(exp(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('v_z', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(U,m)} = \\log{(\\frac{U}{m})} and b{(U,m)} = \\dot{\\mathbf{r}}{(U,m)} - \\log{(\\frac{U}{m})}, then obtain b{(U,m)} = 0", "derivation": "\\dot{\\mathbf{r}}{(U,m)} = \\log{(\\frac{U}{m})} and \\dot{\\mathbf{r}}{(U,m)} - \\log{(\\frac{U}{m})} = 0 and b{(U,m)} = \\dot{\\mathbf{r}}{(U,m)} - \\log{(\\frac{U}{m})} and b{(U,m)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('m', commutative=True)), log(Mul(Symbol('U', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["minus", 1, "log(Mul(Symbol('U', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('U', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))), Integer(0))"], ["renaming_premise", "Equality(Function('b')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('U', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('b')(Symbol('U', commutative=True), Symbol('m', commutative=True)), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(F_{c},u)} = \\frac{F_{c}}{u} and B{(F_{c},g_{\\varepsilon},\\sigma_p,u)} = \\frac{F_{c}}{\\sigma_p u (\\sigma_p + g_{\\varepsilon})}, then obtain 1 + \\frac{\\operatorname{f_{\\mathbf{p}}}{(F_{c},u)}}{\\sigma_p (\\sigma_p + g_{\\varepsilon})} = B{(F_{c},g_{\\varepsilon},\\sigma_p,u)} + 1", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(F_{c},u)} = \\frac{F_{c}}{u} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(F_{c},u)}}{\\sigma_p (\\sigma_p + g_{\\varepsilon})} = \\frac{F_{c}}{\\sigma_p u (\\sigma_p + g_{\\varepsilon})} and 1 + \\frac{\\operatorname{f_{\\mathbf{p}}}{(F_{c},u)}}{\\sigma_p (\\sigma_p + g_{\\varepsilon})} = \\frac{F_{c}}{\\sigma_p u (\\sigma_p + g_{\\varepsilon})} + 1 and B{(F_{c},g_{\\varepsilon},\\sigma_p,u)} = \\frac{F_{c}}{\\sigma_p u (\\sigma_p + g_{\\varepsilon})} and 1 + \\frac{\\operatorname{f_{\\mathbf{p}}}{(F_{c},u)}}{\\sigma_p (\\sigma_p + g_{\\varepsilon})} = B{(F_{c},g_{\\varepsilon},\\sigma_p,u)} + 1", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('F_c', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\sigma_p', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('F_c', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["add", 2, 1], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('B')(Symbol('F_c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('F_c', commutative=True), Symbol('u', commutative=True)))), Add(Function('B')(Symbol('F_c', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('u', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\psi^*)} = e^{\\cos{(\\psi^*)}}, then obtain (1 + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}})^{\\psi^*} = (\\frac{e^{\\cos{(\\psi^*)}}}{\\operatorname{f^{\\prime}}{(\\psi^*)}} + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}})^{\\psi^*}", "derivation": "\\operatorname{f^{\\prime}}{(\\psi^*)} = e^{\\cos{(\\psi^*)}} and 1 = \\frac{e^{\\cos{(\\psi^*)}}}{\\operatorname{f^{\\prime}}{(\\psi^*)}} and 1 + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}} = \\frac{e^{\\cos{(\\psi^*)}}}{\\operatorname{f^{\\prime}}{(\\psi^*)}} + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}} and (1 + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}})^{\\psi^*} = (\\frac{e^{\\cos{(\\psi^*)}}}{\\operatorname{f^{\\prime}}{(\\psi^*)}} + \\frac{1}{\\operatorname{f^{\\prime}}{(\\psi^*)}})^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), exp(cos(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 1, "Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\psi^*', commutative=True)))))"], [["add", 2, "Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\psi^*', commutative=True)))), Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Integer(1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), exp(cos(Symbol('\\\\psi^*', commutative=True)))), Pow(Function('f^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(A_{x},\\phi)} = A_{x} \\phi, then derive \\frac{\\partial}{\\partial A_{x}} \\operatorname{v_{t}}{(A_{x},\\phi)} = \\phi, then derive \\mathbf{J}_M + \\frac{\\phi^{2}}{2} = C_{2} + \\frac{\\phi^{2}}{2}, then obtain \\frac{\\mathbf{J}_M + \\frac{\\phi^{2}}{2}}{\\int \\operatorname{v_{t}}{(A_{x},\\phi)} d\\phi} = \\frac{C_{2} + \\frac{\\phi^{2}}{2}}{\\int \\operatorname{v_{t}}{(A_{x},\\phi)} d\\phi}", "derivation": "\\operatorname{v_{t}}{(A_{x},\\phi)} = A_{x} \\phi and \\frac{\\partial}{\\partial A_{x}} \\operatorname{v_{t}}{(A_{x},\\phi)} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\phi and \\frac{\\partial}{\\partial A_{x}} \\operatorname{v_{t}}{(A_{x},\\phi)} = \\phi and \\frac{\\partial}{\\partial A_{x}} A_{x} \\phi = \\phi and \\int \\frac{\\partial}{\\partial A_{x}} A_{x} \\phi d\\phi = \\int \\phi d\\phi and \\mathbf{J}_M + \\frac{\\phi^{2}}{2} = C_{2} + \\frac{\\phi^{2}}{2} and \\frac{\\mathbf{J}_M + \\frac{\\phi^{2}}{2}}{\\int \\operatorname{v_{t}}{(A_{x},\\phi)} d\\phi} = \\frac{C_{2} + \\frac{\\phi^{2}}{2}}{\\int \\operatorname{v_{t}}{(A_{x},\\phi)} d\\phi}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], [["integrate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))), Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))))"], [["divide", 6, "Integral(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))), Pow(Integral(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))), Mul(Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))), Pow(Integral(Function('v_t')(Symbol('A_x', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\eta)} = \\sin{(\\eta)}, then derive \\int \\bar{\\h}{(\\eta)} \\sin{(\\eta)} d\\eta = \\frac{\\eta}{2} + u - \\frac{\\sin{(\\eta)} \\cos{(\\eta)}}{2}, then obtain \\int \\sin^{2}{(\\eta)} d\\eta = \\frac{\\eta}{2} + u - \\frac{\\sin{(\\eta)} \\cos{(\\eta)}}{2}", "derivation": "\\bar{\\h}{(\\eta)} = \\sin{(\\eta)} and \\bar{\\h}{(\\eta)} \\sin{(\\eta)} = \\sin^{2}{(\\eta)} and \\int \\bar{\\h}{(\\eta)} \\sin{(\\eta)} d\\eta = \\int \\sin^{2}{(\\eta)} d\\eta and \\int \\bar{\\h}{(\\eta)} \\sin{(\\eta)} d\\eta = \\frac{\\eta}{2} + u - \\frac{\\sin{(\\eta)} \\cos{(\\eta)}}{2} and \\int \\sin^{2}{(\\eta)} d\\eta = \\frac{\\eta}{2} + u - \\frac{\\sin{(\\eta)} \\cos{(\\eta)}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\hbar')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\eta', commutative=True)), Symbol('u', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\eta', commutative=True)), Symbol('u', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(J_{\\varepsilon})} = J_{\\varepsilon} and \\hat{\\mathbf{x}}{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})}, then derive \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})} = 1, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{f}} \\log{(\\mathbf{f})} \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(\\mathbf{f})}}", "derivation": "\\operatorname{n_{1}}{(J_{\\varepsilon})} = J_{\\varepsilon} and \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} J_{\\varepsilon} and \\hat{\\mathbf{x}}{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} and \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})} = 1 and 1 = \\frac{\\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})}}{\\hat{\\mathbf{x}}{(\\mathbf{f})}} and \\log{(\\mathbf{f})} \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})} = \\log{(\\mathbf{f})} and 1 = \\frac{\\frac{\\partial}{\\partial \\mathbf{f}} \\log{(\\mathbf{f})} \\frac{d}{d J_{\\varepsilon}} \\operatorname{n_{1}}{(J_{\\varepsilon})}}{\\hat{\\mathbf{x}}{(\\mathbf{f})}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Symbol('J_{\\\\varepsilon}', commutative=True), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{f}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["times", 4, "log(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(log(Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Function('n_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Derivative(Mul(log(Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Function('n_1')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(M_{E})} = \\sin{(\\log{(M_{E})})} and v{(M_{E})} = \\frac{d}{d M_{E}} \\operatorname{m_{s}}{(M_{E})} and \\operatorname{a^{\\dagger}}{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(\\log{(M_{E})})}, then obtain \\operatorname{a^{\\dagger}}{(M_{E})} = v{(M_{E})}", "derivation": "\\operatorname{m_{s}}{(M_{E})} = \\sin{(\\log{(M_{E})})} and \\frac{d}{d M_{E}} \\operatorname{m_{s}}{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(\\log{(M_{E})})} and v{(M_{E})} = \\frac{d}{d M_{E}} \\operatorname{m_{s}}{(M_{E})} and \\operatorname{a^{\\dagger}}{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(\\log{(M_{E})})} and v{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(\\log{(M_{E})})} and \\operatorname{a^{\\dagger}}{(M_{E})} = v{(M_{E})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('M_E', commutative=True)), sin(log(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(sin(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v')(Symbol('M_E', commutative=True)), Derivative(Function('m_s')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('M_E', commutative=True)), Derivative(sin(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v')(Symbol('M_E', commutative=True)), Derivative(sin(log(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('a^{\\\\dagger}')(Symbol('M_E', commutative=True)), Function('v')(Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(A_{2})} = \\cos{(A_{2})}, then obtain - 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} + \\frac{d}{d A_{2}} (\\mathbf{r}{(A_{2})} + \\cos{(A_{2})}) \\mathbf{r}{(A_{2})} = - 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} + \\frac{d}{d A_{2}} 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})}", "derivation": "\\mathbf{r}{(A_{2})} = \\cos{(A_{2})} and \\mathbf{r}{(A_{2})} + \\cos{(A_{2})} = 2 \\cos{(A_{2})} and (\\mathbf{r}{(A_{2})} + \\cos{(A_{2})}) \\mathbf{r}{(A_{2})} = 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} and \\frac{d}{d A_{2}} (\\mathbf{r}{(A_{2})} + \\cos{(A_{2})}) \\mathbf{r}{(A_{2})} = \\frac{d}{d A_{2}} 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} and - 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} + \\frac{d}{d A_{2}} (\\mathbf{r}{(A_{2})} + \\cos{(A_{2})}) \\mathbf{r}{(A_{2})} = - 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})} + \\frac{d}{d A_{2}} 2 \\mathbf{r}{(A_{2})} \\cos{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["add", 1, "cos(Symbol('A_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Mul(Integer(2), cos(Symbol('A_2', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True))), Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Derivative(Mul(Add(Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Derivative(Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(\\psi^*,\\mu)} = - \\psi^* + \\sin{(\\mu)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain - \\psi^* + e^{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})}} - e^{e^{\\mathbf{S}}} + \\sin{(\\mu)} = - \\psi^* + \\sin{(\\mu)}", "derivation": "u{(\\psi^*,\\mu)} = - \\psi^* + \\sin{(\\mu)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})} = e^{\\mathbf{S}} and e^{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})}} = e^{e^{\\mathbf{S}}} and e^{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})}} - e^{e^{\\mathbf{S}}} = 0 and u{(\\psi^*,\\mu)} + e^{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})}} - e^{e^{\\mathbf{S}}} = u{(\\psi^*,\\mu)} and - \\psi^* + e^{\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{S})}} - e^{e^{\\mathbf{S}}} + \\sin{(\\mu)} = - \\psi^* + \\sin{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\mu', commutative=True))))"], ["get_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["exp", 2], "Equality(exp(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{S}', commutative=True))), exp(exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 3, "exp(exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(exp(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{S}', commutative=True))))), Integer(0))"], [["add", 4, "Function('u')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)), exp(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{S}', commutative=True))))), Function('u')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), exp(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{S}', commutative=True)))), sin(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(I)} = \\log{(I)}, then obtain e^{(- \\mathbf{P}^{2}{(I)} + \\mathbf{P}{(I)})^{I}} = e^{(- \\mathbf{P}^{2}{(I)} + \\log{(I)})^{I}}", "derivation": "\\mathbf{P}{(I)} = \\log{(I)} and \\mathbf{P}^{2}{(I)} = \\mathbf{P}{(I)} \\log{(I)} and - \\mathbf{P}{(I)} \\log{(I)} + \\mathbf{P}{(I)} = - \\mathbf{P}{(I)} \\log{(I)} + \\log{(I)} and - \\mathbf{P}^{2}{(I)} + \\mathbf{P}{(I)} = - \\mathbf{P}^{2}{(I)} + \\log{(I)} and (- \\mathbf{P}^{2}{(I)} + \\mathbf{P}{(I)})^{I} = (- \\mathbf{P}^{2}{(I)} + \\log{(I)})^{I} and e^{(- \\mathbf{P}^{2}{(I)} + \\mathbf{P}{(I)})^{I}} = e^{(- \\mathbf{P}^{2}{(I)} + \\log{(I)})^{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{P}')(Symbol('I', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))))"], [["minus", 1, "Mul(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), Function('\\\\mathbf{P}')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True))), log(Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), Function('\\\\mathbf{P}')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), log(Symbol('I', commutative=True))))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), Function('\\\\mathbf{P}')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), log(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), Function('\\\\mathbf{P}')(Symbol('I', commutative=True))), Symbol('I', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True)), Integer(2))), log(Symbol('I', commutative=True))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(v_{x},\\chi)} = \\chi + v_{x} and \\mathbf{A}{(v_{x},\\chi)} = \\frac{\\operatorname{M_{E}}{(v_{x},\\chi)}}{v_{x}}, then obtain \\frac{\\mathbf{A}{(v_{x},\\chi)}}{\\chi (\\chi + v_{x})} = \\frac{1}{\\chi v_{x}}", "derivation": "\\operatorname{M_{E}}{(v_{x},\\chi)} = \\chi + v_{x} and \\frac{\\operatorname{M_{E}}{(v_{x},\\chi)}}{v_{x}} = \\frac{\\chi + v_{x}}{v_{x}} and \\mathbf{A}{(v_{x},\\chi)} = \\frac{\\operatorname{M_{E}}{(v_{x},\\chi)}}{v_{x}} and \\mathbf{A}{(v_{x},\\chi)} = \\frac{\\chi + v_{x}}{v_{x}} and \\frac{\\mathbf{A}{(v_{x},\\chi)}}{\\chi + v_{x}} = \\frac{1}{v_{x}} and \\frac{\\mathbf{A}{(v_{x},\\chi)}}{\\chi (\\chi + v_{x})} = \\frac{1}{\\chi v_{x}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True)))"], [["divide", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('M_E')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('M_E')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True)), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Symbol('v_x', commutative=True), Integer(-1)))"], [["divide", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('v_x', commutative=True)), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('v_x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\psi{(k)} = k, then obtain \\frac{\\psi^{3 k}{(k)}}{k} = \\frac{k^{2 k} \\psi^{k}{(k)}}{k}", "derivation": "\\psi{(k)} = k and \\psi^{k}{(k)} = k^{k} and \\frac{\\psi^{k}{(k)}}{k} = \\frac{k^{k}}{k} and k \\psi^{k}{(k)} = k k^{k} and \\frac{\\psi^{2 k}{(k)}}{k^{2}} = \\frac{k^{k} \\psi^{k}{(k)}}{k^{2}} and k^{k} \\psi^{k}{(k)} = k^{2 k} and \\frac{\\psi^{2 k}{(k)}}{k^{2}} = \\frac{k^{2 k}}{k^{2}} and \\frac{\\psi^{3 k}{(k)}}{k} = \\frac{k^{2 k} \\psi^{k}{(k)}}{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Symbol('k', commutative=True), Symbol('k', commutative=True)))"], [["divide", 2, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Symbol('k', commutative=True))))"], [["divide", 2, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), Pow(Symbol('k', commutative=True), Symbol('k', commutative=True))))"], [["times", 3, "Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-2)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Symbol('k', commutative=True)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Symbol('k', commutative=True), Mul(Integer(2), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-2)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Mul(Integer(2), Symbol('k', commutative=True)))))"], [["times", 7, "Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Mul(Integer(3), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(Symbol('k', commutative=True), Mul(Integer(2), Symbol('k', commutative=True))), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(s,\\mathbf{A})} = \\mathbf{A} + s and \\Psi_{nl}{(E_{n},m)} = \\sin{(\\frac{E_{n}}{m})}, then obtain \\int (\\Psi_{nl}{(E_{n},m)} + \\hat{x}{(s,\\mathbf{A})}) dm = \\int (\\hat{x}{(s,\\mathbf{A})} + \\sin{(\\frac{E_{n}}{m})}) dm", "derivation": "\\hat{x}{(s,\\mathbf{A})} = \\mathbf{A} + s and \\Psi_{nl}{(E_{n},m)} = \\sin{(\\frac{E_{n}}{m})} and \\mathbf{A} + s + \\Psi_{nl}{(E_{n},m)} = \\mathbf{A} + s + \\sin{(\\frac{E_{n}}{m})} and \\Psi_{nl}{(E_{n},m)} + \\hat{x}{(s,\\mathbf{A})} = \\hat{x}{(s,\\mathbf{A})} + \\sin{(\\frac{E_{n}}{m})} and \\int (\\Psi_{nl}{(E_{n},m)} + \\hat{x}{(s,\\mathbf{A})}) dm = \\int (\\hat{x}{(s,\\mathbf{A})} + \\sin{(\\frac{E_{n}}{m})}) dm", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('s', commutative=True)))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["add", 2, "Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('s', commutative=True), Function('\\\\Psi_{nl}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Symbol('s', commutative=True), sin(Mul(Symbol('E_n', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Function('\\\\hat{x}')(Symbol('s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Function('\\\\hat{x}')(Symbol('s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi_{nl}')(Symbol('E_n', commutative=True), Symbol('m', commutative=True)), Function('\\\\hat{x}')(Symbol('s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Function('\\\\hat{x}')(Symbol('s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(v_{1})} = \\int \\log{(v_{1})} dv_{1}, then derive V_{\\mathbf{E}} + v_{1} \\log{(v_{1})} - v_{1} + 2 \\operatorname{E_{n}}{(v_{1})} = 3 V_{\\mathbf{E}} + 3 v_{1} \\log{(v_{1})} - 3 v_{1}, then obtain - \\frac{V_{\\mathbf{E}} + v_{1} \\log{(v_{1})} - v_{1} + 2 \\operatorname{E_{n}}{(v_{1})}}{v_{1}} = - \\frac{3 V_{\\mathbf{E}} + 3 v_{1} \\log{(v_{1})} - 3 v_{1}}{v_{1}}", "derivation": "\\operatorname{E_{n}}{(v_{1})} = \\int \\log{(v_{1})} dv_{1} and \\operatorname{E_{n}}{(v_{1})} + \\int \\log{(v_{1})} dv_{1} = 2 \\int \\log{(v_{1})} dv_{1} and \\operatorname{E_{n}}{(v_{1})} + 2 \\int \\log{(v_{1})} dv_{1} = 3 \\int \\log{(v_{1})} dv_{1} and 2 \\operatorname{E_{n}}{(v_{1})} + \\int \\log{(v_{1})} dv_{1} = 3 \\int \\log{(v_{1})} dv_{1} and V_{\\mathbf{E}} + v_{1} \\log{(v_{1})} - v_{1} + 2 \\operatorname{E_{n}}{(v_{1})} = 3 V_{\\mathbf{E}} + 3 v_{1} \\log{(v_{1})} - 3 v_{1} and - \\frac{V_{\\mathbf{E}} + v_{1} \\log{(v_{1})} - v_{1} + 2 \\operatorname{E_{n}}{(v_{1})}}{v_{1}} = - \\frac{3 V_{\\mathbf{E}} + 3 v_{1} \\log{(v_{1})} - 3 v_{1}}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('v_1', commutative=True)), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["add", 1, "Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Function('E_n')(Symbol('v_1', commutative=True)), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(2), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["add", 2, "Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Function('E_n')(Symbol('v_1', commutative=True)), Mul(Integer(2), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Mul(Integer(3), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('E_n')(Symbol('v_1', commutative=True))), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(3), Integral(log(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('v_1', commutative=True)))), Add(Mul(Integer(3), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(3), Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(3), Symbol('v_1', commutative=True))))"], [["divide", 5, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('v_1', commutative=True))))), Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), Add(Mul(Integer(3), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(3), Symbol('v_1', commutative=True), log(Symbol('v_1', commutative=True))), Mul(Integer(-1), Integer(3), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(v_{x})} = \\log{(v_{x})}, then obtain v_{x} (\\dot{x}{(v_{x})} \\log{(v_{x})})^{2 v_{x}} = v_{x} (\\log{(v_{x})}^{2})^{2 v_{x}}", "derivation": "\\dot{x}{(v_{x})} = \\log{(v_{x})} and \\dot{x}{(v_{x})} \\log{(v_{x})} = \\log{(v_{x})}^{2} and (\\dot{x}{(v_{x})} \\log{(v_{x})})^{v_{x}} = (\\log{(v_{x})}^{2})^{v_{x}} and (\\dot{x}{(v_{x})} \\log{(v_{x})})^{2 v_{x}} = (\\log{(v_{x})}^{2})^{2 v_{x}} and v_{x} (\\dot{x}{(v_{x})} \\log{(v_{x})})^{2 v_{x}} = v_{x} (\\log{(v_{x})}^{2})^{2 v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["times", 1, "log(Symbol('v_x', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True))), Pow(log(Symbol('v_x', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Function('\\\\dot{x}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Pow(log(Symbol('v_x', commutative=True)), Integer(2)), Symbol('v_x', commutative=True)))"], [["power", 3, 2], "Equality(Pow(Mul(Function('\\\\dot{x}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True))), Mul(Integer(2), Symbol('v_x', commutative=True))), Pow(Pow(log(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["times", 4, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Pow(Mul(Function('\\\\dot{x}')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True))), Mul(Integer(2), Symbol('v_x', commutative=True)))), Mul(Symbol('v_x', commutative=True), Pow(Pow(log(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(F_{N})} = F_{N}, then obtain \\frac{d^{2}}{d F_{N}^{2}} \\operatorname{v_{z}}{(F_{N})} = 0", "derivation": "\\operatorname{v_{z}}{(F_{N})} = F_{N} and \\frac{d}{d F_{N}} \\operatorname{v_{z}}{(F_{N})} = \\frac{d}{d F_{N}} F_{N} and \\frac{d^{2}}{d F_{N}^{2}} \\operatorname{v_{z}}{(F_{N})} = \\frac{d^{2}}{d F_{N}^{2}} F_{N} and \\frac{d^{2}}{d F_{N}^{2}} \\operatorname{v_{z}}{(F_{N})} = 0", "srepr_derivation": [["renaming_premise", "Equality(Function('v_z')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(2))), Derivative(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_z')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\hat{p}{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)}, then derive \\hat{p}{(\\Omega)} = \\cos{(\\Omega)}, then obtain \\cos{(\\Omega)} \\cos^{\\Omega}{(\\Omega)} = \\cos^{\\Omega}{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)}", "derivation": "\\hat{p}{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\hat{p}^{\\Omega}{(\\Omega)} = (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega} and \\hat{p}{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega} and \\hat{p}{(\\Omega)} = \\cos{(\\Omega)} and \\hat{p}{(\\Omega)} \\hat{p}^{\\Omega}{(\\Omega)} = \\hat{p}^{\\Omega}{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)} and \\cos{(\\Omega)} \\cos^{\\Omega}{(\\Omega)} = \\cos^{\\Omega}{(\\Omega)} \\frac{d}{d \\Omega} \\sin{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Mul(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(cos(Symbol('\\\\Omega', commutative=True)), Pow(cos(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(cos(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\dot{y})} = \\log{(\\dot{y})}, then derive \\int S{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\lambda, then obtain - \\frac{\\dot{y} \\int S{(\\dot{y})} d\\dot{y}}{S{(\\dot{y})}} = - \\frac{\\dot{y} (\\dot{y} S{(\\dot{y})} - \\dot{y} + \\lambda)}{S{(\\dot{y})}}", "derivation": "S{(\\dot{y})} = \\log{(\\dot{y})} and \\dot{y} S{(\\dot{y})} = \\dot{y} \\log{(\\dot{y})} and \\int S{(\\dot{y})} d\\dot{y} = \\int \\log{(\\dot{y})} d\\dot{y} and \\int S{(\\dot{y})} d\\dot{y} = \\dot{y} \\log{(\\dot{y})} - \\dot{y} + \\lambda and \\int S{(\\dot{y})} d\\dot{y} = \\dot{y} S{(\\dot{y})} - \\dot{y} + \\lambda and \\frac{\\int S{(\\dot{y})} d\\dot{y}}{S{(\\dot{y})}} = \\frac{\\dot{y} S{(\\dot{y})} - \\dot{y} + \\lambda}{S{(\\dot{y})}} and - \\frac{\\dot{y} \\int S{(\\dot{y})} d\\dot{y}}{S{(\\dot{y})}} = - \\frac{\\dot{y} (\\dot{y} S{(\\dot{y})} - \\dot{y} + \\lambda)}{S{(\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Function('S')(Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(log(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), log(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Function('S')(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["divide", 5, "Function('S')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Integral(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Function('S')(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["times", 6, "Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Pow(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Integral(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Function('S')(Symbol('\\\\dot{y}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Function('S')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given B{(\\dot{x})} = \\sin{(e^{\\dot{x}})}, then obtain \\frac{- \\dot{x} + B{(\\dot{x})} + \\sin{(e^{\\dot{x}})}}{\\dot{x}} = \\frac{- \\dot{x} + 2 \\sin{(e^{\\dot{x}})}}{\\dot{x}}", "derivation": "B{(\\dot{x})} = \\sin{(e^{\\dot{x}})} and - \\dot{x} + B{(\\dot{x})} = - \\dot{x} + \\sin{(e^{\\dot{x}})} and - \\dot{x} + B{(\\dot{x})} + \\sin{(e^{\\dot{x}})} = - \\dot{x} + 2 \\sin{(e^{\\dot{x}})} and \\frac{- \\dot{x} + B{(\\dot{x})} + \\sin{(e^{\\dot{x}})}}{\\dot{x}} = \\frac{- \\dot{x} + 2 \\sin{(e^{\\dot{x}})}}{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\dot{x}', commutative=True)), sin(exp(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('B')(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), sin(exp(Symbol('\\\\dot{x}', commutative=True)))))"], [["add", 2, "sin(exp(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('B')(Symbol('\\\\dot{x}', commutative=True)), sin(exp(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(2), sin(exp(Symbol('\\\\dot{x}', commutative=True))))))"], [["divide", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Function('B')(Symbol('\\\\dot{x}', commutative=True)), sin(exp(Symbol('\\\\dot{x}', commutative=True))))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(2), sin(exp(Symbol('\\\\dot{x}', commutative=True)))))))"]]}, {"prompt": "Given \\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda} g_{\\varepsilon} - g, then obtain \\frac{\\partial}{\\partial g} \\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} = -1", "derivation": "\\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda} g_{\\varepsilon} - g and \\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} + 1 = E_{\\lambda} g_{\\varepsilon} - g + 1 and \\frac{\\partial}{\\partial g} (\\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} + 1) = \\frac{\\partial}{\\partial g} (E_{\\lambda} g_{\\varepsilon} - g + 1) and \\frac{\\partial}{\\partial g} \\theta_{1}{(g,g_{\\varepsilon},E_{\\lambda})} = -1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\theta_1')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Integer(1)))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Function('\\\\theta_1')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\theta_1')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{F})} = e^{\\mathbf{F}} and \\operatorname{v_{1}}{(\\mathbf{F})} = \\operatorname{v_{y}}{(\\mathbf{F})} + e^{\\mathbf{F}}, then obtain \\operatorname{v_{1}}{(\\mathbf{F})} e^{\\mathbf{F}} = 2 e^{2 \\mathbf{F}}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{F})} = e^{\\mathbf{F}} and \\operatorname{v_{y}}{(\\mathbf{F})} + e^{\\mathbf{F}} = 2 e^{\\mathbf{F}} and \\operatorname{v_{1}}{(\\mathbf{F})} = \\operatorname{v_{y}}{(\\mathbf{F})} + e^{\\mathbf{F}} and \\operatorname{v_{1}}{(\\mathbf{F})} = 2 e^{\\mathbf{F}} and \\operatorname{v_{1}}{(\\mathbf{F})} \\operatorname{v_{y}}{(\\mathbf{F})} = 2 \\operatorname{v_{y}}{(\\mathbf{F})} e^{\\mathbf{F}} and \\operatorname{v_{1}}{(\\mathbf{F})} e^{\\mathbf{F}} = 2 e^{2 \\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), Add(Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 4, "Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), Function('v_y')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('v_1')(Symbol('\\\\mathbf{F}', commutative=True)), exp(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(J)} = \\sin{(J)}, then obtain \\frac{1}{\\operatorname{V_{\\mathbf{E}}}{(J)} \\sin{(J)}} = \\frac{1}{\\sin^{2}{(J)}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(J)} = \\sin{(J)} and 1 = \\frac{\\sin{(J)}}{\\operatorname{V_{\\mathbf{E}}}{(J)}} and \\operatorname{V_{\\mathbf{E}}}{(J)} \\sin{(J)} = \\sin^{2}{(J)} and \\frac{1}{\\operatorname{V_{\\mathbf{E}}}{(J)} \\sin{(J)}} = \\frac{1}{\\operatorname{V_{\\mathbf{E}}}^{2}{(J)}} and \\frac{1}{\\sin^{2}{(J)}} = \\frac{1}{\\operatorname{V_{\\mathbf{E}}}^{2}{(J)}} and \\frac{1}{\\operatorname{V_{\\mathbf{E}}}{(J)} \\sin{(J)}} = \\frac{1}{\\sin^{2}{(J)}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["divide", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), Integer(-1)), sin(Symbol('J', commutative=True))))"], [["times", 1, "sin(Symbol('J', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True))), Pow(sin(Symbol('J', commutative=True)), Integer(2)))"], [["divide", 2, "Mul(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], "Equality(Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), Integer(-1)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), Integer(-2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(sin(Symbol('J', commutative=True)), Integer(-2)), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), Integer(-2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('J', commutative=True)), Integer(-1)), Pow(sin(Symbol('J', commutative=True)), Integer(-1))), Pow(sin(Symbol('J', commutative=True)), Integer(-2)))"]]}, {"prompt": "Given C{(k,\\psi^*)} = - \\psi^* + k, then obtain \\frac{\\partial}{\\partial \\psi^*} (e^{C{(k,\\psi^*)}})^{k} = \\frac{\\partial}{\\partial \\psi^*} (e^{- \\psi^* + k})^{k}", "derivation": "C{(k,\\psi^*)} = - \\psi^* + k and e^{C{(k,\\psi^*)}} = e^{- \\psi^* + k} and (e^{C{(k,\\psi^*)}})^{k} = (e^{- \\psi^* + k})^{k} and \\frac{\\partial}{\\partial \\psi^*} (e^{C{(k,\\psi^*)}})^{k} = \\frac{\\partial}{\\partial \\psi^*} (e^{- \\psi^* + k})^{k}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('k', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('k', commutative=True)))"], [["exp", 1], "Equality(exp(Function('C')(Symbol('k', commutative=True), Symbol('\\\\psi^*', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('k', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(exp(Function('C')(Symbol('k', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('k', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Pow(exp(Function('C')(Symbol('k', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Pow(exp(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(A_{y})} = \\sin{(A_{y})} and \\hat{p}_0{(A_{y})} = \\frac{d}{d A_{y}} \\frac{\\sin{(A_{y})}}{A_{y}}, then obtain \\frac{\\frac{d}{d A_{y}} \\operatorname{M_{E}}{(A_{y})}}{A_{y}} - \\frac{\\operatorname{M_{E}}{(A_{y})}}{A_{y}^{2}} = \\hat{p}_0{(A_{y})}", "derivation": "\\operatorname{M_{E}}{(A_{y})} = \\sin{(A_{y})} and \\frac{\\operatorname{M_{E}}{(A_{y})}}{A_{y}} = \\frac{\\sin{(A_{y})}}{A_{y}} and \\frac{d}{d A_{y}} \\frac{\\operatorname{M_{E}}{(A_{y})}}{A_{y}} = \\frac{d}{d A_{y}} \\frac{\\sin{(A_{y})}}{A_{y}} and \\hat{p}_0{(A_{y})} = \\frac{d}{d A_{y}} \\frac{\\sin{(A_{y})}}{A_{y}} and \\frac{d}{d A_{y}} \\frac{\\operatorname{M_{E}}{(A_{y})}}{A_{y}} = \\hat{p}_0{(A_{y})} and \\frac{\\frac{d}{d A_{y}} \\operatorname{M_{E}}{(A_{y})}}{A_{y}} - \\frac{\\operatorname{M_{E}}{(A_{y})}}{A_{y}^{2}} = \\hat{p}_0{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["divide", 1, "Symbol('A_y', commutative=True)"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Function('M_E')(Symbol('A_y', commutative=True))), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), sin(Symbol('A_y', commutative=True))))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Function('M_E')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Function('M_E')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Derivative(Function('M_E')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Integer(-2)), Function('M_E')(Symbol('A_y', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given B{(\\psi^*,\\mathbf{f})} = \\mathbf{f} \\psi^*, then obtain \\mathbf{f}^{2} (\\psi^*)^{2} + \\mathbf{f}^{2} + \\cos{(\\mathbf{f} \\psi^* B{(\\psi^*,\\mathbf{f})})} = \\mathbf{f}^{2} (\\psi^*)^{2} + \\mathbf{f}^{2} + \\cos{(\\mathbf{f}^{2} (\\psi^*)^{2})}", "derivation": "B{(\\psi^*,\\mathbf{f})} = \\mathbf{f} \\psi^* and \\mathbf{f} \\psi^* B{(\\psi^*,\\mathbf{f})} = \\mathbf{f}^{2} (\\psi^*)^{2} and \\cos{(\\mathbf{f} \\psi^* B{(\\psi^*,\\mathbf{f})})} = \\cos{(\\mathbf{f}^{2} (\\psi^*)^{2})} and \\mathbf{f}^{2} (\\psi^*)^{2} + \\mathbf{f}^{2} + \\cos{(\\mathbf{f} \\psi^* B{(\\psi^*,\\mathbf{f})})} = \\mathbf{f}^{2} (\\psi^*)^{2} + \\mathbf{f}^{2} + \\cos{(\\mathbf{f}^{2} (\\psi^*)^{2})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('B')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))))"], [["cos", 2], "Equality(cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('B')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), cos(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)))))"], [["add", 3, "Add(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), cos(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\psi^*', commutative=True), Function('B')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))), Add(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), cos(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\varphi^{*}{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\mathbf{H}{(C_{1})} = \\cos{(\\operatorname{n_{2}}{(C_{1})})}, then obtain \\int \\cos{(\\varphi^{*}{(C_{1})})} dC_{1} = \\int \\mathbf{H}{(C_{1})} dC_{1}", "derivation": "\\operatorname{n_{2}}{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\varphi^{*}{(C_{1})} = \\cos{(\\sin{(C_{1})})} and \\varphi^{*}{(C_{1})} = \\operatorname{n_{2}}{(C_{1})} and \\cos{(\\varphi^{*}{(C_{1})})} = \\cos{(\\operatorname{n_{2}}{(C_{1})})} and \\mathbf{H}{(C_{1})} = \\cos{(\\operatorname{n_{2}}{(C_{1})})} and \\int \\cos{(\\varphi^{*}{(C_{1})})} dC_{1} = \\int \\cos{(\\operatorname{n_{2}}{(C_{1})})} dC_{1} and \\int \\mathbf{H}{(C_{1})} dC_{1} = \\int \\cos{(\\operatorname{n_{2}}{(C_{1})})} dC_{1} and \\int \\cos{(\\varphi^{*}{(C_{1})})} dC_{1} = \\int \\mathbf{H}{(C_{1})} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('C_1', commutative=True)), cos(sin(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('C_1', commutative=True)), cos(sin(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\varphi^*')(Symbol('C_1', commutative=True)), Function('n_2')(Symbol('C_1', commutative=True)))"], [["cos", 3], "Equality(cos(Function('\\\\varphi^*')(Symbol('C_1', commutative=True))), cos(Function('n_2')(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True)), cos(Function('n_2')(Symbol('C_1', commutative=True))))"], [["integrate", 4, "Symbol('C_1', commutative=True)"], "Equality(Integral(cos(Function('\\\\varphi^*')(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(cos(Function('n_2')(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(cos(Function('n_2')(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(cos(Function('\\\\varphi^*')(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Function('\\\\mathbf{H}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\dot{z},c_{0})} = - \\dot{z} + \\cos{(c_{0})}, then obtain \\sin^{c_{0}}{(\\operatorname{f_{E}}{(\\dot{z},c_{0})} - 1)} = (- \\sin{(\\dot{z} - \\cos{(c_{0})} + 1)})^{c_{0}}", "derivation": "\\operatorname{f_{E}}{(\\dot{z},c_{0})} = - \\dot{z} + \\cos{(c_{0})} and \\operatorname{f_{E}}{(\\dot{z},c_{0})} - 1 = - \\dot{z} + \\cos{(c_{0})} - 1 and \\sin{(\\operatorname{f_{E}}{(\\dot{z},c_{0})} - 1)} = - \\sin{(\\dot{z} - \\cos{(c_{0})} + 1)} and \\sin^{c_{0}}{(\\operatorname{f_{E}}{(\\dot{z},c_{0})} - 1)} = (- \\sin{(\\dot{z} - \\cos{(c_{0})} + 1)})^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\dot{z}', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('c_0', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f_E')(Symbol('\\\\dot{z}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('c_0', commutative=True)), Integer(-1)))"], [["sin", 2], "Equality(sin(Add(Function('f_E')(Symbol('\\\\dot{z}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Integer(1)))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(sin(Add(Function('f_E')(Symbol('\\\\dot{z}', commutative=True), Symbol('c_0', commutative=True)), Integer(-1))), Symbol('c_0', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Integer(1)))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(F_{g},W)} = \\log{(F_{g} - W)}, then derive \\int \\bar{\\h}{(F_{g},W)} dF_{g} = F_{g} \\log{(F_{g} - W)} - F_{g} - W \\log{(F_{g} - W)} + g, then obtain F_{g} \\bar{\\h}{(F_{g},W)} - F_{g} - W \\bar{\\h}{(F_{g},W)} + g = F_{g} \\log{(F_{g} - W)} - F_{g} - W \\log{(F_{g} - W)} + c_{0}", "derivation": "\\bar{\\h}{(F_{g},W)} = \\log{(F_{g} - W)} and \\int \\bar{\\h}{(F_{g},W)} dF_{g} = \\int \\log{(F_{g} - W)} dF_{g} and \\int \\bar{\\h}{(F_{g},W)} dF_{g} = F_{g} \\log{(F_{g} - W)} - F_{g} - W \\log{(F_{g} - W)} + g and F_{g} \\log{(F_{g} - W)} - F_{g} - W \\log{(F_{g} - W)} + g = \\int \\log{(F_{g} - W)} dF_{g} and F_{g} \\bar{\\h}{(F_{g},W)} - F_{g} - W \\bar{\\h}{(F_{g},W)} + g = \\int \\log{(F_{g} - W)} dF_{g} and F_{g} \\bar{\\h}{(F_{g},W)} - F_{g} - W \\bar{\\h}{(F_{g},W)} + g = F_{g} \\log{(F_{g} - W)} - F_{g} - W \\log{(F_{g} - W)} + c_{0}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('F_g', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Symbol('g', commutative=True)), Integral(log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('F_g', commutative=True), Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True))), Symbol('g', commutative=True)), Integral(log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('F_g', commutative=True), Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), Function('\\\\hbar')(Symbol('F_g', commutative=True), Symbol('W', commutative=True))), Symbol('g', commutative=True)), Add(Mul(Symbol('F_g', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True), log(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given p{(\\dot{x})} = e^{\\dot{x}}, then obtain (p{(\\dot{x})} e^{\\dot{x}} + e^{\\dot{x}} \\frac{d}{d \\dot{x}} p{(\\dot{x})})^{2} e^{- \\dot{x}} = 4 e^{3 \\dot{x}}", "derivation": "p{(\\dot{x})} = e^{\\dot{x}} and p{(\\dot{x})} e^{\\dot{x}} = e^{2 \\dot{x}} and \\frac{d}{d \\dot{x}} p{(\\dot{x})} e^{\\dot{x}} = \\frac{d}{d \\dot{x}} e^{2 \\dot{x}} and (\\frac{d}{d \\dot{x}} p{(\\dot{x})} e^{\\dot{x}})^{2} = (\\frac{d}{d \\dot{x}} e^{2 \\dot{x}})^{2} and e^{- \\dot{x}} (\\frac{d}{d \\dot{x}} p{(\\dot{x})} e^{\\dot{x}})^{2} = e^{- \\dot{x}} (\\frac{d}{d \\dot{x}} e^{2 \\dot{x}})^{2} and (p{(\\dot{x})} e^{\\dot{x}} + e^{\\dot{x}} \\frac{d}{d \\dot{x}} p{(\\dot{x})})^{2} e^{- \\dot{x}} = 4 e^{3 \\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Mul(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(exp(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(2)))"], [["divide", 4, "exp(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Derivative(Mul(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(2))), Mul(exp(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Derivative(exp(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Add(Mul(Function('p')(Symbol('\\\\dot{x}', commutative=True)), exp(Symbol('\\\\dot{x}', commutative=True))), Mul(exp(Symbol('\\\\dot{x}', commutative=True)), Derivative(Function('p')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))), Integer(2)), exp(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))), Mul(Integer(4), exp(Mul(Integer(3), Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\theta_2)} = \\operatorname{c_{0}}{(\\theta_2)} - 1, then derive \\frac{d}{d \\theta_2} \\ddot{x}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\operatorname{c_{0}}{(\\theta_2)}, then obtain \\int \\frac{d}{d \\theta_2} \\ddot{x}{(\\theta_2)} d\\theta_2 = \\int \\frac{d}{d \\theta_2} \\operatorname{c_{0}}{(\\theta_2)} d\\theta_2", "derivation": "\\ddot{x}{(\\theta_2)} = \\operatorname{c_{0}}{(\\theta_2)} - 1 and \\frac{d}{d \\theta_2} \\ddot{x}{(\\theta_2)} = \\frac{d}{d \\theta_2} (\\operatorname{c_{0}}{(\\theta_2)} - 1) and \\frac{d}{d \\theta_2} \\ddot{x}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\operatorname{c_{0}}{(\\theta_2)} and \\int \\frac{d}{d \\theta_2} \\ddot{x}{(\\theta_2)} d\\theta_2 = \\int \\frac{d}{d \\theta_2} \\operatorname{c_{0}}{(\\theta_2)} d\\theta_2", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\theta_2', commutative=True)), Add(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Function('c_0')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f and \\chi{(\\rho_f,\\mathbf{A})} = \\mathbf{A} + \\dot{x}{(\\rho_f,\\mathbf{A})}, then obtain \\frac{\\chi{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f + \\mathbf{A}} = 1", "derivation": "\\dot{x}{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f and \\mathbf{A} + \\dot{x}{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f + \\mathbf{A} and \\chi{(\\rho_f,\\mathbf{A})} = \\mathbf{A} + \\dot{x}{(\\rho_f,\\mathbf{A})} and \\chi{(\\rho_f,\\mathbf{A})} = \\mathbf{A} \\rho_f + \\mathbf{A} and \\frac{\\chi{(\\rho_f,\\mathbf{A})}}{\\mathbf{A} \\rho_f + \\mathbf{A}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 4, "Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('\\\\chi')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\mathbf{J}{(\\rho_f,A,\\rho)} = A - \\rho + \\rho_f, then derive \\frac{\\partial}{\\partial A} \\mathbf{J}{(\\rho_f,A,\\rho)} - 1 = 0, then obtain \\frac{\\partial}{\\partial A} (A - \\rho + \\rho_f) - 1 = 0", "derivation": "\\mathbf{J}{(\\rho_f,A,\\rho)} = A - \\rho + \\rho_f and - A + \\rho - \\rho_f + \\mathbf{J}{(\\rho_f,A,\\rho)} = 0 and \\frac{\\partial}{\\partial A} (- A + \\rho - \\rho_f + \\mathbf{J}{(\\rho_f,A,\\rho)}) = \\frac{d}{d A} 0 and \\frac{\\partial}{\\partial A} \\mathbf{J}{(\\rho_f,A,\\rho)} - 1 = 0 and \\frac{\\partial}{\\partial A} (A - \\rho + \\rho_f) - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('A', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["minus", 1, "Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('A', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('A', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True), Symbol('A', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{r}{(\\eta,n)} = \\eta + n, then obtain \\int (\\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)})^{2} dn = \\int \\frac{\\partial}{\\partial n} (\\eta + n)^{\\eta} \\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)} dn", "derivation": "\\mathbf{r}{(\\eta,n)} = \\eta + n and \\mathbf{r}^{\\eta}{(\\eta,n)} = (\\eta + n)^{\\eta} and \\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)} = \\frac{\\partial}{\\partial n} (\\eta + n)^{\\eta} and (\\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)})^{2} = \\frac{\\partial}{\\partial n} (\\eta + n)^{\\eta} \\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)} and \\int (\\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)})^{2} dn = \\int \\frac{\\partial}{\\partial n} (\\eta + n)^{\\eta} \\frac{\\partial}{\\partial n} \\mathbf{r}^{\\eta}{(\\eta,n)} dn", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Derivative(Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\eta', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given W{(J_{\\varepsilon},u)} = \\frac{u}{J_{\\varepsilon}}, then derive 2 \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du = \\hbar + u + \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du, then obtain 2 \\int 1 du = \\hbar + u + \\int 1 du", "derivation": "W{(J_{\\varepsilon},u)} = \\frac{u}{J_{\\varepsilon}} and \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} = 1 and \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du = \\int 1 du and 2 \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du = \\int 1 du + \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du and 2 \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du = \\hbar + u + \\int \\frac{J_{\\varepsilon} W{(J_{\\varepsilon},u)}}{u} du and 2 \\int 1 du = \\hbar + u + \\int 1 du", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('u', commutative=True))"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Integer(1), Tuple(Symbol('u', commutative=True))))"], [["add", 3, "Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(2), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))), Add(Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)), Function('W')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Integral(Integer(1), Tuple(Symbol('u', commutative=True)))), Add(Symbol('\\\\hbar', commutative=True), Symbol('u', commutative=True), Integral(Integer(1), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(t_{2},\\omega)} = t_{2} + e^{\\omega}, then obtain \\frac{\\partial}{\\partial t_{2}} (- \\int (t_{2} + e^{\\omega}) d\\omega + \\int \\mathbf{A}{(t_{2},\\omega)} d\\omega) = \\frac{d}{d t_{2}} 0", "derivation": "\\mathbf{A}{(t_{2},\\omega)} = t_{2} + e^{\\omega} and \\int \\mathbf{A}{(t_{2},\\omega)} d\\omega = \\int (t_{2} + e^{\\omega}) d\\omega and - \\int (t_{2} + e^{\\omega}) d\\omega + \\int \\mathbf{A}{(t_{2},\\omega)} d\\omega = 0 and \\frac{\\partial}{\\partial t_{2}} (- \\int (t_{2} + e^{\\omega}) d\\omega + \\int \\mathbf{A}{(t_{2},\\omega)} d\\omega) = \\frac{d}{d t_{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('t_2', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('t_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('t_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Symbol('t_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('t_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('t_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(Function('\\\\mathbf{A}')(Symbol('t_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integral(Add(Symbol('t_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Integral(Function('\\\\mathbf{A}')(Symbol('t_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(E,M_{E})} = E - M_{E}, then obtain E - M_{E} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})} + 1 = E - M_{E} + (- E + M_{E} + \\operatorname{L_{\\varepsilon}}{(E,M_{E})})^{M_{E}} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(E,M_{E})} = E - M_{E} and - E + M_{E} + \\operatorname{L_{\\varepsilon}}{(E,M_{E})} = 0 and (- E + M_{E} + \\operatorname{L_{\\varepsilon}}{(E,M_{E})})^{M_{E}} = 0^{M_{E}} and E - M_{E} + (- E + M_{E} + \\operatorname{L_{\\varepsilon}}{(E,M_{E})})^{M_{E}} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})} = 0^{M_{E}} + E - M_{E} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})} and E - M_{E} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})} + 1 = E - M_{E} + (- E + M_{E} + \\operatorname{L_{\\varepsilon}}{(E,M_{E})})^{M_{E}} - \\operatorname{L_{\\varepsilon}}{(E,M_{E})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["minus", 1, "Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('M_E', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Integer(0))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('M_E', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Integer(0), Symbol('M_E', commutative=True)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('M_E', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('M_E', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)))), Add(Pow(Integer(0), Symbol('M_E', commutative=True)), Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Integer(1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('M_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('M_E', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain \\frac{\\int (\\mathbf{J}_f{(\\hat{H}_l)} + \\sin{(\\hat{H}_l)})^{2} d\\hat{H}_l}{\\mathbf{J}_f{(\\hat{H}_l)}} = \\frac{\\int 4 \\sin^{2}{(\\hat{H}_l)} d\\hat{H}_l}{\\mathbf{J}_f{(\\hat{H}_l)}}", "derivation": "\\mathbf{J}_f{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\mathbf{J}_f{(\\hat{H}_l)} + \\sin{(\\hat{H}_l)} = 2 \\sin{(\\hat{H}_l)} and (\\mathbf{J}_f{(\\hat{H}_l)} + \\sin{(\\hat{H}_l)})^{2} = 4 \\sin^{2}{(\\hat{H}_l)} and \\int (\\mathbf{J}_f{(\\hat{H}_l)} + \\sin{(\\hat{H}_l)})^{2} d\\hat{H}_l = \\int 4 \\sin^{2}{(\\hat{H}_l)} d\\hat{H}_l and \\frac{\\int (\\mathbf{J}_f{(\\hat{H}_l)} + \\sin{(\\hat{H}_l)})^{2} d\\hat{H}_l}{\\mathbf{J}_f{(\\hat{H}_l)}} = \\frac{\\int 4 \\sin^{2}{(\\hat{H}_l)} d\\hat{H}_l}{\\mathbf{J}_f{(\\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Mul(Integer(4), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Mul(Integer(4), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 4, "Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Integral(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), Integral(Mul(Integer(4), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given s{(\\mathbf{J})} = e^{\\sin{(\\mathbf{J})}}, then obtain \\cos{(s^{2}{(\\mathbf{J})})} = \\cos{(s{(\\mathbf{J})} e^{\\sin{(\\mathbf{J})}})}", "derivation": "s{(\\mathbf{J})} = e^{\\sin{(\\mathbf{J})}} and s{(\\mathbf{J})} e^{\\sin{(\\mathbf{J})}} = e^{2 \\sin{(\\mathbf{J})}} and \\cos{(s{(\\mathbf{J})} e^{\\sin{(\\mathbf{J})}})} = \\cos{(e^{2 \\sin{(\\mathbf{J})}})} and s{(\\mathbf{J})} e^{- \\sin{(\\mathbf{J})}} = 1 and s^{2}{(\\mathbf{J})} = s{(\\mathbf{J})} e^{\\sin{(\\mathbf{J})}} and \\cos{(s^{2}{(\\mathbf{J})})} = \\cos{(e^{2 \\sin{(\\mathbf{J})}})} and \\cos{(s^{2}{(\\mathbf{J})})} = \\cos{(s{(\\mathbf{J})} e^{\\sin{(\\mathbf{J})}})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 1, "exp(sin(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["cos", 2], "Equality(cos(Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True))))), cos(exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}', commutative=True))))))"], [["divide", 2, "exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))))), Integer(1))"], [["times", 4, "Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2)), Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(cos(Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), cos(exp(Mul(Integer(2), sin(Symbol('\\\\mathbf{J}', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(cos(Pow(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), cos(Mul(Function('s')(Symbol('\\\\mathbf{J}', commutative=True)), exp(sin(Symbol('\\\\mathbf{J}', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(t_{2})} = \\int \\cos{(t_{2})} dt_{2}, then derive \\nabla{(t_{2})} = \\hat{p} + \\sin{(t_{2})}, then obtain \\frac{\\partial}{\\partial t_{2}} \\nabla^{\\hat{p}}{(t_{2})} = \\frac{\\partial}{\\partial t_{2}} (\\hat{p} + \\sin{(t_{2})})^{\\hat{p}}", "derivation": "\\nabla{(t_{2})} = \\int \\cos{(t_{2})} dt_{2} and \\nabla{(t_{2})} = \\hat{p} + \\sin{(t_{2})} and \\nabla^{\\hat{p}}{(t_{2})} = (\\hat{p} + \\sin{(t_{2})})^{\\hat{p}} and \\frac{\\partial}{\\partial t_{2}} \\nabla^{\\hat{p}}{(t_{2})} = \\frac{\\partial}{\\partial t_{2}} (\\hat{p} + \\sin{(t_{2})})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_2', commutative=True)), Integral(cos(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\nabla')(Symbol('t_2', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('t_2', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('t_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('t_2', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\nabla')(Symbol('t_2', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\hat{p}', commutative=True), sin(Symbol('t_2', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{v}{(F_{c})} = e^{e^{F_{c}}} and \\mathbf{E}{(F_{c})} = e^{F_{c}}, then obtain 256 e^{8 e^{F_{c}}} = (e^{\\mathbf{E}{(F_{c})}} + e^{e^{F_{c}}})^{8}", "derivation": "\\mathbf{v}{(F_{c})} = e^{e^{F_{c}}} and \\mathbf{E}{(F_{c})} = e^{F_{c}} and \\mathbf{v}{(F_{c})} = e^{\\mathbf{E}{(F_{c})}} and 2 \\mathbf{v}{(F_{c})} = \\mathbf{v}{(F_{c})} + e^{\\mathbf{E}{(F_{c})}} and 4 \\mathbf{v}^{2}{(F_{c})} = (\\mathbf{v}{(F_{c})} + e^{\\mathbf{E}{(F_{c})}})^{2} and 4 e^{2 e^{F_{c}}} = (e^{\\mathbf{E}{(F_{c})}} + e^{e^{F_{c}}})^{2} and 256 e^{8 e^{F_{c}}} = (e^{\\mathbf{E}{(F_{c})}} + e^{e^{F_{c}}})^{8}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True)), exp(exp(Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True)), exp(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True))))"], [["add", 3, "Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True))), Add(Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True)), exp(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True)), Integer(2))), Pow(Add(Function('\\\\mathbf{v}')(Symbol('F_c', commutative=True)), exp(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(4), exp(Mul(Integer(2), exp(Symbol('F_c', commutative=True))))), Pow(Add(exp(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True))), exp(exp(Symbol('F_c', commutative=True)))), Integer(2)))"], [["power", 6, 4], "Equality(Mul(Integer(256), exp(Mul(Integer(8), exp(Symbol('F_c', commutative=True))))), Pow(Add(exp(Function('\\\\mathbf{E}')(Symbol('F_c', commutative=True))), exp(exp(Symbol('F_c', commutative=True)))), Integer(8)))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{\\mathbf{r}},Q)} = \\hat{\\mathbf{r}} + \\cos{(Q)}, then obtain (2 \\hat{\\mathbf{r}} + 2 \\cos{(Q)}) \\phi_{1}{(\\hat{\\mathbf{r}},Q)} = (\\hat{\\mathbf{r}} + \\cos{(Q)}) (2 \\hat{\\mathbf{r}} + 2 \\cos{(Q)})", "derivation": "\\phi_{1}{(\\hat{\\mathbf{r}},Q)} = \\hat{\\mathbf{r}} + \\cos{(Q)} and \\hat{\\mathbf{r}} + \\phi_{1}{(\\hat{\\mathbf{r}},Q)} + \\cos{(Q)} = 2 \\hat{\\mathbf{r}} + 2 \\cos{(Q)} and (\\hat{\\mathbf{r}} + \\phi_{1}{(\\hat{\\mathbf{r}},Q)} + \\cos{(Q)}) \\phi_{1}{(\\hat{\\mathbf{r}},Q)} = (\\hat{\\mathbf{r}} + \\cos{(Q)}) (\\hat{\\mathbf{r}} + \\phi_{1}{(\\hat{\\mathbf{r}},Q)} + \\cos{(Q)}) and (2 \\hat{\\mathbf{r}} + 2 \\cos{(Q)}) \\phi_{1}{(\\hat{\\mathbf{r}},Q)} = (\\hat{\\mathbf{r}} + \\cos{(Q)}) (2 \\hat{\\mathbf{r}} + 2 \\cos{(Q)})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('Q', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('Q', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(2), cos(Symbol('Q', commutative=True)))))"], [["times", 1, "Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True))), Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('Q', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(2), cos(Symbol('Q', commutative=True)))), Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('Q', commutative=True))), Mul(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('Q', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Integer(2), cos(Symbol('Q', commutative=True))))))"]]}, {"prompt": "Given z{(a)} = e^{a}, then obtain e^{a} + \\frac{e^{a}}{z{(a)}} + \\frac{1}{z{(a)}} = e^{a} - 1 + \\frac{2 e^{a}}{z{(a)}} + \\frac{1}{z{(a)}}", "derivation": "z{(a)} = e^{a} and 1 = \\frac{e^{a}}{z{(a)}} and e^{a} + 1 = e^{a} + \\frac{e^{a}}{z{(a)}} and e^{a} + 1 + \\frac{1}{z{(a)}} = e^{a} + \\frac{e^{a}}{z{(a)}} + \\frac{1}{z{(a)}} and e^{a} + \\frac{e^{a}}{z{(a)}} + \\frac{1}{z{(a)}} = e^{a} - 1 + \\frac{2 e^{a}}{z{(a)}} + \\frac{1}{z{(a)}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["divide", 1, "Function('z')(Symbol('a', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1)), exp(Symbol('a', commutative=True))))"], [["add", 2, "exp(Symbol('a', commutative=True))"], "Equality(Add(exp(Symbol('a', commutative=True)), Integer(1)), Add(exp(Symbol('a', commutative=True)), Mul(Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1)), exp(Symbol('a', commutative=True)))))"], [["add", 3, "Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1))"], "Equality(Add(exp(Symbol('a', commutative=True)), Integer(1), Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1))), Add(exp(Symbol('a', commutative=True)), Mul(Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1)), exp(Symbol('a', commutative=True))), Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(exp(Symbol('a', commutative=True)), Mul(Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1)), exp(Symbol('a', commutative=True))), Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1))), Add(exp(Symbol('a', commutative=True)), Integer(-1), Mul(Integer(2), Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1)), exp(Symbol('a', commutative=True))), Pow(Function('z')(Symbol('a', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\hat{X},F_{x})} = F_{x} \\hat{X}, then derive \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} = \\hat{X}, then derive \\int \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} d\\hat{X} = B + \\frac{\\hat{X}^{2}}{2}, then obtain B + \\frac{\\hat{X}^{2}}{2} = \\int \\hat{X} d\\hat{X}", "derivation": "\\mathbf{J}{(\\hat{X},F_{x})} = F_{x} \\hat{X} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x} \\hat{X} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} = \\hat{X} and \\int \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} d\\hat{X} = \\int \\hat{X} d\\hat{X} and \\int \\frac{\\partial}{\\partial F_{x}} \\mathbf{J}{(\\hat{X},F_{x})} d\\hat{X} = B + \\frac{\\hat{X}^{2}}{2} and B + \\frac{\\hat{X}^{2}}{2} = \\int \\hat{X} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True))"], [["integrate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Symbol('\\\\hat{X}', commutative=True), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\hat{X}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2)))), Integral(Symbol('\\\\hat{X}', commutative=True), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\omega,t)} = \\omega^{t} and H{(\\omega,t)} = \\omega^{t}, then obtain \\frac{\\iint H{(\\omega,t)} d\\omega dt}{H{(\\omega,t)}} = \\frac{\\iint \\omega^{t} d\\omega dt}{H{(\\omega,t)}}", "derivation": "\\operatorname{A_{z}}{(\\omega,t)} = \\omega^{t} and H{(\\omega,t)} = \\omega^{t} and \\int \\operatorname{A_{z}}{(\\omega,t)} d\\omega = \\int \\omega^{t} d\\omega and \\iint \\operatorname{A_{z}}{(\\omega,t)} d\\omega dt = \\iint \\omega^{t} d\\omega dt and \\omega^{- t} \\iint \\operatorname{A_{z}}{(\\omega,t)} d\\omega dt = \\omega^{- t} \\iint \\omega^{t} d\\omega dt and \\frac{\\iint \\operatorname{A_{z}}{(\\omega,t)} d\\omega dt}{\\operatorname{A_{z}}{(\\omega,t)}} = \\frac{\\iint \\omega^{t} d\\omega dt}{\\operatorname{A_{z}}{(\\omega,t)}} and \\operatorname{A_{z}}{(\\omega,t)} = H{(\\omega,t)} and \\frac{\\iint H{(\\omega,t)} d\\omega dt}{H{(\\omega,t)}} = \\frac{\\iint \\omega^{t} d\\omega dt}{H{(\\omega,t)}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integral(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Integral(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('A_z')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Function('H')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Mul(Pow(Function('H')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Integral(Function('H')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Function('H')(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\psi^*,\\hat{p}_0,\\dot{y})} = \\dot{y} + \\hat{p}_0 + \\psi^*, then obtain \\int \\cos{(\\mathbf{g}^{\\hat{p}_0}{(\\psi^*,\\hat{p}_0,\\dot{y})})} d\\dot{y} = \\int \\cos{((\\dot{y} + \\hat{p}_0 + \\psi^*)^{\\hat{p}_0})} d\\dot{y}", "derivation": "\\mathbf{g}{(\\psi^*,\\hat{p}_0,\\dot{y})} = \\dot{y} + \\hat{p}_0 + \\psi^* and \\mathbf{g}^{\\hat{p}_0}{(\\psi^*,\\hat{p}_0,\\dot{y})} = (\\dot{y} + \\hat{p}_0 + \\psi^*)^{\\hat{p}_0} and \\cos{(\\mathbf{g}^{\\hat{p}_0}{(\\psi^*,\\hat{p}_0,\\dot{y})})} = \\cos{((\\dot{y} + \\hat{p}_0 + \\psi^*)^{\\hat{p}_0})} and \\int \\cos{(\\mathbf{g}^{\\hat{p}_0}{(\\psi^*,\\hat{p}_0,\\dot{y})})} d\\dot{y} = \\int \\cos{((\\dot{y} + \\hat{p}_0 + \\psi^*)^{\\hat{p}_0})} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))), cos(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(cos(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(cos(Pow(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given x{(\\hat{H})} = e^{\\hat{H}} and C{(\\hat{H})} = e^{\\hat{H}} - 1, then obtain \\frac{d}{d \\hat{H}} 0 = \\frac{d}{d \\hat{H}} (\\frac{e^{\\hat{H}} - 1}{C{(\\hat{H})}} - 1)", "derivation": "x{(\\hat{H})} = e^{\\hat{H}} and x{(\\hat{H})} - 1 = e^{\\hat{H}} - 1 and C{(\\hat{H})} = e^{\\hat{H}} - 1 and 1 = \\frac{e^{\\hat{H}} - 1}{C{(\\hat{H})}} and C{(\\hat{H})} = x{(\\hat{H})} - 1 and - \\frac{x{(\\hat{H})} - 1}{C{(\\hat{H})}} + 1 = - \\frac{x{(\\hat{H})} - 1}{C{(\\hat{H})}} + \\frac{e^{\\hat{H}} - 1}{C{(\\hat{H})}} and 0 = \\frac{e^{\\hat{H}} - 1}{C{(\\hat{H})}} - 1 and \\frac{d}{d \\hat{H}} 0 = \\frac{d}{d \\hat{H}} (\\frac{e^{\\hat{H}} - 1}{C{(\\hat{H})}} - 1)", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('x')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], [["divide", 3, "Function('C')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(1), Mul(Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Add(Function('x')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], [["minus", 4, "Mul(Add(Function('x')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Add(Function('x')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Integer(1)), Add(Mul(Integer(-1), Add(Function('x')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Mul(Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(0), Add(Mul(Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Integer(-1)))"], [["differentiate", 7, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Add(exp(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Pow(Function('C')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(\\theta_2,\\hat{X})} = \\hat{X} + \\theta_2, then obtain \\delta^{2}{(\\theta_2,\\hat{X})} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2 = (\\hat{X} + \\theta_2) \\delta{(\\theta_2,\\hat{X})} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2", "derivation": "\\delta{(\\theta_2,\\hat{X})} = \\hat{X} + \\theta_2 and \\delta^{2}{(\\theta_2,\\hat{X})} = (\\hat{X} + \\theta_2) \\delta{(\\theta_2,\\hat{X})} and (\\hat{X} + \\theta_2) \\delta{(\\theta_2,\\hat{X})} = (\\hat{X} + \\theta_2)^{2} and \\delta^{2}{(\\theta_2,\\hat{X})} = (\\hat{X} + \\theta_2)^{2} and \\delta^{2}{(\\theta_2,\\hat{X})} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2 = (\\hat{X} + \\theta_2)^{2} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2 and \\delta^{2}{(\\theta_2,\\hat{X})} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2 = (\\hat{X} + \\theta_2) \\delta{(\\theta_2,\\hat{X})} \\int \\delta^{2}{(\\theta_2,\\hat{X})} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)))"], [["times", 4, "Integral(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Integral(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integral(Pow(Function('\\\\delta')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given f{(C_{d})} = C_{d}, then obtain \\sin{(\\frac{d}{d C_{d}} - f^{C_{d}}{(C_{d})})} = \\sin{(\\frac{d}{d C_{d}} - C_{d}^{C_{d}})}", "derivation": "f{(C_{d})} = C_{d} and f^{C_{d}}{(C_{d})} = C_{d}^{C_{d}} and - f^{C_{d}}{(C_{d})} = - C_{d}^{C_{d}} and \\frac{d}{d C_{d}} - f^{C_{d}}{(C_{d})} = \\frac{d}{d C_{d}} - C_{d}^{C_{d}} and \\sin{(\\frac{d}{d C_{d}} - f^{C_{d}}{(C_{d})})} = \\sin{(\\frac{d}{d C_{d}} - C_{d}^{C_{d}})}", "srepr_derivation": [["renaming_premise", "Equality(Function('f')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('f')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('f')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('f')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["sin", 4], "Equality(sin(Derivative(Mul(Integer(-1), Pow(Function('f')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))), sin(Derivative(Mul(Integer(-1), Pow(Symbol('C_d', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(E_{n})} = \\log{(E_{n})} and L{(E_{n})} = \\mathbf{M}{(E_{n})} \\log{(E_{n})}, then obtain E_{n} L{(E_{n})} = E_{n} \\log{(E_{n})}^{2}", "derivation": "\\mathbf{M}{(E_{n})} = \\log{(E_{n})} and L{(E_{n})} = \\mathbf{M}{(E_{n})} \\log{(E_{n})} and E_{n} L{(E_{n})} = E_{n} \\mathbf{M}{(E_{n})} \\log{(E_{n})} and E_{n} L{(E_{n})} = E_{n} \\log{(E_{n})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('E_n', commutative=True)), Mul(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))))"], [["times", 2, "Symbol('E_n', commutative=True)"], "Equality(Mul(Symbol('E_n', commutative=True), Function('L')(Symbol('E_n', commutative=True))), Mul(Symbol('E_n', commutative=True), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('E_n', commutative=True), Function('L')(Symbol('E_n', commutative=True))), Mul(Symbol('E_n', commutative=True), Pow(log(Symbol('E_n', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\sigma_{p}{(\\dot{x},\\mathbf{f})} = \\dot{x} \\mathbf{f} and v{(\\dot{x},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\dot{x} \\mathbf{f}, then derive \\frac{\\partial}{\\partial \\mathbf{f}} \\sigma_{p}{(\\dot{x},\\mathbf{f})} = \\dot{x}, then obtain \\int v{(\\dot{x},\\mathbf{f})} d\\dot{x} = \\int \\dot{x} d\\dot{x}", "derivation": "\\sigma_{p}{(\\dot{x},\\mathbf{f})} = \\dot{x} \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\sigma_{p}{(\\dot{x},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\dot{x} \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\sigma_{p}{(\\dot{x},\\mathbf{f})} = \\dot{x} and v{(\\dot{x},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} \\dot{x} \\mathbf{f} and \\frac{\\partial}{\\partial \\mathbf{f}} \\dot{x} \\mathbf{f} = \\dot{x} and v{(\\dot{x},\\mathbf{f})} = \\dot{x} and \\int v{(\\dot{x},\\mathbf{f})} d\\dot{x} = \\int \\dot{x} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('v')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], [["integrate", 6, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Symbol('\\\\dot{x}', commutative=True), Tuple(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given U{(v_{z},p)} = \\cos{(p v_{z})}, then obtain \\frac{\\partial}{\\partial p} U{(v_{z},p)} - (\\frac{\\partial}{\\partial p} U{(v_{z},p)})^{p} = - (\\frac{\\partial}{\\partial p} U{(v_{z},p)})^{p} + \\frac{\\partial}{\\partial p} \\cos{(p v_{z})}", "derivation": "U{(v_{z},p)} = \\cos{(p v_{z})} and \\frac{\\partial}{\\partial p} U{(v_{z},p)} = \\frac{\\partial}{\\partial p} \\cos{(p v_{z})} and (\\frac{\\partial}{\\partial p} U{(v_{z},p)})^{p} = (\\frac{\\partial}{\\partial p} \\cos{(p v_{z})})^{p} and \\frac{\\partial}{\\partial p} U{(v_{z},p)} - (\\frac{\\partial}{\\partial p} \\cos{(p v_{z})})^{p} = \\frac{\\partial}{\\partial p} \\cos{(p v_{z})} - (\\frac{\\partial}{\\partial p} \\cos{(p v_{z})})^{p} and \\frac{\\partial}{\\partial p} U{(v_{z},p)} - (\\frac{\\partial}{\\partial p} U{(v_{z},p)})^{p} = - (\\frac{\\partial}{\\partial p} U{(v_{z},p)})^{p} + \\frac{\\partial}{\\partial p} \\cos{(p v_{z})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["minus", 2, "Pow(Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True))"], "Equality(Add(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))), Add(Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Pow(Derivative(Function('U')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Symbol('p', commutative=True))), Derivative(cos(Mul(Symbol('p', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{D}{(L)} = \\cos{(L)} and \\Psi_{\\lambda}{(L)} = \\mathbf{D}{(L)} \\cos{(L)}, then obtain (\\mathbf{D}^{2}{(L)} \\cos^{2}{(L)})^{L} = (\\mathbf{D}^{4}{(L)})^{L}", "derivation": "\\mathbf{D}{(L)} = \\cos{(L)} and \\Psi_{\\lambda}{(L)} = \\mathbf{D}{(L)} \\cos{(L)} and \\Psi_{\\lambda}{(L)} = \\mathbf{D}^{2}{(L)} and \\Psi_{\\lambda}{(L)} = \\cos^{2}{(L)} and \\Psi_{\\lambda}^{2}{(L)} = \\cos^{4}{(L)} and \\Psi_{\\lambda}^{2}{(L)} = \\mathbf{D}^{4}{(L)} and (\\Psi_{\\lambda}^{2}{(L)})^{L} = (\\mathbf{D}^{4}{(L)})^{L} and (\\mathbf{D}^{2}{(L)} \\cos^{2}{(L)})^{L} = (\\mathbf{D}^{4}{(L)})^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Mul(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Integer(2)), Pow(cos(Symbol('L', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Integer(2)), Pow(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), Integer(4)))"], [["power", 6, "Symbol('L', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Integer(2)), Symbol('L', commutative=True)), Pow(Pow(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), Integer(4)), Symbol('L', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Pow(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), Integer(2)), Pow(cos(Symbol('L', commutative=True)), Integer(2))), Symbol('L', commutative=True)), Pow(Pow(Function('\\\\mathbf{D}')(Symbol('L', commutative=True)), Integer(4)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given Z{(\\omega,\\tilde{g}^*)} = \\frac{\\log{(\\omega)}}{\\tilde{g}^*}, then obtain \\frac{\\partial}{\\partial \\omega} (\\tilde{g}^* + (\\omega Z{(\\omega,\\tilde{g}^*)})^{\\tilde{g}^*}) = \\frac{\\partial}{\\partial \\omega} (\\tilde{g}^* + (\\frac{\\omega \\log{(\\omega)}}{\\tilde{g}^*})^{\\tilde{g}^*})", "derivation": "Z{(\\omega,\\tilde{g}^*)} = \\frac{\\log{(\\omega)}}{\\tilde{g}^*} and \\omega Z{(\\omega,\\tilde{g}^*)} = \\frac{\\omega \\log{(\\omega)}}{\\tilde{g}^*} and (\\omega Z{(\\omega,\\tilde{g}^*)})^{\\tilde{g}^*} = (\\frac{\\omega \\log{(\\omega)}}{\\tilde{g}^*})^{\\tilde{g}^*} and \\tilde{g}^* + (\\omega Z{(\\omega,\\tilde{g}^*)})^{\\tilde{g}^*} = \\tilde{g}^* + (\\frac{\\omega \\log{(\\omega)}}{\\tilde{g}^*})^{\\tilde{g}^*} and \\frac{\\partial}{\\partial \\omega} (\\tilde{g}^* + (\\omega Z{(\\omega,\\tilde{g}^*)})^{\\tilde{g}^*}) = \\frac{\\partial}{\\partial \\omega} (\\tilde{g}^* + (\\frac{\\omega \\log{(\\omega)}}{\\tilde{g}^*})^{\\tilde{g}^*})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), log(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), log(Symbol('\\\\omega', commutative=True))))"], [["power", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\omega', commutative=True), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), log(Symbol('\\\\omega', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Mul(Symbol('\\\\omega', commutative=True), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), log(Symbol('\\\\omega', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Mul(Symbol('\\\\omega', commutative=True), Function('Z')(Symbol('\\\\omega', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), Pow(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), log(Symbol('\\\\omega', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{x}_0,k)} = \\frac{\\hat{x}_0}{k}, then obtain (2 \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} - \\frac{1}{k} = (\\frac{\\hat{x}_0}{k} + \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} - \\frac{1}{k}", "derivation": "\\operatorname{n_{2}}{(\\hat{x}_0,k)} = \\frac{\\hat{x}_0}{k} and 2 \\operatorname{n_{2}}{(\\hat{x}_0,k)} = \\frac{\\hat{x}_0}{k} + \\operatorname{n_{2}}{(\\hat{x}_0,k)} and 2 \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1 = \\frac{\\hat{x}_0}{k} + \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1 and (2 \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} = (\\frac{\\hat{x}_0}{k} + \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} and (2 \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} - \\frac{1}{k} = (\\frac{\\hat{x}_0}{k} + \\operatorname{n_{2}}{(\\hat{x}_0,k)} - 1)^{2} - \\frac{1}{k}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["add", 1, "Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True)), Integer(-1)))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Integer(2)), Pow(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Integer(2)))"], [["minus", 4, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Add(Pow(Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True))), Integer(-1)), Integer(2)), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))), Add(Pow(Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Function('n_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Integer(2)), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{E}{(v_{x},v_{z})} = (e^{v_{z}})^{v_{x}} and \\hat{x}_0{(v_{x},v_{z})} = \\int (\\mathbf{E}{(v_{x},v_{z})} + (e^{v_{z}})^{v_{x}}) dv_{z}, then obtain \\int 2 (e^{v_{z}})^{v_{x}} dv_{z} = \\int (\\mathbf{E}{(v_{x},v_{z})} + (e^{v_{z}})^{v_{x}}) dv_{z}", "derivation": "\\mathbf{E}{(v_{x},v_{z})} = (e^{v_{z}})^{v_{x}} and \\hat{x}_0{(v_{x},v_{z})} = \\int (\\mathbf{E}{(v_{x},v_{z})} + (e^{v_{z}})^{v_{x}}) dv_{z} and \\hat{x}_0{(v_{x},v_{z})} = \\int 2 (e^{v_{z}})^{v_{x}} dv_{z} and \\int 2 (e^{v_{z}})^{v_{x}} dv_{z} = \\int (\\mathbf{E}{(v_{x},v_{z})} + (e^{v_{z}})^{v_{x}}) dv_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Pow(exp(Symbol('v_z', commutative=True)), Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Integral(Add(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Pow(exp(Symbol('v_z', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Integral(Mul(Integer(2), Pow(exp(Symbol('v_z', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Mul(Integer(2), Pow(exp(Symbol('v_z', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Function('\\\\mathbf{E}')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Pow(exp(Symbol('v_z', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(l,Z)} = - Z + l, then obtain \\eta^{\\prime}{(l,Z)} + \\frac{\\partial}{\\partial l} (Z + e^{\\eta^{\\prime}{(l,Z)}}) = \\eta^{\\prime}{(l,Z)} + \\frac{\\partial}{\\partial l} (Z + e^{- Z + l})", "derivation": "\\eta^{\\prime}{(l,Z)} = - Z + l and e^{\\eta^{\\prime}{(l,Z)}} = e^{- Z + l} and Z + e^{\\eta^{\\prime}{(l,Z)}} = Z + e^{- Z + l} and \\frac{\\partial}{\\partial l} (Z + e^{\\eta^{\\prime}{(l,Z)}}) = \\frac{\\partial}{\\partial l} (Z + e^{- Z + l}) and \\eta^{\\prime}{(l,Z)} + \\frac{\\partial}{\\partial l} (Z + e^{\\eta^{\\prime}{(l,Z)}}) = \\eta^{\\prime}{(l,Z)} + \\frac{\\partial}{\\partial l} (Z + e^{- Z + l})", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('l', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True))), exp(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('l', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Symbol('Z', commutative=True), exp(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)))), Add(Symbol('Z', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('l', commutative=True)))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Symbol('Z', commutative=True), exp(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 4, "Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), exp(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('l', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(\\dot{y},B)} = \\int (B + \\dot{y}) dB, then derive \\dot{y} \\rho_{b}{(\\dot{y},B)} = \\dot{y} (\\frac{B^{2}}{2} + B \\dot{y} + v_{x}), then obtain \\int \\dot{y} \\rho_{b}{(\\dot{y},B)} dv_{x} = \\int \\dot{y} (\\frac{B^{2}}{2} + B \\dot{y} + v_{x}) dv_{x}", "derivation": "\\rho_{b}{(\\dot{y},B)} = \\int (B + \\dot{y}) dB and \\dot{y} \\rho_{b}{(\\dot{y},B)} = \\dot{y} \\int (B + \\dot{y}) dB and \\dot{y} \\rho_{b}{(\\dot{y},B)} = \\dot{y} (\\frac{B^{2}}{2} + B \\dot{y} + v_{x}) and \\int \\dot{y} \\rho_{b}{(\\dot{y},B)} dv_{x} = \\int \\dot{y} (\\frac{B^{2}}{2} + B \\dot{y} + v_{x}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["times", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), Integral(Add(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('v_x', commutative=True))))"], [["integrate", 3, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Function('\\\\rho_b')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(F_{g})} = \\cos{(F_{g})}, then obtain \\operatorname{v_{1}}^{2}{(F_{g})} + \\frac{d}{d F_{g}} \\int \\cos{(F_{g})} dF_{g} = \\operatorname{v_{1}}{(F_{g})} \\cos{(F_{g})} + \\frac{d}{d F_{g}} \\int \\cos{(F_{g})} dF_{g}", "derivation": "\\operatorname{v_{1}}{(F_{g})} = \\cos{(F_{g})} and \\operatorname{v_{1}}^{2}{(F_{g})} = \\operatorname{v_{1}}{(F_{g})} \\cos{(F_{g})} and \\int \\operatorname{v_{1}}{(F_{g})} dF_{g} = \\int \\cos{(F_{g})} dF_{g} and \\operatorname{v_{1}}^{2}{(F_{g})} + \\frac{d}{d F_{g}} \\int \\operatorname{v_{1}}{(F_{g})} dF_{g} = \\operatorname{v_{1}}{(F_{g})} \\cos{(F_{g})} + \\frac{d}{d F_{g}} \\int \\operatorname{v_{1}}{(F_{g})} dF_{g} and \\operatorname{v_{1}}^{2}{(F_{g})} + \\frac{d}{d F_{g}} \\int \\cos{(F_{g})} dF_{g} = \\operatorname{v_{1}}{(F_{g})} \\cos{(F_{g})} + \\frac{d}{d F_{g}} \\int \\cos{(F_{g})} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True)))"], [["times", 1, "Function('v_1')(Symbol('F_g', commutative=True))"], "Equality(Pow(Function('v_1')(Symbol('F_g', commutative=True)), Integer(2)), Mul(Function('v_1')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["add", 2, "Derivative(Integral(Function('v_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('v_1')(Symbol('F_g', commutative=True)), Integer(2)), Derivative(Integral(Function('v_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Function('v_1')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Derivative(Integral(Function('v_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Function('v_1')(Symbol('F_g', commutative=True)), Integer(2)), Derivative(Integral(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Function('v_1')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Derivative(Integral(cos(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then obtain 2 \\mathbf{H}^{\\mathbf{S}}{(\\mathbf{S})} \\sin{(\\mathbf{S})} = 2 \\sin{(\\mathbf{S})} \\sin^{\\mathbf{S}}{(\\mathbf{S})}", "derivation": "\\mathbf{H}{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and \\mathbf{H}^{\\mathbf{S}}{(\\mathbf{S})} = \\sin^{\\mathbf{S}}{(\\mathbf{S})} and \\mathbf{H}{(\\mathbf{S})} + \\sin{(\\mathbf{S})} = 2 \\sin{(\\mathbf{S})} and (\\mathbf{H}{(\\mathbf{S})} + \\sin{(\\mathbf{S})}) \\mathbf{H}^{\\mathbf{S}}{(\\mathbf{S})} = (\\mathbf{H}{(\\mathbf{S})} + \\sin{(\\mathbf{S})}) \\sin^{\\mathbf{S}}{(\\mathbf{S})} and 2 \\mathbf{H}^{\\mathbf{S}}{(\\mathbf{S})} \\sin{(\\mathbf{S})} = 2 \\sin{(\\mathbf{S})} \\sin^{\\mathbf{S}}{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 2, "Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Add(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{S}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(F_{N},G)} = \\frac{\\partial}{\\partial G} \\cos^{G}{(F_{N})}, then derive \\frac{\\partial}{\\partial G} \\dot{\\mathbf{r}}{(F_{N},G)} = \\log{(\\cos{(F_{N})})}^{2} \\cos^{G}{(F_{N})}, then obtain (- F_{N} + \\phi{(F_{N},G)} + 1) \\frac{\\partial}{\\partial G} \\dot{\\mathbf{r}}{(F_{N},G)} = (- F_{N} + \\phi{(F_{N},G)} + 1) \\log{(\\cos{(F_{N})})}^{2} \\cos^{G}{(F_{N})}", "derivation": "\\dot{\\mathbf{r}}{(F_{N},G)} = \\frac{\\partial}{\\partial G} \\cos^{G}{(F_{N})} and \\frac{\\partial}{\\partial G} \\dot{\\mathbf{r}}{(F_{N},G)} = \\frac{\\partial^{2}}{\\partial G^{2}} \\cos^{G}{(F_{N})} and \\frac{\\partial}{\\partial G} \\dot{\\mathbf{r}}{(F_{N},G)} = \\log{(\\cos{(F_{N})})}^{2} \\cos^{G}{(F_{N})} and (- F_{N} + \\phi{(F_{N},G)} + 1) \\frac{\\partial}{\\partial G} \\dot{\\mathbf{r}}{(F_{N},G)} = (- F_{N} + \\phi{(F_{N},G)} + 1) \\log{(\\cos{(F_{N})})}^{2} \\cos^{G}{(F_{N})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Derivative(Pow(cos(Symbol('F_N', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('F_N', commutative=True)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Pow(log(cos(Symbol('F_N', commutative=True))), Integer(2)), Pow(cos(Symbol('F_N', commutative=True)), Symbol('G', commutative=True))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\phi')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\phi')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Integer(1)), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Function('\\\\phi')(Symbol('F_N', commutative=True), Symbol('G', commutative=True)), Integer(1)), Pow(log(cos(Symbol('F_N', commutative=True))), Integer(2)), Pow(cos(Symbol('F_N', commutative=True)), Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\hat{X})} = \\hat{X} and \\mathbf{p}{(\\hat{X})} = \\mathbf{J}_P^{\\hat{X}}{(\\hat{X})}, then obtain \\mathbf{p}{(\\hat{X})} = \\hat{X}^{\\hat{X}}", "derivation": "\\mathbf{J}_P{(\\hat{X})} = \\hat{X} and \\mathbf{J}_P^{\\hat{X}}{(\\hat{X})} = \\hat{X}^{\\hat{X}} and \\mathbf{p}{(\\hat{X})} = \\mathbf{J}_P^{\\hat{X}}{(\\hat{X})} and \\mathbf{p}{(\\hat{X})} = \\hat{X}^{\\hat{X}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{X}', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\pi)} = e^{\\pi}, then obtain \\pi \\hat{p}{(\\pi)} - e^{\\pi} + \\int 0 d\\pi = \\pi \\hat{p}{(\\pi)} - e^{\\pi} + \\int (\\frac{d}{d \\pi} 0 - \\frac{d}{d \\pi} (\\hat{p}{(\\pi)} - e^{\\pi})) d\\pi", "derivation": "\\hat{p}{(\\pi)} = e^{\\pi} and \\hat{p}{(\\pi)} - e^{\\pi} = 0 and \\frac{d}{d \\pi} (\\hat{p}{(\\pi)} - e^{\\pi}) = \\frac{d}{d \\pi} 0 and 0 = \\frac{d}{d \\pi} 0 - \\frac{d}{d \\pi} (\\hat{p}{(\\pi)} - e^{\\pi}) and \\int 0 d\\pi = \\int (\\frac{d}{d \\pi} 0 - \\frac{d}{d \\pi} (\\hat{p}{(\\pi)} - e^{\\pi})) d\\pi and \\pi \\hat{p}{(\\pi)} - e^{\\pi} + \\int 0 d\\pi = \\pi \\hat{p}{(\\pi)} - e^{\\pi} + \\int (\\frac{d}{d \\pi} 0 - \\frac{d}{d \\pi} (\\hat{p}{(\\pi)} - e^{\\pi})) d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 5, "Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Mul(Symbol('\\\\pi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True))), Integral(Add(Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Function('\\\\hat{p}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\sigma_x,f^{\\prime})} = \\sigma_x f^{\\prime}, then derive - f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\ddot{x}{(\\sigma_x,f^{\\prime})} = 0, then obtain (- f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\ddot{x}{(\\sigma_x,f^{\\prime})})^{f^{\\prime}} = 0^{f^{\\prime}}", "derivation": "\\ddot{x}{(\\sigma_x,f^{\\prime})} = \\sigma_x f^{\\prime} and - \\sigma_x f^{\\prime} + \\ddot{x}{(\\sigma_x,f^{\\prime})} = 0 and \\frac{\\partial}{\\partial \\sigma_x} (- \\sigma_x f^{\\prime} + \\ddot{x}{(\\sigma_x,f^{\\prime})}) = \\frac{d}{d \\sigma_x} 0 and - f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\ddot{x}{(\\sigma_x,f^{\\prime})} = 0 and - f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x f^{\\prime} = 0 and (- f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x f^{\\prime})^{f^{\\prime}} = 0^{f^{\\prime}} and (- f^{\\prime} + \\frac{\\partial}{\\partial \\sigma_x} \\ddot{x}{(\\sigma_x,f^{\\prime})})^{f^{\\prime}} = 0^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Integer(0))"], [["power", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\sigma_x', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} = \\hbar + a^{\\dagger}, then obtain \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} + \\frac{\\partial}{\\partial \\hbar} \\int \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} d\\hbar = \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} + \\frac{\\partial}{\\partial \\hbar} \\int (\\hbar + a^{\\dagger}) d\\hbar", "derivation": "\\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} = \\hbar + a^{\\dagger} and \\int \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} d\\hbar = \\int (\\hbar + a^{\\dagger}) d\\hbar and \\frac{\\partial}{\\partial \\hbar} \\int \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} d\\hbar = \\frac{\\partial}{\\partial \\hbar} \\int (\\hbar + a^{\\dagger}) d\\hbar and \\hbar + a^{\\dagger} + \\frac{\\partial}{\\partial \\hbar} \\int \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} d\\hbar = \\hbar + a^{\\dagger} + \\frac{\\partial}{\\partial \\hbar} \\int (\\hbar + a^{\\dagger}) d\\hbar and \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} + \\frac{\\partial}{\\partial \\hbar} \\int \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} d\\hbar = \\operatorname{P_{e}}{(a^{\\dagger},\\hbar)} + \\frac{\\partial}{\\partial \\hbar} \\int (\\hbar + a^{\\dagger}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Integral(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["add", 3, "Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Derivative(Integral(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Derivative(Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Integral(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Function('P_e')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Integral(Add(Symbol('\\\\hbar', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\chi{(P_{e})} = \\sin{(P_{e})}, then derive \\frac{d}{d P_{e}} \\chi{(P_{e})} = \\cos{(P_{e})}, then obtain e^{(\\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{\\cos{(P_{e})}})^{P_{e}}} = e", "derivation": "\\chi{(P_{e})} = \\sin{(P_{e})} and \\frac{d}{d P_{e}} \\chi{(P_{e})} = \\frac{d}{d P_{e}} \\sin{(P_{e})} and \\frac{d}{d P_{e}} \\chi{(P_{e})} = \\cos{(P_{e})} and \\frac{d}{d P_{e}} \\sin{(P_{e})} = \\cos{(P_{e})} and \\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{\\cos{(P_{e})}} = 1 and (\\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{\\cos{(P_{e})}})^{P_{e}} = 1 and e^{(\\frac{\\frac{d}{d P_{e}} \\sin{(P_{e})}}{\\cos{(P_{e})}})^{P_{e}}} = e", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\chi')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), cos(Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), cos(Symbol('P_e', commutative=True)))"], [["divide", 4, "cos(Symbol('P_e', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Integer(1))"], [["power", 5, "Symbol('P_e', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Symbol('P_e', commutative=True)), Integer(1))"], [["exp", 6], "Equality(exp(Pow(Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(sin(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Symbol('P_e', commutative=True))), E)"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{v})} = \\cos{(\\mathbf{v})}, then obtain e^{\\int (\\operatorname{v_{y}}^{\\mathbf{v}}{(\\mathbf{v})} - \\cos{(\\mathbf{v})}) d\\mathbf{v}} = e^{\\int (- \\cos{(\\mathbf{v})} + \\cos^{\\mathbf{v}}{(\\mathbf{v})}) d\\mathbf{v}}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and \\operatorname{v_{y}}^{\\mathbf{v}}{(\\mathbf{v})} = \\cos^{\\mathbf{v}}{(\\mathbf{v})} and \\operatorname{v_{y}}^{\\mathbf{v}}{(\\mathbf{v})} - \\cos{(\\mathbf{v})} = - \\cos{(\\mathbf{v})} + \\cos^{\\mathbf{v}}{(\\mathbf{v})} and \\int (\\operatorname{v_{y}}^{\\mathbf{v}}{(\\mathbf{v})} - \\cos{(\\mathbf{v})}) d\\mathbf{v} = \\int (- \\cos{(\\mathbf{v})} + \\cos^{\\mathbf{v}}{(\\mathbf{v})}) d\\mathbf{v} and e^{\\int (\\operatorname{v_{y}}^{\\mathbf{v}}{(\\mathbf{v})} - \\cos{(\\mathbf{v})}) d\\mathbf{v}} = e^{\\int (- \\cos{(\\mathbf{v})} + \\cos^{\\mathbf{v}}{(\\mathbf{v})}) d\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["minus", 2, "cos(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Pow(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Add(Pow(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Add(Pow(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), exp(Integral(Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given I{(\\varphi)} = \\sin{(\\varphi)}, then obtain \\frac{I{(\\varphi)} (\\iint I{(\\varphi)} d\\varphi d\\varphi)^{\\varphi}}{E_{\\lambda}} = \\frac{I{(\\varphi)} (\\iint \\sin{(\\varphi)} d\\varphi d\\varphi)^{\\varphi}}{E_{\\lambda}}", "derivation": "I{(\\varphi)} = \\sin{(\\varphi)} and \\int I{(\\varphi)} d\\varphi = \\int \\sin{(\\varphi)} d\\varphi and \\iint I{(\\varphi)} d\\varphi d\\varphi = \\iint \\sin{(\\varphi)} d\\varphi d\\varphi and (\\iint I{(\\varphi)} d\\varphi d\\varphi)^{\\varphi} = (\\iint \\sin{(\\varphi)} d\\varphi d\\varphi)^{\\varphi} and I{(\\varphi)} (\\iint I{(\\varphi)} d\\varphi d\\varphi)^{\\varphi} = I{(\\varphi)} (\\iint \\sin{(\\varphi)} d\\varphi d\\varphi)^{\\varphi} and \\frac{I{(\\varphi)} (\\iint I{(\\varphi)} d\\varphi d\\varphi)^{\\varphi}}{E_{\\lambda}} = \\frac{I{(\\varphi)} (\\iint \\sin{(\\varphi)} d\\varphi d\\varphi)^{\\varphi}}{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integral(Function('I')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["times", 4, "Function('I')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\varphi', commutative=True)), Pow(Integral(Function('I')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), Mul(Function('I')(Symbol('\\\\varphi', commutative=True)), Pow(Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"], [["divide", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\varphi', commutative=True)), Pow(Integral(Function('I')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\varphi', commutative=True)), Pow(Integral(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(C)} = e^{C} and W{(C)} = - C, then derive \\frac{d}{d C} W{(C)} + \\frac{d}{d C} \\operatorname{m_{s}}{(C)} = e^{C} + \\frac{d}{d C} W{(C)}, then obtain e^{\\frac{d}{d C} - C + \\frac{d}{d C} \\operatorname{m_{s}}{(C)}} = e^{e^{C} + \\frac{d}{d C} - C}", "derivation": "\\operatorname{m_{s}}{(C)} = e^{C} and - C + \\operatorname{m_{s}}{(C)} = - C + e^{C} and W{(C)} = - C and W{(C)} + \\operatorname{m_{s}}{(C)} = W{(C)} + e^{C} and \\frac{d}{d C} (W{(C)} + \\operatorname{m_{s}}{(C)}) = \\frac{d}{d C} (W{(C)} + e^{C}) and \\frac{d}{d C} W{(C)} + \\frac{d}{d C} \\operatorname{m_{s}}{(C)} = e^{C} + \\frac{d}{d C} W{(C)} and e^{\\frac{d}{d C} W{(C)} + \\frac{d}{d C} \\operatorname{m_{s}}{(C)}} = e^{e^{C} + \\frac{d}{d C} W{(C)}} and e^{\\frac{d}{d C} - C + \\frac{d}{d C} \\operatorname{m_{s}}{(C)}} = e^{e^{C} + \\frac{d}{d C} - C}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('m_s')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('W')(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('W')(Symbol('C', commutative=True)), Function('m_s')(Symbol('C', commutative=True))), Add(Function('W')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('W')(Symbol('C', commutative=True)), Function('m_s')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Function('W')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('W')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('m_s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Add(exp(Symbol('C', commutative=True)), Derivative(Function('W')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["exp", 6], "Equality(exp(Add(Derivative(Function('W')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('m_s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))), exp(Add(exp(Symbol('C', commutative=True)), Derivative(Function('W')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(exp(Add(Derivative(Mul(Integer(-1), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('m_s')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))), exp(Add(exp(Symbol('C', commutative=True)), Derivative(Mul(Integer(-1), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))))"]]}, {"prompt": "Given a{(n_{2},i)} = \\frac{n_{2}}{i} and u{(\\omega)} = \\sin{(\\sin{(\\omega)})}, then obtain - \\sin{(a{(n_{2},i)} - u{(\\omega)})} = - \\sin{(a{(n_{2},i)} - \\sin{(\\sin{(\\omega)})})}", "derivation": "a{(n_{2},i)} = \\frac{n_{2}}{i} and u{(\\omega)} = \\sin{(\\sin{(\\omega)})} and - a{(n_{2},i)} + u{(\\omega)} = - a{(n_{2},i)} + \\sin{(\\sin{(\\omega)})} and u{(\\omega)} - \\frac{n_{2}}{i} = \\sin{(\\sin{(\\omega)})} - \\frac{n_{2}}{i} and \\sin{(u{(\\omega)} - \\frac{n_{2}}{i})} = \\sin{(\\sin{(\\sin{(\\omega)})} - \\frac{n_{2}}{i})} and - \\sin{(a{(n_{2},i)} - u{(\\omega)})} = - \\sin{(a{(n_{2},i)} - \\sin{(\\sin{(\\omega)})})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], ["get_premise", "Equality(Function('u')(Symbol('\\\\omega', commutative=True)), sin(sin(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True))), Function('u')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True))), sin(sin(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('u')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Add(sin(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["sin", 4], "Equality(sin(Add(Function('u')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))), sin(Add(sin(sin(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), sin(Add(Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('\\\\omega', commutative=True)))))), Mul(Integer(-1), sin(Add(Function('a')(Symbol('n_2', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('\\\\omega', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mu_0)} = \\mu_0 and \\operatorname{f_{\\mathbf{v}}}{(\\mu_0)} = \\cos{(\\mu_0)}, then derive \\int (\\operatorname{z^{*}}{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0 = \\mathbb{I} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)}, then obtain \\hat{\\mathbf{r}} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)} = \\mathbb{I} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)}", "derivation": "\\operatorname{z^{*}}{(\\mu_0)} = \\mu_0 and \\operatorname{f_{\\mathbf{v}}}{(\\mu_0)} = \\cos{(\\mu_0)} and \\operatorname{f_{\\mathbf{v}}}{(\\mu_0)} + \\operatorname{z^{*}}{(\\mu_0)} = \\mu_0 + \\operatorname{f_{\\mathbf{v}}}{(\\mu_0)} and \\operatorname{z^{*}}{(\\mu_0)} + \\cos{(\\mu_0)} = \\mu_0 + \\cos{(\\mu_0)} and \\int (\\operatorname{z^{*}}{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0 = \\int (\\mu_0 + \\cos{(\\mu_0)}) d\\mu_0 and \\int (\\operatorname{z^{*}}{(\\mu_0)} + \\cos{(\\mu_0)}) d\\mu_0 = \\mathbb{I} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)} and \\int (\\mu_0 + \\cos{(\\mu_0)}) d\\mu_0 = \\mathbb{I} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)} and \\hat{\\mathbf{r}} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)} = \\mathbb{I} + \\frac{\\mu_0^{2}}{2} + \\sin{(\\mu_0)}", "srepr_derivation": [["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mu_0', commutative=True)), Function('z^*')(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('z^*')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Function('z^*')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Function('z^*')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), sin(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), sin(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), sin(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2))), sin(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(y)} = \\cos{(y)}, then derive - \\int \\hat{p}{(y)} \\int \\cos{(y)} dy dy = - \\dot{z} - \\frac{\\sin^{2}{(y)}}{2}, then obtain - \\dot{z} - \\frac{\\sin^{2}{(y)}}{2} = - \\int \\cos{(y)} \\int \\cos{(y)} dy dy", "derivation": "\\hat{p}{(y)} = \\cos{(y)} and \\int \\hat{p}{(y)} dy = \\int \\cos{(y)} dy and \\hat{p}{(y)} \\int \\hat{p}{(y)} dy = \\cos{(y)} \\int \\hat{p}{(y)} dy and \\hat{p}{(y)} \\int \\cos{(y)} dy = \\cos{(y)} \\int \\cos{(y)} dy and \\int \\hat{p}{(y)} \\int \\cos{(y)} dy dy = \\int \\cos{(y)} \\int \\cos{(y)} dy dy and - \\int \\hat{p}{(y)} \\int \\cos{(y)} dy dy = - \\int \\cos{(y)} \\int \\cos{(y)} dy dy and - \\int \\hat{p}{(y)} \\int \\cos{(y)} dy dy = - \\dot{z} - \\frac{\\sin^{2}{(y)}}{2} and - \\dot{z} - \\frac{\\sin^{2}{(y)}}{2} = - \\int \\cos{(y)} \\int \\cos{(y)} dy dy", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["times", 1, "Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(cos(Symbol('y', commutative=True)), Integral(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(cos(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))), Integral(Mul(cos(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))), Mul(Integer(-1), Integral(Mul(cos(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Integer(-1), Integral(Mul(Function('\\\\hat{p}')(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(sin(Symbol('y', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(sin(Symbol('y', commutative=True)), Integer(2)))), Mul(Integer(-1), Integral(Mul(cos(Symbol('y', commutative=True)), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given U{(F_{g})} = \\sin{(\\cos{(F_{g})})}, then obtain - 2 F_{g} + \\int U{(F_{g})} dF_{g} = - 2 F_{g} + \\int \\sin{(\\cos{(F_{g})})} dF_{g}", "derivation": "U{(F_{g})} = \\sin{(\\cos{(F_{g})})} and \\int U{(F_{g})} dF_{g} = \\int \\sin{(\\cos{(F_{g})})} dF_{g} and - F_{g} + \\int U{(F_{g})} dF_{g} = - F_{g} + \\int \\sin{(\\cos{(F_{g})})} dF_{g} and - 2 F_{g} + \\int U{(F_{g})} dF_{g} = - 2 F_{g} + \\int \\sin{(\\cos{(F_{g})})} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True))))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('U')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["minus", 2, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Integral(Function('U')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"], [["minus", 3, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('F_g', commutative=True)), Integral(Function('U')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('F_g', commutative=True)), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} = A_{z} E_{x} t_{2}, then obtain (A_{z} \\int \\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} dt_{2})^{A_{z}} = (A_{z} \\int A_{z} E_{x} t_{2} dt_{2})^{A_{z}}", "derivation": "\\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} = A_{z} E_{x} t_{2} and \\int \\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} dt_{2} = \\int A_{z} E_{x} t_{2} dt_{2} and A_{z} \\int \\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} dt_{2} = A_{z} \\int A_{z} E_{x} t_{2} dt_{2} and (A_{z} \\int \\dot{\\mathbf{r}}{(t_{2},A_{z},E_{x})} dt_{2})^{A_{z}} = (A_{z} \\int A_{z} E_{x} t_{2} dt_{2})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["times", 2, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Symbol('A_z', commutative=True), Integral(Mul(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Symbol('A_z', commutative=True), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('A_z', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Symbol('A_z', commutative=True)), Pow(Mul(Symbol('A_z', commutative=True), Integral(Mul(Symbol('A_z', commutative=True), Symbol('E_x', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(v)} = \\log{(v)}, then derive \\frac{d}{d v} \\operatorname{C_{1}}{(v)} = \\frac{1}{v}, then derive \\log{(\\delta + \\log{(v)})} = \\log{(b + \\log{(v)})}, then obtain \\log{(\\delta + \\operatorname{C_{1}}{(v)})}^{\\delta} = \\log{(b + \\operatorname{C_{1}}{(v)})}^{\\delta}", "derivation": "\\operatorname{C_{1}}{(v)} = \\log{(v)} and \\frac{d}{d v} \\operatorname{C_{1}}{(v)} = \\frac{d}{d v} \\log{(v)} and \\frac{d}{d v} \\operatorname{C_{1}}{(v)} = \\frac{1}{v} and \\frac{d}{d v} \\log{(v)} = \\frac{1}{v} and \\int \\frac{d}{d v} \\log{(v)} dv = \\int \\frac{1}{v} dv and \\log{(\\int \\frac{d}{d v} \\log{(v)} dv)} = \\log{(\\int \\frac{1}{v} dv)} and \\log{(\\delta + \\log{(v)})} = \\log{(b + \\log{(v)})} and \\log{(\\delta + \\operatorname{C_{1}}{(v)})} = \\log{(b + \\operatorname{C_{1}}{(v)})} and \\log{(\\delta + \\operatorname{C_{1}}{(v)})}^{\\delta} = \\log{(b + \\operatorname{C_{1}}{(v)})}^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Symbol('v', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Pow(Symbol('v', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Integral(Pow(Symbol('v', commutative=True), Integer(-1)), Tuple(Symbol('v', commutative=True))))"], [["log", 5], "Equality(log(Integral(Derivative(log(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True)))), log(Integral(Pow(Symbol('v', commutative=True), Integer(-1)), Tuple(Symbol('v', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(log(Add(Symbol('\\\\delta', commutative=True), log(Symbol('v', commutative=True)))), log(Add(Symbol('b', commutative=True), log(Symbol('v', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(log(Add(Symbol('\\\\delta', commutative=True), Function('C_1')(Symbol('v', commutative=True)))), log(Add(Symbol('b', commutative=True), Function('C_1')(Symbol('v', commutative=True)))))"], [["power", 8, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(log(Add(Symbol('\\\\delta', commutative=True), Function('C_1')(Symbol('v', commutative=True)))), Symbol('\\\\delta', commutative=True)), Pow(log(Add(Symbol('b', commutative=True), Function('C_1')(Symbol('v', commutative=True)))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(m)} = e^{m}, then obtain (2 \\operatorname{v_{z}}{(m)} + e^{m}) \\int \\operatorname{v_{z}}{(m)} dm = (v_{2} + e^{m}) (2 \\operatorname{v_{z}}{(m)} + e^{m})", "derivation": "\\operatorname{v_{z}}{(m)} = e^{m} and 2 \\operatorname{v_{z}}{(m)} = \\operatorname{v_{z}}{(m)} + e^{m} and \\int \\operatorname{v_{z}}{(m)} dm = \\int e^{m} dm and 2 \\operatorname{v_{z}}{(m)} + e^{m} = \\operatorname{v_{z}}{(m)} + 2 e^{m} and (\\operatorname{v_{z}}{(m)} + 2 e^{m}) \\int \\operatorname{v_{z}}{(m)} dm = (\\operatorname{v_{z}}{(m)} + 2 e^{m}) \\int e^{m} dm and (2 \\operatorname{v_{z}}{(m)} + e^{m}) \\int \\operatorname{v_{z}}{(m)} dm = (2 \\operatorname{v_{z}}{(m)} + e^{m}) \\int e^{m} dm and (2 \\operatorname{v_{z}}{(m)} + e^{m}) \\int \\operatorname{v_{z}}{(m)} dm = (v_{2} + e^{m}) (2 \\operatorname{v_{z}}{(m)} + e^{m})", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["add", 1, "Function('v_z')(Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), Add(Function('v_z')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["add", 2, "exp(Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), exp(Symbol('m', commutative=True))), Add(Function('v_z')(Symbol('m', commutative=True)), Mul(Integer(2), exp(Symbol('m', commutative=True)))))"], [["times", 3, "Add(Function('v_z')(Symbol('m', commutative=True)), Mul(Integer(2), exp(Symbol('m', commutative=True))))"], "Equality(Mul(Add(Function('v_z')(Symbol('m', commutative=True)), Mul(Integer(2), exp(Symbol('m', commutative=True)))), Integral(Function('v_z')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Add(Function('v_z')(Symbol('m', commutative=True)), Mul(Integer(2), exp(Symbol('m', commutative=True)))), Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), exp(Symbol('m', commutative=True))), Integral(Function('v_z')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Add(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), exp(Symbol('m', commutative=True))), Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), exp(Symbol('m', commutative=True))), Integral(Function('v_z')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Add(Symbol('v_2', commutative=True), exp(Symbol('m', commutative=True))), Add(Mul(Integer(2), Function('v_z')(Symbol('m', commutative=True))), exp(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given n{(l)} = \\log{(l)}, then obtain (e^{(n^{l}{(l)})^{l}})^{l} = (e^{(\\log{(l)}^{l})^{l}})^{l}", "derivation": "n{(l)} = \\log{(l)} and n^{l}{(l)} = \\log{(l)}^{l} and (n^{l}{(l)})^{l} = (\\log{(l)}^{l})^{l} and e^{(n^{l}{(l)})^{l}} = e^{(\\log{(l)}^{l})^{l}} and (e^{(n^{l}{(l)})^{l}})^{l} = (e^{(\\log{(l)}^{l})^{l}})^{l}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('n')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Function('n')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(log(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Pow(Function('n')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True))), exp(Pow(Pow(log(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(exp(Pow(Pow(Function('n')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True))), Symbol('l', commutative=True)), Pow(exp(Pow(Pow(log(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Symbol('l', commutative=True))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(h,\\mathbb{I})} = \\int (\\mathbb{I} + h) dh, then derive \\operatorname{f^{*}}{(h,\\mathbb{I})} = \\hat{X} + \\mathbb{I} h + \\frac{h^{2}}{2}, then obtain \\hat{X} + \\mathbb{I} h + \\frac{h^{2}}{2} - h = - h + \\int (\\mathbb{I} + h) dh", "derivation": "\\operatorname{f^{*}}{(h,\\mathbb{I})} = \\int (\\mathbb{I} + h) dh and - h + \\operatorname{f^{*}}{(h,\\mathbb{I})} = - h + \\int (\\mathbb{I} + h) dh and \\operatorname{f^{*}}{(h,\\mathbb{I})} = \\hat{X} + \\mathbb{I} h + \\frac{h^{2}}{2} and \\hat{X} + \\mathbb{I} h + \\frac{h^{2}}{2} - h = - h + \\int (\\mathbb{I} + h) dh", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('f^*')(Symbol('h', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('h', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Integral(Add(Symbol('\\\\mathbb{I}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(U)} = \\log{(U)}, then obtain \\int (U^{2} \\mathbf{p}^{3}{(U)})^{3 U} dU = \\int (U^{2} \\mathbf{p}^{2}{(U)} \\log{(U)})^{3 U} dU", "derivation": "\\mathbf{p}{(U)} = \\log{(U)} and U \\mathbf{p}{(U)} = U \\log{(U)} and U^{2} \\mathbf{p}^{2}{(U)} = U^{2} \\mathbf{p}{(U)} \\log{(U)} and U^{2} \\mathbf{p}^{3}{(U)} = U^{2} \\mathbf{p}^{2}{(U)} \\log{(U)} and (U^{2} \\mathbf{p}^{3}{(U)})^{U} = (U^{2} \\mathbf{p}^{2}{(U)} \\log{(U)})^{U} and (U^{2} \\mathbf{p}^{3}{(U)})^{3 U} = (U^{2} \\mathbf{p}^{2}{(U)} \\log{(U)})^{3 U} and \\int (U^{2} \\mathbf{p}^{3}{(U)})^{3 U} dU = \\int (U^{2} \\mathbf{p}^{2}{(U)} \\log{(U)})^{3 U} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('\\\\mathbf{p}')(Symbol('U', commutative=True))), Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))))"], [["times", 2, "Mul(Symbol('U', commutative=True), Function('\\\\mathbf{p}')(Symbol('U', commutative=True)))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(2))), Mul(Pow(Symbol('U', commutative=True), Integer(2)), Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True))))"], [["times", 3, "Function('\\\\mathbf{p}')(Symbol('U', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(3))), Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(3))), Symbol('U', commutative=True)), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["power", 5, 3], "Equality(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(3))), Mul(Integer(3), Symbol('U', commutative=True))), Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Mul(Integer(3), Symbol('U', commutative=True))))"], [["integrate", 6, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(3))), Mul(Integer(3), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Pow(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{p}')(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Mul(Integer(3), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\theta{(A_{x})} = e^{\\sin{(A_{x})}} and \\operatorname{C_{2}}{(A_{x})} = - A_{x} + \\theta{(A_{x})}, then obtain A_{x} e^{\\sin{(A_{x})}} = A_{x} (A_{x} + \\operatorname{C_{2}}{(A_{x})})", "derivation": "\\theta{(A_{x})} = e^{\\sin{(A_{x})}} and A_{x} \\theta{(A_{x})} = A_{x} e^{\\sin{(A_{x})}} and \\operatorname{C_{2}}{(A_{x})} = - A_{x} + \\theta{(A_{x})} and \\operatorname{C_{2}}{(A_{x})} = - A_{x} + e^{\\sin{(A_{x})}} and A_{x} + \\operatorname{C_{2}}{(A_{x})} = e^{\\sin{(A_{x})}} and A_{x} \\theta{(A_{x})} = A_{x} (A_{x} + \\operatorname{C_{2}}{(A_{x})}) and A_{x} e^{\\sin{(A_{x})}} = A_{x} (A_{x} + \\operatorname{C_{2}}{(A_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('A_x', commutative=True)), exp(sin(Symbol('A_x', commutative=True))))"], [["times", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Function('\\\\theta')(Symbol('A_x', commutative=True))), Mul(Symbol('A_x', commutative=True), exp(sin(Symbol('A_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('A_x', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Function('\\\\theta')(Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('C_2')(Symbol('A_x', commutative=True)), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), exp(sin(Symbol('A_x', commutative=True)))))"], [["add", 4, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Function('C_2')(Symbol('A_x', commutative=True))), exp(sin(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Symbol('A_x', commutative=True), Function('\\\\theta')(Symbol('A_x', commutative=True))), Mul(Symbol('A_x', commutative=True), Add(Symbol('A_x', commutative=True), Function('C_2')(Symbol('A_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Symbol('A_x', commutative=True), exp(sin(Symbol('A_x', commutative=True)))), Mul(Symbol('A_x', commutative=True), Add(Symbol('A_x', commutative=True), Function('C_2')(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(h)} = \\int e^{h} dh, then derive 2 \\frac{d}{d h} \\tilde{g}{(h)} = \\frac{d}{d h} (\\tilde{g}{(h)} + \\int e^{h} dh - 1), then obtain - 2 \\frac{d}{d h} \\tilde{g}{(h)} = - \\frac{d}{d h} (2 \\tilde{g}{(h)} - 1)", "derivation": "\\tilde{g}{(h)} = \\int e^{h} dh and 2 \\tilde{g}{(h)} = \\tilde{g}{(h)} + \\int e^{h} dh and 2 \\tilde{g}{(h)} - 1 = \\tilde{g}{(h)} + \\int e^{h} dh - 1 and \\frac{d}{d h} (2 \\tilde{g}{(h)} - 1) = \\frac{d}{d h} (\\tilde{g}{(h)} + \\int e^{h} dh - 1) and 2 \\frac{d}{d h} \\tilde{g}{(h)} = \\frac{d}{d h} (\\tilde{g}{(h)} + \\int e^{h} dh - 1) and 2 \\frac{d}{d h} \\tilde{g}{(h)} = \\frac{d}{d h} (2 \\tilde{g}{(h)} - 1) and - 2 \\frac{d}{d h} \\tilde{g}{(h)} = - \\frac{d}{d h} (2 \\tilde{g}{(h)} - 1)", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["add", 1, "Function('\\\\tilde{g}')(Symbol('h', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('h', commutative=True))), Add(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('h', commutative=True))), Integer(-1)), Add(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('h', commutative=True))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Derivative(Add(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Integral(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Derivative(Add(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('h', commutative=True))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('h', commutative=True))), Integer(-1)), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then derive \\int \\operatorname{a^{\\dagger}}{(\\mathbf{P})} d\\mathbf{P} = v + \\sin{(\\mathbf{P})}, then obtain v + \\sin{(\\mathbf{P})} = \\int \\cos{(\\mathbf{P})} d\\mathbf{P}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\int \\operatorname{a^{\\dagger}}{(\\mathbf{P})} d\\mathbf{P} = \\int \\cos{(\\mathbf{P})} d\\mathbf{P} and \\int \\operatorname{a^{\\dagger}}{(\\mathbf{P})} d\\mathbf{P} = v + \\sin{(\\mathbf{P})} and v + \\sin{(\\mathbf{P})} = \\int \\cos{(\\mathbf{P})} d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('v', commutative=True), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('v', commutative=True), sin(Symbol('\\\\mathbf{P}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given M{(m,\\theta_2)} = \\theta_2 - m, then derive \\int (M{(m,\\theta_2)} + 1) d\\theta_2 = M + \\frac{\\theta_2^{2}}{2} + \\theta_2 (1 - m), then obtain (\\int (M{(m,\\theta_2)} + 1) d\\theta_2)^{\\theta_2} = (M + \\frac{\\theta_2^{2}}{2} + \\theta_2 (1 - m))^{\\theta_2}", "derivation": "M{(m,\\theta_2)} = \\theta_2 - m and M{(m,\\theta_2)} + 1 = \\theta_2 - m + 1 and \\int (M{(m,\\theta_2)} + 1) d\\theta_2 = \\int (\\theta_2 - m + 1) d\\theta_2 and \\int (M{(m,\\theta_2)} + 1) d\\theta_2 = M + \\frac{\\theta_2^{2}}{2} + \\theta_2 (1 - m) and (\\int (M{(m,\\theta_2)} + 1) d\\theta_2)^{\\theta_2} = (M + \\frac{\\theta_2^{2}}{2} + \\theta_2 (1 - m))^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('m', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('M')(Symbol('m', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Function('M')(Symbol('m', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('m', commutative=True)), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('M')(Symbol('m', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('m', commutative=True))))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Integral(Add(Function('M')(Symbol('m', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1)), Tuple(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('m', commutative=True))))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(l)} = \\cos{(\\sin{(l)})}, then obtain 1 = ((\\sigma_{x}{(l)} + 1)^{l})^{- l} ((\\cos{(\\sin{(l)})} + 1)^{l})^{l}", "derivation": "\\sigma_{x}{(l)} = \\cos{(\\sin{(l)})} and \\sigma_{x}{(l)} + 1 = \\cos{(\\sin{(l)})} + 1 and (\\sigma_{x}{(l)} + 1)^{l} = (\\cos{(\\sin{(l)})} + 1)^{l} and ((\\sigma_{x}{(l)} + 1)^{l})^{l} = ((\\cos{(\\sin{(l)})} + 1)^{l})^{l} and 1 = ((\\sigma_{x}{(l)} + 1)^{l})^{- l} ((\\cos{(\\sin{(l)})} + 1)^{l})^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('l', commutative=True)), cos(sin(Symbol('l', commutative=True))))"], [["add", 1, 1], "Equality(Add(Function('\\\\sigma_x')(Symbol('l', commutative=True)), Integer(1)), Add(cos(sin(Symbol('l', commutative=True))), Integer(1)))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Function('\\\\sigma_x')(Symbol('l', commutative=True)), Integer(1)), Symbol('l', commutative=True)), Pow(Add(cos(sin(Symbol('l', commutative=True))), Integer(1)), Symbol('l', commutative=True)))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Pow(Add(Function('\\\\sigma_x')(Symbol('l', commutative=True)), Integer(1)), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Pow(Add(cos(sin(Symbol('l', commutative=True))), Integer(1)), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 4, "Pow(Pow(Add(Function('\\\\sigma_x')(Symbol('l', commutative=True)), Integer(1)), Symbol('l', commutative=True)), Symbol('l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Pow(Add(Function('\\\\sigma_x')(Symbol('l', commutative=True)), Integer(1)), Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Pow(Pow(Add(cos(sin(Symbol('l', commutative=True))), Integer(1)), Symbol('l', commutative=True)), Symbol('l', commutative=True))))"]]}, {"prompt": "Given s{(n)} = \\log{(n)}, then obtain n (n \\log{(n)} + s{(n)}) \\log{(n)} - (n \\log{(n)} + s{(n)}) \\log{(n)} = n (n \\log{(n)} + \\log{(n)}) \\log{(n)} - (n \\log{(n)} + s{(n)}) \\log{(n)}", "derivation": "s{(n)} = \\log{(n)} and n s{(n)} = n \\log{(n)} and n s{(n)} + s{(n)} = n s{(n)} + \\log{(n)} and n \\log{(n)} + s{(n)} = n \\log{(n)} + \\log{(n)} and n (n \\log{(n)} + s{(n)}) \\log{(n)} = n (n \\log{(n)} + \\log{(n)}) \\log{(n)} and n (n \\log{(n)} + s{(n)}) \\log{(n)} - (n s{(n)} + s{(n)}) \\log{(n)} = n (n \\log{(n)} + \\log{(n)}) \\log{(n)} - (n s{(n)} + s{(n)}) \\log{(n)} and n (n \\log{(n)} + s{(n)}) \\log{(n)} - (n \\log{(n)} + s{(n)}) \\log{(n)} = n (n \\log{(n)} + \\log{(n)}) \\log{(n)} - (n \\log{(n)} + s{(n)}) \\log{(n)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["times", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))))"], [["add", 1, "Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True)))"], "Equality(Add(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), Add(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["times", 4, "Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True)))"], "Equality(Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))))"], [["minus", 5, "Mul(Add(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))"], "Equality(Add(Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))), Add(Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Symbol('n', commutative=True), Function('s')(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))), Add(Mul(Symbol('n', commutative=True), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), log(Symbol('n', commutative=True))), Mul(Integer(-1), Add(Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Function('s')(Symbol('n', commutative=True))), log(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})}, then derive (\\int \\operatorname{C_{2}}{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = (A_{1} - \\cos{(\\tilde{g})})^{\\tilde{g}}, then obtain - (\\int \\sin{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = - (A_{1} - \\cos{(\\tilde{g})})^{\\tilde{g}}", "derivation": "\\operatorname{C_{2}}{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\int \\operatorname{C_{2}}{(\\tilde{g})} d\\tilde{g} = \\int \\sin{(\\tilde{g})} d\\tilde{g} and (\\int \\operatorname{C_{2}}{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = (\\int \\sin{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} and (\\int \\operatorname{C_{2}}{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = (A_{1} - \\cos{(\\tilde{g})})^{\\tilde{g}} and (\\int \\sin{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = (A_{1} - \\cos{(\\tilde{g})})^{\\tilde{g}} and - (\\int \\sin{(\\tilde{g})} d\\tilde{g})^{\\tilde{g}} = - (A_{1} - \\cos{(\\tilde{g})})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Integral(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Integral(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('C_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integral(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given n{(\\psi)} = \\cos{(\\psi)}, then derive \\frac{d}{d \\psi} n{(\\psi)} = - \\sin{(\\psi)}, then obtain - \\sin{(\\psi)} = \\frac{d}{d \\psi} \\cos{(\\psi)}", "derivation": "n{(\\psi)} = \\cos{(\\psi)} and \\frac{d}{d \\psi} n{(\\psi)} = \\frac{d}{d \\psi} \\cos{(\\psi)} and \\frac{d}{d \\psi} n{(\\psi)} = - \\sin{(\\psi)} and - \\sin{(\\psi)} = \\frac{d}{d \\psi} \\cos{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} = \\log{(\\phi_2 q)}, then obtain \\frac{\\partial^{2}}{\\partial S\\partial q} - \\frac{\\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} dq}{S} = \\frac{\\partial^{2}}{\\partial S\\partial q} - \\frac{\\int \\log{(\\phi_2 q)} dq}{S}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} = \\log{(\\phi_2 q)} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} dq = \\int \\log{(\\phi_2 q)} dq and - \\frac{\\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} dq}{S} = - \\frac{\\int \\log{(\\phi_2 q)} dq}{S} and \\frac{\\partial}{\\partial q} - \\frac{\\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} dq}{S} = \\frac{\\partial}{\\partial q} - \\frac{\\int \\log{(\\phi_2 q)} dq}{S} and \\frac{\\partial^{2}}{\\partial S\\partial q} - \\frac{\\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\phi_2,q)} dq}{S} = \\frac{\\partial^{2}}{\\partial S\\partial q} - \\frac{\\int \\log{(\\phi_2 q)} dq}{S}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True)), log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('S', commutative=True), Integer(-1)), Integral(log(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1)), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hbar)} = e^{e^{\\hbar}} and \\operatorname{n_{1}}{(\\hbar)} = e^{\\hbar}, then obtain e^{\\operatorname{n_{1}}{(\\hbar)}} - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} = - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} + e^{e^{\\hbar}}", "derivation": "\\tilde{g}^*{(\\hbar)} = e^{e^{\\hbar}} and \\operatorname{n_{1}}{(\\hbar)} = e^{\\hbar} and \\tilde{g}^*{(\\hbar)} = e^{\\operatorname{n_{1}}{(\\hbar)}} and \\tilde{g}^*{(\\hbar)} - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} = e^{\\operatorname{n_{1}}{(\\hbar)}} - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} and e^{\\operatorname{n_{1}}{(\\hbar)}} = e^{e^{\\hbar}} and \\tilde{g}^*{(\\hbar)} - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} = - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} + e^{e^{\\hbar}} and e^{\\operatorname{n_{1}}{(\\hbar)}} - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} = - e^{e^{\\hbar}} \\sin{(e^{\\hat{p}})} + e^{e^{\\hbar}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hbar', commutative=True)), exp(exp(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hbar', commutative=True)), exp(Function('n_1')(Symbol('\\\\hbar', commutative=True))))"], [["minus", 3, "Mul(exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True))))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True))))), Add(exp(Function('n_1')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(exp(Function('n_1')(Symbol('\\\\hbar', commutative=True))), exp(exp(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True))))), Add(Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True)))), exp(exp(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(exp(Function('n_1')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True))))), Add(Mul(Integer(-1), exp(exp(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hat{p}', commutative=True)))), exp(exp(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} = \\sin{(I \\lambda)} and \\operatorname{C_{1}}{(I,\\lambda)} = I \\lambda and \\mathbf{r}{(I,\\lambda)} = \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} - \\sin{(I \\lambda)}, then obtain \\mathbf{r}{(I,\\lambda)} = \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} - \\sin{(\\operatorname{C_{1}}{(I,\\lambda)})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} = \\sin{(I \\lambda)} and \\operatorname{C_{1}}{(I,\\lambda)} = I \\lambda and \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} = \\sin{(\\operatorname{C_{1}}{(I,\\lambda)})} and \\sin{(\\operatorname{C_{1}}{(I,\\lambda)})} = \\sin{(I \\lambda)} and \\mathbf{r}{(I,\\lambda)} = \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} - \\sin{(I \\lambda)} and \\mathbf{r}{(I,\\lambda)} = \\operatorname{f_{\\mathbf{p}}}{(I,\\lambda)} - \\sin{(\\operatorname{C_{1}}{(I,\\lambda)})}", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), sin(Function('C_1')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(sin(Function('C_1')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\mathbf{r}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Function('C_1')(Symbol('I', commutative=True), Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given J{(f^{*})} = \\cos{(f^{*})} and \\mathbf{g}{(f^{*})} = \\cos{(f^{*})}, then derive 2 \\frac{d}{d f^{*}} J{(f^{*})} = - \\sin{(f^{*})} + \\frac{d}{d f^{*}} J{(f^{*})}, then obtain 2 \\frac{d}{d f^{*}} \\mathbf{g}{(f^{*})} = - \\sin{(f^{*})} + \\frac{d}{d f^{*}} \\mathbf{g}{(f^{*})}", "derivation": "J{(f^{*})} = \\cos{(f^{*})} and \\mathbf{g}{(f^{*})} = \\cos{(f^{*})} and \\frac{d}{d f^{*}} J{(f^{*})} = \\frac{d}{d f^{*}} \\cos{(f^{*})} and 2 \\frac{d}{d f^{*}} J{(f^{*})} = \\frac{d}{d f^{*}} J{(f^{*})} + \\frac{d}{d f^{*}} \\cos{(f^{*})} and 2 \\frac{d}{d f^{*}} J{(f^{*})} = - \\sin{(f^{*})} + \\frac{d}{d f^{*}} J{(f^{*})} and 2 \\frac{d}{d f^{*}} \\cos{(f^{*})} = - \\sin{(f^{*})} + \\frac{d}{d f^{*}} \\cos{(f^{*})} and 2 \\frac{d}{d f^{*}} \\mathbf{g}{(f^{*})} = - \\sin{(f^{*})} + \\frac{d}{d f^{*}} \\mathbf{g}{(f^{*})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('f^*', commutative=True))), Derivative(Function('J')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Derivative(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('f^*', commutative=True))), Derivative(cos(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('f^*', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} = \\cos{(C + \\dot{y})}, then obtain - \\rho_f + \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} - \\sin{(C + \\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC = - \\rho_f - \\sin{(C + \\dot{y})} + \\cos{(C + \\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC", "derivation": "\\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} = \\cos{(C + \\dot{y})} and \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC = \\int \\cos{(C + \\dot{y})} dC and \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} + \\int \\cos{(C + \\dot{y})} dC = \\cos{(C + \\dot{y})} + \\int \\cos{(C + \\dot{y})} dC and \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC = \\cos{(C + \\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC and - \\rho_f + \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} - \\sin{(C + \\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC = - \\rho_f - \\sin{(C + \\dot{y})} + \\cos{(C + \\dot{y})} + \\int \\operatorname{J_{\\varepsilon}}{(C,\\dot{y})} dC", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["add", 1, "Integral(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('C', commutative=True)))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integral(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('C', commutative=True)))), Add(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integral(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integral(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integral(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["minus", 4, "Add(Symbol('\\\\rho_f', commutative=True), sin(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Integral(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), cos(Add(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integral(Function('J_{\\\\varepsilon}')(Symbol('C', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(l)} = \\frac{d}{d l} e^{l} and a{(r_{0})} = \\log{(\\cos{(r_{0})})}, then derive l \\operatorname{m_{s}}{(l)} = l e^{l}, then obtain - l \\frac{d}{d l} e^{l} + a^{r_{0}}{(r_{0})} = - l \\frac{d}{d l} e^{l} + \\log{(\\cos{(r_{0})})}^{r_{0}}", "derivation": "\\operatorname{m_{s}}{(l)} = \\frac{d}{d l} e^{l} and l \\operatorname{m_{s}}{(l)} = l \\frac{d}{d l} e^{l} and l \\operatorname{m_{s}}{(l)} = l e^{l} and l e^{l} = l \\frac{d}{d l} e^{l} and a{(r_{0})} = \\log{(\\cos{(r_{0})})} and a^{r_{0}}{(r_{0})} = \\log{(\\cos{(r_{0})})}^{r_{0}} and - l e^{l} + a^{r_{0}}{(r_{0})} = - l e^{l} + \\log{(\\cos{(r_{0})})}^{r_{0}} and - l \\frac{d}{d l} e^{l} + a^{r_{0}}{(r_{0})} = - l \\frac{d}{d l} e^{l} + \\log{(\\cos{(r_{0})})}^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('l', commutative=True)), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('m_s')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('l', commutative=True), Function('m_s')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), exp(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('l', commutative=True), exp(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], ["get_premise", "Equality(Function('a')(Symbol('r_0', commutative=True)), log(cos(Symbol('r_0', commutative=True))))"], [["power", 5, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('a')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["minus", 6, "Mul(Symbol('l', commutative=True), exp(Symbol('l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True), exp(Symbol('l', commutative=True))), Pow(Function('a')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True), exp(Symbol('l', commutative=True))), Pow(log(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Function('a')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True), Derivative(exp(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(log(cos(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given A{(\\varepsilon_0)} = \\log{(\\varepsilon_0)}, then obtain \\frac{(\\frac{\\varepsilon_0 + A{(\\varepsilon_0)}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}})^{\\varepsilon_0}}{\\varepsilon_0 + A{(\\varepsilon_0)}} = \\frac{1}{\\varepsilon_0 + A{(\\varepsilon_0)}}", "derivation": "A{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and \\varepsilon_0 + A{(\\varepsilon_0)} = \\varepsilon_0 + \\log{(\\varepsilon_0)} and \\frac{\\varepsilon_0 + A{(\\varepsilon_0)}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}} = 1 and (\\frac{\\varepsilon_0 + A{(\\varepsilon_0)}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}})^{\\varepsilon_0} = 1 and \\frac{(\\frac{\\varepsilon_0 + A{(\\varepsilon_0)}}{\\varepsilon_0 + \\log{(\\varepsilon_0)}})^{\\varepsilon_0}}{\\varepsilon_0 + A{(\\varepsilon_0)}} = \\frac{1}{\\varepsilon_0 + A{(\\varepsilon_0)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1))), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1))"], [["divide", 4, "Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Pow(Mul(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), log(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1))), Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Function('A')(Symbol('\\\\varepsilon_0', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given u{(I,\\Omega)} = - \\sin{(I - \\Omega)}, then obtain \\log{(- (\\int u{(I,\\Omega)} dI)^{2})} = \\log{(- (\\int - \\sin{(I - \\Omega)} dI)^{2})}", "derivation": "u{(I,\\Omega)} = - \\sin{(I - \\Omega)} and \\int u{(I,\\Omega)} dI = \\int - \\sin{(I - \\Omega)} dI and - \\int u{(I,\\Omega)} dI = - \\int - \\sin{(I - \\Omega)} dI and - (\\int u{(I,\\Omega)} dI) \\int - \\sin{(I - \\Omega)} dI = - (\\int - \\sin{(I - \\Omega)} dI)^{2} and \\frac{\\int u{(I,\\Omega)} dI}{\\int - \\sin{(I - \\Omega)} dI} = 1 and - (\\int u{(I,\\Omega)} dI)^{2} = - (\\int u{(I,\\Omega)} dI) \\int - \\sin{(I - \\Omega)} dI and - (\\int u{(I,\\Omega)} dI)^{2} = - (\\int - \\sin{(I - \\Omega)} dI)^{2} and \\log{(- (\\int u{(I,\\Omega)} dI)^{2})} = \\log{(- (\\int - \\sin{(I - \\Omega)} dI)^{2})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))))"], "Equality(Mul(Integer(-1), Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))), Integer(2))))"], [["divide", 3, "Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))))"], "Equality(Mul(Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))), Integer(-1))), Integer(1))"], [["times", 5, "Mul(Integer(-1), Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(2))), Mul(Integer(-1), Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(-1), Pow(Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(2))), Mul(Integer(-1), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))), Integer(2))))"], [["log", 7], "Equality(log(Mul(Integer(-1), Pow(Integral(Function('u')(Symbol('I', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('I', commutative=True))), Integer(2)))), log(Mul(Integer(-1), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Tuple(Symbol('I', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(k,\\lambda,E_{x})} = \\frac{k}{E_{x} \\lambda} and \\operatorname{L_{\\varepsilon}}{(k,\\lambda,E_{x})} = \\operatorname{A_{z}}^{2}{(k,\\lambda,E_{x})}, then obtain \\frac{k \\operatorname{A_{z}}{(k,\\lambda,E_{x})}}{E_{x} \\lambda} = \\frac{k^{2}}{E_{x}^{2} \\lambda^{2}}", "derivation": "\\operatorname{A_{z}}{(k,\\lambda,E_{x})} = \\frac{k}{E_{x} \\lambda} and \\operatorname{A_{z}}^{2}{(k,\\lambda,E_{x})} = \\frac{k \\operatorname{A_{z}}{(k,\\lambda,E_{x})}}{E_{x} \\lambda} and \\operatorname{L_{\\varepsilon}}{(k,\\lambda,E_{x})} = \\operatorname{A_{z}}^{2}{(k,\\lambda,E_{x})} and \\operatorname{L_{\\varepsilon}}{(k,\\lambda,E_{x})} = \\frac{k^{2}}{E_{x}^{2} \\lambda^{2}} and \\operatorname{A_{z}}^{2}{(k,\\lambda,E_{x})} = \\frac{k^{2}}{E_{x}^{2} \\lambda^{2}} and \\frac{k \\operatorname{A_{z}}{(k,\\lambda,E_{x})}}{E_{x} \\lambda} = \\frac{k^{2}}{E_{x}^{2} \\lambda^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('k', commutative=True)))"], [["times", 1, "Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Pow(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('k', commutative=True), Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Pow(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('k', commutative=True), Function('A_z')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-2)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Pow(Symbol('k', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{v},v_{2})} = e^{\\mathbf{v} - v_{2}}, then derive \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = b + e^{\\mathbf{v} - v_{2}}, then obtain \\frac{\\partial}{\\partial v_{2}} \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = \\frac{\\partial}{\\partial v_{2}} \\int e^{\\mathbf{v} - v_{2}} d\\mathbf{v}", "derivation": "\\eta^{\\prime}{(\\mathbf{v},v_{2})} = e^{\\mathbf{v} - v_{2}} and \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = \\int e^{\\mathbf{v} - v_{2}} d\\mathbf{v} and \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = b + e^{\\mathbf{v} - v_{2}} and \\frac{\\partial}{\\partial v_{2}} \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = \\frac{\\partial}{\\partial v_{2}} (b + e^{\\mathbf{v} - v_{2}}) and b + e^{\\mathbf{v} - v_{2}} = \\int e^{\\mathbf{v} - v_{2}} d\\mathbf{v} and \\frac{\\partial}{\\partial v_{2}} \\int \\eta^{\\prime}{(\\mathbf{v},v_{2})} d\\mathbf{v} = \\frac{\\partial}{\\partial v_{2}} \\int e^{\\mathbf{v} - v_{2}} d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_2', commutative=True)), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('b', commutative=True), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('b', commutative=True), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('b', commutative=True), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Integral(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{p},v_{2})} = \\hat{p} + v_{2}, then obtain e^{2 \\hat{p} + 2 v_{2}} = e^{\\hat{p} + v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})}}", "derivation": "\\operatorname{n_{2}}{(\\hat{p},v_{2})} = \\hat{p} + v_{2} and \\hat{p} + v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})} = 2 \\hat{p} + 2 v_{2} and 2 \\operatorname{n_{2}}{(\\hat{p},v_{2})} = 2 \\hat{p} + 2 v_{2} and \\hat{p} - v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})} = 2 \\hat{p} and 2 \\operatorname{n_{2}}{(\\hat{p},v_{2})} = \\hat{p} + v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})} and e^{2 \\operatorname{n_{2}}{(\\hat{p},v_{2})}} = e^{\\hat{p} + v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})}} and e^{2 \\hat{p} + 2 v_{2}} = e^{\\hat{p} + v_{2} + \\operatorname{n_{2}}{(\\hat{p},v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(Integer(2), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True)))), exp(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(exp(Add(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)))), exp(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True), Function('n_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given G{(\\hat{H}_l,\\theta,\\sigma_p)} = \\hat{H}_l \\sigma_p + \\theta and \\ddot{x}{(\\hat{H}_l,\\sigma_p)} = \\hat{H}_l \\sigma_p, then obtain \\frac{\\partial}{\\partial \\sigma_p} G{(\\hat{H}_l,\\theta,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (\\theta + \\ddot{x}{(\\hat{H}_l,\\sigma_p)})", "derivation": "G{(\\hat{H}_l,\\theta,\\sigma_p)} = \\hat{H}_l \\sigma_p + \\theta and \\ddot{x}{(\\hat{H}_l,\\sigma_p)} = \\hat{H}_l \\sigma_p and G{(\\hat{H}_l,\\theta,\\sigma_p)} = \\theta + \\ddot{x}{(\\hat{H}_l,\\sigma_p)} and \\frac{\\partial}{\\partial \\sigma_p} G{(\\hat{H}_l,\\theta,\\sigma_p)} = \\frac{\\partial}{\\partial \\sigma_p} (\\theta + \\ddot{x}{(\\hat{H}_l,\\sigma_p)})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('G')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta', commutative=True), Function('\\\\ddot{x}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(\\nabla)} = e^{\\nabla} and \\phi_{2}{(\\nabla)} = 2 \\nabla, then obtain \\frac{d}{d \\nabla} \\cos{(\\phi_{2}{(\\nabla)} + u{(\\nabla)})} = \\frac{d}{d \\nabla} \\cos{(\\phi_{2}{(\\nabla)} + e^{\\nabla})}", "derivation": "u{(\\nabla)} = e^{\\nabla} and \\nabla + u{(\\nabla)} = \\nabla + e^{\\nabla} and 2 \\nabla + u{(\\nabla)} = 2 \\nabla + e^{\\nabla} and \\phi_{2}{(\\nabla)} = 2 \\nabla and \\phi_{2}{(\\nabla)} + u{(\\nabla)} = \\phi_{2}{(\\nabla)} + e^{\\nabla} and \\cos{(\\phi_{2}{(\\nabla)} + u{(\\nabla)})} = \\cos{(\\phi_{2}{(\\nabla)} + e^{\\nabla})} and \\frac{d}{d \\nabla} \\cos{(\\phi_{2}{(\\nabla)} + u{(\\nabla)})} = \\frac{d}{d \\nabla} \\cos{(\\phi_{2}{(\\nabla)} + e^{\\nabla})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Function('u')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\nabla', commutative=True)), Function('u')(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), Function('u')(Symbol('\\\\nabla', commutative=True))), Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True))))"], [["cos", 5], "Equality(cos(Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), Function('u')(Symbol('\\\\nabla', commutative=True)))), cos(Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(cos(Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), Function('u')(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(cos(Add(Function('\\\\phi_2')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\mathbf{P},\\mu)} = \\mathbf{P} + \\mu, then derive \\int n{(\\mathbf{P},\\mu)} d\\mu = \\mathbf{P} \\mu + \\frac{\\mu^{2}}{2} + \\tilde{g}^*, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\mu + \\frac{\\mu^{2}}{2} + \\tilde{g}^*) = \\frac{\\partial}{\\partial \\mathbf{P}} \\int (\\mathbf{P} + \\mu) d\\mu", "derivation": "n{(\\mathbf{P},\\mu)} = \\mathbf{P} + \\mu and \\int n{(\\mathbf{P},\\mu)} d\\mu = \\int (\\mathbf{P} + \\mu) d\\mu and \\frac{\\partial}{\\partial \\mathbf{P}} \\int n{(\\mathbf{P},\\mu)} d\\mu = \\frac{\\partial}{\\partial \\mathbf{P}} \\int (\\mathbf{P} + \\mu) d\\mu and \\int n{(\\mathbf{P},\\mu)} d\\mu = \\mathbf{P} \\mu + \\frac{\\mu^{2}}{2} + \\tilde{g}^* and \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\mu + \\frac{\\mu^{2}}{2} + \\tilde{g}^*) = \\frac{\\partial}{\\partial \\mathbf{P}} \\int (\\mathbf{P} + \\mu) d\\mu", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Function('n')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2))), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2))), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(c)} = \\cos{(c)}, then obtain \\frac{d}{d c} (\\operatorname{P_{g}}^{2}{(c)} + \\cos{(c)}) = \\frac{d}{d c} (\\operatorname{P_{g}}{(c)} \\cos{(c)} + \\cos{(c)})", "derivation": "\\operatorname{P_{g}}{(c)} = \\cos{(c)} and \\operatorname{P_{g}}^{2}{(c)} = \\operatorname{P_{g}}{(c)} \\cos{(c)} and \\operatorname{P_{g}}^{2}{(c)} + \\cos{(c)} = \\operatorname{P_{g}}{(c)} \\cos{(c)} + \\cos{(c)} and \\frac{d}{d c} (\\operatorname{P_{g}}^{2}{(c)} + \\cos{(c)}) = \\frac{d}{d c} (\\operatorname{P_{g}}{(c)} \\cos{(c)} + \\cos{(c)})", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True)))"], [["times", 1, "Function('P_g')(Symbol('c', commutative=True))"], "Equality(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(2)), Mul(Function('P_g')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))))"], [["add", 2, "cos(Symbol('c', commutative=True))"], "Equality(Add(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(2)), cos(Symbol('c', commutative=True))), Add(Mul(Function('P_g')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Pow(Function('P_g')(Symbol('c', commutative=True)), Integer(2)), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Function('P_g')(Symbol('c', commutative=True)), cos(Symbol('c', commutative=True))), cos(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(V)} = \\log{(V)}, then derive \\int \\mathbf{F}{(V)} dV = V \\log{(V)} - V + \\mathbf{g}, then obtain 0 = l{(T,F_{x})} \\frac{d}{d \\mathbf{g}} \\int \\log{(V)} dV - l{(T,F_{x})}", "derivation": "\\mathbf{F}{(V)} = \\log{(V)} and \\int \\mathbf{F}{(V)} dV = \\int \\log{(V)} dV and \\int \\mathbf{F}{(V)} dV = V \\log{(V)} - V + \\mathbf{g} and \\int \\log{(V)} dV = V \\log{(V)} - V + \\mathbf{g} and \\frac{d}{d \\mathbf{g}} \\int \\log{(V)} dV = \\frac{\\partial}{\\partial \\mathbf{g}} (V \\log{(V)} - V + \\mathbf{g}) and - l{(T,F_{x})} \\frac{d}{d \\mathbf{g}} \\int \\log{(V)} dV = - l{(T,F_{x})} \\frac{\\partial}{\\partial \\mathbf{g}} (V \\log{(V)} - V + \\mathbf{g}) and 0 = - l{(T,F_{x})} \\frac{\\partial}{\\partial \\mathbf{g}} (V \\log{(V)} - V + \\mathbf{g}) + l{(T,F_{x})} \\frac{d}{d \\mathbf{g}} \\int \\log{(V)} dV and 0 = l{(T,F_{x})} \\frac{d}{d \\mathbf{g}} \\int \\log{(V)} dV - l{(T,F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["times", 5, "Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["minus", 6, "Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Add(Mul(Symbol('V', commutative=True), log(Symbol('V', commutative=True))), Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Add(Mul(Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)), Derivative(Integral(log(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Function('l')(Symbol('T', commutative=True), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given I{(n,\\mathbf{F})} = \\sin{(\\frac{n}{\\mathbf{F}})}, then obtain (((\\frac{I{(n,\\mathbf{F})}}{\\mathbf{F}})^{n})^{\\mathbf{F}})^{n} = (((\\frac{\\sin{(\\frac{n}{\\mathbf{F}})}}{\\mathbf{F}})^{n})^{\\mathbf{F}})^{n}", "derivation": "I{(n,\\mathbf{F})} = \\sin{(\\frac{n}{\\mathbf{F}})} and \\frac{I{(n,\\mathbf{F})}}{\\mathbf{F}} = \\frac{\\sin{(\\frac{n}{\\mathbf{F}})}}{\\mathbf{F}} and (\\frac{I{(n,\\mathbf{F})}}{\\mathbf{F}})^{n} = (\\frac{\\sin{(\\frac{n}{\\mathbf{F}})}}{\\mathbf{F}})^{n} and ((\\frac{I{(n,\\mathbf{F})}}{\\mathbf{F}})^{n})^{\\mathbf{F}} = ((\\frac{\\sin{(\\frac{n}{\\mathbf{F}})}}{\\mathbf{F}})^{n})^{\\mathbf{F}} and (((\\frac{I{(n,\\mathbf{F})}}{\\mathbf{F}})^{n})^{\\mathbf{F}})^{n} = (((\\frac{\\sin{(\\frac{n}{\\mathbf{F}})}}{\\mathbf{F}})^{n})^{\\mathbf{F}})^{n}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('n', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('n', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('n', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 4, "Symbol('n', commutative=True)"], "Equality(Pow(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('n', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('n', commutative=True)), Pow(Pow(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(H)} = \\frac{d}{d H} \\cos{(H)}, then derive \\mathbf{s}{(H)} = - \\sin{(H)}, then derive \\frac{d}{d H} \\mathbf{s}{(H)} = - \\cos{(H)}, then derive \\frac{H^{2}}{2} + \\Psi^{\\dagger} - \\sin{(H)} = \\frac{H^{2}}{2} + \\mathbf{p} - \\sin{(H)}, then obtain \\frac{H^{2}}{2} + \\Psi^{\\dagger} - \\sin{(H)} - \\frac{d}{d H} (H + \\frac{d}{d H} - \\sin{(H)}) = \\frac{H^{2}}{2} + \\mathbf{p} - \\sin{(H)} - \\frac{d}{d H} (H + \\frac{d}{d H} - \\sin{(H)})", "derivation": "\\mathbf{s}{(H)} = \\frac{d}{d H} \\cos{(H)} and \\mathbf{s}{(H)} = - \\sin{(H)} and \\frac{d}{d H} \\mathbf{s}{(H)} = \\frac{d}{d H} - \\sin{(H)} and \\frac{d}{d H} \\mathbf{s}{(H)} = - \\cos{(H)} and \\frac{d}{d H} - \\sin{(H)} = - \\cos{(H)} and H + \\frac{d}{d H} - \\sin{(H)} = H - \\cos{(H)} and \\int (H + \\frac{d}{d H} - \\sin{(H)}) dH = \\int (H - \\cos{(H)}) dH and \\frac{H^{2}}{2} + \\Psi^{\\dagger} - \\sin{(H)} = \\frac{H^{2}}{2} + \\mathbf{p} - \\sin{(H)} and \\frac{H^{2}}{2} + \\Psi^{\\dagger} - \\sin{(H)} - \\frac{d}{d H} (H + \\frac{d}{d H} - \\sin{(H)}) = \\frac{H^{2}}{2} + \\mathbf{p} - \\sin{(H)} - \\frac{d}{d H} (H + \\frac{d}{d H} - \\sin{(H)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('H', commutative=True))))"], [["add", 5, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Add(Symbol('H', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(-1), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), sin(Symbol('H', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), sin(Symbol('H', commutative=True)))))"], [["minus", 8, "Derivative(Add(Symbol('H', commutative=True), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), sin(Symbol('H', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('H', commutative=True), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True), Integer(1))))), Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), sin(Symbol('H', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('H', commutative=True), Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('H', commutative=True), Integer(1))))))"]]}, {"prompt": "Given i{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain i^{2}{(a^{\\dagger})} e^{a^{\\dagger}} + i{(a^{\\dagger})} e^{a^{\\dagger}} = i{(a^{\\dagger})} e^{2 a^{\\dagger}} + i{(a^{\\dagger})} e^{a^{\\dagger}}", "derivation": "i{(a^{\\dagger})} = e^{a^{\\dagger}} and i{(a^{\\dagger})} e^{a^{\\dagger}} = e^{2 a^{\\dagger}} and i^{2}{(a^{\\dagger})} e^{2 a^{\\dagger}} = e^{4 a^{\\dagger}} and i{(a^{\\dagger})} e^{2 a^{\\dagger}} = \\frac{e^{4 a^{\\dagger}}}{i{(a^{\\dagger})}} and i{(a^{\\dagger})} e^{2 a^{\\dagger}} + e^{2 a^{\\dagger}} = e^{2 a^{\\dagger}} + \\frac{e^{4 a^{\\dagger}}}{i{(a^{\\dagger})}} and i^{2}{(a^{\\dagger})} e^{a^{\\dagger}} + i{(a^{\\dagger})} e^{a^{\\dagger}} = i{(a^{\\dagger})} e^{2 a^{\\dagger}} + i{(a^{\\dagger})} e^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 1, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), exp(Mul(Integer(4), Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 3, "Function('i')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(Mul(Integer(4), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["add", 4, "exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), Add(exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Pow(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), exp(Mul(Integer(4), Symbol('a^{\\\\dagger}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Pow(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\eta,\\dot{y})} = \\eta^{\\dot{y}}, then obtain \\int \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\pi{(\\eta,\\dot{y})}}{\\pi{(\\eta,\\dot{y})}} d\\dot{y} = \\int \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\eta^{\\dot{y}}}{\\pi{(\\eta,\\dot{y})}} d\\dot{y}", "derivation": "\\pi{(\\eta,\\dot{y})} = \\eta^{\\dot{y}} and \\frac{\\partial}{\\partial \\dot{y}} \\pi{(\\eta,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} \\eta^{\\dot{y}} and \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\pi{(\\eta,\\dot{y})}}{\\pi{(\\eta,\\dot{y})}} = \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\eta^{\\dot{y}}}{\\pi{(\\eta,\\dot{y})}} and \\int \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\pi{(\\eta,\\dot{y})}}{\\pi{(\\eta,\\dot{y})}} d\\dot{y} = \\int \\frac{\\frac{\\partial}{\\partial \\dot{y}} \\eta^{\\dot{y}}}{\\pi{(\\eta,\\dot{y})}} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Pow(Function('\\\\pi')(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\omega,\\mathbf{J}_M)} = \\omega + \\log{(\\mathbf{J}_M)} and \\hat{H}_l{(\\omega,\\mathbf{J}_M)} = 2 \\mathbf{r}{(\\omega,\\mathbf{J}_M)}, then obtain 2 \\mathbf{J}_M + \\hat{H}_l{(\\omega,\\mathbf{J}_M)} = 2 \\mathbf{J}_M + \\omega + \\mathbf{r}{(\\omega,\\mathbf{J}_M)} + \\log{(\\mathbf{J}_M)}", "derivation": "\\mathbf{r}{(\\omega,\\mathbf{J}_M)} = \\omega + \\log{(\\mathbf{J}_M)} and \\mathbf{J}_M + \\mathbf{r}{(\\omega,\\mathbf{J}_M)} = \\mathbf{J}_M + \\omega + \\log{(\\mathbf{J}_M)} and 2 \\mathbf{J}_M + 2 \\mathbf{r}{(\\omega,\\mathbf{J}_M)} = 2 \\mathbf{J}_M + \\omega + \\mathbf{r}{(\\omega,\\mathbf{J}_M)} + \\log{(\\mathbf{J}_M)} and \\hat{H}_l{(\\omega,\\mathbf{J}_M)} = 2 \\mathbf{r}{(\\omega,\\mathbf{J}_M)} and 2 \\mathbf{J}_M + \\hat{H}_l{(\\omega,\\mathbf{J}_M)} = 2 \\mathbf{J}_M + \\omega + \\mathbf{r}{(\\omega,\\mathbf{J}_M)} + \\log{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\omega', commutative=True), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\mathbf{J}_M', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), log(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given m{(\\varphi^*,Q)} = \\cos{(Q + \\varphi^*)}, then obtain \\frac{m{(\\varphi^*,Q)} + \\int m{(\\varphi^*,Q)} d\\varphi^*}{\\cos{(Q + \\varphi^*)} + \\int m{(\\varphi^*,Q)} d\\varphi^*} = 1", "derivation": "m{(\\varphi^*,Q)} = \\cos{(Q + \\varphi^*)} and \\int m{(\\varphi^*,Q)} d\\varphi^* = \\int \\cos{(Q + \\varphi^*)} d\\varphi^* and m{(\\varphi^*,Q)} + \\int \\cos{(Q + \\varphi^*)} d\\varphi^* = \\cos{(Q + \\varphi^*)} + \\int \\cos{(Q + \\varphi^*)} d\\varphi^* and m{(\\varphi^*,Q)} + \\int m{(\\varphi^*,Q)} d\\varphi^* = \\cos{(Q + \\varphi^*)} + \\int m{(\\varphi^*,Q)} d\\varphi^* and \\frac{m{(\\varphi^*,Q)} + \\int m{(\\varphi^*,Q)} d\\varphi^*}{\\varphi^*} = \\frac{\\cos{(Q + \\varphi^*)} + \\int m{(\\varphi^*,Q)} d\\varphi^*}{\\varphi^*} and \\frac{m{(\\varphi^*,Q)} + \\int m{(\\varphi^*,Q)} d\\varphi^*}{\\cos{(Q + \\varphi^*)} + \\int m{(\\varphi^*,Q)} d\\varphi^*} = 1", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "Integral(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["divide", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))))"], [["divide", 5, "Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], "Equality(Mul(Add(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Pow(Add(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integral(Function('m')(Symbol('\\\\varphi^*', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{g}{(m_{s})} = \\frac{d}{d m_{s}} \\cos{(m_{s})} and \\dot{y}{(m_{s})} = \\mathbf{g}{(m_{s})} + \\sin{(m_{s})}, then obtain \\dot{y}^{m_{s}}{(m_{s})} = 0^{m_{s}}", "derivation": "\\mathbf{g}{(m_{s})} = \\frac{d}{d m_{s}} \\cos{(m_{s})} and \\dot{y}{(m_{s})} = \\mathbf{g}{(m_{s})} + \\sin{(m_{s})} and \\dot{y}^{m_{s}}{(m_{s})} = (\\mathbf{g}{(m_{s})} + \\sin{(m_{s})})^{m_{s}} and \\dot{y}^{m_{s}}{(m_{s})} = (\\sin{(m_{s})} + \\frac{d}{d m_{s}} \\cos{(m_{s})})^{m_{s}} and \\dot{y}^{m_{s}}{(m_{s})} = 0^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)), Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('m_s', commutative=True)), Add(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Add(Function('\\\\mathbf{g}')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\dot{y}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Add(sin(Symbol('m_s', commutative=True)), Derivative(cos(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Function('\\\\dot{y}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Integer(0), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given g{(z)} = e^{\\cos{(z)}}, then obtain - \\frac{\\cos{(z)}}{g{(z)}} = \\frac{- (z g{(z)})^{z} + (z e^{\\cos{(z)}})^{z} - \\cos{(z)}}{g{(z)}}", "derivation": "g{(z)} = e^{\\cos{(z)}} and z g{(z)} = z e^{\\cos{(z)}} and (z g{(z)})^{z} = (z e^{\\cos{(z)}})^{z} and (z g{(z)})^{z} - \\cos{(z)} = (z e^{\\cos{(z)}})^{z} - \\cos{(z)} and - \\cos{(z)} = - (z g{(z)})^{z} + (z e^{\\cos{(z)}})^{z} - \\cos{(z)} and - \\frac{\\cos{(z)}}{g{(z)}} = \\frac{- (z g{(z)})^{z} + (z e^{\\cos{(z)}})^{z} - \\cos{(z)}}{g{(z)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('z', commutative=True)), exp(cos(Symbol('z', commutative=True))))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), exp(cos(Symbol('z', commutative=True)))))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Mul(Symbol('z', commutative=True), exp(cos(Symbol('z', commutative=True)))), Symbol('z', commutative=True)))"], [["minus", 3, "cos(Symbol('z', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Add(Pow(Mul(Symbol('z', commutative=True), exp(cos(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["minus", 4, "Pow(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Mul(Integer(-1), cos(Symbol('z', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Pow(Mul(Symbol('z', commutative=True), exp(cos(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))))"], [["divide", 5, "Function('g')(Symbol('z', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('g')(Symbol('z', commutative=True)), Integer(-1)), cos(Symbol('z', commutative=True))), Mul(Add(Mul(Integer(-1), Pow(Mul(Symbol('z', commutative=True), Function('g')(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Pow(Mul(Symbol('z', commutative=True), exp(cos(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Pow(Function('g')(Symbol('z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(I,P_{g})} = P_{g}^{I} and \\mathbf{J}_P{(I,P_{g})} = \\frac{\\partial}{\\partial I} P_{g}^{I}, then derive (P_{g}^{I} \\log{(P_{g})})^{I} = \\mathbf{J}_P^{I}{(I,P_{g})}, then obtain ((P_{g}^{I} \\log{(P_{g})})^{I})^{I} = (\\mathbf{J}_P^{I}{(I,P_{g})})^{I}", "derivation": "\\operatorname{v_{2}}{(I,P_{g})} = P_{g}^{I} and \\frac{\\partial}{\\partial I} \\operatorname{v_{2}}{(I,P_{g})} = \\frac{\\partial}{\\partial I} P_{g}^{I} and \\mathbf{J}_P{(I,P_{g})} = \\frac{\\partial}{\\partial I} P_{g}^{I} and \\frac{\\partial}{\\partial I} \\operatorname{v_{2}}{(I,P_{g})} = \\mathbf{J}_P{(I,P_{g})} and (\\frac{\\partial}{\\partial I} \\operatorname{v_{2}}{(I,P_{g})})^{I} = \\mathbf{J}_P^{I}{(I,P_{g})} and (\\frac{\\partial}{\\partial I} P_{g}^{I})^{I} = \\mathbf{J}_P^{I}{(I,P_{g})} and (P_{g}^{I} \\log{(P_{g})})^{I} = \\mathbf{J}_P^{I}{(I,P_{g})} and ((P_{g}^{I} \\log{(P_{g})})^{I})^{I} = (\\mathbf{J}_P^{I}{(I,P_{g})})^{I}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Derivative(Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('v_2')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Derivative(Function('v_2')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Derivative(Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Symbol('I', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Pow(Mul(Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)), log(Symbol('P_g', commutative=True))), Symbol('I', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Symbol('I', commutative=True)))"], [["power", 7, "Symbol('I', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('P_g', commutative=True), Symbol('I', commutative=True)), log(Symbol('P_g', commutative=True))), Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Pow(Function('\\\\mathbf{J}_P')(Symbol('I', commutative=True), Symbol('P_g', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(G)} = \\log{(\\cos{(G)})} and \\mathbf{E}{(G)} = \\log{(\\cos{(G)})}, then obtain - \\phi_{2}{(G)} + \\operatorname{f^{\\prime}}{(G)} - \\cos{(G)} = \\mathbf{E}{(G)} - \\phi_{2}{(G)} - \\cos{(G)}", "derivation": "\\operatorname{f^{\\prime}}{(G)} = \\log{(\\cos{(G)})} and \\operatorname{f^{\\prime}}{(G)} - \\cos{(G)} = \\log{(\\cos{(G)})} - \\cos{(G)} and \\mathbf{E}{(G)} = \\log{(\\cos{(G)})} and \\mathbf{E}{(G)} = \\operatorname{f^{\\prime}}{(G)} and \\mathbf{E}{(G)} - \\cos{(G)} = \\log{(\\cos{(G)})} - \\cos{(G)} and \\mathbf{E}{(G)} - \\phi_{2}{(G)} - \\cos{(G)} = - \\phi_{2}{(G)} + \\log{(\\cos{(G)})} - \\cos{(G)} and - \\phi_{2}{(G)} + \\operatorname{f^{\\prime}}{(G)} - \\cos{(G)} = - \\phi_{2}{(G)} + \\log{(\\cos{(G)})} - \\cos{(G)} and - \\phi_{2}{(G)} + \\operatorname{f^{\\prime}}{(G)} - \\cos{(G)} = \\mathbf{E}{(G)} - \\phi_{2}{(G)} - \\cos{(G)}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('G', commutative=True)), log(cos(Symbol('G', commutative=True))))"], [["minus", 1, "cos(Symbol('G', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('G', commutative=True)), Mul(Integer(-1), cos(Symbol('G', commutative=True)))), Add(log(cos(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('G', commutative=True)), log(cos(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{E}')(Symbol('G', commutative=True)), Function('f^{\\\\prime}')(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('G', commutative=True)), Mul(Integer(-1), cos(Symbol('G', commutative=True)))), Add(log(cos(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))))"], [["minus", 5, "Function('\\\\phi_2')(Symbol('G', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('G', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), log(cos(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), Function('f^{\\\\prime}')(Symbol('G', commutative=True)), Mul(Integer(-1), cos(Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), log(cos(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), Function('f^{\\\\prime}')(Symbol('G', commutative=True)), Mul(Integer(-1), cos(Symbol('G', commutative=True)))), Add(Function('\\\\mathbf{E}')(Symbol('G', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('G', commutative=True))), Mul(Integer(-1), cos(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(S,\\omega)} = - S + \\omega, then derive \\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)} = 1, then obtain \\frac{\\partial}{\\partial \\omega} e^{\\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)}} = \\frac{d}{d \\omega} e", "derivation": "\\mathbf{H}{(S,\\omega)} = - S + \\omega and \\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)} = \\frac{\\partial}{\\partial \\omega} (- S + \\omega) and \\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)} = 1 and e^{\\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)}} = e and \\frac{\\partial}{\\partial \\omega} e^{\\frac{\\partial}{\\partial \\omega} \\mathbf{H}{(S,\\omega)}} = \\frac{d}{d \\omega} e", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1))"], [["exp", 3], "Equality(exp(Derivative(Function('\\\\mathbf{H}')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), E)"], [["differentiate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(exp(Derivative(Function('\\\\mathbf{H}')(Symbol('S', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(E, Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)}, then obtain - \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin^{4}{(\\mathbf{J}_f)} = - \\sin^{5}{(\\mathbf{J}_f)}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{J}_f)} = \\sin{(\\mathbf{J}_f)} and \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin{(\\mathbf{J}_f)} = \\sin^{2}{(\\mathbf{J}_f)} and - \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin{(\\mathbf{J}_f)} = - \\sin^{2}{(\\mathbf{J}_f)} and - \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin^{2}{(\\mathbf{J}_f)} = - \\sin^{3}{(\\mathbf{J}_f)} and - \\operatorname{m_{s}}^{2}{(\\mathbf{J}_f)} \\sin{(\\mathbf{J}_f)} = - \\sin^{3}{(\\mathbf{J}_f)} and - \\operatorname{m_{s}}^{3}{(\\mathbf{J}_f)} \\sin^{2}{(\\mathbf{J}_f)} = - \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin^{4}{(\\mathbf{J}_f)} and - \\operatorname{m_{s}}{(\\mathbf{J}_f)} \\sin^{4}{(\\mathbf{J}_f)} = - \\sin^{5}{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))))"], [["times", 1, "Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2)), sin(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(3))))"], [["times", 5, "Mul(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), sin(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(3)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(2))), Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(4))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(4))), Mul(Integer(-1), Pow(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(5))))"]]}, {"prompt": "Given \\varphi{(\\theta_1)} = \\sin{(\\theta_1)} and l{(\\theta_1)} = - \\theta_1, then obtain \\frac{d}{d \\theta_1} (\\varphi{(\\theta_1)} + l{(\\theta_1)}) = \\frac{d}{d \\theta_1} (l{(\\theta_1)} + \\sin{(\\theta_1)})", "derivation": "\\varphi{(\\theta_1)} = \\sin{(\\theta_1)} and - \\theta_1 + \\varphi{(\\theta_1)} = - \\theta_1 + \\sin{(\\theta_1)} and \\frac{d}{d \\theta_1} (- \\theta_1 + \\varphi{(\\theta_1)}) = \\frac{d}{d \\theta_1} (- \\theta_1 + \\sin{(\\theta_1)}) and l{(\\theta_1)} = - \\theta_1 and \\frac{d}{d \\theta_1} (\\varphi{(\\theta_1)} + l{(\\theta_1)}) = \\frac{d}{d \\theta_1} (l{(\\theta_1)} + \\sin{(\\theta_1)})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varphi')(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\varphi')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Function('\\\\varphi')(Symbol('\\\\theta_1', commutative=True)), Function('l')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Function('l')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and c{(\\tilde{g}^*)} = - \\mathbf{J}_f{(\\tilde{g}^*)}, then obtain - \\mathbf{J}_f{(\\tilde{g}^*)} = - 2 \\mathbf{J}_f{(\\tilde{g}^*)} + e^{\\tilde{g}^*}", "derivation": "\\mathbf{J}_f{(\\tilde{g}^*)} = e^{\\tilde{g}^*} and 0 = - \\mathbf{J}_f{(\\tilde{g}^*)} + e^{\\tilde{g}^*} and c{(\\tilde{g}^*)} = - \\mathbf{J}_f{(\\tilde{g}^*)} and 0 = c{(\\tilde{g}^*)} + e^{\\tilde{g}^*} and - \\mathbf{J}_f{(\\tilde{g}^*)} = - \\mathbf{J}_f{(\\tilde{g}^*)} + c{(\\tilde{g}^*)} + e^{\\tilde{g}^*} and - \\mathbf{J}_f{(\\tilde{g}^*)} + c{(\\tilde{g}^*)} = - 2 \\mathbf{J}_f{(\\tilde{g}^*)} and - \\mathbf{J}_f{(\\tilde{g}^*)} = - 2 \\mathbf{J}_f{(\\tilde{g}^*)} + e^{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('c')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), Function('c')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), Function('c')(Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('\\\\tilde{g}^*', commutative=True))), exp(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(T,m)} = \\frac{m}{T}, then obtain \\sin{(\\frac{m}{T})} + \\frac{\\sin{(\\sigma_{x}{(T,m)})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}} = \\sin{(\\frac{m}{T})} + \\frac{\\sin{(\\frac{m}{T})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}}", "derivation": "\\sigma_{x}{(T,m)} = \\frac{m}{T} and \\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)} = \\frac{\\partial}{\\partial m} \\frac{m}{T} and \\sin{(\\sigma_{x}{(T,m)})} = \\sin{(\\frac{m}{T})} and \\frac{\\sin{(\\sigma_{x}{(T,m)})}}{\\frac{\\partial}{\\partial m} \\frac{m}{T}} = \\frac{\\sin{(\\frac{m}{T})}}{\\frac{\\partial}{\\partial m} \\frac{m}{T}} and \\frac{\\sin{(\\sigma_{x}{(T,m)})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}} = \\frac{\\sin{(\\frac{m}{T})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}} and \\sin{(\\frac{m}{T})} + \\frac{\\sin{(\\sigma_{x}{(T,m)})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}} = \\sin{(\\frac{m}{T})} + \\frac{\\sin{(\\frac{m}{T})}}{\\frac{\\partial}{\\partial m} \\sigma_{x}{(T,m)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["sin", 1], "Equality(sin(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True))), sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["divide", 3, "Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Mul(sin(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Pow(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(sin(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Pow(Derivative(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Derivative(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))))"], [["add", 5, "sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], "Equality(Add(sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Mul(sin(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True))), Pow(Derivative(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)))), Add(sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Mul(sin(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Derivative(Function('\\\\sigma_x')(Symbol('T', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(f)} = f, then obtain 1 = (- f (f - \\operatorname{J_{\\varepsilon}}{(f)}))^{f}", "derivation": "\\operatorname{J_{\\varepsilon}}{(f)} = f and - f + \\operatorname{J_{\\varepsilon}}{(f)} = 0 and (- f + \\operatorname{J_{\\varepsilon}}{(f)})^{f} = 0^{f} and 0 = f - \\operatorname{J_{\\varepsilon}}{(f)} and 0 = - f (f - \\operatorname{J_{\\varepsilon}}{(f)}) and 0^{f} = (- f (f - \\operatorname{J_{\\varepsilon}}{(f)}))^{f} and 1 = (- f (f - \\operatorname{J_{\\varepsilon}}{(f)}))^{f}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True)), Symbol('f', commutative=True))"], [["minus", 1, "Symbol('f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))), Integer(0))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integer(0), Symbol('f', commutative=True)))"], [["minus", 1, "Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))"], "Equality(Integer(0), Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Symbol('f', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))))))"], [["power", 5, "Symbol('f', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f', commutative=True)), Pow(Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integer(1), Pow(Mul(Integer(-1), Symbol('f', commutative=True), Add(Symbol('f', commutative=True), Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('f', commutative=True))))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(F_{g})} = e^{F_{g}}, then obtain \\frac{\\frac{d}{d F_{g}} \\phi_{1}{(F_{g})}}{\\phi_{1}{(F_{g})}} = 1", "derivation": "\\phi_{1}{(F_{g})} = e^{F_{g}} and \\log{(\\phi_{1}{(F_{g})})} = \\log{(e^{F_{g}})} and \\frac{d}{d F_{g}} \\log{(\\phi_{1}{(F_{g})})} = \\frac{d}{d F_{g}} \\log{(e^{F_{g}})} and \\frac{\\frac{d}{d F_{g}} \\phi_{1}{(F_{g})}}{\\phi_{1}{(F_{g})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\phi_1')(Symbol('F_g', commutative=True))), log(exp(Symbol('F_g', commutative=True))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(log(Function('\\\\phi_1')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(log(exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\lambda,\\theta_2)} = \\theta_2^{\\lambda} and f{(\\lambda)} = \\lambda, then obtain \\lambda^{2} + f{(\\lambda)} \\operatorname{r_{0}}{(\\lambda,\\theta_2)} = \\lambda^{2} + \\lambda \\operatorname{r_{0}}{(\\lambda,\\theta_2)}", "derivation": "\\operatorname{r_{0}}{(\\lambda,\\theta_2)} = \\theta_2^{\\lambda} and f{(\\lambda)} = \\lambda and \\theta_2^{\\lambda} f{(\\lambda)} = \\lambda \\theta_2^{\\lambda} and f{(\\lambda)} \\operatorname{r_{0}}{(\\lambda,\\theta_2)} = \\lambda \\operatorname{r_{0}}{(\\lambda,\\theta_2)} and \\lambda^{2} + f{(\\lambda)} \\operatorname{r_{0}}{(\\lambda,\\theta_2)} = \\lambda^{2} + \\lambda \\operatorname{r_{0}}{(\\lambda,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["times", 2, "Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\lambda', commutative=True)), Function('f')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('f')(Symbol('\\\\lambda', commutative=True)), Function('r_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Function('r_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["add", 4, "Pow(Symbol('\\\\lambda', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('\\\\lambda', commutative=True), Integer(2)), Mul(Function('f')(Symbol('\\\\lambda', commutative=True)), Function('r_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Pow(Symbol('\\\\lambda', commutative=True), Integer(2)), Mul(Symbol('\\\\lambda', commutative=True), Function('r_0')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(A_{y},\\Psi_{nl})} = \\Psi_{nl} + \\log{(A_{y})}, then derive \\int \\nabla{(A_{y},\\Psi_{nl})} d\\Psi_{nl} = \\frac{\\Psi_{nl}^{2}}{2} + \\Psi_{nl} \\log{(A_{y})} + \\mathbf{s}, then obtain \\frac{\\int \\nabla{(A_{y},\\Psi_{nl})} d\\Psi_{nl}}{\\frac{\\Psi_{nl}^{2}}{2} + \\Psi_{nl} \\log{(A_{y})} + \\mathbf{s}} = 1", "derivation": "\\nabla{(A_{y},\\Psi_{nl})} = \\Psi_{nl} + \\log{(A_{y})} and \\int \\nabla{(A_{y},\\Psi_{nl})} d\\Psi_{nl} = \\int (\\Psi_{nl} + \\log{(A_{y})}) d\\Psi_{nl} and \\int \\nabla{(A_{y},\\Psi_{nl})} d\\Psi_{nl} = \\frac{\\Psi_{nl}^{2}}{2} + \\Psi_{nl} \\log{(A_{y})} + \\mathbf{s} and \\frac{\\int \\nabla{(A_{y},\\Psi_{nl})} d\\Psi_{nl}}{\\frac{\\Psi_{nl}^{2}}{2} + \\Psi_{nl} \\log{(A_{y})} + \\mathbf{s}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), log(Symbol('A_y', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Add(Symbol('\\\\Psi_{nl}', commutative=True), log(Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), log(Symbol('A_y', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 3, "Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), log(Symbol('A_y', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi_{nl}', commutative=True), log(Symbol('A_y', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Function('\\\\nabla')(Symbol('A_y', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\Psi{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain 1 = \\frac{4 \\mathbf{S} + \\Psi{(\\mathbf{S})} + 3 e^{\\mathbf{S}}}{4 \\mathbf{S} + 2 \\Psi{(\\mathbf{S})} + 2 e^{\\mathbf{S}}}", "derivation": "\\Psi{(\\mathbf{S})} = e^{\\mathbf{S}} and \\mathbf{S} + \\Psi{(\\mathbf{S})} = \\mathbf{S} + e^{\\mathbf{S}} and 2 \\mathbf{S} + \\Psi{(\\mathbf{S})} + e^{\\mathbf{S}} = 2 \\mathbf{S} + 2 e^{\\mathbf{S}} and 4 \\mathbf{S} + 2 \\Psi{(\\mathbf{S})} + 2 e^{\\mathbf{S}} = 4 \\mathbf{S} + \\Psi{(\\mathbf{S})} + 3 e^{\\mathbf{S}} and 1 = \\frac{4 \\mathbf{S} + \\Psi{(\\mathbf{S})} + 3 e^{\\mathbf{S}}}{4 \\mathbf{S} + 2 \\Psi{(\\mathbf{S})} + 2 e^{\\mathbf{S}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(4), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(4), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(4), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], "Equality(Integer(1), Mul(Add(Mul(Integer(4), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Pow(Add(Mul(Integer(4), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{S}', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given i{(\\rho_f,F_{H})} = \\frac{F_{H}}{\\rho_f}, then derive \\frac{\\partial}{\\partial \\rho_f} i{(\\rho_f,F_{H})} = - \\frac{F_{H}}{\\rho_f^{2}}, then obtain - \\frac{i{(\\rho_f,F_{H})}}{\\rho_f^{3}} = - \\frac{F_{H}}{\\rho_f^{4}}", "derivation": "i{(\\rho_f,F_{H})} = \\frac{F_{H}}{\\rho_f} and \\frac{\\partial}{\\partial \\rho_f} i{(\\rho_f,F_{H})} = \\frac{\\partial}{\\partial \\rho_f} \\frac{F_{H}}{\\rho_f} and \\frac{\\partial}{\\partial \\rho_f} i{(\\rho_f,F_{H})} = - \\frac{F_{H}}{\\rho_f^{2}} and \\frac{\\partial}{\\partial \\rho_f} i{(\\rho_f,F_{H})} = - \\frac{i{(\\rho_f,F_{H})}}{\\rho_f} and - \\frac{i{(\\rho_f,F_{H})}}{\\rho_f} = - \\frac{F_{H}}{\\rho_f^{2}} and - \\frac{i{(\\rho_f,F_{H})}}{\\rho_f^{3}} = - \\frac{F_{H}}{\\rho_f^{4}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True))), Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-2))))"], [["times", 5, "Pow(Symbol('\\\\rho_f', commutative=True), Integer(-2))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-3)), Function('i')(Symbol('\\\\rho_f', commutative=True), Symbol('F_H', commutative=True))), Mul(Integer(-1), Symbol('F_H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-4))))"]]}, {"prompt": "Given \\theta_{2}{(I,r)} = I + r, then obtain (\\frac{\\theta_{2}{(I,r)}}{r})^{I} (I + r) = (\\frac{I + r}{r})^{I} (I + r)", "derivation": "\\theta_{2}{(I,r)} = I + r and \\frac{\\theta_{2}{(I,r)}}{r} = \\frac{I + r}{r} and (\\frac{\\theta_{2}{(I,r)}}{r})^{I} = (\\frac{I + r}{r})^{I} and (\\frac{\\theta_{2}{(I,r)}}{r})^{I} (I + r) = (\\frac{I + r}{r})^{I} (I + r)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('r', commutative=True)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True)))"], [["divide", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('r', commutative=True))), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True))), Symbol('I', commutative=True)))"], [["times", 3, "Add(Symbol('I', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('I', commutative=True), Symbol('r', commutative=True))), Symbol('I', commutative=True)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True))), Symbol('I', commutative=True)), Add(Symbol('I', commutative=True), Symbol('r', commutative=True))))"]]}, {"prompt": "Given r{(C_{d})} = \\log{(C_{d})}, then derive \\frac{\\frac{d}{d C_{d}} r{(C_{d})}}{r{(C_{d})}} = \\frac{1}{C_{d} r{(C_{d})}}, then obtain \\frac{\\frac{d}{d C_{d}} \\log{(C_{d})}}{r{(C_{d})}} = \\frac{1}{C_{d} r{(C_{d})}}", "derivation": "r{(C_{d})} = \\log{(C_{d})} and \\frac{d}{d C_{d}} r{(C_{d})} = \\frac{d}{d C_{d}} \\log{(C_{d})} and \\frac{\\frac{d}{d C_{d}} r{(C_{d})}}{r{(C_{d})}} = \\frac{\\frac{d}{d C_{d}} \\log{(C_{d})}}{r{(C_{d})}} and \\frac{\\frac{d}{d C_{d}} r{(C_{d})}}{r{(C_{d})}} = \\frac{1}{C_{d} r{(C_{d})}} and \\frac{\\frac{d}{d C_{d}} \\log{(C_{d})}}{r{(C_{d})}} = \\frac{1}{C_{d} r{(C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C_d', commutative=True)), log(Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["divide", 2, "Function('r')(Symbol('C_d', commutative=True))"], "Equality(Mul(Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(Function('r')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(log(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('C_d', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\delta)} = \\int e^{\\delta} d\\delta and J{(\\delta)} = e^{\\delta} + \\int e^{\\delta} d\\delta, then obtain \\frac{\\operatorname{t_{2}}{(\\delta)} + e^{\\delta}}{\\operatorname{t_{2}}{(\\delta)}} - \\int e^{\\delta} d\\delta = \\frac{J{(\\delta)}}{\\operatorname{t_{2}}{(\\delta)}} - \\int e^{\\delta} d\\delta", "derivation": "\\operatorname{t_{2}}{(\\delta)} = \\int e^{\\delta} d\\delta and \\operatorname{t_{2}}{(\\delta)} + e^{\\delta} = e^{\\delta} + \\int e^{\\delta} d\\delta and J{(\\delta)} = e^{\\delta} + \\int e^{\\delta} d\\delta and \\operatorname{t_{2}}{(\\delta)} + e^{\\delta} = J{(\\delta)} and \\frac{\\operatorname{t_{2}}{(\\delta)} + e^{\\delta}}{\\operatorname{t_{2}}{(\\delta)}} = \\frac{J{(\\delta)}}{\\operatorname{t_{2}}{(\\delta)}} and \\frac{\\operatorname{t_{2}}{(\\delta)} + e^{\\delta}}{\\operatorname{t_{2}}{(\\delta)}} - \\int e^{\\delta} d\\delta = \\frac{J{(\\delta)}}{\\operatorname{t_{2}}{(\\delta)}} - \\int e^{\\delta} d\\delta", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\delta', commutative=True)), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["add", 1, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('t_2')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Add(exp(Symbol('\\\\delta', commutative=True)), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\delta', commutative=True)), Add(exp(Symbol('\\\\delta', commutative=True)), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('t_2')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Function('J')(Symbol('\\\\delta', commutative=True)))"], [["divide", 4, "Function('t_2')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Add(Function('t_2')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Pow(Function('t_2')(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Function('J')(Symbol('\\\\delta', commutative=True)), Pow(Function('t_2')(Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["minus", 5, "Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Add(Function('t_2')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Pow(Function('t_2')(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Integer(-1), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Add(Mul(Function('J')(Symbol('\\\\delta', commutative=True)), Pow(Function('t_2')(Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Integer(-1), Integral(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"]]}, {"prompt": "Given s{(\\Omega)} = e^{\\cos{(\\Omega)}}, then obtain \\int (\\frac{d}{d \\Omega} \\frac{s{(\\Omega)}}{\\Omega} + \\int s{(\\Omega)} d\\Omega) d\\Omega = \\int (\\frac{d}{d \\Omega} \\frac{e^{\\cos{(\\Omega)}}}{\\Omega} + \\int s{(\\Omega)} d\\Omega) d\\Omega", "derivation": "s{(\\Omega)} = e^{\\cos{(\\Omega)}} and \\frac{s{(\\Omega)}}{\\Omega} = \\frac{e^{\\cos{(\\Omega)}}}{\\Omega} and \\frac{d}{d \\Omega} \\frac{s{(\\Omega)}}{\\Omega} = \\frac{d}{d \\Omega} \\frac{e^{\\cos{(\\Omega)}}}{\\Omega} and \\frac{d}{d \\Omega} \\frac{s{(\\Omega)}}{\\Omega} + \\int s{(\\Omega)} d\\Omega = \\frac{d}{d \\Omega} \\frac{e^{\\cos{(\\Omega)}}}{\\Omega} + \\int s{(\\Omega)} d\\Omega and \\int (\\frac{d}{d \\Omega} \\frac{s{(\\Omega)}}{\\Omega} + \\int s{(\\Omega)} d\\Omega) d\\Omega = \\int (\\frac{d}{d \\Omega} \\frac{e^{\\cos{(\\Omega)}}}{\\Omega} + \\int s{(\\Omega)} d\\Omega) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\Omega', commutative=True)), exp(cos(Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(cos(Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(cos(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["add", 3, "Integral(Function('s')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Function('s')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(cos(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Function('s')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Function('s')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(cos(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integral(Function('s')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(x^\\prime)} = \\sin{(\\sin{(x^\\prime)})} and \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})}, then obtain - \\mathbf{H}{(x^\\prime)} + \\frac{d}{d x^\\prime} \\mathbf{H}{(x^\\prime)} = - \\mathbf{H}{(x^\\prime)} + \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})}", "derivation": "\\mathbf{H}{(x^\\prime)} = \\sin{(\\sin{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\mathbf{H}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})} and \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})} and \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\mathbf{H}{(x^\\prime)} and - \\mathbf{H}{(x^\\prime)} + \\operatorname{f_{\\mathbf{v}}}{(x^\\prime)} = - \\mathbf{H}{(x^\\prime)} + \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})} and - \\mathbf{H}{(x^\\prime)} + \\frac{d}{d x^\\prime} \\mathbf{H}{(x^\\prime)} = - \\mathbf{H}{(x^\\prime)} + \\frac{d}{d x^\\prime} \\sin{(\\sin{(x^\\prime)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)), sin(sin(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Derivative(sin(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 3, "Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))), Function('f_{\\\\mathbf{v}}')(Symbol('x^\\\\prime', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))), Derivative(sin(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))), Derivative(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True))), Derivative(sin(sin(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(A_{1})} = e^{A_{1}}, then derive (\\frac{\\int \\mathbf{F}{(A_{1})} dA_{1}}{y^{\\prime} + e^{A_{1}}})^{A_{1}} = 1, then obtain (\\frac{\\int e^{A_{1}} dA_{1}}{y^{\\prime} + e^{A_{1}}})^{A_{1}} = 1", "derivation": "\\mathbf{F}{(A_{1})} = e^{A_{1}} and \\int \\mathbf{F}{(A_{1})} dA_{1} = \\int e^{A_{1}} dA_{1} and \\frac{\\int \\mathbf{F}{(A_{1})} dA_{1}}{\\int e^{A_{1}} dA_{1}} = 1 and (\\frac{\\int \\mathbf{F}{(A_{1})} dA_{1}}{\\int e^{A_{1}} dA_{1}})^{A_{1}} = 1 and (\\frac{\\int \\mathbf{F}{(A_{1})} dA_{1}}{y^{\\prime} + e^{A_{1}}})^{A_{1}} = 1 and (\\frac{\\int e^{A_{1}} dA_{1}}{y^{\\prime} + e^{A_{1}}})^{A_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('A_1', commutative=True)), exp(Symbol('A_1', commutative=True)))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["divide", 2, "Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{F}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Pow(Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Integral(Function('\\\\mathbf{F}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Pow(Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integer(-1))), Symbol('A_1', commutative=True)), Integer(1))"], [["evaluate_integrals", 4], "Equality(Pow(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('A_1', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{F}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(Pow(Add(Symbol('y^{\\\\prime}', commutative=True), exp(Symbol('A_1', commutative=True))), Integer(-1)), Integral(exp(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\varepsilon{(m,\\mathbf{J}_P)} = \\mathbf{J}_P + m, then obtain \\mathbf{J}_P + m \\sin{(\\varepsilon{(m,\\mathbf{J}_P)})} = \\mathbf{J}_P + m \\sin{(\\mathbf{J}_P + m)}", "derivation": "\\varepsilon{(m,\\mathbf{J}_P)} = \\mathbf{J}_P + m and \\sin{(\\varepsilon{(m,\\mathbf{J}_P)})} = \\sin{(\\mathbf{J}_P + m)} and m \\sin{(\\varepsilon{(m,\\mathbf{J}_P)})} = m \\sin{(\\mathbf{J}_P + m)} and \\mathbf{J}_P + m \\sin{(\\varepsilon{(m,\\mathbf{J}_P)})} = \\mathbf{J}_P + m \\sin{(\\mathbf{J}_P + m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\varepsilon')(Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m', commutative=True))))"], [["times", 2, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), sin(Function('\\\\varepsilon')(Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Symbol('m', commutative=True), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Symbol('m', commutative=True), sin(Function('\\\\varepsilon')(Symbol('m', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Symbol('m', commutative=True), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(\\mathbf{f},v_{t},L)} = L v_{t} + \\mathbf{f}, then obtain - \\nabla^{L}{(\\mathbf{f},v_{t},L)} + \\int \\frac{\\partial}{\\partial v_{t}} \\nabla^{L}{(\\mathbf{f},v_{t},L)} d\\mathbf{f} = - \\nabla^{L}{(\\mathbf{f},v_{t},L)} + \\int \\frac{\\partial}{\\partial v_{t}} (L v_{t} + \\mathbf{f})^{L} d\\mathbf{f}", "derivation": "\\nabla{(\\mathbf{f},v_{t},L)} = L v_{t} + \\mathbf{f} and \\nabla^{L}{(\\mathbf{f},v_{t},L)} = (L v_{t} + \\mathbf{f})^{L} and \\frac{\\partial}{\\partial v_{t}} \\nabla^{L}{(\\mathbf{f},v_{t},L)} = \\frac{\\partial}{\\partial v_{t}} (L v_{t} + \\mathbf{f})^{L} and \\int \\frac{\\partial}{\\partial v_{t}} \\nabla^{L}{(\\mathbf{f},v_{t},L)} d\\mathbf{f} = \\int \\frac{\\partial}{\\partial v_{t}} (L v_{t} + \\mathbf{f})^{L} d\\mathbf{f} and - \\nabla^{L}{(\\mathbf{f},v_{t},L)} + \\int \\frac{\\partial}{\\partial v_{t}} \\nabla^{L}{(\\mathbf{f},v_{t},L)} d\\mathbf{f} = - \\nabla^{L}{(\\mathbf{f},v_{t},L)} + \\int \\frac{\\partial}{\\partial v_{t}} (L v_{t} + \\mathbf{f})^{L} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Add(Mul(Symbol('L', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Add(Mul(Symbol('L', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('L', commutative=True)))"], [["differentiate", 2, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('L', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Derivative(Pow(Add(Mul(Symbol('L', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 4, "Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Integral(Derivative(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('v_t', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Integral(Derivative(Pow(Add(Mul(Symbol('L', commutative=True), Symbol('v_t', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and k{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})}, then obtain \\frac{d}{d \\Psi_{\\lambda}} \\hat{H}_{\\lambda}{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} k{(\\Psi_{\\lambda})}", "derivation": "\\hat{H}_{\\lambda}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and k{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\hat{H}_{\\lambda}{(\\Psi_{\\lambda})} = k{(\\Psi_{\\lambda})} and \\frac{d}{d \\Psi_{\\lambda}} \\hat{H}_{\\lambda}{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} k{(\\Psi_{\\lambda})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(a)} = \\log{(a)} and \\operatorname{A_{1}}{(a)} = \\log{(a)}, then obtain \\iint \\operatorname{A_{1}}{(a)} da da = \\iint \\log{(a)} da da", "derivation": "\\mathbf{f}{(a)} = \\log{(a)} and \\int \\mathbf{f}{(a)} da = \\int \\log{(a)} da and \\operatorname{A_{1}}{(a)} = \\log{(a)} and \\operatorname{A_{1}}{(a)} = \\mathbf{f}{(a)} and \\iint \\mathbf{f}{(a)} da da = \\iint \\log{(a)} da da and \\iint \\operatorname{A_{1}}{(a)} da da = \\iint \\log{(a)} da da", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_1')(Symbol('a', commutative=True)), Function('\\\\mathbf{f}')(Symbol('a', commutative=True)))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Function('A_1')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(y^{\\prime},L)} = \\cos{(L y^{\\prime})}, then obtain \\mathbf{J}{(y^{\\prime},L)} \\int 1 dL = \\mathbf{J}{(y^{\\prime},L)} \\int \\frac{\\cos{(L y^{\\prime})}}{\\mathbf{J}{(y^{\\prime},L)}} dL", "derivation": "\\mathbf{J}{(y^{\\prime},L)} = \\cos{(L y^{\\prime})} and 1 = \\frac{\\cos{(L y^{\\prime})}}{\\mathbf{J}{(y^{\\prime},L)}} and \\int 1 dL = \\int \\frac{\\cos{(L y^{\\prime})}}{\\mathbf{J}{(y^{\\prime},L)}} dL and \\cos{(L y^{\\prime})} \\int 1 dL = \\cos{(L y^{\\prime})} \\int \\frac{\\cos{(L y^{\\prime})}}{\\mathbf{J}{(y^{\\prime},L)}} dL and \\mathbf{J}{(y^{\\prime},L)} \\int 1 dL = \\mathbf{J}{(y^{\\prime},L)} \\int \\frac{\\cos{(L y^{\\prime})}}{\\mathbf{J}{(y^{\\prime},L)}} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('L', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('L', commutative=True))))"], [["times", 3, "cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Mul(cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Integer(1), Tuple(Symbol('L', commutative=True)))), Mul(cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integral(Integer(1), Tuple(Symbol('L', commutative=True)))), Mul(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Integer(-1)), cos(Mul(Symbol('L', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(P_{e},y)} = e^{P_{e} y}, then obtain \\frac{y \\operatorname{E_{n}}^{y}{(P_{e},y)} \\frac{\\partial}{\\partial P_{e}} \\operatorname{E_{n}}{(P_{e},y)}}{\\operatorname{E_{n}}{(P_{e},y)}} = y^{2} (e^{P_{e} y})^{y}", "derivation": "\\operatorname{E_{n}}{(P_{e},y)} = e^{P_{e} y} and \\operatorname{E_{n}}^{y}{(P_{e},y)} = (e^{P_{e} y})^{y} and \\frac{\\partial}{\\partial P_{e}} \\operatorname{E_{n}}^{y}{(P_{e},y)} = \\frac{\\partial}{\\partial P_{e}} (e^{P_{e} y})^{y} and \\frac{y \\operatorname{E_{n}}^{y}{(P_{e},y)} \\frac{\\partial}{\\partial P_{e}} \\operatorname{E_{n}}{(P_{e},y)}}{\\operatorname{E_{n}}{(P_{e},y)}} = y^{2} (e^{P_{e} y})^{y}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), exp(Mul(Symbol('P_e', commutative=True), Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(Mul(Symbol('P_e', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Pow(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Pow(exp(Mul(Symbol('P_e', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('y', commutative=True), Pow(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Pow(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Derivative(Function('E_n')(Symbol('P_e', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Symbol('y', commutative=True), Integer(2)), Pow(exp(Mul(Symbol('P_e', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then derive \\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger} = V - \\cos{(a^{\\dagger})}, then obtain (- \\frac{\\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger}}{\\cos{(\\tilde{g})}})^{\\tilde{g}} = (\\frac{- V + \\cos{(a^{\\dagger})}}{\\cos{(\\tilde{g})}})^{\\tilde{g}}", "derivation": "\\mathbf{f}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger} = \\int \\sin{(a^{\\dagger})} da^{\\dagger} and \\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger} = V - \\cos{(a^{\\dagger})} and - \\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger} = - V + \\cos{(a^{\\dagger})} and - \\frac{\\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger}}{\\cos{(\\tilde{g})}} = \\frac{- V + \\cos{(a^{\\dagger})}}{\\cos{(\\tilde{g})}} and (- \\frac{\\int \\mathbf{f}{(a^{\\dagger})} da^{\\dagger}}{\\cos{(\\tilde{g})}})^{\\tilde{g}} = (\\frac{- V + \\cos{(a^{\\dagger})}}{\\cos{(\\tilde{g})}})^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 4, "cos(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{f}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))), Pow(cos(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given I{(C)} = \\sin{(C)}, then obtain \\frac{d}{d C} (I^{2}{(C)} + I{(C)} \\sin{(C)} + \\frac{I^{2}{(C)}}{C}) = \\frac{d}{d C} (2 I{(C)} \\sin{(C)} + \\frac{I^{2}{(C)}}{C})", "derivation": "I{(C)} = \\sin{(C)} and I^{2}{(C)} = I{(C)} \\sin{(C)} and I^{2}{(C)} + I{(C)} \\sin{(C)} = 2 I{(C)} \\sin{(C)} and \\frac{I^{2}{(C)}}{C} = \\frac{I{(C)} \\sin{(C)}}{C} and I^{2}{(C)} + I{(C)} \\sin{(C)} + \\frac{I{(C)} \\sin{(C)}}{C} = 2 I{(C)} \\sin{(C)} + \\frac{I{(C)} \\sin{(C)}}{C} and \\frac{d}{d C} (I^{2}{(C)} + I{(C)} \\sin{(C)} + \\frac{I{(C)} \\sin{(C)}}{C}) = \\frac{d}{d C} (2 I{(C)} \\sin{(C)} + \\frac{I{(C)} \\sin{(C)}}{C}) and \\frac{d}{d C} (I^{2}{(C)} + I{(C)} \\sin{(C)} + \\frac{I^{2}{(C)}}{C}) = \\frac{d}{d C} (2 I{(C)} \\sin{(C)} + \\frac{I^{2}{(C)}}{C})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["times", 1, "Function('I')(Symbol('C', commutative=True))"], "Equality(Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["add", 2, "Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], "Equality(Add(Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Mul(Integer(2), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('C', commutative=True)), Integer(2))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], "Equality(Add(Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Add(Mul(Integer(2), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))))"], [["differentiate", 5, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Add(Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Function('I')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Function('I')(Symbol('C', commutative=True)), Integer(2)))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(U)} = \\int \\sin{(U)} dU, then derive n{(U)} = \\mathbf{J} - \\cos{(U)}, then obtain \\frac{\\int (\\mathbf{J} - \\cos{(U)}) dU}{\\cos{(U)}} = \\frac{\\iint \\sin{(U)} dU dU}{\\cos{(U)}}", "derivation": "n{(U)} = \\int \\sin{(U)} dU and \\int n{(U)} dU = \\iint \\sin{(U)} dU dU and n{(U)} = \\mathbf{J} - \\cos{(U)} and - \\frac{\\int n{(U)} dU}{\\cos{(U)}} = - \\frac{\\iint \\sin{(U)} dU dU}{\\cos{(U)}} and \\frac{\\int n{(U)} dU}{\\cos{(U)}} = \\frac{\\iint \\sin{(U)} dU dU}{\\cos{(U)}} and \\frac{\\int (\\mathbf{J} - \\cos{(U)}) dU}{\\cos{(U)}} = \\frac{\\iint \\sin{(U)} dU dU}{\\cos{(U)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('U', commutative=True)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('n')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n')(Symbol('U', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), cos(Symbol('U', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Function('n')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(-1), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Function('n')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), cos(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True)))), Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(G,A_{1})} = - A_{1} + G and t{(G,A_{1})} = A_{1} (- A_{1} + G), then obtain \\int A_{1} \\mathbf{D}{(G,A_{1})} dA_{1} = \\int t{(G,A_{1})} dA_{1}", "derivation": "\\mathbf{D}{(G,A_{1})} = - A_{1} + G and A_{1} \\mathbf{D}{(G,A_{1})} = A_{1} (- A_{1} + G) and t{(G,A_{1})} = A_{1} (- A_{1} + G) and A_{1} \\mathbf{D}{(G,A_{1})} = t{(G,A_{1})} and \\int A_{1} \\mathbf{D}{(G,A_{1})} dA_{1} = \\int t{(G,A_{1})} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('G', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('G', commutative=True)))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\mathbf{D}')(Symbol('G', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('G', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\mathbf{D}')(Symbol('G', commutative=True), Symbol('A_1', commutative=True))), Function('t')(Symbol('G', commutative=True), Symbol('A_1', commutative=True)))"], [["integrate", 4, "Symbol('A_1', commutative=True)"], "Equality(Integral(Mul(Symbol('A_1', commutative=True), Function('\\\\mathbf{D}')(Symbol('G', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(Function('t')(Symbol('G', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given k{(P_{e})} = \\cos{(P_{e})}, then derive \\frac{d}{d P_{e}} k{(P_{e})} = - \\sin{(P_{e})}, then obtain - \\frac{\\sin{(P_{e})}}{\\cos{(P_{e})}} = \\frac{\\frac{d}{d P_{e}} k{(P_{e})}}{\\cos{(P_{e})}}", "derivation": "k{(P_{e})} = \\cos{(P_{e})} and \\frac{d}{d P_{e}} k{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(P_{e})} and \\frac{d}{d P_{e}} k{(P_{e})} = - \\sin{(P_{e})} and \\frac{\\frac{d}{d P_{e}} k{(P_{e})}}{\\cos{(P_{e})}} = \\frac{\\frac{d}{d P_{e}} \\cos{(P_{e})}}{\\cos{(P_{e})}} and - \\frac{\\sin{(P_{e})}}{\\cos{(P_{e})}} = \\frac{\\frac{d}{d P_{e}} \\cos{(P_{e})}}{\\cos{(P_{e})}} and - \\frac{\\sin{(P_{e})}}{\\cos{(P_{e})}} = \\frac{\\frac{d}{d P_{e}} k{(P_{e})}}{\\cos{(P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["differentiate", 1, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('P_e', commutative=True))))"], [["divide", 2, "cos(Symbol('P_e', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(Function('k')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), sin(Symbol('P_e', commutative=True)), Pow(cos(Symbol('P_e', commutative=True)), Integer(-1))), Mul(Pow(cos(Symbol('P_e', commutative=True)), Integer(-1)), Derivative(Function('k')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}^{U}, then obtain \\hat{\\mathbf{x}} i{(U,\\hat{\\mathbf{x}})} + \\hat{\\mathbf{x}}^{U} i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} i{(U,\\hat{\\mathbf{x}})} + \\hat{\\mathbf{x}}^{2 U}", "derivation": "i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}^{U} and \\hat{\\mathbf{x}} i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} \\hat{\\mathbf{x}}^{U} and \\hat{\\mathbf{x}}^{U} i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}}^{2 U} and \\hat{\\mathbf{x}} \\hat{\\mathbf{x}}^{U} + \\hat{\\mathbf{x}}^{U} i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} \\hat{\\mathbf{x}}^{U} + \\hat{\\mathbf{x}}^{2 U} and \\hat{\\mathbf{x}} i{(U,\\hat{\\mathbf{x}})} + \\hat{\\mathbf{x}}^{U} i{(U,\\hat{\\mathbf{x}})} = \\hat{\\mathbf{x}} i{(U,\\hat{\\mathbf{x}})} + \\hat{\\mathbf{x}}^{2 U}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True)), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(2), Symbol('U', commutative=True))))"], [["add", 3, "Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True)), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True))), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(2), Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('U', commutative=True)), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('i')(Symbol('U', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(2), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given g{(\\rho_b)} = \\int \\log{(\\rho_b)} d\\rho_b and \\operatorname{C_{2}}{(\\rho_b)} = - g{(\\rho_b)}, then obtain - g{(\\rho_b)} = - \\rho_b \\log{(\\rho_b)} + \\rho_b - \\tilde{g}^*", "derivation": "g{(\\rho_b)} = \\int \\log{(\\rho_b)} d\\rho_b and \\operatorname{C_{2}}{(\\rho_b)} = - g{(\\rho_b)} and \\operatorname{C_{2}}{(\\rho_b)} = - \\int \\log{(\\rho_b)} d\\rho_b and - g{(\\rho_b)} = - \\int \\log{(\\rho_b)} d\\rho_b and - g{(\\rho_b)} = - \\rho_b \\log{(\\rho_b)} + \\rho_b - \\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\rho_b', commutative=True)), Integral(log(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_2')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Function('g')(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(-1), Function('g')(Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{E_{x}}{(\\sigma_x)} = e^{\\sin{(\\sigma_x)}}, then obtain \\int (\\operatorname{E_{x}}{(\\sigma_x)} - 1) d\\sigma_x = \\int (e^{\\sin{(\\sigma_x)}} - 1) d\\sigma_x", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\operatorname{E_{x}}{(\\sigma_x)} = e^{\\sin{(\\sigma_x)}} and \\operatorname{E_{x}}{(\\sigma_x)} - \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} = - \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{P})}}{\\log{(\\mathbf{P})}} + e^{\\sin{(\\sigma_x)}} and \\operatorname{E_{x}}{(\\sigma_x)} - 1 = e^{\\sin{(\\sigma_x)}} - 1 and \\int (\\operatorname{E_{x}}{(\\sigma_x)} - 1) d\\sigma_x = \\int (e^{\\sin{(\\sigma_x)}} - 1) d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], ["get_premise", "Equality(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), exp(sin(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Mul(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)))"], "Equality(Add(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))), exp(sin(Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Add(exp(sin(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Function('E_x')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(exp(sin(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(M_{E})} = \\sin{(M_{E})}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\lambda - \\cos{(M_{E})}, then obtain - \\psi + \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\mathbf{J} - \\psi - \\cos{(M_{E})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(M_{E})} = \\sin{(M_{E})} and \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\int \\sin{(M_{E})} dM_{E} and \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\lambda - \\cos{(M_{E})} and - \\psi + \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\lambda - \\psi - \\cos{(M_{E})} and - \\psi + \\int \\sin{(M_{E})} dM_{E} = \\lambda - \\psi - \\cos{(M_{E})} and - \\psi + \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = - \\psi + \\int \\sin{(M_{E})} dM_{E} and - \\psi + \\int \\operatorname{f_{\\mathbf{v}}}{(M_{E})} dM_{E} = \\mathbf{J} - \\psi - \\cos{(M_{E})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"], [["minus", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(v_{2},\\rho)} = \\frac{\\rho}{v_{2}}, then obtain - \\frac{v_{2}}{\\frac{\\rho}{v_{2}} - \\lambda{(v_{2},\\rho)}} = \\frac{\\frac{\\rho}{v_{2}} - v_{2} - \\lambda{(v_{2},\\rho)}}{\\frac{\\rho}{v_{2}} - \\lambda{(v_{2},\\rho)}}", "derivation": "\\lambda{(v_{2},\\rho)} = \\frac{\\rho}{v_{2}} and 0 = \\frac{\\rho}{v_{2}} - \\lambda{(v_{2},\\rho)} and - v_{2} = \\frac{\\rho}{v_{2}} - v_{2} - \\lambda{(v_{2},\\rho)} and - \\frac{v_{2}}{\\frac{\\rho}{v_{2}} - \\lambda{(v_{2},\\rho)}} = \\frac{\\frac{\\rho}{v_{2}} - v_{2} - \\lambda{(v_{2},\\rho)}}{\\frac{\\rho}{v_{2}} - \\lambda{(v_{2},\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))"], [["minus", 1, "Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('v_2', commutative=True)), Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["divide", 3, "Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('v_2', commutative=True), Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True)))), Integer(-1))), Mul(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True)))), Integer(-1)), Add(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('v_2', commutative=True), Symbol('\\\\rho', commutative=True))))))"]]}, {"prompt": "Given g{(S,\\eta)} = \\sin{(S \\eta)} and \\theta_{2}{(\\mathbf{f},s)} = \\mathbf{f} + s and A{(S,\\eta)} = \\frac{1}{g{(S,\\eta)}}, then obtain - S + \\theta_{2}{(\\mathbf{f},s)} + \\frac{A{(S,\\eta)}}{S \\eta} = - S + \\theta_{2}{(\\mathbf{f},s)} + \\frac{1}{S \\eta \\sin{(S \\eta)}}", "derivation": "g{(S,\\eta)} = \\sin{(S \\eta)} and \\theta_{2}{(\\mathbf{f},s)} = \\mathbf{f} + s and - S + \\theta_{2}{(\\mathbf{f},s)} = - S + \\mathbf{f} + s and A{(S,\\eta)} = \\frac{1}{g{(S,\\eta)}} and A{(S,\\eta)} = \\frac{1}{\\sin{(S \\eta)}} and \\frac{A{(S,\\eta)}}{S \\eta} = \\frac{1}{S \\eta \\sin{(S \\eta)}} and - S + \\mathbf{f} + s + \\frac{A{(S,\\eta)}}{S \\eta} = - S + \\mathbf{f} + s + \\frac{1}{S \\eta \\sin{(S \\eta)}} and - S + \\theta_{2}{(\\mathbf{f},s)} + \\frac{A{(S,\\eta)}}{S \\eta} = - S + \\theta_{2}{(\\mathbf{f},s)} + \\frac{1}{S \\eta \\sin{(S \\eta)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), sin(Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True)))"], [["minus", 2, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True)))"], ["renaming_premise", "Equality(Function('A')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(Function('g')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('A')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Pow(sin(Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(-1)))"], [["divide", 5, "Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('A')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(sin(Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["add", 6, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('A')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(sin(Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('A')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Pow(sin(Mul(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{x})} = \\cos{(v_{x})}, then obtain \\frac{\\mathbf{H} \\cos{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}{(v_{x})}} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})} = \\frac{\\mathbf{H} \\cos^{2}{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(v_{x})}} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{x})} = \\cos{(v_{x})} and 1 = \\frac{\\cos{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}{(v_{x})}} and \\mathbf{H} = \\frac{\\mathbf{H} \\cos{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}{(v_{x})}} and \\mathbf{H} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})} = \\frac{\\mathbf{H} \\cos{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}{(v_{x})}} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})} and \\frac{\\mathbf{H} \\cos{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}{(v_{x})}} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})} = \\frac{\\mathbf{H} \\cos^{2}{(v_{x})}}{\\operatorname{f_{\\mathbf{p}}}^{2}{(v_{x})}} + \\operatorname{f_{\\mathbf{p}}}{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["divide", 1, "Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), Integer(-1)), cos(Symbol('v_x', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Symbol('\\\\mathbf{H}', commutative=True), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), Integer(-1)), cos(Symbol('v_x', commutative=True))))"], [["add", 3, "Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), Integer(-1)), cos(Symbol('v_x', commutative=True))), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), Integer(-1)), cos(Symbol('v_x', commutative=True))), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))), Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True)), Integer(-2)), Pow(cos(Symbol('v_x', commutative=True)), Integer(2))), Function('f_{\\\\mathbf{p}}')(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{P})} = \\cos{(\\mathbf{P})}, then obtain \\phi_{2}{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} d\\mathbf{P} = \\phi_{2}{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} d\\mathbf{P}", "derivation": "\\phi_{2}{(\\mathbf{P})} = \\cos{(\\mathbf{P})} and \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} and \\int \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} d\\mathbf{P} = \\int \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} d\\mathbf{P} and \\cos{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} d\\mathbf{P} = \\cos{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} d\\mathbf{P} and \\phi_{2}{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} d\\mathbf{P} = \\phi_{2}{(\\mathbf{P})} \\int \\frac{d}{d \\mathbf{P}} \\cos{(\\mathbf{P})} d\\mathbf{P}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), cos(Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["times", 3, "cos(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(cos(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Derivative(cos(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(f)} = \\cos{(f)} and \\operatorname{t_{1}}{(f)} = \\int \\operatorname{t_{2}}{(f)} df, then derive \\int \\operatorname{t_{2}}{(f)} df = y + \\sin{(f)}, then obtain \\operatorname{t_{1}}{(f)} = y + \\sin{(f)}", "derivation": "\\operatorname{t_{2}}{(f)} = \\cos{(f)} and \\int \\operatorname{t_{2}}{(f)} df = \\int \\cos{(f)} df and \\operatorname{t_{1}}{(f)} = \\int \\operatorname{t_{2}}{(f)} df and \\int \\operatorname{t_{2}}{(f)} df = y + \\sin{(f)} and \\operatorname{t_{1}}{(f)} = \\int \\cos{(f)} df and \\int \\cos{(f)} df = y + \\sin{(f)} and \\operatorname{t_{1}}{(f)} = y + \\sin{(f)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('f', commutative=True)), Integral(Function('t_2')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_2')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('y', commutative=True), sin(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('t_1')(Symbol('f', commutative=True)), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Add(Symbol('y', commutative=True), sin(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Function('t_1')(Symbol('f', commutative=True)), Add(Symbol('y', commutative=True), sin(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{H})} = e^{\\cos{(F_{H})}}, then derive \\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})} = - e^{\\cos{(F_{H})}} \\sin{(F_{H})}, then obtain \\frac{(\\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})})^{F_{H}}}{F_{H}} = \\frac{(- \\operatorname{y^{\\prime}}{(F_{H})} \\sin{(F_{H})})^{F_{H}}}{F_{H}}", "derivation": "\\operatorname{y^{\\prime}}{(F_{H})} = e^{\\cos{(F_{H})}} and \\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})} = \\frac{d}{d F_{H}} e^{\\cos{(F_{H})}} and \\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})} = - e^{\\cos{(F_{H})}} \\sin{(F_{H})} and \\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})} = - \\operatorname{y^{\\prime}}{(F_{H})} \\sin{(F_{H})} and (\\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})})^{F_{H}} = (- \\operatorname{y^{\\prime}}{(F_{H})} \\sin{(F_{H})})^{F_{H}} and \\frac{(\\frac{d}{d F_{H}} \\operatorname{y^{\\prime}}{(F_{H})})^{F_{H}}}{F_{H}} = \\frac{(- \\operatorname{y^{\\prime}}{(F_{H})} \\sin{(F_{H})})^{F_{H}}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), exp(cos(Symbol('F_H', commutative=True))))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('F_H', commutative=True))), sin(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))))"], [["power", 4, "Symbol('F_H', commutative=True)"], "Equality(Pow(Derivative(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True)), Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], [["times", 5, "Pow(Symbol('F_H', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(F_{N})} = \\cos{(F_{N})} and \\theta_{2}{(F_{N})} = 2 F_{N} + \\hat{p}_0{(F_{N})} + 2 \\cos{(F_{N})}, then obtain 2 F_{N} (2 F_{N} + 2 \\hat{p}_0{(F_{N})} + \\cos{(F_{N})}) = 2 F_{N} \\theta_{2}{(F_{N})}", "derivation": "\\hat{p}_0{(F_{N})} = \\cos{(F_{N})} and 2 F_{N} + 2 \\hat{p}_0{(F_{N})} + \\cos{(F_{N})} = 2 F_{N} + \\hat{p}_0{(F_{N})} + 2 \\cos{(F_{N})} and 2 F_{N} (2 F_{N} + 2 \\hat{p}_0{(F_{N})} + \\cos{(F_{N})}) = 2 F_{N} (2 F_{N} + \\hat{p}_0{(F_{N})} + 2 \\cos{(F_{N})}) and \\theta_{2}{(F_{N})} = 2 F_{N} + \\hat{p}_0{(F_{N})} + 2 \\cos{(F_{N})} and 2 F_{N} (2 F_{N} + 2 \\hat{p}_0{(F_{N})} + \\cos{(F_{N})}) = 2 F_{N} \\theta_{2}{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["add", 1, "Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True)), Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["times", 2, "Mul(Integer(2), Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Symbol('F_N', commutative=True), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True)))), Mul(Integer(2), Symbol('F_N', commutative=True), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True)), Mul(Integer(2), cos(Symbol('F_N', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('F_N', commutative=True)), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True)), Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Symbol('F_N', commutative=True), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True)))), Mul(Integer(2), Symbol('F_N', commutative=True), Function('\\\\theta_2')(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(P_{e},\\mathbf{H})} = P_{e}^{\\mathbf{H}}, then obtain F_{H} \\mathbf{p} (- (P_{e}^{\\mathbf{H}})^{P_{e}} + \\operatorname{E_{x}}{(P_{e},\\mathbf{H})}) = F_{H} \\mathbf{p} (P_{e}^{\\mathbf{H}} - (P_{e}^{\\mathbf{H}})^{P_{e}})", "derivation": "\\operatorname{E_{x}}{(P_{e},\\mathbf{H})} = P_{e}^{\\mathbf{H}} and \\operatorname{E_{x}}^{P_{e}}{(P_{e},\\mathbf{H})} = (P_{e}^{\\mathbf{H}})^{P_{e}} and \\operatorname{E_{x}}{(P_{e},\\mathbf{H})} - \\operatorname{E_{x}}^{P_{e}}{(P_{e},\\mathbf{H})} = P_{e}^{\\mathbf{H}} - \\operatorname{E_{x}}^{P_{e}}{(P_{e},\\mathbf{H})} and - (P_{e}^{\\mathbf{H}})^{P_{e}} + \\operatorname{E_{x}}{(P_{e},\\mathbf{H})} = P_{e}^{\\mathbf{H}} - (P_{e}^{\\mathbf{H}})^{P_{e}} and F_{H} \\mathbf{p} (- (P_{e}^{\\mathbf{H}})^{P_{e}} + \\operatorname{E_{x}}{(P_{e},\\mathbf{H})}) = F_{H} \\mathbf{p} (P_{e}^{\\mathbf{H}} - (P_{e}^{\\mathbf{H}})^{P_{e}})", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True)), Pow(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True)))"], [["minus", 1, "Pow(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True)))), Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True))), Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True)))))"], [["times", 4, "Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Add(Mul(Integer(-1), Pow(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True))), Function('E_x')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Add(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Pow(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('P_e', commutative=True))))))"]]}, {"prompt": "Given \\varphi^{*}{(t_{1})} = e^{t_{1}} and \\mathbf{r}{(t_{1})} = e^{t_{1}}, then obtain \\frac{1}{\\mathbf{r}{(t_{1})} - \\frac{e^{t_{1}}}{\\varphi^{*}{(t_{1})}}} = \\frac{\\mathbf{r}{(t_{1})} e^{- t_{1}}}{\\mathbf{r}{(t_{1})} - \\frac{e^{t_{1}}}{\\varphi^{*}{(t_{1})}}}", "derivation": "\\varphi^{*}{(t_{1})} = e^{t_{1}} and \\varphi^{*}{(t_{1})} e^{- t_{1}} = 1 and 1 = \\frac{e^{t_{1}}}{\\varphi^{*}{(t_{1})}} and \\mathbf{r}{(t_{1})} = e^{t_{1}} and 1 = \\frac{\\mathbf{r}{(t_{1})}}{\\varphi^{*}{(t_{1})}} and 1 = \\mathbf{r}{(t_{1})} e^{- t_{1}} and \\frac{1}{\\mathbf{r}{(t_{1})} - \\frac{e^{t_{1}}}{\\varphi^{*}{(t_{1})}}} = \\frac{\\mathbf{r}{(t_{1})} e^{- t_{1}}}{\\mathbf{r}{(t_{1})} - \\frac{e^{t_{1}}}{\\varphi^{*}{(t_{1})}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["divide", 1, "exp(Symbol('t_1', commutative=True))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), Symbol('t_1', commutative=True)))), Integer(1))"], [["divide", 2, "Mul(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), Symbol('t_1', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Mul(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), Pow(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Mul(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), Symbol('t_1', commutative=True)))))"], [["divide", 6, "Add(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True))))"], "Equality(Pow(Add(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True)))), Integer(-1)), Mul(Pow(Add(Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\varphi^*')(Symbol('t_1', commutative=True)), Integer(-1)), exp(Symbol('t_1', commutative=True)))), Integer(-1)), Function('\\\\mathbf{r}')(Symbol('t_1', commutative=True)), exp(Mul(Integer(-1), Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given g{(a,H)} = H a, then derive \\frac{\\partial}{\\partial a} g{(a,H)} = H, then obtain \\frac{\\partial}{\\partial a} H^{H} \\frac{\\partial}{\\partial a} H a = \\frac{d}{d a} H H^{H}", "derivation": "g{(a,H)} = H a and \\frac{\\partial}{\\partial a} g{(a,H)} = \\frac{\\partial}{\\partial a} H a and \\frac{\\partial}{\\partial a} g{(a,H)} = H and \\frac{\\partial}{\\partial a} H a = H and H^{H} \\frac{\\partial}{\\partial a} g{(a,H)} = H^{H} \\frac{\\partial}{\\partial a} H a and \\frac{\\partial}{\\partial a} H^{H} \\frac{\\partial}{\\partial a} g{(a,H)} = \\frac{\\partial}{\\partial a} H^{H} \\frac{\\partial}{\\partial a} H a and \\frac{\\partial}{\\partial a} H^{H} \\frac{\\partial}{\\partial a} g{(a,H)} = \\frac{d}{d a} H H^{H} and \\frac{\\partial}{\\partial a} H^{H} \\frac{\\partial}{\\partial a} H a = \\frac{d}{d a} H H^{H}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('H', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('H', commutative=True))"], [["times", 2, "Pow(Symbol('H', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Function('g')(Symbol('a', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Derivative(Mul(Pow(Symbol('H', commutative=True), Symbol('H', commutative=True)), Derivative(Mul(Symbol('H', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Pow(Symbol('H', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(S)} = \\sin{(S)}, then derive \\frac{d}{d S} \\mathbf{F}{(S)} + 1 = \\cos{(S)} + 1, then obtain S (\\frac{d}{d S} \\mathbf{F}{(S)} + 1) = S (\\cos{(S)} + 1)", "derivation": "\\mathbf{F}{(S)} = \\sin{(S)} and \\frac{d}{d S} \\mathbf{F}{(S)} = \\frac{d}{d S} \\sin{(S)} and \\frac{d}{d S} \\mathbf{F}{(S)} + 1 = \\frac{d}{d S} \\sin{(S)} + 1 and \\frac{d}{d S} \\mathbf{F}{(S)} + 1 = \\cos{(S)} + 1 and S (\\frac{d}{d S} \\mathbf{F}{(S)} + 1) = S (\\cos{(S)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Add(Derivative(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Add(cos(Symbol('S', commutative=True)), Integer(1)))"], [["times", 4, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Add(Derivative(Function('\\\\mathbf{F}')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('S', commutative=True), Add(cos(Symbol('S', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\pi)} = \\sin{(\\sin{(\\pi)})} and \\operatorname{a^{\\dagger}}{(\\pi)} = \\sin{(\\sin{(\\pi)})}, then obtain \\frac{d}{d \\pi} (\\operatorname{a^{\\dagger}}{(\\pi)} \\sin{(\\sin{(\\pi)})})^{\\pi} = \\frac{d}{d \\pi} (\\operatorname{a^{\\dagger}}^{2}{(\\pi)})^{\\pi}", "derivation": "\\operatorname{E_{n}}{(\\pi)} = \\sin{(\\sin{(\\pi)})} and \\operatorname{E_{n}}{(\\pi)} \\sin{(\\sin{(\\pi)})} = \\sin^{2}{(\\sin{(\\pi)})} and (\\operatorname{E_{n}}{(\\pi)} \\sin{(\\sin{(\\pi)})})^{\\pi} = (\\sin^{2}{(\\sin{(\\pi)})})^{\\pi} and \\operatorname{a^{\\dagger}}{(\\pi)} = \\sin{(\\sin{(\\pi)})} and \\frac{d}{d \\pi} (\\operatorname{E_{n}}{(\\pi)} \\sin{(\\sin{(\\pi)})})^{\\pi} = \\frac{d}{d \\pi} (\\sin^{2}{(\\sin{(\\pi)})})^{\\pi} and \\frac{d}{d \\pi} (\\operatorname{E_{n}}{(\\pi)} \\operatorname{a^{\\dagger}}{(\\pi)})^{\\pi} = \\frac{d}{d \\pi} (\\operatorname{a^{\\dagger}}^{2}{(\\pi)})^{\\pi} and \\frac{d}{d \\pi} (\\operatorname{a^{\\dagger}}{(\\pi)} \\sin{(\\sin{(\\pi)})})^{\\pi} = \\frac{d}{d \\pi} (\\operatorname{a^{\\dagger}}^{2}{(\\pi)})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True))))"], [["times", 1, "sin(sin(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Function('E_n')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True)))), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Integer(2)))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Mul(Function('E_n')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Pow(Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Integer(2)), Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('E_n')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Integer(2)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Pow(Mul(Function('E_n')(Symbol('\\\\pi', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Integer(2)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Pow(Mul(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\pi', commutative=True)), Integer(2)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(\\eta,A_{1})} = A_{1} + \\cos{(\\eta)} and q{(\\eta,A_{1})} = \\frac{\\partial}{\\partial \\eta} (A_{1} + \\cos{(\\eta)}), then obtain q{(\\eta,A_{1})} = \\frac{\\partial}{\\partial \\eta} \\rho{(\\eta,A_{1})}", "derivation": "\\rho{(\\eta,A_{1})} = A_{1} + \\cos{(\\eta)} and \\frac{\\partial}{\\partial \\eta} \\rho{(\\eta,A_{1})} = \\frac{\\partial}{\\partial \\eta} (A_{1} + \\cos{(\\eta)}) and q{(\\eta,A_{1})} = \\frac{\\partial}{\\partial \\eta} (A_{1} + \\cos{(\\eta)}) and q{(\\eta,A_{1})} = \\frac{\\partial}{\\partial \\eta} \\rho{(\\eta,A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), cos(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('q')(Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Derivative(Function('\\\\rho')(Symbol('\\\\eta', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(V,x^\\prime)} = V - x^\\prime and \\operatorname{C_{1}}{(V,x^\\prime)} = e^{V - x^\\prime} and \\mathbf{J}_P{(V,x^\\prime)} = e^{v{(V,x^\\prime)}}, then obtain (\\operatorname{C_{1}}{(V,x^\\prime)} - e^{V - x^\\prime})^{2} = 0", "derivation": "v{(V,x^\\prime)} = V - x^\\prime and \\operatorname{C_{1}}{(V,x^\\prime)} = e^{V - x^\\prime} and \\mathbf{J}_P{(V,x^\\prime)} = e^{v{(V,x^\\prime)}} and \\mathbf{J}_P{(V,x^\\prime)} - e^{v{(V,x^\\prime)}} = 0 and \\mathbf{J}_P{(V,x^\\prime)} = e^{V - x^\\prime} and \\operatorname{C_{1}}{(V,x^\\prime)} = \\mathbf{J}_P{(V,x^\\prime)} and \\mathbf{J}_P{(V,x^\\prime)} - e^{V - x^\\prime} = 0 and (\\mathbf{J}_P{(V,x^\\prime)} - e^{V - x^\\prime})^{2} = 0 and (\\operatorname{C_{1}}{(V,x^\\prime)} - e^{V - x^\\prime})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Function('v')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 3, "exp(Function('v')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), exp(Function('v')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Function('C_1')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))), Integer(0))"], [["times", 7, "Add(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))))"], "Equality(Pow(Add(Function('\\\\mathbf{J}_P')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))), Integer(2)), Integer(0))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Pow(Add(Function('C_1')(Symbol('V', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))), Integer(2)), Integer(0))"]]}, {"prompt": "Given \\phi{(\\rho_b,\\mathbf{A})} = \\mathbf{A} - \\rho_b, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})} = 1, then obtain \\frac{\\partial}{\\partial \\rho_b} \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})})} = \\frac{d}{d \\rho_b} \\sin{(1)}", "derivation": "\\phi{(\\rho_b,\\mathbf{A})} = \\mathbf{A} - \\rho_b and \\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} (\\mathbf{A} - \\rho_b) and \\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})} = 1 and \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})})} = \\sin{(1)} and \\frac{\\partial}{\\partial \\rho_b} \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\phi{(\\rho_b,\\mathbf{A})})} = \\frac{d}{d \\rho_b} \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integer(1))"], [["sin", 3], "Equality(sin(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), sin(Integer(1)))"], [["differentiate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(sin(Derivative(Function('\\\\phi')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(sin(Integer(1)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(\\rho)} = e^{\\rho} and V{(\\rho)} = - \\hat{p}{(\\rho)} + e^{\\rho}, then obtain \\frac{2 \\hat{p}^{2}{(\\rho)}}{V{(\\rho)}} = \\frac{(\\hat{p}{(\\rho)} + e^{\\rho}) \\hat{p}{(\\rho)}}{V{(\\rho)}}", "derivation": "\\hat{p}{(\\rho)} = e^{\\rho} and 2 \\hat{p}{(\\rho)} = \\hat{p}{(\\rho)} + e^{\\rho} and \\frac{2 \\hat{p}{(\\rho)}}{- \\hat{p}{(\\rho)} + e^{\\rho}} = \\frac{\\hat{p}{(\\rho)} + e^{\\rho}}{- \\hat{p}{(\\rho)} + e^{\\rho}} and V{(\\rho)} = - \\hat{p}{(\\rho)} + e^{\\rho} and \\frac{2 \\hat{p}^{2}{(\\rho)}}{- \\hat{p}{(\\rho)} + e^{\\rho}} = \\frac{(\\hat{p}{(\\rho)} + e^{\\rho}) \\hat{p}{(\\rho)}}{- \\hat{p}{(\\rho)} + e^{\\rho}} and \\frac{2 \\hat{p}^{2}{(\\rho)}}{V{(\\rho)}} = \\frac{(\\hat{p}{(\\rho)} + e^{\\rho}) \\hat{p}{(\\rho)}}{V{(\\rho)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Integer(-1)), Add(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))))"], [["times", 3, "Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Integer(-1)), Pow(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))), exp(Symbol('\\\\rho', commutative=True))), Integer(-1)), Add(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Pow(Function('V')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), Integer(2))), Mul(Add(Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True))), Pow(Function('V')(Symbol('\\\\rho', commutative=True)), Integer(-1)), Function('\\\\hat{p}')(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\varepsilon_0^{\\sigma_p}, then derive \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\frac{\\sigma_p \\varepsilon_0^{\\sigma_p}}{\\varepsilon_0}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\frac{\\sigma_p \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)}}{\\varepsilon_0}", "derivation": "\\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\varepsilon_0^{\\sigma_p} and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0^{\\sigma_p} and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\frac{\\sigma_p \\varepsilon_0^{\\sigma_p}}{\\varepsilon_0} and \\frac{\\partial}{\\partial \\varepsilon_0} \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)} = \\frac{\\sigma_p \\operatorname{A_{2}}{(\\varepsilon_0,\\sigma_p)}}{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given Q{(y)} = \\cos{(y)}, then derive e^{\\int Q{(y)} dy} = e^{\\mu + \\sin{(y)}}, then obtain (e^{\\int Q{(y)} dy}) e^{\\int \\cos{(y)} dy} = e^{2 \\int Q{(y)} dy}", "derivation": "Q{(y)} = \\cos{(y)} and \\int Q{(y)} dy = \\int \\cos{(y)} dy and e^{\\int Q{(y)} dy} = e^{\\int \\cos{(y)} dy} and e^{\\int Q{(y)} dy} = e^{\\mu + \\sin{(y)}} and e^{\\int \\cos{(y)} dy} = e^{\\mu + \\sin{(y)}} and (e^{\\int Q{(y)} dy}) e^{\\int \\cos{(y)} dy} = e^{\\mu + \\sin{(y)}} e^{\\int Q{(y)} dy} and (e^{\\int Q{(y)} dy}) e^{\\int \\cos{(y)} dy} = e^{2 \\int Q{(y)} dy}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('y', commutative=True)))))"], [["times", 5, "exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Mul(exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))), Mul(exp(Add(Symbol('\\\\mu', commutative=True), sin(Symbol('y', commutative=True)))), exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(exp(Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), exp(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))), exp(Mul(Integer(2), Integral(Function('Q')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{z})} = \\sin{(v_{z})}, then derive \\frac{\\frac{d}{d v_{z}} \\operatorname{f_{\\mathbf{p}}}{(v_{z})}}{\\theta_1} = \\frac{\\cos{(v_{z})}}{\\theta_1}, then obtain \\frac{\\frac{d}{d v_{z}} \\sin{(v_{z})}}{\\theta_1} = \\frac{\\cos{(v_{z})}}{\\theta_1}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{z})} = \\sin{(v_{z})} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(v_{z})}}{\\theta_1} = \\frac{\\sin{(v_{z})}}{\\theta_1} and \\frac{\\partial}{\\partial v_{z}} \\frac{\\operatorname{f_{\\mathbf{p}}}{(v_{z})}}{\\theta_1} = \\frac{\\partial}{\\partial v_{z}} \\frac{\\sin{(v_{z})}}{\\theta_1} and \\frac{\\frac{d}{d v_{z}} \\operatorname{f_{\\mathbf{p}}}{(v_{z})}}{\\theta_1} = \\frac{\\cos{(v_{z})}}{\\theta_1} and \\frac{\\frac{d}{d v_{z}} \\sin{(v_{z})}}{\\theta_1} = \\frac{\\cos{(v_{z})}}{\\theta_1}", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["divide", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('v_z', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), cos(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), cos(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\phi_1,E,A)} = A E - \\phi_1 and \\operatorname{E_{n}}{(\\phi_1,E,A)} = A (A E - \\phi_1), then obtain \\operatorname{E_{n}}{(\\phi_1,E,A)} = A \\hat{\\mathbf{x}}{(\\phi_1,E,A)}", "derivation": "\\hat{\\mathbf{x}}{(\\phi_1,E,A)} = A E - \\phi_1 and A \\hat{\\mathbf{x}}{(\\phi_1,E,A)} = A (A E - \\phi_1) and \\operatorname{E_{n}}{(\\phi_1,E,A)} = A (A E - \\phi_1) and \\operatorname{E_{n}}{(\\phi_1,E,A)} = A \\hat{\\mathbf{x}}{(\\phi_1,E,A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True), Symbol('A', commutative=True)), Add(Mul(Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('E_n')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi_1', commutative=True), Symbol('E', commutative=True), Symbol('A', commutative=True))))"]]}, {"prompt": "Given S{(\\phi_1,Z)} = \\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1), then obtain \\int e^{\\int S^{Z}{(\\phi_1,Z)} d\\phi_1 - \\frac{1}{Z}} d\\phi_1 = \\int e^{\\int (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} d\\phi_1 - \\frac{1}{Z}} d\\phi_1", "derivation": "S{(\\phi_1,Z)} = \\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1) and S^{Z}{(\\phi_1,Z)} = (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} and \\int S^{Z}{(\\phi_1,Z)} d\\phi_1 = \\int (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} d\\phi_1 and \\int S^{Z}{(\\phi_1,Z)} d\\phi_1 - \\frac{1}{Z} = \\int (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} d\\phi_1 - \\frac{1}{Z} and e^{\\int S^{Z}{(\\phi_1,Z)} d\\phi_1 - \\frac{1}{Z}} = e^{\\int (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} d\\phi_1 - \\frac{1}{Z}} and \\int e^{\\int S^{Z}{(\\phi_1,Z)} d\\phi_1 - \\frac{1}{Z}} d\\phi_1 = \\int e^{\\int (\\frac{\\partial}{\\partial \\phi_1} (- Z + \\phi_1))^{Z} d\\phi_1 - \\frac{1}{Z}} d\\phi_1", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Pow(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 3, "Pow(Symbol('Z', commutative=True), Integer(-1))"], "Equality(Add(Integral(Pow(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))), Add(Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)))))"], [["exp", 4], "Equality(exp(Add(Integral(Pow(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1))))), exp(Add(Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1))))))"], [["integrate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(exp(Add(Integral(Pow(Function('S')(Symbol('\\\\phi_1', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(exp(Add(Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1))))), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\dot{y},M_{E})} = \\int (M_{E} + \\dot{y}) d\\dot{y}, then derive \\dot{y} + \\mathbf{S}{(\\dot{y},M_{E})} = M_{E} \\dot{y} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} + b, then obtain \\frac{M_{E} \\dot{y} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} + b}{\\mathbf{S}{(\\dot{y},M_{E})}} = \\frac{\\dot{y} + \\int (M_{E} + \\dot{y}) d\\dot{y}}{\\mathbf{S}{(\\dot{y},M_{E})}}", "derivation": "\\mathbf{S}{(\\dot{y},M_{E})} = \\int (M_{E} + \\dot{y}) d\\dot{y} and \\dot{y} + \\mathbf{S}{(\\dot{y},M_{E})} = \\dot{y} + \\int (M_{E} + \\dot{y}) d\\dot{y} and \\frac{\\dot{y} + \\mathbf{S}{(\\dot{y},M_{E})}}{\\mathbf{S}{(\\dot{y},M_{E})}} = \\frac{\\dot{y} + \\int (M_{E} + \\dot{y}) d\\dot{y}}{\\mathbf{S}{(\\dot{y},M_{E})}} and \\dot{y} + \\mathbf{S}{(\\dot{y},M_{E})} = M_{E} \\dot{y} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} + b and \\frac{M_{E} \\dot{y} + \\frac{\\dot{y}^{2}}{2} + \\dot{y} + b}{\\mathbf{S}{(\\dot{y},M_{E})}} = \\frac{\\dot{y} + \\int (M_{E} + \\dot{y}) d\\dot{y}}{\\mathbf{S}{(\\dot{y},M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Integral(Add(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["divide", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Add(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\dot{y}', commutative=True), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Mul(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Symbol('\\\\dot{y}', commutative=True), Symbol('b', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\dot{y}', commutative=True), Integral(Add(Symbol('M_E', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{y}', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given s{(a,t)} = - a + \\cos{(t)}, then derive \\frac{\\partial}{\\partial t} s{(a,t)} = - \\sin{(t)}, then obtain \\int (a - \\cos{(t)}) dt + \\int - \\frac{\\partial}{\\partial t} s{(a,t)} da = \\int (a - \\cos{(t)}) dt + \\int - \\frac{\\partial}{\\partial t} (- a + \\cos{(t)}) da", "derivation": "s{(a,t)} = - a + \\cos{(t)} and \\frac{\\partial}{\\partial t} s{(a,t)} = \\frac{\\partial}{\\partial t} (- a + \\cos{(t)}) and \\frac{\\partial}{\\partial t} s{(a,t)} = - \\sin{(t)} and - \\frac{\\partial}{\\partial t} s{(a,t)} = \\sin{(t)} and \\int - \\frac{\\partial}{\\partial t} s{(a,t)} da = \\int \\sin{(t)} da and - \\frac{\\partial}{\\partial t} (- a + \\cos{(t)}) = \\sin{(t)} and \\int (a - \\cos{(t)}) dt + \\int - \\frac{\\partial}{\\partial t} s{(a,t)} da = \\int (a - \\cos{(t)}) dt + \\int \\sin{(t)} da and \\int (a - \\cos{(t)}) dt + \\int - \\frac{\\partial}{\\partial t} s{(a,t)} da = \\int (a - \\cos{(t)}) dt + \\int - \\frac{\\partial}{\\partial t} (- a + \\cos{(t)}) da", "srepr_derivation": [["get_premise", "Equality(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('t', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('t', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), sin(Symbol('t', commutative=True)))"], [["integrate", 4, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True))), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), sin(Symbol('t', commutative=True)))"], [["add", 5, "Integral(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True)))), Add(Integral(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Integral(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Derivative(Function('s')(Symbol('a', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True)))), Add(Integral(Add(Symbol('a', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(f^{\\prime})} = f^{\\prime}, then derive \\dot{x} - f^{\\prime} + \\mathbf{J}_M{(f^{\\prime})} = A_{x}, then obtain (\\dot{x} - f^{\\prime} + \\mathbf{J}_M{(f^{\\prime})})^{A_{x}} = A_{x}^{A_{x}}", "derivation": "\\mathbf{J}_M{(f^{\\prime})} = f^{\\prime} and \\frac{d}{d f^{\\prime}} \\mathbf{J}_M{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} f^{\\prime} and \\frac{d}{d f^{\\prime}} \\mathbf{J}_M{(f^{\\prime})} - \\frac{1}{\\frac{d}{d f^{\\prime}} f^{\\prime}} = \\frac{d}{d f^{\\prime}} f^{\\prime} - \\frac{1}{\\frac{d}{d f^{\\prime}} f^{\\prime}} and \\int (\\frac{d}{d f^{\\prime}} \\mathbf{J}_M{(f^{\\prime})} - \\frac{1}{\\frac{d}{d f^{\\prime}} f^{\\prime}}) df^{\\prime} = \\int (\\frac{d}{d f^{\\prime}} f^{\\prime} - \\frac{1}{\\frac{d}{d f^{\\prime}} f^{\\prime}}) df^{\\prime} and \\dot{x} - f^{\\prime} + \\mathbf{J}_M{(f^{\\prime})} = A_{x} and (\\dot{x} - f^{\\prime} + \\mathbf{J}_M{(f^{\\prime})})^{A_{x}} = A_{x}^{A_{x}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)))), Add(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)))))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Derivative(Symbol('f^{\\\\prime}', commutative=True), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('A_x', commutative=True))"], [["power", 5, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('f^{\\\\prime}', commutative=True))), Symbol('A_x', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}}{\\cos^{2}{(\\mathbf{p})}} d\\mathbf{p} = \\phi_2 - \\frac{\\log{(\\sin{(\\mathbf{p})} - 1)}}{2} + \\frac{\\log{(\\sin{(\\mathbf{p})} + 1)}}{2}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}}{\\cos{(\\mathbf{p})}} = 1 and \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}}{\\cos^{2}{(\\mathbf{p})}} = \\frac{1}{\\cos{(\\mathbf{p})}} and \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}}{\\cos^{2}{(\\mathbf{p})}} d\\mathbf{p} = \\int \\frac{1}{\\cos{(\\mathbf{p})}} d\\mathbf{p} and \\int \\frac{\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}}{\\cos^{2}{(\\mathbf{p})}} d\\mathbf{p} = \\phi_2 - \\frac{\\log{(\\sin{(\\mathbf{p})} - 1)}}{2} + \\frac{\\log{(\\sin{(\\mathbf{p})} + 1)}}{2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "cos(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-2))), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-2))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Rational(1, 2), log(Add(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)))), Mul(Rational(1, 2), log(Add(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\dot{x})} = \\sin{(\\dot{x})}, then derive \\frac{d}{d \\dot{x}} \\operatorname{a^{\\dagger}}{(\\dot{x})} = \\cos{(\\dot{x})}, then obtain \\int \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} d\\dot{x} = \\int \\cos{(\\dot{x})} d\\dot{x}", "derivation": "\\operatorname{a^{\\dagger}}{(\\dot{x})} = \\sin{(\\dot{x})} and \\frac{d}{d \\dot{x}} \\operatorname{a^{\\dagger}}{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} and \\frac{d}{d \\dot{x}} \\operatorname{a^{\\dagger}}{(\\dot{x})} = \\cos{(\\dot{x})} and \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} = \\cos{(\\dot{x})} and \\int \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} d\\dot{x} = \\int \\cos{(\\dot{x})} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\phi_2,v_{2})} = \\phi_2 - v_{2}, then obtain - \\phi_2 + v_{2} + \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} \\Psi_{nl}{(\\phi_2,v_{2})}) = - \\phi_2 + v_{2} + \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} (\\phi_2 - v_{2}))", "derivation": "\\Psi_{nl}{(\\phi_2,v_{2})} = \\phi_2 - v_{2} and \\frac{\\partial}{\\partial v_{2}} \\Psi_{nl}{(\\phi_2,v_{2})} = \\frac{\\partial}{\\partial v_{2}} (\\phi_2 - v_{2}) and v_{2} + \\frac{\\partial}{\\partial v_{2}} \\Psi_{nl}{(\\phi_2,v_{2})} = v_{2} + \\frac{\\partial}{\\partial v_{2}} (\\phi_2 - v_{2}) and \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} \\Psi_{nl}{(\\phi_2,v_{2})}) = \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} (\\phi_2 - v_{2})) and - \\phi_2 + v_{2} + \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} \\Psi_{nl}{(\\phi_2,v_{2})}) = - \\phi_2 + v_{2} + \\frac{\\partial}{\\partial \\phi_2} (v_{2} + \\frac{\\partial}{\\partial v_{2}} (\\phi_2 - v_{2}))", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Symbol('v_2', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Symbol('v_2', commutative=True), Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Symbol('v_2', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('v_2', commutative=True), Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 4, "Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('v_2', commutative=True), Derivative(Add(Symbol('v_2', commutative=True), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('v_2', commutative=True), Derivative(Add(Symbol('v_2', commutative=True), Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(\\mu,B)} = \\int (B - \\mu) dB and \\Psi_{nl}{(\\mu,B)} = \\hat{H}{(\\mu,B)} + \\int (B - \\mu) dB - 2, then obtain (\\int (B - \\mu) dB - 1) \\Psi_{nl}{(\\mu,B)} = (\\int (B - \\mu) dB - 1) (2 \\int (B - \\mu) dB - 2)", "derivation": "\\hat{H}{(\\mu,B)} = \\int (B - \\mu) dB and \\hat{H}{(\\mu,B)} - 1 = \\int (B - \\mu) dB - 1 and \\Psi_{nl}{(\\mu,B)} = \\hat{H}{(\\mu,B)} + \\int (B - \\mu) dB - 2 and (\\int (B - \\mu) dB - 1) \\Psi_{nl}{(\\mu,B)} = (\\int (B - \\mu) dB - 1) (\\hat{H}{(\\mu,B)} + \\int (B - \\mu) dB - 2) and (\\hat{H}{(\\mu,B)} - 1) \\Psi_{nl}{(\\mu,B)} = (\\hat{H}{(\\mu,B)} - 1) (2 \\hat{H}{(\\mu,B)} - 2) and (\\int (B - \\mu) dB - 1) \\Psi_{nl}{(\\mu,B)} = (\\int (B - \\mu) dB - 1) (2 \\int (B - \\mu) dB - 2)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Add(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-2)))"], [["times", 3, "Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1))"], "Equality(Mul(Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True))), Mul(Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Add(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True))), Mul(Add(Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True))), Integer(-2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True))), Mul(Add(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('B', commutative=True)))), Integer(-2))))"]]}, {"prompt": "Given \\rho_{f}{(\\psi,T)} = T - \\psi and z{(\\psi)} = \\psi^{2}, then obtain \\frac{\\partial}{\\partial \\psi} \\rho_{f}^{2}{(\\psi,T)} z{(\\psi)} = \\frac{\\partial}{\\partial \\psi} (T - \\psi) \\rho_{f}{(\\psi,T)} z{(\\psi)}", "derivation": "\\rho_{f}{(\\psi,T)} = T - \\psi and \\psi \\rho_{f}{(\\psi,T)} = \\psi (T - \\psi) and \\psi^{2} \\rho_{f}^{2}{(\\psi,T)} = \\psi^{2} (T - \\psi) \\rho_{f}{(\\psi,T)} and \\frac{\\partial}{\\partial \\psi} \\psi^{2} \\rho_{f}^{2}{(\\psi,T)} = \\frac{\\partial}{\\partial \\psi} \\psi^{2} (T - \\psi) \\rho_{f}{(\\psi,T)} and z{(\\psi)} = \\psi^{2} and \\frac{\\partial}{\\partial \\psi} \\rho_{f}^{2}{(\\psi,T)} z{(\\psi)} = \\frac{\\partial}{\\partial \\psi} (T - \\psi) \\rho_{f}{(\\psi,T)} z{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))"], [["times", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))))"], [["times", 2, "Mul(Symbol('\\\\psi', commutative=True), Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Pow(Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)), Integer(2))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(2)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)), Integer(2)), Function('z')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Function('\\\\rho_f')(Symbol('\\\\psi', commutative=True), Symbol('T', commutative=True)), Function('z')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(\\rho_b)} = \\sin{(\\sin{(\\rho_b)})} and I{(\\rho_b)} = \\frac{d^{2}}{d \\rho_b^{2}} x{(\\rho_b)}, then obtain I{(\\rho_b)} = - \\sin{(\\rho_b)} \\cos{(\\sin{(\\rho_b)})} - \\sin{(\\sin{(\\rho_b)})} \\cos^{2}{(\\rho_b)}", "derivation": "x{(\\rho_b)} = \\sin{(\\sin{(\\rho_b)})} and \\frac{d}{d \\rho_b} x{(\\rho_b)} = \\frac{d}{d \\rho_b} \\sin{(\\sin{(\\rho_b)})} and \\frac{d^{2}}{d \\rho_b^{2}} x{(\\rho_b)} = \\frac{d^{2}}{d \\rho_b^{2}} \\sin{(\\sin{(\\rho_b)})} and I{(\\rho_b)} = \\frac{d^{2}}{d \\rho_b^{2}} x{(\\rho_b)} and I{(\\rho_b)} = \\frac{d^{2}}{d \\rho_b^{2}} \\sin{(\\sin{(\\rho_b)})} and I{(\\rho_b)} = - \\sin{(\\rho_b)} \\cos{(\\sin{(\\rho_b)})} - \\sin{(\\sin{(\\rho_b)})} \\cos^{2}{(\\rho_b)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\rho_b', commutative=True)), sin(sin(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))), Derivative(sin(sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True)), Derivative(Function('x')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True)), Derivative(sin(sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Add(Mul(sin(Symbol('\\\\rho_b', commutative=True)), cos(sin(Symbol('\\\\rho_b', commutative=True)))), Mul(sin(sin(Symbol('\\\\rho_b', commutative=True))), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Integer(2))))))"]]}, {"prompt": "Given m{(a,\\phi)} = \\frac{a}{\\phi}, then obtain \\frac{m{(a,\\phi)}}{a (\\frac{a}{\\phi} + \\frac{1}{\\phi})} = \\frac{1}{\\phi (\\frac{a}{\\phi} + \\frac{1}{\\phi})}", "derivation": "m{(a,\\phi)} = \\frac{a}{\\phi} and \\frac{m{(a,\\phi)}}{a} = \\frac{1}{\\phi} and m{(a,\\phi)} + \\frac{1}{\\phi} = \\frac{a}{\\phi} + \\frac{1}{\\phi} and \\frac{m{(a,\\phi)}}{a (m{(a,\\phi)} + \\frac{1}{\\phi})} = \\frac{1}{\\phi (m{(a,\\phi)} + \\frac{1}{\\phi})} and \\frac{m{(a,\\phi)}}{a (\\frac{a}{\\phi} + \\frac{1}{\\phi})} = \\frac{1}{\\phi (\\frac{a}{\\phi} + \\frac{1}{\\phi})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True)))"], [["divide", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True))), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))"], [["add", 1, "Pow(Symbol('\\\\phi', commutative=True), Integer(-1))"], "Equality(Add(Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))))"], [["divide", 2, "Add(Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Add(Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Integer(-1)), Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Add(Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Integer(-1)), Function('m')(Symbol('a', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('a', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1))), Integer(-1))))"]]}, {"prompt": "Given H{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain \\mathbf{S} (\\mathbf{S} H{(\\mathbf{S})})^{\\mathbf{S}} H{(\\mathbf{S})} - H{(\\mathbf{S})} = \\mathbf{S} (\\mathbf{S} e^{\\mathbf{S}})^{\\mathbf{S}} H{(\\mathbf{S})} - H{(\\mathbf{S})}", "derivation": "H{(\\mathbf{S})} = e^{\\mathbf{S}} and \\mathbf{S} H{(\\mathbf{S})} = \\mathbf{S} e^{\\mathbf{S}} and (\\mathbf{S} H{(\\mathbf{S})})^{\\mathbf{S}} = (\\mathbf{S} e^{\\mathbf{S}})^{\\mathbf{S}} and \\mathbf{S} (\\mathbf{S} H{(\\mathbf{S})})^{\\mathbf{S}} H{(\\mathbf{S})} = \\mathbf{S} (\\mathbf{S} e^{\\mathbf{S}})^{\\mathbf{S}} H{(\\mathbf{S})} and \\mathbf{S} (\\mathbf{S} H{(\\mathbf{S})})^{\\mathbf{S}} H{(\\mathbf{S})} - H{(\\mathbf{S})} = \\mathbf{S} (\\mathbf{S} e^{\\mathbf{S}})^{\\mathbf{S}} H{(\\mathbf{S})} - H{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 3, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('H')(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 4, "Function('H')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), exp(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Function('H')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given k{(n,v_{z})} = n \\sin{(v_{z})} and \\operatorname{v_{1}}{(n)} = - n, then obtain \\frac{\\partial}{\\partial n} (- n + k{(n,v_{z})} + \\operatorname{v_{1}}{(n)}) = \\frac{\\partial}{\\partial n} (n \\sin{(v_{z})} - n + \\operatorname{v_{1}}{(n)})", "derivation": "k{(n,v_{z})} = n \\sin{(v_{z})} and - n + k{(n,v_{z})} = n \\sin{(v_{z})} - n and \\operatorname{v_{1}}{(n)} = - n and k{(n,v_{z})} + \\operatorname{v_{1}}{(n)} = n \\sin{(v_{z})} + \\operatorname{v_{1}}{(n)} and - n + k{(n,v_{z})} + \\operatorname{v_{1}}{(n)} = n \\sin{(v_{z})} - n + \\operatorname{v_{1}}{(n)} and \\frac{\\partial}{\\partial n} (- n + k{(n,v_{z})} + \\operatorname{v_{1}}{(n)}) = \\frac{\\partial}{\\partial n} (n \\sin{(v_{z})} - n + \\operatorname{v_{1}}{(n)})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('n', commutative=True), sin(Symbol('v_z', commutative=True))))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('k')(Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Symbol('n', commutative=True), sin(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('k')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Function('v_1')(Symbol('n', commutative=True))), Add(Mul(Symbol('n', commutative=True), sin(Symbol('v_z', commutative=True))), Function('v_1')(Symbol('n', commutative=True))))"], [["minus", 4, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('k')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Function('v_1')(Symbol('n', commutative=True))), Add(Mul(Symbol('n', commutative=True), sin(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)), Function('v_1')(Symbol('n', commutative=True))))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('k')(Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Function('v_1')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('n', commutative=True), sin(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)), Function('v_1')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)} = \\hat{\\mathbf{x}} + m, then obtain \\hat{\\mathbf{x}} (- m + \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)}) + m = \\hat{\\mathbf{x}}^{2} + m", "derivation": "\\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)} = \\hat{\\mathbf{x}} + m and - m + \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)} = \\hat{\\mathbf{x}} and \\hat{\\mathbf{x}} (- m + \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)}) = \\hat{\\mathbf{x}}^{2} and \\hat{\\mathbf{x}} (- m + \\operatorname{F_{g}}{(\\hat{\\mathbf{x}},m)}) + m = \\hat{\\mathbf{x}}^{2} + m", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('m', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], [["times", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('m', commutative=True)))), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)))"], [["add", 3, "Symbol('m', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('F_g')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Add(Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(2)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\theta{(t_{2})} = \\log{(e^{t_{2}})}, then obtain 0 = 1 - 0^{t_{2}}", "derivation": "\\theta{(t_{2})} = \\log{(e^{t_{2}})} and 0 = - \\theta{(t_{2})} + \\log{(e^{t_{2}})} and 0^{t_{2}} = (- \\theta{(t_{2})} + \\log{(e^{t_{2}})})^{t_{2}} and 0^{t_{2}} e^{t_{2}} = (- \\theta{(t_{2})} + \\log{(e^{t_{2}})})^{t_{2}} e^{t_{2}} and (- \\theta{(t_{2})} + \\log{(e^{t_{2}})})^{t_{2}} = 1 and 0^{t_{2}} = 1 and 0 = 1 - 0^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True))))"], [["minus", 1, "Function('\\\\theta')(Symbol('t_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('t_2', commutative=True))), log(exp(Symbol('t_2', commutative=True)))))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Integer(0), Symbol('t_2', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('t_2', commutative=True))), log(exp(Symbol('t_2', commutative=True)))), Symbol('t_2', commutative=True)))"], [["times", 3, "exp(Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('t_2', commutative=True))), log(exp(Symbol('t_2', commutative=True)))), Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('t_2', commutative=True))), log(exp(Symbol('t_2', commutative=True)))), Symbol('t_2', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integer(0), Symbol('t_2', commutative=True)), Integer(1))"], [["minus", 6, "Pow(Integer(0), Symbol('t_2', commutative=True))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Integer(0), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(Q)} = \\int \\cos{(Q)} dQ, then derive \\tilde{g}^*{(Q)} = a + \\sin{(Q)}, then derive v + \\sin{(Q)} = a + \\sin{(Q)}, then obtain \\frac{v + \\sin{(Q)}}{Q} = \\frac{\\tilde{g}^*{(Q)}}{Q}", "derivation": "\\tilde{g}^*{(Q)} = \\int \\cos{(Q)} dQ and \\tilde{g}^*{(Q)} = a + \\sin{(Q)} and \\int \\cos{(Q)} dQ = a + \\sin{(Q)} and v + \\sin{(Q)} = a + \\sin{(Q)} and v + \\sin{(Q)} = \\tilde{g}^*{(Q)} and \\frac{v + \\sin{(Q)}}{Q} = \\frac{\\tilde{g}^*{(Q)}}{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True)), Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True)), Add(Symbol('a', commutative=True), sin(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('a', commutative=True), sin(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v', commutative=True), sin(Symbol('Q', commutative=True))), Add(Symbol('a', commutative=True), sin(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('v', commutative=True), sin(Symbol('Q', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True)))"], [["divide", 5, "Symbol('Q', commutative=True)"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Symbol('v', commutative=True), sin(Symbol('Q', commutative=True)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(G)} = e^{G}, then obtain \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(\\operatorname{F_{g}}{(G)})} + \\cos{(e^{G})}) + 1 = \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(e^{G})} + \\cos{(e^{G})}) + 1", "derivation": "\\operatorname{F_{g}}{(G)} = e^{G} and \\cos{(\\operatorname{F_{g}}{(G)})} = \\cos{(e^{G})} and \\operatorname{F_{g}}{(G)} \\cos{(\\operatorname{F_{g}}{(G)})} = \\operatorname{F_{g}}{(G)} \\cos{(e^{G})} and \\operatorname{F_{g}}{(G)} \\cos{(\\operatorname{F_{g}}{(G)})} + \\cos{(e^{G})} = \\operatorname{F_{g}}{(G)} \\cos{(e^{G})} + \\cos{(e^{G})} and \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(\\operatorname{F_{g}}{(G)})} + \\cos{(e^{G})}) = \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(e^{G})} + \\cos{(e^{G})}) and \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(\\operatorname{F_{g}}{(G)})} + \\cos{(e^{G})}) + 1 = \\frac{d}{d G} (\\operatorname{F_{g}}{(G)} \\cos{(e^{G})} + \\cos{(e^{G})}) + 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["cos", 1], "Equality(cos(Function('F_g')(Symbol('G', commutative=True))), cos(exp(Symbol('G', commutative=True))))"], [["times", 2, "Function('F_g')(Symbol('G', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(Function('F_g')(Symbol('G', commutative=True)))), Mul(Function('F_g')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))))"], [["add", 3, "cos(exp(Symbol('G', commutative=True)))"], "Equality(Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(Function('F_g')(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))), Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(Function('F_g')(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["add", 5, 1], "Equality(Add(Derivative(Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(Function('F_g')(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Mul(Function('F_g')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))), cos(exp(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given q{(A_{2})} = \\log{(A_{2})}, then derive \\frac{d}{d A_{2}} q{(A_{2})} = \\frac{1}{A_{2}}, then obtain \\frac{1}{A_{2}} = \\frac{d}{d A_{2}} \\log{(A_{2})}", "derivation": "q{(A_{2})} = \\log{(A_{2})} and \\frac{d}{d A_{2}} q{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and \\frac{d}{d A_{2}} q{(A_{2})} = \\frac{1}{A_{2}} and \\frac{1}{A_{2}} = \\frac{d}{d A_{2}} \\log{(A_{2})}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Pow(Symbol('A_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('A_2', commutative=True), Integer(-1)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(Z,y)} = y \\sin{(Z)}, then obtain 1 = (0^{Z})^{y}", "derivation": "\\operatorname{F_{c}}{(Z,y)} = y \\sin{(Z)} and - y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)} = 0 and (- y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)})^{Z} = 0^{Z} and ((- y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)})^{Z})^{y} = (0^{Z})^{y} and ((- y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)})^{Z})^{2 y} = (0^{Z})^{y} ((- y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)})^{Z})^{y} and 1 = ((- y \\sin{(Z)} + \\operatorname{F_{c}}{(Z,y)})^{Z})^{y} and 1 = (0^{Z})^{y}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))))"], [["minus", 1, "Mul(Symbol('y', commutative=True), sin(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Integer(0))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Pow(Integer(0), Symbol('Z', commutative=True)))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Integer(0), Symbol('Z', commutative=True)), Symbol('y', commutative=True)))"], [["times", 4, "Pow(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Symbol('y', commutative=True))"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))), Mul(Pow(Pow(Integer(0), Symbol('Z', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Pow(Pow(Add(Mul(Integer(-1), Symbol('y', commutative=True), sin(Symbol('Z', commutative=True))), Function('F_c')(Symbol('Z', commutative=True), Symbol('y', commutative=True))), Symbol('Z', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('Z', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given J{(v_{2},\\Psi)} = \\Psi v_{2} and \\operatorname{v_{y}}{(\\mathbf{S},\\eta^{\\prime})} = \\log{(\\mathbf{S}^{\\eta^{\\prime}})}, then obtain (\\Psi v_{2} J{(v_{2},\\Psi)})^{v_{2}} - \\operatorname{v_{y}}^{2}{(\\mathbf{S},\\eta^{\\prime})} = (\\Psi^{2} v_{2}^{2})^{v_{2}} - \\operatorname{v_{y}}^{2}{(\\mathbf{S},\\eta^{\\prime})}", "derivation": "J{(v_{2},\\Psi)} = \\Psi v_{2} and \\Psi v_{2} J{(v_{2},\\Psi)} = \\Psi^{2} v_{2}^{2} and (\\Psi v_{2} J{(v_{2},\\Psi)})^{v_{2}} = (\\Psi^{2} v_{2}^{2})^{v_{2}} and \\operatorname{v_{y}}{(\\mathbf{S},\\eta^{\\prime})} = \\log{(\\mathbf{S}^{\\eta^{\\prime}})} and (\\Psi v_{2} J{(v_{2},\\Psi)})^{v_{2}} - \\log{(\\mathbf{S}^{\\eta^{\\prime}})}^{2} = (\\Psi^{2} v_{2}^{2})^{v_{2}} - \\log{(\\mathbf{S}^{\\eta^{\\prime}})}^{2} and (\\Psi v_{2} J{(v_{2},\\Psi)})^{v_{2}} - \\operatorname{v_{y}}^{2}{(\\mathbf{S},\\eta^{\\prime})} = (\\Psi^{2} v_{2}^{2})^{v_{2}} - \\operatorname{v_{y}}^{2}{(\\mathbf{S},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('J')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Pow(Symbol('v_2', commutative=True), Integer(2))))"], [["power", 2, "Symbol('v_2', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('J')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('v_2', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Pow(Symbol('v_2', commutative=True), Integer(2))), Symbol('v_2', commutative=True)))"], ["get_premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["minus", 3, "Pow(log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(2))"], "Equality(Add(Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('J')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(2)))), Add(Pow(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Pow(Symbol('v_2', commutative=True), Integer(2))), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('v_2', commutative=True), Function('J')(Symbol('v_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)))), Add(Pow(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Pow(Symbol('v_2', commutative=True), Integer(2))), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(t_{1},M_{E})} = M_{E} + t_{1}, then obtain \\frac{\\partial}{\\partial M_{E}} 2 \\operatorname{f_{E}}{(t_{1},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (M_{E} + t_{1} + \\operatorname{f_{E}}{(t_{1},M_{E})})", "derivation": "\\operatorname{f_{E}}{(t_{1},M_{E})} = M_{E} + t_{1} and M_{E} + t_{1} + \\operatorname{f_{E}}{(t_{1},M_{E})} = 2 M_{E} + 2 t_{1} and 2 \\operatorname{f_{E}}{(t_{1},M_{E})} = 2 M_{E} + 2 t_{1} and \\frac{\\partial}{\\partial M_{E}} 2 \\operatorname{f_{E}}{(t_{1},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (2 M_{E} + 2 t_{1}) and \\frac{\\partial}{\\partial M_{E}} 2 \\operatorname{f_{E}}{(t_{1},M_{E})} = \\frac{\\partial}{\\partial M_{E}} (M_{E} + t_{1} + \\operatorname{f_{E}}{(t_{1},M_{E})})", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('t_1', commutative=True)))"], [["add", 1, "Add(Symbol('M_E', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Add(Symbol('M_E', commutative=True), Symbol('t_1', commutative=True), Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(2), Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Integer(2), Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('M_E', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Integer(2), Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Symbol('t_1', commutative=True), Function('f_E')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})} = \\hat{H} \\mathbf{p}, then obtain \\int (\\mathbf{p} + \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})}) d\\hat{H} = \\frac{\\hat{H}^{2} \\mathbf{p}}{2} + \\hat{H} \\mathbf{p} + v_{1}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})} = \\hat{H} \\mathbf{p} and \\mathbf{p} + \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})} = \\hat{H} \\mathbf{p} + \\mathbf{p} and \\int (\\mathbf{p} + \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})}) d\\hat{H} = \\int (\\hat{H} \\mathbf{p} + \\mathbf{p}) d\\hat{H} and \\int (\\mathbf{p} + \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{p})}) d\\hat{H} = \\frac{\\hat{H}^{2} \\mathbf{p}}{2} + \\hat{H} \\mathbf{p} + v_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(M)} = \\int \\sin{(M)} dM, then derive \\operatorname{f_{\\mathbf{p}}}{(M)} = \\delta - \\cos{(M)}, then obtain \\frac{- \\delta + \\int \\sin{(M)} dM}{\\int \\sin{(M)} dM} = - \\frac{\\cos{(M)}}{\\int \\sin{(M)} dM}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(M)} = \\int \\sin{(M)} dM and \\operatorname{f_{\\mathbf{p}}}{(M)} = \\delta - \\cos{(M)} and - \\delta + \\operatorname{f_{\\mathbf{p}}}{(M)} = - \\cos{(M)} and \\frac{- \\delta + \\operatorname{f_{\\mathbf{p}}}{(M)}}{\\delta - \\cos{(M)}} = - \\frac{\\cos{(M)}}{\\delta - \\cos{(M)}} and \\int \\sin{(M)} dM = \\delta - \\cos{(M)} and \\frac{- \\delta + \\operatorname{f_{\\mathbf{p}}}{(M)}}{\\int \\sin{(M)} dM} = - \\frac{\\cos{(M)}}{\\int \\sin{(M)} dM} and \\frac{- \\delta + \\int \\sin{(M)} dM}{\\int \\sin{(M)} dM} = - \\frac{\\cos{(M)}}{\\int \\sin{(M)} dM}", "srepr_derivation": [["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('M', commutative=True)), Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('M', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["minus", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('M', commutative=True))), Mul(Integer(-1), cos(Symbol('M', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('M', commutative=True))), Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Integer(-1))), Mul(Integer(-1), Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Integer(-1)), cos(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('M', commutative=True))), Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(Symbol('M', commutative=True)), Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(-1))), Mul(Integer(-1), cos(Symbol('M', commutative=True)), Pow(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given J{(k,x^\\prime,y)} = (x^\\prime)^{k} + y, then obtain \\frac{\\partial^{2}}{\\partial x^\\prime\\partial y} \\int J{(k,x^\\prime,y)} dk = \\frac{\\partial^{2}}{\\partial x^\\prime\\partial y} (\\int (x^\\prime)^{k} dk + \\int y dk)", "derivation": "J{(k,x^\\prime,y)} = (x^\\prime)^{k} + y and \\int J{(k,x^\\prime,y)} dk = \\int ((x^\\prime)^{k} + y) dk and \\int J{(k,x^\\prime,y)} dk = \\int (x^\\prime)^{k} dk + \\int y dk and \\frac{\\partial}{\\partial y} \\int J{(k,x^\\prime,y)} dk = \\frac{\\partial}{\\partial y} (\\int (x^\\prime)^{k} dk + \\int y dk) and \\frac{\\partial^{2}}{\\partial x^\\prime\\partial y} \\int J{(k,x^\\prime,y)} dk = \\frac{\\partial^{2}}{\\partial x^\\prime\\partial y} (\\int (x^\\prime)^{k} dk + \\int y dk)", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('k', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Add(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('k', commutative=True)), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('J')(Symbol('k', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Add(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('k', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('J')(Symbol('k', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Symbol('y', commutative=True), Tuple(Symbol('k', commutative=True)))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(Function('J')(Symbol('k', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Symbol('y', commutative=True), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('J')(Symbol('k', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('k', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Integral(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Symbol('y', commutative=True), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(\\delta,\\hbar)} = \\delta - \\hbar, then obtain - \\hbar + \\int U{(\\delta,\\hbar)} d\\hbar = \\delta \\hbar + \\hat{p}_0 - \\frac{\\hbar^{2}}{2} - \\hbar", "derivation": "U{(\\delta,\\hbar)} = \\delta - \\hbar and \\int U{(\\delta,\\hbar)} d\\hbar = \\int (\\delta - \\hbar) d\\hbar and - \\hbar + \\int U{(\\delta,\\hbar)} d\\hbar = - \\hbar + \\int (\\delta - \\hbar) d\\hbar and - \\hbar + \\int U{(\\delta,\\hbar)} d\\hbar = \\delta \\hbar + \\hat{p}_0 - \\frac{\\hbar^{2}}{2} - \\hbar", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integral(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integral(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Integral(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(f_{\\mathbf{p}},C_{2})} = C_{2} + f_{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(f_{\\mathbf{p}},C_{2})} = 1, then obtain \\frac{\\partial}{\\partial C_{2}} (C_{2} + f_{\\mathbf{p}}) + \\int \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I} = \\int \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I} + 1", "derivation": "\\dot{y}{(f_{\\mathbf{p}},C_{2})} = C_{2} + f_{\\mathbf{p}} and \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(f_{\\mathbf{p}},C_{2})} = \\frac{\\partial}{\\partial C_{2}} (C_{2} + f_{\\mathbf{p}}) and \\frac{\\partial}{\\partial C_{2}} \\dot{y}{(f_{\\mathbf{p}},C_{2})} = 1 and \\frac{\\partial}{\\partial C_{2}} (C_{2} + f_{\\mathbf{p}}) = 1 and \\frac{\\partial}{\\partial C_{2}} (C_{2} + f_{\\mathbf{p}}) + \\int \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I} = \\int \\cos{(\\sin{(\\mathbb{I})})} d\\mathbb{I} + 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Integral(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Add(Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integral(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Integral(cos(sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(x,F_{x})} = F_{x}^{x}, then obtain 2 = (\\frac{\\int F_{x}^{x} dF_{x}}{\\int \\Psi_{\\lambda}{(x,F_{x})} dF_{x}})^{F_{x}} + 1", "derivation": "\\Psi_{\\lambda}{(x,F_{x})} = F_{x}^{x} and \\int \\Psi_{\\lambda}{(x,F_{x})} dF_{x} = \\int F_{x}^{x} dF_{x} and 1 = \\frac{\\int F_{x}^{x} dF_{x}}{\\int \\Psi_{\\lambda}{(x,F_{x})} dF_{x}} and 1 = (\\frac{\\int F_{x}^{x} dF_{x}}{\\int \\Psi_{\\lambda}{(x,F_{x})} dF_{x}})^{F_{x}} and 2 = (\\frac{\\int F_{x}^{x} dF_{x}}{\\int \\Psi_{\\lambda}{(x,F_{x})} dF_{x}})^{F_{x}} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('F_x', commutative=True), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Pow(Symbol('F_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Pow(Symbol('F_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Integer(1), Pow(Mul(Integral(Pow(Symbol('F_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1))), Symbol('F_x', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Integer(2), Add(Pow(Mul(Integral(Pow(Symbol('F_x', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integer(-1))), Symbol('F_x', commutative=True)), Integer(1)))"]]}, {"prompt": "Given Z{(f^{*},v_{2},\\hat{X})} = \\frac{- f^{*} + v_{2}}{\\hat{X}}, then obtain \\frac{\\partial}{\\partial \\hat{X}} (- f^{*} + v_{2}) (f^{*} - v_{2} + Z{(f^{*},v_{2},\\hat{X})}) = \\frac{\\partial}{\\partial \\hat{X}} (- f^{*} + v_{2}) (f^{*} - v_{2} + \\frac{- f^{*} + v_{2}}{\\hat{X}})", "derivation": "Z{(f^{*},v_{2},\\hat{X})} = \\frac{- f^{*} + v_{2}}{\\hat{X}} and f^{*} - v_{2} + Z{(f^{*},v_{2},\\hat{X})} = f^{*} - v_{2} + \\frac{- f^{*} + v_{2}}{\\hat{X}} and (- f^{*} + v_{2}) (f^{*} - v_{2} + Z{(f^{*},v_{2},\\hat{X})}) = (- f^{*} + v_{2}) (f^{*} - v_{2} + \\frac{- f^{*} + v_{2}}{\\hat{X}}) and \\frac{\\partial}{\\partial \\hat{X}} (- f^{*} + v_{2}) (f^{*} - v_{2} + Z{(f^{*},v_{2},\\hat{X})}) = \\frac{\\partial}{\\partial \\hat{X}} (- f^{*} + v_{2}) (f^{*} - v_{2} + \\frac{- f^{*} + v_{2}}{\\hat{X}})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('f^*', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('Z')(Symbol('f^*', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('Z')(Symbol('f^*', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('Z')(Symbol('f^*', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(a)} = \\log{(\\cos{(a)})} and \\dot{z}{(a)} = \\log{(\\cos{(a)})}, then obtain \\cos{(a)} \\frac{d}{d a} \\log{(\\cos{(a)})} = \\cos{(a)} \\frac{d}{d a} \\operatorname{F_{N}}{(a)}", "derivation": "\\operatorname{F_{N}}{(a)} = \\log{(\\cos{(a)})} and \\dot{z}{(a)} = \\log{(\\cos{(a)})} and \\frac{d}{d a} \\dot{z}{(a)} = \\frac{d}{d a} \\log{(\\cos{(a)})} and \\frac{d}{d a} \\dot{z}{(a)} = \\frac{d}{d a} \\operatorname{F_{N}}{(a)} and \\frac{d}{d a} \\log{(\\cos{(a)})} = \\frac{d}{d a} \\operatorname{F_{N}}{(a)} and \\cos{(a)} \\frac{d}{d a} \\log{(\\cos{(a)})} = \\cos{(a)} \\frac{d}{d a} \\operatorname{F_{N}}{(a)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('a', commutative=True)), log(cos(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('a', commutative=True)), log(cos(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(log(cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Function('F_N')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(log(cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Function('F_N')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 5, "cos(Symbol('a', commutative=True))"], "Equality(Mul(cos(Symbol('a', commutative=True)), Derivative(log(cos(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(cos(Symbol('a', commutative=True)), Derivative(Function('F_N')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\phi_2,\\theta,Z)} = Z \\theta - \\phi_2, then obtain \\phi_2 + \\int \\operatorname{m_{s}}^{Z}{(\\phi_2,\\theta,Z)} d\\theta = \\phi_2 + \\int (Z \\theta - \\phi_2)^{Z} d\\theta", "derivation": "\\operatorname{m_{s}}{(\\phi_2,\\theta,Z)} = Z \\theta - \\phi_2 and \\operatorname{m_{s}}^{Z}{(\\phi_2,\\theta,Z)} = (Z \\theta - \\phi_2)^{Z} and \\int \\operatorname{m_{s}}^{Z}{(\\phi_2,\\theta,Z)} d\\theta = \\int (Z \\theta - \\phi_2)^{Z} d\\theta and \\phi_2 + \\int \\operatorname{m_{s}}^{Z}{(\\phi_2,\\theta,Z)} d\\theta = \\phi_2 + \\int (Z \\theta - \\phi_2)^{Z} d\\theta", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('Z', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Pow(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Pow(Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["add", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Integral(Pow(Function('m_s')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(Symbol('\\\\phi_2', commutative=True), Integral(Pow(Add(Mul(Symbol('Z', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(x,n_{1},\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f n_{1}}{x}, then obtain \\tilde{\\infty}^{n_{1}} ((- \\frac{\\mathbf{J}_f n_{1}}{x} + \\lambda{(x,n_{1},\\mathbf{J}_f)}) \\lambda{(x,n_{1},\\mathbf{J}_f)})^{n_{1}} = 0^{n_{1}} \\tilde{\\infty}^{n_{1}}", "derivation": "\\lambda{(x,n_{1},\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f n_{1}}{x} and - \\frac{\\mathbf{J}_f n_{1}}{x} + \\lambda{(x,n_{1},\\mathbf{J}_f)} = 0 and (- \\frac{\\mathbf{J}_f n_{1}}{x} + \\lambda{(x,n_{1},\\mathbf{J}_f)}) \\lambda{(x,n_{1},\\mathbf{J}_f)} = 0 and ((- \\frac{\\mathbf{J}_f n_{1}}{x} + \\lambda{(x,n_{1},\\mathbf{J}_f)}) \\lambda{(x,n_{1},\\mathbf{J}_f)})^{n_{1}} = 0^{n_{1}} and \\tilde{\\infty}^{n_{1}} ((- \\frac{\\mathbf{J}_f n_{1}}{x} + \\lambda{(x,n_{1},\\mathbf{J}_f)}) \\lambda{(x,n_{1},\\mathbf{J}_f)})^{n_{1}} = 0^{n_{1}} \\tilde{\\infty}^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(0))"], [["times", 2, "Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(0))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('n_1', commutative=True)), Pow(Integer(0), Symbol('n_1', commutative=True)))"], [["divide", 4, "Pow(Integer(0), Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('n_1', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('n_1', commutative=True))), Mul(Pow(Integer(0), Symbol('n_1', commutative=True)), Pow(zoo, Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(u)} = \\cos{(u)}, then obtain \\int \\frac{u \\operatorname{t_{2}}^{8}{(u)}}{\\cos^{7}{(u)}} du = \\int u \\cos{(u)} du", "derivation": "\\operatorname{t_{2}}{(u)} = \\cos{(u)} and u \\operatorname{t_{2}}{(u)} = u \\cos{(u)} and \\frac{u \\operatorname{t_{2}}^{2}{(u)}}{\\cos{(u)}} = u \\operatorname{t_{2}}{(u)} and \\frac{u \\operatorname{t_{2}}^{2}{(u)}}{\\cos{(u)}} = u \\cos{(u)} and \\frac{u \\operatorname{t_{2}}^{4}{(u)}}{\\cos^{3}{(u)}} = u \\cos{(u)} and \\frac{u \\operatorname{t_{2}}^{6}{(u)}}{\\cos^{5}{(u)}} = u \\cos{(u)} and \\frac{u \\operatorname{t_{2}}^{8}{(u)}}{\\cos^{7}{(u)}} = u \\cos{(u)} and \\int \\frac{u \\operatorname{t_{2}}^{8}{(u)}}{\\cos^{7}{(u)}} du = \\int u \\cos{(u)} du", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["divide", 2, "Mul(Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(-1)), cos(Symbol('u', commutative=True)))"], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(2)), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Mul(Symbol('u', commutative=True), Function('t_2')(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(2)), Pow(cos(Symbol('u', commutative=True)), Integer(-1))), Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(4)), Pow(cos(Symbol('u', commutative=True)), Integer(-3))), Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(6)), Pow(cos(Symbol('u', commutative=True)), Integer(-5))), Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(8)), Pow(cos(Symbol('u', commutative=True)), Integer(-7))), Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))))"], [["integrate", 7, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Symbol('u', commutative=True), Pow(Function('t_2')(Symbol('u', commutative=True)), Integer(8)), Pow(cos(Symbol('u', commutative=True)), Integer(-7))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('u', commutative=True), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"]]}, {"prompt": "Given I{(U)} = \\sin{(U)} and f{(F_{N},p)} = \\sin{(F_{N} p)}, then derive \\frac{d}{d U} I{(U)} = \\cos{(U)}, then obtain \\frac{f{(F_{N},p)}}{\\cos{(U)}} = \\frac{\\sin{(F_{N} p)}}{\\cos{(U)}}", "derivation": "I{(U)} = \\sin{(U)} and \\frac{d}{d U} I{(U)} = \\frac{d}{d U} \\sin{(U)} and f{(F_{N},p)} = \\sin{(F_{N} p)} and \\frac{d}{d U} I{(U)} = \\cos{(U)} and \\frac{f{(F_{N},p)}}{\\frac{d}{d U} I{(U)}} = \\frac{\\sin{(F_{N} p)}}{\\frac{d}{d U} I{(U)}} and \\frac{f{(F_{N},p)}}{\\cos{(U)}} = \\frac{\\sin{(F_{N} p)}}{\\cos{(U)}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('f')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Mul(Symbol('F_N', commutative=True), Symbol('p', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), cos(Symbol('U', commutative=True)))"], [["divide", 3, "Derivative(Function('I')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Function('f')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), Pow(Derivative(Function('I')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))), Mul(sin(Mul(Symbol('F_N', commutative=True), Symbol('p', commutative=True))), Pow(Derivative(Function('I')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('f')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), Pow(cos(Symbol('U', commutative=True)), Integer(-1))), Mul(sin(Mul(Symbol('F_N', commutative=True), Symbol('p', commutative=True))), Pow(cos(Symbol('U', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{X}{(y,F_{x})} = \\int \\frac{y}{F_{x}} dy, then obtain \\frac{F_{x} (\\hat{X}^{y}{(y,F_{x})} - (\\int \\frac{y}{F_{x}} dy)^{y})^{2}}{\\hat{X}^{2}{(y,F_{x})} \\int \\frac{y}{F_{x}} dy} = 0", "derivation": "\\hat{X}{(y,F_{x})} = \\int \\frac{y}{F_{x}} dy and \\hat{X}^{y}{(y,F_{x})} = (\\int \\frac{y}{F_{x}} dy)^{y} and \\hat{X}^{y}{(y,F_{x})} - (\\int \\frac{y}{F_{x}} dy)^{y} = 0 and \\frac{\\hat{X}^{y}{(y,F_{x})} - (\\int \\frac{y}{F_{x}} dy)^{y}}{\\hat{X}{(y,F_{x})}} = 0 and \\frac{\\hat{X}^{y}{(y,F_{x})} - (\\int \\frac{y}{F_{x}} dy)^{y}}{\\hat{X}^{2}{(y,F_{x})} \\int \\frac{y}{F_{x}} dy} = 0 and \\frac{F_{x} (\\hat{X}^{y}{(y,F_{x})} - (\\int \\frac{y}{F_{x}} dy)^{y})^{2}}{\\hat{X}^{2}{(y,F_{x})} \\int \\frac{y}{F_{x}} dy} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["minus", 2, "Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Integer(0))"], [["divide", 3, "Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 4, "Mul(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Mul(Add(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Integer(-2)), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Integer(0))"], [["times", 5, "Mul(Symbol('F_x', commutative=True), Add(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))))"], "Equality(Mul(Symbol('F_x', commutative=True), Pow(Add(Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Symbol('y', commutative=True)))), Integer(2)), Pow(Function('\\\\hat{X}')(Symbol('y', commutative=True), Symbol('F_x', commutative=True)), Integer(-2)), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\sigma_{p}{(\\delta,\\mathbf{r})} = \\delta + \\mathbf{r} and \\Psi_{nl}{(\\delta,\\mathbf{r})} = (\\delta + \\mathbf{r})^{\\mathbf{r}} - \\sigma_{p}^{\\mathbf{r}}{(\\delta,\\mathbf{r})}, then obtain \\mathbf{r} \\Psi_{nl}{(\\delta,\\mathbf{r})} = 0", "derivation": "\\sigma_{p}{(\\delta,\\mathbf{r})} = \\delta + \\mathbf{r} and \\sigma_{p}^{\\mathbf{r}}{(\\delta,\\mathbf{r})} = (\\delta + \\mathbf{r})^{\\mathbf{r}} and \\Psi_{nl}{(\\delta,\\mathbf{r})} = (\\delta + \\mathbf{r})^{\\mathbf{r}} - \\sigma_{p}^{\\mathbf{r}}{(\\delta,\\mathbf{r})} and \\mathbf{r} \\Psi_{nl}{(\\delta,\\mathbf{r})} = \\mathbf{r} ((\\delta + \\mathbf{r})^{\\mathbf{r}} - \\sigma_{p}^{\\mathbf{r}}{(\\delta,\\mathbf{r})}) and \\mathbf{r} \\Psi_{nl}{(\\delta,\\mathbf{r})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\sigma_p')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["times", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\sigma_p')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{B}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})}, then derive \\int \\mathbf{B}{(\\Psi_{nl})} d\\Psi_{nl} = A_{y} + \\sin{(\\Psi_{nl})}, then obtain \\frac{A_{x} + \\sin{(\\Psi_{nl})}}{A_{y} + \\sin{(\\Psi_{nl})}} = 1", "derivation": "\\mathbf{B}{(\\Psi_{nl})} = \\cos{(\\Psi_{nl})} and \\int \\mathbf{B}{(\\Psi_{nl})} d\\Psi_{nl} = \\int \\cos{(\\Psi_{nl})} d\\Psi_{nl} and \\int \\mathbf{B}{(\\Psi_{nl})} d\\Psi_{nl} = A_{y} + \\sin{(\\Psi_{nl})} and \\int \\cos{(\\Psi_{nl})} d\\Psi_{nl} = A_{y} + \\sin{(\\Psi_{nl})} and \\frac{\\int \\cos{(\\Psi_{nl})} d\\Psi_{nl}}{A_{y} + \\sin{(\\Psi_{nl})}} = 1 and \\frac{A_{x} + \\sin{(\\Psi_{nl})}}{A_{y} + \\sin{(\\Psi_{nl})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 4, "Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Integral(cos(Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True)))), Integer(1))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Symbol('A_y', commutative=True), sin(Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given A{(u)} = \\frac{d}{d u} \\log{(u)}, then derive \\frac{A{(u)}}{u} = \\frac{1}{u^{2}}, then obtain \\frac{d}{d u} \\frac{1}{u^{2}} = \\frac{d}{d u} \\frac{\\frac{d}{d u} \\log{(u)}}{u}", "derivation": "A{(u)} = \\frac{d}{d u} \\log{(u)} and \\frac{A{(u)}}{u} = \\frac{\\frac{d}{d u} \\log{(u)}}{u} and \\frac{A{(u)}}{u} = \\frac{1}{u^{2}} and \\frac{d}{d u} \\frac{A{(u)}}{u} = \\frac{d}{d u} \\frac{\\frac{d}{d u} \\log{(u)}}{u} and \\frac{d}{d u} \\frac{1}{u^{2}} = \\frac{d}{d u} \\frac{\\frac{d}{d u} \\log{(u)}}{u}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('u', commutative=True)), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True))), Pow(Symbol('u', commutative=True), Integer(-2)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('A')(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Symbol('u', commutative=True), Integer(-2)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(m)} = \\sin{(m)} and \\Omega{(m)} = - \\sin{(m)}, then obtain \\Omega{(m)} \\sin{(m)} - \\sin{(m)} = - \\sin^{2}{(m)} - \\sin{(m)}", "derivation": "A{(m)} = \\sin{(m)} and - A{(m)} \\sin{(m)} = - \\sin^{2}{(m)} and \\Omega{(m)} = - \\sin{(m)} and \\Omega{(m)} = - A{(m)} and - A{(m)} \\sin{(m)} - \\sin{(m)} = - \\sin^{2}{(m)} - \\sin{(m)} and \\Omega{(m)} \\sin{(m)} - \\sin{(m)} = - \\sin^{2}{(m)} - \\sin{(m)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["times", 1, "Mul(Integer(-1), sin(Symbol('m', commutative=True)))"], "Equality(Mul(Integer(-1), Function('A')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\Omega')(Symbol('m', commutative=True)), Mul(Integer(-1), Function('A')(Symbol('m', commutative=True))))"], [["add", 2, "Mul(Integer(-1), sin(Symbol('m', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Function('\\\\Omega')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Pow(sin(Symbol('m', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(v_{x},A_{y},v_{z})} = - A_{y} + v_{x} + v_{z} and A{(A_{y})} = 4 A_{y}, then obtain 4 A_{y} - 2 v_{x} - 2 v_{z} = - 2 v_{x} - 2 v_{z} + A{(A_{y})}", "derivation": "\\rho_{b}{(v_{x},A_{y},v_{z})} = - A_{y} + v_{x} + v_{z} and - A_{y} + \\rho_{b}{(v_{x},A_{y},v_{z})} = - 2 A_{y} + v_{x} + v_{z} and 2 A_{y} - 2 \\rho_{b}{(v_{x},A_{y},v_{z})} = 4 A_{y} - 2 v_{x} - 2 v_{z} and A{(A_{y})} = 4 A_{y} and 2 A_{y} - 2 \\rho_{b}{(v_{x},A_{y},v_{z})} = - 2 v_{x} - 2 v_{z} + A{(A_{y})} and 4 A_{y} - 2 v_{x} - 2 v_{z} = - 2 v_{x} - 2 v_{z} + A{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))"], [["minus", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('\\\\rho_b')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))"], [["divide", 2, "Rational(-1, 2)"], "Equality(Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Integer(4), Symbol('A_y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('A_y', commutative=True)), Mul(Integer(4), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\rho_b')(Symbol('v_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True)), Function('A')(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(4), Symbol('A_y', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('v_x', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True)), Function('A')(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(n_{2},I)} = I n_{2}, then obtain \\frac{\\iint I \\mathbf{H}{(n_{2},I)} dI dn_{2}}{I \\mathbf{H}{(n_{2},I)}} = \\frac{\\iint I^{2} n_{2} dI dn_{2}}{I \\mathbf{H}{(n_{2},I)}}", "derivation": "\\mathbf{H}{(n_{2},I)} = I n_{2} and I \\mathbf{H}{(n_{2},I)} = I^{2} n_{2} and \\int I \\mathbf{H}{(n_{2},I)} dI = \\int I^{2} n_{2} dI and \\iint I \\mathbf{H}{(n_{2},I)} dI dn_{2} = \\iint I^{2} n_{2} dI dn_{2} and \\frac{\\iint I \\mathbf{H}{(n_{2},I)} dI dn_{2}}{I \\mathbf{H}{(n_{2},I)}} = \\frac{\\iint I^{2} n_{2} dI dn_{2}}{I \\mathbf{H}{(n_{2},I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('n_2', commutative=True)))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(2)), Symbol('n_2', commutative=True)))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Symbol('n_2', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Symbol('n_2', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["divide", 4, "Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True)))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True)), Integer(-1)), Integral(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{H}')(Symbol('n_2', commutative=True), Symbol('I', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(2)), Symbol('n_2', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\hat{x}_0,C_{1})} = \\hat{x}_0^{C_{1}}, then obtain \\frac{\\partial}{\\partial \\hat{x}_0} \\log{(\\dot{y}{(\\hat{x}_0,C_{1})})}^{C_{1}} = \\frac{\\partial}{\\partial \\hat{x}_0} \\log{(\\hat{x}_0^{C_{1}})}^{C_{1}}", "derivation": "\\dot{y}{(\\hat{x}_0,C_{1})} = \\hat{x}_0^{C_{1}} and \\log{(\\dot{y}{(\\hat{x}_0,C_{1})})} = \\log{(\\hat{x}_0^{C_{1}})} and \\log{(\\dot{y}{(\\hat{x}_0,C_{1})})}^{C_{1}} = \\log{(\\hat{x}_0^{C_{1}})}^{C_{1}} and \\frac{\\partial}{\\partial \\hat{x}_0} \\log{(\\dot{y}{(\\hat{x}_0,C_{1})})}^{C_{1}} = \\frac{\\partial}{\\partial \\hat{x}_0} \\log{(\\hat{x}_0^{C_{1}})}^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\dot{y}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), log(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(log(Function('\\\\dot{y}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Pow(log(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Pow(log(Function('\\\\dot{y}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Pow(log(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(I,\\mathbf{g})} = \\cos{(I \\mathbf{g})}, then obtain 2 r{(I,\\mathbf{g})} = 2 \\cos{(I \\mathbf{g})}", "derivation": "r{(I,\\mathbf{g})} = \\cos{(I \\mathbf{g})} and 2 r{(I,\\mathbf{g})} = r{(I,\\mathbf{g})} + \\cos{(I \\mathbf{g})} and r{(I,\\mathbf{g})} + \\cos{(I \\mathbf{g})} = 2 \\cos{(I \\mathbf{g})} and 2 r{(I,\\mathbf{g})} = 2 \\cos{(I \\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 1, "Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["add", 1, "cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Integer(2), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Function('r')(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(2), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given z{(f^{*})} = \\int e^{f^{*}} df^{*}, then derive \\frac{d}{d f^{*}} z{(f^{*})} = \\frac{\\partial}{\\partial f^{*}} (E_{x} + e^{f^{*}}), then obtain \\frac{d^{2}}{d (f^{*})^{2}} z{(f^{*})} = \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} (E_{x} + e^{f^{*}})", "derivation": "z{(f^{*})} = \\int e^{f^{*}} df^{*} and \\frac{d}{d f^{*}} z{(f^{*})} = \\frac{d}{d f^{*}} \\int e^{f^{*}} df^{*} and \\frac{d}{d f^{*}} z{(f^{*})} = \\frac{\\partial}{\\partial f^{*}} (E_{x} + e^{f^{*}}) and \\frac{d^{2}}{d (f^{*})^{2}} z{(f^{*})} = \\frac{\\partial^{2}}{\\partial (f^{*})^{2}} (E_{x} + e^{f^{*}})", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('f^*', commutative=True)), Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('z')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('E_x', commutative=True), exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(2))), Derivative(Add(Symbol('E_x', commutative=True), exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given Z{(\\mathbf{E})} = \\mathbf{E}, then derive \\mathbf{f} + \\frac{Z^{2}{(\\mathbf{E})}}{2} = \\int \\mathbf{E} dZ{(\\mathbf{E})}, then obtain \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{f} = \\frac{\\mathbf{E}^{2}}{2} + n_{1}", "derivation": "Z{(\\mathbf{E})} = \\mathbf{E} and \\int Z{(\\mathbf{E})} d\\mathbf{E} = \\int \\mathbf{E} d\\mathbf{E} and \\int Z{(\\mathbf{E})} dZ{(\\mathbf{E})} = \\int \\mathbf{E} dZ{(\\mathbf{E})} and \\mathbf{f} + \\frac{Z^{2}{(\\mathbf{E})}}{2} = \\int \\mathbf{E} dZ{(\\mathbf{E})} and \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{f} = \\int \\mathbf{E} d\\mathbf{E} and \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{f} = \\frac{\\mathbf{E}^{2}}{2} + n_{1}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Rational(1, 2), Pow(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(2)))), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Function('Z')(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Integral(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given z{(C_{1})} = C_{1}, then derive (\\frac{d}{d C_{1}} z{(C_{1})})^{C_{1}} = 1, then obtain (\\frac{d}{d C_{1}} C_{1})^{C_{1}} = 1", "derivation": "z{(C_{1})} = C_{1} and \\frac{d}{d C_{1}} z{(C_{1})} = \\frac{d}{d C_{1}} C_{1} and (\\frac{d}{d C_{1}} z{(C_{1})})^{C_{1}} = (\\frac{d}{d C_{1}} C_{1})^{C_{1}} and (\\frac{d}{d C_{1}} C_{1})^{C_{1}} (\\frac{d}{d C_{1}} z{(C_{1})})^{C_{1}} = (\\frac{d}{d C_{1}} C_{1})^{2 C_{1}} and (\\frac{d}{d C_{1}} z{(C_{1})})^{C_{1}} = 1 and (\\frac{d}{d C_{1}} C_{1})^{C_{1}} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('z')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Derivative(Function('z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], [["times", 3, "Pow(Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Derivative(Function('z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True))), Pow(Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(2), Symbol('C_1', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('z')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Derivative(Symbol('C_1', commutative=True), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(v_{y})} = \\int \\sin{(v_{y})} dv_{y}, then derive \\operatorname{v_{t}}{(v_{y})} = \\mathbf{M} - \\cos{(v_{y})}, then obtain \\frac{d}{d \\mathbf{M}} \\int \\sin{(v_{y})} dv_{y} = \\frac{d}{d \\mathbf{M}} \\operatorname{v_{t}}{(v_{y})}", "derivation": "\\operatorname{v_{t}}{(v_{y})} = \\int \\sin{(v_{y})} dv_{y} and \\operatorname{v_{t}}{(v_{y})} = \\mathbf{M} - \\cos{(v_{y})} and \\frac{d}{d \\mathbf{M}} \\operatorname{v_{t}}{(v_{y})} = \\frac{\\partial}{\\partial \\mathbf{M}} (\\mathbf{M} - \\cos{(v_{y})}) and \\frac{d}{d \\mathbf{M}} \\int \\sin{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial \\mathbf{M}} (\\mathbf{M} - \\cos{(v_{y})}) and \\frac{d}{d \\mathbf{M}} \\int \\sin{(v_{y})} dv_{y} = \\frac{d}{d \\mathbf{M}} \\operatorname{v_{t}}{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('v_y', commutative=True)), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('v_t')(Symbol('v_y', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Function('v_t')(Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\phi_1,l,v)} = (\\frac{v}{l})^{\\phi_1}, then derive \\frac{\\partial}{\\partial l} \\operatorname{f_{E}}{(\\phi_1,l,v)} = - \\frac{\\phi_1 (\\frac{v}{l})^{\\phi_1}}{l}, then obtain (\\frac{\\partial}{\\partial l} (\\frac{v}{l})^{\\phi_1})^{v} = (- \\frac{\\phi_1 (\\frac{v}{l})^{\\phi_1}}{l})^{v}", "derivation": "\\operatorname{f_{E}}{(\\phi_1,l,v)} = (\\frac{v}{l})^{\\phi_1} and \\frac{\\partial}{\\partial l} \\operatorname{f_{E}}{(\\phi_1,l,v)} = \\frac{\\partial}{\\partial l} (\\frac{v}{l})^{\\phi_1} and \\frac{\\partial}{\\partial l} \\operatorname{f_{E}}{(\\phi_1,l,v)} = - \\frac{\\phi_1 (\\frac{v}{l})^{\\phi_1}}{l} and (\\frac{\\partial}{\\partial l} \\operatorname{f_{E}}{(\\phi_1,l,v)})^{v} = (- \\frac{\\phi_1 (\\frac{v}{l})^{\\phi_1}}{l})^{v} and (\\frac{\\partial}{\\partial l} (\\frac{v}{l})^{\\phi_1})^{v} = (- \\frac{\\phi_1 (\\frac{v}{l})^{\\phi_1}}{l})^{v}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Symbol('v', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Derivative(Function('f_E')(Symbol('\\\\phi_1', commutative=True), Symbol('l', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Symbol('v', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\phi_1', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given c{(\\Psi)} = \\log{(\\Psi)} and \\operatorname{r_{0}}{(\\Psi)} = - c{(\\Psi)} + \\log{(\\Psi)}, then obtain - \\operatorname{r_{0}}{(\\Psi)} = 2 c{(\\Psi)} - 2 \\log{(\\Psi)}", "derivation": "c{(\\Psi)} = \\log{(\\Psi)} and c{(\\Psi)} - \\log{(\\Psi)} = 0 and - c{(\\Psi)} = - 2 c{(\\Psi)} + \\log{(\\Psi)} and \\operatorname{r_{0}}{(\\Psi)} = - c{(\\Psi)} + \\log{(\\Psi)} and \\operatorname{r_{0}}{(\\Psi)} = - 2 c{(\\Psi)} + 2 \\log{(\\Psi)} and - \\operatorname{r_{0}}{(\\Psi)} = 2 c{(\\Psi)} - 2 \\log{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('c')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True)))), Integer(0))"], [["minus", 2, "Add(Mul(Integer(2), Function('c')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\Psi', commutative=True))))"], "Equality(Mul(Integer(-1), Function('c')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('c')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\Psi', commutative=True)), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\Psi', commutative=True))), log(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('r_0')(Symbol('\\\\Psi', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('c')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\Psi', commutative=True)))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('r_0')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(2), Function('c')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given i{(z)} = \\log{(\\log{(z)})}, then derive \\frac{d}{d z} i{(z)} = \\frac{1}{z \\log{(z)}}, then obtain z \\log{(z)} \\int \\frac{d}{d z} i{(z)} dz = z \\log{(z)} \\int \\frac{1}{z \\log{(z)}} dz", "derivation": "i{(z)} = \\log{(\\log{(z)})} and \\frac{d}{d z} i{(z)} = \\frac{d}{d z} \\log{(\\log{(z)})} and \\frac{d}{d z} i{(z)} = \\frac{1}{z \\log{(z)}} and \\int \\frac{d}{d z} i{(z)} dz = \\int \\frac{1}{z \\log{(z)}} dz and z \\log{(z)} \\int \\frac{d}{d z} i{(z)} dz = z \\log{(z)} \\int \\frac{1}{z \\log{(z)}} dz", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('z', commutative=True)), log(log(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(log(log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Derivative(Function('i')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True))))"], [["divide", 4, "Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Integer(-1)))"], "Equality(Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True)), Integral(Derivative(Function('i')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True)))), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True)), Integral(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Pow(log(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)}, then obtain 2 \\varepsilon_0 \\rho{(\\varepsilon_0)} + \\varepsilon_0 = 2 \\varepsilon_0 \\cos{(\\varepsilon_0)} + \\varepsilon_0", "derivation": "\\rho{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\varepsilon_0 \\rho{(\\varepsilon_0)} = \\varepsilon_0 \\cos{(\\varepsilon_0)} and \\varepsilon_0 \\rho{(\\varepsilon_0)} + \\varepsilon_0 = \\varepsilon_0 \\cos{(\\varepsilon_0)} + \\varepsilon_0 and 2 \\varepsilon_0 \\rho{(\\varepsilon_0)} + \\varepsilon_0 = \\varepsilon_0 \\rho{(\\varepsilon_0)} + \\varepsilon_0 \\cos{(\\varepsilon_0)} + \\varepsilon_0 and 2 \\varepsilon_0 \\rho{(\\varepsilon_0)} + \\varepsilon_0 = 2 \\varepsilon_0 \\cos{(\\varepsilon_0)} + \\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\rho')(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), cos(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})}, then derive \\sigma_{x}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})}, then obtain \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\sigma_{x}{(f_{\\mathbf{p}})})} = \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})})}", "derivation": "\\sigma_{x}{(f_{\\mathbf{p}})} = \\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})} and \\sigma_{x}{(f_{\\mathbf{p}})} = \\cos{(f_{\\mathbf{p}})} and \\sin{(\\sigma_{x}{(f_{\\mathbf{p}})})} = \\sin{(\\cos{(f_{\\mathbf{p}})})} and \\sin{(\\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})})} = \\sin{(\\cos{(f_{\\mathbf{p}})})} and \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\sigma_{x}{(f_{\\mathbf{p}})})} = \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\cos{(f_{\\mathbf{p}})})} and \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\sigma_{x}{(f_{\\mathbf{p}})})} = \\sigma_{x}{(f_{\\mathbf{p}})} \\sin{(\\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["sin", 2], "Equality(sin(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), sin(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(sin(Derivative(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), sin(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["times", 3, "Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(cos(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Mul(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Derivative(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{H}{(S)} = \\cos{(\\cos{(S)})}, then obtain \\frac{\\frac{d}{d S} S \\hat{H}{(S)}}{\\cos{(S)}} = \\frac{\\frac{d}{d S} S \\cos{(\\cos{(S)})}}{\\cos{(S)}}", "derivation": "\\hat{H}{(S)} = \\cos{(\\cos{(S)})} and S \\hat{H}{(S)} = S \\cos{(\\cos{(S)})} and \\frac{d}{d S} S \\hat{H}{(S)} = \\frac{d}{d S} S \\cos{(\\cos{(S)})} and \\frac{\\frac{d}{d S} S \\hat{H}{(S)}}{\\cos{(S)}} = \\frac{\\frac{d}{d S} S \\cos{(\\cos{(S)})}}{\\cos{(S)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('S', commutative=True)), cos(cos(Symbol('S', commutative=True))))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), cos(cos(Symbol('S', commutative=True)))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Symbol('S', commutative=True), cos(cos(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["divide", 3, "cos(Symbol('S', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('S', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('S', commutative=True), Function('\\\\hat{H}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('S', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('S', commutative=True), cos(cos(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(\\pi,\\mathbf{J}_P)} = \\mathbf{J}_P + \\pi and \\phi_{2}{(\\pi,\\mathbf{J}_P)} = \\mathbf{J}_P + \\pi, then obtain (- H{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} + 1 = (- \\phi_{2}{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} + 1", "derivation": "H{(\\pi,\\mathbf{J}_P)} = \\mathbf{J}_P + \\pi and \\phi_{2}{(\\pi,\\mathbf{J}_P)} = \\mathbf{J}_P + \\pi and - \\mathbf{J}_P - \\pi = - \\phi_{2}{(\\pi,\\mathbf{J}_P)} and (- \\mathbf{J}_P - \\pi)^{\\mathbf{J}_P} = (- \\phi_{2}{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} and \\phi_{2}{(\\pi,\\mathbf{J}_P)} = H{(\\pi,\\mathbf{J}_P)} and (- \\mathbf{J}_P - \\pi)^{\\mathbf{J}_P} + 1 = (- \\phi_{2}{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} + 1 and - \\mathbf{J}_P - \\pi = - H{(\\pi,\\mathbf{J}_P)} and (- H{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} + 1 = (- \\phi_{2}{(\\pi,\\mathbf{J}_P)})^{\\mathbf{J}_P} + 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\pi', commutative=True), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('H')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)), Add(Pow(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('H')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Pow(Mul(Integer(-1), Function('H')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)), Add(Pow(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)))"]]}, {"prompt": "Given p{(\\mu)} = \\cos{(\\mu)}, then obtain \\frac{p{(\\mu)} \\cos{(\\mu)} - \\frac{1}{\\cos{(\\mu)}}}{\\cos{(\\mu)}} = \\frac{\\cos^{2}{(\\mu)} - \\frac{1}{\\cos{(\\mu)}}}{\\cos{(\\mu)}}", "derivation": "p{(\\mu)} = \\cos{(\\mu)} and p{(\\mu)} \\cos{(\\mu)} = \\cos^{2}{(\\mu)} and p{(\\mu)} \\cos{(\\mu)} - \\frac{1}{\\cos{(\\mu)}} = \\cos^{2}{(\\mu)} - \\frac{1}{\\cos{(\\mu)}} and \\frac{p{(\\mu)} \\cos{(\\mu)} - \\frac{1}{\\cos{(\\mu)}}}{\\cos{(\\mu)}} = \\frac{\\cos^{2}{(\\mu)} - \\frac{1}{\\cos{(\\mu)}}}{\\cos{(\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('p')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)))"], [["minus", 2, "Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('p')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1)))), Add(Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1)))))"], [["times", 3, "Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Mul(Function('p')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1)))), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1))), Mul(Add(Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1)))), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\pi)} = e^{\\pi}, then obtain e^{\\pi + 2 e^{\\pi}} e^{\\pi + \\operatorname{t_{2}}{(\\pi)} + e^{\\pi}} = e^{2 \\pi + 4 e^{\\pi}}", "derivation": "\\operatorname{t_{2}}{(\\pi)} = e^{\\pi} and \\operatorname{t_{2}}{(\\pi)} + e^{\\pi} = 2 e^{\\pi} and \\pi + \\operatorname{t_{2}}{(\\pi)} + e^{\\pi} = \\pi + 2 e^{\\pi} and e^{\\pi + \\operatorname{t_{2}}{(\\pi)} + e^{\\pi}} = e^{\\pi + 2 e^{\\pi}} and e^{\\pi + 2 e^{\\pi}} e^{\\pi + \\operatorname{t_{2}}{(\\pi)} + e^{\\pi}} = e^{2 \\pi + 4 e^{\\pi}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('t_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('t_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\pi', commutative=True), Function('t_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))), exp(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))))))"], [["times", 4, "exp(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True)))))"], "Equality(Mul(exp(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))))), exp(Add(Symbol('\\\\pi', commutative=True), Function('t_2')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))))), exp(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(4), exp(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\hat{p})} = e^{\\hat{p}}, then derive \\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})} = e^{\\hat{p}}, then obtain (\\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})})^{\\hat{p}} = (e^{\\hat{p}})^{\\hat{p}}", "derivation": "\\operatorname{m_{s}}{(\\hat{p})} = e^{\\hat{p}} and \\operatorname{m_{s}}^{\\hat{p}}{(\\hat{p})} = (e^{\\hat{p}})^{\\hat{p}} and \\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} e^{\\hat{p}} and \\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})} = e^{\\hat{p}} and \\operatorname{m_{s}}{(\\hat{p})} = \\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})} and (\\frac{d}{d \\hat{p}} \\operatorname{m_{s}}{(\\hat{p})})^{\\hat{p}} = (e^{\\hat{p}})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(exp(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Derivative(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Pow(Derivative(Function('m_s')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Pow(exp(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(t_{1},i,\\mathbf{M})} = \\frac{\\mathbf{M} - i}{t_{1}}, then derive 0 = - \\frac{\\partial}{\\partial \\mathbf{M}} \\hat{X}{(t_{1},i,\\mathbf{M})} + \\frac{1}{t_{1}}, then obtain 0^{t_{1}} = (- \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M} - i}{t_{1}} + \\frac{1}{t_{1}})^{t_{1}}", "derivation": "\\hat{X}{(t_{1},i,\\mathbf{M})} = \\frac{\\mathbf{M} - i}{t_{1}} and 0 = - \\hat{X}{(t_{1},i,\\mathbf{M})} + \\frac{\\mathbf{M} - i}{t_{1}} and \\frac{d}{d \\mathbf{M}} 0 = \\frac{\\partial}{\\partial \\mathbf{M}} (- \\hat{X}{(t_{1},i,\\mathbf{M})} + \\frac{\\mathbf{M} - i}{t_{1}}) and 0 = - \\frac{\\partial}{\\partial \\mathbf{M}} \\hat{X}{(t_{1},i,\\mathbf{M})} + \\frac{1}{t_{1}} and 0^{t_{1}} = (- \\frac{\\partial}{\\partial \\mathbf{M}} \\hat{X}{(t_{1},i,\\mathbf{M})} + \\frac{1}{t_{1}})^{t_{1}} and 0^{t_{1}} = (- \\frac{\\partial}{\\partial \\mathbf{M}} \\frac{\\mathbf{M} - i}{t_{1}} + \\frac{1}{t_{1}})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))))"], [["minus", 1, "Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["power", 4, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('t_1', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('t_1', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Pow(Symbol('t_1', commutative=True), Integer(-1))), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integer(0), Symbol('t_1', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Pow(Symbol('t_1', commutative=True), Integer(-1))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\varphi{(I,v_{2})} = I - v_{2}, then derive \\int \\varphi{(I,v_{2})} dv_{2} = C_{1} + I v_{2} - \\frac{v_{2}^{2}}{2}, then derive I v_{2} + I + \\ddot{x} - \\frac{v_{2}^{2}}{2} = C_{1} + I v_{2} + I - \\frac{v_{2}^{2}}{2}, then obtain I + \\int \\varphi{(I,v_{2})} dv_{2} = I v_{2} + I + \\ddot{x} - \\frac{v_{2}^{2}}{2}", "derivation": "\\varphi{(I,v_{2})} = I - v_{2} and \\int \\varphi{(I,v_{2})} dv_{2} = \\int (I - v_{2}) dv_{2} and \\int \\varphi{(I,v_{2})} dv_{2} = C_{1} + I v_{2} - \\frac{v_{2}^{2}}{2} and \\int (I - v_{2}) dv_{2} = C_{1} + I v_{2} - \\frac{v_{2}^{2}}{2} and I + \\int (I - v_{2}) dv_{2} = C_{1} + I v_{2} + I - \\frac{v_{2}^{2}}{2} and I v_{2} + I + \\ddot{x} - \\frac{v_{2}^{2}}{2} = C_{1} + I v_{2} + I - \\frac{v_{2}^{2}}{2} and I + \\int \\varphi{(I,v_{2})} dv_{2} = C_{1} + I v_{2} + I - \\frac{v_{2}^{2}}{2} and I + \\int \\varphi{(I,v_{2})} dv_{2} = I v_{2} + I + \\ddot{x} - \\frac{v_{2}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"], [["add", 4, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Integral(Add(Symbol('I', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Symbol('I', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Symbol('I', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))), Add(Symbol('C_1', commutative=True), Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Symbol('I', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('I', commutative=True), Integral(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Symbol('I', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Symbol('I', commutative=True), Integral(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Add(Mul(Symbol('I', commutative=True), Symbol('v_2', commutative=True)), Symbol('I', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(v_{x},v_{t})} = \\frac{v_{t}}{v_{x}}, then obtain \\frac{v_{t} \\int v_{x} \\int \\operatorname{A_{y}}{(v_{x},v_{t})} dv_{t} dv_{x}}{v_{x}} = \\frac{v_{t} \\int v_{x} \\int \\frac{v_{t}}{v_{x}} dv_{t} dv_{x}}{v_{x}}", "derivation": "\\operatorname{A_{y}}{(v_{x},v_{t})} = \\frac{v_{t}}{v_{x}} and \\int \\operatorname{A_{y}}{(v_{x},v_{t})} dv_{t} = \\int \\frac{v_{t}}{v_{x}} dv_{t} and v_{x} \\int \\operatorname{A_{y}}{(v_{x},v_{t})} dv_{t} = v_{x} \\int \\frac{v_{t}}{v_{x}} dv_{t} and \\int v_{x} \\int \\operatorname{A_{y}}{(v_{x},v_{t})} dv_{t} dv_{x} = \\int v_{x} \\int \\frac{v_{t}}{v_{x}} dv_{t} dv_{x} and \\frac{v_{t} \\int v_{x} \\int \\operatorname{A_{y}}{(v_{x},v_{t})} dv_{t} dv_{x}}{v_{x}} = \\frac{v_{t} \\int v_{x} \\int \\frac{v_{t}}{v_{x}} dv_{t} dv_{x}}{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True))))"], [["times", 2, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Integral(Function('A_y')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Mul(Symbol('v_x', commutative=True), Integral(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True)))))"], [["integrate", 3, "Symbol('v_x', commutative=True)"], "Equality(Integral(Mul(Symbol('v_x', commutative=True), Integral(Function('A_y')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('v_x', commutative=True), Integral(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["times", 4, "Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)), Integral(Mul(Symbol('v_x', commutative=True), Integral(Function('A_y')(Symbol('v_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))), Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1)), Integral(Mul(Symbol('v_x', commutative=True), Integral(Mul(Symbol('v_t', commutative=True), Pow(Symbol('v_x', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(M)} = \\sin{(\\cos{(M)})}, then obtain \\bar{\\h}{(M)} + \\sin{(M)} \\cos{(\\cos{(M)})} - \\cos{(M)} = \\sin{(M)} \\cos{(\\cos{(M)})} + \\sin{(\\cos{(M)})} - \\cos{(M)}", "derivation": "\\bar{\\h}{(M)} = \\sin{(\\cos{(M)})} and \\bar{\\h}{(M)} - \\cos{(M)} = \\sin{(\\cos{(M)})} - \\cos{(M)} and \\bar{\\h}{(M)} - \\cos{(M)} - \\frac{d}{d M} \\sin{(\\cos{(M)})} = \\sin{(\\cos{(M)})} - \\cos{(M)} - \\frac{d}{d M} \\sin{(\\cos{(M)})} and \\bar{\\h}{(M)} + \\sin{(M)} \\cos{(\\cos{(M)})} - \\cos{(M)} = \\sin{(M)} \\cos{(\\cos{(M)})} + \\sin{(\\cos{(M)})} - \\cos{(M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('M', commutative=True)), sin(cos(Symbol('M', commutative=True))))"], [["minus", 1, "cos(Symbol('M', commutative=True))"], "Equality(Add(Function('\\\\hbar')(Symbol('M', commutative=True)), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Add(sin(cos(Symbol('M', commutative=True))), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"], [["minus", 2, "Derivative(sin(cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hbar')(Symbol('M', commutative=True)), Mul(Integer(-1), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Derivative(sin(cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))), Add(sin(cos(Symbol('M', commutative=True))), Mul(Integer(-1), cos(Symbol('M', commutative=True))), Mul(Integer(-1), Derivative(sin(cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('\\\\hbar')(Symbol('M', commutative=True)), Mul(sin(Symbol('M', commutative=True)), cos(cos(Symbol('M', commutative=True)))), Mul(Integer(-1), cos(Symbol('M', commutative=True)))), Add(Mul(sin(Symbol('M', commutative=True)), cos(cos(Symbol('M', commutative=True)))), sin(cos(Symbol('M', commutative=True))), Mul(Integer(-1), cos(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)} and \\operatorname{E_{n}}{(\\psi^*)} = \\operatorname{g_{\\varepsilon}}{(\\psi^*)} - \\sin{(\\psi^*)}, then obtain \\frac{d}{d \\psi^*} \\operatorname{E_{n}}^{\\psi^*}{(\\psi^*)} = \\frac{d}{d \\psi^*} 0^{\\psi^*}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)} and \\operatorname{g_{\\varepsilon}}{(\\psi^*)} - \\sin{(\\psi^*)} = 0 and (\\operatorname{g_{\\varepsilon}}{(\\psi^*)} - \\sin{(\\psi^*)})^{\\psi^*} = 0^{\\psi^*} and \\operatorname{E_{n}}{(\\psi^*)} = \\operatorname{g_{\\varepsilon}}{(\\psi^*)} - \\sin{(\\psi^*)} and \\frac{d}{d \\psi^*} (\\operatorname{g_{\\varepsilon}}{(\\psi^*)} - \\sin{(\\psi^*)})^{\\psi^*} = \\frac{d}{d \\psi^*} 0^{\\psi^*} and \\frac{d}{d \\psi^*} \\operatorname{E_{n}}^{\\psi^*}{(\\psi^*)} = \\frac{d}{d \\psi^*} 0^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)), Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\psi^*', commutative=True)), Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Pow(Function('E_n')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\Psi)} = \\Psi, then obtain \\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)} + (\\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)})^{\\Psi} = \\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)} + \\Psi^{\\Psi}", "derivation": "t{(\\Psi)} = \\Psi and t^{\\Psi}{(\\Psi)} = \\Psi^{\\Psi} and t{(\\Psi)} t^{\\Psi}{(\\Psi)} = \\Psi^{\\Psi} t{(\\Psi)} and t{(\\Psi)} + t^{\\Psi}{(\\Psi)} = \\Psi^{\\Psi} + t{(\\Psi)} and t{(\\Psi)} = \\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)} and \\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)} + (\\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)})^{\\Psi} = \\Psi^{\\Psi} t{(\\Psi)} t^{- \\Psi}{(\\Psi)} + \\Psi^{\\Psi}", "srepr_derivation": [["renaming_premise", "Equality(Function('t')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["times", 2, "Function('t')(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Function('t')(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True))))"], [["divide", 3, "Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], "Equality(Function('t')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))), Pow(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))), Symbol('\\\\Psi', commutative=True))), Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Function('t')(Symbol('\\\\Psi', commutative=True)), Pow(Function('t')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)))), Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(m)} = \\log{(\\log{(m)})} and \\theta_{2}{(m)} = \\log{(\\log{(m)})}^{m}, then obtain \\theta_{2}{(m)} \\operatorname{v_{2}}{(m)} + \\frac{\\theta_{2}{(m)}}{m} = \\operatorname{v_{2}}{(m)} \\log{(\\log{(m)})}^{m} + \\frac{\\theta_{2}{(m)}}{m}", "derivation": "\\operatorname{v_{2}}{(m)} = \\log{(\\log{(m)})} and \\operatorname{v_{2}}^{m}{(m)} = \\log{(\\log{(m)})}^{m} and \\operatorname{v_{2}}{(m)} \\operatorname{v_{2}}^{m}{(m)} = \\operatorname{v_{2}}{(m)} \\log{(\\log{(m)})}^{m} and \\theta_{2}{(m)} = \\log{(\\log{(m)})}^{m} and \\theta_{2}{(m)} = \\operatorname{v_{2}}^{m}{(m)} and \\theta_{2}{(m)} \\operatorname{v_{2}}{(m)} = \\operatorname{v_{2}}{(m)} \\log{(\\log{(m)})}^{m} and \\theta_{2}{(m)} \\operatorname{v_{2}}{(m)} + \\frac{\\theta_{2}{(m)}}{m} = \\operatorname{v_{2}}{(m)} \\log{(\\log{(m)})}^{m} + \\frac{\\theta_{2}{(m)}}{m}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('m', commutative=True)), log(log(Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(log(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["times", 2, "Function('v_2')(Symbol('m', commutative=True))"], "Equality(Mul(Function('v_2')(Symbol('m', commutative=True)), Pow(Function('v_2')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Function('v_2')(Symbol('m', commutative=True)), Pow(log(log(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('m', commutative=True)), Pow(log(log(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\theta_2')(Symbol('m', commutative=True)), Pow(Function('v_2')(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Function('\\\\theta_2')(Symbol('m', commutative=True)), Function('v_2')(Symbol('m', commutative=True))), Mul(Function('v_2')(Symbol('m', commutative=True)), Pow(log(log(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["add", 6, "Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('m', commutative=True)))"], "Equality(Add(Mul(Function('\\\\theta_2')(Symbol('m', commutative=True)), Function('v_2')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('m', commutative=True)))), Add(Mul(Function('v_2')(Symbol('m', commutative=True)), Pow(log(log(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given f{(a,x)} = a - x, then obtain - \\sin{(x - \\frac{f{(a,x)}}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{1}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{f{(a,x)}}{a - x})} = - \\sin{(x - 1 + \\frac{1}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{f{(a,x)}}{a - x})}", "derivation": "f{(a,x)} = a - x and \\frac{f{(a,x)}}{a - x} = 1 and -1 + \\frac{f{(a,x)}}{a - x} = 0 and x + \\frac{f{(a,x)}}{a - x} = x + 1 and -1 + \\frac{f{(a,x)}}{a - x} - \\frac{1}{a - x} = - \\frac{1}{a - x} and - x - 1 + \\frac{f{(a,x)}}{a - x} - \\frac{1}{a - x} = - x - \\frac{1}{a - x} and - \\sin{(x + 1 - \\frac{f{(a,x)}}{a - x} + \\frac{1}{a - x})} = - \\sin{(x + \\frac{1}{a - x})} and - \\sin{(x - \\frac{f{(a,x)}}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{1}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{f{(a,x)}}{a - x})} = - \\sin{(x - 1 + \\frac{1}{a - x + 1 - \\frac{f{(a,x)}}{a - x}} + \\frac{f{(a,x)}}{a - x})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)), Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["divide", 1, "Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))), Integer(0))"], [["minus", 2, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Add(Symbol('x', commutative=True), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))), Add(Symbol('x', commutative=True), Integer(1)))"], [["minus", 3, "Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))"], "Equality(Add(Integer(-1), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)))), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))))"], [["add", 5, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integer(-1), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)))))"], [["sin", 6], "Equality(Mul(Integer(-1), sin(Add(Symbol('x', commutative=True), Integer(1), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True))), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))))), Mul(Integer(-1), sin(Add(Symbol('x', commutative=True), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(-1), sin(Add(Symbol('x', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True))), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))), Integer(-1)), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))))), Mul(Integer(-1), sin(Add(Symbol('x', commutative=True), Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))), Integer(-1)), Mul(Pow(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Function('f')(Symbol('a', commutative=True), Symbol('x', commutative=True)))))))"]]}, {"prompt": "Given r{(F_{N})} = \\log{(F_{N})} and \\operatorname{A_{y}}{(t_{1},x)} = t_{1} + x, then derive \\frac{\\partial}{\\partial x} \\operatorname{A_{y}}{(t_{1},x)} + 1 = 2, then obtain e^{\\frac{\\partial}{\\partial x} (\\frac{\\partial}{\\partial x} \\operatorname{A_{y}}{(t_{1},x)} + 1)} = e^{\\frac{d}{d x} 2}", "derivation": "r{(F_{N})} = \\log{(F_{N})} and \\operatorname{A_{y}}{(t_{1},x)} = t_{1} + x and x + r{(F_{N})} = x + \\log{(F_{N})} and x + \\operatorname{A_{y}}{(t_{1},x)} + r{(F_{N})} = t_{1} + 2 x + r{(F_{N})} and x + \\operatorname{A_{y}}{(t_{1},x)} + \\log{(F_{N})} = t_{1} + 2 x + r{(F_{N})} and \\frac{\\partial}{\\partial x} (x + \\operatorname{A_{y}}{(t_{1},x)} + \\log{(F_{N})}) = \\frac{\\partial}{\\partial x} (t_{1} + 2 x + r{(F_{N})}) and \\frac{\\partial}{\\partial x} \\operatorname{A_{y}}{(t_{1},x)} + 1 = 2 and \\frac{\\partial}{\\partial x} (\\frac{\\partial}{\\partial x} \\operatorname{A_{y}}{(t_{1},x)} + 1) = \\frac{d}{d x} 2 and e^{\\frac{\\partial}{\\partial x} (\\frac{\\partial}{\\partial x} \\operatorname{A_{y}}{(t_{1},x)} + 1)} = e^{\\frac{d}{d x} 2}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], ["get_premise", "Equality(Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), Add(Symbol('t_1', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('r')(Symbol('F_N', commutative=True))), Add(Symbol('x', commutative=True), log(Symbol('F_N', commutative=True))))"], [["add", 2, "Add(Symbol('x', commutative=True), Function('r')(Symbol('F_N', commutative=True)))"], "Equality(Add(Symbol('x', commutative=True), Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), Function('r')(Symbol('F_N', commutative=True))), Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('x', commutative=True)), Function('r')(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('x', commutative=True), Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), log(Symbol('F_N', commutative=True))), Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('x', commutative=True)), Function('r')(Symbol('F_N', commutative=True))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Symbol('x', commutative=True), Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('x', commutative=True)), Function('r')(Symbol('F_N', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["differentiate", 7, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["exp", 8], "Equality(exp(Derivative(Add(Derivative(Function('A_y')(Symbol('t_1', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1)))), exp(Derivative(Integer(2), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given E{(k,q)} = \\log{(q^{k})}, then obtain \\int E{(k,q)} dq = 2 L - 2 k q + 2 q \\log{(q^{k})} - \\int E{(k,q)} dq", "derivation": "E{(k,q)} = \\log{(q^{k})} and \\int E{(k,q)} dq = \\int \\log{(q^{k})} dq and \\int E{(k,q)} dq + \\int \\log{(q^{k})} dq = 2 \\int \\log{(q^{k})} dq and \\int \\log{(q^{k})} dq = - \\int E{(k,q)} dq + 2 \\int \\log{(q^{k})} dq and \\int E{(k,q)} dq = - \\int E{(k,q)} dq + 2 \\int \\log{(q^{k})} dq and \\int E{(k,q)} dq = 2 L - 2 k q + 2 q \\log{(q^{k})} - \\int E{(k,q)} dq", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["add", 2, "Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True)))), Mul(Integer(2), Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["minus", 3, "Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))"], "Equality(Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(2), Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Mul(Integer(2), Integral(log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('q', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Mul(Integer(2), Symbol('L', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('k', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True), log(Pow(Symbol('q', commutative=True), Symbol('k', commutative=True)))), Mul(Integer(-1), Integral(Function('E')(Symbol('k', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\int (\\mathbf{H}^{2} \\Omega{(\\mathbf{H})} + \\mathbf{H}^{2}) d\\mathbf{H} = \\hat{H}_{\\lambda} + \\frac{\\mathbf{H}^{3}}{3} - \\mathbf{H}^{2} \\cos{(\\mathbf{H})} + 2 \\mathbf{H} \\sin{(\\mathbf{H})} + 2 \\cos{(\\mathbf{H})}", "derivation": "\\Omega{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\mathbf{H} \\Omega{(\\mathbf{H})} = \\mathbf{H} \\sin{(\\mathbf{H})} and \\mathbf{H}^{2} \\Omega{(\\mathbf{H})} = \\mathbf{H}^{2} \\sin{(\\mathbf{H})} and \\mathbf{H}^{2} \\Omega{(\\mathbf{H})} + \\mathbf{H}^{2} = \\mathbf{H}^{2} \\sin{(\\mathbf{H})} + \\mathbf{H}^{2} and \\int (\\mathbf{H}^{2} \\Omega{(\\mathbf{H})} + \\mathbf{H}^{2}) d\\mathbf{H} = \\int (\\mathbf{H}^{2} \\sin{(\\mathbf{H})} + \\mathbf{H}^{2}) d\\mathbf{H} and \\int (\\mathbf{H}^{2} \\Omega{(\\mathbf{H})} + \\mathbf{H}^{2}) d\\mathbf{H} = \\hat{H}_{\\lambda} + \\frac{\\mathbf{H}^{3}}{3} - \\mathbf{H}^{2} \\cos{(\\mathbf{H})} + 2 \\mathbf{H} \\sin{(\\mathbf{H})} + 2 \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 3, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Function('\\\\Omega')(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 3), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(3))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True), sin(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbf{S},y)} = y^{\\mathbf{S}} and k{(\\mathbf{S},y)} = \\operatorname{r_{0}}^{2}{(\\mathbf{S},y)}, then obtain k{(\\mathbf{S},y)} = y^{\\mathbf{S}} \\operatorname{r_{0}}{(\\mathbf{S},y)}", "derivation": "\\operatorname{r_{0}}{(\\mathbf{S},y)} = y^{\\mathbf{S}} and \\operatorname{r_{0}}^{2}{(\\mathbf{S},y)} = y^{\\mathbf{S}} \\operatorname{r_{0}}{(\\mathbf{S},y)} and k{(\\mathbf{S},y)} = \\operatorname{r_{0}}^{2}{(\\mathbf{S},y)} and k{(\\mathbf{S},y)} = y^{2 \\mathbf{S}} and y^{2 \\mathbf{S}} = \\operatorname{r_{0}}^{2}{(\\mathbf{S},y)} and y^{2 \\mathbf{S}} = y^{\\mathbf{S}} \\operatorname{r_{0}}{(\\mathbf{S},y)} and k{(\\mathbf{S},y)} = y^{\\mathbf{S}} \\operatorname{r_{0}}{(\\mathbf{S},y)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Pow(Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('k')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('y', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Pow(Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Symbol('y', commutative=True), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Function('k')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('r_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('y', commutative=True))))"]]}, {"prompt": "Given B{(J,A_{y})} = \\cos{(A_{y} J)} and \\theta{(J,A_{y})} = J \\cos{(A_{y} J)}, then obtain (J B{(J,A_{y})} \\cos{(A_{y} J)} + J \\cos{(A_{y} J)}) \\theta{(J,A_{y})} = J (J B{(J,A_{y})} \\cos{(A_{y} J)} + J \\cos{(A_{y} J)}) B{(J,A_{y})}", "derivation": "B{(J,A_{y})} = \\cos{(A_{y} J)} and \\theta{(J,A_{y})} = J \\cos{(A_{y} J)} and \\theta{(J,A_{y})} = J B{(J,A_{y})} and (J B{(J,A_{y})} \\cos{(A_{y} J)} + J \\cos{(A_{y} J)}) \\theta{(J,A_{y})} = J (J B{(J,A_{y})} \\cos{(A_{y} J)} + J \\cos{(A_{y} J)}) B{(J,A_{y})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('J', commutative=True), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('J', commutative=True), Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True))))"], [["times", 3, "Add(Mul(Symbol('J', commutative=True), Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True)))))"], "Equality(Mul(Add(Mul(Symbol('J', commutative=True), Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True))))), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('A_y', commutative=True))), Mul(Symbol('J', commutative=True), Add(Mul(Symbol('J', commutative=True), Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True)), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), cos(Mul(Symbol('A_y', commutative=True), Symbol('J', commutative=True))))), Function('B')(Symbol('J', commutative=True), Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given T{(n_{2})} = \\log{(n_{2})}, then obtain \\int (T^{2}{(n_{2})} + T{(n_{2})}) dn_{2} = \\int (T{(n_{2})} \\log{(n_{2})} + T{(n_{2})}) dn_{2}", "derivation": "T{(n_{2})} = \\log{(n_{2})} and T^{2}{(n_{2})} = T{(n_{2})} \\log{(n_{2})} and T^{2}{(n_{2})} + T{(n_{2})} = T{(n_{2})} \\log{(n_{2})} + T{(n_{2})} and \\int (T^{2}{(n_{2})} + T{(n_{2})}) dn_{2} = \\int (T{(n_{2})} \\log{(n_{2})} + T{(n_{2})}) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["times", 1, "Function('T')(Symbol('n_2', commutative=True))"], "Equality(Pow(Function('T')(Symbol('n_2', commutative=True)), Integer(2)), Mul(Function('T')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True))))"], [["add", 2, "Function('T')(Symbol('n_2', commutative=True))"], "Equality(Add(Pow(Function('T')(Symbol('n_2', commutative=True)), Integer(2)), Function('T')(Symbol('n_2', commutative=True))), Add(Mul(Function('T')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True))), Function('T')(Symbol('n_2', commutative=True))))"], [["integrate", 3, "Symbol('n_2', commutative=True)"], "Equality(Integral(Add(Pow(Function('T')(Symbol('n_2', commutative=True)), Integer(2)), Function('T')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Function('T')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True))), Function('T')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(m)} = \\cos{(m)} and \\operatorname{f_{E}}{(\\phi,t)} = t + e^{\\phi}, then obtain - t - e^{\\phi} = - t - \\dot{z}{(m)} - e^{\\phi} + \\cos{(m)}", "derivation": "\\dot{z}{(m)} = \\cos{(m)} and 0 = - \\dot{z}{(m)} + \\cos{(m)} and \\operatorname{f_{E}}{(\\phi,t)} = t + e^{\\phi} and - \\operatorname{f_{E}}{(\\phi,t)} = - \\dot{z}{(m)} - \\operatorname{f_{E}}{(\\phi,t)} + \\cos{(m)} and - \\operatorname{f_{E}}{(\\phi,t)} = - t - e^{\\phi} and - t - e^{\\phi} = - t - \\dot{z}{(m)} - e^{\\phi} + \\cos{(m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["minus", 1, "Function('\\\\dot{z}')(Symbol('m', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m', commutative=True))), cos(Symbol('m', commutative=True))))"], ["get_premise", "Equality(Function('f_E')(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Add(Symbol('t', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["minus", 2, "Function('f_E')(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(-1), Function('f_E')(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m', commutative=True))), Mul(Integer(-1), Function('f_E')(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True))), cos(Symbol('m', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_E')(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True))), cos(Symbol('m', commutative=True))))"]]}, {"prompt": "Given E{(y^{\\prime})} = \\cos{(e^{y^{\\prime}})}, then derive \\int E{(y^{\\prime})} dy^{\\prime} = A + \\operatorname{Ci}{(e^{y^{\\prime}})}, then derive \\pi + \\operatorname{Ci}{(e^{y^{\\prime}})} = A + \\operatorname{Ci}{(e^{y^{\\prime}})}, then obtain (\\int E{(y^{\\prime})} dy^{\\prime})^{y^{\\prime}} = (\\pi + \\operatorname{Ci}{(e^{y^{\\prime}})})^{y^{\\prime}}", "derivation": "E{(y^{\\prime})} = \\cos{(e^{y^{\\prime}})} and \\int E{(y^{\\prime})} dy^{\\prime} = \\int \\cos{(e^{y^{\\prime}})} dy^{\\prime} and (\\int E{(y^{\\prime})} dy^{\\prime})^{y^{\\prime}} = (\\int \\cos{(e^{y^{\\prime}})} dy^{\\prime})^{y^{\\prime}} and \\int E{(y^{\\prime})} dy^{\\prime} = A + \\operatorname{Ci}{(e^{y^{\\prime}})} and \\int \\cos{(e^{y^{\\prime}})} dy^{\\prime} = A + \\operatorname{Ci}{(e^{y^{\\prime}})} and \\pi + \\operatorname{Ci}{(e^{y^{\\prime}})} = A + \\operatorname{Ci}{(e^{y^{\\prime}})} and \\pi + \\operatorname{Ci}{(e^{y^{\\prime}})} = \\int \\cos{(e^{y^{\\prime}})} dy^{\\prime} and (\\int E{(y^{\\prime})} dy^{\\prime})^{y^{\\prime}} = (\\pi + \\operatorname{Ci}{(e^{y^{\\prime}})})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), cos(exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Integral(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('A', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('A', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\pi', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))), Add(Symbol('A', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('\\\\pi', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))), Integral(cos(exp(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Pow(Integral(Function('E')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Ci(exp(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(F_{x})} = \\cos{(F_{x})} and \\operatorname{v_{y}}{(F_{x})} = - \\cos{(F_{x})}, then obtain \\frac{d}{d F_{x}} - \\cos{(F_{x})} = \\frac{d}{d F_{x}} - \\operatorname{C_{1}}{(F_{x})}", "derivation": "\\operatorname{C_{1}}{(F_{x})} = \\cos{(F_{x})} and \\operatorname{v_{y}}{(F_{x})} = - \\cos{(F_{x})} and \\operatorname{v_{y}}{(F_{x})} = - \\operatorname{C_{1}}{(F_{x})} and - \\cos{(F_{x})} = - \\operatorname{C_{1}}{(F_{x})} and \\frac{d}{d F_{x}} - \\cos{(F_{x})} = \\frac{d}{d F_{x}} - \\operatorname{C_{1}}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('v_y')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Mul(Integer(-1), Function('C_1')(Symbol('F_x', commutative=True))))"], [["differentiate", 4, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('C_1')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(\\varphi,\\mathbf{A})} = \\varphi + e^{\\mathbf{A}} and \\rho{(\\mathbb{I},\\chi)} = \\chi^{\\mathbb{I}}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} (2 \\varphi + \\rho{(\\mathbb{I},\\chi)} + 2 e^{\\mathbf{A}}) = \\frac{\\partial}{\\partial \\mathbb{I}} (\\chi^{\\mathbb{I}} + 2 \\varphi + 2 e^{\\mathbf{A}})", "derivation": "\\Omega{(\\varphi,\\mathbf{A})} = \\varphi + e^{\\mathbf{A}} and \\rho{(\\mathbb{I},\\chi)} = \\chi^{\\mathbb{I}} and 2 \\Omega{(\\varphi,\\mathbf{A})} + \\rho{(\\mathbb{I},\\chi)} = \\chi^{\\mathbb{I}} + 2 \\Omega{(\\varphi,\\mathbf{A})} and 2 \\varphi + \\rho{(\\mathbb{I},\\chi)} + 2 e^{\\mathbf{A}} = \\chi^{\\mathbb{I}} + 2 \\varphi + 2 e^{\\mathbf{A}} and \\frac{\\partial}{\\partial \\mathbb{I}} (2 \\varphi + \\rho{(\\mathbb{I},\\chi)} + 2 e^{\\mathbf{A}}) = \\frac{\\partial}{\\partial \\mathbb{I}} (\\chi^{\\mathbb{I}} + 2 \\varphi + 2 e^{\\mathbf{A}})", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 2, "Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(2), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mathbf{A}', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(E_{\\lambda},v_{y})} = \\log{(E_{\\lambda} + v_{y})} and \\hat{p}_0{(E_{\\lambda},v_{y})} = \\log{(E_{\\lambda} + v_{y})}, then obtain \\frac{\\partial^{2}}{\\partial v_{y}\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},v_{y})} = \\frac{\\partial^{2}}{\\partial v_{y}\\partial E_{\\lambda}} \\log{(E_{\\lambda} + v_{y})}", "derivation": "\\hat{H}_{\\lambda}{(E_{\\lambda},v_{y})} = \\log{(E_{\\lambda} + v_{y})} and \\frac{\\partial}{\\partial E_{\\lambda}} \\hat{H}_{\\lambda}{(E_{\\lambda},v_{y})} = \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} + v_{y})} and \\hat{p}_0{(E_{\\lambda},v_{y})} = \\log{(E_{\\lambda} + v_{y})} and \\hat{p}_0{(E_{\\lambda},v_{y})} = \\hat{H}_{\\lambda}{(E_{\\lambda},v_{y})} and \\frac{\\partial}{\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},v_{y})} = \\frac{\\partial}{\\partial E_{\\lambda}} \\log{(E_{\\lambda} + v_{y})} and \\frac{\\partial^{2}}{\\partial v_{y}\\partial E_{\\lambda}} \\hat{p}_0{(E_{\\lambda},v_{y})} = \\frac{\\partial^{2}}{\\partial v_{y}\\partial E_{\\lambda}} \\log{(E_{\\lambda} + v_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), log(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), log(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(log(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\mathbf{J},v_{t})} = \\mathbf{J} + v_{t}, then obtain 2 \\mathbf{J} + 2 b{(\\mathbf{J},v_{t})} = 4 \\mathbf{J} + 2 v_{t}", "derivation": "b{(\\mathbf{J},v_{t})} = \\mathbf{J} + v_{t} and \\mathbf{J} + v_{t} + b{(\\mathbf{J},v_{t})} = 2 \\mathbf{J} + 2 v_{t} and 2 b{(\\mathbf{J},v_{t})} = 2 \\mathbf{J} + 2 v_{t} and 2 \\mathbf{J} + 2 b{(\\mathbf{J},v_{t})} = 4 \\mathbf{J} + 2 v_{t}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True), Function('b')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('b')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Function('b')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('v_t', commutative=True)))), Add(Mul(Integer(4), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given k{(U)} = e^{U}, then obtain k{(U)} + \\int k{(U)} dU = 2 z + k{(U)} + 2 e^{U} - \\int k{(U)} dU", "derivation": "k{(U)} = e^{U} and \\int k{(U)} dU = \\int e^{U} dU and e^{U} + \\int k{(U)} dU = e^{U} + \\int e^{U} dU and \\int e^{U} dU = - \\int k{(U)} dU + 2 \\int e^{U} dU and e^{U} + \\int k{(U)} dU = e^{U} - \\int k{(U)} dU + 2 \\int e^{U} dU and k{(U)} + \\int k{(U)} dU = k{(U)} - \\int k{(U)} dU + 2 \\int e^{U} dU and k{(U)} + \\int k{(U)} dU = 2 z + k{(U)} + 2 e^{U} - \\int k{(U)} dU", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["add", 2, "exp(Symbol('U', commutative=True))"], "Equality(Add(exp(Symbol('U', commutative=True)), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(exp(Symbol('U', commutative=True)), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], "Equality(Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(exp(Symbol('U', commutative=True)), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(exp(Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('k')(Symbol('U', commutative=True)), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Function('k')(Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Add(Function('k')(Symbol('U', commutative=True)), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Integer(2), Symbol('z', commutative=True)), Function('k')(Symbol('U', commutative=True)), Mul(Integer(2), exp(Symbol('U', commutative=True))), Mul(Integer(-1), Integral(Function('k')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given n{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}}, then obtain (n{(a^{\\dagger})} + n^{a^{\\dagger}}{(a^{\\dagger})}) (n{(a^{\\dagger})} + (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}}) = (n{(a^{\\dagger})} + (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}})^{2}", "derivation": "n{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and n^{a^{\\dagger}}{(a^{\\dagger})} = (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}} and n{(a^{\\dagger})} + n^{a^{\\dagger}}{(a^{\\dagger})} = n{(a^{\\dagger})} + (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}} and (n{(a^{\\dagger})} + n^{a^{\\dagger}}{(a^{\\dagger})}) (n{(a^{\\dagger})} + (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}}) = (n{(a^{\\dagger})} + (\\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}})^{a^{\\dagger}})^{2}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 2, "Function('n')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 3, "Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True)))), Pow(Add(Function('n')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('a^{\\\\dagger}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\omega{(\\delta)} = e^{\\delta}, then derive \\omega{(\\delta)} e^{\\delta} + e^{\\delta} \\frac{d}{d \\delta} \\omega{(\\delta)} = 2 e^{2 \\delta}, then obtain \\omega^{2}{(\\delta)} + \\omega{(\\delta)} \\frac{d}{d \\delta} \\omega{(\\delta)} = 2 \\omega^{2}{(\\delta)}", "derivation": "\\omega{(\\delta)} = e^{\\delta} and \\omega{(\\delta)} e^{\\delta} = e^{2 \\delta} and \\frac{d}{d \\delta} \\omega{(\\delta)} e^{\\delta} = \\frac{d}{d \\delta} e^{2 \\delta} and \\omega{(\\delta)} e^{\\delta} + e^{\\delta} \\frac{d}{d \\delta} \\omega{(\\delta)} = 2 e^{2 \\delta} and \\omega{(\\delta)} e^{\\delta} + e^{\\delta} \\frac{d}{d \\delta} \\omega{(\\delta)} = 2 \\omega{(\\delta)} e^{\\delta} and \\omega^{2}{(\\delta)} + \\omega{(\\delta)} \\frac{d}{d \\delta} \\omega{(\\delta)} = 2 \\omega^{2}{(\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(exp(Symbol('\\\\delta', commutative=True)), Derivative(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(exp(Symbol('\\\\delta', commutative=True)), Derivative(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Pow(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Derivative(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Mul(Integer(2), Pow(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(I)} = \\cos{(e^{I})} and \\operatorname{m_{s}}{(c)} = e^{c}, then obtain \\operatorname{m_{s}}{(c)} - \\cos^{2}{(e^{I})} = e^{c} - \\cos^{2}{(e^{I})}", "derivation": "\\operatorname{F_{H}}{(I)} = \\cos{(e^{I})} and \\operatorname{m_{s}}{(c)} = e^{c} and - \\operatorname{F_{H}}{(I)} \\cos{(e^{I})} + \\operatorname{m_{s}}{(c)} = - \\operatorname{F_{H}}{(I)} \\cos{(e^{I})} + e^{c} and \\operatorname{m_{s}}{(c)} - \\cos^{2}{(e^{I})} = e^{c} - \\cos^{2}{(e^{I})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('I', commutative=True)), cos(exp(Symbol('I', commutative=True))))"], ["get_premise", "Equality(Function('m_s')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["minus", 2, "Mul(Function('F_H')(Symbol('I', commutative=True)), cos(exp(Symbol('I', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('F_H')(Symbol('I', commutative=True)), cos(exp(Symbol('I', commutative=True)))), Function('m_s')(Symbol('c', commutative=True))), Add(Mul(Integer(-1), Function('F_H')(Symbol('I', commutative=True)), cos(exp(Symbol('I', commutative=True)))), exp(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('m_s')(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(cos(exp(Symbol('I', commutative=True))), Integer(2)))), Add(exp(Symbol('c', commutative=True)), Mul(Integer(-1), Pow(cos(exp(Symbol('I', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{B}{(v_{2})} = \\sin{(\\sin{(v_{2})})} and q{(v_{2})} = \\sin{(v_{2})}, then obtain (- m_{s} - v_{2} + \\mathbf{B}{(v_{2})})^{v_{2}} = (- m_{s} - v_{2} + \\sin{(q{(v_{2})})})^{v_{2}}", "derivation": "\\mathbf{B}{(v_{2})} = \\sin{(\\sin{(v_{2})})} and - v_{2} + \\mathbf{B}{(v_{2})} = - v_{2} + \\sin{(\\sin{(v_{2})})} and q{(v_{2})} = \\sin{(v_{2})} and - m_{s} - v_{2} + \\mathbf{B}{(v_{2})} = - m_{s} - v_{2} + \\sin{(\\sin{(v_{2})})} and (- m_{s} - v_{2} + \\mathbf{B}{(v_{2})})^{v_{2}} = (- m_{s} - v_{2} + \\sin{(\\sin{(v_{2})})})^{v_{2}} and (- m_{s} - v_{2} + \\mathbf{B}{(v_{2})})^{v_{2}} = (- m_{s} - v_{2} + \\sin{(q{(v_{2})})})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True)), sin(sin(Symbol('v_2', commutative=True))))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), sin(sin(Symbol('v_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["minus", 2, "Symbol('m_s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), sin(sin(Symbol('v_2', commutative=True)))))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), sin(sin(Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), Mul(Integer(-1), Symbol('v_2', commutative=True)), sin(Function('q')(Symbol('v_2', commutative=True)))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\psi)} = e^{\\psi}, then obtain \\int \\operatorname{C_{1}}{(\\psi)} e^{- \\psi} d\\psi = \\int 1 d\\psi", "derivation": "\\operatorname{C_{1}}{(\\psi)} = e^{\\psi} and 1 = \\frac{e^{\\psi}}{\\operatorname{C_{1}}{(\\psi)}} and \\operatorname{C_{1}}{(\\psi)} e^{- \\psi} = 1 and \\int \\operatorname{C_{1}}{(\\psi)} e^{- \\psi} d\\psi = \\int 1 d\\psi", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["divide", 1, "Function('C_1')(Symbol('\\\\psi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_1')(Symbol('\\\\psi', commutative=True)), Integer(-1)), exp(Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Mul(Pow(Function('C_1')(Symbol('\\\\psi', commutative=True)), Integer(-1)), exp(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Function('C_1')(Symbol('\\\\psi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))), Integer(1))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Mul(Function('C_1')(Symbol('\\\\psi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given b{(E_{n},L)} = \\log{(L^{E_{n}})}, then obtain \\frac{L^{- E_{n}} b{(E_{n},L)}}{\\sin{(b{(E_{n},L)})}} = \\frac{L^{- E_{n}} \\log{(L^{E_{n}})}}{\\sin{(b{(E_{n},L)})}}", "derivation": "b{(E_{n},L)} = \\log{(L^{E_{n}})} and \\sin{(b{(E_{n},L)})} = \\sin{(\\log{(L^{E_{n}})})} and \\frac{L^{- E_{n}} b{(E_{n},L)}}{\\sin{(\\log{(L^{E_{n}})})}} = \\frac{L^{- E_{n}} \\log{(L^{E_{n}})}}{\\sin{(\\log{(L^{E_{n}})})}} and \\frac{L^{- E_{n}} b{(E_{n},L)}}{\\sin{(b{(E_{n},L)})}} = \\frac{L^{- E_{n}} \\log{(L^{E_{n}})}}{\\sin{(b{(E_{n},L)})}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True))))"], [["sin", 1], "Equality(sin(Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), sin(log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True)))))"], [["divide", 1, "Mul(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True)), sin(log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True)))))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Pow(sin(log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True)))), Integer(-1))), Mul(Pow(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True))), Pow(sin(log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True)), Pow(sin(Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Integer(-1))), Mul(Pow(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('E_n', commutative=True))), log(Pow(Symbol('L', commutative=True), Symbol('E_n', commutative=True))), Pow(sin(Function('b')(Symbol('E_n', commutative=True), Symbol('L', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} = \\Omega - r_{0} - t_{2}, then obtain \\int \\operatorname{f_{E}}^{2}{(t_{2},r_{0},\\Omega)} dr_{0} = \\int (\\Omega \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - r_{0} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - t_{2} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)}) dr_{0}", "derivation": "\\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} = \\Omega - r_{0} - t_{2} and \\operatorname{f_{E}}^{2}{(t_{2},r_{0},\\Omega)} = (\\Omega - r_{0} - t_{2}) \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} and \\operatorname{f_{E}}^{2}{(t_{2},r_{0},\\Omega)} = \\Omega \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - r_{0} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - t_{2} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} and \\int \\operatorname{f_{E}}^{2}{(t_{2},r_{0},\\Omega)} dr_{0} = \\int (\\Omega \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - r_{0} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)} - t_{2} \\operatorname{f_{E}}{(t_{2},r_{0},\\Omega)}) dr_{0}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["times", 1, "Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True))), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["expand", 2], "Equality(Pow(Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Add(Mul(Symbol('\\\\Omega', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('r_0', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(Pow(Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Symbol('\\\\Omega', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('r_0', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('t_2', commutative=True), Function('f_E')(Symbol('t_2', commutative=True), Symbol('r_0', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})} = (\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}}, then obtain (\\hat{\\mathbf{x}} \\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} (\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}})^{\\hat{\\mathbf{x}}}", "derivation": "\\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})} = (\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}} and \\frac{\\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})}}{\\mathbf{A}} = \\frac{(\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}}}{\\mathbf{A}} and \\hat{\\mathbf{x}} \\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})} = \\hat{\\mathbf{x}} (\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}} and (\\hat{\\mathbf{x}} \\operatorname{y^{\\prime}}{(n_{2},\\hat{\\mathbf{x}},\\mathbf{A})})^{\\hat{\\mathbf{x}}} = (\\hat{\\mathbf{x}} (\\hat{\\mathbf{x}} \\mathbf{A})^{n_{2}})^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('n_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('n_2', commutative=True))))"], [["times", 2, "Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('n_2', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('n_2', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given z{(n)} = e^{n}, then obtain \\int (1 - e^{n}) dn = \\sin{(n (-1 + \\frac{e^{n}}{z{(n)}}))} + \\int (- e^{n} + \\frac{e^{n}}{z{(n)}}) dn", "derivation": "z{(n)} = e^{n} and 1 = \\frac{e^{n}}{z{(n)}} and 0 = -1 + \\frac{e^{n}}{z{(n)}} and 0 = n (-1 + \\frac{e^{n}}{z{(n)}}) and 1 - e^{n} = - e^{n} + \\frac{e^{n}}{z{(n)}} and \\int (1 - e^{n}) dn = \\int (- e^{n} + \\frac{e^{n}}{z{(n)}}) dn and 0 = \\sin{(n (-1 + \\frac{e^{n}}{z{(n)}}))} and \\int (- e^{n} + \\frac{e^{n}}{z{(n)}}) dn = \\sin{(n (-1 + \\frac{e^{n}}{z{(n)}}))} + \\int (- e^{n} + \\frac{e^{n}}{z{(n)}}) dn and \\int (1 - e^{n}) dn = \\sin{(n (-1 + \\frac{e^{n}}{z{(n)}}))} + \\int (- e^{n} + \\frac{e^{n}}{z{(n)}}) dn", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["divide", 1, "Function('z')(Symbol('n', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))))"], [["minus", 2, 1], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))"], [["times", 3, "Symbol('n', commutative=True)"], "Equality(Integer(0), Mul(Symbol('n', commutative=True), Add(Integer(-1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True))))))"], [["minus", 2, "exp(Symbol('n', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), exp(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))"], [["integrate", 5, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))))"], [["sin", 4], "Equality(Integer(0), sin(Mul(Symbol('n', commutative=True), Add(Integer(-1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))))"], [["add", 7, "Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))"], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Add(sin(Mul(Symbol('n', commutative=True), Add(Integer(-1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))), Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 8], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True))), Add(sin(Mul(Symbol('n', commutative=True), Add(Integer(-1), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))))), Integral(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(Pow(Function('z')(Symbol('n', commutative=True)), Integer(-1)), exp(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\omega,L)} = \\frac{L}{\\omega} and q{(\\omega,L)} = \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_P{(\\omega,L)}, then obtain \\int q^{2}{(\\omega,L)} d\\omega = \\int q{(\\omega,L)} \\frac{\\partial}{\\partial \\omega} \\frac{L}{\\omega} d\\omega", "derivation": "\\mathbf{J}_P{(\\omega,L)} = \\frac{L}{\\omega} and \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_P{(\\omega,L)} = \\frac{\\partial}{\\partial \\omega} \\frac{L}{\\omega} and q{(\\omega,L)} = \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_P{(\\omega,L)} and q{(\\omega,L)} = \\frac{\\partial}{\\partial \\omega} \\frac{L}{\\omega} and q^{2}{(\\omega,L)} = q{(\\omega,L)} \\frac{\\partial}{\\partial \\omega} \\frac{L}{\\omega} and \\int q^{2}{(\\omega,L)} d\\omega = \\int q{(\\omega,L)} \\frac{\\partial}{\\partial \\omega} \\frac{L}{\\omega} d\\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Derivative(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["times", 4, "Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True))"], "Equality(Pow(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Derivative(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Pow(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Integer(2)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Function('q')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Derivative(Mul(Symbol('L', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\varepsilon_0,Q)} = \\varepsilon_0^{Q}, then obtain Q \\varepsilon_0^{Q} \\iint Q \\mathbb{I}{(\\varepsilon_0,Q)} dQ d\\varepsilon_0 = Q \\varepsilon_0^{Q} \\iint Q \\varepsilon_0^{Q} dQ d\\varepsilon_0", "derivation": "\\mathbb{I}{(\\varepsilon_0,Q)} = \\varepsilon_0^{Q} and Q \\mathbb{I}{(\\varepsilon_0,Q)} = Q \\varepsilon_0^{Q} and \\int Q \\mathbb{I}{(\\varepsilon_0,Q)} dQ = \\int Q \\varepsilon_0^{Q} dQ and \\iint Q \\mathbb{I}{(\\varepsilon_0,Q)} dQ d\\varepsilon_0 = \\iint Q \\varepsilon_0^{Q} dQ d\\varepsilon_0 and Q \\varepsilon_0^{Q} \\iint Q \\mathbb{I}{(\\varepsilon_0,Q)} dQ d\\varepsilon_0 = Q \\varepsilon_0^{Q} \\iint Q \\varepsilon_0^{Q} dQ d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True)))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["times", 4, "Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True)), Integral(Mul(Symbol('Q', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True)), Integral(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\delta{(F_{c},\\ddot{x})} = F_{c} \\ddot{x} + F_{c}, then obtain (F_{c} \\ddot{x} + F_{c}) (\\int \\delta{(F_{c},\\ddot{x})} dF_{c})^{\\ddot{x}} = (F_{c} \\ddot{x} + F_{c}) (\\int (F_{c} \\ddot{x} + F_{c}) dF_{c})^{\\ddot{x}}", "derivation": "\\delta{(F_{c},\\ddot{x})} = F_{c} \\ddot{x} + F_{c} and \\int \\delta{(F_{c},\\ddot{x})} dF_{c} = \\int (F_{c} \\ddot{x} + F_{c}) dF_{c} and (\\int \\delta{(F_{c},\\ddot{x})} dF_{c})^{\\ddot{x}} = (\\int (F_{c} \\ddot{x} + F_{c}) dF_{c})^{\\ddot{x}} and (F_{c} \\ddot{x} + F_{c}) (\\int \\delta{(F_{c},\\ddot{x})} dF_{c})^{\\ddot{x}} = (F_{c} \\ddot{x} + F_{c}) (\\int (F_{c} \\ddot{x} + F_{c}) dF_{c})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Integral(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 3, "Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)), Pow(Integral(Function('\\\\delta')(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))), Mul(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)), Pow(Integral(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{A},M_{E})} = \\mathbf{A}^{M_{E}}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\bar{\\h}{(\\mathbf{A},M_{E})} = \\frac{M_{E} \\mathbf{A}^{M_{E}}}{\\mathbf{A}}, then obtain \\sin{(\\frac{M_{E} \\mathbf{A}^{M_{E}}}{\\mathbf{A}})} = \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{M_{E}})}", "derivation": "\\bar{\\h}{(\\mathbf{A},M_{E})} = \\mathbf{A}^{M_{E}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\bar{\\h}{(\\mathbf{A},M_{E})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{M_{E}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\bar{\\h}{(\\mathbf{A},M_{E})} = \\frac{M_{E} \\mathbf{A}^{M_{E}}}{\\mathbf{A}} and \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\bar{\\h}{(\\mathbf{A},M_{E})})} = \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{M_{E}})} and \\sin{(\\frac{M_{E} \\mathbf{A}^{M_{E}}}{\\mathbf{A}})} = \\sin{(\\frac{\\partial}{\\partial \\mathbf{A}} \\mathbf{A}^{M_{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\hbar')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), sin(Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(sin(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)))), sin(Derivative(Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(r_{0},p)} = r_{0}^{p}, then obtain \\frac{\\partial}{\\partial p} \\sin{(r_{0}^{p} + h{(r_{0},p)})} = \\frac{\\partial}{\\partial p} \\sin{(2 r_{0}^{p})}", "derivation": "h{(r_{0},p)} = r_{0}^{p} and r_{0}^{p} + h{(r_{0},p)} = 2 r_{0}^{p} and \\sin{(r_{0}^{p} + h{(r_{0},p)})} = \\sin{(2 r_{0}^{p})} and \\frac{\\partial}{\\partial p} \\sin{(r_{0}^{p} + h{(r_{0},p)})} = \\frac{\\partial}{\\partial p} \\sin{(2 r_{0}^{p})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('r_0', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)))"], [["add", 1, "Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)), Function('h')(Symbol('r_0', commutative=True), Symbol('p', commutative=True))), Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True))))"], [["sin", 2], "Equality(sin(Add(Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)), Function('h')(Symbol('r_0', commutative=True), Symbol('p', commutative=True)))), sin(Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(sin(Add(Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)), Function('h')(Symbol('r_0', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(sin(Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Symbol('p', commutative=True)))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(h,x)} = h + x, then obtain 12 h \\operatorname{c_{0}}{(h,x)} = 3 h (3 h + 3 x + \\operatorname{c_{0}}{(h,x)})", "derivation": "\\operatorname{c_{0}}{(h,x)} = h + x and 2 \\operatorname{c_{0}}{(h,x)} = h + x + \\operatorname{c_{0}}{(h,x)} and h + x + 3 \\operatorname{c_{0}}{(h,x)} = 2 h + 2 x + 2 \\operatorname{c_{0}}{(h,x)} and 4 \\operatorname{c_{0}}{(h,x)} = 2 h + 2 x + 2 \\operatorname{c_{0}}{(h,x)} and 12 h \\operatorname{c_{0}}{(h,x)} = 3 h (2 h + 2 x + 2 \\operatorname{c_{0}}{(h,x)}) and 6 h (h + x + \\operatorname{c_{0}}{(h,x)}) = 3 h (3 h + 3 x + \\operatorname{c_{0}}{(h,x)}) and 12 h \\operatorname{c_{0}}{(h,x)} = 3 h (3 h + 3 x + \\operatorname{c_{0}}{(h,x)})", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)), Add(Symbol('h', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))), Add(Symbol('h', commutative=True), Symbol('x', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))))"], [["add", 2, "Add(Symbol('h', commutative=True), Symbol('x', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))"], "Equality(Add(Symbol('h', commutative=True), Symbol('x', commutative=True), Mul(Integer(3), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))), Add(Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(4), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))), Add(Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))))"], [["times", 4, "Mul(Integer(3), Symbol('h', commutative=True))"], "Equality(Mul(Integer(12), Symbol('h', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))), Mul(Integer(3), Symbol('h', commutative=True), Add(Mul(Integer(2), Symbol('h', commutative=True)), Mul(Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(6), Symbol('h', commutative=True), Add(Symbol('h', commutative=True), Symbol('x', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))), Mul(Integer(3), Symbol('h', commutative=True), Add(Mul(Integer(3), Symbol('h', commutative=True)), Mul(Integer(3), Symbol('x', commutative=True)), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Integer(12), Symbol('h', commutative=True), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True))), Mul(Integer(3), Symbol('h', commutative=True), Add(Mul(Integer(3), Symbol('h', commutative=True)), Mul(Integer(3), Symbol('x', commutative=True)), Function('c_0')(Symbol('h', commutative=True), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given i{(t)} = t, then derive 2 \\int i{(t)} dt = A_{1} + \\frac{t^{2}}{2} + \\int i{(t)} dt, then obtain 2 \\int t dt = A_{1} + \\frac{t^{2}}{2} + \\int t dt", "derivation": "i{(t)} = t and \\int i{(t)} dt = \\int t dt and 2 \\int i{(t)} dt = \\int t dt + \\int i{(t)} dt and 2 \\int i{(t)} dt = A_{1} + \\frac{t^{2}}{2} + \\int i{(t)} dt and 2 \\int t dt = A_{1} + \\frac{t^{2}}{2} + \\int t dt", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('t', commutative=True)), Symbol('t', commutative=True))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True))))"], [["add", 2, "Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Integral(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True))), Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Integral(Function('i')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Integral(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Integral(Symbol('t', commutative=True), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given M{(\\mathbf{J}_f,Q)} = Q \\mathbf{J}_f, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)} = Q, then obtain Q^{2} (- Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)}) = Q^{2} (- Q^{2} + Q)", "derivation": "M{(\\mathbf{J}_f,Q)} = Q \\mathbf{J}_f and \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} Q \\mathbf{J}_f and - Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)} = - Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} Q \\mathbf{J}_f and Q^{2} (- Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)}) = Q^{2} (- Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} Q \\mathbf{J}_f) and \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)} = Q and - Q^{2} + Q = - Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} Q \\mathbf{J}_f and Q^{2} (- Q^{2} + \\frac{\\partial}{\\partial \\mathbf{J}_f} M{(\\mathbf{J}_f,Q)}) = Q^{2} (- Q^{2} + Q)", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Symbol('Q', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["times", 3, "Pow(Symbol('Q', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))), Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Symbol('Q', commutative=True))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Derivative(Function('M')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))), Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = (V_{\\mathbf{E}}^{c})^{F_{g}}, then derive \\frac{\\partial}{\\partial c} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = F_{g} (V_{\\mathbf{E}}^{c})^{F_{g}} \\log{(V_{\\mathbf{E}})}, then obtain \\frac{\\partial}{\\partial c} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = F_{g} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} \\log{(V_{\\mathbf{E}})}", "derivation": "\\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = (V_{\\mathbf{E}}^{c})^{F_{g}} and \\frac{\\partial}{\\partial c} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial c} (V_{\\mathbf{E}}^{c})^{F_{g}} and \\frac{\\partial}{\\partial c} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = F_{g} (V_{\\mathbf{E}}^{c})^{F_{g}} \\log{(V_{\\mathbf{E}})} and \\frac{\\partial}{\\partial c} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} = F_{g} \\mathbf{A}{(F_{g},c,V_{\\mathbf{E}})} \\log{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('c', commutative=True)), Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('c', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Symbol('F_g', commutative=True), Pow(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('c', commutative=True)), Symbol('F_g', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Symbol('F_g', commutative=True), Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('c', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given B{(x,W)} = e^{W^{x}}, then obtain \\frac{x \\frac{d}{d \\mathbf{J}} \\psi{(\\mathbf{J})} \\iint B{(x,W)} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx} = \\frac{x \\frac{d}{d \\mathbf{J}} \\psi{(\\mathbf{J})} \\iint e^{W^{x}} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx}", "derivation": "B{(x,W)} = e^{W^{x}} and \\int B{(x,W)} dW = \\int e^{W^{x}} dW and \\iint B{(x,W)} dW dx = \\iint e^{W^{x}} dW dx and x \\iint B{(x,W)} dW dx = x \\iint e^{W^{x}} dW dx and \\frac{x \\iint B{(x,W)} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx} = \\frac{x \\iint e^{W^{x}} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx} and \\frac{x \\frac{d}{d \\mathbf{J}} \\psi{(\\mathbf{J})} \\iint B{(x,W)} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx} = \\frac{x \\frac{d}{d \\mathbf{J}} \\psi{(\\mathbf{J})} \\iint e^{W^{x}} dW dx}{\\frac{\\partial}{\\partial W} \\iint e^{W^{x}} dW dx}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["times", 3, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Integral(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Symbol('x', commutative=True), Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["divide", 4, "Derivative(Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('x', commutative=True), Pow(Derivative(Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Integral(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Symbol('x', commutative=True), Pow(Derivative(Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["times", 5, "Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('x', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Pow(Derivative(Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Integral(Function('B')(Symbol('x', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Symbol('x', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Pow(Derivative(Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Integral(exp(Pow(Symbol('W', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(x)} = \\sin{(x)}, then derive \\frac{d}{d x} \\hat{x}_0{(x)} = \\cos{(x)}, then obtain \\sin{(x)} + \\frac{d}{d x} \\hat{x}_0{(x)} = \\sin{(x)} + \\cos{(x)}", "derivation": "\\hat{x}_0{(x)} = \\sin{(x)} and \\frac{d}{d x} \\hat{x}_0{(x)} = \\frac{d}{d x} \\sin{(x)} and \\frac{d}{d x} \\hat{x}_0{(x)} = \\cos{(x)} and \\sin{(x)} + \\frac{d}{d x} \\hat{x}_0{(x)} = \\sin{(x)} + \\frac{d}{d x} \\sin{(x)} and \\sin{(x)} + \\cos{(x)} = \\sin{(x)} + \\frac{d}{d x} \\sin{(x)} and \\sin{(x)} + \\frac{d}{d x} \\hat{x}_0{(x)} = \\sin{(x)} + \\cos{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), cos(Symbol('x', commutative=True)))"], [["minus", 2, "Mul(Integer(-1), sin(Symbol('x', commutative=True)))"], "Equality(Add(sin(Symbol('x', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(sin(Symbol('x', commutative=True)), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(sin(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Add(sin(Symbol('x', commutative=True)), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(sin(Symbol('x', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(sin(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given y{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then derive \\int y{(a^{\\dagger})} da^{\\dagger} = \\phi_1 + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger}, then obtain \\frac{d}{d \\phi_1} \\int y{(a^{\\dagger})} da^{\\dagger} = \\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + a^{\\dagger} y{(a^{\\dagger})} - a^{\\dagger})", "derivation": "y{(a^{\\dagger})} = \\log{(a^{\\dagger})} and \\int y{(a^{\\dagger})} da^{\\dagger} = \\int \\log{(a^{\\dagger})} da^{\\dagger} and \\int y{(a^{\\dagger})} da^{\\dagger} = \\phi_1 + a^{\\dagger} \\log{(a^{\\dagger})} - a^{\\dagger} and \\int y{(a^{\\dagger})} da^{\\dagger} = \\phi_1 + a^{\\dagger} y{(a^{\\dagger})} - a^{\\dagger} and \\frac{d}{d \\phi_1} \\int y{(a^{\\dagger})} da^{\\dagger} = \\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + a^{\\dagger} y{(a^{\\dagger})} - a^{\\dagger})", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('y')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('y')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('y')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Integral(Function('y')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_1', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('y')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(F_{c})} = \\int \\cos{(F_{c})} dF_{c}, then obtain - \\int q{(F_{c})} dF_{c} + (\\int q{(F_{c})} dF_{c})^{F_{c}} = - \\int q{(F_{c})} dF_{c} + (\\iint \\cos{(F_{c})} dF_{c} dF_{c})^{F_{c}}", "derivation": "q{(F_{c})} = \\int \\cos{(F_{c})} dF_{c} and \\int q{(F_{c})} dF_{c} = \\iint \\cos{(F_{c})} dF_{c} dF_{c} and (\\int q{(F_{c})} dF_{c})^{F_{c}} = (\\iint \\cos{(F_{c})} dF_{c} dF_{c})^{F_{c}} and - \\int q{(F_{c})} dF_{c} + (\\int q{(F_{c})} dF_{c})^{F_{c}} = - \\int q{(F_{c})} dF_{c} + (\\iint \\cos{(F_{c})} dF_{c} dF_{c})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('F_c', commutative=True)), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["minus", 3, "Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Pow(Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Integral(Function('q')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Pow(Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(a^{\\dagger},A,a)} = A a - a^{\\dagger}, then obtain \\frac{a (a^{\\dagger} \\eta^{\\prime}{(a^{\\dagger},A,a)})^{A} \\eta^{\\prime}{(a^{\\dagger},A,a)}}{a^{\\dagger}} = \\frac{a (a^{\\dagger} (A a - a^{\\dagger}))^{A} \\eta^{\\prime}{(a^{\\dagger},A,a)}}{a^{\\dagger}}", "derivation": "\\eta^{\\prime}{(a^{\\dagger},A,a)} = A a - a^{\\dagger} and a^{\\dagger} \\eta^{\\prime}{(a^{\\dagger},A,a)} = a^{\\dagger} (A a - a^{\\dagger}) and (a^{\\dagger} \\eta^{\\prime}{(a^{\\dagger},A,a)})^{A} = (a^{\\dagger} (A a - a^{\\dagger}))^{A} and \\frac{(a^{\\dagger} \\eta^{\\prime}{(a^{\\dagger},A,a)})^{A}}{a^{\\dagger}} = \\frac{(a^{\\dagger} (A a - a^{\\dagger}))^{A}}{a^{\\dagger}} and \\frac{a (a^{\\dagger} \\eta^{\\prime}{(a^{\\dagger},A,a)})^{A} \\eta^{\\prime}{(a^{\\dagger},A,a)}}{a^{\\dagger}} = \\frac{a (a^{\\dagger} (A a - a^{\\dagger}))^{A} \\eta^{\\prime}{(a^{\\dagger},A,a)}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True)), Add(Mul(Symbol('A', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('A', commutative=True)))"], [["divide", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))), Symbol('A', commutative=True))), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('A', commutative=True))))"], [["times", 4, "Mul(Symbol('a', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True)))"], "Equality(Mul(Symbol('a', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))), Symbol('A', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Mul(Symbol('A', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('A', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A', commutative=True), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{t})} = e^{e^{v_{t}}}, then obtain \\operatorname{f^{\\prime}}{(v_{t})} e^{- v_{t}} - e^{e^{v_{t}}} \\sin{(s)} = - e^{e^{v_{t}}} \\sin{(s)} + e^{- v_{t}} e^{e^{v_{t}}}", "derivation": "\\operatorname{f^{\\prime}}{(v_{t})} = e^{e^{v_{t}}} and \\operatorname{f^{\\prime}}{(v_{t})} e^{- v_{t}} = e^{- v_{t}} e^{e^{v_{t}}} and \\operatorname{f^{\\prime}}{(v_{t})} \\sin{(s)} = e^{e^{v_{t}}} \\sin{(s)} and - \\operatorname{f^{\\prime}}{(v_{t})} \\sin{(s)} + \\operatorname{f^{\\prime}}{(v_{t})} e^{- v_{t}} = - \\operatorname{f^{\\prime}}{(v_{t})} \\sin{(s)} + e^{- v_{t}} e^{e^{v_{t}}} and \\operatorname{f^{\\prime}}{(v_{t})} e^{- v_{t}} - e^{e^{v_{t}}} \\sin{(s)} = - e^{e^{v_{t}}} \\sin{(s)} + e^{- v_{t}} e^{e^{v_{t}}}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), exp(exp(Symbol('v_t', commutative=True))))"], [["divide", 1, "exp(Symbol('v_t', commutative=True))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('v_t', commutative=True))), exp(exp(Symbol('v_t', commutative=True)))))"], [["times", 1, "sin(Symbol('s', commutative=True))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), sin(Symbol('s', commutative=True))), Mul(exp(exp(Symbol('v_t', commutative=True))), sin(Symbol('s', commutative=True))))"], [["minus", 2, "Mul(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), sin(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), sin(Symbol('s', commutative=True))), Mul(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True))))), Add(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), sin(Symbol('s', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('v_t', commutative=True))), exp(exp(Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Function('f^{\\\\prime}')(Symbol('v_t', commutative=True)), exp(Mul(Integer(-1), Symbol('v_t', commutative=True)))), Mul(Integer(-1), exp(exp(Symbol('v_t', commutative=True))), sin(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), exp(exp(Symbol('v_t', commutative=True))), sin(Symbol('s', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('v_t', commutative=True))), exp(exp(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(t_{2},q)} = \\frac{q}{t_{2}} and \\operatorname{E_{x}}{(t_{2})} = \\frac{1}{t_{2}}, then obtain t_{2} (\\operatorname{E_{x}}{(t_{2})} - \\operatorname{g_{\\varepsilon}}{(t_{2},q)}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)} = t_{2} (- \\operatorname{g_{\\varepsilon}}{(t_{2},q)} + \\frac{1}{t_{2}}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(t_{2},q)} = \\frac{q}{t_{2}} and \\operatorname{E_{x}}{(t_{2})} = \\frac{1}{t_{2}} and - \\frac{q}{t_{2}} + \\operatorname{E_{x}}{(t_{2})} = - \\frac{q}{t_{2}} + \\frac{1}{t_{2}} and t_{2} (- \\frac{q}{t_{2}} + \\operatorname{E_{x}}{(t_{2})}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)} = t_{2} (- \\frac{q}{t_{2}} + \\frac{1}{t_{2}}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)} and t_{2} (\\operatorname{E_{x}}{(t_{2})} - \\operatorname{g_{\\varepsilon}}{(t_{2},q)}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)} = t_{2} (- \\operatorname{g_{\\varepsilon}}{(t_{2},q)} + \\frac{1}{t_{2}}) \\operatorname{g_{\\varepsilon}}{(t_{2},q)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('t_2', commutative=True)), Pow(Symbol('t_2', commutative=True), Integer(-1)))"], [["minus", 2, "Mul(Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Function('E_x')(Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["times", 3, "Mul(Symbol('t_2', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Symbol('t_2', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Function('E_x')(Symbol('t_2', commutative=True))), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True))), Mul(Symbol('t_2', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Pow(Symbol('t_2', commutative=True), Integer(-1))), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('t_2', commutative=True), Add(Function('E_x')(Symbol('t_2', commutative=True)), Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True)))), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True))), Mul(Symbol('t_2', commutative=True), Add(Mul(Integer(-1), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True))), Pow(Symbol('t_2', commutative=True), Integer(-1))), Function('g_{\\\\varepsilon}')(Symbol('t_2', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\hat{x}_0)} = e^{\\sin{(\\hat{x}_0)}}, then derive - e^{\\sin{(\\hat{x}_0)}} \\cos{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\mathbf{S}{(\\hat{x}_0)} = 0, then obtain \\int (- \\mathbf{S}{(\\hat{x}_0)} \\cos{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\mathbf{S}{(\\hat{x}_0)}) d\\hat{x}_0 = \\int 0 d\\hat{x}_0", "derivation": "\\mathbf{S}{(\\hat{x}_0)} = e^{\\sin{(\\hat{x}_0)}} and \\mathbf{S}{(\\hat{x}_0)} - e^{\\sin{(\\hat{x}_0)}} = 0 and \\frac{d}{d \\hat{x}_0} (\\mathbf{S}{(\\hat{x}_0)} - e^{\\sin{(\\hat{x}_0)}}) = \\frac{d}{d \\hat{x}_0} 0 and - e^{\\sin{(\\hat{x}_0)}} \\cos{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\mathbf{S}{(\\hat{x}_0)} = 0 and - \\mathbf{S}{(\\hat{x}_0)} \\cos{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\mathbf{S}{(\\hat{x}_0)} = 0 and \\int (- \\mathbf{S}{(\\hat{x}_0)} \\cos{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\mathbf{S}{(\\hat{x}_0)}) d\\hat{x}_0 = \\int 0 d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(sin(Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 1, "exp(sin(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('\\\\hat{x}_0', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('\\\\hat{x}_0', commutative=True))))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(sin(Symbol('\\\\hat{x}_0', commutative=True))), cos(Symbol('\\\\hat{x}_0', commutative=True))), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 5, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given C{(F_{g},\\dot{x},y)} = \\frac{- F_{g} + \\dot{x}}{y} and \\operatorname{m_{s}}{(F_{g},\\dot{x},y)} = F_{g} (F_{g} + \\frac{- F_{g} + \\dot{x}}{y}), then obtain (F_{g} (F_{g} + C{(F_{g},\\dot{x},y)}))^{F_{g}} = \\operatorname{m_{s}}^{F_{g}}{(F_{g},\\dot{x},y)}", "derivation": "C{(F_{g},\\dot{x},y)} = \\frac{- F_{g} + \\dot{x}}{y} and F_{g} + C{(F_{g},\\dot{x},y)} = F_{g} + \\frac{- F_{g} + \\dot{x}}{y} and F_{g} (F_{g} + C{(F_{g},\\dot{x},y)}) = F_{g} (F_{g} + \\frac{- F_{g} + \\dot{x}}{y}) and (F_{g} (F_{g} + C{(F_{g},\\dot{x},y)}))^{F_{g}} = (F_{g} (F_{g} + \\frac{- F_{g} + \\dot{x}}{y}))^{F_{g}} and \\operatorname{m_{s}}{(F_{g},\\dot{x},y)} = F_{g} (F_{g} + \\frac{- F_{g} + \\dot{x}}{y}) and (F_{g} (F_{g} + C{(F_{g},\\dot{x},y)}))^{F_{g}} = \\operatorname{m_{s}}^{F_{g}}{(F_{g},\\dot{x},y)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('F_g', commutative=True))"], "Equality(Add(Symbol('F_g', commutative=True), Function('C')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))))"], [["times", 2, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Function('C')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)))), Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Function('C')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)))), Symbol('F_g', commutative=True)), Pow(Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))), Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Symbol('F_g', commutative=True), Add(Symbol('F_g', commutative=True), Function('C')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)))), Symbol('F_g', commutative=True)), Pow(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('y', commutative=True)), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})} and C{(\\rho_f,s)} = \\rho_f s, then obtain \\log{((s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}) C{(\\rho_f,s)})} = \\log{(\\rho_f s (s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}))}", "derivation": "\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})} and C{(\\rho_f,s)} = \\rho_f s and s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})} = s + \\sin{(\\dot{\\mathbf{r}})} and (s + \\sin{(\\dot{\\mathbf{r}})}) C{(\\rho_f,s)} = \\rho_f s (s + \\sin{(\\dot{\\mathbf{r}})}) and (s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}) C{(\\rho_f,s)} = \\rho_f s (s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}) and \\log{((s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}) C{(\\rho_f,s)})} = \\log{(\\rho_f s (s + \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}))}", "srepr_derivation": [["get_premise", "Equality(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["get_premise", "Equality(Function('C')(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True)))"], [["add", 1, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('s', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["times", 2, "Add(Symbol('s', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Add(Symbol('s', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('C')(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True), Add(Symbol('s', commutative=True), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('s', commutative=True), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('C')(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True), Add(Symbol('s', commutative=True), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["log", 5], "Equality(log(Mul(Add(Symbol('s', commutative=True), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('C')(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True)))), log(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('s', commutative=True), Add(Symbol('s', commutative=True), Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{J})} = \\cos{(\\cos{(\\mathbf{J})})}, then obtain \\mathbf{J} (\\mathbf{J} + \\varepsilon{(\\mathbf{J})} + \\cos{(\\cos{(\\mathbf{J})})}) = \\mathbf{J} (\\mathbf{J} + 2 \\cos{(\\cos{(\\mathbf{J})})})", "derivation": "\\varepsilon{(\\mathbf{J})} = \\cos{(\\cos{(\\mathbf{J})})} and \\mathbf{J} + \\varepsilon{(\\mathbf{J})} = \\mathbf{J} + \\cos{(\\cos{(\\mathbf{J})})} and \\mathbf{J} + \\varepsilon{(\\mathbf{J})} + \\cos{(\\cos{(\\mathbf{J})})} = \\mathbf{J} + 2 \\cos{(\\cos{(\\mathbf{J})})} and \\mathbf{J} (\\mathbf{J} + \\varepsilon{(\\mathbf{J})} + \\cos{(\\cos{(\\mathbf{J})})}) = \\mathbf{J} (\\mathbf{J} + 2 \\cos{(\\cos{(\\mathbf{J})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}', commutative=True)), cos(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["add", 2, "cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}', commutative=True)), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), cos(cos(Symbol('\\\\mathbf{J}', commutative=True))))))"], [["times", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}', commutative=True)), cos(cos(Symbol('\\\\mathbf{J}', commutative=True))))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(2), cos(cos(Symbol('\\\\mathbf{J}', commutative=True)))))))"]]}, {"prompt": "Given W{(v_{x})} = e^{\\cos{(v_{x})}} and Z{(v_{x})} = \\cos{(v_{x})}, then derive \\frac{d}{d v_{x}} W{(v_{x})} = - e^{\\cos{(v_{x})}} \\sin{(v_{x})}, then obtain \\frac{d^{2}}{d v_{x}^{2}} e^{Z{(v_{x})}} = \\frac{d}{d v_{x}} - e^{Z{(v_{x})}} \\sin{(v_{x})}", "derivation": "W{(v_{x})} = e^{\\cos{(v_{x})}} and \\frac{d}{d v_{x}} W{(v_{x})} = \\frac{d}{d v_{x}} e^{\\cos{(v_{x})}} and \\frac{d}{d v_{x}} W{(v_{x})} = - e^{\\cos{(v_{x})}} \\sin{(v_{x})} and Z{(v_{x})} = \\cos{(v_{x})} and \\frac{d}{d v_{x}} e^{\\cos{(v_{x})}} = - e^{\\cos{(v_{x})}} \\sin{(v_{x})} and \\frac{d}{d v_{x}} e^{Z{(v_{x})}} = - e^{Z{(v_{x})}} \\sin{(v_{x})} and \\frac{d^{2}}{d v_{x}^{2}} e^{Z{(v_{x})}} = \\frac{d}{d v_{x}} - e^{Z{(v_{x})}} \\sin{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('v_x', commutative=True)), exp(cos(Symbol('v_x', commutative=True))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('v_x', commutative=True))), sin(Symbol('v_x', commutative=True))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(cos(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('v_x', commutative=True))), sin(Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(exp(Function('Z')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), exp(Function('Z')(Symbol('v_x', commutative=True))), sin(Symbol('v_x', commutative=True))))"], [["differentiate", 6, "Symbol('v_x', commutative=True)"], "Equality(Derivative(exp(Function('Z')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), exp(Function('Z')(Symbol('v_x', commutative=True))), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\log{(\\mathbf{M} + z)} and m{(z,\\mathbf{M})} = \\mathbf{M} + z, then derive \\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\frac{1}{\\mathbf{M} + z}, then obtain (\\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})})^{\\mathbf{M}} = (\\frac{1}{m{(z,\\mathbf{M})}})^{\\mathbf{M}}", "derivation": "\\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\log{(\\mathbf{M} + z)} and \\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\frac{\\partial}{\\partial z} \\log{(\\mathbf{M} + z)} and m{(z,\\mathbf{M})} = \\mathbf{M} + z and \\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\frac{1}{\\mathbf{M} + z} and \\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})} = \\frac{1}{m{(z,\\mathbf{M})}} and (\\frac{\\partial}{\\partial z} \\operatorname{f^{\\prime}}{(z,\\mathbf{M})})^{\\mathbf{M}} = (\\frac{1}{m{(z,\\mathbf{M})}})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('m')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Pow(Function('m')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)))"], [["power", 5, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Derivative(Function('f^{\\\\prime}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Pow(Function('m')(Symbol('z', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given i{(\\hbar)} = e^{\\cos{(\\hbar)}} and \\mathbf{J}_f{(\\hbar)} = \\frac{e^{\\cos{(\\hbar)}} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})}}{\\cos{(\\hbar)}}, then obtain \\mathbf{J}_f{(\\hbar)} = \\frac{i{(\\hbar)} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})}}{\\cos{(\\hbar)}}", "derivation": "i{(\\hbar)} = e^{\\cos{(\\hbar)}} and i{(\\hbar)} \\cos{(\\hbar)} = e^{\\cos{(\\hbar)}} \\cos{(\\hbar)} and i{(\\hbar)} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})} = e^{\\cos{(\\hbar)}} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})} and \\mathbf{J}_f{(\\hbar)} = \\frac{e^{\\cos{(\\hbar)}} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})}}{\\cos{(\\hbar)}} and \\mathbf{J}_f{(\\hbar)} = \\frac{i{(\\hbar)} \\cos{(\\hbar)} - \\cos{(i{(\\hbar)})}}{\\cos{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\hbar', commutative=True)), exp(cos(Symbol('\\\\hbar', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Function('i')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Mul(exp(cos(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "cos(Function('i')(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Function('i')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Function('i')(Symbol('\\\\hbar', commutative=True))))), Add(Mul(exp(cos(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Function('i')(Symbol('\\\\hbar', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Mul(Add(Mul(exp(cos(Symbol('\\\\hbar', commutative=True))), cos(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Function('i')(Symbol('\\\\hbar', commutative=True))))), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), Mul(Add(Mul(Function('i')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Function('i')(Symbol('\\\\hbar', commutative=True))))), Pow(cos(Symbol('\\\\hbar', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given T{(C_{1},c_{0})} = c_{0}^{C_{1}}, then obtain c_{0}^{C_{1}} T{(C_{1},c_{0})} (\\int - T{(C_{1},c_{0})} dC_{1})^{c_{0}} = c_{0}^{C_{1}} T{(C_{1},c_{0})} (\\int - c_{0}^{C_{1}} dC_{1})^{c_{0}}", "derivation": "T{(C_{1},c_{0})} = c_{0}^{C_{1}} and - T{(C_{1},c_{0})} = - c_{0}^{C_{1}} and \\int - T{(C_{1},c_{0})} dC_{1} = \\int - c_{0}^{C_{1}} dC_{1} and (\\int - T{(C_{1},c_{0})} dC_{1})^{c_{0}} = (\\int - c_{0}^{C_{1}} dC_{1})^{c_{0}} and c_{0}^{C_{1}} T{(C_{1},c_{0})} (\\int - T{(C_{1},c_{0})} dC_{1})^{c_{0}} = c_{0}^{C_{1}} T{(C_{1},c_{0})} (\\int - c_{0}^{C_{1}} dC_{1})^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('c_0', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('c_0', commutative=True)))"], [["times", 4, "Mul(Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True)), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)))"], "Equality(Mul(Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True)), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Pow(Integral(Mul(Integer(-1), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('c_0', commutative=True))), Mul(Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True)), Function('T')(Symbol('C_1', commutative=True), Symbol('c_0', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('c_0', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\phi_1,L)} = L \\phi_1, then obtain \\frac{e^{- L + \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L}}}{\\cos{(L \\phi_1)}} = \\frac{e^{- L + 2 \\phi_1}}{\\cos{(L \\phi_1)}}", "derivation": "\\operatorname{A_{y}}{(\\phi_1,L)} = L \\phi_1 and \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L} = \\phi_1 and \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L} = 2 \\phi_1 and \\cos{(\\operatorname{A_{y}}{(\\phi_1,L)})} = \\cos{(L \\phi_1)} and - L + \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L} = - L + 2 \\phi_1 and e^{- L + \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L}} = e^{- L + 2 \\phi_1} and \\frac{e^{- L + \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L}}}{\\cos{(\\operatorname{A_{y}}{(\\phi_1,L)})}} = \\frac{e^{- L + 2 \\phi_1}}{\\cos{(\\operatorname{A_{y}}{(\\phi_1,L)})}} and \\frac{e^{- L + \\phi_1 + \\frac{\\operatorname{A_{y}}{(\\phi_1,L)}}{L}}}{\\cos{(L \\phi_1)}} = \\frac{e^{- L + 2 \\phi_1}}{\\cos{(L \\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))), Symbol('\\\\phi_1', commutative=True))"], [["add", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True)))), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))"], [["cos", 1], "Equality(cos(Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))), cos(Mul(Symbol('L', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 3, "Symbol('L', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["exp", 5], "Equality(exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))))), exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))))"], [["divide", 6, "cos(Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))))), Pow(cos(Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))), Integer(-1))), Mul(exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))), Pow(cos(Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\phi_1', commutative=True), Symbol('L', commutative=True))))), Pow(cos(Mul(Symbol('L', commutative=True), Symbol('\\\\phi_1', commutative=True))), Integer(-1))), Mul(exp(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))), Pow(cos(Mul(Symbol('L', commutative=True), Symbol('\\\\phi_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\omega{(\\mu_0)} = \\sin{(\\mu_0)} and \\pi{(\\mu_0)} = (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)}, then derive \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = 0, then obtain (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)} + \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)}", "derivation": "\\omega{(\\mu_0)} = \\sin{(\\mu_0)} and \\pi{(\\mu_0)} = (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)} and \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = \\frac{d}{d \\mu_0} (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)} and \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = \\frac{d}{d \\mu_0} 0 and \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = 0 and (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)} + \\frac{d}{d \\mu_0} \\pi{(\\mu_0)} = (- \\omega{(\\mu_0)} + \\sin{(\\mu_0)}) \\sin{(\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True)), Mul(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(0))"], [["add", 5, "Mul(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), Derivative(Function('\\\\pi')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(f_{\\mathbf{v}},f)} = \\cos^{f}{(f_{\\mathbf{v}})}, then obtain \\int \\cos^{f}{(f_{\\mathbf{v}})} df + \\frac{\\int \\rho_{b}{(f_{\\mathbf{v}},f)} df}{f} = \\int \\cos^{f}{(f_{\\mathbf{v}})} df + \\frac{\\int \\cos^{f}{(f_{\\mathbf{v}})} df}{f}", "derivation": "\\rho_{b}{(f_{\\mathbf{v}},f)} = \\cos^{f}{(f_{\\mathbf{v}})} and \\int \\rho_{b}{(f_{\\mathbf{v}},f)} df = \\int \\cos^{f}{(f_{\\mathbf{v}})} df and \\frac{\\int \\rho_{b}{(f_{\\mathbf{v}},f)} df}{f} = \\frac{\\int \\cos^{f}{(f_{\\mathbf{v}})} df}{f} and \\int \\cos^{f}{(f_{\\mathbf{v}})} df + \\frac{\\int \\rho_{b}{(f_{\\mathbf{v}},f)} df}{f} = \\int \\cos^{f}{(f_{\\mathbf{v}})} df + \\frac{\\int \\cos^{f}{(f_{\\mathbf{v}})} df}{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["add", 3, "Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))"], "Equality(Add(Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Function('\\\\rho_b')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Add(Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Integral(Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"]]}, {"prompt": "Given \\lambda{(c_{0},\\ddot{x},\\Omega)} = \\frac{\\ddot{x}^{c_{0}}}{\\Omega}, then obtain - (- \\ddot{x}^{c_{0}} + \\lambda{(c_{0},\\ddot{x},\\Omega)})^{\\ddot{x}} = - (- \\ddot{x}^{c_{0}} + \\frac{\\ddot{x}^{c_{0}}}{\\Omega})^{\\ddot{x}}", "derivation": "\\lambda{(c_{0},\\ddot{x},\\Omega)} = \\frac{\\ddot{x}^{c_{0}}}{\\Omega} and - \\ddot{x}^{c_{0}} + \\lambda{(c_{0},\\ddot{x},\\Omega)} = - \\ddot{x}^{c_{0}} + \\frac{\\ddot{x}^{c_{0}}}{\\Omega} and (- \\ddot{x}^{c_{0}} + \\lambda{(c_{0},\\ddot{x},\\Omega)})^{\\ddot{x}} = (- \\ddot{x}^{c_{0}} + \\frac{\\ddot{x}^{c_{0}}}{\\Omega})^{\\ddot{x}} and - (- \\ddot{x}^{c_{0}} + \\lambda{(c_{0},\\ddot{x},\\Omega)})^{\\ddot{x}} = - (- \\ddot{x}^{c_{0}} + \\frac{\\ddot{x}^{c_{0}}}{\\Omega})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Function('\\\\lambda')(Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True)))))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Function('\\\\lambda')(Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Function('\\\\lambda')(Symbol('c_0', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('c_0', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given t{(f,\\hat{x})} = \\hat{x} f and \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} = (- \\hbar + \\mathbf{M})^{\\theta}, then obtain \\hat{x} f \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} + \\mathbf{P}{(\\Psi)} = \\hat{x} f (- \\hbar + \\mathbf{M})^{\\theta} + \\mathbf{P}{(\\Psi)}", "derivation": "t{(f,\\hat{x})} = \\hat{x} f and \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} = (- \\hbar + \\mathbf{M})^{\\theta} and \\hat{x} f \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} = \\hat{x} f (- \\hbar + \\mathbf{M})^{\\theta} and \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} t{(f,\\hat{x})} = (- \\hbar + \\mathbf{M})^{\\theta} t{(f,\\hat{x})} and \\mathbf{P}{(\\Psi)} + \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} t{(f,\\hat{x})} = (- \\hbar + \\mathbf{M})^{\\theta} t{(f,\\hat{x})} + \\mathbf{P}{(\\Psi)} and \\hat{x} f \\operatorname{f^{*}}{(\\theta,\\mathbf{M},\\hbar)} + \\mathbf{P}{(\\Psi)} = \\hat{x} f (- \\hbar + \\mathbf{M})^{\\theta} + \\mathbf{P}{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('f', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True)))"], ["get_premise", "Equality(Function('f^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["times", 2, "Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True), Function('f^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('f^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('t')(Symbol('f', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\theta', commutative=True)), Function('t')(Symbol('f', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{P}')(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\Psi', commutative=True)), Mul(Function('f^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('t')(Symbol('f', commutative=True), Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\theta', commutative=True)), Function('t')(Symbol('f', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True), Function('f^*')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\hbar', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('f', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\theta', commutative=True))), Function('\\\\mathbf{P}')(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(P_{g},\\hat{H}_{\\lambda})} = P_{g} - \\hat{H}_{\\lambda} and V{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda}, then obtain 0 = (P_{g} - \\hat{H}_{\\lambda}) (\\hat{H}_{\\lambda} - V{(\\hat{H}_{\\lambda})})", "derivation": "\\operatorname{F_{x}}{(P_{g},\\hat{H}_{\\lambda})} = P_{g} - \\hat{H}_{\\lambda} and V{(\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} and 0 = \\hat{H}_{\\lambda} - V{(\\hat{H}_{\\lambda})} and 0 = (\\hat{H}_{\\lambda} - V{(\\hat{H}_{\\lambda})}) \\operatorname{F_{x}}{(P_{g},\\hat{H}_{\\lambda})} and 0 = (P_{g} - \\hat{H}_{\\lambda}) (\\hat{H}_{\\lambda} - V{(\\hat{H}_{\\lambda})})", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('V')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["minus", 2, "Function('V')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('V')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["times", 3, "Function('F_x')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Mul(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('V')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Function('F_x')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Mul(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('V')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} = \\frac{d}{d f} \\sin{(f)}, then derive \\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} = \\cos{(f)}, then obtain f + \\cos{(f)} - \\frac{d}{d f} \\sin{(f)} = f", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} = \\frac{d}{d f} \\sin{(f)} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} - \\frac{d}{d f} \\sin{(f)} = 0 and f + \\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} - \\frac{d}{d f} \\sin{(f)} = f and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f)} = \\cos{(f)} and f + \\cos{(f)} - \\frac{d}{d f} \\sin{(f)} = f", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True)), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), Integer(0))"], [["add", 2, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), Symbol('f', commutative=True))"], [["evaluate_derivatives", 1], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))), Symbol('f', commutative=True))"]]}, {"prompt": "Given U{(\\rho_b)} = e^{\\rho_b}, then derive e^{U{(\\rho_b)}} \\frac{d}{d \\rho_b} U{(\\rho_b)} = e^{\\rho_b} e^{e^{\\rho_b}}, then obtain \\operatorname{m_{s}}{(\\rho_b)} e^{e^{\\rho_b}} \\frac{d}{d \\rho_b} e^{\\rho_b} = \\operatorname{m_{s}}{(\\rho_b)} e^{\\rho_b} e^{e^{\\rho_b}}", "derivation": "U{(\\rho_b)} = e^{\\rho_b} and e^{U{(\\rho_b)}} = e^{e^{\\rho_b}} and \\frac{d}{d \\rho_b} e^{U{(\\rho_b)}} = \\frac{d}{d \\rho_b} e^{e^{\\rho_b}} and e^{U{(\\rho_b)}} \\frac{d}{d \\rho_b} U{(\\rho_b)} = e^{\\rho_b} e^{e^{\\rho_b}} and e^{e^{\\rho_b}} \\frac{d}{d \\rho_b} e^{\\rho_b} = e^{\\rho_b} e^{e^{\\rho_b}} and \\operatorname{m_{s}}{(\\rho_b)} e^{e^{\\rho_b}} \\frac{d}{d \\rho_b} e^{\\rho_b} = \\operatorname{m_{s}}{(\\rho_b)} e^{\\rho_b} e^{e^{\\rho_b}}", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["exp", 1], "Equality(exp(Function('U')(Symbol('\\\\rho_b', commutative=True))), exp(exp(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(exp(Function('U')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('U')(Symbol('\\\\rho_b', commutative=True))), Derivative(Function('U')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(exp(exp(Symbol('\\\\rho_b', commutative=True))), Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True)))))"], [["times", 5, "Function('m_s')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True))), Derivative(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(Function('m_s')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)), exp(exp(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given G{(F_{c})} = \\cos{(F_{c})}, then obtain (G{(F_{c})} \\int \\frac{G{(F_{c})}}{\\cos{(F_{c})}} dF_{c})^{F_{c}} + G{(F_{c})} = (G{(F_{c})} \\int 1 dF_{c})^{F_{c}} + G{(F_{c})}", "derivation": "G{(F_{c})} = \\cos{(F_{c})} and \\frac{G{(F_{c})}}{\\cos{(F_{c})}} = 1 and \\int \\frac{G{(F_{c})}}{\\cos{(F_{c})}} dF_{c} = \\int 1 dF_{c} and G{(F_{c})} \\int \\frac{G{(F_{c})}}{\\cos{(F_{c})}} dF_{c} = G{(F_{c})} \\int 1 dF_{c} and (G{(F_{c})} \\int \\frac{G{(F_{c})}}{\\cos{(F_{c})}} dF_{c})^{F_{c}} = (G{(F_{c})} \\int 1 dF_{c})^{F_{c}} and (G{(F_{c})} \\int \\frac{G{(F_{c})}}{\\cos{(F_{c})}} dF_{c})^{F_{c}} + G{(F_{c})} = (G{(F_{c})} \\int 1 dF_{c})^{F_{c}} + G{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('F_c', commutative=True)), cos(Symbol('F_c', commutative=True)))"], [["divide", 1, "cos(Symbol('F_c', commutative=True))"], "Equality(Mul(Function('G')(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Mul(Function('G')(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Tuple(Symbol('F_c', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True))))"], [["divide", 3, "Pow(Function('G')(Symbol('F_c', commutative=True)), Integer(-1))"], "Equality(Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Mul(Function('G')(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Tuple(Symbol('F_c', commutative=True)))), Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True)))))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Mul(Function('G')(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Tuple(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)), Pow(Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)))"], [["add", 5, "Function('G')(Symbol('F_c', commutative=True))"], "Equality(Add(Pow(Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Mul(Function('G')(Symbol('F_c', commutative=True)), Pow(cos(Symbol('F_c', commutative=True)), Integer(-1))), Tuple(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)), Function('G')(Symbol('F_c', commutative=True))), Add(Pow(Mul(Function('G')(Symbol('F_c', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True)))), Symbol('F_c', commutative=True)), Function('G')(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given H{(n,G)} = \\frac{\\partial}{\\partial n} (G + n), then derive H{(n,G)} = 1, then obtain \\int 1 dn + \\frac{\\frac{\\partial}{\\partial n} (G + n)}{n} = \\int 1 dn + \\frac{1}{n}", "derivation": "H{(n,G)} = \\frac{\\partial}{\\partial n} (G + n) and H{(n,G)} = 1 and \\frac{H{(n,G)}}{n} = \\frac{1}{n} and \\frac{\\partial}{\\partial n} (G + n) = 1 and \\int \\frac{\\partial}{\\partial n} (G + n) dn = \\int 1 dn and \\frac{\\frac{\\partial}{\\partial n} (G + n)}{n} = \\frac{1}{n} and \\int \\frac{\\partial}{\\partial n} (G + n) dn + \\frac{\\frac{\\partial}{\\partial n} (G + n)}{n} = \\int \\frac{\\partial}{\\partial n} (G + n) dn + \\frac{1}{n} and \\int 1 dn + \\frac{\\frac{\\partial}{\\partial n} (G + n)}{n} = \\int 1 dn + \\frac{1}{n}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('H')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Integer(1))"], [["divide", 2, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('H')(Symbol('n', commutative=True), Symbol('G', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(Integer(1), Tuple(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Symbol('n', commutative=True), Integer(-1)))"], [["add", 6, "Integral(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Integral(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Add(Integral(Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Integral(Integer(1), Tuple(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Derivative(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))), Add(Integral(Integer(1), Tuple(Symbol('n', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{H})} = \\sin{(\\cos{(\\mathbf{H})})}, then obtain - \\sin{(\\cos{(\\mathbf{H})})} + \\int \\operatorname{E_{\\lambda}}^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H} = - \\sin{(\\cos{(\\mathbf{H})})} + \\int \\sin^{\\mathbf{H}}{(\\cos{(\\mathbf{H})})} d\\mathbf{H}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{H})} = \\sin{(\\cos{(\\mathbf{H})})} and \\operatorname{E_{\\lambda}}^{\\mathbf{H}}{(\\mathbf{H})} = \\sin^{\\mathbf{H}}{(\\cos{(\\mathbf{H})})} and \\int \\operatorname{E_{\\lambda}}^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H} = \\int \\sin^{\\mathbf{H}}{(\\cos{(\\mathbf{H})})} d\\mathbf{H} and - \\sin{(\\cos{(\\mathbf{H})})} + \\int \\operatorname{E_{\\lambda}}^{\\mathbf{H}}{(\\mathbf{H})} d\\mathbf{H} = - \\sin{(\\cos{(\\mathbf{H})})} + \\int \\sin^{\\mathbf{H}}{(\\cos{(\\mathbf{H})})} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True)), sin(cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(cos(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(sin(cos(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 3, "sin(cos(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(cos(Symbol('\\\\mathbf{H}', commutative=True)))), Integral(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), sin(cos(Symbol('\\\\mathbf{H}', commutative=True)))), Integral(Pow(sin(cos(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given u{(a)} = \\cos{(a)}, then derive \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = 1, then derive \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = 0, then obtain \\Psi_{\\lambda} + \\sin{(a)} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = \\Psi_{\\lambda} + \\sin{(a)}", "derivation": "u{(a)} = \\cos{(a)} and \\int u{(a)} da = \\int \\cos{(a)} da and \\frac{\\int u{(a)} da}{\\int \\cos{(a)} da} = 1 and \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = 1 and \\frac{\\int \\cos{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = 1 and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int \\cos{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = \\frac{d}{d \\hat{\\mathbf{x}}} 1 and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = \\frac{d}{d \\hat{\\mathbf{x}}} 1 and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = 0 and \\Psi_{\\lambda} + \\sin{(a)} + \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\frac{\\int u{(a)} da}{\\hat{\\mathbf{x}} + \\sin{(a)}} = \\Psi_{\\lambda} + \\sin{(a)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["divide", 2, "Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))"], "Equality(Mul(Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Pow(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Integer(1))"], [["differentiate", 5, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Integer(0))"], [["add", 8, "Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Symbol('a', commutative=True)))"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Symbol('a', commutative=True)), Derivative(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(-1)), Integral(Function('u')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(Q,A_{1})} = \\cos{(A_{1} + Q)} and \\operatorname{v_{y}}{(Q,A_{1})} = \\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}, then obtain e e^{A_{1}} = e^{A_{1}} e^{\\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}}", "derivation": "\\mathbf{r}{(Q,A_{1})} = \\cos{(A_{1} + Q)} and \\operatorname{v_{y}}{(Q,A_{1})} = \\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}} and \\operatorname{v_{y}}{(Q,A_{1})} = 1 and e^{\\operatorname{v_{y}}{(Q,A_{1})}} = e and e^{\\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}} = e and e^{\\operatorname{v_{y}}{(Q,A_{1})}} = e^{\\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}} and e^{A_{1}} e^{\\operatorname{v_{y}}{(Q,A_{1})}} = e^{A_{1}} e^{\\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}} and e^{A_{1}} e^{\\operatorname{v_{y}}{(Q,A_{1})}} = e e^{A_{1}} and e e^{A_{1}} = e^{A_{1}} e^{\\frac{\\mathbf{r}{(Q,A_{1})}}{\\cos{(A_{1} + Q)}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Mul(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Integer(1))"], [["exp", 3], "Equality(exp(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True))), E)"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Mul(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))), Integer(-1)))), E)"], [["substitute_RHS_for_LHS", 4, 5], "Equality(exp(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True))), exp(Mul(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))), Integer(-1)))))"], [["times", 6, "exp(Symbol('A_1', commutative=True))"], "Equality(Mul(exp(Symbol('A_1', commutative=True)), exp(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)))), Mul(exp(Symbol('A_1', commutative=True)), exp(Mul(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))), Integer(-1))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(exp(Symbol('A_1', commutative=True)), exp(Function('v_y')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)))), Mul(E, exp(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Mul(E, exp(Symbol('A_1', commutative=True))), Mul(exp(Symbol('A_1', commutative=True)), exp(Mul(Function('\\\\mathbf{r}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Pow(cos(Add(Symbol('A_1', commutative=True), Symbol('Q', commutative=True))), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(i,\\mathbf{g})} = \\log{(\\mathbf{g} + i)} and \\operatorname{v_{1}}{(i,\\mathbf{g})} = \\log{(\\mathbf{g} + i)}, then obtain - \\mathbf{g} + \\log{(\\mathbf{g} + i)} = - \\mathbf{g} + \\operatorname{v_{1}}{(i,\\mathbf{g})}", "derivation": "\\operatorname{f_{E}}{(i,\\mathbf{g})} = \\log{(\\mathbf{g} + i)} and - \\mathbf{g} + \\operatorname{f_{E}}{(i,\\mathbf{g})} = - \\mathbf{g} + \\log{(\\mathbf{g} + i)} and \\operatorname{v_{1}}{(i,\\mathbf{g})} = \\log{(\\mathbf{g} + i)} and - \\mathbf{g} + \\operatorname{f_{E}}{(i,\\mathbf{g})} = - \\mathbf{g} + \\operatorname{v_{1}}{(i,\\mathbf{g})} and - \\mathbf{g} + \\log{(\\mathbf{g} + i)} = - \\mathbf{g} + \\operatorname{v_{1}}{(i,\\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('i', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('f_E')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('i', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('f_E')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('v_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('v_1')(Symbol('i', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\rho{(E)} = \\log{(E)}, then derive 2 \\int \\rho{(E)} dE = E \\log{(E)} - E + \\mathbb{I} + \\int \\rho{(E)} dE, then obtain \\int \\rho{(E)} dE + 3 \\int \\log{(E)} dE = E \\log{(E)} - E + \\mathbb{I} + \\int \\rho{(E)} dE + 2 \\int \\log{(E)} dE", "derivation": "\\rho{(E)} = \\log{(E)} and \\int \\rho{(E)} dE = \\int \\log{(E)} dE and 2 \\int \\rho{(E)} dE = \\int \\rho{(E)} dE + \\int \\log{(E)} dE and 2 \\int \\rho{(E)} dE = E \\log{(E)} - E + \\mathbb{I} + \\int \\rho{(E)} dE and \\int \\rho{(E)} dE + \\int \\log{(E)} dE = E \\log{(E)} - E + \\mathbb{I} + \\int \\rho{(E)} dE and \\int \\rho{(E)} dE + 3 \\int \\log{(E)} dE = E \\log{(E)} - E + \\mathbb{I} + \\int \\rho{(E)} dE + 2 \\int \\log{(E)} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 2, "Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["add", 5, "Mul(Integer(2), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], "Equality(Add(Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Mul(Integer(3), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True), Integral(Function('\\\\rho')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Mul(Integer(2), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\rho_b,l)} = e^{\\rho_b l}, then obtain \\int l^{2} \\operatorname{A_{2}}^{2}{(\\rho_b,l)} d\\rho_b = \\int l^{2} \\operatorname{A_{2}}{(\\rho_b,l)} e^{\\rho_b l} d\\rho_b", "derivation": "\\operatorname{A_{2}}{(\\rho_b,l)} = e^{\\rho_b l} and l \\operatorname{A_{2}}{(\\rho_b,l)} = l e^{\\rho_b l} and l^{2} \\operatorname{A_{2}}^{2}{(\\rho_b,l)} = l^{2} \\operatorname{A_{2}}{(\\rho_b,l)} e^{\\rho_b l} and \\int l^{2} \\operatorname{A_{2}}^{2}{(\\rho_b,l)} d\\rho_b = \\int l^{2} \\operatorname{A_{2}}{(\\rho_b,l)} e^{\\rho_b l} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), exp(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), exp(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)))))"], [["times", 2, "Mul(Symbol('l', commutative=True), Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Integer(2))), Mul(Pow(Symbol('l', commutative=True), Integer(2)), Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), exp(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), Integer(2))), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Function('A_2')(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)), exp(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('l', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{g})} = \\log{(\\sin{(\\mathbf{g})})}, then obtain 1 = \\frac{\\log{(\\sin{(\\mathbf{g})})}}{\\mathbf{f}{(\\mathbf{g})}}", "derivation": "\\mathbf{f}{(\\mathbf{g})} = \\log{(\\sin{(\\mathbf{g})})} and \\mathbf{f}^{2}{(\\mathbf{g})} = \\mathbf{f}{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})} and \\frac{\\mathbf{f}^{2}{(\\mathbf{g})}}{\\mathbf{g}} = \\frac{\\mathbf{f}{(\\mathbf{g})} \\log{(\\sin{(\\mathbf{g})})}}{\\mathbf{g}} and 1 = \\frac{\\log{(\\sin{(\\mathbf{g})})}}{\\mathbf{f}{(\\mathbf{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given B{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})}, then derive - f_{\\mathbf{v}} + B{(f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})} + 1, then obtain - f_{\\mathbf{v}} + \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})} + 1", "derivation": "B{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} and B{(f_{\\mathbf{v}})} + 1 = \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} + 1 and - f_{\\mathbf{v}} + B{(f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} + 1 and - f_{\\mathbf{v}} + B{(f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})} + 1 and - f_{\\mathbf{v}} + \\frac{d}{d f_{\\mathbf{v}}} \\sin{(f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})} + 1", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('B')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Add(Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1)))"], [["minus", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('B')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('B')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(c,\\mathbf{P})} = \\mathbf{P} + c and \\rho_{f}{(c,\\mathbf{P})} = c + \\operatorname{t_{1}}{(c,\\mathbf{P})}, then obtain c + \\rho_{f}{(c,\\mathbf{P})} + \\operatorname{t_{1}}{(c,\\mathbf{P})} = \\mathbf{P} + 3 c + \\operatorname{t_{1}}{(c,\\mathbf{P})}", "derivation": "\\operatorname{t_{1}}{(c,\\mathbf{P})} = \\mathbf{P} + c and c + \\operatorname{t_{1}}{(c,\\mathbf{P})} = \\mathbf{P} + 2 c and \\rho_{f}{(c,\\mathbf{P})} = c + \\operatorname{t_{1}}{(c,\\mathbf{P})} and \\rho_{f}{(c,\\mathbf{P})} = \\mathbf{P} + 2 c and c + \\rho_{f}{(c,\\mathbf{P})} + \\operatorname{t_{1}}{(c,\\mathbf{P})} = \\mathbf{P} + 3 c + \\operatorname{t_{1}}{(c,\\mathbf{P})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('c', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('c', commutative=True), Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), Symbol('c', commutative=True))))"], [["add", 4, "Add(Symbol('c', commutative=True), Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\rho_f')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(3), Symbol('c', commutative=True)), Function('t_1')(Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(A_{1})} = e^{e^{A_{1}}}, then obtain (A_{1} + \\hat{p}_0{(A_{1})} e^{- e^{A_{1}}} + e^{A_{1}})^{A_{1}} = (A_{1} + e^{A_{1}} + 1)^{A_{1}}", "derivation": "\\hat{p}_0{(A_{1})} = e^{e^{A_{1}}} and \\hat{p}_0{(A_{1})} e^{- e^{A_{1}}} = 1 and \\hat{p}_0{(A_{1})} e^{- e^{A_{1}}} + e^{A_{1}} = e^{A_{1}} + 1 and A_{1} + \\hat{p}_0{(A_{1})} e^{- e^{A_{1}}} + e^{A_{1}} = A_{1} + e^{A_{1}} + 1 and (A_{1} + \\hat{p}_0{(A_{1})} e^{- e^{A_{1}}} + e^{A_{1}})^{A_{1}} = (A_{1} + e^{A_{1}} + 1)^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), exp(exp(Symbol('A_1', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('A_1', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))))), Integer(1))"], [["add", 2, "exp(Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))))), exp(Symbol('A_1', commutative=True))), Add(exp(Symbol('A_1', commutative=True)), Integer(1)))"], [["add", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))))), exp(Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), exp(Symbol('A_1', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Symbol('A_1', commutative=True), Mul(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))))), exp(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), exp(Symbol('A_1', commutative=True)), Integer(1)), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(\\varphi)} = \\log{(\\varphi)} and \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\mathbf{g} + (\\varphi + \\log{(\\varphi)})^{\\varphi}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{g}\\partial \\varphi} \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\mathbf{g}\\partial \\varphi} (\\mathbf{g} + (\\varphi + \\hat{p}{(\\varphi)})^{\\varphi})", "derivation": "\\hat{p}{(\\varphi)} = \\log{(\\varphi)} and \\varphi + \\hat{p}{(\\varphi)} = \\varphi + \\log{(\\varphi)} and (\\varphi + \\hat{p}{(\\varphi)})^{\\varphi} = (\\varphi + \\log{(\\varphi)})^{\\varphi} and \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\mathbf{g} + (\\varphi + \\log{(\\varphi)})^{\\varphi} and \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\mathbf{g} + (\\varphi + \\hat{p}{(\\varphi)})^{\\varphi} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{g} + (\\varphi + \\hat{p}{(\\varphi)})^{\\varphi}) and \\frac{\\partial^{2}}{\\partial \\mathbf{g}\\partial \\varphi} \\operatorname{E_{\\lambda}}{(\\varphi,\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\mathbf{g}\\partial \\varphi} (\\mathbf{g} + (\\varphi + \\hat{p}{(\\varphi)})^{\\varphi})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{g}', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(g,\\hat{X})} = g \\sin{(\\hat{X})}, then obtain (g \\sin{(\\hat{X})})^{g} \\sin^{2}{(\\hat{X})} = \\sigma_{p}^{g}{(g,\\hat{X})} \\sin^{2}{(\\hat{X})}", "derivation": "\\sigma_{p}{(g,\\hat{X})} = g \\sin{(\\hat{X})} and g \\sigma_{p}{(g,\\hat{X})} \\sin{(\\hat{X})} = g^{2} \\sin^{2}{(\\hat{X})} and g \\sin{(\\hat{X})} = \\frac{g^{2} \\sin^{2}{(\\hat{X})}}{\\sigma_{p}{(g,\\hat{X})}} and \\sigma_{p}{(g,\\hat{X})} = \\frac{g^{2} \\sin^{2}{(\\hat{X})}}{\\sigma_{p}{(g,\\hat{X})}} and (g \\sin{(\\hat{X})})^{g} = (\\frac{g^{2} \\sin^{2}{(\\hat{X})}}{\\sigma_{p}{(g,\\hat{X})}})^{g} and (g \\sin{(\\hat{X})})^{g} = \\sigma_{p}^{g}{(g,\\hat{X})} and (g \\sin{(\\hat{X})})^{g} \\sin^{2}{(\\hat{X})} = \\sigma_{p}^{g}{(g,\\hat{X})} \\sin^{2}{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 1, "Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Mul(Symbol('g', commutative=True), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(2)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))))"], [["divide", 2, "Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(2)), Pow(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(2)), Pow(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))))"], [["power", 3, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('g', commutative=True), Integer(2)), Pow(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('g', commutative=True)), Pow(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('g', commutative=True)))"], [["times", 6, "Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Mul(Symbol('g', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('g', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('g', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{\\mathbf{x}},y)} = - y + \\cos{(\\hat{\\mathbf{x}})}, then obtain \\frac{\\sin{(y - \\cos{(\\hat{\\mathbf{x}})})} + 1}{- \\frac{y - \\cos{(\\hat{\\mathbf{x}})}}{\\mathbf{p}{(\\hat{\\mathbf{x}},y)}} + \\sin{(y - \\cos{(\\hat{\\mathbf{x}})})}} = 1", "derivation": "\\mathbf{p}{(\\hat{\\mathbf{x}},y)} = - y + \\cos{(\\hat{\\mathbf{x}})} and - \\mathbf{p}{(\\hat{\\mathbf{x}},y)} = y - \\cos{(\\hat{\\mathbf{x}})} and 1 = - \\frac{y - \\cos{(\\hat{\\mathbf{x}})}}{\\mathbf{p}{(\\hat{\\mathbf{x}},y)}} and \\sin{(y - \\cos{(\\hat{\\mathbf{x}})})} + 1 = - \\frac{y - \\cos{(\\hat{\\mathbf{x}})}}{\\mathbf{p}{(\\hat{\\mathbf{x}},y)}} + \\sin{(y - \\cos{(\\hat{\\mathbf{x}})})} and \\frac{\\sin{(y - \\cos{(\\hat{\\mathbf{x}})})} + 1}{- \\frac{y - \\cos{(\\hat{\\mathbf{x}})}}{\\mathbf{p}{(\\hat{\\mathbf{x}},y)}} + \\sin{(y - \\cos{(\\hat{\\mathbf{x}})})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True))), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)))"], "Equality(Integer(1), Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], [["add", 3, "sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], "Equality(Add(sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))), Integer(1)), Add(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)), Integer(-1))), sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))))"], [["divide", 4, "Add(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)), Integer(-1))), sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('y', commutative=True)), Integer(-1))), sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))), Integer(-1)), Add(sin(Add(Symbol('y', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\theta_{1}{(\\theta_2,\\eta^{\\prime})} = \\log{(\\eta^{\\prime} - \\theta_2)}, then obtain \\int \\theta_{1}{(\\theta_2,\\eta^{\\prime})} \\log{(\\eta^{\\prime} - \\theta_2)} d\\theta_2 = 2 \\eta^{\\prime} \\log{(- \\eta^{\\prime} + \\theta_2)} - 2 \\theta_2 \\log{(\\eta^{\\prime} - \\theta_2)} + 2 \\theta_2 + k + (- \\eta^{\\prime} + \\theta_2) \\log{(\\eta^{\\prime} - \\theta_2)}^{2}", "derivation": "\\theta_{1}{(\\theta_2,\\eta^{\\prime})} = \\log{(\\eta^{\\prime} - \\theta_2)} and \\theta_{1}{(\\theta_2,\\eta^{\\prime})} \\log{(\\eta^{\\prime} - \\theta_2)} = \\log{(\\eta^{\\prime} - \\theta_2)}^{2} and \\int \\theta_{1}{(\\theta_2,\\eta^{\\prime})} \\log{(\\eta^{\\prime} - \\theta_2)} d\\theta_2 = \\int \\log{(\\eta^{\\prime} - \\theta_2)}^{2} d\\theta_2 and \\int \\theta_{1}{(\\theta_2,\\eta^{\\prime})} \\log{(\\eta^{\\prime} - \\theta_2)} d\\theta_2 = 2 \\eta^{\\prime} \\log{(- \\eta^{\\prime} + \\theta_2)} - 2 \\theta_2 \\log{(\\eta^{\\prime} - \\theta_2)} + 2 \\theta_2 + k + (- \\eta^{\\prime} + \\theta_2) \\log{(\\eta^{\\prime} - \\theta_2)}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))))"], [["times", 1, "log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Integer(2)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Mul(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\theta_1')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True), log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)), Symbol('k', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(log(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(r_{0})} = \\log{(r_{0})}, then derive 0 = - \\frac{d}{d r_{0}} \\operatorname{C_{d}}{(r_{0})} + \\frac{1}{r_{0}}, then obtain \\frac{\\int 0 dr_{0}}{\\int (- \\frac{d}{d r_{0}} \\log{(r_{0})} + \\frac{1}{r_{0}}) dr_{0}} = 1", "derivation": "\\operatorname{C_{d}}{(r_{0})} = \\log{(r_{0})} and \\frac{d}{d r_{0}} \\operatorname{C_{d}}{(r_{0})} = \\frac{d}{d r_{0}} \\log{(r_{0})} and 0 = - \\frac{d}{d r_{0}} \\operatorname{C_{d}}{(r_{0})} + \\frac{d}{d r_{0}} \\log{(r_{0})} and 0 = - \\frac{d}{d r_{0}} \\operatorname{C_{d}}{(r_{0})} + \\frac{1}{r_{0}} and \\int 0 dr_{0} = \\int (- \\frac{d}{d r_{0}} \\operatorname{C_{d}}{(r_{0})} + \\frac{1}{r_{0}}) dr_{0} and \\int 0 dr_{0} = \\int (- \\frac{d}{d r_{0}} \\log{(r_{0})} + \\frac{1}{r_{0}}) dr_{0} and \\frac{\\int 0 dr_{0}}{\\int (- \\frac{d}{d r_{0}} \\log{(r_{0})} + \\frac{1}{r_{0}}) dr_{0}} = 1", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('C_d')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('C_d')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('C_d')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["integrate", 4, "Symbol('r_0', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Function('C_d')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Integer(0), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True))))"], [["divide", 6, "Integral(Add(Mul(Integer(-1), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True)))"], "Equality(Mul(Integral(Integer(0), Tuple(Symbol('r_0', commutative=True))), Pow(Integral(Add(Mul(Integer(-1), Derivative(log(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given k{(H)} = \\sin{(H)} and b{(H)} = \\frac{d}{d H} \\sin{(H)}, then obtain \\int b{(H)} dH = \\int \\frac{d}{d H} k{(H)} dH", "derivation": "k{(H)} = \\sin{(H)} and b{(H)} = \\frac{d}{d H} \\sin{(H)} and b{(H)} = \\frac{d}{d H} k{(H)} and \\int b{(H)} dH = \\int \\frac{d}{d H} k{(H)} dH", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('H', commutative=True)), sin(Symbol('H', commutative=True)))"], ["renaming_premise", "Equality(Function('b')(Symbol('H', commutative=True)), Derivative(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('b')(Symbol('H', commutative=True)), Derivative(Function('k')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Function('b')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Function('k')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given a{(y)} = \\cos{(\\cos{(y)})} and \\operatorname{f_{\\mathbf{p}}}{(y)} = \\cos{(\\cos{(y)})} and \\operatorname{t_{1}}{(y)} = (\\sin^{y}{(a{(y)})})^{y}, then obtain \\operatorname{t_{1}}{(y)} = (\\sin^{y}{(\\cos{(\\cos{(y)})})})^{y}", "derivation": "a{(y)} = \\cos{(\\cos{(y)})} and \\operatorname{f_{\\mathbf{p}}}{(y)} = \\cos{(\\cos{(y)})} and a{(y)} = \\operatorname{f_{\\mathbf{p}}}{(y)} and \\sin{(a{(y)})} = \\sin{(\\operatorname{f_{\\mathbf{p}}}{(y)})} and \\sin{(a{(y)})} = \\sin{(\\cos{(\\cos{(y)})})} and \\sin^{y}{(a{(y)})} = \\sin^{y}{(\\cos{(\\cos{(y)})})} and (\\sin^{y}{(a{(y)})})^{y} = (\\sin^{y}{(\\cos{(\\cos{(y)})})})^{y} and \\operatorname{t_{1}}{(y)} = (\\sin^{y}{(a{(y)})})^{y} and \\operatorname{t_{1}}{(y)} = (\\sin^{y}{(\\cos{(\\cos{(y)})})})^{y}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('a')(Symbol('y', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('y', commutative=True)))"], [["sin", 3], "Equality(sin(Function('a')(Symbol('y', commutative=True))), sin(Function('f_{\\\\mathbf{p}}')(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(sin(Function('a')(Symbol('y', commutative=True))), sin(cos(cos(Symbol('y', commutative=True)))))"], [["power", 5, "Symbol('y', commutative=True)"], "Equality(Pow(sin(Function('a')(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Pow(sin(cos(cos(Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["power", 6, "Symbol('y', commutative=True)"], "Equality(Pow(Pow(sin(Function('a')(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(sin(cos(cos(Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('y', commutative=True)), Pow(Pow(sin(Function('a')(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Function('t_1')(Symbol('y', commutative=True)), Pow(Pow(sin(cos(cos(Symbol('y', commutative=True)))), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\lambda{(g,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + g and \\rho{(\\Psi^{\\dagger})} = \\frac{(\\Psi^{\\dagger})^{2}}{2}, then derive \\int \\lambda{(g,\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\frac{(\\Psi^{\\dagger})^{2}}{2} + \\Psi^{\\dagger} g + y, then obtain \\int (\\Psi^{\\dagger} + g) d\\Psi^{\\dagger} = \\Psi^{\\dagger} g + y + \\rho{(\\Psi^{\\dagger})}", "derivation": "\\lambda{(g,\\Psi^{\\dagger})} = \\Psi^{\\dagger} + g and \\int \\lambda{(g,\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int (\\Psi^{\\dagger} + g) d\\Psi^{\\dagger} and \\int \\lambda{(g,\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\frac{(\\Psi^{\\dagger})^{2}}{2} + \\Psi^{\\dagger} g + y and \\rho{(\\Psi^{\\dagger})} = \\frac{(\\Psi^{\\dagger})^{2}}{2} and \\int (\\Psi^{\\dagger} + g) d\\Psi^{\\dagger} = \\frac{(\\Psi^{\\dagger})^{2}}{2} + \\Psi^{\\dagger} g + y and \\int (\\Psi^{\\dagger} + g) d\\Psi^{\\dagger} = \\Psi^{\\dagger} g + y + \\rho{(\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Symbol('y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('g', commutative=True)), Symbol('y', commutative=True), Function('\\\\rho')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given x{(n_{1})} = \\cos{(\\sin{(n_{1})})}, then derive \\frac{d}{d n_{1}} x{(n_{1})} = - \\sin{(\\sin{(n_{1})})} \\cos{(n_{1})}, then obtain \\sin{(\\sin{(n_{1})})} \\frac{d}{d n_{1}} \\cos{(\\sin{(n_{1})})} = - \\sin^{2}{(\\sin{(n_{1})})} \\cos{(n_{1})}", "derivation": "x{(n_{1})} = \\cos{(\\sin{(n_{1})})} and \\frac{d}{d n_{1}} x{(n_{1})} = \\frac{d}{d n_{1}} \\cos{(\\sin{(n_{1})})} and \\frac{d}{d n_{1}} x{(n_{1})} = - \\sin{(\\sin{(n_{1})})} \\cos{(n_{1})} and \\frac{d}{d n_{1}} \\cos{(\\sin{(n_{1})})} = - \\sin{(\\sin{(n_{1})})} \\cos{(n_{1})} and \\sin{(\\sin{(n_{1})})} \\frac{d}{d n_{1}} \\cos{(\\sin{(n_{1})})} = - \\sin^{2}{(\\sin{(n_{1})})} \\cos{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('n_1', commutative=True)), cos(sin(Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('n_1', commutative=True))), cos(Symbol('n_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(sin(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('n_1', commutative=True))), cos(Symbol('n_1', commutative=True))))"], [["times", 4, "sin(sin(Symbol('n_1', commutative=True)))"], "Equality(Mul(sin(sin(Symbol('n_1', commutative=True))), Derivative(cos(sin(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(sin(sin(Symbol('n_1', commutative=True))), Integer(2)), cos(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{J})} = \\log{(\\cos{(\\mathbf{J})})}, then obtain - \\cos{(\\mathbf{J})} = - \\hat{\\mathbf{r}}{(\\mathbf{J})} + \\log{(\\cos{(\\mathbf{J})})} - \\cos{(\\mathbf{J})}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{J})} = \\log{(\\cos{(\\mathbf{J})})} and \\hat{\\mathbf{r}}{(\\mathbf{J})} + \\log{(\\cos{(\\mathbf{J})})} = 2 \\log{(\\cos{(\\mathbf{J})})} and \\hat{\\mathbf{r}}{(\\mathbf{J})} + \\log{(\\cos{(\\mathbf{J})})} - \\cos{(\\mathbf{J})} = 2 \\log{(\\cos{(\\mathbf{J})})} - \\cos{(\\mathbf{J})} and - \\cos{(\\mathbf{J})} = - \\hat{\\mathbf{r}}{(\\mathbf{J})} + \\log{(\\cos{(\\mathbf{J})})} - \\cos{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}', commutative=True)), log(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "log(cos(Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}', commutative=True)), log(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Integer(2), log(cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["minus", 2, "cos(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}', commutative=True)), log(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Add(Mul(Integer(2), log(cos(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["minus", 3, "Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}', commutative=True)), log(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], "Equality(Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}', commutative=True))), log(cos(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(g)} = \\log{(g)}, then derive g \\frac{d}{d g} \\operatorname{v_{z}}{(g)} + \\operatorname{v_{z}}{(g)} = \\log{(g)} + 1, then obtain - g \\operatorname{v_{z}}{(g)} + g \\frac{d}{d g} \\log{(g)} + \\log{(g)} = - g \\operatorname{v_{z}}{(g)} + \\log{(g)} + 1", "derivation": "\\operatorname{v_{z}}{(g)} = \\log{(g)} and g \\operatorname{v_{z}}{(g)} = g \\log{(g)} and \\frac{d}{d g} g \\operatorname{v_{z}}{(g)} = \\frac{d}{d g} g \\log{(g)} and g \\frac{d}{d g} \\operatorname{v_{z}}{(g)} + \\operatorname{v_{z}}{(g)} = \\log{(g)} + 1 and g \\frac{d}{d g} \\log{(g)} + \\log{(g)} = \\log{(g)} + 1 and - g \\operatorname{v_{z}}{(g)} + g \\frac{d}{d g} \\log{(g)} + \\log{(g)} = - g \\operatorname{v_{z}}{(g)} + \\log{(g)} + 1", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('v_z')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Symbol('g', commutative=True), Function('v_z')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('g', commutative=True), Derivative(Function('v_z')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Function('v_z')(Symbol('g', commutative=True))), Add(log(Symbol('g', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('g', commutative=True), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), log(Symbol('g', commutative=True))), Add(log(Symbol('g', commutative=True)), Integer(1)))"], [["minus", 5, "Mul(Symbol('g', commutative=True), Function('v_z')(Symbol('g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Function('v_z')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), log(Symbol('g', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True), Function('v_z')(Symbol('g', commutative=True))), log(Symbol('g', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} = \\cos{(F_{g} + \\mu)} and \\mathbf{p}{(\\mathbf{M})} = \\log{(\\mathbf{M})} and \\hat{x}{(F_{g},\\mu,\\mathbf{M})} = \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} + \\log{(\\mathbf{M})}, then obtain - \\hat{x}{(F_{g},\\mu,\\mathbf{M})} + \\mathbf{p}{(\\mathbf{M})} + \\cos{(F_{g} + \\mu)} = 0", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} = \\cos{(F_{g} + \\mu)} and \\mathbf{p}{(\\mathbf{M})} = \\log{(\\mathbf{M})} and \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} + \\mathbf{p}{(\\mathbf{M})} = \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} + \\log{(\\mathbf{M})} and \\hat{x}{(F_{g},\\mu,\\mathbf{M})} = \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} + \\log{(\\mathbf{M})} and \\operatorname{V_{\\mathbf{B}}}{(F_{g},\\mu)} + \\mathbf{p}{(\\mathbf{M})} = \\hat{x}{(F_{g},\\mu,\\mathbf{M})} and \\mathbf{p}{(\\mathbf{M})} + \\cos{(F_{g} + \\mu)} = \\hat{x}{(F_{g},\\mu,\\mathbf{M})} and - \\hat{x}{(F_{g},\\mu,\\mathbf{M})} + \\mathbf{p}{(\\mathbf{M})} + \\cos{(F_{g} + \\mu)} = 0", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 2, "Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)))), Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 6, "Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}')(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('F_g', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(0))"]]}, {"prompt": "Given A{(f_{E})} = \\log{(f_{E})} and \\mathbf{M}{(\\omega,F_{N})} = \\frac{F_{N}}{\\omega}, then obtain (\\frac{\\mathbf{M}{(\\omega,F_{N})}}{\\cos{(\\log{(f_{E})})}})^{\\omega} = (\\frac{F_{N}}{\\omega \\cos{(\\log{(f_{E})})}})^{\\omega}", "derivation": "A{(f_{E})} = \\log{(f_{E})} and \\mathbf{M}{(\\omega,F_{N})} = \\frac{F_{N}}{\\omega} and \\frac{\\mathbf{M}{(\\omega,F_{N})}}{\\cos{(A{(f_{E})})}} = \\frac{F_{N}}{\\omega \\cos{(A{(f_{E})})}} and \\frac{\\mathbf{M}{(\\omega,F_{N})}}{\\cos{(\\log{(f_{E})})}} = \\frac{F_{N}}{\\omega \\cos{(\\log{(f_{E})})}} and (\\frac{\\mathbf{M}{(\\omega,F_{N})}}{\\cos{(\\log{(f_{E})})}})^{\\omega} = (\\frac{F_{N}}{\\omega \\cos{(\\log{(f_{E})})}})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["divide", 2, "cos(Function('A')(Symbol('f_E', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True), Symbol('F_N', commutative=True)), Pow(cos(Function('A')(Symbol('f_E', commutative=True))), Integer(-1))), Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(cos(Function('A')(Symbol('f_E', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True), Symbol('F_N', commutative=True)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(-1))), Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(-1))))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\omega', commutative=True), Symbol('F_N', commutative=True)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(-1))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Symbol('F_N', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(cos(log(Symbol('f_E', commutative=True))), Integer(-1))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(x,m)} = (e^{m})^{x}, then obtain \\frac{\\partial}{\\partial x} \\frac{\\hat{x}_0{(x,m)}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}} = \\frac{\\partial}{\\partial x} \\frac{(e^{m})^{x}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}}", "derivation": "\\hat{x}_0{(x,m)} = (e^{m})^{x} and \\frac{\\partial}{\\partial x} \\hat{x}_0{(x,m)} = \\frac{\\partial}{\\partial x} (e^{m})^{x} and \\frac{\\hat{x}_0{(x,m)}}{\\frac{\\partial}{\\partial x} \\hat{x}_0{(x,m)}} = \\frac{(e^{m})^{x}}{\\frac{\\partial}{\\partial x} \\hat{x}_0{(x,m)}} and \\frac{\\hat{x}_0{(x,m)}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}} = \\frac{(e^{m})^{x}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}} and \\frac{\\partial}{\\partial x} \\frac{\\hat{x}_0{(x,m)}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}} = \\frac{\\partial}{\\partial x} \\frac{(e^{m})^{x}}{\\frac{\\partial}{\\partial x} (e^{m})^{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Pow(Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Pow(Derivative(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Pow(Derivative(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Pow(Derivative(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"], [["differentiate", 4, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{x}_0')(Symbol('x', commutative=True), Symbol('m', commutative=True)), Pow(Derivative(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Pow(Derivative(Pow(exp(Symbol('m', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(h)} = \\cos{(\\sin{(h)})} and \\mathbf{v}{(h)} = \\rho{(h)} - \\log{(\\rho{(h)})} + \\log{(\\cos{(\\sin{(h)})})}, then obtain h (\\mathbf{v}{(h)} + \\log{(\\cos{(\\sin{(h)})})}) = h (\\rho{(h)} + \\log{(\\cos{(\\sin{(h)})})})", "derivation": "\\rho{(h)} = \\cos{(\\sin{(h)})} and \\log{(\\rho{(h)})} = \\log{(\\cos{(\\sin{(h)})})} and \\mathbf{v}{(h)} = \\rho{(h)} - \\log{(\\rho{(h)})} + \\log{(\\cos{(\\sin{(h)})})} and \\mathbf{v}{(h)} = \\rho{(h)} and \\mathbf{v}{(h)} + \\log{(\\cos{(\\sin{(h)})})} = \\rho{(h)} + \\log{(\\cos{(\\sin{(h)})})} and h (\\mathbf{v}{(h)} + \\log{(\\cos{(\\sin{(h)})})}) = h (\\rho{(h)} + \\log{(\\cos{(\\sin{(h)})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('h', commutative=True)), cos(sin(Symbol('h', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\rho')(Symbol('h', commutative=True))), log(cos(sin(Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True)), Add(Function('\\\\rho')(Symbol('h', commutative=True)), Mul(Integer(-1), log(Function('\\\\rho')(Symbol('h', commutative=True)))), log(cos(sin(Symbol('h', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('h', commutative=True)), Function('\\\\rho')(Symbol('h', commutative=True)))"], [["add", 4, "log(cos(sin(Symbol('h', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True)), log(cos(sin(Symbol('h', commutative=True))))), Add(Function('\\\\rho')(Symbol('h', commutative=True)), log(cos(sin(Symbol('h', commutative=True))))))"], [["times", 5, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Add(Function('\\\\mathbf{v}')(Symbol('h', commutative=True)), log(cos(sin(Symbol('h', commutative=True)))))), Mul(Symbol('h', commutative=True), Add(Function('\\\\rho')(Symbol('h', commutative=True)), log(cos(sin(Symbol('h', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} = V - \\psi, then obtain (- \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} dV}{\\psi})^{\\psi} = (- \\frac{\\int (V - \\psi) dV}{\\psi})^{\\psi}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} = V - \\psi and \\int \\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} dV = \\int (V - \\psi) dV and - \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} dV}{\\psi} = - \\frac{\\int (V - \\psi) dV}{\\psi} and (- \\frac{\\int \\operatorname{f_{\\mathbf{v}}}{(V,\\psi)} dV}{\\psi})^{\\psi} = (- \\frac{\\int (V - \\psi) dV}{\\psi})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Integral(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('V', commutative=True)))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('V', commutative=True)))), Symbol('\\\\psi', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Integral(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('V', commutative=True)))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(f,\\sigma_p,B)} = (\\frac{\\sigma_p}{f})^{B} and k{(A_{x})} = \\sin{(\\log{(A_{x})})}, then obtain - \\operatorname{C_{2}}{(f,\\sigma_p,B)} + k{(A_{x})} - \\sin{(\\log{(A_{x})})} = - \\operatorname{C_{2}}{(f,\\sigma_p,B)}", "derivation": "\\operatorname{C_{2}}{(f,\\sigma_p,B)} = (\\frac{\\sigma_p}{f})^{B} and k{(A_{x})} = \\sin{(\\log{(A_{x})})} and k{(A_{x})} - \\sin{(\\log{(A_{x})})} = 0 and - (\\frac{\\sigma_p}{f})^{B} + k{(A_{x})} - \\sin{(\\log{(A_{x})})} = - (\\frac{\\sigma_p}{f})^{B} and - \\operatorname{C_{2}}{(f,\\sigma_p,B)} + k{(A_{x})} - \\sin{(\\log{(A_{x})})} = - \\operatorname{C_{2}}{(f,\\sigma_p,B)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('f', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('B', commutative=True)), Pow(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('B', commutative=True)))"], ["get_premise", "Equality(Function('k')(Symbol('A_x', commutative=True)), sin(log(Symbol('A_x', commutative=True))))"], [["minus", 2, "sin(log(Symbol('A_x', commutative=True)))"], "Equality(Add(Function('k')(Symbol('A_x', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_x', commutative=True))))), Integer(0))"], [["minus", 3, "Pow(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('B', commutative=True))), Function('k')(Symbol('A_x', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_x', commutative=True))))), Mul(Integer(-1), Pow(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('C_2')(Symbol('f', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('B', commutative=True))), Function('k')(Symbol('A_x', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_x', commutative=True))))), Mul(Integer(-1), Function('C_2')(Symbol('f', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)} = \\Omega + \\hat{H}_l, then obtain \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)}}{\\hat{H}_l} = \\frac{1}{\\hat{H}_l}", "derivation": "\\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)} = \\Omega + \\hat{H}_l and \\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)} = \\frac{\\partial}{\\partial \\hat{H}_l} (\\Omega + \\hat{H}_l) and \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)}}{\\hat{H}_l} = \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} (\\Omega + \\hat{H}_l)}{\\hat{H}_l} and \\frac{\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{t_{1}}{(\\hat{H}_l,\\Omega)}}{\\hat{H}_l} = \\frac{1}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Function('t_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Function('t_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(A_{1})} = \\log{(A_{1})}, then derive 0 = - \\frac{\\log{(A_{1})} \\frac{d}{d A_{1}} \\Psi_{\\lambda}{(A_{1})}}{\\Psi_{\\lambda}^{2}{(A_{1})}} + \\frac{1}{A_{1} \\Psi_{\\lambda}{(A_{1})}}, then obtain 0 = - \\frac{\\frac{d}{d A_{1}} \\log{(A_{1})}}{\\log{(A_{1})}} + \\frac{1}{A_{1} \\log{(A_{1})}}", "derivation": "\\Psi_{\\lambda}{(A_{1})} = \\log{(A_{1})} and 1 = \\frac{\\log{(A_{1})}}{\\Psi_{\\lambda}{(A_{1})}} and \\frac{d}{d A_{1}} 1 = \\frac{d}{d A_{1}} \\frac{\\log{(A_{1})}}{\\Psi_{\\lambda}{(A_{1})}} and 0 = - \\frac{\\log{(A_{1})} \\frac{d}{d A_{1}} \\Psi_{\\lambda}{(A_{1})}}{\\Psi_{\\lambda}^{2}{(A_{1})}} + \\frac{1}{A_{1} \\Psi_{\\lambda}{(A_{1})}} and 0 = - \\frac{\\frac{d}{d A_{1}} \\log{(A_{1})}}{\\log{(A_{1})}} + \\frac{1}{A_{1} \\log{(A_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["divide", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), Integer(-1)), log(Symbol('A_1', commutative=True))))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), Integer(-1)), log(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), Integer(-2)), log(Symbol('A_1', commutative=True)), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('A_1', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(log(Symbol('A_1', commutative=True)), Integer(-1)), Derivative(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(log(Symbol('A_1', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(M,\\theta,L)} = \\frac{L M}{\\theta} and \\hat{p}_0{(\\theta)} = \\theta, then obtain \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial L} (\\hat{p}_0{(\\theta)} + \\frac{\\theta}{L M}))^{M} = \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial L} (\\theta + \\frac{\\theta}{L M}))^{M}", "derivation": "\\operatorname{r_{0}}{(M,\\theta,L)} = \\frac{L M}{\\theta} and \\hat{p}_0{(\\theta)} = \\theta and \\hat{p}_0{(\\theta)} + \\frac{1}{\\operatorname{r_{0}}{(M,\\theta,L)}} = \\theta + \\frac{1}{\\operatorname{r_{0}}{(M,\\theta,L)}} and \\hat{p}_0{(\\theta)} + \\frac{\\theta}{L M} = \\theta + \\frac{\\theta}{L M} and \\frac{\\partial}{\\partial L} (\\hat{p}_0{(\\theta)} + \\frac{\\theta}{L M}) = \\frac{\\partial}{\\partial L} (\\theta + \\frac{\\theta}{L M}) and (\\frac{\\partial}{\\partial L} (\\hat{p}_0{(\\theta)} + \\frac{\\theta}{L M}))^{M} = (\\frac{\\partial}{\\partial L} (\\theta + \\frac{\\theta}{L M}))^{M} and \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial L} (\\hat{p}_0{(\\theta)} + \\frac{\\theta}{L M}))^{M} = \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial L} (\\theta + \\frac{\\theta}{L M}))^{M}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('M', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))"], [["add", 2, "Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Add(Symbol('\\\\theta', commutative=True), Pow(Function('r_0')(Symbol('M', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["power", 5, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('M', commutative=True)))"], [["differentiate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Function('\\\\hat{p}_0')(Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('M', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\theta', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('M', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = \\Omega + \\sin{(\\mathbf{r})}, then derive \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = 1, then obtain \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} - 1 = 0", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = \\Omega + \\sin{(\\mathbf{r})} and \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\sin{(\\mathbf{r})}) and \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = 1 and \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\sin{(\\mathbf{r})}) \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\sin{(\\mathbf{r})}) and \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\sin{(\\mathbf{r})}) \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} - 1 = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\sin{(\\mathbf{r})}) - 1 and \\frac{\\partial}{\\partial \\Omega} \\operatorname{x^{{\\}'}}{(\\mathbf{r},\\Omega)} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(Mul(Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given q{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive - q{(\\mathbf{D})} + \\int q{(\\mathbf{D})} d\\mathbf{D} = \\mu_0 - q{(\\mathbf{D})} + e^{\\mathbf{D}}, then obtain \\mu_0 - q{(\\mathbf{D})} + e^{\\mathbf{D}} = \\mu_0", "derivation": "q{(\\mathbf{D})} = e^{\\mathbf{D}} and \\int q{(\\mathbf{D})} d\\mathbf{D} = \\int e^{\\mathbf{D}} d\\mathbf{D} and - q{(\\mathbf{D})} + \\int q{(\\mathbf{D})} d\\mathbf{D} = - q{(\\mathbf{D})} + \\int e^{\\mathbf{D}} d\\mathbf{D} and - q{(\\mathbf{D})} + \\int q{(\\mathbf{D})} d\\mathbf{D} = \\mu_0 - q{(\\mathbf{D})} + e^{\\mathbf{D}} and - e^{\\mathbf{D}} + \\int e^{\\mathbf{D}} d\\mathbf{D} = \\mu_0 and - e^{\\mathbf{D}} + \\int q{(\\mathbf{D})} d\\mathbf{D} = \\mu_0 and - q{(\\mathbf{D})} + \\int q{(\\mathbf{D})} d\\mathbf{D} = \\mu_0 and \\mu_0 - q{(\\mathbf{D})} + e^{\\mathbf{D}} = \\mu_0", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 2, "Function('q')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Function('q')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{D}', commutative=True))), exp(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mu_0', commutative=True))"]]}, {"prompt": "Given S{(b,\\nabla)} = e^{\\nabla + b}, then obtain 2 \\int S{(b,\\nabla)} d\\nabla = C_{1} + e^{\\nabla + b} + \\int S{(b,\\nabla)} d\\nabla", "derivation": "S{(b,\\nabla)} = e^{\\nabla + b} and \\int S{(b,\\nabla)} d\\nabla = \\int e^{\\nabla + b} d\\nabla and 2 \\int S{(b,\\nabla)} d\\nabla = \\int S{(b,\\nabla)} d\\nabla + \\int e^{\\nabla + b} d\\nabla and 2 \\int S{(b,\\nabla)} d\\nabla = C_{1} + e^{\\nabla + b} + \\int S{(b,\\nabla)} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Add(Symbol('\\\\nabla', commutative=True), Symbol('b', commutative=True))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Add(Symbol('\\\\nabla', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 2, "Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Add(Symbol('\\\\nabla', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('C_1', commutative=True), exp(Add(Symbol('\\\\nabla', commutative=True), Symbol('b', commutative=True))), Integral(Function('S')(Symbol('b', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hat{H})} = \\log{(e^{\\hat{H}})} and \\delta{(\\lambda)} = \\log{(e^{\\lambda})}, then obtain (\\operatorname{C_{2}}{(\\hat{H})} + \\delta^{\\lambda}{(\\lambda)})^{\\hat{H}} = (\\delta^{\\lambda}{(\\lambda)} + \\log{(e^{\\hat{H}})})^{\\hat{H}}", "derivation": "\\operatorname{C_{2}}{(\\hat{H})} = \\log{(e^{\\hat{H}})} and \\delta{(\\lambda)} = \\log{(e^{\\lambda})} and \\delta^{\\lambda}{(\\lambda)} = \\log{(e^{\\lambda})}^{\\lambda} and \\operatorname{C_{2}}{(\\hat{H})} + \\delta^{\\lambda}{(\\lambda)} = \\delta^{\\lambda}{(\\lambda)} + \\log{(e^{\\hat{H}})} and \\operatorname{C_{2}}{(\\hat{H})} + \\log{(e^{\\lambda})}^{\\lambda} = \\log{(e^{\\hat{H}})} + \\log{(e^{\\lambda})}^{\\lambda} and (\\operatorname{C_{2}}{(\\hat{H})} + \\log{(e^{\\lambda})}^{\\lambda})^{\\hat{H}} = (\\log{(e^{\\hat{H}})} + \\log{(e^{\\lambda})}^{\\lambda})^{\\hat{H}} and (\\operatorname{C_{2}}{(\\hat{H})} + \\delta^{\\lambda}{(\\lambda)})^{\\hat{H}} = (\\delta^{\\lambda}{(\\lambda)} + \\log{(e^{\\hat{H}})})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), log(exp(Symbol('\\\\hat{H}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), log(exp(Symbol('\\\\lambda', commutative=True))))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Add(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), log(exp(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(log(exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Add(log(exp(Symbol('\\\\hat{H}', commutative=True))), Pow(log(exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], [["power", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Add(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(log(exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(log(exp(Symbol('\\\\hat{H}', commutative=True))), Pow(log(exp(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Add(Function('C_2')(Symbol('\\\\hat{H}', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Pow(Function('\\\\delta')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), log(exp(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(M,\\mathbf{S},E_{x})} = E_{x} - M - \\mathbf{S} and \\mathbf{M}{(M,\\mathbf{S},E_{x})} = E_{x} - M - 2 \\mathbf{S}, then obtain \\mathbf{M}{(M,\\mathbf{S},E_{x})} = - \\mathbf{S} + \\sigma_{x}{(M,\\mathbf{S},E_{x})}", "derivation": "\\sigma_{x}{(M,\\mathbf{S},E_{x})} = E_{x} - M - \\mathbf{S} and - \\mathbf{S} + \\sigma_{x}{(M,\\mathbf{S},E_{x})} = E_{x} - M - 2 \\mathbf{S} and \\mathbf{M}{(M,\\mathbf{S},E_{x})} = E_{x} - M - 2 \\mathbf{S} and \\mathbf{M}{(M,\\mathbf{S},E_{x})} = - \\mathbf{S} + \\sigma_{x}{(M,\\mathbf{S},E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\sigma_x')(Symbol('M', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(Q,B)} = \\frac{Q}{B}, then derive \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} = - \\frac{Q}{B^{2}}, then obtain \\frac{\\partial}{\\partial B} \\frac{Q}{B} + \\frac{\\mathbf{s}{(Q,B)}}{B} = 0", "derivation": "\\mathbf{s}{(Q,B)} = \\frac{Q}{B} and \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} = \\frac{\\partial}{\\partial B} \\frac{Q}{B} and \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} = - \\frac{Q}{B^{2}} and \\frac{\\partial}{\\partial B} \\frac{Q}{B} = - \\frac{Q}{B^{2}} and \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} = - \\frac{\\mathbf{s}{(Q,B)}}{B} and \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} + \\frac{Q}{B} = \\frac{Q}{B} - \\frac{\\mathbf{s}{(Q,B)}}{B} and \\frac{\\partial}{\\partial B} \\mathbf{s}{(Q,B)} + \\frac{\\mathbf{s}{(Q,B)}}{B} = 0 and \\frac{\\partial}{\\partial B} \\frac{Q}{B} + \\frac{\\mathbf{s}{(Q,B)}}{B} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True))))"], [["add", 5, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True))"], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True))), Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)))))"], [["minus", 6, "Add(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True))))"], "Equality(Add(Derivative(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Add(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('Q', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True), Symbol('B', commutative=True)))), Integer(0))"]]}, {"prompt": "Given a{(A_{2},\\Psi_{\\lambda})} = A_{2} \\Psi_{\\lambda}, then obtain \\int A_{2}^{2} d\\Psi_{\\lambda} = \\int \\frac{A_{2}^{3} \\Psi_{\\lambda}}{a{(A_{2},\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "derivation": "a{(A_{2},\\Psi_{\\lambda})} = A_{2} \\Psi_{\\lambda} and A_{2} \\Psi_{\\lambda} a{(A_{2},\\Psi_{\\lambda})} = A_{2}^{2} \\Psi_{\\lambda}^{2} and A_{2}^{3} \\Psi_{\\lambda} a{(A_{2},\\Psi_{\\lambda})} = A_{2}^{4} \\Psi_{\\lambda}^{2} and A_{2}^{2} = \\frac{A_{2}^{3} \\Psi_{\\lambda}}{a{(A_{2},\\Psi_{\\lambda})}} and \\int A_{2}^{2} d\\Psi_{\\lambda} = \\int \\frac{A_{2}^{3} \\Psi_{\\lambda}}{a{(A_{2},\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 1, "Mul(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(2)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(2))))"], [["times", 2, "Pow(Symbol('A_2', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(3)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(4)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(2))))"], [["divide", 3, "Mul(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Pow(Symbol('A_2', commutative=True), Integer(2)), Mul(Pow(Symbol('A_2', commutative=True), Integer(3)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Symbol('A_2', commutative=True), Integer(2)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Mul(Pow(Symbol('A_2', commutative=True), Integer(3)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Function('a')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\chi{(l)} = \\sin{(e^{l})}, then obtain \\frac{l \\chi{(l)} - \\chi^{l}{(l)}}{\\sin{(e^{l})}} = \\frac{l \\sin{(e^{l})} - \\chi^{l}{(l)}}{\\sin{(e^{l})}}", "derivation": "\\chi{(l)} = \\sin{(e^{l})} and \\chi^{l}{(l)} = \\sin^{l}{(e^{l})} and l \\chi{(l)} = l \\sin{(e^{l})} and l \\chi{(l)} - \\sin^{l}{(e^{l})} = l \\sin{(e^{l})} - \\sin^{l}{(e^{l})} and l \\chi{(l)} - \\chi^{l}{(l)} = l \\sin{(e^{l})} - \\chi^{l}{(l)} and \\frac{l \\chi{(l)} - \\chi^{l}{(l)}}{\\sin{(e^{l})}} = \\frac{l \\sin{(e^{l})} - \\chi^{l}{(l)}}{\\sin{(e^{l})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('l', commutative=True)), sin(exp(Symbol('l', commutative=True))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(sin(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\chi')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), sin(exp(Symbol('l', commutative=True)))))"], [["minus", 3, "Pow(sin(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True))"], "Equality(Add(Mul(Symbol('l', commutative=True), Function('\\\\chi')(Symbol('l', commutative=True))), Mul(Integer(-1), Pow(sin(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Add(Mul(Symbol('l', commutative=True), sin(exp(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(sin(exp(Symbol('l', commutative=True))), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('l', commutative=True), Function('\\\\chi')(Symbol('l', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Add(Mul(Symbol('l', commutative=True), sin(exp(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"], [["divide", 5, "sin(exp(Symbol('l', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('l', commutative=True), Function('\\\\chi')(Symbol('l', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Pow(sin(exp(Symbol('l', commutative=True))), Integer(-1))), Mul(Add(Mul(Symbol('l', commutative=True), sin(exp(Symbol('l', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\chi')(Symbol('l', commutative=True)), Symbol('l', commutative=True)))), Pow(sin(exp(Symbol('l', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(\\varphi^*)} = e^{\\varphi^*} and \\varphi{(\\varphi^*)} = 2 \\rho_{f}{(\\varphi^*)}, then obtain \\int \\cos{(2 e^{\\varphi^*})} d\\varphi^* = \\int \\cos{(2 \\rho_{f}{(\\varphi^*)})} d\\varphi^*", "derivation": "\\rho_{f}{(\\varphi^*)} = e^{\\varphi^*} and 2 \\rho_{f}{(\\varphi^*)} = \\rho_{f}{(\\varphi^*)} + e^{\\varphi^*} and \\varphi{(\\varphi^*)} = 2 \\rho_{f}{(\\varphi^*)} and \\cos{(2 \\rho_{f}{(\\varphi^*)})} = \\cos{(\\rho_{f}{(\\varphi^*)} + e^{\\varphi^*})} and \\varphi{(\\varphi^*)} = 2 e^{\\varphi^*} and \\cos{(\\varphi{(\\varphi^*)})} = \\cos{(\\rho_{f}{(\\varphi^*)} + e^{\\varphi^*})} and \\cos{(\\varphi{(\\varphi^*)})} = \\cos{(2 \\rho_{f}{(\\varphi^*)})} and \\cos{(2 e^{\\varphi^*})} = \\cos{(2 \\rho_{f}{(\\varphi^*)})} and \\int \\cos{(2 e^{\\varphi^*})} d\\varphi^* = \\int \\cos{(2 \\rho_{f}{(\\varphi^*)})} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True))), Add(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)))), cos(Add(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\varphi')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(cos(Function('\\\\varphi')(Symbol('\\\\varphi^*', commutative=True))), cos(Add(Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(cos(Function('\\\\varphi')(Symbol('\\\\varphi^*', commutative=True))), cos(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(cos(Mul(Integer(2), exp(Symbol('\\\\varphi^*', commutative=True)))), cos(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)))))"], [["integrate", 8, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(cos(Mul(Integer(2), exp(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Mul(Integer(2), Function('\\\\rho_f')(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given S{(S,y)} = \\sin^{S}{(y)} and \\operatorname{V_{\\mathbf{B}}}{(S,y)} = - \\frac{\\partial}{\\partial S} \\int S{(S,y)} dS, then obtain \\operatorname{V_{\\mathbf{B}}}{(S,y)} + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int S{(S,y)} dS = \\operatorname{V_{\\mathbf{B}}}{(S,y)} + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int \\sin^{S}{(y)} dS", "derivation": "S{(S,y)} = \\sin^{S}{(y)} and \\int S{(S,y)} dS = \\int \\sin^{S}{(y)} dS and \\frac{\\partial}{\\partial S} \\int S{(S,y)} dS = \\frac{\\partial}{\\partial S} \\int \\sin^{S}{(y)} dS and \\frac{\\partial^{2}}{\\partial y\\partial S} \\int S{(S,y)} dS = \\frac{\\partial^{2}}{\\partial y\\partial S} \\int \\sin^{S}{(y)} dS and - \\frac{\\partial}{\\partial S} \\int S{(S,y)} dS + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int S{(S,y)} dS = - \\frac{\\partial}{\\partial S} \\int S{(S,y)} dS + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int \\sin^{S}{(y)} dS and \\operatorname{V_{\\mathbf{B}}}{(S,y)} = - \\frac{\\partial}{\\partial S} \\int S{(S,y)} dS and \\operatorname{V_{\\mathbf{B}}}{(S,y)} + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int S{(S,y)} dS = \\operatorname{V_{\\mathbf{B}}}{(S,y)} + \\frac{\\partial^{2}}{\\partial y\\partial S} \\int \\sin^{S}{(y)} dS", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integral(Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Integral(Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Derivative(Integral(Function('S')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Function('V_{\\\\mathbf{B}}')(Symbol('S', commutative=True), Symbol('y', commutative=True)), Derivative(Integral(Pow(sin(Symbol('y', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{f}{(\\tilde{g})} = \\sin{(\\tilde{g})}, then derive \\frac{d}{d \\tilde{g}} \\rho_{f}{(\\tilde{g})} = \\cos{(\\tilde{g})}, then obtain (- \\tilde{g} + \\rho_{f}{(\\tilde{g})}) \\cos{(\\tilde{g})} = (- \\tilde{g} + \\rho_{f}{(\\tilde{g})}) \\frac{d}{d \\tilde{g}} \\sin{(\\tilde{g})}", "derivation": "\\rho_{f}{(\\tilde{g})} = \\sin{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\rho_{f}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\sin{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\rho_{f}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\cos{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\sin{(\\tilde{g})} and (- \\tilde{g} + \\rho_{f}{(\\tilde{g})}) \\cos{(\\tilde{g})} = (- \\tilde{g} + \\rho_{f}{(\\tilde{g})}) \\frac{d}{d \\tilde{g}} \\sin{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True)), sin(Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), cos(Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\tilde{g}', commutative=True)), Derivative(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True))), cos(Symbol('\\\\tilde{g}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\tilde{g}', commutative=True))), Derivative(sin(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(\\phi,C,y)} = C y + \\phi, then obtain \\frac{d}{d C} 1 = \\frac{\\partial}{\\partial C} \\frac{C y + \\phi + V{(\\phi,C,y)}}{2 V{(\\phi,C,y)}}", "derivation": "V{(\\phi,C,y)} = C y + \\phi and 2 V{(\\phi,C,y)} = C y + \\phi + V{(\\phi,C,y)} and 1 = \\frac{C y + \\phi + V{(\\phi,C,y)}}{2 V{(\\phi,C,y)}} and \\frac{d}{d C} 1 = \\frac{\\partial}{\\partial C} \\frac{C y + \\phi + V{(\\phi,C,y)}}{2 V{(\\phi,C,y)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True)), Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True))), Pow(Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\phi', commutative=True), Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True))), Pow(Function('V')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True), Symbol('y', commutative=True)), Integer(-1))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})} = F_{c} + \\dot{\\mathbf{r}} and \\operatorname{A_{z}}{(F_{c},\\dot{\\mathbf{r}})} = \\log{((F_{c} + \\dot{\\mathbf{r}}) \\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})})}, then obtain \\operatorname{A_{z}}{(F_{c},\\dot{\\mathbf{r}})} = \\log{(\\eta^{\\prime}^{2}{(F_{c},\\dot{\\mathbf{r}})})}", "derivation": "\\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})} = F_{c} + \\dot{\\mathbf{r}} and \\eta^{\\prime}^{2}{(F_{c},\\dot{\\mathbf{r}})} = (F_{c} + \\dot{\\mathbf{r}}) \\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})} and \\log{(\\eta^{\\prime}^{2}{(F_{c},\\dot{\\mathbf{r}})})} = \\log{((F_{c} + \\dot{\\mathbf{r}}) \\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})})} and \\operatorname{A_{z}}{(F_{c},\\dot{\\mathbf{r}})} = \\log{((F_{c} + \\dot{\\mathbf{r}}) \\eta^{\\prime}{(F_{c},\\dot{\\mathbf{r}})})} and \\operatorname{A_{z}}{(F_{c},\\dot{\\mathbf{r}})} = \\log{(\\eta^{\\prime}^{2}{(F_{c},\\dot{\\mathbf{r}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["times", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(2)), Mul(Add(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["log", 2], "Equality(log(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(2))), log(Mul(Add(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Mul(Add(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('A_z')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(t)} = e^{t}, then derive \\frac{d}{d t} \\Psi_{\\lambda}{(t)} = e^{t}, then obtain \\frac{d}{d t} (- \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int \\frac{d}{d t} e^{t} dt) = \\frac{d}{d t} (- \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int e^{t} dt)", "derivation": "\\Psi_{\\lambda}{(t)} = e^{t} and \\frac{d}{d t} \\Psi_{\\lambda}{(t)} = \\frac{d}{d t} e^{t} and \\frac{d}{d t} \\Psi_{\\lambda}{(t)} = e^{t} and \\int \\frac{d}{d t} \\Psi_{\\lambda}{(t)} dt = \\int e^{t} dt and \\int \\frac{d}{d t} e^{t} dt = \\int e^{t} dt and - \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int \\frac{d}{d t} e^{t} dt = - \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int e^{t} dt and \\frac{d}{d t} (- \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int \\frac{d}{d t} e^{t} dt) = \\frac{d}{d t} (- \\frac{d}{d t} \\Psi_{\\lambda}{(t)} + \\int e^{t} dt)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), exp(Symbol('t', commutative=True)))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["minus", 5, "Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["differentiate", 6, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(Derivative(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integral(exp(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\Psi)} = \\cos{(\\Psi)} and \\operatorname{F_{c}}{(\\Psi)} = \\frac{\\cos{(\\Psi)}}{\\Psi}, then obtain \\operatorname{F_{c}}{(\\Psi)} + \\frac{1}{\\Psi} = \\frac{\\cos{(\\Psi)}}{\\Psi} + \\frac{1}{\\Psi}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\Psi)} = \\cos{(\\Psi)} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\Psi)}}{\\Psi} = \\frac{\\cos{(\\Psi)}}{\\Psi} and \\operatorname{F_{c}}{(\\Psi)} = \\frac{\\cos{(\\Psi)}}{\\Psi} and \\operatorname{F_{c}}{(\\Psi)} = \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\Psi)}}{\\Psi} and \\operatorname{F_{c}}{(\\Psi)} + \\frac{1}{\\Psi} = \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\Psi)}}{\\Psi} + \\frac{1}{\\Psi} and \\operatorname{F_{c}}{(\\Psi)} + \\frac{1}{\\Psi} = \\frac{\\cos{(\\Psi)}}{\\Psi} + \\frac{1}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('F_c')(Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True))))"], [["add", 4, "Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))"], "Equality(Add(Function('F_c')(Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\Psi', commutative=True))), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('F_c')(Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), cos(Symbol('\\\\Psi', commutative=True))), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(z,z^{*})} = \\frac{\\partial}{\\partial z} (z - z^{*}), then derive - z^{*} \\ddot{x}{(z,z^{*})} = - z^{*}, then obtain - z^{*} (z - z^{*}) \\ddot{x}{(z,z^{*})} \\frac{\\partial}{\\partial z} (z - z^{*}) = - z^{*} (z - z^{*}) \\frac{\\partial}{\\partial z} (z - z^{*})", "derivation": "\\ddot{x}{(z,z^{*})} = \\frac{\\partial}{\\partial z} (z - z^{*}) and - z^{*} \\ddot{x}{(z,z^{*})} = - z^{*} \\frac{\\partial}{\\partial z} (z - z^{*}) and - z^{*} \\ddot{x}{(z,z^{*})} = - z^{*} and - z^{*} (z - z^{*}) \\ddot{x}{(z,z^{*})} = - z^{*} (z - z^{*}) \\frac{\\partial}{\\partial z} (z - z^{*}) and - z^{*} (z - z^{*}) \\ddot{x}{(z,z^{*})} = - z^{*} (z - z^{*}) and - z^{*} (z - z^{*}) = - z^{*} (z - z^{*}) \\frac{\\partial}{\\partial z} (z - z^{*}) and - z^{*} (z - z^{*}) \\ddot{x}{(z,z^{*})} \\frac{\\partial}{\\partial z} (z - z^{*}) = - z^{*} (z - z^{*}) \\frac{\\partial}{\\partial z} (z - z^{*})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["times", 1, "Mul(Integer(-1), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Symbol('z^*', commutative=True), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Symbol('z^*', commutative=True)))"], [["times", 2, "Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["times", 3, "Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True))), Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True)))), Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Function('\\\\ddot{x}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('z^*', commutative=True), Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Derivative(Add(Symbol('z', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(k,v_{2})} = k - v_{2}, then obtain - \\delta e^{- k + v_{2}} \\frac{\\partial}{\\partial k} S{(k,v_{2})} = - \\delta e^{- k + v_{2}} \\frac{\\partial}{\\partial k} (k - v_{2})", "derivation": "S{(k,v_{2})} = k - v_{2} and \\frac{\\partial}{\\partial k} S{(k,v_{2})} = \\frac{\\partial}{\\partial k} (k - v_{2}) and - \\delta \\frac{\\partial}{\\partial k} S{(k,v_{2})} = - \\delta \\frac{\\partial}{\\partial k} (k - v_{2}) and - \\delta e^{- k + v_{2}} \\frac{\\partial}{\\partial k} S{(k,v_{2})} = - \\delta e^{- k + v_{2}} \\frac{\\partial}{\\partial k} (k - v_{2})", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Derivative(Function('S')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["divide", 3, "exp(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), exp(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('v_2', commutative=True))), Derivative(Function('S')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True), exp(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('v_2', commutative=True))), Derivative(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(h)} = \\log{(\\log{(h)})} and m{(h)} = \\frac{\\int \\log{(\\log{(h)})} dh}{\\log{(h)}}, then obtain \\frac{\\int \\operatorname{F_{H}}{(h)} dh}{\\log{(h)}} = m{(h)}", "derivation": "\\operatorname{F_{H}}{(h)} = \\log{(\\log{(h)})} and \\int \\operatorname{F_{H}}{(h)} dh = \\int \\log{(\\log{(h)})} dh and \\frac{\\int \\operatorname{F_{H}}{(h)} dh}{\\log{(h)}} = \\frac{\\int \\log{(\\log{(h)})} dh}{\\log{(h)}} and m{(h)} = \\frac{\\int \\log{(\\log{(h)})} dh}{\\log{(h)}} and \\frac{\\int \\operatorname{F_{H}}{(h)} dh}{\\log{(h)}} = m{(h)}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('h', commutative=True)), log(log(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(log(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["divide", 2, "log(Symbol('h', commutative=True))"], "Equality(Mul(Pow(log(Symbol('h', commutative=True)), Integer(-1)), Integral(Function('F_H')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(log(Symbol('h', commutative=True)), Integer(-1)), Integral(log(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('h', commutative=True)), Mul(Pow(log(Symbol('h', commutative=True)), Integer(-1)), Integral(log(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(log(Symbol('h', commutative=True)), Integer(-1)), Integral(Function('F_H')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))), Function('m')(Symbol('h', commutative=True)))"]]}, {"prompt": "Given b{(s)} = e^{s}, then derive \\int b{(s)} ds = q + e^{s}, then obtain q + b{(s)} = \\int e^{s} ds", "derivation": "b{(s)} = e^{s} and \\int b{(s)} ds = \\int e^{s} ds and \\int b{(s)} ds = q + e^{s} and q + e^{s} = \\int e^{s} ds and q + b{(s)} = \\int e^{s} ds", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('b')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('q', commutative=True), exp(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('q', commutative=True), exp(Symbol('s', commutative=True))), Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('q', commutative=True), Function('b')(Symbol('s', commutative=True))), Integral(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J,A)} = A J, then obtain A \\int A dJ + J + \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = A \\int A dJ + J + \\int A dJ", "derivation": "\\operatorname{t_{1}}{(J,A)} = A J and \\frac{\\operatorname{t_{1}}{(J,A)}}{J} = A and \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = \\int A dJ and \\frac{\\operatorname{t_{1}}{(J,A)} \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ}{J} = \\frac{\\operatorname{t_{1}}{(J,A)} \\int A dJ}{J} and A \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = A \\int A dJ and A \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ + \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = A \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ + \\int A dJ and A \\int A dJ + \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = A \\int A dJ + \\int A dJ and A \\int A dJ + J + \\int \\frac{\\operatorname{t_{1}}{(J,A)}}{J} dJ = A \\int A dJ + J + \\int A dJ", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('J', commutative=True)))"], [["divide", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Symbol('A', commutative=True))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True))))"], [["times", 3, "Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True)))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True)), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('A', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Symbol('A', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))))"], [["add", 3, "Mul(Symbol('A', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True))))"], "Equality(Add(Mul(Symbol('A', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Symbol('A', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))))"], [["add", 7, "Symbol('J', commutative=True)"], "Equality(Add(Mul(Symbol('A', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('t_1')(Symbol('J', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('J', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True), Integral(Symbol('A', commutative=True), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(l)} = \\sin{(l)}, then obtain \\log{(\\frac{\\operatorname{f_{\\mathbf{v}}}^{l}{(l)}}{\\sin{(l)}})} = \\log{(\\frac{\\sin^{l}{(l)}}{\\sin{(l)}})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(l)} = \\sin{(l)} and \\operatorname{f_{\\mathbf{v}}}^{l}{(l)} = \\sin^{l}{(l)} and \\frac{\\operatorname{f_{\\mathbf{v}}}^{l}{(l)}}{\\sin{(l)}} = \\frac{\\sin^{l}{(l)}}{\\sin{(l)}} and \\log{(\\frac{\\operatorname{f_{\\mathbf{v}}}^{l}{(l)}}{\\sin{(l)}})} = \\log{(\\frac{\\sin^{l}{(l)}}{\\sin{(l)}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["divide", 2, "sin(Symbol('l', commutative=True))"], "Equality(Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(sin(Symbol('l', commutative=True)), Integer(-1))), Mul(Pow(sin(Symbol('l', commutative=True)), Integer(-1)), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True))))"], [["log", 3], "Equality(log(Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(sin(Symbol('l', commutative=True)), Integer(-1)))), log(Mul(Pow(sin(Symbol('l', commutative=True)), Integer(-1)), Pow(sin(Symbol('l', commutative=True)), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{M})} = e^{\\sin{(\\mathbf{M})}}, then obtain \\frac{d}{d \\mathbf{M}} - \\sin{(\\operatorname{F_{x}}{(\\mathbf{M})} + \\sin{(\\mathbf{M})})} = \\frac{d}{d \\mathbf{M}} - \\sin{(2 \\operatorname{F_{x}}{(\\mathbf{M})} - e^{\\sin{(\\mathbf{M})}} + \\sin{(\\mathbf{M})})}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{M})} = e^{\\sin{(\\mathbf{M})}} and - \\sin{(\\mathbf{M})} = - \\operatorname{F_{x}}{(\\mathbf{M})} + e^{\\sin{(\\mathbf{M})}} - \\sin{(\\mathbf{M})} and - \\operatorname{F_{x}}{(\\mathbf{M})} - \\sin{(\\mathbf{M})} = - 2 \\operatorname{F_{x}}{(\\mathbf{M})} + e^{\\sin{(\\mathbf{M})}} - \\sin{(\\mathbf{M})} and - \\sin{(\\operatorname{F_{x}}{(\\mathbf{M})} + \\sin{(\\mathbf{M})})} = - \\sin{(2 \\operatorname{F_{x}}{(\\mathbf{M})} - e^{\\sin{(\\mathbf{M})}} + \\sin{(\\mathbf{M})})} and \\frac{d}{d \\mathbf{M}} - \\sin{(\\operatorname{F_{x}}{(\\mathbf{M})} + \\sin{(\\mathbf{M})})} = \\frac{d}{d \\mathbf{M}} - \\sin{(2 \\operatorname{F_{x}}{(\\mathbf{M})} - e^{\\sin{(\\mathbf{M})}} + \\sin{(\\mathbf{M})})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 1, "Add(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True))), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True))), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["sin", 3], "Equality(Mul(Integer(-1), sin(Add(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('\\\\mathbf{M}', commutative=True)))), sin(Symbol('\\\\mathbf{M}', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Add(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Mul(Integer(2), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('\\\\mathbf{M}', commutative=True)))), sin(Symbol('\\\\mathbf{M}', commutative=True))))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(S,\\theta_1)} = \\cos{(S + \\theta_1)}, then derive \\frac{\\frac{\\partial}{\\partial \\theta_1} \\phi_{1}{(S,\\theta_1)}}{- S + \\cos{(S + \\theta_1)}} = - \\frac{\\sin{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}}, then obtain \\frac{\\frac{\\partial}{\\partial \\theta_1} \\cos{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}} = - \\frac{\\sin{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}}", "derivation": "\\phi_{1}{(S,\\theta_1)} = \\cos{(S + \\theta_1)} and - S + \\phi_{1}{(S,\\theta_1)} = - S + \\cos{(S + \\theta_1)} and \\frac{\\partial}{\\partial \\theta_1} (- S + \\phi_{1}{(S,\\theta_1)}) = \\frac{\\partial}{\\partial \\theta_1} (- S + \\cos{(S + \\theta_1)}) and \\frac{\\frac{\\partial}{\\partial \\theta_1} (- S + \\phi_{1}{(S,\\theta_1)})}{- S + \\cos{(S + \\theta_1)}} = \\frac{\\frac{\\partial}{\\partial \\theta_1} (- S + \\cos{(S + \\theta_1)})}{- S + \\cos{(S + \\theta_1)}} and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\phi_{1}{(S,\\theta_1)}}{- S + \\cos{(S + \\theta_1)}} = - \\frac{\\sin{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}} and \\frac{\\frac{\\partial}{\\partial \\theta_1} \\cos{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}} = - \\frac{\\sin{(S + \\theta_1)}}{- S + \\cos{(S + \\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\phi_1')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\phi_1')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\phi_1')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Derivative(Function('\\\\phi_1')(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), sin(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Derivative(cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True)), cos(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), sin(Add(Symbol('S', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(T)} = \\cos{(e^{T})}, then derive \\frac{d}{d T} \\varphi{(T)} = - e^{T} \\sin{(e^{T})}, then obtain - \\frac{\\cos{(e^{T})} \\frac{d}{d T} \\varphi{(T)}}{\\mathbf{p} \\varphi{(T)}} = \\frac{e^{T} \\sin{(e^{T})} \\cos{(e^{T})}}{\\mathbf{p} \\varphi{(T)}}", "derivation": "\\varphi{(T)} = \\cos{(e^{T})} and \\frac{d}{d T} \\varphi{(T)} = \\frac{d}{d T} \\cos{(e^{T})} and \\frac{d}{d T} \\varphi{(T)} = - e^{T} \\sin{(e^{T})} and \\frac{\\cos{(e^{T})} \\frac{d}{d T} \\varphi{(T)}}{\\varphi{(T)}} = \\frac{\\cos{(e^{T})} \\frac{d}{d T} \\cos{(e^{T})}}{\\varphi{(T)}} and - e^{T} \\sin{(e^{T})} = \\frac{d}{d T} \\cos{(e^{T})} and \\frac{\\cos{(e^{T})} \\frac{d}{d T} \\varphi{(T)}}{\\varphi{(T)}} = - \\frac{e^{T} \\sin{(e^{T})} \\cos{(e^{T})}}{\\varphi{(T)}} and - \\frac{\\cos{(e^{T})} \\frac{d}{d T} \\varphi{(T)}}{\\mathbf{p} \\varphi{(T)}} = \\frac{e^{T} \\sin{(e^{T})} \\cos{(e^{T})}}{\\mathbf{p} \\varphi{(T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('T', commutative=True)), cos(exp(Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('T', commutative=True)), sin(exp(Symbol('T', commutative=True)))))"], [["divide", 2, "Mul(Function('\\\\varphi')(Symbol('T', commutative=True)), Pow(cos(exp(Symbol('T', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), cos(exp(Symbol('T', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), cos(exp(Symbol('T', commutative=True))), Derivative(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(Symbol('T', commutative=True)), sin(exp(Symbol('T', commutative=True)))), Derivative(cos(exp(Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), cos(exp(Symbol('T', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), exp(Symbol('T', commutative=True)), sin(exp(Symbol('T', commutative=True))), cos(exp(Symbol('T', commutative=True)))))"], [["divide", 6, "Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), cos(exp(Symbol('T', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('T', commutative=True)), Integer(-1)), exp(Symbol('T', commutative=True)), sin(exp(Symbol('T', commutative=True))), cos(exp(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} = a n_{1}^{f_{E}} and \\mathbf{g}{(a,n_{1},f_{E})} = \\int \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} da, then obtain - a n_{1}^{f_{E}} \\int \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} da = - a n_{1}^{f_{E}} \\int a n_{1}^{f_{E}} da", "derivation": "\\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} = a n_{1}^{f_{E}} and \\int \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} da = \\int a n_{1}^{f_{E}} da and \\mathbf{g}{(a,n_{1},f_{E})} = \\int \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} da and \\mathbf{g}{(a,n_{1},f_{E})} = \\int a n_{1}^{f_{E}} da and - a n_{1}^{f_{E}} \\mathbf{g}{(a,n_{1},f_{E})} = - a n_{1}^{f_{E}} \\int a n_{1}^{f_{E}} da and - a n_{1}^{f_{E}} \\int \\operatorname{x^{{\\}'}}{(a,n_{1},f_{E})} da = - a n_{1}^{f_{E}} \\int a n_{1}^{f_{E}} da", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Mul(Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Integral(Function('x^\\\\prime')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Function('\\\\mathbf{g}')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Integral(Function('x^\\\\prime')(Symbol('a', commutative=True), Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a', commutative=True)))), Mul(Integer(-1), Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True)), Integral(Mul(Symbol('a', commutative=True), Pow(Symbol('n_1', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(b)} = e^{b}, then derive 2 \\frac{d}{d b} \\dot{y}{(b)} = e^{b} + \\frac{d}{d b} \\dot{y}{(b)}, then obtain 4 (\\frac{d}{d b} e^{b})^{2} = (e^{b} + \\frac{d}{d b} e^{b})^{2}", "derivation": "\\dot{y}{(b)} = e^{b} and 2 \\dot{y}{(b)} = \\dot{y}{(b)} + e^{b} and \\frac{d}{d b} 2 \\dot{y}{(b)} = \\frac{d}{d b} (\\dot{y}{(b)} + e^{b}) and 2 \\frac{d}{d b} \\dot{y}{(b)} = e^{b} + \\frac{d}{d b} \\dot{y}{(b)} and 2 \\frac{d}{d b} \\dot{y}{(b)} = \\dot{y}{(b)} + \\frac{d}{d b} \\dot{y}{(b)} and 2 \\frac{d}{d b} e^{b} = e^{b} + \\frac{d}{d b} e^{b} and 4 (\\frac{d}{d b} e^{b})^{2} = (e^{b} + \\frac{d}{d b} e^{b})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["add", 1, "Function('\\\\dot{y}')(Symbol('b', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('b', commutative=True))), Add(Function('\\\\dot{y}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\dot{y}')(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Function('\\\\dot{y}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\dot{y}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(exp(Symbol('b', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Function('\\\\dot{y}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Function('\\\\dot{y}')(Symbol('b', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(2), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(exp(Symbol('b', commutative=True)), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["power", 6, 2], "Equality(Mul(Integer(4), Pow(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(2))), Pow(Add(exp(Symbol('b', commutative=True)), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{f}{(x)} = \\log{(x)}, then obtain \\mathbf{f}{(x)} \\log{(x)}^{3} \\frac{d}{d x} \\mathbf{f}{(x)} = \\log{(x)}^{4} \\frac{d}{d x} \\mathbf{f}{(x)}", "derivation": "\\mathbf{f}{(x)} = \\log{(x)} and \\frac{d}{d x} \\mathbf{f}{(x)} = \\frac{d}{d x} \\log{(x)} and \\mathbf{f}{(x)} \\log{(x)} = \\log{(x)}^{2} and \\mathbf{f}{(x)} \\log{(x)}^{3} = \\log{(x)}^{4} and \\mathbf{f}{(x)} \\log{(x)}^{3} \\frac{d}{d x} \\log{(x)} = \\log{(x)}^{4} \\frac{d}{d x} \\log{(x)} and \\mathbf{f}{(x)} \\log{(x)}^{3} \\frac{d}{d x} \\mathbf{f}{(x)} = \\log{(x)}^{4} \\frac{d}{d x} \\mathbf{f}{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 1, "log(Symbol('x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), log(Symbol('x', commutative=True))), Pow(log(Symbol('x', commutative=True)), Integer(2)))"], [["times", 3, "Pow(log(Symbol('x', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(3))), Pow(log(Symbol('x', commutative=True)), Integer(4)))"], [["times", 4, "Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(3)), Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('x', commutative=True)), Integer(4)), Derivative(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Pow(log(Symbol('x', commutative=True)), Integer(3)), Derivative(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('x', commutative=True)), Integer(4)), Derivative(Function('\\\\mathbf{f}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\chi,\\dot{y})} = \\sin{(\\frac{\\chi}{\\dot{y}})}, then obtain \\int \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\mathbf{E}{(\\chi,\\dot{y})} d\\chi = B + \\frac{\\chi^{2} \\cos{(\\frac{\\chi}{\\dot{y}})}}{\\dot{y}^{3}}", "derivation": "\\mathbf{E}{(\\chi,\\dot{y})} = \\sin{(\\frac{\\chi}{\\dot{y}})} and \\frac{\\partial}{\\partial \\dot{y}} \\mathbf{E}{(\\chi,\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} \\sin{(\\frac{\\chi}{\\dot{y}})} and \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\mathbf{E}{(\\chi,\\dot{y})} = \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\sin{(\\frac{\\chi}{\\dot{y}})} and \\int \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\mathbf{E}{(\\chi,\\dot{y})} d\\chi = \\int \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\sin{(\\frac{\\chi}{\\dot{y}})} d\\chi and \\int \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} \\mathbf{E}{(\\chi,\\dot{y})} d\\chi = B + \\frac{\\chi^{2} \\cos{(\\frac{\\chi}{\\dot{y}})}}{\\dot{y}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), sin(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Derivative(sin(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Derivative(sin(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('B', commutative=True), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-3)), cos(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given \\varphi{(Z)} = \\sin{(Z)}, then obtain \\varphi^{3 Z}{(Z)} \\sin^{Z}{(Z)} = \\sin^{4 Z}{(Z)}", "derivation": "\\varphi{(Z)} = \\sin{(Z)} and \\varphi^{Z}{(Z)} = \\sin^{Z}{(Z)} and \\varphi^{Z}{(Z)} \\sin^{Z}{(Z)} = \\sin^{2 Z}{(Z)} and - \\varphi^{Z}{(Z)} \\sin^{Z}{(Z)} = - \\sin^{2 Z}{(Z)} and \\varphi^{2 Z}{(Z)} \\sin^{2 Z}{(Z)} = \\sin^{4 Z}{(Z)} and \\varphi^{3 Z}{(Z)} \\sin^{Z}{(Z)} = \\varphi^{2 Z}{(Z)} \\sin^{2 Z}{(Z)} and \\varphi^{3 Z}{(Z)} \\sin^{Z}{(Z)} = \\sin^{4 Z}{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 2, "Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True)))), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(4), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('Z', commutative=True)), Mul(Integer(3), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(sin(Symbol('Z', commutative=True)), Mul(Integer(4), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(f^{*})} = \\sin{(f^{*})}, then obtain 0^{f^{*}} + e^{m_{s}} = (- \\mu_{0}{(f^{*})} + \\sin{(f^{*})})^{f^{*}} + e^{m_{s}}", "derivation": "\\mu_{0}{(f^{*})} = \\sin{(f^{*})} and 0 = - \\mu_{0}{(f^{*})} + \\sin{(f^{*})} and 0^{f^{*}} = (- \\mu_{0}{(f^{*})} + \\sin{(f^{*})})^{f^{*}} and 0^{f^{*}} + e^{m_{s}} = (- \\mu_{0}{(f^{*})} + \\sin{(f^{*})})^{f^{*}} + e^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f^*', commutative=True))), sin(Symbol('f^*', commutative=True))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f^*', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f^*', commutative=True))), sin(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["add", 3, "exp(Symbol('m_s', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('f^*', commutative=True)), exp(Symbol('m_s', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('f^*', commutative=True))), sin(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), exp(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(E_{n},i)} = \\sin{(E_{n} i)}, then obtain - i + 2 \\operatorname{C_{1}}{(E_{n},i)} + 1 = - i + 2 \\sin{(E_{n} i)} + 1", "derivation": "\\operatorname{C_{1}}{(E_{n},i)} = \\sin{(E_{n} i)} and - i + \\operatorname{C_{1}}{(E_{n},i)} = - i + \\sin{(E_{n} i)} and - i = - i - \\operatorname{C_{1}}{(E_{n},i)} + \\sin{(E_{n} i)} and - i + \\sin{(E_{n} i)} = - i - \\operatorname{C_{1}}{(E_{n},i)} + 2 \\sin{(E_{n} i)} and - i + \\operatorname{C_{1}}{(E_{n},i)} = - i - \\operatorname{C_{1}}{(E_{n},i)} + 2 \\sin{(E_{n} i)} and - i + 2 \\operatorname{C_{1}}{(E_{n},i)} = - i + 2 \\sin{(E_{n} i)} and - i + 2 \\operatorname{C_{1}}{(E_{n},i)} + 1 = - i + 2 \\sin{(E_{n} i)} + 1", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True))))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))))"], [["minus", 2, "Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(-1), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True))))))"], [["minus", 5, "Mul(Integer(-1), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True))))))"], [["minus", 6, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), Function('C_1')(Symbol('E_n', commutative=True), Symbol('i', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), sin(Mul(Symbol('E_n', commutative=True), Symbol('i', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(b)} = \\log{(b)}, then obtain (\\int \\operatorname{C_{1}}{(b)} db)^{2 b} = ((\\int \\operatorname{C_{1}}{(b)} db)^{b}) (\\int \\log{(b)} db)^{b}", "derivation": "\\operatorname{C_{1}}{(b)} = \\log{(b)} and \\int \\operatorname{C_{1}}{(b)} db = \\int \\log{(b)} db and (\\int \\operatorname{C_{1}}{(b)} db)^{b} = (\\int \\log{(b)} db)^{b} and (\\int \\operatorname{C_{1}}{(b)} db)^{2 b} = ((\\int \\operatorname{C_{1}}{(b)} db)^{b}) (\\int \\log{(b)} db)^{b}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Function('C_1')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["times", 3, "Pow(Integral(Function('C_1')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))"], "Equality(Pow(Integral(Function('C_1')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Mul(Integer(2), Symbol('b', commutative=True))), Mul(Pow(Integral(Function('C_1')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and r{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})}, then obtain \\frac{\\dot{\\mathbf{r}}{(J_{\\varepsilon})}}{\\log{(J_{\\varepsilon})}} = \\frac{\\dot{\\mathbf{r}}^{2}{(J_{\\varepsilon})}}{r{(J_{\\varepsilon})} \\log{(J_{\\varepsilon})}}", "derivation": "\\dot{\\mathbf{r}}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and r{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and r{(J_{\\varepsilon})} = \\dot{\\mathbf{r}}{(J_{\\varepsilon})} and r{(J_{\\varepsilon})} \\log{(J_{\\varepsilon})} = \\dot{\\mathbf{r}}{(J_{\\varepsilon})} \\log{(J_{\\varepsilon})} and r{(J_{\\varepsilon})} \\log{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})}^{2} and \\dot{\\mathbf{r}}{(J_{\\varepsilon})} r{(J_{\\varepsilon})} = \\dot{\\mathbf{r}}^{2}{(J_{\\varepsilon})} and \\frac{\\dot{\\mathbf{r}}{(J_{\\varepsilon})}}{\\log{(J_{\\varepsilon})}} = \\frac{\\dot{\\mathbf{r}}^{2}{(J_{\\varepsilon})}}{r{(J_{\\varepsilon})} \\log{(J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["times", 3, "log(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Pow(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)))"], [["divide", 6, "Mul(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), Pow(Function('r')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Pow(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given G{(A,\\nabla)} = A - \\nabla, then derive \\int \\cos{(G{(A,\\nabla)})} dA = m_{s} + \\sin{(A - \\nabla)}, then obtain \\cos{(\\int \\cos{(G{(A,\\nabla)})} dA)} = \\cos{(m_{s} + \\sin{(A - \\nabla)})}", "derivation": "G{(A,\\nabla)} = A - \\nabla and \\cos{(G{(A,\\nabla)})} = \\cos{(A - \\nabla)} and \\int \\cos{(G{(A,\\nabla)})} dA = \\int \\cos{(A - \\nabla)} dA and \\int \\cos{(G{(A,\\nabla)})} dA = m_{s} + \\sin{(A - \\nabla)} and \\cos{(\\int \\cos{(G{(A,\\nabla)})} dA)} = \\cos{(m_{s} + \\sin{(A - \\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))"], [["cos", 1], "Equality(cos(Function('G')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True))), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(cos(Function('G')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(cos(Function('G')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('A', commutative=True))), Add(Symbol('m_s', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))))))"], [["cos", 4], "Equality(cos(Integral(cos(Function('G')(Symbol('A', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('A', commutative=True)))), cos(Add(Symbol('m_s', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))))"]]}, {"prompt": "Given E{(f^{*})} = \\sin{(\\log{(f^{*})})} and \\mathbf{M}{(f^{*})} = \\sin^{2}{(\\log{(f^{*})})}, then obtain (\\int (\\mathbf{M}{(f^{*})} - \\sin^{2}{(\\log{(f^{*})})}) df^{*})^{f^{*}} - 1 = (\\int 0 df^{*})^{f^{*}} - 1", "derivation": "E{(f^{*})} = \\sin{(\\log{(f^{*})})} and \\mathbf{M}{(f^{*})} = \\sin^{2}{(\\log{(f^{*})})} and - E{(f^{*})} \\sin{(\\log{(f^{*})})} + \\mathbf{M}{(f^{*})} = - E{(f^{*})} \\sin{(\\log{(f^{*})})} + \\sin^{2}{(\\log{(f^{*})})} and \\mathbf{M}{(f^{*})} - \\sin^{2}{(\\log{(f^{*})})} = 0 and \\int (\\mathbf{M}{(f^{*})} - \\sin^{2}{(\\log{(f^{*})})}) df^{*} = \\int 0 df^{*} and (\\int (\\mathbf{M}{(f^{*})} - \\sin^{2}{(\\log{(f^{*})})}) df^{*})^{f^{*}} = (\\int 0 df^{*})^{f^{*}} and (\\int (\\mathbf{M}{(f^{*})} - \\sin^{2}{(\\log{(f^{*})})}) df^{*})^{f^{*}} - 1 = (\\int 0 df^{*})^{f^{*}} - 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('f^*', commutative=True)), sin(log(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True)), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2)))"], [["minus", 2, "Mul(Function('E')(Symbol('f^*', commutative=True)), sin(log(Symbol('f^*', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('f^*', commutative=True)), sin(log(Symbol('f^*', commutative=True)))), Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True))), Add(Mul(Integer(-1), Function('E')(Symbol('f^*', commutative=True)), sin(log(Symbol('f^*', commutative=True)))), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2)))), Integer(0))"], [["integrate", 4, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2)))), Tuple(Symbol('f^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))))"], [["power", 5, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2)))), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["add", 6, "Integer(-1)"], "Equality(Add(Pow(Integral(Add(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('f^*', commutative=True))), Integer(2)))), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Integer(-1)), Add(Pow(Integral(Integer(0), Tuple(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(f)} = f, then derive \\frac{d}{d f} \\operatorname{z^{*}}{(f)} = 1, then obtain \\int \\frac{d}{d f} f d\\operatorname{z^{*}}{(f)} = \\mathbf{A} + \\operatorname{z^{*}}{(f)}", "derivation": "\\operatorname{z^{*}}{(f)} = f and \\frac{d}{d f} \\operatorname{z^{*}}{(f)} = \\frac{d}{d f} f and \\frac{d}{d f} \\operatorname{z^{*}}{(f)} = 1 and \\frac{d}{d f} f = 1 and \\int \\frac{d}{d f} f df = \\int 1 df and \\int \\frac{d}{d f} f d\\operatorname{z^{*}}{(f)} = \\int 1 d\\operatorname{z^{*}}{(f)} and \\int \\frac{d}{d f} f d\\operatorname{z^{*}}{(f)} = \\mathbf{A} + \\operatorname{z^{*}}{(f)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('f', commutative=True)), Symbol('f', commutative=True))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Derivative(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('f', commutative=True))), Integral(Integer(1), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Derivative(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Function('z^*')(Symbol('f', commutative=True)))), Integral(Integer(1), Tuple(Function('z^*')(Symbol('f', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Integral(Derivative(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Function('z^*')(Symbol('f', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('z^*')(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(J,\\mathbf{H})} = J \\mathbf{H} and \\mathbf{v}{(J,\\mathbf{H})} = J \\mathbf{H} \\operatorname{v_{x}}{(J,\\mathbf{H})} and \\sigma_{p}{(J)} = \\frac{1}{J}, then obtain - J^{2} \\mathbf{H}^{2} + \\sigma_{p}{(J)} = - J^{2} \\mathbf{H}^{2} + \\frac{1}{J}", "derivation": "\\operatorname{v_{x}}{(J,\\mathbf{H})} = J \\mathbf{H} and J \\mathbf{H} \\operatorname{v_{x}}{(J,\\mathbf{H})} = J^{2} \\mathbf{H}^{2} and \\mathbf{v}{(J,\\mathbf{H})} = J \\mathbf{H} \\operatorname{v_{x}}{(J,\\mathbf{H})} and \\mathbf{v}{(J,\\mathbf{H})} = J^{2} \\mathbf{H}^{2} and \\sigma_{p}{(J)} = \\frac{1}{J} and - J \\mathbf{H} \\operatorname{v_{x}}{(J,\\mathbf{H})} + \\sigma_{p}{(J)} = - J \\mathbf{H} \\operatorname{v_{x}}{(J,\\mathbf{H})} + \\frac{1}{J} and - \\mathbf{v}{(J,\\mathbf{H})} + \\sigma_{p}{(J)} = - \\mathbf{v}{(J,\\mathbf{H})} + \\frac{1}{J} and - J^{2} \\mathbf{H}^{2} + \\sigma_{p}{(J)} = - J^{2} \\mathbf{H}^{2} + \\frac{1}{J}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 1, "Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Integer(-1)))"], [["minus", 5, "Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\sigma_p')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Function('v_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\sigma_p')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('J', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Function('\\\\sigma_p')(Symbol('J', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Pow(Symbol('J', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)} = \\Psi \\hat{H}_{\\lambda} - \\varepsilon and \\Omega{(\\Psi,\\varepsilon,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} - \\varepsilon + \\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)}, then obtain \\Omega{(\\Psi,\\varepsilon,\\hat{H}_{\\lambda})} = \\Psi \\hat{H}_{\\lambda} - \\hat{H}_{\\lambda} - 2 \\varepsilon", "derivation": "\\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)} = \\Psi \\hat{H}_{\\lambda} - \\varepsilon and - \\hat{H}_{\\lambda} + \\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)} = \\Psi \\hat{H}_{\\lambda} - \\hat{H}_{\\lambda} - \\varepsilon and - \\hat{H}_{\\lambda} - \\varepsilon + \\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)} = \\Psi \\hat{H}_{\\lambda} - \\hat{H}_{\\lambda} - 2 \\varepsilon and \\Omega{(\\Psi,\\varepsilon,\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} - \\varepsilon + \\theta{(\\hat{H}_{\\lambda},\\Psi,\\varepsilon)} and \\Omega{(\\Psi,\\varepsilon,\\hat{H}_{\\lambda})} = \\Psi \\hat{H}_{\\lambda} - \\hat{H}_{\\lambda} - 2 \\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\theta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\theta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\theta')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\Omega')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given T{(\\mu_0,\\mu)} = \\cos{(\\mu_0^{\\mu})} and \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)} = - \\frac{- T{(\\mu_0,\\mu)} + \\cos{(\\mu_0^{\\mu})}}{T{(\\mu_0,\\mu)}}, then obtain (\\tilde{\\infty} \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)})^{\\mu} + \\cos{(\\mu_0^{\\mu})} = 0^{\\mu} + \\cos{(\\mu_0^{\\mu})}", "derivation": "T{(\\mu_0,\\mu)} = \\cos{(\\mu_0^{\\mu})} and \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)} = - \\frac{- T{(\\mu_0,\\mu)} + \\cos{(\\mu_0^{\\mu})}}{T{(\\mu_0,\\mu)}} and \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)} = 0 and - \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)} = 0 and - \\frac{\\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)}}{- T{(\\mu_0,\\mu)} + \\cos{(\\mu_0^{\\mu})}} = 0 and \\tilde{\\infty} \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)} = 0 and (\\tilde{\\infty} \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)})^{\\mu} = 0^{\\mu} and (\\tilde{\\infty} \\operatorname{x^{{\\}'}}{(\\mu_0,\\mu)})^{\\mu} + \\cos{(\\mu_0^{\\mu})} = 0^{\\mu} + \\cos{(\\mu_0^{\\mu})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Pow(Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(0))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["divide", 4, "Add(Mul(Integer(-1), Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(zoo, Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(0))"], [["power", 6, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(zoo, Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Integer(0), Symbol('\\\\mu', commutative=True)))"], [["add", 7, "cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Pow(Mul(zoo, Function('x^\\\\prime')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)))), Add(Pow(Integer(0), Symbol('\\\\mu', commutative=True)), cos(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given W{(A_{x})} = \\cos{(A_{x})}, then obtain \\int (- A_{x} + \\int W{(A_{x})} dA_{x}) dA_{x} = \\int (- A_{x} + \\int \\cos{(A_{x})} dA_{x}) dA_{x}", "derivation": "W{(A_{x})} = \\cos{(A_{x})} and \\int W{(A_{x})} dA_{x} = \\int \\cos{(A_{x})} dA_{x} and - A_{x} + \\int W{(A_{x})} dA_{x} = - A_{x} + \\int \\cos{(A_{x})} dA_{x} and \\int (- A_{x} + \\int W{(A_{x})} dA_{x}) dA_{x} = \\int (- A_{x} + \\int \\cos{(A_{x})} dA_{x}) dA_{x}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('W')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["minus", 2, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Integral(Function('W')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["integrate", 3, "Symbol('A_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Integral(Function('W')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Integral(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\mathbb{I})} = e^{\\mathbb{I}}, then derive \\frac{d}{d \\mathbb{I}} \\phi{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain \\frac{d}{d \\mathbb{I}} \\phi{(\\mathbb{I})} = \\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}}", "derivation": "\\phi{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} \\phi{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} \\phi{(\\mathbb{I})} = e^{\\mathbb{I}} and e^{\\mathbb{I}} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} \\phi{(\\mathbb{I})} = \\frac{d^{2}}{d \\mathbb{I}^{2}} e^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('\\\\mathbb{I}', commutative=True)), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\dot{x}{(\\hat{p},\\hat{X})} = \\sin{(\\hat{X} + \\hat{p})}, then obtain \\frac{\\hat{p} + \\dot{x}{(\\hat{p},\\hat{X})}}{\\hat{X} + \\dot{x}{(\\hat{p},\\hat{X})}} = \\frac{\\hat{p} + \\sin{(\\hat{X} + \\hat{p})}}{\\hat{X} + \\dot{x}{(\\hat{p},\\hat{X})}}", "derivation": "\\dot{x}{(\\hat{p},\\hat{X})} = \\sin{(\\hat{X} + \\hat{p})} and \\hat{X} + \\dot{x}{(\\hat{p},\\hat{X})} = \\hat{X} + \\sin{(\\hat{X} + \\hat{p})} and \\hat{p} + \\dot{x}{(\\hat{p},\\hat{X})} = \\hat{p} + \\sin{(\\hat{X} + \\hat{p})} and \\frac{\\hat{p} + \\dot{x}{(\\hat{p},\\hat{X})}}{\\hat{X} + \\sin{(\\hat{X} + \\hat{p})}} = \\frac{\\hat{p} + \\sin{(\\hat{X} + \\hat{p})}}{\\hat{X} + \\sin{(\\hat{X} + \\hat{p})}} and \\frac{\\hat{p} + \\dot{x}{(\\hat{p},\\hat{X})}}{\\hat{X} + \\dot{x}{(\\hat{p},\\hat{X})}} = \\frac{\\hat{p} + \\sin{(\\hat{X} + \\hat{p})}}{\\hat{X} + \\dot{x}{(\\hat{p},\\hat{X})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["add", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\hat{X}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), sin(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}', commutative=True))))))"]]}, {"prompt": "Given q{(\\rho_f)} = \\log{(\\rho_f)} and \\omega{(\\rho_f)} = - \\frac{d}{d \\rho_f} q{(\\rho_f)}, then obtain -1 = \\omega{(\\rho_f)} + \\frac{d}{d \\rho_f} \\log{(\\rho_f)} - 1", "derivation": "q{(\\rho_f)} = \\log{(\\rho_f)} and \\frac{d}{d \\rho_f} q{(\\rho_f)} = \\frac{d}{d \\rho_f} \\log{(\\rho_f)} and \\frac{d}{d \\rho_f} q{(\\rho_f)} - 1 = \\frac{d}{d \\rho_f} \\log{(\\rho_f)} - 1 and -1 = - \\frac{d}{d \\rho_f} q{(\\rho_f)} + \\frac{d}{d \\rho_f} \\log{(\\rho_f)} - 1 and \\omega{(\\rho_f)} = - \\frac{d}{d \\rho_f} q{(\\rho_f)} and -1 = \\omega{(\\rho_f)} + \\frac{d}{d \\rho_f} \\log{(\\rho_f)} - 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('q')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 3, "Derivative(Function('q')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Integer(-1), Add(Mul(Integer(-1), Derivative(Function('q')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Derivative(Function('q')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(-1), Add(Function('\\\\omega')(Symbol('\\\\rho_f', commutative=True)), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{p}{(c_{0})} = e^{c_{0}}, then obtain - \\frac{\\int \\mathbf{p}{(c_{0})} dc_{0}}{\\mathbf{p}{(c_{0})}} = - \\frac{\\int e^{c_{0}} dc_{0}}{\\mathbf{p}{(c_{0})}}", "derivation": "\\mathbf{p}{(c_{0})} = e^{c_{0}} and \\int \\mathbf{p}{(c_{0})} dc_{0} = \\int e^{c_{0}} dc_{0} and e^{- c_{0}} \\int \\mathbf{p}{(c_{0})} dc_{0} = e^{- c_{0}} \\int e^{c_{0}} dc_{0} and \\frac{\\int \\mathbf{p}{(c_{0})} dc_{0}}{\\mathbf{p}{(c_{0})}} = \\frac{\\int e^{c_{0}} dc_{0}}{\\mathbf{p}{(c_{0})}} and - \\frac{\\int \\mathbf{p}{(c_{0})} dc_{0}}{\\mathbf{p}{(c_{0})}} = - \\frac{\\int e^{c_{0}} dc_{0}}{\\mathbf{p}{(c_{0})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["divide", 2, "exp(Symbol('c_0', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('c_0', commutative=True))), Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('c_0', commutative=True))), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Integer(-1)), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('c_0', commutative=True)), Integer(-1)), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = \\frac{\\rho_f}{\\mathbf{J}_f c}, then derive \\frac{\\partial}{\\partial c} \\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = - \\frac{\\rho_f}{\\mathbf{J}_f c^{2}}, then obtain \\mathbf{J}_f + \\frac{\\partial}{\\partial c} \\frac{\\rho_f}{\\mathbf{J}_f c} = \\mathbf{J}_f - \\frac{\\rho_f}{\\mathbf{J}_f c^{2}}", "derivation": "\\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = \\frac{\\rho_f}{\\mathbf{J}_f c} and \\frac{\\partial}{\\partial c} \\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = \\frac{\\partial}{\\partial c} \\frac{\\rho_f}{\\mathbf{J}_f c} and \\frac{\\partial}{\\partial c} \\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = - \\frac{\\rho_f}{\\mathbf{J}_f c^{2}} and \\mathbf{J}_f + \\frac{\\partial}{\\partial c} \\operatorname{F_{H}}{(\\rho_f,\\mathbf{J}_f,c)} = \\mathbf{J}_f - \\frac{\\rho_f}{\\mathbf{J}_f c^{2}} and \\mathbf{J}_f + \\frac{\\partial}{\\partial c} \\frac{\\rho_f}{\\mathbf{J}_f c} = \\mathbf{J}_f - \\frac{\\rho_f}{\\mathbf{J}_f c^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2))))"], [["add", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Derivative(Function('F_H')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Pow(Symbol('c', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(F_{H})} = e^{\\cos{(F_{H})}} and \\operatorname{v_{y}}{(F_{H})} = F_{H} + \\operatorname{V_{\\mathbf{B}}}{(F_{H})}, then obtain \\operatorname{v_{y}}{(F_{H})} = F_{H} + e^{\\cos{(F_{H})}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(F_{H})} = e^{\\cos{(F_{H})}} and F_{H} + \\operatorname{V_{\\mathbf{B}}}{(F_{H})} = F_{H} + e^{\\cos{(F_{H})}} and \\operatorname{v_{y}}{(F_{H})} = F_{H} + \\operatorname{V_{\\mathbf{B}}}{(F_{H})} and \\operatorname{v_{y}}{(F_{H})} = F_{H} + e^{\\cos{(F_{H})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_H', commutative=True)), exp(cos(Symbol('F_H', commutative=True))))"], [["add", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), exp(cos(Symbol('F_H', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('v_y')(Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), exp(cos(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given J{(\\mathbf{P},\\mathbf{J}_M)} = \\frac{\\mathbf{P}}{\\mathbf{J}_M}, then derive \\frac{\\partial}{\\partial \\mathbf{P}} J{(\\mathbf{P},\\mathbf{J}_M)} = \\frac{1}{\\mathbf{J}_M}, then obtain \\mathbf{J}_M \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{\\mathbf{J}_M} = 1", "derivation": "J{(\\mathbf{P},\\mathbf{J}_M)} = \\frac{\\mathbf{P}}{\\mathbf{J}_M} and \\frac{\\partial}{\\partial \\mathbf{P}} J{(\\mathbf{P},\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{\\mathbf{J}_M} and \\frac{\\partial}{\\partial \\mathbf{P}} J{(\\mathbf{P},\\mathbf{J}_M)} = \\frac{1}{\\mathbf{J}_M} and \\mathbf{J}_M \\frac{\\partial}{\\partial \\mathbf{P}} J{(\\mathbf{P},\\mathbf{J}_M)} = 1 and \\mathbf{J}_M \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{\\mathbf{J}_M} = 1", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)))"], [["times", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Function('J')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain (\\mathbf{D}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}}) (\\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}}) = (\\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}})^{2}", "derivation": "\\mathbf{D}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\mathbf{D}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = \\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} and \\mathbf{D}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}} = \\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}} and (\\mathbf{D}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}}) (\\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}}) = (\\cos^{\\mathbf{J}_M}{(\\mathbf{J}_M)} - \\frac{1}{\\mathbf{D}{(\\mathbf{J}_M)}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))), Add(Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))))"], [["times", 3, "Add(Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))))"], "Equality(Mul(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))), Add(Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1))))), Pow(Add(Pow(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1)))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\psi,\\hat{H})} = \\hat{H}^{\\psi}, then obtain \\iint (\\mathbf{J}_M^{\\psi}{(\\psi,\\hat{H})})^{\\psi} d\\psi d\\hat{H} = \\iint ((\\hat{H}^{\\psi})^{\\psi})^{\\psi} d\\psi d\\hat{H}", "derivation": "\\mathbf{J}_M{(\\psi,\\hat{H})} = \\hat{H}^{\\psi} and \\mathbf{J}_M^{\\psi}{(\\psi,\\hat{H})} = (\\hat{H}^{\\psi})^{\\psi} and (\\mathbf{J}_M^{\\psi}{(\\psi,\\hat{H})})^{\\psi} = ((\\hat{H}^{\\psi})^{\\psi})^{\\psi} and \\int (\\mathbf{J}_M^{\\psi}{(\\psi,\\hat{H})})^{\\psi} d\\psi = \\int ((\\hat{H}^{\\psi})^{\\psi})^{\\psi} d\\psi and \\iint (\\mathbf{J}_M^{\\psi}{(\\psi,\\hat{H})})^{\\psi} d\\psi d\\hat{H} = \\iint ((\\hat{H}^{\\psi})^{\\psi})^{\\psi} d\\psi d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Pow(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Pow(Pow(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given S{(n,C)} = \\sin^{n}{(C)}, then obtain L{(n,C)} + \\iint (- S{(n,C)} + \\sin{(S{(n,C)})}) dC dn = L{(n,C)} + \\iint (- S{(n,C)} + \\sin{(\\sin^{n}{(C)})}) dC dn", "derivation": "S{(n,C)} = \\sin^{n}{(C)} and \\sin{(S{(n,C)})} = \\sin{(\\sin^{n}{(C)})} and - S{(n,C)} + \\sin{(S{(n,C)})} = - S{(n,C)} + \\sin{(\\sin^{n}{(C)})} and \\int (- S{(n,C)} + \\sin{(S{(n,C)})}) dC = \\int (- S{(n,C)} + \\sin{(\\sin^{n}{(C)})}) dC and \\iint (- S{(n,C)} + \\sin{(S{(n,C)})}) dC dn = \\iint (- S{(n,C)} + \\sin{(\\sin^{n}{(C)})}) dC dn and L{(n,C)} + \\iint (- S{(n,C)} + \\sin{(S{(n,C)})}) dC dn = L{(n,C)} + \\iint (- S{(n,C)} + \\sin{(\\sin^{n}{(C)})}) dC dn", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True)))"], [["sin", 1], "Equality(sin(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True))))"], [["minus", 2, "Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True)))), Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True)))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["add", 5, "Function('L')(Symbol('n', commutative=True), Symbol('C', commutative=True))"], "Equality(Add(Function('L')(Symbol('n', commutative=True), Symbol('C', commutative=True)), Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Function('L')(Symbol('n', commutative=True), Symbol('C', commutative=True)), Integral(Add(Mul(Integer(-1), Function('S')(Symbol('n', commutative=True), Symbol('C', commutative=True))), sin(Pow(sin(Symbol('C', commutative=True)), Symbol('n', commutative=True)))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(E,f)} = f^{E}, then obtain (\\int \\dot{y}^{2}{(E,f)} dE)^{f} = (\\int f^{E} \\dot{y}{(E,f)} dE)^{f}", "derivation": "\\dot{y}{(E,f)} = f^{E} and \\dot{y}^{2}{(E,f)} = f^{E} \\dot{y}{(E,f)} and \\int \\dot{y}^{2}{(E,f)} dE = \\int f^{E} \\dot{y}{(E,f)} dE and (\\int \\dot{y}^{2}{(E,f)} dE)^{f} = (\\int f^{E} \\dot{y}{(E,f)} dE)^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('E', commutative=True)))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True)), Integer(2)), Mul(Pow(Symbol('f', commutative=True), Symbol('E', commutative=True)), Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True)), Integer(2)), Tuple(Symbol('E', commutative=True))), Integral(Mul(Pow(Symbol('f', commutative=True), Symbol('E', commutative=True)), Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True)), Integer(2)), Tuple(Symbol('E', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(Mul(Pow(Symbol('f', commutative=True), Symbol('E', commutative=True)), Function('\\\\dot{y}')(Symbol('E', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(v_{1})} = \\cos{(v_{1})}, then derive c_{0} + \\sin{(v_{1})} + \\int \\mu_{0}{(v_{1})} dv_{1} = 2 c_{0} + 2 \\sin{(v_{1})}, then derive \\int \\mu_{0}{(v_{1})} dv_{1} = \\rho + \\sin{(v_{1})}, then obtain c_{0} + \\sin{(v_{1})} + \\int \\mu_{0}{(v_{1})} dv_{1} = \\rho + c_{0} + 2 \\sin{(v_{1})}", "derivation": "\\mu_{0}{(v_{1})} = \\cos{(v_{1})} and \\int \\mu_{0}{(v_{1})} dv_{1} = \\int \\cos{(v_{1})} dv_{1} and \\int \\mu_{0}{(v_{1})} dv_{1} + \\int \\cos{(v_{1})} dv_{1} = 2 \\int \\cos{(v_{1})} dv_{1} and c_{0} + \\sin{(v_{1})} + \\int \\mu_{0}{(v_{1})} dv_{1} = 2 c_{0} + 2 \\sin{(v_{1})} and \\int \\mu_{0}{(v_{1})} dv_{1} = \\rho + \\sin{(v_{1})} and \\rho + c_{0} + 2 \\sin{(v_{1})} = 2 c_{0} + 2 \\sin{(v_{1})} and c_{0} + \\sin{(v_{1})} + \\int \\mu_{0}{(v_{1})} dv_{1} = \\rho + c_{0} + 2 \\sin{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('c_0', commutative=True), sin(Symbol('v_1', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Mul(Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), sin(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Add(Symbol('\\\\rho', commutative=True), sin(Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True), Mul(Integer(2), sin(Symbol('v_1', commutative=True)))), Add(Mul(Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), sin(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Symbol('c_0', commutative=True), sin(Symbol('v_1', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Symbol('c_0', commutative=True), Mul(Integer(2), sin(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(I,f)} = I f, then derive b + f + \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df = 2 \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df, then obtain b + f + \\int 1 df = 2 \\int 1 df", "derivation": "\\mathbf{P}{(I,f)} = I f and 1 = \\frac{I f}{\\mathbf{P}{(I,f)}} and \\int 1 df = \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df and \\int 1 df + \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df = 2 \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df and b + f + \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df = 2 \\int \\frac{I f}{\\mathbf{P}{(I,f)}} df and b + f + \\int 1 df = 2 \\int 1 df", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True))"], "Equality(Integer(1), Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('f', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True))))"], [["add", 3, "Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True)))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('b', commutative=True), Symbol('f', commutative=True), Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('I', commutative=True), Symbol('f', commutative=True), Pow(Function('\\\\mathbf{P}')(Symbol('I', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Tuple(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('b', commutative=True), Symbol('f', commutative=True), Integral(Integer(1), Tuple(Symbol('f', commutative=True)))), Mul(Integer(2), Integral(Integer(1), Tuple(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\chi,G)} = \\frac{\\chi}{G} and \\hat{H}{(\\chi,G)} = \\frac{\\chi}{G^{2}}, then obtain \\int \\sin{(\\frac{\\phi_{1}{(\\chi,G)}}{G})} d\\chi = \\int \\sin{(\\frac{\\chi}{G^{2}})} d\\chi", "derivation": "\\phi_{1}{(\\chi,G)} = \\frac{\\chi}{G} and \\frac{\\phi_{1}{(\\chi,G)}}{G} = \\frac{\\chi}{G^{2}} and \\hat{H}{(\\chi,G)} = \\frac{\\chi}{G^{2}} and \\sin{(\\hat{H}{(\\chi,G)})} = \\sin{(\\frac{\\chi}{G^{2}})} and \\frac{\\phi_{1}{(\\chi,G)}}{G} = \\hat{H}{(\\chi,G)} and \\sin{(\\frac{\\phi_{1}{(\\chi,G)}}{G})} = \\sin{(\\frac{\\chi}{G^{2}})} and \\int \\sin{(\\frac{\\phi_{1}{(\\chi,G)}}{G})} d\\chi = \\int \\sin{(\\frac{\\chi}{G^{2}})} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\chi', commutative=True)))"], [["sin", 3], "Equality(sin(Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True))), sin(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True))), Function('\\\\hat{H}')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(sin(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True)))), sin(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\chi', commutative=True))))"], [["integrate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(sin(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\chi', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(sin(Mul(Pow(Symbol('G', commutative=True), Integer(-2)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\varphi{(\\varepsilon_0,\\Psi_{nl})} = \\Psi_{nl} \\varepsilon_0, then derive \\Psi_{nl} + \\frac{\\partial}{\\partial \\varepsilon_0} \\varphi{(\\varepsilon_0,\\Psi_{nl})} = 2 \\Psi_{nl}, then obtain \\Psi_{nl} + \\frac{\\partial}{\\partial \\varepsilon_0} \\Psi_{nl} \\varepsilon_0 = 2 \\Psi_{nl}", "derivation": "\\varphi{(\\varepsilon_0,\\Psi_{nl})} = \\Psi_{nl} \\varepsilon_0 and \\Psi_{nl} \\varepsilon_0 + \\varphi{(\\varepsilon_0,\\Psi_{nl})} = 2 \\Psi_{nl} \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} (\\Psi_{nl} \\varepsilon_0 + \\varphi{(\\varepsilon_0,\\Psi_{nl})}) = \\frac{\\partial}{\\partial \\varepsilon_0} 2 \\Psi_{nl} \\varepsilon_0 and \\Psi_{nl} + \\frac{\\partial}{\\partial \\varepsilon_0} \\varphi{(\\varepsilon_0,\\Psi_{nl})} = 2 \\Psi_{nl} and \\Psi_{nl} + \\frac{\\partial}{\\partial \\varepsilon_0} \\Psi_{nl} \\varepsilon_0 = 2 \\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Derivative(Function('\\\\varphi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Derivative(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{F},q)} = \\frac{q}{\\mathbf{F}}, then obtain \\frac{\\partial}{\\partial \\mathbf{F}} (q \\operatorname{n_{1}}{(\\mathbf{F},q)} + \\operatorname{n_{1}}{(\\mathbf{F},q)}) + \\frac{1}{\\mathbf{F}} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\operatorname{n_{1}}{(\\mathbf{F},q)} + \\frac{q^{2}}{\\mathbf{F}}) + \\frac{1}{\\mathbf{F}}", "derivation": "\\operatorname{n_{1}}{(\\mathbf{F},q)} = \\frac{q}{\\mathbf{F}} and q \\operatorname{n_{1}}{(\\mathbf{F},q)} = \\frac{q^{2}}{\\mathbf{F}} and q \\operatorname{n_{1}}{(\\mathbf{F},q)} + \\operatorname{n_{1}}{(\\mathbf{F},q)} = \\operatorname{n_{1}}{(\\mathbf{F},q)} + \\frac{q^{2}}{\\mathbf{F}} and \\frac{\\partial}{\\partial \\mathbf{F}} (q \\operatorname{n_{1}}{(\\mathbf{F},q)} + \\operatorname{n_{1}}{(\\mathbf{F},q)}) = \\frac{\\partial}{\\partial \\mathbf{F}} (\\operatorname{n_{1}}{(\\mathbf{F},q)} + \\frac{q^{2}}{\\mathbf{F}}) and \\frac{\\partial}{\\partial \\mathbf{F}} (q \\operatorname{n_{1}}{(\\mathbf{F},q)} + \\operatorname{n_{1}}{(\\mathbf{F},q)}) + \\frac{1}{\\mathbf{F}} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\operatorname{n_{1}}{(\\mathbf{F},q)} + \\frac{q^{2}}{\\mathbf{F}}) + \\frac{1}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["add", 2, "Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Mul(Symbol('q', commutative=True), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Add(Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(2)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('q', commutative=True), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))"], "Equality(Add(Derivative(Add(Mul(Symbol('q', commutative=True), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))), Add(Derivative(Add(Function('n_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(\\mathbf{H},\\mathbf{f})} = \\mathbf{f}^{\\mathbf{H}}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{f}^{\\mathbf{H}} \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\ddot{x}{(\\mathbf{H},\\mathbf{f})} = (\\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{f}^{\\mathbf{H}})^{2}", "derivation": "\\ddot{x}{(\\mathbf{H},\\mathbf{f})} = \\mathbf{f}^{\\mathbf{H}} and \\frac{\\partial}{\\partial \\mathbf{H}} \\ddot{x}{(\\mathbf{H},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{f}^{\\mathbf{H}} and \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\ddot{x}{(\\mathbf{H},\\mathbf{f})} = \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{f}^{\\mathbf{H}} and \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{f}^{\\mathbf{H}} \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\ddot{x}{(\\mathbf{H},\\mathbf{f})} = (\\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{f}^{\\mathbf{H}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["times", 3, "Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))), Pow(Derivative(Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{t})} = \\cos{(e^{v_{t}})}, then obtain v_{t}^{4} \\operatorname{n_{1}}^{2}{(v_{t})} \\cos^{2}{(e^{v_{t}})} = v_{t}^{4} \\cos^{4}{(e^{v_{t}})}", "derivation": "\\operatorname{n_{1}}{(v_{t})} = \\cos{(e^{v_{t}})} and v_{t} \\operatorname{n_{1}}{(v_{t})} = v_{t} \\cos{(e^{v_{t}})} and v_{t}^{2} \\operatorname{n_{1}}{(v_{t})} \\cos{(e^{v_{t}})} = v_{t}^{2} \\cos^{2}{(e^{v_{t}})} and v_{t}^{4} \\operatorname{n_{1}}^{2}{(v_{t})} \\cos^{2}{(e^{v_{t}})} = v_{t}^{4} \\cos^{4}{(e^{v_{t}})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_t', commutative=True)), cos(exp(Symbol('v_t', commutative=True))))"], [["times", 1, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('n_1')(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), cos(exp(Symbol('v_t', commutative=True)))))"], [["times", 2, "Mul(Symbol('v_t', commutative=True), cos(exp(Symbol('v_t', commutative=True))))"], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(2)), Function('n_1')(Symbol('v_t', commutative=True)), cos(exp(Symbol('v_t', commutative=True)))), Mul(Pow(Symbol('v_t', commutative=True), Integer(2)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(2))))"], [["power", 3, 2], "Equality(Mul(Pow(Symbol('v_t', commutative=True), Integer(4)), Pow(Function('n_1')(Symbol('v_t', commutative=True)), Integer(2)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(2))), Mul(Pow(Symbol('v_t', commutative=True), Integer(4)), Pow(cos(exp(Symbol('v_t', commutative=True))), Integer(4))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(h)} = e^{h}, then obtain \\frac{d}{d h} \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} - \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} = \\frac{d}{d h} 1 - \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}}", "derivation": "\\operatorname{A_{2}}{(h)} = e^{h} and \\frac{d}{d h} \\operatorname{A_{2}}{(h)} = \\frac{d}{d h} e^{h} and h \\frac{d}{d h} \\operatorname{A_{2}}{(h)} = h \\frac{d}{d h} e^{h} and \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} = 1 and \\frac{d}{d h} \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} = \\frac{d}{d h} 1 and \\frac{d}{d h} \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} - \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}} = \\frac{d}{d h} 1 - \\frac{\\frac{d}{d h} \\operatorname{A_{2}}{(h)}}{\\frac{d}{d h} e^{h}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Symbol('h', commutative=True), Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], "Equality(Mul(Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Derivative(Mul(Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)))), Add(Derivative(Integer(1), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('A_2')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\psi^*,M)} = \\psi^* (- M + \\psi^*) and \\operatorname{F_{N}}{(\\psi^*,M)} = \\int \\hat{\\mathbf{r}}{(\\psi^*,M)} dM, then obtain \\operatorname{F_{N}}{(\\psi^*,M)} = \\int \\psi^* (- M + \\psi^*) dM", "derivation": "\\hat{\\mathbf{r}}{(\\psi^*,M)} = \\psi^* (- M + \\psi^*) and \\int \\hat{\\mathbf{r}}{(\\psi^*,M)} dM = \\int \\psi^* (- M + \\psi^*) dM and \\operatorname{F_{N}}{(\\psi^*,M)} = \\int \\hat{\\mathbf{r}}{(\\psi^*,M)} dM and \\operatorname{F_{N}}{(\\psi^*,M)} = \\int \\psi^* (- M + \\psi^*) dM", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\psi^*', commutative=True), Symbol('M', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_N')(Symbol('\\\\psi^*', commutative=True), Symbol('M', commutative=True)), Integral(Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mu{(m_{s})} = \\log{(e^{m_{s}})}, then obtain - 2 \\mu{(m_{s})} = - \\mu{(m_{s})} - \\log{(e^{m_{s}})}", "derivation": "\\mu{(m_{s})} = \\log{(e^{m_{s}})} and m_{s} \\mu{(m_{s})} = m_{s} \\log{(e^{m_{s}})} and - m_{s} \\log{(e^{m_{s}})} + \\mu{(m_{s})} = - m_{s} \\log{(e^{m_{s}})} + \\log{(e^{m_{s}})} and m_{s} \\log{(e^{m_{s}})} - \\mu{(m_{s})} = m_{s} \\log{(e^{m_{s}})} - \\log{(e^{m_{s}})} and - m_{s} \\mu{(m_{s})} + m_{s} \\log{(e^{m_{s}})} - 2 \\mu{(m_{s})} = - m_{s} \\mu{(m_{s})} + m_{s} \\log{(e^{m_{s}})} - \\mu{(m_{s})} - \\log{(e^{m_{s}})} and - 2 \\mu{(m_{s})} = - \\mu{(m_{s})} - \\log{(e^{m_{s}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('m_s', commutative=True)), log(exp(Symbol('m_s', commutative=True))))"], [["times", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Symbol('m_s', commutative=True), Function('\\\\mu')(Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))))"], [["minus", 1, "Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), Function('\\\\mu')(Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), log(exp(Symbol('m_s', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Add(Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('m_s', commutative=True)))), Add(Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), Mul(Integer(-1), log(exp(Symbol('m_s', commutative=True))))))"], [["minus", 4, "Add(Mul(Symbol('m_s', commutative=True), Function('\\\\mu')(Symbol('m_s', commutative=True))), Function('\\\\mu')(Symbol('m_s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('m_s', commutative=True), Function('\\\\mu')(Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Integer(2), Function('\\\\mu')(Symbol('m_s', commutative=True)))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True), Function('\\\\mu')(Symbol('m_s', commutative=True))), Mul(Symbol('m_s', commutative=True), log(exp(Symbol('m_s', commutative=True)))), Mul(Integer(-1), Function('\\\\mu')(Symbol('m_s', commutative=True))), Mul(Integer(-1), log(exp(Symbol('m_s', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\mu')(Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('m_s', commutative=True))), Mul(Integer(-1), log(exp(Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{E},\\mathbf{P},C)} = C \\mathbf{P} - \\mathbf{E} and \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{P},C)} = C \\mathbf{P} - \\mathbf{E}, then obtain \\int \\frac{\\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{P},C)}}{\\mathbf{P}} d\\mathbf{E} = \\int \\frac{C \\mathbf{P} - \\mathbf{E}}{\\mathbf{P}} d\\mathbf{E}", "derivation": "\\hat{H}{(\\mathbf{E},\\mathbf{P},C)} = C \\mathbf{P} - \\mathbf{E} and \\frac{\\hat{H}{(\\mathbf{E},\\mathbf{P},C)}}{\\mathbf{P}} = \\frac{C \\mathbf{P} - \\mathbf{E}}{\\mathbf{P}} and \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{P},C)} = C \\mathbf{P} - \\mathbf{E} and \\int \\frac{\\hat{H}{(\\mathbf{E},\\mathbf{P},C)}}{\\mathbf{P}} d\\mathbf{E} = \\int \\frac{C \\mathbf{P} - \\mathbf{E}}{\\mathbf{P}} d\\mathbf{E} and \\hat{H}{(\\mathbf{E},\\mathbf{P},C)} = \\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{P},C)} and \\int \\frac{\\operatorname{E_{\\lambda}}{(\\mathbf{E},\\mathbf{P},C)}}{\\mathbf{P}} d\\mathbf{E} = \\int \\frac{C \\mathbf{P} - \\mathbf{E}}{\\mathbf{P}} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["divide", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('C', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\phi_1,\\phi)} = - \\phi_1 + e^{\\phi}, then obtain (\\frac{\\partial}{\\partial \\phi} (\\phi_1 + \\Psi_{nl}{(\\phi_1,\\phi)} - e^{\\phi}))^{\\phi_1} = (\\frac{d}{d \\phi} 0)^{\\phi_1}", "derivation": "\\Psi_{nl}{(\\phi_1,\\phi)} = - \\phi_1 + e^{\\phi} and \\phi_1 + \\Psi_{nl}{(\\phi_1,\\phi)} - e^{\\phi} = 0 and \\frac{\\partial}{\\partial \\phi} (\\phi_1 + \\Psi_{nl}{(\\phi_1,\\phi)} - e^{\\phi}) = \\frac{d}{d \\phi} 0 and (\\frac{\\partial}{\\partial \\phi} (\\phi_1 + \\Psi_{nl}{(\\phi_1,\\phi)} - e^{\\phi}))^{\\phi_1} = (\\frac{d}{d \\phi} 0)^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given h{(p)} = \\cos{(p)} and W{(p)} = 2 \\cos{(p)}, then obtain \\log{(2 h{(p)} + 1)} = \\log{(W{(p)} + 1)}", "derivation": "h{(p)} = \\cos{(p)} and h{(p)} + \\cos{(p)} = 2 \\cos{(p)} and h{(p)} + \\cos{(p)} + 1 = 2 \\cos{(p)} + 1 and \\log{(h{(p)} + \\cos{(p)} + 1)} = \\log{(2 \\cos{(p)} + 1)} and W{(p)} = 2 \\cos{(p)} and \\log{(h{(p)} + \\cos{(p)} + 1)} = \\log{(W{(p)} + 1)} and \\log{(2 h{(p)} + 1)} = \\log{(W{(p)} + 1)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["add", 1, "cos(Symbol('p', commutative=True))"], "Equality(Add(Function('h')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True))), Mul(Integer(2), cos(Symbol('p', commutative=True))))"], [["add", 2, 1], "Equality(Add(Function('h')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)), Integer(1)), Add(Mul(Integer(2), cos(Symbol('p', commutative=True))), Integer(1)))"], [["log", 3], "Equality(log(Add(Function('h')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)), Integer(1))), log(Add(Mul(Integer(2), cos(Symbol('p', commutative=True))), Integer(1))))"], ["renaming_premise", "Equality(Function('W')(Symbol('p', commutative=True)), Mul(Integer(2), cos(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Add(Function('h')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)), Integer(1))), log(Add(Function('W')(Symbol('p', commutative=True)), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(log(Add(Mul(Integer(2), Function('h')(Symbol('p', commutative=True))), Integer(1))), log(Add(Function('W')(Symbol('p', commutative=True)), Integer(1))))"]]}, {"prompt": "Given M{(\\pi,\\ddot{x},\\varepsilon)} = - \\ddot{x} + \\pi + \\varepsilon, then derive \\int M{(\\pi,\\ddot{x},\\varepsilon)} d\\ddot{x} = - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon) + v_{1}, then obtain - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon) + v_{1} = G - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon)", "derivation": "M{(\\pi,\\ddot{x},\\varepsilon)} = - \\ddot{x} + \\pi + \\varepsilon and \\int M{(\\pi,\\ddot{x},\\varepsilon)} d\\ddot{x} = \\int (- \\ddot{x} + \\pi + \\varepsilon) d\\ddot{x} and \\int M{(\\pi,\\ddot{x},\\varepsilon)} d\\ddot{x} = - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon) + v_{1} and - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon) + v_{1} = \\int (- \\ddot{x} + \\pi + \\varepsilon) d\\ddot{x} and - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon) + v_{1} = G - \\frac{\\ddot{x}^{2}}{2} + \\ddot{x} (\\pi + \\varepsilon)", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('\\\\pi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('v_1', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('v_1', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(I,\\mathbf{H})} = \\frac{\\mathbf{H}}{I}, then obtain \\frac{e^{\\varepsilon{(I,\\mathbf{H})} + \\frac{1}{I}}}{\\varepsilon{(I,\\mathbf{H})} - \\frac{\\mathbf{H}}{I}} = \\frac{e^{\\frac{\\mathbf{H}}{I} + \\frac{1}{I}}}{\\varepsilon{(I,\\mathbf{H})} - \\frac{\\mathbf{H}}{I}}", "derivation": "\\varepsilon{(I,\\mathbf{H})} = \\frac{\\mathbf{H}}{I} and \\varepsilon{(I,\\mathbf{H})} + \\frac{1}{I} = \\frac{\\mathbf{H}}{I} + \\frac{1}{I} and e^{\\varepsilon{(I,\\mathbf{H})} + \\frac{1}{I}} = e^{\\frac{\\mathbf{H}}{I} + \\frac{1}{I}} and \\frac{e^{\\varepsilon{(I,\\mathbf{H})} + \\frac{1}{I}}}{\\varepsilon{(I,\\mathbf{H})} - \\frac{\\mathbf{H}}{I}} = \\frac{e^{\\frac{\\mathbf{H}}{I} + \\frac{1}{I}}}{\\varepsilon{(I,\\mathbf{H})} - \\frac{\\mathbf{H}}{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["exp", 2], "Equality(exp(Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1)))), exp(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1)))))"], [["divide", 3, "Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1)), exp(Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))))), Mul(Pow(Add(Function('\\\\varepsilon')(Symbol('I', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True))), Integer(-1)), exp(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\phi_{1}{(s,p)} = \\int \\frac{p}{s} ds, then obtain \\frac{\\partial}{\\partial s} \\int \\frac{\\int \\frac{p}{s} ds}{\\phi_{1}{(s,p)}} ds = \\frac{d}{d s} \\int 1 ds", "derivation": "\\phi_{1}{(s,p)} = \\int \\frac{p}{s} ds and \\frac{\\phi_{1}{(s,p)}}{\\int \\frac{p}{s} ds} = 1 and 1 = \\frac{\\int \\frac{p}{s} ds}{\\phi_{1}{(s,p)}} and \\int \\frac{\\phi_{1}{(s,p)}}{\\int \\frac{p}{s} ds} ds = \\int 1 ds and \\frac{\\partial}{\\partial s} \\int \\frac{\\phi_{1}{(s,p)}}{\\int \\frac{p}{s} ds} ds = \\frac{d}{d s} \\int 1 ds and \\int 1 ds = \\int \\frac{\\int \\frac{p}{s} ds}{\\phi_{1}{(s,p)}} ds and \\int \\frac{\\phi_{1}{(s,p)}}{\\int \\frac{p}{s} ds} ds = \\int \\frac{\\int \\frac{p}{s} ds}{\\phi_{1}{(s,p)}} ds and \\frac{\\partial}{\\partial s} \\int \\frac{\\int \\frac{p}{s} ds}{\\phi_{1}{(s,p)}} ds = \\frac{d}{d s} \\int 1 ds", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))))"], [["divide", 1, "Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True)))"], "Equality(Mul(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 1, "Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True)))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integral(Integer(1), Tuple(Symbol('s', commutative=True))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Integral(Mul(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Pow(Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integer(-1))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Derivative(Integral(Mul(Pow(Function('\\\\phi_1')(Symbol('s', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Integral(Mul(Symbol('p', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(n_{1},\\Omega)} = \\Omega + n_{1}, then obtain \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})} = 1 - \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{\\Omega + n_{1}} + \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})}", "derivation": "\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)} = \\Omega + n_{1} and \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{\\Omega + n_{1}} = 1 and \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{\\Omega + n_{1}} + \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})} = 1 + \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})} and \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})} = 1 - \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{\\Omega + n_{1}} + \\frac{\\operatorname{a^{\\dagger}}{(n_{1},\\Omega)}}{n_{1} (\\Omega + n_{1})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(1))"], [["add", 2, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["minus", 3, "Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Integer(1), Mul(Integer(-1), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('n_1', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(U)} = e^{U} and \\mathbf{B}{(U)} = e^{U}, then obtain (\\int 1 dU + \\int e^{U} dU)^{U} = (\\int \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}} dU + \\int e^{U} dU)^{U}", "derivation": "\\hat{\\mathbf{x}}{(U)} = e^{U} and \\mathbf{B}{(U)} = e^{U} and 1 = \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}} and \\mathbf{B}{(U)} + 1 = \\mathbf{B}{(U)} + \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}} and \\int (\\mathbf{B}{(U)} + 1) dU = \\int (\\mathbf{B}{(U)} + \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}}) dU and \\int (e^{U} + 1) dU = \\int (e^{U} + \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}}) dU and \\int 1 dU + \\int e^{U} dU = \\int \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}} dU + \\int e^{U} dU and (\\int 1 dU + \\int e^{U} dU)^{U} = (\\int \\frac{e^{U}}{\\hat{\\mathbf{x}}{(U)}} dU + \\int e^{U} dU)^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["divide", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True))))"], [["add", 3, "Function('\\\\mathbf{B}')(Symbol('U', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), Integer(1)), Add(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True)))))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), Integer(1)), Tuple(Symbol('U', commutative=True))), Integral(Add(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Add(exp(Symbol('U', commutative=True)), Integer(1)), Tuple(Symbol('U', commutative=True))), Integral(Add(exp(Symbol('U', commutative=True)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["expand", 6], "Equality(Add(Integral(Integer(1), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Integral(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["power", 7, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Integral(Integer(1), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Add(Integral(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('U', commutative=True)), Integer(-1)), exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\delta,M_{E})} = \\delta + \\sin{(M_{E})}, then obtain e^{\\frac{\\int \\tilde{g}^*{(\\delta,M_{E})} dM_{E}}{\\int (\\delta + \\sin{(M_{E})}) dM_{E}}} + \\int (\\delta + \\sin{(M_{E})}) dM_{E} = \\int (\\delta + \\sin{(M_{E})}) dM_{E} + e", "derivation": "\\tilde{g}^*{(\\delta,M_{E})} = \\delta + \\sin{(M_{E})} and \\int \\tilde{g}^*{(\\delta,M_{E})} dM_{E} = \\int (\\delta + \\sin{(M_{E})}) dM_{E} and \\frac{\\int \\tilde{g}^*{(\\delta,M_{E})} dM_{E}}{\\int (\\delta + \\sin{(M_{E})}) dM_{E}} = 1 and e^{\\frac{\\int \\tilde{g}^*{(\\delta,M_{E})} dM_{E}}{\\int (\\delta + \\sin{(M_{E})}) dM_{E}}} = e and e^{\\frac{\\int \\tilde{g}^*{(\\delta,M_{E})} dM_{E}}{\\int (\\delta + \\sin{(M_{E})}) dM_{E}}} + \\int (\\delta + \\sin{(M_{E})}) dM_{E} = \\int (\\delta + \\sin{(M_{E})}) dM_{E} + e", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["divide", 2, "Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integer(-1)), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Integer(1))"], [["exp", 3], "Equality(exp(Mul(Pow(Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integer(-1)), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), E)"], [["add", 4, "Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Add(exp(Mul(Pow(Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integer(-1)), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\delta', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))), Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True)))), Add(Integral(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), E))"]]}, {"prompt": "Given b{(V,n)} = e^{n^{V}}, then obtain (- b{(V,n)} + \\frac{b{(V,n)}}{V})^{n} = (- b{(V,n)} + \\frac{e^{n^{V}}}{V})^{n}", "derivation": "b{(V,n)} = e^{n^{V}} and \\frac{b{(V,n)}}{V} = \\frac{e^{n^{V}}}{V} and - b{(V,n)} + \\frac{b{(V,n)}}{V} = - b{(V,n)} + \\frac{e^{n^{V}}}{V} and (- b{(V,n)} + \\frac{b{(V,n)}}{V})^{n} = (- b{(V,n)} + \\frac{e^{n^{V}}}{V})^{n}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True)), exp(Pow(Symbol('n', commutative=True), Symbol('V', commutative=True))))"], [["divide", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), exp(Pow(Symbol('n', commutative=True), Symbol('V', commutative=True)))))"], [["minus", 2, "Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), exp(Pow(Symbol('n', commutative=True), Symbol('V', commutative=True))))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), Function('b')(Symbol('V', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), exp(Pow(Symbol('n', commutative=True), Symbol('V', commutative=True))))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(A,\\theta_2)} = A + \\theta_2, then obtain - 2 \\theta_2 = - A - 3 \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)}", "derivation": "\\hat{\\mathbf{x}}{(A,\\theta_2)} = A + \\theta_2 and A + \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)} = 2 A + 2 \\theta_2 and 2 \\hat{\\mathbf{x}}{(A,\\theta_2)} = 2 A + 2 \\theta_2 and 2 \\hat{\\mathbf{x}}{(A,\\theta_2)} = A + \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)} and - 2 \\theta_2 + 2 \\hat{\\mathbf{x}}{(A,\\theta_2)} = A - \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)} and 2 A = A - \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)} and - 2 \\theta_2 = - A - 3 \\theta_2 + \\hat{\\mathbf{x}}{(A,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["minus", 6, "Add(Mul(Integer(2), Symbol('A', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('A', commutative=True), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given T{(L)} = \\log{(e^{L})} and \\operatorname{P_{g}}{(L)} = \\log{(e^{L})}, then obtain \\frac{d}{d L} \\int T{(L)} dL = \\frac{d}{d L} \\int \\log{(e^{L})} dL", "derivation": "T{(L)} = \\log{(e^{L})} and \\operatorname{P_{g}}{(L)} = \\log{(e^{L})} and T{(L)} = \\operatorname{P_{g}}{(L)} and \\int T{(L)} dL = \\int \\operatorname{P_{g}}{(L)} dL and \\frac{d}{d L} \\int T{(L)} dL = \\frac{d}{d L} \\int \\operatorname{P_{g}}{(L)} dL and \\int \\log{(e^{L})} dL = \\int \\operatorname{P_{g}}{(L)} dL and \\frac{d}{d L} \\int T{(L)} dL = \\frac{d}{d L} \\int \\log{(e^{L})} dL", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('L', commutative=True)), log(exp(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('L', commutative=True)), log(exp(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('T')(Symbol('L', commutative=True)), Function('P_g')(Symbol('L', commutative=True)))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Function('T')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Function('P_g')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Integral(Function('T')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Integral(Function('P_g')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(log(exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Function('P_g')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Integral(Function('T')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Integral(log(exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(h,A_{2})} = - A_{2} + h, then obtain - 2 A_{1} + 2 J_{\\varepsilon} + 2 U^{h}{(h,A_{2})} = - 2 A_{1} + 2 J_{\\varepsilon} + (- A_{2} + h)^{h} + U^{h}{(h,A_{2})}", "derivation": "U{(h,A_{2})} = - A_{2} + h and U^{h}{(h,A_{2})} = (- A_{2} + h)^{h} and - A_{1} + J_{\\varepsilon} + U^{h}{(h,A_{2})} = - A_{1} + J_{\\varepsilon} + (- A_{2} + h)^{h} and - 2 A_{1} + 2 J_{\\varepsilon} + 2 U^{h}{(h,A_{2})} = - 2 A_{1} + 2 J_{\\varepsilon} + (- A_{2} + h)^{h} + U^{h}{(h,A_{2})}", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["minus", 2, "Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Pow(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('A_1', commutative=True)), Mul(Integer(2), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Function('U')(Symbol('h', commutative=True), Symbol('A_2', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given m{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then derive \\int m{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon, then obtain \\frac{d^{2}}{d \\varepsilond \\mathbf{B}} \\int m{(\\mathbf{B})} d\\mathbf{B} = \\frac{\\partial^{2}}{\\partial \\varepsilon\\partial \\mathbf{B}} (\\mathbf{B} m{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon)", "derivation": "m{(\\mathbf{B})} = \\log{(\\mathbf{B})} and \\int m{(\\mathbf{B})} d\\mathbf{B} = \\int \\log{(\\mathbf{B})} d\\mathbf{B} and \\int m{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} \\log{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon and \\int m{(\\mathbf{B})} d\\mathbf{B} = \\mathbf{B} m{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon and \\frac{d}{d \\mathbf{B}} \\int m{(\\mathbf{B})} d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} (\\mathbf{B} m{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon) and \\frac{d^{2}}{d \\varepsilond \\mathbf{B}} \\int m{(\\mathbf{B})} d\\mathbf{B} = \\frac{\\partial^{2}}{\\partial \\varepsilon\\partial \\mathbf{B}} (\\mathbf{B} m{(\\mathbf{B})} - \\mathbf{B} + \\varepsilon)", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), log(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('m')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('m')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integral(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('m')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\phi,P_{e})} = - P_{e} + \\phi, then obtain P_{e} + \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (\\operatorname{v_{1}}{(\\phi,P_{e})} + 1) d\\phi + 1) = P_{e} + \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (- P_{e} + \\phi + 1) d\\phi + 1)", "derivation": "\\operatorname{v_{1}}{(\\phi,P_{e})} = - P_{e} + \\phi and \\operatorname{v_{1}}{(\\phi,P_{e})} + 1 = - P_{e} + \\phi + 1 and \\int (\\operatorname{v_{1}}{(\\phi,P_{e})} + 1) d\\phi = \\int (- P_{e} + \\phi + 1) d\\phi and - P_{e} + \\phi + \\int (\\operatorname{v_{1}}{(\\phi,P_{e})} + 1) d\\phi + 1 = - P_{e} + \\phi + \\int (- P_{e} + \\phi + 1) d\\phi + 1 and \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (\\operatorname{v_{1}}{(\\phi,P_{e})} + 1) d\\phi + 1) = \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (- P_{e} + \\phi + 1) d\\phi + 1) and P_{e} + \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (\\operatorname{v_{1}}{(\\phi,P_{e})} + 1) d\\phi + 1) = P_{e} + \\frac{\\partial}{\\partial \\phi} (- P_{e} + \\phi + \\int (- P_{e} + \\phi + 1) d\\phi + 1)", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)))"], [["differentiate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["add", 5, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Function('v_1')(Symbol('\\\\phi', commutative=True), Symbol('P_e', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Add(Symbol('P_e', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(n)} = \\sin{(n)}, then obtain \\frac{2 (\\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}})}{n} = \\frac{\\frac{\\pi{(n)} + \\sin{(n)}}{\\sin{(n)}} + \\pi{(n)}}{n} + \\frac{\\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}}}{n}", "derivation": "\\pi{(n)} = \\sin{(n)} and 2 \\pi{(n)} = \\pi{(n)} + \\sin{(n)} and \\frac{2 \\pi{(n)}}{\\sin{(n)}} = \\frac{\\pi{(n)} + \\sin{(n)}}{\\sin{(n)}} and \\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}} = \\frac{\\pi{(n)} + \\sin{(n)}}{\\sin{(n)}} + \\pi{(n)} and \\frac{\\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}}}{n} = \\frac{\\frac{\\pi{(n)} + \\sin{(n)}}{\\sin{(n)}} + \\pi{(n)}}{n} and \\frac{2 (\\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}})}{n} = \\frac{\\frac{\\pi{(n)} + \\sin{(n)}}{\\sin{(n)}} + \\pi{(n)}}{n} + \\frac{\\pi{(n)} + \\frac{2 \\pi{(n)}}{\\sin{(n)}}}{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["add", 1, "Function('\\\\pi')(Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True))), Add(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))))"], [["divide", 2, "sin(Symbol('n', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Pow(sin(Symbol('n', commutative=True)), Integer(-1))))"], [["add", 3, "Function('\\\\pi')(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1)))), Add(Mul(Add(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Pow(sin(Symbol('n', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('n', commutative=True))))"], [["divide", 4, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Function('\\\\pi')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1))))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Add(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Pow(sin(Symbol('n', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('n', commutative=True)))))"], [["add", 5, "Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Function('\\\\pi')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1)))))"], "Equality(Mul(Integer(2), Pow(Symbol('n', commutative=True), Integer(-1)), Add(Function('\\\\pi')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1))))), Add(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Add(Function('\\\\pi')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True))), Pow(sin(Symbol('n', commutative=True)), Integer(-1))), Function('\\\\pi')(Symbol('n', commutative=True)))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Function('\\\\pi')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\pi')(Symbol('n', commutative=True)), Pow(sin(Symbol('n', commutative=True)), Integer(-1)))))))"]]}, {"prompt": "Given m{(F_{x})} = e^{e^{F_{x}}}, then obtain m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} = - m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} + 2 (e^{e^{F_{x}}})^{F_{x}}", "derivation": "m{(F_{x})} = e^{e^{F_{x}}} and m^{F_{x}}{(F_{x})} = (e^{e^{F_{x}}})^{F_{x}} and m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} = - e^{e^{F_{x}}} + (e^{e^{F_{x}}})^{F_{x}} and - e^{e^{F_{x}}} = - m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} + (e^{e^{F_{x}}})^{F_{x}} and - e^{e^{F_{x}}} + (e^{e^{F_{x}}})^{F_{x}} = - m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} + 2 (e^{e^{F_{x}}})^{F_{x}} and m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} = - m^{F_{x}}{(F_{x})} - e^{e^{F_{x}}} + 2 (e^{e^{F_{x}}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('F_x', commutative=True)), exp(exp(Symbol('F_x', commutative=True))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["minus", 2, "exp(exp(Symbol('F_x', commutative=True)))"], "Equality(Add(Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True))))), Add(Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["minus", 3, "Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Mul(Integer(2), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True))))), Add(Mul(Integer(-1), Pow(Function('m')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), exp(exp(Symbol('F_x', commutative=True)))), Mul(Integer(2), Pow(exp(exp(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(S)} = \\sin{(S)}, then obtain (\\sin^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\mathbf{f}^{S}{(S)} dS = (\\sin^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\sin^{S}{(S)} dS", "derivation": "\\mathbf{f}{(S)} = \\sin{(S)} and \\mathbf{f}^{S}{(S)} = \\sin^{S}{(S)} and \\int \\mathbf{f}^{S}{(S)} dS = \\int \\sin^{S}{(S)} dS and \\frac{(\\sin^{S}{(S)} - \\frac{1}{\\sin{(S)}}) \\sin{(S)} \\int \\mathbf{f}^{S}{(S)} dS}{\\mathbf{f}{(S)}} = \\frac{(\\sin^{S}{(S)} - \\frac{1}{\\sin{(S)}}) \\sin{(S)} \\int \\sin^{S}{(S)} dS}{\\mathbf{f}{(S)}} and (\\mathbf{f}^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\mathbf{f}^{S}{(S)} dS = (\\mathbf{f}^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\sin^{S}{(S)} dS and (\\sin^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\mathbf{f}^{S}{(S)} dS = (\\sin^{S}{(S)} - \\frac{1}{\\mathbf{f}{(S)}}) \\int \\sin^{S}{(S)} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["times", 3, "Mul(Add(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('S', commutative=True)), Integer(-1)))), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)), sin(Symbol('S', commutative=True)))"], "Equality(Mul(Add(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('S', commutative=True)), Integer(-1)))), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)), sin(Symbol('S', commutative=True)), Integral(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('S', commutative=True)), Integer(-1)))), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)), sin(Symbol('S', commutative=True)), Integral(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)))), Integral(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)))), Integral(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)))), Integral(Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Add(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('S', commutative=True)), Integer(-1)))), Integral(Pow(sin(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\lambda,\\varepsilon)} = \\lambda + \\varepsilon, then derive \\frac{\\partial}{\\partial \\lambda} \\sigma_{x}{(\\lambda,\\varepsilon)} = 1, then obtain \\frac{\\partial}{\\partial \\lambda} (\\lambda + \\varepsilon) = 1", "derivation": "\\sigma_{x}{(\\lambda,\\varepsilon)} = \\lambda + \\varepsilon and \\frac{\\partial}{\\partial \\lambda} \\sigma_{x}{(\\lambda,\\varepsilon)} = \\frac{\\partial}{\\partial \\lambda} (\\lambda + \\varepsilon) and \\frac{\\partial}{\\partial \\lambda} (\\lambda + \\varepsilon) \\frac{\\partial}{\\partial \\lambda} \\sigma_{x}{(\\lambda,\\varepsilon)} = (\\frac{\\partial}{\\partial \\lambda} (\\lambda + \\varepsilon))^{2} and \\frac{\\partial}{\\partial \\lambda} \\sigma_{x}{(\\lambda,\\varepsilon)} = 1 and \\frac{\\partial}{\\partial \\lambda} (\\lambda + \\varepsilon) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v)} = \\cos{(e^{v})}, then obtain \\operatorname{f_{E}}^{v}{(v)} + \\int \\operatorname{f_{E}}^{v}{(v)} dv = \\operatorname{f_{E}}^{v}{(v)} + \\int \\cos^{v}{(e^{v})} dv", "derivation": "\\operatorname{f_{E}}{(v)} = \\cos{(e^{v})} and \\operatorname{f_{E}}^{v}{(v)} = \\cos^{v}{(e^{v})} and \\int \\operatorname{f_{E}}^{v}{(v)} dv = \\int \\cos^{v}{(e^{v})} dv and \\operatorname{f_{E}}^{v}{(v)} + \\int \\operatorname{f_{E}}^{v}{(v)} dv = \\operatorname{f_{E}}^{v}{(v)} + \\int \\cos^{v}{(e^{v})} dv", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v', commutative=True)), cos(exp(Symbol('v', commutative=True))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(cos(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(cos(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 3, "Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True))"], "Equality(Add(Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Integral(Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Pow(Function('f_E')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Integral(Pow(cos(exp(Symbol('v', commutative=True))), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\omega{(r,\\mathbf{H})} = \\log{(r^{\\mathbf{H}})}, then obtain \\tilde{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\omega{(r,\\mathbf{H})} = \\hat{H}", "derivation": "\\omega{(r,\\mathbf{H})} = \\log{(r^{\\mathbf{H}})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\omega{(r,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(r^{\\mathbf{H}})} and \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\omega{(r,\\mathbf{H})} = \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\log{(r^{\\mathbf{H}})} and \\int \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\omega{(r,\\mathbf{H})} d\\mathbf{H} = \\int \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\log{(r^{\\mathbf{H}})} d\\mathbf{H} and \\tilde{g} + \\frac{\\partial}{\\partial \\mathbf{H}} \\omega{(r,\\mathbf{H})} = \\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Pow(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(log(Pow(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\omega')(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Derivative(log(Pow(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Derivative(Function('\\\\omega')(Symbol('r', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Symbol('\\\\hat{H}', commutative=True))"]]}, {"prompt": "Given U{(\\hat{\\mathbf{x}},p)} = p^{\\hat{\\mathbf{x}}} and c{(\\hat{\\mathbf{x}},p)} = \\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}}, then obtain (\\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}})^{p} = c^{p}{(\\hat{\\mathbf{x}},p)}", "derivation": "U{(\\hat{\\mathbf{x}},p)} = p^{\\hat{\\mathbf{x}}} and \\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}} = \\int p^{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} and (\\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}})^{p} = (\\int p^{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}})^{p} and c{(\\hat{\\mathbf{x}},p)} = \\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}} and c{(\\hat{\\mathbf{x}},p)} = \\int p^{\\hat{\\mathbf{x}}} d\\hat{\\mathbf{x}} and (\\int U{(\\hat{\\mathbf{x}},p)} d\\hat{\\mathbf{x}})^{p} = c^{p}{(\\hat{\\mathbf{x}},p)}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Integral(Function('U')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Integral(Function('U')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Integral(Function('U')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('p', commutative=True)), Pow(Function('c')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mu_0)} = \\int e^{\\mu_0} d\\mu_0, then derive \\hat{H}_{\\lambda}{(\\mu_0)} = i + e^{\\mu_0}, then derive \\frac{d}{d \\mu_0} \\hat{H}_{\\lambda}{(\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (f^{*} + e^{\\mu_0}), then obtain \\int \\frac{\\partial}{\\partial \\mu_0} (i + e^{\\mu_0}) di = \\int \\frac{\\partial}{\\partial \\mu_0} (f^{*} + e^{\\mu_0}) di", "derivation": "\\hat{H}_{\\lambda}{(\\mu_0)} = \\int e^{\\mu_0} d\\mu_0 and \\frac{d}{d \\mu_0} \\hat{H}_{\\lambda}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\int e^{\\mu_0} d\\mu_0 and \\hat{H}_{\\lambda}{(\\mu_0)} = i + e^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} (i + e^{\\mu_0}) = \\frac{d}{d \\mu_0} \\int e^{\\mu_0} d\\mu_0 and \\frac{d}{d \\mu_0} \\hat{H}_{\\lambda}{(\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (f^{*} + e^{\\mu_0}) and \\frac{d}{d \\mu_0} \\hat{H}_{\\lambda}{(\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (i + e^{\\mu_0}) and \\frac{\\partial}{\\partial \\mu_0} (i + e^{\\mu_0}) = \\frac{\\partial}{\\partial \\mu_0} (f^{*} + e^{\\mu_0}) and \\int \\frac{\\partial}{\\partial \\mu_0} (i + e^{\\mu_0}) di = \\int \\frac{\\partial}{\\partial \\mu_0} (f^{*} + e^{\\mu_0}) di", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu_0', commutative=True)), Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu_0', commutative=True)), Add(Symbol('i', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('i', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('i', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Add(Symbol('i', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["integrate", 7, "Symbol('i', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('i', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))), Integral(Derivative(Add(Symbol('f^*', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(f_{E})} = \\sin{(f_{E})}, then obtain \\int - (\\frac{\\mathbf{M}{(f_{E})}}{\\sin{(f_{E})}})^{f_{E}} df_{E} = \\int (-1) df_{E}", "derivation": "\\mathbf{M}{(f_{E})} = \\sin{(f_{E})} and \\frac{\\mathbf{M}{(f_{E})}}{\\sin{(f_{E})}} = 1 and (\\frac{\\mathbf{M}{(f_{E})}}{\\sin{(f_{E})}})^{f_{E}} = 1 and - (\\frac{\\mathbf{M}{(f_{E})}}{\\sin{(f_{E})}})^{f_{E}} = -1 and \\int - (\\frac{\\mathbf{M}{(f_{E})}}{\\sin{(f_{E})}})^{f_{E}} df_{E} = \\int (-1) df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["divide", 1, "sin(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{M}')(Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Symbol('f_E', commutative=True)), Integer(1))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Mul(Function('\\\\mathbf{M}')(Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Symbol('f_E', commutative=True))), Integer(-1))"], [["integrate", 4, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Mul(Function('\\\\mathbf{M}')(Symbol('f_E', commutative=True)), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Integer(-1), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} = \\eta^{\\prime} - \\theta_1 + \\varphi^*, then obtain \\cos{(\\iint \\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} d\\theta_1 d\\varphi^*)} = \\cos{(\\iint (\\eta^{\\prime} - \\theta_1 + \\varphi^*) d\\theta_1 d\\varphi^*)}", "derivation": "\\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} = \\eta^{\\prime} - \\theta_1 + \\varphi^* and \\int \\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} d\\theta_1 = \\int (\\eta^{\\prime} - \\theta_1 + \\varphi^*) d\\theta_1 and \\iint \\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} d\\theta_1 d\\varphi^* = \\iint (\\eta^{\\prime} - \\theta_1 + \\varphi^*) d\\theta_1 d\\varphi^* and \\cos{(\\iint \\operatorname{r_{0}}{(\\theta_1,\\varphi^*,\\eta^{\\prime})} d\\theta_1 d\\varphi^*)} = \\cos{(\\iint (\\eta^{\\prime} - \\theta_1 + \\varphi^*) d\\theta_1 d\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Function('r_0')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), cos(Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} = \\mathbf{J}_M^{i} and \\pi{(\\mathbf{J}_M,i)} = \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} \\int \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} d\\mathbf{J}_M, then obtain \\mathbf{J}_M^{i} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M = \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} = \\mathbf{J}_M^{i} and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} d\\mathbf{J}_M = \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M and \\pi{(\\mathbf{J}_M,i)} = \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} \\int \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} d\\mathbf{J}_M and \\pi{(\\mathbf{J}_M,i)} = \\mathbf{J}_M^{i} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M and \\mathbf{J}_M^{i} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M = \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} \\int \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} d\\mathbf{J}_M and \\mathbf{J}_M^{i} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M = \\operatorname{E_{\\lambda}}{(\\mathbf{J}_M,i)} \\int \\mathbf{J}_M^{i} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(Z)} = \\log{(Z)}, then obtain - Z + \\frac{Z - \\mathbf{p}{(Z)} - \\log{(Z)}}{2 \\log{(Z)}} + \\mathbf{p}{(Z)} = - Z + \\frac{Z - \\mathbf{p}{(Z)} - \\log{(Z)}}{2 \\log{(Z)}} + \\log{(Z)}", "derivation": "\\mathbf{p}{(Z)} = \\log{(Z)} and - Z + \\mathbf{p}{(Z)} = - Z + \\log{(Z)} and - Z + \\mathbf{p}{(Z)} + \\log{(Z)} = - Z + 2 \\log{(Z)} and \\frac{- Z + \\mathbf{p}{(Z)} + \\log{(Z)}}{2 \\log{(Z)}} = \\frac{- Z + 2 \\log{(Z)}}{2 \\log{(Z)}} and - Z - \\frac{- Z + 2 \\log{(Z)}}{2 \\log{(Z)}} + \\mathbf{p}{(Z)} = - Z - \\frac{- Z + 2 \\log{(Z)}}{2 \\log{(Z)}} + \\log{(Z)} and - Z - \\frac{- Z + \\mathbf{p}{(Z)} + \\log{(Z)}}{2 \\log{(Z)}} + \\mathbf{p}{(Z)} = - Z - \\frac{- Z + \\mathbf{p}{(Z)} + \\log{(Z)}}{2 \\log{(Z)}} + \\log{(Z)} and - Z + \\frac{Z - \\mathbf{p}{(Z)} - \\log{(Z)}}{2 \\log{(Z)}} + \\mathbf{p}{(Z)} = - Z + \\frac{Z - \\mathbf{p}{(Z)} - \\log{(Z)}}{2 \\log{(Z)}} + \\log{(Z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["minus", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), log(Symbol('Z', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), log(Symbol('Z', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(2), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), log(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Rational(1, 2), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), log(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Rational(1, 2), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Mul(Integer(-1), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Rational(1, 2), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('Z', commutative=True))), Mul(Integer(-1), log(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), log(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(r)} = \\sin{(r)} and \\operatorname{f_{\\mathbf{p}}}{(r)} = \\frac{1}{\\operatorname{F_{N}}{(r)}}, then obtain \\frac{d}{d r} 1 = \\frac{d}{d r} \\frac{\\sin{(r)}}{\\operatorname{F_{N}}{(r)}}", "derivation": "\\operatorname{F_{N}}{(r)} = \\sin{(r)} and \\operatorname{f_{\\mathbf{p}}}{(r)} = \\frac{1}{\\operatorname{F_{N}}{(r)}} and \\operatorname{f_{\\mathbf{p}}}{(r)} = \\frac{1}{\\sin{(r)}} and \\frac{1}{\\sin{(r)}} = \\frac{1}{\\operatorname{F_{N}}{(r)}} and 1 = \\frac{\\sin{(r)}}{\\operatorname{F_{N}}{(r)}} and \\frac{d}{d r} 1 = \\frac{d}{d r} \\frac{\\sin{(r)}}{\\operatorname{F_{N}}{(r)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True)), Pow(Function('F_N')(Symbol('r', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('r', commutative=True)), Pow(sin(Symbol('r', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(sin(Symbol('r', commutative=True)), Integer(-1)), Pow(Function('F_N')(Symbol('r', commutative=True)), Integer(-1)))"], [["divide", 4, "Pow(sin(Symbol('r', commutative=True)), Integer(-1))"], "Equality(Integer(1), Mul(Pow(Function('F_N')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('F_N')(Symbol('r', commutative=True)), Integer(-1)), sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(t_{2})} = e^{t_{2}}, then derive \\int \\operatorname{A_{1}}{(t_{2})} dt_{2} = \\dot{\\mathbf{r}} + e^{t_{2}}, then obtain (\\dot{\\mathbf{r}} + e^{t_{2}})^{t_{2}} = (\\int \\operatorname{A_{1}}{(t_{2})} dt_{2})^{t_{2}}", "derivation": "\\operatorname{A_{1}}{(t_{2})} = e^{t_{2}} and \\int \\operatorname{A_{1}}{(t_{2})} dt_{2} = \\int e^{t_{2}} dt_{2} and (\\int \\operatorname{A_{1}}{(t_{2})} dt_{2})^{t_{2}} = (\\int e^{t_{2}} dt_{2})^{t_{2}} and \\int \\operatorname{A_{1}}{(t_{2})} dt_{2} = \\dot{\\mathbf{r}} + e^{t_{2}} and (\\dot{\\mathbf{r}} + e^{t_{2}})^{t_{2}} = (\\int e^{t_{2}} dt_{2})^{t_{2}} and (\\dot{\\mathbf{r}} + e^{t_{2}})^{t_{2}} = (\\int \\operatorname{A_{1}}{(t_{2})} dt_{2})^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Integral(Function('A_1')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Pow(Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_1')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), exp(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), exp(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Pow(Integral(exp(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), exp(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Pow(Integral(Function('A_1')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\phi{(a^{\\dagger},\\mathbf{M})} = \\mathbf{M} \\log{(a^{\\dagger})} and \\operatorname{t_{1}}{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\log{(a^{\\dagger})})^{\\mathbf{M}} = (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\operatorname{t_{1}}{(a^{\\dagger})})^{\\mathbf{M}}", "derivation": "\\phi{(a^{\\dagger},\\mathbf{M})} = \\mathbf{M} \\log{(a^{\\dagger})} and \\operatorname{t_{1}}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and \\frac{\\partial}{\\partial \\mathbf{M}} \\phi{(a^{\\dagger},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\log{(a^{\\dagger})} and (\\frac{\\partial}{\\partial \\mathbf{M}} \\phi{(a^{\\dagger},\\mathbf{M})})^{\\mathbf{M}} = (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\log{(a^{\\dagger})})^{\\mathbf{M}} and (\\frac{\\partial}{\\partial \\mathbf{M}} \\phi{(a^{\\dagger},\\mathbf{M})})^{\\mathbf{M}} = (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\operatorname{t_{1}}{(a^{\\dagger})})^{\\mathbf{M}} and (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\log{(a^{\\dagger})})^{\\mathbf{M}} = (\\frac{\\partial}{\\partial \\mathbf{M}} \\mathbf{M} \\operatorname{t_{1}}{(a^{\\dagger})})^{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\phi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Derivative(Function('\\\\phi')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('t_1')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{M}', commutative=True), Function('t_1')(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Symbol('\\\\mathbf{M}', commutative=True)))"]]}, {"prompt": "Given h{(F_{H})} = \\sin{(F_{H})} and \\mathbf{s}{(F_{H})} = h{(F_{H})} + \\sin{(F_{H})}, then obtain \\frac{d}{d F_{H}} \\mathbf{s}{(F_{H})} = \\frac{d}{d F_{H}} 2 \\sin{(F_{H})}", "derivation": "h{(F_{H})} = \\sin{(F_{H})} and h{(F_{H})} + \\sin{(F_{H})} = 2 \\sin{(F_{H})} and \\mathbf{s}{(F_{H})} = h{(F_{H})} + \\sin{(F_{H})} and \\mathbf{s}{(F_{H})} = 2 \\sin{(F_{H})} and \\frac{d}{d F_{H}} \\mathbf{s}{(F_{H})} = \\frac{d}{d F_{H}} 2 \\sin{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["add", 1, "sin(Symbol('F_H', commutative=True))"], "Equality(Add(Function('h')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))), Mul(Integer(2), sin(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Add(Function('h')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Mul(Integer(2), sin(Symbol('F_H', commutative=True))))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(\\hat{H})} = \\cos{(\\hat{H})}, then derive \\int \\frac{\\sigma_{x}{(\\hat{H})}}{\\hat{H}} d\\hat{H} = \\varphi - \\log{(\\hat{H})} + \\frac{\\log{(\\hat{H}^{2})}}{2} + \\operatorname{Ci}{(\\hat{H})}, then obtain \\varphi - \\log{(\\hat{H})} + \\frac{\\log{(\\hat{H}^{2})}}{2} + \\operatorname{Ci}{(\\hat{H})} = \\int \\frac{\\cos{(\\hat{H})}}{\\hat{H}} d\\hat{H}", "derivation": "\\sigma_{x}{(\\hat{H})} = \\cos{(\\hat{H})} and \\frac{\\sigma_{x}{(\\hat{H})}}{\\hat{H}} = \\frac{\\cos{(\\hat{H})}}{\\hat{H}} and \\int \\frac{\\sigma_{x}{(\\hat{H})}}{\\hat{H}} d\\hat{H} = \\int \\frac{\\cos{(\\hat{H})}}{\\hat{H}} d\\hat{H} and \\int \\frac{\\sigma_{x}{(\\hat{H})}}{\\hat{H}} d\\hat{H} = \\varphi - \\log{(\\hat{H})} + \\frac{\\log{(\\hat{H}^{2})}}{2} + \\operatorname{Ci}{(\\hat{H})} and \\varphi - \\log{(\\hat{H})} + \\frac{\\log{(\\hat{H}^{2})}}{2} + \\operatorname{Ci}{(\\hat{H})} = \\int \\frac{\\cos{(\\hat{H})}}{\\hat{H}} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), log(Symbol('\\\\hat{H}', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Ci(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), log(Symbol('\\\\hat{H}', commutative=True))), Mul(Rational(1, 2), log(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)))), Ci(Symbol('\\\\hat{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\hat{H})} = \\log{(\\hat{H})}, then obtain 0 = - (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} - \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}", "derivation": "\\operatorname{F_{H}}{(\\hat{H})} = \\log{(\\hat{H})} and 0 = - \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})} and 0 = (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} and - (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} = 0 and - (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} - \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})} = - \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})} and - (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} + \\log{(\\hat{H})} = \\log{(\\hat{H})} and 0 = - (- \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}) \\log{(\\hat{H})} - \\operatorname{F_{H}}{(\\hat{H})} + \\log{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\hat{H}', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 1, "Function('F_H')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 2, "log(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 3, "Mul(Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["add", 4, "Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))), Mul(Integer(-1), Function('F_H')(Symbol('\\\\hat{H}', commutative=True))), log(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given b{(\\pi)} = \\sin{(\\log{(\\pi)})}, then derive \\int b{(\\pi)} d\\pi = \\mathbf{s} + \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2}, then obtain \\sin{(\\int b{(\\pi)} d\\pi)} = \\sin{(\\mathbf{s} + \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2})}", "derivation": "b{(\\pi)} = \\sin{(\\log{(\\pi)})} and \\int b{(\\pi)} d\\pi = \\int \\sin{(\\log{(\\pi)})} d\\pi and \\int b{(\\pi)} d\\pi = \\mathbf{s} + \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2} and \\sin{(\\int b{(\\pi)} d\\pi)} = \\sin{(\\mathbf{s} + \\frac{\\pi \\sin{(\\log{(\\pi)})}}{2} - \\frac{\\pi \\cos{(\\log{(\\pi)})}}{2})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\pi', commutative=True)), sin(log(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(sin(log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\pi', commutative=True), sin(log(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\pi', commutative=True), cos(log(Symbol('\\\\pi', commutative=True))))))"], [["sin", 3], "Equality(sin(Integral(Function('b')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), sin(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\pi', commutative=True), sin(log(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('\\\\pi', commutative=True), cos(log(Symbol('\\\\pi', commutative=True)))))))"]]}, {"prompt": "Given q{(f^{*})} = \\int e^{f^{*}} df^{*}, then derive \\log{(q{(f^{*})})} = \\log{(V_{\\mathbf{B}} + e^{f^{*}})}, then derive \\log{(V_{\\mathbf{B}} + e^{f^{*}})} = \\log{(F_{g} + e^{f^{*}})}, then obtain (\\log{(V_{\\mathbf{B}} + e^{f^{*}})}^{f^{*}})^{V_{\\mathbf{B}}} = (\\log{(F_{g} + e^{f^{*}})}^{f^{*}})^{V_{\\mathbf{B}}}", "derivation": "q{(f^{*})} = \\int e^{f^{*}} df^{*} and \\log{(q{(f^{*})})} = \\log{(\\int e^{f^{*}} df^{*})} and \\log{(q{(f^{*})})} = \\log{(V_{\\mathbf{B}} + e^{f^{*}})} and \\log{(V_{\\mathbf{B}} + e^{f^{*}})} = \\log{(\\int e^{f^{*}} df^{*})} and \\log{(V_{\\mathbf{B}} + e^{f^{*}})} = \\log{(F_{g} + e^{f^{*}})} and \\log{(V_{\\mathbf{B}} + e^{f^{*}})}^{f^{*}} = \\log{(F_{g} + e^{f^{*}})}^{f^{*}} and (\\log{(V_{\\mathbf{B}} + e^{f^{*}})}^{f^{*}})^{V_{\\mathbf{B}}} = (\\log{(F_{g} + e^{f^{*}})}^{f^{*}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('f^*', commutative=True)), Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["log", 1], "Equality(log(Function('q')(Symbol('f^*', commutative=True))), log(Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(log(Function('q')(Symbol('f^*', commutative=True))), log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('f^*', commutative=True)))), log(Integral(exp(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('f^*', commutative=True)))), log(Add(Symbol('F_g', commutative=True), exp(Symbol('f^*', commutative=True)))))"], [["power", 5, "Symbol('f^*', commutative=True)"], "Equality(Pow(log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)), Pow(log(Add(Symbol('F_g', commutative=True), exp(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"], [["power", 6, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Pow(log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Pow(log(Add(Symbol('F_g', commutative=True), exp(Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(W,f_{E})} = f_{E}^{W}, then obtain (- (\\int 2 W f_{E}^{W} df_{E})^{W} + (\\int (W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})}) df_{E})^{W})^{W} = 0^{W}", "derivation": "\\operatorname{y^{\\prime}}{(W,f_{E})} = f_{E}^{W} and W \\operatorname{y^{\\prime}}{(W,f_{E})} = W f_{E}^{W} and W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})} = 2 W f_{E}^{W} and \\int (W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})}) df_{E} = \\int 2 W f_{E}^{W} df_{E} and (\\int (W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})}) df_{E})^{W} = (\\int 2 W f_{E}^{W} df_{E})^{W} and - (\\int 2 W f_{E}^{W} df_{E})^{W} + (\\int (W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})}) df_{E})^{W} = 0 and (- (\\int 2 W f_{E}^{W} df_{E})^{W} + (\\int (W f_{E}^{W} + W \\operatorname{y^{\\prime}}{(W,f_{E})}) df_{E})^{W})^{W} = 0^{W}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True)))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))))"], [["add", 2, "Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)))), Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('f_E', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True)))"], [["minus", 5, "Pow(Integral(Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integral(Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True))), Pow(Integral(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True))), Integer(0))"], [["power", 6, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Integral(Mul(Integer(2), Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True))), Pow(Integral(Add(Mul(Symbol('W', commutative=True), Pow(Symbol('f_E', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Function('y^{\\\\prime}')(Symbol('W', commutative=True), Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Integer(0), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\rho{(t_{1},\\mathbf{D})} = \\mathbf{D} e^{t_{1}}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} \\rho{(t_{1},\\mathbf{D})} = e^{t_{1}}, then obtain (\\frac{\\partial}{\\partial \\mathbf{D}} \\rho{(t_{1},\\mathbf{D})})^{t_{1}} = (e^{t_{1}})^{t_{1}}", "derivation": "\\rho{(t_{1},\\mathbf{D})} = \\mathbf{D} e^{t_{1}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\rho{(t_{1},\\mathbf{D})} = \\frac{\\partial}{\\partial \\mathbf{D}} \\mathbf{D} e^{t_{1}} and \\frac{\\partial}{\\partial \\mathbf{D}} \\rho{(t_{1},\\mathbf{D})} = e^{t_{1}} and (\\frac{\\partial}{\\partial \\mathbf{D}} \\rho{(t_{1},\\mathbf{D})})^{t_{1}} = (e^{t_{1}})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), exp(Symbol('t_1', commutative=True)))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('t_1', commutative=True)), Pow(exp(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given v{(x^\\prime,\\mathbf{H})} = \\int (\\mathbf{H} - x^\\prime) dx^\\prime, then derive v^{x^\\prime}{(x^\\prime,\\mathbf{H})} = (\\mathbf{H} x^\\prime + n_{2} - \\frac{(x^\\prime)^{2}}{2})^{x^\\prime}, then obtain (\\int (\\mathbf{H} - x^\\prime) dx^\\prime)^{x^\\prime} = (\\mathbf{H} x^\\prime + n_{2} - \\frac{(x^\\prime)^{2}}{2})^{x^\\prime}", "derivation": "v{(x^\\prime,\\mathbf{H})} = \\int (\\mathbf{H} - x^\\prime) dx^\\prime and v^{x^\\prime}{(x^\\prime,\\mathbf{H})} = (\\int (\\mathbf{H} - x^\\prime) dx^\\prime)^{x^\\prime} and v^{x^\\prime}{(x^\\prime,\\mathbf{H})} = (\\mathbf{H} x^\\prime + n_{2} - \\frac{(x^\\prime)^{2}}{2})^{x^\\prime} and (\\int (\\mathbf{H} - x^\\prime) dx^\\prime)^{x^\\prime} = (\\mathbf{H} x^\\prime + n_{2} - \\frac{(x^\\prime)^{2}}{2})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('n_2', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('n_2', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given V{(v_{2})} = v_{2}, then derive \\frac{d}{d v_{2}} V{(v_{2})} - 1 = 0, then obtain \\int (\\frac{d}{d v_{2}} v_{2} - 1) dv_{2} = \\int 0 dv_{2}", "derivation": "V{(v_{2})} = v_{2} and - v_{2} + V{(v_{2})} = 0 and \\frac{d}{d v_{2}} (- v_{2} + V{(v_{2})}) = \\frac{d}{d v_{2}} 0 and \\frac{d}{d v_{2}} V{(v_{2})} - 1 = 0 and \\frac{d}{d v_{2}} v_{2} - 1 = 0 and \\int (\\frac{d}{d v_{2}} v_{2} - 1) dv_{2} = \\int 0 dv_{2}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], [["minus", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('V')(Symbol('v_2', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('V')(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["integrate", 5, "Symbol('v_2', commutative=True)"], "Equality(Integral(Add(Derivative(Symbol('v_2', commutative=True), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('v_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(m)} = \\cos{(m)}, then derive \\int \\operatorname{F_{c}}{(m)} dm = J + \\sin{(m)}, then obtain - \\cos{(m)} + \\frac{d}{d \\hat{x}_0} 1 + \\frac{\\int \\cos{(m)} dm}{F_{N}} = - \\cos{(m)} + \\frac{d}{d \\hat{x}_0} 1 + \\frac{J + \\sin{(m)}}{F_{N}}", "derivation": "\\operatorname{F_{c}}{(m)} = \\cos{(m)} and \\int \\operatorname{F_{c}}{(m)} dm = \\int \\cos{(m)} dm and \\int \\operatorname{F_{c}}{(m)} dm = J + \\sin{(m)} and \\int \\cos{(m)} dm = J + \\sin{(m)} and \\frac{\\int \\cos{(m)} dm}{F_{N}} = \\frac{J + \\sin{(m)}}{F_{N}} and - \\cos{(m)} + \\frac{\\int \\cos{(m)} dm}{F_{N}} = - \\cos{(m)} + \\frac{J + \\sin{(m)}}{F_{N}} and - \\cos{(m)} + \\frac{d}{d \\hat{x}_0} 1 + \\frac{\\int \\cos{(m)} dm}{F_{N}} = - \\cos{(m)} + \\frac{d}{d \\hat{x}_0} 1 + \\frac{J + \\sin{(m)}}{F_{N}}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_c')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('m', commutative=True))))"], [["divide", 4, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), sin(Symbol('m', commutative=True)))))"], [["minus", 5, "cos(Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('m', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), cos(Symbol('m', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), sin(Symbol('m', commutative=True))))))"], [["add", 6, "Derivative(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('m', commutative=True))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(Mul(Integer(-1), cos(Symbol('m', commutative=True))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), sin(Symbol('m', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{g}{(\\Psi)} = \\log{(\\Psi)} and x{(\\Psi)} = \\log{(\\Psi)}, then obtain \\int \\frac{d}{d \\Psi} (- \\Psi + x{(\\Psi)}) d\\Psi = \\int \\frac{d}{d \\Psi} (- \\Psi + \\log{(\\Psi)}) d\\Psi", "derivation": "\\mathbf{g}{(\\Psi)} = \\log{(\\Psi)} and x{(\\Psi)} = \\log{(\\Psi)} and - \\Psi + x{(\\Psi)} = - \\Psi + \\log{(\\Psi)} and - \\Psi + x{(\\Psi)} = - \\Psi + \\mathbf{g}{(\\Psi)} and \\frac{d}{d \\Psi} (- \\Psi + x{(\\Psi)}) = \\frac{d}{d \\Psi} (- \\Psi + \\mathbf{g}{(\\Psi)}) and \\int \\frac{d}{d \\Psi} (- \\Psi + x{(\\Psi)}) d\\Psi = \\int \\frac{d}{d \\Psi} (- \\Psi + \\mathbf{g}{(\\Psi)}) d\\Psi and \\int \\frac{d}{d \\Psi} (- \\Psi + x{(\\Psi)}) d\\Psi = \\int \\frac{d}{d \\Psi} (- \\Psi + \\log{(\\Psi)}) d\\Psi", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True)))"], [["minus", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('x')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('x')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('x')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('x')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Function('x')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), log(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given l{(n)} = \\log{(\\cos{(n)})}, then derive x + l{(n)} = p + \\log{(\\cos{(n)})}, then obtain x + l{(n)} + \\iint \\frac{d}{d n} l{(n)} dn dn = x + \\log{(\\cos{(n)})} + \\iint \\frac{d}{d n} l{(n)} dn dn", "derivation": "l{(n)} = \\log{(\\cos{(n)})} and \\frac{d}{d n} l{(n)} = \\frac{d}{d n} \\log{(\\cos{(n)})} and \\int \\frac{d}{d n} l{(n)} dn = \\int \\frac{d}{d n} \\log{(\\cos{(n)})} dn and x + l{(n)} = p + \\log{(\\cos{(n)})} and x + \\log{(\\cos{(n)})} = p + \\log{(\\cos{(n)})} and x + l{(n)} = x + \\log{(\\cos{(n)})} and x + l{(n)} + \\iint \\frac{d}{d n} l{(n)} dn dn = x + \\log{(\\cos{(n)})} + \\iint \\frac{d}{d n} l{(n)} dn dn", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('n', commutative=True)), log(cos(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(log(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Derivative(Function('l')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))), Integral(Derivative(log(cos(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('x', commutative=True), Function('l')(Symbol('n', commutative=True))), Add(Symbol('p', commutative=True), log(cos(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('x', commutative=True), log(cos(Symbol('n', commutative=True)))), Add(Symbol('p', commutative=True), log(cos(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('x', commutative=True), Function('l')(Symbol('n', commutative=True))), Add(Symbol('x', commutative=True), log(cos(Symbol('n', commutative=True)))))"], [["add", 6, "Integral(Derivative(Function('l')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Symbol('x', commutative=True), Function('l')(Symbol('n', commutative=True)), Integral(Derivative(Function('l')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Symbol('x', commutative=True), log(cos(Symbol('n', commutative=True))), Integral(Derivative(Function('l')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Tuple(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(\\mu_0)} = e^{\\mu_0} and \\chi{(\\mu_0)} = \\int e^{2 \\mu_0} d\\mu_0, then obtain 2 \\chi{(\\mu_0)} - 2 \\dot{z}{(\\mu_0)} - 2 e^{\\mu_0} = \\chi{(\\mu_0)} - 2 \\dot{z}{(\\mu_0)} - 2 e^{\\mu_0} + \\int e^{2 \\mu_0} d\\mu_0", "derivation": "\\dot{z}{(\\mu_0)} = e^{\\mu_0} and \\dot{z}{(\\mu_0)} e^{\\mu_0} = e^{2 \\mu_0} and \\int \\dot{z}{(\\mu_0)} e^{\\mu_0} d\\mu_0 = \\int e^{2 \\mu_0} d\\mu_0 and \\chi{(\\mu_0)} = \\int e^{2 \\mu_0} d\\mu_0 and - \\dot{z}{(\\mu_0)} - e^{\\mu_0} + \\int \\dot{z}{(\\mu_0)} e^{\\mu_0} d\\mu_0 = - \\dot{z}{(\\mu_0)} - e^{\\mu_0} + \\int e^{2 \\mu_0} d\\mu_0 and \\chi{(\\mu_0)} = \\int \\dot{z}{(\\mu_0)} e^{\\mu_0} d\\mu_0 and \\chi{(\\mu_0)} - \\dot{z}{(\\mu_0)} - e^{\\mu_0} = - \\dot{z}{(\\mu_0)} - e^{\\mu_0} + \\int e^{2 \\mu_0} d\\mu_0 and 2 \\chi{(\\mu_0)} - 2 \\dot{z}{(\\mu_0)} - 2 e^{\\mu_0} = \\chi{(\\mu_0)} - 2 \\dot{z}{(\\mu_0)} - 2 e^{\\mu_0} + \\int e^{2 \\mu_0} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 3, "Add(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Integral(Mul(Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["add", 7, "Add(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\dot{z}')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given J{(v_{2})} = \\sin{(v_{2})} and \\varphi{(v_{2})} = 8 J^{3}{(v_{2})}, then obtain \\varphi{(v_{2})} = 2 (J{(v_{2})} + \\sin{(v_{2})})^{2} J{(v_{2})}", "derivation": "J{(v_{2})} = \\sin{(v_{2})} and 2 J{(v_{2})} = J{(v_{2})} + \\sin{(v_{2})} and 4 J^{2}{(v_{2})} = (J{(v_{2})} + \\sin{(v_{2})})^{2} and 8 J^{3}{(v_{2})} = 2 (J{(v_{2})} + \\sin{(v_{2})})^{2} J{(v_{2})} and \\varphi{(v_{2})} = 8 J^{3}{(v_{2})} and \\varphi{(v_{2})} = 2 (J{(v_{2})} + \\sin{(v_{2})})^{2} J{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["add", 1, "Function('J')(Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(2), Function('J')(Symbol('v_2', commutative=True))), Add(Function('J')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('J')(Symbol('v_2', commutative=True)), Integer(2))), Pow(Add(Function('J')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(2)))"], [["times", 3, "Mul(Integer(2), Function('J')(Symbol('v_2', commutative=True)))"], "Equality(Mul(Integer(8), Pow(Function('J')(Symbol('v_2', commutative=True)), Integer(3))), Mul(Integer(2), Pow(Add(Function('J')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(2)), Function('J')(Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True)), Mul(Integer(8), Pow(Function('J')(Symbol('v_2', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\varphi')(Symbol('v_2', commutative=True)), Mul(Integer(2), Pow(Add(Function('J')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True))), Integer(2)), Function('J')(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\mu{(F_{g},\\mathbf{H})} = e^{F_{g} + \\mathbf{H}}, then obtain \\frac{\\partial}{\\partial F_{g}} \\int \\mu{(F_{g},\\mathbf{H})} e^{- F_{g} - \\mathbf{H}} dF_{g} = \\frac{\\partial}{\\partial F_{g}} \\int e^{- F_{g} - \\mathbf{H}} e^{F_{g} + \\mathbf{H}} dF_{g}", "derivation": "\\mu{(F_{g},\\mathbf{H})} = e^{F_{g} + \\mathbf{H}} and \\mu{(F_{g},\\mathbf{H})} e^{- F_{g} - \\mathbf{H}} = e^{- F_{g} - \\mathbf{H}} e^{F_{g} + \\mathbf{H}} and \\int \\mu{(F_{g},\\mathbf{H})} e^{- F_{g} - \\mathbf{H}} dF_{g} = \\int e^{- F_{g} - \\mathbf{H}} e^{F_{g} + \\mathbf{H}} dF_{g} and \\frac{\\partial}{\\partial F_{g}} \\int \\mu{(F_{g},\\mathbf{H})} e^{- F_{g} - \\mathbf{H}} dF_{g} = \\frac{\\partial}{\\partial F_{g}} \\int e^{- F_{g} - \\mathbf{H}} e^{F_{g} + \\mathbf{H}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 1, "exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))), Mul(exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mu')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mu')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integral(Mul(exp(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), exp(Add(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(W,v_{z})} = W + v_{z}, then derive \\int \\hat{H}_{\\lambda}{(W,v_{z})} dv_{z} = W v_{z} + v_{y} + \\frac{v_{z}^{2}}{2}, then obtain L + W v_{z} + \\frac{v_{z}^{2}}{2} = W v_{z} + v_{y} + \\frac{v_{z}^{2}}{2}", "derivation": "\\hat{H}_{\\lambda}{(W,v_{z})} = W + v_{z} and \\int \\hat{H}_{\\lambda}{(W,v_{z})} dv_{z} = \\int (W + v_{z}) dv_{z} and \\int \\hat{H}_{\\lambda}{(W,v_{z})} dv_{z} = W v_{z} + v_{y} + \\frac{v_{z}^{2}}{2} and \\int (W + v_{z}) dv_{z} = W v_{z} + v_{y} + \\frac{v_{z}^{2}}{2} and L + W v_{z} + \\frac{v_{z}^{2}}{2} = W v_{z} + v_{y} + \\frac{v_{z}^{2}}{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('W', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Mul(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Mul(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('L', commutative=True), Mul(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))), Add(Mul(Symbol('W', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_y', commutative=True), Mul(Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(r,J_{\\varepsilon})} = \\cos{(J_{\\varepsilon} r)}, then obtain \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r (r + \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r})} = \\frac{\\cos{(J_{\\varepsilon} r)}}{r (r + \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r})}", "derivation": "\\operatorname{v_{2}}{(r,J_{\\varepsilon})} = \\cos{(J_{\\varepsilon} r)} and \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r} = \\frac{\\cos{(J_{\\varepsilon} r)}}{r} and r + \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r} = r + \\frac{\\cos{(J_{\\varepsilon} r)}}{r} and \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r (r + \\frac{\\cos{(J_{\\varepsilon} r)}}{r})} = \\frac{\\cos{(J_{\\varepsilon} r)}}{r (r + \\frac{\\cos{(J_{\\varepsilon} r)}}{r})} and \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r (r + \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r})} = \\frac{\\cos{(J_{\\varepsilon} r)}}{r (r + \\frac{\\operatorname{v_{2}}{(r,J_{\\varepsilon})}}{r})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True))))"], [["divide", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True)))))"], [["add", 2, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True))))))"], [["divide", 2, "Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True)))))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True))))), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True))))), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Pow(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('v_2')(Symbol('r', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(-1)), cos(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = \\mathbf{E}, then derive \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = 1, then obtain 2 \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} + 1", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{E})} = \\mathbf{E} and \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\mathbf{E} and \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = 1 and \\frac{d}{d \\mathbf{E}} \\mathbf{E} = 1 and 2 \\frac{d}{d \\mathbf{E}} \\mathbf{E} = \\frac{d}{d \\mathbf{E}} \\mathbf{E} + 1 and 2 \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\operatorname{x^{{\\}'}}{(\\mathbf{E})} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["add", 4, "Derivative(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Derivative(Symbol('\\\\mathbf{E}', commutative=True), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(2), Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Derivative(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\psi{(C_{2})} = \\log{(\\cos{(C_{2})})}, then obtain 3 \\psi{(C_{2})} = \\psi{(C_{2})} + 2 \\log{(\\cos{(C_{2})})}", "derivation": "\\psi{(C_{2})} = \\log{(\\cos{(C_{2})})} and 2 \\psi{(C_{2})} = \\psi{(C_{2})} + \\log{(\\cos{(C_{2})})} and 3 \\psi{(C_{2})} = 2 \\psi{(C_{2})} + \\log{(\\cos{(C_{2})})} and 3 \\psi{(C_{2})} = \\psi{(C_{2})} + 2 \\log{(\\cos{(C_{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('C_2', commutative=True)), log(cos(Symbol('C_2', commutative=True))))"], [["add", 1, "Function('\\\\psi')(Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi')(Symbol('C_2', commutative=True))), Add(Function('\\\\psi')(Symbol('C_2', commutative=True)), log(cos(Symbol('C_2', commutative=True)))))"], [["add", 2, "Function('\\\\psi')(Symbol('C_2', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\psi')(Symbol('C_2', commutative=True))), Add(Mul(Integer(2), Function('\\\\psi')(Symbol('C_2', commutative=True))), log(cos(Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\psi')(Symbol('C_2', commutative=True))), Add(Function('\\\\psi')(Symbol('C_2', commutative=True)), Mul(Integer(2), log(cos(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\phi{(F_{N})} = \\log{(F_{N})} and \\operatorname{F_{x}}{(F_{N})} = \\log{(F_{N})} and \\operatorname{A_{y}}{(F_{N})} = \\phi{(F_{N})} + \\log{(F_{N})}, then obtain \\frac{d}{d F_{N}} \\operatorname{A_{y}}{(F_{N})} = \\frac{d}{d F_{N}} (\\phi{(F_{N})} + \\log{(F_{N})})", "derivation": "\\phi{(F_{N})} = \\log{(F_{N})} and 2 \\phi{(F_{N})} = \\phi{(F_{N})} + \\log{(F_{N})} and \\operatorname{F_{x}}{(F_{N})} = \\log{(F_{N})} and 2 \\phi{(F_{N})} = \\operatorname{F_{x}}{(F_{N})} + \\phi{(F_{N})} and \\operatorname{A_{y}}{(F_{N})} = \\phi{(F_{N})} + \\log{(F_{N})} and \\frac{d}{d F_{N}} 2 \\phi{(F_{N})} = \\frac{d}{d F_{N}} (\\operatorname{F_{x}}{(F_{N})} + \\phi{(F_{N})}) and 2 \\phi{(F_{N})} = \\operatorname{A_{y}}{(F_{N})} and \\frac{d}{d F_{N}} 2 \\phi{(F_{N})} = \\frac{d}{d F_{N}} (\\phi{(F_{N})} + \\log{(F_{N})}) and \\frac{d}{d F_{N}} \\operatorname{A_{y}}{(F_{N})} = \\frac{d}{d F_{N}} (\\phi{(F_{N})} + \\log{(F_{N})})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["add", 1, "Function('\\\\phi')(Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('F_N', commutative=True))), Add(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('F_N', commutative=True))), Add(Function('F_x')(Symbol('F_N', commutative=True)), Function('\\\\phi')(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('F_N', commutative=True)), Add(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\phi')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Function('F_x')(Symbol('F_N', commutative=True)), Function('\\\\phi')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('F_N', commutative=True))), Function('A_y')(Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Mul(Integer(2), Function('\\\\phi')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Derivative(Function('A_y')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(T)} = \\int e^{T} dT, then derive 0 = m_{s} - g{(T)} + e^{T}, then obtain 0 = - (m_{s} - g{(T)} + e^{T}) g{(T)}", "derivation": "g{(T)} = \\int e^{T} dT and 0 = - g{(T)} + \\int e^{T} dT and 0 = m_{s} - g{(T)} + e^{T} and 0 = - (m_{s} - g{(T)} + e^{T}) g{(T)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('T', commutative=True)), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["minus", 1, "Function('g')(Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('g')(Symbol('T', commutative=True))), Integral(exp(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(0), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Function('g')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Function('g')(Symbol('T', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), Function('g')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True))), Function('g')(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(v_{t})} = \\sin{(e^{v_{t}})} and \\hat{H}_{\\lambda}{(v_{t})} = \\sin{(e^{v_{t}})}, then obtain 2 v_{t} \\mathbf{D}{(v_{t})} = v_{t} (\\mathbf{D}{(v_{t})} + \\sin{(e^{v_{t}})})", "derivation": "\\mathbf{D}{(v_{t})} = \\sin{(e^{v_{t}})} and \\hat{H}_{\\lambda}{(v_{t})} = \\sin{(e^{v_{t}})} and \\hat{H}_{\\lambda}{(v_{t})} + \\mathbf{D}{(v_{t})} = \\mathbf{D}{(v_{t})} + \\sin{(e^{v_{t}})} and v_{t} (\\hat{H}_{\\lambda}{(v_{t})} + \\mathbf{D}{(v_{t})}) = v_{t} (\\mathbf{D}{(v_{t})} + \\sin{(e^{v_{t}})}) and \\mathbf{D}{(v_{t})} = \\hat{H}_{\\lambda}{(v_{t})} and 2 v_{t} \\mathbf{D}{(v_{t})} = v_{t} (\\mathbf{D}{(v_{t})} + \\sin{(e^{v_{t}})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), sin(exp(Symbol('v_t', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), sin(exp(Symbol('v_t', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Add(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), sin(exp(Symbol('v_t', commutative=True)))))"], [["times", 3, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)))), Mul(Symbol('v_t', commutative=True), Add(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), sin(exp(Symbol('v_t', commutative=True))))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Symbol('v_t', commutative=True), Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), Add(Function('\\\\mathbf{D}')(Symbol('v_t', commutative=True)), sin(exp(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)}, then derive \\frac{d}{d \\mathbf{J}_P} \\mathbf{p}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)}, then obtain \\frac{\\frac{d}{d \\mathbf{J}_P} \\sin{(\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{\\cos{(\\mathbf{J}_P)}}{\\mathbf{J}_P}", "derivation": "\\mathbf{p}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\mathbf{p}{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\sin{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\mathbf{p}{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} \\sin{(\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P)} and \\frac{\\frac{d}{d \\mathbf{J}_P} \\sin{(\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{\\cos{(\\mathbf{J}_P)}}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Derivative(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(m)} = \\frac{d}{d m} e^{m}, then derive \\operatorname{f_{\\mathbf{v}}}{(m)} = e^{m}, then obtain \\cos{(1)} = \\cos{(\\frac{\\frac{d}{d m} \\operatorname{f_{\\mathbf{v}}}{(m)}}{\\operatorname{f_{\\mathbf{v}}}{(m)}})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(m)} = \\frac{d}{d m} e^{m} and \\operatorname{f_{\\mathbf{v}}}{(m)} = e^{m} and \\operatorname{f_{\\mathbf{v}}}{(m)} = \\frac{d}{d m} \\operatorname{f_{\\mathbf{v}}}{(m)} and \\operatorname{f_{\\mathbf{v}}}{(m)} e^{- m} = e^{- m} \\frac{d}{d m} \\operatorname{f_{\\mathbf{v}}}{(m)} and 1 = e^{- m} \\frac{d}{d m} e^{m} and 1 = \\frac{\\frac{d}{d m} \\operatorname{f_{\\mathbf{v}}}{(m)}}{\\operatorname{f_{\\mathbf{v}}}{(m)}} and \\cos{(1)} = \\cos{(\\frac{\\frac{d}{d m} \\operatorname{f_{\\mathbf{v}}}{(m)}}{\\operatorname{f_{\\mathbf{v}}}{(m)}})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["divide", 3, "exp(Symbol('m', commutative=True))"], "Equality(Mul(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('m', commutative=True))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(exp(Mul(Integer(-1), Symbol('m', commutative=True))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(1), Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Integer(-1)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["cos", 6], "Equality(cos(Integer(1)), cos(Mul(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Integer(-1)), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{x}{(U,\\eta)} = e^{\\frac{\\eta}{U}}, then obtain \\frac{\\dot{x}^{2}{(U,\\eta)} e^{\\frac{2 \\eta}{U}}}{U} = \\frac{\\dot{x}{(U,\\eta)} e^{\\frac{3 \\eta}{U}}}{U}", "derivation": "\\dot{x}{(U,\\eta)} = e^{\\frac{\\eta}{U}} and \\dot{x}{(U,\\eta)} e^{\\frac{\\eta}{U}} = e^{\\frac{2 \\eta}{U}} and \\frac{\\dot{x}{(U,\\eta)} e^{\\frac{\\eta}{U}}}{U} = \\frac{e^{\\frac{2 \\eta}{U}}}{U} and \\frac{\\dot{x}^{2}{(U,\\eta)} e^{\\frac{2 \\eta}{U}}}{U} = \\frac{\\dot{x}{(U,\\eta)} e^{\\frac{3 \\eta}{U}}}{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True))))"], [["times", 1, "exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))), exp(Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True))))"], [["divide", 2, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), exp(Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))))"], [["times", 3, "Mul(Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True))))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(2)), exp(Mul(Integer(2), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Integer(3), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\phi_2)} = e^{\\phi_2}, then obtain \\mathbf{D}{(\\phi_2)} + \\int (\\mathbf{D}^{\\phi_2}{(\\phi_2)} - (e^{\\phi_2})^{\\phi_2}) d\\phi_2 = \\mathbf{D}{(\\phi_2)} + \\int 0 d\\phi_2", "derivation": "\\mathbf{D}{(\\phi_2)} = e^{\\phi_2} and \\mathbf{D}^{\\phi_2}{(\\phi_2)} = (e^{\\phi_2})^{\\phi_2} and \\mathbf{D}^{\\phi_2}{(\\phi_2)} - (e^{\\phi_2})^{\\phi_2} = 0 and \\int (\\mathbf{D}^{\\phi_2}{(\\phi_2)} - (e^{\\phi_2})^{\\phi_2}) d\\phi_2 = \\int 0 d\\phi_2 and \\mathbf{D}{(\\phi_2)} + \\int (\\mathbf{D}^{\\phi_2}{(\\phi_2)} - (e^{\\phi_2})^{\\phi_2}) d\\phi_2 = \\mathbf{D}{(\\phi_2)} + \\int 0 d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(exp(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 2, "Pow(exp(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["add", 4, "Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Integral(Add(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Function('\\\\mathbf{D}')(Symbol('\\\\phi_2', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(k,A_{x})} = \\frac{k}{A_{x}}, then obtain A_{x} e^{- f^{\\prime}} \\frac{\\partial}{\\partial A_{x}} (- (\\frac{k}{A_{x}})^{k} + \\mathbf{B}^{k}{(k,A_{x})}) = A_{x} e^{- f^{\\prime}} \\frac{d}{d A_{x}} 0", "derivation": "\\mathbf{B}{(k,A_{x})} = \\frac{k}{A_{x}} and \\mathbf{B}^{k}{(k,A_{x})} = (\\frac{k}{A_{x}})^{k} and - (\\frac{k}{A_{x}})^{k} + \\mathbf{B}^{k}{(k,A_{x})} = 0 and \\frac{\\partial}{\\partial A_{x}} (- (\\frac{k}{A_{x}})^{k} + \\mathbf{B}^{k}{(k,A_{x})}) = \\frac{d}{d A_{x}} 0 and e^{- f^{\\prime}} \\frac{\\partial}{\\partial A_{x}} (- (\\frac{k}{A_{x}})^{k} + \\mathbf{B}^{k}{(k,A_{x})}) = e^{- f^{\\prime}} \\frac{d}{d A_{x}} 0 and A_{x} e^{- f^{\\prime}} \\frac{\\partial}{\\partial A_{x}} (- (\\frac{k}{A_{x}})^{k} + \\mathbf{B}^{k}{(k,A_{x})}) = A_{x} e^{- f^{\\prime}} \\frac{d}{d A_{x}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Symbol('k', commutative=True)), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 2, "Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Symbol('k', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 4, "exp(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Derivative(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Derivative(Integer(0), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["divide", 5, "Pow(Symbol('A_x', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('A_x', commutative=True), exp(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Derivative(Add(Mul(Integer(-1), Pow(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Pow(Function('\\\\mathbf{B}')(Symbol('k', commutative=True), Symbol('A_x', commutative=True)), Symbol('k', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Symbol('A_x', commutative=True), exp(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Derivative(Integer(0), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(v_{2},C)} = \\sin^{C}{(v_{2})} and \\psi^{*}{(p,A_{1})} = \\log{(A_{1}^{p})}, then obtain \\frac{\\partial}{\\partial A_{1}} \\frac{\\psi^{*}{(p,A_{1})} \\sin^{- 2 C}{(v_{2})}}{C^{2}} = \\frac{\\partial}{\\partial A_{1}} \\frac{\\log{(A_{1}^{p})} \\sin^{- 2 C}{(v_{2})}}{C^{2}}", "derivation": "l{(v_{2},C)} = \\sin^{C}{(v_{2})} and \\psi^{*}{(p,A_{1})} = \\log{(A_{1}^{p})} and \\frac{\\psi^{*}{(p,A_{1})} \\sin^{- C}{(v_{2})}}{C^{2} l{(v_{2},C)}} = \\frac{\\log{(A_{1}^{p})} \\sin^{- C}{(v_{2})}}{C^{2} l{(v_{2},C)}} and \\frac{\\partial}{\\partial A_{1}} \\frac{\\psi^{*}{(p,A_{1})} \\sin^{- C}{(v_{2})}}{C^{2} l{(v_{2},C)}} = \\frac{\\partial}{\\partial A_{1}} \\frac{\\log{(A_{1}^{p})} \\sin^{- C}{(v_{2})}}{C^{2} l{(v_{2},C)}} and \\frac{\\partial}{\\partial A_{1}} \\frac{\\psi^{*}{(p,A_{1})} \\sin^{- 2 C}{(v_{2})}}{C^{2}} = \\frac{\\partial}{\\partial A_{1}} \\frac{\\log{(A_{1}^{p})} \\sin^{- 2 C}{(v_{2})}}{C^{2}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Pow(sin(Symbol('v_2', commutative=True)), Symbol('C', commutative=True)))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('p', commutative=True), Symbol('A_1', commutative=True)), log(Pow(Symbol('A_1', commutative=True), Symbol('p', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('C', commutative=True), Integer(2)), Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Pow(sin(Symbol('v_2', commutative=True)), Symbol('C', commutative=True)))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-2)), Function('\\\\psi^*')(Symbol('p', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-2)), Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('p', commutative=True))), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))))"], [["differentiate", 3, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-2)), Function('\\\\psi^*')(Symbol('p', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-2)), Pow(Function('l')(Symbol('v_2', commutative=True), Symbol('C', commutative=True)), Integer(-1)), log(Pow(Symbol('A_1', commutative=True), Symbol('p', commutative=True))), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-2)), Function('\\\\psi^*')(Symbol('p', commutative=True), Symbol('A_1', commutative=True)), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('C', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C', commutative=True), Integer(-2)), log(Pow(Symbol('A_1', commutative=True), Symbol('p', commutative=True))), Pow(sin(Symbol('v_2', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('C', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given p{(A_{x})} = e^{A_{x}}, then obtain (e^{2 A_{x}})^{A_{x}} = (p{(A_{x})} e^{A_{x}})^{A_{x}}", "derivation": "p{(A_{x})} = e^{A_{x}} and p{(A_{x})} e^{A_{x}} = e^{2 A_{x}} and p^{2}{(A_{x})} = p{(A_{x})} e^{A_{x}} and (p^{2}{(A_{x})})^{A_{x}} = (p{(A_{x})} e^{A_{x}})^{A_{x}} and (p^{2}{(A_{x})})^{A_{x}} = (e^{2 A_{x}})^{A_{x}} and (e^{2 A_{x}})^{A_{x}} = (p{(A_{x})} e^{A_{x}})^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["times", 1, "exp(Symbol('A_x', commutative=True))"], "Equality(Mul(Function('p')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True))), exp(Mul(Integer(2), Symbol('A_x', commutative=True))))"], [["times", 1, "Function('p')(Symbol('A_x', commutative=True))"], "Equality(Pow(Function('p')(Symbol('A_x', commutative=True)), Integer(2)), Mul(Function('p')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True))))"], [["power", 3, "Symbol('A_x', commutative=True)"], "Equality(Pow(Pow(Function('p')(Symbol('A_x', commutative=True)), Integer(2)), Symbol('A_x', commutative=True)), Pow(Mul(Function('p')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Pow(Function('p')(Symbol('A_x', commutative=True)), Integer(2)), Symbol('A_x', commutative=True)), Pow(exp(Mul(Integer(2), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(exp(Mul(Integer(2), Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Pow(Mul(Function('p')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(P_{g},n)} = P_{g} n, then obtain (P_{g} n + \\mathbf{J}_P{(P_{g},n)})^{4} = 4 P_{g}^{2} n^{2} (P_{g} n + \\mathbf{J}_P{(P_{g},n)})^{2}", "derivation": "\\mathbf{J}_P{(P_{g},n)} = P_{g} n and P_{g} n + \\mathbf{J}_P{(P_{g},n)} = 2 P_{g} n and (P_{g} n + \\mathbf{J}_P{(P_{g},n)})^{2} = 2 P_{g} n (P_{g} n + \\mathbf{J}_P{(P_{g},n)}) and (P_{g} n + \\mathbf{J}_P{(P_{g},n)})^{4} = 4 P_{g}^{2} n^{2} (P_{g} n + \\mathbf{J}_P{(P_{g},n)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Mul(Integer(2), Symbol('P_g', commutative=True), Symbol('n', commutative=True)))"], [["times", 2, "Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))"], "Equality(Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Integer(2)), Mul(Integer(2), Symbol('P_g', commutative=True), Symbol('n', commutative=True), Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Integer(4)), Mul(Integer(4), Pow(Symbol('P_g', commutative=True), Integer(2)), Pow(Symbol('n', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('P_g', commutative=True), Symbol('n', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('P_g', commutative=True), Symbol('n', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\hat{p}{(\\Omega)} = \\log{(\\Omega)} and \\phi{(s)} = \\sin{(s)}, then obtain (- \\Omega + \\hat{p}{(\\Omega)}) \\hat{p}{(\\Omega)} \\phi{(s)} = (- \\Omega + \\hat{p}{(\\Omega)}) \\hat{p}{(\\Omega)} \\sin{(s)}", "derivation": "\\hat{p}{(\\Omega)} = \\log{(\\Omega)} and \\phi{(s)} = \\sin{(s)} and (- \\Omega + \\log{(\\Omega)}) \\hat{p}{(\\Omega)} \\phi{(s)} = (- \\Omega + \\log{(\\Omega)}) \\hat{p}{(\\Omega)} \\sin{(s)} and (- \\Omega + \\hat{p}{(\\Omega)}) \\hat{p}{(\\Omega)} \\phi{(s)} = (- \\Omega + \\hat{p}{(\\Omega)}) \\hat{p}{(\\Omega)} \\sin{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], ["get_premise", "Equality(Function('\\\\phi')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["times", 2, "Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Function('\\\\phi')(Symbol('s', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), Function('\\\\phi')(Symbol('s', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('s', commutative=True))))"]]}, {"prompt": "Given q{(\\lambda)} = e^{\\lambda}, then derive \\int q{(\\lambda)} e^{- \\lambda} d\\lambda = \\hat{X} + \\lambda, then obtain \\frac{d}{d \\lambda} e^{\\int q{(\\lambda)} e^{- \\lambda} d\\lambda} = \\frac{\\partial}{\\partial \\lambda} e^{\\hat{X} + \\lambda}", "derivation": "q{(\\lambda)} = e^{\\lambda} and q{(\\lambda)} e^{- \\lambda} = 1 and \\int q{(\\lambda)} e^{- \\lambda} d\\lambda = \\int 1 d\\lambda and \\int q{(\\lambda)} e^{- \\lambda} d\\lambda = \\hat{X} + \\lambda and e^{\\int q{(\\lambda)} e^{- \\lambda} d\\lambda} = e^{\\int 1 d\\lambda} and \\hat{X} + \\lambda = \\int 1 d\\lambda and \\frac{d}{d \\lambda} e^{\\int q{(\\lambda)} e^{- \\lambda} d\\lambda} = \\frac{d}{d \\lambda} e^{\\int 1 d\\lambda} and \\frac{d}{d \\lambda} e^{\\int q{(\\lambda)} e^{- \\lambda} d\\lambda} = \\frac{\\partial}{\\partial \\lambda} e^{\\hat{X} + \\lambda}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["exp", 3], "Equality(exp(Integral(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))), exp(Integral(Integer(1), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(exp(Integral(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Integral(Integer(1), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(exp(Integral(Mul(Function('q')(Symbol('\\\\lambda', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\omega)} = e^{\\omega}, then obtain \\hat{\\mathbf{x}}^{5}{(\\omega)} = \\hat{\\mathbf{x}}^{3}{(\\omega)} e^{2 \\omega}", "derivation": "\\hat{\\mathbf{x}}{(\\omega)} = e^{\\omega} and \\hat{\\mathbf{x}}^{2}{(\\omega)} = \\hat{\\mathbf{x}}{(\\omega)} e^{\\omega} and \\hat{\\mathbf{x}}^{4}{(\\omega)} = \\hat{\\mathbf{x}}^{3}{(\\omega)} e^{\\omega} and \\hat{\\mathbf{x}}^{5}{(\\omega)} = \\hat{\\mathbf{x}}^{4}{(\\omega)} e^{\\omega} and \\hat{\\mathbf{x}}^{5}{(\\omega)} = \\hat{\\mathbf{x}}^{3}{(\\omega)} e^{2 \\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["times", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(2)), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))))"], [["times", 2, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(3)), exp(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(4))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(5)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(4)), exp(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(5)), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\omega', commutative=True)), Integer(3)), exp(Mul(Integer(2), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\Omega)} = \\cos{(\\Omega)}, then derive (\\int \\rho_{f}{(\\Omega)} d\\Omega)^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega}, then obtain (I + \\sin{(\\Omega)})^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega}", "derivation": "\\rho_{f}{(\\Omega)} = \\cos{(\\Omega)} and \\int \\rho_{f}{(\\Omega)} d\\Omega = \\int \\cos{(\\Omega)} d\\Omega and (\\int \\rho_{f}{(\\Omega)} d\\Omega)^{\\Omega} = (\\int \\cos{(\\Omega)} d\\Omega)^{\\Omega} and (\\int \\rho_{f}{(\\Omega)} d\\Omega)^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega} and (\\int \\cos{(\\Omega)} d\\Omega)^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega} and (I + \\sin{(\\Omega)})^{\\Omega} = (k + \\sin{(\\Omega)})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\rho_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('I', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Symbol('k', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(f^{\\prime})} = \\log{(f^{\\prime})}, then obtain \\int (\\mathbf{M}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} df^{\\prime} = \\int (\\log{(f^{\\prime})}^{f^{\\prime}})^{f^{\\prime}} df^{\\prime}", "derivation": "\\mathbf{M}{(f^{\\prime})} = \\log{(f^{\\prime})} and \\mathbf{M}^{f^{\\prime}}{(f^{\\prime})} = \\log{(f^{\\prime})}^{f^{\\prime}} and (\\mathbf{M}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} = (\\log{(f^{\\prime})}^{f^{\\prime}})^{f^{\\prime}} and \\int (\\mathbf{M}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} df^{\\prime} = \\int (\\log{(f^{\\prime})}^{f^{\\prime}})^{f^{\\prime}} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('\\\\mathbf{M}')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Pow(Pow(log(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given V{(t,p)} = \\frac{\\partial}{\\partial p} t^{p}, then derive 0 = - \\frac{\\partial}{\\partial t} V{(t,p)} + \\frac{t^{p} (p \\log{(t)} + 1)}{t}, then obtain 0 = - \\frac{\\partial^{2}}{\\partial t\\partial p} t^{p} + \\frac{t^{p} (p \\log{(t)} + 1)}{t}", "derivation": "V{(t,p)} = \\frac{\\partial}{\\partial p} t^{p} and \\frac{\\partial}{\\partial t} V{(t,p)} = \\frac{\\partial^{2}}{\\partial t\\partial p} t^{p} and 0 = - \\frac{\\partial}{\\partial t} V{(t,p)} + \\frac{\\partial^{2}}{\\partial t\\partial p} t^{p} and 0 = - \\frac{\\partial}{\\partial t} V{(t,p)} + \\frac{t^{p} (p \\log{(t)} + 1)}{t} and 0 = - \\frac{\\partial^{2}}{\\partial t\\partial p} t^{p} + \\frac{t^{p} (p \\log{(t)} + 1)}{t}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('t', commutative=True), Symbol('p', commutative=True)), Derivative(Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('V')(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('V')(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Derivative(Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('V')(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('p', commutative=True), log(Symbol('t', commutative=True))), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Pow(Symbol('t', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('p', commutative=True), log(Symbol('t', commutative=True))), Integer(1)))))"]]}, {"prompt": "Given \\chi{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain \\frac{d}{d x^\\prime} ((e^{\\int \\chi{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime) = \\frac{d}{d x^\\prime} ((e^{\\int \\cos{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime)", "derivation": "\\chi{(x^\\prime)} = \\cos{(x^\\prime)} and \\int \\chi{(x^\\prime)} dx^\\prime = \\int \\cos{(x^\\prime)} dx^\\prime and e^{\\int \\chi{(x^\\prime)} dx^\\prime} = e^{\\int \\cos{(x^\\prime)} dx^\\prime} and (e^{\\int \\chi{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} = (e^{\\int \\cos{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} and (e^{\\int \\chi{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime = (e^{\\int \\cos{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} ((e^{\\int \\chi{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime) = \\frac{d}{d x^\\prime} ((e^{\\int \\cos{(x^\\prime)} dx^\\prime}) \\cos{(x^\\prime)} - \\int \\cos{(x^\\prime)} dx^\\prime)", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), exp(Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["times", 3, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(exp(Integral(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(exp(Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 4, "Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Mul(exp(Integral(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Add(Mul(exp(Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"], [["differentiate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(exp(Integral(Function('\\\\chi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(exp(Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(F_{c},y)} = \\frac{F_{c}}{y}, then obtain a{(F_{c},y)} + \\frac{\\partial}{\\partial F_{c}} a{(F_{c},y)} = a{(F_{c},y)} + \\frac{1}{y}", "derivation": "a{(F_{c},y)} = \\frac{F_{c}}{y} and \\frac{\\partial}{\\partial F_{c}} a{(F_{c},y)} = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c}}{y} and a{(F_{c},y)} + \\frac{\\partial}{\\partial F_{c}} a{(F_{c},y)} = a{(F_{c},y)} + \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c}}{y} and a{(F_{c},y)} + \\frac{\\partial}{\\partial F_{c}} a{(F_{c},y)} = a{(F_{c},y)} + \\frac{1}{y}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('F_c', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_c', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["add", 2, "Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Derivative(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Derivative(Mul(Symbol('F_c', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Derivative(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Add(Function('a')(Symbol('F_c', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(x)} = \\sin{(x)}, then obtain (\\frac{d}{d x} (\\int \\mathbf{D}{(x)} dx)^{x})^{x} = (\\frac{\\partial}{\\partial x} (\\nabla - \\cos{(x)})^{x})^{x}", "derivation": "\\mathbf{D}{(x)} = \\sin{(x)} and \\int \\mathbf{D}{(x)} dx = \\int \\sin{(x)} dx and (\\int \\mathbf{D}{(x)} dx)^{x} = (\\int \\sin{(x)} dx)^{x} and \\frac{d}{d x} (\\int \\mathbf{D}{(x)} dx)^{x} = \\frac{d}{d x} (\\int \\sin{(x)} dx)^{x} and (\\frac{d}{d x} (\\int \\mathbf{D}{(x)} dx)^{x})^{x} = (\\frac{d}{d x} (\\int \\sin{(x)} dx)^{x})^{x} and (\\frac{d}{d x} (\\int \\mathbf{D}{(x)} dx)^{x})^{x} = (\\frac{\\partial}{\\partial x} (\\nabla - \\cos{(x)})^{x})^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["power", 4, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Pow(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Derivative(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Pow(Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\psi{(\\sigma_x,F_{x})} = F_{x}^{\\sigma_x}, then obtain - \\sigma_x + \\frac{\\partial}{\\partial F_{x}} \\sigma_x \\psi{(\\sigma_x,F_{x})} = - \\sigma_x + \\frac{\\partial}{\\partial F_{x}} F_{x}^{\\sigma_x} \\sigma_x", "derivation": "\\psi{(\\sigma_x,F_{x})} = F_{x}^{\\sigma_x} and \\sigma_x \\psi{(\\sigma_x,F_{x})} = F_{x}^{\\sigma_x} \\sigma_x and \\frac{\\partial}{\\partial F_{x}} \\sigma_x \\psi{(\\sigma_x,F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x}^{\\sigma_x} \\sigma_x and - \\sigma_x + \\frac{\\partial}{\\partial F_{x}} \\sigma_x \\psi{(\\sigma_x,F_{x})} = - \\sigma_x + \\frac{\\partial}{\\partial F_{x}} F_{x}^{\\sigma_x} \\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('F_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(t_{2},C_{d})} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + t_{2}) and C{(t_{2},C_{d})} = t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})}, then derive - t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})} = t_{2}, then obtain \\frac{\\partial}{\\partial C_{d}} - C{(t_{2},C_{d})} = \\frac{d}{d C_{d}} t_{2}", "derivation": "\\operatorname{A_{y}}{(t_{2},C_{d})} = \\frac{\\partial}{\\partial C_{d}} (- C_{d} + t_{2}) and t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})} = t_{2} \\frac{\\partial}{\\partial C_{d}} (- C_{d} + t_{2}) and \\frac{t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})}}{\\frac{\\partial}{\\partial C_{d}} (- C_{d} + t_{2})} = t_{2} and C{(t_{2},C_{d})} = t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})} and - t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})} = t_{2} and \\frac{\\partial}{\\partial C_{d}} - t_{2} \\operatorname{A_{y}}{(t_{2},C_{d})} = \\frac{d}{d C_{d}} t_{2} and \\frac{\\partial}{\\partial C_{d}} - C{(t_{2},C_{d})} = \\frac{d}{d C_{d}} t_{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["times", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True))), Mul(Symbol('t_2', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["divide", 2, "Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('t_2', commutative=True), Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))), Symbol('t_2', commutative=True))"], ["renaming_premise", "Equality(Function('C')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True)), Mul(Symbol('t_2', commutative=True), Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Symbol('t_2', commutative=True), Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True))), Symbol('t_2', commutative=True))"], [["differentiate", 5, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('t_2', commutative=True), Function('A_y')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Symbol('t_2', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(Mul(Integer(-1), Function('C')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Symbol('t_2', commutative=True), Tuple(Symbol('C_d', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(W)} = \\log{(\\log{(W)})}, then derive \\frac{\\int \\mathbf{J}_M{(W)} dW}{W \\log{(\\log{(W)})} + h - \\operatorname{li}{(W)}} = 1, then obtain \\frac{\\int \\log{(\\log{(W)})} dW}{W \\log{(\\log{(W)})} + h - \\operatorname{li}{(W)}} = 1", "derivation": "\\mathbf{J}_M{(W)} = \\log{(\\log{(W)})} and \\int \\mathbf{J}_M{(W)} dW = \\int \\log{(\\log{(W)})} dW and \\frac{\\int \\mathbf{J}_M{(W)} dW}{\\int \\log{(\\log{(W)})} dW} = 1 and \\frac{\\int \\mathbf{J}_M{(W)} dW}{W \\log{(\\log{(W)})} + h - \\operatorname{li}{(W)}} = 1 and \\frac{\\int \\log{(\\log{(W)})} dW}{W \\log{(\\log{(W)})} + h - \\operatorname{li}{(W)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), log(log(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["divide", 2, "Integral(log(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Pow(Integral(log(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Mul(Symbol('W', commutative=True), log(log(Symbol('W', commutative=True)))), Symbol('h', commutative=True), Mul(Integer(-1), li(Symbol('W', commutative=True)))), Integer(-1)), Integral(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Add(Mul(Symbol('W', commutative=True), log(log(Symbol('W', commutative=True)))), Symbol('h', commutative=True), Mul(Integer(-1), li(Symbol('W', commutative=True)))), Integer(-1)), Integral(log(log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Integer(1))"]]}, {"prompt": "Given B{(v_{x},\\eta^{\\prime})} = \\eta^{\\prime} v_{x} and \\operatorname{r_{0}}{(v_{x},\\eta^{\\prime})} = \\eta^{\\prime} v_{x}, then obtain \\operatorname{r_{0}}{(v_{x},\\eta^{\\prime})} = 2 \\eta^{\\prime} v_{x} - B{(v_{x},\\eta^{\\prime})}", "derivation": "B{(v_{x},\\eta^{\\prime})} = \\eta^{\\prime} v_{x} and 0 = \\eta^{\\prime} v_{x} - B{(v_{x},\\eta^{\\prime})} and \\operatorname{r_{0}}{(v_{x},\\eta^{\\prime})} = \\eta^{\\prime} v_{x} and \\eta^{\\prime} v_{x} = 2 \\eta^{\\prime} v_{x} - B{(v_{x},\\eta^{\\prime})} and \\operatorname{r_{0}}{(v_{x},\\eta^{\\prime})} = B{(v_{x},\\eta^{\\prime})} and B{(v_{x},\\eta^{\\prime})} = 2 \\eta^{\\prime} v_{x} - B{(v_{x},\\eta^{\\prime})} and \\operatorname{r_{0}}{(v_{x},\\eta^{\\prime})} = 2 \\eta^{\\prime} v_{x} - B{(v_{x},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)))"], [["minus", 1, "Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 2, "Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True))"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('r_0')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Function('r_0')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('v_x', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(f^{\\prime})} = \\cos{(f^{\\prime})}, then derive \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})} = - \\sin{(f^{\\prime})}, then obtain \\frac{\\frac{\\lambda{(f^{\\prime})}}{\\cos{(f^{\\prime})}} - 1}{2 \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})}} = 0", "derivation": "\\lambda{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})} = - \\sin{(f^{\\prime})} and - \\lambda{(f^{\\prime})} = - \\cos{(f^{\\prime})} and \\frac{\\lambda{(f^{\\prime})}}{\\cos{(f^{\\prime})}} = 1 and \\frac{\\lambda{(f^{\\prime})}}{\\cos{(f^{\\prime})}} - 1 = 0 and \\frac{\\frac{\\lambda{(f^{\\prime})}}{\\cos{(f^{\\prime})}} - 1}{- \\sin{(f^{\\prime})} + \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})}} = 0 and \\frac{\\frac{\\lambda{(f^{\\prime})}}{\\cos{(f^{\\prime})}} - 1}{2 \\frac{d}{d f^{\\prime}} \\lambda{(f^{\\prime})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 5, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Integer(-1)), Integer(0))"], [["divide", 6, "Add(Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], "Equality(Mul(Add(Mul(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Integer(-1)), Pow(Add(Mul(Integer(-1), sin(Symbol('f^{\\\\prime}', commutative=True))), Derivative(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Rational(1, 2), Add(Mul(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1))), Integer(-1)), Pow(Derivative(Function('\\\\lambda')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\theta_1,\\dot{z})} = \\dot{z} \\log{(\\theta_1)}, then obtain (\\frac{\\Psi_{\\lambda}{(\\theta_1,\\dot{z})}}{\\dot{z} \\log{(\\theta_1)}})^{\\theta_1} \\Psi_{\\lambda}{(\\theta_1,\\dot{z})} = \\dot{z} \\log{(\\theta_1)}", "derivation": "\\Psi_{\\lambda}{(\\theta_1,\\dot{z})} = \\dot{z} \\log{(\\theta_1)} and \\frac{\\Psi_{\\lambda}{(\\theta_1,\\dot{z})}}{\\dot{z} \\log{(\\theta_1)}} = 1 and (\\frac{\\Psi_{\\lambda}{(\\theta_1,\\dot{z})}}{\\dot{z} \\log{(\\theta_1)}})^{\\theta_1} = 1 and (\\frac{\\Psi_{\\lambda}{(\\theta_1,\\dot{z})}}{\\dot{z} \\log{(\\theta_1)}})^{\\theta_1} \\Psi_{\\lambda}{(\\theta_1,\\dot{z})} = \\Psi_{\\lambda}{(\\theta_1,\\dot{z})} and (\\frac{\\Psi_{\\lambda}{(\\theta_1,\\dot{z})}}{\\dot{z} \\log{(\\theta_1)}})^{\\theta_1} \\Psi_{\\lambda}{(\\theta_1,\\dot{z})} = \\dot{z} \\log{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Integer(1))"], [["times", 3, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(log(Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Symbol('\\\\theta_1', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(v_{2})} = \\log{(e^{v_{2}})}, then obtain (e^{v_{2}} \\log{(e^{v_{2}})} + e^{v_{2}}) \\operatorname{F_{N}}{(v_{2})} e^{v_{2}} = (e^{v_{2}} \\log{(e^{v_{2}})} + e^{v_{2}}) e^{v_{2}} \\log{(e^{v_{2}})}", "derivation": "\\operatorname{F_{N}}{(v_{2})} = \\log{(e^{v_{2}})} and \\operatorname{F_{N}}{(v_{2})} e^{v_{2}} = e^{v_{2}} \\log{(e^{v_{2}})} and \\operatorname{F_{N}}{(v_{2})} e^{v_{2}} + e^{v_{2}} = e^{v_{2}} \\log{(e^{v_{2}})} + e^{v_{2}} and (\\operatorname{F_{N}}{(v_{2})} e^{v_{2}} + e^{v_{2}}) \\operatorname{F_{N}}{(v_{2})} e^{v_{2}} = (\\operatorname{F_{N}}{(v_{2})} e^{v_{2}} + e^{v_{2}}) e^{v_{2}} \\log{(e^{v_{2}})} and (e^{v_{2}} \\log{(e^{v_{2}})} + e^{v_{2}}) \\operatorname{F_{N}}{(v_{2})} e^{v_{2}} = (e^{v_{2}} \\log{(e^{v_{2}})} + e^{v_{2}}) e^{v_{2}} \\log{(e^{v_{2}})}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True))))"], [["times", 1, "exp(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Mul(exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))))"], [["add", 2, "exp(Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True))), Add(Mul(exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))), exp(Symbol('v_2', commutative=True))))"], [["times", 2, "Add(Mul(Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True)))"], "Equality(Mul(Add(Mul(Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True))), Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Mul(Add(Mul(Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))), exp(Symbol('v_2', commutative=True))), Function('F_N')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), Mul(Add(Mul(exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))), exp(Symbol('v_2', commutative=True))), exp(Symbol('v_2', commutative=True)), log(exp(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\pi{(F_{g},\\tilde{g},Z)} = \\frac{F_{g}}{Z \\tilde{g}}, then derive \\frac{\\partial}{\\partial F_{g}} \\pi{(F_{g},\\tilde{g},Z)} = \\frac{1}{Z \\tilde{g}}, then obtain \\frac{\\partial}{\\partial F_{g}} \\frac{F_{g}}{Z \\tilde{g}} = \\frac{1}{Z \\tilde{g}}", "derivation": "\\pi{(F_{g},\\tilde{g},Z)} = \\frac{F_{g}}{Z \\tilde{g}} and \\frac{\\partial}{\\partial F_{g}} \\pi{(F_{g},\\tilde{g},Z)} = \\frac{\\partial}{\\partial F_{g}} \\frac{F_{g}}{Z \\tilde{g}} and \\frac{\\partial}{\\partial F_{g}} \\pi{(F_{g},\\tilde{g},Z)} = \\frac{1}{Z \\tilde{g}} and \\frac{\\partial}{\\partial F_{g}} \\frac{F_{g}}{Z \\tilde{g}} = \\frac{1}{Z \\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('F_g', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given l{(g_{\\varepsilon},\\ddot{x})} = \\sin{(\\ddot{x} g_{\\varepsilon})} and U{(g_{\\varepsilon},\\ddot{x})} = g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})}, then obtain U^{\\ddot{x}}{(g_{\\varepsilon},\\ddot{x})} = (g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})})^{\\ddot{x}}", "derivation": "l{(g_{\\varepsilon},\\ddot{x})} = \\sin{(\\ddot{x} g_{\\varepsilon})} and g_{\\varepsilon} l{(g_{\\varepsilon},\\ddot{x})} = g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})} and (g_{\\varepsilon} l{(g_{\\varepsilon},\\ddot{x})})^{\\ddot{x}} = (g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})})^{\\ddot{x}} and U{(g_{\\varepsilon},\\ddot{x})} = g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})} and U{(g_{\\varepsilon},\\ddot{x})} = g_{\\varepsilon} l{(g_{\\varepsilon},\\ddot{x})} and U^{\\ddot{x}}{(g_{\\varepsilon},\\ddot{x})} = (g_{\\varepsilon} \\sin{(\\ddot{x} g_{\\varepsilon})})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('U')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('U')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('U')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(v_{t})} = \\frac{d}{d v_{t}} \\sin{(v_{t})}, then derive \\hat{p}_0{(v_{t})} - \\cos{(v_{t})} = 0, then obtain (\\hat{p}_0{(v_{t})} - \\cos{(v_{t})})^{2} = 0", "derivation": "\\hat{p}_0{(v_{t})} = \\frac{d}{d v_{t}} \\sin{(v_{t})} and \\hat{p}_0{(v_{t})} - \\frac{d}{d v_{t}} \\sin{(v_{t})} = 0 and \\hat{p}_0{(v_{t})} - \\cos{(v_{t})} = 0 and - \\cos{(v_{t})} + \\frac{d}{d v_{t}} \\sin{(v_{t})} = 0 and (- \\cos{(v_{t})} + \\frac{d}{d v_{t}} \\sin{(v_{t})})^{2} = 0 and (\\hat{p}_0{(v_{t})} - \\cos{(v_{t})})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('v_t', commutative=True)), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('v_t', commutative=True)), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integer(0))"], [["times", 4, "Add(Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], "Equality(Pow(Add(Mul(Integer(-1), cos(Symbol('v_t', commutative=True))), Derivative(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Integer(2)), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Function('\\\\hat{p}_0')(Symbol('v_t', commutative=True)), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Integer(2)), Integer(0))"]]}, {"prompt": "Given Q{(f^{\\prime},\\mathbf{r},m_{s})} = \\frac{\\mathbf{r}}{f^{\\prime} m_{s}}, then obtain \\frac{d}{d f^{\\prime}} 1 = \\frac{\\partial}{\\partial f^{\\prime}} \\frac{\\mathbf{r}}{f^{\\prime} m_{s} Q{(f^{\\prime},\\mathbf{r},m_{s})}}", "derivation": "Q{(f^{\\prime},\\mathbf{r},m_{s})} = \\frac{\\mathbf{r}}{f^{\\prime} m_{s}} and - Q{(f^{\\prime},\\mathbf{r},m_{s})} = - \\frac{\\mathbf{r}}{f^{\\prime} m_{s}} and 1 = \\frac{\\mathbf{r}}{f^{\\prime} m_{s} Q{(f^{\\prime},\\mathbf{r},m_{s})}} and \\frac{d}{d f^{\\prime}} 1 = \\frac{\\partial}{\\partial f^{\\prime}} \\frac{\\mathbf{r}}{f^{\\prime} m_{s} Q{(f^{\\prime},\\mathbf{r},m_{s})}}", "srepr_derivation": [["get_premise", "Equality(Function('Q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('Q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["divide", 2, "Mul(Integer(-1), Function('Q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('Q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Function('Q')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(r_{0})} = e^{\\sin{(r_{0})}}, then obtain \\frac{d^{2}}{d r_{0}^{2}} (\\operatorname{v_{z}}{(r_{0})} + \\frac{1}{r_{0}}) = \\frac{d^{2}}{d r_{0}^{2}} (e^{\\sin{(r_{0})}} + \\frac{1}{r_{0}})", "derivation": "\\operatorname{v_{z}}{(r_{0})} = e^{\\sin{(r_{0})}} and \\operatorname{v_{z}}{(r_{0})} + \\frac{1}{r_{0}} = e^{\\sin{(r_{0})}} + \\frac{1}{r_{0}} and \\frac{d}{d r_{0}} (\\operatorname{v_{z}}{(r_{0})} + \\frac{1}{r_{0}}) = \\frac{d}{d r_{0}} (e^{\\sin{(r_{0})}} + \\frac{1}{r_{0}}) and \\frac{d^{2}}{d r_{0}^{2}} (\\operatorname{v_{z}}{(r_{0})} + \\frac{1}{r_{0}}) = \\frac{d^{2}}{d r_{0}^{2}} (e^{\\sin{(r_{0})}} + \\frac{1}{r_{0}})", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('r_0', commutative=True)), exp(sin(Symbol('r_0', commutative=True))))"], [["add", 1, "Pow(Symbol('r_0', commutative=True), Integer(-1))"], "Equality(Add(Function('v_z')(Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Add(exp(sin(Symbol('r_0', commutative=True))), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Function('v_z')(Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(exp(sin(Symbol('r_0', commutative=True))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Function('v_z')(Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(2))), Derivative(Add(exp(sin(Symbol('r_0', commutative=True))), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(2))))"]]}, {"prompt": "Given E{(C_{1})} = e^{C_{1}}, then obtain - C_{1} e^{C_{1}} + \\frac{e^{E^{C_{1}}{(C_{1})}}}{E{(C_{1})}} = - C_{1} e^{C_{1}} + \\frac{e^{(e^{C_{1}})^{C_{1}}}}{E{(C_{1})}}", "derivation": "E{(C_{1})} = e^{C_{1}} and C_{1} E{(C_{1})} = C_{1} e^{C_{1}} and E^{C_{1}}{(C_{1})} = (e^{C_{1}})^{C_{1}} and e^{E^{C_{1}}{(C_{1})}} = e^{(e^{C_{1}})^{C_{1}}} and \\frac{e^{E^{C_{1}}{(C_{1})}}}{E{(C_{1})}} = \\frac{e^{(e^{C_{1}})^{C_{1}}}}{E{(C_{1})}} and - C_{1} E{(C_{1})} + \\frac{e^{E^{C_{1}}{(C_{1})}}}{E{(C_{1})}} = - C_{1} E{(C_{1})} + \\frac{e^{(e^{C_{1}})^{C_{1}}}}{E{(C_{1})}} and - C_{1} e^{C_{1}} + \\frac{e^{E^{C_{1}}{(C_{1})}}}{E{(C_{1})}} = - C_{1} e^{C_{1}} + \\frac{e^{(e^{C_{1}})^{C_{1}}}}{E{(C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["times", 1, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Function('E')(Symbol('C_1', commutative=True))), Mul(Symbol('C_1', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('E')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Function('E')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))), exp(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))))"], [["divide", 4, "Function('E')(Symbol('C_1', commutative=True))"], "Equality(Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(Function('E')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))), Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))))"], [["minus", 5, "Mul(Symbol('C_1', commutative=True), Function('E')(Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True), Function('E')(Symbol('C_1', commutative=True))), Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(Function('E')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True), Function('E')(Symbol('C_1', commutative=True))), Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True), exp(Symbol('C_1', commutative=True))), Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(Function('E')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True), exp(Symbol('C_1', commutative=True))), Mul(Pow(Function('E')(Symbol('C_1', commutative=True)), Integer(-1)), exp(Pow(exp(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given t{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and q{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} t{(J_{\\varepsilon})}, then obtain - J_{\\varepsilon} + q{(J_{\\varepsilon})} = - J_{\\varepsilon} + \\frac{d}{d J_{\\varepsilon}} t{(J_{\\varepsilon})}", "derivation": "t{(J_{\\varepsilon})} = \\sin{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} t{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and q{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} t{(J_{\\varepsilon})} and q{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and - J_{\\varepsilon} + q{(J_{\\varepsilon})} = - J_{\\varepsilon} + \\frac{d}{d J_{\\varepsilon}} \\sin{(J_{\\varepsilon})} and - J_{\\varepsilon} + q{(J_{\\varepsilon})} = - J_{\\varepsilon} + \\frac{d}{d J_{\\varepsilon}} t{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(sin(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given n{(\\mathbf{D})} = e^{\\mathbf{D}}, then derive \\int n{(\\mathbf{D})} d\\mathbf{D} = c_{0} + e^{\\mathbf{D}}, then obtain \\mathbf{D} + \\int n{(\\mathbf{D})} d\\mathbf{D} = \\mathbf{D} + c_{0} + e^{\\mathbf{D}}", "derivation": "n{(\\mathbf{D})} = e^{\\mathbf{D}} and \\int n{(\\mathbf{D})} d\\mathbf{D} = \\int e^{\\mathbf{D}} d\\mathbf{D} and \\mathbf{D} + \\int n{(\\mathbf{D})} d\\mathbf{D} = \\mathbf{D} + \\int e^{\\mathbf{D}} d\\mathbf{D} and \\int n{(\\mathbf{D})} d\\mathbf{D} = c_{0} + e^{\\mathbf{D}} and \\int e^{\\mathbf{D}} d\\mathbf{D} = c_{0} + e^{\\mathbf{D}} and \\mathbf{D} + \\int n{(\\mathbf{D})} d\\mathbf{D} = \\mathbf{D} + c_{0} + e^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Symbol('\\\\mathbf{D}', commutative=True), Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Integral(Function('n')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('c_0', commutative=True), exp(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})}, then obtain \\dot{\\mathbf{r}} - \\frac{\\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)}}{\\tilde{g}^*} = \\dot{\\mathbf{r}} - \\frac{\\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})}}{\\tilde{g}^*}", "derivation": "\\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)} = \\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})} and \\frac{\\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{\\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})}}{\\tilde{g}^*} and - \\frac{\\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)}}{\\tilde{g}^*} = - \\frac{\\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})}}{\\tilde{g}^*} and \\dot{\\mathbf{r}} - \\frac{\\operatorname{y^{\\prime}}{(\\dot{\\mathbf{r}},\\tilde{g}^*)}}{\\tilde{g}^*} = \\dot{\\mathbf{r}} - \\frac{\\sin{(\\dot{\\mathbf{r}}^{\\tilde{g}^*})}}{\\tilde{g}^*}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["divide", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["add", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(A)} = \\log{(A)} and h{(A)} = \\frac{d}{d A} \\log{(A)}, then obtain h{(A)} \\int 1 dA = \\frac{\\int 1 dA}{A}", "derivation": "\\operatorname{F_{N}}{(A)} = \\log{(A)} and \\frac{d}{d A} \\operatorname{F_{N}}{(A)} = \\frac{d}{d A} \\log{(A)} and h{(A)} = \\frac{d}{d A} \\log{(A)} and h{(A)} = \\frac{d}{d A} \\operatorname{F_{N}}{(A)} and h{(A)} \\int 1 dA = \\frac{d}{d A} \\operatorname{F_{N}}{(A)} \\int 1 dA and h{(A)} \\int 1 dA = \\frac{d}{d A} \\log{(A)} \\int 1 dA and h{(A)} \\int 1 dA = \\frac{\\int 1 dA}{A}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('h')(Symbol('A', commutative=True)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('h')(Symbol('A', commutative=True)), Derivative(Function('F_N')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 4, "Integral(Integer(1), Tuple(Symbol('A', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('A', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))), Mul(Derivative(Function('F_N')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('h')(Symbol('A', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))), Mul(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))))"], [["evaluate_derivatives", 6], "Equality(Mul(Function('h')(Symbol('A', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\hat{H}_l)} = e^{\\hat{H}_l} and T{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l}, then obtain - T{(\\hat{H}_l)} + \\varphi{(\\hat{H}_l)} = - T{(\\hat{H}_l)} + e^{\\hat{H}_l}", "derivation": "\\varphi{(\\hat{H}_l)} = e^{\\hat{H}_l} and \\varphi{(\\hat{H}_l)} - \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} = e^{\\hat{H}_l} - \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and T{(\\hat{H}_l)} = \\frac{d}{d \\hat{H}_l} e^{\\hat{H}_l} and - T{(\\hat{H}_l)} + \\varphi{(\\hat{H}_l)} = - T{(\\hat{H}_l)} + e^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\hat{H}_l', commutative=True)), Derivative(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('\\\\hat{H}_l', commutative=True))), Function('\\\\varphi')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\hat{H}_l', commutative=True))), exp(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given B{(M_{E})} = e^{M_{E}} and A{(M_{E})} = B{(M_{E})} + e^{M_{E}}, then obtain A{(M_{E})} B{(M_{E})} = 2 B^{2}{(M_{E})}", "derivation": "B{(M_{E})} = e^{M_{E}} and B{(M_{E})} + e^{M_{E}} = 2 e^{M_{E}} and A{(M_{E})} = B{(M_{E})} + e^{M_{E}} and A{(M_{E})} B{(M_{E})} = (B{(M_{E})} + e^{M_{E}}) B{(M_{E})} and A{(M_{E})} B{(M_{E})} = 2 B{(M_{E})} e^{M_{E}} and A{(M_{E})} B{(M_{E})} = 2 B^{2}{(M_{E})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["add", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Function('B')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Mul(Integer(2), exp(Symbol('M_E', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('M_E', commutative=True)), Add(Function('B')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))))"], [["times", 3, "Function('B')(Symbol('M_E', commutative=True))"], "Equality(Mul(Function('A')(Symbol('M_E', commutative=True)), Function('B')(Symbol('M_E', commutative=True))), Mul(Add(Function('B')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Function('B')(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('A')(Symbol('M_E', commutative=True)), Function('B')(Symbol('M_E', commutative=True))), Mul(Integer(2), Function('B')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('A')(Symbol('M_E', commutative=True)), Function('B')(Symbol('M_E', commutative=True))), Mul(Integer(2), Pow(Function('B')(Symbol('M_E', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{M}{(v,G)} = \\int (G + v) dG and \\mathbf{s}{(v,G)} = \\int (G + v) dG, then obtain \\int v \\mathbf{M}{(v,G)} dv = \\frac{G^{2} v^{2}}{4} + \\frac{G v^{3}}{3} + f_{E}", "derivation": "\\mathbf{M}{(v,G)} = \\int (G + v) dG and \\mathbf{s}{(v,G)} = \\int (G + v) dG and \\mathbf{M}{(v,G)} = \\mathbf{s}{(v,G)} and v \\mathbf{M}{(v,G)} = v \\mathbf{s}{(v,G)} and \\int v \\mathbf{M}{(v,G)} dv = \\int v \\mathbf{s}{(v,G)} dv and \\int v \\mathbf{M}{(v,G)} dv = \\int v \\int (G + v) dG dv and \\int v \\mathbf{M}{(v,G)} dv = \\frac{G^{2} v^{2}}{4} + \\frac{G v^{3}}{3} + f_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True)), Integral(Add(Symbol('G', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('v', commutative=True), Symbol('G', commutative=True)), Integral(Add(Symbol('G', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True)), Function('\\\\mathbf{s}')(Symbol('v', commutative=True), Symbol('G', commutative=True)))"], [["times", 3, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True))), Mul(Symbol('v', commutative=True), Function('\\\\mathbf{s}')(Symbol('v', commutative=True), Symbol('G', commutative=True))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{s}')(Symbol('v', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Integral(Add(Symbol('G', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{M}')(Symbol('v', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('v', commutative=True))), Add(Mul(Rational(1, 4), Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Rational(1, 3), Symbol('G', commutative=True), Pow(Symbol('v', commutative=True), Integer(3))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\sigma_x)} = \\cos{(\\sin{(\\sigma_x)})}, then obtain 1 = \\frac{\\iint \\cos{(\\sin{(\\sigma_x)})} d\\sigma_x d\\sigma_x}{\\iint \\hat{\\mathbf{x}}{(\\sigma_x)} d\\sigma_x d\\sigma_x}", "derivation": "\\hat{\\mathbf{x}}{(\\sigma_x)} = \\cos{(\\sin{(\\sigma_x)})} and \\int \\hat{\\mathbf{x}}{(\\sigma_x)} d\\sigma_x = \\int \\cos{(\\sin{(\\sigma_x)})} d\\sigma_x and \\iint \\hat{\\mathbf{x}}{(\\sigma_x)} d\\sigma_x d\\sigma_x = \\iint \\cos{(\\sin{(\\sigma_x)})} d\\sigma_x d\\sigma_x and 1 = \\frac{\\iint \\cos{(\\sin{(\\sigma_x)})} d\\sigma_x d\\sigma_x}{\\iint \\hat{\\mathbf{x}}{(\\sigma_x)} d\\sigma_x d\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_x', commutative=True)), cos(sin(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(cos(sin(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(cos(sin(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 3, "Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Integral(cos(sin(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\Psi,A_{z})} = A_{z}^{\\Psi} and \\operatorname{x^{{\\}'}}{(\\Psi,A_{z})} = - A_{z}^{\\Psi}, then obtain - \\mathbf{M}{(\\Psi,A_{z})} = \\operatorname{x^{{\\}'}}{(\\Psi,A_{z})}", "derivation": "\\mathbf{M}{(\\Psi,A_{z})} = A_{z}^{\\Psi} and - \\mathbf{M}{(\\Psi,A_{z})} = - A_{z}^{\\Psi} and \\operatorname{x^{{\\}'}}{(\\Psi,A_{z})} = - A_{z}^{\\Psi} and - \\mathbf{M}{(\\Psi,A_{z})} = \\operatorname{x^{{\\}'}}{(\\Psi,A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\Psi', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\Psi', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\Psi', commutative=True), Symbol('A_z', commutative=True))), Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain (e^{(2 \\hat{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}}})^{\\mathbf{r}} = (e^{(\\hat{\\mathbf{r}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})})^{\\mathbf{r}}})^{\\mathbf{r}}", "derivation": "\\hat{\\mathbf{r}}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and 2 \\hat{\\mathbf{r}}{(\\mathbf{r})} = \\hat{\\mathbf{r}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})} and (2 \\hat{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}} = (\\hat{\\mathbf{r}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})})^{\\mathbf{r}} and e^{(2 \\hat{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}}} = e^{(\\hat{\\mathbf{r}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})})^{\\mathbf{r}}} and (e^{(2 \\hat{\\mathbf{r}}{(\\mathbf{r})})^{\\mathbf{r}}})^{\\mathbf{r}} = (e^{(\\hat{\\mathbf{r}}{(\\mathbf{r})} + \\cos{(\\mathbf{r})})^{\\mathbf{r}}})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), exp(Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(exp(Pow(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(exp(Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(h,m)} = - h + \\sin{(m)}, then derive - h + (\\int \\operatorname{m_{s}}{(h,m)} dh)^{h} = - h + (\\tilde{g}^* - \\frac{h^{2}}{2} + h \\sin{(m)})^{h}, then obtain - h + (\\int (- h + \\sin{(m)}) dh)^{h} = - h + (\\tilde{g}^* - \\frac{h^{2}}{2} + h \\sin{(m)})^{h}", "derivation": "\\operatorname{m_{s}}{(h,m)} = - h + \\sin{(m)} and \\int \\operatorname{m_{s}}{(h,m)} dh = \\int (- h + \\sin{(m)}) dh and (\\int \\operatorname{m_{s}}{(h,m)} dh)^{h} = (\\int (- h + \\sin{(m)}) dh)^{h} and - h + (\\int \\operatorname{m_{s}}{(h,m)} dh)^{h} = - h + (\\int (- h + \\sin{(m)}) dh)^{h} and - h + (\\int \\operatorname{m_{s}}{(h,m)} dh)^{h} = - h + (\\tilde{g}^* - \\frac{h^{2}}{2} + h \\sin{(m)})^{h} and - h + (\\int (- h + \\sin{(m)}) dh)^{h} = - h + (\\tilde{g}^* - \\frac{h^{2}}{2} + h \\sin{(m)})^{h}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('h', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('m', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('m', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["minus", 3, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('m', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), sin(Symbol('m', commutative=True)))), Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), sin(Symbol('m', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), sin(Symbol('m', commutative=True)))), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then derive \\frac{d}{d \\mathbf{A}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}, then obtain \\int - (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\sin{(\\mathbf{A})} d\\mathbf{A} = \\int (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} d\\mathbf{A}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\operatorname{f_{\\mathbf{v}}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} and - \\sin{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and - (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\sin{(\\mathbf{A})} = (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\int - (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\sin{(\\mathbf{A})} d\\mathbf{A} = \\int (- \\sin{(\\mathbf{A})})^{\\mathbf{A}} \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["times", 4, "Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Pow(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{r},G)} = \\frac{\\partial}{\\partial G} G \\mathbf{r}, then derive \\operatorname{F_{H}}{(\\mathbf{r},G)} = \\mathbf{r}, then derive \\frac{\\partial}{\\partial G} \\operatorname{F_{H}}{(\\mathbf{r},G)} = 0, then obtain \\frac{d}{d G} \\mathbf{r} + \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r} = \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{r},G)} = \\frac{\\partial}{\\partial G} G \\mathbf{r} and \\operatorname{F_{H}}{(\\mathbf{r},G)} = \\mathbf{r} and \\frac{\\partial}{\\partial G} G \\mathbf{r} = \\mathbf{r} and \\operatorname{F_{H}}{(\\frac{\\partial}{\\partial G} G \\mathbf{r},G)} = \\frac{\\partial}{\\partial G} G \\mathbf{r} and \\frac{\\partial}{\\partial G} \\operatorname{F_{H}}{(\\frac{\\partial}{\\partial G} G \\mathbf{r},G)} = \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r} and \\frac{\\partial}{\\partial G} \\operatorname{F_{H}}{(\\mathbf{r},G)} = 0 and \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r} + \\frac{\\partial}{\\partial G} \\operatorname{F_{H}}{(\\mathbf{r},G)} = \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r} and \\frac{d}{d G} \\mathbf{r} + \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r} = \\frac{\\partial^{2}}{\\partial G^{2}} G \\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('G', commutative=True)), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_H')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('G', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('\\\\mathbf{r}', commutative=True))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('F_H')(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('F_H')(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('F_H')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(0))"], [["add", 6, "Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Function('F_H')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Add(Derivative(Symbol('\\\\mathbf{r}', commutative=True), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))), Derivative(Mul(Symbol('G', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(C_{2},n,\\varepsilon)} = C_{2} (- \\varepsilon + n) and \\dot{y}{(\\varepsilon)} = - \\varepsilon, then obtain \\varepsilon + \\operatorname{v_{x}}{(C_{2},n,\\varepsilon)} = C_{2} (n + \\dot{y}{(\\varepsilon)}) + \\varepsilon", "derivation": "\\operatorname{v_{x}}{(C_{2},n,\\varepsilon)} = C_{2} (- \\varepsilon + n) and \\varepsilon + \\operatorname{v_{x}}{(C_{2},n,\\varepsilon)} = C_{2} (- \\varepsilon + n) + \\varepsilon and \\dot{y}{(\\varepsilon)} = - \\varepsilon and \\varepsilon + \\operatorname{v_{x}}{(C_{2},n,\\varepsilon)} = C_{2} (n + \\dot{y}{(\\varepsilon)}) + \\varepsilon", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('C_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('C_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True))))"], [["add", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('v_x')(Symbol('C_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Symbol('n', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('v_x')(Symbol('C_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Add(Symbol('n', commutative=True), Function('\\\\dot{y}')(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given L{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then derive \\int L{(L_{\\varepsilon})} dL_{\\varepsilon} = p + e^{L_{\\varepsilon}}, then obtain (\\int L{(L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} = (\\int e^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}}", "derivation": "L{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\int L{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int e^{L_{\\varepsilon}} dL_{\\varepsilon} and \\int L{(L_{\\varepsilon})} dL_{\\varepsilon} = p + e^{L_{\\varepsilon}} and (\\int L{(L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} = (p + e^{L_{\\varepsilon}})^{L_{\\varepsilon}} and (\\int e^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}} = (p + e^{L_{\\varepsilon}})^{L_{\\varepsilon}} and (\\int L{(L_{\\varepsilon})} dL_{\\varepsilon})^{L_{\\varepsilon}} = (\\int e^{L_{\\varepsilon}} dL_{\\varepsilon})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('L')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('p', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Function('L')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('p', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('p', commutative=True), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(Function('L')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integral(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\chi)} = \\cos{(\\chi)}, then obtain \\frac{d}{d \\chi} (- \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\operatorname{A_{z}}^{\\chi}{(\\chi)} - 1) = \\frac{d}{d \\chi} (- \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\cos^{\\chi}{(\\chi)} - 1)", "derivation": "\\operatorname{A_{z}}{(\\chi)} = \\cos{(\\chi)} and \\operatorname{A_{z}}^{\\chi}{(\\chi)} = \\cos^{\\chi}{(\\chi)} and - \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\operatorname{A_{z}}^{\\chi}{(\\chi)} - 1 = - \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\cos^{\\chi}{(\\chi)} - 1 and \\frac{d}{d \\chi} (- \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\operatorname{A_{z}}^{\\chi}{(\\chi)} - 1) = \\frac{d}{d \\chi} (- \\operatorname{A_{z}}{(\\chi)} \\cos{(\\chi)} + \\cos^{\\chi}{(\\chi)} - 1)", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Pow(Function('A_z')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Pow(Function('A_z')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(t_{2},k)} = \\frac{t_{2}}{k}, then obtain \\int (t_{2} + v{(t_{2},k)}) dk = \\int (t_{2} + \\frac{t_{2}}{k}) dk", "derivation": "v{(t_{2},k)} = \\frac{t_{2}}{k} and k v{(t_{2},k)} = t_{2} and k v{(t_{2},k)} + v{(t_{2},k)} = k v{(t_{2},k)} + \\frac{t_{2}}{k} and t_{2} + v{(t_{2},k)} = t_{2} + \\frac{t_{2}}{k} and \\int (t_{2} + v{(t_{2},k)}) dk = \\int (t_{2} + \\frac{t_{2}}{k}) dk", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["divide", 1, "Pow(Symbol('k', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('k', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Symbol('t_2', commutative=True))"], [["add", 1, "Mul(Symbol('k', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True)))"], "Equality(Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Add(Mul(Symbol('k', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('t_2', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Add(Symbol('t_2', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["integrate", 4, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Symbol('t_2', commutative=True), Function('v')(Symbol('t_2', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Symbol('t_2', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(T)} = \\frac{d}{d T} \\cos{(T)}, then derive \\operatorname{F_{c}}{(T)} = - \\sin{(T)}, then obtain \\frac{\\frac{d}{d T} \\cos{(T)}}{\\sin{(T)}} = -1", "derivation": "\\operatorname{F_{c}}{(T)} = \\frac{d}{d T} \\cos{(T)} and \\operatorname{F_{c}}{(T)} = - \\sin{(T)} and \\frac{\\operatorname{F_{c}}{(T)}}{\\sin{(T)}} = -1 and \\frac{\\frac{d}{d T} \\cos{(T)}}{\\sin{(T)}} = -1", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_c')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True))))"], [["divide", 2, "sin(Symbol('T', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('T', commutative=True)), Pow(sin(Symbol('T', commutative=True)), Integer(-1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(sin(Symbol('T', commutative=True)), Integer(-1)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Integer(-1))"]]}, {"prompt": "Given \\delta{(\\hat{x}_0,\\phi)} = - \\hat{x}_0 + \\phi, then obtain \\int (- \\frac{\\delta{(\\hat{x}_0,\\phi)}}{\\hat{x}_0})^{\\hat{x}_0} d\\phi = \\int (- \\frac{- \\hat{x}_0 + \\phi}{\\hat{x}_0})^{\\hat{x}_0} d\\phi", "derivation": "\\delta{(\\hat{x}_0,\\phi)} = - \\hat{x}_0 + \\phi and - \\frac{\\delta{(\\hat{x}_0,\\phi)}}{\\hat{x}_0} = - \\frac{- \\hat{x}_0 + \\phi}{\\hat{x}_0} and (- \\frac{\\delta{(\\hat{x}_0,\\phi)}}{\\hat{x}_0})^{\\hat{x}_0} = (- \\frac{- \\hat{x}_0 + \\phi}{\\hat{x}_0})^{\\hat{x}_0} and \\int (- \\frac{\\delta{(\\hat{x}_0,\\phi)}}{\\hat{x}_0})^{\\hat{x}_0} d\\phi = \\int (- \\frac{- \\hat{x}_0 + \\phi}{\\hat{x}_0})^{\\hat{x}_0} d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\eta{(H)} = \\log{(H)}, then derive \\frac{d}{d H} \\eta{(H)} = \\frac{1}{H}, then obtain \\frac{d}{d H} \\log{(H)} = \\frac{1}{H}", "derivation": "\\eta{(H)} = \\log{(H)} and \\frac{d}{d H} \\eta{(H)} = \\frac{d}{d H} \\log{(H)} and \\frac{d}{d H} \\eta{(H)} = \\frac{1}{H} and \\frac{d}{d H} \\log{(H)} = \\frac{1}{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Symbol('H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Pow(Symbol('H', commutative=True), Integer(-1)))"]]}, {"prompt": "Given g{(I,\\hat{\\mathbf{x}})} = \\cos{(I \\hat{\\mathbf{x}})}, then derive \\frac{\\partial}{\\partial I} g{(I,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} \\sin{(I \\hat{\\mathbf{x}})}, then obtain \\frac{\\frac{\\partial}{\\partial I} \\cos{(I \\hat{\\mathbf{x}})}}{I \\hat{\\mathbf{x}}} = - \\frac{\\sin{(I \\hat{\\mathbf{x}})}}{I}", "derivation": "g{(I,\\hat{\\mathbf{x}})} = \\cos{(I \\hat{\\mathbf{x}})} and \\frac{\\partial}{\\partial I} g{(I,\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial I} \\cos{(I \\hat{\\mathbf{x}})} and \\frac{\\partial}{\\partial I} g{(I,\\hat{\\mathbf{x}})} = - \\hat{\\mathbf{x}} \\sin{(I \\hat{\\mathbf{x}})} and \\frac{\\frac{\\partial}{\\partial I} g{(I,\\hat{\\mathbf{x}})}}{I \\hat{\\mathbf{x}}} = - \\frac{\\sin{(I \\hat{\\mathbf{x}})}}{I} and \\frac{\\frac{\\partial}{\\partial I} \\cos{(I \\hat{\\mathbf{x}})}}{I \\hat{\\mathbf{x}}} = - \\frac{\\sin{(I \\hat{\\mathbf{x}})}}{I}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["divide", 3, "Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Derivative(Function('g')(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(-1)), Derivative(cos(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(M,A)} = \\log{(\\frac{A}{M})} and \\operatorname{V_{\\mathbf{E}}}{(M,A)} = \\frac{A}{M}, then obtain (\\frac{\\partial}{\\partial A} - \\log{(\\frac{A}{M})})^{M} = (\\frac{\\partial}{\\partial A} - \\log{(\\operatorname{V_{\\mathbf{E}}}{(M,A)})})^{M}", "derivation": "\\operatorname{C_{2}}{(M,A)} = \\log{(\\frac{A}{M})} and - \\operatorname{C_{2}}{(M,A)} = - \\log{(\\frac{A}{M})} and \\operatorname{V_{\\mathbf{E}}}{(M,A)} = \\frac{A}{M} and - \\operatorname{C_{2}}{(M,A)} = - \\log{(\\operatorname{V_{\\mathbf{E}}}{(M,A)})} and \\frac{\\partial}{\\partial A} - \\operatorname{C_{2}}{(M,A)} = \\frac{\\partial}{\\partial A} - \\log{(\\operatorname{V_{\\mathbf{E}}}{(M,A)})} and \\frac{\\partial}{\\partial A} - \\log{(\\frac{A}{M})} = \\frac{\\partial}{\\partial A} - \\log{(\\operatorname{V_{\\mathbf{E}}}{(M,A)})} and (\\frac{\\partial}{\\partial A} - \\log{(\\frac{A}{M})})^{M} = (\\frac{\\partial}{\\partial A} - \\log{(\\operatorname{V_{\\mathbf{E}}}{(M,A)})})^{M}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('M', commutative=True), Symbol('A', commutative=True)), log(Mul(Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('C_2')(Symbol('M', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), log(Mul(Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Function('C_2')(Symbol('M', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), log(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True), Symbol('A', commutative=True)))))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('C_2')(Symbol('M', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Mul(Integer(-1), log(Mul(Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 6, "Symbol('M', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(-1), log(Mul(Symbol('A', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Integer(-1), log(Function('V_{\\\\mathbf{E}}')(Symbol('M', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('M', commutative=True)))"]]}, {"prompt": "Given W{(\\psi^*)} = \\sin{(\\cos{(\\psi^*)})}, then derive \\frac{d}{d \\psi^*} W{(\\psi^*)} = - \\sin{(\\psi^*)} \\cos{(\\cos{(\\psi^*)})}, then obtain - \\sin{(\\psi^*)} \\cos{(\\cos{(\\psi^*)})} = \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})}", "derivation": "W{(\\psi^*)} = \\sin{(\\cos{(\\psi^*)})} and \\frac{d}{d \\psi^*} W{(\\psi^*)} = \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})} and \\frac{d}{d \\psi^*} W{(\\psi^*)} = - \\sin{(\\psi^*)} \\cos{(\\cos{(\\psi^*)})} and - \\sin{(\\psi^*)} \\cos{(\\cos{(\\psi^*)})} = \\frac{d}{d \\psi^*} \\sin{(\\cos{(\\psi^*)})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\psi^*', commutative=True)), sin(cos(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('W')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)), cos(cos(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)), cos(cos(Symbol('\\\\psi^*', commutative=True)))), Derivative(sin(cos(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{1}{(F_{g},E_{x})} = - F_{g} + e^{E_{x}}, then obtain \\sin^{F_{g}}{(\\theta_{1}^{E_{x}}{(F_{g},E_{x})})} = \\sin^{F_{g}}{((- F_{g} + e^{E_{x}})^{E_{x}})}", "derivation": "\\theta_{1}{(F_{g},E_{x})} = - F_{g} + e^{E_{x}} and \\theta_{1}^{E_{x}}{(F_{g},E_{x})} = (- F_{g} + e^{E_{x}})^{E_{x}} and \\sin{(\\theta_{1}^{E_{x}}{(F_{g},E_{x})})} = \\sin{((- F_{g} + e^{E_{x}})^{E_{x}})} and \\sin^{F_{g}}{(\\theta_{1}^{E_{x}}{(F_{g},E_{x})})} = \\sin^{F_{g}}{((- F_{g} + e^{E_{x}})^{E_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('E_x', commutative=True))))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\theta_1')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), sin(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(sin(Pow(Function('\\\\theta_1')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Symbol('F_g', commutative=True)), Pow(sin(Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), exp(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{J},\\mathbf{J}_M)} = \\mathbf{J} - \\mathbf{J}_M, then obtain 4 \\operatorname{C_{2}}^{2}{(\\mathbf{J},\\mathbf{J}_M)} = (2 \\mathbf{J} - 2 \\mathbf{J}_M)^{2}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{J},\\mathbf{J}_M)} = \\mathbf{J} - \\mathbf{J}_M and 2 \\operatorname{C_{2}}{(\\mathbf{J},\\mathbf{J}_M)} = \\mathbf{J} - \\mathbf{J}_M + \\operatorname{C_{2}}{(\\mathbf{J},\\mathbf{J}_M)} and 4 \\operatorname{C_{2}}^{2}{(\\mathbf{J},\\mathbf{J}_M)} = (\\mathbf{J} - \\mathbf{J}_M + \\operatorname{C_{2}}{(\\mathbf{J},\\mathbf{J}_M)})^{2} and 4 (\\mathbf{J} - \\mathbf{J}_M)^{2} = (2 \\mathbf{J} - 2 \\mathbf{J}_M)^{2} and 4 \\operatorname{C_{2}}^{2}{(\\mathbf{J},\\mathbf{J}_M)} = (2 \\mathbf{J} - 2 \\mathbf{J}_M)^{2}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('C_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\nabla{(f^{*},\\mu_0)} = \\mu_0 + \\cos{(f^{*})}, then derive 0 = 1 - (\\frac{\\partial}{\\partial \\mu_0} \\nabla{(f^{*},\\mu_0)})^{f^{*}}, then obtain \\frac{d}{d \\mu_0} 0 = \\frac{\\partial}{\\partial \\mu_0} (1 - (\\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}))^{f^{*}})", "derivation": "\\nabla{(f^{*},\\mu_0)} = \\mu_0 + \\cos{(f^{*})} and \\frac{\\partial}{\\partial \\mu_0} \\nabla{(f^{*},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}) and (\\frac{\\partial}{\\partial \\mu_0} \\nabla{(f^{*},\\mu_0)})^{f^{*}} = (\\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}))^{f^{*}} and 0 = (\\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}))^{f^{*}} - (\\frac{\\partial}{\\partial \\mu_0} \\nabla{(f^{*},\\mu_0)})^{f^{*}} and 0 = 1 - (\\frac{\\partial}{\\partial \\mu_0} \\nabla{(f^{*},\\mu_0)})^{f^{*}} and 0 = 1 - (\\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}))^{f^{*}} and \\frac{d}{d \\mu_0} 0 = \\frac{\\partial}{\\partial \\mu_0} (1 - (\\frac{\\partial}{\\partial \\mu_0} (\\mu_0 + \\cos{(f^{*})}))^{f^{*}})", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))"], [["minus", 3, "Pow(Derivative(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Pow(Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Mul(Integer(-1), Pow(Derivative(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Derivative(Function('\\\\nabla')(Symbol('f^*', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Pow(Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(\\dot{z},\\hat{p}_0)} = \\dot{z}^{\\hat{p}_0}, then obtain \\frac{\\partial}{\\partial \\hat{p}_0} (- \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} + \\int \\hat{p}{(\\dot{z},\\hat{p}_0)} d\\dot{z}) = \\frac{d}{d \\hat{p}_0} 0", "derivation": "\\hat{p}{(\\dot{z},\\hat{p}_0)} = \\dot{z}^{\\hat{p}_0} and \\int \\hat{p}{(\\dot{z},\\hat{p}_0)} d\\dot{z} = \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} and \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} + \\int \\hat{p}{(\\dot{z},\\hat{p}_0)} d\\dot{z} = 2 \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} and - \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} + \\int \\hat{p}{(\\dot{z},\\hat{p}_0)} d\\dot{z} = 0 and \\frac{\\partial}{\\partial \\hat{p}_0} (- \\int \\dot{z}^{\\hat{p}_0} d\\dot{z} + \\int \\hat{p}{(\\dot{z},\\hat{p}_0)} d\\dot{z}) = \\frac{d}{d \\hat{p}_0} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 2, "Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(2), Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["minus", 3, "Mul(Integer(2), Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Integral(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integral(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Integral(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{2})} = \\log{(\\sin{(v_{2})})}, then derive \\sin{(v_{2})} \\frac{d}{d v_{2}} \\operatorname{t_{2}}{(v_{2})} = \\cos{(v_{2})}, then obtain (\\sin{(v_{2})} \\frac{d}{d v_{2}} \\log{(\\sin{(v_{2})})})^{v_{2}} = \\cos^{v_{2}}{(v_{2})}", "derivation": "\\operatorname{t_{2}}{(v_{2})} = \\log{(\\sin{(v_{2})})} and \\frac{d}{d v_{2}} \\operatorname{t_{2}}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(\\sin{(v_{2})})} and \\sin{(v_{2})} \\frac{d}{d v_{2}} \\operatorname{t_{2}}{(v_{2})} = \\sin{(v_{2})} \\frac{d}{d v_{2}} \\log{(\\sin{(v_{2})})} and \\sin{(v_{2})} \\frac{d}{d v_{2}} \\operatorname{t_{2}}{(v_{2})} = \\cos{(v_{2})} and \\sin{(v_{2})} \\frac{d}{d v_{2}} \\log{(\\sin{(v_{2})})} = \\cos{(v_{2})} and (\\sin{(v_{2})} \\frac{d}{d v_{2}} \\log{(\\sin{(v_{2})})})^{v_{2}} = \\cos^{v_{2}}{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_2', commutative=True)), log(sin(Symbol('v_2', commutative=True))))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(log(sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["times", 2, "sin(Symbol('v_2', commutative=True))"], "Equality(Mul(sin(Symbol('v_2', commutative=True)), Derivative(Function('t_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(sin(Symbol('v_2', commutative=True)), Derivative(log(sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(sin(Symbol('v_2', commutative=True)), Derivative(Function('t_2')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), cos(Symbol('v_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(sin(Symbol('v_2', commutative=True)), Derivative(log(sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), cos(Symbol('v_2', commutative=True)))"], [["power", 5, "Symbol('v_2', commutative=True)"], "Equality(Pow(Mul(sin(Symbol('v_2', commutative=True)), Derivative(log(sin(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Symbol('v_2', commutative=True)), Pow(cos(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain \\mathbf{J} \\operatorname{C_{d}}^{2}{(\\mathbf{J})} = \\mathbf{J} e^{2 \\mathbf{J}}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\mathbf{J} \\operatorname{C_{d}}{(\\mathbf{J})} = \\mathbf{J} e^{\\mathbf{J}} and \\mathbf{J} \\operatorname{C_{d}}^{2}{(\\mathbf{J})} = \\mathbf{J} \\operatorname{C_{d}}{(\\mathbf{J})} e^{\\mathbf{J}} and \\mathbf{J} \\operatorname{C_{d}}{(\\mathbf{J})} e^{\\mathbf{J}} = \\mathbf{J} e^{2 \\mathbf{J}} and \\mathbf{J} \\operatorname{C_{d}}^{2}{(\\mathbf{J})} = \\mathbf{J} e^{2 \\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 2, "Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{J}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{J}', commutative=True), exp(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given Q{(\\delta,l)} = \\delta + l and \\operatorname{E_{\\lambda}}{(\\delta,l)} = \\frac{Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta} and q{(\\delta,l)} = (\\frac{Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta})^{l}, then obtain q{(\\delta,l)} = (\\frac{(\\delta + l) \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta})^{l}", "derivation": "Q{(\\delta,l)} = \\delta + l and Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)} = (\\delta + l) \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)} and \\frac{Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta} = \\frac{(\\delta + l) \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta} and \\operatorname{E_{\\lambda}}{(\\delta,l)} = \\frac{Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta} and \\operatorname{E_{\\lambda}}{(\\delta,l)} = \\frac{(\\delta + l) \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta} and q{(\\delta,l)} = (\\frac{Q{(\\delta,l)} \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta})^{l} and q{(\\delta,l)} = \\operatorname{E_{\\lambda}}^{l}{(\\delta,l)} and q{(\\delta,l)} = (\\frac{(\\delta + l) \\frac{\\partial}{\\partial \\delta} Q{(\\delta,l)}}{\\delta})^{l}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Mul(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["divide", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Function('q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Function('q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Derivative(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\mathbf{E},\\mu)} = \\mathbf{E} + \\mu, then derive \\int \\rho{(\\mathbf{E},\\mu)} d\\mathbf{E} = J_{\\varepsilon} + \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{E} \\mu, then obtain (\\int (\\mathbf{E} + \\mu) d\\mathbf{E})^{3} = (J_{\\varepsilon} + \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{E} \\mu) (\\int (\\mathbf{E} + \\mu) d\\mathbf{E})^{2}", "derivation": "\\rho{(\\mathbf{E},\\mu)} = \\mathbf{E} + \\mu and \\int \\rho{(\\mathbf{E},\\mu)} d\\mathbf{E} = \\int (\\mathbf{E} + \\mu) d\\mathbf{E} and \\int \\rho{(\\mathbf{E},\\mu)} d\\mathbf{E} = J_{\\varepsilon} + \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{E} \\mu and \\int (\\mathbf{E} + \\mu) d\\mathbf{E} = J_{\\varepsilon} + \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{E} \\mu and (\\int (\\mathbf{E} + \\mu) d\\mathbf{E})^{3} = (J_{\\varepsilon} + \\frac{\\mathbf{E}^{2}}{2} + \\mathbf{E} \\mu) (\\int (\\mathbf{E} + \\mu) d\\mathbf{E})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["times", 4, "Pow(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integer(2))"], "Equality(Pow(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integer(3)), Mul(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\hat{x}{(n)} = \\cos{(n)}, then derive \\frac{d}{d n} \\hat{x}{(n)} = - \\sin{(n)}, then obtain n + \\frac{d}{d n} \\cos{(n)} = n - \\sin{(n)}", "derivation": "\\hat{x}{(n)} = \\cos{(n)} and \\frac{d}{d n} \\hat{x}{(n)} = \\frac{d}{d n} \\cos{(n)} and \\frac{d}{d n} \\hat{x}{(n)} = - \\sin{(n)} and n + \\frac{d}{d n} \\hat{x}{(n)} = n + \\frac{d}{d n} \\cos{(n)} and \\frac{d}{d n} \\cos{(n)} = - \\sin{(n)} and n + \\frac{d}{d n} \\hat{x}{(n)} = n - \\sin{(n)} and n + \\frac{d}{d n} \\cos{(n)} = n - \\sin{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n', commutative=True))))"], [["add", 2, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Derivative(Function('\\\\hat{x}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Symbol('n', commutative=True), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('n', commutative=True), Derivative(Function('\\\\hat{x}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Symbol('n', commutative=True), Mul(Integer(-1), sin(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Symbol('n', commutative=True), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Symbol('n', commutative=True), Mul(Integer(-1), sin(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(T,E_{\\lambda})} = E_{\\lambda} + T and \\theta_{2}{(T,E_{\\lambda})} = E_{\\lambda} + T, then obtain \\frac{\\hat{x}^{4}{(T,E_{\\lambda})}}{\\theta_{2}^{2}{(T,E_{\\lambda})}} = (E_{\\lambda} + T)^{2}", "derivation": "\\hat{x}{(T,E_{\\lambda})} = E_{\\lambda} + T and \\hat{x}^{2}{(T,E_{\\lambda})} = (E_{\\lambda} + T) \\hat{x}{(T,E_{\\lambda})} and \\frac{\\hat{x}^{2}{(T,E_{\\lambda})}}{E_{\\lambda} + T} = \\hat{x}{(T,E_{\\lambda})} and \\theta_{2}{(T,E_{\\lambda})} = E_{\\lambda} + T and \\frac{\\hat{x}^{2}{(T,E_{\\lambda})}}{\\theta_{2}{(T,E_{\\lambda})}} = \\hat{x}{(T,E_{\\lambda})} and \\frac{\\hat{x}^{2}{(T,E_{\\lambda})}}{\\theta_{2}{(T,E_{\\lambda})}} = E_{\\lambda} + T and \\frac{\\hat{x}^{4}{(T,E_{\\lambda})}}{\\theta_{2}^{2}{(T,E_{\\lambda})}} = (E_{\\lambda} + T)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)))"], [["times", 1, "Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)), Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 2, "Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2))), Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Pow(Function('\\\\theta_2')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(2)), Pow(Function('\\\\theta_2')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)))"], [["power", 6, 2], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(4)), Pow(Function('\\\\theta_2')(Symbol('T', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-2))), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('T', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{E}{(C_{2},\\hat{x})} = - C_{2} + \\hat{x}, then derive \\frac{\\partial}{\\partial C_{2}} \\mathbf{E}{(C_{2},\\hat{x})} = -1, then obtain \\frac{\\partial^{- \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})}}{\\partial C_{2}^{- \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})}} \\mathbf{E}{(C_{2},\\hat{x})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})", "derivation": "\\mathbf{E}{(C_{2},\\hat{x})} = - C_{2} + \\hat{x} and \\frac{\\partial}{\\partial C_{2}} \\mathbf{E}{(C_{2},\\hat{x})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x}) and \\frac{\\partial}{\\partial C_{2}} \\mathbf{E}{(C_{2},\\hat{x})} = -1 and -1 = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x}) and \\frac{\\partial^{- \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})}}{\\partial C_{2}^{- \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})}} \\mathbf{E}{(C_{2},\\hat{x})} = \\frac{\\partial}{\\partial C_{2}} (- C_{2} + \\hat{x})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('C_2', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(E)} = \\frac{d}{d E} e^{E} and \\hat{H}_l{(E)} = \\frac{d}{d E} e^{E}, then obtain \\hat{H}_l^{E}{(E)} - (e^{E})^{E} = - (e^{E})^{E} + (\\frac{d}{d E} e^{E})^{E}", "derivation": "\\mathbf{r}{(E)} = \\frac{d}{d E} e^{E} and \\hat{H}_l{(E)} = \\frac{d}{d E} e^{E} and \\mathbf{r}^{E}{(E)} = (\\frac{d}{d E} e^{E})^{E} and \\mathbf{r}^{E}{(E)} - (e^{E})^{E} = - (e^{E})^{E} + (\\frac{d}{d E} e^{E})^{E} and \\mathbf{r}^{E}{(E)} = \\hat{H}_l^{E}{(E)} and \\hat{H}_l^{E}{(E)} - (e^{E})^{E} = - (e^{E})^{E} + (\\frac{d}{d E} e^{E})^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["minus", 3, "Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Pow(Function('\\\\hat{H}_l')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True))), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"]]}, {"prompt": "Given z{(\\Omega)} = \\Omega, then obtain \\frac{d}{d \\Omega} 0 = \\frac{d}{d \\Omega} ((\\Omega^{\\Omega})^{\\Omega} - (z^{\\Omega}{(\\Omega)})^{\\Omega})", "derivation": "z{(\\Omega)} = \\Omega and z^{\\Omega}{(\\Omega)} = \\Omega^{\\Omega} and (z^{\\Omega}{(\\Omega)})^{\\Omega} = (\\Omega^{\\Omega})^{\\Omega} and 0 = (\\Omega^{\\Omega})^{\\Omega} - (z^{\\Omega}{(\\Omega)})^{\\Omega} and \\frac{d}{d \\Omega} 0 = \\frac{d}{d \\Omega} ((\\Omega^{\\Omega})^{\\Omega} - (z^{\\Omega}{(\\Omega)})^{\\Omega})", "srepr_derivation": [["renaming_premise", "Equality(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Pow(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["minus", 3, "Pow(Pow(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(0), Add(Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(Pow(Function('z')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(\\rho)} = \\rho, then derive \\frac{d}{d \\rho} \\hat{x}{(\\rho)} = 1, then obtain \\cos{(\\int \\frac{d}{d \\hat{x}{(\\rho)}} \\hat{x}{(\\rho)} d\\rho)} = \\cos{(\\int 1 d\\rho)}", "derivation": "\\hat{x}{(\\rho)} = \\rho and \\frac{d}{d \\rho} \\hat{x}{(\\rho)} = \\frac{d}{d \\rho} \\rho and \\frac{d}{d \\rho} \\hat{x}{(\\rho)} = 1 and \\frac{d}{d \\rho} \\rho = 1 and \\frac{d}{d \\hat{x}{(\\rho)}} \\hat{x}{(\\rho)} = 1 and \\int \\frac{d}{d \\hat{x}{(\\rho)}} \\hat{x}{(\\rho)} d\\rho = \\int 1 d\\rho and \\cos{(\\int \\frac{d}{d \\hat{x}{(\\rho)}} \\hat{x}{(\\rho)} d\\rho)} = \\cos{(\\int 1 d\\rho)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Symbol('\\\\rho', commutative=True), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Integer(1))), Integer(1))"], [["integrate", 5, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\rho', commutative=True))))"], [["cos", 6], "Equality(cos(Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Tuple(Function('\\\\hat{x}')(Symbol('\\\\rho', commutative=True)), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True)))), cos(Integral(Integer(1), Tuple(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given M{(z^{*},\\varepsilon)} = \\varepsilon + \\cos{(z^{*})}, then obtain \\iint \\frac{\\partial}{\\partial \\varepsilon} (M{(z^{*},\\varepsilon)} - \\cos{(z^{*})}) d\\varepsilon d\\varepsilon = \\iint \\frac{d}{d \\varepsilon} \\varepsilon d\\varepsilon d\\varepsilon", "derivation": "M{(z^{*},\\varepsilon)} = \\varepsilon + \\cos{(z^{*})} and M{(z^{*},\\varepsilon)} - \\cos{(z^{*})} = \\varepsilon and \\frac{\\partial}{\\partial \\varepsilon} (M{(z^{*},\\varepsilon)} - \\cos{(z^{*})}) = \\frac{d}{d \\varepsilon} \\varepsilon and \\int \\frac{\\partial}{\\partial \\varepsilon} (M{(z^{*},\\varepsilon)} - \\cos{(z^{*})}) d\\varepsilon = \\int \\frac{d}{d \\varepsilon} \\varepsilon d\\varepsilon and \\iint \\frac{\\partial}{\\partial \\varepsilon} (M{(z^{*},\\varepsilon)} - \\cos{(z^{*})}) d\\varepsilon d\\varepsilon = \\iint \\frac{d}{d \\varepsilon} \\varepsilon d\\varepsilon d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\varepsilon', commutative=True), cos(Symbol('z^*', commutative=True))))"], [["minus", 1, "cos(Symbol('z^*', commutative=True))"], "Equality(Add(Function('M')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Symbol('\\\\varepsilon', commutative=True))"], [["differentiate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Add(Function('M')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Add(Function('M')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Add(Function('M')(Symbol('z^*', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Symbol('\\\\varepsilon', commutative=True), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(E,\\Psi_{\\lambda})} = e^{E - \\Psi_{\\lambda}}, then derive \\int \\mathbf{A}{(E,\\Psi_{\\lambda})} dE = \\theta_1 + e^{E - \\Psi_{\\lambda}}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\theta_1 + \\mathbf{A}{(E,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\int e^{E - \\Psi_{\\lambda}} dE", "derivation": "\\mathbf{A}{(E,\\Psi_{\\lambda})} = e^{E - \\Psi_{\\lambda}} and \\int \\mathbf{A}{(E,\\Psi_{\\lambda})} dE = \\int e^{E - \\Psi_{\\lambda}} dE and \\int \\mathbf{A}{(E,\\Psi_{\\lambda})} dE = \\theta_1 + e^{E - \\Psi_{\\lambda}} and \\theta_1 + e^{E - \\Psi_{\\lambda}} = \\int e^{E - \\Psi_{\\lambda}} dE and \\theta_1 + \\mathbf{A}{(E,\\Psi_{\\lambda})} = \\int e^{E - \\Psi_{\\lambda}} dE and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\theta_1 + \\mathbf{A}{(E,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\int e^{E - \\Psi_{\\lambda}} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\theta_1', commutative=True), exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Integral(exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{A}')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{A}')(Symbol('E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integral(exp(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{c},\\hbar)} = F_{c} + \\hbar, then derive \\frac{\\partial}{\\partial \\hbar} \\eta^{\\prime}{(F_{c},\\hbar)} = 1, then obtain \\frac{\\partial}{\\partial \\hbar} (F_{c} + \\hbar) = 1", "derivation": "\\eta^{\\prime}{(F_{c},\\hbar)} = F_{c} + \\hbar and \\frac{\\partial}{\\partial \\hbar} \\eta^{\\prime}{(F_{c},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (F_{c} + \\hbar) and \\frac{\\partial}{\\partial \\hbar} \\eta^{\\prime}{(F_{c},\\hbar)} = 1 and \\frac{\\partial}{\\partial \\hbar} (F_{c} + \\hbar) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{s}{(F_{g},u)} = \\frac{\\cos{(u)}}{F_{g}}, then obtain \\frac{u \\mathbf{s}^{u}{(F_{g},u)} \\frac{\\partial}{\\partial F_{g}} \\mathbf{s}{(F_{g},u)}}{\\mathbf{s}{(F_{g},u)}} - \\frac{u (\\frac{\\cos{(u)}}{F_{g}})^{u}}{F_{g}} = - \\frac{2 u (\\frac{\\cos{(u)}}{F_{g}})^{u}}{F_{g}}", "derivation": "\\mathbf{s}{(F_{g},u)} = \\frac{\\cos{(u)}}{F_{g}} and \\mathbf{s}^{u}{(F_{g},u)} = (\\frac{\\cos{(u)}}{F_{g}})^{u} and (\\frac{\\cos{(u)}}{F_{g}})^{u} + \\mathbf{s}^{u}{(F_{g},u)} = 2 (\\frac{\\cos{(u)}}{F_{g}})^{u} and \\frac{\\partial}{\\partial F_{g}} ((\\frac{\\cos{(u)}}{F_{g}})^{u} + \\mathbf{s}^{u}{(F_{g},u)}) = \\frac{\\partial}{\\partial F_{g}} 2 (\\frac{\\cos{(u)}}{F_{g}})^{u} and \\frac{u \\mathbf{s}^{u}{(F_{g},u)} \\frac{\\partial}{\\partial F_{g}} \\mathbf{s}{(F_{g},u)}}{\\mathbf{s}{(F_{g},u)}} - \\frac{u (\\frac{\\cos{(u)}}{F_{g}})^{u}}{F_{g}} = - \\frac{2 u (\\frac{\\cos{(u)}}{F_{g}})^{u}}{F_{g}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["add", 2, "Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(2), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('u', commutative=True), Pow(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('u', commutative=True), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('u', commutative=True), Pow(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\hat{X},\\Omega)} = \\log{(\\Omega - \\hat{X})} and l{(\\hat{X},\\Omega)} = \\varepsilon{(\\hat{X},\\Omega)} + \\log{(\\Omega - \\hat{X})}, then obtain \\varepsilon{(\\hat{X},\\Omega)} + \\log{(\\Omega - \\hat{X})} = 2 \\varepsilon{(\\hat{X},\\Omega)}", "derivation": "\\varepsilon{(\\hat{X},\\Omega)} = \\log{(\\Omega - \\hat{X})} and \\varepsilon{(\\hat{X},\\Omega)} + \\log{(\\Omega - \\hat{X})} = 2 \\log{(\\Omega - \\hat{X})} and l{(\\hat{X},\\Omega)} = \\varepsilon{(\\hat{X},\\Omega)} + \\log{(\\Omega - \\hat{X})} and l{(\\hat{X},\\Omega)} = 2 \\varepsilon{(\\hat{X},\\Omega)} and l{(\\hat{X},\\Omega)} = 2 \\log{(\\Omega - \\hat{X})} and 2 \\log{(\\Omega - \\hat{X})} = 2 \\varepsilon{(\\hat{X},\\Omega)} and \\varepsilon{(\\hat{X},\\Omega)} + \\log{(\\Omega - \\hat{X})} = 2 \\varepsilon{(\\hat{X},\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)))))"], [["minus", 1, "Mul(Integer(-1), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)))))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))), Mul(Integer(2), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('l')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))))), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\mathbf{g}^{\\phi_1}, then derive \\frac{\\partial}{\\partial \\phi_1} \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\mathbf{g}^{\\phi_1} \\log{(\\mathbf{g})}, then obtain (\\frac{\\partial}{\\partial \\phi_1} \\mathbf{g}^{\\phi_1})^{\\phi_1} = (\\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} \\log{(\\mathbf{g})})^{\\phi_1}", "derivation": "\\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\mathbf{g}^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\frac{\\partial}{\\partial \\phi_1} \\mathbf{g}^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\mathbf{g}^{\\phi_1} \\log{(\\mathbf{g})} and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} = \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} \\log{(\\mathbf{g})} and (\\frac{\\partial}{\\partial \\phi_1} \\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})})^{\\phi_1} = (\\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} \\log{(\\mathbf{g})})^{\\phi_1} and (\\frac{\\partial}{\\partial \\phi_1} \\mathbf{g}^{\\phi_1})^{\\phi_1} = (\\operatorname{f^{*}}{(\\phi_1,\\mathbf{g})} \\log{(\\mathbf{g})})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Derivative(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Function('f^*')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given x{(E_{x})} = E_{x}, then derive \\hat{p} + x{(E_{x})} = E_{x} + \\mathbf{J}_P, then obtain \\frac{\\hat{p} + x{(E_{x})}}{\\mathbf{J}_P} = \\frac{E_{x} + \\mathbf{J}_P}{\\mathbf{J}_P}", "derivation": "x{(E_{x})} = E_{x} and \\frac{d}{d E_{x}} x{(E_{x})} = \\frac{d}{d E_{x}} E_{x} and \\int \\frac{d}{d E_{x}} x{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} E_{x} dE_{x} and \\hat{p} + x{(E_{x})} = E_{x} + \\mathbf{J}_P and \\frac{\\hat{p} + x{(E_{x})}}{\\mathbf{J}_P} = \\frac{E_{x} + \\mathbf{J}_P}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Symbol('E_x', commutative=True), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Function('x')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(Symbol('E_x', commutative=True), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('x')(Symbol('E_x', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Function('x')(Symbol('E_x', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given k{(L)} = \\log{(L)}, then derive \\log{(L)} \\frac{d}{d L} k{(L)} = \\frac{\\log{(L)}}{L}, then obtain (2 \\log{(L)} \\frac{d}{d L} \\log{(L)})^{L} = (\\log{(L)} \\frac{d}{d L} \\log{(L)} + \\frac{\\log{(L)}}{L})^{L}", "derivation": "k{(L)} = \\log{(L)} and \\frac{d}{d L} k{(L)} = \\frac{d}{d L} \\log{(L)} and \\log{(L)} \\frac{d}{d L} k{(L)} = \\log{(L)} \\frac{d}{d L} \\log{(L)} and \\log{(L)} \\frac{d}{d L} k{(L)} = \\frac{\\log{(L)}}{L} and \\log{(L)} \\frac{d}{d L} k{(L)} + \\log{(L)} \\frac{d}{d L} \\log{(L)} = \\log{(L)} \\frac{d}{d L} \\log{(L)} + \\frac{\\log{(L)}}{L} and (\\log{(L)} \\frac{d}{d L} k{(L)} + \\log{(L)} \\frac{d}{d L} \\log{(L)})^{L} = (\\log{(L)} \\frac{d}{d L} \\log{(L)} + \\frac{\\log{(L)}}{L})^{L} and (2 \\log{(L)} \\frac{d}{d L} \\log{(L)})^{L} = (\\log{(L)} \\frac{d}{d L} \\log{(L)} + \\frac{\\log{(L)}}{L})^{L}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["times", 2, "log(Symbol('L', commutative=True))"], "Equality(Mul(log(Symbol('L', commutative=True)), Derivative(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(log(Symbol('L', commutative=True)), Derivative(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True))))"], [["add", 4, "Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], "Equality(Add(Mul(log(Symbol('L', commutative=True)), Derivative(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True)))))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Mul(log(Symbol('L', commutative=True)), Derivative(Function('k')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Symbol('L', commutative=True)), Pow(Add(Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True)))), Symbol('L', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Mul(Integer(2), log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Symbol('L', commutative=True)), Pow(Add(Mul(log(Symbol('L', commutative=True)), Derivative(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), log(Symbol('L', commutative=True)))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(n)} = \\cos{(n)}, then derive - \\sin{(n)} \\frac{d}{d n} \\phi_{1}{(n)} = \\sin^{2}{(n)}, then obtain - \\sin{(n)} \\frac{d}{d n} \\cos{(n)} + (\\frac{d}{d n} \\cos{(n)})^{n} = \\sin^{2}{(n)} + (\\frac{d}{d n} \\cos{(n)})^{n}", "derivation": "\\phi_{1}{(n)} = \\cos{(n)} and \\frac{d}{d n} \\phi_{1}{(n)} = \\frac{d}{d n} \\cos{(n)} and \\frac{d}{d n} \\phi_{1}{(n)} \\frac{d}{d n} \\cos{(n)} = (\\frac{d}{d n} \\cos{(n)})^{2} and - \\sin{(n)} \\frac{d}{d n} \\phi_{1}{(n)} = \\sin^{2}{(n)} and - \\sin{(n)} \\frac{d}{d n} \\cos{(n)} = \\sin^{2}{(n)} and - \\sin{(n)} \\frac{d}{d n} \\cos{(n)} + (\\frac{d}{d n} \\cos{(n)})^{n} = \\sin^{2}{(n)} + (\\frac{d}{d n} \\cos{(n)})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 2, "Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\phi_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), sin(Symbol('n', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(sin(Symbol('n', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), sin(Symbol('n', commutative=True)), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(sin(Symbol('n', commutative=True)), Integer(2)))"], [["add", 5, "Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('n', commutative=True)), Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True))), Add(Pow(sin(Symbol('n', commutative=True)), Integer(2)), Pow(Derivative(cos(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(B)} = \\sin{(B)}, then obtain (\\operatorname{v_{y}}{(B)} - 3 \\sin{(B)})^{2} \\sin^{2}{(B)} = 4 \\sin^{4}{(B)}", "derivation": "\\operatorname{v_{y}}{(B)} = \\sin{(B)} and \\operatorname{v_{y}}{(B)} + \\sin{(B)} = 2 \\sin{(B)} and \\operatorname{v_{y}}{(B)} - 3 \\sin{(B)} = - 2 \\sin{(B)} and - (\\operatorname{v_{y}}{(B)} - 3 \\sin{(B)}) \\sin{(B)} = 2 \\sin^{2}{(B)} and (\\operatorname{v_{y}}{(B)} - 3 \\sin{(B)})^{2} \\sin^{2}{(B)} = 4 \\sin^{4}{(B)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["add", 1, "sin(Symbol('B', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True))), Mul(Integer(2), sin(Symbol('B', commutative=True))))"], [["minus", 2, "Mul(Integer(4), sin(Symbol('B', commutative=True)))"], "Equality(Add(Function('v_y')(Symbol('B', commutative=True)), Mul(Integer(-1), Integer(3), sin(Symbol('B', commutative=True)))), Mul(Integer(-1), Integer(2), sin(Symbol('B', commutative=True))))"], [["times", 3, "Mul(Integer(-1), sin(Symbol('B', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('v_y')(Symbol('B', commutative=True)), Mul(Integer(-1), Integer(3), sin(Symbol('B', commutative=True)))), sin(Symbol('B', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('B', commutative=True)), Integer(2))))"], [["power", 4, 2], "Equality(Mul(Pow(Add(Function('v_y')(Symbol('B', commutative=True)), Mul(Integer(-1), Integer(3), sin(Symbol('B', commutative=True)))), Integer(2)), Pow(sin(Symbol('B', commutative=True)), Integer(2))), Mul(Integer(4), Pow(sin(Symbol('B', commutative=True)), Integer(4))))"]]}, {"prompt": "Given A{(E_{x})} = \\sin{(E_{x})}, then obtain A^{2}{(E_{x})} \\sin{(E_{x})} = A{(E_{x})} \\sin^{2}{(E_{x})}", "derivation": "A{(E_{x})} = \\sin{(E_{x})} and A^{2}{(E_{x})} = A{(E_{x})} \\sin{(E_{x})} and A^{3}{(E_{x})} = A^{2}{(E_{x})} \\sin{(E_{x})} and A^{3}{(E_{x})} = A{(E_{x})} \\sin^{2}{(E_{x})} and A^{2}{(E_{x})} \\sin{(E_{x})} = A{(E_{x})} \\sin^{2}{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], [["times", 1, "Function('A')(Symbol('E_x', commutative=True))"], "Equality(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Function('A')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))))"], [["times", 1, "Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2))"], "Equality(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(3)), Mul(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2)), sin(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(3)), Mul(Function('A')(Symbol('E_x', commutative=True)), Pow(sin(Symbol('E_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2)), sin(Symbol('E_x', commutative=True))), Mul(Function('A')(Symbol('E_x', commutative=True)), Pow(sin(Symbol('E_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\Omega{(v_{t})} = \\int \\sin{(v_{t})} dv_{t}, then derive \\Omega{(v_{t})} = \\Psi_{nl} - \\cos{(v_{t})}, then derive (\\Psi_{nl} - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)} = (\\psi^* - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)}, then obtain \\operatorname{M_{E}}{(\\varphi)} \\int \\sin{(v_{t})} dv_{t} = (\\psi^* - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)}", "derivation": "\\Omega{(v_{t})} = \\int \\sin{(v_{t})} dv_{t} and \\Omega{(v_{t})} = \\Psi_{nl} - \\cos{(v_{t})} and \\operatorname{M_{E}}{(\\varphi)} \\Omega{(v_{t})} = \\operatorname{M_{E}}{(\\varphi)} \\int \\sin{(v_{t})} dv_{t} and (\\Psi_{nl} - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)} = \\operatorname{M_{E}}{(\\varphi)} \\int \\sin{(v_{t})} dv_{t} and (\\Psi_{nl} - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)} = (\\psi^* - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)} and \\int \\sin{(v_{t})} dv_{t} = \\Psi_{nl} - \\cos{(v_{t})} and \\operatorname{M_{E}}{(\\varphi)} \\int \\sin{(v_{t})} dv_{t} = (\\psi^* - \\cos{(v_{t})}) \\operatorname{M_{E}}{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\Omega')(Symbol('v_t', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["times", 1, "Function('M_E')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Function('\\\\Omega')(Symbol('v_t', commutative=True))), Mul(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Function('M_E')(Symbol('\\\\varphi', commutative=True))), Mul(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Function('M_E')(Symbol('\\\\varphi', commutative=True))), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Function('M_E')(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Function('M_E')(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\theta_1}, then obtain e^{- \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\mathbf{v}{(\\theta_1)} d\\theta_1 + 1} = e^{- \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\frac{d}{d \\theta_1} e^{\\theta_1} d\\theta_1 + 1}", "derivation": "\\mathbf{v}{(\\theta_1)} = \\frac{d}{d \\theta_1} e^{\\theta_1} and \\int \\mathbf{v}{(\\theta_1)} d\\theta_1 = \\int \\frac{d}{d \\theta_1} e^{\\theta_1} d\\theta_1 and - \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\mathbf{v}{(\\theta_1)} d\\theta_1 = - \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\frac{d}{d \\theta_1} e^{\\theta_1} d\\theta_1 and - \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\mathbf{v}{(\\theta_1)} d\\theta_1 + 1 = - \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\frac{d}{d \\theta_1} e^{\\theta_1} d\\theta_1 + 1 and e^{- \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\mathbf{v}{(\\theta_1)} d\\theta_1 + 1} = e^{- \\hat{\\mathbf{x}}{(\\theta_1)} \\int \\frac{d}{d \\theta_1} e^{\\theta_1} d\\theta_1 + 1}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\theta_1', commutative=True)), Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["add", 3, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Integer(1)), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Integer(1)))"], [["exp", 4], "Equality(exp(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Function('\\\\mathbf{v}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Integer(1))), exp(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\theta_1', commutative=True)), Integral(Derivative(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then obtain 3 \\operatorname{n_{2}}{(\\mathbf{E})} + \\log{(\\mathbf{E})} = \\operatorname{n_{2}}{(\\mathbf{E})} + 3 \\log{(\\mathbf{E})}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and 2 \\operatorname{n_{2}}{(\\mathbf{E})} = \\operatorname{n_{2}}{(\\mathbf{E})} + \\log{(\\mathbf{E})} and 3 \\operatorname{n_{2}}{(\\mathbf{E})} + \\log{(\\mathbf{E})} = 2 \\operatorname{n_{2}}{(\\mathbf{E})} + 2 \\log{(\\mathbf{E})} and 3 \\operatorname{n_{2}}{(\\mathbf{E})} + \\log{(\\mathbf{E})} = \\operatorname{n_{2}}{(\\mathbf{E})} + 3 \\log{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["add", 1, "Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 2, "Add(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True))), log(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True))), log(Symbol('\\\\mathbf{E}', commutative=True))), Add(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(3), log(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given C{(\\hat{x}_0)} = \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0, then derive C{(\\hat{x}_0)} \\cos{(Z)} = (\\dot{z} - \\cos{(\\hat{x}_0)}) \\cos{(Z)}, then obtain C{(\\hat{x}_0)} + \\cos{(Z)} \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0 = (\\dot{z} - \\cos{(\\hat{x}_0)}) \\cos{(Z)} + C{(\\hat{x}_0)}", "derivation": "C{(\\hat{x}_0)} = \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0 and C{(\\hat{x}_0)} \\cos{(Z)} = \\cos{(Z)} \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0 and C{(\\hat{x}_0)} \\cos{(Z)} = (\\dot{z} - \\cos{(\\hat{x}_0)}) \\cos{(Z)} and \\cos{(Z)} \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0 = (\\dot{z} - \\cos{(\\hat{x}_0)}) \\cos{(Z)} and C{(\\hat{x}_0)} + \\cos{(Z)} \\int \\sin{(\\hat{x}_0)} d\\hat{x}_0 = (\\dot{z} - \\cos{(\\hat{x}_0)}) \\cos{(Z)} + C{(\\hat{x}_0)}", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('\\\\hat{x}_0', commutative=True)), Integral(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 1, "cos(Symbol('Z', commutative=True))"], "Equality(Mul(Function('C')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('Z', commutative=True))), Mul(cos(Symbol('Z', commutative=True)), Integral(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('C')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('Z', commutative=True))), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}_0', commutative=True)))), cos(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(cos(Symbol('Z', commutative=True)), Integral(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}_0', commutative=True)))), cos(Symbol('Z', commutative=True))))"], [["add", 4, "Function('C')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Function('C')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(cos(Symbol('Z', commutative=True)), Integral(sin(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))), Add(Mul(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}_0', commutative=True)))), cos(Symbol('Z', commutative=True))), Function('C')(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)} = L^{\\theta}, then obtain \\frac{\\partial}{\\partial L} \\log{(\\sin^{\\theta}{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})})}^{L} = \\frac{\\partial}{\\partial L} \\log{(\\sin^{\\theta}{(L L^{\\theta})})}^{L}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\theta,L)} = L^{\\theta} and L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)} = L L^{\\theta} and \\sin{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})} = \\sin{(L L^{\\theta})} and \\sin^{\\theta}{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})} = \\sin^{\\theta}{(L L^{\\theta})} and \\log{(\\sin^{\\theta}{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})})} = \\log{(\\sin^{\\theta}{(L L^{\\theta})})} and \\log{(\\sin^{\\theta}{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})})}^{L} = \\log{(\\sin^{\\theta}{(L L^{\\theta})})}^{L} and \\frac{\\partial}{\\partial L} \\log{(\\sin^{\\theta}{(L \\operatorname{V_{\\mathbf{E}}}{(\\theta,L)})})}^{L} = \\frac{\\partial}{\\partial L} \\log{(\\sin^{\\theta}{(L L^{\\theta})})}^{L}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)))), sin(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(sin(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(sin(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)))"], [["log", 4], "Equality(log(Pow(sin(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)))), Symbol('\\\\theta', commutative=True))), log(Pow(sin(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True))))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(log(Pow(sin(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)))), Symbol('\\\\theta', commutative=True))), Symbol('L', commutative=True)), Pow(log(Pow(sin(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True))), Symbol('L', commutative=True)))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(log(Pow(sin(Mul(Symbol('L', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\theta', commutative=True), Symbol('L', commutative=True)))), Symbol('\\\\theta', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(log(Pow(sin(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(z)} = \\log{(e^{z})}, then obtain (z + \\operatorname{F_{N}}{(z)} + \\log{(e^{z})})^{z} = (z + 2 \\log{(e^{z})})^{z}", "derivation": "\\operatorname{F_{N}}{(z)} = \\log{(e^{z})} and \\operatorname{F_{N}}{(z)} + \\log{(e^{z})} = 2 \\log{(e^{z})} and z + \\operatorname{F_{N}}{(z)} + \\log{(e^{z})} = z + 2 \\log{(e^{z})} and (z + \\operatorname{F_{N}}{(z)} + \\log{(e^{z})})^{z} = (z + 2 \\log{(e^{z})})^{z}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('z', commutative=True)), log(exp(Symbol('z', commutative=True))))"], [["add", 1, "log(exp(Symbol('z', commutative=True)))"], "Equality(Add(Function('F_N')(Symbol('z', commutative=True)), log(exp(Symbol('z', commutative=True)))), Mul(Integer(2), log(exp(Symbol('z', commutative=True)))))"], [["add", 2, "Symbol('z', commutative=True)"], "Equality(Add(Symbol('z', commutative=True), Function('F_N')(Symbol('z', commutative=True)), log(exp(Symbol('z', commutative=True)))), Add(Symbol('z', commutative=True), Mul(Integer(2), log(exp(Symbol('z', commutative=True))))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Add(Symbol('z', commutative=True), Function('F_N')(Symbol('z', commutative=True)), log(exp(Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Add(Symbol('z', commutative=True), Mul(Integer(2), log(exp(Symbol('z', commutative=True))))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\Psi,B)} = \\sin^{B}{(\\Psi)}, then obtain - 2 B - \\sin{(\\Psi)} + 2 - \\frac{2 \\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} = - 2 B - \\sin{(\\Psi)} + 1 - \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}}", "derivation": "\\rho{(\\Psi,B)} = \\sin^{B}{(\\Psi)} and 1 = \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} and 1 - \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} = 0 and - B + 1 - \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} = - B and - 2 B + 2 - \\frac{2 \\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} = - 2 B + 1 - \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} and - 2 B - \\sin{(\\Psi)} + 2 - \\frac{2 \\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}} = - 2 B - \\sin{(\\Psi)} + 1 - \\frac{\\sin^{B}{(\\Psi)}}{\\rho{(\\Psi,B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))"], [["divide", 1, "Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))), Integer(0))"], [["minus", 3, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))), Mul(Integer(-1), Symbol('B', commutative=True)))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Integer(2), Mul(Integer(-1), Integer(2), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Integer(1), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))))"], [["minus", 5, "sin(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Integer(2), Mul(Integer(-1), Integer(2), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\Psi', commutative=True))), Integer(1), Mul(Integer(-1), Pow(Function('\\\\rho')(Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\Psi', commutative=True)), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain - \\sin{(\\psi^*)} + \\frac{\\eta^{\\prime}^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}} = - \\sin{(\\psi^*)} + \\frac{\\sin^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}}", "derivation": "\\eta^{\\prime}{(\\psi^*)} = \\sin{(\\psi^*)} and \\eta^{\\prime}^{\\psi^*}{(\\psi^*)} = \\sin^{\\psi^*}{(\\psi^*)} and \\frac{\\eta^{\\prime}^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}} = \\frac{\\sin^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}} and - \\sin{(\\psi^*)} + \\frac{\\eta^{\\prime}^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}} = - \\sin{(\\psi^*)} + \\frac{\\sin^{\\psi^*}{(\\psi^*)}}{\\eta^{\\prime}{(\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["divide", 2, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 3, "sin(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given Z{(\\theta_2)} = \\log{(\\cos{(\\theta_2)})} and \\hat{H}_l{(\\theta_2)} = - \\theta_2 + Z{(\\theta_2)}, then obtain \\frac{d}{d \\theta_2} \\hat{H}_l^{2}{(\\theta_2)} = \\frac{d}{d \\theta_2} (- \\theta_2 + \\log{(\\cos{(\\theta_2)})}) \\hat{H}_l{(\\theta_2)}", "derivation": "Z{(\\theta_2)} = \\log{(\\cos{(\\theta_2)})} and \\hat{H}_l{(\\theta_2)} = - \\theta_2 + Z{(\\theta_2)} and \\hat{H}_l^{2}{(\\theta_2)} = (- \\theta_2 + Z{(\\theta_2)}) \\hat{H}_l{(\\theta_2)} and \\hat{H}_l^{2}{(\\theta_2)} = (- \\theta_2 + \\log{(\\cos{(\\theta_2)})}) \\hat{H}_l{(\\theta_2)} and \\frac{d}{d \\theta_2} \\hat{H}_l^{2}{(\\theta_2)} = \\frac{d}{d \\theta_2} (- \\theta_2 + \\log{(\\cos{(\\theta_2)})}) \\hat{H}_l{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\theta_2', commutative=True)), log(cos(Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('Z')(Symbol('\\\\theta_2', commutative=True))))"], [["times", 2, "Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('Z')(Symbol('\\\\theta_2', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(cos(Symbol('\\\\theta_2', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), log(cos(Symbol('\\\\theta_2', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(\\Psi^{\\dagger},f_{E})} = \\cos^{f_{E}}{(\\Psi^{\\dagger})} and \\mathbf{r}{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})}, then obtain (\\Psi^{\\dagger} \\mathbf{r}^{f_{E}}{(\\Psi^{\\dagger})})^{\\Psi^{\\dagger}} = (\\Psi^{\\dagger} \\cos^{f_{E}}{(\\Psi^{\\dagger})})^{\\Psi^{\\dagger}}", "derivation": "\\sigma_{x}{(\\Psi^{\\dagger},f_{E})} = \\cos^{f_{E}}{(\\Psi^{\\dagger})} and \\mathbf{r}{(\\Psi^{\\dagger})} = \\cos{(\\Psi^{\\dagger})} and \\Psi^{\\dagger} \\sigma_{x}{(\\Psi^{\\dagger},f_{E})} = \\Psi^{\\dagger} \\cos^{f_{E}}{(\\Psi^{\\dagger})} and \\Psi^{\\dagger} \\sigma_{x}{(\\Psi^{\\dagger},f_{E})} = \\Psi^{\\dagger} \\mathbf{r}^{f_{E}}{(\\Psi^{\\dagger})} and (\\Psi^{\\dagger} \\sigma_{x}{(\\Psi^{\\dagger},f_{E})})^{\\Psi^{\\dagger}} = (\\Psi^{\\dagger} \\cos^{f_{E}}{(\\Psi^{\\dagger})})^{\\Psi^{\\dagger}} and (\\Psi^{\\dagger} \\mathbf{r}^{f_{E}}{(\\Psi^{\\dagger})})^{\\Psi^{\\dagger}} = (\\Psi^{\\dagger} \\cos^{f_{E}}{(\\Psi^{\\dagger})})^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Pow(cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(cos(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('f_E', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(J)} = \\cos{(e^{J})}, then obtain e^{J} \\sin{(e^{J})} + \\frac{d}{d J} \\mathbf{S}{(J)} = 0", "derivation": "\\mathbf{S}{(J)} = \\cos{(e^{J})} and \\frac{d}{d J} \\mathbf{S}{(J)} = \\frac{d}{d J} \\cos{(e^{J})} and \\frac{d}{d J} \\mathbf{S}{(J)} - \\frac{d}{d J} \\cos{(e^{J})} = 0 and e^{J} \\sin{(e^{J})} + \\frac{d}{d J} \\mathbf{S}{(J)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), cos(exp(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(cos(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(exp(Symbol('J', commutative=True)), sin(exp(Symbol('J', commutative=True)))), Derivative(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\dot{z}{(C)} = \\log{(C)}, then derive \\int \\dot{z}{(C)} dC = C \\log{(C)} - C + n_{1}, then obtain C + \\int \\dot{z}{(C)} dC = C \\dot{z}{(C)} + n_{1}", "derivation": "\\dot{z}{(C)} = \\log{(C)} and \\int \\dot{z}{(C)} dC = \\int \\log{(C)} dC and \\int \\dot{z}{(C)} dC = C \\log{(C)} - C + n_{1} and \\int \\dot{z}{(C)} dC = C \\dot{z}{(C)} - C + n_{1} and C + \\int \\dot{z}{(C)} dC = C \\dot{z}{(C)} + n_{1}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(log(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Mul(Symbol('C', commutative=True), Function('\\\\dot{z}')(Symbol('C', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('n_1', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), Symbol('C', commutative=True))"], "Equality(Add(Symbol('C', commutative=True), Integral(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Mul(Symbol('C', commutative=True), Function('\\\\dot{z}')(Symbol('C', commutative=True))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{x}_0,T)} = T \\hat{x}_0 and \\omega{(T)} = T^{2}, then obtain T (- 2 T \\hat{x}_0 + \\operatorname{P_{g}}{(\\hat{x}_0,T)}) = - \\hat{x}_0 \\omega{(T)}", "derivation": "\\operatorname{P_{g}}{(\\hat{x}_0,T)} = T \\hat{x}_0 and - 2 T \\hat{x}_0 + \\operatorname{P_{g}}{(\\hat{x}_0,T)} = - T \\hat{x}_0 and T (- 2 T \\hat{x}_0 + \\operatorname{P_{g}}{(\\hat{x}_0,T)}) = - T^{2} \\hat{x}_0 and \\omega{(T)} = T^{2} and T (- 2 T \\hat{x}_0 + \\operatorname{P_{g}}{(\\hat{x}_0,T)}) = - \\hat{x}_0 \\omega{(T)}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 2, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(2)), Symbol('\\\\hat{x}_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('T', commutative=True)), Pow(Symbol('T', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('T', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('T', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('P_g')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('T', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\omega')(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(P_{g})} = \\sin{(P_{g})}, then derive \\int \\operatorname{t_{1}}{(P_{g})} dP_{g} = A_{1} - \\cos{(P_{g})}, then obtain - \\cos{(P_{g})} + \\frac{d}{d A_{1}} \\int \\sin{(P_{g})} dP_{g} = - \\cos{(P_{g})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\cos{(P_{g})})", "derivation": "\\operatorname{t_{1}}{(P_{g})} = \\sin{(P_{g})} and \\int \\operatorname{t_{1}}{(P_{g})} dP_{g} = \\int \\sin{(P_{g})} dP_{g} and \\int \\operatorname{t_{1}}{(P_{g})} dP_{g} = A_{1} - \\cos{(P_{g})} and \\int \\sin{(P_{g})} dP_{g} = A_{1} - \\cos{(P_{g})} and \\frac{d}{d A_{1}} \\int \\sin{(P_{g})} dP_{g} = \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\cos{(P_{g})}) and - \\cos{(P_{g})} + \\frac{d}{d A_{1}} \\int \\sin{(P_{g})} dP_{g} = - \\cos{(P_{g})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\cos{(P_{g})})", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(sin(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_1')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('P_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('P_g', commutative=True)))))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('P_g', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["add", 5, "Mul(Integer(-1), cos(Symbol('P_g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Derivative(Integral(sin(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), cos(Symbol('P_g', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(a^{\\dagger},W)} = - W + a^{\\dagger}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} (e^{(- \\operatorname{y^{\\prime}}{(a^{\\dagger},W)})^{a^{\\dagger}}})^{W} = \\frac{\\partial}{\\partial a^{\\dagger}} (e^{(W - a^{\\dagger})^{a^{\\dagger}}})^{W}", "derivation": "\\operatorname{y^{\\prime}}{(a^{\\dagger},W)} = - W + a^{\\dagger} and - \\operatorname{y^{\\prime}}{(a^{\\dagger},W)} = W - a^{\\dagger} and (- \\operatorname{y^{\\prime}}{(a^{\\dagger},W)})^{a^{\\dagger}} = (W - a^{\\dagger})^{a^{\\dagger}} and e^{(- \\operatorname{y^{\\prime}}{(a^{\\dagger},W)})^{a^{\\dagger}}} = e^{(W - a^{\\dagger})^{a^{\\dagger}}} and (e^{(- \\operatorname{y^{\\prime}}{(a^{\\dagger},W)})^{a^{\\dagger}}})^{W} = (e^{(W - a^{\\dagger})^{a^{\\dagger}}})^{W} and \\frac{\\partial}{\\partial a^{\\dagger}} (e^{(- \\operatorname{y^{\\prime}}{(a^{\\dagger},W)})^{a^{\\dagger}}})^{W} = \\frac{\\partial}{\\partial a^{\\dagger}} (e^{(W - a^{\\dagger})^{a^{\\dagger}}})^{W}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), exp(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(exp(Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('W', commutative=True)), Pow(exp(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('W', commutative=True)))"], [["differentiate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Pow(exp(Pow(Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('W', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(exp(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('W', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{S},F_{x})} = \\log{(\\mathbf{S}^{F_{x}})}, then obtain (\\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\mathbf{r}{(\\mathbf{S},F_{x})}))^{\\mathbf{S}} = (\\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\log{(\\mathbf{S}^{F_{x}})}))^{\\mathbf{S}}", "derivation": "\\mathbf{r}{(\\mathbf{S},F_{x})} = \\log{(\\mathbf{S}^{F_{x}})} and - \\mathbf{S}^{F_{x}} + \\mathbf{r}{(\\mathbf{S},F_{x})} = - \\mathbf{S}^{F_{x}} + \\log{(\\mathbf{S}^{F_{x}})} and \\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\mathbf{r}{(\\mathbf{S},F_{x})}) = \\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\log{(\\mathbf{S}^{F_{x}})}) and (\\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\mathbf{r}{(\\mathbf{S},F_{x})}))^{\\mathbf{S}} = (\\frac{\\partial}{\\partial \\mathbf{S}} (- \\mathbf{S}^{F_{x}} + \\log{(\\mathbf{S}^{F_{x}})}))^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)), log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True))), log(Pow(Symbol('\\\\mathbf{S}', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(F_{c},f)} = F_{c} + f, then obtain e^{\\frac{d}{d F_{c}} 1} = e^{\\frac{\\partial}{\\partial F_{c}} e^{- F_{c} - f} e^{F_{c} + f}}", "derivation": "\\mathbf{g}{(F_{c},f)} = F_{c} + f and e^{\\mathbf{g}{(F_{c},f)}} = e^{F_{c} + f} and 1 = e^{F_{c} + f} e^{- \\mathbf{g}{(F_{c},f)}} and 1 = e^{- F_{c} - f} e^{F_{c} + f} and \\frac{d}{d F_{c}} 1 = \\frac{\\partial}{\\partial F_{c}} e^{- F_{c} - f} e^{F_{c} + f} and e^{\\frac{d}{d F_{c}} 1} = e^{\\frac{\\partial}{\\partial F_{c}} e^{- F_{c} - f} e^{F_{c} + f}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('F_c', commutative=True), Symbol('f', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{g}')(Symbol('F_c', commutative=True), Symbol('f', commutative=True))), exp(Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True))))"], [["divide", 2, "exp(Function('\\\\mathbf{g}')(Symbol('F_c', commutative=True), Symbol('f', commutative=True)))"], "Equality(Integer(1), Mul(exp(Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True))), exp(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('F_c', commutative=True), Symbol('f', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Mul(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True)))), exp(Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True)))))"], [["differentiate", 4, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True)))), exp(Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["exp", 5], "Equality(exp(Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1)))), exp(Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True)))), exp(Add(Symbol('F_c', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\dot{x})} = \\sin{(\\dot{x})}, then obtain \\frac{d}{d \\dot{x}} \\bar{\\h}{(\\dot{x})} - \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} - 1 = -1", "derivation": "\\bar{\\h}{(\\dot{x})} = \\sin{(\\dot{x})} and \\frac{d}{d \\dot{x}} \\bar{\\h}{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} and \\frac{d}{d \\dot{x}} \\bar{\\h}{(\\dot{x})} - \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} = 0 and \\frac{d}{d \\dot{x}} \\bar{\\h}{(\\dot{x})} - \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True)), sin(Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))), Integer(0))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('\\\\hbar')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given \\varphi{(W)} = \\sin{(W)}, then derive \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\varphi{(W)}}{h^{2}} = \\frac{(\\Psi^{\\dagger})^{4} \\cos{(W)}}{h^{2}}, then obtain \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\sin{(W)}}{h^{2}} = \\frac{(\\Psi^{\\dagger})^{4} \\cos{(W)}}{h^{2}}", "derivation": "\\varphi{(W)} = \\sin{(W)} and \\frac{d}{d W} \\varphi{(W)} = \\frac{d}{d W} \\sin{(W)} and (\\Psi^{\\dagger})^{2} \\frac{d}{d W} \\varphi{(W)} = (\\Psi^{\\dagger})^{2} \\frac{d}{d W} \\sin{(W)} and \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\varphi{(W)}}{h^{2}} = \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\sin{(W)}}{h^{2}} and \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\varphi{(W)}}{h^{2}} = \\frac{(\\Psi^{\\dagger})^{4} \\cos{(W)}}{h^{2}} and \\frac{(\\Psi^{\\dagger})^{4} \\frac{d}{d W} \\sin{(W)}}{h^{2}} = \\frac{(\\Psi^{\\dagger})^{4} \\cos{(W)}}{h^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2)), Derivative(Function('\\\\varphi')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(2)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-2)), Pow(Symbol('h', commutative=True), Integer(2)))"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), Derivative(Function('\\\\varphi')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), Derivative(Function('\\\\varphi')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), cos(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(4)), Pow(Symbol('h', commutative=True), Integer(-2)), cos(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(f)} = \\cos{(f)}, then obtain \\int (- \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\operatorname{t_{1}}{(f)} df}{2 \\cos{(f)}}) df = \\int (- \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\cos{(f)} df}{2 \\cos{(f)}}) df", "derivation": "\\operatorname{t_{1}}{(f)} = \\cos{(f)} and \\int \\operatorname{t_{1}}{(f)} df = \\int \\cos{(f)} df and \\frac{\\int \\operatorname{t_{1}}{(f)} df}{2 \\cos{(f)}} = \\frac{\\int \\cos{(f)} df}{2 \\cos{(f)}} and - \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\operatorname{t_{1}}{(f)} df}{2 \\cos{(f)}} = - \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\cos{(f)} df}{2 \\cos{(f)}} and \\int (- \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\operatorname{t_{1}}{(f)} df}{2 \\cos{(f)}}) df = \\int (- \\frac{d}{d f} 2 \\cos{(f)} + \\frac{\\int \\cos{(f)} df}{2 \\cos{(f)}}) df", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["divide", 2, "Mul(Integer(2), cos(Symbol('f', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('t_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["minus", 3, "Derivative(Mul(Integer(2), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('t_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Add(Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(Function('t_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True))), Integral(Add(Mul(Integer(-1), Derivative(Mul(Integer(2), cos(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('f', commutative=True)), Integer(-1)), Integral(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(t_{2},\\hat{H})} = \\hat{H} - t_{2}, then obtain \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} + \\int \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} dt_{2} = (\\hat{H} - t_{2})^{t_{2}} + \\int \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} dt_{2}", "derivation": "\\operatorname{t_{1}}{(t_{2},\\hat{H})} = \\hat{H} - t_{2} and \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} = (\\hat{H} - t_{2})^{t_{2}} and \\int \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} dt_{2} = \\int (\\hat{H} - t_{2})^{t_{2}} dt_{2} and \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} + \\int (\\hat{H} - t_{2})^{t_{2}} dt_{2} = (\\hat{H} - t_{2})^{t_{2}} + \\int (\\hat{H} - t_{2})^{t_{2}} dt_{2} and \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} + \\int \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} dt_{2} = (\\hat{H} - t_{2})^{t_{2}} + \\int \\operatorname{t_{1}}^{t_{2}}{(t_{2},\\hat{H})} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["add", 2, "Integral(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Add(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Integral(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Integral(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Integral(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Add(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Integral(Pow(Function('t_1')(Symbol('t_2', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given i{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\operatorname{A_{1}}{(\\sigma_x)} = (\\sigma_x + i{(\\sigma_x)})^{\\sigma_x}, then obtain \\operatorname{A_{1}}^{\\sigma_x}{(\\sigma_x)} = ((\\sigma_x + \\cos{(\\sigma_x)})^{\\sigma_x})^{\\sigma_x}", "derivation": "i{(\\sigma_x)} = \\cos{(\\sigma_x)} and \\sigma_x + i{(\\sigma_x)} = \\sigma_x + \\cos{(\\sigma_x)} and (\\sigma_x + i{(\\sigma_x)})^{\\sigma_x} = (\\sigma_x + \\cos{(\\sigma_x)})^{\\sigma_x} and \\operatorname{A_{1}}{(\\sigma_x)} = (\\sigma_x + i{(\\sigma_x)})^{\\sigma_x} and \\operatorname{A_{1}}{(\\sigma_x)} = (\\sigma_x + \\cos{(\\sigma_x)})^{\\sigma_x} and \\operatorname{A_{1}}^{\\sigma_x}{(\\sigma_x)} = ((\\sigma_x + \\cos{(\\sigma_x)})^{\\sigma_x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('i')(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\sigma_x', commutative=True), Function('i')(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), Function('i')(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('A_1')(Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(z,E_{x})} = e^{- E_{x} + z}, then obtain - z + e^{- e^{- E_{x} + z} - 1} \\sin{(\\operatorname{L_{\\varepsilon}}{(z,E_{x})} + 1)} = - z + e^{- e^{- E_{x} + z} - 1} \\sin{(e^{- E_{x} + z} + 1)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(z,E_{x})} = e^{- E_{x} + z} and \\operatorname{L_{\\varepsilon}}{(z,E_{x})} + 1 = e^{- E_{x} + z} + 1 and \\sin{(\\operatorname{L_{\\varepsilon}}{(z,E_{x})} + 1)} = \\sin{(e^{- E_{x} + z} + 1)} and e^{- e^{- E_{x} + z} - 1} \\sin{(\\operatorname{L_{\\varepsilon}}{(z,E_{x})} + 1)} = e^{- e^{- E_{x} + z} - 1} \\sin{(e^{- E_{x} + z} + 1)} and - z + e^{- e^{- E_{x} + z} - 1} \\sin{(\\operatorname{L_{\\varepsilon}}{(z,E_{x})} + 1)} = - z + e^{- e^{- E_{x} + z} - 1} \\sin{(e^{- E_{x} + z} + 1)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('E_x', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('E_x', commutative=True)), Integer(1)), Add(exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))), Integer(1)))"], [["sin", 2], "Equality(sin(Add(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('E_x', commutative=True)), Integer(1))), sin(Add(exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))), Integer(1))))"], [["divide", 3, "exp(Add(exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))), Integer(1)))"], "Equality(Mul(exp(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True)))), Integer(-1))), sin(Add(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('E_x', commutative=True)), Integer(1)))), Mul(exp(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True)))), Integer(-1))), sin(Add(exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))), Integer(1)))))"], [["minus", 4, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(exp(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True)))), Integer(-1))), sin(Add(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('E_x', commutative=True)), Integer(1))))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Mul(exp(Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True)))), Integer(-1))), sin(Add(exp(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Symbol('z', commutative=True))), Integer(1))))))"]]}, {"prompt": "Given \\lambda{(\\phi)} = \\phi, then obtain \\int \\frac{\\mathbf{A} \\phi \\int \\lambda{(\\phi)} d\\phi}{\\int \\phi d\\lambda{(\\phi)}} d\\phi = \\int \\frac{\\mathbf{A} \\phi \\int \\phi d\\phi}{\\int \\phi d\\lambda{(\\phi)}} d\\phi", "derivation": "\\lambda{(\\phi)} = \\phi and \\int \\lambda{(\\phi)} d\\phi = \\int \\phi d\\phi and \\int \\lambda{(\\phi)} d\\lambda{(\\phi)} = \\int \\phi d\\lambda{(\\phi)} and \\frac{\\mathbf{A} \\phi \\int \\lambda{(\\phi)} d\\phi}{\\int \\lambda{(\\phi)} d\\lambda{(\\phi)}} = \\frac{\\mathbf{A} \\phi \\int \\phi d\\phi}{\\int \\lambda{(\\phi)} d\\lambda{(\\phi)}} and \\frac{\\mathbf{A} \\phi \\int \\lambda{(\\phi)} d\\phi}{\\int \\phi d\\lambda{(\\phi)}} = \\frac{\\mathbf{A} \\phi \\int \\phi d\\phi}{\\int \\phi d\\lambda{(\\phi)}} and \\int \\frac{\\mathbf{A} \\phi \\int \\lambda{(\\phi)} d\\phi}{\\int \\phi d\\lambda{(\\phi)}} d\\phi = \\int \\frac{\\mathbf{A} \\phi \\int \\phi d\\phi}{\\int \\phi d\\lambda{(\\phi)}} d\\phi", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integral(Symbol('\\\\phi', commutative=True), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))))"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Pow(Integral(Symbol('\\\\phi', commutative=True), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Symbol('\\\\phi', commutative=True), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1))))"], [["integrate", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Pow(Integral(Symbol('\\\\phi', commutative=True), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\phi', commutative=True), Integral(Symbol('\\\\phi', commutative=True), Tuple(Symbol('\\\\phi', commutative=True))), Pow(Integral(Symbol('\\\\phi', commutative=True), Tuple(Function('\\\\lambda')(Symbol('\\\\phi', commutative=True)))), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(q)} = \\cos{(q)}, then obtain (\\mathbf{r}{(q)} + \\frac{1}{q}) \\int (\\mathbf{r}{(q)} + \\frac{1}{q}) dq = (\\mathbf{r}{(q)} + \\frac{1}{q}) \\int (\\cos{(q)} + \\frac{1}{q}) dq", "derivation": "\\mathbf{r}{(q)} = \\cos{(q)} and \\mathbf{r}{(q)} + \\frac{1}{q} = \\cos{(q)} + \\frac{1}{q} and \\int (\\mathbf{r}{(q)} + \\frac{1}{q}) dq = \\int (\\cos{(q)} + \\frac{1}{q}) dq and (\\cos{(q)} + \\frac{1}{q}) \\int (\\mathbf{r}{(q)} + \\frac{1}{q}) dq = (\\cos{(q)} + \\frac{1}{q}) \\int (\\cos{(q)} + \\frac{1}{q}) dq and (\\mathbf{r}{(q)} + \\frac{1}{q}) \\int (\\mathbf{r}{(q)} + \\frac{1}{q}) dq = (\\mathbf{r}{(q)} + \\frac{1}{q}) \\int (\\cos{(q)} + \\frac{1}{q}) dq", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["add", 1, "Pow(Symbol('q', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True))), Integral(Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True))))"], [["times", 3, "Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1)))"], "Equality(Mul(Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Integral(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Mul(Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Integral(Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Integral(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Mul(Add(Function('\\\\mathbf{r}')(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Integral(Add(cos(Symbol('q', commutative=True)), Pow(Symbol('q', commutative=True), Integer(-1))), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(x,\\Psi,B)} = \\frac{\\Psi x}{B}, then derive \\frac{\\partial}{\\partial B} \\operatorname{c_{0}}{(x,\\Psi,B)} = - \\frac{\\Psi x}{B^{2}}, then obtain - \\frac{\\operatorname{c_{0}}{(x,\\Psi,B)}}{B} = \\frac{\\partial}{\\partial B} \\frac{\\Psi x}{B}", "derivation": "\\operatorname{c_{0}}{(x,\\Psi,B)} = \\frac{\\Psi x}{B} and \\operatorname{c_{0}}{(x,\\Psi,B)} + 1 = 1 + \\frac{\\Psi x}{B} and \\frac{\\partial}{\\partial B} (\\operatorname{c_{0}}{(x,\\Psi,B)} + 1) = \\frac{\\partial}{\\partial B} (1 + \\frac{\\Psi x}{B}) and \\frac{\\partial}{\\partial B} \\operatorname{c_{0}}{(x,\\Psi,B)} = - \\frac{\\Psi x}{B^{2}} and \\frac{\\partial}{\\partial B} \\frac{\\Psi x}{B} = - \\frac{\\Psi x}{B^{2}} and \\frac{\\partial}{\\partial B} \\operatorname{c_{0}}{(x,\\Psi,B)} = \\frac{\\partial}{\\partial B} \\frac{\\Psi x}{B} and \\frac{\\partial}{\\partial B} \\operatorname{c_{0}}{(x,\\Psi,B)} = - \\frac{\\operatorname{c_{0}}{(x,\\Psi,B)}}{B} and - \\frac{\\operatorname{c_{0}}{(x,\\Psi,B)}}{B} = \\frac{\\partial}{\\partial B} \\frac{\\Psi x}{B}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-2)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('c_0')(Symbol('x', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('B', commutative=True))), Derivative(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(c,\\tilde{g})} = - c + e^{\\tilde{g}}, then obtain \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + \\int (c + \\mathbf{D}{(c,\\tilde{g})}) dc) = \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + \\int e^{\\tilde{g}} dc)", "derivation": "\\mathbf{D}{(c,\\tilde{g})} = - c + e^{\\tilde{g}} and c + \\mathbf{D}{(c,\\tilde{g})} = e^{\\tilde{g}} and \\int (c + \\mathbf{D}{(c,\\tilde{g})}) dc = \\int e^{\\tilde{g}} dc and \\tilde{g} + \\int (c + \\mathbf{D}{(c,\\tilde{g})}) dc = \\tilde{g} + \\int e^{\\tilde{g}} dc and \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + \\int (c + \\mathbf{D}{(c,\\tilde{g})}) dc) = \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + \\int e^{\\tilde{g}} dc)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('c', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), Function('\\\\mathbf{D}')(Symbol('c', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Symbol('c', commutative=True), Function('\\\\mathbf{D}')(Symbol('c', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Integral(Add(Symbol('c', commutative=True), Function('\\\\mathbf{D}')(Symbol('c', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('c', commutative=True)))), Add(Symbol('\\\\tilde{g}', commutative=True), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Integral(Add(Symbol('c', commutative=True), Function('\\\\mathbf{D}')(Symbol('c', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(l,\\Omega)} = l^{\\Omega} and \\mathbf{A}{(l,\\Omega)} = l^{\\Omega} + 2 \\operatorname{a^{\\dagger}}{(l,\\Omega)}, then obtain 2 l^{\\Omega} + \\operatorname{a^{\\dagger}}{(l,\\Omega)} = 3 \\operatorname{a^{\\dagger}}{(l,\\Omega)}", "derivation": "\\operatorname{a^{\\dagger}}{(l,\\Omega)} = l^{\\Omega} and l^{\\Omega} + \\operatorname{a^{\\dagger}}{(l,\\Omega)} = 2 l^{\\Omega} and l^{\\Omega} + 2 \\operatorname{a^{\\dagger}}{(l,\\Omega)} = 2 l^{\\Omega} + \\operatorname{a^{\\dagger}}{(l,\\Omega)} and \\mathbf{A}{(l,\\Omega)} = l^{\\Omega} + 2 \\operatorname{a^{\\dagger}}{(l,\\Omega)} and \\mathbf{A}{(l,\\Omega)} = 3 \\operatorname{a^{\\dagger}}{(l,\\Omega)} and \\mathbf{A}{(l,\\Omega)} = 2 l^{\\Omega} + \\operatorname{a^{\\dagger}}{(l,\\Omega)} and 2 l^{\\Omega} + \\operatorname{a^{\\dagger}}{(l,\\Omega)} = 3 \\operatorname{a^{\\dagger}}{(l,\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(2), Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Integer(3), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(2), Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(2), Pow(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(3), Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(A_{y})} = \\sin{(\\log{(A_{y})})} and \\operatorname{V_{\\mathbf{B}}}{(L)} = \\sin{(L)}, then obtain e^{- \\operatorname{A_{2}}{(A_{y})} + \\operatorname{V_{\\mathbf{B}}}{(L)} + 1} = e^{- \\operatorname{A_{2}}{(A_{y})} + \\sin{(L)} + 1}", "derivation": "\\operatorname{A_{2}}{(A_{y})} = \\sin{(\\log{(A_{y})})} and \\operatorname{V_{\\mathbf{B}}}{(L)} = \\sin{(L)} and \\operatorname{V_{\\mathbf{B}}}{(L)} + \\frac{\\sin{(\\log{(A_{y})})}}{\\operatorname{A_{2}}{(A_{y})}} = \\sin{(L)} + \\frac{\\sin{(\\log{(A_{y})})}}{\\operatorname{A_{2}}{(A_{y})}} and \\operatorname{V_{\\mathbf{B}}}{(L)} + 1 = \\sin{(L)} + 1 and \\operatorname{V_{\\mathbf{B}}}{(L)} - \\sin{(\\log{(A_{y})})} + 1 = \\sin{(L)} - \\sin{(\\log{(A_{y})})} + 1 and e^{\\operatorname{V_{\\mathbf{B}}}{(L)} - \\sin{(\\log{(A_{y})})} + 1} = e^{\\sin{(L)} - \\sin{(\\log{(A_{y})})} + 1} and e^{- \\operatorname{A_{2}}{(A_{y})} + \\operatorname{V_{\\mathbf{B}}}{(L)} + 1} = e^{- \\operatorname{A_{2}}{(A_{y})} + \\sin{(L)} + 1}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('A_y', commutative=True)), sin(log(Symbol('A_y', commutative=True))))"], ["get_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["add", 2, "Mul(Pow(Function('A_2')(Symbol('A_y', commutative=True)), Integer(-1)), sin(log(Symbol('A_y', commutative=True))))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), Mul(Pow(Function('A_2')(Symbol('A_y', commutative=True)), Integer(-1)), sin(log(Symbol('A_y', commutative=True))))), Add(sin(Symbol('L', commutative=True)), Mul(Pow(Function('A_2')(Symbol('A_y', commutative=True)), Integer(-1)), sin(log(Symbol('A_y', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), Integer(1)), Add(sin(Symbol('L', commutative=True)), Integer(1)))"], [["minus", 4, "sin(log(Symbol('A_y', commutative=True)))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_y', commutative=True)))), Integer(1)), Add(sin(Symbol('L', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_y', commutative=True)))), Integer(1)))"], [["exp", 5], "Equality(exp(Add(Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_y', commutative=True)))), Integer(1))), exp(Add(sin(Symbol('L', commutative=True)), Mul(Integer(-1), sin(log(Symbol('A_y', commutative=True)))), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(exp(Add(Mul(Integer(-1), Function('A_2')(Symbol('A_y', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('L', commutative=True)), Integer(1))), exp(Add(Mul(Integer(-1), Function('A_2')(Symbol('A_y', commutative=True))), sin(Symbol('L', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_l{(v)} = \\cos{(v)}, then obtain \\iint (2 \\hat{H}_l{(v)} - \\cos{(v)} + \\frac{1}{\\cos{(v)}}) dv dv = \\iint (\\hat{H}_l{(v)} + \\frac{1}{\\cos{(v)}}) dv dv", "derivation": "\\hat{H}_l{(v)} = \\cos{(v)} and \\hat{H}_l{(v)} + \\frac{1}{\\cos{(v)}} = \\cos{(v)} + \\frac{1}{\\cos{(v)}} and \\hat{H}_l{(v)} - \\cos{(v)} + \\frac{1}{\\cos{(v)}} = \\frac{1}{\\cos{(v)}} and 2 \\hat{H}_l{(v)} - \\cos{(v)} + \\frac{1}{\\cos{(v)}} = \\hat{H}_l{(v)} + \\frac{1}{\\cos{(v)}} and \\int (2 \\hat{H}_l{(v)} - \\cos{(v)} + \\frac{1}{\\cos{(v)}}) dv = \\int (\\hat{H}_l{(v)} + \\frac{1}{\\cos{(v)}}) dv and \\iint (2 \\hat{H}_l{(v)} - \\cos{(v)} + \\frac{1}{\\cos{(v)}}) dv dv = \\iint (\\hat{H}_l{(v)} + \\frac{1}{\\cos{(v)}}) dv dv", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["add", 1, "Pow(cos(Symbol('v', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Add(cos(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))))"], [["minus", 2, "cos(Symbol('v', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), Mul(Integer(-1), cos(Symbol('v', commutative=True))), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Pow(cos(Symbol('v', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('v', commutative=True))), Mul(Integer(-1), cos(Symbol('v', commutative=True))), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Add(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('v', commutative=True))), Mul(Integer(-1), cos(Symbol('v', commutative=True))), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True))), Integral(Add(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True))))"], [["integrate", 5, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\hat{H}_l')(Symbol('v', commutative=True))), Mul(Integer(-1), cos(Symbol('v', commutative=True))), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Add(Function('\\\\hat{H}_l')(Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given S{(\\Omega)} = \\Omega, then obtain \\sin{((\\int 1 dS{(\\Omega)})^{S{(\\Omega)}})} = \\sin{((\\int \\frac{\\Omega}{S{(\\Omega)}} dS{(\\Omega)})^{S{(\\Omega)}})}", "derivation": "S{(\\Omega)} = \\Omega and 1 = \\frac{\\Omega}{S{(\\Omega)}} and \\int 1 d\\Omega = \\int \\frac{\\Omega}{S{(\\Omega)}} d\\Omega and (\\int 1 d\\Omega)^{\\Omega} = (\\int \\frac{\\Omega}{S{(\\Omega)}} d\\Omega)^{\\Omega} and (\\int 1 dS{(\\Omega)})^{S{(\\Omega)}} = (\\int \\frac{\\Omega}{S{(\\Omega)}} dS{(\\Omega)})^{S{(\\Omega)}} and \\sin{((\\int 1 dS{(\\Omega)})^{S{(\\Omega)}})} = \\sin{((\\int \\frac{\\Omega}{S{(\\Omega)}} dS{(\\Omega)})^{S{(\\Omega)}})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["divide", 1, "Function('S')(Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Integer(1), Tuple(Function('S')(Symbol('\\\\Omega', commutative=True)))), Function('S')(Symbol('\\\\Omega', commutative=True))), Pow(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Function('S')(Symbol('\\\\Omega', commutative=True)))), Function('S')(Symbol('\\\\Omega', commutative=True))))"], [["sin", 5], "Equality(sin(Pow(Integral(Integer(1), Tuple(Function('S')(Symbol('\\\\Omega', commutative=True)))), Function('S')(Symbol('\\\\Omega', commutative=True)))), sin(Pow(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Function('S')(Symbol('\\\\Omega', commutative=True)), Integer(-1))), Tuple(Function('S')(Symbol('\\\\Omega', commutative=True)))), Function('S')(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\phi,\\mathbf{S})} = \\int \\mathbf{S} \\phi d\\phi and \\varphi{(\\phi,\\mathbf{S})} = \\operatorname{M_{E}}^{\\mathbf{S}}{(\\phi,\\mathbf{S})}, then obtain \\varphi^{\\mathbf{S}}{(\\phi,\\mathbf{S})} = ((\\int \\mathbf{S} \\phi d\\phi)^{\\mathbf{S}})^{\\mathbf{S}}", "derivation": "\\operatorname{M_{E}}{(\\phi,\\mathbf{S})} = \\int \\mathbf{S} \\phi d\\phi and \\operatorname{M_{E}}^{\\mathbf{S}}{(\\phi,\\mathbf{S})} = (\\int \\mathbf{S} \\phi d\\phi)^{\\mathbf{S}} and (\\operatorname{M_{E}}^{\\mathbf{S}}{(\\phi,\\mathbf{S})})^{\\mathbf{S}} = ((\\int \\mathbf{S} \\phi d\\phi)^{\\mathbf{S}})^{\\mathbf{S}} and \\varphi{(\\phi,\\mathbf{S})} = \\operatorname{M_{E}}^{\\mathbf{S}}{(\\phi,\\mathbf{S})} and \\varphi^{\\mathbf{S}}{(\\phi,\\mathbf{S})} = ((\\int \\mathbf{S} \\phi d\\phi)^{\\mathbf{S}})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Integral(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(v_{2})} = e^{e^{v_{2}}}, then obtain 0 = (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}}) \\frac{d}{d v_{2}} (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}})", "derivation": "\\theta_{2}{(v_{2})} = e^{e^{v_{2}}} and 0 = - \\theta_{2}{(v_{2})} + e^{e^{v_{2}}} and \\frac{d}{d v_{2}} 0 = \\frac{d}{d v_{2}} (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}}) and 0 = (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}}) \\frac{d}{d v_{2}} 0 and 0 = (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}}) \\frac{d}{d v_{2}} (- \\theta_{2}{(v_{2})} + e^{e^{v_{2}}})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_2', commutative=True)), exp(exp(Symbol('v_2', commutative=True))))"], [["minus", 1, "Function('\\\\theta_2')(Symbol('v_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Derivative(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('v_2', commutative=True))), exp(exp(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(n)} = e^{n}, then derive \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} = e^{n}, then obtain \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} + 2 \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)} = 3 \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(n)} = e^{n} and \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} = \\frac{d}{d n} e^{n} and \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} = e^{n} and e^{n} = \\frac{d}{d n} e^{n} and e^{n} + \\frac{d}{d n} e^{n} = 2 \\frac{d}{d n} e^{n} and \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} + \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)} = 2 \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)} and \\frac{d}{d n} \\operatorname{f_{\\mathbf{v}}}{(n)} + 2 \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)} = 3 \\frac{d^{2}}{d n^{2}} \\operatorname{f_{\\mathbf{v}}}{(n)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), exp(Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["add", 4, "Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Add(exp(Symbol('n', commutative=True)), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2)))))"], [["add", 6, "Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2)))"], "Equality(Add(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Integer(2), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2))))), Mul(Integer(3), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\theta,f_{E})} = \\theta + f_{E}, then obtain - \\frac{- 2 \\theta + \\hat{\\mathbf{r}}{(\\theta,f_{E})}}{\\theta (\\theta + f_{E})} = - \\frac{- \\theta + f_{E}}{\\theta (\\theta + f_{E})}", "derivation": "\\hat{\\mathbf{r}}{(\\theta,f_{E})} = \\theta + f_{E} and - \\theta + \\hat{\\mathbf{r}}{(\\theta,f_{E})} = f_{E} and - 2 \\theta + \\hat{\\mathbf{r}}{(\\theta,f_{E})} = - \\theta + f_{E} and \\frac{- 2 \\theta + \\hat{\\mathbf{r}}{(\\theta,f_{E})}}{\\theta + f_{E}} = \\frac{- \\theta + f_{E}}{\\theta + f_{E}} and - \\frac{- 2 \\theta + \\hat{\\mathbf{r}}{(\\theta,f_{E})}}{\\theta (\\theta + f_{E})} = - \\frac{- \\theta + f_{E}}{\\theta (\\theta + f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))"], [["minus", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_E', commutative=True)))"], [["divide", 3, "Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True))), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_E', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True))), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Symbol('f_E', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})} = \\mathbf{H} + \\varepsilon_0, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} (\\frac{\\partial}{\\partial \\varepsilon_0} (- \\mathbf{H} + \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})}) + 1) = \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\varepsilon_0} \\varepsilon_0 + 1)", "derivation": "\\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})} = \\mathbf{H} + \\varepsilon_0 and - \\mathbf{H} + \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})} = \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} (- \\mathbf{H} + \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})}) = \\frac{d}{d \\varepsilon_0} \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} (- \\mathbf{H} + \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})}) + 1 = \\frac{d}{d \\varepsilon_0} \\varepsilon_0 + 1 and \\frac{\\partial}{\\partial \\mathbf{H}} (\\frac{\\partial}{\\partial \\varepsilon_0} (- \\mathbf{H} + \\operatorname{t_{2}}{(\\varepsilon_0,\\mathbf{H})}) + 1) = \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\varepsilon_0} \\varepsilon_0 + 1)", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True))"], [["differentiate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Function('t_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Derivative(Symbol('\\\\varepsilon_0', commutative=True), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(z,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z), then obtain \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\dot{\\mathbf{r}}^{2}{(z,\\mathbf{f})} dz = \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\dot{\\mathbf{r}}{(z,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z) dz", "derivation": "\\dot{\\mathbf{r}}{(z,\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z) and \\dot{\\mathbf{r}}^{2}{(z,\\mathbf{f})} = \\dot{\\mathbf{r}}{(z,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z) and \\int \\dot{\\mathbf{r}}^{2}{(z,\\mathbf{f})} dz = \\int \\dot{\\mathbf{r}}{(z,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z) dz and \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\dot{\\mathbf{r}}^{2}{(z,\\mathbf{f})} dz = \\frac{\\partial}{\\partial \\mathbf{f}} \\int \\dot{\\mathbf{r}}{(z,\\mathbf{f})} \\frac{\\partial}{\\partial \\mathbf{f}} (\\mathbf{f} - z) dz", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True))), Integral(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Integral(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\delta, then derive \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda}, then obtain \\frac{\\partial}{\\partial \\delta} \\Psi_{\\lambda} \\delta = \\Psi_{\\lambda}", "derivation": "\\operatorname{z^{*}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\delta and \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(\\delta,\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\delta} \\Psi_{\\lambda} \\delta and \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\delta} \\Psi_{\\lambda} \\delta = \\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"]]}, {"prompt": "Given C{(\\hat{x})} = \\cos{(\\sin{(\\hat{x})})}, then obtain \\frac{d}{d \\hat{x}} \\sin{(\\hat{x} C{(\\hat{x})})} = \\frac{d}{d \\hat{x}} \\sin{(\\hat{x} \\cos{(\\sin{(\\hat{x})})})}", "derivation": "C{(\\hat{x})} = \\cos{(\\sin{(\\hat{x})})} and \\hat{x} C{(\\hat{x})} = \\hat{x} \\cos{(\\sin{(\\hat{x})})} and \\sin{(\\hat{x} C{(\\hat{x})})} = \\sin{(\\hat{x} \\cos{(\\sin{(\\hat{x})})})} and \\frac{d}{d \\hat{x}} \\sin{(\\hat{x} C{(\\hat{x})})} = \\frac{d}{d \\hat{x}} \\sin{(\\hat{x} \\cos{(\\sin{(\\hat{x})})})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\hat{x}', commutative=True)), cos(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Function('C')(Symbol('\\\\hat{x}', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), cos(sin(Symbol('\\\\hat{x}', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Symbol('\\\\hat{x}', commutative=True), Function('C')(Symbol('\\\\hat{x}', commutative=True)))), sin(Mul(Symbol('\\\\hat{x}', commutative=True), cos(sin(Symbol('\\\\hat{x}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(sin(Mul(Symbol('\\\\hat{x}', commutative=True), Function('C')(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\hat{x}', commutative=True), cos(sin(Symbol('\\\\hat{x}', commutative=True))))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(y)} = \\cos{(e^{y})} and \\hat{H}_l{(y)} = \\cos{(e^{y})}, then obtain - \\hat{H}_l{(y)} + \\int \\operatorname{c_{0}}{(y)} dy = \\eta^{\\prime} - \\hat{H}_l{(y)} + \\operatorname{Ci}{(e^{y})}", "derivation": "\\operatorname{c_{0}}{(y)} = \\cos{(e^{y})} and \\hat{H}_l{(y)} = \\cos{(e^{y})} and \\operatorname{c_{0}}{(y)} = \\hat{H}_l{(y)} and \\int \\hat{H}_l{(y)} dy = \\int \\cos{(e^{y})} dy and \\int \\operatorname{c_{0}}{(y)} dy = \\int \\cos{(e^{y})} dy and - \\hat{H}_l{(y)} + \\int \\operatorname{c_{0}}{(y)} dy = - \\hat{H}_l{(y)} + \\int \\cos{(e^{y})} dy and - \\hat{H}_l{(y)} + \\int \\operatorname{c_{0}}{(y)} dy = \\eta^{\\prime} - \\hat{H}_l{(y)} + \\operatorname{Ci}{(e^{y})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('y', commutative=True)), cos(exp(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('y', commutative=True)), cos(exp(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('c_0')(Symbol('y', commutative=True)), Function('\\\\hat{H}_l')(Symbol('y', commutative=True)))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(exp(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Function('c_0')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(exp(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["add", 5, "Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Integral(Function('c_0')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Integral(cos(exp(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Integral(Function('c_0')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('y', commutative=True))), Ci(exp(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\varepsilon_{0}{(\\varepsilon)} = \\log{(\\varepsilon)} + \\iint \\operatorname{f^{*}}{(\\varepsilon)} d\\varepsilon d\\varepsilon, then obtain \\varepsilon_{0}{(\\varepsilon)} = \\log{(\\varepsilon)} + \\iint \\log{(\\varepsilon)} d\\varepsilon d\\varepsilon", "derivation": "\\operatorname{f^{*}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\int \\operatorname{f^{*}}{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\varepsilon)} d\\varepsilon and \\iint \\operatorname{f^{*}}{(\\varepsilon)} d\\varepsilon d\\varepsilon = \\iint \\log{(\\varepsilon)} d\\varepsilon d\\varepsilon and \\log{(\\varepsilon)} + \\iint \\operatorname{f^{*}}{(\\varepsilon)} d\\varepsilon d\\varepsilon = \\log{(\\varepsilon)} + \\iint \\log{(\\varepsilon)} d\\varepsilon d\\varepsilon and \\varepsilon_{0}{(\\varepsilon)} = \\log{(\\varepsilon)} + \\iint \\operatorname{f^{*}}{(\\varepsilon)} d\\varepsilon d\\varepsilon and \\varepsilon_{0}{(\\varepsilon)} = \\log{(\\varepsilon)} + \\iint \\log{(\\varepsilon)} d\\varepsilon d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 3, "log(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(log(Symbol('\\\\varepsilon', commutative=True)), Integral(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(log(Symbol('\\\\varepsilon', commutative=True)), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True)), Add(log(Symbol('\\\\varepsilon', commutative=True)), Integral(Function('f^*')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\varepsilon', commutative=True)), Add(log(Symbol('\\\\varepsilon', commutative=True)), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given i{(V_{\\mathbf{B}},f^{\\prime})} = \\cos{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})}, then obtain 2 i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} = 2 \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})}", "derivation": "i{(V_{\\mathbf{B}},f^{\\prime})} = \\cos{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})} and i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} = \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})} and i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} + \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})} = 2 \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})} and 2 i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} = i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} + \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})} and 2 i^{V_{\\mathbf{B}}}{(V_{\\mathbf{B}},f^{\\prime})} = 2 \\cos^{V_{\\mathbf{B}}}{(\\frac{f^{\\prime}}{V_{\\mathbf{B}}})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["add", 2, "Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 2, "Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('i')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(2), Pow(cos(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} = \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}} and \\mathbf{g}{(\\pi,\\mu)} = \\frac{\\cos{(\\pi)}}{\\mu}, then obtain - \\mathbf{g}{(\\pi,\\mu)} = - \\mathbf{g}{(\\pi,\\mu)} - \\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} + \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}}", "derivation": "\\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} = \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}} and 0 = - \\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} + \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}} and \\mathbf{g}{(\\pi,\\mu)} = \\frac{\\cos{(\\pi)}}{\\mu} and - \\frac{\\cos{(\\pi)}}{\\mu} = - \\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} + \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}} - \\frac{\\cos{(\\pi)}}{\\mu} and - \\mathbf{g}{(\\pi,\\mu)} = - \\mathbf{g}{(\\pi,\\mu)} - \\psi^{*}{(f_{\\mathbf{p}},g_{\\varepsilon})} + \\frac{\\sin{(f_{\\mathbf{p}})}}{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 1, "Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), cos(Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(F_{g},F_{N})} = - F_{N} + F_{g} and \\operatorname{A_{y}}{(F_{N})} = - F_{N}, then obtain \\int F_{N} \\operatorname{z^{*}}{(F_{g},F_{N})} dF_{g} = \\int F_{N} (- F_{N} + F_{g}) dF_{g}", "derivation": "\\operatorname{z^{*}}{(F_{g},F_{N})} = - F_{N} + F_{g} and \\operatorname{A_{y}}{(F_{N})} = - F_{N} and \\operatorname{z^{*}}{(F_{g},F_{N})} = F_{g} + \\operatorname{A_{y}}{(F_{N})} and F_{N} \\operatorname{z^{*}}{(F_{g},F_{N})} = F_{N} (F_{g} + \\operatorname{A_{y}}{(F_{N})}) and - F_{N} + F_{g} = F_{g} + \\operatorname{A_{y}}{(F_{N})} and F_{N} \\operatorname{z^{*}}{(F_{g},F_{N})} = F_{N} (- F_{N} + F_{g}) and \\int F_{N} \\operatorname{z^{*}}{(F_{g},F_{N})} dF_{g} = \\int F_{N} (- F_{N} + F_{g}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('F_g', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('z^*')(Symbol('F_g', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_g', commutative=True), Function('A_y')(Symbol('F_N', commutative=True))))"], [["times", 3, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Function('z^*')(Symbol('F_g', commutative=True), Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), Add(Symbol('F_g', commutative=True), Function('A_y')(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('F_g', commutative=True)), Add(Symbol('F_g', commutative=True), Function('A_y')(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('F_N', commutative=True), Function('z^*')(Symbol('F_g', commutative=True), Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('F_g', commutative=True))))"], [["integrate", 6, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Symbol('F_N', commutative=True), Function('z^*')(Symbol('F_g', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Symbol('F_N', commutative=True), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(r)} = e^{r}, then obtain (\\psi^{*}{(r)} \\psi^{*}^{2 r}{(r)})^{r} = (\\psi^{*}^{2 r}{(r)} e^{r})^{r}", "derivation": "\\psi^{*}{(r)} = e^{r} and \\psi^{*}^{r}{(r)} = (e^{r})^{r} and \\psi^{*}{(r)} \\psi^{*}^{r}{(r)} = \\psi^{*}^{r}{(r)} e^{r} and \\psi^{*}{(r)} (e^{r})^{r} = e^{r} (e^{r})^{r} and \\psi^{*}{(r)} \\psi^{*}^{r}{(r)} (e^{r})^{r} = \\psi^{*}^{r}{(r)} e^{r} (e^{r})^{r} and \\psi^{*}{(r)} \\psi^{*}^{2 r}{(r)} = \\psi^{*}^{2 r}{(r)} e^{r} and (\\psi^{*}{(r)} \\psi^{*}^{2 r}{(r)})^{r} = (\\psi^{*}^{2 r}{(r)} e^{r})^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["times", 1, "Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('r', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Mul(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\psi^*')(Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Mul(exp(Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], [["times", 4, "Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('r', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))), Mul(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)), Pow(exp(Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('\\\\psi^*')(Symbol('r', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True)))), Mul(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True))), exp(Symbol('r', commutative=True))))"], [["power", 6, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Function('\\\\psi^*')(Symbol('r', commutative=True)), Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Pow(Mul(Pow(Function('\\\\psi^*')(Symbol('r', commutative=True)), Mul(Integer(2), Symbol('r', commutative=True))), exp(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(A_{2})} = \\log{(A_{2})}, then derive 2 \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} = \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} + \\frac{1}{A_{2}}, then obtain \\frac{d}{d A_{2}} (\\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})}) + 2 \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} = \\frac{d}{d A_{2}} (\\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})}) + \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} + \\frac{1}{A_{2}}", "derivation": "\\hat{H}_{\\lambda}{(A_{2})} = \\log{(A_{2})} and 2 \\hat{H}_{\\lambda}{(A_{2})} = \\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})} and \\frac{d}{d A_{2}} 2 \\hat{H}_{\\lambda}{(A_{2})} = \\frac{d}{d A_{2}} (\\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})}) and 2 \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} = \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} + \\frac{1}{A_{2}} and \\frac{d}{d A_{2}} (\\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})}) + 2 \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} = \\frac{d}{d A_{2}} (\\hat{H}_{\\lambda}{(A_{2})} + \\log{(A_{2})}) + \\frac{d}{d A_{2}} \\hat{H}_{\\lambda}{(A_{2})} + \\frac{1}{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Pow(Symbol('A_2', commutative=True), Integer(-1))))"], [["add", 4, "Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(2), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))), Add(Derivative(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Pow(Symbol('A_2', commutative=True), Integer(-1))))"]]}, {"prompt": "Given i{(\\pi)} = \\log{(\\pi)} and \\mathbf{D}{(\\pi)} = \\log{(\\pi)}, then obtain \\mathbf{D}^{- 2 \\pi}{(\\pi)} \\log{(\\mathbf{D}^{\\pi}{(\\pi)})} = \\mathbf{D}^{- 2 \\pi}{(\\pi)} \\log{(\\log{(\\pi)}^{\\pi})}", "derivation": "i{(\\pi)} = \\log{(\\pi)} and i^{\\pi}{(\\pi)} = \\log{(\\pi)}^{\\pi} and \\mathbf{D}{(\\pi)} = \\log{(\\pi)} and i^{\\pi}{(\\pi)} = \\mathbf{D}^{\\pi}{(\\pi)} and \\mathbf{D}^{\\pi}{(\\pi)} = \\log{(\\pi)}^{\\pi} and \\log{(\\mathbf{D}^{\\pi}{(\\pi)})} = \\log{(\\log{(\\pi)}^{\\pi})} and \\mathbf{D}^{- \\pi}{(\\pi)} \\log{(\\mathbf{D}^{\\pi}{(\\pi)})} = \\mathbf{D}^{- \\pi}{(\\pi)} \\log{(\\log{(\\pi)}^{\\pi})} and \\mathbf{D}^{- 2 \\pi}{(\\pi)} \\log{(\\mathbf{D}^{\\pi}{(\\pi)})} = \\mathbf{D}^{- 2 \\pi}{(\\pi)} \\log{(\\log{(\\pi)}^{\\pi})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('i')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["log", 5], "Equality(log(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))), log(Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["divide", 6, "Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))), log(Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))))"], [["divide", 7, "Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))), log(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))), Mul(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\pi', commutative=True))), log(Pow(log(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\delta)} = \\log{(\\delta)}, then derive \\int \\Omega{(\\delta)} d\\delta = J + \\delta \\log{(\\delta)} - \\delta, then obtain \\frac{\\int \\log{(\\delta)} d\\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} = \\frac{J + \\delta \\Omega{(\\delta)} - \\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}}", "derivation": "\\Omega{(\\delta)} = \\log{(\\delta)} and \\int \\Omega{(\\delta)} d\\delta = \\int \\log{(\\delta)} d\\delta and \\int \\Omega{(\\delta)} d\\delta = J + \\delta \\log{(\\delta)} - \\delta and \\frac{\\int \\Omega{(\\delta)} d\\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} = \\frac{J + \\delta \\log{(\\delta)} - \\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} and \\frac{\\int \\Omega{(\\delta)} d\\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} = \\frac{J + \\delta \\Omega{(\\delta)} - \\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} and \\frac{\\int \\log{(\\delta)} d\\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}} = \\frac{J + \\delta \\Omega{(\\delta)} - \\delta}{\\operatorname{C_{2}}{(\\pi,\\mathbf{S},\\sigma_p)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), log(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(log(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('J', commutative=True), Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))"], [["divide", 3, "Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Symbol('\\\\delta', commutative=True), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\Omega')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\Omega')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(S,z)} = \\log{(S^{z})}, then obtain \\int (\\int \\frac{\\operatorname{L_{\\varepsilon}}{(S,z)}}{\\log{(S^{z})}} dz)^{z} dz = \\int (\\int 1 dz)^{z} dz", "derivation": "\\operatorname{L_{\\varepsilon}}{(S,z)} = \\log{(S^{z})} and \\frac{\\operatorname{L_{\\varepsilon}}{(S,z)}}{\\log{(S^{z})}} = 1 and \\int \\frac{\\operatorname{L_{\\varepsilon}}{(S,z)}}{\\log{(S^{z})}} dz = \\int 1 dz and (\\int \\frac{\\operatorname{L_{\\varepsilon}}{(S,z)}}{\\log{(S^{z})}} dz)^{z} = (\\int 1 dz)^{z} and \\int (\\int \\frac{\\operatorname{L_{\\varepsilon}}{(S,z)}}{\\log{(S^{z})}} dz)^{z} dz = \\int (\\int 1 dz)^{z} dz", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True))))"], [["divide", 1, "log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True)))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Pow(log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Function('L_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Pow(log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integer(-1))), Tuple(Symbol('z', commutative=True))), Integral(Integer(1), Tuple(Symbol('z', commutative=True))))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Integral(Mul(Function('L_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Pow(log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integer(-1))), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["integrate", 4, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(Integral(Mul(Function('L_{\\\\varepsilon}')(Symbol('S', commutative=True), Symbol('z', commutative=True)), Pow(log(Pow(Symbol('S', commutative=True), Symbol('z', commutative=True))), Integer(-1))), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(Integral(Integer(1), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(z^{*})} = \\log{(z^{*})}, then derive \\frac{d}{d z^{*}} \\operatorname{v_{1}}{(z^{*})} = \\frac{1}{z^{*}}, then obtain \\operatorname{v_{1}}{(z^{*})} + \\frac{1}{z^{*}} = \\operatorname{v_{1}}{(z^{*})} + \\frac{d}{d z^{*}} \\log{(z^{*})}", "derivation": "\\operatorname{v_{1}}{(z^{*})} = \\log{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{v_{1}}{(z^{*})} = \\frac{d}{d z^{*}} \\log{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{v_{1}}{(z^{*})} = \\frac{1}{z^{*}} and \\operatorname{v_{1}}{(z^{*})} + \\frac{d}{d z^{*}} \\operatorname{v_{1}}{(z^{*})} = \\operatorname{v_{1}}{(z^{*})} + \\frac{d}{d z^{*}} \\log{(z^{*})} and \\operatorname{v_{1}}{(z^{*})} + \\frac{1}{z^{*}} = \\operatorname{v_{1}}{(z^{*})} + \\frac{d}{d z^{*}} \\log{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Pow(Symbol('z^*', commutative=True), Integer(-1)))"], [["add", 2, "Function('v_1')(Symbol('z^*', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('z^*', commutative=True)), Derivative(Function('v_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(Function('v_1')(Symbol('z^*', commutative=True)), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('v_1')(Symbol('z^*', commutative=True)), Pow(Symbol('z^*', commutative=True), Integer(-1))), Add(Function('v_1')(Symbol('z^*', commutative=True)), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta}, then derive \\operatorname{v_{z}}{(\\eta)} = e^{\\eta}, then obtain - 2 \\eta + 2 \\operatorname{v_{z}}{(\\eta)} = - 2 \\eta + \\operatorname{v_{z}}{(\\eta)} + e^{\\eta}", "derivation": "\\operatorname{v_{z}}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\operatorname{v_{z}}{(\\eta)} = e^{\\eta} and \\frac{d}{d \\eta} e^{\\eta} = e^{\\eta} and - \\eta + \\frac{d}{d \\eta} e^{\\eta} = - \\eta + e^{\\eta} and - \\eta + \\operatorname{v_{z}}{(\\eta)} = - \\eta + e^{\\eta} and - 2 \\eta + 2 \\operatorname{v_{z}}{(\\eta)} = - 2 \\eta + \\operatorname{v_{z}}{(\\eta)} + e^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('v_z')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), exp(Symbol('\\\\eta', commutative=True)))"], [["minus", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('v_z')(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('v_z')(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Mul(Integer(2), Function('v_z')(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\eta', commutative=True)), Function('v_z')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given r{(L)} = e^{\\cos{(L)}}, then obtain \\frac{d}{d L} r{(L)} \\int \\log{(\\frac{d}{d L} r{(L)})} dL = \\frac{d}{d L} e^{\\cos{(L)}} \\int \\log{(\\frac{d}{d L} r{(L)})} dL", "derivation": "r{(L)} = e^{\\cos{(L)}} and \\frac{d}{d L} r{(L)} = \\frac{d}{d L} e^{\\cos{(L)}} and \\log{(\\frac{d}{d L} r{(L)})} = \\log{(\\frac{d}{d L} e^{\\cos{(L)}})} and \\int \\log{(\\frac{d}{d L} r{(L)})} dL = \\int \\log{(\\frac{d}{d L} e^{\\cos{(L)}})} dL and \\frac{d}{d L} r{(L)} \\int \\log{(\\frac{d}{d L} e^{\\cos{(L)}})} dL = \\frac{d}{d L} e^{\\cos{(L)}} \\int \\log{(\\frac{d}{d L} e^{\\cos{(L)}})} dL and \\frac{d}{d L} r{(L)} \\int \\log{(\\frac{d}{d L} r{(L)})} dL = \\frac{d}{d L} e^{\\cos{(L)}} \\int \\log{(\\frac{d}{d L} r{(L)})} dL", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(log(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True))), Integral(log(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True))))"], [["times", 2, "Integral(log(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True)))"], "Equality(Mul(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integral(log(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True)))), Mul(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integral(log(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integral(log(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True)))), Mul(Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integral(log(Derivative(Function('r')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)} = \\frac{\\mathbf{s}}{\\Psi \\phi_1}, then obtain - \\Psi + \\mathbf{r} + \\frac{\\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)}}{\\phi_1} = - \\Psi + \\mathbf{r} + \\frac{\\mathbf{s}}{\\Psi \\phi_1^{2}}", "derivation": "\\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)} = \\frac{\\mathbf{s}}{\\Psi \\phi_1} and \\frac{\\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)}}{\\phi_1} = \\frac{\\mathbf{s}}{\\Psi \\phi_1^{2}} and - \\Psi + \\frac{\\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)}}{\\phi_1} = - \\Psi + \\frac{\\mathbf{s}}{\\Psi \\phi_1^{2}} and - \\Psi + \\mathbf{r} + \\frac{\\mathbf{E}{(\\phi_1,\\mathbf{s},\\Psi)}}{\\phi_1} = - \\Psi + \\mathbf{r} + \\frac{\\mathbf{s}}{\\Psi \\phi_1^{2}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2))))"], [["minus", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)))))"], [["add", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given \\hat{p}{(\\dot{z})} = \\sin{(\\dot{z})} and \\operatorname{F_{H}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain \\tilde{\\infty} \\operatorname{F_{H}}{(V_{\\mathbf{B}})} = \\tilde{\\infty} \\sin{(V_{\\mathbf{B}})}", "derivation": "\\hat{p}{(\\dot{z})} = \\sin{(\\dot{z})} and \\operatorname{F_{H}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\frac{\\operatorname{F_{H}}{(V_{\\mathbf{B}})}}{- \\hat{p}{(\\dot{z})} + \\sin{(\\dot{z})}} = \\frac{\\sin{(V_{\\mathbf{B}})}}{- \\hat{p}{(\\dot{z})} + \\sin{(\\dot{z})}} and \\tilde{\\infty} \\operatorname{F_{H}}{(V_{\\mathbf{B}})} = \\tilde{\\infty} \\sin{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], ["get_premise", "Equality(Function('F_H')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 2, "Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True))), sin(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True))), sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), Function('F_H')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\dot{z}', commutative=True))), sin(Symbol('\\\\dot{z}', commutative=True))), Integer(-1)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(zoo, Function('F_H')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(zoo, sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(v_{t})} = e^{v_{t}}, then derive \\log{(e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t})} = \\log{(\\mathbf{D} + 2 e^{v_{t}})}, then obtain \\int \\log{(e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t})} d\\mathbf{D} = \\int \\log{(\\mathbf{D} + 2 e^{v_{t}})} d\\mathbf{D}", "derivation": "\\theta_{2}{(v_{t})} = e^{v_{t}} and \\int \\theta_{2}{(v_{t})} dv_{t} = \\int e^{v_{t}} dv_{t} and e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t} = e^{v_{t}} + \\int e^{v_{t}} dv_{t} and \\log{(e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t})} = \\log{(e^{v_{t}} + \\int e^{v_{t}} dv_{t})} and \\log{(e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t})} = \\log{(\\mathbf{D} + 2 e^{v_{t}})} and \\int \\log{(e^{v_{t}} + \\int \\theta_{2}{(v_{t})} dv_{t})} d\\mathbf{D} = \\int \\log{(\\mathbf{D} + 2 e^{v_{t}})} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["add", 2, "exp(Symbol('v_t', commutative=True))"], "Equality(Add(exp(Symbol('v_t', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))), Add(exp(Symbol('v_t', commutative=True)), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["log", 3], "Equality(log(Add(exp(Symbol('v_t', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), log(Add(exp(Symbol('v_t', commutative=True)), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(log(Add(exp(Symbol('v_t', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(2), exp(Symbol('v_t', commutative=True))))))"], [["integrate", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(log(Add(exp(Symbol('v_t', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(log(Add(Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(2), exp(Symbol('v_t', commutative=True))))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(H)} = e^{e^{H}}, then derive \\frac{d}{d H} \\eta^{\\prime}{(H)} = e^{H} e^{e^{H}}, then obtain e^{2 H} e^{2 e^{H}} = \\eta^{\\prime}{(H)} e^{2 H} e^{e^{H}}", "derivation": "\\eta^{\\prime}{(H)} = e^{e^{H}} and \\frac{d}{d H} \\eta^{\\prime}{(H)} = \\frac{d}{d H} e^{e^{H}} and \\frac{d}{d H} \\eta^{\\prime}{(H)} = e^{H} e^{e^{H}} and \\frac{d}{d H} \\eta^{\\prime}{(H)} = \\eta^{\\prime}{(H)} e^{H} and \\frac{d}{d H} \\eta^{\\prime}{(H)} \\frac{d}{d H} e^{e^{H}} = \\eta^{\\prime}{(H)} e^{H} \\frac{d}{d H} e^{e^{H}} and (\\frac{d}{d H} \\eta^{\\prime}{(H)})^{2} = \\eta^{\\prime}{(H)} e^{H} \\frac{d}{d H} \\eta^{\\prime}{(H)} and e^{2 H} e^{2 e^{H}} = \\eta^{\\prime}{(H)} e^{2 H} e^{e^{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(exp(Symbol('H', commutative=True)), exp(exp(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True))))"], [["times", 4, "Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Derivative(exp(exp(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2)), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), exp(Symbol('H', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(exp(Mul(Integer(2), Symbol('H', commutative=True))), exp(Mul(Integer(2), exp(Symbol('H', commutative=True))))), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('H', commutative=True)), exp(Mul(Integer(2), Symbol('H', commutative=True))), exp(exp(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\eta,v_{z})} = \\eta v_{z}, then derive \\cos{(\\frac{\\partial}{\\partial \\eta} \\mathbf{g}{(\\eta,v_{z})})} = \\cos{(v_{z})}, then obtain \\int \\cos{(\\frac{\\partial}{\\partial \\eta} \\eta v_{z})} dv_{z} = \\int \\cos{(v_{z})} dv_{z}", "derivation": "\\mathbf{g}{(\\eta,v_{z})} = \\eta v_{z} and \\frac{\\partial}{\\partial \\eta} \\mathbf{g}{(\\eta,v_{z})} = \\frac{\\partial}{\\partial \\eta} \\eta v_{z} and \\cos{(\\frac{\\partial}{\\partial \\eta} \\mathbf{g}{(\\eta,v_{z})})} = \\cos{(\\frac{\\partial}{\\partial \\eta} \\eta v_{z})} and \\cos{(\\frac{\\partial}{\\partial \\eta} \\mathbf{g}{(\\eta,v_{z})})} = \\cos{(v_{z})} and \\cos{(\\frac{\\partial}{\\partial \\eta} \\eta v_{z})} = \\cos{(v_{z})} and \\int \\cos{(\\frac{\\partial}{\\partial \\eta} \\eta v_{z})} dv_{z} = \\int \\cos{(v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), cos(Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), cos(Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(cos(Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), cos(Symbol('v_z', commutative=True)))"], [["integrate", 5, "Symbol('v_z', commutative=True)"], "Equality(Integral(cos(Derivative(Mul(Symbol('\\\\eta', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mu)} = \\cos{(\\mu)} and \\operatorname{A_{x}}{(\\sigma_x)} = \\log{(\\sigma_x)}, then obtain \\int \\cos{(\\mu)} d\\mu = - \\operatorname{A_{x}}{(\\sigma_x)} + \\log{(\\sigma_x)} + \\int \\cos{(\\mu)} d\\mu", "derivation": "\\operatorname{C_{2}}{(\\mu)} = \\cos{(\\mu)} and \\operatorname{A_{x}}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\operatorname{A_{x}}{(\\sigma_x)} + \\int \\operatorname{C_{2}}{(\\mu)} d\\mu = \\log{(\\sigma_x)} + \\int \\operatorname{C_{2}}{(\\mu)} d\\mu and \\operatorname{A_{x}}{(\\sigma_x)} + \\int \\cos{(\\mu)} d\\mu = \\log{(\\sigma_x)} + \\int \\cos{(\\mu)} d\\mu and \\int \\cos{(\\mu)} d\\mu = - \\operatorname{A_{x}}{(\\sigma_x)} + \\log{(\\sigma_x)} + \\int \\cos{(\\mu)} d\\mu", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], ["get_premise", "Equality(Function('A_x')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["add", 2, "Integral(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('A_x')(Symbol('\\\\sigma_x', commutative=True)), Integral(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(log(Symbol('\\\\sigma_x', commutative=True)), Integral(Function('C_2')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('A_x')(Symbol('\\\\sigma_x', commutative=True)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(log(Symbol('\\\\sigma_x', commutative=True)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["minus", 4, "Function('A_x')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\sigma_x', commutative=True))), log(Symbol('\\\\sigma_x', commutative=True)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})}, then obtain (- \\Psi^{\\dagger} + \\int \\operatorname{r_{0}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (- \\Psi^{\\dagger} + \\int \\log{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}}", "derivation": "\\operatorname{r_{0}}{(\\Psi^{\\dagger})} = \\log{(\\Psi^{\\dagger})} and \\int \\operatorname{r_{0}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int \\log{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and - \\Psi^{\\dagger} + \\int \\operatorname{r_{0}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = - \\Psi^{\\dagger} + \\int \\log{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} and (- \\Psi^{\\dagger} + \\int \\operatorname{r_{0}}{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (- \\Psi^{\\dagger} + \\int \\log{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["minus", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Function('r_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["power", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Function('r_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\theta_2)} = \\sin{(\\theta_2)} and t{(\\theta_2)} = \\sin{(\\sin{(\\theta_2)})} and \\operatorname{n_{2}}{(\\theta_2,Z)} = e^{Z} + \\log{(\\sin^{\\theta_2}{(\\operatorname{f^{\\prime}}{(\\theta_2)})})}, then obtain \\mathbf{P}{(r_{0})} + \\operatorname{n_{2}}{(\\theta_2,Z)} = \\mathbf{P}{(r_{0})} + e^{Z} + \\log{(\\sin^{\\theta_2}{(\\sin{(\\theta_2)})})}", "derivation": "\\operatorname{f^{\\prime}}{(\\theta_2)} = \\sin{(\\theta_2)} and t{(\\theta_2)} = \\sin{(\\sin{(\\theta_2)})} and t{(\\theta_2)} = \\sin{(\\operatorname{f^{\\prime}}{(\\theta_2)})} and t^{\\theta_2}{(\\theta_2)} = \\sin^{\\theta_2}{(\\operatorname{f^{\\prime}}{(\\theta_2)})} and \\operatorname{n_{2}}{(\\theta_2,Z)} = e^{Z} + \\log{(\\sin^{\\theta_2}{(\\operatorname{f^{\\prime}}{(\\theta_2)})})} and \\sin^{\\theta_2}{(\\sin{(\\theta_2)})} = \\sin^{\\theta_2}{(\\operatorname{f^{\\prime}}{(\\theta_2)})} and \\operatorname{n_{2}}{(\\theta_2,Z)} = e^{Z} + \\log{(\\sin^{\\theta_2}{(\\sin{(\\theta_2)})})} and \\mathbf{P}{(r_{0})} + \\operatorname{n_{2}}{(\\theta_2,Z)} = \\mathbf{P}{(r_{0})} + e^{Z} + \\log{(\\sin^{\\theta_2}{(\\sin{(\\theta_2)})})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), sin(sin(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), sin(Function('f^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))))"], [["power", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('t')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(sin(Function('f^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('\\\\theta_2', commutative=True), Symbol('Z', commutative=True)), Add(exp(Symbol('Z', commutative=True)), log(Pow(sin(Function('f^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(sin(Function('f^{\\\\prime}')(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('n_2')(Symbol('\\\\theta_2', commutative=True), Symbol('Z', commutative=True)), Add(exp(Symbol('Z', commutative=True)), log(Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))))"], [["add", 7, "Function('\\\\mathbf{P}')(Symbol('r_0', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('r_0', commutative=True)), Function('n_2')(Symbol('\\\\theta_2', commutative=True), Symbol('Z', commutative=True))), Add(Function('\\\\mathbf{P}')(Symbol('r_0', commutative=True)), exp(Symbol('Z', commutative=True)), log(Pow(sin(sin(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given B{(C_{2},m_{s})} = C_{2} + m_{s}, then obtain \\frac{2 C_{2} B{(C_{2},m_{s})}}{\\psi} = \\frac{C_{2} (2 C_{2} + 2 m_{s})}{\\psi}", "derivation": "B{(C_{2},m_{s})} = C_{2} + m_{s} and C_{2} + m_{s} + B{(C_{2},m_{s})} = 2 C_{2} + 2 m_{s} and 2 B{(C_{2},m_{s})} = 2 C_{2} + 2 m_{s} and C_{2} (C_{2} + m_{s} + B{(C_{2},m_{s})}) = C_{2} (2 C_{2} + 2 m_{s}) and C_{2} (C_{2} + m_{s} + B{(C_{2},m_{s})}) = 2 C_{2} B{(C_{2},m_{s})} and 2 C_{2} B{(C_{2},m_{s})} = C_{2} (2 C_{2} + 2 m_{s}) and \\frac{2 C_{2} B{(C_{2},m_{s})}}{\\psi} = \\frac{C_{2} (2 C_{2} + 2 m_{s})}{\\psi}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True)))"], [["add", 1, "Add(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))"], "Equality(Add(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True))))"], [["times", 2, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True)))), Mul(Symbol('C_2', commutative=True), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True)))), Mul(Integer(2), Symbol('C_2', commutative=True), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Symbol('C_2', commutative=True), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))), Mul(Symbol('C_2', commutative=True), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True)))))"], [["divide", 6, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Integer(2), Symbol('C_2', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('B')(Symbol('C_2', commutative=True), Symbol('m_s', commutative=True))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(2), Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given I{(r_{0},\\mu)} = \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}}, then derive I{(r_{0},\\mu)} = - \\frac{\\mu}{r_{0}^{2}}, then obtain \\frac{\\mu}{r_{0}^{2}} + \\int (- \\frac{\\mu}{r_{0}^{2}} + r_{0}) d\\mu = \\frac{\\mu}{r_{0}^{2}} + \\int (r_{0} + \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}}) d\\mu", "derivation": "I{(r_{0},\\mu)} = \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}} and r_{0} + I{(r_{0},\\mu)} = r_{0} + \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}} and I{(r_{0},\\mu)} = - \\frac{\\mu}{r_{0}^{2}} and - \\frac{\\mu}{r_{0}^{2}} + r_{0} = r_{0} + \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}} and \\int (- \\frac{\\mu}{r_{0}^{2}} + r_{0}) d\\mu = \\int (r_{0} + \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}}) d\\mu and \\frac{\\mu}{r_{0}^{2}} + \\int (- \\frac{\\mu}{r_{0}^{2}} + r_{0}) d\\mu = \\frac{\\mu}{r_{0}^{2}} + \\int (r_{0} + \\frac{\\partial}{\\partial r_{0}} \\frac{\\mu}{r_{0}}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('r_0', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["add", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Function('I')(Symbol('r_0', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('I')(Symbol('r_0', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))), Symbol('r_0', commutative=True)), Add(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Add(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2)))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-2))), Integral(Add(Symbol('r_0', commutative=True), Derivative(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(E_{x})} = \\sin{(E_{x})}, then derive \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} - 1 = \\cos{(E_{x})} - 1, then obtain (\\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} - 1) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = (\\cos{(E_{x})} - 1) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})}", "derivation": "\\mathbf{J}_M{(E_{x})} = \\sin{(E_{x})} and \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = \\frac{d}{d E_{x}} \\sin{(E_{x})} and \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} - 1 = \\frac{d}{d E_{x}} \\sin{(E_{x})} - 1 and \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} - 1 = \\cos{(E_{x})} - 1 and (\\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} - 1) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = (\\cos{(E_{x})} - 1) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)), Add(cos(Symbol('E_x', commutative=True)), Integer(-1)))"], [["times", 4, "Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Add(cos(Symbol('E_x', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho{(\\chi,z)} = \\chi z, then obtain \\int \\chi \\rho{(\\chi,z)} dz + 1 = \\int \\chi^{2} z dz + 1", "derivation": "\\rho{(\\chi,z)} = \\chi z and \\chi \\rho{(\\chi,z)} = \\chi^{2} z and \\int \\chi \\rho{(\\chi,z)} dz = \\int \\chi^{2} z dz and \\int \\chi \\rho{(\\chi,z)} dz + 1 = \\int \\chi^{2} z dz + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('z', commutative=True)))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('z', commutative=True)))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\rho')(Symbol('\\\\chi', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(2)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = \\dot{\\mathbf{r}} + \\hat{H}_l, then derive \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = 1, then obtain 0 = 1 - \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)}", "derivation": "\\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = \\dot{\\mathbf{r}} + \\hat{H}_l and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\hat{H}_l) and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = 1 and - \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\hat{H}_l) + \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = 1 - \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} (\\dot{\\mathbf{r}} + \\hat{H}_l) and 0 = 1 - \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\tilde{g}{(\\dot{\\mathbf{r}},\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(C)} = \\log{(C)} and \\operatorname{t_{2}}{(\\phi,C)} = \\frac{\\partial}{\\partial \\phi} \\phi \\operatorname{A_{y}}^{C}{(C)}, then derive \\operatorname{t_{2}}{(\\phi,C)} = \\log{(C)}^{C}, then obtain \\frac{\\phi \\operatorname{A_{y}}^{C}{(C)} \\log{(C)}^{- C}}{2} = \\frac{\\phi \\operatorname{t_{2}}{(\\phi,C)} \\log{(C)}^{- C}}{2}", "derivation": "\\operatorname{A_{y}}{(C)} = \\log{(C)} and \\operatorname{A_{y}}^{C}{(C)} = \\log{(C)}^{C} and \\phi \\operatorname{A_{y}}^{C}{(C)} = \\phi \\log{(C)}^{C} and \\frac{\\partial}{\\partial \\phi} \\phi \\operatorname{A_{y}}^{C}{(C)} = \\frac{\\partial}{\\partial \\phi} \\phi \\log{(C)}^{C} and \\operatorname{t_{2}}{(\\phi,C)} = \\frac{\\partial}{\\partial \\phi} \\phi \\operatorname{A_{y}}^{C}{(C)} and \\operatorname{t_{2}}{(\\phi,C)} = \\frac{\\partial}{\\partial \\phi} \\phi \\log{(C)}^{C} and \\operatorname{t_{2}}{(\\phi,C)} = \\log{(C)}^{C} and \\phi \\operatorname{A_{y}}^{C}{(C)} = \\phi \\operatorname{t_{2}}{(\\phi,C)} and \\frac{\\phi \\operatorname{A_{y}}^{C}{(C)} \\log{(C)}^{- C}}{2} = \\frac{\\phi \\operatorname{t_{2}}{(\\phi,C)} \\log{(C)}^{- C}}{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["times", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True))))"], [["divide", 8, "Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], "Equality(Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True), Pow(Function('A_y')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))), Mul(Rational(1, 2), Symbol('\\\\phi', commutative=True), Function('t_2')(Symbol('\\\\phi', commutative=True), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(x,r_{0})} = \\frac{x}{r_{0}}, then obtain - (\\dot{\\mathbf{r}}^{2}{(x,r_{0})})^{- r_{0}} \\dot{\\mathbf{r}}^{2}{(x,r_{0})} = - \\frac{x (\\dot{\\mathbf{r}}^{2}{(x,r_{0})})^{- r_{0}} \\dot{\\mathbf{r}}{(x,r_{0})}}{r_{0}}", "derivation": "\\dot{\\mathbf{r}}{(x,r_{0})} = \\frac{x}{r_{0}} and \\dot{\\mathbf{r}}^{2}{(x,r_{0})} = \\frac{x \\dot{\\mathbf{r}}{(x,r_{0})}}{r_{0}} and - \\dot{\\mathbf{r}}^{2}{(x,r_{0})} = - \\frac{x \\dot{\\mathbf{r}}{(x,r_{0})}}{r_{0}} and - (\\dot{\\mathbf{r}}^{2}{(x,r_{0})})^{- r_{0}} \\dot{\\mathbf{r}}^{2}{(x,r_{0})} = - \\frac{x (\\dot{\\mathbf{r}}^{2}{(x,r_{0})})^{- r_{0}} \\dot{\\mathbf{r}}{(x,r_{0})}}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["times", 1, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True))))"], [["divide", 3, "Pow(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('r_0', commutative=True))), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1)), Symbol('x', commutative=True), Pow(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('r_0', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('x', commutative=True), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(F_{g},h)} = F_{g} + h, then obtain - \\frac{- F_{g} - h}{\\dot{x}^{3}{(F_{g},h)}} - \\frac{1}{\\dot{x}^{2}{(F_{g},h)}} = 0", "derivation": "\\dot{x}{(F_{g},h)} = F_{g} + h and \\dot{x}^{2}{(F_{g},h)} = (F_{g} + h) \\dot{x}{(F_{g},h)} and 1 = \\frac{F_{g} + h}{\\dot{x}{(F_{g},h)}} and - \\frac{1}{\\dot{x}^{2}{(F_{g},h)}} = - \\frac{F_{g} + h}{\\dot{x}^{3}{(F_{g},h)}} and - \\frac{1}{(F_{g} + h) \\dot{x}{(F_{g},h)}} = - \\frac{F_{g} + h}{\\dot{x}^{3}{(F_{g},h)}} and - \\frac{1}{\\dot{x}^{2}{(F_{g},h)}} = \\frac{- F_{g} - h}{\\dot{x}^{3}{(F_{g},h)}} and - \\frac{- F_{g} - h}{\\dot{x}^{3}{(F_{g},h)}} - \\frac{1}{\\dot{x}^{2}{(F_{g},h)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)))"], [["times", 1, "Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True))"], "Equality(Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(2)), Mul(Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-2))), Mul(Integer(-1), Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-3))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-3))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-2))), Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-3))))"], [["minus", 6, "Mul(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-3)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True))), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-3))), Mul(Integer(-1), Pow(Function('\\\\dot{x}')(Symbol('F_g', commutative=True), Symbol('h', commutative=True)), Integer(-2)))), Integer(0))"]]}, {"prompt": "Given S{(L)} = \\sin{(L)}, then obtain \\int (\\mathbf{s} + \\frac{d}{d L} S{(L)}) d\\mathbf{s} = \\int (\\mathbf{s} + \\cos{(L)}) d\\mathbf{s}", "derivation": "S{(L)} = \\sin{(L)} and \\frac{d}{d L} S{(L)} = \\frac{d}{d L} \\sin{(L)} and \\mathbf{s} + \\frac{d}{d L} S{(L)} = \\mathbf{s} + \\frac{d}{d L} \\sin{(L)} and \\int (\\mathbf{s} + \\frac{d}{d L} S{(L)}) d\\mathbf{s} = \\int (\\mathbf{s} + \\frac{d}{d L} \\sin{(L)}) d\\mathbf{s} and \\int (\\mathbf{s} + \\frac{d}{d L} S{(L)}) d\\mathbf{s} = \\int (\\mathbf{s} + \\cos{(L)}) d\\mathbf{s}", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('S')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('S')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Function('S')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), cos(Symbol('L', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given x{(\\omega,\\varphi)} = \\frac{\\varphi}{\\omega}, then obtain x{(\\omega,\\varphi)} + \\frac{\\partial}{\\partial \\omega} \\frac{\\omega x{(\\omega,\\varphi)}}{\\varphi} = x{(\\omega,\\varphi)} + \\frac{d}{d \\omega} 1", "derivation": "x{(\\omega,\\varphi)} = \\frac{\\varphi}{\\omega} and \\frac{\\omega x{(\\omega,\\varphi)}}{\\varphi} = 1 and \\frac{\\partial}{\\partial \\omega} \\frac{\\omega x{(\\omega,\\varphi)}}{\\varphi} = \\frac{d}{d \\omega} 1 and x{(\\omega,\\varphi)} + \\frac{\\partial}{\\partial \\omega} \\frac{\\omega x{(\\omega,\\varphi)}}{\\varphi} = x{(\\omega,\\varphi)} + \\frac{d}{d \\omega} 1", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 3, "Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Function('x')(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)), Derivative(Integer(1), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(t_{1},\\hat{x})} = \\hat{x} + t_{1}, then obtain 0^{t_{1}} \\tilde{\\infty}^{t_{1}} = 1", "derivation": "\\nabla{(t_{1},\\hat{x})} = \\hat{x} + t_{1} and 1 = \\frac{\\hat{x} + t_{1}}{\\nabla{(t_{1},\\hat{x})}} and 0 = \\frac{\\hat{x} + t_{1}}{\\nabla{(t_{1},\\hat{x})}} - 1 and 0 = \\hat{x} (\\frac{\\hat{x} + t_{1}}{\\nabla{(t_{1},\\hat{x})}} - 1) and 0^{t_{1}} = (\\hat{x} (\\frac{\\hat{x} + t_{1}}{\\nabla{(t_{1},\\hat{x})}} - 1))^{t_{1}} and 0^{t_{1}} (\\hat{x} (\\frac{\\hat{x} + t_{1}}{\\nabla{(t_{1},\\hat{x})}} - 1))^{- t_{1}} = 1 and 0^{t_{1}} \\tilde{\\infty}^{t_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)))"], [["divide", 1, "Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))))"], [["minus", 2, 1], "Equality(Integer(0), Add(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Integer(-1)))"], [["times", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Integer(-1))))"], [["power", 4, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('t_1', commutative=True)), Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Integer(-1))), Symbol('t_1', commutative=True)))"], [["divide", 5, "Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Integer(-1))), Symbol('t_1', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('t_1', commutative=True)), Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t_1', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Integer(-1))), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Integer(0), Symbol('t_1', commutative=True)), Pow(zoo, Symbol('t_1', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\mathbf{P}{(v,n)} = e^{n - v}, then derive \\phi_2 - \\int - n \\mathbf{P}{(v,n)} dn - \\int v \\mathbf{P}{(v,n)} dn = \\sigma_p + (n - v - 1) e^{n - v}, then obtain \\phi_2 - g^{\\prime}_{\\varepsilon} - (1 - n) e^{n - v} - \\int v e^{n - v} dn = \\sigma_p + (n - v - 1) e^{n - v}", "derivation": "\\mathbf{P}{(v,n)} = e^{n - v} and (n - v) \\mathbf{P}{(v,n)} = (n - v) e^{n - v} and \\int (n - v) \\mathbf{P}{(v,n)} dn = \\int (n - v) e^{n - v} dn and \\phi_2 - \\int - n \\mathbf{P}{(v,n)} dn - \\int v \\mathbf{P}{(v,n)} dn = \\sigma_p + (n - v - 1) e^{n - v} and \\phi_2 - \\int - n e^{n - v} dn - \\int v e^{n - v} dn = \\sigma_p + (n - v - 1) e^{n - v} and \\phi_2 - g^{\\prime}_{\\varepsilon} - (1 - n) e^{n - v} - \\int v e^{n - v} dn = \\sigma_p + (n - v - 1) e^{n - v}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('v', commutative=True), Symbol('n', commutative=True)), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["times", 1, "Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))"], "Equality(Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{P}')(Symbol('v', commutative=True), Symbol('n', commutative=True))), Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Function('\\\\mathbf{P}')(Symbol('v', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('n', commutative=True), Function('\\\\mathbf{P}')(Symbol('v', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('v', commutative=True), Function('\\\\mathbf{P}')(Symbol('v', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(-1)), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('n', commutative=True), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))), Tuple(Symbol('n', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('v', commutative=True), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))), Tuple(Symbol('n', commutative=True))))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(-1)), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Add(Integer(1), Mul(Integer(-1), Symbol('n', commutative=True))), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))), Mul(Integer(-1), Integral(Mul(Symbol('v', commutative=True), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))), Tuple(Symbol('n', commutative=True))))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(-1)), exp(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))))"]]}, {"prompt": "Given q{(\\eta,m_{s})} = (e^{\\eta})^{m_{s}}, then obtain - \\frac{\\frac{\\partial}{\\partial \\eta} q{(\\eta,m_{s})}}{m_{s}} = - (e^{\\eta})^{m_{s}}", "derivation": "q{(\\eta,m_{s})} = (e^{\\eta})^{m_{s}} and \\frac{q{(\\eta,m_{s})}}{m_{s}} = \\frac{(e^{\\eta})^{m_{s}}}{m_{s}} and \\frac{\\partial}{\\partial \\eta} \\frac{q{(\\eta,m_{s})}}{m_{s}} = \\frac{\\partial}{\\partial \\eta} \\frac{(e^{\\eta})^{m_{s}}}{m_{s}} and - \\frac{\\partial}{\\partial \\eta} \\frac{q{(\\eta,m_{s})}}{m_{s}} = - \\frac{\\partial}{\\partial \\eta} \\frac{(e^{\\eta})^{m_{s}}}{m_{s}} and - \\frac{\\frac{\\partial}{\\partial \\eta} q{(\\eta,m_{s})}}{m_{s}} = - (e^{\\eta})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\eta', commutative=True), Symbol('m_s', commutative=True)), Pow(exp(Symbol('\\\\eta', commutative=True)), Symbol('m_s', commutative=True)))"], [["divide", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\eta', commutative=True), Symbol('m_s', commutative=True))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(exp(Symbol('\\\\eta', commutative=True)), Symbol('m_s', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\eta', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(exp(Symbol('\\\\eta', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\eta', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(exp(Symbol('\\\\eta', commutative=True)), Symbol('m_s', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Derivative(Function('q')(Symbol('\\\\eta', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(exp(Symbol('\\\\eta', commutative=True)), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})} = \\theta_1 i m_{s}, then obtain \\frac{(\\theta_1 i m_{s})^{\\theta_1}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} - \\frac{\\operatorname{y^{\\prime}}^{\\theta_1}{(\\theta_1,i,m_{s})}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} = 0", "derivation": "\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})} = \\theta_1 i m_{s} and \\operatorname{y^{\\prime}}^{\\theta_1}{(\\theta_1,i,m_{s})} = (\\theta_1 i m_{s})^{\\theta_1} and - \\frac{\\operatorname{y^{\\prime}}^{\\theta_1}{(\\theta_1,i,m_{s})}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} = - \\frac{(\\theta_1 i m_{s})^{\\theta_1}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} and \\frac{(\\theta_1 i m_{s})^{\\theta_1}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} - \\frac{\\operatorname{y^{\\prime}}^{\\theta_1}{(\\theta_1,i,m_{s})}}{\\operatorname{y^{\\prime}}{(\\theta_1,i,m_{s})}} = 0", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))))"], [["add", 3, "Mul(Pow(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Pow(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Pow(Function('y^{\\\\prime}')(Symbol('\\\\theta_1', commutative=True), Symbol('i', commutative=True), Symbol('m_s', commutative=True)), Symbol('\\\\theta_1', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{J}_P{(i)} = \\cos{(i)}, then derive \\frac{d}{d i} \\mathbf{J}_P{(i)} = - \\sin{(i)}, then obtain 0 = \\frac{\\frac{\\sin{(i)}}{\\frac{d}{d i} \\mathbf{J}_P{(i)}} + 1}{\\frac{d}{d i} \\cos{(i)}}", "derivation": "\\mathbf{J}_P{(i)} = \\cos{(i)} and \\frac{d}{d i} \\mathbf{J}_P{(i)} = \\frac{d}{d i} \\cos{(i)} and \\frac{d}{d i} \\mathbf{J}_P{(i)} = - \\sin{(i)} and \\frac{\\frac{d}{d i} \\mathbf{J}_P{(i)}}{\\frac{d}{d i} \\cos{(i)}} = 1 and - \\frac{\\sin{(i)}}{\\frac{d}{d i} \\cos{(i)}} = 1 and 0 = \\frac{\\sin{(i)}}{\\frac{d}{d i} \\cos{(i)}} + 1 and 0 = \\frac{\\sin{(i)}}{\\frac{d}{d i} \\mathbf{J}_P{(i)}} + 1 and 0 = \\frac{\\frac{\\sin{(i)}}{\\frac{d}{d i} \\mathbf{J}_P{(i)}} + 1}{\\frac{d}{d i} \\cos{(i)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('i', commutative=True))))"], [["divide", 2, "Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), sin(Symbol('i', commutative=True)), Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["minus", 5, "Mul(Integer(-1), sin(Symbol('i', commutative=True)), Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(sin(Symbol('i', commutative=True)), Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Add(Mul(sin(Symbol('i', commutative=True)), Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(1)))"], [["times", 7, "Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))"], "Equality(Integer(0), Mul(Add(Mul(sin(Symbol('i', commutative=True)), Pow(Derivative(Function('\\\\mathbf{J}_P')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))), Integer(1)), Pow(Derivative(cos(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\lambda{(\\mu)} = e^{\\mu}, then derive - e^{\\mu} + \\int \\lambda{(\\mu)} d\\mu = A_{x}, then derive e^{B - \\lambda{(\\mu)} + e^{\\mu}} = e^{A_{x}}, then obtain e^{B} = e^{A_{x}}", "derivation": "\\lambda{(\\mu)} = e^{\\mu} and \\int \\lambda{(\\mu)} d\\mu = \\int e^{\\mu} d\\mu and - e^{\\mu} + \\int \\lambda{(\\mu)} d\\mu = - e^{\\mu} + \\int e^{\\mu} d\\mu and - e^{\\mu} + \\int \\lambda{(\\mu)} d\\mu = A_{x} and e^{- e^{\\mu} + \\int \\lambda{(\\mu)} d\\mu} = e^{A_{x}} and e^{- e^{\\mu} + \\int e^{\\mu} d\\mu} = e^{A_{x}} and e^{- \\lambda{(\\mu)} + \\int e^{\\mu} d\\mu} = e^{A_{x}} and e^{B - \\lambda{(\\mu)} + e^{\\mu}} = e^{A_{x}} and e^{B} = e^{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Integral(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Integral(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Symbol('A_x', commutative=True))"], [["exp", 4], "Equality(exp(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Integral(Function('\\\\lambda')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), exp(Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(exp(Add(Mul(Integer(-1), exp(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), exp(Symbol('A_x', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(exp(Add(Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mu', commutative=True))), Integral(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))), exp(Symbol('A_x', commutative=True)))"], [["evaluate_integrals", 7], "Equality(exp(Add(Symbol('B', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mu', commutative=True))), exp(Symbol('\\\\mu', commutative=True)))), exp(Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(exp(Symbol('B', commutative=True)), exp(Symbol('A_x', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(r)} = \\log{(r)}, then derive \\int \\psi^{*}{(r)} dr = A + r \\log{(r)} - r, then obtain \\log{(r)} + \\int \\psi^{*}{(r)} dr = A + r \\log{(r)} - r + \\log{(r)}", "derivation": "\\psi^{*}{(r)} = \\log{(r)} and \\int \\psi^{*}{(r)} dr = \\int \\log{(r)} dr and \\int \\psi^{*}{(r)} dr = A + r \\log{(r)} - r and \\int \\psi^{*}{(r)} dr = A + r \\psi^{*}{(r)} - r and \\log{(r)} + \\int \\psi^{*}{(r)} dr = A + r \\psi^{*}{(r)} - r + \\log{(r)} and \\log{(r)} + \\int \\log{(r)} dr = A + r \\log{(r)} - r + \\log{(r)} and \\log{(r)} + \\int \\psi^{*}{(r)} dr = A + r \\log{(r)} - r + \\log{(r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('A', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\psi^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('A', commutative=True), Mul(Symbol('r', commutative=True), Function('\\\\psi^*')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["add", 4, "log(Symbol('r', commutative=True))"], "Equality(Add(log(Symbol('r', commutative=True)), Integral(Function('\\\\psi^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Symbol('A', commutative=True), Mul(Symbol('r', commutative=True), Function('\\\\psi^*')(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(log(Symbol('r', commutative=True)), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Symbol('A', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(log(Symbol('r', commutative=True)), Integral(Function('\\\\psi^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(Symbol('A', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(B,n_{1},p)} = p (B + n_{1}), then obtain \\frac{n_{1}^{2} p (\\int \\Psi_{nl}{(B,n_{1},p)} dn_{1})^{B}}{2} = \\frac{n_{1}^{2} p (B n_{1} p + \\phi_2 + \\frac{n_{1}^{2} p}{2})^{B}}{2}", "derivation": "\\Psi_{nl}{(B,n_{1},p)} = p (B + n_{1}) and \\int \\Psi_{nl}{(B,n_{1},p)} dn_{1} = \\int p (B + n_{1}) dn_{1} and (\\int \\Psi_{nl}{(B,n_{1},p)} dn_{1})^{B} = (\\int p (B + n_{1}) dn_{1})^{B} and \\frac{n_{1}^{2} p (\\int \\Psi_{nl}{(B,n_{1},p)} dn_{1})^{B}}{2} = \\frac{n_{1}^{2} p (\\int p (B + n_{1}) dn_{1})^{B}}{2} and \\frac{n_{1}^{2} p (\\int \\Psi_{nl}{(B,n_{1},p)} dn_{1})^{B}}{2} = \\frac{n_{1}^{2} p (B n_{1} p + \\phi_2 + \\frac{n_{1}^{2} p}{2})^{B}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Mul(Symbol('p', commutative=True), Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Mul(Symbol('p', commutative=True), Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Symbol('B', commutative=True)))"], [["times", 3, "Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True), Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('B', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True), Pow(Integral(Mul(Symbol('p', commutative=True), Add(Symbol('B', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True))), Symbol('B', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True), Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Symbol('B', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True), Pow(Add(Mul(Symbol('B', commutative=True), Symbol('n_1', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\phi_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)), Symbol('p', commutative=True))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\chi,A_{1})} = e^{\\frac{A_{1}}{\\chi}}, then derive \\int \\mathbf{J}_f{(\\chi,A_{1})} dA_{1} = \\chi e^{\\frac{A_{1}}{\\chi}} + \\hbar, then obtain (Q \\int e^{\\frac{A_{1}}{\\chi}} dA_{1})^{A_{1}} = (Q (\\chi \\mathbf{J}_f{(\\chi,A_{1})} + \\hbar))^{A_{1}}", "derivation": "\\mathbf{J}_f{(\\chi,A_{1})} = e^{\\frac{A_{1}}{\\chi}} and \\int \\mathbf{J}_f{(\\chi,A_{1})} dA_{1} = \\int e^{\\frac{A_{1}}{\\chi}} dA_{1} and \\int \\mathbf{J}_f{(\\chi,A_{1})} dA_{1} = \\chi e^{\\frac{A_{1}}{\\chi}} + \\hbar and \\int \\mathbf{J}_f{(\\chi,A_{1})} dA_{1} = \\chi \\mathbf{J}_f{(\\chi,A_{1})} + \\hbar and \\int e^{\\frac{A_{1}}{\\chi}} dA_{1} = \\chi \\mathbf{J}_f{(\\chi,A_{1})} + \\hbar and Q \\int e^{\\frac{A_{1}}{\\chi}} dA_{1} = Q (\\chi \\mathbf{J}_f{(\\chi,A_{1})} + \\hbar) and (Q \\int e^{\\frac{A_{1}}{\\chi}} dA_{1})^{A_{1}} = (Q (\\chi \\mathbf{J}_f{(\\chi,A_{1})} + \\hbar))^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Mul(Symbol('\\\\chi', commutative=True), exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True))), Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["times", 5, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Integral(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True)))), Mul(Symbol('Q', commutative=True), Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hbar', commutative=True))))"], [["power", 6, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Symbol('Q', commutative=True), Integral(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Pow(Mul(Symbol('Q', commutative=True), Add(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\chi', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hbar', commutative=True))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(C)} = e^{C} and \\operatorname{C_{2}}{(C)} = \\frac{d}{d C} \\hat{x}_0{(C)}, then obtain \\int \\frac{d^{2}}{d C^{2}} \\operatorname{C_{2}}{(C)} dC = \\int \\frac{d^{3}}{d C^{3}} e^{C} dC", "derivation": "\\hat{x}_0{(C)} = e^{C} and \\operatorname{C_{2}}{(C)} = \\frac{d}{d C} \\hat{x}_0{(C)} and \\frac{d}{d C} \\operatorname{C_{2}}{(C)} = \\frac{d^{2}}{d C^{2}} \\hat{x}_0{(C)} and \\frac{d^{2}}{d C^{2}} \\operatorname{C_{2}}{(C)} = \\frac{d^{3}}{d C^{3}} \\hat{x}_0{(C)} and \\int \\frac{d^{2}}{d C^{2}} \\operatorname{C_{2}}{(C)} dC = \\int \\frac{d^{3}}{d C^{3}} \\hat{x}_0{(C)} dC and \\int \\frac{d^{2}}{d C^{2}} \\operatorname{C_{2}}{(C)} dC = \\int \\frac{d^{3}}{d C^{3}} e^{C} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('C', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('\\\\hat{x}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(Function('\\\\hat{x}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(3))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Function('C_2')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(3))), Tuple(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(Function('C_2')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(3))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given H{(n_{2})} = \\int \\cos{(n_{2})} dn_{2}, then derive H{(n_{2})} = r_{0} + \\sin{(n_{2})}, then obtain (H{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) (H{(n_{2})} + \\int \\cos{(n_{2})} dn_{2}) = 2 (H{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) \\int \\cos{(n_{2})} dn_{2}", "derivation": "H{(n_{2})} = \\int \\cos{(n_{2})} dn_{2} and H{(n_{2})} = r_{0} + \\sin{(n_{2})} and r_{0} + \\sin{(n_{2})} = \\int \\cos{(n_{2})} dn_{2} and H{(n_{2})} + \\int \\cos{(n_{2})} dn_{2} = r_{0} + \\sin{(n_{2})} + \\int \\cos{(n_{2})} dn_{2} and H{(n_{2})} + \\int \\cos{(n_{2})} dn_{2} = 2 \\int \\cos{(n_{2})} dn_{2} and (H{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) (H{(n_{2})} + \\int \\cos{(n_{2})} dn_{2}) = 2 (H{(n_{2})} - \\int \\cos{(n_{2})} dn_{2}) \\int \\cos{(n_{2})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('H')(Symbol('n_2', commutative=True)), Add(Symbol('r_0', commutative=True), sin(Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('r_0', commutative=True), sin(Symbol('n_2', commutative=True))), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], "Equality(Add(Function('H')(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Symbol('r_0', commutative=True), sin(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('H')(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["times", 5, "Add(Function('H')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], "Equality(Mul(Add(Function('H')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Add(Function('H')(Symbol('n_2', commutative=True)), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Mul(Integer(2), Add(Function('H')(Symbol('n_2', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))), Integral(cos(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given h{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{v_{t}}{(a^{\\dagger})} = a^{\\dagger} h{(a^{\\dagger})}, then obtain - (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} + \\int \\operatorname{v_{t}}^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} = - (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} + \\int (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger}", "derivation": "h{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{v_{t}}{(a^{\\dagger})} = a^{\\dagger} h{(a^{\\dagger})} and \\operatorname{v_{t}}^{a^{\\dagger}}{(a^{\\dagger})} = (a^{\\dagger} h{(a^{\\dagger})})^{a^{\\dagger}} and \\operatorname{v_{t}}^{a^{\\dagger}}{(a^{\\dagger})} = (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} and \\int \\operatorname{v_{t}}^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} = \\int (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger} and - (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} + \\int \\operatorname{v_{t}}^{a^{\\dagger}}{(a^{\\dagger})} da^{\\dagger} = - (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} + \\int (a^{\\dagger} e^{a^{\\dagger}})^{a^{\\dagger}} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('h')(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('h')(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('v_t')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Function('v_t')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 5, "Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Function('v_t')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given v{(n_{1},\\mu)} = \\mu + n_{1}, then obtain \\frac{\\mu (\\mu + v{(n_{1},\\mu)}) - \\mu - n_{1}}{\\mu + v{(n_{1},\\mu)}} = \\frac{\\mu (2 \\mu + n_{1}) - \\mu - n_{1}}{\\mu + v{(n_{1},\\mu)}}", "derivation": "v{(n_{1},\\mu)} = \\mu + n_{1} and \\mu + v{(n_{1},\\mu)} = 2 \\mu + n_{1} and \\mu (\\mu + v{(n_{1},\\mu)}) = \\mu (2 \\mu + n_{1}) and \\mu (\\mu + v{(n_{1},\\mu)}) - \\mu - n_{1} = \\mu (2 \\mu + n_{1}) - \\mu - n_{1} and \\frac{\\mu (\\mu + v{(n_{1},\\mu)}) - \\mu - n_{1}}{\\mu + v{(n_{1},\\mu)}} = \\frac{\\mu (2 \\mu + n_{1}) - \\mu - n_{1}}{\\mu + v{(n_{1},\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Symbol('n_1', commutative=True)))"], [["times", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Symbol('n_1', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\mu', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True))), Add(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Symbol('n_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\mu', commutative=True), Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mu', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Symbol('n_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(F_{c})} = e^{F_{c}}, then obtain 1 = \\frac{F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int e^{F_{c}} dF_{c}}{F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int \\operatorname{L_{\\varepsilon}}{(F_{c})} dF_{c}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(F_{c})} = e^{F_{c}} and \\int \\operatorname{L_{\\varepsilon}}{(F_{c})} dF_{c} = \\int e^{F_{c}} dF_{c} and F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int \\operatorname{L_{\\varepsilon}}{(F_{c})} dF_{c} = F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int e^{F_{c}} dF_{c} and 1 = \\frac{F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int e^{F_{c}} dF_{c}}{F_{c} + \\operatorname{L_{\\varepsilon}}{(F_{c})} + \\int \\operatorname{L_{\\varepsilon}}{(F_{c})} dF_{c}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["add", 2, "Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)))"], "Equality(Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["divide", 3, "Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))), Integer(-1)), Add(Symbol('F_c', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('F_c', commutative=True)), Integral(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} = (Q + \\Psi)^{\\mathbf{p}}, then obtain Q (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{\\mathbf{p}} \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} d\\mathbf{p} = Q (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{2 \\mathbf{p}} d\\mathbf{p}", "derivation": "\\mathbf{J}{(\\mathbf{p},\\Psi,Q)} = (Q + \\Psi)^{\\mathbf{p}} and (Q + \\Psi)^{\\mathbf{p}} \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} = (Q + \\Psi)^{2 \\mathbf{p}} and \\int (Q + \\Psi)^{\\mathbf{p}} \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} d\\mathbf{p} = \\int (Q + \\Psi)^{2 \\mathbf{p}} d\\mathbf{p} and (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{\\mathbf{p}} \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} d\\mathbf{p} = (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{2 \\mathbf{p}} d\\mathbf{p} and Q (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{\\mathbf{p}} \\mathbf{J}{(\\mathbf{p},\\Psi,Q)} d\\mathbf{p} = Q (Q + \\Psi)^{- 2 \\mathbf{p}} \\int (Q + \\Psi)^{2 \\mathbf{p}} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True))), Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["divide", 3, "Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["times", 4, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Symbol('Q', commutative=True), Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Add(Symbol('Q', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given S{(F_{N})} = \\cos{(F_{N})}, then obtain \\frac{d}{d F_{N}} \\frac{0^{F_{N}}}{S{(F_{N})}} = \\frac{d}{d F_{N}} \\frac{1}{S{(F_{N})}}", "derivation": "S{(F_{N})} = \\cos{(F_{N})} and S{(F_{N})} + \\cos{(F_{N})} = 2 \\cos{(F_{N})} and 0 = - S{(F_{N})} + \\cos{(F_{N})} and 0^{F_{N}} = (- S{(F_{N})} + \\cos{(F_{N})})^{F_{N}} and \\frac{0^{F_{N}}}{S{(F_{N})}} = \\frac{(- S{(F_{N})} + \\cos{(F_{N})})^{F_{N}}}{S{(F_{N})}} and \\frac{(- S{(F_{N})} + \\cos{(F_{N})})^{F_{N}}}{S{(F_{N})}} = \\frac{1}{S{(F_{N})}} and \\frac{d}{d F_{N}} \\frac{(- S{(F_{N})} + \\cos{(F_{N})})^{F_{N}}}{S{(F_{N})}} = \\frac{d}{d F_{N}} \\frac{1}{S{(F_{N})}} and \\frac{d}{d F_{N}} \\frac{0^{F_{N}}}{S{(F_{N})}} = \\frac{d}{d F_{N}} \\frac{1}{S{(F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], [["add", 1, "cos(Symbol('F_N', commutative=True))"], "Equality(Add(Function('S')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True))), Mul(Integer(2), cos(Symbol('F_N', commutative=True))))"], [["minus", 2, "Add(Function('S')(Symbol('F_N', commutative=True)), cos(Symbol('F_N', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('S')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Integer(0), Symbol('F_N', commutative=True)), Pow(Add(Mul(Integer(-1), Function('S')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"], [["divide", 4, "Function('S')(Symbol('F_N', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('F_N', commutative=True)), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Function('S')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('S')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1))), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1)))"], [["differentiate", 6, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Function('S')(Symbol('F_N', commutative=True))), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Derivative(Mul(Pow(Integer(0), Symbol('F_N', commutative=True)), Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Function('S')(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hat{H},n)} = \\int (\\hat{H} - n) dn, then derive \\hat{H} \\mathbf{H}{(\\hat{H},n)} = \\hat{H} (\\hat{H} n + \\theta_1 - \\frac{n^{2}}{2}), then obtain \\hat{H} \\mathbf{H}{(\\hat{H},n)} + 1 = \\hat{H} (\\hat{H} n + \\theta_1 - \\frac{n^{2}}{2}) + 1", "derivation": "\\mathbf{H}{(\\hat{H},n)} = \\int (\\hat{H} - n) dn and \\hat{H} \\mathbf{H}{(\\hat{H},n)} = \\hat{H} \\int (\\hat{H} - n) dn and \\hat{H} \\mathbf{H}{(\\hat{H},n)} = \\hat{H} (\\hat{H} n + \\theta_1 - \\frac{n^{2}}{2}) and \\hat{H} \\mathbf{H}{(\\hat{H},n)} + 1 = \\hat{H} \\int (\\hat{H} - n) dn + 1 and \\hat{H} (\\hat{H} n + \\theta_1 - \\frac{n^{2}}{2}) = \\hat{H} \\int (\\hat{H} - n) dn and \\hat{H} \\mathbf{H}{(\\hat{H},n)} + 1 = \\hat{H} (\\hat{H} n + \\theta_1 - \\frac{n^{2}}{2}) + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), Integer(1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))))), Mul(Symbol('\\\\hat{H}', commutative=True), Integral(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\theta_1', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})}, then derive \\int \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\delta - \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain - \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} + \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\delta - \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} - \\cos{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\mathbf{g}{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and \\int \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\int \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\delta - \\cos{(g^{\\prime}_{\\varepsilon})} and \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\delta - \\cos{(g^{\\prime}_{\\varepsilon})} and - \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} + \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\delta - \\mathbf{g}{(g^{\\prime}_{\\varepsilon})} - \\cos{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["minus", 4, "Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(z,t)} = \\frac{\\partial}{\\partial z} (t + z) and \\operatorname{A_{y}}{(z,t)} = t + z, then derive \\operatorname{L_{\\varepsilon}}^{z}{(z,t)} = 1, then obtain - z + (\\frac{\\partial}{\\partial z} \\operatorname{A_{y}}{(z,t)})^{z} = 1 - z", "derivation": "\\operatorname{L_{\\varepsilon}}{(z,t)} = \\frac{\\partial}{\\partial z} (t + z) and \\operatorname{L_{\\varepsilon}}^{z}{(z,t)} = (\\frac{\\partial}{\\partial z} (t + z))^{z} and \\operatorname{L_{\\varepsilon}}^{z}{(z,t)} = 1 and \\operatorname{A_{y}}{(z,t)} = t + z and \\operatorname{L_{\\varepsilon}}{(z,t)} = \\frac{\\partial}{\\partial z} \\operatorname{A_{y}}{(z,t)} and - z + \\operatorname{L_{\\varepsilon}}^{z}{(z,t)} = 1 - z and - z + (\\frac{\\partial}{\\partial z} \\operatorname{A_{y}}{(z,t)})^{z} = 1 - z", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Derivative(Add(Symbol('t', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Symbol('z', commutative=True)), Pow(Derivative(Add(Symbol('t', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Symbol('z', commutative=True)), Integer(1))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Add(Symbol('t', commutative=True), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Derivative(Function('A_y')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Symbol('z', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Derivative(Function('A_y')(Symbol('z', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(m_{s},v_{z})} = \\frac{e^{v_{z}}}{m_{s}} and \\rho_{f}{(m_{s},v_{z})} = \\frac{e^{v_{z}} + \\frac{e^{v_{z}}}{m_{s}}}{\\mathbf{J}_P{(m_{s},v_{z})} + e^{v_{z}}}, then obtain 1 = \\rho_{f}{(m_{s},v_{z})}", "derivation": "\\mathbf{J}_P{(m_{s},v_{z})} = \\frac{e^{v_{z}}}{m_{s}} and \\mathbf{J}_P{(m_{s},v_{z})} + e^{v_{z}} = e^{v_{z}} + \\frac{e^{v_{z}}}{m_{s}} and 1 = \\frac{e^{v_{z}} + \\frac{e^{v_{z}}}{m_{s}}}{\\mathbf{J}_P{(m_{s},v_{z})} + e^{v_{z}}} and \\rho_{f}{(m_{s},v_{z})} = \\frac{e^{v_{z}} + \\frac{e^{v_{z}}}{m_{s}}}{\\mathbf{J}_P{(m_{s},v_{z})} + e^{v_{z}}} and 1 = \\rho_{f}{(m_{s},v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), exp(Symbol('v_z', commutative=True))))"], [["add", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Add(exp(Symbol('v_z', commutative=True)), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), exp(Symbol('v_z', commutative=True)))))"], [["divide", 2, "Add(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Integer(-1)), Add(exp(Symbol('v_z', commutative=True)), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), exp(Symbol('v_z', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Add(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Integer(-1)), Add(exp(Symbol('v_z', commutative=True)), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), exp(Symbol('v_z', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Function('\\\\rho_f')(Symbol('m_s', commutative=True), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})}, then derive \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\cos{(\\mu_0)} \\cos{(\\sin{(\\mu_0)})}, then obtain \\frac{d}{d \\mu_0} \\sin{(\\sin{(\\mu_0)})} = \\cos{(\\mu_0)} \\cos{(\\sin{(\\mu_0)})}", "derivation": "\\operatorname{v_{z}}{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})} and \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\sin{(\\sin{(\\mu_0)})} and \\frac{d}{d \\mu_0} \\operatorname{v_{z}}{(\\mu_0)} = \\cos{(\\mu_0)} \\cos{(\\sin{(\\mu_0)})} and \\frac{d}{d \\mu_0} \\sin{(\\sin{(\\mu_0)})} = \\cos{(\\mu_0)} \\cos{(\\sin{(\\mu_0)})}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), sin(sin(Symbol('\\\\mu_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mu_0', commutative=True)), cos(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Mul(cos(Symbol('\\\\mu_0', commutative=True)), cos(sin(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given M{(\\phi_2,I,\\hbar)} = - I + \\hbar^{\\phi_2}, then obtain e^{(\\hbar^{\\phi_2} M{(\\phi_2,I,\\hbar)})^{I}} = e^{(\\hbar^{\\phi_2} (- I + \\hbar^{\\phi_2}))^{I}}", "derivation": "M{(\\phi_2,I,\\hbar)} = - I + \\hbar^{\\phi_2} and \\hbar^{\\phi_2} M{(\\phi_2,I,\\hbar)} = \\hbar^{\\phi_2} (- I + \\hbar^{\\phi_2}) and (\\hbar^{\\phi_2} M{(\\phi_2,I,\\hbar)})^{I} = (\\hbar^{\\phi_2} (- I + \\hbar^{\\phi_2}))^{I} and e^{(\\hbar^{\\phi_2} M{(\\phi_2,I,\\hbar)})^{I}} = e^{(\\hbar^{\\phi_2} (- I + \\hbar^{\\phi_2}))^{I}}", "srepr_derivation": [["get_premise", "Equality(Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('I', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Symbol('I', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Function('M')(Symbol('\\\\phi_2', commutative=True), Symbol('I', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('I', commutative=True))), exp(Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(a^{\\dagger},J)} = \\log{(J + a^{\\dagger})}, then derive \\int \\psi^{*}{(a^{\\dagger},J)} da^{\\dagger} = J \\log{(J + a^{\\dagger})} + J_{\\varepsilon} + a^{\\dagger} \\log{(J + a^{\\dagger})} - a^{\\dagger}, then obtain \\frac{\\int \\psi^{*}{(a^{\\dagger},J)} da^{\\dagger}}{\\psi^{*}{(a^{\\dagger},J)}} = \\frac{J \\log{(J + a^{\\dagger})} + J_{\\varepsilon} + a^{\\dagger} \\log{(J + a^{\\dagger})} - a^{\\dagger}}{\\psi^{*}{(a^{\\dagger},J)}}", "derivation": "\\psi^{*}{(a^{\\dagger},J)} = \\log{(J + a^{\\dagger})} and \\int \\psi^{*}{(a^{\\dagger},J)} da^{\\dagger} = \\int \\log{(J + a^{\\dagger})} da^{\\dagger} and \\int \\psi^{*}{(a^{\\dagger},J)} da^{\\dagger} = J \\log{(J + a^{\\dagger})} + J_{\\varepsilon} + a^{\\dagger} \\log{(J + a^{\\dagger})} - a^{\\dagger} and \\frac{\\int \\psi^{*}{(a^{\\dagger},J)} da^{\\dagger}}{\\psi^{*}{(a^{\\dagger},J)}} = \\frac{J \\log{(J + a^{\\dagger})} + J_{\\varepsilon} + a^{\\dagger} \\log{(J + a^{\\dagger})} - a^{\\dagger}}{\\psi^{*}{(a^{\\dagger},J)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('J', commutative=True), log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 3, "Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True))"], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Add(Mul(Symbol('J', commutative=True), log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Add(Symbol('J', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Pow(Function('\\\\psi^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(g,\\Psi^{\\dagger})} = g \\log{(\\Psi^{\\dagger})}, then derive - \\frac{\\partial}{\\partial g} \\mathbf{g}{(g,\\Psi^{\\dagger})} = - \\log{(\\Psi^{\\dagger})}, then obtain g \\log{(\\Psi^{\\dagger})} - \\frac{\\partial}{\\partial g} g \\log{(\\Psi^{\\dagger})} = g \\log{(\\Psi^{\\dagger})} - \\log{(\\Psi^{\\dagger})}", "derivation": "\\mathbf{g}{(g,\\Psi^{\\dagger})} = g \\log{(\\Psi^{\\dagger})} and \\frac{\\partial}{\\partial g} \\mathbf{g}{(g,\\Psi^{\\dagger})} = \\frac{\\partial}{\\partial g} g \\log{(\\Psi^{\\dagger})} and - \\frac{\\partial}{\\partial g} \\mathbf{g}{(g,\\Psi^{\\dagger})} = - \\frac{\\partial}{\\partial g} g \\log{(\\Psi^{\\dagger})} and - \\frac{\\partial}{\\partial g} \\mathbf{g}{(g,\\Psi^{\\dagger})} = - \\log{(\\Psi^{\\dagger})} and - \\frac{\\partial}{\\partial g} g \\log{(\\Psi^{\\dagger})} = - \\log{(\\Psi^{\\dagger})} and g \\log{(\\Psi^{\\dagger})} - \\frac{\\partial}{\\partial g} g \\log{(\\Psi^{\\dagger})} = g \\log{(\\Psi^{\\dagger})} - \\log{(\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 5, "Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Add(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))), Add(Mul(Symbol('g', commutative=True), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given M{(\\hat{x}_0,\\hat{H})} = \\hat{H}^{\\hat{x}_0}, then obtain (2 M{(\\hat{x}_0,\\hat{H})} - 1)^{2} = (\\hat{H}^{\\hat{x}_0} + M{(\\hat{x}_0,\\hat{H})} - 1)^{2}", "derivation": "M{(\\hat{x}_0,\\hat{H})} = \\hat{H}^{\\hat{x}_0} and - \\hat{H}^{\\hat{x}_0} + M{(\\hat{x}_0,\\hat{H})} = 0 and 2 M{(\\hat{x}_0,\\hat{H})} = \\hat{H}^{\\hat{x}_0} + M{(\\hat{x}_0,\\hat{H})} and 2 M{(\\hat{x}_0,\\hat{H})} - 1 = \\hat{H}^{\\hat{x}_0} + M{(\\hat{x}_0,\\hat{H})} - 1 and (2 M{(\\hat{x}_0,\\hat{H})} - 1)^{2} = (\\hat{H}^{\\hat{x}_0} + M{(\\hat{x}_0,\\hat{H})} - 1)^{2}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(0))"], [["add", 2, "Add(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Integer(2), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Add(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(2), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(-1)), Add(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Integer(2), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Integer(-1)), Integer(2)), Pow(Add(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{E}{(a)} = e^{a} and \\operatorname{V_{\\mathbf{E}}}{(a)} = - a + \\mathbf{E}{(a)} e^{- a}, then obtain \\operatorname{V_{\\mathbf{E}}}{(a)} + \\mathbf{E}{(a)} e^{- a} - 1 = \\operatorname{V_{\\mathbf{E}}}{(a)}", "derivation": "\\mathbf{E}{(a)} = e^{a} and \\mathbf{E}{(a)} e^{- a} = 1 and - a + \\mathbf{E}{(a)} e^{- a} = 1 - a and \\operatorname{V_{\\mathbf{E}}}{(a)} = - a + \\mathbf{E}{(a)} e^{- a} and \\operatorname{V_{\\mathbf{E}}}{(a)} = 1 - a and \\operatorname{V_{\\mathbf{E}}}{(a)} + \\mathbf{E}{(a)} e^{- a} - 1 = \\operatorname{V_{\\mathbf{E}}}{(a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["divide", 1, "exp(Symbol('a', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integer(1))"], [["add", 2, "Mul(Integer(-1), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True)), Add(Integer(1), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True)), Mul(Function('\\\\mathbf{E}')(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('a', commutative=True)))), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(x,r)} = \\frac{e^{r}}{x}, then obtain \\int (- \\operatorname{z^{*}}{(x,r)} + \\cos{(\\operatorname{z^{*}}{(x,r)} + \\frac{e^{r}}{x})} - \\frac{e^{r}}{x}) dr = \\int (- \\operatorname{z^{*}}{(x,r)} + \\cos{(\\frac{2 e^{r}}{x})} - \\frac{e^{r}}{x}) dr", "derivation": "\\operatorname{z^{*}}{(x,r)} = \\frac{e^{r}}{x} and \\operatorname{z^{*}}{(x,r)} + \\frac{e^{r}}{x} = \\frac{2 e^{r}}{x} and \\cos{(\\operatorname{z^{*}}{(x,r)} + \\frac{e^{r}}{x})} = \\cos{(\\frac{2 e^{r}}{x})} and - \\operatorname{z^{*}}{(x,r)} + \\cos{(\\operatorname{z^{*}}{(x,r)} + \\frac{e^{r}}{x})} - \\frac{e^{r}}{x} = - \\operatorname{z^{*}}{(x,r)} + \\cos{(\\frac{2 e^{r}}{x})} - \\frac{e^{r}}{x} and \\int (- \\operatorname{z^{*}}{(x,r)} + \\cos{(\\operatorname{z^{*}}{(x,r)} + \\frac{e^{r}}{x})} - \\frac{e^{r}}{x}) dr = \\int (- \\operatorname{z^{*}}{(x,r)} + \\cos{(\\frac{2 e^{r}}{x})} - \\frac{e^{r}}{x}) dr", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))"], "Equality(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Mul(Integer(2), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))), cos(Mul(Integer(2), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))))"], [["minus", 3, "Add(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True))), cos(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Add(Mul(Integer(-1), Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True))), cos(Mul(Integer(2), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True))), cos(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True))))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Integer(-1), Function('z^*')(Symbol('x', commutative=True), Symbol('r', commutative=True))), cos(Mul(Integer(2), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(v_{z})} = e^{v_{z}}, then obtain \\frac{d}{d v_{z}} 2 e^{v_{z}} = - \\frac{d}{d v_{z}} (\\hat{p}{(v_{z})} + e^{v_{z}}) + 2 \\frac{d}{d v_{z}} 2 e^{v_{z}}", "derivation": "\\hat{p}{(v_{z})} = e^{v_{z}} and \\hat{p}{(v_{z})} + e^{v_{z}} = 2 e^{v_{z}} and \\frac{d}{d v_{z}} (\\hat{p}{(v_{z})} + e^{v_{z}}) = \\frac{d}{d v_{z}} 2 e^{v_{z}} and \\frac{d}{d v_{z}} (\\hat{p}{(v_{z})} + e^{v_{z}}) + \\frac{d}{d v_{z}} 2 e^{v_{z}} = 2 \\frac{d}{d v_{z}} 2 e^{v_{z}} and \\frac{d}{d v_{z}} 2 e^{v_{z}} = - \\frac{d}{d v_{z}} (\\hat{p}{(v_{z})} + e^{v_{z}}) + 2 \\frac{d}{d v_{z}} 2 e^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["add", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Mul(Integer(2), exp(Symbol('v_z', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["minus", 4, "Derivative(Add(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Add(Mul(Integer(-1), Derivative(Add(Function('\\\\hat{p}')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Integer(2), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{F})} = \\cos{(\\mathbf{F} - a^{\\dagger})}, then derive \\int \\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{F})} da^{\\dagger} = \\hat{\\mathbf{x}} - \\sin{(\\mathbf{F} - a^{\\dagger})}, then obtain \\int \\cos{(\\mathbf{F} - a^{\\dagger})} da^{\\dagger} = \\hat{\\mathbf{x}} - \\sin{(\\mathbf{F} - a^{\\dagger})}", "derivation": "\\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{F})} = \\cos{(\\mathbf{F} - a^{\\dagger})} and \\int \\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{F})} da^{\\dagger} = \\int \\cos{(\\mathbf{F} - a^{\\dagger})} da^{\\dagger} and \\int \\operatorname{C_{2}}{(a^{\\dagger},\\mathbf{F})} da^{\\dagger} = \\hat{\\mathbf{x}} - \\sin{(\\mathbf{F} - a^{\\dagger})} and \\int \\cos{(\\mathbf{F} - a^{\\dagger})} da^{\\dagger} = \\hat{\\mathbf{x}} - \\sin{(\\mathbf{F} - a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})}, then derive v_{2} + \\operatorname{f_{E}}{(E_{n})} + \\sin{(E_{n})} = v_{2}, then obtain v_{2} + \\sin{(E_{n})} + \\frac{d}{d E_{n}} \\cos{(E_{n})} = v_{2}", "derivation": "\\operatorname{f_{E}}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})} and v_{2} + \\operatorname{f_{E}}{(E_{n})} = v_{2} + \\frac{d}{d E_{n}} \\cos{(E_{n})} and v_{2} + \\operatorname{f_{E}}{(E_{n})} + \\sin{(E_{n})} = v_{2} + \\sin{(E_{n})} + \\frac{d}{d E_{n}} \\cos{(E_{n})} and v_{2} + \\operatorname{f_{E}}{(E_{n})} + \\sin{(E_{n})} = v_{2} and v_{2} + \\sin{(E_{n})} + \\frac{d}{d E_{n}} \\cos{(E_{n})} = v_{2}", "srepr_derivation": [["get_premise", "Equality(Function('f_E')(Symbol('E_n', commutative=True)), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["add", 1, "Symbol('v_2', commutative=True)"], "Equality(Add(Symbol('v_2', commutative=True), Function('f_E')(Symbol('E_n', commutative=True))), Add(Symbol('v_2', commutative=True), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["add", 2, "sin(Symbol('E_n', commutative=True))"], "Equality(Add(Symbol('v_2', commutative=True), Function('f_E')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Add(Symbol('v_2', commutative=True), sin(Symbol('E_n', commutative=True)), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('v_2', commutative=True), Function('f_E')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Symbol('v_2', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('v_2', commutative=True), sin(Symbol('E_n', commutative=True)), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('v_2', commutative=True))"]]}, {"prompt": "Given I{(\\mathbf{v})} = \\cos{(\\cos{(\\mathbf{v})})}, then obtain \\frac{I{(\\mathbf{v})}}{(\\mathbf{v} + I{(\\mathbf{v})}) \\cos{(\\cos{(\\mathbf{v})})}} = \\frac{1}{\\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})}}", "derivation": "I{(\\mathbf{v})} = \\cos{(\\cos{(\\mathbf{v})})} and \\mathbf{v} + I{(\\mathbf{v})} = \\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})} and (\\mathbf{v} + I{(\\mathbf{v})}) \\cos{(\\cos{(\\mathbf{v})})} = (\\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})}) \\cos{(\\cos{(\\mathbf{v})})} and \\frac{I{(\\mathbf{v})}}{(\\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})}) \\cos{(\\cos{(\\mathbf{v})})}} = \\frac{1}{\\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})}} and \\frac{I{(\\mathbf{v})}}{(\\mathbf{v} + I{(\\mathbf{v})}) \\cos{(\\cos{(\\mathbf{v})})}} = \\frac{1}{\\mathbf{v} + \\cos{(\\cos{(\\mathbf{v})})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{v}', commutative=True)), cos(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["times", 2, "cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["divide", 1, "Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), cos(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), Integer(-1)), Function('I')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(cos(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), Function('I')(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1)), Function('I')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(cos(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Pow(Add(Symbol('\\\\mathbf{v}', commutative=True), cos(cos(Symbol('\\\\mathbf{v}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)} = \\frac{f_{\\mathbf{v}}^{\\tilde{g}^*}}{v_{t}}, then obtain 1 = f_{\\mathbf{v}}^{- \\tilde{g}^*} v_{t} \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)}", "derivation": "\\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)} = \\frac{f_{\\mathbf{v}}^{\\tilde{g}^*}}{v_{t}} and 1 = \\frac{f_{\\mathbf{v}}^{\\tilde{g}^*}}{v_{t} \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)}} and f_{\\mathbf{v}}^{\\tilde{g}^*} = \\frac{f_{\\mathbf{v}}^{2 \\tilde{g}^*}}{v_{t} \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)}} and \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)} = \\frac{f_{\\mathbf{v}}^{2 \\tilde{g}^*}}{v_{t}^{2} \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)}} and 1 = f_{\\mathbf{v}}^{- \\tilde{g}^*} v_{t} \\psi{(f_{\\mathbf{v}},v_{t},\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["divide", 1, "Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["times", 2, "Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Symbol('v_t', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(2), Symbol('\\\\tilde{g}^*', commutative=True))), Pow(Symbol('v_t', commutative=True), Integer(-2)), Pow(Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(1), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('v_t', commutative=True), Function('\\\\psi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{v})} = \\int e^{\\mathbf{v}} d\\mathbf{v} and m{(\\mathbf{v})} = - (\\int e^{\\mathbf{v}} d\\mathbf{v})^{\\mathbf{v}}, then obtain - (s + e^{\\mathbf{v}})^{\\mathbf{v}} = - \\hat{H}_l^{\\mathbf{v}}{(\\mathbf{v})}", "derivation": "\\hat{H}_l{(\\mathbf{v})} = \\int e^{\\mathbf{v}} d\\mathbf{v} and m{(\\mathbf{v})} = - (\\int e^{\\mathbf{v}} d\\mathbf{v})^{\\mathbf{v}} and m{(\\mathbf{v})} = - \\hat{H}_l^{\\mathbf{v}}{(\\mathbf{v})} and - (\\int e^{\\mathbf{v}} d\\mathbf{v})^{\\mathbf{v}} = - \\hat{H}_l^{\\mathbf{v}}{(\\mathbf{v})} and - (s + e^{\\mathbf{v}})^{\\mathbf{v}} = - \\hat{H}_l^{\\mathbf{v}}{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{v}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Mul(Integer(-1), Pow(Add(Symbol('s', commutative=True), exp(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given k{(U,\\ddot{x})} = \\ddot{x}^{U}, then obtain \\frac{\\frac{\\partial}{\\partial \\ddot{x}} k{(U,\\ddot{x})}}{U} = \\frac{\\ddot{x}^{U}}{\\ddot{x}}", "derivation": "k{(U,\\ddot{x})} = \\ddot{x}^{U} and \\frac{k{(U,\\ddot{x})}}{U} = \\frac{\\ddot{x}^{U}}{U} and \\frac{\\partial}{\\partial \\ddot{x}} \\frac{k{(U,\\ddot{x})}}{U} = \\frac{\\partial}{\\partial \\ddot{x}} \\frac{\\ddot{x}^{U}}{U} and \\frac{\\frac{\\partial}{\\partial \\ddot{x}} k{(U,\\ddot{x})}}{U} = \\frac{\\ddot{x}^{U}}{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('U', commutative=True)))"], [["divide", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('k')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('k')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Derivative(Function('k')(Symbol('U', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(J,\\mathbf{J}_f,\\chi)} = \\frac{J \\mathbf{J}_f}{\\chi}, then obtain ((\\chi (\\phi_{1}{(J,\\mathbf{J}_f,\\chi)} + \\frac{1}{\\chi}))^{\\chi})^{J} = ((\\chi (\\frac{J \\mathbf{J}_f}{\\chi} + \\frac{1}{\\chi}))^{\\chi})^{J}", "derivation": "\\phi_{1}{(J,\\mathbf{J}_f,\\chi)} = \\frac{J \\mathbf{J}_f}{\\chi} and \\phi_{1}{(J,\\mathbf{J}_f,\\chi)} + \\frac{1}{\\chi} = \\frac{J \\mathbf{J}_f}{\\chi} + \\frac{1}{\\chi} and \\chi (\\phi_{1}{(J,\\mathbf{J}_f,\\chi)} + \\frac{1}{\\chi}) = \\chi (\\frac{J \\mathbf{J}_f}{\\chi} + \\frac{1}{\\chi}) and (\\chi (\\phi_{1}{(J,\\mathbf{J}_f,\\chi)} + \\frac{1}{\\chi}))^{\\chi} = (\\chi (\\frac{J \\mathbf{J}_f}{\\chi} + \\frac{1}{\\chi}))^{\\chi} and ((\\chi (\\phi_{1}{(J,\\mathbf{J}_f,\\chi)} + \\frac{1}{\\chi}))^{\\chi})^{J} = ((\\chi (\\frac{J \\mathbf{J}_f}{\\chi} + \\frac{1}{\\chi}))^{\\chi})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\chi', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\phi_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["divide", 2, "Pow(Symbol('\\\\chi', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Add(Function('\\\\phi_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Mul(Symbol('\\\\chi', commutative=True), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Function('\\\\phi_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Symbol('\\\\chi', commutative=True)))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Function('\\\\phi_1')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Symbol('\\\\chi', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))), Symbol('\\\\chi', commutative=True)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(F_{c})} = \\log{(F_{c})} and \\dot{z}{(U)} = e^{U}, then obtain \\frac{\\dot{z}^{U}{(U)}}{\\log{(F_{c})}} = \\frac{(e^{U})^{U}}{\\log{(F_{c})}}", "derivation": "\\bar{\\h}{(F_{c})} = \\log{(F_{c})} and \\dot{z}{(U)} = e^{U} and \\dot{z}^{U}{(U)} = (e^{U})^{U} and \\frac{\\dot{z}^{U}{(U)}}{\\bar{\\h}{(F_{c})}} = \\frac{(e^{U})^{U}}{\\bar{\\h}{(F_{c})}} and \\frac{\\dot{z}^{U}{(U)}}{\\log{(F_{c})}} = \\frac{(e^{U})^{U}}{\\log{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["divide", 3, "Function('\\\\hbar')(Symbol('F_c', commutative=True))"], "Equality(Mul(Pow(Function('\\\\dot{z}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('\\\\hbar')(Symbol('F_c', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\hbar')(Symbol('F_c', commutative=True)), Integer(-1)), Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Function('\\\\dot{z}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1))), Mul(Pow(exp(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(f)} = \\sin{(\\sin{(f)})}, then obtain \\frac{d^{2}}{d f^{2}} (\\int \\hat{H}_{\\lambda}{(f)} df)^{f} = \\frac{d^{2}}{d f^{2}} (\\int \\sin{(\\sin{(f)})} df)^{f}", "derivation": "\\hat{H}_{\\lambda}{(f)} = \\sin{(\\sin{(f)})} and \\int \\hat{H}_{\\lambda}{(f)} df = \\int \\sin{(\\sin{(f)})} df and (\\int \\hat{H}_{\\lambda}{(f)} df)^{f} = (\\int \\sin{(\\sin{(f)})} df)^{f} and \\frac{d}{d f} (\\int \\hat{H}_{\\lambda}{(f)} df)^{f} = \\frac{d}{d f} (\\int \\sin{(\\sin{(f)})} df)^{f} and \\frac{d^{2}}{d f^{2}} (\\int \\hat{H}_{\\lambda}{(f)} df)^{f} = \\frac{d^{2}}{d f^{2}} (\\int \\sin{(\\sin{(f)})} df)^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f', commutative=True)), sin(sin(Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(sin(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(sin(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Pow(Integral(sin(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\hat{x}{(x,G)} = G \\sin{(x)}, then obtain (\\hat{x}^{2}{(x,G)})^{- G} (\\frac{\\partial}{\\partial G} \\hat{x}{(x,G)})^{x} = (\\hat{x}^{2}{(x,G)})^{- G} (\\frac{\\partial}{\\partial G} G \\sin{(x)})^{x}", "derivation": "\\hat{x}{(x,G)} = G \\sin{(x)} and \\frac{\\partial}{\\partial G} \\hat{x}{(x,G)} = \\frac{\\partial}{\\partial G} G \\sin{(x)} and \\hat{x}^{2}{(x,G)} = G \\hat{x}{(x,G)} \\sin{(x)} and (\\frac{\\partial}{\\partial G} \\hat{x}{(x,G)})^{x} = (\\frac{\\partial}{\\partial G} G \\sin{(x)})^{x} and (G \\hat{x}{(x,G)} \\sin{(x)})^{- G} (\\frac{\\partial}{\\partial G} \\hat{x}{(x,G)})^{x} = (G \\hat{x}{(x,G)} \\sin{(x)})^{- G} (\\frac{\\partial}{\\partial G} G \\sin{(x)})^{x} and (\\hat{x}^{2}{(x,G)})^{- G} (\\frac{\\partial}{\\partial G} \\hat{x}{(x,G)})^{x} = (\\hat{x}^{2}{(x,G)})^{- G} (\\frac{\\partial}{\\partial G} G \\sin{(x)})^{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('G', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Integer(2)), Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), sin(Symbol('x', commutative=True))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Derivative(Mul(Symbol('G', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True)))"], [["divide", 4, "Pow(Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), sin(Symbol('x', commutative=True))), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True))), Pow(Derivative(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True))), Mul(Pow(Mul(Symbol('G', commutative=True), Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), sin(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True))), Pow(Derivative(Mul(Symbol('G', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Pow(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('G', commutative=True))), Pow(Derivative(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True))), Mul(Pow(Pow(Function('\\\\hat{x}')(Symbol('x', commutative=True), Symbol('G', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('G', commutative=True))), Pow(Derivative(Mul(Symbol('G', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} = \\frac{l}{\\eta^{\\prime}}, then obtain \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} (\\frac{\\partial}{\\partial l} \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})})^{- l} = \\frac{l (\\frac{\\partial}{\\partial l} \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})})^{- l}}{\\eta^{\\prime}}", "derivation": "\\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} = \\frac{l}{\\eta^{\\prime}} and \\frac{\\partial}{\\partial l} \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} = \\frac{\\partial}{\\partial l} \\frac{l}{\\eta^{\\prime}} and \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} (\\frac{\\partial}{\\partial l} \\frac{l}{\\eta^{\\prime}})^{- l} = \\frac{l (\\frac{\\partial}{\\partial l} \\frac{l}{\\eta^{\\prime}})^{- l}}{\\eta^{\\prime}} and \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})} (\\frac{\\partial}{\\partial l} \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})})^{- l} = \\frac{l (\\frac{\\partial}{\\partial l} \\operatorname{a^{\\dagger}}{(l,\\eta^{\\prime})})^{- l}}{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 1, "Pow(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Symbol('l', commutative=True))"], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True), Pow(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('l', commutative=True), Pow(Derivative(Function('a^{\\\\dagger}')(Symbol('l', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})} and \\varepsilon{(\\dot{\\mathbf{r}})} = \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}}, then obtain \\varepsilon{(\\dot{\\mathbf{r}})} + \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}} = 2 \\varepsilon{(\\dot{\\mathbf{r}})}", "derivation": "\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})} = \\log{(\\dot{\\mathbf{r}})} and \\varepsilon{(\\dot{\\mathbf{r}})} = \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}} and \\varepsilon{(\\dot{\\mathbf{r}})} = \\frac{1}{\\log{(\\dot{\\mathbf{r}})}} and \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}} = \\frac{1}{\\log{(\\dot{\\mathbf{r}})}} and \\frac{1}{\\log{(\\dot{\\mathbf{r}})}} + \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}} = \\frac{2}{\\log{(\\dot{\\mathbf{r}})}} and \\varepsilon{(\\dot{\\mathbf{r}})} + \\frac{1}{\\operatorname{r_{0}}{(\\dot{\\mathbf{r}})}} = 2 \\varepsilon{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Function('r_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\varepsilon')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('r_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"], [["add", 4, "Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))"], "Equality(Add(Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), Pow(Function('r_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Function('r_0')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1))), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given C{(h,v_{x},f)} = (h v_{x})^{f}, then obtain C{(h,v_{x},f)} \\frac{\\partial}{\\partial v_{x}} C{(h,v_{x},f)} = (h v_{x})^{f} \\frac{\\partial}{\\partial v_{x}} C{(h,v_{x},f)}", "derivation": "C{(h,v_{x},f)} = (h v_{x})^{f} and \\frac{\\partial}{\\partial v_{x}} C{(h,v_{x},f)} = \\frac{\\partial}{\\partial v_{x}} (h v_{x})^{f} and C{(h,v_{x},f)} \\frac{\\partial}{\\partial v_{x}} (h v_{x})^{f} = (h v_{x})^{f} \\frac{\\partial}{\\partial v_{x}} (h v_{x})^{f} and C{(h,v_{x},f)} \\frac{\\partial}{\\partial v_{x}} C{(h,v_{x},f)} = (h v_{x})^{f} \\frac{\\partial}{\\partial v_{x}} C{(h,v_{x},f)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Mul(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Derivative(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Derivative(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Derivative(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Pow(Mul(Symbol('h', commutative=True), Symbol('v_x', commutative=True)), Symbol('f', commutative=True)), Derivative(Function('C')(Symbol('h', commutative=True), Symbol('v_x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(V_{\\mathbf{B}},a,t)} = \\frac{V_{\\mathbf{B}} + a}{t} and \\hat{H}_l{(V_{\\mathbf{B}},a,t)} = \\frac{- V_{\\mathbf{B}} - a}{t}, then obtain \\hat{H}_l{(V_{\\mathbf{B}},a,t)} = - \\frac{V_{\\mathbf{B}} + a}{t}", "derivation": "\\varphi^{*}{(V_{\\mathbf{B}},a,t)} = \\frac{V_{\\mathbf{B}} + a}{t} and - \\varphi^{*}{(V_{\\mathbf{B}},a,t)} = - \\frac{V_{\\mathbf{B}} + a}{t} and - \\varphi^{*}{(V_{\\mathbf{B}},a,t)} = \\frac{- V_{\\mathbf{B}} - a}{t} and - \\frac{V_{\\mathbf{B}} + a}{t} = \\frac{- V_{\\mathbf{B}} - a}{t} and \\hat{H}_l{(V_{\\mathbf{B}},a,t)} = \\frac{- V_{\\mathbf{B}} - a}{t} and \\hat{H}_l{(V_{\\mathbf{B}},a,t)} = - \\frac{V_{\\mathbf{B}} + a}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True), Symbol('t', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\hat{H}_l')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(C)} = \\log{(e^{C})}, then derive \\frac{d}{d C} \\tilde{g}^*{(C)} = 1, then obtain \\tilde{g}^*{(C)} - \\frac{d}{d C} \\tilde{g}^*{(C)} + \\frac{d}{d C} \\log{(e^{C})} = \\tilde{g}^*{(C)} - \\frac{d}{d C} \\tilde{g}^*{(C)} + 1", "derivation": "\\tilde{g}^*{(C)} = \\log{(e^{C})} and \\frac{d}{d C} \\tilde{g}^*{(C)} = \\frac{d}{d C} \\log{(e^{C})} and \\frac{d}{d C} \\tilde{g}^*{(C)} = 1 and \\frac{d}{d C} \\log{(e^{C})} = 1 and - \\frac{d}{d C} \\tilde{g}^*{(C)} + \\frac{d}{d C} \\log{(e^{C})} = 1 - \\frac{d}{d C} \\tilde{g}^*{(C)} and \\tilde{g}^*{(C)} - \\frac{d}{d C} \\tilde{g}^*{(C)} + \\frac{d}{d C} \\log{(e^{C})} = \\tilde{g}^*{(C)} - \\frac{d}{d C} \\tilde{g}^*{(C)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), log(exp(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Derivative(log(exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))))"], [["add", 5, "Function('\\\\tilde{g}^*')(Symbol('C', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Derivative(log(exp(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Add(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\tilde{g}^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(y)} = \\cos{(y)}, then obtain \\frac{d}{d y} \\int \\operatorname{n_{2}}{(y)} dy = \\frac{\\partial}{\\partial y} (A_{x} + \\sin{(y)})", "derivation": "\\operatorname{n_{2}}{(y)} = \\cos{(y)} and \\int \\operatorname{n_{2}}{(y)} dy = \\int \\cos{(y)} dy and \\frac{d}{d y} \\int \\operatorname{n_{2}}{(y)} dy = \\frac{d}{d y} \\int \\cos{(y)} dy and \\frac{d}{d y} \\int \\operatorname{n_{2}}{(y)} dy = \\frac{\\partial}{\\partial y} (A_{x} + \\sin{(y)})", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(Function('n_2')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('n_2')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\nabla)} = \\cos{(\\nabla)}, then derive \\int \\tilde{g}^*{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + \\sin{(\\nabla)}, then obtain 2 \\tilde{g}^*{(\\nabla)} + \\int \\cos{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + 2 \\tilde{g}^*{(\\nabla)} + \\sin{(\\nabla)}", "derivation": "\\tilde{g}^*{(\\nabla)} = \\cos{(\\nabla)} and \\int \\tilde{g}^*{(\\nabla)} d\\nabla = \\int \\cos{(\\nabla)} d\\nabla and \\int \\tilde{g}^*{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + \\sin{(\\nabla)} and \\int \\cos{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + \\sin{(\\nabla)} and 2 \\tilde{g}^*{(\\nabla)} = \\tilde{g}^*{(\\nabla)} + \\cos{(\\nabla)} and \\tilde{g}^*{(\\nabla)} + \\cos{(\\nabla)} + \\int \\cos{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + \\tilde{g}^*{(\\nabla)} + \\sin{(\\nabla)} + \\cos{(\\nabla)} and 2 \\tilde{g}^*{(\\nabla)} + \\int \\cos{(\\nabla)} d\\nabla = V_{\\mathbf{E}} + 2 \\tilde{g}^*{(\\nabla)} + \\sin{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["add", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True))), Add(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True))))"], [["add", 4, "Add(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True))), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\nabla', commutative=True))), sin(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\mathbf{H},\\varphi)} = \\mathbf{H}^{\\varphi} and \\operatorname{E_{n}}{(\\varphi)} = \\varphi, then obtain \\operatorname{E_{n}}{(\\varphi)} + \\frac{\\int \\mathbf{H}^{\\varphi} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}} = \\varphi + \\frac{\\int \\mathbf{H}^{\\varphi} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}}", "derivation": "\\rho{(\\mathbf{H},\\varphi)} = \\mathbf{H}^{\\varphi} and \\int \\rho{(\\mathbf{H},\\varphi)} d\\mathbf{H} = \\int \\mathbf{H}^{\\varphi} d\\mathbf{H} and \\operatorname{E_{n}}{(\\varphi)} = \\varphi and \\operatorname{E_{n}}{(\\varphi)} + \\frac{\\int \\rho{(\\mathbf{H},\\varphi)} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}} = \\varphi + \\frac{\\int \\rho{(\\mathbf{H},\\varphi)} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}} and \\operatorname{E_{n}}{(\\varphi)} + \\frac{\\int \\mathbf{H}^{\\varphi} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}} = \\varphi + \\frac{\\int \\mathbf{H}^{\\varphi} d\\mathbf{H}}{\\rho{(\\mathbf{H},\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], [["add", 3, "Mul(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Add(Function('E_n')(Symbol('\\\\varphi', commutative=True)), Mul(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Symbol('\\\\varphi', commutative=True), Mul(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integral(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('E_n')(Symbol('\\\\varphi', commutative=True)), Mul(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Symbol('\\\\varphi', commutative=True), Mul(Pow(Function('\\\\rho')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"]]}, {"prompt": "Given f{(\\mathbf{J}_M,v)} = \\mathbf{J}_M^{v} and L{(\\mathbf{J}_M,v)} = (f^{v}{(\\mathbf{J}_M,v)})^{v}, then obtain \\mathbf{J}_M^{v} L{(\\mathbf{J}_M,v)} = \\mathbf{J}_M^{v} ((\\mathbf{J}_M^{v})^{v})^{v}", "derivation": "f{(\\mathbf{J}_M,v)} = \\mathbf{J}_M^{v} and L{(\\mathbf{J}_M,v)} = (f^{v}{(\\mathbf{J}_M,v)})^{v} and L{(\\mathbf{J}_M,v)} = ((\\mathbf{J}_M^{v})^{v})^{v} and L{(\\mathbf{J}_M,v)} f{(\\mathbf{J}_M,v)} = ((\\mathbf{J}_M^{v})^{v})^{v} f{(\\mathbf{J}_M,v)} and \\mathbf{J}_M^{v} L{(\\mathbf{J}_M,v)} = \\mathbf{J}_M^{v} ((\\mathbf{J}_M^{v})^{v})^{v}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Pow(Pow(Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('L')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["times", 3, "Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Function('L')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Pow(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Function('f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Function('L')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then derive \\int \\operatorname{A_{x}}{(g_{\\varepsilon})} dg_{\\varepsilon} = F_{g} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}, then obtain \\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon} = F_{g} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}", "derivation": "\\operatorname{A_{x}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\int \\operatorname{A_{x}}{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon} and \\int \\operatorname{A_{x}}{(g_{\\varepsilon})} dg_{\\varepsilon} = F_{g} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon} and \\int \\log{(g_{\\varepsilon})} dg_{\\varepsilon} = F_{g} + g_{\\varepsilon} \\log{(g_{\\varepsilon})} - g_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_x')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(z)} = \\int \\log{(z)} dz, then derive \\operatorname{z^{*}}{(z)} = \\psi + z \\log{(z)} - z, then obtain ((\\psi + z \\log{(z)} - z)^{z})^{z} = (\\operatorname{z^{*}}^{z}{(z)})^{z}", "derivation": "\\operatorname{z^{*}}{(z)} = \\int \\log{(z)} dz and \\operatorname{z^{*}}^{z}{(z)} = (\\int \\log{(z)} dz)^{z} and (\\operatorname{z^{*}}^{z}{(z)})^{z} = ((\\int \\log{(z)} dz)^{z})^{z} and \\operatorname{z^{*}}{(z)} = \\psi + z \\log{(z)} - z and ((\\psi + z \\log{(z)} - z)^{z})^{z} = ((\\int \\log{(z)} dz)^{z})^{z} and ((\\psi + z \\log{(z)} - z)^{z})^{z} = (\\operatorname{z^{*}}^{z}{(z)})^{z}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('z', commutative=True)), Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Function('z^*')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('z^*')(Symbol('z', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Function('z^*')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(f^{*})} = e^{\\cos{(f^{*})}}, then obtain -1 = - \\operatorname{f_{E}}^{f^{*}}{(f^{*})} + (e^{\\cos{(f^{*})}})^{f^{*}} - 1", "derivation": "\\operatorname{f_{E}}{(f^{*})} = e^{\\cos{(f^{*})}} and \\operatorname{f_{E}}^{f^{*}}{(f^{*})} = (e^{\\cos{(f^{*})}})^{f^{*}} and 0 = - \\operatorname{f_{E}}^{f^{*}}{(f^{*})} + (e^{\\cos{(f^{*})}})^{f^{*}} and -1 = - \\operatorname{f_{E}}^{f^{*}}{(f^{*})} + (e^{\\cos{(f^{*})}})^{f^{*}} - 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('f^*', commutative=True)), exp(cos(Symbol('f^*', commutative=True))))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(exp(cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)))"], [["minus", 2, "Pow(Function('f_E')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('f_E')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Pow(exp(cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Pow(Function('f_E')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Pow(exp(cos(Symbol('f^*', commutative=True))), Symbol('f^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(l,F_{N})} = \\frac{\\partial}{\\partial l} F_{N} l, then derive \\operatorname{r_{0}}{(l,F_{N})} = F_{N}, then obtain \\frac{1}{l} = \\frac{\\frac{\\partial}{\\partial l} F_{N} l}{F_{N} l}", "derivation": "\\operatorname{r_{0}}{(l,F_{N})} = \\frac{\\partial}{\\partial l} F_{N} l and \\operatorname{r_{0}}{(l,F_{N})} = F_{N} and F_{N} = \\frac{\\partial}{\\partial l} F_{N} l and \\frac{1}{l} = \\frac{\\frac{\\partial}{\\partial l} F_{N} l}{F_{N} l}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('r_0')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Symbol('F_N', commutative=True), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Symbol('F_N', commutative=True), Symbol('l', commutative=True))"], "Equality(Pow(Symbol('l', commutative=True), Integer(-1)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given k{(b,F_{x})} = \\cos{(F_{x}^{b})}, then obtain \\frac{(F_{x}^{b} + \\cos{(k{(b,F_{x})})})^{b}}{F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})}} = \\frac{(F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})})^{b}}{F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})}}", "derivation": "k{(b,F_{x})} = \\cos{(F_{x}^{b})} and \\cos{(k{(b,F_{x})})} = \\cos{(\\cos{(F_{x}^{b})})} and F_{x}^{b} + \\cos{(k{(b,F_{x})})} = F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})} and (F_{x}^{b} + \\cos{(k{(b,F_{x})})})^{b} = (F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})})^{b} and \\frac{(F_{x}^{b} + \\cos{(k{(b,F_{x})})})^{b}}{F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})}} = \\frac{(F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})})^{b}}{F_{x}^{b} + \\cos{(\\cos{(F_{x}^{b})})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('b', commutative=True), Symbol('F_x', commutative=True)), cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))"], [["cos", 1], "Equality(cos(Function('k')(Symbol('b', commutative=True), Symbol('F_x', commutative=True))), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)))))"], [["add", 2, "Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))"], "Equality(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(Function('k')(Symbol('b', commutative=True), Symbol('F_x', commutative=True)))), Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(Function('k')(Symbol('b', commutative=True), Symbol('F_x', commutative=True)))), Symbol('b', commutative=True)), Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))), Symbol('b', commutative=True)))"], [["divide", 4, "Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)))))"], "Equality(Mul(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(Function('k')(Symbol('b', commutative=True), Symbol('F_x', commutative=True)))), Symbol('b', commutative=True)), Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))), Integer(-1))), Mul(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))), Integer(-1)), Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True)), cos(cos(Pow(Symbol('F_x', commutative=True), Symbol('b', commutative=True))))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(h,\\eta)} = \\frac{\\partial}{\\partial h} \\frac{\\eta}{h}, then derive \\int \\Psi_{nl}{(h,\\eta)} dh = F_{g} + \\frac{\\eta}{h}, then obtain \\iint \\frac{\\partial}{\\partial h} \\frac{\\eta}{h} dh dF_{g} = \\int (F_{g} + \\frac{\\eta}{h}) dF_{g}", "derivation": "\\Psi_{nl}{(h,\\eta)} = \\frac{\\partial}{\\partial h} \\frac{\\eta}{h} and \\int \\Psi_{nl}{(h,\\eta)} dh = \\int \\frac{\\partial}{\\partial h} \\frac{\\eta}{h} dh and \\int \\Psi_{nl}{(h,\\eta)} dh = F_{g} + \\frac{\\eta}{h} and \\iint \\Psi_{nl}{(h,\\eta)} dh dF_{g} = \\int (F_{g} + \\frac{\\eta}{h}) dF_{g} and \\iint \\frac{\\partial}{\\partial h} \\frac{\\eta}{h} dh dF_{g} = \\int (F_{g} + \\frac{\\eta}{h}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Symbol('F_g', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Tuple(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Symbol('F_g', commutative=True), Mul(Symbol('\\\\eta', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(U,v_{t})} = \\frac{\\sin{(U)}}{v_{t}}, then obtain \\int v_{t} \\operatorname{M_{E}}{(U,v_{t})} dv_{t} + \\frac{\\operatorname{M_{E}}{(U,v_{t})}}{U} = \\int \\sin{(U)} dv_{t} + \\frac{\\operatorname{M_{E}}{(U,v_{t})}}{U}", "derivation": "\\operatorname{M_{E}}{(U,v_{t})} = \\frac{\\sin{(U)}}{v_{t}} and v_{t} \\operatorname{M_{E}}{(U,v_{t})} = \\sin{(U)} and \\int v_{t} \\operatorname{M_{E}}{(U,v_{t})} dv_{t} = \\int \\sin{(U)} dv_{t} and \\int v_{t} \\operatorname{M_{E}}{(U,v_{t})} dv_{t} + \\frac{\\operatorname{M_{E}}{(U,v_{t})}}{U} = \\int \\sin{(U)} dv_{t} + \\frac{\\operatorname{M_{E}}{(U,v_{t})}}{U}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), sin(Symbol('U', commutative=True))))"], [["divide", 1, "Pow(Symbol('v_t', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True))), sin(Symbol('U', commutative=True)))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('v_t', commutative=True), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)))), Add(Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Function('M_E')(Symbol('U', commutative=True), Symbol('v_t', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})} and C{(\\sigma_x)} = \\sigma_x + \\mathbf{D}{(\\sigma_x)}, then obtain C{(\\sigma_x)} = \\sigma_x + \\sin{(\\cos{(\\sigma_x)})}", "derivation": "\\mathbf{D}{(\\sigma_x)} = \\sin{(\\cos{(\\sigma_x)})} and \\sigma_x + \\mathbf{D}{(\\sigma_x)} = \\sigma_x + \\sin{(\\cos{(\\sigma_x)})} and C{(\\sigma_x)} = \\sigma_x + \\mathbf{D}{(\\sigma_x)} and C{(\\sigma_x)} = \\sigma_x + \\sin{(\\cos{(\\sigma_x)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\sigma_x', commutative=True)), sin(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('C')(Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), sin(cos(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\psi,t)} = - \\psi + t, then derive \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}} = \\frac{\\psi (- \\psi + t)^{\\psi}}{- \\psi + t}, then obtain \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}} = \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}}", "derivation": "\\operatorname{z^{*}}{(\\psi,t)} = - \\psi + t and \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} = (- \\psi + t)^{\\psi} and \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} = \\frac{\\partial}{\\partial t} (- \\psi + t)^{\\psi} and \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}} = \\frac{\\psi (- \\psi + t)^{\\psi}}{- \\psi + t} and \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)} \\frac{\\partial}{\\partial t} \\operatorname{z^{*}}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}} = \\frac{\\psi \\operatorname{z^{*}}^{\\psi}{(\\psi,t)}}{\\operatorname{z^{*}}{(\\psi,t)}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('t', commutative=True)))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('t', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True)), Derivative(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Integer(-1)), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\mu{(f^{\\prime})} = \\cos{(e^{f^{\\prime}})}, then obtain \\frac{\\mu{(f^{\\prime})} \\cos^{3}{(e^{f^{\\prime}})}}{\\cos{(\\cos{(e^{f^{\\prime}})})}} = \\frac{\\cos^{4}{(e^{f^{\\prime}})}}{\\cos{(\\cos{(e^{f^{\\prime}})})}}", "derivation": "\\mu{(f^{\\prime})} = \\cos{(e^{f^{\\prime}})} and \\mu{(f^{\\prime})} \\cos{(e^{f^{\\prime}})} = \\cos^{2}{(e^{f^{\\prime}})} and \\mu{(f^{\\prime})} \\cos^{3}{(e^{f^{\\prime}})} = \\cos^{4}{(e^{f^{\\prime}})} and \\frac{\\mu{(f^{\\prime})} \\cos^{3}{(e^{f^{\\prime}})}}{\\cos{(\\cos{(e^{f^{\\prime}})})}} = \\frac{\\cos^{4}{(e^{f^{\\prime}})}}{\\cos{(\\cos{(e^{f^{\\prime}})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True)), cos(exp(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 1, "cos(exp(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True)), cos(exp(Symbol('f^{\\\\prime}', commutative=True)))), Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))"], [["times", 2, "Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(2))"], "Equality(Mul(Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(3))), Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(4)))"], [["divide", 3, "cos(cos(exp(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Mul(Function('\\\\mu')(Symbol('f^{\\\\prime}', commutative=True)), Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(3)), Pow(cos(cos(exp(Symbol('f^{\\\\prime}', commutative=True)))), Integer(-1))), Mul(Pow(cos(exp(Symbol('f^{\\\\prime}', commutative=True))), Integer(4)), Pow(cos(cos(exp(Symbol('f^{\\\\prime}', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given H{(G)} = \\int \\sin{(G)} dG, then obtain \\iint G \\sin{(G)} dG dG + \\iint H{(G)} \\sin{(G)} dG dG = \\iint G \\sin{(G)} dG dG + \\iint \\sin{(G)} \\int \\sin{(G)} dG dG dG", "derivation": "H{(G)} = \\int \\sin{(G)} dG and G + H{(G)} = G + \\int \\sin{(G)} dG and (G + H{(G)}) \\sin{(G)} = (G + \\int \\sin{(G)} dG) \\sin{(G)} and \\int (G + H{(G)}) \\sin{(G)} dG = \\int (G + \\int \\sin{(G)} dG) \\sin{(G)} dG and \\iint (G + H{(G)}) \\sin{(G)} dG dG = \\iint (G + \\int \\sin{(G)} dG) \\sin{(G)} dG dG and \\iint G \\sin{(G)} dG dG + \\iint H{(G)} \\sin{(G)} dG dG = \\iint G \\sin{(G)} dG dG + \\iint \\sin{(G)} \\int \\sin{(G)} dG dG dG", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('G', commutative=True)), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('H')(Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["times", 2, "sin(Symbol('G', commutative=True))"], "Equality(Mul(Add(Symbol('G', commutative=True), Function('H')(Symbol('G', commutative=True))), sin(Symbol('G', commutative=True))), Mul(Add(Symbol('G', commutative=True), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('G', commutative=True), Function('H')(Symbol('G', commutative=True))), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Add(Symbol('G', commutative=True), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('G', commutative=True), Function('H')(Symbol('G', commutative=True))), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Add(Symbol('G', commutative=True), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["expand", 5], "Equality(Add(Integral(Mul(Symbol('G', commutative=True), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Function('H')(Symbol('G', commutative=True)), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Add(Integral(Mul(Symbol('G', commutative=True), sin(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Mul(sin(Symbol('G', commutative=True)), Integral(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(M_{E})} = e^{M_{E}} and m{(M_{E})} = e^{M_{E}}, then obtain \\frac{d}{d M_{E}} \\mathbf{v}{(M_{E})} = \\frac{d}{d M_{E}} m{(M_{E})}", "derivation": "\\mathbf{v}{(M_{E})} = e^{M_{E}} and m{(M_{E})} = e^{M_{E}} and \\mathbf{v}{(M_{E})} = m{(M_{E})} and \\frac{d}{d M_{E}} \\mathbf{v}{(M_{E})} = \\frac{d}{d M_{E}} m{(M_{E})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('M_E', commutative=True)), Function('m')(Symbol('M_E', commutative=True)))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Function('m')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{E},T,b)} = (T \\mathbf{E})^{b}, then obtain (\\iint T \\mathbf{E} (- (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)}) d\\mathbf{E} db)^{b} = (\\iint 0 d\\mathbf{E} db)^{b}", "derivation": "\\phi_{1}{(\\mathbf{E},T,b)} = (T \\mathbf{E})^{b} and - (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)} = 0 and T \\mathbf{E} (- (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)}) = 0 and \\int T \\mathbf{E} (- (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)}) d\\mathbf{E} = \\int 0 d\\mathbf{E} and \\iint T \\mathbf{E} (- (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)}) d\\mathbf{E} db = \\iint 0 d\\mathbf{E} db and (\\iint T \\mathbf{E} (- (T \\mathbf{E})^{b} + \\phi_{1}{(\\mathbf{E},T,b)}) d\\mathbf{E} db)^{b} = (\\iint 0 d\\mathbf{E} db)^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True)))"], [["minus", 1, "Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True))), Integer(0))"], [["times", 2, "Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["power", 5, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Add(Mul(Integer(-1), Pow(Mul(Symbol('T', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('b', commutative=True))), Function('\\\\phi_1')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('T', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\rho{(u)} = \\cos{(u)}, then obtain 0 = -1 + \\frac{\\cos{(u)}}{\\rho{(u)}}", "derivation": "\\rho{(u)} = \\cos{(u)} and 0 = - \\rho{(u)} + \\cos{(u)} and 0 = \\frac{- \\rho{(u)} + \\cos{(u)}}{\\rho{(u)}} and 0 = -1 + \\frac{\\cos{(u)}}{\\rho{(u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["minus", 1, "Function('\\\\rho')(Symbol('u', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('u', commutative=True))), cos(Symbol('u', commutative=True))))"], [["divide", 2, "Function('\\\\rho')(Symbol('u', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('u', commutative=True))), cos(Symbol('u', commutative=True))), Pow(Function('\\\\rho')(Symbol('u', commutative=True)), Integer(-1))))"], [["expand", 3], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('\\\\rho')(Symbol('u', commutative=True)), Integer(-1)), cos(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(k)} = \\cos{(k)}, then obtain (0^{k})^{k} = 1", "derivation": "\\dot{z}{(k)} = \\cos{(k)} and 0 = - \\dot{z}{(k)} + \\cos{(k)} and 0^{k} = (- \\dot{z}{(k)} + \\cos{(k)})^{k} and (0^{k})^{k} = ((- \\dot{z}{(k)} + \\cos{(k)})^{k})^{k} and ((- \\dot{z}{(k)} + \\cos{(k)})^{k})^{k} = 1 and (0^{k})^{k} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["minus", 1, "Function('\\\\dot{z}')(Symbol('k', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Integer(0), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('k', commutative=True))), cos(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Pow(Integer(0), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\theta_{2}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})}, then obtain - V_{\\mathbf{B}} \\frac{d}{d V_{\\mathbf{B}}} \\theta_{2}{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})}", "derivation": "\\theta_{2}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} and \\theta_{2}{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} = \\cos{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} and \\frac{d}{d V_{\\mathbf{B}}} \\theta_{2}{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} and - V_{\\mathbf{B}} \\frac{d}{d V_{\\mathbf{B}}} \\theta_{2}{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})} = - V_{\\mathbf{B}} \\frac{d}{d V_{\\mathbf{B}}} \\cos{(V_{\\mathbf{B}})} \\cos^{- V_{\\mathbf{B}}}{(V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 1, "Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Mul(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\theta_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Mul(Function('\\\\theta_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Mul(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(cos(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbf{H})} = e^{\\mathbf{H}}, then derive 1 = \\frac{e^{\\mathbf{H}}}{\\frac{d^{2}}{d \\mathbf{H}^{2}} \\eta^{\\prime}{(\\mathbf{H})}}, then obtain 1 = \\frac{e^{\\mathbf{H}}}{\\frac{d^{2}}{d \\mathbf{H}^{2}} e^{\\mathbf{H}}}", "derivation": "\\eta^{\\prime}{(\\mathbf{H})} = e^{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\eta^{\\prime}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} e^{\\mathbf{H}} and \\frac{d^{2}}{d \\mathbf{H}^{2}} \\eta^{\\prime}{(\\mathbf{H})} = \\frac{d^{2}}{d \\mathbf{H}^{2}} e^{\\mathbf{H}} and 1 = \\frac{\\frac{d^{2}}{d \\mathbf{H}^{2}} e^{\\mathbf{H}}}{\\frac{d^{2}}{d \\mathbf{H}^{2}} \\eta^{\\prime}{(\\mathbf{H})}} and 1 = \\frac{e^{\\mathbf{H}}}{\\frac{d^{2}}{d \\mathbf{H}^{2}} \\eta^{\\prime}{(\\mathbf{H})}} and 1 = \\frac{e^{\\mathbf{H}}}{\\frac{d^{2}}{d \\mathbf{H}^{2}} e^{\\mathbf{H}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"], [["divide", 3, "Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(-1)), Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Mul(exp(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Mul(exp(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(G,t,v_{t})} = (\\frac{t}{v_{t}})^{G}, then obtain \\frac{\\partial}{\\partial t} (- G + \\operatorname{x^{{\\}'}}{(G,t,v_{t})})^{G} = \\frac{\\partial}{\\partial t} (- G + (\\frac{t}{v_{t}})^{G})^{G}", "derivation": "\\operatorname{x^{{\\}'}}{(G,t,v_{t})} = (\\frac{t}{v_{t}})^{G} and - G + \\operatorname{x^{{\\}'}}{(G,t,v_{t})} = - G + (\\frac{t}{v_{t}})^{G} and (- G + \\operatorname{x^{{\\}'}}{(G,t,v_{t})})^{G} = (- G + (\\frac{t}{v_{t}})^{G})^{G} and \\frac{\\partial}{\\partial t} (- G + \\operatorname{x^{{\\}'}}{(G,t,v_{t})})^{G} = \\frac{\\partial}{\\partial t} (- G + (\\frac{t}{v_{t}})^{G})^{G}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('t', commutative=True), Symbol('v_t', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Symbol('G', commutative=True)))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Symbol('G', commutative=True))))"], [["power", 2, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Symbol('G', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('x^\\\\prime')(Symbol('G', commutative=True), Symbol('t', commutative=True), Symbol('v_t', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Mul(Symbol('t', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\dot{z},q)} = \\int (\\dot{z} + q) dq, then derive 1 = \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\mathbf{S}{(\\dot{z},q)}}, then obtain \\cos{(\\frac{d}{d q} 1)} = \\cos{(\\frac{\\partial}{\\partial q} \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\int (\\dot{z} + q) dq})}", "derivation": "\\mathbf{S}{(\\dot{z},q)} = \\int (\\dot{z} + q) dq and 1 = \\frac{\\int (\\dot{z} + q) dq}{\\mathbf{S}{(\\dot{z},q)}} and 1 = \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\mathbf{S}{(\\dot{z},q)}} and \\frac{d}{d q} 1 = \\frac{\\partial}{\\partial q} \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\mathbf{S}{(\\dot{z},q)}} and \\cos{(\\frac{d}{d q} 1)} = \\cos{(\\frac{\\partial}{\\partial q} \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\mathbf{S}{(\\dot{z},q)}})} and \\cos{(\\frac{d}{d q} 1)} = \\cos{(\\frac{\\partial}{\\partial q} \\frac{\\dot{z} q + \\lambda + \\frac{q^{2}}{2}}{\\int (\\dot{z} + q) dq})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Integral(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Integral(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(1), Mul(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('q', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1)))), cos(Derivative(Mul(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Pow(Function('\\\\mathbf{S}')(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(cos(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1)))), cos(Derivative(Mul(Add(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\lambda', commutative=True), Mul(Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2)))), Pow(Integral(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Tuple(Symbol('q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(x,v_{x})} = \\frac{v_{x}}{x}, then derive \\int 0 dv_{x} = \\phi_2 - \\frac{\\int - v_{x} dv_{x} + \\int x \\operatorname{t_{1}}{(x,v_{x})} dv_{x}}{x}, then obtain \\int (\\frac{v_{x}}{x} - \\operatorname{t_{1}}{(x,v_{x})}) dv_{x} = \\phi_2 - \\frac{\\int - v_{x} dv_{x} + \\int v_{x} dv_{x}}{x}", "derivation": "\\operatorname{t_{1}}{(x,v_{x})} = \\frac{v_{x}}{x} and 0 = \\frac{v_{x}}{x} - \\operatorname{t_{1}}{(x,v_{x})} and \\int 0 dv_{x} = \\int (\\frac{v_{x}}{x} - \\operatorname{t_{1}}{(x,v_{x})}) dv_{x} and \\int 0 dv_{x} = \\phi_2 - \\frac{\\int - v_{x} dv_{x} + \\int x \\operatorname{t_{1}}{(x,v_{x})} dv_{x}}{x} and \\int 0 dv_{x} = \\phi_2 - \\frac{\\int - v_{x} dv_{x} + \\int v_{x} dv_{x}}{x} and \\int (\\frac{v_{x}}{x} - \\operatorname{t_{1}}{(x,v_{x})}) dv_{x} = \\phi_2 - \\frac{\\int - v_{x} dv_{x} + \\int v_{x} dv_{x}}{x}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)), Mul(Symbol('v_x', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Integer(-1), Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('v_x', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Mul(Symbol('v_x', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Symbol('v_x', commutative=True), Tuple(Symbol('v_x', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Mul(Symbol('v_x', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t_1')(Symbol('x', commutative=True), Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Add(Integral(Mul(Integer(-1), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Symbol('v_x', commutative=True), Tuple(Symbol('v_x', commutative=True)))))))"]]}, {"prompt": "Given \\psi{(q)} = \\log{(q)} and \\operatorname{g_{\\varepsilon}}{(q)} = \\log{(q)}^{q}, then obtain \\frac{\\operatorname{g_{\\varepsilon}}{(q)}}{q} = \\frac{\\psi^{q}{(q)}}{q}", "derivation": "\\psi{(q)} = \\log{(q)} and \\psi^{q}{(q)} = \\log{(q)}^{q} and \\operatorname{g_{\\varepsilon}}{(q)} = \\log{(q)}^{q} and \\frac{\\operatorname{g_{\\varepsilon}}{(q)}}{q} = \\frac{\\log{(q)}^{q}}{q} and \\frac{\\operatorname{g_{\\varepsilon}}{(q)}}{q} = \\frac{\\psi^{q}{(q)}}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True)), Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["divide", 3, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(log(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('q', commutative=True))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(\\nabla)} = e^{\\nabla}, then obtain \\nabla + (\\frac{d}{d \\nabla} \\hat{H}{(\\nabla)})^{\\nabla} = \\nabla + (\\frac{d}{d \\nabla} e^{\\nabla})^{\\nabla}", "derivation": "\\hat{H}{(\\nabla)} = e^{\\nabla} and \\frac{d}{d \\nabla} \\hat{H}{(\\nabla)} = \\frac{d}{d \\nabla} e^{\\nabla} and (\\frac{d}{d \\nabla} \\hat{H}{(\\nabla)})^{\\nabla} = (\\frac{d}{d \\nabla} e^{\\nabla})^{\\nabla} and \\nabla + (\\frac{d}{d \\nabla} \\hat{H}{(\\nabla)})^{\\nabla} = \\nabla + (\\frac{d}{d \\nabla} e^{\\nabla})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True)), Pow(Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True)))"], [["add", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Symbol('\\\\nabla', commutative=True), Pow(Derivative(Function('\\\\hat{H}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Pow(Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mu,P_{e})} = \\sin{(P_{e} + \\mu)} and \\operatorname{v_{x}}{(\\mu,P_{e})} = \\int \\sin{(P_{e} + \\mu)} d\\mu, then derive \\operatorname{v_{x}}{(\\mu,P_{e})} = b - \\cos{(P_{e} + \\mu)}, then obtain \\int \\operatorname{A_{z}}{(\\mu,P_{e})} d\\mu = b - \\cos{(P_{e} + \\mu)}", "derivation": "\\operatorname{A_{z}}{(\\mu,P_{e})} = \\sin{(P_{e} + \\mu)} and \\int \\operatorname{A_{z}}{(\\mu,P_{e})} d\\mu = \\int \\sin{(P_{e} + \\mu)} d\\mu and \\operatorname{v_{x}}{(\\mu,P_{e})} = \\int \\sin{(P_{e} + \\mu)} d\\mu and \\operatorname{v_{x}}{(\\mu,P_{e})} = b - \\cos{(P_{e} + \\mu)} and \\int \\sin{(P_{e} + \\mu)} d\\mu = b - \\cos{(P_{e} + \\mu)} and \\int \\operatorname{A_{z}}{(\\mu,P_{e})} d\\mu = b - \\cos{(P_{e} + \\mu)}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mu', commutative=True), Symbol('P_e', commutative=True)), sin(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('\\\\mu', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(sin(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('P_e', commutative=True)), Integral(sin(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Function('v_x')(Symbol('\\\\mu', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(sin(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('A_z')(Symbol('\\\\mu', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), cos(Add(Symbol('P_e', commutative=True), Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given g{(f,M_{E})} = M_{E} + f and \\phi_{1}{(\\rho_f,\\mu,r)} = \\frac{\\rho_f r}{\\mu}, then derive \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})} = 1, then obtain \\phi_{1}{(\\rho_f,\\mu,r)} = \\frac{\\rho_f r \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})}}{\\mu}", "derivation": "g{(f,M_{E})} = M_{E} + f and \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})} = \\frac{\\partial}{\\partial M_{E}} (M_{E} + f) and \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})} = 1 and \\phi_{1}{(\\rho_f,\\mu,r)} = \\frac{\\rho_f r}{\\mu} and \\frac{\\rho_f r \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})}}{\\mu} = \\frac{\\rho_f r}{\\mu} and \\phi_{1}{(\\rho_f,\\mu,r)} = \\frac{\\rho_f r \\frac{\\partial}{\\partial M_{E}} g{(f,M_{E})}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('f', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('f', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('f', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(1))"], ["get_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Symbol('r', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Symbol('r', commutative=True), Derivative(Function('g')(Symbol('f', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\phi_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True), Symbol('r', commutative=True), Derivative(Function('g')(Symbol('f', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(v_{2})} = \\cos{(v_{2})} and b{(v_{2})} = v_{2}, then obtain \\int f{(v_{2})} db{(v_{2})} = \\int \\cos{(v_{2})} db{(v_{2})}", "derivation": "f{(v_{2})} = \\cos{(v_{2})} and b{(v_{2})} = v_{2} and \\int f{(v_{2})} dv_{2} = \\int \\cos{(v_{2})} dv_{2} and \\int f{(v_{2})} db{(v_{2})} = \\int \\cos{(v_{2})} db{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], ["renaming_premise", "Equality(Function('b')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('f')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Function('f')(Symbol('v_2', commutative=True)), Tuple(Function('b')(Symbol('v_2', commutative=True)))), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Function('b')(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{y})} = \\sin{(v_{y})} and \\dot{\\mathbf{r}}{(C_{2},v_{z})} = C_{2} + v_{z}, then derive \\int \\operatorname{F_{c}}{(v_{y})} dv_{y} = z - \\cos{(v_{y})}, then obtain \\dot{\\mathbf{r}}{(C_{2},v_{z})} \\int \\sin{(v_{y})} dv_{y} = (C_{2} + v_{z}) \\int \\sin{(v_{y})} dv_{y}", "derivation": "\\operatorname{F_{c}}{(v_{y})} = \\sin{(v_{y})} and \\int \\operatorname{F_{c}}{(v_{y})} dv_{y} = \\int \\sin{(v_{y})} dv_{y} and \\int \\operatorname{F_{c}}{(v_{y})} dv_{y} = z - \\cos{(v_{y})} and \\dot{\\mathbf{r}}{(C_{2},v_{z})} = C_{2} + v_{z} and \\int \\sin{(v_{y})} dv_{y} = z - \\cos{(v_{y})} and (z - \\cos{(v_{y})}) \\dot{\\mathbf{r}}{(C_{2},v_{z})} = (C_{2} + v_{z}) (z - \\cos{(v_{y})}) and \\dot{\\mathbf{r}}{(C_{2},v_{z})} \\int \\sin{(v_{y})} dv_{y} = (C_{2} + v_{z}) \\int \\sin{(v_{y})} dv_{y}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_c')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))))"], [["times", 4, "Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True))))"], "Equality(Mul(Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True))), Mul(Add(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('z', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))), Mul(Add(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given f{(\\mathbf{H})} = e^{\\mathbf{H}} and \\mathbf{r}{(\\mathbf{H})} = \\int e^{\\mathbf{H}} d\\mathbf{H}, then obtain - \\frac{(\\int f{(\\mathbf{H})} d\\mathbf{H}) (\\int e^{\\mathbf{H}} d\\mathbf{H})^{\\mathbf{H}}}{\\mathbf{H}} = - \\frac{\\mathbf{r}^{\\mathbf{H}}{(\\mathbf{H})} \\int f{(\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H}}", "derivation": "f{(\\mathbf{H})} = e^{\\mathbf{H}} and \\int f{(\\mathbf{H})} d\\mathbf{H} = \\int e^{\\mathbf{H}} d\\mathbf{H} and (\\int f{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\int e^{\\mathbf{H}} d\\mathbf{H})^{\\mathbf{H}} and \\mathbf{r}{(\\mathbf{H})} = \\int e^{\\mathbf{H}} d\\mathbf{H} and (\\int f{(\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = \\mathbf{r}^{\\mathbf{H}}{(\\mathbf{H})} and (\\int e^{\\mathbf{H}} d\\mathbf{H})^{\\mathbf{H}} = \\mathbf{r}^{\\mathbf{H}}{(\\mathbf{H})} and - \\frac{(\\int f{(\\mathbf{H})} d\\mathbf{H}) (\\int e^{\\mathbf{H}} d\\mathbf{H})^{\\mathbf{H}}}{\\mathbf{H}} = - \\frac{\\mathbf{r}^{\\mathbf{H}}{(\\mathbf{H})} \\int f{(\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 6, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Pow(Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Function('f')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(g)} = \\cos{(g)}, then obtain 0 = \\frac{d}{d g} (\\cos{(g)} + 2) - \\frac{d}{d g} (\\frac{\\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} + 1)", "derivation": "\\mathbf{v}{(g)} = \\cos{(g)} and \\frac{\\mathbf{v}{(g)}}{\\cos{(g)}} = 1 and \\frac{\\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} = \\cos{(g)} + 1 and \\frac{2 \\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} = \\frac{\\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} + 1 and \\frac{2 \\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} = \\cos{(g)} + 2 and \\frac{d}{d g} (\\frac{2 \\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)}) = \\frac{d}{d g} (\\cos{(g)} + 2) and 0 = - \\frac{d}{d g} (\\frac{2 \\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)}) + \\frac{d}{d g} (\\cos{(g)} + 2) and 0 = \\frac{d}{d g} (\\cos{(g)} + 2) - \\frac{d}{d g} (\\frac{\\mathbf{v}{(g)}}{\\cos{(g)}} + \\cos{(g)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["divide", 1, "cos(Symbol('g', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "cos(Symbol('g', commutative=True))"], "Equality(Add(Mul(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Add(cos(Symbol('g', commutative=True)), Integer(1)))"], [["add", 3, "Mul(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Add(Mul(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Add(cos(Symbol('g', commutative=True)), Integer(2)))"], [["differentiate", 5, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 6, "Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Derivative(Add(cos(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Integer(0), Add(Derivative(Add(cos(Symbol('g', commutative=True)), Integer(2)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Mul(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1))), cos(Symbol('g', commutative=True)), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given Z{(F_{H})} = \\cos{(F_{H})}, then obtain - \\mathbf{J}{(F_{H})} \\sin{(\\frac{Z^{F_{H}}{(F_{H})}}{Z{(F_{H})}})} = - \\mathbf{J}{(F_{H})} \\sin{(\\frac{\\cos^{F_{H}}{(F_{H})}}{Z{(F_{H})}})}", "derivation": "Z{(F_{H})} = \\cos{(F_{H})} and Z^{F_{H}}{(F_{H})} = \\cos^{F_{H}}{(F_{H})} and \\frac{Z^{F_{H}}{(F_{H})}}{Z{(F_{H})}} = \\frac{\\cos^{F_{H}}{(F_{H})}}{Z{(F_{H})}} and - \\frac{Z^{F_{H}}{(F_{H})}}{Z{(F_{H})}} = - \\frac{\\cos^{F_{H}}{(F_{H})}}{Z{(F_{H})}} and - \\sin{(\\frac{Z^{F_{H}}{(F_{H})}}{Z{(F_{H})}})} = - \\sin{(\\frac{\\cos^{F_{H}}{(F_{H})}}{Z{(F_{H})}})} and - \\mathbf{J}{(F_{H})} \\sin{(\\frac{Z^{F_{H}}{(F_{H})}}{Z{(F_{H})}})} = - \\mathbf{J}{(F_{H})} \\sin{(\\frac{\\cos^{F_{H}}{(F_{H})}}{Z{(F_{H})}})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["divide", 2, "Function('Z')(Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Integer(-1), Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))), Mul(Integer(-1), sin(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))))"], [["times", 5, "Function('\\\\mathbf{J}')(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('F_H', commutative=True)), sin(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('Z')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))), Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('F_H', commutative=True)), sin(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given I{(M)} = \\log{(e^{M})}, then derive ((\\dot{z} + I{(M)})^{M})^{M} = ((M + n_{2})^{M})^{M}, then obtain ((\\dot{z} + \\log{(e^{M})})^{M})^{M} = ((M + n_{2})^{M})^{M}", "derivation": "I{(M)} = \\log{(e^{M})} and \\frac{d}{d M} I{(M)} = \\frac{d}{d M} \\log{(e^{M})} and \\int \\frac{d}{d M} I{(M)} dM = \\int \\frac{d}{d M} \\log{(e^{M})} dM and (\\int \\frac{d}{d M} I{(M)} dM)^{M} = (\\int \\frac{d}{d M} \\log{(e^{M})} dM)^{M} and ((\\int \\frac{d}{d M} I{(M)} dM)^{M})^{M} = ((\\int \\frac{d}{d M} \\log{(e^{M})} dM)^{M})^{M} and ((\\dot{z} + I{(M)})^{M})^{M} = ((M + n_{2})^{M})^{M} and ((\\dot{z} + \\log{(e^{M})})^{M})^{M} = ((M + n_{2})^{M})^{M}", "srepr_derivation": [["get_premise", "Equality(Function('I')(Symbol('M', commutative=True)), log(exp(Symbol('M', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(log(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Function('I')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(log(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('I')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Integral(Derivative(log(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Integral(Derivative(Function('I')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Integral(Derivative(log(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Pow(Add(Symbol('\\\\dot{z}', commutative=True), Function('I')(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Add(Symbol('M', commutative=True), Symbol('n_2', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Pow(Add(Symbol('\\\\dot{z}', commutative=True), log(exp(Symbol('M', commutative=True)))), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Add(Symbol('M', commutative=True), Symbol('n_2', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(f_{\\mathbf{p}},v_{1})} = \\int f_{\\mathbf{p}} v_{1} dv_{1} and b{(f_{\\mathbf{p}},v_{1})} = v_{1} \\int f_{\\mathbf{p}} v_{1} dv_{1}, then obtain - \\frac{b{(f_{\\mathbf{p}},v_{1})}}{v_{1} (- v_{1} + \\int f_{\\mathbf{p}} v_{1} dv_{1})} = - \\frac{\\mathbf{F}{(f_{\\mathbf{p}},v_{1})}}{- v_{1} + \\int f_{\\mathbf{p}} v_{1} dv_{1}}", "derivation": "\\mathbf{F}{(f_{\\mathbf{p}},v_{1})} = \\int f_{\\mathbf{p}} v_{1} dv_{1} and v_{1} \\mathbf{F}{(f_{\\mathbf{p}},v_{1})} = v_{1} \\int f_{\\mathbf{p}} v_{1} dv_{1} and b{(f_{\\mathbf{p}},v_{1})} = v_{1} \\int f_{\\mathbf{p}} v_{1} dv_{1} and b{(f_{\\mathbf{p}},v_{1})} = v_{1} \\mathbf{F}{(f_{\\mathbf{p}},v_{1})} and - \\frac{b{(f_{\\mathbf{p}},v_{1})}}{v_{1} (- v_{1} + \\int f_{\\mathbf{p}} v_{1} dv_{1})} = - \\frac{\\mathbf{F}{(f_{\\mathbf{p}},v_{1})}}{- v_{1} + \\int f_{\\mathbf{p}} v_{1} dv_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["times", 1, "Symbol('v_1', commutative=True)"], "Equality(Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('v_1', commutative=True), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('v_1', commutative=True), Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Integer(-1)), Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Integer(-1)), Function('\\\\mathbf{F}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\rho_f,Z)} = Z \\cos{(\\rho_f)}, then obtain (- Z \\cos{(\\rho_f)} + \\frac{\\operatorname{F_{c}}{(\\rho_f,Z)}}{Z \\cos{(\\rho_f)}})^{\\rho_f} = (- Z \\cos{(\\rho_f)} + 1)^{\\rho_f}", "derivation": "\\operatorname{F_{c}}{(\\rho_f,Z)} = Z \\cos{(\\rho_f)} and \\frac{\\operatorname{F_{c}}{(\\rho_f,Z)}}{Z \\cos{(\\rho_f)}} = 1 and - Z \\cos{(\\rho_f)} + \\frac{\\operatorname{F_{c}}{(\\rho_f,Z)}}{Z \\cos{(\\rho_f)}} = - Z \\cos{(\\rho_f)} + 1 and (- Z \\cos{(\\rho_f)} + \\frac{\\operatorname{F_{c}}{(\\rho_f,Z)}}{Z \\cos{(\\rho_f)}})^{\\rho_f} = (- Z \\cos{(\\rho_f)} + 1)^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 1, "Mul(Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('F_c')(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('F_c')(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))), Integer(1)))"], [["power", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Function('F_c')(Symbol('\\\\rho_f', commutative=True), Symbol('Z', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(-1)))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))), Integer(1)), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(l,\\mathbf{S},\\rho)} = \\mathbf{S} \\rho l, then obtain \\int \\frac{\\partial}{\\partial \\rho} \\frac{- \\mathbf{S} \\rho l + \\varphi^{*}{(l,\\mathbf{S},\\rho)}}{\\mathbf{S} \\rho l} dl = \\int \\frac{d}{d \\rho} 0 dl", "derivation": "\\varphi^{*}{(l,\\mathbf{S},\\rho)} = \\mathbf{S} \\rho l and - \\mathbf{S} \\rho l + \\varphi^{*}{(l,\\mathbf{S},\\rho)} = 0 and \\frac{- \\mathbf{S} \\rho l + \\varphi^{*}{(l,\\mathbf{S},\\rho)}}{\\mathbf{S} \\rho l} = 0 and \\frac{\\partial}{\\partial \\rho} \\frac{- \\mathbf{S} \\rho l + \\varphi^{*}{(l,\\mathbf{S},\\rho)}}{\\mathbf{S} \\rho l} = \\frac{d}{d \\rho} 0 and \\int \\frac{\\partial}{\\partial \\rho} \\frac{- \\mathbf{S} \\rho l + \\varphi^{*}{(l,\\mathbf{S},\\rho)}}{\\mathbf{S} \\rho l} dl = \\int \\frac{d}{d \\rho} 0 dl", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True)), Function('\\\\varphi^*')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(0))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True)), Function('\\\\varphi^*')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True)), Function('\\\\varphi^*')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('l', commutative=True)), Function('\\\\varphi^*')(Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(f)} = e^{f} and \\mathbf{F}{(f)} = e^{f} and \\theta{(f)} = e^{f}, then obtain 2 \\rho_{f}{(f)} e^{f} + \\rho_{f}{(f)} - 2 e^{f} = (\\rho_{f}{(f)} + e^{f}) e^{f} + \\rho_{f}{(f)} - 2 e^{f}", "derivation": "\\rho_{f}{(f)} = e^{f} and 2 \\rho_{f}{(f)} = \\rho_{f}{(f)} + e^{f} and \\mathbf{F}{(f)} = e^{f} and \\theta{(f)} = e^{f} and 2 \\rho_{f}{(f)} = \\mathbf{F}{(f)} + \\rho_{f}{(f)} and 2 \\rho_{f}{(f)} \\theta{(f)} = (\\mathbf{F}{(f)} + \\rho_{f}{(f)}) \\theta{(f)} and 2 \\rho_{f}{(f)} \\theta{(f)} = (\\rho_{f}{(f)} + e^{f}) \\theta{(f)} and 2 \\rho_{f}{(f)} e^{f} = (\\rho_{f}{(f)} + e^{f}) e^{f} and 2 \\rho_{f}{(f)} e^{f} + \\rho_{f}{(f)} - 2 e^{f} = (\\rho_{f}{(f)} + e^{f}) e^{f} + \\rho_{f}{(f)} - 2 e^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True)))"], [["add", 1, "Function('\\\\rho_f')(Symbol('f', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True))), Add(Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Function('\\\\rho_f')(Symbol('f', commutative=True))))"], [["times", 5, "Function('\\\\theta')(Symbol('f', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True)), Function('\\\\theta')(Symbol('f', commutative=True))), Mul(Add(Function('\\\\mathbf{F}')(Symbol('f', commutative=True)), Function('\\\\rho_f')(Symbol('f', commutative=True))), Function('\\\\theta')(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True)), Function('\\\\theta')(Symbol('f', commutative=True))), Mul(Add(Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))), Function('\\\\theta')(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))), Mul(Add(Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))), exp(Symbol('f', commutative=True))))"], [["minus", 8, "Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f', commutative=True))), Mul(Integer(2), exp(Symbol('f', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))), Function('\\\\rho_f')(Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('f', commutative=True)))), Add(Mul(Add(Function('\\\\rho_f')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True))), exp(Symbol('f', commutative=True))), Function('\\\\rho_f')(Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given q{(J_{\\varepsilon},\\tilde{g})} = J_{\\varepsilon} - \\tilde{g}, then derive \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})} = 1, then obtain \\frac{J_{\\varepsilon} \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})}}{2} = \\frac{J_{\\varepsilon}}{2}", "derivation": "q{(J_{\\varepsilon},\\tilde{g})} = J_{\\varepsilon} - \\tilde{g} and \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} (J_{\\varepsilon} - \\tilde{g}) and \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})} = 1 and \\frac{J_{\\varepsilon}^{2} \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})}}{2} = \\frac{J_{\\varepsilon}^{2}}{2} and \\frac{J_{\\varepsilon} \\frac{\\partial}{\\partial J_{\\varepsilon}} q{(J_{\\varepsilon},\\tilde{g})}}{2} = \\frac{J_{\\varepsilon}}{2}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), Derivative(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["divide", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Rational(1, 2), Symbol('J_{\\\\varepsilon}', commutative=True), Derivative(Function('q')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(Rational(1, 2), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given B{(\\hat{H},\\mathbf{J}_f,\\hbar)} = \\hat{H} - \\hbar - \\mathbf{J}_f and \\mathbf{r}{(\\hat{H})} = \\hat{H}, then obtain \\frac{B{(\\hat{H},\\mathbf{J}_f,\\hbar)}}{\\hat{H}^{2} \\hbar} = \\frac{\\hat{H} - \\hbar - \\mathbf{J}_f}{\\hat{H}^{2} \\hbar}", "derivation": "B{(\\hat{H},\\mathbf{J}_f,\\hbar)} = \\hat{H} - \\hbar - \\mathbf{J}_f and \\frac{B{(\\hat{H},\\mathbf{J}_f,\\hbar)}}{\\hbar} = \\frac{\\hat{H} - \\hbar - \\mathbf{J}_f}{\\hbar} and \\frac{B{(\\hat{H},\\mathbf{J}_f,\\hbar)}}{\\hat{H} \\hbar} = \\frac{\\hat{H} - \\hbar - \\mathbf{J}_f}{\\hat{H} \\hbar} and \\mathbf{r}{(\\hat{H})} = \\hat{H} and \\frac{B{(\\hat{H},\\mathbf{J}_f,\\hbar)}}{\\hat{H} \\hbar \\mathbf{r}{(\\hat{H})}} = \\frac{\\hat{H} - \\hbar - \\mathbf{J}_f}{\\hat{H} \\hbar \\mathbf{r}{(\\hat{H})}} and \\frac{B{(\\hat{H},\\mathbf{J}_f,\\hbar)}}{\\hat{H}^{2} \\hbar} = \\frac{\\hat{H} - \\hbar - \\mathbf{J}_f}{\\hat{H}^{2} \\hbar}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["divide", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], [["divide", 3, "Function('\\\\mathbf{r}')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('B')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\eta{(J,v_{1})} = \\frac{v_{1}}{J} and \\mathbf{E}{(J,v_{1})} = \\int \\eta{(J,v_{1})} dJ, then obtain \\frac{\\partial}{\\partial J} (e^{\\int \\eta{(J,v_{1})} dJ})^{J} = \\frac{\\partial}{\\partial J} (e^{\\mathbf{E}{(J,v_{1})}})^{J}", "derivation": "\\eta{(J,v_{1})} = \\frac{v_{1}}{J} and \\int \\eta{(J,v_{1})} dJ = \\int \\frac{v_{1}}{J} dJ and \\mathbf{E}{(J,v_{1})} = \\int \\eta{(J,v_{1})} dJ and \\mathbf{E}{(J,v_{1})} = \\int \\frac{v_{1}}{J} dJ and e^{\\int \\eta{(J,v_{1})} dJ} = e^{\\int \\frac{v_{1}}{J} dJ} and (e^{\\int \\eta{(J,v_{1})} dJ})^{J} = (e^{\\int \\frac{v_{1}}{J} dJ})^{J} and \\frac{\\partial}{\\partial J} (e^{\\int \\eta{(J,v_{1})} dJ})^{J} = \\frac{\\partial}{\\partial J} (e^{\\int \\frac{v_{1}}{J} dJ})^{J} and \\frac{\\partial}{\\partial J} (e^{\\int \\eta{(J,v_{1})} dJ})^{J} = \\frac{\\partial}{\\partial J} (e^{\\mathbf{E}{(J,v_{1})}})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), exp(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(exp(Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Pow(exp(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"], [["differentiate", 6, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(exp(Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(exp(Integral(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Derivative(Pow(exp(Integral(Function('\\\\eta')(Symbol('J', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(exp(Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('v_1', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(\\mathbf{J},\\mathbf{J}_P)} = \\mathbf{J} - \\mathbf{J}_P, then obtain \\int (\\mathbf{J}_P H{(\\mathbf{J},\\mathbf{J}_P)})^{\\mathbf{J}_P} d\\mathbf{J}_P = \\int (\\mathbf{J}_P (\\mathbf{J} - \\mathbf{J}_P))^{\\mathbf{J}_P} d\\mathbf{J}_P", "derivation": "H{(\\mathbf{J},\\mathbf{J}_P)} = \\mathbf{J} - \\mathbf{J}_P and \\mathbf{J}_P H{(\\mathbf{J},\\mathbf{J}_P)} = \\mathbf{J}_P (\\mathbf{J} - \\mathbf{J}_P) and (\\mathbf{J}_P H{(\\mathbf{J},\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\mathbf{J}_P (\\mathbf{J} - \\mathbf{J}_P))^{\\mathbf{J}_P} and \\int (\\mathbf{J}_P H{(\\mathbf{J},\\mathbf{J}_P)})^{\\mathbf{J}_P} d\\mathbf{J}_P = \\int (\\mathbf{J}_P (\\mathbf{J} - \\mathbf{J}_P))^{\\mathbf{J}_P} d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('H')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{f},x,z^{*})} = - \\mathbf{f} + x + z^{*}, then obtain \\int - \\frac{\\int (\\mathbf{f} - x - z^{*} + \\mathbf{P}{(\\mathbf{f},x,z^{*})}) d\\mathbf{f}}{x} d\\mathbf{f} = \\int - \\frac{\\int 0 d\\mathbf{f}}{x} d\\mathbf{f}", "derivation": "\\mathbf{P}{(\\mathbf{f},x,z^{*})} = - \\mathbf{f} + x + z^{*} and 0 = - \\mathbf{f} + x + z^{*} - \\mathbf{P}{(\\mathbf{f},x,z^{*})} and \\mathbf{f} - x - z^{*} + \\mathbf{P}{(\\mathbf{f},x,z^{*})} = 0 and \\int (\\mathbf{f} - x - z^{*} + \\mathbf{P}{(\\mathbf{f},x,z^{*})}) d\\mathbf{f} = \\int 0 d\\mathbf{f} and - \\frac{\\int (\\mathbf{f} - x - z^{*} + \\mathbf{P}{(\\mathbf{f},x,z^{*})}) d\\mathbf{f}}{x} = - \\frac{\\int 0 d\\mathbf{f}}{x} and \\int - \\frac{\\int (\\mathbf{f} - x - z^{*} + \\mathbf{P}{(\\mathbf{f},x,z^{*})}) d\\mathbf{f}}{x} d\\mathbf{f} = \\int - \\frac{\\int 0 d\\mathbf{f}}{x} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('x', commutative=True), Symbol('z^*', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('x', commutative=True), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('x', commutative=True), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))))"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('x', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(L)} = \\sin{(L)}, then obtain (\\int \\frac{d}{d L} (\\operatorname{C_{2}}{(L)} - 1)^{L} dL)^{L} = (\\int \\frac{d}{d L} (\\sin{(L)} - 1)^{L} dL)^{L}", "derivation": "\\operatorname{C_{2}}{(L)} = \\sin{(L)} and \\operatorname{C_{2}}{(L)} - 1 = \\sin{(L)} - 1 and (\\operatorname{C_{2}}{(L)} - 1)^{L} = (\\sin{(L)} - 1)^{L} and \\frac{d}{d L} (\\operatorname{C_{2}}{(L)} - 1)^{L} = \\frac{d}{d L} (\\sin{(L)} - 1)^{L} and \\int \\frac{d}{d L} (\\operatorname{C_{2}}{(L)} - 1)^{L} dL = \\int \\frac{d}{d L} (\\sin{(L)} - 1)^{L} dL and (\\int \\frac{d}{d L} (\\operatorname{C_{2}}{(L)} - 1)^{L} dL)^{L} = (\\int \\frac{d}{d L} (\\sin{(L)} - 1)^{L} dL)^{L}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('C_2')(Symbol('L', commutative=True)), Integer(-1)), Add(sin(Symbol('L', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Pow(Add(sin(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Add(sin(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Pow(Add(sin(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"], [["power", 5, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Derivative(Pow(Add(Function('C_2')(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Derivative(Pow(Add(sin(Symbol('L', commutative=True)), Integer(-1)), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\eta,Z)} = \\frac{\\eta}{Z}, then obtain (\\frac{\\partial}{\\partial Z} Z \\theta_{1}{(\\eta,Z)})^{Z} = (\\frac{d}{d Z} \\eta)^{Z}", "derivation": "\\theta_{1}{(\\eta,Z)} = \\frac{\\eta}{Z} and \\frac{Z \\theta_{1}{(\\eta,Z)}}{\\eta} = 1 and Z \\theta_{1}{(\\eta,Z)} = \\eta and \\frac{\\partial}{\\partial Z} Z \\theta_{1}{(\\eta,Z)} = \\frac{d}{d Z} \\eta and (\\frac{\\partial}{\\partial Z} Z \\theta_{1}{(\\eta,Z)})^{Z} = (\\frac{d}{d Z} \\eta)^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\eta', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('\\\\eta', commutative=True), Symbol('Z', commutative=True))), Integer(1))"], [["times", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Function('\\\\theta_1')(Symbol('\\\\eta', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\eta', commutative=True))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Symbol('Z', commutative=True), Function('\\\\theta_1')(Symbol('\\\\eta', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('Z', commutative=True), Function('\\\\theta_1')(Symbol('\\\\eta', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('Z', commutative=True)), Pow(Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M}, then derive \\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M}, then obtain (\\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} = (\\frac{d^{2}}{d \\mathbf{J}_M^{2}} e^{\\mathbf{J}_M})^{\\mathbf{J}_M}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M} and (\\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} = (\\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M})^{\\mathbf{J}_M} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)} = e^{\\mathbf{J}_M} and e^{\\mathbf{J}_M} = \\frac{d}{d \\mathbf{J}_M} e^{\\mathbf{J}_M} and (\\frac{d}{d \\mathbf{J}_M} \\operatorname{v_{2}}{(\\mathbf{J}_M)})^{\\mathbf{J}_M} = (\\frac{d^{2}}{d \\mathbf{J}_M^{2}} e^{\\mathbf{J}_M})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), exp(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Derivative(Function('v_2')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(2))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\chi{(\\hat{X},u)} = u^{\\hat{X}} and \\operatorname{f^{\\prime}}{(\\hat{X},u)} = u^{\\hat{X}}, then obtain \\chi{(\\hat{X},u)} - \\operatorname{f^{\\prime}}{(\\hat{X},u)} = 0", "derivation": "\\chi{(\\hat{X},u)} = u^{\\hat{X}} and \\operatorname{f^{\\prime}}{(\\hat{X},u)} = u^{\\hat{X}} and \\chi{(\\hat{X},u)} = \\operatorname{f^{\\prime}}{(\\hat{X},u)} and - u^{\\hat{X}} + \\chi{(\\hat{X},u)} = - u^{\\hat{X}} + \\operatorname{f^{\\prime}}{(\\hat{X},u)} and \\chi{(\\hat{X},u)} - \\operatorname{f^{\\prime}}{(\\hat{X},u)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)))"], [["minus", 3, "Pow(Symbol('u', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\chi')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\varepsilon_0,h)} = \\varepsilon_0 h, then obtain (\\int \\frac{\\hat{H}_{\\lambda}^{h}{(\\varepsilon_0,h)}}{h} d\\varepsilon_0)^{h} = (\\int \\frac{(\\varepsilon_0 h)^{h}}{h} d\\varepsilon_0)^{h}", "derivation": "\\hat{H}_{\\lambda}{(\\varepsilon_0,h)} = \\varepsilon_0 h and \\hat{H}_{\\lambda}^{h}{(\\varepsilon_0,h)} = (\\varepsilon_0 h)^{h} and \\frac{\\hat{H}_{\\lambda}^{h}{(\\varepsilon_0,h)}}{h} = \\frac{(\\varepsilon_0 h)^{h}}{h} and \\int \\frac{\\hat{H}_{\\lambda}^{h}{(\\varepsilon_0,h)}}{h} d\\varepsilon_0 = \\int \\frac{(\\varepsilon_0 h)^{h}}{h} d\\varepsilon_0 and (\\int \\frac{\\hat{H}_{\\lambda}^{h}{(\\varepsilon_0,h)}}{h} d\\varepsilon_0)^{h} = (\\int \\frac{(\\varepsilon_0 h)^{h}}{h} d\\varepsilon_0)^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["divide", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(U,x^\\prime)} = \\cos{(U - x^\\prime)}, then derive \\int \\mathbb{I}{(U,x^\\prime)} dU = W + \\sin{(U - x^\\prime)}, then obtain - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} (W + \\sin{(U - x^\\prime)}) = - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} \\int \\cos{(U - x^\\prime)} dU", "derivation": "\\mathbb{I}{(U,x^\\prime)} = \\cos{(U - x^\\prime)} and \\int \\mathbb{I}{(U,x^\\prime)} dU = \\int \\cos{(U - x^\\prime)} dU and \\int \\mathbb{I}{(U,x^\\prime)} dU = W + \\sin{(U - x^\\prime)} and \\frac{\\partial}{\\partial U} \\int \\mathbb{I}{(U,x^\\prime)} dU = \\frac{\\partial}{\\partial U} \\int \\cos{(U - x^\\prime)} dU and \\frac{\\partial}{\\partial U} (W + \\sin{(U - x^\\prime)}) = \\frac{\\partial}{\\partial U} \\int \\cos{(U - x^\\prime)} dU and - Z - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} (W + \\sin{(U - x^\\prime)}) = - Z - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} \\int \\cos{(U - x^\\prime)} dU and - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} (W + \\sin{(U - x^\\prime)}) = - \\cos{(U - x^\\prime)} + \\frac{\\partial}{\\partial U} \\int \\cos{(U - x^\\prime)} dU", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('x^\\\\prime', commutative=True)), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('W', commutative=True), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Symbol('W', commutative=True), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Integral(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 5, "Add(Symbol('Z', commutative=True), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Derivative(Add(Symbol('W', commutative=True), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Derivative(Integral(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["add", 6, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Derivative(Add(Symbol('W', commutative=True), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))))), Derivative(Integral(cos(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\delta,r)} = \\delta r, then derive 2 \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\delta + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)}, then obtain \\delta + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\frac{\\partial}{\\partial r} \\delta r + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)}", "derivation": "\\operatorname{F_{c}}{(\\delta,r)} = \\delta r and \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\frac{\\partial}{\\partial r} \\delta r and 2 \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\frac{\\partial}{\\partial r} \\delta r + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} and 2 \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\delta + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} and \\delta + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)} = \\frac{\\partial}{\\partial r} \\delta r + \\frac{\\partial}{\\partial r} \\operatorname{F_{c}}{(\\delta,r)}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi^{*}{(F_{H})} = \\sin{(F_{H})}, then obtain \\cos{(\\int - \\sin{(F_{H})} dF_{H} + \\int \\sin{(F_{H})} dF_{H})} = 1", "derivation": "\\psi^{*}{(F_{H})} = \\sin{(F_{H})} and \\psi^{*}{(F_{H})} - \\sin{(F_{H})} = 0 and \\int (\\psi^{*}{(F_{H})} - \\sin{(F_{H})}) dF_{H} = \\int 0 dF_{H} and \\cos{(\\int (\\psi^{*}{(F_{H})} - \\sin{(F_{H})}) dF_{H})} = 1 and \\cos{(\\int \\psi^{*}{(F_{H})} dF_{H} + \\int - \\sin{(F_{H})} dF_{H})} = 1 and \\cos{(\\int - \\sin{(F_{H})} dF_{H} + \\int \\sin{(F_{H})} dF_{H})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["minus", 1, "sin(Symbol('F_H', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(Symbol('F_H', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Add(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True)))), Integer(1))"], [["expand", 4], "Equality(cos(Add(Integral(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(cos(Add(Integral(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(sin(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Integer(1))"]]}, {"prompt": "Given \\sigma_{x}{(\\rho_b)} = \\cos{(\\log{(\\rho_b)})}, then obtain (\\cos^{\\rho_b}{(\\log{(\\rho_b)})})^{\\rho_b} \\sigma_{x}^{\\rho_b}{(\\rho_b)} = (\\cos^{\\rho_b}{(\\log{(\\rho_b)})})^{\\rho_b} \\cos^{\\rho_b}{(\\log{(\\rho_b)})}", "derivation": "\\sigma_{x}{(\\rho_b)} = \\cos{(\\log{(\\rho_b)})} and \\sigma_{x}^{\\rho_b}{(\\rho_b)} = \\cos^{\\rho_b}{(\\log{(\\rho_b)})} and (\\sigma_{x}^{\\rho_b}{(\\rho_b)})^{\\rho_b} = (\\cos^{\\rho_b}{(\\log{(\\rho_b)})})^{\\rho_b} and (\\sigma_{x}^{\\rho_b}{(\\rho_b)})^{\\rho_b} \\sigma_{x}^{\\rho_b}{(\\rho_b)} = (\\sigma_{x}^{\\rho_b}{(\\rho_b)})^{\\rho_b} \\cos^{\\rho_b}{(\\log{(\\rho_b)})} and (\\cos^{\\rho_b}{(\\log{(\\rho_b)})})^{\\rho_b} \\sigma_{x}^{\\rho_b}{(\\rho_b)} = (\\cos^{\\rho_b}{(\\log{(\\rho_b)})})^{\\rho_b} \\cos^{\\rho_b}{(\\log{(\\rho_b)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), cos(log(Symbol('\\\\rho_b', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["times", 2, "Pow(Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(cos(log(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(k,v_{2})} = k + v_{2}, then obtain \\frac{3 \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{2 k + 2 v_{2}} = \\frac{2 k + 2 v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{2 k + 2 v_{2}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(k,v_{2})} = k + v_{2} and k + v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})} = 2 k + 2 v_{2} and k + v_{2} + 2 \\operatorname{L_{\\varepsilon}}{(k,v_{2})} = 2 k + 2 v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})} and 3 \\operatorname{L_{\\varepsilon}}{(k,v_{2})} = 2 k + 2 v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})} and \\frac{3 \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{k + v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})}} = \\frac{2 k + 2 v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{k + v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})}} and \\frac{3 \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{2 k + 2 v_{2}} = \\frac{2 k + 2 v_{2} + \\operatorname{L_{\\varepsilon}}{(k,v_{2})}}{2 k + 2 v_{2}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True)))"], [["add", 1, "Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["add", 2, "Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True), Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(3), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))))"], [["divide", 4, "Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)))"], "Equality(Mul(Integer(3), Pow(Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Mul(Pow(Add(Symbol('k', commutative=True), Symbol('v_2', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(3), Pow(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True))), Mul(Pow(Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('v_2', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(E_{x})} = \\sin{(e^{E_{x}})}, then obtain 1 = 0^{E_{x}} (\\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x} - \\operatorname{F_{x}}{(E_{x})} + \\sin{(e^{E_{x}})}})})^{- E_{x}}", "derivation": "\\operatorname{F_{x}}{(E_{x})} = \\sin{(e^{E_{x}})} and \\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x}})} = 0 and - E_{x} + \\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x}})} = - E_{x} and (\\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x}})})^{E_{x}} = 0^{E_{x}} and 1 = 0^{E_{x}} (\\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x}})})^{- E_{x}} and 1 = 0^{E_{x}} (\\operatorname{F_{x}}{(E_{x})} - \\sin{(e^{E_{x} - \\operatorname{F_{x}}{(E_{x})} + \\sin{(e^{E_{x}})}})})^{- E_{x}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('E_x', commutative=True)), sin(exp(Symbol('E_x', commutative=True))))"], [["minus", 1, "sin(exp(Symbol('E_x', commutative=True)))"], "Equality(Add(Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('E_x', commutative=True))))), Integer(0))"], [["minus", 2, "Symbol('E_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('E_x', commutative=True))))), Mul(Integer(-1), Symbol('E_x', commutative=True)))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Add(Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('E_x', commutative=True))))), Symbol('E_x', commutative=True)), Pow(Integer(0), Symbol('E_x', commutative=True)))"], [["divide", 4, "Pow(Add(Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('E_x', commutative=True))))), Symbol('E_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integer(0), Symbol('E_x', commutative=True)), Pow(Add(Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('E_x', commutative=True))))), Mul(Integer(-1), Symbol('E_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Mul(Pow(Integer(0), Symbol('E_x', commutative=True)), Pow(Add(Function('F_x')(Symbol('E_x', commutative=True)), Mul(Integer(-1), sin(exp(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Function('F_x')(Symbol('E_x', commutative=True))), sin(exp(Symbol('E_x', commutative=True)))))))), Mul(Integer(-1), Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given J{(r)} = \\int \\log{(r)} dr, then derive J{(r)} = n_{2} + r \\log{(r)} - r, then derive - r + J{(r)} = \\sigma_p + r \\log{(r)} - 2 r, then obtain (\\sigma_p + r \\log{(r)} - 2 r) (n_{2} + r \\log{(r)} - 2 r) = (\\sigma_p + r \\log{(r)} - 2 r)^{2}", "derivation": "J{(r)} = \\int \\log{(r)} dr and J{(r)} = n_{2} + r \\log{(r)} - r and - r + J{(r)} = - r + \\int \\log{(r)} dr and - r + J{(r)} = \\sigma_p + r \\log{(r)} - 2 r and n_{2} + r \\log{(r)} - 2 r = \\sigma_p + r \\log{(r)} - 2 r and (\\sigma_p + r \\log{(r)} - 2 r) (n_{2} + r \\log{(r)} - 2 r) = (\\sigma_p + r \\log{(r)} - 2 r)^{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('r', commutative=True)), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('J')(Symbol('r', commutative=True)), Add(Symbol('n_2', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('J')(Symbol('r', commutative=True))), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Integral(log(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('J')(Symbol('r', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('n_2', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["times", 5, "Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True)))), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Symbol('r', commutative=True), log(Symbol('r', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{f}{(g)} = \\frac{d}{d g} \\sin{(g)}, then obtain (\\sin{(g)} + \\frac{d}{d g} \\mathbf{f}{(g)})^{2} = 0", "derivation": "\\mathbf{f}{(g)} = \\frac{d}{d g} \\sin{(g)} and \\frac{d}{d g} \\mathbf{f}{(g)} = \\frac{d^{2}}{d g^{2}} \\sin{(g)} and \\sin{(g)} + \\frac{d}{d g} \\mathbf{f}{(g)} = \\sin{(g)} + \\frac{d^{2}}{d g^{2}} \\sin{(g)} and (\\sin{(g)} + \\frac{d}{d g} \\mathbf{f}{(g)})^{2} = (\\sin{(g)} + \\frac{d^{2}}{d g^{2}} \\sin{(g)})^{2} and (\\sin{(g)} + \\frac{d}{d g} \\mathbf{f}{(g)})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('g', commutative=True)), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))))"], [["add", 2, "sin(Symbol('g', commutative=True))"], "Equality(Add(sin(Symbol('g', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(sin(Symbol('g', commutative=True)), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2)))))"], [["power", 3, 2], "Equality(Pow(Add(sin(Symbol('g', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Integer(2)), Pow(Add(sin(Symbol('g', commutative=True)), Derivative(sin(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2)))), Integer(2)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(sin(Symbol('g', commutative=True)), Derivative(Function('\\\\mathbf{f}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Integer(2)), Integer(0))"]]}, {"prompt": "Given L{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi}, then derive L{(\\varphi)} e^{\\varphi} = e^{2 \\varphi}, then derive L{(\\varphi)} e^{\\varphi} - e^{2 \\varphi} = 0, then obtain - e^{2 \\varphi} + e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} = 0", "derivation": "L{(\\varphi)} = \\frac{d}{d \\varphi} e^{\\varphi} and L{(\\varphi)} \\frac{d}{d \\varphi} e^{\\varphi} = (\\frac{d}{d \\varphi} e^{\\varphi})^{2} and L{(\\varphi)} e^{\\varphi} = e^{2 \\varphi} and e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} = e^{2 \\varphi} and L{(\\varphi)} e^{\\varphi} = e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} and L{(\\varphi)} e^{\\varphi} - e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} = 0 and L{(\\varphi)} e^{\\varphi} - e^{2 \\varphi} = 0 and - e^{2 \\varphi} + e^{\\varphi} \\frac{d}{d \\varphi} e^{\\varphi} = 0", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["times", 1, "Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Mul(Function('L')(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 2], "Equality(Mul(Function('L')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('L')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True))), Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["minus", 5, "Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], "Equality(Add(Mul(Function('L')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Function('L')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)))), Mul(exp(Symbol('\\\\varphi', commutative=True)), Derivative(exp(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{M}{(I)} = \\sin{(e^{I})} and \\operatorname{C_{2}}{(I)} = \\int \\sin{(e^{I})} dI, then derive \\int \\mathbf{M}{(I)} dI = \\mathbf{H} + \\operatorname{Si}{(e^{I})}, then obtain \\iint \\mathbf{M}{(I)} dI dI = \\int (\\mathbf{H} + \\operatorname{Si}{(e^{I})}) dI", "derivation": "\\mathbf{M}{(I)} = \\sin{(e^{I})} and \\int \\mathbf{M}{(I)} dI = \\int \\sin{(e^{I})} dI and \\int \\mathbf{M}{(I)} dI = \\mathbf{H} + \\operatorname{Si}{(e^{I})} and \\operatorname{C_{2}}{(I)} = \\int \\sin{(e^{I})} dI and \\int \\sin{(e^{I})} dI = \\mathbf{H} + \\operatorname{Si}{(e^{I})} and \\int \\mathbf{M}{(I)} dI = \\operatorname{C_{2}}{(I)} and \\operatorname{C_{2}}{(I)} = \\mathbf{H} + \\operatorname{Si}{(e^{I})} and \\iint \\mathbf{M}{(I)} dI dI = \\int \\operatorname{C_{2}}{(I)} dI and \\iint \\mathbf{M}{(I)} dI dI = \\int (\\mathbf{H} + \\operatorname{Si}{(e^{I})}) dI", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), sin(exp(Symbol('I', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Si(exp(Symbol('I', commutative=True)))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('I', commutative=True)), Integral(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Si(exp(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Function('C_2')(Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('C_2')(Symbol('I', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Si(exp(Symbol('I', commutative=True)))))"], [["integrate", 6, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Function('C_2')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Si(exp(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given B{(v)} = \\sin{(\\sin{(v)})} and \\hat{X}{(v)} = \\int B{(v)} dv, then obtain \\frac{d}{d v} 1 + 1 = \\frac{d}{d v} \\frac{\\int \\sin{(\\sin{(v)})} dv}{\\hat{X}{(v)}} + 1", "derivation": "B{(v)} = \\sin{(\\sin{(v)})} and \\int B{(v)} dv = \\int \\sin{(\\sin{(v)})} dv and \\hat{X}{(v)} = \\int B{(v)} dv and 1 = \\frac{\\int B{(v)} dv}{\\hat{X}{(v)}} and \\frac{d}{d v} 1 = \\frac{d}{d v} \\frac{\\int B{(v)} dv}{\\hat{X}{(v)}} and \\frac{d}{d v} 1 = \\frac{d}{d v} \\frac{\\int \\sin{(\\sin{(v)})} dv}{\\hat{X}{(v)}} and \\frac{d}{d v} 1 + 1 = \\frac{d}{d v} \\frac{\\int \\sin{(\\sin{(v)})} dv}{\\hat{X}{(v)}} + 1", "srepr_derivation": [["renaming_premise", "Equality(Function('B')(Symbol('v', commutative=True)), sin(sin(Symbol('v', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('B')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('v', commutative=True)), Integral(Function('B')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 3, "Function('\\\\hat{X}')(Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('v', commutative=True)), Integer(-1)), Integral(Function('B')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["differentiate", 4, "Symbol('v', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('v', commutative=True)), Integer(-1)), Integral(Function('B')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('v', commutative=True)), Integer(-1)), Integral(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["minus", 6, "Integer(-1)"], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Pow(Function('\\\\hat{X}')(Symbol('v', commutative=True)), Integer(-1)), Integral(sin(sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{f}{(q,v_{z})} = \\frac{v_{z}}{q}, then obtain 2 (\\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q})^{q} = (\\frac{v_{z}^{2}}{q^{2}})^{q} + (\\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q})^{q}", "derivation": "\\mathbf{f}{(q,v_{z})} = \\frac{v_{z}}{q} and \\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q} = \\frac{v_{z}^{2}}{q^{2}} and (\\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q})^{q} = (\\frac{v_{z}^{2}}{q^{2}})^{q} and 2 (\\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q})^{q} = (\\frac{v_{z}^{2}}{q^{2}})^{q} + (\\frac{v_{z} \\mathbf{f}{(q,v_{z})}}{q})^{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True), Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Pow(Symbol('v_z', commutative=True), Integer(2))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True), Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Symbol('q', commutative=True)), Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Pow(Symbol('v_z', commutative=True), Integer(2))), Symbol('q', commutative=True)))"], [["add", 3, "Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True), Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True), Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Symbol('q', commutative=True))), Add(Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-2)), Pow(Symbol('v_z', commutative=True), Integer(2))), Symbol('q', commutative=True)), Pow(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_z', commutative=True), Function('\\\\mathbf{f}')(Symbol('q', commutative=True), Symbol('v_z', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given c{(\\theta_2)} = \\sin{(\\theta_2)}, then obtain \\int (- e^{c^{\\theta_2}{(\\theta_2)}})^{\\theta_2} d\\theta_2 = \\int (- e^{\\sin^{\\theta_2}{(\\theta_2)}})^{\\theta_2} d\\theta_2", "derivation": "c{(\\theta_2)} = \\sin{(\\theta_2)} and c^{\\theta_2}{(\\theta_2)} = \\sin^{\\theta_2}{(\\theta_2)} and e^{c^{\\theta_2}{(\\theta_2)}} = e^{\\sin^{\\theta_2}{(\\theta_2)}} and - e^{c^{\\theta_2}{(\\theta_2)}} = - e^{\\sin^{\\theta_2}{(\\theta_2)}} and (- e^{c^{\\theta_2}{(\\theta_2)}})^{\\theta_2} = (- e^{\\sin^{\\theta_2}{(\\theta_2)}})^{\\theta_2} and \\int (- e^{c^{\\theta_2}{(\\theta_2)}})^{\\theta_2} d\\theta_2 = \\int (- e^{\\sin^{\\theta_2}{(\\theta_2)}})^{\\theta_2} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('c')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), exp(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Pow(Function('c')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), exp(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Mul(Integer(-1), exp(Pow(Function('c')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Integer(-1), exp(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 5, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), exp(Pow(Function('c')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(Mul(Integer(-1), exp(Pow(sin(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\chi{(v,F_{H})} = F_{H}^{v}, then obtain \\frac{F_{H}^{v} \\chi{(v,F_{H})}}{v} = \\frac{F_{H}^{2 v}}{v}", "derivation": "\\chi{(v,F_{H})} = F_{H}^{v} and \\frac{\\chi{(v,F_{H})}}{v} = \\frac{F_{H}^{v}}{v} and \\frac{\\chi^{2}{(v,F_{H})}}{v} = \\frac{F_{H}^{v} \\chi{(v,F_{H})}}{v} and \\frac{F_{H}^{v} \\chi{(v,F_{H})}}{v} = \\frac{F_{H}^{2 v}}{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Symbol('v', commutative=True)))"], [["divide", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["times", 2, "Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True)), Integer(2))), Mul(Pow(Symbol('F_H', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('v', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Mul(Integer(2), Symbol('v', commutative=True))), Pow(Symbol('v', commutative=True), Integer(-1))))"]]}, {"prompt": "Given g{(r)} = e^{r} and v{(\\phi_1)} = \\sin{(\\cos{(\\phi_1)})}, then obtain (\\phi_1 v{(\\phi_1)} - e^{r}) e^{- r} = (\\phi_1 \\sin{(\\cos{(\\phi_1)})} - e^{r}) e^{- r}", "derivation": "g{(r)} = e^{r} and v{(\\phi_1)} = \\sin{(\\cos{(\\phi_1)})} and \\phi_1 v{(\\phi_1)} = \\phi_1 \\sin{(\\cos{(\\phi_1)})} and \\phi_1 v{(\\phi_1)} - g{(r)} = \\phi_1 \\sin{(\\cos{(\\phi_1)})} - g{(r)} and \\phi_1 v{(\\phi_1)} - e^{r} = \\phi_1 \\sin{(\\cos{(\\phi_1)})} - e^{r} and (\\phi_1 v{(\\phi_1)} - e^{r}) e^{- r} = (\\phi_1 \\sin{(\\cos{(\\phi_1)})} - e^{r}) e^{- r}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], ["get_premise", "Equality(Function('v')(Symbol('\\\\phi_1', commutative=True)), sin(cos(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('v')(Symbol('\\\\phi_1', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), sin(cos(Symbol('\\\\phi_1', commutative=True)))))"], [["minus", 3, "Function('g')(Symbol('r', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), Function('v')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('r', commutative=True)))), Add(Mul(Symbol('\\\\phi_1', commutative=True), sin(cos(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(-1), Function('g')(Symbol('r', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), Function('v')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), exp(Symbol('r', commutative=True)))), Add(Mul(Symbol('\\\\phi_1', commutative=True), sin(cos(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(-1), exp(Symbol('r', commutative=True)))))"], [["divide", 5, "exp(Symbol('r', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\phi_1', commutative=True), Function('v')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), exp(Symbol('r', commutative=True)))), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Mul(Add(Mul(Symbol('\\\\phi_1', commutative=True), sin(cos(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(-1), exp(Symbol('r', commutative=True)))), exp(Mul(Integer(-1), Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\delta)} = \\delta, then obtain \\delta \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} - \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\delta \\delta^{\\delta} \\hat{\\mathbf{x}}{(\\delta)} - \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)}", "derivation": "\\hat{\\mathbf{x}}{(\\delta)} = \\delta and \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\delta^{\\delta} and \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\delta^{\\delta} \\hat{\\mathbf{x}}{(\\delta)} and \\delta \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\delta \\delta^{\\delta} \\hat{\\mathbf{x}}{(\\delta)} and \\delta \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} - \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\delta \\delta^{\\delta} \\hat{\\mathbf{x}}{(\\delta)} - \\hat{\\mathbf{x}}{(\\delta)} \\hat{\\mathbf{x}}^{\\delta}{(\\delta)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["times", 2, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True))))"], [["times", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True))))"], [["minus", 4, "Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\delta', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(A)} = e^{A}, then derive \\int \\mathbf{F}{(A)} dA = \\varepsilon_0 + e^{A}, then obtain \\int \\mathbf{F}{(A)} dA = \\varepsilon_0 + \\mathbf{F}{(A)}", "derivation": "\\mathbf{F}{(A)} = e^{A} and \\int \\mathbf{F}{(A)} dA = \\int e^{A} dA and \\int \\mathbf{F}{(A)} dA = \\varepsilon_0 + e^{A} and \\int \\mathbf{F}{(A)} dA = \\varepsilon_0 + \\mathbf{F}{(A)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), Function('\\\\mathbf{F}')(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} = \\sigma_x (\\delta + \\theta_2) and \\hat{x}{(\\delta,\\sigma_x,\\theta_2)} = \\iint \\sigma_x (\\delta + \\theta_2) d\\sigma_x d\\delta, then obtain \\iint \\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} d\\sigma_x d\\delta = \\hat{x}{(\\delta,\\sigma_x,\\theta_2)}", "derivation": "\\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} = \\sigma_x (\\delta + \\theta_2) and \\int \\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} d\\sigma_x = \\int \\sigma_x (\\delta + \\theta_2) d\\sigma_x and \\iint \\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} d\\sigma_x d\\delta = \\iint \\sigma_x (\\delta + \\theta_2) d\\sigma_x d\\delta and \\hat{x}{(\\delta,\\sigma_x,\\theta_2)} = \\iint \\sigma_x (\\delta + \\theta_2) d\\sigma_x d\\delta and \\iint \\mathbf{J}_M{(\\delta,\\sigma_x,\\theta_2)} d\\sigma_x d\\delta = \\hat{x}{(\\delta,\\sigma_x,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integral(Mul(Symbol('\\\\sigma_x', commutative=True), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Function('\\\\hat{x}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(h,\\mathbf{S})} = \\frac{\\log{(h)}}{\\mathbf{S}}, then derive \\frac{\\int \\Psi_{nl}{(h,\\mathbf{S})} dh}{\\mathbf{S}} = \\frac{M + \\frac{h \\log{(h)}}{\\mathbf{S}} - \\frac{h}{\\mathbf{S}}}{\\mathbf{S}}, then obtain \\frac{\\int \\frac{\\log{(h)}}{\\mathbf{S}} dh}{\\mathbf{S}} = \\frac{M + \\frac{h \\log{(h)}}{\\mathbf{S}} - \\frac{h}{\\mathbf{S}}}{\\mathbf{S}}", "derivation": "\\Psi_{nl}{(h,\\mathbf{S})} = \\frac{\\log{(h)}}{\\mathbf{S}} and \\int \\Psi_{nl}{(h,\\mathbf{S})} dh = \\int \\frac{\\log{(h)}}{\\mathbf{S}} dh and \\frac{\\int \\Psi_{nl}{(h,\\mathbf{S})} dh}{\\mathbf{S}} = \\frac{\\int \\frac{\\log{(h)}}{\\mathbf{S}} dh}{\\mathbf{S}} and \\frac{\\int \\Psi_{nl}{(h,\\mathbf{S})} dh}{\\mathbf{S}} = \\frac{M + \\frac{h \\log{(h)}}{\\mathbf{S}} - \\frac{h}{\\mathbf{S}}}{\\mathbf{S}} and \\frac{\\int \\frac{\\log{(h)}}{\\mathbf{S}} dh}{\\mathbf{S}} = \\frac{M + \\frac{h \\log{(h)}}{\\mathbf{S}} - \\frac{h}{\\mathbf{S}}}{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('h', commutative=True), log(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\omega,A_{x})} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\omega, then derive \\mathbf{E}{(\\omega,A_{x})} = \\omega, then obtain \\mathbf{E}{(\\frac{\\partial}{\\partial A_{x}} A_{x} \\omega,A_{x})} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\omega", "derivation": "\\mathbf{E}{(\\omega,A_{x})} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\omega and \\mathbf{E}{(\\omega,A_{x})} = \\omega and \\omega = \\frac{\\partial}{\\partial A_{x}} A_{x} \\omega and \\mathbf{E}{(\\frac{\\partial}{\\partial A_{x}} A_{x} \\omega,A_{x})} = \\frac{\\partial}{\\partial A_{x}} A_{x} \\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\omega', commutative=True), Symbol('A_x', commutative=True)), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\omega', commutative=True), Symbol('A_x', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Symbol('\\\\omega', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Function('\\\\mathbf{E}')(Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(M,z^{*})} = - M + z^{*}, then obtain - z^{*} + U{(M,z^{*})} - 1 = - M - 1", "derivation": "U{(M,z^{*})} = - M + z^{*} and - M + U{(M,z^{*})} = - 2 M + z^{*} and - M + U{(M,z^{*})} - 1 = - 2 M + z^{*} - 1 and - z^{*} + U{(M,z^{*})} - 1 = - M - 1", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('z^*', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('U')(Symbol('M', commutative=True), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Symbol('z^*', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('U')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Symbol('z^*', commutative=True), Integer(-1)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('z^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('U')(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mu_{0}{(\\phi_2,v_{2})} = \\phi_2 v_{2}, then obtain (v_{2} \\mu_{0}^{v_{2}}{(\\phi_2,v_{2})} - v_{2})^{v_{2}} = (v_{2} (\\phi_2 v_{2})^{v_{2}} - v_{2})^{v_{2}}", "derivation": "\\mu_{0}{(\\phi_2,v_{2})} = \\phi_2 v_{2} and \\mu_{0}^{v_{2}}{(\\phi_2,v_{2})} = (\\phi_2 v_{2})^{v_{2}} and v_{2} \\mu_{0}^{v_{2}}{(\\phi_2,v_{2})} = v_{2} (\\phi_2 v_{2})^{v_{2}} and v_{2} \\mu_{0}^{v_{2}}{(\\phi_2,v_{2})} - v_{2} = v_{2} (\\phi_2 v_{2})^{v_{2}} - v_{2} and (v_{2} \\mu_{0}^{v_{2}}{(\\phi_2,v_{2})} - v_{2})^{v_{2}} = (v_{2} (\\phi_2 v_{2})^{v_{2}} - v_{2})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], [["times", 2, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Symbol('v_2', commutative=True), Pow(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))))"], [["minus", 3, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Symbol('v_2', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True))), Add(Mul(Symbol('v_2', commutative=True), Pow(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('v_2', commutative=True), Pow(Function('\\\\mu_0')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Add(Mul(Symbol('v_2', commutative=True), Pow(Mul(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True))), Symbol('v_2', commutative=True)))"]]}, {"prompt": "Given s{(\\sigma_p)} = \\cos{(\\sigma_p)}, then obtain \\log{(s{(\\sigma_p)} + 3 \\cos{(\\sigma_p)})} = \\log{(2 s{(\\sigma_p)} + 2 \\cos{(\\sigma_p)})}", "derivation": "s{(\\sigma_p)} = \\cos{(\\sigma_p)} and 2 s{(\\sigma_p)} = s{(\\sigma_p)} + \\cos{(\\sigma_p)} and 3 s{(\\sigma_p)} = 2 s{(\\sigma_p)} + \\cos{(\\sigma_p)} and 3 s{(\\sigma_p)} + \\cos{(\\sigma_p)} = 2 s{(\\sigma_p)} + 2 \\cos{(\\sigma_p)} and \\log{(3 s{(\\sigma_p)} + \\cos{(\\sigma_p)})} = \\log{(2 s{(\\sigma_p)} + 2 \\cos{(\\sigma_p)})} and 3 s{(\\sigma_p)} + \\cos{(\\sigma_p)} = s{(\\sigma_p)} + 3 \\cos{(\\sigma_p)} and \\log{(s{(\\sigma_p)} + 3 \\cos{(\\sigma_p)})} = \\log{(2 s{(\\sigma_p)} + 2 \\cos{(\\sigma_p)})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "Function('s')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('s')(Symbol('\\\\sigma_p', commutative=True))), Add(Function('s')(Symbol('\\\\sigma_p', commutative=True)), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Function('s')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(3), Function('s')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(2), Function('s')(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 3, "cos(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(3), Function('s')(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(2), Function('s')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["log", 4], "Equality(log(Add(Mul(Integer(3), Function('s')(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True)))), log(Add(Mul(Integer(2), Function('s')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(3), Function('s')(Symbol('\\\\sigma_p', commutative=True))), cos(Symbol('\\\\sigma_p', commutative=True))), Add(Function('s')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(3), cos(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(log(Add(Function('s')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(3), cos(Symbol('\\\\sigma_p', commutative=True))))), log(Add(Mul(Integer(2), Function('s')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mu,\\tilde{g})} = \\cos{(\\mu + \\tilde{g})}, then obtain ((\\mu + \\operatorname{c_{0}}{(\\mu,\\tilde{g})})^{\\mu})^{\\mu} = ((\\mu + \\cos{(\\mu + \\tilde{g})})^{\\mu})^{\\mu}", "derivation": "\\operatorname{c_{0}}{(\\mu,\\tilde{g})} = \\cos{(\\mu + \\tilde{g})} and \\mu + \\operatorname{c_{0}}{(\\mu,\\tilde{g})} = \\mu + \\cos{(\\mu + \\tilde{g})} and (\\mu + \\operatorname{c_{0}}{(\\mu,\\tilde{g})})^{\\mu} = (\\mu + \\cos{(\\mu + \\tilde{g})})^{\\mu} and ((\\mu + \\operatorname{c_{0}}{(\\mu,\\tilde{g})})^{\\mu})^{\\mu} = ((\\mu + \\cos{(\\mu + \\tilde{g})})^{\\mu})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('\\\\mu', commutative=True), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Add(Symbol('\\\\mu', commutative=True), cos(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(l,\\mathbf{A})} = \\mathbf{A} - l, then obtain - \\int \\operatorname{f^{\\prime}}{(l,\\mathbf{A})} d\\mathbf{A} = - \\frac{\\mathbf{A}^{2}}{2} + \\mathbf{A} l - p", "derivation": "\\operatorname{f^{\\prime}}{(l,\\mathbf{A})} = \\mathbf{A} - l and \\int \\operatorname{f^{\\prime}}{(l,\\mathbf{A})} d\\mathbf{A} = \\int (\\mathbf{A} - l) d\\mathbf{A} and - \\int \\operatorname{f^{\\prime}}{(l,\\mathbf{A})} d\\mathbf{A} = - \\int (\\mathbf{A} - l) d\\mathbf{A} and - \\int \\operatorname{f^{\\prime}}{(l,\\mathbf{A})} d\\mathbf{A} = - \\frac{\\mathbf{A}^{2}}{2} + \\mathbf{A} l - p", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('f^{\\\\prime}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('f^{\\\\prime}')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{f})} = \\log{(\\mathbf{f})}, then derive \\frac{d}{d \\mathbf{f}} \\varphi{(\\mathbf{f})} = \\frac{1}{\\mathbf{f}}, then obtain (\\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - \\frac{1}{\\mathbf{f}})^{\\mathbf{f}} = 0^{\\mathbf{f}}", "derivation": "\\varphi{(\\mathbf{f})} = \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} \\varphi{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} \\varphi{(\\mathbf{f})} = \\frac{1}{\\mathbf{f}} and \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} = \\frac{1}{\\mathbf{f}} and - \\mathbf{f} + \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} = - \\mathbf{f} + \\frac{1}{\\mathbf{f}} and \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - \\frac{1}{\\mathbf{f}} = 0 and (\\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - \\frac{1}{\\mathbf{f}})^{\\mathbf{f}} = 0^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))"], [["minus", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))"], "Equality(Add(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Integer(0))"], [["power", 6, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Add(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(G)} = \\cos{(G)}, then obtain \\frac{\\sigma_{x}{(G)} + \\sin{(\\sigma_{x}{(G)})} + \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} + \\frac{1}{\\log{(t_{1})}} = \\frac{\\sigma_{x}{(G)} + 2 \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} + \\frac{1}{\\log{(t_{1})}}", "derivation": "\\sigma_{x}{(G)} = \\cos{(G)} and \\sin{(\\sigma_{x}{(G)})} = \\sin{(\\cos{(G)})} and \\sin{(\\sigma_{x}{(G)})} + \\sin{(\\cos{(G)})} = 2 \\sin{(\\cos{(G)})} and \\sigma_{x}{(G)} + \\sin{(\\sigma_{x}{(G)})} + \\sin{(\\cos{(G)})} = \\sigma_{x}{(G)} + 2 \\sin{(\\cos{(G)})} and \\frac{\\sigma_{x}{(G)} + \\sin{(\\sigma_{x}{(G)})} + \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} = \\frac{\\sigma_{x}{(G)} + 2 \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} and \\frac{\\sigma_{x}{(G)} + \\sin{(\\sigma_{x}{(G)})} + \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} + \\frac{1}{\\log{(t_{1})}} = \\frac{\\sigma_{x}{(G)} + 2 \\sin{(\\cos{(G)})}}{\\log{(t_{1})}} + \\frac{1}{\\log{(t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\sigma_x')(Symbol('G', commutative=True))), sin(cos(Symbol('G', commutative=True))))"], [["add", 2, "sin(cos(Symbol('G', commutative=True)))"], "Equality(Add(sin(Function('\\\\sigma_x')(Symbol('G', commutative=True))), sin(cos(Symbol('G', commutative=True)))), Mul(Integer(2), sin(cos(Symbol('G', commutative=True)))))"], [["add", 3, "Function('\\\\sigma_x')(Symbol('G', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), sin(Function('\\\\sigma_x')(Symbol('G', commutative=True))), sin(cos(Symbol('G', commutative=True)))), Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Mul(Integer(2), sin(cos(Symbol('G', commutative=True))))))"], [["divide", 4, "log(Symbol('t_1', commutative=True))"], "Equality(Mul(Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), sin(Function('\\\\sigma_x')(Symbol('G', commutative=True))), sin(cos(Symbol('G', commutative=True)))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Mul(Integer(2), sin(cos(Symbol('G', commutative=True))))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))))"], [["add", 5, "Pow(log(Symbol('t_1', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), sin(Function('\\\\sigma_x')(Symbol('G', commutative=True))), sin(cos(Symbol('G', commutative=True)))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))), Add(Mul(Add(Function('\\\\sigma_x')(Symbol('G', commutative=True)), Mul(Integer(2), sin(cos(Symbol('G', commutative=True))))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))), Pow(log(Symbol('t_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then derive (\\int \\mathbf{D}{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (c_{0} - \\cos{(\\varphi^*)})^{\\varphi^*}, then obtain \\int \\mathbf{D}{(\\varphi^*)} d\\varphi^* + (\\int \\sin{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (c_{0} - \\cos{(\\varphi^*)})^{\\varphi^*} + \\int \\mathbf{D}{(\\varphi^*)} d\\varphi^*", "derivation": "\\mathbf{D}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\int \\mathbf{D}{(\\varphi^*)} d\\varphi^* = \\int \\sin{(\\varphi^*)} d\\varphi^* and (\\int \\mathbf{D}{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (\\int \\sin{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} and (\\int \\mathbf{D}{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (c_{0} - \\cos{(\\varphi^*)})^{\\varphi^*} and (\\int \\sin{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (c_{0} - \\cos{(\\varphi^*)})^{\\varphi^*} and \\int \\mathbf{D}{(\\varphi^*)} d\\varphi^* + (\\int \\sin{(\\varphi^*)} d\\varphi^*)^{\\varphi^*} = (c_{0} - \\cos{(\\varphi^*)})^{\\varphi^*} + \\int \\mathbf{D}{(\\varphi^*)} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)), Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 5, "Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Pow(Integral(sin(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))), Add(Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\varphi^*', commutative=True)))), Symbol('\\\\varphi^*', commutative=True)), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(U,y,\\hat{X})} = (U \\hat{X})^{y} and \\mathbb{I}{(U,y,\\hat{X})} = \\frac{\\frac{\\partial}{\\partial \\hat{X}} (U \\hat{X})^{y}}{U \\hat{X} y}, then obtain \\frac{\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{v_{2}}{(U,y,\\hat{X})}}{U \\hat{X} y} = \\mathbb{I}{(U,y,\\hat{X})}", "derivation": "\\operatorname{v_{2}}{(U,y,\\hat{X})} = (U \\hat{X})^{y} and \\frac{\\partial}{\\partial \\hat{X}} \\operatorname{v_{2}}{(U,y,\\hat{X})} = \\frac{\\partial}{\\partial \\hat{X}} (U \\hat{X})^{y} and \\frac{\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{v_{2}}{(U,y,\\hat{X})}}{y} = \\frac{\\frac{\\partial}{\\partial \\hat{X}} (U \\hat{X})^{y}}{y} and \\frac{\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{v_{2}}{(U,y,\\hat{X})}}{U \\hat{X} y} = \\frac{\\frac{\\partial}{\\partial \\hat{X}} (U \\hat{X})^{y}}{U \\hat{X} y} and \\mathbb{I}{(U,y,\\hat{X})} = \\frac{\\frac{\\partial}{\\partial \\hat{X}} (U \\hat{X})^{y}}{U \\hat{X} y} and \\frac{\\frac{\\partial}{\\partial \\hat{X}} \\operatorname{v_{2}}{(U,y,\\hat{X})}}{U \\hat{X} y} = \\mathbb{I}{(U,y,\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Function('v_2')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Function('v_2')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Integer(-1)), Derivative(Function('v_2')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Function('\\\\mathbb{I}')(Symbol('U', commutative=True), Symbol('y', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given L{(g,f_{E})} = e^{f_{E} + g}, then derive \\frac{\\partial}{\\partial f_{E}} L{(g,f_{E})} = e^{f_{E} + g}, then obtain \\frac{\\partial}{\\partial f_{E}} L{(g,f_{E})} = L{(g,f_{E})}", "derivation": "L{(g,f_{E})} = e^{f_{E} + g} and \\frac{\\partial}{\\partial f_{E}} L{(g,f_{E})} = \\frac{\\partial}{\\partial f_{E}} e^{f_{E} + g} and \\frac{\\partial}{\\partial f_{E}} L{(g,f_{E})} = e^{f_{E} + g} and \\frac{\\partial}{\\partial f_{E}} L{(g,f_{E})} = L{(g,f_{E})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('g', commutative=True), Symbol('f_E', commutative=True)), exp(Add(Symbol('f_E', commutative=True), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('f_E', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), exp(Add(Symbol('f_E', commutative=True), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('L')(Symbol('g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Function('L')(Symbol('g', commutative=True), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\eta,B)} = e^{B \\eta} and \\mathbb{I}{(\\eta,B)} = B \\eta and \\hat{p}_0{(\\eta,B)} = \\mathbf{s}^{2}{(\\eta,B)}, then obtain \\hat{p}_0{(\\eta,B)} = e^{2 \\mathbb{I}{(\\eta,B)}}", "derivation": "\\mathbf{s}{(\\eta,B)} = e^{B \\eta} and \\mathbb{I}{(\\eta,B)} = B \\eta and \\mathbf{s}{(\\eta,B)} = e^{\\mathbb{I}{(\\eta,B)}} and \\hat{p}_0{(\\eta,B)} = \\mathbf{s}^{2}{(\\eta,B)} and \\hat{p}_0{(\\eta,B)} = e^{2 \\mathbb{I}{(\\eta,B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), exp(Mul(Symbol('B', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), exp(Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)), exp(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\eta', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given l{(I)} = \\sin{(I)} and \\operatorname{C_{1}}{(I)} = \\frac{d}{d I} l{(I)}, then obtain \\int \\operatorname{C_{1}}{(I)} dI = \\int \\frac{d}{d I} \\sin{(I)} dI", "derivation": "l{(I)} = \\sin{(I)} and \\frac{d}{d I} l{(I)} = \\frac{d}{d I} \\sin{(I)} and \\operatorname{C_{1}}{(I)} = \\frac{d}{d I} l{(I)} and \\operatorname{C_{1}}{(I)} = \\frac{d}{d I} \\sin{(I)} and \\int \\operatorname{C_{1}}{(I)} dI = \\int \\frac{d}{d I} \\sin{(I)} dI", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('I', commutative=True)), Derivative(Function('l')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('C_1')(Symbol('I', commutative=True)), Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('I', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Derivative(sin(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\pi,\\mu)} = \\mu + \\pi and \\rho_{f}{(\\pi)} = \\pi, then obtain - \\int (- \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)}) d\\mu + \\frac{(\\mu + \\pi) \\rho_{f}{(\\pi)}}{\\pi} = \\mu + \\pi - \\int (- \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)}) d\\mu", "derivation": "\\operatorname{n_{2}}{(\\pi,\\mu)} = \\mu + \\pi and - \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)} = \\mu and \\rho_{f}{(\\pi)} = \\pi and \\frac{(\\mu + \\pi) \\rho_{f}{(\\pi)}}{\\pi} = \\mu + \\pi and \\int (- \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)}) d\\mu = \\int \\mu d\\mu and - \\int \\mu d\\mu + \\frac{(\\mu + \\pi) \\rho_{f}{(\\pi)}}{\\pi} = \\mu + \\pi - \\int \\mu d\\mu and - \\int (- \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)}) d\\mu + \\frac{(\\mu + \\pi) \\rho_{f}{(\\pi)}}{\\pi} = \\mu + \\pi - \\int (- \\pi + \\operatorname{n_{2}}{(\\pi,\\mu)}) d\\mu", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('n_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["times", 3, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('n_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 4, "Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('n_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\pi', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('n_2')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(b)} = \\cos{(b)}, then derive \\frac{d}{d b} \\operatorname{f^{\\prime}}{(b)} = - \\sin{(b)}, then obtain \\sin{(b)} + 1 = \\sin{(b)} - \\frac{\\sin{(b)}}{\\frac{d}{d b} \\cos{(b)}}", "derivation": "\\operatorname{f^{\\prime}}{(b)} = \\cos{(b)} and \\frac{d}{d b} \\operatorname{f^{\\prime}}{(b)} = \\frac{d}{d b} \\cos{(b)} and \\frac{d}{d b} \\operatorname{f^{\\prime}}{(b)} = - \\sin{(b)} and \\operatorname{f^{\\prime}}{(b)} \\frac{d}{d b} \\operatorname{f^{\\prime}}{(b)} = - \\operatorname{f^{\\prime}}{(b)} \\sin{(b)} and 1 = - \\frac{\\sin{(b)}}{\\frac{d}{d b} \\operatorname{f^{\\prime}}{(b)}} and 1 = - \\frac{\\sin{(b)}}{\\frac{d}{d b} \\cos{(b)}} and \\sin{(b)} + 1 = \\sin{(b)} - \\frac{\\sin{(b)}}{\\frac{d}{d b} \\cos{(b)}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('b', commutative=True))))"], [["times", 3, "Function('f^{\\\\prime}')(Symbol('b', commutative=True))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))))"], [["divide", 4, "Mul(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Integer(-1), sin(Symbol('b', commutative=True)), Pow(Derivative(Function('f^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Mul(Integer(-1), sin(Symbol('b', commutative=True)), Pow(Derivative(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1))))"], [["add", 6, "sin(Symbol('b', commutative=True))"], "Equality(Add(sin(Symbol('b', commutative=True)), Integer(1)), Add(sin(Symbol('b', commutative=True)), Mul(Integer(-1), sin(Symbol('b', commutative=True)), Pow(Derivative(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})} and \\operatorname{r_{0}}{(\\varepsilon)} = \\frac{\\log{(\\cos{(\\varepsilon)})}}{\\operatorname{a^{\\dagger}}{(\\varepsilon)}}, then derive 0 = \\frac{d}{d \\varepsilon} \\operatorname{r_{0}}{(\\varepsilon)}, then obtain \\int 0 d\\varepsilon = \\frac{d}{d \\varepsilon} 1 + \\int 0 d\\varepsilon", "derivation": "\\operatorname{a^{\\dagger}}{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})} and 1 = \\frac{\\log{(\\cos{(\\varepsilon)})}}{\\operatorname{a^{\\dagger}}{(\\varepsilon)}} and \\operatorname{r_{0}}{(\\varepsilon)} = \\frac{\\log{(\\cos{(\\varepsilon)})}}{\\operatorname{a^{\\dagger}}{(\\varepsilon)}} and 1 = \\operatorname{r_{0}}{(\\varepsilon)} and \\frac{d}{d \\varepsilon} 1 = \\frac{d}{d \\varepsilon} \\operatorname{r_{0}}{(\\varepsilon)} and 0 = \\frac{d}{d \\varepsilon} \\operatorname{r_{0}}{(\\varepsilon)} and 0 = \\frac{d}{d \\varepsilon} 1 and \\int 0 d\\varepsilon = \\frac{d}{d \\varepsilon} 1 + \\int 0 d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True)), log(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\varepsilon', commutative=True)))))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Function('r_0')(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Function('r_0')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Derivative(Function('r_0')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(0), Derivative(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["add", 7, "Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True))), Add(Derivative(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Integral(Integer(0), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(S,x^\\prime,Z)} = - S + Z + x^\\prime and \\operatorname{v_{t}}{(S,x^\\prime,Z)} = - 2 S + Z + x^\\prime, then obtain \\iint \\operatorname{v_{t}}{(S,x^\\prime,Z)} dZ dZ = \\iint (- S + \\hat{\\mathbf{x}}{(S,x^\\prime,Z)}) dZ dZ", "derivation": "\\hat{\\mathbf{x}}{(S,x^\\prime,Z)} = - S + Z + x^\\prime and - S + \\hat{\\mathbf{x}}{(S,x^\\prime,Z)} = - 2 S + Z + x^\\prime and \\operatorname{v_{t}}{(S,x^\\prime,Z)} = - 2 S + Z + x^\\prime and \\operatorname{v_{t}}{(S,x^\\prime,Z)} = - S + \\hat{\\mathbf{x}}{(S,x^\\prime,Z)} and \\int \\operatorname{v_{t}}{(S,x^\\prime,Z)} dZ = \\int (- S + \\hat{\\mathbf{x}}{(S,x^\\prime,Z)}) dZ and \\iint \\operatorname{v_{t}}{(S,x^\\prime,Z)} dZ dZ = \\iint (- S + \\hat{\\mathbf{x}}{(S,x^\\prime,Z)}) dZ dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('v_t')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))))"], [["integrate", 4, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given z{(f^{\\prime})} = \\log{(f^{\\prime})}, then derive \\frac{d}{d f^{\\prime}} z{(f^{\\prime})} = \\frac{1}{f^{\\prime}}, then obtain - G^{\\nabla} \\nabla + \\frac{1}{f^{\\prime}} = - G^{\\nabla} \\nabla + \\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})}", "derivation": "z{(f^{\\prime})} = \\log{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} z{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} z{(f^{\\prime})} = \\frac{1}{f^{\\prime}} and \\frac{1}{f^{\\prime}} = \\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})} and - G^{\\nabla} \\nabla + \\frac{1}{f^{\\prime}} = - G^{\\nabla} \\nabla + \\frac{d}{d f^{\\prime}} \\log{(f^{\\prime})}", "srepr_derivation": [["get_premise", "Equality(Function('z')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["minus", 4, "Mul(Pow(Symbol('G', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('G', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Derivative(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(v)} = \\sin{(v)} and k{(v)} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)}}{\\sin{(v)}}, then obtain \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)} k{(v)}}{\\sin{(v)}} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)}}{\\sin{(v)}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)} = \\sin{(v)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)}}{\\sin{(v)}} = 1 and k{(v)} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)}}{\\sin{(v)}} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)} k{(v)}}{\\sin{(v)}} = k{(v)} and \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)} k{(v)}}{\\sin{(v)}} = \\frac{\\operatorname{g^{\\prime}_{\\varepsilon}}{(v)}}{\\sin{(v)}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["divide", 1, "sin(Symbol('v', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('k')(Symbol('v', commutative=True)), Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))))"], [["times", 2, "Function('k')(Symbol('v', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), Function('k')(Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Function('k')(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), Function('k')(Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))), Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('v', commutative=True)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}^*{(J,v_{z})} = J v_{z}, then obtain \\tilde{g}^*{(J,v_{z})} + \\frac{\\partial}{\\partial v_{z}} \\int J v_{z} dJ = J v_{z} + \\frac{\\partial}{\\partial v_{z}} \\int J v_{z} dJ", "derivation": "\\tilde{g}^*{(J,v_{z})} = J v_{z} and \\int \\tilde{g}^*{(J,v_{z})} dJ = \\int J v_{z} dJ and \\frac{\\partial}{\\partial v_{z}} \\int \\tilde{g}^*{(J,v_{z})} dJ = \\frac{\\partial}{\\partial v_{z}} \\int J v_{z} dJ and \\tilde{g}^*{(J,v_{z})} + \\frac{\\partial}{\\partial v_{z}} \\int \\tilde{g}^*{(J,v_{z})} dJ = J v_{z} + \\frac{\\partial}{\\partial v_{z}} \\int \\tilde{g}^*{(J,v_{z})} dJ and \\tilde{g}^*{(J,v_{z})} + \\frac{\\partial}{\\partial v_{z}} \\int J v_{z} dJ = J v_{z} + \\frac{\\partial}{\\partial v_{z}} \\int J v_{z} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Derivative(Integral(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Derivative(Integral(Mul(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given G{(F_{g},\\rho_f,\\pi)} = \\rho_f (F_{g} + \\pi) and \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} = \\hat{H}_l \\hat{p}_0, then obtain - F_{g} - \\pi + \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} = - F_{g} + \\hat{H}_l \\hat{p}_0 - \\pi", "derivation": "G{(F_{g},\\rho_f,\\pi)} = \\rho_f (F_{g} + \\pi) and \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} = \\hat{H}_l \\hat{p}_0 and \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} - \\frac{\\partial}{\\partial \\rho_f} G{(F_{g},\\rho_f,\\pi)} = \\hat{H}_l \\hat{p}_0 - \\frac{\\partial}{\\partial \\rho_f} G{(F_{g},\\rho_f,\\pi)} and \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} - \\frac{\\partial}{\\partial \\rho_f} \\rho_f (F_{g} + \\pi) = \\hat{H}_l \\hat{p}_0 - \\frac{\\partial}{\\partial \\rho_f} \\rho_f (F_{g} + \\pi) and - F_{g} - \\pi + \\operatorname{P_{e}}{(\\hat{H}_l,\\hat{p}_0)} = - F_{g} + \\hat{H}_l \\hat{p}_0 - \\pi", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('F_g', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True))))"], ["get_premise", "Equality(Function('P_e')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 2, "Derivative(Function('G')(Symbol('F_g', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Add(Function('P_e')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('F_g', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Derivative(Function('G')(Symbol('F_g', commutative=True), Symbol('\\\\rho_f', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('P_e')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\rho_f', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\rho_f', commutative=True), Add(Symbol('F_g', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Function('P_e')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mu,\\Omega)} = \\Omega \\mu, then obtain \\lambda + \\mathbf{F}{(\\mu,\\Omega)} = \\Omega \\mu + \\hbar", "derivation": "\\mathbf{F}{(\\mu,\\Omega)} = \\Omega \\mu and \\frac{\\partial}{\\partial \\Omega} \\mathbf{F}{(\\mu,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\Omega \\mu and \\int \\frac{\\partial}{\\partial \\Omega} \\mathbf{F}{(\\mu,\\Omega)} d\\Omega = \\int \\frac{\\partial}{\\partial \\Omega} \\Omega \\mu d\\Omega and \\lambda + \\mathbf{F}{(\\mu,\\Omega)} = \\Omega \\mu + \\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\delta)} = e^{\\cos{(\\delta)}} and \\operatorname{F_{g}}{(\\delta)} = \\operatorname{M_{E}}^{\\delta}{(\\delta)}, then obtain e^{\\cos{(\\delta)}} (e^{\\cos{(\\delta)}})^{\\delta} = \\operatorname{F_{g}}{(\\delta)} e^{\\cos{(\\delta)}}", "derivation": "\\operatorname{M_{E}}{(\\delta)} = e^{\\cos{(\\delta)}} and \\operatorname{M_{E}}^{\\delta}{(\\delta)} = (e^{\\cos{(\\delta)}})^{\\delta} and \\operatorname{M_{E}}^{\\delta}{(\\delta)} e^{\\cos{(\\delta)}} = e^{\\cos{(\\delta)}} (e^{\\cos{(\\delta)}})^{\\delta} and \\operatorname{F_{g}}{(\\delta)} = \\operatorname{M_{E}}^{\\delta}{(\\delta)} and \\operatorname{F_{g}}{(\\delta)} = (e^{\\cos{(\\delta)}})^{\\delta} and \\operatorname{M_{E}}^{\\delta}{(\\delta)} e^{\\cos{(\\delta)}} = \\operatorname{F_{g}}{(\\delta)} e^{\\cos{(\\delta)}} and e^{\\cos{(\\delta)}} (e^{\\cos{(\\delta)}})^{\\delta} = \\operatorname{F_{g}}{(\\delta)} e^{\\cos{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["times", 2, "exp(cos(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Function('M_E')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True)))), Mul(exp(cos(Symbol('\\\\delta', commutative=True))), Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\delta', commutative=True)), Pow(Function('M_E')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('F_g')(Symbol('\\\\delta', commutative=True)), Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Function('M_E')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True)))), Mul(Function('F_g')(Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(exp(cos(Symbol('\\\\delta', commutative=True))), Pow(exp(cos(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Mul(Function('F_g')(Symbol('\\\\delta', commutative=True)), exp(cos(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(u)} = \\int \\cos{(u)} du, then derive \\dot{z}{(u)} = \\eta + \\sin{(u)}, then obtain \\frac{\\partial}{\\partial u} (- \\eta - \\sin{(u)} + (\\int \\cos{(u)} du)^{\\eta}) = \\frac{\\partial}{\\partial u} (- \\eta + (\\eta + \\sin{(u)})^{\\eta} - \\sin{(u)})", "derivation": "\\dot{z}{(u)} = \\int \\cos{(u)} du and \\dot{z}{(u)} = \\eta + \\sin{(u)} and \\int \\cos{(u)} du = \\eta + \\sin{(u)} and (\\int \\cos{(u)} du)^{\\eta} = (\\eta + \\sin{(u)})^{\\eta} and - \\eta - \\sin{(u)} + (\\int \\cos{(u)} du)^{\\eta} = - \\eta + (\\eta + \\sin{(u)})^{\\eta} - \\sin{(u)} and \\frac{\\partial}{\\partial u} (- \\eta - \\sin{(u)} + (\\int \\cos{(u)} du)^{\\eta}) = \\frac{\\partial}{\\partial u} (- \\eta + (\\eta + \\sin{(u)})^{\\eta} - \\sin{(u)})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('u', commutative=True)), Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{z}')(Symbol('u', commutative=True)), Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True))))"], [["power", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["minus", 4, "Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True))), Pow(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True))), Pow(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), sin(Symbol('u', commutative=True))), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{s}{(Q)} = e^{Q} and \\mu{(Q)} = e^{Q}, then obtain e^{2 \\mathbf{s}{(Q)}} = e^{\\mathbf{s}{(Q)} + \\mu{(Q)}}", "derivation": "\\mathbf{s}{(Q)} = e^{Q} and 2 \\mathbf{s}{(Q)} = \\mathbf{s}{(Q)} + e^{Q} and 2 \\mathbf{s}{(Q)} e^{Q} = (\\mathbf{s}{(Q)} + e^{Q}) e^{Q} and \\mu{(Q)} = e^{Q} and \\frac{2 \\mathbf{s}{(Q)} e^{Q}}{\\mu{(Q)}} = \\frac{(\\mathbf{s}{(Q)} + e^{Q}) e^{Q}}{\\mu{(Q)}} and e^{\\frac{2 \\mathbf{s}{(Q)} e^{Q}}{\\mu{(Q)}}} = e^{\\frac{(\\mathbf{s}{(Q)} + e^{Q}) e^{Q}}{\\mu{(Q)}}} and e^{2 \\mathbf{s}{(Q)}} = e^{\\mathbf{s}{(Q)} + e^{Q}} and e^{2 \\mathbf{s}{(Q)}} = e^{\\mathbf{s}{(Q)} + \\mu{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{s}')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True))), Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["times", 2, "exp(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Mul(Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), exp(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["divide", 3, "Function('\\\\mu')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)), exp(Symbol('Q', commutative=True))), Mul(Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)), exp(Symbol('Q', commutative=True))))"], [["exp", 5], "Equality(exp(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)), exp(Symbol('Q', commutative=True)))), exp(Mul(Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Pow(Function('\\\\mu')(Symbol('Q', commutative=True)), Integer(-1)), exp(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(exp(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)))), exp(Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(exp(Mul(Integer(2), Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)))), exp(Add(Function('\\\\mathbf{s}')(Symbol('Q', commutative=True)), Function('\\\\mu')(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)}, then obtain \\int \\frac{d}{d \\tilde{g}^*} (\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} - \\cos{(\\tilde{g}^*)}) d\\tilde{g}^* = \\int \\frac{d}{d \\tilde{g}^*} 0 d\\tilde{g}^*", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} = \\cos{(\\tilde{g}^*)} and \\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} - \\cos{(\\tilde{g}^*)} = 0 and \\frac{d}{d \\tilde{g}^*} (\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} - \\cos{(\\tilde{g}^*)}) = \\frac{d}{d \\tilde{g}^*} 0 and \\int \\frac{d}{d \\tilde{g}^*} (\\operatorname{V_{\\mathbf{B}}}{(\\tilde{g}^*)} - \\cos{(\\tilde{g}^*)}) d\\tilde{g}^* = \\int \\frac{d}{d \\tilde{g}^*} 0 d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Derivative(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given z{(\\phi_1,\\Psi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\Psi \\phi_1 and \\operatorname{x^{{\\}'}}{(\\Psi)} = \\Psi, then obtain (\\Psi \\phi_1 + z{(\\phi_1,\\Psi,V_{\\mathbf{E}})}) \\operatorname{x^{{\\}'}}^{\\Psi}{(\\Psi)} = (V_{\\mathbf{E}} + 2 \\Psi \\phi_1) \\operatorname{x^{{\\}'}}^{\\Psi}{(\\Psi)}", "derivation": "z{(\\phi_1,\\Psi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\Psi \\phi_1 and \\operatorname{x^{{\\}'}}{(\\Psi)} = \\Psi and \\operatorname{x^{{\\}'}}^{\\Psi}{(\\Psi)} = \\Psi^{\\Psi} and \\Psi \\phi_1 + z{(\\phi_1,\\Psi,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + 2 \\Psi \\phi_1 and \\Psi^{\\Psi} (\\Psi \\phi_1 + z{(\\phi_1,\\Psi,V_{\\mathbf{E}})}) = \\Psi^{\\Psi} (V_{\\mathbf{E}} + 2 \\Psi \\phi_1) and (\\Psi \\phi_1 + z{(\\phi_1,\\Psi,V_{\\mathbf{E}})}) \\operatorname{x^{{\\}'}}^{\\Psi}{(\\Psi)} = (V_{\\mathbf{E}} + 2 \\Psi \\phi_1) \\operatorname{x^{{\\}'}}^{\\Psi}{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))"], [["power", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["times", 4, "Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('z')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))), Mul(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(2), Symbol('\\\\Psi', commutative=True), Symbol('\\\\phi_1', commutative=True))), Pow(Function('x^\\\\prime')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given k{(Z)} = \\log{(Z)}, then derive \\frac{d}{d Z} k{(Z)} = \\frac{1}{Z}, then derive x^\\prime + k{(Z)} = A + \\log{(Z)}, then obtain x^\\prime + k{(Z)} = A + k{(Z)}", "derivation": "k{(Z)} = \\log{(Z)} and \\frac{d}{d Z} k{(Z)} = \\frac{d}{d Z} \\log{(Z)} and \\frac{d}{d Z} k{(Z)} = \\frac{1}{Z} and \\int \\frac{d}{d Z} k{(Z)} dZ = \\int \\frac{1}{Z} dZ and x^\\prime + k{(Z)} = A + \\log{(Z)} and x^\\prime + \\log{(Z)} = A + \\log{(Z)} and x^\\prime + k{(Z)} = A + k{(Z)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Symbol('Z', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Function('k')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Symbol('Z', commutative=True), Integer(-1)), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('k')(Symbol('Z', commutative=True))), Add(Symbol('A', commutative=True), log(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('x^\\\\prime', commutative=True), log(Symbol('Z', commutative=True))), Add(Symbol('A', commutative=True), log(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('k')(Symbol('Z', commutative=True))), Add(Symbol('A', commutative=True), Function('k')(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(T,F_{N},\\dot{z})} = F_{N} (T + \\dot{z}), then obtain \\dot{z} + \\frac{\\partial}{\\partial T} \\int \\tilde{g}^*{(T,F_{N},\\dot{z})} dF_{N} = \\dot{z} + \\frac{\\partial}{\\partial T} \\int F_{N} (T + \\dot{z}) dF_{N}", "derivation": "\\tilde{g}^*{(T,F_{N},\\dot{z})} = F_{N} (T + \\dot{z}) and \\int \\tilde{g}^*{(T,F_{N},\\dot{z})} dF_{N} = \\int F_{N} (T + \\dot{z}) dF_{N} and \\frac{\\partial}{\\partial T} \\int \\tilde{g}^*{(T,F_{N},\\dot{z})} dF_{N} = \\frac{\\partial}{\\partial T} \\int F_{N} (T + \\dot{z}) dF_{N} and \\dot{z} + \\frac{\\partial}{\\partial T} \\int \\tilde{g}^*{(T,F_{N},\\dot{z})} dF_{N} = \\dot{z} + \\frac{\\partial}{\\partial T} \\int F_{N} (T + \\dot{z}) dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('T', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('F_N', commutative=True), Add(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('T', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Symbol('F_N', commutative=True), Add(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('T', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('F_N', commutative=True), Add(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{z}', commutative=True), Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('T', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{z}', commutative=True), Derivative(Integral(Mul(Symbol('F_N', commutative=True), Add(Symbol('T', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(r_{0},\\dot{z})} = \\log{(\\frac{r_{0}}{\\dot{z}})} and E{(\\mathbf{J}_M,A_{z})} = \\sin{(A_{z} - \\mathbf{J}_M)}, then obtain E{(\\mathbf{J}_M,A_{z})} - s{(r_{0},\\dot{z})} = - s{(r_{0},\\dot{z})} + \\sin{(A_{z} - \\mathbf{J}_M)}", "derivation": "s{(r_{0},\\dot{z})} = \\log{(\\frac{r_{0}}{\\dot{z}})} and E{(\\mathbf{J}_M,A_{z})} = \\sin{(A_{z} - \\mathbf{J}_M)} and E{(\\mathbf{J}_M,A_{z})} - \\log{(\\frac{r_{0}}{\\dot{z}})} = - \\log{(\\frac{r_{0}}{\\dot{z}})} + \\sin{(A_{z} - \\mathbf{J}_M)} and E{(\\mathbf{J}_M,A_{z})} - s{(r_{0},\\dot{z})} = - s{(r_{0},\\dot{z})} + \\sin{(A_{z} - \\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('r_0', commutative=True), Symbol('\\\\dot{z}', commutative=True)), log(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('A_z', commutative=True)), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["minus", 2, "log(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], "Equality(Add(Function('E')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), log(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))), Add(Mul(Integer(-1), log(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('E')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('r_0', commutative=True), Symbol('\\\\dot{z}', commutative=True)))), Add(Mul(Integer(-1), Function('s')(Symbol('r_0', commutative=True), Symbol('\\\\dot{z}', commutative=True))), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))))))"]]}, {"prompt": "Given L{(r_{0})} = \\log{(r_{0})}, then obtain 1 = \\frac{\\frac{d}{d r_{0}} 1}{\\frac{d}{d r_{0}} 0^{r_{0}}}", "derivation": "L{(r_{0})} = \\log{(r_{0})} and 0 = - L{(r_{0})} + \\log{(r_{0})} and 0^{r_{0}} = (- L{(r_{0})} + \\log{(r_{0})})^{r_{0}} and \\frac{d}{d r_{0}} 0^{r_{0}} = \\frac{d}{d r_{0}} (- L{(r_{0})} + \\log{(r_{0})})^{r_{0}} and \\frac{d}{d r_{0}} (- L{(r_{0})} + \\log{(r_{0})})^{r_{0}} = \\frac{d}{d r_{0}} 1 and 1 = \\frac{\\frac{d}{d r_{0}} 1}{\\frac{d}{d r_{0}} (- L{(r_{0})} + \\log{(r_{0})})^{r_{0}}} and 1 = \\frac{\\frac{d}{d r_{0}} 1}{\\frac{d}{d r_{0}} 0^{r_{0}}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["minus", 1, "Function('L')(Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r_0', commutative=True)), Pow(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))), Pow(Derivative(Pow(Add(Mul(Integer(-1), Function('L')(Symbol('r_0', commutative=True))), log(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(1), Mul(Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))), Pow(Derivative(Pow(Integer(0), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(g)} = \\sin{(e^{g})}, then derive \\frac{d}{d g} \\mathbf{J}_P{(g)} = e^{g} \\cos{(e^{g})}, then obtain \\cos{(e^{g})} \\frac{d}{d g} \\sin{(e^{g})} = e^{g} \\cos^{2}{(e^{g})}", "derivation": "\\mathbf{J}_P{(g)} = \\sin{(e^{g})} and \\frac{d}{d g} \\mathbf{J}_P{(g)} = \\frac{d}{d g} \\sin{(e^{g})} and \\frac{d}{d g} \\mathbf{J}_P{(g)} = e^{g} \\cos{(e^{g})} and \\cos{(e^{g})} \\frac{d}{d g} \\mathbf{J}_P{(g)} = e^{g} \\cos^{2}{(e^{g})} and \\cos{(e^{g})} \\frac{d}{d g} \\sin{(e^{g})} = e^{g} \\cos^{2}{(e^{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('g', commutative=True)), sin(exp(Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(exp(Symbol('g', commutative=True)), cos(exp(Symbol('g', commutative=True)))))"], [["times", 3, "cos(exp(Symbol('g', commutative=True)))"], "Equality(Mul(cos(exp(Symbol('g', commutative=True))), Derivative(Function('\\\\mathbf{J}_P')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(exp(Symbol('g', commutative=True)), Pow(cos(exp(Symbol('g', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(cos(exp(Symbol('g', commutative=True))), Derivative(sin(exp(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(exp(Symbol('g', commutative=True)), Pow(cos(exp(Symbol('g', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(\\rho_b)} = \\cos{(\\cos{(\\rho_b)})}, then obtain \\frac{\\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)}}{\\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)} + \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})}} = \\frac{1}{2}", "derivation": "\\theta_{1}{(\\rho_b)} = \\cos{(\\cos{(\\rho_b)})} and \\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)} = \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})} and \\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)} + \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})} = 2 \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})} and \\frac{\\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)}}{2 \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})}} = \\frac{1}{2} and \\frac{\\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)}}{\\frac{d}{d \\rho_b} \\theta_{1}{(\\rho_b)} + \\frac{d}{d \\rho_b} \\cos{(\\cos{(\\rho_b)})}} = \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["divide", 2, "Mul(Integer(2), Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], "Equality(Mul(Rational(1, 2), Derivative(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Pow(Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Derivative(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Integer(-1)), Derivative(Function('\\\\theta_1')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Rational(1, 2))"]]}, {"prompt": "Given \\rho{(a,\\mu)} = \\frac{\\mu}{a}, then obtain - \\int \\rho{(a,\\mu)} d\\mu - \\frac{1}{a} = - \\int \\frac{\\mu}{a} d\\mu - \\frac{1}{a}", "derivation": "\\rho{(a,\\mu)} = \\frac{\\mu}{a} and \\int \\rho{(a,\\mu)} d\\mu = \\int \\frac{\\mu}{a} d\\mu and - \\int \\rho{(a,\\mu)} d\\mu = - \\int \\frac{\\mu}{a} d\\mu and - \\int \\rho{(a,\\mu)} d\\mu - \\frac{1}{a} = - \\int \\frac{\\mu}{a} d\\mu - \\frac{1}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["minus", 3, "Pow(Symbol('a', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Integral(Function('\\\\rho')(Symbol('a', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Integral(Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given M{(J_{\\varepsilon},\\Psi^{\\dagger})} = \\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}}, then obtain \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\Psi^{\\dagger} + M{(J_{\\varepsilon},\\Psi^{\\dagger})} - \\frac{1}{\\Psi^{\\dagger}}) = \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}} + \\Psi^{\\dagger} - \\frac{1}{\\Psi^{\\dagger}})", "derivation": "M{(J_{\\varepsilon},\\Psi^{\\dagger})} = \\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}} and \\Psi^{\\dagger} + M{(J_{\\varepsilon},\\Psi^{\\dagger})} = \\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}} + \\Psi^{\\dagger} and \\Psi^{\\dagger} + M{(J_{\\varepsilon},\\Psi^{\\dagger})} - \\frac{1}{\\Psi^{\\dagger}} = \\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}} + \\Psi^{\\dagger} - \\frac{1}{\\Psi^{\\dagger}} and \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\Psi^{\\dagger} + M{(J_{\\varepsilon},\\Psi^{\\dagger})} - \\frac{1}{\\Psi^{\\dagger}}) = \\frac{\\partial}{\\partial J_{\\varepsilon}} (\\frac{J_{\\varepsilon}}{\\Psi^{\\dagger}} + \\Psi^{\\dagger} - \\frac{1}{\\Psi^{\\dagger}})", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["minus", 2, "Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))))"], [["differentiate", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('M')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(m)} = e^{m}, then derive (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = (g + e^{m})^{m}, then obtain (g + \\operatorname{A_{2}}{(m)})^{m} - e^{m} = - e^{m} + (\\int e^{m} dm)^{m}", "derivation": "\\operatorname{A_{2}}{(m)} = e^{m} and \\int \\operatorname{A_{2}}{(m)} dm = \\int e^{m} dm and (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = (\\int e^{m} dm)^{m} and (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = (g + e^{m})^{m} and (g + e^{m})^{m} = (\\int e^{m} dm)^{m} and - e^{m} + (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = (g + e^{m})^{m} - e^{m} and (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = (g + \\operatorname{A_{2}}{(m)})^{m} and - e^{m} + (\\int \\operatorname{A_{2}}{(m)} dm)^{m} = - e^{m} + (\\int e^{m} dm)^{m} and (g + \\operatorname{A_{2}}{(m)})^{m} - e^{m} = - e^{m} + (\\int e^{m} dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Add(Symbol('g', commutative=True), exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('g', commutative=True), exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["minus", 4, "exp(Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Pow(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Add(Pow(Add(Symbol('g', commutative=True), exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Add(Symbol('g', commutative=True), Function('A_2')(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Pow(Integral(Function('A_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Add(Pow(Add(Symbol('g', commutative=True), Function('A_2')(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('m', commutative=True))), Pow(Integral(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given C{(C_{2})} = \\cos{(C_{2})}, then obtain \\frac{C^{C_{2}}{(C_{2})} \\cos^{- C_{2}}{(C_{2})} + 1}{C{(C_{2})} \\cos^{C_{2}}{(C_{2})} - 1} = \\frac{2}{C{(C_{2})} \\cos^{C_{2}}{(C_{2})} - 1}", "derivation": "C{(C_{2})} = \\cos{(C_{2})} and C^{C_{2}}{(C_{2})} = \\cos^{C_{2}}{(C_{2})} and C{(C_{2})} C^{C_{2}}{(C_{2})} = C{(C_{2})} \\cos^{C_{2}}{(C_{2})} and C^{C_{2}}{(C_{2})} \\cos^{- C_{2}}{(C_{2})} = 1 and C^{C_{2}}{(C_{2})} \\cos^{- C_{2}}{(C_{2})} + 1 = 2 and \\frac{C^{C_{2}}{(C_{2})} \\cos^{- C_{2}}{(C_{2})} + 1}{C{(C_{2})} \\cos^{C_{2}}{(C_{2})} - 1} = \\frac{2}{C{(C_{2})} \\cos^{C_{2}}{(C_{2})} - 1}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], [["times", 2, "Function('C')(Symbol('C_2', commutative=True))"], "Equality(Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))))"], [["divide", 3, "Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)))"], "Equality(Mul(Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Integer(1))"], [["add", 4, 1], "Equality(Add(Mul(Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Integer(1)), Integer(2))"], [["divide", 5, "Add(Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Integer(-1)), Integer(-1)), Add(Mul(Pow(Function('C')(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Integer(1))), Mul(Integer(2), Pow(Add(Mul(Function('C')(Symbol('C_2', commutative=True)), Pow(cos(Symbol('C_2', commutative=True)), Symbol('C_2', commutative=True))), Integer(-1)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})}, then obtain \\frac{d}{d \\mathbf{p}} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} \\sin{(\\mathbf{p})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} = \\sin^{\\mathbf{p}}{(\\mathbf{p})} and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} \\sin^{\\mathbf{p}}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} \\sin^{\\mathbf{p}}{(\\mathbf{p})} and \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} = \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} \\sin{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\operatorname{V_{\\mathbf{B}}}^{\\mathbf{p}}{(\\mathbf{p})} \\sin{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(sin(Symbol('\\\\mathbf{p}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(T)} = \\cos{(T)}, then derive T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\mathbf{A}{(T)} = T + \\mathbf{A}{(T)} - \\sin{(T)}, then obtain T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\cos{(T)} = T + \\mathbf{A}{(T)} - \\sin{(T)}", "derivation": "\\mathbf{A}{(T)} = \\cos{(T)} and \\frac{d}{d T} \\mathbf{A}{(T)} = \\frac{d}{d T} \\cos{(T)} and \\mathbf{A}{(T)} + \\frac{d}{d T} \\mathbf{A}{(T)} = \\mathbf{A}{(T)} + \\frac{d}{d T} \\cos{(T)} and T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\mathbf{A}{(T)} = T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\cos{(T)} and T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\mathbf{A}{(T)} = T + \\mathbf{A}{(T)} - \\sin{(T)} and T + \\mathbf{A}{(T)} + \\frac{d}{d T} \\cos{(T)} = T + \\mathbf{A}{(T)} - \\sin{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\mathbf{A}')(Symbol('T', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["add", 3, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Derivative(cos(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Symbol('T', commutative=True), Function('\\\\mathbf{A}')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{g})} = \\log{(F_{g})}, then obtain \\frac{\\partial}{\\partial F_{g}} (- F_{g} - I + \\int \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} dF_{g}) = \\frac{d}{d F_{g}} 0", "derivation": "\\operatorname{x^{{\\}'}}{(F_{g})} = \\log{(F_{g})} and \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} = 1 and \\int \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} dF_{g} = \\int 1 dF_{g} and - \\int 1 dF_{g} + \\int \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} dF_{g} = 0 and \\frac{d}{d F_{g}} (- \\int 1 dF_{g} + \\int \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} dF_{g}) = \\frac{d}{d F_{g}} 0 and \\frac{\\partial}{\\partial F_{g}} (- F_{g} - I + \\int \\frac{\\operatorname{x^{{\\}'}}{(F_{g})}}{\\log{(F_{g})}} dF_{g}) = \\frac{d}{d F_{g}} 0", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))"], [["divide", 1, "log(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True))))"], [["minus", 3, "Integral(Integer(1), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True)))), Integral(Mul(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('F_g', commutative=True)))), Integral(Mul(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Integral(Mul(Function('x^\\\\prime')(Symbol('F_g', commutative=True)), Pow(log(Symbol('F_g', commutative=True)), Integer(-1))), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(C_{1},x^\\prime)} = (x^\\prime)^{C_{1}} and \\varepsilon{(C_{1},x^\\prime)} = - \\frac{1}{\\int (- C_{1} + (x^\\prime)^{C_{1}}) dC_{1}}, then obtain C_{1} - S{(C_{1},x^\\prime)} + \\varepsilon{(C_{1},x^\\prime)} = C_{1} - S{(C_{1},x^\\prime)} - \\frac{1}{\\int (- C_{1} + S{(C_{1},x^\\prime)}) dC_{1}}", "derivation": "S{(C_{1},x^\\prime)} = (x^\\prime)^{C_{1}} and - C_{1} + S{(C_{1},x^\\prime)} = - C_{1} + (x^\\prime)^{C_{1}} and \\int (- C_{1} + S{(C_{1},x^\\prime)}) dC_{1} = \\int (- C_{1} + (x^\\prime)^{C_{1}}) dC_{1} and \\varepsilon{(C_{1},x^\\prime)} = - \\frac{1}{\\int (- C_{1} + (x^\\prime)^{C_{1}}) dC_{1}} and \\varepsilon{(C_{1},x^\\prime)} = - \\frac{1}{\\int (- C_{1} + S{(C_{1},x^\\prime)}) dC_{1}} and C_{1} - S{(C_{1},x^\\prime)} + \\varepsilon{(C_{1},x^\\prime)} = C_{1} - S{(C_{1},x^\\prime)} - \\frac{1}{\\int (- C_{1} + S{(C_{1},x^\\prime)}) dC_{1}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C_1', commutative=True)))"], [["minus", 1, "Symbol('C_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C_1', commutative=True))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(-1))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Function('\\\\varepsilon')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Function('S')(Symbol('C_1', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\sigma_p)} = \\sin{(\\sigma_p)}, then derive 0 = \\cos{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\operatorname{F_{N}}{(\\sigma_p)}, then obtain (- \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)})^{\\sigma_p} = (\\cos{(\\sigma_p)} - 2 \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)})^{\\sigma_p}", "derivation": "\\operatorname{F_{N}}{(\\sigma_p)} = \\sin{(\\sigma_p)} and 0 = - \\operatorname{F_{N}}{(\\sigma_p)} + \\sin{(\\sigma_p)} and \\frac{d}{d \\sigma_p} 0 = \\frac{d}{d \\sigma_p} (- \\operatorname{F_{N}}{(\\sigma_p)} + \\sin{(\\sigma_p)}) and 0 = \\cos{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\operatorname{F_{N}}{(\\sigma_p)} and 0 = \\cos{(\\sigma_p)} - \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)} and - \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)} = \\cos{(\\sigma_p)} - 2 \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)} and (- \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)})^{\\sigma_p} = (\\cos{(\\sigma_p)} - 2 \\frac{d}{d \\sigma_p} \\sin{(\\sigma_p)})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Function('F_N')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_N')(Symbol('\\\\sigma_p', commutative=True))), sin(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(cos(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(cos(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"], [["minus", 5, "Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(cos(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))))"], [["power", 6, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Symbol('\\\\sigma_p', commutative=True)), Pow(Add(cos(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integer(2), Derivative(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(\\varphi,\\Psi)} = \\Psi - \\varphi, then derive \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} = 1, then obtain - \\varphi (\\varphi + 2 \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} - 1)^{\\varphi} = - \\varphi (\\varphi + \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)})^{\\varphi}", "derivation": "\\dot{z}{(\\varphi,\\Psi)} = \\Psi - \\varphi and \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} = \\frac{\\partial}{\\partial \\Psi} (\\Psi - \\varphi) and \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} = 1 and \\varphi + \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} = \\varphi + 1 and (\\varphi + \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)})^{\\varphi} = (\\varphi + 1)^{\\varphi} and (\\varphi + 2 \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} - 1)^{\\varphi} = (\\varphi + \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)})^{\\varphi} and - \\varphi (\\varphi + 2 \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)} - 1)^{\\varphi} = - \\varphi (\\varphi + \\frac{\\partial}{\\partial \\Psi} \\dot{z}{(\\varphi,\\Psi)})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(Symbol('\\\\varphi', commutative=True), Integer(1)))"], [["power", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\varphi', commutative=True), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), Integer(1)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\varphi', commutative=True)))"], [["times", 6, "Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Integer(-1)), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Pow(Add(Symbol('\\\\varphi', commutative=True), Derivative(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(b)} = \\log{(b)} and \\psi^{*}{(\\theta_1,J_{\\varepsilon})} = \\theta_1^{J_{\\varepsilon}}, then derive \\int \\operatorname{E_{x}}{(b)} db = \\rho_f + b \\log{(b)} - b, then obtain \\frac{\\psi^{*}{(\\theta_1,J_{\\varepsilon})}}{\\int \\operatorname{E_{x}}{(b)} db} = \\frac{\\theta_1^{J_{\\varepsilon}}}{\\int \\operatorname{E_{x}}{(b)} db}", "derivation": "\\operatorname{E_{x}}{(b)} = \\log{(b)} and \\int \\operatorname{E_{x}}{(b)} db = \\int \\log{(b)} db and \\int \\operatorname{E_{x}}{(b)} db = \\rho_f + b \\log{(b)} - b and \\psi^{*}{(\\theta_1,J_{\\varepsilon})} = \\theta_1^{J_{\\varepsilon}} and \\frac{\\psi^{*}{(\\theta_1,J_{\\varepsilon})}}{\\rho_f + b \\log{(b)} - b} = \\frac{\\theta_1^{J_{\\varepsilon}}}{\\rho_f + b \\log{(b)} - b} and \\frac{\\psi^{*}{(\\theta_1,J_{\\varepsilon})}}{\\int \\operatorname{E_{x}}{(b)} db} = \\frac{\\theta_1^{J_{\\varepsilon}}}{\\int \\operatorname{E_{x}}{(b)} db}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('b', commutative=True)), log(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(log(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))))"], ["get_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 4, "Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Integer(-1)), Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('b', commutative=True), log(Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Integral(Function('E_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Integral(Function('E_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given S{(C_{1})} = \\log{(C_{1})}, then obtain C_{1} + (\\frac{S{(C_{1})}}{\\log{(C_{1})}})^{C_{1}} = C_{1} + 1", "derivation": "S{(C_{1})} = \\log{(C_{1})} and \\frac{S{(C_{1})}}{\\log{(C_{1})}} = 1 and (\\frac{S{(C_{1})}}{\\log{(C_{1})}})^{C_{1}} = 1 and C_{1} + (\\frac{S{(C_{1})}}{\\log{(C_{1})}})^{C_{1}} = C_{1} + 1", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('C_1', commutative=True)), log(Symbol('C_1', commutative=True)))"], [["divide", 1, "log(Symbol('C_1', commutative=True))"], "Equality(Mul(Function('S')(Symbol('C_1', commutative=True)), Pow(log(Symbol('C_1', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Mul(Function('S')(Symbol('C_1', commutative=True)), Pow(log(Symbol('C_1', commutative=True)), Integer(-1))), Symbol('C_1', commutative=True)), Integer(1))"], [["add", 3, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Pow(Mul(Function('S')(Symbol('C_1', commutative=True)), Pow(log(Symbol('C_1', commutative=True)), Integer(-1))), Symbol('C_1', commutative=True))), Add(Symbol('C_1', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{p})} = \\log{(\\log{(\\hat{p})})}, then obtain \\int \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} d\\hat{p} + 1 = \\int 1 d\\hat{p} + 1", "derivation": "\\mathbf{A}{(\\hat{p})} = \\log{(\\log{(\\hat{p})})} and \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} = 1 and \\int \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} d\\hat{p} = \\int 1 d\\hat{p} and \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} + \\int \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} d\\hat{p} = \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} + \\int 1 d\\hat{p} and \\int \\frac{\\mathbf{A}{(\\hat{p})}}{\\log{(\\log{(\\hat{p})})}} d\\hat{p} + 1 = \\int 1 d\\hat{p} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), log(log(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "log(log(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 3, "Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integral(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True)), Pow(log(log(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(1)), Add(Integral(Integer(1), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(a^{\\dagger},\\psi^*)} = (a^{\\dagger})^{\\psi^*} and \\operatorname{A_{z}}{(\\psi^*)} = - \\psi^*, then obtain ((a^{\\dagger})^{\\psi^*})^{\\operatorname{A_{z}}{(\\psi^*)}} \\hat{H}_{\\lambda}^{\\psi^*}{(a^{\\dagger},\\psi^*)} = 1", "derivation": "\\hat{H}_{\\lambda}{(a^{\\dagger},\\psi^*)} = (a^{\\dagger})^{\\psi^*} and \\hat{H}_{\\lambda}^{\\psi^*}{(a^{\\dagger},\\psi^*)} = ((a^{\\dagger})^{\\psi^*})^{\\psi^*} and ((a^{\\dagger})^{\\psi^*})^{- \\psi^*} \\hat{H}_{\\lambda}^{\\psi^*}{(a^{\\dagger},\\psi^*)} = 1 and \\operatorname{A_{z}}{(\\psi^*)} = - \\psi^* and ((a^{\\dagger})^{\\psi^*})^{\\operatorname{A_{z}}{(\\psi^*)}} \\hat{H}_{\\lambda}^{\\psi^*}{(a^{\\dagger},\\psi^*)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["divide", 2, "Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Function('A_z')(Symbol('\\\\psi^*', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Integer(1))"]]}, {"prompt": "Given I{(\\varepsilon)} = \\varepsilon, then obtain - \\varepsilon + 2 I{(\\varepsilon)} - 1 = \\varepsilon - 1", "derivation": "I{(\\varepsilon)} = \\varepsilon and - \\varepsilon + I{(\\varepsilon)} = 0 and - \\varepsilon + I{(\\varepsilon)} - 1 = -1 and I{(\\varepsilon)} - 1 = \\varepsilon - 1 and - \\varepsilon + 2 I{(\\varepsilon)} - 1 = I{(\\varepsilon)} - 1 and - \\varepsilon + 2 I{(\\varepsilon)} - 1 = \\varepsilon - 1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))"], [["minus", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('I')(Symbol('\\\\varepsilon', commutative=True))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Function('I')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Integer(-1))"], [["add", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Integer(-1)))"], [["add", 3, "Function('I')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(2), Function('I')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Add(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(2), Function('I')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)), Add(Symbol('\\\\varepsilon', commutative=True), Integer(-1)))"]]}, {"prompt": "Given b{(c,L)} = \\log{(L - c)} and m{(c,L)} = b{(c,L)} + \\log{(L - c)}, then obtain - b{(c,L)} - \\log{(L - c)} + \\int 2 b{(c,L)} dc = - b{(c,L)} - \\log{(L - c)} + \\int (b{(c,L)} + \\log{(L - c)}) dc", "derivation": "b{(c,L)} = \\log{(L - c)} and 2 b{(c,L)} = b{(c,L)} + \\log{(L - c)} and m{(c,L)} = b{(c,L)} + \\log{(L - c)} and \\int m{(c,L)} dc = \\int (b{(c,L)} + \\log{(L - c)}) dc and - 2 b{(c,L)} + \\int m{(c,L)} dc = - 2 b{(c,L)} + \\int (b{(c,L)} + \\log{(L - c)}) dc and m{(c,L)} = 2 b{(c,L)} and - m{(c,L)} + \\int m{(c,L)} dc = - m{(c,L)} + \\int (b{(c,L)} + \\log{(L - c)}) dc and - 2 b{(c,L)} + \\int 2 b{(c,L)} dc = - 2 b{(c,L)} + \\int (b{(c,L)} + \\log{(L - c)}) dc and - b{(c,L)} - \\log{(L - c)} + \\int 2 b{(c,L)} dc = - b{(c,L)} - \\log{(L - c)} + \\int (b{(c,L)} + \\log{(L - c)}) dc", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["add", 1, "Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], ["renaming_premise", "Equality(Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True)), Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))))"], [["integrate", 3, "Symbol('c', commutative=True)"], "Equality(Integral(Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True)), Mul(Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Mul(Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Integral(Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Add(Mul(Integer(-1), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Integral(Mul(Integer(2), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Integral(Add(Function('b')(Symbol('c', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\pi)} = \\log{(\\pi)} and \\Omega{(\\pi)} = - \\operatorname{t_{1}}{(\\pi)} + \\log{(\\pi)}, then obtain (\\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)})^{2} = - (\\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)}) \\operatorname{t_{1}}{(\\pi)}", "derivation": "\\operatorname{t_{1}}{(\\pi)} = \\log{(\\pi)} and \\Omega{(\\pi)} = - \\operatorname{t_{1}}{(\\pi)} + \\log{(\\pi)} and \\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)} = - 2 \\operatorname{t_{1}}{(\\pi)} + \\log{(\\pi)} and \\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)} = - \\operatorname{t_{1}}{(\\pi)} and (\\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)})^{2} = - (\\Omega{(\\pi)} - \\operatorname{t_{1}}{(\\pi)}) \\operatorname{t_{1}}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Add(Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True))), log(Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('t_1')(Symbol('\\\\pi', commutative=True))), log(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True))))"], [["times", 4, "Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True))))"], "Equality(Pow(Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True)))), Integer(2)), Mul(Integer(-1), Add(Function('\\\\Omega')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('t_1')(Symbol('\\\\pi', commutative=True)))), Function('t_1')(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(k,\\theta)} = \\log{(\\frac{\\theta}{k})}, then obtain \\int \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\dot{x}{(k,\\theta)})^{k} dk = \\int \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\log{(\\frac{\\theta}{k})})^{k} dk", "derivation": "\\dot{x}{(k,\\theta)} = \\log{(\\frac{\\theta}{k})} and \\theta \\dot{x}{(k,\\theta)} = \\theta \\log{(\\frac{\\theta}{k})} and \\frac{\\partial}{\\partial k} \\theta \\dot{x}{(k,\\theta)} = \\frac{\\partial}{\\partial k} \\theta \\log{(\\frac{\\theta}{k})} and (\\frac{\\partial}{\\partial k} \\theta \\dot{x}{(k,\\theta)})^{k} = (\\frac{\\partial}{\\partial k} \\theta \\log{(\\frac{\\theta}{k})})^{k} and \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\dot{x}{(k,\\theta)})^{k} = \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\log{(\\frac{\\theta}{k})})^{k} and \\int \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\dot{x}{(k,\\theta)})^{k} dk = \\int \\frac{\\partial}{\\partial \\theta} (\\frac{\\partial}{\\partial k} \\theta \\log{(\\frac{\\theta}{k})})^{k} dk", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1)))))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Derivative(Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\dot{x}')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))), Integral(Derivative(Pow(Derivative(Mul(Symbol('\\\\theta', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\psi^*,F_{H})} = F_{H} + \\psi^*, then derive \\int (\\psi^* + \\theta{(\\psi^*,F_{H})}) dF_{H} = C_{1} + \\frac{F_{H}^{2}}{2} + 2 F_{H} \\psi^*, then obtain \\frac{\\int (\\psi^* + \\theta{(\\psi^*,F_{H})}) dF_{H}}{F_{H}^{2}} = \\frac{C_{1} + \\frac{F_{H}^{2}}{2} + 2 F_{H} \\psi^*}{F_{H}^{2}}", "derivation": "\\theta{(\\psi^*,F_{H})} = F_{H} + \\psi^* and \\psi^* + \\theta{(\\psi^*,F_{H})} = F_{H} + 2 \\psi^* and \\int (\\psi^* + \\theta{(\\psi^*,F_{H})}) dF_{H} = \\int (F_{H} + 2 \\psi^*) dF_{H} and \\int (\\psi^* + \\theta{(\\psi^*,F_{H})}) dF_{H} = C_{1} + \\frac{F_{H}^{2}}{2} + 2 F_{H} \\psi^* and \\frac{\\int (\\psi^* + \\theta{(\\psi^*,F_{H})}) dF_{H}}{F_{H}^{2}} = \\frac{C_{1} + \\frac{F_{H}^{2}}{2} + 2 F_{H} \\psi^*}{F_{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Integer(2), Symbol('F_H', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["divide", 4, "Pow(Symbol('F_H', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-2)), Integral(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\theta')(Symbol('\\\\psi^*', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-2)), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Mul(Integer(2), Symbol('F_H', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(F_{x})} = \\log{(F_{x})} and \\omega{(F_{x})} = \\frac{1}{F_{x}}, then obtain \\omega{(F_{x})} \\int \\omega{(F_{x})} \\log{(F_{x})} dF_{x} = \\omega{(F_{x})} \\int \\frac{\\log{(F_{x})}}{F_{x}} dF_{x}", "derivation": "\\mathbf{M}{(F_{x})} = \\log{(F_{x})} and \\omega{(F_{x})} = \\frac{1}{F_{x}} and \\mathbf{M}{(F_{x})} \\omega{(F_{x})} = \\frac{\\mathbf{M}{(F_{x})}}{F_{x}} and \\int \\mathbf{M}{(F_{x})} \\omega{(F_{x})} dF_{x} = \\int \\frac{\\mathbf{M}{(F_{x})}}{F_{x}} dF_{x} and \\omega{(F_{x})} \\int \\mathbf{M}{(F_{x})} \\omega{(F_{x})} dF_{x} = \\omega{(F_{x})} \\int \\frac{\\mathbf{M}{(F_{x})}}{F_{x}} dF_{x} and \\omega{(F_{x})} \\int \\omega{(F_{x})} \\log{(F_{x})} dF_{x} = \\omega{(F_{x})} \\int \\frac{\\log{(F_{x})}}{F_{x}} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('F_x', commutative=True)), Pow(Symbol('F_x', commutative=True), Integer(-1)))"], [["times", 2, "Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True)), Function('\\\\omega')(Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True)), Function('\\\\omega')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["times", 4, "Function('\\\\omega')(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('\\\\omega')(Symbol('F_x', commutative=True)), Integral(Mul(Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True)), Function('\\\\omega')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))), Mul(Function('\\\\omega')(Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Function('\\\\omega')(Symbol('F_x', commutative=True)), Integral(Mul(Function('\\\\omega')(Symbol('F_x', commutative=True)), log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))), Mul(Function('\\\\omega')(Symbol('F_x', commutative=True)), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(l)} = \\log{(l)}, then derive \\chi = \\int (- \\operatorname{A_{y}}{(l)} + \\log{(l)})^{l} dl, then obtain \\frac{E}{\\operatorname{A_{y}}{(l)}} = \\frac{\\chi}{\\operatorname{A_{y}}{(l)}}", "derivation": "\\operatorname{A_{y}}{(l)} = \\log{(l)} and 0 = - \\operatorname{A_{y}}{(l)} + \\log{(l)} and 0^{l} = (- \\operatorname{A_{y}}{(l)} + \\log{(l)})^{l} and \\int 0^{l} dl = \\int (- \\operatorname{A_{y}}{(l)} + \\log{(l)})^{l} dl and \\chi = \\int (- \\operatorname{A_{y}}{(l)} + \\log{(l)})^{l} dl and \\int 0^{l} dl = \\chi and \\frac{\\int 0^{l} dl}{\\operatorname{A_{y}}{(l)}} = \\frac{\\chi}{\\operatorname{A_{y}}{(l)}} and \\frac{E}{\\operatorname{A_{y}}{(l)}} = \\frac{\\chi}{\\operatorname{A_{y}}{(l)}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["minus", 1, "Function('A_y')(Symbol('l', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True))), log(Symbol('l', commutative=True))))"], [["power", 2, "Symbol('l', commutative=True)"], "Equality(Pow(Integer(0), Symbol('l', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True))), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["integrate", 3, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True))), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Symbol('\\\\chi', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Function('A_y')(Symbol('l', commutative=True))), log(Symbol('l', commutative=True))), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Pow(Integer(0), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Symbol('\\\\chi', commutative=True))"], [["times", 6, "Pow(Function('A_y')(Symbol('l', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Function('A_y')(Symbol('l', commutative=True)), Integer(-1)), Integral(Pow(Integer(0), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), Pow(Function('A_y')(Symbol('l', commutative=True)), Integer(-1))))"], [["evaluate_integrals", 7], "Equality(Mul(Symbol('E', commutative=True), Pow(Function('A_y')(Symbol('l', commutative=True)), Integer(-1))), Mul(Symbol('\\\\chi', commutative=True), Pow(Function('A_y')(Symbol('l', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C)} = \\sin{(C)} and \\omega{(C)} = 2 \\operatorname{M_{E}}{(C)}, then obtain - \\operatorname{M_{E}}{(C)} - \\omega{(C)} - \\sin{(C)} + 1 = - \\operatorname{M_{E}}{(C)} - \\omega{(C)} - \\sin{(C)} + \\frac{\\sin{(C)}}{\\operatorname{M_{E}}{(C)}}", "derivation": "\\operatorname{M_{E}}{(C)} = \\sin{(C)} and 2 \\operatorname{M_{E}}{(C)} = \\operatorname{M_{E}}{(C)} + \\sin{(C)} and \\omega{(C)} = 2 \\operatorname{M_{E}}{(C)} and 2 \\omega{(C)} = 2 \\operatorname{M_{E}}{(C)} + \\omega{(C)} and 1 = \\frac{\\sin{(C)}}{\\operatorname{M_{E}}{(C)}} and 2 \\omega{(C)} = \\operatorname{M_{E}}{(C)} + \\omega{(C)} + \\sin{(C)} and 1 - 2 \\omega{(C)} = - 2 \\omega{(C)} + \\frac{\\sin{(C)}}{\\operatorname{M_{E}}{(C)}} and - \\operatorname{M_{E}}{(C)} - \\omega{(C)} - \\sin{(C)} + 1 = - \\operatorname{M_{E}}{(C)} - \\omega{(C)} - \\sin{(C)} + \\frac{\\sin{(C)}}{\\operatorname{M_{E}}{(C)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["add", 1, "Function('M_E')(Symbol('C', commutative=True))"], "Equality(Mul(Integer(2), Function('M_E')(Symbol('C', commutative=True))), Add(Function('M_E')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('C', commutative=True)), Mul(Integer(2), Function('M_E')(Symbol('C', commutative=True))))"], [["add", 3, "Function('\\\\omega')(Symbol('C', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('C', commutative=True))), Add(Mul(Integer(2), Function('M_E')(Symbol('C', commutative=True))), Function('\\\\omega')(Symbol('C', commutative=True))))"], [["divide", 1, "Function('M_E')(Symbol('C', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Integer(-1)), sin(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('C', commutative=True))), Add(Function('M_E')(Symbol('C', commutative=True)), Function('\\\\omega')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["minus", 5, "Mul(Integer(2), Function('\\\\omega')(Symbol('C', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integer(2), Function('\\\\omega')(Symbol('C', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\omega')(Symbol('C', commutative=True))), Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Integer(-1)), sin(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Mul(Integer(-1), Function('M_E')(Symbol('C', commutative=True))), Mul(Integer(-1), Function('\\\\omega')(Symbol('C', commutative=True))), Mul(Integer(-1), sin(Symbol('C', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Function('M_E')(Symbol('C', commutative=True))), Mul(Integer(-1), Function('\\\\omega')(Symbol('C', commutative=True))), Mul(Integer(-1), sin(Symbol('C', commutative=True))), Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Integer(-1)), sin(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{X})} = \\log{(\\sin{(\\hat{X})})}, then obtain \\frac{d^{2}}{d \\hat{X}^{2}} \\operatorname{P_{g}}^{\\hat{X}}{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\sin{(\\hat{X})})}^{\\hat{X}}", "derivation": "\\operatorname{P_{g}}{(\\hat{X})} = \\log{(\\sin{(\\hat{X})})} and \\operatorname{P_{g}}^{\\hat{X}}{(\\hat{X})} = \\log{(\\sin{(\\hat{X})})}^{\\hat{X}} and \\frac{d}{d \\hat{X}} \\operatorname{P_{g}}^{\\hat{X}}{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\log{(\\sin{(\\hat{X})})}^{\\hat{X}} and \\frac{d^{2}}{d \\hat{X}^{2}} \\operatorname{P_{g}}^{\\hat{X}}{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\sin{(\\hat{X})})}^{\\hat{X}}", "srepr_derivation": [["get_premise", "Equality(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), log(sin(Symbol('\\\\hat{X}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Pow(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Pow(log(sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Pow(Function('P_g')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))), Derivative(Pow(log(sin(Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"]]}, {"prompt": "Given u{(\\mu_0)} = e^{\\mu_0}, then obtain \\log{((2 e^{\\mu_0})^{\\mu_0} (u{(\\mu_0)} + e^{\\mu_0})^{\\mu_0})} = \\log{((2 e^{\\mu_0})^{2 \\mu_0})}", "derivation": "u{(\\mu_0)} = e^{\\mu_0} and u{(\\mu_0)} + e^{\\mu_0} = 2 e^{\\mu_0} and (u{(\\mu_0)} + e^{\\mu_0})^{\\mu_0} = (2 e^{\\mu_0})^{\\mu_0} and (2 e^{\\mu_0})^{\\mu_0} (u{(\\mu_0)} + e^{\\mu_0})^{\\mu_0} = (2 e^{\\mu_0})^{2 \\mu_0} and \\log{((2 e^{\\mu_0})^{\\mu_0} (u{(\\mu_0)} + e^{\\mu_0})^{\\mu_0})} = \\log{((2 e^{\\mu_0})^{2 \\mu_0})}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Add(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["times", 3, "Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True))), Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))))"], [["log", 4], "Equality(log(Mul(Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Function('u')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))), log(Pow(Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} = \\hat{\\mathbf{r}} + \\log{(A_{x})}, then obtain - \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} + \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})}) = - \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} + \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\hat{\\mathbf{r}} + \\log{(A_{x})})", "derivation": "\\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} = \\hat{\\mathbf{r}} + \\log{(A_{x})} and A_{x} + \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} = A_{x} + \\hat{\\mathbf{r}} + \\log{(A_{x})} and \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})}) = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\hat{\\mathbf{r}} + \\log{(A_{x})}) and - \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} + \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})}) = - \\mathbf{J}_M{(\\hat{\\mathbf{r}},A_{x})} + \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\hat{\\mathbf{r}} + \\log{(A_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('A_x', commutative=True))))"], [["add", 1, "Symbol('A_x', commutative=True)"], "Equality(Add(Symbol('A_x', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))), Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('A_x', commutative=True))))"], [["differentiate", 2, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["minus", 3, "Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))), Derivative(Add(Symbol('A_x', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_x', commutative=True))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), log(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})}, then derive \\frac{d}{d a^{\\dagger}} \\operatorname{v_{2}}{(a^{\\dagger})} = - \\sin{(a^{\\dagger})}, then obtain (\\int \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = (\\int - \\sin{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}}", "derivation": "\\operatorname{v_{2}}{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\operatorname{v_{2}}{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})} and \\frac{d}{d a^{\\dagger}} \\operatorname{v_{2}}{(a^{\\dagger})} = - \\sin{(a^{\\dagger})} and \\int \\frac{d}{d a^{\\dagger}} \\operatorname{v_{2}}{(a^{\\dagger})} da^{\\dagger} = \\int - \\sin{(a^{\\dagger})} da^{\\dagger} and (\\int \\frac{d}{d a^{\\dagger}} \\operatorname{v_{2}}{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = (\\int - \\sin{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} and (\\int \\frac{d}{d a^{\\dagger}} \\cos{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}} = (\\int - \\sin{(a^{\\dagger})} da^{\\dagger})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Derivative(Function('v_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('v_2')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(Derivative(cos(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given B{(C_{d},r)} = \\cos{(r^{C_{d}})}, then obtain r^{C_{d}} + B{(C_{d},r)} + \\int (r^{C_{d}} + B{(C_{d},r)}) dC_{d} = r^{C_{d}} + B{(C_{d},r)} + \\int (r^{C_{d}} + \\cos{(r^{C_{d}})}) dC_{d}", "derivation": "B{(C_{d},r)} = \\cos{(r^{C_{d}})} and 2 B{(C_{d},r)} = B{(C_{d},r)} + \\cos{(r^{C_{d}})} and r^{C_{d}} + B{(C_{d},r)} = r^{C_{d}} + \\cos{(r^{C_{d}})} and \\int (r^{C_{d}} + B{(C_{d},r)}) dC_{d} = \\int (r^{C_{d}} + \\cos{(r^{C_{d}})}) dC_{d} and r^{C_{d}} + B{(C_{d},r)} + \\int (r^{C_{d}} + B{(C_{d},r)}) dC_{d} = r^{C_{d}} + B{(C_{d},r)} + \\int (r^{C_{d}} + \\cos{(r^{C_{d}})}) dC_{d}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)), cos(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True))))"], [["add", 1, "Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Integer(2), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True))), Add(Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)), cos(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True))), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)))"], "Equality(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True))), Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), cos(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), cos(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"], [["add", 4, "Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)))"], "Equality(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Integral(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('C_d', commutative=True)))), Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), Function('B')(Symbol('C_d', commutative=True), Symbol('r', commutative=True)), Integral(Add(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)), cos(Pow(Symbol('r', commutative=True), Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given \\chi{(i,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} + i), then derive \\int 0 dA_{1} = \\hat{x}_0 + \\int (1 - \\chi{(i,A_{1})}) dA_{1}, then obtain A_{1} (\\hat{x}_0 + \\int (1 - \\chi{(i,A_{1})}) dA_{1}) = A_{1} \\int (- \\chi{(i,A_{1})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + i)) dA_{1}", "derivation": "\\chi{(i,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} + i) and 0 = - \\chi{(i,A_{1})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + i) and \\int 0 dA_{1} = \\int (- \\chi{(i,A_{1})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + i)) dA_{1} and A_{1} \\int 0 dA_{1} = A_{1} \\int (- \\chi{(i,A_{1})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + i)) dA_{1} and \\int 0 dA_{1} = \\hat{x}_0 + \\int (1 - \\chi{(i,A_{1})}) dA_{1} and A_{1} (\\hat{x}_0 + \\int (1 - \\chi{(i,A_{1})}) dA_{1}) = A_{1} \\int (- \\chi{(i,A_{1})} + \\frac{\\partial}{\\partial A_{1}} (A_{1} + i)) dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Tuple(Symbol('A_1', commutative=True))))"], [["times", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Integral(Integer(0), Tuple(Symbol('A_1', commutative=True)))), Mul(Symbol('A_1', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Tuple(Symbol('A_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Integer(0), Tuple(Symbol('A_1', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Integral(Add(Integer(1), Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Symbol('A_1', commutative=True), Add(Symbol('\\\\hat{x}_0', commutative=True), Integral(Add(Integer(1), Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True))))), Mul(Symbol('A_1', commutative=True), Integral(Add(Mul(Integer(-1), Function('\\\\chi')(Symbol('i', commutative=True), Symbol('A_1', commutative=True))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Tuple(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(x)} = e^{\\sin{(x)}}, then derive \\frac{d}{d x} \\varepsilon_{0}{(x)} = e^{\\sin{(x)}} \\cos{(x)}, then obtain \\frac{d}{d x} e^{\\sin{(x)}} = e^{\\sin{(x)}} \\cos{(x)}", "derivation": "\\varepsilon_{0}{(x)} = e^{\\sin{(x)}} and \\frac{d}{d x} \\varepsilon_{0}{(x)} = \\frac{d}{d x} e^{\\sin{(x)}} and \\frac{d}{d x} \\varepsilon_{0}{(x)} = e^{\\sin{(x)}} \\cos{(x)} and \\frac{d}{d x} e^{\\sin{(x)}} = e^{\\sin{(x)}} \\cos{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('x', commutative=True)), exp(sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(exp(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(exp(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\Omega,A_{1})} = A_{1}^{\\Omega}, then obtain \\frac{\\operatorname{v_{2}}^{A_{1}}{(\\Omega,A_{1})}}{\\Omega + \\operatorname{v_{2}}{(\\Omega,A_{1})}} = \\frac{(A_{1}^{\\Omega})^{A_{1}}}{\\Omega + \\operatorname{v_{2}}{(\\Omega,A_{1})}}", "derivation": "\\operatorname{v_{2}}{(\\Omega,A_{1})} = A_{1}^{\\Omega} and \\Omega + \\operatorname{v_{2}}{(\\Omega,A_{1})} = A_{1}^{\\Omega} + \\Omega and \\operatorname{v_{2}}^{A_{1}}{(\\Omega,A_{1})} = (A_{1}^{\\Omega})^{A_{1}} and \\frac{\\operatorname{v_{2}}^{A_{1}}{(\\Omega,A_{1})}}{A_{1}^{\\Omega} + \\Omega} = \\frac{(A_{1}^{\\Omega})^{A_{1}}}{A_{1}^{\\Omega} + \\Omega} and \\frac{\\operatorname{v_{2}}^{A_{1}}{(\\Omega,A_{1})}}{\\Omega + \\operatorname{v_{2}}{(\\Omega,A_{1})}} = \\frac{(A_{1}^{\\Omega})^{A_{1}}}{\\Omega + \\operatorname{v_{2}}{(\\Omega,A_{1})}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))), Add(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('A_1', commutative=True)))"], [["divide", 3, "Add(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Add(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Pow(Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Pow(Add(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))), Integer(-1)), Pow(Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), Function('v_2')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))), Integer(-1)), Pow(Pow(Symbol('A_1', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(M,A_{2})} = A_{2} - M, then obtain 1 + (\\hat{p}_0{(M,A_{2})} + 1)^{- M} = (\\hat{p}_0{(M,A_{2})} + 1)^{- M} + (\\hat{p}_0{(M,A_{2})} + 1)^{- 2 M} (A_{2} - M + 1)^{2 M}", "derivation": "\\hat{p}_0{(M,A_{2})} = A_{2} - M and \\hat{p}_0{(M,A_{2})} + 1 = A_{2} - M + 1 and (\\hat{p}_0{(M,A_{2})} + 1)^{M} = (A_{2} - M + 1)^{M} and 1 = (\\hat{p}_0{(M,A_{2})} + 1)^{- M} (A_{2} - M + 1)^{M} and 1 + (\\hat{p}_0{(M,A_{2})} + 1)^{- M} = (\\hat{p}_0{(M,A_{2})} + 1)^{- M} (A_{2} - M + 1)^{M} + (\\hat{p}_0{(M,A_{2})} + 1)^{- M} and (A_{2} - M + 1)^{M} = (\\hat{p}_0{(M,A_{2})} + 1)^{- M} (A_{2} - M + 1)^{2 M} and 1 + (\\hat{p}_0{(M,A_{2})} + 1)^{- M} = (\\hat{p}_0{(M,A_{2})} + 1)^{- M} + (\\hat{p}_0{(M,A_{2})} + 1)^{- 2 M} (A_{2} - M + 1)^{2 M}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Symbol('M', commutative=True)))"], [["divide", 3, "Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Symbol('M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Symbol('M', commutative=True))))"], [["add", 4, "Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True)))"], "Equality(Add(Integer(1), Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True)))), Add(Mul(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Symbol('M', commutative=True))), Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True)))))"], [["times", 4, "Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Symbol('M', commutative=True))"], "Equality(Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Symbol('M', commutative=True)), Mul(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Mul(Integer(2), Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Integer(1), Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True)))), Add(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Symbol('M', commutative=True))), Mul(Pow(Add(Function('\\\\hat{p}_0')(Symbol('M', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Mul(Integer(-1), Integer(2), Symbol('M', commutative=True))), Pow(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True)), Integer(1)), Mul(Integer(2), Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(n)} = \\sin{(n)}, then obtain \\frac{d}{d n} (\\log{(\\sin{(n)})} + \\frac{d}{d n} \\tilde{g}^*{(n)}) = \\frac{d}{d n} (\\log{(\\sin{(n)})} + \\frac{d}{d n} \\sin{(n)})", "derivation": "\\tilde{g}^*{(n)} = \\sin{(n)} and \\frac{d}{d n} \\tilde{g}^*{(n)} = \\frac{d}{d n} \\sin{(n)} and \\log{(\\tilde{g}^*{(n)})} = \\log{(\\sin{(n)})} and \\log{(\\tilde{g}^*{(n)})} + \\frac{d}{d n} \\tilde{g}^*{(n)} = \\log{(\\tilde{g}^*{(n)})} + \\frac{d}{d n} \\sin{(n)} and \\log{(\\sin{(n)})} + \\frac{d}{d n} \\tilde{g}^*{(n)} = \\log{(\\sin{(n)})} + \\frac{d}{d n} \\sin{(n)} and \\frac{d}{d n} (\\log{(\\sin{(n)})} + \\frac{d}{d n} \\tilde{g}^*{(n)}) = \\frac{d}{d n} (\\log{(\\sin{(n)})} + \\frac{d}{d n} \\sin{(n)})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)), sin(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True))), log(sin(Symbol('n', commutative=True))))"], [["add", 2, "log(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)))"], "Equality(Add(log(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(log(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(log(sin(Symbol('n', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(log(sin(Symbol('n', commutative=True))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(log(sin(Symbol('n', commutative=True))), Derivative(Function('\\\\tilde{g}^*')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(log(sin(Symbol('n', commutative=True))), Derivative(sin(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(A_{z},\\dot{y})} = \\dot{y}^{A_{z}}, then derive \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{z}}{(A_{z},\\dot{y})} = \\frac{A_{z} \\dot{y}^{A_{z}}}{\\dot{y}}, then obtain - A_{z} + \\frac{\\partial}{\\partial \\dot{y}} \\dot{y}^{A_{z}} = - A_{z} + \\frac{A_{z} \\dot{y}^{A_{z}}}{\\dot{y}}", "derivation": "\\operatorname{v_{z}}{(A_{z},\\dot{y})} = \\dot{y}^{A_{z}} and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{z}}{(A_{z},\\dot{y})} = \\frac{\\partial}{\\partial \\dot{y}} \\dot{y}^{A_{z}} and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{v_{z}}{(A_{z},\\dot{y})} = \\frac{A_{z} \\dot{y}^{A_{z}}}{\\dot{y}} and \\frac{\\partial}{\\partial \\dot{y}} \\dot{y}^{A_{z}} = \\frac{A_{z} \\dot{y}^{A_{z}}}{\\dot{y}} and - A_{z} + \\frac{\\partial}{\\partial \\dot{y}} \\dot{y}^{A_{z}} = - A_{z} + \\frac{A_{z} \\dot{y}^{A_{z}}}{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('A_z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('A_z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('A_z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Derivative(Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{y}', commutative=True), Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\dot{x},\\mathbf{S})} = \\dot{x}^{\\mathbf{S}} and \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})} = \\dot{x}^{\\mathbf{S}} + \\mathbf{P}{(\\dot{x},\\mathbf{S})}, then obtain (- \\mathbf{S} + \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})})^{\\mathbf{S}} = (2 \\dot{x}^{\\mathbf{S}} - \\mathbf{S})^{\\mathbf{S}}", "derivation": "\\mathbf{P}{(\\dot{x},\\mathbf{S})} = \\dot{x}^{\\mathbf{S}} and \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})} = \\dot{x}^{\\mathbf{S}} + \\mathbf{P}{(\\dot{x},\\mathbf{S})} and \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})} = 2 \\mathbf{P}{(\\dot{x},\\mathbf{S})} and \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})} = 2 \\dot{x}^{\\mathbf{S}} and - \\mathbf{S} + \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})} = 2 \\dot{x}^{\\mathbf{S}} - \\mathbf{S} and (- \\mathbf{S} + \\operatorname{z^{*}}{(\\dot{x},\\mathbf{S})})^{\\mathbf{S}} = (2 \\dot{x}^{\\mathbf{S}} - \\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('z^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('z^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('z^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Function('z^*')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(2), Pow(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(n_{1})} = \\cos{(n_{1})}, then obtain - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\int \\operatorname{V_{\\mathbf{B}}}{(n_{1})} dn_{1} = \\nabla - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\sin{(n_{1})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(n_{1})} = \\cos{(n_{1})} and \\int \\operatorname{V_{\\mathbf{B}}}{(n_{1})} dn_{1} = \\int \\cos{(n_{1})} dn_{1} and - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\int \\operatorname{V_{\\mathbf{B}}}{(n_{1})} dn_{1} = - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\int \\cos{(n_{1})} dn_{1} and - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\int \\operatorname{V_{\\mathbf{B}}}{(n_{1})} dn_{1} = \\nabla - \\operatorname{V_{\\mathbf{B}}}^{n_{1}}{(n_{1})} + \\sin{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(cos(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["minus", 2, "Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Integral(cos(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), Add(Symbol('\\\\nabla', commutative=True), Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), sin(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(t_{1},\\sigma_x)} = \\frac{\\sigma_x}{t_{1}}, then obtain t_{1}^{2} \\operatorname{g_{\\varepsilon}}^{4}{(t_{1},\\sigma_x)} = \\sigma_x^{2} \\operatorname{g_{\\varepsilon}}^{2}{(t_{1},\\sigma_x)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(t_{1},\\sigma_x)} = \\frac{\\sigma_x}{t_{1}} and \\frac{\\operatorname{g_{\\varepsilon}}{(t_{1},\\sigma_x)}}{t_{1}} = \\frac{\\sigma_x}{t_{1}^{2}} and \\operatorname{g_{\\varepsilon}}^{2}{(t_{1},\\sigma_x)} = \\frac{\\sigma_x \\operatorname{g_{\\varepsilon}}{(t_{1},\\sigma_x)}}{t_{1}} and \\operatorname{g_{\\varepsilon}}^{2}{(t_{1},\\sigma_x)} = \\frac{\\sigma_x^{2}}{t_{1}^{2}} and t_{1} \\operatorname{g_{\\varepsilon}}^{2}{(t_{1},\\sigma_x)} = \\frac{\\sigma_x^{2}}{t_{1}} and t_{1}^{2} \\operatorname{g_{\\varepsilon}}^{4}{(t_{1},\\sigma_x)} = \\sigma_x^{2} \\operatorname{g_{\\varepsilon}}^{2}{(t_{1},\\sigma_x)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('t_1', commutative=True)"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-2))))"], [["times", 1, "Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Symbol('\\\\sigma_x', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('t_1', commutative=True), Integer(-2))))"], [["times", 4, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["times", 5, "Mul(Symbol('t_1', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(2)), Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(4))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(2)), Pow(Function('g_{\\\\varepsilon}')(Symbol('t_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\Omega{(\\lambda)} = \\sin{(\\lambda)} and c{(\\lambda)} = \\int \\Omega{(\\lambda)} d\\lambda, then derive \\int \\Omega{(\\lambda)} d\\lambda = \\mu_0 - \\cos{(\\lambda)}, then obtain 2 c{(\\lambda)} = \\mu_0 + c{(\\lambda)} - \\cos{(\\lambda)}", "derivation": "\\Omega{(\\lambda)} = \\sin{(\\lambda)} and \\int \\Omega{(\\lambda)} d\\lambda = \\int \\sin{(\\lambda)} d\\lambda and \\int \\Omega{(\\lambda)} d\\lambda = \\mu_0 - \\cos{(\\lambda)} and c{(\\lambda)} = \\int \\Omega{(\\lambda)} d\\lambda and c{(\\lambda)} = \\int \\sin{(\\lambda)} d\\lambda and c{(\\lambda)} = \\mu_0 - \\cos{(\\lambda)} and c{(\\lambda)} + \\int \\sin{(\\lambda)} d\\lambda = \\mu_0 - \\cos{(\\lambda)} + \\int \\sin{(\\lambda)} d\\lambda and 2 c{(\\lambda)} = \\mu_0 + c{(\\lambda)} - \\cos{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\lambda', commutative=True)), Integral(Function('\\\\Omega')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('c')(Symbol('\\\\lambda', commutative=True)), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('c')(Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))))"], [["add", 6, "Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('c')(Symbol('\\\\lambda', commutative=True)), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Integer(2), Function('c')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Function('c')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\operatorname{A_{y}}{(\\mathbf{H})} + 1 = 1 + \\frac{1}{\\mathbf{H}}, then obtain 1 = \\frac{\\frac{d}{d \\mathbf{H}} (1 + \\frac{1}{\\mathbf{H}})}{\\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\operatorname{A_{y}}{(\\mathbf{H})} + 1)}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{H})} = \\log{(\\mathbf{H})} and \\mathbf{H} + \\operatorname{A_{y}}{(\\mathbf{H})} = \\mathbf{H} + \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} (\\mathbf{H} + \\operatorname{A_{y}}{(\\mathbf{H})}) = \\frac{d}{d \\mathbf{H}} (\\mathbf{H} + \\log{(\\mathbf{H})}) and \\frac{d}{d \\mathbf{H}} \\operatorname{A_{y}}{(\\mathbf{H})} + 1 = 1 + \\frac{1}{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\operatorname{A_{y}}{(\\mathbf{H})} + 1) = \\frac{d}{d \\mathbf{H}} (1 + \\frac{1}{\\mathbf{H}}) and 1 = \\frac{\\frac{d}{d \\mathbf{H}} (1 + \\frac{1}{\\mathbf{H}})}{\\frac{d}{d \\mathbf{H}} (\\frac{d}{d \\mathbf{H}} \\operatorname{A_{y}}{(\\mathbf{H})} + 1)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{H}', commutative=True), log(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Add(Derivative(Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Add(Integer(1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Derivative(Add(Derivative(Function('A_y')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{f}{(h,\\hat{x})} = \\int \\frac{\\hat{x}}{h} d\\hat{x}, then obtain \\frac{\\frac{\\partial}{\\partial h} \\mathbf{f}{(h,\\hat{x})}}{G{(n_{1})}} = \\frac{\\partial}{\\partial h} \\frac{\\int \\frac{\\hat{x}}{h} d\\hat{x}}{G{(n_{1})}}", "derivation": "\\mathbf{f}{(h,\\hat{x})} = \\int \\frac{\\hat{x}}{h} d\\hat{x} and \\frac{\\mathbf{f}{(h,\\hat{x})}}{G{(n_{1})}} = \\frac{\\int \\frac{\\hat{x}}{h} d\\hat{x}}{G{(n_{1})}} and \\frac{\\partial}{\\partial h} \\frac{\\mathbf{f}{(h,\\hat{x})}}{G{(n_{1})}} = \\frac{\\partial}{\\partial h} \\frac{\\int \\frac{\\hat{x}}{h} d\\hat{x}}{G{(n_{1})}} and \\frac{\\frac{\\partial}{\\partial h} \\mathbf{f}{(h,\\hat{x})}}{G{(n_{1})}} = \\frac{\\partial}{\\partial h} \\frac{\\int \\frac{\\hat{x}}{h} d\\hat{x}}{G{(n_{1})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 1, "Function('G')(Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{f}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Derivative(Mul(Pow(Function('G')(Symbol('n_1', commutative=True)), Integer(-1)), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{s},Q)} = Q + \\mathbf{s} and \\phi_{2}{(S,F_{c})} = F_{c} S, then obtain Q + \\mathbf{s} + \\phi_{2}{(S,F_{c})} + \\int (Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)}) dQ = F_{c} S + Q + \\mathbf{s} + \\int (Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)}) dQ", "derivation": "\\theta_{1}{(\\mathbf{s},Q)} = Q + \\mathbf{s} and 0 = Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)} and \\int 0 dQ = \\int (Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)}) dQ and \\phi_{2}{(S,F_{c})} = F_{c} S and Q + \\mathbf{s} + \\phi_{2}{(S,F_{c})} = F_{c} S + Q + \\mathbf{s} and Q + \\mathbf{s} + \\phi_{2}{(S,F_{c})} + \\int 0 dQ = F_{c} S + Q + \\mathbf{s} + \\int 0 dQ and Q + \\mathbf{s} + \\phi_{2}{(S,F_{c})} + \\int (Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)}) dQ = F_{c} S + Q + \\mathbf{s} + \\int (Q + \\mathbf{s} - \\theta_{1}{(\\mathbf{s},Q)}) dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True)))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], ["get_premise", "Equality(Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('S', commutative=True)))"], [["add", 4, "Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 5, "Integral(Integer(0), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Symbol('F_c', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Integral(Integer(0), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\phi_2')(Symbol('S', commutative=True), Symbol('F_c', commutative=True)), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))), Add(Mul(Symbol('F_c', commutative=True), Symbol('S', commutative=True)), Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Integral(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} = \\rho_f^{\\tilde{g}}, then obtain \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} \\int \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} d\\rho_f = \\rho_f^{\\tilde{g}} \\int \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} d\\rho_f", "derivation": "\\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} = \\rho_f^{\\tilde{g}} and \\int \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} d\\rho_f = \\int \\rho_f^{\\tilde{g}} d\\rho_f and \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} \\int \\rho_f^{\\tilde{g}} d\\rho_f = \\rho_f^{\\tilde{g}} \\int \\rho_f^{\\tilde{g}} d\\rho_f and \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} \\int \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} d\\rho_f = \\rho_f^{\\tilde{g}} \\int \\operatorname{f^{\\prime}}{(\\rho_f,\\tilde{g})} d\\rho_f", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "Integral(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\operatorname{F_{c}}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain - \\frac{- \\operatorname{F_{c}}{(\\mathbf{p})} + \\varphi^{*}{(\\mathbf{p})}}{\\operatorname{F_{c}}{(\\mathbf{p})} W{(\\mathbf{p})}} = 0", "derivation": "\\varphi^{*}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\varphi^{*}{(\\mathbf{p})} - e^{\\mathbf{p}} = 0 and \\operatorname{F_{c}}{(\\mathbf{p})} = e^{\\mathbf{p}} and - \\operatorname{F_{c}}{(\\mathbf{p})} + \\varphi^{*}{(\\mathbf{p})} = 0 and \\frac{- \\operatorname{F_{c}}{(\\mathbf{p})} + \\varphi^{*}{(\\mathbf{p})}}{W{(\\mathbf{p})}} = 0 and - \\frac{- \\operatorname{F_{c}}{(\\mathbf{p})} + \\varphi^{*}{(\\mathbf{p})}}{\\operatorname{F_{c}}{(\\mathbf{p})} W{(\\mathbf{p})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{p}', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True))), Integer(0))"], [["divide", 4, "Function('W')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True))), Pow(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 5, "Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\mathbf{p}', commutative=True))), Pow(Function('F_c')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Pow(Function('W')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\mu_{0}{(\\delta,E_{\\lambda},q)} = q (E_{\\lambda} + \\delta) and \\operatorname{F_{c}}{(\\delta,E_{\\lambda},q)} = \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)}, then obtain \\frac{\\partial}{\\partial q} \\operatorname{F_{c}}{(\\delta,E_{\\lambda},q)} = \\frac{\\partial}{\\partial q} \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)}", "derivation": "\\mu_{0}{(\\delta,E_{\\lambda},q)} = q (E_{\\lambda} + \\delta) and \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)} = (q (E_{\\lambda} + \\delta))^{q} and \\frac{\\partial}{\\partial q} \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)} = \\frac{\\partial}{\\partial q} (q (E_{\\lambda} + \\delta))^{q} and \\operatorname{F_{c}}{(\\delta,E_{\\lambda},q)} = \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)} and \\frac{\\partial}{\\partial q} \\operatorname{F_{c}}{(\\delta,E_{\\lambda},q)} = \\frac{\\partial}{\\partial q} (q (E_{\\lambda} + \\delta))^{q} and \\frac{\\partial}{\\partial q} \\operatorname{F_{c}}{(\\delta,E_{\\lambda},q)} = \\frac{\\partial}{\\partial q} \\mu_{0}^{q}{(\\delta,E_{\\lambda},q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Mul(Symbol('q', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('q', commutative=True)))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('q', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('q', commutative=True), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Function('F_c')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mu_0')(Symbol('\\\\delta', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given Q{(m,Z,A_{1})} = \\frac{A_{1}}{Z m}, then obtain \\frac{A_{1} (\\frac{A_{1}}{Z m})^{A_{1}} Q{(m,Z,A_{1})}}{Z m} = \\frac{A_{1}^{2} (\\frac{A_{1}}{Z m})^{A_{1}}}{Z^{2} m^{2}}", "derivation": "Q{(m,Z,A_{1})} = \\frac{A_{1}}{Z m} and Q^{A_{1}}{(m,Z,A_{1})} = (\\frac{A_{1}}{Z m})^{A_{1}} and (\\frac{A_{1}}{Z m})^{A_{1}} Q{(m,Z,A_{1})} = \\frac{A_{1} (\\frac{A_{1}}{Z m})^{A_{1}}}{Z m} and Q{(m,Z,A_{1})} Q^{A_{1}}{(m,Z,A_{1})} = \\frac{A_{1} Q^{A_{1}}{(m,Z,A_{1})}}{Z m} and \\frac{A_{1} Q{(m,Z,A_{1})} Q^{A_{1}}{(m,Z,A_{1})}}{Z m} = \\frac{A_{1}^{2} Q^{A_{1}}{(m,Z,A_{1})}}{Z^{2} m^{2}} and \\frac{A_{1} (\\frac{A_{1}}{Z m})^{A_{1}} Q{(m,Z,A_{1})}}{Z m} = \\frac{A_{1}^{2} (\\frac{A_{1}}{Z m})^{A_{1}}}{Z^{2} m^{2}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True)))"], [["times", 1, "Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True)), Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"], [["times", 4, "Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('Z', commutative=True), Integer(-2)), Pow(Symbol('m', commutative=True), Integer(-2)), Pow(Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True)), Function('Q')(Symbol('m', commutative=True), Symbol('Z', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Symbol('Z', commutative=True), Integer(-2)), Pow(Symbol('m', commutative=True), Integer(-2)), Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given E{(\\theta_2,J_{\\varepsilon})} = - J_{\\varepsilon} + \\theta_2, then obtain \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} E{(\\theta_2,J_{\\varepsilon})} d\\theta_2 + 1 = \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} (- J_{\\varepsilon} + \\theta_2) d\\theta_2 + 1", "derivation": "E{(\\theta_2,J_{\\varepsilon})} = - J_{\\varepsilon} + \\theta_2 and J_{\\varepsilon} E{(\\theta_2,J_{\\varepsilon})} = J_{\\varepsilon} (- J_{\\varepsilon} + \\theta_2) and \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} E{(\\theta_2,J_{\\varepsilon})} = \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} (- J_{\\varepsilon} + \\theta_2) and \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} E{(\\theta_2,J_{\\varepsilon})} d\\theta_2 = \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} (- J_{\\varepsilon} + \\theta_2) d\\theta_2 and \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} E{(\\theta_2,J_{\\varepsilon})} d\\theta_2 + 1 = \\int \\frac{\\partial}{\\partial J_{\\varepsilon}} J_{\\varepsilon} (- J_{\\varepsilon} + \\theta_2) d\\theta_2 + 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["add", 4, 1], "Equality(Add(Integral(Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('E')(Symbol('\\\\theta_2', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)), Add(Integral(Derivative(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{B}{(\\varepsilon,B)} = B + \\varepsilon and m{(\\varepsilon,B)} = B + \\varepsilon and \\Psi_{nl}{(B,\\varepsilon)} = \\mathbf{B}^{\\varepsilon}{(\\varepsilon,B)}, then obtain \\Psi_{nl}{(B,\\varepsilon)} = m^{\\varepsilon}{(\\varepsilon,B)}", "derivation": "\\mathbf{B}{(\\varepsilon,B)} = B + \\varepsilon and \\mathbf{B}^{\\varepsilon}{(\\varepsilon,B)} = (B + \\varepsilon)^{\\varepsilon} and m{(\\varepsilon,B)} = B + \\varepsilon and \\Psi_{nl}{(B,\\varepsilon)} = \\mathbf{B}^{\\varepsilon}{(\\varepsilon,B)} and \\Psi_{nl}{(B,\\varepsilon)} = (B + \\varepsilon)^{\\varepsilon} and \\Psi_{nl}{(B,\\varepsilon)} = m^{\\varepsilon}{(\\varepsilon,B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\varepsilon', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\varepsilon', commutative=True), Symbol('B', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{B}')(Symbol('\\\\varepsilon', commutative=True), Symbol('B', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('m')(Symbol('\\\\varepsilon', commutative=True), Symbol('B', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\hat{x})} = \\cos{(\\hat{x})}, then obtain \\int \\frac{d}{d \\hat{x}} (\\operatorname{g_{\\varepsilon}}{(\\hat{x})} - \\cos{(\\hat{x})}) d\\hat{x} = \\int \\frac{d}{d \\hat{x}} 0 d\\hat{x}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\hat{x})} = \\cos{(\\hat{x})} and \\operatorname{g_{\\varepsilon}}{(\\hat{x})} - \\cos{(\\hat{x})} = 0 and \\frac{d}{d \\hat{x}} (\\operatorname{g_{\\varepsilon}}{(\\hat{x})} - \\cos{(\\hat{x})}) = \\frac{d}{d \\hat{x}} 0 and \\int \\frac{d}{d \\hat{x}} (\\operatorname{g_{\\varepsilon}}{(\\hat{x})} - \\cos{(\\hat{x})}) d\\hat{x} = \\int \\frac{d}{d \\hat{x}} 0 d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Derivative(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\Omega)} = e^{\\Omega}, then derive \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} = e^{\\Omega}, then obtain - e^{\\Omega} + \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} = 0", "derivation": "\\operatorname{v_{2}}{(\\Omega)} = e^{\\Omega} and \\operatorname{v_{2}}{(\\Omega)} - e^{\\Omega} = 0 and \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\Omega} and \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} = e^{\\Omega} and \\operatorname{v_{2}}{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} and - e^{\\Omega} + \\frac{d}{d \\Omega} \\operatorname{v_{2}}{(\\Omega)} = 0", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))), Integer(0))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), exp(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Derivative(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Derivative(Function('v_2')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}^*{(Q)} = \\log{(Q)}, then obtain (\\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\tilde{g}^*{(Q)}))^{Q})^{Q} = (\\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\log{(Q)}))^{Q})^{Q}", "derivation": "\\tilde{g}^*{(Q)} = \\log{(Q)} and - Q + \\tilde{g}^*{(Q)} = - Q + \\log{(Q)} and \\frac{d}{d Q} (- Q + \\tilde{g}^*{(Q)}) = \\frac{d}{d Q} (- Q + \\log{(Q)}) and (\\frac{d}{d Q} (- Q + \\tilde{g}^*{(Q)}))^{Q} = (\\frac{d}{d Q} (- Q + \\log{(Q)}))^{Q} and \\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\tilde{g}^*{(Q)}))^{Q} = \\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\log{(Q)}))^{Q} and (\\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\tilde{g}^*{(Q)}))^{Q})^{Q} = (\\frac{d}{d Q} (\\frac{d}{d Q} (- Q + \\log{(Q)}))^{Q})^{Q}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True)))"], [["minus", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 3, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 5, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), log(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\dot{z},v_{1})} = - v_{1} + \\sin{(\\dot{z})}, then derive \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f^{*}}{(\\dot{z},v_{1})} = \\cos{(\\dot{z})}, then obtain 0 = \\cos{(\\dot{z})} - \\frac{\\partial}{\\partial \\dot{z}} (- v_{1} + \\sin{(\\dot{z})})", "derivation": "\\operatorname{f^{*}}{(\\dot{z},v_{1})} = - v_{1} + \\sin{(\\dot{z})} and \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f^{*}}{(\\dot{z},v_{1})} = \\frac{\\partial}{\\partial \\dot{z}} (- v_{1} + \\sin{(\\dot{z})}) and \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f^{*}}{(\\dot{z},v_{1})} = \\cos{(\\dot{z})} and - \\frac{\\partial}{\\partial \\dot{z}} (- v_{1} + \\sin{(\\dot{z})}) + \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f^{*}}{(\\dot{z},v_{1})} = \\cos{(\\dot{z})} - \\frac{\\partial}{\\partial \\dot{z}} (- v_{1} + \\sin{(\\dot{z})}) and 0 = \\cos{(\\dot{z})} - \\frac{\\partial}{\\partial \\dot{z}} \\operatorname{f^{*}}{(\\dot{z},v_{1})} and 0 = \\cos{(\\dot{z})} - \\frac{\\partial}{\\partial \\dot{z}} (- v_{1} + \\sin{(\\dot{z})})", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["minus", 3, "Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Derivative(Function('f^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Add(cos(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(cos(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Derivative(Function('f^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(cos(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(v,u)} = u + v, then obtain \\int (- u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int \\operatorname{c_{0}}{(v,u)} du dv) dv = \\int (- u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int (u + v) du dv) dv", "derivation": "\\operatorname{c_{0}}{(v,u)} = u + v and \\int \\operatorname{c_{0}}{(v,u)} du = \\int (u + v) du and \\operatorname{c_{0}}{(v,u)} \\int \\operatorname{c_{0}}{(v,u)} du = \\operatorname{c_{0}}{(v,u)} \\int (u + v) du and \\int \\operatorname{c_{0}}{(v,u)} \\int \\operatorname{c_{0}}{(v,u)} du dv = \\int \\operatorname{c_{0}}{(v,u)} \\int (u + v) du dv and - u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int \\operatorname{c_{0}}{(v,u)} du dv = - u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int (u + v) du dv and \\int (- u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int \\operatorname{c_{0}}{(v,u)} du dv) dv = \\int (- u \\operatorname{c_{0}}{(v,u)} + \\int \\operatorname{c_{0}}{(v,u)} \\int (u + v) du dv) dv", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('u', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["times", 2, "Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('u', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True))), Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('u', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["minus", 4, "Mul(Symbol('u', commutative=True), Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True))), Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('u', commutative=True), Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True))), Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('u', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True)))))"], [["integrate", 5, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('u', commutative=True), Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True))), Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('u', commutative=True), Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True))), Integral(Mul(Function('c_0')(Symbol('v', commutative=True), Symbol('u', commutative=True)), Integral(Add(Symbol('u', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('u', commutative=True)))), Tuple(Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mu_0)} = \\log{(\\sin{(\\mu_0)})}, then derive \\cos{(\\hat{x}_0{(\\mu_0)})} \\frac{d}{d \\mu_0} \\hat{x}_0{(\\mu_0)} + 1 = 1 + \\frac{\\cos{(\\mu_0)} \\cos{(\\log{(\\sin{(\\mu_0)})})}}{\\sin{(\\mu_0)}}, then obtain \\cos{(\\log{(\\sin{(\\mu_0)})})} \\frac{d}{d \\mu_0} \\log{(\\sin{(\\mu_0)})} + 1 = 1 + \\frac{\\cos{(\\mu_0)} \\cos{(\\log{(\\sin{(\\mu_0)})})}}{\\sin{(\\mu_0)}}", "derivation": "\\hat{x}_0{(\\mu_0)} = \\log{(\\sin{(\\mu_0)})} and \\sin{(\\hat{x}_0{(\\mu_0)})} = \\sin{(\\log{(\\sin{(\\mu_0)})})} and \\mu_0 + \\sin{(\\hat{x}_0{(\\mu_0)})} = \\mu_0 + \\sin{(\\log{(\\sin{(\\mu_0)})})} and \\frac{d}{d \\mu_0} (\\mu_0 + \\sin{(\\hat{x}_0{(\\mu_0)})}) = \\frac{d}{d \\mu_0} (\\mu_0 + \\sin{(\\log{(\\sin{(\\mu_0)})})}) and \\cos{(\\hat{x}_0{(\\mu_0)})} \\frac{d}{d \\mu_0} \\hat{x}_0{(\\mu_0)} + 1 = 1 + \\frac{\\cos{(\\mu_0)} \\cos{(\\log{(\\sin{(\\mu_0)})})}}{\\sin{(\\mu_0)}} and \\cos{(\\log{(\\sin{(\\mu_0)})})} \\frac{d}{d \\mu_0} \\log{(\\sin{(\\mu_0)})} + 1 = 1 + \\frac{\\cos{(\\mu_0)} \\cos{(\\log{(\\sin{(\\mu_0)})})}}{\\sin{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), log(sin(Symbol('\\\\mu_0', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True))), sin(log(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["add", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), sin(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), sin(log(sin(Symbol('\\\\mu_0', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mu_0', commutative=True), sin(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu_0', commutative=True), sin(log(sin(Symbol('\\\\mu_0', commutative=True))))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(cos(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True))), Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Integer(1)), Add(Integer(1), Mul(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True)), cos(log(sin(Symbol('\\\\mu_0', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(cos(log(sin(Symbol('\\\\mu_0', commutative=True)))), Derivative(log(sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Integer(1)), Add(Integer(1), Mul(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True)), cos(log(sin(Symbol('\\\\mu_0', commutative=True)))))))"]]}, {"prompt": "Given \\ddot{x}{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}, then obtain \\int e^{\\frac{\\ddot{x}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} d\\Psi_{\\lambda} = \\int e^{\\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} d\\Psi_{\\lambda}", "derivation": "\\ddot{x}{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})} and \\frac{\\ddot{x}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} = \\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}} and e^{\\frac{\\ddot{x}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} = e^{\\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} and \\int e^{\\frac{\\ddot{x}{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} d\\Psi_{\\lambda} = \\int e^{\\frac{\\frac{d}{d \\Psi_{\\lambda}} \\cos{(\\Psi_{\\lambda})}}{\\Psi_{\\lambda}}} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1)))))"], [["exp", 2], "Equality(exp(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), exp(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))))"], [["integrate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(exp(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(exp(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(A_{z})} = e^{A_{z}}, then obtain \\int (\\frac{1}{\\varepsilon{(A_{z})}} + \\frac{\\varepsilon{(A_{z})}}{A_{z}}) dA_{z} = \\int (\\frac{1}{\\varepsilon{(A_{z})}} + \\frac{e^{A_{z}}}{A_{z}}) dA_{z}", "derivation": "\\varepsilon{(A_{z})} = e^{A_{z}} and \\frac{\\varepsilon{(A_{z})}}{A_{z}} = \\frac{e^{A_{z}}}{A_{z}} and \\frac{1}{\\varepsilon{(A_{z})}} + \\frac{\\varepsilon{(A_{z})}}{A_{z}} = \\frac{1}{\\varepsilon{(A_{z})}} + \\frac{e^{A_{z}}}{A_{z}} and \\int (\\frac{1}{\\varepsilon{(A_{z})}} + \\frac{\\varepsilon{(A_{z})}}{A_{z}}) dA_{z} = \\int (\\frac{1}{\\varepsilon{(A_{z})}} + \\frac{e^{A_{z}}}{A_{z}}) dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), exp(Symbol('A_z', commutative=True)))"], [["divide", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), exp(Symbol('A_z', commutative=True))))"], [["add", 2, "Pow(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('A_z', commutative=True)))), Add(Pow(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), exp(Symbol('A_z', commutative=True)))))"], [["integrate", 3, "Symbol('A_z', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Pow(Function('\\\\varepsilon')(Symbol('A_z', commutative=True)), Integer(-1)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), exp(Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(b,Q)} = Q - b, then obtain \\iint (Q - b + \\log{(\\cos{(\\operatorname{r_{0}}{(b,Q)})})}) db dQ = \\iint (Q - b + \\log{(\\cos{(Q - b)})}) db dQ", "derivation": "\\operatorname{r_{0}}{(b,Q)} = Q - b and \\cos{(\\operatorname{r_{0}}{(b,Q)})} = \\cos{(Q - b)} and \\log{(\\cos{(\\operatorname{r_{0}}{(b,Q)})})} = \\log{(\\cos{(Q - b)})} and Q - b + \\log{(\\cos{(\\operatorname{r_{0}}{(b,Q)})})} = Q - b + \\log{(\\cos{(Q - b)})} and \\int (Q - b + \\log{(\\cos{(\\operatorname{r_{0}}{(b,Q)})})}) db = \\int (Q - b + \\log{(\\cos{(Q - b)})}) db and \\iint (Q - b + \\log{(\\cos{(\\operatorname{r_{0}}{(b,Q)})})}) db dQ = \\iint (Q - b + \\log{(\\cos{(Q - b)})}) db dQ", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["cos", 1], "Equality(cos(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), cos(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["log", 2], "Equality(log(cos(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True)))), log(cos(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))))"], [["add", 3, "Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))"], "Equality(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True))))), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True))))), Tuple(Symbol('b', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))), Tuple(Symbol('b', commutative=True))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Function('r_0')(Symbol('b', commutative=True), Symbol('Q', commutative=True))))), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)), log(cos(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))), Tuple(Symbol('b', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given V{(\\psi)} = \\sin{(\\psi)}, then obtain \\frac{(\\frac{V{(\\psi)}}{\\sin{(\\psi)}} - \\sin{(\\psi)}) \\sin{(\\psi)}}{V{(\\psi)}} = \\frac{(1 - \\sin{(\\psi)}) \\sin{(\\psi)}}{V{(\\psi)}}", "derivation": "V{(\\psi)} = \\sin{(\\psi)} and - V{(\\psi)} = - \\sin{(\\psi)} and \\frac{V{(\\psi)}}{\\sin{(\\psi)}} = 1 and - V{(\\psi)} + \\frac{V{(\\psi)}}{\\sin{(\\psi)}} = 1 - V{(\\psi)} and \\frac{V{(\\psi)}}{\\sin{(\\psi)}} - \\sin{(\\psi)} = 1 - \\sin{(\\psi)} and \\frac{(\\frac{V{(\\psi)}}{\\sin{(\\psi)}} - \\sin{(\\psi)}) \\sin{(\\psi)}}{V{(\\psi)}} = \\frac{(1 - \\sin{(\\psi)}) \\sin{(\\psi)}}{V{(\\psi)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\psi', commutative=True)), sin(Symbol('\\\\psi', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('V')(Symbol('\\\\psi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Function('V')(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Integer(-1))), Integer(1))"], [["add", 3, "Mul(Integer(-1), Function('V')(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('\\\\psi', commutative=True))), Mul(Function('V')(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('V')(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Function('V')(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True)))), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True)))))"], [["divide", 5, "Mul(Function('V')(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Integer(-1)))"], "Equality(Mul(Add(Mul(Function('V')(Symbol('\\\\psi', commutative=True)), Pow(sin(Symbol('\\\\psi', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True)))), Pow(Function('V')(Symbol('\\\\psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\psi', commutative=True))), Mul(Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True)))), Pow(Function('V')(Symbol('\\\\psi', commutative=True)), Integer(-1)), sin(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\lambda{(P_{e},C,u)} = (C u)^{P_{e}}, then obtain \\lambda^{4}{(P_{e},C,u)} = (C u)^{2 P_{e}} \\lambda^{2}{(P_{e},C,u)}", "derivation": "\\lambda{(P_{e},C,u)} = (C u)^{P_{e}} and \\lambda^{2}{(P_{e},C,u)} = (C u)^{P_{e}} \\lambda{(P_{e},C,u)} and \\lambda^{4}{(P_{e},C,u)} = (C u)^{P_{e}} \\lambda^{3}{(P_{e},C,u)} and \\lambda^{3}{(P_{e},C,u)} = (C u)^{P_{e}} \\lambda^{2}{(P_{e},C,u)} and \\lambda^{4}{(P_{e},C,u)} = (C u)^{2 P_{e}} \\lambda^{2}{(P_{e},C,u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Symbol('P_e', commutative=True)))"], [["times", 1, "Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True))"], "Equality(Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(2)), Mul(Pow(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Symbol('P_e', commutative=True)), Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True))))"], [["times", 2, "Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(4)), Mul(Pow(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Symbol('P_e', commutative=True)), Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(3))))"], [["divide", 3, "Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True))"], "Equality(Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(3)), Mul(Pow(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Symbol('P_e', commutative=True)), Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(4)), Mul(Pow(Mul(Symbol('C', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('P_e', commutative=True))), Pow(Function('\\\\lambda')(Symbol('P_e', commutative=True), Symbol('C', commutative=True), Symbol('u', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A}, then derive \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = (L + e^{\\mathbf{A}}) e^{- \\mathbf{A}}, then obtain (- L - e^{\\mathbf{A}}) e^{- \\mathbf{A}} + \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = 0", "derivation": "\\operatorname{A_{2}}{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A} and \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = e^{- \\mathbf{A}} \\int e^{\\mathbf{A}} d\\mathbf{A} and \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = (L + e^{\\mathbf{A}}) e^{- \\mathbf{A}} and - (L + e^{\\mathbf{A}}) e^{- \\mathbf{A}} + \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = 0 and - (L + e^{\\mathbf{A}}) e^{- \\mathbf{A}} + e^{- \\mathbf{A}} \\int e^{\\mathbf{A}} d\\mathbf{A} = 0 and (- L - e^{\\mathbf{A}}) e^{- \\mathbf{A}} + \\operatorname{A_{2}}{(\\mathbf{A})} e^{- \\mathbf{A}} = 0", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 1, "exp(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Add(Symbol('L', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["minus", 3, "Mul(Add(Symbol('L', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('L', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Add(Symbol('L', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('L', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{A}', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Function('A_2')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(M)} = \\cos{(M)}, then obtain \\frac{d}{d M} \\int ((\\operatorname{P_{e}}^{M}{(M)})^{M})^{M} dM = \\frac{d}{d M} \\int ((\\cos^{M}{(M)})^{M})^{M} dM", "derivation": "\\operatorname{P_{e}}{(M)} = \\cos{(M)} and \\operatorname{P_{e}}^{M}{(M)} = \\cos^{M}{(M)} and (\\operatorname{P_{e}}^{M}{(M)})^{M} = (\\cos^{M}{(M)})^{M} and ((\\operatorname{P_{e}}^{M}{(M)})^{M})^{M} = ((\\cos^{M}{(M)})^{M})^{M} and \\int ((\\operatorname{P_{e}}^{M}{(M)})^{M})^{M} dM = \\int ((\\cos^{M}{(M)})^{M})^{M} dM and \\frac{d}{d M} \\int ((\\operatorname{P_{e}}^{M}{(M)})^{M})^{M} dM = \\frac{d}{d M} \\int ((\\cos^{M}{(M)})^{M})^{M} dM", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Function('P_e')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["power", 3, "Symbol('M', commutative=True)"], "Equality(Pow(Pow(Pow(Function('P_e')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Pow(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Pow(Pow(Pow(Function('P_e')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(Pow(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 5, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Pow(Pow(Pow(Function('P_e')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(Pow(Pow(Pow(cos(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(B)} = \\log{(e^{B})} and U{(\\mu)} = \\cos{(\\mu)}, then obtain \\operatorname{A_{y}}^{2}{(B)} \\iint U{(\\mu)} d\\mu d\\mu = \\operatorname{A_{y}}^{2}{(B)} \\iint \\cos{(\\mu)} d\\mu d\\mu", "derivation": "\\operatorname{A_{y}}{(B)} = \\log{(e^{B})} and U{(\\mu)} = \\cos{(\\mu)} and \\int U{(\\mu)} d\\mu = \\int \\cos{(\\mu)} d\\mu and \\iint U{(\\mu)} d\\mu d\\mu = \\iint \\cos{(\\mu)} d\\mu d\\mu and \\log{(e^{B})}^{2} \\iint U{(\\mu)} d\\mu d\\mu = \\log{(e^{B})}^{2} \\iint \\cos{(\\mu)} d\\mu d\\mu and \\operatorname{A_{y}}^{2}{(B)} \\iint U{(\\mu)} d\\mu d\\mu = \\operatorname{A_{y}}^{2}{(B)} \\iint \\cos{(\\mu)} d\\mu d\\mu", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('B', commutative=True)), log(exp(Symbol('B', commutative=True))))"], ["get_premise", "Equality(Function('U')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('U')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["times", 4, "Pow(log(exp(Symbol('B', commutative=True))), Integer(2))"], "Equality(Mul(Pow(log(exp(Symbol('B', commutative=True))), Integer(2)), Integral(Function('U')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Pow(log(exp(Symbol('B', commutative=True))), Integer(2)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('A_y')(Symbol('B', commutative=True)), Integer(2)), Integral(Function('U')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Pow(Function('A_y')(Symbol('B', commutative=True)), Integer(2)), Integral(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(L)} = \\log{(L)}, then derive 0 = - (\\frac{L \\frac{d}{d L} \\mathbf{A}{(L)}}{\\mathbf{A}{(L)}} + \\log{(\\mathbf{A}{(L)})}) \\mathbf{A}^{L}{(L)} + (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L}, then obtain 0 = - (\\frac{L \\frac{d}{d L} \\mathbf{A}{(L)}}{\\mathbf{A}{(L)}} + \\log{(\\mathbf{A}{(L)})}) \\log{(L)}^{L} + (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L}", "derivation": "\\mathbf{A}{(L)} = \\log{(L)} and \\mathbf{A}^{L}{(L)} = \\log{(L)}^{L} and 0 = - \\mathbf{A}^{L}{(L)} + \\log{(L)}^{L} and \\frac{d}{d L} 0 = \\frac{d}{d L} (- \\mathbf{A}^{L}{(L)} + \\log{(L)}^{L}) and 0 = - (\\frac{L \\frac{d}{d L} \\mathbf{A}{(L)}}{\\mathbf{A}{(L)}} + \\log{(\\mathbf{A}{(L)})}) \\mathbf{A}^{L}{(L)} + (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L} and 0 = - (\\frac{L \\frac{d}{d L} \\mathbf{A}{(L)}}{\\mathbf{A}{(L)}} + \\log{(\\mathbf{A}{(L)})}) \\log{(L)}^{L} + (\\log{(\\log{(L)})} + \\frac{1}{\\log{(L)}}) \\log{(L)}^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True))))"], [["differentiate", 3, "Symbol('L', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Mul(Symbol('L', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)))), Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Add(log(log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Mul(Symbol('L', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), log(Function('\\\\mathbf{A}')(Symbol('L', commutative=True)))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Add(log(log(Symbol('L', commutative=True))), Pow(log(Symbol('L', commutative=True)), Integer(-1))), Pow(log(Symbol('L', commutative=True)), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\lambda)} = \\log{(\\lambda)}, then obtain - \\frac{\\operatorname{A_{y}}{(\\lambda)}}{\\lambda} + \\frac{1}{\\lambda} = - \\frac{\\log{(\\lambda)}}{\\lambda} + \\frac{1}{\\lambda}", "derivation": "\\operatorname{A_{y}}{(\\lambda)} = \\log{(\\lambda)} and \\frac{\\operatorname{A_{y}}{(\\lambda)}}{\\lambda} = \\frac{\\log{(\\lambda)}}{\\lambda} and - \\frac{\\operatorname{A_{y}}{(\\lambda)}}{\\lambda} = - \\frac{\\log{(\\lambda)}}{\\lambda} and - \\frac{\\operatorname{A_{y}}{(\\lambda)}}{\\lambda} + \\frac{1}{\\lambda} = - \\frac{\\log{(\\lambda)}}{\\lambda} + \\frac{1}{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))))"], [["add", 3, "Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\lambda', commutative=True))), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), log(Symbol('\\\\lambda', commutative=True))), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(r,z^{*})} = \\frac{\\partial}{\\partial r} (r - z^{*}), then derive (\\operatorname{P_{e}}{(r,z^{*})} + 1)^{r} = 2^{r}, then obtain ((\\frac{\\partial}{\\partial r} (r - z^{*}) + 1)^{r})^{z^{*}} = (2^{r})^{z^{*}}", "derivation": "\\operatorname{P_{e}}{(r,z^{*})} = \\frac{\\partial}{\\partial r} (r - z^{*}) and \\operatorname{P_{e}}{(r,z^{*})} + 1 = \\frac{\\partial}{\\partial r} (r - z^{*}) + 1 and (\\operatorname{P_{e}}{(r,z^{*})} + 1)^{r} = (\\frac{\\partial}{\\partial r} (r - z^{*}) + 1)^{r} and (\\operatorname{P_{e}}{(r,z^{*})} + 1)^{r} = 2^{r} and ((\\operatorname{P_{e}}{(r,z^{*})} + 1)^{r})^{z^{*}} = (2^{r})^{z^{*}} and ((\\frac{\\partial}{\\partial r} (r - z^{*}) + 1)^{r})^{z^{*}} = (2^{r})^{z^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('P_e')(Symbol('r', commutative=True), Symbol('z^*', commutative=True)), Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('P_e')(Symbol('r', commutative=True), Symbol('z^*', commutative=True)), Integer(1)), Add(Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Add(Function('P_e')(Symbol('r', commutative=True), Symbol('z^*', commutative=True)), Integer(1)), Symbol('r', commutative=True)), Pow(Add(Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Add(Function('P_e')(Symbol('r', commutative=True), Symbol('z^*', commutative=True)), Integer(1)), Symbol('r', commutative=True)), Pow(Integer(2), Symbol('r', commutative=True)))"], [["power", 4, "Symbol('z^*', commutative=True)"], "Equality(Pow(Pow(Add(Function('P_e')(Symbol('r', commutative=True), Symbol('z^*', commutative=True)), Integer(1)), Symbol('r', commutative=True)), Symbol('z^*', commutative=True)), Pow(Pow(Integer(2), Symbol('r', commutative=True)), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Pow(Add(Derivative(Add(Symbol('r', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Symbol('r', commutative=True)), Symbol('z^*', commutative=True)), Pow(Pow(Integer(2), Symbol('r', commutative=True)), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(\\theta_1)} = e^{\\theta_1}, then obtain \\frac{d}{d \\theta_1} (\\frac{\\theta_1 + \\hat{X}{(\\theta_1)}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1) = \\frac{d}{d \\theta_1} (\\frac{\\theta_1 + e^{\\theta_1}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1)", "derivation": "\\hat{X}{(\\theta_1)} = e^{\\theta_1} and \\theta_1 + \\hat{X}{(\\theta_1)} = \\theta_1 + e^{\\theta_1} and \\frac{\\theta_1 + \\hat{X}{(\\theta_1)}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} = \\frac{\\theta_1 + e^{\\theta_1}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} and \\frac{\\theta_1 + \\hat{X}{(\\theta_1)}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1 = \\frac{\\theta_1 + e^{\\theta_1}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1 and \\frac{d}{d \\theta_1} (\\frac{\\theta_1 + \\hat{X}{(\\theta_1)}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1) = \\frac{d}{d \\theta_1} (\\frac{\\theta_1 + e^{\\theta_1}}{\\frac{d}{d \\theta_1} \\hat{X}{(\\theta_1)}} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 2, "Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))), Integer(-1)), Add(Mul(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Pow(Derivative(Function('\\\\hat{X}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(n)} = \\log{(n)}, then derive \\frac{d}{d n} i{(n)} = \\frac{1}{n}, then obtain \\cos^{n}{((\\frac{d}{d n} \\log{(n)})^{n} + \\frac{1}{n})} = \\cos^{n}{((\\frac{1}{n})^{n} + \\frac{1}{n})}", "derivation": "i{(n)} = \\log{(n)} and \\frac{d}{d n} i{(n)} = \\frac{d}{d n} \\log{(n)} and \\frac{d}{d n} i{(n)} = \\frac{1}{n} and (\\frac{d}{d n} i{(n)})^{n} = (\\frac{1}{n})^{n} and (\\frac{d}{d n} i{(n)})^{n} + \\frac{1}{n} = (\\frac{1}{n})^{n} + \\frac{1}{n} and \\cos{((\\frac{d}{d n} i{(n)})^{n} + \\frac{1}{n})} = \\cos{((\\frac{1}{n})^{n} + \\frac{1}{n})} and \\cos{((\\frac{d}{d n} \\log{(n)})^{n} + \\frac{1}{n})} = \\cos{((\\frac{1}{n})^{n} + \\frac{1}{n})} and \\cos^{n}{((\\frac{d}{d n} \\log{(n)})^{n} + \\frac{1}{n})} = \\cos^{n}{((\\frac{1}{n})^{n} + \\frac{1}{n})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Pow(Symbol('n', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('i')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["add", 4, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Pow(Derivative(Function('i')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["cos", 5], "Equality(cos(Add(Pow(Derivative(Function('i')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))), cos(Add(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(cos(Add(Pow(Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))), cos(Add(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["power", 7, "Symbol('n', commutative=True)"], "Equality(Pow(cos(Add(Pow(Derivative(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('n', commutative=True)), Pow(cos(Add(Pow(Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} = \\frac{- \\hat{H}_l + v_{1}}{I}, then obtain (\\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} + \\frac{\\hat{H}_l - v_{1}}{I})^{I} = 0^{I}", "derivation": "\\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} = \\frac{- \\hat{H}_l + v_{1}}{I} and \\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} - \\frac{- \\hat{H}_l + v_{1}}{I} = 0 and \\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} + \\frac{\\hat{H}_l - v_{1}}{I} = 0 and I \\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} = - \\hat{H}_l + v_{1} and \\frac{- \\hat{H}_l + v_{1}}{I} + \\frac{\\hat{H}_l - v_{1}}{I} = 0 and I \\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} - v_{1} = - \\hat{H}_l and (\\frac{- \\hat{H}_l + v_{1}}{I} + \\frac{\\hat{H}_l - v_{1}}{I})^{I} = 0^{I} and (\\operatorname{c_{0}}{(I,\\hat{H}_l,v_{1})} + \\frac{\\hat{H}_l - v_{1}}{I})^{I} = 0^{I}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True)))"], "Equality(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Integer(0))"], [["divide", 1, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('I', commutative=True), Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Integer(0))"], [["add", 4, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Symbol('I', commutative=True), Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 5, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Symbol('I', commutative=True)), Pow(Integer(0), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Add(Function('c_0')(Symbol('I', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Symbol('I', commutative=True)), Pow(Integer(0), Symbol('I', commutative=True)))"]]}, {"prompt": "Given Z{(M)} = \\sin{(M)}, then derive \\Psi + Z{(M)} = f_{\\mathbf{v}} + \\sin{(M)}, then obtain \\frac{\\Psi + \\sin{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}} = \\frac{f_{\\mathbf{v}} + Z{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}}", "derivation": "Z{(M)} = \\sin{(M)} and \\frac{d}{d M} Z{(M)} = \\frac{d}{d M} \\sin{(M)} and \\int \\frac{d}{d M} Z{(M)} dM = \\int \\frac{d}{d M} \\sin{(M)} dM and \\Psi + Z{(M)} = f_{\\mathbf{v}} + \\sin{(M)} and \\Psi + \\sin{(M)} = f_{\\mathbf{v}} + \\sin{(M)} and \\Psi + Z{(M)} = f_{\\mathbf{v}} + Z{(M)} and \\frac{\\Psi + Z{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}} = \\frac{f_{\\mathbf{v}} + Z{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}} and \\Psi + Z{(M)} = \\Psi + \\sin{(M)} and \\frac{\\Psi + \\sin{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}} = \\frac{f_{\\mathbf{v}} + Z{(M)}}{f_{\\mathbf{v}} + \\sin{(M)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Function('Z')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('Z')(Symbol('M', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('M', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('Z')(Symbol('M', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('Z')(Symbol('M', commutative=True))))"], [["divide", 6, "Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), Function('Z')(Symbol('M', commutative=True))), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))), Integer(-1))), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('Z')(Symbol('M', commutative=True))), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('Z')(Symbol('M', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('M', commutative=True))), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))), Integer(-1))), Mul(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('Z')(Symbol('M', commutative=True))), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('M', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\phi_2,F_{g})} = \\phi_2^{F_{g}}, then obtain \\frac{\\partial}{\\partial F_{g}} (\\int - \\phi_2 dF_{g} + \\int \\mathbf{J}_M{(\\phi_2,F_{g})} dF_{g}) = \\frac{\\partial}{\\partial F_{g}} (\\int - \\phi_2 dF_{g} + \\int \\phi_2^{F_{g}} dF_{g})", "derivation": "\\mathbf{J}_M{(\\phi_2,F_{g})} = \\phi_2^{F_{g}} and - \\phi_2 + \\mathbf{J}_M{(\\phi_2,F_{g})} = - \\phi_2 + \\phi_2^{F_{g}} and \\int (- \\phi_2 + \\mathbf{J}_M{(\\phi_2,F_{g})}) dF_{g} = \\int (- \\phi_2 + \\phi_2^{F_{g}}) dF_{g} and \\frac{\\partial}{\\partial F_{g}} \\int (- \\phi_2 + \\mathbf{J}_M{(\\phi_2,F_{g})}) dF_{g} = \\frac{\\partial}{\\partial F_{g}} \\int (- \\phi_2 + \\phi_2^{F_{g}}) dF_{g} and \\frac{\\partial}{\\partial F_{g}} (\\int - \\phi_2 dF_{g} + \\int \\mathbf{J}_M{(\\phi_2,F_{g})} dF_{g}) = \\frac{\\partial}{\\partial F_{g}} (\\int - \\phi_2 dF_{g} + \\int \\phi_2^{F_{g}} dF_{g})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["expand", 4], "Equality(Derivative(Add(Integral(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)} = \\mathbf{v} + \\varepsilon^{v_{t}}, then obtain 0 = \\frac{\\mathbf{v} (\\mathbf{v} + \\varepsilon^{v_{t}})}{\\varepsilon} - \\frac{\\mathbf{v} \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)}}{\\varepsilon}", "derivation": "\\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)} = \\mathbf{v} + \\varepsilon^{v_{t}} and \\mathbf{v} \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)} = \\mathbf{v} (\\mathbf{v} + \\varepsilon^{v_{t}}) and \\frac{\\mathbf{v} \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)}}{\\varepsilon} = \\frac{\\mathbf{v} (\\mathbf{v} + \\varepsilon^{v_{t}})}{\\varepsilon} and \\frac{\\mathbf{v} \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)}}{\\varepsilon} + v_{t} = \\frac{\\mathbf{v} (\\mathbf{v} + \\varepsilon^{v_{t}})}{\\varepsilon} + v_{t} and 0 = \\frac{\\mathbf{v} (\\mathbf{v} + \\varepsilon^{v_{t}})}{\\varepsilon} - \\frac{\\mathbf{v} \\mathbf{P}{(\\mathbf{v},v_{t},\\varepsilon)}}{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_t', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Add(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_t', commutative=True)))))"], [["divide", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_t', commutative=True)))))"], [["add", 3, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('v_t', commutative=True)), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_t', commutative=True)))), Symbol('v_t', commutative=True)))"], [["minus", 4, "Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Symbol('v_t', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_t', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('v_t', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given l{(h,Q)} = e^{- Q + h}, then derive \\int l{(h,Q)} dh = V_{\\mathbf{B}} + e^{- Q + h}, then obtain - 2 \\sin{(Q + c{(h,Q)})} + 2 \\int l{(h,Q)} dh = V_{\\mathbf{B}} + e^{- Q + h} - 2 \\sin{(Q + c{(h,Q)})} + \\int l{(h,Q)} dh", "derivation": "l{(h,Q)} = e^{- Q + h} and \\int l{(h,Q)} dh = \\int e^{- Q + h} dh and \\int l{(h,Q)} dh = V_{\\mathbf{B}} + e^{- Q + h} and - \\sin{(Q + c{(h,Q)})} + \\int l{(h,Q)} dh = V_{\\mathbf{B}} + e^{- Q + h} - \\sin{(Q + c{(h,Q)})} and - 2 \\sin{(Q + c{(h,Q)})} + 2 \\int l{(h,Q)} dh = V_{\\mathbf{B}} + e^{- Q + h} - 2 \\sin{(Q + c{(h,Q)})} + \\int l{(h,Q)} dh", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('h', commutative=True)))))"], [["minus", 3, "sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True))))"], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True))))), Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True)))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('h', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True)))))))"], [["add", 4, "Add(Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True))))), Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True))))), Mul(Integer(2), Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True))))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('h', commutative=True))), Mul(Integer(-1), Integer(2), sin(Add(Symbol('Q', commutative=True), Function('c')(Symbol('h', commutative=True), Symbol('Q', commutative=True))))), Integral(Function('l')(Symbol('h', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(z^{*})} = \\sin{(z^{*})}, then derive \\frac{d}{d z^{*}} \\mathbf{g}{(z^{*})} = \\cos{(z^{*})}, then obtain (\\frac{d}{d z^{*}} \\mathbf{g}{(z^{*})})^{z^{*}} = \\cos^{z^{*}}{(z^{*})}", "derivation": "\\mathbf{g}{(z^{*})} = \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\mathbf{g}{(z^{*})} = \\frac{d}{d z^{*}} \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\mathbf{g}{(z^{*})} = \\cos{(z^{*})} and (\\frac{d}{d z^{*}} \\mathbf{g}{(z^{*})})^{z^{*}} = \\cos^{z^{*}}{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), cos(Symbol('z^*', commutative=True)))"], [["power", 3, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{g}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(cos(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} = J + \\mathbf{J}_M, then derive \\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ = \\frac{J^{2}}{2} + J \\mathbf{J}_M + f_{\\mathbf{v}}, then obtain 1 = \\frac{\\int (J + \\mathbf{J}_M) dJ}{\\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ}", "derivation": "\\operatorname{A_{x}}{(J,\\mathbf{J}_M)} = J + \\mathbf{J}_M and \\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ = \\int (J + \\mathbf{J}_M) dJ and \\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ = \\frac{J^{2}}{2} + J \\mathbf{J}_M + f_{\\mathbf{v}} and \\frac{\\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ}{\\frac{J^{2}}{2} + J \\mathbf{J}_M + f_{\\mathbf{v}}} = \\frac{\\int (J + \\mathbf{J}_M) dJ}{\\frac{J^{2}}{2} + J \\mathbf{J}_M + f_{\\mathbf{v}}} and 1 = \\frac{\\int (J + \\mathbf{J}_M) dJ}{\\int \\operatorname{A_{x}}{(J,\\mathbf{J}_M)} dJ}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 2, "Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), Integral(Function('A_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('J', commutative=True), Integer(2))), Mul(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Integral(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True))), Pow(Integral(Function('A_x')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{P}{(x^\\prime)} = \\sin{(\\log{(x^\\prime)})} and \\varphi^{*}{(x^\\prime)} = \\int \\mathbf{P}{(x^\\prime)} dx^\\prime, then obtain x^\\prime \\varphi^{*}{(x^\\prime)} = x^\\prime \\int \\sin{(\\log{(x^\\prime)})} dx^\\prime", "derivation": "\\mathbf{P}{(x^\\prime)} = \\sin{(\\log{(x^\\prime)})} and \\int \\mathbf{P}{(x^\\prime)} dx^\\prime = \\int \\sin{(\\log{(x^\\prime)})} dx^\\prime and x^\\prime \\int \\mathbf{P}{(x^\\prime)} dx^\\prime = x^\\prime \\int \\sin{(\\log{(x^\\prime)})} dx^\\prime and \\varphi^{*}{(x^\\prime)} = \\int \\mathbf{P}{(x^\\prime)} dx^\\prime and x^\\prime \\varphi^{*}{(x^\\prime)} = x^\\prime \\int \\sin{(\\log{(x^\\prime)})} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), sin(log(Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Integral(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Symbol('x^\\\\prime', commutative=True), Integral(sin(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), Integral(sin(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(b)} = e^{\\cos{(b)}} and \\mathbf{D}{(\\mathbf{J}_M)} = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M, then obtain \\mathbf{D}{(\\mathbf{J}_M)} - (\\frac{d}{d b} 0)^{b} = - \\operatorname{A_{z}}{(b)} + \\mathbf{D}{(\\mathbf{J}_M)} + e^{\\cos{(b)}} - (\\frac{d}{d b} 0)^{b}", "derivation": "\\operatorname{A_{z}}{(b)} = e^{\\cos{(b)}} and 0 = - \\operatorname{A_{z}}{(b)} + e^{\\cos{(b)}} and \\mathbf{D}{(\\mathbf{J}_M)} = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M = - \\operatorname{A_{z}}{(b)} + e^{\\cos{(b)}} + \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\mathbf{D}{(\\mathbf{J}_M)} = - \\operatorname{A_{z}}{(b)} + e^{\\cos{(b)}} + \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\mathbf{D}{(\\mathbf{J}_M)} = - \\operatorname{A_{z}}{(b)} + \\mathbf{D}{(\\mathbf{J}_M)} + e^{\\cos{(b)}} and \\mathbf{D}{(\\mathbf{J}_M)} - (\\frac{d}{d b} 0)^{b} = - \\operatorname{A_{z}}{(b)} + \\mathbf{D}{(\\mathbf{J}_M)} + e^{\\cos{(b)}} - (\\frac{d}{d b} 0)^{b}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('b', commutative=True)), exp(cos(Symbol('b', commutative=True))))"], [["minus", 1, "Function('A_z')(Symbol('b', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_z')(Symbol('b', commutative=True))), exp(cos(Symbol('b', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 2, "Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Integer(-1), Function('A_z')(Symbol('b', commutative=True))), exp(cos(Symbol('b', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Function('A_z')(Symbol('b', commutative=True))), exp(cos(Symbol('b', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Mul(Integer(-1), Function('A_z')(Symbol('b', commutative=True))), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(cos(Symbol('b', commutative=True)))))"], [["minus", 6, "Pow(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Pow(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Function('A_z')(Symbol('b', commutative=True))), Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(cos(Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(r,\\dot{y})} = r^{\\dot{y}}, then obtain (\\int \\theta_{1}^{r}{(r,\\dot{y})} d\\dot{y})^{\\dot{y}} = (\\int (r^{\\dot{y}})^{r} d\\dot{y})^{\\dot{y}}", "derivation": "\\theta_{1}{(r,\\dot{y})} = r^{\\dot{y}} and \\theta_{1}^{r}{(r,\\dot{y})} = (r^{\\dot{y}})^{r} and \\int \\theta_{1}^{r}{(r,\\dot{y})} d\\dot{y} = \\int (r^{\\dot{y}})^{r} d\\dot{y} and (\\int \\theta_{1}^{r}{(r,\\dot{y})} d\\dot{y})^{\\dot{y}} = (\\int (r^{\\dot{y}})^{r} d\\dot{y})^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)), Pow(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\theta_1')(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)), Pow(Integral(Pow(Pow(Symbol('r', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\dot{y}', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\omega,t_{1})} = \\frac{\\partial}{\\partial \\omega} \\omega t_{1}, then derive \\frac{\\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})}}{\\mu{(\\omega,t_{1})}} = \\frac{1}{\\mu{(\\omega,t_{1})}}, then obtain \\frac{\\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})}}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}} = \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}}", "derivation": "\\mu{(\\omega,t_{1})} = \\frac{\\partial}{\\partial \\omega} \\omega t_{1} and \\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})} = \\frac{\\partial^{2}}{\\partial t_{1}\\partial \\omega} \\omega t_{1} and \\frac{\\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})}}{\\mu{(\\omega,t_{1})}} = \\frac{\\frac{\\partial^{2}}{\\partial t_{1}\\partial \\omega} \\omega t_{1}}{\\mu{(\\omega,t_{1})}} and \\frac{\\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})}}{\\mu{(\\omega,t_{1})}} = \\frac{1}{\\mu{(\\omega,t_{1})}} and \\frac{\\frac{\\partial^{2}}{\\partial t_{1}\\partial \\omega} \\omega t_{1}}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}} = \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}} and \\frac{\\frac{\\partial}{\\partial t_{1}} \\mu{(\\omega,t_{1})}}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}} = \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["divide", 2, "Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)), Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Pow(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\phi_{2}{(A_{2},s)} = \\log{(- A_{2} + s)}, then obtain A_{2} (A_{2} + \\phi_{2}{(A_{2},s)}) \\int \\phi_{2}{(A_{2},s)} ds = A_{2} (A_{2} + \\log{(- A_{2} + s)}) \\int \\phi_{2}{(A_{2},s)} ds", "derivation": "\\phi_{2}{(A_{2},s)} = \\log{(- A_{2} + s)} and \\int \\phi_{2}{(A_{2},s)} ds = \\int \\log{(- A_{2} + s)} ds and A_{2} + \\phi_{2}{(A_{2},s)} = A_{2} + \\log{(- A_{2} + s)} and (A_{2} + \\phi_{2}{(A_{2},s)}) \\int \\log{(- A_{2} + s)} ds = (A_{2} + \\log{(- A_{2} + s)}) \\int \\log{(- A_{2} + s)} ds and (A_{2} + \\phi_{2}{(A_{2},s)}) \\int \\phi_{2}{(A_{2},s)} ds = (A_{2} + \\log{(- A_{2} + s)}) \\int \\phi_{2}{(A_{2},s)} ds and A_{2} (A_{2} + \\phi_{2}{(A_{2},s)}) \\int \\phi_{2}{(A_{2},s)} ds = A_{2} (A_{2} + \\log{(- A_{2} + s)}) \\int \\phi_{2}{(A_{2},s)} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('A_2', commutative=True))"], "Equality(Add(Symbol('A_2', commutative=True), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Add(Symbol('A_2', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True)))))"], [["times", 3, "Integral(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))"], "Equality(Mul(Add(Symbol('A_2', commutative=True), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Mul(Add(Symbol('A_2', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True)))), Integral(log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Symbol('A_2', commutative=True), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Integral(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Add(Symbol('A_2', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True)))), Integral(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["times", 5, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Add(Symbol('A_2', commutative=True), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True))), Integral(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Symbol('A_2', commutative=True), Add(Symbol('A_2', commutative=True), log(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('s', commutative=True)))), Integral(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain \\omega \\cos{(\\frac{\\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})}}{\\sin{(1)}})} = \\omega \\cos{(1)}", "derivation": "\\phi_{2}{(\\mathbf{p})} = e^{\\mathbf{p}} and \\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}} = 1 and \\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})} = \\sin{(1)} and - \\frac{\\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})}}{\\mathbf{p}} = - \\frac{\\sin{(1)}}{\\mathbf{p}} and \\frac{\\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})}}{\\sin{(1)}} = 1 and \\cos{(\\frac{\\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})}}{\\sin{(1)}})} = \\cos{(1)} and \\omega \\cos{(\\frac{\\sin{(\\phi_{2}{(\\mathbf{p})} e^{- \\mathbf{p}})}}{\\sin{(1)}})} = \\omega \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))), Integer(1))"], [["sin", 2], "Equality(sin(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))), sin(Integer(1)))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), sin(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), sin(Integer(1))))"], [["divide", 4, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), sin(Integer(1)))"], "Equality(Mul(Pow(sin(Integer(1)), Integer(-1)), sin(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))), Integer(1))"], [["cos", 5], "Equality(cos(Mul(Pow(sin(Integer(1)), Integer(-1)), sin(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))))))), cos(Integer(1)))"], [["times", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), cos(Mul(Pow(sin(Integer(1)), Integer(-1)), sin(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))))))), Mul(Symbol('\\\\omega', commutative=True), cos(Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(A_{2})} = \\sin{(A_{2})}, then obtain \\frac{d}{d A_{2}} ((\\psi^{*}^{A_{2}}{(A_{2})})^{A_{2}} + (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2) = \\frac{d}{d A_{2}} (2 (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2)", "derivation": "\\psi^{*}{(A_{2})} = \\sin{(A_{2})} and \\psi^{*}^{A_{2}}{(A_{2})} = \\sin^{A_{2}}{(A_{2})} and (\\psi^{*}^{A_{2}}{(A_{2})})^{A_{2}} = (\\sin^{A_{2}}{(A_{2})})^{A_{2}} and (\\psi^{*}^{A_{2}}{(A_{2})})^{A_{2}} - 1 = (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 1 and (\\psi^{*}^{A_{2}}{(A_{2})})^{A_{2}} + (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2 = 2 (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2 and \\frac{d}{d A_{2}} ((\\psi^{*}^{A_{2}}{(A_{2})})^{A_{2}} + (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2) = \\frac{d}{d A_{2}} (2 (\\sin^{A_{2}}{(A_{2})})^{A_{2}} - 2)", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["power", 1, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Pow(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Integer(-1)), Add(Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Integer(-1)))"], [["add", 4, "Add(Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Pow(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Integer(-2)), Add(Mul(Integer(2), Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), Integer(-2)))"], [["differentiate", 5, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Pow(Pow(Function('\\\\psi^*')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Integer(-2)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Pow(Pow(sin(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), Integer(-2)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain \\tilde{\\infty}^{\\theta_2} \\int \\nabla{(\\theta_2)} d\\theta_2 - \\int \\nabla{(\\theta_2)} d\\theta_2 = \\tilde{\\infty}^{\\theta_2} \\int \\cos{(\\theta_2)} d\\theta_2 - \\int \\nabla{(\\theta_2)} d\\theta_2", "derivation": "\\nabla{(\\theta_2)} = \\cos{(\\theta_2)} and \\int \\nabla{(\\theta_2)} d\\theta_2 = \\int \\cos{(\\theta_2)} d\\theta_2 and \\tilde{\\infty}^{\\theta_2} \\int \\nabla{(\\theta_2)} d\\theta_2 = \\tilde{\\infty}^{\\theta_2} \\int \\cos{(\\theta_2)} d\\theta_2 and \\tilde{\\infty}^{\\theta_2} \\int \\nabla{(\\theta_2)} d\\theta_2 - \\int \\nabla{(\\theta_2)} d\\theta_2 = \\tilde{\\infty}^{\\theta_2} \\int \\cos{(\\theta_2)} d\\theta_2 - \\int \\nabla{(\\theta_2)} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 2, "Pow(Integer(0), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('\\\\theta_2', commutative=True)), Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(zoo, Symbol('\\\\theta_2', commutative=True)), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["minus", 3, "Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Mul(Pow(zoo, Symbol('\\\\theta_2', commutative=True)), Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))), Add(Mul(Pow(zoo, Symbol('\\\\theta_2', commutative=True)), Integral(cos(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(a,p)} = \\frac{\\partial}{\\partial a} \\frac{p}{a}, then derive \\int \\mathbf{J}_f{(a,p)} da = f^{*} + \\frac{p}{a}, then derive (\\tilde{g}^* + \\frac{p}{a})^{f^{*}} = (f^{*} + \\frac{p}{a})^{f^{*}}, then obtain \\frac{(\\tilde{g}^* + \\frac{p}{a})^{f^{*}}}{\\frac{\\partial}{\\partial a} \\frac{p}{a}} = \\frac{(f^{*} + \\frac{p}{a})^{f^{*}}}{\\frac{\\partial}{\\partial a} \\frac{p}{a}}", "derivation": "\\mathbf{J}_f{(a,p)} = \\frac{\\partial}{\\partial a} \\frac{p}{a} and \\int \\mathbf{J}_f{(a,p)} da = \\int \\frac{\\partial}{\\partial a} \\frac{p}{a} da and \\int \\mathbf{J}_f{(a,p)} da = f^{*} + \\frac{p}{a} and \\int \\frac{\\partial}{\\partial a} \\frac{p}{a} da = f^{*} + \\frac{p}{a} and (\\int \\frac{\\partial}{\\partial a} \\frac{p}{a} da)^{f^{*}} = (f^{*} + \\frac{p}{a})^{f^{*}} and (\\tilde{g}^* + \\frac{p}{a})^{f^{*}} = (f^{*} + \\frac{p}{a})^{f^{*}} and \\frac{(\\tilde{g}^* + \\frac{p}{a})^{f^{*}}}{\\frac{\\partial}{\\partial a} \\frac{p}{a}} = \\frac{(f^{*} + \\frac{p}{a})^{f^{*}}}{\\frac{\\partial}{\\partial a} \\frac{p}{a}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["power", 4, "Symbol('f^*', commutative=True)"], "Equality(Pow(Integral(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Tuple(Symbol('a', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Symbol('f^*', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Symbol('f^*', commutative=True)))"], [["divide", 6, "Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Symbol('f^*', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Symbol('f^*', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\varphi^{*}{(\\psi)} = \\cos{(\\psi)}, then obtain \\operatorname{f_{E}}^{t_{2}}{(t_{2})} (\\frac{d}{d \\psi} \\varphi^{*}{(\\psi)})^{\\psi} = \\operatorname{f_{E}}^{t_{2}}{(t_{2})} (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi}", "derivation": "\\varphi^{*}{(\\psi)} = \\cos{(\\psi)} and \\frac{d}{d \\psi} \\varphi^{*}{(\\psi)} = \\frac{d}{d \\psi} \\cos{(\\psi)} and (\\frac{d}{d \\psi} \\varphi^{*}{(\\psi)})^{\\psi} = (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi} and \\operatorname{f_{E}}^{t_{2}}{(t_{2})} (\\frac{d}{d \\psi} \\varphi^{*}{(\\psi)})^{\\psi} = \\operatorname{f_{E}}^{t_{2}}{(t_{2})} (\\frac{d}{d \\psi} \\cos{(\\psi)})^{\\psi}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)), Pow(Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True)))"], [["times", 3, "Pow(Function('f_E')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Function('f_E')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True))), Mul(Pow(Function('f_E')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\phi_2)} = e^{\\phi_2}, then obtain \\frac{d}{d \\phi_2} \\mathbf{F}{(\\phi_2)} + 1 = e^{\\phi_2} + 1", "derivation": "\\mathbf{F}{(\\phi_2)} = e^{\\phi_2} and \\phi_2 + \\mathbf{F}{(\\phi_2)} = \\phi_2 + e^{\\phi_2} and \\frac{d}{d \\phi_2} (\\phi_2 + \\mathbf{F}{(\\phi_2)}) = \\frac{d}{d \\phi_2} (\\phi_2 + e^{\\phi_2}) and \\frac{d}{d \\phi_2} \\mathbf{F}{(\\phi_2)} + 1 = e^{\\phi_2} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["add", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\phi_2', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{B},a)} = \\sin{(\\mathbf{B} a)}, then obtain - \\frac{(\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B})^{3}}{a^{3}} = - \\frac{((\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B})^{2}) \\int \\sin{(\\mathbf{B} a)} d\\mathbf{B}}{a^{3}}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{B},a)} = \\sin{(\\mathbf{B} a)} and \\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B} = \\int \\sin{(\\mathbf{B} a)} d\\mathbf{B} and \\frac{\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B}}{a} = \\frac{\\int \\sin{(\\mathbf{B} a)} d\\mathbf{B}}{a} and \\frac{(\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B})^{2}}{a^{2}} = \\frac{(\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B}) \\int \\sin{(\\mathbf{B} a)} d\\mathbf{B}}{a^{2}} and - \\frac{(\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B})^{3}}{a^{3}} = - \\frac{((\\int \\operatorname{z^{*}}{(\\mathbf{B},a)} d\\mathbf{B})^{2}) \\int \\sin{(\\mathbf{B} a)} d\\mathbf{B}}{a^{3}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Pow(Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2))), Mul(Pow(Symbol('a', commutative=True), Integer(-2)), Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-3)), Pow(Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integer(3))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-3)), Pow(Integral(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2)), Integral(sin(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(B)} = \\int \\cos{(B)} dB, then derive \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\frac{\\partial}{\\partial B} (A_{1} + \\sin{(B)}), then derive \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\operatorname{E_{\\lambda}}{(B)} \\frac{\\partial}{\\partial B} (u + \\sin{(B)}), then derive \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\operatorname{E_{\\lambda}}{(B)} \\cos{(B)}, then obtain \\operatorname{E_{\\lambda}}{(B)} \\frac{\\partial}{\\partial B} (A_{1} + \\sin{(B)}) = \\operatorname{E_{\\lambda}}{(B)} \\cos{(B)}", "derivation": "\\operatorname{E_{\\lambda}}{(B)} = \\int \\cos{(B)} dB and \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\frac{d}{d B} \\int \\cos{(B)} dB and \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\int \\cos{(B)} dB and \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\frac{\\partial}{\\partial B} (A_{1} + \\sin{(B)}) and \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\operatorname{E_{\\lambda}}{(B)} \\frac{\\partial}{\\partial B} (u + \\sin{(B)}) and \\operatorname{E_{\\lambda}}{(B)} \\frac{d}{d B} \\operatorname{E_{\\lambda}}{(B)} = \\operatorname{E_{\\lambda}}{(B)} \\cos{(B)} and \\operatorname{E_{\\lambda}}{(B)} \\frac{\\partial}{\\partial B} (A_{1} + \\sin{(B)}) = \\operatorname{E_{\\lambda}}{(B)} \\cos{(B)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["times", 2, "Function('E_{\\\\lambda}')(Symbol('B', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Add(Symbol('u', commutative=True), sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(V_{\\mathbf{B}})} = \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}}, then derive \\phi_{2}{(V_{\\mathbf{B}})} = v_{1} + e^{V_{\\mathbf{B}}}, then obtain \\int (v_{1} + e^{V_{\\mathbf{B}}})^{v_{1}} dv_{1} = \\int (\\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}})^{v_{1}} dv_{1}", "derivation": "\\phi_{2}{(V_{\\mathbf{B}})} = \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} and \\phi_{2}{(V_{\\mathbf{B}})} = v_{1} + e^{V_{\\mathbf{B}}} and v_{1} + e^{V_{\\mathbf{B}}} = \\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} and (v_{1} + e^{V_{\\mathbf{B}}})^{v_{1}} = (\\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}})^{v_{1}} and \\int (v_{1} + e^{V_{\\mathbf{B}}})^{v_{1}} dv_{1} = \\int (\\int e^{V_{\\mathbf{B}}} dV_{\\mathbf{B}})^{v_{1}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\phi_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('v_1', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('v_1', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Symbol('v_1', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('v_1', commutative=True)), Pow(Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('v_1', commutative=True)))"], [["integrate", 4, "Symbol('v_1', commutative=True)"], "Equality(Integral(Pow(Add(Symbol('v_1', commutative=True), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Pow(Integral(exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\varphi{(\\mu_0)} = \\sin{(\\mu_0)}, then obtain 1 = ((\\int 0 d\\mu_0)^{\\mu_0})^{\\mu_0}", "derivation": "\\varphi{(\\mu_0)} = \\sin{(\\mu_0)} and \\varphi^{\\mu_0}{(\\mu_0)} = \\sin^{\\mu_0}{(\\mu_0)} and \\varphi^{\\mu_0}{(\\mu_0)} - \\sin^{\\mu_0}{(\\mu_0)} = 0 and \\int (\\varphi^{\\mu_0}{(\\mu_0)} - \\sin^{\\mu_0}{(\\mu_0)}) d\\mu_0 = \\int 0 d\\mu_0 and (\\int (\\varphi^{\\mu_0}{(\\mu_0)} - \\sin^{\\mu_0}{(\\mu_0)}) d\\mu_0)^{\\mu_0} = (\\int 0 d\\mu_0)^{\\mu_0} and ((\\int (\\varphi^{\\mu_0}{(\\mu_0)} - \\sin^{\\mu_0}{(\\mu_0)}) d\\mu_0)^{\\mu_0})^{\\mu_0} = ((\\int 0 d\\mu_0)^{\\mu_0})^{\\mu_0} and 1 = ((\\int (\\varphi^{\\mu_0}{(\\mu_0)} - \\sin^{\\mu_0}{(\\mu_0)}) d\\mu_0)^{\\mu_0})^{\\mu_0} and 1 = ((\\int 0 d\\mu_0)^{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["minus", 2, "Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["power", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Pow(Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(1), Pow(Pow(Integral(Add(Pow(Function('\\\\varphi')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integer(1), Pow(Pow(Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})} = \\log{(\\mathbf{r} \\varepsilon_0)}, then obtain \\frac{\\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})}}{\\mathbf{r}{(x,L_{\\varepsilon})}} = \\frac{\\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\log{(\\mathbf{r} \\varepsilon_0)}}{\\mathbf{r}{(x,L_{\\varepsilon})}}", "derivation": "\\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})} = \\log{(\\mathbf{r} \\varepsilon_0)} and \\mathbf{r} \\varepsilon_0 + \\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})} = \\mathbf{r} \\varepsilon_0 + \\log{(\\mathbf{r} \\varepsilon_0)} and \\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})} = \\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\log{(\\mathbf{r} \\varepsilon_0)} and \\frac{\\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\operatorname{m_{s}}{(\\varepsilon_0,\\mathbf{r})}}{\\mathbf{r}{(x,L_{\\varepsilon})}} = \\frac{\\mathbf{r} \\varepsilon_0 - \\varepsilon_0 + \\log{(\\mathbf{r} \\varepsilon_0)}}{\\mathbf{r}{(x,L_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["minus", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["divide", 3, "Function('\\\\mathbf{r}')(Symbol('x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), Function('m_s')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Function('\\\\mathbf{r}')(Symbol('x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), log(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))), Pow(Function('\\\\mathbf{r}')(Symbol('x', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(A_{y})} = \\sin{(A_{y})} and \\operatorname{V_{\\mathbf{E}}}{(A_{y})} = \\sin{(A_{y})}, then obtain (- A_{y} + \\operatorname{V_{\\mathbf{E}}}{(A_{y})})^{A_{y}} (- A_{y} + \\operatorname{v_{2}}{(A_{y})})^{A_{y}} = (- A_{y} + \\operatorname{V_{\\mathbf{E}}}{(A_{y})})^{2 A_{y}}", "derivation": "\\operatorname{v_{2}}{(A_{y})} = \\sin{(A_{y})} and - A_{y} + \\operatorname{v_{2}}{(A_{y})} = - A_{y} + \\sin{(A_{y})} and \\operatorname{V_{\\mathbf{E}}}{(A_{y})} = \\sin{(A_{y})} and (- A_{y} + \\operatorname{v_{2}}{(A_{y})})^{A_{y}} = (- A_{y} + \\sin{(A_{y})})^{A_{y}} and (- A_{y} + \\operatorname{v_{2}}{(A_{y})})^{A_{y}} = (- A_{y} + \\operatorname{V_{\\mathbf{E}}}{(A_{y})})^{A_{y}} and (- A_{y} + \\operatorname{V_{\\mathbf{E}}}{(A_{y})})^{A_{y}} (- A_{y} + \\operatorname{v_{2}}{(A_{y})})^{A_{y}} = (- A_{y} + \\operatorname{V_{\\mathbf{E}}}{(A_{y})})^{2 A_{y}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["minus", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('v_2')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('v_2')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('v_2')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["times", 5, "Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('v_2')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('A_y', commutative=True))), Mul(Integer(2), Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\psi^*)} = \\psi^*, then derive \\int \\tilde{g}^*{(\\psi^*)} d\\psi^* = \\frac{(\\psi^*)^{2}}{2} + t_{1}, then obtain \\frac{\\frac{\\partial}{\\partial t_{1}} (\\hat{\\mathbf{x}} + \\frac{(\\psi^*)^{2}}{2})}{\\frac{\\partial}{\\partial t_{1}} (\\frac{(\\psi^*)^{2}}{2} + t_{1})} = 1", "derivation": "\\tilde{g}^*{(\\psi^*)} = \\psi^* and \\int \\tilde{g}^*{(\\psi^*)} d\\psi^* = \\int \\psi^* d\\psi^* and \\int \\tilde{g}^*{(\\psi^*)} d\\psi^* = \\frac{(\\psi^*)^{2}}{2} + t_{1} and \\int \\psi^* d\\psi^* = \\frac{(\\psi^*)^{2}}{2} + t_{1} and \\frac{d}{d t_{1}} \\int \\psi^* d\\psi^* = \\frac{\\partial}{\\partial t_{1}} (\\frac{(\\psi^*)^{2}}{2} + t_{1}) and \\frac{\\frac{d}{d t_{1}} \\int \\psi^* d\\psi^*}{\\frac{\\partial}{\\partial t_{1}} (\\frac{(\\psi^*)^{2}}{2} + t_{1})} = 1 and \\frac{\\frac{\\partial}{\\partial t_{1}} (\\hat{\\mathbf{x}} + \\frac{(\\psi^*)^{2}}{2})}{\\frac{\\partial}{\\partial t_{1}} (\\frac{(\\psi^*)^{2}}{2} + t_{1})} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)))"], [["differentiate", 4, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1)), Derivative(Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_integrals", 6], "Equality(Mul(Derivative(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Pow(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\psi^*', commutative=True), Integer(2))), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\omega{(z)} = \\log{(z)}, then obtain - \\omega{(z)} + (\\frac{d}{d z} (\\omega{(z)} - \\log{(z)}) \\log{(z)})^{z} = - \\omega{(z)} + (\\frac{d}{d z} 0)^{z}", "derivation": "\\omega{(z)} = \\log{(z)} and \\omega{(z)} - \\log{(z)} = 0 and (\\omega{(z)} - \\log{(z)}) \\log{(z)} = 0 and \\frac{d}{d z} (\\omega{(z)} - \\log{(z)}) \\log{(z)} = \\frac{d}{d z} 0 and (\\frac{d}{d z} (\\omega{(z)} - \\log{(z)}) \\log{(z)})^{z} = (\\frac{d}{d z} 0)^{z} and - \\omega{(z)} + (\\frac{d}{d z} (\\omega{(z)} - \\log{(z)}) \\log{(z)})^{z} = - \\omega{(z)} + (\\frac{d}{d z} 0)^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["minus", 1, "log(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), Integer(0))"], [["times", 2, "log(Symbol('z', commutative=True))"], "Equality(Mul(Add(Function('\\\\omega')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), log(Symbol('z', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Add(Function('\\\\omega')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Function('\\\\omega')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True)))"], [["minus", 5, "Function('\\\\omega')(Symbol('z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('z', commutative=True))), Pow(Derivative(Mul(Add(Function('\\\\omega')(Symbol('z', commutative=True)), Mul(Integer(-1), log(Symbol('z', commutative=True)))), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('z', commutative=True))), Pow(Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\theta_1)} = \\log{(\\theta_1)}, then derive (\\rho_f + \\theta_1 \\log{(\\theta_1)} - \\theta_1) \\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1 = (\\rho_f + \\theta_1 \\log{(\\theta_1)} - \\theta_1)^{2}, then obtain (\\rho_f + \\theta_1 \\operatorname{E_{\\lambda}}{(\\theta_1)} - \\theta_1) \\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1 = (\\rho_f + \\theta_1 \\operatorname{E_{\\lambda}}{(\\theta_1)} - \\theta_1)^{2}", "derivation": "\\operatorname{E_{\\lambda}}{(\\theta_1)} = \\log{(\\theta_1)} and \\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1 = \\int \\log{(\\theta_1)} d\\theta_1 and (\\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1) \\int \\log{(\\theta_1)} d\\theta_1 = (\\int \\log{(\\theta_1)} d\\theta_1)^{2} and (\\rho_f + \\theta_1 \\log{(\\theta_1)} - \\theta_1) \\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1 = (\\rho_f + \\theta_1 \\log{(\\theta_1)} - \\theta_1)^{2} and (\\rho_f + \\theta_1 \\operatorname{E_{\\lambda}}{(\\theta_1)} - \\theta_1) \\int \\operatorname{E_{\\lambda}}{(\\theta_1)} d\\theta_1 = (\\rho_f + \\theta_1 \\operatorname{E_{\\lambda}}{(\\theta_1)} - \\theta_1)^{2}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), log(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Pow(Integral(log(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\theta_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given I{(\\mathbf{B},u)} = \\frac{\\mathbf{B}}{u} and \\operatorname{t_{2}}{(\\mathbf{B},u)} = \\frac{\\partial}{\\partial u} \\frac{\\mathbf{B}}{u}, then derive \\frac{\\partial}{\\partial u} I{(\\mathbf{B},u)} = - \\frac{\\mathbf{B}}{u^{2}}, then obtain \\operatorname{t_{2}}{(\\mathbf{B},u)} = - \\frac{\\mathbf{B}}{u^{2}}", "derivation": "I{(\\mathbf{B},u)} = \\frac{\\mathbf{B}}{u} and \\frac{\\partial}{\\partial u} I{(\\mathbf{B},u)} = \\frac{\\partial}{\\partial u} \\frac{\\mathbf{B}}{u} and \\operatorname{t_{2}}{(\\mathbf{B},u)} = \\frac{\\partial}{\\partial u} \\frac{\\mathbf{B}}{u} and \\frac{\\partial}{\\partial u} I{(\\mathbf{B},u)} = - \\frac{\\mathbf{B}}{u^{2}} and - \\frac{\\mathbf{B}}{u^{2}} = \\frac{\\partial}{\\partial u} \\frac{\\mathbf{B}}{u} and \\operatorname{t_{2}}{(\\mathbf{B},u)} = - \\frac{\\mathbf{B}}{u^{2}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-2))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('t_2')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) and T{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l), then obtain - \\mathbf{P} + T{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) - \\mathbf{P}", "derivation": "\\hat{x}_0{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) and - \\mathbf{P} + \\hat{x}_0{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) - \\mathbf{P} and T{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) and \\hat{x}_0{(\\mathbf{P},l,\\mathbf{S})} = T{(\\mathbf{P},l,\\mathbf{S})} and - \\mathbf{P} + T{(\\mathbf{P},l,\\mathbf{S})} = \\mathbf{P} (\\mathbf{S} + l) - \\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('l', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('T')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('l', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then derive 0^{\\mathbf{s}} = (- \\sin{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\mathbf{B}{(\\mathbf{s})})^{\\mathbf{s}}, then obtain (- \\sin{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\mathbf{B}{(\\mathbf{s})})^{\\mathbf{s}} = 1", "derivation": "\\mathbf{B}{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and 0 = - \\mathbf{B}{(\\mathbf{s})} + \\cos{(\\mathbf{s})} and 0^{\\mathbf{s}} = (- \\mathbf{B}{(\\mathbf{s})} + \\cos{(\\mathbf{s})})^{\\mathbf{s}} and \\frac{d}{d \\mathbf{s}} 0 = \\frac{d}{d \\mathbf{s}} (- \\mathbf{B}{(\\mathbf{s})} + \\cos{(\\mathbf{s})}) and (\\frac{d}{d \\mathbf{s}} 0)^{\\mathbf{s}} = (\\frac{d}{d \\mathbf{s}} (- \\mathbf{B}{(\\mathbf{s})} + \\cos{(\\mathbf{s})}))^{\\mathbf{s}} and 0^{\\mathbf{s}} = (- \\sin{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\mathbf{B}{(\\mathbf{s})})^{\\mathbf{s}} and (- \\sin{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\mathbf{B}{(\\mathbf{s})})^{\\mathbf{s}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True))), cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Pow(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\hat{X}{(\\phi_2)} = \\cos{(\\phi_2)}, then derive \\int \\hat{X}{(\\phi_2)} d\\phi_2 = A_{x} + \\sin{(\\phi_2)}, then derive A_{x} + \\sin{(\\phi_2)} = P_{g} + \\sin{(\\phi_2)}, then obtain 2 A_{x} + \\sin{(\\phi_2)} = A_{x} + P_{g} + \\sin{(\\phi_2)}", "derivation": "\\hat{X}{(\\phi_2)} = \\cos{(\\phi_2)} and \\int \\hat{X}{(\\phi_2)} d\\phi_2 = \\int \\cos{(\\phi_2)} d\\phi_2 and \\int \\hat{X}{(\\phi_2)} d\\phi_2 = A_{x} + \\sin{(\\phi_2)} and A_{x} + \\sin{(\\phi_2)} = \\int \\cos{(\\phi_2)} d\\phi_2 and A_{x} + \\sin{(\\phi_2)} = P_{g} + \\sin{(\\phi_2)} and 2 A_{x} + \\sin{(\\phi_2)} = A_{x} + P_{g} + \\sin{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\phi_2', commutative=True))), Integral(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('P_g', commutative=True), sin(Symbol('\\\\phi_2', commutative=True))))"], [["add", 5, "Symbol('A_x', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('A_x', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('A_x', commutative=True), Symbol('P_g', commutative=True), sin(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(U)} = \\frac{d}{d U} \\cos{(U)}, then derive \\operatorname{x^{{\\}'}}{(U)} = - \\sin{(U)}, then obtain (\\operatorname{x^{{\\}'}}{(U)} + \\frac{d}{d U} \\cos{(U)}) \\frac{d}{d U} \\cos{(U)} = - 2 \\sin{(U)} \\frac{d}{d U} \\cos{(U)}", "derivation": "\\operatorname{x^{{\\}'}}{(U)} = \\frac{d}{d U} \\cos{(U)} and 2 \\operatorname{x^{{\\}'}}{(U)} = \\operatorname{x^{{\\}'}}{(U)} + \\frac{d}{d U} \\cos{(U)} and \\operatorname{x^{{\\}'}}{(U)} = - \\sin{(U)} and 2 \\operatorname{x^{{\\}'}}{(U)} \\frac{d}{d U} \\cos{(U)} = - 2 \\sin{(U)} \\frac{d}{d U} \\cos{(U)} and (\\operatorname{x^{{\\}'}}{(U)} + \\frac{d}{d U} \\cos{(U)}) \\frac{d}{d U} \\cos{(U)} = - 2 \\sin{(U)} \\frac{d}{d U} \\cos{(U)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["add", 1, "Function('x^\\\\prime')(Symbol('U', commutative=True))"], "Equality(Mul(Integer(2), Function('x^\\\\prime')(Symbol('U', commutative=True))), Add(Function('x^\\\\prime')(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('x^\\\\prime')(Symbol('U', commutative=True)), Mul(Integer(-1), sin(Symbol('U', commutative=True))))"], [["times", 3, "Mul(Integer(2), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), Function('x^\\\\prime')(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Function('x^\\\\prime')(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('U', commutative=True)), Derivative(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(F_{x})} = e^{F_{x}}, then derive (\\frac{d}{d F_{x}} \\hat{H}{(F_{x})})^{F_{x}} = (e^{F_{x}})^{F_{x}}, then obtain (\\frac{d}{d F_{x}} \\hat{H}{(F_{x})})^{F_{x}} = \\hat{H}^{F_{x}}{(F_{x})}", "derivation": "\\hat{H}{(F_{x})} = e^{F_{x}} and \\frac{d}{d F_{x}} \\hat{H}{(F_{x})} = \\frac{d}{d F_{x}} e^{F_{x}} and (\\frac{d}{d F_{x}} \\hat{H}{(F_{x})})^{F_{x}} = (\\frac{d}{d F_{x}} e^{F_{x}})^{F_{x}} and (\\frac{d}{d F_{x}} \\hat{H}{(F_{x})})^{F_{x}} = (e^{F_{x}})^{F_{x}} and (\\frac{d}{d F_{x}} \\hat{H}{(F_{x})})^{F_{x}} = \\hat{H}^{F_{x}}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Derivative(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(exp(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Function('\\\\hat{H}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given y{(\\hat{x}_0,v_{y})} = \\frac{v_{y}}{\\hat{x}_0} and \\mathbf{P}{(\\hat{x}_0,v_{y})} = \\frac{v_{y}}{\\hat{x}_0}, then obtain \\mathbf{P}^{v_{y}}{(\\hat{x}_0,v_{y})} = y^{v_{y}}{(\\hat{x}_0,v_{y})}", "derivation": "y{(\\hat{x}_0,v_{y})} = \\frac{v_{y}}{\\hat{x}_0} and \\mathbf{P}{(\\hat{x}_0,v_{y})} = \\frac{v_{y}}{\\hat{x}_0} and \\mathbf{P}{(\\hat{x}_0,v_{y})} = y{(\\hat{x}_0,v_{y})} and \\mathbf{P}^{v_{y}}{(\\hat{x}_0,v_{y})} = y^{v_{y}}{(\\hat{x}_0,v_{y})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(Function('y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given s{(\\sigma_x)} = \\log{(\\sigma_x)}, then obtain \\frac{(s{(\\sigma_x)} - \\frac{1}{\\sigma_x}) \\log{(\\sigma_x)}}{\\sigma_x} = \\frac{(\\log{(\\sigma_x)} - \\frac{1}{\\sigma_x}) \\log{(\\sigma_x)}}{\\sigma_x}", "derivation": "s{(\\sigma_x)} = \\log{(\\sigma_x)} and \\frac{s{(\\sigma_x)}}{\\sigma_x} = \\frac{\\log{(\\sigma_x)}}{\\sigma_x} and s{(\\sigma_x)} - \\frac{1}{\\sigma_x} = \\log{(\\sigma_x)} - \\frac{1}{\\sigma_x} and \\frac{(s{(\\sigma_x)} - \\frac{1}{\\sigma_x}) s{(\\sigma_x)}}{\\sigma_x} = \\frac{(\\log{(\\sigma_x)} - \\frac{1}{\\sigma_x}) s{(\\sigma_x)}}{\\sigma_x} and \\frac{(s{(\\sigma_x)} - \\frac{1}{\\sigma_x}) \\log{(\\sigma_x)}}{\\sigma_x} = \\frac{(\\log{(\\sigma_x)} - \\frac{1}{\\sigma_x}) \\log{(\\sigma_x)}}{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))"], "Equality(Add(Function('s')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Add(log(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))))"], [["times", 3, "Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Function('s')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Function('s')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(log(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), Function('s')(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Function('s')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), log(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(log(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))), log(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\eta{(x^\\prime)} = \\cos{(x^\\prime)}, then derive \\int \\eta{(x^\\prime)} dx^\\prime = U + \\sin{(x^\\prime)}, then obtain (U + \\sin{(x^\\prime)})^{2} = (U + \\sin{(x^\\prime)}) \\int \\cos{(x^\\prime)} dx^\\prime", "derivation": "\\eta{(x^\\prime)} = \\cos{(x^\\prime)} and \\int \\eta{(x^\\prime)} dx^\\prime = \\int \\cos{(x^\\prime)} dx^\\prime and (\\int \\eta{(x^\\prime)} dx^\\prime)^{2} = (\\int \\eta{(x^\\prime)} dx^\\prime) \\int \\cos{(x^\\prime)} dx^\\prime and \\int \\eta{(x^\\prime)} dx^\\prime = U + \\sin{(x^\\prime)} and (U + \\sin{(x^\\prime)})^{2} = (U + \\sin{(x^\\prime)}) \\int \\cos{(x^\\prime)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(2)), Mul(Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('U', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('U', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))), Integer(2)), Mul(Add(Symbol('U', commutative=True), sin(Symbol('x^\\\\prime', commutative=True))), Integral(cos(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})}, then derive \\int \\frac{\\operatorname{n_{1}}{(\\mathbf{D})}}{\\cos{(\\mathbf{D})}} d\\mathbf{D} = \\mathbf{D} + n_{1}, then obtain (\\mathbf{D} + n_{1})^{\\mathbf{D}} = (C_{1} + \\mathbf{D})^{\\mathbf{D}}", "derivation": "\\operatorname{n_{1}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and \\frac{\\operatorname{n_{1}}{(\\mathbf{D})}}{\\cos{(\\mathbf{D})}} = 1 and \\int \\frac{\\operatorname{n_{1}}{(\\mathbf{D})}}{\\cos{(\\mathbf{D})}} d\\mathbf{D} = \\int 1 d\\mathbf{D} and \\int \\frac{\\operatorname{n_{1}}{(\\mathbf{D})}}{\\cos{(\\mathbf{D})}} d\\mathbf{D} = \\mathbf{D} + n_{1} and \\mathbf{D} + n_{1} = \\int 1 d\\mathbf{D} and (\\mathbf{D} + n_{1})^{\\mathbf{D}} = (\\int 1 d\\mathbf{D})^{\\mathbf{D}} and (\\mathbf{D} + n_{1})^{\\mathbf{D}} = (C_{1} + \\mathbf{D})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Mul(Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('n_1')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(A_{x},C_{2})} = \\log{(A_{x} - C_{2})}, then derive \\int \\varphi^{*}{(A_{x},C_{2})} dA_{x} = A_{x} \\log{(A_{x} - C_{2})} - A_{x} - C_{2} \\log{(A_{x} - C_{2})} + \\psi^*, then obtain A_{x} \\log{(A_{x} - C_{2})} - A_{x} - C_{2} \\log{(A_{x} - C_{2})} + \\psi^* = A_{x} \\log{(A_{x} - C_{2})} - A_{x} + B - C_{2} \\log{(A_{x} - C_{2})}", "derivation": "\\varphi^{*}{(A_{x},C_{2})} = \\log{(A_{x} - C_{2})} and \\int \\varphi^{*}{(A_{x},C_{2})} dA_{x} = \\int \\log{(A_{x} - C_{2})} dA_{x} and \\int \\varphi^{*}{(A_{x},C_{2})} dA_{x} = A_{x} \\log{(A_{x} - C_{2})} - A_{x} - C_{2} \\log{(A_{x} - C_{2})} + \\psi^* and A_{x} \\log{(A_{x} - C_{2})} - A_{x} - C_{2} \\log{(A_{x} - C_{2})} + \\psi^* = \\int \\log{(A_{x} - C_{2})} dA_{x} and A_{x} \\log{(A_{x} - C_{2})} - A_{x} - C_{2} \\log{(A_{x} - C_{2})} + \\psi^* = A_{x} \\log{(A_{x} - C_{2})} - A_{x} + B - C_{2} \\log{(A_{x} - C_{2})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('C_2', commutative=True)), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True)))))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Mul(Symbol('A_x', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('A_x', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Symbol('\\\\psi^*', commutative=True)), Integral(log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True)))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('A_x', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Symbol('\\\\psi^*', commutative=True)), Add(Mul(Symbol('A_x', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True))))), Mul(Integer(-1), Symbol('A_x', commutative=True)), Symbol('B', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True), log(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('C_2', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = \\frac{e^{\\mathbf{B}}}{\\omega} and n{(\\mathbf{B})} = e^{\\mathbf{B}}, then obtain \\frac{\\partial}{\\partial \\omega} \\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = - \\frac{n{(\\mathbf{B})}}{\\omega^{2}}", "derivation": "\\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = \\frac{e^{\\mathbf{B}}}{\\omega} and \\frac{\\partial}{\\partial \\omega} \\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = \\frac{\\partial}{\\partial \\omega} \\frac{e^{\\mathbf{B}}}{\\omega} and n{(\\mathbf{B})} = e^{\\mathbf{B}} and \\frac{\\partial}{\\partial \\omega} \\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = \\frac{\\partial}{\\partial \\omega} \\frac{n{(\\mathbf{B})}}{\\omega} and \\frac{\\partial}{\\partial \\omega} \\operatorname{t_{1}}{(\\omega,\\mathbf{B})} = - \\frac{n{(\\mathbf{B})}}{\\omega^{2}}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('t_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('n')(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('t_1')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Integer(-2)), Function('n')(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\ddot{x},v_{y})} = - \\ddot{x} + \\sin{(v_{y})} and \\sigma_{p}{(v_{y})} = \\sin{(v_{y})}, then obtain \\frac{\\partial}{\\partial \\ddot{x}} \\operatorname{F_{N}}{(\\ddot{x},v_{y})} = \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} + \\sigma_{p}{(v_{y})})", "derivation": "\\operatorname{F_{N}}{(\\ddot{x},v_{y})} = - \\ddot{x} + \\sin{(v_{y})} and \\frac{\\partial}{\\partial \\ddot{x}} \\operatorname{F_{N}}{(\\ddot{x},v_{y})} = \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} + \\sin{(v_{y})}) and \\sigma_{p}{(v_{y})} = \\sin{(v_{y})} and \\frac{\\partial}{\\partial \\ddot{x}} \\operatorname{F_{N}}{(\\ddot{x},v_{y})} = \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} + \\sigma_{p}{(v_{y})})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('F_N')(Symbol('\\\\ddot{x}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\sigma_p')(Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then obtain \\cos{((- \\mathbf{B})^{\\mathbf{B}} + \\sin{(\\mathbf{B})})} = \\cos{((- \\mathbf{B} - \\phi_{2}{(\\mathbf{B})} + \\sin{(\\mathbf{B})})^{\\mathbf{B}} + \\sin{(\\mathbf{B})})}", "derivation": "\\phi_{2}{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and - \\mathbf{B} + \\phi_{2}{(\\mathbf{B})} = - \\mathbf{B} + \\sin{(\\mathbf{B})} and - \\mathbf{B} = - \\mathbf{B} - \\phi_{2}{(\\mathbf{B})} + \\sin{(\\mathbf{B})} and (- \\mathbf{B})^{\\mathbf{B}} = (- \\mathbf{B} - \\phi_{2}{(\\mathbf{B})} + \\sin{(\\mathbf{B})})^{\\mathbf{B}} and (- \\mathbf{B})^{\\mathbf{B}} + \\sin{(\\mathbf{B})} = (- \\mathbf{B} - \\phi_{2}{(\\mathbf{B})} + \\sin{(\\mathbf{B})})^{\\mathbf{B}} + \\sin{(\\mathbf{B})} and \\cos{((- \\mathbf{B})^{\\mathbf{B}} + \\sin{(\\mathbf{B})})} = \\cos{((- \\mathbf{B} - \\phi_{2}{(\\mathbf{B})} + \\sin{(\\mathbf{B})})^{\\mathbf{B}} + \\sin{(\\mathbf{B})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["minus", 2, "Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 4, "Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["cos", 5], "Equality(cos(Add(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))), cos(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('\\\\mathbf{B}', commutative=True))), sin(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(v_{x},\\dot{x})} = \\log{(v_{x}^{\\dot{x}})}, then obtain \\frac{d}{d v_{x}} (\\dot{x} + 1) = \\frac{\\partial}{\\partial v_{x}} (\\dot{x} + \\frac{\\log{(v_{x}^{\\dot{x}})}}{\\ddot{x}{(v_{x},\\dot{x})}})", "derivation": "\\ddot{x}{(v_{x},\\dot{x})} = \\log{(v_{x}^{\\dot{x}})} and 1 = \\frac{\\log{(v_{x}^{\\dot{x}})}}{\\ddot{x}{(v_{x},\\dot{x})}} and \\dot{x} + 1 = \\dot{x} + \\frac{\\log{(v_{x}^{\\dot{x}})}}{\\ddot{x}{(v_{x},\\dot{x})}} and \\frac{d}{d v_{x}} (\\dot{x} + 1) = \\frac{\\partial}{\\partial v_{x}} (\\dot{x} + \\frac{\\log{(v_{x}^{\\dot{x}})}}{\\ddot{x}{(v_{x},\\dot{x})}})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), log(Pow(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 1, "Function('\\\\ddot{x}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\ddot{x}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), log(Pow(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["add", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Pow(Function('\\\\ddot{x}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), log(Pow(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))))))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Pow(Function('\\\\ddot{x}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), log(Pow(Symbol('v_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} = C_{d}^{E_{\\lambda}} and \\mathbf{H}{(C_{d},E_{\\lambda})} = C_{d}^{- E_{\\lambda}} \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})}, then obtain \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + \\mathbf{H}{(C_{d},E_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + 1", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} = C_{d}^{E_{\\lambda}} and C_{d}^{- E_{\\lambda}} \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} = 1 and \\mathbf{H}{(C_{d},E_{\\lambda})} = C_{d}^{- E_{\\lambda}} \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} and \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + C_{d}^{- E_{\\lambda}} \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + 1 and \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + \\mathbf{H}{(C_{d},E_{\\lambda})} = \\operatorname{V_{\\mathbf{E}}}{(C_{d},E_{\\lambda})} + 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Pow(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 2, "Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Function('V_{\\\\mathbf{E}}')(Symbol('C_d', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mu_{0}{(a,T)} = T a, then obtain \\frac{\\mu_{0}^{T}{(a,T)}}{- T a + \\mu_{0}^{T}{(a,T)}} = \\frac{(T a)^{T}}{- T a + \\mu_{0}^{T}{(a,T)}}", "derivation": "\\mu_{0}{(a,T)} = T a and \\mu_{0}^{T}{(a,T)} = (T a)^{T} and - T a + \\mu_{0}^{T}{(a,T)} = - T a + (T a)^{T} and \\frac{\\mu_{0}^{T}{(a,T)}}{- T a + (T a)^{T}} = \\frac{(T a)^{T}}{- T a + (T a)^{T}} and \\frac{\\mu_{0}^{T}{(a,T)}}{- T a + \\mu_{0}^{T}{(a,T)}} = \\frac{(T a)^{T}}{- T a + \\mu_{0}^{T}{(a,T)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True)))"], [["minus", 2, "Mul(Symbol('T', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True))), Integer(-1)), Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Integer(-1)), Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Pow(Mul(Symbol('T', commutative=True), Symbol('a', commutative=True)), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('a', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('a', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given H{(\\nabla)} = \\log{(\\nabla)}, then obtain \\frac{\\nabla \\int (- \\nabla + H{(\\nabla)}) d\\nabla}{h} = \\frac{\\nabla (G - \\frac{\\nabla^{2}}{2} + \\nabla \\log{(\\nabla)} - \\nabla)}{h}", "derivation": "H{(\\nabla)} = \\log{(\\nabla)} and - \\nabla + H{(\\nabla)} = - \\nabla + \\log{(\\nabla)} and \\int (- \\nabla + H{(\\nabla)}) d\\nabla = \\int (- \\nabla + \\log{(\\nabla)}) d\\nabla and - \\frac{\\int (- \\nabla + H{(\\nabla)}) d\\nabla}{h} = - \\frac{\\int (- \\nabla + \\log{(\\nabla)}) d\\nabla}{h} and \\frac{\\nabla \\int (- \\nabla + H{(\\nabla)}) d\\nabla}{h} = \\frac{\\nabla \\int (- \\nabla + \\log{(\\nabla)}) d\\nabla}{h} and \\frac{\\nabla \\int (- \\nabla + H{(\\nabla)}) d\\nabla}{h} = \\frac{\\nabla (G - \\frac{\\nabla^{2}}{2} + \\nabla \\log{(\\nabla)} - \\nabla)}{h}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["minus", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('H')(Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('H')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('H')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('H')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('H')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\nabla', commutative=True), Integer(2))), Mul(Symbol('\\\\nabla', commutative=True), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\pi{(V)} = \\log{(V)}, then obtain \\frac{d}{d V} ((- V + \\pi{(V)}) \\pi{(V)})^{V} = \\frac{d}{d V} ((- V + \\log{(V)}) \\pi{(V)})^{V}", "derivation": "\\pi{(V)} = \\log{(V)} and - V + \\pi{(V)} = - V + \\log{(V)} and (- V + \\pi{(V)}) \\pi{(V)} = (- V + \\log{(V)}) \\pi{(V)} and ((- V + \\pi{(V)}) \\pi{(V)})^{V} = ((- V + \\log{(V)}) \\pi{(V)})^{V} and \\frac{d}{d V} ((- V + \\pi{(V)}) \\pi{(V)})^{V} = \\frac{d}{d V} ((- V + \\log{(V)}) \\pi{(V)})^{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["minus", 1, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\pi')(Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('V', commutative=True))))"], [["times", 2, "Function('\\\\pi')(Symbol('V', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\pi')(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\pi')(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\pi')(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Mul(Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Function('\\\\pi')(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(\\eta)} = \\cos{(\\eta)}, then obtain \\int (\\int \\mu{(\\eta)} d\\eta)^{\\eta} d\\eta = \\int (\\int \\cos{(\\eta)} d\\eta)^{\\eta} d\\eta", "derivation": "\\mu{(\\eta)} = \\cos{(\\eta)} and \\int \\mu{(\\eta)} d\\eta = \\int \\cos{(\\eta)} d\\eta and (\\int \\mu{(\\eta)} d\\eta)^{\\eta} = (\\int \\cos{(\\eta)} d\\eta)^{\\eta} and \\int (\\int \\mu{(\\eta)} d\\eta)^{\\eta} d\\eta = \\int (\\int \\cos{(\\eta)} d\\eta)^{\\eta} d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mu')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\mu')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Pow(Integral(cos(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\mathbf{J})} = \\log{(\\cos{(\\mathbf{J})})}, then obtain (\\frac{d}{d \\mathbf{J}} \\theta^{\\mathbf{J}}{(\\mathbf{J})})^{\\mathbf{J}} = (\\frac{d}{d \\mathbf{J}} \\log{(\\cos{(\\mathbf{J})})}^{\\mathbf{J}})^{\\mathbf{J}}", "derivation": "\\theta{(\\mathbf{J})} = \\log{(\\cos{(\\mathbf{J})})} and \\theta^{\\mathbf{J}}{(\\mathbf{J})} = \\log{(\\cos{(\\mathbf{J})})}^{\\mathbf{J}} and \\frac{d}{d \\mathbf{J}} \\theta^{\\mathbf{J}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\cos{(\\mathbf{J})})}^{\\mathbf{J}} and (\\frac{d}{d \\mathbf{J}} \\theta^{\\mathbf{J}}{(\\mathbf{J})})^{\\mathbf{J}} = (\\frac{d}{d \\mathbf{J}} \\log{(\\cos{(\\mathbf{J})})}^{\\mathbf{J}})^{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{J}', commutative=True)), log(cos(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\theta')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Pow(log(cos(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\theta')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Derivative(Pow(log(cos(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = \\log{(e^{g^{\\prime}_{\\varepsilon}})}, then derive \\int \\mathbf{A}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\frac{(g^{\\prime}_{\\varepsilon})^{2}}{2} + m_{s}, then obtain \\int \\log{(e^{g^{\\prime}_{\\varepsilon}})} dg^{\\prime}_{\\varepsilon} = \\frac{(g^{\\prime}_{\\varepsilon})^{2}}{2} + m_{s}", "derivation": "\\mathbf{A}{(g^{\\prime}_{\\varepsilon})} = \\log{(e^{g^{\\prime}_{\\varepsilon}})} and \\int \\mathbf{A}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int \\log{(e^{g^{\\prime}_{\\varepsilon}})} dg^{\\prime}_{\\varepsilon} and \\int \\mathbf{A}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\frac{(g^{\\prime}_{\\varepsilon})^{2}}{2} + m_{s} and \\int \\log{(e^{g^{\\prime}_{\\varepsilon}})} dg^{\\prime}_{\\varepsilon} = \\frac{(g^{\\prime}_{\\varepsilon})^{2}}{2} + m_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(log(exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2))), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(exp(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(2))), Symbol('m_s', commutative=True)))"]]}, {"prompt": "Given L{(\\chi,L_{\\varepsilon})} = \\cos^{L_{\\varepsilon}}{(\\chi)}, then derive \\frac{\\partial}{\\partial L_{\\varepsilon}} L{(\\chi,L_{\\varepsilon})} = \\log{(\\cos{(\\chi)})} \\cos^{L_{\\varepsilon}}{(\\chi)}, then obtain \\frac{\\partial}{\\partial L_{\\varepsilon}} \\cos^{L_{\\varepsilon}}{(\\chi)} = \\log{(\\cos{(\\chi)})} \\cos^{L_{\\varepsilon}}{(\\chi)}", "derivation": "L{(\\chi,L_{\\varepsilon})} = \\cos^{L_{\\varepsilon}}{(\\chi)} and \\frac{\\partial}{\\partial L_{\\varepsilon}} L{(\\chi,L_{\\varepsilon})} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\cos^{L_{\\varepsilon}}{(\\chi)} and \\frac{\\partial}{\\partial L_{\\varepsilon}} L{(\\chi,L_{\\varepsilon})} = \\log{(\\cos{(\\chi)})} \\cos^{L_{\\varepsilon}}{(\\chi)} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\cos^{L_{\\varepsilon}}{(\\chi)} = \\log{(\\cos{(\\chi)})} \\cos^{L_{\\varepsilon}}{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(log(cos(Symbol('\\\\chi', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(log(cos(Symbol('\\\\chi', commutative=True))), Pow(cos(Symbol('\\\\chi', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\delta,\\varphi^*)} = \\cos^{\\delta}{(\\varphi^*)}, then obtain \\frac{\\delta^{2} \\sin{(\\varphi^*)} \\cos^{\\delta}{(\\varphi^*)}}{\\cos{(\\varphi^*)}} + \\delta \\nabla{(\\delta,\\varphi^*)} = \\frac{\\delta^{2} \\sin{(\\varphi^*)} \\cos^{\\delta}{(\\varphi^*)}}{\\cos{(\\varphi^*)}} + \\delta \\cos^{\\delta}{(\\varphi^*)}", "derivation": "\\nabla{(\\delta,\\varphi^*)} = \\cos^{\\delta}{(\\varphi^*)} and \\delta \\nabla{(\\delta,\\varphi^*)} = \\delta \\cos^{\\delta}{(\\varphi^*)} and \\delta \\nabla{(\\delta,\\varphi^*)} - \\frac{\\partial}{\\partial \\varphi^*} \\delta \\cos^{\\delta}{(\\varphi^*)} = \\delta \\cos^{\\delta}{(\\varphi^*)} - \\frac{\\partial}{\\partial \\varphi^*} \\delta \\cos^{\\delta}{(\\varphi^*)} and \\frac{\\delta^{2} \\sin{(\\varphi^*)} \\cos^{\\delta}{(\\varphi^*)}}{\\cos{(\\varphi^*)}} + \\delta \\nabla{(\\delta,\\varphi^*)} = \\frac{\\delta^{2} \\sin{(\\varphi^*)} \\cos^{\\delta}{(\\varphi^*)}}{\\cos{(\\varphi^*)}} + \\delta \\cos^{\\delta}{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\delta', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\nabla')(Symbol('\\\\delta', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\nabla')(Symbol('\\\\delta', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))), Add(Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), sin(Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\nabla')(Symbol('\\\\delta', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), sin(Symbol('\\\\varphi^*', commutative=True)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(f)} = \\log{(f)} and \\theta_{1}{(f)} = \\operatorname{c_{0}}^{f}{(f)}, then obtain \\frac{d}{d f} \\cos{(\\log{(\\int \\operatorname{c_{0}}^{f}{(f)} df)})} = \\frac{d}{d f} \\cos{(\\log{(\\int \\log{(f)}^{f} df)})}", "derivation": "\\operatorname{c_{0}}{(f)} = \\log{(f)} and \\theta_{1}{(f)} = \\operatorname{c_{0}}^{f}{(f)} and \\int \\theta_{1}{(f)} df = \\int \\operatorname{c_{0}}^{f}{(f)} df and \\log{(\\int \\theta_{1}{(f)} df)} = \\log{(\\int \\operatorname{c_{0}}^{f}{(f)} df)} and \\log{(\\int \\theta_{1}{(f)} df)} = \\log{(\\int \\log{(f)}^{f} df)} and \\log{(\\int \\operatorname{c_{0}}^{f}{(f)} df)} = \\log{(\\int \\log{(f)}^{f} df)} and \\cos{(\\log{(\\int \\operatorname{c_{0}}^{f}{(f)} df)})} = \\cos{(\\log{(\\int \\log{(f)}^{f} df)})} and \\frac{d}{d f} \\cos{(\\log{(\\int \\operatorname{c_{0}}^{f}{(f)} df)})} = \\frac{d}{d f} \\cos{(\\log{(\\int \\log{(f)}^{f} df)})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('f', commutative=True)), Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["integrate", 2, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["log", 3], "Equality(log(Integral(Function('\\\\theta_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), log(Integral(Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(log(Integral(Function('\\\\theta_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), log(Integral(Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(log(Integral(Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), log(Integral(Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["cos", 6], "Equality(cos(log(Integral(Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), cos(log(Integral(Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"], [["differentiate", 7, "Symbol('f', commutative=True)"], "Equality(Derivative(cos(log(Integral(Pow(Function('c_0')(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(cos(log(Integral(Pow(log(Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(M)} = \\log{(\\cos{(M)})}, then derive M + m = \\int \\frac{\\log{(\\cos{(M)})}}{\\psi{(M)}} dM, then derive M + m + \\log{(\\cos{(M)})} = M + y + \\log{(\\cos{(M)})}, then obtain \\frac{M + m + \\log{(\\cos{(M)})}}{\\phi_1} = \\frac{M + y + \\log{(\\cos{(M)})}}{\\phi_1}", "derivation": "\\psi{(M)} = \\log{(\\cos{(M)})} and \\psi^{2}{(M)} = \\psi{(M)} \\log{(\\cos{(M)})} and 1 = \\frac{\\log{(\\cos{(M)})}}{\\psi{(M)}} and \\int 1 dM = \\int \\frac{\\log{(\\cos{(M)})}}{\\psi{(M)}} dM and M + m = \\int \\frac{\\log{(\\cos{(M)})}}{\\psi{(M)}} dM and M + m = \\int 1 dM and M + m + \\log{(\\cos{(M)})} = \\log{(\\cos{(M)})} + \\int 1 dM and M + m + \\log{(\\cos{(M)})} = M + y + \\log{(\\cos{(M)})} and \\frac{M + m + \\log{(\\cos{(M)})}}{\\phi_1} = \\frac{M + y + \\log{(\\cos{(M)})}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('M', commutative=True)), log(cos(Symbol('M', commutative=True))))"], [["times", 1, "Function('\\\\psi')(Symbol('M', commutative=True))"], "Equality(Pow(Function('\\\\psi')(Symbol('M', commutative=True)), Integer(2)), Mul(Function('\\\\psi')(Symbol('M', commutative=True)), log(cos(Symbol('M', commutative=True)))))"], [["divide", 2, "Pow(Function('\\\\psi')(Symbol('M', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('\\\\psi')(Symbol('M', commutative=True)), Integer(-1)), log(cos(Symbol('M', commutative=True)))))"], [["integrate", 3, "Symbol('M', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('M', commutative=True))), Integral(Mul(Pow(Function('\\\\psi')(Symbol('M', commutative=True)), Integer(-1)), log(cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('M', commutative=True), Symbol('m', commutative=True)), Integral(Mul(Pow(Function('\\\\psi')(Symbol('M', commutative=True)), Integer(-1)), log(cos(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('M', commutative=True), Symbol('m', commutative=True)), Integral(Integer(1), Tuple(Symbol('M', commutative=True))))"], [["add", 6, "log(cos(Symbol('M', commutative=True)))"], "Equality(Add(Symbol('M', commutative=True), Symbol('m', commutative=True), log(cos(Symbol('M', commutative=True)))), Add(log(cos(Symbol('M', commutative=True))), Integral(Integer(1), Tuple(Symbol('M', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('M', commutative=True), Symbol('m', commutative=True), log(cos(Symbol('M', commutative=True)))), Add(Symbol('M', commutative=True), Symbol('y', commutative=True), log(cos(Symbol('M', commutative=True)))))"], [["divide", 8, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('m', commutative=True), log(cos(Symbol('M', commutative=True))))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Symbol('M', commutative=True), Symbol('y', commutative=True), log(cos(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given r{(g,p)} = g - p, then obtain 2 p r{(g,p)} - p = 2 p (g - p) - p", "derivation": "r{(g,p)} = g - p and p r{(g,p)} = p (g - p) and 2 p r{(g,p)} = p (g - p) + p r{(g,p)} and 2 p r{(g,p)} - p = p (g - p) + p r{(g,p)} - p and p r{(g,p)} - p = p (g - p) - p and 2 p r{(g,p)} - p = 2 p (g - p) - p", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True)), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))))"], [["add", 2, "Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('p', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True)))))"], [["minus", 3, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Symbol('p', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["minus", 4, "Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True)))"], "Equality(Add(Mul(Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Symbol('p', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(2), Symbol('p', commutative=True), Function('r')(Symbol('g', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Add(Mul(Integer(2), Symbol('p', commutative=True), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)))), Mul(Integer(-1), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(q,L)} = L q, then obtain 2 q (2 L q + 2 q + \\operatorname{y^{\\prime}}{(q,L)}) = 2 q (L q + 2 q + 2 \\operatorname{y^{\\prime}}{(q,L)})", "derivation": "\\operatorname{y^{\\prime}}{(q,L)} = L q and q + \\operatorname{y^{\\prime}}{(q,L)} = L q + q and L q + 2 q + \\operatorname{y^{\\prime}}{(q,L)} = 2 L q + 2 q and 2 L q + 2 q + \\operatorname{y^{\\prime}}{(q,L)} = 3 L q + 2 q and L q + 2 q + 2 \\operatorname{y^{\\prime}}{(q,L)} = 3 L q + 2 q and 2 L q + 2 q + \\operatorname{y^{\\prime}}{(q,L)} = L q + 2 q + 2 \\operatorname{y^{\\prime}}{(q,L)} and 2 q (2 L q + 2 q + \\operatorname{y^{\\prime}}{(q,L)}) = 2 q (L q + 2 q + 2 \\operatorname{y^{\\prime}}{(q,L)})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["add", 2, "Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(2), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))))"], [["add", 1, "Add(Mul(Integer(2), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(3), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True)))), Add(Mul(Integer(3), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(2), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True)))))"], [["times", 6, "Mul(Integer(2), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Symbol('q', commutative=True), Add(Mul(Integer(2), Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True)))), Mul(Integer(2), Symbol('q', commutative=True), Add(Mul(Symbol('L', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('q', commutative=True), Symbol('L', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}}, then obtain \\frac{\\operatorname{f_{E}}{(\\Psi^{\\dagger})} e^{\\Psi^{\\dagger}}}{\\Psi^{\\dagger}} = \\frac{e^{2 \\Psi^{\\dagger}}}{\\Psi^{\\dagger}}", "derivation": "\\operatorname{f_{E}}{(\\Psi^{\\dagger})} = \\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}} and \\operatorname{f_{E}}{(\\Psi^{\\dagger})} \\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}} = (\\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}})^{2} and \\frac{\\operatorname{f_{E}}{(\\Psi^{\\dagger})} \\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}}}{\\Psi^{\\dagger}} = \\frac{(\\frac{d}{d \\Psi^{\\dagger}} e^{\\Psi^{\\dagger}})^{2}}{\\Psi^{\\dagger}} and \\frac{\\operatorname{f_{E}}{(\\Psi^{\\dagger})} e^{\\Psi^{\\dagger}}}{\\Psi^{\\dagger}} = \\frac{e^{2 \\Psi^{\\dagger}}}{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Mul(Function('f_E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Pow(Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(2)))"], [["divide", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Derivative(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given H{(F_{c},v_{z})} = - F_{c} + v_{z}, then obtain (\\int \\frac{-1 + \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}}}{- F_{c} + v_{z}} dv_{z})^{F_{c}} = (\\int 0 dv_{z})^{F_{c}}", "derivation": "H{(F_{c},v_{z})} = - F_{c} + v_{z} and \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}} = 1 and -1 + \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}} = 0 and \\frac{-1 + \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}}}{- F_{c} + v_{z}} = 0 and \\int \\frac{-1 + \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}}}{- F_{c} + v_{z}} dv_{z} = \\int 0 dv_{z} and (\\int \\frac{-1 + \\frac{H{(F_{c},v_{z})}}{- F_{c} + v_{z}}}{- F_{c} + v_{z}} dv_{z})^{F_{c}} = (\\int 0 dv_{z})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True))), Integer(1))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True)))), Integer(0))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1))), Integer(0))"], [["integrate", 4, "Symbol('v_z', commutative=True)"], "Equality(Integral(Mul(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True))), Integral(Integer(0), Tuple(Symbol('v_z', commutative=True))))"], [["power", 5, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integral(Mul(Add(Integer(-1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1)), Function('H')(Symbol('F_c', commutative=True), Symbol('v_z', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('v_z', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('v_z', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given k{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})}, then obtain \\int (2 k^{2}{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})}) d\\hat{\\mathbf{r}} = \\int ((k{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})}) k{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})}) d\\hat{\\mathbf{r}}", "derivation": "k{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and 2 k{(\\hat{\\mathbf{r}})} = k{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})} and 2 k^{2}{(\\hat{\\mathbf{r}})} = (k{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})}) k{(\\hat{\\mathbf{r}})} and 2 k^{2}{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})} = (k{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})}) k{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})} and \\int (2 k^{2}{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})}) d\\hat{\\mathbf{r}} = \\int ((k{(\\hat{\\mathbf{r}})} + \\sin{(\\hat{\\mathbf{r}})}) k{(\\hat{\\mathbf{r}})} + 2 k{(\\hat{\\mathbf{r}})}) d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["add", 1, "Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 2, "Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Add(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["add", 3, "Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Add(Mul(Add(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Pow(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Add(Mul(Add(Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(2), Function('k')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(U)} = \\sin{(U)}, then derive - \\sin{(U)} + \\int \\Omega{(U)} dU = \\lambda - \\sin{(U)} - \\cos{(U)}, then obtain (- \\Omega{(U)} + \\int \\Omega{(U)} dU)^{U} = (\\lambda - \\Omega{(U)} - \\cos{(U)})^{U}", "derivation": "\\Omega{(U)} = \\sin{(U)} and \\int \\Omega{(U)} dU = \\int \\sin{(U)} dU and - \\sin{(U)} + \\int \\Omega{(U)} dU = - \\sin{(U)} + \\int \\sin{(U)} dU and - \\Omega{(U)} + \\int \\Omega{(U)} dU = - \\Omega{(U)} + \\int \\sin{(U)} dU and - \\sin{(U)} + \\int \\Omega{(U)} dU = \\lambda - \\sin{(U)} - \\cos{(U)} and (- \\Omega{(U)} + \\int \\Omega{(U)} dU)^{U} = (- \\Omega{(U)} + \\int \\sin{(U)} dU)^{U} and - \\Omega{(U)} + \\int \\Omega{(U)} dU = \\lambda - \\Omega{(U)} - \\cos{(U)} and \\lambda - \\Omega{(U)} - \\cos{(U)} = - \\Omega{(U)} + \\int \\sin{(U)} dU and (\\lambda - \\Omega{(U)} - \\cos{(U)})^{U} = (- \\Omega{(U)} + \\int \\sin{(U)} dU)^{U} and (- \\Omega{(U)} + \\int \\Omega{(U)} dU)^{U} = (\\lambda - \\Omega{(U)} - \\cos{(U)})^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["minus", 2, "sin(Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), sin(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["power", 8, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 9], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Integral(Function('\\\\Omega')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('U', commutative=True))), Mul(Integer(-1), cos(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given Q{(t_{2},\\hat{x},\\mathbf{f})} = \\hat{x} (- \\mathbf{f} + t_{2}), then derive \\int Q{(t_{2},\\hat{x},\\mathbf{f})} dt_{2} = - \\hat{x} \\mathbf{f} t_{2} + \\frac{\\hat{x} t_{2}^{2}}{2} + i, then obtain (\\hat{x} (- \\mathbf{f} + t_{2}))^{t_{2}} \\int Q{(t_{2},\\hat{x},\\mathbf{f})} dt_{2} = (\\hat{x} (- \\mathbf{f} + t_{2}))^{t_{2}} (- \\hat{x} \\mathbf{f} t_{2} + \\frac{\\hat{x} t_{2}^{2}}{2} + i)", "derivation": "Q{(t_{2},\\hat{x},\\mathbf{f})} = \\hat{x} (- \\mathbf{f} + t_{2}) and \\int Q{(t_{2},\\hat{x},\\mathbf{f})} dt_{2} = \\int \\hat{x} (- \\mathbf{f} + t_{2}) dt_{2} and \\int Q{(t_{2},\\hat{x},\\mathbf{f})} dt_{2} = - \\hat{x} \\mathbf{f} t_{2} + \\frac{\\hat{x} t_{2}^{2}}{2} + i and (\\hat{x} (- \\mathbf{f} + t_{2}))^{t_{2}} \\int Q{(t_{2},\\hat{x},\\mathbf{f})} dt_{2} = (\\hat{x} (- \\mathbf{f} + t_{2}))^{t_{2}} (- \\hat{x} \\mathbf{f} t_{2} + \\frac{\\hat{x} t_{2}^{2}}{2} + i)", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('t_2', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))))"], [["integrate", 1, "Symbol('t_2', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('t_2', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Q')(Symbol('t_2', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Mul(Rational(1, 2), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(2))), Symbol('i', commutative=True)))"], [["times", 3, "Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Integral(Function('Q')(Symbol('t_2', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\hat{x}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('t_2', commutative=True)), Mul(Rational(1, 2), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(2))), Symbol('i', commutative=True))))"]]}, {"prompt": "Given r{(\\mathbf{B},A_{x})} = - \\mathbf{B} + e^{A_{x}} and \\mathbf{g}{(A_{x})} = e^{A_{x}}, then derive \\frac{\\partial}{\\partial A_{x}} r{(\\mathbf{B},A_{x})} = e^{A_{x}}, then obtain \\mathbf{g}{(A_{x})} = \\frac{\\partial}{\\partial A_{x}} (- \\mathbf{B} + e^{A_{x}})", "derivation": "r{(\\mathbf{B},A_{x})} = - \\mathbf{B} + e^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} r{(\\mathbf{B},A_{x})} = \\frac{\\partial}{\\partial A_{x}} (- \\mathbf{B} + e^{A_{x}}) and \\mathbf{g}{(A_{x})} = e^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} r{(\\mathbf{B},A_{x})} = e^{A_{x}} and \\frac{\\partial}{\\partial A_{x}} (- \\mathbf{B} + e^{A_{x}}) = e^{A_{x}} and \\mathbf{g}{(A_{x})} = \\frac{\\partial}{\\partial A_{x}} (- \\mathbf{B} + e^{A_{x}})", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('A_x', commutative=True))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A_x', commutative=True)), exp(Symbol('A_x', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), exp(Symbol('A_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), exp(Symbol('A_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\mathbf{g}')(Symbol('A_x', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(F_{H})} = e^{e^{F_{H}}}, then obtain \\frac{\\rho^{F_{H}}{(F_{H})} + e^{e^{F_{H}}} - (e^{e^{F_{H}}})^{F_{H}}}{F_{H}} = \\frac{e^{e^{F_{H}}}}{F_{H}}", "derivation": "\\rho{(F_{H})} = e^{e^{F_{H}}} and \\frac{\\rho{(F_{H})}}{F_{H}} = \\frac{e^{e^{F_{H}}}}{F_{H}} and \\rho^{F_{H}}{(F_{H})} = (e^{e^{F_{H}}})^{F_{H}} and \\rho^{F_{H}}{(F_{H})} + e^{e^{F_{H}}} = e^{e^{F_{H}}} + (e^{e^{F_{H}}})^{F_{H}} and \\rho^{F_{H}}{(F_{H})} + e^{e^{F_{H}}} - (e^{e^{F_{H}}})^{F_{H}} = e^{e^{F_{H}}} and \\rho{(F_{H})} = \\rho^{F_{H}}{(F_{H})} + e^{e^{F_{H}}} - (e^{e^{F_{H}}})^{F_{H}} and \\frac{\\rho^{F_{H}}{(F_{H})} + e^{e^{F_{H}}} - (e^{e^{F_{H}}})^{F_{H}}}{F_{H}} = \\frac{e^{e^{F_{H}}}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('F_H', commutative=True)), exp(exp(Symbol('F_H', commutative=True))))"], [["divide", 1, "Symbol('F_H', commutative=True)"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\rho')(Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), exp(exp(Symbol('F_H', commutative=True)))))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], [["add", 3, "exp(exp(Symbol('F_H', commutative=True)))"], "Equality(Add(Pow(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), exp(exp(Symbol('F_H', commutative=True)))), Add(exp(exp(Symbol('F_H', commutative=True))), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))"], "Equality(Add(Pow(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), exp(exp(Symbol('F_H', commutative=True))), Mul(Integer(-1), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))), exp(exp(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('\\\\rho')(Symbol('F_H', commutative=True)), Add(Pow(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), exp(exp(Symbol('F_H', commutative=True))), Mul(Integer(-1), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Pow(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), exp(exp(Symbol('F_H', commutative=True))), Mul(Integer(-1), Pow(exp(exp(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), exp(exp(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given G{(I)} = \\sin{(e^{I})} and \\mathbf{H}{(I)} = 2 \\frac{d}{d I} G{(I)}, then obtain I \\mathbf{H}{(I)} - \\sin{(e^{I})} = 2 I \\frac{d}{d I} G{(I)} - \\sin{(e^{I})}", "derivation": "G{(I)} = \\sin{(e^{I})} and \\frac{d}{d I} G{(I)} = \\frac{d}{d I} \\sin{(e^{I})} and \\mathbf{H}{(I)} = 2 \\frac{d}{d I} G{(I)} and I \\mathbf{H}{(I)} = 2 I \\frac{d}{d I} G{(I)} and I \\mathbf{H}{(I)} = 2 I \\frac{d}{d I} \\sin{(e^{I})} and 2 I \\frac{d}{d I} \\sin{(e^{I})} = 2 I \\frac{d}{d I} G{(I)} and 2 I \\frac{d}{d I} \\sin{(e^{I})} - \\sin{(e^{I})} = 2 I \\frac{d}{d I} G{(I)} - \\sin{(e^{I})} and I \\mathbf{H}{(I)} - \\sin{(e^{I})} = 2 I \\frac{d}{d I} G{(I)} - \\sin{(e^{I})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('I', commutative=True)), sin(exp(Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('I', commutative=True)), Mul(Integer(2), Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('I', commutative=True))), Mul(Integer(2), Symbol('I', commutative=True), Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('I', commutative=True))), Mul(Integer(2), Symbol('I', commutative=True), Derivative(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Symbol('I', commutative=True), Derivative(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('I', commutative=True), Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["minus", 6, "sin(exp(Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('I', commutative=True), Derivative(sin(exp(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), sin(exp(Symbol('I', commutative=True))))), Add(Mul(Integer(2), Symbol('I', commutative=True), Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), sin(exp(Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Mul(Symbol('I', commutative=True), Function('\\\\mathbf{H}')(Symbol('I', commutative=True))), Mul(Integer(-1), sin(exp(Symbol('I', commutative=True))))), Add(Mul(Integer(2), Symbol('I', commutative=True), Derivative(Function('G')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), sin(exp(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(h,x^\\prime,F_{x})} = h (x^\\prime)^{F_{x}} and \\chi{(h,x^\\prime,F_{x})} = \\frac{h (x^\\prime)^{F_{x}}}{F_{x}}, then obtain \\frac{\\operatorname{F_{H}}{(h,x^\\prime,F_{x})}}{F_{x} h} = \\frac{(x^\\prime)^{F_{x}}}{F_{x}}", "derivation": "\\operatorname{F_{H}}{(h,x^\\prime,F_{x})} = h (x^\\prime)^{F_{x}} and \\frac{\\operatorname{F_{H}}{(h,x^\\prime,F_{x})}}{F_{x}} = \\frac{h (x^\\prime)^{F_{x}}}{F_{x}} and \\chi{(h,x^\\prime,F_{x})} = \\frac{h (x^\\prime)^{F_{x}}}{F_{x}} and \\frac{\\operatorname{F_{H}}{(h,x^\\prime,F_{x})}}{F_{x}} = \\chi{(h,x^\\prime,F_{x})} and \\frac{\\operatorname{F_{H}}{(h,x^\\prime,F_{x})}}{F_{x} h} = \\frac{\\chi{(h,x^\\prime,F_{x})}}{h} and \\frac{\\operatorname{F_{H}}{(h,x^\\prime,F_{x})}}{F_{x} h} = \\frac{(x^\\prime)^{F_{x}}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('h', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"], [["divide", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('F_H')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('h', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Symbol('h', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('F_H')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Function('\\\\chi')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True)))"], [["divide", 4, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('F_H')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\chi')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('F_H')(Symbol('h', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then derive \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\varphi{(\\mathbf{r})} - 1 = -1, then obtain \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} - 1 = -1", "derivation": "\\varphi{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\varphi{(\\mathbf{r})} - \\cos{(\\mathbf{r})} = 0 and - \\mathbf{r} + \\varphi{(\\mathbf{r})} - \\cos{(\\mathbf{r})} = - \\mathbf{r} and \\frac{d}{d \\mathbf{r}} (- \\mathbf{r} + \\varphi{(\\mathbf{r})} - \\cos{(\\mathbf{r})}) = \\frac{d}{d \\mathbf{r}} - \\mathbf{r} and \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\varphi{(\\mathbf{r})} - 1 = -1 and \\sin{(\\mathbf{r})} + \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))), Integer(0))"], [["minus", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\varphi')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})} = \\frac{\\sin{(y^{\\prime})}}{V}, then obtain \\mathbf{f}^{C}{(C)} + \\int ((V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})})^{V})^{V} dV = \\mathbf{f}^{C}{(C)} + \\int ((V + \\frac{\\sin{(y^{\\prime})}}{V})^{V})^{V} dV", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})} = \\frac{\\sin{(y^{\\prime})}}{V} and V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})} = V + \\frac{\\sin{(y^{\\prime})}}{V} and (V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})})^{V} = (V + \\frac{\\sin{(y^{\\prime})}}{V})^{V} and ((V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})})^{V})^{V} = ((V + \\frac{\\sin{(y^{\\prime})}}{V})^{V})^{V} and \\int ((V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})})^{V})^{V} dV = \\int ((V + \\frac{\\sin{(y^{\\prime})}}{V})^{V})^{V} dV and \\mathbf{f}^{C}{(C)} + \\int ((V + \\operatorname{f_{\\mathbf{v}}}{(V,y^{\\prime})})^{V})^{V} dV = \\mathbf{f}^{C}{(C)} + \\int ((V + \\frac{\\sin{(y^{\\prime})}}{V})^{V})^{V} dV", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('V', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["power", 2, "Symbol('V', commutative=True)"], "Equality(Pow(Add(Symbol('V', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('V', commutative=True)), Pow(Add(Symbol('V', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('V', commutative=True)))"], [["power", 3, "Symbol('V', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('V', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Pow(Add(Symbol('V', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Pow(Pow(Add(Symbol('V', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(Pow(Add(Symbol('V', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 5, "Pow(Function('\\\\mathbf{f}')(Symbol('C', commutative=True)), Symbol('C', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Pow(Pow(Add(Symbol('V', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('V', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Pow(Function('\\\\mathbf{f}')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Pow(Pow(Add(Symbol('V', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), sin(Symbol('y^{\\\\prime}', commutative=True)))), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(C_{2},E_{x})} = C_{2} E_{x}, then obtain \\frac{\\frac{\\partial}{\\partial E_{x}} (C_{2} E_{x} + 2 C_{2} + \\operatorname{z^{*}}{(C_{2},E_{x})})}{E_{x}} = \\frac{\\frac{\\partial}{\\partial E_{x}} (2 C_{2} E_{x} + 2 C_{2})}{E_{x}}", "derivation": "\\operatorname{z^{*}}{(C_{2},E_{x})} = C_{2} E_{x} and C_{2} + \\operatorname{z^{*}}{(C_{2},E_{x})} = C_{2} E_{x} + C_{2} and C_{2} E_{x} + 2 C_{2} + \\operatorname{z^{*}}{(C_{2},E_{x})} = 2 C_{2} E_{x} + 2 C_{2} and \\frac{\\partial}{\\partial E_{x}} (C_{2} E_{x} + 2 C_{2} + \\operatorname{z^{*}}{(C_{2},E_{x})}) = \\frac{\\partial}{\\partial E_{x}} (2 C_{2} E_{x} + 2 C_{2}) and \\frac{\\frac{\\partial}{\\partial E_{x}} (C_{2} E_{x} + 2 C_{2} + \\operatorname{z^{*}}{(C_{2},E_{x})})}{E_{x}} = \\frac{\\frac{\\partial}{\\partial E_{x}} (2 C_{2} E_{x} + 2 C_{2})}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)))"], [["add", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Function('z^*')(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Symbol('C_2', commutative=True)))"], [["add", 2, "Add(Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True)), Function('z^*')(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True))), Add(Mul(Integer(2), Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True)), Function('z^*')(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["divide", 4, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True)), Function('z^*')(Symbol('C_2', commutative=True), Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(2), Symbol('C_2', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbb{I}{(A_{2},Z)} = A_{2} - Z and k{(A_{2},Z)} = A_{2} Z, then obtain k{(A_{2},Z)} = Z (Z + \\mathbb{I}{(A_{2},Z)})", "derivation": "\\mathbb{I}{(A_{2},Z)} = A_{2} - Z and Z + \\mathbb{I}{(A_{2},Z)} = A_{2} and Z (Z + \\mathbb{I}{(A_{2},Z)}) = A_{2} Z and k{(A_{2},Z)} = A_{2} Z and k{(A_{2},Z)} = Z (Z + \\mathbb{I}{(A_{2},Z)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True))), Symbol('A_2', commutative=True))"], [["times", 2, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Add(Symbol('Z', commutative=True), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)))), Mul(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('k')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Add(Symbol('Z', commutative=True), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(E_{n},x^\\prime)} = x^\\prime + \\cos{(E_{n})} and \\mathbb{I}{(x^\\prime)} = x^\\prime, then obtain (0^{E_{n}} \\mathbb{I}{(x^\\prime)})^{x^\\prime} = (0^{E_{n}} x^\\prime)^{x^\\prime}", "derivation": "\\mathbf{v}{(E_{n},x^\\prime)} = x^\\prime + \\cos{(E_{n})} and \\mathbb{I}{(x^\\prime)} = x^\\prime and (x^\\prime - \\mathbf{v}{(E_{n},x^\\prime)} + \\cos{(E_{n})})^{E_{n}} \\mathbb{I}{(x^\\prime)} = x^\\prime (x^\\prime - \\mathbf{v}{(E_{n},x^\\prime)} + \\cos{(E_{n})})^{E_{n}} and 0^{E_{n}} \\mathbb{I}{(x^\\prime)} = 0^{E_{n}} x^\\prime and (0^{E_{n}} \\mathbb{I}{(x^\\prime)})^{x^\\prime} = (0^{E_{n}} x^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('x^\\\\prime', commutative=True), cos(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["times", 2, "Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('x^\\\\prime', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('x^\\\\prime', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Function('\\\\mathbb{I}')(Symbol('x^\\\\prime', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('E_n', commutative=True), Symbol('x^\\\\prime', commutative=True))), cos(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Integer(0), Symbol('E_n', commutative=True)), Function('\\\\mathbb{I}')(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Integer(0), Symbol('E_n', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(Pow(Integer(0), Symbol('E_n', commutative=True)), Function('\\\\mathbb{I}')(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Pow(Integer(0), Symbol('E_n', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given B{(\\theta_2)} = \\theta_2, then derive \\frac{d}{d \\theta_2} B{(\\theta_2)} = 1, then derive \\theta_2 + p = \\mathbf{v} + \\theta_2, then obtain \\frac{\\int (\\theta_2 + p) d\\mathbf{v}}{u} = \\frac{\\int (\\mathbf{v} + \\theta_2) d\\mathbf{v}}{u}", "derivation": "B{(\\theta_2)} = \\theta_2 and \\frac{d}{d \\theta_2} B{(\\theta_2)} = \\frac{d}{d \\theta_2} \\theta_2 and \\frac{d}{d \\theta_2} B{(\\theta_2)} = 1 and \\int \\frac{d}{d \\theta_2} B{(\\theta_2)} d\\theta_2 = \\int 1 d\\theta_2 and \\int \\frac{d}{d \\theta_2} \\theta_2 d\\theta_2 = \\int 1 d\\theta_2 and \\theta_2 + p = \\mathbf{v} + \\theta_2 and \\int (\\theta_2 + p) d\\mathbf{v} = \\int (\\mathbf{v} + \\theta_2) d\\mathbf{v} and \\frac{\\int (\\theta_2 + p) d\\mathbf{v}}{u} = \\frac{\\int (\\mathbf{v} + \\theta_2) d\\mathbf{v}}{u}", "srepr_derivation": [["renaming_premise", "Equality(Function('B')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Symbol('\\\\theta_2', commutative=True), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Symbol('\\\\theta_2', commutative=True), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\theta_2', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 6, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\theta_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 7, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\theta_2', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\hbar,p)} = \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar}, then derive \\operatorname{f^{\\prime}}{(\\hbar,p)} = - \\frac{p}{\\hbar^{2}}, then obtain \\frac{\\int \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar} dp}{\\mathbf{A}{(\\hbar)}} = \\frac{\\int - \\frac{p}{\\hbar^{2}} dp}{\\mathbf{A}{(\\hbar)}}", "derivation": "\\operatorname{f^{\\prime}}{(\\hbar,p)} = \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar} and \\operatorname{f^{\\prime}}{(\\hbar,p)} = - \\frac{p}{\\hbar^{2}} and \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar} = - \\frac{p}{\\hbar^{2}} and \\int \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar} dp = \\int - \\frac{p}{\\hbar^{2}} dp and \\frac{\\int \\frac{\\partial}{\\partial \\hbar} \\frac{p}{\\hbar} dp}{\\mathbf{A}{(\\hbar)}} = \\frac{\\int - \\frac{p}{\\hbar^{2}} dp}{\\mathbf{A}{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Symbol('p', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Symbol('p', commutative=True)))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["divide", 4, "Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Derivative(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-2)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain 0 = \\cos{(\\theta_1)} - \\frac{d}{d \\theta_1} \\operatorname{F_{x}}{(\\theta_1)}", "derivation": "\\operatorname{F_{x}}{(\\theta_1)} = \\sin{(\\theta_1)} and 0 = - \\operatorname{F_{x}}{(\\theta_1)} + \\sin{(\\theta_1)} and \\frac{d}{d \\theta_1} 0 = \\frac{d}{d \\theta_1} (- \\operatorname{F_{x}}{(\\theta_1)} + \\sin{(\\theta_1)}) and 0 = \\cos{(\\theta_1)} - \\frac{d}{d \\theta_1} \\operatorname{F_{x}}{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Function('F_x')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('F_x')(Symbol('\\\\theta_1', commutative=True))), sin(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(cos(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Derivative(Function('F_x')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given E{(V)} = e^{V} and \\operatorname{F_{N}}{(V)} = (e^{V})^{V}, then obtain \\frac{\\operatorname{F_{N}}{(V)}}{E^{2}{(V)}} = \\frac{E^{V}{(V)}}{E^{2}{(V)}}", "derivation": "E{(V)} = e^{V} and E^{V}{(V)} = (e^{V})^{V} and \\frac{E^{V}{(V)}}{E{(V)}} = \\frac{(e^{V})^{V}}{E{(V)}} and \\frac{E^{V}{(V)}}{E^{2}{(V)}} = \\frac{(e^{V})^{V}}{E^{2}{(V)}} and \\operatorname{F_{N}}{(V)} = (e^{V})^{V} and E^{V}{(V)} = \\operatorname{F_{N}}{(V)} and \\frac{\\operatorname{F_{N}}{(V)}}{E^{2}{(V)}} = \\frac{(e^{V})^{V}}{E^{2}{(V)}} and \\frac{\\operatorname{F_{N}}{(V)}}{E^{2}{(V)}} = \\frac{E^{V}{(V)}}{E^{2}{(V)}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('E')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["divide", 2, "Function('E')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-1)), Pow(Function('E')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-1)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True))))"], [["divide", 3, "Function('E')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Pow(Function('E')(Symbol('V', commutative=True)), Symbol('V', commutative=True))), Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('V', commutative=True)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('E')(Symbol('V', commutative=True)), Symbol('V', commutative=True)), Function('F_N')(Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Function('F_N')(Symbol('V', commutative=True))), Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Pow(exp(Symbol('V', commutative=True)), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Function('F_N')(Symbol('V', commutative=True))), Mul(Pow(Function('E')(Symbol('V', commutative=True)), Integer(-2)), Pow(Function('E')(Symbol('V', commutative=True)), Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\phi_1,\\theta_2)} = e^{\\phi_1 + \\theta_2}, then derive \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = e^{\\phi_1 + \\theta_2}, then obtain 2 \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} + \\frac{\\partial^{2}}{\\partial \\theta_2^{2}} \\theta{(\\phi_1,\\theta_2)}", "derivation": "\\theta{(\\phi_1,\\theta_2)} = e^{\\phi_1 + \\theta_2} and \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} e^{\\phi_1 + \\theta_2} and \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = e^{\\phi_1 + \\theta_2} and e^{\\phi_1 + \\theta_2} = \\frac{\\partial}{\\partial \\theta_2} e^{\\phi_1 + \\theta_2} and e^{\\phi_1 + \\theta_2} + \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} + \\frac{\\partial}{\\partial \\theta_2} e^{\\phi_1 + \\theta_2} and 2 \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\theta{(\\phi_1,\\theta_2)} + \\frac{\\partial^{2}}{\\partial \\theta_2^{2}} \\theta{(\\phi_1,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Derivative(exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Add(exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2)))))"]]}, {"prompt": "Given L{(F_{g})} = e^{F_{g}}, then obtain F_{g} L{(F_{g})} - F_{g} + L{(F_{g})} = F_{g} e^{F_{g}} - F_{g} + L{(F_{g})}", "derivation": "L{(F_{g})} = e^{F_{g}} and F_{g} L{(F_{g})} = F_{g} e^{F_{g}} and F_{g} L{(F_{g})} + L{(F_{g})} = F_{g} e^{F_{g}} + L{(F_{g})} and F_{g} L{(F_{g})} - F_{g} + L{(F_{g})} = F_{g} e^{F_{g}} - F_{g} + L{(F_{g})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["times", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Function('L')(Symbol('F_g', commutative=True))), Mul(Symbol('F_g', commutative=True), exp(Symbol('F_g', commutative=True))))"], [["add", 2, "Function('L')(Symbol('F_g', commutative=True))"], "Equality(Add(Mul(Symbol('F_g', commutative=True), Function('L')(Symbol('F_g', commutative=True))), Function('L')(Symbol('F_g', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), exp(Symbol('F_g', commutative=True))), Function('L')(Symbol('F_g', commutative=True))))"], [["minus", 3, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Symbol('F_g', commutative=True), Function('L')(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('L')(Symbol('F_g', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), exp(Symbol('F_g', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('L')(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(F_{c},\\mathbf{J}_P)} = F_{c}^{\\mathbf{J}_P}, then derive \\frac{\\frac{\\partial}{\\partial F_{c}} \\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{F_{c}^{\\mathbf{J}_P}}{F_{c}}, then obtain \\frac{\\frac{\\partial}{\\partial F_{c}} \\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{\\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{F_{c}}", "derivation": "\\mathbf{s}{(F_{c},\\mathbf{J}_P)} = F_{c}^{\\mathbf{J}_P} and \\frac{\\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{F_{c}^{\\mathbf{J}_P}}{\\mathbf{J}_P} and \\frac{\\partial}{\\partial F_{c}} \\frac{\\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{\\partial}{\\partial F_{c}} \\frac{F_{c}^{\\mathbf{J}_P}}{\\mathbf{J}_P} and \\frac{\\frac{\\partial}{\\partial F_{c}} \\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{F_{c}^{\\mathbf{J}_P}}{F_{c}} and \\frac{\\frac{\\partial}{\\partial F_{c}} \\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{\\mathbf{J}_P} = \\frac{\\mathbf{s}{(F_{c},\\mathbf{J}_P)}}{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given z{(\\phi,q)} = \\phi q and \\mathbf{J}{(\\phi,q)} = \\phi^{2} q^{2}, then obtain \\frac{\\phi^{4} + \\phi^{3} q z{(\\phi,q)}}{\\phi q z{(\\phi,q)}} = \\frac{\\phi^{4} q^{2} + \\phi^{4}}{\\phi q z{(\\phi,q)}}", "derivation": "z{(\\phi,q)} = \\phi q and \\phi q z{(\\phi,q)} = \\phi^{2} q^{2} and \\mathbf{J}{(\\phi,q)} = \\phi^{2} q^{2} and \\mathbf{J}{(\\phi,q)} = \\phi q z{(\\phi,q)} and \\phi^{3} q z{(\\phi,q)} = \\phi^{4} q^{2} and \\phi^{4} + \\phi^{3} q z{(\\phi,q)} = \\phi^{4} q^{2} + \\phi^{4} and \\frac{\\phi^{4} + \\phi^{3} q z{(\\phi,q)}}{\\mathbf{J}{(\\phi,q)}} = \\frac{\\phi^{4} q^{2} + \\phi^{4}}{\\mathbf{J}{(\\phi,q)}} and \\frac{\\phi^{4} + \\phi^{3} q z{(\\phi,q)}}{\\phi q z{(\\phi,q)}} = \\frac{\\phi^{4} q^{2} + \\phi^{4}}{\\phi q z{(\\phi,q)}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True))))"], [["times", 2, "Pow(Symbol('\\\\phi', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('q', commutative=True), Integer(2))))"], [["add", 5, "Pow(Symbol('\\\\phi', commutative=True), Integer(4))"], "Equality(Add(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('q', commutative=True), Integer(2))), Pow(Symbol('\\\\phi', commutative=True), Integer(4))))"], [["divide", 6, "Function('\\\\mathbf{J}')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Add(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Mul(Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('q', commutative=True), Integer(2))), Pow(Symbol('\\\\phi', commutative=True), Integer(4))), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Symbol('q', commutative=True), Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)))), Pow(Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('q', commutative=True), Integer(2))), Pow(Symbol('\\\\phi', commutative=True), Integer(4))), Pow(Function('z')(Symbol('\\\\phi', commutative=True), Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given v{(\\hat{\\mathbf{x}},Z)} = \\sin{(Z + \\hat{\\mathbf{x}})}, then obtain \\frac{Z \\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)}}{\\cos{(Z + \\hat{\\mathbf{x}})}} = Z", "derivation": "v{(\\hat{\\mathbf{x}},Z)} = \\sin{(Z + \\hat{\\mathbf{x}})} and \\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)} = \\frac{\\partial}{\\partial Z} \\sin{(Z + \\hat{\\mathbf{x}})} and \\frac{\\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)}}{Z + \\hat{\\mathbf{x}}} = \\frac{\\frac{\\partial}{\\partial Z} \\sin{(Z + \\hat{\\mathbf{x}})}}{Z + \\hat{\\mathbf{x}}} and \\frac{Z \\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)}}{Z + \\hat{\\mathbf{x}}} = \\frac{Z \\frac{\\partial}{\\partial Z} \\sin{(Z + \\hat{\\mathbf{x}})}}{Z + \\hat{\\mathbf{x}}} and \\frac{Z \\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)}}{\\frac{\\partial}{\\partial Z} \\sin{(Z + \\hat{\\mathbf{x}})}} = Z and \\frac{Z \\frac{\\partial}{\\partial Z} v{(\\hat{\\mathbf{x}},Z)}}{\\cos{(Z + \\hat{\\mathbf{x}})}} = Z", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["divide", 2, "Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), Derivative(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Symbol('Z', commutative=True), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), Derivative(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["divide", 4, "Mul(Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integer(-1)), Derivative(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('Z', commutative=True), Derivative(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Derivative(sin(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))), Symbol('Z', commutative=True))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('Z', commutative=True), Pow(cos(Add(Symbol('Z', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integer(-1)), Derivative(Function('v')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Symbol('Z', commutative=True))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\phi_1,\\chi)} = \\phi_1 \\log{(\\chi)}, then obtain (\\int \\frac{\\partial}{\\partial \\chi} 2 \\operatorname{v_{2}}{(\\phi_1,\\chi)} d\\phi_1)^{\\chi} = (\\int \\frac{\\partial}{\\partial \\chi} (\\phi_1 \\log{(\\chi)} + \\operatorname{v_{2}}{(\\phi_1,\\chi)}) d\\phi_1)^{\\chi}", "derivation": "\\operatorname{v_{2}}{(\\phi_1,\\chi)} = \\phi_1 \\log{(\\chi)} and 2 \\operatorname{v_{2}}{(\\phi_1,\\chi)} = \\phi_1 \\log{(\\chi)} + \\operatorname{v_{2}}{(\\phi_1,\\chi)} and \\frac{\\partial}{\\partial \\chi} 2 \\operatorname{v_{2}}{(\\phi_1,\\chi)} = \\frac{\\partial}{\\partial \\chi} (\\phi_1 \\log{(\\chi)} + \\operatorname{v_{2}}{(\\phi_1,\\chi)}) and \\int \\frac{\\partial}{\\partial \\chi} 2 \\operatorname{v_{2}}{(\\phi_1,\\chi)} d\\phi_1 = \\int \\frac{\\partial}{\\partial \\chi} (\\phi_1 \\log{(\\chi)} + \\operatorname{v_{2}}{(\\phi_1,\\chi)}) d\\phi_1 and (\\int \\frac{\\partial}{\\partial \\chi} 2 \\operatorname{v_{2}}{(\\phi_1,\\chi)} d\\phi_1)^{\\chi} = (\\int \\frac{\\partial}{\\partial \\chi} (\\phi_1 \\log{(\\chi)} + \\operatorname{v_{2}}{(\\phi_1,\\chi)}) d\\phi_1)^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\chi', commutative=True))))"], [["add", 1, "Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Integer(2), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\chi', commutative=True))), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\chi', commutative=True))), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\chi', commutative=True))), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["power", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Derivative(Mul(Integer(2), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integral(Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), log(Symbol('\\\\chi', commutative=True))), Function('v_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given \\mu{(I,n)} = \\log{(I n)}, then derive \\int \\mu{(I,n)} dI = I \\log{(I n)} - I + \\sigma_p, then obtain \\int \\mu{(I,n)} dI = I \\mu{(I,n)} - I + \\sigma_p", "derivation": "\\mu{(I,n)} = \\log{(I n)} and \\int \\mu{(I,n)} dI = \\int \\log{(I n)} dI and \\int \\mu{(I,n)} dI = I \\log{(I n)} - I + \\sigma_p and \\int \\log{(I n)} dI = I \\log{(I n)} - I + \\sigma_p and \\int \\log{(I n)} dI = I \\mu{(I,n)} - I + \\sigma_p and I \\log{(I n)} - I + \\sigma_p = I \\mu{(I,n)} - I + \\sigma_p and \\int \\mu{(I,n)} dI = I \\mu{(I,n)} - I + \\sigma_p", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True)), log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Mul(Symbol('I', commutative=True), log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('I', commutative=True))), Add(Mul(Symbol('I', commutative=True), log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('I', commutative=True))), Add(Mul(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('I', commutative=True), log(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integral(Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Mul(Symbol('I', commutative=True), Function('\\\\mu')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(F_{H})} = \\sin{(\\log{(F_{H})})}, then derive \\frac{d}{d F_{H}} \\psi^{*}{(F_{H})} = \\frac{\\cos{(\\log{(F_{H})})}}{F_{H}}, then obtain - \\cos{(\\log{(F_{H})})} + \\frac{d}{d F_{H}} \\psi^{*}{(F_{H})} = - \\cos{(\\log{(F_{H})})} + \\frac{\\cos{(\\log{(F_{H})})}}{F_{H}}", "derivation": "\\psi^{*}{(F_{H})} = \\sin{(\\log{(F_{H})})} and \\frac{d}{d F_{H}} \\psi^{*}{(F_{H})} = \\frac{d}{d F_{H}} \\sin{(\\log{(F_{H})})} and \\frac{d}{d F_{H}} \\psi^{*}{(F_{H})} = \\frac{\\cos{(\\log{(F_{H})})}}{F_{H}} and - \\cos{(\\log{(F_{H})})} + \\frac{d}{d F_{H}} \\psi^{*}{(F_{H})} = - \\cos{(\\log{(F_{H})})} + \\frac{\\cos{(\\log{(F_{H})})}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), sin(log(Symbol('F_H', commutative=True))))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(sin(log(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), cos(log(Symbol('F_H', commutative=True)))))"], [["minus", 3, "cos(log(Symbol('F_H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(log(Symbol('F_H', commutative=True)))), Derivative(Function('\\\\psi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(log(Symbol('F_H', commutative=True)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), cos(log(Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(J,W,y^{\\prime})} = (J^{y^{\\prime}})^{W} and q{(J,y^{\\prime})} = J^{y^{\\prime}}, then obtain (\\eta^{\\prime}^{y^{\\prime}}{(J,W,y^{\\prime})})^{W} - \\frac{\\partial}{\\partial J} q^{W}{(J,y^{\\prime})} = ((q^{W}{(J,y^{\\prime})})^{y^{\\prime}})^{W} - \\frac{\\partial}{\\partial J} q^{W}{(J,y^{\\prime})}", "derivation": "\\eta^{\\prime}{(J,W,y^{\\prime})} = (J^{y^{\\prime}})^{W} and \\eta^{\\prime}^{y^{\\prime}}{(J,W,y^{\\prime})} = ((J^{y^{\\prime}})^{W})^{y^{\\prime}} and q{(J,y^{\\prime})} = J^{y^{\\prime}} and (\\eta^{\\prime}^{y^{\\prime}}{(J,W,y^{\\prime})})^{W} = (((J^{y^{\\prime}})^{W})^{y^{\\prime}})^{W} and (\\eta^{\\prime}^{y^{\\prime}}{(J,W,y^{\\prime})})^{W} = ((q^{W}{(J,y^{\\prime})})^{y^{\\prime}})^{W} and (\\eta^{\\prime}^{y^{\\prime}}{(J,W,y^{\\prime})})^{W} - \\frac{\\partial}{\\partial J} q^{W}{(J,y^{\\prime})} = ((q^{W}{(J,y^{\\prime})})^{y^{\\prime}})^{W} - \\frac{\\partial}{\\partial J} q^{W}{(J,y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('W', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('W', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(Pow(Pow(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('W', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Pow(Pow(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('W', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Pow(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)))"], [["minus", 5, "Derivative(Pow(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('J', commutative=True), Symbol('W', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))), Add(Pow(Pow(Pow(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Derivative(Pow(Function('q')(Symbol('J', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(m)} = \\cos{(\\sin{(m)})}, then obtain 3 (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} = (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} + 2 (\\sin{(m)} \\cos{(\\sin{(m)})})^{m}", "derivation": "\\operatorname{v_{y}}{(m)} = \\cos{(\\sin{(m)})} and \\operatorname{v_{y}}{(m)} \\sin{(m)} = \\sin{(m)} \\cos{(\\sin{(m)})} and (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} = (\\sin{(m)} \\cos{(\\sin{(m)})})^{m} and 2 (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} = (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} + (\\sin{(m)} \\cos{(\\sin{(m)})})^{m} and 3 (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} = 2 (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} + (\\sin{(m)} \\cos{(\\sin{(m)})})^{m} and 3 (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} = (\\operatorname{v_{y}}{(m)} \\sin{(m)})^{m} + 2 (\\sin{(m)} \\cos{(\\sin{(m)})})^{m}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True))))"], [["times", 1, "sin(Symbol('m', commutative=True))"], "Equality(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Mul(sin(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True)))))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(sin(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Add(Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(sin(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True))))"], [["add", 4, "Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True))"], "Equality(Mul(Integer(3), Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Add(Mul(Integer(2), Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Pow(Mul(sin(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(3), Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True))), Add(Pow(Mul(Function('v_y')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Mul(Integer(2), Pow(Mul(sin(Symbol('m', commutative=True)), cos(sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(n,\\hat{X})} = \\hat{X} n, then obtain \\frac{\\partial}{\\partial \\hat{X}} \\int \\Psi{(n,\\hat{X})} dn + \\frac{\\int \\Psi{(n,\\hat{X})} dn}{\\hat{X}^{2}} = \\frac{\\partial}{\\partial \\hat{X}} \\int \\Psi{(n,\\hat{X})} dn + \\frac{\\int \\hat{X} n dn}{\\hat{X}^{2}}", "derivation": "\\Psi{(n,\\hat{X})} = \\hat{X} n and \\int \\Psi{(n,\\hat{X})} dn = \\int \\hat{X} n dn and \\frac{\\int \\Psi{(n,\\hat{X})} dn}{\\hat{X}} = \\frac{\\int \\hat{X} n dn}{\\hat{X}} and \\frac{1}{\\hat{X}} = \\frac{\\int \\hat{X} n dn}{\\hat{X} \\int \\Psi{(n,\\hat{X})} dn} and \\frac{\\int \\Psi{(n,\\hat{X})} dn}{\\hat{X}^{2}} = \\frac{\\int \\hat{X} n dn}{\\hat{X}^{2}} and \\frac{\\partial}{\\partial \\hat{X}} \\int \\Psi{(n,\\hat{X})} dn + \\frac{\\int \\Psi{(n,\\hat{X})} dn}{\\hat{X}^{2}} = \\frac{\\partial}{\\partial \\hat{X}} \\int \\Psi{(n,\\hat{X})} dn + \\frac{\\int \\hat{X} n dn}{\\hat{X}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["divide", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Pow(Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1))))"], [["times", 4, "Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))))"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2)), Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["add", 5, "Derivative(Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2)), Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))))), Add(Derivative(Integral(Function('\\\\Psi')(Symbol('n', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-2)), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(b,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} + b)} and \\phi_{1}{(\\hat{\\mathbf{x}})} = \\cos{(\\cos{(\\hat{\\mathbf{x}})})}, then obtain \\sin{(L_{\\varepsilon} + b)} \\int \\phi_{1}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\sin{(L_{\\varepsilon} + b)} \\int \\cos{(\\cos{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}}", "derivation": "\\varepsilon{(b,L_{\\varepsilon})} = \\sin{(L_{\\varepsilon} + b)} and \\phi_{1}{(\\hat{\\mathbf{x}})} = \\cos{(\\cos{(\\hat{\\mathbf{x}})})} and \\int \\phi_{1}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\cos{(\\cos{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}} and \\varepsilon{(b,L_{\\varepsilon})} \\int \\phi_{1}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\varepsilon{(b,L_{\\varepsilon})} \\int \\cos{(\\cos{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}} and \\sin{(L_{\\varepsilon} + b)} \\int \\phi_{1}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\sin{(L_{\\varepsilon} + b)} \\int \\cos{(\\cos{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('b', commutative=True))))"], ["get_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(cos(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 3, "Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(cos(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('b', commutative=True))), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(sin(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('b', commutative=True))), Integral(cos(cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given n{(I)} = \\sin{(I)}, then obtain (\\sin^{I}{(I)} + \\int (n{(I)} - \\sin^{I}{(I)}) dI)^{I} = (\\sin^{I}{(I)} + \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI)^{I}", "derivation": "n{(I)} = \\sin{(I)} and n^{I}{(I)} = \\sin^{I}{(I)} and n{(I)} - \\sin^{I}{(I)} = \\sin{(I)} - \\sin^{I}{(I)} and \\int (n{(I)} - \\sin^{I}{(I)}) dI = \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI and \\sin^{I}{(I)} + \\int (n{(I)} - \\sin^{I}{(I)}) dI = \\sin^{I}{(I)} + \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI and n^{I}{(I)} + \\int (n{(I)} - \\sin^{I}{(I)}) dI = n^{I}{(I)} + \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI and (n^{I}{(I)} + \\int (n{(I)} - \\sin^{I}{(I)}) dI)^{I} = (n^{I}{(I)} + \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI)^{I} and (\\sin^{I}{(I)} + \\int (n{(I)} - \\sin^{I}{(I)}) dI)^{I} = (\\sin^{I}{(I)} + \\int (\\sin{(I)} - \\sin^{I}{(I)}) dI)^{I}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('n')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["minus", 1, "Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))))"], [["add", 4, "Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Add(Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Add(Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Pow(Function('n')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Add(Pow(Function('n')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))))"], [["power", 6, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Pow(Function('n')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Pow(Function('n')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Pow(Add(Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(Function('n')(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Integral(Add(sin(Symbol('I', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given G{(\\psi^*,\\mathbf{J}_P)} = \\frac{\\psi^*}{\\mathbf{J}_P}, then obtain (\\frac{\\psi^*}{\\mathbf{J}_P})^{- \\psi^*} G^{\\psi^*}{(\\psi^*,\\mathbf{J}_P)} = 1", "derivation": "G{(\\psi^*,\\mathbf{J}_P)} = \\frac{\\psi^*}{\\mathbf{J}_P} and G^{\\psi^*}{(\\psi^*,\\mathbf{J}_P)} = (\\frac{\\psi^*}{\\mathbf{J}_P})^{\\psi^*} and \\frac{G^{\\psi^*}{(\\psi^*,\\mathbf{J}_P)}}{G{(\\psi^*,\\mathbf{J}_P)}} = \\frac{(\\frac{\\psi^*}{\\mathbf{J}_P})^{\\psi^*}}{G{(\\psi^*,\\mathbf{J}_P)}} and (\\frac{\\psi^*}{\\mathbf{J}_P})^{- \\psi^*} G^{\\psi^*}{(\\psi^*,\\mathbf{J}_P)} = 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["divide", 2, "Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Pow(Function('G')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\mu{(C)} = \\log{(C)} and \\hat{H}{(C)} = 2 \\log{(C)}^{2}, then derive \\frac{4 \\log{(C)}}{C} = 4 \\mu{(C)} \\frac{d}{d C} \\mu{(C)}, then obtain \\frac{4 \\log{(C)} (\\frac{d}{d C} 2 \\log{(C)}^{2})^{C}}{C} = 4 \\mu{(C)} \\frac{d}{d C} \\mu{(C)} (\\frac{d}{d C} 2 \\log{(C)}^{2})^{C}", "derivation": "\\mu{(C)} = \\log{(C)} and \\hat{H}{(C)} = 2 \\log{(C)}^{2} and \\hat{H}{(C)} = 2 \\mu^{2}{(C)} and \\frac{d}{d C} \\hat{H}{(C)} = \\frac{d}{d C} 2 \\mu^{2}{(C)} and \\frac{d}{d C} 2 \\log{(C)}^{2} = \\frac{d}{d C} 2 \\mu^{2}{(C)} and \\frac{4 \\log{(C)}}{C} = 4 \\mu{(C)} \\frac{d}{d C} \\mu{(C)} and \\frac{4 \\log{(C)} (\\frac{d}{d C} 2 \\log{(C)}^{2})^{C}}{C} = 4 \\mu{(C)} \\frac{d}{d C} \\mu{(C)} (\\frac{d}{d C} 2 \\log{(C)}^{2})^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('C', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(4), Pow(Symbol('C', commutative=True), Integer(-1)), log(Symbol('C', commutative=True))), Mul(Integer(4), Function('\\\\mu')(Symbol('C', commutative=True)), Derivative(Function('\\\\mu')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["times", 6, "Pow(Derivative(Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))"], "Equality(Mul(Integer(4), Pow(Symbol('C', commutative=True), Integer(-1)), log(Symbol('C', commutative=True)), Pow(Derivative(Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))), Mul(Integer(4), Function('\\\\mu')(Symbol('C', commutative=True)), Derivative(Function('\\\\mu')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Derivative(Mul(Integer(2), Pow(log(Symbol('C', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True))))"]]}, {"prompt": "Given r{(U)} = \\cos{(e^{U})}, then derive - e^{U} \\sin{(e^{U})} + \\frac{d}{d U} r{(U)} = - 2 e^{U} \\sin{(e^{U})}, then derive \\frac{d}{d U} r{(U)} = - e^{U} \\sin{(e^{U})}, then obtain \\frac{d}{d U} r{(U)} + \\frac{d}{d U} \\cos{(e^{U})} = 2 \\frac{d}{d U} r{(U)}", "derivation": "r{(U)} = \\cos{(e^{U})} and \\frac{d}{d U} r{(U)} = \\frac{d}{d U} \\cos{(e^{U})} and \\frac{d}{d U} r{(U)} + \\frac{d}{d U} \\cos{(e^{U})} = 2 \\frac{d}{d U} \\cos{(e^{U})} and - e^{U} \\sin{(e^{U})} + \\frac{d}{d U} r{(U)} = - 2 e^{U} \\sin{(e^{U})} and - e^{U} \\sin{(e^{U})} + \\frac{d}{d U} \\cos{(e^{U})} = - 2 e^{U} \\sin{(e^{U})} and e^{U} \\sin{(e^{U})} + \\frac{d}{d U} r{(U)} + \\frac{d}{d U} \\cos{(e^{U})} = e^{U} \\sin{(e^{U})} + 2 \\frac{d}{d U} \\cos{(e^{U})} and \\frac{d}{d U} r{(U)} = - e^{U} \\sin{(e^{U})} and \\frac{d}{d U} r{(U)} + \\frac{d}{d U} \\cos{(e^{U})} = 2 \\frac{d}{d U} r{(U)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('U', commutative=True)), cos(exp(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True))))"], "Equality(Add(Mul(exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Add(Mul(exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Mul(Integer(2), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('r')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(\\phi_2,\\delta)} = \\frac{\\sin{(\\delta)}}{\\phi_2}, then obtain \\frac{\\phi_2 + a{(\\phi_2,\\delta)} - \\frac{\\sin{(\\delta)}}{\\phi_2}}{\\phi_2^{2} (- \\phi_2 + \\frac{\\sin{(\\delta)}}{\\phi_2})} = \\frac{1}{\\phi_2 (- \\phi_2 + \\frac{\\sin{(\\delta)}}{\\phi_2})}", "derivation": "a{(\\phi_2,\\delta)} = \\frac{\\sin{(\\delta)}}{\\phi_2} and - \\phi_2 + a{(\\phi_2,\\delta)} = - \\phi_2 + \\frac{\\sin{(\\delta)}}{\\phi_2} and \\phi_2 + a{(\\phi_2,\\delta)} - \\frac{\\sin{(\\delta)}}{\\phi_2} = \\phi_2 and \\frac{\\phi_2 + a{(\\phi_2,\\delta)} - \\frac{\\sin{(\\delta)}}{\\phi_2}}{\\phi_2^{2}} = \\frac{1}{\\phi_2} and \\frac{\\phi_2 + a{(\\phi_2,\\delta)} - \\frac{\\sin{(\\delta)}}{\\phi_2}}{\\phi_2^{2} (- \\phi_2 + a{(\\phi_2,\\delta)})} = \\frac{1}{\\phi_2 (- \\phi_2 + a{(\\phi_2,\\delta)})} and \\frac{\\phi_2 + a{(\\phi_2,\\delta)} - \\frac{\\sin{(\\delta)}}{\\phi_2}}{\\phi_2^{2} (- \\phi_2 + \\frac{\\sin{(\\delta)}}{\\phi_2})} = \\frac{1}{\\phi_2 (- \\phi_2 + \\frac{\\sin{(\\delta)}}{\\phi_2})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True)))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True))))"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True)))), Symbol('\\\\phi_2', commutative=True))"], [["times", 3, "Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Add(Symbol('\\\\phi_2', commutative=True), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True))))), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True))))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True)))), Integer(-1)), Add(Symbol('\\\\phi_2', commutative=True), Function('a')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True))))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), sin(Symbol('\\\\delta', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\hat{x},\\hbar)} = \\hat{x} + \\hbar, then derive \\int \\operatorname{v_{z}}{(\\hat{x},\\hbar)} d\\hbar = \\hat{x} \\hbar + \\frac{\\hbar^{2}}{2} + n_{1}, then obtain \\int (\\hat{x} + \\hbar) d\\hbar = \\hat{x} \\hbar + \\frac{\\hbar^{2}}{2} + n_{1}", "derivation": "\\operatorname{v_{z}}{(\\hat{x},\\hbar)} = \\hat{x} + \\hbar and \\int \\operatorname{v_{z}}{(\\hat{x},\\hbar)} d\\hbar = \\int (\\hat{x} + \\hbar) d\\hbar and \\int \\operatorname{v_{z}}{(\\hat{x},\\hbar)} d\\hbar = \\hat{x} \\hbar + \\frac{\\hbar^{2}}{2} + n_{1} and \\int (\\hat{x} + \\hbar) d\\hbar = \\hat{x} \\hbar + \\frac{\\hbar^{2}}{2} + n_{1}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_z')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(J,\\hat{H}_l)} = J + \\hat{H}_l, then derive \\frac{\\partial}{\\partial J} \\operatorname{F_{c}}{(J,\\hat{H}_l)} = 1, then obtain \\frac{\\frac{\\partial}{\\partial J} \\operatorname{F_{c}}{(J,\\hat{H}_l)}}{\\hat{H}_l} = \\frac{1}{\\hat{H}_l}", "derivation": "\\operatorname{F_{c}}{(J,\\hat{H}_l)} = J + \\hat{H}_l and \\frac{\\partial}{\\partial J} \\operatorname{F_{c}}{(J,\\hat{H}_l)} = \\frac{\\partial}{\\partial J} (J + \\hat{H}_l) and \\frac{\\partial}{\\partial J} \\operatorname{F_{c}}{(J,\\hat{H}_l)} = 1 and \\frac{\\frac{\\partial}{\\partial J} \\operatorname{F_{c}}{(J,\\hat{H}_l)}}{\\hat{H}_l} = \\frac{1}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_c')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Derivative(Function('F_c')(Symbol('J', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)))"]]}, {"prompt": "Given Z{(S,\\psi^*)} = (e^{S})^{\\psi^*}, then obtain - \\sin{(\\sin{(F_{g})})} \\cos{(F_{g})} \\int Z^{S}{(S,\\psi^*)} d\\psi^* = - \\sin{(\\sin{(F_{g})})} \\cos{(F_{g})} \\int ((e^{S})^{\\psi^*})^{S} d\\psi^*", "derivation": "Z{(S,\\psi^*)} = (e^{S})^{\\psi^*} and Z^{S}{(S,\\psi^*)} = ((e^{S})^{\\psi^*})^{S} and \\int Z^{S}{(S,\\psi^*)} d\\psi^* = \\int ((e^{S})^{\\psi^*})^{S} d\\psi^* and \\frac{d}{d F_{g}} \\cos{(\\sin{(F_{g})})} \\int Z^{S}{(S,\\psi^*)} d\\psi^* = \\frac{d}{d F_{g}} \\cos{(\\sin{(F_{g})})} \\int ((e^{S})^{\\psi^*})^{S} d\\psi^* and - \\sin{(\\sin{(F_{g})})} \\cos{(F_{g})} \\int Z^{S}{(S,\\psi^*)} d\\psi^* = - \\sin{(\\sin{(F_{g})})} \\cos{(F_{g})} \\int ((e^{S})^{\\psi^*})^{S} d\\psi^*", "srepr_derivation": [["get_premise", "Equality(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(exp(Symbol('S', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(exp(Symbol('S', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Pow(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(Pow(exp(Symbol('S', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["times", 3, "Derivative(cos(sin(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(cos(sin(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integral(Pow(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Derivative(cos(sin(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Integral(Pow(Pow(exp(Symbol('S', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), sin(sin(Symbol('F_g', commutative=True))), cos(Symbol('F_g', commutative=True)), Integral(Pow(Function('Z')(Symbol('S', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(-1), sin(sin(Symbol('F_g', commutative=True))), cos(Symbol('F_g', commutative=True)), Integral(Pow(Pow(exp(Symbol('S', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given S{(\\rho_f,G)} = \\cos{(G - \\rho_f)}, then derive 2 \\frac{\\partial}{\\partial G} S{(\\rho_f,G)} = - \\sin{(G - \\rho_f)} + \\frac{\\partial}{\\partial G} S{(\\rho_f,G)}, then obtain 2 \\frac{\\partial}{\\partial G} \\cos{(G - \\rho_f)} = - \\sin{(G - \\rho_f)} + \\frac{\\partial}{\\partial G} \\cos{(G - \\rho_f)}", "derivation": "S{(\\rho_f,G)} = \\cos{(G - \\rho_f)} and 2 S{(\\rho_f,G)} = S{(\\rho_f,G)} + \\cos{(G - \\rho_f)} and \\frac{\\partial}{\\partial G} 2 S{(\\rho_f,G)} = \\frac{\\partial}{\\partial G} (S{(\\rho_f,G)} + \\cos{(G - \\rho_f)}) and 2 \\frac{\\partial}{\\partial G} S{(\\rho_f,G)} = - \\sin{(G - \\rho_f)} + \\frac{\\partial}{\\partial G} S{(\\rho_f,G)} and 2 \\frac{\\partial}{\\partial G} \\cos{(G - \\rho_f)} = - \\sin{(G - \\rho_f)} + \\frac{\\partial}{\\partial G} \\cos{(G - \\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))))"], [["add", 1, "Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True))), Add(Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))), Derivative(Function('S')(Symbol('\\\\rho_f', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))), Derivative(cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\omega,\\hat{X})} = \\omega + \\sin{(\\hat{X})}, then derive \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_f{(\\omega,\\hat{X})} = 1, then obtain (- \\hat{X} + \\omega (\\omega + \\sin{(\\hat{X})})) \\sin{(\\hat{X})} (\\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}))^{\\hat{X}} = (- \\hat{X} + \\omega (\\omega + \\sin{(\\hat{X})})) \\sin{(\\hat{X})}", "derivation": "\\mathbf{J}_f{(\\omega,\\hat{X})} = \\omega + \\sin{(\\hat{X})} and \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_f{(\\omega,\\hat{X})} = \\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}) and \\frac{\\partial}{\\partial \\omega} \\mathbf{J}_f{(\\omega,\\hat{X})} = 1 and \\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}) = 1 and (\\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}))^{\\hat{X}} = 1 and \\sin{(\\hat{X})} (\\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}))^{\\hat{X}} = \\sin{(\\hat{X})} and (- \\hat{X} + \\omega (\\omega + \\sin{(\\hat{X})})) \\sin{(\\hat{X})} (\\frac{\\partial}{\\partial \\omega} (\\omega + \\sin{(\\hat{X})}))^{\\hat{X}} = (- \\hat{X} + \\omega (\\omega + \\sin{(\\hat{X})})) \\sin{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\omega', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True)), Integer(1))"], [["times", 5, "sin(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True))), sin(Symbol('\\\\hat{X}', commutative=True)))"], [["times", 6, "Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))))), sin(Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\hat{X}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), sin(Symbol('\\\\hat{X}', commutative=True))))), sin(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} = \\omega^{E_{\\lambda}} + n_{1}, then derive e^{\\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1}} = e^{\\hat{H}_{\\lambda} + \\omega^{E_{\\lambda}} n_{1} + \\frac{n_{1}^{2}}{2}}, then obtain - n_{1} + e^{\\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1}} = - n_{1} + e^{\\hat{H}_{\\lambda} + \\omega^{E_{\\lambda}} n_{1} + \\frac{n_{1}^{2}}{2}}", "derivation": "\\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} = \\omega^{E_{\\lambda}} + n_{1} and \\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1} = \\int (\\omega^{E_{\\lambda}} + n_{1}) dn_{1} and e^{\\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1}} = e^{\\int (\\omega^{E_{\\lambda}} + n_{1}) dn_{1}} and e^{\\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1}} = e^{\\hat{H}_{\\lambda} + \\omega^{E_{\\lambda}} n_{1} + \\frac{n_{1}^{2}}{2}} and - n_{1} + e^{\\int \\operatorname{f^{*}}{(\\omega,n_{1},E_{\\lambda})} dn_{1}} = - n_{1} + e^{\\hat{H}_{\\lambda} + \\omega^{E_{\\lambda}} n_{1} + \\frac{n_{1}^{2}}{2}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)))"], [["integrate", 1, "Symbol('n_1', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_1', commutative=True))), Integral(Add(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), exp(Integral(Add(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_1', commutative=True)))), exp(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2))))))"], [["minus", 4, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Integral(Function('f^*')(Symbol('\\\\omega', commutative=True), Symbol('n_1', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), exp(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('\\\\omega', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('n_1', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n_1', commutative=True), Integer(2)))))))"]]}, {"prompt": "Given A{(\\mathbf{H},c_{0})} = \\mathbf{H} c_{0}, then derive \\frac{\\partial}{\\partial c_{0}} A{(\\mathbf{H},c_{0})} = \\mathbf{H}, then obtain A^{\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0}}{(\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0},c_{0})} - 1 = (c_{0} \\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0})^{\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0}} - 1", "derivation": "A{(\\mathbf{H},c_{0})} = \\mathbf{H} c_{0} and \\frac{\\partial}{\\partial c_{0}} A{(\\mathbf{H},c_{0})} = \\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0} and \\frac{\\partial}{\\partial c_{0}} A{(\\mathbf{H},c_{0})} = \\mathbf{H} and A^{\\mathbf{H}}{(\\mathbf{H},c_{0})} = (\\mathbf{H} c_{0})^{\\mathbf{H}} and A^{\\mathbf{H}}{(\\mathbf{H},c_{0})} - 1 = (\\mathbf{H} c_{0})^{\\mathbf{H}} - 1 and \\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0} = \\mathbf{H} and A^{\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0}}{(\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0},c_{0})} - 1 = (c_{0} \\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0})^{\\frac{\\partial}{\\partial c_{0}} \\mathbf{H} c_{0}} - 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Add(Pow(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Pow(Function('A')(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Symbol('c_0', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(-1)), Add(Pow(Mul(Symbol('c_0', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(n)} = \\cos{(n)} and \\Psi_{nl}{(\\mathbb{I})} = - \\mathbb{I}, then obtain - \\operatorname{A_{1}}{(n)} \\operatorname{A_{1}}^{n}{(n)} + \\int \\Psi_{nl}{(\\mathbb{I})} d\\mathbb{I} = - \\operatorname{A_{1}}{(n)} \\operatorname{A_{1}}^{n}{(n)} + \\int - \\mathbb{I} d\\mathbb{I}", "derivation": "\\operatorname{A_{1}}{(n)} = \\cos{(n)} and \\operatorname{A_{1}}^{n}{(n)} = \\cos^{n}{(n)} and \\Psi_{nl}{(\\mathbb{I})} = - \\mathbb{I} and \\int \\Psi_{nl}{(\\mathbb{I})} d\\mathbb{I} = \\int - \\mathbb{I} d\\mathbb{I} and - \\operatorname{A_{1}}{(n)} \\cos^{n}{(n)} + \\int \\Psi_{nl}{(\\mathbb{I})} d\\mathbb{I} = - \\operatorname{A_{1}}{(n)} \\cos^{n}{(n)} + \\int - \\mathbb{I} d\\mathbb{I} and - \\operatorname{A_{1}}{(n)} \\operatorname{A_{1}}^{n}{(n)} + \\int \\Psi_{nl}{(\\mathbb{I})} d\\mathbb{I} = - \\operatorname{A_{1}}{(n)} \\operatorname{A_{1}}^{n}{(n)} + \\int - \\mathbb{I} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('n', commutative=True)), cos(Symbol('n', commutative=True)))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["minus", 4, "Mul(Function('A_1')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True)), Pow(cos(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True)), Pow(Function('A_1')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Integer(-1), Function('A_1')(Symbol('n', commutative=True)), Pow(Function('A_1')(Symbol('n', commutative=True)), Symbol('n', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(A)} = \\cos{(A)}, then obtain ((\\tilde{g}^*^{A}{(A)} \\cos^{- A}{(A)})^{A})^{A} = 1", "derivation": "\\tilde{g}^*{(A)} = \\cos{(A)} and \\tilde{g}^*^{A}{(A)} = \\cos^{A}{(A)} and \\tilde{g}^*^{A}{(A)} \\cos^{- A}{(A)} = 1 and (\\tilde{g}^*^{A}{(A)} \\cos^{- A}{(A)})^{A} = 1 and ((\\tilde{g}^*^{A}{(A)} \\cos^{- A}{(A)})^{A})^{A} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Symbol('A', commutative=True)))"], [["divide", 2, "Pow(cos(Symbol('A', commutative=True)), Symbol('A', commutative=True))"], "Equality(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Integer(1))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{J}_f{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})}, then obtain (\\mathbf{J}_f{(x^\\prime)} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime})^{x^\\prime} = (\\log{(\\cos{(x^\\prime)})} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime})^{x^\\prime}", "derivation": "\\mathbf{J}_f{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})} and \\mathbf{J}_f^{x^\\prime}{(x^\\prime)} = \\log{(\\cos{(x^\\prime)})}^{x^\\prime} and \\mathbf{J}_f{(x^\\prime)} - \\mathbf{J}_f^{x^\\prime}{(x^\\prime)} = - \\mathbf{J}_f^{x^\\prime}{(x^\\prime)} + \\log{(\\cos{(x^\\prime)})} and \\mathbf{J}_f{(x^\\prime)} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime} = \\log{(\\cos{(x^\\prime)})} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime} and (\\mathbf{J}_f{(x^\\prime)} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime})^{x^\\prime} = (\\log{(\\cos{(x^\\prime)})} - \\log{(\\cos{(x^\\prime)})}^{x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), log(cos(Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), log(cos(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))), Add(log(cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{J}_f')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(log(cos(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(log(cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given k{(y^{\\prime})} = \\sin{(y^{\\prime})}, then obtain (k{(y^{\\prime})} + 1)^{2} (\\sin{(y^{\\prime})} + 1) = (\\sin{(y^{\\prime})} + 1)^{3}", "derivation": "k{(y^{\\prime})} = \\sin{(y^{\\prime})} and k{(y^{\\prime})} + 1 = \\sin{(y^{\\prime})} + 1 and (k{(y^{\\prime})} + 1) (\\sin{(y^{\\prime})} + 1) = (\\sin{(y^{\\prime})} + 1)^{2} and (k{(y^{\\prime})} + 1) (\\sin{(y^{\\prime})} + 1)^{2} = (\\sin{(y^{\\prime})} + 1)^{3} and (k{(y^{\\prime})} + 1)^{2} (\\sin{(y^{\\prime})} + 1) = (\\sin{(y^{\\prime})} + 1)^{3}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('y^{\\\\prime}', commutative=True)), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('k')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"], [["times", 2, "Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1))"], "Equality(Mul(Add(Function('k')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1))), Pow(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(2)))"], [["times", 2, "Pow(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(2))"], "Equality(Mul(Add(Function('k')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Pow(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(2))), Pow(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(3)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Function('k')(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(2)), Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1))), Pow(Add(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(1)), Integer(3)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\theta_2,\\varphi)} = \\theta_2 \\varphi, then obtain \\log{((\\varphi + \\operatorname{P_{g}}{(\\theta_2,\\varphi)})^{\\varphi})} = \\log{((\\theta_2 \\varphi + \\varphi)^{\\varphi})}", "derivation": "\\operatorname{P_{g}}{(\\theta_2,\\varphi)} = \\theta_2 \\varphi and \\varphi + \\operatorname{P_{g}}{(\\theta_2,\\varphi)} = \\theta_2 \\varphi + \\varphi and (\\varphi + \\operatorname{P_{g}}{(\\theta_2,\\varphi)})^{\\varphi} = (\\theta_2 \\varphi + \\varphi)^{\\varphi} and \\log{((\\varphi + \\operatorname{P_{g}}{(\\theta_2,\\varphi)})^{\\varphi})} = \\log{((\\theta_2 \\varphi + \\varphi)^{\\varphi})}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["power", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["log", 3], "Equality(log(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('P_g')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))), log(Pow(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\operatorname{n_{1}}{(\\dot{z})} = \\frac{\\cos^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}}, then obtain \\operatorname{n_{1}}{(\\dot{z})} = \\frac{\\operatorname{v_{x}}^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}}", "derivation": "\\operatorname{v_{x}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\operatorname{v_{x}}^{\\dot{z}}{(\\dot{z})} = \\cos^{\\dot{z}}{(\\dot{z})} and \\frac{\\operatorname{v_{x}}^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}} = \\frac{\\cos^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}} and \\operatorname{n_{1}}{(\\dot{z})} = \\frac{\\cos^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}} and \\operatorname{n_{1}}{(\\dot{z})} = \\frac{\\operatorname{v_{x}}^{\\dot{z}}{(\\dot{z})}}{\\operatorname{v_{x}}{(\\dot{z})}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Function('v_x')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Mul(Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('n_1')(Symbol('\\\\dot{z}', commutative=True)), Mul(Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), Pow(Function('v_x')(Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{r})} = e^{\\mathbf{r}}, then obtain 2 \\operatorname{F_{g}}{(\\mathbf{r})} - 3 e^{\\mathbf{r}} = - e^{\\mathbf{r}}", "derivation": "\\operatorname{F_{g}}{(\\mathbf{r})} = e^{\\mathbf{r}} and 2 \\operatorname{F_{g}}{(\\mathbf{r})} = \\operatorname{F_{g}}{(\\mathbf{r})} + e^{\\mathbf{r}} and 2 \\operatorname{F_{g}}{(\\mathbf{r})} - e^{\\mathbf{r}} = \\operatorname{F_{g}}{(\\mathbf{r})} and \\operatorname{F_{g}}{(\\mathbf{r})} - 2 e^{\\mathbf{r}} = - e^{\\mathbf{r}} and 2 \\operatorname{F_{g}}{(\\mathbf{r})} - 3 e^{\\mathbf{r}} = - e^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["add", 1, "Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Integer(2), Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{r}', commutative=True)))), Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 3, "Add(Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('F_g')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Integer(3), exp(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(E)} = e^{E}, then derive E + \\frac{d}{d E} \\hat{p}_0{(E)} = E + e^{E}, then obtain E + \\hat{p}_0{(E)} + \\frac{d}{d E} e^{E} = E + 2 \\frac{d}{d E} e^{E}", "derivation": "\\hat{p}_0{(E)} = e^{E} and \\frac{d}{d E} \\hat{p}_0{(E)} = \\frac{d}{d E} e^{E} and E + \\frac{d}{d E} \\hat{p}_0{(E)} = E + \\frac{d}{d E} e^{E} and E + \\frac{d}{d E} \\hat{p}_0{(E)} = E + e^{E} and E + \\frac{d}{d E} \\hat{p}_0{(E)} + \\frac{d}{d E} e^{E} = E + 2 \\frac{d}{d E} e^{E} and E + e^{E} + \\frac{d}{d E} e^{E} = E + 2 \\frac{d}{d E} e^{E} and E + \\frac{d}{d E} \\hat{p}_0{(E)} = E + \\hat{p}_0{(E)} and E + e^{E} = E + \\hat{p}_0{(E)} and E + \\hat{p}_0{(E)} + \\frac{d}{d E} e^{E} = E + 2 \\frac{d}{d E} e^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('E', commutative=True))"], "Equality(Add(Symbol('E', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('E', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), exp(Symbol('E', commutative=True))))"], [["add", 3, "Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Add(Symbol('E', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), Mul(Integer(2), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('E', commutative=True), exp(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), Mul(Integer(2), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('E', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), Function('\\\\hat{p}_0')(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Symbol('E', commutative=True), exp(Symbol('E', commutative=True))), Add(Symbol('E', commutative=True), Function('\\\\hat{p}_0')(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 8], "Equality(Add(Symbol('E', commutative=True), Function('\\\\hat{p}_0')(Symbol('E', commutative=True)), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Symbol('E', commutative=True), Mul(Integer(2), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\psi{(U,c_{0})} = \\frac{U}{c_{0}} and \\operatorname{C_{1}}{(U,c_{0})} = \\frac{U}{c_{0}}, then obtain \\psi^{U}{(U,c_{0})} - \\int \\psi{(U,c_{0})} dU = (\\frac{U}{c_{0}})^{U} - \\int \\psi{(U,c_{0})} dU", "derivation": "\\psi{(U,c_{0})} = \\frac{U}{c_{0}} and \\operatorname{C_{1}}{(U,c_{0})} = \\frac{U}{c_{0}} and \\operatorname{C_{1}}^{U}{(U,c_{0})} = (\\frac{U}{c_{0}})^{U} and \\psi{(U,c_{0})} = \\operatorname{C_{1}}{(U,c_{0})} and \\operatorname{C_{1}}^{U}{(U,c_{0})} - \\int \\operatorname{C_{1}}{(U,c_{0})} dU = (\\frac{U}{c_{0}})^{U} - \\int \\operatorname{C_{1}}{(U,c_{0})} dU and \\psi^{U}{(U,c_{0})} - \\int \\psi{(U,c_{0})} dU = (\\frac{U}{c_{0}})^{U} - \\int \\psi{(U,c_{0})} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Symbol('U', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\psi')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)))"], [["minus", 3, "Integral(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Pow(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True))))), Add(Pow(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('C_1')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Function('\\\\psi')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\psi')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True))))), Add(Pow(Mul(Symbol('U', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\psi')(Symbol('U', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\hat{X},L)} = L^{\\hat{X}}, then obtain (\\iint \\eta^{\\prime}{(\\hat{X},L)} d\\hat{X} dL)^{\\hat{X}} = (\\iint L^{\\hat{X}} d\\hat{X} dL)^{\\hat{X}}", "derivation": "\\eta^{\\prime}{(\\hat{X},L)} = L^{\\hat{X}} and \\int \\eta^{\\prime}{(\\hat{X},L)} d\\hat{X} = \\int L^{\\hat{X}} d\\hat{X} and \\iint \\eta^{\\prime}{(\\hat{X},L)} d\\hat{X} dL = \\iint L^{\\hat{X}} d\\hat{X} dL and (\\iint \\eta^{\\prime}{(\\hat{X},L)} d\\hat{X} dL)^{\\hat{X}} = (\\iint L^{\\hat{X}} d\\hat{X} dL)^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{X}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('\\\\hat{X}', commutative=True)), Pow(Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('L', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(A_{x},l)} = A_{x} l, then derive \\frac{\\partial}{\\partial A_{x}} \\mathbf{r}{(A_{x},l)} = l, then obtain \\mathbf{r}{(A_{x},\\frac{\\partial}{\\partial A_{x}} A_{x} l)} + \\frac{\\partial}{\\partial A_{x}} A_{x} l = A_{x} \\frac{\\partial}{\\partial A_{x}} A_{x} l + \\frac{\\partial}{\\partial A_{x}} A_{x} l", "derivation": "\\mathbf{r}{(A_{x},l)} = A_{x} l and \\frac{\\partial}{\\partial A_{x}} \\mathbf{r}{(A_{x},l)} = \\frac{\\partial}{\\partial A_{x}} A_{x} l and \\frac{\\partial}{\\partial A_{x}} \\mathbf{r}{(A_{x},l)} = l and \\frac{\\partial}{\\partial A_{x}} A_{x} l = l and \\mathbf{r}{(A_{x},\\frac{\\partial}{\\partial A_{x}} A_{x} l)} = A_{x} \\frac{\\partial}{\\partial A_{x}} A_{x} l and \\mathbf{r}{(A_{x},\\frac{\\partial}{\\partial A_{x}} A_{x} l)} + \\frac{\\partial}{\\partial A_{x}} A_{x} l = A_{x} \\frac{\\partial}{\\partial A_{x}} A_{x} l + \\frac{\\partial}{\\partial A_{x}} A_{x} l", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{r}')(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('l', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Symbol('l', commutative=True))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{r}')(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["add", 5, "Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Mul(Symbol('A_x', commutative=True), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Derivative(Mul(Symbol('A_x', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given c{(\\dot{\\mathbf{r}},F_{N})} = \\log{(F_{N} + \\dot{\\mathbf{r}})}, then obtain (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} c^{F_{N}}{(\\dot{\\mathbf{r}},F_{N})})^{\\dot{\\mathbf{r}}} = (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\log{(F_{N} + \\dot{\\mathbf{r}})}^{F_{N}})^{\\dot{\\mathbf{r}}}", "derivation": "c{(\\dot{\\mathbf{r}},F_{N})} = \\log{(F_{N} + \\dot{\\mathbf{r}})} and c^{F_{N}}{(\\dot{\\mathbf{r}},F_{N})} = \\log{(F_{N} + \\dot{\\mathbf{r}})}^{F_{N}} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} c^{F_{N}}{(\\dot{\\mathbf{r}},F_{N})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\log{(F_{N} + \\dot{\\mathbf{r}})}^{F_{N}} and (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} c^{F_{N}}{(\\dot{\\mathbf{r}},F_{N})})^{\\dot{\\mathbf{r}}} = (\\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\log{(F_{N} + \\dot{\\mathbf{r}})}^{F_{N}})^{\\dot{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), log(Add(Symbol('F_N', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(log(Add(Symbol('F_N', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('F_N', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Pow(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(log(Add(Symbol('F_N', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('c')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Derivative(Pow(log(Add(Symbol('F_N', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(\\phi)} = \\cos{(\\sin{(\\phi)})}, then obtain \\int (\\phi + \\mathbb{I}{(\\phi)} + \\cos^{2}{(\\sin{(\\phi)})}) d\\phi = \\int (\\phi + \\cos^{2}{(\\sin{(\\phi)})} + \\cos{(\\sin{(\\phi)})}) d\\phi", "derivation": "\\mathbb{I}{(\\phi)} = \\cos{(\\sin{(\\phi)})} and \\mathbb{I}{(\\phi)} \\cos{(\\sin{(\\phi)})} = \\cos^{2}{(\\sin{(\\phi)})} and \\mathbb{I}{(\\phi)} \\cos{(\\sin{(\\phi)})} + \\mathbb{I}{(\\phi)} = \\mathbb{I}{(\\phi)} \\cos{(\\sin{(\\phi)})} + \\cos{(\\sin{(\\phi)})} and \\phi + \\mathbb{I}{(\\phi)} \\cos{(\\sin{(\\phi)})} + \\mathbb{I}{(\\phi)} = \\phi + \\mathbb{I}{(\\phi)} \\cos{(\\sin{(\\phi)})} + \\cos{(\\sin{(\\phi)})} and \\phi + \\mathbb{I}{(\\phi)} + \\cos^{2}{(\\sin{(\\phi)})} = \\phi + \\cos^{2}{(\\sin{(\\phi)})} + \\cos{(\\sin{(\\phi)})} and \\int (\\phi + \\mathbb{I}{(\\phi)} + \\cos^{2}{(\\sin{(\\phi)})}) d\\phi = \\int (\\phi + \\cos^{2}{(\\sin{(\\phi)})} + \\cos{(\\sin{(\\phi)})}) d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True))))"], [["times", 1, "cos(sin(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True)))), Pow(cos(sin(Symbol('\\\\phi', commutative=True))), Integer(2)))"], [["add", 1, "Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True))))"], "Equality(Add(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True)))), Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True))), Add(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True)))), cos(sin(Symbol('\\\\phi', commutative=True)))))"], [["add", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True)))), Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), cos(sin(Symbol('\\\\phi', commutative=True)))), cos(sin(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), Pow(cos(sin(Symbol('\\\\phi', commutative=True))), Integer(2))), Add(Symbol('\\\\phi', commutative=True), Pow(cos(sin(Symbol('\\\\phi', commutative=True))), Integer(2)), cos(sin(Symbol('\\\\phi', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\phi', commutative=True), Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), Pow(cos(sin(Symbol('\\\\phi', commutative=True))), Integer(2))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Symbol('\\\\phi', commutative=True), Pow(cos(sin(Symbol('\\\\phi', commutative=True))), Integer(2)), cos(sin(Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given H{(g)} = \\log{(g)}, then obtain H^{2}{(g)} \\log{(g)}^{2} + (e^{P_{g} \\mathbf{g}})^{\\mathbf{g}} = H{(g)} \\log{(g)}^{3} + (e^{P_{g} \\mathbf{g}})^{\\mathbf{g}}", "derivation": "H{(g)} = \\log{(g)} and H^{2}{(g)} = H{(g)} \\log{(g)} and H^{4}{(g)} = H^{2}{(g)} \\log{(g)}^{2} and H^{2}{(g)} \\log{(g)}^{2} = H{(g)} \\log{(g)}^{3} and H^{2}{(g)} \\log{(g)}^{2} + (e^{P_{g} \\mathbf{g}})^{\\mathbf{g}} = H{(g)} \\log{(g)}^{3} + (e^{P_{g} \\mathbf{g}})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["times", 1, "Function('H')(Symbol('g', commutative=True))"], "Equality(Pow(Function('H')(Symbol('g', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('H')(Symbol('g', commutative=True)), Integer(4)), Mul(Pow(Function('H')(Symbol('g', commutative=True)), Integer(2)), Pow(log(Symbol('g', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('H')(Symbol('g', commutative=True)), Integer(2)), Pow(log(Symbol('g', commutative=True)), Integer(2))), Mul(Function('H')(Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(3))))"], [["add", 4, "Pow(exp(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Pow(Function('H')(Symbol('g', commutative=True)), Integer(2)), Pow(log(Symbol('g', commutative=True)), Integer(2))), Pow(exp(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Function('H')(Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Integer(3))), Pow(exp(Mul(Symbol('P_g', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\Psi{(x,C,\\tilde{g})} = C \\tilde{g} - x, then obtain - \\frac{0^{x}}{x} = - \\frac{(C \\tilde{g} - x - \\Psi{(x,C,\\tilde{g})})^{x}}{x}", "derivation": "\\Psi{(x,C,\\tilde{g})} = C \\tilde{g} - x and 0 = C \\tilde{g} - x - \\Psi{(x,C,\\tilde{g})} and 0^{x} = (C \\tilde{g} - x - \\Psi{(x,C,\\tilde{g})})^{x} and - \\frac{0^{x}}{x} = - \\frac{(C \\tilde{g} - x - \\Psi{(x,C,\\tilde{g})})^{x}}{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["minus", 1, "Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))))"], [["power", 2, "Symbol('x', commutative=True)"], "Equality(Pow(Integer(0), Symbol('x', commutative=True)), Pow(Add(Mul(Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('x', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Symbol('x', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('x', commutative=True)), Pow(Symbol('x', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(-1), Function('\\\\Psi')(Symbol('x', commutative=True), Symbol('C', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(y^{\\prime})} = \\sin{(\\cos{(y^{\\prime})})} and \\hat{x}{(y^{\\prime})} = \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime}, then obtain \\hat{x}{(y^{\\prime})} = \\int \\sin{(\\cos{(y^{\\prime})})} dy^{\\prime}", "derivation": "\\psi^{*}{(y^{\\prime})} = \\sin{(\\cos{(y^{\\prime})})} and \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime} = \\int \\sin{(\\cos{(y^{\\prime})})} dy^{\\prime} and \\hat{x}{(y^{\\prime})} = \\int \\psi^{*}{(y^{\\prime})} dy^{\\prime} and \\hat{x}{(y^{\\prime})} = \\int \\sin{(\\cos{(y^{\\prime})})} dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), sin(cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(sin(cos(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('y^{\\\\prime}', commutative=True)), Integral(Function('\\\\psi^*')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{x}')(Symbol('y^{\\\\prime}', commutative=True)), Integral(sin(cos(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\dot{z},\\mathbf{A})} = \\dot{z} + \\log{(\\mathbf{A})}, then obtain 2 (2 \\dot{z} + 2 \\log{(\\mathbf{A})}) \\hat{p}_0{(\\dot{z},\\mathbf{A})} = (2 \\dot{z} + 2 \\log{(\\mathbf{A})})^{2}", "derivation": "\\hat{p}_0{(\\dot{z},\\mathbf{A})} = \\dot{z} + \\log{(\\mathbf{A})} and 2 \\hat{p}_0{(\\dot{z},\\mathbf{A})} = \\dot{z} + \\hat{p}_0{(\\dot{z},\\mathbf{A})} + \\log{(\\mathbf{A})} and 2 (\\dot{z} + \\hat{p}_0{(\\dot{z},\\mathbf{A})} + \\log{(\\mathbf{A})}) \\hat{p}_0{(\\dot{z},\\mathbf{A})} = (\\dot{z} + \\hat{p}_0{(\\dot{z},\\mathbf{A})} + \\log{(\\mathbf{A})})^{2} and 2 (\\dot{z} + \\log{(\\mathbf{A})}) (2 \\dot{z} + 2 \\log{(\\mathbf{A})}) = (2 \\dot{z} + 2 \\log{(\\mathbf{A})})^{2} and 2 (2 \\dot{z} + 2 \\log{(\\mathbf{A})}) \\hat{p}_0{(\\dot{z},\\mathbf{A})} = (2 \\dot{z} + 2 \\log{(\\mathbf{A})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 1, "Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Integer(2), Add(Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True))), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Pow(Add(Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Add(Symbol('\\\\dot{z}', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{A}', commutative=True))))), Pow(Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{A}', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(2), log(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{H}_l,v)} = \\hat{H}_l + \\log{(v)} and \\Psi^{\\dagger}{(v)} = \\log{(v)}, then obtain 1 - \\hat{H}_l = - \\hat{H}_l + \\frac{2 \\hat{H}_l + \\Psi^{\\dagger}{(v)}}{2 \\hat{H}_l + \\log{(v)}}", "derivation": "\\operatorname{E_{x}}{(\\hat{H}_l,v)} = \\hat{H}_l + \\log{(v)} and \\hat{H}_l + \\operatorname{E_{x}}{(\\hat{H}_l,v)} = 2 \\hat{H}_l + \\log{(v)} and 1 = \\frac{2 \\hat{H}_l + \\log{(v)}}{\\hat{H}_l + \\operatorname{E_{x}}{(\\hat{H}_l,v)}} and \\Psi^{\\dagger}{(v)} = \\log{(v)} and 1 = \\frac{2 \\hat{H}_l + \\Psi^{\\dagger}{(v)}}{\\hat{H}_l + \\operatorname{E_{x}}{(\\hat{H}_l,v)}} and 1 = \\frac{2 \\hat{H}_l + \\Psi^{\\dagger}{(v)}}{2 \\hat{H}_l + \\log{(v)}} and 1 - \\hat{H}_l = - \\hat{H}_l + \\frac{2 \\hat{H}_l + \\Psi^{\\dagger}{(v)}}{2 \\hat{H}_l + \\log{(v)}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('v', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('v', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\hat{H}_l', commutative=True), Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('v', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), log(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('E_x')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Mul(Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('v', commutative=True))), Integer(-1))))"], [["minus", 6, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True))), Pow(Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('v', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given z{(\\mathbf{r})} = \\sin{(\\mathbf{r})} and \\mathbf{F}{(\\mathbf{r})} = z^{2}{(\\mathbf{r})}, then obtain \\mathbf{F}^{\\mathbf{r}}{(\\mathbf{r})} = (z{(\\mathbf{r})} \\sin{(\\mathbf{r})})^{\\mathbf{r}}", "derivation": "z{(\\mathbf{r})} = \\sin{(\\mathbf{r})} and z^{2}{(\\mathbf{r})} = z{(\\mathbf{r})} \\sin{(\\mathbf{r})} and (z^{2}{(\\mathbf{r})})^{\\mathbf{r}} = (z{(\\mathbf{r})} \\sin{(\\mathbf{r})})^{\\mathbf{r}} and \\mathbf{F}{(\\mathbf{r})} = z^{2}{(\\mathbf{r})} and \\mathbf{F}^{\\mathbf{r}}{(\\mathbf{r})} = (z{(\\mathbf{r})} \\sin{(\\mathbf{r})})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "Function('z')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Pow(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Mul(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Mul(Function('z')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given E{(\\varepsilon_0,\\varphi)} = \\log{(\\varepsilon_0 + \\varphi)}, then derive \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} E{(\\varepsilon_0,\\varphi)})} = \\cos{(\\frac{1}{\\varepsilon_0 + \\varphi})}, then obtain \\cos^{\\varphi}{(\\frac{\\partial}{\\partial \\varepsilon_0} \\log{(\\varepsilon_0 + \\varphi)})} = \\cos^{\\varphi}{(\\frac{1}{\\varepsilon_0 + \\varphi})}", "derivation": "E{(\\varepsilon_0,\\varphi)} = \\log{(\\varepsilon_0 + \\varphi)} and \\frac{\\partial}{\\partial \\varepsilon_0} E{(\\varepsilon_0,\\varphi)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\log{(\\varepsilon_0 + \\varphi)} and \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} E{(\\varepsilon_0,\\varphi)})} = \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} \\log{(\\varepsilon_0 + \\varphi)})} and \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} E{(\\varepsilon_0,\\varphi)})} = \\cos{(\\frac{1}{\\varepsilon_0 + \\varphi})} and \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} \\log{(\\varepsilon_0 + \\varphi)})} = \\cos{(\\frac{1}{\\varepsilon_0 + \\varphi})} and \\cos^{\\varphi}{(\\frac{\\partial}{\\partial \\varepsilon_0} \\log{(\\varepsilon_0 + \\varphi)})} = \\cos^{\\varphi}{(\\frac{1}{\\varepsilon_0 + \\varphi})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Derivative(log(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('E')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(cos(Derivative(log(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(cos(Derivative(log(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Symbol('\\\\varphi', commutative=True)), Pow(cos(Pow(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given b{(E_{x})} = e^{e^{E_{x}}} and \\mathbf{F}{(E_{x})} = e^{e^{E_{x}}}, then obtain (\\frac{d}{d E_{x}} b{(E_{x})})^{E_{x}} = (\\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})})^{E_{x}}", "derivation": "b{(E_{x})} = e^{e^{E_{x}}} and \\frac{d}{d E_{x}} b{(E_{x})} = \\frac{d}{d E_{x}} e^{e^{E_{x}}} and (\\frac{d}{d E_{x}} b{(E_{x})})^{E_{x}} = (\\frac{d}{d E_{x}} e^{e^{E_{x}}})^{E_{x}} and \\mathbf{F}{(E_{x})} = e^{e^{E_{x}}} and (\\frac{d}{d E_{x}} b{(E_{x})})^{E_{x}} = (\\frac{d}{d E_{x}} \\mathbf{F}{(E_{x})})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E_x', commutative=True)), exp(exp(Symbol('E_x', commutative=True))))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E_x', commutative=True)"], "Equality(Pow(Derivative(Function('b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)), Pow(Derivative(exp(exp(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), exp(exp(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Derivative(Function('b')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)), Pow(Derivative(Function('\\\\mathbf{F}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} = \\frac{C_{d}}{\\hat{p}_0 t_{2}}, then obtain \\int \\frac{\\hat{p}_0 t_{2} (\\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} - 1)}{C_{d}} dC_{d} = \\int \\frac{\\hat{p}_0 t_{2} (\\frac{C_{d}}{\\hat{p}_0 t_{2}} - 1)}{C_{d}} dC_{d}", "derivation": "\\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} = \\frac{C_{d}}{\\hat{p}_0 t_{2}} and \\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} - 1 = \\frac{C_{d}}{\\hat{p}_0 t_{2}} - 1 and \\frac{\\hat{p}_0 t_{2} (\\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} - 1)}{C_{d}} = \\frac{\\hat{p}_0 t_{2} (\\frac{C_{d}}{\\hat{p}_0 t_{2}} - 1)}{C_{d}} and \\int \\frac{\\hat{p}_0 t_{2} (\\operatorname{f^{*}}{(t_{2},C_{d},\\hat{p}_0)} - 1)}{C_{d}} dC_{d} = \\int \\frac{\\hat{p}_0 t_{2} (\\frac{C_{d}}{\\hat{p}_0 t_{2}} - 1)}{C_{d}} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f^*')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(-1)))"], [["divide", 2, "Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('t_2', commutative=True), Add(Function('f^*')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('t_2', commutative=True), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(-1))))"], [["integrate", 3, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('t_2', commutative=True), Add(Function('f^*')(Symbol('t_2', commutative=True), Symbol('C_d', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('t_2', commutative=True), Add(Mul(Symbol('C_d', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))), Integer(-1))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v_{x},y)} = v_{x} \\cos{(y)} and \\mathbf{J}{(v_{x},y)} = v_{x} \\cos{(y)}, then obtain - v_{x} + (v_{x} \\cos{(y)})^{y} + \\mathbf{J}_P{(v_{x},y)} = - v_{x} + \\mathbf{J}_P{(v_{x},y)} + \\mathbf{J}_P^{y}{(v_{x},y)}", "derivation": "\\mathbf{J}_P{(v_{x},y)} = v_{x} \\cos{(y)} and \\mathbf{J}{(v_{x},y)} = v_{x} \\cos{(y)} and \\mathbf{J}{(v_{x},y)} = \\mathbf{J}_P{(v_{x},y)} and \\mathbf{J}^{y}{(v_{x},y)} = \\mathbf{J}_P^{y}{(v_{x},y)} and - v_{x} + \\mathbf{J}^{y}{(v_{x},y)} = - v_{x} + \\mathbf{J}_P^{y}{(v_{x},y)} and - v_{x} + \\mathbf{J}^{y}{(v_{x},y)} + \\mathbf{J}_P{(v_{x},y)} = - v_{x} + \\mathbf{J}_P{(v_{x},y)} + \\mathbf{J}_P^{y}{(v_{x},y)} and - v_{x} + (v_{x} \\cos{(y)})^{y} + \\mathbf{J}_P{(v_{x},y)} = - v_{x} + \\mathbf{J}_P{(v_{x},y)} + \\mathbf{J}_P^{y}{(v_{x},y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('v_x', commutative=True), cos(Symbol('y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('v_x', commutative=True), cos(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))"], [["power", 3, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["minus", 4, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["add", 5, "Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Pow(Mul(Symbol('v_x', commutative=True), cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))))"]]}, {"prompt": "Given g{(x,c)} = x^{c}, then derive \\frac{c g^{c}{(x,c)} \\frac{\\partial}{\\partial x} g{(x,c)}}{g{(x,c)}} + 1 = \\frac{c^{2} (x^{c})^{c}}{x} + 1, then obtain \\frac{c g^{c}{(x,c)} \\frac{\\partial}{\\partial x} g{(x,c)}}{g{(x,c)}} + 1 = \\frac{c^{2} g^{c}{(x,c)}}{x} + 1", "derivation": "g{(x,c)} = x^{c} and g^{c}{(x,c)} = (x^{c})^{c} and \\frac{\\partial}{\\partial x} g^{c}{(x,c)} = \\frac{\\partial}{\\partial x} (x^{c})^{c} and \\frac{\\partial}{\\partial x} g^{c}{(x,c)} + 1 = \\frac{\\partial}{\\partial x} (x^{c})^{c} + 1 and \\frac{c g^{c}{(x,c)} \\frac{\\partial}{\\partial x} g{(x,c)}}{g{(x,c)}} + 1 = \\frac{c^{2} (x^{c})^{c}}{x} + 1 and c x^{- c} (x^{c})^{c} \\frac{\\partial}{\\partial x} x^{c} + 1 = \\frac{c^{2} (x^{c})^{c}}{x} + 1 and \\frac{c g^{c}{(x,c)} \\frac{\\partial}{\\partial x} g{(x,c)}}{g{(x,c)}} + 1 = \\frac{c^{2} g^{c}{(x,c)}}{x} + 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('c', commutative=True), Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Integer(-1)), Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('c', commutative=True), Pow(Symbol('x', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Pow(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Symbol('c', commutative=True), Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Integer(-1)), Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(1)), Add(Mul(Pow(Symbol('c', commutative=True), Integer(2)), Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('g')(Symbol('x', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\rho_{b}{(E,\\chi)} = \\log{(E \\chi)}, then obtain - \\frac{(\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) \\int (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) d\\chi}{\\log{(E \\chi)}} = 0", "derivation": "\\rho_{b}{(E,\\chi)} = \\log{(E \\chi)} and \\rho_{b}{(E,\\chi)} - \\log{(E \\chi)} = 0 and \\int (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) d\\chi = \\int 0 d\\chi and (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) \\int (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) d\\chi = 0 and (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) \\int 0 d\\chi = 0 and - \\frac{(\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) \\int 0 d\\chi}{\\log{(E \\chi)}} = 0 and - \\frac{(\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) \\int (\\rho_{b}{(E,\\chi)} - \\log{(E \\chi)}) d\\chi}{\\log{(E \\chi)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["minus", 1, "log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Integral(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Integral(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["divide", 5, "Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Pow(log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Add(Function('\\\\rho_b')(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('E', commutative=True), Symbol('\\\\chi', commutative=True))))), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{E}{(H)} = \\cos{(H)} and \\hat{p}{(H)} = \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH, then obtain \\int - 2 \\hat{p}{(H)} \\sin{(t_{2})} \\cos{(H)} dt_{2} = \\int - 2 \\sin{(t_{2})} \\cos{(H)} \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH dt_{2}", "derivation": "\\mathbf{E}{(H)} = \\cos{(H)} and H + \\mathbf{E}{(H)} = H + \\cos{(H)} and H + \\mathbf{E}{(H)} + \\cos{(H)} = H + 2 \\cos{(H)} and \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH = \\int (H + 2 \\cos{(H)}) dH and \\hat{p}{(H)} = \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH and \\hat{p}{(H)} = \\int (H + 2 \\cos{(H)}) dH and - 2 \\hat{p}{(H)} \\sin{(t_{2})} \\cos{(H)} = - 2 \\sin{(t_{2})} \\cos{(H)} \\int (H + 2 \\cos{(H)}) dH and - 2 \\hat{p}{(H)} \\sin{(t_{2})} \\cos{(H)} = - 2 \\sin{(t_{2})} \\cos{(H)} \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH and \\int - 2 \\hat{p}{(H)} \\sin{(t_{2})} \\cos{(H)} dt_{2} = \\int - 2 \\sin{(t_{2})} \\cos{(H)} \\int (H + \\mathbf{E}{(H)} + \\cos{(H)}) dH dt_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True)))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), cos(Symbol('H', commutative=True))))"], [["add", 2, "cos(Symbol('H', commutative=True))"], "Equality(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(2), cos(Symbol('H', commutative=True)))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\hat{p}')(Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))"], [["times", 6, "Mul(Integer(-1), Integer(2), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True)), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Mul(Integer(2), cos(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(-1), Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True)), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))))"], [["integrate", 8, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integer(2), Function('\\\\hat{p}')(Symbol('H', commutative=True)), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Integer(-1), Integer(2), sin(Symbol('t_2', commutative=True)), cos(Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Function('\\\\mathbf{E}')(Symbol('H', commutative=True)), cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} = \\varepsilon_0 f, then obtain \\int (\\int \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} df - 1) d\\varepsilon_0 = \\Psi^{\\dagger} + \\frac{\\varepsilon_0^{2} f^{2}}{4} - \\varepsilon_0", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} = \\varepsilon_0 f and \\int \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} df = \\int \\varepsilon_0 f df and \\int \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} df - 1 = \\int \\varepsilon_0 f df - 1 and \\int (\\int \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} df - 1) d\\varepsilon_0 = \\int (\\int \\varepsilon_0 f df - 1) d\\varepsilon_0 and \\int (\\int \\operatorname{J_{\\varepsilon}}{(\\varepsilon_0,f)} df - 1) d\\varepsilon_0 = \\Psi^{\\dagger} + \\frac{\\varepsilon_0^{2} f^{2}}{4} - \\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)), Add(Integral(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Add(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Add(Integral(Mul(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Rational(1, 4), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(2)), Pow(Symbol('f', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given H{(r_{0})} = \\cos{(r_{0})} and \\rho_{b}{(r_{0})} = e^{H^{2}{(r_{0})}}, then obtain \\rho_{b}{(r_{0})} = e^{H{(r_{0})} \\cos{(r_{0})}}", "derivation": "H{(r_{0})} = \\cos{(r_{0})} and H^{2}{(r_{0})} = H{(r_{0})} \\cos{(r_{0})} and e^{H^{2}{(r_{0})}} = e^{H{(r_{0})} \\cos{(r_{0})}} and \\rho_{b}{(r_{0})} = e^{H^{2}{(r_{0})}} and \\rho_{b}{(r_{0})} = e^{H{(r_{0})} \\cos{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], [["times", 1, "Function('H')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('H')(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('H')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True))))"], [["exp", 2], "Equality(exp(Pow(Function('H')(Symbol('r_0', commutative=True)), Integer(2))), exp(Mul(Function('H')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('r_0', commutative=True)), exp(Pow(Function('H')(Symbol('r_0', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\rho_b')(Symbol('r_0', commutative=True)), exp(Mul(Function('H')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(g,a^{\\dagger})} = \\frac{a^{\\dagger}}{g}, then obtain \\frac{\\partial}{\\partial g} \\frac{g^{2} \\operatorname{A_{1}}{(g,a^{\\dagger})}}{a^{\\dagger}} = \\frac{d}{d g} g", "derivation": "\\operatorname{A_{1}}{(g,a^{\\dagger})} = \\frac{a^{\\dagger}}{g} and g \\operatorname{A_{1}}{(g,a^{\\dagger})} = a^{\\dagger} and \\frac{g^{2} \\operatorname{A_{1}}{(g,a^{\\dagger})}}{a^{\\dagger}} = g and \\frac{\\partial}{\\partial g} \\frac{g^{2} \\operatorname{A_{1}}{(g,a^{\\dagger})}}{a^{\\dagger}} = \\frac{d}{d g} g", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('g', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('g', commutative=True), Function('A_1')(Symbol('g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True))"], [["divide", 2, "Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(2)), Function('A_1')(Symbol('g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Symbol('g', commutative=True))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Integer(2)), Function('A_1')(Symbol('g', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Symbol('g', commutative=True), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\mathbf{E},U,f)} = \\frac{U + \\mathbf{E}}{f}, then derive \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\operatorname{A_{x}}{(\\mathbf{E},U,f)} = 0, then obtain - \\operatorname{A_{x}}{(\\mathbf{E},U,f)} + \\cos{(\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\frac{U + \\mathbf{E}}{f})} = 1 - \\operatorname{A_{x}}{(\\mathbf{E},U,f)}", "derivation": "\\operatorname{A_{x}}{(\\mathbf{E},U,f)} = \\frac{U + \\mathbf{E}}{f} and \\frac{\\partial}{\\partial U} \\operatorname{A_{x}}{(\\mathbf{E},U,f)} = \\frac{\\partial}{\\partial U} \\frac{U + \\mathbf{E}}{f} and \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\operatorname{A_{x}}{(\\mathbf{E},U,f)} = \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\frac{U + \\mathbf{E}}{f} and \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\operatorname{A_{x}}{(\\mathbf{E},U,f)} = 0 and \\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\frac{U + \\mathbf{E}}{f} = 0 and \\cos{(\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\frac{U + \\mathbf{E}}{f})} = 1 and - \\operatorname{A_{x}}{(\\mathbf{E},U,f)} + \\cos{(\\frac{\\partial^{2}}{\\partial \\mathbf{E}\\partial U} \\frac{U + \\mathbf{E}}{f})} = 1 - \\operatorname{A_{x}}{(\\mathbf{E},U,f)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(0))"], [["cos", 5], "Equality(cos(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Integer(1))"], [["add", 6, "Mul(Integer(-1), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True))), cos(Derivative(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Symbol('U', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))), Add(Integer(1), Mul(Integer(-1), Function('A_x')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('U', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given G{(\\lambda)} = e^{\\lambda} and \\operatorname{A_{1}}{(\\lambda)} = \\frac{d^{2}}{d \\lambda^{2}} (e^{\\lambda} - 1), then obtain \\log{(\\operatorname{A_{1}}^{\\lambda}{(\\lambda)})} = \\log{((\\frac{d^{2}}{d \\lambda^{2}} (G{(\\lambda)} - 1))^{\\lambda})}", "derivation": "G{(\\lambda)} = e^{\\lambda} and G{(\\lambda)} - 1 = e^{\\lambda} - 1 and \\frac{d}{d \\lambda} (G{(\\lambda)} - 1) = \\frac{d}{d \\lambda} (e^{\\lambda} - 1) and \\frac{d^{2}}{d \\lambda^{2}} (G{(\\lambda)} - 1) = \\frac{d^{2}}{d \\lambda^{2}} (e^{\\lambda} - 1) and \\operatorname{A_{1}}{(\\lambda)} = \\frac{d^{2}}{d \\lambda^{2}} (e^{\\lambda} - 1) and \\operatorname{A_{1}}^{\\lambda}{(\\lambda)} = (\\frac{d^{2}}{d \\lambda^{2}} (e^{\\lambda} - 1))^{\\lambda} and \\operatorname{A_{1}}^{\\lambda}{(\\lambda)} = (\\frac{d^{2}}{d \\lambda^{2}} (G{(\\lambda)} - 1))^{\\lambda} and \\log{(\\operatorname{A_{1}}^{\\lambda}{(\\lambda)})} = \\log{((\\frac{d^{2}}{d \\lambda^{2}} (G{(\\lambda)} - 1))^{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["minus", 1, 1], "Equality(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Add(exp(Symbol('\\\\lambda', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Derivative(Add(exp(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\lambda', commutative=True)), Derivative(Add(exp(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))))"], [["power", 5, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(Add(exp(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Function('A_1')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Derivative(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True)))"], [["log", 7], "Equality(log(Pow(Function('A_1')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), log(Pow(Derivative(Add(Function('G')(Symbol('\\\\lambda', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))), Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given r{(A_{2})} = \\log{(A_{2})}, then derive A_{2} (\\frac{d}{d A_{2}} r{(A_{2})} - \\frac{1}{A_{2}}) + r{(A_{2})} - \\log{(A_{2})} = 0, then obtain A_{2} (\\frac{d}{d A_{2}} \\log{(A_{2})} - \\frac{1}{A_{2}}) = 0", "derivation": "r{(A_{2})} = \\log{(A_{2})} and r{(A_{2})} - \\log{(A_{2})} = 0 and A_{2} (r{(A_{2})} - \\log{(A_{2})}) = 0 and \\frac{d}{d A_{2}} A_{2} (r{(A_{2})} - \\log{(A_{2})}) = \\frac{d}{d A_{2}} 0 and A_{2} (\\frac{d}{d A_{2}} r{(A_{2})} - \\frac{1}{A_{2}}) + r{(A_{2})} - \\log{(A_{2})} = 0 and A_{2} (\\frac{d}{d A_{2}} r{(A_{2})} - \\frac{1}{A_{2}}) = 0 and A_{2} (\\frac{d}{d A_{2}} \\log{(A_{2})} - \\frac{1}{A_{2}}) = 0", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["minus", 1, "log(Symbol('A_2', commutative=True))"], "Equality(Add(Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), log(Symbol('A_2', commutative=True)))), Integer(0))"], [["times", 2, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Add(Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), log(Symbol('A_2', commutative=True))))), Integer(0))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('A_2', commutative=True), Add(Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), log(Symbol('A_2', commutative=True))))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('A_2', commutative=True), Add(Derivative(Function('r')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1))))), Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), log(Symbol('A_2', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('A_2', commutative=True), Add(Derivative(Function('r')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1))))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('A_2', commutative=True), Add(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1))))), Integer(0))"]]}, {"prompt": "Given Z{(A_{1})} = A_{1}, then derive c + \\frac{Z^{2}{(A_{1})}}{2} = \\int A_{1} dZ{(A_{1})}, then derive - \\frac{A_{1}^{2}}{2} - c = - \\frac{A_{1}^{2}}{2} - A_{z}, then obtain \\int \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - c) dc = \\int \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - A_{z}) dc", "derivation": "Z{(A_{1})} = A_{1} and \\int Z{(A_{1})} dA_{1} = \\int A_{1} dA_{1} and \\int Z{(A_{1})} dZ{(A_{1})} = \\int A_{1} dZ{(A_{1})} and c + \\frac{Z^{2}{(A_{1})}}{2} = \\int A_{1} dZ{(A_{1})} and \\frac{A_{1}^{2}}{2} + c = \\int A_{1} dA_{1} and - \\frac{A_{1}^{2}}{2} - c = - \\int A_{1} dA_{1} and - \\frac{A_{1}^{2}}{2} - c = - \\frac{A_{1}^{2}}{2} - A_{z} and \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - c) = \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - A_{z}) and \\int \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - c) dc = \\int \\frac{\\partial}{\\partial c} (- \\frac{A_{1}^{2}}{2} - A_{z}) dc", "srepr_derivation": [["renaming_premise", "Equality(Function('Z')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Symbol('A_1', commutative=True), Tuple(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('Z')(Symbol('A_1', commutative=True)), Tuple(Function('Z')(Symbol('A_1', commutative=True)))), Integral(Symbol('A_1', commutative=True), Tuple(Function('Z')(Symbol('A_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('c', commutative=True), Mul(Rational(1, 2), Pow(Function('Z')(Symbol('A_1', commutative=True)), Integer(2)))), Integral(Symbol('A_1', commutative=True), Tuple(Function('Z')(Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Symbol('c', commutative=True)), Integral(Symbol('A_1', commutative=True), Tuple(Symbol('A_1', commutative=True))))"], [["times", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True))), Mul(Integer(-1), Integral(Symbol('A_1', commutative=True), Tuple(Symbol('A_1', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_z', commutative=True))))"], [["differentiate", 7, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["integrate", 8, "Symbol('c', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_z', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Tuple(Symbol('c', commutative=True))))"]]}, {"prompt": "Given A{(\\mathbf{H})} = \\int e^{\\mathbf{H}} d\\mathbf{H} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{f})} = \\mathbf{f}, then derive A^{\\mathbf{H}}{(\\mathbf{H})} = (\\mathbf{f} + e^{\\mathbf{H}})^{\\mathbf{H}}, then obtain A^{2 \\mathbf{H}}{(\\mathbf{H})} = (\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{f})} + e^{\\mathbf{H}})^{\\mathbf{H}} A^{\\mathbf{H}}{(\\mathbf{H})}", "derivation": "A{(\\mathbf{H})} = \\int e^{\\mathbf{H}} d\\mathbf{H} and A^{\\mathbf{H}}{(\\mathbf{H})} = (\\int e^{\\mathbf{H}} d\\mathbf{H})^{\\mathbf{H}} and A^{\\mathbf{H}}{(\\mathbf{H})} = (\\mathbf{f} + e^{\\mathbf{H}})^{\\mathbf{H}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{f})} = \\mathbf{f} and A^{2 \\mathbf{H}}{(\\mathbf{H})} = (\\mathbf{f} + e^{\\mathbf{H}})^{\\mathbf{H}} A^{\\mathbf{H}}{(\\mathbf{H})} and A^{2 \\mathbf{H}}{(\\mathbf{H})} = (\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{f})} + e^{\\mathbf{H}})^{\\mathbf{H}} A^{\\mathbf{H}}{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(exp(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], [["times", 3, "Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), exp(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{f}', commutative=True)), exp(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('A')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given J{(E,\\hbar)} = E^{\\hbar}, then obtain J{(E,\\hbar)} \\frac{\\partial}{\\partial E} E^{\\hbar} = E^{\\hbar} \\frac{\\partial}{\\partial E} E^{\\hbar}", "derivation": "J{(E,\\hbar)} = E^{\\hbar} and \\frac{\\partial}{\\partial E} J{(E,\\hbar)} = \\frac{\\partial}{\\partial E} E^{\\hbar} and J{(E,\\hbar)} \\frac{\\partial}{\\partial E} J{(E,\\hbar)} = E^{\\hbar} \\frac{\\partial}{\\partial E} J{(E,\\hbar)} and J{(E,\\hbar)} \\frac{\\partial}{\\partial E} E^{\\hbar} = E^{\\hbar} \\frac{\\partial}{\\partial E} E^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Mul(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('J')(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Pow(Symbol('E', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(Z,u)} = Z^{u}, then obtain - \\int (Z^{u})^{Z} du + \\iint \\operatorname{P_{g}}^{Z}{(Z,u)} du dZ = - \\int (Z^{u})^{Z} du + \\iint (Z^{u})^{Z} du dZ", "derivation": "\\operatorname{P_{g}}{(Z,u)} = Z^{u} and \\operatorname{P_{g}}^{Z}{(Z,u)} = (Z^{u})^{Z} and \\int \\operatorname{P_{g}}^{Z}{(Z,u)} du = \\int (Z^{u})^{Z} du and \\iint \\operatorname{P_{g}}^{Z}{(Z,u)} du dZ = \\iint (Z^{u})^{Z} du dZ and - \\int (Z^{u})^{Z} du + \\iint \\operatorname{P_{g}}^{Z}{(Z,u)} du dZ = - \\int (Z^{u})^{Z} du + \\iint (Z^{u})^{Z} du dZ", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(Function('P_g')(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Pow(Function('P_g')(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["minus", 4, "Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)))), Integral(Pow(Function('P_g')(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)))), Integral(Pow(Pow(Symbol('Z', commutative=True), Symbol('u', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\pi,\\theta)} = (e^{\\pi})^{\\theta} and \\operatorname{v_{1}}{(\\pi,\\theta)} = (e^{\\pi})^{\\theta}, then obtain \\operatorname{E_{\\lambda}}{(\\pi,\\theta)} \\operatorname{v_{1}}{(\\pi,\\theta)} = \\operatorname{v_{1}}^{2}{(\\pi,\\theta)}", "derivation": "\\operatorname{E_{\\lambda}}{(\\pi,\\theta)} = (e^{\\pi})^{\\theta} and \\operatorname{v_{1}}{(\\pi,\\theta)} = (e^{\\pi})^{\\theta} and \\operatorname{E_{\\lambda}}{(\\pi,\\theta)} (e^{\\pi})^{\\theta} = (e^{\\pi})^{2 \\theta} and \\operatorname{E_{\\lambda}}{(\\pi,\\theta)} \\operatorname{v_{1}}{(\\pi,\\theta)} = \\operatorname{v_{1}}^{2}{(\\pi,\\theta)}", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Pow(exp(Symbol('\\\\pi', commutative=True)), Symbol('\\\\theta', commutative=True))), Pow(exp(Symbol('\\\\pi', commutative=True)), Mul(Integer(2), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Function('v_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True))), Pow(Function('v_1')(Symbol('\\\\pi', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\eta^{\\prime}{(g,s)} = e^{g s} and y{(g,s)} = e^{g s}, then obtain y{(g,s)} - \\frac{\\eta^{\\prime}{(g,s)}}{g s} = \\eta^{\\prime}{(g,s)} - \\frac{\\eta^{\\prime}{(g,s)}}{g s}", "derivation": "\\eta^{\\prime}{(g,s)} = e^{g s} and y{(g,s)} = e^{g s} and \\frac{\\eta^{\\prime}{(g,s)}}{g s} = \\frac{e^{g s}}{g s} and \\frac{\\eta^{\\prime}{(g,s)}}{g s} = \\frac{y{(g,s)}}{g s} and y{(g,s)} = \\eta^{\\prime}{(g,s)} and y{(g,s)} - \\frac{y{(g,s)}}{g s} = \\eta^{\\prime}{(g,s)} - \\frac{y{(g,s)}}{g s} and y{(g,s)} - \\frac{\\eta^{\\prime}{(g,s)}}{g s} = \\eta^{\\prime}{(g,s)} - \\frac{\\eta^{\\prime}{(g,s)}}{g s}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('g', commutative=True), Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('g', commutative=True), Symbol('s', commutative=True))))"], [["divide", 1, "Mul(Symbol('g', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)))"], [["minus", 5, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)))"], "Equality(Add(Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('y')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(r)} = \\sin{(\\sin{(r)})}, then obtain \\int (\\varepsilon_{0}^{r}{(r)} \\sin{(r)} - \\sin^{r}{(\\sin{(r)})}) dr = \\int (\\sin{(r)} \\sin^{r}{(\\sin{(r)})} - \\sin^{r}{(\\sin{(r)})}) dr", "derivation": "\\varepsilon_{0}{(r)} = \\sin{(\\sin{(r)})} and \\varepsilon_{0}^{r}{(r)} = \\sin^{r}{(\\sin{(r)})} and \\varepsilon_{0}^{r}{(r)} \\sin{(r)} = \\sin{(r)} \\sin^{r}{(\\sin{(r)})} and \\varepsilon_{0}^{r}{(r)} \\sin{(r)} - \\sin^{r}{(\\sin{(r)})} = \\sin{(r)} \\sin^{r}{(\\sin{(r)})} - \\sin^{r}{(\\sin{(r)})} and \\int (\\varepsilon_{0}^{r}{(r)} \\sin{(r)} - \\sin^{r}{(\\sin{(r)})}) dr = \\int (\\sin{(r)} \\sin^{r}{(\\sin{(r)})} - \\sin^{r}{(\\sin{(r)})}) dr", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('r', commutative=True)), sin(sin(Symbol('r', commutative=True))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["times", 2, "sin(Symbol('r', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Mul(sin(Symbol('r', commutative=True)), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))))"], [["minus", 3, "Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))), Add(Mul(sin(Symbol('r', commutative=True)), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('r', commutative=True)), Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(sin(Symbol('r', commutative=True)), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Mul(Integer(-1), Pow(sin(sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\psi{(V,\\mathbf{J})} = V - \\mathbf{J} and \\varphi^{*}{(V,\\mathbf{J})} = \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})}, then obtain - V + \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})} = - V + \\varphi^{*}{(V,\\mathbf{J})}", "derivation": "\\psi{(V,\\mathbf{J})} = V - \\mathbf{J} and \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})} = \\frac{\\partial}{\\partial V} (V - \\mathbf{J}) and \\varphi^{*}{(V,\\mathbf{J})} = \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})} and \\varphi^{*}{(V,\\mathbf{J})} = \\frac{\\partial}{\\partial V} (V - \\mathbf{J}) and - V + \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})} = - V + \\frac{\\partial}{\\partial V} (V - \\mathbf{J}) and - V + \\frac{\\partial}{\\partial V} \\psi{(V,\\mathbf{J})} = - V + \\varphi^{*}{(V,\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('V', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Function('\\\\psi')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Function('\\\\varphi^*')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(M,\\omega)} = M \\omega, then derive \\frac{\\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)}}{M \\omega} = \\frac{1}{M}, then obtain \\frac{\\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)}}{\\hat{\\mathbf{r}}{(M,\\omega)}} = \\frac{1}{M}", "derivation": "\\hat{\\mathbf{r}}{(M,\\omega)} = M \\omega and \\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)} = \\frac{\\partial}{\\partial M} M \\omega and \\frac{\\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)}}{M \\omega} = \\frac{\\frac{\\partial}{\\partial M} M \\omega}{M \\omega} and \\frac{\\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)}}{M \\omega} = \\frac{1}{M} and \\frac{\\frac{\\partial}{\\partial M} \\hat{\\mathbf{r}}{(M,\\omega)}}{\\hat{\\mathbf{r}}{(M,\\omega)}} = \\frac{1}{M}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Mul(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Pow(Symbol('M', commutative=True), Integer(-1)))"]]}, {"prompt": "Given E{(T,\\hat{H}_l)} = - T + \\hat{H}_l, then derive - \\int E{(T,\\hat{H}_l)} dT = \\frac{T^{2}}{2} - T \\hat{H}_l - g, then obtain - \\int (- T + \\hat{H}_l) dT = \\frac{T^{2}}{2} - T \\hat{H}_l - g", "derivation": "E{(T,\\hat{H}_l)} = - T + \\hat{H}_l and \\int E{(T,\\hat{H}_l)} dT = \\int (- T + \\hat{H}_l) dT and - \\int E{(T,\\hat{H}_l)} dT = - \\int (- T + \\hat{H}_l) dT and - \\int E{(T,\\hat{H}_l)} dT = \\frac{T^{2}}{2} - T \\hat{H}_l - g and - \\int (- T + \\hat{H}_l) dT = \\frac{T^{2}}{2} - T \\hat{H}_l - g", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('E')(Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('E')(Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Integral(Function('E')(Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(p)} = \\log{(p)}, then derive \\int \\operatorname{f^{*}}{(p)} dp = V_{\\mathbf{B}} + p \\log{(p)} - p, then obtain V_{\\mathbf{B}} + p \\operatorname{f^{*}}{(p)} - p = \\int \\log{(p)} dp", "derivation": "\\operatorname{f^{*}}{(p)} = \\log{(p)} and \\int \\operatorname{f^{*}}{(p)} dp = \\int \\log{(p)} dp and \\int \\operatorname{f^{*}}{(p)} dp = V_{\\mathbf{B}} + p \\log{(p)} - p and \\int \\operatorname{f^{*}}{(p)} dp = V_{\\mathbf{B}} + p \\operatorname{f^{*}}{(p)} - p and V_{\\mathbf{B}} + p \\operatorname{f^{*}}{(p)} - p = \\int \\log{(p)} dp", "srepr_derivation": [["get_premise", "Equality(Function('f^*')(Symbol('p', commutative=True)), log(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('p', commutative=True), log(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('f^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('p', commutative=True), Function('f^*')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Symbol('p', commutative=True), Function('f^*')(Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('p', commutative=True))), Integral(log(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(k)} = \\int \\log{(k)} dk, then derive \\operatorname{n_{2}}{(k)} = E_{x} + k \\log{(k)} - k, then obtain E_{x} + k \\log{(k)} - k + 1 = \\int \\log{(k)} dk + 1", "derivation": "\\operatorname{n_{2}}{(k)} = \\int \\log{(k)} dk and \\operatorname{n_{2}}{(k)} = E_{x} + k \\log{(k)} - k and E_{x} + k \\log{(k)} - k = \\int \\log{(k)} dk and E_{x} + k \\log{(k)} - k + 1 = \\int \\log{(k)} dk + 1", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('k', commutative=True)), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n_2')(Symbol('k', commutative=True)), Add(Symbol('E_x', commutative=True), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('E_x', commutative=True), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True))), Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Symbol('E_x', commutative=True), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))), Mul(Integer(-1), Symbol('k', commutative=True)), Integer(1)), Add(Integral(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\hat{x}{(T,\\chi)} = (e^{\\chi})^{T}, then obtain - V_{\\mathbf{E}} + (e^{\\chi})^{T} + ((e^{\\chi})^{T})^{- T} \\hat{x}^{T}{(T,\\chi)} = - V_{\\mathbf{E}} + (e^{\\chi})^{T} + 1", "derivation": "\\hat{x}{(T,\\chi)} = (e^{\\chi})^{T} and \\hat{x}^{T}{(T,\\chi)} = ((e^{\\chi})^{T})^{T} and ((e^{\\chi})^{T})^{- T} \\hat{x}^{T}{(T,\\chi)} = 1 and - V_{\\mathbf{E}} + (e^{\\chi})^{T} + ((e^{\\chi})^{T})^{- T} \\hat{x}^{T}{(T,\\chi)} = - V_{\\mathbf{E}} + (e^{\\chi})^{T} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["divide", 2, "Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Symbol('T', commutative=True))"], "Equality(Mul(Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True))), Integer(1))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Mul(Pow(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Pow(Function('\\\\hat{x}')(Symbol('T', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('T', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(b,\\mathbb{I})} = \\cos{(\\mathbb{I} b)}, then obtain \\int (\\int \\operatorname{c_{0}}^{b}{(b,\\mathbb{I})} d\\mathbb{I} - 1) db = \\int (\\int \\cos^{b}{(\\mathbb{I} b)} d\\mathbb{I} - 1) db", "derivation": "\\operatorname{c_{0}}{(b,\\mathbb{I})} = \\cos{(\\mathbb{I} b)} and \\operatorname{c_{0}}^{b}{(b,\\mathbb{I})} = \\cos^{b}{(\\mathbb{I} b)} and \\int \\operatorname{c_{0}}^{b}{(b,\\mathbb{I})} d\\mathbb{I} = \\int \\cos^{b}{(\\mathbb{I} b)} d\\mathbb{I} and \\int \\operatorname{c_{0}}^{b}{(b,\\mathbb{I})} d\\mathbb{I} - 1 = \\int \\cos^{b}{(\\mathbb{I} b)} d\\mathbb{I} - 1 and \\int (\\int \\operatorname{c_{0}}^{b}{(b,\\mathbb{I})} d\\mathbb{I} - 1) db = \\int (\\int \\cos^{b}{(\\mathbb{I} b)} d\\mathbb{I} - 1) db", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('b', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('b', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('b', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('b', commutative=True)), Pow(cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Pow(Function('c_0')(Symbol('b', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Pow(cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Pow(Function('c_0')(Symbol('b', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Add(Integral(Pow(cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Integral(Pow(Function('c_0')(Symbol('b', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True))), Integral(Add(Integral(Pow(cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given H{(\\mu)} = \\sin{(\\log{(\\mu)})}, then obtain \\frac{d}{d \\mu} ((- \\sin{(\\log{(\\mu)})})^{\\mu} - H{(\\mu)}) = \\frac{d}{d \\mu} ((- H{(\\mu)})^{\\mu} - H{(\\mu)})", "derivation": "H{(\\mu)} = \\sin{(\\log{(\\mu)})} and H{(\\mu)} - \\sin{(\\log{(\\mu)})} = 0 and - \\sin{(\\log{(\\mu)})} = - H{(\\mu)} and (- \\sin{(\\log{(\\mu)})})^{\\mu} = (- H{(\\mu)})^{\\mu} and (- \\sin{(\\log{(\\mu)})})^{\\mu} - H{(\\mu)} = (- H{(\\mu)})^{\\mu} - H{(\\mu)} and \\frac{d}{d \\mu} ((- \\sin{(\\log{(\\mu)})})^{\\mu} - H{(\\mu)}) = \\frac{d}{d \\mu} ((- H{(\\mu)})^{\\mu} - H{(\\mu)})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mu', commutative=True)), sin(log(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "sin(log(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('H')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\mu', commutative=True))))), Integer(0))"], [["minus", 2, "Function('H')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), sin(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True))))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Integer(-1), sin(log(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["minus", 4, "Function('H')(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Mul(Integer(-1), sin(log(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True)))), Add(Pow(Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Pow(Mul(Integer(-1), sin(log(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(\\eta)} = \\log{(\\sin{(\\eta)})}, then obtain \\frac{\\eta}{\\int \\log{(\\sin{(\\eta)})} d\\eta} = \\frac{\\eta - T{(\\eta)} + \\log{(\\sin{(\\eta)})}}{\\int \\log{(\\sin{(\\eta)})} d\\eta}", "derivation": "T{(\\eta)} = \\log{(\\sin{(\\eta)})} and \\eta + T{(\\eta)} = \\eta + \\log{(\\sin{(\\eta)})} and \\eta = \\eta - T{(\\eta)} + \\log{(\\sin{(\\eta)})} and \\int T{(\\eta)} d\\eta = \\int \\log{(\\sin{(\\eta)})} d\\eta and \\frac{\\eta}{\\int T{(\\eta)} d\\eta} = \\frac{\\eta - T{(\\eta)} + \\log{(\\sin{(\\eta)})}}{\\int T{(\\eta)} d\\eta} and \\frac{\\eta}{\\int \\log{(\\sin{(\\eta)})} d\\eta} = \\frac{\\eta - T{(\\eta)} + \\log{(\\sin{(\\eta)})}}{\\int \\log{(\\sin{(\\eta)})} d\\eta}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\eta', commutative=True)), log(sin(Symbol('\\\\eta', commutative=True))))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('T')(Symbol('\\\\eta', commutative=True))), Add(Symbol('\\\\eta', commutative=True), log(sin(Symbol('\\\\eta', commutative=True)))))"], [["minus", 2, "Function('T')(Symbol('\\\\eta', commutative=True))"], "Equality(Symbol('\\\\eta', commutative=True), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True))), log(sin(Symbol('\\\\eta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('T')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(log(sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["divide", 3, "Integral(Function('T')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Pow(Integral(Function('T')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True))), log(sin(Symbol('\\\\eta', commutative=True)))), Pow(Integral(Function('T')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\eta', commutative=True), Pow(Integral(log(sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\eta', commutative=True))), log(sin(Symbol('\\\\eta', commutative=True)))), Pow(Integral(log(sin(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\psi,\\omega,\\mathbf{f})} = \\mathbf{f} + \\omega^{\\psi} and m{(\\psi,\\omega,\\mathbf{f})} = 2 \\mathbf{f} + \\psi (\\mathbf{f} + \\omega^{\\psi})^{\\omega}, then obtain m{(\\psi,\\omega,\\mathbf{f})} = 2 \\mathbf{f} + \\psi \\operatorname{z^{*}}^{\\omega}{(\\psi,\\omega,\\mathbf{f})}", "derivation": "\\operatorname{z^{*}}{(\\psi,\\omega,\\mathbf{f})} = \\mathbf{f} + \\omega^{\\psi} and \\operatorname{z^{*}}^{\\omega}{(\\psi,\\omega,\\mathbf{f})} = (\\mathbf{f} + \\omega^{\\psi})^{\\omega} and \\psi \\operatorname{z^{*}}^{\\omega}{(\\psi,\\omega,\\mathbf{f})} = \\psi (\\mathbf{f} + \\omega^{\\psi})^{\\omega} and 2 \\mathbf{f} + \\psi \\operatorname{z^{*}}^{\\omega}{(\\psi,\\omega,\\mathbf{f})} = 2 \\mathbf{f} + \\psi (\\mathbf{f} + \\omega^{\\psi})^{\\omega} and m{(\\psi,\\omega,\\mathbf{f})} = 2 \\mathbf{f} + \\psi (\\mathbf{f} + \\omega^{\\psi})^{\\omega} and m{(\\psi,\\omega,\\mathbf{f})} = 2 \\mathbf{f} + \\psi \\operatorname{z^{*}}^{\\omega}{(\\psi,\\omega,\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["times", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Add(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\psi', commutative=True))), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('m')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Pow(Function('z^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(t_{1},A_{2})} = A_{2} t_{1} and \\rho{(t_{1},A_{2})} = (- A_{2} + \\theta_{2}{(t_{1},A_{2})})^{A_{2}}, then obtain - t_{1} + \\rho{(t_{1},A_{2})} = - t_{1} + (- A_{2} + \\theta_{2}{(t_{1},A_{2})})^{A_{2}}", "derivation": "\\theta_{2}{(t_{1},A_{2})} = A_{2} t_{1} and - A_{2} + \\theta_{2}{(t_{1},A_{2})} = A_{2} t_{1} - A_{2} and (- A_{2} + \\theta_{2}{(t_{1},A_{2})})^{A_{2}} = (A_{2} t_{1} - A_{2})^{A_{2}} and \\rho{(t_{1},A_{2})} = (- A_{2} + \\theta_{2}{(t_{1},A_{2})})^{A_{2}} and \\rho{(t_{1},A_{2})} = (A_{2} t_{1} - A_{2})^{A_{2}} and - t_{1} + \\rho{(t_{1},A_{2})} = - t_{1} + (A_{2} t_{1} - A_{2})^{A_{2}} and - t_{1} + \\rho{(t_{1},A_{2})} = - t_{1} + (- A_{2} + \\theta_{2}{(t_{1},A_{2})})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True)))"], [["minus", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\theta_2')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\theta_2')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\theta_2')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))"], [["minus", 5, "Symbol('t_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True)), Mul(Integer(-1), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('\\\\rho')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Function('\\\\theta_2')(Symbol('t_1', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(\\theta,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta and \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = (\\mathbf{J}_P \\theta)^{\\theta}, then obtain \\theta \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = \\theta \\hat{x}_0^{\\theta}{(\\theta,\\mathbf{J}_P)}", "derivation": "\\hat{x}_0{(\\theta,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta and \\hat{x}_0^{\\theta}{(\\theta,\\mathbf{J}_P)} = (\\mathbf{J}_P \\theta)^{\\theta} and \\theta \\hat{x}_0^{\\theta}{(\\theta,\\mathbf{J}_P)} = \\theta (\\mathbf{J}_P \\theta)^{\\theta} and \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = (\\mathbf{J}_P \\theta)^{\\theta} and \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = \\hat{x}_0^{\\theta}{(\\theta,\\mathbf{J}_P)} and \\theta \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = \\theta (\\mathbf{J}_P \\theta)^{\\theta} and \\theta \\operatorname{A_{x}}{(\\theta,\\mathbf{J}_P)} = \\theta \\hat{x}_0^{\\theta}{(\\theta,\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["times", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('A_x')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mu_0,r)} = \\int \\frac{\\mu_0}{r} dr, then obtain \\frac{\\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mu_0,r)} \\int \\frac{\\mu_0}{r} dr}{\\operatorname{v_{t}}{(\\mu_0,r)}} = \\frac{(\\frac{\\partial}{\\partial \\mu_0} \\int \\frac{\\mu_0}{r} dr) \\int \\frac{\\mu_0}{r} dr}{\\operatorname{v_{t}}{(\\mu_0,r)}}", "derivation": "\\operatorname{v_{t}}{(\\mu_0,r)} = \\int \\frac{\\mu_0}{r} dr and \\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mu_0,r)} = \\frac{\\partial}{\\partial \\mu_0} \\int \\frac{\\mu_0}{r} dr and \\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mu_0,r)} \\int \\frac{\\mu_0}{r} dr = (\\frac{\\partial}{\\partial \\mu_0} \\int \\frac{\\mu_0}{r} dr) \\int \\frac{\\mu_0}{r} dr and \\frac{\\frac{\\partial}{\\partial \\mu_0} \\operatorname{v_{t}}{(\\mu_0,r)} \\int \\frac{\\mu_0}{r} dr}{\\operatorname{v_{t}}{(\\mu_0,r)}} = \\frac{(\\frac{\\partial}{\\partial \\mu_0} \\int \\frac{\\mu_0}{r} dr) \\int \\frac{\\mu_0}{r} dr}{\\operatorname{v_{t}}{(\\mu_0,r)}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["times", 2, "Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Derivative(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True)))), Mul(Derivative(Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True)))))"], [["divide", 3, "Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Pow(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True)))), Mul(Pow(Function('v_t')(Symbol('\\\\mu_0', commutative=True), Symbol('r', commutative=True)), Integer(-1)), Derivative(Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('r', commutative=True), Integer(-1))), Tuple(Symbol('r', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)}, then obtain \\mathbf{F}{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\mathbf{F}{(\\tilde{g}^*)}) d\\tilde{g}^* = \\mathbf{F}{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\sin{(\\tilde{g}^*)}) d\\tilde{g}^*", "derivation": "\\mathbf{F}{(\\tilde{g}^*)} = \\sin{(\\tilde{g}^*)} and \\tilde{g}^* + \\mathbf{F}{(\\tilde{g}^*)} = \\tilde{g}^* + \\sin{(\\tilde{g}^*)} and \\int (\\tilde{g}^* + \\mathbf{F}{(\\tilde{g}^*)}) d\\tilde{g}^* = \\int (\\tilde{g}^* + \\sin{(\\tilde{g}^*)}) d\\tilde{g}^* and \\sin{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\mathbf{F}{(\\tilde{g}^*)}) d\\tilde{g}^* = \\sin{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\sin{(\\tilde{g}^*)}) d\\tilde{g}^* and \\mathbf{F}{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\mathbf{F}{(\\tilde{g}^*)}) d\\tilde{g}^* = \\mathbf{F}{(\\tilde{g}^*)} + \\int (\\tilde{g}^* + \\sin{(\\tilde{g}^*)}) d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True))), Add(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 3, "sin(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(sin(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Add(Function('\\\\mathbf{F}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given A{(\\mathbf{M},v_{t})} = \\frac{\\mathbf{M}}{v_{t}}, then obtain ((\\int A{(\\mathbf{M},v_{t})} dv_{t})^{\\mathbf{M}})^{v_{t}} = ((\\int \\frac{\\mathbf{M}}{v_{t}} dv_{t})^{\\mathbf{M}})^{v_{t}}", "derivation": "A{(\\mathbf{M},v_{t})} = \\frac{\\mathbf{M}}{v_{t}} and \\int A{(\\mathbf{M},v_{t})} dv_{t} = \\int \\frac{\\mathbf{M}}{v_{t}} dv_{t} and (\\int A{(\\mathbf{M},v_{t})} dv_{t})^{\\mathbf{M}} = (\\int \\frac{\\mathbf{M}}{v_{t}} dv_{t})^{\\mathbf{M}} and ((\\int A{(\\mathbf{M},v_{t})} dv_{t})^{\\mathbf{M}})^{v_{t}} = ((\\int \\frac{\\mathbf{M}}{v_{t}} dv_{t})^{\\mathbf{M}})^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Integral(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Pow(Integral(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('v_t', commutative=True)), Pow(Pow(Integral(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('v_t', commutative=True), Integer(-1))), Tuple(Symbol('v_t', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given f{(\\hat{p},\\mathbf{H})} = \\hat{p} \\mathbf{H} and \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = \\int f{(\\hat{p},\\mathbf{H})} d\\hat{p}, then obtain (\\int \\hat{p} \\mathbf{H} d\\hat{p})^{\\mathbf{H}} = (\\int f{(\\hat{p},\\mathbf{H})} d\\hat{p})^{\\mathbf{H}}", "derivation": "f{(\\hat{p},\\mathbf{H})} = \\hat{p} \\mathbf{H} and \\int f{(\\hat{p},\\mathbf{H})} d\\hat{p} = \\int \\hat{p} \\mathbf{H} d\\hat{p} and \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = \\int f{(\\hat{p},\\mathbf{H})} d\\hat{p} and \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = \\int \\hat{p} \\mathbf{H} d\\hat{p} and \\operatorname{A_{y}}^{\\mathbf{H}}{(\\hat{p},\\mathbf{H})} = (\\int f{(\\hat{p},\\mathbf{H})} d\\hat{p})^{\\mathbf{H}} and (\\int \\hat{p} \\mathbf{H} d\\hat{p})^{\\mathbf{H}} = (\\int f{(\\hat{p},\\mathbf{H})} d\\hat{p})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Function('f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Function('f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Function('f')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(r)} = \\cos{(e^{r})}, then derive \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = - (e^{r} \\cos{(e^{r})} + \\sin{(e^{r})}) e^{r}, then obtain 2 \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = - (\\operatorname{c_{0}}{(r)} e^{r} + \\sin{(e^{r})}) e^{r} + \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)}", "derivation": "\\operatorname{c_{0}}{(r)} = \\cos{(e^{r})} and \\frac{d}{d r} \\operatorname{c_{0}}{(r)} = \\frac{d}{d r} \\cos{(e^{r})} and \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = \\frac{d^{2}}{d r^{2}} \\cos{(e^{r})} and \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = - (e^{r} \\cos{(e^{r})} + \\sin{(e^{r})}) e^{r} and \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = - (\\operatorname{c_{0}}{(r)} e^{r} + \\sin{(e^{r})}) e^{r} and 2 \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)} = - (\\operatorname{c_{0}}{(r)} e^{r} + \\sin{(e^{r})}) e^{r} + \\frac{d^{2}}{d r^{2}} \\operatorname{c_{0}}{(r)}", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('r', commutative=True)), cos(exp(Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(cos(exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Add(Mul(exp(Symbol('r', commutative=True)), cos(exp(Symbol('r', commutative=True)))), sin(exp(Symbol('r', commutative=True)))), exp(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Add(Mul(Function('c_0')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))), exp(Symbol('r', commutative=True))))"], [["add", 5, "Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Add(Mul(Function('c_0')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True))), sin(exp(Symbol('r', commutative=True)))), exp(Symbol('r', commutative=True))), Derivative(Function('c_0')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))))"]]}, {"prompt": "Given l{(\\omega)} = \\cos{(\\omega)}, then obtain (3 l{(\\omega)} + \\cos{(\\omega)})^{2} \\cos{(\\omega)} - l{(\\omega)} - \\cos{(\\omega)} = (2 l{(\\omega)} + 2 \\cos{(\\omega)})^{2} \\cos{(\\omega)} - l{(\\omega)} - \\cos{(\\omega)}", "derivation": "l{(\\omega)} = \\cos{(\\omega)} and 2 l{(\\omega)} = l{(\\omega)} + \\cos{(\\omega)} and 3 l{(\\omega)} + \\cos{(\\omega)} = 2 l{(\\omega)} + 2 \\cos{(\\omega)} and (3 l{(\\omega)} + \\cos{(\\omega)})^{2} = (2 l{(\\omega)} + 2 \\cos{(\\omega)})^{2} and (3 l{(\\omega)} + \\cos{(\\omega)})^{2} \\cos{(\\omega)} = (2 l{(\\omega)} + 2 \\cos{(\\omega)})^{2} \\cos{(\\omega)} and (3 l{(\\omega)} + \\cos{(\\omega)})^{2} \\cos{(\\omega)} - l{(\\omega)} - \\cos{(\\omega)} = (2 l{(\\omega)} + 2 \\cos{(\\omega)})^{2} \\cos{(\\omega)} - l{(\\omega)} - \\cos{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["add", 1, "Function('l')(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(2), Function('l')(Symbol('\\\\omega', commutative=True))), Add(Function('l')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True))))"], [["add", 2, "Add(Function('l')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('l')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(2), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Mul(Integer(3), Function('l')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(2), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))), Integer(2)))"], [["times", 4, "cos(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(3), Function('l')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Integer(2)), cos(Symbol('\\\\omega', commutative=True))), Mul(Pow(Add(Mul(Integer(2), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))), Integer(2)), cos(Symbol('\\\\omega', commutative=True))))"], [["minus", 5, "Add(Function('l')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Pow(Add(Mul(Integer(3), Function('l')(Symbol('\\\\omega', commutative=True))), cos(Symbol('\\\\omega', commutative=True))), Integer(2)), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Add(Mul(Pow(Add(Mul(Integer(2), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\omega', commutative=True)))), Integer(2)), cos(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Function('l')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\pi,\\chi)} = \\chi + \\pi, then obtain - 2^{\\pi} V{(\\sigma_p)} = - ((\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} + 1)^{\\pi} V{(\\sigma_p)}", "derivation": "\\mathbf{J}{(\\pi,\\chi)} = \\chi + \\pi and \\mathbf{J}^{\\chi}{(\\pi,\\chi)} = (\\chi + \\pi)^{\\chi} and 1 = (\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} and 2 = (\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} + 1 and 2^{\\pi} = ((\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} + 1)^{\\pi} and - 2^{\\pi} = - ((\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} + 1)^{\\pi} and - 2^{\\pi} V{(\\sigma_p)} = - ((\\chi + \\pi)^{\\chi} \\mathbf{J}^{- \\chi}{(\\pi,\\chi)} + 1)^{\\pi} V{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["divide", 2, "Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1)))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integer(2), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1)), Symbol('\\\\pi', commutative=True)))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integer(2), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1)), Symbol('\\\\pi', commutative=True))))"], [["times", 6, "Function('V')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Integer(2), Symbol('\\\\pi', commutative=True)), Function('V')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\pi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Integer(1)), Symbol('\\\\pi', commutative=True)), Function('V')(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(V,p)} = V p, then obtain \\frac{\\partial}{\\partial p} \\frac{(V (V p + \\operatorname{r_{0}}{(V,p)}))^{p}}{2 V^{2} p} = \\frac{\\partial}{\\partial p} \\frac{(2 V^{2} p)^{p}}{2 V^{2} p}", "derivation": "\\operatorname{r_{0}}{(V,p)} = V p and V p + \\operatorname{r_{0}}{(V,p)} = 2 V p and V (V p + \\operatorname{r_{0}}{(V,p)}) = 2 V^{2} p and (V (V p + \\operatorname{r_{0}}{(V,p)}))^{p} = (2 V^{2} p)^{p} and \\frac{(V (V p + \\operatorname{r_{0}}{(V,p)}))^{p}}{2 V^{2} p} = \\frac{(2 V^{2} p)^{p}}{2 V^{2} p} and \\frac{\\partial}{\\partial p} \\frac{(V (V p + \\operatorname{r_{0}}{(V,p)}))^{p}}{2 V^{2} p} = \\frac{\\partial}{\\partial p} \\frac{(2 V^{2} p)^{p}}{2 V^{2} p}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)))"], [["add", 1, "Mul(Symbol('V', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True))), Mul(Integer(2), Symbol('V', commutative=True), Symbol('p', commutative=True)))"], [["times", 2, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Add(Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True)))), Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(2)), Symbol('p', commutative=True)))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Mul(Symbol('V', commutative=True), Add(Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(2)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["divide", 4, "Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(2)), Symbol('p', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(-2)), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Mul(Symbol('V', commutative=True), Add(Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(-2)), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(2)), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(-2)), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Mul(Symbol('V', commutative=True), Add(Mul(Symbol('V', commutative=True), Symbol('p', commutative=True)), Function('r_0')(Symbol('V', commutative=True), Symbol('p', commutative=True)))), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(-2)), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(2)), Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(m)} = \\sin{(\\sin{(m)})}, then obtain \\frac{\\sin{(m)} - \\int \\hat{H}_{\\lambda}{(m)} dm}{\\hat{H}_{\\lambda}{(m)}} = \\frac{\\sin{(m)} - \\int \\sin{(\\sin{(m)})} dm}{\\hat{H}_{\\lambda}{(m)}}", "derivation": "\\hat{H}_{\\lambda}{(m)} = \\sin{(\\sin{(m)})} and \\int \\hat{H}_{\\lambda}{(m)} dm = \\int \\sin{(\\sin{(m)})} dm and - \\sin{(m)} + \\int \\hat{H}_{\\lambda}{(m)} dm = - \\sin{(m)} + \\int \\sin{(\\sin{(m)})} dm and \\sin{(m)} - \\int \\hat{H}_{\\lambda}{(m)} dm = \\sin{(m)} - \\int \\sin{(\\sin{(m)})} dm and \\frac{\\sin{(m)} - \\int \\hat{H}_{\\lambda}{(m)} dm}{\\sin{(\\sin{(m)})}} = \\frac{\\sin{(m)} - \\int \\sin{(\\sin{(m)})} dm}{\\sin{(\\sin{(m)})}} and \\frac{\\sin{(m)} - \\int \\hat{H}_{\\lambda}{(m)} dm}{\\hat{H}_{\\lambda}{(m)}} = \\frac{\\sin{(m)} - \\int \\sin{(\\sin{(m)})} dm}{\\hat{H}_{\\lambda}{(m)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), sin(sin(Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(sin(sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["minus", 2, "sin(Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integral(sin(sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(sin(sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))))"], [["divide", 4, "sin(sin(Symbol('m', commutative=True)))"], "Equality(Mul(Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Pow(sin(sin(Symbol('m', commutative=True))), Integer(-1))), Mul(Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(sin(sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))), Pow(sin(sin(Symbol('m', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Integer(-1))), Mul(Add(sin(Symbol('m', commutative=True)), Mul(Integer(-1), Integral(sin(sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('m', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\dot{x},y^{\\prime})} = e^{(y^{\\prime})^{\\dot{x}}} and \\dot{z}{(y^{\\prime})} = y^{\\prime}, then obtain - \\dot{x} - \\dot{z}{(y^{\\prime})} - e^{(y^{\\prime})^{\\dot{x}}} = - \\dot{x} - y^{\\prime} - e^{(y^{\\prime})^{\\dot{x}}}", "derivation": "\\operatorname{F_{g}}{(\\dot{x},y^{\\prime})} = e^{(y^{\\prime})^{\\dot{x}}} and \\dot{z}{(y^{\\prime})} = y^{\\prime} and - \\dot{z}{(y^{\\prime})} = - y^{\\prime} and - \\dot{x} - \\operatorname{F_{g}}{(\\dot{x},y^{\\prime})} - \\dot{z}{(y^{\\prime})} = - \\dot{x} - y^{\\prime} - \\operatorname{F_{g}}{(\\dot{x},y^{\\prime})} and - \\dot{x} - \\dot{z}{(y^{\\prime})} - e^{(y^{\\prime})^{\\dot{x}}} = - \\dot{x} - y^{\\prime} - e^{(y^{\\prime})^{\\dot{x}}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\dot{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), exp(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 3, "Add(Symbol('\\\\dot{x}', commutative=True), Function('F_g')(Symbol('\\\\dot{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\dot{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\dot{x}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), exp(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon_{0}{(A_{x},y)} = \\sin{(A_{x}^{y})}, then derive \\frac{\\partial}{\\partial y} \\varepsilon_{0}{(A_{x},y)} = A_{x}^{y} \\log{(A_{x})} \\cos{(A_{x}^{y})}, then obtain \\frac{\\partial^{2}}{\\partial A_{x}\\partial y} \\varepsilon_{0}{(A_{x},y)} = \\frac{\\partial}{\\partial A_{x}} A_{x}^{y} \\log{(A_{x})} \\cos{(A_{x}^{y})}", "derivation": "\\varepsilon_{0}{(A_{x},y)} = \\sin{(A_{x}^{y})} and \\frac{\\partial}{\\partial y} \\varepsilon_{0}{(A_{x},y)} = \\frac{\\partial}{\\partial y} \\sin{(A_{x}^{y})} and \\frac{\\partial}{\\partial y} \\varepsilon_{0}{(A_{x},y)} = A_{x}^{y} \\log{(A_{x})} \\cos{(A_{x}^{y})} and \\frac{\\partial^{2}}{\\partial A_{x}\\partial y} \\varepsilon_{0}{(A_{x},y)} = \\frac{\\partial}{\\partial A_{x}} A_{x}^{y} \\log{(A_{x})} \\cos{(A_{x}^{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), sin(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), log(Symbol('A_x', commutative=True)), cos(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)), log(Symbol('A_x', commutative=True)), cos(Pow(Symbol('A_x', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(\\mathbf{J},A_{1})} = \\mathbf{J}^{A_{1}}, then obtain \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} + 2 A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} + 2 A_{1})}", "derivation": "\\mu{(\\mathbf{J},A_{1})} = \\mathbf{J}^{A_{1}} and \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} = (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} and A_{1} \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} = A_{1} (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} and A_{1} \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} + 2 A_{1} = A_{1} (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} + 2 A_{1} and \\sin{(A_{1} \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} + 2 A_{1})} = \\sin{(A_{1} (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} + 2 A_{1})} and \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} \\mu^{\\mathbf{J}}{(\\mathbf{J},A_{1})} + 2 A_{1})} = \\frac{\\partial}{\\partial A_{1}} \\sin{(A_{1} (\\mathbf{J}^{A_{1}})^{\\mathbf{J}} + 2 A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 2, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True))))"], [["sin", 4], "Equality(sin(Add(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True)))), sin(Add(Mul(Symbol('A_1', commutative=True), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True)))))"], [["differentiate", 5, "Symbol('A_1', commutative=True)"], "Equality(Derivative(sin(Add(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\mu')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(sin(Add(Mul(Symbol('A_1', commutative=True), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('A_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(2), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(z^{*},\\eta^{\\prime})} = \\eta^{\\prime} + z^{*} and \\operatorname{F_{x}}{(z^{*})} = z^{*}, then obtain (\\operatorname{F_{x}}{(z^{*})} + \\hat{x}_0{(z^{*},\\eta^{\\prime})})^{\\eta^{\\prime}} = (\\eta^{\\prime} + 2 z^{*})^{\\eta^{\\prime}}", "derivation": "\\hat{x}_0{(z^{*},\\eta^{\\prime})} = \\eta^{\\prime} + z^{*} and \\operatorname{F_{x}}{(z^{*})} = z^{*} and \\operatorname{F_{x}}{(z^{*})} + \\hat{x}_0{(z^{*},\\eta^{\\prime})} = z^{*} + \\hat{x}_0{(z^{*},\\eta^{\\prime})} and \\operatorname{F_{x}}{(z^{*})} + \\hat{x}_0{(z^{*},\\eta^{\\prime})} = \\eta^{\\prime} + z^{*} + \\operatorname{F_{x}}{(z^{*})} and \\eta^{\\prime} + z^{*} + \\operatorname{F_{x}}{(z^{*})} = \\eta^{\\prime} + 2 z^{*} and \\operatorname{F_{x}}{(z^{*})} + \\hat{x}_0{(z^{*},\\eta^{\\prime})} = \\eta^{\\prime} + 2 z^{*} and (\\operatorname{F_{x}}{(z^{*})} + \\hat{x}_0{(z^{*},\\eta^{\\prime})})^{\\eta^{\\prime}} = (\\eta^{\\prime} + 2 z^{*})^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))"], [["add", 2, "Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('z^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('z^*', commutative=True), Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 1, "Function('F_x')(Symbol('z^*', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('z^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z^*', commutative=True), Function('F_x')(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z^*', commutative=True), Function('F_x')(Symbol('z^*', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Function('F_x')(Symbol('z^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(2), Symbol('z^*', commutative=True))))"], [["power", 6, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Add(Function('F_x')(Symbol('z^*', commutative=True)), Function('\\\\hat{x}_0')(Symbol('z^*', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(2), Symbol('z^*', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}}, then obtain \\eta^{\\prime} + e^{\\eta^{\\prime}} + \\int e^{\\eta^{\\prime}} e^{\\operatorname{t_{1}}{(\\eta^{\\prime})}} d\\eta^{\\prime} = \\eta^{\\prime} + e^{\\eta^{\\prime}} + \\int e^{\\eta^{\\prime}} e^{e^{\\eta^{\\prime}}} d\\eta^{\\prime}", "derivation": "\\operatorname{t_{1}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}} and e^{\\operatorname{t_{1}}{(\\eta^{\\prime})}} = e^{e^{\\eta^{\\prime}}} and e^{\\eta^{\\prime}} e^{\\operatorname{t_{1}}{(\\eta^{\\prime})}} = e^{\\eta^{\\prime}} e^{e^{\\eta^{\\prime}}} and \\int e^{\\eta^{\\prime}} e^{\\operatorname{t_{1}}{(\\eta^{\\prime})}} d\\eta^{\\prime} = \\int e^{\\eta^{\\prime}} e^{e^{\\eta^{\\prime}}} d\\eta^{\\prime} and \\eta^{\\prime} + e^{\\eta^{\\prime}} + \\int e^{\\eta^{\\prime}} e^{\\operatorname{t_{1}}{(\\eta^{\\prime})}} d\\eta^{\\prime} = \\eta^{\\prime} + e^{\\eta^{\\prime}} + \\int e^{\\eta^{\\prime}} e^{e^{\\eta^{\\prime}}} d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["add", 4, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Function('t_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integral(Mul(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\theta_1,\\Psi^{\\dagger})} = \\Psi^{\\dagger} \\theta_1, then obtain (2 \\theta_1 + 2 \\rho{(\\theta_1,\\Psi^{\\dagger})})^{\\theta_1} = (\\Psi^{\\dagger} \\theta_1 + 2 \\theta_1 + \\rho{(\\theta_1,\\Psi^{\\dagger})})^{\\theta_1}", "derivation": "\\rho{(\\theta_1,\\Psi^{\\dagger})} = \\Psi^{\\dagger} \\theta_1 and \\theta_1 + \\rho{(\\theta_1,\\Psi^{\\dagger})} = \\Psi^{\\dagger} \\theta_1 + \\theta_1 and 2 \\theta_1 + 2 \\rho{(\\theta_1,\\Psi^{\\dagger})} = \\Psi^{\\dagger} \\theta_1 + 2 \\theta_1 + \\rho{(\\theta_1,\\Psi^{\\dagger})} and (2 \\theta_1 + 2 \\rho{(\\theta_1,\\Psi^{\\dagger})})^{\\theta_1} = (\\Psi^{\\dagger} \\theta_1 + 2 \\theta_1 + \\rho{(\\theta_1,\\Psi^{\\dagger})})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["add", 2, "Add(Symbol('\\\\theta_1', commutative=True), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_1', commutative=True)), Function('\\\\rho')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = \\frac{\\dot{x}}{\\sigma_x}, then obtain - \\frac{\\dot{x}}{\\sigma_x} + \\hat{x} + \\phi - \\sigma_x + \\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = \\hat{x} + \\phi - \\sigma_x", "derivation": "\\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = \\frac{\\dot{x}}{\\sigma_x} and - \\sigma_x + \\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = \\frac{\\dot{x}}{\\sigma_x} - \\sigma_x and - \\frac{\\dot{x}}{\\sigma_x} - \\sigma_x + \\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = - \\sigma_x and - \\frac{\\dot{x}}{\\sigma_x} + \\hat{x} + \\phi - \\sigma_x + \\operatorname{F_{g}}{(\\sigma_x,\\dot{x})} = \\hat{x} + \\phi - \\sigma_x", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"], [["add", 3, "Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then derive \\int \\operatorname{v_{t}}{(\\hat{H}_l)} d\\hat{H}_l = \\hat{H}_l \\log{(\\hat{H}_l)} - \\hat{H}_l + \\mathbf{J}_M, then obtain \\int \\log{(\\hat{H}_l)} d\\hat{H}_l = \\hat{H}_l \\log{(\\hat{H}_l)} - \\hat{H}_l + \\mathbf{J}_M", "derivation": "\\operatorname{v_{t}}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\int \\operatorname{v_{t}}{(\\hat{H}_l)} d\\hat{H}_l = \\int \\log{(\\hat{H}_l)} d\\hat{H}_l and \\int \\operatorname{v_{t}}{(\\hat{H}_l)} d\\hat{H}_l = \\hat{H}_l \\log{(\\hat{H}_l)} - \\hat{H}_l + \\mathbf{J}_M and \\int \\log{(\\hat{H}_l)} d\\hat{H}_l = \\hat{H}_l \\log{(\\hat{H}_l)} - \\hat{H}_l + \\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(log(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given V{(A_{x})} = e^{\\cos{(A_{x})}}, then obtain \\frac{\\frac{d}{d A_{x}} V^{2}{(A_{x})}}{V{(A_{x})}} = \\frac{\\frac{d}{d A_{x}} V{(A_{x})} e^{\\cos{(A_{x})}}}{V{(A_{x})}}", "derivation": "V{(A_{x})} = e^{\\cos{(A_{x})}} and V^{2}{(A_{x})} = V{(A_{x})} e^{\\cos{(A_{x})}} and \\frac{d}{d A_{x}} V^{2}{(A_{x})} = \\frac{d}{d A_{x}} V{(A_{x})} e^{\\cos{(A_{x})}} and \\frac{\\frac{d}{d A_{x}} V^{2}{(A_{x})}}{V{(A_{x})}} = \\frac{\\frac{d}{d A_{x}} V{(A_{x})} e^{\\cos{(A_{x})}}}{V{(A_{x})}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True))))"], [["times", 1, "Function('V')(Symbol('A_x', commutative=True))"], "Equality(Pow(Function('V')(Symbol('A_x', commutative=True)), Integer(2)), Mul(Function('V')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True)))))"], [["differentiate", 2, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Pow(Function('V')(Symbol('A_x', commutative=True)), Integer(2)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Function('V')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["divide", 3, "Function('V')(Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Function('V')(Symbol('A_x', commutative=True)), Integer(-1)), Derivative(Pow(Function('V')(Symbol('A_x', commutative=True)), Integer(2)), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Pow(Function('V')(Symbol('A_x', commutative=True)), Integer(-1)), Derivative(Mul(Function('V')(Symbol('A_x', commutative=True)), exp(cos(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(v_{z},\\dot{y})} = (e^{\\dot{y}})^{v_{z}}, then obtain \\cos{(0^{\\dot{y}})} = \\cos{((- \\Psi^{\\dagger}{(v_{z},\\dot{y})} + (e^{\\dot{y}})^{v_{z}})^{\\dot{y}})}", "derivation": "\\Psi^{\\dagger}{(v_{z},\\dot{y})} = (e^{\\dot{y}})^{v_{z}} and 0 = - \\Psi^{\\dagger}{(v_{z},\\dot{y})} + (e^{\\dot{y}})^{v_{z}} and 0^{\\dot{y}} = (- \\Psi^{\\dagger}{(v_{z},\\dot{y})} + (e^{\\dot{y}})^{v_{z}})^{\\dot{y}} and \\cos{(0^{\\dot{y}})} = \\cos{((- \\Psi^{\\dagger}{(v_{z},\\dot{y})} + (e^{\\dot{y}})^{v_{z}})^{\\dot{y}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('v_z', commutative=True)))"], [["minus", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('v_z', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('v_z', commutative=True))), Symbol('\\\\dot{y}', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Integer(0), Symbol('\\\\dot{y}', commutative=True))), cos(Pow(Add(Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('v_z', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('v_z', commutative=True))), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(\\psi,f_{\\mathbf{p}})} = \\psi + f_{\\mathbf{p}}, then obtain 2 \\hat{p}{(\\psi,f_{\\mathbf{p}})} = \\psi + f_{\\mathbf{p}} + \\hat{p}{(\\psi,f_{\\mathbf{p}})}", "derivation": "\\hat{p}{(\\psi,f_{\\mathbf{p}})} = \\psi + f_{\\mathbf{p}} and \\psi + f_{\\mathbf{p}} + \\hat{p}{(\\psi,f_{\\mathbf{p}})} = 2 \\psi + 2 f_{\\mathbf{p}} and 2 \\hat{p}{(\\psi,f_{\\mathbf{p}})} = 2 \\psi + 2 f_{\\mathbf{p}} and 2 \\hat{p}{(\\psi,f_{\\mathbf{p}})} = \\psi + f_{\\mathbf{p}} + \\hat{p}{(\\psi,f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('\\\\hat{p}')(Symbol('\\\\psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(r_{0},q)} = \\int r_{0}^{q} dr_{0}, then obtain \\iint \\sin{(\\operatorname{m_{s}}^{r_{0}}{(r_{0},q)})} dr_{0} dq = \\iint \\sin{((\\int r_{0}^{q} dr_{0})^{r_{0}})} dr_{0} dq", "derivation": "\\operatorname{m_{s}}{(r_{0},q)} = \\int r_{0}^{q} dr_{0} and \\operatorname{m_{s}}^{r_{0}}{(r_{0},q)} = (\\int r_{0}^{q} dr_{0})^{r_{0}} and \\sin{(\\operatorname{m_{s}}^{r_{0}}{(r_{0},q)})} = \\sin{((\\int r_{0}^{q} dr_{0})^{r_{0}})} and \\int \\sin{(\\operatorname{m_{s}}^{r_{0}}{(r_{0},q)})} dr_{0} = \\int \\sin{((\\int r_{0}^{q} dr_{0})^{r_{0}})} dr_{0} and \\iint \\sin{(\\operatorname{m_{s}}^{r_{0}}{(r_{0},q)})} dr_{0} dq = \\iint \\sin{((\\int r_{0}^{q} dr_{0})^{r_{0}})} dr_{0} dq", "srepr_derivation": [["renaming_premise", "Equality(Function('m_s')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Integral(Pow(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Symbol('r_0', commutative=True)), Pow(Integral(Pow(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Symbol('r_0', commutative=True))), sin(Pow(Integral(Pow(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(sin(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(sin(Pow(Integral(Pow(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))))"], [["integrate", 4, "Symbol('q', commutative=True)"], "Equality(Integral(sin(Pow(Function('m_s')(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(sin(Pow(Integral(Pow(Symbol('r_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True)), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(g,\\mathbf{F})} = g^{\\mathbf{F}}, then obtain - \\hat{H}{(g,\\mathbf{F})} + \\frac{\\hat{H}{(g,\\mathbf{F})}}{g} = - g^{\\mathbf{F}} + \\frac{\\hat{H}{(g,\\mathbf{F})}}{g}", "derivation": "\\hat{H}{(g,\\mathbf{F})} = g^{\\mathbf{F}} and \\frac{\\hat{H}{(g,\\mathbf{F})}}{g} = \\frac{g^{\\mathbf{F}}}{g} and \\hat{H}{(g,\\mathbf{F})} - \\frac{g^{\\mathbf{F}}}{g} = g^{\\mathbf{F}} - \\frac{g^{\\mathbf{F}}}{g} and \\hat{H}{(g,\\mathbf{F})} - \\frac{\\hat{H}{(g,\\mathbf{F})}}{g} = g^{\\mathbf{F}} - \\frac{\\hat{H}{(g,\\mathbf{F})}}{g} and - \\hat{H}{(g,\\mathbf{F})} + \\frac{\\hat{H}{(g,\\mathbf{F})}}{g} = - g^{\\mathbf{F}} + \\frac{\\hat{H}{(g,\\mathbf{F})}}{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('\\\\hat{H}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{P},u)} = e^{\\mathbf{P} u}, then derive (\\int \\mathbf{P} \\operatorname{E_{x}}{(\\mathbf{P},u)} du)^{u} = (\\mu_0 + e^{\\mathbf{P} u})^{u}, then obtain (\\mu + e^{\\mathbf{P} u})^{u} = (\\mu_0 + e^{\\mathbf{P} u})^{u}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{P},u)} = e^{\\mathbf{P} u} and \\mathbf{P} \\operatorname{E_{x}}{(\\mathbf{P},u)} = \\mathbf{P} e^{\\mathbf{P} u} and \\int \\mathbf{P} \\operatorname{E_{x}}{(\\mathbf{P},u)} du = \\int \\mathbf{P} e^{\\mathbf{P} u} du and (\\int \\mathbf{P} \\operatorname{E_{x}}{(\\mathbf{P},u)} du)^{u} = (\\int \\mathbf{P} e^{\\mathbf{P} u} du)^{u} and (\\int \\mathbf{P} \\operatorname{E_{x}}{(\\mathbf{P},u)} du)^{u} = (\\mu_0 + e^{\\mathbf{P} u})^{u} and (\\int \\mathbf{P} e^{\\mathbf{P} u} du)^{u} = (\\mu_0 + e^{\\mathbf{P} u})^{u} and (\\mu + e^{\\mathbf{P} u})^{u} = (\\mu_0 + e^{\\mathbf{P} u})^{u}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('E_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('E_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('E_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), Function('E_x')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Integral(Mul(Symbol('\\\\mathbf{P}', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)), Pow(Add(Symbol('\\\\mu_0', commutative=True), exp(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('u', commutative=True)))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\varepsilon_0)} = e^{\\varepsilon_0}, then derive \\frac{d}{d \\varepsilon_0} \\mathbf{J}_M{(\\varepsilon_0)} = e^{\\varepsilon_0}, then obtain 2 \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} = e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0}", "derivation": "\\mathbf{J}_M{(\\varepsilon_0)} = e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} \\mathbf{J}_M{(\\varepsilon_0)} = \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} \\mathbf{J}_M{(\\varepsilon_0)} = e^{\\varepsilon_0} and \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} = e^{\\varepsilon_0} and 2 \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0} = e^{\\varepsilon_0} + \\frac{d}{d \\varepsilon_0} e^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True)), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), exp(Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 4, "Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(exp(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(W,\\eta)} = \\cos{(W^{\\eta})} and \\operatorname{v_{1}}{(W,\\eta)} = \\cos{(W^{\\eta})}, then obtain - W^{\\eta} + \\cos{(W^{\\eta})} = - W^{\\eta} + \\operatorname{v_{1}}{(W,\\eta)}", "derivation": "\\mathbf{M}{(W,\\eta)} = \\cos{(W^{\\eta})} and \\operatorname{v_{1}}{(W,\\eta)} = \\cos{(W^{\\eta})} and \\mathbf{M}{(W,\\eta)} = \\operatorname{v_{1}}{(W,\\eta)} and - W^{\\eta} + \\mathbf{M}{(W,\\eta)} = - W^{\\eta} + \\operatorname{v_{1}}{(W,\\eta)} and - W^{\\eta} + \\cos{(W^{\\eta})} = - W^{\\eta} + \\operatorname{v_{1}}{(W,\\eta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{M}')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True)), Function('v_1')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["minus", 3, "Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))), Function('\\\\mathbf{M}')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))), Function('v_1')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))), cos(Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))), Function('v_1')(Symbol('W', commutative=True), Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(n_{2},y^{\\prime},\\tilde{g}^*)} = (\\tilde{g}^*)^{n_{2}} - y^{\\prime} and \\theta{(n_{2},\\tilde{g}^*)} = (\\tilde{g}^*)^{n_{2}}, then obtain \\mu_{0}{(n_{2},y^{\\prime},\\tilde{g}^*)} + 1 = - y^{\\prime} + \\theta{(n_{2},\\tilde{g}^*)} + 1", "derivation": "\\mu_{0}{(n_{2},y^{\\prime},\\tilde{g}^*)} = (\\tilde{g}^*)^{n_{2}} - y^{\\prime} and \\mu_{0}{(n_{2},y^{\\prime},\\tilde{g}^*)} + 1 = (\\tilde{g}^*)^{n_{2}} - y^{\\prime} + 1 and \\theta{(n_{2},\\tilde{g}^*)} = (\\tilde{g}^*)^{n_{2}} and \\mu_{0}{(n_{2},y^{\\prime},\\tilde{g}^*)} + 1 = - y^{\\prime} + \\theta{(n_{2},\\tilde{g}^*)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mu_0')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1)), Add(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('n_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mu_0')(Symbol('n_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\theta')(Symbol('n_2', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\Omega{(\\hat{x}_0,\\mathbf{r})} = - \\mathbf{r} + \\log{(\\hat{x}_0)}, then obtain \\log{(\\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)}) \\Omega{(\\hat{x}_0,\\mathbf{r})})})} = \\log{(\\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)})^{2})})}", "derivation": "\\Omega{(\\hat{x}_0,\\mathbf{r})} = - \\mathbf{r} + \\log{(\\hat{x}_0)} and (- \\mathbf{r} + \\log{(\\hat{x}_0)}) \\Omega{(\\hat{x}_0,\\mathbf{r})} = (- \\mathbf{r} + \\log{(\\hat{x}_0)})^{2} and \\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)}) \\Omega{(\\hat{x}_0,\\mathbf{r})})} = \\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)})^{2})} and \\log{(\\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)}) \\Omega{(\\hat{x}_0,\\mathbf{r})})})} = \\log{(\\log{((- \\mathbf{r} + \\log{(\\hat{x}_0)})^{2})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Integer(2)))"], [["log", 2], "Equality(log(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), log(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Integer(2))))"], [["log", 3], "Equality(log(log(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Function('\\\\Omega')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))), log(log(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\hat{x}_0', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given W{(v_{1})} = \\sin{(\\log{(v_{1})})}, then obtain (\\int W{(v_{1})} dv_{1} + \\int \\sin{(\\log{(v_{1})})} dv_{1}) W{(v_{1})} = 2 W{(v_{1})} \\int \\sin{(\\log{(v_{1})})} dv_{1}", "derivation": "W{(v_{1})} = \\sin{(\\log{(v_{1})})} and \\int W{(v_{1})} dv_{1} = \\int \\sin{(\\log{(v_{1})})} dv_{1} and \\int W{(v_{1})} dv_{1} + \\int \\sin{(\\log{(v_{1})})} dv_{1} = 2 \\int \\sin{(\\log{(v_{1})})} dv_{1} and (\\int W{(v_{1})} dv_{1} + \\int \\sin{(\\log{(v_{1})})} dv_{1}) W{(v_{1})} = 2 W{(v_{1})} \\int \\sin{(\\log{(v_{1})})} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('v_1', commutative=True)), sin(log(Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('W')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True))))"], [["add", 2, "Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Integral(Function('W')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(2), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))))"], [["times", 3, "Function('W')(Symbol('v_1', commutative=True))"], "Equality(Mul(Add(Integral(Function('W')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))), Function('W')(Symbol('v_1', commutative=True))), Mul(Integer(2), Function('W')(Symbol('v_1', commutative=True)), Integral(sin(log(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given i{(E)} = \\cos{(E)} and u{(E)} = E + \\cos{(E)}, then obtain \\int (\\frac{E + \\cos{(E)}}{i{(E)}})^{E} dE = \\int (\\frac{E + i{(E)}}{i{(E)}})^{E} dE", "derivation": "i{(E)} = \\cos{(E)} and u{(E)} = E + \\cos{(E)} and u{(E)} = E + i{(E)} and \\frac{u{(E)}}{i{(E)}} = \\frac{E + i{(E)}}{i{(E)}} and (\\frac{u{(E)}}{i{(E)}})^{E} = (\\frac{E + i{(E)}}{i{(E)}})^{E} and (\\frac{E + \\cos{(E)}}{i{(E)}})^{E} = (\\frac{E + i{(E)}}{i{(E)}})^{E} and \\int (\\frac{E + \\cos{(E)}}{i{(E)}})^{E} dE = \\int (\\frac{E + i{(E)}}{i{(E)}})^{E} dE", "srepr_derivation": [["renaming_premise", "Equality(Function('i')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('u')(Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Function('i')(Symbol('E', commutative=True))))"], [["divide", 3, "Function('i')(Symbol('E', commutative=True))"], "Equality(Mul(Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1)), Function('u')(Symbol('E', commutative=True))), Mul(Add(Symbol('E', commutative=True), Function('i')(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1)), Function('u')(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Mul(Add(Symbol('E', commutative=True), Function('i')(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Add(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Pow(Mul(Add(Symbol('E', commutative=True), Function('i')(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)))"], [["integrate", 6, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Mul(Add(Symbol('E', commutative=True), cos(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(Mul(Add(Symbol('E', commutative=True), Function('i')(Symbol('E', commutative=True))), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(x,\\mathbf{r})} = \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r}, then obtain \\int - \\mathbf{r} \\operatorname{P_{g}}{(x,\\mathbf{r})} d\\mathbf{r} = \\int - \\mathbf{r} \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r} d\\mathbf{r}", "derivation": "\\operatorname{P_{g}}{(x,\\mathbf{r})} = \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r} and \\mathbf{r} \\operatorname{P_{g}}{(x,\\mathbf{r})} = \\mathbf{r} \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r} and - \\mathbf{r} \\operatorname{P_{g}}{(x,\\mathbf{r})} = - \\mathbf{r} \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r} and \\int - \\mathbf{r} \\operatorname{P_{g}}{(x,\\mathbf{r})} d\\mathbf{r} = \\int - \\mathbf{r} \\int \\frac{\\mathbf{r}}{x} d\\mathbf{r} d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Function('P_g')(Symbol('x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(I,\\mathbf{D})} = I^{\\mathbf{D}} and \\mathbf{E}{(I,\\mathbf{D})} = \\sin{(\\frac{\\partial}{\\partial I} \\mathbf{J}_f{(I,\\mathbf{D})})}, then obtain (\\frac{\\partial}{\\partial I} \\mathbf{E}{(I,\\mathbf{D})})^{I} = (\\frac{\\partial}{\\partial I} \\sin{(\\frac{\\partial}{\\partial I} I^{\\mathbf{D}})})^{I}", "derivation": "\\mathbf{J}_f{(I,\\mathbf{D})} = I^{\\mathbf{D}} and \\frac{\\partial}{\\partial I} \\mathbf{J}_f{(I,\\mathbf{D})} = \\frac{\\partial}{\\partial I} I^{\\mathbf{D}} and \\sin{(\\frac{\\partial}{\\partial I} \\mathbf{J}_f{(I,\\mathbf{D})})} = \\sin{(\\frac{\\partial}{\\partial I} I^{\\mathbf{D}})} and \\mathbf{E}{(I,\\mathbf{D})} = \\sin{(\\frac{\\partial}{\\partial I} \\mathbf{J}_f{(I,\\mathbf{D})})} and \\mathbf{E}{(I,\\mathbf{D})} = \\sin{(\\frac{\\partial}{\\partial I} I^{\\mathbf{D}})} and \\frac{\\partial}{\\partial I} \\mathbf{E}{(I,\\mathbf{D})} = \\frac{\\partial}{\\partial I} \\sin{(\\frac{\\partial}{\\partial I} I^{\\mathbf{D}})} and (\\frac{\\partial}{\\partial I} \\mathbf{E}{(I,\\mathbf{D})})^{I} = (\\frac{\\partial}{\\partial I} \\sin{(\\frac{\\partial}{\\partial I} I^{\\mathbf{D}})})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{J}_f')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), sin(Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), sin(Derivative(Function('\\\\mathbf{J}_f')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), sin(Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["power", 6, "Symbol('I', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)), Pow(Derivative(sin(Derivative(Pow(Symbol('I', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(E_{n},\\mathbf{J}_P)} = - E_{n} + \\mathbf{J}_P, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{B}{(E_{n},\\mathbf{J}_P)} + 1)^{E_{n}} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- E_{n} + \\mathbf{J}_P + 1)^{E_{n}}", "derivation": "\\mathbf{B}{(E_{n},\\mathbf{J}_P)} = - E_{n} + \\mathbf{J}_P and \\mathbf{B}{(E_{n},\\mathbf{J}_P)} + 1 = - E_{n} + \\mathbf{J}_P + 1 and (\\mathbf{B}{(E_{n},\\mathbf{J}_P)} + 1)^{E_{n}} = (- E_{n} + \\mathbf{J}_P + 1)^{E_{n}} and \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{B}{(E_{n},\\mathbf{J}_P)} + 1)^{E_{n}} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- E_{n} + \\mathbf{J}_P + 1)^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))"], [["power", 2, "Symbol('E_n', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)), Symbol('E_n', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)), Symbol('E_n', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\mathbf{B}')(Symbol('E_n', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(1)), Symbol('E_n', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)), Symbol('E_n', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(T)} = e^{T}, then obtain \\int (-1) dT - \\frac{1}{T} = \\int (- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1) dT - \\frac{1}{T}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(T)} = e^{T} and \\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T} = 0 and (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} = 0 and 0 = - (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} and -1 = - (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1 and \\int (-1) dT = \\int (- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1) dT and \\int (-1) dT + \\frac{- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1}{T} = \\int (- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1) dT + \\frac{- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1}{T} and \\int (-1) dT - \\frac{1}{T} = \\int (- (\\operatorname{f_{\\mathbf{p}}}{(T)} - e^{T})^{2} - 1) dT - \\frac{1}{T}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["minus", 1, "exp(Symbol('T', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(0))"], [["times", 2, "Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True))))"], "Equality(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2)), Integer(0))"], [["minus", 3, "Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))))"], [["add", 4, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)))"], [["integrate", 5, "Symbol('T', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)), Tuple(Symbol('T', commutative=True))))"], [["add", 6, "Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)))"], "Equality(Add(Integral(Integer(-1), Tuple(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)))), Add(Integral(Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)), Tuple(Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Integral(Integer(-1), Tuple(Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))), Add(Integral(Add(Mul(Integer(-1), Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(2))), Integer(-1)), Tuple(Symbol('T', commutative=True))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\theta{(C_{1})} = \\cos{(C_{1})}, then obtain \\frac{d}{d C_{1}} \\int \\theta^{C_{1}}{(C_{1})} dC_{1} = \\frac{d}{d C_{1}} \\int \\cos^{C_{1}}{(C_{1})} dC_{1}", "derivation": "\\theta{(C_{1})} = \\cos{(C_{1})} and \\theta^{C_{1}}{(C_{1})} = \\cos^{C_{1}}{(C_{1})} and \\int \\theta^{C_{1}}{(C_{1})} dC_{1} = \\int \\cos^{C_{1}}{(C_{1})} dC_{1} and \\frac{d}{d C_{1}} \\int \\theta^{C_{1}}{(C_{1})} dC_{1} = \\frac{d}{d C_{1}} \\int \\cos^{C_{1}}{(C_{1})} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Pow(cos(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(cos(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\theta')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(x^\\prime)} = \\log{(x^\\prime)} and \\operatorname{v_{y}}{(x^\\prime)} = \\log{(x^\\prime)}^{x^\\prime}, then obtain (\\frac{d}{d x^\\prime} \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)})^{x^\\prime} = (\\frac{d}{d x^\\prime} \\operatorname{v_{y}}{(x^\\prime)})^{x^\\prime}", "derivation": "\\operatorname{v_{2}}{(x^\\prime)} = \\log{(x^\\prime)} and \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} = \\log{(x^\\prime)}^{x^\\prime} and \\frac{d}{d x^\\prime} \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)}^{x^\\prime} and (\\frac{d}{d x^\\prime} \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)})^{x^\\prime} = (\\frac{d}{d x^\\prime} \\log{(x^\\prime)}^{x^\\prime})^{x^\\prime} and \\operatorname{v_{y}}{(x^\\prime)} = \\log{(x^\\prime)}^{x^\\prime} and (\\frac{d}{d x^\\prime} \\operatorname{v_{2}}^{x^\\prime}{(x^\\prime)})^{x^\\prime} = (\\frac{d}{d x^\\prime} \\operatorname{v_{y}}{(x^\\prime)})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('x^\\\\prime', commutative=True)), Pow(log(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Pow(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Function('v_y')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\psi,z)} = \\sin{(\\psi^{z})}, then obtain \\cos{(\\frac{\\psi \\hat{x}{(\\psi,z)}}{z})} = \\cos{(\\frac{\\psi \\sin{(\\psi^{z})}}{z})}", "derivation": "\\hat{x}{(\\psi,z)} = \\sin{(\\psi^{z})} and \\psi \\hat{x}{(\\psi,z)} = \\psi \\sin{(\\psi^{z})} and \\frac{\\psi \\hat{x}{(\\psi,z)}}{z} = \\frac{\\psi \\sin{(\\psi^{z})}}{z} and \\cos{(\\frac{\\psi \\hat{x}{(\\psi,z)}}{z})} = \\cos{(\\frac{\\psi \\sin{(\\psi^{z})}}{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True))))"], [["times", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True)))))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True)))))"], [["cos", 3], "Equality(cos(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True)))), cos(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), sin(Pow(Symbol('\\\\psi', commutative=True), Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mathbf{H},y)} = \\mathbf{H} + y, then obtain - \\mathbf{H} + e^{\\operatorname{a^{\\dagger}}^{2}{(\\mathbf{H},y)}} = - \\mathbf{H} + e^{(\\mathbf{H} + y) \\operatorname{a^{\\dagger}}{(\\mathbf{H},y)}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mathbf{H},y)} = \\mathbf{H} + y and \\operatorname{a^{\\dagger}}^{2}{(\\mathbf{H},y)} = (\\mathbf{H} + y) \\operatorname{a^{\\dagger}}{(\\mathbf{H},y)} and e^{\\operatorname{a^{\\dagger}}^{2}{(\\mathbf{H},y)}} = e^{(\\mathbf{H} + y) \\operatorname{a^{\\dagger}}{(\\mathbf{H},y)}} and - \\mathbf{H} + e^{\\operatorname{a^{\\dagger}}^{2}{(\\mathbf{H},y)}} = - \\mathbf{H} + e^{(\\mathbf{H} + y) \\operatorname{a^{\\dagger}}{(\\mathbf{H},y)}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True))))"], [["exp", 2], "Equality(exp(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Integer(2))), exp(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)))))"], [["minus", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Pow(Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), exp(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('y', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(\\rho_f)} = \\log{(\\rho_f)}, then obtain \\frac{\\dot{y}{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\dot{y}{(\\rho_f)}} = \\frac{\\log{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\dot{y}{(\\rho_f)}}", "derivation": "\\dot{y}{(\\rho_f)} = \\log{(\\rho_f)} and \\frac{d}{d \\rho_f} \\dot{y}{(\\rho_f)} = \\frac{d}{d \\rho_f} \\log{(\\rho_f)} and \\frac{\\dot{y}{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\log{(\\rho_f)}} = \\frac{\\log{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\log{(\\rho_f)}} and \\frac{\\dot{y}{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\dot{y}{(\\rho_f)}} = \\frac{\\log{(\\rho_f)}}{\\frac{d}{d \\rho_f} \\dot{y}{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), log(Symbol('\\\\rho_f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('\\\\rho_f', commutative=True)), Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(S,\\hat{H}_{\\lambda})} = \\frac{S}{\\hat{H}_{\\lambda}} and \\operatorname{P_{e}}{(S,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} \\operatorname{v_{2}}{(S,\\hat{H}_{\\lambda})}, then obtain \\frac{S}{\\hat{H}_{\\lambda}} + \\operatorname{P_{e}}{(S,\\hat{H}_{\\lambda})} = \\frac{S}{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} S", "derivation": "\\operatorname{v_{2}}{(S,\\hat{H}_{\\lambda})} = \\frac{S}{\\hat{H}_{\\lambda}} and \\operatorname{P_{e}}{(S,\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{H}_{\\lambda} \\operatorname{v_{2}}{(S,\\hat{H}_{\\lambda})} and \\operatorname{P_{e}}{(S,\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} S and \\frac{S}{\\hat{H}_{\\lambda}} + \\operatorname{P_{e}}{(S,\\hat{H}_{\\lambda})} = \\frac{S}{\\hat{H}_{\\lambda}} + \\frac{d}{d \\hat{H}_{\\lambda}} S", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('v_2')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Derivative(Symbol('S', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 3, "Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Function('P_e')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1))), Derivative(Symbol('S', commutative=True), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(V_{\\mathbf{E}})} = \\log{(\\sin{(V_{\\mathbf{E}})})}, then obtain \\int (2 \\hat{p}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} dV_{\\mathbf{E}} = \\int (\\hat{p}{(V_{\\mathbf{E}})} + \\log{(\\sin{(V_{\\mathbf{E}})})})^{V_{\\mathbf{E}}} dV_{\\mathbf{E}}", "derivation": "\\hat{p}{(V_{\\mathbf{E}})} = \\log{(\\sin{(V_{\\mathbf{E}})})} and 2 \\hat{p}{(V_{\\mathbf{E}})} = \\hat{p}{(V_{\\mathbf{E}})} + \\log{(\\sin{(V_{\\mathbf{E}})})} and (2 \\hat{p}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (\\hat{p}{(V_{\\mathbf{E}})} + \\log{(\\sin{(V_{\\mathbf{E}})})})^{V_{\\mathbf{E}}} and \\int (2 \\hat{p}{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} dV_{\\mathbf{E}} = \\int (\\hat{p}{(V_{\\mathbf{E}})} + \\log{(\\sin{(V_{\\mathbf{E}})})})^{V_{\\mathbf{E}}} dV_{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["power", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integral(Pow(Add(Function('\\\\hat{p}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\varepsilon)} = e^{\\varepsilon}, then obtain 0 = 1 - (- \\theta{(\\varepsilon)} + e^{\\varepsilon})^{\\varepsilon}", "derivation": "\\theta{(\\varepsilon)} = e^{\\varepsilon} and 0 = - \\theta{(\\varepsilon)} + e^{\\varepsilon} and 0^{\\varepsilon} = (- \\theta{(\\varepsilon)} + e^{\\varepsilon})^{\\varepsilon} and 0 = - 0^{\\varepsilon} + (- \\theta{(\\varepsilon)} + e^{\\varepsilon})^{\\varepsilon} and 0 = 1 - (- \\theta{(\\varepsilon)} + e^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 1, "Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True))), exp(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True))), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 3, "Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\varepsilon', commutative=True))), Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True))), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('\\\\varepsilon', commutative=True))), exp(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} = \\eta^{\\prime} \\mathbb{I}, then obtain \\eta^{\\prime} + \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbb{I} + \\eta^{\\prime} - \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})}", "derivation": "\\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} = \\eta^{\\prime} \\mathbb{I} and 0 = \\eta^{\\prime} \\mathbb{I} - \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} and \\eta^{\\prime} + \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} = \\eta^{\\prime} \\mathbb{I} + \\eta^{\\prime} and \\eta^{\\prime} \\mathbb{I} + \\eta^{\\prime} = 2 \\eta^{\\prime} \\mathbb{I} + \\eta^{\\prime} - \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} and \\eta^{\\prime} + \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})} = 2 \\eta^{\\prime} \\mathbb{I} + \\eta^{\\prime} - \\operatorname{n_{2}}{(\\mathbb{I},\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["add", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["add", 2, "Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(L)} = \\sin{(\\sin{(L)})}, then obtain \\iint 2 \\eta^{\\prime}{(L)} dL dL = \\iint (\\eta^{\\prime}{(L)} + \\sin{(\\sin{(L)})}) dL dL", "derivation": "\\eta^{\\prime}{(L)} = \\sin{(\\sin{(L)})} and 2 \\eta^{\\prime}{(L)} = \\eta^{\\prime}{(L)} + \\sin{(\\sin{(L)})} and \\int 2 \\eta^{\\prime}{(L)} dL = \\int (\\eta^{\\prime}{(L)} + \\sin{(\\sin{(L)})}) dL and \\iint 2 \\eta^{\\prime}{(L)} dL dL = \\iint (\\eta^{\\prime}{(L)} + \\sin{(\\sin{(L)})}) dL dL", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), sin(sin(Symbol('L', commutative=True))))"], [["add", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), sin(sin(Symbol('L', commutative=True)))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), sin(sin(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('L', commutative=True)), sin(sin(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\dot{z})} = \\sin{(\\dot{z})}, then derive \\cos{(\\dot{z})} + \\frac{d}{d \\dot{z}} \\mathbf{g}{(\\dot{z})} = 2 \\cos{(\\dot{z})}, then obtain \\cos{(\\dot{z})} + \\frac{d}{d \\dot{z}} \\sin{(\\dot{z})} = 2 \\cos{(\\dot{z})}", "derivation": "\\mathbf{g}{(\\dot{z})} = \\sin{(\\dot{z})} and \\mathbf{g}{(\\dot{z})} + \\sin{(\\dot{z})} = 2 \\sin{(\\dot{z})} and \\frac{d}{d \\dot{z}} (\\mathbf{g}{(\\dot{z})} + \\sin{(\\dot{z})}) = \\frac{d}{d \\dot{z}} 2 \\sin{(\\dot{z})} and \\cos{(\\dot{z})} + \\frac{d}{d \\dot{z}} \\mathbf{g}{(\\dot{z})} = 2 \\cos{(\\dot{z})} and \\cos{(\\dot{z})} + \\frac{d}{d \\dot{z}} \\sin{(\\dot{z})} = 2 \\cos{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('\\\\dot{z}', commutative=True)), Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('\\\\dot{z}', commutative=True)), Derivative(sin(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\psi,v,\\sigma_p)} = (\\frac{\\psi}{\\sigma_p})^{v}, then obtain - \\sigma_p + (\\operatorname{v_{z}}^{\\psi}{(\\psi,v,\\sigma_p)})^{\\psi} = - \\sigma_p + (((\\frac{\\psi}{\\sigma_p})^{v})^{\\psi})^{\\psi}", "derivation": "\\operatorname{v_{z}}{(\\psi,v,\\sigma_p)} = (\\frac{\\psi}{\\sigma_p})^{v} and \\operatorname{v_{z}}^{\\psi}{(\\psi,v,\\sigma_p)} = ((\\frac{\\psi}{\\sigma_p})^{v})^{\\psi} and (\\operatorname{v_{z}}^{\\psi}{(\\psi,v,\\sigma_p)})^{\\psi} = (((\\frac{\\psi}{\\sigma_p})^{v})^{\\psi})^{\\psi} and - \\sigma_p + (\\operatorname{v_{z}}^{\\psi}{(\\psi,v,\\sigma_p)})^{\\psi} = - \\sigma_p + (((\\frac{\\psi}{\\sigma_p})^{v})^{\\psi})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('v', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Pow(Function('v_z')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Pow(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('v', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["minus", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Function('v_z')(Symbol('\\\\psi', commutative=True), Symbol('v', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Pow(Pow(Pow(Mul(Symbol('\\\\psi', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Symbol('v', commutative=True)), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\chi,\\rho)} = \\sin{(\\rho^{\\chi})}, then obtain \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} = (\\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} - \\sin^{\\chi}{(\\rho^{\\chi})})^{\\rho} \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)}", "derivation": "\\operatorname{P_{e}}{(\\chi,\\rho)} = \\sin{(\\rho^{\\chi})} and \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} = \\sin^{\\chi}{(\\rho^{\\chi})} and \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} - \\sin^{\\chi}{(\\rho^{\\chi})} = 0 and (\\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} - \\sin^{\\chi}{(\\rho^{\\chi})})^{\\rho} = 0^{\\rho} and (\\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} - \\sin^{\\chi}{(\\rho^{\\chi})})^{\\rho} \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} = 0^{\\rho} \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} and \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} = (\\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)} - \\sin^{\\chi}{(\\rho^{\\chi})})^{\\rho} \\operatorname{P_{e}}^{\\chi}{(\\chi,\\rho)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["minus", 2, "Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(Integer(0), Symbol('\\\\rho', commutative=True)))"], [["times", 4, "Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Add(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\rho', commutative=True)), Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Pow(Add(Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(sin(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(Function('P_e')(Symbol('\\\\chi', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(G,\\hbar)} = \\log{(\\frac{\\hbar}{G})}, then obtain \\frac{\\int (- \\hbar + \\hat{H}_{\\lambda}{(G,\\hbar)}) dG}{\\hat{H}_{\\lambda}{(G,\\hbar)}} = \\frac{\\int (- \\hbar + \\log{(\\frac{\\hbar}{G})}) dG}{\\hat{H}_{\\lambda}{(G,\\hbar)}}", "derivation": "\\hat{H}_{\\lambda}{(G,\\hbar)} = \\log{(\\frac{\\hbar}{G})} and - \\hbar + \\hat{H}_{\\lambda}{(G,\\hbar)} = - \\hbar + \\log{(\\frac{\\hbar}{G})} and \\int (- \\hbar + \\hat{H}_{\\lambda}{(G,\\hbar)}) dG = \\int (- \\hbar + \\log{(\\frac{\\hbar}{G})}) dG and \\frac{\\int (- \\hbar + \\hat{H}_{\\lambda}{(G,\\hbar)}) dG}{\\log{(\\frac{\\hbar}{G})}} = \\frac{\\int (- \\hbar + \\log{(\\frac{\\hbar}{G})}) dG}{\\log{(\\frac{\\hbar}{G})}} and \\frac{\\int (- \\hbar + \\hat{H}_{\\lambda}{(G,\\hbar)}) dG}{\\hat{H}_{\\lambda}{(G,\\hbar)}} = \\frac{\\int (- \\hbar + \\log{(\\frac{\\hbar}{G})}) dG}{\\hat{H}_{\\lambda}{(G,\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True))))"], [["minus", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["divide", 3, "log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Pow(log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('G', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), log(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(U)} = \\log{(U)}, then obtain U - \\log{(U)} + \\int (2 \\hat{p}_0{(U)} - \\log{(U)}) dU = U - \\log{(U)} + \\int \\hat{p}_0{(U)} dU", "derivation": "\\hat{p}_0{(U)} = \\log{(U)} and - U + 2 \\hat{p}_0{(U)} = - U + \\hat{p}_0{(U)} + \\log{(U)} and 2 \\hat{p}_0{(U)} - \\log{(U)} = \\hat{p}_0{(U)} and \\int (2 \\hat{p}_0{(U)} - \\log{(U)}) dU = \\int \\hat{p}_0{(U)} dU and U - \\log{(U)} + \\int (2 \\hat{p}_0{(U)} - \\log{(U)}) dU = U - \\log{(U)} + \\int \\hat{p}_0{(U)} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True))), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('U', commutative=True)))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True))), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(-1), log(Symbol('U', commutative=True))), Integral(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('U', commutative=True))), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True)))), Add(Symbol('U', commutative=True), Mul(Integer(-1), log(Symbol('U', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{P})} = \\mathbf{P}, then obtain \\cos{(\\sin{(\\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})})})} = \\cos{(\\sin{(\\frac{d}{d \\mathbf{P}} \\mathbf{P})})}", "derivation": "\\phi_{2}{(\\mathbf{P})} = \\mathbf{P} and \\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} \\mathbf{P} and \\sin{(\\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})})} = \\sin{(\\frac{d}{d \\mathbf{P}} \\mathbf{P})} and \\cos{(\\sin{(\\frac{d}{d \\mathbf{P}} \\phi_{2}{(\\mathbf{P})})})} = \\cos{(\\sin{(\\frac{d}{d \\mathbf{P}} \\mathbf{P})})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), sin(Derivative(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(sin(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))), cos(sin(Derivative(Symbol('\\\\mathbf{P}', commutative=True), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Omega{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\hat{x}{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}}, then obtain \\Omega{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}} = \\hat{x}{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}}", "derivation": "\\Omega{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\Omega{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}} = e^{2 \\hat{\\mathbf{r}}} and \\hat{x}{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\hat{x}{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}} = e^{2 \\hat{\\mathbf{r}}} and \\Omega{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}} = \\hat{x}{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["times", 3, "exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Function('\\\\hat{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\pi,z^{*})} = - \\pi + z^{*} and \\omega{(\\pi,z^{*})} = \\operatorname{V_{\\mathbf{B}}}{(\\pi,z^{*})} - \\operatorname{r_{0}}{(z^{*})}, then obtain \\omega{(\\pi,z^{*})} = - \\pi + z^{*} - \\operatorname{r_{0}}{(z^{*})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\pi,z^{*})} = - \\pi + z^{*} and \\operatorname{V_{\\mathbf{B}}}{(\\pi,z^{*})} - \\operatorname{r_{0}}{(z^{*})} = - \\pi + z^{*} - \\operatorname{r_{0}}{(z^{*})} and \\omega{(\\pi,z^{*})} = \\operatorname{V_{\\mathbf{B}}}{(\\pi,z^{*})} - \\operatorname{r_{0}}{(z^{*})} and \\omega{(\\pi,z^{*})} = - \\pi + z^{*} - \\operatorname{r_{0}}{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('z^*', commutative=True)))"], [["minus", 1, "Function('r_0')(Symbol('z^*', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('r_0')(Symbol('z^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('z^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), Function('r_0')(Symbol('z^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\omega')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given y{(v_{x},f_{E})} = f_{E}^{v_{x}} and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain \\frac{v_{x} y{(v_{x},f_{E})}}{\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})}} = \\frac{f_{E}^{v_{x}} v_{x}}{\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})}}", "derivation": "y{(v_{x},f_{E})} = f_{E}^{v_{x}} and v_{x} y{(v_{x},f_{E})} = f_{E}^{v_{x}} v_{x} and \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{v_{x} y{(v_{x},f_{E})}}{\\cos{(\\mathbf{A})}} = \\frac{f_{E}^{v_{x}} v_{x}}{\\cos{(\\mathbf{A})}} and \\frac{v_{x} y{(v_{x},f_{E})}}{\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})}} = \\frac{f_{E}^{v_{x}} v_{x}}{\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_x', commutative=True), Symbol('f_E', commutative=True)), Pow(Symbol('f_E', commutative=True), Symbol('v_x', commutative=True)))"], [["times", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Function('y')(Symbol('v_x', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["times", 2, "Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))"], "Equality(Mul(Symbol('v_x', commutative=True), Function('y')(Symbol('v_x', commutative=True), Symbol('f_E', commutative=True)), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('f_E', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True), Pow(cos(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('v_x', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Function('y')(Symbol('v_x', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('f_E', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(t_{2},E_{n},f)} = \\frac{E_{n}}{f} - t_{2} and Q{(t_{2},E_{n},f)} = \\frac{\\partial}{\\partial t_{2}} (\\frac{E_{n}}{f} - t_{2}), then obtain (\\frac{\\partial}{\\partial t_{2}} \\Psi^{\\dagger}{(t_{2},E_{n},f)})^{E_{n}} = Q^{E_{n}}{(t_{2},E_{n},f)}", "derivation": "\\Psi^{\\dagger}{(t_{2},E_{n},f)} = \\frac{E_{n}}{f} - t_{2} and \\frac{\\partial}{\\partial t_{2}} \\Psi^{\\dagger}{(t_{2},E_{n},f)} = \\frac{\\partial}{\\partial t_{2}} (\\frac{E_{n}}{f} - t_{2}) and (\\frac{\\partial}{\\partial t_{2}} \\Psi^{\\dagger}{(t_{2},E_{n},f)})^{E_{n}} = (\\frac{\\partial}{\\partial t_{2}} (\\frac{E_{n}}{f} - t_{2}))^{E_{n}} and Q{(t_{2},E_{n},f)} = \\frac{\\partial}{\\partial t_{2}} (\\frac{E_{n}}{f} - t_{2}) and (\\frac{\\partial}{\\partial t_{2}} \\Psi^{\\dagger}{(t_{2},E_{n},f)})^{E_{n}} = Q^{E_{n}}{(t_{2},E_{n},f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E_n', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Symbol('E_n', commutative=True)), Pow(Derivative(Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Derivative(Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Symbol('E_n', commutative=True)), Pow(Function('Q')(Symbol('t_2', commutative=True), Symbol('E_n', commutative=True), Symbol('f', commutative=True)), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(\\phi)} = \\cos{(\\cos{(\\phi)})}, then obtain (\\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\ddot{x}{(\\phi)})^{\\phi} = (\\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})})^{\\phi}", "derivation": "\\ddot{x}{(\\phi)} = \\cos{(\\cos{(\\phi)})} and \\frac{d}{d \\phi} \\ddot{x}{(\\phi)} = \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} and \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\ddot{x}{(\\phi)} = \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} and \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\ddot{x}{(\\phi)} = \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} and (\\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\ddot{x}{(\\phi)})^{\\phi} = (\\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})} \\frac{d}{d \\phi} \\cos{(\\cos{(\\phi)})})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\phi', commutative=True)), cos(cos(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["times", 2, "cos(cos(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(cos(cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(cos(cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Derivative(Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(Mul(cos(cos(Symbol('\\\\phi', commutative=True))), Derivative(cos(cos(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\nabla,G)} = \\nabla^{G} and \\hat{p}_0{(\\nabla,G)} = - G + \\nabla^{G}, then obtain (\\frac{\\hat{p}_0{(\\nabla,G)}}{\\nabla})^{\\nabla} = (\\frac{- G + \\operatorname{z^{*}}{(\\nabla,G)}}{\\nabla})^{\\nabla}", "derivation": "\\operatorname{z^{*}}{(\\nabla,G)} = \\nabla^{G} and \\hat{p}_0{(\\nabla,G)} = - G + \\nabla^{G} and \\frac{\\hat{p}_0{(\\nabla,G)}}{\\nabla} = \\frac{- G + \\nabla^{G}}{\\nabla} and \\frac{\\hat{p}_0{(\\nabla,G)}}{\\nabla} = \\frac{- G + \\operatorname{z^{*}}{(\\nabla,G)}}{\\nabla} and (\\frac{\\hat{p}_0{(\\nabla,G)}}{\\nabla})^{\\nabla} = (\\frac{- G + \\operatorname{z^{*}}{(\\nabla,G)}}{\\nabla})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True))))"], [["divide", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('z^*')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)))))"], [["power", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('z^*')(Symbol('\\\\nabla', commutative=True), Symbol('G', commutative=True)))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given k{(g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda} + g_{\\varepsilon}, then obtain 0 = - \\frac{(E_{\\lambda} + g_{\\varepsilon}) \\frac{\\partial}{\\partial g_{\\varepsilon}} k{(g_{\\varepsilon},E_{\\lambda})}}{k^{2}{(g_{\\varepsilon},E_{\\lambda})}} + \\frac{1}{k{(g_{\\varepsilon},E_{\\lambda})}}", "derivation": "k{(g_{\\varepsilon},E_{\\lambda})} = E_{\\lambda} + g_{\\varepsilon} and 1 = \\frac{E_{\\lambda} + g_{\\varepsilon}}{k{(g_{\\varepsilon},E_{\\lambda})}} and \\frac{d}{d g_{\\varepsilon}} 1 = \\frac{\\partial}{\\partial g_{\\varepsilon}} \\frac{E_{\\lambda} + g_{\\varepsilon}}{k{(g_{\\varepsilon},E_{\\lambda})}} and 0 = - \\frac{(E_{\\lambda} + g_{\\varepsilon}) \\frac{\\partial}{\\partial g_{\\varepsilon}} k{(g_{\\varepsilon},E_{\\lambda})}}{k^{2}{(g_{\\varepsilon},E_{\\lambda})}} + \\frac{1}{k{(g_{\\varepsilon},E_{\\lambda})}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-2)), Derivative(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)))), Pow(Function('k')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{1}{(\\theta)} = \\log{(\\theta)} and \\sigma_{x}{(\\theta)} = - \\theta + \\log{(\\theta)}, then obtain \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} \\sigma_{x}^{\\theta}{(\\theta)} = \\frac{d}{d \\theta} (- \\theta + \\log{(\\theta)})^{\\theta} \\sigma_{x}{(\\theta)}", "derivation": "\\phi_{1}{(\\theta)} = \\log{(\\theta)} and - \\theta + \\phi_{1}{(\\theta)} = - \\theta + \\log{(\\theta)} and (- \\theta + \\phi_{1}{(\\theta)})^{\\theta} = (- \\theta + \\log{(\\theta)})^{\\theta} and \\sigma_{x}{(\\theta)} = - \\theta + \\log{(\\theta)} and - \\theta + \\phi_{1}{(\\theta)} = \\sigma_{x}{(\\theta)} and \\sigma_{x}^{\\theta}{(\\theta)} = (- \\theta + \\log{(\\theta)})^{\\theta} and \\sigma_{x}{(\\theta)} \\sigma_{x}^{\\theta}{(\\theta)} = (- \\theta + \\log{(\\theta)})^{\\theta} \\sigma_{x}{(\\theta)} and \\frac{d}{d \\theta} \\sigma_{x}{(\\theta)} \\sigma_{x}^{\\theta}{(\\theta)} = \\frac{d}{d \\theta} (- \\theta + \\log{(\\theta)})^{\\theta} \\sigma_{x}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))))"], [["power", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\phi_1')(Symbol('\\\\theta', commutative=True))), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["times", 6, "Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 7, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(C_{d})} = \\cos{(C_{d})} and \\operatorname{y^{\\prime}}{(Z)} = \\sin{(Z)}, then derive \\int \\operatorname{y^{\\prime}}{(Z)} dZ = A_{y} - \\cos{(Z)}, then obtain \\int \\frac{C_{d} l{(C_{d})}}{A_{y} - \\cos{(Z)}} dC_{d} = \\int \\frac{C_{d} \\cos{(C_{d})}}{A_{y} - \\cos{(Z)}} dC_{d}", "derivation": "l{(C_{d})} = \\cos{(C_{d})} and \\operatorname{y^{\\prime}}{(Z)} = \\sin{(Z)} and \\int \\operatorname{y^{\\prime}}{(Z)} dZ = \\int \\sin{(Z)} dZ and \\frac{C_{d} l{(C_{d})}}{\\int \\operatorname{y^{\\prime}}{(Z)} dZ} = \\frac{C_{d} \\cos{(C_{d})}}{\\int \\operatorname{y^{\\prime}}{(Z)} dZ} and \\int \\operatorname{y^{\\prime}}{(Z)} dZ = A_{y} - \\cos{(Z)} and \\int \\frac{C_{d} l{(C_{d})}}{\\int \\operatorname{y^{\\prime}}{(Z)} dZ} dC_{d} = \\int \\frac{C_{d} \\cos{(C_{d})}}{\\int \\operatorname{y^{\\prime}}{(Z)} dZ} dC_{d} and \\int \\frac{C_{d} l{(C_{d})}}{A_{y} - \\cos{(Z)}} dC_{d} = \\int \\frac{C_{d} \\cos{(C_{d})}}{A_{y} - \\cos{(Z)}} dC_{d}", "srepr_derivation": [["get_premise", "Equality(Function('l')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], ["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(sin(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], "Equality(Mul(Symbol('C_d', commutative=True), Function('l')(Symbol('C_d', commutative=True)), Pow(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Mul(Symbol('C_d', commutative=True), cos(Symbol('C_d', commutative=True)), Pow(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Symbol('C_d', commutative=True), Function('l')(Symbol('C_d', commutative=True)), Pow(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Symbol('C_d', commutative=True), cos(Symbol('C_d', commutative=True)), Pow(Integral(Function('y^{\\\\prime}')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Mul(Symbol('C_d', commutative=True), Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Integer(-1)), Function('l')(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Symbol('C_d', commutative=True), Pow(Add(Symbol('A_y', commutative=True), Mul(Integer(-1), cos(Symbol('Z', commutative=True)))), Integer(-1)), cos(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given S{(y^{\\prime})} = \\log{(y^{\\prime})}, then derive \\frac{d}{d y^{\\prime}} S{(y^{\\prime})} = \\frac{1}{y^{\\prime}}, then obtain \\log{(y^{\\prime})}^{- y^{\\prime}} \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})} = \\frac{\\log{(y^{\\prime})}^{- y^{\\prime}}}{y^{\\prime}}", "derivation": "S{(y^{\\prime})} = \\log{(y^{\\prime})} and \\frac{d}{d y^{\\prime}} S{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})} and \\frac{d}{d y^{\\prime}} S{(y^{\\prime})} = \\frac{1}{y^{\\prime}} and S^{- y^{\\prime}}{(y^{\\prime})} \\frac{d}{d y^{\\prime}} S{(y^{\\prime})} = \\frac{S^{- y^{\\prime}}{(y^{\\prime})}}{y^{\\prime}} and \\log{(y^{\\prime})}^{- y^{\\prime}} \\frac{d}{d y^{\\prime}} \\log{(y^{\\prime})} = \\frac{\\log{(y^{\\prime})}^{- y^{\\prime}}}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)))"], [["divide", 3, "Pow(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Derivative(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Function('S')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Derivative(log(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(log(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(t_{2},\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (- \\dot{x} + t_{2}), then derive \\frac{\\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = - \\frac{1}{t_{2}}, then obtain \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x} \\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = \\frac{\\partial}{\\partial \\dot{x}} - \\frac{\\dot{x}}{t_{2}}", "derivation": "\\operatorname{A_{y}}{(t_{2},\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (- \\dot{x} + t_{2}) and \\frac{\\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = \\frac{\\frac{\\partial}{\\partial \\dot{x}} (- \\dot{x} + t_{2})}{t_{2}} and \\frac{\\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = - \\frac{1}{t_{2}} and \\frac{\\dot{x} \\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = - \\frac{\\dot{x}}{t_{2}} and \\frac{\\partial}{\\partial \\dot{x}} \\frac{\\dot{x} \\operatorname{A_{y}}{(t_{2},\\dot{x})}}{t_{2}} = \\frac{\\partial}{\\partial \\dot{x}} - \\frac{\\dot{x}}{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('t_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["times", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('A_y')(Symbol('t_2', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('t_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(\\tilde{g}^*)} = e^{e^{\\tilde{g}^*}} and l{(\\tilde{g}^*)} = \\int q{(\\tilde{g}^*)} d\\tilde{g}^*, then obtain \\iint e^{e^{\\tilde{g}^*}} d\\tilde{g}^* d\\tilde{g}^* = \\iint q{(\\tilde{g}^*)} d\\tilde{g}^* d\\tilde{g}^*", "derivation": "q{(\\tilde{g}^*)} = e^{e^{\\tilde{g}^*}} and \\int q{(\\tilde{g}^*)} d\\tilde{g}^* = \\int e^{e^{\\tilde{g}^*}} d\\tilde{g}^* and l{(\\tilde{g}^*)} = \\int q{(\\tilde{g}^*)} d\\tilde{g}^* and l{(\\tilde{g}^*)} = \\int e^{e^{\\tilde{g}^*}} d\\tilde{g}^* and \\int l{(\\tilde{g}^*)} d\\tilde{g}^* = \\iint q{(\\tilde{g}^*)} d\\tilde{g}^* d\\tilde{g}^* and \\iint e^{e^{\\tilde{g}^*}} d\\tilde{g}^* d\\tilde{g}^* = \\iint q{(\\tilde{g}^*)} d\\tilde{g}^* d\\tilde{g}^*", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(exp(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(exp(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(Function('q')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('l')(Symbol('\\\\tilde{g}^*', commutative=True)), Integral(exp(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["integrate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('q')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(exp(exp(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Function('q')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\rho_f)} = \\cos{(\\sin{(\\rho_f)})}, then derive \\frac{d}{d \\rho_f} \\mathbf{J}{(\\rho_f)} = - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)}, then obtain \\int - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)} d\\rho_f = \\sigma_x + \\cos{(\\sin{(\\rho_f)})}", "derivation": "\\mathbf{J}{(\\rho_f)} = \\cos{(\\sin{(\\rho_f)})} and \\frac{d}{d \\rho_f} \\mathbf{J}{(\\rho_f)} = \\frac{d}{d \\rho_f} \\cos{(\\sin{(\\rho_f)})} and \\frac{d}{d \\rho_f} \\mathbf{J}{(\\rho_f)} = - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)} and - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)} = \\frac{d}{d \\rho_f} \\cos{(\\sin{(\\rho_f)})} and \\int - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)} d\\rho_f = \\int \\frac{d}{d \\rho_f} \\cos{(\\sin{(\\rho_f)})} d\\rho_f and \\int - \\sin{(\\sin{(\\rho_f)})} \\cos{(\\rho_f)} d\\rho_f = \\sigma_x + \\cos{(\\sin{(\\rho_f)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True)), cos(sin(Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))), Derivative(cos(sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Derivative(cos(sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), sin(sin(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), cos(sin(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given s{(\\hat{H}_l,v,\\mathbf{F})} = \\hat{H}_l + \\mathbf{F} - v, then obtain (\\frac{\\hat{H}_l + s{(\\hat{H}_l,v,\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + q{(v_{1})} = (\\frac{2 \\hat{H}_l + \\mathbf{F} - v}{\\mathbf{F}})^{\\mathbf{F}} + q{(v_{1})}", "derivation": "s{(\\hat{H}_l,v,\\mathbf{F})} = \\hat{H}_l + \\mathbf{F} - v and \\hat{H}_l + s{(\\hat{H}_l,v,\\mathbf{F})} = 2 \\hat{H}_l + \\mathbf{F} - v and \\frac{\\hat{H}_l + s{(\\hat{H}_l,v,\\mathbf{F})}}{\\mathbf{F}} = \\frac{2 \\hat{H}_l + \\mathbf{F} - v}{\\mathbf{F}} and (\\frac{\\hat{H}_l + s{(\\hat{H}_l,v,\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} = (\\frac{2 \\hat{H}_l + \\mathbf{F} - v}{\\mathbf{F}})^{\\mathbf{F}} and (\\frac{\\hat{H}_l + s{(\\hat{H}_l,v,\\mathbf{F})}}{\\mathbf{F}})^{\\mathbf{F}} + q{(v_{1})} = (\\frac{2 \\hat{H}_l + \\mathbf{F} - v}{\\mathbf{F}})^{\\mathbf{F}} + q{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('s')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Function('s')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Function('s')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 4, "Function('q')(Symbol('v_1', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}_l', commutative=True), Function('s')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('v', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)), Function('q')(Symbol('v_1', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)), Function('q')(Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(v_{t})} = \\sin{(v_{t})}, then derive \\int \\tilde{g}^*{(v_{t})} dv_{t} = v_{1} - \\cos{(v_{t})}, then obtain \\frac{d}{d v_{t}} \\int \\tilde{g}^*{(v_{t})} dv_{t} = \\frac{d}{d v_{t}} \\int \\sin{(v_{t})} dv_{t}", "derivation": "\\tilde{g}^*{(v_{t})} = \\sin{(v_{t})} and \\int \\tilde{g}^*{(v_{t})} dv_{t} = \\int \\sin{(v_{t})} dv_{t} and \\int \\tilde{g}^*{(v_{t})} dv_{t} = v_{1} - \\cos{(v_{t})} and v_{1} - \\cos{(v_{t})} = \\int \\sin{(v_{t})} dv_{t} and \\frac{d}{d v_{t}} \\int \\tilde{g}^*{(v_{t})} dv_{t} = \\frac{\\partial}{\\partial v_{t}} (v_{1} - \\cos{(v_{t})}) and \\frac{d}{d v_{t}} \\int \\tilde{g}^*{(v_{t})} dv_{t} = \\frac{d}{d v_{t}} \\int \\sin{(v_{t})} dv_{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integral(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} = A_{z}^{\\delta} - n_{2}, then obtain - n_{2} + \\frac{\\iint \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} d\\delta dA_{z}}{A_{z}} = - n_{2} + \\frac{\\iint (A_{z}^{\\delta} - n_{2}) d\\delta dA_{z}}{A_{z}}", "derivation": "\\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} = A_{z}^{\\delta} - n_{2} and \\int \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} d\\delta = \\int (A_{z}^{\\delta} - n_{2}) d\\delta and \\iint \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} d\\delta dA_{z} = \\iint (A_{z}^{\\delta} - n_{2}) d\\delta dA_{z} and \\frac{\\iint \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} d\\delta dA_{z}}{A_{z}} = \\frac{\\iint (A_{z}^{\\delta} - n_{2}) d\\delta dA_{z}}{A_{z}} and - n_{2} + \\frac{\\iint \\operatorname{C_{1}}{(n_{2},\\delta,A_{z})} d\\delta dA_{z}}{A_{z}} = - n_{2} + \\frac{\\iint (A_{z}^{\\delta} - n_{2}) d\\delta dA_{z}}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('A_z', commutative=True)), Add(Pow(Symbol('A_z', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Pow(Symbol('A_z', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(Add(Pow(Symbol('A_z', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["divide", 3, "Symbol('A_z', commutative=True)"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Function('C_1')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('A_z', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Function('C_1')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True))))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Integral(Add(Pow(Symbol('A_z', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given i{(r_{0},\\mathbf{P})} = \\mathbf{P} + r_{0}, then obtain \\frac{\\partial}{\\partial r_{0}} (- \\frac{\\partial}{\\partial r_{0}} (\\mathbf{P} + r_{0}) + \\frac{\\partial}{\\partial r_{0}} i{(r_{0},\\mathbf{P})}) = \\frac{d}{d r_{0}} 0", "derivation": "i{(r_{0},\\mathbf{P})} = \\mathbf{P} + r_{0} and \\frac{\\partial}{\\partial r_{0}} i{(r_{0},\\mathbf{P})} = \\frac{\\partial}{\\partial r_{0}} (\\mathbf{P} + r_{0}) and - \\frac{\\partial}{\\partial r_{0}} (\\mathbf{P} + r_{0}) + \\frac{\\partial}{\\partial r_{0}} i{(r_{0},\\mathbf{P})} = 0 and \\frac{\\partial}{\\partial r_{0}} (- \\frac{\\partial}{\\partial r_{0}} (\\mathbf{P} + r_{0}) + \\frac{\\partial}{\\partial r_{0}} i{(r_{0},\\mathbf{P})}) = \\frac{d}{d r_{0}} 0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Derivative(Function('i')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Derivative(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Derivative(Function('i')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(\\sigma_x)} = e^{\\sigma_x}, then derive - e^{\\sigma_x} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)} = 0, then obtain \\tilde{\\infty} \\iint (- x{(\\sigma_x)} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)}) d\\sigma_x d\\sigma_x = \\tilde{\\infty} \\iint 0 d\\sigma_x d\\sigma_x", "derivation": "x{(\\sigma_x)} = e^{\\sigma_x} and x{(\\sigma_x)} - e^{\\sigma_x} = 0 and \\frac{d}{d \\sigma_x} (x{(\\sigma_x)} - e^{\\sigma_x}) = \\frac{d}{d \\sigma_x} 0 and - e^{\\sigma_x} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)} = 0 and - x{(\\sigma_x)} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)} = 0 and \\int (- x{(\\sigma_x)} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)}) d\\sigma_x = \\int 0 d\\sigma_x and \\iint (- x{(\\sigma_x)} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)}) d\\sigma_x d\\sigma_x = \\iint 0 d\\sigma_x d\\sigma_x and \\tilde{\\infty} \\iint (- x{(\\sigma_x)} + \\frac{d}{d \\sigma_x} x{(\\sigma_x)}) d\\sigma_x d\\sigma_x = \\tilde{\\infty} \\iint 0 d\\sigma_x d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_x', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 6, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 7, 0], "Equality(Mul(zoo, Integral(Add(Mul(Integer(-1), Function('x')(Symbol('\\\\sigma_x', commutative=True))), Derivative(Function('x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(zoo, Integral(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(k,J)} = - J + k, then obtain - 2 J + k + (- k + \\mathbf{M}{(k,J)} - 1)^{k} - 2 = - 2 J + k + (- J - 1)^{k} - 2", "derivation": "\\mathbf{M}{(k,J)} = - J + k and \\mathbf{M}{(k,J)} - 1 = - J + k - 1 and - k + \\mathbf{M}{(k,J)} - 1 = - J - 1 and (- k + \\mathbf{M}{(k,J)} - 1)^{k} = (- J - 1)^{k} and - J + (- k + \\mathbf{M}{(k,J)} - 1)^{k} - 1 = - J + (- J - 1)^{k} - 1 and - 2 J + k + (- k + \\mathbf{M}{(k,J)} - 1)^{k} - 2 = - 2 J + k + (- J - 1)^{k} - 2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('k', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('k', commutative=True), Integer(-1)))"], [["minus", 2, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integer(-1)))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)), Integer(-1)))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('k', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Symbol('k', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\mathbf{M}')(Symbol('k', commutative=True), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)), Integer(-2)), Add(Mul(Integer(-1), Integer(2), Symbol('J', commutative=True)), Symbol('k', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Integer(-1)), Symbol('k', commutative=True)), Integer(-2)))"]]}, {"prompt": "Given \\dot{y}{(U,\\mathbf{M})} = U^{\\mathbf{M}}, then obtain \\log{(U U^{- \\mathbf{M}} \\dot{y}{(U,\\mathbf{M})})} = \\log{(U)}", "derivation": "\\dot{y}{(U,\\mathbf{M})} = U^{\\mathbf{M}} and U^{- \\mathbf{M}} \\dot{y}{(U,\\mathbf{M})} = 1 and U U^{- \\mathbf{M}} \\dot{y}{(U,\\mathbf{M})} = U and \\log{(U U^{- \\mathbf{M}} \\dot{y}{(U,\\mathbf{M})})} = \\log{(U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "Pow(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\dot{y}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Integer(1))"], [["times", 2, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\dot{y}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('U', commutative=True))"], [["log", 3], "Equality(log(Mul(Symbol('U', commutative=True), Pow(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\dot{y}')(Symbol('U', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), log(Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\tilde{g})} = \\log{(\\tilde{g})}, then obtain \\frac{\\tilde{g}^{3} \\mathbf{P}^{3}{(\\tilde{g})}}{\\log{(\\tilde{g})}^{3}} = \\tilde{g}^{3}", "derivation": "\\mathbf{P}{(\\tilde{g})} = \\log{(\\tilde{g})} and \\tilde{g} \\mathbf{P}{(\\tilde{g})} = \\tilde{g} \\log{(\\tilde{g})} and \\tilde{g}^{3} \\mathbf{P}{(\\tilde{g})} = \\tilde{g}^{3} \\log{(\\tilde{g})} and \\frac{\\tilde{g}^{3} \\mathbf{P}{(\\tilde{g})}}{\\log{(\\tilde{g})}} = \\tilde{g}^{3} and \\frac{\\tilde{g}^{3} \\mathbf{P}^{2}{(\\tilde{g})}}{\\log{(\\tilde{g})}} = \\tilde{g}^{3} \\mathbf{P}{(\\tilde{g})} and \\frac{\\tilde{g}^{3} \\mathbf{P}^{3}{(\\tilde{g})}}{\\log{(\\tilde{g})}^{3}} = \\tilde{g}^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Symbol('\\\\tilde{g}', commutative=True), log(Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), log(Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 3, "log(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True)), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}', commutative=True)), Integer(3)), Pow(log(Symbol('\\\\tilde{g}', commutative=True)), Integer(-3))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(3)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(V_{\\mathbf{B}},t_{2})} = - V_{\\mathbf{B}} + \\log{(t_{2})}, then obtain - \\frac{- V_{\\mathbf{B}} + \\log{(t_{2})}}{V_{\\mathbf{B}}} = \\frac{V_{\\mathbf{B}} - \\log{(t_{2})}}{V_{\\mathbf{B}}}", "derivation": "\\Psi_{\\lambda}{(V_{\\mathbf{B}},t_{2})} = - V_{\\mathbf{B}} + \\log{(t_{2})} and - \\frac{\\Psi_{\\lambda}{(V_{\\mathbf{B}},t_{2})}}{V_{\\mathbf{B}}} = - \\frac{- V_{\\mathbf{B}} + \\log{(t_{2})}}{V_{\\mathbf{B}}} and - \\frac{\\Psi_{\\lambda}{(V_{\\mathbf{B}},t_{2})}}{V_{\\mathbf{B}}} = 1 - \\frac{\\log{(t_{2})}}{V_{\\mathbf{B}}} and 1 - \\frac{\\log{(t_{2})}}{V_{\\mathbf{B}}} = \\frac{V_{\\mathbf{B}} - \\log{(t_{2})}}{V_{\\mathbf{B}}} and - \\frac{- V_{\\mathbf{B}} + \\log{(t_{2})}}{V_{\\mathbf{B}}} = 1 - \\frac{\\log{(t_{2})}}{V_{\\mathbf{B}}} and - \\frac{- V_{\\mathbf{B}} + \\log{(t_{2})}}{V_{\\mathbf{B}}} = \\frac{V_{\\mathbf{B}} - \\log{(t_{2})}}{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('t_2', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('t_2', commutative=True)))))"], [["expand", 2], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_2', commutative=True))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), log(Symbol('t_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), log(Symbol('t_2', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), log(Symbol('t_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('t_2', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), log(Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('t_2', commutative=True)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), log(Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\phi_1,\\mu)} = \\frac{\\partial}{\\partial \\phi_1} (- \\mu + \\phi_1) and \\varphi{(y)} = \\frac{d}{d y} \\log{(y)} and Q{(y)} = \\frac{d}{d y} \\log{(y)}, then obtain \\cos{(Q{(y)} \\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)})} = \\cos{(\\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)} \\varphi{(y)})}", "derivation": "\\hat{\\mathbf{r}}{(\\phi_1,\\mu)} = \\frac{\\partial}{\\partial \\phi_1} (- \\mu + \\phi_1) and \\varphi{(y)} = \\frac{d}{d y} \\log{(y)} and Q{(y)} = \\frac{d}{d y} \\log{(y)} and Q{(y)} = \\varphi{(y)} and Q{(y)} (\\frac{\\partial}{\\partial \\phi_1} (- \\mu + \\phi_1))^{- \\mu} = \\varphi{(y)} (\\frac{\\partial}{\\partial \\phi_1} (- \\mu + \\phi_1))^{- \\mu} and Q{(y)} \\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)} = \\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)} \\varphi{(y)} and \\cos{(Q{(y)} \\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)})} = \\cos{(\\hat{\\mathbf{r}}^{- \\mu}{(\\phi_1,\\mu)} \\varphi{(y)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\varphi')(Symbol('y', commutative=True)), Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('y', commutative=True)), Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('Q')(Symbol('y', commutative=True)), Function('\\\\varphi')(Symbol('y', commutative=True)))"], [["divide", 4, "Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('y', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Mul(Function('\\\\varphi')(Symbol('y', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('Q')(Symbol('y', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True)))), Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Function('\\\\varphi')(Symbol('y', commutative=True))))"], [["cos", 6], "Equality(cos(Mul(Function('Q')(Symbol('y', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))))), cos(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True))), Function('\\\\varphi')(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\mu{(h)} = \\cos{(h)}, then derive \\frac{d}{d h} \\mu{(h)} = - \\sin{(h)}, then obtain \\int - \\sin{(h)} dh = Z + \\cos{(h)}", "derivation": "\\mu{(h)} = \\cos{(h)} and \\frac{d}{d h} \\mu{(h)} = \\frac{d}{d h} \\cos{(h)} and \\frac{d}{d h} \\mu{(h)} = - \\sin{(h)} and - \\sin{(h)} = \\frac{d}{d h} \\cos{(h)} and \\int - \\sin{(h)} dh = \\int \\frac{d}{d h} \\cos{(h)} dh and \\int - \\sin{(h)} dh = Z + \\cos{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mu')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Derivative(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Derivative(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Add(Symbol('Z', commutative=True), cos(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\phi_1,\\sigma_p)} = - \\phi_1 + \\sigma_p, then derive \\frac{\\partial}{\\partial \\phi_1} \\operatorname{n_{2}}{(\\phi_1,\\sigma_p)} = -1, then obtain -1 = \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + \\sigma_p)", "derivation": "\\operatorname{n_{2}}{(\\phi_1,\\sigma_p)} = - \\phi_1 + \\sigma_p and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{n_{2}}{(\\phi_1,\\sigma_p)} = \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + \\sigma_p) and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{n_{2}}{(\\phi_1,\\sigma_p)} = -1 and -1 = \\frac{\\partial}{\\partial \\phi_1} (- \\phi_1 + \\sigma_p)", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(\\mathbf{v})} = e^{\\mathbf{v}}, then obtain m^{\\mathbf{v}}{(\\mathbf{v})} + \\int m{(\\mathbf{v})} d\\mathbf{v} = m^{\\mathbf{v}}{(\\mathbf{v})} + \\int e^{\\mathbf{v}} d\\mathbf{v}", "derivation": "m{(\\mathbf{v})} = e^{\\mathbf{v}} and \\int m{(\\mathbf{v})} d\\mathbf{v} = \\int e^{\\mathbf{v}} d\\mathbf{v} and m^{\\mathbf{v}}{(\\mathbf{v})} = (e^{\\mathbf{v}})^{\\mathbf{v}} and (e^{\\mathbf{v}})^{\\mathbf{v}} + \\int m{(\\mathbf{v})} d\\mathbf{v} = (e^{\\mathbf{v}})^{\\mathbf{v}} + \\int e^{\\mathbf{v}} d\\mathbf{v} and m^{\\mathbf{v}}{(\\mathbf{v})} + \\int m{(\\mathbf{v})} d\\mathbf{v} = m^{\\mathbf{v}}{(\\mathbf{v})} + \\int e^{\\mathbf{v}} d\\mathbf{v}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["add", 2, "Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Add(Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Pow(exp(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Pow(Function('m')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(x)} = \\cos{(x)}, then derive 1 = - \\frac{\\sin{(x)}}{\\frac{d}{d x} \\rho_{f}{(x)}}, then obtain \\cos{(x)} + \\frac{\\frac{d}{d x} \\rho_{f}{(x)}}{\\frac{d}{d x} \\cos{(x)}} = - \\frac{\\sin{(x)}}{\\frac{d}{d x} \\cos{(x)}} + \\cos{(x)}", "derivation": "\\rho_{f}{(x)} = \\cos{(x)} and \\frac{d}{d x} \\rho_{f}{(x)} = \\frac{d}{d x} \\cos{(x)} and 1 = \\frac{\\frac{d}{d x} \\cos{(x)}}{\\frac{d}{d x} \\rho_{f}{(x)}} and 1 = - \\frac{\\sin{(x)}}{\\frac{d}{d x} \\rho_{f}{(x)}} and \\frac{\\frac{d}{d x} \\rho_{f}{(x)}}{\\frac{d}{d x} \\cos{(x)}} = - \\frac{\\sin{(x)}}{\\frac{d}{d x} \\cos{(x)}} and \\cos{(x)} + \\frac{\\frac{d}{d x} \\rho_{f}{(x)}}{\\frac{d}{d x} \\cos{(x)}} = - \\frac{\\sin{(x)}}{\\frac{d}{d x} \\cos{(x)}} + \\cos{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), Mul(Integer(-1), sin(Symbol('x', commutative=True)), Pow(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 4, "Mul(Pow(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], "Equality(Mul(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), sin(Symbol('x', commutative=True)), Pow(Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))))"], [["add", 5, "cos(Symbol('x', commutative=True))"], "Equality(Add(cos(Symbol('x', commutative=True)), Mul(Derivative(Function('\\\\rho_f')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))), Add(Mul(Integer(-1), sin(Symbol('x', commutative=True)), Pow(Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))), cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\delta,y)} = \\delta + y, then obtain \\sin{(\\int \\mathbf{g}^{4}{(\\delta,y)} dy)} = \\sin{(\\int (\\delta + y) \\mathbf{g}^{3}{(\\delta,y)} dy)}", "derivation": "\\mathbf{g}{(\\delta,y)} = \\delta + y and \\mathbf{g}^{2}{(\\delta,y)} = (\\delta + y) \\mathbf{g}{(\\delta,y)} and \\mathbf{g}^{4}{(\\delta,y)} = (\\delta + y) \\mathbf{g}^{3}{(\\delta,y)} and \\int \\mathbf{g}^{4}{(\\delta,y)} dy = \\int (\\delta + y) \\mathbf{g}^{3}{(\\delta,y)} dy and \\sin{(\\int \\mathbf{g}^{4}{(\\delta,y)} dy)} = \\sin{(\\int (\\delta + y) \\mathbf{g}^{3}{(\\delta,y)} dy)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(4)), Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(3))))"], [["integrate", 3, "Symbol('y', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(4)), Tuple(Symbol('y', commutative=True))), Integral(Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(3))), Tuple(Symbol('y', commutative=True))))"], [["sin", 4], "Equality(sin(Integral(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(4)), Tuple(Symbol('y', commutative=True)))), sin(Integral(Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\delta', commutative=True), Symbol('y', commutative=True)), Integer(3))), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given g{(S)} = \\log{(S)} and \\mathbf{g}{(S)} = \\frac{d}{d S} \\int e^{g{(S)}} \\log{(S)} dS, then obtain \\frac{\\mathbf{g}{(S)} - 1}{\\mathbf{g}{(S)}} = \\frac{\\frac{\\partial}{\\partial S} (\\frac{S^{2} \\log{(S)}}{2} - \\frac{S^{2}}{4} + \\mathbf{J}_P) - 1}{\\mathbf{g}{(S)}}", "derivation": "g{(S)} = \\log{(S)} and e^{g{(S)}} = S and e^{g{(S)}} \\log{(S)} = S \\log{(S)} and \\int e^{g{(S)}} \\log{(S)} dS = \\int S \\log{(S)} dS and \\mathbf{g}{(S)} = \\frac{d}{d S} \\int e^{g{(S)}} \\log{(S)} dS and \\mathbf{g}{(S)} = \\frac{d}{d S} \\int S \\log{(S)} dS and \\mathbf{g}{(S)} - 1 = \\frac{d}{d S} \\int S \\log{(S)} dS - 1 and \\frac{\\mathbf{g}{(S)} - 1}{\\mathbf{g}{(S)}} = \\frac{\\frac{d}{d S} \\int S \\log{(S)} dS - 1}{\\mathbf{g}{(S)}} and \\frac{\\mathbf{g}{(S)} - 1}{\\mathbf{g}{(S)}} = \\frac{\\frac{\\partial}{\\partial S} (\\frac{S^{2} \\log{(S)}}{2} - \\frac{S^{2}}{4} + \\mathbf{J}_P) - 1}{\\mathbf{g}{(S)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["exp", 1], "Equality(exp(Function('g')(Symbol('S', commutative=True))), Symbol('S', commutative=True))"], [["times", 2, "log(Symbol('S', commutative=True))"], "Equality(Mul(exp(Function('g')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(exp(Function('g')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Derivative(Integral(Mul(exp(Function('g')(Symbol('S', commutative=True))), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Derivative(Integral(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 6, 1], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1)), Add(Derivative(Integral(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 7, "Function('\\\\mathbf{g}')(Symbol('S', commutative=True))"], "Equality(Mul(Add(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1))), Mul(Add(Derivative(Integral(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1))))"], [["evaluate_integrals", 8], "Equality(Mul(Add(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1))), Mul(Add(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2)), log(Symbol('S', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Pow(Function('\\\\mathbf{g}')(Symbol('S', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(V,F_{H})} = \\frac{F_{H}}{V}, then derive F_{H} + \\int V \\operatorname{A_{1}}{(V,F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} + v_{y}, then obtain F_{H} + \\int F_{H} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} + v_{y}", "derivation": "\\operatorname{A_{1}}{(V,F_{H})} = \\frac{F_{H}}{V} and V \\operatorname{A_{1}}{(V,F_{H})} = F_{H} and \\int V \\operatorname{A_{1}}{(V,F_{H})} dF_{H} = \\int F_{H} dF_{H} and F_{H} + \\int V \\operatorname{A_{1}}{(V,F_{H})} dF_{H} = F_{H} + \\int F_{H} dF_{H} and F_{H} + \\int V \\operatorname{A_{1}}{(V,F_{H})} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} + v_{y} and F_{H} + \\int F_{H} dF_{H} = \\frac{F_{H}^{2}}{2} + F_{H} + v_{y}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('V', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('V', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('A_1')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Symbol('V', commutative=True), Function('A_1')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True))))"], [["add", 3, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Integral(Mul(Symbol('V', commutative=True), Function('A_1')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)))), Add(Symbol('F_H', commutative=True), Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_H', commutative=True), Integral(Mul(Symbol('V', commutative=True), Function('A_1')(Symbol('V', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('F_H', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('F_H', commutative=True), Integral(Symbol('F_H', commutative=True), Tuple(Symbol('F_H', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_H', commutative=True), Integer(2))), Symbol('F_H', commutative=True), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\nabla)} = \\sin{(\\sin{(\\nabla)})}, then obtain \\frac{d}{d \\nabla} - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})} = \\frac{d}{d \\nabla} (- \\varphi{(\\nabla)} + \\sin{(\\sin{(\\nabla)})} - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})})", "derivation": "\\varphi{(\\nabla)} = \\sin{(\\sin{(\\nabla)})} and \\frac{d}{d \\nabla} \\varphi{(\\nabla)} = \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})} and - \\frac{d}{d \\nabla} \\varphi{(\\nabla)} = - \\varphi{(\\nabla)} + \\sin{(\\sin{(\\nabla)})} - \\frac{d}{d \\nabla} \\varphi{(\\nabla)} and - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})} = - \\varphi{(\\nabla)} + \\sin{(\\sin{(\\nabla)})} - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})} and \\frac{d}{d \\nabla} - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})} = \\frac{d}{d \\nabla} (- \\varphi{(\\nabla)} + \\sin{(\\sin{(\\nabla)})} - \\frac{d}{d \\nabla} \\sin{(\\sin{(\\nabla)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), sin(sin(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["minus", 1, "Add(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), Derivative(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True))), sin(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Derivative(sin(sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True))), sin(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Derivative(sin(sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))))"], [["differentiate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Derivative(sin(sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\nabla', commutative=True))), sin(sin(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Derivative(sin(sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(G)} = \\frac{d}{d G} e^{G}, then derive \\frac{d}{d G} v{(G)} = e^{G}, then obtain e^{G} \\frac{d^{2}}{d G^{2}} e^{G} = (\\frac{d^{2}}{d G^{2}} e^{G})^{2}", "derivation": "v{(G)} = \\frac{d}{d G} e^{G} and \\frac{d}{d G} v{(G)} = \\frac{d^{2}}{d G^{2}} e^{G} and \\frac{d}{d G} v{(G)} = e^{G} and e^{G} = \\frac{d^{2}}{d G^{2}} e^{G} and e^{G} \\frac{d^{2}}{d G^{2}} e^{G} = (\\frac{d^{2}}{d G^{2}} e^{G})^{2}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('G', commutative=True)), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), exp(Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('G', commutative=True)), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["times", 4, "Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))"], "Equality(Mul(exp(Symbol('G', commutative=True)), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2)))), Pow(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(t_{1})} = \\sin{(t_{1})}, then derive \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} = z^{*} - \\cos{(t_{1})}, then obtain (c_{0} - \\cos{(t_{1})}) \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} = (z^{*} - \\cos{(t_{1})}) \\int \\operatorname{v_{1}}{(t_{1})} dt_{1}", "derivation": "\\operatorname{v_{1}}{(t_{1})} = \\sin{(t_{1})} and \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} = \\int \\sin{(t_{1})} dt_{1} and \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} = z^{*} - \\cos{(t_{1})} and \\int \\sin{(t_{1})} dt_{1} = z^{*} - \\cos{(t_{1})} and (\\int \\operatorname{v_{1}}{(t_{1})} dt_{1}) \\int \\sin{(t_{1})} dt_{1} = (z^{*} - \\cos{(t_{1})}) \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} and (c_{0} - \\cos{(t_{1})}) \\int \\operatorname{v_{1}}{(t_{1})} dt_{1} = (z^{*} - \\cos{(t_{1})}) \\int \\operatorname{v_{1}}{(t_{1})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(sin(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))))"], [["times", 4, "Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(sin(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Mul(Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Mul(Add(Symbol('z^*', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Integral(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)} = \\frac{\\phi_1}{\\varepsilon_0}, then obtain (\\frac{\\frac{\\partial}{\\partial \\phi_1} \\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)}}{\\varepsilon_0})^{\\phi_1} = (\\frac{\\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{\\varepsilon_0}}{\\varepsilon_0})^{\\phi_1}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)} = \\frac{\\phi_1}{\\varepsilon_0} and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{\\varepsilon_0} and \\frac{\\frac{\\partial}{\\partial \\phi_1} \\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)}}{\\varepsilon_0} = \\frac{\\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{\\varepsilon_0}}{\\varepsilon_0} and (\\frac{\\frac{\\partial}{\\partial \\phi_1} \\operatorname{L_{\\varepsilon}}{(\\phi_1,\\varepsilon_0)}}{\\varepsilon_0})^{\\phi_1} = (\\frac{\\frac{\\partial}{\\partial \\phi_1} \\frac{\\phi_1}{\\varepsilon_0}}{\\varepsilon_0})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["times", 2, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given l{(u)} = \\cos{(u)}, then obtain - l{(u)} + (\\frac{d}{d u} l{(u)})^{2} = - l{(u)} + \\frac{d}{d u} l{(u)} \\frac{d}{d u} \\cos{(u)}", "derivation": "l{(u)} = \\cos{(u)} and \\frac{d}{d u} l{(u)} = \\frac{d}{d u} \\cos{(u)} and (\\frac{d}{d u} l{(u)})^{2} = \\frac{d}{d u} l{(u)} \\frac{d}{d u} \\cos{(u)} and - l{(u)} + (\\frac{d}{d u} l{(u)})^{2} = - l{(u)} + \\frac{d}{d u} l{(u)} \\frac{d}{d u} \\cos{(u)}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["minus", 3, "Function('l')(Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('u', commutative=True))), Pow(Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(2))), Add(Mul(Integer(-1), Function('l')(Symbol('u', commutative=True))), Mul(Derivative(Function('l')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"]]}, {"prompt": "Given J{(t_{1})} = \\sin{(e^{t_{1}})}, then obtain \\operatorname{a^{\\dagger}}{(n)} = \\frac{(- J{(t_{1})} + \\frac{\\sin{(e^{t_{1}})}}{t_{1}}) \\operatorname{a^{\\dagger}}{(n)}}{- J{(t_{1})} + \\frac{J{(t_{1})}}{t_{1}}}", "derivation": "J{(t_{1})} = \\sin{(e^{t_{1}})} and \\frac{J{(t_{1})}}{t_{1}} = \\frac{\\sin{(e^{t_{1}})}}{t_{1}} and - J{(t_{1})} + \\frac{J{(t_{1})}}{t_{1}} = - J{(t_{1})} + \\frac{\\sin{(e^{t_{1}})}}{t_{1}} and (- J{(t_{1})} + \\frac{J{(t_{1})}}{t_{1}}) \\operatorname{a^{\\dagger}}{(n)} = (- J{(t_{1})} + \\frac{\\sin{(e^{t_{1}})}}{t_{1}}) \\operatorname{a^{\\dagger}}{(n)} and \\operatorname{a^{\\dagger}}{(n)} = \\frac{(- J{(t_{1})} + \\frac{\\sin{(e^{t_{1}})}}{t_{1}}) \\operatorname{a^{\\dagger}}{(n)}}{- J{(t_{1})} + \\frac{J{(t_{1})}}{t_{1}}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('t_1', commutative=True)), sin(exp(Symbol('t_1', commutative=True))))"], [["divide", 1, "Symbol('t_1', commutative=True)"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), sin(exp(Symbol('t_1', commutative=True)))))"], [["minus", 2, "Function('J')(Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('J')(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), sin(exp(Symbol('t_1', commutative=True))))))"], [["times", 3, "Function('a^{\\\\dagger}')(Symbol('n', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('J')(Symbol('t_1', commutative=True)))), Function('a^{\\\\dagger}')(Symbol('n', commutative=True))), Mul(Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), sin(exp(Symbol('t_1', commutative=True))))), Function('a^{\\\\dagger}')(Symbol('n', commutative=True))))"], [["divide", 4, "Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('J')(Symbol('t_1', commutative=True))))"], "Equality(Function('a^{\\\\dagger}')(Symbol('n', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('J')(Symbol('t_1', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Function('J')(Symbol('t_1', commutative=True))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), sin(exp(Symbol('t_1', commutative=True))))), Function('a^{\\\\dagger}')(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} = \\sin{(\\log{(\\phi_2)})}, then obtain \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} + \\log{(\\phi_2)}}{\\int \\sin{(\\log{(\\phi_2)})} d\\phi_2} = \\frac{\\log{(\\phi_2)} + \\sin{(\\log{(\\phi_2)})}}{\\int \\sin{(\\log{(\\phi_2)})} d\\phi_2}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} = \\sin{(\\log{(\\phi_2)})} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} d\\phi_2 = \\int \\sin{(\\log{(\\phi_2)})} d\\phi_2 and \\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} + \\log{(\\phi_2)} = \\log{(\\phi_2)} + \\sin{(\\log{(\\phi_2)})} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} + \\log{(\\phi_2)}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} d\\phi_2} = \\frac{\\log{(\\phi_2)} + \\sin{(\\log{(\\phi_2)})}}{\\int \\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} d\\phi_2} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\phi_2)} + \\log{(\\phi_2)}}{\\int \\sin{(\\log{(\\phi_2)})} d\\phi_2} = \\frac{\\log{(\\phi_2)} + \\sin{(\\log{(\\phi_2)})}}{\\int \\sin{(\\log{(\\phi_2)})} d\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), sin(log(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(sin(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["add", 1, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))), Add(log(Symbol('\\\\phi_2', commutative=True)), sin(log(Symbol('\\\\phi_2', commutative=True)))))"], [["divide", 3, "Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Mul(Add(log(Symbol('\\\\phi_2', commutative=True)), sin(log(Symbol('\\\\phi_2', commutative=True)))), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True))), Pow(Integral(sin(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Mul(Add(log(Symbol('\\\\phi_2', commutative=True)), sin(log(Symbol('\\\\phi_2', commutative=True)))), Pow(Integral(sin(log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\nabla{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})}, then obtain - \\Psi_{\\lambda} + \\int \\frac{\\sin{(\\Psi_{\\lambda})}}{\\nabla{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} = - \\Psi_{\\lambda} + \\int \\frac{\\sin^{2}{(\\Psi_{\\lambda})}}{\\nabla^{2}{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "derivation": "\\nabla{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})} and 1 = \\frac{\\sin{(\\Psi_{\\lambda})}}{\\nabla{(\\Psi_{\\lambda})}} and \\frac{\\sin{(\\Psi_{\\lambda})}}{\\nabla{(\\Psi_{\\lambda})}} = \\frac{\\sin^{2}{(\\Psi_{\\lambda})}}{\\nabla^{2}{(\\Psi_{\\lambda})}} and \\int \\frac{\\sin{(\\Psi_{\\lambda})}}{\\nabla{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} = \\int \\frac{\\sin^{2}{(\\Psi_{\\lambda})}}{\\nabla^{2}{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} and - \\Psi_{\\lambda} + \\int \\frac{\\sin{(\\Psi_{\\lambda})}}{\\nabla{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} = - \\Psi_{\\lambda} + \\int \\frac{\\sin^{2}{(\\Psi_{\\lambda})}}{\\nabla^{2}{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 2, "Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integral(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integral(Mul(Pow(Function('\\\\nabla')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-2)), Pow(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given a{(E_{n},E_{\\lambda})} = \\cos{(E_{n} + E_{\\lambda})} and \\operatorname{V_{\\mathbf{B}}}{(E_{n},E_{\\lambda})} = 2 a{(E_{n},E_{\\lambda})}, then obtain E_{n} + E_{\\lambda} + 2 a{(E_{n},E_{\\lambda})} = E_{n} + E_{\\lambda} + a{(E_{n},E_{\\lambda})} + \\cos{(E_{n} + E_{\\lambda})}", "derivation": "a{(E_{n},E_{\\lambda})} = \\cos{(E_{n} + E_{\\lambda})} and 2 a{(E_{n},E_{\\lambda})} = a{(E_{n},E_{\\lambda})} + \\cos{(E_{n} + E_{\\lambda})} and \\operatorname{V_{\\mathbf{B}}}{(E_{n},E_{\\lambda})} = 2 a{(E_{n},E_{\\lambda})} and \\operatorname{V_{\\mathbf{B}}}{(E_{n},E_{\\lambda})} = a{(E_{n},E_{\\lambda})} + \\cos{(E_{n} + E_{\\lambda})} and E_{n} + E_{\\lambda} + \\operatorname{V_{\\mathbf{B}}}{(E_{n},E_{\\lambda})} = E_{n} + E_{\\lambda} + a{(E_{n},E_{\\lambda})} + \\cos{(E_{n} + E_{\\lambda})} and E_{n} + E_{\\lambda} + 2 a{(E_{n},E_{\\lambda})} = E_{n} + E_{\\lambda} + a{(E_{n},E_{\\lambda})} + \\cos{(E_{n} + E_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["add", 1, "Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["add", 4, "Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(2), Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Function('a')(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given c{(\\phi_1,\\mathbf{p})} = \\frac{\\log{(\\mathbf{p})}}{\\phi_1}, then derive \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = - \\frac{\\log{(\\mathbf{p})}}{\\phi_1^{2}}, then obtain \\phi_1 \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = - \\frac{\\log{(\\mathbf{p})}}{\\phi_1}", "derivation": "c{(\\phi_1,\\mathbf{p})} = \\frac{\\log{(\\mathbf{p})}}{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\log{(\\mathbf{p})}}{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = - \\frac{\\log{(\\mathbf{p})}}{\\phi_1^{2}} and - \\frac{\\log{(\\mathbf{p})}}{\\phi_1^{2}} = \\frac{\\partial}{\\partial \\phi_1} \\frac{\\log{(\\mathbf{p})}}{\\phi_1} and \\phi_1 \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = \\phi_1 \\frac{\\partial}{\\partial \\phi_1} \\frac{\\log{(\\mathbf{p})}}{\\phi_1} and \\phi_1 \\frac{\\partial}{\\partial \\phi_1} c{(\\phi_1,\\mathbf{p})} = - \\frac{\\log{(\\mathbf{p})}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), log(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-2)), log(Symbol('\\\\mathbf{p}', commutative=True))), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Function('c')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Derivative(Function('c')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given g{(\\phi_2,F_{c})} = \\frac{e^{\\phi_2}}{F_{c}} and \\nabla{(\\phi_2,F_{c})} = \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{F_{c}}, then obtain \\frac{\\partial}{\\partial \\phi_2} g{(\\phi_2,F_{c})} = \\nabla{(\\phi_2,F_{c})}", "derivation": "g{(\\phi_2,F_{c})} = \\frac{e^{\\phi_2}}{F_{c}} and \\frac{\\partial}{\\partial \\phi_2} g{(\\phi_2,F_{c})} = \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{F_{c}} and \\nabla{(\\phi_2,F_{c})} = \\frac{\\partial}{\\partial \\phi_2} \\frac{e^{\\phi_2}}{F_{c}} and \\frac{\\partial}{\\partial \\phi_2} g{(\\phi_2,F_{c})} = \\nabla{(\\phi_2,F_{c})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Derivative(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('g')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Function('\\\\nabla')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(T,\\dot{x})} = - T + \\dot{x}, then derive \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(T,\\dot{x})} = 1, then obtain \\iint 1 dT dT = \\iint \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} dT dT", "derivation": "\\operatorname{C_{d}}{(T,\\dot{x})} = - T + \\dot{x} and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(T,\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x}) and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(T,\\dot{x})} = 1 and \\frac{\\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(T,\\dot{x})}}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} = \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} and 1 = \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} \\operatorname{C_{d}}{(T,\\dot{x})}} and 1 = \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} and \\int 1 dT = \\int \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} dT and \\iint 1 dT dT = \\iint \\frac{1}{\\frac{\\partial}{\\partial \\dot{x}} (- T + \\dot{x})} dT dT", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Pow(Derivative(Function('C_d')(Symbol('T', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 6, "Symbol('T', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('T', commutative=True))), Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('T', commutative=True))))"], [["integrate", 7, "Symbol('T', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Pow(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} = \\sin{(\\pi^{b})}, then obtain \\int \\frac{d}{d b} 0 db = \\int \\frac{\\partial}{\\partial b} (- \\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} + \\sin{(\\pi^{b})}) db", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} = \\sin{(\\pi^{b})} and 0 = - \\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} + \\sin{(\\pi^{b})} and \\frac{d}{d b} 0 = \\frac{\\partial}{\\partial b} (- \\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} + \\sin{(\\pi^{b})}) and \\int \\frac{d}{d b} 0 db = \\int \\frac{\\partial}{\\partial b} (- \\operatorname{f_{\\mathbf{v}}}{(b,\\pi)} + \\sin{(\\pi^{b})}) db", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True))))"], [["minus", 1, "Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), sin(Pow(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} - \\hat{x}_0, then derive \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} = -1, then obtain \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} + \\frac{\\partial}{\\partial \\hat{x}_0} (\\dot{\\mathbf{r}} - \\hat{x}_0) = \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} - 1", "derivation": "\\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} - \\hat{x}_0 and \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\dot{\\mathbf{r}} - \\hat{x}_0) and \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} = -1 and \\frac{\\partial}{\\partial \\hat{x}_0} (\\dot{\\mathbf{r}} - \\hat{x}_0) = -1 and \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} + \\frac{\\partial}{\\partial \\hat{x}_0} (\\dot{\\mathbf{r}} - \\hat{x}_0) = \\mathbf{J}_P{(\\hat{x}_0,\\dot{\\mathbf{r}})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(-1))"], [["add", 4, "Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{E}{(r_{0},\\mathbf{g})} = \\cos{(\\mathbf{g}^{r_{0}})} and \\mathbf{M}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\mathbf{E}{(r_{0},\\mathbf{g})} \\cos{(\\mathbf{g}^{r_{0}})}, then obtain \\mathbf{g}^{r_{0}} \\mathbf{E}^{2}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\cos^{2}{(\\mathbf{g}^{r_{0}})}", "derivation": "\\mathbf{E}{(r_{0},\\mathbf{g})} = \\cos{(\\mathbf{g}^{r_{0}})} and \\mathbf{M}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\mathbf{E}{(r_{0},\\mathbf{g})} \\cos{(\\mathbf{g}^{r_{0}})} and \\mathbf{M}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\cos^{2}{(\\mathbf{g}^{r_{0}})} and \\mathbf{M}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\mathbf{E}^{2}{(r_{0},\\mathbf{g})} and \\mathbf{g}^{r_{0}} \\mathbf{E}^{2}{(r_{0},\\mathbf{g})} = \\mathbf{g}^{r_{0}} \\cos^{2}{(\\mathbf{g}^{r_{0}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)), Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{E}')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True)), Pow(cos(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('r_0', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}_f{(V_{\\mathbf{E}})} = e^{\\sin{(V_{\\mathbf{E}})}}, then obtain (\\mathbf{J}_f{(V_{\\mathbf{E}})} e^{\\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})}) e^{- 2 \\sin{(V_{\\mathbf{E}})}} = (e^{2 \\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})}) e^{- 2 \\sin{(V_{\\mathbf{E}})}}", "derivation": "\\mathbf{J}_f{(V_{\\mathbf{E}})} = e^{\\sin{(V_{\\mathbf{E}})}} and \\mathbf{J}_f{(V_{\\mathbf{E}})} e^{\\sin{(V_{\\mathbf{E}})}} = e^{2 \\sin{(V_{\\mathbf{E}})}} and \\mathbf{J}_f{(V_{\\mathbf{E}})} e^{\\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})} = e^{2 \\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})} and (\\mathbf{J}_f{(V_{\\mathbf{E}})} e^{\\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})}) e^{- 2 \\sin{(V_{\\mathbf{E}})}} = (e^{2 \\sin{(V_{\\mathbf{E}})}} + 2 \\sin{(V_{\\mathbf{E}})}) e^{- 2 \\sin{(V_{\\mathbf{E}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 1, "exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["add", 2, "Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(exp(Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["divide", 3, "exp(Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], "Equality(Mul(Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), exp(Mul(Integer(-1), Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))), Mul(Add(exp(Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), exp(Mul(Integer(-1), Integer(2), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mu)} = e^{\\mu}, then obtain \\frac{\\operatorname{x^{{\\}'}}{(\\mu)} - 1 - \\cos{(1)}}{\\mu} = \\frac{e^{\\mu} - 1 - \\cos{(1)}}{\\mu}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mu)} = e^{\\mu} and 1 = \\frac{e^{\\mu}}{\\operatorname{x^{{\\}'}}{(\\mu)}} and \\cos{(1)} = \\cos{(\\frac{e^{\\mu}}{\\operatorname{x^{{\\}'}}{(\\mu)}})} and \\operatorname{x^{{\\}'}}{(\\mu)} - \\cos{(\\frac{e^{\\mu}}{\\operatorname{x^{{\\}'}}{(\\mu)}})} - 1 = e^{\\mu} - \\cos{(\\frac{e^{\\mu}}{\\operatorname{x^{{\\}'}}{(\\mu)}})} - 1 and \\operatorname{x^{{\\}'}}{(\\mu)} - 1 - \\cos{(1)} = e^{\\mu} - 1 - \\cos{(1)} and \\frac{\\operatorname{x^{{\\}'}}{(\\mu)} - 1 - \\cos{(1)}}{\\mu} = \\frac{e^{\\mu} - 1 - \\cos{(1)}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True))))"], [["cos", 2], "Equality(cos(Integer(1)), cos(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True)))))"], [["minus", 1, "Add(cos(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True)))), Integer(1))"], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True))))), Integer(-1)), Add(exp(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1)), exp(Symbol('\\\\mu', commutative=True))))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1), Mul(Integer(-1), cos(Integer(1)))), Add(exp(Symbol('\\\\mu', commutative=True)), Integer(-1), Mul(Integer(-1), cos(Integer(1)))))"], [["divide", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Function('x^\\\\prime')(Symbol('\\\\mu', commutative=True)), Integer(-1), Mul(Integer(-1), cos(Integer(1))))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(exp(Symbol('\\\\mu', commutative=True)), Integer(-1), Mul(Integer(-1), cos(Integer(1))))))"]]}, {"prompt": "Given H{(v_{z})} = \\log{(v_{z})}, then obtain - \\log{(v_{z})}^{v_{z}} + \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu H{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} = - \\log{(v_{z})}^{v_{z}} + \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu \\log{(v_{z})}}{\\cos{(\\mathbf{J}_f)}}", "derivation": "H{(v_{z})} = \\log{(v_{z})} and \\frac{H{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} = \\frac{\\log{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} and \\frac{\\mu H{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} = \\frac{\\mu \\log{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} and \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu H{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} = \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu \\log{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} and - \\log{(v_{z})}^{v_{z}} + \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu H{(v_{z})}}{\\cos{(\\mathbf{J}_f)}} = - \\log{(v_{z})}^{v_{z}} + \\frac{\\partial}{\\partial v_{z}} \\frac{\\mu \\log{(v_{z})}}{\\cos{(\\mathbf{J}_f)}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('H')(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Mul(log(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["divide", 2, "Pow(Symbol('\\\\mu', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('H')(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mu', commutative=True), log(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mu', commutative=True), Function('H')(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu', commutative=True), log(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["minus", 4, "Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Derivative(Mul(Symbol('\\\\mu', commutative=True), Function('H')(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Derivative(Mul(Symbol('\\\\mu', commutative=True), log(Symbol('v_z', commutative=True)), Pow(cos(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)}, then derive \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)}, then obtain - \\frac{\\sin{(\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{\\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)}}{\\mathbf{J}_M}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)} and \\frac{\\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{\\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)}}{\\mathbf{J}_M} and \\operatorname{L_{\\varepsilon}}{(\\mathbf{J}_M)} = - \\sin{(\\mathbf{J}_M)} and - \\frac{\\sin{(\\mathbf{J}_M)}}{\\mathbf{J}_M} = \\frac{\\frac{d}{d \\mathbf{J}_M} \\cos{(\\mathbf{J}_M)}}{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Derivative(cos(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\sigma_x)} = \\cos{(\\cos{(\\sigma_x)})}, then obtain \\int \\tilde{g}^*^{4}{(\\sigma_x)} d\\sigma_x = \\int \\tilde{g}^*^{2}{(\\sigma_x)} \\cos^{2}{(\\cos{(\\sigma_x)})} d\\sigma_x", "derivation": "\\tilde{g}^*{(\\sigma_x)} = \\cos{(\\cos{(\\sigma_x)})} and \\tilde{g}^*^{2}{(\\sigma_x)} = \\tilde{g}^*{(\\sigma_x)} \\cos{(\\cos{(\\sigma_x)})} and \\tilde{g}^*^{4}{(\\sigma_x)} = \\tilde{g}^*^{2}{(\\sigma_x)} \\cos^{2}{(\\cos{(\\sigma_x)})} and \\int \\tilde{g}^*^{4}{(\\sigma_x)} d\\sigma_x = \\int \\tilde{g}^*^{2}{(\\sigma_x)} \\cos^{2}{(\\cos{(\\sigma_x)})} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), cos(cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Pow(cos(cos(Symbol('\\\\sigma_x', commutative=True))), Integer(2))))"], [["integrate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(4)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Pow(cos(cos(Symbol('\\\\sigma_x', commutative=True))), Integer(2))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given r{(W,n)} = n \\cos{(W)} and \\pi{(W,n)} = \\frac{W \\frac{\\partial}{\\partial W} n \\cos{(W)}}{\\frac{\\partial}{\\partial W} r{(W,n)}}, then obtain \\int \\frac{\\partial}{\\partial W} r{(W,n)} d\\pi{(W,n)} = \\int \\frac{\\partial}{\\partial W} n \\cos{(W)} d\\pi{(W,n)}", "derivation": "r{(W,n)} = n \\cos{(W)} and \\frac{\\partial}{\\partial W} r{(W,n)} = \\frac{\\partial}{\\partial W} n \\cos{(W)} and W \\frac{\\partial}{\\partial W} r{(W,n)} = W \\frac{\\partial}{\\partial W} n \\cos{(W)} and \\int \\frac{\\partial}{\\partial W} r{(W,n)} dW = \\int \\frac{\\partial}{\\partial W} n \\cos{(W)} dW and \\pi{(W,n)} = \\frac{W \\frac{\\partial}{\\partial W} n \\cos{(W)}}{\\frac{\\partial}{\\partial W} r{(W,n)}} and \\pi{(W,n)} = W and \\int \\frac{\\partial}{\\partial W} r{(W,n)} d\\pi{(W,n)} = \\int \\frac{\\partial}{\\partial W} n \\cos{(W)} d\\pi{(W,n)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["times", 2, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Derivative(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Symbol('W', commutative=True), Derivative(Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Derivative(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))), Integral(Derivative(Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('W', commutative=True), Derivative(Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Pow(Derivative(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\pi')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Symbol('W', commutative=True))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integral(Derivative(Function('r')(Symbol('W', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Function('\\\\pi')(Symbol('W', commutative=True), Symbol('n', commutative=True)))), Integral(Derivative(Mul(Symbol('n', commutative=True), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Function('\\\\pi')(Symbol('W', commutative=True), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given B{(M,P_{e})} = \\log{(M P_{e})}, then derive \\int \\frac{B{(M,P_{e})}}{\\log{(M P_{e})}} dP_{e} = L_{\\varepsilon} + P_{e}, then obtain \\int 1 dP_{e} = L_{\\varepsilon} + P_{e}", "derivation": "B{(M,P_{e})} = \\log{(M P_{e})} and \\frac{B{(M,P_{e})}}{\\log{(M P_{e})}} = 1 and \\int \\frac{B{(M,P_{e})}}{\\log{(M P_{e})}} dP_{e} = \\int 1 dP_{e} and \\int \\frac{B{(M,P_{e})}}{\\log{(M P_{e})}} dP_{e} = L_{\\varepsilon} + P_{e} and \\int 1 dP_{e} = L_{\\varepsilon} + P_{e}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), log(Mul(Symbol('M', commutative=True), Symbol('P_e', commutative=True))))"], [["divide", 1, "log(Mul(Symbol('M', commutative=True), Symbol('P_e', commutative=True)))"], "Equality(Mul(Function('B')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Pow(log(Mul(Symbol('M', commutative=True), Symbol('P_e', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('P_e', commutative=True)"], "Equality(Integral(Mul(Function('B')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Pow(log(Mul(Symbol('M', commutative=True), Symbol('P_e', commutative=True))), Integer(-1))), Tuple(Symbol('P_e', commutative=True))), Integral(Integer(1), Tuple(Symbol('P_e', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('B')(Symbol('M', commutative=True), Symbol('P_e', commutative=True)), Pow(log(Mul(Symbol('M', commutative=True), Symbol('P_e', commutative=True))), Integer(-1))), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('P_e', commutative=True))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then obtain \\operatorname{F_{c}}{(f_{\\mathbf{v}})} \\int \\operatorname{F_{c}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = e^{f_{\\mathbf{v}}} \\int \\operatorname{F_{c}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}}", "derivation": "\\operatorname{F_{c}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\int \\operatorname{F_{c}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\operatorname{F_{c}}{(f_{\\mathbf{v}})} \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} = e^{f_{\\mathbf{v}}} \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\operatorname{F_{c}}{(f_{\\mathbf{v}})} \\int \\operatorname{F_{c}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = e^{f_{\\mathbf{v}}} \\int \\operatorname{F_{c}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Mul(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Function('F_c')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(Q)} = \\sin{(Q)}, then derive \\int \\operatorname{V_{\\mathbf{E}}}{(Q)} dQ = \\hat{p}_0 - \\cos{(Q)}, then obtain \\int \\sin{(Q)} dQ = \\hat{p}_0 - \\cos{(Q)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(Q)} = \\sin{(Q)} and \\int \\operatorname{V_{\\mathbf{E}}}{(Q)} dQ = \\int \\sin{(Q)} dQ and \\int \\operatorname{V_{\\mathbf{E}}}{(Q)} dQ = \\hat{p}_0 - \\cos{(Q)} and \\int \\sin{(Q)} dQ = \\hat{p}_0 - \\cos{(Q)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given I{(L_{\\varepsilon},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{F}^{L_{\\varepsilon}}, then derive I^{\\mathbf{F}}{(L_{\\varepsilon},\\mathbf{F})} = (\\frac{L_{\\varepsilon} \\mathbf{F}^{L_{\\varepsilon}}}{\\mathbf{F}})^{\\mathbf{F}}, then obtain (\\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{F}^{L_{\\varepsilon}})^{\\mathbf{F}} = (\\frac{L_{\\varepsilon} \\mathbf{F}^{L_{\\varepsilon}}}{\\mathbf{F}})^{\\mathbf{F}}", "derivation": "I{(L_{\\varepsilon},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{F}^{L_{\\varepsilon}} and I^{\\mathbf{F}}{(L_{\\varepsilon},\\mathbf{F})} = (\\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{F}^{L_{\\varepsilon}})^{\\mathbf{F}} and I^{\\mathbf{F}}{(L_{\\varepsilon},\\mathbf{F})} = (\\frac{L_{\\varepsilon} \\mathbf{F}^{L_{\\varepsilon}}}{\\mathbf{F}})^{\\mathbf{F}} and (\\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{F}^{L_{\\varepsilon}})^{\\mathbf{F}} = (\\frac{L_{\\varepsilon} \\mathbf{F}^{L_{\\varepsilon}}}{\\mathbf{F}})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given y{(h,\\rho_f)} = h + \\log{(\\rho_f)}, then derive \\int y{(h,\\rho_f)} dh = C_{1} + \\frac{h^{2}}{2} + h \\log{(\\rho_f)}, then obtain \\int (h + \\log{(\\rho_f)}) dh = C_{1} + \\frac{h^{2}}{2} + h \\log{(\\rho_f)}", "derivation": "y{(h,\\rho_f)} = h + \\log{(\\rho_f)} and \\int y{(h,\\rho_f)} dh = \\int (h + \\log{(\\rho_f)}) dh and \\int y{(h,\\rho_f)} dh = C_{1} + \\frac{h^{2}}{2} + h \\log{(\\rho_f)} and \\int (h + \\log{(\\rho_f)}) dh = C_{1} + \\frac{h^{2}}{2} + h \\log{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('h', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('h', commutative=True), log(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('y')(Symbol('h', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Symbol('h', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('h', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), log(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('h', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('h', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Symbol('h', commutative=True), log(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(V,t_{2})} = \\frac{\\partial}{\\partial V} (V + t_{2}) and \\mathbf{J}_f{(V,t_{2})} = \\mathbf{s}{(V,t_{2})} - 1, then derive \\mathbf{J}_f{(V,t_{2})} = 0, then obtain - V + \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial V} (V + t_{2}) - 1) = - V + \\frac{d}{d t_{2}} 0", "derivation": "\\mathbf{s}{(V,t_{2})} = \\frac{\\partial}{\\partial V} (V + t_{2}) and \\mathbf{s}{(V,t_{2})} - 1 = \\frac{\\partial}{\\partial V} (V + t_{2}) - 1 and \\mathbf{J}_f{(V,t_{2})} = \\mathbf{s}{(V,t_{2})} - 1 and \\mathbf{J}_f{(V,t_{2})} = \\frac{\\partial}{\\partial V} (V + t_{2}) - 1 and \\mathbf{J}_f{(V,t_{2})} = 0 and \\frac{\\partial}{\\partial V} (V + t_{2}) - 1 = 0 and \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial V} (V + t_{2}) - 1) = \\frac{d}{d t_{2}} 0 and - V + \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial V} (V + t_{2}) - 1) = - V + \\frac{d}{d t_{2}} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Add(Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Add(Function('\\\\mathbf{s}')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Add(Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["differentiate", 6, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["add", 7, "Mul(Integer(-1), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Add(Derivative(Add(Symbol('V', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Derivative(Integer(0), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{E},f_{E})} = \\frac{f_{E}}{\\mathbf{E}}, then obtain \\int (\\operatorname{A_{y}}{(\\mathbf{E},f_{E})} - 1)^{f_{E}} d\\mathbf{E} = \\int (-1 + \\frac{f_{E}}{\\mathbf{E}})^{f_{E}} d\\mathbf{E}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{E},f_{E})} = \\frac{f_{E}}{\\mathbf{E}} and \\operatorname{A_{y}}{(\\mathbf{E},f_{E})} - 1 = -1 + \\frac{f_{E}}{\\mathbf{E}} and (\\operatorname{A_{y}}{(\\mathbf{E},f_{E})} - 1)^{f_{E}} = (-1 + \\frac{f_{E}}{\\mathbf{E}})^{f_{E}} and \\int (\\operatorname{A_{y}}{(\\mathbf{E},f_{E})} - 1)^{f_{E}} d\\mathbf{E} = \\int (-1 + \\frac{f_{E}}{\\mathbf{E}})^{f_{E}} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Symbol('f_E', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Pow(Add(Function('A_y')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Pow(Add(Integer(-1), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} = (\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}}, then obtain \\mathbf{F} (\\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} - 1 - \\frac{t_{2}}{\\mathbf{F}}) = \\mathbf{F} ((\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}} - 1 - \\frac{t_{2}}{\\mathbf{F}})", "derivation": "\\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} = (\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}} and \\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} - \\frac{t_{2}}{\\mathbf{F}} = (\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}} - \\frac{t_{2}}{\\mathbf{F}} and \\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} - 1 - \\frac{t_{2}}{\\mathbf{F}} = (\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}} - 1 - \\frac{t_{2}}{\\mathbf{F}} and \\mathbf{F} (\\tilde{g}^*{(t_{2},y^{\\prime},\\mathbf{F})} - 1 - \\frac{t_{2}}{\\mathbf{F}}) = \\mathbf{F} ((\\frac{t_{2}}{\\mathbf{F}})^{y^{\\prime}} - 1 - \\frac{t_{2}}{\\mathbf{F}})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True))))"], [["times", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Add(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Add(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given m{(A_{x})} = A_{x}, then derive \\sigma_x + \\frac{m^{2}{(A_{x})}}{2} = \\int A_{x} dm{(A_{x})}, then obtain \\sigma_x - \\frac{m^{2}{(A_{x})}}{2} = - m^{2}{(A_{x})} + \\int A_{x} dm{(A_{x})}", "derivation": "m{(A_{x})} = A_{x} and \\int m{(A_{x})} dA_{x} = \\int A_{x} dA_{x} and \\int m{(A_{x})} dm{(A_{x})} = \\int A_{x} dm{(A_{x})} and \\sigma_x + \\frac{m^{2}{(A_{x})}}{2} = \\int A_{x} dm{(A_{x})} and \\sigma_x - \\frac{m^{2}{(A_{x})}}{2} = - m^{2}{(A_{x})} + \\int A_{x} dm{(A_{x})}", "srepr_derivation": [["renaming_premise", "Equality(Function('m')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('m')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Symbol('A_x', commutative=True), Tuple(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('m')(Symbol('A_x', commutative=True)), Tuple(Function('m')(Symbol('A_x', commutative=True)))), Integral(Symbol('A_x', commutative=True), Tuple(Function('m')(Symbol('A_x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Rational(1, 2), Pow(Function('m')(Symbol('A_x', commutative=True)), Integer(2)))), Integral(Symbol('A_x', commutative=True), Tuple(Function('m')(Symbol('A_x', commutative=True)))))"], [["minus", 4, "Pow(Function('m')(Symbol('A_x', commutative=True)), Integer(2))"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Function('m')(Symbol('A_x', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(Function('m')(Symbol('A_x', commutative=True)), Integer(2))), Integral(Symbol('A_x', commutative=True), Tuple(Function('m')(Symbol('A_x', commutative=True))))))"]]}, {"prompt": "Given \\mathbb{I}{(P_{e},Z)} = e^{P_{e} + Z} and \\operatorname{v_{2}}{(P_{e},Z)} = e^{P_{e} + Z}, then obtain 0 = - \\operatorname{v_{2}}^{Z}{(P_{e},Z)} + (e^{P_{e} + Z})^{Z}", "derivation": "\\mathbb{I}{(P_{e},Z)} = e^{P_{e} + Z} and \\mathbb{I}^{Z}{(P_{e},Z)} = (e^{P_{e} + Z})^{Z} and 0 = - \\mathbb{I}^{Z}{(P_{e},Z)} + (e^{P_{e} + Z})^{Z} and \\operatorname{v_{2}}{(P_{e},Z)} = e^{P_{e} + Z} and \\operatorname{v_{2}}{(P_{e},Z)} = \\mathbb{I}{(P_{e},Z)} and 0 = - \\operatorname{v_{2}}^{Z}{(P_{e},Z)} + (e^{P_{e} + Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_e', commutative=True), Symbol('Z', commutative=True))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(exp(Add(Symbol('P_e', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mathbb{I}')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(exp(Add(Symbol('P_e', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('P_e', commutative=True), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('v_2')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), Function('\\\\mathbb{I}')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('v_2')(Symbol('P_e', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Pow(exp(Add(Symbol('P_e', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(a,f^{\\prime})} = \\log{(\\frac{a}{f^{\\prime}})} and \\operatorname{f_{\\mathbf{p}}}{(a,f^{\\prime})} = \\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime}, then obtain \\cos{(\\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime})} = \\cos{(\\operatorname{f_{\\mathbf{p}}}{(a,f^{\\prime})})}", "derivation": "\\operatorname{n_{1}}{(a,f^{\\prime})} = \\log{(\\frac{a}{f^{\\prime}})} and \\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime} = \\int \\log{(\\frac{a}{f^{\\prime}})} df^{\\prime} and \\operatorname{f_{\\mathbf{p}}}{(a,f^{\\prime})} = \\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime} and \\operatorname{f_{\\mathbf{p}}}{(a,f^{\\prime})} = \\int \\log{(\\frac{a}{f^{\\prime}})} df^{\\prime} and \\cos{(\\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime})} = \\cos{(\\int \\log{(\\frac{a}{f^{\\prime}})} df^{\\prime})} and \\cos{(\\int \\operatorname{n_{1}}{(a,f^{\\prime})} df^{\\prime})} = \\cos{(\\operatorname{f_{\\mathbf{p}}}{(a,f^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), log(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integral(Function('n_1')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integral(log(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('n_1')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), cos(Integral(log(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(cos(Integral(Function('n_1')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), cos(Function('f_{\\\\mathbf{p}}')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(n_{2},z^{*},u)} = \\frac{n_{2}^{z^{*}}}{u}, then derive \\frac{\\partial}{\\partial u} \\hat{p}_0{(n_{2},z^{*},u)} = - \\frac{n_{2}^{z^{*}}}{u^{2}}, then obtain \\int (- \\frac{n_{2}^{z^{*}}}{u^{2}})^{u} dn_{2} = \\int (\\frac{\\partial}{\\partial u} \\frac{n_{2}^{z^{*}}}{u})^{u} dn_{2}", "derivation": "\\hat{p}_0{(n_{2},z^{*},u)} = \\frac{n_{2}^{z^{*}}}{u} and \\frac{\\partial}{\\partial u} \\hat{p}_0{(n_{2},z^{*},u)} = \\frac{\\partial}{\\partial u} \\frac{n_{2}^{z^{*}}}{u} and \\frac{\\partial}{\\partial u} \\hat{p}_0{(n_{2},z^{*},u)} = - \\frac{n_{2}^{z^{*}}}{u^{2}} and (\\frac{\\partial}{\\partial u} \\hat{p}_0{(n_{2},z^{*},u)})^{u} = (\\frac{\\partial}{\\partial u} \\frac{n_{2}^{z^{*}}}{u})^{u} and (- \\frac{n_{2}^{z^{*}}}{u^{2}})^{u} = (\\frac{\\partial}{\\partial u} \\frac{n_{2}^{z^{*}}}{u})^{u} and \\int (- \\frac{n_{2}^{z^{*}}}{u^{2}})^{u} dn_{2} = \\int (\\frac{\\partial}{\\partial u} \\frac{n_{2}^{z^{*}}}{u})^{u} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-2))))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{p}_0')(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-2))), Symbol('u', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)))"], [["integrate", 5, "Symbol('n_2', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-2))), Symbol('u', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Pow(Derivative(Mul(Pow(Symbol('n_2', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(v_{x},y)} = e^{y^{v_{x}}}, then obtain - \\operatorname{F_{c}}^{2}{(v_{x},y)} + \\operatorname{F_{c}}{(v_{x},y)} = - \\operatorname{F_{c}}^{2}{(v_{x},y)} + e^{y^{v_{x}}}", "derivation": "\\operatorname{F_{c}}{(v_{x},y)} = e^{y^{v_{x}}} and \\operatorname{F_{c}}^{2}{(v_{x},y)} = \\operatorname{F_{c}}{(v_{x},y)} e^{y^{v_{x}}} and - \\operatorname{F_{c}}{(v_{x},y)} e^{y^{v_{x}}} + \\operatorname{F_{c}}{(v_{x},y)} = - \\operatorname{F_{c}}{(v_{x},y)} e^{y^{v_{x}}} + e^{y^{v_{x}}} and - \\operatorname{F_{c}}^{2}{(v_{x},y)} + \\operatorname{F_{c}}{(v_{x},y)} = - \\operatorname{F_{c}}^{2}{(v_{x},y)} + e^{y^{v_{x}}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True))))"], [["times", 1, "Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))"], "Equality(Pow(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Integer(2)), Mul(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True)))))"], [["minus", 1, "Mul(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True)))), Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True)))), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Integer(2))), Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Pow(Function('F_c')(Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Integer(2))), exp(Pow(Symbol('y', commutative=True), Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given y{(C_{d},L)} = L^{C_{d}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P \\sin{(L^{C_{d}} - y{(C_{d},L)})} = \\frac{d}{d \\mathbf{J}_P} 0", "derivation": "y{(C_{d},L)} = L^{C_{d}} and - L^{C_{d}} + y{(C_{d},L)} = 0 and - \\sin{(L^{C_{d}} - y{(C_{d},L)})} = 0 and \\mathbf{J}_P \\sin{(L^{C_{d}} - y{(C_{d},L)})} = 0 and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P \\sin{(L^{C_{d}} - y{(C_{d},L)})} = \\frac{d}{d \\mathbf{J}_P} 0", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True)))"], [["minus", 1, "Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True))), Function('y')(Symbol('C_d', commutative=True), Symbol('L', commutative=True))), Integer(0))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Add(Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))))), Integer(0))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Add(Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), sin(Add(Pow(Symbol('L', commutative=True), Symbol('C_d', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('C_d', commutative=True), Symbol('L', commutative=True)))))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(P_{e})} = \\sin{(P_{e})} and \\pi{(P_{e})} = \\sin{(P_{e})}, then obtain \\frac{a^{2}{(P_{e})}}{P_{e}} = \\frac{\\pi{(P_{e})} a{(P_{e})}}{P_{e}}", "derivation": "a{(P_{e})} = \\sin{(P_{e})} and a^{2}{(P_{e})} = a{(P_{e})} \\sin{(P_{e})} and \\pi{(P_{e})} = \\sin{(P_{e})} and a^{2}{(P_{e})} = \\pi{(P_{e})} a{(P_{e})} and \\frac{a^{2}{(P_{e})}}{P_{e}} = \\frac{\\pi{(P_{e})} a{(P_{e})}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["times", 1, "Function('a')(Symbol('P_e', commutative=True))"], "Equality(Pow(Function('a')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('a')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('P_e', commutative=True)), sin(Symbol('P_e', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('a')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('\\\\pi')(Symbol('P_e', commutative=True)), Function('a')(Symbol('P_e', commutative=True))))"], [["divide", 4, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Pow(Function('a')(Symbol('P_e', commutative=True)), Integer(2))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('P_e', commutative=True)), Function('a')(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\delta{(\\mathbf{M})} = \\mathbf{M}, then obtain (\\mathbf{M} + \\delta{(\\mathbf{M})})^{2} \\delta{(\\mathbf{M})} = \\mathbf{M} (\\mathbf{M} + \\delta{(\\mathbf{M})})^{2}", "derivation": "\\delta{(\\mathbf{M})} = \\mathbf{M} and 2 \\delta{(\\mathbf{M})} = \\mathbf{M} + \\delta{(\\mathbf{M})} and 2 \\delta^{2}{(\\mathbf{M})} = 2 \\mathbf{M} \\delta{(\\mathbf{M})} and (\\mathbf{M} + \\delta{(\\mathbf{M})}) \\delta{(\\mathbf{M})} = \\mathbf{M} (\\mathbf{M} + \\delta{(\\mathbf{M})}) and (\\mathbf{M} + \\delta{(\\mathbf{M})})^{2} \\delta{(\\mathbf{M})} = \\mathbf{M} (\\mathbf{M} + \\delta{(\\mathbf{M})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], [["add", 1, "Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 1, "Mul(Integer(2), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["times", 4, "Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2)), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('\\\\delta')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2))))"]]}, {"prompt": "Given a{(\\eta)} = \\cos{(\\eta)}, then obtain \\int a^{\\eta}{(\\eta)} \\cos^{\\eta}{(\\eta)} d\\eta = \\int \\cos^{2 \\eta}{(\\eta)} d\\eta", "derivation": "a{(\\eta)} = \\cos{(\\eta)} and a^{\\eta}{(\\eta)} = \\cos^{\\eta}{(\\eta)} and a^{\\eta}{(\\eta)} \\cos^{\\eta}{(\\eta)} = \\cos^{2 \\eta}{(\\eta)} and \\int a^{\\eta}{(\\eta)} \\cos^{\\eta}{(\\eta)} d\\eta = \\int \\cos^{2 \\eta}{(\\eta)} d\\eta", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('a')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["times", 2, "Pow(cos(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Function('a')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Pow(cos(Symbol('\\\\eta', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Pow(Function('a')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Pow(cos(Symbol('\\\\eta', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given q{(\\rho_f)} = \\cos{(\\rho_f)}, then obtain \\frac{q^{2}{(\\rho_f)}}{\\rho_f} = \\frac{\\cos^{2}{(\\rho_f)}}{\\rho_f}", "derivation": "q{(\\rho_f)} = \\cos{(\\rho_f)} and \\frac{q{(\\rho_f)}}{\\rho_f} = \\frac{\\cos{(\\rho_f)}}{\\rho_f} and \\frac{q{(\\rho_f)} \\cos{(\\rho_f)}}{\\rho_f} = \\frac{\\cos^{2}{(\\rho_f)}}{\\rho_f} and q{(\\rho_f)} \\cos{(\\rho_f)} = \\cos^{2}{(\\rho_f)} and \\frac{q^{2}{(\\rho_f)}}{\\rho_f} = \\frac{q{(\\rho_f)} \\cos{(\\rho_f)}}{\\rho_f} and \\frac{q^{2}{(\\rho_f)}}{\\rho_f} = \\frac{\\cos^{2}{(\\rho_f)}}{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), cos(Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), cos(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(2))))"], [["divide", 3, "Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))"], "Equality(Mul(Function('q')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('q')(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('q')(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(2))))"]]}, {"prompt": "Given k{(n_{1})} = \\cos{(\\cos{(n_{1})})}, then obtain \\frac{d^{2}}{d n_{1}^{2}} 1 = \\frac{d^{2}}{d n_{1}^{2}} (\\frac{\\cos{(\\cos{(n_{1})})}}{k{(n_{1})}})^{n_{1}}", "derivation": "k{(n_{1})} = \\cos{(\\cos{(n_{1})})} and 1 = \\frac{\\cos{(\\cos{(n_{1})})}}{k{(n_{1})}} and 1 = (\\frac{\\cos{(\\cos{(n_{1})})}}{k{(n_{1})}})^{n_{1}} and \\frac{d}{d n_{1}} 1 = \\frac{d}{d n_{1}} (\\frac{\\cos{(\\cos{(n_{1})})}}{k{(n_{1})}})^{n_{1}} and \\frac{d^{2}}{d n_{1}^{2}} 1 = \\frac{d^{2}}{d n_{1}^{2}} (\\frac{\\cos{(\\cos{(n_{1})})}}{k{(n_{1})}})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('n_1', commutative=True)), cos(cos(Symbol('n_1', commutative=True))))"], [["divide", 1, "Function('k')(Symbol('n_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('k')(Symbol('n_1', commutative=True)), Integer(-1)), cos(cos(Symbol('n_1', commutative=True)))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('k')(Symbol('n_1', commutative=True)), Integer(-1)), cos(cos(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('k')(Symbol('n_1', commutative=True)), Integer(-1)), cos(cos(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('n_1', commutative=True), Integer(2))), Derivative(Pow(Mul(Pow(Function('k')(Symbol('n_1', commutative=True)), Integer(-1)), cos(cos(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\phi{(\\Psi,M_{E})} = \\frac{\\partial}{\\partial \\Psi} \\Psi^{M_{E}} and \\delta{(M_{E})} = M_{E}, then derive \\log{(\\phi{(\\Psi,M_{E})})} = \\log{(\\frac{M_{E} \\Psi^{M_{E}}}{\\Psi})}, then obtain \\int \\log{(\\frac{\\partial}{\\partial \\Psi} \\Psi^{\\delta{(M_{E})}})} d\\Psi = \\int \\log{(\\frac{\\Psi^{\\delta{(M_{E})}} \\delta{(M_{E})}}{\\Psi})} d\\Psi", "derivation": "\\phi{(\\Psi,M_{E})} = \\frac{\\partial}{\\partial \\Psi} \\Psi^{M_{E}} and \\log{(\\phi{(\\Psi,M_{E})})} = \\log{(\\frac{\\partial}{\\partial \\Psi} \\Psi^{M_{E}})} and \\log{(\\phi{(\\Psi,M_{E})})} = \\log{(\\frac{M_{E} \\Psi^{M_{E}}}{\\Psi})} and \\delta{(M_{E})} = M_{E} and \\log{(\\frac{\\partial}{\\partial \\Psi} \\Psi^{M_{E}})} = \\log{(\\frac{M_{E} \\Psi^{M_{E}}}{\\Psi})} and \\log{(\\frac{\\partial}{\\partial \\Psi} \\Psi^{\\delta{(M_{E})}})} = \\log{(\\frac{\\Psi^{\\delta{(M_{E})}} \\delta{(M_{E})}}{\\Psi})} and \\int \\log{(\\frac{\\partial}{\\partial \\Psi} \\Psi^{\\delta{(M_{E})}})} d\\Psi = \\int \\log{(\\frac{\\Psi^{\\delta{(M_{E})}} \\delta{(M_{E})}}{\\Psi})} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)), Derivative(Pow(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True))), log(Derivative(Pow(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(log(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True))), log(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(log(Derivative(Pow(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), log(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Symbol('M_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(log(Derivative(Pow(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), log(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M_E', commutative=True))), Function('\\\\delta')(Symbol('M_E', commutative=True)))))"], [["integrate", 6, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(log(Derivative(Pow(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M_E', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi', commutative=True), Function('\\\\delta')(Symbol('M_E', commutative=True))), Function('\\\\delta')(Symbol('M_E', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{J})} = \\sin{(\\mathbf{J})}, then obtain \\frac{d}{d \\mathbf{J}} \\int (- \\mathbf{J} + \\Psi_{\\lambda}{(\\mathbf{J})}) d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\int (- \\mathbf{J} + \\sin{(\\mathbf{J})}) d\\mathbf{J}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and - \\mathbf{J} + \\Psi_{\\lambda}{(\\mathbf{J})} = - \\mathbf{J} + \\sin{(\\mathbf{J})} and \\int (- \\mathbf{J} + \\Psi_{\\lambda}{(\\mathbf{J})}) d\\mathbf{J} = \\int (- \\mathbf{J} + \\sin{(\\mathbf{J})}) d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\int (- \\mathbf{J} + \\Psi_{\\lambda}{(\\mathbf{J})}) d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\int (- \\mathbf{J} + \\sin{(\\mathbf{J})}) d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(f^{*})} = \\sin{(f^{*})}, then obtain - I^{f^{*}}{(f^{*})} - 2 = - I^{f^{*}}{(f^{*})} - 1 - \\frac{\\sin{(f^{*})}}{I{(f^{*})}}", "derivation": "I{(f^{*})} = \\sin{(f^{*})} and 1 = \\frac{\\sin{(f^{*})}}{I{(f^{*})}} and 2 = 1 + \\frac{\\sin{(f^{*})}}{I{(f^{*})}} and I^{f^{*}}{(f^{*})} + 2 = I^{f^{*}}{(f^{*})} + 1 + \\frac{\\sin{(f^{*})}}{I{(f^{*})}} and - I^{f^{*}}{(f^{*})} - 2 = - I^{f^{*}}{(f^{*})} - 1 - \\frac{\\sin{(f^{*})}}{I{(f^{*})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["divide", 1, "Function('I')(Symbol('f^*', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('I')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Symbol('f^*', commutative=True))))"], [["add", 2, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('I')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Symbol('f^*', commutative=True)))))"], [["add", 3, "Pow(Function('I')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Add(Pow(Function('I')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Integer(2)), Add(Pow(Function('I')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Integer(1), Mul(Pow(Function('I')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Symbol('f^*', commutative=True)))))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('I')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integer(-2)), Add(Mul(Integer(-1), Pow(Function('I')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integer(-1), Mul(Integer(-1), Pow(Function('I')(Symbol('f^*', commutative=True)), Integer(-1)), sin(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given u{(\\mathbf{v})} = \\sin{(\\cos{(\\mathbf{v})})} and \\Omega{(\\mathbf{v})} = \\cos{(\\mathbf{v})}, then obtain 1 = e^{- \\frac{\\sin{(\\Omega{(\\mathbf{v})})}}{\\sin{(\\cos{(\\mathbf{v})})}} + 1}", "derivation": "u{(\\mathbf{v})} = \\sin{(\\cos{(\\mathbf{v})})} and \\frac{u{(\\mathbf{v})}}{\\sin{(\\cos{(\\mathbf{v})})}} = 1 and 0 = - \\frac{u{(\\mathbf{v})}}{\\sin{(\\cos{(\\mathbf{v})})}} + 1 and \\Omega{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and u{(\\mathbf{v})} = \\sin{(\\Omega{(\\mathbf{v})})} and 0 = - \\frac{\\sin{(\\Omega{(\\mathbf{v})})}}{\\sin{(\\cos{(\\mathbf{v})})}} + 1 and 1 = e^{- \\frac{\\sin{(\\Omega{(\\mathbf{v})})}}{\\sin{(\\cos{(\\mathbf{v})})}} + 1}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\mathbf{v}', commutative=True)), sin(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 1, "sin(cos(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Function('u')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Function('u')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('u')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('u')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Function('\\\\Omega')(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Function('\\\\Omega')(Symbol('\\\\mathbf{v}', commutative=True))), Pow(sin(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Integer(1)))"], [["exp", 6], "Equality(Integer(1), exp(Add(Mul(Integer(-1), sin(Function('\\\\Omega')(Symbol('\\\\mathbf{v}', commutative=True))), Pow(sin(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{J}_f,y)} = \\frac{\\mathbf{J}_f}{y} and \\varepsilon{(\\mathbf{J}_f,y)} = - \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f and \\sigma_{x}{(\\mathbf{J}_f,y)} = - \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f, then obtain \\sigma_{x}{(\\mathbf{J}_f,y)} = \\varepsilon{(\\mathbf{J}_f,y)}", "derivation": "\\dot{y}{(\\mathbf{J}_f,y)} = \\frac{\\mathbf{J}_f}{y} and \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f = \\int \\frac{\\mathbf{J}_f}{y} d\\mathbf{J}_f and - \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f = - \\int \\frac{\\mathbf{J}_f}{y} d\\mathbf{J}_f and \\varepsilon{(\\mathbf{J}_f,y)} = - \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f and \\varepsilon{(\\mathbf{J}_f,y)} = - \\int \\frac{\\mathbf{J}_f}{y} d\\mathbf{J}_f and \\sigma_{x}{(\\mathbf{J}_f,y)} = - \\int \\dot{y}{(\\mathbf{J}_f,y)} d\\mathbf{J}_f and \\sigma_{x}{(\\mathbf{J}_f,y)} = - \\int \\frac{\\mathbf{J}_f}{y} d\\mathbf{J}_f and \\sigma_{x}{(\\mathbf{J}_f,y)} = \\varepsilon{(\\mathbf{J}_f,y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(f^{*},\\nabla)} = \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}), then obtain - \\frac{\\int \\operatorname{v_{y}}{(f^{*},\\nabla)} d\\nabla}{\\nabla (- \\nabla + f^{*})} = - \\frac{\\int \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) d\\nabla}{\\nabla (- \\nabla + f^{*})}", "derivation": "\\operatorname{v_{y}}{(f^{*},\\nabla)} = \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) and \\int \\operatorname{v_{y}}{(f^{*},\\nabla)} d\\nabla = \\int \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) d\\nabla and \\frac{\\int \\operatorname{v_{y}}{(f^{*},\\nabla)} d\\nabla}{\\nabla} = \\frac{\\int \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) d\\nabla}{\\nabla} and - \\frac{\\int \\operatorname{v_{y}}{(f^{*},\\nabla)} d\\nabla}{\\nabla} = - \\frac{\\int \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) d\\nabla}{\\nabla} and - \\frac{\\int \\operatorname{v_{y}}{(f^{*},\\nabla)} d\\nabla}{\\nabla (- \\nabla + f^{*})} = - \\frac{\\int \\frac{\\partial}{\\partial f^{*}} (- \\nabla + f^{*}) d\\nabla}{\\nabla (- \\nabla + f^{*})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('f^*', commutative=True), Symbol('\\\\nabla', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('f^*', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["divide", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Function('v_y')(Symbol('f^*', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Function('v_y')(Symbol('f^*', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('v_y')(Symbol('f^*', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(v_{z})} = \\log{(v_{z})}, then obtain \\frac{d}{d v_{z}} (\\mathbf{f}{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}}) = \\frac{d}{d v_{z}} (\\log{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}})", "derivation": "\\mathbf{f}{(v_{z})} = \\log{(v_{z})} and \\mathbf{f}^{v_{z}}{(v_{z})} = \\log{(v_{z})}^{v_{z}} and \\mathbf{f}{(v_{z})} - \\mathbf{f}^{v_{z}}{(v_{z})} = - \\mathbf{f}^{v_{z}}{(v_{z})} + \\log{(v_{z})} and \\mathbf{f}{(v_{z})} - \\log{(v_{z})}^{v_{z}} = \\log{(v_{z})} - \\log{(v_{z})}^{v_{z}} and \\mathbf{f}{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}} = \\log{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}} and \\frac{d}{d v_{z}} (\\mathbf{f}{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}}) = \\frac{d}{d v_{z}} (\\log{(v_{z})} - 2 \\log{(v_{z})}^{v_{z}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))), Add(log(Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))))"], [["minus", 4, "Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Integer(2), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))), Add(log(Symbol('v_z', commutative=True)), Mul(Integer(-1), Integer(2), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))))"], [["differentiate", 5, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{f}')(Symbol('v_z', commutative=True)), Mul(Integer(-1), Integer(2), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(log(Symbol('v_z', commutative=True)), Mul(Integer(-1), Integer(2), Pow(log(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\delta,A_{y},u)} = A_{y} u + \\delta and g{(\\delta,A_{y},u)} = \\delta + \\varepsilon{(\\delta,A_{y},u)}, then obtain - \\delta - u + (u + g{(\\delta,A_{y},u)}) (A_{y} u + 2 \\delta + u) - \\varepsilon{(\\delta,A_{y},u)} = - \\delta - u + (A_{y} u + 2 \\delta + u)^{2} - \\varepsilon{(\\delta,A_{y},u)}", "derivation": "\\varepsilon{(\\delta,A_{y},u)} = A_{y} u + \\delta and g{(\\delta,A_{y},u)} = \\delta + \\varepsilon{(\\delta,A_{y},u)} and u + g{(\\delta,A_{y},u)} = \\delta + u + \\varepsilon{(\\delta,A_{y},u)} and u + g{(\\delta,A_{y},u)} = A_{y} u + 2 \\delta + u and (u + g{(\\delta,A_{y},u)}) (\\delta + u + \\varepsilon{(\\delta,A_{y},u)}) = (\\delta + u + \\varepsilon{(\\delta,A_{y},u)}) (A_{y} u + 2 \\delta + u) and (u + g{(\\delta,A_{y},u)}) (A_{y} u + 2 \\delta + u) = (A_{y} u + 2 \\delta + u)^{2} and - \\delta - u + (u + g{(\\delta,A_{y},u)}) (A_{y} u + 2 \\delta + u) - \\varepsilon{(\\delta,A_{y},u)} = - \\delta - u + (A_{y} u + 2 \\delta + u)^{2} - \\varepsilon{(\\delta,A_{y},u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))))"], [["add", 2, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('u', commutative=True), Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)))"], [["times", 4, "Add(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Add(Symbol('u', commutative=True), Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)))), Mul(Add(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Symbol('u', commutative=True), Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True))), Pow(Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Integer(2)))"], [["minus", 6, "Add(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Add(Symbol('u', commutative=True), Function('g')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Pow(Add(Mul(Symbol('A_y', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('\\\\delta', commutative=True), Symbol('A_y', commutative=True), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given m{(\\sigma_p)} = \\sin{(\\sigma_p)}, then obtain \\cos{((\\frac{\\int m{(\\sigma_p)} d\\sigma_p}{\\int \\sin{(\\sigma_p)} d\\sigma_p})^{\\sigma_p})} = \\cos{(1)}", "derivation": "m{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\int m{(\\sigma_p)} d\\sigma_p = \\int \\sin{(\\sigma_p)} d\\sigma_p and \\frac{\\int m{(\\sigma_p)} d\\sigma_p}{\\int \\sin{(\\sigma_p)} d\\sigma_p} = 1 and (\\frac{\\int m{(\\sigma_p)} d\\sigma_p}{\\int \\sin{(\\sigma_p)} d\\sigma_p})^{\\sigma_p} = 1 and \\cos{((\\frac{\\int m{(\\sigma_p)} d\\sigma_p}{\\int \\sin{(\\sigma_p)} d\\sigma_p})^{\\sigma_p})} = \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Integral(Function('m')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Integral(Function('m')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\sigma_p', commutative=True)), Integer(1))"], [["cos", 4], "Equality(cos(Pow(Mul(Integral(Function('m')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(sin(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integer(-1))), Symbol('\\\\sigma_p', commutative=True))), cos(Integer(1)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})} = \\dot{y} + \\mathbf{A}, then obtain \\frac{d^{2}}{d \\dot{y}^{2}} 0 = \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} (\\dot{y} + \\mathbf{A} - \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})})", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})} = \\dot{y} + \\mathbf{A} and - \\dot{y} + \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})} = \\mathbf{A} and 0 = \\dot{y} + \\mathbf{A} - \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})} and \\frac{d}{d \\dot{y}} 0 = \\frac{\\partial}{\\partial \\dot{y}} (\\dot{y} + \\mathbf{A} - \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})}) and \\frac{d^{2}}{d \\dot{y}^{2}} 0 = \\frac{\\partial^{2}}{\\partial \\dot{y}^{2}} (\\dot{y} + \\mathbf{A} - \\operatorname{E_{\\lambda}}{(\\mathbf{A},\\dot{y})})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{S}{(g_{\\varepsilon})} = \\log{(\\sin{(g_{\\varepsilon})})} and u{(n_{2})} = \\frac{1}{n_{2}}, then obtain \\frac{n_{2} \\mathbf{S}{(g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{n_{2} \\log{(\\sin{(g_{\\varepsilon})})}}{g_{\\varepsilon}}", "derivation": "\\mathbf{S}{(g_{\\varepsilon})} = \\log{(\\sin{(g_{\\varepsilon})})} and u{(n_{2})} = \\frac{1}{n_{2}} and \\frac{\\mathbf{S}{(g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{\\log{(\\sin{(g_{\\varepsilon})})}}{g_{\\varepsilon}} and \\frac{\\mathbf{S}{(g_{\\varepsilon})}}{g_{\\varepsilon} u{(n_{2})}} = \\frac{\\log{(\\sin{(g_{\\varepsilon})})}}{g_{\\varepsilon} u{(n_{2})}} and \\frac{n_{2} \\mathbf{S}{(g_{\\varepsilon})}}{g_{\\varepsilon}} = \\frac{n_{2} \\log{(\\sin{(g_{\\varepsilon})})}}{g_{\\varepsilon}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('n_2', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1)))"], [["divide", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["divide", 3, "Function('u')(Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Function('u')(Symbol('n_2', commutative=True)), Integer(-1))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('u')(Symbol('n_2', commutative=True)), Integer(-1)), log(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), Function('\\\\mathbf{S}')(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True), log(sin(Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})} = \\hat{x} b, then obtain b (\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})})^{2} = b \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} b \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})} = \\hat{x} b and \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} b and (\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})})^{2} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} b \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})} and b (\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})})^{2} = b \\frac{\\partial}{\\partial \\hat{x}} \\hat{x} b \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{f_{\\mathbf{v}}}{(b,\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(2))), Mul(Symbol('b', commutative=True), Derivative(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(E)} = \\cos{(E)}, then obtain \\frac{\\int (- E + \\operatorname{A_{2}}{(E)}) dE}{- \\operatorname{A_{2}}{(E)} + \\cos{(E)}} = \\frac{\\int (- E + \\cos{(E)}) dE}{- \\operatorname{A_{2}}{(E)} + \\cos{(E)}}", "derivation": "\\operatorname{A_{2}}{(E)} = \\cos{(E)} and 0 = - \\operatorname{A_{2}}{(E)} + \\cos{(E)} and - E + \\operatorname{A_{2}}{(E)} = - E + \\cos{(E)} and \\int (- E + \\operatorname{A_{2}}{(E)}) dE = \\int (- E + \\cos{(E)}) dE and - \\operatorname{A_{2}}{(E)} + \\cos{(E)} = - 2 \\operatorname{A_{2}}{(E)} + 2 \\cos{(E)} and \\frac{\\int (- E + \\operatorname{A_{2}}{(E)}) dE}{- 2 \\operatorname{A_{2}}{(E)} + 2 \\cos{(E)}} = \\frac{\\int (- E + \\cos{(E)}) dE}{- 2 \\operatorname{A_{2}}{(E)} + 2 \\cos{(E)}} and \\frac{\\int (- E + \\operatorname{A_{2}}{(E)}) dE}{- \\operatorname{A_{2}}{(E)} + \\cos{(E)}} = \\frac{\\int (- E + \\cos{(E)}) dE}{- \\operatorname{A_{2}}{(E)} + \\cos{(E)}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["minus", 1, "Function('A_2')(Symbol('E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_2')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))))"], [["minus", 1, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('A_2')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))))"], [["integrate", 3, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('A_2')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('A_2')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('A_2')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('E', commutative=True))), Mul(Integer(2), cos(Symbol('E', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('E', commutative=True))), Mul(Integer(2), cos(Symbol('E', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('E', commutative=True))), Mul(Integer(2), cos(Symbol('E', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('A_2')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('E', commutative=True))), Mul(Integer(2), cos(Symbol('E', commutative=True)))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('A_2')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('A_2')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('A_2')(Symbol('E', commutative=True))), cos(Symbol('E', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), cos(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f^{\\prime})} = \\int \\sin{(f^{\\prime})} df^{\\prime}, then obtain \\int - \\frac{f^{\\prime} + \\operatorname{F_{H}}{(f^{\\prime})} - \\int \\sin{(f^{\\prime})} df^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} df^{\\prime} = \\int - \\frac{f^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} df^{\\prime}", "derivation": "\\operatorname{F_{H}}{(f^{\\prime})} = \\int \\sin{(f^{\\prime})} df^{\\prime} and f^{\\prime} + \\operatorname{F_{H}}{(f^{\\prime})} = f^{\\prime} + \\int \\sin{(f^{\\prime})} df^{\\prime} and f^{\\prime} + \\operatorname{F_{H}}{(f^{\\prime})} - \\int \\sin{(f^{\\prime})} df^{\\prime} = f^{\\prime} and - \\frac{f^{\\prime} + \\operatorname{F_{H}}{(f^{\\prime})} - \\int \\sin{(f^{\\prime})} df^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} = - \\frac{f^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} and \\int - \\frac{f^{\\prime} + \\operatorname{F_{H}}{(f^{\\prime})} - \\int \\sin{(f^{\\prime})} df^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} df^{\\prime} = \\int - \\frac{f^{\\prime}}{\\int \\sin{(f^{\\prime})} df^{\\prime}} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f^{\\\\prime}', commutative=True)), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('F_H')(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 2, "Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('F_H')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))), Symbol('f^{\\\\prime}', commutative=True))"], [["divide", 3, "Mul(Integer(-1), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Symbol('f^{\\\\prime}', commutative=True), Function('F_H')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Add(Symbol('f^{\\\\prime}', commutative=True), Function('F_H')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given Z{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\log{(\\hat{X})}, then derive Z{(\\hat{X})} = \\frac{1}{\\hat{X}}, then obtain \\frac{d}{d \\hat{X}} \\frac{1}{\\hat{X}} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})}", "derivation": "Z{(\\hat{X})} = \\frac{d}{d \\hat{X}} \\log{(\\hat{X})} and Z{(\\hat{X})} = \\frac{1}{\\hat{X}} and \\frac{d}{d \\hat{X}} Z{(\\hat{X})} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})} and \\frac{d}{d \\hat{X}} \\frac{1}{\\hat{X}} = \\frac{d^{2}}{d \\hat{X}^{2}} \\log{(\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\hat{X}', commutative=True)), Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Z')(Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)))"], [["differentiate", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\ddot{x}{(m,A_{y})} = A_{y} + e^{m}, then obtain - m (\\ddot{x}{(m,A_{y})} e^{- m})^{A_{y}} = - m ((A_{y} + e^{m}) e^{- m})^{A_{y}}", "derivation": "\\ddot{x}{(m,A_{y})} = A_{y} + e^{m} and \\ddot{x}{(m,A_{y})} e^{- m} = (A_{y} + e^{m}) e^{- m} and (\\ddot{x}{(m,A_{y})} e^{- m})^{A_{y}} = ((A_{y} + e^{m}) e^{- m})^{A_{y}} and - m (\\ddot{x}{(m,A_{y})} e^{- m})^{A_{y}} = - m ((A_{y} + e^{m}) e^{- m})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('m', commutative=True), Symbol('A_y', commutative=True)), Add(Symbol('A_y', commutative=True), exp(Symbol('m', commutative=True))))"], [["divide", 1, "exp(Symbol('m', commutative=True))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('m', commutative=True), Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Mul(Add(Symbol('A_y', commutative=True), exp(Symbol('m', commutative=True))), exp(Mul(Integer(-1), Symbol('m', commutative=True)))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(Mul(Function('\\\\ddot{x}')(Symbol('m', commutative=True), Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('A_y', commutative=True)), Pow(Mul(Add(Symbol('A_y', commutative=True), exp(Symbol('m', commutative=True))), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('A_y', commutative=True)))"], [["times", 3, "Mul(Integer(-1), Symbol('m', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('m', commutative=True), Pow(Mul(Function('\\\\ddot{x}')(Symbol('m', commutative=True), Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('A_y', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True), Pow(Mul(Add(Symbol('A_y', commutative=True), exp(Symbol('m', commutative=True))), exp(Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(p)} = \\cos{(p)}, then derive p (\\varphi^* + \\sin{(p)} + \\int \\operatorname{A_{x}}{(p)} dp) = 2 p (\\varphi^* + \\sin{(p)}), then obtain p (\\varphi^* + \\sin{(p)} + \\int \\cos{(p)} dp) = 2 p (\\varphi^* + \\sin{(p)})", "derivation": "\\operatorname{A_{x}}{(p)} = \\cos{(p)} and \\int \\operatorname{A_{x}}{(p)} dp = \\int \\cos{(p)} dp and \\int \\operatorname{A_{x}}{(p)} dp + \\int \\cos{(p)} dp = 2 \\int \\cos{(p)} dp and p (\\int \\operatorname{A_{x}}{(p)} dp + \\int \\cos{(p)} dp) = 2 p \\int \\cos{(p)} dp and p (\\varphi^* + \\sin{(p)} + \\int \\operatorname{A_{x}}{(p)} dp) = 2 p (\\varphi^* + \\sin{(p)}) and p (\\varphi^* + \\sin{(p)} + \\int \\cos{(p)} dp) = 2 p (\\varphi^* + \\sin{(p)})", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Integral(Function('A_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["times", 3, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Add(Integral(Function('A_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Integer(2), Symbol('p', commutative=True), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('p', commutative=True)), Integral(Function('A_x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Integer(2), Symbol('p', commutative=True), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('p', commutative=True), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('p', commutative=True)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Integer(2), Symbol('p', commutative=True), Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(T,v_{2})} = \\cos{(v_{2}^{T})} and x{(T,v_{2})} = \\ddot{x}{(T,v_{2})} - \\cos{(v_{2}^{T})}, then obtain \\frac{\\partial}{\\partial T} - \\frac{x{(T,v_{2})}}{\\ddot{x}{(T,v_{2})}} = \\frac{d}{d T} 0", "derivation": "\\ddot{x}{(T,v_{2})} = \\cos{(v_{2}^{T})} and \\ddot{x}{(T,v_{2})} - \\cos{(v_{2}^{T})} = 0 and - \\frac{\\ddot{x}{(T,v_{2})} - \\cos{(v_{2}^{T})}}{\\cos{(v_{2}^{T})}} = 0 and \\frac{\\partial}{\\partial T} - \\frac{\\ddot{x}{(T,v_{2})} - \\cos{(v_{2}^{T})}}{\\cos{(v_{2}^{T})}} = \\frac{d}{d T} 0 and x{(T,v_{2})} = \\ddot{x}{(T,v_{2})} - \\cos{(v_{2}^{T})} and \\frac{\\partial}{\\partial T} - \\frac{x{(T,v_{2})}}{\\cos{(v_{2}^{T})}} = \\frac{d}{d T} 0 and \\frac{\\partial}{\\partial T} - \\frac{x{(T,v_{2})}}{\\ddot{x}{(T,v_{2})}} = \\frac{d}{d T} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))"], [["minus", 1, "cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))), Pow(cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Integer(-1))), Integer(0))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))), Pow(cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('x')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Add(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Integer(-1), Function('x')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Pow(cos(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Integer(-1), Pow(Function('\\\\ddot{x}')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Integer(-1)), Function('x')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(k)} = e^{k} and \\operatorname{C_{2}}{(k)} = e^{k}, then obtain \\int 0 dk = \\int (- \\operatorname{C_{2}}{(k)} + e^{k}) dk", "derivation": "\\eta^{\\prime}{(k)} = e^{k} and 0 = - \\eta^{\\prime}{(k)} + e^{k} and \\operatorname{C_{2}}{(k)} = e^{k} and \\operatorname{C_{2}}{(k)} = \\eta^{\\prime}{(k)} and 0 = \\operatorname{C_{2}}{(k)} - \\eta^{\\prime}{(k)} and \\int 0 dk = \\int (\\operatorname{C_{2}}{(k)} - \\eta^{\\prime}{(k)}) dk and \\int 0 dk = \\int (- \\eta^{\\prime}{(k)} + e^{k}) dk and \\int 0 dk = \\int (- \\operatorname{C_{2}}{(k)} + e^{k}) dk", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], [["minus", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('C_2')(Symbol('k', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('C_2')(Symbol('k', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True)))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('k', commutative=True))), Integral(Add(Function('C_2')(Symbol('k', commutative=True)), Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integral(Integer(0), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Integral(Integer(0), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Function('C_2')(Symbol('k', commutative=True))), exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\phi{(f^{\\prime},F_{H})} = \\log{(F_{H} f^{\\prime})} and \\mathbf{r}{(f^{\\prime},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\phi{(f^{\\prime},F_{H})}, then derive \\frac{\\partial}{\\partial F_{H}} \\phi{(f^{\\prime},F_{H})} = \\frac{1}{F_{H}}, then obtain 1 + \\frac{1}{F_{H}} = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} f^{\\prime})} + 1", "derivation": "\\phi{(f^{\\prime},F_{H})} = \\log{(F_{H} f^{\\prime})} and \\frac{\\partial}{\\partial F_{H}} \\phi{(f^{\\prime},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} f^{\\prime})} and \\frac{\\partial}{\\partial F_{H}} \\phi{(f^{\\prime},F_{H})} = \\frac{1}{F_{H}} and \\mathbf{r}{(f^{\\prime},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\phi{(f^{\\prime},F_{H})} and \\mathbf{r}{(f^{\\prime},F_{H})} = \\frac{1}{F_{H}} and \\mathbf{r}{(f^{\\prime},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} f^{\\prime})} and \\mathbf{r}{(f^{\\prime},F_{H})} + 1 = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} f^{\\prime})} + 1 and 1 + \\frac{1}{F_{H}} = \\frac{\\partial}{\\partial F_{H}} \\log{(F_{H} f^{\\prime})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('F_H', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Derivative(Function('\\\\phi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Derivative(log(Mul(Symbol('F_H', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["add", 6, 1], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('F_H', commutative=True)), Integer(1)), Add(Derivative(log(Mul(Symbol('F_H', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Integer(1), Pow(Symbol('F_H', commutative=True), Integer(-1))), Add(Derivative(log(Mul(Symbol('F_H', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given C{(\\dot{x})} = \\log{(\\dot{x})}, then derive \\int C{(\\dot{x})} d\\dot{x} = \\dot{x} \\log{(\\dot{x})} - \\dot{x} + \\rho_f, then obtain \\frac{\\dot{x} C{(\\dot{x})} - \\dot{x} + \\rho_f}{\\dot{x}} = \\frac{\\int \\log{(\\dot{x})} d\\dot{x}}{\\dot{x}}", "derivation": "C{(\\dot{x})} = \\log{(\\dot{x})} and \\int C{(\\dot{x})} d\\dot{x} = \\int \\log{(\\dot{x})} d\\dot{x} and \\frac{\\int C{(\\dot{x})} d\\dot{x}}{\\dot{x}} = \\frac{\\int \\log{(\\dot{x})} d\\dot{x}}{\\dot{x}} and \\int C{(\\dot{x})} d\\dot{x} = \\dot{x} \\log{(\\dot{x})} - \\dot{x} + \\rho_f and \\int C{(\\dot{x})} d\\dot{x} = \\dot{x} C{(\\dot{x})} - \\dot{x} + \\rho_f and \\frac{\\dot{x} C{(\\dot{x})} - \\dot{x} + \\rho_f}{\\dot{x}} = \\frac{\\int \\log{(\\dot{x})} d\\dot{x}}{\\dot{x}}", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Integral(Function('C')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('C')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Function('C')(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Function('C')(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(E)} = \\cos{(e^{E})} and I{(E)} = e^{E} + \\cos{(e^{E})}, then obtain \\frac{d}{d E} (e^{E} + \\cos{(e^{E})}) = \\frac{d}{d E} I{(E)}", "derivation": "\\operatorname{P_{e}}{(E)} = \\cos{(e^{E})} and \\operatorname{P_{e}}{(E)} + e^{E} = e^{E} + \\cos{(e^{E})} and I{(E)} = e^{E} + \\cos{(e^{E})} and \\frac{d}{d E} (\\operatorname{P_{e}}{(E)} + e^{E}) = \\frac{d}{d E} (e^{E} + \\cos{(e^{E})}) and \\frac{d}{d E} (\\operatorname{P_{e}}{(E)} + e^{E}) = \\frac{d}{d E} I{(E)} and \\frac{d}{d E} (e^{E} + \\cos{(e^{E})}) = \\frac{d}{d E} I{(E)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True))))"], [["add", 1, "exp(Symbol('E', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Add(exp(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True)))))"], ["renaming_premise", "Equality(Function('I')(Symbol('E', commutative=True)), Add(exp(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True)))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Function('P_e')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Function('P_e')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Function('I')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Add(exp(Symbol('E', commutative=True)), cos(exp(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Function('I')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\delta{(v_{t},G)} = G^{v_{t}} and \\mathbf{J}_P{(G,v_{t})} = \\frac{1}{\\delta{(v_{t},G)}}, then obtain \\log{(\\delta{(v_{t},G)} \\mathbf{J}_P{(G,v_{t})})} = 0", "derivation": "\\delta{(v_{t},G)} = G^{v_{t}} and G^{- v_{t}} \\delta{(v_{t},G)} = 1 and \\log{(G^{- v_{t}} \\delta{(v_{t},G)})} = 0 and \\mathbf{J}_P{(G,v_{t})} = \\frac{1}{\\delta{(v_{t},G)}} and \\mathbf{J}_P{(G,v_{t})} = G^{- v_{t}} and \\log{(\\delta{(v_{t},G)} \\mathbf{J}_P{(G,v_{t})})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('v_t', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('v_t', commutative=True)))"], [["divide", 1, "Pow(Symbol('G', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Function('\\\\delta')(Symbol('v_t', commutative=True), Symbol('G', commutative=True))), Integer(1))"], [["log", 2], "Equality(log(Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Function('\\\\delta')(Symbol('v_t', commutative=True), Symbol('G', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Pow(Function('\\\\delta')(Symbol('v_t', commutative=True), Symbol('G', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\mathbf{J}_P')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(log(Mul(Function('\\\\delta')(Symbol('v_t', commutative=True), Symbol('G', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('G', commutative=True), Symbol('v_t', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{v})} = \\log{(\\cos{(\\mathbf{v})})} and \\operatorname{v_{1}}{(\\hat{X},\\rho)} = \\rho^{\\hat{X}}, then obtain \\frac{\\partial}{\\partial \\hat{X}} (\\operatorname{v_{1}}{(\\hat{X},\\rho)} - \\operatorname{v_{x}}{(\\mathbf{v})}) = \\frac{\\partial}{\\partial \\hat{X}} (\\rho^{\\hat{X}} - \\operatorname{v_{x}}{(\\mathbf{v})})", "derivation": "\\operatorname{v_{x}}{(\\mathbf{v})} = \\log{(\\cos{(\\mathbf{v})})} and \\operatorname{v_{1}}{(\\hat{X},\\rho)} = \\rho^{\\hat{X}} and \\operatorname{v_{1}}{(\\hat{X},\\rho)} - \\log{(\\cos{(\\mathbf{v})})} = \\rho^{\\hat{X}} - \\log{(\\cos{(\\mathbf{v})})} and \\operatorname{v_{1}}{(\\hat{X},\\rho)} - \\operatorname{v_{x}}{(\\mathbf{v})} = \\rho^{\\hat{X}} - \\operatorname{v_{x}}{(\\mathbf{v})} and \\frac{\\partial}{\\partial \\hat{X}} (\\operatorname{v_{1}}{(\\hat{X},\\rho)} - \\operatorname{v_{x}}{(\\mathbf{v})}) = \\frac{\\partial}{\\partial \\hat{X}} (\\rho^{\\hat{X}} - \\operatorname{v_{x}}{(\\mathbf{v})})", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{v}', commutative=True)), log(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], ["get_premise", "Equality(Function('v_1')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["minus", 2, "log(cos(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Function('v_1')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{v}', commutative=True))))), Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{v}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('v_1')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Add(Function('v_1')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('\\\\mathbf{v}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\mathbf{S})} = \\sin{(\\mathbf{S})}, then derive \\int b{(\\mathbf{S})} d\\mathbf{S} = - \\cos{(\\mathbf{S})}, then obtain (\\int b{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (\\int \\sin{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}}", "derivation": "b{(\\mathbf{S})} = \\sin{(\\mathbf{S})} and \\int b{(\\mathbf{S})} d\\mathbf{S} = \\int \\sin{(\\mathbf{S})} d\\mathbf{S} and \\iint b{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\iint \\sin{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} and \\frac{d}{d \\mathbf{S}} \\iint b{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\frac{d}{d \\mathbf{S}} \\iint \\sin{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} and \\int b{(\\mathbf{S})} d\\mathbf{S} = - \\cos{(\\mathbf{S})} and (\\int b{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (- \\cos{(\\mathbf{S})})^{\\mathbf{S}} and - \\cos{(\\mathbf{S})} = \\int \\sin{(\\mathbf{S})} d\\mathbf{S} and (\\int b{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (\\int \\sin{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Integer(-1), cos(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Pow(Integral(Function('b')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(h,F_{N})} = \\sin{(h^{F_{N}})} and \\operatorname{f_{\\mathbf{v}}}{(h,F_{N})} = 2 \\operatorname{x^{{\\}'}}{(h,F_{N})}, then obtain \\frac{\\operatorname{f_{\\mathbf{v}}}{(h,F_{N})}}{2 \\sin{(h^{F_{N}})}} = 1", "derivation": "\\operatorname{x^{{\\}'}}{(h,F_{N})} = \\sin{(h^{F_{N}})} and 2 \\operatorname{x^{{\\}'}}{(h,F_{N})} = \\operatorname{x^{{\\}'}}{(h,F_{N})} + \\sin{(h^{F_{N}})} and \\operatorname{f_{\\mathbf{v}}}{(h,F_{N})} = 2 \\operatorname{x^{{\\}'}}{(h,F_{N})} and \\operatorname{f_{\\mathbf{v}}}{(h,F_{N})} = \\operatorname{x^{{\\}'}}{(h,F_{N})} + \\sin{(h^{F_{N}})} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(h,F_{N})}}{2 \\operatorname{x^{{\\}'}}{(h,F_{N})}} = \\frac{\\operatorname{x^{{\\}'}}{(h,F_{N})} + \\sin{(h^{F_{N}})}}{2 \\operatorname{x^{{\\}'}}{(h,F_{N})}} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(h,F_{N})}}{2 \\sin{(h^{F_{N}})}} = 1", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), sin(Pow(Symbol('h', commutative=True), Symbol('F_N', commutative=True))))"], [["add", 1, "Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True))), Add(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), sin(Pow(Symbol('h', commutative=True), Symbol('F_N', commutative=True)))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Add(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), sin(Pow(Symbol('h', commutative=True), Symbol('F_N', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Pow(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), sin(Pow(Symbol('h', commutative=True), Symbol('F_N', commutative=True)))), Pow(Function('x^\\\\prime')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Rational(1, 2), Function('f_{\\\\mathbf{v}}')(Symbol('h', commutative=True), Symbol('F_N', commutative=True)), Pow(sin(Pow(Symbol('h', commutative=True), Symbol('F_N', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given f{(A_{2},\\pi)} = A_{2} \\cos{(\\pi)}, then obtain \\frac{\\partial}{\\partial \\pi} (\\frac{f^{2}{(A_{2},\\pi)}}{A_{2}^{2} \\cos^{2}{(\\pi)}})^{\\pi} = \\frac{d}{d \\pi} 1", "derivation": "f{(A_{2},\\pi)} = A_{2} \\cos{(\\pi)} and \\frac{f{(A_{2},\\pi)}}{A_{2} \\cos{(\\pi)}} = 1 and \\frac{f^{2}{(A_{2},\\pi)}}{A_{2}^{2} \\cos^{2}{(\\pi)}} = \\frac{f{(A_{2},\\pi)}}{A_{2} \\cos{(\\pi)}} and \\frac{f^{2}{(A_{2},\\pi)}}{A_{2}^{2} \\cos^{2}{(\\pi)}} = 1 and (\\frac{f^{2}{(A_{2},\\pi)}}{A_{2}^{2} \\cos^{2}{(\\pi)}})^{\\pi} = 1 and (\\frac{f{(A_{2},\\pi)}}{A_{2} \\cos{(\\pi)}})^{\\pi} = 1 and \\frac{\\partial}{\\partial \\pi} (\\frac{f{(A_{2},\\pi)}}{A_{2} \\cos{(\\pi)}})^{\\pi} = \\frac{d}{d \\pi} 1 and \\frac{\\partial}{\\partial \\pi} (\\frac{f^{2}{(A_{2},\\pi)}}{A_{2}^{2} \\cos^{2}{(\\pi)}})^{\\pi} = \\frac{d}{d \\pi} 1", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('A_2', commutative=True), cos(Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Mul(Symbol('A_2', commutative=True), cos(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-2))), Integer(1))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-2))), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1))), Symbol('\\\\pi', commutative=True)), Integer(1))"], [["differentiate", 6, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-1))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(Pow(Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Function('f')(Symbol('A_2', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\pi', commutative=True)), Integer(-2))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{x})} = \\int \\sin{(F_{x})} dF_{x}, then derive \\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})} = \\frac{\\partial}{\\partial F_{x}} (n - \\cos{(F_{x})}), then obtain 1 = \\frac{\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x}}{\\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})}}", "derivation": "\\eta^{\\prime}{(F_{x})} = \\int \\sin{(F_{x})} dF_{x} and \\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})} = \\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x} and \\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})} = \\frac{\\partial}{\\partial F_{x}} (n - \\cos{(F_{x})}) and \\frac{\\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})}}{\\frac{\\partial}{\\partial F_{x}} (n - \\cos{(F_{x})})} = \\frac{\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x}}{\\frac{\\partial}{\\partial F_{x}} (n - \\cos{(F_{x})})} and 1 = \\frac{\\frac{d}{d F_{x}} \\int \\sin{(F_{x})} dF_{x}}{\\frac{d}{d F_{x}} \\eta^{\\prime}{(F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_x', commutative=True)), Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Derivative(Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Pow(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Integer(-1)), Derivative(Integral(sin(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(p)} = \\sin{(p)}, then obtain - \\frac{\\cos{(p)}}{\\sin^{2}{(p)}} = - \\frac{\\frac{d}{d p} x{(p)}}{x^{2}{(p)}}", "derivation": "x{(p)} = \\sin{(p)} and \\frac{1}{\\sin{(p)}} = \\frac{1}{x{(p)}} and \\frac{d}{d p} \\frac{1}{\\sin{(p)}} = \\frac{d}{d p} \\frac{1}{x{(p)}} and - \\frac{\\cos{(p)}}{\\sin^{2}{(p)}} = - \\frac{\\frac{d}{d p} x{(p)}}{x^{2}{(p)}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["divide", 1, "Mul(Function('x')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], "Equality(Pow(sin(Symbol('p', commutative=True)), Integer(-1)), Pow(Function('x')(Symbol('p', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(sin(Symbol('p', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Function('x')(Symbol('p', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(sin(Symbol('p', commutative=True)), Integer(-2)), cos(Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Function('x')(Symbol('p', commutative=True)), Integer(-2)), Derivative(Function('x')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and v{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)}, then derive \\phi + v{(\\phi)} = \\phi + \\cos{(\\phi)}, then obtain (\\phi + v{(\\phi)}) \\sin{(\\operatorname{A_{1}}{(\\phi)})} = (\\phi + \\cos{(\\phi)}) \\sin{(\\operatorname{A_{1}}{(\\phi)})}", "derivation": "\\operatorname{A_{1}}{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and \\phi + \\operatorname{A_{1}}{(\\phi)} = \\phi + \\frac{d}{d \\phi} \\sin{(\\phi)} and v{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and \\operatorname{A_{1}}{(\\phi)} = v{(\\phi)} and \\phi + v{(\\phi)} = \\phi + \\frac{d}{d \\phi} \\sin{(\\phi)} and \\phi + v{(\\phi)} = \\phi + \\cos{(\\phi)} and (\\phi + v{(\\phi)}) \\sin{(\\operatorname{A_{1}}{(\\phi)})} = (\\phi + \\cos{(\\phi)}) \\sin{(\\operatorname{A_{1}}{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('A_1')(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\phi', commutative=True)), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('A_1')(Symbol('\\\\phi', commutative=True)), Function('v')(Symbol('\\\\phi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('v')(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('v')(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\phi', commutative=True))))"], [["times", 6, "sin(Function('A_1')(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\phi', commutative=True), Function('v')(Symbol('\\\\phi', commutative=True))), sin(Function('A_1')(Symbol('\\\\phi', commutative=True)))), Mul(Add(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\phi', commutative=True))), sin(Function('A_1')(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given B{(\\chi,\\hat{x})} = - \\chi + \\hat{x}, then obtain \\frac{B{(\\chi,\\hat{x})} \\int B{(\\chi,\\hat{x})} d\\chi}{\\int (- \\chi + \\hat{x}) d\\chi} = B{(\\chi,\\hat{x})}", "derivation": "B{(\\chi,\\hat{x})} = - \\chi + \\hat{x} and \\int B{(\\chi,\\hat{x})} d\\chi = \\int (- \\chi + \\hat{x}) d\\chi and \\frac{\\int B{(\\chi,\\hat{x})} d\\chi}{\\int (- \\chi + \\hat{x}) d\\chi} = 1 and \\frac{(- \\chi + \\hat{x}) \\int B{(\\chi,\\hat{x})} d\\chi}{\\int (- \\chi + \\hat{x}) d\\chi} = - \\chi + \\hat{x} and \\frac{B{(\\chi,\\hat{x})} \\int B{(\\chi,\\hat{x})} d\\chi}{\\int (- \\chi + \\hat{x}) d\\chi} = B{(\\chi,\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(1))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Function('B')(Symbol('\\\\chi', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given a{(C)} = \\sin{(C)}, then obtain \\log{(- v + \\sin{(\\frac{d}{d C} a{(C)})})} = \\log{(- v + \\sin{(\\cos{(C)})})}", "derivation": "a{(C)} = \\sin{(C)} and \\frac{d}{d C} a{(C)} = \\frac{d}{d C} \\sin{(C)} and \\sin{(\\frac{d}{d C} a{(C)})} = \\sin{(\\frac{d}{d C} \\sin{(C)})} and - v + \\sin{(\\frac{d}{d C} a{(C)})} = - v + \\sin{(\\frac{d}{d C} \\sin{(C)})} and \\log{(- v + \\sin{(\\frac{d}{d C} a{(C)})})} = \\log{(- v + \\sin{(\\frac{d}{d C} \\sin{(C)})})} and \\log{(- v + \\sin{(\\frac{d}{d C} a{(C)})})} = \\log{(- v + \\sin{(\\cos{(C)})})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('a')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), sin(Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["minus", 3, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Derivative(Function('a')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))))"], [["log", 4], "Equality(log(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Derivative(Function('a')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))), log(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))))"], [["evaluate_derivatives", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(Derivative(Function('a')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))), log(Add(Mul(Integer(-1), Symbol('v', commutative=True)), sin(cos(Symbol('C', commutative=True))))))"]]}, {"prompt": "Given \\psi{(\\psi^*,c,\\mathbf{P})} = \\mathbf{P} \\psi^* - c, then derive \\psi^* + \\frac{\\partial}{\\partial \\mathbf{P}} \\psi{(\\psi^*,c,\\mathbf{P})} = 2 \\psi^*, then obtain \\psi^* + \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\psi^* - c) = 2 \\psi^*", "derivation": "\\psi{(\\psi^*,c,\\mathbf{P})} = \\mathbf{P} \\psi^* - c and - c + \\psi{(\\psi^*,c,\\mathbf{P})} = \\mathbf{P} \\psi^* - 2 c and \\mathbf{P} \\psi^* - 2 c + \\psi{(\\psi^*,c,\\mathbf{P})} - 1 = 2 \\mathbf{P} \\psi^* - 3 c - 1 and \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\psi^* - 2 c + \\psi{(\\psi^*,c,\\mathbf{P})} - 1) = \\frac{\\partial}{\\partial \\mathbf{P}} (2 \\mathbf{P} \\psi^* - 3 c - 1) and \\psi^* + \\frac{\\partial}{\\partial \\mathbf{P}} \\psi{(\\psi^*,c,\\mathbf{P})} = 2 \\psi^* and \\psi^* + \\frac{\\partial}{\\partial \\mathbf{P}} (\\mathbf{P} \\psi^* - c) = 2 \\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["minus", 1, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\psi')(Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True))))"], [["add", 2, "Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Function('\\\\psi')(Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('c', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('c', commutative=True)), Function('\\\\psi')(Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Integer(3), Symbol('c', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Derivative(Function('\\\\psi')(Symbol('\\\\psi^*', commutative=True), Symbol('c', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Derivative(Add(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\mathbf{s},\\pi)} = \\mathbf{s} + \\pi, then obtain \\frac{\\partial}{\\partial \\pi} \\frac{\\hat{H}{(\\mathbf{s},\\pi)}}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}} = \\frac{\\partial}{\\partial \\pi} \\frac{\\mathbf{s} + \\pi}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}}", "derivation": "\\hat{H}{(\\mathbf{s},\\pi)} = \\mathbf{s} + \\pi and \\pi + \\hat{H}{(\\mathbf{s},\\pi)} = \\mathbf{s} + 2 \\pi and \\frac{\\hat{H}{(\\mathbf{s},\\pi)}}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}} = \\frac{\\mathbf{s} + \\pi}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}} and \\frac{\\hat{H}{(\\mathbf{s},\\pi)}}{\\mathbf{s} + 2 \\pi} = \\frac{\\mathbf{s} + \\pi}{\\mathbf{s} + 2 \\pi} and \\frac{\\partial}{\\partial \\pi} \\frac{\\hat{H}{(\\mathbf{s},\\pi)}}{\\mathbf{s} + 2 \\pi} = \\frac{\\partial}{\\partial \\pi} \\frac{\\mathbf{s} + \\pi}{\\mathbf{s} + 2 \\pi} and \\frac{\\partial}{\\partial \\pi} \\frac{\\hat{H}{(\\mathbf{s},\\pi)}}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}} = \\frac{\\partial}{\\partial \\pi} \\frac{\\mathbf{s} + \\pi}{\\pi + \\hat{H}{(\\mathbf{s},\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["add", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(2), Symbol('\\\\pi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1)), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True)), Pow(Add(Symbol('\\\\pi', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(y)} = e^{y}, then obtain \\int \\frac{h{(y)}}{y} dy = V_{\\mathbf{B}} + \\operatorname{Ei}{(y)}", "derivation": "h{(y)} = e^{y} and \\frac{h{(y)}}{y} = \\frac{e^{y}}{y} and \\int \\frac{h{(y)}}{y} dy = \\int \\frac{e^{y}}{y} dy and \\int \\frac{h{(y)}}{y} dy = V_{\\mathbf{B}} + \\operatorname{Ei}{(y)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('h')(Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), exp(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('h')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), exp(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('h')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Ei(Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})}, then obtain (\\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda})^{2} = (t_{1} - \\cos{(\\Psi_{\\lambda})}) \\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}", "derivation": "\\mathbf{J}_M{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})} and \\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} and (\\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda})^{2} = (\\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}) \\int \\sin{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} and (\\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda})^{2} = (t_{1} - \\cos{(\\Psi_{\\lambda})}) \\int \\mathbf{J}_M{(\\Psi_{\\lambda})} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 2, "Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(2)), Mul(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(2)), Mul(Add(Symbol('t_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(h,\\hat{x})} = \\sin{(\\hat{x}^{h})}, then derive \\frac{\\partial}{\\partial h} \\dot{z}{(h,\\hat{x})} = \\hat{x}^{h} \\log{(\\hat{x})} \\cos{(\\hat{x}^{h})}, then obtain \\frac{\\partial}{\\partial h} \\sin{(\\hat{x}^{h})} = \\hat{x}^{h} \\log{(\\hat{x})} \\cos{(\\hat{x}^{h})}", "derivation": "\\dot{z}{(h,\\hat{x})} = \\sin{(\\hat{x}^{h})} and \\frac{\\partial}{\\partial h} \\dot{z}{(h,\\hat{x})} = \\frac{\\partial}{\\partial h} \\sin{(\\hat{x}^{h})} and \\frac{\\partial}{\\partial h} \\dot{z}{(h,\\hat{x})} = \\hat{x}^{h} \\log{(\\hat{x})} \\cos{(\\hat{x}^{h})} and \\frac{\\partial}{\\partial h} \\sin{(\\hat{x}^{h})} = \\hat{x}^{h} \\log{(\\hat{x})} \\cos{(\\hat{x}^{h})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), sin(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('h', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)), cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)), log(Symbol('\\\\hat{x}', commutative=True)), cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(b,c_{0})} = - b + e^{c_{0}} and \\chi{(b,c_{0})} = - b + e^{c_{0}}, then obtain \\chi^{2}{(b,c_{0})} + \\frac{\\dot{\\mathbf{r}}{(b,c_{0})} - e^{c_{0}}}{b} = \\chi{(b,c_{0})} \\dot{\\mathbf{r}}{(b,c_{0})} + \\frac{\\dot{\\mathbf{r}}{(b,c_{0})} - e^{c_{0}}}{b}", "derivation": "\\dot{\\mathbf{r}}{(b,c_{0})} = - b + e^{c_{0}} and \\dot{\\mathbf{r}}{(b,c_{0})} - e^{c_{0}} = - b and \\chi{(b,c_{0})} = - b + e^{c_{0}} and \\chi{(b,c_{0})} = \\dot{\\mathbf{r}}{(b,c_{0})} and \\chi^{2}{(b,c_{0})} = \\chi{(b,c_{0})} \\dot{\\mathbf{r}}{(b,c_{0})} and \\chi^{2}{(b,c_{0})} + \\frac{\\dot{\\mathbf{r}}{(b,c_{0})} - e^{c_{0}}}{b} = \\chi{(b,c_{0})} \\dot{\\mathbf{r}}{(b,c_{0})} + \\frac{\\dot{\\mathbf{r}}{(b,c_{0})} - e^{c_{0}}}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), exp(Symbol('c_0', commutative=True))))"], [["minus", 1, "exp(Symbol('c_0', commutative=True))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))), Mul(Integer(-1), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), exp(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)))"], [["times", 4, "Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Pow(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Integer(2)), Mul(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True))))"], [["minus", 5, "Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(-1)), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))))"], "Equality(Add(Pow(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Integer(2)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))))), Add(Mul(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('b', commutative=True), Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{J}_M,\\chi)} = \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi, then obtain - \\chi + \\int \\lambda{(\\mathbf{J}_M,\\chi)} \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi d\\chi = - \\chi + \\int (\\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi)^{2} d\\chi", "derivation": "\\lambda{(\\mathbf{J}_M,\\chi)} = \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi and \\lambda{(\\mathbf{J}_M,\\chi)} \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi = (\\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi)^{2} and \\int \\lambda{(\\mathbf{J}_M,\\chi)} \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi d\\chi = \\int (\\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi)^{2} d\\chi and - \\chi + \\int \\lambda{(\\mathbf{J}_M,\\chi)} \\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi d\\chi = - \\chi + \\int (\\int \\frac{\\chi}{\\mathbf{J}_M} d\\chi)^{2} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 1, "Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["minus", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integral(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(c,A)} = A^{c}, then obtain A^{c} (A^{c} \\hat{H}_l{(c,A)} - c) = A^{c} (A^{2 c} - c)", "derivation": "\\hat{H}_l{(c,A)} = A^{c} and A^{c} \\hat{H}_l{(c,A)} = A^{2 c} and A^{c} \\hat{H}_l{(c,A)} - c = A^{2 c} - c and A^{c} (A^{c} \\hat{H}_l{(c,A)} - c) = A^{c} (A^{2 c} - c)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('c', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)))"], [["times", 1, "Pow(Symbol('A', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)), Function('\\\\hat{H}_l')(Symbol('c', commutative=True), Symbol('A', commutative=True))), Pow(Symbol('A', commutative=True), Mul(Integer(2), Symbol('c', commutative=True))))"], [["minus", 2, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)), Function('\\\\hat{H}_l')(Symbol('c', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Pow(Symbol('A', commutative=True), Mul(Integer(2), Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["times", 3, "Pow(Symbol('A', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)), Add(Mul(Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)), Function('\\\\hat{H}_l')(Symbol('c', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Pow(Symbol('A', commutative=True), Symbol('c', commutative=True)), Add(Pow(Symbol('A', commutative=True), Mul(Integer(2), Symbol('c', commutative=True))), Mul(Integer(-1), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(i,r)} = i \\cos{(r)} and \\Psi^{\\dagger}{(\\hat{H},\\phi)} = \\hat{H} \\phi, then obtain ((\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}) \\Psi^{\\dagger}{(\\hat{H},\\phi)})^{r} = (\\hat{H} \\phi (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}))^{r}", "derivation": "\\operatorname{F_{N}}{(i,r)} = i \\cos{(r)} and \\Psi^{\\dagger}{(\\hat{H},\\phi)} = \\hat{H} \\phi and (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} i \\cos{(r)}) \\Psi^{\\dagger}{(\\hat{H},\\phi)} = \\hat{H} \\phi (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} i \\cos{(r)}) and (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}) \\Psi^{\\dagger}{(\\hat{H},\\phi)} = \\hat{H} \\phi (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}) and ((\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}) \\Psi^{\\dagger}{(\\hat{H},\\phi)})^{r} = (\\hat{H} \\phi (\\operatorname{F_{N}}{(i,r)} + \\frac{\\partial}{\\partial r} \\operatorname{F_{N}}{(i,r)}))^{r}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Mul(Symbol('i', commutative=True), cos(Symbol('r', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["times", 2, "Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], "Equality(Mul(Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True), Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Mul(Symbol('i', commutative=True), cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True), Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Symbol('r', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True), Add(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Derivative(Function('F_N')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{H},\\varphi^*)} = e^{\\mathbf{H} - \\varphi^*}, then obtain \\dot{y}{(\\mathbf{H},\\varphi^*)} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}{(\\mathbf{H},\\varphi^*)} e^{\\mathbf{H} - \\varphi^*} = e^{\\mathbf{H} - \\varphi^*} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}{(\\mathbf{H},\\varphi^*)} e^{\\mathbf{H} - \\varphi^*}", "derivation": "\\dot{y}{(\\mathbf{H},\\varphi^*)} = e^{\\mathbf{H} - \\varphi^*} and \\dot{y}^{2}{(\\mathbf{H},\\varphi^*)} = \\dot{y}{(\\mathbf{H},\\varphi^*)} e^{\\mathbf{H} - \\varphi^*} and \\dot{y}{(\\mathbf{H},\\varphi^*)} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}^{2}{(\\mathbf{H},\\varphi^*)} = e^{\\mathbf{H} - \\varphi^*} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}^{2}{(\\mathbf{H},\\varphi^*)} and \\dot{y}{(\\mathbf{H},\\varphi^*)} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}{(\\mathbf{H},\\varphi^*)} e^{\\mathbf{H} - \\varphi^*} = e^{\\mathbf{H} - \\varphi^*} \\frac{\\partial}{\\partial \\varphi^*} \\dot{y}{(\\mathbf{H},\\varphi^*)} e^{\\mathbf{H} - \\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))))"], [["times", 1, "Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))))"], [["times", 1, "Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Mul(exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Derivative(Pow(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Mul(exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)))), Derivative(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(W)} = \\sin{(W)}, then obtain \\iint \\eta{(W)} \\eta^{W}{(W)} dW dW = \\iint \\eta^{W}{(W)} \\sin{(W)} dW dW", "derivation": "\\eta{(W)} = \\sin{(W)} and \\eta^{W}{(W)} = \\sin^{W}{(W)} and \\eta{(W)} \\sin^{W}{(W)} = \\sin{(W)} \\sin^{W}{(W)} and \\eta{(W)} \\eta^{W}{(W)} = \\eta^{W}{(W)} \\sin{(W)} and \\int \\eta{(W)} \\eta^{W}{(W)} dW = \\int \\eta^{W}{(W)} \\sin{(W)} dW and \\iint \\eta{(W)} \\eta^{W}{(W)} dW dW = \\iint \\eta^{W}{(W)} \\sin{(W)} dW dW", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["times", 1, "Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(sin(Symbol('W', commutative=True)), Pow(sin(Symbol('W', commutative=True)), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\eta')(Symbol('W', commutative=True)), Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))))"], [["integrate", 4, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Function('\\\\eta')(Symbol('W', commutative=True)), Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integral(Mul(Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["integrate", 5, "Symbol('W', commutative=True)"], "Equality(Integral(Mul(Function('\\\\eta')(Symbol('W', commutative=True)), Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Pow(Function('\\\\eta')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(C_{1},\\hbar)} = \\sin{(C_{1}^{\\hbar})} and \\operatorname{t_{1}}{(C_{1},\\hbar)} = C_{1}^{\\hbar}, then obtain \\operatorname{E_{x}}^{\\hbar}{(C_{1},\\hbar)} = \\sin^{\\hbar}{(\\operatorname{t_{1}}{(C_{1},\\hbar)})}", "derivation": "\\operatorname{E_{x}}{(C_{1},\\hbar)} = \\sin{(C_{1}^{\\hbar})} and \\operatorname{t_{1}}{(C_{1},\\hbar)} = C_{1}^{\\hbar} and \\operatorname{E_{x}}{(C_{1},\\hbar)} = \\sin{(\\operatorname{t_{1}}{(C_{1},\\hbar)})} and \\operatorname{E_{x}}^{\\hbar}{(C_{1},\\hbar)} = \\sin^{\\hbar}{(\\operatorname{t_{1}}{(C_{1},\\hbar)})}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Pow(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E_x')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(sin(Function('t_1')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(C_{2},f_{E})} = C_{2} + f_{E}, then derive \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} \\operatorname{f^{*}}{(C_{2},f_{E})} = 0, then obtain \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} (C_{2} + f_{E}) = 0", "derivation": "\\operatorname{f^{*}}{(C_{2},f_{E})} = C_{2} + f_{E} and \\frac{\\partial}{\\partial C_{2}} \\operatorname{f^{*}}{(C_{2},f_{E})} = \\frac{\\partial}{\\partial C_{2}} (C_{2} + f_{E}) and \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} \\operatorname{f^{*}}{(C_{2},f_{E})} = \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} (C_{2} + f_{E}) and \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} \\operatorname{f^{*}}{(C_{2},f_{E})} = 0 and \\frac{\\partial^{2}}{\\partial f_{E}\\partial C_{2}} (C_{2} + f_{E}) = 0", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('f^*')(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('C_2', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\mathbf{P}{(\\delta,\\hat{H}_l)} = \\delta + \\hat{H}_l, then obtain \\int (- \\delta - \\hat{H}_l + \\frac{\\mathbf{P}{(\\delta,\\hat{H}_l)}}{\\hat{H}_l}) d\\hat{H}_l = \\int (- \\delta - \\hat{H}_l + \\frac{\\delta + \\hat{H}_l}{\\hat{H}_l}) d\\hat{H}_l", "derivation": "\\mathbf{P}{(\\delta,\\hat{H}_l)} = \\delta + \\hat{H}_l and \\frac{\\mathbf{P}{(\\delta,\\hat{H}_l)}}{\\hat{H}_l} = \\frac{\\delta + \\hat{H}_l}{\\hat{H}_l} and - \\delta - \\hat{H}_l + \\frac{\\mathbf{P}{(\\delta,\\hat{H}_l)}}{\\hat{H}_l} = - \\delta - \\hat{H}_l + \\frac{\\delta + \\hat{H}_l}{\\hat{H}_l} and \\int (- \\delta - \\hat{H}_l + \\frac{\\mathbf{P}{(\\delta,\\hat{H}_l)}}{\\hat{H}_l}) d\\hat{H}_l = \\int (- \\delta - \\hat{H}_l + \\frac{\\delta + \\hat{H}_l}{\\hat{H}_l}) d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(a)} = \\log{(a)}, then derive \\frac{d}{d a} \\int \\Psi^{\\dagger}{(a)} da = \\frac{\\partial}{\\partial a} (\\mathbf{J} + a \\log{(a)} - a), then obtain \\frac{d}{d a} \\int \\Psi^{\\dagger}{(a)} da = \\frac{\\partial}{\\partial a} (\\mathbf{J} + a \\Psi^{\\dagger}{(a)} - a)", "derivation": "\\Psi^{\\dagger}{(a)} = \\log{(a)} and \\int \\Psi^{\\dagger}{(a)} da = \\int \\log{(a)} da and \\frac{d}{d a} \\int \\Psi^{\\dagger}{(a)} da = \\frac{d}{d a} \\int \\log{(a)} da and \\frac{d}{d a} \\int \\Psi^{\\dagger}{(a)} da = \\frac{\\partial}{\\partial a} (\\mathbf{J} + a \\log{(a)} - a) and \\frac{d}{d a} \\int \\Psi^{\\dagger}{(a)} da = \\frac{\\partial}{\\partial a} (\\mathbf{J} + a \\Psi^{\\dagger}{(a)} - a)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Symbol('a', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(f_{E},\\Omega)} = \\Omega + f_{E}, then obtain 1 - 2 \\theta^{\\Omega}{(f_{E},\\Omega)} = \\frac{f_{E}}{- \\Omega + \\theta{(f_{E},\\Omega)}} - 2 \\theta^{\\Omega}{(f_{E},\\Omega)}", "derivation": "\\theta{(f_{E},\\Omega)} = \\Omega + f_{E} and - \\Omega + \\theta{(f_{E},\\Omega)} = f_{E} and 1 = \\frac{f_{E}}{- \\Omega + \\theta{(f_{E},\\Omega)}} and 1 - 2 \\theta^{\\Omega}{(f_{E},\\Omega)} = \\frac{f_{E}}{- \\Omega + \\theta{(f_{E},\\Omega)}} - 2 \\theta^{\\Omega}{(f_{E},\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('f_E', commutative=True)))"], [["minus", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('f_E', commutative=True))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Symbol('f_E', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1))))"], [["minus", 3, "Mul(Integer(2), Pow(Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integer(2), Pow(Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))), Add(Mul(Symbol('f_E', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(Function('\\\\theta')(Symbol('f_E', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\omega)} = e^{\\omega}, then derive \\int \\phi_{1}{(\\omega)} d\\omega = \\phi_2 + e^{\\omega}, then obtain e^{- \\omega} \\int e^{\\omega} d\\omega = (\\phi_2 + e^{\\omega}) e^{- \\omega}", "derivation": "\\phi_{1}{(\\omega)} = e^{\\omega} and \\int \\phi_{1}{(\\omega)} d\\omega = \\int e^{\\omega} d\\omega and \\int \\phi_{1}{(\\omega)} d\\omega = \\phi_2 + e^{\\omega} and \\int e^{\\omega} d\\omega = \\phi_2 + e^{\\omega} and e^{- \\omega} \\int e^{\\omega} d\\omega = (\\phi_2 + e^{\\omega}) e^{- \\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))))"], [["divide", 4, "exp(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Add(Symbol('\\\\phi_2', commutative=True), exp(Symbol('\\\\omega', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(\\chi)} = e^{\\chi} and \\phi{(\\chi)} = \\frac{d}{d \\chi} \\psi^{*}{(\\chi)}, then derive \\phi{(\\chi)} = e^{\\chi}, then obtain \\frac{d}{d \\chi} e^{\\chi} - 1 = \\frac{d}{d \\chi} \\psi^{*}{(\\chi)} - 1", "derivation": "\\psi^{*}{(\\chi)} = e^{\\chi} and \\phi{(\\chi)} = \\frac{d}{d \\chi} \\psi^{*}{(\\chi)} and \\phi{(\\chi)} = \\frac{d}{d \\chi} e^{\\chi} and \\phi{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\psi^{*}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\psi^{*}{(\\chi)} - 1 = e^{\\chi} - 1 and \\frac{d}{d \\chi} e^{\\chi} - 1 = e^{\\chi} - 1 and \\frac{d}{d \\chi} e^{\\chi} - 1 = \\frac{d}{d \\chi} \\psi^{*}{(\\chi)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\psi^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\phi')(Symbol('\\\\chi', commutative=True)), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('\\\\phi')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), exp(Symbol('\\\\chi', commutative=True)))"], [["minus", 5, 1], "Equality(Add(Derivative(Function('\\\\psi^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(exp(Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(exp(Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('\\\\psi^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{v})} = \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\rho{(\\mathbf{v},W)} = (W - \\cos{(\\mathbf{v})})^{\\mathbf{v}}, then derive \\mathbf{S}{(\\mathbf{v})} = W - \\cos{(\\mathbf{v})}, then obtain \\mathbf{S}^{\\mathbf{v}}{(\\mathbf{v})} = \\rho{(\\mathbf{v},W)}", "derivation": "\\mathbf{S}{(\\mathbf{v})} = \\int \\sin{(\\mathbf{v})} d\\mathbf{v} and \\mathbf{S}^{\\mathbf{v}}{(\\mathbf{v})} = (\\int \\sin{(\\mathbf{v})} d\\mathbf{v})^{\\mathbf{v}} and \\mathbf{S}{(\\mathbf{v})} = W - \\cos{(\\mathbf{v})} and (W - \\cos{(\\mathbf{v})})^{\\mathbf{v}} = (\\int \\sin{(\\mathbf{v})} d\\mathbf{v})^{\\mathbf{v}} and \\mathbf{S}^{\\mathbf{v}}{(\\mathbf{v})} = (W - \\cos{(\\mathbf{v})})^{\\mathbf{v}} and \\rho{(\\mathbf{v},W)} = (W - \\cos{(\\mathbf{v})})^{\\mathbf{v}} and \\mathbf{S}^{\\mathbf{v}}{(\\mathbf{v})} = \\rho{(\\mathbf{v},W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{v}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Integral(sin(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\rho')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\hbar)} = \\log{(\\hbar)}, then derive \\int \\hat{H}{(\\hbar)} d\\hbar = E + \\hbar \\log{(\\hbar)} - \\hbar, then obtain \\frac{E + \\hbar \\hat{H}{(\\hbar)} - \\hbar}{v_{x}} = \\frac{\\int \\hat{H}{(\\hbar)} d\\hbar}{v_{x}}", "derivation": "\\hat{H}{(\\hbar)} = \\log{(\\hbar)} and \\int \\hat{H}{(\\hbar)} d\\hbar = \\int \\log{(\\hbar)} d\\hbar and \\int \\hat{H}{(\\hbar)} d\\hbar = E + \\hbar \\log{(\\hbar)} - \\hbar and \\int \\hat{H}{(\\hbar)} d\\hbar = E + \\hbar \\hat{H}{(\\hbar)} - \\hbar and E + \\hbar \\hat{H}{(\\hbar)} - \\hbar = E + \\hbar \\log{(\\hbar)} - \\hbar and \\frac{E + \\hbar \\hat{H}{(\\hbar)} - \\hbar}{v_{x}} = \\frac{E + \\hbar \\log{(\\hbar)} - \\hbar}{v_{x}} and \\frac{E + \\hbar \\hat{H}{(\\hbar)} - \\hbar}{v_{x}} = \\frac{\\int \\hat{H}{(\\hbar)} d\\hbar}{v_{x}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["divide", 5, "Symbol('v_x', commutative=True)"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Integral(Function('\\\\hat{H}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(v_{1},A_{1})} = A_{1} + v_{1}, then obtain \\frac{\\partial^{2}}{\\partial A_{1}\\partial v_{1}} (- A_{1} + \\operatorname{C_{2}}{(v_{1},A_{1})}) = \\frac{d^{2}}{d A_{1}d v_{1}} v_{1}", "derivation": "\\operatorname{C_{2}}{(v_{1},A_{1})} = A_{1} + v_{1} and - A_{1} + \\operatorname{C_{2}}{(v_{1},A_{1})} = v_{1} and \\frac{\\partial}{\\partial v_{1}} (- A_{1} + \\operatorname{C_{2}}{(v_{1},A_{1})}) = \\frac{d}{d v_{1}} v_{1} and \\frac{\\partial^{2}}{\\partial A_{1}\\partial v_{1}} (- A_{1} + \\operatorname{C_{2}}{(v_{1},A_{1})}) = \\frac{d^{2}}{d A_{1}d v_{1}} v_{1}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('v_1', commutative=True)))"], [["minus", 1, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('C_2')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Symbol('v_1', commutative=True))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('C_2')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('C_2')(Symbol('v_1', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Symbol('v_1', commutative=True), Tuple(Symbol('v_1', commutative=True), Integer(1)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} = A_{1} - \\sigma_p - v_{x}, then obtain \\int \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} \\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} dA_{1} = \\int \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} (A_{1} - \\sigma_p - v_{x}) dA_{1}", "derivation": "\\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} = A_{1} - \\sigma_p - v_{x} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\sigma_p - v_{x}) and \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} \\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} = \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} (A_{1} - \\sigma_p - v_{x}) and \\int \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} \\operatorname{f^{\\prime}}{(v_{x},\\sigma_p,A_{1})} dA_{1} = \\int \\frac{\\partial^{2}}{\\partial v_{x}\\partial A_{1}} (A_{1} - \\sigma_p - v_{x}) dA_{1}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('A_1', commutative=True)"], "Equality(Integral(Derivative(Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('A_1', commutative=True))), Integral(Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\Psi{(i)} = \\log{(e^{i})}, then obtain 0 = (- \\frac{\\Psi{(i)}}{i} + \\frac{\\log{(e^{i})}}{i}) \\log{(e^{i})}", "derivation": "\\Psi{(i)} = \\log{(e^{i})} and \\frac{\\Psi{(i)}}{i} = \\frac{\\log{(e^{i})}}{i} and 0 = - \\frac{\\Psi{(i)}}{i} + \\frac{\\log{(e^{i})}}{i} and 0 = (- \\frac{\\Psi{(i)}}{i} + \\frac{\\log{(e^{i})}}{i}) \\log{(e^{i})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('i', commutative=True)), log(exp(Symbol('i', commutative=True))))"], [["divide", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(exp(Symbol('i', commutative=True)))))"], [["minus", 2, "Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('i', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(exp(Symbol('i', commutative=True))))))"], [["times", 3, "log(exp(Symbol('i', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), Function('\\\\Psi')(Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), log(exp(Symbol('i', commutative=True))))), log(exp(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(f^{\\prime},\\mathbf{r})} = \\log{(\\mathbf{r} + f^{\\prime})} and \\operatorname{v_{1}}{(f^{\\prime},\\mathbf{r})} = \\log{(\\mathbf{r} + f^{\\prime})}, then obtain f^{\\prime} + \\log{(\\mathbf{r} + f^{\\prime})} = f^{\\prime} + \\operatorname{v_{1}}{(f^{\\prime},\\mathbf{r})}", "derivation": "\\varphi{(f^{\\prime},\\mathbf{r})} = \\log{(\\mathbf{r} + f^{\\prime})} and \\operatorname{v_{1}}{(f^{\\prime},\\mathbf{r})} = \\log{(\\mathbf{r} + f^{\\prime})} and f^{\\prime} + \\varphi{(f^{\\prime},\\mathbf{r})} = f^{\\prime} + \\log{(\\mathbf{r} + f^{\\prime})} and f^{\\prime} + \\varphi{(f^{\\prime},\\mathbf{r})} = f^{\\prime} + \\operatorname{v_{1}}{(f^{\\prime},\\mathbf{r})} and f^{\\prime} + \\log{(\\mathbf{r} + f^{\\prime})} = f^{\\prime} + \\operatorname{v_{1}}{(f^{\\prime},\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('\\\\varphi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Function('v_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), log(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))), Add(Symbol('f^{\\\\prime}', commutative=True), Function('v_1')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given z{(E_{n})} = \\sin{(E_{n})} and u{(E_{n})} = z^{2}{(E_{n})}, then obtain \\frac{u{(E_{n})}}{\\int z^{2}{(E_{n})} dE_{n}} = \\frac{z{(E_{n})} \\sin{(E_{n})}}{\\int z^{2}{(E_{n})} dE_{n}}", "derivation": "z{(E_{n})} = \\sin{(E_{n})} and z^{2}{(E_{n})} = z{(E_{n})} \\sin{(E_{n})} and u{(E_{n})} = z^{2}{(E_{n})} and \\int z^{2}{(E_{n})} dE_{n} = \\int z{(E_{n})} \\sin{(E_{n})} dE_{n} and u{(E_{n})} = z{(E_{n})} \\sin{(E_{n})} and \\frac{u{(E_{n})}}{\\int z{(E_{n})} \\sin{(E_{n})} dE_{n}} = \\frac{z{(E_{n})} \\sin{(E_{n})}}{\\int z{(E_{n})} \\sin{(E_{n})} dE_{n}} and \\frac{u{(E_{n})}}{\\int z^{2}{(E_{n})} dE_{n}} = \\frac{z{(E_{n})} \\sin{(E_{n})}}{\\int z^{2}{(E_{n})} dE_{n}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["times", 1, "Function('z')(Symbol('E_n', commutative=True))"], "Equality(Pow(Function('z')(Symbol('E_n', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('E_n', commutative=True)), Pow(Function('z')(Symbol('E_n', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Pow(Function('z')(Symbol('E_n', commutative=True)), Integer(2)), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('u')(Symbol('E_n', commutative=True)), Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))))"], [["divide", 5, "Integral(Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Mul(Function('u')(Symbol('E_n', commutative=True)), Pow(Integral(Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integer(-1))), Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)), Pow(Integral(Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Function('u')(Symbol('E_n', commutative=True)), Pow(Integral(Pow(Function('z')(Symbol('E_n', commutative=True)), Integer(2)), Tuple(Symbol('E_n', commutative=True))), Integer(-1))), Mul(Function('z')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)), Pow(Integral(Pow(Function('z')(Symbol('E_n', commutative=True)), Integer(2)), Tuple(Symbol('E_n', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{x},F_{H})} = \\sin{(F_{H} \\hat{x})} and s{(\\hat{x},F_{H})} = \\hat{x} + e^{\\operatorname{n_{2}}{(\\hat{x},F_{H})}}, then obtain s{(\\hat{x},F_{H})} = \\hat{x} + e^{\\sin{(F_{H} \\hat{x})}}", "derivation": "\\operatorname{n_{2}}{(\\hat{x},F_{H})} = \\sin{(F_{H} \\hat{x})} and e^{\\operatorname{n_{2}}{(\\hat{x},F_{H})}} = e^{\\sin{(F_{H} \\hat{x})}} and \\hat{x} + e^{\\operatorname{n_{2}}{(\\hat{x},F_{H})}} = \\hat{x} + e^{\\sin{(F_{H} \\hat{x})}} and s{(\\hat{x},F_{H})} = \\hat{x} + e^{\\operatorname{n_{2}}{(\\hat{x},F_{H})}} and s{(\\hat{x},F_{H})} = \\hat{x} + e^{\\sin{(F_{H} \\hat{x})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True)), sin(Mul(Symbol('F_H', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('n_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True))), exp(sin(Mul(Symbol('F_H', commutative=True), Symbol('\\\\hat{x}', commutative=True)))))"], [["add", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{x}', commutative=True), exp(Function('n_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True)))), Add(Symbol('\\\\hat{x}', commutative=True), exp(sin(Mul(Symbol('F_H', commutative=True), Symbol('\\\\hat{x}', commutative=True))))))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), exp(Function('n_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('s')(Symbol('\\\\hat{x}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), exp(sin(Mul(Symbol('F_H', commutative=True), Symbol('\\\\hat{x}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(I)} = e^{I}, then obtain ((\\frac{d^{2}}{d I^{2}} \\operatorname{C_{d}}{(I)})^{2})^{I} = ((\\frac{d^{2}}{d I^{2}} e^{I})^{2})^{I}", "derivation": "\\operatorname{C_{d}}{(I)} = e^{I} and \\frac{d}{d I} \\operatorname{C_{d}}{(I)} = \\frac{d}{d I} e^{I} and \\frac{d^{2}}{d I^{2}} \\operatorname{C_{d}}{(I)} = \\frac{d^{2}}{d I^{2}} e^{I} and (\\frac{d^{2}}{d I^{2}} \\operatorname{C_{d}}{(I)})^{2} = (\\frac{d^{2}}{d I^{2}} e^{I})^{2} and ((\\frac{d^{2}}{d I^{2}} \\operatorname{C_{d}}{(I)})^{2})^{I} = ((\\frac{d^{2}}{d I^{2}} e^{I})^{2})^{I}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('C_d')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Integer(2)))"], [["power", 4, "Symbol('I', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('C_d')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Integer(2)), Symbol('I', commutative=True)), Pow(Pow(Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Integer(2)), Symbol('I', commutative=True)))"]]}, {"prompt": "Given y{(Z,\\hat{X})} = \\hat{X}^{Z}, then obtain \\frac{\\partial}{\\partial Z} (Z + 2 \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1) = \\frac{d}{d Z} (Z - 1)", "derivation": "y{(Z,\\hat{X})} = \\hat{X}^{Z} and \\hat{X}^{- Z} y{(Z,\\hat{X})} = 1 and \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} = 0 and \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1 = -1 and Z + \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1 = Z - 1 and \\frac{\\partial}{\\partial Z} (Z + \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1) = \\frac{d}{d Z} (Z - 1) and \\frac{\\partial}{\\partial Z} (Z + 2 \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1) = \\frac{\\partial}{\\partial Z} (Z + \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1) and \\frac{\\partial}{\\partial Z} (Z + 2 \\log{(\\hat{X}^{- Z} y{(Z,\\hat{X})})} - 1) = \\frac{d}{d Z} (Z - 1)", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('Z', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(1))"], [["log", 2], "Equality(log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Integer(0))"], [["minus", 3, 1], "Equality(Add(log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Integer(-1)), Integer(-1))"], [["minus", 4, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Symbol('Z', commutative=True), log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Integer(-1)), Add(Symbol('Z', commutative=True), Integer(-1)))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Symbol('Z', commutative=True), log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(2), log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(2), log(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('y')(Symbol('Z', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), Integer(-1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{s})} = \\mathbf{s}, then derive \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\frac{\\mathbf{s}^{2}}{2} + \\mu_0, then obtain \\int \\mathbf{s} d\\mathbf{s} = \\frac{\\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})}}{2} + \\mu_0", "derivation": "\\operatorname{v_{2}}{(\\mathbf{s})} = \\mathbf{s} and \\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})} = \\mathbf{s}^{2} and \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\int \\mathbf{s} d\\mathbf{s} and \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\frac{\\mathbf{s}^{2}}{2} + \\mu_0 and \\int \\mathbf{s} d\\mathbf{s} = \\frac{\\mathbf{s}^{2}}{2} + \\mu_0 and \\int \\mathbf{s} d\\mathbf{s} = \\frac{\\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})}}{2} + \\mu_0", "srepr_derivation": [["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True))), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Rational(1, 2), Symbol('\\\\mathbf{s}', commutative=True), Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given A{(z)} = \\sin{(z)}, then obtain 2 A{(z)} \\int \\sin{(z)} dz + 2 \\int \\sin{(z)} dz = A{(z)} \\int \\sin{(z)} dz + \\sin{(z)} \\int \\sin{(z)} dz + 2 \\int \\sin{(z)} dz", "derivation": "A{(z)} = \\sin{(z)} and \\int A{(z)} dz = \\int \\sin{(z)} dz and A{(z)} \\int \\sin{(z)} dz = \\sin{(z)} \\int \\sin{(z)} dz and A{(z)} \\int \\sin{(z)} dz + \\int A{(z)} dz = \\sin{(z)} \\int \\sin{(z)} dz + \\int A{(z)} dz and A{(z)} \\int \\sin{(z)} dz + \\int \\sin{(z)} dz = \\sin{(z)} \\int \\sin{(z)} dz + \\int \\sin{(z)} dz and 2 A{(z)} \\int \\sin{(z)} dz + 2 \\int \\sin{(z)} dz = A{(z)} \\int \\sin{(z)} dz + \\sin{(z)} \\int \\sin{(z)} dz + 2 \\int \\sin{(z)} dz", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('A')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 1, "Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Mul(Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(sin(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["add", 3, "Integral(Function('A')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(Function('A')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(sin(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(Function('A')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(sin(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["add", 5, "Add(Mul(Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(Mul(Function('A')(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(sin(Symbol('z', commutative=True)), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\mu_{0}{(A_{2},C)} = A_{2}^{C}, then obtain A_{2} + \\hat{x}_0 = \\int \\frac{A_{2}^{C}}{\\mu_{0}{(A_{2},C)}} dA_{2}", "derivation": "\\mu_{0}{(A_{2},C)} = A_{2}^{C} and 1 = \\frac{A_{2}^{C}}{\\mu_{0}{(A_{2},C)}} and \\int 1 dA_{2} = \\int \\frac{A_{2}^{C}}{\\mu_{0}{(A_{2},C)}} dA_{2} and A_{2} + \\hat{x}_0 = \\int \\frac{A_{2}^{C}}{\\mu_{0}{(A_{2},C)}} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('A_2', commutative=True), Symbol('C', commutative=True)))"], [["divide", 1, "Function('\\\\mu_0')(Symbol('A_2', commutative=True), Symbol('C', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('A_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('A_2', commutative=True))), Integral(Mul(Pow(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_2', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Integral(Mul(Pow(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('A_2', commutative=True), Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given U{(\\theta_2)} = e^{\\theta_2} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain - \\sin{(\\psi^*)} + \\frac{d}{d \\theta_2} U{(\\theta_2)} = - \\sin{(\\psi^*)} + \\frac{d}{d \\theta_2} e^{\\theta_2}", "derivation": "U{(\\theta_2)} = e^{\\theta_2} and \\frac{d}{d \\theta_2} U{(\\theta_2)} = \\frac{d}{d \\theta_2} e^{\\theta_2} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\psi^*)} = \\sin{(\\psi^*)} and - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\psi^*)} + \\frac{d}{d \\theta_2} U{(\\theta_2)} = - \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\psi^*)} + \\frac{d}{d \\theta_2} e^{\\theta_2} and - \\sin{(\\psi^*)} + \\frac{d}{d \\theta_2} U{(\\theta_2)} = - \\sin{(\\psi^*)} + \\frac{d}{d \\theta_2} e^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True))), Derivative(Function('U')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True))), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Derivative(Function('U')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{x})} = \\sin{(\\hat{x})}, then derive \\frac{d}{d \\hat{x}} \\operatorname{E_{x}}{(\\hat{x})} = \\cos{(\\hat{x})}, then obtain \\frac{d^{2}}{d \\hat{x}^{2}} \\operatorname{E_{x}}{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\cos{(\\hat{x})}", "derivation": "\\operatorname{E_{x}}{(\\hat{x})} = \\sin{(\\hat{x})} and \\frac{d}{d \\hat{x}} \\operatorname{E_{x}}{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\sin{(\\hat{x})} and \\frac{d}{d \\hat{x}} \\operatorname{E_{x}}{(\\hat{x})} = \\cos{(\\hat{x})} and \\frac{d^{2}}{d \\hat{x}^{2}} \\operatorname{E_{x}}{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\cos{(\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\mathbf{H},\\varphi,G)} = G \\varphi^{\\mathbf{H}}, then obtain (\\int (- G + I{(\\mathbf{H},\\varphi,G)}) d\\varphi)^{\\varphi} = (\\int (G \\varphi^{\\mathbf{H}} - G) d\\varphi)^{\\varphi}", "derivation": "I{(\\mathbf{H},\\varphi,G)} = G \\varphi^{\\mathbf{H}} and - G + I{(\\mathbf{H},\\varphi,G)} = G \\varphi^{\\mathbf{H}} - G and \\int (- G + I{(\\mathbf{H},\\varphi,G)}) d\\varphi = \\int (G \\varphi^{\\mathbf{H}} - G) d\\varphi and (\\int (- G + I{(\\mathbf{H},\\varphi,G)}) d\\varphi)^{\\varphi} = (\\int (G \\varphi^{\\mathbf{H}} - G) d\\varphi)^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True))), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('I')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Integral(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given s{(n_{2})} = e^{n_{2}} and \\operatorname{v_{z}}{(n_{2})} = e^{n_{2}}, then obtain s^{n_{2}}{(n_{2})} e^{- n_{2}} = \\operatorname{v_{z}}^{n_{2}}{(n_{2})} e^{- n_{2}}", "derivation": "s{(n_{2})} = e^{n_{2}} and s^{n_{2}}{(n_{2})} = (e^{n_{2}})^{n_{2}} and \\operatorname{v_{z}}{(n_{2})} = e^{n_{2}} and s^{n_{2}}{(n_{2})} = \\operatorname{v_{z}}^{n_{2}}{(n_{2})} and s^{n_{2}}{(n_{2})} e^{- n_{2}} = \\operatorname{v_{z}}^{n_{2}}{(n_{2})} e^{- n_{2}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('s')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(exp(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('s')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Function('v_z')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["divide", 4, "exp(Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Function('s')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Mul(Pow(Function('v_z')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\varepsilon)} = \\log{(\\varepsilon)}, then obtain \\cos{(\\frac{\\phi_{1}{(\\varepsilon)}}{\\varepsilon})} = \\cos{(\\frac{\\log{(\\varepsilon)}}{\\varepsilon})}", "derivation": "\\phi_{1}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\frac{\\phi_{1}{(\\varepsilon)}}{\\varepsilon} = \\frac{\\log{(\\varepsilon)}}{\\varepsilon} and - \\frac{\\phi_{1}{(\\varepsilon)}}{\\varepsilon} = - \\frac{\\log{(\\varepsilon)}}{\\varepsilon} and \\cos{(\\frac{\\phi_{1}{(\\varepsilon)}}{\\varepsilon})} = \\cos{(\\frac{\\log{(\\varepsilon)}}{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\varepsilon', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\varepsilon', commutative=True))))"], [["cos", 3], "Equality(cos(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\varepsilon', commutative=True)))), cos(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1)), log(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(W)} = \\sin{(\\log{(W)})} and M{(W)} = \\log{(W)}, then obtain \\frac{d}{d W} (0^{W})^{W} = \\frac{d}{d W} ((- \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(M{(W)})})^{W})^{W}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(W)} = \\sin{(\\log{(W)})} and M{(W)} = \\log{(W)} and 0 = - \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(\\log{(W)})} and 0^{W} = (- \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(\\log{(W)})})^{W} and (0^{W})^{W} = ((- \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(\\log{(W)})})^{W})^{W} and (0^{W})^{W} = ((- \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(M{(W)})})^{W})^{W} and \\frac{d}{d W} (0^{W})^{W} = \\frac{d}{d W} ((- \\operatorname{f_{\\mathbf{p}}}{(W)} + \\sin{(M{(W)})})^{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True)), sin(log(Symbol('W', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["minus", 1, "Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))), sin(log(Symbol('W', commutative=True)))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Integer(0), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))), sin(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))), sin(log(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))), sin(Function('M')(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Pow(Integer(0), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('W', commutative=True))), sin(Function('M')(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}}, then obtain e^{- f_{\\mathbf{p}}} \\frac{d}{d f_{\\mathbf{p}}} \\operatorname{P_{e}}{(f_{\\mathbf{p}})} e^{f_{\\mathbf{p}}} = e^{- f_{\\mathbf{p}}} \\frac{d}{d f_{\\mathbf{p}}} e^{2 f_{\\mathbf{p}}}", "derivation": "\\operatorname{P_{e}}{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and \\operatorname{P_{e}}{(f_{\\mathbf{p}})} e^{f_{\\mathbf{p}}} = e^{2 f_{\\mathbf{p}}} and \\frac{d}{d f_{\\mathbf{p}}} \\operatorname{P_{e}}{(f_{\\mathbf{p}})} e^{f_{\\mathbf{p}}} = \\frac{d}{d f_{\\mathbf{p}}} e^{2 f_{\\mathbf{p}}} and e^{- f_{\\mathbf{p}}} \\frac{d}{d f_{\\mathbf{p}}} \\operatorname{P_{e}}{(f_{\\mathbf{p}})} e^{f_{\\mathbf{p}}} = e^{- f_{\\mathbf{p}}} \\frac{d}{d f_{\\mathbf{p}}} e^{2 f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('P_e')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), exp(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Mul(Function('P_e')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["divide", 3, "exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Derivative(Mul(Function('P_e')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Derivative(exp(Mul(Integer(2), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain (\\hat{p}{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)}) \\cos{(x^\\prime)} = (\\cos{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)}) \\cos{(x^\\prime)}", "derivation": "\\hat{p}{(x^\\prime)} = \\cos{(x^\\prime)} and \\hat{p}^{x^\\prime}{(x^\\prime)} = \\cos^{x^\\prime}{(x^\\prime)} and \\hat{p}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)} = \\cos{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)} and \\hat{p}{(x^\\prime)} + \\hat{p}^{x^\\prime}{(x^\\prime)} = \\hat{p}^{x^\\prime}{(x^\\prime)} + \\cos{(x^\\prime)} and \\hat{p}{(x^\\prime)} + 2 \\hat{p}^{x^\\prime}{(x^\\prime)} = 2 \\hat{p}^{x^\\prime}{(x^\\prime)} + \\cos{(x^\\prime)} and \\hat{p}{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)} = \\cos{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)} and (\\hat{p}{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)}) \\cos{(x^\\prime)} = (\\cos{(x^\\prime)} + 2 \\cos^{x^\\prime}{(x^\\prime)}) \\cos{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(cos(Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Add(Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True))))"], [["add", 4, "Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(2), Pow(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))), Add(cos(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))))"], [["times", 6, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))), Mul(Add(cos(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))), cos(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(V,m)} = \\cos{(V + m)}, then obtain (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV)^{2} + \\tilde{\\infty} = (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV) \\int \\cos^{m}{(V + m)} dV + \\tilde{\\infty}", "derivation": "\\operatorname{v_{2}}{(V,m)} = \\cos{(V + m)} and \\operatorname{v_{2}}^{m}{(V,m)} = \\cos^{m}{(V + m)} and \\int \\operatorname{v_{2}}^{m}{(V,m)} dV = \\int \\cos^{m}{(V + m)} dV and (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV)^{2} = (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV) \\int \\cos^{m}{(V + m)} dV and (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV)^{2} + \\tilde{\\infty} = (\\int \\operatorname{v_{2}}^{m}{(V,m)} dV) \\int \\cos^{m}{(V + m)} dV + \\tilde{\\infty}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('V', commutative=True), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(cos(Add(Symbol('V', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(cos(Add(Symbol('V', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["times", 3, "Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True)))"], "Equality(Pow(Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(2)), Mul(Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(cos(Add(Symbol('V', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True)))))"], [["minus", 4, "zoo"], "Equality(Add(Pow(Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(2)), zoo), Add(Mul(Integral(Pow(Function('v_2')(Symbol('V', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(cos(Add(Symbol('V', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('V', commutative=True)))), zoo))"]]}, {"prompt": "Given \\Psi_{nl}{(\\rho)} = \\log{(\\log{(\\rho)})}, then derive \\int \\Psi_{nl}{(\\rho)} d\\rho = S + \\rho \\log{(\\log{(\\rho)})} - \\operatorname{li}{(\\rho)}, then obtain (S + \\rho \\log{(\\log{(\\rho)})} - \\operatorname{li}{(\\rho)})^{\\rho} = (\\int \\log{(\\log{(\\rho)})} d\\rho)^{\\rho}", "derivation": "\\Psi_{nl}{(\\rho)} = \\log{(\\log{(\\rho)})} and \\int \\Psi_{nl}{(\\rho)} d\\rho = \\int \\log{(\\log{(\\rho)})} d\\rho and (\\int \\Psi_{nl}{(\\rho)} d\\rho)^{\\rho} = (\\int \\log{(\\log{(\\rho)})} d\\rho)^{\\rho} and \\int \\Psi_{nl}{(\\rho)} d\\rho = S + \\rho \\log{(\\log{(\\rho)})} - \\operatorname{li}{(\\rho)} and (S + \\rho \\log{(\\log{(\\rho)})} - \\operatorname{li}{(\\rho)})^{\\rho} = (\\int \\log{(\\log{(\\rho)})} d\\rho)^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\rho', commutative=True)), log(log(Symbol('\\\\rho', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Add(Symbol('S', commutative=True), Mul(Symbol('\\\\rho', commutative=True), log(log(Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), li(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('S', commutative=True), Mul(Symbol('\\\\rho', commutative=True), log(log(Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), li(Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)), Pow(Integral(log(log(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{x})} = e^{F_{x}}, then derive \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = P_{g} + e^{F_{x}}, then obtain \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = P_{g} + \\operatorname{x^{{\\}'}}{(F_{x})}", "derivation": "\\operatorname{x^{{\\}'}}{(F_{x})} = e^{F_{x}} and \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = \\int e^{F_{x}} dF_{x} and \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = P_{g} + e^{F_{x}} and \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = P_{g} + \\operatorname{x^{{\\}'}}{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(exp(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('P_g', commutative=True), exp(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('P_g', commutative=True), Function('x^\\\\prime')(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\phi_2,\\rho)} = \\frac{\\phi_2}{\\rho}, then obtain \\frac{d}{d \\rho} \\frac{1}{\\rho} + 1 = \\frac{\\partial}{\\partial \\rho} \\frac{\\phi_2}{\\rho^{2} \\operatorname{P_{e}}{(\\phi_2,\\rho)}} + 1", "derivation": "\\operatorname{P_{e}}{(\\phi_2,\\rho)} = \\frac{\\phi_2}{\\rho} and \\frac{1}{\\rho} = \\frac{\\phi_2}{\\rho^{2} \\operatorname{P_{e}}{(\\phi_2,\\rho)}} and \\frac{d}{d \\rho} \\frac{1}{\\rho} = \\frac{\\partial}{\\partial \\rho} \\frac{\\phi_2}{\\rho^{2} \\operatorname{P_{e}}{(\\phi_2,\\rho)}} and \\frac{d}{d \\rho} \\frac{1}{\\rho} + 1 = \\frac{\\partial}{\\partial \\rho} \\frac{\\phi_2}{\\rho^{2} \\operatorname{P_{e}}{(\\phi_2,\\rho)}} + 1", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\rho', commutative=True), Function('P_e')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given q{(\\psi^*)} = \\psi^*, then derive \\frac{d}{d \\psi^*} q{(\\psi^*)} + 1 = 2, then obtain \\frac{d}{d \\psi^*} \\psi^* + 1 = 2", "derivation": "q{(\\psi^*)} = \\psi^* and \\frac{d}{d \\psi^*} q{(\\psi^*)} = \\frac{d}{d \\psi^*} \\psi^* and \\frac{d}{d \\psi^*} \\psi^* + \\frac{d}{d \\psi^*} q{(\\psi^*)} = 2 \\frac{d}{d \\psi^*} \\psi^* and \\frac{d}{d \\psi^*} q{(\\psi^*)} + 1 = 2 and \\frac{d}{d \\psi^*} \\psi^* + 1 = 2", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Function('q')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('q')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(1)), Integer(2))"]]}, {"prompt": "Given L{(\\varphi)} = \\log{(\\varphi)}, then obtain \\frac{(L^{\\varphi}{(\\varphi)} + \\log{(\\varphi)}) L^{\\varphi}{(\\varphi)}}{L{(\\varphi)}} = \\frac{(L^{\\varphi}{(\\varphi)} + \\log{(\\varphi)}) \\log{(\\varphi)}^{\\varphi}}{L{(\\varphi)}}", "derivation": "L{(\\varphi)} = \\log{(\\varphi)} and L^{\\varphi}{(\\varphi)} = \\log{(\\varphi)}^{\\varphi} and \\frac{L^{\\varphi}{(\\varphi)}}{L{(\\varphi)}} = \\frac{\\log{(\\varphi)}^{\\varphi}}{L{(\\varphi)}} and L^{\\varphi}{(\\varphi)} + \\log{(\\varphi)} = \\log{(\\varphi)} + \\log{(\\varphi)}^{\\varphi} and \\frac{(\\log{(\\varphi)} + \\log{(\\varphi)}^{\\varphi}) L^{\\varphi}{(\\varphi)}}{L{(\\varphi)}} = \\frac{(\\log{(\\varphi)} + \\log{(\\varphi)}^{\\varphi}) \\log{(\\varphi)}^{\\varphi}}{L{(\\varphi)}} and \\frac{(L^{\\varphi}{(\\varphi)} + \\log{(\\varphi)}) L^{\\varphi}{(\\varphi)}}{L{(\\varphi)}} = \\frac{(L^{\\varphi}{(\\varphi)} + \\log{(\\varphi)}) \\log{(\\varphi)}^{\\varphi}}{L{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["divide", 2, "Function('L')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "log(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Add(log(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["times", 3, "Add(log(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Add(log(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Add(log(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Mul(Add(Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Pow(Function('L')(Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{A},Q)} = \\cos{(Q + \\mathbf{A})}, then obtain - \\mathbf{A} - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\dot{x}{(\\mathbf{A},Q)} = - \\mathbf{A} - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\cos{(Q + \\mathbf{A})}", "derivation": "\\dot{x}{(\\mathbf{A},Q)} = \\cos{(Q + \\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\dot{x}{(\\mathbf{A},Q)} = \\frac{\\partial}{\\partial \\mathbf{A}} \\cos{(Q + \\mathbf{A})} and \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\dot{x}{(\\mathbf{A},Q)} = \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\cos{(Q + \\mathbf{A})} and - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\dot{x}{(\\mathbf{A},Q)} = - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\cos{(Q + \\mathbf{A})} and - \\mathbf{A} - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\dot{x}{(\\mathbf{A},Q)} = - \\mathbf{A} - \\frac{\\partial^{2}}{\\partial Q\\partial \\mathbf{A}} \\cos{(Q + \\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Q', commutative=True)), cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["minus", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(cos(Add(Symbol('Q', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\phi)} = \\cos{(\\phi)}, then derive \\log{(\\int \\operatorname{F_{g}}{(\\phi)} d\\phi)} = \\log{(\\theta_2 + \\sin{(\\phi)})}, then obtain \\log{(\\int \\cos{(\\phi)} d\\phi)} = \\log{(\\theta_2 + \\sin{(\\phi)})}", "derivation": "\\operatorname{F_{g}}{(\\phi)} = \\cos{(\\phi)} and \\int \\operatorname{F_{g}}{(\\phi)} d\\phi = \\int \\cos{(\\phi)} d\\phi and \\log{(\\int \\operatorname{F_{g}}{(\\phi)} d\\phi)} = \\log{(\\int \\cos{(\\phi)} d\\phi)} and \\log{(\\int \\operatorname{F_{g}}{(\\phi)} d\\phi)} = \\log{(\\theta_2 + \\sin{(\\phi)})} and \\log{(\\int \\cos{(\\phi)} d\\phi)} = \\log{(\\theta_2 + \\sin{(\\phi)})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('F_g')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), log(Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('F_g')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), log(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(log(Integral(cos(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), log(Add(Symbol('\\\\theta_2', commutative=True), sin(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given n{(\\pi)} = e^{\\pi}, then obtain (n{(\\pi)} + e^{\\pi})^{(n{(\\pi)} + e^{\\pi}) e^{- \\pi}} = 4 e^{\\pi (n{(\\pi)} + e^{\\pi}) e^{- \\pi}}", "derivation": "n{(\\pi)} = e^{\\pi} and n{(\\pi)} + e^{\\pi} = 2 e^{\\pi} and (n{(\\pi)} + e^{\\pi})^{2} = 4 e^{2 \\pi} and (n{(\\pi)} + e^{\\pi}) e^{2 \\pi} = 2 e^{3 \\pi} and (n{(\\pi)} + e^{\\pi}) e^{- \\pi} = 2 and (n{(\\pi)} + e^{\\pi})^{(n{(\\pi)} + e^{\\pi}) e^{- \\pi}} = 4 e^{\\pi (n{(\\pi)} + e^{\\pi}) e^{- \\pi}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Integer(2)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True)))))"], [["times", 2, "exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\pi', commutative=True)))), Mul(Integer(2), exp(Mul(Integer(3), Symbol('\\\\pi', commutative=True)))))"], [["divide", 4, "exp(Mul(Integer(3), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Integer(2))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Mul(Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True))))), Mul(Integer(4), exp(Mul(Symbol('\\\\pi', commutative=True), Add(Function('n')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))))"]]}, {"prompt": "Given \\rho_{f}{(m,\\mathbf{g},p)} = \\mathbf{g} m p and M{(m,\\mathbf{g},p)} = p^{2} (- \\mathbf{g} + \\rho_{f}{(m,\\mathbf{g},p)}) (\\mathbf{g} m p - \\mathbf{g}), then obtain m M{(m,\\mathbf{g},p)} \\rho_{f}{(m,\\mathbf{g},p)} = m p^{2} (\\mathbf{g} m p - \\mathbf{g})^{2} \\rho_{f}{(m,\\mathbf{g},p)}", "derivation": "\\rho_{f}{(m,\\mathbf{g},p)} = \\mathbf{g} m p and - \\mathbf{g} + \\rho_{f}{(m,\\mathbf{g},p)} = \\mathbf{g} m p - \\mathbf{g} and M{(m,\\mathbf{g},p)} = p^{2} (- \\mathbf{g} + \\rho_{f}{(m,\\mathbf{g},p)}) (\\mathbf{g} m p - \\mathbf{g}) and M{(m,\\mathbf{g},p)} = p^{2} (\\mathbf{g} m p - \\mathbf{g})^{2} and m M{(m,\\mathbf{g},p)} = m p^{2} (\\mathbf{g} m p - \\mathbf{g})^{2} and m M{(m,\\mathbf{g},p)} \\rho_{f}{(m,\\mathbf{g},p)} = m p^{2} (\\mathbf{g} m p - \\mathbf{g})^{2} \\rho_{f}{(m,\\mathbf{g},p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('M')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('p', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Integer(2))))"], [["times", 4, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Function('M')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Integer(2))))"], [["times", 5, "Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Symbol('m', commutative=True), Function('M')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True)), Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('m', commutative=True), Pow(Symbol('p', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('m', commutative=True), Symbol('p', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True))), Integer(2)), Function('\\\\rho_f')(Symbol('m', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(z)} = \\log{(z)} and \\eta{(z)} = \\operatorname{v_{z}}^{z}{(z)}, then obtain \\int - (\\log{(z)}^{z})^{z} dz = \\int - \\eta^{z}{(z)} dz", "derivation": "\\operatorname{v_{z}}{(z)} = \\log{(z)} and \\operatorname{v_{z}}^{z}{(z)} = \\log{(z)}^{z} and \\eta{(z)} = \\operatorname{v_{z}}^{z}{(z)} and (\\operatorname{v_{z}}^{z}{(z)})^{z} = (\\log{(z)}^{z})^{z} and \\eta{(z)} = \\log{(z)}^{z} and (\\operatorname{v_{z}}^{z}{(z)})^{z} = \\eta^{z}{(z)} and (\\log{(z)}^{z})^{z} = \\eta^{z}{(z)} and - (\\log{(z)}^{z})^{z} = - \\eta^{z}{(z)} and \\int - (\\log{(z)}^{z})^{z} dz = \\int - \\eta^{z}{(z)} dz", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('z', commutative=True)), Pow(Function('v_z')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Function('v_z')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\eta')(Symbol('z', commutative=True)), Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Pow(Function('v_z')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Function('\\\\eta')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Function('\\\\eta')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["times", 7, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('z', commutative=True)), Symbol('z', commutative=True))))"], [["integrate", 8, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Pow(log(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Mul(Integer(-1), Pow(Function('\\\\eta')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbf{J}_f,v_{z})} = \\mathbf{J}_f - v_{z}, then derive \\mathbf{J}_f - v_{z} + \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\mathbf{J}_f v_{z} + \\mathbf{J}_f + \\mathbf{P} - \\frac{v_{z}^{2}}{2} - v_{z}, then obtain p{(\\mathbf{J}_f,v_{z})} + \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\mathbf{J}_f v_{z} + \\mathbf{P} - \\frac{v_{z}^{2}}{2} + p{(\\mathbf{J}_f,v_{z})}", "derivation": "p{(\\mathbf{J}_f,v_{z})} = \\mathbf{J}_f - v_{z} and \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\int (\\mathbf{J}_f - v_{z}) dv_{z} and \\mathbf{J}_f - v_{z} + \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\mathbf{J}_f - v_{z} + \\int (\\mathbf{J}_f - v_{z}) dv_{z} and \\mathbf{J}_f - v_{z} + \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\mathbf{J}_f v_{z} + \\mathbf{J}_f + \\mathbf{P} - \\frac{v_{z}^{2}}{2} - v_{z} and p{(\\mathbf{J}_f,v_{z})} + \\int p{(\\mathbf{J}_f,v_{z})} dv_{z} = \\mathbf{J}_f v_{z} + \\mathbf{P} - \\frac{v_{z}^{2}}{2} + p{(\\mathbf{J}_f,v_{z})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Integral(Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_z', commutative=True), Integer(2))), Function('p')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\eta{(i,z^{*})} = i^{z^{*}}, then obtain \\frac{\\partial}{\\partial i} 2 \\eta^{3}{(i,z^{*})} = \\frac{\\partial}{\\partial i} 2 i^{z^{*}} \\eta^{2}{(i,z^{*})}", "derivation": "\\eta{(i,z^{*})} = i^{z^{*}} and 2 \\eta{(i,z^{*})} = i^{z^{*}} + \\eta{(i,z^{*})} and (i^{z^{*}} + \\eta{(i,z^{*})}) \\eta{(i,z^{*})} = i^{z^{*}} (i^{z^{*}} + \\eta{(i,z^{*})}) and 2 \\eta^{2}{(i,z^{*})} = 2 i^{z^{*}} \\eta{(i,z^{*})} and 2 \\eta^{3}{(i,z^{*})} = 2 i^{z^{*}} \\eta^{2}{(i,z^{*})} and \\frac{\\partial}{\\partial i} 2 \\eta^{3}{(i,z^{*})} = \\frac{\\partial}{\\partial i} 2 i^{z^{*}} \\eta^{2}{(i,z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)))"], [["add", 1, "Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))), Add(Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))))"], [["times", 1, "Add(Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)))"], "Equality(Mul(Add(Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Add(Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Pow(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Mul(Integer(2), Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))))"], [["times", 4, "Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Integer(3))), Mul(Integer(2), Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Pow(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Integer(2))))"], [["differentiate", 5, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Integer(3))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Pow(Function('\\\\eta')(Symbol('i', commutative=True), Symbol('z^*', commutative=True)), Integer(2))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} = V_{\\mathbf{B}}^{\\phi_1}, then obtain \\iint V_{\\mathbf{B}}^{\\phi_1} \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} d\\phi_1 d\\phi_1 = \\iint V_{\\mathbf{B}}^{2 \\phi_1} d\\phi_1 d\\phi_1", "derivation": "\\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} = V_{\\mathbf{B}}^{\\phi_1} and V_{\\mathbf{B}}^{\\phi_1} \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} = V_{\\mathbf{B}}^{2 \\phi_1} and \\int V_{\\mathbf{B}}^{\\phi_1} \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} d\\phi_1 = \\int V_{\\mathbf{B}}^{2 \\phi_1} d\\phi_1 and \\iint V_{\\mathbf{B}}^{\\phi_1} \\operatorname{C_{2}}{(V_{\\mathbf{B}},\\phi_1)} d\\phi_1 d\\phi_1 = \\iint V_{\\mathbf{B}}^{2 \\phi_1} d\\phi_1 d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then obtain (- \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})})^{\\mathbf{v}} = 1", "derivation": "\\operatorname{A_{1}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and 0 = - \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})} and 0^{\\mathbf{v}} = (- \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})})^{\\mathbf{v}} and - \\frac{0^{\\mathbf{v}}}{\\operatorname{A_{1}}{(\\mathbf{v})}} = - \\frac{(- \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})})^{\\mathbf{v}}}{\\operatorname{A_{1}}{(\\mathbf{v})}} and - \\frac{(- \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})})^{\\mathbf{v}}}{\\operatorname{A_{1}}{(\\mathbf{v})}} = - \\frac{1}{\\operatorname{A_{1}}{(\\mathbf{v})}} and (- \\operatorname{A_{1}}{(\\mathbf{v})} + \\sin{(\\mathbf{v})})^{\\mathbf{v}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["minus", 1, "Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))), sin(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 3, "Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))), sin(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))), sin(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))))"], [["times", 5, "Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{v}', commutative=True))), sin(Symbol('\\\\mathbf{v}', commutative=True))), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1))"]]}, {"prompt": "Given G{(a^{\\dagger},\\mathbf{E})} = \\sin{(\\mathbf{E}^{a^{\\dagger}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} e^{\\mathbf{E}^{- a^{\\dagger}} G{(a^{\\dagger},\\mathbf{E})}} = \\frac{\\partial}{\\partial \\mathbf{E}} e^{\\mathbf{E}^{- a^{\\dagger}} \\sin{(\\mathbf{E}^{a^{\\dagger}})}}", "derivation": "G{(a^{\\dagger},\\mathbf{E})} = \\sin{(\\mathbf{E}^{a^{\\dagger}})} and \\mathbf{E}^{- a^{\\dagger}} G{(a^{\\dagger},\\mathbf{E})} = \\mathbf{E}^{- a^{\\dagger}} \\sin{(\\mathbf{E}^{a^{\\dagger}})} and e^{\\mathbf{E}^{- a^{\\dagger}} G{(a^{\\dagger},\\mathbf{E})}} = e^{\\mathbf{E}^{- a^{\\dagger}} \\sin{(\\mathbf{E}^{a^{\\dagger}})}} and \\frac{\\partial}{\\partial \\mathbf{E}} e^{\\mathbf{E}^{- a^{\\dagger}} G{(a^{\\dagger},\\mathbf{E})}} = \\frac{\\partial}{\\partial \\mathbf{E}} e^{\\mathbf{E}^{- a^{\\dagger}} \\sin{(\\mathbf{E}^{a^{\\dagger}})}}", "srepr_derivation": [["get_premise", "Equality(Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), exp(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('G')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), sin(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(\\mathbf{B})} = \\log{(\\mathbf{B})}, then obtain 3 y{(\\mathbf{B})} + \\log{(\\mathbf{B})} = y{(\\mathbf{B})} + 3 \\log{(\\mathbf{B})}", "derivation": "y{(\\mathbf{B})} = \\log{(\\mathbf{B})} and 2 y{(\\mathbf{B})} = y{(\\mathbf{B})} + \\log{(\\mathbf{B})} and 3 y{(\\mathbf{B})} + \\log{(\\mathbf{B})} = 2 y{(\\mathbf{B})} + 2 \\log{(\\mathbf{B})} and 3 y{(\\mathbf{B})} + \\log{(\\mathbf{B})} = y{(\\mathbf{B})} + 3 \\log{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "Function('y')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(2), Function('y')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Function('y')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 2, "Add(Function('y')(Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('y')(Symbol('\\\\mathbf{B}', commutative=True))), log(Symbol('\\\\mathbf{B}', commutative=True))), Add(Mul(Integer(2), Function('y')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('y')(Symbol('\\\\mathbf{B}', commutative=True))), log(Symbol('\\\\mathbf{B}', commutative=True))), Add(Function('y')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(3), log(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(J)} = \\cos{(J)} and \\mathbf{S}{(J)} = - \\cos{(J)} + \\frac{d}{d J} \\cos{(J)}, then derive - \\cos{(J)} + \\frac{d}{d J} \\operatorname{J_{\\varepsilon}}{(J)} = - \\sin{(J)} - \\cos{(J)}, then obtain \\mathbf{S}^{J}{(J)} = (- \\sin{(J)} - \\cos{(J)})^{J}", "derivation": "\\operatorname{J_{\\varepsilon}}{(J)} = \\cos{(J)} and \\frac{d}{d J} \\operatorname{J_{\\varepsilon}}{(J)} = \\frac{d}{d J} \\cos{(J)} and - \\cos{(J)} + \\frac{d}{d J} \\operatorname{J_{\\varepsilon}}{(J)} = - \\cos{(J)} + \\frac{d}{d J} \\cos{(J)} and - \\cos{(J)} + \\frac{d}{d J} \\operatorname{J_{\\varepsilon}}{(J)} = - \\sin{(J)} - \\cos{(J)} and - \\cos{(J)} + \\frac{d}{d J} \\cos{(J)} = - \\sin{(J)} - \\cos{(J)} and \\mathbf{S}{(J)} = - \\cos{(J)} + \\frac{d}{d J} \\cos{(J)} and \\mathbf{S}{(J)} = - \\sin{(J)} - \\cos{(J)} and \\mathbf{S}^{J}{(J)} = (- \\sin{(J)} - \\cos{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["add", 2, "Mul(Integer(-1), cos(Symbol('J', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(Function('J_{\\\\varepsilon}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(Function('J_{\\\\varepsilon}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Mul(Integer(-1), cos(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Mul(Integer(-1), cos(Symbol('J', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Add(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Add(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Mul(Integer(-1), cos(Symbol('J', commutative=True)))))"], [["power", 7, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Mul(Integer(-1), cos(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)} = - C + \\frac{\\mathbf{E}}{p}, then obtain \\frac{\\mathbf{E} (- C + \\frac{\\mathbf{E}}{p})}{p} + p = \\frac{\\mathbf{E} \\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)}}{p} + p", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)} = - C + \\frac{\\mathbf{E}}{p} and - \\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)} = C - \\frac{\\mathbf{E}}{p} and \\frac{\\mathbf{E} \\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)}}{p} = - \\frac{\\mathbf{E} (C - \\frac{\\mathbf{E}}{p})}{p} and \\frac{\\mathbf{E} (- C + \\frac{\\mathbf{E}}{p})}{p} = - \\frac{\\mathbf{E} (C - \\frac{\\mathbf{E}}{p})}{p} and \\frac{\\mathbf{E} (- C + \\frac{\\mathbf{E}}{p})}{p} + p = - \\frac{\\mathbf{E} (C - \\frac{\\mathbf{E}}{p})}{p} + p and \\frac{\\mathbf{E} (- C + \\frac{\\mathbf{E}}{p})}{p} + p = \\frac{\\mathbf{E} \\operatorname{x^{{\\}'}}{(\\mathbf{E},C,p)}}{p} + p", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('C', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('C', commutative=True), Symbol('p', commutative=True))), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))))"], [["minus", 4, "Mul(Integer(-1), Symbol('p', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))))), Symbol('p', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = (\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}}, then obtain - A_{1} + \\frac{\\partial}{\\partial A_{1}} 2 \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = - A_{1} + \\frac{\\partial}{\\partial A_{1}} ((\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}} + \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})})", "derivation": "\\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = (\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}} and 2 \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = (\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}} + \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} and \\frac{\\partial}{\\partial A_{1}} 2 \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = \\frac{\\partial}{\\partial A_{1}} ((\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}} + \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})}) and - A_{1} + \\frac{\\partial}{\\partial A_{1}} 2 \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})} = - A_{1} + \\frac{\\partial}{\\partial A_{1}} ((\\frac{\\sigma_p}{A_{1}})^{\\Psi_{nl}} + \\sigma_{x}{(\\Psi_{nl},\\sigma_p,A_{1})})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True)), Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["add", 1, "Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))), Add(Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Mul(Integer(2), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Add(Pow(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\sigma_x')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(C_{2})} = e^{C_{2}}, then obtain (C_{2} (C_{2} + \\theta{(C_{2})}) - C_{2}) \\theta{(C_{2})} = (C_{2} (C_{2} + e^{C_{2}}) - C_{2}) \\theta{(C_{2})}", "derivation": "\\theta{(C_{2})} = e^{C_{2}} and C_{2} + \\theta{(C_{2})} = C_{2} + e^{C_{2}} and C_{2} (C_{2} + \\theta{(C_{2})}) = C_{2} (C_{2} + e^{C_{2}}) and C_{2} (C_{2} + \\theta{(C_{2})}) - C_{2} = C_{2} (C_{2} + e^{C_{2}}) - C_{2} and (C_{2} (C_{2} + \\theta{(C_{2})}) - C_{2}) \\theta{(C_{2})} = (C_{2} (C_{2} + e^{C_{2}}) - C_{2}) \\theta{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["add", 1, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Function('\\\\theta')(Symbol('C_2', commutative=True))), Add(Symbol('C_2', commutative=True), exp(Symbol('C_2', commutative=True))))"], [["times", 2, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Function('\\\\theta')(Symbol('C_2', commutative=True)))), Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), exp(Symbol('C_2', commutative=True)))))"], [["minus", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Function('\\\\theta')(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Symbol('C_2', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), exp(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Symbol('C_2', commutative=True))))"], [["times", 4, "Function('\\\\theta')(Symbol('C_2', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), Function('\\\\theta')(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Symbol('C_2', commutative=True))), Function('\\\\theta')(Symbol('C_2', commutative=True))), Mul(Add(Mul(Symbol('C_2', commutative=True), Add(Symbol('C_2', commutative=True), exp(Symbol('C_2', commutative=True)))), Mul(Integer(-1), Symbol('C_2', commutative=True))), Function('\\\\theta')(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(M,z)} = M + z, then obtain z \\mathbf{p}{(M,z)} - \\int (M + z) dz = z (M + z) - \\int (M + z) dz", "derivation": "\\mathbf{p}{(M,z)} = M + z and \\int \\mathbf{p}{(M,z)} dz = \\int (M + z) dz and z \\mathbf{p}{(M,z)} = z (M + z) and z \\mathbf{p}{(M,z)} - \\int \\mathbf{p}{(M,z)} dz = z (M + z) - \\int \\mathbf{p}{(M,z)} dz and z \\mathbf{p}{(M,z)} - \\int (M + z) dz = z (M + z) - \\int (M + z) dz", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True)), Add(Symbol('M', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), Add(Symbol('M', commutative=True), Symbol('z', commutative=True))))"], [["minus", 3, "Integral(Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(Mul(Symbol('z', commutative=True), Add(Symbol('M', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{p}')(Symbol('M', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(Mul(Symbol('z', commutative=True), Add(Symbol('M', commutative=True), Symbol('z', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('M', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{p}{(\\psi^*)} = e^{\\psi^*}, then derive \\int \\mathbf{p}{(\\psi^*)} d\\psi^* = \\hat{x} + e^{\\psi^*}, then obtain - \\mathbf{r} + \\frac{d}{d \\hat{x}} \\int e^{\\psi^*} d\\psi^* = - \\mathbf{r} + \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} + \\mathbf{p}{(\\psi^*)})", "derivation": "\\mathbf{p}{(\\psi^*)} = e^{\\psi^*} and \\int \\mathbf{p}{(\\psi^*)} d\\psi^* = \\int e^{\\psi^*} d\\psi^* and \\int \\mathbf{p}{(\\psi^*)} d\\psi^* = \\hat{x} + e^{\\psi^*} and \\int e^{\\psi^*} d\\psi^* = \\hat{x} + e^{\\psi^*} and \\int e^{\\psi^*} d\\psi^* = \\hat{x} + \\mathbf{p}{(\\psi^*)} and \\frac{d}{d \\hat{x}} \\int e^{\\psi^*} d\\psi^* = \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} + \\mathbf{p}{(\\psi^*)}) and - \\mathbf{r} + \\frac{d}{d \\hat{x}} \\int e^{\\psi^*} d\\psi^* = - \\mathbf{r} + \\frac{\\partial}{\\partial \\hat{x}} (\\hat{x} + \\mathbf{p}{(\\psi^*)})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["minus", 6, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\Omega)} = \\sin{(\\Omega)}, then derive \\int \\hat{p}_0{(\\Omega)} d\\Omega = \\mathbf{s} - \\cos{(\\Omega)}, then derive 1 = \\frac{\\mathbf{s} - \\cos{(\\Omega)}}{\\tilde{g} - \\cos{(\\Omega)}}, then obtain (\\frac{\\tilde{g} - \\cos{(\\Omega)}}{\\mathbf{s} - \\cos{(\\Omega)}})^{\\mathbf{s}} = 1", "derivation": "\\hat{p}_0{(\\Omega)} = \\sin{(\\Omega)} and \\int \\hat{p}_0{(\\Omega)} d\\Omega = \\int \\sin{(\\Omega)} d\\Omega and \\int \\hat{p}_0{(\\Omega)} d\\Omega = \\mathbf{s} - \\cos{(\\Omega)} and 1 = \\frac{\\mathbf{s} - \\cos{(\\Omega)}}{\\int \\hat{p}_0{(\\Omega)} d\\Omega} and 1 = \\frac{\\mathbf{s} - \\cos{(\\Omega)}}{\\int \\sin{(\\Omega)} d\\Omega} and 1 = \\frac{\\mathbf{s} - \\cos{(\\Omega)}}{\\tilde{g} - \\cos{(\\Omega)}} and \\frac{\\tilde{g} - \\cos{(\\Omega)}}{\\mathbf{s} - \\cos{(\\Omega)}} = 1 and (\\frac{\\tilde{g} - \\cos{(\\Omega)}}{\\mathbf{s} - \\cos{(\\Omega)}})^{\\mathbf{s}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\hat{p}_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Pow(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Pow(Integral(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 5], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Integer(-1))))"], [["divide", 6, "Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))))), Integer(1))"], [["power", 7, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Omega', commutative=True))))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{H}{(\\eta,\\rho)} = \\cos{(\\rho^{\\eta})} and \\operatorname{M_{E}}{(\\eta,\\rho)} = \\rho^{\\eta}, then obtain \\eta + \\cos{(\\operatorname{M_{E}}{(\\eta,\\rho)})} = \\eta + \\mathbf{H}{(\\eta,\\rho)}", "derivation": "\\mathbf{H}{(\\eta,\\rho)} = \\cos{(\\rho^{\\eta})} and \\operatorname{M_{E}}{(\\eta,\\rho)} = \\rho^{\\eta} and \\mathbf{H}{(\\eta,\\rho)} = \\cos{(\\operatorname{M_{E}}{(\\eta,\\rho)})} and \\eta + \\mathbf{H}{(\\eta,\\rho)} = \\eta + \\cos{(\\rho^{\\eta})} and \\eta + \\cos{(\\operatorname{M_{E}}{(\\eta,\\rho)})} = \\eta + \\cos{(\\rho^{\\eta})} and \\eta + \\cos{(\\operatorname{M_{E}}{(\\eta,\\rho)})} = \\eta + \\mathbf{H}{(\\eta,\\rho)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Function('M_E')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["add", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\eta', commutative=True), cos(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\eta', commutative=True), cos(Function('M_E')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), cos(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\eta', commutative=True), cos(Function('M_E')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\eta', commutative=True), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given E{(V_{\\mathbf{B}},k,y)} = V_{\\mathbf{B}} + k + y, then derive \\frac{\\partial}{\\partial V_{\\mathbf{B}}} E{(V_{\\mathbf{B}},k,y)} = 1, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + k + y) = 1", "derivation": "E{(V_{\\mathbf{B}},k,y)} = V_{\\mathbf{B}} + k + y and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} E{(V_{\\mathbf{B}},k,y)} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + k + y) and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} E{(V_{\\mathbf{B}},k,y)} = 1 and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} (V_{\\mathbf{B}} + k + y) = 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given r{(t_{2})} = \\sin{(t_{2})} and \\mu{(t_{2})} = r^{t_{2}}{(t_{2})}, then obtain \\frac{d}{d t_{2}} \\frac{\\mu{(t_{2})}}{r{(t_{2})}} = \\frac{d}{d t_{2}} \\frac{\\sin^{t_{2}}{(t_{2})}}{r{(t_{2})}}", "derivation": "r{(t_{2})} = \\sin{(t_{2})} and r^{t_{2}}{(t_{2})} = \\sin^{t_{2}}{(t_{2})} and \\mu{(t_{2})} = r^{t_{2}}{(t_{2})} and \\mu{(t_{2})} = \\sin^{t_{2}}{(t_{2})} and \\frac{\\mu{(t_{2})}}{r{(t_{2})}} = \\frac{\\sin^{t_{2}}{(t_{2})}}{r{(t_{2})}} and \\frac{d}{d t_{2}} \\frac{\\mu{(t_{2})}}{r{(t_{2})}} = \\frac{d}{d t_{2}} \\frac{\\sin^{t_{2}}{(t_{2})}}{r{(t_{2})}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('r')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('t_2', commutative=True)), Pow(Function('r')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mu')(Symbol('t_2', commutative=True)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["divide", 4, "Function('r')(Symbol('t_2', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('t_2', commutative=True)), Pow(Function('r')(Symbol('t_2', commutative=True)), Integer(-1))), Mul(Pow(Function('r')(Symbol('t_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))))"], [["differentiate", 5, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\mu')(Symbol('t_2', commutative=True)), Pow(Function('r')(Symbol('t_2', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('r')(Symbol('t_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\rho_f,H)} = \\frac{H}{\\rho_f}, then obtain \\frac{\\partial}{\\partial \\rho_f} (- H + \\operatorname{A_{y}}{(\\rho_f,H)})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} = \\frac{\\partial}{\\partial \\rho_f} (- H + \\frac{H}{\\rho_f})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})}", "derivation": "\\operatorname{A_{y}}{(\\rho_f,H)} = \\frac{H}{\\rho_f} and - H + \\operatorname{A_{y}}{(\\rho_f,H)} = - H + \\frac{H}{\\rho_f} and (- H + \\operatorname{A_{y}}{(\\rho_f,H)})^{H} = (- H + \\frac{H}{\\rho_f})^{H} and (- H + \\operatorname{A_{y}}{(\\rho_f,H)})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} = (- H + \\frac{H}{\\rho_f})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} and \\frac{\\partial}{\\partial \\rho_f} (- H + \\operatorname{A_{y}}{(\\rho_f,H)})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})} = \\frac{\\partial}{\\partial \\rho_f} (- H + \\frac{H}{\\rho_f})^{H} \\frac{d}{d \\hat{p}} \\sin{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('A_y')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))))"], [["power", 2, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('A_y')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))), Symbol('H', commutative=True)))"], [["times", 3, "Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('A_y')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))), Symbol('H', commutative=True)), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('A_y')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))), Symbol('H', commutative=True)), Derivative(sin(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(A)} = \\cos{(A)}, then obtain \\int \\frac{\\int M{(A)} dA}{\\int \\cos{(A)} dA} dA = \\int 1 dA", "derivation": "M{(A)} = \\cos{(A)} and \\int M{(A)} dA = \\int \\cos{(A)} dA and \\frac{\\int M{(A)} dA}{\\int \\cos{(A)} dA} = 1 and \\int \\frac{\\int M{(A)} dA}{\\int \\cos{(A)} dA} dA = \\int 1 dA", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('M')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["divide", 2, "Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Mul(Integral(Function('M')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Pow(Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Integral(Function('M')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Pow(Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1))), Tuple(Symbol('A', commutative=True))), Integral(Integer(1), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(E_{n})} = \\sin{(E_{n})}, then derive \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} = E_{n} + \\eta^{\\prime}, then obtain \\int 1 dE_{n} + 1 = \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} + 1", "derivation": "\\operatorname{v_{t}}{(E_{n})} = \\sin{(E_{n})} and \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} = 1 and \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} = \\int 1 dE_{n} and \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} = E_{n} + \\eta^{\\prime} and \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} + 1 = E_{n} + \\eta^{\\prime} + 1 and \\int 1 dE_{n} + 1 = E_{n} + \\eta^{\\prime} + 1 and \\int 1 dE_{n} + 1 = \\int \\frac{\\operatorname{v_{t}}{(E_{n})}}{\\sin{(E_{n})}} dE_{n} + 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["divide", 1, "sin(Symbol('E_n', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('E_n', commutative=True)), Pow(sin(Symbol('E_n', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Function('v_t')(Symbol('E_n', commutative=True)), Pow(sin(Symbol('E_n', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))), Integral(Integer(1), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('v_t')(Symbol('E_n', commutative=True)), Pow(sin(Symbol('E_n', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))), Add(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(Mul(Function('v_t')(Symbol('E_n', commutative=True)), Pow(sin(Symbol('E_n', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))), Integer(1)), Add(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True))), Integer(1)), Add(Symbol('E_n', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Integral(Integer(1), Tuple(Symbol('E_n', commutative=True))), Integer(1)), Add(Integral(Mul(Function('v_t')(Symbol('E_n', commutative=True)), Pow(sin(Symbol('E_n', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} = \\cos^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} and \\mathbf{D}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})}, then obtain \\Psi^{\\dagger} + \\int \\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\int \\mathbf{D}^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} d\\Psi^{\\dagger}", "derivation": "\\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} = \\cos^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} and \\int \\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int \\cos^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} d\\Psi^{\\dagger} and \\Psi^{\\dagger} + \\int \\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\int \\cos^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} d\\Psi^{\\dagger} and \\mathbf{D}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and \\Psi^{\\dagger} + \\int \\mathbf{D}{(\\Psi_{\\lambda},\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\Psi^{\\dagger} + \\int \\mathbf{D}^{\\Psi^{\\dagger}}{(\\Psi_{\\lambda})} d\\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\Psi)} = e^{\\Psi}, then obtain \\frac{2 (\\bar{\\h}{(\\Psi)} - e^{\\Psi})}{\\bar{\\h}{(\\Psi)}} = 0", "derivation": "\\bar{\\h}{(\\Psi)} = e^{\\Psi} and 0 = - \\bar{\\h}{(\\Psi)} + e^{\\Psi} and \\bar{\\h}{(\\Psi)} - e^{\\Psi} = 0 and \\frac{\\bar{\\h}{(\\Psi)} - e^{\\Psi}}{\\bar{\\h}{(\\Psi)}} = 0 and \\frac{2 (\\bar{\\h}{(\\Psi)} - e^{\\Psi})}{\\bar{\\h}{(\\Psi)}} = \\frac{\\bar{\\h}{(\\Psi)} - e^{\\Psi}}{\\bar{\\h}{(\\Psi)}} and \\frac{2 (\\bar{\\h}{(\\Psi)} - e^{\\Psi})}{\\bar{\\h}{(\\Psi)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Integer(0))"], [["divide", 3, "Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Pow(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Integer(0))"], [["add", 4, "Mul(Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Pow(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(2), Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Pow(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Mul(Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Pow(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Add(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Pow(Function('\\\\hbar')(Symbol('\\\\Psi', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)} = \\frac{H}{\\theta_1}, then obtain - \\frac{\\frac{\\partial}{\\partial \\theta_1} \\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1} + \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1^{2}} = \\frac{2 H}{\\theta_1^{3}}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)} = \\frac{H}{\\theta_1} and - \\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)} = - \\frac{H}{\\theta_1} and - \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1} = - \\frac{H}{\\theta_1^{2}} and \\frac{\\partial}{\\partial \\theta_1} - \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1} = \\frac{\\partial}{\\partial \\theta_1} - \\frac{H}{\\theta_1^{2}} and - \\frac{\\frac{\\partial}{\\partial \\theta_1} \\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1} + \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\theta_1,H)}}{\\theta_1^{2}} = \\frac{2 H}{\\theta_1^{3}}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))))"], [["times", 2, "Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2))))"], [["differentiate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('H', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-2)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True)))), Mul(Integer(2), Symbol('H', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-3))))"]]}, {"prompt": "Given \\phi_{1}{(T,Q)} = \\cos^{Q}{(T)}, then derive \\frac{\\partial^{2}}{\\partial T\\partial Q} \\phi_{1}{(T,Q)} = - \\frac{(Q \\log{(\\cos{(T)})} + 1) \\sin{(T)} \\cos^{Q}{(T)}}{\\cos{(T)}}, then obtain \\frac{\\partial^{2}}{\\partial T\\partial Q} \\phi_{1}{(T,Q)} = \\frac{\\partial^{2}}{\\partial T\\partial Q} \\cos^{Q}{(T)}", "derivation": "\\phi_{1}{(T,Q)} = \\cos^{Q}{(T)} and - T + \\phi_{1}{(T,Q)} = - T + \\cos^{Q}{(T)} and \\frac{\\partial}{\\partial Q} (- T + \\phi_{1}{(T,Q)}) = \\frac{\\partial}{\\partial Q} (- T + \\cos^{Q}{(T)}) and \\frac{\\partial^{2}}{\\partial T\\partial Q} (- T + \\phi_{1}{(T,Q)}) = \\frac{\\partial^{2}}{\\partial T\\partial Q} (- T + \\cos^{Q}{(T)}) and \\frac{\\partial^{2}}{\\partial T\\partial Q} \\phi_{1}{(T,Q)} = - \\frac{(Q \\log{(\\cos{(T)})} + 1) \\sin{(T)} \\cos^{Q}{(T)}}{\\cos{(T)}} and \\frac{\\partial^{2}}{\\partial T\\partial Q} \\cos^{Q}{(T)} = - \\frac{(Q \\log{(\\cos{(T)})} + 1) \\sin{(T)} \\cos^{Q}{(T)}}{\\cos{(T)}} and \\frac{\\partial^{2}}{\\partial T\\partial Q} \\phi_{1}{(T,Q)} = \\frac{\\partial^{2}}{\\partial T\\partial Q} \\cos^{Q}{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True)))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Add(Mul(Symbol('Q', commutative=True), log(cos(Symbol('T', commutative=True)))), Integer(1)), sin(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(Integer(-1), Add(Mul(Symbol('Q', commutative=True), log(cos(Symbol('T', commutative=True)))), Integer(1)), sin(Symbol('T', commutative=True)), Pow(cos(Symbol('T', commutative=True)), Integer(-1)), Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('T', commutative=True)), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(\\lambda,t_{1},H)} = \\frac{H \\lambda}{t_{1}}, then obtain \\int \\frac{\\partial}{\\partial t_{1}} t{(\\lambda,t_{1},H)} dH = - \\frac{H^{2} \\lambda}{2 t_{1}^{2}} + \\mathbf{E}", "derivation": "t{(\\lambda,t_{1},H)} = \\frac{H \\lambda}{t_{1}} and \\frac{\\partial}{\\partial t_{1}} t{(\\lambda,t_{1},H)} = \\frac{\\partial}{\\partial t_{1}} \\frac{H \\lambda}{t_{1}} and \\int \\frac{\\partial}{\\partial t_{1}} t{(\\lambda,t_{1},H)} dH = \\int \\frac{\\partial}{\\partial t_{1}} \\frac{H \\lambda}{t_{1}} dH and \\int \\frac{\\partial}{\\partial t_{1}} t{(\\lambda,t_{1},H)} dH = - \\frac{H^{2} \\lambda}{2 t_{1}^{2}} + \\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('t')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Mul(Symbol('H', commutative=True), Symbol('\\\\lambda', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-1))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('t')(Symbol('\\\\lambda', commutative=True), Symbol('t_1', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2)), Symbol('\\\\lambda', commutative=True), Pow(Symbol('t_1', commutative=True), Integer(-2))), Symbol('\\\\mathbf{E}', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)} = (- \\phi + \\phi_1)^{\\hat{x}_0}, then obtain \\iint (- \\phi + \\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)}) d\\hat{x}_0 d\\phi = \\iint (- \\phi + (- \\phi + \\phi_1)^{\\hat{x}_0}) d\\hat{x}_0 d\\phi", "derivation": "\\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)} = (- \\phi + \\phi_1)^{\\hat{x}_0} and - \\phi + \\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)} = - \\phi + (- \\phi + \\phi_1)^{\\hat{x}_0} and \\int (- \\phi + \\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)}) d\\hat{x}_0 = \\int (- \\phi + (- \\phi + \\phi_1)^{\\hat{x}_0}) d\\hat{x}_0 and \\iint (- \\phi + \\phi_{2}{(\\phi_1,\\hat{x}_0,\\phi)}) d\\hat{x}_0 d\\phi = \\iint (- \\phi + (- \\phi + \\phi_1)^{\\hat{x}_0}) d\\hat{x}_0 d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('\\\\phi_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\chi{(n_{1},A_{2})} = \\log{(\\frac{A_{2}}{n_{1}})}, then obtain 2 \\chi{(n_{1},A_{2})} + \\chi^{n_{1}}{(n_{1},A_{2})} = 2 \\chi{(n_{1},A_{2})} + \\log{(\\frac{A_{2}}{n_{1}})}^{n_{1}}", "derivation": "\\chi{(n_{1},A_{2})} = \\log{(\\frac{A_{2}}{n_{1}})} and \\chi^{n_{1}}{(n_{1},A_{2})} = \\log{(\\frac{A_{2}}{n_{1}})}^{n_{1}} and 2 \\chi{(n_{1},A_{2})} = \\chi{(n_{1},A_{2})} + \\log{(\\frac{A_{2}}{n_{1}})} and \\chi{(n_{1},A_{2})} + \\chi^{n_{1}}{(n_{1},A_{2})} + \\log{(\\frac{A_{2}}{n_{1}})} = \\chi{(n_{1},A_{2})} + \\log{(\\frac{A_{2}}{n_{1}})} + \\log{(\\frac{A_{2}}{n_{1}})}^{n_{1}} and 2 \\chi{(n_{1},A_{2})} + \\chi^{n_{1}}{(n_{1},A_{2})} = 2 \\chi{(n_{1},A_{2})} + \\log{(\\frac{A_{2}}{n_{1}})}^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), Symbol('n_1', commutative=True)), Pow(log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Symbol('n_1', commutative=True)))"], [["add", 1, "Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True))), Add(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))))"], [["add", 2, "Add(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"], "Equality(Add(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), Pow(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), Symbol('n_1', commutative=True)), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))), Add(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Pow(log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True))), Pow(Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True)), Symbol('n_1', commutative=True))), Add(Mul(Integer(2), Function('\\\\chi')(Symbol('n_1', commutative=True), Symbol('A_2', commutative=True))), Pow(log(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(b)} = \\sin{(b)}, then derive \\int \\sigma_{x}{(b)} db = \\phi - \\cos{(b)}, then obtain \\frac{\\partial}{\\partial b} (- \\phi + \\cos{(b)} + \\int \\sigma_{x}{(b)} db)^{\\phi} = \\frac{d}{d b} 0^{\\phi}", "derivation": "\\sigma_{x}{(b)} = \\sin{(b)} and \\int \\sigma_{x}{(b)} db = \\int \\sin{(b)} db and \\int \\sigma_{x}{(b)} db - \\int \\sin{(b)} db = 0 and \\int \\sigma_{x}{(b)} db = \\phi - \\cos{(b)} and \\int \\sin{(b)} db = \\phi - \\cos{(b)} and - \\phi + \\cos{(b)} + \\int \\sigma_{x}{(b)} db = 0 and (- \\phi + \\cos{(b)} + \\int \\sigma_{x}{(b)} db)^{\\phi} = 0^{\\phi} and \\frac{\\partial}{\\partial b} (- \\phi + \\cos{(b)} + \\int \\sigma_{x}{(b)} db)^{\\phi} = \\frac{d}{d b} 0^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 2, "Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), cos(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('b', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Integer(0))"], [["power", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('b', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Symbol('\\\\phi', commutative=True)), Pow(Integer(0), Symbol('\\\\phi', commutative=True)))"], [["differentiate", 7, "Symbol('b', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), cos(Symbol('b', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(\\mu)} = \\cos{(\\log{(\\mu)})}, then obtain - S{(\\mu)} \\cos{(\\log{(\\mu)})} + 2 S{(\\mu)} + 2 \\cos{(\\log{(\\mu)})} = - S{(\\mu)} \\cos{(\\log{(\\mu)})} + S{(\\mu)} + 3 \\cos{(\\log{(\\mu)})}", "derivation": "S{(\\mu)} = \\cos{(\\log{(\\mu)})} and S^{2}{(\\mu)} = S{(\\mu)} \\cos{(\\log{(\\mu)})} and S{(\\mu)} + \\cos{(\\log{(\\mu)})} = 2 \\cos{(\\log{(\\mu)})} and - S{(\\mu)} \\cos{(\\log{(\\mu)})} + S{(\\mu)} + \\cos{(\\log{(\\mu)})} = - S{(\\mu)} \\cos{(\\log{(\\mu)})} + 2 \\cos{(\\log{(\\mu)})} and - S^{2}{(\\mu)} + S{(\\mu)} + \\cos{(\\log{(\\mu)})} = - S^{2}{(\\mu)} + 2 \\cos{(\\log{(\\mu)})} and - S^{2}{(\\mu)} + 2 S{(\\mu)} + 2 \\cos{(\\log{(\\mu)})} = - S^{2}{(\\mu)} + S{(\\mu)} + 3 \\cos{(\\log{(\\mu)})} and - S{(\\mu)} \\cos{(\\log{(\\mu)})} + 2 S{(\\mu)} + 2 \\cos{(\\log{(\\mu)})} = - S{(\\mu)} \\cos{(\\log{(\\mu)})} + S{(\\mu)} + 3 \\cos{(\\log{(\\mu)})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Function('S')(Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('S')(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))))"], [["add", 1, "cos(log(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), cos(log(Symbol('\\\\mu', commutative=True)))))"], [["minus", 3, "Mul(Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), cos(log(Symbol('\\\\mu', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mu', commutative=True)), Integer(2))), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Integer(2), cos(log(Symbol('\\\\mu', commutative=True))))))"], [["add", 5, "Add(Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Integer(2), Function('S')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\mu', commutative=True))))), Add(Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mu', commutative=True)), Integer(2))), Function('S')(Symbol('\\\\mu', commutative=True)), Mul(Integer(3), cos(log(Symbol('\\\\mu', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(-1), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), Function('S')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\mu', commutative=True))))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\mu', commutative=True)), cos(log(Symbol('\\\\mu', commutative=True)))), Function('S')(Symbol('\\\\mu', commutative=True)), Mul(Integer(3), cos(log(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\delta{(F_{g},W,\\Psi_{nl})} = \\frac{F_{g} - W}{\\Psi_{nl}}, then obtain \\frac{W \\Psi_{nl}^{2} \\delta{(F_{g},W,\\Psi_{nl})}}{F_{g} - W} = W \\Psi_{nl}", "derivation": "\\delta{(F_{g},W,\\Psi_{nl})} = \\frac{F_{g} - W}{\\Psi_{nl}} and W \\delta{(F_{g},W,\\Psi_{nl})} = \\frac{W (F_{g} - W)}{\\Psi_{nl}} and \\frac{W \\Psi_{nl} \\delta{(F_{g},W,\\Psi_{nl})}}{F_{g} - W} = W and \\frac{W \\Psi_{nl}^{2} \\delta{(F_{g},W,\\Psi_{nl})}}{F_{g} - W} = W \\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], "Equality(Mul(Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True), Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Integer(-1)), Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Symbol('W', commutative=True))"], [["times", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)), Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))), Integer(-1)), Function('\\\\delta')(Symbol('F_g', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Symbol('W', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(\\phi_2,u)} = \\frac{u}{\\phi_2}, then obtain \\sin{(\\frac{\\partial}{\\partial u} (\\bar{\\h}{(\\phi_2,u)} - \\frac{u}{\\phi_2}))} = \\sin{(\\frac{d}{d u} 0)}", "derivation": "\\bar{\\h}{(\\phi_2,u)} = \\frac{u}{\\phi_2} and - u^{2} + \\bar{\\h}{(\\phi_2,u)} = - u^{2} + \\frac{u}{\\phi_2} and 0 = - \\bar{\\h}{(\\phi_2,u)} + \\frac{u}{\\phi_2} and \\bar{\\h}{(\\phi_2,u)} - \\frac{u}{\\phi_2} = 0 and \\frac{\\partial}{\\partial u} (\\bar{\\h}{(\\phi_2,u)} - \\frac{u}{\\phi_2}) = \\frac{d}{d u} 0 and \\sin{(\\frac{\\partial}{\\partial u} (\\bar{\\h}{(\\phi_2,u)} - \\frac{u}{\\phi_2}))} = \\sin{(\\frac{d}{d u} 0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["minus", 1, "Pow(Symbol('u', commutative=True), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(2))), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(2))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(2))), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('u', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["sin", 5], "Equality(sin(Derivative(Add(Function('\\\\hbar')(Symbol('\\\\phi_2', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), sin(Derivative(Integer(0), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(\\mathbf{s},G)} = \\frac{\\mathbf{s}}{G}, then derive \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} = \\frac{1}{G}, then obtain \\frac{\\partial}{\\partial \\mathbf{s}} \\frac{\\mathbf{s}}{G} + \\frac{1}{G} = \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} + \\frac{1}{G}", "derivation": "r{(\\mathbf{s},G)} = \\frac{\\mathbf{s}}{G} and \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} = \\frac{\\partial}{\\partial \\mathbf{s}} \\frac{\\mathbf{s}}{G} and \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} = \\frac{1}{G} and \\frac{1}{G} = \\frac{\\partial}{\\partial \\mathbf{s}} \\frac{\\mathbf{s}}{G} and \\frac{2}{G} = \\frac{\\partial}{\\partial \\mathbf{s}} \\frac{\\mathbf{s}}{G} + \\frac{1}{G} and \\frac{2}{G} = \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} + \\frac{1}{G} and \\frac{\\partial}{\\partial \\mathbf{s}} \\frac{\\mathbf{s}}{G} + \\frac{1}{G} = \\frac{\\partial}{\\partial \\mathbf{s}} r{(\\mathbf{s},G)} + \\frac{1}{G}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('G', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["add", 4, "Pow(Symbol('G', commutative=True), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Derivative(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Derivative(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1))), Add(Derivative(Function('r')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Pow(Symbol('G', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(t_{1})} = \\log{(e^{t_{1}})}, then derive \\int \\mathbf{g}{(t_{1})} dt_{1} = C_{d} + \\frac{t_{1}^{2}}{2}, then derive - \\mathbf{J}_M - \\frac{t_{1}^{2}}{2} + \\int \\mathbf{g}{(t_{1})} dt_{1} = 0, then obtain C_{d} - \\mathbf{J}_M = 0", "derivation": "\\mathbf{g}{(t_{1})} = \\log{(e^{t_{1}})} and \\int \\mathbf{g}{(t_{1})} dt_{1} = \\int \\log{(e^{t_{1}})} dt_{1} and \\int \\mathbf{g}{(t_{1})} dt_{1} - \\int \\log{(e^{t_{1}})} dt_{1} = 0 and \\int \\mathbf{g}{(t_{1})} dt_{1} = C_{d} + \\frac{t_{1}^{2}}{2} and - \\mathbf{J}_M - \\frac{t_{1}^{2}}{2} + \\int \\mathbf{g}{(t_{1})} dt_{1} = 0 and C_{d} - \\mathbf{J}_M = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True)), log(exp(Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(log(exp(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"], [["minus", 2, "Integral(log(exp(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Mul(Integer(-1), Integral(log(exp(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t_1', commutative=True), Integer(2))), Integral(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\chi{(\\psi,\\mathbb{I})} = \\sin{(\\mathbb{I} \\psi)}, then obtain \\psi \\cos{(\\mathbb{I} \\psi)} - 1 = 2 \\psi \\cos{(\\mathbb{I} \\psi)} - \\frac{\\partial}{\\partial \\mathbb{I}} \\chi{(\\psi,\\mathbb{I})} - 1", "derivation": "\\chi{(\\psi,\\mathbb{I})} = \\sin{(\\mathbb{I} \\psi)} and - \\mathbb{I} + \\chi{(\\psi,\\mathbb{I})} = - \\mathbb{I} + \\sin{(\\mathbb{I} \\psi)} and - \\mathbb{I} = - \\mathbb{I} - \\chi{(\\psi,\\mathbb{I})} + \\sin{(\\mathbb{I} \\psi)} and - \\mathbb{I} + \\sin{(\\mathbb{I} \\psi)} = - \\mathbb{I} - \\chi{(\\psi,\\mathbb{I})} + 2 \\sin{(\\mathbb{I} \\psi)} and \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} + \\sin{(\\mathbb{I} \\psi)}) = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\mathbb{I} - \\chi{(\\psi,\\mathbb{I})} + 2 \\sin{(\\mathbb{I} \\psi)}) and \\psi \\cos{(\\mathbb{I} \\psi)} - 1 = 2 \\psi \\cos{(\\mathbb{I} \\psi)} - \\frac{\\partial}{\\partial \\mathbb{I}} \\chi{(\\psi,\\mathbb{I})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))))"], [["minus", 2, "Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(2), sin(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True))))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Symbol('\\\\psi', commutative=True), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\psi', commutative=True), cos(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\psi', commutative=True)))), Mul(Integer(-1), Derivative(Function('\\\\chi')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given u{(I,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (C_{2} - I), then derive u{(I,C_{2})} = 1, then derive \\frac{\\partial}{\\partial C_{2}} u{(I,C_{2})} = 0, then obtain \\frac{d}{d C_{2}} 1 = 0", "derivation": "u{(I,C_{2})} = \\frac{\\partial}{\\partial C_{2}} (C_{2} - I) and u{(I,C_{2})} = 1 and \\frac{\\partial}{\\partial C_{2}} u{(I,C_{2})} = \\frac{d}{d C_{2}} 1 and \\frac{\\partial}{\\partial C_{2}} u{(I,C_{2})} = 0 and \\frac{d}{d C_{2}} 1 = 0", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Derivative(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('u')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Integer(1))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('u')(Symbol('I', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)} = (Q - \\theta_1)^{T}, then obtain \\int \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)}) dT dQ = \\int \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + (Q - \\theta_1)^{T}) dT dQ", "derivation": "\\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)} = (Q - \\theta_1)^{T} and Q - \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)} = Q - \\theta_1 + (Q - \\theta_1)^{T} and - \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)} = - \\theta_1 + (Q - \\theta_1)^{T} and \\int (- \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)}) dT = \\int (- \\theta_1 + (Q - \\theta_1)^{T}) dT and \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)}) dT = \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + (Q - \\theta_1)^{T}) dT and \\int \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + \\operatorname{x^{{\\}'}}{(\\theta_1,T,Q)}) dT dQ = \\int \\frac{\\partial}{\\partial Q} \\int (- \\theta_1 + (Q - \\theta_1)^{T}) dT dQ", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True)))"], [["add", 1, "Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True))))"], [["minus", 2, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('x^\\\\prime')(Symbol('\\\\theta_1', commutative=True), Symbol('T', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given h{(g,v)} = e^{g v}, then obtain \\frac{e^{3 g v}}{g v h^{3}{(g,v)}} = \\frac{e^{4 g v}}{g v h^{4}{(g,v)}}", "derivation": "h{(g,v)} = e^{g v} and 1 = \\frac{e^{g v}}{h{(g,v)}} and \\frac{1}{g v} = \\frac{e^{g v}}{g v h{(g,v)}} and \\frac{e^{g v}}{g v h{(g,v)}} = \\frac{e^{2 g v}}{g v h^{2}{(g,v)}} and \\frac{1}{g v} = \\frac{e^{2 g v}}{g v h^{2}{(g,v)}} and \\frac{e^{g v}}{g v h{(g,v)}} = \\frac{e^{3 g v}}{g v h^{3}{(g,v)}} and \\frac{e^{3 g v}}{g v h^{3}{(g,v)}} = \\frac{e^{4 g v}}{g v h^{4}{(g,v)}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True))))"], [["divide", 1, "Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True)))))"], [["divide", 2, "Mul(Symbol('g', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True)))))"], [["times", 2, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True))))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('g', commutative=True), Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-2)), exp(Mul(Integer(2), Symbol('g', commutative=True), Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-1)), exp(Mul(Symbol('g', commutative=True), Symbol('v', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-3)), exp(Mul(Integer(3), Symbol('g', commutative=True), Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-3)), exp(Mul(Integer(3), Symbol('g', commutative=True), Symbol('v', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), Pow(Function('h')(Symbol('g', commutative=True), Symbol('v', commutative=True)), Integer(-4)), exp(Mul(Integer(4), Symbol('g', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})}, then derive \\frac{d}{d \\mathbf{D}} \\operatorname{r_{0}}{(\\mathbf{D})} + 1 = 1 - \\sin{(\\mathbf{D})}, then obtain e^{\\mathbf{D} (\\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} + 1)} = e^{\\mathbf{D} (1 - \\sin{(\\mathbf{D})})}", "derivation": "\\operatorname{r_{0}}{(\\mathbf{D})} = \\cos{(\\mathbf{D})} and \\mathbf{D} + \\operatorname{r_{0}}{(\\mathbf{D})} = \\mathbf{D} + \\cos{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} (\\mathbf{D} + \\operatorname{r_{0}}{(\\mathbf{D})}) = \\frac{d}{d \\mathbf{D}} (\\mathbf{D} + \\cos{(\\mathbf{D})}) and \\frac{d}{d \\mathbf{D}} \\operatorname{r_{0}}{(\\mathbf{D})} + 1 = 1 - \\sin{(\\mathbf{D})} and \\mathbf{D} (\\frac{d}{d \\mathbf{D}} \\operatorname{r_{0}}{(\\mathbf{D})} + 1) = \\mathbf{D} (1 - \\sin{(\\mathbf{D})}) and \\mathbf{D} (\\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} + 1) = \\mathbf{D} (1 - \\sin{(\\mathbf{D})}) and e^{\\mathbf{D} (\\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} + 1)} = e^{\\mathbf{D} (1 - \\sin{(\\mathbf{D})})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{D}', commutative=True)), cos(Symbol('\\\\mathbf{D}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), Function('r_0')(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), cos(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('r_0')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["times", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Derivative(Function('r_0')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))))))"], [["exp", 6], "Equality(exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(1)))), exp(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True)))))))"]]}, {"prompt": "Given \\Omega{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} and \\mu_{0}{(x^\\prime)} = e^{x^\\prime}, then derive \\Omega{(x^\\prime)} = e^{x^\\prime}, then obtain \\int \\mu_{0}{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} \\mu_{0}{(x^\\prime)} dx^\\prime", "derivation": "\\Omega{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{x^\\prime} and \\mu_{0}{(x^\\prime)} = e^{x^\\prime} and \\Omega{(x^\\prime)} = \\frac{d}{d x^\\prime} \\mu_{0}{(x^\\prime)} and \\Omega{(x^\\prime)} = e^{x^\\prime} and \\mu_{0}{(x^\\prime)} = \\Omega{(x^\\prime)} and \\mu_{0}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\mu_{0}{(x^\\prime)} and \\int \\mu_{0}{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} \\mu_{0}{(x^\\prime)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True)), Derivative(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Function('\\\\Omega')(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(Function('\\\\mu_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{g})} = \\cos{(\\mathbf{g})}, then derive \\int \\mathbf{F}{(\\mathbf{g})} d\\mathbf{g} = \\mathbf{B} + \\sin{(\\mathbf{g})}, then obtain e^{\\mathbf{B} + \\sin{(\\mathbf{g})}} = e^{\\psi^* + \\sin{(\\mathbf{g})}}", "derivation": "\\mathbf{F}{(\\mathbf{g})} = \\cos{(\\mathbf{g})} and \\int \\mathbf{F}{(\\mathbf{g})} d\\mathbf{g} = \\int \\cos{(\\mathbf{g})} d\\mathbf{g} and \\int \\mathbf{F}{(\\mathbf{g})} d\\mathbf{g} = \\mathbf{B} + \\sin{(\\mathbf{g})} and e^{\\int \\mathbf{F}{(\\mathbf{g})} d\\mathbf{g}} = e^{\\int \\cos{(\\mathbf{g})} d\\mathbf{g}} and e^{\\mathbf{B} + \\sin{(\\mathbf{g})}} = e^{\\int \\cos{(\\mathbf{g})} d\\mathbf{g}} and e^{\\mathbf{B} + \\sin{(\\mathbf{g})}} = e^{\\psi^* + \\sin{(\\mathbf{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\mathbf{g}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))), exp(Integral(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True)))), exp(Integral(cos(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(exp(Add(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True)))), exp(Add(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True)))))"]]}, {"prompt": "Given v{(I)} = \\frac{d}{d I} \\log{(I)}, then derive v^{I}{(I)} = (\\frac{1}{I})^{I}, then obtain U + v^{I}{(I)} = U + (\\frac{d}{d I} \\log{(I)})^{I}", "derivation": "v{(I)} = \\frac{d}{d I} \\log{(I)} and v^{I}{(I)} = (\\frac{d}{d I} \\log{(I)})^{I} and v^{I}{(I)} = (\\frac{1}{I})^{I} and U + v^{I}{(I)} = U + (\\frac{1}{I})^{I} and U + (\\frac{d}{d I} \\log{(I)})^{I} = U + (\\frac{1}{I})^{I} and U + v^{I}{(I)} = U + (\\frac{d}{d I} \\log{(I)})^{I}", "srepr_derivation": [["get_premise", "Equality(Function('v')(Symbol('I', commutative=True)), Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('v')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('v')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('I', commutative=True)))"], [["add", 3, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Pow(Function('v')(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Add(Symbol('U', commutative=True), Pow(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('U', commutative=True), Pow(Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))), Add(Symbol('U', commutative=True), Pow(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('U', commutative=True), Pow(Function('v')(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Add(Symbol('U', commutative=True), Pow(Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\theta_1)} = \\log{(e^{\\theta_1})}, then derive \\int \\sigma_{x}{(\\theta_1)} e^{\\theta_1} d\\theta_1 = C + (\\theta_1 - 1) e^{\\theta_1}, then obtain C (\\delta + (\\theta_1 - 1) e^{\\theta_1}) = C (C + (\\theta_1 - 1) e^{\\theta_1})", "derivation": "\\sigma_{x}{(\\theta_1)} = \\log{(e^{\\theta_1})} and \\sigma_{x}{(\\theta_1)} e^{\\theta_1} = e^{\\theta_1} \\log{(e^{\\theta_1})} and \\int \\sigma_{x}{(\\theta_1)} e^{\\theta_1} d\\theta_1 = \\int e^{\\theta_1} \\log{(e^{\\theta_1})} d\\theta_1 and \\int \\sigma_{x}{(\\theta_1)} e^{\\theta_1} d\\theta_1 = C + (\\theta_1 - 1) e^{\\theta_1} and \\int e^{\\theta_1} \\log{(e^{\\theta_1})} d\\theta_1 = C + (\\theta_1 - 1) e^{\\theta_1} and C \\int e^{\\theta_1} \\log{(e^{\\theta_1})} d\\theta_1 = C (C + (\\theta_1 - 1) e^{\\theta_1}) and C (\\delta + (\\theta_1 - 1) e^{\\theta_1}) = C (C + (\\theta_1 - 1) e^{\\theta_1})", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "exp(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Mul(exp(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(exp(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\sigma_x')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('C', commutative=True), Mul(Add(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(exp(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('C', commutative=True), Mul(Add(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)))))"], [["times", 5, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Integral(Mul(exp(Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Mul(Add(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))))"], [["evaluate_integrals", 6], "Equality(Mul(Symbol('C', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Add(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))), Mul(Symbol('C', commutative=True), Add(Symbol('C', commutative=True), Mul(Add(Symbol('\\\\theta_1', commutative=True), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\varepsilon_0,\\mathbf{f})} = \\sin{(\\mathbf{f} \\varepsilon_0)} and \\phi{(\\varepsilon_0,\\mathbf{f})} = \\mathbf{A}^{2}{(\\varepsilon_0,\\mathbf{f})} and \\operatorname{V_{\\mathbf{E}}}{(\\varepsilon_0,\\mathbf{f})} = \\frac{\\partial}{\\partial \\varepsilon_0} \\phi{(\\varepsilon_0,\\mathbf{f})}, then obtain \\operatorname{V_{\\mathbf{E}}}{(\\varepsilon_0,\\mathbf{f})} = \\frac{\\partial}{\\partial \\varepsilon_0} \\sin^{2}{(\\mathbf{f} \\varepsilon_0)}", "derivation": "\\mathbf{A}{(\\varepsilon_0,\\mathbf{f})} = \\sin{(\\mathbf{f} \\varepsilon_0)} and \\phi{(\\varepsilon_0,\\mathbf{f})} = \\mathbf{A}^{2}{(\\varepsilon_0,\\mathbf{f})} and \\phi{(\\varepsilon_0,\\mathbf{f})} = \\sin^{2}{(\\mathbf{f} \\varepsilon_0)} and \\operatorname{V_{\\mathbf{E}}}{(\\varepsilon_0,\\mathbf{f})} = \\frac{\\partial}{\\partial \\varepsilon_0} \\phi{(\\varepsilon_0,\\mathbf{f})} and \\operatorname{V_{\\mathbf{E}}}{(\\varepsilon_0,\\mathbf{f})} = \\frac{\\partial}{\\partial \\varepsilon_0} \\sin^{2}{(\\mathbf{f} \\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\phi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Function('\\\\phi')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Derivative(Pow(sin(Mul(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(V)} = \\frac{d}{d V} e^{V}, then obtain \\frac{\\operatorname{t_{2}}{(V)}}{\\int \\operatorname{t_{2}}{(V)} dV} = \\frac{\\frac{d}{d V} e^{V}}{\\int \\operatorname{t_{2}}{(V)} dV}", "derivation": "\\operatorname{t_{2}}{(V)} = \\frac{d}{d V} e^{V} and \\int \\operatorname{t_{2}}{(V)} dV = \\int \\frac{d}{d V} e^{V} dV and \\frac{\\operatorname{t_{2}}{(V)}}{\\int \\frac{d}{d V} e^{V} dV} = \\frac{\\frac{d}{d V} e^{V}}{\\int \\frac{d}{d V} e^{V} dV} and \\frac{\\operatorname{t_{2}}{(V)}}{\\int \\operatorname{t_{2}}{(V)} dV} = \\frac{\\frac{d}{d V} e^{V}}{\\int \\operatorname{t_{2}}{(V)} dV}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('V', commutative=True)), Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))))"], [["divide", 1, "Integral(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True)))"], "Equality(Mul(Function('t_2')(Symbol('V', commutative=True)), Pow(Integral(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integer(-1))), Mul(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Pow(Integral(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Tuple(Symbol('V', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('t_2')(Symbol('V', commutative=True)), Pow(Integral(Function('t_2')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1))), Mul(Derivative(exp(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Pow(Integral(Function('t_2')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given k{(\\sigma_p)} = \\log{(\\sigma_p)} and \\operatorname{F_{g}}{(\\sigma_p)} = \\sigma_p, then derive \\frac{d}{d \\sigma_p} k{(\\sigma_p)} = \\frac{1}{\\sigma_p}, then obtain \\frac{1}{\\operatorname{F_{g}}{(\\sigma_p)}} = \\frac{d}{d \\operatorname{F_{g}}{(\\sigma_p)}} \\log{(\\operatorname{F_{g}}{(\\sigma_p)})}", "derivation": "k{(\\sigma_p)} = \\log{(\\sigma_p)} and \\operatorname{F_{g}}{(\\sigma_p)} = \\sigma_p and \\frac{d}{d \\sigma_p} k{(\\sigma_p)} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\frac{d}{d \\sigma_p} k{(\\sigma_p)} = \\frac{1}{\\sigma_p} and \\frac{1}{\\sigma_p} = \\frac{d}{d \\sigma_p} \\log{(\\sigma_p)} and \\frac{1}{\\operatorname{F_{g}}{(\\sigma_p)}} = \\frac{d}{d \\operatorname{F_{g}}{(\\sigma_p)}} \\log{(\\operatorname{F_{g}}{(\\sigma_p)})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('k')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('F_g')(Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Derivative(log(Function('F_g')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Function('F_g')(Symbol('\\\\sigma_p', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\omega{(z,\\mathbf{f})} = - \\mathbf{f} + \\log{(z)}, then obtain \\int \\omega{(z,\\mathbf{f})} dz - \\frac{z + \\omega{(z,\\mathbf{f})}}{\\mathbf{f}} = \\int (- \\mathbf{f} + \\log{(z)}) dz - \\frac{z + \\omega{(z,\\mathbf{f})}}{\\mathbf{f}}", "derivation": "\\omega{(z,\\mathbf{f})} = - \\mathbf{f} + \\log{(z)} and \\int \\omega{(z,\\mathbf{f})} dz = \\int (- \\mathbf{f} + \\log{(z)}) dz and \\int \\omega{(z,\\mathbf{f})} dz - \\frac{- \\mathbf{f} + z + \\log{(z)}}{\\mathbf{f}} = \\int (- \\mathbf{f} + \\log{(z)}) dz - \\frac{- \\mathbf{f} + z + \\log{(z)}}{\\mathbf{f}} and \\int \\omega{(z,\\mathbf{f})} dz - \\frac{z + \\omega{(z,\\mathbf{f})}}{\\mathbf{f}} = \\int (- \\mathbf{f} + \\log{(z)}) dz - \\frac{z + \\omega{(z,\\mathbf{f})}}{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('z', commutative=True), log(Symbol('z', commutative=True))))"], "Equality(Add(Integral(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('z', commutative=True), log(Symbol('z', commutative=True))))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('z', commutative=True), log(Symbol('z', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Integral(Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Symbol('z', commutative=True), Function('\\\\omega')(Symbol('z', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain \\operatorname{J_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\operatorname{J_{\\varepsilon}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\operatorname{J_{\\varepsilon}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\operatorname{J_{\\varepsilon}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}^{\\hat{H}_{\\lambda}} and \\operatorname{J_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}^{\\hat{H}_{\\lambda}} = \\log{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}^{\\hat{H}_{\\lambda}} and \\operatorname{J_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\operatorname{J_{\\varepsilon}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} = \\operatorname{J_{\\varepsilon}}^{\\hat{H}_{\\lambda}}{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["add", 1, "Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given M{(\\omega,\\pi)} = \\omega \\pi and L{(\\omega,\\pi)} = \\frac{\\partial}{\\partial \\omega} \\omega \\pi, then obtain (\\frac{\\partial}{\\partial \\omega} M{(\\omega,\\pi)})^{\\omega} - \\frac{1}{L{(\\omega,\\pi)}} = (\\frac{\\partial}{\\partial \\omega} \\omega \\pi)^{\\omega} - \\frac{1}{L{(\\omega,\\pi)}}", "derivation": "M{(\\omega,\\pi)} = \\omega \\pi and \\frac{\\partial}{\\partial \\omega} M{(\\omega,\\pi)} = \\frac{\\partial}{\\partial \\omega} \\omega \\pi and L{(\\omega,\\pi)} = \\frac{\\partial}{\\partial \\omega} \\omega \\pi and (\\frac{\\partial}{\\partial \\omega} M{(\\omega,\\pi)})^{\\omega} = (\\frac{\\partial}{\\partial \\omega} \\omega \\pi)^{\\omega} and 1 = \\frac{\\frac{\\partial}{\\partial \\omega} \\omega \\pi}{L{(\\omega,\\pi)}} and \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega \\pi} = \\frac{1}{L{(\\omega,\\pi)}} and (\\frac{\\partial}{\\partial \\omega} M{(\\omega,\\pi)})^{\\omega} - \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega \\pi} = (\\frac{\\partial}{\\partial \\omega} \\omega \\pi)^{\\omega} - \\frac{1}{\\frac{\\partial}{\\partial \\omega} \\omega \\pi} and (\\frac{\\partial}{\\partial \\omega} M{(\\omega,\\pi)})^{\\omega} - \\frac{1}{L{(\\omega,\\pi)}} = (\\frac{\\partial}{\\partial \\omega} \\omega \\pi)^{\\omega} - \\frac{1}{L{(\\omega,\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["divide", 3, "Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["divide", 5, "Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)), Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)))"], [["minus", 4, "Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Pow(Derivative(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))), Add(Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Pow(Derivative(Function('M')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)))), Add(Pow(Derivative(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Function('L')(Symbol('\\\\omega', commutative=True), Symbol('\\\\pi', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given s{(h,T)} = \\cos{(T h)}, then obtain \\int (- h s{(h,T)} + \\frac{s{(h,T)}}{\\cos{(T h)}}) dh = \\int (- h s{(h,T)} + 1) dh", "derivation": "s{(h,T)} = \\cos{(T h)} and h s{(h,T)} = h \\cos{(T h)} and \\frac{s{(h,T)}}{\\cos{(T h)}} = 1 and - h \\cos{(T h)} + \\frac{s{(h,T)}}{\\cos{(T h)}} = - h \\cos{(T h)} + 1 and - h s{(h,T)} + \\frac{s{(h,T)}}{\\cos{(T h)}} = - h s{(h,T)} + 1 and \\int (- h s{(h,T)} + \\frac{s{(h,T)}}{\\cos{(T h)}}) dh = \\int (- h s{(h,T)} + 1) dh", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True)), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Mul(Symbol('h', commutative=True), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True)))))"], [["divide", 2, "Mul(Symbol('h', commutative=True), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))))"], "Equality(Mul(Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Pow(cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 3, "Mul(Symbol('h', commutative=True), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True)))), Mul(Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Pow(cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('h', commutative=True), cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True)))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True), Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Mul(Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Pow(cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('h', commutative=True), Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Integer(1)))"], [["integrate", 5, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True), Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Mul(Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True)), Pow(cos(Mul(Symbol('T', commutative=True), Symbol('h', commutative=True))), Integer(-1)))), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True), Function('s')(Symbol('h', commutative=True), Symbol('T', commutative=True))), Integer(1)), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given y{(\\chi,\\mathbf{H})} = \\frac{\\chi}{\\mathbf{H}}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{H} y{(\\chi,\\mathbf{H})} = \\frac{d^{2}}{d \\mathbf{H}^{2}} \\chi", "derivation": "y{(\\chi,\\mathbf{H})} = \\frac{\\chi}{\\mathbf{H}} and \\mathbf{H} y{(\\chi,\\mathbf{H})} = \\chi and \\frac{\\partial}{\\partial \\mathbf{H}} \\mathbf{H} y{(\\chi,\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\chi and \\frac{\\partial^{2}}{\\partial \\mathbf{H}^{2}} \\mathbf{H} y{(\\chi,\\mathbf{H})} = \\frac{d^{2}}{d \\mathbf{H}^{2}} \\chi", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\chi', commutative=True))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('y')(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))))"]]}, {"prompt": "Given t{(\\phi,p)} = \\phi^{p}, then obtain \\frac{\\partial}{\\partial \\phi} (\\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int t{(\\phi,p)} dp}{p}) = \\frac{\\partial}{\\partial \\phi} (\\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int \\phi^{p} dp}{p})", "derivation": "t{(\\phi,p)} = \\phi^{p} and \\int t{(\\phi,p)} dp = \\int \\phi^{p} dp and \\phi + \\int t{(\\phi,p)} dp = \\phi + \\int \\phi^{p} dp and \\frac{\\phi + \\int t{(\\phi,p)} dp}{p} = \\frac{\\phi + \\int \\phi^{p} dp}{p} and \\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int t{(\\phi,p)} dp}{p} = \\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int \\phi^{p} dp}{p} and \\frac{\\partial}{\\partial \\phi} (\\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int t{(\\phi,p)} dp}{p}) = \\frac{\\partial}{\\partial \\phi} (\\phi + \\int t{(\\phi,p)} dp + \\frac{\\phi + \\int \\phi^{p} dp}{p})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Symbol('\\\\phi', commutative=True), Integral(Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["divide", 3, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"], [["add", 4, "Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))), Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))))"], [["differentiate", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), Integral(Function('t')(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Integral(Pow(Symbol('\\\\phi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mu,A)} = \\mu e^{A}, then obtain \\frac{\\mu (- A + \\int \\mathbf{S}{(\\mu,A)} dA) e^{A}}{\\int \\mu e^{A} dA} = \\frac{\\mu (- A + \\int \\mu e^{A} dA) e^{A}}{\\int \\mu e^{A} dA}", "derivation": "\\mathbf{S}{(\\mu,A)} = \\mu e^{A} and \\int \\mathbf{S}{(\\mu,A)} dA = \\int \\mu e^{A} dA and - A + \\int \\mathbf{S}{(\\mu,A)} dA = - A + \\int \\mu e^{A} dA and \\frac{\\mathbf{S}{(\\mu,A)}}{\\int \\mu e^{A} dA} = \\frac{\\mu e^{A}}{\\int \\mu e^{A} dA} and \\frac{(- A + \\int \\mathbf{S}{(\\mu,A)} dA) \\mathbf{S}{(\\mu,A)}}{\\int \\mu e^{A} dA} = \\frac{(- A + \\int \\mu e^{A} dA) \\mathbf{S}{(\\mu,A)}}{\\int \\mu e^{A} dA} and \\frac{\\mu (- A + \\int \\mathbf{S}{(\\mu,A)} dA) e^{A}}{\\int \\mu e^{A} dA} = \\frac{\\mu (- A + \\int \\mu e^{A} dA) e^{A}}{\\int \\mu e^{A} dA}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["minus", 2, "Symbol('A', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))))"], [["divide", 1, "Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))), Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))))"], [["times", 3, "Mul(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mu', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), exp(Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))), Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('A', commutative=True)), Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True)))), exp(Symbol('A', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mu', commutative=True), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}{(m_{s},\\dot{y})} = \\dot{y} m_{s}, then derive \\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})} = \\dot{y}, then obtain - \\frac{\\dot{y}}{\\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})}} + 1 = 0", "derivation": "\\mathbf{J}{(m_{s},\\dot{y})} = \\dot{y} m_{s} and \\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})} = \\frac{\\partial}{\\partial m_{s}} \\dot{y} m_{s} and \\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})} = \\dot{y} and 1 = \\frac{\\dot{y}}{\\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})}} and 1 - \\dot{y} = - \\dot{y} + \\frac{\\dot{y}}{\\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})}} and - \\frac{\\dot{y}}{\\frac{\\partial}{\\partial m_{s}} \\mathbf{J}{(m_{s},\\dot{y})}} + 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('m_s', commutative=True)))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('\\\\dot{y}', commutative=True))"], [["divide", 3, "Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 4, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1)))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Pow(Derivative(Function('\\\\mathbf{J}')(Symbol('m_s', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Integer(-1))), Integer(1)), Integer(0))"]]}, {"prompt": "Given \\varphi^{*}{(H)} = \\log{(\\log{(H)})}, then obtain \\int (H + \\frac{\\varphi^{*}{(H)}}{H}) dH = \\int (H + \\frac{\\log{(\\log{(H)})}}{H}) dH", "derivation": "\\varphi^{*}{(H)} = \\log{(\\log{(H)})} and \\frac{\\varphi^{*}{(H)}}{H} = \\frac{\\log{(\\log{(H)})}}{H} and H + \\frac{\\varphi^{*}{(H)}}{H} = H + \\frac{\\log{(\\log{(H)})}}{H} and \\int (H + \\frac{\\varphi^{*}{(H)}}{H}) dH = \\int (H + \\frac{\\log{(\\log{(H)})}}{H}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), log(log(Symbol('H', commutative=True)))))"], [["add", 2, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('H', commutative=True)))), Add(Symbol('H', commutative=True), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), log(log(Symbol('H', commutative=True))))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Symbol('H', commutative=True), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\varphi^*')(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), log(log(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\psi)} = e^{\\psi} and \\operatorname{a^{\\dagger}}{(\\psi)} = (\\lambda{(\\psi)} + e^{\\psi})^{2}, then obtain 2 \\operatorname{a^{\\dagger}}{(\\psi)} + \\int 2 e^{\\psi} d\\psi = 8 e^{2 \\psi} + \\int 2 e^{\\psi} d\\psi", "derivation": "\\lambda{(\\psi)} = e^{\\psi} and \\lambda{(\\psi)} + e^{\\psi} = 2 e^{\\psi} and \\operatorname{a^{\\dagger}}{(\\psi)} = (\\lambda{(\\psi)} + e^{\\psi})^{2} and \\operatorname{a^{\\dagger}}{(\\psi)} = 4 e^{2 \\psi} and \\operatorname{a^{\\dagger}}{(\\psi)} + \\int 2 e^{\\psi} d\\psi = 4 e^{2 \\psi} + \\int 2 e^{\\psi} d\\psi and 2 \\operatorname{a^{\\dagger}}{(\\psi)} + \\int 2 e^{\\psi} d\\psi = \\operatorname{a^{\\dagger}}{(\\psi)} + 4 e^{2 \\psi} + \\int 2 e^{\\psi} d\\psi and 2 \\operatorname{a^{\\dagger}}{(\\psi)} + \\int 2 e^{\\psi} d\\psi = 8 e^{2 \\psi} + \\int 2 e^{\\psi} d\\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Pow(Add(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True)))))"], [["add", 4, "Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(4), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True)))), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["add", 4, "Add(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True)), Mul(Integer(4), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True)))), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('\\\\psi', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(8), exp(Mul(Integer(2), Symbol('\\\\psi', commutative=True)))), Integral(Mul(Integer(2), exp(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(A_{2})} = e^{A_{2}}, then obtain \\iint (A_{2} + \\cos{(\\dot{z}{(A_{2})})}) dA_{2} dA_{2} = \\iint (A_{2} + \\cos{(e^{A_{2}})}) dA_{2} dA_{2}", "derivation": "\\dot{z}{(A_{2})} = e^{A_{2}} and \\cos{(\\dot{z}{(A_{2})})} = \\cos{(e^{A_{2}})} and A_{2} + \\cos{(\\dot{z}{(A_{2})})} = A_{2} + \\cos{(e^{A_{2}})} and \\int (A_{2} + \\cos{(\\dot{z}{(A_{2})})}) dA_{2} = \\int (A_{2} + \\cos{(e^{A_{2}})}) dA_{2} and \\iint (A_{2} + \\cos{(\\dot{z}{(A_{2})})}) dA_{2} dA_{2} = \\iint (A_{2} + \\cos{(e^{A_{2}})}) dA_{2} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\dot{z}')(Symbol('A_2', commutative=True))), cos(exp(Symbol('A_2', commutative=True))))"], [["add", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), cos(Function('\\\\dot{z}')(Symbol('A_2', commutative=True)))), Add(Symbol('A_2', commutative=True), cos(exp(Symbol('A_2', commutative=True)))))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Symbol('A_2', commutative=True), cos(Function('\\\\dot{z}')(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), cos(exp(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"], [["integrate", 4, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Symbol('A_2', commutative=True), cos(Function('\\\\dot{z}')(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), cos(exp(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(y^{\\prime},\\mathbf{D})} = \\mathbf{D} y^{\\prime}, then obtain (\\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{1}{(y^{\\prime},\\mathbf{D})})^{\\mathbf{D}} = (\\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{D} y^{\\prime})^{\\mathbf{D}}", "derivation": "\\phi_{1}{(y^{\\prime},\\mathbf{D})} = \\mathbf{D} y^{\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{1}{(y^{\\prime},\\mathbf{D})} = \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{D} y^{\\prime} and \\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{1}{(y^{\\prime},\\mathbf{D})} = \\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{D} y^{\\prime} and (\\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\phi_{1}{(y^{\\prime},\\mathbf{D})})^{\\mathbf{D}} = (\\phi_{1}{(y^{\\prime},\\mathbf{D})} + \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{D} y^{\\prime})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Add(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Function('\\\\phi_1')(Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(m)} = \\sin{(m)}, then obtain m \\sin{(m)} + (m \\frac{d}{d m} \\mathbf{D}{(m)})^{m} = m \\sin{(m)} + (m \\cos{(m)})^{m}", "derivation": "\\mathbf{D}{(m)} = \\sin{(m)} and \\frac{d}{d m} \\mathbf{D}{(m)} = \\frac{d}{d m} \\sin{(m)} and m \\frac{d}{d m} \\mathbf{D}{(m)} = m \\frac{d}{d m} \\sin{(m)} and (m \\frac{d}{d m} \\mathbf{D}{(m)})^{m} = (m \\frac{d}{d m} \\sin{(m)})^{m} and m \\sin{(m)} + (m \\frac{d}{d m} \\mathbf{D}{(m)})^{m} = m \\sin{(m)} + (m \\frac{d}{d m} \\sin{(m)})^{m} and m \\sin{(m)} + (m \\frac{d}{d m} \\mathbf{D}{(m)})^{m} = m \\sin{(m)} + (m \\cos{(m)})^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["times", 2, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Symbol('m', commutative=True), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Symbol('m', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True)), Pow(Mul(Symbol('m', commutative=True), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True)))"], [["add", 4, "Mul(Symbol('m', commutative=True), sin(Symbol('m', commutative=True)))"], "Equality(Add(Mul(Symbol('m', commutative=True), sin(Symbol('m', commutative=True))), Pow(Mul(Symbol('m', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True))), Add(Mul(Symbol('m', commutative=True), sin(Symbol('m', commutative=True))), Pow(Mul(Symbol('m', commutative=True), Derivative(sin(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Symbol('m', commutative=True), sin(Symbol('m', commutative=True))), Pow(Mul(Symbol('m', commutative=True), Derivative(Function('\\\\mathbf{D}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True))), Add(Mul(Symbol('m', commutative=True), sin(Symbol('m', commutative=True))), Pow(Mul(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given W{(C_{2},\\eta)} = \\int (- C_{2} + \\eta) d\\eta, then derive W{(C_{2},\\eta)} = - C_{2} \\eta + \\frac{\\eta^{2}}{2} + z, then obtain e^{\\int (- C_{2} + \\eta) d\\eta} = e^{- C_{2} \\eta + \\frac{\\eta^{2}}{2} + z}", "derivation": "W{(C_{2},\\eta)} = \\int (- C_{2} + \\eta) d\\eta and W{(C_{2},\\eta)} = - C_{2} \\eta + \\frac{\\eta^{2}}{2} + z and \\int (- C_{2} + \\eta) d\\eta = - C_{2} \\eta + \\frac{\\eta^{2}}{2} + z and e^{\\int (- C_{2} + \\eta) d\\eta} = e^{- C_{2} \\eta + \\frac{\\eta^{2}}{2} + z}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\eta', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('W')(Symbol('C_2', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('z', commutative=True)))"], [["exp", 3], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), exp(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\nabla,n)} = \\nabla + n and \\hat{H}_l{(\\nabla,n)} = - \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n}, then obtain (- \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n})^{n} = \\hat{H}_l^{n}{(\\nabla,n)}", "derivation": "\\tilde{g}{(\\nabla,n)} = \\nabla + n and \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n} = 1 and - \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n} = -1 and \\hat{H}_l{(\\nabla,n)} = - \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n} and \\hat{H}_l{(\\nabla,n)} = -1 and \\hat{H}_l^{n}{(\\nabla,n)} = (-1)^{n} and (- \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n})^{n} = (-1)^{n} and (- \\frac{\\tilde{g}{(\\nabla,n)}}{\\nabla + n})^{n} = \\hat{H}_l^{n}{(\\nabla,n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))), Integer(-1))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1))"], [["power", 5, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Integer(-1), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Mul(Integer(-1), Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Integer(-1), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Mul(Integer(-1), Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)}, then obtain \\mathbf{M}{(\\varepsilon_0)} \\cos{(\\varepsilon_0)} = \\cos^{\\frac{\\mathbf{M}{(\\varepsilon_0)}}{\\cos{(\\varepsilon_0)}} + 1}{(\\varepsilon_0)}", "derivation": "\\mathbf{M}{(\\varepsilon_0)} = \\cos{(\\varepsilon_0)} and \\frac{\\mathbf{M}{(\\varepsilon_0)}}{\\cos{(\\varepsilon_0)}} = 1 and \\frac{\\mathbf{M}{(\\varepsilon_0)}}{\\cos{(\\varepsilon_0)}} + 1 = 2 and \\mathbf{M}{(\\varepsilon_0)} \\cos{(\\varepsilon_0)} = \\cos^{2}{(\\varepsilon_0)} and \\mathbf{M}{(\\varepsilon_0)} \\cos{(\\varepsilon_0)} = \\cos^{\\frac{\\mathbf{M}{(\\varepsilon_0)}}{\\cos{(\\varepsilon_0)}} + 1}{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Integer(1)), Integer(2))"], [["divide", 1, "Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True))), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), cos(Symbol('\\\\varepsilon_0', commutative=True))), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Add(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(cos(Symbol('\\\\varepsilon_0', commutative=True)), Integer(-1))), Integer(1))))"]]}, {"prompt": "Given \\Omega{(n_{2})} = \\log{(n_{2})}, then derive - \\Omega{(n_{2})} + \\frac{d}{d n_{2}} \\Omega{(n_{2})} = - \\Omega{(n_{2})} + \\frac{1}{n_{2}}, then obtain - \\log{(n_{2})} + \\frac{1}{n_{2}} = - \\log{(n_{2})} + \\frac{d}{d n_{2}} \\Omega{(n_{2})}", "derivation": "\\Omega{(n_{2})} = \\log{(n_{2})} and \\frac{d}{d n_{2}} \\Omega{(n_{2})} = \\frac{d}{d n_{2}} \\log{(n_{2})} and - \\Omega{(n_{2})} + \\frac{d}{d n_{2}} \\Omega{(n_{2})} = - \\Omega{(n_{2})} + \\frac{d}{d n_{2}} \\log{(n_{2})} and - \\Omega{(n_{2})} + \\frac{d}{d n_{2}} \\Omega{(n_{2})} = - \\Omega{(n_{2})} + \\frac{1}{n_{2}} and - \\Omega{(n_{2})} + \\frac{1}{n_{2}} = - \\Omega{(n_{2})} + \\frac{d}{d n_{2}} \\log{(n_{2})} and - \\log{(n_{2})} + \\frac{1}{n_{2}} = - \\log{(n_{2})} + \\frac{d}{d n_{2}} \\log{(n_{2})} and - \\log{(n_{2})} + \\frac{1}{n_{2}} = - \\log{(n_{2})} + \\frac{d}{d n_{2}} \\Omega{(n_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\Omega')(Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Derivative(Function('\\\\Omega')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Derivative(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Derivative(Function('\\\\Omega')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('n_2', commutative=True))), Derivative(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), log(Symbol('n_2', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1))), Add(Mul(Integer(-1), log(Symbol('n_2', commutative=True))), Derivative(log(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Mul(Integer(-1), log(Symbol('n_2', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1))), Add(Mul(Integer(-1), log(Symbol('n_2', commutative=True))), Derivative(Function('\\\\Omega')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)} and J{(\\varepsilon_0)} = \\varepsilon_0, then derive \\varepsilon_0 h^{\\phi_2}{(\\phi_2)} = \\varepsilon_0 \\cos^{\\phi_2}{(\\phi_2)}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 h^{\\phi_2}{(\\phi_2)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 \\cos^{\\phi_2}{(\\phi_2)}", "derivation": "h{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)} and h^{\\phi_2}{(\\phi_2)} = (\\frac{d}{d \\phi_2} \\sin{(\\phi_2)})^{\\phi_2} and J{(\\varepsilon_0)} = \\varepsilon_0 and J{(\\varepsilon_0)} h^{\\phi_2}{(\\phi_2)} = J{(\\varepsilon_0)} (\\frac{d}{d \\phi_2} \\sin{(\\phi_2)})^{\\phi_2} and \\varepsilon_0 h^{\\phi_2}{(\\phi_2)} = \\varepsilon_0 (\\frac{d}{d \\phi_2} \\sin{(\\phi_2)})^{\\phi_2} and \\varepsilon_0 h^{\\phi_2}{(\\phi_2)} = \\varepsilon_0 \\cos^{\\phi_2}{(\\phi_2)} and \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 h^{\\phi_2}{(\\phi_2)} = \\frac{\\partial}{\\partial \\varepsilon_0} \\varepsilon_0 \\cos^{\\phi_2}{(\\phi_2)}", "srepr_derivation": [["get_premise", "Equality(Function('h')(Symbol('\\\\phi_2', commutative=True)), Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True)))"], ["renaming_premise", "Equality(Function('J')(Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))"], [["times", 2, "Function('J')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Function('J')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('h')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Function('J')(Symbol('\\\\varepsilon_0', commutative=True)), Pow(Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Function('h')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Function('h')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(Function('h')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon_0', commutative=True), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(v_{1},G)} = G + v_{1} and \\mathbf{S}{(n_{1},\\Psi)} = \\Psi + n_{1}, then obtain - \\frac{\\mathbf{S}{(n_{1},\\Psi)}}{(G + v_{1}) \\mathbf{g}{(v_{1},G)}} = - \\frac{\\Psi + n_{1}}{(G + v_{1}) \\mathbf{g}{(v_{1},G)}}", "derivation": "\\mathbf{g}{(v_{1},G)} = G + v_{1} and \\mathbf{S}{(n_{1},\\Psi)} = \\Psi + n_{1} and - \\frac{\\mathbf{S}{(n_{1},\\Psi)}}{\\mathbf{g}{(v_{1},G)}} = - \\frac{\\Psi + n_{1}}{\\mathbf{g}{(v_{1},G)}} and - \\frac{\\mathbf{S}{(n_{1},\\Psi)}}{G + v_{1}} = - \\frac{\\Psi + n_{1}}{G + v_{1}} and - \\frac{\\mathbf{S}{(n_{1},\\Psi)}}{(G + v_{1}) \\mathbf{g}{(v_{1},G)}} = - \\frac{\\Psi + n_{1}}{(G + v_{1}) \\mathbf{g}{(v_{1},G)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('v_1', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('n_1', commutative=True)))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('\\\\Psi', commutative=True), Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), Symbol('n_1', commutative=True))))"], [["times", 4, "Pow(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('n_1', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Add(Symbol('G', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), Add(Symbol('\\\\Psi', commutative=True), Symbol('n_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('v_1', commutative=True), Symbol('G', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\chi)} = e^{\\chi}, then derive (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} = (\\chi + e^{\\chi})^{\\chi}, then obtain (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} \\cos^{- q}{(\\sin{(q)})} = (\\chi + \\operatorname{z^{*}}{(\\chi)})^{\\chi} \\cos^{- q}{(\\sin{(q)})}", "derivation": "\\operatorname{z^{*}}{(\\chi)} = e^{\\chi} and \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)} = \\frac{d}{d \\chi} e^{\\chi} and \\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)} = \\chi + \\frac{d}{d \\chi} e^{\\chi} and (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} = (\\chi + \\frac{d}{d \\chi} e^{\\chi})^{\\chi} and (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} = (\\chi + e^{\\chi})^{\\chi} and (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} = (\\chi + \\operatorname{z^{*}}{(\\chi)})^{\\chi} and (\\chi + \\frac{d}{d \\chi} \\operatorname{z^{*}}{(\\chi)})^{\\chi} \\cos^{- q}{(\\sin{(q)})} = (\\chi + \\operatorname{z^{*}}{(\\chi)})^{\\chi} \\cos^{- q}{(\\sin{(q)})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Symbol('\\\\chi', commutative=True), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\chi', commutative=True), Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Derivative(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Symbol('\\\\chi', commutative=True), Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Symbol('\\\\chi', commutative=True), Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Function('z^*')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["divide", 6, "Pow(cos(sin(Symbol('q', commutative=True))), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Derivative(Function('z^*')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Symbol('\\\\chi', commutative=True)), Pow(cos(sin(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)))), Mul(Pow(Add(Symbol('\\\\chi', commutative=True), Function('z^*')(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(cos(sin(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(k)} = e^{k}, then derive \\frac{d}{d k} \\operatorname{v_{y}}{(k)} = e^{k}, then obtain \\frac{d}{d k} \\operatorname{v_{y}}{(k)} = \\frac{d^{2}}{d k^{2}} e^{k}", "derivation": "\\operatorname{v_{y}}{(k)} = e^{k} and \\frac{d}{d k} \\operatorname{v_{y}}{(k)} = \\frac{d}{d k} e^{k} and \\frac{d}{d k} \\operatorname{v_{y}}{(k)} = e^{k} and \\operatorname{v_{y}}{(k)} = \\frac{d}{d k} \\operatorname{v_{y}}{(k)} and \\operatorname{v_{y}}{(k)} = \\frac{d}{d k} e^{k} and \\frac{d}{d k} e^{k} = \\frac{d^{2}}{d k^{2}} e^{k} and \\frac{d}{d k} \\operatorname{v_{y}}{(k)} = \\frac{d^{2}}{d k^{2}} e^{k}", "srepr_derivation": [["get_premise", "Equality(Function('v_y')(Symbol('k', commutative=True)), exp(Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), exp(Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_y')(Symbol('k', commutative=True)), Derivative(Function('v_y')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('v_y')(Symbol('k', commutative=True)), Derivative(exp(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(exp(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Derivative(Function('v_y')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}_M{(c,v,\\eta^{\\prime})} = \\eta^{\\prime} + c + v and m{(c,v,\\eta^{\\prime})} = \\eta^{\\prime} + c + v, then obtain 0 = \\hat{H}_l (\\mathbf{J}_M{(c,v,\\eta^{\\prime})} - m{(c,v,\\eta^{\\prime})})", "derivation": "\\mathbf{J}_M{(c,v,\\eta^{\\prime})} = \\eta^{\\prime} + c + v and m{(c,v,\\eta^{\\prime})} = \\eta^{\\prime} + c + v and m{(c,v,\\eta^{\\prime})} = \\mathbf{J}_M{(c,v,\\eta^{\\prime})} and 0 = \\mathbf{J}_M{(c,v,\\eta^{\\prime})} - m{(c,v,\\eta^{\\prime})} and 0 = \\hat{H}_l (\\mathbf{J}_M{(c,v,\\eta^{\\prime})} - m{(c,v,\\eta^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('c', commutative=True), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('c', commutative=True), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 3, "Function('m')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Function('\\\\mathbf{J}_M')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["times", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\hat{H}_l', commutative=True), Add(Function('\\\\mathbf{J}_M')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('c', commutative=True), Symbol('v', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{y},\\mathbf{f},\\varepsilon)} = - \\mathbf{f} + \\varepsilon - v_{y}, then obtain \\varepsilon - v_{y} \\sigma_{p}^{\\varepsilon}{(v_{y},\\mathbf{f},\\varepsilon)} = \\varepsilon - v_{y} (- \\mathbf{f} + \\varepsilon - v_{y})^{\\varepsilon}", "derivation": "\\sigma_{p}{(v_{y},\\mathbf{f},\\varepsilon)} = - \\mathbf{f} + \\varepsilon - v_{y} and \\sigma_{p}^{\\varepsilon}{(v_{y},\\mathbf{f},\\varepsilon)} = (- \\mathbf{f} + \\varepsilon - v_{y})^{\\varepsilon} and - v_{y} \\sigma_{p}^{\\varepsilon}{(v_{y},\\mathbf{f},\\varepsilon)} = - v_{y} (- \\mathbf{f} + \\varepsilon - v_{y})^{\\varepsilon} and \\varepsilon - v_{y} \\sigma_{p}^{\\varepsilon}{(v_{y},\\mathbf{f},\\varepsilon)} = \\varepsilon - v_{y} (- \\mathbf{f} + \\varepsilon - v_{y})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(Function('\\\\sigma_p')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Symbol('\\\\varepsilon', commutative=True))))"], [["add", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(Function('\\\\sigma_p')(Symbol('v_y', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)}, then derive \\mathbf{p}{(\\omega)} = - \\sin{(\\omega)}, then derive - y^{\\prime} - \\cos{(\\omega)} = - \\int - \\sin{(\\omega)} d\\omega, then obtain \\int (- y^{\\prime} - \\cos{(\\omega)}) dy^{\\prime} = \\int - \\int \\frac{d}{d \\omega} \\cos{(\\omega)} d\\omega dy^{\\prime}", "derivation": "\\mathbf{p}{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\omega)} and \\mathbf{p}{(\\omega)} = - \\sin{(\\omega)} and \\int \\mathbf{p}{(\\omega)} d\\omega = \\int - \\sin{(\\omega)} d\\omega and \\int \\frac{d}{d \\omega} \\cos{(\\omega)} d\\omega = \\int - \\sin{(\\omega)} d\\omega and - \\int \\frac{d}{d \\omega} \\cos{(\\omega)} d\\omega = - \\int - \\sin{(\\omega)} d\\omega and - y^{\\prime} - \\cos{(\\omega)} = - \\int - \\sin{(\\omega)} d\\omega and \\int (- y^{\\prime} - \\cos{(\\omega)}) dy^{\\prime} = \\int - \\int - \\sin{(\\omega)} d\\omega dy^{\\prime} and \\int (- y^{\\prime} - \\cos{(\\omega)}) dy^{\\prime} = \\int - \\int \\frac{d}{d \\omega} \\cos{(\\omega)} d\\omega dy^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\omega', commutative=True)), Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["integrate", 6, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Integral(Mul(Integer(-1), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Integral(Derivative(cos(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(g,\\nabla)} = - \\nabla + g, then derive 2 \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} - 1, then obtain \\frac{\\partial}{\\partial \\nabla} 2 \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} (\\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} - 1)", "derivation": "\\mathbf{J}{(g,\\nabla)} = - \\nabla + g and 2 \\mathbf{J}{(g,\\nabla)} = - \\nabla + g + \\mathbf{J}{(g,\\nabla)} and \\frac{\\partial}{\\partial \\nabla} 2 \\mathbf{J}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} (- \\nabla + g + \\mathbf{J}{(g,\\nabla)}) and 2 \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} - 1 and 2 \\frac{\\partial}{\\partial \\nabla} (- \\nabla + g) = \\frac{\\partial}{\\partial \\nabla} (- \\nabla + g) - 1 and \\frac{\\partial}{\\partial \\nabla} 2 \\frac{\\partial}{\\partial \\nabla} (- \\nabla + g) = \\frac{\\partial}{\\partial \\nabla} (\\frac{\\partial}{\\partial \\nabla} (- \\nabla + g) - 1) and \\frac{\\partial}{\\partial \\nabla} 2 \\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} (\\frac{\\partial}{\\partial \\nabla} \\mathbf{J}{(g,\\nabla)} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True), Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Mul(Integer(2), Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Derivative(Function('\\\\mathbf{J}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(\\mathbf{F})} = \\sin{(\\sin{(\\mathbf{F})})} and \\hat{p}_0{(\\mathbf{F})} = (2 \\eta{(\\mathbf{F})})^{\\mathbf{F}}, then obtain \\eta{(\\mathbf{F})} + \\hat{p}_0{(\\mathbf{F})} = (\\eta{(\\mathbf{F})} + \\sin{(\\sin{(\\mathbf{F})})})^{\\mathbf{F}} + \\eta{(\\mathbf{F})}", "derivation": "\\eta{(\\mathbf{F})} = \\sin{(\\sin{(\\mathbf{F})})} and 2 \\eta{(\\mathbf{F})} = \\eta{(\\mathbf{F})} + \\sin{(\\sin{(\\mathbf{F})})} and (2 \\eta{(\\mathbf{F})})^{\\mathbf{F}} = (\\eta{(\\mathbf{F})} + \\sin{(\\sin{(\\mathbf{F})})})^{\\mathbf{F}} and \\hat{p}_0{(\\mathbf{F})} = (2 \\eta{(\\mathbf{F})})^{\\mathbf{F}} and \\hat{p}_0{(\\mathbf{F})} = (\\eta{(\\mathbf{F})} + \\sin{(\\sin{(\\mathbf{F})})})^{\\mathbf{F}} and \\eta{(\\mathbf{F})} + \\hat{p}_0{(\\mathbf{F})} = (\\eta{(\\mathbf{F})} + \\sin{(\\sin{(\\mathbf{F})})})^{\\mathbf{F}} + \\eta{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), sin(sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 1, "Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), sin(sin(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), sin(sin(Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), sin(sin(Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["add", 5, "Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Pow(Add(Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True)), sin(sin(Symbol('\\\\mathbf{F}', commutative=True)))), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\eta')(Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(h,F_{g})} = h^{F_{g}}, then obtain (0^{h} \\mathbf{v} \\bar{\\h}{(h,F_{g})})^{F_{g}} = (\\mathbf{v} \\bar{\\h}{(h,F_{g})})^{F_{g}}", "derivation": "\\bar{\\h}{(h,F_{g})} = h^{F_{g}} and 0 = h^{F_{g}} - \\bar{\\h}{(h,F_{g})} and 0^{h} = (h^{F_{g}} - \\bar{\\h}{(h,F_{g})})^{h} and 0^{h} \\bar{\\h}{(h,F_{g})} = (h^{F_{g}} - \\bar{\\h}{(h,F_{g})})^{h} \\bar{\\h}{(h,F_{g})} and (h^{F_{g}} - \\bar{\\h}{(h,F_{g})})^{h} \\bar{\\h}{(h,F_{g})} = \\bar{\\h}{(h,F_{g})} and 0^{h} \\bar{\\h}{(h,F_{g})} = \\bar{\\h}{(h,F_{g})} and 0^{h} \\mathbf{v} \\bar{\\h}{(h,F_{g})} = \\mathbf{v} \\bar{\\h}{(h,F_{g})} and (0^{h} \\mathbf{v} \\bar{\\h}{(h,F_{g})})^{F_{g}} = (\\mathbf{v} \\bar{\\h}{(h,F_{g})})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('h', commutative=True), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Integer(0), Symbol('h', commutative=True)), Pow(Add(Pow(Symbol('h', commutative=True), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))), Symbol('h', commutative=True)))"], [["times", 3, "Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('h', commutative=True)), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Mul(Pow(Add(Pow(Symbol('h', commutative=True), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))), Symbol('h', commutative=True)), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Pow(Symbol('h', commutative=True), Symbol('F_g', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))), Symbol('h', commutative=True)), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Integer(0), Symbol('h', commutative=True)), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True)))"], [["times", 6, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Integer(0), Symbol('h', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))))"], [["power", 7, "Symbol('F_g', commutative=True)"], "Equality(Pow(Mul(Pow(Integer(0), Symbol('h', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('F_g', commutative=True))), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(v_{y})} = \\sin{(v_{y})}, then derive \\frac{d}{d v_{y}} \\int \\operatorname{V_{\\mathbf{E}}}{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} (M_{E} - \\cos{(v_{y})}), then obtain \\frac{\\partial}{\\partial v_{y}} (t - \\cos{(v_{y})}) = \\frac{\\partial}{\\partial v_{y}} (M_{E} - \\cos{(v_{y})})", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(v_{y})} = \\sin{(v_{y})} and \\int \\operatorname{V_{\\mathbf{E}}}{(v_{y})} dv_{y} = \\int \\sin{(v_{y})} dv_{y} and \\frac{d}{d v_{y}} \\int \\operatorname{V_{\\mathbf{E}}}{(v_{y})} dv_{y} = \\frac{d}{d v_{y}} \\int \\sin{(v_{y})} dv_{y} and \\frac{d}{d v_{y}} \\int \\operatorname{V_{\\mathbf{E}}}{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} (M_{E} - \\cos{(v_{y})}) and \\frac{d}{d v_{y}} \\int \\sin{(v_{y})} dv_{y} = \\frac{\\partial}{\\partial v_{y}} (M_{E} - \\cos{(v_{y})}) and \\frac{\\partial}{\\partial v_{y}} (t - \\cos{(v_{y})}) = \\frac{\\partial}{\\partial v_{y}} (M_{E} - \\cos{(v_{y})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(sin(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), cos(Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(t)} = \\sin{(e^{t})} and \\tilde{g}^*{(n_{2},\\delta)} = \\delta n_{2}, then obtain \\frac{\\tilde{g}^*{(n_{2},\\delta)} - \\sin{(e^{t})}}{- \\tilde{g}^*{(n_{2},\\delta)} + 2 \\operatorname{v_{x}}{(t)}} = \\frac{\\delta n_{2} - \\sin{(e^{t})}}{- \\tilde{g}^*{(n_{2},\\delta)} + 2 \\operatorname{v_{x}}{(t)}}", "derivation": "\\operatorname{v_{x}}{(t)} = \\sin{(e^{t})} and \\tilde{g}^*{(n_{2},\\delta)} = \\delta n_{2} and \\tilde{g}^*{(n_{2},\\delta)} - \\operatorname{v_{x}}{(t)} = \\delta n_{2} - \\operatorname{v_{x}}{(t)} and \\tilde{g}^*{(n_{2},\\delta)} - \\sin{(e^{t})} = \\delta n_{2} - \\sin{(e^{t})} and \\frac{\\tilde{g}^*{(n_{2},\\delta)} - \\sin{(e^{t})}}{- \\tilde{g}^*{(n_{2},\\delta)} + 2 \\operatorname{v_{x}}{(t)}} = \\frac{\\delta n_{2} - \\sin{(e^{t})}}{- \\tilde{g}^*{(n_{2},\\delta)} + 2 \\operatorname{v_{x}}{(t)}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('t', commutative=True)), sin(exp(Symbol('t', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 2, "Function('v_x')(Symbol('t', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('t', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Function('v_x')(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('t', commutative=True))))), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('t', commutative=True))))))"], [["divide", 4, "Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Function('v_x')(Symbol('t', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Function('v_x')(Symbol('t', commutative=True)))), Integer(-1)), Add(Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('t', commutative=True)))))), Mul(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('t', commutative=True))))), Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('n_2', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Function('v_x')(Symbol('t', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given y{(g_{\\varepsilon},v_{t})} = g_{\\varepsilon} + e^{v_{t}}, then obtain \\frac{\\partial^{2}}{\\partial v_{t}\\partial g_{\\varepsilon}} y^{g_{\\varepsilon}}{(g_{\\varepsilon},v_{t})} = \\frac{\\partial^{2}}{\\partial v_{t}\\partial g_{\\varepsilon}} (g_{\\varepsilon} + e^{v_{t}})^{g_{\\varepsilon}}", "derivation": "y{(g_{\\varepsilon},v_{t})} = g_{\\varepsilon} + e^{v_{t}} and y^{g_{\\varepsilon}}{(g_{\\varepsilon},v_{t})} = (g_{\\varepsilon} + e^{v_{t}})^{g_{\\varepsilon}} and \\frac{\\partial}{\\partial g_{\\varepsilon}} y^{g_{\\varepsilon}}{(g_{\\varepsilon},v_{t})} = \\frac{\\partial}{\\partial g_{\\varepsilon}} (g_{\\varepsilon} + e^{v_{t}})^{g_{\\varepsilon}} and \\frac{\\partial^{2}}{\\partial v_{t}\\partial g_{\\varepsilon}} y^{g_{\\varepsilon}}{(g_{\\varepsilon},v_{t})} = \\frac{\\partial^{2}}{\\partial v_{t}\\partial g_{\\varepsilon}} (g_{\\varepsilon} + e^{v_{t}})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_t', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('v_t', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Pow(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_t', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('v_t', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Pow(Function('y')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('v_t', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('v_t', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(x,\\Omega,F_{H})} = (F_{H}^{\\Omega})^{x}, then obtain \\sin{(\\cos{(\\Omega - \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\Omega,F_{H})})})} = \\sin{(\\cos{(\\Omega - \\frac{\\partial}{\\partial x} (F_{H}^{\\Omega})^{x})})}", "derivation": "\\dot{x}{(x,\\Omega,F_{H})} = (F_{H}^{\\Omega})^{x} and \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\Omega,F_{H})} = \\frac{\\partial}{\\partial x} (F_{H}^{\\Omega})^{x} and - \\Omega + \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\Omega,F_{H})} = - \\Omega + \\frac{\\partial}{\\partial x} (F_{H}^{\\Omega})^{x} and \\cos{(\\Omega - \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\Omega,F_{H})})} = \\cos{(\\Omega - \\frac{\\partial}{\\partial x} (F_{H}^{\\Omega})^{x})} and \\sin{(\\cos{(\\Omega - \\frac{\\partial}{\\partial x} \\dot{x}{(x,\\Omega,F_{H})})})} = \\sin{(\\cos{(\\Omega - \\frac{\\partial}{\\partial x} (F_{H}^{\\Omega})^{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('F_H', commutative=True)), Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))), cos(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))))"], [["sin", 4], "Equality(sin(cos(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\dot{x}')(Symbol('x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))), sin(cos(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Derivative(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = \\Psi g_{\\varepsilon} - \\theta_2, then derive - \\frac{\\partial}{\\partial \\Psi} \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = - g_{\\varepsilon}, then obtain \\frac{\\frac{\\partial}{\\partial \\Psi} \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)}}{g_{\\varepsilon}} = 1", "derivation": "\\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = \\Psi g_{\\varepsilon} - \\theta_2 and - \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = - \\Psi g_{\\varepsilon} + \\theta_2 and \\frac{\\partial}{\\partial \\Psi} - \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = \\frac{\\partial}{\\partial \\Psi} (- \\Psi g_{\\varepsilon} + \\theta_2) and - \\frac{\\partial}{\\partial \\Psi} \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)} = - g_{\\varepsilon} and \\frac{\\frac{\\partial}{\\partial \\Psi} \\mathbf{J}_P{(g_{\\varepsilon},\\theta_2,\\Psi)}}{g_{\\varepsilon}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_P')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 4, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\theta{(f_{\\mathbf{p}})} = \\sin{(f_{\\mathbf{p}})}, then derive \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}})} = \\mathbf{g} \\cos{(f_{\\mathbf{p}})}, then obtain \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}})} = \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})}", "derivation": "\\theta{(f_{\\mathbf{p}})} = \\sin{(f_{\\mathbf{p}})} and \\mathbf{g} \\theta{(f_{\\mathbf{p}})} = \\mathbf{g} \\sin{(f_{\\mathbf{p}})} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\mathbf{g} \\theta{(f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\mathbf{g} \\sin{(f_{\\mathbf{p}})} and \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}})} = \\mathbf{g} \\cos{(f_{\\mathbf{p}})} and \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})} = \\mathbf{g} \\cos{(f_{\\mathbf{p}})} and \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\theta{(f_{\\mathbf{p}})} = \\mathbf{g} \\frac{d}{d f_{\\mathbf{p}}} \\sin{(f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('\\\\mathbf{g}', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Derivative(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), cos(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Function('\\\\theta')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Derivative(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{F}{(x)} = \\cos{(x)}, then derive \\int \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} dx + \\frac{1}{\\mathbf{F}{(x)}} = p + x + \\frac{1}{\\mathbf{F}{(x)}}, then obtain p + x + \\frac{1}{\\mathbf{F}{(x)}} = \\int 1 dx + \\frac{1}{\\mathbf{F}{(x)}}", "derivation": "\\mathbf{F}{(x)} = \\cos{(x)} and \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} = 1 and \\int \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} dx = \\int 1 dx and \\int \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} dx + \\frac{1}{\\cos{(x)}} = \\int 1 dx + \\frac{1}{\\cos{(x)}} and \\int \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} dx + \\frac{1}{\\mathbf{F}{(x)}} = \\int 1 dx + \\frac{1}{\\mathbf{F}{(x)}} and \\int \\frac{\\mathbf{F}{(x)}}{\\cos{(x)}} dx + \\frac{1}{\\mathbf{F}{(x)}} = p + x + \\frac{1}{\\mathbf{F}{(x)}} and p + x + \\frac{1}{\\mathbf{F}{(x)}} = \\int 1 dx + \\frac{1}{\\mathbf{F}{(x)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["divide", 1, "cos(Symbol('x', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))), Integral(Integer(1), Tuple(Symbol('x', commutative=True))))"], [["add", 3, "Pow(cos(Symbol('x', commutative=True)), Integer(-1))"], "Equality(Add(Integral(Mul(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Add(Integral(Integer(1), Tuple(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Mul(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))), Add(Integral(Integer(1), Tuple(Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))))"], [["evaluate_integrals", 5], "Equality(Add(Integral(Mul(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Tuple(Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))), Add(Symbol('p', commutative=True), Symbol('x', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('p', commutative=True), Symbol('x', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))), Add(Integral(Integer(1), Tuple(Symbol('x', commutative=True))), Pow(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)} = (C \\phi_1)^{\\phi_2} and \\mu{(g)} = \\log{(g)}, then obtain \\frac{\\mu^{g}{(g)}}{\\int \\log{(- C + (C \\phi_1)^{\\phi_2})} dC} = \\frac{\\log{(g)}^{g}}{\\int \\log{(- C + (C \\phi_1)^{\\phi_2})} dC}", "derivation": "\\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)} = (C \\phi_1)^{\\phi_2} and - C + \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)} = - C + (C \\phi_1)^{\\phi_2} and \\log{(- C + \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)})} = \\log{(- C + (C \\phi_1)^{\\phi_2})} and \\mu{(g)} = \\log{(g)} and \\int \\log{(- C + \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)})} dC = \\int \\log{(- C + (C \\phi_1)^{\\phi_2})} dC and \\mu^{g}{(g)} = \\log{(g)}^{g} and \\frac{\\mu^{g}{(g)}}{\\int \\log{(- C + \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)})} dC} = \\frac{\\log{(g)}^{g}}{\\int \\log{(- C + \\operatorname{v_{t}}{(\\phi_1,C,\\phi_2)})} dC} and \\frac{\\mu^{g}{(g)}}{\\int \\log{(- C + (C \\phi_1)^{\\phi_2})} dC} = \\frac{\\log{(g)}^{g}}{\\int \\log{(- C + (C \\phi_1)^{\\phi_2})} dC}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["log", 2], "Equality(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["divide", 6, "Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integer(-1))), Mul(Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('v_t')(Symbol('\\\\phi_1', commutative=True), Symbol('C', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integer(-1))), Mul(Pow(log(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(log(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given E{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}}, then derive \\frac{d}{d V_{\\mathbf{E}}} E{(V_{\\mathbf{E}})} - 1 = e^{V_{\\mathbf{E}}} - 1, then obtain \\frac{\\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} - 1}{V_{\\mathbf{E}} \\nabla} = \\frac{e^{V_{\\mathbf{E}}} - 1}{V_{\\mathbf{E}} \\nabla}", "derivation": "E{(V_{\\mathbf{E}})} = e^{V_{\\mathbf{E}}} and \\frac{d}{d V_{\\mathbf{E}}} E{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} and \\frac{d}{d V_{\\mathbf{E}}} E{(V_{\\mathbf{E}})} - 1 = \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} - 1 and \\frac{d}{d V_{\\mathbf{E}}} E{(V_{\\mathbf{E}})} - 1 = e^{V_{\\mathbf{E}}} - 1 and \\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} - 1 = e^{V_{\\mathbf{E}}} - 1 and \\frac{\\frac{d}{d V_{\\mathbf{E}}} e^{V_{\\mathbf{E}}} - 1}{V_{\\mathbf{E}} \\nabla} = \\frac{e^{V_{\\mathbf{E}}} - 1}{V_{\\mathbf{E}} \\nabla}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('E')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('E')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)), Add(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)), Add(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)))"], [["divide", 5, "Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Derivative(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(exp(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = J \\varphi^*, then obtain \\frac{\\partial}{\\partial J} - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = \\frac{\\partial}{\\partial J} (J^{2} \\varphi^* - J \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = J \\varphi^* and J \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = J^{2} \\varphi^* and - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = J^{2} \\varphi^* - J \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} and \\frac{\\partial}{\\partial J} - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} = \\frac{\\partial}{\\partial J} (J^{2} \\varphi^* - J \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)} - \\operatorname{f_{\\mathbf{p}}}{(J,\\varphi^*)})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 2, "Add(Mul(Symbol('J', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('J', commutative=True), Integer(2)), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('J', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given q{(x^\\prime,v_{1})} = x^\\prime e^{v_{1}} and \\phi_{1}{(v_{1})} = - v_{1}, then obtain \\log{(\\phi_{1}{(v_{1})} + q{(x^\\prime,v_{1})} e^{- v_{1}})} = \\log{(- v_{1} + q{(x^\\prime,v_{1})} e^{- v_{1}})}", "derivation": "q{(x^\\prime,v_{1})} = x^\\prime e^{v_{1}} and q{(x^\\prime,v_{1})} e^{- v_{1}} = x^\\prime and \\phi_{1}{(v_{1})} = - v_{1} and x^\\prime + \\phi_{1}{(v_{1})} = - v_{1} + x^\\prime and \\phi_{1}{(v_{1})} + q{(x^\\prime,v_{1})} e^{- v_{1}} = - v_{1} + q{(x^\\prime,v_{1})} e^{- v_{1}} and \\log{(\\phi_{1}{(v_{1})} + q{(x^\\prime,v_{1})} e^{- v_{1}})} = \\log{(- v_{1} + q{(x^\\prime,v_{1})} e^{- v_{1}})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), exp(Symbol('v_1', commutative=True))))"], [["divide", 1, "exp(Symbol('v_1', commutative=True))"], "Equality(Mul(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))), Symbol('x^\\\\prime', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))"], [["add", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('\\\\phi_1')(Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\phi_1')(Symbol('v_1', commutative=True)), Mul(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True))))))"], [["log", 5], "Equality(log(Add(Function('\\\\phi_1')(Symbol('v_1', commutative=True)), Mul(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))))), log(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Function('q')(Symbol('x^\\\\prime', commutative=True), Symbol('v_1', commutative=True)), exp(Mul(Integer(-1), Symbol('v_1', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(L)} = \\sin{(L)}, then derive - \\operatorname{v_{t}}{(L)} + \\int \\operatorname{v_{t}}{(L)} dL = \\Psi_{\\lambda} - \\operatorname{v_{t}}{(L)} - \\cos{(L)}, then obtain (\\Psi_{\\lambda} - \\operatorname{v_{t}}{(L)} - \\cos{(L)})^{\\Psi_{\\lambda}} = (- \\operatorname{v_{t}}{(L)} + \\int \\sin{(L)} dL)^{\\Psi_{\\lambda}}", "derivation": "\\operatorname{v_{t}}{(L)} = \\sin{(L)} and \\int \\operatorname{v_{t}}{(L)} dL = \\int \\sin{(L)} dL and - \\operatorname{v_{t}}{(L)} + \\int \\operatorname{v_{t}}{(L)} dL = - \\operatorname{v_{t}}{(L)} + \\int \\sin{(L)} dL and - \\operatorname{v_{t}}{(L)} + \\int \\operatorname{v_{t}}{(L)} dL = \\Psi_{\\lambda} - \\operatorname{v_{t}}{(L)} - \\cos{(L)} and \\Psi_{\\lambda} - \\operatorname{v_{t}}{(L)} - \\cos{(L)} = - \\operatorname{v_{t}}{(L)} + \\int \\sin{(L)} dL and (\\Psi_{\\lambda} - \\operatorname{v_{t}}{(L)} - \\cos{(L)})^{\\Psi_{\\lambda}} = (- \\operatorname{v_{t}}{(L)} + \\int \\sin{(L)} dL)^{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["minus", 2, "Function('v_t')(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Integral(Function('v_t')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Integral(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Integral(Function('v_t')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Mul(Integer(-1), cos(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Mul(Integer(-1), cos(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Integral(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["power", 5, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Mul(Integer(-1), cos(Symbol('L', commutative=True)))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('v_t')(Symbol('L', commutative=True))), Integral(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(l)} = \\sin{(\\log{(l)})} and \\varphi^{*}{(l)} = \\operatorname{n_{1}}{(l)} + \\log{(l)}, then obtain \\varphi^{*}{(l)} + \\operatorname{n_{1}}{(l)} \\sin{(\\log{(l)})} = \\varphi^{*}{(l)} + \\sin^{2}{(\\log{(l)})}", "derivation": "\\operatorname{n_{1}}{(l)} = \\sin{(\\log{(l)})} and \\operatorname{n_{1}}{(l)} \\sin{(\\log{(l)})} = \\sin^{2}{(\\log{(l)})} and \\operatorname{n_{1}}{(l)} \\sin{(\\log{(l)})} + \\log{(l)} + \\sin{(\\log{(l)})} = \\log{(l)} + \\sin^{2}{(\\log{(l)})} + \\sin{(\\log{(l)})} and \\varphi^{*}{(l)} = \\operatorname{n_{1}}{(l)} + \\log{(l)} and \\varphi^{*}{(l)} = \\log{(l)} + \\sin{(\\log{(l)})} and \\varphi^{*}{(l)} + \\operatorname{n_{1}}{(l)} \\sin{(\\log{(l)})} = \\varphi^{*}{(l)} + \\sin^{2}{(\\log{(l)})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True))))"], [["times", 1, "sin(log(Symbol('l', commutative=True)))"], "Equality(Mul(Function('n_1')(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True)))), Pow(sin(log(Symbol('l', commutative=True))), Integer(2)))"], [["add", 2, "Add(log(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True))))"], "Equality(Add(Mul(Function('n_1')(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True)))), log(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True)))), Add(log(Symbol('l', commutative=True)), Pow(sin(log(Symbol('l', commutative=True))), Integer(2)), sin(log(Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('l', commutative=True)), Add(Function('n_1')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\varphi^*')(Symbol('l', commutative=True)), Add(log(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Function('\\\\varphi^*')(Symbol('l', commutative=True)), Mul(Function('n_1')(Symbol('l', commutative=True)), sin(log(Symbol('l', commutative=True))))), Add(Function('\\\\varphi^*')(Symbol('l', commutative=True)), Pow(sin(log(Symbol('l', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{g}{(A_{1})} = \\sin{(A_{1})} and \\Omega{(A_{1})} = A_{1}^{2} \\mathbf{g}{(A_{1})} \\sin{(A_{1})}, then obtain \\frac{\\Omega{(A_{1})}}{\\mathbf{g}^{2}{(A_{1})}} = A_{1}^{2}", "derivation": "\\mathbf{g}{(A_{1})} = \\sin{(A_{1})} and A_{1} \\mathbf{g}{(A_{1})} = A_{1} \\sin{(A_{1})} and A_{1}^{2} \\mathbf{g}^{2}{(A_{1})} = A_{1}^{2} \\mathbf{g}{(A_{1})} \\sin{(A_{1})} and \\Omega{(A_{1})} = A_{1}^{2} \\mathbf{g}{(A_{1})} \\sin{(A_{1})} and \\Omega{(A_{1})} = A_{1}^{2} \\sin^{2}{(A_{1})} and \\frac{\\Omega{(A_{1})}}{\\mathbf{g}^{2}{(A_{1})}} = \\frac{A_{1}^{2} \\sin^{2}{(A_{1})}}{\\mathbf{g}^{2}{(A_{1})}} and A_{1}^{2} \\mathbf{g}^{2}{(A_{1})} = \\Omega{(A_{1})} and A_{1}^{2} \\mathbf{g}^{2}{(A_{1})} = A_{1}^{2} \\sin^{2}{(A_{1})} and \\frac{\\Omega{(A_{1})}}{\\mathbf{g}^{2}{(A_{1})}} = A_{1}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), sin(Symbol('A_1', commutative=True))))"], [["times", 2, "Mul(Symbol('A_1', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(sin(Symbol('A_1', commutative=True)), Integer(2))))"], [["divide", 5, "Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(2))"], "Equality(Mul(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(-2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(-2)), Pow(sin(Symbol('A_1', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(2))), Function('\\\\Omega')(Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Pow(sin(Symbol('A_1', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Mul(Function('\\\\Omega')(Symbol('A_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('A_1', commutative=True)), Integer(-2))), Pow(Symbol('A_1', commutative=True), Integer(2)))"]]}, {"prompt": "Given \\hat{H}_l{(\\psi^*,Q)} = \\cos{(Q \\psi^*)}, then obtain \\frac{\\partial}{\\partial Q} \\int Q \\hat{H}_l{(\\psi^*,Q)} dQ + 1 = \\frac{\\partial}{\\partial Q} \\int Q \\cos{(Q \\psi^*)} dQ + 1", "derivation": "\\hat{H}_l{(\\psi^*,Q)} = \\cos{(Q \\psi^*)} and Q \\hat{H}_l{(\\psi^*,Q)} = Q \\cos{(Q \\psi^*)} and \\int Q \\hat{H}_l{(\\psi^*,Q)} dQ = \\int Q \\cos{(Q \\psi^*)} dQ and \\frac{\\partial}{\\partial Q} \\int Q \\hat{H}_l{(\\psi^*,Q)} dQ = \\frac{\\partial}{\\partial Q} \\int Q \\cos{(Q \\psi^*)} dQ and \\frac{\\partial}{\\partial Q} \\int Q \\hat{H}_l{(\\psi^*,Q)} dQ + 1 = \\frac{\\partial}{\\partial Q} \\int Q \\cos{(Q \\psi^*)} dQ + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\psi^*', commutative=True), Symbol('Q', commutative=True)), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\psi^*', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\psi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\psi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('Q', commutative=True), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\psi^*', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Mul(Symbol('Q', commutative=True), cos(Mul(Symbol('Q', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given H{(C_{2})} = \\sin{(C_{2})} and \\mathbf{J}_P{(C_{2})} = \\frac{d}{d \\lambda} \\int \\sin{(C_{2})} dC_{2} - 1, then derive \\int H{(C_{2})} dC_{2} = \\lambda - \\cos{(C_{2})}, then derive \\frac{d}{d \\lambda} \\int \\sin{(C_{2})} dC_{2} - 1 = 0, then obtain \\mathbf{J}_P{(C_{2})} = 0", "derivation": "H{(C_{2})} = \\sin{(C_{2})} and \\int H{(C_{2})} dC_{2} = \\int \\sin{(C_{2})} dC_{2} and \\int H{(C_{2})} dC_{2} = \\lambda - \\cos{(C_{2})} and \\frac{d}{d \\lambda} \\int H{(C_{2})} dC_{2} = \\frac{\\partial}{\\partial \\lambda} (\\lambda - \\cos{(C_{2})}) and - \\frac{\\partial}{\\partial \\lambda} (\\lambda - \\cos{(C_{2})}) + \\frac{d}{d \\lambda} \\int H{(C_{2})} dC_{2} = 0 and - \\frac{\\partial}{\\partial \\lambda} (\\lambda - \\cos{(C_{2})}) + \\frac{d}{d \\lambda} \\int \\sin{(C_{2})} dC_{2} = 0 and \\frac{d}{d \\lambda} \\int \\sin{(C_{2})} dC_{2} - 1 = 0 and \\mathbf{J}_P{(C_{2})} = \\frac{d}{d \\lambda} \\int \\sin{(C_{2})} dC_{2} - 1 and \\mathbf{J}_P{(C_{2})} = 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('H')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('H')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Function('H')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Derivative(Integral(Function('H')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Derivative(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Integer(0))"], [["evaluate_derivatives", 6], "Equality(Add(Derivative(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True)), Add(Derivative(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 7, 8], "Equality(Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True)), Integer(0))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(H)} = \\sin{(\\sin{(H)})} and J{(H)} = \\sin{(\\sin{(H)})}, then obtain \\int (\\frac{d}{d H} \\int J{(H)} dH - \\int \\sin{(\\sin{(H)})} dH) dH = \\int (\\frac{d}{d H} \\int \\sin{(\\sin{(H)})} dH - \\int \\sin{(\\sin{(H)})} dH) dH", "derivation": "\\operatorname{E_{x}}{(H)} = \\sin{(\\sin{(H)})} and \\int \\operatorname{E_{x}}{(H)} dH = \\int \\sin{(\\sin{(H)})} dH and J{(H)} = \\sin{(\\sin{(H)})} and J{(H)} = \\operatorname{E_{x}}{(H)} and \\int J{(H)} dH = \\int \\sin{(\\sin{(H)})} dH and \\frac{d}{d H} \\int J{(H)} dH = \\frac{d}{d H} \\int \\sin{(\\sin{(H)})} dH and \\frac{d}{d H} \\int J{(H)} dH - \\int \\sin{(\\sin{(H)})} dH = \\frac{d}{d H} \\int \\sin{(\\sin{(H)})} dH - \\int \\sin{(\\sin{(H)})} dH and \\int (\\frac{d}{d H} \\int J{(H)} dH - \\int \\sin{(\\sin{(H)})} dH) dH = \\int (\\frac{d}{d H} \\int \\sin{(\\sin{(H)})} dH - \\int \\sin{(\\sin{(H)})} dH) dH", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('J')(Symbol('H', commutative=True)), Function('E_x')(Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('J')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Integral(Function('J')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["minus", 6, "Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Derivative(Integral(Function('J')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))), Add(Derivative(Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))))"], [["integrate", 7, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Derivative(Integral(Function('J')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True))), Integral(Add(Derivative(Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given m{(v_{1})} = \\cos{(v_{1})}, then obtain - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\cos{(v_{1})} + \\frac{d}{d v_{1}} 0 = - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\cos{(v_{1})} + \\frac{d}{d v_{1}} - \\sin{(m{(v_{1})} - \\cos{(v_{1})})}", "derivation": "m{(v_{1})} = \\cos{(v_{1})} and 0 = - m{(v_{1})} + \\cos{(v_{1})} and 0 = - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} and \\frac{d}{d v_{1}} 0 = \\frac{d}{d v_{1}} - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} and - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\frac{d}{d v_{1}} 0 = - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\frac{d}{d v_{1}} - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} and - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\cos{(v_{1})} + \\frac{d}{d v_{1}} 0 = - \\sin{(m{(v_{1})} - \\cos{(v_{1})})} + \\cos{(v_{1})} + \\frac{d}{d v_{1}} - \\sin{(m{(v_{1})} - \\cos{(v_{1})})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["minus", 1, "Function('m')(Symbol('v_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m')(Symbol('v_1', commutative=True))), cos(Symbol('v_1', commutative=True))))"], [["sin", 2], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["add", 4, "Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True))))))"], "Equality(Add(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), Derivative(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["minus", 5, "Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), cos(Symbol('v_1', commutative=True)), Derivative(Integer(0), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), cos(Symbol('v_1', commutative=True)), Derivative(Mul(Integer(-1), sin(Add(Function('m')(Symbol('v_1', commutative=True)), Mul(Integer(-1), cos(Symbol('v_1', commutative=True)))))), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(\\theta_2,E_{\\lambda})} = \\frac{\\theta_2}{E_{\\lambda}}, then obtain \\int (E_{\\lambda} \\pi{(\\theta_2,E_{\\lambda})} - E_{\\lambda} + \\theta_2) dE_{\\lambda} = \\int (- E_{\\lambda} + 2 \\theta_2) dE_{\\lambda}", "derivation": "\\pi{(\\theta_2,E_{\\lambda})} = \\frac{\\theta_2}{E_{\\lambda}} and E_{\\lambda} \\pi{(\\theta_2,E_{\\lambda})} = \\theta_2 and E_{\\lambda} \\pi{(\\theta_2,E_{\\lambda})} + \\theta_2 = 2 \\theta_2 and E_{\\lambda} \\pi{(\\theta_2,E_{\\lambda})} - E_{\\lambda} + \\theta_2 = - E_{\\lambda} + 2 \\theta_2 and \\int (E_{\\lambda} \\pi{(\\theta_2,E_{\\lambda})} - E_{\\lambda} + \\theta_2) dE_{\\lambda} = \\int (- E_{\\lambda} + 2 \\theta_2) dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], [["add", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)))"], [["minus", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('E_{\\\\lambda}', commutative=True), Function('\\\\pi')(Symbol('\\\\theta_2', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\rho_f,\\hat{X})} = \\rho_f^{\\hat{X}} and \\pi{(\\rho_f,\\hat{X})} = \\frac{\\rho_f^{\\hat{X}}}{\\hat{X}}, then obtain \\frac{\\bar{\\h}{(\\rho_f,\\hat{X})}}{\\hat{X}} = \\pi{(\\rho_f,\\hat{X})}", "derivation": "\\bar{\\h}{(\\rho_f,\\hat{X})} = \\rho_f^{\\hat{X}} and \\frac{\\bar{\\h}{(\\rho_f,\\hat{X})}}{\\hat{X}} = \\frac{\\rho_f^{\\hat{X}}}{\\hat{X}} and \\pi{(\\rho_f,\\hat{X})} = \\frac{\\rho_f^{\\hat{X}}}{\\hat{X}} and \\frac{\\bar{\\h}{(\\rho_f,\\hat{X})}}{\\hat{X}} = \\pi{(\\rho_f,\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and x{(a^{\\dagger})} = \\varepsilon^{a^{\\dagger}}{(a^{\\dagger})}, then obtain 1 = \\frac{\\cos^{a^{\\dagger}}{(a^{\\dagger})}}{x{(a^{\\dagger})}}", "derivation": "\\varepsilon{(a^{\\dagger})} = \\cos{(a^{\\dagger})} and \\varepsilon^{a^{\\dagger}}{(a^{\\dagger})} = \\cos^{a^{\\dagger}}{(a^{\\dagger})} and x{(a^{\\dagger})} = \\varepsilon^{a^{\\dagger}}{(a^{\\dagger})} and 1 = \\frac{\\varepsilon^{a^{\\dagger}}{(a^{\\dagger})}}{x{(a^{\\dagger})}} and 1 = \\frac{\\cos^{a^{\\dagger}}{(a^{\\dagger})}}{x{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('a^{\\\\dagger}', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 3, "Function('x')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Pow(Function('x')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Pow(cos(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(M_{E})} = \\log{(\\log{(M_{E})})}, then obtain \\frac{d}{d M_{E}} \\int (\\Psi^{\\dagger}{(M_{E})} - \\log{(M_{E})}) dM_{E} = \\frac{d}{d M_{E}} \\int (- \\log{(M_{E})} + \\log{(\\log{(M_{E})})}) dM_{E}", "derivation": "\\Psi^{\\dagger}{(M_{E})} = \\log{(\\log{(M_{E})})} and \\Psi^{\\dagger}{(M_{E})} - \\log{(M_{E})} = - \\log{(M_{E})} + \\log{(\\log{(M_{E})})} and \\int (\\Psi^{\\dagger}{(M_{E})} - \\log{(M_{E})}) dM_{E} = \\int (- \\log{(M_{E})} + \\log{(\\log{(M_{E})})}) dM_{E} and \\frac{d}{d M_{E}} \\int (\\Psi^{\\dagger}{(M_{E})} - \\log{(M_{E})}) dM_{E} = \\frac{d}{d M_{E}} \\int (- \\log{(M_{E})} + \\log{(\\log{(M_{E})})}) dM_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), log(log(Symbol('M_E', commutative=True))))"], [["minus", 1, "log(Symbol('M_E', commutative=True))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), log(Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('M_E', commutative=True))), log(log(Symbol('M_E', commutative=True)))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), log(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Integral(Add(Mul(Integer(-1), log(Symbol('M_E', commutative=True))), log(log(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), log(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), log(Symbol('M_E', commutative=True))), log(log(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(r)} = e^{r}, then derive \\int i{(r)} dr = F_{N} + e^{r}, then obtain (F_{N} + i{(r)})^{r} = (\\int e^{r} dr)^{r}", "derivation": "i{(r)} = e^{r} and \\int i{(r)} dr = \\int e^{r} dr and (\\int i{(r)} dr)^{r} = (\\int e^{r} dr)^{r} and \\int i{(r)} dr = F_{N} + e^{r} and \\int i{(r)} dr = F_{N} + i{(r)} and (F_{N} + i{(r)})^{r} = (\\int e^{r} dr)^{r}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Integral(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('i')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('F_N', commutative=True), Function('i')(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Symbol('F_N', commutative=True), Function('i')(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\pi{(x^\\prime)} = e^{\\cos{(x^\\prime)}}, then derive \\frac{d}{d x^\\prime} \\pi{(x^\\prime)} = - e^{\\cos{(x^\\prime)}} \\sin{(x^\\prime)}, then obtain (\\frac{d}{d x^\\prime} \\pi{(x^\\prime)})^{x^\\prime} = (- e^{\\cos{(x^\\prime)}} \\sin{(x^\\prime)})^{x^\\prime}", "derivation": "\\pi{(x^\\prime)} = e^{\\cos{(x^\\prime)}} and \\frac{d}{d x^\\prime} \\pi{(x^\\prime)} = \\frac{d}{d x^\\prime} e^{\\cos{(x^\\prime)}} and \\frac{d}{d x^\\prime} \\pi{(x^\\prime)} = - e^{\\cos{(x^\\prime)}} \\sin{(x^\\prime)} and \\frac{d}{d x^\\prime} \\pi{(x^\\prime)} = - \\pi{(x^\\prime)} \\sin{(x^\\prime)} and (\\frac{d}{d x^\\prime} \\pi{(x^\\prime)})^{x^\\prime} = (- \\pi{(x^\\prime)} \\sin{(x^\\prime)})^{x^\\prime} and - e^{\\cos{(x^\\prime)}} \\sin{(x^\\prime)} = - \\pi{(x^\\prime)} \\sin{(x^\\prime)} and (\\frac{d}{d x^\\prime} \\pi{(x^\\prime)})^{x^\\prime} = (- e^{\\cos{(x^\\prime)}} \\sin{(x^\\prime)})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), exp(cos(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Derivative(Function('\\\\pi')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Integer(-1), exp(cos(Symbol('x^\\\\prime', commutative=True))), sin(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} = f_{\\mathbf{v}}^{\\phi}, then obtain 1 = \\frac{\\log{(\\int f_{\\mathbf{v}}^{\\phi} d\\phi)}}{\\log{(\\int \\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} d\\phi)}}", "derivation": "\\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} = f_{\\mathbf{v}}^{\\phi} and \\int \\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} d\\phi = \\int f_{\\mathbf{v}}^{\\phi} d\\phi and \\log{(\\int \\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} d\\phi)} = \\log{(\\int f_{\\mathbf{v}}^{\\phi} d\\phi)} and 1 = \\frac{\\log{(\\int f_{\\mathbf{v}}^{\\phi} d\\phi)}}{\\log{(\\int \\hat{H}_{\\lambda}{(f_{\\mathbf{v}},\\phi)} d\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), log(Integral(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["divide", 3, "log(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], "Equality(Integer(1), Mul(log(Integral(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Pow(log(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{E}{(x,\\Psi^{\\dagger})} = \\Psi^{\\dagger} - x, then obtain \\frac{(- 2 \\Psi^{\\dagger} + \\mathbf{E}{(x,\\Psi^{\\dagger})})^{\\Psi^{\\dagger}}}{- \\Psi^{\\dagger} - x} = \\frac{(- \\Psi^{\\dagger} - x)^{\\Psi^{\\dagger}}}{- \\Psi^{\\dagger} - x}", "derivation": "\\mathbf{E}{(x,\\Psi^{\\dagger})} = \\Psi^{\\dagger} - x and - \\Psi^{\\dagger} + \\mathbf{E}{(x,\\Psi^{\\dagger})} = - x and - 2 \\Psi^{\\dagger} + \\mathbf{E}{(x,\\Psi^{\\dagger})} = - \\Psi^{\\dagger} - x and (- 2 \\Psi^{\\dagger} + \\mathbf{E}{(x,\\Psi^{\\dagger})})^{\\Psi^{\\dagger}} = (- \\Psi^{\\dagger} - x)^{\\Psi^{\\dagger}} and \\frac{(- 2 \\Psi^{\\dagger} + \\mathbf{E}{(x,\\Psi^{\\dagger})})^{\\Psi^{\\dagger}}}{- \\Psi^{\\dagger} - x} = \\frac{(- \\Psi^{\\dagger} - x)^{\\Psi^{\\dagger}}}{- \\Psi^{\\dagger} - x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["minus", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True)))"], [["minus", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Symbol('x', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given v{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\Omega{(\\theta_2)} = \\cos{(\\theta_2)}, then obtain \\Omega{(\\theta_2)} + v{(\\mathbf{p})} = v{(\\mathbf{p})} + \\cos{(\\theta_2)}", "derivation": "v{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\Omega{(\\theta_2)} = \\cos{(\\theta_2)} and \\Omega{(\\theta_2)} + \\sin{(\\mathbf{p})} = \\sin{(\\mathbf{p})} + \\cos{(\\theta_2)} and \\Omega{(\\theta_2)} + v{(\\mathbf{p})} = v{(\\mathbf{p})} + \\cos{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["add", 2, "sin(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True))), Add(sin(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True)), Function('v')(Symbol('\\\\mathbf{p}', commutative=True))), Add(Function('v')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given u{(p)} = \\cos{(p)}, then obtain \\frac{d}{d p} u^{3}{(p)} = \\frac{d}{d p} u{(p)} \\cos^{2}{(p)}", "derivation": "u{(p)} = \\cos{(p)} and u{(p)} \\cos{(p)} = \\cos^{2}{(p)} and u^{2}{(p)} \\cos{(p)} = u{(p)} \\cos^{2}{(p)} and u^{2}{(p)} \\cos^{2}{(p)} = \\cos^{4}{(p)} and u^{3}{(p)} \\cos{(p)} = u^{2}{(p)} \\cos^{2}{(p)} and u^{3}{(p)} = u^{2}{(p)} \\cos{(p)} and u^{3}{(p)} = u{(p)} \\cos^{2}{(p)} and \\frac{d}{d p} u^{3}{(p)} = \\frac{d}{d p} u{(p)} \\cos^{2}{(p)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["times", 1, "cos(Symbol('p', commutative=True))"], "Equality(Mul(Function('u')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True))), Pow(cos(Symbol('p', commutative=True)), Integer(2)))"], [["times", 1, "Mul(Function('u')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Function('u')(Symbol('p', commutative=True)), Integer(2)), cos(Symbol('p', commutative=True))), Mul(Function('u')(Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('u')(Symbol('p', commutative=True)), Integer(2)), Pow(cos(Symbol('p', commutative=True)), Integer(2))), Pow(cos(Symbol('p', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('u')(Symbol('p', commutative=True)), Integer(3)), cos(Symbol('p', commutative=True))), Mul(Pow(Function('u')(Symbol('p', commutative=True)), Integer(2)), Pow(cos(Symbol('p', commutative=True)), Integer(2))))"], [["divide", 5, "cos(Symbol('p', commutative=True))"], "Equality(Pow(Function('u')(Symbol('p', commutative=True)), Integer(3)), Mul(Pow(Function('u')(Symbol('p', commutative=True)), Integer(2)), cos(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Function('u')(Symbol('p', commutative=True)), Integer(3)), Mul(Function('u')(Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Integer(2))))"], [["differentiate", 7, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Function('u')(Symbol('p', commutative=True)), Integer(3)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Function('u')(Symbol('p', commutative=True)), Pow(cos(Symbol('p', commutative=True)), Integer(2))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(\\dot{y})} = \\log{(\\dot{y})} and c{(\\dot{y})} = \\frac{\\psi^{*}{(\\dot{y})}}{\\log{(\\dot{y})}}, then obtain c^{\\dot{y}}{(\\dot{y})} = 1", "derivation": "\\psi^{*}{(\\dot{y})} = \\log{(\\dot{y})} and c{(\\dot{y})} = \\frac{\\psi^{*}{(\\dot{y})}}{\\log{(\\dot{y})}} and c^{\\dot{y}}{(\\dot{y})} = (\\frac{\\psi^{*}{(\\dot{y})}}{\\log{(\\dot{y})}})^{\\dot{y}} and c^{\\dot{y}}{(\\dot{y})} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\dot{y}', commutative=True)), log(Symbol('\\\\dot{y}', commutative=True)))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\dot{y}', commutative=True)), Mul(Function('\\\\psi^*')(Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Function('\\\\psi^*')(Symbol('\\\\dot{y}', commutative=True)), Pow(log(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('c')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\sigma_{p}{(\\sigma_x,F_{H})} = F_{H}^{\\sigma_x}, then obtain F_{H}^{\\sigma_x} + (\\sigma_{p}^{\\sigma_x}{(\\sigma_x,F_{H})})^{\\sigma_x} - \\sigma_{p}{(\\sigma_x,F_{H})} = F_{H}^{\\sigma_x} + ((F_{H}^{\\sigma_x})^{\\sigma_x})^{\\sigma_x} - \\sigma_{p}{(\\sigma_x,F_{H})}", "derivation": "\\sigma_{p}{(\\sigma_x,F_{H})} = F_{H}^{\\sigma_x} and \\sigma_{p}^{\\sigma_x}{(\\sigma_x,F_{H})} = (F_{H}^{\\sigma_x})^{\\sigma_x} and (\\sigma_{p}^{\\sigma_x}{(\\sigma_x,F_{H})})^{\\sigma_x} = ((F_{H}^{\\sigma_x})^{\\sigma_x})^{\\sigma_x} and F_{H}^{\\sigma_x} + (\\sigma_{p}^{\\sigma_x}{(\\sigma_x,F_{H})})^{\\sigma_x} = F_{H}^{\\sigma_x} + ((F_{H}^{\\sigma_x})^{\\sigma_x})^{\\sigma_x} and F_{H}^{\\sigma_x} + (\\sigma_{p}^{\\sigma_x}{(\\sigma_x,F_{H})})^{\\sigma_x} - \\sigma_{p}{(\\sigma_x,F_{H})} = F_{H}^{\\sigma_x} + ((F_{H}^{\\sigma_x})^{\\sigma_x})^{\\sigma_x} - \\sigma_{p}{(\\sigma_x,F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Pow(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["add", 3, "Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Add(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 4, "Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Add(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)))), Add(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Pow(Symbol('F_H', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\sigma_x', commutative=True), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given f{(\\mathbf{J},i)} = (e^{\\mathbf{J}})^{i} and \\mu_{0}{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain (\\mu_{0}^{i}{(\\mathbf{J})} - (e^{\\mathbf{J}})^{i})^{i} = 0^{i}", "derivation": "f{(\\mathbf{J},i)} = (e^{\\mathbf{J}})^{i} and f{(\\mathbf{J},i)} - (e^{\\mathbf{J}})^{i} = 0 and - f{(\\mathbf{J},i)} + (e^{\\mathbf{J}})^{i} = 0 and \\mu_{0}{(\\mathbf{J})} = e^{\\mathbf{J}} and \\mu_{0}^{i}{(\\mathbf{J})} - f{(\\mathbf{J},i)} = 0 and \\mu_{0}^{i}{(\\mathbf{J})} - (e^{\\mathbf{J}})^{i} = 0 and (\\mu_{0}^{i}{(\\mathbf{J})} - (e^{\\mathbf{J}})^{i})^{i} = 0^{i}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('i', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)))"], [["minus", 1, "Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True))"], "Equality(Add(Function('f')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)))), Integer(0))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('f')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('i', commutative=True))), Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)), Mul(Integer(-1), Function('f')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('i', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)))), Integer(0))"], [["power", 6, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Pow(Function('\\\\mu_0')(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{J}', commutative=True)), Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Integer(0), Symbol('i', commutative=True)))"]]}, {"prompt": "Given M{(\\delta,B)} = \\int (B - \\delta) dB, then derive M{(\\delta,B)} = A_{1} + \\frac{B^{2}}{2} - B \\delta, then obtain M^{A_{1}}{(\\delta,B)} = (A_{1} + \\frac{B^{2}}{2} - B \\delta)^{A_{1}}", "derivation": "M{(\\delta,B)} = \\int (B - \\delta) dB and M{(\\delta,B)} = A_{1} + \\frac{B^{2}}{2} - B \\delta and \\int (B - \\delta) dB = A_{1} + \\frac{B^{2}}{2} - B \\delta and (\\int (B - \\delta) dB)^{A_{1}} = (A_{1} + \\frac{B^{2}}{2} - B \\delta)^{A_{1}} and M^{A_{1}}{(\\delta,B)} = (A_{1} + \\frac{B^{2}}{2} - B \\delta)^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('M')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('B', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('M')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(t,\\dot{z},\\mathbf{r})} = \\mathbf{r} (- \\dot{z} + t), then obtain t \\int \\mathbb{I}{(t,\\dot{z},\\mathbf{r})} dt = t \\int - \\dot{z} \\mathbf{r} dt + t \\int \\mathbf{r} t dt", "derivation": "\\mathbb{I}{(t,\\dot{z},\\mathbf{r})} = \\mathbf{r} (- \\dot{z} + t) and \\int \\mathbb{I}{(t,\\dot{z},\\mathbf{r})} dt = \\int \\mathbf{r} (- \\dot{z} + t) dt and t \\int \\mathbb{I}{(t,\\dot{z},\\mathbf{r})} dt = t \\int \\mathbf{r} (- \\dot{z} + t) dt and t \\int \\mathbb{I}{(t,\\dot{z},\\mathbf{r})} dt = t \\int - \\dot{z} \\mathbf{r} dt + t \\int \\mathbf{r} t dt", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["times", 2, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["expand", 3], "Equality(Mul(Symbol('t', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('t', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Mul(Symbol('t', commutative=True), Integral(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\rho{(a^{\\dagger},\\rho_f)} = - \\rho_f + a^{\\dagger}, then derive \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} = -1, then obtain - a^{\\dagger} + \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} = - a^{\\dagger} - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} - 1", "derivation": "\\rho{(a^{\\dagger},\\rho_f)} = - \\rho_f + a^{\\dagger} and \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} = \\frac{\\partial}{\\partial \\rho_f} (- \\rho_f + a^{\\dagger}) and \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} = -1 and \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} = - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} - 1 and - a^{\\dagger} + \\frac{\\partial}{\\partial \\rho_f} \\rho{(a^{\\dagger},\\rho_f)} - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} = - a^{\\dagger} - \\frac{\\partial}{\\partial a^{\\dagger}} \\rho{(a^{\\dagger},\\rho_f)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))"], [["minus", 3, "Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Integer(-1)))"], [["minus", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Integer(-1)))"]]}, {"prompt": "Given M{(V,\\omega)} = \\log{(\\frac{V}{\\omega})} and \\operatorname{v_{1}}{(V,\\omega)} = \\frac{V}{\\omega}, then obtain \\int \\frac{\\log{(\\frac{V}{\\omega})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} dV = \\int \\frac{\\log{(\\operatorname{v_{1}}{(V,\\omega)})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} dV", "derivation": "M{(V,\\omega)} = \\log{(\\frac{V}{\\omega})} and \\operatorname{v_{1}}{(V,\\omega)} = \\frac{V}{\\omega} and M{(V,\\omega)} = \\log{(\\operatorname{v_{1}}{(V,\\omega)})} and M{(V,\\omega)} - 1 = \\log{(\\operatorname{v_{1}}{(V,\\omega)})} - 1 and \\log{(\\frac{V}{\\omega})} - 1 = \\log{(\\operatorname{v_{1}}{(V,\\omega)})} - 1 and \\frac{\\log{(\\frac{V}{\\omega})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} = \\frac{\\log{(\\operatorname{v_{1}}{(V,\\omega)})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} and \\int \\frac{\\log{(\\frac{V}{\\omega})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} dV = \\int \\frac{\\log{(\\operatorname{v_{1}}{(V,\\omega)})} - 1}{\\frac{\\log{(\\frac{V}{\\omega})}}{\\omega} + \\frac{1}{\\omega}} dV", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('M')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), log(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('M')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1)), Add(log(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Integer(-1)), Add(log(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1)))"], [["divide", 5, "Add(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Integer(-1)), Add(log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Integer(-1))), Mul(Pow(Add(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Integer(-1)), Add(log(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1))))"], [["integrate", 6, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Integer(-1)), Add(log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integral(Mul(Pow(Add(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Integer(-1)), Add(log(Function('v_1')(Symbol('V', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(-1))), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\rho_f,H)} = \\cos{(H \\rho_f)} and \\ddot{x}{(a,f^{*})} = - a + \\cos{(f^{*})}, then obtain (- a - \\tilde{g}^*{(\\rho_f,H)} + \\cos{(f^{*})})^{\\rho_f} + \\ddot{x}{(a,f^{*})} - \\cos{(H \\rho_f)} = - a + (- a - \\tilde{g}^*{(\\rho_f,H)} + \\cos{(f^{*})})^{\\rho_f} + \\cos{(f^{*})} - \\cos{(H \\rho_f)}", "derivation": "\\tilde{g}^*{(\\rho_f,H)} = \\cos{(H \\rho_f)} and \\ddot{x}{(a,f^{*})} = - a + \\cos{(f^{*})} and \\ddot{x}{(a,f^{*})} - \\tilde{g}^*{(\\rho_f,H)} = - a - \\tilde{g}^*{(\\rho_f,H)} + \\cos{(f^{*})} and \\ddot{x}{(a,f^{*})} - \\cos{(H \\rho_f)} = - a + \\cos{(f^{*})} - \\cos{(H \\rho_f)} and (- a - \\tilde{g}^*{(\\rho_f,H)} + \\cos{(f^{*})})^{\\rho_f} + \\ddot{x}{(a,f^{*})} - \\cos{(H \\rho_f)} = - a + (- a - \\tilde{g}^*{(\\rho_f,H)} + \\cos{(f^{*})})^{\\rho_f} + \\cos{(f^{*})} - \\cos{(H \\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True)), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('f^*', commutative=True))))"], [["minus", 2, "Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True)))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), cos(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_f', commutative=True))))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), cos(Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_f', commutative=True))))))"], [["add", 4, "Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), cos(Symbol('f^*', commutative=True))), Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), cos(Symbol('f^*', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Function('\\\\ddot{x}')(Symbol('a', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_f', commutative=True))))), Add(Mul(Integer(-1), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\rho_f', commutative=True), Symbol('H', commutative=True))), cos(Symbol('f^*', commutative=True))), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('f^*', commutative=True)), Mul(Integer(-1), cos(Mul(Symbol('H', commutative=True), Symbol('\\\\rho_f', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(a,\\mathbf{A})} = a^{\\mathbf{A}} and \\psi{(a,\\mathbf{A})} = a^{- \\mathbf{A}}, then obtain (e^{\\psi{(a,\\mathbf{A})} \\operatorname{t_{1}}{(a,\\mathbf{A})}})^{a} = e^{a}", "derivation": "\\operatorname{t_{1}}{(a,\\mathbf{A})} = a^{\\mathbf{A}} and a^{- \\mathbf{A}} \\operatorname{t_{1}}{(a,\\mathbf{A})} = 1 and \\psi{(a,\\mathbf{A})} = a^{- \\mathbf{A}} and e^{a^{- \\mathbf{A}} \\operatorname{t_{1}}{(a,\\mathbf{A})}} = e and e^{\\psi{(a,\\mathbf{A})} \\operatorname{t_{1}}{(a,\\mathbf{A})}} = e and e^{\\psi{(a,\\mathbf{A})} \\operatorname{t_{1}}{(a,\\mathbf{A})}} = e^{a^{- \\mathbf{A}} \\operatorname{t_{1}}{(a,\\mathbf{A})}} and (e^{\\psi{(a,\\mathbf{A})} \\operatorname{t_{1}}{(a,\\mathbf{A})}})^{a} = (e^{a^{- \\mathbf{A}} \\operatorname{t_{1}}{(a,\\mathbf{A})}})^{a} and (e^{\\psi{(a,\\mathbf{A})} \\operatorname{t_{1}}{(a,\\mathbf{A})}})^{a} = e^{a}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 1, "Pow(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"], [["exp", 2], "Equality(exp(Mul(Pow(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), E)"], [["substitute_RHS_for_LHS", 4, 3], "Equality(exp(Mul(Function('\\\\psi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), E)"], [["substitute_RHS_for_LHS", 5, 4], "Equality(exp(Mul(Function('\\\\psi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), exp(Mul(Pow(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Pow(exp(Mul(Function('\\\\psi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('a', commutative=True)), Pow(exp(Mul(Pow(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(exp(Mul(Function('\\\\psi')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Function('t_1')(Symbol('a', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})} = \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})}, then derive \\frac{\\partial}{\\partial M} \\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})} = - \\frac{\\hat{\\mathbf{x}} \\cos{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M^{2}}, then obtain - \\frac{\\hat{\\mathbf{x}} \\cos{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M^{3}} = \\frac{\\frac{\\partial}{\\partial M} \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M}", "derivation": "\\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})} = \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})} and \\frac{\\partial}{\\partial M} \\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial M} \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})} and \\frac{\\partial}{\\partial M} \\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})} = - \\frac{\\hat{\\mathbf{x}} \\cos{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M^{2}} and \\frac{\\frac{\\partial}{\\partial M} \\operatorname{F_{H}}{(M,\\hat{\\mathbf{x}})}}{M} = \\frac{\\frac{\\partial}{\\partial M} \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M} and - \\frac{\\hat{\\mathbf{x}} \\cos{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M^{3}} = \\frac{\\frac{\\partial}{\\partial M} \\sin{(\\frac{\\hat{\\mathbf{x}}}{M})}}{M}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('M', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('M', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('M', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-2)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["times", 2, "Pow(Symbol('M', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(Function('F_H')(Symbol('M', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('M', commutative=True), Integer(-3)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), cos(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(sin(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given T{(s)} = \\log{(s)}, then obtain - \\frac{\\log{(s)}^{s}}{\\log{(s)}} + \\frac{\\frac{d}{d s} T{(s)}}{\\log{(s)}} = - \\frac{\\log{(s)}^{s}}{\\log{(s)}} + \\frac{\\frac{d}{d s} \\log{(s)}}{\\log{(s)}}", "derivation": "T{(s)} = \\log{(s)} and \\frac{d}{d s} T{(s)} = \\frac{d}{d s} \\log{(s)} and \\frac{\\frac{d}{d s} T{(s)}}{\\log{(s)}} = \\frac{\\frac{d}{d s} \\log{(s)}}{\\log{(s)}} and - \\frac{\\log{(s)}^{s}}{\\log{(s)}} + \\frac{\\frac{d}{d s} T{(s)}}{\\log{(s)}} = - \\frac{\\log{(s)}^{s}}{\\log{(s)}} + \\frac{\\frac{d}{d s} \\log{(s)}}{\\log{(s)}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('s', commutative=True))"], "Equality(Mul(Pow(log(Symbol('s', commutative=True)), Integer(-1)), Derivative(Function('T')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('s', commutative=True)), Integer(-1)), Derivative(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["minus", 3, "Mul(Pow(log(Symbol('s', commutative=True)), Integer(-1)), Pow(log(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('s', commutative=True)), Integer(-1)), Pow(log(Symbol('s', commutative=True)), Symbol('s', commutative=True))), Mul(Pow(log(Symbol('s', commutative=True)), Integer(-1)), Derivative(Function('T')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(log(Symbol('s', commutative=True)), Integer(-1)), Pow(log(Symbol('s', commutative=True)), Symbol('s', commutative=True))), Mul(Pow(log(Symbol('s', commutative=True)), Integer(-1)), Derivative(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mu_{0}{(S)} = \\log{(\\cos{(S)})} and \\tilde{g}{(S)} = \\log{(\\cos{(S)})}^{S}, then obtain 2 \\mu_{0}^{S}{(S)} \\tilde{g}{(S)} = (\\mu_{0}^{S}{(S)} + \\tilde{g}{(S)}) \\tilde{g}{(S)}", "derivation": "\\mu_{0}{(S)} = \\log{(\\cos{(S)})} and \\mu_{0}^{S}{(S)} = \\log{(\\cos{(S)})}^{S} and \\tilde{g}{(S)} = \\log{(\\cos{(S)})}^{S} and 2 \\mu_{0}^{S}{(S)} = \\mu_{0}^{S}{(S)} + \\log{(\\cos{(S)})}^{S} and 2 \\mu_{0}^{S}{(S)} = \\mu_{0}^{S}{(S)} + \\tilde{g}{(S)} and 2 \\mu_{0}^{S}{(S)} \\tilde{g}{(S)} = (\\mu_{0}^{S}{(S)} + \\tilde{g}{(S)}) \\tilde{g}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('S', commutative=True)), log(cos(Symbol('S', commutative=True))))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(log(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('S', commutative=True)), Pow(log(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["add", 2, "Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Add(Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(log(cos(Symbol('S', commutative=True))), Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True))), Add(Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Function('\\\\tilde{g}')(Symbol('S', commutative=True))))"], [["times", 5, "Function('\\\\tilde{g}')(Symbol('S', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Function('\\\\tilde{g}')(Symbol('S', commutative=True))), Mul(Add(Pow(Function('\\\\mu_0')(Symbol('S', commutative=True)), Symbol('S', commutative=True)), Function('\\\\tilde{g}')(Symbol('S', commutative=True))), Function('\\\\tilde{g}')(Symbol('S', commutative=True))))"]]}, {"prompt": "Given k{(\\sigma_x,u)} = \\sigma_x u, then obtain k{(\\sigma_x,u)} - k^{\\sigma_x}{(\\sigma_x,u)} = \\sigma_x u - k^{\\sigma_x}{(\\sigma_x,u)}", "derivation": "k{(\\sigma_x,u)} = \\sigma_x u and k^{\\sigma_x}{(\\sigma_x,u)} = (\\sigma_x u)^{\\sigma_x} and - (\\sigma_x u)^{\\sigma_x} + k{(\\sigma_x,u)} = \\sigma_x u - (\\sigma_x u)^{\\sigma_x} and k{(\\sigma_x,u)} - k^{\\sigma_x}{(\\sigma_x,u)} = \\sigma_x u - k^{\\sigma_x}{(\\sigma_x,u)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True))), Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Function('k')(Symbol('\\\\sigma_x', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(v_{y})} = e^{\\cos{(v_{y})}}, then derive \\frac{d}{d v_{y}} \\phi_{2}{(v_{y})} = - e^{\\cos{(v_{y})}} \\sin{(v_{y})}, then obtain \\frac{d}{d v_{y}} e^{\\cos{(v_{y})}} = - \\phi_{2}{(v_{y})} \\sin{(v_{y})}", "derivation": "\\phi_{2}{(v_{y})} = e^{\\cos{(v_{y})}} and \\frac{d}{d v_{y}} \\phi_{2}{(v_{y})} = \\frac{d}{d v_{y}} e^{\\cos{(v_{y})}} and \\frac{d}{d v_{y}} \\phi_{2}{(v_{y})} = - e^{\\cos{(v_{y})}} \\sin{(v_{y})} and \\frac{d}{d v_{y}} \\phi_{2}{(v_{y})} = - \\phi_{2}{(v_{y})} \\sin{(v_{y})} and - e^{\\cos{(v_{y})}} \\sin{(v_{y})} = - \\phi_{2}{(v_{y})} \\sin{(v_{y})} and - e^{\\cos{(v_{y})}} \\sin{(v_{y})} = \\frac{d}{d v_{y}} e^{\\cos{(v_{y})}} and \\frac{d}{d v_{y}} e^{\\cos{(v_{y})}} = - \\phi_{2}{(v_{y})} \\sin{(v_{y})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('v_y', commutative=True)), exp(cos(Symbol('v_y', commutative=True))))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('v_y', commutative=True))), sin(Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\phi_2')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('v_y', commutative=True))), sin(Symbol('v_y', commutative=True))), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), exp(cos(Symbol('v_y', commutative=True))), sin(Symbol('v_y', commutative=True))), Derivative(exp(cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(exp(cos(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\phi_2')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\Omega{(t,\\mathbf{r})} = \\mathbf{r} t, then derive \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\Omega{(t,\\mathbf{r})} dt = a^{\\dagger} + \\frac{t^{2}}{2}, then obtain t \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} t dt = t (a^{\\dagger} + \\frac{t^{2}}{2})", "derivation": "\\Omega{(t,\\mathbf{r})} = \\mathbf{r} t and \\frac{\\partial}{\\partial \\mathbf{r}} \\Omega{(t,\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} t and \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\Omega{(t,\\mathbf{r})} dt = \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} t dt and \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\Omega{(t,\\mathbf{r})} dt = a^{\\dagger} + \\frac{t^{2}}{2} and t \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\Omega{(t,\\mathbf{r})} dt = t (a^{\\dagger} + \\frac{t^{2}}{2}) and t \\int \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} t dt = t (a^{\\dagger} + \\frac{t^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Omega')(Symbol('t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\Omega')(Symbol('t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["times", 4, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Integral(Derivative(Function('\\\\Omega')(Symbol('t', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('t', commutative=True), Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('t', commutative=True)))), Mul(Symbol('t', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\varphi^{*}{(f^{\\prime},\\rho)} = \\frac{f^{\\prime}}{\\rho}, then obtain \\int (\\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\varphi^{*}{(f^{\\prime},\\rho)} df^{\\prime}) df^{\\prime} = \\int (\\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\frac{f^{\\prime}}{\\rho} df^{\\prime}) df^{\\prime}", "derivation": "\\varphi^{*}{(f^{\\prime},\\rho)} = \\frac{f^{\\prime}}{\\rho} and \\int \\varphi^{*}{(f^{\\prime},\\rho)} df^{\\prime} = \\int \\frac{f^{\\prime}}{\\rho} df^{\\prime} and \\int \\varphi^{*}{(f^{\\prime},\\rho)} df^{\\prime} + \\frac{f^{\\prime}}{\\rho} = \\int \\frac{f^{\\prime}}{\\rho} df^{\\prime} + \\frac{f^{\\prime}}{\\rho} and \\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\varphi^{*}{(f^{\\prime},\\rho)} df^{\\prime} = \\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\frac{f^{\\prime}}{\\rho} df^{\\prime} and \\int (\\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\varphi^{*}{(f^{\\prime},\\rho)} df^{\\prime}) df^{\\prime} = \\int (\\varphi^{*}{(f^{\\prime},\\rho)} + \\int \\frac{f^{\\prime}}{\\rho} df^{\\prime}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Integral(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))), Add(Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Add(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Add(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Function('\\\\varphi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A_{2},v_{1})} = - v_{1} + e^{A_{2}} and \\operatorname{A_{x}}{(A_{2},v_{1})} = \\frac{\\partial}{\\partial A_{2}} (- v_{1} + e^{A_{2}}), then derive \\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{g}}{(A_{2},v_{1})} = e^{A_{2}}, then obtain - v_{1} + \\operatorname{A_{x}}{(A_{2},v_{1})} = - v_{1} + e^{A_{2}}", "derivation": "\\operatorname{P_{g}}{(A_{2},v_{1})} = - v_{1} + e^{A_{2}} and v_{1} + \\operatorname{P_{g}}{(A_{2},v_{1})} = e^{A_{2}} and \\frac{\\partial}{\\partial A_{2}} (v_{1} + \\operatorname{P_{g}}{(A_{2},v_{1})}) = \\frac{d}{d A_{2}} e^{A_{2}} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{g}}{(A_{2},v_{1})} = e^{A_{2}} and \\frac{\\partial}{\\partial A_{2}} (- v_{1} + e^{A_{2}}) = e^{A_{2}} and \\operatorname{A_{x}}{(A_{2},v_{1})} = \\frac{\\partial}{\\partial A_{2}} (- v_{1} + e^{A_{2}}) and - v_{1} + \\operatorname{A_{x}}{(A_{2},v_{1})} = - v_{1} + \\frac{\\partial}{\\partial A_{2}} (- v_{1} + e^{A_{2}}) and - v_{1} + \\operatorname{A_{x}}{(A_{2},v_{1})} = - v_{1} + e^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Symbol('A_2', commutative=True))))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('P_g')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True))), exp(Symbol('A_2', commutative=True)))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Symbol('v_1', commutative=True), Function('P_g')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(exp(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('P_g')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), exp(Symbol('A_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), exp(Symbol('A_2', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["minus", 6, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('A_x')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('A_x')(Symbol('A_2', commutative=True), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), exp(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\phi{(F_{H})} = \\cos{(\\sin{(F_{H})})} and \\mathbf{J}_M{(F_{H})} = F_{H}, then obtain \\frac{\\mathbf{J}_M{(F_{H})} \\sin{(\\phi{(F_{H})})}}{F_{H}} = \\sin{(\\phi{(F_{H})})}", "derivation": "\\phi{(F_{H})} = \\cos{(\\sin{(F_{H})})} and \\mathbf{J}_M{(F_{H})} = F_{H} and \\sin{(\\phi{(F_{H})})} = \\sin{(\\cos{(\\sin{(F_{H})})})} and \\frac{\\mathbf{J}_M{(F_{H})}}{F_{H}} = 1 and \\frac{\\mathbf{J}_M{(F_{H})} \\sin{(\\cos{(\\sin{(F_{H})})})}}{F_{H}} = \\sin{(\\cos{(\\sin{(F_{H})})})} and \\frac{\\mathbf{J}_M{(F_{H})} \\sin{(\\phi{(F_{H})})}}{F_{H}} = \\sin{(\\phi{(F_{H})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi')(Symbol('F_H', commutative=True)), cos(sin(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["sin", 1], "Equality(sin(Function('\\\\phi')(Symbol('F_H', commutative=True))), sin(cos(sin(Symbol('F_H', commutative=True)))))"], [["divide", 2, "Symbol('F_H', commutative=True)"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('F_H', commutative=True))), Integer(1))"], [["times", 4, "sin(cos(sin(Symbol('F_H', commutative=True))))"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('F_H', commutative=True)), sin(cos(sin(Symbol('F_H', commutative=True))))), sin(cos(sin(Symbol('F_H', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('F_H', commutative=True)), sin(Function('\\\\phi')(Symbol('F_H', commutative=True)))), sin(Function('\\\\phi')(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given U{(I)} = e^{I}, then obtain (U{(I)} + \\int U^{I}{(I)} dI)^{I} = (U{(I)} + \\int (e^{I})^{I} dI)^{I}", "derivation": "U{(I)} = e^{I} and U^{I}{(I)} = (e^{I})^{I} and \\int U^{I}{(I)} dI = \\int (e^{I})^{I} dI and e^{I} + \\int U^{I}{(I)} dI = e^{I} + \\int (e^{I})^{I} dI and U{(I)} + \\int U^{I}{(I)} dI = U{(I)} + \\int (e^{I})^{I} dI and (U{(I)} + \\int U^{I}{(I)} dI)^{I} = (U{(I)} + \\int (e^{I})^{I} dI)^{I}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('U')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(exp(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Function('U')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(exp(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 3, "exp(Symbol('I', commutative=True))"], "Equality(Add(exp(Symbol('I', commutative=True)), Integral(Pow(Function('U')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(exp(Symbol('I', commutative=True)), Integral(Pow(exp(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('U')(Symbol('I', commutative=True)), Integral(Pow(Function('U')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Add(Function('U')(Symbol('I', commutative=True)), Integral(Pow(exp(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["power", 5, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Function('U')(Symbol('I', commutative=True)), Integral(Pow(Function('U')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Function('U')(Symbol('I', commutative=True)), Integral(Pow(exp(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\pi,\\mathbf{M})} = \\log{(\\frac{\\mathbf{M}}{\\pi})} and y{(C_{1})} = e^{C_{1}}, then obtain \\frac{y{(C_{1})} - \\log{(\\frac{\\mathbf{M}}{\\pi})}}{e^{C_{1}} - \\log{(\\frac{\\mathbf{M}}{\\pi})}} = 1", "derivation": "\\mathbf{P}{(\\pi,\\mathbf{M})} = \\log{(\\frac{\\mathbf{M}}{\\pi})} and - \\mathbf{P}{(\\pi,\\mathbf{M})} = - \\log{(\\frac{\\mathbf{M}}{\\pi})} and y{(C_{1})} = e^{C_{1}} and y{(C_{1})} - \\log{(\\frac{\\mathbf{M}}{\\pi})} = e^{C_{1}} - \\log{(\\frac{\\mathbf{M}}{\\pi})} and - \\mathbf{P}{(\\pi,\\mathbf{M})} + y{(C_{1})} = - \\mathbf{P}{(\\pi,\\mathbf{M})} + e^{C_{1}} and \\frac{- \\mathbf{P}{(\\pi,\\mathbf{M})} + y{(C_{1})}}{- \\mathbf{P}{(\\pi,\\mathbf{M})} + e^{C_{1}}} = 1 and \\frac{y{(C_{1})} - \\log{(\\frac{\\mathbf{M}}{\\pi})}}{e^{C_{1}} - \\log{(\\frac{\\mathbf{M}}{\\pi})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))))"], ["get_premise", "Equality(Function('y')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["minus", 3, "log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], "Equality(Add(Function('y')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))), Add(exp(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('y')(Symbol('C_1', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('C_1', commutative=True))))"], [["divide", 5, "Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('C_1', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('y')(Symbol('C_1', commutative=True))), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), exp(Symbol('C_1', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Add(Function('y')(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))), Pow(Add(exp(Symbol('C_1', commutative=True)), Mul(Integer(-1), log(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A_{2})} = \\cos{(A_{2})}, then obtain e^{\\frac{d}{d A_{2}} (2 \\operatorname{E_{n}}{(A_{2})})^{A_{2}}} = e^{\\frac{d}{d A_{2}} (\\operatorname{E_{n}}{(A_{2})} + \\cos{(A_{2})})^{A_{2}}}", "derivation": "\\operatorname{E_{n}}{(A_{2})} = \\cos{(A_{2})} and 2 \\operatorname{E_{n}}{(A_{2})} = \\operatorname{E_{n}}{(A_{2})} + \\cos{(A_{2})} and (2 \\operatorname{E_{n}}{(A_{2})})^{A_{2}} = (\\operatorname{E_{n}}{(A_{2})} + \\cos{(A_{2})})^{A_{2}} and \\frac{d}{d A_{2}} (2 \\operatorname{E_{n}}{(A_{2})})^{A_{2}} = \\frac{d}{d A_{2}} (\\operatorname{E_{n}}{(A_{2})} + \\cos{(A_{2})})^{A_{2}} and e^{\\frac{d}{d A_{2}} (2 \\operatorname{E_{n}}{(A_{2})})^{A_{2}}} = e^{\\frac{d}{d A_{2}} (\\operatorname{E_{n}}{(A_{2})} + \\cos{(A_{2})})^{A_{2}}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["add", 1, "Function('E_n')(Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(2), Function('E_n')(Symbol('A_2', commutative=True))), Add(Function('E_n')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('E_n')(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Pow(Add(Function('E_n')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('E_n')(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Pow(Add(Function('E_n')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(Derivative(Pow(Mul(Integer(2), Function('E_n')(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), exp(Derivative(Pow(Add(Function('E_n')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given y{(k,n_{1},\\dot{y})} = \\frac{\\dot{y} + n_{1}}{k}, then obtain \\dot{y} + \\frac{\\partial}{\\partial k} y^{n_{1}}{(k,n_{1},\\dot{y})} = \\dot{y} + \\frac{\\partial}{\\partial k} (\\frac{\\dot{y} + n_{1}}{k})^{n_{1}}", "derivation": "y{(k,n_{1},\\dot{y})} = \\frac{\\dot{y} + n_{1}}{k} and y^{n_{1}}{(k,n_{1},\\dot{y})} = (\\frac{\\dot{y} + n_{1}}{k})^{n_{1}} and \\frac{\\partial}{\\partial k} y^{n_{1}}{(k,n_{1},\\dot{y})} = \\frac{\\partial}{\\partial k} (\\frac{\\dot{y} + n_{1}}{k})^{n_{1}} and \\dot{y} + \\frac{\\partial}{\\partial k} y^{n_{1}}{(k,n_{1},\\dot{y})} = \\dot{y} + \\frac{\\partial}{\\partial k} (\\frac{\\dot{y} + n_{1}}{k})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('k', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('n_1', commutative=True))))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('y')(Symbol('k', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('n_1', commutative=True)), Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Pow(Function('y')(Symbol('k', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), Derivative(Pow(Function('y')(Symbol('k', commutative=True), Symbol('n_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Symbol('\\\\dot{y}', commutative=True), Derivative(Pow(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}_0{(U,\\dot{z})} = \\dot{z} \\cos{(U)}, then obtain \\frac{\\partial}{\\partial U} (\\int - \\dot{z} \\hat{x}_0{(U,\\dot{z})} d\\dot{z} + \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z}) = \\frac{\\partial}{\\partial U} 2 \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z}", "derivation": "\\hat{x}_0{(U,\\dot{z})} = \\dot{z} \\cos{(U)} and - \\hat{x}_0{(U,\\dot{z})} = - \\dot{z} \\cos{(U)} and - \\dot{z} \\hat{x}_0{(U,\\dot{z})} = - \\dot{z}^{2} \\cos{(U)} and \\int - \\dot{z} \\hat{x}_0{(U,\\dot{z})} d\\dot{z} = \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z} and \\int - \\dot{z} \\hat{x}_0{(U,\\dot{z})} d\\dot{z} + \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z} = 2 \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z} and \\frac{\\partial}{\\partial U} (\\int - \\dot{z} \\hat{x}_0{(U,\\dot{z})} d\\dot{z} + \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z}) = \\frac{\\partial}{\\partial U} 2 \\int - \\dot{z}^{2} \\cos{(U)} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), cos(Symbol('U', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), cos(Symbol('U', commutative=True))))"], [["times", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["add", 4, "Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Integral(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(2), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"], [["differentiate", 5, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Integral(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True), Function('\\\\hat{x}_0')(Symbol('U', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(Mul(Integer(-1), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2)), cos(Symbol('U', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(E_{x})} = e^{E_{x}} and k{(E_{x})} = E_{x}, then obtain k{(E_{x})} - \\int \\operatorname{f_{\\mathbf{v}}}{(E_{x})} dE_{x} = E_{x} - \\int \\operatorname{f_{\\mathbf{v}}}{(E_{x})} dE_{x}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(E_{x})} = e^{E_{x}} and k{(E_{x})} = E_{x} and \\int \\operatorname{f_{\\mathbf{v}}}{(E_{x})} dE_{x} = \\int e^{E_{x}} dE_{x} and k{(E_{x})} - \\int e^{E_{x}} dE_{x} = E_{x} - \\int e^{E_{x}} dE_{x} and k{(E_{x})} - \\int \\operatorname{f_{\\mathbf{v}}}{(E_{x})} dE_{x} = E_{x} - \\int \\operatorname{f_{\\mathbf{v}}}{(E_{x})} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], ["renaming_premise", "Equality(Function('k')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["minus", 2, "Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))"], "Equality(Add(Function('k')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Integral(exp(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('k')(Symbol('E_x', commutative=True)), Mul(Integer(-1), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Integral(Function('f_{\\\\mathbf{v}}')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(\\nabla,z^{*})} = \\log{((z^{*})^{\\nabla})}, then derive \\frac{\\frac{\\partial}{\\partial z^{*}} \\varepsilon{(\\nabla,z^{*})}}{\\nabla} = \\frac{1}{z^{*}}, then obtain \\frac{(z^{*})^{- \\nabla}}{z^{*}} = \\frac{(z^{*})^{- \\nabla} \\frac{\\partial}{\\partial z^{*}} \\log{((z^{*})^{\\nabla})}}{\\nabla}", "derivation": "\\varepsilon{(\\nabla,z^{*})} = \\log{((z^{*})^{\\nabla})} and \\frac{\\partial}{\\partial z^{*}} \\varepsilon{(\\nabla,z^{*})} = \\frac{\\partial}{\\partial z^{*}} \\log{((z^{*})^{\\nabla})} and \\frac{\\frac{\\partial}{\\partial z^{*}} \\varepsilon{(\\nabla,z^{*})}}{\\nabla} = \\frac{\\frac{\\partial}{\\partial z^{*}} \\log{((z^{*})^{\\nabla})}}{\\nabla} and \\frac{\\frac{\\partial}{\\partial z^{*}} \\varepsilon{(\\nabla,z^{*})}}{\\nabla} = \\frac{1}{z^{*}} and \\frac{1}{z^{*}} = \\frac{\\frac{\\partial}{\\partial z^{*}} \\log{((z^{*})^{\\nabla})}}{\\nabla} and \\frac{(z^{*})^{- \\nabla}}{z^{*}} = \\frac{(z^{*})^{- \\nabla} \\frac{\\partial}{\\partial z^{*}} \\log{((z^{*})^{\\nabla})}}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)), log(Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(log(Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\nabla', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Pow(Symbol('z^*', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('z^*', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Derivative(log(Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["divide", 5, "Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True))), Derivative(log(Pow(Symbol('z^*', commutative=True), Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{X}{(h,\\lambda)} = \\lambda + h, then derive \\int \\hat{X}{(h,\\lambda)} d\\lambda = \\frac{\\lambda^{2}}{2} + \\lambda h + x, then derive \\frac{\\lambda^{2}}{2} + \\lambda h + \\mathbf{A} = \\frac{\\lambda^{2}}{2} + \\lambda h + x, then obtain \\frac{\\lambda^{2}}{2} + \\lambda h + \\mathbf{A} + \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda + \\int \\hat{X}{(h,\\lambda)} d\\lambda", "derivation": "\\hat{X}{(h,\\lambda)} = \\lambda + h and \\int \\hat{X}{(h,\\lambda)} d\\lambda = \\int (\\lambda + h) d\\lambda and \\int \\hat{X}{(h,\\lambda)} d\\lambda = \\frac{\\lambda^{2}}{2} + \\lambda h + x and \\int (\\lambda + h) d\\lambda = \\frac{\\lambda^{2}}{2} + \\lambda h + x and \\frac{\\lambda^{2}}{2} + \\lambda h + \\mathbf{A} = \\frac{\\lambda^{2}}{2} + \\lambda h + x and \\frac{\\lambda^{2}}{2} + \\lambda h + \\mathbf{A} + \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda = \\frac{\\lambda^{2}}{2} + \\lambda h + x + \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda and \\frac{\\lambda^{2}}{2} + \\lambda h + \\mathbf{A} + \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int (\\lambda + h) d\\lambda + \\int \\hat{X}{(h,\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('x', commutative=True)))"], [["add", 5, "Derivative(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Derivative(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('x', commutative=True), Derivative(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\lambda', commutative=True), Integer(2))), Mul(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True), Derivative(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Derivative(Integral(Add(Symbol('\\\\lambda', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Integral(Function('\\\\hat{X}')(Symbol('h', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(n_{1},E_{x})} = \\sin{(E_{x} + n_{1})}, then obtain \\frac{\\partial}{\\partial E_{x}} \\iint \\hat{p}_0{(n_{1},E_{x})} dE_{x} dE_{x} = \\frac{\\partial}{\\partial E_{x}} \\iint \\sin{(E_{x} + n_{1})} dE_{x} dE_{x}", "derivation": "\\hat{p}_0{(n_{1},E_{x})} = \\sin{(E_{x} + n_{1})} and \\int \\hat{p}_0{(n_{1},E_{x})} dE_{x} = \\int \\sin{(E_{x} + n_{1})} dE_{x} and \\iint \\hat{p}_0{(n_{1},E_{x})} dE_{x} dE_{x} = \\iint \\sin{(E_{x} + n_{1})} dE_{x} dE_{x} and \\frac{\\partial}{\\partial E_{x}} \\iint \\hat{p}_0{(n_{1},E_{x})} dE_{x} dE_{x} = \\frac{\\partial}{\\partial E_{x}} \\iint \\sin{(E_{x} + n_{1})} dE_{x} dE_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('E_x', commutative=True)), sin(Add(Symbol('E_x', commutative=True), Symbol('n_1', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(sin(Add(Symbol('E_x', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["integrate", 2, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(sin(Add(Symbol('E_x', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{p}_0')(Symbol('n_1', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Integral(sin(Add(Symbol('E_x', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(T)} = \\log{(T)} and \\mathbf{v}{(T)} = \\frac{T \\frac{d}{d T} \\mathbf{p}{(T)}}{\\mathbf{p}{(T)}} + \\log{(\\mathbf{p}{(T)})}, then derive (\\frac{T \\frac{d}{d T} \\mathbf{p}{(T)}}{\\mathbf{p}{(T)}} + \\log{(\\mathbf{p}{(T)})}) \\mathbf{p}^{T}{(T)} - \\log{(T)} = (\\log{(\\log{(T)})} + \\frac{1}{\\log{(T)}}) \\log{(T)}^{T} - \\log{(T)}, then obtain \\mathbf{p}^{T}{(T)} \\mathbf{v}{(T)} - \\log{(T)} = (\\log{(\\log{(T)})} + \\frac{1}{\\log{(T)}}) \\log{(T)}^{T} - \\log{(T)}", "derivation": "\\mathbf{p}{(T)} = \\log{(T)} and \\mathbf{p}^{T}{(T)} = \\log{(T)}^{T} and \\frac{d}{d T} \\mathbf{p}^{T}{(T)} = \\frac{d}{d T} \\log{(T)}^{T} and - \\log{(T)} + \\frac{d}{d T} \\mathbf{p}^{T}{(T)} = - \\log{(T)} + \\frac{d}{d T} \\log{(T)}^{T} and (\\frac{T \\frac{d}{d T} \\mathbf{p}{(T)}}{\\mathbf{p}{(T)}} + \\log{(\\mathbf{p}{(T)})}) \\mathbf{p}^{T}{(T)} - \\log{(T)} = (\\log{(\\log{(T)})} + \\frac{1}{\\log{(T)}}) \\log{(T)}^{T} - \\log{(T)} and \\mathbf{v}{(T)} = \\frac{T \\frac{d}{d T} \\mathbf{p}{(T)}}{\\mathbf{p}{(T)}} + \\log{(\\mathbf{p}{(T)})} and \\mathbf{p}^{T}{(T)} \\mathbf{v}{(T)} - \\log{(T)} = (\\log{(\\log{(T)})} + \\frac{1}{\\log{(T)}}) \\log{(T)}^{T} - \\log{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True)))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["minus", 3, "log(Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Derivative(Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Derivative(Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Mul(Symbol('T', commutative=True), Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), log(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)))), Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Add(Mul(Add(log(log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Integer(-1))), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('T', commutative=True)), Add(Mul(Symbol('T', commutative=True), Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), log(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Pow(Function('\\\\mathbf{p}')(Symbol('T', commutative=True)), Symbol('T', commutative=True)), Function('\\\\mathbf{v}')(Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))), Add(Mul(Add(log(log(Symbol('T', commutative=True))), Pow(log(Symbol('T', commutative=True)), Integer(-1))), Pow(log(Symbol('T', commutative=True)), Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(I)} = e^{I}, then obtain \\Omega + \\int \\frac{\\operatorname{E_{n}}{(I)} - e^{I}}{\\operatorname{E_{n}}{(I)}} dI = \\int 0 dI", "derivation": "\\operatorname{E_{n}}{(I)} = e^{I} and 1 = \\frac{e^{I}}{\\operatorname{E_{n}}{(I)}} and 1 - \\frac{e^{I}}{\\operatorname{E_{n}}{(I)}} = 0 and \\int (1 - \\frac{e^{I}}{\\operatorname{E_{n}}{(I)}}) dI = \\int 0 dI and \\Omega + \\int \\frac{\\operatorname{E_{n}}{(I)} - e^{I}}{\\operatorname{E_{n}}{(I)}} dI = \\int 0 dI", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["divide", 1, "Function('E_n')(Symbol('I', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_n')(Symbol('I', commutative=True)), Integer(-1)), exp(Symbol('I', commutative=True))))"], [["minus", 2, "Mul(Pow(Function('E_n')(Symbol('I', commutative=True)), Integer(-1)), exp(Symbol('I', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('E_n')(Symbol('I', commutative=True)), Integer(-1)), exp(Symbol('I', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Pow(Function('E_n')(Symbol('I', commutative=True)), Integer(-1)), exp(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Omega', commutative=True), Integral(Mul(Add(Function('E_n')(Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('I', commutative=True)))), Pow(Function('E_n')(Symbol('I', commutative=True)), Integer(-1))), Tuple(Symbol('I', commutative=True)))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(W)} = \\log{(W)}, then obtain - W \\log{(W)} - \\phi_{1}{(W)} + \\frac{d}{d W} (W \\phi_{1}{(W)} + \\phi_{1}{(W)} + \\log{(W)}) = - W \\log{(W)} - \\phi_{1}{(W)} + \\frac{d}{d W} (W \\log{(W)} + \\phi_{1}{(W)} + \\log{(W)})", "derivation": "\\phi_{1}{(W)} = \\log{(W)} and W \\phi_{1}{(W)} = W \\log{(W)} and W \\phi_{1}{(W)} + \\phi_{1}{(W)} = W \\log{(W)} + \\phi_{1}{(W)} and W \\phi_{1}{(W)} + \\phi_{1}{(W)} + \\log{(W)} = W \\log{(W)} + \\phi_{1}{(W)} + \\log{(W)} and \\frac{d}{d W} (W \\phi_{1}{(W)} + \\phi_{1}{(W)} + \\log{(W)}) = \\frac{d}{d W} (W \\log{(W)} + \\phi_{1}{(W)} + \\log{(W)}) and - W \\log{(W)} - \\phi_{1}{(W)} + \\frac{d}{d W} (W \\phi_{1}{(W)} + \\phi_{1}{(W)} + \\log{(W)}) = - W \\log{(W)} - \\phi_{1}{(W)} + \\frac{d}{d W} (W \\log{(W)} + \\phi_{1}{(W)} + \\log{(W)})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('\\\\phi_1')(Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))))"], [["add", 2, "Function('\\\\phi_1')(Symbol('W', commutative=True))"], "Equality(Add(Mul(Symbol('W', commutative=True), Function('\\\\phi_1')(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True))))"], [["add", 3, "log(Symbol('W', commutative=True))"], "Equality(Add(Mul(Symbol('W', commutative=True), Function('\\\\phi_1')(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))))"], [["differentiate", 4, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('W', commutative=True), Function('\\\\phi_1')(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["minus", 5, "Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('W', commutative=True))), Derivative(Add(Mul(Symbol('W', commutative=True), Function('\\\\phi_1')(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('W', commutative=True))), Derivative(Add(Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Function('\\\\phi_1')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\operatorname{C_{1}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}, then obtain \\frac{\\operatorname{C_{1}}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\operatorname{F_{H}}{(\\mathbf{J})}}{\\mathbf{J}}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{J})} = \\log{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\operatorname{F_{H}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} and \\operatorname{C_{1}}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})} and \\frac{\\operatorname{C_{1}}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\log{(\\mathbf{J})}}{\\mathbf{J}} and \\frac{\\operatorname{C_{1}}{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\frac{d}{d \\mathbf{J}} \\operatorname{F_{H}}{(\\mathbf{J})}}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), log(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Function('C_1')(Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Derivative(Function('F_H')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\varepsilon)} = e^{\\varepsilon} and \\delta{(\\varepsilon)} = \\log{(e^{\\varepsilon})}, then obtain - \\sin{(\\operatorname{F_{H}}{(\\varepsilon)})} = - \\sin{(\\operatorname{F_{H}}{(\\varepsilon)} + \\delta{(\\varepsilon)} - \\log{(\\operatorname{F_{H}}{(\\varepsilon)})})}", "derivation": "\\operatorname{F_{H}}{(\\varepsilon)} = e^{\\varepsilon} and \\log{(\\operatorname{F_{H}}{(\\varepsilon)})} = \\log{(e^{\\varepsilon})} and \\delta{(\\varepsilon)} = \\log{(e^{\\varepsilon})} and \\delta{(\\varepsilon)} = \\log{(\\operatorname{F_{H}}{(\\varepsilon)})} and \\delta{(\\varepsilon)} - 1 = \\log{(\\operatorname{F_{H}}{(\\varepsilon)})} - 1 and 0 = - \\delta{(\\varepsilon)} + \\log{(\\operatorname{F_{H}}{(\\varepsilon)})} and - \\operatorname{F_{H}}{(\\varepsilon)} = - \\operatorname{F_{H}}{(\\varepsilon)} - \\delta{(\\varepsilon)} + \\log{(\\operatorname{F_{H}}{(\\varepsilon)})} and - \\sin{(\\operatorname{F_{H}}{(\\varepsilon)})} = - \\sin{(\\operatorname{F_{H}}{(\\varepsilon)} + \\delta{(\\varepsilon)} - \\log{(\\operatorname{F_{H}}{(\\varepsilon)})})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], [["log", 1], "Equality(log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True))), log(exp(Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True)), log(exp(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True)), log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1)), Add(log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True))), Integer(-1)))"], [["minus", 5, "Add(Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True)), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True))), log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)))))"], [["minus", 6, "Function('F_H')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(-1), Function('F_H')(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Function('F_H')(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True))), log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)))))"], [["sin", 7], "Equality(Mul(Integer(-1), sin(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), sin(Add(Function('F_H')(Symbol('\\\\varepsilon', commutative=True)), Function('\\\\delta')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), log(Function('F_H')(Symbol('\\\\varepsilon', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\theta_1)} = \\cos{(\\theta_1)}, then derive 2 \\operatorname{C_{2}}{(\\theta_1)} \\frac{d}{d \\theta_1} \\operatorname{C_{2}}{(\\theta_1)} = - \\operatorname{C_{2}}{(\\theta_1)} \\sin{(\\theta_1)} + \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\operatorname{C_{2}}{(\\theta_1)}, then obtain 2 \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} = - \\sin{(\\theta_1)} \\cos{(\\theta_1)} + \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\cos{(\\theta_1)}", "derivation": "\\operatorname{C_{2}}{(\\theta_1)} = \\cos{(\\theta_1)} and \\operatorname{C_{2}}^{2}{(\\theta_1)} = \\operatorname{C_{2}}{(\\theta_1)} \\cos{(\\theta_1)} and \\frac{d}{d \\theta_1} \\operatorname{C_{2}}^{2}{(\\theta_1)} = \\frac{d}{d \\theta_1} \\operatorname{C_{2}}{(\\theta_1)} \\cos{(\\theta_1)} and 2 \\operatorname{C_{2}}{(\\theta_1)} \\frac{d}{d \\theta_1} \\operatorname{C_{2}}{(\\theta_1)} = - \\operatorname{C_{2}}{(\\theta_1)} \\sin{(\\theta_1)} + \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\operatorname{C_{2}}{(\\theta_1)} and 2 \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\cos{(\\theta_1)} = - \\sin{(\\theta_1)} \\cos{(\\theta_1)} + \\cos{(\\theta_1)} \\frac{d}{d \\theta_1} \\cos{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Function('C_2')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Pow(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Mul(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Pow(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Integer(2)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Derivative(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(cos(Symbol('\\\\theta_1', commutative=True)), Derivative(Function('C_2')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), cos(Symbol('\\\\theta_1', commutative=True)), Derivative(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True))), Mul(cos(Symbol('\\\\theta_1', commutative=True)), Derivative(cos(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given r{(c_{0},J_{\\varepsilon})} = \\cos^{J_{\\varepsilon}}{(c_{0})}, then obtain - (r{(c_{0},J_{\\varepsilon})} - \\cos{(c_{0})} - \\cos^{J_{\\varepsilon}}{(c_{0})}) \\cos^{J_{\\varepsilon}}{(c_{0})} = \\cos{(c_{0})} \\cos^{J_{\\varepsilon}}{(c_{0})}", "derivation": "r{(c_{0},J_{\\varepsilon})} = \\cos^{J_{\\varepsilon}}{(c_{0})} and r{(c_{0},J_{\\varepsilon})} - \\cos^{J_{\\varepsilon}}{(c_{0})} = 0 and r{(c_{0},J_{\\varepsilon})} - \\cos{(c_{0})} - \\cos^{J_{\\varepsilon}}{(c_{0})} = - \\cos{(c_{0})} and - (r{(c_{0},J_{\\varepsilon})} - \\cos{(c_{0})} - \\cos^{J_{\\varepsilon}}{(c_{0})}) \\cos^{J_{\\varepsilon}}{(c_{0})} = \\cos{(c_{0})} \\cos^{J_{\\varepsilon}}{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('c_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('r')(Symbol('c_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))), Integer(0))"], [["minus", 2, "cos(Symbol('c_0', commutative=True))"], "Equality(Add(Function('r')(Symbol('c_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('r')(Symbol('c_0', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(cos(Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(F_{g})} = e^{F_{g}} and I{(F_{g})} = 2 \\phi_{1}{(F_{g})}, then derive \\hat{H} + I{(F_{g})} = f + \\phi_{1}{(F_{g})} + e^{F_{g}}, then obtain \\hat{H} + 2 I{(F_{g})} = f + I{(F_{g})} + \\phi_{1}{(F_{g})} + e^{F_{g}}", "derivation": "\\phi_{1}{(F_{g})} = e^{F_{g}} and 2 \\phi_{1}{(F_{g})} = \\phi_{1}{(F_{g})} + e^{F_{g}} and \\frac{d}{d F_{g}} 2 \\phi_{1}{(F_{g})} = \\frac{d}{d F_{g}} (\\phi_{1}{(F_{g})} + e^{F_{g}}) and I{(F_{g})} = 2 \\phi_{1}{(F_{g})} and \\frac{d}{d F_{g}} I{(F_{g})} = \\frac{d}{d F_{g}} (\\phi_{1}{(F_{g})} + e^{F_{g}}) and \\int \\frac{d}{d F_{g}} I{(F_{g})} dF_{g} = \\int \\frac{d}{d F_{g}} (\\phi_{1}{(F_{g})} + e^{F_{g}}) dF_{g} and \\hat{H} + I{(F_{g})} = f + \\phi_{1}{(F_{g})} + e^{F_{g}} and \\hat{H} + 2 I{(F_{g})} = f + I{(F_{g})} + \\phi_{1}{(F_{g})} + e^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["add", 1, "Function('\\\\phi_1')(Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('F_g', commutative=True))), Add(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\phi_1')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('I')(Symbol('F_g', commutative=True)), Mul(Integer(2), Function('\\\\phi_1')(Symbol('F_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('I')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('F_g', commutative=True)"], "Equality(Integral(Derivative(Function('I')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))), Integral(Derivative(Add(Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Function('I')(Symbol('F_g', commutative=True))), Add(Symbol('f', commutative=True), Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))))"], [["add", 7, "Function('I')(Symbol('F_g', commutative=True))"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(2), Function('I')(Symbol('F_g', commutative=True)))), Add(Symbol('f', commutative=True), Function('I')(Symbol('F_g', commutative=True)), Function('\\\\phi_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(M,\\mathbf{g})} = e^{M^{\\mathbf{g}}} and \\operatorname{E_{n}}{(M,\\mathbf{g})} = \\mathbf{J}_M^{M}{(M,\\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} (e^{M^{\\mathbf{g}}})^{M}, then obtain \\operatorname{E_{n}}^{\\mathbf{g}}{(M,\\mathbf{g})} = (\\mathbf{J}_M^{M}{(M,\\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_M^{M}{(M,\\mathbf{g})})^{\\mathbf{g}}", "derivation": "\\mathbf{J}_M{(M,\\mathbf{g})} = e^{M^{\\mathbf{g}}} and \\mathbf{J}_M^{M}{(M,\\mathbf{g})} = (e^{M^{\\mathbf{g}}})^{M} and \\operatorname{E_{n}}{(M,\\mathbf{g})} = \\mathbf{J}_M^{M}{(M,\\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} (e^{M^{\\mathbf{g}}})^{M} and \\operatorname{E_{n}}^{\\mathbf{g}}{(M,\\mathbf{g})} = (\\mathbf{J}_M^{M}{(M,\\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} (e^{M^{\\mathbf{g}}})^{M})^{\\mathbf{g}} and \\operatorname{E_{n}}^{\\mathbf{g}}{(M,\\mathbf{g})} = (\\mathbf{J}_M^{M}{(M,\\mathbf{g})} + \\frac{\\partial}{\\partial \\mathbf{g}} \\mathbf{J}_M^{M}{(M,\\mathbf{g})})^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Pow(exp(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Pow(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Derivative(Pow(exp(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Add(Pow(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Derivative(Pow(exp(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('E_n')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Add(Pow(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Derivative(Pow(Function('\\\\mathbf{J}_M')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\psi{(r_{0},s)} = \\cos{(\\frac{r_{0}}{s})} and \\sigma_{x}{(r_{0},s)} = \\frac{\\partial}{\\partial s} \\psi{(r_{0},s)}, then derive \\frac{\\partial}{\\partial s} \\psi{(r_{0},s)} = \\frac{r_{0} \\sin{(\\frac{r_{0}}{s})}}{s^{2}}, then obtain \\sigma_{x}{(r_{0},s)} = \\frac{r_{0} \\sin{(\\frac{r_{0}}{s})}}{s^{2}}", "derivation": "\\psi{(r_{0},s)} = \\cos{(\\frac{r_{0}}{s})} and \\frac{\\partial}{\\partial s} \\psi{(r_{0},s)} = \\frac{\\partial}{\\partial s} \\cos{(\\frac{r_{0}}{s})} and \\sigma_{x}{(r_{0},s)} = \\frac{\\partial}{\\partial s} \\psi{(r_{0},s)} and \\frac{\\partial}{\\partial s} \\psi{(r_{0},s)} = \\frac{r_{0} \\sin{(\\frac{r_{0}}{s})}}{s^{2}} and \\sigma_{x}{(r_{0},s)} = \\frac{r_{0} \\sin{(\\frac{r_{0}}{s})}}{s^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), cos(Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))), Tuple(Symbol('s', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Derivative(Function('\\\\psi')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-2)), sin(Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\sigma_x')(Symbol('r_0', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-2)), sin(Mul(Symbol('r_0', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\eta,A_{z})} = A_{z} + \\eta, then derive 1 - \\frac{\\partial}{\\partial A_{z}} \\Psi_{\\lambda}{(\\eta,A_{z})} = 0, then obtain \\frac{\\partial}{\\partial \\eta} (1 - \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\eta)) = \\frac{d}{d \\eta} 0", "derivation": "\\Psi_{\\lambda}{(\\eta,A_{z})} = A_{z} + \\eta and - \\eta + \\Psi_{\\lambda}{(\\eta,A_{z})} = A_{z} and \\eta - \\Psi_{\\lambda}{(\\eta,A_{z})} = - A_{z} and 2 \\eta - \\Psi_{\\lambda}{(\\eta,A_{z})} = - A_{z} + \\eta and A_{z} + 3 \\eta - \\Psi_{\\lambda}{(\\eta,A_{z})} = 2 \\eta and \\frac{\\partial}{\\partial A_{z}} (A_{z} + 3 \\eta - \\Psi_{\\lambda}{(\\eta,A_{z})}) = \\frac{d}{d A_{z}} 2 \\eta and 1 - \\frac{\\partial}{\\partial A_{z}} \\Psi_{\\lambda}{(\\eta,A_{z})} = 0 and 1 - \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\eta) = 0 and \\frac{\\partial}{\\partial \\eta} (1 - \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\eta)) = \\frac{d}{d \\eta} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)))), Mul(Integer(-1), Symbol('A_z', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["add", 4, "Add(Symbol('A_z', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Symbol('A_z', commutative=True), Mul(Integer(3), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)))), Mul(Integer(2), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 5, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(3), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\eta', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))), Integer(0))"], [["differentiate", 8, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} = \\cos{(\\mathbf{J} - \\mathbf{f})}, then obtain 0 = (- \\mathbf{f} + \\cos{(\\mathbf{J} - \\mathbf{f})}) (- \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} + \\cos{(\\mathbf{J} - \\mathbf{f})})", "derivation": "\\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} = \\cos{(\\mathbf{J} - \\mathbf{f})} and - \\mathbf{f} + \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} = - \\mathbf{f} + \\cos{(\\mathbf{J} - \\mathbf{f})} and 0 = - \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} + \\cos{(\\mathbf{J} - \\mathbf{f})} and 0 = (- \\mathbf{f} + \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})}) (- \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} + \\cos{(\\mathbf{J} - \\mathbf{f})}) and 0 = (- \\mathbf{f} + \\cos{(\\mathbf{J} - \\mathbf{f})}) (- \\operatorname{v_{t}}{(\\mathbf{J},\\mathbf{f})} + \\cos{(\\mathbf{J} - \\mathbf{f})})", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), cos(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))))"]]}, {"prompt": "Given m{(\\dot{x})} = e^{\\sin{(\\dot{x})}}, then obtain 2 \\dot{x} m^{2}{(\\dot{x})} = 2 \\dot{x} m{(\\dot{x})} e^{\\sin{(\\dot{x})}}", "derivation": "m{(\\dot{x})} = e^{\\sin{(\\dot{x})}} and 2 m{(\\dot{x})} = m{(\\dot{x})} + e^{\\sin{(\\dot{x})}} and (m{(\\dot{x})} + e^{\\sin{(\\dot{x})}}) m{(\\dot{x})} = (m{(\\dot{x})} + e^{\\sin{(\\dot{x})}}) e^{\\sin{(\\dot{x})}} and 2 m^{2}{(\\dot{x})} = 2 m{(\\dot{x})} e^{\\sin{(\\dot{x})}} and 2 \\dot{x} m^{2}{(\\dot{x})} = 2 \\dot{x} m{(\\dot{x})} e^{\\sin{(\\dot{x})}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True))))"], [["add", 1, "Function('m')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('m')(Symbol('\\\\dot{x}', commutative=True))), Add(Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))))"], [["times", 1, "Add(Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True))))"], "Equality(Mul(Add(Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))), Function('m')(Symbol('\\\\dot{x}', commutative=True))), Mul(Add(Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))), exp(sin(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Pow(Function('m')(Symbol('\\\\dot{x}', commutative=True)), Integer(2))), Mul(Integer(2), Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))))"], [["times", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True), Pow(Function('m')(Symbol('\\\\dot{x}', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True), Function('m')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(v_{x})} = \\sin{(v_{x})}, then derive \\frac{d}{d v_{x}} \\operatorname{r_{0}}{(v_{x})} = \\cos{(v_{x})}, then obtain - \\cos{(v_{x})} + \\frac{d^{2}}{d v_{x}^{2}} \\operatorname{r_{0}}{(v_{x})} = - \\cos{(v_{x})} + \\frac{d}{d v_{x}} \\cos{(v_{x})}", "derivation": "\\operatorname{r_{0}}{(v_{x})} = \\sin{(v_{x})} and \\frac{d}{d v_{x}} \\operatorname{r_{0}}{(v_{x})} = \\frac{d}{d v_{x}} \\sin{(v_{x})} and \\frac{d}{d v_{x}} \\operatorname{r_{0}}{(v_{x})} = \\cos{(v_{x})} and \\frac{d^{2}}{d v_{x}^{2}} \\operatorname{r_{0}}{(v_{x})} = \\frac{d}{d v_{x}} \\cos{(v_{x})} and - \\cos{(v_{x})} + \\frac{d^{2}}{d v_{x}^{2}} \\operatorname{r_{0}}{(v_{x})} = - \\cos{(v_{x})} + \\frac{d}{d v_{x}} \\cos{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), cos(Symbol('v_x', commutative=True)))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(2))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["minus", 4, "cos(Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('v_x', commutative=True))), Derivative(Function('r_0')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(2)))), Add(Mul(Integer(-1), cos(Symbol('v_x', commutative=True))), Derivative(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(f^{\\prime})} = e^{f^{\\prime}}, then obtain 0^{f^{\\prime}} = (- (v_{1} \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} + (v_{1} (e^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}}", "derivation": "\\operatorname{r_{0}}{(f^{\\prime})} = e^{f^{\\prime}} and \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})} = (e^{f^{\\prime}})^{f^{\\prime}} and v_{1} \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})} = v_{1} (e^{f^{\\prime}})^{f^{\\prime}} and (v_{1} \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} = (v_{1} (e^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}} and 0 = - (v_{1} \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} + (v_{1} (e^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}} and 0^{f^{\\prime}} = (- (v_{1} \\operatorname{r_{0}}^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} + (v_{1} (e^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}})^{f^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["divide", 2, "Pow(Symbol('v_1', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('v_1', commutative=True), Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Mul(Symbol('v_1', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Symbol('v_1', commutative=True), Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('v_1', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 4, "Pow(Mul(Symbol('v_1', commutative=True), Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Mul(Symbol('v_1', commutative=True), Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))), Pow(Mul(Symbol('v_1', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Mul(Symbol('v_1', commutative=True), Pow(Function('r_0')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))), Pow(Mul(Symbol('v_1', commutative=True), Pow(exp(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(b,t_{1})} = \\frac{t_{1}}{b}, then derive \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = \\frac{b^{2}}{2} + r_{0} + t_{1} \\log{(b)}, then obtain \\frac{\\partial}{\\partial b} \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = b + \\frac{t_{1}}{b}", "derivation": "\\operatorname{E_{n}}{(b,t_{1})} = \\frac{t_{1}}{b} and b + \\operatorname{E_{n}}{(b,t_{1})} = b + \\frac{t_{1}}{b} and \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = \\int (b + \\frac{t_{1}}{b}) db and \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = \\frac{b^{2}}{2} + r_{0} + t_{1} \\log{(b)} and \\frac{\\partial}{\\partial b} \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = \\frac{\\partial}{\\partial b} (\\frac{b^{2}}{2} + r_{0} + t_{1} \\log{(b)}) and \\frac{\\partial}{\\partial b} \\int (b + \\operatorname{E_{n}}{(b,t_{1})}) db = b + \\frac{t_{1}}{b}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('t_1', commutative=True)))"], [["add", 1, "Symbol('b', commutative=True)"], "Equality(Add(Symbol('b', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('b', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Symbol('b', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Add(Symbol('b', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('b', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('b', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('r_0', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('b', commutative=True)))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('b', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Symbol('r_0', commutative=True), Mul(Symbol('t_1', commutative=True), log(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Add(Symbol('b', commutative=True), Function('E_n')(Symbol('b', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Add(Symbol('b', commutative=True), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\theta{(r_{0})} = \\cos{(r_{0})} and \\mathbf{J}_f{(r_{0})} = \\sin{(\\frac{\\cos{(r_{0})}}{\\theta{(r_{0})}})}, then obtain \\frac{\\mathbf{J}_f^{r_{0}}{(r_{0})}}{c_{0} + e^{C_{d}}} = \\frac{\\sin^{r_{0}}{(1)}}{c_{0} + e^{C_{d}}}", "derivation": "\\theta{(r_{0})} = \\cos{(r_{0})} and \\mathbf{J}_f{(r_{0})} = \\sin{(\\frac{\\cos{(r_{0})}}{\\theta{(r_{0})}})} and \\mathbf{J}_f{(r_{0})} = \\sin{(1)} and \\mathbf{J}_f^{r_{0}}{(r_{0})} = \\sin^{r_{0}}{(1)} and \\frac{\\mathbf{J}_f^{r_{0}}{(r_{0})}}{c_{0} + e^{C_{d}}} = \\frac{\\sin^{r_{0}}{(1)}}{c_{0} + e^{C_{d}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('r_0', commutative=True)), cos(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('r_0', commutative=True)), sin(Mul(Pow(Function('\\\\theta')(Symbol('r_0', commutative=True)), Integer(-1)), cos(Symbol('r_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('r_0', commutative=True)), sin(Integer(1)))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(sin(Integer(1)), Symbol('r_0', commutative=True)))"], [["divide", 4, "Add(Symbol('c_0', commutative=True), exp(Symbol('C_d', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('c_0', commutative=True), exp(Symbol('C_d', commutative=True))), Integer(-1)), Pow(Function('\\\\mathbf{J}_f')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), Mul(Pow(Add(Symbol('c_0', commutative=True), exp(Symbol('C_d', commutative=True))), Integer(-1)), Pow(sin(Integer(1)), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(L)} = \\sin{(L)} and \\hat{H}{(L)} = \\frac{d}{d L} \\psi^{*}{(L)}, then obtain \\hat{H}^{L}{(L)} = \\cos^{L}{(L)}", "derivation": "\\psi^{*}{(L)} = \\sin{(L)} and \\frac{d}{d L} \\psi^{*}{(L)} = \\frac{d}{d L} \\sin{(L)} and \\hat{H}{(L)} = \\frac{d}{d L} \\psi^{*}{(L)} and \\hat{H}{(L)} = \\frac{d}{d L} \\sin{(L)} and \\hat{H}^{L}{(L)} = (\\frac{d}{d L} \\sin{(L)})^{L} and \\hat{H}^{L}{(L)} = \\cos^{L}{(L)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Derivative(Function('\\\\psi^*')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["power", 4, "Symbol('L', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Derivative(sin(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('L', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Function('\\\\hat{H}')(Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(cos(Symbol('L', commutative=True)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given \\pi{(x)} = \\sin{(x)}, then derive \\frac{d}{d x} \\pi{(x)} = \\cos{(x)}, then obtain \\frac{d}{d x} \\frac{- \\cos{(x)} + \\frac{d}{d x} \\pi{(x)}}{\\sin{(x)} + \\frac{d}{d x} \\pi{(x)}} = \\frac{d}{d x} 0", "derivation": "\\pi{(x)} = \\sin{(x)} and \\frac{d}{d x} \\pi{(x)} = \\frac{d}{d x} \\sin{(x)} and \\frac{d}{d x} \\pi{(x)} = \\cos{(x)} and - \\cos{(x)} + \\frac{d}{d x} \\pi{(x)} = 0 and \\frac{- \\cos{(x)} + \\frac{d}{d x} \\pi{(x)}}{\\sin{(x)} + \\frac{d}{d x} \\pi{(x)}} = 0 and \\frac{d}{d x} \\frac{- \\cos{(x)} + \\frac{d}{d x} \\pi{(x)}}{\\sin{(x)} + \\frac{d}{d x} \\pi{(x)}} = \\frac{d}{d x} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), cos(Symbol('x', commutative=True)))"], [["minus", 3, "cos(Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(0))"], [["divide", 4, "Add(sin(Symbol('x', commutative=True)), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(sin(Symbol('x', commutative=True)), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Integer(0))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(sin(Symbol('x', commutative=True)), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('x', commutative=True))), Derivative(Function('\\\\pi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(n,G)} = G + n, then obtain \\int (G + n) dG + \\iint b{(n,G)} dG dG = \\int (G + n) dG + \\iint (G + n) dG dG", "derivation": "b{(n,G)} = G + n and \\int b{(n,G)} dG = \\int (G + n) dG and \\iint b{(n,G)} dG dG = \\iint (G + n) dG dG and \\int (G + n) dG + \\iint b{(n,G)} dG dG = \\int (G + n) dG + \\iint (G + n) dG dG", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('b')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Function('b')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["add", 3, "Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True)))"], "Equality(Add(Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Function('b')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Add(Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(g,a)} = \\frac{\\log{(g)}}{a}, then obtain (\\frac{2 \\log{(g)}}{a})^{g} (\\operatorname{P_{e}}{(g,a)} + \\frac{\\log{(g)}}{a})^{g} = (\\frac{2 \\log{(g)}}{a})^{2 g}", "derivation": "\\operatorname{P_{e}}{(g,a)} = \\frac{\\log{(g)}}{a} and \\operatorname{P_{e}}{(g,a)} + \\frac{\\log{(g)}}{a} = \\frac{2 \\log{(g)}}{a} and (\\operatorname{P_{e}}{(g,a)} + \\frac{\\log{(g)}}{a})^{g} = (\\frac{2 \\log{(g)}}{a})^{g} and (\\frac{2 \\log{(g)}}{a})^{g} (\\operatorname{P_{e}}{(g,a)} + \\frac{\\log{(g)}}{a})^{g} = (\\frac{2 \\log{(g)}}{a})^{2 g}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('g', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)))"], "Equality(Add(Function('P_e')(Symbol('g', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)))), Mul(Integer(2), Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Function('P_e')(Symbol('g', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["times", 3, "Pow(Mul(Integer(2), Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Add(Function('P_e')(Symbol('g', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)))), Symbol('g', commutative=True))), Pow(Mul(Integer(2), Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Mul(Integer(2), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\pi{(P_{e})} = \\cos{(e^{P_{e}})}, then obtain \\sin{(2 \\pi{(P_{e})} + 2 e^{P_{e}})} = \\sin{(2 e^{P_{e}} + 2 \\cos{(e^{P_{e}})})}", "derivation": "\\pi{(P_{e})} = \\cos{(e^{P_{e}})} and \\pi{(P_{e})} + e^{P_{e}} = e^{P_{e}} + \\cos{(e^{P_{e}})} and 2 \\pi{(P_{e})} + e^{P_{e}} = \\pi{(P_{e})} + e^{P_{e}} + \\cos{(e^{P_{e}})} and 2 \\pi{(P_{e})} + e^{P_{e}} = e^{P_{e}} + 2 \\cos{(e^{P_{e}})} and 2 \\pi{(P_{e})} + 2 e^{P_{e}} = 2 e^{P_{e}} + 2 \\cos{(e^{P_{e}})} and \\sin{(2 \\pi{(P_{e})} + 2 e^{P_{e}})} = \\sin{(2 e^{P_{e}} + 2 \\cos{(e^{P_{e}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('P_e', commutative=True)), cos(exp(Symbol('P_e', commutative=True))))"], [["add", 1, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Add(exp(Symbol('P_e', commutative=True)), cos(exp(Symbol('P_e', commutative=True)))))"], [["add", 2, "Function('\\\\pi')(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\pi')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))), Add(Function('\\\\pi')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)), cos(exp(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\pi')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))), Add(exp(Symbol('P_e', commutative=True)), Mul(Integer(2), cos(exp(Symbol('P_e', commutative=True))))))"], [["minus", 4, "Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\pi')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True)))), Add(Mul(Integer(2), exp(Symbol('P_e', commutative=True))), Mul(Integer(2), cos(exp(Symbol('P_e', commutative=True))))))"], [["sin", 5], "Equality(sin(Add(Mul(Integer(2), Function('\\\\pi')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True))))), sin(Add(Mul(Integer(2), exp(Symbol('P_e', commutative=True))), Mul(Integer(2), cos(exp(Symbol('P_e', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} = \\log{(\\frac{L_{\\varepsilon}}{\\mu_0})}, then obtain \\int (\\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} - 1) dL_{\\varepsilon} = L_{\\varepsilon} \\log{(\\frac{L_{\\varepsilon}}{\\mu_0})} - 2 L_{\\varepsilon} + \\hat{H}", "derivation": "\\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} = \\log{(\\frac{L_{\\varepsilon}}{\\mu_0})} and \\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} - 1 = \\log{(\\frac{L_{\\varepsilon}}{\\mu_0})} - 1 and \\int (\\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} - 1) dL_{\\varepsilon} = \\int (\\log{(\\frac{L_{\\varepsilon}}{\\mu_0})} - 1) dL_{\\varepsilon} and \\int (\\operatorname{C_{d}}{(\\mu_0,L_{\\varepsilon})} - 1) dL_{\\varepsilon} = L_{\\varepsilon} \\log{(\\frac{L_{\\varepsilon}}{\\mu_0})} - 2 L_{\\varepsilon} + \\hat{H}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mu_0', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('C_d')(Symbol('\\\\mu_0', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(log(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))), Integer(-1)))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Function('C_d')(Symbol('\\\\mu_0', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Add(log(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))), Integer(-1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('C_d')(Symbol('\\\\mu_0', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), log(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))), Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(G)} = \\int \\log{(G)} dG, then derive \\frac{d^{2}}{d G^{2}} \\theta_{2}{(G)} = \\frac{\\partial^{2}}{\\partial G^{2}} (G \\log{(G)} - G + \\mathbf{A}), then obtain \\frac{\\partial^{2}}{\\partial G^{2}} (G \\log{(G)} - G + \\mathbf{A}) = \\frac{d^{2}}{d G^{2}} \\int \\log{(G)} dG", "derivation": "\\theta_{2}{(G)} = \\int \\log{(G)} dG and \\frac{d}{d G} \\theta_{2}{(G)} = \\frac{d}{d G} \\int \\log{(G)} dG and \\frac{d^{2}}{d G^{2}} \\theta_{2}{(G)} = \\frac{d^{2}}{d G^{2}} \\int \\log{(G)} dG and \\frac{d^{2}}{d G^{2}} \\theta_{2}{(G)} = \\frac{\\partial^{2}}{\\partial G^{2}} (G \\log{(G)} - G + \\mathbf{A}) and \\frac{\\partial^{2}}{\\partial G^{2}} (G \\log{(G)} - G + \\mathbf{A}) = \\frac{d^{2}}{d G^{2}} \\int \\log{(G)} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('G', commutative=True)), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["evaluate_integrals", 3], "Equality(Derivative(Function('\\\\theta_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))), Derivative(Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(2))))"]]}, {"prompt": "Given n{(v_{x})} = \\int \\log{(v_{x})} dv_{x}, then derive n{(v_{x})} = Q + v_{x} \\log{(v_{x})} - v_{x}, then obtain - (\\int \\log{(v_{x})} dv_{x})^{v_{x}} = - (Q + v_{x} \\log{(v_{x})} - v_{x})^{v_{x}}", "derivation": "n{(v_{x})} = \\int \\log{(v_{x})} dv_{x} and n{(v_{x})} = Q + v_{x} \\log{(v_{x})} - v_{x} and \\int \\log{(v_{x})} dv_{x} = Q + v_{x} \\log{(v_{x})} - v_{x} and (\\int \\log{(v_{x})} dv_{x})^{v_{x}} = (Q + v_{x} \\log{(v_{x})} - v_{x})^{v_{x}} and - (\\int \\log{(v_{x})} dv_{x})^{v_{x}} = - (Q + v_{x} \\log{(v_{x})} - v_{x})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('v_x', commutative=True)), Integral(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('n')(Symbol('v_x', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('v_x', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('v_x', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Integral(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Add(Symbol('Q', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('v_x', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integral(log(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('Q', commutative=True), Mul(Symbol('v_x', commutative=True), log(Symbol('v_x', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(v)} = \\sin{(v)}, then derive \\cos{(v)} \\frac{d}{d v} \\mathbf{J}_P{(v)} = \\cos^{2}{(v)}, then obtain \\cos{(v)} \\frac{d}{d v} \\sin{(v)} = \\cos^{2}{(v)}", "derivation": "\\mathbf{J}_P{(v)} = \\sin{(v)} and \\frac{d}{d v} \\mathbf{J}_P{(v)} = \\frac{d}{d v} \\sin{(v)} and \\frac{d}{d v} \\mathbf{J}_P{(v)} \\frac{d}{d v} \\sin{(v)} = (\\frac{d}{d v} \\sin{(v)})^{2} and \\cos{(v)} \\frac{d}{d v} \\mathbf{J}_P{(v)} = \\cos^{2}{(v)} and \\cos{(v)} \\frac{d}{d v} \\sin{(v)} = \\cos^{2}{(v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 2, "Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\mathbf{J}_P')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Pow(Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Symbol('v', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Pow(cos(Symbol('v', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(cos(Symbol('v', commutative=True)), Derivative(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Pow(cos(Symbol('v', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(F_{g})} = \\sin{(F_{g})}, then obtain \\frac{\\frac{d^{2}}{d F_{g}^{2}} \\operatorname{V_{\\mathbf{B}}}{(F_{g})}}{\\operatorname{V_{\\mathbf{B}}}{(F_{g})}} = \\frac{\\frac{d^{2}}{d F_{g}^{2}} \\sin{(F_{g})}}{\\operatorname{V_{\\mathbf{B}}}{(F_{g})}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(F_{g})} = \\sin{(F_{g})} and \\frac{d}{d F_{g}} \\operatorname{V_{\\mathbf{B}}}{(F_{g})} = \\frac{d}{d F_{g}} \\sin{(F_{g})} and \\frac{d^{2}}{d F_{g}^{2}} \\operatorname{V_{\\mathbf{B}}}{(F_{g})} = \\frac{d^{2}}{d F_{g}^{2}} \\sin{(F_{g})} and \\frac{\\frac{d^{2}}{d F_{g}^{2}} \\operatorname{V_{\\mathbf{B}}}{(F_{g})}}{\\operatorname{V_{\\mathbf{B}}}{(F_{g})}} = \\frac{\\frac{d^{2}}{d F_{g}^{2}} \\sin{(F_{g})}}{\\operatorname{V_{\\mathbf{B}}}{(F_{g})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), sin(Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))), Derivative(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))))"], [["divide", 3, "Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), Integer(-1)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2)))), Mul(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('F_g', commutative=True)), Integer(-1)), Derivative(sin(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\rho{(t_{2},\\Psi)} = \\Psi + t_{2}, then obtain \\frac{2 \\rho{(t_{2},\\Psi)}}{2 \\Psi + 2 t_{2}} = 1", "derivation": "\\rho{(t_{2},\\Psi)} = \\Psi + t_{2} and 2 \\rho{(t_{2},\\Psi)} = \\Psi + t_{2} + \\rho{(t_{2},\\Psi)} and \\frac{2 \\rho{(t_{2},\\Psi)}}{\\Psi + t_{2} + \\rho{(t_{2},\\Psi)}} = 1 and \\frac{2 (\\Psi + t_{2})}{2 \\Psi + 2 t_{2}} = 1 and \\frac{2 \\rho{(t_{2},\\Psi)}}{2 \\Psi + 2 t_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 1, "Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Symbol('t_2', commutative=True), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\Psi', commutative=True), Symbol('t_2', commutative=True), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('t_2', commutative=True), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(-1)), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Add(Symbol('\\\\Psi', commutative=True), Symbol('t_2', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Mul(Integer(2), Symbol('t_2', commutative=True))), Integer(-1)), Function('\\\\rho')(Symbol('t_2', commutative=True), Symbol('\\\\Psi', commutative=True))), Integer(1))"]]}, {"prompt": "Given h{(B)} = \\sin{(\\cos{(B)})} and \\operatorname{L_{\\varepsilon}}{(B)} = - \\sin{(\\cos{(B)})} + 1 + \\frac{h{(B)}}{B}, then obtain \\operatorname{L_{\\varepsilon}}^{B}{(B)} = (- \\sin{(\\cos{(B)})} + 1 + \\frac{\\sin{(\\cos{(B)})}}{B})^{B}", "derivation": "h{(B)} = \\sin{(\\cos{(B)})} and \\frac{h{(B)}}{B} = \\frac{\\sin{(\\cos{(B)})}}{B} and 1 + \\frac{h{(B)}}{B} = 1 + \\frac{\\sin{(\\cos{(B)})}}{B} and - \\sin{(\\cos{(B)})} + 1 + \\frac{h{(B)}}{B} = - \\sin{(\\cos{(B)})} + 1 + \\frac{\\sin{(\\cos{(B)})}}{B} and \\operatorname{L_{\\varepsilon}}{(B)} = - \\sin{(\\cos{(B)})} + 1 + \\frac{h{(B)}}{B} and \\operatorname{L_{\\varepsilon}}{(B)} = - \\sin{(\\cos{(B)})} + 1 + \\frac{\\sin{(\\cos{(B)})}}{B} and \\operatorname{L_{\\varepsilon}}^{B}{(B)} = (- \\sin{(\\cos{(B)})} + 1 + \\frac{\\sin{(\\cos{(B)})}}{B})^{B}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('B', commutative=True)), sin(cos(Symbol('B', commutative=True))))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('h')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(cos(Symbol('B', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('h')(Symbol('B', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(cos(Symbol('B', commutative=True))))))"], [["minus", 3, "sin(cos(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(cos(Symbol('B', commutative=True)))), Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('h')(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), sin(cos(Symbol('B', commutative=True)))), Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(cos(Symbol('B', commutative=True))))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Add(Mul(Integer(-1), sin(cos(Symbol('B', commutative=True)))), Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('h')(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Add(Mul(Integer(-1), sin(cos(Symbol('B', commutative=True)))), Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(cos(Symbol('B', commutative=True))))))"], [["power", 6, "Symbol('B', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), sin(cos(Symbol('B', commutative=True)))), Integer(1), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(cos(Symbol('B', commutative=True))))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(\\hat{p})} = e^{e^{\\hat{p}}} and i{(\\hat{p})} = e^{\\hat{p}}, then obtain (\\dot{y}{(\\hat{p})} + e^{i{(\\hat{p})}}) e^{\\hat{p}} = 2 e^{\\hat{p}} e^{i{(\\hat{p})}}", "derivation": "\\dot{y}{(\\hat{p})} = e^{e^{\\hat{p}}} and i{(\\hat{p})} = e^{\\hat{p}} and \\dot{y}{(\\hat{p})} = e^{i{(\\hat{p})}} and \\dot{y}{(\\hat{p})} + e^{e^{\\hat{p}}} = e^{i{(\\hat{p})}} + e^{e^{\\hat{p}}} and e^{i{(\\hat{p})}} = e^{e^{\\hat{p}}} and (\\dot{y}{(\\hat{p})} + e^{e^{\\hat{p}}}) e^{\\hat{p}} = (e^{i{(\\hat{p})}} + e^{e^{\\hat{p}}}) e^{\\hat{p}} and (\\dot{y}{(\\hat{p})} + e^{i{(\\hat{p})}}) e^{\\hat{p}} = 2 e^{\\hat{p}} e^{i{(\\hat{p})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\dot{y}')(Symbol('\\\\hat{p}', commutative=True)), exp(Function('i')(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 3, "exp(exp(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Symbol('\\\\hat{p}', commutative=True)))), Add(exp(Function('i')(Symbol('\\\\hat{p}', commutative=True))), exp(exp(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(exp(Function('i')(Symbol('\\\\hat{p}', commutative=True))), exp(exp(Symbol('\\\\hat{p}', commutative=True))))"], [["times", 4, "exp(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Add(Function('\\\\dot{y}')(Symbol('\\\\hat{p}', commutative=True)), exp(exp(Symbol('\\\\hat{p}', commutative=True)))), exp(Symbol('\\\\hat{p}', commutative=True))), Mul(Add(exp(Function('i')(Symbol('\\\\hat{p}', commutative=True))), exp(exp(Symbol('\\\\hat{p}', commutative=True)))), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Add(Function('\\\\dot{y}')(Symbol('\\\\hat{p}', commutative=True)), exp(Function('i')(Symbol('\\\\hat{p}', commutative=True)))), exp(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\hat{p}', commutative=True)), exp(Function('i')(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(b)} = \\cos{(b)}, then derive \\frac{d}{d b} \\int \\mathbf{M}{(b)} db = \\frac{\\partial}{\\partial b} (\\mathbf{S} + \\sin{(b)}), then obtain \\frac{d^{2}}{d b^{2}} \\int \\mathbf{M}{(b)} db = \\frac{\\partial^{2}}{\\partial b^{2}} (\\mathbf{S} + \\sin{(b)})", "derivation": "\\mathbf{M}{(b)} = \\cos{(b)} and \\int \\mathbf{M}{(b)} db = \\int \\cos{(b)} db and \\frac{d}{d b} \\int \\mathbf{M}{(b)} db = \\frac{d}{d b} \\int \\cos{(b)} db and \\frac{d}{d b} \\int \\mathbf{M}{(b)} db = \\frac{\\partial}{\\partial b} (\\mathbf{S} + \\sin{(b)}) and \\frac{d^{2}}{d b^{2}} \\int \\mathbf{M}{(b)} db = \\frac{\\partial^{2}}{\\partial b^{2}} (\\mathbf{S} + \\sin{(b)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{J}{(M,\\mathbf{g})} = M^{\\mathbf{g}}, then obtain M \\int 2 \\mathbf{J}{(M,\\mathbf{g})} dM - \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM = M \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM - \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM", "derivation": "\\mathbf{J}{(M,\\mathbf{g})} = M^{\\mathbf{g}} and 2 \\mathbf{J}{(M,\\mathbf{g})} = M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})} and \\int 2 \\mathbf{J}{(M,\\mathbf{g})} dM = \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM and M \\int 2 \\mathbf{J}{(M,\\mathbf{g})} dM = M \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM and M \\int 2 \\mathbf{J}{(M,\\mathbf{g})} dM - \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM = M \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM - \\int (M^{\\mathbf{g}} + \\mathbf{J}{(M,\\mathbf{g})}) dM", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["times", 3, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Integral(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Symbol('M', commutative=True), Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True)))))"], [["minus", 4, "Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Add(Mul(Symbol('M', commutative=True), Integral(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True))))), Add(Mul(Symbol('M', commutative=True), Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), Integral(Add(Pow(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\rho_{b}{(I,A)} = \\frac{A}{I}, then derive \\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)} = - \\frac{A}{I^{2}}, then obtain \\frac{\\partial}{\\partial A} (\\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)})^{A} = \\frac{\\partial}{\\partial A} (- \\frac{\\rho_{b}{(I,A)}}{I})^{A}", "derivation": "\\rho_{b}{(I,A)} = \\frac{A}{I} and \\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)} = \\frac{\\partial}{\\partial I} \\frac{A}{I} and \\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)} = - \\frac{A}{I^{2}} and \\frac{\\partial}{\\partial I} \\frac{A}{I} = - \\frac{A}{I^{2}} and \\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)} = - \\frac{\\rho_{b}{(I,A)}}{I} and (\\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)})^{A} = (- \\frac{\\rho_{b}{(I,A)}}{I})^{A} and \\frac{\\partial}{\\partial A} (\\frac{\\partial}{\\partial I} \\rho_{b}{(I,A)})^{A} = \\frac{\\partial}{\\partial A} (- \\frac{\\rho_{b}{(I,A)}}{I})^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Symbol('A', commutative=True), Pow(Symbol('I', commutative=True), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A', commutative=True), Pow(Symbol('I', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('A', commutative=True), Pow(Symbol('I', commutative=True), Integer(-1))), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('A', commutative=True), Pow(Symbol('I', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('A', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["differentiate", 6, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(z^{*})} = z^{*}, then obtain (\\frac{d}{d z^{*}} \\operatorname{f_{\\mathbf{p}}}{(z^{*})} - 1) e^{- z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = 0", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(z^{*})} = z^{*} and - z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*})} = 0 and e^{- z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = 1 and \\frac{d}{d z^{*}} e^{- z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = \\frac{d}{d z^{*}} 1 and (\\frac{d}{d z^{*}} \\operatorname{f_{\\mathbf{p}}}{(z^{*})} - 1) e^{- z^{*} + \\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = 0", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))"], [["minus", 1, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)))), Integer(1))"], [["differentiate", 3, "Symbol('z^*', commutative=True)"], "Equality(Derivative(exp(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(F_{c})} = \\log{(e^{F_{c}})}, then derive \\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})} = 1, then obtain 1 = \\frac{1}{\\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})}}", "derivation": "\\operatorname{P_{g}}{(F_{c})} = \\log{(e^{F_{c}})} and \\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})} = \\frac{d}{d F_{c}} \\log{(e^{F_{c}})} and \\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})} = 1 and \\frac{\\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})}}{\\frac{d}{d F_{c}} \\log{(e^{F_{c}})}} = \\frac{1}{\\frac{d}{d F_{c}} \\log{(e^{F_{c}})}} and 1 = \\frac{1}{\\frac{d}{d F_{c}} \\operatorname{P_{g}}{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('F_c', commutative=True)), log(exp(Symbol('F_c', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(log(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(1))"], [["divide", 3, "Derivative(log(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('P_g')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Pow(Derivative(log(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1))), Pow(Derivative(log(exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Pow(Derivative(Function('P_g')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\varphi^*)} = \\log{(\\sin{(\\varphi^*)})}, then derive \\frac{d}{d \\varphi^*} \\operatorname{F_{g}}{(\\varphi^*)} = \\frac{\\cos{(\\varphi^*)}}{\\sin{(\\varphi^*)}}, then derive 0 = - \\nabla + p - \\operatorname{F_{g}}{(\\varphi^*)} + \\log{(\\sin{(\\varphi^*)})}, then obtain 1 = - \\nabla + p + 1", "derivation": "\\operatorname{F_{g}}{(\\varphi^*)} = \\log{(\\sin{(\\varphi^*)})} and \\frac{d}{d \\varphi^*} \\operatorname{F_{g}}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} \\log{(\\sin{(\\varphi^*)})} and \\frac{d}{d \\varphi^*} \\operatorname{F_{g}}{(\\varphi^*)} = \\frac{\\cos{(\\varphi^*)}}{\\sin{(\\varphi^*)}} and \\int \\frac{d}{d \\varphi^*} \\operatorname{F_{g}}{(\\varphi^*)} d\\varphi^* = \\int \\frac{\\cos{(\\varphi^*)}}{\\sin{(\\varphi^*)}} d\\varphi^* and 0 = \\int \\frac{\\cos{(\\varphi^*)}}{\\sin{(\\varphi^*)}} d\\varphi^* - \\int \\frac{d}{d \\varphi^*} \\operatorname{F_{g}}{(\\varphi^*)} d\\varphi^* and 0 = - \\nabla + p - \\operatorname{F_{g}}{(\\varphi^*)} + \\log{(\\sin{(\\varphi^*)})} and 0 = - \\nabla + p and 1 = - \\nabla + p + 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), log(sin(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Derivative(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 4, "Integral(Derivative(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Pow(sin(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Integral(Derivative(Function('F_g')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Tuple(Symbol('\\\\varphi^*', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('p', commutative=True), Mul(Integer(-1), Function('F_g')(Symbol('\\\\varphi^*', commutative=True))), log(sin(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('p', commutative=True)))"], [["minus", 7, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('p', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\varphi{(u)} = \\sin{(\\sin{(u)})}, then obtain (u + \\varphi{(u)}) \\varphi{(u)} \\sin{(u)} \\sin{(\\sin{(u)})} = (u + \\varphi{(u)}) \\sin{(u)} \\sin^{2}{(\\sin{(u)})}", "derivation": "\\varphi{(u)} = \\sin{(\\sin{(u)})} and u + \\varphi{(u)} = u + \\sin{(\\sin{(u)})} and (u + \\sin{(\\sin{(u)})}) \\varphi{(u)} = (u + \\sin{(\\sin{(u)})}) \\sin{(\\sin{(u)})} and \\varphi{(u)} \\sin{(u)} = \\sin{(u)} \\sin{(\\sin{(u)})} and (u + \\varphi{(u)}) \\varphi{(u)} = (u + \\varphi{(u)}) \\sin{(\\sin{(u)})} and (u + \\varphi{(u)}) \\varphi^{2}{(u)} \\sin{(u)} = (u + \\varphi{(u)}) \\varphi{(u)} \\sin{(u)} \\sin{(\\sin{(u)})} and (u + \\varphi{(u)}) \\varphi{(u)} \\sin{(u)} \\sin{(\\sin{(u)})} = (u + \\varphi{(u)}) \\sin{(u)} \\sin^{2}{(\\sin{(u)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('u', commutative=True)), sin(sin(Symbol('u', commutative=True))))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), sin(sin(Symbol('u', commutative=True)))))"], [["times", 1, "Add(Symbol('u', commutative=True), sin(sin(Symbol('u', commutative=True))))"], "Equality(Mul(Add(Symbol('u', commutative=True), sin(sin(Symbol('u', commutative=True)))), Function('\\\\varphi')(Symbol('u', commutative=True))), Mul(Add(Symbol('u', commutative=True), sin(sin(Symbol('u', commutative=True)))), sin(sin(Symbol('u', commutative=True)))))"], [["times", 1, "sin(Symbol('u', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True))), Mul(sin(Symbol('u', commutative=True)), sin(sin(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), Function('\\\\varphi')(Symbol('u', commutative=True))), Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), sin(sin(Symbol('u', commutative=True)))))"], [["times", 5, "Mul(Function('\\\\varphi')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], "Equality(Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), Pow(Function('\\\\varphi')(Symbol('u', commutative=True)), Integer(2)), sin(Symbol('u', commutative=True))), Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), Function('\\\\varphi')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)), sin(sin(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), Function('\\\\varphi')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)), sin(sin(Symbol('u', commutative=True)))), Mul(Add(Symbol('u', commutative=True), Function('\\\\varphi')(Symbol('u', commutative=True))), sin(Symbol('u', commutative=True)), Pow(sin(sin(Symbol('u', commutative=True))), Integer(2))))"]]}, {"prompt": "Given h{(n_{1})} = e^{n_{1}}, then obtain \\frac{\\partial}{\\partial n_{2}} \\int (- \\phi + h{(n_{1})} + \\frac{1}{n_{2}}) d\\phi = \\frac{\\partial}{\\partial n_{2}} \\int (- \\phi + e^{n_{1}} + \\frac{1}{n_{2}}) d\\phi", "derivation": "h{(n_{1})} = e^{n_{1}} and h{(n_{1})} + \\frac{1}{n_{2}} = e^{n_{1}} + \\frac{1}{n_{2}} and - \\phi + h{(n_{1})} + \\frac{1}{n_{2}} = - \\phi + e^{n_{1}} + \\frac{1}{n_{2}} and \\int (- \\phi + h{(n_{1})} + \\frac{1}{n_{2}}) d\\phi = \\int (- \\phi + e^{n_{1}} + \\frac{1}{n_{2}}) d\\phi and \\frac{\\partial}{\\partial n_{2}} \\int (- \\phi + h{(n_{1})} + \\frac{1}{n_{2}}) d\\phi = \\frac{\\partial}{\\partial n_{2}} \\int (- \\phi + e^{n_{1}} + \\frac{1}{n_{2}}) d\\phi", "srepr_derivation": [["get_premise", "Equality(Function('h')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["add", 1, "Pow(Symbol('n_2', commutative=True), Integer(-1))"], "Equality(Add(Function('h')(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Add(exp(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["minus", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 4, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('h')(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), exp(Symbol('n_1', commutative=True)), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(U)} = \\sin{(U)}, then obtain \\log{(2 \\varepsilon_0 \\int \\hat{H}{(U)} dU)} = \\log{(2 \\varepsilon_0 \\int \\sin{(U)} dU)}", "derivation": "\\hat{H}{(U)} = \\sin{(U)} and \\int \\hat{H}{(U)} dU = \\int \\sin{(U)} dU and 2 \\varepsilon_0 \\int \\hat{H}{(U)} dU = 2 \\varepsilon_0 \\int \\sin{(U)} dU and \\log{(2 \\varepsilon_0 \\int \\hat{H}{(U)} dU)} = \\log{(2 \\varepsilon_0 \\int \\sin{(U)} dU)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["times", 2, "Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Integral(Function('\\\\hat{H}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["log", 3], "Equality(log(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Integral(Function('\\\\hat{H}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))), log(Mul(Integer(2), Symbol('\\\\varepsilon_0', commutative=True), Integral(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})} = \\ddot{x} + e^{J_{\\varepsilon}}, then obtain \\int J_{\\varepsilon}^{2} (- \\ddot{x} + \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})}) d\\ddot{x} = \\int J_{\\varepsilon}^{2} e^{J_{\\varepsilon}} d\\ddot{x}", "derivation": "\\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})} = \\ddot{x} + e^{J_{\\varepsilon}} and - \\ddot{x} + \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})} = e^{J_{\\varepsilon}} and J_{\\varepsilon} (- \\ddot{x} + \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})}) = J_{\\varepsilon} e^{J_{\\varepsilon}} and J_{\\varepsilon}^{2} (- \\ddot{x} + \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})}) = J_{\\varepsilon}^{2} e^{J_{\\varepsilon}} and \\int J_{\\varepsilon}^{2} (- \\ddot{x} + \\operatorname{A_{z}}{(J_{\\varepsilon},\\ddot{x})}) d\\ddot{x} = \\int J_{\\varepsilon}^{2} e^{J_{\\varepsilon}} d\\ddot{x}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["times", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('A_z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(2)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given t{(\\psi,A_{2})} = A_{2} \\psi, then derive A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} t{(\\psi,A_{2})} = A_{2}, then obtain A_{2} = A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} A_{2} \\psi", "derivation": "t{(\\psi,A_{2})} = A_{2} \\psi and \\frac{\\partial}{\\partial A_{2}} t{(\\psi,A_{2})} = \\frac{\\partial}{\\partial A_{2}} A_{2} \\psi and \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} t{(\\psi,A_{2})} = \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} A_{2} \\psi and A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} t{(\\psi,A_{2})} = A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} A_{2} \\psi and A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} t{(\\psi,A_{2})} = A_{2} and A_{2} = A_{2} \\frac{\\partial^{2}}{\\partial \\psi\\partial A_{2}} A_{2} \\psi", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["times", 3, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Derivative(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Mul(Symbol('A_2', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('A_2', commutative=True), Derivative(Function('t')(Symbol('\\\\psi', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Symbol('A_2', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Symbol('A_2', commutative=True), Mul(Symbol('A_2', commutative=True), Derivative(Mul(Symbol('A_2', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(i,n)} = \\sin{(i n)} and q{(i,n)} = \\sin{(i n)}, then obtain 0 = y (q{(i,n)} - \\sin{(i n)})", "derivation": "\\operatorname{c_{0}}{(i,n)} = \\sin{(i n)} and i n + \\operatorname{c_{0}}{(i,n)} = i n + \\sin{(i n)} and 0 = - \\operatorname{c_{0}}{(i,n)} + \\sin{(i n)} and q{(i,n)} = \\sin{(i n)} and 0 = - \\operatorname{c_{0}}{(i,n)} + q{(i,n)} and 0 = q{(i,n)} - \\sin{(i n)} and 0 = y (q{(i,n)} - \\sin{(i n)})", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('i', commutative=True), Symbol('n', commutative=True)), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True))))"], [["add", 1, "Mul(Symbol('i', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)), Function('c_0')(Symbol('i', commutative=True), Symbol('n', commutative=True))), Add(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)))))"], [["minus", 2, "Add(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)), Function('c_0')(Symbol('i', commutative=True), Symbol('n', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('i', commutative=True), Symbol('n', commutative=True))), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('q')(Symbol('i', commutative=True), Symbol('n', commutative=True)), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('i', commutative=True), Symbol('n', commutative=True))), Function('q')(Symbol('i', commutative=True), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Function('q')(Symbol('i', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True))))))"], [["times", 6, "Symbol('y', commutative=True)"], "Equality(Integer(0), Mul(Symbol('y', commutative=True), Add(Function('q')(Symbol('i', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), sin(Mul(Symbol('i', commutative=True), Symbol('n', commutative=True)))))))"]]}, {"prompt": "Given v{(F_{N})} = e^{F_{N}}, then derive \\int v{(F_{N})} dF_{N} = f^{\\prime} + e^{F_{N}}, then obtain \\int v{(F_{N})} dF_{N} + 1 = f^{\\prime} + v{(F_{N})} + 1", "derivation": "v{(F_{N})} = e^{F_{N}} and \\int v{(F_{N})} dF_{N} = \\int e^{F_{N}} dF_{N} and \\int v{(F_{N})} dF_{N} = f^{\\prime} + e^{F_{N}} and f^{\\prime} + e^{F_{N}} = \\int e^{F_{N}} dF_{N} and \\int v{(F_{N})} dF_{N} = f^{\\prime} + v{(F_{N})} and \\int v{(F_{N})} dF_{N} + \\frac{\\int e^{F_{N}} dF_{N}}{f^{\\prime} + e^{F_{N}}} = f^{\\prime} + v{(F_{N})} + \\frac{\\int e^{F_{N}} dF_{N}}{f^{\\prime} + e^{F_{N}}} and \\int v{(F_{N})} dF_{N} + 1 = f^{\\prime} + v{(F_{N})} + 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('F_N', commutative=True))), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), Function('v')(Symbol('F_N', commutative=True))))"], [["add", 5, "Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('F_N', commutative=True))), Integer(-1)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], "Equality(Add(Integral(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('F_N', commutative=True))), Integer(-1)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))), Add(Symbol('f^{\\\\prime}', commutative=True), Function('v')(Symbol('F_N', commutative=True)), Mul(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), exp(Symbol('F_N', commutative=True))), Integer(-1)), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Integral(Function('v')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integer(1)), Add(Symbol('f^{\\\\prime}', commutative=True), Function('v')(Symbol('F_N', commutative=True)), Integer(1)))"]]}, {"prompt": "Given L{(n,\\dot{y})} = n^{\\dot{y}} and Z{(n,\\dot{y})} = L^{\\dot{y}}{(n,\\dot{y})}, then obtain n^{\\dot{y}} (n^{\\dot{y}})^{\\dot{y}} = n^{\\dot{y}} L^{\\dot{y}}{(n,\\dot{y})}", "derivation": "L{(n,\\dot{y})} = n^{\\dot{y}} and L^{\\dot{y}}{(n,\\dot{y})} = (n^{\\dot{y}})^{\\dot{y}} and Z{(n,\\dot{y})} = L^{\\dot{y}}{(n,\\dot{y})} and Z{(n,\\dot{y})} = (n^{\\dot{y}})^{\\dot{y}} and n^{\\dot{y}} Z{(n,\\dot{y})} = n^{\\dot{y}} (n^{\\dot{y}})^{\\dot{y}} and n^{\\dot{y}} Z{(n,\\dot{y})} = n^{\\dot{y}} L^{\\dot{y}}{(n,\\dot{y})} and n^{\\dot{y}} (n^{\\dot{y}})^{\\dot{y}} = n^{\\dot{y}} L^{\\dot{y}}{(n,\\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('L')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["renaming_premise", "Equality(Function('Z')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('L')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('Z')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["times", 4, "Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('Z')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Function('Z')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('L')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('L')(Symbol('n', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given k{(\\phi_1,J_{\\varepsilon})} = \\phi_1 \\cos{(J_{\\varepsilon})}, then obtain \\phi_1 k{(\\phi_1,J_{\\varepsilon})} \\cos^{2}{(J_{\\varepsilon})} = \\phi_1^{2} \\cos^{3}{(J_{\\varepsilon})}", "derivation": "k{(\\phi_1,J_{\\varepsilon})} = \\phi_1 \\cos{(J_{\\varepsilon})} and \\phi_1 k{(\\phi_1,J_{\\varepsilon})} \\cos{(J_{\\varepsilon})} = \\phi_1^{2} \\cos^{2}{(J_{\\varepsilon})} and \\phi_1 k^{2}{(\\phi_1,J_{\\varepsilon})} \\cos{(J_{\\varepsilon})} = \\phi_1^{2} k{(\\phi_1,J_{\\varepsilon})} \\cos^{2}{(J_{\\varepsilon})} and k^{2}{(\\phi_1,J_{\\varepsilon})} \\cos{(J_{\\varepsilon})} = \\phi_1 k{(\\phi_1,J_{\\varepsilon})} \\cos^{2}{(J_{\\varepsilon})} and k^{2}{(\\phi_1,J_{\\varepsilon})} \\cos{(J_{\\varepsilon})} = \\phi_1^{2} \\cos^{3}{(J_{\\varepsilon})} and \\phi_1 k{(\\phi_1,J_{\\varepsilon})} \\cos^{2}{(J_{\\varepsilon})} = \\phi_1^{2} \\cos^{3}{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), cos(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\phi_1', commutative=True), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(2)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))))"], [["times", 2, "Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(2)), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))))"], [["divide", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2)), cos(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(2)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('k')(Symbol('\\\\phi_1', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(2)), Pow(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)} = \\mathbf{J}_f^{\\phi_2} g_{\\varepsilon}, then obtain \\int (g_{\\varepsilon} + \\mathbf{J}_f^{- \\phi_2} \\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)}) dg_{\\varepsilon} = \\int 2 g_{\\varepsilon} dg_{\\varepsilon}", "derivation": "\\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)} = \\mathbf{J}_f^{\\phi_2} g_{\\varepsilon} and \\mathbf{J}_f^{- \\phi_2} \\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)} = g_{\\varepsilon} and g_{\\varepsilon} + \\mathbf{J}_f^{- \\phi_2} \\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)} = 2 g_{\\varepsilon} and \\int (g_{\\varepsilon} + \\mathbf{J}_f^{- \\phi_2} \\tilde{g}^*{(\\phi_2,g_{\\varepsilon},\\mathbf{J}_f)}) dg_{\\varepsilon} = \\int 2 g_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi_2', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["add", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\phi_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and \\mu{(\\varphi,u)} = \\varphi + u, then obtain (\\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\mu{(\\varphi,u)})})^{u} = (\\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\varphi + u)})^{u}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and \\mu{(\\varphi,u)} = \\varphi + u and \\sin{(\\mu{(\\varphi,u)})} = \\sin{(\\varphi + u)} and \\frac{\\sin{(\\mu{(\\varphi,u)})}}{- \\Psi_{\\lambda}{(\\mathbf{F})} + \\log{(\\mathbf{F})}} = \\frac{\\sin{(\\varphi + u)}}{- \\Psi_{\\lambda}{(\\mathbf{F})} + \\log{(\\mathbf{F})}} and \\tilde{\\infty} \\sin{(\\mu{(\\varphi,u)})} = \\tilde{\\infty} \\sin{(\\varphi + u)} and \\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\mu{(\\varphi,u)})} = \\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\varphi + u)} and (\\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\mu{(\\varphi,u)})})^{u} = (\\frac{\\partial}{\\partial u} \\tilde{\\infty} \\sin{(\\varphi + u)})^{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))"], [["sin", 2], "Equality(sin(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))), sin(Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True))), log(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True))), log(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), sin(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{F}', commutative=True))), log(Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), sin(Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(zoo, sin(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Mul(zoo, sin(Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(zoo, sin(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(zoo, sin(Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Derivative(Mul(zoo, sin(Function('\\\\mu')(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)), Pow(Derivative(Mul(zoo, sin(Add(Symbol('\\\\varphi', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given r{(v_{t})} = \\int \\sin{(v_{t})} dv_{t}, then obtain - r{(v_{t})} + \\frac{\\int r{(v_{t})} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}} = - r{(v_{t})} + \\frac{\\iint \\sin{(v_{t})} dv_{t} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}}", "derivation": "r{(v_{t})} = \\int \\sin{(v_{t})} dv_{t} and \\int r{(v_{t})} dv_{t} = \\iint \\sin{(v_{t})} dv_{t} dv_{t} and \\frac{\\int r{(v_{t})} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}} = \\frac{\\iint \\sin{(v_{t})} dv_{t} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}} and - r{(v_{t})} + \\frac{\\int r{(v_{t})} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}} = - r{(v_{t})} + \\frac{\\iint \\sin{(v_{t})} dv_{t} dv_{t}}{\\int \\sin{(v_{t})} dv_{t}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('v_t', commutative=True)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('r')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Mul(Integral(Function('r')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Pow(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1))), Mul(Pow(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))))"], [["minus", 3, "Function('r')(Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('r')(Symbol('v_t', commutative=True))), Mul(Integral(Function('r')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Pow(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Function('r')(Symbol('v_t', commutative=True))), Mul(Pow(Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integer(-1)), Integral(sin(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given c{(C_{1})} = \\cos{(e^{C_{1}})}, then obtain \\int C_{1} \\frac{d^{3}}{d C_{1}^{3}} c{(C_{1})} dC_{1} = \\int C_{1} \\frac{d^{3}}{d C_{1}^{3}} \\cos{(e^{C_{1}})} dC_{1}", "derivation": "c{(C_{1})} = \\cos{(e^{C_{1}})} and \\frac{d}{d C_{1}} c{(C_{1})} = \\frac{d}{d C_{1}} \\cos{(e^{C_{1}})} and \\frac{d^{2}}{d C_{1}^{2}} c{(C_{1})} = \\frac{d^{2}}{d C_{1}^{2}} \\cos{(e^{C_{1}})} and \\frac{d^{3}}{d C_{1}^{3}} c{(C_{1})} = \\frac{d^{3}}{d C_{1}^{3}} \\cos{(e^{C_{1}})} and C_{1} \\frac{d^{3}}{d C_{1}^{3}} c{(C_{1})} = C_{1} \\frac{d^{3}}{d C_{1}^{3}} \\cos{(e^{C_{1}})} and \\int C_{1} \\frac{d^{3}}{d C_{1}^{3}} c{(C_{1})} dC_{1} = \\int C_{1} \\frac{d^{3}}{d C_{1}^{3}} \\cos{(e^{C_{1}})} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('C_1', commutative=True)), cos(exp(Symbol('C_1', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(2))), Derivative(cos(exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(3))), Derivative(cos(exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(3))))"], [["times", 4, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Derivative(Function('c')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(3)))), Mul(Symbol('C_1', commutative=True), Derivative(cos(exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(3)))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Symbol('C_1', commutative=True), Derivative(Function('c')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(3)))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Symbol('C_1', commutative=True), Derivative(cos(exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(3)))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(J,\\mathbf{E},s)} = \\frac{J - s}{\\mathbf{E}}, then obtain J - s = \\frac{(J - s) (- \\mathbf{E} + \\frac{- J + s}{\\mathbf{E}})}{- \\mathbf{E} - \\operatorname{F_{g}}{(J,\\mathbf{E},s)}}", "derivation": "\\operatorname{F_{g}}{(J,\\mathbf{E},s)} = \\frac{J - s}{\\mathbf{E}} and \\mathbf{E} + \\operatorname{F_{g}}{(J,\\mathbf{E},s)} = \\mathbf{E} + \\frac{J - s}{\\mathbf{E}} and - \\mathbf{E} - \\operatorname{F_{g}}{(J,\\mathbf{E},s)} = - \\mathbf{E} - \\frac{J - s}{\\mathbf{E}} and - \\mathbf{E} - \\operatorname{F_{g}}{(J,\\mathbf{E},s)} = - \\mathbf{E} + \\frac{- J + s}{\\mathbf{E}} and J - s = \\frac{(J - s) (- \\mathbf{E} + \\frac{- J + s}{\\mathbf{E}})}{- \\mathbf{E} - \\operatorname{F_{g}}{(J,\\mathbf{E},s)}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)))))"], [["add", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('s', commutative=True)))))"], [["divide", 4, "Mul(Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True)))))"], "Equality(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Mul(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('s', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('J', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('s', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given x{(r)} = e^{r}, then derive 0 = e^{r} - \\frac{d}{d r} x{(r)}, then obtain 0^{r} = (e^{r} - \\frac{d}{d r} e^{r})^{r}", "derivation": "x{(r)} = e^{r} and 0 = - x{(r)} + e^{r} and \\frac{d}{d r} 0 = \\frac{d}{d r} (- x{(r)} + e^{r}) and 0 = e^{r} - \\frac{d}{d r} x{(r)} and 0 = e^{r} - \\frac{d}{d r} e^{r} and 0 = x{(r)} - \\frac{d}{d r} x{(r)} and 0^{r} = (x{(r)} - \\frac{d}{d r} x{(r)})^{r} and 0^{r} = (e^{r} - \\frac{d}{d r} e^{r})^{r}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["minus", 1, "Function('x')(Symbol('r', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x')(Symbol('r', commutative=True))), exp(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('x')(Symbol('r', commutative=True))), exp(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(exp(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(exp(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Function('x')(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))))"], [["power", 6, "Symbol('r', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r', commutative=True)), Pow(Add(Function('x')(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(Function('x')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Pow(Integer(0), Symbol('r', commutative=True)), Pow(Add(exp(Symbol('r', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Symbol('r', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\dot{z},\\mathbf{f})} = \\dot{z} + \\mathbf{f}, then derive \\int \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\dot{z} \\mathbf{f} + \\psi, then obtain \\iint \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} d\\psi = \\int \\frac{\\dot{z}^{2}}{2} d\\psi + \\int \\psi d\\psi + \\int \\dot{z} \\mathbf{f} d\\psi", "derivation": "\\varphi^{*}{(\\dot{z},\\mathbf{f})} = \\dot{z} + \\mathbf{f} and \\int \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} = \\int (\\dot{z} + \\mathbf{f}) d\\dot{z} and \\int \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} = \\frac{\\dot{z}^{2}}{2} + \\dot{z} \\mathbf{f} + \\psi and \\iint \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} d\\psi = \\int (\\frac{\\dot{z}^{2}}{2} + \\dot{z} \\mathbf{f} + \\psi) d\\psi and \\iint \\varphi^{*}{(\\dot{z},\\mathbf{f})} d\\dot{z} d\\psi = \\int \\frac{\\dot{z}^{2}}{2} d\\psi + \\int \\psi d\\psi + \\int \\dot{z} \\mathbf{f} d\\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Add(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["expand", 4], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Integral(Mul(Rational(1, 2), Pow(Symbol('\\\\dot{z}', commutative=True), Integer(2))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\varphi{(I)} = e^{I}, then obtain \\operatorname{c_{0}}{(V_{\\mathbf{E}})} e^{- I} = e^{- I} \\sin{(V_{\\mathbf{E}})}", "derivation": "\\operatorname{c_{0}}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\varphi{(I)} = e^{I} and \\frac{\\operatorname{c_{0}}{(V_{\\mathbf{E}})}}{\\varphi{(I)}} = \\frac{\\sin{(V_{\\mathbf{E}})}}{\\varphi{(I)}} and \\operatorname{c_{0}}{(V_{\\mathbf{E}})} e^{- I} = e^{- I} \\sin{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varphi')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["divide", 1, "Function('\\\\varphi')(Symbol('I', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('I', commutative=True)), Integer(-1)), Function('c_0')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Pow(Function('\\\\varphi')(Symbol('I', commutative=True)), Integer(-1)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('c_0')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('I', commutative=True))), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(l,\\theta)} = \\theta e^{l}, then derive \\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)} = e^{l}, then obtain \\int \\frac{e^{l}}{\\theta} d\\theta = \\int \\frac{\\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)}}{\\theta} d\\theta", "derivation": "\\hat{X}{(l,\\theta)} = \\theta e^{l} and \\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\theta e^{l} and \\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)} = e^{l} and e^{l} = \\frac{\\partial}{\\partial \\theta} \\theta e^{l} and \\frac{e^{l}}{\\theta} = \\frac{\\frac{\\partial}{\\partial \\theta} \\theta e^{l}}{\\theta} and \\frac{e^{l}}{\\theta} = \\frac{\\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)}}{\\theta} and \\int \\frac{e^{l}}{\\theta} d\\theta = \\int \\frac{\\frac{\\partial}{\\partial \\theta} \\hat{X}{(l,\\theta)}}{\\theta} d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), exp(Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('l', commutative=True)), Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["divide", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["integrate", 6, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{X}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\lambda)} = \\sin{(\\lambda)} and \\operatorname{y^{\\prime}}{(\\lambda)} = \\lambda, then obtain \\frac{\\operatorname{y^{\\prime}}{(\\lambda)} - \\int \\sin{(\\lambda)} d\\lambda}{\\lambda - \\int \\sin{(\\lambda)} d\\lambda} = 1", "derivation": "\\sigma_{x}{(\\lambda)} = \\sin{(\\lambda)} and \\int \\sigma_{x}{(\\lambda)} d\\lambda = \\int \\sin{(\\lambda)} d\\lambda and \\operatorname{y^{\\prime}}{(\\lambda)} = \\lambda and \\operatorname{y^{\\prime}}{(\\lambda)} - \\int \\sin{(\\lambda)} d\\lambda = \\lambda - \\int \\sin{(\\lambda)} d\\lambda and \\operatorname{y^{\\prime}}{(\\lambda)} - \\int \\sigma_{x}{(\\lambda)} d\\lambda = \\lambda - \\int \\sigma_{x}{(\\lambda)} d\\lambda and \\frac{\\operatorname{y^{\\prime}}{(\\lambda)} - \\int \\sigma_{x}{(\\lambda)} d\\lambda}{\\lambda - \\int \\sigma_{x}{(\\lambda)} d\\lambda} = 1 and \\frac{\\operatorname{y^{\\prime}}{(\\lambda)} - \\int \\sin{(\\lambda)} d\\lambda}{\\lambda - \\int \\sin{(\\lambda)} d\\lambda} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["minus", 3, "Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"], [["divide", 5, "Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Integer(-1)), Add(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\sigma_x')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Integer(-1)), Add(Function('y^{\\\\prime}')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))), Integer(1))"]]}, {"prompt": "Given W{(\\lambda,t)} = \\log{(t^{\\lambda})} and A{(\\lambda)} = \\lambda, then obtain (\\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int W{(\\lambda,t)} dA{(\\lambda)}}{2})^{\\lambda} = (\\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int \\log{(t^{\\lambda})} dA{(\\lambda)}}{2})^{\\lambda}", "derivation": "W{(\\lambda,t)} = \\log{(t^{\\lambda})} and \\int W{(\\lambda,t)} d\\lambda = \\int \\log{(t^{\\lambda})} d\\lambda and \\frac{\\partial}{\\partial \\lambda} \\int W{(\\lambda,t)} d\\lambda = \\frac{\\partial}{\\partial \\lambda} \\int \\log{(t^{\\lambda})} d\\lambda and A{(\\lambda)} = \\lambda and \\frac{\\partial}{\\partial A{(\\lambda)}} \\int W{(\\lambda,t)} dA{(\\lambda)} = \\frac{\\partial}{\\partial A{(\\lambda)}} \\int \\log{(t^{\\lambda})} dA{(\\lambda)} and \\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int W{(\\lambda,t)} dA{(\\lambda)}}{2} = \\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int \\log{(t^{\\lambda})} dA{(\\lambda)}}{2} and (\\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int W{(\\lambda,t)} dA{(\\lambda)}}{2})^{\\lambda} = (\\frac{\\frac{\\partial}{\\partial A{(\\lambda)}} \\int \\log{(t^{\\lambda})} dA{(\\lambda)}}{2})^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1))), Derivative(Integral(log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1))))"], [["times", 5, "Rational(1, 2)"], "Equality(Mul(Rational(1, 2), Derivative(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1)))), Mul(Rational(1, 2), Derivative(Integral(log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1)))))"], [["power", 6, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Derivative(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('t', commutative=True)), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1)))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Rational(1, 2), Derivative(Integral(log(Pow(Symbol('t', commutative=True), Symbol('\\\\lambda', commutative=True))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)))), Tuple(Function('A')(Symbol('\\\\lambda', commutative=True)), Integer(1)))), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given q{(\\mathbf{J},\\mathbf{D})} = \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})} and \\mathbf{M}{(\\mathbf{D})} = \\frac{1}{\\mathbf{D}}, then obtain - \\mathbf{D}^{2} q{(\\mathbf{J},\\mathbf{D})} = - \\mathbf{D}^{2} \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})}", "derivation": "q{(\\mathbf{J},\\mathbf{D})} = \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})} and \\mathbf{D} q{(\\mathbf{J},\\mathbf{D})} = \\mathbf{D} \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})} and \\mathbf{M}{(\\mathbf{D})} = \\frac{1}{\\mathbf{D}} and - \\mathbf{D} q{(\\mathbf{J},\\mathbf{D})} = - \\mathbf{D} \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})} and - \\frac{\\mathbf{D} q{(\\mathbf{J},\\mathbf{D})}}{\\mathbf{M}{(\\mathbf{D})}} = - \\frac{\\mathbf{D} \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})}}{\\mathbf{M}{(\\mathbf{D})}} and - \\mathbf{D}^{2} q{(\\mathbf{J},\\mathbf{D})} = - \\mathbf{D}^{2} \\log{(\\frac{\\mathbf{J}}{\\mathbf{D}})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), log(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('q')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Function('q')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), log(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 4, "Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Function('q')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), log(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Function('q')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), log(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(\\mathbf{J}_P,v)} = - \\mathbf{J}_P + v, then derive \\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)} = -1, then obtain - v = \\frac{v}{\\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)}}", "derivation": "\\varphi{(\\mathbf{J}_P,v)} = - \\mathbf{J}_P + v and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\mathbf{J}_P + v) and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)} = -1 and \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\mathbf{J}_P + v) = -1 and v \\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)} = v \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\mathbf{J}_P + v) and v \\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\mathbf{J}_P + v) = \\frac{v (\\frac{\\partial}{\\partial \\mathbf{J}_P} (- \\mathbf{J}_P + v))^{2}}{\\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)}} and - v = \\frac{v}{\\frac{\\partial}{\\partial \\mathbf{J}_P} \\varphi{(\\mathbf{J}_P,v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1))"], [["times", 2, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Symbol('v', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["times", 5, "Mul(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Mul(Symbol('v', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Symbol('v', commutative=True), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Derivative(Function('\\\\varphi')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\phi,y)} = \\cos{(\\frac{y}{\\phi})}, then obtain \\frac{\\partial}{\\partial y} (\\varepsilon_{0}{(\\phi,y)} + \\cos{(\\frac{y}{\\phi})})^{2} = \\frac{\\partial}{\\partial y} 4 \\cos^{2}{(\\frac{y}{\\phi})}", "derivation": "\\varepsilon_{0}{(\\phi,y)} = \\cos{(\\frac{y}{\\phi})} and \\varepsilon_{0}{(\\phi,y)} + \\cos{(\\frac{y}{\\phi})} = 2 \\cos{(\\frac{y}{\\phi})} and (\\varepsilon_{0}{(\\phi,y)} + \\cos{(\\frac{y}{\\phi})})^{2} = 4 \\cos^{2}{(\\frac{y}{\\phi})} and \\frac{\\partial}{\\partial y} (\\varepsilon_{0}{(\\phi,y)} + \\cos{(\\frac{y}{\\phi})})^{2} = \\frac{\\partial}{\\partial y} 4 \\cos^{2}{(\\frac{y}{\\phi})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["add", 1, "cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Mul(Integer(2), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))))"], [["power", 2, 2], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Integer(2)), Mul(Integer(4), Pow(cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True)))), Integer(2)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Integer(4), Pow(cos(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(b,f_{\\mathbf{p}})} = b + f_{\\mathbf{p}} and \\theta_{1}{(b,f_{\\mathbf{p}})} = g{(b,f_{\\mathbf{p}})} g^{f_{\\mathbf{p}}}{(b,f_{\\mathbf{p}})}, then obtain \\theta_{1}{(b,f_{\\mathbf{p}})} = (b + f_{\\mathbf{p}})^{f_{\\mathbf{p}}} g{(b,f_{\\mathbf{p}})}", "derivation": "g{(b,f_{\\mathbf{p}})} = b + f_{\\mathbf{p}} and g^{f_{\\mathbf{p}}}{(b,f_{\\mathbf{p}})} = (b + f_{\\mathbf{p}})^{f_{\\mathbf{p}}} and g{(b,f_{\\mathbf{p}})} g^{f_{\\mathbf{p}}}{(b,f_{\\mathbf{p}})} = (b + f_{\\mathbf{p}})^{f_{\\mathbf{p}}} g{(b,f_{\\mathbf{p}})} and \\theta_{1}{(b,f_{\\mathbf{p}})} = g{(b,f_{\\mathbf{p}})} g^{f_{\\mathbf{p}}}{(b,f_{\\mathbf{p}})} and \\theta_{1}{(b,f_{\\mathbf{p}})} = (b + f_{\\mathbf{p}})^{f_{\\mathbf{p}}} g{(b,f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["power", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Add(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["times", 2, "Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Mul(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Pow(Add(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\theta_1')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Pow(Add(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Function('g')(Symbol('b', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\pi,F_{x})} = F_{x} \\pi, then derive \\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})} = \\pi, then obtain ((\\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})})^{F_{x}} + 1)^{F_{x}} = (\\pi^{F_{x}} + 1)^{F_{x}}", "derivation": "\\operatorname{A_{1}}{(\\pi,F_{x})} = F_{x} \\pi and \\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})} = \\frac{\\partial}{\\partial F_{x}} F_{x} \\pi and \\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})} = \\pi and (\\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})})^{F_{x}} = (\\frac{\\partial}{\\partial F_{x}} F_{x} \\pi)^{F_{x}} and \\frac{\\partial}{\\partial F_{x}} F_{x} \\pi = \\pi and (\\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})})^{F_{x}} = \\pi^{F_{x}} and (\\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})})^{F_{x}} + 1 = \\pi^{F_{x}} + 1 and ((\\frac{\\partial}{\\partial F_{x}} \\operatorname{A_{1}}{(\\pi,F_{x})})^{F_{x}} + 1)^{F_{x}} = (\\pi^{F_{x}} + 1)^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)))"], [["add", 6, 1], "Equality(Add(Pow(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Integer(1)), Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Integer(1)))"], [["power", 7, "Symbol('F_x', commutative=True)"], "Equality(Pow(Add(Pow(Derivative(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Symbol('F_x', commutative=True)), Integer(1)), Symbol('F_x', commutative=True)), Pow(Add(Pow(Symbol('\\\\pi', commutative=True), Symbol('F_x', commutative=True)), Integer(1)), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given k{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})}, then obtain \\int \\frac{1}{k{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} = \\int \\frac{\\sin{(\\Psi_{\\lambda})}}{k^{2}{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "derivation": "k{(\\Psi_{\\lambda})} = \\sin{(\\Psi_{\\lambda})} and 1 = \\frac{\\sin{(\\Psi_{\\lambda})}}{k{(\\Psi_{\\lambda})}} and \\frac{1}{k{(\\Psi_{\\lambda})}} = \\frac{\\sin{(\\Psi_{\\lambda})}}{k^{2}{(\\Psi_{\\lambda})}} and \\int \\frac{1}{k{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda} = \\int \\frac{\\sin{(\\Psi_{\\lambda})}}{k^{2}{(\\Psi_{\\lambda})}} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 2, "Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))"], "Equality(Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Mul(Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-2)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Mul(Pow(Function('k')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-2)), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given L{(\\theta_2,H)} = H \\theta_2 and \\dot{x}{(F_{c})} = \\int \\cos{(F_{c})} dF_{c}, then obtain \\frac{\\dot{x}{(F_{c})}}{H L{(\\theta_2,H)}} = \\frac{\\int \\cos{(F_{c})} dF_{c}}{H L{(\\theta_2,H)}}", "derivation": "L{(\\theta_2,H)} = H \\theta_2 and H L{(\\theta_2,H)} = H^{2} \\theta_2 and \\dot{x}{(F_{c})} = \\int \\cos{(F_{c})} dF_{c} and \\frac{\\dot{x}{(F_{c})}}{H^{2} \\theta_2} = \\frac{\\int \\cos{(F_{c})} dF_{c}}{H^{2} \\theta_2} and \\frac{\\dot{x}{(F_{c})}}{H L{(\\theta_2,H)}} = \\frac{\\int \\cos{(F_{c})} dF_{c}}{H L{(\\theta_2,H)}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Function('L')(Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(2)), Symbol('\\\\theta_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('F_c', commutative=True)), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('H', commutative=True), Integer(2)), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\dot{x}')(Symbol('F_c', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('L')(Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Function('\\\\dot{x}')(Symbol('F_c', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('L')(Symbol('\\\\theta_2', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Integral(cos(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\sigma_x)} = \\sin{(e^{\\sigma_x})}, then obtain \\operatorname{A_{y}}{(\\sigma_x)} + \\sin{(e^{\\sigma_x})} + \\int \\operatorname{A_{y}}{(\\sigma_x)} d\\sigma_x = \\operatorname{A_{y}}{(\\sigma_x)} + \\sin{(e^{\\sigma_x})} + \\int \\sin{(e^{\\sigma_x})} d\\sigma_x", "derivation": "\\operatorname{A_{y}}{(\\sigma_x)} = \\sin{(e^{\\sigma_x})} and \\int \\operatorname{A_{y}}{(\\sigma_x)} d\\sigma_x = \\int \\sin{(e^{\\sigma_x})} d\\sigma_x and \\operatorname{A_{y}}{(\\sigma_x)} + \\int \\operatorname{A_{y}}{(\\sigma_x)} d\\sigma_x = \\operatorname{A_{y}}{(\\sigma_x)} + \\int \\sin{(e^{\\sigma_x})} d\\sigma_x and \\operatorname{A_{y}}{(\\sigma_x)} + \\sin{(e^{\\sigma_x})} + \\int \\operatorname{A_{y}}{(\\sigma_x)} d\\sigma_x = \\operatorname{A_{y}}{(\\sigma_x)} + \\sin{(e^{\\sigma_x})} + \\int \\sin{(e^{\\sigma_x})} d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), sin(exp(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 2, "Function('A_y')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), Integral(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), Integral(sin(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"], [["add", 3, "sin(exp(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Add(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), sin(exp(Symbol('\\\\sigma_x', commutative=True))), Integral(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Add(Function('A_y')(Symbol('\\\\sigma_x', commutative=True)), sin(exp(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(exp(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given T{(f^{\\prime})} = \\sin{(f^{\\prime})}, then obtain (\\frac{d}{d f^{\\prime}} T^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} = (\\frac{d}{d f^{\\prime}} \\sin^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}}", "derivation": "T{(f^{\\prime})} = \\sin{(f^{\\prime})} and T^{f^{\\prime}}{(f^{\\prime})} = \\sin^{f^{\\prime}}{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} T^{f^{\\prime}}{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\sin^{f^{\\prime}}{(f^{\\prime})} and (\\frac{d}{d f^{\\prime}} T^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}} = (\\frac{d}{d f^{\\prime}} \\sin^{f^{\\prime}}{(f^{\\prime})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('T')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Pow(Function('T')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('T')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Derivative(Pow(sin(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given u{(V_{\\mathbf{B}},t_{1})} = \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}}, then obtain t_{1} \\int (- V_{\\mathbf{B}} + u{(V_{\\mathbf{B}},t_{1})}) dV_{\\mathbf{B}} = t_{1} \\int (- V_{\\mathbf{B}} + \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}}) dV_{\\mathbf{B}}", "derivation": "u{(V_{\\mathbf{B}},t_{1})} = \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}} and - V_{\\mathbf{B}} + u{(V_{\\mathbf{B}},t_{1})} = - V_{\\mathbf{B}} + \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}} and \\int (- V_{\\mathbf{B}} + u{(V_{\\mathbf{B}},t_{1})}) dV_{\\mathbf{B}} = \\int (- V_{\\mathbf{B}} + \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}}) dV_{\\mathbf{B}} and t_{1} \\int (- V_{\\mathbf{B}} + u{(V_{\\mathbf{B}},t_{1})}) dV_{\\mathbf{B}} = t_{1} \\int (- V_{\\mathbf{B}} + \\frac{V_{\\mathbf{B}}^{t_{1}}}{V_{\\mathbf{B}}}) dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True))))"], [["minus", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 3, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Mul(Symbol('t_1', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('t_1', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given n{(P_{e})} = e^{P_{e}} and \\hat{X}{(P_{e})} = n{(P_{e})} n^{P_{e}}{(P_{e})} - (e^{P_{e}})^{P_{e}}, then obtain \\hat{X}{(P_{e})} = e^{P_{e}} (e^{P_{e}})^{P_{e}} - (e^{P_{e}})^{P_{e}}", "derivation": "n{(P_{e})} = e^{P_{e}} and n^{P_{e}}{(P_{e})} = (e^{P_{e}})^{P_{e}} and n{(P_{e})} n^{P_{e}}{(P_{e})} = n^{P_{e}}{(P_{e})} e^{P_{e}} and n{(P_{e})} n^{P_{e}}{(P_{e})} - (e^{P_{e}})^{P_{e}} = n^{P_{e}}{(P_{e})} e^{P_{e}} - (e^{P_{e}})^{P_{e}} and \\hat{X}{(P_{e})} = n{(P_{e})} n^{P_{e}}{(P_{e})} - (e^{P_{e}})^{P_{e}} and \\hat{X}{(P_{e})} = n^{P_{e}}{(P_{e})} e^{P_{e}} - (e^{P_{e}})^{P_{e}} and \\hat{X}{(P_{e})} = e^{P_{e}} (e^{P_{e}})^{P_{e}} - (e^{P_{e}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))"], [["times", 1, "Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], "Equality(Mul(Function('n')(Symbol('P_e', commutative=True)), Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))))"], [["minus", 3, "Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Function('n')(Symbol('P_e', commutative=True)), Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))), Add(Mul(Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('P_e', commutative=True)), Add(Mul(Function('n')(Symbol('P_e', commutative=True)), Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\hat{X}')(Symbol('P_e', commutative=True)), Add(Mul(Pow(Function('n')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Function('\\\\hat{X}')(Symbol('P_e', commutative=True)), Add(Mul(exp(Symbol('P_e', commutative=True)), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given C{(\\mathbf{B})} = \\log{(e^{\\mathbf{B}})}, then obtain \\sin{((\\frac{d^{2}}{d \\mathbf{B}^{2}} C{(\\mathbf{B})})^{\\mathbf{B}})} = \\sin{((\\frac{d^{2}}{d \\mathbf{B}^{2}} \\log{(e^{\\mathbf{B}})})^{\\mathbf{B}})}", "derivation": "C{(\\mathbf{B})} = \\log{(e^{\\mathbf{B}})} and \\frac{d}{d \\mathbf{B}} C{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(e^{\\mathbf{B}})} and \\frac{d^{2}}{d \\mathbf{B}^{2}} C{(\\mathbf{B})} = \\frac{d^{2}}{d \\mathbf{B}^{2}} \\log{(e^{\\mathbf{B}})} and (\\frac{d^{2}}{d \\mathbf{B}^{2}} C{(\\mathbf{B})})^{\\mathbf{B}} = (\\frac{d^{2}}{d \\mathbf{B}^{2}} \\log{(e^{\\mathbf{B}})})^{\\mathbf{B}} and \\sin{((\\frac{d^{2}}{d \\mathbf{B}^{2}} C{(\\mathbf{B})})^{\\mathbf{B}})} = \\sin{((\\frac{d^{2}}{d \\mathbf{B}^{2}} \\log{(e^{\\mathbf{B}})})^{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mathbf{B}', commutative=True)), log(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Derivative(log(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Derivative(Function('C')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Derivative(log(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Derivative(Function('C')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Symbol('\\\\mathbf{B}', commutative=True))), sin(Pow(Derivative(log(exp(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))), Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(v_{z},\\Psi_{nl})} = \\cos{(\\Psi_{nl} v_{z})}, then obtain \\int \\mathbf{S}^{2}{(v_{z},\\Psi_{nl})} \\cos{(\\Psi_{nl} v_{z})} d\\Psi_{nl} = \\int \\mathbf{S}{(v_{z},\\Psi_{nl})} \\cos^{2}{(\\Psi_{nl} v_{z})} d\\Psi_{nl}", "derivation": "\\mathbf{S}{(v_{z},\\Psi_{nl})} = \\cos{(\\Psi_{nl} v_{z})} and - \\mathbf{S}{(v_{z},\\Psi_{nl})} = - \\cos{(\\Psi_{nl} v_{z})} and - \\mathbf{S}{(v_{z},\\Psi_{nl})} \\cos{(\\Psi_{nl} v_{z})} = - \\cos^{2}{(\\Psi_{nl} v_{z})} and \\mathbf{S}^{2}{(v_{z},\\Psi_{nl})} \\cos{(\\Psi_{nl} v_{z})} = \\mathbf{S}{(v_{z},\\Psi_{nl})} \\cos^{2}{(\\Psi_{nl} v_{z})} and \\int \\mathbf{S}^{2}{(v_{z},\\Psi_{nl})} \\cos{(\\Psi_{nl} v_{z})} d\\Psi_{nl} = \\int \\mathbf{S}{(v_{z},\\Psi_{nl})} \\cos^{2}{(\\Psi_{nl} v_{z})} d\\Psi_{nl}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Mul(Integer(-1), cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True)))))"], [["times", 2, "cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True)))), Mul(Integer(-1), Pow(cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True))), Integer(2))))"], [["times", 3, "Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True)))), Mul(Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True))), Integer(2))))"], [["integrate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(2)), cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Function('\\\\mathbf{S}')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(cos(Mul(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('v_z', commutative=True))), Integer(2))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"]]}, {"prompt": "Given E{(M_{E})} = \\sin{(M_{E})}, then obtain - M_{E} + \\frac{d}{d M_{E}} E{(M_{E})} = - M_{E} + \\cos{(M_{E})}", "derivation": "E{(M_{E})} = \\sin{(M_{E})} and \\frac{d}{d M_{E}} E{(M_{E})} = \\frac{d}{d M_{E}} \\sin{(M_{E})} and - M_{E} + \\frac{d}{d M_{E}} E{(M_{E})} = - M_{E} + \\frac{d}{d M_{E}} \\sin{(M_{E})} and - M_{E} + \\frac{d}{d M_{E}} E{(M_{E})} = - M_{E} + \\cos{(M_{E})}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('M_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(Function('E')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(sin(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Derivative(Function('E')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given i{(M)} = \\cos{(M)}, then derive (\\int i{(M)} dM)^{M} = (F_{c} + \\sin{(M)})^{M}, then obtain \\frac{(\\mathbf{J}_f + \\sin{(M)})^{M}}{i{(M)}} = \\frac{(F_{c} + \\sin{(M)})^{M}}{i{(M)}}", "derivation": "i{(M)} = \\cos{(M)} and \\int i{(M)} dM = \\int \\cos{(M)} dM and (\\int i{(M)} dM)^{M} = (\\int \\cos{(M)} dM)^{M} and (\\int i{(M)} dM)^{M} = (F_{c} + \\sin{(M)})^{M} and \\frac{(\\int i{(M)} dM)^{M}}{i{(M)}} = \\frac{(\\int \\cos{(M)} dM)^{M}}{i{(M)}} and (\\int \\cos{(M)} dM)^{M} = (F_{c} + \\sin{(M)})^{M} and \\frac{(\\int i{(M)} dM)^{M}}{i{(M)}} = \\frac{(F_{c} + \\sin{(M)})^{M}}{i{(M)}} and \\frac{(\\int \\cos{(M)} dM)^{M}}{i{(M)}} = \\frac{(F_{c} + \\sin{(M)})^{M}}{i{(M)}} and \\frac{(\\mathbf{J}_f + \\sin{(M)})^{M}}{i{(M)}} = \\frac{(F_{c} + \\sin{(M)})^{M}}{i{(M)}}", "srepr_derivation": [["get_premise", "Equality(Function('i')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('i')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Integral(Function('i')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('i')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["divide", 3, "Function('i')(Symbol('M', commutative=True))"], "Equality(Mul(Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1)), Pow(Integral(Function('i')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Mul(Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1)), Pow(Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1)), Pow(Integral(Function('i')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Mul(Pow(Add(Symbol('F_c', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1)), Pow(Integral(cos(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('M', commutative=True))), Mul(Pow(Add(Symbol('F_c', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1))))"], [["evaluate_integrals", 8], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1))), Mul(Pow(Add(Symbol('F_c', commutative=True), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Function('i')(Symbol('M', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mu,i)} = - \\mu + i, then obtain (- \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{M}{(\\mu,i)}) \\frac{\\partial}{\\partial i} (- \\mu + i) = (- \\mu + \\frac{\\partial}{\\partial \\mu} (- \\mu + i)) \\frac{\\partial}{\\partial i} (- \\mu + i)", "derivation": "\\mathbf{M}{(\\mu,i)} = - \\mu + i and \\frac{\\partial}{\\partial \\mu} \\mathbf{M}{(\\mu,i)} = \\frac{\\partial}{\\partial \\mu} (- \\mu + i) and - \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{M}{(\\mu,i)} = - \\mu + \\frac{\\partial}{\\partial \\mu} (- \\mu + i) and (- \\mu + \\frac{\\partial}{\\partial \\mu} \\mathbf{M}{(\\mu,i)}) \\frac{\\partial}{\\partial i} (- \\mu + i) = (- \\mu + \\frac{\\partial}{\\partial \\mu} (- \\mu + i)) \\frac{\\partial}{\\partial i} (- \\mu + i)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["times", 3, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\mu', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(H)} = \\sin{(\\cos{(H)})}, then obtain \\frac{d}{d H} (\\operatorname{C_{2}}^{H}{(H)} - 1) = \\frac{d}{d H} (\\sin^{H}{(\\cos{(H)})} - 1)", "derivation": "\\operatorname{C_{2}}{(H)} = \\sin{(\\cos{(H)})} and \\operatorname{C_{2}}^{H}{(H)} = \\sin^{H}{(\\cos{(H)})} and \\operatorname{C_{2}}^{H}{(H)} - 1 = \\sin^{H}{(\\cos{(H)})} - 1 and \\frac{d}{d H} (\\operatorname{C_{2}}^{H}{(H)} - 1) = \\frac{d}{d H} (\\sin^{H}{(\\cos{(H)})} - 1)", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('H', commutative=True)), sin(cos(Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(sin(cos(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Pow(Function('C_2')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Add(Pow(sin(cos(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Pow(Function('C_2')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Integer(-1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Pow(sin(cos(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Integer(-1)), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(E)} = e^{E}, then obtain - e^{E} + \\int 2 \\hat{\\mathbf{x}}{(E)} dE = - e^{E} + \\int (\\hat{\\mathbf{x}}{(E)} + e^{E}) dE", "derivation": "\\hat{\\mathbf{x}}{(E)} = e^{E} and 2 \\hat{\\mathbf{x}}{(E)} = \\hat{\\mathbf{x}}{(E)} + e^{E} and \\int 2 \\hat{\\mathbf{x}}{(E)} dE = \\int (\\hat{\\mathbf{x}}{(E)} + e^{E}) dE and - e^{E} = - \\hat{\\mathbf{x}}{(E)} and - \\hat{\\mathbf{x}}{(E)} + \\int 2 \\hat{\\mathbf{x}}{(E)} dE = - \\hat{\\mathbf{x}}{(E)} + \\int (\\hat{\\mathbf{x}}{(E)} + e^{E}) dE and - e^{E} + \\int 2 \\hat{\\mathbf{x}}{(E)} dE = - e^{E} + \\int (\\hat{\\mathbf{x}}{(E)} + e^{E}) dE", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["minus", 1, "Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Symbol('E', commutative=True))), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Integral(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('E', commutative=True))), Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('E', commutative=True))), Integral(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{v})} = \\log{(\\mathbf{v})}, then obtain - \\mathbf{v} \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} + 1 = - \\mathbf{v} \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} + \\frac{\\log{(\\mathbf{v})}}{\\lambda{(\\mathbf{v})}}", "derivation": "\\lambda{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} = \\log{(\\mathbf{v})}^{2} and \\mathbf{v} \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} = \\mathbf{v} \\log{(\\mathbf{v})}^{2} and 1 = \\frac{\\log{(\\mathbf{v})}}{\\lambda{(\\mathbf{v})}} and - \\mathbf{v} \\log{(\\mathbf{v})}^{2} + 1 = - \\mathbf{v} \\log{(\\mathbf{v})}^{2} + \\frac{\\log{(\\mathbf{v})}}{\\lambda{(\\mathbf{v})}} and - \\mathbf{v} \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} + 1 = - \\mathbf{v} \\lambda{(\\mathbf{v})} \\log{(\\mathbf{v})} + \\frac{\\log{(\\mathbf{v})}}{\\lambda{(\\mathbf{v})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 1, "log(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))))"], [["divide", 3, "Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Function('\\\\lambda')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given S{(a,n)} = e^{a - n}, then derive \\frac{a S^{a}{(a,n)} \\frac{\\partial}{\\partial n} S{(a,n)}}{S{(a,n)}} = - a (e^{a} e^{- n})^{a}, then obtain \\int - a S^{a}{(a,n)} dn = \\int - a (e^{a} e^{- n})^{a} dn", "derivation": "S{(a,n)} = e^{a - n} and S^{a}{(a,n)} = (e^{a - n})^{a} and S^{a}{(a,n)} = (e^{a} e^{- n})^{a} and \\frac{\\partial}{\\partial n} S^{a}{(a,n)} = \\frac{\\partial}{\\partial n} (e^{a} e^{- n})^{a} and \\frac{a S^{a}{(a,n)} \\frac{\\partial}{\\partial n} S{(a,n)}}{S{(a,n)}} = - a (e^{a} e^{- n})^{a} and \\frac{a S^{a}{(a,n)} \\frac{\\partial}{\\partial n} S{(a,n)}}{S{(a,n)}} = - a S^{a}{(a,n)} and - a S^{a}{(a,n)} = - a (e^{a} e^{- n})^{a} and \\int - a S^{a}{(a,n)} dn = \\int - a (e^{a} e^{- n})^{a} dn", "srepr_derivation": [["get_premise", "Equality(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), exp(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True)), Pow(exp(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True)))"], [["expand", 2], "Equality(Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(exp(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Mul(exp(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True)), Derivative(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('a', commutative=True), Pow(Mul(exp(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True)), Derivative(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('a', commutative=True), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True))), Mul(Integer(-1), Symbol('a', commutative=True), Pow(Mul(exp(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True))))"], [["integrate", 7, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Function('S')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Symbol('a', commutative=True))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Integer(-1), Symbol('a', commutative=True), Pow(Mul(exp(Symbol('a', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Symbol('a', commutative=True))), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(z,Q)} = Q^{z} and \\hat{\\mathbf{r}}{(z,Q)} = (Q^{z})^{z}, then obtain \\frac{\\partial}{\\partial Q} - \\hat{\\mathbf{r}}{(z,Q)} = \\frac{\\partial}{\\partial Q} - \\operatorname{C_{1}}^{z}{(z,Q)}", "derivation": "\\operatorname{C_{1}}{(z,Q)} = Q^{z} and \\operatorname{C_{1}}^{z}{(z,Q)} = (Q^{z})^{z} and \\hat{\\mathbf{r}}{(z,Q)} = (Q^{z})^{z} and \\hat{\\mathbf{r}}{(z,Q)} = \\operatorname{C_{1}}^{z}{(z,Q)} and - \\hat{\\mathbf{r}}{(z,Q)} = - \\operatorname{C_{1}}^{z}{(z,Q)} and \\frac{\\partial}{\\partial Q} - \\hat{\\mathbf{r}}{(z,Q)} = \\frac{\\partial}{\\partial Q} - \\operatorname{C_{1}}^{z}{(z,Q)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('C_1')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Symbol('Q', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Pow(Pow(Symbol('Q', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Pow(Function('C_1')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Symbol('z', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Function('C_1')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Symbol('z', commutative=True))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Function('C_1')(Symbol('z', commutative=True), Symbol('Q', commutative=True)), Symbol('z', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(\\pi)} = \\pi, then obtain - P_{e} - \\pi T{(\\pi)} + \\frac{T^{2}{(\\pi)}}{2} - \\sin{(P_{e} + \\pi T{(\\pi)} - \\frac{T^{2}{(\\pi)}}{2} - \\int 0 dT{(\\pi)})} + \\int 0 dT{(\\pi)} = - P_{e} - \\pi T{(\\pi)} + \\frac{T^{2}{(\\pi)}}{2} + \\int 0 dT{(\\pi)}", "derivation": "T{(\\pi)} = \\pi and 0 = \\pi - T{(\\pi)} and \\int 0 d\\pi = \\int (\\pi - T{(\\pi)}) d\\pi and \\int 0 d\\pi - \\int (\\pi - T{(\\pi)}) d\\pi = 0 and \\sin{(\\int 0 d\\pi - \\int (\\pi - T{(\\pi)}) d\\pi)} = 0 and \\sin{(\\int 0 dT{(\\pi)} - \\int (\\pi - T{(\\pi)}) dT{(\\pi)})} = 0 and \\sin{(\\int 0 dT{(\\pi)} - \\int (\\pi - T{(\\pi)}) dT{(\\pi)})} + \\int 0 dT{(\\pi)} - \\int (\\pi - T{(\\pi)}) dT{(\\pi)} = \\int 0 dT{(\\pi)} - \\int (\\pi - T{(\\pi)}) dT{(\\pi)} and - P_{e} - \\pi T{(\\pi)} + \\frac{T^{2}{(\\pi)}}{2} - \\sin{(P_{e} + \\pi T{(\\pi)} - \\frac{T^{2}{(\\pi)}}{2} - \\int 0 dT{(\\pi)})} + \\int 0 dT{(\\pi)} = - P_{e} - \\pi T{(\\pi)} + \\frac{T^{2}{(\\pi)}}{2} + \\int 0 dT{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["minus", 1, "Function('T')(Symbol('\\\\pi', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["minus", 3, "Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))), Integer(0))"], [["sin", 4], "Equality(sin(Add(Integral(Integer(0), Tuple(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(sin(Add(Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Function('T')(Symbol('\\\\pi', commutative=True))))))), Integer(0))"], [["add", 6, "Add(Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Function('T')(Symbol('\\\\pi', commutative=True))))))"], "Equality(Add(sin(Add(Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Function('T')(Symbol('\\\\pi', commutative=True))))))), Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))))), Add(Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Function('T')(Symbol('\\\\pi', commutative=True)))), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))))))"], [["evaluate_integrals", 7], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('T')(Symbol('\\\\pi', commutative=True))), Mul(Rational(1, 2), Pow(Function('T')(Symbol('\\\\pi', commutative=True)), Integer(2))), Mul(Integer(-1), sin(Add(Symbol('P_e', commutative=True), Mul(Symbol('\\\\pi', commutative=True), Function('T')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Function('T')(Symbol('\\\\pi', commutative=True)), Integer(2))), Mul(Integer(-1), Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True)))))))), Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True))))), Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('T')(Symbol('\\\\pi', commutative=True))), Mul(Rational(1, 2), Pow(Function('T')(Symbol('\\\\pi', commutative=True)), Integer(2))), Integral(Integer(0), Tuple(Function('T')(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given W{(\\lambda,\\hbar)} = - \\hbar + \\lambda, then obtain ((\\int W{(\\lambda,\\hbar)} d\\hbar)^{\\hbar})^{\\lambda} = ((- \\frac{\\hbar^{2}}{2} + \\hbar \\lambda + f^{\\prime})^{\\hbar})^{\\lambda}", "derivation": "W{(\\lambda,\\hbar)} = - \\hbar + \\lambda and \\int W{(\\lambda,\\hbar)} d\\hbar = \\int (- \\hbar + \\lambda) d\\hbar and (\\int W{(\\lambda,\\hbar)} d\\hbar)^{\\hbar} = (\\int (- \\hbar + \\lambda) d\\hbar)^{\\hbar} and ((\\int W{(\\lambda,\\hbar)} d\\hbar)^{\\hbar})^{\\lambda} = ((\\int (- \\hbar + \\lambda) d\\hbar)^{\\hbar})^{\\lambda} and ((\\int W{(\\lambda,\\hbar)} d\\hbar)^{\\hbar})^{\\lambda} = ((- \\frac{\\hbar^{2}}{2} + \\hbar \\lambda + f^{\\prime})^{\\hbar})^{\\lambda}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["power", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Pow(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Pow(Integral(Function('W')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\hbar', commutative=True), Integer(2))), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{p},A_{2})} = A_{2} + \\hat{p}, then derive \\frac{\\partial}{\\partial A_{2}} \\operatorname{z^{*}}{(\\hat{p},A_{2})} = 1, then obtain \\frac{\\partial}{\\partial A_{2}} (A_{2} + \\hat{p}) = 1", "derivation": "\\operatorname{z^{*}}{(\\hat{p},A_{2})} = A_{2} + \\hat{p} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{z^{*}}{(\\hat{p},A_{2})} = \\frac{\\partial}{\\partial A_{2}} (A_{2} + \\hat{p}) and \\frac{\\partial}{\\partial A_{2}} \\operatorname{z^{*}}{(\\hat{p},A_{2})} = 1 and \\frac{\\partial}{\\partial A_{2}} (A_{2} + \\hat{p}) = 1", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\hat{p}', commutative=True), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('A_2', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then derive \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)}, then obtain (\\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)}) \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} = (\\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)}) \\cos{(\\mathbf{J}_M)}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)} = \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} = \\cos{(\\mathbf{J}_M)} and (\\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)}) \\frac{d}{d \\mathbf{J}_M} \\sin{(\\mathbf{J}_M)} = (\\sin{(\\mathbf{J}_M)} + \\frac{d}{d \\mathbf{J}_M} \\operatorname{A_{y}}{(\\mathbf{J}_M)}) \\cos{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 4, "Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], "Equality(Mul(Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Derivative(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Mul(Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('A_y')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(I,F_{H})} = \\frac{\\partial}{\\partial I} (F_{H} + I), then derive \\dot{z}{(I,F_{H})} = 1, then obtain \\frac{\\cos{(\\frac{\\partial}{\\partial I} (F_{H} + I))}}{I^{2}} = \\frac{\\cos{(1)}}{I^{2}}", "derivation": "\\dot{z}{(I,F_{H})} = \\frac{\\partial}{\\partial I} (F_{H} + I) and \\dot{z}{(I,F_{H})} = 1 and \\cos{(\\dot{z}{(I,F_{H})})} = \\cos{(1)} and \\frac{\\cos{(\\dot{z}{(I,F_{H})})}}{I} = \\frac{\\cos{(1)}}{I} and \\frac{\\cos{(\\frac{\\partial}{\\partial I} (F_{H} + I))}}{I} = \\frac{\\cos{(1)}}{I} and \\frac{\\cos{(\\frac{\\partial}{\\partial I} (F_{H} + I))}}{I} = \\frac{\\cos{(\\dot{z}{(I,F_{H})})}}{I} and \\frac{\\cos{(\\frac{\\partial}{\\partial I} (F_{H} + I))}}{I^{2}} = \\frac{\\cos{(\\dot{z}{(I,F_{H})})}}{I^{2}} and \\frac{\\cos{(\\frac{\\partial}{\\partial I} (F_{H} + I))}}{I^{2}} = \\frac{\\cos{(1)}}{I^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Derivative(Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)), Integer(1))"], [["cos", 2], "Equality(cos(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True))), cos(Integer(1)))"], [["divide", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Derivative(Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Derivative(Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)))))"], [["times", 6, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-2)), cos(Derivative(Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), cos(Function('\\\\dot{z}')(Symbol('I', commutative=True), Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-2)), cos(Derivative(Add(Symbol('F_H', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))), Mul(Pow(Symbol('I', commutative=True), Integer(-2)), cos(Integer(1))))"]]}, {"prompt": "Given q{(\\pi,\\rho_b,v_{t})} = \\pi \\rho_b v_{t}, then derive -1 + \\frac{\\frac{\\partial}{\\partial \\pi} q{(\\pi,\\rho_b,v_{t})}}{\\pi \\rho_b v_{t}} - \\frac{q{(\\pi,\\rho_b,v_{t})}}{\\pi^{2} \\rho_b v_{t}} = -1, then obtain -1 - \\frac{1}{\\pi} + \\frac{\\frac{\\partial}{\\partial \\pi} \\pi \\rho_b v_{t}}{\\pi \\rho_b v_{t}} = -1", "derivation": "q{(\\pi,\\rho_b,v_{t})} = \\pi \\rho_b v_{t} and \\frac{q{(\\pi,\\rho_b,v_{t})}}{\\pi \\rho_b v_{t}} = 1 and \\frac{\\partial}{\\partial \\pi} \\frac{q{(\\pi,\\rho_b,v_{t})}}{\\pi \\rho_b v_{t}} = \\frac{d}{d \\pi} 1 and \\frac{\\partial}{\\partial \\pi} \\frac{q{(\\pi,\\rho_b,v_{t})}}{\\pi \\rho_b v_{t}} - 1 = \\frac{d}{d \\pi} 1 - 1 and -1 + \\frac{\\frac{\\partial}{\\partial \\pi} q{(\\pi,\\rho_b,v_{t})}}{\\pi \\rho_b v_{t}} - \\frac{q{(\\pi,\\rho_b,v_{t})}}{\\pi^{2} \\rho_b v_{t}} = -1 and -1 - \\frac{1}{\\pi} + \\frac{\\frac{\\partial}{\\partial \\pi} \\pi \\rho_b v_{t}}{\\pi \\rho_b v_{t}} = -1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integer(1), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Integer(-1), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Derivative(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)))), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\pi', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(h,\\mathbf{H})} = \\mathbf{H} h, then obtain - \\mathbf{H} h + (\\int \\operatorname{m_{s}}{(h,\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = - \\mathbf{H} h + (\\int \\mathbf{H} h d\\mathbf{H})^{\\mathbf{H}}", "derivation": "\\operatorname{m_{s}}{(h,\\mathbf{H})} = \\mathbf{H} h and \\int \\operatorname{m_{s}}{(h,\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H} h d\\mathbf{H} and (\\int \\operatorname{m_{s}}{(h,\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = (\\int \\mathbf{H} h d\\mathbf{H})^{\\mathbf{H}} and - \\mathbf{H} h + (\\int \\operatorname{m_{s}}{(h,\\mathbf{H})} d\\mathbf{H})^{\\mathbf{H}} = - \\mathbf{H} h + (\\int \\mathbf{H} h d\\mathbf{H})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('h', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 3, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)), Pow(Integral(Function('m_s')(Symbol('h', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}})} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}})}, then obtain \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} - \\cos{(\\hat{\\mathbf{x}})} = 0", "derivation": "\\varphi^{*}{(\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}})} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} = \\cos{(\\hat{\\mathbf{x}})} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} - \\varphi^{*}{(\\hat{\\mathbf{x}})} = - \\varphi^{*}{(\\hat{\\mathbf{x}})} + \\cos{(\\hat{\\mathbf{x}})} and \\operatorname{V_{\\mathbf{E}}}{(\\hat{\\mathbf{x}})} - \\cos{(\\hat{\\mathbf{x}})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["minus", 2, "Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(F_{x})} = \\cos{(F_{x})}, then obtain (\\cos^{2}{(F_{x})})^{F_{x}} = (\\frac{\\cos^{4}{(F_{x})}}{\\operatorname{m_{s}}^{2}{(F_{x})}})^{F_{x}}", "derivation": "\\operatorname{m_{s}}{(F_{x})} = \\cos{(F_{x})} and \\operatorname{m_{s}}{(F_{x})} \\cos{(F_{x})} = \\cos^{2}{(F_{x})} and (\\operatorname{m_{s}}{(F_{x})} \\cos{(F_{x})})^{F_{x}} = (\\cos^{2}{(F_{x})})^{F_{x}} and \\cos{(F_{x})} = \\frac{\\cos^{2}{(F_{x})}}{\\operatorname{m_{s}}{(F_{x})}} and (\\cos^{2}{(F_{x})})^{F_{x}} = (\\frac{\\cos^{4}{(F_{x})}}{\\operatorname{m_{s}}^{2}{(F_{x})}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["times", 1, "cos(Symbol('F_x', commutative=True))"], "Equality(Mul(Function('m_s')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Pow(cos(Symbol('F_x', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Mul(Function('m_s')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Pow(cos(Symbol('F_x', commutative=True)), Integer(2)), Symbol('F_x', commutative=True)))"], [["divide", 2, "Function('m_s')(Symbol('F_x', commutative=True))"], "Equality(cos(Symbol('F_x', commutative=True)), Mul(Pow(Function('m_s')(Symbol('F_x', commutative=True)), Integer(-1)), Pow(cos(Symbol('F_x', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Pow(cos(Symbol('F_x', commutative=True)), Integer(2)), Symbol('F_x', commutative=True)), Pow(Mul(Pow(Function('m_s')(Symbol('F_x', commutative=True)), Integer(-2)), Pow(cos(Symbol('F_x', commutative=True)), Integer(4))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\omega{(E_{n},M)} = \\frac{E_{n}}{M} and \\eta^{\\prime}{(E_{n},M)} = \\frac{E_{n}}{M}, then obtain (E_{n} + \\eta^{\\prime}{(E_{n},M)} + 1) \\omega{(E_{n},M)} = (E_{n} + \\omega{(E_{n},M)} + 1) \\omega{(E_{n},M)}", "derivation": "\\omega{(E_{n},M)} = \\frac{E_{n}}{M} and \\omega{(E_{n},M)} + 1 = \\frac{E_{n}}{M} + 1 and \\eta^{\\prime}{(E_{n},M)} = \\frac{E_{n}}{M} and \\omega{(E_{n},M)} = \\eta^{\\prime}{(E_{n},M)} and E_{n} + \\omega{(E_{n},M)} + 1 = E_{n} + \\frac{E_{n}}{M} + 1 and E_{n} + \\eta^{\\prime}{(E_{n},M)} + 1 = E_{n} + \\frac{E_{n}}{M} + 1 and E_{n} + \\eta^{\\prime}{(E_{n},M)} + 1 = E_{n} + \\omega{(E_{n},M)} + 1 and (E_{n} + \\eta^{\\prime}{(E_{n},M)} + 1) \\omega{(E_{n},M)} = (E_{n} + \\omega{(E_{n},M)} + 1) \\omega{(E_{n},M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)))"], [["add", 2, "Symbol('E_n', commutative=True)"], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Add(Symbol('E_n', commutative=True), Mul(Symbol('E_n', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Add(Symbol('E_n', commutative=True), Mul(Symbol('E_n', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Integer(1)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('E_n', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Add(Symbol('E_n', commutative=True), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)))"], [["times", 7, "Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Add(Symbol('E_n', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True))), Mul(Add(Symbol('E_n', commutative=True), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True)), Integer(1)), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(v_{x},\\Psi^{\\dagger})} = \\Psi^{\\dagger} + v_{x}, then obtain (\\Psi^{\\dagger} + v_{x})^{v_{x}} \\int \\mathbf{B}{(v_{x},\\Psi^{\\dagger})} dv_{x} = (\\Psi^{\\dagger} + v_{x})^{v_{x}} \\int (\\Psi^{\\dagger} + v_{x}) dv_{x}", "derivation": "\\mathbf{B}{(v_{x},\\Psi^{\\dagger})} = \\Psi^{\\dagger} + v_{x} and \\int \\mathbf{B}{(v_{x},\\Psi^{\\dagger})} dv_{x} = \\int (\\Psi^{\\dagger} + v_{x}) dv_{x} and \\mathbf{B}^{v_{x}}{(v_{x},\\Psi^{\\dagger})} = (\\Psi^{\\dagger} + v_{x})^{v_{x}} and \\mathbf{B}^{v_{x}}{(v_{x},\\Psi^{\\dagger})} \\int \\mathbf{B}{(v_{x},\\Psi^{\\dagger})} dv_{x} = \\mathbf{B}^{v_{x}}{(v_{x},\\Psi^{\\dagger})} \\int (\\Psi^{\\dagger} + v_{x}) dv_{x} and (\\Psi^{\\dagger} + v_{x})^{v_{x}} \\int \\mathbf{B}{(v_{x},\\Psi^{\\dagger})} dv_{x} = (\\Psi^{\\dagger} + v_{x})^{v_{x}} \\int (\\Psi^{\\dagger} + v_{x}) dv_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_x', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["times", 2, "Pow(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_x', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('v_x', commutative=True)), Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Integral(Function('\\\\mathbf{B}')(Symbol('v_x', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Mul(Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Integral(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{p}_0)} = e^{\\hat{p}_0}, then obtain (\\frac{d}{d \\hat{p}_0} \\int \\mathbf{r}{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{H} + e^{\\hat{p}_0}))^{\\hat{p}_0}", "derivation": "\\mathbf{r}{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\int \\mathbf{r}{(\\hat{p}_0)} d\\hat{p}_0 = \\int e^{\\hat{p}_0} d\\hat{p}_0 and \\frac{d}{d \\hat{p}_0} \\int \\mathbf{r}{(\\hat{p}_0)} d\\hat{p}_0 = \\frac{d}{d \\hat{p}_0} \\int e^{\\hat{p}_0} d\\hat{p}_0 and (\\frac{d}{d \\hat{p}_0} \\int \\mathbf{r}{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\frac{d}{d \\hat{p}_0} \\int e^{\\hat{p}_0} d\\hat{p}_0)^{\\hat{p}_0} and (\\frac{d}{d \\hat{p}_0} \\int \\mathbf{r}{(\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{H} + e^{\\hat{p}_0}))^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Integral(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given y{(\\psi,A_{z})} = e^{- A_{z} + \\psi}, then derive \\int y{(\\psi,A_{z})} d\\psi = n_{2} + e^{- A_{z} + \\psi}, then obtain \\frac{n_{2} + y{(\\psi,A_{z})}}{y{(\\psi,A_{z})}} = \\frac{\\int y{(\\psi,A_{z})} d\\psi}{y{(\\psi,A_{z})}}", "derivation": "y{(\\psi,A_{z})} = e^{- A_{z} + \\psi} and \\int y{(\\psi,A_{z})} d\\psi = \\int e^{- A_{z} + \\psi} d\\psi and \\int y{(\\psi,A_{z})} d\\psi = n_{2} + e^{- A_{z} + \\psi} and n_{2} + e^{- A_{z} + \\psi} = \\int e^{- A_{z} + \\psi} d\\psi and n_{2} + y{(\\psi,A_{z})} = \\int e^{- A_{z} + \\psi} d\\psi and n_{2} + y{(\\psi,A_{z})} = \\int y{(\\psi,A_{z})} d\\psi and \\frac{n_{2} + y{(\\psi,A_{z})}}{y{(\\psi,A_{z})}} = \\frac{\\int y{(\\psi,A_{z})} d\\psi}{y{(\\psi,A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Add(Symbol('n_2', commutative=True), exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('n_2', commutative=True), exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True)))), Integral(exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('n_2', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Integral(exp(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('n_2', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Integral(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 6, "Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Mul(Add(Symbol('n_2', commutative=True), Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True))), Pow(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))), Mul(Pow(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Integer(-1)), Integral(Function('y')(Symbol('\\\\psi', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given f{(J,\\mathbf{J}_M)} = J \\sin{(\\mathbf{J}_M)}, then derive 2 \\frac{\\partial}{\\partial J} f{(J,\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} + \\frac{\\partial}{\\partial J} f{(J,\\mathbf{J}_M)}, then obtain 2 \\frac{\\partial}{\\partial J} J \\sin{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} + \\frac{\\partial}{\\partial J} J \\sin{(\\mathbf{J}_M)}", "derivation": "f{(J,\\mathbf{J}_M)} = J \\sin{(\\mathbf{J}_M)} and 2 f{(J,\\mathbf{J}_M)} = J \\sin{(\\mathbf{J}_M)} + f{(J,\\mathbf{J}_M)} and \\frac{\\partial}{\\partial J} 2 f{(J,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial J} (J \\sin{(\\mathbf{J}_M)} + f{(J,\\mathbf{J}_M)}) and 2 \\frac{\\partial}{\\partial J} f{(J,\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} + \\frac{\\partial}{\\partial J} f{(J,\\mathbf{J}_M)} and 2 \\frac{\\partial}{\\partial J} J \\sin{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} + \\frac{\\partial}{\\partial J} J \\sin{(\\mathbf{J}_M)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('J', commutative=True), sin(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["add", 1, "Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Integer(2), Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Symbol('J', commutative=True), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('J', commutative=True), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('f')(Symbol('J', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Mul(Symbol('J', commutative=True), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Mul(Symbol('J', commutative=True), sin(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given J{(z)} = \\cos{(z)}, then obtain \\frac{d}{d z} (J{(z)} - 2 \\cos{(z)}) = \\frac{d}{d z} - \\cos{(z)}", "derivation": "J{(z)} = \\cos{(z)} and J{(z)} - \\cos{(z)} = 0 and J{(z)} - 2 \\cos{(z)} = - \\cos{(z)} and \\frac{d}{d z} (J{(z)} - 2 \\cos{(z)}) = \\frac{d}{d z} - \\cos{(z)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["minus", 1, "cos(Symbol('z', commutative=True))"], "Equality(Add(Function('J')(Symbol('z', commutative=True)), Mul(Integer(-1), cos(Symbol('z', commutative=True)))), Integer(0))"], [["add", 2, "Mul(Integer(-1), cos(Symbol('z', commutative=True)))"], "Equality(Add(Function('J')(Symbol('z', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True)))), Mul(Integer(-1), cos(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('z', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(\\phi_2,\\mu)} = - \\mu + \\phi_2 and \\varepsilon_{0}{(v_{t})} = \\cos{(v_{t})}, then obtain \\frac{\\varepsilon_{0}{(v_{t})}}{\\mu \\tilde{g}{(\\phi_2,\\mu)}} = \\frac{\\cos{(v_{t})}}{\\mu \\tilde{g}{(\\phi_2,\\mu)}}", "derivation": "\\tilde{g}{(\\phi_2,\\mu)} = - \\mu + \\phi_2 and \\mu \\tilde{g}{(\\phi_2,\\mu)} = \\mu (- \\mu + \\phi_2) and \\varepsilon_{0}{(v_{t})} = \\cos{(v_{t})} and \\frac{\\varepsilon_{0}{(v_{t})}}{\\mu (- \\mu + \\phi_2)} = \\frac{\\cos{(v_{t})}}{\\mu (- \\mu + \\phi_2)} and \\frac{\\varepsilon_{0}{(v_{t})}}{\\mu \\tilde{g}{(\\phi_2,\\mu)}} = \\frac{\\cos{(v_{t})}}{\\mu \\tilde{g}{(\\phi_2,\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["divide", 3, "Mul(Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Integer(-1)), cos(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('v_t', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), cos(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(s)} = e^{\\sin{(s)}}, then obtain 6 \\theta_{2}{(s)} + 3 e^{\\sin{(s)}} = 4 \\theta_{2}{(s)} + 5 e^{\\sin{(s)}}", "derivation": "\\theta_{2}{(s)} = e^{\\sin{(s)}} and \\theta_{2}{(s)} + e^{\\sin{(s)}} = 2 e^{\\sin{(s)}} and 2 \\theta_{2}{(s)} + e^{\\sin{(s)}} = \\theta_{2}{(s)} + 2 e^{\\sin{(s)}} and 3 \\theta_{2}{(s)} = 2 \\theta_{2}{(s)} + e^{\\sin{(s)}} and 3 \\theta_{2}{(s)} = \\theta_{2}{(s)} + 2 e^{\\sin{(s)}} and 4 \\theta_{2}{(s)} + 2 e^{\\sin{(s)}} = 2 \\theta_{2}{(s)} + 4 e^{\\sin{(s)}} and 6 \\theta_{2}{(s)} + 3 e^{\\sin{(s)}} = 4 \\theta_{2}{(s)} + 5 e^{\\sin{(s)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('s', commutative=True)), exp(sin(Symbol('s', commutative=True))))"], [["add", 1, "exp(sin(Symbol('s', commutative=True)))"], "Equality(Add(Function('\\\\theta_2')(Symbol('s', commutative=True)), exp(sin(Symbol('s', commutative=True)))), Mul(Integer(2), exp(sin(Symbol('s', commutative=True)))))"], [["add", 2, "Function('\\\\theta_2')(Symbol('s', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('s', commutative=True))), exp(sin(Symbol('s', commutative=True)))), Add(Function('\\\\theta_2')(Symbol('s', commutative=True)), Mul(Integer(2), exp(sin(Symbol('s', commutative=True))))))"], [["add", 1, "Mul(Integer(2), Function('\\\\theta_2')(Symbol('s', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\theta_2')(Symbol('s', commutative=True))), Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('s', commutative=True))), exp(sin(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(3), Function('\\\\theta_2')(Symbol('s', commutative=True))), Add(Function('\\\\theta_2')(Symbol('s', commutative=True)), Mul(Integer(2), exp(sin(Symbol('s', commutative=True))))))"], [["add", 5, "Add(Function('\\\\theta_2')(Symbol('s', commutative=True)), Mul(Integer(2), exp(sin(Symbol('s', commutative=True)))))"], "Equality(Add(Mul(Integer(4), Function('\\\\theta_2')(Symbol('s', commutative=True))), Mul(Integer(2), exp(sin(Symbol('s', commutative=True))))), Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('s', commutative=True))), Mul(Integer(4), exp(sin(Symbol('s', commutative=True))))))"], [["add", 6, "Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('s', commutative=True))), exp(sin(Symbol('s', commutative=True))))"], "Equality(Add(Mul(Integer(6), Function('\\\\theta_2')(Symbol('s', commutative=True))), Mul(Integer(3), exp(sin(Symbol('s', commutative=True))))), Add(Mul(Integer(4), Function('\\\\theta_2')(Symbol('s', commutative=True))), Mul(Integer(5), exp(sin(Symbol('s', commutative=True))))))"]]}, {"prompt": "Given \\omega{(f)} = \\log{(f)}, then obtain 0 = f (- \\frac{\\omega{(f)}}{\\int \\log{(f)} df} + \\frac{\\log{(f)}}{\\int \\log{(f)} df})", "derivation": "\\omega{(f)} = \\log{(f)} and \\int \\omega{(f)} df = \\int \\log{(f)} df and \\frac{\\omega{(f)}}{\\int \\omega{(f)} df} = \\frac{\\log{(f)}}{\\int \\omega{(f)} df} and 0 = - \\frac{\\omega{(f)}}{\\int \\omega{(f)} df} + \\frac{\\log{(f)}}{\\int \\omega{(f)} df} and 0 = - \\frac{\\omega{(f)}}{\\int \\log{(f)} df} + \\frac{\\log{(f)}}{\\int \\log{(f)} df} and 0 = f (- \\frac{\\omega{(f)}}{\\int \\log{(f)} df} + \\frac{\\log{(f)}}{\\int \\log{(f)} df})", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["divide", 1, "Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('f', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))), Mul(log(Symbol('f', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))))"], [["minus", 3, "Mul(Function('\\\\omega')(Symbol('f', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('f', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))), Mul(log(Symbol('f', commutative=True)), Pow(Integral(Function('\\\\omega')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('f', commutative=True)), Pow(Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))), Mul(log(Symbol('f', commutative=True)), Pow(Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1)))))"], [["times", 5, "Symbol('f', commutative=True)"], "Equality(Integer(0), Mul(Symbol('f', commutative=True), Add(Mul(Integer(-1), Function('\\\\omega')(Symbol('f', commutative=True)), Pow(Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))), Mul(log(Symbol('f', commutative=True)), Pow(Integral(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} = \\frac{A_{x} v}{A_{y}}, then obtain \\int \\frac{\\partial}{\\partial v} \\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} dA_{x} = \\frac{A_{x}^{2}}{2 A_{y}} + \\eta^{\\prime}", "derivation": "\\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} = \\frac{A_{x} v}{A_{y}} and \\frac{\\partial}{\\partial v} \\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} = \\frac{\\partial}{\\partial v} \\frac{A_{x} v}{A_{y}} and \\int \\frac{\\partial}{\\partial v} \\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} dA_{x} = \\int \\frac{\\partial}{\\partial v} \\frac{A_{x} v}{A_{y}} dA_{x} and \\int \\frac{\\partial}{\\partial v} \\operatorname{J_{\\varepsilon}}{(A_{x},A_{y},v)} dA_{x} = \\frac{A_{x}^{2}}{2 A_{y}} + \\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A_x', commutative=True)"], "Equality(Integral(Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Integral(Derivative(Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('J_{\\\\varepsilon}')(Symbol('A_x', commutative=True), Symbol('A_y', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2)), Pow(Symbol('A_y', commutative=True), Integer(-1))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(A_{2},i)} = A_{2}^{i}, then derive \\frac{\\partial}{\\partial A_{2}} \\operatorname{y^{\\prime}}{(A_{2},i)} = \\frac{A_{2}^{i} i}{A_{2}}, then obtain \\frac{A_{2}^{i} i}{A_{2}} = \\frac{\\partial}{\\partial A_{2}} A_{2}^{i}", "derivation": "\\operatorname{y^{\\prime}}{(A_{2},i)} = A_{2}^{i} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{y^{\\prime}}{(A_{2},i)} = \\frac{\\partial}{\\partial A_{2}} A_{2}^{i} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{y^{\\prime}}{(A_{2},i)} = \\frac{A_{2}^{i} i}{A_{2}} and \\frac{A_{2}^{i} i}{A_{2}} = \\frac{\\partial}{\\partial A_{2}} A_{2}^{i}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('A_2', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Symbol('i', commutative=True)), Derivative(Pow(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(t,J_{\\varepsilon})} = J_{\\varepsilon}^{t}, then obtain (J_{\\varepsilon} H{(t,J_{\\varepsilon})} - J_{\\varepsilon} + J_{\\varepsilon}^{t})^{t} = (J_{\\varepsilon} J_{\\varepsilon}^{t} - J_{\\varepsilon} + J_{\\varepsilon}^{t})^{t}", "derivation": "H{(t,J_{\\varepsilon})} = J_{\\varepsilon}^{t} and J_{\\varepsilon} H{(t,J_{\\varepsilon})} = J_{\\varepsilon} J_{\\varepsilon}^{t} and J_{\\varepsilon} H{(t,J_{\\varepsilon})} - J_{\\varepsilon} + J_{\\varepsilon}^{t} = J_{\\varepsilon} J_{\\varepsilon}^{t} - J_{\\varepsilon} + J_{\\varepsilon}^{t} and (J_{\\varepsilon} H{(t,J_{\\varepsilon})} - J_{\\varepsilon} + J_{\\varepsilon}^{t})^{t} = (J_{\\varepsilon} J_{\\varepsilon}^{t} - J_{\\varepsilon} + J_{\\varepsilon}^{t})^{t}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True)))"], [["times", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('H')(Symbol('t', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True)))"], "Equality(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('H')(Symbol('t', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('H')(Symbol('t', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\psi^*,t)} = \\psi^* - t and \\operatorname{F_{H}}{(h,k)} = h + k, then derive \\int \\eta{(\\psi^*,t)} dt = \\psi^* t + n - \\frac{t^{2}}{2}, then obtain \\operatorname{F_{H}}^{k}{(h,k)} \\iint \\eta{(\\psi^*,t)} dt d\\psi^* = \\operatorname{F_{H}}^{k}{(h,k)} \\int (\\psi^* t + n - \\frac{t^{2}}{2}) d\\psi^*", "derivation": "\\eta{(\\psi^*,t)} = \\psi^* - t and \\operatorname{F_{H}}{(h,k)} = h + k and \\operatorname{F_{H}}^{k}{(h,k)} = (h + k)^{k} and \\int \\eta{(\\psi^*,t)} dt = \\int (\\psi^* - t) dt and \\int \\eta{(\\psi^*,t)} dt = \\psi^* t + n - \\frac{t^{2}}{2} and \\iint \\eta{(\\psi^*,t)} dt d\\psi^* = \\int (\\psi^* t + n - \\frac{t^{2}}{2}) d\\psi^* and (h + k)^{k} \\iint \\eta{(\\psi^*,t)} dt d\\psi^* = (h + k)^{k} \\int (\\psi^* t + n - \\frac{t^{2}}{2}) d\\psi^* and \\operatorname{F_{H}}^{k}{(h,k)} \\iint \\eta{(\\psi^*,t)} dt d\\psi^* = \\operatorname{F_{H}}^{k}{(h,k)} \\int (\\psi^* t + n - \\frac{t^{2}}{2}) d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], ["get_premise", "Equality(Function('F_H')(Symbol('h', commutative=True), Symbol('k', commutative=True)), Add(Symbol('h', commutative=True), Symbol('k', commutative=True)))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["integrate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["times", 6, "Pow(Add(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Pow(Add(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Mul(Pow(Function('F_H')(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Pow(Function('F_H')(Symbol('h', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Add(Mul(Symbol('\\\\psi^*', commutative=True), Symbol('t', commutative=True)), Symbol('n', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})}, then derive \\psi{(\\mathbf{v})} - \\sin{(\\mathbf{v})} = - 2 \\sin{(\\mathbf{v})}, then obtain - \\sin{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} = - 2 \\sin{(\\mathbf{v})}", "derivation": "\\psi{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} and \\psi{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} = 2 \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} and \\psi{(\\mathbf{v})} - \\sin{(\\mathbf{v})} = - 2 \\sin{(\\mathbf{v})} and - \\sin{(\\mathbf{v})} + \\frac{d}{d \\mathbf{v}} \\cos{(\\mathbf{v})} = - 2 \\sin{(\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\psi')(Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{v}', commutative=True)))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{v}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given \\omega{(E_{n},B)} = \\sin{(B + E_{n})}, then obtain (\\int (- E_{n} + \\omega{(E_{n},B)}) dE_{n})^{B} = (\\int (- E_{n} + \\sin{(B + E_{n})}) dE_{n})^{B}", "derivation": "\\omega{(E_{n},B)} = \\sin{(B + E_{n})} and - E_{n} + \\omega{(E_{n},B)} = - E_{n} + \\sin{(B + E_{n})} and \\int (- E_{n} + \\omega{(E_{n},B)}) dE_{n} = \\int (- E_{n} + \\sin{(B + E_{n})}) dE_{n} and (\\int (- E_{n} + \\omega{(E_{n},B)}) dE_{n})^{B} = (\\int (- E_{n} + \\sin{(B + E_{n})}) dE_{n})^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('B', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True))))"], [["minus", 1, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\omega')(Symbol('E_n', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('E_n', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), sin(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\varphi{(\\rho)} = \\log{(\\rho)} and \\mathbf{f}{(V,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + e^{V} and \\mathbb{I}{(V,f_{\\mathbf{v}})} = \\mathbf{f}{(V,f_{\\mathbf{v}})} + 1, then obtain (- f_{\\mathbf{v}} + e^{V}) \\mathbb{I}{(V,f_{\\mathbf{v}})} = (- f_{\\mathbf{v}} + e^{V}) (- f_{\\mathbf{v}} + e^{V} + 1)", "derivation": "\\varphi{(\\rho)} = \\log{(\\rho)} and \\mathbf{f}{(V,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + e^{V} and \\mathbf{f}{(V,f_{\\mathbf{v}})} + \\frac{\\log{(\\rho)}}{\\varphi{(\\rho)}} = - f_{\\mathbf{v}} + e^{V} + \\frac{\\log{(\\rho)}}{\\varphi{(\\rho)}} and \\mathbf{f}{(V,f_{\\mathbf{v}})} + 1 = - f_{\\mathbf{v}} + e^{V} + 1 and \\mathbb{I}{(V,f_{\\mathbf{v}})} = \\mathbf{f}{(V,f_{\\mathbf{v}})} + 1 and \\mathbb{I}{(V,f_{\\mathbf{v}})} = - f_{\\mathbf{v}} + e^{V} + 1 and (- f_{\\mathbf{v}} + e^{V}) \\mathbb{I}{(V,f_{\\mathbf{v}})} = (- f_{\\mathbf{v}} + e^{V}) (- f_{\\mathbf{v}} + e^{V} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\rho', commutative=True)), log(Symbol('\\\\rho', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True))))"], [["add", 2, "Mul(Pow(Function('\\\\varphi')(Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Function('\\\\varphi')(Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Symbol('\\\\rho', commutative=True)))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True)), Mul(Pow(Function('\\\\varphi')(Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{f}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Function('\\\\mathbf{f}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True)), Integer(1)))"], [["times", 6, "Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True))), Function('\\\\mathbb{I}')(Symbol('V', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('V', commutative=True)), Integer(1))))"]]}, {"prompt": "Given s{(t_{1},m)} = m + \\sin{(t_{1})} and \\operatorname{v_{x}}{(t_{1},m)} = \\frac{\\partial}{\\partial m} s{(t_{1},m)}, then obtain 2 \\operatorname{v_{x}}{(t_{1},m)} = \\operatorname{v_{x}}{(t_{1},m)} + \\frac{\\partial}{\\partial m} (m + \\sin{(t_{1})})", "derivation": "s{(t_{1},m)} = m + \\sin{(t_{1})} and \\frac{\\partial}{\\partial m} s{(t_{1},m)} = \\frac{\\partial}{\\partial m} (m + \\sin{(t_{1})}) and 2 \\frac{\\partial}{\\partial m} s{(t_{1},m)} = \\frac{\\partial}{\\partial m} (m + \\sin{(t_{1})}) + \\frac{\\partial}{\\partial m} s{(t_{1},m)} and \\operatorname{v_{x}}{(t_{1},m)} = \\frac{\\partial}{\\partial m} s{(t_{1},m)} and 2 \\operatorname{v_{x}}{(t_{1},m)} = \\operatorname{v_{x}}{(t_{1},m)} + \\frac{\\partial}{\\partial m} (m + \\sin{(t_{1})})", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), sin(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('m', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Derivative(Function('s')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(2), Function('v_x')(Symbol('t_1', commutative=True), Symbol('m', commutative=True))), Add(Function('v_x')(Symbol('t_1', commutative=True), Symbol('m', commutative=True)), Derivative(Add(Symbol('m', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(x^\\prime)} = x^\\prime, then derive \\int \\operatorname{f_{E}}{(x^\\prime)} dx^\\prime = \\hat{H}_{\\lambda} + \\frac{(x^\\prime)^{2}}{2}, then obtain \\int \\operatorname{f_{E}}{(x^\\prime)} d\\operatorname{f_{E}}{(x^\\prime)} = \\hat{H}_{\\lambda} + \\frac{\\operatorname{f_{E}}^{2}{(x^\\prime)}}{2}", "derivation": "\\operatorname{f_{E}}{(x^\\prime)} = x^\\prime and \\int \\operatorname{f_{E}}{(x^\\prime)} dx^\\prime = \\int x^\\prime dx^\\prime and \\int \\operatorname{f_{E}}{(x^\\prime)} dx^\\prime = \\hat{H}_{\\lambda} + \\frac{(x^\\prime)^{2}}{2} and \\int \\operatorname{f_{E}}{(x^\\prime)} d\\operatorname{f_{E}}{(x^\\prime)} = \\hat{H}_{\\lambda} + \\frac{\\operatorname{f_{E}}^{2}{(x^\\prime)}}{2}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), Tuple(Function('f_E')(Symbol('x^\\\\prime', commutative=True)))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Rational(1, 2), Pow(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\theta_{2}{(P_{e})} = e^{P_{e}}, then obtain \\theta_{2}{(P_{e})} e^{- P_{e}} = 1", "derivation": "\\theta_{2}{(P_{e})} = e^{P_{e}} and \\frac{\\theta_{2}{(P_{e})} e^{- P_{e}}}{P_{e}} = \\frac{1}{P_{e}} and 1 = \\frac{e^{P_{e}}}{\\theta_{2}{(P_{e})}} and \\theta_{2}{(P_{e})} e^{- P_{e}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["divide", 1, "Mul(Symbol('P_e', commutative=True), exp(Symbol('P_e', commutative=True)))"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), Pow(Symbol('P_e', commutative=True), Integer(-1)))"], [["divide", 2, "Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_2')(Symbol('P_e', commutative=True)), Integer(-1)), exp(Symbol('P_e', commutative=True))))"], [["divide", 3, "Mul(Pow(Function('\\\\theta_2')(Symbol('P_e', commutative=True)), Integer(-1)), exp(Symbol('P_e', commutative=True)))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{g})} = \\sin{(\\cos{(\\mathbf{g})})}, then obtain \\frac{d^{2}}{d \\mathbf{g}^{2}} \\tilde{g}^*^{\\mathbf{g}}{(\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin^{\\mathbf{g}}{(\\cos{(\\mathbf{g})})}", "derivation": "\\tilde{g}^*{(\\mathbf{g})} = \\sin{(\\cos{(\\mathbf{g})})} and \\tilde{g}^*^{\\mathbf{g}}{(\\mathbf{g})} = \\sin^{\\mathbf{g}}{(\\cos{(\\mathbf{g})})} and \\frac{d}{d \\mathbf{g}} \\tilde{g}^*^{\\mathbf{g}}{(\\mathbf{g})} = \\frac{d}{d \\mathbf{g}} \\sin^{\\mathbf{g}}{(\\cos{(\\mathbf{g})})} and \\frac{d^{2}}{d \\mathbf{g}^{2}} \\tilde{g}^*^{\\mathbf{g}}{(\\mathbf{g})} = \\frac{d^{2}}{d \\mathbf{g}^{2}} \\sin^{\\mathbf{g}}{(\\cos{(\\mathbf{g})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{g}', commutative=True)), sin(cos(Symbol('\\\\mathbf{g}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Pow(sin(cos(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Pow(sin(cos(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Derivative(Pow(sin(cos(Symbol('\\\\mathbf{g}', commutative=True))), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(n_{2},\\dot{x})} = \\dot{x} - n_{2} and \\operatorname{t_{1}}{(n_{2},\\dot{x})} = \\int (\\dot{x} - n_{2})^{\\dot{x}} d\\dot{x}, then obtain \\int \\operatorname{y^{\\prime}}^{\\dot{x}}{(n_{2},\\dot{x})} d\\dot{x} = \\operatorname{t_{1}}{(n_{2},\\dot{x})}", "derivation": "\\operatorname{y^{\\prime}}{(n_{2},\\dot{x})} = \\dot{x} - n_{2} and \\operatorname{y^{\\prime}}^{\\dot{x}}{(n_{2},\\dot{x})} = (\\dot{x} - n_{2})^{\\dot{x}} and \\int \\operatorname{y^{\\prime}}^{\\dot{x}}{(n_{2},\\dot{x})} d\\dot{x} = \\int (\\dot{x} - n_{2})^{\\dot{x}} d\\dot{x} and \\operatorname{t_{1}}{(n_{2},\\dot{x})} = \\int (\\dot{x} - n_{2})^{\\dot{x}} d\\dot{x} and \\int \\operatorname{y^{\\prime}}^{\\dot{x}}{(n_{2},\\dot{x})} d\\dot{x} = \\operatorname{t_{1}}{(n_{2},\\dot{x})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Pow(Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integral(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Pow(Function('y^{\\\\prime}')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Function('t_1')(Symbol('n_2', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain \\int \\hat{p}_0{(\\mathbf{P})} \\log{(\\mathbf{P})} d\\mathbf{P} = P_{e} + \\mathbf{P} \\log{(\\mathbf{P})}^{2} - 2 \\mathbf{P} \\log{(\\mathbf{P})} + 2 \\mathbf{P}", "derivation": "\\hat{p}_0{(\\mathbf{P})} = \\log{(\\mathbf{P})} and \\hat{p}_0{(\\mathbf{P})} \\log{(\\mathbf{P})} = \\log{(\\mathbf{P})}^{2} and \\int \\hat{p}_0{(\\mathbf{P})} \\log{(\\mathbf{P})} d\\mathbf{P} = \\int \\log{(\\mathbf{P})}^{2} d\\mathbf{P} and \\int \\hat{p}_0{(\\mathbf{P})} \\log{(\\mathbf{P})} d\\mathbf{P} = P_{e} + \\mathbf{P} \\log{(\\mathbf{P})}^{2} - 2 \\mathbf{P} \\log{(\\mathbf{P})} + 2 \\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["times", 1, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Integer(2))), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given h{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and S{(c)} = \\int \\log{(c)} dc, then obtain (h{(\\mathbb{I})} \\int S{(c)} dc)^{c} = (h{(\\mathbb{I})} \\iint \\log{(c)} dc dc)^{c}", "derivation": "h{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and S{(c)} = \\int \\log{(c)} dc and \\int S{(c)} dc = \\iint \\log{(c)} dc dc and \\cos{(\\mathbb{I})} \\int S{(c)} dc = \\cos{(\\mathbb{I})} \\iint \\log{(c)} dc dc and (\\cos{(\\mathbb{I})} \\int S{(c)} dc)^{c} = (\\cos{(\\mathbb{I})} \\iint \\log{(c)} dc dc)^{c} and (h{(\\mathbb{I})} \\int S{(c)} dc)^{c} = (h{(\\mathbb{I})} \\iint \\log{(c)} dc dc)^{c}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], ["get_premise", "Equality(Function('S')(Symbol('c', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["times", 3, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["power", 4, "Symbol('c', commutative=True)"], "Equality(Pow(Mul(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(cos(Symbol('\\\\mathbb{I}', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Mul(Function('h')(Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('S')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)), Pow(Mul(Function('h')(Symbol('\\\\mathbb{I}', commutative=True)), Integral(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(p,F_{x})} = F_{x} + p and \\operatorname{f^{\\prime}}{(F_{x},\\hat{x},p)} = F_{x} p + \\hat{x} + \\frac{p^{2}}{2}, then derive \\int \\mathbf{M}{(p,F_{x})} dp = F_{x} p + \\hat{x} + \\frac{p^{2}}{2}, then obtain \\operatorname{f^{\\prime}}^{F_{x}}{(F_{x},\\hat{x},p)} = (F_{x} p + \\hat{X} + \\frac{p^{2}}{2})^{F_{x}}", "derivation": "\\mathbf{M}{(p,F_{x})} = F_{x} + p and \\int \\mathbf{M}{(p,F_{x})} dp = \\int (F_{x} + p) dp and \\int \\mathbf{M}{(p,F_{x})} dp = F_{x} p + \\hat{x} + \\frac{p^{2}}{2} and F_{x} p + \\hat{x} + \\frac{p^{2}}{2} = \\int (F_{x} + p) dp and \\operatorname{f^{\\prime}}{(F_{x},\\hat{x},p)} = F_{x} p + \\hat{x} + \\frac{p^{2}}{2} and \\operatorname{f^{\\prime}}{(F_{x},\\hat{x},p)} = \\int (F_{x} + p) dp and \\operatorname{f^{\\prime}}^{F_{x}}{(F_{x},\\hat{x},p)} = (\\int (F_{x} + p) dp)^{F_{x}} and \\operatorname{f^{\\prime}}^{F_{x}}{(F_{x},\\hat{x},p)} = (F_{x} p + \\hat{X} + \\frac{p^{2}}{2})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('p', commutative=True), Symbol('F_x', commutative=True)), Add(Symbol('F_x', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('p', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('p', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('p', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2)))), Integral(Add(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('f^{\\\\prime}')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Integral(Add(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["power", 6, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Symbol('F_x', commutative=True)), Pow(Integral(Add(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('F_x', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('F_x', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('p', commutative=True)), Symbol('F_x', commutative=True)), Pow(Add(Mul(Symbol('F_x', commutative=True), Symbol('p', commutative=True)), Symbol('\\\\hat{X}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2)))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(C)} = \\cos{(C)} and \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} = z^{\\hat{H}_l}, then obtain \\frac{\\int \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} dz}{\\sin{(C)} + \\int \\mathbf{s}{(C)} dC} = \\frac{\\int z^{\\hat{H}_l} dz}{\\sin{(C)} + \\int \\mathbf{s}{(C)} dC}", "derivation": "\\mathbf{s}{(C)} = \\cos{(C)} and \\int \\mathbf{s}{(C)} dC = \\int \\cos{(C)} dC and \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} = z^{\\hat{H}_l} and \\int \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} dz = \\int z^{\\hat{H}_l} dz and \\sin{(C)} + \\int \\mathbf{s}{(C)} dC = \\sin{(C)} + \\int \\cos{(C)} dC and \\frac{\\int \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} dz}{\\sin{(C)} + \\int \\cos{(C)} dC} = \\frac{\\int z^{\\hat{H}_l} dz}{\\sin{(C)} + \\int \\cos{(C)} dC} and \\frac{\\int \\dot{\\mathbf{r}}{(\\hat{H}_l,z)} dz}{\\sin{(C)} + \\int \\mathbf{s}{(C)} dC} = \\frac{\\int z^{\\hat{H}_l} dz}{\\sin{(C)} + \\int \\mathbf{s}{(C)} dC}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(Symbol('z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["add", 2, "sin(Symbol('C', commutative=True))"], "Equality(Add(sin(Symbol('C', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(sin(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["divide", 4, "Add(sin(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], "Equality(Mul(Pow(Add(sin(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integer(-1)), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Pow(Add(sin(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integer(-1)), Integral(Pow(Symbol('z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Add(sin(Symbol('C', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integer(-1)), Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Pow(Add(sin(Symbol('C', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integer(-1)), Integral(Pow(Symbol('z', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)} = - F_{g} - \\hat{p} + \\rho, then obtain ((\\hat{p} - \\rho + \\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)})^{\\hat{p}})^{F_{g}} = ((- F_{g})^{\\hat{p}})^{F_{g}}", "derivation": "\\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)} = - F_{g} - \\hat{p} + \\rho and \\hat{p} - \\rho + \\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)} = - F_{g} and (\\hat{p} - \\rho + \\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)})^{\\hat{p}} = (- F_{g})^{\\hat{p}} and ((\\hat{p} - \\rho + \\dot{\\mathbf{r}}{(F_{g},\\hat{p},\\rho)})^{\\hat{p}})^{F_{g}} = ((- F_{g})^{\\hat{p}})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Symbol('F_g', commutative=True)), Pow(Pow(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given u{(y)} = \\log{(y)} and \\operatorname{F_{x}}{(V_{\\mathbf{B}})} = e^{\\sin{(V_{\\mathbf{B}})}}, then obtain \\frac{\\operatorname{F_{x}}{(V_{\\mathbf{B}})} \\int u{(y)} dy}{\\log{(y)}} = \\frac{e^{\\sin{(V_{\\mathbf{B}})}} \\int u{(y)} dy}{\\log{(y)}}", "derivation": "u{(y)} = \\log{(y)} and \\int u{(y)} dy = \\int \\log{(y)} dy and \\operatorname{F_{x}}{(V_{\\mathbf{B}})} = e^{\\sin{(V_{\\mathbf{B}})}} and \\frac{\\operatorname{F_{x}}{(V_{\\mathbf{B}})} \\int \\log{(y)} dy}{\\log{(y)}} = \\frac{e^{\\sin{(V_{\\mathbf{B}})}} \\int \\log{(y)} dy}{\\log{(y)}} and \\frac{\\operatorname{F_{x}}{(V_{\\mathbf{B}})} \\int u{(y)} dy}{\\log{(y)}} = \\frac{e^{\\sin{(V_{\\mathbf{B}})}} \\int u{(y)} dy}{\\log{(y)}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('u')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], ["get_premise", "Equality(Function('F_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), exp(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 3, "Mul(Pow(log(Symbol('y', commutative=True)), Integer(-1)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Mul(Function('F_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(log(Symbol('y', commutative=True)), Integer(-1)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(exp(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(log(Symbol('y', commutative=True)), Integer(-1)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('F_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(log(Symbol('y', commutative=True)), Integer(-1)), Integral(Function('u')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Mul(exp(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Pow(log(Symbol('y', commutative=True)), Integer(-1)), Integral(Function('u')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given g{(E)} = \\sin{(E)} and \\eta^{\\prime}{(E)} = \\frac{g{(E)}}{\\sin{(E)}}, then obtain \\frac{\\eta^{\\prime}{(E)} - \\sin^{2}{(E)}}{\\sin{(E)}} = \\frac{1 - \\sin^{2}{(E)}}{\\sin{(E)}}", "derivation": "g{(E)} = \\sin{(E)} and \\eta^{\\prime}{(E)} = \\frac{g{(E)}}{\\sin{(E)}} and \\eta^{\\prime}{(E)} - \\sin^{2}{(E)} = \\frac{g{(E)}}{\\sin{(E)}} - \\sin^{2}{(E)} and \\eta^{\\prime}{(E)} - g^{2}{(E)} = 1 - g^{2}{(E)} and \\eta^{\\prime}{(E)} - \\sin^{2}{(E)} = 1 - \\sin^{2}{(E)} and \\frac{\\eta^{\\prime}{(E)} - \\sin^{2}{(E)}}{\\sin{(E)}} = \\frac{1 - \\sin^{2}{(E)}}{\\sin{(E)}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True)), Mul(Function('g')(Symbol('E', commutative=True)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))))"], [["minus", 2, "Pow(sin(Symbol('E', commutative=True)), Integer(2))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))), Add(Mul(Function('g')(Symbol('E', commutative=True)), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('g')(Symbol('E', commutative=True)), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))))"], [["times", 5, "Pow(sin(Symbol('E', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('\\\\eta^{\\\\prime}')(Symbol('E', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))), Pow(sin(Symbol('E', commutative=True)), Integer(-1))), Mul(Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('E', commutative=True)), Integer(2)))), Pow(sin(Symbol('E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given S{(\\delta)} = \\delta, then obtain \\frac{\\delta \\int \\frac{S{(\\delta)}}{\\delta} d\\delta}{S{(\\delta)}} = \\frac{\\delta \\int 1 d\\delta}{S{(\\delta)}}", "derivation": "S{(\\delta)} = \\delta and \\frac{S{(\\delta)}}{\\delta} = 1 and \\int \\frac{S{(\\delta)}}{\\delta} d\\delta = \\int 1 d\\delta and \\frac{\\delta \\int \\frac{S{(\\delta)}}{\\delta} d\\delta}{S{(\\delta)}} = \\frac{\\delta \\int 1 d\\delta}{S{(\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["divide", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\delta', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Pow(Function('S')(Symbol('\\\\delta', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Symbol('\\\\delta', commutative=True), Pow(Function('S')(Symbol('\\\\delta', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given T{(\\Psi_{\\lambda},v_{y})} = \\cos{(\\Psi_{\\lambda} v_{y})} and \\hat{H}{(M_{E})} = e^{M_{E}}, then obtain \\frac{\\hat{H}{(M_{E})}}{\\cos{(\\Psi_{\\lambda} v_{y})}} = \\frac{e^{M_{E}}}{\\cos{(\\Psi_{\\lambda} v_{y})}}", "derivation": "T{(\\Psi_{\\lambda},v_{y})} = \\cos{(\\Psi_{\\lambda} v_{y})} and \\hat{H}{(M_{E})} = e^{M_{E}} and \\frac{\\hat{H}{(M_{E})}}{T{(\\Psi_{\\lambda},v_{y})}} = \\frac{e^{M_{E}}}{T{(\\Psi_{\\lambda},v_{y})}} and \\frac{\\hat{H}{(M_{E})}}{\\cos{(\\Psi_{\\lambda} v_{y})}} = \\frac{e^{M_{E}}}{\\cos{(\\Psi_{\\lambda} v_{y})}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), cos(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["divide", 2, "Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('\\\\hat{H}')(Symbol('M_E', commutative=True))), Mul(Pow(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), exp(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\hat{H}')(Symbol('M_E', commutative=True)), Pow(cos(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))), Integer(-1))), Mul(exp(Symbol('M_E', commutative=True)), Pow(cos(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given i{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain \\frac{a^{\\dagger} i{(a^{\\dagger})}}{a^{\\dagger} \\log{(a^{\\dagger})} - \\log{(a^{\\dagger})}} = \\frac{a^{\\dagger} \\log{(a^{\\dagger})}}{a^{\\dagger} \\log{(a^{\\dagger})} - \\log{(a^{\\dagger})}}", "derivation": "i{(a^{\\dagger})} = \\log{(a^{\\dagger})} and a^{\\dagger} i{(a^{\\dagger})} = a^{\\dagger} \\log{(a^{\\dagger})} and a^{\\dagger} i{(a^{\\dagger})} - \\log{(a^{\\dagger})} = a^{\\dagger} \\log{(a^{\\dagger})} - \\log{(a^{\\dagger})} and \\frac{a^{\\dagger} i{(a^{\\dagger})}}{a^{\\dagger} i{(a^{\\dagger})} - \\log{(a^{\\dagger})}} = \\frac{a^{\\dagger} \\log{(a^{\\dagger})}}{a^{\\dagger} i{(a^{\\dagger})} - \\log{(a^{\\dagger})}} and \\frac{a^{\\dagger} i{(a^{\\dagger})}}{a^{\\dagger} \\log{(a^{\\dagger})} - \\log{(a^{\\dagger})}} = \\frac{a^{\\dagger} \\log{(a^{\\dagger})}}{a^{\\dagger} \\log{(a^{\\dagger})} - \\log{(a^{\\dagger})}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 2, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["divide", 2, "Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True))))"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(-1)), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(-1)), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(-1)), Function('i')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Pow(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), log(Symbol('a^{\\\\dagger}', commutative=True)))), Integer(-1)), log(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(m_{s},v_{y})} = m_{s} v_{y}, then obtain \\frac{(v_{y} - \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}})^{v_{y}}}{m_{s}} = \\frac{1}{m_{s}}", "derivation": "\\operatorname{M_{E}}{(m_{s},v_{y})} = m_{s} v_{y} and \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}} = v_{y} and 0 = v_{y} - \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}} and 0^{v_{y}} = (v_{y} - \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}})^{v_{y}} and \\frac{0^{v_{y}}}{m_{s}} = \\frac{(v_{y} - \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}})^{v_{y}}}{m_{s}} and \\frac{(v_{y} - \\frac{\\operatorname{M_{E}}{(m_{s},v_{y})}}{m_{s}})^{v_{y}}}{m_{s}} = \\frac{1}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))"], [["divide", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))"], [["minus", 2, "Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))"], "Equality(Integer(0), Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_y', commutative=True)), Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))), Symbol('v_y', commutative=True)))"], [["times", 4, "Pow(Symbol('m_s', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Integer(0), Symbol('v_y', commutative=True)), Pow(Symbol('m_s', commutative=True), Integer(-1))), Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))), Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Pow(Add(Symbol('v_y', commutative=True), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('M_E')(Symbol('m_s', commutative=True), Symbol('v_y', commutative=True)))), Symbol('v_y', commutative=True))), Pow(Symbol('m_s', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(x,u)} = u x and \\operatorname{y^{\\prime}}{(I,\\hat{X})} = I + \\hat{X} and n{(I,x,\\hat{X},u)} = (u x + \\operatorname{y^{\\prime}}{(I,\\hat{X})})^{\\hat{X}}, then obtain n{(I,x,\\hat{X},u)} = (I + \\hat{X} + \\Psi^{\\dagger}{(x,u)})^{\\hat{X}}", "derivation": "\\Psi^{\\dagger}{(x,u)} = u x and \\operatorname{y^{\\prime}}{(I,\\hat{X})} = I + \\hat{X} and u x + \\operatorname{y^{\\prime}}{(I,\\hat{X})} = I + \\hat{X} + u x and \\Psi^{\\dagger}{(x,u)} + \\operatorname{y^{\\prime}}{(I,\\hat{X})} = I + \\hat{X} + \\Psi^{\\dagger}{(x,u)} and n{(I,x,\\hat{X},u)} = (u x + \\operatorname{y^{\\prime}}{(I,\\hat{X})})^{\\hat{X}} and n{(I,x,\\hat{X},u)} = (\\Psi^{\\dagger}{(x,u)} + \\operatorname{y^{\\prime}}{(I,\\hat{X})})^{\\hat{X}} and n{(I,x,\\hat{X},u)} = (I + \\hat{X} + \\Psi^{\\dagger}{(x,u)})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('u', commutative=True), Symbol('x', commutative=True)))"], ["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 2, "Mul(Symbol('u', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Mul(Symbol('u', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('u', commutative=True), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True), Symbol('u', commutative=True)), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Pow(Add(Mul(Symbol('u', commutative=True), Symbol('x', commutative=True)), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('n')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Pow(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True), Symbol('u', commutative=True)), Function('y^{\\\\prime}')(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Function('n')(Symbol('I', commutative=True), Symbol('x', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('u', commutative=True)), Pow(Add(Symbol('I', commutative=True), Symbol('\\\\hat{X}', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('x', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(\\Psi)} = \\sin{(\\Psi)}, then obtain \\frac{\\hat{p}_0{(\\Psi)}}{\\Psi} + \\frac{2 \\sin{(\\Psi)}}{\\Psi} = \\frac{3 \\sin{(\\Psi)}}{\\Psi}", "derivation": "\\hat{p}_0{(\\Psi)} = \\sin{(\\Psi)} and \\frac{\\hat{p}_0{(\\Psi)}}{\\Psi} = \\frac{\\sin{(\\Psi)}}{\\Psi} and \\frac{\\hat{p}_0{(\\Psi)}}{\\Psi} + \\frac{\\sin{(\\Psi)}}{\\Psi} = \\frac{2 \\sin{(\\Psi)}}{\\Psi} and \\frac{\\hat{p}_0{(\\Psi)}}{\\Psi} + \\frac{2 \\sin{(\\Psi)}}{\\Psi} = \\frac{3 \\sin{(\\Psi)}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\hat{p}_0')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(3), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), sin(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = \\frac{A_{x} \\theta_2}{A_{z}}, then obtain - e^{\\frac{A_{x} \\theta_2}{A_{z}}} + \\frac{\\partial}{\\partial \\theta_2} A_{x} \\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = - e^{\\frac{A_{x} \\theta_2}{A_{z}}} + \\frac{\\partial}{\\partial \\theta_2} \\frac{A_{x}^{2} \\theta_2}{A_{z}}", "derivation": "\\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = \\frac{A_{x} \\theta_2}{A_{z}} and A_{x} \\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = \\frac{A_{x}^{2} \\theta_2}{A_{z}} and \\frac{\\partial}{\\partial \\theta_2} A_{x} \\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = \\frac{\\partial}{\\partial \\theta_2} \\frac{A_{x}^{2} \\theta_2}{A_{z}} and - e^{\\frac{A_{x} \\theta_2}{A_{z}}} + \\frac{\\partial}{\\partial \\theta_2} A_{x} \\eta^{\\prime}{(A_{x},\\theta_2,A_{z})} = - e^{\\frac{A_{x} \\theta_2}{A_{z}}} + \\frac{\\partial}{\\partial \\theta_2} \\frac{A_{x}^{2} \\theta_2}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('A_x', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Symbol('A_x', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('A_x', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(2)), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('A_x', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('A_x', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(2)), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["minus", 3, "exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))), Derivative(Mul(Symbol('A_x', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('A_x', commutative=True), Symbol('\\\\theta_2', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Mul(Symbol('A_x', commutative=True), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Integer(2)), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta_{1}{(t)} = e^{t}, then obtain (t + 2 \\theta_{1}{(t)}) \\theta_{1}{(t)} = (t + 2 e^{t}) \\theta_{1}{(t)}", "derivation": "\\theta_{1}{(t)} = e^{t} and t + \\theta_{1}{(t)} = t + e^{t} and t + 2 \\theta_{1}{(t)} = t + \\theta_{1}{(t)} + e^{t} and (t + 2 \\theta_{1}{(t)}) \\theta_{1}{(t)} = (t + \\theta_{1}{(t)} + e^{t}) \\theta_{1}{(t)} and (t + 2 \\theta_{1}{(t)}) \\theta_{1}{(t)} = (t + 2 e^{t}) \\theta_{1}{(t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True)))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\theta_1')(Symbol('t', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('t', commutative=True))))"], [["add", 2, "Function('\\\\theta_1')(Symbol('t', commutative=True))"], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(2), Function('\\\\theta_1')(Symbol('t', commutative=True)))), Add(Symbol('t', commutative=True), Function('\\\\theta_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))))"], [["times", 3, "Function('\\\\theta_1')(Symbol('t', commutative=True))"], "Equality(Mul(Add(Symbol('t', commutative=True), Mul(Integer(2), Function('\\\\theta_1')(Symbol('t', commutative=True)))), Function('\\\\theta_1')(Symbol('t', commutative=True))), Mul(Add(Symbol('t', commutative=True), Function('\\\\theta_1')(Symbol('t', commutative=True)), exp(Symbol('t', commutative=True))), Function('\\\\theta_1')(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('t', commutative=True), Mul(Integer(2), Function('\\\\theta_1')(Symbol('t', commutative=True)))), Function('\\\\theta_1')(Symbol('t', commutative=True))), Mul(Add(Symbol('t', commutative=True), Mul(Integer(2), exp(Symbol('t', commutative=True)))), Function('\\\\theta_1')(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\delta{(J,T,\\Psi^{\\dagger})} = T (J - \\Psi^{\\dagger}), then obtain \\frac{\\partial}{\\partial T} - \\frac{\\delta^{2}{(J,T,\\Psi^{\\dagger})}}{J} = \\frac{\\partial}{\\partial T} - \\frac{T^{2} (J - \\Psi^{\\dagger})^{2}}{J}", "derivation": "\\delta{(J,T,\\Psi^{\\dagger})} = T (J - \\Psi^{\\dagger}) and - \\delta{(J,T,\\Psi^{\\dagger})} = - T (J - \\Psi^{\\dagger}) and \\frac{\\delta{(J,T,\\Psi^{\\dagger})}}{J} = \\frac{T (J - \\Psi^{\\dagger})}{J} and - \\frac{\\delta^{2}{(J,T,\\Psi^{\\dagger})}}{J} = - \\frac{T (J - \\Psi^{\\dagger}) \\delta{(J,T,\\Psi^{\\dagger})}}{J} and - \\frac{T (J - \\Psi^{\\dagger}) \\delta{(J,T,\\Psi^{\\dagger})}}{J} = - \\frac{T^{2} (J - \\Psi^{\\dagger})^{2}}{J} and - \\frac{\\delta^{2}{(J,T,\\Psi^{\\dagger})}}{J} = - \\frac{T^{2} (J - \\Psi^{\\dagger})^{2}}{J} and \\frac{\\partial}{\\partial T} - \\frac{\\delta^{2}{(J,T,\\Psi^{\\dagger})}}{J} = \\frac{\\partial}{\\partial T} - \\frac{T^{2} (J - \\Psi^{\\dagger})^{2}}{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Mul(Symbol('T', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["divide", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('T', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["times", 2, "Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('T', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('T', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Integer(2)), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Integer(2)), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(2))))"], [["differentiate", 6, "Symbol('T', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('\\\\delta')(Symbol('J', commutative=True), Symbol('T', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(2))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Integer(2)), Pow(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(2))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then obtain \\varphi^* \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} + \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\varphi^* \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} + \\sin{(\\varphi^*)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\sin{(\\varphi^*)} and \\varphi^* \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\varphi^* \\sin{(\\varphi^*)} and \\varphi^* \\sin{(\\varphi^*)} + \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\varphi^* \\sin{(\\varphi^*)} + \\sin{(\\varphi^*)} and \\varphi^* \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} + \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} = \\varphi^* \\operatorname{f_{\\mathbf{p}}}{(\\varphi^*)} + \\sin{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True))), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given b{(M,F_{c})} = \\sin^{F_{c}}{(M)} and \\hat{\\mathbf{r}}{(M)} = \\sin{(M)}, then obtain - \\hat{\\mathbf{r}}{(M)} + b{(M,F_{c})} = - \\hat{\\mathbf{r}}{(M)} + \\hat{\\mathbf{r}}^{F_{c}}{(M)}", "derivation": "b{(M,F_{c})} = \\sin^{F_{c}}{(M)} and \\hat{\\mathbf{r}}{(M)} = \\sin{(M)} and - \\hat{\\mathbf{r}}{(M)} + b{(M,F_{c})} = - \\hat{\\mathbf{r}}{(M)} + \\sin^{F_{c}}{(M)} and - \\hat{\\mathbf{r}}{(M)} + b{(M,F_{c})} = - \\hat{\\mathbf{r}}{(M)} + \\hat{\\mathbf{r}}^{F_{c}}{(M)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('M', commutative=True), Symbol('F_c', commutative=True)), Pow(sin(Symbol('M', commutative=True)), Symbol('F_c', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["minus", 1, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True))), Function('b')(Symbol('M', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True))), Pow(sin(Symbol('M', commutative=True)), Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True))), Function('b')(Symbol('M', commutative=True), Symbol('F_c', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True))), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('M', commutative=True)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(k,\\mathbf{H})} = \\mathbf{H}^{k} and f{(k,\\mathbf{H})} = \\mathbf{H}^{k} \\dot{y}{(k,\\mathbf{H})} \\dot{y}^{- \\mathbf{H}}{(k,\\mathbf{H})}, then obtain \\int f{(k,\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H}^{2 k} (\\mathbf{H}^{k})^{- \\mathbf{H}} d\\mathbf{H}", "derivation": "\\dot{y}{(k,\\mathbf{H})} = \\mathbf{H}^{k} and f{(k,\\mathbf{H})} = \\mathbf{H}^{k} \\dot{y}{(k,\\mathbf{H})} \\dot{y}^{- \\mathbf{H}}{(k,\\mathbf{H})} and \\int f{(k,\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H}^{k} \\dot{y}{(k,\\mathbf{H})} \\dot{y}^{- \\mathbf{H}}{(k,\\mathbf{H})} d\\mathbf{H} and \\int f{(k,\\mathbf{H})} d\\mathbf{H} = \\int \\mathbf{H}^{2 k} (\\mathbf{H}^{k})^{- \\mathbf{H}} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('k', commutative=True)), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('k', commutative=True)), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Function('f')(Symbol('k', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Symbol('k', commutative=True))), Pow(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(I,\\hat{H})} = \\cos^{I}{(\\hat{H})} and \\operatorname{C_{2}}{(I,\\hat{H})} = \\hat{H} + \\operatorname{v_{z}}{(I,\\hat{H})}, then obtain \\operatorname{C_{2}}^{I}{(I,\\hat{H})} = (\\hat{H} + \\cos^{I}{(\\hat{H})})^{I}", "derivation": "\\operatorname{v_{z}}{(I,\\hat{H})} = \\cos^{I}{(\\hat{H})} and \\operatorname{C_{2}}{(I,\\hat{H})} = \\hat{H} + \\operatorname{v_{z}}{(I,\\hat{H})} and \\operatorname{C_{2}}{(I,\\hat{H})} = \\hat{H} + \\cos^{I}{(\\hat{H})} and \\operatorname{C_{2}}^{I}{(I,\\hat{H})} = (\\hat{H} + \\operatorname{v_{z}}{(I,\\hat{H})})^{I} and (\\hat{H} + \\cos^{I}{(\\hat{H})})^{I} = (\\hat{H} + \\operatorname{v_{z}}{(I,\\hat{H})})^{I} and \\operatorname{C_{2}}^{I}{(I,\\hat{H})} = (\\hat{H} + \\cos^{I}{(\\hat{H})})^{I}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Function('v_z')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('C_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Function('v_z')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('I', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Symbol('\\\\hat{H}', commutative=True), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Function('v_z')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('C_2')(Symbol('I', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True)), Pow(Add(Symbol('\\\\hat{H}', commutative=True), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Symbol('I', commutative=True))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(r_{0},c_{0})} = c_{0} - r_{0}, then obtain - r_{0} - \\operatorname{z^{*}}{(r_{0},c_{0})} = - c_{0}", "derivation": "\\operatorname{z^{*}}{(r_{0},c_{0})} = c_{0} - r_{0} and - \\operatorname{z^{*}}{(r_{0},c_{0})} = - c_{0} + r_{0} and r_{0} - \\operatorname{z^{*}}{(r_{0},c_{0})} = - c_{0} + 2 r_{0} and - r_{0} - \\operatorname{z^{*}}{(r_{0},c_{0})} = - c_{0}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('r_0', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('z^*')(Symbol('r_0', commutative=True), Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["add", 2, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Function('z^*')(Symbol('r_0', commutative=True), Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(2), Symbol('r_0', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Integer(-1), Function('z^*')(Symbol('r_0', commutative=True), Symbol('c_0', commutative=True)))), Mul(Integer(-1), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given k{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})}, then obtain k^{\\dot{x}}{(\\dot{x})} - \\cos^{\\dot{x}}{(\\dot{x})} = 0", "derivation": "k{(\\dot{x})} = \\frac{d}{d \\dot{x}} \\sin{(\\dot{x})} and k^{\\dot{x}}{(\\dot{x})} = (\\frac{d}{d \\dot{x}} \\sin{(\\dot{x})})^{\\dot{x}} and k^{\\dot{x}}{(\\dot{x})} - (\\frac{d}{d \\dot{x}} \\sin{(\\dot{x})})^{\\dot{x}} = 0 and k^{\\dot{x}}{(\\dot{x})} - \\cos^{\\dot{x}}{(\\dot{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\dot{x}', commutative=True)), Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('k')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 2, "Pow(Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Pow(Function('k')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Pow(Derivative(sin(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Symbol('\\\\dot{x}', commutative=True)))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Pow(Function('k')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{P},P_{e},\\mathbf{f})} = \\frac{P_{e} + \\mathbf{f}}{\\mathbf{P}} and \\pi{(P_{e},\\mathbf{f})} = P_{e} + \\mathbf{f}, then obtain \\mathbf{f} \\hat{\\mathbf{x}}^{P_{e}}{(\\mathbf{P},P_{e},\\mathbf{f})} = \\mathbf{f} (\\frac{\\pi{(P_{e},\\mathbf{f})}}{\\mathbf{P}})^{P_{e}}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{P},P_{e},\\mathbf{f})} = \\frac{P_{e} + \\mathbf{f}}{\\mathbf{P}} and \\hat{\\mathbf{x}}^{P_{e}}{(\\mathbf{P},P_{e},\\mathbf{f})} = (\\frac{P_{e} + \\mathbf{f}}{\\mathbf{P}})^{P_{e}} and \\pi{(P_{e},\\mathbf{f})} = P_{e} + \\mathbf{f} and \\hat{\\mathbf{x}}^{P_{e}}{(\\mathbf{P},P_{e},\\mathbf{f})} = (\\frac{\\pi{(P_{e},\\mathbf{f})}}{\\mathbf{P}})^{P_{e}} and \\mathbf{f} \\hat{\\mathbf{x}}^{P_{e}}{(\\mathbf{P},P_{e},\\mathbf{f})} = \\mathbf{f} (\\frac{\\pi{(P_{e},\\mathbf{f})}}{\\mathbf{P}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('P_e', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('P_e', commutative=True)))"], [["times", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('P_e', commutative=True))), Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\phi,\\mathbb{I})} = \\frac{\\phi}{\\mathbb{I}} and \\operatorname{v_{t}}{(\\phi,\\mathbb{I})} = \\frac{\\phi}{\\mathbb{I}}, then obtain \\mathbb{I} \\operatorname{v_{t}}{(\\phi,\\mathbb{I})} = \\phi", "derivation": "\\operatorname{n_{1}}{(\\phi,\\mathbb{I})} = \\frac{\\phi}{\\mathbb{I}} and \\operatorname{v_{t}}{(\\phi,\\mathbb{I})} = \\frac{\\phi}{\\mathbb{I}} and \\mathbb{I} \\operatorname{n_{1}}{(\\phi,\\mathbb{I})} = \\phi and \\operatorname{n_{1}}{(\\phi,\\mathbb{I})} = \\operatorname{v_{t}}{(\\phi,\\mathbb{I})} and \\mathbb{I} \\operatorname{v_{t}}{(\\phi,\\mathbb{I})} = \\phi", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\phi', commutative=True))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('v_t')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Function('v_t')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\phi', commutative=True))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then obtain \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\phi_{2}^{2}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\phi_{2}{(\\mathbf{F})} \\sin{(\\mathbf{F})}", "derivation": "\\phi_{2}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\mathbf{F} \\phi_{2}{(\\mathbf{F})} = \\mathbf{F} \\sin{(\\mathbf{F})} and \\mathbf{F} \\phi_{2}{(\\mathbf{F})} \\sin{(\\mathbf{F})} = \\mathbf{F} \\sin^{2}{(\\mathbf{F})} and \\mathbf{F} \\phi_{2}^{2}{(\\mathbf{F})} = \\mathbf{F} \\phi_{2}{(\\mathbf{F})} \\sin{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\phi_{2}^{2}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\mathbf{F} \\phi_{2}{(\\mathbf{F})} \\sin{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{F}', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(sin(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(v_{1},\\eta^{\\prime})} = \\eta^{\\prime} - v_{1}, then derive \\int \\mathbf{p}{(v_{1},\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{(\\eta^{\\prime})^{2}}{2} - \\eta^{\\prime} v_{1} + p, then obtain \\frac{\\partial}{\\partial p} \\int \\mathbf{p}{(v_{1},\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{\\partial}{\\partial p} (\\frac{(\\eta^{\\prime})^{2}}{2} - \\eta^{\\prime} v_{1} + p)", "derivation": "\\mathbf{p}{(v_{1},\\eta^{\\prime})} = \\eta^{\\prime} - v_{1} and \\int \\mathbf{p}{(v_{1},\\eta^{\\prime})} d\\eta^{\\prime} = \\int (\\eta^{\\prime} - v_{1}) d\\eta^{\\prime} and \\int \\mathbf{p}{(v_{1},\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{(\\eta^{\\prime})^{2}}{2} - \\eta^{\\prime} v_{1} + p and \\frac{\\partial}{\\partial p} \\int \\mathbf{p}{(v_{1},\\eta^{\\prime})} d\\eta^{\\prime} = \\frac{\\partial}{\\partial p} (\\frac{(\\eta^{\\prime})^{2}}{2} - \\eta^{\\prime} v_{1} + p)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('v_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('v_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{p}')(Symbol('v_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('v_1', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given p{(k,A_{z})} = \\sin{(k^{A_{z}})}, then obtain - A_{z} p{(k,A_{z})} + p^{2}{(k,A_{z})} = - A_{z} p{(k,A_{z})} + p{(k,A_{z})} \\sin{(k^{A_{z}})}", "derivation": "p{(k,A_{z})} = \\sin{(k^{A_{z}})} and p^{2}{(k,A_{z})} = p{(k,A_{z})} \\sin{(k^{A_{z}})} and A_{z} p{(k,A_{z})} = A_{z} \\sin{(k^{A_{z}})} and - A_{z} \\sin{(k^{A_{z}})} + p^{2}{(k,A_{z})} = - A_{z} \\sin{(k^{A_{z}})} + p{(k,A_{z})} \\sin{(k^{A_{z}})} and - A_{z} p{(k,A_{z})} + p^{2}{(k,A_{z})} = - A_{z} p{(k,A_{z})} + p{(k,A_{z})} \\sin{(k^{A_{z}})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True))))"], [["times", 1, "Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Pow(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True)))))"], [["times", 1, "Symbol('A_z', commutative=True)"], "Equality(Mul(Symbol('A_z', commutative=True), Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))), Mul(Symbol('A_z', commutative=True), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True)))))"], [["minus", 2, "Mul(Symbol('A_z', commutative=True), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True)))), Pow(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True)))), Mul(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True), Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))), Pow(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True), Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True))), Mul(Function('p')(Symbol('k', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('k', commutative=True), Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given r{(C,\\eta^{\\prime})} = C + \\eta^{\\prime}, then obtain (0^{C})^{\\eta^{\\prime}} = 1", "derivation": "r{(C,\\eta^{\\prime})} = C + \\eta^{\\prime} and 0 = C + \\eta^{\\prime} - r{(C,\\eta^{\\prime})} and 0^{C} = (C + \\eta^{\\prime} - r{(C,\\eta^{\\prime})})^{C} and (0^{C})^{\\eta^{\\prime}} = ((C + \\eta^{\\prime} - r{(C,\\eta^{\\prime})})^{C})^{\\eta^{\\prime}} and ((C + \\eta^{\\prime} - r{(C,\\eta^{\\prime})})^{C})^{\\eta^{\\prime}} = 1 and (0^{C})^{\\eta^{\\prime}} = 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["minus", 1, "Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Integer(0), Add(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Integer(0), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Symbol('C', commutative=True)))"], [["power", 3, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Pow(Add(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Pow(Add(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Function('r')(Symbol('C', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Pow(Integer(0), Symbol('C', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Integer(1))"]]}, {"prompt": "Given q{(\\delta,\\ddot{x})} = \\ddot{x}^{\\delta} and \\dot{z}{(\\delta)} = \\delta, then obtain \\delta + \\dot{z}^{2}{(\\delta)} + 1 = \\delta \\dot{z}{(\\delta)} + \\delta + 1", "derivation": "q{(\\delta,\\ddot{x})} = \\ddot{x}^{\\delta} and \\dot{z}{(\\delta)} = \\delta and \\dot{z}^{2}{(\\delta)} = \\delta \\dot{z}{(\\delta)} and \\dot{z}^{2}{(\\delta)} + \\ddot{x}^{- \\delta} q{(\\delta,\\ddot{x})} = \\delta \\dot{z}{(\\delta)} + \\ddot{x}^{- \\delta} q{(\\delta,\\ddot{x})} and \\dot{z}^{2}{(\\delta)} + 1 = \\delta \\dot{z}{(\\delta)} + 1 and \\delta + \\dot{z}^{2}{(\\delta)} + 1 = \\delta \\dot{z}{(\\delta)} + \\delta + 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["times", 2, "Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Pow(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True)), Integer(2)), Integer(1)), Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True))), Integer(1)))"], [["add", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Pow(Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True)), Integer(2)), Integer(1)), Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A_{2},C_{2})} = \\sin^{A_{2}}{(C_{2})}, then obtain (e^{\\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{e}}{(A_{2},C_{2})}})^{A_{2}} = (e^{\\frac{\\partial}{\\partial A_{2}} \\sin^{A_{2}}{(C_{2})}})^{A_{2}}", "derivation": "\\operatorname{P_{e}}{(A_{2},C_{2})} = \\sin^{A_{2}}{(C_{2})} and \\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{e}}{(A_{2},C_{2})} = \\frac{\\partial}{\\partial A_{2}} \\sin^{A_{2}}{(C_{2})} and e^{\\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{e}}{(A_{2},C_{2})}} = e^{\\frac{\\partial}{\\partial A_{2}} \\sin^{A_{2}}{(C_{2})}} and (e^{\\frac{\\partial}{\\partial A_{2}} \\operatorname{P_{e}}{(A_{2},C_{2})}})^{A_{2}} = (e^{\\frac{\\partial}{\\partial A_{2}} \\sin^{A_{2}}{(C_{2})}})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A_2', commutative=True), Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('A_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('C_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('P_e')(Symbol('A_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), exp(Derivative(Pow(sin(Symbol('C_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('A_2', commutative=True)"], "Equality(Pow(exp(Derivative(Function('P_e')(Symbol('A_2', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Symbol('A_2', commutative=True)), Pow(exp(Derivative(Pow(sin(Symbol('C_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(v_{z})} = e^{v_{z}} and \\phi_{2}{(v_{z})} = v_{z}, then obtain \\int (\\operatorname{v_{1}}{(v_{z})} - e^{v_{z}}) d\\phi_{2}{(v_{z})} = \\int 0 d\\phi_{2}{(v_{z})}", "derivation": "\\operatorname{v_{1}}{(v_{z})} = e^{v_{z}} and \\operatorname{v_{1}}{(v_{z})} - e^{v_{z}} = 0 and \\int (\\operatorname{v_{1}}{(v_{z})} - e^{v_{z}}) dv_{z} = \\int 0 dv_{z} and \\phi_{2}{(v_{z})} = v_{z} and \\int (\\operatorname{v_{1}}{(v_{z})} - e^{v_{z}}) d\\phi_{2}{(v_{z})} = \\int 0 d\\phi_{2}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('v_z', commutative=True)), exp(Symbol('v_z', commutative=True)))"], [["minus", 1, "exp(Symbol('v_z', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Function('v_1')(Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True))), Integral(Integer(0), Tuple(Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Function('v_1')(Symbol('v_z', commutative=True)), Mul(Integer(-1), exp(Symbol('v_z', commutative=True)))), Tuple(Function('\\\\phi_2')(Symbol('v_z', commutative=True)))), Integral(Integer(0), Tuple(Function('\\\\phi_2')(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given B{(A,F_{H})} = \\cos{(A - F_{H})}, then derive \\int B{(A,F_{H})} dA = f_{\\mathbf{p}} + \\sin{(A - F_{H})}, then derive \\mathbf{S} + \\sin{(A - F_{H})} = f_{\\mathbf{p}} + \\sin{(A - F_{H})}, then obtain (\\mathbf{S} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})} = (\\eta^{\\prime} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})}", "derivation": "B{(A,F_{H})} = \\cos{(A - F_{H})} and \\int B{(A,F_{H})} dA = \\int \\cos{(A - F_{H})} dA and \\int B{(A,F_{H})} dA = f_{\\mathbf{p}} + \\sin{(A - F_{H})} and \\int \\cos{(A - F_{H})} dA = f_{\\mathbf{p}} + \\sin{(A - F_{H})} and \\mathbf{S} + \\sin{(A - F_{H})} = f_{\\mathbf{p}} + \\sin{(A - F_{H})} and (\\mathbf{S} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})} = (f_{\\mathbf{p}} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})} and (\\mathbf{S} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})} = \\sin{(A - F_{H})} \\int \\cos{(A - F_{H})} dA and (\\mathbf{S} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})} = (\\eta^{\\prime} + \\sin{(A - F_{H})}) \\sin{(A - F_{H})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('A', commutative=True), Symbol('F_H', commutative=True)), cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('B')(Symbol('A', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('A', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))), Tuple(Symbol('A', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"], [["times", 5, "sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Mul(sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))), Integral(cos(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True)))), Tuple(Symbol('A', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Mul(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), Mul(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))), sin(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given U{(b,f^{*})} = b + \\sin{(f^{*})}, then obtain - \\frac{b + 4 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} + \\frac{U{(b,f^{*})} + 3 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} = 0", "derivation": "U{(b,f^{*})} = b + \\sin{(f^{*})} and U{(b,f^{*})} + \\sin{(f^{*})} = b + 2 \\sin{(f^{*})} and U{(b,f^{*})} + 3 \\sin{(f^{*})} = b + 4 \\sin{(f^{*})} and \\frac{U{(b,f^{*})} + 3 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} = \\frac{b + 4 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} and - \\frac{b + 4 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} + \\frac{U{(b,f^{*})} + 3 \\sin{(f^{*})}}{b + \\sin{(f^{*})}} = 0", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))))"], [["add", 1, "sin(Symbol('f^*', commutative=True))"], "Equality(Add(Function('U')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(2), sin(Symbol('f^*', commutative=True)))))"], [["add", 2, "Mul(Integer(2), sin(Symbol('f^*', commutative=True)))"], "Equality(Add(Function('U')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(3), sin(Symbol('f^*', commutative=True)))), Add(Symbol('b', commutative=True), Mul(Integer(4), sin(Symbol('f^*', commutative=True)))))"], [["times", 3, "Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1)), Add(Function('U')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(3), sin(Symbol('f^*', commutative=True))))), Mul(Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1)), Add(Symbol('b', commutative=True), Mul(Integer(4), sin(Symbol('f^*', commutative=True))))))"], [["minus", 4, "Mul(Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1)), Add(Symbol('b', commutative=True), Mul(Integer(4), sin(Symbol('f^*', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1)), Add(Symbol('b', commutative=True), Mul(Integer(4), sin(Symbol('f^*', commutative=True))))), Mul(Pow(Add(Symbol('b', commutative=True), sin(Symbol('f^*', commutative=True))), Integer(-1)), Add(Function('U')(Symbol('b', commutative=True), Symbol('f^*', commutative=True)), Mul(Integer(3), sin(Symbol('f^*', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given T{(\\mathbf{s},\\tilde{g}^*)} = \\mathbf{s} \\tilde{g}^*, then obtain - \\int \\frac{\\mathbf{s} \\tilde{g}^* + T{(\\mathbf{s},\\tilde{g}^*)}}{\\mathbf{s} \\tilde{g}^*} d\\mathbf{s} = - \\int 2 d\\mathbf{s}", "derivation": "T{(\\mathbf{s},\\tilde{g}^*)} = \\mathbf{s} \\tilde{g}^* and \\mathbf{s} \\tilde{g}^* + T{(\\mathbf{s},\\tilde{g}^*)} = 2 \\mathbf{s} \\tilde{g}^* and \\frac{\\mathbf{s} \\tilde{g}^* + T{(\\mathbf{s},\\tilde{g}^*)}}{\\mathbf{s} \\tilde{g}^*} = 2 and \\int \\frac{\\mathbf{s} \\tilde{g}^* + T{(\\mathbf{s},\\tilde{g}^*)}}{\\mathbf{s} \\tilde{g}^*} d\\mathbf{s} = \\int 2 d\\mathbf{s} and - \\int \\frac{\\mathbf{s} \\tilde{g}^* + T{(\\mathbf{s},\\tilde{g}^*)}}{\\mathbf{s} \\tilde{g}^*} d\\mathbf{s} = - \\int 2 d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('T')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('T')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Integer(2))"], [["integrate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('T')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Integer(2), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Function('T')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Integer(-1), Integral(Integer(2), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{H},\\mu_0,H)} = \\frac{- \\hat{H} + \\mu_0}{H}, then obtain (- \\frac{- \\hat{H} + \\mu_0}{H})^{\\mu_0} = (\\frac{\\hat{H} - \\mu_0}{H})^{\\mu_0}", "derivation": "\\operatorname{v_{2}}{(\\hat{H},\\mu_0,H)} = \\frac{- \\hat{H} + \\mu_0}{H} and - \\operatorname{v_{2}}{(\\hat{H},\\mu_0,H)} = - \\frac{- \\hat{H} + \\mu_0}{H} and - \\operatorname{v_{2}}{(\\hat{H},\\mu_0,H)} = \\frac{\\hat{H} - \\mu_0}{H} and - \\frac{- \\hat{H} + \\mu_0}{H} = \\frac{\\hat{H} - \\mu_0}{H} and (- \\frac{- \\hat{H} + \\mu_0}{H})^{\\mu_0} = (\\frac{\\hat{H} - \\mu_0}{H})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\mu_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given L{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0, then obtain \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0) L{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0)^{2}", "derivation": "L{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0 and \\mathbf{r} L{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0) and \\mathbf{r} L^{2}{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0) L{(\\mathbf{M},\\mu_0,\\mathbf{r})} and \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0) L{(\\mathbf{M},\\mu_0,\\mathbf{r})} = \\mathbf{r} (\\frac{\\mathbf{M}}{\\mathbf{r}} + \\mu_0)^{2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)), Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)), Function('L')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Symbol('\\\\mu_0', commutative=True)), Integer(2))))"]]}, {"prompt": "Given U{(F_{c},v_{y})} = F_{c}^{v_{y}}, then obtain \\frac{\\partial}{\\partial v_{y}} (\\frac{\\partial}{\\partial F_{c}} (- b + U{(F_{c},v_{y})}))^{F_{c}} = \\frac{\\partial}{\\partial v_{y}} (\\frac{\\partial}{\\partial F_{c}} (F_{c}^{v_{y}} - b))^{F_{c}}", "derivation": "U{(F_{c},v_{y})} = F_{c}^{v_{y}} and - b + U{(F_{c},v_{y})} = F_{c}^{v_{y}} - b and \\frac{\\partial}{\\partial F_{c}} (- b + U{(F_{c},v_{y})}) = \\frac{\\partial}{\\partial F_{c}} (F_{c}^{v_{y}} - b) and (\\frac{\\partial}{\\partial F_{c}} (- b + U{(F_{c},v_{y})}))^{F_{c}} = (\\frac{\\partial}{\\partial F_{c}} (F_{c}^{v_{y}} - b))^{F_{c}} and \\frac{\\partial}{\\partial v_{y}} (\\frac{\\partial}{\\partial F_{c}} (- b + U{(F_{c},v_{y})}))^{F_{c}} = \\frac{\\partial}{\\partial v_{y}} (\\frac{\\partial}{\\partial F_{c}} (F_{c}^{v_{y}} - b))^{F_{c}}", "srepr_derivation": [["get_premise", "Equality(Function('U')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('U')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Add(Pow(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('U')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_c', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('U')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Pow(Derivative(Add(Pow(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)))"], [["differentiate", 4, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('U')(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Pow(Symbol('F_c', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('b', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Symbol('F_c', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(H)} = \\log{(H)}, then obtain \\frac{2 \\bar{\\h}{(H)}}{\\log{(H)}} + \\frac{4 \\bar{\\h}^{2}{(H)}}{H^{2} \\log{(H)}^{2}} = \\frac{2 \\bar{\\h}{(H)}}{\\log{(H)}} + \\frac{(\\bar{\\h}{(H)} + \\log{(H)})^{2}}{H^{2} \\log{(H)}^{2}}", "derivation": "\\bar{\\h}{(H)} = \\log{(H)} and 2 \\bar{\\h}{(H)} = \\bar{\\h}{(H)} + \\log{(H)} and \\frac{2 \\bar{\\h}{(H)}}{H} = \\frac{\\bar{\\h}{(H)} + \\log{(H)}}{H} and \\frac{2 \\bar{\\h}{(H)}}{H \\log{(H)}} = \\frac{\\bar{\\h}{(H)} + \\log{(H)}}{H \\log{(H)}} and \\frac{4 \\bar{\\h}^{2}{(H)}}{H^{2} \\log{(H)}^{2}} = \\frac{(\\bar{\\h}{(H)} + \\log{(H)})^{2}}{H^{2} \\log{(H)}^{2}} and \\frac{2 \\bar{\\h}{(H)}}{\\log{(H)}} + \\frac{4 \\bar{\\h}^{2}{(H)}}{H^{2} \\log{(H)}^{2}} = \\frac{2 \\bar{\\h}{(H)}}{\\log{(H)}} + \\frac{(\\bar{\\h}{(H)} + \\log{(H)})^{2}}{H^{2} \\log{(H)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["add", 1, "Function('\\\\hbar')(Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hbar')(Symbol('H', commutative=True))), Add(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True))))"], [["divide", 2, "Symbol('H', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))))"], [["times", 3, "Pow(log(Symbol('H', commutative=True)), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\hbar')(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True))), Pow(log(Symbol('H', commutative=True)), Integer(-1))))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True)), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(-2))), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Add(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True))), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(-2))))"], [["add", 5, "Mul(Integer(2), Function('\\\\hbar')(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(2), Function('\\\\hbar')(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Mul(Integer(4), Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Function('\\\\hbar')(Symbol('H', commutative=True)), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(-2)))), Add(Mul(Integer(2), Function('\\\\hbar')(Symbol('H', commutative=True)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Mul(Pow(Symbol('H', commutative=True), Integer(-2)), Pow(Add(Function('\\\\hbar')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True))), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given A{(\\mathbf{H},v_{t})} = e^{\\mathbf{H} v_{t}}, then obtain \\frac{\\partial}{\\partial v_{t}} \\iint A{(\\mathbf{H},v_{t})} d\\mathbf{H} dv_{t} - 1 = \\frac{\\partial}{\\partial v_{t}} \\iint e^{\\mathbf{H} v_{t}} d\\mathbf{H} dv_{t} - 1", "derivation": "A{(\\mathbf{H},v_{t})} = e^{\\mathbf{H} v_{t}} and \\int A{(\\mathbf{H},v_{t})} d\\mathbf{H} = \\int e^{\\mathbf{H} v_{t}} d\\mathbf{H} and \\iint A{(\\mathbf{H},v_{t})} d\\mathbf{H} dv_{t} = \\iint e^{\\mathbf{H} v_{t}} d\\mathbf{H} dv_{t} and \\frac{\\partial}{\\partial v_{t}} \\iint A{(\\mathbf{H},v_{t})} d\\mathbf{H} dv_{t} = \\frac{\\partial}{\\partial v_{t}} \\iint e^{\\mathbf{H} v_{t}} d\\mathbf{H} dv_{t} and \\frac{\\partial}{\\partial v_{t}} \\iint A{(\\mathbf{H},v_{t})} d\\mathbf{H} dv_{t} - 1 = \\frac{\\partial}{\\partial v_{t}} \\iint e^{\\mathbf{H} v_{t}} d\\mathbf{H} dv_{t} - 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True)), exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Integral(exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Derivative(Integral(Function('A')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(exp(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_t', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\psi^{*}{(\\phi_2)} = \\phi_2 and G{(\\phi_2)} = - \\psi^{*}{(\\phi_2)}, then obtain \\frac{d}{d \\phi_2} G^{\\phi_2}{(\\phi_2)} = \\frac{d}{d \\phi_2} (- \\phi_2)^{\\phi_2}", "derivation": "\\psi^{*}{(\\phi_2)} = \\phi_2 and - \\psi^{*}{(\\phi_2)} = - \\phi_2 and (- \\psi^{*}{(\\phi_2)})^{\\phi_2} = (- \\phi_2)^{\\phi_2} and \\frac{d}{d \\phi_2} (- \\psi^{*}{(\\phi_2)})^{\\phi_2} = \\frac{d}{d \\phi_2} (- \\phi_2)^{\\phi_2} and G{(\\phi_2)} = - \\psi^{*}{(\\phi_2)} and \\frac{d}{d \\phi_2} G^{\\phi_2}{(\\phi_2)} = \\frac{d}{d \\phi_2} (- \\phi_2)^{\\phi_2}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Pow(Function('G')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{1}{(Z)} = \\sin{(Z)}, then obtain \\int \\frac{d}{d Z} (2 Z \\phi_{1}^{Z}{(Z)} - \\phi_{1}{(Z)}) dZ = \\int \\frac{d}{d Z} (2 Z \\sin^{Z}{(Z)} - \\phi_{1}{(Z)}) dZ", "derivation": "\\phi_{1}{(Z)} = \\sin{(Z)} and \\phi_{1}^{Z}{(Z)} = \\sin^{Z}{(Z)} and 2 Z \\phi_{1}^{Z}{(Z)} = 2 Z \\sin^{Z}{(Z)} and 2 Z \\phi_{1}^{Z}{(Z)} - \\phi_{1}{(Z)} = 2 Z \\sin^{Z}{(Z)} - \\phi_{1}{(Z)} and \\frac{d}{d Z} (2 Z \\phi_{1}^{Z}{(Z)} - \\phi_{1}{(Z)}) = \\frac{d}{d Z} (2 Z \\sin^{Z}{(Z)} - \\phi_{1}{(Z)}) and \\int \\frac{d}{d Z} (2 Z \\phi_{1}^{Z}{(Z)} - \\phi_{1}{(Z)}) dZ = \\int \\frac{d}{d Z} (2 Z \\sin^{Z}{(Z)} - \\phi_{1}{(Z)}) dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 2, "Mul(Integer(2), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Symbol('Z', commutative=True), Pow(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(2), Symbol('Z', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["minus", 3, "Function('\\\\phi_1')(Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))), Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(Function('\\\\phi_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Add(Mul(Integer(2), Symbol('Z', commutative=True), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(m,Z)} = Z m, then obtain \\mathbf{A}{(m,Z)} + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)}) = \\frac{Z^{2} m^{2}}{\\mathbf{A}{(m,Z)}} + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)})", "derivation": "\\mathbf{A}{(m,Z)} = Z m and Z \\mathbf{A}{(m,Z)} = Z^{2} m and \\mathbf{A}{(m,Z)} + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)}) = Z m + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)}) and Z m = \\frac{Z^{2} m^{2}}{\\mathbf{A}{(m,Z)}} and \\mathbf{A}{(m,Z)} + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)}) = \\frac{Z^{2} m^{2}}{\\mathbf{A}{(m,Z)}} + \\frac{\\partial}{\\partial Z} (Z - \\mathbf{A}{(m,Z)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True)))"], [["times", 1, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(2)), Symbol('m', commutative=True)))"], [["add", 1, "Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["divide", 2, "Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Mul(Symbol('Z', commutative=True), Symbol('m', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(2)), Pow(Symbol('m', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('Z', commutative=True), Integer(2)), Pow(Symbol('m', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(C_{d})} = \\cos{(C_{d})} and \\operatorname{A_{2}}{(C_{d})} = \\int \\operatorname{E_{n}}{(C_{d})} dC_{d}, then derive \\operatorname{A_{2}}{(C_{d})} = n_{2} + \\sin{(C_{d})}, then obtain (n_{2} + \\sin{(C_{d})})^{C_{d}} = (\\int \\operatorname{E_{n}}{(C_{d})} dC_{d})^{C_{d}}", "derivation": "\\operatorname{E_{n}}{(C_{d})} = \\cos{(C_{d})} and \\int \\operatorname{E_{n}}{(C_{d})} dC_{d} = \\int \\cos{(C_{d})} dC_{d} and \\operatorname{A_{2}}{(C_{d})} = \\int \\operatorname{E_{n}}{(C_{d})} dC_{d} and \\operatorname{A_{2}}{(C_{d})} = \\int \\cos{(C_{d})} dC_{d} and \\operatorname{A_{2}}{(C_{d})} = n_{2} + \\sin{(C_{d})} and \\operatorname{A_{2}}^{C_{d}}{(C_{d})} = (\\int \\operatorname{E_{n}}{(C_{d})} dC_{d})^{C_{d}} and (n_{2} + \\sin{(C_{d})})^{C_{d}} = (\\int \\operatorname{E_{n}}{(C_{d})} dC_{d})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('C_d', commutative=True)), Integral(Function('E_n')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('A_2')(Symbol('C_d', commutative=True)), Integral(cos(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Function('A_2')(Symbol('C_d', commutative=True)), Add(Symbol('n_2', commutative=True), sin(Symbol('C_d', commutative=True))))"], [["power", 3, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Integral(Function('E_n')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Add(Symbol('n_2', commutative=True), sin(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)), Pow(Integral(Function('E_n')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given s{(b,h)} = b + h and \\operatorname{F_{x}}{(b,h)} = b + h, then obtain h = h + \\operatorname{F_{x}}{(b,h)} - s{(b,h)}", "derivation": "s{(b,h)} = b + h and \\operatorname{F_{x}}{(b,h)} = b + h and s{(b,h)} = \\operatorname{F_{x}}{(b,h)} and h + s{(b,h)} = h + \\operatorname{F_{x}}{(b,h)} and h = h + \\operatorname{F_{x}}{(b,h)} - s{(b,h)}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('b', commutative=True), Symbol('h', commutative=True)), Add(Symbol('b', commutative=True), Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('b', commutative=True), Symbol('h', commutative=True)), Add(Symbol('b', commutative=True), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('s')(Symbol('b', commutative=True), Symbol('h', commutative=True)), Function('F_x')(Symbol('b', commutative=True), Symbol('h', commutative=True)))"], [["add", 3, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('s')(Symbol('b', commutative=True), Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), Function('F_x')(Symbol('b', commutative=True), Symbol('h', commutative=True))))"], [["minus", 4, "Function('s')(Symbol('b', commutative=True), Symbol('h', commutative=True))"], "Equality(Symbol('h', commutative=True), Add(Symbol('h', commutative=True), Function('F_x')(Symbol('b', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('b', commutative=True), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(F_{c},\\mathbf{J})} = \\int F_{c} \\mathbf{J} d\\mathbf{J} and b{(E_{n},i)} = (e^{i})^{E_{n}}, then obtain \\frac{b{(E_{n},i)} \\int F_{c} \\mathbf{J} d\\mathbf{J}}{F_{c}} = \\frac{(e^{i})^{E_{n}} \\int F_{c} \\mathbf{J} d\\mathbf{J}}{F_{c}}", "derivation": "\\operatorname{f_{E}}{(F_{c},\\mathbf{J})} = \\int F_{c} \\mathbf{J} d\\mathbf{J} and \\frac{\\operatorname{f_{E}}{(F_{c},\\mathbf{J})}}{F_{c}} = \\frac{\\int F_{c} \\mathbf{J} d\\mathbf{J}}{F_{c}} and b{(E_{n},i)} = (e^{i})^{E_{n}} and \\frac{b{(E_{n},i)} \\operatorname{f_{E}}{(F_{c},\\mathbf{J})}}{F_{c}} = \\frac{\\operatorname{f_{E}}{(F_{c},\\mathbf{J})} (e^{i})^{E_{n}}}{F_{c}} and \\frac{b{(E_{n},i)} \\int F_{c} \\mathbf{J} d\\mathbf{J}}{F_{c}} = \\frac{(e^{i})^{E_{n}} \\int F_{c} \\mathbf{J} d\\mathbf{J}}{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 1, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('f_E')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], ["get_premise", "Equality(Function('b')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Pow(exp(Symbol('i', commutative=True)), Symbol('E_n', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('f_E')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('b')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Function('f_E')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('f_E')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(exp(Symbol('i', commutative=True)), Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('b')(Symbol('E_n', commutative=True), Symbol('i', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(exp(Symbol('i', commutative=True)), Symbol('E_n', commutative=True)), Integral(Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(h)} = \\log{(\\log{(h)})}, then derive \\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh = f_{\\mathbf{p}} + h \\log{(\\log{(h)})} - \\operatorname{li}{(h)}, then obtain (\\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh)^{h} = (f_{\\mathbf{p}} + h \\operatorname{V_{\\mathbf{E}}}{(h)} - \\operatorname{li}{(h)})^{h}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(h)} = \\log{(\\log{(h)})} and \\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh = \\int \\log{(\\log{(h)})} dh and \\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh = f_{\\mathbf{p}} + h \\log{(\\log{(h)})} - \\operatorname{li}{(h)} and (\\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh)^{h} = (f_{\\mathbf{p}} + h \\log{(\\log{(h)})} - \\operatorname{li}{(h)})^{h} and (\\int \\operatorname{V_{\\mathbf{E}}}{(h)} dh)^{h} = (f_{\\mathbf{p}} + h \\operatorname{V_{\\mathbf{E}}}{(h)} - \\operatorname{li}{(h)})^{h}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), log(log(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(log(log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Symbol('h', commutative=True), log(log(Symbol('h', commutative=True)))), Mul(Integer(-1), li(Symbol('h', commutative=True)))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Symbol('h', commutative=True), log(log(Symbol('h', commutative=True)))), Mul(Integer(-1), li(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Mul(Symbol('h', commutative=True), Function('V_{\\\\mathbf{E}}')(Symbol('h', commutative=True))), Mul(Integer(-1), li(Symbol('h', commutative=True)))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given h{(\\Omega,v_{z})} = v_{z}^{\\Omega}, then obtain \\frac{\\partial}{\\partial \\Omega} (- v_{z} + h{(\\Omega,v_{z})}) = \\frac{\\partial}{\\partial \\Omega} (- v_{z} + v_{z}^{\\Omega})", "derivation": "h{(\\Omega,v_{z})} = v_{z}^{\\Omega} and h^{v_{z}}{(\\Omega,v_{z})} = (v_{z}^{\\Omega})^{v_{z}} and - v_{z} + 2 h{(\\Omega,v_{z})} + h^{v_{z}}{(\\Omega,v_{z})} = - v_{z} + v_{z}^{\\Omega} + h{(\\Omega,v_{z})} + h^{v_{z}}{(\\Omega,v_{z})} and - v_{z} + (v_{z}^{\\Omega})^{v_{z}} + 2 h{(\\Omega,v_{z})} = - v_{z} + v_{z}^{\\Omega} + (v_{z}^{\\Omega})^{v_{z}} + h{(\\Omega,v_{z})} and - v_{z} + h{(\\Omega,v_{z})} = - v_{z} + v_{z}^{\\Omega} and \\frac{\\partial}{\\partial \\Omega} (- v_{z} + h{(\\Omega,v_{z})}) = \\frac{\\partial}{\\partial \\Omega} (- v_{z} + v_{z}^{\\Omega})", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('v_z', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Mul(Integer(2), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Pow(Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Pow(Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('v_z', commutative=True)), Mul(Integer(2), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('v_z', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))))"], [["minus", 4, "Add(Pow(Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('v_z', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('h')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(W,\\varepsilon)} = \\frac{\\partial}{\\partial W} W^{\\varepsilon}, then derive \\Psi_{nl}{(W,\\varepsilon)} = \\frac{W^{\\varepsilon} \\varepsilon}{W}, then obtain e^{\\Psi_{nl}^{\\varepsilon}{(W,\\varepsilon)}} = e^{(\\frac{\\partial}{\\partial W} W^{\\varepsilon})^{\\varepsilon}}", "derivation": "\\Psi_{nl}{(W,\\varepsilon)} = \\frac{\\partial}{\\partial W} W^{\\varepsilon} and \\Psi_{nl}{(W,\\varepsilon)} = \\frac{W^{\\varepsilon} \\varepsilon}{W} and \\frac{W^{\\varepsilon} \\varepsilon}{W} = \\frac{\\partial}{\\partial W} W^{\\varepsilon} and \\Psi_{nl}^{\\varepsilon}{(W,\\varepsilon)} = (\\frac{W^{\\varepsilon} \\varepsilon}{W})^{\\varepsilon} and \\Psi_{nl}^{\\varepsilon}{(W,\\varepsilon)} = (\\frac{\\partial}{\\partial W} W^{\\varepsilon})^{\\varepsilon} and e^{\\Psi_{nl}^{\\varepsilon}{(W,\\varepsilon)}} = e^{(\\frac{\\partial}{\\partial W} W^{\\varepsilon})^{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Derivative(Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Derivative(Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Function('\\\\Psi_{nl}')(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), exp(Pow(Derivative(Pow(Symbol('W', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\pi{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})}, then obtain f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})} + \\log{((f_{\\mathbf{v}} \\pi{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})} = f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})} + \\log{((f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})}", "derivation": "\\pi{(f_{\\mathbf{v}})} = \\sin{(f_{\\mathbf{v}})} and f_{\\mathbf{v}} \\pi{(f_{\\mathbf{v}})} = f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})} and (f_{\\mathbf{v}} \\pi{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}} and \\log{((f_{\\mathbf{v}} \\pi{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})} = \\log{((f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})} and f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})} + \\log{((f_{\\mathbf{v}} \\pi{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})} = f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})} + \\log{((f_{\\mathbf{v}} \\sin{(f_{\\mathbf{v}})})^{f_{\\mathbf{v}}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["log", 3], "Equality(log(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 4, "Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('\\\\pi')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Pow(Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbb{I},\\Psi)} = \\Psi^{\\mathbb{I}}, then obtain \\Psi^{\\mathbb{I}} = \\Psi^{\\mathbb{I}} + \\frac{\\Psi^{\\mathbb{I}}}{\\mathbb{I}} - \\frac{\\Psi_{\\lambda}{(\\mathbb{I},\\Psi)}}{\\mathbb{I}}", "derivation": "\\Psi_{\\lambda}{(\\mathbb{I},\\Psi)} = \\Psi^{\\mathbb{I}} and \\frac{\\Psi_{\\lambda}{(\\mathbb{I},\\Psi)}}{\\mathbb{I}} = \\frac{\\Psi^{\\mathbb{I}}}{\\mathbb{I}} and 0 = \\frac{\\Psi^{\\mathbb{I}}}{\\mathbb{I}} - \\frac{\\Psi_{\\lambda}{(\\mathbb{I},\\Psi)}}{\\mathbb{I}} and \\Psi^{\\mathbb{I}} = \\Psi^{\\mathbb{I}} + \\frac{\\Psi^{\\mathbb{I}}}{\\mathbb{I}} - \\frac{\\Psi_{\\lambda}{(\\mathbb{I},\\Psi)}}{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi', commutative=True)))))"], [["add", 3, "Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{A})} = \\sin{(\\mathbf{A})}, then obtain \\hat{X} + \\mathbf{E}{(\\mathbf{A})} = G + \\sin{(\\mathbf{A})}", "derivation": "\\mathbf{E}{(\\mathbf{A})} = \\sin{(\\mathbf{A})} and \\frac{d}{d \\mathbf{A}} \\mathbf{E}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} and \\int \\frac{d}{d \\mathbf{A}} \\mathbf{E}{(\\mathbf{A})} d\\mathbf{A} = \\int \\frac{d}{d \\mathbf{A}} \\sin{(\\mathbf{A})} d\\mathbf{A} and \\hat{X} + \\mathbf{E}{(\\mathbf{A})} = G + \\sin{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Derivative(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('G', commutative=True), sin(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{B},l)} = - \\mathbf{B} + l, then derive \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + \\frac{\\partial}{\\partial l} \\mathbf{S}{(\\mathbf{B},l)} = \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + 1, then obtain l + \\frac{\\partial}{\\partial l} (- \\mathbf{B} + l) = l + 1", "derivation": "\\mathbf{S}{(\\mathbf{B},l)} = - \\mathbf{B} + l and \\frac{\\partial}{\\partial l} \\mathbf{S}{(\\mathbf{B},l)} = \\frac{\\partial}{\\partial l} (- \\mathbf{B} + l) and \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + \\frac{\\partial}{\\partial l} \\mathbf{S}{(\\mathbf{B},l)} = \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + \\frac{\\partial}{\\partial l} (- \\mathbf{B} + l) and \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + \\frac{\\partial}{\\partial l} \\mathbf{S}{(\\mathbf{B},l)} = \\mathbf{B} + \\mathbf{S}{(\\mathbf{B},l)} + 1 and l + \\frac{\\partial}{\\partial l} (- \\mathbf{B} + l) = l + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 2, "Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('l', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('l', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('l', commutative=True), Integer(1)))"]]}, {"prompt": "Given \\rho{(\\mathbf{S})} = \\log{(\\mathbf{S})}, then obtain (\\mathbf{S}^{2} \\rho^{2}{(\\mathbf{S})} \\log{(\\mathbf{S})}^{2})^{\\mathbf{S}} = (\\mathbf{S}^{2} \\log{(\\mathbf{S})}^{4})^{\\mathbf{S}}", "derivation": "\\rho{(\\mathbf{S})} = \\log{(\\mathbf{S})} and \\mathbf{S} \\rho{(\\mathbf{S})} = \\mathbf{S} \\log{(\\mathbf{S})} and \\mathbf{S} \\rho^{2}{(\\mathbf{S})} = \\mathbf{S} \\rho{(\\mathbf{S})} \\log{(\\mathbf{S})} and \\mathbf{S} \\rho{(\\mathbf{S})} \\log{(\\mathbf{S})} = \\mathbf{S} \\log{(\\mathbf{S})}^{2} and \\mathbf{S}^{2} \\rho^{2}{(\\mathbf{S})} \\log{(\\mathbf{S})}^{2} = \\mathbf{S}^{2} \\log{(\\mathbf{S})}^{4} and (\\mathbf{S}^{2} \\rho^{2}{(\\mathbf{S})} \\log{(\\mathbf{S})}^{2})^{\\mathbf{S}} = (\\mathbf{S}^{2} \\log{(\\mathbf{S})}^{4})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))))"], [["power", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4))))"], [["power", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(Function('\\\\rho')(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(4))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{f}{(A_{2},E_{x})} = A_{2} + E_{x}, then obtain \\int 0 dA_{2} = \\int (\\int 0^{A_{2}} dA_{2} - \\int (- A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})})^{A_{2}} dA_{2}) dA_{2}", "derivation": "\\mathbf{f}{(A_{2},E_{x})} = A_{2} + E_{x} and - A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})} = 0 and (- A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})})^{A_{2}} = 0^{A_{2}} and \\int (- A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})})^{A_{2}} dA_{2} = \\int 0^{A_{2}} dA_{2} and 0 = \\int 0^{A_{2}} dA_{2} - \\int (- A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})})^{A_{2}} dA_{2} and \\int 0 dA_{2} = \\int (\\int 0^{A_{2}} dA_{2} - \\int (- A_{2} - E_{x} + \\mathbf{f}{(A_{2},E_{x})})^{A_{2}} dA_{2}) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True)))"], [["minus", 1, "Add(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Integer(0))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('A_2', commutative=True)), Pow(Integer(0), Symbol('A_2', commutative=True)))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Pow(Integer(0), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["minus", 4, "Integral(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))"], "Equality(Integer(0), Add(Integral(Pow(Integer(0), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Mul(Integer(-1), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))))"], [["integrate", 5, "Symbol('A_2', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Integral(Pow(Integer(0), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Mul(Integer(-1), Integral(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True)), Function('\\\\mathbf{f}')(Symbol('A_2', commutative=True), Symbol('E_x', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(U,\\delta)} = \\delta^{U}, then derive \\delta^{U} \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} + 2 \\delta^{U} = \\delta^{2 U} \\log{(\\delta)} + 2 \\delta^{U}, then obtain \\delta^{U} \\frac{\\partial}{\\partial U} \\delta^{U} + 2 \\delta^{U} = \\delta^{2 U} \\log{(\\delta)} + 2 \\delta^{U}", "derivation": "\\mu_{0}{(U,\\delta)} = \\delta^{U} and \\delta^{U} + \\mu_{0}{(U,\\delta)} = 2 \\delta^{U} and \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} = \\frac{\\partial}{\\partial U} \\delta^{U} and \\delta^{U} \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} = \\delta^{U} \\frac{\\partial}{\\partial U} \\delta^{U} and \\delta^{U} \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} + \\delta^{U} + \\mu_{0}{(U,\\delta)} = \\delta^{U} \\frac{\\partial}{\\partial U} \\delta^{U} + \\delta^{U} + \\mu_{0}{(U,\\delta)} and \\delta^{U} \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} + 2 \\delta^{U} = \\delta^{U} \\frac{\\partial}{\\partial U} \\delta^{U} + 2 \\delta^{U} and \\delta^{U} \\frac{\\partial}{\\partial U} \\mu_{0}{(U,\\delta)} + 2 \\delta^{U} = \\delta^{2 U} \\log{(\\delta)} + 2 \\delta^{U} and \\delta^{U} \\frac{\\partial}{\\partial U} \\delta^{U} + 2 \\delta^{U} = \\delta^{2 U} \\log{(\\delta)} + 2 \\delta^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["times", 3, "Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["add", 4, "Add(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(2), Symbol('U', commutative=True))), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Derivative(Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(2), Symbol('U', commutative=True))), log(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\delta', commutative=True), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(x,M)} = x \\cos{(M)} and \\lambda{(x,M)} = (x \\cos{(M)})^{M}, then obtain \\frac{\\int \\frac{\\partial}{\\partial x} \\lambda{(x,M)} dM}{\\mathbf{A}{(x,M)}} = \\frac{\\int \\frac{\\partial}{\\partial x} (x \\cos{(M)})^{M} dM}{\\mathbf{A}{(x,M)}}", "derivation": "\\mathbf{A}{(x,M)} = x \\cos{(M)} and \\lambda{(x,M)} = (x \\cos{(M)})^{M} and \\frac{\\partial}{\\partial x} \\lambda{(x,M)} = \\frac{\\partial}{\\partial x} (x \\cos{(M)})^{M} and \\frac{\\partial}{\\partial x} \\lambda{(x,M)} = \\frac{\\partial}{\\partial x} \\mathbf{A}^{M}{(x,M)} and \\int \\frac{\\partial}{\\partial x} \\lambda{(x,M)} dM = \\int \\frac{\\partial}{\\partial x} \\mathbf{A}^{M}{(x,M)} dM and \\int \\frac{\\partial}{\\partial x} \\lambda{(x,M)} dM = \\int \\frac{\\partial}{\\partial x} (x \\cos{(M)})^{M} dM and \\frac{\\int \\frac{\\partial}{\\partial x} \\lambda{(x,M)} dM}{\\mathbf{A}{(x,M)}} = \\frac{\\int \\frac{\\partial}{\\partial x} (x \\cos{(M)})^{M} dM}{\\mathbf{A}{(x,M)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('x', commutative=True), cos(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Pow(Mul(Symbol('x', commutative=True), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('x', commutative=True), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(Pow(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Derivative(Pow(Mul(Symbol('x', commutative=True), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"], [["divide", 6, "Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Integral(Derivative(Function('\\\\lambda')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('x', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Integral(Derivative(Pow(Mul(Symbol('x', commutative=True), cos(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})} = \\frac{1}{a^{\\dagger}}, then obtain - \\mu (a^{\\dagger} + \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})}) - (\\frac{\\varphi}{a^{\\dagger}})^{\\varphi} = - \\mu (a^{\\dagger} + \\frac{1}{a^{\\dagger}}) - (\\frac{\\varphi}{a^{\\dagger}})^{\\varphi}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})} = \\frac{1}{a^{\\dagger}} and a^{\\dagger} + \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})} = a^{\\dagger} + \\frac{1}{a^{\\dagger}} and - \\mu (a^{\\dagger} + \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})}) = - \\mu (a^{\\dagger} + \\frac{1}{a^{\\dagger}}) and - \\mu (a^{\\dagger} + \\operatorname{f_{\\mathbf{v}}}{(a^{\\dagger})}) - (\\frac{\\varphi}{a^{\\dagger}})^{\\varphi} = - \\mu (a^{\\dagger} + \\frac{1}{a^{\\dagger}}) - (\\frac{\\varphi}{a^{\\dagger}})^{\\varphi}", "srepr_derivation": [["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))))"], [["minus", 3, "Pow(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('f_{\\\\mathbf{v}}')(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Pow(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Mul(Symbol('\\\\varphi', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1))), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(\\hbar)} = \\frac{d}{d \\hbar} \\sin{(\\hbar)}, then obtain e^{\\mathbf{H}^{\\hbar}{(\\hbar)}} (\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar} = e^{(\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar}} (\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar}", "derivation": "\\mathbf{H}{(\\hbar)} = \\frac{d}{d \\hbar} \\sin{(\\hbar)} and \\mathbf{H}^{\\hbar}{(\\hbar)} = (\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar} and e^{\\mathbf{H}^{\\hbar}{(\\hbar)}} = e^{(\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar}} and e^{\\mathbf{H}^{\\hbar}{(\\hbar)}} (\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar} = e^{(\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar}} (\\frac{d}{d \\hbar} \\sin{(\\hbar)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), exp(Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))))"], [["times", 3, "Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(exp(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))), Mul(exp(Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))), Pow(Derivative(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{x}_0{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})}, then obtain \\hat{\\mathbf{r}}{(L_{\\varepsilon})} - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\hat{x}_0{(L_{\\varepsilon})})} = \\hat{\\mathbf{r}}{(L_{\\varepsilon})} - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\cos{(L_{\\varepsilon})})}", "derivation": "\\hat{x}_0{(L_{\\varepsilon})} = \\cos{(L_{\\varepsilon})} and \\log{(\\hat{x}_0{(L_{\\varepsilon})})} = \\log{(\\cos{(L_{\\varepsilon})})} and - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\hat{x}_0{(L_{\\varepsilon})})} = - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\cos{(L_{\\varepsilon})})} and \\hat{\\mathbf{r}}{(L_{\\varepsilon})} - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\hat{x}_0{(L_{\\varepsilon})})} = \\hat{\\mathbf{r}}{(L_{\\varepsilon})} - \\hat{x}_0{(L_{\\varepsilon})} + \\log{(\\cos{(L_{\\varepsilon})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"], [["add", 3, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(cos(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given r{(v_{x})} = e^{v_{x}}, then obtain 2 \\frac{d}{d v_{x}} r{(v_{x})} = e^{v_{x}} + \\frac{d}{d v_{x}} r{(v_{x})}", "derivation": "r{(v_{x})} = e^{v_{x}} and 2 r{(v_{x})} = r{(v_{x})} + e^{v_{x}} and \\frac{d}{d v_{x}} 2 r{(v_{x})} = \\frac{d}{d v_{x}} (r{(v_{x})} + e^{v_{x}}) and 2 \\frac{d}{d v_{x}} r{(v_{x})} = e^{v_{x}} + \\frac{d}{d v_{x}} r{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["add", 1, "Function('r')(Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Function('r')(Symbol('v_x', commutative=True))), Add(Function('r')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('r')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Function('r')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('r')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Add(exp(Symbol('v_x', commutative=True)), Derivative(Function('r')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)} = \\mu + \\varepsilon_0, then obtain 0 = \\mu + \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)}", "derivation": "\\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)} = \\mu + \\varepsilon_0 and \\mu + \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)} = 2 \\mu + \\varepsilon_0 and 0 = \\mu + \\varepsilon_0 - \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)} and 0 = (\\mu + \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)}) (\\mu + \\varepsilon_0 - \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)}) and 0 = \\mu + \\dot{\\mathbf{r}}{(\\varepsilon_0,\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["minus", 2, "Add(Symbol('\\\\mu', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["times", 3, "Add(Symbol('\\\\mu', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Integer(0), Mul(Add(Symbol('\\\\mu', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))))))"], [["divide", 4, "Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))))"], "Equality(Integer(0), Add(Symbol('\\\\mu', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then obtain \\bar{\\h}^{6}{(\\mathbf{r})} + \\frac{\\bar{\\h}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}}{\\mathbf{r}} = \\bar{\\h}^{4}{(\\mathbf{r})} \\log{(\\mathbf{r})}^{2} + \\frac{\\bar{\\h}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}}{\\mathbf{r}}", "derivation": "\\bar{\\h}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\bar{\\h}^{2}{(\\mathbf{r})} = \\bar{\\h}{(\\mathbf{r})} \\log{(\\mathbf{r})} and \\bar{\\h}^{3}{(\\mathbf{r})} = \\bar{\\h}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})} and \\bar{\\h}^{6}{(\\mathbf{r})} = \\bar{\\h}^{4}{(\\mathbf{r})} \\log{(\\mathbf{r})}^{2} and \\bar{\\h}^{6}{(\\mathbf{r})} + \\frac{\\bar{\\h}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}}{\\mathbf{r}} = \\bar{\\h}^{4}{(\\mathbf{r})} \\log{(\\mathbf{r})}^{2} + \\frac{\\bar{\\h}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}}{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Mul(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 2, "Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(6)), Mul(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(6)), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{r}', commutative=True)))), Add(Mul(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(4)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(f,Q)} = \\int Q^{f} dQ, then obtain \\frac{d}{d f} 0 = \\frac{\\partial}{\\partial f} (- \\tilde{g}{(f,Q)} + \\int Q^{f} dQ)", "derivation": "\\tilde{g}{(f,Q)} = \\int Q^{f} dQ and Q^{f} + \\tilde{g}{(f,Q)} = Q^{f} + \\int Q^{f} dQ and 0 = - \\tilde{g}{(f,Q)} + \\int Q^{f} dQ and \\frac{d}{d f} 0 = \\frac{\\partial}{\\partial f} (- \\tilde{g}{(f,Q)} + \\int Q^{f} dQ)", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)), Integral(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["add", 1, "Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True))"], "Equality(Add(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Function('\\\\tilde{g}')(Symbol('f', commutative=True), Symbol('Q', commutative=True))), Add(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Integral(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["minus", 2, "Add(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Function('\\\\tilde{g}')(Symbol('f', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('f', commutative=True), Symbol('Q', commutative=True))), Integral(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('f', commutative=True), Symbol('Q', commutative=True))), Integral(Pow(Symbol('Q', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})} and m{(\\mathbf{M})} = \\operatorname{P_{e}}^{\\mathbf{M}}{(\\mathbf{M})} + \\cos{(\\mathbf{M})}, then obtain m{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{\\mathbf{M}} + \\cos{(\\mathbf{M})}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})} and \\operatorname{P_{e}}^{\\mathbf{M}}{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{\\mathbf{M}} and \\operatorname{P_{e}}^{\\mathbf{M}}{(\\mathbf{M})} + \\cos{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{\\mathbf{M}} + \\cos{(\\mathbf{M})} and m{(\\mathbf{M})} = \\operatorname{P_{e}}^{\\mathbf{M}}{(\\mathbf{M})} + \\cos{(\\mathbf{M})} and m{(\\mathbf{M})} = \\log{(\\cos{(\\mathbf{M})})}^{\\mathbf{M}} + \\cos{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{M}', commutative=True)), log(cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 2, "cos(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Pow(Function('P_e')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True))), Add(Pow(log(cos(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\mathbf{M}', commutative=True)), Add(Pow(Function('P_e')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('m')(Symbol('\\\\mathbf{M}', commutative=True)), Add(Pow(log(cos(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\nabla{(f,n)} = f^{n}, then obtain ((\\frac{\\partial}{\\partial f} \\nabla{(f,n)})^{n})^{f} = ((\\frac{\\partial}{\\partial f} f^{n})^{n})^{f}", "derivation": "\\nabla{(f,n)} = f^{n} and \\frac{\\partial}{\\partial f} \\nabla{(f,n)} = \\frac{\\partial}{\\partial f} f^{n} and (\\frac{\\partial}{\\partial f} \\nabla{(f,n)})^{n} = (\\frac{\\partial}{\\partial f} f^{n})^{n} and ((\\frac{\\partial}{\\partial f} \\nabla{(f,n)})^{n})^{f} = ((\\frac{\\partial}{\\partial f} f^{n})^{n})^{f}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\nabla')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\nabla')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('n', commutative=True)), Pow(Derivative(Pow(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('\\\\nabla')(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('n', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Derivative(Pow(Symbol('f', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('n', commutative=True)), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(n_{1},c)} = \\log{(c + n_{1})}, then obtain \\varepsilon{(n_{1},c)} - 2 \\log{(c + n_{1})} = - \\log{(c + n_{1})}", "derivation": "\\varepsilon{(n_{1},c)} = \\log{(c + n_{1})} and - c + \\varepsilon{(n_{1},c)} = - c + \\log{(c + n_{1})} and \\varepsilon{(n_{1},c)} - \\log{(c + n_{1})} = 0 and \\varepsilon{(n_{1},c)} - 2 \\log{(c + n_{1})} = - \\log{(c + n_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))"], [["minus", 1, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('c', commutative=True)), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))), Integer(0))"], [["add", 3, "Mul(Integer(-1), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Integer(2), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True))))), Mul(Integer(-1), log(Add(Symbol('c', commutative=True), Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(n_{1},z)} = e^{n_{1} + z}, then derive \\frac{\\partial}{\\partial n_{1}} \\phi_{1}{(n_{1},z)} = e^{n_{1} + z}, then obtain - \\phi_{1}^{2}{(n_{1},z)} = - 2 \\phi_{1}^{2}{(n_{1},z)} + \\phi_{1}{(n_{1},z)} \\frac{\\partial}{\\partial n_{1}} e^{n_{1} + z}", "derivation": "\\phi_{1}{(n_{1},z)} = e^{n_{1} + z} and \\frac{\\partial}{\\partial n_{1}} \\phi_{1}{(n_{1},z)} = \\frac{\\partial}{\\partial n_{1}} e^{n_{1} + z} and \\phi_{1}^{2}{(n_{1},z)} = \\phi_{1}{(n_{1},z)} e^{n_{1} + z} and \\frac{\\partial}{\\partial n_{1}} \\phi_{1}{(n_{1},z)} = e^{n_{1} + z} and \\frac{\\partial}{\\partial n_{1}} e^{n_{1} + z} = e^{n_{1} + z} and \\phi_{1}^{2}{(n_{1},z)} = \\phi_{1}{(n_{1},z)} \\frac{\\partial}{\\partial n_{1}} e^{n_{1} + z} and - \\phi_{1}^{2}{(n_{1},z)} = - 2 \\phi_{1}^{2}{(n_{1},z)} + \\phi_{1}{(n_{1},z)} \\frac{\\partial}{\\partial n_{1}} e^{n_{1} + z}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 1, "Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True))"], "Equality(Pow(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(2)), Mul(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(2)), Mul(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Derivative(exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["minus", 6, "Mul(Integer(2), Pow(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(2)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Integer(2), Pow(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Integer(2))), Mul(Function('\\\\phi_1')(Symbol('n_1', commutative=True), Symbol('z', commutative=True)), Derivative(exp(Add(Symbol('n_1', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\psi{(c_{0})} = \\cos{(\\sin{(c_{0})})}, then obtain \\frac{d^{2}}{d c_{0}^{2}} (c_{0} + \\int \\psi{(c_{0})} dc_{0}) = \\frac{d^{2}}{d c_{0}^{2}} (c_{0} + \\int \\cos{(\\sin{(c_{0})})} dc_{0})", "derivation": "\\psi{(c_{0})} = \\cos{(\\sin{(c_{0})})} and \\int \\psi{(c_{0})} dc_{0} = \\int \\cos{(\\sin{(c_{0})})} dc_{0} and c_{0} + \\int \\psi{(c_{0})} dc_{0} = c_{0} + \\int \\cos{(\\sin{(c_{0})})} dc_{0} and \\frac{d}{d c_{0}} (c_{0} + \\int \\psi{(c_{0})} dc_{0}) = \\frac{d}{d c_{0}} (c_{0} + \\int \\cos{(\\sin{(c_{0})})} dc_{0}) and \\frac{d^{2}}{d c_{0}^{2}} (c_{0} + \\int \\psi{(c_{0})} dc_{0}) = \\frac{d^{2}}{d c_{0}^{2}} (c_{0} + \\int \\cos{(\\sin{(c_{0})})} dc_{0})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('c_0', commutative=True)), cos(sin(Symbol('c_0', commutative=True))))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(cos(sin(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True))))"], [["add", 2, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Integral(Function('\\\\psi')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Add(Symbol('c_0', commutative=True), Integral(cos(sin(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True)))))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Symbol('c_0', commutative=True), Integral(Function('\\\\psi')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Symbol('c_0', commutative=True), Integral(cos(sin(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Symbol('c_0', commutative=True), Integral(Function('\\\\psi')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(2))), Derivative(Add(Symbol('c_0', commutative=True), Integral(cos(sin(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{H}{(J_{\\varepsilon})} = \\sin{(e^{J_{\\varepsilon}})} and \\hat{x}{(J_{\\varepsilon})} = \\mathbf{H}{(J_{\\varepsilon})} + \\sin{(e^{J_{\\varepsilon}})}, then obtain 1 = \\frac{\\hat{x}{(J_{\\varepsilon})}}{2 \\mathbf{H}{(J_{\\varepsilon})}}", "derivation": "\\mathbf{H}{(J_{\\varepsilon})} = \\sin{(e^{J_{\\varepsilon}})} and 2 \\mathbf{H}{(J_{\\varepsilon})} = \\mathbf{H}{(J_{\\varepsilon})} + \\sin{(e^{J_{\\varepsilon}})} and 1 = \\frac{\\mathbf{H}{(J_{\\varepsilon})} + \\sin{(e^{J_{\\varepsilon}})}}{2 \\mathbf{H}{(J_{\\varepsilon})}} and \\hat{x}{(J_{\\varepsilon})} = \\mathbf{H}{(J_{\\varepsilon})} + \\sin{(e^{J_{\\varepsilon}})} and 1 = \\frac{\\hat{x}{(J_{\\varepsilon})}}{2 \\mathbf{H}{(J_{\\varepsilon})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["divide", 2, "Mul(Integer(2), Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))), Pow(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Add(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), sin(exp(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Mul(Rational(1, 2), Function('\\\\hat{x}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{x}{(\\varphi)} = \\sin{(\\varphi)}, then obtain \\sin^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sigma_{x}{(\\varphi)})^{\\varphi} = \\sin^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sin{(\\varphi)})^{\\varphi}", "derivation": "\\sigma_{x}{(\\varphi)} = \\sin{(\\varphi)} and \\sigma_{x}^{\\varphi}{(\\varphi)} = \\sin^{\\varphi}{(\\varphi)} and \\frac{d}{d \\varphi} \\sigma_{x}{(\\varphi)} = \\frac{d}{d \\varphi} \\sin{(\\varphi)} and (\\frac{d}{d \\varphi} \\sigma_{x}{(\\varphi)})^{\\varphi} = (\\frac{d}{d \\varphi} \\sin{(\\varphi)})^{\\varphi} and \\sigma_{x}^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sigma_{x}{(\\varphi)})^{\\varphi} = \\sigma_{x}^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sin{(\\varphi)})^{\\varphi} and \\sin^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sigma_{x}{(\\varphi)})^{\\varphi} = \\sin^{\\varphi}{(\\varphi)} + (\\frac{d}{d \\varphi} \\sin{(\\varphi)})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)))"], [["add", 4, "Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))), Add(Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))), Add(Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(E_{n})} = \\log{(E_{n})}, then obtain \\cos{(4 \\mathbf{M}{(E_{n})} \\sin{(4 \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})})})} = \\cos{(4 \\mathbf{M}{(E_{n})} \\sin{(3 \\mathbf{M}{(E_{n})} + 3 \\log{(E_{n})})})}", "derivation": "\\mathbf{M}{(E_{n})} = \\log{(E_{n})} and 2 \\mathbf{M}{(E_{n})} + \\log{(E_{n})} = \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})} and 4 \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})} = 3 \\mathbf{M}{(E_{n})} + 3 \\log{(E_{n})} and \\sin{(4 \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})})} = \\sin{(3 \\mathbf{M}{(E_{n})} + 3 \\log{(E_{n})})} and 4 \\mathbf{M}{(E_{n})} \\sin{(4 \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})})} = 4 \\mathbf{M}{(E_{n})} \\sin{(3 \\mathbf{M}{(E_{n})} + 3 \\log{(E_{n})})} and \\cos{(4 \\mathbf{M}{(E_{n})} \\sin{(4 \\mathbf{M}{(E_{n})} + 2 \\log{(E_{n})})})} = \\cos{(4 \\mathbf{M}{(E_{n})} \\sin{(3 \\mathbf{M}{(E_{n})} + 3 \\log{(E_{n})})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], [["add", 1, "Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), log(Symbol('E_n', commutative=True))), Add(Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), Mul(Integer(2), log(Symbol('E_n', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), log(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True)))), Add(Mul(Integer(3), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(3), log(Symbol('E_n', commutative=True)))))"], [["sin", 3], "Equality(sin(Add(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True))))), sin(Add(Mul(Integer(3), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(3), log(Symbol('E_n', commutative=True))))))"], [["times", 4, "Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)))"], "Equality(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), sin(Add(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True)))))), Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), sin(Add(Mul(Integer(3), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(3), log(Symbol('E_n', commutative=True)))))))"], [["cos", 5], "Equality(cos(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), sin(Add(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True))))))), cos(Mul(Integer(4), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True)), sin(Add(Mul(Integer(3), Function('\\\\mathbf{M}')(Symbol('E_n', commutative=True))), Mul(Integer(3), log(Symbol('E_n', commutative=True))))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(S)} = \\sin{(S)} and \\theta_{1}{(S)} = \\operatorname{F_{x}}^{2}{(S)} \\sin{(S)}, then obtain \\theta_{1}{(S)} = \\operatorname{F_{x}}{(S)} \\sin^{2}{(S)}", "derivation": "\\operatorname{F_{x}}{(S)} = \\sin{(S)} and \\operatorname{F_{x}}^{2}{(S)} = \\operatorname{F_{x}}{(S)} \\sin{(S)} and \\theta_{1}{(S)} = \\operatorname{F_{x}}^{2}{(S)} \\sin{(S)} and \\theta_{1}{(S)} = \\operatorname{F_{x}}{(S)} \\sin^{2}{(S)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["times", 1, "Function('F_x')(Symbol('S', commutative=True))"], "Equality(Pow(Function('F_x')(Symbol('S', commutative=True)), Integer(2)), Mul(Function('F_x')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('S', commutative=True)), Mul(Pow(Function('F_x')(Symbol('S', commutative=True)), Integer(2)), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\theta_1')(Symbol('S', commutative=True)), Mul(Function('F_x')(Symbol('S', commutative=True)), Pow(sin(Symbol('S', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(L)} = \\log{(e^{L})}, then obtain \\int \\frac{d}{d L} L \\operatorname{A_{x}}{(L)} dL = \\int \\frac{d}{d L} L \\log{(e^{L})} dL", "derivation": "\\operatorname{A_{x}}{(L)} = \\log{(e^{L})} and L \\operatorname{A_{x}}{(L)} = L \\log{(e^{L})} and \\frac{d}{d L} L \\operatorname{A_{x}}{(L)} = \\frac{d}{d L} L \\log{(e^{L})} and \\int \\frac{d}{d L} L \\operatorname{A_{x}}{(L)} dL = \\int \\frac{d}{d L} L \\log{(e^{L})} dL", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('L', commutative=True)), log(exp(Symbol('L', commutative=True))))"], [["times", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('A_x')(Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), log(exp(Symbol('L', commutative=True)))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Symbol('L', commutative=True), Function('A_x')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Symbol('L', commutative=True), log(exp(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('L', commutative=True), Function('A_x')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Mul(Symbol('L', commutative=True), log(exp(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)} = V (- \\mathbf{J}_M + \\psi) and \\operatorname{f_{\\mathbf{v}}}{(\\psi,V,\\mathbf{J}_M)} = V^{2} \\psi (- \\mathbf{J}_M + \\psi), then obtain \\frac{\\partial}{\\partial \\psi} \\operatorname{f_{\\mathbf{v}}}{(\\psi,V,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\psi} V \\psi \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)}", "derivation": "\\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)} = V (- \\mathbf{J}_M + \\psi) and \\psi \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)} = V \\psi (- \\mathbf{J}_M + \\psi) and V \\psi \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)} = V^{2} \\psi (- \\mathbf{J}_M + \\psi) and \\operatorname{f_{\\mathbf{v}}}{(\\psi,V,\\mathbf{J}_M)} = V^{2} \\psi (- \\mathbf{J}_M + \\psi) and \\operatorname{f_{\\mathbf{v}}}{(\\psi,V,\\mathbf{J}_M)} = V \\psi \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)} and \\frac{\\partial}{\\partial \\psi} \\operatorname{f_{\\mathbf{v}}}{(\\psi,V,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\psi} V \\psi \\Psi_{\\lambda}{(V,\\mathbf{J}_M,\\psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Symbol('V', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["times", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Symbol('\\\\psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\psi', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(2)), Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\psi', commutative=True))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True), Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(2)), Symbol('\\\\psi', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True), Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\psi', commutative=True), Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\psi', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('V', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\phi_1,\\rho)} = \\cos{(\\phi_1 + \\rho)}, then obtain 2 \\phi_1 + 2 \\rho + \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho - \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)})} = 2 \\phi_1 + 2 \\rho + 2 \\cos{(\\phi_1 + \\rho - \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)})}", "derivation": "\\operatorname{P_{e}}{(\\phi_1,\\rho)} = \\cos{(\\phi_1 + \\rho)} and \\phi_1 + \\rho + \\operatorname{P_{e}}{(\\phi_1,\\rho)} = \\phi_1 + \\rho + \\cos{(\\phi_1 + \\rho)} and \\phi_1 + \\rho = \\phi_1 + \\rho - \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)} and 2 \\phi_1 + 2 \\rho + \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)} = 2 \\phi_1 + 2 \\rho + 2 \\cos{(\\phi_1 + \\rho)} and 2 \\phi_1 + 2 \\rho + \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho - \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)})} = 2 \\phi_1 + 2 \\rho + 2 \\cos{(\\phi_1 + \\rho - \\operatorname{P_{e}}{(\\phi_1,\\rho)} + \\cos{(\\phi_1 + \\rho)})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["minus", 2, "Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)))))"], [["add", 2, "Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True)))))), Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('\\\\rho', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Add(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\rho', commutative=True))))))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{x}_0,\\theta_1)} = \\hat{x}_0^{\\theta_1}, then obtain - 2 \\theta_1 = \\hat{x}_0^{\\theta_1} - 2 \\theta_1 - \\mathbb{I}{(\\hat{x}_0,\\theta_1)}", "derivation": "\\mathbb{I}{(\\hat{x}_0,\\theta_1)} = \\hat{x}_0^{\\theta_1} and - \\theta_1 + \\mathbb{I}{(\\hat{x}_0,\\theta_1)} = \\hat{x}_0^{\\theta_1} - \\theta_1 and - 2 \\theta_1 + \\mathbb{I}{(\\hat{x}_0,\\theta_1)} = \\hat{x}_0^{\\theta_1} - 2 \\theta_1 and - 2 \\theta_1 = \\hat{x}_0^{\\theta_1} - 2 \\theta_1 - \\mathbb{I}{(\\hat{x}_0,\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 3, "Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), Add(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\Psi)} = \\cos{(\\Psi)}, then obtain 0 = - (- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\tilde{g}{(\\Psi)}", "derivation": "\\tilde{g}{(\\Psi)} = \\cos{(\\Psi)} and 0 = - \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)} and 0 = (- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\cos{(\\Psi)} and - \\tilde{g}{(\\Psi)} = (- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\cos{(\\Psi)} - \\tilde{g}{(\\Psi)} and 0 = (- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\cos{(\\Psi)} - \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)} and 0 = ((- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\cos{(\\Psi)} - \\tilde{g}{(\\Psi)}) ((- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\cos{(\\Psi)} - \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) and 0 = - (- \\tilde{g}{(\\Psi)} + \\cos{(\\Psi)}) \\tilde{g}{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["times", 2, "cos(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Add(Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["times", 5, "Add(Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))))"], "Equality(Integer(0), Mul(Add(Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Function('\\\\tilde{g}')(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(n_{1})} = e^{n_{1}} and W{(n_{1})} = \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})}, then obtain - n_{1} + \\frac{d}{d n_{1}} \\sin{(\\operatorname{c_{0}}{(n_{1})})} = - n_{1} + W{(n_{1})}", "derivation": "\\operatorname{c_{0}}{(n_{1})} = e^{n_{1}} and \\sin{(\\operatorname{c_{0}}{(n_{1})})} = \\sin{(e^{n_{1}})} and \\frac{d}{d n_{1}} \\sin{(\\operatorname{c_{0}}{(n_{1})})} = \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} and - n_{1} + \\frac{d}{d n_{1}} \\sin{(\\operatorname{c_{0}}{(n_{1})})} = - n_{1} + \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} and W{(n_{1})} = \\frac{d}{d n_{1}} \\sin{(e^{n_{1}})} and - n_{1} + \\frac{d}{d n_{1}} \\sin{(\\operatorname{c_{0}}{(n_{1})})} = - n_{1} + W{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["sin", 1], "Equality(sin(Function('c_0')(Symbol('n_1', commutative=True))), sin(exp(Symbol('n_1', commutative=True))))"], [["differentiate", 2, "Symbol('n_1', commutative=True)"], "Equality(Derivative(sin(Function('c_0')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(sin(Function('c_0')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(sin(exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('W')(Symbol('n_1', commutative=True)), Derivative(sin(exp(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(sin(Function('c_0')(Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('W')(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given L{(c,\\mathbf{E})} = \\mathbf{E}^{c}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{E}^{2}} \\int (L{(c,\\mathbf{E})} + \\int \\mathbf{E}^{c} dc) dc = \\frac{\\partial^{2}}{\\partial \\mathbf{E}^{2}} \\int (\\mathbf{E}^{c} + \\int \\mathbf{E}^{c} dc) dc", "derivation": "L{(c,\\mathbf{E})} = \\mathbf{E}^{c} and L{(c,\\mathbf{E})} + \\int \\mathbf{E}^{c} dc = \\mathbf{E}^{c} + \\int \\mathbf{E}^{c} dc and \\int (L{(c,\\mathbf{E})} + \\int \\mathbf{E}^{c} dc) dc = \\int (\\mathbf{E}^{c} + \\int \\mathbf{E}^{c} dc) dc and \\frac{\\partial}{\\partial \\mathbf{E}} \\int (L{(c,\\mathbf{E})} + \\int \\mathbf{E}^{c} dc) dc = \\frac{\\partial}{\\partial \\mathbf{E}} \\int (\\mathbf{E}^{c} + \\int \\mathbf{E}^{c} dc) dc and \\frac{\\partial^{2}}{\\partial \\mathbf{E}^{2}} \\int (L{(c,\\mathbf{E})} + \\int \\mathbf{E}^{c} dc) dc = \\frac{\\partial^{2}}{\\partial \\mathbf{E}^{2}} \\int (\\mathbf{E}^{c} + \\int \\mathbf{E}^{c} dc) dc", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('c', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)))"], [["add", 1, "Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Function('L')(Symbol('c', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Function('L')(Symbol('c', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Integral(Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Integral(Add(Function('L')(Symbol('c', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Integral(Add(Function('L')(Symbol('c', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))), Derivative(Integral(Add(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\lambda{(W,C)} = W + \\log{(C)}, then derive \\frac{\\partial}{\\partial C} \\lambda{(W,C)} = \\frac{1}{C}, then obtain (\\frac{\\partial}{\\partial C} \\lambda{(W,C)})^{W} = (\\frac{1}{C})^{W}", "derivation": "\\lambda{(W,C)} = W + \\log{(C)} and \\frac{\\partial}{\\partial C} \\lambda{(W,C)} = \\frac{\\partial}{\\partial C} (W + \\log{(C)}) and \\frac{\\partial}{\\partial C} \\lambda{(W,C)} = \\frac{1}{C} and (\\frac{\\partial}{\\partial C} \\lambda{(W,C)})^{W} = (\\frac{\\partial}{\\partial C} (W + \\log{(C)}))^{W} and \\frac{\\partial}{\\partial C} (W + \\log{(C)}) = \\frac{1}{C} and (\\frac{\\partial}{\\partial C} \\lambda{(W,C)})^{W} = (\\frac{1}{C})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Add(Symbol('W', commutative=True), log(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Symbol('C', commutative=True), Integer(-1)))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\lambda')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Add(Symbol('W', commutative=True), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('W', commutative=True), log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Symbol('C', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\lambda')(Symbol('W', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\hat{p}_0,\\Omega)} = \\log{(- \\Omega + \\hat{p}_0)}, then obtain \\hbar + \\mathbf{J}_P{(\\hat{p}_0,\\Omega)} = l + \\log{(\\Omega - \\hat{p}_0)}", "derivation": "\\mathbf{J}_P{(\\hat{p}_0,\\Omega)} = \\log{(- \\Omega + \\hat{p}_0)} and \\frac{\\partial}{\\partial \\Omega} \\mathbf{J}_P{(\\hat{p}_0,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\log{(- \\Omega + \\hat{p}_0)} and \\int \\frac{\\partial}{\\partial \\Omega} \\mathbf{J}_P{(\\hat{p}_0,\\Omega)} d\\Omega = \\int \\frac{\\partial}{\\partial \\Omega} \\log{(- \\Omega + \\hat{p}_0)} d\\Omega and \\hbar + \\mathbf{J}_P{(\\hat{p}_0,\\Omega)} = l + \\log{(\\Omega - \\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('l', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{H}{(\\varepsilon)} = e^{\\varepsilon} and l{(\\varepsilon)} = e^{\\varepsilon}, then obtain (e^{\\varepsilon})^{\\varepsilon} = \\mathbf{H}^{\\varepsilon}{(\\varepsilon)}", "derivation": "\\mathbf{H}{(\\varepsilon)} = e^{\\varepsilon} and l{(\\varepsilon)} = e^{\\varepsilon} and l{(\\varepsilon)} = \\mathbf{H}{(\\varepsilon)} and l^{\\varepsilon}{(\\varepsilon)} = \\mathbf{H}^{\\varepsilon}{(\\varepsilon)} and (e^{\\varepsilon})^{\\varepsilon} = \\mathbf{H}^{\\varepsilon}{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\varepsilon', commutative=True)), exp(Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('l')(Symbol('\\\\varepsilon', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\varepsilon', commutative=True)))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('l')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(exp(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given H{(\\mathbf{E},c_{0})} = \\mathbf{E} + c_{0} and c{(\\mathbf{E},c_{0})} = \\int H{(\\mathbf{E},c_{0})} dc_{0}, then derive c{(\\mathbf{E},c_{0})} = \\mathbf{E} c_{0} + \\mu + \\frac{c_{0}^{2}}{2}, then obtain \\int H{(\\mathbf{E},c_{0})} dc_{0} = \\mathbf{E} c_{0} + \\mu + \\frac{c_{0}^{2}}{2}", "derivation": "H{(\\mathbf{E},c_{0})} = \\mathbf{E} + c_{0} and \\int H{(\\mathbf{E},c_{0})} dc_{0} = \\int (\\mathbf{E} + c_{0}) dc_{0} and c{(\\mathbf{E},c_{0})} = \\int H{(\\mathbf{E},c_{0})} dc_{0} and c{(\\mathbf{E},c_{0})} = \\int (\\mathbf{E} + c_{0}) dc_{0} and c{(\\mathbf{E},c_{0})} = \\mathbf{E} c_{0} + \\mu + \\frac{c_{0}^{2}}{2} and \\int H{(\\mathbf{E},c_{0})} dc_{0} = \\mathbf{E} c_{0} + \\mu + \\frac{c_{0}^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Integral(Function('H')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Function('c')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Function('H')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{H}{(x^\\prime,z,\\Psi^{\\dagger})} = (z^{\\Psi^{\\dagger}})^{x^\\prime}, then obtain \\mathbf{H}{(x^\\prime,z,\\Psi^{\\dagger})} \\mathbf{H}^{z}{(x^\\prime,z,\\Psi^{\\dagger})} = (z^{\\Psi^{\\dagger}})^{x^\\prime} \\mathbf{H}^{z}{(x^\\prime,z,\\Psi^{\\dagger})}", "derivation": "\\mathbf{H}{(x^\\prime,z,\\Psi^{\\dagger})} = (z^{\\Psi^{\\dagger}})^{x^\\prime} and \\mathbf{H}^{z}{(x^\\prime,z,\\Psi^{\\dagger})} = ((z^{\\Psi^{\\dagger}})^{x^\\prime})^{z} and ((z^{\\Psi^{\\dagger}})^{x^\\prime})^{z} \\mathbf{H}{(x^\\prime,z,\\Psi^{\\dagger})} = (z^{\\Psi^{\\dagger}})^{x^\\prime} ((z^{\\Psi^{\\dagger}})^{x^\\prime})^{z} and \\mathbf{H}{(x^\\prime,z,\\Psi^{\\dagger})} \\mathbf{H}^{z}{(x^\\prime,z,\\Psi^{\\dagger})} = (z^{\\Psi^{\\dagger}})^{x^\\prime} \\mathbf{H}^{z}{(x^\\prime,z,\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('z', commutative=True)))"], [["times", 1, "Pow(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('z', commutative=True)), Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Pow(Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mu,\\mathbf{r})} = \\int \\frac{\\mu}{\\mathbf{r}} d\\mu, then obtain \\frac{\\mathbf{s}^{\\mathbf{r}}{(\\mu,\\mathbf{r})}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} - \\frac{1}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} = \\frac{(\\int \\frac{\\mu}{\\mathbf{r}} d\\mu)^{\\mathbf{r}}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} - \\frac{1}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu}", "derivation": "\\mathbf{s}{(\\mu,\\mathbf{r})} = \\int \\frac{\\mu}{\\mathbf{r}} d\\mu and \\mathbf{s}^{\\mathbf{r}}{(\\mu,\\mathbf{r})} = (\\int \\frac{\\mu}{\\mathbf{r}} d\\mu)^{\\mathbf{r}} and \\frac{\\mathbf{s}^{\\mathbf{r}}{(\\mu,\\mathbf{r})}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} = \\frac{(\\int \\frac{\\mu}{\\mathbf{r}} d\\mu)^{\\mathbf{r}}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} and \\frac{\\mathbf{s}^{\\mathbf{r}}{(\\mu,\\mathbf{r})}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} - \\frac{1}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} = \\frac{(\\int \\frac{\\mu}{\\mathbf{r}} d\\mu)^{\\mathbf{r}}}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu} - \\frac{1}{\\int \\frac{\\mu}{\\mathbf{r}} d\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["divide", 2, "Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 3, "Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1))"], "Equality(Add(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1)))), Add(Mul(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{g}{(v_{x})} = e^{e^{v_{x}}}, then obtain (e^{v_{x}} + 1)^{v_{x}} (e^{v_{x}} + \\frac{e^{e^{v_{x}}}}{\\mathbf{g}{(v_{x})}})^{- v_{x}} = 1", "derivation": "\\mathbf{g}{(v_{x})} = e^{e^{v_{x}}} and 1 = \\frac{e^{e^{v_{x}}}}{\\mathbf{g}{(v_{x})}} and e^{v_{x}} + 1 = e^{v_{x}} + \\frac{e^{e^{v_{x}}}}{\\mathbf{g}{(v_{x})}} and (e^{v_{x}} + 1)^{v_{x}} = (e^{v_{x}} + \\frac{e^{e^{v_{x}}}}{\\mathbf{g}{(v_{x})}})^{v_{x}} and (e^{v_{x}} + 1)^{v_{x}} (e^{v_{x}} + \\frac{e^{e^{v_{x}}}}{\\mathbf{g}{(v_{x})}})^{- v_{x}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), exp(exp(Symbol('v_x', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Integer(-1)), exp(exp(Symbol('v_x', commutative=True)))))"], [["add", 2, "exp(Symbol('v_x', commutative=True))"], "Equality(Add(exp(Symbol('v_x', commutative=True)), Integer(1)), Add(exp(Symbol('v_x', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Integer(-1)), exp(exp(Symbol('v_x', commutative=True))))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Add(exp(Symbol('v_x', commutative=True)), Integer(1)), Symbol('v_x', commutative=True)), Pow(Add(exp(Symbol('v_x', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Integer(-1)), exp(exp(Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True)))"], [["divide", 4, "Pow(Add(exp(Symbol('v_x', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Integer(-1)), exp(exp(Symbol('v_x', commutative=True))))), Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Add(exp(Symbol('v_x', commutative=True)), Integer(1)), Symbol('v_x', commutative=True)), Pow(Add(exp(Symbol('v_x', commutative=True)), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('v_x', commutative=True)), Integer(-1)), exp(exp(Symbol('v_x', commutative=True))))), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\psi^{*}{(\\theta_2,\\Psi^{\\dagger})} = \\log{((\\Psi^{\\dagger})^{\\theta_2})}, then obtain (\\frac{\\Psi^{\\dagger}}{\\theta_2})^{\\Psi^{\\dagger}} = (\\frac{\\Psi^{\\dagger} \\log{((\\Psi^{\\dagger})^{\\theta_2})}}{\\theta_2 \\psi^{*}{(\\theta_2,\\Psi^{\\dagger})}})^{\\Psi^{\\dagger}}", "derivation": "\\psi^{*}{(\\theta_2,\\Psi^{\\dagger})} = \\log{((\\Psi^{\\dagger})^{\\theta_2})} and \\frac{\\psi^{*}{(\\theta_2,\\Psi^{\\dagger})}}{\\theta_2} = \\frac{\\log{((\\Psi^{\\dagger})^{\\theta_2})}}{\\theta_2} and \\frac{\\Psi^{\\dagger} \\psi^{*}{(\\theta_2,\\Psi^{\\dagger})}}{\\theta_2} = \\frac{\\Psi^{\\dagger} \\log{((\\Psi^{\\dagger})^{\\theta_2})}}{\\theta_2} and \\frac{\\Psi^{\\dagger}}{\\theta_2} = \\frac{\\Psi^{\\dagger} \\log{((\\Psi^{\\dagger})^{\\theta_2})}}{\\theta_2 \\psi^{*}{(\\theta_2,\\Psi^{\\dagger})}} and (\\frac{\\Psi^{\\dagger}}{\\theta_2})^{\\Psi^{\\dagger}} = (\\frac{\\Psi^{\\dagger} \\log{((\\Psi^{\\dagger})^{\\theta_2})}}{\\theta_2 \\psi^{*}{(\\theta_2,\\Psi^{\\dagger})}})^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), log(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["times", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), log(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["divide", 3, "Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["power", 4, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} = \\psi^* - f_{\\mathbf{p}}, then obtain (\\frac{\\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)})^{f_{\\mathbf{p}}} = (\\frac{\\psi^* - f_{\\mathbf{p}} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)})^{f_{\\mathbf{p}}}", "derivation": "\\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} = \\psi^* - f_{\\mathbf{p}} and \\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} - 1 = \\psi^* - f_{\\mathbf{p}} - 1 and \\frac{\\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)} = \\frac{\\psi^* - f_{\\mathbf{p}} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)} and (\\frac{\\operatorname{f_{E}}{(f_{\\mathbf{p}},\\psi^*)} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)})^{f_{\\mathbf{p}}} = (\\frac{\\psi^* - f_{\\mathbf{p}} - 1}{\\frac{\\partial}{\\partial \\psi^*} (\\psi^* - f_{\\mathbf{p}} - 1)})^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('f_E')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)))"], [["divide", 2, "Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('f_E')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Pow(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))))"], [["power", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Mul(Add(Function('f_E')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Pow(Derivative(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(W,\\mathbf{A})} = - W + \\mathbf{A}, then derive - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})} = - W + \\mathbf{A}, then obtain \\mathbf{J}_M{(W,\\mathbf{A})} = - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})}", "derivation": "\\mathbf{J}_M{(W,\\mathbf{A})} = - W + \\mathbf{A} and \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})} = \\frac{\\partial}{\\partial W} (- W + \\mathbf{A}) and (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})} = (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} (- W + \\mathbf{A}) and - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})} = - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} (- W + \\mathbf{A}) and - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})} = - W + \\mathbf{A} and \\mathbf{J}_M{(W,\\mathbf{A})} = - (- W + \\mathbf{A}) \\frac{\\partial}{\\partial W} \\mathbf{J}_M{(W,\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 5], "Equality(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Function('\\\\mathbf{J}_M')(Symbol('W', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{H}_l,\\hat{x})} = \\hat{H}_l \\sin{(\\hat{x})} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\cos{(\\hat{x})}, then derive \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{J_{\\varepsilon}}{(\\hat{H}_l,\\hat{x})} = \\hat{H}_l \\cos{(\\hat{x})}, then obtain \\hat{H}_l \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\hat{H}_l \\cos{(\\hat{x})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{H}_l,\\hat{x})} = \\hat{H}_l \\sin{(\\hat{x})} and \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{J_{\\varepsilon}}{(\\hat{H}_l,\\hat{x})} = \\frac{\\partial}{\\partial \\hat{x}} \\hat{H}_l \\sin{(\\hat{x})} and \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{J_{\\varepsilon}}{(\\hat{H}_l,\\hat{x})} = \\hat{H}_l \\cos{(\\hat{x})} and \\frac{\\partial}{\\partial \\hat{x}} \\hat{H}_l \\sin{(\\hat{x})} = \\hat{H}_l \\cos{(\\hat{x})} and \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\cos{(\\hat{x})} and \\frac{\\partial}{\\partial \\hat{x}} \\hat{H}_l \\sin{(\\hat{x})} = \\hat{H}_l \\operatorname{J_{\\varepsilon}}{(\\hat{x})} and \\hat{H}_l \\operatorname{J_{\\varepsilon}}{(\\hat{x})} = \\hat{H}_l \\cos{(\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{H}_l', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{x}', commutative=True))), Mul(Symbol('\\\\hat{H}_l', commutative=True), cos(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given g{(S,\\hat{H}_l)} = S e^{\\hat{H}_l}, then obtain 4 S^{2} e^{2 \\hat{H}_l} = 4 g^{2}{(S,\\hat{H}_l)}", "derivation": "g{(S,\\hat{H}_l)} = S e^{\\hat{H}_l} and S e^{\\hat{H}_l} + g{(S,\\hat{H}_l)} = 2 S e^{\\hat{H}_l} and (S e^{\\hat{H}_l} + g{(S,\\hat{H}_l)})^{2} = 4 S^{2} e^{2 \\hat{H}_l} and (S e^{\\hat{H}_l} + g{(S,\\hat{H}_l)})^{2} e^{- 2 \\hat{H}_l} = 4 S^{2} and 4 g^{2}{(S,\\hat{H}_l)} e^{- 2 \\hat{H}_l} = 4 S^{2} and (S e^{\\hat{H}_l} + g{(S,\\hat{H}_l)})^{2} = 4 g^{2}{(S,\\hat{H}_l)} and 4 S^{2} e^{2 \\hat{H}_l} = 4 g^{2}{(S,\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 1, "Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Symbol('S', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["divide", 3, "exp(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Integer(4), Pow(Symbol('S', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Integer(4), Pow(Symbol('S', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Add(Mul(Symbol('S', commutative=True), exp(Symbol('\\\\hat{H}_l', commutative=True))), Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(4), Pow(Symbol('S', commutative=True), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Integer(4), Pow(Function('g')(Symbol('S', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(t,\\Omega)} = \\Omega^{t} and \\mathbf{M}{(\\Omega)} = \\Omega, then obtain (\\int \\operatorname{F_{x}}{(t,\\Omega)} d\\mathbf{M}{(\\Omega)})^{\\Omega - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)}} = (\\int \\Omega^{t} d\\mathbf{M}{(\\Omega)})^{\\Omega - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)}}", "derivation": "\\operatorname{F_{x}}{(t,\\Omega)} = \\Omega^{t} and - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)} = 0 and \\int \\operatorname{F_{x}}{(t,\\Omega)} d\\Omega = \\int \\Omega^{t} d\\Omega and \\mathbf{M}{(\\Omega)} = \\Omega and \\int \\operatorname{F_{x}}{(t,\\Omega)} d\\mathbf{M}{(\\Omega)} = \\int \\Omega^{t} d\\mathbf{M}{(\\Omega)} and \\Omega - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)} = \\Omega and (\\int \\operatorname{F_{x}}{(t,\\Omega)} d\\mathbf{M}{(\\Omega)})^{\\Omega} = (\\int \\Omega^{t} d\\mathbf{M}{(\\Omega)})^{\\Omega} and (\\int \\operatorname{F_{x}}{(t,\\Omega)} d\\mathbf{M}{(\\Omega)})^{\\Omega - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)}} = (\\int \\Omega^{t} d\\mathbf{M}{(\\Omega)})^{\\Omega - \\Omega^{t} + \\operatorname{F_{x}}{(t,\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True))), Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(0))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))), Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))))"], [["add", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True))), Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integral(Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)), Pow(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Pow(Integral(Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True))), Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)))), Pow(Integral(Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True)), Tuple(Function('\\\\mathbf{M}')(Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Symbol('t', commutative=True))), Function('F_x')(Symbol('t', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then obtain \\mathbf{J}_f^{2 \\mathbf{p}}{(\\mathbf{p})} = \\cos^{2 \\mathbf{p}}{(\\mathbf{p})}", "derivation": "\\mathbf{J}_f{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\mathbf{J}_f^{\\mathbf{p}}{(\\mathbf{p})} = \\cos^{\\mathbf{p}}{(\\mathbf{p})} and \\mathbf{J}_f^{\\mathbf{p}}{(\\mathbf{p})} \\cos^{\\mathbf{p}}{(\\mathbf{p})} = \\cos^{2 \\mathbf{p}}{(\\mathbf{p})} and \\mathbf{J}_f^{2 \\mathbf{p}}{(\\mathbf{p})} = \\mathbf{J}_f^{\\mathbf{p}}{(\\mathbf{p})} \\cos^{\\mathbf{p}}{(\\mathbf{p})} and \\mathbf{J}_f^{2 \\mathbf{p}}{(\\mathbf{p})} = \\cos^{2 \\mathbf{p}}{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 2, "Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))))"], [["times", 2, "Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(C_{2},v_{y})} = C_{2}^{v_{y}} and \\mathbf{s}{(\\psi^*,P_{g})} = \\frac{P_{g}}{\\psi^*}, then obtain (C_{2}^{v_{y}})^{C_{2}} \\mathbf{s}{(\\psi^*,P_{g})} = \\frac{P_{g} (C_{2}^{v_{y}})^{C_{2}}}{\\psi^*}", "derivation": "\\hat{p}_0{(C_{2},v_{y})} = C_{2}^{v_{y}} and \\hat{p}_0^{C_{2}}{(C_{2},v_{y})} = (C_{2}^{v_{y}})^{C_{2}} and \\mathbf{s}{(\\psi^*,P_{g})} = \\frac{P_{g}}{\\psi^*} and \\hat{p}_0^{C_{2}}{(C_{2},v_{y})} \\mathbf{s}{(\\psi^*,P_{g})} = \\frac{P_{g} \\hat{p}_0^{C_{2}}{(C_{2},v_{y})}}{\\psi^*} and (C_{2}^{v_{y}})^{C_{2}} \\mathbf{s}{(\\psi^*,P_{g})} = \\frac{P_{g} (C_{2}^{v_{y}})^{C_{2}}}{\\psi^*}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True)), Pow(Pow(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True), Symbol('P_g', commutative=True)), Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))))"], [["times", 3, "Pow(Function('\\\\hat{p}_0')(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True), Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Pow(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\psi^*', commutative=True), Symbol('P_g', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Pow(Pow(Symbol('C_2', commutative=True), Symbol('v_y', commutative=True)), Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\Omega{(\\varepsilon_0)} = \\int \\varepsilon{(\\varepsilon_0)} d\\varepsilon_0, then obtain \\Omega{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0", "derivation": "\\varepsilon{(\\varepsilon_0)} = \\sin{(\\varepsilon_0)} and \\int \\varepsilon{(\\varepsilon_0)} d\\varepsilon_0 = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0 and \\Omega{(\\varepsilon_0)} = \\int \\varepsilon{(\\varepsilon_0)} d\\varepsilon_0 and \\Omega{(\\varepsilon_0)} = \\int \\sin{(\\varepsilon_0)} d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('\\\\varepsilon_0', commutative=True)), Integral(Function('\\\\varepsilon')(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\Omega')(Symbol('\\\\varepsilon_0', commutative=True)), Integral(sin(Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given x{(C_{1})} = \\sin{(C_{1})}, then obtain e^{x{(C_{1})} \\sin{(C_{1})}} e^{x^{2}{(C_{1})}} = e^{2 x{(C_{1})} \\sin{(C_{1})}}", "derivation": "x{(C_{1})} = \\sin{(C_{1})} and x^{2}{(C_{1})} = x{(C_{1})} \\sin{(C_{1})} and e^{x^{2}{(C_{1})}} = e^{x{(C_{1})} \\sin{(C_{1})}} and e^{x{(C_{1})} \\sin{(C_{1})}} e^{x^{2}{(C_{1})}} = e^{2 x{(C_{1})} \\sin{(C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["times", 1, "Function('x')(Symbol('C_1', commutative=True))"], "Equality(Pow(Function('x')(Symbol('C_1', commutative=True)), Integer(2)), Mul(Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True))))"], [["exp", 2], "Equality(exp(Pow(Function('x')(Symbol('C_1', commutative=True)), Integer(2))), exp(Mul(Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))))"], [["times", 3, "exp(Mul(Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True))))"], "Equality(Mul(exp(Mul(Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))), exp(Pow(Function('x')(Symbol('C_1', commutative=True)), Integer(2)))), exp(Mul(Integer(2), Function('x')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given p{(\\ddot{x})} = \\sin{(\\ddot{x})}, then obtain p^{3}{(\\ddot{x})} \\sin{(\\ddot{x})} = p{(\\ddot{x})} \\sin^{3}{(\\ddot{x})}", "derivation": "p{(\\ddot{x})} = \\sin{(\\ddot{x})} and p{(\\ddot{x})} \\sin{(\\ddot{x})} = \\sin^{2}{(\\ddot{x})} and p^{2}{(\\ddot{x})} \\sin^{2}{(\\ddot{x})} = p{(\\ddot{x})} \\sin^{3}{(\\ddot{x})} and p^{3}{(\\ddot{x})} \\sin{(\\ddot{x})} = p{(\\ddot{x})} \\sin^{3}{(\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True))), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(2)))"], [["times", 2, "Mul(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Mul(Pow(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(2))), Mul(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), Integer(3)), sin(Symbol('\\\\ddot{x}', commutative=True))), Mul(Function('p')(Symbol('\\\\ddot{x}', commutative=True)), Pow(sin(Symbol('\\\\ddot{x}', commutative=True)), Integer(3))))"]]}, {"prompt": "Given h{(n_{1})} = \\cos{(n_{1})}, then obtain n_{1} \\frac{d}{d n_{1}} h{(n_{1})} = - n_{1} \\sin{(n_{1})}", "derivation": "h{(n_{1})} = \\cos{(n_{1})} and \\frac{d}{d n_{1}} h{(n_{1})} = \\frac{d}{d n_{1}} \\cos{(n_{1})} and n_{1} \\frac{d}{d n_{1}} h{(n_{1})} = n_{1} \\frac{d}{d n_{1}} \\cos{(n_{1})} and n_{1} \\frac{d}{d n_{1}} h{(n_{1})} = - n_{1} \\sin{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('n_1', commutative=True)), cos(Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(cos(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 2, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Derivative(Function('h')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Symbol('n_1', commutative=True), Derivative(cos(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('n_1', commutative=True), Derivative(Function('h')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('n_1', commutative=True), sin(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given s{(f^{*})} = \\cos{(f^{*})} and \\psi^{*}{(f^{*})} = f^{*}, then obtain f^{*} (\\frac{s{(f^{*})}}{\\cos^{2}{(f^{*})}} - 1) = f^{*} (-1 + \\frac{1}{\\cos{(f^{*})}})", "derivation": "s{(f^{*})} = \\cos{(f^{*})} and \\frac{s{(f^{*})}}{\\cos{(f^{*})}} = 1 and \\frac{s{(f^{*})}}{\\cos^{2}{(f^{*})}} = \\frac{1}{\\cos{(f^{*})}} and \\psi^{*}{(f^{*})} = f^{*} and \\frac{s{(f^{*})}}{\\cos^{2}{(f^{*})}} - 1 = -1 + \\frac{1}{\\cos{(f^{*})}} and (\\frac{s{(f^{*})}}{\\cos^{2}{(f^{*})}} - 1) \\psi^{*}{(f^{*})} = (-1 + \\frac{1}{\\cos{(f^{*})}}) \\psi^{*}{(f^{*})} and f^{*} (\\frac{s{(f^{*})}}{\\cos^{2}{(f^{*})}} - 1) = f^{*} (-1 + \\frac{1}{\\cos{(f^{*})}})", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('f^*', commutative=True)), cos(Symbol('f^*', commutative=True)))"], [["divide", 1, "cos(Symbol('f^*', commutative=True))"], "Equality(Mul(Function('s')(Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Pow(cos(Symbol('f^*', commutative=True)), Integer(-1))"], "Equality(Mul(Function('s')(Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-2))), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Function('s')(Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-2))), Integer(-1)), Add(Integer(-1), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1))))"], [["times", 5, "Function('\\\\psi^*')(Symbol('f^*', commutative=True))"], "Equality(Mul(Add(Mul(Function('s')(Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-2))), Integer(-1)), Function('\\\\psi^*')(Symbol('f^*', commutative=True))), Mul(Add(Integer(-1), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1))), Function('\\\\psi^*')(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Symbol('f^*', commutative=True), Add(Mul(Function('s')(Symbol('f^*', commutative=True)), Pow(cos(Symbol('f^*', commutative=True)), Integer(-2))), Integer(-1))), Mul(Symbol('f^*', commutative=True), Add(Integer(-1), Pow(cos(Symbol('f^*', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\lambda{(n_{2},A_{2})} = \\frac{A_{2}}{n_{2}}, then obtain \\frac{(- n_{2} + \\lambda{(n_{2},A_{2})})^{A_{2}}}{A_{2}} = \\frac{(\\frac{A_{2}}{n_{2}} - n_{2})^{A_{2}}}{A_{2}}", "derivation": "\\lambda{(n_{2},A_{2})} = \\frac{A_{2}}{n_{2}} and - n_{2} + \\lambda{(n_{2},A_{2})} = \\frac{A_{2}}{n_{2}} - n_{2} and (- n_{2} + \\lambda{(n_{2},A_{2})})^{A_{2}} = (\\frac{A_{2}}{n_{2}} - n_{2})^{A_{2}} and \\frac{(- n_{2} + \\lambda{(n_{2},A_{2})})^{A_{2}}}{A_{2}} = \\frac{(\\frac{A_{2}}{n_{2}} - n_{2})^{A_{2}}}{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('A_2', commutative=True)))"], [["divide", 3, "Symbol('A_2', commutative=True)"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('\\\\lambda')(Symbol('n_2', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given z{(M,L_{\\varepsilon})} = \\frac{\\sin{(L_{\\varepsilon})}}{M}, then obtain \\int (\\iint z{(M,L_{\\varepsilon})} dM dL_{\\varepsilon})^{L_{\\varepsilon}} dM = \\int (\\iint \\frac{\\sin{(L_{\\varepsilon})}}{M} dM dL_{\\varepsilon})^{L_{\\varepsilon}} dM", "derivation": "z{(M,L_{\\varepsilon})} = \\frac{\\sin{(L_{\\varepsilon})}}{M} and \\int z{(M,L_{\\varepsilon})} dM = \\int \\frac{\\sin{(L_{\\varepsilon})}}{M} dM and \\iint z{(M,L_{\\varepsilon})} dM dL_{\\varepsilon} = \\iint \\frac{\\sin{(L_{\\varepsilon})}}{M} dM dL_{\\varepsilon} and (\\iint z{(M,L_{\\varepsilon})} dM dL_{\\varepsilon})^{L_{\\varepsilon}} = (\\iint \\frac{\\sin{(L_{\\varepsilon})}}{M} dM dL_{\\varepsilon})^{L_{\\varepsilon}} and \\int (\\iint z{(M,L_{\\varepsilon})} dM dL_{\\varepsilon})^{L_{\\varepsilon}} dM = \\int (\\iint \\frac{\\sin{(L_{\\varepsilon})}}{M} dM dL_{\\varepsilon})^{L_{\\varepsilon}} dM", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('M', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('z')(Symbol('M', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('z')(Symbol('M', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Function('z')(Symbol('M', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Pow(Integral(Function('z')(Symbol('M', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(F_{c})} = \\sin{(\\cos{(F_{c})})} and \\operatorname{v_{t}}{(F_{c})} = \\sin{(\\cos{(F_{c})})}, then obtain \\frac{\\frac{d}{d F_{c}} 1}{\\operatorname{v_{t}}{(F_{c})}} = \\frac{\\frac{d}{d F_{c}} \\frac{\\operatorname{v_{t}}{(F_{c})}}{\\mathbf{H}{(F_{c})}}}{\\operatorname{v_{t}}{(F_{c})}}", "derivation": "\\mathbf{H}{(F_{c})} = \\sin{(\\cos{(F_{c})})} and 1 = \\frac{\\sin{(\\cos{(F_{c})})}}{\\mathbf{H}{(F_{c})}} and \\frac{d}{d F_{c}} 1 = \\frac{d}{d F_{c}} \\frac{\\sin{(\\cos{(F_{c})})}}{\\mathbf{H}{(F_{c})}} and \\operatorname{v_{t}}{(F_{c})} = \\sin{(\\cos{(F_{c})})} and \\frac{\\frac{d}{d F_{c}} 1}{\\sin{(\\cos{(F_{c})})}} = \\frac{\\frac{d}{d F_{c}} \\frac{\\sin{(\\cos{(F_{c})})}}{\\mathbf{H}{(F_{c})}}}{\\sin{(\\cos{(F_{c})})}} and \\frac{\\frac{d}{d F_{c}} 1}{\\operatorname{v_{t}}{(F_{c})}} = \\frac{\\frac{d}{d F_{c}} \\frac{\\operatorname{v_{t}}{(F_{c})}}{\\mathbf{H}{(F_{c})}}}{\\operatorname{v_{t}}{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True)), sin(cos(Symbol('F_c', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True)), Integer(-1)), sin(cos(Symbol('F_c', commutative=True)))))"], [["differentiate", 2, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True)), Integer(-1)), sin(cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('F_c', commutative=True)), sin(cos(Symbol('F_c', commutative=True))))"], [["divide", 3, "sin(cos(Symbol('F_c', commutative=True)))"], "Equality(Mul(Pow(sin(cos(Symbol('F_c', commutative=True))), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(sin(cos(Symbol('F_c', commutative=True))), Integer(-1)), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True)), Integer(-1)), sin(cos(Symbol('F_c', commutative=True)))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('v_t')(Symbol('F_c', commutative=True)), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('F_c', commutative=True), Integer(1)))), Mul(Pow(Function('v_t')(Symbol('F_c', commutative=True)), Integer(-1)), Derivative(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('F_c', commutative=True)), Integer(-1)), Function('v_t')(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(\\tilde{g},A_{z})} = e^{A_{z} + \\tilde{g}}, then obtain \\frac{\\partial}{\\partial \\tilde{g}} (V^{A_{z}}{(\\tilde{g},A_{z})})^{A_{z}} = \\frac{\\partial}{\\partial \\tilde{g}} ((e^{A_{z} + \\tilde{g}})^{A_{z}})^{A_{z}}", "derivation": "V{(\\tilde{g},A_{z})} = e^{A_{z} + \\tilde{g}} and V^{A_{z}}{(\\tilde{g},A_{z})} = (e^{A_{z} + \\tilde{g}})^{A_{z}} and (V^{A_{z}}{(\\tilde{g},A_{z})})^{A_{z}} = ((e^{A_{z} + \\tilde{g}})^{A_{z}})^{A_{z}} and \\frac{\\partial}{\\partial \\tilde{g}} (V^{A_{z}}{(\\tilde{g},A_{z})})^{A_{z}} = \\frac{\\partial}{\\partial \\tilde{g}} ((e^{A_{z} + \\tilde{g}})^{A_{z}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)), exp(Add(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('V')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(exp(Add(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('A_z', commutative=True)))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Pow(Function('V')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(Pow(exp(Add(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('V')(Symbol('\\\\tilde{g}', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Pow(Pow(exp(Add(Symbol('A_z', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\hat{H}_l)} = \\int e^{\\hat{H}_l} d\\hat{H}_l, then obtain f^{\\sigma_p}{(\\theta_1,\\sigma_p)} \\int (I{(\\hat{H}_l)} + e^{\\hat{H}_l}) d\\hat{H}_l = f^{\\sigma_p}{(\\theta_1,\\sigma_p)} \\int (e^{\\hat{H}_l} + \\int e^{\\hat{H}_l} d\\hat{H}_l) d\\hat{H}_l", "derivation": "I{(\\hat{H}_l)} = \\int e^{\\hat{H}_l} d\\hat{H}_l and I{(\\hat{H}_l)} + e^{\\hat{H}_l} = e^{\\hat{H}_l} + \\int e^{\\hat{H}_l} d\\hat{H}_l and \\int (I{(\\hat{H}_l)} + e^{\\hat{H}_l}) d\\hat{H}_l = \\int (e^{\\hat{H}_l} + \\int e^{\\hat{H}_l} d\\hat{H}_l) d\\hat{H}_l and f^{\\sigma_p}{(\\theta_1,\\sigma_p)} \\int (I{(\\hat{H}_l)} + e^{\\hat{H}_l}) d\\hat{H}_l = f^{\\sigma_p}{(\\theta_1,\\sigma_p)} \\int (e^{\\hat{H}_l} + \\int e^{\\hat{H}_l} d\\hat{H}_l) d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["add", 1, "exp(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('I')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))), Add(exp(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Add(Function('I')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Add(exp(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 3, "Pow(Function('f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Function('f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Function('I')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Pow(Function('f')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(exp(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\phi_2,E_{n})} = E_{n} \\phi_2 and L{(\\phi_2,E_{n})} = E_{n} \\phi_2 \\operatorname{f_{\\mathbf{p}}}{(\\phi_2,E_{n})}, then obtain \\frac{\\partial}{\\partial \\phi_2} L{(\\phi_2,E_{n})} = \\frac{\\partial}{\\partial \\phi_2} \\operatorname{f_{\\mathbf{p}}}^{2}{(\\phi_2,E_{n})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\phi_2,E_{n})} = E_{n} \\phi_2 and L{(\\phi_2,E_{n})} = E_{n} \\phi_2 \\operatorname{f_{\\mathbf{p}}}{(\\phi_2,E_{n})} and L{(\\phi_2,E_{n})} = \\operatorname{f_{\\mathbf{p}}}^{2}{(\\phi_2,E_{n})} and \\frac{\\partial}{\\partial \\phi_2} L{(\\phi_2,E_{n})} = \\frac{\\partial}{\\partial \\phi_2} \\operatorname{f_{\\mathbf{p}}}^{2}{(\\phi_2,E_{n})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('\\\\phi_2', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Integer(2)))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\phi_2', commutative=True), Symbol('E_n', commutative=True)), Integer(2)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(p)} = p, then obtain \\hat{\\mathbf{x}}^{p}{(p)} \\frac{d}{d p} 1 = p^{p} \\frac{d}{d p} 1", "derivation": "\\hat{\\mathbf{x}}{(p)} = p and \\frac{\\hat{\\mathbf{x}}{(p)}}{p} = 1 and \\hat{\\mathbf{x}}^{p}{(p)} = p^{p} and \\frac{d}{d p} \\frac{\\hat{\\mathbf{x}}{(p)}}{p} = \\frac{d}{d p} 1 and \\hat{\\mathbf{x}}^{p}{(p)} \\frac{d}{d p} \\frac{\\hat{\\mathbf{x}}{(p)}}{p} = p^{p} \\frac{d}{d p} \\frac{\\hat{\\mathbf{x}}{(p)}}{p} and \\hat{\\mathbf{x}}^{p}{(p)} \\frac{d}{d p} 1 = p^{p} \\frac{d}{d p} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], [["divide", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Integer(1))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 2, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Pow(Symbol('p', commutative=True), Symbol('p', commutative=True)), Derivative(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Integer(1), Tuple(Symbol('p', commutative=True), Integer(1)))), Mul(Pow(Symbol('p', commutative=True), Symbol('p', commutative=True)), Derivative(Integer(1), Tuple(Symbol('p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\mathbf{f},H)} = \\int (H + \\mathbf{f}) d\\mathbf{f}, then derive \\frac{S{(\\mathbf{f},H)}}{H} = \\frac{H \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\rho}{H}, then obtain \\frac{S{(\\mathbf{f},H)} S^{- H}{(\\mathbf{f},H)}}{H} = \\frac{(H \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\rho) S^{- H}{(\\mathbf{f},H)}}{H}", "derivation": "S{(\\mathbf{f},H)} = \\int (H + \\mathbf{f}) d\\mathbf{f} and \\frac{S{(\\mathbf{f},H)}}{H} = \\frac{\\int (H + \\mathbf{f}) d\\mathbf{f}}{H} and \\frac{S{(\\mathbf{f},H)}}{H} = \\frac{H \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\rho}{H} and \\frac{S{(\\mathbf{f},H)} S^{- H}{(\\mathbf{f},H)}}{H} = \\frac{(H \\mathbf{f} + \\frac{\\mathbf{f}^{2}}{2} + \\rho) S^{- H}{(\\mathbf{f},H)}}{H}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True)), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True))))"], [["divide", 3, "Pow(Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True)), Pow(Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True)), Pow(Function('S')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(l,\\Psi_{nl})} = \\log{(- \\Psi_{nl} + l)}, then obtain - \\log{(\\log{(- \\Psi_{nl} + l)})} - \\frac{- \\Psi_{nl} + \\dot{\\mathbf{r}}{(l,\\Psi_{nl})}}{\\Psi_{nl} (- \\Psi_{nl} + \\log{(- \\Psi_{nl} + l)})} = - \\log{(\\log{(- \\Psi_{nl} + l)})} - \\frac{1}{\\Psi_{nl}}", "derivation": "\\dot{\\mathbf{r}}{(l,\\Psi_{nl})} = \\log{(- \\Psi_{nl} + l)} and - \\Psi_{nl} + \\dot{\\mathbf{r}}{(l,\\Psi_{nl})} = - \\Psi_{nl} + \\log{(- \\Psi_{nl} + l)} and \\frac{- \\Psi_{nl} + \\dot{\\mathbf{r}}{(l,\\Psi_{nl})}}{- \\Psi_{nl} + \\log{(- \\Psi_{nl} + l)}} = 1 and - \\frac{- \\Psi_{nl} + \\dot{\\mathbf{r}}{(l,\\Psi_{nl})}}{\\Psi_{nl} (- \\Psi_{nl} + \\log{(- \\Psi_{nl} + l)})} = - \\frac{1}{\\Psi_{nl}} and - \\log{(\\log{(- \\Psi_{nl} + l)})} - \\frac{- \\Psi_{nl} + \\dot{\\mathbf{r}}{(l,\\Psi_{nl})}}{\\Psi_{nl} (- \\Psi_{nl} + \\log{(- \\Psi_{nl} + l)})} = - \\log{(\\log{(- \\Psi_{nl} + l)})} - \\frac{1}{\\Psi_{nl}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True)))), Integer(-1))), Integer(1))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True)))), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1))))"], [["minus", 4, "log(log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True))))"], "Equality(Add(Mul(Integer(-1), log(log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('l', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True)))), Integer(-1)))), Add(Mul(Integer(-1), log(log(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('l', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} = e^{e^{\\mathbf{J}_M}}, then obtain \\mathbf{J}_M \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} + \\operatorname{y^{\\prime}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = \\mathbf{J}_M \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} + (e^{e^{\\mathbf{J}_M}})^{\\mathbf{J}_M}", "derivation": "\\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} = e^{e^{\\mathbf{J}_M}} and \\mathbf{J}_M \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} = \\mathbf{J}_M e^{e^{\\mathbf{J}_M}} and \\operatorname{y^{\\prime}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = (e^{e^{\\mathbf{J}_M}})^{\\mathbf{J}_M} and \\mathbf{J}_M e^{e^{\\mathbf{J}_M}} + \\operatorname{y^{\\prime}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = \\mathbf{J}_M e^{e^{\\mathbf{J}_M}} + (e^{e^{\\mathbf{J}_M}})^{\\mathbf{J}_M} and \\mathbf{J}_M \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} + \\operatorname{y^{\\prime}}^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = \\mathbf{J}_M \\operatorname{y^{\\prime}}{(\\mathbf{J}_M)} + (e^{e^{\\mathbf{J}_M}})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Pow(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True)))), Pow(exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(exp(exp(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain (\\operatorname{F_{N}}^{\\sigma_p}{(\\sigma_p)} + \\log{(\\sigma_p)}) \\log{(\\sigma_p)} = (\\log{(\\sigma_p)} + \\log{(\\sigma_p)}^{\\sigma_p}) \\log{(\\sigma_p)}", "derivation": "\\operatorname{F_{N}}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\operatorname{F_{N}}^{\\sigma_p}{(\\sigma_p)} = \\log{(\\sigma_p)}^{\\sigma_p} and \\operatorname{F_{N}}^{\\sigma_p}{(\\sigma_p)} + \\log{(\\sigma_p)} = \\log{(\\sigma_p)} + \\log{(\\sigma_p)}^{\\sigma_p} and (\\operatorname{F_{N}}^{\\sigma_p}{(\\sigma_p)} + \\log{(\\sigma_p)}) \\log{(\\sigma_p)} = (\\log{(\\sigma_p)} + \\log{(\\sigma_p)}^{\\sigma_p}) \\log{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 2, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Pow(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), Add(log(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["times", 3, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Add(Pow(Function('F_N')(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Add(log(Symbol('\\\\sigma_p', commutative=True)), Pow(log(Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\nabla)} = \\nabla and \\rho_{f}{(f)} = \\frac{1}{f}, then obtain \\hat{H}_l{(\\nabla)} \\rho_{f}{(f)} = \\frac{\\hat{H}_l{(\\nabla)}}{f}", "derivation": "\\hat{H}_l{(\\nabla)} = \\nabla and \\rho_{f}{(f)} = \\frac{1}{f} and \\nabla \\rho_{f}{(f)} = \\frac{\\nabla}{f} and \\hat{H}_l{(\\nabla)} \\rho_{f}{(f)} = \\frac{\\hat{H}_l{(\\nabla)}}{f}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('f', commutative=True)), Pow(Symbol('f', commutative=True), Integer(-1)))"], [["times", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('\\\\rho_f')(Symbol('f', commutative=True))), Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True)), Function('\\\\rho_f')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\lambda)} = \\cos{(\\lambda)}, then derive \\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)} = - \\sin{(\\lambda)}, then obtain \\sin{(\\lambda)} + \\cos{(\\sin{(\\lambda)})} = \\sin{(\\lambda)} + \\cos{(\\frac{d}{d \\lambda} \\cos{(\\lambda)})}", "derivation": "\\operatorname{x^{{\\}'}}{(\\lambda)} = \\cos{(\\lambda)} and \\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)} = \\frac{d}{d \\lambda} \\cos{(\\lambda)} and \\cos{(\\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)})} = \\cos{(\\frac{d}{d \\lambda} \\cos{(\\lambda)})} and \\cos{(\\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)})} - \\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)} = \\cos{(\\frac{d}{d \\lambda} \\cos{(\\lambda)})} - \\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)} and \\frac{d}{d \\lambda} \\operatorname{x^{{\\}'}}{(\\lambda)} = - \\sin{(\\lambda)} and \\sin{(\\lambda)} + \\cos{(\\sin{(\\lambda)})} = \\sin{(\\lambda)} + \\cos{(\\frac{d}{d \\lambda} \\cos{(\\lambda)})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), cos(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["minus", 3, "Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(cos(Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Add(cos(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(sin(Symbol('\\\\lambda', commutative=True)), cos(sin(Symbol('\\\\lambda', commutative=True)))), Add(sin(Symbol('\\\\lambda', commutative=True)), cos(Derivative(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\tilde{g}{(x,a^{\\dagger},\\rho)} = (- a^{\\dagger} + x)^{\\rho}, then derive \\frac{\\partial^{2}}{\\partial x\\partial a^{\\dagger}} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\rho (1 - \\rho) (- a^{\\dagger} + x)^{\\rho}}{(a^{\\dagger} - x)^{2}}, then obtain \\frac{\\partial^{2}}{\\partial x\\partial a^{\\dagger}} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\rho (1 - \\rho) \\tilde{g}{(x,a^{\\dagger},\\rho)}}{(a^{\\dagger} - x)^{2}}", "derivation": "\\tilde{g}{(x,a^{\\dagger},\\rho)} = (- a^{\\dagger} + x)^{\\rho} and \\frac{\\partial}{\\partial x} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\partial}{\\partial x} (- a^{\\dagger} + x)^{\\rho} and \\frac{\\partial^{2}}{\\partial a^{\\dagger}\\partial x} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\partial^{2}}{\\partial a^{\\dagger}\\partial x} (- a^{\\dagger} + x)^{\\rho} and \\frac{\\partial^{2}}{\\partial x\\partial a^{\\dagger}} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\rho (1 - \\rho) (- a^{\\dagger} + x)^{\\rho}}{(a^{\\dagger} - x)^{2}} and \\frac{\\partial^{2}}{\\partial x\\partial a^{\\dagger}} \\tilde{g}{(x,a^{\\dagger},\\rho)} = \\frac{\\rho (1 - \\rho) \\tilde{g}{(x,a^{\\dagger},\\rho)}}{(a^{\\dagger} - x)^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Symbol('\\\\rho', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('x', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Symbol('\\\\rho', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(-2)), Function('\\\\tilde{g}')(Symbol('x', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given H{(n_{1})} = e^{n_{1}}, then obtain \\frac{H^{n_{1}}{(n_{1})}}{\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}} = \\frac{(e^{n_{1}})^{n_{1}}}{\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}}", "derivation": "H{(n_{1})} = e^{n_{1}} and H^{n_{1}}{(n_{1})} = (e^{n_{1}})^{n_{1}} and - H^{n_{1}}{(n_{1})} = - (e^{n_{1}})^{n_{1}} and \\frac{d}{d n_{1}} - H^{n_{1}}{(n_{1})} = \\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}} and \\frac{H^{n_{1}}{(n_{1})}}{\\frac{d}{d n_{1}} - H^{n_{1}}{(n_{1})}} = \\frac{(e^{n_{1}})^{n_{1}}}{\\frac{d}{d n_{1}} - H^{n_{1}}{(n_{1})}} and \\frac{H^{n_{1}}{(n_{1})}}{\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}} = \\frac{(e^{n_{1}})^{n_{1}}}{\\frac{d}{d n_{1}} - (e^{n_{1}})^{n_{1}}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Mul(Integer(-1), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Mul(Integer(-1), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Mul(Integer(-1), Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Function('H')(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Mul(Integer(-1), Pow(exp(Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(a)} = \\cos{(a)} and \\Psi_{\\lambda}{(a)} = \\frac{\\int \\cos{(a)} da}{\\sin{(a)}} and Q{(a)} = \\frac{\\int \\cos{(a)} da}{\\sin{(a)}}, then derive \\int \\operatorname{v_{2}}{(a)} da = \\sigma_x + \\sin{(a)}, then obtain \\Psi_{\\lambda}{(a)} = Q{(a)}", "derivation": "\\operatorname{v_{2}}{(a)} = \\cos{(a)} and \\int \\operatorname{v_{2}}{(a)} da = \\int \\cos{(a)} da and \\int \\operatorname{v_{2}}{(a)} da = \\sigma_x + \\sin{(a)} and \\frac{\\int \\operatorname{v_{2}}{(a)} da}{\\sin{(a)}} = \\frac{\\sigma_x + \\sin{(a)}}{\\sin{(a)}} and \\int \\cos{(a)} da = \\sigma_x + \\sin{(a)} and \\frac{\\int \\operatorname{v_{2}}{(a)} da}{\\sin{(a)}} = \\frac{\\int \\cos{(a)} da}{\\sin{(a)}} and \\Psi_{\\lambda}{(a)} = \\frac{\\int \\cos{(a)} da}{\\sin{(a)}} and Q{(a)} = \\frac{\\int \\cos{(a)} da}{\\sin{(a)}} and \\frac{\\int \\operatorname{v_{2}}{(a)} da}{\\sin{(a)}} = Q{(a)} and \\frac{\\int \\operatorname{v_{2}}{(a)} da}{\\sin{(a)}} = \\Psi_{\\lambda}{(a)} and \\Psi_{\\lambda}{(a)} = Q{(a)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('a', commutative=True))))"], [["divide", 3, "sin(Symbol('a', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Mul(Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('a', commutative=True))), Pow(sin(Symbol('a', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('a', commutative=True)), Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Function('Q')(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Pow(sin(Symbol('a', commutative=True)), Integer(-1)), Integral(Function('v_2')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)))"], [["substitute_LHS_for_RHS", 9, "10"], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), Function('Q')(Symbol('a', commutative=True)))"]]}, {"prompt": "Given h{(\\mathbf{P},\\chi)} = \\log{(\\chi^{\\mathbf{P}})}, then derive \\int h{(\\mathbf{P},\\chi)} d\\chi = U - \\chi \\mathbf{P} + \\chi \\log{(\\chi^{\\mathbf{P}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{P}} \\int h{(\\mathbf{P},\\chi)} d\\chi = \\frac{\\partial}{\\partial \\mathbf{P}} (U - \\chi \\mathbf{P} + \\chi \\log{(\\chi^{\\mathbf{P}})})", "derivation": "h{(\\mathbf{P},\\chi)} = \\log{(\\chi^{\\mathbf{P}})} and \\int h{(\\mathbf{P},\\chi)} d\\chi = \\int \\log{(\\chi^{\\mathbf{P}})} d\\chi and \\int h{(\\mathbf{P},\\chi)} d\\chi = U - \\chi \\mathbf{P} + \\chi \\log{(\\chi^{\\mathbf{P}})} and \\frac{\\partial}{\\partial \\mathbf{P}} \\int h{(\\mathbf{P},\\chi)} d\\chi = \\frac{\\partial}{\\partial \\mathbf{P}} (U - \\chi \\mathbf{P} + \\chi \\log{(\\chi^{\\mathbf{P}})})", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\chi', commutative=True)), log(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(log(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('h')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), log(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Function('h')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), log(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(I)} = e^{I}, then derive \\int \\operatorname{E_{\\lambda}}{(I)} dI = F_{x} + e^{I}, then obtain (\\int (F_{x} + \\operatorname{E_{\\lambda}}{(I)}) dF_{x}) \\int (F_{x} + e^{I}) dF_{x} = (\\int (F_{x} + e^{I}) dF_{x})^{2}", "derivation": "\\operatorname{E_{\\lambda}}{(I)} = e^{I} and \\int \\operatorname{E_{\\lambda}}{(I)} dI = \\int e^{I} dI and \\int \\operatorname{E_{\\lambda}}{(I)} dI = F_{x} + e^{I} and \\iint \\operatorname{E_{\\lambda}}{(I)} dI dF_{x} = \\int (F_{x} + e^{I}) dF_{x} and \\iint \\operatorname{E_{\\lambda}}{(I)} dI dF_{x} = \\int (F_{x} + \\operatorname{E_{\\lambda}}{(I)}) dF_{x} and \\int (F_{x} + \\operatorname{E_{\\lambda}}{(I)}) dF_{x} = \\int (F_{x} + e^{I}) dF_{x} and (\\int (F_{x} + \\operatorname{E_{\\lambda}}{(I)}) dF_{x}) \\int (F_{x} + e^{I}) dF_{x} = (\\int (F_{x} + e^{I}) dF_{x})^{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integral(Add(Symbol('F_x', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["times", 6, "Integral(Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Mul(Integral(Add(Symbol('F_x', commutative=True), Function('E_{\\\\lambda}')(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True)))), Pow(Integral(Add(Symbol('F_x', commutative=True), exp(Symbol('I', commutative=True))), Tuple(Symbol('F_x', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})} = \\frac{\\hat{x}_0}{\\eta^{\\prime}} and u{(\\hat{x}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})}, then obtain \\cos{(u{(\\hat{x}_0,\\eta^{\\prime})})} = \\cos{(\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\hat{x}_0}{\\eta^{\\prime}})}", "derivation": "\\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})} = \\frac{\\hat{x}_0}{\\eta^{\\prime}} and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\hat{x}_0}{\\eta^{\\prime}} and u{(\\hat{x}_0,\\eta^{\\prime})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})} and \\cos{(\\frac{\\partial}{\\partial \\eta^{\\prime}} \\mathbf{s}{(\\hat{x}_0,\\eta^{\\prime})})} = \\cos{(\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\hat{x}_0}{\\eta^{\\prime}})} and \\cos{(u{(\\hat{x}_0,\\eta^{\\prime})})} = \\cos{(\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{\\hat{x}_0}{\\eta^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), cos(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(cos(Function('u')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), cos(Derivative(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(M)} = e^{M}, then obtain - s{(M)} \\int (s{(M)} - e^{M}) dM - e^{M} \\int 0 dM = - s{(M)} \\int 0 dM - e^{M} \\int 0 dM", "derivation": "s{(M)} = e^{M} and s{(M)} - e^{M} = 0 and \\int (s{(M)} - e^{M}) dM = \\int 0 dM and - e^{M} \\int (s{(M)} - e^{M}) dM = - e^{M} \\int 0 dM and - s{(M)} \\int (s{(M)} - e^{M}) dM = - s{(M)} \\int 0 dM and - s{(M)} \\int (s{(M)} - e^{M}) dM - e^{M} \\int 0 dM = - s{(M)} \\int 0 dM - e^{M} \\int 0 dM", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["minus", 1, "exp(Symbol('M', commutative=True))"], "Equality(Add(Function('s')(Symbol('M', commutative=True)), Mul(Integer(-1), exp(Symbol('M', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Add(Function('s')(Symbol('M', commutative=True)), Mul(Integer(-1), exp(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integral(Integer(0), Tuple(Symbol('M', commutative=True))))"], [["times", 3, "Mul(Integer(-1), exp(Symbol('M', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Symbol('M', commutative=True)), Integral(Add(Function('s')(Symbol('M', commutative=True)), Mul(Integer(-1), exp(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), exp(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Function('s')(Symbol('M', commutative=True)), Integral(Add(Function('s')(Symbol('M', commutative=True)), Mul(Integer(-1), exp(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), Function('s')(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True)))))"], [["add", 5, "Mul(Integer(-1), exp(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('s')(Symbol('M', commutative=True)), Integral(Add(Function('s')(Symbol('M', commutative=True)), Mul(Integer(-1), exp(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), exp(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True))))), Add(Mul(Integer(-1), Function('s')(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True)))), Mul(Integer(-1), exp(Symbol('M', commutative=True)), Integral(Integer(0), Tuple(Symbol('M', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{H}{(\\delta,\\eta)} = \\delta \\eta and \\mathbf{r}{(\\delta,\\eta)} = \\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta \\eta \\mathbf{H}{(\\delta,\\eta)}, then obtain \\delta + \\mathbf{r}^{\\delta}{(\\delta,\\eta)} = \\delta + (\\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta^{2} \\eta^{2})^{\\delta}", "derivation": "\\mathbf{H}{(\\delta,\\eta)} = \\delta \\eta and \\delta \\eta \\mathbf{H}{(\\delta,\\eta)} = \\delta^{2} \\eta^{2} and \\mathbf{r}{(\\delta,\\eta)} = \\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta \\eta \\mathbf{H}{(\\delta,\\eta)} and \\mathbf{r}{(\\delta,\\eta)} = \\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta^{2} \\eta^{2} and \\mathbf{r}^{\\delta}{(\\delta,\\eta)} = (\\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta^{2} \\eta^{2})^{\\delta} and \\delta + \\mathbf{r}^{\\delta}{(\\delta,\\eta)} = \\delta + (\\delta \\eta + \\frac{\\partial}{\\partial \\delta} \\delta^{2} \\eta^{2})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(Symbol('\\\\eta', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Derivative(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('\\\\delta', commutative=True)))"], [["add", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\eta', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(Symbol('\\\\eta', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(f_{\\mathbf{v}},r_{0})} = f_{\\mathbf{v}} \\sin{(r_{0})}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int (- \\frac{f_{\\mathbf{v}} \\sin{(r_{0})}}{\\hat{x}{(f_{\\mathbf{v}},r_{0})}} + 1) dr_{0} = \\frac{d}{d f_{\\mathbf{v}}} \\int 0 dr_{0}", "derivation": "\\hat{x}{(f_{\\mathbf{v}},r_{0})} = f_{\\mathbf{v}} \\sin{(r_{0})} and 1 = \\frac{f_{\\mathbf{v}} \\sin{(r_{0})}}{\\hat{x}{(f_{\\mathbf{v}},r_{0})}} and - \\frac{f_{\\mathbf{v}} \\sin{(r_{0})}}{\\hat{x}{(f_{\\mathbf{v}},r_{0})}} + 1 = 0 and \\int (- \\frac{f_{\\mathbf{v}} \\sin{(r_{0})}}{\\hat{x}{(f_{\\mathbf{v}},r_{0})}} + 1) dr_{0} = \\int 0 dr_{0} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\int (- \\frac{f_{\\mathbf{v}} \\sin{(r_{0})}}{\\hat{x}{(f_{\\mathbf{v}},r_{0})}} + 1) dr_{0} = \\frac{d}{d f_{\\mathbf{v}}} \\int 0 dr_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), sin(Symbol('r_0', commutative=True))))"], [["divide", 1, "Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Integer(1), Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True))))"], [["minus", 2, "Mul(Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True))), Integer(1)), Integer(0))"], [["integrate", 3, "Symbol('r_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True))), Integer(1)), Tuple(Symbol('r_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('r_0', commutative=True))))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True))), Integer(1)), Tuple(Symbol('r_0', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Integral(Integer(0), Tuple(Symbol('r_0', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(f^{*},E_{x})} = f^{*} \\cos{(E_{x})}, then obtain \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x} + \\int \\mathbf{H}{(f^{*},E_{x})} dE_{x}}{E_{x}} = \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x} + \\int f^{*} \\cos{(E_{x})} dE_{x}}{E_{x}}", "derivation": "\\mathbf{H}{(f^{*},E_{x})} = f^{*} \\cos{(E_{x})} and \\int \\mathbf{H}{(f^{*},E_{x})} dE_{x} = \\int f^{*} \\cos{(E_{x})} dE_{x} and E_{x} + \\int \\mathbf{H}{(f^{*},E_{x})} dE_{x} = E_{x} + \\int f^{*} \\cos{(E_{x})} dE_{x} and \\frac{E_{x} + \\int \\mathbf{H}{(f^{*},E_{x})} dE_{x}}{E_{x}} = \\frac{E_{x} + \\int f^{*} \\cos{(E_{x})} dE_{x}}{E_{x}} and \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x} + \\int \\mathbf{H}{(f^{*},E_{x})} dE_{x}}{E_{x}} = \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x} + \\int f^{*} \\cos{(E_{x})} dE_{x}}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('f^*', commutative=True), cos(Symbol('E_x', commutative=True))))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(Symbol('f^*', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))"], [["add", 2, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Integral(Function('\\\\mathbf{H}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(Symbol('E_x', commutative=True), Integral(Mul(Symbol('f^*', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True)))))"], [["divide", 3, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Integral(Function('\\\\mathbf{H}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Integral(Mul(Symbol('f^*', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))))"], [["differentiate", 4, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Integral(Function('\\\\mathbf{H}')(Symbol('f^*', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Symbol('E_x', commutative=True), Integral(Mul(Symbol('f^*', commutative=True), cos(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True))))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(\\phi_1,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\phi_1, then obtain \\int \\frac{\\partial}{\\partial \\phi_1} \\theta_{2}^{\\phi_1}{(\\phi_1,\\Psi_{\\lambda})} d\\phi_1 = \\int \\frac{\\partial}{\\partial \\phi_1} (\\Psi_{\\lambda} \\phi_1)^{\\phi_1} d\\phi_1", "derivation": "\\theta_{2}{(\\phi_1,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\phi_1 and \\theta_{2}^{\\phi_1}{(\\phi_1,\\Psi_{\\lambda})} = (\\Psi_{\\lambda} \\phi_1)^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} \\theta_{2}^{\\phi_1}{(\\phi_1,\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial \\phi_1} (\\Psi_{\\lambda} \\phi_1)^{\\phi_1} and \\int \\frac{\\partial}{\\partial \\phi_1} \\theta_{2}^{\\phi_1}{(\\phi_1,\\Psi_{\\lambda})} d\\phi_1 = \\int \\frac{\\partial}{\\partial \\phi_1} (\\Psi_{\\lambda} \\phi_1)^{\\phi_1} d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Derivative(Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given H{(f_{\\mathbf{p}},E_{x})} = \\frac{E_{x}}{f_{\\mathbf{p}}}, then obtain 0 = - \\frac{E_{x}}{f_{\\mathbf{p}} H{(f_{\\mathbf{p}},E_{x})}} + 1", "derivation": "H{(f_{\\mathbf{p}},E_{x})} = \\frac{E_{x}}{f_{\\mathbf{p}}} and 1 = \\frac{E_{x}}{f_{\\mathbf{p}} H{(f_{\\mathbf{p}},E_{x})}} and -1 = - \\frac{E_{x}}{f_{\\mathbf{p}} H{(f_{\\mathbf{p}},E_{x})}} and - \\frac{E_{x}}{f_{\\mathbf{p}}} - 1 = - \\frac{E_{x}}{f_{\\mathbf{p}}} - \\frac{E_{x}}{f_{\\mathbf{p}} H{(f_{\\mathbf{p}},E_{x})}} and 0 = - \\frac{E_{x}}{f_{\\mathbf{p}} H{(f_{\\mathbf{p}},E_{x})}} + 1", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))))"], [["divide", 1, "Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Integer(1), Mul(Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))))"], [["minus", 3, "Mul(Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Pow(Function('H')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given h{(\\mu_0,y)} = \\log{(\\mu_0 y)}, then obtain \\frac{(- \\frac{- y + h{(\\mu_0,y)}}{y})^{\\mu_0}}{(y - \\log{(\\mu_0 y)}) \\log{(\\mu_0 y)}} = \\frac{(- \\frac{- y + \\log{(\\mu_0 y)}}{y})^{\\mu_0}}{(y - \\log{(\\mu_0 y)}) \\log{(\\mu_0 y)}}", "derivation": "h{(\\mu_0,y)} = \\log{(\\mu_0 y)} and - y + h{(\\mu_0,y)} = - y + \\log{(\\mu_0 y)} and - \\frac{- y + h{(\\mu_0,y)}}{y} = - \\frac{- y + \\log{(\\mu_0 y)}}{y} and (- \\frac{- y + h{(\\mu_0,y)}}{y})^{\\mu_0} = (- \\frac{- y + \\log{(\\mu_0 y)}}{y})^{\\mu_0} and \\frac{(- \\frac{- y + h{(\\mu_0,y)}}{y})^{\\mu_0}}{y - \\log{(\\mu_0 y)}} = \\frac{(- \\frac{- y + \\log{(\\mu_0 y)}}{y})^{\\mu_0}}{y - \\log{(\\mu_0 y)}} and \\frac{(- \\frac{- y + h{(\\mu_0,y)}}{y})^{\\mu_0}}{(y - \\log{(\\mu_0 y)}) \\log{(\\mu_0 y)}} = \\frac{(- \\frac{- y + \\log{(\\mu_0 y)}}{y})^{\\mu_0}}{(y - \\log{(\\mu_0 y)}) \\log{(\\mu_0 y)}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Symbol('y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))))"], [["power", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Symbol('\\\\mu_0', commutative=True)))"], [["divide", 4, "Add(Symbol('y', commutative=True), Mul(Integer(-1), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))))"], "Equality(Mul(Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('y', commutative=True), Mul(Integer(-1), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Integer(-1))), Mul(Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('y', commutative=True), Mul(Integer(-1), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Integer(-1))))"], [["divide", 5, "log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('h')(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True)))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('y', commutative=True), Mul(Integer(-1), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Integer(-1)), Pow(log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))), Integer(-1))), Mul(Pow(Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('y', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Symbol('y', commutative=True), Mul(Integer(-1), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))))), Integer(-1)), Pow(log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(x^\\prime,C)} = (x^\\prime)^{C}, then obtain n + x^\\prime = \\int \\frac{(x^\\prime)^{C}}{\\operatorname{A_{y}}{(x^\\prime,C)}} dx^\\prime", "derivation": "\\operatorname{A_{y}}{(x^\\prime,C)} = (x^\\prime)^{C} and 1 = \\frac{(x^\\prime)^{C}}{\\operatorname{A_{y}}{(x^\\prime,C)}} and \\int 1 dx^\\prime = \\int \\frac{(x^\\prime)^{C}}{\\operatorname{A_{y}}{(x^\\prime,C)}} dx^\\prime and n + x^\\prime = \\int \\frac{(x^\\prime)^{C}}{\\operatorname{A_{y}}{(x^\\prime,C)}} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)))"], [["divide", 1, "Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Pow(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Pow(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integral(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Pow(Function('A_y')(Symbol('x^\\\\prime', commutative=True), Symbol('C', commutative=True)), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(n_{2},f^{*})} = f^{*} - n_{2}, then derive \\frac{\\partial}{\\partial f^{*}} \\operatorname{E_{\\lambda}}{(n_{2},f^{*})} = 1, then obtain \\frac{\\partial}{\\partial f^{*}} (f^{*} - n_{2}) = 1", "derivation": "\\operatorname{E_{\\lambda}}{(n_{2},f^{*})} = f^{*} - n_{2} and \\frac{\\partial}{\\partial f^{*}} \\operatorname{E_{\\lambda}}{(n_{2},f^{*})} = \\frac{\\partial}{\\partial f^{*}} (f^{*} - n_{2}) and \\frac{\\frac{\\partial}{\\partial f^{*}} \\operatorname{E_{\\lambda}}{(n_{2},f^{*})}}{\\frac{\\partial}{\\partial f^{*}} (f^{*} - n_{2})} = 1 and \\frac{\\partial}{\\partial f^{*}} \\operatorname{E_{\\lambda}}{(n_{2},f^{*})} = 1 and \\frac{\\partial}{\\partial f^{*}} (f^{*} - n_{2}) = 1", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('f^*', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})}, then derive \\int \\operatorname{f_{\\mathbf{v}}}{(\\eta^{\\prime})} d\\eta^{\\prime} = t_{2} + \\sin{(\\eta^{\\prime})}, then obtain - \\int \\cos{(\\eta^{\\prime})} d\\eta^{\\prime} = - t_{2} - \\sin{(\\eta^{\\prime})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\eta^{\\prime})} = \\cos{(\\eta^{\\prime})} and \\int \\operatorname{f_{\\mathbf{v}}}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\int \\cos{(\\eta^{\\prime})} d\\eta^{\\prime} and \\int \\operatorname{f_{\\mathbf{v}}}{(\\eta^{\\prime})} d\\eta^{\\prime} = t_{2} + \\sin{(\\eta^{\\prime})} and \\int \\cos{(\\eta^{\\prime})} d\\eta^{\\prime} = t_{2} + \\sin{(\\eta^{\\prime})} and - \\int \\cos{(\\eta^{\\prime})} d\\eta^{\\prime} = - t_{2} - \\sin{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('t_2', commutative=True), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(t_{1})} = \\log{(t_{1})}, then obtain t_{1} \\log{(t_{1})} = \\frac{t_{1} \\log{(t_{1})}^{2}}{\\operatorname{F_{N}}{(t_{1})}}", "derivation": "\\operatorname{F_{N}}{(t_{1})} = \\log{(t_{1})} and 1 = \\frac{\\log{(t_{1})}}{\\operatorname{F_{N}}{(t_{1})}} and t_{1} = \\frac{t_{1} \\log{(t_{1})}}{\\operatorname{F_{N}}{(t_{1})}} and t_{1} \\log{(t_{1})} = \\frac{t_{1} \\log{(t_{1})}^{2}}{\\operatorname{F_{N}}{(t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('t_1', commutative=True)), log(Symbol('t_1', commutative=True)))"], [["divide", 1, "Function('F_N')(Symbol('t_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('F_N')(Symbol('t_1', commutative=True)), Integer(-1)), log(Symbol('t_1', commutative=True))))"], [["times", 2, "Symbol('t_1', commutative=True)"], "Equality(Symbol('t_1', commutative=True), Mul(Symbol('t_1', commutative=True), Pow(Function('F_N')(Symbol('t_1', commutative=True)), Integer(-1)), log(Symbol('t_1', commutative=True))))"], [["times", 3, "log(Symbol('t_1', commutative=True))"], "Equality(Mul(Symbol('t_1', commutative=True), log(Symbol('t_1', commutative=True))), Mul(Symbol('t_1', commutative=True), Pow(Function('F_N')(Symbol('t_1', commutative=True)), Integer(-1)), Pow(log(Symbol('t_1', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(f^{\\prime},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime}), then derive \\operatorname{F_{N}}{(f^{\\prime},\\mathbf{M})} = -1, then obtain - \\frac{-1 + (-1)^{- \\mathbf{M}} \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime})}{2 \\mathbf{M}} = - \\frac{-1 - (-1)^{- \\mathbf{M}}}{2 \\mathbf{M}}", "derivation": "\\operatorname{F_{N}}{(f^{\\prime},\\mathbf{M})} = \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime}) and \\operatorname{F_{N}}{(f^{\\prime},\\mathbf{M})} = -1 and (-1)^{- \\mathbf{M}} \\operatorname{F_{N}}{(f^{\\prime},\\mathbf{M})} = - (-1)^{- \\mathbf{M}} and (-1)^{- \\mathbf{M}} \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime}) = - (-1)^{- \\mathbf{M}} and -1 + (-1)^{- \\mathbf{M}} \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime}) = -1 - (-1)^{- \\mathbf{M}} and - \\frac{-1 + (-1)^{- \\mathbf{M}} \\frac{\\partial}{\\partial \\mathbf{M}} (- \\mathbf{M} + f^{\\prime})}{2 \\mathbf{M}} = - \\frac{-1 - (-1)^{- \\mathbf{M}}}{2 \\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_N')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))"], [["divide", 2, "Pow(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Function('F_N')(Symbol('f^{\\\\prime}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))), Add(Integer(-1), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))))"], [["divide", 5, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Integer(-1), Mul(Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Integer(-1), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))))"]]}, {"prompt": "Given S{(\\mathbf{P},z^{*})} = \\mathbf{P} z^{*}, then obtain \\frac{z^{*} - S^{\\mathbf{P}}{(\\mathbf{P},z^{*})}}{\\mathbf{P} z^{*}} = \\frac{z^{*} - (\\mathbf{P} z^{*})^{\\mathbf{P}}}{\\mathbf{P} z^{*}}", "derivation": "S{(\\mathbf{P},z^{*})} = \\mathbf{P} z^{*} and S^{\\mathbf{P}}{(\\mathbf{P},z^{*})} = (\\mathbf{P} z^{*})^{\\mathbf{P}} and - z^{*} + S^{\\mathbf{P}}{(\\mathbf{P},z^{*})} = - z^{*} + (\\mathbf{P} z^{*})^{\\mathbf{P}} and z^{*} - S^{\\mathbf{P}}{(\\mathbf{P},z^{*})} = z^{*} - (\\mathbf{P} z^{*})^{\\mathbf{P}} and \\frac{z^{*} - S^{\\mathbf{P}}{(\\mathbf{P},z^{*})}}{\\mathbf{P} z^{*}} = \\frac{z^{*} - (\\mathbf{P} z^{*})^{\\mathbf{P}}}{\\mathbf{P} z^{*}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Symbol('z^*', commutative=True), Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 4, "Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))))), Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given s{(F_{c})} = F_{c}, then derive \\int (s{(F_{c})} + 1) dF_{c} = \\frac{F_{c}^{2}}{2} + F_{c} + \\hbar, then obtain \\frac{\\partial}{\\partial z} (\\int (s{(F_{c})} + 1) dF_{c} - \\frac{z}{F_{c}}) = \\frac{\\partial}{\\partial z} (\\frac{F_{c}^{2}}{2} + F_{c} + \\hbar - \\frac{z}{F_{c}})", "derivation": "s{(F_{c})} = F_{c} and s{(F_{c})} + 1 = F_{c} + 1 and \\int (s{(F_{c})} + 1) dF_{c} = \\int (F_{c} + 1) dF_{c} and \\int (s{(F_{c})} + 1) dF_{c} = \\frac{F_{c}^{2}}{2} + F_{c} + \\hbar and \\int (s{(F_{c})} + 1) dF_{c} - \\frac{z}{F_{c}} = \\frac{F_{c}^{2}}{2} + F_{c} + \\hbar - \\frac{z}{F_{c}} and \\frac{\\partial}{\\partial z} (\\int (s{(F_{c})} + 1) dF_{c} - \\frac{z}{F_{c}}) = \\frac{\\partial}{\\partial z} (\\frac{F_{c}^{2}}{2} + F_{c} + \\hbar - \\frac{z}{F_{c}})", "srepr_derivation": [["renaming_premise", "Equality(Function('s')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('s')(Symbol('F_c', commutative=True)), Integer(1)), Add(Symbol('F_c', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Function('s')(Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Integer(1)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('s')(Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["minus", 4, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('z', commutative=True))"], "Equality(Add(Integral(Add(Function('s')(Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["differentiate", 5, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Integral(Add(Function('s')(Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Symbol('F_c', commutative=True), Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and I{(\\mathbf{f})} = \\mathbf{f} and \\operatorname{v_{1}}{(\\mathbf{f})} = \\mathbf{f}^{\\mathbf{f}}, then obtain - \\mathbf{f}^{\\mathbf{f}} + \\operatorname{v_{1}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})} = - \\cos{(\\mathbf{f})}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and I{(\\mathbf{f})} = \\mathbf{f} and I^{\\mathbf{f}}{(\\mathbf{f})} = \\mathbf{f}^{\\mathbf{f}} and \\operatorname{v_{1}}{(\\mathbf{f})} = \\mathbf{f}^{\\mathbf{f}} and - \\mathbf{f}^{\\mathbf{f}} + I^{\\mathbf{f}}{(\\mathbf{f})} = 0 and \\operatorname{v_{1}}{(\\mathbf{f})} = I^{\\mathbf{f}}{(\\mathbf{f})} and - \\mathbf{f}^{\\mathbf{f}} + \\operatorname{v_{1}}{(\\mathbf{f})} = 0 and - \\mathbf{f}^{\\mathbf{f}} - \\operatorname{A_{1}}{(\\mathbf{f})} + \\operatorname{v_{1}}{(\\mathbf{f})} = - \\operatorname{A_{1}}{(\\mathbf{f})} and - \\mathbf{f}^{\\mathbf{f}} + \\operatorname{v_{1}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})} = - \\cos{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))"], [["power", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 3, "Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Pow(Function('I')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Pow(Function('I')(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"], [["minus", 7, "Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))), Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Function('v_1')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\sigma_p,G)} = \\sin{(G^{\\sigma_p})}, then obtain - \\frac{\\partial}{\\partial G} \\sin{(G^{\\sigma_p})} + \\frac{\\mathbf{E}{(\\sigma_p,G)}}{\\sigma_p} = - \\frac{\\partial}{\\partial G} \\sin{(G^{\\sigma_p})} + \\frac{\\sin{(G^{\\sigma_p})}}{\\sigma_p}", "derivation": "\\mathbf{E}{(\\sigma_p,G)} = \\sin{(G^{\\sigma_p})} and \\frac{\\mathbf{E}{(\\sigma_p,G)}}{\\sigma_p} = \\frac{\\sin{(G^{\\sigma_p})}}{\\sigma_p} and \\frac{\\partial}{\\partial G} \\mathbf{E}{(\\sigma_p,G)} = \\frac{\\partial}{\\partial G} \\sin{(G^{\\sigma_p})} and - \\frac{\\partial}{\\partial G} \\mathbf{E}{(\\sigma_p,G)} + \\frac{\\mathbf{E}{(\\sigma_p,G)}}{\\sigma_p} = - \\frac{\\partial}{\\partial G} \\mathbf{E}{(\\sigma_p,G)} + \\frac{\\sin{(G^{\\sigma_p})}}{\\sigma_p} and - \\frac{\\partial}{\\partial G} \\sin{(G^{\\sigma_p})} + \\frac{\\mathbf{E}{(\\sigma_p,G)}}{\\sigma_p} = - \\frac{\\partial}{\\partial G} \\sin{(G^{\\sigma_p})} + \\frac{\\sin{(G^{\\sigma_p})}}{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{E}')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Derivative(sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), sin(Pow(Symbol('G', commutative=True), Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given \\bar{\\h}{(h,C_{1})} = - C_{1} + h, then derive \\frac{\\partial}{\\partial C_{1}} (M_{E} + \\frac{h^{2}}{2} + \\int - C_{1} dh) = \\frac{\\partial}{\\partial C_{1}} (- C_{1} h + Q + \\frac{h^{2}}{2}), then obtain \\frac{\\partial}{\\partial C_{1}} (M_{E} + \\frac{h^{2}}{2} + \\int - C_{1} dh) = - h", "derivation": "\\bar{\\h}{(h,C_{1})} = - C_{1} + h and \\int \\bar{\\h}{(h,C_{1})} dh = \\int (- C_{1} + h) dh and \\int \\bar{\\h}{(h,C_{1})} dh = \\int - C_{1} dh + \\int h dh and \\frac{\\partial}{\\partial C_{1}} \\int \\bar{\\h}{(h,C_{1})} dh = \\frac{\\partial}{\\partial C_{1}} \\int (- C_{1} + h) dh and \\frac{\\partial}{\\partial C_{1}} (\\int - C_{1} dh + \\int h dh) = \\frac{\\partial}{\\partial C_{1}} \\int (- C_{1} + h) dh and \\frac{\\partial}{\\partial C_{1}} (M_{E} + \\frac{h^{2}}{2} + \\int - C_{1} dh) = \\frac{\\partial}{\\partial C_{1}} (- C_{1} h + Q + \\frac{h^{2}}{2}) and \\frac{\\partial}{\\partial C_{1}} (M_{E} + \\frac{h^{2}}{2} + \\int - C_{1} dh) = - h", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True))), Add(Integral(Mul(Integer(-1), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Symbol('h', commutative=True), Tuple(Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Integral(Mul(Integer(-1), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Symbol('h', commutative=True), Tuple(Symbol('h', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Integral(Mul(Integer(-1), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C_1', commutative=True), Symbol('h', commutative=True)), Symbol('Q', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2)))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Integral(Mul(Integer(-1), Symbol('C_1', commutative=True)), Tuple(Symbol('h', commutative=True)))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('h', commutative=True)))"]]}, {"prompt": "Given v{(\\rho,p)} = e^{\\rho^{p}}, then obtain \\int (- p + (\\int v{(\\rho,p)} d\\rho)^{\\rho}) d\\rho = \\int (- p + (\\int e^{\\rho^{p}} d\\rho)^{\\rho}) d\\rho", "derivation": "v{(\\rho,p)} = e^{\\rho^{p}} and \\int v{(\\rho,p)} d\\rho = \\int e^{\\rho^{p}} d\\rho and (\\int v{(\\rho,p)} d\\rho)^{\\rho} = (\\int e^{\\rho^{p}} d\\rho)^{\\rho} and - p + (\\int v{(\\rho,p)} d\\rho)^{\\rho} = - p + (\\int e^{\\rho^{p}} d\\rho)^{\\rho} and \\int (- p + (\\int v{(\\rho,p)} d\\rho)^{\\rho}) d\\rho = \\int (- p + (\\int e^{\\rho^{p}} d\\rho)^{\\rho}) d\\rho", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('v')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Function('v')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["minus", 3, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(Function('v')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))))"], [["integrate", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(Function('v')(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(exp(Pow(Symbol('\\\\rho', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})} = \\cos{(f^{\\prime} y^{\\prime})}, then obtain \\frac{- f^{\\prime} - y^{\\prime} + \\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}}{\\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}} = \\frac{- f^{\\prime} - y^{\\prime} + \\cos{(f^{\\prime} y^{\\prime})}}{\\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}}", "derivation": "\\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})} = \\cos{(f^{\\prime} y^{\\prime})} and - f^{\\prime} + \\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})} = - f^{\\prime} + \\cos{(f^{\\prime} y^{\\prime})} and - f^{\\prime} - y^{\\prime} + \\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})} = - f^{\\prime} - y^{\\prime} + \\cos{(f^{\\prime} y^{\\prime})} and \\frac{- f^{\\prime} - y^{\\prime} + \\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}}{\\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}} = \\frac{- f^{\\prime} - y^{\\prime} + \\cos{(f^{\\prime} y^{\\prime})}}{\\operatorname{A_{z}}{(f^{\\prime},y^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["minus", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["divide", 3, "Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), cos(Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Pow(Function('A_z')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mu{(\\nabla,v)} = \\nabla + v, then obtain \\mu^{v}{(\\nabla,v)} + \\frac{\\partial}{\\partial \\nabla} (\\nabla + v) = (\\nabla + v)^{v} + \\frac{\\partial}{\\partial \\nabla} (\\nabla + v)", "derivation": "\\mu{(\\nabla,v)} = \\nabla + v and \\frac{\\partial}{\\partial \\nabla} \\mu{(\\nabla,v)} = \\frac{\\partial}{\\partial \\nabla} (\\nabla + v) and \\mu^{v}{(\\nabla,v)} = (\\nabla + v)^{v} and \\mu^{v}{(\\nabla,v)} + \\frac{\\partial}{\\partial \\nabla} \\mu{(\\nabla,v)} = (\\nabla + v)^{v} + \\frac{\\partial}{\\partial \\nabla} \\mu{(\\nabla,v)} and \\mu^{v}{(\\nabla,v)} + \\frac{\\partial}{\\partial \\nabla} (\\nabla + v) = (\\nabla + v)^{v} + \\frac{\\partial}{\\partial \\nabla} (\\nabla + v)", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["add", 3, "Derivative(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Derivative(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\mu')(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Derivative(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Add(Pow(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Derivative(Add(Symbol('\\\\nabla', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(b,\\pi)} = \\pi + b, then derive b + \\int (- b + \\dot{y}{(b,\\pi)})^{2} d\\pi = b + c - \\int \\pi b d\\pi - \\int - \\pi \\dot{y}{(b,\\pi)} d\\pi, then obtain - \\pi (- b + \\dot{y}{(b,\\pi)}) + b + \\int \\pi^{2} d\\pi = - \\pi (- b + \\dot{y}{(b,\\pi)}) + b + c - \\int \\pi b d\\pi - \\int - \\pi (\\pi + b) d\\pi", "derivation": "\\dot{y}{(b,\\pi)} = \\pi + b and - b + \\dot{y}{(b,\\pi)} = \\pi and (- b + \\dot{y}{(b,\\pi)})^{2} = \\pi (- b + \\dot{y}{(b,\\pi)}) and \\int (- b + \\dot{y}{(b,\\pi)})^{2} d\\pi = \\int \\pi (- b + \\dot{y}{(b,\\pi)}) d\\pi and b + \\int (- b + \\dot{y}{(b,\\pi)})^{2} d\\pi = b + \\int \\pi (- b + \\dot{y}{(b,\\pi)}) d\\pi and b + \\int (- b + \\dot{y}{(b,\\pi)})^{2} d\\pi = b + c - \\int \\pi b d\\pi - \\int - \\pi \\dot{y}{(b,\\pi)} d\\pi and b + \\int \\pi^{2} d\\pi = b + c - \\int \\pi b d\\pi - \\int - \\pi (\\pi + b) d\\pi and - \\pi (- b + \\dot{y}{(b,\\pi)}) + b + \\int \\pi^{2} d\\pi = - \\pi (- b + \\dot{y}{(b,\\pi)}) + b + c - \\int \\pi b d\\pi - \\int - \\pi (\\pi + b) d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(2)), Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Add(Symbol('b', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Symbol('b', commutative=True), Integral(Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('b', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Integer(2)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Symbol('b', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('b', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(2)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Symbol('b', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))))"], [["minus", 7, "Mul(Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))), Symbol('b', commutative=True), Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(2)), Tuple(Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\dot{y}')(Symbol('b', commutative=True), Symbol('\\\\pi', commutative=True)))), Symbol('b', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Symbol('\\\\pi', commutative=True), Add(Symbol('\\\\pi', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))))"]]}, {"prompt": "Given T{(\\psi,\\hat{H},G)} = G^{\\hat{H}} - \\psi, then obtain (- (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} T{(\\psi,\\hat{H},G)})^{\\hat{H}} = (- (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} (G^{\\hat{H}} - \\psi))^{\\hat{H}}", "derivation": "T{(\\psi,\\hat{H},G)} = G^{\\hat{H}} - \\psi and G^{- \\hat{H}} T{(\\psi,\\hat{H},G)} = G^{- \\hat{H}} (G^{\\hat{H}} - \\psi) and - (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} T{(\\psi,\\hat{H},G)} = - (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} (G^{\\hat{H}} - \\psi) and (- (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} T{(\\psi,\\hat{H},G)})^{\\hat{H}} = (- (G^{\\hat{H}} - \\psi)^{G} + G^{- \\hat{H}} (G^{\\hat{H}} - \\psi))^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True)), Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))"], [["divide", 1, "Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Function('T')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)))))"], [["minus", 2, "Pow(Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Function('T')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Function('T')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('G', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Add(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\mathbb{I}{(a,c)} = \\cos{(a c)}, then obtain (c + \\frac{\\cos{(a c)}}{c}) (\\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc - 1) = (c + \\frac{\\cos{(a c)}}{c}) (\\eta^{\\prime} + c - 1)", "derivation": "\\mathbb{I}{(a,c)} = \\cos{(a c)} and \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} = 1 and \\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc = \\int 1 dc and - \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} + \\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc = - \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} + \\int 1 dc and \\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc - 1 = \\int 1 dc - 1 and (c + \\frac{\\cos{(a c)}}{c}) (\\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc - 1) = (c + \\frac{\\cos{(a c)}}{c}) (\\int 1 dc - 1) and (c + \\frac{\\cos{(a c)}}{c}) (\\int \\frac{\\mathbb{I}{(a,c)}}{\\cos{(a c)}} dc - 1) = (c + \\frac{\\cos{(a c)}}{c}) (\\eta^{\\prime} + c - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))))"], [["divide", 1, "cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True)))"], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integral(Integer(1), Tuple(Symbol('c', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Integral(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Integral(Integer(1), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1)))"], [["times", 5, "Add(Symbol('c', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True)))))"], "Equality(Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))))), Add(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integer(-1))), Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))))), Add(Integral(Integer(1), Tuple(Symbol('c', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 6], "Equality(Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))))), Add(Integral(Mul(Function('\\\\mathbb{I}')(Symbol('a', commutative=True), Symbol('c', commutative=True)), Pow(cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))), Integer(-1))), Tuple(Symbol('c', commutative=True))), Integer(-1))), Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), cos(Mul(Symbol('a', commutative=True), Symbol('c', commutative=True))))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('c', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}{(v_{z})} = \\sin{(v_{z})}, then derive \\int \\hat{H}{(v_{z})} dv_{z} = P_{e} - \\cos{(v_{z})}, then derive \\frac{\\int \\hat{H}{(v_{z})} dv_{z}}{v_{z}} = \\frac{n - \\cos{(v_{z})}}{v_{z}}, then obtain \\frac{P_{e} - \\cos{(v_{z})}}{v_{z}} = \\frac{n - \\cos{(v_{z})}}{v_{z}}", "derivation": "\\hat{H}{(v_{z})} = \\sin{(v_{z})} and \\int \\hat{H}{(v_{z})} dv_{z} = \\int \\sin{(v_{z})} dv_{z} and \\int \\hat{H}{(v_{z})} dv_{z} = P_{e} - \\cos{(v_{z})} and \\frac{\\int \\hat{H}{(v_{z})} dv_{z}}{v_{z}} = \\frac{\\int \\sin{(v_{z})} dv_{z}}{v_{z}} and \\frac{\\int \\hat{H}{(v_{z})} dv_{z}}{v_{z}} = \\frac{n - \\cos{(v_{z})}}{v_{z}} and \\frac{P_{e} - \\cos{(v_{z})}}{v_{z}} = \\frac{n - \\cos{(v_{z})}}{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True)))))"], [["divide", 2, "Symbol('v_z', commutative=True)"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Integral(Function('\\\\hat{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Integral(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Integral(Function('\\\\hat{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True))))), Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('v_z', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(n,\\sigma_p)} = \\sin{(\\sigma_p + n)}, then obtain \\phi_{2}{(n,\\sigma_p)} \\frac{\\partial}{\\partial n} (\\phi_{2}^{n}{(n,\\sigma_p)})^{n} = \\phi_{2}{(n,\\sigma_p)} \\frac{\\partial}{\\partial n} (\\sin^{n}{(\\sigma_p + n)})^{n}", "derivation": "\\phi_{2}{(n,\\sigma_p)} = \\sin{(\\sigma_p + n)} and \\phi_{2}^{n}{(n,\\sigma_p)} = \\sin^{n}{(\\sigma_p + n)} and (\\phi_{2}^{n}{(n,\\sigma_p)})^{n} = (\\sin^{n}{(\\sigma_p + n)})^{n} and \\frac{\\partial}{\\partial n} (\\phi_{2}^{n}{(n,\\sigma_p)})^{n} = \\frac{\\partial}{\\partial n} (\\sin^{n}{(\\sigma_p + n)})^{n} and \\phi_{2}{(n,\\sigma_p)} \\frac{\\partial}{\\partial n} (\\phi_{2}^{n}{(n,\\sigma_p)})^{n} = \\phi_{2}{(n,\\sigma_p)} \\frac{\\partial}{\\partial n} (\\sin^{n}{(\\sigma_p + n)})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), sin(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('n', commutative=True)), Pow(sin(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Pow(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Pow(sin(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Symbol('n', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["times", 4, "Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Pow(Pow(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(Function('\\\\phi_2')(Symbol('n', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Derivative(Pow(Pow(sin(Add(Symbol('\\\\sigma_p', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(m,E_{\\lambda})} = \\sin{(E_{\\lambda} m)} and \\operatorname{F_{g}}{(m,E_{\\lambda})} = e^{\\sin{(E_{\\lambda} m)}} + \\sin{(E_{\\lambda} m)}, then obtain - E_{\\lambda} m (e^{\\sin{(E_{\\lambda} m)}} + \\sin{(E_{\\lambda} m)}) = - E_{\\lambda} m (\\operatorname{t_{2}}{(m,E_{\\lambda})} + e^{\\operatorname{t_{2}}{(m,E_{\\lambda})}})", "derivation": "\\operatorname{t_{2}}{(m,E_{\\lambda})} = \\sin{(E_{\\lambda} m)} and \\operatorname{F_{g}}{(m,E_{\\lambda})} = e^{\\sin{(E_{\\lambda} m)}} + \\sin{(E_{\\lambda} m)} and \\operatorname{F_{g}}{(m,E_{\\lambda})} = \\operatorname{t_{2}}{(m,E_{\\lambda})} + e^{\\operatorname{t_{2}}{(m,E_{\\lambda})}} and e^{\\sin{(E_{\\lambda} m)}} + \\sin{(E_{\\lambda} m)} = \\operatorname{t_{2}}{(m,E_{\\lambda})} + e^{\\operatorname{t_{2}}{(m,E_{\\lambda})}} and - E_{\\lambda} m (e^{\\sin{(E_{\\lambda} m)}} + \\sin{(E_{\\lambda} m)}) = - E_{\\lambda} m (\\operatorname{t_{2}}{(m,E_{\\lambda})} + e^{\\operatorname{t_{2}}{(m,E_{\\lambda})}})", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(exp(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_g')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), exp(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(exp(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))), Add(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), exp(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True), Add(exp(sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))), sin(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True))))), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Symbol('m', commutative=True), Add(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), exp(Function('t_2')(Symbol('m', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given Q{(\\hat{H}_l,\\Psi)} = \\Psi \\hat{H}_l, then obtain \\Psi Q^{2}{(\\hat{H}_l,\\Psi)} = \\Psi^{3} \\hat{H}_l^{2}", "derivation": "Q{(\\hat{H}_l,\\Psi)} = \\Psi \\hat{H}_l and \\Psi Q{(\\hat{H}_l,\\Psi)} = \\Psi^{2} \\hat{H}_l and \\Psi^{2} \\hat{H}_l Q{(\\hat{H}_l,\\Psi)} = \\Psi^{3} \\hat{H}_l^{2} and \\Psi Q^{2}{(\\hat{H}_l,\\Psi)} = \\Psi^{2} \\hat{H}_l Q{(\\hat{H}_l,\\Psi)} and \\Psi Q^{2}{(\\hat{H}_l,\\Psi)} = \\Psi^{3} \\hat{H}_l^{2}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True), Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(3)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Pow(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(2)), Symbol('\\\\hat{H}_l', commutative=True), Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Pow(Function('Q')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(3)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbb{I}{(z)} = \\cos{(z)}, then obtain \\int 0 dz = \\int (- \\mathbb{I}^{2}{(z)} + \\mathbb{I}{(z)} \\cos{(z)}) dz", "derivation": "\\mathbb{I}{(z)} = \\cos{(z)} and \\mathbb{I}^{2}{(z)} = \\mathbb{I}{(z)} \\cos{(z)} and 0 = - \\mathbb{I}^{2}{(z)} + \\mathbb{I}{(z)} \\cos{(z)} and \\int 0 dz = \\int (- \\mathbb{I}^{2}{(z)} + \\mathbb{I}{(z)} \\cos{(z)}) dz", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["times", 1, "Function('\\\\mathbb{I}')(Symbol('z', commutative=True))"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Integer(2)), Mul(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))))"], [["minus", 2, "Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Integer(2))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Integer(2))), Mul(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('z', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), Integer(2))), Mul(Function('\\\\mathbb{I}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\hat{x}_0,E,r)} = E r + \\hat{x}_0, then derive (\\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE)^{r} = (\\frac{E^{2} r^{2}}{2} + E \\hat{x}_0 r + \\mathbf{J}_P)^{r}, then obtain \\frac{(\\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE)^{r}}{E r + \\hat{x}_0} = \\frac{(\\frac{E^{2} r^{2}}{2} + E \\hat{x}_0 r + \\mathbf{J}_P)^{r}}{E r + \\hat{x}_0}", "derivation": "\\mathbf{r}{(\\hat{x}_0,E,r)} = E r + \\hat{x}_0 and r \\mathbf{r}{(\\hat{x}_0,E,r)} = r (E r + \\hat{x}_0) and \\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE = \\int r (E r + \\hat{x}_0) dE and (\\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE)^{r} = (\\int r (E r + \\hat{x}_0) dE)^{r} and (\\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE)^{r} = (\\frac{E^{2} r^{2}}{2} + E \\hat{x}_0 r + \\mathbf{J}_P)^{r} and \\frac{(\\int r \\mathbf{r}{(\\hat{x}_0,E,r)} dE)^{r}}{E r + \\hat{x}_0} = \\frac{(\\frac{E^{2} r^{2}}{2} + E \\hat{x}_0 r + \\mathbf{J}_P)^{r}}{E r + \\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True)), Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Symbol('r', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('r', commutative=True), Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Symbol('r', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('r', commutative=True), Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('r', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('r', commutative=True)), Pow(Integral(Mul(Symbol('r', commutative=True), Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('r', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Symbol('r', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('r', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('r', commutative=True)))"], [["times", 5, "Pow(Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Pow(Integral(Mul(Symbol('r', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('E', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('r', commutative=True))), Mul(Pow(Add(Mul(Symbol('E', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2)), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('r', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(q)} = \\log{(\\log{(q)})} and \\dot{\\mathbf{r}}{(q)} = - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\operatorname{x^{{\\}'}}{(q)}, then obtain \\dot{\\mathbf{r}}{(q)} = - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\log{(\\log{(q)})}", "derivation": "\\operatorname{x^{{\\}'}}{(q)} = \\log{(\\log{(q)})} and \\frac{d}{d q} \\operatorname{x^{{\\}'}}{(q)} = \\frac{d}{d q} \\log{(\\log{(q)})} and - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\operatorname{x^{{\\}'}}{(q)} = - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\log{(\\log{(q)})} and \\dot{\\mathbf{r}}{(q)} = - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\operatorname{x^{{\\}'}}{(q)} and \\dot{\\mathbf{r}}{(q)} = - \\operatorname{x^{{\\}'}}{(q)} + \\frac{d}{d q} \\log{(\\log{(q)})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('q', commutative=True)), log(log(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(log(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 2, "Function('x^\\\\prime')(Symbol('q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('q', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('q', commutative=True))), Derivative(log(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True)), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('q', commutative=True))), Derivative(Function('x^\\\\prime')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('q', commutative=True)), Add(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('q', commutative=True))), Derivative(log(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{g}{(M)} = \\sin{(M)}, then obtain \\sin{(\\frac{d}{d M} \\mathbf{g}{(M)})} + \\sin{(\\frac{d}{d M} \\sin{(M)})} = 2 \\sin{(\\frac{d}{d M} \\sin{(M)})}", "derivation": "\\mathbf{g}{(M)} = \\sin{(M)} and \\frac{d}{d M} \\mathbf{g}{(M)} = \\frac{d}{d M} \\sin{(M)} and \\sin{(\\frac{d}{d M} \\mathbf{g}{(M)})} = \\sin{(\\frac{d}{d M} \\sin{(M)})} and \\sin{(\\frac{d}{d M} \\mathbf{g}{(M)})} + \\sin{(\\frac{d}{d M} \\sin{(M)})} = 2 \\sin{(\\frac{d}{d M} \\sin{(M)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{g}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), sin(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["add", 3, "sin(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], "Equality(Add(sin(Derivative(Function('\\\\mathbf{g}')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), sin(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))), Mul(Integer(2), sin(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(v_{1},u)} = u + v_{1} and \\hat{X}{(v_{1},u)} = u + v_{1}, then obtain u (- u - v_{1}) (u + v_{1}) = - u (u + v_{1}) \\hat{X}{(v_{1},u)}", "derivation": "\\operatorname{c_{0}}{(v_{1},u)} = u + v_{1} and u \\operatorname{c_{0}}{(v_{1},u)} = u (u + v_{1}) and \\hat{X}{(v_{1},u)} = u + v_{1} and \\operatorname{c_{0}}{(v_{1},u)} = \\hat{X}{(v_{1},u)} and - \\operatorname{c_{0}}{(v_{1},u)} = - \\hat{X}{(v_{1},u)} and u \\hat{X}{(v_{1},u)} = u (u + v_{1}) and - u - v_{1} = - \\hat{X}{(v_{1},u)} and u (- u - v_{1}) \\hat{X}{(v_{1},u)} = - u \\hat{X}^{2}{(v_{1},u)} and u (- u - v_{1}) (u + v_{1}) = - u (u + v_{1}) \\hat{X}{(v_{1},u)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('c_0')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('c_0')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c_0')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('u', commutative=True), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))), Mul(Symbol('u', commutative=True), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))))"], [["times", 7, "Mul(Symbol('u', commutative=True), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)))"], "Equality(Mul(Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Mul(Symbol('u', commutative=True), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True), Add(Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Function('\\\\hat{X}')(Symbol('v_1', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given g{(\\hat{p}_0,V_{\\mathbf{E}})} = \\cos{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})}, then derive \\frac{\\partial}{\\partial V_{\\mathbf{E}}} g{(\\hat{p}_0,V_{\\mathbf{E}})} = - \\frac{\\sin{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})}}{\\hat{p}_0}, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})} = - \\frac{\\sin{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})}}{\\hat{p}_0}", "derivation": "g{(\\hat{p}_0,V_{\\mathbf{E}})} = \\cos{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} g{(\\hat{p}_0,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} g{(\\hat{p}_0,V_{\\mathbf{E}})} = - \\frac{\\sin{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})}}{\\hat{p}_0} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\cos{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})} = - \\frac{\\sin{(\\frac{V_{\\mathbf{E}}}{\\hat{p}_0})}}{\\hat{p}_0}", "srepr_derivation": [["get_premise", "Equality(Function('g')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), cos(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), sin(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), sin(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given J{(h)} = \\sin{(h)}, then obtain \\frac{J^{2}{(h)}}{2 \\sin^{2}{(h)}} - \\frac{d}{d h} J{(h)} = \\frac{1}{2} - \\frac{d}{d h} J{(h)}", "derivation": "J{(h)} = \\sin{(h)} and J{(h)} + \\sin{(h)} = 2 \\sin{(h)} and \\frac{J{(h)}}{J{(h)} + \\sin{(h)}} = \\frac{\\sin{(h)}}{J{(h)} + \\sin{(h)}} and \\frac{J{(h)}}{2 \\sin{(h)}} = \\frac{1}{2} and \\frac{J{(h)}}{J{(h)} + \\sin{(h)}} = \\frac{1}{2} and \\frac{J^{2}{(h)}}{(J{(h)} + \\sin{(h)}) \\sin{(h)}} = \\frac{J{(h)}}{J{(h)} + \\sin{(h)}} and \\frac{J^{2}{(h)}}{(J{(h)} + \\sin{(h)}) \\sin{(h)}} = \\frac{1}{2} and - \\frac{d}{d h} J{(h)} + \\frac{J^{2}{(h)}}{(J{(h)} + \\sin{(h)}) \\sin{(h)}} = \\frac{1}{2} - \\frac{d}{d h} J{(h)} and \\frac{J^{2}{(h)}}{2 \\sin^{2}{(h)}} - \\frac{d}{d h} J{(h)} = \\frac{1}{2} - \\frac{d}{d h} J{(h)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, "sin(Symbol('h', commutative=True))"], "Equality(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Mul(Integer(2), sin(Symbol('h', commutative=True))))"], [["divide", 1, "Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], "Equality(Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Function('J')(Symbol('h', commutative=True))), Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), sin(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Rational(1, 2), Function('J')(Symbol('h', commutative=True)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Function('J')(Symbol('h', commutative=True))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Pow(Function('J')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Function('J')(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Pow(Function('J')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-1))), Rational(1, 2))"], [["minus", 7, "Derivative(Function('J')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('J')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Pow(Add(Function('J')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(-1)), Pow(Function('J')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-1)))), Add(Rational(1, 2), Mul(Integer(-1), Derivative(Function('J')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Add(Mul(Rational(1, 2), Pow(Function('J')(Symbol('h', commutative=True)), Integer(2)), Pow(sin(Symbol('h', commutative=True)), Integer(-2))), Mul(Integer(-1), Derivative(Function('J')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))), Add(Rational(1, 2), Mul(Integer(-1), Derivative(Function('J')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(M,t_{1})} = M t_{1}, then obtain (\\int \\operatorname{v_{y}}^{M}{(M,t_{1})} dM)^{t_{1}} = (\\int (M t_{1})^{M} dM)^{t_{1}}", "derivation": "\\operatorname{v_{y}}{(M,t_{1})} = M t_{1} and \\operatorname{v_{y}}^{M}{(M,t_{1})} = (M t_{1})^{M} and \\int \\operatorname{v_{y}}^{M}{(M,t_{1})} dM = \\int (M t_{1})^{M} dM and (\\int \\operatorname{v_{y}}^{M}{(M,t_{1})} dM)^{t_{1}} = (\\int (M t_{1})^{M} dM)^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Pow(Function('v_y')(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(Mul(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integral(Pow(Function('v_y')(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('t_1', commutative=True)), Pow(Integral(Pow(Mul(Symbol('M', commutative=True), Symbol('t_1', commutative=True)), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given Z{(U)} = \\log{(U)}, then obtain - (Z{(U)} - \\log{(U)})^{U} + (0^{U})^{U} + 1 = - (Z{(U)} - \\log{(U)})^{U} + 2 (0^{U})^{U}", "derivation": "Z{(U)} = \\log{(U)} and Z{(U)} - \\log{(U)} = 0 and (Z{(U)} - \\log{(U)})^{U} = 0^{U} and ((Z{(U)} - \\log{(U)})^{U})^{U} = (0^{U})^{U} and 1 = ((Z{(U)} - \\log{(U)})^{U})^{U} and ((Z{(U)} - \\log{(U)})^{U})^{U} + 1 = 2 ((Z{(U)} - \\log{(U)})^{U})^{U} and (0^{U})^{U} + 1 = 2 (0^{U})^{U} and - (Z{(U)} - \\log{(U)})^{U} + (0^{U})^{U} + 1 = - (Z{(U)} - \\log{(U)})^{U} + 2 (0^{U})^{U}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["minus", 1, "log(Symbol('U', commutative=True))"], "Equality(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Integer(0), Symbol('U', commutative=True)))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(Integer(0), Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["add", 5, "Pow(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Add(Pow(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Integer(1)), Mul(Integer(2), Pow(Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Pow(Pow(Integer(0), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Integer(1)), Mul(Integer(2), Pow(Pow(Integer(0), Symbol('U', commutative=True)), Symbol('U', commutative=True))))"], [["minus", 7, "Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True))), Pow(Pow(Integer(0), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Pow(Add(Function('Z')(Symbol('U', commutative=True)), Mul(Integer(-1), log(Symbol('U', commutative=True)))), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Pow(Integer(0), Symbol('U', commutative=True)), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(s)} = \\sin{(s)}, then derive \\frac{\\dot{\\mathbf{r}} + s}{s} = \\frac{\\int \\frac{\\operatorname{P_{g}}{(s)} + \\sin{(s)}}{2 \\operatorname{P_{g}}{(s)}} ds}{s}, then obtain \\frac{\\dot{\\mathbf{r}} + s}{s} = \\frac{\\int 1 ds}{s}", "derivation": "\\operatorname{P_{g}}{(s)} = \\sin{(s)} and 2 \\operatorname{P_{g}}{(s)} = \\operatorname{P_{g}}{(s)} + \\sin{(s)} and 1 = \\frac{\\operatorname{P_{g}}{(s)} + \\sin{(s)}}{2 \\operatorname{P_{g}}{(s)}} and \\int 1 ds = \\int \\frac{\\operatorname{P_{g}}{(s)} + \\sin{(s)}}{2 \\operatorname{P_{g}}{(s)}} ds and \\frac{\\int 1 ds}{s} = \\frac{\\int \\frac{\\operatorname{P_{g}}{(s)} + \\sin{(s)}}{2 \\operatorname{P_{g}}{(s)}} ds}{s} and \\frac{\\dot{\\mathbf{r}} + s}{s} = \\frac{\\int \\frac{\\operatorname{P_{g}}{(s)} + \\sin{(s)}}{2 \\operatorname{P_{g}}{(s)}} ds}{s} and \\frac{\\dot{\\mathbf{r}} + s}{s} = \\frac{\\int 1 ds}{s}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["add", 1, "Function('P_g')(Symbol('s', commutative=True))"], "Equality(Mul(Integer(2), Function('P_g')(Symbol('s', commutative=True))), Add(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Function('P_g')(Symbol('s', commutative=True)))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(Function('P_g')(Symbol('s', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Integral(Mul(Rational(1, 2), Add(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(Function('P_g')(Symbol('s', commutative=True)), Integer(-1))), Tuple(Symbol('s', commutative=True))))"], [["times", 4, "Pow(Symbol('s', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Integer(1), Tuple(Symbol('s', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Mul(Rational(1, 2), Add(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(Function('P_g')(Symbol('s', commutative=True)), Integer(-1))), Tuple(Symbol('s', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Mul(Rational(1, 2), Add(Function('P_g')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True))), Pow(Function('P_g')(Symbol('s', commutative=True)), Integer(-1))), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Integral(Integer(1), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\pi{(m)} = e^{m}, then obtain \\int \\sin{((\\pi^{m}{(m)})^{m} + \\pi{(m)})} dm - \\int \\sin{(((e^{m})^{m})^{m} + \\pi{(m)})} dm = 0", "derivation": "\\pi{(m)} = e^{m} and \\pi^{m}{(m)} = (e^{m})^{m} and (\\pi^{m}{(m)})^{m} = ((e^{m})^{m})^{m} and (\\pi^{m}{(m)})^{m} + \\pi{(m)} = ((e^{m})^{m})^{m} + \\pi{(m)} and \\sin{((\\pi^{m}{(m)})^{m} + \\pi{(m)})} = \\sin{(((e^{m})^{m})^{m} + \\pi{(m)})} and \\int \\sin{((\\pi^{m}{(m)})^{m} + \\pi{(m)})} dm = \\int \\sin{(((e^{m})^{m})^{m} + \\pi{(m)})} dm and \\int \\sin{((\\pi^{m}{(m)})^{m} + \\pi{(m)})} dm - \\int \\sin{(((e^{m})^{m})^{m} + \\pi{(m)})} dm = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["add", 3, "Function('\\\\pi')(Symbol('m', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True))), Add(Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True))))"], [["sin", 4], "Equality(sin(Add(Pow(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), sin(Add(Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(sin(Add(Pow(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Integral(sin(Add(Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))"], [["minus", 6, "Integral(sin(Add(Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True)))"], "Equality(Add(Integral(sin(Add(Pow(Pow(Function('\\\\pi')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))), Mul(Integer(-1), Integral(sin(Add(Pow(Pow(exp(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\pi')(Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\ddot{x}{(F_{H})} = \\log{(F_{H})} and \\rho_{f}{(F_{H})} = F_{H} \\log{(F_{H})}^{F_{H}}, then obtain F_{H} \\ddot{x}^{F_{H}}{(F_{H})} = \\rho_{f}{(F_{H})}", "derivation": "\\ddot{x}{(F_{H})} = \\log{(F_{H})} and \\ddot{x}^{F_{H}}{(F_{H})} = \\log{(F_{H})}^{F_{H}} and F_{H} \\ddot{x}^{F_{H}}{(F_{H})} = F_{H} \\log{(F_{H})}^{F_{H}} and \\rho_{f}{(F_{H})} = F_{H} \\log{(F_{H})}^{F_{H}} and F_{H} \\ddot{x}^{F_{H}}{(F_{H})} = \\rho_{f}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(log(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["times", 2, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(log(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(log(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('F_H', commutative=True), Pow(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))), Function('\\\\rho_f')(Symbol('F_H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\hat{H}_l)} = \\cos{(\\sin{(\\hat{H}_l)})} and \\mathbf{J}{(c,r)} = (e^{c})^{r}, then obtain \\operatorname{L_{\\varepsilon}}{(\\hat{H}_l)} (e^{c})^{r} = (e^{c})^{r} \\cos{(\\sin{(\\hat{H}_l)})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\hat{H}_l)} = \\cos{(\\sin{(\\hat{H}_l)})} and \\mathbf{J}{(c,r)} = (e^{c})^{r} and \\operatorname{L_{\\varepsilon}}{(\\hat{H}_l)} \\mathbf{J}{(c,r)} = \\mathbf{J}{(c,r)} \\cos{(\\sin{(\\hat{H}_l)})} and \\operatorname{L_{\\varepsilon}}{(\\hat{H}_l)} (e^{c})^{r} = (e^{c})^{r} \\cos{(\\sin{(\\hat{H}_l)})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True)), cos(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('c', commutative=True), Symbol('r', commutative=True)), Pow(exp(Symbol('c', commutative=True)), Symbol('r', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}')(Symbol('c', commutative=True), Symbol('r', commutative=True))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{J}')(Symbol('c', commutative=True), Symbol('r', commutative=True))), Mul(Function('\\\\mathbf{J}')(Symbol('c', commutative=True), Symbol('r', commutative=True)), cos(sin(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True)), Pow(exp(Symbol('c', commutative=True)), Symbol('r', commutative=True))), Mul(Pow(exp(Symbol('c', commutative=True)), Symbol('r', commutative=True)), cos(sin(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given u{(\\Psi,p)} = p^{\\Psi}, then obtain \\log{(\\int u{(\\Psi,p)} d\\Psi)} = \\log{(\\int p^{\\Psi} d\\Psi)}", "derivation": "u{(\\Psi,p)} = p^{\\Psi} and \\frac{u{(\\Psi,p)}}{p} = \\frac{p^{\\Psi}}{p} and \\int u{(\\Psi,p)} d\\Psi = \\int p^{\\Psi} d\\Psi and \\int u{(\\Psi,p)} d\\Psi - \\frac{p^{\\Psi}}{p} + \\frac{u{(\\Psi,p)}}{p} = \\int p^{\\Psi} d\\Psi - \\frac{p^{\\Psi}}{p} + \\frac{u{(\\Psi,p)}}{p} and \\log{(\\int u{(\\Psi,p)} d\\Psi - \\frac{p^{\\Psi}}{p} + \\frac{u{(\\Psi,p)}}{p})} = \\log{(\\int p^{\\Psi} d\\Psi - \\frac{p^{\\Psi}}{p} + \\frac{u{(\\Psi,p)}}{p})} and \\log{(\\int u{(\\Psi,p)} d\\Psi)} = \\log{(\\int p^{\\Psi} d\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["divide", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Add(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True))))"], "Equality(Add(Integral(Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)))), Add(Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)))))"], [["log", 4], "Equality(log(Add(Integral(Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True))))), log(Add(Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Pow(Symbol('p', commutative=True), Integer(-1)), Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(log(Integral(Function('u')(Symbol('\\\\Psi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), log(Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(x)} = \\int \\sin{(x)} dx and \\eta{(x)} = \\frac{d}{d x} \\mathbf{B}{(x)}, then derive \\frac{d}{d x} \\mathbf{B}{(x)} = \\frac{\\partial}{\\partial x} (v_{t} - \\cos{(x)}), then obtain \\eta{(x)} = \\frac{\\partial}{\\partial x} (v_{t} - \\cos{(x)})", "derivation": "\\mathbf{B}{(x)} = \\int \\sin{(x)} dx and \\frac{d}{d x} \\mathbf{B}{(x)} = \\frac{d}{d x} \\int \\sin{(x)} dx and \\frac{d}{d x} \\mathbf{B}{(x)} = \\frac{\\partial}{\\partial x} (v_{t} - \\cos{(x)}) and \\eta{(x)} = \\frac{d}{d x} \\mathbf{B}{(x)} and \\eta{(x)} = \\frac{\\partial}{\\partial x} (v_{t} - \\cos{(x)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('x', commutative=True)), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('v_t', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('x', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\eta')(Symbol('x', commutative=True)), Derivative(Add(Symbol('v_t', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(h)} = \\log{(h)} and p{(h)} = - \\frac{d}{d h} \\log{(h)}, then obtain 1 = (\\frac{\\frac{d}{d h} \\operatorname{A_{1}}{(h)}}{\\frac{d}{d h} \\log{(h)}})^{h}", "derivation": "\\operatorname{A_{1}}{(h)} = \\log{(h)} and \\frac{d}{d h} \\operatorname{A_{1}}{(h)} = \\frac{d}{d h} \\log{(h)} and \\frac{d}{d h} \\operatorname{A_{1}}{(h)} - \\frac{d}{d h} \\log{(h)} = 0 and p{(h)} = - \\frac{d}{d h} \\log{(h)} and - \\frac{d}{d h} \\log{(h)} = - \\frac{d}{d h} \\operatorname{A_{1}}{(h)} and p{(h)} = - \\frac{d}{d h} \\operatorname{A_{1}}{(h)} and 1 = - \\frac{\\frac{d}{d h} \\operatorname{A_{1}}{(h)}}{p{(h)}} and 1 = (- \\frac{\\frac{d}{d h} \\operatorname{A_{1}}{(h)}}{p{(h)}})^{h} and 1 = (\\frac{\\frac{d}{d h} \\operatorname{A_{1}}{(h)}}{\\frac{d}{d h} \\log{(h)}})^{h}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))), Integer(0))"], ["renaming_premise", "Equality(Function('p')(Symbol('h', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["minus", 3, "Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Mul(Integer(-1), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('p')(Symbol('h', commutative=True)), Mul(Integer(-1), Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["divide", 6, "Function('p')(Symbol('h', commutative=True))"], "Equality(Integer(1), Mul(Integer(-1), Pow(Function('p')(Symbol('h', commutative=True)), Integer(-1)), Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["power", 7, "Symbol('h', commutative=True)"], "Equality(Integer(1), Pow(Mul(Integer(-1), Pow(Function('p')(Symbol('h', commutative=True)), Integer(-1)), Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Integer(1), Pow(Mul(Derivative(Function('A_1')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(E_{\\lambda},Q)} = E_{\\lambda} + Q, then obtain E_{\\lambda} + Q + \\hat{x}{(E_{\\lambda},Q)} = 2 \\hat{x}{(E_{\\lambda},Q)}", "derivation": "\\hat{x}{(E_{\\lambda},Q)} = E_{\\lambda} + Q and E_{\\lambda} + Q + \\hat{x}{(E_{\\lambda},Q)} = 2 E_{\\lambda} + 2 Q and 2 \\hat{x}{(E_{\\lambda},Q)} = 2 E_{\\lambda} + 2 Q and E_{\\lambda} + Q + \\hat{x}{(E_{\\lambda},Q)} = 2 \\hat{x}{(E_{\\lambda},Q)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)))"], [["add", 1, "Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True), Function('\\\\hat{x}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(2), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(2), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True), Function('\\\\hat{x}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(2), Function('\\\\hat{x}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\delta)} = e^{\\delta}, then derive \\frac{d}{d \\delta} \\operatorname{v_{y}}{(\\delta)} = e^{\\delta}, then obtain 2 e^{\\delta} = e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} e^{\\delta}", "derivation": "\\operatorname{v_{y}}{(\\delta)} = e^{\\delta} and 2 \\operatorname{v_{y}}{(\\delta)} = \\operatorname{v_{y}}{(\\delta)} + e^{\\delta} and \\frac{d}{d \\delta} \\operatorname{v_{y}}{(\\delta)} = \\frac{d}{d \\delta} e^{\\delta} and \\frac{d}{d \\delta} \\operatorname{v_{y}}{(\\delta)} = e^{\\delta} and \\frac{d}{d \\delta} e^{\\delta} = e^{\\delta} and 2 \\operatorname{v_{y}}{(\\delta)} = \\operatorname{v_{y}}{(\\delta)} + \\frac{d}{d \\delta} e^{\\delta} and 2 \\operatorname{v_{y}}{(\\delta)} = \\operatorname{v_{y}}{(\\delta)} + \\frac{d^{2}}{d \\delta^{2}} e^{\\delta} and 2 e^{\\delta} = e^{\\delta} + \\frac{d^{2}}{d \\delta^{2}} e^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Function('v_y')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(2), Function('v_y')(Symbol('\\\\delta', commutative=True))), Add(Function('v_y')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_y')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), exp(Symbol('\\\\delta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), exp(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Mul(Integer(2), Function('v_y')(Symbol('\\\\delta', commutative=True))), Add(Function('v_y')(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integer(2), Function('v_y')(Symbol('\\\\delta', commutative=True))), Add(Function('v_y')(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Integer(2), exp(Symbol('\\\\delta', commutative=True))), Add(exp(Symbol('\\\\delta', commutative=True)), Derivative(exp(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(2)))))"]]}, {"prompt": "Given A{(U)} = e^{U}, then obtain (\\frac{d}{d U} e^{U} \\int \\frac{d}{d U} A{(U)} dU)^{U} = (\\frac{d}{d U} e^{U} \\int \\frac{d}{d U} e^{U} dU)^{U}", "derivation": "A{(U)} = e^{U} and \\frac{d}{d U} A{(U)} = \\frac{d}{d U} e^{U} and \\int \\frac{d}{d U} A{(U)} dU = \\int \\frac{d}{d U} e^{U} dU and \\frac{d}{d U} A{(U)} \\int \\frac{d}{d U} A{(U)} dU = \\frac{d}{d U} A{(U)} \\int \\frac{d}{d U} e^{U} dU and (\\frac{d}{d U} A{(U)} \\int \\frac{d}{d U} A{(U)} dU)^{U} = (\\frac{d}{d U} A{(U)} \\int \\frac{d}{d U} e^{U} dU)^{U} and (\\frac{d}{d U} e^{U} \\int \\frac{d}{d U} A{(U)} dU)^{U} = (\\frac{d}{d U} e^{U} \\int \\frac{d}{d U} e^{U} dU)^{U}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True))))"], [["times", 3, "Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))), Mul(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))))"], [["power", 4, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Mul(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(Function('A')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Mul(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integral(Derivative(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{p})} = \\cos{(\\cos{(\\mathbf{p})})} and \\operatorname{C_{2}}{(E)} = \\sin{(E)}, then obtain \\frac{\\int \\phi_{1}{(\\mathbf{p})} d\\mathbf{p}}{\\sin{(E)}} = \\frac{\\int \\cos{(\\cos{(\\mathbf{p})})} d\\mathbf{p}}{\\sin{(E)}}", "derivation": "\\phi_{1}{(\\mathbf{p})} = \\cos{(\\cos{(\\mathbf{p})})} and \\int \\phi_{1}{(\\mathbf{p})} d\\mathbf{p} = \\int \\cos{(\\cos{(\\mathbf{p})})} d\\mathbf{p} and \\operatorname{C_{2}}{(E)} = \\sin{(E)} and \\frac{\\int \\phi_{1}{(\\mathbf{p})} d\\mathbf{p}}{\\operatorname{C_{2}}{(E)}} = \\frac{\\int \\cos{(\\cos{(\\mathbf{p})})} d\\mathbf{p}}{\\operatorname{C_{2}}{(E)}} and \\frac{\\int \\phi_{1}{(\\mathbf{p})} d\\mathbf{p}}{\\sin{(E)}} = \\frac{\\int \\cos{(\\cos{(\\mathbf{p})})} d\\mathbf{p}}{\\sin{(E)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), cos(cos(Symbol('\\\\mathbf{p}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(cos(cos(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], ["get_premise", "Equality(Function('C_2')(Symbol('E', commutative=True)), sin(Symbol('E', commutative=True)))"], [["divide", 2, "Function('C_2')(Symbol('E', commutative=True))"], "Equality(Mul(Pow(Function('C_2')(Symbol('E', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(Function('C_2')(Symbol('E', commutative=True)), Integer(-1)), Integral(cos(cos(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(sin(Symbol('E', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_1')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Pow(sin(Symbol('E', commutative=True)), Integer(-1)), Integral(cos(cos(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\hbar)} = e^{\\hbar} and U{(B)} = e^{B}, then obtain (\\int (U{(B)} + e^{- B} e^{\\hbar}) dB)^{B} = (\\int (e^{B} + e^{- B} e^{\\hbar}) dB)^{B}", "derivation": "\\mathbf{J}_f{(\\hbar)} = e^{\\hbar} and U{(B)} = e^{B} and U{(B)} + \\mathbf{J}_f{(\\hbar)} e^{- B} = \\mathbf{J}_f{(\\hbar)} e^{- B} + e^{B} and \\int (U{(B)} + \\mathbf{J}_f{(\\hbar)} e^{- B}) dB = \\int (\\mathbf{J}_f{(\\hbar)} e^{- B} + e^{B}) dB and \\int (U{(B)} + e^{- B} e^{\\hbar}) dB = \\int (e^{B} + e^{- B} e^{\\hbar}) dB and (\\int (U{(B)} + e^{- B} e^{\\hbar}) dB)^{B} = (\\int (e^{B} + e^{- B} e^{\\hbar}) dB)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], ["get_premise", "Equality(Function('U')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["add", 2, "Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))))"], "Equality(Add(Function('U')(Symbol('B', commutative=True)), Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))))), Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True)))), exp(Symbol('B', commutative=True))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Add(Function('U')(Symbol('B', commutative=True)), Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True))), Integral(Add(Mul(Function('\\\\mathbf{J}_f')(Symbol('\\\\hbar', commutative=True)), exp(Mul(Integer(-1), Symbol('B', commutative=True)))), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Add(Function('U')(Symbol('B', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('B', commutative=True))), exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('B', commutative=True))), Integral(Add(exp(Symbol('B', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('B', commutative=True))), exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('B', commutative=True))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Add(Function('U')(Symbol('B', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('B', commutative=True))), exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(Add(exp(Symbol('B', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('B', commutative=True))), exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(Z)} = \\log{(Z)}, then derive \\int Z \\operatorname{P_{g}}{(Z)} \\log{(Z)} dZ = \\frac{Z^{2} \\log{(Z)}^{2}}{2} - \\frac{Z^{2} \\log{(Z)}}{2} + \\frac{Z^{2}}{4} + \\mathbf{P}, then obtain \\iint Z \\log{(Z)}^{2} dZ dZ = \\int (\\frac{Z^{2} \\log{(Z)}^{2}}{2} - \\frac{Z^{2} \\log{(Z)}}{2} + \\frac{Z^{2}}{4} + \\mathbf{P}) dZ", "derivation": "\\operatorname{P_{g}}{(Z)} = \\log{(Z)} and \\operatorname{P_{g}}{(Z)} \\log{(Z)} = \\log{(Z)}^{2} and Z \\operatorname{P_{g}}{(Z)} \\log{(Z)} = Z \\log{(Z)}^{2} and \\int Z \\operatorname{P_{g}}{(Z)} \\log{(Z)} dZ = \\int Z \\log{(Z)}^{2} dZ and \\int Z \\operatorname{P_{g}}{(Z)} \\log{(Z)} dZ = \\frac{Z^{2} \\log{(Z)}^{2}}{2} - \\frac{Z^{2} \\log{(Z)}}{2} + \\frac{Z^{2}}{4} + \\mathbf{P} and \\iint Z \\operatorname{P_{g}}{(Z)} \\log{(Z)} dZ dZ = \\int (\\frac{Z^{2} \\log{(Z)}^{2}}{2} - \\frac{Z^{2} \\log{(Z)}}{2} + \\frac{Z^{2}}{4} + \\mathbf{P}) dZ and \\iint Z \\log{(Z)}^{2} dZ dZ = \\int (\\frac{Z^{2} \\log{(Z)}^{2}}{2} - \\frac{Z^{2} \\log{(Z)}}{2} + \\frac{Z^{2}}{4} + \\mathbf{P}) dZ", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["times", 1, "log(Symbol('Z', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('Z', commutative=True)"], "Equality(Mul(Symbol('Z', commutative=True), Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Mul(Symbol('Z', commutative=True), Pow(log(Symbol('Z', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Symbol('Z', commutative=True), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), log(Symbol('Z', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('P_g')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), log(Symbol('Z', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Mul(Symbol('Z', commutative=True), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), Pow(log(Symbol('Z', commutative=True)), Integer(2))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2)), log(Symbol('Z', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('Z', commutative=True))))"]]}, {"prompt": "Given c{(I,\\Psi_{\\lambda},\\hat{H}_l)} = I + \\Psi_{\\lambda} - \\hat{H}_l and \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} = I + \\Psi_{\\lambda} - \\hat{H}_l, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} c{(I,\\Psi_{\\lambda},\\hat{H}_l)} + 1 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} + 1", "derivation": "c{(I,\\Psi_{\\lambda},\\hat{H}_l)} = I + \\Psi_{\\lambda} - \\hat{H}_l and \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} = I + \\Psi_{\\lambda} - \\hat{H}_l and c{(I,\\Psi_{\\lambda},\\hat{H}_l)} = \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} c{(I,\\Psi_{\\lambda},\\hat{H}_l)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} c{(I,\\Psi_{\\lambda},\\hat{H}_l)} + 1 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\mathbf{s}{(I,\\Psi_{\\lambda},\\hat{H}_l)} + 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('c')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Function('c')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Function('\\\\mathbf{s}')(Symbol('I', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\omega{(E)} = \\log{(E)}, then derive \\int \\omega{(E)} dE = E \\log{(E)} - E + \\phi, then obtain 2 \\int \\omega{(E)} dE + 1 = E \\log{(E)} - E + \\phi + \\int \\omega{(E)} dE + 1", "derivation": "\\omega{(E)} = \\log{(E)} and \\int \\omega{(E)} dE = \\int \\log{(E)} dE and \\int \\omega{(E)} dE + 1 = \\int \\log{(E)} dE + 1 and \\int \\omega{(E)} dE = E \\log{(E)} - E + \\phi and \\int \\omega{(E)} dE + \\int \\log{(E)} dE + 1 = E \\log{(E)} - E + \\phi + \\int \\log{(E)} dE + 1 and 2 \\int \\omega{(E)} dE + 1 = E \\log{(E)} - E + \\phi + \\int \\omega{(E)} dE + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 2, 1], "Equality(Add(Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1)), Add(Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["add", 4, "Add(Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1))"], "Equality(Add(Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1)), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(2), Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Integer(1)), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\phi', commutative=True), Integral(Function('\\\\omega')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\eta{(b)} = \\sin{(b)} and \\operatorname{y^{\\prime}}{(b)} = \\sin{(b)}, then obtain \\frac{d}{d b} \\eta{(b)} = \\frac{d}{d b} \\operatorname{y^{\\prime}}{(b)}", "derivation": "\\eta{(b)} = \\sin{(b)} and \\operatorname{y^{\\prime}}{(b)} = \\sin{(b)} and \\eta{(b)} = \\operatorname{y^{\\prime}}{(b)} and \\frac{d}{d b} \\eta{(b)} = \\frac{d}{d b} \\operatorname{y^{\\prime}}{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\eta')(Symbol('b', commutative=True)), Function('y^{\\\\prime}')(Symbol('b', commutative=True)))"], [["differentiate", 3, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Function('y^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given g{(\\mathbf{f})} = \\log{(\\mathbf{f})}, then derive \\frac{d}{d \\mathbf{f}} g{(\\mathbf{f})} - 1 = -1 + \\frac{1}{\\mathbf{f}}, then obtain \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - 1 = -1 + \\frac{1}{\\mathbf{f}}", "derivation": "g{(\\mathbf{f})} = \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} g{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} g{(\\mathbf{f})} - 1 = \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - 1 and \\frac{d}{d \\mathbf{f}} g{(\\mathbf{f})} - 1 = -1 + \\frac{1}{\\mathbf{f}} and \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - 1 = -1 + \\frac{1}{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('g')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('g')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given I{(y)} = \\log{(y)}, then obtain - I{(y)} = - \\frac{(A + y \\log{(y)} - y) I{(y)}}{\\int I{(y)} dy}", "derivation": "I{(y)} = \\log{(y)} and \\int I{(y)} dy = \\int \\log{(y)} dy and (\\int I{(y)} dy) \\int \\log{(y)} dy = (\\int \\log{(y)} dy)^{2} and 1 = \\frac{\\int \\log{(y)} dy}{\\int I{(y)} dy} and - I{(y)} = - \\frac{I{(y)} \\int \\log{(y)} dy}{\\int I{(y)} dy} and - I{(y)} = - \\frac{(A + y \\log{(y)} - y) I{(y)}}{\\int I{(y)} dy}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["times", 2, "Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Pow(Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(2)))"], [["divide", 3, "Mul(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Function('I')(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), Function('I')(Symbol('y', commutative=True))), Mul(Integer(-1), Function('I')(Symbol('y', commutative=True)), Pow(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1)), Integral(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Integer(-1), Function('I')(Symbol('y', commutative=True))), Mul(Integer(-1), Add(Symbol('A', commutative=True), Mul(Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))), Function('I')(Symbol('y', commutative=True)), Pow(Integral(Function('I')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{B},\\mathbf{r})} = \\mathbf{B} \\sin{(\\mathbf{r})}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{f^{\\prime}}{(\\mathbf{B},\\mathbf{r})} = \\sin{(\\mathbf{r})}, then obtain \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{r})} = \\frac{\\partial^{2}}{\\partial \\mathbf{B}^{2}} \\mathbf{B} \\sin{(\\mathbf{r})}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{B},\\mathbf{r})} = \\mathbf{B} \\sin{(\\mathbf{r})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{f^{\\prime}}{(\\mathbf{B},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\sin{(\\mathbf{r})} and \\frac{\\partial}{\\partial \\mathbf{B}} \\operatorname{f^{\\prime}}{(\\mathbf{B},\\mathbf{r})} = \\sin{(\\mathbf{r})} and \\sin{(\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\sin{(\\mathbf{r})} and \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{r})} = \\frac{\\partial^{2}}{\\partial \\mathbf{B}^{2}} \\mathbf{B} \\sin{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), sin(Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(sin(Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(sin(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), sin(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(2))))"]]}, {"prompt": "Given r{(\\hat{p},C)} = \\frac{C}{\\hat{p}}, then obtain 0 = - (\\frac{C}{\\hat{p}})^{C} + (\\frac{4 C}{\\hat{p}} - 3 r{(\\hat{p},C)})^{C}", "derivation": "r{(\\hat{p},C)} = \\frac{C}{\\hat{p}} and \\frac{C}{\\hat{p}} = \\frac{2 C}{\\hat{p}} - r{(\\hat{p},C)} and r{(\\hat{p},C)} = \\frac{2 C}{\\hat{p}} - r{(\\hat{p},C)} and (\\frac{C}{\\hat{p}})^{C} = (\\frac{2 C}{\\hat{p}} - r{(\\hat{p},C)})^{C} and r{(\\hat{p},C)} = \\frac{4 C}{\\hat{p}} - 3 r{(\\hat{p},C)} and (\\frac{C}{\\hat{p}})^{C} = r^{C}{(\\hat{p},C)} and 0 = - (\\frac{C}{\\hat{p}})^{C} + r^{C}{(\\hat{p},C)} and 0 = - (\\frac{C}{\\hat{p}})^{C} + (\\frac{4 C}{\\hat{p}} - 3 r{(\\hat{p},C)})^{C}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))"], "Equality(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Add(Mul(Integer(2), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)), Add(Mul(Integer(2), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Symbol('C', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)), Add(Mul(Integer(4), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(3), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Symbol('C', commutative=True)), Pow(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["minus", 6, "Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Symbol('C', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Symbol('C', commutative=True))), Pow(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Symbol('C', commutative=True))), Pow(Add(Mul(Integer(4), Symbol('C', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(3), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('C', commutative=True)))), Symbol('C', commutative=True))))"]]}, {"prompt": "Given W{(r,E_{\\lambda})} = E_{\\lambda} + r, then obtain \\int \\frac{- r + W{(r,E_{\\lambda})}}{E_{\\lambda}} dr = \\int 1 dr", "derivation": "W{(r,E_{\\lambda})} = E_{\\lambda} + r and - r + W{(r,E_{\\lambda})} = E_{\\lambda} and \\frac{- r + W{(r,E_{\\lambda})}}{E_{\\lambda}} = 1 and \\int \\frac{- r + W{(r,E_{\\lambda})}}{E_{\\lambda}} dr = \\int 1 dr", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('r', commutative=True)))"], [["minus", 1, "Symbol('r', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('W')(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True))"], [["divide", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('W')(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Integer(1))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('r', commutative=True)), Function('W')(Symbol('r', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('r', commutative=True))), Integral(Integer(1), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\nabla,\\varepsilon)} = \\frac{\\varepsilon}{\\nabla} and \\mathbf{J}{(\\nabla,\\varepsilon)} = - \\varepsilon \\mathbf{D}{(\\nabla,\\varepsilon)}, then obtain - \\mathbf{J}{(\\nabla,\\varepsilon)} = \\frac{\\varepsilon^{2}}{\\nabla}", "derivation": "\\mathbf{D}{(\\nabla,\\varepsilon)} = \\frac{\\varepsilon}{\\nabla} and - \\mathbf{D}{(\\nabla,\\varepsilon)} = - \\frac{\\varepsilon}{\\nabla} and - \\varepsilon \\mathbf{D}{(\\nabla,\\varepsilon)} = - \\frac{\\varepsilon^{2}}{\\nabla} and \\mathbf{J}{(\\nabla,\\varepsilon)} = - \\varepsilon \\mathbf{D}{(\\nabla,\\varepsilon)} and \\mathbf{J}{(\\nabla,\\varepsilon)} = - \\frac{\\varepsilon^{2}}{\\nabla} and - \\mathbf{J}{(\\nabla,\\varepsilon)} = \\frac{\\varepsilon^{2}}{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('\\\\mathbf{D}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{J}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"]]}, {"prompt": "Given g{(f^{*},\\tilde{g})} = \\int (\\tilde{g} + f^{*}) df^{*}, then derive g{(f^{*},\\tilde{g})} = \\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}}, then obtain - f^{*} + \\iint (\\tilde{g} + f^{*}) df^{*} d\\tilde{g} = - f^{*} + \\int (\\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}}) d\\tilde{g}", "derivation": "g{(f^{*},\\tilde{g})} = \\int (\\tilde{g} + f^{*}) df^{*} and g{(f^{*},\\tilde{g})} = \\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}} and \\int (\\tilde{g} + f^{*}) df^{*} = \\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}} and \\iint (\\tilde{g} + f^{*}) df^{*} d\\tilde{g} = \\int (\\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}}) d\\tilde{g} and - f^{*} + \\iint (\\tilde{g} + f^{*}) df^{*} d\\tilde{g} = - f^{*} + \\int (\\tilde{g} f^{*} + \\frac{(f^{*})^{2}}{2} + f_{\\mathbf{p}}) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('f^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('g')(Symbol('f^*', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 4, "Symbol('f^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Integral(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))), Add(Mul(Integer(-1), Symbol('f^*', commutative=True)), Integral(Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Symbol('f^*', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f^*', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True)))))"]]}, {"prompt": "Given s{(\\tilde{g}^*)} = \\log{(\\sin{(\\tilde{g}^*)})} and \\ddot{x}{(V_{\\mathbf{E}},\\dot{y})} = \\frac{\\dot{y}}{V_{\\mathbf{E}}}, then obtain \\frac{\\ddot{x}^{\\dot{y}}{(V_{\\mathbf{E}},\\dot{y})}}{\\log{(\\sin{(\\tilde{g}^*)})}} = \\frac{(\\frac{\\dot{y}}{V_{\\mathbf{E}}})^{\\dot{y}}}{\\log{(\\sin{(\\tilde{g}^*)})}}", "derivation": "s{(\\tilde{g}^*)} = \\log{(\\sin{(\\tilde{g}^*)})} and \\ddot{x}{(V_{\\mathbf{E}},\\dot{y})} = \\frac{\\dot{y}}{V_{\\mathbf{E}}} and \\ddot{x}^{\\dot{y}}{(V_{\\mathbf{E}},\\dot{y})} = (\\frac{\\dot{y}}{V_{\\mathbf{E}}})^{\\dot{y}} and \\frac{\\ddot{x}^{\\dot{y}}{(V_{\\mathbf{E}},\\dot{y})}}{s{(\\tilde{g}^*)}} = \\frac{(\\frac{\\dot{y}}{V_{\\mathbf{E}}})^{\\dot{y}}}{s{(\\tilde{g}^*)}} and \\frac{\\ddot{x}^{\\dot{y}}{(V_{\\mathbf{E}},\\dot{y})}}{\\log{(\\sin{(\\tilde{g}^*)})}} = \\frac{(\\frac{\\dot{y}}{V_{\\mathbf{E}}})^{\\dot{y}}}{\\log{(\\sin{(\\tilde{g}^*)})}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\tilde{g}^*', commutative=True)), log(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)))"], [["power", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["divide", 3, "Function('s')(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('s')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(Function('s')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1))), Mul(Pow(Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(log(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given T{(y^{\\prime},Z)} = e^{Z^{y^{\\prime}}}, then obtain T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}} + \\int \\frac{\\partial}{\\partial Z} (2 T{(y^{\\prime},Z)})^{Z} dZ = T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}} + \\int \\frac{\\partial}{\\partial Z} (T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}})^{Z} dZ", "derivation": "T{(y^{\\prime},Z)} = e^{Z^{y^{\\prime}}} and 2 T{(y^{\\prime},Z)} = T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}} and (2 T{(y^{\\prime},Z)})^{Z} = (T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}})^{Z} and \\frac{\\partial}{\\partial Z} (2 T{(y^{\\prime},Z)})^{Z} = \\frac{\\partial}{\\partial Z} (T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}})^{Z} and \\int \\frac{\\partial}{\\partial Z} (2 T{(y^{\\prime},Z)})^{Z} dZ = \\int \\frac{\\partial}{\\partial Z} (T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}})^{Z} dZ and T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}} + \\int \\frac{\\partial}{\\partial Z} (2 T{(y^{\\prime},Z)})^{Z} dZ = T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}} + \\int \\frac{\\partial}{\\partial Z} (T{(y^{\\prime},Z)} + e^{Z^{y^{\\prime}}})^{Z} dZ", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 1, "Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))), Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('Z', commutative=True)))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('Z', commutative=True)"], "Equality(Integral(Derivative(Pow(Mul(Integer(2), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))), Integral(Derivative(Pow(Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True))))"], [["add", 5, "Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Pow(Mul(Integer(2), Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True)))), Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Derivative(Pow(Add(Function('T')(Symbol('y^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), exp(Pow(Symbol('Z', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(l)} = \\cos{(l)}, then obtain \\mathbf{H}^{2}{(l)} \\cos^{2}{(l)} = \\mathbf{H}{(l)} \\cos^{3}{(l)}", "derivation": "\\mathbf{H}{(l)} = \\cos{(l)} and \\mathbf{H}^{2}{(l)} = \\mathbf{H}{(l)} \\cos{(l)} and \\mathbf{H}^{4}{(l)} = \\mathbf{H}^{2}{(l)} \\cos^{2}{(l)} and \\mathbf{H}^{2}{(l)} \\cos^{2}{(l)} = \\mathbf{H}{(l)} \\cos^{3}{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{H}')(Symbol('l', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), Integer(2)), Pow(cos(Symbol('l', commutative=True)), Integer(2))), Mul(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), Pow(cos(Symbol('l', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} = A_{1} + \\tilde{g}^*, then derive \\frac{\\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1}}{\\tilde{g}^*} = \\frac{\\frac{A_{1}^{2}}{2} + A_{1} \\tilde{g}^* + b}{\\tilde{g}^*}, then obtain \\frac{\\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1}}{(\\tilde{g}^*)^{2}} = \\frac{\\frac{A_{1}^{2}}{2} + A_{1} \\tilde{g}^* + b}{(\\tilde{g}^*)^{2}}", "derivation": "\\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} = A_{1} + \\tilde{g}^* and \\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1} = \\int (A_{1} + \\tilde{g}^*) dA_{1} and \\frac{\\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1}}{\\tilde{g}^*} = \\frac{\\int (A_{1} + \\tilde{g}^*) dA_{1}}{\\tilde{g}^*} and \\frac{\\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1}}{\\tilde{g}^*} = \\frac{\\frac{A_{1}^{2}}{2} + A_{1} \\tilde{g}^* + b}{\\tilde{g}^*} and \\frac{\\int \\operatorname{z^{*}}{(\\tilde{g}^*,A_{1})} dA_{1}}{(\\tilde{g}^*)^{2}} = \\frac{\\frac{A_{1}^{2}}{2} + A_{1} \\tilde{g}^* + b}{(\\tilde{g}^*)^{2}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["integrate", 1, "Symbol('A_1', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(Add(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["divide", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Integral(Function('z^*')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Integral(Add(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Integral(Function('z^*')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('b', commutative=True))))"], [["divide", 4, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-2)), Integral(Function('z^*')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-2)), Add(Mul(Rational(1, 2), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Symbol('A_1', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\theta{(A)} = \\log{(A)} and H{(A)} = A \\theta{(A)}, then obtain H^{A}{(A)} = (A \\theta{(A)})^{A}", "derivation": "\\theta{(A)} = \\log{(A)} and A \\theta{(A)} = A \\log{(A)} and H{(A)} = A \\theta{(A)} and H{(A)} = A \\log{(A)} and H^{A}{(A)} = (A \\log{(A)})^{A} and H^{A}{(A)} = (A \\theta{(A)})^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["times", 1, "Symbol('A', commutative=True)"], "Equality(Mul(Symbol('A', commutative=True), Function('\\\\theta')(Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Function('\\\\theta')(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('H')(Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Function('H')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Symbol('A', commutative=True), log(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('H')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Function('\\\\theta')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\pi,\\mathbf{s},\\mathbf{A})} = \\mathbf{A} - \\mathbf{s} + \\pi, then obtain (- \\frac{\\mathbf{A} - \\mathbf{s} + \\pi}{\\mathbf{s}})^{\\mathbf{A}} = (\\frac{- \\mathbf{A} + \\mathbf{s} - \\pi}{\\mathbf{s}})^{\\mathbf{A}}", "derivation": "\\mu{(\\pi,\\mathbf{s},\\mathbf{A})} = \\mathbf{A} - \\mathbf{s} + \\pi and - \\frac{\\mu{(\\pi,\\mathbf{s},\\mathbf{A})}}{\\mathbf{s}} = - \\frac{\\mathbf{A} - \\mathbf{s} + \\pi}{\\mathbf{s}} and (- \\frac{\\mu{(\\pi,\\mathbf{s},\\mathbf{A})}}{\\mathbf{s}})^{\\mathbf{A}} = (- \\frac{\\mathbf{A} - \\mathbf{s} + \\pi}{\\mathbf{s}})^{\\mathbf{A}} and (- \\frac{\\mathbf{A} - \\mathbf{s} + \\pi}{\\mathbf{s}})^{\\mathbf{A}} = (\\frac{- \\mathbf{A} + \\mathbf{s} - \\pi}{\\mathbf{s}})^{\\mathbf{A}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\pi', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})}, then derive \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = \\frac{1}{\\dot{\\mathbf{r}}}, then obtain \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = - \\frac{1}{\\dot{\\mathbf{r}}^{2}}", "derivation": "\\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\log{(\\dot{\\mathbf{r}})} and \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = \\frac{1}{\\dot{\\mathbf{r}}} and \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = \\frac{d}{d \\dot{\\mathbf{r}}} \\frac{1}{\\dot{\\mathbf{r}}} and \\frac{d}{d \\dot{\\mathbf{r}}} \\operatorname{A_{y}}{(\\dot{\\mathbf{r}})} = - \\frac{1}{\\dot{\\mathbf{r}}^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Derivative(log(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_y')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\mu{(k)} = \\cos{(\\cos{(k)})}, then obtain ((\\frac{d}{d k} (\\mu{(k)} \\cos{(\\cos{(k)})} + \\mu{(k)}))^{k})^{k} = ((\\frac{d}{d k} (\\mu{(k)} + \\cos^{2}{(\\cos{(k)})}))^{k})^{k}", "derivation": "\\mu{(k)} = \\cos{(\\cos{(k)})} and \\mu{(k)} \\cos{(\\cos{(k)})} = \\cos^{2}{(\\cos{(k)})} and \\mu{(k)} \\cos{(\\cos{(k)})} + \\mu{(k)} = \\mu{(k)} + \\cos^{2}{(\\cos{(k)})} and \\frac{d}{d k} (\\mu{(k)} \\cos{(\\cos{(k)})} + \\mu{(k)}) = \\frac{d}{d k} (\\mu{(k)} + \\cos^{2}{(\\cos{(k)})}) and (\\frac{d}{d k} (\\mu{(k)} \\cos{(\\cos{(k)})} + \\mu{(k)}))^{k} = (\\frac{d}{d k} (\\mu{(k)} + \\cos^{2}{(\\cos{(k)})}))^{k} and ((\\frac{d}{d k} (\\mu{(k)} \\cos{(\\cos{(k)})} + \\mu{(k)}))^{k})^{k} = ((\\frac{d}{d k} (\\mu{(k)} + \\cos^{2}{(\\cos{(k)})}))^{k})^{k}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True))))"], [["times", 1, "cos(cos(Symbol('k', commutative=True)))"], "Equality(Mul(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True)))), Pow(cos(cos(Symbol('k', commutative=True))), Integer(2)))"], [["add", 2, "Function('\\\\mu')(Symbol('k', commutative=True))"], "Equality(Add(Mul(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True)))), Function('\\\\mu')(Symbol('k', commutative=True))), Add(Function('\\\\mu')(Symbol('k', commutative=True)), Pow(cos(cos(Symbol('k', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Mul(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True)))), Function('\\\\mu')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mu')(Symbol('k', commutative=True)), Pow(cos(cos(Symbol('k', commutative=True))), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True)))), Function('\\\\mu')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(Add(Function('\\\\mu')(Symbol('k', commutative=True)), Pow(cos(cos(Symbol('k', commutative=True))), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["power", 5, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Derivative(Add(Mul(Function('\\\\mu')(Symbol('k', commutative=True)), cos(cos(Symbol('k', commutative=True)))), Function('\\\\mu')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Derivative(Add(Function('\\\\mu')(Symbol('k', commutative=True)), Pow(cos(cos(Symbol('k', commutative=True))), Integer(2))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} = \\log{(- V_{\\mathbf{B}} + \\mathbf{B})}, then obtain E_{x} + \\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} = A_{x} + \\log{(V_{\\mathbf{B}} - \\mathbf{B})}", "derivation": "\\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} = \\log{(- V_{\\mathbf{B}} + \\mathbf{B})} and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\log{(- V_{\\mathbf{B}} + \\mathbf{B})} and \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} dV_{\\mathbf{B}} = \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\log{(- V_{\\mathbf{B}} + \\mathbf{B})} dV_{\\mathbf{B}} and E_{x} + \\mathbf{S}{(V_{\\mathbf{B}},\\mathbf{B})} = A_{x} + \\log{(V_{\\mathbf{B}} - \\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{S}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Derivative(log(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('E_x', commutative=True), Function('\\\\mathbf{S}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('A_x', commutative=True), log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))))))"]]}, {"prompt": "Given L{(g_{\\varepsilon},A_{y},x^\\prime)} = g_{\\varepsilon} (A_{y} + x^\\prime), then obtain (\\sin{(g_{\\varepsilon} (A_{y} + x^\\prime))} \\sin{(L{(g_{\\varepsilon},A_{y},x^\\prime)})})^{x^\\prime} = (\\sin^{2}{(g_{\\varepsilon} (A_{y} + x^\\prime))})^{x^\\prime}", "derivation": "L{(g_{\\varepsilon},A_{y},x^\\prime)} = g_{\\varepsilon} (A_{y} + x^\\prime) and \\sin{(L{(g_{\\varepsilon},A_{y},x^\\prime)})} = \\sin{(g_{\\varepsilon} (A_{y} + x^\\prime))} and \\sin{(g_{\\varepsilon} (A_{y} + x^\\prime))} \\sin{(L{(g_{\\varepsilon},A_{y},x^\\prime)})} = \\sin^{2}{(g_{\\varepsilon} (A_{y} + x^\\prime))} and (\\sin{(g_{\\varepsilon} (A_{y} + x^\\prime))} \\sin{(L{(g_{\\varepsilon},A_{y},x^\\prime)})})^{x^\\prime} = (\\sin^{2}{(g_{\\varepsilon} (A_{y} + x^\\prime))})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["sin", 1], "Equality(sin(Function('L')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True))), sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["times", 2, "sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], "Equality(Mul(sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), sin(Function('L')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Pow(sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Integer(2)))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), sin(Function('L')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(sin(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Add(Symbol('A_y', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Integer(2)), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(A,P_{e})} = e^{A P_{e}}, then derive (\\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})})^{P_{e}} = (P_{e} e^{A P_{e}})^{P_{e}}, then obtain \\int (\\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})})^{P_{e}} dP_{e} = \\int (P_{e} e^{A P_{e}})^{P_{e}} dP_{e}", "derivation": "\\mathbf{A}{(A,P_{e})} = e^{A P_{e}} and \\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})} = \\frac{\\partial}{\\partial A} e^{A P_{e}} and (\\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})})^{P_{e}} = (\\frac{\\partial}{\\partial A} e^{A P_{e}})^{P_{e}} and (\\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})})^{P_{e}} = (P_{e} e^{A P_{e}})^{P_{e}} and \\int (\\frac{\\partial}{\\partial A} \\mathbf{A}{(A,P_{e})})^{P_{e}} dP_{e} = \\int (P_{e} e^{A P_{e}})^{P_{e}} dP_{e}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('P_e', commutative=True)), exp(Mul(Symbol('A', commutative=True), Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('A', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Pow(Derivative(exp(Mul(Symbol('A', commutative=True), Symbol('P_e', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('P_e', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Pow(Mul(Symbol('P_e', commutative=True), exp(Mul(Symbol('A', commutative=True), Symbol('P_e', commutative=True)))), Symbol('P_e', commutative=True)))"], [["integrate", 4, "Symbol('P_e', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('A', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Pow(Mul(Symbol('P_e', commutative=True), exp(Mul(Symbol('A', commutative=True), Symbol('P_e', commutative=True)))), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given x{(U,F_{c})} = F_{c}^{U}, then derive \\frac{\\partial}{\\partial U} x{(U,F_{c})} = F_{c}^{U} \\log{(F_{c})}, then obtain \\int \\log{(F_{c})} dF_{c} = \\int F_{c}^{- U} \\frac{\\partial}{\\partial U} F_{c}^{U} dF_{c}", "derivation": "x{(U,F_{c})} = F_{c}^{U} and \\frac{\\partial}{\\partial U} x{(U,F_{c})} = \\frac{\\partial}{\\partial U} F_{c}^{U} and F_{c}^{- U} \\frac{\\partial}{\\partial U} x{(U,F_{c})} = F_{c}^{- U} \\frac{\\partial}{\\partial U} F_{c}^{U} and \\frac{\\partial}{\\partial U} x{(U,F_{c})} = F_{c}^{U} \\log{(F_{c})} and \\log{(F_{c})} = F_{c}^{- U} \\frac{\\partial}{\\partial U} F_{c}^{U} and \\int \\log{(F_{c})} dF_{c} = \\int F_{c}^{- U} \\frac{\\partial}{\\partial U} F_{c}^{U} dF_{c}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('U', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('U', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Function('x')(Symbol('U', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('U', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)), log(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(log(Symbol('F_c', commutative=True)), Mul(Pow(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('F_c', commutative=True)"], "Equality(Integral(log(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Pow(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('U', commutative=True))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given y{(x^\\prime)} = \\log{(x^\\prime)}, then derive \\frac{d}{d x^\\prime} y{(x^\\prime)} = \\frac{1}{x^\\prime}, then obtain ((\\frac{d}{d x^\\prime} y{(x^\\prime)})^{x^\\prime})^{x^\\prime} = ((\\frac{1}{x^\\prime})^{x^\\prime})^{x^\\prime}", "derivation": "y{(x^\\prime)} = \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} y{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} y{(x^\\prime)} = \\frac{1}{x^\\prime} and (\\frac{d}{d x^\\prime} y{(x^\\prime)})^{x^\\prime} = (\\frac{1}{x^\\prime})^{x^\\prime} and ((\\frac{d}{d x^\\prime} y{(x^\\prime)})^{x^\\prime})^{x^\\prime} = ((\\frac{1}{x^\\prime})^{x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Function('y')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('y')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(Pow(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\phi,\\theta_2)} = \\phi + \\theta_2 and \\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\hat{H}{(\\phi,\\theta_2)}, then derive \\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)} = 1, then obtain \\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2) = 1", "derivation": "\\hat{H}{(\\phi,\\theta_2)} = \\phi + \\theta_2 and \\frac{\\partial}{\\partial \\theta_2} \\hat{H}{(\\phi,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2) and \\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\hat{H}{(\\phi,\\theta_2)} and \\frac{\\frac{\\partial}{\\partial \\theta_2} \\hat{H}{(\\phi,\\theta_2)}}{\\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2)} = 1 and \\frac{\\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)}}{\\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2)} = 1 and \\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2) and \\operatorname{E_{\\lambda}}{(\\phi,\\theta_2)} = 1 and \\frac{\\partial}{\\partial \\theta_2} (\\phi + \\theta_2) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given L{(E_{\\lambda},B)} = \\log{(\\frac{B}{E_{\\lambda}})} and \\hat{x}_0{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\phi{(\\omega)})}, then obtain \\frac{B L^{B}{(E_{\\lambda},B)}}{\\hat{x}_0{(\\omega)}} = \\frac{B \\log{(\\frac{B}{E_{\\lambda}})}^{B}}{\\hat{x}_0{(\\omega)}}", "derivation": "L{(E_{\\lambda},B)} = \\log{(\\frac{B}{E_{\\lambda}})} and L^{B}{(E_{\\lambda},B)} = \\log{(\\frac{B}{E_{\\lambda}})}^{B} and B L^{B}{(E_{\\lambda},B)} = B \\log{(\\frac{B}{E_{\\lambda}})}^{B} and \\frac{B L^{B}{(E_{\\lambda},B)}}{\\frac{d}{d \\omega} \\cos{(\\phi{(\\omega)})}} = \\frac{B \\log{(\\frac{B}{E_{\\lambda}})}^{B}}{\\frac{d}{d \\omega} \\cos{(\\phi{(\\omega)})}} and \\hat{x}_0{(\\omega)} = \\frac{d}{d \\omega} \\cos{(\\phi{(\\omega)})} and \\frac{B L^{B}{(E_{\\lambda},B)}}{\\hat{x}_0{(\\omega)}} = \\frac{B \\log{(\\frac{B}{E_{\\lambda}})}^{B}}{\\hat{x}_0{(\\omega)}}", "srepr_derivation": [["get_premise", "Equality(Function('L')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), log(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('L')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(log(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Symbol('B', commutative=True)))"], [["times", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('L')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Mul(Symbol('B', commutative=True), Pow(log(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Symbol('B', commutative=True))))"], [["divide", 3, "Derivative(cos(Function('\\\\phi')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('L')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(cos(Function('\\\\phi')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('B', commutative=True), Pow(log(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Symbol('B', commutative=True)), Pow(Derivative(cos(Function('\\\\phi')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\omega', commutative=True)), Derivative(cos(Function('\\\\phi')(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('B', commutative=True), Pow(Function('L')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\omega', commutative=True)), Integer(-1))), Mul(Symbol('B', commutative=True), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\omega', commutative=True)), Integer(-1)), Pow(log(Mul(Symbol('B', commutative=True), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{J},E_{x})} = \\log{(\\frac{E_{x}}{\\mathbf{J}})}, then obtain \\iint (- g + \\tilde{g}^{\\mathbf{J}}{(\\mathbf{J},E_{x})}) d\\mathbf{J} dE_{x} = \\iint (- g + \\log{(\\frac{E_{x}}{\\mathbf{J}})}^{\\mathbf{J}}) d\\mathbf{J} dE_{x}", "derivation": "\\tilde{g}{(\\mathbf{J},E_{x})} = \\log{(\\frac{E_{x}}{\\mathbf{J}})} and \\tilde{g}^{\\mathbf{J}}{(\\mathbf{J},E_{x})} = \\log{(\\frac{E_{x}}{\\mathbf{J}})}^{\\mathbf{J}} and - g + \\tilde{g}^{\\mathbf{J}}{(\\mathbf{J},E_{x})} = - g + \\log{(\\frac{E_{x}}{\\mathbf{J}})}^{\\mathbf{J}} and \\int (- g + \\tilde{g}^{\\mathbf{J}}{(\\mathbf{J},E_{x})}) d\\mathbf{J} = \\int (- g + \\log{(\\frac{E_{x}}{\\mathbf{J}})}^{\\mathbf{J}}) d\\mathbf{J} and \\iint (- g + \\tilde{g}^{\\mathbf{J}}{(\\mathbf{J},E_{x})}) d\\mathbf{J} dE_{x} = \\iint (- g + \\log{(\\frac{E_{x}}{\\mathbf{J}})}^{\\mathbf{J}}) d\\mathbf{J} dE_{x}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_x', commutative=True)), log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["minus", 2, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 4, "Symbol('E_x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Pow(log(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given J{(\\mathbb{I},\\dot{z})} = e^{\\frac{\\dot{z}}{\\mathbb{I}}}, then obtain - \\int (\\frac{J{(\\mathbb{I},\\dot{z})}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z} = - \\int (\\frac{e^{\\frac{\\dot{z}}{\\mathbb{I}}}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z}", "derivation": "J{(\\mathbb{I},\\dot{z})} = e^{\\frac{\\dot{z}}{\\mathbb{I}}} and \\frac{J{(\\mathbb{I},\\dot{z})}}{\\mathbb{I}} = \\frac{e^{\\frac{\\dot{z}}{\\mathbb{I}}}}{\\mathbb{I}} and (\\frac{J{(\\mathbb{I},\\dot{z})}}{\\mathbb{I}})^{\\dot{z}} = (\\frac{e^{\\frac{\\dot{z}}{\\mathbb{I}}}}{\\mathbb{I}})^{\\dot{z}} and \\int (\\frac{J{(\\mathbb{I},\\dot{z})}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z} = \\int (\\frac{e^{\\frac{\\dot{z}}{\\mathbb{I}}}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z} and - \\int (\\frac{J{(\\mathbb{I},\\dot{z})}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z} = - \\int (\\frac{e^{\\frac{\\dot{z}}{\\mathbb{I}}}}{\\mathbb{I}})^{\\dot{z}} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))))"], [["times", 1, "Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))))"], [["power", 2, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))), Symbol('\\\\dot{z}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))), Mul(Integer(-1), Integral(Pow(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), exp(Mul(Symbol('\\\\dot{z}', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(l)} = \\sin{(l)}, then obtain \\frac{\\sin{(l)} \\cos{(\\sigma_{p}{(l)})} + \\frac{1}{\\mathbf{r}}}{\\sigma_{p}{(l)}} = \\frac{\\sin{(l)} \\cos{(\\sin{(l)})} + \\frac{1}{\\mathbf{r}}}{\\sigma_{p}{(l)}}", "derivation": "\\sigma_{p}{(l)} = \\sin{(l)} and \\cos{(\\sigma_{p}{(l)})} = \\cos{(\\sin{(l)})} and \\sin{(l)} \\cos{(\\sigma_{p}{(l)})} = \\sin{(l)} \\cos{(\\sin{(l)})} and \\sin{(l)} \\cos{(\\sigma_{p}{(l)})} + \\frac{1}{\\mathbf{r}} = \\sin{(l)} \\cos{(\\sin{(l)})} + \\frac{1}{\\mathbf{r}} and \\frac{\\sin{(l)} \\cos{(\\sigma_{p}{(l)})} + \\frac{1}{\\mathbf{r}}}{\\sigma_{p}{(l)}} = \\frac{\\sin{(l)} \\cos{(\\sin{(l)})} + \\frac{1}{\\mathbf{r}}}{\\sigma_{p}{(l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\sigma_p')(Symbol('l', commutative=True))), cos(sin(Symbol('l', commutative=True))))"], [["times", 2, "sin(Symbol('l', commutative=True))"], "Equality(Mul(sin(Symbol('l', commutative=True)), cos(Function('\\\\sigma_p')(Symbol('l', commutative=True)))), Mul(sin(Symbol('l', commutative=True)), cos(sin(Symbol('l', commutative=True)))))"], [["add", 3, "Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))"], "Equality(Add(Mul(sin(Symbol('l', commutative=True)), cos(Function('\\\\sigma_p')(Symbol('l', commutative=True)))), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Add(Mul(sin(Symbol('l', commutative=True)), cos(sin(Symbol('l', commutative=True)))), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))))"], [["divide", 4, "Function('\\\\sigma_p')(Symbol('l', commutative=True))"], "Equality(Mul(Add(Mul(sin(Symbol('l', commutative=True)), cos(Function('\\\\sigma_p')(Symbol('l', commutative=True)))), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Pow(Function('\\\\sigma_p')(Symbol('l', commutative=True)), Integer(-1))), Mul(Add(Mul(sin(Symbol('l', commutative=True)), cos(sin(Symbol('l', commutative=True)))), Pow(Symbol('\\\\mathbf{r}', commutative=True), Integer(-1))), Pow(Function('\\\\sigma_p')(Symbol('l', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(\\Psi,\\rho)} = \\rho^{\\Psi}, then derive (\\delta{(\\Psi,\\rho)} - 1) \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = \\rho^{\\Psi} (\\delta{(\\Psi,\\rho)} - 1) \\log{(\\rho)}, then obtain (\\rho^{\\Psi} - 1) \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = \\rho^{\\Psi} (\\rho^{\\Psi} - 1) \\log{(\\rho)}", "derivation": "\\delta{(\\Psi,\\rho)} = \\rho^{\\Psi} and \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = \\frac{\\partial}{\\partial \\Psi} \\rho^{\\Psi} and \\delta{(\\Psi,\\rho)} - 1 = \\rho^{\\Psi} - 1 and (\\delta{(\\Psi,\\rho)} - 1) \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = (\\delta{(\\Psi,\\rho)} - 1) \\frac{\\partial}{\\partial \\Psi} \\rho^{\\Psi} and (\\delta{(\\Psi,\\rho)} - 1) \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = \\rho^{\\Psi} (\\delta{(\\Psi,\\rho)} - 1) \\log{(\\rho)} and (\\rho^{\\Psi} - 1) \\frac{\\partial}{\\partial \\Psi} \\delta{(\\Psi,\\rho)} = \\rho^{\\Psi} (\\rho^{\\Psi} - 1) \\log{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)))"], [["times", 2, "Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), log(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi', commutative=True)), Integer(-1)), log(Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\hat{x}_0,r_{0})} = \\cos{(\\hat{x}_0 + r_{0})}, then obtain \\frac{\\sin{(\\operatorname{v_{y}}^{\\hat{x}_0}{(\\hat{x}_0,r_{0})})}}{\\hat{x}_0} = \\frac{\\sin{(\\cos^{\\hat{x}_0}{(\\hat{x}_0 + r_{0})})}}{\\hat{x}_0}", "derivation": "\\operatorname{v_{y}}{(\\hat{x}_0,r_{0})} = \\cos{(\\hat{x}_0 + r_{0})} and \\operatorname{v_{y}}^{\\hat{x}_0}{(\\hat{x}_0,r_{0})} = \\cos^{\\hat{x}_0}{(\\hat{x}_0 + r_{0})} and \\sin{(\\operatorname{v_{y}}^{\\hat{x}_0}{(\\hat{x}_0,r_{0})})} = \\sin{(\\cos^{\\hat{x}_0}{(\\hat{x}_0 + r_{0})})} and \\frac{\\sin{(\\operatorname{v_{y}}^{\\hat{x}_0}{(\\hat{x}_0,r_{0})})}}{\\hat{x}_0} = \\frac{\\sin{(\\cos^{\\hat{x}_0}{(\\hat{x}_0 + r_{0})})}}{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), cos(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(cos(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('v_y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), sin(Pow(cos(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), sin(Pow(Function('v_y')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), sin(Pow(cos(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(h,\\mathbf{A})} = \\frac{\\mathbf{A}}{h}, then obtain - \\sin{(Q - h \\psi^{*}^{2}{(h,\\mathbf{A})})} = - \\sin{(Q - \\frac{\\mathbf{A}^{2}}{h})}", "derivation": "\\psi^{*}{(h,\\mathbf{A})} = \\frac{\\mathbf{A}}{h} and \\psi^{*}^{2}{(h,\\mathbf{A})} = \\frac{\\mathbf{A} \\psi^{*}{(h,\\mathbf{A})}}{h} and \\mathbf{A} \\psi^{*}{(h,\\mathbf{A})} = \\frac{\\mathbf{A}^{2}}{h} and h \\psi^{*}^{2}{(h,\\mathbf{A})} = \\mathbf{A} \\psi^{*}{(h,\\mathbf{A})} and h \\psi^{*}^{2}{(h,\\mathbf{A})} = \\frac{\\mathbf{A}^{2}}{h} and - Q + h \\psi^{*}^{2}{(h,\\mathbf{A})} = - Q + \\frac{\\mathbf{A}^{2}}{h} and - \\sin{(Q - h \\psi^{*}^{2}{(h,\\mathbf{A})})} = - \\sin{(Q - \\frac{\\mathbf{A}^{2}}{h})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["times", 1, "Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["divide", 2, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('h', commutative=True), Pow(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["minus", 5, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Symbol('h', commutative=True), Pow(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(-1)))))"], [["sin", 6], "Equality(Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('h', commutative=True), Pow(Function('\\\\psi^*')(Symbol('h', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)))))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(2)), Pow(Symbol('h', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\chi)} = \\log{(\\chi)}, then obtain (\\chi + \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}}) \\operatorname{x^{{\\}'}}{(\\chi)} = (\\chi + \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}}) \\log{(\\chi)}", "derivation": "\\operatorname{x^{{\\}'}}{(\\chi)} = \\log{(\\chi)} and \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}} = 1 and \\chi + \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}} = \\chi + 1 and (\\chi + 1) \\operatorname{x^{{\\}'}}{(\\chi)} = (\\chi + 1) \\log{(\\chi)} and (\\chi + \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}}) \\operatorname{x^{{\\}'}}{(\\chi)} = (\\chi + \\frac{\\operatorname{x^{{\\}'}}{(\\chi)}}{\\log{(\\chi)}}) \\log{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Mul(Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1)))), Add(Symbol('\\\\chi', commutative=True), Integer(1)))"], [["times", 1, "Add(Symbol('\\\\chi', commutative=True), Integer(1))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Integer(1)), Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True))), Mul(Add(Symbol('\\\\chi', commutative=True), Integer(1)), log(Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1)))), Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True))), Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Function('x^\\\\prime')(Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Integer(-1)))), log(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(U)} = \\cos{(\\log{(U)})}, then obtain \\operatorname{f_{E}}{(U)} \\operatorname{f_{E}}^{U}{(U)} = \\operatorname{f_{E}}^{U}{(U)} \\cos{(\\log{(U)})}", "derivation": "\\operatorname{f_{E}}{(U)} = \\cos{(\\log{(U)})} and \\operatorname{f_{E}}^{U}{(U)} = \\cos^{U}{(\\log{(U)})} and \\operatorname{f_{E}}{(U)} \\cos^{U}{(\\log{(U)})} = \\cos{(\\log{(U)})} \\cos^{U}{(\\log{(U)})} and \\operatorname{f_{E}}{(U)} \\operatorname{f_{E}}^{U}{(U)} = \\operatorname{f_{E}}^{U}{(U)} \\cos{(\\log{(U)})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('U', commutative=True)), cos(log(Symbol('U', commutative=True))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(cos(log(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["times", 1, "Pow(cos(log(Symbol('U', commutative=True))), Symbol('U', commutative=True))"], "Equality(Mul(Function('f_E')(Symbol('U', commutative=True)), Pow(cos(log(Symbol('U', commutative=True))), Symbol('U', commutative=True))), Mul(cos(log(Symbol('U', commutative=True))), Pow(cos(log(Symbol('U', commutative=True))), Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('f_E')(Symbol('U', commutative=True)), Pow(Function('f_E')(Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Function('f_E')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), cos(log(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(Z,A_{z})} = \\sin{(A_{z}^{Z})}, then obtain - \\frac{\\partial}{\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} + \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\operatorname{t_{1}}^{A_{z}}{(Z,A_{z})} = - \\frac{\\partial}{\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} + \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})}", "derivation": "\\operatorname{t_{1}}{(Z,A_{z})} = \\sin{(A_{z}^{Z})} and \\operatorname{t_{1}}^{A_{z}}{(Z,A_{z})} = \\sin^{A_{z}}{(A_{z}^{Z})} and \\frac{\\partial}{\\partial A_{z}} \\operatorname{t_{1}}^{A_{z}}{(Z,A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} and \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\operatorname{t_{1}}^{A_{z}}{(Z,A_{z})} = \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} and - \\frac{\\partial}{\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} + \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\operatorname{t_{1}}^{A_{z}}{(Z,A_{z})} = - \\frac{\\partial}{\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})} + \\frac{\\partial^{2}}{\\partial Z\\partial A_{z}} \\sin^{A_{z}}{(A_{z}^{Z})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)))"], [["differentiate", 2, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Pow(Function('t_1')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Function('t_1')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Derivative(Pow(Function('t_1')(Symbol('Z', commutative=True), Symbol('A_z', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Derivative(Pow(sin(Pow(Symbol('A_z', commutative=True), Symbol('Z', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{H}{(t_{2},\\phi)} = \\cos{(\\phi - t_{2})} and \\mathbb{I}{(B)} = e^{B} and \\operatorname{m_{s}}{(t_{2},\\phi)} = \\cos{(\\phi - t_{2})}, then obtain (\\frac{\\mathbb{I}{(B)}}{\\operatorname{m_{s}}{(t_{2},\\phi)}})^{\\phi} = (\\frac{e^{B}}{\\operatorname{m_{s}}{(t_{2},\\phi)}})^{\\phi}", "derivation": "\\hat{H}{(t_{2},\\phi)} = \\cos{(\\phi - t_{2})} and \\mathbb{I}{(B)} = e^{B} and \\frac{\\mathbb{I}{(B)}}{\\hat{H}{(t_{2},\\phi)}} = \\frac{e^{B}}{\\hat{H}{(t_{2},\\phi)}} and \\operatorname{m_{s}}{(t_{2},\\phi)} = \\cos{(\\phi - t_{2})} and \\operatorname{m_{s}}{(t_{2},\\phi)} = \\hat{H}{(t_{2},\\phi)} and \\frac{\\mathbb{I}{(B)}}{\\operatorname{m_{s}}{(t_{2},\\phi)}} = \\frac{e^{B}}{\\operatorname{m_{s}}{(t_{2},\\phi)}} and (\\frac{\\mathbb{I}{(B)}}{\\operatorname{m_{s}}{(t_{2},\\phi)}})^{\\phi} = (\\frac{e^{B}}{\\operatorname{m_{s}}{(t_{2},\\phi)}})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["divide", 2, "Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('B', commutative=True))), Mul(Pow(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Function('\\\\mathbb{I}')(Symbol('B', commutative=True)), Pow(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Mul(Pow(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))))"], [["power", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbb{I}')(Symbol('B', commutative=True)), Pow(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('\\\\phi', commutative=True)), Pow(Mul(Pow(Function('m_s')(Symbol('t_2', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(-1)), exp(Symbol('B', commutative=True))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(z^{*})} = \\cos{(z^{*})}, then obtain \\frac{d}{d z^{*}} \\operatorname{E_{n}}{(z^{*})} \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})}", "derivation": "\\operatorname{E_{n}}{(z^{*})} = \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{E_{n}}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{d^{2}}{d (z^{*})^{2}} \\operatorname{E_{n}}{(z^{*})} = \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{E_{n}}{(z^{*})} \\frac{d^{2}}{d (z^{*})^{2}} \\operatorname{E_{n}}{(z^{*})} = \\frac{d^{2}}{d (z^{*})^{2}} \\operatorname{E_{n}}{(z^{*})} \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{E_{n}}{(z^{*})} \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} \\frac{d^{2}}{d (z^{*})^{2}} \\cos{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))))"], [["times", 2, "Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2)))), Mul(Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Derivative(Function('E_n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2)))), Mul(Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(f_{\\mathbf{v}})} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}, then derive \\eta^{\\prime}{(f_{\\mathbf{v}})} = \\mathbf{J}_P + e^{f_{\\mathbf{v}}}, then obtain 1 = \\frac{\\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}}{\\mathbf{J}_P + e^{f_{\\mathbf{v}}}}", "derivation": "\\eta^{\\prime}{(f_{\\mathbf{v}})} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\eta^{\\prime}{(f_{\\mathbf{v}})} = \\mathbf{J}_P + e^{f_{\\mathbf{v}}} and \\mathbf{J}_P + e^{f_{\\mathbf{v}}} = \\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and 1 = \\frac{\\int e^{f_{\\mathbf{v}}} df_{\\mathbf{v}}}{\\mathbf{J}_P + e^{f_{\\mathbf{v}}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), Integral(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(H)} = \\log{(H)}, then obtain \\int \\frac{d}{d H} \\operatorname{P_{e}}^{4}{(H)} dH = \\int \\frac{d}{d H} \\operatorname{P_{e}}^{2}{(H)} \\log{(H)}^{2} dH", "derivation": "\\operatorname{P_{e}}{(H)} = \\log{(H)} and \\operatorname{P_{e}}^{2}{(H)} = \\operatorname{P_{e}}{(H)} \\log{(H)} and \\operatorname{P_{e}}^{4}{(H)} = \\operatorname{P_{e}}^{2}{(H)} \\log{(H)}^{2} and \\frac{d}{d H} \\operatorname{P_{e}}^{4}{(H)} = \\frac{d}{d H} \\operatorname{P_{e}}^{2}{(H)} \\log{(H)}^{2} and \\int \\frac{d}{d H} \\operatorname{P_{e}}^{4}{(H)} dH = \\int \\frac{d}{d H} \\operatorname{P_{e}}^{2}{(H)} \\log{(H)}^{2} dH", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('H', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(2)), Mul(Function('P_e')(Symbol('H', commutative=True)), log(Symbol('H', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(4)), Mul(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(4)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(2))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(4)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Mul(Pow(Function('P_e')(Symbol('H', commutative=True)), Integer(2)), Pow(log(Symbol('H', commutative=True)), Integer(2))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(h,\\omega)} = \\cos{(\\omega + h)}, then derive \\frac{\\partial}{\\partial h} \\bar{\\h}{(h,\\omega)} = - \\sin{(\\omega + h)}, then obtain \\frac{\\frac{\\partial^{2}}{\\partial h^{2}} \\bar{\\h}{(h,\\omega)}}{\\sin{(\\omega + h)}} = \\frac{\\frac{\\partial}{\\partial h} - \\sin{(\\omega + h)}}{\\sin{(\\omega + h)}}", "derivation": "\\bar{\\h}{(h,\\omega)} = \\cos{(\\omega + h)} and \\frac{\\partial}{\\partial h} \\bar{\\h}{(h,\\omega)} = \\frac{\\partial}{\\partial h} \\cos{(\\omega + h)} and \\frac{\\partial}{\\partial h} \\bar{\\h}{(h,\\omega)} = - \\sin{(\\omega + h)} and \\frac{\\partial^{2}}{\\partial h^{2}} \\bar{\\h}{(h,\\omega)} = \\frac{\\partial}{\\partial h} - \\sin{(\\omega + h)} and \\frac{\\frac{\\partial^{2}}{\\partial h^{2}} \\bar{\\h}{(h,\\omega)}}{\\sin{(\\omega + h)}} = \\frac{\\frac{\\partial}{\\partial h} - \\sin{(\\omega + h)}}{\\sin{(\\omega + h)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Integer(-1), sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 4, "sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Pow(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Derivative(Function('\\\\hbar')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(2)))), Mul(Pow(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{F_{H} \\varphi}{\\ddot{x}}, then obtain \\frac{\\partial}{\\partial F_{H}} F_{H} \\varphi \\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H}^{2} \\varphi^{2}}{\\ddot{x}}", "derivation": "\\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{F_{H} \\varphi}{\\ddot{x}} and \\varphi \\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{F_{H} \\varphi^{2}}{\\ddot{x}} and F_{H} \\varphi \\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{F_{H}^{2} \\varphi^{2}}{\\ddot{x}} and \\frac{\\partial}{\\partial F_{H}} F_{H} \\varphi \\pi{(\\varphi,\\ddot{x},F_{H})} = \\frac{\\partial}{\\partial F_{H}} \\frac{F_{H}^{2} \\varphi^{2}}{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('\\\\pi')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))))"], [["times", 2, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Symbol('\\\\varphi', commutative=True), Function('\\\\pi')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('F_H', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(2)), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Mul(Symbol('F_H', commutative=True), Symbol('\\\\varphi', commutative=True), Function('\\\\pi')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\ddot{x}', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_H', commutative=True), Integer(2)), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(M)} = e^{M} and \\operatorname{t_{2}}{(Z)} = \\sin{(Z)}, then obtain \\frac{\\partial}{\\partial Z} (- M + \\operatorname{C_{2}}^{M}{(M)} + \\operatorname{t_{2}}{(Z)}) = \\frac{\\partial}{\\partial Z} (- M + \\operatorname{t_{2}}{(Z)} + (e^{M})^{M})", "derivation": "\\operatorname{C_{2}}{(M)} = e^{M} and \\operatorname{C_{2}}^{M}{(M)} = (e^{M})^{M} and - M + \\operatorname{C_{2}}^{M}{(M)} = - M + (e^{M})^{M} and \\operatorname{t_{2}}{(Z)} = \\sin{(Z)} and - M + \\operatorname{C_{2}}^{M}{(M)} + \\sin{(Z)} = - M + (e^{M})^{M} + \\sin{(Z)} and - M + \\operatorname{C_{2}}^{M}{(M)} + \\operatorname{t_{2}}{(Z)} = - M + \\operatorname{t_{2}}{(Z)} + (e^{M})^{M} and \\frac{\\partial}{\\partial Z} (- M + \\operatorname{C_{2}}^{M}{(M)} + \\operatorname{t_{2}}{(Z)}) = \\frac{\\partial}{\\partial Z} (- M + \\operatorname{t_{2}}{(Z)} + (e^{M})^{M})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["minus", 2, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], ["get_premise", "Equality(Function('t_2')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["add", 3, "sin(Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), sin(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True)), sin(Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Function('t_2')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('t_2')(Symbol('Z', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], [["differentiate", 6, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Pow(Function('C_2')(Symbol('M', commutative=True)), Symbol('M', commutative=True)), Function('t_2')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('t_2')(Symbol('Z', commutative=True)), Pow(exp(Symbol('M', commutative=True)), Symbol('M', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} = \\sin{(\\log{(i)})}, then obtain \\frac{- i + 2 \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(i)}} = \\frac{- i + \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di + \\int \\sin{(\\log{(i)})} di}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(i)}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} = \\sin{(\\log{(i)})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di = \\int \\sin{(\\log{(i)})} di and - i + 2 \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di = - i + \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di + \\int \\sin{(\\log{(i)})} di and \\frac{- i + 2 \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di}{\\sin{(\\log{(i)})}} = \\frac{- i + \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di + \\int \\sin{(\\log{(i)})} di}{\\sin{(\\log{(i)})}} and \\frac{- i + 2 \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(i)}} = \\frac{- i + \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(i)} di + \\int \\sin{(\\log{(i)})} di}{\\operatorname{g^{\\prime}_{\\varepsilon}}{(i)}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('i', commutative=True)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))))"], [["divide", 3, "sin(log(Symbol('i', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Pow(sin(log(Symbol('i', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Integer(2), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(log(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('i', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given L{(\\hat{p},\\varepsilon)} = \\sin{(\\frac{\\varepsilon}{\\hat{p}})} and \\delta{(x^\\prime)} = \\sin{(x^\\prime)}, then obtain x^\\prime + \\delta{(x^\\prime)} \\sin{(\\frac{\\varepsilon}{\\hat{p}})} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})} = x^\\prime + \\sin{(x^\\prime)} \\sin{(\\frac{\\varepsilon}{\\hat{p}})} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})}", "derivation": "L{(\\hat{p},\\varepsilon)} = \\sin{(\\frac{\\varepsilon}{\\hat{p}})} and \\delta{(x^\\prime)} = \\sin{(x^\\prime)} and L{(\\hat{p},\\varepsilon)} \\delta{(x^\\prime)} = L{(\\hat{p},\\varepsilon)} \\sin{(x^\\prime)} and x^\\prime + L{(\\hat{p},\\varepsilon)} \\delta{(x^\\prime)} = x^\\prime + L{(\\hat{p},\\varepsilon)} \\sin{(x^\\prime)} and x^\\prime + L{(\\hat{p},\\varepsilon)} \\delta{(x^\\prime)} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})} = x^\\prime + L{(\\hat{p},\\varepsilon)} \\sin{(x^\\prime)} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})} and x^\\prime + \\delta{(x^\\prime)} \\sin{(\\frac{\\varepsilon}{\\hat{p}})} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})} = x^\\prime + \\sin{(x^\\prime)} \\sin{(\\frac{\\varepsilon}{\\hat{p}})} + \\sin{(\\frac{\\varepsilon}{\\hat{p}})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["times", 2, "Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\delta')(Symbol('x^\\\\prime', commutative=True))), Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))))"], [["add", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\delta')(Symbol('x^\\\\prime', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))))"], [["add", 4, "sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\delta')(Symbol('x^\\\\prime', commutative=True))), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Function('L')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True))), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Mul(Function('\\\\delta')(Symbol('x^\\\\prime', commutative=True)), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), Add(Symbol('x^\\\\prime', commutative=True), Mul(sin(Symbol('x^\\\\prime', commutative=True)), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))), sin(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(P_{g},M_{E})} = \\frac{P_{g}}{M_{E}}, then obtain - \\operatorname{F_{g}}{(P_{g},M_{E})} + \\int \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} dM_{E} - \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} = - \\operatorname{F_{g}}{(P_{g},M_{E})} + \\int \\frac{1}{M_{E}} dM_{E} - \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}}", "derivation": "\\operatorname{F_{g}}{(P_{g},M_{E})} = \\frac{P_{g}}{M_{E}} and \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} = \\frac{1}{M_{E}} and \\int \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} dM_{E} = \\int \\frac{1}{M_{E}} dM_{E} and - \\operatorname{F_{g}}{(P_{g},M_{E})} + \\int \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} dM_{E} - \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}} = - \\operatorname{F_{g}}{(P_{g},M_{E})} + \\int \\frac{1}{M_{E}} dM_{E} - \\frac{\\operatorname{F_{g}}{(P_{g},M_{E})}}{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('P_g', commutative=True)))"], [["divide", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Integral(Pow(Symbol('M_E', commutative=True), Integer(-1)), Tuple(Symbol('M_E', commutative=True))))"], [["minus", 3, "Add(Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))), Integral(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True))), Integral(Pow(Symbol('M_E', commutative=True), Integer(-1)), Tuple(Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('P_g', commutative=True), Integer(-1)), Function('F_g')(Symbol('P_g', commutative=True), Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\psi)} = \\log{(\\psi)} and \\rho{(\\theta_2)} = \\sin{(\\theta_2)}, then obtain \\tilde{\\infty} (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} + \\rho{(\\theta_2)} = \\tilde{\\infty} (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} + \\sin{(\\theta_2)}", "derivation": "\\mathbf{v}{(\\psi)} = \\log{(\\psi)} and 0 = - \\mathbf{v}{(\\psi)} + \\log{(\\psi)} and 0^{\\psi} = (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} and 0^{\\psi} \\tilde{\\infty} = \\tilde{\\infty} (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} and \\rho{(\\theta_2)} = \\sin{(\\theta_2)} and 0^{\\psi} \\tilde{\\infty} + \\rho{(\\theta_2)} = 0^{\\psi} \\tilde{\\infty} + \\sin{(\\theta_2)} and \\tilde{\\infty} (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} + \\rho{(\\theta_2)} = \\tilde{\\infty} (- \\mathbf{v}{(\\psi)} + \\log{(\\psi)})^{\\psi} + \\sin{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["divide", 3, 0], "Equality(Mul(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), zoo), Mul(zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["add", 5, "Mul(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), zoo)"], "Equality(Add(Mul(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), zoo), Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), zoo), sin(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True))), Function('\\\\rho')(Symbol('\\\\theta_2', commutative=True))), Add(Mul(zoo, Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\psi', commutative=True))), log(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True))), sin(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given p{(g)} = \\cos{(g)}, then obtain (\\frac{p{(g)} p^{g}{(g)}}{g})^{g} = (\\frac{p{(g)} \\cos^{g}{(g)}}{g})^{g}", "derivation": "p{(g)} = \\cos{(g)} and \\frac{p{(g)}}{g} = \\frac{\\cos{(g)}}{g} and p^{g}{(g)} = \\cos^{g}{(g)} and \\frac{p^{g}{(g)} \\cos{(g)}}{g} = \\frac{\\cos{(g)} \\cos^{g}{(g)}}{g} and (\\frac{p^{g}{(g)} \\cos{(g)}}{g})^{g} = (\\frac{\\cos{(g)} \\cos^{g}{(g)}}{g})^{g} and (\\frac{p{(g)} p^{g}{(g)}}{g})^{g} = (\\frac{p{(g)} \\cos^{g}{(g)}}{g})^{g}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('p')(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('p')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('p')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('p')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), cos(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('p')(Symbol('g', commutative=True)), Pow(Function('p')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('p')(Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = \\frac{\\mathbf{g}}{\\hat{x}_0}, then obtain - \\mathbf{J}_P + \\mathbf{g} \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = - \\mathbf{J}_P + \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\frac{\\mathbf{g}^{2}}{\\hat{x}_0}", "derivation": "\\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = \\frac{\\mathbf{g}}{\\hat{x}_0} and \\mathbf{g} \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = \\frac{\\mathbf{g}^{2}}{\\hat{x}_0} and \\mathbf{g} \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\frac{\\mathbf{g}^{2}}{\\hat{x}_0} and - \\mathbf{J}_P + \\mathbf{g} \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} = - \\mathbf{J}_P + \\mathbf{A}{(\\hat{x}_0,\\mathbf{g})} + \\frac{\\mathbf{g}^{2}}{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"], [["add", 2, "Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)))))"], [["minus", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\hat{p}{(U,z)} = U + z, then derive \\int \\hat{p}{(U,z)} dU = \\frac{U^{2}}{2} + U z + \\rho_b, then derive \\frac{\\partial}{\\partial \\rho_b} (\\frac{U^{2}}{2} + U z + \\rho_b)^{\\rho_b} = \\frac{\\partial}{\\partial \\rho_b} (E_{n} + \\frac{U^{2}}{2} + U z)^{\\rho_b}, then obtain \\frac{\\partial}{\\partial \\rho_b} (\\int (U + z) dU)^{\\rho_b} = \\frac{\\partial}{\\partial \\rho_b} (E_{n} + \\frac{U^{2}}{2} + U z)^{\\rho_b}", "derivation": "\\hat{p}{(U,z)} = U + z and \\int \\hat{p}{(U,z)} dU = \\int (U + z) dU and \\int \\hat{p}{(U,z)} dU = \\frac{U^{2}}{2} + U z + \\rho_b and \\frac{U^{2}}{2} + U z + \\rho_b = \\int (U + z) dU and (\\frac{U^{2}}{2} + U z + \\rho_b)^{\\rho_b} = (\\int (U + z) dU)^{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} (\\frac{U^{2}}{2} + U z + \\rho_b)^{\\rho_b} = \\frac{\\partial}{\\partial \\rho_b} (\\int (U + z) dU)^{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} (\\frac{U^{2}}{2} + U z + \\rho_b)^{\\rho_b} = \\frac{\\partial}{\\partial \\rho_b} (E_{n} + \\frac{U^{2}}{2} + U z)^{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} (\\int (U + z) dU)^{\\rho_b} = \\frac{\\partial}{\\partial \\rho_b} (E_{n} + \\frac{U^{2}}{2} + U z)^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('z', commutative=True)), Add(Symbol('U', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{p}')(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Integral(Add(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["power", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(Add(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Pow(Integral(Add(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(Pow(Integral(Add(Symbol('U', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('U', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(2))), Mul(Symbol('U', commutative=True), Symbol('z', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(n_{2},\\mathbf{J}_P)} = \\mathbf{J}_P n_{2}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (n_{2} + \\phi{(n_{2},\\mathbf{J}_P)}) d\\mathbf{J}_P = \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P n_{2} + n_{2}) d\\mathbf{J}_P", "derivation": "\\phi{(n_{2},\\mathbf{J}_P)} = \\mathbf{J}_P n_{2} and n_{2} + \\phi{(n_{2},\\mathbf{J}_P)} = \\mathbf{J}_P n_{2} + n_{2} and \\frac{\\partial}{\\partial \\mathbf{J}_P} (n_{2} + \\phi{(n_{2},\\mathbf{J}_P)}) = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P n_{2} + n_{2}) and \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (n_{2} + \\phi{(n_{2},\\mathbf{J}_P)}) d\\mathbf{J}_P = \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P n_{2} + n_{2}) d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n_2', commutative=True)))"], [["add", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Symbol('n_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Symbol('n_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('n_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Derivative(Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\phi_1)} = e^{\\phi_1}, then obtain \\int (- \\phi_1 + (\\phi_1 + \\mathbf{J}_M{(\\phi_1)}) (\\phi_1 + e^{\\phi_1})) d\\phi_1 = \\int (- \\phi_1 + (\\phi_1 + e^{\\phi_1})^{2}) d\\phi_1", "derivation": "\\mathbf{J}_M{(\\phi_1)} = e^{\\phi_1} and \\phi_1 + \\mathbf{J}_M{(\\phi_1)} = \\phi_1 + e^{\\phi_1} and (\\phi_1 + \\mathbf{J}_M{(\\phi_1)}) (\\phi_1 + e^{\\phi_1}) = (\\phi_1 + e^{\\phi_1})^{2} and - \\phi_1 + (\\phi_1 + \\mathbf{J}_M{(\\phi_1)}) (\\phi_1 + e^{\\phi_1}) = - \\phi_1 + (\\phi_1 + e^{\\phi_1})^{2} and \\int (- \\phi_1 + (\\phi_1 + \\mathbf{J}_M{(\\phi_1)}) (\\phi_1 + e^{\\phi_1})) d\\phi_1 = \\int (- \\phi_1 + (\\phi_1 + e^{\\phi_1})^{2}) d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["add", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True)))), Pow(Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))), Integer(2)))"], [["minus", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))), Integer(2))))"], [["integrate", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))), Integer(2))), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given E{(F_{c})} = \\sin{(F_{c})} and \\tilde{g}^*{(F_{c})} = \\sin{(E{(F_{c})})}, then obtain \\hat{\\mathbf{r}} x^\\prime + \\sin^{F_{c}}{(E{(F_{c})})} = \\hat{\\mathbf{r}} x^\\prime + \\tilde{g}^*^{F_{c}}{(F_{c})}", "derivation": "E{(F_{c})} = \\sin{(F_{c})} and \\sin{(E{(F_{c})})} = \\sin{(\\sin{(F_{c})})} and \\tilde{g}^*{(F_{c})} = \\sin{(E{(F_{c})})} and \\tilde{g}^*{(F_{c})} = \\sin{(\\sin{(F_{c})})} and \\tilde{g}^*^{F_{c}}{(F_{c})} = \\sin^{F_{c}}{(\\sin{(F_{c})})} and \\sin^{F_{c}}{(E{(F_{c})})} = \\sin^{F_{c}}{(\\sin{(F_{c})})} and \\hat{\\mathbf{r}} x^\\prime + \\sin^{F_{c}}{(E{(F_{c})})} = \\hat{\\mathbf{r}} x^\\prime + \\sin^{F_{c}}{(\\sin{(F_{c})})} and \\hat{\\mathbf{r}} x^\\prime + \\sin^{F_{c}}{(E{(F_{c})})} = \\hat{\\mathbf{r}} x^\\prime + \\tilde{g}^*^{F_{c}}{(F_{c})}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["sin", 1], "Equality(sin(Function('E')(Symbol('F_c', commutative=True))), sin(sin(Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), sin(Function('E')(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), sin(sin(Symbol('F_c', commutative=True))))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(sin(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(sin(Function('E')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(sin(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["add", 6, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(sin(Function('E')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(sin(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(sin(Function('E')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given J{(p)} = \\sin{(p)}, then obtain 4 J^{3 p}{(p)} \\sin^{p}{(p)} = 4 J^{2 p}{(p)} \\sin^{2 p}{(p)}", "derivation": "J{(p)} = \\sin{(p)} and J^{p}{(p)} = \\sin^{p}{(p)} and J^{p}{(p)} \\sin^{p}{(p)} = \\sin^{2 p}{(p)} and 2 J^{p}{(p)} \\sin^{p}{(p)} = J^{p}{(p)} \\sin^{p}{(p)} + \\sin^{2 p}{(p)} and 4 J^{2 p}{(p)} \\sin^{2 p}{(p)} = (J^{p}{(p)} \\sin^{p}{(p)} + \\sin^{2 p}{(p)})^{2} and 4 J^{3 p}{(p)} \\sin^{p}{(p)} = 4 J^{2 p}{(p)} \\sin^{2 p}{(p)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["times", 2, "Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))))"], [["add", 3, "Mul(Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Mul(Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Integer(4), Pow(Function('J')(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True)))), Pow(Add(Mul(Pow(Function('J')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(4), Pow(Function('J')(Symbol('p', commutative=True)), Mul(Integer(3), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Mul(Integer(4), Pow(Function('J')(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given U{(P_{g})} = P_{g}, then obtain 1 = e^{\\frac{d^{2}}{d P_{g}^{2}} P_{g}} e^{- \\frac{d^{2}}{d P_{g}^{2}} U{(P_{g})}}", "derivation": "U{(P_{g})} = P_{g} and \\frac{d}{d P_{g}} U{(P_{g})} = \\frac{d}{d P_{g}} P_{g} and \\frac{d^{2}}{d P_{g}^{2}} U{(P_{g})} = \\frac{d^{2}}{d P_{g}^{2}} P_{g} and e^{\\frac{d^{2}}{d P_{g}^{2}} U{(P_{g})}} = e^{\\frac{d^{2}}{d P_{g}^{2}} P_{g}} and 1 = e^{\\frac{d^{2}}{d P_{g}^{2}} P_{g}} e^{- \\frac{d^{2}}{d P_{g}^{2}} U{(P_{g})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('U')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2))), Derivative(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True), Integer(2))))"], [["exp", 3], "Equality(exp(Derivative(Function('U')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2)))), exp(Derivative(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True), Integer(2)))))"], [["divide", 4, "exp(Derivative(Function('U')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2))))"], "Equality(Integer(1), Mul(exp(Derivative(Symbol('P_g', commutative=True), Tuple(Symbol('P_g', commutative=True), Integer(2)))), exp(Mul(Integer(-1), Derivative(Function('U')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(2)))))))"]]}, {"prompt": "Given G{(\\psi,\\mathbf{v},y)} = \\mathbf{v} + \\psi + y, then obtain \\frac{G{(\\psi,\\mathbf{v},y)}}{\\int (\\mathbf{v} + \\psi + y) dy} = \\frac{\\mathbf{v} + \\psi + y}{\\int (\\mathbf{v} + \\psi + y) dy}", "derivation": "G{(\\psi,\\mathbf{v},y)} = \\mathbf{v} + \\psi + y and \\int G{(\\psi,\\mathbf{v},y)} dy = \\int (\\mathbf{v} + \\psi + y) dy and \\frac{G{(\\psi,\\mathbf{v},y)}}{\\int G{(\\psi,\\mathbf{v},y)} dy} = \\frac{\\mathbf{v} + \\psi + y}{\\int G{(\\psi,\\mathbf{v},y)} dy} and \\frac{G{(\\psi,\\mathbf{v},y)}}{\\int (\\mathbf{v} + \\psi + y) dy} = \\frac{\\mathbf{v} + \\psi + y}{\\int (\\mathbf{v} + \\psi + y) dy}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["divide", 1, "Integral(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))"], "Equality(Mul(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('G')(Symbol('\\\\psi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbb{I},\\varphi)} = \\varphi + \\sin{(\\mathbb{I})}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi} + \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{f^{\\prime}}^{\\varphi}{(\\mathbb{I},\\varphi)} = 2 \\frac{\\partial}{\\partial \\mathbb{I}} (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbb{I},\\varphi)} = \\varphi + \\sin{(\\mathbb{I})} and \\operatorname{f^{\\prime}}^{\\varphi}{(\\mathbb{I},\\varphi)} = (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi} and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{f^{\\prime}}^{\\varphi}{(\\mathbb{I},\\varphi)} = \\frac{\\partial}{\\partial \\mathbb{I}} (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi} and \\frac{\\partial}{\\partial \\mathbb{I}} (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi} + \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{f^{\\prime}}^{\\varphi}{(\\mathbb{I},\\varphi)} = 2 \\frac{\\partial}{\\partial \\mathbb{I}} (\\varphi + \\sin{(\\mathbb{I})})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given p{(i,v_{2})} = i + \\log{(v_{2})} and \\hat{x}_0{(i,v_{2})} = i \\log{(v_{2})}, then derive \\int p{(i,v_{2})} di = \\mathbf{v} + \\frac{i^{2}}{2} + i \\log{(v_{2})}, then obtain \\mathbf{v} + \\frac{i^{2}}{2} + \\hat{x}_0{(i,v_{2})} = \\int (i + \\log{(v_{2})}) di", "derivation": "p{(i,v_{2})} = i + \\log{(v_{2})} and \\int p{(i,v_{2})} di = \\int (i + \\log{(v_{2})}) di and \\int p{(i,v_{2})} di = \\mathbf{v} + \\frac{i^{2}}{2} + i \\log{(v_{2})} and \\mathbf{v} + \\frac{i^{2}}{2} + i \\log{(v_{2})} = \\int (i + \\log{(v_{2})}) di and \\hat{x}_0{(i,v_{2})} = i \\log{(v_{2})} and \\mathbf{v} + \\frac{i^{2}}{2} + \\hat{x}_0{(i,v_{2})} = \\int (i + \\log{(v_{2})}) di", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('i', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('p')(Symbol('i', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('i', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True)))), Integral(Add(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2))), Function('\\\\hat{x}_0')(Symbol('i', commutative=True), Symbol('v_2', commutative=True))), Integral(Add(Symbol('i', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\lambda)} = \\log{(\\lambda)}, then obtain \\operatorname{v_{x}}{(\\lambda)} \\log{(\\lambda)}^{2} = \\log{(\\lambda)}^{3}", "derivation": "\\operatorname{v_{x}}{(\\lambda)} = \\log{(\\lambda)} and \\operatorname{v_{x}}{(\\lambda)} \\log{(\\lambda)} = \\log{(\\lambda)}^{2} and \\operatorname{v_{x}}{(\\lambda)} \\log{(\\lambda)}^{3} = \\log{(\\lambda)}^{4} and \\operatorname{v_{x}}{(\\lambda)} \\log{(\\lambda)}^{2} = \\log{(\\lambda)}^{3}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('v_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2)))"], [["times", 2, "Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))"], "Equality(Mul(Function('v_x')(Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(3))), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(4)))"], [["divide", 3, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('v_x')(Symbol('\\\\lambda', commutative=True)), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(3)))"]]}, {"prompt": "Given I{(h,\\mathbf{r})} = \\frac{\\mathbf{r}}{h} and \\mathbf{J}_P{(f)} = \\int e^{f} df, then obtain \\log{(- I{(h,\\mathbf{r})} + \\mathbf{J}_P{(f)})} = \\log{(- \\frac{\\mathbf{r}}{h} + \\mathbf{J}_P{(f)})}", "derivation": "I{(h,\\mathbf{r})} = \\frac{\\mathbf{r}}{h} and - I{(h,\\mathbf{r})} = - \\frac{\\mathbf{r}}{h} and \\mathbf{J}_P{(f)} = \\int e^{f} df and - I{(h,\\mathbf{r})} + \\mathbf{J}_P{(f)} = - \\frac{\\mathbf{r}}{h} + \\mathbf{J}_P{(f)} and - I{(h,\\mathbf{r})} + \\int e^{f} df = - \\frac{\\mathbf{r}}{h} + \\int e^{f} df and \\log{(- I{(h,\\mathbf{r})} + \\int e^{f} df)} = \\log{(- \\frac{\\mathbf{r}}{h} + \\int e^{f} df)} and \\log{(- I{(h,\\mathbf{r})} + \\mathbf{J}_P{(f)})} = \\log{(- \\frac{\\mathbf{r}}{h} + \\mathbf{J}_P{(f)})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(log(Add(Mul(Integer(-1), Function('I')(Symbol('h', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1))), Function('\\\\mathbf{J}_P')(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(F_{g})} = e^{F_{g}}, then obtain \\frac{\\partial^{2}}{\\partial \\hat{H}_l^{2}} (\\log{(\\hat{H}_l)} + \\int \\operatorname{A_{1}}{(F_{g})} dF_{g}) = \\frac{\\partial^{2}}{\\partial \\hat{H}_l^{2}} (\\log{(\\hat{H}_l)} + \\int e^{F_{g}} dF_{g})", "derivation": "\\operatorname{A_{1}}{(F_{g})} = e^{F_{g}} and \\int \\operatorname{A_{1}}{(F_{g})} dF_{g} = \\int e^{F_{g}} dF_{g} and \\log{(\\hat{H}_l)} + \\int \\operatorname{A_{1}}{(F_{g})} dF_{g} = \\log{(\\hat{H}_l)} + \\int e^{F_{g}} dF_{g} and \\frac{\\partial}{\\partial \\hat{H}_l} (\\log{(\\hat{H}_l)} + \\int \\operatorname{A_{1}}{(F_{g})} dF_{g}) = \\frac{\\partial}{\\partial \\hat{H}_l} (\\log{(\\hat{H}_l)} + \\int e^{F_{g}} dF_{g}) and \\frac{\\partial^{2}}{\\partial \\hat{H}_l^{2}} (\\log{(\\hat{H}_l)} + \\int \\operatorname{A_{1}}{(F_{g})} dF_{g}) = \\frac{\\partial^{2}}{\\partial \\hat{H}_l^{2}} (\\log{(\\hat{H}_l)} + \\int e^{F_{g}} dF_{g})", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["add", 2, "log(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(Function('A_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(Function('A_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(Function('A_1')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))), Derivative(Add(log(Symbol('\\\\hat{H}_l', commutative=True)), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{B},\\mathbf{p})} = \\frac{\\mathbf{B}}{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{D}{(\\mathbf{B},\\mathbf{p})} = \\frac{1}{\\mathbf{p}}, then obtain \\frac{\\mathbf{B}}{\\mathbf{p}} + \\omega = \\int \\frac{1}{\\mathbf{p}} d\\mathbf{B}", "derivation": "\\mathbf{D}{(\\mathbf{B},\\mathbf{p})} = \\frac{\\mathbf{B}}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{D}{(\\mathbf{B},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{D}{(\\mathbf{B},\\mathbf{p})} = \\frac{1}{\\mathbf{p}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\mathbf{p}} = \\frac{1}{\\mathbf{p}} and \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\mathbf{p}} d\\mathbf{B} = \\int \\frac{1}{\\mathbf{p}} d\\mathbf{B} and \\frac{\\mathbf{B}}{\\mathbf{p}} + \\omega = \\int \\frac{1}{\\mathbf{p}} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1))), Symbol('\\\\omega', commutative=True)), Integral(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given m{(\\phi,y)} = \\phi y, then obtain - \\phi^{4} y^{3} - \\frac{\\frac{\\partial}{\\partial y} \\phi m^{2}{(\\phi,y)}}{\\phi^{2}} = - \\phi^{4} y^{3} - \\frac{\\frac{\\partial}{\\partial y} \\phi^{3} y^{2}}{\\phi^{2}}", "derivation": "m{(\\phi,y)} = \\phi y and \\phi m{(\\phi,y)} = \\phi^{2} y and \\phi^{2} y m{(\\phi,y)} = \\phi^{3} y^{2} and \\phi m^{2}{(\\phi,y)} = \\phi^{2} y m{(\\phi,y)} and \\frac{\\partial}{\\partial y} \\phi m^{2}{(\\phi,y)} = \\frac{\\partial}{\\partial y} \\phi^{2} y m{(\\phi,y)} and \\frac{\\partial}{\\partial y} \\phi m^{2}{(\\phi,y)} = \\frac{\\partial}{\\partial y} \\phi^{3} y^{2} and - \\frac{\\frac{\\partial}{\\partial y} \\phi m^{2}{(\\phi,y)}}{\\phi^{2}} = - \\frac{\\frac{\\partial}{\\partial y} \\phi^{3} y^{2}}{\\phi^{2}} and - \\phi^{4} y^{3} - \\frac{\\frac{\\partial}{\\partial y} \\phi m^{2}{(\\phi,y)}}{\\phi^{2}} = - \\phi^{4} y^{3} - \\frac{\\frac{\\partial}{\\partial y} \\phi^{3} y^{2}}{\\phi^{2}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Symbol('y', commutative=True)))"], [["times", 1, "Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Symbol('y', commutative=True), Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Pow(Symbol('y', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Symbol('y', commutative=True), Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True))))"], [["differentiate", 4, "Symbol('y', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Symbol('y', commutative=True), Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Pow(Symbol('y', commutative=True), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 6, "Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(2)))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Pow(Symbol('y', commutative=True), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["minus", 7, "Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('y', commutative=True), Integer(3)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('y', commutative=True), Integer(3))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Function('m')(Symbol('\\\\phi', commutative=True), Symbol('y', commutative=True)), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(4)), Pow(Symbol('y', commutative=True), Integer(3))), Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Integer(-2)), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(3)), Pow(Symbol('y', commutative=True), Integer(2))), Tuple(Symbol('y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\ddot{x}{(n_{2},\\mathbf{p})} = \\sin{(\\mathbf{p} n_{2})}, then obtain ((\\ddot{x}^{2}{(n_{2},\\mathbf{p})} \\sin^{2}{(\\mathbf{p} n_{2})})^{n_{2}})^{n_{2}} = ((\\sin^{4}{(\\mathbf{p} n_{2})})^{n_{2}})^{n_{2}}", "derivation": "\\ddot{x}{(n_{2},\\mathbf{p})} = \\sin{(\\mathbf{p} n_{2})} and \\ddot{x}{(n_{2},\\mathbf{p})} \\sin{(\\mathbf{p} n_{2})} = \\sin^{2}{(\\mathbf{p} n_{2})} and \\ddot{x}^{2}{(n_{2},\\mathbf{p})} \\sin^{2}{(\\mathbf{p} n_{2})} = \\sin^{4}{(\\mathbf{p} n_{2})} and (\\ddot{x}^{2}{(n_{2},\\mathbf{p})} \\sin^{2}{(\\mathbf{p} n_{2})})^{n_{2}} = (\\sin^{4}{(\\mathbf{p} n_{2})})^{n_{2}} and ((\\ddot{x}^{2}{(n_{2},\\mathbf{p})} \\sin^{2}{(\\mathbf{p} n_{2})})^{n_{2}})^{n_{2}} = ((\\sin^{4}{(\\mathbf{p} n_{2})})^{n_{2}})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))))"], [["times", 1, "sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True)))), Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\ddot{x}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(2))), Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(4)))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\ddot{x}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(2))), Symbol('n_2', commutative=True)), Pow(Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(4)), Symbol('n_2', commutative=True)))"], [["power", 4, "Symbol('n_2', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Function('\\\\ddot{x}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(2))), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Pow(Pow(sin(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('n_2', commutative=True))), Integer(4)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(I,\\mu,\\eta)} = I + \\eta^{\\mu}, then obtain - \\eta^{\\mu} - \\int 0 d\\eta = - \\eta^{\\mu} + (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} - \\int 0 d\\eta - 1", "derivation": "\\mathbf{D}{(I,\\mu,\\eta)} = I + \\eta^{\\mu} and - I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)} = 0 and (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} = 0^{I} and 0 = 0^{I} - (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} and 0 = (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} - 1 and - \\eta^{\\mu} = - \\eta^{\\mu} + (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} - 1 and - \\eta^{\\mu} - \\int 0 d\\eta = - \\eta^{\\mu} + (- I - \\eta^{\\mu} + \\mathbf{D}{(I,\\mu,\\eta)})^{I} - \\int 0 d\\eta - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('I', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Add(Symbol('I', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Integer(0))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True)), Pow(Integer(0), Symbol('I', commutative=True)))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Pow(Integer(0), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True)), Integer(-1)))"], [["minus", 5, "Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True)), Integer(-1)))"], [["minus", 6, "Integral(Integer(0), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('\\\\eta', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\mathbf{D}')(Symbol('I', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('I', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('\\\\eta', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given c{(h,W,v_{1})} = \\frac{W + v_{1}}{h}, then obtain (c{(h,W,v_{1})} + \\frac{W + v_{1}}{h})^{h} + \\frac{2 (W + v_{1})}{h} = (\\frac{2 (W + v_{1})}{h})^{h} + \\frac{2 (W + v_{1})}{h}", "derivation": "c{(h,W,v_{1})} = \\frac{W + v_{1}}{h} and c{(h,W,v_{1})} + \\frac{W + v_{1}}{h} = \\frac{2 (W + v_{1})}{h} and (c{(h,W,v_{1})} + \\frac{W + v_{1}}{h})^{h} = (\\frac{2 (W + v_{1})}{h})^{h} and (c{(h,W,v_{1})} + \\frac{W + v_{1}}{h})^{h} + \\frac{2 (W + v_{1})}{h} = (\\frac{2 (W + v_{1})}{h})^{h} + \\frac{2 (W + v_{1})}{h}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('h', commutative=True), Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Add(Function('c')(Symbol('h', commutative=True), Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))), Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Function('c')(Symbol('h', commutative=True), Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))), Symbol('h', commutative=True)), Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True))), Symbol('h', commutative=True)))"], [["add", 3, "Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))"], "Equality(Add(Pow(Add(Function('c')(Symbol('h', commutative=True), Symbol('W', commutative=True), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))), Symbol('h', commutative=True)), Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))), Add(Pow(Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True))), Symbol('h', commutative=True)), Mul(Integer(2), Pow(Symbol('h', commutative=True), Integer(-1)), Add(Symbol('W', commutative=True), Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{N})} = e^{\\cos{(F_{N})}}, then obtain \\operatorname{x^{{\\}'}}^{3}{(F_{N})} e^{- \\cos{(F_{N})}} = e^{2 \\cos{(F_{N})}}", "derivation": "\\operatorname{x^{{\\}'}}{(F_{N})} = e^{\\cos{(F_{N})}} and 1 = \\frac{e^{\\cos{(F_{N})}}}{\\operatorname{x^{{\\}'}}{(F_{N})}} and \\operatorname{x^{{\\}'}}^{2}{(F_{N})} = \\operatorname{x^{{\\}'}}{(F_{N})} e^{\\cos{(F_{N})}} and \\operatorname{x^{{\\}'}}{(F_{N})} e^{\\cos{(F_{N})}} = e^{2 \\cos{(F_{N})}} and \\operatorname{x^{{\\}'}}^{2}{(F_{N})} = e^{2 \\cos{(F_{N})}} and \\operatorname{x^{{\\}'}}^{3}{(F_{N})} e^{- \\cos{(F_{N})}} = \\operatorname{x^{{\\}'}}{(F_{N})} e^{\\cos{(F_{N})}} and \\operatorname{x^{{\\}'}}^{3}{(F_{N})} e^{- \\cos{(F_{N})}} = e^{2 \\cos{(F_{N})}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True))))"], [["divide", 1, "Function('x^\\\\prime')(Symbol('F_N', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(-1)), exp(cos(Symbol('F_N', commutative=True)))))"], [["divide", 1, "Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(-1))"], "Equality(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(2)), Mul(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True)))))"], [["times", 2, "Mul(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True))))"], "Equality(Mul(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(2)), exp(Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"], [["divide", 5, "Mul(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(-1)), exp(cos(Symbol('F_N', commutative=True))))"], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(3)), exp(Mul(Integer(-1), cos(Symbol('F_N', commutative=True))))), Mul(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), exp(cos(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Function('x^\\\\prime')(Symbol('F_N', commutative=True)), Integer(3)), exp(Mul(Integer(-1), cos(Symbol('F_N', commutative=True))))), exp(Mul(Integer(2), cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(a)} = e^{a} and y{(a)} = \\frac{d}{d a} \\mathbf{F}{(a)}, then obtain - e^{a} + \\frac{d}{d a} \\mathbf{F}{(a)} = y{(a)} - e^{a}", "derivation": "\\mathbf{F}{(a)} = e^{a} and \\frac{d}{d a} \\mathbf{F}{(a)} = \\frac{d}{d a} e^{a} and - e^{a} + \\frac{d}{d a} \\mathbf{F}{(a)} = - e^{a} + \\frac{d}{d a} e^{a} and y{(a)} = \\frac{d}{d a} \\mathbf{F}{(a)} and y{(a)} = \\frac{d}{d a} e^{a} and - e^{a} + \\frac{d}{d a} \\mathbf{F}{(a)} = y{(a)} - e^{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('a', commutative=True)), exp(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), Derivative(Function('\\\\mathbf{F}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('y')(Symbol('a', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('y')(Symbol('a', commutative=True)), Derivative(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Add(Mul(Integer(-1), exp(Symbol('a', commutative=True))), Derivative(Function('\\\\mathbf{F}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Function('y')(Symbol('a', commutative=True)), Mul(Integer(-1), exp(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(B)} = \\cos{(B)} and \\operatorname{E_{\\lambda}}{(B)} = \\mathbf{J}_f^{4}{(B)}, then obtain 4 \\cos^{4}{(B)} - \\cos{(B)} = (\\mathbf{J}_f^{2}{(B)} + \\mathbf{J}_f{(B)} \\cos{(B)})^{2} - \\cos{(B)}", "derivation": "\\mathbf{J}_f{(B)} = \\cos{(B)} and \\mathbf{J}_f^{2}{(B)} = \\mathbf{J}_f{(B)} \\cos{(B)} and 2 \\mathbf{J}_f^{2}{(B)} = \\mathbf{J}_f^{2}{(B)} + \\mathbf{J}_f{(B)} \\cos{(B)} and 4 \\mathbf{J}_f^{4}{(B)} = (\\mathbf{J}_f^{2}{(B)} + \\mathbf{J}_f{(B)} \\cos{(B)})^{2} and 4 \\mathbf{J}_f^{4}{(B)} - \\cos{(B)} = (\\mathbf{J}_f^{2}{(B)} + \\mathbf{J}_f{(B)} \\cos{(B)})^{2} - \\cos{(B)} and \\operatorname{E_{\\lambda}}{(B)} = \\mathbf{J}_f^{4}{(B)} and \\operatorname{E_{\\lambda}}{(B)} = \\cos^{4}{(B)} and \\mathbf{J}_f^{4}{(B)} = \\cos^{4}{(B)} and 4 \\cos^{4}{(B)} - \\cos{(B)} = (\\mathbf{J}_f^{2}{(B)} + \\mathbf{J}_f{(B)} \\cos{(B)})^{2} - \\cos{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))))"], [["add", 2, "Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2))), Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(4))), Pow(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))), Integer(2)))"], [["minus", 4, "cos(Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(4), Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(4))), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Add(Pow(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))), Integer(2)), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(4)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Function('E_{\\\\lambda}')(Symbol('B', commutative=True)), Pow(cos(Symbol('B', commutative=True)), Integer(4)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(4)), Pow(cos(Symbol('B', commutative=True)), Integer(4)))"], [["substitute_LHS_for_RHS", 5, 8], "Equality(Add(Mul(Integer(4), Pow(cos(Symbol('B', commutative=True)), Integer(4))), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Add(Pow(Add(Pow(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))), Integer(2)), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given r{(\\hat{p},\\hat{x})} = \\hat{p} - \\hat{x}, then obtain - \\hat{p} + 2 \\hat{x} + \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (- \\hat{x} + r{(\\hat{p},\\hat{x})}) = - \\hat{p} + 2 \\hat{x} + \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (\\hat{p} - 2 \\hat{x})", "derivation": "r{(\\hat{p},\\hat{x})} = \\hat{p} - \\hat{x} and - \\hat{x} + r{(\\hat{p},\\hat{x})} = \\hat{p} - 2 \\hat{x} and \\frac{\\partial}{\\partial \\hat{p}} (- \\hat{x} + r{(\\hat{p},\\hat{x})}) = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} - 2 \\hat{x}) and \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (- \\hat{x} + r{(\\hat{p},\\hat{x})}) = \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (\\hat{p} - 2 \\hat{x}) and - \\hat{p} + 2 \\hat{x} + \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (- \\hat{x} + r{(\\hat{p},\\hat{x})}) = - \\hat{p} + 2 \\hat{x} + \\frac{\\partial^{2}}{\\partial \\hat{p}^{2}} (\\hat{p} - 2 \\hat{x})", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2))))"], [["minus", 4, "Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Function('r')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{x}', commutative=True)), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given Q{(\\mathbf{P},L_{\\varepsilon})} = - L_{\\varepsilon} + \\log{(\\mathbf{P})}, then obtain \\int (\\mathbf{P} + Q{(\\mathbf{P},L_{\\varepsilon})}) dL_{\\varepsilon} = F_{H} - \\frac{L_{\\varepsilon}^{2}}{2} + L_{\\varepsilon} (\\mathbf{P} + \\log{(\\mathbf{P})})", "derivation": "Q{(\\mathbf{P},L_{\\varepsilon})} = - L_{\\varepsilon} + \\log{(\\mathbf{P})} and \\mathbf{P} + Q{(\\mathbf{P},L_{\\varepsilon})} = - L_{\\varepsilon} + \\mathbf{P} + \\log{(\\mathbf{P})} and \\int (\\mathbf{P} + Q{(\\mathbf{P},L_{\\varepsilon})}) dL_{\\varepsilon} = \\int (- L_{\\varepsilon} + \\mathbf{P} + \\log{(\\mathbf{P})}) dL_{\\varepsilon} and \\int (\\mathbf{P} + Q{(\\mathbf{P},L_{\\varepsilon})}) dL_{\\varepsilon} = F_{H} - \\frac{L_{\\varepsilon}^{2}}{2} + L_{\\varepsilon} (\\mathbf{P} + \\log{(\\mathbf{P})})", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('Q')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('Q')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\mathbf{P}', commutative=True), Function('Q')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{s}{(T,c_{0},q)} = - T + c_{0} q, then obtain (\\int (c_{0} + \\mathbf{s}{(T,c_{0},q)}) dq)^{c_{0}} = (\\int (- T + c_{0} q + c_{0}) dq)^{c_{0}}", "derivation": "\\mathbf{s}{(T,c_{0},q)} = - T + c_{0} q and c_{0} + \\mathbf{s}{(T,c_{0},q)} = - T + c_{0} q + c_{0} and \\int (c_{0} + \\mathbf{s}{(T,c_{0},q)}) dq = \\int (- T + c_{0} q + c_{0}) dq and (\\int (c_{0} + \\mathbf{s}{(T,c_{0},q)}) dq)^{c_{0}} = (\\int (- T + c_{0} q + c_{0}) dq)^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('q', commutative=True))))"], [["add", 1, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('c_0', commutative=True)))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('c_0', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(i,z)} = i + z and \\hat{p}_0{(i,z)} = \\frac{1}{\\int (i + z - \\mu_{0}{(i,z)}) di}, then obtain \\frac{(1 + \\frac{1}{\\int (i + z - \\mu_{0}{(i,z)}) di})^{i}}{1 + \\frac{1}{\\int 0 di}} = \\frac{(1 + \\frac{1}{\\int 0 di})^{i}}{1 + \\frac{1}{\\int 0 di}}", "derivation": "\\mu_{0}{(i,z)} = i + z and 0 = i + z - \\mu_{0}{(i,z)} and \\int 0 di = \\int (i + z - \\mu_{0}{(i,z)}) di and \\hat{p}_0{(i,z)} = \\frac{1}{\\int (i + z - \\mu_{0}{(i,z)}) di} and \\hat{p}_0{(i,z)} + 1 = 1 + \\frac{1}{\\int (i + z - \\mu_{0}{(i,z)}) di} and \\hat{p}_0{(i,z)} + 1 = 1 + \\frac{1}{\\int 0 di} and (\\hat{p}_0{(i,z)} + 1)^{i} = (1 + \\frac{1}{\\int 0 di})^{i} and \\frac{(\\hat{p}_0{(i,z)} + 1)^{i}}{1 + \\frac{1}{\\int 0 di}} = \\frac{(1 + \\frac{1}{\\int 0 di})^{i}}{1 + \\frac{1}{\\int 0 di}} and \\frac{(1 + \\frac{1}{\\int (i + z - \\mu_{0}{(i,z)}) di})^{i}}{1 + \\frac{1}{\\int 0 di}} = \\frac{(1 + \\frac{1}{\\int 0 di})^{i}}{1 + \\frac{1}{\\int 0 di}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))"], [["minus", 1, "Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True))"], "Equality(Integer(0), Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Pow(Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integer(-1)))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(1)), Add(Integer(1), Pow(Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(1)), Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))))"], [["power", 6, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(1)), Symbol('i', commutative=True)), Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Symbol('i', commutative=True)))"], [["divide", 7, "Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1)), Pow(Add(Function('\\\\hat{p}_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)), Integer(1)), Symbol('i', commutative=True))), Mul(Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1)), Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Mul(Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1)), Pow(Add(Integer(1), Pow(Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\mu_0')(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integer(-1))), Symbol('i', commutative=True))), Mul(Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1)), Pow(Add(Integer(1), Pow(Integral(Integer(0), Tuple(Symbol('i', commutative=True))), Integer(-1))), Symbol('i', commutative=True))))"]]}, {"prompt": "Given c{(a^{\\dagger},f_{E})} = a^{\\dagger} f_{E}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} c{(a^{\\dagger},f_{E})} = f_{E}, then obtain \\mathbf{g} + f_{E} \\int c{(a^{\\dagger},f_{E})} da^{\\dagger} = \\int f_{E} c{(a^{\\dagger},f_{E})} da^{\\dagger}", "derivation": "c{(a^{\\dagger},f_{E})} = a^{\\dagger} f_{E} and \\frac{\\partial}{\\partial a^{\\dagger}} c{(a^{\\dagger},f_{E})} = \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} f_{E} and \\frac{\\partial}{\\partial a^{\\dagger}} c{(a^{\\dagger},f_{E})} = f_{E} and \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} f_{E} = f_{E} and c{(a^{\\dagger},f_{E})} \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} f_{E} = f_{E} c{(a^{\\dagger},f_{E})} and \\int c{(a^{\\dagger},f_{E})} \\frac{\\partial}{\\partial a^{\\dagger}} a^{\\dagger} f_{E} da^{\\dagger} = \\int f_{E} c{(a^{\\dagger},f_{E})} da^{\\dagger} and \\mathbf{g} + f_{E} \\int c{(a^{\\dagger},f_{E})} da^{\\dagger} = \\int f_{E} c{(a^{\\dagger},f_{E})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('f_E', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Symbol('f_E', commutative=True))"], [["times", 4, "Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Mul(Symbol('f_E', commutative=True), Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))))"], [["integrate", 5, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Mul(Symbol('f_E', commutative=True), Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Mul(Symbol('f_E', commutative=True), Integral(Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Integral(Mul(Symbol('f_E', commutative=True), Function('c')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\rho{(M,J_{\\varepsilon})} = \\frac{\\partial}{\\partial M} M^{J_{\\varepsilon}}, then obtain \\cos{(\\rho{(M,J_{\\varepsilon})} - \\frac{\\partial}{\\partial J_{\\varepsilon}} \\rho{(M,J_{\\varepsilon})})} = \\cos{(\\rho{(M,J_{\\varepsilon})} - \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial M} M^{J_{\\varepsilon}})}", "derivation": "\\rho{(M,J_{\\varepsilon})} = \\frac{\\partial}{\\partial M} M^{J_{\\varepsilon}} and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\rho{(M,J_{\\varepsilon})} = \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial M} M^{J_{\\varepsilon}} and - \\rho{(M,J_{\\varepsilon})} + \\frac{\\partial}{\\partial J_{\\varepsilon}} \\rho{(M,J_{\\varepsilon})} = - \\rho{(M,J_{\\varepsilon})} + \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial M} M^{J_{\\varepsilon}} and \\cos{(\\rho{(M,J_{\\varepsilon})} - \\frac{\\partial}{\\partial J_{\\varepsilon}} \\rho{(M,J_{\\varepsilon})})} = \\cos{(\\rho{(M,J_{\\varepsilon})} - \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial M} M^{J_{\\varepsilon}})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Pow(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Pow(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Derivative(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Derivative(Pow(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["cos", 3], "Equality(cos(Add(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))), cos(Add(Function('\\\\rho')(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(Pow(Symbol('M', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\theta_2)} = \\sin{(\\theta_2)}, then derive 1 = \\frac{v_{1} - \\cos{(\\theta_2)}}{\\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2}, then obtain 1 = \\frac{(v_{1} - \\cos{(\\theta_2)}) \\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2}{(\\int \\sin{(\\theta_2)} d\\theta_2)^{2}}", "derivation": "\\operatorname{F_{c}}{(\\theta_2)} = \\sin{(\\theta_2)} and \\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2 = \\int \\sin{(\\theta_2)} d\\theta_2 and 1 = \\frac{\\int \\sin{(\\theta_2)} d\\theta_2}{\\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2} and 1 = \\frac{v_{1} - \\cos{(\\theta_2)}}{\\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2} and \\int \\sin{(\\theta_2)} d\\theta_2 = \\frac{(\\int \\sin{(\\theta_2)} d\\theta_2)^{2}}{\\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2} and \\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2 = \\frac{(\\int \\sin{(\\theta_2)} d\\theta_2)^{2}}{\\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2} and 1 = \\frac{(v_{1} - \\cos{(\\theta_2)}) \\int \\operatorname{F_{c}}{(\\theta_2)} d\\theta_2}{(\\int \\sin{(\\theta_2)} d\\theta_2)^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 2, "Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Pow(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1))))"], [["times", 2, "Mul(Pow(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], "Equality(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Pow(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-1)), Pow(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Integer(1), Mul(Add(Symbol('v_1', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Integral(Function('F_c')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Pow(Integral(sin(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\mathbf{s})} = \\log{(\\mathbf{s})}, then derive \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\eta^{\\prime} + \\mathbf{s} \\log{(\\mathbf{s})} - \\mathbf{s}, then obtain \\eta^{\\prime} + \\mathbf{s} \\log{(\\mathbf{s})} - \\mathbf{s} = \\eta^{\\prime} + \\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})} - \\mathbf{s}", "derivation": "\\operatorname{v_{2}}{(\\mathbf{s})} = \\log{(\\mathbf{s})} and \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\int \\log{(\\mathbf{s})} d\\mathbf{s} and \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\eta^{\\prime} + \\mathbf{s} \\log{(\\mathbf{s})} - \\mathbf{s} and \\int \\operatorname{v_{2}}{(\\mathbf{s})} d\\mathbf{s} = \\eta^{\\prime} + \\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})} - \\mathbf{s} and \\eta^{\\prime} + \\mathbf{s} \\log{(\\mathbf{s})} - \\mathbf{s} = \\eta^{\\prime} + \\mathbf{s} \\operatorname{v_{2}}{(\\mathbf{s})} - \\mathbf{s}", "srepr_derivation": [["renaming_premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), log(Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('v_2')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given z{(S,E)} = E + S, then derive \\frac{\\partial}{\\partial S} z{(S,E)} + 1 = 2, then obtain (\\frac{\\partial}{\\partial S} (E + S) + 1) \\frac{\\partial}{\\partial S} z{(S,E)} = 2 \\frac{\\partial}{\\partial S} z{(S,E)}", "derivation": "z{(S,E)} = E + S and - E + z{(S,E)} = S and \\frac{\\partial}{\\partial S} (- E + z{(S,E)}) = \\frac{d}{d S} S and \\frac{\\partial}{\\partial S} (- E + z{(S,E)}) + 1 = \\frac{d}{d S} S + 1 and \\frac{\\partial}{\\partial S} z{(S,E)} + 1 = 2 and \\frac{\\partial}{\\partial S} (E + S) + 1 = 2 and \\frac{\\partial}{\\partial S} (E + S) + 1 = \\frac{\\partial}{\\partial S} z{(S,E)} + 1 and (\\frac{\\partial}{\\partial S} (E + S) + 1) \\frac{\\partial}{\\partial S} z{(S,E)} = (\\frac{\\partial}{\\partial S} z{(S,E)} + 1) \\frac{\\partial}{\\partial S} z{(S,E)} and (\\frac{\\partial}{\\partial S} (E + S) + 1) \\frac{\\partial}{\\partial S} z{(S,E)} = 2 \\frac{\\partial}{\\partial S} z{(S,E)}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Add(Symbol('E', commutative=True), Symbol('S', commutative=True)))"], [["minus", 1, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True))), Symbol('S', commutative=True))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Symbol('S', commutative=True), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Symbol('S', commutative=True), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(Add(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Derivative(Add(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)))"], [["times", 7, "Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Add(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Add(Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Mul(Add(Derivative(Add(Symbol('E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Function('z')(Symbol('S', commutative=True), Symbol('E', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(b,\\sigma_p)} = \\sigma_p - b, then obtain \\frac{\\partial}{\\partial \\sigma_p} \\iint b \\lambda{(b,\\sigma_p)} d\\sigma_p d\\sigma_p = \\frac{\\partial}{\\partial \\sigma_p} \\iint b (\\sigma_p - b) d\\sigma_p d\\sigma_p", "derivation": "\\lambda{(b,\\sigma_p)} = \\sigma_p - b and b \\lambda{(b,\\sigma_p)} = b (\\sigma_p - b) and \\int b \\lambda{(b,\\sigma_p)} d\\sigma_p = \\int b (\\sigma_p - b) d\\sigma_p and \\iint b \\lambda{(b,\\sigma_p)} d\\sigma_p d\\sigma_p = \\iint b (\\sigma_p - b) d\\sigma_p d\\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} \\iint b \\lambda{(b,\\sigma_p)} d\\sigma_p d\\sigma_p = \\frac{\\partial}{\\partial \\sigma_p} \\iint b (\\sigma_p - b) d\\sigma_p d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\lambda')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Mul(Symbol('b', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Symbol('b', commutative=True), Function('\\\\lambda')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Symbol('b', commutative=True), Function('\\\\lambda')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('b', commutative=True), Function('\\\\lambda')(Symbol('b', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('b', commutative=True), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(l,F_{N})} = \\cos{(F_{N} + l)}, then obtain (\\int (\\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} - \\cos^{l}{(F_{N} + l)})^{F_{N}} dF_{N})^{l} = (\\int 0^{F_{N}} dF_{N})^{l}", "derivation": "\\operatorname{E_{\\lambda}}{(l,F_{N})} = \\cos{(F_{N} + l)} and \\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} = \\cos^{l}{(F_{N} + l)} and \\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} - \\cos^{l}{(F_{N} + l)} = 0 and (\\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} - \\cos^{l}{(F_{N} + l)})^{F_{N}} = 0^{F_{N}} and \\int (\\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} - \\cos^{l}{(F_{N} + l)})^{F_{N}} dF_{N} = \\int 0^{F_{N}} dF_{N} and (\\int (\\operatorname{E_{\\lambda}}^{l}{(l,F_{N})} - \\cos^{l}{(F_{N} + l)})^{F_{N}} dF_{N})^{l} = (\\int 0^{F_{N}} dF_{N})^{l}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('l', commutative=True)), Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)))"], [["minus", 2, "Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True))"], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Add(Pow(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Symbol('F_N', commutative=True)), Pow(Integer(0), Symbol('F_N', commutative=True)))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Pow(Add(Pow(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(Integer(0), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["power", 5, "Symbol('l', commutative=True)"], "Equality(Pow(Integral(Pow(Add(Pow(Function('E_{\\\\lambda}')(Symbol('l', commutative=True), Symbol('F_N', commutative=True)), Symbol('l', commutative=True)), Mul(Integer(-1), Pow(cos(Add(Symbol('F_N', commutative=True), Symbol('l', commutative=True))), Symbol('l', commutative=True)))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('l', commutative=True)), Pow(Integral(Pow(Integer(0), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\mathbf{M})} = e^{\\sin{(\\mathbf{M})}}, then obtain - \\operatorname{v_{t}}{(\\mathbf{M})} \\sin{(\\mathbf{M})} + \\operatorname{v_{t}}{(\\mathbf{M})} = - \\operatorname{v_{t}}{(\\mathbf{M})} \\sin{(\\mathbf{M})} + e^{\\sin{(\\mathbf{M})}}", "derivation": "\\operatorname{v_{t}}{(\\mathbf{M})} = e^{\\sin{(\\mathbf{M})}} and \\operatorname{v_{t}}{(\\mathbf{M})} \\sin{(\\mathbf{M})} = e^{\\sin{(\\mathbf{M})}} \\sin{(\\mathbf{M})} and \\operatorname{v_{t}}{(\\mathbf{M})} - e^{\\sin{(\\mathbf{M})}} \\sin{(\\mathbf{M})} = - e^{\\sin{(\\mathbf{M})}} \\sin{(\\mathbf{M})} + e^{\\sin{(\\mathbf{M})}} and - \\operatorname{v_{t}}{(\\mathbf{M})} \\sin{(\\mathbf{M})} + \\operatorname{v_{t}}{(\\mathbf{M})} = - \\operatorname{v_{t}}{(\\mathbf{M})} \\sin{(\\mathbf{M})} + e^{\\sin{(\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True)), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Mul(exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 1, "Mul(exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), sin(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), exp(sin(Symbol('\\\\mathbf{M}', commutative=True))), sin(Symbol('\\\\mathbf{M}', commutative=True))), exp(sin(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(-1), Function('v_t')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))), exp(sin(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})} = \\cos{(\\mathbf{D}^{\\mathbf{S}})} and \\operatorname{x^{{\\}'}}{(\\mathbf{D},\\mathbf{S})} = \\int (\\mathbf{D}^{\\mathbf{S}} + \\cos{(\\mathbf{D}^{\\mathbf{S}})}) d\\mathbf{S}, then obtain \\int (\\mathbf{D}^{\\mathbf{S}} + \\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})}) d\\mathbf{S} = \\operatorname{x^{{\\}'}}{(\\mathbf{D},\\mathbf{S})}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})} = \\cos{(\\mathbf{D}^{\\mathbf{S}})} and \\mathbf{D}^{\\mathbf{S}} + \\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})} = \\mathbf{D}^{\\mathbf{S}} + \\cos{(\\mathbf{D}^{\\mathbf{S}})} and \\int (\\mathbf{D}^{\\mathbf{S}} + \\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})}) d\\mathbf{S} = \\int (\\mathbf{D}^{\\mathbf{S}} + \\cos{(\\mathbf{D}^{\\mathbf{S}})}) d\\mathbf{S} and \\operatorname{x^{{\\}'}}{(\\mathbf{D},\\mathbf{S})} = \\int (\\mathbf{D}^{\\mathbf{S}} + \\cos{(\\mathbf{D}^{\\mathbf{S}})}) d\\mathbf{S} and \\int (\\mathbf{D}^{\\mathbf{S}} + \\operatorname{E_{\\lambda}}{(\\mathbf{D},\\mathbf{S})}) d\\mathbf{S} = \\operatorname{x^{{\\}'}}{(\\mathbf{D},\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 1, "Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Function('x^\\\\prime')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given Q{(M_{E})} = e^{M_{E}}, then obtain \\cos{(\\frac{d}{d M_{E}} 0)} - \\frac{Q{(M_{E})}}{M_{E}} = \\cos{(\\frac{d}{d M_{E}} (- \\frac{Q{(M_{E})}}{M_{E}} + \\frac{e^{M_{E}}}{M_{E}}))} - \\frac{Q{(M_{E})}}{M_{E}}", "derivation": "Q{(M_{E})} = e^{M_{E}} and \\frac{Q{(M_{E})}}{M_{E}} = \\frac{e^{M_{E}}}{M_{E}} and 0 = - \\frac{Q{(M_{E})}}{M_{E}} + \\frac{e^{M_{E}}}{M_{E}} and \\frac{d}{d M_{E}} 0 = \\frac{d}{d M_{E}} (- \\frac{Q{(M_{E})}}{M_{E}} + \\frac{e^{M_{E}}}{M_{E}}) and \\cos{(\\frac{d}{d M_{E}} 0)} = \\cos{(\\frac{d}{d M_{E}} (- \\frac{Q{(M_{E})}}{M_{E}} + \\frac{e^{M_{E}}}{M_{E}}))} and \\cos{(\\frac{d}{d M_{E}} 0)} - \\frac{Q{(M_{E})}}{M_{E}} = \\cos{(\\frac{d}{d M_{E}} (- \\frac{Q{(M_{E})}}{M_{E}} + \\frac{e^{M_{E}}}{M_{E}}))} - \\frac{Q{(M_{E})}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["divide", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('M_E', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('M_E', commutative=True)))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Integer(0), Tuple(Symbol('M_E', commutative=True), Integer(1)))), cos(Derivative(Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["add", 5, "Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True)))"], "Equality(Add(cos(Derivative(Integer(0), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True)))), Add(cos(Derivative(Add(Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('Q')(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(Q)} = \\sin{(Q)}, then obtain - Q (- 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\phi_{2}{(Q)}) = - Q (- 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\sin{(Q)})", "derivation": "\\phi_{2}{(Q)} = \\sin{(Q)} and - Q + \\phi_{2}{(Q)} = - Q + \\sin{(Q)} and - Q - (2 \\phi_{2}{(Q)})^{Q} + \\phi_{2}{(Q)} = - Q - (2 \\phi_{2}{(Q)})^{Q} + \\sin{(Q)} and - 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\phi_{2}{(Q)} = - 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\sin{(Q)} and - Q (- 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\phi_{2}{(Q)}) = - Q (- 2^{Q} \\phi_{2}^{Q}{(Q)} - Q + \\sin{(Q)})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], [["minus", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["minus", 2, "Pow(Mul(Integer(2), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(2), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Mul(Integer(2), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), sin(Symbol('Q', commutative=True))))"], [["expand", 3], "Equality(Add(Mul(Integer(-1), Pow(Integer(2), Symbol('Q', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\phi_2')(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Pow(Integer(2), Symbol('Q', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('Q', commutative=True), Add(Mul(Integer(-1), Pow(Integer(2), Symbol('Q', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\phi_2')(Symbol('Q', commutative=True)))), Mul(Integer(-1), Symbol('Q', commutative=True), Add(Mul(Integer(-1), Pow(Integer(2), Symbol('Q', commutative=True)), Pow(Function('\\\\phi_2')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))), Mul(Integer(-1), Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} = \\sin{(f_{\\mathbf{p}} u)} and C{(f_{\\mathbf{p}},u)} = \\sin^{u}{(f_{\\mathbf{p}} u)}, then obtain (\\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} + \\sin^{u}{(f_{\\mathbf{p}} u)})^{f_{\\mathbf{p}}} = (\\sin{(f_{\\mathbf{p}} u)} + \\sin^{u}{(f_{\\mathbf{p}} u)})^{f_{\\mathbf{p}}}", "derivation": "\\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} = \\sin{(f_{\\mathbf{p}} u)} and C{(f_{\\mathbf{p}},u)} = \\sin^{u}{(f_{\\mathbf{p}} u)} and C{(f_{\\mathbf{p}},u)} + \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} = C{(f_{\\mathbf{p}},u)} + \\sin{(f_{\\mathbf{p}} u)} and \\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} + \\sin^{u}{(f_{\\mathbf{p}} u)} = \\sin{(f_{\\mathbf{p}} u)} + \\sin^{u}{(f_{\\mathbf{p}} u)} and (\\hat{\\mathbf{r}}{(f_{\\mathbf{p}},u)} + \\sin^{u}{(f_{\\mathbf{p}} u)})^{f_{\\mathbf{p}}} = (\\sin{(f_{\\mathbf{p}} u)} + \\sin^{u}{(f_{\\mathbf{p}} u)})^{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), Pow(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["add", 1, "Function('C')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Function('C')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Add(Function('C')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), Pow(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True))), Add(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Pow(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True))))"], [["power", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True)), Pow(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Add(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Pow(sin(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('u', commutative=True))), Symbol('u', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\Omega)} = \\log{(\\Omega)}, then derive \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = F_{c} + \\Omega \\log{(\\Omega)} - \\Omega, then obtain (\\int - \\int \\log{(\\Omega)} d\\Omega dF_{c})^{F_{c}} = (\\int (- F_{c} - \\Omega \\operatorname{E_{x}}{(\\Omega)} + \\Omega) dF_{c})^{F_{c}}", "derivation": "\\operatorname{E_{x}}{(\\Omega)} = \\log{(\\Omega)} and \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = \\int \\log{(\\Omega)} d\\Omega and \\int \\operatorname{E_{x}}{(\\Omega)} d\\Omega = F_{c} + \\Omega \\log{(\\Omega)} - \\Omega and \\int \\log{(\\Omega)} d\\Omega = F_{c} + \\Omega \\log{(\\Omega)} - \\Omega and - \\int \\log{(\\Omega)} d\\Omega = - F_{c} - \\Omega \\log{(\\Omega)} + \\Omega and \\int - \\int \\log{(\\Omega)} d\\Omega dF_{c} = \\int (- F_{c} - \\Omega \\log{(\\Omega)} + \\Omega) dF_{c} and (\\int - \\int \\log{(\\Omega)} d\\Omega dF_{c})^{F_{c}} = (\\int (- F_{c} - \\Omega \\log{(\\Omega)} + \\Omega) dF_{c})^{F_{c}} and (\\int - \\int \\log{(\\Omega)} d\\Omega dF_{c})^{F_{c}} = (\\int (- F_{c} - \\Omega \\operatorname{E_{x}}{(\\Omega)} + \\Omega) dF_{c})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 5, "Symbol('F_c', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["power", 6, "Symbol('F_c', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Pow(Integral(Mul(Integer(-1), Integral(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Function('E_x')(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given H{(g)} = \\cos{(g)} and \\operatorname{A_{z}}{(g)} = - H{(g)}, then derive \\frac{d}{d g} \\operatorname{A_{z}}{(g)} = - \\frac{d}{d g} H{(g)}, then obtain (\\frac{d}{d g} - \\cos{(g)})^{g} = (- \\frac{d}{d g} H{(g)})^{g}", "derivation": "H{(g)} = \\cos{(g)} and \\operatorname{A_{z}}{(g)} = - H{(g)} and \\operatorname{A_{z}}{(g)} = - \\cos{(g)} and \\frac{d}{d g} \\operatorname{A_{z}}{(g)} = \\frac{d}{d g} - H{(g)} and \\frac{d}{d g} \\operatorname{A_{z}}{(g)} = - \\frac{d}{d g} H{(g)} and (\\frac{d}{d g} \\operatorname{A_{z}}{(g)})^{g} = (- \\frac{d}{d g} H{(g)})^{g} and (\\frac{d}{d g} - \\cos{(g)})^{g} = (- \\frac{d}{d g} H{(g)})^{g}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('g', commutative=True)), Mul(Integer(-1), Function('H')(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_z')(Symbol('g', commutative=True)), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('H')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('A_z')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('H')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('g', commutative=True)"], "Equality(Pow(Derivative(Function('A_z')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Derivative(Function('H')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Derivative(Mul(Integer(-1), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Symbol('g', commutative=True)), Pow(Mul(Integer(-1), Derivative(Function('H')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(y)} = \\cos{(\\cos{(y)})}, then derive \\omega + \\operatorname{f_{E}}{(y)} = \\theta_2 + \\cos{(\\cos{(y)})}, then derive \\frac{d}{d y} \\operatorname{f_{E}}{(y)} = \\sin{(y)} \\sin{(\\cos{(y)})}, then obtain \\frac{\\partial}{\\partial y} (\\omega - \\theta_2 + \\operatorname{f_{E}}{(y)}) = \\sin{(y)} \\sin{(\\cos{(y)})}", "derivation": "\\operatorname{f_{E}}{(y)} = \\cos{(\\cos{(y)})} and \\frac{d}{d y} \\operatorname{f_{E}}{(y)} = \\frac{d}{d y} \\cos{(\\cos{(y)})} and \\int \\frac{d}{d y} \\operatorname{f_{E}}{(y)} dy = \\int \\frac{d}{d y} \\cos{(\\cos{(y)})} dy and \\omega + \\operatorname{f_{E}}{(y)} = \\theta_2 + \\cos{(\\cos{(y)})} and \\frac{d}{d y} \\operatorname{f_{E}}{(y)} = \\sin{(y)} \\sin{(\\cos{(y)})} and \\omega + \\operatorname{f_{E}}{(y)} = \\theta_2 + \\operatorname{f_{E}}{(y)} and \\omega - \\theta_2 + \\operatorname{f_{E}}{(y)} = \\operatorname{f_{E}}{(y)} and \\frac{\\partial}{\\partial y} (\\omega - \\theta_2 + \\operatorname{f_{E}}{(y)}) = \\sin{(y)} \\sin{(\\cos{(y)})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Function('f_E')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(cos(cos(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('f_E')(Symbol('y', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), cos(cos(Symbol('y', commutative=True)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(sin(Symbol('y', commutative=True)), sin(cos(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('f_E')(Symbol('y', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Function('f_E')(Symbol('y', commutative=True))))"], [["minus", 6, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('f_E')(Symbol('y', commutative=True))), Function('f_E')(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Derivative(Add(Symbol('\\\\omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('f_E')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(sin(Symbol('y', commutative=True)), sin(cos(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(l)} = \\log{(\\log{(l)})}, then obtain \\int \\operatorname{F_{x}}{(l)} dl = \\int \\log{(\\frac{\\log{(l)} \\log{(\\log{(l)})}}{\\operatorname{F_{x}}{(l)}})} dl", "derivation": "\\operatorname{F_{x}}{(l)} = \\log{(\\log{(l)})} and \\operatorname{F_{x}}{(l)} \\log{(l)} = \\log{(l)} \\log{(\\log{(l)})} and \\log{(l)} = \\frac{\\log{(l)} \\log{(\\log{(l)})}}{\\operatorname{F_{x}}{(l)}} and \\operatorname{F_{x}}{(l)} = \\log{(\\frac{\\log{(l)} \\log{(\\log{(l)})}}{\\operatorname{F_{x}}{(l)}})} and \\int \\operatorname{F_{x}}{(l)} dl = \\int \\log{(\\frac{\\log{(l)} \\log{(\\log{(l)})}}{\\operatorname{F_{x}}{(l)}})} dl", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))"], [["times", 1, "log(Symbol('l', commutative=True))"], "Equality(Mul(Function('F_x')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Mul(log(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))))"], [["divide", 2, "Function('F_x')(Symbol('l', commutative=True))"], "Equality(log(Symbol('l', commutative=True)), Mul(Pow(Function('F_x')(Symbol('l', commutative=True)), Integer(-1)), log(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('F_x')(Symbol('l', commutative=True)), log(Mul(Pow(Function('F_x')(Symbol('l', commutative=True)), Integer(-1)), log(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(log(Mul(Pow(Function('F_x')(Symbol('l', commutative=True)), Integer(-1)), log(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given r{(S)} = \\log{(\\cos{(S)})}, then obtain \\int \\frac{d}{d S} (\\int r{(S)} dS + \\frac{r{(S)}}{S}) dS = \\int \\frac{d}{d S} (\\int r{(S)} dS + \\frac{\\log{(\\cos{(S)})}}{S}) dS", "derivation": "r{(S)} = \\log{(\\cos{(S)})} and \\frac{r{(S)}}{S} = \\frac{\\log{(\\cos{(S)})}}{S} and \\int r{(S)} dS = \\int \\log{(\\cos{(S)})} dS and \\int \\log{(\\cos{(S)})} dS + \\frac{r{(S)}}{S} = \\int \\log{(\\cos{(S)})} dS + \\frac{\\log{(\\cos{(S)})}}{S} and \\int r{(S)} dS + \\frac{r{(S)}}{S} = \\int r{(S)} dS + \\frac{\\log{(\\cos{(S)})}}{S} and \\frac{d}{d S} (\\int r{(S)} dS + \\frac{r{(S)}}{S}) = \\frac{d}{d S} (\\int r{(S)} dS + \\frac{\\log{(\\cos{(S)})}}{S}) and \\int \\frac{d}{d S} (\\int r{(S)} dS + \\frac{r{(S)}}{S}) dS = \\int \\frac{d}{d S} (\\int r{(S)} dS + \\frac{\\log{(\\cos{(S)})}}{S}) dS", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('S', commutative=True)), log(cos(Symbol('S', commutative=True))))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), log(cos(Symbol('S', commutative=True)))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["add", 2, "Integral(log(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))"], "Equality(Add(Integral(log(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True)))), Add(Integral(log(cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), log(cos(Symbol('S', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True)))), Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), log(cos(Symbol('S', commutative=True))))))"], [["differentiate", 5, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), log(cos(Symbol('S', commutative=True))))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('S', commutative=True)"], "Equality(Integral(Derivative(Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('r')(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))), Integral(Derivative(Add(Integral(Function('r')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), log(cos(Symbol('S', commutative=True))))), Tuple(Symbol('S', commutative=True), Integer(1))), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = \\dot{x} + f, then derive \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = 1, then obtain \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} df = \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} df", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = \\dot{x} + f and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = \\frac{\\partial}{\\partial \\dot{x}} (\\dot{x} + f) and \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = 1 and \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} \\frac{\\partial}{\\partial \\dot{x}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} df = \\int \\operatorname{g^{\\prime}_{\\varepsilon}}{(f,\\dot{x})} df", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Tuple(Symbol('f', commutative=True))), Integral(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(x^\\prime)} = \\sin{(e^{x^\\prime})} and \\rho_{b}{(x^\\prime)} = (\\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})})^{x^\\prime}, then obtain \\frac{d}{d x^\\prime} \\rho_{b}{(x^\\prime)} = \\frac{d}{d x^\\prime} (\\frac{d}{d x^\\prime} \\sigma_{x}{(x^\\prime)})^{x^\\prime}", "derivation": "\\sigma_{x}{(x^\\prime)} = \\sin{(e^{x^\\prime})} and \\frac{d}{d x^\\prime} \\sigma_{x}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})} and (\\frac{d}{d x^\\prime} \\sigma_{x}{(x^\\prime)})^{x^\\prime} = (\\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})})^{x^\\prime} and \\rho_{b}{(x^\\prime)} = (\\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})})^{x^\\prime} and \\frac{d}{d x^\\prime} \\rho_{b}{(x^\\prime)} = \\frac{d}{d x^\\prime} (\\frac{d}{d x^\\prime} \\sin{(e^{x^\\prime})})^{x^\\prime} and \\frac{d}{d x^\\prime} \\rho_{b}{(x^\\prime)} = \\frac{d}{d x^\\prime} (\\frac{d}{d x^\\prime} \\sigma_{x}{(x^\\prime)})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('x^\\\\prime', commutative=True)), sin(exp(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\sigma_x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(sin(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(sin(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Derivative(sin(exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('\\\\rho_b')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Derivative(Function('\\\\sigma_x')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(u)} = \\log{(u)} and b{(u)} = u + \\int \\operatorname{y^{\\prime}}{(u)} du, then derive b{(u)} = r + u \\log{(u)}, then obtain u + \\int \\operatorname{y^{\\prime}}{(u)} du = r + u \\log{(u)}", "derivation": "\\operatorname{y^{\\prime}}{(u)} = \\log{(u)} and \\int \\operatorname{y^{\\prime}}{(u)} du = \\int \\log{(u)} du and u + \\int \\operatorname{y^{\\prime}}{(u)} du = u + \\int \\log{(u)} du and b{(u)} = u + \\int \\operatorname{y^{\\prime}}{(u)} du and b{(u)} = u + \\int \\log{(u)} du and b{(u)} = r + u \\log{(u)} and u + \\int \\log{(u)} du = r + u \\log{(u)} and u + \\int \\operatorname{y^{\\prime}}{(u)} du = r + u \\log{(u)}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["add", 2, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Symbol('u', commutative=True), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], ["renaming_premise", "Equality(Function('b')(Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('b')(Symbol('u', commutative=True)), Add(Symbol('u', commutative=True), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Function('b')(Symbol('u', commutative=True)), Add(Symbol('r', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Symbol('u', commutative=True), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Symbol('u', commutative=True), Integral(Function('y^{\\\\prime}')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(F_{g})} = e^{F_{g}}, then obtain \\frac{d}{d F_{g}} (2 \\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}} - 1) = \\frac{d}{d F_{g}} (\\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}})", "derivation": "\\operatorname{P_{g}}{(F_{g})} = e^{F_{g}} and \\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} = 1 and \\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}} = 1 - e^{F_{g}} and \\frac{d}{d F_{g}} (\\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}}) = \\frac{d}{d F_{g}} (1 - e^{F_{g}}) and \\frac{d}{d F_{g}} (2 \\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}} - 1) = \\frac{d}{d F_{g}} (\\operatorname{P_{g}}{(F_{g})} e^{- F_{g}} - e^{F_{g}})", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["divide", 1, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Integer(1))"], [["minus", 2, "exp(Symbol('F_g', commutative=True))"], "Equality(Add(Mul(Function('P_g')(Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))))"], [["differentiate", 3, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Add(Mul(Function('P_g')(Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(2), Function('P_g')(Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True))), Integer(-1)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Mul(Function('P_g')(Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Mul(Integer(-1), exp(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(T,\\hat{X})} = T \\hat{X} and Q{(S,L)} = L^{S}, then obtain \\frac{\\partial}{\\partial T} (- T^{2} \\hat{X} \\int v{(S,\\eta)} dS + Q{(S,L)} + v{(S,\\eta)}) = \\frac{\\partial}{\\partial T} (L^{S} - T^{2} \\hat{X} \\int v{(S,\\eta)} dS + v{(S,\\eta)})", "derivation": "\\operatorname{n_{2}}{(T,\\hat{X})} = T \\hat{X} and Q{(S,L)} = L^{S} and Q{(S,L)} + v{(S,\\eta)} = L^{S} + v{(S,\\eta)} and - T \\operatorname{n_{2}}{(T,\\hat{X})} \\int v{(S,\\eta)} dS + Q{(S,L)} + v{(S,\\eta)} = L^{S} - T \\operatorname{n_{2}}{(T,\\hat{X})} \\int v{(S,\\eta)} dS + v{(S,\\eta)} and - T^{2} \\hat{X} \\int v{(S,\\eta)} dS + Q{(S,L)} + v{(S,\\eta)} = L^{S} - T^{2} \\hat{X} \\int v{(S,\\eta)} dS + v{(S,\\eta)} and \\frac{\\partial}{\\partial T} (- T^{2} \\hat{X} \\int v{(S,\\eta)} dS + Q{(S,L)} + v{(S,\\eta)}) = \\frac{\\partial}{\\partial T} (L^{S} - T^{2} \\hat{X} \\int v{(S,\\eta)} dS + v{(S,\\eta)})", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["get_premise", "Equality(Function('Q')(Symbol('S', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('S', commutative=True)))"], [["add", 2, "Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('Q')(Symbol('S', commutative=True), Symbol('L', commutative=True)), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('S', commutative=True)), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["minus", 3, "Mul(Symbol('T', commutative=True), Function('n_2')(Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Function('n_2')(Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('Q')(Symbol('S', commutative=True), Symbol('L', commutative=True)), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True), Function('n_2')(Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(2)), Symbol('\\\\hat{X}', commutative=True), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('Q')(Symbol('S', commutative=True), Symbol('L', commutative=True)), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Pow(Symbol('L', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(2)), Symbol('\\\\hat{X}', commutative=True), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["differentiate", 5, "Symbol('T', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(2)), Symbol('\\\\hat{X}', commutative=True), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('Q')(Symbol('S', commutative=True), Symbol('L', commutative=True)), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('L', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(2)), Symbol('\\\\hat{X}', commutative=True), Integral(Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('S', commutative=True)))), Function('v')(Symbol('S', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('T', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} = - \\hat{\\mathbf{x}} + \\frac{\\Psi}{W}, then obtain \\Psi \\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} (\\Psi - \\hat{\\mathbf{x}} - \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} + \\frac{\\Psi}{W})", "derivation": "\\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} = - \\hat{\\mathbf{x}} + \\frac{\\Psi}{W} and \\Psi + \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} = \\Psi - \\hat{\\mathbf{x}} + \\frac{\\Psi}{W} and \\Psi + 2 \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} = \\Psi - \\hat{\\mathbf{x}} + \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} + \\frac{\\Psi}{W} and \\Psi = \\Psi - \\hat{\\mathbf{x}} - \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} + \\frac{\\Psi}{W} and \\Psi \\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} (\\Psi - \\hat{\\mathbf{x}} - \\mathbf{J}_M{(\\hat{\\mathbf{x}},W,\\Psi)} + \\frac{\\Psi}{W})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))))"], [["add", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True)))), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True)))"], "Equality(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True))))"], [["times", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\Psi', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = f^{\\prime} - f_{E}, then derive \\frac{\\partial}{\\partial f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = -1, then obtain - f^{\\prime} + f_{E} - 1 = - f^{\\prime} + f_{E} + \\frac{\\partial}{\\partial f_{E}} (f^{\\prime} - f_{E})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = f^{\\prime} - f_{E} and \\frac{\\partial}{\\partial f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = \\frac{\\partial}{\\partial f_{E}} (f^{\\prime} - f_{E}) and \\frac{\\partial}{\\partial f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = -1 and - f^{\\prime} + f_{E} + \\frac{\\partial}{\\partial f_{E}} \\operatorname{f_{\\mathbf{p}}}{(f^{\\prime},f_{E})} = - f^{\\prime} + f_{E} + \\frac{\\partial}{\\partial f_{E}} (f^{\\prime} - f_{E}) and - f^{\\prime} + f_{E} - 1 = - f^{\\prime} + f_{E} + \\frac{\\partial}{\\partial f_{E}} (f^{\\prime} - f_{E})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1))"], [["minus", 2, "Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True), Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Symbol('f_E', commutative=True), Derivative(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\varepsilon_0,z)} = e^{\\varepsilon_0 + z}, then derive \\int \\dot{x}{(\\varepsilon_0,z)} d\\varepsilon_0 = c + e^{\\varepsilon_0 + z}, then derive \\hat{X} + e^{\\varepsilon_0 + z} = c + e^{\\varepsilon_0 + z}, then obtain \\hat{X} + e^{\\varepsilon_0 + z} = c + \\dot{x}{(\\varepsilon_0,z)}", "derivation": "\\dot{x}{(\\varepsilon_0,z)} = e^{\\varepsilon_0 + z} and \\int \\dot{x}{(\\varepsilon_0,z)} d\\varepsilon_0 = \\int e^{\\varepsilon_0 + z} d\\varepsilon_0 and \\int \\dot{x}{(\\varepsilon_0,z)} d\\varepsilon_0 = c + e^{\\varepsilon_0 + z} and \\int \\dot{x}{(\\varepsilon_0,z)} d\\varepsilon_0 = c + \\dot{x}{(\\varepsilon_0,z)} and \\int e^{\\varepsilon_0 + z} d\\varepsilon_0 = c + e^{\\varepsilon_0 + z} and \\hat{X} + e^{\\varepsilon_0 + z} = c + e^{\\varepsilon_0 + z} and \\hat{X} + e^{\\varepsilon_0 + z} = \\int \\dot{x}{(\\varepsilon_0,z)} d\\varepsilon_0 and \\hat{X} + e^{\\varepsilon_0 + z} = c + \\dot{x}{(\\varepsilon_0,z)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('c', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('c', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Add(Symbol('c', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))), Add(Symbol('c', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))), Integral(Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 7], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), exp(Add(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))), Add(Symbol('c', commutative=True), Function('\\\\dot{x}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given q{(V_{\\mathbf{E}},M_{E})} = \\cos{(M_{E} V_{\\mathbf{E}})}, then obtain (V_{\\mathbf{E}} q{(V_{\\mathbf{E}},M_{E})} + 1)^{V_{\\mathbf{E}}} = (V_{\\mathbf{E}} \\cos{(M_{E} V_{\\mathbf{E}})} + 1)^{V_{\\mathbf{E}}}", "derivation": "q{(V_{\\mathbf{E}},M_{E})} = \\cos{(M_{E} V_{\\mathbf{E}})} and V_{\\mathbf{E}} q{(V_{\\mathbf{E}},M_{E})} = V_{\\mathbf{E}} \\cos{(M_{E} V_{\\mathbf{E}})} and V_{\\mathbf{E}} q{(V_{\\mathbf{E}},M_{E})} + 1 = V_{\\mathbf{E}} \\cos{(M_{E} V_{\\mathbf{E}})} + 1 and (V_{\\mathbf{E}} q{(V_{\\mathbf{E}},M_{E})} + 1)^{V_{\\mathbf{E}}} = (V_{\\mathbf{E}} \\cos{(M_{E} V_{\\mathbf{E}})} + 1)^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('M_E', commutative=True)), cos(Mul(Symbol('M_E', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["times", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('q')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Mul(Symbol('M_E', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('q')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('M_E', commutative=True))), Integer(1)), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Mul(Symbol('M_E', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Integer(1)))"], [["power", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('q')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('M_E', commutative=True))), Integer(1)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), cos(Mul(Symbol('M_E', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Integer(1)), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(v,\\varphi^*)} = \\frac{\\sin{(v)}}{\\varphi^*}, then derive \\frac{\\partial}{\\partial v} \\operatorname{v_{2}}{(v,\\varphi^*)} = \\frac{\\cos{(v)}}{\\varphi^*}, then obtain \\frac{\\cos{(v)}}{\\varphi^*} = \\frac{\\partial}{\\partial v} \\frac{\\sin{(v)}}{\\varphi^*}", "derivation": "\\operatorname{v_{2}}{(v,\\varphi^*)} = \\frac{\\sin{(v)}}{\\varphi^*} and \\frac{\\partial}{\\partial v} \\operatorname{v_{2}}{(v,\\varphi^*)} = \\frac{\\partial}{\\partial v} \\frac{\\sin{(v)}}{\\varphi^*} and \\frac{\\partial}{\\partial v} \\operatorname{v_{2}}{(v,\\varphi^*)} = \\frac{\\cos{(v)}}{\\varphi^*} and \\frac{\\cos{(v)}}{\\varphi^*} = \\frac{\\partial}{\\partial v} \\frac{\\sin{(v)}}{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), sin(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_2')(Symbol('v', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), cos(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), cos(Symbol('v', commutative=True))), Derivative(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), sin(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(S,U,\\nabla)} = \\frac{S}{U \\nabla}, then obtain (\\frac{\\frac{S \\operatorname{v_{t}}{(S,U,\\nabla)}}{U \\nabla} - 1}{\\nabla})^{\\nabla} = (\\frac{\\frac{S^{2}}{U^{2} \\nabla^{2}} - 1}{\\nabla})^{\\nabla}", "derivation": "\\operatorname{v_{t}}{(S,U,\\nabla)} = \\frac{S}{U \\nabla} and \\frac{S \\operatorname{v_{t}}{(S,U,\\nabla)}}{U \\nabla} = \\frac{S^{2}}{U^{2} \\nabla^{2}} and \\frac{S \\operatorname{v_{t}}{(S,U,\\nabla)}}{U \\nabla} - 1 = \\frac{S^{2}}{U^{2} \\nabla^{2}} - 1 and \\frac{\\frac{S \\operatorname{v_{t}}{(S,U,\\nabla)}}{U \\nabla} - 1}{\\nabla} = \\frac{\\frac{S^{2}}{U^{2} \\nabla^{2}} - 1}{\\nabla} and (\\frac{\\frac{S \\operatorname{v_{t}}{(S,U,\\nabla)}}{U \\nabla} - 1}{\\nabla})^{\\nabla} = (\\frac{\\frac{S^{2}}{U^{2} \\nabla^{2}} - 1}{\\nabla})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('S', commutative=True), Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["times", 1, "Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('v_t')(Symbol('S', commutative=True), Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('v_t')(Symbol('S', commutative=True), Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True))), Integer(-1)), Add(Mul(Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Integer(-1)))"], [["divide", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('v_t')(Symbol('S', commutative=True), Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True))), Integer(-1))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Integer(-1))))"], [["power", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Symbol('S', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Function('v_t')(Symbol('S', commutative=True), Symbol('U', commutative=True), Symbol('\\\\nabla', commutative=True))), Integer(-1))), Symbol('\\\\nabla', commutative=True)), Pow(Mul(Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('S', commutative=True), Integer(2)), Pow(Symbol('U', commutative=True), Integer(-2)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-2))), Integer(-1))), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(W,\\hat{x})} = \\frac{\\cos{(\\hat{x})}}{W}, then obtain \\frac{\\mathbf{p}{(W,\\hat{x})} \\mathbf{p}^{W}{(W,\\hat{x})}}{\\mathbb{I}} = \\frac{\\mathbf{p}^{W}{(W,\\hat{x})} \\cos{(\\hat{x})}}{W \\mathbb{I}}", "derivation": "\\mathbf{p}{(W,\\hat{x})} = \\frac{\\cos{(\\hat{x})}}{W} and \\mathbf{p}^{W}{(W,\\hat{x})} = (\\frac{\\cos{(\\hat{x})}}{W})^{W} and (\\frac{\\cos{(\\hat{x})}}{W})^{W} \\mathbf{p}{(W,\\hat{x})} = \\frac{(\\frac{\\cos{(\\hat{x})}}{W})^{W} \\cos{(\\hat{x})}}{W} and \\mathbf{p}{(W,\\hat{x})} \\mathbf{p}^{W}{(W,\\hat{x})} = \\frac{\\mathbf{p}^{W}{(W,\\hat{x})} \\cos{(\\hat{x})}}{W} and \\frac{\\mathbf{p}{(W,\\hat{x})} \\mathbf{p}^{W}{(W,\\hat{x})}}{\\mathbb{I}} = \\frac{\\mathbf{p}^{W}{(W,\\hat{x})} \\cos{(\\hat{x})}}{W \\mathbb{I}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('W', commutative=True)), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('W', commutative=True)))"], [["times", 1, "Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('W', commutative=True))"], "Equality(Mul(Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('W', commutative=True)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Symbol('\\\\hat{x}', commutative=True))), Symbol('W', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('W', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Symbol('W', commutative=True)), cos(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(g_{\\varepsilon},F_{H})} = F_{H} - g_{\\varepsilon}, then obtain \\int\\limits^{g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}} (- F_{H} + g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}) dF_{H} = \\int\\limits^{g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}} 0 dF_{H}", "derivation": "\\mathbf{P}{(g_{\\varepsilon},F_{H})} = F_{H} - g_{\\varepsilon} and g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})} = F_{H} and - F_{H} + g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})} = 0 and \\int (- F_{H} + g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}) dF_{H} = \\int 0 dF_{H} and \\int\\limits^{g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}} (- F_{H} + g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}) dF_{H} = \\int\\limits^{g_{\\varepsilon} + \\mathbf{P}{(g_{\\varepsilon},F_{H})}} 0 dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))"], [["minus", 1, "Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{P}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(\\Psi_{\\lambda},\\mu)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\mu), then derive \\dot{y}{(\\Psi_{\\lambda},\\mu)} = 1, then obtain \\Psi_{\\lambda} + \\mathbf{r} = \\Psi_{\\lambda} + m", "derivation": "\\dot{y}{(\\Psi_{\\lambda},\\mu)} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\mu) and \\dot{y}{(\\Psi_{\\lambda},\\mu)} = 1 and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\mu) = 1 and \\int \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} + \\mu) d\\Psi_{\\lambda} = \\int 1 d\\Psi_{\\lambda} and \\Psi_{\\lambda} + \\mathbf{r} = \\Psi_{\\lambda} + m", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{y}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(I,l,p)} = \\frac{p}{I l}, then derive \\frac{\\partial}{\\partial I} \\phi_{2}{(I,l,p)} = - \\frac{p}{I^{2} l}, then obtain \\pi + \\frac{2 p}{I l} = \\int - \\frac{p}{I^{2} l} dI + \\frac{p}{I l}", "derivation": "\\phi_{2}{(I,l,p)} = \\frac{p}{I l} and \\frac{\\partial}{\\partial I} \\phi_{2}{(I,l,p)} = \\frac{\\partial}{\\partial I} \\frac{p}{I l} and \\frac{\\partial}{\\partial I} \\phi_{2}{(I,l,p)} = - \\frac{p}{I^{2} l} and \\int \\frac{\\partial}{\\partial I} \\phi_{2}{(I,l,p)} dI = \\int - \\frac{p}{I^{2} l} dI and \\int \\frac{\\partial}{\\partial I} \\frac{p}{I l} dI = \\int - \\frac{p}{I^{2} l} dI and \\int \\frac{\\partial}{\\partial I} \\frac{p}{I l} dI + \\frac{p}{I l} = \\int - \\frac{p}{I^{2} l} dI + \\frac{p}{I l} and \\pi + \\frac{2 p}{I l} = \\int - \\frac{p}{I^{2} l} dI + \\frac{p}{I l}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('I', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('I', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('I', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('I', commutative=True), Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["add", 5, "Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True))"], "Equality(Add(Integral(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Add(Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(2), Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True))), Add(Integral(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-2)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True)), Tuple(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Symbol('l', commutative=True), Integer(-1)), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\psi{(\\Omega)} = e^{\\Omega} and \\hat{H}_{\\lambda}{(\\Omega)} = \\frac{e^{\\Omega}}{\\Omega}, then obtain 0 = - \\Omega \\hat{H}_{\\lambda}{(\\Omega)} + \\psi{(\\Omega)}", "derivation": "\\psi{(\\Omega)} = e^{\\Omega} and \\frac{\\psi{(\\Omega)}}{\\Omega} = \\frac{e^{\\Omega}}{\\Omega} and \\hat{H}_{\\lambda}{(\\Omega)} = \\frac{e^{\\Omega}}{\\Omega} and \\hat{H}_{\\lambda}{(\\Omega)} = \\frac{\\psi{(\\Omega)}}{\\Omega} and \\Omega \\hat{H}_{\\lambda}{(\\Omega)} = \\psi{(\\Omega)} and 0 = - \\Omega \\hat{H}_{\\lambda}{(\\Omega)} + \\psi{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), exp(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True))), Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)))"], [["minus", 5, "Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\Omega', commutative=True))), Function('\\\\psi')(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(k,\\hat{\\mathbf{r}},\\mathbf{D})} = \\hat{\\mathbf{r}} \\mathbf{D} - k, then derive \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{L_{\\varepsilon}}{(k,\\hat{\\mathbf{r}},\\mathbf{D})} = \\mathbf{D}, then obtain (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} \\mathbf{D} - k))^{\\hat{\\mathbf{r}}} = \\mathbf{D}^{\\hat{\\mathbf{r}}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(k,\\hat{\\mathbf{r}},\\mathbf{D})} = \\hat{\\mathbf{r}} \\mathbf{D} - k and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{L_{\\varepsilon}}{(k,\\hat{\\mathbf{r}},\\mathbf{D})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} \\mathbf{D} - k) and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\operatorname{L_{\\varepsilon}}{(k,\\hat{\\mathbf{r}},\\mathbf{D})} = \\mathbf{D} and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} \\mathbf{D} - k) = \\mathbf{D} and (\\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} (\\hat{\\mathbf{r}} \\mathbf{D} - k))^{\\hat{\\mathbf{r}}} = \\mathbf{D}^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True))"], [["power", 4, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{B},q)} = e^{\\mathbf{B} - q} and \\bar{\\h}{(\\mathbf{B},q)} = \\mathbf{B} - q, then obtain \\int (\\operatorname{P_{e}}{(\\mathbf{B},q)} - 1) d\\mathbf{B} = \\int (e^{\\mathbf{B} - q} - 1) d\\mathbf{B}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{B},q)} = e^{\\mathbf{B} - q} and \\operatorname{P_{e}}{(\\mathbf{B},q)} - 1 = e^{\\mathbf{B} - q} - 1 and \\bar{\\h}{(\\mathbf{B},q)} = \\mathbf{B} - q and \\operatorname{P_{e}}{(\\mathbf{B},q)} - 1 = e^{\\bar{\\h}{(\\mathbf{B},q)}} - 1 and \\int (\\operatorname{P_{e}}{(\\mathbf{B},q)} - 1) d\\mathbf{B} = \\int (e^{\\bar{\\h}{(\\mathbf{B},q)}} - 1) d\\mathbf{B} and \\int (\\operatorname{P_{e}}{(\\mathbf{B},q)} - 1) d\\mathbf{B} = \\int (e^{\\mathbf{B} - q} - 1) d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), exp(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_e')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Add(exp(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('P_e')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Add(exp(Function('\\\\hbar')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True))), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Add(Function('P_e')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(exp(Function('\\\\hbar')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Add(Function('P_e')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(exp(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Integer(-1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\theta{(F_{g},u)} = - F_{g} + \\cos{(u)}, then derive \\frac{\\partial}{\\partial u} \\theta{(F_{g},u)} = - \\sin{(u)}, then obtain \\int - \\sin{(u)} dF_{g} = \\int \\frac{\\partial}{\\partial u} (- F_{g} + \\cos{(u)}) dF_{g}", "derivation": "\\theta{(F_{g},u)} = - F_{g} + \\cos{(u)} and \\frac{\\partial}{\\partial u} \\theta{(F_{g},u)} = \\frac{\\partial}{\\partial u} (- F_{g} + \\cos{(u)}) and \\frac{\\partial}{\\partial u} \\theta{(F_{g},u)} = - \\sin{(u)} and - \\sin{(u)} = \\frac{\\partial}{\\partial u} (- F_{g} + \\cos{(u)}) and \\int - \\sin{(u)} dF_{g} = \\int \\frac{\\partial}{\\partial u} (- F_{g} + \\cos{(u)}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('F_g', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('u', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Symbol('u', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\hat{H}_l)} = e^{\\sin{(\\hat{H}_l)}}, then obtain \\operatorname{F_{c}}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}} \\cos{(\\hat{H}_l)} + e^{\\sin{(\\hat{H}_l)}} \\frac{d}{d \\hat{H}_l} \\operatorname{F_{c}}{(\\hat{H}_l)} = 2 e^{2 \\sin{(\\hat{H}_l)}} \\cos{(\\hat{H}_l)}", "derivation": "\\operatorname{F_{c}}{(\\hat{H}_l)} = e^{\\sin{(\\hat{H}_l)}} and \\operatorname{F_{c}}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}} = e^{2 \\sin{(\\hat{H}_l)}} and \\frac{d}{d \\hat{H}_l} \\operatorname{F_{c}}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}} = \\frac{d}{d \\hat{H}_l} e^{2 \\sin{(\\hat{H}_l)}} and \\operatorname{F_{c}}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}} \\cos{(\\hat{H}_l)} + e^{\\sin{(\\hat{H}_l)}} \\frac{d}{d \\hat{H}_l} \\operatorname{F_{c}}{(\\hat{H}_l)} = 2 e^{2 \\sin{(\\hat{H}_l)}} \\cos{(\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 1, "exp(sin(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Mul(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True))), cos(Symbol('\\\\hat{H}_l', commutative=True))), Mul(exp(sin(Symbol('\\\\hat{H}_l', commutative=True))), Derivative(Function('F_c')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))), Mul(Integer(2), exp(Mul(Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)))), cos(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{p}_0)} = \\log{(e^{\\hat{p}_0})}, then obtain \\frac{e^{- \\hat{p}_0 + \\operatorname{E_{x}}{(\\hat{p}_0)}}}{\\operatorname{E_{x}}{(\\hat{p}_0)}} = \\frac{1}{\\operatorname{E_{x}}{(\\hat{p}_0)}}", "derivation": "\\operatorname{E_{x}}{(\\hat{p}_0)} = \\log{(e^{\\hat{p}_0})} and \\operatorname{E_{x}}{(\\hat{p}_0)} - \\log{(e^{\\hat{p}_0})} = 0 and e^{- \\hat{p}_0 + \\operatorname{E_{x}}{(\\hat{p}_0)}} = 1 and \\frac{e^{- \\hat{p}_0 + \\operatorname{E_{x}}{(\\hat{p}_0)}}}{\\log{(e^{\\hat{p}_0})}} = \\frac{1}{\\log{(e^{\\hat{p}_0})}} and \\frac{e^{- \\hat{p}_0 + \\operatorname{E_{x}}{(\\hat{p}_0)}}}{\\operatorname{E_{x}}{(\\hat{p}_0)}} = \\frac{1}{\\operatorname{E_{x}}{(\\hat{p}_0)}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)), log(exp(Symbol('\\\\hat{p}_0', commutative=True))))"], [["minus", 1, "log(exp(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Add(Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\hat{p}_0', commutative=True))))), Integer(0))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)))), Integer(1))"], [["divide", 3, "log(exp(Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)))), Pow(log(exp(Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Pow(log(exp(Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True))))), Pow(Function('E_x')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{D}{(\\lambda,\\Omega)} = - \\Omega + \\lambda and \\operatorname{C_{2}}{(\\lambda,\\Omega)} = - 2 \\Omega + 2 \\lambda - \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} + 2, then obtain \\int \\operatorname{C_{2}}{(\\lambda,\\Omega)} d\\Omega = \\int (- 2 \\Omega + 2 \\lambda - (- \\Omega + \\lambda)^{\\Omega} + 2) d\\Omega", "derivation": "\\mathbf{D}{(\\lambda,\\Omega)} = - \\Omega + \\lambda and \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} = (- \\Omega + \\lambda)^{\\Omega} and \\Omega - \\lambda + \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} - 1 = \\Omega - \\lambda + (- \\Omega + \\lambda)^{\\Omega} - 1 and 2 \\Omega - 2 \\lambda + \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} - 2 = 2 \\Omega - 2 \\lambda + (- \\Omega + \\lambda)^{\\Omega} - 2 and - 2 \\Omega + 2 \\lambda - \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} + 2 = - 2 \\Omega + 2 \\lambda - (- \\Omega + \\lambda)^{\\Omega} + 2 and \\operatorname{C_{2}}{(\\lambda,\\Omega)} = - 2 \\Omega + 2 \\lambda - \\mathbf{D}^{\\Omega}{(\\lambda,\\Omega)} + 2 and \\operatorname{C_{2}}{(\\lambda,\\Omega)} = - 2 \\Omega + 2 \\lambda - (- \\Omega + \\lambda)^{\\Omega} + 2 and \\int \\operatorname{C_{2}}{(\\lambda,\\Omega)} d\\Omega = \\int (- 2 \\Omega + 2 \\lambda - (- \\Omega + \\lambda)^{\\Omega} + 2) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True), Integer(1))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True), Integer(1))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-2)), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\lambda', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True)), Integer(-2)))"], [["times", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(2)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('C_2')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(2)))"], [["integrate", 7, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\Omega', commutative=True))), Integer(2)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given U{(\\mu_0)} = e^{\\mu_0} and \\sigma_{p}{(\\mu_0)} = e^{\\mu_0}, then obtain - 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0} = \\frac{(- \\mu_0 + 2 e^{\\mu_0}) (- 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0})}{- \\mu_0 + U{(\\mu_0)} + e^{\\mu_0}}", "derivation": "U{(\\mu_0)} = e^{\\mu_0} and - \\mu_0 + U{(\\mu_0)} = - \\mu_0 + e^{\\mu_0} and - \\mu_0 + U{(\\mu_0)} + e^{\\mu_0} = - \\mu_0 + 2 e^{\\mu_0} and 1 = \\frac{- \\mu_0 + 2 e^{\\mu_0}}{- \\mu_0 + U{(\\mu_0)} + e^{\\mu_0}} and \\sigma_{p}{(\\mu_0)} = e^{\\mu_0} and \\sigma_{p}{(\\mu_0)} = U{(\\mu_0)} and 1 = \\frac{- \\mu_0 + 2 e^{\\mu_0}}{- \\mu_0 + \\sigma_{p}{(\\mu_0)} + e^{\\mu_0}} and - 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0} = \\frac{(- \\mu_0 + 2 e^{\\mu_0}) (- 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0})}{- \\mu_0 + \\sigma_{p}{(\\mu_0)} + e^{\\mu_0}} and - 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0} = \\frac{(- \\mu_0 + 2 e^{\\mu_0}) (- 2 \\mu_0 + U{(\\mu_0)} + 2 e^{\\mu_0})}{- \\mu_0 + U{(\\mu_0)} + e^{\\mu_0}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "exp(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\sigma_p')(Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\sigma_p')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], [["times", 7, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\sigma_p')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 8, 6], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\mu_0', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('U')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(t_{1})} = \\cos{(t_{1})} and \\tilde{g}^*{(t_{1})} = \\frac{d}{d t_{1}} \\cos{(t_{1})}, then derive \\tilde{g}^*{(t_{1})} = - \\sin{(t_{1})}, then obtain (\\sin{(t_{1})} + \\frac{d}{d t_{1}} \\operatorname{v_{1}}{(t_{1})}) (- \\tilde{g}^*{(t_{1})} - \\sin{(t_{1})} + (\\int - \\sin{(t_{1})} dt_{1})^{t_{1}}) = 0", "derivation": "\\operatorname{v_{1}}{(t_{1})} = \\cos{(t_{1})} and \\tilde{g}^*{(t_{1})} = \\frac{d}{d t_{1}} \\cos{(t_{1})} and \\tilde{g}^*{(t_{1})} = - \\sin{(t_{1})} and \\tilde{g}^*{(t_{1})} + \\sin{(t_{1})} = 0 and \\sin{(t_{1})} + \\frac{d}{d t_{1}} \\cos{(t_{1})} = 0 and (\\sin{(t_{1})} + \\frac{d}{d t_{1}} \\cos{(t_{1})}) (- \\tilde{g}^*{(t_{1})} - \\sin{(t_{1})} + (\\int - \\sin{(t_{1})} dt_{1})^{t_{1}}) = 0 and (\\sin{(t_{1})} + \\frac{d}{d t_{1}} \\operatorname{v_{1}}{(t_{1})}) (- \\tilde{g}^*{(t_{1})} - \\sin{(t_{1})} + (\\int - \\sin{(t_{1})} dt_{1})^{t_{1}}) = 0", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True)), Derivative(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True)), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), sin(Symbol('t_1', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(sin(Symbol('t_1', commutative=True)), Derivative(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Integer(0))"], [["times", 5, "Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True))), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Pow(Integral(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"], "Equality(Mul(Add(sin(Symbol('t_1', commutative=True)), Derivative(cos(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True))), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Pow(Integral(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Add(sin(Symbol('t_1', commutative=True)), Derivative(Function('v_1')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('t_1', commutative=True))), Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Pow(Integral(Mul(Integer(-1), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}{(g)} = \\sin{(\\log{(g)})}, then obtain \\frac{\\tilde{g}^{g}{(g)}}{g + \\tilde{g}^{g}{(g)}} = \\frac{\\sin^{g}{(\\log{(g)})}}{g + \\tilde{g}^{g}{(g)}}", "derivation": "\\tilde{g}{(g)} = \\sin{(\\log{(g)})} and \\tilde{g}^{g}{(g)} = \\sin^{g}{(\\log{(g)})} and g + \\tilde{g}^{g}{(g)} = g + \\sin^{g}{(\\log{(g)})} and \\frac{\\tilde{g}^{g}{(g)}}{g + \\sin^{g}{(\\log{(g)})}} = \\frac{\\sin^{g}{(\\log{(g)})}}{g + \\sin^{g}{(\\log{(g)})}} and \\frac{\\tilde{g}^{g}{(g)}}{g + \\tilde{g}^{g}{(g)}} = \\frac{\\sin^{g}{(\\log{(g)})}}{g + \\tilde{g}^{g}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), sin(log(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["add", 2, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Add(Symbol('g', commutative=True), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"], [["divide", 2, "Add(Symbol('g', commutative=True), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True))), Integer(-1)), Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Add(Symbol('g', commutative=True), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True))), Integer(-1)), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Integer(-1)), Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Mul(Pow(Add(Symbol('g', commutative=True), Pow(Function('\\\\tilde{g}')(Symbol('g', commutative=True)), Symbol('g', commutative=True))), Integer(-1)), Pow(sin(log(Symbol('g', commutative=True))), Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{S})} = \\cos{(\\cos{(\\mathbf{S})})}, then obtain \\frac{d}{d \\mathbf{S}} 0^{\\mathbf{S}} = \\frac{d}{d \\mathbf{S}} (- \\int \\operatorname{A_{y}}{(\\mathbf{S})} d\\mathbf{S} + \\int \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S})^{\\mathbf{S}}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{S})} = \\cos{(\\cos{(\\mathbf{S})})} and \\int \\operatorname{A_{y}}{(\\mathbf{S})} d\\mathbf{S} = \\int \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S} and 0 = - \\int \\operatorname{A_{y}}{(\\mathbf{S})} d\\mathbf{S} + \\int \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S} and 0^{\\mathbf{S}} = (- \\int \\operatorname{A_{y}}{(\\mathbf{S})} d\\mathbf{S} + \\int \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S})^{\\mathbf{S}} and \\frac{d}{d \\mathbf{S}} 0^{\\mathbf{S}} = \\frac{d}{d \\mathbf{S}} (- \\int \\operatorname{A_{y}}{(\\mathbf{S})} d\\mathbf{S} + \\int \\cos{(\\cos{(\\mathbf{S})})} d\\mathbf{S})^{\\mathbf{S}}", "srepr_derivation": [["get_premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), cos(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 2, "Integral(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integral(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Mul(Integer(-1), Integral(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integral(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), Integral(Function('A_y')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Integral(cos(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\chi)} = e^{\\chi}, then obtain \\int 2 \\int \\operatorname{M_{E}}{(\\chi)} d\\chi d\\chi = \\int (\\int \\operatorname{M_{E}}{(\\chi)} d\\chi + \\int e^{\\chi} d\\chi) d\\chi", "derivation": "\\operatorname{M_{E}}{(\\chi)} = e^{\\chi} and \\int \\operatorname{M_{E}}{(\\chi)} d\\chi = \\int e^{\\chi} d\\chi and 2 \\int \\operatorname{M_{E}}{(\\chi)} d\\chi = \\int \\operatorname{M_{E}}{(\\chi)} d\\chi + \\int e^{\\chi} d\\chi and \\int 2 \\int \\operatorname{M_{E}}{(\\chi)} d\\chi d\\chi = \\int (\\int \\operatorname{M_{E}}{(\\chi)} d\\chi + \\int e^{\\chi} d\\chi) d\\chi", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Integer(2), Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Integral(Function('M_E')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain (\\theta_{2}^{3}{(\\mathbf{r})})^{\\mathbf{r}} = (\\cos^{3}{(\\mathbf{r})})^{\\mathbf{r}}", "derivation": "\\theta_{2}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\theta_{2}^{2}{(\\mathbf{r})} = \\theta_{2}{(\\mathbf{r})} \\cos{(\\mathbf{r})} and \\theta_{2}^{3}{(\\mathbf{r})} = \\theta_{2}^{2}{(\\mathbf{r})} \\cos{(\\mathbf{r})} and \\theta_{2}{(\\mathbf{r})} \\cos{(\\mathbf{r})} = \\cos^{2}{(\\mathbf{r})} and \\theta_{2}^{2}{(\\mathbf{r})} = \\cos^{2}{(\\mathbf{r})} and (\\theta_{2}^{3}{(\\mathbf{r})})^{\\mathbf{r}} = (\\theta_{2}^{2}{(\\mathbf{r})} \\cos{(\\mathbf{r})})^{\\mathbf{r}} and (\\theta_{2}^{3}{(\\mathbf{r})})^{\\mathbf{r}} = (\\cos^{3}{(\\mathbf{r})})^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 2, "Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)))"], [["power", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), cos(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Pow(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Pow(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}}, then obtain - 2 (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} + 2 \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = 0", "derivation": "\\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} and - (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} + \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = 0 and - 2 (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} + 2 \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = - (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} + \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} and - 2 (\\hat{p}^{\\Psi_{\\lambda}})^{\\mathbf{p}} + 2 \\operatorname{v_{2}}{(\\hat{p},\\Psi_{\\lambda},\\mathbf{p})} = 0", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Integer(0))"], [["add", 2, "Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Integer(2), Function('v_2')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}^*{(f^{\\prime})} = \\int \\sin{(f^{\\prime})} df^{\\prime}, then derive \\tilde{g}^*{(f^{\\prime})} = F_{H} - \\cos{(f^{\\prime})}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} \\frac{F_{H} - \\cos{(f^{\\prime})}}{f^{\\prime}} = \\frac{d}{d f^{\\prime}} \\frac{\\int \\sin{(f^{\\prime})} df^{\\prime}}{f^{\\prime}}", "derivation": "\\tilde{g}^*{(f^{\\prime})} = \\int \\sin{(f^{\\prime})} df^{\\prime} and \\tilde{g}^*{(f^{\\prime})} = F_{H} - \\cos{(f^{\\prime})} and \\frac{\\tilde{g}^*{(f^{\\prime})}}{f^{\\prime}} = \\frac{\\int \\sin{(f^{\\prime})} df^{\\prime}}{f^{\\prime}} and \\frac{F_{H} - \\cos{(f^{\\prime})}}{f^{\\prime}} = \\frac{\\int \\sin{(f^{\\prime})} df^{\\prime}}{f^{\\prime}} and \\frac{\\partial}{\\partial f^{\\prime}} \\frac{F_{H} - \\cos{(f^{\\prime})}}{f^{\\prime}} = \\frac{d}{d f^{\\prime}} \\frac{\\int \\sin{(f^{\\prime})} df^{\\prime}}{f^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True)), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))))"], [["divide", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True))))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["differentiate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True))))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{f}{(m,y)} = m^{y}, then obtain (m^{y})^{y} + (\\int \\mathbf{f}{(m,y)} dm)^{y} = (m^{y})^{y} + (\\int m^{y} dm)^{y}", "derivation": "\\mathbf{f}{(m,y)} = m^{y} and \\int \\mathbf{f}{(m,y)} dm = \\int m^{y} dm and (\\int \\mathbf{f}{(m,y)} dm)^{y} = (\\int m^{y} dm)^{y} and \\mathbf{f}^{y}{(m,y)} = (m^{y})^{y} and \\mathbf{f}^{y}{(m,y)} + (\\int \\mathbf{f}{(m,y)} dm)^{y} = \\mathbf{f}^{y}{(m,y)} + (\\int m^{y} dm)^{y} and (m^{y})^{y} + (\\int \\mathbf{f}{(m,y)} dm)^{y} = (m^{y})^{y} + (\\int m^{y} dm)^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["add", 3, "Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True))), Add(Pow(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Function('\\\\mathbf{f}')(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True))), Add(Pow(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Integral(Pow(Symbol('m', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('y', commutative=True))))"]]}, {"prompt": "Given y{(M,F_{N})} = \\sin{(\\frac{M}{F_{N}})}, then obtain \\frac{\\partial}{\\partial M} (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int y{(M,F_{N})} dM = \\frac{\\partial}{\\partial M} (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int \\sin{(\\frac{M}{F_{N}})} dM", "derivation": "y{(M,F_{N})} = \\sin{(\\frac{M}{F_{N}})} and \\int y{(M,F_{N})} dM = \\int \\sin{(\\frac{M}{F_{N}})} dM and 2 y{(M,F_{N})} = y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})} and 2 y{(M,F_{N})} \\int y{(M,F_{N})} dM = 2 y{(M,F_{N})} \\int \\sin{(\\frac{M}{F_{N}})} dM and (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int y{(M,F_{N})} dM = (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int \\sin{(\\frac{M}{F_{N}})} dM and \\frac{\\partial}{\\partial M} (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int y{(M,F_{N})} dM = \\frac{\\partial}{\\partial M} (y{(M,F_{N})} + \\sin{(\\frac{M}{F_{N}})}) \\int \\sin{(\\frac{M}{F_{N}})} dM", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True))))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True))))"], [["add", 1, "Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True))), Add(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True)))))"], [["times", 2, "Mul(Integer(2), Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Mul(Integer(2), Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Integral(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Integer(2), Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Integral(sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Integral(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Add(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Integral(sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"], [["differentiate", 5, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Add(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Integral(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Add(Function('y')(Symbol('M', commutative=True), Symbol('F_N', commutative=True)), sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True)))), Integral(sin(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(\\Psi)} = \\sin{(\\Psi)}, then obtain (2 \\Psi + W{(\\Psi)})^{\\Psi} = (2 \\Psi + \\sin{(\\Psi)})^{\\Psi}", "derivation": "W{(\\Psi)} = \\sin{(\\Psi)} and \\Psi + W{(\\Psi)} = \\Psi + \\sin{(\\Psi)} and 2 \\Psi + W{(\\Psi)} = 2 \\Psi + \\sin{(\\Psi)} and (2 \\Psi + W{(\\Psi)})^{\\Psi} = (2 \\Psi + \\sin{(\\Psi)})^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["add", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('W')(Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Function('W')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), Function('W')(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(Q)} = \\cos{(Q)}, then derive \\int Q \\sigma_{p}{(Q)} dQ = Q \\sin{(Q)} + \\varphi + \\cos{(Q)}, then obtain (Q \\sin{(Q)} + \\varphi + \\cos{(Q)})^{\\varphi} = (Q \\sin{(Q)} + \\varphi + \\sigma_{p}{(Q)})^{\\varphi}", "derivation": "\\sigma_{p}{(Q)} = \\cos{(Q)} and Q \\sigma_{p}{(Q)} = Q \\cos{(Q)} and \\int Q \\sigma_{p}{(Q)} dQ = \\int Q \\cos{(Q)} dQ and \\int Q \\sigma_{p}{(Q)} dQ = Q \\sin{(Q)} + \\varphi + \\cos{(Q)} and \\int Q \\sigma_{p}{(Q)} dQ = Q \\sin{(Q)} + \\varphi + \\sigma_{p}{(Q)} and Q \\sin{(Q)} + \\varphi + \\cos{(Q)} = Q \\sin{(Q)} + \\varphi + \\sigma_{p}{(Q)} and (Q \\sin{(Q)} + \\varphi + \\cos{(Q)})^{\\varphi} = (Q \\sin{(Q)} + \\varphi + \\sigma_{p}{(Q)})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["times", 1, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), cos(Symbol('Q', commutative=True))))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), cos(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), cos(Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))))"], [["power", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), cos(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Symbol('Q', commutative=True), sin(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True), Function('\\\\sigma_p')(Symbol('Q', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(z^{*},F_{H})} = \\cos{(\\frac{z^{*}}{F_{H}})}, then obtain \\frac{\\partial^{4}}{\\partial F_{H}^{3}\\partial z^{*}} \\mathbf{H}{(z^{*},F_{H})} = \\frac{\\partial^{4}}{\\partial F_{H}^{3}\\partial z^{*}} \\cos{(\\frac{z^{*}}{F_{H}})}", "derivation": "\\mathbf{H}{(z^{*},F_{H})} = \\cos{(\\frac{z^{*}}{F_{H}})} and \\frac{\\partial}{\\partial z^{*}} \\mathbf{H}{(z^{*},F_{H})} = \\frac{\\partial}{\\partial z^{*}} \\cos{(\\frac{z^{*}}{F_{H}})} and \\frac{\\partial^{2}}{\\partial F_{H}\\partial z^{*}} \\mathbf{H}{(z^{*},F_{H})} = \\frac{\\partial^{2}}{\\partial F_{H}\\partial z^{*}} \\cos{(\\frac{z^{*}}{F_{H}})} and \\frac{\\partial^{3}}{\\partial F_{H}^{2}\\partial z^{*}} \\mathbf{H}{(z^{*},F_{H})} = \\frac{\\partial^{3}}{\\partial F_{H}^{2}\\partial z^{*}} \\cos{(\\frac{z^{*}}{F_{H}})} and \\frac{\\partial^{4}}{\\partial F_{H}^{3}\\partial z^{*}} \\mathbf{H}{(z^{*},F_{H})} = \\frac{\\partial^{4}}{\\partial F_{H}^{3}\\partial z^{*}} \\cos{(\\frac{z^{*}}{F_{H}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('z^*', commutative=True), Symbol('F_H', commutative=True)), cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('z^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('z^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('z^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(2))), Derivative(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('z^*', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(3))), Derivative(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1)), Tuple(Symbol('F_H', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(U,A_{y},\\sigma_p)} = A_{y}^{U} + \\sigma_p and \\theta{(U,A_{y},\\sigma_p)} = A_{y}^{U} + \\sigma_p, then obtain \\theta{(U,A_{y},\\sigma_p)} + \\frac{\\theta{(U,A_{y},\\sigma_p)}}{\\operatorname{v_{t}}{(U,A_{y},\\sigma_p)}} = \\theta{(U,A_{y},\\sigma_p)} + 1", "derivation": "\\operatorname{v_{t}}{(U,A_{y},\\sigma_p)} = A_{y}^{U} + \\sigma_p and \\theta{(U,A_{y},\\sigma_p)} = A_{y}^{U} + \\sigma_p and \\frac{\\theta{(U,A_{y},\\sigma_p)}}{A_{y}^{U} + \\sigma_p} = 1 and \\theta{(U,A_{y},\\sigma_p)} + \\frac{\\theta{(U,A_{y},\\sigma_p)}}{A_{y}^{U} + \\sigma_p} = \\theta{(U,A_{y},\\sigma_p)} + 1 and \\theta{(U,A_{y},\\sigma_p)} + \\frac{\\theta{(U,A_{y},\\sigma_p)}}{\\operatorname{v_{t}}{(U,A_{y},\\sigma_p)}} = \\theta{(U,A_{y},\\sigma_p)} + 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Pow(Symbol('A_y', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Pow(Symbol('A_y', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 2, "Add(Pow(Symbol('A_y', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Pow(Add(Pow(Symbol('A_y', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Integer(1))"], [["add", 3, "Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Add(Pow(Symbol('A_y', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)), Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Pow(Function('v_t')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(-1)))), Add(Function('\\\\theta')(Symbol('U', commutative=True), Symbol('A_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integer(1)))"]]}, {"prompt": "Given H{(m)} = \\cos{(m)} and \\sigma_{x}{(m)} = \\frac{H{(m)}}{\\cos{(m)}}, then obtain \\frac{H{(m)}}{\\cos{(m)}} - 1 = 0", "derivation": "H{(m)} = \\cos{(m)} and \\sigma_{x}{(m)} = \\frac{H{(m)}}{\\cos{(m)}} and \\sigma_{x}{(m)} - 1 = \\frac{H{(m)}}{\\cos{(m)}} - 1 and \\sigma_{x}{(m)} - 1 = 0 and \\frac{H{(m)}}{\\cos{(m)}} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('m', commutative=True)), Mul(Function('H')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))))"], [["minus", 2, 1], "Equality(Add(Function('\\\\sigma_x')(Symbol('m', commutative=True)), Integer(-1)), Add(Mul(Function('H')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\sigma_x')(Symbol('m', commutative=True)), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Function('H')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(i,s)} = - i + s, then obtain \\int (\\int (- s + \\operatorname{m_{s}}{(i,s)}) ds)^{i} ds = \\int (\\int - i ds)^{i} ds", "derivation": "\\operatorname{m_{s}}{(i,s)} = - i + s and i - s + \\operatorname{m_{s}}{(i,s)} = 0 and - s + \\operatorname{m_{s}}{(i,s)} = - i and \\int (- s + \\operatorname{m_{s}}{(i,s)}) ds = \\int - i ds and (\\int (- s + \\operatorname{m_{s}}{(i,s)}) ds)^{i} = (\\int - i ds)^{i} and \\int (\\int (- s + \\operatorname{m_{s}}{(i,s)}) ds)^{i} ds = \\int (\\int - i ds)^{i} ds", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('s', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Symbol('i', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["minus", 2, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["integrate", 3, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('i', commutative=True)))"], [["integrate", 5, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('m_s')(Symbol('i', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(z)} = \\log{(\\cos{(z)})} and M{(z)} = \\log{(\\cos{(z)})}, then obtain (\\frac{M{(z)}}{\\cos{(z)}})^{z} e^{- r_{0}} = (\\frac{\\mathbf{H}{(z)}}{\\cos{(z)}})^{z} e^{- r_{0}}", "derivation": "\\mathbf{H}{(z)} = \\log{(\\cos{(z)})} and M{(z)} = \\log{(\\cos{(z)})} and \\frac{\\mathbf{H}{(z)}}{\\cos{(z)}} = \\frac{\\log{(\\cos{(z)})}}{\\cos{(z)}} and \\mathbf{H}{(z)} = M{(z)} and \\frac{M{(z)}}{\\cos{(z)}} = \\frac{\\log{(\\cos{(z)})}}{\\cos{(z)}} and (\\frac{M{(z)}}{\\cos{(z)}})^{z} = (\\frac{\\log{(\\cos{(z)})}}{\\cos{(z)}})^{z} and (\\frac{M{(z)}}{\\cos{(z)}})^{z} = (\\frac{\\mathbf{H}{(z)}}{\\cos{(z)}})^{z} and (\\frac{M{(z)}}{\\cos{(z)}})^{z} e^{- r_{0}} = (\\frac{\\mathbf{H}{(z)}}{\\cos{(z)}})^{z} e^{- r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), log(cos(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('z', commutative=True)), log(cos(Symbol('z', commutative=True))))"], [["divide", 1, "cos(Symbol('z', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Mul(log(cos(Symbol('z', commutative=True))), Pow(cos(Symbol('z', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Function('M')(Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Function('M')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Mul(log(cos(Symbol('z', commutative=True))), Pow(cos(Symbol('z', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('z', commutative=True)"], "Equality(Pow(Mul(Function('M')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)), Pow(Mul(log(cos(Symbol('z', commutative=True))), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Mul(Function('M')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)), Pow(Mul(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)))"], [["divide", 7, "exp(Symbol('r_0', commutative=True))"], "Equality(Mul(Pow(Mul(Function('M')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))), Mul(Pow(Mul(Function('\\\\mathbf{H}')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), Symbol('z', commutative=True)), exp(Mul(Integer(-1), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(C,y)} = C y, then obtain y + \\Psi{(C,y)} + \\int (y + \\Psi{(C,y)}) dy = y + \\Psi{(C,y)} + \\int (C y + y) dy", "derivation": "\\Psi{(C,y)} = C y and y + \\Psi{(C,y)} = C y + y and \\int (y + \\Psi{(C,y)}) dy = \\int (C y + y) dy and y + \\Psi{(C,y)} + \\int (y + \\Psi{(C,y)}) dy = y + \\Psi{(C,y)} + \\int (C y + y) dy", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True))), Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["add", 3, "Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True)), Integral(Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('y', commutative=True)), Integral(Add(Mul(Symbol('C', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given B{(\\omega)} = \\log{(\\omega)}, then derive \\int B{(\\omega)} d\\omega = \\omega \\log{(\\omega)} - \\omega + v_{y}, then obtain \\omega \\log{(\\omega)} - \\omega + v_{y} + \\cos{(\\omega B{(\\omega)} - \\omega + v_{y})} = \\omega \\log{(\\omega)} - \\omega + v_{y} + \\cos{(\\int \\log{(\\omega)} d\\omega)}", "derivation": "B{(\\omega)} = \\log{(\\omega)} and \\int B{(\\omega)} d\\omega = \\int \\log{(\\omega)} d\\omega and \\int B{(\\omega)} d\\omega = \\omega \\log{(\\omega)} - \\omega + v_{y} and \\omega \\log{(\\omega)} - \\omega + v_{y} = \\int \\log{(\\omega)} d\\omega and \\omega B{(\\omega)} - \\omega + v_{y} = \\int \\log{(\\omega)} d\\omega and \\cos{(\\omega B{(\\omega)} - \\omega + v_{y})} = \\cos{(\\int \\log{(\\omega)} d\\omega)} and \\omega \\log{(\\omega)} - \\omega + v_{y} + \\cos{(\\omega B{(\\omega)} - \\omega + v_{y})} = \\omega \\log{(\\omega)} - \\omega + v_{y} + \\cos{(\\int \\log{(\\omega)} d\\omega)}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('B')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('B')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True)), Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), Function('B')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True)), Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["cos", 5], "Equality(cos(Add(Mul(Symbol('\\\\omega', commutative=True), Function('B')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True))), cos(Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["add", 6, "Add(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True), cos(Add(Mul(Symbol('\\\\omega', commutative=True), Function('B')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True)))), Add(Mul(Symbol('\\\\omega', commutative=True), log(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('v_y', commutative=True), cos(Integral(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(f)} = e^{\\sin{(f)}} and x{(f)} = \\frac{1}{\\operatorname{v_{1}}{(f)}}, then obtain - \\frac{1}{\\operatorname{v_{1}}{(f)} \\sin{(f)}} = - \\frac{e^{- \\sin{(f)}}}{\\sin{(f)}}", "derivation": "\\operatorname{v_{1}}{(f)} = e^{\\sin{(f)}} and x{(f)} = \\frac{1}{\\operatorname{v_{1}}{(f)}} and x{(f)} = e^{- \\sin{(f)}} and - \\frac{x{(f)}}{\\sin{(f)}} = - \\frac{e^{- \\sin{(f)}}}{\\sin{(f)}} and - \\frac{1}{\\operatorname{v_{1}}{(f)} \\sin{(f)}} = - \\frac{e^{- \\sin{(f)}}}{\\sin{(f)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('f', commutative=True)), exp(sin(Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('f', commutative=True)), Pow(Function('v_1')(Symbol('f', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('x')(Symbol('f', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('f', commutative=True)))))"], [["divide", 3, "Mul(Integer(-1), sin(Symbol('f', commutative=True)))"], "Equality(Mul(Integer(-1), Function('x')(Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Integer(-1))), Mul(Integer(-1), exp(Mul(Integer(-1), sin(Symbol('f', commutative=True)))), Pow(sin(Symbol('f', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Function('v_1')(Symbol('f', commutative=True)), Integer(-1)), Pow(sin(Symbol('f', commutative=True)), Integer(-1))), Mul(Integer(-1), exp(Mul(Integer(-1), sin(Symbol('f', commutative=True)))), Pow(sin(Symbol('f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{f}{(z^{*})} = \\log{(z^{*})}, then obtain - z^{*} + (\\frac{d}{d z^{*}} \\mathbf{f}{(z^{*})})^{z^{*}} = - z^{*} + (\\frac{1}{z^{*}})^{z^{*}}", "derivation": "\\mathbf{f}{(z^{*})} = \\log{(z^{*})} and \\frac{d}{d z^{*}} \\mathbf{f}{(z^{*})} = \\frac{d}{d z^{*}} \\log{(z^{*})} and (\\frac{d}{d z^{*}} \\mathbf{f}{(z^{*})})^{z^{*}} = (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} and - z^{*} + (\\frac{d}{d z^{*}} \\mathbf{f}{(z^{*})})^{z^{*}} = - z^{*} + (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} and - z^{*} + (\\frac{d}{d z^{*}} \\mathbf{f}{(z^{*})})^{z^{*}} = - z^{*} + (\\frac{1}{z^{*}})^{z^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["power", 2, "Symbol('z^*', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"], [["minus", 3, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Derivative(Function('\\\\mathbf{f}')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), Pow(Pow(Symbol('z^*', commutative=True), Integer(-1)), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\lambda,\\mathbf{P})} = \\frac{\\mathbf{P}}{\\lambda} and \\hat{H}{(\\lambda,\\mathbf{P})} = -1 + \\frac{\\mathbf{P}}{\\lambda}, then obtain \\hat{H}{(\\lambda,\\mathbf{P})} = \\mathbf{v}{(\\lambda,\\mathbf{P})} - 1", "derivation": "\\mathbf{v}{(\\lambda,\\mathbf{P})} = \\frac{\\mathbf{P}}{\\lambda} and \\mathbf{v}{(\\lambda,\\mathbf{P})} - 1 = -1 + \\frac{\\mathbf{P}}{\\lambda} and \\hat{H}{(\\lambda,\\mathbf{P})} = -1 + \\frac{\\mathbf{P}}{\\lambda} and \\hat{H}{(\\lambda,\\mathbf{P})} = \\mathbf{v}{(\\lambda,\\mathbf{P})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Integer(-1), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{H}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Function('\\\\mathbf{v}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given J{(r_{0},\\psi^*)} = \\frac{r_{0}}{\\psi^*}, then derive \\frac{\\partial}{\\partial \\psi^*} J{(r_{0},\\psi^*)} + \\frac{r_{0}}{(\\psi^*)^{2}} = 0, then obtain \\frac{\\partial}{\\partial \\psi^*} J{(r_{0},\\psi^*)} + \\frac{J{(r_{0},\\psi^*)}}{\\psi^*} + \\frac{1}{(\\psi^*)^{2}} = \\frac{1}{(\\psi^*)^{2}}", "derivation": "J{(r_{0},\\psi^*)} = \\frac{r_{0}}{\\psi^*} and J{(r_{0},\\psi^*)} - \\frac{r_{0}}{\\psi^*} = 0 and \\frac{\\partial}{\\partial \\psi^*} (J{(r_{0},\\psi^*)} - \\frac{r_{0}}{\\psi^*}) = \\frac{d}{d \\psi^*} 0 and \\frac{\\partial}{\\partial \\psi^*} J{(r_{0},\\psi^*)} + \\frac{r_{0}}{(\\psi^*)^{2}} = 0 and \\frac{\\partial}{\\partial \\psi^*} J{(r_{0},\\psi^*)} + \\frac{J{(r_{0},\\psi^*)}}{\\psi^*} = 0 and \\frac{\\partial}{\\partial \\psi^*} J{(r_{0},\\psi^*)} + \\frac{J{(r_{0},\\psi^*)}}{\\psi^*} + \\frac{1}{(\\psi^*)^{2}} = \\frac{1}{(\\psi^*)^{2}}", "srepr_derivation": [["renaming_premise", "Equality(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))"], "Equality(Add(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)), Symbol('r_0', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Integer(0))"], [["add", 5, "Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2))"], "Equality(Add(Derivative(Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Function('J')(Symbol('r_0', commutative=True), Symbol('\\\\psi^*', commutative=True))), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2))), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-2)))"]]}, {"prompt": "Given \\tilde{g}^*{(H,\\hbar)} = H \\hbar, then obtain (\\tilde{g}^*{(H,\\hbar)} + \\log{(H \\hbar)}) \\log{(\\tilde{g}^*{(H,\\hbar)})}^{\\hbar} = (\\tilde{g}^*{(H,\\hbar)} + \\log{(H \\hbar)}) \\log{(H \\hbar)}^{\\hbar}", "derivation": "\\tilde{g}^*{(H,\\hbar)} = H \\hbar and \\log{(\\tilde{g}^*{(H,\\hbar)})} = \\log{(H \\hbar)} and \\tilde{g}^*{(H,\\hbar)} + \\log{(\\tilde{g}^*{(H,\\hbar)})} = \\tilde{g}^*{(H,\\hbar)} + \\log{(H \\hbar)} and \\log{(\\tilde{g}^*{(H,\\hbar)})}^{\\hbar} = \\log{(H \\hbar)}^{\\hbar} and (\\tilde{g}^*{(H,\\hbar)} + \\log{(\\tilde{g}^*{(H,\\hbar)})}) \\log{(\\tilde{g}^*{(H,\\hbar)})}^{\\hbar} = (\\tilde{g}^*{(H,\\hbar)} + \\log{(\\tilde{g}^*{(H,\\hbar)})}) \\log{(H \\hbar)}^{\\hbar} and (\\tilde{g}^*{(H,\\hbar)} + \\log{(H \\hbar)}) \\log{(\\tilde{g}^*{(H,\\hbar)})}^{\\hbar} = (\\tilde{g}^*{(H,\\hbar)} + \\log{(H \\hbar)}) \\log{(H \\hbar)}^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["times", 4, "Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))))"], "Equality(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))), Pow(log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Mul(Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))), Pow(log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))), Pow(log(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Mul(Add(Function('\\\\tilde{g}^*')(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True)))), Pow(log(Mul(Symbol('H', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given L{(\\theta_2)} = \\log{(\\theta_2)}, then obtain (e^{\\frac{d}{d \\theta_2} (L{(\\theta_2)} - \\log{(\\theta_2)})})^{\\theta_2} = (e^{\\frac{d}{d \\theta_2} 0})^{\\theta_2}", "derivation": "L{(\\theta_2)} = \\log{(\\theta_2)} and L{(\\theta_2)} - \\log{(\\theta_2)} = 0 and \\frac{d}{d \\theta_2} (L{(\\theta_2)} - \\log{(\\theta_2)}) = \\frac{d}{d \\theta_2} 0 and e^{\\frac{d}{d \\theta_2} (L{(\\theta_2)} - \\log{(\\theta_2)})} = e^{\\frac{d}{d \\theta_2} 0} and (e^{\\frac{d}{d \\theta_2} (L{(\\theta_2)} - \\log{(\\theta_2)})})^{\\theta_2} = (e^{\\frac{d}{d \\theta_2} 0})^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('L')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta_2', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Add(Function('L')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["exp", 3], "Equality(exp(Derivative(Add(Function('L')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), exp(Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(exp(Derivative(Add(Function('L')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Symbol('\\\\theta_2', commutative=True)), Pow(exp(Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given k{(h,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\frac{h}{g^{\\prime}_{\\varepsilon}}, then derive k^{h}{(h,g^{\\prime}_{\\varepsilon})} = (- \\frac{h}{(g^{\\prime}_{\\varepsilon})^{2}})^{h}, then obtain \\frac{h k^{h}{(h,g^{\\prime}_{\\varepsilon})}}{g^{\\prime}_{\\varepsilon}} = \\frac{h (- \\frac{h}{(g^{\\prime}_{\\varepsilon})^{2}})^{h}}{g^{\\prime}_{\\varepsilon}}", "derivation": "k{(h,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\frac{h}{g^{\\prime}_{\\varepsilon}} and k^{h}{(h,g^{\\prime}_{\\varepsilon})} = (\\frac{\\partial}{\\partial g^{\\prime}_{\\varepsilon}} \\frac{h}{g^{\\prime}_{\\varepsilon}})^{h} and k^{h}{(h,g^{\\prime}_{\\varepsilon})} = (- \\frac{h}{(g^{\\prime}_{\\varepsilon})^{2}})^{h} and \\frac{h k^{h}{(h,g^{\\prime}_{\\varepsilon})}}{g^{\\prime}_{\\varepsilon}} = \\frac{h (- \\frac{h}{(g^{\\prime}_{\\varepsilon})^{2}})^{h}}{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('h', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Derivative(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('k')(Symbol('h', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('h', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('k')(Symbol('h', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('h', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-2)), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('h', commutative=True), Pow(Function('k')(Symbol('h', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('h', commutative=True))), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('h', commutative=True), Pow(Mul(Integer(-1), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-2)), Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(h)} = \\sin{(h)}, then obtain (\\int (1 - h) dh)^{h} = (\\int (- h + \\frac{h + \\sin{(h)}}{h + \\operatorname{A_{1}}{(h)}}) dh)^{h}", "derivation": "\\operatorname{A_{1}}{(h)} = \\sin{(h)} and h + \\operatorname{A_{1}}{(h)} = h + \\sin{(h)} and 1 = \\frac{h + \\sin{(h)}}{h + \\operatorname{A_{1}}{(h)}} and 1 - h = - h + \\frac{h + \\sin{(h)}}{h + \\operatorname{A_{1}}{(h)}} and \\int (1 - h) dh = \\int (- h + \\frac{h + \\sin{(h)}}{h + \\operatorname{A_{1}}{(h)}}) dh and (\\int (1 - h) dh)^{h} = (\\int (- h + \\frac{h + \\sin{(h)}}{h + \\operatorname{A_{1}}{(h)}}) dh)^{h}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), sin(Symbol('h', commutative=True))))"], [["divide", 2, "Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('h', commutative=True), sin(Symbol('h', commutative=True)))))"], [["minus", 3, "Symbol('h', commutative=True)"], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Pow(Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('h', commutative=True), sin(Symbol('h', commutative=True))))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Pow(Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('h', commutative=True), sin(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True))))"], [["power", 5, "Symbol('h', commutative=True)"], "Equality(Pow(Integral(Add(Integer(1), Mul(Integer(-1), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Pow(Add(Symbol('h', commutative=True), Function('A_1')(Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('h', commutative=True), sin(Symbol('h', commutative=True))))), Tuple(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(f,F_{N})} = f^{F_{N}}, then obtain ((\\frac{\\partial}{\\partial f} \\int \\operatorname{A_{2}}{(f,F_{N})} dF_{N})^{f})^{f} = ((\\frac{\\partial}{\\partial f} \\int f^{F_{N}} dF_{N})^{f})^{f}", "derivation": "\\operatorname{A_{2}}{(f,F_{N})} = f^{F_{N}} and \\int \\operatorname{A_{2}}{(f,F_{N})} dF_{N} = \\int f^{F_{N}} dF_{N} and \\frac{\\partial}{\\partial f} \\int \\operatorname{A_{2}}{(f,F_{N})} dF_{N} = \\frac{\\partial}{\\partial f} \\int f^{F_{N}} dF_{N} and (\\frac{\\partial}{\\partial f} \\int \\operatorname{A_{2}}{(f,F_{N})} dF_{N})^{f} = (\\frac{\\partial}{\\partial f} \\int f^{F_{N}} dF_{N})^{f} and ((\\frac{\\partial}{\\partial f} \\int \\operatorname{A_{2}}{(f,F_{N})} dF_{N})^{f})^{f} = ((\\frac{\\partial}{\\partial f} \\int f^{F_{N}} dF_{N})^{f})^{f}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Pow(Symbol('f', commutative=True), Symbol('F_N', commutative=True)))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Function('A_2')(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('A_2')(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Derivative(Integral(Pow(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Derivative(Integral(Function('A_2')(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Pow(Derivative(Integral(Pow(Symbol('f', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\eta{(\\phi_1)} = e^{\\phi_1}, then obtain \\eta^{2}{(\\phi_1)} \\int \\eta{(\\phi_1)} d\\phi_1 = (\\sigma_x + e^{\\phi_1}) \\eta^{2}{(\\phi_1)}", "derivation": "\\eta{(\\phi_1)} = e^{\\phi_1} and \\int \\eta{(\\phi_1)} d\\phi_1 = \\int e^{\\phi_1} d\\phi_1 and \\eta{(\\phi_1)} \\int \\eta{(\\phi_1)} d\\phi_1 = \\eta{(\\phi_1)} \\int e^{\\phi_1} d\\phi_1 and \\eta^{2}{(\\phi_1)} \\int \\eta{(\\phi_1)} d\\phi_1 = \\eta^{2}{(\\phi_1)} \\int e^{\\phi_1} d\\phi_1 and \\eta^{2}{(\\phi_1)} \\int \\eta{(\\phi_1)} d\\phi_1 = (\\sigma_x + e^{\\phi_1}) \\eta^{2}{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integral(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integral(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["times", 3, "Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Integral(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Pow(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Integral(exp(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integer(2)), Integral(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Add(Symbol('\\\\sigma_x', commutative=True), exp(Symbol('\\\\phi_1', commutative=True))), Pow(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Integer(2))))"]]}, {"prompt": "Given k{(V)} = \\cos{(V)}, then obtain 0 = (\\frac{d}{d V} \\frac{k{(V)} + \\cos{(V)}}{V})^{2} - (\\frac{d}{d V} \\frac{2 k{(V)}}{V})^{2}", "derivation": "k{(V)} = \\cos{(V)} and 2 k{(V)} = k{(V)} + \\cos{(V)} and \\frac{2 k{(V)}}{V} = \\frac{k{(V)} + \\cos{(V)}}{V} and \\frac{d}{d V} \\frac{2 k{(V)}}{V} = \\frac{d}{d V} \\frac{k{(V)} + \\cos{(V)}}{V} and (\\frac{d}{d V} \\frac{2 k{(V)}}{V})^{2} = (\\frac{d}{d V} \\frac{k{(V)} + \\cos{(V)}}{V})^{2} and 2 (\\frac{d}{d V} \\frac{2 k{(V)}}{V})^{2} = (\\frac{d}{d V} \\frac{k{(V)} + \\cos{(V)}}{V})^{2} + (\\frac{d}{d V} \\frac{2 k{(V)}}{V})^{2} and 0 = (\\frac{d}{d V} \\frac{k{(V)} + \\cos{(V)}}{V})^{2} - (\\frac{d}{d V} \\frac{2 k{(V)}}{V})^{2}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["add", 1, "Function('k')(Symbol('V', commutative=True))"], "Equality(Mul(Integer(2), Function('k')(Symbol('V', commutative=True))), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))))"], [["divide", 2, "Symbol('V', commutative=True)"], "Equality(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)))"], [["add", 5, "Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2))"], "Equality(Mul(Integer(2), Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2))), Add(Pow(Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2))))"], [["minus", 6, "Mul(Integer(2), Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)))"], "Equality(Integer(0), Add(Pow(Derivative(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Function('k')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), Pow(Derivative(Mul(Integer(2), Pow(Symbol('V', commutative=True), Integer(-1)), Function('k')(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain - \\operatorname{A_{2}}{(\\hat{H}_l)} + \\sin{(\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)})} = - \\operatorname{A_{2}}{(\\hat{H}_l)} - \\sin{(\\sin{(\\hat{H}_l)} - \\sin^{\\hat{H}_l}{(\\hat{H}_l)})}", "derivation": "\\operatorname{A_{2}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} = \\sin^{\\hat{H}_l}{(\\hat{H}_l)} and \\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} = - \\sin{(\\hat{H}_l)} + \\sin^{\\hat{H}_l}{(\\hat{H}_l)} and \\sin{(\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)})} = - \\sin{(\\sin{(\\hat{H}_l)} - \\sin^{\\hat{H}_l}{(\\hat{H}_l)})} and - \\operatorname{A_{2}}{(\\hat{H}_l)} + \\sin{(\\operatorname{A_{2}}^{\\hat{H}_l}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)})} = - \\operatorname{A_{2}}{(\\hat{H}_l)} - \\sin{(\\sin{(\\hat{H}_l)} - \\sin^{\\hat{H}_l}{(\\hat{H}_l)})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 2, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["sin", 3], "Equality(sin(Add(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))))), Mul(Integer(-1), sin(Add(sin(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))))))"], [["minus", 4, "Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True))), sin(Add(Pow(Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))))), Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), sin(Add(sin(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))))))"]]}, {"prompt": "Given y{(\\varepsilon_0,z)} = e^{\\frac{z}{\\varepsilon_0}}, then obtain \\frac{\\frac{\\partial}{\\partial z} y{(\\varepsilon_0,z)}}{z} - \\frac{y{(\\varepsilon_0,z)}}{z^{2}} = - \\frac{e^{\\frac{z}{\\varepsilon_0}}}{z^{2}} + \\frac{e^{\\frac{z}{\\varepsilon_0}}}{\\varepsilon_0 z}", "derivation": "y{(\\varepsilon_0,z)} = e^{\\frac{z}{\\varepsilon_0}} and \\frac{y{(\\varepsilon_0,z)}}{z} = \\frac{e^{\\frac{z}{\\varepsilon_0}}}{z} and \\frac{\\partial}{\\partial z} \\frac{y{(\\varepsilon_0,z)}}{z} = \\frac{\\partial}{\\partial z} \\frac{e^{\\frac{z}{\\varepsilon_0}}}{z} and \\frac{\\frac{\\partial}{\\partial z} y{(\\varepsilon_0,z)}}{z} - \\frac{y{(\\varepsilon_0,z)}}{z^{2}} = - \\frac{e^{\\frac{z}{\\varepsilon_0}}}{z^{2}} + \\frac{e^{\\frac{z}{\\varepsilon_0}}}{\\varepsilon_0 z}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["divide", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Function('y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-2)), Function('y')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-2)), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})}, then obtain (\\operatorname{C_{2}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})})^{\\mathbf{f}} = 0^{\\mathbf{f}}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{f})} = \\cos{(\\mathbf{f})} and \\operatorname{C_{2}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})} = 0 and 2 \\operatorname{C_{2}}{(\\mathbf{f})} - 2 \\cos{(\\mathbf{f})} = \\operatorname{C_{2}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})} and 2 \\operatorname{C_{2}}{(\\mathbf{f})} - 2 \\cos{(\\mathbf{f})} = 0 and (2 \\operatorname{C_{2}}{(\\mathbf{f})} - 2 \\cos{(\\mathbf{f})})^{\\mathbf{f}} = 0^{\\mathbf{f}} and (\\operatorname{C_{2}}{(\\mathbf{f})} - \\cos{(\\mathbf{f})})^{\\mathbf{f}} = 0^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True)), cos(Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["add", 2, "Add(Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["power", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('C_2')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\chi)}, then derive \\sigma_{p}{(\\chi)} = - \\sin{(\\chi)}, then obtain 0 = - \\frac{(\\sin{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}) \\sin{(\\chi)}}{- \\sigma_{p}{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}}", "derivation": "\\sigma_{p}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\chi)} and 0 = - \\sigma_{p}{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)} and 0 = (- \\sigma_{p}{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}) \\sigma_{p}{(\\chi)} and \\sigma_{p}{(\\chi)} = - \\sin{(\\chi)} and 0 = - (\\sin{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}) \\sin{(\\chi)} and 0 = - \\frac{(\\sin{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}) \\sin{(\\chi)}}{- \\sigma_{p}{(\\chi)} + \\frac{d}{d \\chi} \\cos{(\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True)), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 1, "Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["times", 2, "Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Integer(-1), Add(sin(Symbol('\\\\chi', commutative=True)), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), sin(Symbol('\\\\chi', commutative=True))))"], [["divide", 5, "Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('\\\\chi', commutative=True))), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Integer(-1)), Add(sin(Symbol('\\\\chi', commutative=True)), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), sin(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given c{(\\rho_f)} = \\rho_f, then derive \\frac{d}{d \\rho_f} c{(\\rho_f)} = 1, then obtain \\frac{\\int (\\frac{d}{d \\rho_f} \\rho_f)^{2} d\\rho_f}{\\rho_f c{(\\rho_f)}} = \\frac{\\int \\frac{d}{d \\rho_f} \\rho_f d\\rho_f}{\\rho_f c{(\\rho_f)}}", "derivation": "c{(\\rho_f)} = \\rho_f and \\frac{d}{d \\rho_f} c{(\\rho_f)} = \\frac{d}{d \\rho_f} \\rho_f and \\frac{d}{d \\rho_f} c{(\\rho_f)} = 1 and \\frac{d}{d \\rho_f} \\rho_f \\frac{d}{d \\rho_f} c{(\\rho_f)} = \\frac{d}{d \\rho_f} \\rho_f and (\\frac{d}{d \\rho_f} \\rho_f)^{2} = \\frac{d}{d \\rho_f} \\rho_f and \\int (\\frac{d}{d \\rho_f} \\rho_f)^{2} d\\rho_f = \\int \\frac{d}{d \\rho_f} \\rho_f d\\rho_f and \\frac{\\int (\\frac{d}{d \\rho_f} \\rho_f)^{2} d\\rho_f}{\\rho_f c{(\\rho_f)}} = \\frac{\\int \\frac{d}{d \\rho_f} \\rho_f d\\rho_f}{\\rho_f c{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Function('c')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(2)), Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Pow(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 6, "Mul(Symbol('\\\\rho_f', commutative=True), Function('c')(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('c')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Integral(Pow(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Pow(Function('c')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Integral(Derivative(Symbol('\\\\rho_f', commutative=True), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\rho_b)} = \\log{(\\sin{(\\rho_b)})}, then obtain 2 \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} = \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} + \\frac{\\cos{(\\rho_b)}}{\\sin{(\\rho_b)}}", "derivation": "\\theta{(\\rho_b)} = \\log{(\\sin{(\\rho_b)})} and \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} = \\frac{d}{d \\rho_b} \\log{(\\sin{(\\rho_b)})} and 2 \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} = \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} + \\frac{d}{d \\rho_b} \\log{(\\sin{(\\rho_b)})} and 2 \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} = \\frac{d}{d \\rho_b} \\theta{(\\rho_b)} + \\frac{\\cos{(\\rho_b)}}{\\sin{(\\rho_b)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), log(sin(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(log(sin(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\theta')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Pow(sin(Symbol('\\\\rho_b', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(t_{1},\\rho_f)} = \\rho_f + t_{1}, then derive \\frac{\\partial}{\\partial \\rho_f} \\hat{H}_l{(t_{1},\\rho_f)} - 1 = 0, then obtain \\frac{\\partial}{\\partial \\rho_f} (\\rho_f + t_{1}) - 1 = 0", "derivation": "\\hat{H}_l{(t_{1},\\rho_f)} = \\rho_f + t_{1} and - \\rho_f - t_{1} + \\hat{H}_l{(t_{1},\\rho_f)} = 0 and \\frac{\\partial}{\\partial \\rho_f} (- \\rho_f - t_{1} + \\hat{H}_l{(t_{1},\\rho_f)}) = \\frac{d}{d \\rho_f} 0 and \\frac{\\partial}{\\partial \\rho_f} \\hat{H}_l{(t_{1},\\rho_f)} - 1 = 0 and \\frac{\\partial}{\\partial \\rho_f} (\\rho_f + t_{1}) - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('t_1', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\rho_f', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Symbol('t_1', commutative=True)), Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Add(Symbol('\\\\rho_f', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\mathbf{A}{(l,\\theta)} = \\int (\\theta - l) d\\theta, then obtain \\frac{\\partial}{\\partial \\theta} \\frac{\\int \\mathbf{A}{(l,\\theta)} d\\theta}{\\theta - l} = \\frac{\\partial}{\\partial \\theta} \\frac{\\iint (\\theta - l) d\\theta d\\theta}{\\theta - l}", "derivation": "\\mathbf{A}{(l,\\theta)} = \\int (\\theta - l) d\\theta and \\int \\mathbf{A}{(l,\\theta)} d\\theta = \\iint (\\theta - l) d\\theta d\\theta and \\frac{\\int \\mathbf{A}{(l,\\theta)} d\\theta}{\\theta - l} = \\frac{\\iint (\\theta - l) d\\theta d\\theta}{\\theta - l} and \\frac{\\partial}{\\partial \\theta} \\frac{\\int \\mathbf{A}{(l,\\theta)} d\\theta}{\\theta - l} = \\frac{\\partial}{\\partial \\theta} \\frac{\\iint (\\theta - l) d\\theta d\\theta}{\\theta - l}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Integral(Function('\\\\mathbf{A}')(Symbol('l', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(a,\\delta)} = \\delta - a, then obtain - \\delta + a + S{(a,\\delta)} \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}} = - \\delta + a + (\\delta - a) \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}}", "derivation": "S{(a,\\delta)} = \\delta - a and e^{S{(a,\\delta)}} = e^{\\delta - a} and S{(a,\\delta)} \\frac{\\partial}{\\partial \\delta} e^{\\delta - a} = (\\delta - a) \\frac{\\partial}{\\partial \\delta} e^{\\delta - a} and S{(a,\\delta)} \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}} = (\\delta - a) \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}} and - \\delta + a + S{(a,\\delta)} \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}} = - \\delta + a + (\\delta - a) \\frac{\\partial}{\\partial \\delta} e^{S{(a,\\delta)}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["exp", 1], "Equality(exp(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True))), exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))))"], [["times", 1, "Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Mul(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Derivative(exp(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(exp(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Derivative(exp(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["minus", 4, "Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a', commutative=True), Mul(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(exp(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('a', commutative=True), Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))), Derivative(exp(Function('S')(Symbol('a', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\lambda,\\Omega)} = \\Omega^{\\lambda}, then obtain \\lambda \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\eta^{\\prime}{(\\lambda,\\Omega)}) d\\lambda = \\lambda \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\Omega^{\\lambda}) d\\lambda", "derivation": "\\eta^{\\prime}{(\\lambda,\\Omega)} = \\Omega^{\\lambda} and \\frac{\\partial}{\\partial \\lambda} \\eta^{\\prime}{(\\lambda,\\Omega)} = \\frac{\\partial}{\\partial \\lambda} \\Omega^{\\lambda} and \\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\eta^{\\prime}{(\\lambda,\\Omega)} = \\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\Omega^{\\lambda} and \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\eta^{\\prime}{(\\lambda,\\Omega)}) d\\lambda = \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\Omega^{\\lambda}) d\\lambda and \\lambda \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\eta^{\\prime}{(\\lambda,\\Omega)}) d\\lambda = \\lambda \\int (\\eta^{\\prime}{(\\lambda,\\Omega)} + \\frac{\\partial}{\\partial \\lambda} \\Omega^{\\lambda}) d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["times", 4, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(Symbol('\\\\lambda', commutative=True), Integral(Add(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\Omega', commutative=True)), Derivative(Pow(Symbol('\\\\Omega', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(l)} = \\frac{d}{d l} \\sin{(l)}, then derive \\operatorname{n_{2}}{(l)} = \\cos{(l)}, then obtain 1 - \\sin{(l)} = - \\sin{(l)} + \\frac{\\cos{(l)}}{\\operatorname{n_{2}}{(l)}}", "derivation": "\\operatorname{n_{2}}{(l)} = \\frac{d}{d l} \\sin{(l)} and 1 = \\frac{\\frac{d}{d l} \\sin{(l)}}{\\operatorname{n_{2}}{(l)}} and \\operatorname{n_{2}}{(l)} = \\cos{(l)} and 1 = \\frac{\\frac{d}{d l} \\sin{(l)}}{\\cos{(l)}} and 1 - \\sin{(l)} = - \\sin{(l)} + \\frac{\\frac{d}{d l} \\sin{(l)}}{\\cos{(l)}} and 1 - \\sin{(l)} = - \\sin{(l)} + \\frac{\\frac{d}{d l} \\sin{(l)}}{\\operatorname{n_{2}}{(l)}} and 1 - \\sin{(l)} = - \\sin{(l)} + \\frac{\\cos{(l)}}{\\operatorname{n_{2}}{(l)}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('l', commutative=True)), Derivative(sin(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 1, "Function('n_2')(Symbol('l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('n_2')(Symbol('l', commutative=True)), Integer(-1)), Derivative(sin(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('n_2')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(1), Mul(Pow(cos(Symbol('l', commutative=True)), Integer(-1)), Derivative(sin(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["minus", 4, "sin(Symbol('l', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), sin(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Pow(cos(Symbol('l', commutative=True)), Integer(-1)), Derivative(sin(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integer(1), Mul(Integer(-1), sin(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Pow(Function('n_2')(Symbol('l', commutative=True)), Integer(-1)), Derivative(sin(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 6], "Equality(Add(Integer(1), Mul(Integer(-1), sin(Symbol('l', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('l', commutative=True))), Mul(Pow(Function('n_2')(Symbol('l', commutative=True)), Integer(-1)), cos(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given C{(z^{*})} = \\frac{d}{d z^{*}} e^{z^{*}} and \\delta{(z^{*})} = e^{z^{*}}, then obtain e^{2 (C{(z^{*})} + \\frac{d}{d z^{*}} \\delta{(z^{*})}) \\frac{d}{d z^{*}} \\delta{(z^{*})}} = e^{4 (\\frac{d}{d z^{*}} \\delta{(z^{*})})^{2}}", "derivation": "C{(z^{*})} = \\frac{d}{d z^{*}} e^{z^{*}} and C{(z^{*})} + \\frac{d}{d z^{*}} e^{z^{*}} = 2 \\frac{d}{d z^{*}} e^{z^{*}} and 2 (C{(z^{*})} + \\frac{d}{d z^{*}} e^{z^{*}}) \\frac{d}{d z^{*}} e^{z^{*}} = 4 (\\frac{d}{d z^{*}} e^{z^{*}})^{2} and e^{2 (C{(z^{*})} + \\frac{d}{d z^{*}} e^{z^{*}}) \\frac{d}{d z^{*}} e^{z^{*}}} = e^{4 (\\frac{d}{d z^{*}} e^{z^{*}})^{2}} and \\delta{(z^{*})} = e^{z^{*}} and e^{2 (C{(z^{*})} + \\frac{d}{d z^{*}} \\delta{(z^{*})}) \\frac{d}{d z^{*}} \\delta{(z^{*})}} = e^{4 (\\frac{d}{d z^{*}} \\delta{(z^{*})})^{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('z^*', commutative=True)), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["add", 1, "Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))"], "Equality(Add(Function('C')(Symbol('z^*', commutative=True)), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["times", 2, "Mul(Integer(2), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), Add(Function('C')(Symbol('z^*', commutative=True)), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Integer(4), Pow(Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(2))))"], [["exp", 3], "Equality(exp(Mul(Integer(2), Add(Function('C')(Symbol('z^*', commutative=True)), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), exp(Mul(Integer(4), Pow(Derivative(exp(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(2)))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('z^*', commutative=True)), exp(Symbol('z^*', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(exp(Mul(Integer(2), Add(Function('C')(Symbol('z^*', commutative=True)), Derivative(Function('\\\\delta')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Derivative(Function('\\\\delta')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), exp(Mul(Integer(4), Pow(Derivative(Function('\\\\delta')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Integer(2)))))"]]}, {"prompt": "Given W{(F_{g})} = \\log{(\\log{(F_{g})})}, then obtain - F_{g} (- F_{g} + W{(F_{g})} \\log{(F_{g})}) = - F_{g} (- F_{g} + \\log{(F_{g})} \\log{(\\log{(F_{g})})})", "derivation": "W{(F_{g})} = \\log{(\\log{(F_{g})})} and W{(F_{g})} \\log{(F_{g})} = \\log{(F_{g})} \\log{(\\log{(F_{g})})} and - F_{g} + W{(F_{g})} \\log{(F_{g})} = - F_{g} + \\log{(F_{g})} \\log{(\\log{(F_{g})})} and - F_{g} (- F_{g} + W{(F_{g})} \\log{(F_{g})}) = - F_{g} (- F_{g} + \\log{(F_{g})} \\log{(\\log{(F_{g})})})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('F_g', commutative=True)), log(log(Symbol('F_g', commutative=True))))"], [["times", 1, "log(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('W')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))), Mul(log(Symbol('F_g', commutative=True)), log(log(Symbol('F_g', commutative=True)))))"], [["minus", 2, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Function('W')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(log(Symbol('F_g', commutative=True)), log(log(Symbol('F_g', commutative=True))))))"], [["times", 3, "Mul(Integer(-1), Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(Function('W')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True))))), Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Mul(log(Symbol('F_g', commutative=True)), log(log(Symbol('F_g', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{X},M_{E})} = \\frac{\\partial}{\\partial \\hat{X}} (M_{E} + \\hat{X}), then derive \\operatorname{E_{n}}{(\\hat{X},M_{E})} = 1, then obtain \\hat{X} e^{\\frac{\\partial}{\\partial \\hat{X}} (M_{E} + \\hat{X})} = e \\hat{X}", "derivation": "\\operatorname{E_{n}}{(\\hat{X},M_{E})} = \\frac{\\partial}{\\partial \\hat{X}} (M_{E} + \\hat{X}) and \\operatorname{E_{n}}{(\\hat{X},M_{E})} = 1 and e^{\\operatorname{E_{n}}{(\\hat{X},M_{E})}} = e and e^{\\frac{\\partial}{\\partial \\hat{X}} (M_{E} + \\hat{X})} = e and \\hat{X} e^{\\frac{\\partial}{\\partial \\hat{X}} (M_{E} + \\hat{X})} = e \\hat{X}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{X}', commutative=True), Symbol('M_E', commutative=True)), Derivative(Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('E_n')(Symbol('\\\\hat{X}', commutative=True), Symbol('M_E', commutative=True)), Integer(1))"], [["exp", 2], "Equality(exp(Function('E_n')(Symbol('\\\\hat{X}', commutative=True), Symbol('M_E', commutative=True))), E)"], [["substitute_LHS_for_RHS", 3, 1], "Equality(exp(Derivative(Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), E)"], [["times", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), exp(Derivative(Add(Symbol('M_E', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))), Mul(E, Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(Z,\\tilde{g}^*)} = Z - \\tilde{g}^* and \\pi{(\\hat{H})} = \\sin{(\\sin{(\\hat{H})})}, then obtain \\frac{\\int \\pi{(\\hat{H})} d\\hat{H}}{\\operatorname{A_{2}}{(Z,\\tilde{g}^*)}} = \\frac{\\int \\sin{(\\sin{(\\hat{H})})} d\\hat{H}}{\\operatorname{A_{2}}{(Z,\\tilde{g}^*)}}", "derivation": "\\operatorname{A_{2}}{(Z,\\tilde{g}^*)} = Z - \\tilde{g}^* and \\tilde{g}^* + \\operatorname{A_{2}}{(Z,\\tilde{g}^*)} = Z and \\pi{(\\hat{H})} = \\sin{(\\sin{(\\hat{H})})} and \\int \\pi{(\\hat{H})} d\\hat{H} = \\int \\sin{(\\sin{(\\hat{H})})} d\\hat{H} and \\frac{\\int \\pi{(\\hat{H})} d\\hat{H}}{Z - \\tilde{g}^*} = \\frac{\\int \\sin{(\\sin{(\\hat{H})})} d\\hat{H}}{Z - \\tilde{g}^*} and \\frac{\\int \\pi{(\\hat{H})} d\\hat{H}}{\\operatorname{A_{2}}{(Z,\\tilde{g}^*)}} = \\frac{\\int \\sin{(\\sin{(\\hat{H})})} d\\hat{H}}{\\operatorname{A_{2}}{(Z,\\tilde{g}^*)}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('A_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('Z', commutative=True))"], ["get_premise", "Equality(Function('\\\\pi')(Symbol('\\\\hat{H}', commutative=True)), sin(sin(Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(sin(sin(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 4, "Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Integral(Function('\\\\pi')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)), Integral(sin(sin(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('A_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Integral(Function('\\\\pi')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))), Mul(Pow(Function('A_2')(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Integral(sin(sin(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(h,W)} = h^{W} and \\operatorname{v_{2}}{(h,W)} = (\\frac{\\partial}{\\partial h} h^{W})^{h}, then derive \\frac{\\partial}{\\partial h} \\mathbf{E}{(h,W)} = \\frac{W h^{W}}{h}, then obtain \\frac{\\partial}{\\partial W} \\operatorname{v_{2}}{(h,W)} = \\frac{\\partial}{\\partial W} (\\frac{W h^{W}}{h})^{h}", "derivation": "\\mathbf{E}{(h,W)} = h^{W} and \\frac{\\partial}{\\partial h} \\mathbf{E}{(h,W)} = \\frac{\\partial}{\\partial h} h^{W} and \\frac{\\partial}{\\partial h} \\mathbf{E}{(h,W)} = \\frac{W h^{W}}{h} and \\frac{\\partial}{\\partial h} h^{W} = \\frac{W h^{W}}{h} and (\\frac{\\partial}{\\partial h} h^{W})^{h} = (\\frac{W h^{W}}{h})^{h} and \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial h} h^{W})^{h} = \\frac{\\partial}{\\partial W} (\\frac{W h^{W}}{h})^{h} and \\operatorname{v_{2}}{(h,W)} = (\\frac{\\partial}{\\partial h} h^{W})^{h} and \\frac{\\partial}{\\partial W} \\operatorname{v_{2}}{(h,W)} = \\frac{\\partial}{\\partial W} (\\frac{W h^{W}}{h})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('W', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Mul(Symbol('W', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True))))"], [["power", 4, "Symbol('h', commutative=True)"], "Equality(Pow(Derivative(Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True))), Symbol('h', commutative=True)))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Derivative(Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True))), Symbol('h', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_2')(Symbol('h', commutative=True), Symbol('W', commutative=True)), Pow(Derivative(Pow(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Function('v_2')(Symbol('h', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('W', commutative=True))), Symbol('h', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(v_{z})} = \\log{(v_{z})}, then obtain - \\log{(v_{z})} = - 2 r{(v_{z})} + \\log{(v_{z})}", "derivation": "r{(v_{z})} = \\log{(v_{z})} and 0 = - r{(v_{z})} + \\log{(v_{z})} and - r{(v_{z})} = - 2 r{(v_{z})} + \\log{(v_{z})} and 0 = - 2 r{(v_{z})} + 2 \\log{(v_{z})} and - \\log{(v_{z})} = - 2 r{(v_{z})} + \\log{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["minus", 1, "Function('r')(Symbol('v_z', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('r')(Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Function('r')(Symbol('v_z', commutative=True)))"], "Equality(Mul(Integer(-1), Function('r')(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('r')(Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('r')(Symbol('v_z', commutative=True))), Mul(Integer(2), log(Symbol('v_z', commutative=True)))))"], [["minus", 4, "log(Symbol('v_z', commutative=True))"], "Equality(Mul(Integer(-1), log(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('r')(Symbol('v_z', commutative=True))), log(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} = L_{\\varepsilon} + \\ddot{x}, then obtain \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon} = \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int (L_{\\varepsilon} + \\ddot{x}) dL_{\\varepsilon} dL_{\\varepsilon}", "derivation": "\\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} = L_{\\varepsilon} + \\ddot{x} and \\int \\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} dL_{\\varepsilon} = \\int (L_{\\varepsilon} + \\ddot{x}) dL_{\\varepsilon} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} dL_{\\varepsilon} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int (L_{\\varepsilon} + \\ddot{x}) dL_{\\varepsilon} and \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int \\operatorname{z^{*}}{(\\ddot{x},L_{\\varepsilon})} dL_{\\varepsilon} dL_{\\varepsilon} = \\int \\frac{\\partial}{\\partial L_{\\varepsilon}} \\int (L_{\\varepsilon} + \\ddot{x}) dL_{\\varepsilon} dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\ddot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\ddot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Function('z^*')(Symbol('\\\\ddot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('z^*')(Symbol('\\\\ddot{x}', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Derivative(Integral(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given I{(a,\\hat{H}_l,i)} = \\hat{H}_l + a - i, then obtain \\hat{H}_l + a + \\frac{(i + I{(a,\\hat{H}_l,i)}) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i} = \\hat{H}_l + a + \\frac{(\\hat{H}_l + a) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i}", "derivation": "I{(a,\\hat{H}_l,i)} = \\hat{H}_l + a - i and i + I{(a,\\hat{H}_l,i)} = \\hat{H}_l + a and \\frac{i + I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i} = \\frac{\\hat{H}_l + a}{\\hat{H}_l + a - i} and \\frac{(i + I{(a,\\hat{H}_l,i)}) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i} = \\frac{(\\hat{H}_l + a) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i} and \\hat{H}_l + a + \\frac{(i + I{(a,\\hat{H}_l,i)}) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i} = \\hat{H}_l + a + \\frac{(\\hat{H}_l + a) I{(a,\\hat{H}_l,i)}}{\\hat{H}_l + a - i}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True)), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Add(Symbol('i', commutative=True), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True)))"], [["divide", 2, "Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)))"], "Equality(Mul(Add(Symbol('i', commutative=True), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1))), Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1))))"], [["times", 3, "Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Add(Symbol('i', commutative=True), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))), Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))))"], [["add", 4, "Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Add(Symbol('i', commutative=True), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True))), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True)))), Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True))), Integer(-1)), Function('I')(Symbol('a', commutative=True), Symbol('\\\\hat{H}_l', commutative=True), Symbol('i', commutative=True)))))"]]}, {"prompt": "Given S{(\\hat{p},E)} = E \\hat{p}, then obtain (E \\hat{p})^{E} + ((E \\hat{p})^{- E} S{(\\hat{p},E)})^{E} = (E \\hat{p})^{E} + (E \\hat{p} (E \\hat{p})^{- E})^{E}", "derivation": "S{(\\hat{p},E)} = E \\hat{p} and S^{E}{(\\hat{p},E)} = (E \\hat{p})^{E} and S{(\\hat{p},E)} S^{- E}{(\\hat{p},E)} = E \\hat{p} S^{- E}{(\\hat{p},E)} and (S{(\\hat{p},E)} S^{- E}{(\\hat{p},E)})^{E} = (E \\hat{p} S^{- E}{(\\hat{p},E)})^{E} and (S{(\\hat{p},E)} S^{- E}{(\\hat{p},E)})^{E} + S^{E}{(\\hat{p},E)} = (E \\hat{p} S^{- E}{(\\hat{p},E)})^{E} + S^{E}{(\\hat{p},E)} and (E \\hat{p})^{E} + ((E \\hat{p})^{- E} S{(\\hat{p},E)})^{E} = (E \\hat{p})^{E} + (E \\hat{p} (E \\hat{p})^{- E})^{E}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E', commutative=True)))"], [["divide", 1, "Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True))"], "Equality(Mul(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)))"], [["add", 4, "Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True))"], "Equality(Add(Pow(Mul(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True))), Add(Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Pow(Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True)), Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True))), Function('S')(Symbol('\\\\hat{p}', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))), Add(Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Symbol('E', commutative=True)), Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True), Pow(Mul(Symbol('E', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))), Symbol('E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(u,\\mathbf{F})} = \\mathbf{F} + u, then derive \\int \\frac{\\mathbf{S}{(u,\\mathbf{F})}}{u} du = J_{\\varepsilon} + \\mathbf{F} \\log{(u)} + u, then obtain F_{H} + \\mathbf{F} \\log{(u)} + u = J_{\\varepsilon} + \\mathbf{F} \\log{(u)} + u", "derivation": "\\mathbf{S}{(u,\\mathbf{F})} = \\mathbf{F} + u and \\frac{\\mathbf{S}{(u,\\mathbf{F})}}{u} = \\frac{\\mathbf{F} + u}{u} and \\int \\frac{\\mathbf{S}{(u,\\mathbf{F})}}{u} du = \\int \\frac{\\mathbf{F} + u}{u} du and \\int \\frac{\\mathbf{S}{(u,\\mathbf{F})}}{u} du = J_{\\varepsilon} + \\mathbf{F} \\log{(u)} + u and \\int \\frac{\\mathbf{F} + u}{u} du = J_{\\varepsilon} + \\mathbf{F} \\log{(u)} + u and F_{H} + \\mathbf{F} \\log{(u)} + u = J_{\\varepsilon} + \\mathbf{F} \\log{(u)} + u", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('u', commutative=True)))"], [["divide", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('u', commutative=True))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{S}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_H', commutative=True), Mul(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{F}', commutative=True), log(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given J{(\\phi_2,B)} = B \\phi_2, then obtain \\phi_2 (- \\phi_2 + \\frac{J{(\\phi_2,B)}}{B \\phi_2}) = \\phi_2 (1 - \\phi_2)", "derivation": "J{(\\phi_2,B)} = B \\phi_2 and \\frac{J{(\\phi_2,B)}}{B \\phi_2} = 1 and - \\phi_2 + \\frac{J{(\\phi_2,B)}}{B \\phi_2} = 1 - \\phi_2 and \\phi_2 (- \\phi_2 + \\frac{J{(\\phi_2,B)}}{B \\phi_2}) = \\phi_2 (1 - \\phi_2)", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\phi_2', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Mul(Symbol('B', commutative=True), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_2', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["minus", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_2', commutative=True), Symbol('B', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["times", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\phi_2', commutative=True), Symbol('B', commutative=True))))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{\\psi}, then obtain \\int \\frac{\\mathbf{E}^{2}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} d\\hat{\\mathbf{r}} = \\int \\frac{\\hat{\\mathbf{r}}^{\\psi} \\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} d\\hat{\\mathbf{r}}", "derivation": "\\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{\\psi} and \\mathbf{E}^{2}{(\\psi,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{\\psi} \\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})} and \\frac{\\mathbf{E}^{2}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{\\hat{\\mathbf{r}}^{\\psi} \\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} and \\int \\frac{\\mathbf{E}^{2}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} d\\hat{\\mathbf{r}} = \\int \\frac{\\hat{\\mathbf{r}}^{\\psi} \\mathbf{E}{(\\psi,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2)), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\psi', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and v{(\\mathbf{J}_f)} = \\operatorname{P_{e}}{(\\mathbf{J}_f)} - 1, then obtain \\frac{v{(\\mathbf{J}_f)}}{\\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}} = \\frac{e^{\\mathbf{J}_f} - 1}{\\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} and v{(\\mathbf{J}_f)} = \\operatorname{P_{e}}{(\\mathbf{J}_f)} - 1 and v{(\\mathbf{J}_f)} = e^{\\mathbf{J}_f} - 1 and \\frac{v{(\\mathbf{J}_f)}}{\\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}} = \\frac{e^{\\mathbf{J}_f} - 1}{\\frac{d}{d \\mathbf{J}_f} e^{\\mathbf{J}_f}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('\\\\mathbf{J}_f', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Function('P_e')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Add(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)))"], [["divide", 3, "Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))"], "Equality(Mul(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), Pow(Derivative(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given b{(\\hat{p},\\psi^*)} = e^{\\hat{p} \\psi^*}, then obtain - \\hat{p} \\psi^* + b{(\\hat{p},\\psi^*)} e^{2 \\hat{p} \\psi^*} = - \\hat{p} \\psi^* + e^{3 \\hat{p} \\psi^*}", "derivation": "b{(\\hat{p},\\psi^*)} = e^{\\hat{p} \\psi^*} and b{(\\hat{p},\\psi^*)} e^{\\hat{p} \\psi^*} = e^{2 \\hat{p} \\psi^*} and b{(\\hat{p},\\psi^*)} e^{2 \\hat{p} \\psi^*} = e^{3 \\hat{p} \\psi^*} and - \\hat{p} \\psi^* + b{(\\hat{p},\\psi^*)} e^{2 \\hat{p} \\psi^*} = - \\hat{p} \\psi^* + e^{3 \\hat{p} \\psi^*}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["times", 1, "exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Function('b')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))), exp(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["times", 2, "exp(Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Function('b')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))), exp(Mul(Integer(3), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Function('b')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Mul(Integer(3), Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(C_{d},H)} = C_{d}^{H}, then derive \\frac{\\partial^{2}}{\\partial H^{2}} \\sigma_{x}{(C_{d},H)} = C_{d}^{H} \\log{(C_{d})}^{2}, then obtain (\\frac{\\partial^{2}}{\\partial H^{2}} \\sigma_{x}{(C_{d},H)})^{H} = (C_{d}^{H} \\log{(C_{d})}^{2})^{H}", "derivation": "\\sigma_{x}{(C_{d},H)} = C_{d}^{H} and \\frac{\\partial}{\\partial H} \\sigma_{x}{(C_{d},H)} = \\frac{\\partial}{\\partial H} C_{d}^{H} and \\frac{\\partial}{\\partial H} \\sigma_{x}{(C_{d},H)} - 1 = \\frac{\\partial}{\\partial H} C_{d}^{H} - 1 and \\frac{\\partial}{\\partial H} (\\frac{\\partial}{\\partial H} \\sigma_{x}{(C_{d},H)} - 1) = \\frac{\\partial}{\\partial H} (\\frac{\\partial}{\\partial H} C_{d}^{H} - 1) and \\frac{\\partial^{2}}{\\partial H^{2}} \\sigma_{x}{(C_{d},H)} = C_{d}^{H} \\log{(C_{d})}^{2} and (\\frac{\\partial^{2}}{\\partial H^{2}} \\sigma_{x}{(C_{d},H)})^{H} = (C_{d}^{H} \\log{(C_{d})}^{2})^{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Derivative(Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Mul(Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(2))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\sigma_x')(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(2))), Symbol('H', commutative=True)), Pow(Mul(Pow(Symbol('C_d', commutative=True), Symbol('H', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(2))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(A_{z})} = \\sin{(A_{z})}, then obtain \\sin{(A_{z})} + \\frac{d}{d A_{z}} \\operatorname{y^{\\prime}}{(A_{z})} = \\sin{(A_{z})} + \\cos{(A_{z})}", "derivation": "\\operatorname{y^{\\prime}}{(A_{z})} = \\sin{(A_{z})} and \\frac{d}{d A_{z}} \\operatorname{y^{\\prime}}{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})} and \\sin{(A_{z})} + \\frac{d}{d A_{z}} \\operatorname{y^{\\prime}}{(A_{z})} = \\sin{(A_{z})} + \\frac{d}{d A_{z}} \\sin{(A_{z})} and \\sin{(A_{z})} + \\frac{d}{d A_{z}} \\operatorname{y^{\\prime}}{(A_{z})} = \\sin{(A_{z})} + \\cos{(A_{z})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["add", 2, "sin(Symbol('A_z', commutative=True))"], "Equality(Add(sin(Symbol('A_z', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(sin(Symbol('A_z', commutative=True)), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(sin(Symbol('A_z', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Add(sin(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbf{F})} = \\log{(\\mathbf{F})}, then derive \\frac{d}{d \\mathbf{F}} \\operatorname{r_{0}}{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}}, then obtain \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\frac{1}{\\mathbf{F}}", "derivation": "\\operatorname{r_{0}}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\operatorname{r_{0}}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\operatorname{r_{0}}{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}} and \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}} and \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\frac{1}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(S)} = \\cos{(S)}, then derive \\int \\Psi{(S)} dS = \\mathbf{r} + \\sin{(S)}, then derive \\Omega + \\sin{(S)} = \\mathbf{r} + \\sin{(S)}, then obtain (\\frac{\\iint \\Psi{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}})^{S} = (\\frac{\\iint \\cos{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}})^{S}", "derivation": "\\Psi{(S)} = \\cos{(S)} and \\int \\Psi{(S)} dS = \\int \\cos{(S)} dS and \\int \\Psi{(S)} dS = \\mathbf{r} + \\sin{(S)} and \\int \\cos{(S)} dS = \\mathbf{r} + \\sin{(S)} and \\Omega + \\sin{(S)} = \\mathbf{r} + \\sin{(S)} and \\iint \\Psi{(S)} dS dS = \\iint \\cos{(S)} dS dS and \\frac{\\iint \\Psi{(S)} dS dS}{\\Omega + \\sin{(S)}} = \\frac{\\iint \\cos{(S)} dS dS}{\\Omega + \\sin{(S)}} and \\frac{\\iint \\Psi{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}} = \\frac{\\iint \\cos{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}} and (\\frac{\\iint \\Psi{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}})^{S} = (\\frac{\\iint \\cos{(S)} dS dS}{\\mathbf{r} + \\sin{(S)}})^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('S', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["divide", 6, "Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('S', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Pow(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["power", 8, "Symbol('S', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(Function('\\\\Psi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\mathbf{r}', commutative=True), sin(Symbol('S', commutative=True))), Integer(-1)), Integral(cos(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\delta{(\\ddot{x})} = \\sin{(\\ddot{x})}, then derive \\int \\delta{(\\ddot{x})} d\\ddot{x} = l - \\cos{(\\ddot{x})}, then obtain \\frac{l - \\cos{(\\ddot{x})}}{(\\int \\delta{(\\ddot{x})} d\\ddot{x})^{2}} = \\frac{\\int \\sin{(\\ddot{x})} d\\ddot{x}}{(\\int \\delta{(\\ddot{x})} d\\ddot{x})^{2}}", "derivation": "\\delta{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\int \\delta{(\\ddot{x})} d\\ddot{x} = \\int \\sin{(\\ddot{x})} d\\ddot{x} and \\int \\delta{(\\ddot{x})} d\\ddot{x} = l - \\cos{(\\ddot{x})} and l - \\cos{(\\ddot{x})} = \\int \\sin{(\\ddot{x})} d\\ddot{x} and \\frac{l - \\cos{(\\ddot{x})}}{(\\int \\delta{(\\ddot{x})} d\\ddot{x})^{2}} = \\frac{\\int \\sin{(\\ddot{x})} d\\ddot{x}}{(\\int \\delta{(\\ddot{x})} d\\ddot{x})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["divide", 4, "Pow(Integral(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(2))"], "Equality(Mul(Add(Symbol('l', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Pow(Integral(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-2))), Mul(Pow(Integral(Function('\\\\delta')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integer(-2)), Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"]]}, {"prompt": "Given l{(M,A_{y})} = A_{y} M, then derive A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})} = A_{y}^{2}, then obtain - A_{y} M = - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} A_{y} M + A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})}", "derivation": "l{(M,A_{y})} = A_{y} M and \\frac{\\partial}{\\partial M} l{(M,A_{y})} = \\frac{\\partial}{\\partial M} A_{y} M and A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})} = A_{y} \\frac{\\partial}{\\partial M} A_{y} M and A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})} = A_{y}^{2} and - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} A_{y} M + A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})} = A_{y}^{2} - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} A_{y} M and - A_{y} M = A_{y}^{2} - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})} and - A_{y} M = A_{y}^{2} - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} A_{y} M and - A_{y} M = - A_{y} M - A_{y} \\frac{\\partial}{\\partial M} A_{y} M + A_{y} \\frac{\\partial}{\\partial M} l{(M,A_{y})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["times", 2, "Symbol('A_y', commutative=True)"], "Equality(Mul(Symbol('A_y', commutative=True), Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('A_y', commutative=True), Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Pow(Symbol('A_y', commutative=True), Integer(2)))"], [["minus", 4, "Add(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Symbol('A_y', commutative=True), Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))), Add(Pow(Symbol('A_y', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Pow(Symbol('A_y', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True), Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Pow(Symbol('A_y', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Mul(Symbol('A_y', commutative=True), Derivative(Function('l')(Symbol('M', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"]]}, {"prompt": "Given c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} = f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})}, then obtain - \\hat{\\mathbf{r}}{(Z)} + \\iint c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}} = - \\hat{\\mathbf{r}}{(Z)} + \\iint f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}}", "derivation": "c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} = f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})} and \\int c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} df_{\\mathbf{p}} = \\int f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})} df_{\\mathbf{p}} and \\iint c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}} and - \\hat{\\mathbf{r}}{(Z)} + \\iint c{(f_{\\mathbf{p}},\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}} = - \\hat{\\mathbf{r}}{(Z)} + \\iint f_{\\mathbf{p}} \\cos{(\\hat{\\mathbf{r}})} df_{\\mathbf{p}} df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 3, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Z', commutative=True))), Integral(Function('c')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('Z', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"]]}, {"prompt": "Given u{(\\hat{H},\\phi)} = \\phi^{\\hat{H}}, then obtain \\frac{\\partial^{2}}{\\partial \\hat{H}\\partial \\phi} \\frac{u{(\\hat{H},\\phi)}}{\\phi} = \\frac{\\partial^{2}}{\\partial \\hat{H}\\partial \\phi} \\frac{\\phi^{\\hat{H}}}{\\phi}", "derivation": "u{(\\hat{H},\\phi)} = \\phi^{\\hat{H}} and \\frac{u{(\\hat{H},\\phi)}}{\\phi} = \\frac{\\phi^{\\hat{H}}}{\\phi} and \\frac{\\partial}{\\partial \\phi} \\frac{u{(\\hat{H},\\phi)}}{\\phi} = \\frac{\\partial}{\\partial \\phi} \\frac{\\phi^{\\hat{H}}}{\\phi} and \\frac{\\partial^{2}}{\\partial \\hat{H}\\partial \\phi} \\frac{u{(\\hat{H},\\phi)}}{\\phi} = \\frac{\\partial^{2}}{\\partial \\hat{H}\\partial \\phi} \\frac{\\phi^{\\hat{H}}}{\\phi}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)}, then obtain (\\cos{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\operatorname{a^{\\dagger}}{(\\hat{p}_0)})^{\\hat{p}_0} = (2 \\cos{(\\hat{p}_0)})^{\\hat{p}_0}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{p}_0)} = \\sin{(\\hat{p}_0)} and \\operatorname{a^{\\dagger}}{(\\hat{p}_0)} + \\sin{(\\hat{p}_0)} = 2 \\sin{(\\hat{p}_0)} and \\frac{d}{d \\hat{p}_0} (\\operatorname{a^{\\dagger}}{(\\hat{p}_0)} + \\sin{(\\hat{p}_0)}) = \\frac{d}{d \\hat{p}_0} 2 \\sin{(\\hat{p}_0)} and (\\frac{d}{d \\hat{p}_0} (\\operatorname{a^{\\dagger}}{(\\hat{p}_0)} + \\sin{(\\hat{p}_0)}))^{\\hat{p}_0} = (\\frac{d}{d \\hat{p}_0} 2 \\sin{(\\hat{p}_0)})^{\\hat{p}_0} and (\\cos{(\\hat{p}_0)} + \\frac{d}{d \\hat{p}_0} \\operatorname{a^{\\dagger}}{(\\hat{p}_0)})^{\\hat{p}_0} = (2 \\cos{(\\hat{p}_0)})^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\hat{p}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Derivative(Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}_0', commutative=True)), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Mul(Integer(2), sin(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(cos(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Mul(Integer(2), cos(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given E{(\\mathbf{g})} = \\mathbf{g}, then derive \\frac{\\partial}{\\partial \\mathbf{g}} (n + \\frac{E^{2}{(\\mathbf{g})}}{2}) = \\frac{\\partial}{\\partial \\mathbf{g}} \\int \\mathbf{g} dE{(\\mathbf{g})}, then obtain \\mathbf{g} = \\frac{d}{d \\mathbf{g}} \\int \\mathbf{g} d\\mathbf{g}", "derivation": "E{(\\mathbf{g})} = \\mathbf{g} and \\int E{(\\mathbf{g})} d\\mathbf{g} = \\int \\mathbf{g} d\\mathbf{g} and \\int E{(\\mathbf{g})} dE{(\\mathbf{g})} = \\int \\mathbf{g} dE{(\\mathbf{g})} and \\frac{\\partial}{\\partial \\mathbf{g}} \\int E{(\\mathbf{g})} dE{(\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} \\int \\mathbf{g} dE{(\\mathbf{g})} and \\frac{\\partial}{\\partial \\mathbf{g}} (n + \\frac{E^{2}{(\\mathbf{g})}}{2}) = \\frac{\\partial}{\\partial \\mathbf{g}} \\int \\mathbf{g} dE{(\\mathbf{g})} and \\frac{\\partial}{\\partial \\mathbf{g}} (\\frac{\\mathbf{g}^{2}}{2} + n) = \\frac{d}{d \\mathbf{g}} \\int \\mathbf{g} d\\mathbf{g} and \\mathbf{g} = \\frac{d}{d \\mathbf{g}} \\int \\mathbf{g} d\\mathbf{g}", "srepr_derivation": [["renaming_premise", "Equality(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)))), Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Integral(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('n', commutative=True), Mul(Rational(1, 2), Pow(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Function('E')(Symbol('\\\\mathbf{g}', commutative=True)))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Symbol('n', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Integral(Symbol('\\\\mathbf{g}', commutative=True), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\omega)} = \\sin{(\\omega)} and \\psi^{*}{(\\omega)} = 1 - \\dot{y}{(\\omega)}, then obtain e^{\\frac{\\dot{y}{(\\omega)}}{\\sin{(\\omega)}} + \\psi^{*}{(\\omega)} - 1} = e^{\\psi^{*}{(\\omega)}}", "derivation": "\\dot{y}{(\\omega)} = \\sin{(\\omega)} and \\frac{\\dot{y}{(\\omega)}}{\\sin{(\\omega)}} = 1 and - \\dot{y}{(\\omega)} + \\frac{\\dot{y}{(\\omega)}}{\\sin{(\\omega)}} = 1 - \\dot{y}{(\\omega)} and e^{- \\dot{y}{(\\omega)} + \\frac{\\dot{y}{(\\omega)}}{\\sin{(\\omega)}}} = e^{1 - \\dot{y}{(\\omega)}} and \\psi^{*}{(\\omega)} = 1 - \\dot{y}{(\\omega)} and e^{\\frac{\\dot{y}{(\\omega)}}{\\sin{(\\omega)}} + \\psi^{*}{(\\omega)} - 1} = e^{\\psi^{*}{(\\omega)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True))), Mul(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True))), Mul(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))))), exp(Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\omega', commutative=True)), Add(Integer(1), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(exp(Add(Mul(Function('\\\\dot{y}')(Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Integer(-1))), Function('\\\\psi^*')(Symbol('\\\\omega', commutative=True)), Integer(-1))), exp(Function('\\\\psi^*')(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}}, then obtain \\frac{3 \\operatorname{P_{e}}{(\\Psi_{\\lambda})} - 4 e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} + e^{\\Psi_{\\lambda}}} = - \\frac{- \\operatorname{P_{e}}{(\\Psi_{\\lambda})} + 2 e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} - \\operatorname{P_{e}}{(\\Psi_{\\lambda})} + 2 e^{\\Psi_{\\lambda}}}", "derivation": "\\operatorname{P_{e}}{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\operatorname{P_{e}}{(\\Psi_{\\lambda})} - e^{\\Psi_{\\lambda}} = 0 and \\operatorname{P_{e}}{(\\Psi_{\\lambda})} - 2 e^{\\Psi_{\\lambda}} = - e^{\\Psi_{\\lambda}} and \\frac{\\operatorname{P_{e}}{(\\Psi_{\\lambda})} - 2 e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} + e^{\\Psi_{\\lambda}}} = - \\frac{e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} + e^{\\Psi_{\\lambda}}} and \\frac{3 \\operatorname{P_{e}}{(\\Psi_{\\lambda})} - 4 e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} + e^{\\Psi_{\\lambda}}} = - \\frac{- \\operatorname{P_{e}}{(\\Psi_{\\lambda})} + 2 e^{\\Psi_{\\lambda}}}{\\Psi_{\\lambda} - \\operatorname{P_{e}}{(\\Psi_{\\lambda})} + 2 e^{\\Psi_{\\lambda}}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(0))"], [["add", 2, "Mul(Integer(-1), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Add(Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Mul(Integer(-1), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Add(Mul(Integer(3), Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Integer(4), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Mul(Integer(-1), Function('P_e')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given f{(r,C)} = C - r, then derive (\\frac{\\partial}{\\partial C} f{(r,C)})^{r} = 1, then obtain \\cos{((\\frac{\\partial}{\\partial C} (C - r))^{r})} - 1 = -1 + \\cos{(1)}", "derivation": "f{(r,C)} = C - r and \\frac{\\partial}{\\partial C} f{(r,C)} = \\frac{\\partial}{\\partial C} (C - r) and (\\frac{\\partial}{\\partial C} f{(r,C)})^{r} = (\\frac{\\partial}{\\partial C} (C - r))^{r} and (\\frac{\\partial}{\\partial C} f{(r,C)})^{r} = 1 and \\cos{((\\frac{\\partial}{\\partial C} f{(r,C)})^{r})} = \\cos{(1)} and \\cos{((\\frac{\\partial}{\\partial C} (C - r))^{r})} = \\cos{(1)} and \\cos{((\\frac{\\partial}{\\partial C} (C - r))^{r})} - 1 = -1 + \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True)), Integer(1))"], [["cos", 4], "Equality(cos(Pow(Derivative(Function('f')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True))), cos(Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(cos(Pow(Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True))), cos(Integer(1)))"], [["minus", 6, 1], "Equality(Add(cos(Pow(Derivative(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('r', commutative=True))), Integer(-1)), Add(Integer(-1), cos(Integer(1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})} = A_{y} \\dot{x}, then obtain \\int \\frac{\\dot{x}}{A_{y}} d\\dot{x} = \\int \\frac{\\dot{x}^{2}}{\\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})}} d\\dot{x}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})} = A_{y} \\dot{x} and A_{y} \\dot{x} \\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})} = A_{y}^{2} \\dot{x}^{2} and \\frac{\\dot{x} \\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})}}{A_{y}} = \\dot{x}^{2} and \\frac{\\dot{x}}{A_{y}} = \\frac{\\dot{x}^{2}}{\\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})}} and \\int \\frac{\\dot{x}}{A_{y}} d\\dot{x} = \\int \\frac{\\dot{x}^{2}}{\\operatorname{L_{\\varepsilon}}{(\\dot{x},A_{y})}} d\\dot{x}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "Mul(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Symbol('A_y', commutative=True), Symbol('\\\\dot{x}', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))), Mul(Pow(Symbol('A_y', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2))))"], [["divide", 2, "Pow(Symbol('A_y', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)))"], [["divide", 3, "Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True))"], "Equality(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)), Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))))"], [["integrate", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)), Pow(Function('L_{\\\\varepsilon}')(Symbol('\\\\dot{x}', commutative=True), Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\varphi^*)} = \\cos{(\\varphi^*)}, then derive \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = z^{*} + \\sin{(\\varphi^*)}, then obtain \\frac{d}{d \\varphi^*} 2 \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = \\frac{d}{d \\varphi^*} (\\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* + \\int \\cos{(\\varphi^*)} d\\varphi^*)", "derivation": "\\hat{H}_l{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = \\int \\cos{(\\varphi^*)} d\\varphi^* and \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = z^{*} + \\sin{(\\varphi^*)} and z^{*} + \\sin{(\\varphi^*)} + \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = z^{*} + \\sin{(\\varphi^*)} + \\int \\cos{(\\varphi^*)} d\\varphi^* and 2 \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* + \\int \\cos{(\\varphi^*)} d\\varphi^* and \\frac{d}{d \\varphi^*} 2 \\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* = \\frac{d}{d \\varphi^*} (\\int \\hat{H}_l{(\\varphi^*)} d\\varphi^* + \\int \\cos{(\\varphi^*)} d\\varphi^*)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 2, "Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Symbol('z^*', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True)), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Add(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Add(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(C)} = \\sin{(C)}, then obtain (f^{2}{(C)} - \\sin{(C)})^{2} = (f{(C)} \\sin{(C)} - \\sin{(C)})^{2}", "derivation": "f{(C)} = \\sin{(C)} and f^{2}{(C)} = f{(C)} \\sin{(C)} and f^{2}{(C)} - \\sin{(C)} = f{(C)} \\sin{(C)} - \\sin{(C)} and (f^{2}{(C)} - \\sin{(C)})^{2} = (f{(C)} \\sin{(C)} - \\sin{(C)})^{2}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True)))"], [["times", 1, "Function('f')(Symbol('C', commutative=True))"], "Equality(Pow(Function('f')(Symbol('C', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))))"], [["minus", 2, "sin(Symbol('C', commutative=True))"], "Equality(Add(Pow(Function('f')(Symbol('C', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('C', commutative=True)))), Add(Mul(Function('f')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Integer(-1), sin(Symbol('C', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Pow(Function('f')(Symbol('C', commutative=True)), Integer(2)), Mul(Integer(-1), sin(Symbol('C', commutative=True)))), Integer(2)), Pow(Add(Mul(Function('f')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Mul(Integer(-1), sin(Symbol('C', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(G)} = \\log{(G)}, then derive \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} = \\frac{G \\log{(G)} - G + \\Psi_{nl}}{G}, then obtain \\int \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} d\\Psi_{nl} = Q + \\Psi_{nl} (\\operatorname{C_{2}}{(G)} - 1) + \\frac{\\Psi_{nl}^{2}}{2 G}", "derivation": "\\operatorname{C_{2}}{(G)} = \\log{(G)} and \\int \\operatorname{C_{2}}{(G)} dG = \\int \\log{(G)} dG and \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} = \\frac{\\int \\log{(G)} dG}{G} and \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} = \\frac{G \\log{(G)} - G + \\Psi_{nl}}{G} and \\int \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} d\\Psi_{nl} = \\int \\frac{G \\log{(G)} - G + \\Psi_{nl}}{G} d\\Psi_{nl} and \\int \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} d\\Psi_{nl} = \\int \\frac{G \\operatorname{C_{2}}{(G)} - G + \\Psi_{nl}}{G} d\\Psi_{nl} and \\int \\frac{\\int \\operatorname{C_{2}}{(G)} dG}{G} d\\Psi_{nl} = Q + \\Psi_{nl} (\\operatorname{C_{2}}{(G)} - 1) + \\frac{\\Psi_{nl}^{2}}{2 G}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["divide", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(log(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Symbol('G', commutative=True), log(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Add(Mul(Symbol('G', commutative=True), Function('C_2')(Symbol('G', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Integral(Function('C_2')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True))), Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\Psi_{nl}', commutative=True), Add(Function('C_2')(Symbol('G', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Symbol('G', commutative=True), Integer(-1)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(n_{2},a)} = \\cos{(a n_{2})}, then obtain \\int \\frac{\\operatorname{F_{H}}{(n_{2},a)}}{\\int \\cos{(a n_{2})} dn_{2}} dn_{2} = \\int \\frac{\\cos{(a n_{2})}}{\\int \\cos{(a n_{2})} dn_{2}} dn_{2}", "derivation": "\\operatorname{F_{H}}{(n_{2},a)} = \\cos{(a n_{2})} and \\int \\operatorname{F_{H}}{(n_{2},a)} dn_{2} = \\int \\cos{(a n_{2})} dn_{2} and \\frac{\\operatorname{F_{H}}{(n_{2},a)}}{\\int \\operatorname{F_{H}}{(n_{2},a)} dn_{2}} = \\frac{\\cos{(a n_{2})}}{\\int \\operatorname{F_{H}}{(n_{2},a)} dn_{2}} and \\frac{\\operatorname{F_{H}}{(n_{2},a)}}{\\int \\cos{(a n_{2})} dn_{2}} = \\frac{\\cos{(a n_{2})}}{\\int \\cos{(a n_{2})} dn_{2}} and \\int \\frac{\\operatorname{F_{H}}{(n_{2},a)}}{\\int \\cos{(a n_{2})} dn_{2}} dn_{2} = \\int \\frac{\\cos{(a n_{2})}}{\\int \\cos{(a n_{2})} dn_{2}} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["divide", 1, "Integral(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Mul(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Pow(Integral(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Mul(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Pow(Integral(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Pow(Integral(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Mul(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Pow(Integral(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('n_2', commutative=True)"], "Equality(Integral(Mul(Function('F_H')(Symbol('n_2', commutative=True), Symbol('a', commutative=True)), Pow(Integral(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Pow(Integral(cos(Mul(Symbol('a', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integer(-1))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(P_{g},f)} = - P_{g} + e^{f} and \\operatorname{A_{z}}{(f)} = e^{f}, then obtain e^{\\mathbb{I}^{P_{g}}{(P_{g},f)}} = e^{(- P_{g} + \\operatorname{A_{z}}{(f)})^{P_{g}}}", "derivation": "\\mathbb{I}{(P_{g},f)} = - P_{g} + e^{f} and \\mathbb{I}^{P_{g}}{(P_{g},f)} = (- P_{g} + e^{f})^{P_{g}} and e^{\\mathbb{I}^{P_{g}}{(P_{g},f)}} = e^{(- P_{g} + e^{f})^{P_{g}}} and \\operatorname{A_{z}}{(f)} = e^{f} and e^{\\mathbb{I}^{P_{g}}{(P_{g},f)}} = e^{(- P_{g} + \\operatorname{A_{z}}{(f)})^{P_{g}}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), exp(Symbol('f', commutative=True))))"], [["power", 1, "Symbol('P_g', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Symbol('P_g', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), exp(Symbol('f', commutative=True))), Symbol('P_g', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Symbol('P_g', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), exp(Symbol('f', commutative=True))), Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('f', commutative=True)), exp(Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(exp(Pow(Function('\\\\mathbb{I}')(Symbol('P_g', commutative=True), Symbol('f', commutative=True)), Symbol('P_g', commutative=True))), exp(Pow(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Function('A_z')(Symbol('f', commutative=True))), Symbol('P_g', commutative=True))))"]]}, {"prompt": "Given M{(A_{y})} = e^{A_{y}}, then derive \\int M{(A_{y})} dA_{y} = E + e^{A_{y}}, then obtain \\int e^{A_{y}} dA_{y} + \\iint M{(A_{y})} dA_{y} dA_{y} = \\int e^{A_{y}} dA_{y} + \\iint e^{A_{y}} dA_{y} dA_{y}", "derivation": "M{(A_{y})} = e^{A_{y}} and \\int M{(A_{y})} dA_{y} = \\int e^{A_{y}} dA_{y} and \\int M{(A_{y})} dA_{y} = E + e^{A_{y}} and \\iint M{(A_{y})} dA_{y} dA_{y} = \\iint e^{A_{y}} dA_{y} dA_{y} and \\int M{(A_{y})} dA_{y} = E + M{(A_{y})} and \\int e^{A_{y}} dA_{y} = E + M{(A_{y})} and E + M{(A_{y})} + \\iint M{(A_{y})} dA_{y} dA_{y} = E + M{(A_{y})} + \\iint e^{A_{y}} dA_{y} dA_{y} and \\int e^{A_{y}} dA_{y} + \\iint M{(A_{y})} dA_{y} dA_{y} = \\int e^{A_{y}} dA_{y} + \\iint e^{A_{y}} dA_{y} dA_{y}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('E', commutative=True), exp(Symbol('A_y', commutative=True))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('E', commutative=True), Function('M')(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Add(Symbol('E', commutative=True), Function('M')(Symbol('A_y', commutative=True))))"], [["add", 4, "Add(Symbol('E', commutative=True), Function('M')(Symbol('A_y', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Function('M')(Symbol('A_y', commutative=True)), Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Symbol('E', commutative=True), Function('M')(Symbol('A_y', commutative=True)), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(Function('M')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(b)} = e^{b}, then obtain 0 = \\operatorname{f^{*}}{(b)} \\int \\operatorname{f^{*}}{(b)} db - \\operatorname{f^{*}}{(b)} \\int e^{b} db", "derivation": "\\operatorname{f^{*}}{(b)} = e^{b} and \\int \\operatorname{f^{*}}{(b)} db = \\int e^{b} db and - \\operatorname{f^{*}}{(b)} \\int \\operatorname{f^{*}}{(b)} db = - \\operatorname{f^{*}}{(b)} \\int e^{b} db and 0 = \\operatorname{f^{*}}{(b)} \\int \\operatorname{f^{*}}{(b)} db - \\operatorname{f^{*}}{(b)} \\int e^{b} db", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('f^*')(Symbol('b', commutative=True)))"], "Equality(Mul(Integer(-1), Function('f^*')(Symbol('b', commutative=True)), Integral(Function('f^*')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Mul(Integer(-1), Function('f^*')(Symbol('b', commutative=True)), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Function('f^*')(Symbol('b', commutative=True)), Integral(Function('f^*')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], "Equality(Integer(0), Add(Mul(Function('f^*')(Symbol('b', commutative=True)), Integral(Function('f^*')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Mul(Integer(-1), Function('f^*')(Symbol('b', commutative=True)), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))))"]]}, {"prompt": "Given \\ddot{x}{(\\mu_0)} = e^{\\mu_0}, then derive \\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)} = e^{\\mu_0}, then obtain \\ddot{x}^{\\mu_0}{(\\mu_0)} = (\\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)})^{\\mu_0}", "derivation": "\\ddot{x}{(\\mu_0)} = e^{\\mu_0} and \\ddot{x}^{\\mu_0}{(\\mu_0)} = (e^{\\mu_0})^{\\mu_0} and \\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)} = e^{\\mu_0} and (\\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)})^{\\mu_0} = (e^{\\mu_0})^{\\mu_0} and \\ddot{x}^{\\mu_0}{(\\mu_0)} = (\\frac{d}{d \\mu_0} \\ddot{x}{(\\mu_0)})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), exp(Symbol('\\\\mu_0', commutative=True)))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)), Pow(exp(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\chi)} = e^{\\chi}, then obtain ((\\int \\Omega{(\\chi)} d\\chi)^{5}) \\int e^{\\chi} d\\chi = ((\\int \\Omega{(\\chi)} d\\chi)^{2}) (\\int e^{\\chi} d\\chi)^{4}", "derivation": "\\Omega{(\\chi)} = e^{\\chi} and \\int \\Omega{(\\chi)} d\\chi = \\int e^{\\chi} d\\chi and (\\int \\Omega{(\\chi)} d\\chi) \\int e^{\\chi} d\\chi = (\\int e^{\\chi} d\\chi)^{2} and ((\\int \\Omega{(\\chi)} d\\chi)^{2}) \\int e^{\\chi} d\\chi = (\\int \\Omega{(\\chi)} d\\chi) (\\int e^{\\chi} d\\chi)^{2} and ((\\int \\Omega{(\\chi)} d\\chi)^{4}) (\\int e^{\\chi} d\\chi)^{2} = ((\\int \\Omega{(\\chi)} d\\chi)^{2}) (\\int e^{\\chi} d\\chi)^{4} and ((\\int \\Omega{(\\chi)} d\\chi)^{5}) \\int e^{\\chi} d\\chi = ((\\int \\Omega{(\\chi)} d\\chi)^{4}) (\\int e^{\\chi} d\\chi)^{2} and ((\\int \\Omega{(\\chi)} d\\chi)^{5}) \\int e^{\\chi} d\\chi = ((\\int \\Omega{(\\chi)} d\\chi)^{2}) (\\int e^{\\chi} d\\chi)^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["times", 2, "Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)))"], [["times", 3, "Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2))))"], [["power", 4, 2], "Equality(Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(4)), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2))), Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(4))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(5)), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(4)), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(5)), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Mul(Pow(Integral(Function('\\\\Omega')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(2)), Pow(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integer(4))))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{p},H)} = \\frac{\\mathbf{p}}{H}, then obtain \\int \\sin{(\\mathbf{f}^{6}{(\\mathbf{p},H)})} d\\mathbf{p} = \\int \\sin{(\\frac{\\mathbf{p} \\mathbf{f}^{5}{(\\mathbf{p},H)}}{H})} d\\mathbf{p}", "derivation": "\\mathbf{f}{(\\mathbf{p},H)} = \\frac{\\mathbf{p}}{H} and \\mathbf{f}^{2}{(\\mathbf{p},H)} = \\frac{\\mathbf{p} \\mathbf{f}{(\\mathbf{p},H)}}{H} and \\mathbf{f}^{3}{(\\mathbf{p},H)} = \\frac{\\mathbf{p} \\mathbf{f}^{2}{(\\mathbf{p},H)}}{H} and \\mathbf{f}^{6}{(\\mathbf{p},H)} = \\frac{\\mathbf{p} \\mathbf{f}^{5}{(\\mathbf{p},H)}}{H} and \\sin{(\\mathbf{f}^{6}{(\\mathbf{p},H)})} = \\sin{(\\frac{\\mathbf{p} \\mathbf{f}^{5}{(\\mathbf{p},H)}}{H})} and \\int \\sin{(\\mathbf{f}^{6}{(\\mathbf{p},H)})} d\\mathbf{p} = \\int \\sin{(\\frac{\\mathbf{p} \\mathbf{f}^{5}{(\\mathbf{p},H)}}{H})} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(2)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(3)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(2))))"], [["times", 3, "Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(3))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(6)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(5))))"], [["sin", 4], "Equality(sin(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(6))), sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(5)))))"], [["integrate", 5, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(sin(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(6))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(sin(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('\\\\mathbf{p}', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('H', commutative=True)), Integer(5)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given J{(\\psi^*)} = e^{\\psi^*}, then obtain \\frac{d}{d \\psi^*} J{(\\psi^*)} + \\frac{J{(\\psi^*)}}{M_{E}} = \\frac{d}{d \\psi^*} J{(\\psi^*)} + \\frac{e^{\\psi^*}}{M_{E}}", "derivation": "J{(\\psi^*)} = e^{\\psi^*} and \\frac{d}{d \\psi^*} J{(\\psi^*)} = \\frac{d}{d \\psi^*} e^{\\psi^*} and \\frac{J{(\\psi^*)}}{M_{E}} = \\frac{e^{\\psi^*}}{M_{E}} and \\frac{d}{d \\psi^*} e^{\\psi^*} + \\frac{J{(\\psi^*)}}{M_{E}} = \\frac{d}{d \\psi^*} e^{\\psi^*} + \\frac{e^{\\psi^*}}{M_{E}} and \\frac{d}{d \\psi^*} J{(\\psi^*)} + \\frac{J{(\\psi^*)}}{M_{E}} = \\frac{d}{d \\psi^*} J{(\\psi^*)} + \\frac{e^{\\psi^*}}{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('M_E', commutative=True)"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["add", 3, "Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Add(Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\psi^*', commutative=True)))), Add(Derivative(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('J')(Symbol('\\\\psi^*', commutative=True)))), Add(Derivative(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(T,r)} = - T + r, then derive - T + \\int \\mathbf{S}{(T,r)} dT = - \\frac{T^{2}}{2} + T r - T + \\sigma_x, then obtain (- T + r) (- T + \\int \\mathbf{S}{(T,r)} dT) (- \\frac{T^{2}}{2} + T r - T + \\sigma_x) \\mathbf{S}{(T,r)} = (- T + r)^{2} (- T + \\int \\mathbf{S}{(T,r)} dT) (- \\frac{T^{2}}{2} + T r - T + \\sigma_x)", "derivation": "\\mathbf{S}{(T,r)} = - T + r and \\int \\mathbf{S}{(T,r)} dT = \\int (- T + r) dT and - T + \\int \\mathbf{S}{(T,r)} dT = - T + \\int (- T + r) dT and - T + \\int \\mathbf{S}{(T,r)} dT = - \\frac{T^{2}}{2} + T r - T + \\sigma_x and (- \\frac{T^{2}}{2} + T r - T + \\sigma_x) \\mathbf{S}{(T,r)} = (- T + r) (- \\frac{T^{2}}{2} + T r - T + \\sigma_x) and (- T + \\int \\mathbf{S}{(T,r)} dT) \\mathbf{S}{(T,r)} = (- T + r) (- T + \\int \\mathbf{S}{(T,r)} dT) and (- T + r) (- T + \\int \\mathbf{S}{(T,r)} dT) (- \\frac{T^{2}}{2} + T r - T + \\sigma_x) \\mathbf{S}{(T,r)} = (- T + r)^{2} (- T + \\int \\mathbf{S}{(T,r)} dT) (- \\frac{T^{2}}{2} + T r - T + \\sigma_x)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True))))))"], [["times", 6, "Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('r', commutative=True)), Integer(2)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Integral(Function('\\\\mathbf{S}')(Symbol('T', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\sigma_x', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\phi)} = \\sin{(\\phi)}, then derive \\int \\mathbb{I}{(\\phi)} d\\phi = \\hat{p} - \\cos{(\\phi)}, then obtain - \\cos{(\\phi)} + \\int \\sin{(\\phi)} d\\phi = \\hat{p} - 2 \\cos{(\\phi)}", "derivation": "\\mathbb{I}{(\\phi)} = \\sin{(\\phi)} and \\int \\mathbb{I}{(\\phi)} d\\phi = \\int \\sin{(\\phi)} d\\phi and \\int \\mathbb{I}{(\\phi)} d\\phi = \\hat{p} - \\cos{(\\phi)} and \\int \\sin{(\\phi)} d\\phi = \\hat{p} - \\cos{(\\phi)} and - \\cos{(\\phi)} + \\int \\sin{(\\phi)} d\\phi = \\hat{p} - 2 \\cos{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\phi', commutative=True))), Integral(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{X})} = \\hat{X}, then obtain (- \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}})^{\\hat{X}} = (- \\frac{\\hat{X}^{2} \\operatorname{A_{2}}^{- \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}}}{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})}})^{\\hat{X}}", "derivation": "\\operatorname{A_{2}}{(\\hat{X})} = \\hat{X} and -1 = - \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}} and (-1)^{\\hat{X}} = (- \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}})^{\\hat{X}} and (- \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}})^{\\hat{X}} = (- \\frac{\\hat{X}^{2} \\operatorname{A_{2}}^{- \\frac{\\hat{X}}{\\operatorname{A_{2}}{(\\hat{X})}}}{(\\hat{X})}}{\\operatorname{A_{2}}{(\\hat{X})}})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))"], [["divide", 1, "Mul(Integer(-1), Function('A_2')(Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(-1), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Symbol('\\\\hat{X}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Symbol('\\\\hat{X}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(2)), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True), Pow(Function('A_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} = \\hat{p}_0 + \\sin{(\\mathbf{g})}, then obtain \\frac{(\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int \\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} d\\hat{p}_0}{\\mathbf{g}} = \\frac{(\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int (\\hat{p}_0 + \\sin{(\\mathbf{g})}) d\\hat{p}_0}{\\mathbf{g}}", "derivation": "\\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} = \\hat{p}_0 + \\sin{(\\mathbf{g})} and \\int \\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} d\\hat{p}_0 = \\int (\\hat{p}_0 + \\sin{(\\mathbf{g})}) d\\hat{p}_0 and (\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int \\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} d\\hat{p}_0 = (\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int (\\hat{p}_0 + \\sin{(\\mathbf{g})}) d\\hat{p}_0 and \\frac{(\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int \\operatorname{m_{s}}{(\\hat{p}_0,\\mathbf{g})} d\\hat{p}_0}{\\mathbf{g}} = \\frac{(\\hat{p}_0 + \\sin{(\\mathbf{g})}) \\int (\\hat{p}_0 + \\sin{(\\mathbf{g})}) d\\hat{p}_0}{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["divide", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Function('m_s')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Integral(Add(Symbol('\\\\hat{p}_0', commutative=True), sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given V{(I,\\rho_b,v)} = I \\rho_b v, then obtain \\frac{\\partial}{\\partial \\rho_b} \\int \\frac{\\partial}{\\partial v} (I + V{(I,\\rho_b,v)}) dv = \\frac{\\partial}{\\partial \\rho_b} \\int \\frac{\\partial}{\\partial v} (I \\rho_b v + I) dv", "derivation": "V{(I,\\rho_b,v)} = I \\rho_b v and I + V{(I,\\rho_b,v)} = I \\rho_b v + I and \\frac{\\partial}{\\partial v} (I + V{(I,\\rho_b,v)}) = \\frac{\\partial}{\\partial v} (I \\rho_b v + I) and \\int \\frac{\\partial}{\\partial v} (I + V{(I,\\rho_b,v)}) dv = \\int \\frac{\\partial}{\\partial v} (I \\rho_b v + I) dv and \\frac{\\partial}{\\partial \\rho_b} \\int \\frac{\\partial}{\\partial v} (I + V{(I,\\rho_b,v)}) dv = \\frac{\\partial}{\\partial \\rho_b} \\int \\frac{\\partial}{\\partial v} (I \\rho_b v + I) dv", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('V')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)), Symbol('I', commutative=True)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Symbol('I', commutative=True), Function('V')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('I', commutative=True), Function('V')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Integral(Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Integral(Derivative(Add(Symbol('I', commutative=True), Function('V')(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\rho_b', commutative=True), Symbol('v', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(A,\\varphi^*)} = \\int A \\varphi^* d\\varphi^*, then obtain \\frac{\\partial}{\\partial A} (A \\varphi^* + \\varphi^* \\mathbf{r}{(A,\\varphi^*)}) = \\frac{\\partial}{\\partial A} (A \\varphi^* + \\varphi^* \\int A \\varphi^* d\\varphi^*)", "derivation": "\\mathbf{r}{(A,\\varphi^*)} = \\int A \\varphi^* d\\varphi^* and \\varphi^* \\mathbf{r}{(A,\\varphi^*)} = \\varphi^* \\int A \\varphi^* d\\varphi^* and A \\varphi^* + \\varphi^* \\mathbf{r}{(A,\\varphi^*)} = A \\varphi^* + \\varphi^* \\int A \\varphi^* d\\varphi^* and \\frac{\\partial}{\\partial A} (A \\varphi^* + \\varphi^* \\mathbf{r}{(A,\\varphi^*)}) = \\frac{\\partial}{\\partial A} (A \\varphi^* + \\varphi^* \\int A \\varphi^* d\\varphi^*)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["add", 2, "Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Mul(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(F_{c},\\mu_0)} = \\mu_0 \\log{(F_{c})} and \\hat{H}{(\\phi)} = \\log{(\\phi)}, then derive \\frac{\\partial}{\\partial \\mu_0} Z{(F_{c},\\mu_0)} = \\log{(F_{c})}, then obtain \\frac{\\hat{H}{(\\phi)}}{\\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})}} = \\frac{\\log{(\\phi)}}{\\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})}}", "derivation": "Z{(F_{c},\\mu_0)} = \\mu_0 \\log{(F_{c})} and \\frac{\\partial}{\\partial \\mu_0} Z{(F_{c},\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})} and \\hat{H}{(\\phi)} = \\log{(\\phi)} and \\frac{\\partial}{\\partial \\mu_0} Z{(F_{c},\\mu_0)} = \\log{(F_{c})} and \\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})} = \\log{(F_{c})} and \\frac{\\hat{H}{(\\phi)}}{\\log{(F_{c})}} = \\frac{\\log{(\\phi)}}{\\log{(F_{c})}} and \\frac{\\hat{H}{(\\phi)}}{\\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})}} = \\frac{\\log{(\\phi)}}{\\frac{\\partial}{\\partial \\mu_0} \\mu_0 \\log{(F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('F_c', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('F_c', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('F_c', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('F_c', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), log(Symbol('F_c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), log(Symbol('F_c', commutative=True)))"], [["divide", 3, "log(Symbol('F_c', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('F_c', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('F_c', commutative=True)), Integer(-1)), log(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Function('\\\\hat{H}')(Symbol('\\\\phi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))), Mul(log(Symbol('\\\\phi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('F_c', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\pi)} = e^{\\pi}, then obtain (\\int (\\operatorname{v_{x}}{(\\pi)} + e^{\\pi}) d\\pi)^{2} = (\\int 2 e^{\\pi} d\\pi)^{2}", "derivation": "\\operatorname{v_{x}}{(\\pi)} = e^{\\pi} and \\operatorname{v_{x}}{(\\pi)} + e^{\\pi} = 2 e^{\\pi} and \\int (\\operatorname{v_{x}}{(\\pi)} + e^{\\pi}) d\\pi = \\int 2 e^{\\pi} d\\pi and (\\int (\\operatorname{v_{x}}{(\\pi)} + e^{\\pi}) d\\pi)^{2} = (\\int 2 e^{\\pi} d\\pi)^{2}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Function('v_x')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Add(Function('v_x')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), exp(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\mu{(\\phi,z^{*},G)} = \\frac{G + \\phi}{z^{*}} and \\tilde{g}^*{(\\phi,z^{*},G)} = \\frac{- G - \\phi}{z^{*}}, then obtain \\frac{\\partial}{\\partial \\phi} \\tilde{g}^*{(\\phi,z^{*},G)} = - \\frac{1}{z^{*}}", "derivation": "\\mu{(\\phi,z^{*},G)} = \\frac{G + \\phi}{z^{*}} and - \\mu{(\\phi,z^{*},G)} = - \\frac{G + \\phi}{z^{*}} and - \\mu{(\\phi,z^{*},G)} = \\frac{- G - \\phi}{z^{*}} and \\frac{\\partial}{\\partial \\phi} - \\mu{(\\phi,z^{*},G)} = \\frac{\\partial}{\\partial \\phi} \\frac{- G - \\phi}{z^{*}} and \\tilde{g}^*{(\\phi,z^{*},G)} = \\frac{- G - \\phi}{z^{*}} and \\tilde{g}^*{(\\phi,z^{*},G)} = - \\mu{(\\phi,z^{*},G)} and - \\tilde{g}^*{(\\phi,z^{*},G)} = \\mu{(\\phi,z^{*},G)} and \\frac{\\partial}{\\partial \\phi} \\tilde{g}^*{(\\phi,z^{*},G)} = \\frac{\\partial}{\\partial \\phi} \\frac{- G - \\phi}{z^{*}} and \\frac{\\partial}{\\partial \\phi} \\tilde{g}^*{(\\phi,z^{*},G)} = - \\frac{1}{z^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Symbol('G', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True))))"], [["times", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True))), Function('\\\\mu')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 7], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('z^*', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 8], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\phi', commutative=True), Symbol('z^*', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('z^*', commutative=True), Integer(-1))))"]]}, {"prompt": "Given A{(\\mathbf{P},f^{\\prime})} = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}}, then derive A{(\\mathbf{P},f^{\\prime})} = \\frac{1}{f^{\\prime}}, then obtain - \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} - 1 = - A{(\\mathbf{P},\\frac{1}{\\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}}})} - 1", "derivation": "A{(\\mathbf{P},f^{\\prime})} = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} and A{(\\mathbf{P},f^{\\prime})} + 1 = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} + 1 and A{(\\mathbf{P},f^{\\prime})} = \\frac{1}{f^{\\prime}} and 1 + \\frac{1}{f^{\\prime}} = \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} + 1 and \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} = \\frac{1}{f^{\\prime}} and 1 + \\frac{1}{f^{\\prime}} = A{(\\mathbf{P},f^{\\prime})} + 1 and -1 - \\frac{1}{f^{\\prime}} = - A{(\\mathbf{P},f^{\\prime})} - 1 and - \\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}} - 1 = - A{(\\mathbf{P},\\frac{1}{\\frac{\\partial}{\\partial \\mathbf{P}} \\frac{\\mathbf{P}}{f^{\\prime}}})} - 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integer(1)), Add(Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Integer(1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Add(Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Integer(1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Add(Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integer(1)))"], [["divide", 6, "Integer(-1)"], "Equality(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Function('A')(Symbol('\\\\mathbf{P}', commutative=True), Pow(Derivative(Mul(Symbol('\\\\mathbf{P}', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Integer(-1)))), Integer(-1)))"]]}, {"prompt": "Given \\varphi^{*}{(y)} = e^{y} and \\mu{(y)} = e^{y}, then obtain e^{y} + \\frac{d}{d y} \\mu{(y)} = 2 e^{y}", "derivation": "\\varphi^{*}{(y)} = e^{y} and \\frac{d}{d y} \\varphi^{*}{(y)} = \\frac{d}{d y} e^{y} and \\mu{(y)} = e^{y} and \\varphi^{*}{(y)} = \\mu{(y)} and \\frac{d}{d y} \\varphi^{*}{(y)} + \\frac{d}{d y} e^{y} = 2 \\frac{d}{d y} e^{y} and \\frac{d}{d y} \\mu{(y)} + \\frac{d}{d y} e^{y} = 2 \\frac{d}{d y} e^{y} and e^{y} + \\frac{d}{d y} \\mu{(y)} = 2 e^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\varphi^*')(Symbol('y', commutative=True)), Function('\\\\mu')(Symbol('y', commutative=True)))"], [["add", 2, "Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\varphi^*')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Function('\\\\mu')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(exp(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(Add(exp(Symbol('y', commutative=True)), Derivative(Function('\\\\mu')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('y', commutative=True))))"]]}, {"prompt": "Given c{(v_{2})} = \\cos{(v_{2})}, then derive \\frac{d}{d v_{2}} c{(v_{2})} = - \\sin{(v_{2})}, then obtain \\frac{d}{d v_{2}} \\cos{(v_{2})} - 1 = - \\sin{(v_{2})} - 1", "derivation": "c{(v_{2})} = \\cos{(v_{2})} and \\frac{d}{d v_{2}} c{(v_{2})} = \\frac{d}{d v_{2}} \\cos{(v_{2})} and \\frac{d}{d v_{2}} c{(v_{2})} = - \\sin{(v_{2})} and \\frac{d}{d v_{2}} c{(v_{2})} - 1 = - \\sin{(v_{2})} - 1 and \\frac{d}{d v_{2}} \\cos{(v_{2})} - 1 = - \\sin{(v_{2})} - 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v_2', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('c')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('v_2', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('v_2', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{M}{(M,\\mathbf{J}_P)} = \\frac{\\cos{(\\mathbf{J}_P)}}{M}, then obtain \\frac{\\partial}{\\partial \\sigma_p} \\iint \\sigma_p \\mathbf{M}{(M,\\mathbf{J}_P)} d\\mathbf{J}_P d\\sigma_p = \\frac{\\partial}{\\partial \\sigma_p} \\iint \\frac{\\sigma_p \\cos{(\\mathbf{J}_P)}}{M} d\\mathbf{J}_P d\\sigma_p", "derivation": "\\mathbf{M}{(M,\\mathbf{J}_P)} = \\frac{\\cos{(\\mathbf{J}_P)}}{M} and \\sigma_p \\mathbf{M}{(M,\\mathbf{J}_P)} = \\frac{\\sigma_p \\cos{(\\mathbf{J}_P)}}{M} and \\int \\sigma_p \\mathbf{M}{(M,\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\frac{\\sigma_p \\cos{(\\mathbf{J}_P)}}{M} d\\mathbf{J}_P and \\iint \\sigma_p \\mathbf{M}{(M,\\mathbf{J}_P)} d\\mathbf{J}_P d\\sigma_p = \\iint \\frac{\\sigma_p \\cos{(\\mathbf{J}_P)}}{M} d\\mathbf{J}_P d\\sigma_p and \\frac{\\partial}{\\partial \\sigma_p} \\iint \\sigma_p \\mathbf{M}{(M,\\mathbf{J}_P)} d\\mathbf{J}_P d\\sigma_p = \\frac{\\partial}{\\partial \\sigma_p} \\iint \\frac{\\sigma_p \\cos{(\\mathbf{J}_P)}}{M} d\\mathbf{J}_P d\\sigma_p", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{M}')(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\sigma_p', commutative=True), cos(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{P}{(C_{1})} = e^{C_{1}} and \\phi{(C_{1})} = 2 e^{C_{1}}, then obtain \\int \\frac{d}{d C_{1}} \\phi{(C_{1})} dC_{1} = \\int \\frac{d}{d C_{1}} (\\mathbf{P}{(C_{1})} + e^{C_{1}}) dC_{1}", "derivation": "\\mathbf{P}{(C_{1})} = e^{C_{1}} and \\mathbf{P}{(C_{1})} + e^{C_{1}} = 2 e^{C_{1}} and \\phi{(C_{1})} = 2 e^{C_{1}} and \\phi{(C_{1})} = \\mathbf{P}{(C_{1})} + e^{C_{1}} and \\frac{d}{d C_{1}} \\phi{(C_{1})} = \\frac{d}{d C_{1}} (\\mathbf{P}{(C_{1})} + e^{C_{1}}) and \\int \\frac{d}{d C_{1}} \\phi{(C_{1})} dC_{1} = \\int \\frac{d}{d C_{1}} (\\mathbf{P}{(C_{1})} + e^{C_{1}}) dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True)))"], [["add", 1, "exp(Symbol('C_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))), Mul(Integer(2), exp(Symbol('C_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\phi')(Symbol('C_1', commutative=True)), Add(Function('\\\\mathbf{P}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))))"], [["differentiate", 4, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{P}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Derivative(Add(Function('\\\\mathbf{P}')(Symbol('C_1', commutative=True)), exp(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\phi)} = \\sin{(\\phi)}, then obtain \\sin{(\\phi)} + \\int \\frac{d}{d \\phi} (\\varphi^{*}{(\\phi)} + \\sin{(\\phi)}) d\\phi = \\sin{(\\phi)} + \\int \\frac{d}{d \\phi} 2 \\sin{(\\phi)} d\\phi", "derivation": "\\varphi^{*}{(\\phi)} = \\sin{(\\phi)} and \\varphi^{*}{(\\phi)} + \\sin{(\\phi)} = 2 \\sin{(\\phi)} and \\frac{d}{d \\phi} (\\varphi^{*}{(\\phi)} + \\sin{(\\phi)}) = \\frac{d}{d \\phi} 2 \\sin{(\\phi)} and \\int \\frac{d}{d \\phi} (\\varphi^{*}{(\\phi)} + \\sin{(\\phi)}) d\\phi = \\int \\frac{d}{d \\phi} 2 \\sin{(\\phi)} d\\phi and \\sin{(\\phi)} + \\int \\frac{d}{d \\phi} (\\varphi^{*}{(\\phi)} + \\sin{(\\phi)}) d\\phi = \\sin{(\\phi)} + \\int \\frac{d}{d \\phi} 2 \\sin{(\\phi)} d\\phi", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Derivative(Mul(Integer(2), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 4, "sin(Symbol('\\\\phi', commutative=True))"], "Equality(Add(sin(Symbol('\\\\phi', commutative=True)), Integral(Derivative(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True)))), Add(sin(Symbol('\\\\phi', commutative=True)), Integral(Derivative(Mul(Integer(2), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(r_{0})} = \\log{(r_{0})} and V{(r_{0})} = \\log{(r_{0})}, then obtain \\log{(r_{0})}^{2} = \\hat{X}{(r_{0})} \\log{(r_{0})}", "derivation": "\\hat{X}{(r_{0})} = \\log{(r_{0})} and V{(r_{0})} = \\log{(r_{0})} and V{(r_{0})} = \\hat{X}{(r_{0})} and V^{2}{(r_{0})} = V{(r_{0})} \\hat{X}{(r_{0})} and \\log{(r_{0})}^{2} = \\hat{X}{(r_{0})} \\log{(r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('V')(Symbol('r_0', commutative=True)), Function('\\\\hat{X}')(Symbol('r_0', commutative=True)))"], [["times", 3, "Function('V')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('V')(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('V')(Symbol('r_0', commutative=True)), Function('\\\\hat{X}')(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(log(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('\\\\hat{X}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given s{(Q)} = \\cos{(Q)}, then obtain \\int \\log{(\\frac{\\frac{d}{d Q} s{(Q)}}{s{(Q)}})} dQ = \\int \\log{(\\frac{\\frac{d}{d Q} \\cos{(Q)}}{s{(Q)}})} dQ", "derivation": "s{(Q)} = \\cos{(Q)} and \\frac{d}{d Q} s{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\frac{\\frac{d}{d Q} s{(Q)}}{s{(Q)}} = \\frac{\\frac{d}{d Q} \\cos{(Q)}}{s{(Q)}} and \\log{(\\frac{\\frac{d}{d Q} s{(Q)}}{s{(Q)}})} = \\log{(\\frac{\\frac{d}{d Q} \\cos{(Q)}}{s{(Q)}})} and \\int \\log{(\\frac{\\frac{d}{d Q} s{(Q)}}{s{(Q)}})} dQ = \\int \\log{(\\frac{\\frac{d}{d Q} \\cos{(Q)}}{s{(Q)}})} dQ", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["divide", 2, "Function('s')(Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Function('s')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["log", 3], "Equality(log(Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Function('s')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), log(Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(log(Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(Function('s')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Tuple(Symbol('Q', commutative=True))), Integral(log(Mul(Pow(Function('s')(Symbol('Q', commutative=True)), Integer(-1)), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given c{(W)} = \\sin{(W)}, then obtain \\frac{c{(W)} + \\sin{(W)} - \\iint 2 \\sin{(W)} dW dW}{2 \\sin{(W)}} = \\frac{2 \\sin{(W)} - \\iint 2 \\sin{(W)} dW dW}{2 \\sin{(W)}}", "derivation": "c{(W)} = \\sin{(W)} and c{(W)} + \\sin{(W)} = 2 \\sin{(W)} and \\int (c{(W)} + \\sin{(W)}) dW = \\int 2 \\sin{(W)} dW and \\iint (c{(W)} + \\sin{(W)}) dW dW = \\iint 2 \\sin{(W)} dW dW and c{(W)} + \\sin{(W)} - \\iint (c{(W)} + \\sin{(W)}) dW dW = 2 \\sin{(W)} - \\iint (c{(W)} + \\sin{(W)}) dW dW and \\frac{c{(W)} + \\sin{(W)} - \\iint (c{(W)} + \\sin{(W)}) dW dW}{2 \\sin{(W)}} = \\frac{2 \\sin{(W)} - \\iint (c{(W)} + \\sin{(W)}) dW dW}{2 \\sin{(W)}} and \\frac{c{(W)} + \\sin{(W)} - \\iint 2 \\sin{(W)} dW dW}{2 \\sin{(W)}} = \\frac{2 \\sin{(W)} - \\iint 2 \\sin{(W)} dW dW}{2 \\sin{(W)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["add", 1, "sin(Symbol('W', commutative=True))"], "Equality(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Mul(Integer(2), sin(Symbol('W', commutative=True))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["minus", 2, "Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Add(Mul(Integer(2), sin(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"], [["divide", 5, "Mul(Integer(2), sin(Symbol('W', commutative=True)))"], "Equality(Mul(Rational(1, 2), Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Pow(sin(Symbol('W', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Integer(2), sin(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Pow(sin(Symbol('W', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Rational(1, 2), Add(Function('c')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Pow(sin(Symbol('W', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Add(Mul(Integer(2), sin(Symbol('W', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))), Pow(sin(Symbol('W', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(E_{n},v_{t})} = E_{n} - v_{t} and W{(E_{n},v_{t})} = - \\operatorname{F_{H}}{(E_{n},v_{t})}, then obtain (- \\operatorname{F_{H}}{(E_{n},v_{t})})^{v_{t}} = W^{v_{t}}{(E_{n},v_{t})}", "derivation": "\\operatorname{F_{H}}{(E_{n},v_{t})} = E_{n} - v_{t} and - \\operatorname{F_{H}}{(E_{n},v_{t})} = - E_{n} + v_{t} and W{(E_{n},v_{t})} = - \\operatorname{F_{H}}{(E_{n},v_{t})} and (- \\operatorname{F_{H}}{(E_{n},v_{t})})^{v_{t}} = (- E_{n} + v_{t})^{v_{t}} and W{(E_{n},v_{t})} = - E_{n} + v_{t} and (- \\operatorname{F_{H}}{(E_{n},v_{t})})^{v_{t}} = W^{v_{t}}{(E_{n},v_{t})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('F_H')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('v_t', commutative=True)))"], ["renaming_premise", "Equality(Function('W')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('F_H')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True))))"], [["power", 2, "Symbol('v_t', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('F_H')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('W')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Symbol('v_t', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Integer(-1), Function('F_H')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Function('W')(Symbol('E_n', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(v_{z})} = \\log{(e^{v_{z}})}, then obtain - \\int 0 dv_{z} + \\int \\operatorname{t_{2}}{(v_{z})} dv_{z} = - \\int 0 dv_{z} + \\int \\log{(e^{v_{z}})} dv_{z}", "derivation": "\\operatorname{t_{2}}{(v_{z})} = \\log{(e^{v_{z}})} and \\operatorname{t_{2}}{(v_{z})} - \\log{(e^{v_{z}})} = 0 and \\int (\\operatorname{t_{2}}{(v_{z})} - \\log{(e^{v_{z}})}) dv_{z} = \\int 0 dv_{z} and \\int \\operatorname{t_{2}}{(v_{z})} dv_{z} = \\int \\log{(e^{v_{z}})} dv_{z} and - \\int (\\operatorname{t_{2}}{(v_{z})} - \\log{(e^{v_{z}})}) dv_{z} + \\int \\operatorname{t_{2}}{(v_{z})} dv_{z} = - \\int (\\operatorname{t_{2}}{(v_{z})} - \\log{(e^{v_{z}})}) dv_{z} + \\int \\log{(e^{v_{z}})} dv_{z} and - \\int 0 dv_{z} + \\int \\operatorname{t_{2}}{(v_{z})} dv_{z} = - \\int 0 dv_{z} + \\int \\log{(e^{v_{z}})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('v_z', commutative=True)), log(exp(Symbol('v_z', commutative=True))))"], [["minus", 1, "log(exp(Symbol('v_z', commutative=True)))"], "Equality(Add(Function('t_2')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(exp(Symbol('v_z', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Function('t_2')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(exp(Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True))), Integral(Integer(0), Tuple(Symbol('v_z', commutative=True))))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(log(exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True))))"], [["minus", 4, "Integral(Add(Function('t_2')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(exp(Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Function('t_2')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(exp(Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True)))), Integral(Function('t_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Integral(Add(Function('t_2')(Symbol('v_z', commutative=True)), Mul(Integer(-1), log(exp(Symbol('v_z', commutative=True))))), Tuple(Symbol('v_z', commutative=True)))), Integral(log(exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('v_z', commutative=True)))), Integral(Function('t_2')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('v_z', commutative=True)))), Integral(log(exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}{(g,\\mathbf{D})} = \\log{(\\mathbf{D} + g)} and \\varepsilon_{0}{(g,\\mathbf{D})} = \\mathbf{D} \\hat{x}{(g,\\mathbf{D})}, then obtain \\mathbf{D} \\log{(\\mathbf{D} + g)} + \\varepsilon_{0}{(g,\\mathbf{D})} = 2 \\mathbf{D} \\log{(\\mathbf{D} + g)}", "derivation": "\\hat{x}{(g,\\mathbf{D})} = \\log{(\\mathbf{D} + g)} and \\mathbf{D} \\hat{x}{(g,\\mathbf{D})} = \\mathbf{D} \\log{(\\mathbf{D} + g)} and \\varepsilon_{0}{(g,\\mathbf{D})} = \\mathbf{D} \\hat{x}{(g,\\mathbf{D})} and \\varepsilon_{0}{(g,\\mathbf{D})} = \\mathbf{D} \\log{(\\mathbf{D} + g)} and \\mathbf{D} \\log{(\\mathbf{D} + g)} + \\varepsilon_{0}{(g,\\mathbf{D})} = 2 \\mathbf{D} \\log{(\\mathbf{D} + g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('\\\\hat{x}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('\\\\hat{x}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)))))"], [["add", 4, "Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{D}', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(u)} = \\cos{(e^{u})} and \\bar{\\h}{(u)} = e^{u}, then obtain \\sin{(\\hat{p} \\cos{(\\bar{\\h}{(u)})})} = \\sin{(\\hat{p} \\cos{(e^{u})})}", "derivation": "\\operatorname{z^{*}}{(u)} = \\cos{(e^{u})} and \\hat{p} \\operatorname{z^{*}}{(u)} = \\hat{p} \\cos{(e^{u})} and \\sin{(\\hat{p} \\operatorname{z^{*}}{(u)})} = \\sin{(\\hat{p} \\cos{(e^{u})})} and \\bar{\\h}{(u)} = e^{u} and \\operatorname{z^{*}}{(u)} = \\cos{(\\bar{\\h}{(u)})} and \\sin{(\\hat{p} \\cos{(\\bar{\\h}{(u)})})} = \\sin{(\\hat{p} \\cos{(e^{u})})}", "srepr_derivation": [["get_premise", "Equality(Function('z^*')(Symbol('u', commutative=True)), cos(exp(Symbol('u', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{p}', commutative=True), Function('z^*')(Symbol('u', commutative=True))), Mul(Symbol('\\\\hat{p}', commutative=True), cos(exp(Symbol('u', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Symbol('\\\\hat{p}', commutative=True), Function('z^*')(Symbol('u', commutative=True)))), sin(Mul(Symbol('\\\\hat{p}', commutative=True), cos(exp(Symbol('u', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('z^*')(Symbol('u', commutative=True)), cos(Function('\\\\hbar')(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(sin(Mul(Symbol('\\\\hat{p}', commutative=True), cos(Function('\\\\hbar')(Symbol('u', commutative=True))))), sin(Mul(Symbol('\\\\hat{p}', commutative=True), cos(exp(Symbol('u', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} = (\\Psi_{\\lambda} + t_{2})^{\\mu}, then obtain (- \\mu \\int \\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} d\\mu)^{t_{2}} = (- \\mu \\int (\\Psi_{\\lambda} + t_{2})^{\\mu} d\\mu)^{t_{2}}", "derivation": "\\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} = (\\Psi_{\\lambda} + t_{2})^{\\mu} and \\int \\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} d\\mu = \\int (\\Psi_{\\lambda} + t_{2})^{\\mu} d\\mu and - \\mu \\int \\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} d\\mu = - \\mu \\int (\\Psi_{\\lambda} + t_{2})^{\\mu} d\\mu and (- \\mu \\int \\nabla{(t_{2},\\Psi_{\\lambda},\\mu)} d\\mu)^{t_{2}} = (- \\mu \\int (\\Psi_{\\lambda} + t_{2})^{\\mu} d\\mu)^{t_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Integral(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Integral(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["power", 3, "Symbol('t_2', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Integral(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Symbol('t_2', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Integral(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), Symbol('t_2', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(r_{0},\\mathbf{A})} = \\mathbf{A}^{r_{0}} and \\phi{(r_{0},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + \\rho_{f}{(r_{0},\\mathbf{A})}}, then obtain (\\frac{1}{\\mathbf{A} + \\rho_{f}{(r_{0},\\mathbf{A})}})^{r_{0}} = (\\frac{1}{\\mathbf{A} + \\mathbf{A}^{r_{0}}})^{r_{0}}", "derivation": "\\rho_{f}{(r_{0},\\mathbf{A})} = \\mathbf{A}^{r_{0}} and \\mathbf{A} + \\rho_{f}{(r_{0},\\mathbf{A})} = \\mathbf{A} + \\mathbf{A}^{r_{0}} and \\phi{(r_{0},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + \\rho_{f}{(r_{0},\\mathbf{A})}} and \\phi{(r_{0},\\mathbf{A})} = \\frac{1}{\\mathbf{A} + \\mathbf{A}^{r_{0}}} and \\phi^{r_{0}}{(r_{0},\\mathbf{A})} = (\\frac{1}{\\mathbf{A} + \\mathbf{A}^{r_{0}}})^{r_{0}} and (\\frac{1}{\\mathbf{A} + \\rho_{f}{(r_{0},\\mathbf{A})}})^{r_{0}} = (\\frac{1}{\\mathbf{A} + \\mathbf{A}^{r_{0}}})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('r_0', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\rho_f')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('r_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\rho_f')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('r_0', commutative=True))), Integer(-1)))"], [["power", 4, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('r_0', commutative=True))), Integer(-1)), Symbol('r_0', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\rho_f')(Symbol('r_0', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)), Symbol('r_0', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Symbol('r_0', commutative=True))), Integer(-1)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given n{(C_{d})} = \\cos{(\\log{(C_{d})})} and \\mathbf{J}_M{(C_{d})} = - C_{d}, then obtain (\\mathbf{J}_M{(C_{d})} \\cos^{- C_{d}}{(\\log{(C_{d})})})^{C_{d}} = (- C_{d} \\cos^{- C_{d}}{(\\log{(C_{d})})})^{C_{d}}", "derivation": "n{(C_{d})} = \\cos{(\\log{(C_{d})})} and \\mathbf{J}_M{(C_{d})} = - C_{d} and \\mathbf{J}_M{(C_{d})} n^{- C_{d}}{(C_{d})} = - C_{d} n^{- C_{d}}{(C_{d})} and \\mathbf{J}_M{(C_{d})} \\cos^{- C_{d}}{(\\log{(C_{d})})} = - C_{d} \\cos^{- C_{d}}{(\\log{(C_{d})})} and (\\mathbf{J}_M{(C_{d})} \\cos^{- C_{d}}{(\\log{(C_{d})})})^{C_{d}} = (- C_{d} \\cos^{- C_{d}}{(\\log{(C_{d})})})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('C_d', commutative=True)), cos(log(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True)))"], [["divide", 2, "Pow(Function('n')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('C_d', commutative=True)), Pow(Function('n')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True)))), Mul(Integer(-1), Symbol('C_d', commutative=True), Pow(Function('n')(Symbol('C_d', commutative=True)), Mul(Integer(-1), Symbol('C_d', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{J}_M')(Symbol('C_d', commutative=True)), Pow(cos(log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)))), Mul(Integer(-1), Symbol('C_d', commutative=True), Pow(cos(log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)))))"], [["power", 4, "Symbol('C_d', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{J}_M')(Symbol('C_d', commutative=True)), Pow(cos(log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)), Pow(Mul(Integer(-1), Symbol('C_d', commutative=True), Pow(cos(log(Symbol('C_d', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(x)} = e^{x} and \\operatorname{P_{e}}{(x)} = \\frac{e^{x}}{- \\operatorname{v_{z}}{(x)} + e^{x}}, then obtain e^{\\operatorname{P_{e}}{(x)}} = e^{\\tilde{\\infty} e^{x}}", "derivation": "\\operatorname{v_{z}}{(x)} = e^{x} and \\operatorname{P_{e}}{(x)} = \\frac{e^{x}}{- \\operatorname{v_{z}}{(x)} + e^{x}} and \\operatorname{P_{e}}{(x)} = \\tilde{\\infty} e^{x} and e^{\\operatorname{P_{e}}{(x)}} = e^{\\tilde{\\infty} e^{x}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], ["renaming_premise", "Equality(Function('P_e')(Symbol('x', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Function('v_z')(Symbol('x', commutative=True))), exp(Symbol('x', commutative=True))), Integer(-1)), exp(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_e')(Symbol('x', commutative=True)), Mul(zoo, exp(Symbol('x', commutative=True))))"], [["exp", 3], "Equality(exp(Function('P_e')(Symbol('x', commutative=True))), exp(Mul(zoo, exp(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(C_{2})} = e^{C_{2}}, then derive \\int \\nabla{(C_{2})} dC_{2} = v_{z} + e^{C_{2}}, then obtain (C_{2} + \\int \\nabla{(C_{2})} dC_{2}) e^{C_{2}} = (C_{2} + \\mathbf{J}_f + e^{C_{2}}) e^{C_{2}}", "derivation": "\\nabla{(C_{2})} = e^{C_{2}} and \\int \\nabla{(C_{2})} dC_{2} = \\int e^{C_{2}} dC_{2} and \\int \\nabla{(C_{2})} dC_{2} = v_{z} + e^{C_{2}} and \\int e^{C_{2}} dC_{2} = v_{z} + e^{C_{2}} and C_{2} + \\int \\nabla{(C_{2})} dC_{2} = C_{2} + v_{z} + e^{C_{2}} and C_{2} + \\int \\nabla{(C_{2})} dC_{2} = C_{2} + \\int e^{C_{2}} dC_{2} and (C_{2} + \\int \\nabla{(C_{2})} dC_{2}) e^{C_{2}} = (C_{2} + \\int e^{C_{2}} dC_{2}) e^{C_{2}} and (C_{2} + \\int \\nabla{(C_{2})} dC_{2}) e^{C_{2}} = (C_{2} + \\mathbf{J}_f + e^{C_{2}}) e^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('v_z', commutative=True), exp(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('v_z', commutative=True), exp(Symbol('C_2', commutative=True))))"], [["add", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True), exp(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"], [["times", 6, "exp(Symbol('C_2', commutative=True))"], "Equality(Mul(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), exp(Symbol('C_2', commutative=True))), Mul(Add(Symbol('C_2', commutative=True), Integral(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), exp(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Mul(Add(Symbol('C_2', commutative=True), Integral(Function('\\\\nabla')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), exp(Symbol('C_2', commutative=True))), Mul(Add(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), exp(Symbol('C_2', commutative=True))), exp(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(E_{n})} = e^{e^{E_{n}}}, then obtain \\frac{d}{d E_{n}} (\\ddot{x}{(E_{n})} + \\ddot{x}^{E_{n}}{(E_{n})}) = \\frac{d}{d E_{n}} (\\ddot{x}{(E_{n})} + (e^{e^{E_{n}}})^{E_{n}})", "derivation": "\\ddot{x}{(E_{n})} = e^{e^{E_{n}}} and \\ddot{x}^{E_{n}}{(E_{n})} = (e^{e^{E_{n}}})^{E_{n}} and \\ddot{x}{(E_{n})} + \\ddot{x}^{E_{n}}{(E_{n})} = \\ddot{x}{(E_{n})} + (e^{e^{E_{n}}})^{E_{n}} and \\frac{d}{d E_{n}} (\\ddot{x}{(E_{n})} + \\ddot{x}^{E_{n}}{(E_{n})}) = \\frac{d}{d E_{n}} (\\ddot{x}{(E_{n})} + (e^{e^{E_{n}}})^{E_{n}})", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), exp(exp(Symbol('E_n', commutative=True))))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(exp(exp(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)))"], [["add", 2, "Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Pow(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))), Add(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Pow(exp(exp(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))))"], [["differentiate", 3, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Add(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Pow(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Add(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), Pow(exp(exp(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(b,l,\\Omega)} = (\\Omega^{l})^{b}, then obtain - \\Omega^{- l} \\operatorname{f^{*}}{(b,l,\\Omega)} \\log{(\\Omega)} + \\Omega^{- l} \\frac{\\partial}{\\partial l} \\operatorname{f^{*}}{(b,l,\\Omega)} = \\Omega^{- l} b (\\Omega^{l})^{b} \\log{(\\Omega)} - \\Omega^{- l} (\\Omega^{l})^{b} \\log{(\\Omega)}", "derivation": "\\operatorname{f^{*}}{(b,l,\\Omega)} = (\\Omega^{l})^{b} and \\Omega^{- l} \\operatorname{f^{*}}{(b,l,\\Omega)} = \\Omega^{- l} (\\Omega^{l})^{b} and \\frac{\\partial}{\\partial l} \\Omega^{- l} \\operatorname{f^{*}}{(b,l,\\Omega)} = \\frac{\\partial}{\\partial l} \\Omega^{- l} (\\Omega^{l})^{b} and - \\Omega^{- l} \\operatorname{f^{*}}{(b,l,\\Omega)} \\log{(\\Omega)} + \\Omega^{- l} \\frac{\\partial}{\\partial l} \\operatorname{f^{*}}{(b,l,\\Omega)} = \\Omega^{- l} b (\\Omega^{l})^{b} \\log{(\\Omega)} - \\Omega^{- l} (\\Omega^{l})^{b} \\log{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('b', commutative=True), Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Function('f^*')(Symbol('b', commutative=True), Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Function('f^*')(Symbol('b', commutative=True), Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Function('f^*')(Symbol('b', commutative=True), Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Derivative(Function('f^*')(Symbol('b', commutative=True), Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))), Add(Mul(Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('b', commutative=True), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Pow(Pow(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)), Symbol('b', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(q)} = \\int \\log{(q)} dq, then derive 1 = \\frac{\\hat{X} + q \\log{(q)} - q}{\\mu_{0}{(q)}}, then obtain 1 = \\frac{\\hat{X} + q \\log{(q)} - q}{\\int \\log{(q)} dq}", "derivation": "\\mu_{0}{(q)} = \\int \\log{(q)} dq and 1 = \\frac{\\int \\log{(q)} dq}{\\mu_{0}{(q)}} and 1 = \\frac{\\hat{X} + q \\log{(q)} - q}{\\mu_{0}{(q)}} and 1 = \\frac{\\hat{X} + q \\log{(q)} - q}{\\int \\log{(q)} dq}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('q', commutative=True)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["divide", 1, "Function('\\\\mu_0')(Symbol('q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mu_0')(Symbol('q', commutative=True)), Integer(-1)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Function('\\\\mu_0')(Symbol('q', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Pow(Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})}, then obtain 0 = - \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{J}_P{(f_{\\mathbf{v}})} + \\frac{d}{d f_{\\mathbf{v}}} \\log{(f_{\\mathbf{v}})}", "derivation": "\\mathbf{J}_P{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{J}_P{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\log{(f_{\\mathbf{v}})} and \\mathbf{J}_P{(f_{\\mathbf{v}})} + \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{J}_P{(f_{\\mathbf{v}})} = \\mathbf{J}_P{(f_{\\mathbf{v}})} + \\frac{d}{d f_{\\mathbf{v}}} \\log{(f_{\\mathbf{v}})} and 0 = - \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{J}_P{(f_{\\mathbf{v}})} + \\frac{d}{d f_{\\mathbf{v}}} \\log{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["add", 2, "Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))))"], [["minus", 3, "Add(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_P')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), Derivative(log(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi{(\\chi,p)} = \\sin^{\\chi}{(p)}, then derive \\frac{\\partial}{\\partial \\chi} \\phi{(\\chi,p)} = \\log{(\\sin{(p)})} \\sin^{\\chi}{(p)}, then obtain \\chi \\frac{\\partial}{\\partial \\chi} \\sin^{\\chi}{(p)} = \\chi \\phi{(\\chi,p)} \\log{(\\sin{(p)})}", "derivation": "\\phi{(\\chi,p)} = \\sin^{\\chi}{(p)} and \\frac{\\partial}{\\partial \\chi} \\phi{(\\chi,p)} = \\frac{\\partial}{\\partial \\chi} \\sin^{\\chi}{(p)} and \\frac{\\partial}{\\partial \\chi} \\phi{(\\chi,p)} = \\log{(\\sin{(p)})} \\sin^{\\chi}{(p)} and \\frac{\\partial}{\\partial \\chi} \\phi{(\\chi,p)} = \\phi{(\\chi,p)} \\log{(\\sin{(p)})} and \\frac{\\partial}{\\partial \\chi} \\sin^{\\chi}{(p)} = \\phi{(\\chi,p)} \\log{(\\sin{(p)})} and \\chi \\frac{\\partial}{\\partial \\chi} \\sin^{\\chi}{(p)} = \\chi \\phi{(\\chi,p)} \\log{(\\sin{(p)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('p', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(log(sin(Symbol('p', commutative=True))), Pow(sin(Symbol('p', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), log(sin(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Pow(sin(Symbol('p', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), log(sin(Symbol('p', commutative=True)))))"], [["times", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Derivative(Pow(sin(Symbol('p', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Symbol('\\\\chi', commutative=True), Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('p', commutative=True)), log(sin(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(v_{y})} = \\cos{(v_{y})}, then obtain \\cos{((2 \\dot{x}{(v_{y})})^{v_{y}})} = \\cos{(2^{v_{y}} \\dot{x}^{v_{y}}{(v_{y})})}", "derivation": "\\dot{x}{(v_{y})} = \\cos{(v_{y})} and \\dot{x}{(v_{y})} + \\cos{(v_{y})} = 2 \\cos{(v_{y})} and (\\dot{x}{(v_{y})} + \\cos{(v_{y})})^{v_{y}} = (2 \\cos{(v_{y})})^{v_{y}} and \\cos{((\\dot{x}{(v_{y})} + \\cos{(v_{y})})^{v_{y}})} = \\cos{((2 \\cos{(v_{y})})^{v_{y}})} and \\cos{((\\dot{x}{(v_{y})} + \\cos{(v_{y})})^{v_{y}})} = \\cos{(2^{v_{y}} \\cos^{v_{y}}{(v_{y})})} and \\cos{((2 \\dot{x}{(v_{y})})^{v_{y}})} = \\cos{(2^{v_{y}} \\dot{x}^{v_{y}}{(v_{y})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True)))"], [["add", 1, "cos(Symbol('v_y', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Mul(Integer(2), cos(Symbol('v_y', commutative=True))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(Mul(Integer(2), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Add(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))), cos(Pow(Mul(Integer(2), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))))"], [["expand", 4], "Equality(cos(Pow(Add(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), cos(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))), cos(Mul(Pow(Integer(2), Symbol('v_y', commutative=True)), Pow(cos(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(cos(Pow(Mul(Integer(2), Function('\\\\dot{x}')(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))), cos(Mul(Pow(Integer(2), Symbol('v_y', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(m_{s})} = e^{m_{s}}, then derive \\int \\Psi^{\\dagger}{(m_{s})} dm_{s} = v_{z} + e^{m_{s}}, then obtain - \\frac{v_{z}}{m_{s}} = - \\frac{- e^{m_{s}} + \\int e^{m_{s}} dm_{s}}{m_{s}}", "derivation": "\\Psi^{\\dagger}{(m_{s})} = e^{m_{s}} and \\int \\Psi^{\\dagger}{(m_{s})} dm_{s} = \\int e^{m_{s}} dm_{s} and - e^{m_{s}} + \\int \\Psi^{\\dagger}{(m_{s})} dm_{s} = - e^{m_{s}} + \\int e^{m_{s}} dm_{s} and \\int \\Psi^{\\dagger}{(m_{s})} dm_{s} = v_{z} + e^{m_{s}} and v_{z} = - e^{m_{s}} + \\int e^{m_{s}} dm_{s} and - \\frac{v_{z}}{m_{s}} = - \\frac{- e^{m_{s}} + \\int e^{m_{s}} dm_{s}}{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["minus", 2, "exp(Symbol('m_s', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('m_s', commutative=True))), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('v_z', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Symbol('v_z', commutative=True), Add(Mul(Integer(-1), exp(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["divide", 5, "Mul(Integer(-1), Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Mul(Integer(-1), Pow(Symbol('m_s', commutative=True), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given k{(A_{1})} = \\sin{(\\sin{(A_{1})})} and \\theta{(A_{1})} = \\sin{(\\sin{(A_{1})})}, then obtain k{(A_{1})} - \\sin{(A_{1})} = \\theta{(A_{1})} - \\sin{(A_{1})}", "derivation": "k{(A_{1})} = \\sin{(\\sin{(A_{1})})} and k{(A_{1})} - \\sin{(A_{1})} = - \\sin{(A_{1})} + \\sin{(\\sin{(A_{1})})} and \\theta{(A_{1})} = \\sin{(\\sin{(A_{1})})} and k{(A_{1})} - \\sin{(A_{1})} = \\theta{(A_{1})} - \\sin{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('A_1', commutative=True)), sin(sin(Symbol('A_1', commutative=True))))"], [["minus", 1, "sin(Symbol('A_1', commutative=True))"], "Equality(Add(Function('k')(Symbol('A_1', commutative=True)), Mul(Integer(-1), sin(Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('A_1', commutative=True))), sin(sin(Symbol('A_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('A_1', commutative=True)), sin(sin(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('k')(Symbol('A_1', commutative=True)), Mul(Integer(-1), sin(Symbol('A_1', commutative=True)))), Add(Function('\\\\theta')(Symbol('A_1', commutative=True)), Mul(Integer(-1), sin(Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given U{(\\delta,u)} = \\sin^{u}{(\\delta)}, then obtain \\frac{\\partial}{\\partial u} (\\int ((U^{\\delta}{(\\delta,u)})^{\\delta})^{\\delta} du)^{u} = \\frac{\\partial}{\\partial u} (\\int (((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta})^{\\delta} du)^{u}", "derivation": "U{(\\delta,u)} = \\sin^{u}{(\\delta)} and U^{\\delta}{(\\delta,u)} = (\\sin^{u}{(\\delta)})^{\\delta} and (U^{\\delta}{(\\delta,u)})^{\\delta} = ((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta} and ((U^{\\delta}{(\\delta,u)})^{\\delta})^{\\delta} = (((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta})^{\\delta} and \\int ((U^{\\delta}{(\\delta,u)})^{\\delta})^{\\delta} du = \\int (((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta})^{\\delta} du and (\\int ((U^{\\delta}{(\\delta,u)})^{\\delta})^{\\delta} du)^{u} = (\\int (((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta})^{\\delta} du)^{u} and \\frac{\\partial}{\\partial u} (\\int ((U^{\\delta}{(\\delta,u)})^{\\delta})^{\\delta} du)^{u} = \\frac{\\partial}{\\partial u} (\\int (((\\sin^{u}{(\\delta)})^{\\delta})^{\\delta})^{\\delta} du)^{u}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Pow(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["integrate", 4, "Symbol('u', commutative=True)"], "Equality(Integral(Pow(Pow(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Pow(Pow(Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["power", 5, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(Pow(Pow(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(Pow(Pow(Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["differentiate", 6, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Integral(Pow(Pow(Pow(Function('U')(Symbol('\\\\delta', commutative=True), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(Pow(Pow(Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('u', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{x}{(\\hat{\\mathbf{r}})} = \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}, then derive \\dot{x}{(\\hat{\\mathbf{r}})} = U - \\cos{(\\hat{\\mathbf{r}})}, then obtain (\\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} = \\dot{x}^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})}", "derivation": "\\dot{x}{(\\hat{\\mathbf{r}})} = \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} and \\dot{x}{(\\hat{\\mathbf{r}})} = U - \\cos{(\\hat{\\mathbf{r}})} and \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} = U - \\cos{(\\hat{\\mathbf{r}})} and (\\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} = (U - \\cos{(\\hat{\\mathbf{r}})})^{\\hat{\\mathbf{r}}} and (\\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}})^{\\hat{\\mathbf{r}}} = \\dot{x}^{\\hat{\\mathbf{r}}}{(\\hat{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Add(Symbol('U', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given V{(\\sigma_p)} = e^{\\sigma_p}, then derive V + V{(\\sigma_p)} = \\hat{H}_{\\lambda} + e^{\\sigma_p}, then obtain V + V{(\\sigma_p)} = \\hat{H}_{\\lambda} + V{(\\sigma_p)}", "derivation": "V{(\\sigma_p)} = e^{\\sigma_p} and \\frac{d}{d \\sigma_p} V{(\\sigma_p)} = \\frac{d}{d \\sigma_p} e^{\\sigma_p} and \\int \\frac{d}{d \\sigma_p} V{(\\sigma_p)} d\\sigma_p = \\int \\frac{d}{d \\sigma_p} e^{\\sigma_p} d\\sigma_p and V + V{(\\sigma_p)} = \\hat{H}_{\\lambda} + e^{\\sigma_p} and V + V{(\\sigma_p)} = \\hat{H}_{\\lambda} + V{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Derivative(Function('V')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Derivative(exp(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('V', commutative=True), Function('V')(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('V', commutative=True), Function('V')(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('V')(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(A_{1})} = \\cos{(A_{1})}, then derive \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})} = - \\sin{(A_{1})}, then obtain - A_{1} \\sin{(A_{1})} = A_{1} \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})}", "derivation": "\\hat{p}_0{(A_{1})} = \\cos{(A_{1})} and \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})} = \\frac{d}{d A_{1}} \\cos{(A_{1})} and A_{1} \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})} = A_{1} \\frac{d}{d A_{1}} \\cos{(A_{1})} and \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})} = - \\sin{(A_{1})} and - A_{1} \\sin{(A_{1})} = A_{1} \\frac{d}{d A_{1}} \\cos{(A_{1})} and - A_{1} \\sin{(A_{1})} = A_{1} \\frac{d}{d A_{1}} \\hat{p}_0{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["times", 2, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Symbol('A_1', commutative=True), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('A_1', commutative=True), sin(Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Derivative(cos(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Symbol('A_1', commutative=True), sin(Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Derivative(Function('\\\\hat{p}_0')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})} = \\mathbf{D} + x^\\prime, then obtain 1 = (0^{\\mathbf{D}})^{\\mathbf{D}}", "derivation": "\\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})} = \\mathbf{D} + x^\\prime and - \\mathbf{D} - x^\\prime + \\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})} = 0 and (- \\mathbf{D} - x^\\prime + \\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})})^{\\mathbf{D}} = 0^{\\mathbf{D}} and ((- \\mathbf{D} - x^\\prime + \\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}} = (0^{\\mathbf{D}})^{\\mathbf{D}} and 1 = ((- \\mathbf{D} - x^\\prime + \\operatorname{y^{\\prime}}{(x^\\prime,\\mathbf{D})})^{\\mathbf{D}})^{\\mathbf{D}} and 1 = (0^{\\mathbf{D}})^{\\mathbf{D}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('y^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Pow(Pow(Integer(0), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given Q{(t_{1})} = \\cos{(t_{1})} and \\operatorname{P_{g}}{(\\mathbf{v},\\dot{z})} = \\dot{z} \\mathbf{v}, then obtain \\operatorname{P_{g}}{(\\mathbf{v},\\dot{z})} \\cos{(t_{1})} = \\dot{z} \\mathbf{v} \\cos{(t_{1})}", "derivation": "Q{(t_{1})} = \\cos{(t_{1})} and \\operatorname{P_{g}}{(\\mathbf{v},\\dot{z})} = \\dot{z} \\mathbf{v} and \\operatorname{P_{g}}{(\\mathbf{v},\\dot{z})} Q{(t_{1})} = \\dot{z} \\mathbf{v} Q{(t_{1})} and \\operatorname{P_{g}}{(\\mathbf{v},\\dot{z})} \\cos{(t_{1})} = \\dot{z} \\mathbf{v} \\cos{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 2, "Function('Q')(Symbol('t_1', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Function('Q')(Symbol('t_1', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), Function('Q')(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('P_g')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('t_1', commutative=True))), Mul(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True), cos(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(h)} = \\sin{(h)}, then obtain 1 - \\mathbf{F}^{2}{(h)} = (\\frac{\\sin{(h)}}{\\mathbf{F}{(h)}})^{h} - \\mathbf{F}^{2}{(h)}", "derivation": "\\mathbf{F}{(h)} = \\sin{(h)} and \\mathbf{F}^{2}{(h)} = \\mathbf{F}{(h)} \\sin{(h)} and 1 = \\frac{\\sin{(h)}}{\\mathbf{F}{(h)}} and 1 = (\\frac{\\sin{(h)}}{\\mathbf{F}{(h)}})^{h} and - \\mathbf{F}{(h)} \\sin{(h)} + 1 = (\\frac{\\sin{(h)}}{\\mathbf{F}{(h)}})^{h} - \\mathbf{F}{(h)} \\sin{(h)} and 1 - \\mathbf{F}^{2}{(h)} = (\\frac{\\sin{(h)}}{\\mathbf{F}{(h)}})^{h} - \\mathbf{F}^{2}{(h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{F}')(Symbol('h', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(-1)), sin(Symbol('h', commutative=True))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(-1)), sin(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["minus", 4, "Mul(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Integer(1)), Add(Pow(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(-1)), sin(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(2)))), Add(Pow(Mul(Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(-1)), sin(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{F}')(Symbol('h', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(x^\\prime)} = \\log{(x^\\prime)}, then obtain (\\frac{d}{d x^\\prime} (x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} + \\log{(x^\\prime)})^{x^\\prime})^{2} = (\\frac{d}{d x^\\prime} (x^\\prime + 2 \\log{(x^\\prime)})^{x^\\prime})^{2}", "derivation": "\\operatorname{f_{E}}{(x^\\prime)} = \\log{(x^\\prime)} and x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} = x^\\prime + \\log{(x^\\prime)} and x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} + \\log{(x^\\prime)} = x^\\prime + 2 \\log{(x^\\prime)} and (x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} + \\log{(x^\\prime)})^{x^\\prime} = (x^\\prime + 2 \\log{(x^\\prime)})^{x^\\prime} and \\frac{d}{d x^\\prime} (x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} + \\log{(x^\\prime)})^{x^\\prime} = \\frac{d}{d x^\\prime} (x^\\prime + 2 \\log{(x^\\prime)})^{x^\\prime} and (\\frac{d}{d x^\\prime} (x^\\prime + \\operatorname{f_{E}}{(x^\\prime)} + \\log{(x^\\prime)})^{x^\\prime})^{2} = (\\frac{d}{d x^\\prime} (x^\\prime + 2 \\log{(x^\\prime)})^{x^\\prime})^{2}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('f_E')(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), log(Symbol('x^\\\\prime', commutative=True))))"], [["add", 2, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Symbol('x^\\\\prime', commutative=True), Function('f_E')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Add(Symbol('x^\\\\prime', commutative=True), Function('f_E')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('x^\\\\prime', commutative=True), Function('f_E')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 5, 2], "Equality(Pow(Derivative(Pow(Add(Symbol('x^\\\\prime', commutative=True), Function('f_E')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Pow(Add(Symbol('x^\\\\prime', commutative=True), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{A}{(v_{x})} = e^{v_{x}}, then obtain \\log{(\\int \\frac{d}{d v_{x}} (2 \\mathbf{A}{(v_{x})} - e^{v_{x}}) dv_{x})} = \\log{(\\int \\frac{d}{d v_{x}} \\mathbf{A}{(v_{x})} dv_{x})}", "derivation": "\\mathbf{A}{(v_{x})} = e^{v_{x}} and 2 \\mathbf{A}{(v_{x})} = \\mathbf{A}{(v_{x})} + e^{v_{x}} and 2 \\mathbf{A}{(v_{x})} - e^{v_{x}} = \\mathbf{A}{(v_{x})} and \\frac{d}{d v_{x}} (2 \\mathbf{A}{(v_{x})} - e^{v_{x}}) = \\frac{d}{d v_{x}} \\mathbf{A}{(v_{x})} and \\int \\frac{d}{d v_{x}} (2 \\mathbf{A}{(v_{x})} - e^{v_{x}}) dv_{x} = \\int \\frac{d}{d v_{x}} \\mathbf{A}{(v_{x})} dv_{x} and \\log{(\\int \\frac{d}{d v_{x}} (2 \\mathbf{A}{(v_{x})} - e^{v_{x}}) dv_{x})} = \\log{(\\int \\frac{d}{d v_{x}} \\mathbf{A}{(v_{x})} dv_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Add(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["minus", 2, "exp(Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))), Integral(Derivative(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True))))"], [["log", 5], "Equality(log(Integral(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True))), Mul(Integer(-1), exp(Symbol('v_x', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True)))), log(Integral(Derivative(Function('\\\\mathbf{A}')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(b)} = e^{b} and U{(b)} = \\frac{d}{d b} \\operatorname{y^{\\prime}}{(b)}, then obtain U{(b)} = \\frac{d}{d b} e^{b}", "derivation": "\\operatorname{y^{\\prime}}{(b)} = e^{b} and \\frac{d}{d b} \\operatorname{y^{\\prime}}{(b)} = \\frac{d}{d b} e^{b} and U{(b)} = \\frac{d}{d b} \\operatorname{y^{\\prime}}{(b)} and U{(b)} = \\frac{d}{d b} e^{b}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('U')(Symbol('b', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('U')(Symbol('b', commutative=True)), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{1})} = \\sin{(\\sin{(A_{1})})}, then obtain x^\\prime (\\operatorname{F_{x}}^{A_{1}}{(A_{1})})^{A_{1}} = x^\\prime (\\sin^{A_{1}}{(\\sin{(A_{1})})})^{A_{1}}", "derivation": "\\operatorname{F_{x}}{(A_{1})} = \\sin{(\\sin{(A_{1})})} and \\operatorname{F_{x}}^{A_{1}}{(A_{1})} = \\sin^{A_{1}}{(\\sin{(A_{1})})} and (\\operatorname{F_{x}}^{A_{1}}{(A_{1})})^{A_{1}} = (\\sin^{A_{1}}{(\\sin{(A_{1})})})^{A_{1}} and x^\\prime (\\operatorname{F_{x}}^{A_{1}}{(A_{1})})^{A_{1}} = x^\\prime (\\sin^{A_{1}}{(\\sin{(A_{1})})})^{A_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('F_x')(Symbol('A_1', commutative=True)), sin(sin(Symbol('A_1', commutative=True))))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(sin(sin(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Pow(Function('F_x')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Pow(sin(sin(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["times", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Pow(Pow(Function('F_x')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), Pow(Pow(sin(sin(Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\eta{(\\phi_1)} = \\log{(\\phi_1)}, then obtain (2 \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)}) \\int - \\log{(\\phi_1)} d\\phi_1 = 0", "derivation": "\\eta{(\\phi_1)} = \\log{(\\phi_1)} and \\eta{(\\phi_1)} - \\log{(\\phi_1)} = 0 and \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)} = - \\log{(\\phi_1)} and 2 \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)} = \\eta{(\\phi_1)} - \\log{(\\phi_1)} and \\int (\\eta{(\\phi_1)} - 2 \\log{(\\phi_1)}) d\\phi_1 = \\int - \\log{(\\phi_1)} d\\phi_1 and 2 \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)} = 0 and (2 \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)}) \\int (\\eta{(\\phi_1)} - 2 \\log{(\\phi_1)}) d\\phi_1 = 0 and (2 \\eta{(\\phi_1)} - 2 \\log{(\\phi_1)}) \\int - \\log{(\\phi_1)} d\\phi_1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"], [["add", 2, "Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True))))"], [["add", 2, "Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Add(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"], [["times", 6, "Integral(Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Integral(Add(Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\eta')(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\phi_1', commutative=True)))), Integral(Mul(Integer(-1), log(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\varepsilon_{0}{(f_{E})} = \\sin{(f_{E})} and \\operatorname{F_{c}}{(f_{E})} = \\frac{\\sin{(f_{E})}}{\\varepsilon_{0}{(f_{E})}}, then obtain 1 = \\operatorname{F_{c}}{(f_{E})}", "derivation": "\\varepsilon_{0}{(f_{E})} = \\sin{(f_{E})} and 1 = \\frac{\\sin{(f_{E})}}{\\varepsilon_{0}{(f_{E})}} and \\operatorname{F_{c}}{(f_{E})} = \\frac{\\sin{(f_{E})}}{\\varepsilon_{0}{(f_{E})}} and 1 = \\operatorname{F_{c}}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["divide", 1, "Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), Integer(-1)), sin(Symbol('f_E', commutative=True))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('f_E', commutative=True)), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('f_E', commutative=True)), Integer(-1)), sin(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Function('F_c')(Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\Psi_{\\lambda}{(\\dot{z})} = - \\operatorname{E_{\\lambda}}^{2}{(\\dot{z})}, then obtain \\Psi_{\\lambda}{(\\dot{z})} = - \\operatorname{E_{\\lambda}}{(\\dot{z})} \\cos{(\\dot{z})}", "derivation": "\\operatorname{E_{\\lambda}}{(\\dot{z})} = \\cos{(\\dot{z})} and \\operatorname{E_{\\lambda}}^{2}{(\\dot{z})} = \\operatorname{E_{\\lambda}}{(\\dot{z})} \\cos{(\\dot{z})} and \\operatorname{E_{\\lambda}}^{2}{(\\dot{z})} - \\operatorname{E_{\\lambda}}{(\\dot{z})} \\cos{(\\dot{z})} = 0 and - \\operatorname{E_{\\lambda}}{(\\dot{z})} \\cos{(\\dot{z})} = - \\operatorname{E_{\\lambda}}^{2}{(\\dot{z})} and \\Psi_{\\lambda}{(\\dot{z})} = - \\operatorname{E_{\\lambda}}^{2}{(\\dot{z})} and \\Psi_{\\lambda}{(\\dot{z})} = - \\operatorname{E_{\\lambda}}{(\\dot{z})} \\cos{(\\dot{z})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["times", 1, "Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), Mul(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["minus", 2, "Mul(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], "Equality(Add(Pow(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2)), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))), Integer(0))"], [["minus", 3, "Pow(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2))"], "Equality(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{J}_M,v_{z})} = \\frac{v_{z}}{\\mathbf{J}_M}, then obtain \\int (- \\mathbf{J}_M + \\rho_{f}{(\\mathbf{J}_M,v_{z})} - 1)^{v_{z}} d\\mathbf{J}_M = \\int (- \\mathbf{J}_M - 1 + \\frac{v_{z}}{\\mathbf{J}_M})^{v_{z}} d\\mathbf{J}_M", "derivation": "\\rho_{f}{(\\mathbf{J}_M,v_{z})} = \\frac{v_{z}}{\\mathbf{J}_M} and - \\mathbf{J}_M + \\rho_{f}{(\\mathbf{J}_M,v_{z})} = - \\mathbf{J}_M + \\frac{v_{z}}{\\mathbf{J}_M} and - \\mathbf{J}_M + \\rho_{f}{(\\mathbf{J}_M,v_{z})} - 1 = - \\mathbf{J}_M - 1 + \\frac{v_{z}}{\\mathbf{J}_M} and (- \\mathbf{J}_M + \\rho_{f}{(\\mathbf{J}_M,v_{z})} - 1)^{v_{z}} = (- \\mathbf{J}_M - 1 + \\frac{v_{z}}{\\mathbf{J}_M})^{v_{z}} and \\int (- \\mathbf{J}_M + \\rho_{f}{(\\mathbf{J}_M,v_{z})} - 1)^{v_{z}} d\\mathbf{J}_M = \\int (- \\mathbf{J}_M - 1 + \\frac{v_{z}}{\\mathbf{J}_M})^{v_{z}} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["power", 3, "Symbol('v_z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(-1), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given m{(z^{*},A_{z})} = (e^{z^{*}})^{A_{z}}, then obtain - \\frac{\\partial}{\\partial A_{z}} (A_{z} + 2 m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}} - 1) = - \\frac{\\partial}{\\partial A_{z}} (A_{z} + m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}})", "derivation": "m{(z^{*},A_{z})} = (e^{z^{*}})^{A_{z}} and m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}} = 1 and A_{z} + m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}} = A_{z} + 1 and \\frac{\\partial}{\\partial A_{z}} (A_{z} + m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}}) = \\frac{d}{d A_{z}} (A_{z} + 1) and - \\frac{\\partial}{\\partial A_{z}} (A_{z} + m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}}) = - \\frac{d}{d A_{z}} (A_{z} + 1) and - \\frac{\\partial}{\\partial A_{z}} (A_{z} + 2 m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}} - 1) = - \\frac{\\partial}{\\partial A_{z}} (A_{z} + m{(z^{*},A_{z})} (e^{z^{*}})^{- A_{z}})", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Symbol('A_z', commutative=True)))"], [["divide", 1, "Pow(exp(Symbol('z^*', commutative=True)), Symbol('A_z', commutative=True))"], "Equality(Mul(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True)))), Integer(1))"], [["minus", 2, "Mul(Integer(-1), Symbol('A_z', commutative=True))"], "Equality(Add(Symbol('A_z', commutative=True), Mul(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True))))), Add(Symbol('A_z', commutative=True), Integer(1)))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Symbol('A_z', commutative=True), Mul(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True))))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Mul(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True))))), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Mul(Integer(2), Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True)))), Integer(-1)), Tuple(Symbol('A_z', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('A_z', commutative=True), Mul(Function('m')(Symbol('z^*', commutative=True), Symbol('A_z', commutative=True)), Pow(exp(Symbol('z^*', commutative=True)), Mul(Integer(-1), Symbol('A_z', commutative=True))))), Tuple(Symbol('A_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given B{(W,l)} = - \\sin{(W - l)} and \\dot{z}{(W,l)} = - B{(W,l)} \\sin{(W - l)}, then obtain e^{\\dot{z}{(W,l)}} = e^{- B{(W,l)} \\sin{(W - l)}}", "derivation": "B{(W,l)} = - \\sin{(W - l)} and - B{(W,l)} \\sin{(W - l)} = \\sin^{2}{(W - l)} and e^{- B{(W,l)} \\sin{(W - l)}} = e^{\\sin^{2}{(W - l)}} and \\dot{z}{(W,l)} = - B{(W,l)} \\sin{(W - l)} and e^{\\dot{z}{(W,l)}} = e^{\\sin^{2}{(W - l)}} and e^{\\dot{z}{(W,l)}} = e^{- B{(W,l)} \\sin{(W - l)}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["times", 1, "Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))"], "Equality(Mul(Integer(-1), Function('B')(Symbol('W', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Pow(sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(2)))"], [["exp", 2], "Equality(exp(Mul(Integer(-1), Function('B')(Symbol('W', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))), exp(Pow(sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('W', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(exp(Function('\\\\dot{z}')(Symbol('W', commutative=True), Symbol('l', commutative=True))), exp(Pow(sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(exp(Function('\\\\dot{z}')(Symbol('W', commutative=True), Symbol('l', commutative=True))), exp(Mul(Integer(-1), Function('B')(Symbol('W', commutative=True), Symbol('l', commutative=True)), sin(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))))"]]}, {"prompt": "Given G{(v_{y})} = e^{v_{y}}, then obtain (G{(v_{y})} - e^{v_{y}}) G^{2}{(v_{y})} = 0", "derivation": "G{(v_{y})} = e^{v_{y}} and G{(v_{y})} - e^{v_{y}} = 0 and (G{(v_{y})} - e^{v_{y}}) G{(v_{y})} = 0 and (G{(v_{y})} - e^{v_{y}}) G^{2}{(v_{y})} = 0", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('v_y', commutative=True)), exp(Symbol('v_y', commutative=True)))"], [["minus", 1, "exp(Symbol('v_y', commutative=True))"], "Equality(Add(Function('G')(Symbol('v_y', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))), Integer(0))"], [["times", 2, "Function('G')(Symbol('v_y', commutative=True))"], "Equality(Mul(Add(Function('G')(Symbol('v_y', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))), Function('G')(Symbol('v_y', commutative=True))), Integer(0))"], [["times", 3, "Function('G')(Symbol('v_y', commutative=True))"], "Equality(Mul(Add(Function('G')(Symbol('v_y', commutative=True)), Mul(Integer(-1), exp(Symbol('v_y', commutative=True)))), Pow(Function('G')(Symbol('v_y', commutative=True)), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\chi)} = \\sin{(\\chi)}, then obtain 3 \\chi + 3 \\operatorname{v_{2}}{(\\chi)} = 3 \\chi + \\operatorname{v_{2}}{(\\chi)} + 2 \\sin{(\\chi)}", "derivation": "\\operatorname{v_{2}}{(\\chi)} = \\sin{(\\chi)} and \\chi + \\operatorname{v_{2}}{(\\chi)} = \\chi + \\sin{(\\chi)} and 2 \\chi + 2 \\operatorname{v_{2}}{(\\chi)} = 2 \\chi + \\operatorname{v_{2}}{(\\chi)} + \\sin{(\\chi)} and 3 \\chi + 3 \\operatorname{v_{2}}{(\\chi)} = 3 \\chi + 2 \\operatorname{v_{2}}{(\\chi)} + \\sin{(\\chi)} and 4 \\chi + 2 \\operatorname{v_{2}}{(\\chi)} = 4 \\chi + \\operatorname{v_{2}}{(\\chi)} + \\sin{(\\chi)} and 3 \\chi + 2 \\operatorname{v_{2}}{(\\chi)} = 3 \\chi + \\operatorname{v_{2}}{(\\chi)} + \\sin{(\\chi)} and 3 \\chi + 3 \\operatorname{v_{2}}{(\\chi)} = 3 \\chi + \\operatorname{v_{2}}{(\\chi)} + 2 \\sin{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Function('v_2')(Symbol('\\\\chi', commutative=True))), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\chi', commutative=True), Function('v_2')(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\chi', commutative=True)), Function('v_2')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\chi', commutative=True), Function('v_2')(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Mul(Integer(3), Function('v_2')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('\\\\chi', commutative=True))), sin(Symbol('\\\\chi', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(4), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(4), Symbol('\\\\chi', commutative=True)), Function('v_2')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True))))"], [["minus", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Function('v_2')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Function('v_2')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Mul(Integer(3), Function('v_2')(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(3), Symbol('\\\\chi', commutative=True)), Function('v_2')(Symbol('\\\\chi', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(\\mathbf{r})} = \\sin{(e^{\\mathbf{r}})}, then obtain \\sin{(e^{\\mathbf{r}})} + \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\mathbf{g}{(\\mathbf{r})} d\\mathbf{r} = \\sin{(e^{\\mathbf{r}})} + \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\sin{(e^{\\mathbf{r}})} d\\mathbf{r}", "derivation": "\\mathbf{g}{(\\mathbf{r})} = \\sin{(e^{\\mathbf{r}})} and \\mathbf{r} \\mathbf{g}{(\\mathbf{r})} = \\mathbf{r} \\sin{(e^{\\mathbf{r}})} and \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\mathbf{g}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\sin{(e^{\\mathbf{r}})} and \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\mathbf{g}{(\\mathbf{r})} d\\mathbf{r} = \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\sin{(e^{\\mathbf{r}})} d\\mathbf{r} and \\sin{(e^{\\mathbf{r}})} + \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\mathbf{g}{(\\mathbf{r})} d\\mathbf{r} = \\sin{(e^{\\mathbf{r}})} + \\int \\frac{d}{d \\mathbf{r}} \\mathbf{r} \\sin{(e^{\\mathbf{r}})} d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{r}', commutative=True)), sin(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 4, "sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(sin(exp(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Add(sin(exp(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\lambda)} = \\sin{(\\lambda)}, then obtain 2 \\int \\sin{(\\lambda)} d\\lambda + \\sin{(2)} = \\sin{(\\frac{\\varepsilon{(\\lambda)} + \\sin{(\\lambda)}}{\\varepsilon{(\\lambda)}})} + 2 \\int \\sin{(\\lambda)} d\\lambda", "derivation": "\\varepsilon{(\\lambda)} = \\sin{(\\lambda)} and 2 \\varepsilon{(\\lambda)} = \\varepsilon{(\\lambda)} + \\sin{(\\lambda)} and 2 = \\frac{\\varepsilon{(\\lambda)} + \\sin{(\\lambda)}}{\\varepsilon{(\\lambda)}} and \\sin{(2)} = \\sin{(\\frac{\\varepsilon{(\\lambda)} + \\sin{(\\lambda)}}{\\varepsilon{(\\lambda)}})} and 2 \\int \\sin{(\\lambda)} d\\lambda + \\sin{(2)} = \\sin{(\\frac{\\varepsilon{(\\lambda)} + \\sin{(\\lambda)}}{\\varepsilon{(\\lambda)}})} + 2 \\int \\sin{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True))), Add(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))))"], [["divide", 2, "Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True))"], "Equality(Integer(2), Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), Integer(-1))))"], [["sin", 3], "Equality(sin(Integer(2)), sin(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), Integer(-1)))))"], [["add", 4, "Mul(Integer(2), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], "Equality(Add(Mul(Integer(2), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), sin(Integer(2))), Add(sin(Mul(Add(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Pow(Function('\\\\varepsilon')(Symbol('\\\\lambda', commutative=True)), Integer(-1)))), Mul(Integer(2), Integral(sin(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\varphi{(F_{x})} = \\log{(\\cos{(F_{x})})}, then obtain F_{x} + \\varphi{(F_{x})} - \\frac{\\varphi{(F_{x})}}{F_{x}} = F_{x} + \\log{(\\cos{(F_{x})})} - \\frac{\\varphi{(F_{x})}}{F_{x}}", "derivation": "\\varphi{(F_{x})} = \\log{(\\cos{(F_{x})})} and \\frac{\\varphi{(F_{x})}}{F_{x}} = \\frac{\\log{(\\cos{(F_{x})})}}{F_{x}} and F_{x} + \\varphi{(F_{x})} = F_{x} + \\log{(\\cos{(F_{x})})} and F_{x} + \\varphi{(F_{x})} - \\frac{\\log{(\\cos{(F_{x})})}}{F_{x}} = F_{x} + \\log{(\\cos{(F_{x})})} - \\frac{\\log{(\\cos{(F_{x})})}}{F_{x}} and F_{x} + \\varphi{(F_{x})} - \\frac{\\varphi{(F_{x})}}{F_{x}} = F_{x} + \\log{(\\cos{(F_{x})})} - \\frac{\\varphi{(F_{x})}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('F_x', commutative=True)), log(cos(Symbol('F_x', commutative=True))))"], [["divide", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), log(cos(Symbol('F_x', commutative=True)))))"], [["add", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\varphi')(Symbol('F_x', commutative=True))), Add(Symbol('F_x', commutative=True), log(cos(Symbol('F_x', commutative=True)))))"], [["minus", 3, "Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), log(cos(Symbol('F_x', commutative=True))))"], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\varphi')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), log(cos(Symbol('F_x', commutative=True))))), Add(Symbol('F_x', commutative=True), log(cos(Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), log(cos(Symbol('F_x', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('F_x', commutative=True), Function('\\\\varphi')(Symbol('F_x', commutative=True)), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True)))), Add(Symbol('F_x', commutative=True), log(cos(Symbol('F_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given a{(\\lambda)} = \\cos{(\\sin{(\\lambda)})} and \\operatorname{t_{2}}{(\\lambda)} = \\frac{d}{d \\lambda} a{(\\lambda)}, then derive \\operatorname{t_{2}}{(\\lambda)} = - \\sin{(\\sin{(\\lambda)})} \\cos{(\\lambda)}, then obtain \\int - \\sin{(\\sin{(\\lambda)})} \\cos{(\\lambda)} d\\lambda = \\int \\frac{d}{d \\lambda} a{(\\lambda)} d\\lambda", "derivation": "a{(\\lambda)} = \\cos{(\\sin{(\\lambda)})} and \\operatorname{t_{2}}{(\\lambda)} = \\frac{d}{d \\lambda} a{(\\lambda)} and \\operatorname{t_{2}}{(\\lambda)} = \\frac{d}{d \\lambda} \\cos{(\\sin{(\\lambda)})} and \\operatorname{t_{2}}{(\\lambda)} = - \\sin{(\\sin{(\\lambda)})} \\cos{(\\lambda)} and \\int \\operatorname{t_{2}}{(\\lambda)} d\\lambda = \\int \\frac{d}{d \\lambda} a{(\\lambda)} d\\lambda and \\int - \\sin{(\\sin{(\\lambda)})} \\cos{(\\lambda)} d\\lambda = \\int \\frac{d}{d \\lambda} a{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\lambda', commutative=True)), cos(sin(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\lambda', commutative=True)), Derivative(Function('a')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('t_2')(Symbol('\\\\lambda', commutative=True)), Derivative(cos(sin(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Function('t_2')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('\\\\lambda', commutative=True))), cos(Symbol('\\\\lambda', commutative=True))))"], [["integrate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Derivative(Function('a')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Mul(Integer(-1), sin(sin(Symbol('\\\\lambda', commutative=True))), cos(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Derivative(Function('a')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Tuple(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given h{(E_{n})} = e^{E_{n}}, then obtain h^{E_{n}}{(E_{n})} + \\int h{(E_{n})} \\sin{(h{(E_{n})})} \\sin{(e^{E_{n}})} dE_{n} = h^{E_{n}}{(E_{n})} + \\int h{(E_{n})} \\sin^{2}{(e^{E_{n}})} dE_{n}", "derivation": "h{(E_{n})} = e^{E_{n}} and \\sin{(h{(E_{n})})} = \\sin{(e^{E_{n}})} and \\sin{(h{(E_{n})})} \\sin{(e^{E_{n}})} = \\sin^{2}{(e^{E_{n}})} and h{(E_{n})} \\sin{(h{(E_{n})})} \\sin{(e^{E_{n}})} = h{(E_{n})} \\sin^{2}{(e^{E_{n}})} and \\int h{(E_{n})} \\sin{(h{(E_{n})})} \\sin{(e^{E_{n}})} dE_{n} = \\int h{(E_{n})} \\sin^{2}{(e^{E_{n}})} dE_{n} and h^{E_{n}}{(E_{n})} + \\int h{(E_{n})} \\sin{(h{(E_{n})})} \\sin{(e^{E_{n}})} dE_{n} = h^{E_{n}}{(E_{n})} + \\int h{(E_{n})} \\sin^{2}{(e^{E_{n}})} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["sin", 1], "Equality(sin(Function('h')(Symbol('E_n', commutative=True))), sin(exp(Symbol('E_n', commutative=True))))"], [["times", 2, "sin(exp(Symbol('E_n', commutative=True)))"], "Equality(Mul(sin(Function('h')(Symbol('E_n', commutative=True))), sin(exp(Symbol('E_n', commutative=True)))), Pow(sin(exp(Symbol('E_n', commutative=True))), Integer(2)))"], [["times", 3, "Function('h')(Symbol('E_n', commutative=True))"], "Equality(Mul(Function('h')(Symbol('E_n', commutative=True)), sin(Function('h')(Symbol('E_n', commutative=True))), sin(exp(Symbol('E_n', commutative=True)))), Mul(Function('h')(Symbol('E_n', commutative=True)), Pow(sin(exp(Symbol('E_n', commutative=True))), Integer(2))))"], [["integrate", 4, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Function('h')(Symbol('E_n', commutative=True)), sin(Function('h')(Symbol('E_n', commutative=True))), sin(exp(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True))), Integral(Mul(Function('h')(Symbol('E_n', commutative=True)), Pow(sin(exp(Symbol('E_n', commutative=True))), Integer(2))), Tuple(Symbol('E_n', commutative=True))))"], [["add", 5, "Pow(Function('h')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True))"], "Equality(Add(Pow(Function('h')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Integral(Mul(Function('h')(Symbol('E_n', commutative=True)), sin(Function('h')(Symbol('E_n', commutative=True))), sin(exp(Symbol('E_n', commutative=True)))), Tuple(Symbol('E_n', commutative=True)))), Add(Pow(Function('h')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Integral(Mul(Function('h')(Symbol('E_n', commutative=True)), Pow(sin(exp(Symbol('E_n', commutative=True))), Integer(2))), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given h{(l,r)} = e^{l - r}, then derive \\frac{\\partial}{\\partial r} h{(l,r)} = - e^{l - r}, then obtain \\frac{\\partial}{\\partial r} e^{l - r} = - e^{l - r}", "derivation": "h{(l,r)} = e^{l - r} and \\frac{\\partial}{\\partial r} h{(l,r)} = \\frac{\\partial}{\\partial r} e^{l - r} and \\frac{\\partial}{\\partial r} h{(l,r)} = - e^{l - r} and \\frac{\\partial}{\\partial r} h{(l,r)} = - h{(l,r)} and \\frac{\\partial}{\\partial r} e^{l - r} = - e^{l - r}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('l', commutative=True), Symbol('r', commutative=True)), exp(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('h')(Symbol('l', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Integer(-1), Function('h')(Symbol('l', commutative=True), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\hat{H})} = \\log{(\\cos{(\\hat{H})})}, then obtain \\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H} + (\\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H})^{\\hat{H}} = \\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H} + (\\iint \\log{(\\cos{(\\hat{H})})} d\\hat{H} d\\hat{H})^{\\hat{H}}", "derivation": "\\operatorname{P_{g}}{(\\hat{H})} = \\log{(\\cos{(\\hat{H})})} and \\int \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} = \\int \\log{(\\cos{(\\hat{H})})} d\\hat{H} and \\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H} = \\iint \\log{(\\cos{(\\hat{H})})} d\\hat{H} d\\hat{H} and (\\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H})^{\\hat{H}} = (\\iint \\log{(\\cos{(\\hat{H})})} d\\hat{H} d\\hat{H})^{\\hat{H}} and \\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H} + (\\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H})^{\\hat{H}} = \\iint \\operatorname{P_{g}}{(\\hat{H})} d\\hat{H} d\\hat{H} + (\\iint \\log{(\\cos{(\\hat{H})})} d\\hat{H} d\\hat{H})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), log(cos(Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(log(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(log(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(log(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["add", 4, "Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Add(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Pow(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), Add(Integral(Function('P_g')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Pow(Integral(log(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\mathbf{J}_P,F_{H})} = F_{H} + \\mathbf{J}_P, then obtain 2 (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} - \\mathbf{M}{(\\mathbf{J}_P,F_{H})} = (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} + (2 F_{H} + \\mathbf{J}_P)^{F_{H}} - \\mathbf{M}{(\\mathbf{J}_P,F_{H})}", "derivation": "\\mathbf{M}{(\\mathbf{J}_P,F_{H})} = F_{H} + \\mathbf{J}_P and F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})} = 2 F_{H} + \\mathbf{J}_P and (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} = (2 F_{H} + \\mathbf{J}_P)^{F_{H}} and 2 (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} = (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} + (2 F_{H} + \\mathbf{J}_P)^{F_{H}} and 2 (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} - \\mathbf{M}{(\\mathbf{J}_P,F_{H})} = (F_{H} + \\mathbf{M}{(\\mathbf{J}_P,F_{H})})^{F_{H}} + (2 F_{H} + \\mathbf{J}_P)^{F_{H}} - \\mathbf{M}{(\\mathbf{J}_P,F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('F_H', commutative=True)))"], [["add", 3, "Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(2), Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))), Add(Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('F_H', commutative=True))))"], [["minus", 4, "Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True)))), Add(Pow(Add(Symbol('F_H', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('F_H', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(k)} = \\cos{(k)} and \\operatorname{f^{*}}{(k)} = \\sigma_{x}{(k)} \\cos{(k)} and \\operatorname{t_{1}}{(k)} = \\sigma_{x}{(k)} \\cos{(k)}, then obtain \\operatorname{t_{1}}^{k}{(k)} = \\operatorname{f^{*}}^{k}{(k)}", "derivation": "\\sigma_{x}{(k)} = \\cos{(k)} and \\operatorname{f^{*}}{(k)} = \\sigma_{x}{(k)} \\cos{(k)} and \\operatorname{f^{*}}{(k)} = \\cos^{2}{(k)} and \\operatorname{t_{1}}{(k)} = \\sigma_{x}{(k)} \\cos{(k)} and \\operatorname{t_{1}}{(k)} = \\cos^{2}{(k)} and \\operatorname{t_{1}}^{k}{(k)} = (\\cos^{2}{(k)})^{k} and \\operatorname{t_{1}}^{k}{(k)} = \\operatorname{f^{*}}^{k}{(k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('k', commutative=True)), Mul(Function('\\\\sigma_x')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f^*')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('k', commutative=True)), Mul(Function('\\\\sigma_x')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('t_1')(Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Integer(2)))"], [["power", 5, "Symbol('k', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(cos(Symbol('k', commutative=True)), Integer(2)), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Function('t_1')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Function('f^*')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a n}{f^{\\prime}}, then derive \\frac{\\partial}{\\partial n} \\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a}{f^{\\prime}}, then obtain \\cos{(\\frac{a n}{f^{\\prime}})} + \\frac{\\partial}{\\partial n} \\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a}{f^{\\prime}} + \\cos{(\\frac{a n}{f^{\\prime}})}", "derivation": "\\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a n}{f^{\\prime}} and \\hat{H}_l{(a,f^{\\prime},n)} - \\frac{1}{f^{\\prime}} = \\frac{a n}{f^{\\prime}} - \\frac{1}{f^{\\prime}} and \\frac{\\partial}{\\partial n} (\\hat{H}_l{(a,f^{\\prime},n)} - \\frac{1}{f^{\\prime}}) = \\frac{\\partial}{\\partial n} (\\frac{a n}{f^{\\prime}} - \\frac{1}{f^{\\prime}}) and \\frac{\\partial}{\\partial n} \\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a}{f^{\\prime}} and \\cos{(\\frac{a n}{f^{\\prime}})} + \\frac{\\partial}{\\partial n} \\hat{H}_l{(a,f^{\\prime},n)} = \\frac{a}{f^{\\prime}} + \\cos{(\\frac{a n}{f^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], [["minus", 1, "Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Add(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["add", 4, "cos(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))"], "Equality(Add(cos(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True))), Derivative(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), cos(Mul(Symbol('a', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(g,\\phi)} = \\phi^{g} and C{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})}, then obtain \\frac{\\frac{\\partial}{\\partial \\phi} \\varepsilon_{0}{(g,\\phi)}}{\\log{(\\cos{(\\tilde{g})})}} = \\frac{\\frac{\\partial}{\\partial \\phi} \\phi^{g}}{\\log{(\\cos{(\\tilde{g})})}}", "derivation": "\\varepsilon_{0}{(g,\\phi)} = \\phi^{g} and C{(\\tilde{g})} = \\log{(\\cos{(\\tilde{g})})} and \\frac{\\partial}{\\partial \\phi} \\varepsilon_{0}{(g,\\phi)} = \\frac{\\partial}{\\partial \\phi} \\phi^{g} and \\frac{\\frac{\\partial}{\\partial \\phi} \\varepsilon_{0}{(g,\\phi)}}{C{(\\tilde{g})}} = \\frac{\\frac{\\partial}{\\partial \\phi} \\phi^{g}}{C{(\\tilde{g})}} and \\frac{\\frac{\\partial}{\\partial \\phi} \\varepsilon_{0}{(g,\\phi)}}{\\log{(\\cos{(\\tilde{g})})}} = \\frac{\\frac{\\partial}{\\partial \\phi} \\phi^{g}}{\\log{(\\cos{(\\tilde{g})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)))"], ["get_premise", "Equality(Function('C')(Symbol('\\\\tilde{g}', commutative=True)), log(cos(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["divide", 3, "Function('C')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Pow(Function('C')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(Function('C')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(log(cos(Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Derivative(Function('\\\\varepsilon_0')(Symbol('g', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Pow(log(cos(Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Derivative(Pow(Symbol('\\\\phi', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given U{(\\mu)} = \\cos{(\\cos{(\\mu)})}, then obtain \\int U^{3}{(\\mu)} d\\mu = \\int U{(\\mu)} \\cos^{2}{(\\cos{(\\mu)})} d\\mu", "derivation": "U{(\\mu)} = \\cos{(\\cos{(\\mu)})} and U^{2}{(\\mu)} = U{(\\mu)} \\cos{(\\cos{(\\mu)})} and U^{3}{(\\mu)} = U^{2}{(\\mu)} \\cos{(\\cos{(\\mu)})} and U^{3}{(\\mu)} = U{(\\mu)} \\cos^{2}{(\\cos{(\\mu)})} and \\int U^{3}{(\\mu)} d\\mu = \\int U{(\\mu)} \\cos^{2}{(\\cos{(\\mu)})} d\\mu", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Function('U')(Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('U')(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Function('U')(Symbol('\\\\mu', commutative=True)), cos(cos(Symbol('\\\\mu', commutative=True)))))"], [["times", 2, "Function('U')(Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('U')(Symbol('\\\\mu', commutative=True)), Integer(3)), Mul(Pow(Function('U')(Symbol('\\\\mu', commutative=True)), Integer(2)), cos(cos(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('U')(Symbol('\\\\mu', commutative=True)), Integer(3)), Mul(Function('U')(Symbol('\\\\mu', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(2))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Pow(Function('U')(Symbol('\\\\mu', commutative=True)), Integer(3)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Function('U')(Symbol('\\\\mu', commutative=True)), Pow(cos(cos(Symbol('\\\\mu', commutative=True))), Integer(2))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given b{(x,i)} = i^{x}, then derive - i + \\frac{\\partial}{\\partial i} b{(x,i)} = - i + \\frac{i^{x} x}{i}, then obtain \\log{(- i + \\frac{i^{x} x}{i})} = \\log{(- i + \\frac{x b{(x,i)}}{i})}", "derivation": "b{(x,i)} = i^{x} and \\frac{\\partial}{\\partial i} b{(x,i)} = \\frac{\\partial}{\\partial i} i^{x} and - i + \\frac{\\partial}{\\partial i} b{(x,i)} = - i + \\frac{\\partial}{\\partial i} i^{x} and - i + \\frac{\\partial}{\\partial i} b{(x,i)} = - i + \\frac{i^{x} x}{i} and - i + \\frac{\\partial}{\\partial i} b{(x,i)} = - i + \\frac{x b{(x,i)}}{i} and \\log{(- i + \\frac{\\partial}{\\partial i} b{(x,i)})} = \\log{(- i + \\frac{x b{(x,i)}}{i})} and \\log{(- i + \\frac{\\partial}{\\partial i} i^{x})} = \\log{(- i + \\frac{x b{(x,i)}}{i})} and \\log{(- i + \\frac{i^{x} x}{i})} = \\log{(- i + \\frac{x b{(x,i)}}{i})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)))))"], [["log", 5], "Equality(log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Derivative(Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True))))))"], [["evaluate_derivatives", 7], "Equality(log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Symbol('x', commutative=True), Function('b')(Symbol('x', commutative=True), Symbol('i', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(E_{n},F_{H})} = \\sin{(E_{n} F_{H})} and \\operatorname{c_{0}}{(E_{n},F_{H})} = E_{n} F_{H}, then obtain \\int \\operatorname{f_{\\mathbf{v}}}{(E_{n},F_{H})} dE_{n} = \\int \\sin{(\\operatorname{c_{0}}{(E_{n},F_{H})})} dE_{n}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(E_{n},F_{H})} = \\sin{(E_{n} F_{H})} and \\operatorname{c_{0}}{(E_{n},F_{H})} = E_{n} F_{H} and \\operatorname{f_{\\mathbf{v}}}{(E_{n},F_{H})} = \\sin{(\\operatorname{c_{0}}{(E_{n},F_{H})})} and \\int \\operatorname{f_{\\mathbf{v}}}{(E_{n},F_{H})} dE_{n} = \\int \\sin{(\\operatorname{c_{0}}{(E_{n},F_{H})})} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True)), sin(Mul(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True)), sin(Function('c_0')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(sin(Function('c_0')(Symbol('E_n', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(S,z^{*})} = - z^{*} + \\log{(S)}, then obtain - S + \\int \\frac{\\operatorname{A_{1}}{(S,z^{*})}}{S} dS = - S + \\int \\frac{- z^{*} + \\log{(S)}}{S} dS", "derivation": "\\operatorname{A_{1}}{(S,z^{*})} = - z^{*} + \\log{(S)} and \\frac{\\operatorname{A_{1}}{(S,z^{*})}}{S} = \\frac{- z^{*} + \\log{(S)}}{S} and \\int \\frac{\\operatorname{A_{1}}{(S,z^{*})}}{S} dS = \\int \\frac{- z^{*} + \\log{(S)}}{S} dS and - S + \\int \\frac{\\operatorname{A_{1}}{(S,z^{*})}}{S} dS = - S + \\int \\frac{- z^{*} + \\log{(S)}}{S} dS", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('S', commutative=True), Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('S', commutative=True))))"], [["divide", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('A_1')(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('S', commutative=True)))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('A_1')(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))"], [["minus", 3, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Function('A_1')(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), log(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)))))"]]}, {"prompt": "Given i{(l,v_{z})} = l + v_{z}, then derive i{(l,v_{z})} + \\frac{\\partial}{\\partial l} i{(l,v_{z})} = i{(l,v_{z})} + 1, then obtain (i{(l,v_{z})} + \\frac{\\partial}{\\partial l} i{(l,v_{z})})^{l} = (i{(l,v_{z})} + 1)^{l}", "derivation": "i{(l,v_{z})} = l + v_{z} and \\frac{\\partial}{\\partial l} i{(l,v_{z})} = \\frac{\\partial}{\\partial l} (l + v_{z}) and i{(l,v_{z})} + \\frac{\\partial}{\\partial l} i{(l,v_{z})} = i{(l,v_{z})} + \\frac{\\partial}{\\partial l} (l + v_{z}) and i{(l,v_{z})} + \\frac{\\partial}{\\partial l} i{(l,v_{z})} = i{(l,v_{z})} + 1 and l + v_{z} + \\frac{\\partial}{\\partial l} (l + v_{z}) = l + v_{z} + 1 and (l + v_{z} + \\frac{\\partial}{\\partial l} (l + v_{z}))^{l} = (l + v_{z} + 1)^{l} and (i{(l,v_{z})} + \\frac{\\partial}{\\partial l} i{(l,v_{z})})^{l} = (i{(l,v_{z})} + 1)^{l}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["add", 2, "Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Derivative(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Derivative(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Derivative(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True), Derivative(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True), Integer(1)))"], [["power", 5, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True), Derivative(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)), Pow(Add(Symbol('l', commutative=True), Symbol('v_z', commutative=True), Integer(1)), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Derivative(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Symbol('l', commutative=True)), Pow(Add(Function('i')(Symbol('l', commutative=True), Symbol('v_z', commutative=True)), Integer(1)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given E{(r)} = r and C{(r)} = \\frac{d}{d r} E{(r)}, then derive (\\frac{d}{d r} \\int C{(r)} dr)^{r} = (\\frac{\\partial}{\\partial r} (G + r))^{r}, then obtain \\frac{(\\frac{d}{d r} \\int C{(r)} dr)^{r}}{\\int C{(r)} dr} = \\frac{(\\frac{\\partial}{\\partial r} (G + r))^{r}}{\\int C{(r)} dr}", "derivation": "E{(r)} = r and \\frac{d}{d r} E{(r)} = \\frac{d}{d r} r and C{(r)} = \\frac{d}{d r} E{(r)} and C{(r)} = \\frac{d}{d r} r and \\int C{(r)} dr = \\int \\frac{d}{d r} r dr and \\frac{d}{d r} \\int C{(r)} dr = \\frac{d}{d r} \\int \\frac{d}{d r} r dr and (\\frac{d}{d r} \\int C{(r)} dr)^{r} = (\\frac{d}{d r} \\int \\frac{d}{d r} r dr)^{r} and (\\frac{d}{d r} \\int C{(r)} dr)^{r} = (\\frac{\\partial}{\\partial r} (G + r))^{r} and \\frac{(\\frac{d}{d r} \\int C{(r)} dr)^{r}}{\\int C{(r)} dr} = \\frac{(\\frac{\\partial}{\\partial r} (G + r))^{r}}{\\int C{(r)} dr}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('r', commutative=True)), Symbol('r', commutative=True))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C')(Symbol('r', commutative=True)), Derivative(Function('E')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('C')(Symbol('r', commutative=True)), Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integral(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 6, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Integral(Derivative(Symbol('r', commutative=True), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["evaluate_integrals", 7], "Equality(Pow(Derivative(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Add(Symbol('G', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["divide", 8, "Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Derivative(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))), Mul(Pow(Derivative(Add(Symbol('G', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Integral(Function('C')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\sigma_{p}{(M_{E})} = \\sin{(\\sin{(M_{E})})}, then obtain (\\int \\frac{\\sigma_{p}{(M_{E})}}{\\sin{(\\sin{(M_{E})})}} dM_{E})^{M_{E}} = (M + M_{E})^{M_{E}}", "derivation": "\\sigma_{p}{(M_{E})} = \\sin{(\\sin{(M_{E})})} and \\frac{\\sigma_{p}{(M_{E})}}{\\sin{(\\sin{(M_{E})})}} = 1 and \\int \\frac{\\sigma_{p}{(M_{E})}}{\\sin{(\\sin{(M_{E})})}} dM_{E} = \\int 1 dM_{E} and (\\int \\frac{\\sigma_{p}{(M_{E})}}{\\sin{(\\sin{(M_{E})})}} dM_{E})^{M_{E}} = (\\int 1 dM_{E})^{M_{E}} and (\\int \\frac{\\sigma_{p}{(M_{E})}}{\\sin{(\\sin{(M_{E})})}} dM_{E})^{M_{E}} = (M + M_{E})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('M_E', commutative=True)), sin(sin(Symbol('M_E', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('M_E', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('M_E', commutative=True)), Pow(sin(sin(Symbol('M_E', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Function('\\\\sigma_p')(Symbol('M_E', commutative=True)), Pow(sin(sin(Symbol('M_E', commutative=True))), Integer(-1))), Tuple(Symbol('M_E', commutative=True))), Integral(Integer(1), Tuple(Symbol('M_E', commutative=True))))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Integral(Mul(Function('\\\\sigma_p')(Symbol('M_E', commutative=True)), Pow(sin(sin(Symbol('M_E', commutative=True))), Integer(-1))), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Function('\\\\sigma_p')(Symbol('M_E', commutative=True)), Pow(sin(sin(Symbol('M_E', commutative=True))), Integer(-1))), Tuple(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Add(Symbol('M', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})} = \\hat{p}_0 \\mathbf{F}, then derive \\int (- \\hat{p}_0 \\mathbf{F} + \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})})^{\\mathbf{F}} d\\mathbf{F} = A_{x}, then obtain \\int 0^{\\mathbf{F}} d\\mathbf{F} = A_{x}", "derivation": "\\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})} = \\hat{p}_0 \\mathbf{F} and - \\hat{p}_0 \\mathbf{F} + \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})} = 0 and (- \\hat{p}_0 \\mathbf{F} + \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})})^{\\mathbf{F}} = 0^{\\mathbf{F}} and \\int (- \\hat{p}_0 \\mathbf{F} + \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})})^{\\mathbf{F}} d\\mathbf{F} = \\int 0^{\\mathbf{F}} d\\mathbf{F} and \\int (- \\hat{p}_0 \\mathbf{F} + \\operatorname{c_{0}}{(\\hat{p}_0,\\mathbf{F})})^{\\mathbf{F}} d\\mathbf{F} = A_{x} and \\int 0^{\\mathbf{F}} d\\mathbf{F} = A_{x}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('c_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('c_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('c_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('c_0')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('A_x', commutative=True))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Pow(Integer(0), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('A_x', commutative=True))"]]}, {"prompt": "Given \\hat{x}_0{(v_{y})} = v_{y}, then derive \\frac{d}{d v_{y}} \\hat{x}_0{(v_{y})} = 1, then obtain (\\frac{d}{d v_{y}} v_{y} - \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS)^{S} = (- \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS + 1)^{S}", "derivation": "\\hat{x}_0{(v_{y})} = v_{y} and \\frac{d}{d v_{y}} \\hat{x}_0{(v_{y})} = \\frac{d}{d v_{y}} v_{y} and \\frac{d}{d v_{y}} \\hat{x}_0{(v_{y})} = 1 and \\frac{d}{d v_{y}} \\hat{x}_0{(v_{y})} - \\int \\sin{(S)} dS = 1 - \\int \\sin{(S)} dS and \\frac{d}{d v_{y}} v_{y} - \\int \\sin{(S)} dS = 1 - \\int \\sin{(S)} dS and \\frac{d}{d v_{y}} v_{y} - \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS = - \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS + 1 and (\\frac{d}{d v_{y}} v_{y} - \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS)^{S} = (- \\int \\sin{(S)} dS - \\int \\sin{(\\int \\sin{(S)} dS)} dS + 1)^{S}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Symbol('v_y', commutative=True), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\hat{x}_0')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Symbol('v_y', commutative=True), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))))"], [["minus", 5, "Integral(sin(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)))"], "Equality(Add(Derivative(Symbol('v_y', commutative=True), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Integral(sin(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))), Add(Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Integral(sin(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)))), Integer(1)))"], [["power", 6, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Derivative(Symbol('v_y', commutative=True), Tuple(Symbol('v_y', commutative=True), Integer(1))), Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Integral(sin(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))))), Symbol('S', commutative=True)), Pow(Add(Mul(Integer(-1), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Integral(sin(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)))), Integer(1)), Symbol('S', commutative=True)))"]]}, {"prompt": "Given E{(\\Omega)} = e^{\\Omega}, then obtain \\log{(E^{\\Omega}{(\\Omega)} - e^{\\Omega} - (e^{\\Omega})^{\\Omega})} = \\log{(- e^{\\Omega})}", "derivation": "E{(\\Omega)} = e^{\\Omega} and E^{\\Omega}{(\\Omega)} = (e^{\\Omega})^{\\Omega} and E^{\\Omega}{(\\Omega)} - (e^{\\Omega})^{\\Omega} = 0 and E^{\\Omega}{(\\Omega)} - e^{\\Omega} - (e^{\\Omega})^{\\Omega} = - e^{\\Omega} and \\log{(E^{\\Omega}{(\\Omega)} - e^{\\Omega} - (e^{\\Omega})^{\\Omega})} = \\log{(- e^{\\Omega})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["minus", 2, "Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Function('E')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))), Integer(0))"], [["minus", 3, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Function('E')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))))"], [["log", 4], "Equality(log(Add(Pow(Function('E')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))), log(Mul(Integer(-1), exp(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\ddot{x})} = e^{\\ddot{x}}, then obtain (2 e^{\\ddot{x}})^{\\ddot{x}} (2 \\mathbf{M}{(\\ddot{x})} + e^{\\ddot{x}})^{\\ddot{x}} = (2 e^{\\ddot{x}})^{\\ddot{x}} (\\mathbf{M}{(\\ddot{x})} + 2 e^{\\ddot{x}})^{\\ddot{x}}", "derivation": "\\mathbf{M}{(\\ddot{x})} = e^{\\ddot{x}} and \\mathbf{M}{(\\ddot{x})} + e^{\\ddot{x}} = 2 e^{\\ddot{x}} and 2 \\mathbf{M}{(\\ddot{x})} + e^{\\ddot{x}} = \\mathbf{M}{(\\ddot{x})} + 2 e^{\\ddot{x}} and (2 \\mathbf{M}{(\\ddot{x})} + e^{\\ddot{x}})^{\\ddot{x}} = (\\mathbf{M}{(\\ddot{x})} + 2 e^{\\ddot{x}})^{\\ddot{x}} and (2 e^{\\ddot{x}})^{\\ddot{x}} (2 \\mathbf{M}{(\\ddot{x})} + e^{\\ddot{x}})^{\\ddot{x}} = (2 e^{\\ddot{x}})^{\\ddot{x}} (\\mathbf{M}{(\\ddot{x})} + 2 e^{\\ddot{x}})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('\\\\ddot{x}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True))), exp(Symbol('\\\\ddot{x}', commutative=True))), Add(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True)))))"], [["power", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True))), exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 4, "Pow(Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True))), exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Pow(Add(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\ddot{x}', commutative=True)))), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given b{(A_{2},g_{\\varepsilon},u)} = A_{2}^{u} g_{\\varepsilon} and \\mathbf{J}_f{(A_{2},g_{\\varepsilon},u)} = \\frac{1}{b{(A_{2},g_{\\varepsilon},u)}}, then obtain \\frac{\\partial}{\\partial u} (\\frac{A_{2}^{- u}}{g_{\\varepsilon}})^{u} = \\frac{\\partial}{\\partial u} (\\frac{1}{b{(A_{2},g_{\\varepsilon},u)}})^{u}", "derivation": "b{(A_{2},g_{\\varepsilon},u)} = A_{2}^{u} g_{\\varepsilon} and \\mathbf{J}_f{(A_{2},g_{\\varepsilon},u)} = \\frac{1}{b{(A_{2},g_{\\varepsilon},u)}} and \\mathbf{J}_f{(A_{2},g_{\\varepsilon},u)} = \\frac{A_{2}^{- u}}{g_{\\varepsilon}} and \\frac{A_{2}^{- u}}{g_{\\varepsilon}} = \\frac{1}{b{(A_{2},g_{\\varepsilon},u)}} and (\\frac{A_{2}^{- u}}{g_{\\varepsilon}})^{u} = (\\frac{1}{b{(A_{2},g_{\\varepsilon},u)}})^{u} and \\frac{\\partial}{\\partial u} (\\frac{A_{2}^{- u}}{g_{\\varepsilon}})^{u} = \\frac{\\partial}{\\partial u} (\\frac{1}{b{(A_{2},g_{\\varepsilon},u)}})^{u}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Symbol('u', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Pow(Function('b')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1))), Pow(Function('b')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Integer(-1)))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1))), Symbol('u', commutative=True)), Pow(Pow(Function('b')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Symbol('u', commutative=True)))"], [["differentiate", 5, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Pow(Function('b')(Symbol('A_2', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(x^\\prime)} = e^{x^\\prime}, then obtain \\frac{d}{d x^\\prime} \\int \\operatorname{F_{c}}{(x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} (\\rho_b + e^{x^\\prime})", "derivation": "\\operatorname{F_{c}}{(x^\\prime)} = e^{x^\\prime} and \\int \\operatorname{F_{c}}{(x^\\prime)} dx^\\prime = \\int e^{x^\\prime} dx^\\prime and \\frac{d}{d x^\\prime} \\int \\operatorname{F_{c}}{(x^\\prime)} dx^\\prime = \\frac{d}{d x^\\prime} \\int e^{x^\\prime} dx^\\prime and \\frac{d}{d x^\\prime} \\int \\operatorname{F_{c}}{(x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} (\\rho_b + e^{x^\\prime})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('F_c')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('F_c')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('F_c')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_b', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(u)} = u, then derive \\int \\operatorname{v_{t}}{(u)} du = \\varepsilon + \\frac{u^{2}}{2}, then obtain \\iiint \\operatorname{v_{t}}{(u)} du du d\\varepsilon = \\iint (\\varepsilon + \\frac{u^{2}}{2}) du d\\varepsilon", "derivation": "\\operatorname{v_{t}}{(u)} = u and \\int \\operatorname{v_{t}}{(u)} du = \\int u du and \\int \\operatorname{v_{t}}{(u)} du = \\varepsilon + \\frac{u^{2}}{2} and \\iint \\operatorname{v_{t}}{(u)} du du = \\int (\\varepsilon + \\frac{u^{2}}{2}) du and \\iiint \\operatorname{v_{t}}{(u)} du du d\\varepsilon = \\iint (\\varepsilon + \\frac{u^{2}}{2}) du d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Symbol('u', commutative=True), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('u', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('u', commutative=True), Integer(2)))), Tuple(Symbol('u', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(Q)} = \\sin{(Q)} and \\operatorname{E_{\\lambda}}{(Q)} = \\int \\sin{(Q)} dQ, then derive \\operatorname{E_{\\lambda}}{(Q)} = \\mathbf{f} - \\cos{(Q)}, then obtain \\operatorname{E_{\\lambda}}{(Q)} - \\mathbf{P}{(Q)} \\sin{(Q)} + \\int \\sin{(Q)} dQ = \\mathbf{f} - \\mathbf{P}{(Q)} \\sin{(Q)} - \\cos{(Q)} + \\int \\sin{(Q)} dQ", "derivation": "\\mathbf{P}{(Q)} = \\sin{(Q)} and \\operatorname{E_{\\lambda}}{(Q)} = \\int \\sin{(Q)} dQ and \\operatorname{E_{\\lambda}}{(Q)} = \\mathbf{f} - \\cos{(Q)} and \\operatorname{E_{\\lambda}}{(Q)} + \\int \\mathbf{P}{(Q)} dQ = \\mathbf{f} - \\cos{(Q)} + \\int \\mathbf{P}{(Q)} dQ and \\operatorname{E_{\\lambda}}{(Q)} + \\int \\sin{(Q)} dQ = \\mathbf{f} - \\cos{(Q)} + \\int \\sin{(Q)} dQ and \\operatorname{E_{\\lambda}}{(Q)} - \\mathbf{P}{(Q)} \\sin{(Q)} + \\int \\sin{(Q)} dQ = \\mathbf{f} - \\mathbf{P}{(Q)} \\sin{(Q)} - \\cos{(Q)} + \\int \\sin{(Q)} dQ", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True)))))"], [["add", 3, "Integral(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True))), Integral(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), cos(Symbol('Q', commutative=True))), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["minus", 5, "Mul(Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True)))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('Q', commutative=True)), sin(Symbol('Q', commutative=True))), Mul(Integer(-1), cos(Symbol('Q', commutative=True))), Integral(sin(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\Omega)} = \\log{(\\Omega)}, then derive t_{1} + \\mathbf{J}_f{(\\Omega)} = f + \\log{(\\Omega)}, then obtain t_{1} + \\log{(\\Omega)} = t_{1} + \\mathbf{J}_f{(\\Omega)}", "derivation": "\\mathbf{J}_f{(\\Omega)} = \\log{(\\Omega)} and \\frac{d}{d \\Omega} \\mathbf{J}_f{(\\Omega)} = \\frac{d}{d \\Omega} \\log{(\\Omega)} and \\int \\frac{d}{d \\Omega} \\mathbf{J}_f{(\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} \\log{(\\Omega)} d\\Omega and t_{1} + \\mathbf{J}_f{(\\Omega)} = f + \\log{(\\Omega)} and t_{1} + \\mathbf{J}_f{(\\Omega)} = f + \\mathbf{J}_f{(\\Omega)} and t_{1} + \\log{(\\Omega)} = f + \\log{(\\Omega)} and f + \\log{(\\Omega)} = f + \\mathbf{J}_f{(\\Omega)} and t_{1} + \\log{(\\Omega)} = f + \\mathbf{J}_f{(\\Omega)} and t_{1} + \\log{(\\Omega)} = t_{1} + \\mathbf{J}_f{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(log(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('f', commutative=True), log(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('f', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Add(Symbol('f', commutative=True), log(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Symbol('f', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Add(Symbol('f', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Add(Symbol('f', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Add(Symbol('t_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Add(Symbol('t_1', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\nabla{(A)} = \\sin{(A)}, then derive \\frac{d}{d A} - \\int \\nabla{(A)} dA = \\frac{\\partial}{\\partial A} (- \\mu_0 + \\cos{(A)}), then obtain \\frac{d}{d A} - \\int \\nabla{(A)} dA = - \\sin{(A)}", "derivation": "\\nabla{(A)} = \\sin{(A)} and \\int \\nabla{(A)} dA = \\int \\sin{(A)} dA and - \\int \\nabla{(A)} dA = - \\int \\sin{(A)} dA and \\frac{d}{d A} - \\int \\nabla{(A)} dA = \\frac{d}{d A} - \\int \\sin{(A)} dA and \\frac{d}{d A} - \\int \\nabla{(A)} dA = \\frac{\\partial}{\\partial A} (- \\mu_0 + \\cos{(A)}) and \\frac{d}{d A} - \\int \\nabla{(A)} dA = - \\sin{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Integer(-1), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Mul(Integer(-1), Integral(Function('\\\\nabla')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(T)} = \\log{(T)} and \\mathbf{S}{(T)} = \\log{(T)}, then obtain \\int \\frac{\\psi^* \\cos{(\\frac{\\hat{p}{(T)}}{\\log{(T)}})}}{\\cos{(1)}} dT = \\int \\psi^* dT", "derivation": "\\hat{p}{(T)} = \\log{(T)} and \\frac{\\hat{p}{(T)}}{\\log{(T)}} = 1 and \\mathbf{S}{(T)} = \\log{(T)} and \\cos{(\\frac{\\hat{p}{(T)}}{\\log{(T)}})} = \\cos{(1)} and \\cos{(\\frac{\\hat{p}{(T)}}{\\mathbf{S}{(T)}})} = \\cos{(1)} and \\frac{\\psi^* \\cos{(\\frac{\\hat{p}{(T)}}{\\mathbf{S}{(T)}})}}{\\cos{(1)}} = \\psi^* and \\frac{\\psi^* \\cos{(\\frac{\\hat{p}{(T)}}{\\log{(T)}})}}{\\cos{(1)}} = \\psi^* and \\int \\frac{\\psi^* \\cos{(\\frac{\\hat{p}{(T)}}{\\log{(T)}})}}{\\cos{(1)}} dT = \\int \\psi^* dT", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["divide", 1, "log(Symbol('T', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["cos", 2], "Equality(cos(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Integer(-1)))), cos(Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(cos(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('T', commutative=True)), Integer(-1)))), cos(Integer(1)))"], [["divide", 5, "Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), cos(Integer(1)))"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Pow(cos(Integer(1)), Integer(-1)), cos(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(Function('\\\\mathbf{S}')(Symbol('T', commutative=True)), Integer(-1))))), Symbol('\\\\psi^*', commutative=True))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Pow(cos(Integer(1)), Integer(-1)), cos(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Integer(-1))))), Symbol('\\\\psi^*', commutative=True))"], [["integrate", 7, "Symbol('T', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\psi^*', commutative=True), Pow(cos(Integer(1)), Integer(-1)), cos(Mul(Function('\\\\hat{p}')(Symbol('T', commutative=True)), Pow(log(Symbol('T', commutative=True)), Integer(-1))))), Tuple(Symbol('T', commutative=True))), Integral(Symbol('\\\\psi^*', commutative=True), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given A{(\\nabla,F_{x})} = \\frac{\\log{(\\nabla)}}{F_{x}}, then obtain \\frac{A^{2}{(\\nabla,F_{x})} - \\frac{1}{F_{x}}}{F_{x}} = \\frac{\\frac{A{(\\nabla,F_{x})} \\log{(\\nabla)}}{F_{x}} - \\frac{1}{F_{x}}}{F_{x}}", "derivation": "A{(\\nabla,F_{x})} = \\frac{\\log{(\\nabla)}}{F_{x}} and A^{2}{(\\nabla,F_{x})} = \\frac{A{(\\nabla,F_{x})} \\log{(\\nabla)}}{F_{x}} and A^{2}{(\\nabla,F_{x})} - \\frac{1}{F_{x}} = \\frac{A{(\\nabla,F_{x})} \\log{(\\nabla)}}{F_{x}} - \\frac{1}{F_{x}} and \\frac{A^{2}{(\\nabla,F_{x})} - \\frac{1}{F_{x}}}{F_{x}} = \\frac{\\frac{A{(\\nabla,F_{x})} \\log{(\\nabla)}}{F_{x}} - \\frac{1}{F_{x}}}{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), log(Symbol('\\\\nabla', commutative=True))))"], [["times", 1, "Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Pow(Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))"], [["minus", 2, "Pow(Symbol('F_x', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1)))))"], [["divide", 3, "Symbol('F_x', commutative=True)"], "Equality(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Add(Pow(Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), Integer(2)), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1))))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\nabla', commutative=True), Symbol('F_x', commutative=True)), log(Symbol('\\\\nabla', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(t_{1},A)} = A + t_{1}, then obtain (\\frac{\\partial}{\\partial A} \\int \\operatorname{P_{g}}{(t_{1},A)} dt_{1})^{t_{1}} = (\\frac{\\partial}{\\partial A} \\int (A + t_{1}) dt_{1})^{t_{1}}", "derivation": "\\operatorname{P_{g}}{(t_{1},A)} = A + t_{1} and \\int \\operatorname{P_{g}}{(t_{1},A)} dt_{1} = \\int (A + t_{1}) dt_{1} and \\frac{\\partial}{\\partial A} \\int \\operatorname{P_{g}}{(t_{1},A)} dt_{1} = \\frac{\\partial}{\\partial A} \\int (A + t_{1}) dt_{1} and (\\frac{\\partial}{\\partial A} \\int \\operatorname{P_{g}}{(t_{1},A)} dt_{1})^{t_{1}} = (\\frac{\\partial}{\\partial A} \\int (A + t_{1}) dt_{1})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('t_1', commutative=True)))"], [["integrate", 1, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Add(Symbol('A', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Integral(Function('P_g')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('A', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('P_g')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('t_1', commutative=True)), Pow(Derivative(Integral(Add(Symbol('A', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{D}{(V,l,\\psi)} = V - \\psi - l and \\operatorname{f_{E}}{(V,l,\\psi)} = \\frac{\\partial}{\\partial V} (V - \\psi - l)^{V}, then obtain \\frac{\\partial}{\\partial V} \\mathbf{D}^{V}{(V,l,\\psi)} = \\operatorname{f_{E}}{(V,l,\\psi)}", "derivation": "\\mathbf{D}{(V,l,\\psi)} = V - \\psi - l and \\mathbf{D}^{V}{(V,l,\\psi)} = (V - \\psi - l)^{V} and \\frac{\\partial}{\\partial V} \\mathbf{D}^{V}{(V,l,\\psi)} = \\frac{\\partial}{\\partial V} (V - \\psi - l)^{V} and \\operatorname{f_{E}}{(V,l,\\psi)} = \\frac{\\partial}{\\partial V} (V - \\psi - l)^{V} and \\frac{\\partial}{\\partial V} \\mathbf{D}^{V}{(V,l,\\psi)} = \\operatorname{f_{E}}{(V,l,\\psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('V', commutative=True)), Pow(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('V', commutative=True)))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)), Derivative(Pow(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Function('f_E')(Symbol('V', commutative=True), Symbol('l', commutative=True), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given \\Omega{(c,\\sigma_p)} = \\frac{\\sigma_p}{c}, then obtain - c + \\frac{\\int \\Omega{(c,\\sigma_p)} d\\sigma_p}{c} = - c + \\frac{\\int \\frac{\\sigma_p}{c} d\\sigma_p}{c}", "derivation": "\\Omega{(c,\\sigma_p)} = \\frac{\\sigma_p}{c} and \\int \\Omega{(c,\\sigma_p)} d\\sigma_p = \\int \\frac{\\sigma_p}{c} d\\sigma_p and \\frac{\\int \\Omega{(c,\\sigma_p)} d\\sigma_p}{c} = \\frac{\\int \\frac{\\sigma_p}{c} d\\sigma_p}{c} and - c + \\frac{\\int \\Omega{(c,\\sigma_p)} d\\sigma_p}{c} = - c + \\frac{\\int \\frac{\\sigma_p}{c} d\\sigma_p}{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 2, "Symbol('c', commutative=True)"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_p', commutative=True)))))"], [["minus", 3, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Function('\\\\Omega')(Symbol('c', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))))"]]}, {"prompt": "Given H{(\\mathbf{M})} = \\sin{(\\mathbf{M})}, then obtain \\frac{2 H{(\\mathbf{M})}}{\\mathbf{J}_P^{3}} = \\frac{2 \\sin{(\\mathbf{M})}}{\\mathbf{J}_P^{3}}", "derivation": "H{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\frac{H{(\\mathbf{M})}}{\\mathbf{J}_P} = \\frac{\\sin{(\\mathbf{M})}}{\\mathbf{J}_P} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\frac{H{(\\mathbf{M})}}{\\mathbf{J}_P} = \\frac{\\partial}{\\partial \\mathbf{J}_P} \\frac{\\sin{(\\mathbf{M})}}{\\mathbf{J}_P} and \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\frac{H{(\\mathbf{M})}}{\\mathbf{J}_P} = \\frac{\\partial^{2}}{\\partial \\mathbf{J}_P^{2}} \\frac{\\sin{(\\mathbf{M})}}{\\mathbf{J}_P} and \\frac{2 H{(\\mathbf{M})}}{\\mathbf{J}_P^{3}} = \\frac{2 \\sin{(\\mathbf{M})}}{\\mathbf{J}_P^{3}}", "srepr_derivation": [["get_premise", "Equality(Function('H')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Function('H')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Function('H')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Function('H')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-3)), Function('H')(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-3)), sin(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\omega)} = \\int e^{\\omega} d\\omega, then derive e^{- \\omega + \\pi{(\\omega)}} = e^{\\hat{H}_l - \\omega + e^{\\omega}}, then obtain e^{- \\omega + \\int e^{\\omega} d\\omega} = e^{\\hat{H}_l - \\omega + e^{\\omega}}", "derivation": "\\pi{(\\omega)} = \\int e^{\\omega} d\\omega and - \\omega + \\pi{(\\omega)} = - \\omega + \\int e^{\\omega} d\\omega and e^{- \\omega + \\pi{(\\omega)}} = e^{- \\omega + \\int e^{\\omega} d\\omega} and e^{- \\omega + \\pi{(\\omega)}} = e^{\\hat{H}_l - \\omega + e^{\\omega}} and e^{- \\omega + \\int e^{\\omega} d\\omega} = e^{\\hat{H}_l - \\omega + e^{\\omega}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\pi')(Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["exp", 2], "Equality(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\pi')(Symbol('\\\\omega', commutative=True)))), exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"], [["evaluate_integrals", 3], "Equality(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\pi')(Symbol('\\\\omega', commutative=True)))), exp(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(exp(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), exp(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given y{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} and \\theta_{1}{(C_{2})} = \\log{(C_{2})}, then obtain l + \\frac{\\theta_{1}{(C_{2})}}{l (- A_{1} + \\dot{y})} = l + \\frac{\\log{(C_{2})}}{l (- A_{1} + \\dot{y})}", "derivation": "y{(A_{1},\\dot{y})} = - A_{1} + \\dot{y} and \\theta_{1}{(C_{2})} = \\log{(C_{2})} and \\frac{\\theta_{1}{(C_{2})}}{y{(A_{1},\\dot{y})}} = \\frac{\\log{(C_{2})}}{y{(A_{1},\\dot{y})}} and \\frac{\\theta_{1}{(C_{2})}}{l y{(A_{1},\\dot{y})}} = \\frac{\\log{(C_{2})}}{l y{(A_{1},\\dot{y})}} and \\frac{\\theta_{1}{(C_{2})}}{l (- A_{1} + \\dot{y})} = \\frac{\\log{(C_{2})}}{l (- A_{1} + \\dot{y})} and l + \\frac{\\theta_{1}{(C_{2})}}{l (- A_{1} + \\dot{y})} = l + \\frac{\\log{(C_{2})}}{l (- A_{1} + \\dot{y})}", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["times", 2, "Pow(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Pow(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Mul(Pow(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), log(Symbol('C_2', commutative=True))))"], [["divide", 3, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Pow(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Function('y')(Symbol('A_1', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), log(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), log(Symbol('C_2', commutative=True))))"], [["add", 5, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), Function('\\\\theta_1')(Symbol('C_2', commutative=True)))), Add(Symbol('l', commutative=True), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), log(Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})}, then derive \\operatorname{t_{1}}{(v_{2})} = \\frac{1}{v_{2}}, then obtain \\frac{d}{d v_{2}} (- v_{2} + \\operatorname{t_{1}}{(v_{2})}) = \\frac{d}{d v_{2}} (- v_{2} + \\frac{1}{v_{2}})", "derivation": "\\operatorname{t_{1}}{(v_{2})} = \\frac{d}{d v_{2}} \\log{(v_{2})} and \\operatorname{t_{1}}{(v_{2})} = \\frac{1}{v_{2}} and - v_{2} + \\operatorname{t_{1}}{(v_{2})} = - v_{2} + \\frac{1}{v_{2}} and \\frac{d}{d v_{2}} (- v_{2} + \\operatorname{t_{1}}{(v_{2})}) = \\frac{d}{d v_{2}} (- v_{2} + \\frac{1}{v_{2}})", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('v_2', commutative=True)), Derivative(log(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t_1')(Symbol('v_2', commutative=True)), Pow(Symbol('v_2', commutative=True), Integer(-1)))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('t_1')(Symbol('v_2', commutative=True))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Symbol('v_2', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Function('t_1')(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Pow(Symbol('v_2', commutative=True), Integer(-1))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{c})} = e^{F_{c}}, then obtain 0 = - F_{c} \\eta^{\\prime}^{F_{c}}{(F_{c})} + F_{c} (e^{F_{c}})^{F_{c}}", "derivation": "\\eta^{\\prime}{(F_{c})} = e^{F_{c}} and \\eta^{\\prime}^{F_{c}}{(F_{c})} = (e^{F_{c}})^{F_{c}} and F_{c} \\eta^{\\prime}^{F_{c}}{(F_{c})} = F_{c} (e^{F_{c}})^{F_{c}} and F_{c} \\eta^{\\prime}^{F_{c}}{(F_{c})} + F_{c} = F_{c} (e^{F_{c}})^{F_{c}} + F_{c} and 0 = - F_{c} \\eta^{\\prime}^{F_{c}}{(F_{c})} + F_{c} (e^{F_{c}})^{F_{c}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["power", 1, "Symbol('F_c', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)), Pow(exp(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))"], [["times", 2, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(exp(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))))"], [["add", 3, "Symbol('F_c', commutative=True)"], "Equality(Add(Mul(Symbol('F_c', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Add(Mul(Symbol('F_c', commutative=True), Pow(exp(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["minus", 4, "Add(Mul(Symbol('F_c', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(exp(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(m_{s})} = e^{m_{s}}, then obtain \\int 0^{m_{s}} dm_{s} = \\int (- \\Psi_{\\lambda}{(m_{s})} + e^{m_{s}})^{m_{s}} dm_{s}", "derivation": "\\Psi_{\\lambda}{(m_{s})} = e^{m_{s}} and 0 = - \\Psi_{\\lambda}{(m_{s})} + e^{m_{s}} and 0^{m_{s}} = (- \\Psi_{\\lambda}{(m_{s})} + e^{m_{s}})^{m_{s}} and \\int 0^{m_{s}} dm_{s} = \\int (- \\Psi_{\\lambda}{(m_{s})} + e^{m_{s}})^{m_{s}} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))), exp(Symbol('m_s', commutative=True))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Integer(0), Symbol('m_s', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))), exp(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["integrate", 3, "Symbol('m_s', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True))), exp(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(n_{1},P_{g},F_{x})} = F_{x}^{n_{1}} + P_{g}, then obtain ((F_{x}^{n_{1}} + P_{g})^{F_{x}} \\theta_{2}^{F_{x}}{(n_{1},P_{g},F_{x})})^{F_{x}} = ((F_{x}^{n_{1}} + P_{g})^{2 F_{x}})^{F_{x}}", "derivation": "\\theta_{2}{(n_{1},P_{g},F_{x})} = F_{x}^{n_{1}} + P_{g} and \\theta_{2}^{F_{x}}{(n_{1},P_{g},F_{x})} = (F_{x}^{n_{1}} + P_{g})^{F_{x}} and (F_{x}^{n_{1}} + P_{g})^{F_{x}} \\theta_{2}^{F_{x}}{(n_{1},P_{g},F_{x})} = (F_{x}^{n_{1}} + P_{g})^{2 F_{x}} and ((F_{x}^{n_{1}} + P_{g})^{F_{x}} \\theta_{2}^{F_{x}}{(n_{1},P_{g},F_{x})})^{F_{x}} = ((F_{x}^{n_{1}} + P_{g})^{2 F_{x}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('n_1', commutative=True), Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('n_1', commutative=True), Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Symbol('F_x', commutative=True)))"], [["times", 2, "Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Symbol('F_x', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('n_1', commutative=True), Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Mul(Integer(2), Symbol('F_x', commutative=True))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Symbol('F_x', commutative=True)), Pow(Function('\\\\theta_2')(Symbol('n_1', commutative=True), Symbol('P_g', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Pow(Add(Pow(Symbol('F_x', commutative=True), Symbol('n_1', commutative=True)), Symbol('P_g', commutative=True)), Mul(Integer(2), Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(f^{*},u)} = \\frac{\\sin{(f^{*})}}{u}, then derive \\frac{\\partial}{\\partial f^{*}} \\eta^{\\prime}{(f^{*},u)} = \\frac{\\cos{(f^{*})}}{u}, then obtain \\frac{\\partial}{\\partial f^{*}} \\frac{\\sin{(f^{*})}}{u} = \\frac{\\cos{(f^{*})}}{u}", "derivation": "\\eta^{\\prime}{(f^{*},u)} = \\frac{\\sin{(f^{*})}}{u} and \\frac{\\partial}{\\partial f^{*}} \\eta^{\\prime}{(f^{*},u)} = \\frac{\\partial}{\\partial f^{*}} \\frac{\\sin{(f^{*})}}{u} and \\frac{\\partial}{\\partial f^{*}} \\eta^{\\prime}{(f^{*},u)} = \\frac{\\cos{(f^{*})}}{u} and \\frac{\\partial}{\\partial f^{*}} \\frac{\\sin{(f^{*})}}{u} = \\frac{\\cos{(f^{*})}}{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('f^*', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), cos(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), sin(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), cos(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\Psi)} = \\int e^{\\Psi} d\\Psi, then derive \\hat{X}{(\\Psi)} = \\Omega + e^{\\Psi}, then derive (\\Omega + e^{\\Psi})^{\\Psi} = (c_{0} + e^{\\Psi})^{\\Psi}, then obtain (\\Omega + e^{\\Psi})^{\\Psi} - e^{\\Psi} = - e^{\\Psi} + (\\int e^{\\Psi} d\\Psi)^{\\Psi}", "derivation": "\\hat{X}{(\\Psi)} = \\int e^{\\Psi} d\\Psi and \\hat{X}^{\\Psi}{(\\Psi)} = (\\int e^{\\Psi} d\\Psi)^{\\Psi} and \\hat{X}{(\\Psi)} = \\Omega + e^{\\Psi} and (\\Omega + e^{\\Psi})^{\\Psi} = (\\int e^{\\Psi} d\\Psi)^{\\Psi} and (\\Omega + e^{\\Psi})^{\\Psi} = (c_{0} + e^{\\Psi})^{\\Psi} and (\\Omega + e^{\\Psi})^{\\Psi} - e^{\\Psi} = (c_{0} + e^{\\Psi})^{\\Psi} - e^{\\Psi} and (\\int e^{\\Psi} d\\Psi)^{\\Psi} = (c_{0} + e^{\\Psi})^{\\Psi} and (\\Omega + e^{\\Psi})^{\\Psi} - e^{\\Psi} = - e^{\\Psi} + (\\int e^{\\Psi} d\\Psi)^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\Psi', commutative=True)), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\Psi', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{X}')(Symbol('\\\\Psi', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\Psi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["minus", 5, "exp(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Pow(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Add(Pow(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('c_0', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Add(Pow(Add(Symbol('\\\\Omega', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\Psi', commutative=True))), Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(f,T)} = T + f, then obtain \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)} + \\frac{\\partial}{\\partial f} 2 \\hat{x}^{2}{(f,T)} = 2 \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)}", "derivation": "\\hat{x}{(f,T)} = T + f and 2 \\hat{x}{(f,T)} = T + f + \\hat{x}{(f,T)} and 2 \\hat{x}^{2}{(f,T)} = (T + f + \\hat{x}{(f,T)}) \\hat{x}{(f,T)} and 2 (T + f)^{2} = (T + f) (2 T + 2 f) and \\frac{\\partial}{\\partial f} 2 (T + f)^{2} = \\frac{\\partial}{\\partial f} (T + f) (2 T + 2 f) and \\frac{\\partial}{\\partial f} 2 \\hat{x}^{2}{(f,T)} = \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)} and \\frac{\\partial}{\\partial f} (T + f) (2 T + 2 f) + \\frac{\\partial}{\\partial f} 2 \\hat{x}^{2}{(f,T)} = \\frac{\\partial}{\\partial f} (T + f) (2 T + 2 f) + \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)} and \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)} + \\frac{\\partial}{\\partial f} 2 \\hat{x}^{2}{(f,T)} = 2 \\frac{\\partial}{\\partial f} (2 T + 2 f) \\hat{x}{(f,T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('f', commutative=True)))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), Symbol('f', commutative=True), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))))"], [["times", 2, "Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(2))), Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True))), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["add", 6, "Derivative(Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Derivative(Mul(Add(Symbol('T', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True))), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Derivative(Mul(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True))), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True)), Integer(2))), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('f', commutative=True))), Function('\\\\hat{x}')(Symbol('f', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\delta,\\mathbf{S})} = \\delta + \\mathbf{S} and \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} = \\frac{1}{\\operatorname{F_{x}}{(\\delta,\\mathbf{S})}}, then obtain \\frac{\\partial}{\\partial \\mathbf{S}} \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} - 1 = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{1}{\\delta + \\mathbf{S}} - 1", "derivation": "\\operatorname{F_{x}}{(\\delta,\\mathbf{S})} = \\delta + \\mathbf{S} and \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} = \\frac{1}{\\operatorname{F_{x}}{(\\delta,\\mathbf{S})}} and \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} = \\frac{1}{\\delta + \\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{1}{\\delta + \\mathbf{S}} and \\frac{\\partial}{\\partial \\mathbf{S}} \\operatorname{f_{E}}{(\\delta,\\mathbf{S})} - 1 = \\frac{\\partial}{\\partial \\mathbf{S}} \\frac{1}{\\delta + \\mathbf{S}} - 1", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('F_x')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('f_E')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(Derivative(Function('f_E')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Pow(Add(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J)} = e^{J} and \\theta_{1}{(J)} = e^{J}, then derive J + n = \\int \\operatorname{t_{1}}{(J)} e^{- J} dJ, then obtain \\int (J + n) dJ = \\iint \\operatorname{t_{1}}{(J)} e^{- J} dJ dJ", "derivation": "\\operatorname{t_{1}}{(J)} = e^{J} and \\theta_{1}{(J)} = e^{J} and 1 = \\frac{e^{J}}{\\theta_{1}{(J)}} and 1 = \\frac{\\operatorname{t_{1}}{(J)}}{\\theta_{1}{(J)}} and 1 = \\operatorname{t_{1}}{(J)} e^{- J} and \\int 1 dJ = \\int \\operatorname{t_{1}}{(J)} e^{- J} dJ and J + n = \\int \\operatorname{t_{1}}{(J)} e^{- J} dJ and \\int (J + n) dJ = \\iint \\operatorname{t_{1}}{(J)} e^{- J} dJ dJ", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_1')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["divide", 2, "Function('\\\\theta_1')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_1')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_1')(Symbol('J', commutative=True)), Integer(-1)), Function('t_1')(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(1), Mul(Function('t_1')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))))"], [["integrate", 5, "Symbol('J', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Integral(Mul(Function('t_1')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('J', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Function('t_1')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True))))"], [["integrate", 7, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Symbol('J', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Function('t_1')(Symbol('J', commutative=True)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(V)} = \\cos{(V)}, then obtain \\frac{\\cos^{3}{(V)}}{2} = \\frac{\\cos^{4}{(V)}}{\\hat{H}_{\\lambda}{(V)} + \\cos{(V)}}", "derivation": "\\hat{H}_{\\lambda}{(V)} = \\cos{(V)} and 2 \\hat{H}_{\\lambda}{(V)} = \\hat{H}_{\\lambda}{(V)} + \\cos{(V)} and \\hat{H}_{\\lambda}{(V)} \\cos{(V)} = \\cos^{2}{(V)} and \\frac{\\cos{(V)}}{2} = \\frac{\\cos^{2}{(V)}}{2 \\hat{H}_{\\lambda}{(V)}} and \\frac{\\cos^{3}{(V)}}{2} = \\frac{\\cos^{4}{(V)}}{2 \\hat{H}_{\\lambda}{(V)}} and \\frac{\\cos^{3}{(V)}}{2} = \\frac{\\cos^{4}{(V)}}{\\hat{H}_{\\lambda}{(V)} + \\cos{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["add", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True))), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))))"], [["times", 1, "cos(Symbol('V', commutative=True))"], "Equality(Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Pow(cos(Symbol('V', commutative=True)), Integer(2)))"], [["divide", 3, "Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)))"], "Equality(Mul(Rational(1, 2), cos(Symbol('V', commutative=True))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), Integer(-1)), Pow(cos(Symbol('V', commutative=True)), Integer(2))))"], [["times", 4, "Pow(cos(Symbol('V', commutative=True)), Integer(2))"], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('V', commutative=True)), Integer(3))), Mul(Rational(1, 2), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), Integer(-1)), Pow(cos(Symbol('V', commutative=True)), Integer(4))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('V', commutative=True)), Integer(3))), Mul(Pow(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))), Integer(-1)), Pow(cos(Symbol('V', commutative=True)), Integer(4))))"]]}, {"prompt": "Given z{(\\varphi^*,A_{z})} = - A_{z} + \\varphi^* and \\mathbf{r}{(E)} = \\cos{(E)}, then obtain \\frac{\\partial^{2}}{\\partial E\\partial A_{z}} (- A_{z} + \\varphi^* + \\mathbf{r}{(E)}) = \\frac{\\partial^{2}}{\\partial E\\partial A_{z}} (- A_{z} + \\varphi^* + \\cos{(E)})", "derivation": "z{(\\varphi^*,A_{z})} = - A_{z} + \\varphi^* and \\mathbf{r}{(E)} = \\cos{(E)} and \\mathbf{r}{(E)} + z{(\\varphi^*,A_{z})} = z{(\\varphi^*,A_{z})} + \\cos{(E)} and - A_{z} + \\varphi^* + \\mathbf{r}{(E)} = - A_{z} + \\varphi^* + \\cos{(E)} and \\frac{\\partial}{\\partial A_{z}} (- A_{z} + \\varphi^* + \\mathbf{r}{(E)}) = \\frac{\\partial}{\\partial A_{z}} (- A_{z} + \\varphi^* + \\cos{(E)}) and \\frac{\\partial^{2}}{\\partial E\\partial A_{z}} (- A_{z} + \\varphi^* + \\mathbf{r}{(E)}) = \\frac{\\partial^{2}}{\\partial E\\partial A_{z}} (- A_{z} + \\varphi^* + \\cos{(E)})", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["add", 2, "Function('z')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Add(Function('\\\\mathbf{r}')(Symbol('E', commutative=True)), Function('z')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_z', commutative=True))), Add(Function('z')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_z', commutative=True)), cos(Symbol('E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), cos(Symbol('E', commutative=True))))"], [["differentiate", 4, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('E', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), cos(Symbol('E', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('E', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Symbol('\\\\varphi^*', commutative=True), cos(Symbol('E', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1)), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})}, then derive \\frac{d}{d L_{\\varepsilon}} I{(L_{\\varepsilon})} = - \\sin{(L_{\\varepsilon})}, then obtain - \\sin{(L_{\\varepsilon})} - 1 = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})} - 1", "derivation": "I{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} I{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} I{(L_{\\varepsilon})} = - \\sin{(L_{\\varepsilon})} and - \\sin{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})} and - \\sin{(L_{\\varepsilon})} - 1 = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})} - 1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["minus", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)), Add(Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = C_{2}^{\\mathbf{f}}, then obtain - C_{2}^{\\mathbf{f}} - \\mathbf{f} + 2 \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = - \\mathbf{f} + \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = C_{2}^{\\mathbf{f}} and - \\mathbf{f} + \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = C_{2}^{\\mathbf{f}} - \\mathbf{f} and - C_{2}^{\\mathbf{f}} - \\mathbf{f} + \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = - \\mathbf{f} and - C_{2}^{\\mathbf{f}} - \\mathbf{f} + 2 \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})} = - \\mathbf{f} + \\operatorname{f_{\\mathbf{p}}}{(C_{2},\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 2, "Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('f_{\\\\mathbf{p}}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given I{(\\phi)} = \\log{(\\cos{(\\phi)})}, then obtain - I{(\\phi)} + \\frac{\\log{(\\cos{(\\phi)})}}{\\phi} = - \\log{(\\cos{(\\phi)})} + \\frac{\\log{(\\cos{(\\phi)})}}{\\phi}", "derivation": "I{(\\phi)} = \\log{(\\cos{(\\phi)})} and \\frac{I{(\\phi)}}{\\phi} = \\frac{\\log{(\\cos{(\\phi)})}}{\\phi} and - I{(\\phi)} = - \\log{(\\cos{(\\phi)})} and - I{(\\phi)} + \\frac{I{(\\phi)}}{\\phi} = - \\log{(\\cos{(\\phi)})} + \\frac{I{(\\phi)}}{\\phi} and - I{(\\phi)} + \\frac{\\log{(\\cos{(\\phi)})}}{\\phi} = - \\log{(\\cos{(\\phi)})} + \\frac{\\log{(\\cos{(\\phi)})}}{\\phi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\phi', commutative=True)), log(cos(Symbol('\\\\phi', commutative=True))))"], [["divide", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), log(cos(Symbol('\\\\phi', commutative=True)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('I')(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), log(cos(Symbol('\\\\phi', commutative=True)))))"], [["add", 3, "Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(-1), log(cos(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('I')(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('I')(Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), log(cos(Symbol('\\\\phi', commutative=True))))), Add(Mul(Integer(-1), log(cos(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), log(cos(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(m)} = \\log{(m)}, then derive \\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} = \\frac{1}{m}, then obtain \\frac{d}{d m} \\log{(m)} + \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} \\log{(m)}} = \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} \\log{(m)}} + \\frac{1}{m}", "derivation": "\\operatorname{y^{\\prime}}{(m)} = \\log{(m)} and \\operatorname{y^{\\prime}}^{2}{(m)} = \\operatorname{y^{\\prime}}{(m)} \\log{(m)} and \\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} = \\frac{d}{d m} \\log{(m)} and \\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} = \\frac{1}{m} and \\frac{d}{d m} \\log{(m)} = \\frac{1}{m} and \\frac{d}{d m} \\log{(m)} + \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}^{2}{(m)}} = \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}^{2}{(m)}} + \\frac{1}{m} and \\frac{d}{d m} \\log{(m)} + \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} \\log{(m)}} = \\frac{1}{\\frac{d}{d m} \\operatorname{y^{\\prime}}{(m)} \\log{(m)}} + \\frac{1}{m}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('m', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), Mul(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["add", 5, "Pow(Derivative(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Derivative(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))), Add(Pow(Derivative(Pow(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), Integer(2)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Derivative(Mul(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))), Add(Pow(Derivative(Mul(Function('y^{\\\\prime}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = \\mathbf{D}^{\\mathbf{S}} - m_{s}, then obtain - \\mathbf{D} + \\frac{\\partial}{\\partial m_{s}} \\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = - \\mathbf{D} - 1", "derivation": "\\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = \\mathbf{D}^{\\mathbf{S}} - m_{s} and \\frac{\\partial}{\\partial m_{s}} \\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = \\frac{\\partial}{\\partial m_{s}} (\\mathbf{D}^{\\mathbf{S}} - m_{s}) and - \\mathbf{D} + \\frac{\\partial}{\\partial m_{s}} \\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = - \\mathbf{D} + \\frac{\\partial}{\\partial m_{s}} (\\mathbf{D}^{\\mathbf{S}} - m_{s}) and - \\mathbf{D} + \\frac{\\partial}{\\partial m_{s}} \\mu_{0}{(\\mathbf{D},\\mathbf{S},m_{s})} = - \\mathbf{D} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('m_s', commutative=True)), Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Add(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\omega{(\\Psi_{\\lambda},\\mathbf{J}_M)} = \\Psi_{\\lambda} + \\mathbf{J}_M, then obtain (\\Psi_{\\lambda} + \\mathbf{J}_M)^{3} \\omega^{2}{(\\Psi_{\\lambda},\\mathbf{J}_M)} = (\\Psi_{\\lambda} + \\mathbf{J}_M)^{5}", "derivation": "\\omega{(\\Psi_{\\lambda},\\mathbf{J}_M)} = \\Psi_{\\lambda} + \\mathbf{J}_M and (\\Psi_{\\lambda} + \\mathbf{J}_M) \\omega{(\\Psi_{\\lambda},\\mathbf{J}_M)} = (\\Psi_{\\lambda} + \\mathbf{J}_M)^{2} and (\\Psi_{\\lambda} + \\mathbf{J}_M)^{3} \\omega{(\\Psi_{\\lambda},\\mathbf{J}_M)} = (\\Psi_{\\lambda} + \\mathbf{J}_M)^{4} and (\\Psi_{\\lambda} + \\mathbf{J}_M)^{4} \\omega{(\\Psi_{\\lambda},\\mathbf{J}_M)} = (\\Psi_{\\lambda} + \\mathbf{J}_M)^{5} and (\\Psi_{\\lambda} + \\mathbf{J}_M)^{3} \\omega^{2}{(\\Psi_{\\lambda},\\mathbf{J}_M)} = (\\Psi_{\\lambda} + \\mathbf{J}_M)^{5}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2)))"], [["times", 2, "Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(3)), Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(4)))"], [["times", 2, "Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(3))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(4)), Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(5)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(3)), Pow(Function('\\\\omega')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(5)))"]]}, {"prompt": "Given s{(\\mathbf{S},v_{y})} = \\mathbf{S} v_{y}, then obtain \\frac{\\partial}{\\partial v_{y}} \\mathbf{S} (v_{y} + s{(\\mathbf{S},v_{y})}) = \\frac{\\partial}{\\partial v_{y}} \\mathbf{S} (\\mathbf{S} v_{y} + v_{y})", "derivation": "s{(\\mathbf{S},v_{y})} = \\mathbf{S} v_{y} and v_{y} + s{(\\mathbf{S},v_{y})} = \\mathbf{S} v_{y} + v_{y} and \\mathbf{S} (v_{y} + s{(\\mathbf{S},v_{y})}) = \\mathbf{S} (\\mathbf{S} v_{y} + v_{y}) and \\frac{\\partial}{\\partial v_{y}} \\mathbf{S} (v_{y} + s{(\\mathbf{S},v_{y})}) = \\frac{\\partial}{\\partial v_{y}} \\mathbf{S} (\\mathbf{S} v_{y} + v_{y})", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)))"], [["add", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Symbol('v_y', commutative=True), Function('s')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))"], [["times", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('v_y', commutative=True), Function('s')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))))"], [["differentiate", 3, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Symbol('v_y', commutative=True), Function('s')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{S}', commutative=True), Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(a^{\\dagger})} = \\log{(\\sin{(a^{\\dagger})})} and \\varepsilon{(z)} = \\int \\log{(z)} dz, then derive \\varepsilon{(z)} = m + z \\log{(z)} - z, then obtain - \\hat{\\mathbf{r}}{(a^{\\dagger})} + \\varepsilon{(z)} = m + z \\log{(z)} - z - \\hat{\\mathbf{r}}{(a^{\\dagger})}", "derivation": "\\hat{\\mathbf{r}}{(a^{\\dagger})} = \\log{(\\sin{(a^{\\dagger})})} and \\varepsilon{(z)} = \\int \\log{(z)} dz and \\varepsilon{(z)} = m + z \\log{(z)} - z and \\varepsilon{(z)} - \\log{(\\sin{(a^{\\dagger})})} = m + z \\log{(z)} - z - \\log{(\\sin{(a^{\\dagger})})} and - \\hat{\\mathbf{r}}{(a^{\\dagger})} + \\varepsilon{(z)} = m + z \\log{(z)} - z - \\hat{\\mathbf{r}}{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True)), log(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Add(Symbol('m', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["minus", 3, "log(sin(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Mul(Integer(-1), log(sin(Symbol('a^{\\\\dagger}', commutative=True))))), Add(Symbol('m', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), log(sin(Symbol('a^{\\\\dagger}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True))), Function('\\\\varepsilon')(Symbol('z', commutative=True))), Add(Symbol('m', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(l)} = \\log{(l)}, then obtain \\frac{\\partial}{\\partial l} \\frac{l (- k + \\bar{\\h}{(l)})}{V_{\\mathbf{E}} + k} = \\frac{\\partial}{\\partial l} \\frac{l (- k + \\log{(l)})}{V_{\\mathbf{E}} + k}", "derivation": "\\bar{\\h}{(l)} = \\log{(l)} and - k + \\bar{\\h}{(l)} = - k + \\log{(l)} and l (- k + \\bar{\\h}{(l)}) = l (- k + \\log{(l)}) and \\frac{l (- k + \\bar{\\h}{(l)})}{V_{\\mathbf{E}} + k} = \\frac{l (- k + \\log{(l)})}{V_{\\mathbf{E}} + k} and \\frac{\\partial}{\\partial l} \\frac{l (- k + \\bar{\\h}{(l)})}{V_{\\mathbf{E}} + k} = \\frac{\\partial}{\\partial l} \\frac{l (- k + \\log{(l)})}{V_{\\mathbf{E}} + k}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hbar')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\hbar')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('l', commutative=True))))"], [["times", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\hbar')(Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('l', commutative=True)))))"], [["divide", 3, "Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Symbol('l', commutative=True), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\hbar')(Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('l', commutative=True)))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Symbol('l', commutative=True), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\hbar')(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Pow(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + v_{2} and \\mathbf{M}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + v_{2}, then obtain - \\hat{H}_{\\lambda} \\operatorname{C_{2}}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} \\mathbf{M}{(v_{2},\\hat{H}_{\\lambda})}", "derivation": "\\operatorname{C_{2}}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + v_{2} and - \\hat{H}_{\\lambda} \\operatorname{C_{2}}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} (- \\hat{H}_{\\lambda} + v_{2}) and \\mathbf{M}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} + v_{2} and - \\hat{H}_{\\lambda} \\operatorname{C_{2}}{(v_{2},\\hat{H}_{\\lambda})} = - \\hat{H}_{\\lambda} \\mathbf{M}{(v_{2},\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('v_2', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('C_2')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Symbol('v_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('C_2')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('\\\\mathbf{M}')(Symbol('v_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(q)} = \\sin{(q)}, then obtain - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\mathbf{g}{(q)} dq + \\iint \\mathbf{g}{(q)} dq dq = - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\mathbf{g}{(q)} dq + \\iint \\sin{(q)} dq dq", "derivation": "\\mathbf{g}{(q)} = \\sin{(q)} and \\int \\mathbf{g}{(q)} dq = \\int \\sin{(q)} dq and \\iint \\mathbf{g}{(q)} dq dq = \\iint \\sin{(q)} dq dq and \\frac{d}{d q} \\int \\mathbf{g}{(q)} dq = \\frac{d}{d q} \\int \\sin{(q)} dq and - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\sin{(q)} dq + \\iint \\mathbf{g}{(q)} dq dq = - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\sin{(q)} dq + \\iint \\sin{(q)} dq dq and - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\mathbf{g}{(q)} dq + \\iint \\mathbf{g}{(q)} dq dq = - \\mathbf{g}{(q)} \\frac{d}{d q} \\int \\mathbf{g}{(q)} dq + \\iint \\sin{(q)} dq dq", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 3, "Mul(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Derivative(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Derivative(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Derivative(Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Derivative(Integral(Function('\\\\mathbf{g}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Integral(sin(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\cos{(\\log{(\\hat{H}_{\\lambda})})}, then obtain 3 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\cos{(\\log{(\\hat{H}_{\\lambda})})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + 3 \\cos{(\\log{(\\hat{H}_{\\lambda})})}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\cos{(\\log{(\\hat{H}_{\\lambda})})} and 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\cos{(\\log{(\\hat{H}_{\\lambda})})} and 3 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\cos{(\\log{(\\hat{H}_{\\lambda})})} = 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + 2 \\cos{(\\log{(\\hat{H}_{\\lambda})})} and 3 \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + \\cos{(\\log{(\\hat{H}_{\\lambda})})} = \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\hat{H}_{\\lambda})} + 3 \\cos{(\\log{(\\hat{H}_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["add", 1, "Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["add", 2, "Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], "Equality(Add(Mul(Integer(3), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(3), cos(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(A,F_{N})} = \\frac{\\partial}{\\partial A} A F_{N}, then derive \\sin{(A + \\nabla{(A,F_{N})})} = \\sin{(A + F_{N})}, then obtain \\log{(- F_{N} + \\sin{(A + \\nabla{(A,F_{N})})})} = \\log{(- F_{N} + \\sin{(A + \\frac{\\partial}{\\partial A} A F_{N})})}", "derivation": "\\nabla{(A,F_{N})} = \\frac{\\partial}{\\partial A} A F_{N} and A + \\nabla{(A,F_{N})} = A + \\frac{\\partial}{\\partial A} A F_{N} and \\sin{(A + \\nabla{(A,F_{N})})} = \\sin{(A + \\frac{\\partial}{\\partial A} A F_{N})} and \\sin{(A + \\nabla{(A,F_{N})})} = \\sin{(A + F_{N})} and \\sin{(A + \\frac{\\partial}{\\partial A} A F_{N})} = \\sin{(A + F_{N})} and - F_{N} + \\sin{(A + \\frac{\\partial}{\\partial A} A F_{N})} = - F_{N} + \\sin{(A + F_{N})} and - F_{N} + \\sin{(A + \\nabla{(A,F_{N})})} = - F_{N} + \\sin{(A + F_{N})} and \\log{(- F_{N} + \\sin{(A + \\nabla{(A,F_{N})})})} = \\log{(- F_{N} + \\sin{(A + F_{N})})} and \\log{(- F_{N} + \\sin{(A + \\nabla{(A,F_{N})})})} = \\log{(- F_{N} + \\sin{(A + \\frac{\\partial}{\\partial A} A F_{N})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["add", 1, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["sin", 2], "Equality(sin(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))), sin(Add(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(sin(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))), sin(Add(Symbol('A', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(sin(Add(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), sin(Add(Symbol('A', commutative=True), Symbol('F_N', commutative=True))))"], [["minus", 5, "Symbol('F_N', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))))"], [["log", 7], "Equality(log(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))))), log(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Symbol('F_N', commutative=True))))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(log(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Function('\\\\nabla')(Symbol('A', commutative=True), Symbol('F_N', commutative=True)))))), log(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), sin(Add(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given A{(M)} = \\sin{(M)} and T{(M)} = (A{(M)} + \\sin{(M)})^{M} and \\operatorname{g_{\\varepsilon}}{(M)} = (A{(M)} + \\sin{(M)})^{M}, then obtain \\frac{\\operatorname{g_{\\varepsilon}}{(M)}}{\\cos{(M)}} = \\frac{T{(M)}}{\\cos{(M)}}", "derivation": "A{(M)} = \\sin{(M)} and 2 A{(M)} = A{(M)} + \\sin{(M)} and (2 A{(M)})^{M} = (A{(M)} + \\sin{(M)})^{M} and T{(M)} = (A{(M)} + \\sin{(M)})^{M} and \\operatorname{g_{\\varepsilon}}{(M)} = (A{(M)} + \\sin{(M)})^{M} and \\frac{\\operatorname{g_{\\varepsilon}}{(M)}}{\\cos{(M)}} = \\frac{(A{(M)} + \\sin{(M)})^{M}}{\\cos{(M)}} and T{(M)} = (2 A{(M)})^{M} and \\frac{\\operatorname{g_{\\varepsilon}}{(M)}}{\\cos{(M)}} = \\frac{(2 A{(M)})^{M}}{\\cos{(M)}} and \\frac{\\operatorname{g_{\\varepsilon}}{(M)}}{\\cos{(M)}} = \\frac{T{(M)}}{\\cos{(M)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["add", 1, "Function('A')(Symbol('M', commutative=True))"], "Equality(Mul(Integer(2), Function('A')(Symbol('M', commutative=True))), Add(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('A')(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Add(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('M', commutative=True)), Pow(Add(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(Add(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["divide", 5, "cos(Symbol('M', commutative=True))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))), Mul(Pow(Add(Function('A')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('T')(Symbol('M', commutative=True)), Pow(Mul(Integer(2), Function('A')(Symbol('M', commutative=True))), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))), Mul(Pow(Mul(Integer(2), Function('A')(Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))), Mul(Function('T')(Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\nabla{(\\mathbf{p},Q)} = Q \\mathbf{p} and \\dot{x}{(\\mathbf{p},Q)} = Q \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)}, then derive Q \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)} = Q^{2}, then obtain \\dot{x}{(\\mathbf{p},Q)} = Q^{2}", "derivation": "\\nabla{(\\mathbf{p},Q)} = Q \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)} = \\frac{\\partial}{\\partial \\mathbf{p}} Q \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} Q \\mathbf{p} \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)} = (\\frac{\\partial}{\\partial \\mathbf{p}} Q \\mathbf{p})^{2} and Q \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)} = Q^{2} and Q \\frac{\\partial}{\\partial \\mathbf{p}} Q \\mathbf{p} = Q^{2} and \\dot{x}{(\\mathbf{p},Q)} = Q \\frac{\\partial}{\\partial \\mathbf{p}} \\nabla{(\\mathbf{p},Q)} and \\dot{x}{(\\mathbf{p},Q)} = Q \\frac{\\partial}{\\partial \\mathbf{p}} Q \\mathbf{p} and \\dot{x}{(\\mathbf{p},Q)} = Q^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Function('\\\\nabla')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Pow(Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Pow(Symbol('Q', commutative=True), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('Q', commutative=True), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Pow(Symbol('Q', commutative=True), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Derivative(Function('\\\\nabla')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Derivative(Mul(Symbol('Q', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(2)))"]]}, {"prompt": "Given M{(v_{1})} = \\log{(v_{1})}, then obtain \\frac{1}{\\log{(v_{1})}} + \\frac{1}{v_{1} \\log{(v_{1})}^{2}} = \\frac{1}{M{(v_{1})}} + \\frac{1}{v_{1} \\log{(v_{1})}^{2}}", "derivation": "M{(v_{1})} = \\log{(v_{1})} and \\frac{1}{\\log{(v_{1})}} = \\frac{1}{M{(v_{1})}} and \\frac{d}{d v_{1}} \\frac{1}{\\log{(v_{1})}} = \\frac{d}{d v_{1}} \\frac{1}{M{(v_{1})}} and - \\frac{d}{d v_{1}} \\frac{1}{M{(v_{1})}} + \\frac{1}{\\log{(v_{1})}} = - \\frac{d}{d v_{1}} \\frac{1}{M{(v_{1})}} + \\frac{1}{M{(v_{1})}} and - \\frac{d}{d v_{1}} \\frac{1}{\\log{(v_{1})}} + \\frac{1}{\\log{(v_{1})}} = - \\frac{d}{d v_{1}} \\frac{1}{\\log{(v_{1})}} + \\frac{1}{M{(v_{1})}} and \\frac{1}{\\log{(v_{1})}} + \\frac{1}{v_{1} \\log{(v_{1})}^{2}} = \\frac{1}{M{(v_{1})}} + \\frac{1}{v_{1} \\log{(v_{1})}^{2}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], [["divide", 1, "Mul(Function('M')(Symbol('v_1', commutative=True)), log(Symbol('v_1', commutative=True)))"], "Equality(Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(log(Symbol('v_1', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Derivative(Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(log(Symbol('v_1', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Derivative(Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Add(Pow(log(Symbol('v_1', commutative=True)), Integer(-1)), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(log(Symbol('v_1', commutative=True)), Integer(-2)))), Add(Pow(Function('M')(Symbol('v_1', commutative=True)), Integer(-1)), Mul(Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(log(Symbol('v_1', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(t_{1},M_{E})} = t_{1}^{M_{E}}, then obtain (2 \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E})^{t_{1}} = (\\iint t_{1}^{M_{E}} dM_{E} dM_{E} + \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E})^{t_{1}}", "derivation": "\\operatorname{y^{\\prime}}{(t_{1},M_{E})} = t_{1}^{M_{E}} and \\int \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} = \\int t_{1}^{M_{E}} dM_{E} and \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E} = \\iint t_{1}^{M_{E}} dM_{E} dM_{E} and 2 \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E} = \\iint t_{1}^{M_{E}} dM_{E} dM_{E} + \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E} and (2 \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E})^{t_{1}} = (\\iint t_{1}^{M_{E}} dM_{E} dM_{E} + \\iint \\operatorname{y^{\\prime}}{(t_{1},M_{E})} dM_{E} dM_{E})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Pow(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Pow(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["add", 3, "Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Add(Integral(Pow(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))))"], [["power", 4, "Symbol('t_1', commutative=True)"], "Equality(Pow(Mul(Integer(2), Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Symbol('t_1', commutative=True)), Pow(Add(Integral(Pow(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(Function('y^{\\\\prime}')(Symbol('t_1', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\varphi{(z)} = \\log{(\\sin{(z)})}, then obtain - \\frac{d}{d z} \\varphi{(z)} + \\int \\frac{\\varphi{(z)} - \\sin{(z)}}{\\sin{(z)}} dz = - \\frac{d}{d z} \\varphi{(z)} + \\int \\frac{\\log{(\\sin{(z)})} - \\sin{(z)}}{\\sin{(z)}} dz", "derivation": "\\varphi{(z)} = \\log{(\\sin{(z)})} and \\varphi{(z)} - \\sin{(z)} = \\log{(\\sin{(z)})} - \\sin{(z)} and \\frac{\\varphi{(z)} - \\sin{(z)}}{\\sin{(z)}} = \\frac{\\log{(\\sin{(z)})} - \\sin{(z)}}{\\sin{(z)}} and \\int \\frac{\\varphi{(z)} - \\sin{(z)}}{\\sin{(z)}} dz = \\int \\frac{\\log{(\\sin{(z)})} - \\sin{(z)}}{\\sin{(z)}} dz and - \\frac{d}{d z} \\varphi{(z)} + \\int \\frac{\\varphi{(z)} - \\sin{(z)}}{\\sin{(z)}} dz = - \\frac{d}{d z} \\varphi{(z)} + \\int \\frac{\\log{(\\sin{(z)})} - \\sin{(z)}}{\\sin{(z)}} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('z', commutative=True)), log(sin(Symbol('z', commutative=True))))"], [["minus", 1, "sin(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Add(log(sin(Symbol('z', commutative=True))), Mul(Integer(-1), sin(Symbol('z', commutative=True)))))"], [["divide", 2, "sin(Symbol('z', commutative=True))"], "Equality(Mul(Add(Function('\\\\varphi')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Mul(Add(log(sin(Symbol('z', commutative=True))), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\varphi')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True))), Integral(Mul(Add(log(sin(Symbol('z', commutative=True))), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True))))"], [["minus", 4, "Derivative(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integral(Mul(Add(Function('\\\\varphi')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Derivative(Function('\\\\varphi')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Integral(Mul(Add(log(sin(Symbol('z', commutative=True))), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Pow(sin(Symbol('z', commutative=True)), Integer(-1))), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\dot{y},B)} = \\frac{\\partial}{\\partial \\dot{y}} B \\dot{y}, then derive \\mathbf{J}{(\\dot{y},B)} = B, then obtain \\frac{\\partial}{\\partial B} \\mathbf{J}^{B}{(\\dot{y},B)} = \\frac{\\partial}{\\partial B} \\frac{B \\mathbf{J}^{B}{(\\dot{y},B)}}{\\mathbf{J}{(\\dot{y},B)}}", "derivation": "\\mathbf{J}{(\\dot{y},B)} = \\frac{\\partial}{\\partial \\dot{y}} B \\dot{y} and \\mathbf{J}^{B}{(\\dot{y},B)} = (\\frac{\\partial}{\\partial \\dot{y}} B \\dot{y})^{B} and \\mathbf{J}{(\\dot{y},B)} = B and (\\frac{\\partial}{\\partial \\dot{y}} B \\dot{y})^{B} = \\frac{B (\\frac{\\partial}{\\partial \\dot{y}} B \\dot{y})^{B}}{\\mathbf{J}{(\\dot{y},B)}} and \\mathbf{J}^{B}{(\\dot{y},B)} = \\frac{B (\\frac{\\partial}{\\partial \\dot{y}} B \\dot{y})^{B}}{\\mathbf{J}{(\\dot{y},B)}} and \\mathbf{J}^{B}{(\\dot{y},B)} = \\frac{B \\mathbf{J}^{B}{(\\dot{y},B)}}{\\mathbf{J}{(\\dot{y},B)}} and \\frac{\\partial}{\\partial B} \\mathbf{J}^{B}{(\\dot{y},B)} = \\frac{\\partial}{\\partial B} \\frac{B \\mathbf{J}^{B}{(\\dot{y},B)}}{\\mathbf{J}{(\\dot{y},B)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('B', commutative=True)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))"], [["divide", 3, "Mul(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('B', commutative=True))))"], "Equality(Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Derivative(Mul(Symbol('B', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["differentiate", 6, "Symbol('B', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\dot{y}', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(i,r)} = \\sin{(i + r)} and \\mathbf{v}{(i,r)} = i + r, then obtain \\frac{\\partial}{\\partial i} \\sin{(\\mathbf{v}{(i,r)})} = \\frac{\\partial}{\\partial i} \\sin{(i + r)}", "derivation": "\\mathbf{B}{(i,r)} = \\sin{(i + r)} and \\mathbf{v}{(i,r)} = i + r and \\mathbf{B}{(i,r)} = \\sin{(\\mathbf{v}{(i,r)})} and \\frac{\\partial}{\\partial i} \\mathbf{B}{(i,r)} = \\frac{\\partial}{\\partial i} \\sin{(i + r)} and \\frac{\\partial}{\\partial i} \\sin{(\\mathbf{v}{(i,r)})} = \\frac{\\partial}{\\partial i} \\sin{(i + r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Add(Symbol('i', commutative=True), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{B}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), sin(Function('\\\\mathbf{v}')(Symbol('i', commutative=True), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('i', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('i', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(sin(Function('\\\\mathbf{v}')(Symbol('i', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('i', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi_{2}{(r)} = \\cos{(\\sin{(r)})}, then obtain \\phi_{2}{(r)} + \\frac{\\int \\cos{(\\sin{(r)})} dr}{\\phi_{2}{(r)}} = \\cos{(\\sin{(r)})} + \\frac{\\int \\cos{(\\sin{(r)})} dr}{\\phi_{2}{(r)}}", "derivation": "\\phi_{2}{(r)} = \\cos{(\\sin{(r)})} and \\int \\phi_{2}{(r)} dr = \\int \\cos{(\\sin{(r)})} dr and \\frac{\\int \\phi_{2}{(r)} dr}{\\phi_{2}{(r)}} = \\frac{\\int \\cos{(\\sin{(r)})} dr}{\\phi_{2}{(r)}} and \\phi_{2}{(r)} + \\frac{\\int \\phi_{2}{(r)} dr}{\\phi_{2}{(r)}} = \\cos{(\\sin{(r)})} + \\frac{\\int \\phi_{2}{(r)} dr}{\\phi_{2}{(r)}} and \\phi_{2}{(r)} + \\frac{\\int \\cos{(\\sin{(r)})} dr}{\\phi_{2}{(r)}} = \\cos{(\\sin{(r)})} + \\frac{\\int \\cos{(\\sin{(r)})} dr}{\\phi_{2}{(r)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('r', commutative=True)), cos(sin(Symbol('r', commutative=True))))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(cos(sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["divide", 2, "Function('\\\\phi_2')(Symbol('r', commutative=True))"], "Equality(Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(cos(sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True)))))"], [["add", 1, "Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], "Equality(Add(Function('\\\\phi_2')(Symbol('r', commutative=True)), Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))), Add(cos(sin(Symbol('r', commutative=True))), Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(Function('\\\\phi_2')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\phi_2')(Symbol('r', commutative=True)), Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(cos(sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))), Add(cos(sin(Symbol('r', commutative=True))), Mul(Pow(Function('\\\\phi_2')(Symbol('r', commutative=True)), Integer(-1)), Integral(cos(sin(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given U{(C,E_{x})} = C + E_{x}, then derive \\frac{\\partial}{\\partial C} U{(C,E_{x})} = 1, then obtain \\frac{\\frac{\\partial}{\\partial C} U{(C,E_{x})}}{(C + E_{x}) U^{2}{(C,E_{x})}} = \\frac{1}{(C + E_{x}) U^{2}{(C,E_{x})}}", "derivation": "U{(C,E_{x})} = C + E_{x} and \\frac{\\partial}{\\partial C} U{(C,E_{x})} = \\frac{\\partial}{\\partial C} (C + E_{x}) and \\frac{\\partial}{\\partial C} U{(C,E_{x})} = 1 and \\frac{\\partial}{\\partial C} (C + E_{x}) = 1 and \\frac{\\frac{\\partial}{\\partial C} (C + E_{x})}{C + E_{x}} = \\frac{1}{C + E_{x}} and \\frac{\\frac{\\partial}{\\partial C} U{(C,E_{x})}}{U{(C,E_{x})}} = \\frac{1}{U{(C,E_{x})}} and \\frac{\\frac{\\partial}{\\partial C} U{(C,E_{x})}}{U^{2}{(C,E_{x})}} = \\frac{1}{U^{2}{(C,E_{x})}} and \\frac{\\frac{\\partial}{\\partial C} U{(C,E_{x})}}{(C + E_{x}) U^{2}{(C,E_{x})}} = \\frac{1}{(C + E_{x}) U^{2}{(C,E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Derivative(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Derivative(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)))"], [["divide", 6, "Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Mul(Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-2)), Derivative(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-2)))"], [["times", 7, "Pow(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-2)), Derivative(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-1)), Pow(Function('U')(Symbol('C', commutative=True), Symbol('E_x', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\hat{X}{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain (- 2 \\mu_0 + \\hat{X}{(\\mu_0)}) (- \\mu_0 + \\hat{X}{(\\mu_0)}) = (- 2 \\mu_0 + \\hat{X}{(\\mu_0)}) (- \\mu_0 + \\cos{(\\mu_0)})", "derivation": "\\hat{X}{(\\mu_0)} = \\cos{(\\mu_0)} and - \\mu_0 + \\hat{X}{(\\mu_0)} = - \\mu_0 + \\cos{(\\mu_0)} and - 2 \\mu_0 + \\hat{X}{(\\mu_0)} = - 2 \\mu_0 + \\cos{(\\mu_0)} and (- 2 \\mu_0 + \\cos{(\\mu_0)}) (- \\mu_0 + \\hat{X}{(\\mu_0)}) = (- 2 \\mu_0 + \\cos{(\\mu_0)}) (- \\mu_0 + \\cos{(\\mu_0)}) and (- 2 \\mu_0 + \\hat{X}{(\\mu_0)}) (- \\mu_0 + \\hat{X}{(\\mu_0)}) = (- 2 \\mu_0 + \\hat{X}{(\\mu_0)}) (- \\mu_0 + \\cos{(\\mu_0)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["minus", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hbar)} = \\sin{(\\hbar)}, then derive 2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar = \\mathbf{J}_M + 2 \\mathbb{I}{(\\hbar)} - \\cos{(\\hbar)}, then obtain \\frac{d}{d \\mathbf{J}_M} (2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar) = 1", "derivation": "\\mathbb{I}{(\\hbar)} = \\sin{(\\hbar)} and \\int \\mathbb{I}{(\\hbar)} d\\hbar = \\int \\sin{(\\hbar)} d\\hbar and 2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar = 2 \\mathbb{I}{(\\hbar)} + \\int \\sin{(\\hbar)} d\\hbar and 2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar = \\mathbf{J}_M + 2 \\mathbb{I}{(\\hbar)} - \\cos{(\\hbar)} and \\frac{d}{d \\mathbf{J}_M} (2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar) = \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M + 2 \\mathbb{I}{(\\hbar)} - \\cos{(\\hbar)}) and \\frac{d}{d \\mathbf{J}_M} (2 \\mathbb{I}{(\\hbar)} + \\int \\mathbb{I}{(\\hbar)} d\\hbar) = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), sin(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Integral(sin(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True))), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\lambda{(\\mathbf{s})} = \\sin{(e^{\\mathbf{s}})}, then obtain (\\frac{(\\mathbf{s} + \\lambda{(\\mathbf{s})})^{\\mathbf{s}}}{\\mathbf{s}})^{\\mathbf{s}} = (\\frac{(\\mathbf{s} + \\sin{(e^{\\mathbf{s}})})^{\\mathbf{s}}}{\\mathbf{s}})^{\\mathbf{s}}", "derivation": "\\lambda{(\\mathbf{s})} = \\sin{(e^{\\mathbf{s}})} and \\mathbf{s} + \\lambda{(\\mathbf{s})} = \\mathbf{s} + \\sin{(e^{\\mathbf{s}})} and (\\mathbf{s} + \\lambda{(\\mathbf{s})})^{\\mathbf{s}} = (\\mathbf{s} + \\sin{(e^{\\mathbf{s}})})^{\\mathbf{s}} and \\frac{(\\mathbf{s} + \\lambda{(\\mathbf{s})})^{\\mathbf{s}}}{\\mathbf{s}} = \\frac{(\\mathbf{s} + \\sin{(e^{\\mathbf{s}})})^{\\mathbf{s}}}{\\mathbf{s}} and (\\frac{(\\mathbf{s} + \\lambda{(\\mathbf{s})})^{\\mathbf{s}}}{\\mathbf{s}})^{\\mathbf{s}} = (\\frac{(\\mathbf{s} + \\sin{(e^{\\mathbf{s}})})^{\\mathbf{s}}}{\\mathbf{s}})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), sin(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})} = \\psi^{\\hat{\\mathbf{x}}}, then derive \\frac{\\partial}{\\partial \\psi} \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})} = \\frac{\\hat{\\mathbf{x}} \\psi^{\\hat{\\mathbf{x}}}}{\\psi}, then obtain \\frac{\\partial}{\\partial \\psi} \\psi^{\\hat{\\mathbf{x}}} = \\frac{\\hat{\\mathbf{x}} \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})}}{\\psi}", "derivation": "\\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})} = \\psi^{\\hat{\\mathbf{x}}} and \\frac{\\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})}}{\\psi} = \\frac{\\psi^{\\hat{\\mathbf{x}}}}{\\psi} and \\frac{\\partial}{\\partial \\psi} \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial \\psi} \\psi^{\\hat{\\mathbf{x}}} and \\frac{\\partial}{\\partial \\psi} \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})} = \\frac{\\hat{\\mathbf{x}} \\psi^{\\hat{\\mathbf{x}}}}{\\psi} and \\frac{\\partial}{\\partial \\psi} \\psi^{\\hat{\\mathbf{x}}} = \\frac{\\hat{\\mathbf{x}} \\psi^{\\hat{\\mathbf{x}}}}{\\psi} and \\frac{\\partial}{\\partial \\psi} \\psi^{\\hat{\\mathbf{x}}} = \\frac{\\hat{\\mathbf{x}} \\Psi^{\\dagger}{(\\psi,\\hat{\\mathbf{x}})}}{\\psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(C_{d},g_{\\varepsilon})} = g_{\\varepsilon} + e^{C_{d}} and \\mu{(C_{d})} = e^{C_{d}}, then obtain g_{\\varepsilon} + \\mu{(C_{d})} = g_{\\varepsilon} + e^{C_{d}}", "derivation": "\\operatorname{F_{c}}{(C_{d},g_{\\varepsilon})} = g_{\\varepsilon} + e^{C_{d}} and \\mu{(C_{d})} = e^{C_{d}} and \\operatorname{F_{c}}{(C_{d},g_{\\varepsilon})} = g_{\\varepsilon} + \\mu{(C_{d})} and g_{\\varepsilon} + \\mu{(C_{d})} = g_{\\varepsilon} + e^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('C_d', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_c')(Symbol('C_d', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mu')(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mu')(Symbol('C_d', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(z)} = \\log{(z)}, then derive \\int \\rho_{f}{(z)} dz = b + z \\log{(z)} - z, then obtain \\int \\log{(z)} dz = b + z \\log{(z)} - z", "derivation": "\\rho_{f}{(z)} = \\log{(z)} and \\int \\rho_{f}{(z)} dz = \\int \\log{(z)} dz and \\int \\rho_{f}{(z)} dz = b + z \\log{(z)} - z and \\int \\log{(z)} dz = b + z \\log{(z)} - z", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\rho_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho_f')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('b', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('b', commutative=True), Mul(Symbol('z', commutative=True), log(Symbol('z', commutative=True))), Mul(Integer(-1), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} = \\mathbf{J}_P \\mathbf{M} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{M},\\mathbf{J}_P)} = (\\int \\mathbf{J}_P \\mathbf{M} d\\mathbf{J}_P)^{\\mathbf{J}_P}, then obtain (\\int \\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} d\\mathbf{J}_P)^{\\mathbf{J}_P} = \\operatorname{g_{\\varepsilon}}{(\\mathbf{M},\\mathbf{J}_P)}", "derivation": "\\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} = \\mathbf{J}_P \\mathbf{M} and \\int \\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\mathbf{J}_P \\mathbf{M} d\\mathbf{J}_P and (\\int \\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} d\\mathbf{J}_P)^{\\mathbf{J}_P} = (\\int \\mathbf{J}_P \\mathbf{M} d\\mathbf{J}_P)^{\\mathbf{J}_P} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{M},\\mathbf{J}_P)} = (\\int \\mathbf{J}_P \\mathbf{M} d\\mathbf{J}_P)^{\\mathbf{J}_P} and (\\int \\operatorname{x^{{\\}'}}{(\\mathbf{M},\\mathbf{J}_P)} d\\mathbf{J}_P)^{\\mathbf{J}_P} = \\operatorname{g_{\\varepsilon}}{(\\mathbf{M},\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Integral(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given p{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\omega)}, then derive \\int p{(\\omega)} d\\omega = \\mathbf{P} + \\log{(\\omega)}, then obtain - \\mathbf{P} - \\log{(\\omega)} + \\int \\frac{d}{d \\omega} \\log{(\\omega)} d\\omega = 0", "derivation": "p{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\omega)} and \\int p{(\\omega)} d\\omega = \\int \\frac{d}{d \\omega} \\log{(\\omega)} d\\omega and \\int p{(\\omega)} d\\omega = \\mathbf{P} + \\log{(\\omega)} and \\int \\frac{d}{d \\omega} \\log{(\\omega)} d\\omega = \\mathbf{P} + \\log{(\\omega)} and - \\mathbf{P} - \\log{(\\omega)} + \\int \\frac{d}{d \\omega} \\log{(\\omega)} d\\omega = 0", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\omega', commutative=True)), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\omega', commutative=True))))"], [["minus", 4, "Add(Symbol('\\\\mathbf{P}', commutative=True), log(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\omega', commutative=True))), Integral(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\Psi{(\\delta)} = \\sin{(\\log{(\\delta)})}, then obtain \\log{(\\Psi^{\\delta}{(\\delta)} + 1)} = \\log{(\\sin^{\\delta}{(\\log{(\\delta)})} + 1)}", "derivation": "\\Psi{(\\delta)} = \\sin{(\\log{(\\delta)})} and \\Psi^{\\delta}{(\\delta)} = \\sin^{\\delta}{(\\log{(\\delta)})} and \\Psi^{\\delta}{(\\delta)} + 1 = \\sin^{\\delta}{(\\log{(\\delta)})} + 1 and \\log{(\\Psi^{\\delta}{(\\delta)} + 1)} = \\log{(\\sin^{\\delta}{(\\log{(\\delta)})} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True)), sin(log(Symbol('\\\\delta', commutative=True))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Pow(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(1)), Add(Pow(sin(log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Integer(1)))"], [["log", 3], "Equality(log(Add(Pow(Function('\\\\Psi')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(1))), log(Add(Pow(sin(log(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(r_{0},\\theta)} = \\cos{(\\theta + r_{0})}, then obtain \\theta \\frac{\\partial}{\\partial r_{0}} \\varepsilon_{0}{(r_{0},\\theta)} = - \\theta \\sin{(\\theta + r_{0})}", "derivation": "\\varepsilon_{0}{(r_{0},\\theta)} = \\cos{(\\theta + r_{0})} and \\theta \\varepsilon_{0}{(r_{0},\\theta)} = \\theta \\cos{(\\theta + r_{0})} and \\theta \\varepsilon_{0}{(r_{0},\\theta)} - \\theta = \\theta \\cos{(\\theta + r_{0})} - \\theta and \\frac{\\partial}{\\partial r_{0}} (\\theta \\varepsilon_{0}{(r_{0},\\theta)} - \\theta) = \\frac{\\partial}{\\partial r_{0}} (\\theta \\cos{(\\theta + r_{0})} - \\theta) and \\theta \\frac{\\partial}{\\partial r_{0}} \\varepsilon_{0}{(r_{0},\\theta)} = - \\theta \\sin{(\\theta + r_{0})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True))))"], [["times", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))))"], [["minus", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('\\\\theta', commutative=True), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\theta', commutative=True), Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\theta', commutative=True), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('\\\\theta', commutative=True), Derivative(Function('\\\\varepsilon_0')(Symbol('r_0', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), sin(Add(Symbol('\\\\theta', commutative=True), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(C_{d},p)} = - p + \\log{(C_{d})} and \\mathbf{f}{(C_{1},v)} = v e^{C_{1}}, then obtain C_{d} \\mathbf{f}{(C_{1},v)} e^{1 - p} = C_{d} v e^{C_{1}} e^{1 - p}", "derivation": "\\mathbf{F}{(C_{d},p)} = - p + \\log{(C_{d})} and \\mathbf{F}{(C_{d},p)} + 1 = - p + \\log{(C_{d})} + 1 and \\mathbf{f}{(C_{1},v)} = v e^{C_{1}} and \\mathbf{f}{(C_{1},v)} e^{\\mathbf{F}{(C_{d},p)} + 1} = v e^{C_{1}} e^{\\mathbf{F}{(C_{d},p)} + 1} and C_{d} \\mathbf{f}{(C_{1},v)} e^{1 - p} = C_{d} v e^{C_{1}} e^{1 - p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), log(Symbol('C_d', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), log(Symbol('C_d', commutative=True)), Integer(1)))"], ["get_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('C_1', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), exp(Symbol('C_1', commutative=True))))"], [["times", 3, "exp(Add(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{f}')(Symbol('C_1', commutative=True), Symbol('v', commutative=True)), exp(Add(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Integer(1)))), Mul(Symbol('v', commutative=True), exp(Symbol('C_1', commutative=True)), exp(Add(Function('\\\\mathbf{F}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('C_d', commutative=True), Function('\\\\mathbf{f}')(Symbol('C_1', commutative=True), Symbol('v', commutative=True)), exp(Add(Integer(1), Mul(Integer(-1), Symbol('p', commutative=True))))), Mul(Symbol('C_d', commutative=True), Symbol('v', commutative=True), exp(Symbol('C_1', commutative=True)), exp(Add(Integer(1), Mul(Integer(-1), Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} = - n_{2} + \\sin{(\\hat{H}_{\\lambda})}, then obtain (\\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2})^{2} = (- \\frac{n_{2}^{2}}{2} + n_{2} \\sin{(\\hat{H}_{\\lambda})} + q) \\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2}", "derivation": "\\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} = - n_{2} + \\sin{(\\hat{H}_{\\lambda})} and \\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2} = \\int (- n_{2} + \\sin{(\\hat{H}_{\\lambda})}) dn_{2} and (\\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2})^{2} = (\\int (- n_{2} + \\sin{(\\hat{H}_{\\lambda})}) dn_{2}) \\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2} and (\\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2})^{2} = (- \\frac{n_{2}^{2}}{2} + n_{2} \\sin{(\\hat{H}_{\\lambda})} + q) \\int \\mathbf{E}{(n_{2},\\hat{H}_{\\lambda})} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["times", 2, "Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(2)), Mul(Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Mul(Symbol('n_2', commutative=True), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('q', commutative=True)), Integral(Function('\\\\mathbf{E}')(Symbol('n_2', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} = \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H} and \\mathbf{B}{(g,b)} = g^{b}, then obtain g^{b} + \\mathbf{B}{(g,b)} - 2 \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H} = 2 g^{b} - 2 \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H}", "derivation": "\\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} = \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H} and \\mathbf{B}{(g,b)} = g^{b} and \\mathbf{B}{(g,b)} - \\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} = g^{b} - \\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} and g^{b} + \\mathbf{B}{(g,b)} - 2 \\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} = 2 g^{b} - 2 \\tilde{g}{(f_{\\mathbf{p}},\\mathbf{H})} and g^{b} + \\mathbf{B}{(g,b)} - 2 \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H} = 2 g^{b} - 2 \\int (\\mathbf{H} + f_{\\mathbf{p}}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Pow(Symbol('g', commutative=True), Symbol('b', commutative=True)))"], [["minus", 2, "Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Pow(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["add", 3, "Add(Pow(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], "Equality(Add(Pow(Symbol('g', commutative=True), Symbol('b', commutative=True)), Function('\\\\mathbf{B}')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(2), Pow(Symbol('g', commutative=True), Symbol('b', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Symbol('g', commutative=True), Symbol('b', commutative=True)), Function('\\\\mathbf{B}')(Symbol('g', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Integer(2), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))), Add(Mul(Integer(2), Pow(Symbol('g', commutative=True), Symbol('b', commutative=True))), Mul(Integer(-1), Integer(2), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))))"]]}, {"prompt": "Given \\rho_{b}{(W)} = \\log{(W)}, then derive \\int \\rho_{b}{(W)} dW = F_{x} + W \\log{(W)} - W, then obtain \\frac{\\partial}{\\partial W} (F_{x} + W \\log{(W)} - W) = \\frac{d}{d W} \\int \\log{(W)} dW", "derivation": "\\rho_{b}{(W)} = \\log{(W)} and \\int \\rho_{b}{(W)} dW = \\int \\log{(W)} dW and \\int \\rho_{b}{(W)} dW = F_{x} + W \\log{(W)} - W and F_{x} + W \\log{(W)} - W = \\int \\log{(W)} dW and F_{x} + W \\rho_{b}{(W)} - W = \\int \\log{(W)} dW and F_{x} + W \\log{(W)} - W = F_{x} + W \\rho_{b}{(W)} - W and \\frac{\\partial}{\\partial W} (F_{x} + W \\log{(W)} - W) = \\frac{\\partial}{\\partial W} (F_{x} + W \\rho_{b}{(W)} - W) and \\frac{\\partial}{\\partial W} (F_{x} + W \\log{(W)} - W) = \\frac{d}{d W} \\int \\log{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\rho_b')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), Function('\\\\rho_b')(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), Function('\\\\rho_b')(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))))"], [["differentiate", 6, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), Function('\\\\rho_b')(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(Add(Symbol('F_x', commutative=True), Mul(Symbol('W', commutative=True), log(Symbol('W', commutative=True))), Mul(Integer(-1), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{p}{(k)} = \\cos{(k)} and m{(k)} = - k + \\sigma_{p}{(k)}, then obtain \\sin{(k)} \\int m{(k)} dk = \\sin{(k)} \\int (- k + \\cos{(k)}) dk", "derivation": "\\sigma_{p}{(k)} = \\cos{(k)} and - k + \\sigma_{p}{(k)} = - k + \\cos{(k)} and m{(k)} = - k + \\sigma_{p}{(k)} and \\int m{(k)} dk = \\int (- k + \\sigma_{p}{(k)}) dk and m{(k)} = - k + \\cos{(k)} and \\sin{(k)} \\int m{(k)} dk = \\sin{(k)} \\int (- k + \\sigma_{p}{(k)}) dk and \\sin{(k)} \\int (- k + \\cos{(k)}) dk = \\sin{(k)} \\int (- k + \\sigma_{p}{(k)}) dk and \\sin{(k)} \\int m{(k)} dk = \\sin{(k)} \\int (- k + \\cos{(k)}) dk", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('k', commutative=True)), cos(Symbol('k', commutative=True)))"], [["minus", 1, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\sigma_p')(Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('k', commutative=True)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\sigma_p')(Symbol('k', commutative=True))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\sigma_p')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('m')(Symbol('k', commutative=True)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))))"], [["times", 4, "sin(Symbol('k', commutative=True))"], "Equality(Mul(sin(Symbol('k', commutative=True)), Integral(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\sigma_p')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\sigma_p')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(sin(Symbol('k', commutative=True)), Integral(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Mul(sin(Symbol('k', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('k', commutative=True)), cos(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(E_{\\lambda})} = \\log{(\\cos{(E_{\\lambda})})}, then obtain \\int (\\operatorname{n_{1}}^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}} dE_{\\lambda} = \\int (\\log{(\\cos{(E_{\\lambda})})}^{E_{\\lambda}})^{E_{\\lambda}} dE_{\\lambda}", "derivation": "\\operatorname{n_{1}}{(E_{\\lambda})} = \\log{(\\cos{(E_{\\lambda})})} and \\operatorname{n_{1}}^{E_{\\lambda}}{(E_{\\lambda})} = \\log{(\\cos{(E_{\\lambda})})}^{E_{\\lambda}} and (\\operatorname{n_{1}}^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}} = (\\log{(\\cos{(E_{\\lambda})})}^{E_{\\lambda}})^{E_{\\lambda}} and \\int (\\operatorname{n_{1}}^{E_{\\lambda}}{(E_{\\lambda})})^{E_{\\lambda}} dE_{\\lambda} = \\int (\\log{(\\cos{(E_{\\lambda})})}^{E_{\\lambda}})^{E_{\\lambda}} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), log(cos(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(log(cos(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Pow(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Pow(log(cos(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('n_1')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Pow(Pow(log(cos(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(S)} = \\sin{(S)}, then derive \\frac{\\int \\tilde{g}^*{(S)} dS}{\\Psi_{nl} - \\cos{(S)}} = 1, then obtain \\frac{\\cos{(S)} \\int \\tilde{g}^*{(S)} dS}{\\Psi_{nl} - \\cos{(S)}} = \\cos{(S)}", "derivation": "\\tilde{g}^*{(S)} = \\sin{(S)} and \\int \\tilde{g}^*{(S)} dS = \\int \\sin{(S)} dS and \\frac{\\int \\tilde{g}^*{(S)} dS}{\\int \\sin{(S)} dS} = 1 and \\frac{\\int \\tilde{g}^*{(S)} dS}{\\Psi_{nl} - \\cos{(S)}} = 1 and \\frac{\\cos{(S)} \\int \\tilde{g}^*{(S)} dS}{\\Psi_{nl} - \\cos{(S)}} = \\cos{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["divide", 2, "Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))"], "Equality(Mul(Integral(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Pow(Integral(sin(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Integer(-1)), Integral(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Integer(1))"], [["times", 4, "cos(Symbol('S', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Mul(Integer(-1), cos(Symbol('S', commutative=True)))), Integer(-1)), cos(Symbol('S', commutative=True)), Integral(Function('\\\\tilde{g}^*')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), cos(Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(S,J,\\chi)} = J \\chi + S, then derive - J \\chi + \\int \\mathbf{J}_M{(S,J,\\chi)} dS = J S \\chi - J \\chi + \\frac{S^{2}}{2} + T, then obtain \\frac{- J \\chi + \\int \\mathbf{J}_M{(S,J,\\chi)} dS}{S} = \\frac{J S \\chi - J \\chi + \\frac{S^{2}}{2} + T}{S}", "derivation": "\\mathbf{J}_M{(S,J,\\chi)} = J \\chi + S and \\int \\mathbf{J}_M{(S,J,\\chi)} dS = \\int (J \\chi + S) dS and - J \\chi + \\int \\mathbf{J}_M{(S,J,\\chi)} dS = - J \\chi + \\int (J \\chi + S) dS and - J \\chi + \\int \\mathbf{J}_M{(S,J,\\chi)} dS = J S \\chi - J \\chi + \\frac{S^{2}}{2} + T and \\frac{- J \\chi + \\int \\mathbf{J}_M{(S,J,\\chi)} dS}{S} = \\frac{J S \\chi - J \\chi + \\frac{S^{2}}{2} + T}{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Mul(Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["minus", 2, "Mul(Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Add(Mul(Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Symbol('J', commutative=True), Symbol('S', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('T', commutative=True)))"], [["divide", 4, "Symbol('S', commutative=True)"], "Equality(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Integral(Function('\\\\mathbf{J}_M')(Symbol('S', commutative=True), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('S', commutative=True))))), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Add(Mul(Symbol('J', commutative=True), Symbol('S', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mu{(E_{n},\\omega)} = \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\omega}, then derive \\mu{(E_{n},\\omega)} = \\frac{1}{\\omega}, then obtain 0 = - \\mu{(E_{n},\\omega)} + \\frac{1}{\\omega}", "derivation": "\\mu{(E_{n},\\omega)} = \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\omega} and - \\mu{(E_{n},\\omega)} = - \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\omega} and 0 = \\mu{(E_{n},\\omega)} - \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\omega} and \\mu{(E_{n},\\omega)} = \\frac{1}{\\omega} and 0 = - \\frac{\\partial}{\\partial E_{n}} \\frac{E_{n}}{\\omega} + \\frac{1}{\\omega} and 0 = - \\mu{(E_{n},\\omega)} + \\frac{1}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["minus", 2, "Mul(Integer(-1), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Integer(0), Add(Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Mul(Symbol('E_n', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('E_n', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} = \\log{(\\mathbf{J} - \\mathbf{P})}, then obtain 2 \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} + \\log{(e^{\\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})}})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} = \\log{(\\mathbf{J} - \\mathbf{P})} and 2 \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} + \\log{(\\mathbf{J} - \\mathbf{P})} and e^{\\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})}} = \\mathbf{J} - \\mathbf{P} and 2 \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} = \\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})} + \\log{(e^{\\operatorname{t_{2}}{(\\mathbf{J},\\mathbf{P})}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["add", 1, "Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Integer(2), Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))))"], [["exp", 1], "Equality(exp(Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Integer(2), Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), log(exp(Function('t_2')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(V,F_{x})} = V \\sin{(F_{x})} and \\operatorname{F_{x}}{(V)} = V^{2}, then derive \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = V \\cos{(F_{x})}, then obtain V \\sin{(F_{x})} \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = \\operatorname{F_{x}}{(V)} \\sin{(F_{x})} \\cos{(F_{x})}", "derivation": "\\mathbf{F}{(V,F_{x})} = V \\sin{(F_{x})} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = \\frac{\\partial}{\\partial F_{x}} V \\sin{(F_{x})} and V \\sin{(F_{x})} \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = V \\sin{(F_{x})} \\frac{\\partial}{\\partial F_{x}} V \\sin{(F_{x})} and \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = V \\cos{(F_{x})} and V \\cos{(F_{x})} = \\frac{\\partial}{\\partial F_{x}} V \\sin{(F_{x})} and V \\sin{(F_{x})} \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = V^{2} \\sin{(F_{x})} \\cos{(F_{x})} and \\operatorname{F_{x}}{(V)} = V^{2} and V \\sin{(F_{x})} \\frac{\\partial}{\\partial F_{x}} \\mathbf{F}{(V,F_{x})} = \\operatorname{F_{x}}{(V)} \\sin{(F_{x})} \\cos{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["times", 2, "Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True)))"], "Equality(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True)), Derivative(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Symbol('V', commutative=True), cos(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('V', commutative=True), cos(Symbol('F_x', commutative=True))), Derivative(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('V', commutative=True), Integer(2)), sin(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('V', commutative=True)), Pow(Symbol('V', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Symbol('V', commutative=True), sin(Symbol('F_x', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('V', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Function('F_x')(Symbol('V', commutative=True)), sin(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbb{I})} = e^{\\mathbb{I}}, then derive \\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} + 1 = e^{\\mathbb{I}} + 1, then obtain (\\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} + 1)^{\\mathbb{I}} = (e^{\\mathbb{I}} + 1)^{\\mathbb{I}}", "derivation": "p{(\\mathbb{I})} = e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} and \\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} + 1 = \\frac{d}{d \\mathbb{I}} e^{\\mathbb{I}} + 1 and \\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} + 1 = e^{\\mathbb{I}} + 1 and (\\frac{d}{d \\mathbb{I}} p{(\\mathbb{I})} + 1)^{\\mathbb{I}} = (e^{\\mathbb{I}} + 1)^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\mathbb{I}', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Add(Derivative(Function('p')(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Add(exp(Symbol('\\\\mathbb{I}', commutative=True)), Integer(1)), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(\\mu,l)} = \\mu l, then obtain \\mathbf{v}{(\\mu,l)} \\int \\mathbf{v}^{l}{(\\mu,l)} dl = \\mu l \\int \\mathbf{v}^{l}{(\\mu,l)} dl", "derivation": "\\mathbf{v}{(\\mu,l)} = \\mu l and \\mathbf{v}^{l}{(\\mu,l)} = (\\mu l)^{l} and \\int \\mathbf{v}^{l}{(\\mu,l)} dl = \\int (\\mu l)^{l} dl and \\mathbf{v}{(\\mu,l)} \\int (\\mu l)^{l} dl = \\mu l \\int (\\mu l)^{l} dl and \\mathbf{v}{(\\mu,l)} \\int \\mathbf{v}^{l}{(\\mu,l)} dl = \\mu l \\int \\mathbf{v}^{l}{(\\mu,l)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 1, "Integral(Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True), Integral(Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True), Integral(Pow(Function('\\\\mathbf{v}')(Symbol('\\\\mu', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\psi{(v_{2},A_{y})} = - A_{y} + v_{2}, then obtain - (- A_{y} + v_{2})^{A_{y}} - \\frac{- A_{y} + v_{2}}{v_{2}} = - (- A_{y} + v_{2})^{A_{y}} + \\frac{A_{y} - v_{2}}{v_{2}}", "derivation": "\\psi{(v_{2},A_{y})} = - A_{y} + v_{2} and - \\psi{(v_{2},A_{y})} = A_{y} - v_{2} and - \\frac{\\psi{(v_{2},A_{y})}}{v_{2}} = \\frac{A_{y} - v_{2}}{v_{2}} and - \\frac{- A_{y} + v_{2}}{v_{2}} = \\frac{A_{y} - v_{2}}{v_{2}} and - \\psi^{A_{y}}{(v_{2},A_{y})} - \\frac{- A_{y} + v_{2}}{v_{2}} = - \\psi^{A_{y}}{(v_{2},A_{y})} + \\frac{A_{y} - v_{2}}{v_{2}} and - (- A_{y} + v_{2})^{A_{y}} - \\frac{- A_{y} + v_{2}}{v_{2}} = - (- A_{y} + v_{2})^{A_{y}} + \\frac{A_{y} - v_{2}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["divide", 2, "Symbol('v_2', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True)))))"], [["minus", 4, "Pow(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('v_2', commutative=True), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True)), Symbol('A_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Symbol('v_2', commutative=True)), Symbol('A_y', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Symbol('A_y', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})} and \\lambda{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})}, then obtain \\int 2 \\lambda{(\\mathbf{s})} d\\mathbf{s} = \\int (\\lambda{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})}) d\\mathbf{s}", "derivation": "\\Omega{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})} and 2 \\Omega{(\\mathbf{s})} = \\Omega{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})} and \\int 2 \\Omega{(\\mathbf{s})} d\\mathbf{s} = \\int (\\Omega{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})}) d\\mathbf{s} and \\lambda{(\\mathbf{s})} = \\log{(e^{\\mathbf{s}})} and \\Omega{(\\mathbf{s})} = \\lambda{(\\mathbf{s})} and \\int 2 \\lambda{(\\mathbf{s})} d\\mathbf{s} = \\int (\\lambda{(\\mathbf{s})} + \\log{(e^{\\mathbf{s}})}) d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 1, "Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Integral(Mul(Integer(2), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), log(exp(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(f_{\\mathbf{v}},l)} = \\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l) and \\mathbf{g}{(g^{\\prime}_{\\varepsilon},C_{1},z)} = C_{1} + g^{\\prime}_{\\varepsilon} + z, then derive \\operatorname{f^{\\prime}}{(f_{\\mathbf{v}},l)} = 1, then obtain \\frac{\\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l)}{C_{1} + g^{\\prime}_{\\varepsilon} + z} = \\frac{1}{C_{1} + g^{\\prime}_{\\varepsilon} + z}", "derivation": "\\operatorname{f^{\\prime}}{(f_{\\mathbf{v}},l)} = \\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l) and \\operatorname{f^{\\prime}}{(f_{\\mathbf{v}},l)} = 1 and \\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l) = 1 and \\mathbf{g}{(g^{\\prime}_{\\varepsilon},C_{1},z)} = C_{1} + g^{\\prime}_{\\varepsilon} + z and \\frac{\\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l)}{\\mathbf{g}{(g^{\\prime}_{\\varepsilon},C_{1},z)}} = \\frac{1}{\\mathbf{g}{(g^{\\prime}_{\\varepsilon},C_{1},z)}} and \\frac{\\frac{\\partial}{\\partial l} (- f_{\\mathbf{v}} + l)}{C_{1} + g^{\\prime}_{\\varepsilon} + z} = \\frac{1}{C_{1} + g^{\\prime}_{\\varepsilon} + z}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(1))"], ["get_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('C_1', commutative=True), Symbol('z', commutative=True)), Add(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('z', commutative=True)))"], [["divide", 3, "Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('C_1', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('C_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Function('\\\\mathbf{g}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('C_1', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Pow(Add(Symbol('C_1', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('z', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mu{(E,\\mu_0,\\nabla)} = E + \\frac{\\mu_0}{\\nabla}, then obtain \\frac{\\partial}{\\partial E} ((E + \\frac{\\mu_0}{\\nabla})^{E} - \\mu^{E}{(E,\\mu_0,\\nabla)})^{E} = \\frac{d}{d E} 1", "derivation": "\\mu{(E,\\mu_0,\\nabla)} = E + \\frac{\\mu_0}{\\nabla} and \\mu^{E}{(E,\\mu_0,\\nabla)} = (E + \\frac{\\mu_0}{\\nabla})^{E} and 0 = (E + \\frac{\\mu_0}{\\nabla})^{E} - \\mu^{E}{(E,\\mu_0,\\nabla)} and 0^{E} = ((E + \\frac{\\mu_0}{\\nabla})^{E} - \\mu^{E}{(E,\\mu_0,\\nabla)})^{E} and \\frac{d}{d E} 0^{E} = \\frac{\\partial}{\\partial E} ((E + \\frac{\\mu_0}{\\nabla})^{E} - \\mu^{E}{(E,\\mu_0,\\nabla)})^{E} and \\frac{\\partial}{\\partial E} ((E + \\frac{\\mu_0}{\\nabla})^{E} - \\mu^{E}{(E,\\mu_0,\\nabla)})^{E} = \\frac{d}{d E} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True)), Pow(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('E', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True)))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E', commutative=True)), Pow(Add(Pow(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True)))), Symbol('E', commutative=True)))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Pow(Integer(0), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Add(Pow(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Pow(Add(Pow(Add(Symbol('E', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1)))), Symbol('E', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('E', commutative=True), Symbol('\\\\mu_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('E', commutative=True)))), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(\\theta_2)} = \\theta_2, then obtain \\frac{\\partial}{\\partial v_{y}} (- \\theta_2 - v_{y} + 2 \\hat{X}{(\\theta_2)}) = \\frac{\\partial}{\\partial v_{y}} (\\theta_2 - v_{y})", "derivation": "\\hat{X}{(\\theta_2)} = \\theta_2 and - \\theta_2 - v_{y} + \\hat{X}{(\\theta_2)} = - v_{y} and - v_{y} + \\hat{X}{(\\theta_2)} = \\theta_2 - v_{y} and - \\theta_2 - v_{y} + 2 \\hat{X}{(\\theta_2)} = - v_{y} + \\hat{X}{(\\theta_2)} and - \\theta_2 - v_{y} + 2 \\hat{X}{(\\theta_2)} = \\theta_2 - v_{y} and \\frac{\\partial}{\\partial v_{y}} (- \\theta_2 - v_{y} + 2 \\hat{X}{(\\theta_2)}) = \\frac{\\partial}{\\partial v_{y}} (\\theta_2 - v_{y})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))"], [["minus", 1, "Add(Symbol('\\\\theta_2', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('v_y', commutative=True)))"], [["add", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["differentiate", 5, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True)), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(\\hat{x}_0,r_{0})} = \\hat{x}_0^{r_{0}} and t{(S)} = \\log{(S)}, then obtain \\frac{- t{(S)} + \\frac{E{(\\hat{x}_0,r_{0})}}{r_{0}}}{\\hat{x}_0} = \\frac{\\frac{\\hat{x}_0^{r_{0}}}{r_{0}} - t{(S)}}{\\hat{x}_0}", "derivation": "E{(\\hat{x}_0,r_{0})} = \\hat{x}_0^{r_{0}} and \\frac{E{(\\hat{x}_0,r_{0})}}{r_{0}} = \\frac{\\hat{x}_0^{r_{0}}}{r_{0}} and t{(S)} = \\log{(S)} and - \\log{(S)} + \\frac{E{(\\hat{x}_0,r_{0})}}{r_{0}} = \\frac{\\hat{x}_0^{r_{0}}}{r_{0}} - \\log{(S)} and \\frac{- \\log{(S)} + \\frac{E{(\\hat{x}_0,r_{0})}}{r_{0}}}{\\hat{x}_0} = \\frac{\\frac{\\hat{x}_0^{r_{0}}}{r_{0}} - \\log{(S)}}{\\hat{x}_0} and \\frac{- t{(S)} + \\frac{E{(\\hat{x}_0,r_{0})}}{r_{0}}}{\\hat{x}_0} = \\frac{\\frac{\\hat{x}_0^{r_{0}}}{r_{0}} - t{(S)}}{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('t')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["minus", 2, "log(Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('S', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)))), Add(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Mul(Integer(-1), log(Symbol('S', commutative=True)))))"], [["divide", 4, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('S', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Mul(Integer(-1), log(Symbol('S', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('t')(Symbol('S', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True))))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Integer(-1))), Mul(Integer(-1), Function('t')(Symbol('S', commutative=True))))))"]]}, {"prompt": "Given W{(\\mathbf{A},G)} = \\frac{G}{\\mathbf{A}} and \\operatorname{C_{1}}{(\\rho_f,\\varphi)} = e^{\\frac{\\varphi}{\\rho_f}}, then obtain 1 = \\frac{\\frac{G}{\\mathbf{A}} - W{(\\mathbf{A},G)} + e^{\\frac{\\varphi}{\\rho_f}}}{\\frac{G}{\\mathbf{A}} + \\operatorname{C_{1}}{(\\rho_f,\\varphi)} - W{(\\mathbf{A},G)}}", "derivation": "W{(\\mathbf{A},G)} = \\frac{G}{\\mathbf{A}} and \\operatorname{C_{1}}{(\\rho_f,\\varphi)} = e^{\\frac{\\varphi}{\\rho_f}} and \\operatorname{C_{1}}{(\\rho_f,\\varphi)} + W{(\\mathbf{A},G)} = W{(\\mathbf{A},G)} + e^{\\frac{\\varphi}{\\rho_f}} and \\frac{G}{\\mathbf{A}} + \\operatorname{C_{1}}{(\\rho_f,\\varphi)} = \\frac{G}{\\mathbf{A}} + e^{\\frac{\\varphi}{\\rho_f}} and \\frac{G}{\\mathbf{A}} + \\operatorname{C_{1}}{(\\rho_f,\\varphi)} - W{(\\mathbf{A},G)} = \\frac{G}{\\mathbf{A}} - W{(\\mathbf{A},G)} + e^{\\frac{\\varphi}{\\rho_f}} and 1 = \\frac{\\frac{G}{\\mathbf{A}} - W{(\\mathbf{A},G)} + e^{\\frac{\\varphi}{\\rho_f}}}{\\frac{G}{\\mathbf{A}} + \\operatorname{C_{1}}{(\\rho_f,\\varphi)} - W{(\\mathbf{A},G)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"], ["get_premise", "Equality(Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True)), exp(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))), Add(Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True)), exp(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), exp(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))))"], [["minus", 4, "Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))"], "Equality(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True)))), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))), exp(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))))"], [["divide", 5, "Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('C_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True)))), Integer(-1)), Add(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('G', commutative=True))), exp(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))))"]]}, {"prompt": "Given \\theta_{2}{(\\psi,f_{E})} = \\psi f_{E}, then obtain \\frac{\\partial^{2}}{\\partial f_{E}\\partial \\psi} \\frac{\\theta_{2}{(\\psi,f_{E})}}{\\psi} = \\frac{d^{2}}{d f_{E}d \\psi} f_{E}", "derivation": "\\theta_{2}{(\\psi,f_{E})} = \\psi f_{E} and \\frac{\\theta_{2}{(\\psi,f_{E})}}{\\psi} = f_{E} and \\frac{\\partial}{\\partial \\psi} \\frac{\\theta_{2}{(\\psi,f_{E})}}{\\psi} = \\frac{d}{d \\psi} f_{E} and \\frac{\\partial^{2}}{\\partial f_{E}\\partial \\psi} \\frac{\\theta_{2}{(\\psi,f_{E})}}{\\psi} = \\frac{d^{2}}{d f_{E}d \\psi} f_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)))"], [["divide", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Symbol('f_E', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Function('\\\\theta_2')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Symbol('f_E', commutative=True), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(P_{g},m)} = \\frac{P_{g}}{m}, then derive \\frac{\\partial}{\\partial m} T{(P_{g},m)} = - \\frac{P_{g}}{m^{2}}, then obtain ((\\frac{\\partial}{\\partial m} T{(P_{g},m)})^{m})^{m} = ((- \\frac{T{(P_{g},m)}}{m})^{m})^{m}", "derivation": "T{(P_{g},m)} = \\frac{P_{g}}{m} and \\frac{\\partial}{\\partial m} T{(P_{g},m)} = \\frac{\\partial}{\\partial m} \\frac{P_{g}}{m} and \\frac{\\partial}{\\partial m} T{(P_{g},m)} = - \\frac{P_{g}}{m^{2}} and \\frac{\\partial}{\\partial m} T{(P_{g},m)} = - \\frac{T{(P_{g},m)}}{m} and (\\frac{\\partial}{\\partial m} T{(P_{g},m)})^{m} = (- \\frac{T{(P_{g},m)}}{m})^{m} and ((\\frac{\\partial}{\\partial m} T{(P_{g},m)})^{m})^{m} = ((- \\frac{T{(P_{g},m)}}{m})^{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('P_g', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_g', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('P_g', commutative=True), Pow(Symbol('m', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)), Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True))))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Derivative(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)), Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["power", 5, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Mul(Integer(-1), Pow(Symbol('m', commutative=True), Integer(-1)), Function('T')(Symbol('P_g', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given Q{(\\omega,\\Psi_{\\lambda},H)} = (- \\Psi_{\\lambda} + \\omega)^{H} and \\operatorname{y^{\\prime}}{(z,z^{*})} = z z^{*}, then obtain - \\Psi_{\\lambda} + \\omega + (- \\Psi_{\\lambda} + \\omega)^{H} \\operatorname{y^{\\prime}}^{z}{(z,z^{*})} = - \\Psi_{\\lambda} + \\omega + (z z^{*})^{z} (- \\Psi_{\\lambda} + \\omega)^{H}", "derivation": "Q{(\\omega,\\Psi_{\\lambda},H)} = (- \\Psi_{\\lambda} + \\omega)^{H} and \\operatorname{y^{\\prime}}{(z,z^{*})} = z z^{*} and \\operatorname{y^{\\prime}}^{z}{(z,z^{*})} = (z z^{*})^{z} and Q{(\\omega,\\Psi_{\\lambda},H)} \\operatorname{y^{\\prime}}^{z}{(z,z^{*})} = (z z^{*})^{z} Q{(\\omega,\\Psi_{\\lambda},H)} and (- \\Psi_{\\lambda} + \\omega)^{H} \\operatorname{y^{\\prime}}^{z}{(z,z^{*})} = (z z^{*})^{z} (- \\Psi_{\\lambda} + \\omega)^{H} and - \\Psi_{\\lambda} + \\omega + (- \\Psi_{\\lambda} + \\omega)^{H} \\operatorname{y^{\\prime}}^{z}{(z,z^{*})} = - \\Psi_{\\lambda} + \\omega + (z z^{*})^{z} (- \\Psi_{\\lambda} + \\omega)^{H}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('H', commutative=True)))"], ["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Mul(Symbol('z', commutative=True), Symbol('z^*', commutative=True)))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)), Pow(Mul(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)))"], [["times", 3, "Function('Q')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('H', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Mul(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)), Function('Q')(Symbol('\\\\omega', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('H', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True))), Mul(Pow(Mul(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('H', commutative=True))))"], [["add", 5, "Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('H', commutative=True)), Pow(Function('y^{\\\\prime}')(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Pow(Mul(Symbol('z', commutative=True), Symbol('z^*', commutative=True)), Symbol('z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('H', commutative=True)))))"]]}, {"prompt": "Given q{(v_{1})} = \\cos{(v_{1})}, then derive \\frac{\\int q{(v_{1})} dv_{1}}{v_{1} + \\cos{(v_{1})}} = \\frac{C_{2} + \\sin{(v_{1})}}{v_{1} + \\cos{(v_{1})}}, then obtain \\frac{\\int \\cos{(v_{1})} dv_{1}}{v_{1} + q{(v_{1})}} = \\frac{C_{2} + \\sin{(v_{1})}}{v_{1} + q{(v_{1})}}", "derivation": "q{(v_{1})} = \\cos{(v_{1})} and v_{1} + q{(v_{1})} = v_{1} + \\cos{(v_{1})} and \\int q{(v_{1})} dv_{1} = \\int \\cos{(v_{1})} dv_{1} and \\frac{\\int q{(v_{1})} dv_{1}}{v_{1} + \\cos{(v_{1})}} = \\frac{\\int \\cos{(v_{1})} dv_{1}}{v_{1} + \\cos{(v_{1})}} and \\frac{\\int q{(v_{1})} dv_{1}}{v_{1} + \\cos{(v_{1})}} = \\frac{C_{2} + \\sin{(v_{1})}}{v_{1} + \\cos{(v_{1})}} and \\frac{\\int \\cos{(v_{1})} dv_{1}}{v_{1} + \\cos{(v_{1})}} = \\frac{C_{2} + \\sin{(v_{1})}}{v_{1} + \\cos{(v_{1})}} and \\frac{\\int \\cos{(v_{1})} dv_{1}}{v_{1} + q{(v_{1})}} = \\frac{C_{2} + \\sin{(v_{1})}}{v_{1} + q{(v_{1})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('v_1', commutative=True)), cos(Symbol('v_1', commutative=True)))"], [["add", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('q')(Symbol('v_1', commutative=True))), Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('q')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["divide", 3, "Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1)), Integral(Function('q')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1)), Integral(Function('q')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Add(Symbol('C_2', commutative=True), sin(Symbol('v_1', commutative=True))), Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Add(Symbol('C_2', commutative=True), sin(Symbol('v_1', commutative=True))), Pow(Add(Symbol('v_1', commutative=True), cos(Symbol('v_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(Add(Symbol('v_1', commutative=True), Function('q')(Symbol('v_1', commutative=True))), Integer(-1)), Integral(cos(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Add(Symbol('C_2', commutative=True), sin(Symbol('v_1', commutative=True))), Pow(Add(Symbol('v_1', commutative=True), Function('q')(Symbol('v_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(t_{1},\\mu)} = \\cos{(\\mu t_{1})}, then obtain \\int \\mu \\operatorname{c_{0}}{(t_{1},\\mu)} \\cos{(\\mu t_{1})} d\\mu = \\int \\mu \\cos^{2}{(\\mu t_{1})} d\\mu", "derivation": "\\operatorname{c_{0}}{(t_{1},\\mu)} = \\cos{(\\mu t_{1})} and \\mu \\operatorname{c_{0}}{(t_{1},\\mu)} = \\mu \\cos{(\\mu t_{1})} and \\mu \\operatorname{c_{0}}^{2}{(t_{1},\\mu)} = \\mu \\operatorname{c_{0}}{(t_{1},\\mu)} \\cos{(\\mu t_{1})} and \\mu \\operatorname{c_{0}}{(t_{1},\\mu)} \\cos{(\\mu t_{1})} = \\mu \\cos^{2}{(\\mu t_{1})} and \\int \\mu \\operatorname{c_{0}}{(t_{1},\\mu)} \\cos{(\\mu t_{1})} d\\mu = \\int \\mu \\cos^{2}{(\\mu t_{1})} d\\mu", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True))))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True)))))"], [["times", 2, "Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True)))), Mul(Symbol('\\\\mu', commutative=True), Pow(cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True))), Integer(2))))"], [["integrate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\mu', commutative=True), Function('c_0')(Symbol('t_1', commutative=True), Symbol('\\\\mu', commutative=True)), cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Mul(Symbol('\\\\mu', commutative=True), Pow(cos(Mul(Symbol('\\\\mu', commutative=True), Symbol('t_1', commutative=True))), Integer(2))), Tuple(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given a{(E,\\varepsilon)} = E \\cos{(\\varepsilon)}, then obtain a{(E,\\varepsilon)} = 3 E \\cos{(\\varepsilon)} - 2 a{(E,\\varepsilon)}", "derivation": "a{(E,\\varepsilon)} = E \\cos{(\\varepsilon)} and E \\cos{(\\varepsilon)} = 2 E \\cos{(\\varepsilon)} - a{(E,\\varepsilon)} and 2 E \\cos{(\\varepsilon)} = 3 E \\cos{(\\varepsilon)} - a{(E,\\varepsilon)} and E \\cos{(\\varepsilon)} = 3 E \\cos{(\\varepsilon)} - 2 a{(E,\\varepsilon)} and a{(E,\\varepsilon)} = 3 E \\cos{(\\varepsilon)} - 2 a{(E,\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "Add(Mul(Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], "Equality(Mul(Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 2, "Mul(Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(3), Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(3), Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Integer(2), Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Integer(3), Symbol('E', commutative=True), cos(Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Integer(2), Function('a')(Symbol('E', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(n_{2})} = \\log{(\\cos{(n_{2})})} and \\eta{(n_{2})} = n_{2}^{2} \\log{(\\cos{(n_{2})})}, then obtain \\sin{(\\eta{(n_{2})})} - \\frac{1}{2} = \\sin{(n_{2}^{2} \\mathbf{s}{(n_{2})})} - \\frac{1}{2}", "derivation": "\\mathbf{s}{(n_{2})} = \\log{(\\cos{(n_{2})})} and n_{2} \\mathbf{s}{(n_{2})} = n_{2} \\log{(\\cos{(n_{2})})} and n_{2}^{2} \\mathbf{s}{(n_{2})} = n_{2}^{2} \\log{(\\cos{(n_{2})})} and \\eta{(n_{2})} = n_{2}^{2} \\log{(\\cos{(n_{2})})} and \\sin{(\\eta{(n_{2})})} = \\sin{(n_{2}^{2} \\log{(\\cos{(n_{2})})})} and \\sin{(\\eta{(n_{2})})} - 1 = \\sin{(n_{2}^{2} \\log{(\\cos{(n_{2})})})} - 1 and \\sin{(\\eta{(n_{2})})} - 1 = \\sin{(n_{2}^{2} \\mathbf{s}{(n_{2})})} - 1 and \\sin{(\\eta{(n_{2})})} - \\frac{1}{2} = \\sin{(n_{2}^{2} \\mathbf{s}{(n_{2})})} - \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('n_2', commutative=True)), log(cos(Symbol('n_2', commutative=True))))"], [["times", 1, "Symbol('n_2', commutative=True)"], "Equality(Mul(Symbol('n_2', commutative=True), Function('\\\\mathbf{s}')(Symbol('n_2', commutative=True))), Mul(Symbol('n_2', commutative=True), log(cos(Symbol('n_2', commutative=True)))))"], [["times", 2, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), Function('\\\\mathbf{s}')(Symbol('n_2', commutative=True))), Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), log(cos(Symbol('n_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('n_2', commutative=True)), Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), log(cos(Symbol('n_2', commutative=True)))))"], [["sin", 4], "Equality(sin(Function('\\\\eta')(Symbol('n_2', commutative=True))), sin(Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), log(cos(Symbol('n_2', commutative=True))))))"], [["minus", 5, 1], "Equality(Add(sin(Function('\\\\eta')(Symbol('n_2', commutative=True))), Integer(-1)), Add(sin(Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), log(cos(Symbol('n_2', commutative=True))))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(sin(Function('\\\\eta')(Symbol('n_2', commutative=True))), Integer(-1)), Add(sin(Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), Function('\\\\mathbf{s}')(Symbol('n_2', commutative=True)))), Integer(-1)))"], [["add", 7, "Rational(1, 2)"], "Equality(Add(sin(Function('\\\\eta')(Symbol('n_2', commutative=True))), Rational(-1, 2)), Add(sin(Mul(Pow(Symbol('n_2', commutative=True), Integer(2)), Function('\\\\mathbf{s}')(Symbol('n_2', commutative=True)))), Rational(-1, 2)))"]]}, {"prompt": "Given \\phi_{1}{(T,v_{2})} = v_{2}^{T}, then obtain - \\sin{(v_{2}^{T} - \\frac{v_{2}^{T} + \\phi_{1}{(T,v_{2})}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}})} = - \\sin{(v_{2}^{T} - \\frac{2 v_{2}^{T}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}})}", "derivation": "\\phi_{1}{(T,v_{2})} = v_{2}^{T} and v_{2}^{T} + \\phi_{1}{(T,v_{2})} = 2 v_{2}^{T} and \\frac{v_{2}^{T} + \\phi_{1}{(T,v_{2})}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}} = \\frac{2 v_{2}^{T}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}} and - v_{2}^{T} + \\frac{v_{2}^{T} + \\phi_{1}{(T,v_{2})}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}} = - v_{2}^{T} + \\frac{2 v_{2}^{T}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}} and - \\sin{(v_{2}^{T} - \\frac{v_{2}^{T} + \\phi_{1}{(T,v_{2})}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}})} = - \\sin{(v_{2}^{T} - \\frac{2 v_{2}^{T}}{- v_{2}^{T} + \\phi_{1}{(T,v_{2})}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)))"], [["add", 1, "Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))"], "Equality(Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True)))), Mul(Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1))))"], [["add", 3, "Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))))), Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Mul(Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))))))), Mul(Integer(-1), sin(Add(Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('T', commutative=True))), Function('\\\\phi_1')(Symbol('T', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)))))))"]]}, {"prompt": "Given J{(M_{E})} = \\log{(e^{M_{E}})}, then obtain \\int \\frac{d}{d M_{E}} (J{(M_{E})} + e^{M_{E}}) dM_{E} = \\int \\frac{d}{d M_{E}} (e^{M_{E}} + \\log{(e^{M_{E}})}) dM_{E}", "derivation": "J{(M_{E})} = \\log{(e^{M_{E}})} and J{(M_{E})} + e^{M_{E}} = e^{M_{E}} + \\log{(e^{M_{E}})} and \\frac{d}{d M_{E}} (J{(M_{E})} + e^{M_{E}}) = \\frac{d}{d M_{E}} (e^{M_{E}} + \\log{(e^{M_{E}})}) and \\int \\frac{d}{d M_{E}} (J{(M_{E})} + e^{M_{E}}) dM_{E} = \\int \\frac{d}{d M_{E}} (e^{M_{E}} + \\log{(e^{M_{E}})}) dM_{E}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('M_E', commutative=True)), log(exp(Symbol('M_E', commutative=True))))"], [["add", 1, "exp(Symbol('M_E', commutative=True))"], "Equality(Add(Function('J')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Add(exp(Symbol('M_E', commutative=True)), log(exp(Symbol('M_E', commutative=True)))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('M_E', commutative=True)), log(exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('M_E', commutative=True)"], "Equality(Integral(Derivative(Add(Function('J')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))), Integral(Derivative(Add(exp(Symbol('M_E', commutative=True)), log(exp(Symbol('M_E', commutative=True)))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(A_{1})} = \\log{(A_{1})} and M{(A_{1})} = \\log{(A_{1})}, then obtain \\frac{\\log{(A_{1})}^{A_{1}}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} = \\frac{\\mathbf{H}^{A_{1}}{(A_{1})}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}}", "derivation": "\\mathbf{H}{(A_{1})} = \\log{(A_{1})} and M{(A_{1})} = \\log{(A_{1})} and M^{A_{1}}{(A_{1})} = \\log{(A_{1})}^{A_{1}} and \\frac{M^{A_{1}}{(A_{1})}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} = \\frac{\\log{(A_{1})}^{A_{1}}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} and \\frac{M^{A_{1}}{(A_{1})}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} = \\frac{\\mathbf{H}^{A_{1}}{(A_{1})}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} and \\frac{\\log{(A_{1})}^{A_{1}}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}} = \\frac{\\mathbf{H}^{A_{1}}{(A_{1})}}{\\frac{\\partial}{\\partial \\omega} \\mu{(\\omega,C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], ["renaming_premise", "Equality(Function('M')(Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('M')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["divide", 3, "Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('M')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('M')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(log(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Function('\\\\mathbf{H}')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Derivative(Function('\\\\mu')(Symbol('\\\\omega', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given q{(g,\\sigma_p)} = \\sigma_p + 2 g, then obtain q{(g,\\sigma_p)} + \\int (2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)}) dg = \\sigma_p + 2 g + \\int (2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)}) dg", "derivation": "q{(g,\\sigma_p)} = \\sigma_p + 2 g and (\\sigma_p + 2 g) q{(g,\\sigma_p)} = (\\sigma_p + 2 g)^{2} and 2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)} = 2 g + (\\sigma_p + 2 g)^{2} and \\int (2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)}) dg = \\int (2 g + (\\sigma_p + 2 g)^{2}) dg and q{(g,\\sigma_p)} + \\int (2 g + (\\sigma_p + 2 g)^{2}) dg = \\sigma_p + 2 g + \\int (2 g + (\\sigma_p + 2 g)^{2}) dg and q{(g,\\sigma_p)} + \\int (2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)}) dg = \\sigma_p + 2 g + \\int (2 g + (\\sigma_p + 2 g) q{(g,\\sigma_p)}) dg", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2)))"], [["add", 2, "Mul(Integer(2), Symbol('g', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(2), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))), Tuple(Symbol('g', commutative=True))))"], [["add", 1, "Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))), Tuple(Symbol('g', commutative=True)))"], "Equality(Add(Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Pow(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Integer(2))), Tuple(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('g', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), Symbol('g', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(A_{y})} = \\sin{(\\cos{(A_{y})})}, then obtain (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})})^{2} = 2 (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})}) \\operatorname{L_{\\varepsilon}}{(A_{y})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(A_{y})} = \\sin{(\\cos{(A_{y})})} and 2 \\operatorname{L_{\\varepsilon}}{(A_{y})} = \\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})} and 4 \\operatorname{L_{\\varepsilon}}^{2}{(A_{y})} = 2 (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})}) \\operatorname{L_{\\varepsilon}}{(A_{y})} and 4 \\operatorname{L_{\\varepsilon}}^{2}{(A_{y})} = (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})})^{2} and (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})})^{2} = 2 (\\operatorname{L_{\\varepsilon}}{(A_{y})} + \\sin{(\\cos{(A_{y})})}) \\operatorname{L_{\\varepsilon}}{(A_{y})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True))))"], [["add", 1, "Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True))"], "Equality(Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True))), Add(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))))"], [["times", 2, "Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), Integer(2))), Pow(Add(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Integer(2)), Mul(Integer(2), Add(Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Function('L_{\\\\varepsilon}')(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given c{(b,Q)} = \\sin{(Q - b)}, then obtain \\frac{2 c^{3}{(b,Q)}}{\\sin^{3}{(Q - b)}} = - \\frac{- c{(b,Q)} - \\sin{(Q - b)}}{\\sin{(Q - b)}}", "derivation": "c{(b,Q)} = \\sin{(Q - b)} and - c{(b,Q)} = - \\sin{(Q - b)} and - 2 c{(b,Q)} = - c{(b,Q)} - \\sin{(Q - b)} and \\frac{2 c{(b,Q)}}{\\sin{(Q - b)}} = - \\frac{- c{(b,Q)} - \\sin{(Q - b)}}{\\sin{(Q - b)}} and \\frac{2 c{(b,Q)}}{\\sin{(Q - b)}} = 2 and \\frac{2 c^{2}{(b,Q)}}{\\sin^{2}{(Q - b)}} = - \\frac{- c{(b,Q)} - \\sin{(Q - b)}}{\\sin{(Q - b)}} and \\frac{2 c^{2}{(b,Q)}}{\\sin^{2}{(Q - b)}} = \\frac{2 c{(b,Q)}}{\\sin{(Q - b)}} and \\frac{2 c^{2}{(b,Q)}}{\\sin^{2}{(Q - b)}} = 2 and \\frac{2 c^{3}{(b,Q)}}{\\sin^{3}{(Q - b)}} = - \\frac{- c{(b,Q)} - \\sin{(Q - b)}}{\\sin{(Q - b)}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True))))))"], [["add", 2, "Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))))"], [["divide", 3, "Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))"], "Equality(Mul(Integer(2), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(2), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))), Integer(2))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Integer(2)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-2))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Integer(2)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-2))), Mul(Integer(2), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Integer(2)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-2))), Integer(2))"], [["substitute_RHS_for_LHS", 4, 8], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True)), Integer(3)), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-3))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('c')(Symbol('b', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))))), Pow(sin(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('b', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given B{(\\Omega)} = \\log{(\\log{(\\Omega)})}, then derive 0 = - \\frac{\\log{(\\log{(\\Omega)})} \\frac{d}{d \\Omega} B{(\\Omega)}}{B^{2}{(\\Omega)}} + \\frac{1}{\\Omega B{(\\Omega)} \\log{(\\Omega)}}, then obtain 0^{\\Omega} = (- \\frac{\\frac{d}{d \\Omega} \\log{(\\log{(\\Omega)})}}{\\log{(\\log{(\\Omega)})}} + \\frac{1}{\\Omega \\log{(\\Omega)} \\log{(\\log{(\\Omega)})}})^{\\Omega}", "derivation": "B{(\\Omega)} = \\log{(\\log{(\\Omega)})} and 1 = \\frac{\\log{(\\log{(\\Omega)})}}{B{(\\Omega)}} and \\frac{d}{d \\Omega} 1 = \\frac{d}{d \\Omega} \\frac{\\log{(\\log{(\\Omega)})}}{B{(\\Omega)}} and 0 = - \\frac{\\log{(\\log{(\\Omega)})} \\frac{d}{d \\Omega} B{(\\Omega)}}{B^{2}{(\\Omega)}} + \\frac{1}{\\Omega B{(\\Omega)} \\log{(\\Omega)}} and 0 = - \\frac{\\frac{d}{d \\Omega} \\log{(\\log{(\\Omega)})}}{\\log{(\\log{(\\Omega)})}} + \\frac{1}{\\Omega \\log{(\\Omega)} \\log{(\\log{(\\Omega)})}} and 0^{\\Omega} = (- \\frac{\\frac{d}{d \\Omega} \\log{(\\log{(\\Omega)})}}{\\log{(\\log{(\\Omega)})}} + \\frac{1}{\\Omega \\log{(\\Omega)} \\log{(\\log{(\\Omega)})}})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Omega', commutative=True)), log(log(Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "Function('B')(Symbol('\\\\Omega', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('B')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), log(log(Symbol('\\\\Omega', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('B')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), log(log(Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('B')(Symbol('\\\\Omega', commutative=True)), Integer(-2)), log(log(Symbol('\\\\Omega', commutative=True))), Derivative(Function('B')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('B')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\Omega', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(log(log(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Derivative(log(log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Pow(log(log(Symbol('\\\\Omega', commutative=True))), Integer(-1)))))"], [["power", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(log(log(Symbol('\\\\Omega', commutative=True))), Integer(-1)), Derivative(log(log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Pow(log(log(Symbol('\\\\Omega', commutative=True))), Integer(-1)))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then obtain - 2 \\mathbf{r} + \\int (- \\mathbf{r} + \\mathbf{P}{(\\mathbf{r})}) d\\mathbf{r} = - 2 \\mathbf{r} + \\int (- \\mathbf{r} + \\log{(\\mathbf{r})}) d\\mathbf{r}", "derivation": "\\mathbf{P}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and - \\mathbf{r} + \\mathbf{P}{(\\mathbf{r})} = - \\mathbf{r} + \\log{(\\mathbf{r})} and \\int (- \\mathbf{r} + \\mathbf{P}{(\\mathbf{r})}) d\\mathbf{r} = \\int (- \\mathbf{r} + \\log{(\\mathbf{r})}) d\\mathbf{r} and - 2 \\mathbf{r} + \\int (- \\mathbf{r} + \\mathbf{P}{(\\mathbf{r})}) d\\mathbf{r} = - 2 \\mathbf{r} + \\int (- \\mathbf{r} + \\log{(\\mathbf{r})}) d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given a{(l)} = e^{\\sin{(l)}} and \\operatorname{L_{\\varepsilon}}{(l)} = \\frac{2 a{(l)}}{a{(l)} + e^{\\sin{(l)}}}, then obtain \\frac{d}{d l} (- l + \\operatorname{L_{\\varepsilon}}{(l)}) = \\frac{d}{d l} (1 - l)", "derivation": "a{(l)} = e^{\\sin{(l)}} and 2 a{(l)} = a{(l)} + e^{\\sin{(l)}} and \\frac{2 a{(l)}}{a{(l)} + e^{\\sin{(l)}}} = 1 and - l + \\frac{2 a{(l)}}{a{(l)} + e^{\\sin{(l)}}} = 1 - l and \\operatorname{L_{\\varepsilon}}{(l)} = \\frac{2 a{(l)}}{a{(l)} + e^{\\sin{(l)}}} and - l + \\operatorname{L_{\\varepsilon}}{(l)} = 1 - l and \\frac{d}{d l} (- l + \\operatorname{L_{\\varepsilon}}{(l)}) = \\frac{d}{d l} (1 - l)", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True))))"], [["add", 1, "Function('a')(Symbol('l', commutative=True))"], "Equality(Mul(Integer(2), Function('a')(Symbol('l', commutative=True))), Add(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True)))))"], [["divide", 2, "Add(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Add(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True)))), Integer(-1)), Function('a')(Symbol('l', commutative=True))), Integer(1))"], [["minus", 3, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Mul(Integer(2), Pow(Add(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True)))), Integer(-1)), Function('a')(Symbol('l', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Symbol('l', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True)), Mul(Integer(2), Pow(Add(Function('a')(Symbol('l', commutative=True)), exp(sin(Symbol('l', commutative=True)))), Integer(-1)), Function('a')(Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["differentiate", 6, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(r)} = e^{r}, then obtain r + \\frac{d}{d r} \\int \\operatorname{x^{{\\}'}}{(r)} dr = r + \\frac{d}{d r} \\int e^{r} dr", "derivation": "\\operatorname{x^{{\\}'}}{(r)} = e^{r} and \\int \\operatorname{x^{{\\}'}}{(r)} dr = \\int e^{r} dr and \\frac{d}{d r} \\int \\operatorname{x^{{\\}'}}{(r)} dr = \\frac{d}{d r} \\int e^{r} dr and r + \\frac{d}{d r} \\int \\operatorname{x^{{\\}'}}{(r)} dr = r + \\frac{d}{d r} \\int e^{r} dr", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Integral(Function('x^\\\\prime')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["add", 3, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Derivative(Integral(Function('x^\\\\prime')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))), Add(Symbol('r', commutative=True), Derivative(Integral(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\Omega)} = \\cos{(\\Omega)}, then derive e^{\\operatorname{A_{1}}{(\\Omega)}} \\frac{d}{d \\Omega} \\operatorname{A_{1}}{(\\Omega)} = - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)}, then obtain \\log{(\\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} \\frac{d}{d \\Omega} \\cos{(\\Omega)})} = \\log{(\\frac{d}{d \\Omega} - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)})}", "derivation": "\\operatorname{A_{1}}{(\\Omega)} = \\cos{(\\Omega)} and e^{\\operatorname{A_{1}}{(\\Omega)}} = e^{\\cos{(\\Omega)}} and \\frac{d}{d \\Omega} e^{\\operatorname{A_{1}}{(\\Omega)}} = \\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} and e^{\\operatorname{A_{1}}{(\\Omega)}} \\frac{d}{d \\Omega} \\operatorname{A_{1}}{(\\Omega)} = - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)} and e^{\\cos{(\\Omega)}} \\frac{d}{d \\Omega} \\cos{(\\Omega)} = - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} \\frac{d}{d \\Omega} \\cos{(\\Omega)} = \\frac{d}{d \\Omega} - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)} and \\log{(\\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} \\frac{d}{d \\Omega} \\cos{(\\Omega)})} = \\log{(\\frac{d}{d \\Omega} - e^{\\cos{(\\Omega)}} \\sin{(\\Omega)})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["exp", 1], "Equality(exp(Function('A_1')(Symbol('\\\\Omega', commutative=True))), exp(cos(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(exp(Function('A_1')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(exp(Function('A_1')(Symbol('\\\\Omega', commutative=True))), Derivative(Function('A_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(-1), exp(cos(Symbol('\\\\Omega', commutative=True))), sin(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(exp(cos(Symbol('\\\\Omega', commutative=True))), Derivative(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(-1), exp(cos(Symbol('\\\\Omega', commutative=True))), sin(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(exp(cos(Symbol('\\\\Omega', commutative=True))), Derivative(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), exp(cos(Symbol('\\\\Omega', commutative=True))), sin(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["log", 6], "Equality(log(Derivative(Mul(exp(cos(Symbol('\\\\Omega', commutative=True))), Derivative(cos(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), log(Derivative(Mul(Integer(-1), exp(cos(Symbol('\\\\Omega', commutative=True))), sin(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(r,C)} = C + r, then obtain \\frac{\\partial}{\\partial r} (\\operatorname{f_{\\mathbf{v}}}{(r,C)} + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})}) = \\frac{\\partial}{\\partial r} (C + r + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})})", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(r,C)} = C + r and \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})} = \\cos{(C + r)} and \\operatorname{f_{\\mathbf{v}}}{(r,C)} + \\cos{(C + r)} = C + r + \\cos{(C + r)} and \\operatorname{f_{\\mathbf{v}}}{(r,C)} + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})} = C + r + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})} and \\frac{\\partial}{\\partial r} (\\operatorname{f_{\\mathbf{v}}}{(r,C)} + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})}) = \\frac{\\partial}{\\partial r} (C + r + \\cos{(\\operatorname{f_{\\mathbf{v}}}{(r,C)})})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))"], [["cos", 1], "Equality(cos(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True))), cos(Add(Symbol('C', commutative=True), Symbol('r', commutative=True))))"], [["add", 1, "cos(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)), cos(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))), Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Add(Symbol('C', commutative=True), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)), cos(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)))), Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)))))"], [["differentiate", 4, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)), cos(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('C', commutative=True), Symbol('r', commutative=True), cos(Function('f_{\\\\mathbf{v}}')(Symbol('r', commutative=True), Symbol('C', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})}, then obtain \\frac{\\operatorname{A_{x}}{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})}} = \\frac{\\cos{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})}}", "derivation": "\\operatorname{A_{x}}{(f_{\\mathbf{v}})} = \\cos{(f_{\\mathbf{v}})} and f_{\\mathbf{v}} + \\operatorname{A_{x}}{(f_{\\mathbf{v}})} = f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})} and \\frac{\\operatorname{A_{x}}{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\operatorname{A_{x}}{(f_{\\mathbf{v}})}} = \\frac{\\cos{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\operatorname{A_{x}}{(f_{\\mathbf{v}})}} and \\frac{\\operatorname{A_{x}}{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})}} = \\frac{\\cos{(f_{\\mathbf{v}})}}{f_{\\mathbf{v}} + \\cos{(f_{\\mathbf{v}})}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["divide", 1, "Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), Function('A_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integer(-1)), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s}, then obtain \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} + \\operatorname{a^{\\dagger}}^{2}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} + \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})}", "derivation": "\\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s} and \\mathbf{s} + \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s} + \\mathbf{s} and (\\mathbf{s} + \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})}) \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} = (C_{2} \\mathbf{s} + \\mathbf{s}) \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} and \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} + \\operatorname{a^{\\dagger}}^{2}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})} + \\mathbf{s} \\operatorname{a^{\\dagger}}{(C_{2},\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 2, "Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["expand", 3], "Equality(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2))), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('a^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(\\psi,f_{E})} = \\psi f_{E} and \\operatorname{n_{1}}{(\\psi,f_{E})} = \\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} + 1, then obtain \\int \\operatorname{n_{1}}{(\\psi,f_{E})} d\\psi = \\int (\\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} + 1) d\\psi", "derivation": "\\hat{H}{(\\psi,f_{E})} = \\psi f_{E} and 1 = \\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} and 2 = \\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} + 1 and \\operatorname{n_{1}}{(\\psi,f_{E})} = \\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} + 1 and \\operatorname{n_{1}}{(\\psi,f_{E})} = 2 and \\int \\operatorname{n_{1}}{(\\psi,f_{E})} d\\psi = \\int 2 d\\psi and \\int \\operatorname{n_{1}}{(\\psi,f_{E})} d\\psi = \\int (\\frac{\\psi f_{E}}{\\hat{H}{(\\psi,f_{E})}} + 1) d\\psi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('n_1')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Integer(2))"], [["integrate", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Integer(2), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integral(Function('n_1')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Mul(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True), Pow(Function('\\\\hat{H}')(Symbol('\\\\psi', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(E)} = \\sin{(\\log{(E)})}, then derive - \\log{(E)} + \\int \\operatorname{C_{2}}{(E)} dE = \\frac{E \\sin{(\\log{(E)})}}{2} - \\frac{E \\cos{(\\log{(E)})}}{2} + v_{1} - \\log{(E)}, then obtain (- \\log{(E)} + \\int \\operatorname{C_{2}}{(E)} dE)^{v_{1}} = (\\frac{E \\sin{(\\log{(E)})}}{2} - \\frac{E \\cos{(\\log{(E)})}}{2} + v_{1} - \\log{(E)})^{v_{1}}", "derivation": "\\operatorname{C_{2}}{(E)} = \\sin{(\\log{(E)})} and \\int \\operatorname{C_{2}}{(E)} dE = \\int \\sin{(\\log{(E)})} dE and - \\log{(E)} + \\int \\operatorname{C_{2}}{(E)} dE = - \\log{(E)} + \\int \\sin{(\\log{(E)})} dE and - \\log{(E)} + \\int \\operatorname{C_{2}}{(E)} dE = \\frac{E \\sin{(\\log{(E)})}}{2} - \\frac{E \\cos{(\\log{(E)})}}{2} + v_{1} - \\log{(E)} and (- \\log{(E)} + \\int \\operatorname{C_{2}}{(E)} dE)^{v_{1}} = (\\frac{E \\sin{(\\log{(E)})}}{2} - \\frac{E \\cos{(\\log{(E)})}}{2} + v_{1} - \\log{(E)})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('E', commutative=True)), sin(log(Symbol('E', commutative=True))))"], [["integrate", 1, "Symbol('E', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(sin(log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["minus", 2, "log(Symbol('E', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('E', commutative=True))), Integral(Function('C_2')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('E', commutative=True))), Integral(sin(log(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), log(Symbol('E', commutative=True))), Integral(Function('C_2')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('E', commutative=True), sin(log(Symbol('E', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('E', commutative=True), cos(log(Symbol('E', commutative=True)))), Symbol('v_1', commutative=True), Mul(Integer(-1), log(Symbol('E', commutative=True)))))"], [["power", 4, "Symbol('v_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('E', commutative=True))), Integral(Function('C_2')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), Symbol('v_1', commutative=True)), Pow(Add(Mul(Rational(1, 2), Symbol('E', commutative=True), sin(log(Symbol('E', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('E', commutative=True), cos(log(Symbol('E', commutative=True)))), Symbol('v_1', commutative=True), Mul(Integer(-1), log(Symbol('E', commutative=True)))), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} = \\frac{\\hat{\\mathbf{r}} - v}{A_{2}}, then obtain \\int \\frac{\\partial}{\\partial A_{2}} \\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} d\\hat{\\mathbf{r}} = \\dot{\\mathbf{r}} - \\frac{\\hat{\\mathbf{r}}^{2}}{2 A_{2}^{2}} + \\frac{\\hat{\\mathbf{r}} v}{A_{2}^{2}}", "derivation": "\\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} = \\frac{\\hat{\\mathbf{r}} - v}{A_{2}} and \\frac{\\partial}{\\partial A_{2}} \\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} = \\frac{\\partial}{\\partial A_{2}} \\frac{\\hat{\\mathbf{r}} - v}{A_{2}} and \\int \\frac{\\partial}{\\partial A_{2}} \\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} d\\hat{\\mathbf{r}} = \\int \\frac{\\partial}{\\partial A_{2}} \\frac{\\hat{\\mathbf{r}} - v}{A_{2}} d\\hat{\\mathbf{r}} and \\int \\frac{\\partial}{\\partial A_{2}} \\varphi^{*}{(\\hat{\\mathbf{r}},A_{2},v)} d\\hat{\\mathbf{r}} = \\dot{\\mathbf{r}} - \\frac{\\hat{\\mathbf{r}}^{2}}{2 A_{2}^{2}} + \\frac{\\hat{\\mathbf{r}} v}{A_{2}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('A_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(-2)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-2)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain \\sigma_{x}{(a^{\\dagger})} \\int \\log{(a^{\\dagger})} da^{\\dagger} = \\log{(a^{\\dagger})} \\int \\log{(a^{\\dagger})} da^{\\dagger}", "derivation": "\\sigma_{x}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and \\int \\sigma_{x}{(a^{\\dagger})} da^{\\dagger} = \\int \\log{(a^{\\dagger})} da^{\\dagger} and \\sigma_{x}{(a^{\\dagger})} \\int \\sigma_{x}{(a^{\\dagger})} da^{\\dagger} = \\log{(a^{\\dagger})} \\int \\sigma_{x}{(a^{\\dagger})} da^{\\dagger} and \\sigma_{x}{(a^{\\dagger})} \\int \\log{(a^{\\dagger})} da^{\\dagger} = \\log{(a^{\\dagger})} \\int \\log{(a^{\\dagger})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Integral(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(log(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\sigma_x')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(log(Symbol('a^{\\\\dagger}', commutative=True)), Integral(log(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(V,r)} = e^{\\frac{r}{V}}, then derive \\frac{\\partial}{\\partial r} \\hat{\\mathbf{x}}{(V,r)} = \\frac{e^{\\frac{r}{V}}}{V}, then obtain \\frac{\\partial}{\\partial r} \\hat{\\mathbf{x}}{(V,r)} = \\frac{\\hat{\\mathbf{x}}{(V,r)}}{V}", "derivation": "\\hat{\\mathbf{x}}{(V,r)} = e^{\\frac{r}{V}} and \\frac{\\partial}{\\partial r} \\hat{\\mathbf{x}}{(V,r)} = \\frac{\\partial}{\\partial r} e^{\\frac{r}{V}} and \\frac{\\partial}{\\partial r} \\hat{\\mathbf{x}}{(V,r)} = \\frac{e^{\\frac{r}{V}}}{V} and \\frac{\\partial}{\\partial r} \\hat{\\mathbf{x}}{(V,r)} = \\frac{\\hat{\\mathbf{x}}{(V,r)}}{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), exp(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Symbol('r', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('V', commutative=True), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(q)} = e^{q}, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq = \\dot{\\mathbf{r}} + e^{q}, then obtain 0 = (\\dot{\\mathbf{r}} + \\operatorname{f_{\\mathbf{p}}}{(q)})^{q} - (\\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq)^{q}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(q)} = e^{q} and \\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq = \\int e^{q} dq and \\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq = \\dot{\\mathbf{r}} + e^{q} and (\\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq)^{q} = (\\dot{\\mathbf{r}} + e^{q})^{q} and (\\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq)^{q} = (\\dot{\\mathbf{r}} + \\operatorname{f_{\\mathbf{p}}}{(q)})^{q} and 0 = (\\dot{\\mathbf{r}} + \\operatorname{f_{\\mathbf{p}}}{(q)})^{q} - (\\int \\operatorname{f_{\\mathbf{p}}}{(q)} dq)^{q}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), exp(Symbol('q', commutative=True))))"], [["power", 3, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), exp(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["minus", 5, "Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Pow(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hbar)} = \\sin{(e^{\\hbar})}, then obtain \\int 0 d\\hbar = \\int (- \\operatorname{C_{2}}{(\\hbar)} + \\sin{(e^{\\hbar})}) d\\hbar", "derivation": "\\operatorname{C_{2}}{(\\hbar)} = \\sin{(e^{\\hbar})} and \\hbar \\operatorname{C_{2}}{(\\hbar)} = \\hbar \\sin{(e^{\\hbar})} and \\hbar \\sin{(e^{\\hbar})} + \\operatorname{C_{2}}{(\\hbar)} + \\frac{\\operatorname{C_{2}}{(\\hbar)}}{\\sin{(e^{\\hbar})}} = \\hbar \\sin{(e^{\\hbar})} + \\frac{\\operatorname{C_{2}}{(\\hbar)}}{\\sin{(e^{\\hbar})}} + \\sin{(e^{\\hbar})} and - \\hbar \\operatorname{C_{2}}{(\\hbar)} + \\hbar \\sin{(e^{\\hbar})} = - \\hbar \\operatorname{C_{2}}{(\\hbar)} + \\hbar \\sin{(e^{\\hbar})} - \\operatorname{C_{2}}{(\\hbar)} + \\sin{(e^{\\hbar})} and 0 = - \\operatorname{C_{2}}{(\\hbar)} + \\sin{(e^{\\hbar})} and \\int 0 d\\hbar = \\int (- \\operatorname{C_{2}}{(\\hbar)} + \\sin{(e^{\\hbar})}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hbar', commutative=True)), sin(exp(Symbol('\\\\hbar', commutative=True))))"], [["times", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Function('C_2')(Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))))"], [["add", 1, "Add(Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))), Mul(Function('C_2')(Symbol('\\\\hbar', commutative=True)), Pow(sin(exp(Symbol('\\\\hbar', commutative=True))), Integer(-1))))"], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))), Function('C_2')(Symbol('\\\\hbar', commutative=True)), Mul(Function('C_2')(Symbol('\\\\hbar', commutative=True)), Pow(sin(exp(Symbol('\\\\hbar', commutative=True))), Integer(-1)))), Add(Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))), Mul(Function('C_2')(Symbol('\\\\hbar', commutative=True)), Pow(sin(exp(Symbol('\\\\hbar', commutative=True))), Integer(-1))), sin(exp(Symbol('\\\\hbar', commutative=True)))))"], [["minus", 3, "Add(Mul(Symbol('\\\\hbar', commutative=True), Function('C_2')(Symbol('\\\\hbar', commutative=True))), Function('C_2')(Symbol('\\\\hbar', commutative=True)), Mul(Function('C_2')(Symbol('\\\\hbar', commutative=True)), Pow(sin(exp(Symbol('\\\\hbar', commutative=True))), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('C_2')(Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('C_2')(Symbol('\\\\hbar', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), sin(exp(Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Function('C_2')(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hbar', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(-1), Function('C_2')(Symbol('\\\\hbar', commutative=True))), sin(exp(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\phi{(t_{2},v_{t})} = \\cos^{t_{2}}{(v_{t})}, then obtain v_{t} \\phi^{2}{(t_{2},v_{t})} \\cos^{2}{(v_{t})} = v_{t} \\phi{(t_{2},v_{t})} \\cos^{2}{(v_{t})} \\cos^{t_{2}}{(v_{t})}", "derivation": "\\phi{(t_{2},v_{t})} = \\cos^{t_{2}}{(v_{t})} and \\phi{(t_{2},v_{t})} \\cos{(v_{t})} = \\cos{(v_{t})} \\cos^{t_{2}}{(v_{t})} and v_{t} \\phi{(t_{2},v_{t})} \\cos{(v_{t})} = v_{t} \\cos{(v_{t})} \\cos^{t_{2}}{(v_{t})} and v_{t} \\phi^{2}{(t_{2},v_{t})} \\cos^{2}{(v_{t})} = v_{t} \\phi{(t_{2},v_{t})} \\cos^{2}{(v_{t})} \\cos^{t_{2}}{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Symbol('t_2', commutative=True)))"], [["times", 1, "cos(Symbol('v_t', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True))), Mul(cos(Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Symbol('t_2', commutative=True))))"], [["times", 2, "Symbol('v_t', commutative=True)"], "Equality(Mul(Symbol('v_t', commutative=True), Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True))), Mul(Symbol('v_t', commutative=True), cos(Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Symbol('t_2', commutative=True))))"], [["times", 3, "Mul(Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], "Equality(Mul(Symbol('v_t', commutative=True), Pow(Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Integer(2)), Pow(cos(Symbol('v_t', commutative=True)), Integer(2))), Mul(Symbol('v_t', commutative=True), Function('\\\\phi')(Symbol('t_2', commutative=True), Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Integer(2)), Pow(cos(Symbol('v_t', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given H{(\\lambda,g_{\\varepsilon})} = \\lambda e^{g_{\\varepsilon}} and S{(g_{\\varepsilon})} = e^{g_{\\varepsilon}}, then obtain S{(g_{\\varepsilon})} + \\int H{(\\lambda,g_{\\varepsilon})} d\\lambda = S{(g_{\\varepsilon})} + \\int \\lambda S{(g_{\\varepsilon})} d\\lambda", "derivation": "H{(\\lambda,g_{\\varepsilon})} = \\lambda e^{g_{\\varepsilon}} and \\int H{(\\lambda,g_{\\varepsilon})} d\\lambda = \\int \\lambda e^{g_{\\varepsilon}} d\\lambda and S{(g_{\\varepsilon})} = e^{g_{\\varepsilon}} and \\int H{(\\lambda,g_{\\varepsilon})} d\\lambda = \\int \\lambda S{(g_{\\varepsilon})} d\\lambda and S{(g_{\\varepsilon})} + \\int H{(\\lambda,g_{\\varepsilon})} d\\lambda = S{(g_{\\varepsilon})} + \\int \\lambda S{(g_{\\varepsilon})} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["add", 4, "Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Function('H')(Symbol('\\\\lambda', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Mul(Symbol('\\\\lambda', commutative=True), Function('S')(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\Omega)} = \\Omega, then derive e^{\\eta^{\\prime} + \\frac{\\operatorname{C_{1}}^{2}{(\\Omega)}}{2}} e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega} = (e^{\\int \\Omega d\\operatorname{C_{1}}{(\\Omega)}}) e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega}, then obtain e^{\\frac{\\Omega^{2}}{2} + \\eta^{\\prime}} e^{\\int \\Omega d\\Omega} = e^{2 \\int \\Omega d\\Omega}", "derivation": "\\operatorname{C_{1}}{(\\Omega)} = \\Omega and \\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega = \\int \\Omega d\\Omega and e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega} = e^{\\int \\Omega d\\Omega} and e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\operatorname{C_{1}}{(\\Omega)}} = e^{\\int \\Omega d\\operatorname{C_{1}}{(\\Omega)}} and (e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega}) e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\operatorname{C_{1}}{(\\Omega)}} = (e^{\\int \\Omega d\\operatorname{C_{1}}{(\\Omega)}}) e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega} and e^{\\eta^{\\prime} + \\frac{\\operatorname{C_{1}}^{2}{(\\Omega)}}{2}} e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega} = (e^{\\int \\Omega d\\operatorname{C_{1}}{(\\Omega)}}) e^{\\int \\operatorname{C_{1}}{(\\Omega)} d\\Omega} and e^{\\frac{\\Omega^{2}}{2} + \\eta^{\\prime}} e^{\\int \\Omega d\\Omega} = e^{2 \\int \\Omega d\\Omega}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), exp(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Function('C_1')(Symbol('\\\\Omega', commutative=True))))), exp(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Function('C_1')(Symbol('\\\\Omega', commutative=True))))))"], [["times", 4, "exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Function('C_1')(Symbol('\\\\Omega', commutative=True)))))), Mul(exp(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Function('C_1')(Symbol('\\\\Omega', commutative=True))))), exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Mul(exp(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Rational(1, 2), Pow(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Integer(2))))), exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))), Mul(exp(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Function('C_1')(Symbol('\\\\Omega', commutative=True))))), exp(Integral(Function('C_1')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Symbol('\\\\eta^{\\\\prime}', commutative=True))), exp(Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))), exp(Mul(Integer(2), Integral(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{D}{(r,n)} = \\frac{\\partial}{\\partial r} n r and \\mathbf{f}{(r,n)} = (\\frac{\\partial}{\\partial r} n r)^{n}, then obtain \\frac{\\partial}{\\partial r} \\mathbf{f}{(r,n)} = \\frac{n \\mathbf{D}^{n}{(r,n)} \\frac{\\partial}{\\partial r} \\mathbf{D}{(r,n)}}{\\mathbf{D}{(r,n)}}", "derivation": "\\mathbf{D}{(r,n)} = \\frac{\\partial}{\\partial r} n r and \\mathbf{D}^{n}{(r,n)} = (\\frac{\\partial}{\\partial r} n r)^{n} and \\frac{\\partial}{\\partial r} \\mathbf{D}^{n}{(r,n)} = \\frac{\\partial}{\\partial r} (\\frac{\\partial}{\\partial r} n r)^{n} and \\mathbf{f}{(r,n)} = (\\frac{\\partial}{\\partial r} n r)^{n} and \\mathbf{D}^{n}{(r,n)} = \\mathbf{f}{(r,n)} and \\frac{\\partial}{\\partial r} \\mathbf{f}{(r,n)} = \\frac{\\partial}{\\partial r} (\\frac{\\partial}{\\partial r} n r)^{n} and \\frac{\\partial}{\\partial r} \\mathbf{f}{(r,n)} = \\frac{\\partial}{\\partial r} \\mathbf{D}^{n}{(r,n)} and \\frac{\\partial}{\\partial r} \\mathbf{f}{(r,n)} = \\frac{n \\mathbf{D}^{n}{(r,n)} \\frac{\\partial}{\\partial r} \\mathbf{D}{(r,n)}}{\\mathbf{D}{(r,n)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Derivative(Mul(Symbol('n', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('n', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Function('\\\\mathbf{f}')(Symbol('r', commutative=True), Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Symbol('n', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Mul(Symbol('n', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Derivative(Function('\\\\mathbf{D}')(Symbol('r', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(C,v)} = e^{- C + v}, then obtain - C e^{- C + v} + (C s{(C,v)})^{v} - (C e^{- C + v})^{v} = - C e^{- C + v}", "derivation": "s{(C,v)} = e^{- C + v} and C s{(C,v)} = C e^{- C + v} and s^{v}{(C,v)} = (e^{- C + v})^{v} and (C s{(C,v)})^{v} = (C e^{- C + v})^{v} and (C s{(C,v)})^{v} + s^{v}{(C,v)} = (C e^{- C + v})^{v} + s^{v}{(C,v)} and (C s{(C,v)})^{v} + (e^{- C + v})^{v} = (C e^{- C + v})^{v} + (e^{- C + v})^{v} and - C e^{- C + v} + (C s{(C,v)})^{v} + (e^{- C + v})^{v} = - C e^{- C + v} + (C e^{- C + v})^{v} + (e^{- C + v})^{v} and - C e^{- C + v} + (C s{(C,v)})^{v} - (C e^{- C + v})^{v} = - C e^{- C + v}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)))"], [["add", 4, "Pow(Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))), Add(Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Add(Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["minus", 6, "Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Pow(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["minus", 7, "Add(Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Pow(Mul(Symbol('C', commutative=True), Function('s')(Symbol('C', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))), Symbol('v', commutative=True)))), Mul(Integer(-1), Symbol('C', commutative=True), exp(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then derive 2 \\Psi{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\Psi{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\Psi{(\\mathbf{B})}, then obtain 2 \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} = \\sin{(\\mathbf{B})} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})}", "derivation": "\\Psi{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and \\Psi^{2}{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} \\sin{(\\mathbf{B})} and \\frac{d}{d \\mathbf{B}} \\Psi^{2}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\Psi{(\\mathbf{B})} \\sin{(\\mathbf{B})} and 2 \\Psi{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\Psi{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\Psi{(\\mathbf{B})} and 2 \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})} = \\sin{(\\mathbf{B})} \\cos{(\\mathbf{B})} + \\sin{(\\mathbf{B})} \\frac{d}{d \\mathbf{B}} \\sin{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 1, "Pow(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Mul(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(Mul(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Mul(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Add(Mul(sin(Symbol('\\\\mathbf{B}', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Mul(sin(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given s{(v_{z},p)} = p + \\log{(v_{z})}, then obtain - v_{z} + \\int s{(v_{z},p)} dp = G + \\frac{p^{2}}{2} + p \\log{(v_{z})} - v_{z}", "derivation": "s{(v_{z},p)} = p + \\log{(v_{z})} and \\int s{(v_{z},p)} dp = \\int (p + \\log{(v_{z})}) dp and - v_{z} + \\int s{(v_{z},p)} dp = - v_{z} + \\int (p + \\log{(v_{z})}) dp and - v_{z} + \\int s{(v_{z},p)} dp = G + \\frac{p^{2}}{2} + p \\log{(v_{z})} - v_{z}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Add(Symbol('p', commutative=True), log(Symbol('v_z', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('s')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Add(Symbol('p', commutative=True), log(Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Function('s')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Add(Symbol('p', commutative=True), log(Symbol('v_z', commutative=True))), Tuple(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Integral(Function('s')(Symbol('v_z', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Rational(1, 2), Pow(Symbol('p', commutative=True), Integer(2))), Mul(Symbol('p', commutative=True), log(Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(U)} = \\log{(U)} and \\operatorname{t_{1}}{(U)} = 4 \\log{(U)}^{2}, then obtain (\\mathbf{P}{(U)} + \\log{(U)}^{2} + \\log{(U)}) \\operatorname{t_{1}}{(U)} = 4 (\\mathbf{P}{(U)} + \\log{(U)}^{2} + \\log{(U)}) \\log{(U)}^{2}", "derivation": "\\mathbf{P}{(U)} = \\log{(U)} and \\mathbf{P}{(U)} + \\log{(U)} = 2 \\log{(U)} and \\mathbf{P}{(U)} + \\log{(U)}^{2} + \\log{(U)} = \\log{(U)}^{2} + 2 \\log{(U)} and \\operatorname{t_{1}}{(U)} = 4 \\log{(U)}^{2} and (\\log{(U)}^{2} + 2 \\log{(U)}) \\operatorname{t_{1}}{(U)} = 4 (\\log{(U)}^{2} + 2 \\log{(U)}) \\log{(U)}^{2} and (\\mathbf{P}{(U)} + \\log{(U)}^{2} + \\log{(U)}) \\operatorname{t_{1}}{(U)} = 4 (\\mathbf{P}{(U)} + \\log{(U)}^{2} + \\log{(U)}) \\log{(U)}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["add", 1, "log(Symbol('U', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Mul(Integer(2), log(Symbol('U', commutative=True))))"], [["add", 2, "Pow(log(Symbol('U', commutative=True)), Integer(2))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Add(Pow(log(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(2), log(Symbol('U', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('U', commutative=True)), Mul(Integer(4), Pow(log(Symbol('U', commutative=True)), Integer(2))))"], [["times", 4, "Add(Pow(log(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(2), log(Symbol('U', commutative=True))))"], "Equality(Mul(Add(Pow(log(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(2), log(Symbol('U', commutative=True)))), Function('t_1')(Symbol('U', commutative=True))), Mul(Integer(4), Add(Pow(log(Symbol('U', commutative=True)), Integer(2)), Mul(Integer(2), log(Symbol('U', commutative=True)))), Pow(log(Symbol('U', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Function('t_1')(Symbol('U', commutative=True))), Mul(Integer(4), Add(Function('\\\\mathbf{P}')(Symbol('U', commutative=True)), Pow(log(Symbol('U', commutative=True)), Integer(2)), log(Symbol('U', commutative=True))), Pow(log(Symbol('U', commutative=True)), Integer(2))))"]]}, {"prompt": "Given A{(M)} = \\cos{(e^{M})}, then derive \\frac{d}{d M} A{(M)} = - e^{M} \\sin{(e^{M})}, then obtain \\frac{d}{d M} \\cos{(e^{M})} = - e^{M} \\sin{(e^{M})}", "derivation": "A{(M)} = \\cos{(e^{M})} and \\frac{d}{d M} A{(M)} = \\frac{d}{d M} \\cos{(e^{M})} and \\frac{d}{d M} A{(M)} = - e^{M} \\sin{(e^{M})} and \\frac{d}{d M} \\cos{(e^{M})} = - e^{M} \\sin{(e^{M})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('M', commutative=True)), cos(exp(Symbol('M', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('M', commutative=True)), sin(exp(Symbol('M', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(exp(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('M', commutative=True)), sin(exp(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\tilde{g})} = \\tilde{g}, then obtain \\int \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g} \\operatorname{f_{E}}{(\\tilde{g})}}{\\operatorname{f_{E}}{(\\tilde{g})}} d\\tilde{g} = \\int \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g}^{2}}{\\operatorname{f_{E}}{(\\tilde{g})}} d\\tilde{g}", "derivation": "\\operatorname{f_{E}}{(\\tilde{g})} = \\tilde{g} and \\tilde{g} \\operatorname{f_{E}}{(\\tilde{g})} = \\tilde{g}^{2} and \\frac{d}{d \\tilde{g}} \\tilde{g} \\operatorname{f_{E}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\tilde{g}^{2} and \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g} \\operatorname{f_{E}}{(\\tilde{g})}}{\\operatorname{f_{E}}{(\\tilde{g})}} = \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g}^{2}}{\\operatorname{f_{E}}{(\\tilde{g})}} and \\int \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g} \\operatorname{f_{E}}{(\\tilde{g})}}{\\operatorname{f_{E}}{(\\tilde{g})}} d\\tilde{g} = \\int \\frac{\\frac{d}{d \\tilde{g}} \\tilde{g}^{2}}{\\operatorname{f_{E}}{(\\tilde{g})}} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))"], [["times", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('f_E')(Symbol('\\\\tilde{g}', commutative=True))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('f_E')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["divide", 3, "Function('f_E')(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Pow(Function('f_E')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('f_E')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Pow(Function('f_E')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('f_E')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\tilde{g}', commutative=True), Function('f_E')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Pow(Function('f_E')(Symbol('\\\\tilde{g}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\phi_2,n)} = \\int \\frac{\\phi_2}{n} dn, then obtain - \\int \\frac{\\phi_2}{n} dn = \\sigma_{x}{(\\phi_2,n)} - 2 \\int \\frac{\\phi_2}{n} dn", "derivation": "\\sigma_{x}{(\\phi_2,n)} = \\int \\frac{\\phi_2}{n} dn and - \\sigma_{x}{(\\phi_2,n)} = - \\int \\frac{\\phi_2}{n} dn and - \\sigma_{x}{(\\phi_2,n)} - \\int \\frac{\\phi_2}{n} dn = - 2 \\int \\frac{\\phi_2}{n} dn and - \\int \\frac{\\phi_2}{n} dn = \\sigma_{x}{(\\phi_2,n)} - 2 \\int \\frac{\\phi_2}{n} dn", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\phi_2', commutative=True), Symbol('n', commutative=True)), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\phi_2', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\phi_2', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True))))), Mul(Integer(-1), Integer(2), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\phi_2', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True)))), Add(Function('\\\\sigma_x')(Symbol('\\\\phi_2', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), Integral(Mul(Symbol('\\\\phi_2', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('n', commutative=True))))))"]]}, {"prompt": "Given U{(t,u)} = \\frac{u}{t}, then obtain ((\\frac{t u \\frac{\\partial}{\\partial t} U{(t,u)}}{U{(t,u)}} + \\log{(U^{u}{(t,u)})}) (U^{u}{(t,u)})^{t})^{t} = ((- u + \\log{((\\frac{u}{t})^{u})}) ((\\frac{u}{t})^{u})^{t})^{t}", "derivation": "U{(t,u)} = \\frac{u}{t} and U^{u}{(t,u)} = (\\frac{u}{t})^{u} and (U^{u}{(t,u)})^{t} = ((\\frac{u}{t})^{u})^{t} and \\frac{\\partial}{\\partial t} (U^{u}{(t,u)})^{t} = \\frac{\\partial}{\\partial t} ((\\frac{u}{t})^{u})^{t} and (\\frac{\\partial}{\\partial t} (U^{u}{(t,u)})^{t})^{t} = (\\frac{\\partial}{\\partial t} ((\\frac{u}{t})^{u})^{t})^{t} and ((\\frac{t u \\frac{\\partial}{\\partial t} U{(t,u)}}{U{(t,u)}} + \\log{(U^{u}{(t,u)})}) (U^{u}{(t,u)})^{t})^{t} = ((- u + \\log{((\\frac{u}{t})^{u})}) ((\\frac{u}{t})^{u})^{t})^{t}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)), Pow(Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 4, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Pow(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Pow(Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Mul(Add(Mul(Symbol('t', commutative=True), Symbol('u', commutative=True), Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Integer(-1)), Derivative(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), log(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))), Pow(Pow(Function('U')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)))), Pow(Pow(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(I,n)} = I n and \\hat{X}{(I,n)} = - I n - I + \\frac{\\partial}{\\partial n} (I + \\operatorname{F_{x}}{(I,n)}), then obtain \\frac{\\partial}{\\partial n} \\hat{X}{(I,n)} = \\frac{\\partial}{\\partial n} (- I n - I + \\frac{\\partial}{\\partial n} (I n + I))", "derivation": "\\operatorname{F_{x}}{(I,n)} = I n and I + \\operatorname{F_{x}}{(I,n)} = I n + I and \\frac{\\partial}{\\partial n} (I + \\operatorname{F_{x}}{(I,n)}) = \\frac{\\partial}{\\partial n} (I n + I) and - I n - I + \\frac{\\partial}{\\partial n} (I + \\operatorname{F_{x}}{(I,n)}) = - I n - I + \\frac{\\partial}{\\partial n} (I n + I) and \\frac{\\partial}{\\partial n} (- I n - I + \\frac{\\partial}{\\partial n} (I + \\operatorname{F_{x}}{(I,n)})) = \\frac{\\partial}{\\partial n} (- I n - I + \\frac{\\partial}{\\partial n} (I n + I)) and \\hat{X}{(I,n)} = - I n - I + \\frac{\\partial}{\\partial n} (I + \\operatorname{F_{x}}{(I,n)}) and \\frac{\\partial}{\\partial n} \\hat{X}{(I,n)} = \\frac{\\partial}{\\partial n} (- I n - I + \\frac{\\partial}{\\partial n} (I n + I))", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True)))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('I', commutative=True), Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 3, "Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Function('F_x')(Symbol('I', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('I', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('n', commutative=True)), Symbol('I', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(G)} = \\log{(\\cos{(G)})} and \\Psi_{nl}{(s)} = \\cos{(s)}, then obtain \\Psi_{nl}{(G)} \\cos^{- s}{(s)} = \\log{(\\cos{(G)})} \\cos^{- s}{(s)}", "derivation": "\\Psi_{nl}{(G)} = \\log{(\\cos{(G)})} and \\Psi_{nl}{(s)} = \\cos{(s)} and \\Psi_{nl}{(G)} \\Psi_{nl}^{- s}{(s)} = \\Psi_{nl}^{- s}{(s)} \\log{(\\cos{(G)})} and \\Psi_{nl}{(G)} \\cos^{- s}{(s)} = \\log{(\\cos{(G)})} \\cos^{- s}{(s)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True)), log(cos(Symbol('G', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["divide", 1, "Pow(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True)), Pow(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)))), Mul(Pow(Function('\\\\Psi_{nl}')(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), log(cos(Symbol('G', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('G', commutative=True)), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)))), Mul(log(cos(Symbol('G', commutative=True))), Pow(cos(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\eta)} = \\sin{(\\eta)}, then obtain \\operatorname{z^{*}}{(\\eta)} + \\frac{d}{d \\eta} \\sin^{\\eta}{(\\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)})} = \\operatorname{z^{*}}{(\\eta)} + \\frac{d}{d \\eta} 0^{\\eta}", "derivation": "\\operatorname{z^{*}}{(\\eta)} = \\sin{(\\eta)} and \\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)} = 0 and \\sin{(\\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)})} = 0 and \\sin^{\\eta}{(\\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)})} = 0^{\\eta} and \\frac{d}{d \\eta} \\sin^{\\eta}{(\\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)})} = \\frac{d}{d \\eta} 0^{\\eta} and \\operatorname{z^{*}}{(\\eta)} + \\frac{d}{d \\eta} \\sin^{\\eta}{(\\operatorname{z^{*}}{(\\eta)} - \\sin{(\\eta)})} = \\operatorname{z^{*}}{(\\eta)} + \\frac{d}{d \\eta} 0^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Integer(0))"], [["sin", 2], "Equality(sin(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))), Integer(0))"], [["power", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(sin(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))), Symbol('\\\\eta', commutative=True)), Pow(Integer(0), Symbol('\\\\eta', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Pow(sin(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Integer(-1), Function('z^*')(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Derivative(Pow(sin(Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True))))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Function('z^*')(Symbol('\\\\eta', commutative=True)), Derivative(Pow(Integer(0), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\lambda,\\hat{H}_l)} = \\cos{(\\hat{H}_l - \\lambda)}, then obtain \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l e^{\\varphi^{*}{(\\lambda,\\hat{H}_l)}} = \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l e^{\\cos{(\\hat{H}_l - \\lambda)}}", "derivation": "\\varphi^{*}{(\\lambda,\\hat{H}_l)} = \\cos{(\\hat{H}_l - \\lambda)} and e^{\\varphi^{*}{(\\lambda,\\hat{H}_l)}} = e^{\\cos{(\\hat{H}_l - \\lambda)}} and \\hat{H}_l e^{\\varphi^{*}{(\\lambda,\\hat{H}_l)}} = \\hat{H}_l e^{\\cos{(\\hat{H}_l - \\lambda)}} and \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l e^{\\varphi^{*}{(\\lambda,\\hat{H}_l)}} = \\frac{\\partial}{\\partial \\hat{H}_l} \\hat{H}_l e^{\\cos{(\\hat{H}_l - \\lambda)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))"], [["exp", 1], "Equality(exp(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), exp(cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True))))))"], [["times", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))))"], [["differentiate", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{H}_l', commutative=True), exp(cos(Add(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(n_{2},\\theta_2)} = - n_{2} + \\log{(\\theta_2)} and \\operatorname{t_{1}}{(n_{2})} = e^{- n_{2}}, then obtain e^{\\theta_2 e^{- n_{2}} - n_{2}} = e^{\\theta_2 \\operatorname{t_{1}}{(n_{2})} - n_{2}}", "derivation": "\\mathbf{J}_f{(n_{2},\\theta_2)} = - n_{2} + \\log{(\\theta_2)} and e^{\\mathbf{J}_f{(n_{2},\\theta_2)}} = \\theta_2 e^{- n_{2}} and \\operatorname{t_{1}}{(n_{2})} = e^{- n_{2}} and e^{\\mathbf{J}_f{(n_{2},\\theta_2)}} = \\theta_2 \\operatorname{t_{1}}{(n_{2})} and - n_{2} + e^{\\mathbf{J}_f{(n_{2},\\theta_2)}} = \\theta_2 \\operatorname{t_{1}}{(n_{2})} - n_{2} and \\theta_2 e^{- n_{2}} - n_{2} = \\theta_2 \\operatorname{t_{1}}{(n_{2})} - n_{2} and e^{\\theta_2 e^{- n_{2}} - n_{2}} = e^{\\theta_2 \\operatorname{t_{1}}{(n_{2})} - n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('n_2', commutative=True)), exp(Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(exp(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Function('t_1')(Symbol('n_2', commutative=True))))"], [["minus", 4, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), exp(Function('\\\\mathbf{J}_f')(Symbol('n_2', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Function('t_1')(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('\\\\theta_2', commutative=True), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Function('t_1')(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True))))"], [["exp", 6], "Equality(exp(Add(Mul(Symbol('\\\\theta_2', commutative=True), exp(Mul(Integer(-1), Symbol('n_2', commutative=True)))), Mul(Integer(-1), Symbol('n_2', commutative=True)))), exp(Add(Mul(Symbol('\\\\theta_2', commutative=True), Function('t_1')(Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given m{(\\psi^*)} = \\cos{(\\psi^*)}, then derive 0 = \\frac{d}{d \\psi^*} (- \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^*), then obtain 0 = \\frac{d}{d \\psi^*} 0 \\frac{d}{d \\psi^*} (- \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^*)", "derivation": "m{(\\psi^*)} = \\cos{(\\psi^*)} and \\int m{(\\psi^*)} d\\psi^* = \\int \\cos{(\\psi^*)} d\\psi^* and 0 = - \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^* and \\frac{d}{d \\psi^*} 0 = \\frac{d}{d \\psi^*} (- \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^*) and 0 = \\frac{d}{d \\psi^*} (- \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^*) and 0 = \\frac{d}{d \\psi^*} 0 and 0 = \\frac{d}{d \\psi^*} 0 \\frac{d}{d \\psi^*} (- \\int m{(\\psi^*)} d\\psi^* + \\int \\cos{(\\psi^*)} d\\psi^*)", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["minus", 2, "Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Derivative(Add(Mul(Integer(-1), Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(0), Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["times", 6, "Derivative(Add(Mul(Integer(-1), Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Derivative(Integer(0), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integral(Function('m')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Integral(cos(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{z}{(E,G)} = \\frac{\\partial}{\\partial G} \\frac{G}{E}, then derive \\dot{z}{(E,G)} = \\frac{1}{E}, then obtain \\frac{\\partial}{\\partial E} \\frac{\\dot{z}{(E,G)}}{E} = \\frac{d}{d E} \\frac{1}{E^{2}}", "derivation": "\\dot{z}{(E,G)} = \\frac{\\partial}{\\partial G} \\frac{G}{E} and \\dot{z}{(E,G)} = \\frac{1}{E} and \\frac{\\dot{z}{(E,G)}}{E} = \\frac{1}{E^{2}} and \\frac{\\partial}{\\partial E} \\frac{\\dot{z}{(E,G)}}{E} = \\frac{d}{d E} \\frac{1}{E^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('E', commutative=True), Symbol('G', commutative=True)), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{z}')(Symbol('E', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('E', commutative=True), Integer(-1)))"], [["divide", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('E', commutative=True), Symbol('G', commutative=True))), Pow(Symbol('E', commutative=True), Integer(-2)))"], [["differentiate", 3, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Function('\\\\dot{z}')(Symbol('E', commutative=True), Symbol('G', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Pow(Symbol('E', commutative=True), Integer(-2)), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{x},p)} = - p + v_{x}, then derive \\frac{\\partial}{\\partial v_{x}} \\operatorname{F_{H}}{(v_{x},p)} = 1, then obtain \\frac{\\partial^{2}}{\\partial v_{x}^{2}} \\operatorname{F_{H}}{(v_{x},p)} = \\frac{d}{d v_{x}} 1", "derivation": "\\operatorname{F_{H}}{(v_{x},p)} = - p + v_{x} and \\frac{\\partial}{\\partial v_{x}} \\operatorname{F_{H}}{(v_{x},p)} = \\frac{\\partial}{\\partial v_{x}} (- p + v_{x}) and \\frac{\\partial}{\\partial v_{x}} \\operatorname{F_{H}}{(v_{x},p)} = 1 and \\frac{\\partial}{\\partial v_{x}} (- p + v_{x}) = 1 and \\frac{\\partial^{2}}{\\partial v_{x}^{2}} (- p + v_{x}) = \\frac{d}{d v_{x}} 1 and \\frac{\\partial^{2}}{\\partial v_{x}^{2}} \\operatorname{F_{H}}{(v_{x},p)} = \\frac{d}{d v_{x}} 1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('F_H')(Symbol('v_x', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('v_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(I,A_{x})} = A_{x}^{I}, then obtain e^{\\frac{A_{x}^{I}}{I} + \\frac{\\frac{\\partial}{\\partial A_{x}} \\varphi{(I,A_{x})}}{I}} = e^{\\frac{A_{x}^{I}}{I} + \\frac{A_{x}^{I}}{A_{x}}}", "derivation": "\\varphi{(I,A_{x})} = A_{x}^{I} and \\frac{\\varphi{(I,A_{x})}}{I} = \\frac{A_{x}^{I}}{I} and \\frac{\\partial}{\\partial A_{x}} \\frac{\\varphi{(I,A_{x})}}{I} = \\frac{\\partial}{\\partial A_{x}} \\frac{A_{x}^{I}}{I} and \\frac{A_{x}^{I}}{I} + \\frac{\\partial}{\\partial A_{x}} \\frac{\\varphi{(I,A_{x})}}{I} = \\frac{A_{x}^{I}}{I} + \\frac{\\partial}{\\partial A_{x}} \\frac{A_{x}^{I}}{I} and e^{\\frac{A_{x}^{I}}{I} + \\frac{\\partial}{\\partial A_{x}} \\frac{\\varphi{(I,A_{x})}}{I}} = e^{\\frac{A_{x}^{I}}{I} + \\frac{\\partial}{\\partial A_{x}} \\frac{A_{x}^{I}}{I}} and e^{\\frac{A_{x}^{I}}{I} + \\frac{\\frac{\\partial}{\\partial A_{x}} \\varphi{(I,A_{x})}}{I}} = e^{\\frac{A_{x}^{I}}{I} + \\frac{A_{x}^{I}}{A_{x}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["add", 3, "Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"], [["exp", 4], "Equality(exp(Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))), exp(Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 5], "Equality(exp(Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Derivative(Function('\\\\varphi')(Symbol('I', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)))))), exp(Add(Mul(Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('I', commutative=True), Integer(-1))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Pow(Symbol('A_x', commutative=True), Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(A_{2},A)} = A A_{2}, then obtain - A A_{2}^{2} + (\\int (A_{2} \\theta_{1}{(A_{2},A)} + A_{2}) dA)^{2} = - A A_{2}^{2} + (\\int (A A_{2}^{2} + A_{2}) dA)^{2}", "derivation": "\\theta_{1}{(A_{2},A)} = A A_{2} and A_{2} \\theta_{1}{(A_{2},A)} = A A_{2}^{2} and A_{2} \\theta_{1}{(A_{2},A)} + A_{2} = A A_{2}^{2} + A_{2} and \\int (A_{2} \\theta_{1}{(A_{2},A)} + A_{2}) dA = \\int (A A_{2}^{2} + A_{2}) dA and (\\int (A_{2} \\theta_{1}{(A_{2},A)} + A_{2}) dA)^{2} = (\\int (A A_{2}^{2} + A_{2}) dA)^{2} and - A A_{2}^{2} + (\\int (A_{2} \\theta_{1}{(A_{2},A)} + A_{2}) dA)^{2} = - A A_{2}^{2} + (\\int (A A_{2}^{2} + A_{2}) dA)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('A_2', commutative=True)))"], [["times", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))))"], [["add", 2, "Symbol('A_2', commutative=True)"], "Equality(Add(Mul(Symbol('A_2', commutative=True), Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Symbol('A_2', commutative=True)), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('A_2', commutative=True)))"], [["integrate", 3, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('A_2', commutative=True), Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Add(Mul(Symbol('A_2', commutative=True), Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2)), Pow(Integral(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2)))"], [["minus", 5, "Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Pow(Integral(Add(Mul(Symbol('A_2', commutative=True), Function('\\\\theta_1')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2))), Add(Mul(Integer(-1), Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Pow(Integral(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('A_2', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(m_{s})} = \\cos{(e^{m_{s}})}, then derive \\frac{d}{d m_{s}} \\dot{\\mathbf{r}}{(m_{s})} = - e^{m_{s}} \\sin{(e^{m_{s}})}, then obtain \\dot{y} + \\cos{(e^{m_{s}})} = \\int - e^{m_{s}} \\sin{(e^{m_{s}})} dm_{s}", "derivation": "\\dot{\\mathbf{r}}{(m_{s})} = \\cos{(e^{m_{s}})} and \\frac{d}{d m_{s}} \\dot{\\mathbf{r}}{(m_{s})} = \\frac{d}{d m_{s}} \\cos{(e^{m_{s}})} and \\frac{d}{d m_{s}} \\dot{\\mathbf{r}}{(m_{s})} = - e^{m_{s}} \\sin{(e^{m_{s}})} and \\int \\frac{d}{d m_{s}} \\dot{\\mathbf{r}}{(m_{s})} dm_{s} = \\int - e^{m_{s}} \\sin{(e^{m_{s}})} dm_{s} and \\int \\frac{d}{d m_{s}} \\cos{(e^{m_{s}})} dm_{s} = \\int - e^{m_{s}} \\sin{(e^{m_{s}})} dm_{s} and \\dot{y} + \\cos{(e^{m_{s}})} = \\int - e^{m_{s}} \\sin{(e^{m_{s}})} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('m_s', commutative=True)), cos(exp(Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('m_s', commutative=True)), sin(exp(Symbol('m_s', commutative=True)))))"], [["integrate", 3, "Symbol('m_s', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('m_s', commutative=True)), sin(exp(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(cos(exp(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Integer(-1), exp(Symbol('m_s', commutative=True)), sin(exp(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\dot{y}', commutative=True), cos(exp(Symbol('m_s', commutative=True)))), Integral(Mul(Integer(-1), exp(Symbol('m_s', commutative=True)), sin(exp(Symbol('m_s', commutative=True)))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given q{(r_{0})} = \\cos{(\\sin{(r_{0})})}, then obtain \\frac{d^{2}}{d r_{0}^{2}} 1 = \\frac{d^{2}}{d r_{0}^{2}} \\frac{\\cos{(\\sin{(r_{0})})}}{q{(r_{0})}}", "derivation": "q{(r_{0})} = \\cos{(\\sin{(r_{0})})} and 1 = \\frac{\\cos{(\\sin{(r_{0})})}}{q{(r_{0})}} and \\frac{d}{d r_{0}} 1 = \\frac{d}{d r_{0}} \\frac{\\cos{(\\sin{(r_{0})})}}{q{(r_{0})}} and \\frac{d^{2}}{d r_{0}^{2}} 1 = \\frac{d^{2}}{d r_{0}^{2}} \\frac{\\cos{(\\sin{(r_{0})})}}{q{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('r_0', commutative=True)), cos(sin(Symbol('r_0', commutative=True))))"], [["divide", 1, "Function('q')(Symbol('r_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('q')(Symbol('r_0', commutative=True)), Integer(-1)), cos(sin(Symbol('r_0', commutative=True)))))"], [["differentiate", 2, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('q')(Symbol('r_0', commutative=True)), Integer(-1)), cos(sin(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('r_0', commutative=True), Integer(2))), Derivative(Mul(Pow(Function('q')(Symbol('r_0', commutative=True)), Integer(-1)), cos(sin(Symbol('r_0', commutative=True)))), Tuple(Symbol('r_0', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\rho_{f}{(\\dot{z})} = e^{\\dot{z}} and \\theta{(\\dot{z})} = e^{\\dot{z}}, then obtain \\frac{d}{d \\dot{z}} 1 = \\frac{d}{d \\dot{z}} (\\frac{e^{\\dot{z}}}{\\rho_{f}{(\\dot{z})}})^{\\dot{z}}", "derivation": "\\rho_{f}{(\\dot{z})} = e^{\\dot{z}} and \\theta{(\\dot{z})} = e^{\\dot{z}} and \\rho_{f}{(\\dot{z})} = \\theta{(\\dot{z})} and 1 = \\frac{e^{\\dot{z}}}{\\theta{(\\dot{z})}} and 1 = (\\frac{e^{\\dot{z}}}{\\theta{(\\dot{z})}})^{\\dot{z}} and 1 = (\\frac{e^{\\dot{z}}}{\\rho_{f}{(\\dot{z})}})^{\\dot{z}} and \\frac{d}{d \\dot{z}} 1 = \\frac{d}{d \\dot{z}} (\\frac{e^{\\dot{z}}}{\\rho_{f}{(\\dot{z})}})^{\\dot{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\dot{z}', commutative=True)), exp(Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\rho_f')(Symbol('\\\\dot{z}', commutative=True)), Function('\\\\theta')(Symbol('\\\\dot{z}', commutative=True)))"], [["divide", 2, "Function('\\\\theta')(Symbol('\\\\dot{z}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))))"], [["power", 4, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\theta')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 6, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\dot{z}', commutative=True)), Integer(-1)), exp(Symbol('\\\\dot{z}', commutative=True))), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})}, then obtain e^{- \\mathbf{v} - \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}} e^{\\mathbf{v} + \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\mathbf{v}}} = e^{- \\mathbf{v} - \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}} e^{\\mathbf{v} + \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{v})} = \\sin{(\\mathbf{v})} and \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\mathbf{v}} = \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}} and \\mathbf{v} + \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\mathbf{v}} = \\mathbf{v} + \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}} and e^{\\mathbf{v} + \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\mathbf{v}}} = e^{\\mathbf{v} + \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}} and e^{- \\mathbf{v} - \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}} e^{\\mathbf{v} + \\frac{\\operatorname{v_{y}}{(\\mathbf{v})}}{\\mathbf{v}}} = e^{- \\mathbf{v} - \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}} e^{\\mathbf{v} + \\frac{\\sin{(\\mathbf{v})}}{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)), sin(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True))))), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))))"], [["divide", 4, "exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True)))))"], "Equality(Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\mathbf{v}', commutative=True)))))), Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True))))), exp(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbf{v}', commutative=True)))))))"]]}, {"prompt": "Given \\Psi{(\\phi_2)} = \\log{(\\log{(\\phi_2)})}, then obtain 1 = (\\frac{\\Psi{(\\phi_2)}}{\\log{(\\log{(\\phi_2)})}})^{- \\phi_2}", "derivation": "\\Psi{(\\phi_2)} = \\log{(\\log{(\\phi_2)})} and \\frac{\\Psi{(\\phi_2)}}{\\log{(\\log{(\\phi_2)})}} = 1 and (\\frac{\\Psi{(\\phi_2)}}{\\log{(\\log{(\\phi_2)})}})^{\\phi_2} = 1 and 1 = (\\frac{\\Psi{(\\phi_2)}}{\\log{(\\log{(\\phi_2)})}})^{- \\phi_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\phi_2', commutative=True)), log(log(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 1, "log(log(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Function('\\\\Psi')(Symbol('\\\\phi_2', commutative=True)), Pow(log(log(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Function('\\\\Psi')(Symbol('\\\\phi_2', commutative=True)), Pow(log(log(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Symbol('\\\\phi_2', commutative=True)), Integer(1))"], [["divide", 3, "Pow(Mul(Function('\\\\Psi')(Symbol('\\\\phi_2', commutative=True)), Pow(log(log(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(1), Pow(Mul(Function('\\\\Psi')(Symbol('\\\\phi_2', commutative=True)), Pow(log(log(Symbol('\\\\phi_2', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(F_{g},\\varphi^*)} = F_{g} \\varphi^*, then derive \\frac{\\partial}{\\partial F_{g}} \\rho_{b}{(F_{g},\\varphi^*)} = \\varphi^*, then obtain \\varphi^* = \\frac{\\partial}{\\partial F_{g}} F_{g} \\varphi^*", "derivation": "\\rho_{b}{(F_{g},\\varphi^*)} = F_{g} \\varphi^* and \\frac{\\partial}{\\partial F_{g}} \\rho_{b}{(F_{g},\\varphi^*)} = \\frac{\\partial}{\\partial F_{g}} F_{g} \\varphi^* and \\frac{\\partial}{\\partial F_{g}} \\rho_{b}{(F_{g},\\varphi^*)} = \\varphi^* and \\varphi^* = \\frac{\\partial}{\\partial F_{g}} F_{g} \\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\varphi^*', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('\\\\varphi^*', commutative=True), Derivative(Mul(Symbol('F_g', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(H)} = \\sin{(\\cos{(H)})} and I{(H)} = \\frac{d}{d H} t{(H)} \\frac{d}{d H} \\sin{(\\cos{(H)})}, then obtain \\frac{(\\frac{d}{d H} t{(H)})^{2} (\\frac{d}{d H} \\sin{(\\cos{(H)})})^{2}}{\\sin^{2}{(H)}} = \\frac{(\\frac{d}{d H} t{(H)})^{4}}{\\sin^{2}{(H)}}", "derivation": "t{(H)} = \\sin{(\\cos{(H)})} and \\frac{d}{d H} t{(H)} = \\frac{d}{d H} \\sin{(\\cos{(H)})} and I{(H)} = \\frac{d}{d H} t{(H)} \\frac{d}{d H} \\sin{(\\cos{(H)})} and I{(H)} = (\\frac{d}{d H} \\sin{(\\cos{(H)})})^{2} and I^{2}{(H)} = (\\frac{d}{d H} \\sin{(\\cos{(H)})})^{4} and I^{2}{(H)} = (\\frac{d}{d H} t{(H)})^{4} and (\\frac{d}{d H} t{(H)})^{2} (\\frac{d}{d H} \\sin{(\\cos{(H)})})^{2} = (\\frac{d}{d H} t{(H)})^{4} and \\frac{(\\frac{d}{d H} t{(H)})^{2} (\\frac{d}{d H} \\sin{(\\cos{(H)})})^{2}}{\\sin^{2}{(H)}} = \\frac{(\\frac{d}{d H} t{(H)})^{4}}{\\sin^{2}{(H)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('H', commutative=True)), sin(cos(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('I')(Symbol('H', commutative=True)), Mul(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('I')(Symbol('H', commutative=True)), Pow(Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2)))"], [["power", 4, 2], "Equality(Pow(Function('I')(Symbol('H', commutative=True)), Integer(2)), Pow(Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(4)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('I')(Symbol('H', commutative=True)), Integer(2)), Pow(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(4)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(4)))"], [["divide", 7, "Pow(sin(Symbol('H', commutative=True)), Integer(2))"], "Equality(Mul(Pow(sin(Symbol('H', commutative=True)), Integer(-2)), Pow(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(sin(cos(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(2))), Mul(Pow(sin(Symbol('H', commutative=True)), Integer(-2)), Pow(Derivative(Function('t')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(4))))"]]}, {"prompt": "Given h{(P_{g})} = e^{e^{P_{g}}}, then obtain 4^{P_{g}} (h^{2}{(P_{g})})^{P_{g}} = (h^{2}{(P_{g})} + 2 h{(P_{g})} e^{e^{P_{g}}} + e^{2 e^{P_{g}}})^{P_{g}}", "derivation": "h{(P_{g})} = e^{e^{P_{g}}} and 2 h{(P_{g})} = h{(P_{g})} + e^{e^{P_{g}}} and 4 h^{2}{(P_{g})} = (h{(P_{g})} + e^{e^{P_{g}}})^{2} and (4 h^{2}{(P_{g})})^{P_{g}} = ((h{(P_{g})} + e^{e^{P_{g}}})^{2})^{P_{g}} and 4^{P_{g}} (h^{2}{(P_{g})})^{P_{g}} = (h^{2}{(P_{g})} + 2 h{(P_{g})} e^{e^{P_{g}}} + e^{2 e^{P_{g}}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True))))"], [["add", 1, "Function('h')(Symbol('P_g', commutative=True))"], "Equality(Mul(Integer(2), Function('h')(Symbol('P_g', commutative=True))), Add(Function('h')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('h')(Symbol('P_g', commutative=True)), Integer(2))), Pow(Add(Function('h')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True)))), Integer(2)))"], [["power", 3, "Symbol('P_g', commutative=True)"], "Equality(Pow(Mul(Integer(4), Pow(Function('h')(Symbol('P_g', commutative=True)), Integer(2))), Symbol('P_g', commutative=True)), Pow(Pow(Add(Function('h')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True)))), Integer(2)), Symbol('P_g', commutative=True)))"], [["expand", 4], "Equality(Mul(Pow(Integer(4), Symbol('P_g', commutative=True)), Pow(Pow(Function('h')(Symbol('P_g', commutative=True)), Integer(2)), Symbol('P_g', commutative=True))), Pow(Add(Pow(Function('h')(Symbol('P_g', commutative=True)), Integer(2)), Mul(Integer(2), Function('h')(Symbol('P_g', commutative=True)), exp(exp(Symbol('P_g', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('P_g', commutative=True))))), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} = e^{\\dot{x} + \\mathbf{J}_f}, then obtain \\dot{x} e^{\\dot{x} + \\mathbf{J}_f} \\iint \\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} d\\dot{x} d\\mathbf{J}_f = \\dot{x} e^{\\dot{x} + \\mathbf{J}_f} \\iint e^{\\dot{x} + \\mathbf{J}_f} d\\dot{x} d\\mathbf{J}_f", "derivation": "\\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} = e^{\\dot{x} + \\mathbf{J}_f} and \\int \\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} d\\dot{x} = \\int e^{\\dot{x} + \\mathbf{J}_f} d\\dot{x} and \\iint \\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} d\\dot{x} d\\mathbf{J}_f = \\iint e^{\\dot{x} + \\mathbf{J}_f} d\\dot{x} d\\mathbf{J}_f and \\dot{x} e^{\\dot{x} + \\mathbf{J}_f} \\iint \\operatorname{m_{s}}{(\\mathbf{J}_f,\\dot{x})} d\\dot{x} d\\mathbf{J}_f = \\dot{x} e^{\\dot{x} + \\mathbf{J}_f} \\iint e^{\\dot{x} + \\mathbf{J}_f} d\\dot{x} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["times", 3, "Mul(Symbol('\\\\dot{x}', commutative=True), exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Function('m_s')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Symbol('\\\\dot{x}', commutative=True), exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(exp(Add(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\phi_2)} = \\log{(\\phi_2)}, then obtain \\log{(\\phi_2)} \\frac{d}{d \\phi_2} \\operatorname{A_{z}}^{\\phi_2}{(\\phi_2)} = \\log{(\\phi_2)} \\frac{d}{d \\phi_2} \\log{(\\phi_2)}^{\\phi_2}", "derivation": "\\operatorname{A_{z}}{(\\phi_2)} = \\log{(\\phi_2)} and \\operatorname{A_{z}}^{\\phi_2}{(\\phi_2)} = \\log{(\\phi_2)}^{\\phi_2} and \\frac{d}{d \\phi_2} \\operatorname{A_{z}}^{\\phi_2}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\log{(\\phi_2)}^{\\phi_2} and \\log{(\\phi_2)} \\frac{d}{d \\phi_2} \\operatorname{A_{z}}^{\\phi_2}{(\\phi_2)} = \\log{(\\phi_2)} \\frac{d}{d \\phi_2} \\log{(\\phi_2)}^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Pow(Function('A_z')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["times", 3, "log(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(log(Symbol('\\\\phi_2', commutative=True)), Derivative(Pow(Function('A_z')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Mul(log(Symbol('\\\\phi_2', commutative=True)), Derivative(Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(x,t)} = - \\sin{(t - x)}, then obtain \\frac{\\partial}{\\partial x} \\sin{(t - x)} = \\frac{\\partial}{\\partial x} - \\rho_{b}{(x,t)}", "derivation": "\\rho_{b}{(x,t)} = - \\sin{(t - x)} and \\rho_{b}{(x,t)} + \\sin{(t - x)} = 0 and \\sin{(t - x)} = - \\rho_{b}{(x,t)} and \\frac{\\partial}{\\partial x} \\sin{(t - x)} = \\frac{\\partial}{\\partial x} - \\rho_{b}{(x,t)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))))"], [["minus", 1, "Mul(Integer(-1), sin(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))"], "Equality(Add(Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('t', commutative=True)), sin(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))), Integer(0))"], [["minus", 2, "Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('t', commutative=True))"], "Equality(sin(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('t', commutative=True))))"], [["differentiate", 3, "Symbol('x', commutative=True)"], "Equality(Derivative(sin(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('x', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(k,n_{1})} = e^{k^{n_{1}}} and \\hat{\\mathbf{r}}{(k,n_{1})} = e^{k^{n_{1}}}, then obtain e^{k^{n_{1}}} + \\frac{\\partial}{\\partial k} e^{k^{n_{1}}} = e^{k^{n_{1}}} + \\frac{\\partial}{\\partial k} b{(k,n_{1})}", "derivation": "b{(k,n_{1})} = e^{k^{n_{1}}} and \\hat{\\mathbf{r}}{(k,n_{1})} = e^{k^{n_{1}}} and \\frac{\\partial}{\\partial k} \\hat{\\mathbf{r}}{(k,n_{1})} = \\frac{\\partial}{\\partial k} e^{k^{n_{1}}} and \\hat{\\mathbf{r}}{(k,n_{1})} + \\frac{\\partial}{\\partial k} \\hat{\\mathbf{r}}{(k,n_{1})} = \\hat{\\mathbf{r}}{(k,n_{1})} + \\frac{\\partial}{\\partial k} e^{k^{n_{1}}} and \\hat{\\mathbf{r}}{(k,n_{1})} + \\frac{\\partial}{\\partial k} \\hat{\\mathbf{r}}{(k,n_{1})} = \\hat{\\mathbf{r}}{(k,n_{1})} + \\frac{\\partial}{\\partial k} b{(k,n_{1})} and e^{k^{n_{1}}} + \\frac{\\partial}{\\partial k} e^{k^{n_{1}}} = e^{k^{n_{1}}} + \\frac{\\partial}{\\partial k} b{(k,n_{1})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Derivative(Function('b')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))), Derivative(exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1)))), Add(exp(Pow(Symbol('k', commutative=True), Symbol('n_1', commutative=True))), Derivative(Function('b')(Symbol('k', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(C_{2},\\sigma_x,H)} = \\sigma_x (- C_{2} + H), then obtain (\\sigma_x (- C_{2} + H))^{H} = (- C_{2} \\sigma_x + H \\sigma_x)^{H}", "derivation": "\\operatorname{F_{c}}{(C_{2},\\sigma_x,H)} = \\sigma_x (- C_{2} + H) and \\operatorname{F_{c}}^{H}{(C_{2},\\sigma_x,H)} = (\\sigma_x (- C_{2} + H))^{H} and \\operatorname{F_{c}}^{H}{(C_{2},\\sigma_x,H)} = (- C_{2} \\sigma_x + H \\sigma_x)^{H} and (\\sigma_x (- C_{2} + H))^{H} = (- C_{2} \\sigma_x + H \\sigma_x)^{H}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('C_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('C_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["expand", 2], "Equality(Pow(Function('F_c')(Symbol('C_2', commutative=True), Symbol('\\\\sigma_x', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('H', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Mul(Symbol('\\\\sigma_x', commutative=True), Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Symbol('H', commutative=True))), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('C_2', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(z^{*},E_{\\lambda})} = E_{\\lambda} z^{*}, then obtain \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} \\operatorname{A_{z}}{(z^{*},E_{\\lambda})} dE_{\\lambda} = \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} E_{\\lambda} z^{*} dE_{\\lambda}", "derivation": "\\operatorname{A_{z}}{(z^{*},E_{\\lambda})} = E_{\\lambda} z^{*} and \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{A_{z}}{(z^{*},E_{\\lambda})} = \\frac{\\partial}{\\partial E_{\\lambda}} E_{\\lambda} z^{*} and \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} \\operatorname{A_{z}}{(z^{*},E_{\\lambda})} = \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} E_{\\lambda} z^{*} and \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} \\operatorname{A_{z}}{(z^{*},E_{\\lambda})} dE_{\\lambda} = \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}^{2}} E_{\\lambda} z^{*} dE_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('z^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('z^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('z^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))))"], [["integrate", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Derivative(Function('A_z')(Symbol('z^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then obtain - \\mathbf{r} + \\operatorname{m_{s}}{(\\mathbf{r})} - \\int \\operatorname{m_{s}}{(\\mathbf{r})} d\\mathbf{r} = - \\mathbf{r} + \\log{(\\mathbf{r})} - \\int \\operatorname{m_{s}}{(\\mathbf{r})} d\\mathbf{r}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\int \\operatorname{m_{s}}{(\\mathbf{r})} d\\mathbf{r} = \\int \\log{(\\mathbf{r})} d\\mathbf{r} and - \\mathbf{r} + \\operatorname{m_{s}}{(\\mathbf{r})} = - \\mathbf{r} + \\log{(\\mathbf{r})} and - \\mathbf{r} + \\operatorname{m_{s}}{(\\mathbf{r})} - \\int \\log{(\\mathbf{r})} d\\mathbf{r} = - \\mathbf{r} + \\log{(\\mathbf{r})} - \\int \\log{(\\mathbf{r})} d\\mathbf{r} and - \\mathbf{r} + \\operatorname{m_{s}}{(\\mathbf{r})} - \\int \\operatorname{m_{s}}{(\\mathbf{r})} d\\mathbf{r} = - \\mathbf{r} + \\log{(\\mathbf{r})} - \\int \\operatorname{m_{s}}{(\\mathbf{r})} d\\mathbf{r}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))))"], [["minus", 3, "Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Integral(Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Integral(Function('m_s')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))))"]]}, {"prompt": "Given \\varphi^{*}{(Z)} = e^{Z}, then derive \\frac{d}{d Z} \\varphi^{*}{(Z)} = e^{Z}, then obtain e^{Z} = \\frac{d}{d Z} e^{Z}", "derivation": "\\varphi^{*}{(Z)} = e^{Z} and \\frac{d}{d Z} \\varphi^{*}{(Z)} = \\frac{d}{d Z} e^{Z} and \\frac{d}{d Z} \\varphi^{*}{(Z)} = e^{Z} and e^{Z} = \\frac{d}{d Z} e^{Z}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varphi^*')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), exp(Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('Z', commutative=True)), Derivative(exp(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given J{(V)} = \\log{(V)}, then obtain \\frac{J{(V)} (\\int \\frac{J{(V)}}{\\log{(V)}} dV)^{\\frac{J{(V)}}{\\log{(V)}} + 1}}{\\log{(V)}} = \\frac{J{(V)} (\\int 1 dV) \\int \\frac{J{(V)}}{\\log{(V)}} dV}{\\log{(V)}}", "derivation": "J{(V)} = \\log{(V)} and \\frac{J{(V)}}{\\log{(V)}} = 1 and \\int \\frac{J{(V)}}{\\log{(V)}} dV = \\int 1 dV and (\\int \\frac{J{(V)}}{\\log{(V)}} dV)^{2} = (\\int 1 dV) \\int \\frac{J{(V)}}{\\log{(V)}} dV and \\frac{J{(V)}}{\\log{(V)}} + 1 = 2 and \\frac{J{(V)} (\\int \\frac{J{(V)}}{\\log{(V)}} dV)^{2}}{\\log{(V)}} = \\frac{J{(V)} (\\int 1 dV) \\int \\frac{J{(V)}}{\\log{(V)}} dV}{\\log{(V)}} and \\frac{J{(V)} (\\int \\frac{J{(V)}}{\\log{(V)}} dV)^{\\frac{J{(V)}}{\\log{(V)}} + 1}}{\\log{(V)}} = \\frac{J{(V)} (\\int 1 dV) \\int \\frac{J{(V)}}{\\log{(V)}} dV}{\\log{(V)}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["divide", 1, "log(Symbol('V', commutative=True))"], "Equality(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integral(Integer(1), Tuple(Symbol('V', commutative=True))))"], [["times", 3, "Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True)))"], "Equality(Pow(Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integer(2)), Mul(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True)))))"], [["add", 2, 1], "Equality(Add(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Integer(1)), Integer(2))"], [["times", 4, "Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1)), Pow(Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Integer(2))), Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1)), Pow(Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True))), Add(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Integer(1)))), Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Mul(Function('J')(Symbol('V', commutative=True)), Pow(log(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\rho{(C_{d},i)} = - i + \\sin{(C_{d})}, then derive - \\frac{\\partial}{\\partial C_{d}} \\rho{(C_{d},i)} = - \\cos{(C_{d})}, then obtain - ((I + v_{z})^{I})^{I} - \\cos{(C_{d})} = - ((I + v_{z})^{I})^{I} - \\frac{\\partial}{\\partial C_{d}} (- i + \\sin{(C_{d})})", "derivation": "\\rho{(C_{d},i)} = - i + \\sin{(C_{d})} and \\frac{\\partial}{\\partial C_{d}} \\rho{(C_{d},i)} = \\frac{\\partial}{\\partial C_{d}} (- i + \\sin{(C_{d})}) and - \\frac{\\partial}{\\partial C_{d}} \\rho{(C_{d},i)} = - \\frac{\\partial}{\\partial C_{d}} (- i + \\sin{(C_{d})}) and - \\frac{\\partial}{\\partial C_{d}} \\rho{(C_{d},i)} = - \\cos{(C_{d})} and - \\cos{(C_{d})} = - \\frac{\\partial}{\\partial C_{d}} (- i + \\sin{(C_{d})}) and - ((I + v_{z})^{I})^{I} - \\cos{(C_{d})} = - ((I + v_{z})^{I})^{I} - \\frac{\\partial}{\\partial C_{d}} (- i + \\sin{(C_{d})})", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('C_d', commutative=True))))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('\\\\rho')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), cos(Symbol('C_d', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1)))))"], [["minus", 5, "Pow(Pow(Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True))), Mul(Integer(-1), cos(Symbol('C_d', commutative=True)))), Add(Mul(Integer(-1), Pow(Pow(Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('I', commutative=True)), Symbol('I', commutative=True))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), sin(Symbol('C_d', commutative=True))), Tuple(Symbol('C_d', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(U)} = \\log{(U)}, then obtain U \\frac{d^{2}}{d U^{2}} \\operatorname{n_{1}}{(U)} + 2 \\frac{d}{d U} \\operatorname{n_{1}}{(U)} = \\frac{1}{U}", "derivation": "\\operatorname{n_{1}}{(U)} = \\log{(U)} and U \\operatorname{n_{1}}{(U)} = U \\log{(U)} and \\frac{d}{d U} U \\operatorname{n_{1}}{(U)} = \\frac{d}{d U} U \\log{(U)} and \\frac{d^{2}}{d U^{2}} U \\operatorname{n_{1}}{(U)} = \\frac{d^{2}}{d U^{2}} U \\log{(U)} and U \\frac{d^{2}}{d U^{2}} \\operatorname{n_{1}}{(U)} + 2 \\frac{d}{d U} \\operatorname{n_{1}}{(U)} = \\frac{1}{U}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('n_1')(Symbol('U', commutative=True))), Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Mul(Symbol('U', commutative=True), Function('n_1')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Mul(Symbol('U', commutative=True), Function('n_1')(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(2))), Derivative(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Symbol('U', commutative=True), Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(2)))), Mul(Integer(2), Derivative(Function('n_1')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))), Pow(Symbol('U', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} = - F_{H} + M_{E} + \\mathbf{g}, then obtain \\int - 2 F_{H} (\\mathbf{g} \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} - 1) dM_{E} = \\int - 2 F_{H} (\\mathbf{g} (- F_{H} + M_{E} + \\mathbf{g}) - 1) dM_{E}", "derivation": "\\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} = - F_{H} + M_{E} + \\mathbf{g} and \\mathbf{g} \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} = \\mathbf{g} (- F_{H} + M_{E} + \\mathbf{g}) and \\mathbf{g} \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} - 1 = \\mathbf{g} (- F_{H} + M_{E} + \\mathbf{g}) - 1 and - 2 F_{H} (\\mathbf{g} \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} - 1) = - 2 F_{H} (\\mathbf{g} (- F_{H} + M_{E} + \\mathbf{g}) - 1) and \\int - 2 F_{H} (\\mathbf{g} \\operatorname{x^{{\\}'}}{(F_{H},\\mathbf{g},M_{E})} - 1) dM_{E} = \\int - 2 F_{H} (\\mathbf{g} (- F_{H} + M_{E} + \\mathbf{g}) - 1) dM_{E}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('x^\\\\prime')(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('M_E', commutative=True))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["minus", 2, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('x^\\\\prime')(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('M_E', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1)))"], [["times", 3, "Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('x^\\\\prime')(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('M_E', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Function('x^\\\\prime')(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('M_E', commutative=True))), Integer(-1))), Tuple(Symbol('M_E', commutative=True))), Integral(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('M_E', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Integer(-1))), Tuple(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\mu,v)} = \\frac{v}{\\mu}, then obtain (\\frac{v (- v + \\theta{(\\mu,v)})^{v}}{\\mu})^{v} = (\\frac{v (- v + \\frac{v}{\\mu})^{v}}{\\mu})^{v}", "derivation": "\\theta{(\\mu,v)} = \\frac{v}{\\mu} and - v + \\theta{(\\mu,v)} = - v + \\frac{v}{\\mu} and (- v + \\theta{(\\mu,v)})^{v} = (- v + \\frac{v}{\\mu})^{v} and \\frac{(- v + \\theta{(\\mu,v)})^{v}}{\\mu} = \\frac{(- v + \\frac{v}{\\mu})^{v}}{\\mu} and \\frac{v (- v + \\theta{(\\mu,v)})^{v}}{\\mu} = \\frac{v (- v + \\frac{v}{\\mu})^{v}}{\\mu} and (\\frac{v (- v + \\theta{(\\mu,v)})^{v}}{\\mu})^{v} = (\\frac{v (- v + \\frac{v}{\\mu})^{v}}{\\mu})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True)))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True))))"], [["power", 2, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"], [["divide", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["times", 4, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True))))"], [["power", 5, "Symbol('v', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\theta')(Symbol('\\\\mu', commutative=True), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Symbol('v', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Symbol('v', commutative=True))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given \\Omega{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})}, then obtain 0 = - 2 \\Omega{(\\theta_2)} + 2 \\cos{(\\log{(\\theta_2)})}", "derivation": "\\Omega{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})} and 0 = - \\Omega{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})} and - \\Omega{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})} = - 2 \\Omega{(\\theta_2)} + 2 \\cos{(\\log{(\\theta_2)})} and 0 = - 2 \\Omega{(\\theta_2)} + 2 \\cos{(\\log{(\\theta_2)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True))))"], [["minus", 1, "Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))), cos(log(Symbol('\\\\theta_2', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))), cos(log(Symbol('\\\\theta_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))), cos(log(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\theta_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\Omega')(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given \\phi_{2}{(m,\\chi)} = \\chi^{m}, then obtain \\iint (- f_{\\mathbf{v}} + \\frac{\\phi_{2}{(m,\\chi)}}{\\chi}) d\\chi df_{\\mathbf{v}} = \\iint (- f_{\\mathbf{v}} + \\frac{\\chi^{m}}{\\chi}) d\\chi df_{\\mathbf{v}}", "derivation": "\\phi_{2}{(m,\\chi)} = \\chi^{m} and \\frac{\\phi_{2}{(m,\\chi)}}{\\chi} = \\frac{\\chi^{m}}{\\chi} and - f_{\\mathbf{v}} + \\frac{\\phi_{2}{(m,\\chi)}}{\\chi} = - f_{\\mathbf{v}} + \\frac{\\chi^{m}}{\\chi} and \\int (- f_{\\mathbf{v}} + \\frac{\\phi_{2}{(m,\\chi)}}{\\chi}) d\\chi = \\int (- f_{\\mathbf{v}} + \\frac{\\chi^{m}}{\\chi}) d\\chi and \\iint (- f_{\\mathbf{v}} + \\frac{\\phi_{2}{(m,\\chi)}}{\\chi}) d\\chi df_{\\mathbf{v}} = \\iint (- f_{\\mathbf{v}} + \\frac{\\chi^{m}}{\\chi}) d\\chi df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('m', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('m', commutative=True), Symbol('\\\\chi', commutative=True))), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('m', commutative=True))))"], [["minus", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('m', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('m', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('m', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["integrate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Function('\\\\phi_2')(Symbol('m', commutative=True), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Symbol('\\\\chi', commutative=True), Integer(-1)), Pow(Symbol('\\\\chi', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given x{(\\rho_b)} = e^{\\rho_b} and \\operatorname{v_{x}}{(\\rho_b)} = - x{(\\rho_b)}, then obtain 0 = \\frac{(\\operatorname{v_{x}}{(\\rho_b)} + x{(\\rho_b)}) \\cos{(V)}}{\\sin{(C_{d})}}", "derivation": "x{(\\rho_b)} = e^{\\rho_b} and 0 = - x{(\\rho_b)} + e^{\\rho_b} and \\operatorname{v_{x}}{(\\rho_b)} = - x{(\\rho_b)} and 0 = \\operatorname{v_{x}}{(\\rho_b)} + e^{\\rho_b} and 0 = \\operatorname{v_{x}}{(\\rho_b)} + x{(\\rho_b)} and 0 = (\\operatorname{v_{x}}{(\\rho_b)} + x{(\\rho_b)}) \\cos{(V)} and 0 = \\frac{(\\operatorname{v_{x}}{(\\rho_b)} + x{(\\rho_b)}) \\cos{(V)}}{\\sin{(C_{d})}}", "srepr_derivation": [["get_premise", "Equality(Function('x')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["minus", 1, "Function('x')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x')(Symbol('\\\\rho_b', commutative=True))), exp(Symbol('\\\\rho_b', commutative=True))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Function('x')(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(0), Add(Function('v_x')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Function('v_x')(Symbol('\\\\rho_b', commutative=True)), Function('x')(Symbol('\\\\rho_b', commutative=True))))"], [["times", 5, "cos(Symbol('V', commutative=True))"], "Equality(Integer(0), Mul(Add(Function('v_x')(Symbol('\\\\rho_b', commutative=True)), Function('x')(Symbol('\\\\rho_b', commutative=True))), cos(Symbol('V', commutative=True))))"], [["divide", 6, "sin(Symbol('C_d', commutative=True))"], "Equality(Integer(0), Mul(Add(Function('v_x')(Symbol('\\\\rho_b', commutative=True)), Function('x')(Symbol('\\\\rho_b', commutative=True))), Pow(sin(Symbol('C_d', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))))"]]}, {"prompt": "Given U{(g)} = \\cos{(g)}, then derive \\int U{(g)} dg = \\mathbb{I} + \\sin{(g)}, then obtain e^{F_{N} (\\sin{(g)} \\int \\cos{(g)} dg)^{g}} = e^{F_{N} (\\sin{(g)} \\int U{(g)} dg)^{g}}", "derivation": "U{(g)} = \\cos{(g)} and \\int U{(g)} dg = \\int \\cos{(g)} dg and \\int U{(g)} dg = \\mathbb{I} + \\sin{(g)} and \\int \\cos{(g)} dg = \\mathbb{I} + \\sin{(g)} and \\sin{(g)} \\int \\cos{(g)} dg = (\\mathbb{I} + \\sin{(g)}) \\sin{(g)} and \\sin{(g)} \\int \\cos{(g)} dg = \\sin{(g)} \\int U{(g)} dg and (\\sin{(g)} \\int \\cos{(g)} dg)^{g} = (\\sin{(g)} \\int U{(g)} dg)^{g} and F_{N} (\\sin{(g)} \\int \\cos{(g)} dg)^{g} = F_{N} (\\sin{(g)} \\int U{(g)} dg)^{g} and e^{F_{N} (\\sin{(g)} \\int \\cos{(g)} dg)^{g}} = e^{F_{N} (\\sin{(g)} \\int U{(g)} dg)^{g}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('g', commutative=True))))"], [["times", 4, "sin(Symbol('g', commutative=True))"], "Equality(Mul(sin(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Add(Symbol('\\\\mathbb{I}', commutative=True), sin(Symbol('g', commutative=True))), sin(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(sin(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(sin(Symbol('g', commutative=True)), Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["power", 6, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(sin(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Mul(sin(Symbol('g', commutative=True)), Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["times", 7, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Pow(Mul(sin(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True))), Mul(Symbol('F_N', commutative=True), Pow(Mul(sin(Symbol('g', commutative=True)), Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True))))"], [["exp", 8], "Equality(exp(Mul(Symbol('F_N', commutative=True), Pow(Mul(sin(Symbol('g', commutative=True)), Integral(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))), exp(Mul(Symbol('F_N', commutative=True), Pow(Mul(sin(Symbol('g', commutative=True)), Integral(Function('U')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(C_{d},s)} = \\frac{\\partial}{\\partial s} C_{d}^{s}, then obtain 2 \\operatorname{n_{1}}{(C_{d},s)} \\log{(C_{d})}^{2} \\frac{\\partial}{\\partial s} C_{d}^{s} = (\\operatorname{n_{1}}{(C_{d},s)} \\frac{\\partial}{\\partial s} C_{d}^{s} + (\\frac{\\partial}{\\partial s} C_{d}^{s})^{2}) \\log{(C_{d})}^{2}", "derivation": "\\operatorname{n_{1}}{(C_{d},s)} = \\frac{\\partial}{\\partial s} C_{d}^{s} and \\operatorname{n_{1}}{(C_{d},s)} \\frac{\\partial}{\\partial s} C_{d}^{s} = (\\frac{\\partial}{\\partial s} C_{d}^{s})^{2} and 2 \\operatorname{n_{1}}{(C_{d},s)} \\frac{\\partial}{\\partial s} C_{d}^{s} = \\operatorname{n_{1}}{(C_{d},s)} \\frac{\\partial}{\\partial s} C_{d}^{s} + (\\frac{\\partial}{\\partial s} C_{d}^{s})^{2} and 2 \\operatorname{n_{1}}{(C_{d},s)} \\log{(C_{d})}^{2} \\frac{\\partial}{\\partial s} C_{d}^{s} = (\\operatorname{n_{1}}{(C_{d},s)} \\frac{\\partial}{\\partial s} C_{d}^{s} + (\\frac{\\partial}{\\partial s} C_{d}^{s})^{2}) \\log{(C_{d})}^{2}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Mul(Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Pow(Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(2)))"], [["add", 2, "Mul(Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Pow(Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(2))))"], [["times", 3, "Pow(log(Symbol('C_d', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Pow(log(Symbol('C_d', commutative=True)), Integer(2)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Add(Mul(Function('n_1')(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Pow(Derivative(Pow(Symbol('C_d', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Integer(2))), Pow(log(Symbol('C_d', commutative=True)), Integer(2))))"]]}, {"prompt": "Given x{(y)} = \\log{(\\log{(y)})}, then derive \\frac{d}{d y} x{(y)} = \\frac{1}{y \\log{(y)}}, then obtain v_{t} + \\frac{d}{d y} x{(y)} = F_{x} + \\frac{1}{y \\log{(y)}}", "derivation": "x{(y)} = \\log{(\\log{(y)})} and \\frac{d}{d y} x{(y)} = \\frac{d}{d y} \\log{(\\log{(y)})} and \\frac{d}{d y} x{(y)} = \\frac{1}{y \\log{(y)}} and \\frac{d^{2}}{d y^{2}} x{(y)} = \\frac{d}{d y} \\frac{1}{y \\log{(y)}} and \\int \\frac{d^{2}}{d y^{2}} x{(y)} dy = \\int \\frac{d}{d y} \\frac{1}{y \\log{(y)}} dy and v_{t} + \\frac{d}{d y} x{(y)} = F_{x} + \\frac{1}{y \\log{(y)}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('y', commutative=True)), log(log(Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(log(log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(log(Symbol('y', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(log(Symbol('y', commutative=True)), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(2))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(log(Symbol('y', commutative=True)), Integer(-1))), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('v_t', commutative=True), Derivative(Function('x')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Add(Symbol('F_x', commutative=True), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(log(Symbol('y', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(a,n)} = a n, then obtain (\\frac{\\operatorname{r_{0}}{(a,n)}}{n})^{n} + \\operatorname{r_{0}}{(a,n)} = a^{n} + \\operatorname{r_{0}}{(a,n)}", "derivation": "\\operatorname{r_{0}}{(a,n)} = a n and \\frac{\\operatorname{r_{0}}{(a,n)}}{n} = a and (\\frac{\\operatorname{r_{0}}{(a,n)}}{n})^{n} = a^{n} and (\\frac{\\operatorname{r_{0}}{(a,n)}}{n})^{n} + \\operatorname{r_{0}}{(a,n)} = a^{n} + \\operatorname{r_{0}}{(a,n)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('a', commutative=True), Symbol('n', commutative=True)))"], [["divide", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('a', commutative=True))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)))"], [["add", 3, "Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))), Add(Pow(Symbol('a', commutative=True), Symbol('n', commutative=True)), Function('r_0')(Symbol('a', commutative=True), Symbol('n', commutative=True))))"]]}, {"prompt": "Given I{(\\phi,x^\\prime)} = \\frac{\\partial}{\\partial \\phi} (\\phi + x^\\prime), then derive I{(\\phi,x^\\prime)} = 1, then derive \\frac{\\partial}{\\partial x^\\prime} I{(\\phi,x^\\prime)} = 0, then obtain \\frac{\\frac{\\partial}{\\partial x^\\prime} I{(\\phi,x^\\prime)}}{\\frac{\\partial}{\\partial \\phi} (\\phi + x^\\prime)} = 0", "derivation": "I{(\\phi,x^\\prime)} = \\frac{\\partial}{\\partial \\phi} (\\phi + x^\\prime) and I{(\\phi,x^\\prime)} = 1 and \\frac{\\partial}{\\partial x^\\prime} I{(\\phi,x^\\prime)} = \\frac{d}{d x^\\prime} 1 and \\frac{\\partial}{\\partial x^\\prime} I{(\\phi,x^\\prime)} = 0 and \\frac{d}{d x^\\prime} 1 = 0 and \\frac{\\frac{d}{d x^\\prime} 1}{\\frac{\\partial}{\\partial \\phi} (\\phi + x^\\prime)} = 0 and \\frac{\\frac{\\partial}{\\partial x^\\prime} I{(\\phi,x^\\prime)}}{\\frac{\\partial}{\\partial \\phi} (\\phi + x^\\prime)} = 0", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('I')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(1))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('I')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(0))"], [["divide", 5, "Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Pow(Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Pow(Derivative(Add(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('I')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{D},\\hat{X})} = \\hat{X} + \\mathbf{D}, then obtain \\hat{X} + y^{\\prime} = \\int \\frac{\\hat{X} + \\mathbf{D}}{\\phi_{1}{(\\mathbf{D},\\hat{X})}} d\\hat{X}", "derivation": "\\phi_{1}{(\\mathbf{D},\\hat{X})} = \\hat{X} + \\mathbf{D} and \\frac{\\phi_{1}{(\\mathbf{D},\\hat{X})}}{\\hat{X} + \\mathbf{D}} = 1 and 1 = \\frac{\\hat{X} + \\mathbf{D}}{\\phi_{1}{(\\mathbf{D},\\hat{X})}} and \\int 1 d\\hat{X} = \\int \\frac{\\hat{X} + \\mathbf{D}}{\\phi_{1}{(\\mathbf{D},\\hat{X})}} d\\hat{X} and \\hat{X} + y^{\\prime} = \\int \\frac{\\hat{X} + \\mathbf{D}}{\\phi_{1}{(\\mathbf{D},\\hat{X})}} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Integer(1))"], [["divide", 2, "Mul(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\phi_1')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(u)} = \\cos{(\\sin{(u)})}, then obtain \\frac{d}{d u} \\int \\mathbf{A}{(u)} \\cos{(\\sin{(u)})} du = \\frac{d}{d u} \\int \\cos^{2}{(\\sin{(u)})} du", "derivation": "\\mathbf{A}{(u)} = \\cos{(\\sin{(u)})} and \\mathbf{A}{(u)} \\cos{(\\sin{(u)})} = \\cos^{2}{(\\sin{(u)})} and \\int \\mathbf{A}{(u)} \\cos{(\\sin{(u)})} du = \\int \\cos^{2}{(\\sin{(u)})} du and \\frac{d}{d u} \\int \\mathbf{A}{(u)} \\cos{(\\sin{(u)})} du = \\frac{d}{d u} \\int \\cos^{2}{(\\sin{(u)})} du", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True))))"], [["times", 1, "cos(sin(Symbol('u', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Pow(cos(sin(Symbol('u', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Pow(cos(sin(Symbol('u', commutative=True))), Integer(2)), Tuple(Symbol('u', commutative=True))))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('\\\\mathbf{A}')(Symbol('u', commutative=True)), cos(sin(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(sin(Symbol('u', commutative=True))), Integer(2)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given i{(C_{1})} = e^{\\cos{(C_{1})}}, then obtain (2 i{(C_{1})} - e^{\\cos{(C_{1})}}) \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}} = i{(C_{1})} \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}}", "derivation": "i{(C_{1})} = e^{\\cos{(C_{1})}} and 2 i{(C_{1})} = i{(C_{1})} + e^{\\cos{(C_{1})}} and \\frac{d}{d C_{1}} i{(C_{1})} = \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}} and 2 i{(C_{1})} - e^{\\cos{(C_{1})}} = i{(C_{1})} and (2 i{(C_{1})} - e^{\\cos{(C_{1})}}) \\frac{d}{d C_{1}} i{(C_{1})} = i{(C_{1})} \\frac{d}{d C_{1}} i{(C_{1})} and (2 i{(C_{1})} - e^{\\cos{(C_{1})}}) \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}} = i{(C_{1})} \\frac{d}{d C_{1}} e^{\\cos{(C_{1})}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('C_1', commutative=True)), exp(cos(Symbol('C_1', commutative=True))))"], [["add", 1, "Function('i')(Symbol('C_1', commutative=True))"], "Equality(Mul(Integer(2), Function('i')(Symbol('C_1', commutative=True))), Add(Function('i')(Symbol('C_1', commutative=True)), exp(cos(Symbol('C_1', commutative=True)))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 2, "exp(cos(Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('i')(Symbol('C_1', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('C_1', commutative=True))))), Function('i')(Symbol('C_1', commutative=True)))"], [["times", 4, "Derivative(Function('i')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(2), Function('i')(Symbol('C_1', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('C_1', commutative=True))))), Derivative(Function('i')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Mul(Function('i')(Symbol('C_1', commutative=True)), Derivative(Function('i')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Mul(Integer(2), Function('i')(Symbol('C_1', commutative=True))), Mul(Integer(-1), exp(cos(Symbol('C_1', commutative=True))))), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Mul(Function('i')(Symbol('C_1', commutative=True)), Derivative(exp(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(a^{\\dagger},m)} = \\sin{(a^{\\dagger} m)} and \\Psi_{nl}{(a^{\\dagger},m)} = \\sin{(a^{\\dagger} m)}, then obtain \\int \\frac{\\partial}{\\partial m} \\int \\Psi_{nl}{(a^{\\dagger},m)} da^{\\dagger} dm = \\int \\frac{\\partial}{\\partial m} \\int \\sin{(a^{\\dagger} m)} da^{\\dagger} dm", "derivation": "\\operatorname{P_{g}}{(a^{\\dagger},m)} = \\sin{(a^{\\dagger} m)} and \\int \\operatorname{P_{g}}{(a^{\\dagger},m)} da^{\\dagger} = \\int \\sin{(a^{\\dagger} m)} da^{\\dagger} and \\frac{\\partial}{\\partial m} \\int \\operatorname{P_{g}}{(a^{\\dagger},m)} da^{\\dagger} = \\frac{\\partial}{\\partial m} \\int \\sin{(a^{\\dagger} m)} da^{\\dagger} and \\Psi_{nl}{(a^{\\dagger},m)} = \\sin{(a^{\\dagger} m)} and \\Psi_{nl}{(a^{\\dagger},m)} = \\operatorname{P_{g}}{(a^{\\dagger},m)} and \\frac{\\partial}{\\partial m} \\int \\Psi_{nl}{(a^{\\dagger},m)} da^{\\dagger} = \\frac{\\partial}{\\partial m} \\int \\sin{(a^{\\dagger} m)} da^{\\dagger} and \\int \\frac{\\partial}{\\partial m} \\int \\Psi_{nl}{(a^{\\dagger},m)} da^{\\dagger} dm = \\int \\frac{\\partial}{\\partial m} \\int \\sin{(a^{\\dagger} m)} da^{\\dagger} dm", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Integral(Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integral(sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), Function('P_g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integral(sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('m', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))), Integral(Derivative(Integral(sin(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(\\phi_2,A)} = \\phi_2^{A}, then obtain 2 A \\phi_2^{A} - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} dA = 2 A \\phi_2^{A} - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} dA", "derivation": "\\mathbf{g}{(\\phi_2,A)} = \\phi_2^{A} and \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} = \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} and \\int \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} dA = \\int \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} dA and - \\phi_2^{A} + \\int \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} dA = - \\phi_2^{A} + \\int \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} dA and - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} dA = - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} dA and 2 A \\phi_2^{A} - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\mathbf{g}{(\\phi_2,A)} dA = 2 A \\phi_2^{A} - \\mathbf{g}{(\\phi_2,A)} + \\int \\frac{\\partial}{\\partial \\phi_2} \\phi_2^{A} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))"], [["minus", 3, "Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))))"], [["add", 5, "Mul(Integer(2), Symbol('A', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('A', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))), Add(Mul(Integer(2), Symbol('A', commutative=True), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\phi_2', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\ddot{x})} = \\sin{(\\ddot{x})}, then obtain \\frac{d}{d \\ddot{x}} - \\sin{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} - \\theta_{2}{(\\ddot{x})}", "derivation": "\\theta_{2}{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\theta_{2}{(\\ddot{x})} - \\sin{(\\ddot{x})} = 0 and - \\sin{(\\ddot{x})} = - \\theta_{2}{(\\ddot{x})} and \\frac{d}{d \\ddot{x}} - \\sin{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} - \\theta_{2}{(\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True)))), Integer(0))"], [["minus", 2, "Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True))), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(f^{\\prime},m)} = - m + \\sin{(f^{\\prime})} and \\psi^{*}{(f^{\\prime},m)} = (- m + \\sin{(f^{\\prime})})^{m}, then obtain \\int \\Psi^{m}{(f^{\\prime},m)} df^{\\prime} = \\int (- m + \\sin{(f^{\\prime})})^{m} df^{\\prime}", "derivation": "\\Psi{(f^{\\prime},m)} = - m + \\sin{(f^{\\prime})} and \\Psi^{m}{(f^{\\prime},m)} = (- m + \\sin{(f^{\\prime})})^{m} and \\psi^{*}{(f^{\\prime},m)} = (- m + \\sin{(f^{\\prime})})^{m} and \\Psi^{m}{(f^{\\prime},m)} = \\psi^{*}{(f^{\\prime},m)} and \\int \\Psi^{m}{(f^{\\prime},m)} df^{\\prime} = \\int \\psi^{*}{(f^{\\prime},m)} df^{\\prime} and \\int \\Psi^{m}{(f^{\\prime},m)} df^{\\prime} = \\int (- m + \\sin{(f^{\\prime})})^{m} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\Psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Function('\\\\psi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\Psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Pow(Function('\\\\Psi')(Symbol('f^{\\\\prime}', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Symbol('m', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given f{(\\mathbf{F})} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F}, then derive f^{\\mathbf{F}}{(\\mathbf{F})} = (m + \\sin{(\\mathbf{F})})^{\\mathbf{F}}, then obtain (m + \\sin{(\\mathbf{F})})^{\\mathbf{F}} = (\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}}", "derivation": "f{(\\mathbf{F})} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and f^{\\mathbf{F}}{(\\mathbf{F})} = (\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}} and f^{\\mathbf{F}}{(\\mathbf{F})} = (m + \\sin{(\\mathbf{F})})^{\\mathbf{F}} and (m + \\sin{(\\mathbf{F})})^{\\mathbf{F}} = (\\int \\cos{(\\mathbf{F})} d\\mathbf{F})^{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\mathbf{F}', commutative=True)), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('f')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('f')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('m', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('m', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(n)} = \\log{(\\log{(n)})}, then derive \\int \\operatorname{V_{\\mathbf{E}}}{(n)} dn = P_{e} + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)}, then obtain P_{e} + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)} = \\rho + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(n)} = \\log{(\\log{(n)})} and \\int \\operatorname{V_{\\mathbf{E}}}{(n)} dn = \\int \\log{(\\log{(n)})} dn and \\int \\operatorname{V_{\\mathbf{E}}}{(n)} dn = P_{e} + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)} and P_{e} + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)} = \\int \\log{(\\log{(n)})} dn and P_{e} + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)} = \\rho + n \\log{(\\log{(n)})} - \\operatorname{li}{(n)}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), log(log(Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(log(log(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('V_{\\\\mathbf{E}}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Symbol('n', commutative=True), log(log(Symbol('n', commutative=True)))), Mul(Integer(-1), li(Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('P_e', commutative=True), Mul(Symbol('n', commutative=True), log(log(Symbol('n', commutative=True)))), Mul(Integer(-1), li(Symbol('n', commutative=True)))), Integral(log(log(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('P_e', commutative=True), Mul(Symbol('n', commutative=True), log(log(Symbol('n', commutative=True)))), Mul(Integer(-1), li(Symbol('n', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Mul(Symbol('n', commutative=True), log(log(Symbol('n', commutative=True)))), Mul(Integer(-1), li(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(B)} = \\sin{(B)}, then obtain \\rho_{f}{(B)} \\rho_{f}^{B}{(B)} \\sin^{B}{(B)} + \\rho_{f}{(B)} - 1 = \\rho_{f}{(B)} + \\rho_{f}^{B}{(B)} \\sin{(B)} \\sin^{B}{(B)} - 1", "derivation": "\\rho_{f}{(B)} = \\sin{(B)} and \\rho_{f}^{B}{(B)} = \\sin^{B}{(B)} and \\rho_{f}^{2 B}{(B)} = \\rho_{f}^{B}{(B)} \\sin^{B}{(B)} and \\rho_{f}{(B)} \\rho_{f}^{2 B}{(B)} = \\rho_{f}^{2 B}{(B)} \\sin{(B)} and \\rho_{f}{(B)} \\rho_{f}^{2 B}{(B)} + \\rho_{f}{(B)} = \\rho_{f}{(B)} + \\rho_{f}^{2 B}{(B)} \\sin{(B)} and \\rho_{f}{(B)} \\rho_{f}^{2 B}{(B)} + \\rho_{f}{(B)} - 1 = \\rho_{f}{(B)} + \\rho_{f}^{2 B}{(B)} \\sin{(B)} - 1 and \\rho_{f}{(B)} \\rho_{f}^{B}{(B)} \\sin^{B}{(B)} + \\rho_{f}{(B)} - 1 = \\rho_{f}{(B)} + \\rho_{f}^{B}{(B)} \\sin{(B)} \\sin^{B}{(B)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(sin(Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["times", 2, "Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Symbol('B', commutative=True))"], "Equality(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))), Mul(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(sin(Symbol('B', commutative=True)), Symbol('B', commutative=True))))"], [["times", 1, "Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True)))"], "Equality(Mul(Function('\\\\rho_f')(Symbol('B', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True)))), Mul(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))), sin(Symbol('B', commutative=True))))"], [["add", 4, "Function('\\\\rho_f')(Symbol('B', commutative=True))"], "Equality(Add(Mul(Function('\\\\rho_f')(Symbol('B', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True)))), Function('\\\\rho_f')(Symbol('B', commutative=True))), Add(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))), sin(Symbol('B', commutative=True)))))"], [["minus", 5, 1], "Equality(Add(Mul(Function('\\\\rho_f')(Symbol('B', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True)))), Function('\\\\rho_f')(Symbol('B', commutative=True)), Integer(-1)), Add(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Integer(2), Symbol('B', commutative=True))), sin(Symbol('B', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Function('\\\\rho_f')(Symbol('B', commutative=True)), Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(sin(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Function('\\\\rho_f')(Symbol('B', commutative=True)), Integer(-1)), Add(Function('\\\\rho_f')(Symbol('B', commutative=True)), Mul(Pow(Function('\\\\rho_f')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)), Pow(sin(Symbol('B', commutative=True)), Symbol('B', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}{(F_{N},Q)} = e^{F_{N} - Q}, then obtain \\frac{e^{F_{N} - Q} \\frac{d}{d F_{N}} 1}{\\hat{x}{(F_{N},Q)}} = \\frac{e^{F_{N} - Q} \\frac{\\partial}{\\partial F_{N}} e^{- F_{N} + Q} e^{F_{N} - Q}}{\\hat{x}{(F_{N},Q)}}", "derivation": "\\hat{x}{(F_{N},Q)} = e^{F_{N} - Q} and 1 = \\frac{e^{F_{N} - Q}}{\\hat{x}{(F_{N},Q)}} and 1 = e^{- F_{N} + Q} e^{F_{N} - Q} and \\frac{d}{d F_{N}} 1 = \\frac{\\partial}{\\partial F_{N}} e^{- F_{N} + Q} e^{F_{N} - Q} and \\frac{e^{F_{N} - Q} \\frac{d}{d F_{N}} 1}{\\hat{x}{(F_{N},Q)}} = \\frac{e^{F_{N} - Q} \\frac{\\partial}{\\partial F_{N}} e^{- F_{N} + Q} e^{F_{N} - Q}}{\\hat{x}{(F_{N},Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True)))))"], [["divide", 1, "Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integer(1), Mul(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Q', commutative=True))), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True))))))"], [["differentiate", 3, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Q', commutative=True))), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["times", 4, "Mul(Pow(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True)))))"], "Equality(Mul(Pow(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True)))), Derivative(Integer(1), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\hat{x}')(Symbol('F_N', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True)))), Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Q', commutative=True))), exp(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Q', commutative=True))))), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(n)} = \\log{(n)} and \\mu{(n)} = \\log{(n)}, then obtain \\mu{(n)} - 1 = \\log{(n)} - 1", "derivation": "\\operatorname{v_{2}}{(n)} = \\log{(n)} and \\operatorname{v_{2}}{(n)} - 1 = \\log{(n)} - 1 and \\mu{(n)} = \\log{(n)} and \\operatorname{v_{2}}{(n)} = \\mu{(n)} and \\mu{(n)} - 1 = \\log{(n)} - 1", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_2')(Symbol('n', commutative=True)), Integer(-1)), Add(log(Symbol('n', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('v_2')(Symbol('n', commutative=True)), Function('\\\\mu')(Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('\\\\mu')(Symbol('n', commutative=True)), Integer(-1)), Add(log(Symbol('n', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given L{(x^\\prime)} = e^{x^\\prime} and \\hat{H}_l{(x^\\prime)} = (x^\\prime)^{2} L{(x^\\prime)} e^{x^\\prime} - L^{2}{(x^\\prime)}, then obtain ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)}) \\hat{H}_l{(x^\\prime)} = ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)})^{2}", "derivation": "L{(x^\\prime)} = e^{x^\\prime} and x^\\prime L{(x^\\prime)} = x^\\prime e^{x^\\prime} and (x^\\prime)^{2} L^{2}{(x^\\prime)} = (x^\\prime)^{2} L{(x^\\prime)} e^{x^\\prime} and (x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)} = (x^\\prime)^{2} L{(x^\\prime)} e^{x^\\prime} - L^{2}{(x^\\prime)} and \\hat{H}_l{(x^\\prime)} = (x^\\prime)^{2} L{(x^\\prime)} e^{x^\\prime} - L^{2}{(x^\\prime)} and ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)}) \\hat{H}_l{(x^\\prime)} = ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)}) ((x^\\prime)^{2} L{(x^\\prime)} e^{x^\\prime} - L^{2}{(x^\\prime)}) and ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)}) \\hat{H}_l{(x^\\prime)} = ((x^\\prime)^{2} L^{2}{(x^\\prime)} - L^{2}{(x^\\prime)})^{2}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Function('L')(Symbol('x^\\\\prime', commutative=True))), Mul(Symbol('x^\\\\prime', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "Mul(Symbol('x^\\\\prime', commutative=True), Function('L')(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('L')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 3, "Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))"], "Equality(Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))), Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('L')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('x^\\\\prime', commutative=True)), Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('L')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))))"], [["times", 5, "Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))))"], "Equality(Mul(Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))), Function('\\\\hat{H}_l')(Symbol('x^\\\\prime', commutative=True))), Mul(Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))), Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Function('L')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))), Function('\\\\hat{H}_l')(Symbol('x^\\\\prime', commutative=True))), Pow(Add(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Function('L')(Symbol('x^\\\\prime', commutative=True)), Integer(2)))), Integer(2)))"]]}, {"prompt": "Given \\omega{(\\mathbf{H})} = \\cos{(\\sin{(\\mathbf{H})})} and \\mathbf{p}{(\\mathbf{H})} = \\mathbf{H}, then obtain \\mathbf{p}{(\\mathbf{H})} - \\cos^{\\mathbf{H}}{(\\sin{(\\mathbf{H})})} = \\mathbf{H} - \\cos^{\\mathbf{H}}{(\\sin{(\\mathbf{H})})}", "derivation": "\\omega{(\\mathbf{H})} = \\cos{(\\sin{(\\mathbf{H})})} and \\mathbf{p}{(\\mathbf{H})} = \\mathbf{H} and \\mathbf{p}{(\\mathbf{H})} - \\omega^{\\mathbf{H}}{(\\mathbf{H})} = \\mathbf{H} - \\omega^{\\mathbf{H}}{(\\mathbf{H})} and \\mathbf{p}{(\\mathbf{H})} - \\cos^{\\mathbf{H}}{(\\sin{(\\mathbf{H})})} = \\mathbf{H} - \\cos^{\\mathbf{H}}{(\\sin{(\\mathbf{H})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{H}', commutative=True)), cos(sin(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["minus", 2, "Pow(Function('\\\\omega')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(cos(sin(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Pow(cos(sin(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\mu{(\\mathbf{p})} = e^{\\mathbf{p}}, then derive 0 = e^{\\mathbf{p}} - \\frac{d}{d \\mathbf{p}} \\mu{(\\mathbf{p})}, then obtain 0 = (\\mu{(\\mathbf{p})} - \\frac{d}{d \\mathbf{p}} \\mu{(\\mathbf{p})}) \\frac{d}{d \\mathbf{p}} (- \\mu{(\\mathbf{p})} + e^{\\mathbf{p}})", "derivation": "\\mu{(\\mathbf{p})} = e^{\\mathbf{p}} and 0 = - \\mu{(\\mathbf{p})} + e^{\\mathbf{p}} and \\frac{d}{d \\mathbf{p}} 0 = \\frac{d}{d \\mathbf{p}} (- \\mu{(\\mathbf{p})} + e^{\\mathbf{p}}) and 0 = e^{\\mathbf{p}} - \\frac{d}{d \\mathbf{p}} \\mu{(\\mathbf{p})} and 0 = \\mu{(\\mathbf{p})} - \\frac{d}{d \\mathbf{p}} \\mu{(\\mathbf{p})} and 0 = (\\mu{(\\mathbf{p})} - \\frac{d}{d \\mathbf{p}} \\mu{(\\mathbf{p})}) \\frac{d}{d \\mathbf{p}} (- \\mu{(\\mathbf{p})} + e^{\\mathbf{p}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["minus", 1, "Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(exp(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))))"], [["times", 5, "Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))"], "Equality(Integer(0), Mul(Add(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))), Derivative(Add(Mul(Integer(-1), Function('\\\\mu')(Symbol('\\\\mathbf{p}', commutative=True))), exp(Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(z)} = \\log{(z)}, then derive \\frac{d}{d z} \\operatorname{n_{1}}{(z)} = \\frac{1}{z}, then obtain (- \\log{(z)} + \\frac{d}{d z} \\log{(z)}) \\frac{d}{d z} \\log{(z)} = \\frac{- \\log{(z)} + \\frac{d}{d z} \\log{(z)}}{z}", "derivation": "\\operatorname{n_{1}}{(z)} = \\log{(z)} and \\frac{d}{d z} \\operatorname{n_{1}}{(z)} = \\frac{d}{d z} \\log{(z)} and \\frac{d}{d z} \\operatorname{n_{1}}{(z)} = \\frac{1}{z} and \\frac{d}{d z} \\log{(z)} = \\frac{1}{z} and (- \\log{(z)} + \\frac{d}{d z} \\operatorname{n_{1}}{(z)}) \\frac{d}{d z} \\log{(z)} = \\frac{- \\log{(z)} + \\frac{d}{d z} \\operatorname{n_{1}}{(z)}}{z} and (- \\log{(z)} + \\frac{d}{d z} \\log{(z)}) \\frac{d}{d z} \\log{(z)} = \\frac{- \\log{(z)} + \\frac{d}{d z} \\log{(z)}}{z}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Pow(Symbol('z', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Pow(Symbol('z', commutative=True), Integer(-1)))"], [["times", 4, "Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Derivative(Function('n_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], "Equality(Mul(Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Derivative(Function('n_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Derivative(Function('n_1')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Add(Mul(Integer(-1), log(Symbol('z', commutative=True))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\delta{(L)} = \\cos{(L)}, then obtain \\frac{\\delta{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\delta{(L)}}{t_{2}}} = \\frac{\\cos{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\delta{(L)}}{t_{2}}}", "derivation": "\\delta{(L)} = \\cos{(L)} and \\frac{\\delta{(L)}}{t_{2}} = \\frac{\\cos{(L)}}{t_{2}} and \\frac{\\partial}{\\partial L} \\frac{\\delta{(L)}}{t_{2}} = \\frac{\\partial}{\\partial L} \\frac{\\cos{(L)}}{t_{2}} and \\frac{\\delta{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\cos{(L)}}{t_{2}}} = \\frac{\\cos{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\cos{(L)}}{t_{2}}} and \\frac{\\delta{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\delta{(L)}}{t_{2}}} = \\frac{\\cos{(L)}}{t_{2} \\frac{\\partial}{\\partial L} \\frac{\\delta{(L)}}{t_{2}}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\delta')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["divide", 1, "Symbol('t_2', commutative=True)"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Function('\\\\delta')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(B)} = \\cos{(\\cos{(B)})}, then derive \\lambda + \\operatorname{L_{\\varepsilon}}{(B)} = Z + \\cos{(\\cos{(B)})}, then obtain \\frac{\\partial}{\\partial Z} (\\lambda + \\operatorname{L_{\\varepsilon}}{(B)}) = \\frac{\\partial}{\\partial Z} (Z + \\cos{(\\cos{(B)})})", "derivation": "\\operatorname{L_{\\varepsilon}}{(B)} = \\cos{(\\cos{(B)})} and \\int \\operatorname{L_{\\varepsilon}}{(B)} dB = \\int \\cos{(\\cos{(B)})} dB and \\frac{d}{d B} \\int \\operatorname{L_{\\varepsilon}}{(B)} dB = \\frac{d}{d B} \\int \\cos{(\\cos{(B)})} dB and \\frac{d^{2}}{d B^{2}} \\int \\operatorname{L_{\\varepsilon}}{(B)} dB = \\frac{d^{2}}{d B^{2}} \\int \\cos{(\\cos{(B)})} dB and \\int \\frac{d^{2}}{d B^{2}} \\int \\operatorname{L_{\\varepsilon}}{(B)} dB dB = \\int \\frac{d^{2}}{d B^{2}} \\int \\cos{(\\cos{(B)})} dB dB and \\lambda + \\operatorname{L_{\\varepsilon}}{(B)} = Z + \\cos{(\\cos{(B)})} and \\frac{\\partial}{\\partial Z} (\\lambda + \\operatorname{L_{\\varepsilon}}{(B)}) = \\frac{\\partial}{\\partial Z} (Z + \\cos{(\\cos{(B)})})", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), cos(cos(Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Integral(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Integral(cos(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(Integral(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(2))), Derivative(Integral(cos(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(2))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Integral(cos(cos(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(2))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True))), Add(Symbol('Z', commutative=True), cos(cos(Symbol('B', commutative=True)))))"], [["differentiate", 6, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\lambda', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('B', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), cos(cos(Symbol('B', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(M)} = \\log{(\\sin{(M)})}, then obtain \\frac{4 \\Omega^{2}{(M)}}{(- M + 2 \\Omega{(M)})^{2}} = \\frac{(\\Omega{(M)} + \\log{(\\sin{(M)})})^{2}}{(- M + 2 \\Omega{(M)})^{2}}", "derivation": "\\Omega{(M)} = \\log{(\\sin{(M)})} and 2 \\Omega{(M)} = \\Omega{(M)} + \\log{(\\sin{(M)})} and - M + 2 \\Omega{(M)} = - M + \\Omega{(M)} + \\log{(\\sin{(M)})} and \\frac{2 \\Omega{(M)}}{- M + \\Omega{(M)} + \\log{(\\sin{(M)})}} = \\frac{\\Omega{(M)} + \\log{(\\sin{(M)})}}{- M + \\Omega{(M)} + \\log{(\\sin{(M)})}} and \\frac{2 \\Omega{(M)}}{- M + 2 \\Omega{(M)}} = \\frac{\\Omega{(M)} + \\log{(\\sin{(M)})}}{- M + 2 \\Omega{(M)}} and \\frac{4 \\Omega^{2}{(M)}}{(- M + 2 \\Omega{(M)})^{2}} = \\frac{(\\Omega{(M)} + \\log{(\\sin{(M)})})^{2}}{(- M + 2 \\Omega{(M)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True))))"], [["add", 1, "Function('\\\\Omega')(Symbol('M', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True))), Add(Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))))"], [["minus", 2, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True)))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))), Integer(-1)), Function('\\\\Omega')(Symbol('M', commutative=True))), Mul(Add(Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True)))), Integer(-1)), Function('\\\\Omega')(Symbol('M', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True)))), Integer(-1)), Add(Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True))))))"], [["power", 5, 2], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True)))), Integer(-2)), Pow(Function('\\\\Omega')(Symbol('M', commutative=True)), Integer(2))), Mul(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Mul(Integer(2), Function('\\\\Omega')(Symbol('M', commutative=True)))), Integer(-2)), Pow(Add(Function('\\\\Omega')(Symbol('M', commutative=True)), log(sin(Symbol('M', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{P}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\operatorname{F_{x}}{(\\sigma_x)} = \\mathbf{P}^{\\sigma_x}{(\\sigma_x)}, then obtain \\frac{d}{d \\sigma_x} \\operatorname{F_{x}}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}^{\\sigma_x}", "derivation": "\\mathbf{P}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\mathbf{P}^{\\sigma_x}{(\\sigma_x)} = \\log{(\\sigma_x)}^{\\sigma_x} and \\frac{d}{d \\sigma_x} \\mathbf{P}^{\\sigma_x}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}^{\\sigma_x} and \\operatorname{F_{x}}{(\\sigma_x)} = \\mathbf{P}^{\\sigma_x}{(\\sigma_x)} and \\frac{d}{d \\sigma_x} \\operatorname{F_{x}}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(log(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\sigma_x', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('F_x')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(E_{x})} = e^{E_{x}}, then obtain (\\eta^{\\prime}^{2}{(E_{x})})^{E_{x}} = (e^{2 E_{x}})^{E_{x}}", "derivation": "\\eta^{\\prime}{(E_{x})} = e^{E_{x}} and \\eta^{\\prime}^{2}{(E_{x})} = \\eta^{\\prime}{(E_{x})} e^{E_{x}} and \\eta^{\\prime}{(E_{x})} e^{E_{x}} = e^{2 E_{x}} and \\eta^{\\prime}^{2}{(E_{x})} = e^{2 E_{x}} and (\\eta^{\\prime}^{2}{(E_{x})})^{E_{x}} = (e^{2 E_{x}})^{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], [["times", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True))"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True))))"], [["times", 1, "exp(Symbol('E_x', commutative=True))"], "Equality(Mul(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True))), exp(Mul(Integer(2), Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('E_x', commutative=True))))"], [["power", 4, "Symbol('E_x', commutative=True)"], "Equality(Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('E_x', commutative=True)), Integer(2)), Symbol('E_x', commutative=True)), Pow(exp(Mul(Integer(2), Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(t,M_{E})} = M_{E} - t, then obtain \\frac{\\partial}{\\partial t} - \\frac{\\eta^{\\prime}{(t,M_{E})}}{t} = \\frac{\\partial}{\\partial t} \\frac{- M_{E} + t}{t}", "derivation": "\\eta^{\\prime}{(t,M_{E})} = M_{E} - t and - \\frac{\\eta^{\\prime}{(t,M_{E})}}{t} = - \\frac{M_{E} - t}{t} and \\frac{\\partial}{\\partial t} - \\frac{\\eta^{\\prime}{(t,M_{E})}}{t} = \\frac{\\partial}{\\partial t} - \\frac{M_{E} - t}{t} and \\frac{\\partial}{\\partial t} - \\frac{M_{E} - t}{t} = \\frac{\\partial}{\\partial t} \\frac{- M_{E} + t}{t} and \\frac{\\partial}{\\partial t} - \\frac{\\eta^{\\prime}{(t,M_{E})}}{t} = \\frac{\\partial}{\\partial t} \\frac{- M_{E} + t}{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('t', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True), Symbol('M_E', commutative=True))), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\eta^{\\\\prime}')(Symbol('t', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(n_{2},\\hat{X})} = \\frac{\\hat{X}}{n_{2}}, then obtain \\frac{\\partial}{\\partial \\hat{X}} \\iint t{(n_{2},\\hat{X})} dn_{2} dn_{2} = \\frac{\\partial}{\\partial \\hat{X}} \\iint \\frac{\\hat{X}}{n_{2}} dn_{2} dn_{2}", "derivation": "t{(n_{2},\\hat{X})} = \\frac{\\hat{X}}{n_{2}} and \\int t{(n_{2},\\hat{X})} dn_{2} = \\int \\frac{\\hat{X}}{n_{2}} dn_{2} and \\iint t{(n_{2},\\hat{X})} dn_{2} dn_{2} = \\iint \\frac{\\hat{X}}{n_{2}} dn_{2} dn_{2} and \\frac{\\partial}{\\partial \\hat{X}} \\iint t{(n_{2},\\hat{X})} dn_{2} dn_{2} = \\frac{\\partial}{\\partial \\hat{X}} \\iint \\frac{\\hat{X}}{n_{2}} dn_{2} dn_{2}", "srepr_derivation": [["renaming_premise", "Equality(Function('t')(Symbol('n_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('t')(Symbol('n_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('n_2', commutative=True))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('t')(Symbol('n_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Integral(Function('t')(Symbol('n_2', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('n_2', commutative=True), Integer(-1))), Tuple(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(t)} = \\cos{(\\sin{(t)})} and \\omega{(t)} = \\sin{(t)}, then obtain P_{g} (\\sin{(t)} \\cos{(\\omega{(t)})})^{t} = P_{g} (\\sin{(t)} \\cos{(\\sin{(t)})})^{t}", "derivation": "\\mathbf{g}{(t)} = \\cos{(\\sin{(t)})} and \\mathbf{g}{(t)} \\sin{(t)} = \\sin{(t)} \\cos{(\\sin{(t)})} and \\omega{(t)} = \\sin{(t)} and (\\mathbf{g}{(t)} \\sin{(t)})^{t} = (\\sin{(t)} \\cos{(\\sin{(t)})})^{t} and P_{g} (\\mathbf{g}{(t)} \\sin{(t)})^{t} = P_{g} (\\sin{(t)} \\cos{(\\sin{(t)})})^{t} and \\mathbf{g}{(t)} = \\cos{(\\omega{(t)})} and P_{g} (\\sin{(t)} \\cos{(\\omega{(t)})})^{t} = P_{g} (\\sin{(t)} \\cos{(\\sin{(t)})})^{t}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), cos(sin(Symbol('t', commutative=True))))"], [["times", 1, "sin(Symbol('t', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Mul(sin(Symbol('t', commutative=True)), cos(sin(Symbol('t', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Mul(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Mul(sin(Symbol('t', commutative=True)), cos(sin(Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["times", 4, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Pow(Mul(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True))), Symbol('t', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Mul(sin(Symbol('t', commutative=True)), cos(sin(Symbol('t', commutative=True)))), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{g}')(Symbol('t', commutative=True)), cos(Function('\\\\omega')(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Symbol('P_g', commutative=True), Pow(Mul(sin(Symbol('t', commutative=True)), cos(Function('\\\\omega')(Symbol('t', commutative=True)))), Symbol('t', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Mul(sin(Symbol('t', commutative=True)), cos(sin(Symbol('t', commutative=True)))), Symbol('t', commutative=True))))"]]}, {"prompt": "Given U{(\\nabla)} = \\cos{(\\nabla)}, then obtain \\cos{(U{(\\nabla)})} \\cos^{\\nabla}{(U{(\\nabla)})} = \\cos{(U{(\\nabla)})} \\cos^{\\nabla}{(\\cos{(\\nabla)})}", "derivation": "U{(\\nabla)} = \\cos{(\\nabla)} and \\cos{(U{(\\nabla)})} = \\cos{(\\cos{(\\nabla)})} and \\cos^{\\nabla}{(U{(\\nabla)})} = \\cos^{\\nabla}{(\\cos{(\\nabla)})} and \\cos{(U{(\\nabla)})} \\cos^{\\nabla}{(U{(\\nabla)})} = \\cos{(U{(\\nabla)})} \\cos^{\\nabla}{(\\cos{(\\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["cos", 1], "Equality(cos(Function('U')(Symbol('\\\\nabla', commutative=True))), cos(cos(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(cos(Function('U')(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)), Pow(cos(cos(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True)))"], [["times", 3, "cos(Function('U')(Symbol('\\\\nabla', commutative=True)))"], "Equality(Mul(cos(Function('U')(Symbol('\\\\nabla', commutative=True))), Pow(cos(Function('U')(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))), Mul(cos(Function('U')(Symbol('\\\\nabla', commutative=True))), Pow(cos(cos(Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(g^{\\prime}_{\\varepsilon},H)} = H^{g^{\\prime}_{\\varepsilon}} and \\dot{z}{(g^{\\prime}_{\\varepsilon},H)} = H^{g^{\\prime}_{\\varepsilon}}, then obtain \\operatorname{F_{g}}^{H}{(g^{\\prime}_{\\varepsilon},H)} = \\dot{z}^{H}{(g^{\\prime}_{\\varepsilon},H)}", "derivation": "\\operatorname{F_{g}}{(g^{\\prime}_{\\varepsilon},H)} = H^{g^{\\prime}_{\\varepsilon}} and \\dot{z}{(g^{\\prime}_{\\varepsilon},H)} = H^{g^{\\prime}_{\\varepsilon}} and \\operatorname{F_{g}}^{H}{(g^{\\prime}_{\\varepsilon},H)} = (H^{g^{\\prime}_{\\varepsilon}})^{H} and \\operatorname{F_{g}}^{H}{(g^{\\prime}_{\\varepsilon},H)} = \\dot{z}^{H}{(g^{\\prime}_{\\varepsilon},H)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Pow(Symbol('H', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('F_g')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\Psi,\\mathbf{A})} = \\Psi \\log{(\\mathbf{A})}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\sigma_{x}{(\\Psi,\\mathbf{A})} = \\frac{\\Psi}{\\mathbf{A}}, then obtain \\frac{1}{\\sigma_{x}{(\\Psi,\\mathbf{A})}} = \\frac{\\mathbf{A} \\frac{\\partial}{\\partial \\mathbf{A}} \\Psi \\log{(\\mathbf{A})}}{\\Psi \\sigma_{x}{(\\Psi,\\mathbf{A})}}", "derivation": "\\sigma_{x}{(\\Psi,\\mathbf{A})} = \\Psi \\log{(\\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\sigma_{x}{(\\Psi,\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\Psi \\log{(\\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} \\sigma_{x}{(\\Psi,\\mathbf{A})} = \\frac{\\Psi}{\\mathbf{A}} and \\frac{\\mathbf{A} \\frac{\\partial}{\\partial \\mathbf{A}} \\sigma_{x}{(\\Psi,\\mathbf{A})}}{\\Psi} = \\frac{\\mathbf{A} \\frac{\\partial}{\\partial \\mathbf{A}} \\Psi \\log{(\\mathbf{A})}}{\\Psi} and 1 = \\frac{\\mathbf{A} \\frac{\\partial}{\\partial \\mathbf{A}} \\Psi \\log{(\\mathbf{A})}}{\\Psi} and \\frac{1}{\\sigma_{x}{(\\Psi,\\mathbf{A})}} = \\frac{\\mathbf{A} \\frac{\\partial}{\\partial \\mathbf{A}} \\Psi \\log{(\\mathbf{A})}}{\\Psi \\sigma_{x}{(\\Psi,\\mathbf{A})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"], [["divide", 2, "Mul(Symbol('\\\\Psi', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True), Derivative(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True), Derivative(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(1), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True), Derivative(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["divide", 5, "Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\mathbf{A}', commutative=True), Pow(Function('\\\\sigma_x')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\varphi)} = \\sin{(\\varphi)} and i{(\\varphi)} = \\varphi (2 \\mathbf{P}{(\\varphi)} + 2 \\sin{(\\varphi)}), then obtain 3 \\varphi \\mathbf{P}{(\\varphi)} - 3 \\varphi \\sin{(\\varphi)} + i{(\\varphi)} = 3 \\varphi \\mathbf{P}{(\\varphi)} + \\varphi \\sin{(\\varphi)}", "derivation": "\\mathbf{P}{(\\varphi)} = \\sin{(\\varphi)} and \\mathbf{P}{(\\varphi)} + \\sin{(\\varphi)} = 2 \\sin{(\\varphi)} and i{(\\varphi)} = \\varphi (2 \\mathbf{P}{(\\varphi)} + 2 \\sin{(\\varphi)}) and i{(\\varphi)} = \\varphi (3 \\mathbf{P}{(\\varphi)} + \\sin{(\\varphi)}) and i{(\\varphi)} = 4 \\varphi \\mathbf{P}{(\\varphi)} and i{(\\varphi)} = 4 \\varphi \\sin{(\\varphi)} and 3 \\varphi \\mathbf{P}{(\\varphi)} + \\varphi \\sin{(\\varphi)} + i{(\\varphi)} = 3 \\varphi \\mathbf{P}{(\\varphi)} + 5 \\varphi \\sin{(\\varphi)} and 3 \\varphi \\mathbf{P}{(\\varphi)} - 3 \\varphi \\sin{(\\varphi)} + i{(\\varphi)} = 3 \\varphi \\mathbf{P}{(\\varphi)} + \\varphi \\sin{(\\varphi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('i')(Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(3), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), sin(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('i')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(4), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('i')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(4), Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], [["add", 6, "Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True))))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Function('i')(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(5), Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True)))))"], [["minus", 7, "Mul(Integer(4), Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Integer(3), Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True))), Function('i')(Symbol('\\\\varphi', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), sin(Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\tilde{g})} = \\cos{(e^{\\tilde{g}})}, then obtain \\frac{\\frac{d}{d \\tilde{g}} \\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}^{2}} = - \\frac{e^{\\tilde{g}} \\sin{(e^{\\tilde{g}})}}{\\tilde{g}} - \\frac{\\cos{(e^{\\tilde{g}})}}{\\tilde{g}^{2}}", "derivation": "\\operatorname{n_{2}}{(\\tilde{g})} = \\cos{(e^{\\tilde{g}})} and \\frac{\\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}} = \\frac{\\cos{(e^{\\tilde{g}})}}{\\tilde{g}} and \\frac{d}{d \\tilde{g}} \\frac{\\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}} = \\frac{d}{d \\tilde{g}} \\frac{\\cos{(e^{\\tilde{g}})}}{\\tilde{g}} and \\frac{\\frac{d}{d \\tilde{g}} \\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}} - \\frac{\\operatorname{n_{2}}{(\\tilde{g})}}{\\tilde{g}^{2}} = - \\frac{e^{\\tilde{g}} \\sin{(e^{\\tilde{g}})}}{\\tilde{g}} - \\frac{\\cos{(e^{\\tilde{g}})}}{\\tilde{g}^{2}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), cos(exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["divide", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('n_2')(Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\tilde{g}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Function('n_2')(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), cos(exp(Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Derivative(Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Function('n_2')(Symbol('\\\\tilde{g}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), exp(Symbol('\\\\tilde{g}', commutative=True)), sin(exp(Symbol('\\\\tilde{g}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), cos(exp(Symbol('\\\\tilde{g}', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\phi_{2}{(\\mathbf{H})} = \\mathbf{H}, then obtain \\frac{d}{d \\mathbf{H}} (\\mathbf{H} + \\cos{(\\mathbf{H})}) \\phi_{2}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\mathbf{H} (\\mathbf{H} + \\cos{(\\mathbf{H})})", "derivation": "\\Psi_{nl}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\phi_{2}{(\\mathbf{H})} = \\mathbf{H} and (\\mathbf{H} + \\Psi_{nl}{(\\mathbf{H})}) \\phi_{2}{(\\mathbf{H})} = \\mathbf{H} (\\mathbf{H} + \\Psi_{nl}{(\\mathbf{H})}) and (\\mathbf{H} + \\cos{(\\mathbf{H})}) \\phi_{2}{(\\mathbf{H})} = \\mathbf{H} (\\mathbf{H} + \\cos{(\\mathbf{H})}) and \\frac{d}{d \\mathbf{H}} (\\mathbf{H} + \\cos{(\\mathbf{H})}) \\phi_{2}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\mathbf{H} (\\mathbf{H} + \\cos{(\\mathbf{H})})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["times", 2, "Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\phi_2')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\phi_2')(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Function('\\\\phi_2')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{H}', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(x,A)} = A + x, then derive \\int \\mathbf{E}{(x,A)} dA = \\frac{A^{2}}{2} + A x + f_{E}, then obtain (\\frac{A^{2}}{2} + A x + f_{E}) a{(x,A)} = a{(x,A)} \\int (A + x) dA", "derivation": "\\mathbf{E}{(x,A)} = A + x and \\int \\mathbf{E}{(x,A)} dA = \\int (A + x) dA and \\int \\mathbf{E}{(x,A)} dA = \\frac{A^{2}}{2} + A x + f_{E} and \\frac{A^{2}}{2} + A x + f_{E} = \\int (A + x) dA and (\\frac{A^{2}}{2} + A x + f_{E}) a{(x,A)} = a{(x,A)} \\int (A + x) dA", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('A', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Mul(Symbol('A', commutative=True), Symbol('x', commutative=True)), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Mul(Symbol('A', commutative=True), Symbol('x', commutative=True)), Symbol('f_E', commutative=True)), Integral(Add(Symbol('A', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["times", 4, "Function('a')(Symbol('x', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Mul(Symbol('A', commutative=True), Symbol('x', commutative=True)), Symbol('f_E', commutative=True)), Function('a')(Symbol('x', commutative=True), Symbol('A', commutative=True))), Mul(Function('a')(Symbol('x', commutative=True), Symbol('A', commutative=True)), Integral(Add(Symbol('A', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\delta, then derive \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} - 1 = \\delta - 1, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\delta - 1 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} - 1", "derivation": "\\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\delta and - \\Psi_{\\lambda} + \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} = \\Psi_{\\lambda} \\delta - \\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (- \\Psi_{\\lambda} + \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})}) = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} (\\Psi_{\\lambda} \\delta - \\Psi_{\\lambda}) and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} - 1 = \\delta - 1 and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\delta - 1 = \\delta - 1 and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\delta - 1 = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\operatorname{r_{0}}{(\\delta,\\Psi_{\\lambda})} - 1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('r_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('r_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('r_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('r_0')(Symbol('\\\\delta', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mu)} = \\cos{(\\mu)}, then obtain \\frac{\\operatorname{F_{N}}^{4}{(\\mu)}}{\\mu + \\operatorname{F_{N}}{(\\mu)}} = \\frac{\\operatorname{F_{N}}^{2}{(\\mu)} \\cos^{2}{(\\mu)}}{\\mu + \\operatorname{F_{N}}{(\\mu)}}", "derivation": "\\operatorname{F_{N}}{(\\mu)} = \\cos{(\\mu)} and \\mu + \\operatorname{F_{N}}{(\\mu)} = \\mu + \\cos{(\\mu)} and \\operatorname{F_{N}}^{2}{(\\mu)} = \\operatorname{F_{N}}{(\\mu)} \\cos{(\\mu)} and - \\operatorname{F_{N}}^{2}{(\\mu)} = - \\operatorname{F_{N}}{(\\mu)} \\cos{(\\mu)} and \\operatorname{F_{N}}^{4}{(\\mu)} = \\operatorname{F_{N}}^{2}{(\\mu)} \\cos^{2}{(\\mu)} and \\frac{\\operatorname{F_{N}}^{4}{(\\mu)}}{\\mu + \\cos{(\\mu)}} = \\frac{\\operatorname{F_{N}}^{2}{(\\mu)} \\cos^{2}{(\\mu)}}{\\mu + \\cos{(\\mu)}} and \\frac{\\operatorname{F_{N}}^{4}{(\\mu)}}{\\mu + \\operatorname{F_{N}}{(\\mu)}} = \\frac{\\operatorname{F_{N}}^{2}{(\\mu)} \\cos^{2}{(\\mu)}}{\\mu + \\operatorname{F_{N}}{(\\mu)}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Function('F_N')(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Function('F_N')(Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Function('F_N')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(2))), Mul(Integer(-1), Function('F_N')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(4)), Mul(Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2))))"], [["divide", 5, "Add(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(4))), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), cos(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Function('F_N')(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(4))), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Function('F_N')(Symbol('\\\\mu', commutative=True))), Integer(-1)), Pow(Function('F_N')(Symbol('\\\\mu', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mu', commutative=True)), Integer(2))))"]]}, {"prompt": "Given f{(A_{2},i)} = A_{2} + i, then derive \\frac{\\partial}{\\partial i} f{(A_{2},i)} = 1, then obtain \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int \\frac{\\partial}{\\partial i} (A_{2} + i) dA_{2} = \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int 1 dA_{2}", "derivation": "f{(A_{2},i)} = A_{2} + i and \\frac{\\partial}{\\partial i} f{(A_{2},i)} = \\frac{\\partial}{\\partial i} (A_{2} + i) and \\frac{\\partial}{\\partial i} f{(A_{2},i)} = 1 and \\int \\frac{\\partial}{\\partial i} f{(A_{2},i)} dA_{2} = \\int 1 dA_{2} and \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int \\frac{\\partial}{\\partial i} f{(A_{2},i)} dA_{2} = \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int 1 dA_{2} and \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int \\frac{\\partial}{\\partial i} (A_{2} + i) dA_{2} = \\frac{\\partial}{\\partial i} (A_{2} + i) + \\int 1 dA_{2}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Derivative(Function('f')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_2', commutative=True))))"], [["add", 4, "Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Derivative(Function('f')(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True)))), Add(Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True)))), Add(Derivative(Add(Symbol('A_2', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(v_{1},A_{z})} = A_{z}^{v_{1}}, then obtain 1 = \\frac{- A_{z}^{v_{1}} + \\frac{A_{z}^{v_{1}}}{v_{1} \\varphi{(v_{1},A_{z})}}}{- A_{z}^{v_{1}} + \\frac{1}{v_{1}}}", "derivation": "\\varphi{(v_{1},A_{z})} = A_{z}^{v_{1}} and \\frac{1}{v_{1}} = \\frac{A_{z}^{v_{1}}}{v_{1} \\varphi{(v_{1},A_{z})}} and - A_{z}^{v_{1}} + \\frac{1}{v_{1}} = - A_{z}^{v_{1}} + \\frac{A_{z}^{v_{1}}}{v_{1} \\varphi{(v_{1},A_{z})}} and 1 = \\frac{- A_{z}^{v_{1}} + \\frac{A_{z}^{v_{1}}}{v_{1} \\varphi{(v_{1},A_{z})}}}{- A_{z}^{v_{1}} + \\frac{1}{v_{1}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)))"], [["divide", 1, "Mul(Symbol('v_1', commutative=True), Function('\\\\varphi')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)))"], "Equality(Pow(Symbol('v_1', commutative=True), Integer(-1)), Mul(Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))))"], [["minus", 2, "Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Integer(-1)))))"], [["divide", 3, "Add(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Pow(Symbol('v_1', commutative=True), Integer(-1))), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True))), Mul(Pow(Symbol('A_z', commutative=True), Symbol('v_1', commutative=True)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Pow(Function('\\\\varphi')(Symbol('v_1', commutative=True), Symbol('A_z', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given t{(\\rho)} = \\cos{(\\rho)}, then obtain 2 t{(\\rho)} - \\cos{(\\rho)} + \\frac{\\cos{(\\rho)}}{\\rho} = t{(\\rho)} + \\frac{t{(\\rho)}}{\\rho}", "derivation": "t{(\\rho)} = \\cos{(\\rho)} and \\frac{t{(\\rho)}}{\\rho} = \\frac{\\cos{(\\rho)}}{\\rho} and t{(\\rho)} - \\frac{t{(\\rho)}}{\\rho} = \\cos{(\\rho)} - \\frac{t{(\\rho)}}{\\rho} and t{(\\rho)} + \\frac{t{(\\rho)}}{\\rho} = t{(\\rho)} + \\frac{\\cos{(\\rho)}}{\\rho} and t{(\\rho)} - \\frac{\\cos{(\\rho)}}{\\rho} = \\cos{(\\rho)} - \\frac{\\cos{(\\rho)}}{\\rho} and 2 t{(\\rho)} - \\cos{(\\rho)} + \\frac{\\cos{(\\rho)}}{\\rho} = t{(\\rho)} + \\frac{\\cos{(\\rho)}}{\\rho} and 2 t{(\\rho)} - \\cos{(\\rho)} + \\frac{\\cos{(\\rho)}}{\\rho} = t{(\\rho)} + \\frac{t{(\\rho)}}{\\rho}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["divide", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True)))"], "Equality(Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True)))), Add(cos(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True)))))"], [["add", 2, "Function('t')(Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True)))), Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))), Add(cos(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))))"], [["minus", 5, "Add(Mul(Integer(-1), Function('t')(Symbol('\\\\rho', commutative=True))), cos(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('t')(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))), Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(2), Function('t')(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))), Add(Function('t')(Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\omega)} = e^{\\sin{(\\omega)}} and l{(\\omega)} = 2 \\operatorname{F_{H}}{(\\omega)}, then obtain \\frac{d}{d \\omega} \\frac{(l{(\\omega)} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1} = \\frac{d}{d \\omega} \\frac{(\\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1}", "derivation": "\\operatorname{F_{H}}{(\\omega)} = e^{\\sin{(\\omega)}} and \\operatorname{F_{H}}{(\\omega)} + 1 = e^{\\sin{(\\omega)}} + 1 and 2 \\operatorname{F_{H}}{(\\omega)} + 2 = \\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2 and l{(\\omega)} = 2 \\operatorname{F_{H}}{(\\omega)} and l{(\\omega)} + 2 = \\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2 and (l{(\\omega)} + 2)^{2} = (\\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2)^{2} and \\frac{(l{(\\omega)} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1} = \\frac{(\\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1} and \\frac{d}{d \\omega} \\frac{(l{(\\omega)} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1} = \\frac{d}{d \\omega} \\frac{(\\operatorname{F_{H}}{(\\omega)} + e^{\\sin{(\\omega)}} + 2)^{2}}{e^{\\sin{(\\omega)}} + 1}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), Integer(1)), Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)))"], [["add", 2, "Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('\\\\omega', commutative=True))), Integer(2)), Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\omega', commutative=True)), Mul(Integer(2), Function('F_H')(Symbol('\\\\omega', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('l')(Symbol('\\\\omega', commutative=True)), Integer(2)), Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))), Integer(2)))"], [["power", 5, 2], "Equality(Pow(Add(Function('l')(Symbol('\\\\omega', commutative=True)), Integer(2)), Integer(2)), Pow(Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))), Integer(2)), Integer(2)))"], [["times", 6, "Pow(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)), Integer(-1))"], "Equality(Mul(Pow(Add(Function('l')(Symbol('\\\\omega', commutative=True)), Integer(2)), Integer(2)), Pow(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)), Integer(-1))), Mul(Pow(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)), Integer(-1)), Pow(Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))), Integer(2)), Integer(2))))"], [["differentiate", 7, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Function('l')(Symbol('\\\\omega', commutative=True)), Integer(2)), Integer(2)), Pow(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)), Integer(-1))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Add(exp(sin(Symbol('\\\\omega', commutative=True))), Integer(1)), Integer(-1)), Pow(Add(Function('F_H')(Symbol('\\\\omega', commutative=True)), exp(sin(Symbol('\\\\omega', commutative=True))), Integer(2)), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(F_{N},H)} = \\frac{\\cos{(F_{N})}}{H}, then obtain \\frac{\\partial}{\\partial F_{N}} (\\frac{\\cos{(F_{N})}}{H})^{F_{N}} + \\frac{\\partial}{\\partial F_{N}} u^{F_{N}}{(F_{N},H)} = 2 \\frac{\\partial}{\\partial F_{N}} (\\frac{\\cos{(F_{N})}}{H})^{F_{N}}", "derivation": "u{(F_{N},H)} = \\frac{\\cos{(F_{N})}}{H} and u^{F_{N}}{(F_{N},H)} = (\\frac{\\cos{(F_{N})}}{H})^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} u^{F_{N}}{(F_{N},H)} = \\frac{\\partial}{\\partial F_{N}} (\\frac{\\cos{(F_{N})}}{H})^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} (\\frac{\\cos{(F_{N})}}{H})^{F_{N}} + \\frac{\\partial}{\\partial F_{N}} u^{F_{N}}{(F_{N},H)} = 2 \\frac{\\partial}{\\partial F_{N}} (\\frac{\\cos{(F_{N})}}{H})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('F_N', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('u')(Symbol('F_N', commutative=True), Symbol('H', commutative=True)), Symbol('F_N', commutative=True)), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Pow(Function('u')(Symbol('F_N', commutative=True), Symbol('H', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Function('u')(Symbol('F_N', commutative=True), Symbol('H', commutative=True)), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), cos(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(v_{2},\\varphi^*)} = - \\sin{(\\varphi^* - v_{2})}, then obtain - \\frac{(\\int \\operatorname{f_{E}}{(v_{2},\\varphi^*)} d\\varphi^*) \\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^*}{v_{2}} = - \\frac{(\\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^*)^{2}}{v_{2}}", "derivation": "\\operatorname{f_{E}}{(v_{2},\\varphi^*)} = - \\sin{(\\varphi^* - v_{2})} and \\int \\operatorname{f_{E}}{(v_{2},\\varphi^*)} d\\varphi^* = \\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^* and (\\int \\operatorname{f_{E}}{(v_{2},\\varphi^*)} d\\varphi^*) \\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^* = (\\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^*)^{2} and - \\frac{(\\int \\operatorname{f_{E}}{(v_{2},\\varphi^*)} d\\varphi^*) \\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^*}{v_{2}} = - \\frac{(\\int - \\sin{(\\varphi^* - v_{2})} d\\varphi^*)^{2}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('v_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('v_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 2, "Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Integral(Function('f_E')(Symbol('v_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2)))"], [["divide", 3, "Mul(Integer(-1), Symbol('v_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Integral(Function('f_E')(Symbol('v_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('\\\\varphi^*', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{D}{(n_{2})} = \\log{(n_{2})}, then obtain (n_{2} \\mathbf{D}^{n_{2}}{(n_{2})})^{n_{2}} (n_{2} \\log{(n_{2})}^{n_{2}})^{- n_{2}} = 1", "derivation": "\\mathbf{D}{(n_{2})} = \\log{(n_{2})} and \\mathbf{D}^{n_{2}}{(n_{2})} = \\log{(n_{2})}^{n_{2}} and n_{2} \\mathbf{D}^{n_{2}}{(n_{2})} = n_{2} \\log{(n_{2})}^{n_{2}} and (n_{2} \\mathbf{D}^{n_{2}}{(n_{2})})^{n_{2}} = (n_{2} \\log{(n_{2})}^{n_{2}})^{n_{2}} and (n_{2} \\mathbf{D}^{n_{2}}{(n_{2})})^{n_{2}} (n_{2} \\log{(n_{2})}^{n_{2}})^{- n_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), log(Symbol('n_2', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(log(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["divide", 2, "Pow(Symbol('n_2', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('n_2', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Mul(Symbol('n_2', commutative=True), Pow(log(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Mul(Symbol('n_2', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Mul(Symbol('n_2', commutative=True), Pow(log(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)))"], [["divide", 4, "Pow(Mul(Symbol('n_2', commutative=True), Pow(log(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('n_2', commutative=True), Pow(Function('\\\\mathbf{D}')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Symbol('n_2', commutative=True)), Pow(Mul(Symbol('n_2', commutative=True), Pow(log(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))), Mul(Integer(-1), Symbol('n_2', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\ddot{x}{(L)} = e^{\\sin{(L)}} and v{(L)} = - \\sin{(L)}, then obtain \\ddot{x}{(L)} e^{v{(L)}} = 1", "derivation": "\\ddot{x}{(L)} = e^{\\sin{(L)}} and \\ddot{x}{(L)} e^{- \\sin{(L)}} = 1 and v{(L)} = - \\sin{(L)} and \\ddot{x}{(L)} e^{v{(L)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('L', commutative=True)), exp(sin(Symbol('L', commutative=True))))"], [["divide", 1, "exp(sin(Symbol('L', commutative=True)))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('L', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('L', commutative=True))))), Integer(1))"], ["renaming_premise", "Equality(Function('v')(Symbol('L', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('L', commutative=True)), exp(Function('v')(Symbol('L', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(l,v_{1})} = l v_{1}, then derive \\frac{l (-1 + \\frac{\\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{n}}{(l,v_{1})}}{l}) (- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l})^{l}}{- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l}} = 0, then obtain 0^{l} \\tilde{\\infty} l (-1 + \\frac{\\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{n}}{(l,v_{1})}}{l}) = 0", "derivation": "\\operatorname{E_{n}}{(l,v_{1})} = l v_{1} and \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l} = v_{1} and - v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l} = 0 and (- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l})^{l} = 0^{l} and \\frac{\\partial}{\\partial v_{1}} (- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l})^{l} = \\frac{d}{d v_{1}} 0^{l} and \\frac{l (-1 + \\frac{\\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{n}}{(l,v_{1})}}{l}) (- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l})^{l}}{- v_{1} + \\frac{\\operatorname{E_{n}}{(l,v_{1})}}{l}} = 0 and 0^{l} \\tilde{\\infty} l (-1 + \\frac{\\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{n}}{(l,v_{1})}}{l}) = 0", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))"], [["divide", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True))), Symbol('v_1', commutative=True))"], [["minus", 2, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))), Symbol('l', commutative=True)), Pow(Integer(0), Symbol('l', commutative=True)))"], [["differentiate", 4, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))), Symbol('l', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('l', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('l', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Derivative(Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))), Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)))), Symbol('l', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Integer(0), Symbol('l', commutative=True)), zoo, Symbol('l', commutative=True), Add(Integer(-1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Derivative(Function('E_n')(Symbol('l', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given \\phi{(y,\\eta^{\\prime})} = \\eta^{\\prime} + y, then obtain - \\eta^{\\prime} + y + \\phi{(y,\\eta^{\\prime})} = 2 y", "derivation": "\\phi{(y,\\eta^{\\prime})} = \\eta^{\\prime} + y and \\eta^{\\prime} + y + \\phi{(y,\\eta^{\\prime})} = 2 \\eta^{\\prime} + 2 y and 2 \\phi{(y,\\eta^{\\prime})} = 2 \\eta^{\\prime} + 2 y and - 2 \\eta^{\\prime} + 2 \\phi{(y,\\eta^{\\prime})} = 2 y and \\eta^{\\prime} + y + \\phi{(y,\\eta^{\\prime})} = 2 \\phi{(y,\\eta^{\\prime})} and - \\eta^{\\prime} + y + \\phi{(y,\\eta^{\\prime})} = 2 y", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y', commutative=True), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('y', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Integer(2), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y', commutative=True), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('y', commutative=True), Function('\\\\phi')(Symbol('y', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Integer(2), Symbol('y', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\lambda,s)} = s + \\cos{(\\lambda)}, then obtain \\frac{\\partial}{\\partial s} \\frac{\\mathbf{J}_P^{2}{(\\lambda,s)}}{\\lambda^{2}} = \\frac{\\partial}{\\partial s} \\frac{(s + \\cos{(\\lambda)}) \\mathbf{J}_P{(\\lambda,s)}}{\\lambda^{2}}", "derivation": "\\mathbf{J}_P{(\\lambda,s)} = s + \\cos{(\\lambda)} and \\frac{\\mathbf{J}_P{(\\lambda,s)}}{\\lambda} = \\frac{s + \\cos{(\\lambda)}}{\\lambda} and \\frac{\\mathbf{J}_P^{2}{(\\lambda,s)}}{\\lambda^{2}} = \\frac{(s + \\cos{(\\lambda)}) \\mathbf{J}_P{(\\lambda,s)}}{\\lambda^{2}} and \\frac{\\partial}{\\partial s} \\frac{\\mathbf{J}_P^{2}{(\\lambda,s)}}{\\lambda^{2}} = \\frac{\\partial}{\\partial s} \\frac{(s + \\cos{(\\lambda)}) \\mathbf{J}_P{(\\lambda,s)}}{\\lambda^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True)), Add(Symbol('s', commutative=True), cos(Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), cos(Symbol('\\\\lambda', commutative=True)))))"], [["times", 2, "Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Add(Symbol('s', commutative=True), cos(Symbol('\\\\lambda', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True))))"], [["differentiate", 3, "Symbol('s', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True)), Integer(2))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Add(Symbol('s', commutative=True), cos(Symbol('\\\\lambda', commutative=True))), Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = e^{C_{d}}, then derive \\frac{d^{2}}{d C_{d}^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = e^{C_{d}}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{d}}{(C_{d})} = (\\frac{d^{2}}{d C_{d}^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})})^{C_{d}}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = e^{C_{d}} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{d}}{(C_{d})} = (e^{C_{d}})^{C_{d}} and \\frac{d}{d C_{d}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = \\frac{d}{d C_{d}} e^{C_{d}} and \\frac{d^{2}}{d C_{d}^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = \\frac{d^{2}}{d C_{d}^{2}} e^{C_{d}} and \\frac{d^{2}}{d C_{d}^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})} = e^{C_{d}} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{C_{d}}{(C_{d})} = (\\frac{d^{2}}{d C_{d}^{2}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(C_{d})})^{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(exp(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], [["differentiate", 1, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))), Derivative(exp(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))), exp(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(2))), Symbol('C_d', commutative=True)))"]]}, {"prompt": "Given k{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\theta_{2}{(f^{\\prime})} = (\\int \\sin{(f^{\\prime})} df^{\\prime})^{f^{\\prime}}, then derive \\theta_{2}{(f^{\\prime})} = (\\Psi^{\\dagger} - \\cos{(f^{\\prime})})^{f^{\\prime}}, then obtain (\\int k{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} = (\\Psi^{\\dagger} - \\cos{(f^{\\prime})})^{f^{\\prime}}", "derivation": "k{(f^{\\prime})} = \\sin{(f^{\\prime})} and \\int k{(f^{\\prime})} df^{\\prime} = \\int \\sin{(f^{\\prime})} df^{\\prime} and (\\int k{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} = (\\int \\sin{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} and \\theta_{2}{(f^{\\prime})} = (\\int \\sin{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} and \\theta_{2}{(f^{\\prime})} = (\\Psi^{\\dagger} - \\cos{(f^{\\prime})})^{f^{\\prime}} and (\\int k{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} = \\theta_{2}{(f^{\\prime})} and (\\int k{(f^{\\prime})} df^{\\prime})^{f^{\\prime}} = (\\Psi^{\\dagger} - \\cos{(f^{\\prime})})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('f^{\\\\prime}', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Integral(Function('k')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True)), Pow(Integral(sin(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Integral(Function('k')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\theta_2')(Symbol('f^{\\\\prime}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Integral(Function('k')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(L)} = \\log{(L)} and \\phi_{1}{(L)} = \\operatorname{C_{d}}{(L)} \\log{(L)} + \\log{(L)}, then obtain \\phi_{1}{(L)} = \\operatorname{C_{d}}{(L)} \\log{(L)} + \\operatorname{C_{d}}{(L)}", "derivation": "\\operatorname{C_{d}}{(L)} = \\log{(L)} and \\operatorname{C_{d}}{(L)} \\log{(L)} + \\operatorname{C_{d}}{(L)} = \\operatorname{C_{d}}{(L)} \\log{(L)} + \\log{(L)} and \\phi_{1}{(L)} = \\operatorname{C_{d}}{(L)} \\log{(L)} + \\log{(L)} and \\phi_{1}{(L)} = \\operatorname{C_{d}}{(L)} \\log{(L)} + \\operatorname{C_{d}}{(L)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["add", 1, "Mul(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Function('C_d')(Symbol('L', commutative=True))), Add(Mul(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), log(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('L', commutative=True)), Add(Mul(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), log(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\phi_1')(Symbol('L', commutative=True)), Add(Mul(Function('C_d')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True))), Function('C_d')(Symbol('L', commutative=True))))"]]}, {"prompt": "Given l{(t_{1},u)} = \\log{(\\frac{u}{t_{1}})}, then obtain - u + 2 l{(t_{1},u)} - \\frac{u}{t_{1}} = - u + l{(t_{1},u)} + \\log{(\\frac{u}{t_{1}})} - \\frac{u}{t_{1}}", "derivation": "l{(t_{1},u)} = \\log{(\\frac{u}{t_{1}})} and - u + l{(t_{1},u)} = - u + \\log{(\\frac{u}{t_{1}})} and - u + l{(t_{1},u)} - \\frac{u}{t_{1}} = - u + \\log{(\\frac{u}{t_{1}})} - \\frac{u}{t_{1}} and - u + 2 l{(t_{1},u)} - \\frac{u}{t_{1}} = - u + l{(t_{1},u)} + \\log{(\\frac{u}{t_{1}})} - \\frac{u}{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["minus", 1, "Symbol('u', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True)))))"], [["minus", 2, "Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True)), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"], [["add", 3, "Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(2), Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('l')(Symbol('t_1', commutative=True), Symbol('u', commutative=True)), log(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(U)} = \\frac{d}{d U} \\sin{(U)}, then derive \\mathbf{M}{(U)} = \\cos{(U)}, then obtain \\cos{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})} = \\frac{d}{d U} \\sin{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})}", "derivation": "\\mathbf{M}{(U)} = \\frac{d}{d U} \\sin{(U)} and \\mathbf{M}{(U)} = \\cos{(U)} and \\mathbf{M}{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})} = \\frac{d}{d U} \\sin{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})} and \\cos{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})} = \\frac{d}{d U} \\sin{(U)} - \\frac{d^{2}}{d \\mathbf{f}^{2}} \\cos{(\\sin{(\\mathbf{f})})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["minus", 1, "Derivative(cos(sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2)))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))))), Add(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(cos(Symbol('U', commutative=True)), Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))))), Add(Derivative(sin(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(sin(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(2))))))"]]}, {"prompt": "Given y{(\\mathbf{J}_P)} = \\cos{(e^{\\mathbf{J}_P})}, then obtain e^{\\mathbf{J}_P} + \\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)} = - e^{\\mathbf{J}_P} \\sin{(e^{\\mathbf{J}_P})} + e^{\\mathbf{J}_P}", "derivation": "y{(\\mathbf{J}_P)} = \\cos{(e^{\\mathbf{J}_P})} and y{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} = e^{\\mathbf{J}_P} + \\cos{(e^{\\mathbf{J}_P})} and \\frac{d}{d \\mathbf{J}_P} (y{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P}) = \\frac{d}{d \\mathbf{J}_P} (e^{\\mathbf{J}_P} + \\cos{(e^{\\mathbf{J}_P})}) and e^{\\mathbf{J}_P} + \\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)} = - e^{\\mathbf{J}_P} \\sin{(e^{\\mathbf{J}_P})} + e^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 1, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given i{(\\theta_1)} = \\cos{(\\theta_1)}, then obtain i{(\\theta_1)} - \\sin{(i{(\\theta_1)})} = - \\sin{(i{(\\theta_1)})} + \\cos{(\\theta_1)}", "derivation": "i{(\\theta_1)} = \\cos{(\\theta_1)} and \\sin{(i{(\\theta_1)})} = \\sin{(\\cos{(\\theta_1)})} and i{(\\theta_1)} - \\sin{(\\cos{(\\theta_1)})} = - \\sin{(\\cos{(\\theta_1)})} + \\cos{(\\theta_1)} and i{(\\theta_1)} - \\sin{(i{(\\theta_1)})} = - \\sin{(i{(\\theta_1)})} + \\cos{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["sin", 1], "Equality(sin(Function('i')(Symbol('\\\\theta_1', commutative=True))), sin(cos(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "sin(cos(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Function('i')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\theta_1', commutative=True))))), Add(Mul(Integer(-1), sin(cos(Symbol('\\\\theta_1', commutative=True)))), cos(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('i')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), sin(Function('i')(Symbol('\\\\theta_1', commutative=True))))), Add(Mul(Integer(-1), sin(Function('i')(Symbol('\\\\theta_1', commutative=True)))), cos(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given s{(\\rho_b,\\Psi)} = \\Psi \\rho_b, then obtain \\rho_b + (\\Psi \\rho_b)^{\\rho_b} + s{(\\rho_b,\\Psi)} = \\Psi \\rho_b + \\rho_b + (\\Psi \\rho_b)^{\\rho_b}", "derivation": "s{(\\rho_b,\\Psi)} = \\Psi \\rho_b and s^{\\rho_b}{(\\rho_b,\\Psi)} = (\\Psi \\rho_b)^{\\rho_b} and s{(\\rho_b,\\Psi)} + s^{\\rho_b}{(\\rho_b,\\Psi)} = \\Psi \\rho_b + s^{\\rho_b}{(\\rho_b,\\Psi)} and \\rho_b + s{(\\rho_b,\\Psi)} + s^{\\rho_b}{(\\rho_b,\\Psi)} = \\Psi \\rho_b + \\rho_b + s^{\\rho_b}{(\\rho_b,\\Psi)} and \\rho_b + (\\Psi \\rho_b)^{\\rho_b} + s{(\\rho_b,\\Psi)} = \\Psi \\rho_b + \\rho_b + (\\Psi \\rho_b)^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["add", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True), Pow(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('\\\\rho_b', commutative=True), Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Psi', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True), Pow(Mul(Symbol('\\\\Psi', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given m{(\\theta_1,P_{e})} = P_{e} + \\theta_1 and \\bar{\\h}{(H)} = \\int \\sin{(H)} dH, then obtain (- P_{e} - \\theta_1 + m{(\\theta_1,P_{e})})^{P_{e}} - \\int \\sin{(H)} dH = 0^{P_{e}} - \\int \\sin{(H)} dH", "derivation": "m{(\\theta_1,P_{e})} = P_{e} + \\theta_1 and - P_{e} - \\theta_1 + m{(\\theta_1,P_{e})} = 0 and \\bar{\\h}{(H)} = \\int \\sin{(H)} dH and (- P_{e} - \\theta_1 + m{(\\theta_1,P_{e})})^{P_{e}} = 0^{P_{e}} and (- P_{e} - \\theta_1 + m{(\\theta_1,P_{e})})^{P_{e}} - \\bar{\\h}{(H)} = 0^{P_{e}} - \\bar{\\h}{(H)} and (- P_{e} - \\theta_1 + m{(\\theta_1,P_{e})})^{P_{e}} - \\int \\sin{(H)} dH = 0^{P_{e}} - \\int \\sin{(H)} dH", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Add(Symbol('P_e', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True))), Integer(0))"], ["get_premise", "Equality(Function('\\\\hbar')(Symbol('H', commutative=True)), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Pow(Integer(0), Symbol('P_e', commutative=True)))"], [["minus", 4, "Function('\\\\hbar')(Symbol('H', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('H', commutative=True)))), Add(Pow(Integer(0), Symbol('P_e', commutative=True)), Mul(Integer(-1), Function('\\\\hbar')(Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('m')(Symbol('\\\\theta_1', commutative=True), Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(Pow(Integer(0), Symbol('P_e', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = \\dot{\\mathbf{r}} + \\hat{H}_l, then derive \\int \\operatorname{f^{\\prime}}{(\\dot{\\mathbf{r}},\\hat{H}_l)} d\\hat{H}_l = F_{x} + \\dot{\\mathbf{r}} \\hat{H}_l + \\frac{\\hat{H}_l^{2}}{2}, then obtain \\int (\\dot{\\mathbf{r}} + \\hat{H}_l) d\\hat{H}_l = F_{x} + \\dot{\\mathbf{r}} \\hat{H}_l + \\frac{\\hat{H}_l^{2}}{2}", "derivation": "\\operatorname{f^{\\prime}}{(\\dot{\\mathbf{r}},\\hat{H}_l)} = \\dot{\\mathbf{r}} + \\hat{H}_l and \\int \\operatorname{f^{\\prime}}{(\\dot{\\mathbf{r}},\\hat{H}_l)} d\\hat{H}_l = \\int (\\dot{\\mathbf{r}} + \\hat{H}_l) d\\hat{H}_l and \\int \\operatorname{f^{\\prime}}{(\\dot{\\mathbf{r}},\\hat{H}_l)} d\\hat{H}_l = F_{x} + \\dot{\\mathbf{r}} \\hat{H}_l + \\frac{\\hat{H}_l^{2}}{2} and \\int (\\dot{\\mathbf{r}} + \\hat{H}_l) d\\hat{H}_l = F_{x} + \\dot{\\mathbf{r}} \\hat{H}_l + \\frac{\\hat{H}_l^{2}}{2}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^{\\\\prime}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('F_x', commutative=True), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(2)))))"]]}, {"prompt": "Given Q{(t)} = \\log{(t)}, then obtain Q{(t)} \\int Q{(t)} dt - \\log{(t)} = Q{(t)} \\int \\log{(t)} dt - \\log{(t)}", "derivation": "Q{(t)} = \\log{(t)} and \\int Q{(t)} dt = \\int \\log{(t)} dt and Q{(t)} \\int Q{(t)} dt = Q{(t)} \\int \\log{(t)} dt and Q{(t)} \\int Q{(t)} dt - \\log{(t)} = Q{(t)} \\int \\log{(t)} dt - \\log{(t)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 2, "Function('Q')(Symbol('t', commutative=True))"], "Equality(Mul(Function('Q')(Symbol('t', commutative=True)), Integral(Function('Q')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Function('Q')(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["minus", 3, "log(Symbol('t', commutative=True))"], "Equality(Add(Mul(Function('Q')(Symbol('t', commutative=True)), Integral(Function('Q')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Add(Mul(Function('Q')(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Integer(-1), log(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\phi_2)} = \\log{(\\phi_2)}, then derive V_{\\mathbf{B}} - \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)} = \\pi - 2 \\Psi_{\\lambda}{(\\phi_2)} + 2 \\log{(\\phi_2)}, then obtain V_{\\mathbf{B}} = \\pi", "derivation": "\\Psi_{\\lambda}{(\\phi_2)} = \\log{(\\phi_2)} and 0 = - \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)} and - \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)} = - 2 \\Psi_{\\lambda}{(\\phi_2)} + 2 \\log{(\\phi_2)} and \\frac{d}{d \\phi_2} (- \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)}) = \\frac{d}{d \\phi_2} (- 2 \\Psi_{\\lambda}{(\\phi_2)} + 2 \\log{(\\phi_2)}) and \\int \\frac{d}{d \\phi_2} (- \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)}) d\\phi_2 = \\int \\frac{d}{d \\phi_2} (- 2 \\Psi_{\\lambda}{(\\phi_2)} + 2 \\log{(\\phi_2)}) d\\phi_2 and V_{\\mathbf{B}} - \\Psi_{\\lambda}{(\\phi_2)} + \\log{(\\phi_2)} = \\pi - 2 \\Psi_{\\lambda}{(\\phi_2)} + 2 \\log{(\\phi_2)} and V_{\\mathbf{B}} = \\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True)), log(Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), log(Symbol('\\\\phi_2', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(2), log(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\pi', commutative=True))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(s,B)} = \\int B^{s} dB, then obtain - s \\iint (\\operatorname{v_{t}}^{s}{(s,B)})^{s} dB ds = - s \\iint ((\\int B^{s} dB)^{s})^{s} dB ds", "derivation": "\\operatorname{v_{t}}{(s,B)} = \\int B^{s} dB and \\operatorname{v_{t}}^{s}{(s,B)} = (\\int B^{s} dB)^{s} and (\\operatorname{v_{t}}^{s}{(s,B)})^{s} = ((\\int B^{s} dB)^{s})^{s} and \\int (\\operatorname{v_{t}}^{s}{(s,B)})^{s} dB = \\int ((\\int B^{s} dB)^{s})^{s} dB and \\iint (\\operatorname{v_{t}}^{s}{(s,B)})^{s} dB ds = \\iint ((\\int B^{s} dB)^{s})^{s} dB ds and - s \\iint (\\operatorname{v_{t}}^{s}{(s,B)})^{s} dB ds = - s \\iint ((\\int B^{s} dB)^{s})^{s} dB ds", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Pow(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('s', commutative=True)))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Pow(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Pow(Pow(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Pow(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Pow(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Pow(Pow(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Symbol('s', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('s', commutative=True), Integral(Pow(Pow(Function('v_t')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Integer(-1), Symbol('s', commutative=True), Integral(Pow(Pow(Integral(Pow(Symbol('B', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('s', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given V{(x)} = \\sin{(x)}, then obtain V{(x)} + \\int V{(x)} dx = \\sin{(x)} + \\int V{(x)} dx", "derivation": "V{(x)} = \\sin{(x)} and \\int V{(x)} dx = \\int \\sin{(x)} dx and V{(x)} + \\int \\sin{(x)} dx = \\sin{(x)} + \\int \\sin{(x)} dx and V{(x)} + \\int V{(x)} dx = \\sin{(x)} + \\int V{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('V')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["add", 1, "Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Add(Function('V')(Symbol('x', commutative=True)), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(sin(Symbol('x', commutative=True)), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('V')(Symbol('x', commutative=True)), Integral(Function('V')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(sin(Symbol('x', commutative=True)), Integral(Function('V')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given y{(t)} = \\int \\cos{(t)} dt, then derive y^{t}{(t)} = (r_{0} + \\sin{(t)})^{t}, then obtain r_{0} (r_{0} + \\sin{(t)})^{t} = r_{0} (\\int \\cos{(t)} dt)^{t}", "derivation": "y{(t)} = \\int \\cos{(t)} dt and y^{t}{(t)} = (\\int \\cos{(t)} dt)^{t} and y^{t}{(t)} = (r_{0} + \\sin{(t)})^{t} and (r_{0} + \\sin{(t)})^{t} = (\\int \\cos{(t)} dt)^{t} and r_{0} (r_{0} + \\sin{(t)})^{t} = r_{0} (\\int \\cos{(t)} dt)^{t}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('t', commutative=True)), Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('y')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('y')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["times", 4, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), Pow(Add(Symbol('r_0', commutative=True), sin(Symbol('t', commutative=True))), Symbol('t', commutative=True))), Mul(Symbol('r_0', commutative=True), Pow(Integral(cos(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(t_{1})} = \\cos{(t_{1})} and \\mathbf{B}{(\\psi,V_{\\mathbf{E}})} = - V_{\\mathbf{E}} + \\psi, then obtain \\mathbf{B}{(\\psi,V_{\\mathbf{E}})} - \\cos{(t_{1})} = - V_{\\mathbf{E}} + \\psi - \\cos{(t_{1})}", "derivation": "\\rho_{b}{(t_{1})} = \\cos{(t_{1})} and \\mathbf{B}{(\\psi,V_{\\mathbf{E}})} = - V_{\\mathbf{E}} + \\psi and \\mathbf{B}{(\\psi,V_{\\mathbf{E}})} - \\rho_{b}{(t_{1})} = - V_{\\mathbf{E}} + \\psi - \\rho_{b}{(t_{1})} and \\mathbf{B}{(\\psi,V_{\\mathbf{E}})} - \\cos{(t_{1})} = - V_{\\mathbf{E}} + \\psi - \\cos{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\psi', commutative=True)))"], [["minus", 2, "Function('\\\\rho_b')(Symbol('t_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\psi', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\psi', commutative=True), Mul(Integer(-1), cos(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(I)} = \\log{(I)}, then derive \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} = \\frac{1}{I}, then obtain (\\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} + \\int \\frac{1}{I} dI) \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} = (\\int \\frac{1}{I} dI + \\frac{1}{I}) \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)}", "derivation": "\\hat{H}_{\\lambda}{(I)} = \\log{(I)} and \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} = \\frac{d}{d I} \\log{(I)} and \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} = \\frac{1}{I} and \\frac{d}{d I} \\log{(I)} = \\frac{1}{I} and \\frac{d}{d I} \\log{(I)} + \\int \\frac{1}{I} dI = \\int \\frac{1}{I} dI + \\frac{1}{I} and \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} + \\int \\frac{1}{I} dI = \\int \\frac{1}{I} dI + \\frac{1}{I} and (\\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} + \\int \\frac{1}{I} dI) \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)} = (\\int \\frac{1}{I} dI + \\frac{1}{I}) \\frac{d}{d I} \\hat{H}_{\\lambda}{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Symbol('I', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Pow(Symbol('I', commutative=True), Integer(-1)))"], [["add", 4, "Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True)))"], "Equality(Add(Derivative(log(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True)))), Add(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True))), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True)))), Add(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True))), Pow(Symbol('I', commutative=True), Integer(-1))))"], [["times", 6, "Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True)))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Add(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True))), Pow(Symbol('I', commutative=True), Integer(-1))), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(f_{\\mathbf{p}},p)} = p^{f_{\\mathbf{p}}}, then obtain \\iint - f_{\\mathbf{p}} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint (- f_{\\mathbf{p}} + \\int p^{f_{\\mathbf{p}}} dp - \\int b{(f_{\\mathbf{p}},p)} dp) df_{\\mathbf{p}} df_{\\mathbf{p}}", "derivation": "b{(f_{\\mathbf{p}},p)} = p^{f_{\\mathbf{p}}} and \\int b{(f_{\\mathbf{p}},p)} dp = \\int p^{f_{\\mathbf{p}}} dp and - f_{\\mathbf{p}} = - f_{\\mathbf{p}} + \\int p^{f_{\\mathbf{p}}} dp - \\int b{(f_{\\mathbf{p}},p)} dp and \\int - f_{\\mathbf{p}} df_{\\mathbf{p}} = \\int (- f_{\\mathbf{p}} + \\int p^{f_{\\mathbf{p}}} dp - \\int b{(f_{\\mathbf{p}},p)} dp) df_{\\mathbf{p}} and \\iint - f_{\\mathbf{p}} df_{\\mathbf{p}} df_{\\mathbf{p}} = \\iint (- f_{\\mathbf{p}} + \\int p^{f_{\\mathbf{p}}} dp - \\int b{(f_{\\mathbf{p}},p)} dp) df_{\\mathbf{p}} df_{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["minus", 2, "Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["integrate", 4, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(h,\\mu)} = \\int (\\mu + h) d\\mu, then derive \\mathbb{I}{(h,\\mu)} = \\frac{\\mu^{2}}{2} + \\mu h + \\sigma_p, then obtain \\sin{(\\int (\\mu + h) d\\mu)} = \\sin{(\\frac{\\mu^{2}}{2} + \\mu h + \\sigma_p)}", "derivation": "\\mathbb{I}{(h,\\mu)} = \\int (\\mu + h) d\\mu and \\mathbb{I}{(h,\\mu)} = \\frac{\\mu^{2}}{2} + \\mu h + \\sigma_p and \\sin{(\\mathbb{I}{(h,\\mu)})} = \\sin{(\\frac{\\mu^{2}}{2} + \\mu h + \\sigma_p)} and \\sin{(\\int (\\mu + h) d\\mu)} = \\sin{(\\frac{\\mu^{2}}{2} + \\mu h + \\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Integral(Add(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["sin", 2], "Equality(sin(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mu', commutative=True))), sin(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(sin(Integral(Add(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True)))), sin(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2))), Mul(Symbol('\\\\mu', commutative=True), Symbol('h', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\Omega)} = \\sin{(\\Omega)} and \\operatorname{v_{y}}{(\\Omega)} = \\Omega, then obtain \\int \\frac{d}{d \\Omega} \\Omega \\Psi_{nl}{(\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} \\Omega \\sin{(\\Omega)} d\\Omega", "derivation": "\\Psi_{nl}{(\\Omega)} = \\sin{(\\Omega)} and \\operatorname{v_{y}}{(\\Omega)} = \\Omega and \\Psi_{nl}{(\\Omega)} \\operatorname{v_{y}}{(\\Omega)} = \\operatorname{v_{y}}{(\\Omega)} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\Psi_{nl}{(\\Omega)} \\operatorname{v_{y}}{(\\Omega)} = \\frac{d}{d \\Omega} \\operatorname{v_{y}}{(\\Omega)} \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\Omega \\Psi_{nl}{(\\Omega)} = \\frac{d}{d \\Omega} \\Omega \\sin{(\\Omega)} and \\int \\frac{d}{d \\Omega} \\Omega \\Psi_{nl}{(\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} \\Omega \\sin{(\\Omega)} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))"], [["times", 1, "Function('v_y')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\Omega', commutative=True)), Function('v_y')(Symbol('\\\\Omega', commutative=True))), Mul(Function('v_y')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\Omega', commutative=True)), Function('v_y')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Function('v_y')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), sin(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\Omega{(k)} = e^{e^{k}}, then derive \\frac{d^{2}}{d k^{2}} \\Omega{(k)} = (e^{k} + 1) e^{k} e^{e^{k}}, then obtain e^{2 k} e^{e^{k}} + e^{k} e^{e^{k}} = (e^{k} + 1) e^{k} e^{e^{k}}", "derivation": "\\Omega{(k)} = e^{e^{k}} and \\frac{d}{d k} \\Omega{(k)} = \\frac{d}{d k} e^{e^{k}} and \\frac{d^{2}}{d k^{2}} \\Omega{(k)} = \\frac{d^{2}}{d k^{2}} e^{e^{k}} and \\frac{d^{2}}{d k^{2}} \\Omega{(k)} = (e^{k} + 1) e^{k} e^{e^{k}} and \\frac{d^{2}}{d k^{2}} e^{e^{k}} = (e^{k} + 1) e^{k} e^{e^{k}} and \\frac{d^{2}}{d k^{2}} e^{e^{k}} = e^{2 k} e^{e^{k}} + e^{k} e^{e^{k}} and e^{2 k} e^{e^{k}} + e^{k} e^{e^{k}} = (e^{k} + 1) e^{k} e^{e^{k}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))), Derivative(exp(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Omega')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(2))), Mul(Add(exp(Symbol('k', commutative=True)), Integer(1)), exp(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))), Mul(Add(exp(Symbol('k', commutative=True)), Integer(1)), exp(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))))"], [["expand", 5], "Equality(Derivative(exp(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(2))), Add(Mul(exp(Mul(Integer(2), Symbol('k', commutative=True))), exp(exp(Symbol('k', commutative=True)))), Mul(exp(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(exp(Mul(Integer(2), Symbol('k', commutative=True))), exp(exp(Symbol('k', commutative=True)))), Mul(exp(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True))))), Mul(Add(exp(Symbol('k', commutative=True)), Integer(1)), exp(Symbol('k', commutative=True)), exp(exp(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\phi_{2}{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})}, then obtain \\frac{\\mu_0 + \\phi_{2}{(\\mu_0)} + \\sin{(\\cos{(\\mu_0)})}}{\\cos{(\\mu_0)}} = \\frac{\\mu_0 + 2 \\sin{(\\cos{(\\mu_0)})}}{\\cos{(\\mu_0)}}", "derivation": "\\phi_{2}{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})} and \\mu_0 + \\phi_{2}{(\\mu_0)} = \\mu_0 + \\sin{(\\cos{(\\mu_0)})} and \\mu_0 + \\phi_{2}{(\\mu_0)} + \\sin{(\\cos{(\\mu_0)})} = \\mu_0 + 2 \\sin{(\\cos{(\\mu_0)})} and \\frac{\\mu_0 + \\phi_{2}{(\\mu_0)} + \\sin{(\\cos{(\\mu_0)})}}{\\cos{(\\mu_0)}} = \\frac{\\mu_0 + 2 \\sin{(\\cos{(\\mu_0)})}}{\\cos{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mu_0', commutative=True)), sin(cos(Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), sin(cos(Symbol('\\\\mu_0', commutative=True)))))"], [["add", 2, "sin(cos(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mu_0', commutative=True)), sin(cos(Symbol('\\\\mu_0', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(2), sin(cos(Symbol('\\\\mu_0', commutative=True))))))"], [["divide", 3, "cos(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mu_0', commutative=True), Function('\\\\phi_2')(Symbol('\\\\mu_0', commutative=True)), sin(cos(Symbol('\\\\mu_0', commutative=True)))), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(2), sin(cos(Symbol('\\\\mu_0', commutative=True))))), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given b{(I,v_{z})} = I + v_{z}, then derive \\int b{(I,v_{z})} dI = \\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger}, then derive \\frac{\\frac{I^{2}}{2} + I v_{z} + n}{\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger}} = 1, then obtain \\frac{(\\frac{I^{2}}{2} + I v_{z} + n)^{2}}{(\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger})^{2}} = 1", "derivation": "b{(I,v_{z})} = I + v_{z} and \\int b{(I,v_{z})} dI = \\int (I + v_{z}) dI and \\int b{(I,v_{z})} dI = \\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger} and \\frac{\\int b{(I,v_{z})} dI}{\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger}} = 1 and \\frac{\\int (I + v_{z}) dI}{\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger}} = 1 and \\frac{\\frac{I^{2}}{2} + I v_{z} + n}{\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger}} = 1 and \\frac{(\\frac{I^{2}}{2} + I v_{z} + n)^{2}}{(\\frac{I^{2}}{2} + I v_{z} + \\Psi^{\\dagger})^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('b')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('b')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["divide", 3, "Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Function('b')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Integral(Add(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True)))), Integer(1))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('n', commutative=True))), Integer(1))"], [["power", 6, 2], "Equality(Mul(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integer(-2)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('I', commutative=True), Integer(2))), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('n', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\hat{x}_0{(\\mu,s)} = \\mu + s, then obtain \\log{((\\mu + s) (\\int \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} ds)^{s})} = \\log{((\\mu + s) (\\int 1 ds)^{s})}", "derivation": "\\hat{x}_0{(\\mu,s)} = \\mu + s and \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} = 1 and \\int \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} ds = \\int 1 ds and (\\int \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} ds)^{s} = (\\int 1 ds)^{s} and (\\mu + s) (\\int \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} ds)^{s} = (\\mu + s) (\\int 1 ds)^{s} and \\log{((\\mu + s) (\\int \\frac{\\hat{x}_0{(\\mu,s)}}{\\mu + s} ds)^{s})} = \\log{((\\mu + s) (\\int 1 ds)^{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Integer(1), Tuple(Symbol('s', commutative=True))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["times", 4, "Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))), Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"], [["log", 5], "Equality(log(Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))), log(Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('s', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(x,\\varphi)} = x^{\\varphi}, then derive - x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi} = - x^{\\varphi} + \\frac{x^{\\varphi} \\log{(x)}}{\\varphi}, then obtain \\int (- x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi}) dx = \\int (- x^{\\varphi} + \\frac{x^{\\varphi} \\log{(x)}}{\\varphi}) dx", "derivation": "\\mathbf{F}{(x,\\varphi)} = x^{\\varphi} and \\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)} = \\frac{\\partial}{\\partial \\varphi} x^{\\varphi} and \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi} = \\frac{\\frac{\\partial}{\\partial \\varphi} x^{\\varphi}}{\\varphi} and - x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi} = - x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} x^{\\varphi}}{\\varphi} and - x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi} = - x^{\\varphi} + \\frac{x^{\\varphi} \\log{(x)}}{\\varphi} and \\int (- x^{\\varphi} + \\frac{\\frac{\\partial}{\\partial \\varphi} \\mathbf{F}{(x,\\varphi)}}{\\varphi}) dx = \\int (- x^{\\varphi} + \\frac{x^{\\varphi} \\log{(x)}}{\\varphi}) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"], [["minus", 3, "Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('x', commutative=True)))))"], [["integrate", 5, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{F}')(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Tuple(Symbol('x', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Symbol('x', commutative=True), Symbol('\\\\varphi', commutative=True)), log(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))))"]]}, {"prompt": "Given a{(\\nabla,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla}, then obtain \\int \\frac{\\int (a{(\\nabla,\\chi)} + \\frac{1}{\\nabla}) d\\chi}{\\int (\\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} + \\frac{1}{\\nabla}) d\\chi} d\\chi = \\int 1 d\\chi", "derivation": "a{(\\nabla,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} and a{(\\nabla,\\chi)} + \\frac{1}{\\nabla} = \\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} + \\frac{1}{\\nabla} and \\int (a{(\\nabla,\\chi)} + \\frac{1}{\\nabla}) d\\chi = \\int (\\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} + \\frac{1}{\\nabla}) d\\chi and \\frac{\\int (a{(\\nabla,\\chi)} + \\frac{1}{\\nabla}) d\\chi}{\\int (\\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} + \\frac{1}{\\nabla}) d\\chi} = 1 and \\int \\frac{\\int (a{(\\nabla,\\chi)} + \\frac{1}{\\nabla}) d\\chi}{\\int (\\frac{\\partial}{\\partial \\chi} \\frac{\\chi}{\\nabla} + \\frac{1}{\\nabla}) d\\chi} d\\chi = \\int 1 d\\chi", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True)), Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 1, "Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))"], "Equality(Add(Function('a')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Function('a')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 3, "Integral(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integral(Add(Function('a')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Pow(Integral(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Integral(Add(Function('a')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Pow(Integral(Add(Derivative(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\nabla', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(h)} = e^{h}, then obtain \\frac{d}{d h} \\mathbf{A}{(h)} - 1 = e^{h} - 1", "derivation": "\\mathbf{A}{(h)} = e^{h} and - h + \\mathbf{A}{(h)} = - h + e^{h} and \\frac{d}{d h} (- h + \\mathbf{A}{(h)}) = \\frac{d}{d h} (- h + e^{h}) and \\frac{d}{d h} \\mathbf{A}{(h)} - 1 = e^{h} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\mathbf{A}')(Symbol('h', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('\\\\mathbf{A}')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\mathbf{A}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1)), Add(exp(Symbol('h', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\hat{H}{(T,\\hbar)} = \\frac{T}{\\hbar} and \\mathbf{s}{(T,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\hat{H}{(T,\\hbar)}, then obtain - \\frac{T}{\\hbar} + \\frac{\\partial}{\\partial u} \\frac{\\mathbf{s}{(T,\\hbar)}}{u} = - \\frac{T}{\\hbar} + \\frac{\\partial}{\\partial u} \\frac{\\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\frac{T}{\\hbar}}{u}", "derivation": "\\hat{H}{(T,\\hbar)} = \\frac{T}{\\hbar} and \\mathbf{s}{(T,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\hat{H}{(T,\\hbar)} and \\frac{\\mathbf{s}{(T,\\hbar)}}{u} = \\frac{\\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\hat{H}{(T,\\hbar)}}{u} and \\frac{\\mathbf{s}{(T,\\hbar)}}{u} = \\frac{\\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\frac{T}{\\hbar}}{u} and \\frac{\\partial}{\\partial u} \\frac{\\mathbf{s}{(T,\\hbar)}}{u} = \\frac{\\partial}{\\partial u} \\frac{\\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\frac{T}{\\hbar}}{u} and - \\frac{T}{\\hbar} + \\frac{\\partial}{\\partial u} \\frac{\\mathbf{s}{(T,\\hbar)}}{u} = - \\frac{T}{\\hbar} + \\frac{\\partial}{\\partial u} \\frac{\\frac{\\partial^{2}}{\\partial \\hbar\\partial T} \\frac{T}{\\hbar}}{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('u', commutative=True)"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{H}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["minus", 5, "Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Function('\\\\mathbf{s}')(Symbol('T', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Derivative(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Derivative(Mul(Symbol('T', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Tuple(Symbol('u', commutative=True), Integer(1)))))"]]}, {"prompt": "Given z{(x,\\pi)} = \\pi - x, then obtain x (- \\pi + x + \\frac{z{(x,\\pi)}}{x}) = x (- \\pi + x + \\frac{\\pi - x}{x})", "derivation": "z{(x,\\pi)} = \\pi - x and \\frac{z{(x,\\pi)}}{x} = \\frac{\\pi - x}{x} and - \\pi + x + \\frac{z{(x,\\pi)}}{x} = - \\pi + x + \\frac{\\pi - x}{x} and x (- \\pi + x + \\frac{z{(x,\\pi)}}{x}) = x (- \\pi + x + \\frac{\\pi - x}{x})", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["divide", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('z')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))"], [["minus", 2, "Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('x', commutative=True), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('z')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('x', commutative=True), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))))"], [["times", 3, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('x', commutative=True), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Function('z')(Symbol('x', commutative=True), Symbol('\\\\pi', commutative=True))))), Mul(Symbol('x', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('x', commutative=True), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))))))"]]}, {"prompt": "Given \\omega{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain - \\frac{\\int (\\omega^{\\psi^*}{(\\psi^*)} - \\sin{(\\psi^*)}) d\\psi^*}{\\psi^*} = - \\frac{\\int (- \\sin{(\\psi^*)} + \\sin^{\\psi^*}{(\\psi^*)}) d\\psi^*}{\\psi^*}", "derivation": "\\omega{(\\psi^*)} = \\sin{(\\psi^*)} and \\omega^{\\psi^*}{(\\psi^*)} = \\sin^{\\psi^*}{(\\psi^*)} and \\omega^{\\psi^*}{(\\psi^*)} - \\sin{(\\psi^*)} = - \\sin{(\\psi^*)} + \\sin^{\\psi^*}{(\\psi^*)} and \\int (\\omega^{\\psi^*}{(\\psi^*)} - \\sin{(\\psi^*)}) d\\psi^* = \\int (- \\sin{(\\psi^*)} + \\sin^{\\psi^*}{(\\psi^*)}) d\\psi^* and - \\frac{\\int (\\omega^{\\psi^*}{(\\psi^*)} - \\sin{(\\psi^*)}) d\\psi^*}{\\psi^*} = - \\frac{\\int (- \\sin{(\\psi^*)} + \\sin^{\\psi^*}{(\\psi^*)}) d\\psi^*}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "sin(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Pow(Function('\\\\omega')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Pow(Function('\\\\omega')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Integral(Add(Pow(Function('\\\\omega')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), sin(Symbol('\\\\psi^*', commutative=True))), Pow(sin(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{H})} = \\cos{(\\cos{(\\hat{H})})}, then obtain - \\hat{H} + (\\int \\operatorname{z^{*}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = - \\hat{H} + (\\int \\cos{(\\cos{(\\hat{H})})} d\\hat{H})^{\\hat{H}}", "derivation": "\\operatorname{z^{*}}{(\\hat{H})} = \\cos{(\\cos{(\\hat{H})})} and \\int \\operatorname{z^{*}}{(\\hat{H})} d\\hat{H} = \\int \\cos{(\\cos{(\\hat{H})})} d\\hat{H} and (\\int \\operatorname{z^{*}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = (\\int \\cos{(\\cos{(\\hat{H})})} d\\hat{H})^{\\hat{H}} and - \\hat{H} + (\\int \\operatorname{z^{*}}{(\\hat{H})} d\\hat{H})^{\\hat{H}} = - \\hat{H} + (\\int \\cos{(\\cos{(\\hat{H})})} d\\hat{H})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{H}', commutative=True)), cos(cos(Symbol('\\\\hat{H}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(cos(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Integral(Function('z^*')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(cos(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["minus", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(Function('z^*')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Pow(Integral(cos(cos(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then derive \\int \\mathbf{E}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\mathbb{I} - \\cos{(\\eta^{\\prime})}, then obtain \\mathbb{I} = \\frac{\\mathbb{I} \\int \\sin{(\\eta^{\\prime})} d\\eta^{\\prime}}{\\mathbb{I} - \\cos{(\\eta^{\\prime})}}", "derivation": "\\mathbf{E}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and \\int \\mathbf{E}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\int \\sin{(\\eta^{\\prime})} d\\eta^{\\prime} and \\int \\mathbf{E}{(\\eta^{\\prime})} d\\eta^{\\prime} = \\mathbb{I} - \\cos{(\\eta^{\\prime})} and \\mathbb{I} - \\cos{(\\eta^{\\prime})} = \\int \\sin{(\\eta^{\\prime})} d\\eta^{\\prime} and 1 = \\frac{\\int \\sin{(\\eta^{\\prime})} d\\eta^{\\prime}}{\\mathbb{I} - \\cos{(\\eta^{\\prime})}} and \\mathbb{I} = \\frac{\\mathbb{I} \\int \\sin{(\\eta^{\\prime})} d\\eta^{\\prime}}{\\mathbb{I} - \\cos{(\\eta^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integral(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integer(-1)), Integral(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["times", 5, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Symbol('\\\\mathbb{I}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Integer(-1)), Integral(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given p{(\\lambda,\\mathbf{E})} = \\cos{(\\lambda \\mathbf{E})}, then obtain \\log{(\\cos^{\\mathbf{E}}{(\\lambda \\mathbf{E})})} + \\int p{(\\lambda,\\mathbf{E})} d\\mathbf{E} = \\log{(\\cos^{\\mathbf{E}}{(\\lambda \\mathbf{E})})} + \\int \\cos{(\\lambda \\mathbf{E})} d\\mathbf{E}", "derivation": "p{(\\lambda,\\mathbf{E})} = \\cos{(\\lambda \\mathbf{E})} and p^{\\mathbf{E}}{(\\lambda,\\mathbf{E})} = \\cos^{\\mathbf{E}}{(\\lambda \\mathbf{E})} and \\int p{(\\lambda,\\mathbf{E})} d\\mathbf{E} = \\int \\cos{(\\lambda \\mathbf{E})} d\\mathbf{E} and \\log{(p^{\\mathbf{E}}{(\\lambda,\\mathbf{E})})} + \\int p{(\\lambda,\\mathbf{E})} d\\mathbf{E} = \\log{(p^{\\mathbf{E}}{(\\lambda,\\mathbf{E})})} + \\int \\cos{(\\lambda \\mathbf{E})} d\\mathbf{E} and \\log{(\\cos^{\\mathbf{E}}{(\\lambda \\mathbf{E})})} + \\int p{(\\lambda,\\mathbf{E})} d\\mathbf{E} = \\log{(\\cos^{\\mathbf{E}}{(\\lambda \\mathbf{E})})} + \\int \\cos{(\\lambda \\mathbf{E})} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 3, "log(Pow(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(log(Pow(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Integral(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Add(log(Pow(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Integral(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(log(Pow(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))), Integral(Function('p')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))), Add(log(Pow(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True))), Integral(cos(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(B)} = \\sin{(\\log{(B)})}, then obtain \\int \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\operatorname{z^{*}}{(B)} dB = \\int \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\sin{(\\log{(B)})} dB", "derivation": "\\operatorname{z^{*}}{(B)} = \\sin{(\\log{(B)})} and \\operatorname{z^{*}}^{2}{(B)} = \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} and \\frac{d}{d B} \\operatorname{z^{*}}{(B)} = \\frac{d}{d B} \\sin{(\\log{(B)})} and \\operatorname{z^{*}}^{2}{(B)} \\frac{d}{d B} \\operatorname{z^{*}}{(B)} = \\operatorname{z^{*}}^{2}{(B)} \\frac{d}{d B} \\sin{(\\log{(B)})} and \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\operatorname{z^{*}}{(B)} = \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\sin{(\\log{(B)})} and \\int \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\operatorname{z^{*}}{(B)} dB = \\int \\operatorname{z^{*}}{(B)} \\sin{(\\log{(B)})} \\frac{d}{d B} \\sin{(\\log{(B)})} dB", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))))"], [["times", 1, "Function('z^*')(Symbol('B', commutative=True))"], "Equality(Pow(Function('z^*')(Symbol('B', commutative=True)), Integer(2)), Mul(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True)))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["times", 3, "Pow(Function('z^*')(Symbol('B', commutative=True)), Integer(2))"], "Equality(Mul(Pow(Function('z^*')(Symbol('B', commutative=True)), Integer(2)), Derivative(Function('z^*')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Pow(Function('z^*')(Symbol('B', commutative=True)), Integer(2)), Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))), Derivative(Function('z^*')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Mul(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))), Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))), Derivative(Function('z^*')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Tuple(Symbol('B', commutative=True))), Integral(Mul(Function('z^*')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))), Derivative(sin(log(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(a)} = \\sin{(a)}, then derive \\int \\operatorname{g_{\\varepsilon}}{(a)} da = F_{H} - \\cos{(a)}, then obtain \\tilde{\\infty} (\\mathbf{P} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da)^{F_{H}} = \\tilde{\\infty} (F_{H} + \\mathbf{P} - \\cos{(a)})^{F_{H}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(a)} = \\sin{(a)} and \\int \\operatorname{g_{\\varepsilon}}{(a)} da = \\int \\sin{(a)} da and \\int \\operatorname{g_{\\varepsilon}}{(a)} da = F_{H} - \\cos{(a)} and \\mathbf{P} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da = F_{H} + \\mathbf{P} - \\cos{(a)} and (\\mathbf{P} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da)^{F_{H}} = (F_{H} + \\mathbf{P} - \\cos{(a)})^{F_{H}} and \\frac{(\\mathbf{P} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da)^{F_{H}}}{\\operatorname{g_{\\varepsilon}}{(a)} - \\sin{(a)}} = \\frac{(F_{H} + \\mathbf{P} - \\cos{(a)})^{F_{H}}}{\\operatorname{g_{\\varepsilon}}{(a)} - \\sin{(a)}} and \\tilde{\\infty} (\\mathbf{P} + \\int \\operatorname{g_{\\varepsilon}}{(a)} da)^{F_{H}} = \\tilde{\\infty} (F_{H} + \\mathbf{P} - \\cos{(a)})^{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('a', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), cos(Symbol('a', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('a', commutative=True)))))"], [["power", 4, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True)), Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True)))"], [["divide", 5, "Add(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True)), Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Integer(-1))), Mul(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Mul(Integer(-1), sin(Symbol('a', commutative=True)))), Integer(-1)), Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(zoo, Pow(Add(Symbol('\\\\mathbf{P}', commutative=True), Integral(Function('g_{\\\\varepsilon}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True))), Mul(zoo, Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), cos(Symbol('a', commutative=True)))), Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\varphi^*,s)} = e^{\\varphi^* + s}, then obtain \\int (\\frac{- \\varphi^* + \\operatorname{F_{H}}{(\\varphi^*,s)}}{- \\varphi^* + e^{\\varphi^* + s}})^{s} ds = \\int 1 ds", "derivation": "\\operatorname{F_{H}}{(\\varphi^*,s)} = e^{\\varphi^* + s} and - \\varphi^* + \\operatorname{F_{H}}{(\\varphi^*,s)} = - \\varphi^* + e^{\\varphi^* + s} and \\frac{- \\varphi^* + \\operatorname{F_{H}}{(\\varphi^*,s)}}{- \\varphi^* + e^{\\varphi^* + s}} = 1 and (\\frac{- \\varphi^* + \\operatorname{F_{H}}{(\\varphi^*,s)}}{- \\varphi^* + e^{\\varphi^* + s}})^{s} = 1 and \\int (\\frac{- \\varphi^* + \\operatorname{F_{H}}{(\\varphi^*,s)}}{- \\varphi^* + e^{\\varphi^* + s}})^{s} ds = \\int 1 ds", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('F_H')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('F_H')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('F_H')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))), Integer(-1))), Symbol('s', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('s', commutative=True)"], "Equality(Integral(Pow(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('F_H')(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True)))), Integer(-1))), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Integer(1), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\pi{(v_{1},q)} = \\frac{v_{1}}{q}, then obtain v_{1} + \\pi{(v_{1},q)} + 1 = v_{1} - \\pi{(v_{1},q)} + 1 + \\frac{2 v_{1}}{q}", "derivation": "\\pi{(v_{1},q)} = \\frac{v_{1}}{q} and \\pi{(v_{1},q)} + 1 = 1 + \\frac{v_{1}}{q} and v_{1} + \\pi{(v_{1},q)} + 1 = v_{1} + 1 + \\frac{v_{1}}{q} and 0 = - \\pi{(v_{1},q)} + \\frac{v_{1}}{q} and 1 + \\frac{v_{1}}{q} = - \\pi{(v_{1},q)} + 1 + \\frac{2 v_{1}}{q} and v_{1} + \\pi{(v_{1},q)} + 1 = v_{1} - \\pi{(v_{1},q)} + 1 + \\frac{2 v_{1}}{q}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["add", 2, "Symbol('v_1', commutative=True)"], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Integer(1)), Add(Symbol('v_1', commutative=True), Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["minus", 1, "Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["add", 4, "Add(Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True)))"], "Equality(Add(Integer(1), Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Integer(1), Mul(Integer(2), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Symbol('v_1', commutative=True), Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True)), Integer(1)), Add(Symbol('v_1', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('v_1', commutative=True), Symbol('q', commutative=True))), Integer(1), Mul(Integer(2), Pow(Symbol('q', commutative=True), Integer(-1)), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given p{(v_{z})} = v_{z}, then obtain \\frac{d}{d v_{z}} p{(v_{z})} \\frac{d}{d v_{z}} v_{z} = \\frac{d}{d v_{z}} v_{z} \\frac{d}{d v_{z}} v_{z}", "derivation": "p{(v_{z})} = v_{z} and \\frac{d}{d v_{z}} p{(v_{z})} = \\frac{d}{d v_{z}} v_{z} and p{(v_{z})} \\frac{d}{d v_{z}} p{(v_{z})} = v_{z} \\frac{d}{d v_{z}} p{(v_{z})} and p{(v_{z})} \\frac{d}{d v_{z}} v_{z} = v_{z} \\frac{d}{d v_{z}} v_{z} and \\frac{d}{d v_{z}} p{(v_{z})} \\frac{d}{d v_{z}} v_{z} = \\frac{d}{d v_{z}} v_{z} \\frac{d}{d v_{z}} v_{z}", "srepr_derivation": [["renaming_premise", "Equality(Function('p')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('p')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Mul(Function('p')(Symbol('v_z', commutative=True)), Derivative(Function('p')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Symbol('v_z', commutative=True), Derivative(Function('p')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('p')(Symbol('v_z', commutative=True)), Derivative(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Symbol('v_z', commutative=True), Derivative(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Function('p')(Symbol('v_z', commutative=True)), Derivative(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_z', commutative=True), Derivative(Symbol('v_z', commutative=True), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given B{(S)} = \\frac{d}{d S} e^{S}, then obtain B{(S)} - e^{S} + 1 = 1", "derivation": "B{(S)} = \\frac{d}{d S} e^{S} and B{(S)} - e^{S} = - e^{S} + \\frac{d}{d S} e^{S} and B{(S)} - e^{S} + 1 = - e^{S} + \\frac{d}{d S} e^{S} + 1 and B{(S)} - e^{S} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('S', commutative=True)), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 1, "exp(Symbol('S', commutative=True))"], "Equality(Add(Function('B')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('B')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True))), Integer(1)), Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), Derivative(exp(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Function('B')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True))), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\varphi{(f^{*},\\mathbb{I},\\rho)} = \\frac{\\mathbb{I} - f^{*}}{\\rho}, then obtain \\frac{\\varphi{(f^{*},\\mathbb{I},\\rho)}}{\\varphi{(f^{*},\\mathbb{I},\\rho)} - \\frac{1}{\\rho}} = \\frac{\\mathbb{I} - f^{*}}{\\rho (\\varphi{(f^{*},\\mathbb{I},\\rho)} - \\frac{1}{\\rho})}", "derivation": "\\varphi{(f^{*},\\mathbb{I},\\rho)} = \\frac{\\mathbb{I} - f^{*}}{\\rho} and \\varphi{(f^{*},\\mathbb{I},\\rho)} - \\frac{1}{\\rho} = \\frac{\\mathbb{I} - f^{*}}{\\rho} - \\frac{1}{\\rho} and \\frac{\\varphi{(f^{*},\\mathbb{I},\\rho)}}{\\frac{\\mathbb{I} - f^{*}}{\\rho} - \\frac{1}{\\rho}} = \\frac{\\mathbb{I} - f^{*}}{\\rho (\\frac{\\mathbb{I} - f^{*}}{\\rho} - \\frac{1}{\\rho})} and \\frac{\\varphi{(f^{*},\\mathbb{I},\\rho)}}{\\varphi{(f^{*},\\mathbb{I},\\rho)} - \\frac{1}{\\rho}} = \\frac{\\mathbb{I} - f^{*}}{\\rho (\\varphi{(f^{*},\\mathbb{I},\\rho)} - \\frac{1}{\\rho})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["minus", 1, "Pow(Symbol('\\\\rho', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))))"], [["divide", 1, "Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], "Equality(Mul(Pow(Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Integer(-1)), Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Pow(Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Integer(-1)), Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Pow(Add(Function('\\\\varphi')(Symbol('f^*', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)))), Integer(-1))))"]]}, {"prompt": "Given \\phi{(p)} = e^{p}, then derive (\\frac{p \\frac{d}{d p} \\phi{(p)}}{\\phi{(p)}} + \\log{(\\phi{(p)})}) \\phi^{p}{(p)} - 1 = (p + \\log{(e^{p})}) (e^{p})^{p} - 1, then obtain (\\frac{p \\frac{d}{d p} \\phi{(p)}}{\\phi{(p)}} + \\log{(\\phi{(p)})}) (e^{p})^{p} - 1 = (p + \\log{(e^{p})}) (e^{p})^{p} - 1", "derivation": "\\phi{(p)} = e^{p} and \\phi^{p}{(p)} = (e^{p})^{p} and - p + \\phi^{p}{(p)} = - p + (e^{p})^{p} and \\frac{d}{d p} (- p + \\phi^{p}{(p)}) = \\frac{d}{d p} (- p + (e^{p})^{p}) and (\\frac{p \\frac{d}{d p} \\phi{(p)}}{\\phi{(p)}} + \\log{(\\phi{(p)})}) \\phi^{p}{(p)} - 1 = (p + \\log{(e^{p})}) (e^{p})^{p} - 1 and (\\frac{p \\frac{d}{d p} \\phi{(p)}}{\\phi{(p)}} + \\log{(\\phi{(p)})}) (e^{p})^{p} - 1 = (p + \\log{(e^{p})}) (e^{p})^{p} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["minus", 2, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Add(Mul(Symbol('p', commutative=True), Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), log(Function('\\\\phi')(Symbol('p', commutative=True)))), Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(-1)), Add(Mul(Add(Symbol('p', commutative=True), log(exp(Symbol('p', commutative=True)))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Add(Mul(Symbol('p', commutative=True), Pow(Function('\\\\phi')(Symbol('p', commutative=True)), Integer(-1)), Derivative(Function('\\\\phi')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), log(Function('\\\\phi')(Symbol('p', commutative=True)))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(-1)), Add(Mul(Add(Symbol('p', commutative=True), log(exp(Symbol('p', commutative=True)))), Pow(exp(Symbol('p', commutative=True)), Symbol('p', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\varphi,L)} = \\varphi^{L}, then derive \\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)} = \\varphi^{L} \\log{(\\varphi)}, then obtain (\\varepsilon_{0}{(\\varphi,L)} \\log{(\\varphi)})^{\\varphi} + (\\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)})^{\\varphi} = 2 (\\varepsilon_{0}{(\\varphi,L)} \\log{(\\varphi)})^{\\varphi}", "derivation": "\\varepsilon_{0}{(\\varphi,L)} = \\varphi^{L} and \\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)} = \\frac{\\partial}{\\partial L} \\varphi^{L} and \\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)} = \\varphi^{L} \\log{(\\varphi)} and (\\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)})^{\\varphi} = (\\varphi^{L} \\log{(\\varphi)})^{\\varphi} and (\\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)})^{\\varphi} = (\\varepsilon_{0}{(\\varphi,L)} \\log{(\\varphi)})^{\\varphi} and (\\varepsilon_{0}{(\\varphi,L)} \\log{(\\varphi)})^{\\varphi} + (\\frac{\\partial}{\\partial L} \\varepsilon_{0}{(\\varphi,L)})^{\\varphi} = 2 (\\varepsilon_{0}{(\\varphi,L)} \\log{(\\varphi)})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["add", 5, "Pow(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Pow(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Symbol('\\\\varphi', commutative=True))), Mul(Integer(2), Pow(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\varphi', commutative=True), Symbol('L', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\hat{H},T)} = \\hat{H}^{T}, then obtain \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\operatorname{M_{E}}{(\\hat{H},T)}}{T} \\int \\hat{H}^{T} dT = \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\hat{H}^{T}}{T} \\int \\hat{H}^{T} dT", "derivation": "\\operatorname{M_{E}}{(\\hat{H},T)} = \\hat{H}^{T} and \\frac{\\operatorname{M_{E}}{(\\hat{H},T)}}{T} = \\frac{\\hat{H}^{T}}{T} and \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\operatorname{M_{E}}{(\\hat{H},T)}}{T} = \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\hat{H}^{T}}{T} and \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\operatorname{M_{E}}{(\\hat{H},T)}}{T} \\int \\hat{H}^{T} dT = \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\hat{H}^{T}}{T} \\int \\hat{H}^{T} dT", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True)))"], [["divide", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('M_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('M_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["times", 3, "Integral(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Function('M_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integral(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Derivative(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Integral(Pow(Symbol('\\\\hat{H}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(t)} = \\sin{(\\cos{(t)})}, then derive \\int \\frac{\\operatorname{f_{E}}{(t)}}{\\sin{(\\cos{(t)})}} dt = \\Psi_{nl} + t, then derive \\theta + t = \\Psi_{nl} + t, then obtain \\int 1 dt = \\theta + t", "derivation": "\\operatorname{f_{E}}{(t)} = \\sin{(\\cos{(t)})} and \\frac{\\operatorname{f_{E}}{(t)}}{\\sin{(\\cos{(t)})}} = 1 and \\int \\frac{\\operatorname{f_{E}}{(t)}}{\\sin{(\\cos{(t)})}} dt = \\int 1 dt and \\int \\frac{\\operatorname{f_{E}}{(t)}}{\\sin{(\\cos{(t)})}} dt = \\Psi_{nl} + t and \\int 1 dt = \\Psi_{nl} + t and \\theta + t = \\Psi_{nl} + t and \\int 1 dt = \\theta + t", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('t', commutative=True)), sin(cos(Symbol('t', commutative=True))))"], [["divide", 1, "sin(cos(Symbol('t', commutative=True)))"], "Equality(Mul(Function('f_E')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Function('f_E')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('t', commutative=True))), Integral(Integer(1), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('f_E')(Symbol('t', commutative=True)), Pow(sin(cos(Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\theta', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Integer(1), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Symbol('t', commutative=True)))"]]}, {"prompt": "Given B{(W,\\hbar)} = \\cos{(W + \\hbar)} and \\mathbf{S}{(W,\\hbar)} = \\frac{B{(W,\\hbar)}}{W}, then obtain \\mathbf{S}{(W,\\hbar)} + \\int \\cos{(W + \\hbar)} dW = \\int \\cos{(W + \\hbar)} dW + \\frac{\\cos{(W + \\hbar)}}{W}", "derivation": "B{(W,\\hbar)} = \\cos{(W + \\hbar)} and \\frac{B{(W,\\hbar)}}{W} = \\frac{\\cos{(W + \\hbar)}}{W} and \\int \\cos{(W + \\hbar)} dW + \\frac{B{(W,\\hbar)}}{W} = \\int \\cos{(W + \\hbar)} dW + \\frac{\\cos{(W + \\hbar)}}{W} and \\mathbf{S}{(W,\\hbar)} = \\frac{B{(W,\\hbar)}}{W} and \\mathbf{S}{(W,\\hbar)} + \\int \\cos{(W + \\hbar)} dW = \\int \\cos{(W + \\hbar)} dW + \\frac{\\cos{(W + \\hbar)}}{W}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["divide", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('B')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["add", 2, "Integral(cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Integral(cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('B')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True)))), Add(Integral(cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('B')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('W', commutative=True)))), Add(Integral(cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), cos(Add(Symbol('W', commutative=True), Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = (B + \\phi_1)^{\\mathbf{J}}, then obtain - B + \\phi_1 \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}^{2}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = - B + \\phi_1 (B + \\phi_1)^{\\mathbf{J}} - \\Psi_{\\lambda}^{2}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)}", "derivation": "\\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = (B + \\phi_1)^{\\mathbf{J}} and \\phi_1 \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = \\phi_1 (B + \\phi_1)^{\\mathbf{J}} and - B + \\phi_1 \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = - B + \\phi_1 (B + \\phi_1)^{\\mathbf{J}} and - B + \\phi_1 \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}^{2}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)} = - B + \\phi_1 (B + \\phi_1)^{\\mathbf{J}} - \\Psi_{\\lambda}^{2}{(\\phi_1,\\mathbf{J},B)} - \\Psi_{\\lambda}{(\\phi_1,\\mathbf{J},B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 2, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["minus", 3, "Add(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Integer(2)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\theta)} = \\cos{(\\theta)}, then obtain \\int \\mathbf{J}_f{(\\theta)} d\\theta + \\iint \\cos{(\\theta)} d\\theta d\\theta = \\int \\cos{(\\theta)} d\\theta + \\iint \\cos{(\\theta)} d\\theta d\\theta", "derivation": "\\mathbf{J}_f{(\\theta)} = \\cos{(\\theta)} and \\int \\mathbf{J}_f{(\\theta)} d\\theta = \\int \\cos{(\\theta)} d\\theta and \\iint \\mathbf{J}_f{(\\theta)} d\\theta d\\theta = \\iint \\cos{(\\theta)} d\\theta d\\theta and \\int \\mathbf{J}_f{(\\theta)} d\\theta + \\iint \\mathbf{J}_f{(\\theta)} d\\theta d\\theta = \\int \\cos{(\\theta)} d\\theta + \\iint \\mathbf{J}_f{(\\theta)} d\\theta d\\theta and \\int \\mathbf{J}_f{(\\theta)} d\\theta + \\iint \\cos{(\\theta)} d\\theta d\\theta = \\int \\cos{(\\theta)} d\\theta + \\iint \\cos{(\\theta)} d\\theta d\\theta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], [["add", 2, "Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Function('\\\\mathbf{J}_f')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Add(Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(cos(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given n{(E,\\mu)} = \\log{(E \\mu)}, then derive E \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} n{(E,\\mu)} = \\frac{E \\log{(E \\mu)}}{\\mu}, then obtain E \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} \\log{(E \\mu)} = \\frac{E \\log{(E \\mu)}}{\\mu}", "derivation": "n{(E,\\mu)} = \\log{(E \\mu)} and E + n{(E,\\mu)} = E + \\log{(E \\mu)} and E (E + n{(E,\\mu)}) = E (E + \\log{(E \\mu)}) and \\frac{\\partial}{\\partial \\mu} E (E + n{(E,\\mu)}) = \\frac{\\partial}{\\partial \\mu} E (E + \\log{(E \\mu)}) and \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} E (E + n{(E,\\mu)}) = \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} E (E + \\log{(E \\mu)}) and E \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} n{(E,\\mu)} = \\frac{E \\log{(E \\mu)}}{\\mu} and E \\log{(E \\mu)} \\frac{\\partial}{\\partial \\mu} \\log{(E \\mu)} = \\frac{E \\log{(E \\mu)}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["add", 1, "Symbol('E', commutative=True)"], "Equality(Add(Symbol('E', commutative=True), Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["times", 2, "Symbol('E', commutative=True)"], "Equality(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))), Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["times", 4, "log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))"], "Equality(Mul(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Derivative(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Derivative(Mul(Symbol('E', commutative=True), Add(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Derivative(Function('n')(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('E', commutative=True), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Derivative(log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Mul(Symbol('E', commutative=True), Symbol('\\\\mu', commutative=True)))))"]]}, {"prompt": "Given f{(\\eta^{\\prime},f_{E})} = \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})}, then obtain 0 = - \\eta^{\\prime} f{(\\eta^{\\prime},f_{E})} + \\eta^{\\prime} \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})}", "derivation": "f{(\\eta^{\\prime},f_{E})} = \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})} and \\eta^{\\prime} f{(\\eta^{\\prime},f_{E})} = \\eta^{\\prime} \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})} and \\eta^{\\prime} f{(\\eta^{\\prime},f_{E})} - 1 = \\eta^{\\prime} \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})} - 1 and 0 = - \\eta^{\\prime} f{(\\eta^{\\prime},f_{E})} + \\eta^{\\prime} \\sin{(\\frac{f_{E}}{\\eta^{\\prime}})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True)), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))"], [["times", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))), Integer(-1)))"], [["minus", 3, "Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))), Integer(-1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True), Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), sin(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given M{(V,\\mathbf{s})} = \\frac{V}{\\mathbf{s}} and q{(\\pi,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi, then obtain \\frac{q^{\\pi}{(\\pi,\\mathbf{J}_M)}}{\\int M{(V,\\mathbf{s})} d\\mathbf{s}} = \\frac{(\\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi)^{\\pi}}{\\int M{(V,\\mathbf{s})} d\\mathbf{s}}", "derivation": "M{(V,\\mathbf{s})} = \\frac{V}{\\mathbf{s}} and \\int M{(V,\\mathbf{s})} d\\mathbf{s} = \\int \\frac{V}{\\mathbf{s}} d\\mathbf{s} and q{(\\pi,\\mathbf{J}_M)} = \\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi and q^{\\pi}{(\\pi,\\mathbf{J}_M)} = (\\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi)^{\\pi} and \\frac{q^{\\pi}{(\\pi,\\mathbf{J}_M)}}{\\int \\frac{V}{\\mathbf{s}} d\\mathbf{s}} = \\frac{(\\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi)^{\\pi}}{\\int \\frac{V}{\\mathbf{s}} d\\mathbf{s}} and \\frac{q^{\\pi}{(\\pi,\\mathbf{J}_M)}}{\\int M{(V,\\mathbf{s})} d\\mathbf{s}} = \\frac{(\\frac{\\partial}{\\partial \\pi} \\mathbf{J}_M \\pi)^{\\pi}}{\\int M{(V,\\mathbf{s})} d\\mathbf{s}}", "srepr_derivation": [["get_premise", "Equality(Function('M')(Symbol('V', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('M')(Symbol('V', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], ["get_premise", "Equality(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)))"], [["divide", 4, "Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Mul(Pow(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Mul(Symbol('V', commutative=True), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Integral(Function('M')(Symbol('V', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))), Mul(Pow(Derivative(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Function('M')(Symbol('V', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(\\phi_1,S)} = \\int \\frac{\\phi_1}{S} dS, then obtain \\frac{S (\\theta_{2}^{2}{(\\phi_1,S)} - \\theta_{2}{(\\phi_1,S)})}{\\mathbf{J}_P} = \\frac{S (\\theta_{2}{(\\phi_1,S)} \\int \\frac{\\phi_1}{S} dS - \\theta_{2}{(\\phi_1,S)})}{\\mathbf{J}_P}", "derivation": "\\theta_{2}{(\\phi_1,S)} = \\int \\frac{\\phi_1}{S} dS and \\theta_{2}^{2}{(\\phi_1,S)} = \\theta_{2}{(\\phi_1,S)} \\int \\frac{\\phi_1}{S} dS and \\theta_{2}^{2}{(\\phi_1,S)} - \\theta_{2}{(\\phi_1,S)} = \\theta_{2}{(\\phi_1,S)} \\int \\frac{\\phi_1}{S} dS - \\theta_{2}{(\\phi_1,S)} and S (\\theta_{2}^{2}{(\\phi_1,S)} - \\theta_{2}{(\\phi_1,S)}) = S (\\theta_{2}{(\\phi_1,S)} \\int \\frac{\\phi_1}{S} dS - \\theta_{2}{(\\phi_1,S)}) and \\frac{S (\\theta_{2}^{2}{(\\phi_1,S)} - \\theta_{2}{(\\phi_1,S)})}{\\mathbf{J}_P} = \\frac{S (\\theta_{2}{(\\phi_1,S)} \\int \\frac{\\phi_1}{S} dS - \\theta_{2}{(\\phi_1,S)})}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["times", 1, "Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["minus", 2, "Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)))), Add(Mul(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)))))"], [["divide", 3, "Pow(Symbol('S', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('S', commutative=True), Add(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))))), Mul(Symbol('S', commutative=True), Add(Mul(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))))))"], [["divide", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Pow(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))))), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Add(Mul(Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True)), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('S', commutative=True)))), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\phi_1', commutative=True), Symbol('S', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{M},g)} = \\mathbf{M} \\cos{(g)}, then obtain - g + \\operatorname{C_{2}}{(\\mathbf{M},g)} - \\int \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} d\\mathbf{M} = \\mathbf{M} \\cos{(g)} - g - \\int \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} d\\mathbf{M}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{M},g)} = \\mathbf{M} \\cos{(g)} and \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} = (\\mathbf{M} \\cos{(g)})^{g} and \\int \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} d\\mathbf{M} = \\int (\\mathbf{M} \\cos{(g)})^{g} d\\mathbf{M} and - g + \\operatorname{C_{2}}{(\\mathbf{M},g)} - \\int (\\mathbf{M} \\cos{(g)})^{g} d\\mathbf{M} = \\mathbf{M} \\cos{(g)} - g - \\int (\\mathbf{M} \\cos{(g)})^{g} d\\mathbf{M} and - g + \\operatorname{C_{2}}{(\\mathbf{M},g)} - \\int \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} d\\mathbf{M} = \\mathbf{M} \\cos{(g)} - g - \\int \\operatorname{C_{2}}^{g}{(\\mathbf{M},g)} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Pow(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 1, "Add(Symbol('g', commutative=True), Integral(Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integral(Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(-1), Integral(Pow(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(-1), Integral(Pow(Function('C_2')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{x},\\dot{\\mathbf{r}})} = \\sin{(\\frac{\\dot{\\mathbf{r}}}{\\hat{x}})} and \\mathbf{J}_M{(\\hat{x},\\dot{\\mathbf{r}})} = \\frac{\\dot{\\mathbf{r}}}{\\hat{x}}, then obtain \\sin{(\\mathbf{J}_M{(\\hat{x},\\dot{\\mathbf{r}})})} - \\frac{1}{\\hat{x}} = \\sin{(\\frac{\\dot{\\mathbf{r}}}{\\hat{x}})} - \\frac{1}{\\hat{x}}", "derivation": "\\phi_{1}{(\\hat{x},\\dot{\\mathbf{r}})} = \\sin{(\\frac{\\dot{\\mathbf{r}}}{\\hat{x}})} and \\phi_{1}{(\\hat{x},\\dot{\\mathbf{r}})} - \\frac{1}{\\hat{x}} = \\sin{(\\frac{\\dot{\\mathbf{r}}}{\\hat{x}})} - \\frac{1}{\\hat{x}} and \\mathbf{J}_M{(\\hat{x},\\dot{\\mathbf{r}})} = \\frac{\\dot{\\mathbf{r}}}{\\hat{x}} and \\phi_{1}{(\\hat{x},\\dot{\\mathbf{r}})} = \\sin{(\\mathbf{J}_M{(\\hat{x},\\dot{\\mathbf{r}})})} and \\sin{(\\mathbf{J}_M{(\\hat{x},\\dot{\\mathbf{r}})})} - \\frac{1}{\\hat{x}} = \\sin{(\\frac{\\dot{\\mathbf{r}}}{\\hat{x}})} - \\frac{1}{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))))"], [["minus", 1, "Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))), Add(sin(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(sin(Function('\\\\mathbf{J}_M')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))), Add(sin(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given T{(\\mu,\\varphi^*)} = - \\mu + \\varphi^*, then obtain e^{- \\mu T^{2}{(\\mu,\\varphi^*)}} = e^{- \\mu (- \\mu + \\varphi^*)^{2}}", "derivation": "T{(\\mu,\\varphi^*)} = - \\mu + \\varphi^* and - \\mu T{(\\mu,\\varphi^*)} = - \\mu (- \\mu + \\varphi^*) and - \\mu T^{2}{(\\mu,\\varphi^*)} = - \\mu (- \\mu + \\varphi^*) T{(\\mu,\\varphi^*)} and - \\mu (- \\mu + \\varphi^*) T{(\\mu,\\varphi^*)} = - \\mu (- \\mu + \\varphi^*)^{2} and - \\mu T^{2}{(\\mu,\\varphi^*)} = - \\mu (- \\mu + \\varphi^*)^{2} and e^{- \\mu T^{2}{(\\mu,\\varphi^*)}} = e^{- \\mu (- \\mu + \\varphi^*)^{2}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True))))"], [["times", 2, "Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["exp", 5], "Equality(exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Function('T')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2)))), exp(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\mathbf{p}{(h)} = e^{h}, then obtain \\int \\frac{e^{4 h}}{h^{2} \\mathbf{p}^{2}{(h)}} dh = \\int \\frac{e^{2 h}}{h^{2}} dh", "derivation": "\\mathbf{p}{(h)} = e^{h} and h \\mathbf{p}{(h)} = h e^{h} and h \\mathbf{p}{(h)} e^{- 2 h} = h e^{- h} and \\frac{e^{4 h}}{h^{2} \\mathbf{p}^{2}{(h)}} = \\frac{e^{2 h}}{h^{2}} and \\int \\frac{e^{4 h}}{h^{2} \\mathbf{p}^{2}{(h)}} dh = \\int \\frac{e^{2 h}}{h^{2}} dh", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["times", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\mathbf{p}')(Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), exp(Symbol('h', commutative=True))))"], [["divide", 2, "exp(Mul(Integer(2), Symbol('h', commutative=True)))"], "Equality(Mul(Symbol('h', commutative=True), Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('h', commutative=True)))), Mul(Symbol('h', commutative=True), exp(Mul(Integer(-1), Symbol('h', commutative=True)))))"], [["power", 3, "Integer(-2)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), Integer(-2)), exp(Mul(Integer(4), Symbol('h', commutative=True)))), Mul(Pow(Symbol('h', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('h', commutative=True)))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{p}')(Symbol('h', commutative=True)), Integer(-2)), exp(Mul(Integer(4), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(C,r_{0})} = \\sin^{r_{0}}{(C)}, then obtain 4 \\sin^{r_{0}}{(C)} \\int \\sin{(\\mathbf{H}{(C,r_{0})} + \\sin^{r_{0}}{(C)})} dC = 4 \\sin^{r_{0}}{(C)} \\int \\sin{(2 \\sin^{r_{0}}{(C)})} dC", "derivation": "\\mathbf{H}{(C,r_{0})} = \\sin^{r_{0}}{(C)} and \\mathbf{H}{(C,r_{0})} + \\sin^{r_{0}}{(C)} = 2 \\sin^{r_{0}}{(C)} and \\sin{(\\mathbf{H}{(C,r_{0})} + \\sin^{r_{0}}{(C)})} = \\sin{(2 \\sin^{r_{0}}{(C)})} and \\int \\sin{(\\mathbf{H}{(C,r_{0})} + \\sin^{r_{0}}{(C)})} dC = \\int \\sin{(2 \\sin^{r_{0}}{(C)})} dC and 4 \\sin^{r_{0}}{(C)} \\int \\sin{(\\mathbf{H}{(C,r_{0})} + \\sin^{r_{0}}{(C)})} dC = 4 \\sin^{r_{0}}{(C)} \\int \\sin{(2 \\sin^{r_{0}}{(C)})} dC", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('C', commutative=True), Symbol('r_0', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('C', commutative=True), Symbol('r_0', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True))), Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True))))"], [["sin", 2], "Equality(sin(Add(Function('\\\\mathbf{H}')(Symbol('C', commutative=True), Symbol('r_0', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))), sin(Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(sin(Add(Function('\\\\mathbf{H}')(Symbol('C', commutative=True), Symbol('r_0', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(sin(Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["times", 4, "Mul(Integer(4), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))"], "Equality(Mul(Integer(4), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)), Integral(sin(Add(Function('\\\\mathbf{H}')(Symbol('C', commutative=True), Symbol('r_0', commutative=True)), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))), Tuple(Symbol('C', commutative=True)))), Mul(Integer(4), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)), Integral(sin(Mul(Integer(2), Pow(sin(Symbol('C', commutative=True)), Symbol('r_0', commutative=True)))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given k{(\\rho,A_{x})} = \\cos{(A_{x} - \\rho)}, then obtain (k{(\\rho,A_{x})} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})})^{\\rho} = (\\cos{(A_{x} - \\rho)} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})})^{\\rho}", "derivation": "k{(\\rho,A_{x})} = \\cos{(A_{x} - \\rho)} and \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})} = \\frac{\\partial}{\\partial A_{x}} \\cos{(A_{x} - \\rho)} and k{(\\rho,A_{x})} - \\frac{\\partial}{\\partial A_{x}} \\cos{(A_{x} - \\rho)} = \\cos{(A_{x} - \\rho)} - \\frac{\\partial}{\\partial A_{x}} \\cos{(A_{x} - \\rho)} and k{(\\rho,A_{x})} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})} = \\cos{(A_{x} - \\rho)} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})} and (k{(\\rho,A_{x})} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})})^{\\rho} = (\\cos{(A_{x} - \\rho)} - \\frac{\\partial}{\\partial A_{x}} k{(\\rho,A_{x})})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1)))"], "Equality(Add(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Mul(Integer(-1), Derivative(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))), Add(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), Derivative(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Mul(Integer(-1), Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))), Add(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Mul(Integer(-1), Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))), Symbol('\\\\rho', commutative=True)), Pow(Add(cos(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Mul(Integer(-1), Derivative(Function('k')(Symbol('\\\\rho', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given i{(A_{1},v_{t})} = A_{1}^{v_{t}}, then obtain \\frac{d}{d v_{t}} \\int 0 dv_{t} = \\frac{\\partial}{\\partial v_{t}} \\int (A_{1}^{v_{t}} - i{(A_{1},v_{t})}) dv_{t}", "derivation": "i{(A_{1},v_{t})} = A_{1}^{v_{t}} and 0 = A_{1}^{v_{t}} - i{(A_{1},v_{t})} and \\int 0 dv_{t} = \\int (A_{1}^{v_{t}} - i{(A_{1},v_{t})}) dv_{t} and \\frac{d}{d v_{t}} \\int 0 dv_{t} = \\frac{\\partial}{\\partial v_{t}} \\int (A_{1}^{v_{t}} - i{(A_{1},v_{t})}) dv_{t}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Function('i')(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)))))"], [["integrate", 2, "Symbol('v_t', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_t', commutative=True))), Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Integral(Add(Pow(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('i')(Symbol('A_1', commutative=True), Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(v_{x},\\sigma_x)} = - \\sigma_x + v_{x}, then derive \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} \\operatorname{n_{2}}{(v_{x},\\sigma_x)} = 0, then obtain - \\sigma_x + \\sin{(\\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} (- \\sigma_x + v_{x}))} = - \\sigma_x", "derivation": "\\operatorname{n_{2}}{(v_{x},\\sigma_x)} = - \\sigma_x + v_{x} and \\frac{\\partial}{\\partial v_{x}} \\operatorname{n_{2}}{(v_{x},\\sigma_x)} = \\frac{\\partial}{\\partial v_{x}} (- \\sigma_x + v_{x}) and \\frac{\\partial^{2}}{\\partial \\sigma_x\\partial v_{x}} \\operatorname{n_{2}}{(v_{x},\\sigma_x)} = \\frac{\\partial^{2}}{\\partial \\sigma_x\\partial v_{x}} (- \\sigma_x + v_{x}) and \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} \\operatorname{n_{2}}{(v_{x},\\sigma_x)} = 0 and \\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} (- \\sigma_x + v_{x}) = 0 and \\sin{(\\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} (- \\sigma_x + v_{x}))} = 0 and - \\sigma_x + \\sin{(\\frac{\\partial^{2}}{\\partial v_{x}\\partial \\sigma_x} (- \\sigma_x + v_{x}))} = - \\sigma_x", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('n_2')(Symbol('v_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(0))"], [["sin", 5], "Equality(sin(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Integer(0))"], [["add", 6, "Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), sin(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)), Tuple(Symbol('v_x', commutative=True), Integer(1))))), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given B{(\\mathbf{s})} = \\mathbf{s} and \\mathbf{B}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s}, then obtain \\mathbf{s} (B{(\\mathbf{s})} + \\mathbf{B}{(C_{2},\\mathbf{s})}) = \\mathbf{s} (\\mathbf{s} + \\mathbf{B}{(C_{2},\\mathbf{s})})", "derivation": "B{(\\mathbf{s})} = \\mathbf{s} and C_{2} \\mathbf{s} + B{(\\mathbf{s})} = C_{2} \\mathbf{s} + \\mathbf{s} and \\mathbf{B}{(C_{2},\\mathbf{s})} = C_{2} \\mathbf{s} and B{(\\mathbf{s})} + \\mathbf{B}{(C_{2},\\mathbf{s})} = \\mathbf{s} + \\mathbf{B}{(C_{2},\\mathbf{s})} and \\mathbf{s} (B{(\\mathbf{s})} + \\mathbf{B}{(C_{2},\\mathbf{s})}) = \\mathbf{s} (\\mathbf{s} + \\mathbf{B}{(C_{2},\\mathbf{s})})", "srepr_derivation": [["renaming_premise", "Equality(Function('B')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["add", 1, "Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Function('B')(Symbol('\\\\mathbf{s}', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('B')(Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\mathbf{B}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["times", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Add(Function('B')(Symbol('\\\\mathbf{s}', commutative=True)), Function('\\\\mathbf{B}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\mathbf{B}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(z)} = \\log{(z)}, then derive \\frac{d}{d z} \\hat{\\mathbf{r}}{(z)} = \\frac{1}{z}, then obtain \\frac{\\log{(z)} + \\frac{d}{d z} \\log{(z)}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} + \\log{(z)} = \\frac{\\log{(z)} + \\frac{1}{z}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} + \\log{(z)}", "derivation": "\\hat{\\mathbf{r}}{(z)} = \\log{(z)} and \\frac{d}{d z} \\hat{\\mathbf{r}}{(z)} = \\frac{d}{d z} \\log{(z)} and \\frac{d}{d z} \\hat{\\mathbf{r}}{(z)} = \\frac{1}{z} and \\hat{\\mathbf{r}}{(z)} + \\frac{d}{d z} \\hat{\\mathbf{r}}{(z)} = \\hat{\\mathbf{r}}{(z)} + \\frac{1}{z} and \\log{(z)} + \\frac{d}{d z} \\log{(z)} = \\log{(z)} + \\frac{1}{z} and \\frac{\\log{(z)} + \\frac{d}{d z} \\log{(z)}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} = \\frac{\\log{(z)} + \\frac{1}{z}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} and \\frac{\\log{(z)} + \\frac{d}{d z} \\log{(z)}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} + \\log{(z)} = \\frac{\\log{(z)} + \\frac{1}{z}}{\\sin{(\\sigma_x^{\\mathbf{F}})}} + \\log{(z)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Pow(Symbol('z', commutative=True), Integer(-1)))"], [["add", 3, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('z', commutative=True)), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(log(Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["divide", 5, "sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Add(log(Symbol('z', commutative=True)), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))), Mul(Add(log(Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))), Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"], [["add", 6, "log(Symbol('z', commutative=True))"], "Equality(Add(Mul(Add(log(Symbol('z', commutative=True)), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))), log(Symbol('z', commutative=True))), Add(Mul(Add(log(Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1))), Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))), log(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(f_{E})} = \\sin{(\\sin{(f_{E})})} and r{(f_{E})} = \\hat{x}{(f_{E})} + \\sin{(\\sin{(f_{E})})} and \\rho_{f}{(f_{E})} = \\sin{(\\sin{(f_{E})})}, then obtain (- \\rho_{f}{(f_{E})} + r{(f_{E})})^{2} = \\rho_{f}^{2}{(f_{E})}", "derivation": "\\hat{x}{(f_{E})} = \\sin{(\\sin{(f_{E})})} and \\hat{x}{(f_{E})} + \\sin{(\\sin{(f_{E})})} = 2 \\sin{(\\sin{(f_{E})})} and r{(f_{E})} = \\hat{x}{(f_{E})} + \\sin{(\\sin{(f_{E})})} and r{(f_{E})} = 2 \\sin{(\\sin{(f_{E})})} and \\rho_{f}{(f_{E})} = \\sin{(\\sin{(f_{E})})} and - \\rho_{f}{(f_{E})} + r{(f_{E})} = - \\rho_{f}{(f_{E})} + 2 \\sin{(\\sin{(f_{E})})} and (- \\rho_{f}{(f_{E})} + r{(f_{E})})^{2} = (- \\rho_{f}{(f_{E})} + 2 \\sin{(\\sin{(f_{E})})})^{2} and (- \\rho_{f}{(f_{E})} + r{(f_{E})})^{2} = \\rho_{f}^{2}{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), sin(sin(Symbol('f_E', commutative=True))))"], [["add", 1, "sin(sin(Symbol('f_E', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), sin(sin(Symbol('f_E', commutative=True)))), Mul(Integer(2), sin(sin(Symbol('f_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('r')(Symbol('f_E', commutative=True)), Add(Function('\\\\hat{x}')(Symbol('f_E', commutative=True)), sin(sin(Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('r')(Symbol('f_E', commutative=True)), Mul(Integer(2), sin(sin(Symbol('f_E', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('f_E', commutative=True)), sin(sin(Symbol('f_E', commutative=True))))"], [["minus", 4, "Function('\\\\rho_f')(Symbol('f_E', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f_E', commutative=True))), Function('r')(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f_E', commutative=True))), Mul(Integer(2), sin(sin(Symbol('f_E', commutative=True))))))"], [["power", 6, 2], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f_E', commutative=True))), Function('r')(Symbol('f_E', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f_E', commutative=True))), Mul(Integer(2), sin(sin(Symbol('f_E', commutative=True))))), Integer(2)))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\rho_f')(Symbol('f_E', commutative=True))), Function('r')(Symbol('f_E', commutative=True))), Integer(2)), Pow(Function('\\\\rho_f')(Symbol('f_E', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})} = - \\mathbf{J}_M + \\sin{(f^{\\prime})}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})} = -1, then obtain \\cos{(1)} = \\cos{(\\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})})}", "derivation": "\\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})} = - \\mathbf{J}_M + \\sin{(f^{\\prime})} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})} = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\sin{(f^{\\prime})}) and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})} = -1 and -1 = \\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\sin{(f^{\\prime})}) and \\cos{(1)} = \\cos{(\\frac{\\partial}{\\partial \\mathbf{J}_M} (- \\mathbf{J}_M + \\sin{(f^{\\prime})}))} and \\cos{(1)} = \\cos{(\\frac{\\partial}{\\partial \\mathbf{J}_M} \\mathbf{f}{(\\mathbf{J}_M,f^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Integer(1)), cos(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(cos(Integer(1)), cos(Derivative(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{s})} = e^{\\mathbf{s}}, then derive \\int \\operatorname{c_{0}}{(\\mathbf{s})} d\\mathbf{s} = a^{\\dagger} + e^{\\mathbf{s}}, then obtain \\frac{d}{d a^{\\dagger}} \\int e^{\\mathbf{s}} d\\mathbf{s} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} + e^{\\mathbf{s}})", "derivation": "\\operatorname{c_{0}}{(\\mathbf{s})} = e^{\\mathbf{s}} and \\int \\operatorname{c_{0}}{(\\mathbf{s})} d\\mathbf{s} = \\int e^{\\mathbf{s}} d\\mathbf{s} and \\int \\operatorname{c_{0}}{(\\mathbf{s})} d\\mathbf{s} = a^{\\dagger} + e^{\\mathbf{s}} and \\int e^{\\mathbf{s}} d\\mathbf{s} = a^{\\dagger} + e^{\\mathbf{s}} and \\frac{d}{d a^{\\dagger}} \\int e^{\\mathbf{s}} d\\mathbf{s} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} + e^{\\mathbf{s}})", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)), exp(Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c_0')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 4, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\lambda,C)} = \\frac{\\lambda}{C}, then obtain \\frac{(\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda}}{\\int (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} dC} = \\frac{(-1 + \\frac{\\lambda}{C})^{\\lambda}}{\\int (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} dC}", "derivation": "\\varepsilon_{0}{(\\lambda,C)} = \\frac{\\lambda}{C} and \\varepsilon_{0}{(\\lambda,C)} - 1 = -1 + \\frac{\\lambda}{C} and (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} = (-1 + \\frac{\\lambda}{C})^{\\lambda} and \\int (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} dC = \\int (-1 + \\frac{\\lambda}{C})^{\\lambda} dC and \\frac{(\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda}}{\\int (-1 + \\frac{\\lambda}{C})^{\\lambda} dC} = \\frac{(-1 + \\frac{\\lambda}{C})^{\\lambda}}{\\int (-1 + \\frac{\\lambda}{C})^{\\lambda} dC} and \\frac{(\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda}}{\\int (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} dC} = \\frac{(-1 + \\frac{\\lambda}{C})^{\\lambda}}{\\int (\\varepsilon_{0}{(\\lambda,C)} - 1)^{\\lambda} dC}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["divide", 3, "Integral(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1))), Mul(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1))), Mul(Pow(Add(Integer(-1), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Integral(Pow(Add(Function('\\\\varepsilon_0')(Symbol('\\\\lambda', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('C', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(L)} = \\log{(L)} and T{(n,g^{\\prime}_{\\varepsilon})} = n^{g^{\\prime}_{\\varepsilon}}, then obtain e^{- (\\Psi_{\\lambda}{(L)} - \\log{(L)})^{L}} \\frac{\\partial}{\\partial n} T{(n,g^{\\prime}_{\\varepsilon})} = e^{- (\\Psi_{\\lambda}{(L)} - \\log{(L)})^{L}} \\frac{\\partial}{\\partial n} n^{g^{\\prime}_{\\varepsilon}}", "derivation": "\\Psi_{\\lambda}{(L)} = \\log{(L)} and \\Psi_{\\lambda}{(L)} - \\log{(L)} = 0 and T{(n,g^{\\prime}_{\\varepsilon})} = n^{g^{\\prime}_{\\varepsilon}} and \\frac{\\partial}{\\partial n} T{(n,g^{\\prime}_{\\varepsilon})} = \\frac{\\partial}{\\partial n} n^{g^{\\prime}_{\\varepsilon}} and (\\Psi_{\\lambda}{(L)} - \\log{(L)})^{L} = 0^{L} and e^{- 0^{L}} \\frac{\\partial}{\\partial n} T{(n,g^{\\prime}_{\\varepsilon})} = e^{- 0^{L}} \\frac{\\partial}{\\partial n} n^{g^{\\prime}_{\\varepsilon}} and e^{- (\\Psi_{\\lambda}{(L)} - \\log{(L)})^{L}} \\frac{\\partial}{\\partial n} T{(n,g^{\\prime}_{\\varepsilon})} = e^{- (\\Psi_{\\lambda}{(L)} - \\log{(L)})^{L}} \\frac{\\partial}{\\partial n} n^{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["minus", 1, "log(Symbol('L', commutative=True))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True)))), Integer(0))"], ["get_premise", "Equality(Function('T')(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True)))), Symbol('L', commutative=True)), Pow(Integer(0), Symbol('L', commutative=True)))"], [["divide", 4, "exp(Pow(Integer(0), Symbol('L', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Pow(Integer(0), Symbol('L', commutative=True)))), Derivative(Function('T')(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Pow(Integer(0), Symbol('L', commutative=True)))), Derivative(Pow(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(exp(Mul(Integer(-1), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True)))), Symbol('L', commutative=True)))), Derivative(Function('T')(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('L', commutative=True)), Mul(Integer(-1), log(Symbol('L', commutative=True)))), Symbol('L', commutative=True)))), Derivative(Pow(Symbol('n', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\ddot{x})} = \\log{(\\log{(\\ddot{x})})}, then derive \\frac{d}{d \\ddot{x}} \\mathbf{M}{(\\ddot{x})} = \\frac{1}{\\ddot{x} \\log{(\\ddot{x})}}, then obtain \\frac{d^{2}}{d \\ddot{x}^{2}} \\mathbf{M}{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} \\frac{1}{\\ddot{x} \\log{(\\ddot{x})}}", "derivation": "\\mathbf{M}{(\\ddot{x})} = \\log{(\\log{(\\ddot{x})})} and \\frac{d}{d \\ddot{x}} \\mathbf{M}{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} \\log{(\\log{(\\ddot{x})})} and \\frac{d}{d \\ddot{x}} \\mathbf{M}{(\\ddot{x})} = \\frac{1}{\\ddot{x} \\log{(\\ddot{x})}} and \\frac{d^{2}}{d \\ddot{x}^{2}} \\mathbf{M}{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} \\frac{1}{\\ddot{x} \\log{(\\ddot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), log(log(Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(2))), Derivative(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{s})} = \\mathbf{s}, then obtain \\frac{\\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{A}{(\\mathbf{s})})^{\\mathbf{s}})}}{\\mathbf{s}} = \\frac{\\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{s})^{\\mathbf{s}})}}{\\mathbf{s}}", "derivation": "\\mathbf{A}{(\\mathbf{s})} = \\mathbf{s} and \\frac{d}{d \\mathbf{s}} \\mathbf{A}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\mathbf{s} and (\\frac{d}{d \\mathbf{s}} \\mathbf{A}{(\\mathbf{s})})^{\\mathbf{s}} = (\\frac{d}{d \\mathbf{s}} \\mathbf{s})^{\\mathbf{s}} and \\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{A}{(\\mathbf{s})})^{\\mathbf{s}})} = \\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{s})^{\\mathbf{s}})} and \\frac{\\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{A}{(\\mathbf{s})})^{\\mathbf{s}})}}{\\mathbf{s}} = \\frac{\\sin{((\\frac{d}{d \\mathbf{s}} \\mathbf{s})^{\\mathbf{s}})}}{\\mathbf{s}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True))), sin(Pow(Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), sin(Pow(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), sin(Pow(Derivative(Symbol('\\\\mathbf{s}', commutative=True), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(\\lambda)} = \\log{(\\sin{(\\lambda)})}, then obtain (\\hat{p}_0{(\\lambda)} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} - (\\log{(\\sin{(\\lambda)})} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} = 0", "derivation": "\\hat{p}_0{(\\lambda)} = \\log{(\\sin{(\\lambda)})} and \\hat{p}_0{(\\lambda)} + \\sin{(\\lambda)} = \\log{(\\sin{(\\lambda)})} + \\sin{(\\lambda)} and (\\hat{p}_0{(\\lambda)} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} = (\\log{(\\sin{(\\lambda)})} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} and (\\hat{p}_0{(\\lambda)} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} - (\\log{(\\sin{(\\lambda)})} + \\sin{(\\lambda)}) \\log{(\\sin{(\\lambda)})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), log(sin(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), Add(log(sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "log(sin(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), log(sin(Symbol('\\\\lambda', commutative=True)))), Mul(Add(log(sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))), log(sin(Symbol('\\\\lambda', commutative=True)))))"], [["minus", 3, "Mul(Add(log(sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))), log(sin(Symbol('\\\\lambda', commutative=True))))"], "Equality(Add(Mul(Add(Function('\\\\hat{p}_0')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True))), log(sin(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Add(log(sin(Symbol('\\\\lambda', commutative=True))), sin(Symbol('\\\\lambda', commutative=True))), log(sin(Symbol('\\\\lambda', commutative=True))))), Integer(0))"]]}, {"prompt": "Given l{(P_{e},g)} = \\log{(P_{e} + g)}, then obtain P_{e} + g + \\log{(P_{e} + g)} = P_{e} + g - l{(P_{e},g)} + \\log{(P_{e} + g)} + \\log{(P_{e} + g - l{(P_{e},g)} + \\log{(P_{e} + g)})}", "derivation": "l{(P_{e},g)} = \\log{(P_{e} + g)} and P_{e} + g + l{(P_{e},g)} = P_{e} + g + \\log{(P_{e} + g)} and P_{e} + g = P_{e} + g - l{(P_{e},g)} + \\log{(P_{e} + g)} and P_{e} + g + \\log{(P_{e} + g)} = P_{e} + g - l{(P_{e},g)} + \\log{(P_{e} + g)} + \\log{(P_{e} + g - l{(P_{e},g)} + \\log{(P_{e} + g)})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True))))"], [["add", 1, "Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True))), Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True)))))"], [["minus", 2, "Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True))"], "Equality(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True)), Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True))), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True)))), Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True))), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True))), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Function('l')(Symbol('P_e', commutative=True), Symbol('g', commutative=True))), log(Add(Symbol('P_e', commutative=True), Symbol('g', commutative=True)))))))"]]}, {"prompt": "Given \\Psi{(u)} = \\log{(\\cos{(u)})}, then obtain \\frac{\\Psi{(u)} - 2 \\log{(\\cos{(u)})}}{\\Psi{(u)}} = - \\frac{\\log{(\\cos{(u)})}}{\\Psi{(u)}}", "derivation": "\\Psi{(u)} = \\log{(\\cos{(u)})} and \\Psi{(u)} - \\log{(\\cos{(u)})} = 0 and \\Psi{(u)} - 2 \\log{(\\cos{(u)})} = - \\log{(\\cos{(u)})} and \\frac{\\Psi{(u)} - 2 \\log{(\\cos{(u)})}}{\\Psi{(u)}} = - \\frac{\\log{(\\cos{(u)})}}{\\Psi{(u)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('u', commutative=True)), log(cos(Symbol('u', commutative=True))))"], [["minus", 1, "log(cos(Symbol('u', commutative=True)))"], "Equality(Add(Function('\\\\Psi')(Symbol('u', commutative=True)), Mul(Integer(-1), log(cos(Symbol('u', commutative=True))))), Integer(0))"], [["minus", 2, "log(cos(Symbol('u', commutative=True)))"], "Equality(Add(Function('\\\\Psi')(Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), log(cos(Symbol('u', commutative=True))))), Mul(Integer(-1), log(cos(Symbol('u', commutative=True)))))"], [["divide", 3, "Function('\\\\Psi')(Symbol('u', commutative=True))"], "Equality(Mul(Add(Function('\\\\Psi')(Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), log(cos(Symbol('u', commutative=True))))), Pow(Function('\\\\Psi')(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\Psi')(Symbol('u', commutative=True)), Integer(-1)), log(cos(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{J}_M)} = \\log{(\\cos{(\\mathbf{J}_M)})} and \\nabla{(\\mathbf{J}_M)} = \\varepsilon{(\\mathbf{J}_M)} - \\log{(\\cos{(\\mathbf{J}_M)})}, then obtain 1 = 0^{\\mathbf{J}_M}", "derivation": "\\varepsilon{(\\mathbf{J}_M)} = \\log{(\\cos{(\\mathbf{J}_M)})} and \\varepsilon{(\\mathbf{J}_M)} - \\log{(\\cos{(\\mathbf{J}_M)})} = 0 and \\nabla{(\\mathbf{J}_M)} = \\varepsilon{(\\mathbf{J}_M)} - \\log{(\\cos{(\\mathbf{J}_M)})} and \\nabla{(\\mathbf{J}_M)} = 0 and \\nabla^{\\mathbf{J}_M}{(\\mathbf{J}_M)} = 0^{\\mathbf{J}_M} and (\\varepsilon{(\\mathbf{J}_M)} - \\log{(\\cos{(\\mathbf{J}_M)})})^{\\mathbf{J}_M} = 0^{\\mathbf{J}_M} and 1 = \\nabla^{\\mathbf{J}_M}{(\\mathbf{J}_M)} and 1 = 0^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 1, "log(cos(Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{J}_M', commutative=True)), Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\nabla')(Symbol('\\\\mathbf{J}_M', commutative=True)), Integer(0))"], [["power", 4, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{J}_M', commutative=True))))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(1), Pow(Function('\\\\nabla')(Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integer(1), Pow(Integer(0), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(L_{\\varepsilon})} = \\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}}, then obtain -1 = \\log{(\\frac{\\log{(\\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}})}}{\\log{(\\mathbf{s}{(L_{\\varepsilon})})}})} - 1", "derivation": "\\mathbf{s}{(L_{\\varepsilon})} = \\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}} and \\log{(\\mathbf{s}{(L_{\\varepsilon})})} = \\log{(\\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}})} and 1 = \\frac{\\log{(\\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}})}}{\\log{(\\mathbf{s}{(L_{\\varepsilon})})}} and 0 = \\log{(\\frac{\\log{(\\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}})}}{\\log{(\\mathbf{s}{(L_{\\varepsilon})})}})} and -1 = \\log{(\\frac{\\log{(\\frac{1}{\\sin{(\\cos{(L_{\\varepsilon})})}})}}{\\log{(\\mathbf{s}{(L_{\\varepsilon})})}})} - 1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), log(Pow(sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1))))"], [["divide", 2, "log(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(log(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)), log(Pow(sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)))))"], [["log", 3], "Equality(Integer(0), log(Mul(Pow(log(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)), log(Pow(sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1))))))"], [["minus", 4, 1], "Equality(Integer(-1), Add(log(Mul(Pow(log(Function('\\\\mathbf{s}')(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1)), log(Pow(sin(cos(Symbol('L_{\\\\varepsilon}', commutative=True))), Integer(-1))))), Integer(-1)))"]]}, {"prompt": "Given \\hat{x}_0{(\\Omega,\\theta)} = \\Omega - \\theta and \\operatorname{a^{\\dagger}}{(\\Omega,\\theta)} = \\int (\\Omega - \\theta) d\\Omega, then obtain (\\int (\\Omega - \\theta) d\\Omega)^{\\theta} = \\operatorname{a^{\\dagger}}^{\\theta}{(\\Omega,\\theta)}", "derivation": "\\hat{x}_0{(\\Omega,\\theta)} = \\Omega - \\theta and \\int \\hat{x}_0{(\\Omega,\\theta)} d\\Omega = \\int (\\Omega - \\theta) d\\Omega and \\operatorname{a^{\\dagger}}{(\\Omega,\\theta)} = \\int (\\Omega - \\theta) d\\Omega and \\int \\hat{x}_0{(\\Omega,\\theta)} d\\Omega = \\operatorname{a^{\\dagger}}{(\\Omega,\\theta)} and (\\int \\hat{x}_0{(\\Omega,\\theta)} d\\Omega)^{\\theta} = \\operatorname{a^{\\dagger}}^{\\theta}{(\\Omega,\\theta)} and (\\int (\\Omega - \\theta) d\\Omega)^{\\theta} = \\operatorname{a^{\\dagger}}^{\\theta}{(\\Omega,\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["power", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Integral(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\theta', commutative=True)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\mathbf{P}{(\\rho_b)} = e^{\\rho_b} and I{(\\rho_b)} = \\mathbf{P}{(\\rho_b)} + e^{\\rho_b}, then obtain e^{\\rho_b} (\\frac{d}{d \\rho_b} I{(\\rho_b)})^{2} = e^{\\rho_b} (\\frac{d}{d \\rho_b} 2 \\mathbf{P}{(\\rho_b)})^{2}", "derivation": "\\mathbf{P}{(\\rho_b)} = e^{\\rho_b} and \\mathbf{P}{(\\rho_b)} + e^{\\rho_b} = 2 e^{\\rho_b} and I{(\\rho_b)} = \\mathbf{P}{(\\rho_b)} + e^{\\rho_b} and I{(\\rho_b)} = 2 e^{\\rho_b} and \\frac{d}{d \\rho_b} I{(\\rho_b)} = \\frac{d}{d \\rho_b} 2 e^{\\rho_b} and (\\frac{d}{d \\rho_b} I{(\\rho_b)})^{2} = (\\frac{d}{d \\rho_b} 2 e^{\\rho_b})^{2} and (\\frac{d}{d \\rho_b} I{(\\rho_b)})^{2} = (\\frac{d}{d \\rho_b} 2 \\mathbf{P}{(\\rho_b)})^{2} and e^{\\rho_b} (\\frac{d}{d \\rho_b} I{(\\rho_b)})^{2} = e^{\\rho_b} (\\frac{d}{d \\rho_b} 2 \\mathbf{P}{(\\rho_b)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\rho_b', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True)), Add(Function('\\\\mathbf{P}')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('I')(Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["power", 5, 2], "Equality(Pow(Derivative(Function('I')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Integer(2), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Derivative(Function('I')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2)))"], [["times", 7, "exp(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\rho_b', commutative=True)), Pow(Derivative(Function('I')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2))), Mul(exp(Symbol('\\\\rho_b', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('\\\\mathbf{P}')(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(G)} = e^{G} and \\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})} = (\\Psi^{\\dagger})^{\\mathbf{g}}, then obtain \\int \\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})} e^{- G} d\\Psi^{\\dagger} = \\int (\\Psi^{\\dagger})^{\\mathbf{g}} e^{- G} d\\Psi^{\\dagger}", "derivation": "\\operatorname{A_{1}}{(G)} = e^{G} and \\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})} = (\\Psi^{\\dagger})^{\\mathbf{g}} and \\frac{\\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})}}{\\operatorname{A_{1}}{(G)}} = \\frac{(\\Psi^{\\dagger})^{\\mathbf{g}}}{\\operatorname{A_{1}}{(G)}} and \\int \\frac{\\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})}}{\\operatorname{A_{1}}{(G)}} d\\Psi^{\\dagger} = \\int \\frac{(\\Psi^{\\dagger})^{\\mathbf{g}}}{\\operatorname{A_{1}}{(G)}} d\\Psi^{\\dagger} and \\int \\operatorname{v_{x}}{(\\Psi^{\\dagger},\\mathbf{g})} e^{- G} d\\Psi^{\\dagger} = \\int (\\Psi^{\\dagger})^{\\mathbf{g}} e^{- G} d\\Psi^{\\dagger}", "srepr_derivation": [["get_premise", "Equality(Function('A_1')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], ["get_premise", "Equality(Function('v_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 2, "Function('A_1')(Symbol('G', commutative=True))"], "Equality(Mul(Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(-1)), Function('v_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Mul(Function('v_x')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), exp(Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(\\phi,\\mathbf{s})} = - \\mathbf{s} + \\phi, then obtain \\int 0 d\\mathbf{s} = \\int - 2 (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} d\\mathbf{s}", "derivation": "\\mathbf{r}{(\\phi,\\mathbf{s})} = - \\mathbf{s} + \\phi and 0 = - \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})} and 0 = - (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} and - (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} = - 2 (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} and 0 = - 2 (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} and \\int 0 d\\mathbf{s} = \\int - 2 (- \\mathbf{s} + \\phi - \\mathbf{r}{(\\phi,\\mathbf{s})}) \\mathbf{r}{(\\phi,\\mathbf{s})} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 3, "Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Integer(-1), Integer(2), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))), Function('\\\\mathbf{r}')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(A_{1},y)} = A_{1} + y and \\operatorname{v_{2}}{(m)} = \\cos{(m)}, then obtain (- (A_{1} + y) \\frac{d}{d m} \\operatorname{v_{2}}{(m)})^{A_{1}} = (- (A_{1} + y) \\frac{d}{d m} \\cos{(m)})^{A_{1}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(A_{1},y)} = A_{1} + y and \\operatorname{v_{2}}{(m)} = \\cos{(m)} and \\frac{d}{d m} \\operatorname{v_{2}}{(m)} = \\frac{d}{d m} \\cos{(m)} and - \\operatorname{L_{\\varepsilon}}{(A_{1},y)} \\frac{d}{d m} \\operatorname{v_{2}}{(m)} = - \\operatorname{L_{\\varepsilon}}{(A_{1},y)} \\frac{d}{d m} \\cos{(m)} and (- \\operatorname{L_{\\varepsilon}}{(A_{1},y)} \\frac{d}{d m} \\operatorname{v_{2}}{(m)})^{A_{1}} = (- \\operatorname{L_{\\varepsilon}}{(A_{1},y)} \\frac{d}{d m} \\cos{(m)})^{A_{1}} and (- (A_{1} + y) \\frac{d}{d m} \\operatorname{v_{2}}{(m)})^{A_{1}} = (- (A_{1} + y) \\frac{d}{d m} \\cos{(m)})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('y', commutative=True)))"], ["get_premise", "Equality(Function('v_2')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(Function('v_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(Function('v_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('A_1', commutative=True)), Pow(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('A_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(Integer(-1), Add(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(Function('v_2')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('A_1', commutative=True)), Pow(Mul(Integer(-1), Add(Symbol('A_1', commutative=True), Symbol('y', commutative=True)), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mathbf{J},\\rho_b)} = \\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b and \\dot{x}{(\\mathbf{J},\\rho_b)} = \\hat{H}_{\\lambda}^{\\rho_b}{(\\mathbf{J},\\rho_b)}, then obtain \\dot{x}^{\\rho_b}{(\\mathbf{J},\\rho_b)} = ((\\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b)^{\\rho_b})^{\\rho_b}", "derivation": "\\hat{H}_{\\lambda}{(\\mathbf{J},\\rho_b)} = \\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b and \\hat{H}_{\\lambda}^{\\rho_b}{(\\mathbf{J},\\rho_b)} = (\\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b)^{\\rho_b} and (\\hat{H}_{\\lambda}^{\\rho_b}{(\\mathbf{J},\\rho_b)})^{\\rho_b} = ((\\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b)^{\\rho_b})^{\\rho_b} and \\dot{x}{(\\mathbf{J},\\rho_b)} = \\hat{H}_{\\lambda}^{\\rho_b}{(\\mathbf{J},\\rho_b)} and \\dot{x}^{\\rho_b}{(\\mathbf{J},\\rho_b)} = ((\\int \\frac{\\rho_b}{\\mathbf{J}} d\\rho_b)^{\\rho_b})^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\dot{x}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(Z)} = \\log{(Z)}, then obtain - \\operatorname{n_{2}}{(Z)} + 2 \\int \\operatorname{n_{2}}{(Z)} dZ = - \\operatorname{n_{2}}{(Z)} + 2 \\int \\log{(Z)} dZ", "derivation": "\\operatorname{n_{2}}{(Z)} = \\log{(Z)} and \\int \\operatorname{n_{2}}{(Z)} dZ = \\int \\log{(Z)} dZ and - \\log{(Z)} + \\int \\operatorname{n_{2}}{(Z)} dZ = - \\log{(Z)} + \\int \\log{(Z)} dZ and - \\operatorname{n_{2}}{(Z)} + \\int \\operatorname{n_{2}}{(Z)} dZ = - \\operatorname{n_{2}}{(Z)} + \\int \\log{(Z)} dZ and - \\log{(Z)} + 2 \\int \\operatorname{n_{2}}{(Z)} dZ = - \\log{(Z)} + \\int \\operatorname{n_{2}}{(Z)} dZ + \\int \\log{(Z)} dZ and - \\operatorname{n_{2}}{(Z)} + 2 \\int \\operatorname{n_{2}}{(Z)} dZ = - \\operatorname{n_{2}}{(Z)} + \\int \\operatorname{n_{2}}{(Z)} dZ + \\int \\log{(Z)} dZ and - \\operatorname{n_{2}}{(Z)} + 2 \\int \\operatorname{n_{2}}{(Z)} dZ = - \\operatorname{n_{2}}{(Z)} + 2 \\int \\log{(Z)} dZ", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('Z', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["minus", 2, "log(Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('Z', commutative=True))), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["add", 3, "Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Symbol('Z', commutative=True))), Mul(Integer(2), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Add(Mul(Integer(-1), log(Symbol('Z', commutative=True))), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Mul(Integer(2), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Mul(Integer(2), Integral(Function('n_2')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Add(Mul(Integer(-1), Function('n_2')(Symbol('Z', commutative=True))), Mul(Integer(2), Integral(log(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_l{(\\phi,\\mathbf{A})} = \\frac{e^{\\phi}}{\\mathbf{A}}, then obtain (1 + \\frac{e^{\\phi}}{\\mathbf{A}}) e^{\\phi} + \\hat{H}_l{(\\phi,\\mathbf{A})} = (1 + \\frac{e^{\\phi}}{\\mathbf{A}}) e^{\\phi} + \\frac{e^{\\phi}}{\\mathbf{A}}", "derivation": "\\hat{H}_l{(\\phi,\\mathbf{A})} = \\frac{e^{\\phi}}{\\mathbf{A}} and \\hat{H}_l{(\\phi,\\mathbf{A})} + 1 = 1 + \\frac{e^{\\phi}}{\\mathbf{A}} and (\\hat{H}_l{(\\phi,\\mathbf{A})} + 1) e^{\\phi} = (1 + \\frac{e^{\\phi}}{\\mathbf{A}}) e^{\\phi} and (\\hat{H}_l{(\\phi,\\mathbf{A})} + 1) e^{\\phi} + \\hat{H}_l{(\\phi,\\mathbf{A})} = (\\hat{H}_l{(\\phi,\\mathbf{A})} + 1) e^{\\phi} + \\frac{e^{\\phi}}{\\mathbf{A}} and (1 + \\frac{e^{\\phi}}{\\mathbf{A}}) e^{\\phi} + \\hat{H}_l{(\\phi,\\mathbf{A})} = (1 + \\frac{e^{\\phi}}{\\mathbf{A}}) e^{\\phi} + \\frac{e^{\\phi}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))))"], [["times", 2, "exp(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Add(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), exp(Symbol('\\\\phi', commutative=True))), Mul(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Mul(Add(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), exp(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Add(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), exp(Symbol('\\\\phi', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Add(Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), exp(Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True))), Function('\\\\hat{H}_l')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))), exp(Symbol('\\\\phi', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), exp(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{M},v_{t})} = \\mathbf{M} v_{t}, then obtain (- \\mathbf{M} - f^{*} + \\mathbf{E}{(\\mathbf{M},v_{t})})^{v_{t}} = (\\mathbf{M} v_{t} - \\mathbf{M} - f^{*})^{v_{t}}", "derivation": "\\mathbf{E}{(\\mathbf{M},v_{t})} = \\mathbf{M} v_{t} and - \\mathbf{M} + \\mathbf{E}{(\\mathbf{M},v_{t})} = \\mathbf{M} v_{t} - \\mathbf{M} and - 2 A_{z} - \\mathbf{M} - f^{*} + \\mathbf{E}{(\\mathbf{M},v_{t})} = - 2 A_{z} + \\mathbf{M} v_{t} - \\mathbf{M} - f^{*} and - \\mathbf{M} - f^{*} + \\mathbf{E}{(\\mathbf{M},v_{t})} = \\mathbf{M} v_{t} - \\mathbf{M} - f^{*} and (- \\mathbf{M} - f^{*} + \\mathbf{E}{(\\mathbf{M},v_{t})})^{v_{t}} = (\\mathbf{M} v_{t} - \\mathbf{M} - f^{*})^{v_{t}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(2), Symbol('A_z', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('A_z', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["power", 4, "Symbol('v_t', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Pow(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('f^*', commutative=True))), Symbol('v_t', commutative=True)))"]]}, {"prompt": "Given V{(\\delta)} = \\int \\sin{(\\delta)} d\\delta, then derive V{(\\delta)} = V - \\cos{(\\delta)}, then obtain \\delta + \\frac{d}{d V} V{(\\delta)} + 1 = \\delta + \\frac{\\partial}{\\partial V} (\\mathbf{M} - \\cos{(\\delta)}) + 1", "derivation": "V{(\\delta)} = \\int \\sin{(\\delta)} d\\delta and V{(\\delta)} = V - \\cos{(\\delta)} and \\int \\sin{(\\delta)} d\\delta = V - \\cos{(\\delta)} and \\frac{d}{d V} V{(\\delta)} = \\frac{\\partial}{\\partial V} (V - \\cos{(\\delta)}) and \\delta + \\frac{d}{d V} V{(\\delta)} = \\delta + \\frac{\\partial}{\\partial V} (V - \\cos{(\\delta)}) and \\delta + \\frac{d}{d V} V{(\\delta)} = \\delta + \\frac{d}{d V} \\int \\sin{(\\delta)} d\\delta and \\delta + \\frac{d}{d V} V{(\\delta)} + 1 = \\delta + \\frac{d}{d V} \\int \\sin{(\\delta)} d\\delta + 1 and \\delta + \\frac{d}{d V} V{(\\delta)} + 1 = \\delta + \\frac{\\partial}{\\partial V} (\\mathbf{M} - \\cos{(\\delta)}) + 1", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\delta', commutative=True)), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('V')(Symbol('\\\\delta', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 2, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('V')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('V', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('V')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Symbol('\\\\delta', commutative=True), Derivative(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["add", 6, 1], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('V')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\delta', commutative=True), Derivative(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\delta', commutative=True), Derivative(Function('V')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)), Add(Symbol('\\\\delta', commutative=True), Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given b{(F_{g},\\mathbf{f})} = e^{F_{g} \\mathbf{f}}, then obtain - F_{g} \\mathbf{f} + 2 e^{F_{g} \\mathbf{f}} = - F_{g} \\mathbf{f} + b{(F_{g},\\mathbf{f})} + e^{F_{g} \\mathbf{f}}", "derivation": "b{(F_{g},\\mathbf{f})} = e^{F_{g} \\mathbf{f}} and - F_{g} \\mathbf{f} + b{(F_{g},\\mathbf{f})} = - F_{g} \\mathbf{f} + e^{F_{g} \\mathbf{f}} and - F_{g} \\mathbf{f} + 2 b{(F_{g},\\mathbf{f})} = - F_{g} \\mathbf{f} + b{(F_{g},\\mathbf{f})} + e^{F_{g} \\mathbf{f}} and - F_{g} \\mathbf{f} + 2 b{(F_{g},\\mathbf{f})} = - F_{g} \\mathbf{f} + 2 e^{F_{g} \\mathbf{f}} and - F_{g} \\mathbf{f} + 2 e^{F_{g} \\mathbf{f}} = - F_{g} \\mathbf{f} + b{(F_{g},\\mathbf{f})} + e^{F_{g} \\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 1, "Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Function('b')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(b)} = \\sin{(b)} and a{(b)} = \\sin{(b)}, then obtain 1 = \\frac{\\cos{(\\operatorname{z^{*}}{(b)})}}{\\cos{(a{(b)})}}", "derivation": "\\operatorname{z^{*}}{(b)} = \\sin{(b)} and a{(b)} = \\sin{(b)} and \\cos{(a{(b)})} = \\cos{(\\sin{(b)})} and 1 = \\frac{\\cos{(\\sin{(b)})}}{\\cos{(a{(b)})}} and \\cos{(a{(b)})} = \\cos{(\\operatorname{z^{*}}{(b)})} and \\cos{(\\operatorname{z^{*}}{(b)})} = \\cos{(\\sin{(b)})} and 1 = \\frac{\\cos{(\\operatorname{z^{*}}{(b)})}}{\\cos{(a{(b)})}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["cos", 2], "Equality(cos(Function('a')(Symbol('b', commutative=True))), cos(sin(Symbol('b', commutative=True))))"], [["divide", 3, "cos(Function('a')(Symbol('b', commutative=True)))"], "Equality(Integer(1), Mul(Pow(cos(Function('a')(Symbol('b', commutative=True))), Integer(-1)), cos(sin(Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(cos(Function('a')(Symbol('b', commutative=True))), cos(Function('z^*')(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(cos(Function('z^*')(Symbol('b', commutative=True))), cos(sin(Symbol('b', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Integer(1), Mul(Pow(cos(Function('a')(Symbol('b', commutative=True))), Integer(-1)), cos(Function('z^*')(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given B{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\mathbf{H}{(\\psi^*)} = e^{e^{\\psi^*}}, then obtain \\int \\tilde{\\infty} \\mathbf{H}{(\\psi^*)} d\\psi^* = \\int \\tilde{\\infty} e^{e^{\\psi^*}} d\\psi^*", "derivation": "B{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\mathbf{H}{(\\psi^*)} = e^{e^{\\psi^*}} and \\frac{\\mathbf{H}{(\\psi^*)}}{- B{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}} = \\frac{e^{e^{\\psi^*}}}{- B{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}} and \\tilde{\\infty} \\mathbf{H}{(\\psi^*)} = \\tilde{\\infty} e^{e^{\\psi^*}} and \\int \\tilde{\\infty} \\mathbf{H}{(\\psi^*)} d\\psi^* = \\int \\tilde{\\infty} e^{e^{\\psi^*}} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\psi^*', commutative=True)), exp(exp(Symbol('\\\\psi^*', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Function('B')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(-1)), exp(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(zoo, Function('\\\\mathbf{H}')(Symbol('\\\\psi^*', commutative=True))), Mul(zoo, exp(exp(Symbol('\\\\psi^*', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Mul(zoo, Function('\\\\mathbf{H}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(zoo, exp(exp(Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given g{(\\varepsilon_0,F_{N})} = F_{N} \\varepsilon_0 and \\mu_{0}{(\\varepsilon_0,F_{N})} = \\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})}, then obtain \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})})} = \\cos{(F_{N})}", "derivation": "g{(\\varepsilon_0,F_{N})} = F_{N} \\varepsilon_0 and \\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})} = \\frac{\\partial}{\\partial \\varepsilon_0} F_{N} \\varepsilon_0 and \\mu_{0}{(\\varepsilon_0,F_{N})} = \\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})} and \\mu_{0}{(\\varepsilon_0,F_{N})} = \\frac{\\partial}{\\partial \\varepsilon_0} F_{N} \\varepsilon_0 and \\cos{(\\mu_{0}{(\\varepsilon_0,F_{N})})} = \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} F_{N} \\varepsilon_0)} and \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})})} = \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} F_{N} \\varepsilon_0)} and \\cos{(\\frac{\\partial}{\\partial \\varepsilon_0} g{(\\varepsilon_0,F_{N})})} = \\cos{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Derivative(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mu_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Function('\\\\mu_0')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True))), cos(Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(cos(Derivative(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Derivative(Mul(Symbol('F_N', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 6], "Equality(cos(Derivative(Function('g')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1)))), cos(Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\varepsilon)} = \\log{(\\varepsilon)}, then obtain (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) \\int (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) d\\varepsilon = (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) \\int (\\varepsilon + \\log{(\\varepsilon)}) d\\varepsilon", "derivation": "\\operatorname{y^{\\prime}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)} = \\varepsilon + \\log{(\\varepsilon)} and \\int (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) d\\varepsilon = \\int (\\varepsilon + \\log{(\\varepsilon)}) d\\varepsilon and (\\varepsilon + \\log{(\\varepsilon)}) \\int (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) d\\varepsilon = (\\varepsilon + \\log{(\\varepsilon)}) \\int (\\varepsilon + \\log{(\\varepsilon)}) d\\varepsilon and (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) \\int (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) d\\varepsilon = (\\varepsilon + \\operatorname{y^{\\prime}}{(\\varepsilon)}) \\int (\\varepsilon + \\log{(\\varepsilon)}) d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["add", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 3, "Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Add(Symbol('\\\\varepsilon', commutative=True), Function('y^{\\\\prime}')(Symbol('\\\\varepsilon', commutative=True))), Integral(Add(Symbol('\\\\varepsilon', commutative=True), log(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(n)} = e^{n}, then derive \\frac{d}{d n} \\mathbf{M}{(n)} = e^{n}, then obtain \\frac{d}{d n} (n + e^{n}) - 1 = \\frac{d}{d n} (n + \\frac{d}{d n} e^{n}) - 1", "derivation": "\\mathbf{M}{(n)} = e^{n} and n + \\mathbf{M}{(n)} = n + e^{n} and \\frac{d}{d n} (n + \\mathbf{M}{(n)}) = \\frac{d}{d n} (n + e^{n}) and \\frac{d}{d n} (n + \\mathbf{M}{(n)}) - 1 = \\frac{d}{d n} (n + e^{n}) - 1 and \\frac{d}{d n} \\mathbf{M}{(n)} = e^{n} and \\frac{d}{d n} e^{n} = e^{n} and \\frac{d}{d n} (n + \\mathbf{M}{(n)}) - 1 = \\frac{d}{d n} (n + \\frac{d}{d n} e^{n}) - 1 and \\frac{d}{d n} (n + e^{n}) - 1 = \\frac{d}{d n} (n + \\frac{d}{d n} e^{n}) - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["add", 1, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Symbol('n', commutative=True), Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Add(Symbol('n', commutative=True), Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), exp(Symbol('n', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), exp(Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Derivative(Add(Symbol('n', commutative=True), Function('\\\\mathbf{M}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('n', commutative=True), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Add(Derivative(Add(Symbol('n', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('n', commutative=True), Derivative(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)} = z^{V_{\\mathbf{E}}}, then derive \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)} - 1 = z^{V_{\\mathbf{E}}} \\log{(z)} - 1, then obtain \\frac{\\partial}{\\partial V_{\\mathbf{E}}} z^{V_{\\mathbf{E}}} - 1 = z^{V_{\\mathbf{E}}} \\log{(z)} - 1", "derivation": "\\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)} = z^{V_{\\mathbf{E}}} and - V_{\\mathbf{E}} + \\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)} = - V_{\\mathbf{E}} + z^{V_{\\mathbf{E}}} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- V_{\\mathbf{E}} + \\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)}) = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- V_{\\mathbf{E}} + z^{V_{\\mathbf{E}}}) and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\hat{\\mathbf{r}}{(V_{\\mathbf{E}},z)} - 1 = z^{V_{\\mathbf{E}}} \\log{(z)} - 1 and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} z^{V_{\\mathbf{E}}} - 1 = z^{V_{\\mathbf{E}}} \\log{(z)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["minus", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('z', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('z', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{S},A_{z})} = A_{z} + \\log{(\\mathbf{S})}, then derive \\frac{\\partial}{\\partial A_{z}} \\operatorname{t_{2}}{(\\mathbf{S},A_{z})} = 1, then obtain 1 = \\frac{1}{\\frac{\\partial}{\\partial A_{z}} (A_{z} + \\log{(\\mathbf{S})})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{S},A_{z})} = A_{z} + \\log{(\\mathbf{S})} and \\frac{\\partial}{\\partial A_{z}} \\operatorname{t_{2}}{(\\mathbf{S},A_{z})} = \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\log{(\\mathbf{S})}) and \\frac{\\partial}{\\partial A_{z}} \\operatorname{t_{2}}{(\\mathbf{S},A_{z})} = 1 and \\frac{\\partial}{\\partial A_{z}} (A_{z} + \\log{(\\mathbf{S})}) = 1 and 1 = \\frac{1}{\\frac{\\partial}{\\partial A_{z}} (A_{z} + \\log{(\\mathbf{S})})}", "srepr_derivation": [["get_premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Derivative(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Integer(1), Pow(Derivative(Add(Symbol('A_z', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(p,f,t_{1})} = \\frac{f^{p}}{t_{1}} and \\mathbf{g}{(p,f,t_{1})} = \\frac{f^{p}}{t_{1}}, then obtain \\operatorname{x^{{\\}'}}^{t_{1}}{(p,f,t_{1})} = \\mathbf{g}^{t_{1}}{(p,f,t_{1})}", "derivation": "\\operatorname{x^{{\\}'}}{(p,f,t_{1})} = \\frac{f^{p}}{t_{1}} and \\mathbf{g}{(p,f,t_{1})} = \\frac{f^{p}}{t_{1}} and \\operatorname{x^{{\\}'}}{(p,f,t_{1})} = \\mathbf{g}{(p,f,t_{1})} and \\operatorname{x^{{\\}'}}^{t_{1}}{(p,f,t_{1})} = \\mathbf{g}^{t_{1}}{(p,f,t_{1})}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('t_1', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('x^\\\\prime')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)), Function('\\\\mathbf{g}')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Function('\\\\mathbf{g}')(Symbol('p', commutative=True), Symbol('f', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(f_{E},\\hat{x}_0)} = e^{\\frac{f_{E}}{\\hat{x}_0}}, then derive \\int (- f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)}) df_{E} = U + \\hat{x}_0 e^{\\frac{f_{E}}{\\hat{x}_0}} - \\frac{f_{E}^{2}}{2}, then obtain \\int (- f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)}) df_{E} = U + \\hat{x}_0 \\phi_{1}{(f_{E},\\hat{x}_0)} - \\frac{f_{E}^{2}}{2}", "derivation": "\\phi_{1}{(f_{E},\\hat{x}_0)} = e^{\\frac{f_{E}}{\\hat{x}_0}} and - f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)} = - f_{E} + e^{\\frac{f_{E}}{\\hat{x}_0}} and \\int (- f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)}) df_{E} = \\int (- f_{E} + e^{\\frac{f_{E}}{\\hat{x}_0}}) df_{E} and \\int (- f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)}) df_{E} = U + \\hat{x}_0 e^{\\frac{f_{E}}{\\hat{x}_0}} - \\frac{f_{E}^{2}}{2} and \\int (- f_{E} + \\phi_{1}{(f_{E},\\hat{x}_0)}) df_{E} = U + \\hat{x}_0 \\phi_{1}{(f_{E},\\hat{x}_0)} - \\frac{f_{E}^{2}}{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), exp(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))))"], [["minus", 1, "Symbol('f_E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), exp(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('U', commutative=True), Mul(Symbol('\\\\hat{x}_0', commutative=True), exp(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f_E', commutative=True)), Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('U', commutative=True), Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\phi_1')(Symbol('f_E', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\phi_{1}{(\\eta^{\\prime},y^{\\prime})} = \\eta^{\\prime} e^{y^{\\prime}} and \\theta_{2}{(\\eta^{\\prime},y^{\\prime})} = \\eta^{\\prime} e^{y^{\\prime}}, then obtain \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\phi_{1}{(\\eta^{\\prime},y^{\\prime})}) = \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\theta_{2}{(\\eta^{\\prime},y^{\\prime})})", "derivation": "\\phi_{1}{(\\eta^{\\prime},y^{\\prime})} = \\eta^{\\prime} e^{y^{\\prime}} and y^{\\prime} + \\phi_{1}{(\\eta^{\\prime},y^{\\prime})} = \\eta^{\\prime} e^{y^{\\prime}} + y^{\\prime} and \\theta_{2}{(\\eta^{\\prime},y^{\\prime})} = \\eta^{\\prime} e^{y^{\\prime}} and y^{\\prime} + \\phi_{1}{(\\eta^{\\prime},y^{\\prime})} = y^{\\prime} + \\theta_{2}{(\\eta^{\\prime},y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\phi_{1}{(\\eta^{\\prime},y^{\\prime})}) = \\frac{\\partial}{\\partial y^{\\prime}} (y^{\\prime} + \\theta_{2}{(\\eta^{\\prime},y^{\\prime})})", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\theta_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\phi_1')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\theta_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f^{*},\\rho_b)} = e^{\\rho_b^{f^{*}}}, then obtain \\tilde{\\infty} (2 \\operatorname{n_{1}}{(f^{*},\\rho_b)})^{f^{*}} = \\tilde{\\infty} (\\operatorname{n_{1}}{(f^{*},\\rho_b)} + e^{\\rho_b^{f^{*}}})^{f^{*}}", "derivation": "\\operatorname{n_{1}}{(f^{*},\\rho_b)} = e^{\\rho_b^{f^{*}}} and 2 \\operatorname{n_{1}}{(f^{*},\\rho_b)} = \\operatorname{n_{1}}{(f^{*},\\rho_b)} + e^{\\rho_b^{f^{*}}} and (2 \\operatorname{n_{1}}{(f^{*},\\rho_b)})^{f^{*}} = (\\operatorname{n_{1}}{(f^{*},\\rho_b)} + e^{\\rho_b^{f^{*}}})^{f^{*}} and \\tilde{\\infty} (2 \\operatorname{n_{1}}{(f^{*},\\rho_b)})^{f^{*}} = \\tilde{\\infty} (\\operatorname{n_{1}}{(f^{*},\\rho_b)} + e^{\\rho_b^{f^{*}}})^{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('f^*', commutative=True))))"], [["add", 1, "Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True))), Add(Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('f^*', commutative=True)))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('f^*', commutative=True)), Pow(Add(Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True)))"], [["divide", 3, 0], "Equality(Mul(zoo, Pow(Mul(Integer(2), Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True))), Symbol('f^*', commutative=True))), Mul(zoo, Pow(Add(Function('n_1')(Symbol('f^*', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Pow(Symbol('\\\\rho_b', commutative=True), Symbol('f^*', commutative=True)))), Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given c{(\\mathbf{J}_M)} = \\log{(\\log{(\\mathbf{J}_M)})}, then obtain c{(\\mathbf{J}_M)} - \\log{(\\log{(\\mathbf{J}_M)})} + \\log{(\\log{(\\mathbf{J}_M)})}^{\\mathbf{J}_M} = \\log{(\\log{(\\mathbf{J}_M)})}^{\\mathbf{J}_M}", "derivation": "c{(\\mathbf{J}_M)} = \\log{(\\log{(\\mathbf{J}_M)})} and 0 = - c{(\\mathbf{J}_M)} + \\log{(\\log{(\\mathbf{J}_M)})} and c{(\\mathbf{J}_M)} - \\log{(\\log{(\\mathbf{J}_M)})} = 0 and c{(\\mathbf{J}_M)} - \\log{(\\log{(\\mathbf{J}_M)})} + \\log{(\\log{(\\mathbf{J}_M)})}^{\\mathbf{J}_M} = \\log{(\\log{(\\mathbf{J}_M)})}^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True)), log(log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 1, "Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True))), log(log(Symbol('\\\\mathbf{J}_M', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True))), log(log(Symbol('\\\\mathbf{J}_M', commutative=True))))"], "Equality(Add(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\mathbf{J}_M', commutative=True))))), Integer(0))"], [["add", 3, "Pow(log(log(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\mathbf{J}_M', commutative=True)))), Pow(log(log(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))), Pow(log(log(Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\mathbf{M}{(V_{\\mathbf{B}})}, then obtain \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} - \\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})} = 0", "derivation": "\\mathbf{M}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\mathbf{M}{(V_{\\mathbf{B}})} and \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})} and \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} + 1 = \\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})} + 1 and \\hat{H}_{\\lambda}{(V_{\\mathbf{B}})} - \\frac{d}{d V_{\\mathbf{B}}} \\log{(V_{\\mathbf{B}})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Derivative(Function('\\\\mathbf{M}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["add", 3, 1], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(1)), Add(Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1)))"], [["minus", 4, "Add(Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), Derivative(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(C_{1},v_{x})} = C_{1}^{v_{x}}, then derive \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{x}}{(C_{1},v_{x})} = \\frac{C_{1}^{v_{x}} v_{x}}{C_{1}}, then obtain \\frac{\\partial}{\\partial C_{1}} C_{1}^{v_{x}} = \\frac{C_{1}^{v_{x}} v_{x}}{C_{1}}", "derivation": "\\operatorname{A_{x}}{(C_{1},v_{x})} = C_{1}^{v_{x}} and v_{x} + \\operatorname{A_{x}}{(C_{1},v_{x})} = C_{1}^{v_{x}} + v_{x} and \\frac{\\partial}{\\partial C_{1}} (v_{x} + \\operatorname{A_{x}}{(C_{1},v_{x})}) = \\frac{\\partial}{\\partial C_{1}} (C_{1}^{v_{x}} + v_{x}) and \\frac{\\partial}{\\partial C_{1}} \\operatorname{A_{x}}{(C_{1},v_{x})} = \\frac{C_{1}^{v_{x}} v_{x}}{C_{1}} and \\frac{\\partial}{\\partial C_{1}} C_{1}^{v_{x}} = \\frac{C_{1}^{v_{x}} v_{x}}{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Function('A_x')(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True))), Add(Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Symbol('v_x', commutative=True), Function('A_x')(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('A_x')(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('C_1', commutative=True), Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\psi)} = \\cos{(\\psi)}, then obtain \\frac{\\operatorname{C_{d}}{(\\psi)}}{2 \\cos{(\\psi)}} - \\cos{(\\psi)} = \\frac{1}{2} - \\cos{(\\psi)}", "derivation": "\\operatorname{C_{d}}{(\\psi)} = \\cos{(\\psi)} and \\operatorname{C_{d}}{(\\psi)} + \\cos{(\\psi)} = 2 \\cos{(\\psi)} and \\frac{\\operatorname{C_{d}}{(\\psi)}}{\\operatorname{C_{d}}{(\\psi)} + \\cos{(\\psi)}} = \\frac{\\cos{(\\psi)}}{\\operatorname{C_{d}}{(\\psi)} + \\cos{(\\psi)}} and - \\cos{(\\psi)} + \\frac{\\operatorname{C_{d}}{(\\psi)}}{\\operatorname{C_{d}}{(\\psi)} + \\cos{(\\psi)}} = - \\cos{(\\psi)} + \\frac{\\cos{(\\psi)}}{\\operatorname{C_{d}}{(\\psi)} + \\cos{(\\psi)}} and \\frac{\\operatorname{C_{d}}{(\\psi)}}{2 \\cos{(\\psi)}} - \\cos{(\\psi)} = \\frac{1}{2} - \\cos{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\psi', commutative=True))))"], [["divide", 1, "Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Pow(Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Integer(-1)), Function('C_d')(Symbol('\\\\psi', commutative=True))), Mul(Pow(Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Integer(-1)), cos(Symbol('\\\\psi', commutative=True))))"], [["minus", 3, "cos(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True))), Mul(Pow(Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Integer(-1)), Function('C_d')(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True))), Mul(Pow(Add(Function('C_d')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Integer(-1)), cos(Symbol('\\\\psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Rational(1, 2), Function('C_d')(Symbol('\\\\psi', commutative=True)), Pow(cos(Symbol('\\\\psi', commutative=True)), Integer(-1))), Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True)))), Add(Rational(1, 2), Mul(Integer(-1), cos(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(y,Q)} = \\log{(Q y)}, then obtain (- Q y + \\operatorname{m_{s}}{(y,Q)})^{Q} (- Q y + \\operatorname{m_{s}}{(y,Q)})^{y} = (- Q y + \\operatorname{m_{s}}{(y,Q)})^{Q} (- Q y + \\log{(Q y)})^{y}", "derivation": "\\operatorname{m_{s}}{(y,Q)} = \\log{(Q y)} and - Q y + \\operatorname{m_{s}}{(y,Q)} = - Q y + \\log{(Q y)} and (- Q y + \\operatorname{m_{s}}{(y,Q)})^{y} = (- Q y + \\log{(Q y)})^{y} and (- Q y + \\operatorname{m_{s}}{(y,Q)})^{Q} (- Q y + \\operatorname{m_{s}}{(y,Q)})^{y} = (- Q y + \\operatorname{m_{s}}{(y,Q)})^{Q} (- Q y + \\log{(Q y)})^{y}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True)), log(Mul(Symbol('Q', commutative=True), Symbol('y', commutative=True))))"], [["minus", 1, "Mul(Symbol('Q', commutative=True), Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), log(Mul(Symbol('Q', commutative=True), Symbol('y', commutative=True)))))"], [["power", 2, "Symbol('y', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Symbol('y', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), log(Mul(Symbol('Q', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True)))"], [["times", 3, "Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Symbol('y', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), Function('m_s')(Symbol('y', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('y', commutative=True)), log(Mul(Symbol('Q', commutative=True), Symbol('y', commutative=True)))), Symbol('y', commutative=True))))"]]}, {"prompt": "Given l{(\\mathbf{r})} = e^{\\mathbf{r}} and \\sigma_{p}{(\\mathbf{r})} = \\mathbf{r} e^{\\mathbf{r}}, then obtain \\mathbf{r} e^{\\mathbf{r}} \\sin{(l{(\\mathbf{r})})} = \\mathbf{r} e^{\\mathbf{r}} \\sin{(e^{\\mathbf{r}})}", "derivation": "l{(\\mathbf{r})} = e^{\\mathbf{r}} and \\sin{(l{(\\mathbf{r})})} = \\sin{(e^{\\mathbf{r}})} and \\sigma_{p}{(\\mathbf{r})} = \\mathbf{r} e^{\\mathbf{r}} and \\sigma_{p}{(\\mathbf{r})} \\sin{(l{(\\mathbf{r})})} = \\sigma_{p}{(\\mathbf{r})} \\sin{(e^{\\mathbf{r}})} and \\mathbf{r} e^{\\mathbf{r}} \\sin{(l{(\\mathbf{r})})} = \\mathbf{r} e^{\\mathbf{r}} \\sin{(e^{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('l')(Symbol('\\\\mathbf{r}', commutative=True))), sin(exp(Symbol('\\\\mathbf{r}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\mathbf{r}', commutative=True))))"], [["times", 2, "Function('\\\\sigma_p')(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbf{r}', commutative=True)), sin(Function('l')(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Function('\\\\sigma_p')(Symbol('\\\\mathbf{r}', commutative=True)), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\mathbf{r}', commutative=True)), sin(Function('l')(Symbol('\\\\mathbf{r}', commutative=True)))), Mul(Symbol('\\\\mathbf{r}', commutative=True), exp(Symbol('\\\\mathbf{r}', commutative=True)), sin(exp(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given G{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})}, then derive \\int G{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = t - \\cos{(\\dot{\\mathbf{r}})}, then obtain \\cos{(\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})} = \\cos{(t - \\cos{(\\dot{\\mathbf{r}})})}", "derivation": "G{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})} and \\int G{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = \\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\int G{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} = t - \\cos{(\\dot{\\mathbf{r}})} and t - \\cos{(\\dot{\\mathbf{r}})} = \\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}} and \\cos{(\\int G{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})} = \\cos{(\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})} and \\cos{(\\int G{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})} = \\cos{(t - \\cos{(\\dot{\\mathbf{r}})})} and \\cos{(\\int \\sin{(\\dot{\\mathbf{r}})} d\\dot{\\mathbf{r}})} = \\cos{(t - \\cos{(\\dot{\\mathbf{r}})})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(cos(Integral(Function('G')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(cos(Integral(sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), cos(Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))))"]]}, {"prompt": "Given \\mu{(Z,Q)} = Q - Z, then derive \\int \\mu{(Z,Q)} dQ = B + \\frac{Q^{2}}{2} - Q Z, then obtain \\frac{\\partial}{\\partial Z} (\\mu{(Z,Q)} (\\int \\mu{(Z,Q)} dQ)^{B} + \\int (Q - Z) dQ) = \\frac{\\partial}{\\partial Z} (B + \\frac{Q^{2}}{2} - Q Z + \\mu{(Z,Q)} (\\int \\mu{(Z,Q)} dQ)^{B})", "derivation": "\\mu{(Z,Q)} = Q - Z and \\int \\mu{(Z,Q)} dQ = \\int (Q - Z) dQ and \\int \\mu{(Z,Q)} dQ = B + \\frac{Q^{2}}{2} - Q Z and \\int (Q - Z) dQ = B + \\frac{Q^{2}}{2} - Q Z and (Q - Z) (\\int \\mu{(Z,Q)} dQ)^{B} + \\int (Q - Z) dQ = B + \\frac{Q^{2}}{2} - Q Z + (Q - Z) (\\int \\mu{(Z,Q)} dQ)^{B} and \\frac{\\partial}{\\partial Z} ((Q - Z) (\\int \\mu{(Z,Q)} dQ)^{B} + \\int (Q - Z) dQ) = \\frac{\\partial}{\\partial Z} (B + \\frac{Q^{2}}{2} - Q Z + (Q - Z) (\\int \\mu{(Z,Q)} dQ)^{B}) and \\frac{\\partial}{\\partial Z} (\\mu{(Z,Q)} (\\int \\mu{(Z,Q)} dQ)^{B} + \\int (Q - Z) dQ) = \\frac{\\partial}{\\partial Z} (B + \\frac{Q^{2}}{2} - Q Z + \\mu{(Z,Q)} (\\int \\mu{(Z,Q)} dQ)^{B})", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Q', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('Z', commutative=True))))"], [["add", 4, "Mul(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True)))"], "Equality(Add(Mul(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Mul(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True)))))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Mul(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Derivative(Add(Mul(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Mul(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Pow(Integral(Function('\\\\mu')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('B', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(C,\\mathbf{v})} = \\sin{(C \\mathbf{v})}, then obtain \\iint 2 m^{2}{(C,\\mathbf{v})} dC dC = \\iint (m{(C,\\mathbf{v})} + \\sin{(C \\mathbf{v})}) m{(C,\\mathbf{v})} dC dC", "derivation": "m{(C,\\mathbf{v})} = \\sin{(C \\mathbf{v})} and 2 m{(C,\\mathbf{v})} = m{(C,\\mathbf{v})} + \\sin{(C \\mathbf{v})} and 2 m^{2}{(C,\\mathbf{v})} = (m{(C,\\mathbf{v})} + \\sin{(C \\mathbf{v})}) m{(C,\\mathbf{v})} and \\int 2 m^{2}{(C,\\mathbf{v})} dC = \\int (m{(C,\\mathbf{v})} + \\sin{(C \\mathbf{v})}) m{(C,\\mathbf{v})} dC and \\iint 2 m^{2}{(C,\\mathbf{v})} dC dC = \\iint (m{(C,\\mathbf{v})} + \\sin{(C \\mathbf{v})}) m{(C,\\mathbf{v})} dC dC", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 1, "Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(2), Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Add(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))))"], [["times", 2, "Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))), Mul(Add(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Add(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(2))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Mul(Add(Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Mul(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), Function('m')(Symbol('C', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given J{(A_{2},Q)} = - A_{2} + Q, then obtain \\int \\frac{\\partial}{\\partial A_{2}} (J{(A_{2},Q)} + 1) dA_{2} = \\int \\frac{\\partial}{\\partial A_{2}} (- A_{2} + Q + 1) dA_{2}", "derivation": "J{(A_{2},Q)} = - A_{2} + Q and J{(A_{2},Q)} + 1 = - A_{2} + Q + 1 and \\frac{\\partial}{\\partial A_{2}} (J{(A_{2},Q)} + 1) = \\frac{\\partial}{\\partial A_{2}} (- A_{2} + Q + 1) and \\int \\frac{\\partial}{\\partial A_{2}} (J{(A_{2},Q)} + 1) dA_{2} = \\int \\frac{\\partial}{\\partial A_{2}} (- A_{2} + Q + 1) dA_{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Q', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('J')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Q', commutative=True), Integer(1)))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Derivative(Add(Function('J')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('Q', commutative=True), Integer(1)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Tuple(Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(J)} = \\log{(J)}, then derive J + \\mathbf{r} = \\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ, then obtain \\frac{d}{d \\mathbf{r}} (\\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ)^{J} = \\frac{\\partial}{\\partial \\mathbf{r}} (J + \\mathbf{r})^{J}", "derivation": "\\operatorname{P_{e}}{(J)} = \\log{(J)} and \\operatorname{P_{e}}^{2}{(J)} = \\operatorname{P_{e}}{(J)} \\log{(J)} and 1 = \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} and \\int 1 dJ = \\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ and J + \\mathbf{r} = \\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ and (\\int 1 dJ)^{J} = (\\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ)^{J} and (\\int 1 dJ)^{J} = (J + \\mathbf{r})^{J} and (\\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ)^{J} = (J + \\mathbf{r})^{J} and \\frac{d}{d \\mathbf{r}} (\\int \\frac{\\log{(J)}}{\\operatorname{P_{e}}{(J)}} dJ)^{J} = \\frac{\\partial}{\\partial \\mathbf{r}} (J + \\mathbf{r})^{J}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('J', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(2)), Mul(Function('P_e')(Symbol('J', commutative=True)), log(Symbol('J', commutative=True))))"], [["divide", 2, "Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(2))"], "Equality(Integer(1), Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Integral(Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Pow(Integral(Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('J', commutative=True)))"], [["differentiate", 8, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Pow(Integral(Mul(Pow(Function('P_e')(Symbol('J', commutative=True)), Integer(-1)), log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C_{2})} = e^{C_{2}}, then derive \\frac{d}{d C_{2}} \\operatorname{M_{E}}{(C_{2})} = e^{C_{2}}, then obtain \\frac{\\frac{d}{d C_{2}} \\operatorname{M_{E}}{(C_{2})}}{\\operatorname{M_{E}}{(C_{2})} \\frac{d}{d C_{2}} e^{C_{2}}} = \\frac{e^{C_{2}}}{\\operatorname{M_{E}}{(C_{2})} \\frac{d}{d C_{2}} e^{C_{2}}}", "derivation": "\\operatorname{M_{E}}{(C_{2})} = e^{C_{2}} and \\frac{d}{d C_{2}} \\operatorname{M_{E}}{(C_{2})} = \\frac{d}{d C_{2}} e^{C_{2}} and \\frac{d}{d C_{2}} \\operatorname{M_{E}}{(C_{2})} = e^{C_{2}} and \\frac{\\frac{d}{d C_{2}} \\operatorname{M_{E}}{(C_{2})}}{\\operatorname{M_{E}}{(C_{2})} \\frac{d}{d C_{2}} e^{C_{2}}} = \\frac{e^{C_{2}}}{\\operatorname{M_{E}}{(C_{2})} \\frac{d}{d C_{2}} e^{C_{2}}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C_2', commutative=True)), exp(Symbol('C_2', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M_E')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), exp(Symbol('C_2', commutative=True)))"], [["divide", 3, "Mul(Function('M_E')(Symbol('C_2', commutative=True)), Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Function('M_E')(Symbol('C_2', commutative=True)), Integer(-1)), Derivative(Function('M_E')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Function('M_E')(Symbol('C_2', commutative=True)), Integer(-1)), exp(Symbol('C_2', commutative=True)), Pow(Derivative(exp(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(u)} = e^{u}, then obtain (\\operatorname{f_{\\mathbf{p}}}{(u)} + 1)^{u} = (e^{u} + 1)^{u}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(u)} = e^{u} and \\operatorname{f_{\\mathbf{p}}}^{u}{(u)} = (e^{u})^{u} and \\operatorname{f_{\\mathbf{p}}}{(u)} + \\operatorname{f_{\\mathbf{p}}}^{u}{(u)} (e^{u})^{- u} = \\operatorname{f_{\\mathbf{p}}}^{u}{(u)} (e^{u})^{- u} + e^{u} and \\operatorname{f_{\\mathbf{p}}}{(u)} + 1 = e^{u} + 1 and (\\operatorname{f_{\\mathbf{p}}}{(u)} + 1)^{u} = (e^{u} + 1)^{u}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True))))), Add(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Integer(1)), Add(exp(Symbol('u', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('u', commutative=True)), Integer(1)), Symbol('u', commutative=True)), Pow(Add(exp(Symbol('u', commutative=True)), Integer(1)), Symbol('u', commutative=True)))"]]}, {"prompt": "Given z{(\\mathbf{E},A_{1})} = A_{1} \\mathbf{E}, then derive - A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} z{(\\mathbf{E},A_{1})} = 0, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} (- A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} A_{1} \\mathbf{E}) = \\frac{d}{d \\mathbf{E}} 0", "derivation": "z{(\\mathbf{E},A_{1})} = A_{1} \\mathbf{E} and \\frac{\\partial}{\\partial \\mathbf{E}} z{(\\mathbf{E},A_{1})} = \\frac{\\partial}{\\partial \\mathbf{E}} A_{1} \\mathbf{E} and - A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} z{(\\mathbf{E},A_{1})} = - A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} A_{1} \\mathbf{E} and - A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} z{(\\mathbf{E},A_{1})} = 0 and - A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} A_{1} \\mathbf{E} = 0 and \\frac{\\partial}{\\partial \\mathbf{E}} (- A_{1} + \\frac{\\partial}{\\partial \\mathbf{E}} A_{1} \\mathbf{E}) = \\frac{d}{d \\mathbf{E}} 0", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Derivative(Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(F_{g},s)} = - F_{g} + s, then obtain F_{g} - s + U{(F_{g},s)} + U^{F_{g}}{(F_{g},s)} = (- F_{g} + s)^{F_{g}}", "derivation": "U{(F_{g},s)} = - F_{g} + s and F_{g} - s + U{(F_{g},s)} = 0 and U^{F_{g}}{(F_{g},s)} = (- F_{g} + s)^{F_{g}} and F_{g} - s + U{(F_{g},s)} + U^{F_{g}}{(F_{g},s)} = U^{F_{g}}{(F_{g},s)} and F_{g} - s + U{(F_{g},s)} + U^{F_{g}}{(F_{g},s)} = (- F_{g} + s)^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('s', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True))), Integer(0))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Symbol('F_g', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('s', commutative=True)), Symbol('F_g', commutative=True)))"], [["add", 2, "Pow(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Symbol('F_g', commutative=True))"], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Pow(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Symbol('F_g', commutative=True))), Pow(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Pow(Function('U')(Symbol('F_g', commutative=True), Symbol('s', commutative=True)), Symbol('F_g', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Symbol('s', commutative=True)), Symbol('F_g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(E,L)} = \\frac{\\cos{(L)}}{E}, then obtain \\frac{\\partial}{\\partial E} (\\operatorname{A_{y}}{(E,L)} - \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)}) = \\frac{\\partial}{\\partial E} (- \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)} + \\frac{\\cos{(L)}}{E})", "derivation": "\\operatorname{A_{y}}{(E,L)} = \\frac{\\cos{(L)}}{E} and \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)} = \\frac{\\partial}{\\partial E} \\frac{\\cos{(L)}}{E} and \\operatorname{A_{y}}{(E,L)} - \\frac{\\partial}{\\partial E} \\frac{\\cos{(L)}}{E} = - \\frac{\\partial}{\\partial E} \\frac{\\cos{(L)}}{E} + \\frac{\\cos{(L)}}{E} and \\operatorname{A_{y}}{(E,L)} - \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)} = - \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)} + \\frac{\\cos{(L)}}{E} and \\frac{\\partial}{\\partial E} (\\operatorname{A_{y}}{(E,L)} - \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)}) = \\frac{\\partial}{\\partial E} (- \\frac{\\partial}{\\partial E} \\operatorname{A_{y}}{(E,L)} + \\frac{\\cos{(L)}}{E})", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))"], "Equality(Add(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)))))"], [["differentiate", 4, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Function('A_y')(Symbol('E', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), cos(Symbol('L', commutative=True)))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\operatorname{z^{*}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then obtain \\operatorname{z^{*}}^{\\mathbf{S}}{(\\mathbf{S})} + \\sin{(\\mathbb{I})} = \\sin{(\\mathbb{I})} + \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "derivation": "\\rho_{f}{(\\mathbb{I})} = \\sin{(\\mathbb{I})} and \\operatorname{z^{*}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\operatorname{z^{*}}^{\\mathbf{S}}{(\\mathbf{S})} = \\cos^{\\mathbf{S}}{(\\mathbf{S})} and \\rho_{f}{(\\mathbb{I})} + \\operatorname{z^{*}}^{\\mathbf{S}}{(\\mathbf{S})} = \\rho_{f}{(\\mathbb{I})} + \\cos^{\\mathbf{S}}{(\\mathbf{S})} and \\operatorname{z^{*}}^{\\mathbf{S}}{(\\mathbf{S})} + \\sin{(\\mathbb{I})} = \\sin{(\\mathbb{I})} + \\cos^{\\mathbf{S}}{(\\mathbf{S})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True)))"], ["get_premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 3, "Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))), Add(Function('\\\\rho_f')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Function('z^*')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbb{I}', commutative=True))), Add(sin(Symbol('\\\\mathbb{I}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(M,\\hat{p}_0)} = M \\hat{p}_0, then derive M \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{H}{(M,\\hat{p}_0)} = M^{2}, then derive \\int M \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{H}{(M,\\hat{p}_0)} dM = \\frac{M^{3}}{3} + \\varepsilon, then obtain \\int M^{2} dM = \\frac{M^{3}}{3} + \\varepsilon", "derivation": "\\mathbf{H}{(M,\\hat{p}_0)} = M \\hat{p}_0 and M \\mathbf{H}{(M,\\hat{p}_0)} = M^{2} \\hat{p}_0 and \\frac{\\partial}{\\partial \\hat{p}_0} M \\mathbf{H}{(M,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} M^{2} \\hat{p}_0 and M \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{H}{(M,\\hat{p}_0)} = M^{2} and \\int M \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{H}{(M,\\hat{p}_0)} dM = \\int M^{2} dM and \\int M \\frac{\\partial}{\\partial \\hat{p}_0} \\mathbf{H}{(M,\\hat{p}_0)} dM = \\frac{M^{3}}{3} + \\varepsilon and \\int M^{2} dM = \\frac{M^{3}}{3} + \\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(2)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('M', commutative=True), Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('M', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Pow(Symbol('M', commutative=True), Integer(2)))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Symbol('M', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True))), Integral(Pow(Symbol('M', commutative=True), Integer(2)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Symbol('M', commutative=True), Derivative(Function('\\\\mathbf{H}')(Symbol('M', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Tuple(Symbol('M', commutative=True))), Add(Mul(Rational(1, 3), Pow(Symbol('M', commutative=True), Integer(3))), Symbol('\\\\varepsilon', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Pow(Symbol('M', commutative=True), Integer(2)), Tuple(Symbol('M', commutative=True))), Add(Mul(Rational(1, 3), Pow(Symbol('M', commutative=True), Integer(3))), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} = \\cos{(A_{y} + f^{\\prime})}, then obtain \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} df^{\\prime} = \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} \\int \\cos{(A_{y} + f^{\\prime})} df^{\\prime}", "derivation": "\\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} = \\cos{(A_{y} + f^{\\prime})} and \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} df^{\\prime} = \\int \\cos{(A_{y} + f^{\\prime})} df^{\\prime} and \\cos{(A_{y} + f^{\\prime})} \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} df^{\\prime} = \\cos{(A_{y} + f^{\\prime})} \\int \\cos{(A_{y} + f^{\\prime})} df^{\\prime} and \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} \\int \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} df^{\\prime} = \\operatorname{g_{\\varepsilon}}{(f^{\\prime},A_{y})} \\int \\cos{(A_{y} + f^{\\prime})} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 2, "cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Mul(cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integral(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Integral(cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integral(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Mul(Function('g_{\\\\varepsilon}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('A_y', commutative=True)), Integral(cos(Add(Symbol('A_y', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(f_{E})} = \\log{(f_{E})} and \\psi{(f_{E})} = \\frac{d}{d f_{E}} \\operatorname{t_{1}}{(f_{E})}, then obtain \\frac{f_{E} \\psi{(f_{E})} \\log{(f_{E})}}{\\log{(f_{E})} + 1} = \\frac{f_{E} \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(f_{E})}}{\\log{(f_{E})} + 1}", "derivation": "\\operatorname{t_{1}}{(f_{E})} = \\log{(f_{E})} and \\psi{(f_{E})} = \\frac{d}{d f_{E}} \\operatorname{t_{1}}{(f_{E})} and f_{E} \\psi{(f_{E})} \\log{(f_{E})} = f_{E} \\log{(f_{E})} \\frac{d}{d f_{E}} \\operatorname{t_{1}}{(f_{E})} and f_{E} \\psi{(f_{E})} \\log{(f_{E})} = f_{E} \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(f_{E})} and \\frac{f_{E} \\psi{(f_{E})} \\log{(f_{E})}}{\\log{(f_{E})} + 1} = \\frac{f_{E} \\log{(f_{E})} \\frac{d}{d f_{E}} \\log{(f_{E})}}{\\log{(f_{E})} + 1}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('f_E', commutative=True)), Derivative(Function('t_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["times", 2, "Mul(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True)))"], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\psi')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True)), Derivative(Function('t_1')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('f_E', commutative=True), Function('\\\\psi')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), log(Symbol('f_E', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"], [["divide", 4, "Add(log(Symbol('f_E', commutative=True)), Integer(1))"], "Equality(Mul(Symbol('f_E', commutative=True), Pow(Add(log(Symbol('f_E', commutative=True)), Integer(1)), Integer(-1)), Function('\\\\psi')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True))), Mul(Symbol('f_E', commutative=True), Pow(Add(log(Symbol('f_E', commutative=True)), Integer(1)), Integer(-1)), log(Symbol('f_E', commutative=True)), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{D}{(g,U)} = e^{U - g}, then obtain - g (\\mathbf{D}{(g,U)} - e^{U - g}) - g = - g", "derivation": "\\mathbf{D}{(g,U)} = e^{U - g} and \\mathbf{D}{(g,U)} - e^{U - g} = 0 and - g (\\mathbf{D}{(g,U)} - e^{U - g}) = 0 and - g (\\mathbf{D}{(g,U)} - e^{U - g}) - g = - g", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('g', commutative=True), Symbol('U', commutative=True)), exp(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))"], [["minus", 1, "exp(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('g', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True)))))), Integer(0))"], [["times", 2, "Mul(Integer(-1), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('g', commutative=True), Add(Function('\\\\mathbf{D}')(Symbol('g', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))))), Integer(0))"], [["minus", 3, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True), Add(Function('\\\\mathbf{D}')(Symbol('g', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), exp(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))))))), Mul(Integer(-1), Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(l,\\psi^*)} = \\log{(\\psi^* - l)}, then obtain l \\hat{x}{(l,\\psi^*)} (\\frac{\\partial}{\\partial \\psi^*} l \\hat{x}{(l,\\psi^*)})^{l} = l \\hat{x}{(l,\\psi^*)} (\\frac{\\partial}{\\partial \\psi^*} l \\log{(\\psi^* - l)})^{l}", "derivation": "\\hat{x}{(l,\\psi^*)} = \\log{(\\psi^* - l)} and l \\hat{x}{(l,\\psi^*)} = l \\log{(\\psi^* - l)} and \\frac{\\partial}{\\partial \\psi^*} l \\hat{x}{(l,\\psi^*)} = \\frac{\\partial}{\\partial \\psi^*} l \\log{(\\psi^* - l)} and (\\frac{\\partial}{\\partial \\psi^*} l \\hat{x}{(l,\\psi^*)})^{l} = (\\frac{\\partial}{\\partial \\psi^*} l \\log{(\\psi^* - l)})^{l} and l \\hat{x}{(l,\\psi^*)} (\\frac{\\partial}{\\partial \\psi^*} l \\hat{x}{(l,\\psi^*)})^{l} = l \\hat{x}{(l,\\psi^*)} (\\frac{\\partial}{\\partial \\psi^*} l \\log{(\\psi^* - l)})^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True)), log(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["power", 3, "Symbol('l', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('l', commutative=True)), Pow(Derivative(Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('l', commutative=True)))"], [["times", 4, "Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Mul(Symbol('l', commutative=True), log(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then derive \\int \\dot{\\mathbf{r}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = v_{x} - \\cos{(\\hat{\\mathbf{x}})}, then obtain \\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = v_{x} - \\cos{(\\hat{\\mathbf{x}})}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\int \\dot{\\mathbf{r}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} and \\int \\dot{\\mathbf{r}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = v_{x} - \\cos{(\\hat{\\mathbf{x}})} and \\int \\sin{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = v_{x} - \\cos{(\\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('v_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Symbol('v_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given M{(\\eta)} = \\cos{(\\eta)}, then obtain (M{(\\eta)} + \\int \\frac{M{(\\eta)}}{\\cos{(\\eta)}} d\\eta) \\cos{(\\eta)} = (G + \\eta + M{(\\eta)}) \\cos{(\\eta)}", "derivation": "M{(\\eta)} = \\cos{(\\eta)} and \\frac{M{(\\eta)}}{\\cos{(\\eta)}} = 1 and \\int \\frac{M{(\\eta)}}{\\cos{(\\eta)}} d\\eta = \\int 1 d\\eta and M{(\\eta)} + \\int \\frac{M{(\\eta)}}{\\cos{(\\eta)}} d\\eta = M{(\\eta)} + \\int 1 d\\eta and (M{(\\eta)} + \\int \\frac{M{(\\eta)}}{\\cos{(\\eta)}} d\\eta) \\cos{(\\eta)} = (M{(\\eta)} + \\int 1 d\\eta) \\cos{(\\eta)} and (M{(\\eta)} + \\int \\frac{M{(\\eta)}}{\\cos{(\\eta)}} d\\eta) \\cos{(\\eta)} = (G + \\eta + M{(\\eta)}) \\cos{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Function('M')(Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Mul(Function('M')(Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))))"], [["add", 3, "Function('M')(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('M')(Symbol('\\\\eta', commutative=True)), Integral(Mul(Function('M')(Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Function('M')(Symbol('\\\\eta', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["divide", 4, "Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Function('M')(Symbol('\\\\eta', commutative=True)), Integral(Mul(Function('M')(Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True)))), cos(Symbol('\\\\eta', commutative=True))), Mul(Add(Function('M')(Symbol('\\\\eta', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True)))), cos(Symbol('\\\\eta', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Function('M')(Symbol('\\\\eta', commutative=True)), Integral(Mul(Function('M')(Symbol('\\\\eta', commutative=True)), Pow(cos(Symbol('\\\\eta', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\eta', commutative=True)))), cos(Symbol('\\\\eta', commutative=True))), Mul(Add(Symbol('G', commutative=True), Symbol('\\\\eta', commutative=True), Function('M')(Symbol('\\\\eta', commutative=True))), cos(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(z^{*})} = \\cos{(z^{*})}, then derive \\int \\operatorname{t_{1}}{(z^{*})} dz^{*} = \\chi + \\sin{(z^{*})}, then obtain \\int (\\chi \\int \\operatorname{t_{1}}{(z^{*})} dz^{*})^{\\chi} dz^{*} = \\int (\\chi (\\chi + \\sin{(z^{*})}))^{\\chi} dz^{*}", "derivation": "\\operatorname{t_{1}}{(z^{*})} = \\cos{(z^{*})} and \\int \\operatorname{t_{1}}{(z^{*})} dz^{*} = \\int \\cos{(z^{*})} dz^{*} and \\int \\operatorname{t_{1}}{(z^{*})} dz^{*} = \\chi + \\sin{(z^{*})} and \\int \\cos{(z^{*})} dz^{*} = \\chi + \\sin{(z^{*})} and \\chi \\int \\operatorname{t_{1}}{(z^{*})} dz^{*} = \\chi \\int \\cos{(z^{*})} dz^{*} and (\\chi \\int \\operatorname{t_{1}}{(z^{*})} dz^{*})^{\\chi} = (\\chi \\int \\cos{(z^{*})} dz^{*})^{\\chi} and (\\chi \\int \\operatorname{t_{1}}{(z^{*})} dz^{*})^{\\chi} = (\\chi (\\chi + \\sin{(z^{*})}))^{\\chi} and \\int (\\chi \\int \\operatorname{t_{1}}{(z^{*})} dz^{*})^{\\chi} dz^{*} = \\int (\\chi (\\chi + \\sin{(z^{*})}))^{\\chi} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('z^*', commutative=True))))"], [["times", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Mul(Symbol('\\\\chi', commutative=True), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"], [["power", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Integral(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Pow(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)))"], [["integrate", 7, "Symbol('z^*', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Integral(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Add(Symbol('\\\\chi', commutative=True), sin(Symbol('z^*', commutative=True)))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{J}_M,I)} = I \\mathbf{J}_M, then obtain \\int \\frac{d}{d \\mathbf{J}_M} 0 dI = \\int \\frac{\\partial}{\\partial \\mathbf{J}_M} (I \\mathbf{J}_M - \\operatorname{v_{z}}{(\\mathbf{J}_M,I)}) dI", "derivation": "\\operatorname{v_{z}}{(\\mathbf{J}_M,I)} = I \\mathbf{J}_M and 0 = I \\mathbf{J}_M - \\operatorname{v_{z}}{(\\mathbf{J}_M,I)} and \\frac{d}{d \\mathbf{J}_M} 0 = \\frac{\\partial}{\\partial \\mathbf{J}_M} (I \\mathbf{J}_M - \\operatorname{v_{z}}{(\\mathbf{J}_M,I)}) and \\int \\frac{d}{d \\mathbf{J}_M} 0 dI = \\int \\frac{\\partial}{\\partial \\mathbf{J}_M} (I \\mathbf{J}_M - \\operatorname{v_{z}}{(\\mathbf{J}_M,I)}) dI", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["minus", 1, "Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('I', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('I', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('I', commutative=True)"], "Equality(Integral(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Add(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('I', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(p)} = e^{p}, then derive \\frac{d}{d p} \\hat{H}_{\\lambda}{(p)} = e^{p}, then obtain p e^{p} = p \\frac{d^{2}}{d p^{2}} \\hat{H}_{\\lambda}{(p)}", "derivation": "\\hat{H}_{\\lambda}{(p)} = e^{p} and \\frac{d}{d p} \\hat{H}_{\\lambda}{(p)} = \\frac{d}{d p} e^{p} and \\frac{d}{d p} \\hat{H}_{\\lambda}{(p)} = e^{p} and p \\hat{H}_{\\lambda}{(p)} = p e^{p} and p \\hat{H}_{\\lambda}{(p)} = p \\frac{d}{d p} \\hat{H}_{\\lambda}{(p)} and \\frac{d^{2}}{d p^{2}} \\hat{H}_{\\lambda}{(p)} = \\frac{d}{d p} e^{p} and p e^{p} = p \\frac{d}{d p} e^{p} and p e^{p} = p \\frac{d^{2}}{d p^{2}} \\hat{H}_{\\lambda}{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), exp(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), exp(Symbol('p', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(2))), Derivative(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Derivative(exp(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Symbol('p', commutative=True), exp(Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mathbb{I}{(\\varphi^*,H)} = H \\varphi^*, then obtain \\frac{\\partial}{\\partial H} \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{(\\mathbb{I}^{H}{(\\varphi^*,H)})} = \\frac{\\partial}{\\partial H} \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{((H \\varphi^*)^{H})}", "derivation": "\\mathbb{I}{(\\varphi^*,H)} = H \\varphi^* and \\mathbb{I}^{H}{(\\varphi^*,H)} = (H \\varphi^*)^{H} and \\log{(\\mathbb{I}^{H}{(\\varphi^*,H)})} = \\log{((H \\varphi^*)^{H})} and \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{(\\mathbb{I}^{H}{(\\varphi^*,H)})} = \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{((H \\varphi^*)^{H})} and \\frac{\\partial}{\\partial H} \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{(\\mathbb{I}^{H}{(\\varphi^*,H)})} = \\frac{\\partial}{\\partial H} \\mathbb{I}^{H}{(\\varphi^*,H)} \\log{((H \\varphi^*)^{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('H', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))), log(Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('H', commutative=True))))"], [["times", 3, "Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), log(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)))), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), log(Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('H', commutative=True)))))"], [["differentiate", 4, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), log(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), log(Pow(Mul(Symbol('H', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\lambda,a^{\\dagger})} = \\cos^{a^{\\dagger}}{(\\lambda)}, then obtain C{(\\lambda,a^{\\dagger})} + \\frac{\\partial}{\\partial \\lambda} \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger} = \\cos^{a^{\\dagger}}{(\\lambda)} + \\frac{\\partial}{\\partial \\lambda} \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger}", "derivation": "C{(\\lambda,a^{\\dagger})} = \\cos^{a^{\\dagger}}{(\\lambda)} and \\int C{(\\lambda,a^{\\dagger})} da^{\\dagger} = \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger} and \\frac{\\partial}{\\partial \\lambda} \\int C{(\\lambda,a^{\\dagger})} da^{\\dagger} = \\frac{\\partial}{\\partial \\lambda} \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger} and C{(\\lambda,a^{\\dagger})} + \\frac{\\partial}{\\partial \\lambda} \\int C{(\\lambda,a^{\\dagger})} da^{\\dagger} = \\cos^{a^{\\dagger}}{(\\lambda)} + \\frac{\\partial}{\\partial \\lambda} \\int C{(\\lambda,a^{\\dagger})} da^{\\dagger} and C{(\\lambda,a^{\\dagger})} + \\frac{\\partial}{\\partial \\lambda} \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger} = \\cos^{a^{\\dagger}}{(\\lambda)} + \\frac{\\partial}{\\partial \\lambda} \\int \\cos^{a^{\\dagger}}{(\\lambda)} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Integral(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integral(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integral(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('C')(Symbol('\\\\lambda', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integral(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Integral(Pow(cos(Symbol('\\\\lambda', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given I{(n_{1})} = e^{n_{1}}, then obtain (\\frac{I{(n_{1})} e^{n_{1}}}{n_{1}^{2}})^{n_{1}} = (\\frac{e^{2 n_{1}}}{n_{1}^{2}})^{n_{1}}", "derivation": "I{(n_{1})} = e^{n_{1}} and \\frac{I{(n_{1})}}{n_{1}} = \\frac{e^{n_{1}}}{n_{1}} and \\frac{I{(n_{1})} e^{n_{1}}}{n_{1}^{2}} = \\frac{e^{2 n_{1}}}{n_{1}^{2}} and (\\frac{I{(n_{1})} e^{n_{1}}}{n_{1}^{2}})^{n_{1}} = (\\frac{e^{2 n_{1}}}{n_{1}^{2}})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["divide", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('I')(Symbol('n_1', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Symbol('n_1', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Symbol('n_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Function('I')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('n_1', commutative=True)))))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), Function('I')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True))), Symbol('n_1', commutative=True)), Pow(Mul(Pow(Symbol('n_1', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(n_{2},r_{0})} = n_{2} - r_{0}, then obtain (n_{2} - r_{0})^{2} = - (- n_{2} + r_{0}) (n_{2} - r_{0})", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(n_{2},r_{0})} = n_{2} - r_{0} and - \\operatorname{V_{\\mathbf{E}}}{(n_{2},r_{0})} = - n_{2} + r_{0} and \\operatorname{V_{\\mathbf{E}}}^{2}{(n_{2},r_{0})} = - (- n_{2} + r_{0}) \\operatorname{V_{\\mathbf{E}}}{(n_{2},r_{0})} and (n_{2} - r_{0})^{2} = - (- n_{2} + r_{0}) (n_{2} - r_{0})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('r_0', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('r_0', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_2', commutative=True), Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(2)), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Symbol('r_0', commutative=True)), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{J}_M,E_{x})} = E_{x} + \\mathbf{J}_M, then obtain \\frac{\\partial}{\\partial E_{x}} \\int \\dot{\\mathbf{r}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{x})} d\\mathbf{J}_M = \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + \\mathbf{J}_M)^{\\mathbf{J}_M} d\\mathbf{J}_M", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{J}_M,E_{x})} = E_{x} + \\mathbf{J}_M and \\dot{\\mathbf{r}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{x})} = (E_{x} + \\mathbf{J}_M)^{\\mathbf{J}_M} and \\int \\dot{\\mathbf{r}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{x})} d\\mathbf{J}_M = \\int (E_{x} + \\mathbf{J}_M)^{\\mathbf{J}_M} d\\mathbf{J}_M and \\frac{\\partial}{\\partial E_{x}} \\int \\dot{\\mathbf{r}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{x})} d\\mathbf{J}_M = \\frac{\\partial}{\\partial E_{x}} \\int (E_{x} + \\mathbf{J}_M)^{\\mathbf{J}_M} d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_x', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\Psi,P_{g})} = P_{g} + \\Psi, then obtain \\int \\frac{\\partial}{\\partial P_{g}} \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi^{2}} d\\Psi = E_{\\lambda} - \\frac{1}{\\Psi}", "derivation": "\\phi{(\\Psi,P_{g})} = P_{g} + \\Psi and \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi} = \\frac{P_{g} + \\Psi}{\\Psi} and \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi^{2}} = \\frac{P_{g} + \\Psi}{\\Psi^{2}} and \\frac{\\partial}{\\partial P_{g}} \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi^{2}} = \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g} + \\Psi}{\\Psi^{2}} and \\int \\frac{\\partial}{\\partial P_{g}} \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi^{2}} d\\Psi = \\int \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g} + \\Psi}{\\Psi^{2}} d\\Psi and \\int \\frac{\\partial}{\\partial P_{g}} \\frac{\\phi{(\\Psi,P_{g})}}{\\Psi^{2}} d\\Psi = E_{\\lambda} - \\frac{1}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True)), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi', commutative=True)))"], [["divide", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["divide", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 3, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Add(Symbol('P_g', commutative=True), Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Derivative(Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-2)), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('P_g', commutative=True))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\eta^{\\prime},z)} = \\frac{\\eta^{\\prime}}{z} and \\operatorname{J_{\\varepsilon}}{(\\eta^{\\prime},z)} = \\int \\frac{\\eta^{\\prime}}{z} dz, then obtain \\operatorname{J_{\\varepsilon}}{(\\eta^{\\prime},z)} + 1 = \\int \\operatorname{r_{0}}{(\\eta^{\\prime},z)} dz + 1", "derivation": "\\operatorname{r_{0}}{(\\eta^{\\prime},z)} = \\frac{\\eta^{\\prime}}{z} and \\int \\operatorname{r_{0}}{(\\eta^{\\prime},z)} dz = \\int \\frac{\\eta^{\\prime}}{z} dz and \\operatorname{J_{\\varepsilon}}{(\\eta^{\\prime},z)} = \\int \\frac{\\eta^{\\prime}}{z} dz and \\operatorname{J_{\\varepsilon}}{(\\eta^{\\prime},z)} + 1 = \\int \\frac{\\eta^{\\prime}}{z} dz + 1 and \\operatorname{J_{\\varepsilon}}{(\\eta^{\\prime},z)} + 1 = \\int \\operatorname{r_{0}}{(\\eta^{\\prime},z)} dz + 1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Integral(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Integer(1)), Add(Integral(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Tuple(Symbol('z', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Integer(1)), Add(Integral(Function('r_0')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(q)} = \\cos{(\\log{(q)})}, then derive (\\int \\operatorname{J_{\\varepsilon}}{(q)} dq)^{q} = (\\hat{p} + \\frac{q \\sin{(\\log{(q)})}}{2} + \\frac{q \\cos{(\\log{(q)})}}{2})^{q}, then obtain (\\hat{p} + \\frac{q \\operatorname{J_{\\varepsilon}}{(q)}}{2} + \\frac{q \\sin{(\\log{(q)})}}{2})^{q} = (\\int \\cos{(\\log{(q)})} dq)^{q}", "derivation": "\\operatorname{J_{\\varepsilon}}{(q)} = \\cos{(\\log{(q)})} and \\int \\operatorname{J_{\\varepsilon}}{(q)} dq = \\int \\cos{(\\log{(q)})} dq and (\\int \\operatorname{J_{\\varepsilon}}{(q)} dq)^{q} = (\\int \\cos{(\\log{(q)})} dq)^{q} and (\\int \\operatorname{J_{\\varepsilon}}{(q)} dq)^{q} = (\\hat{p} + \\frac{q \\sin{(\\log{(q)})}}{2} + \\frac{q \\cos{(\\log{(q)})}}{2})^{q} and (\\hat{p} + \\frac{q \\sin{(\\log{(q)})}}{2} + \\frac{q \\cos{(\\log{(q)})}}{2})^{q} = (\\int \\cos{(\\log{(q)})} dq)^{q} and (\\hat{p} + \\frac{q \\operatorname{J_{\\varepsilon}}{(q)}}{2} + \\frac{q \\sin{(\\log{(q)})}}{2})^{q} = (\\int \\cos{(\\log{(q)})} dq)^{q}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Integral(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('q', commutative=True), sin(log(Symbol('q', commutative=True)))), Mul(Rational(1, 2), Symbol('q', commutative=True), cos(log(Symbol('q', commutative=True))))), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('q', commutative=True), sin(log(Symbol('q', commutative=True)))), Mul(Rational(1, 2), Symbol('q', commutative=True), cos(log(Symbol('q', commutative=True))))), Symbol('q', commutative=True)), Pow(Integral(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Rational(1, 2), Symbol('q', commutative=True), Function('J_{\\\\varepsilon}')(Symbol('q', commutative=True))), Mul(Rational(1, 2), Symbol('q', commutative=True), sin(log(Symbol('q', commutative=True))))), Symbol('q', commutative=True)), Pow(Integral(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\Psi{(x)} = \\cos{(x)}, then derive - \\sin{(x)} + \\frac{d}{d x} \\Psi{(x)} = - 2 \\sin{(x)}, then obtain - \\frac{(\\Psi{(x)} + \\cos{(x)}) \\sin{(x)}}{\\cos{(x)}} = - \\sin{(x)} + \\frac{d}{d x} \\cos{(x)}", "derivation": "\\Psi{(x)} = \\cos{(x)} and \\Psi{(x)} + \\cos{(x)} = 2 \\cos{(x)} and \\frac{\\Psi{(x)} + \\cos{(x)}}{\\cos{(x)}} = 2 and \\frac{d}{d x} (\\Psi{(x)} + \\cos{(x)}) = \\frac{d}{d x} 2 \\cos{(x)} and - \\sin{(x)} + \\frac{d}{d x} \\Psi{(x)} = - 2 \\sin{(x)} and - \\sin{(x)} + \\frac{d}{d x} \\Psi{(x)} = - \\frac{(\\Psi{(x)} + \\cos{(x)}) \\sin{(x)}}{\\cos{(x)}} and - \\sin{(x)} + \\frac{d}{d x} \\cos{(x)} = - 2 \\sin{(x)} and - \\frac{(\\Psi{(x)} + \\cos{(x)}) \\sin{(x)}}{\\cos{(x)}} = - 2 \\sin{(x)} and - \\frac{(\\Psi{(x)} + \\cos{(x)}) \\sin{(x)}}{\\cos{(x)}} = - \\sin{(x)} + \\frac{d}{d x} \\cos{(x)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["add", 1, "cos(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Mul(Integer(2), cos(Symbol('x', commutative=True))))"], [["divide", 2, "cos(Symbol('x', commutative=True))"], "Equality(Mul(Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Integer(2))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Derivative(Function('\\\\Psi')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), sin(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(-1), Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), sin(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), sin(Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Mul(Integer(-1), Add(Function('\\\\Psi')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), sin(Symbol('x', commutative=True)), Pow(cos(Symbol('x', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), sin(Symbol('x', commutative=True))), Derivative(cos(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{H},\\hbar)} = - \\hbar + \\cos{(\\mathbf{H})}, then obtain (\\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}))^{\\hbar} (\\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{2}}{(\\mathbf{H},\\hbar)})^{\\hbar} = (\\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}))^{2 \\hbar}", "derivation": "\\operatorname{C_{2}}{(\\mathbf{H},\\hbar)} = - \\hbar + \\cos{(\\mathbf{H})} and \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{2}}{(\\mathbf{H},\\hbar)} = \\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}) and (\\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{2}}{(\\mathbf{H},\\hbar)})^{\\hbar} = (\\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}))^{\\hbar} and (\\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}))^{\\hbar} (\\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{2}}{(\\mathbf{H},\\hbar)})^{\\hbar} = (\\frac{\\partial}{\\partial \\hbar} (- \\hbar + \\cos{(\\mathbf{H})}))^{2 \\hbar}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Derivative(Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)))"], [["times", 3, "Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True)), Pow(Derivative(Function('C_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\hbar', commutative=True))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(2), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)}, then derive \\hat{\\mathbf{r}}{(\\psi^*)} = \\frac{1}{\\psi^*}, then obtain 2 \\hat{\\mathbf{r}}{(\\psi^*)} = \\hat{\\mathbf{r}}{(\\psi^*)} + \\hat{\\mathbf{r}}{(\\frac{1}{\\frac{d}{d \\psi^*} \\log{(\\psi^*)}})}", "derivation": "\\hat{\\mathbf{r}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and \\hat{\\mathbf{r}}{(\\psi^*)} = \\frac{1}{\\psi^*} and \\frac{d}{d \\psi^*} \\log{(\\psi^*)} = \\frac{1}{\\psi^*} and \\hat{\\mathbf{r}}{(\\frac{1}{\\frac{d}{d \\psi^*} \\log{(\\psi^*)}})} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and 2 \\hat{\\mathbf{r}}{(\\psi^*)} = \\hat{\\mathbf{r}}{(\\psi^*)} + \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and 2 \\hat{\\mathbf{r}}{(\\psi^*)} = \\hat{\\mathbf{r}}{(\\psi^*)} + \\hat{\\mathbf{r}}{(\\frac{1}{\\frac{d}{d \\psi^*} \\log{(\\psi^*)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Pow(Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1))), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["minus", 1, "Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True))), Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\psi^*', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Pow(Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} = \\mathbf{J}_P^{\\varepsilon}, then obtain \\iint \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} d\\mathbf{J}_P d\\mathbf{J}_P = \\iint \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P^{\\varepsilon} d\\mathbf{J}_P d\\mathbf{J}_P", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} = \\mathbf{J}_P^{\\varepsilon} and \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P^{\\varepsilon} and \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} d\\mathbf{J}_P = \\int \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P^{\\varepsilon} d\\mathbf{J}_P and \\iint \\frac{\\partial}{\\partial \\mathbf{J}_P} \\operatorname{g_{\\varepsilon}}{(\\mathbf{J}_P,\\varepsilon)} d\\mathbf{J}_P d\\mathbf{J}_P = \\iint \\frac{\\partial}{\\partial \\mathbf{J}_P} \\mathbf{J}_P^{\\varepsilon} d\\mathbf{J}_P d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(m_{s})} = e^{e^{m_{s}}}, then obtain e^{2 \\dot{z}{(m_{s})}} + e^{- \\dot{z}{(m_{s})}} = e^{2 \\dot{z}{(m_{s})}} + e^{- 3 \\dot{z}{(m_{s})}} e^{2 e^{e^{m_{s}}}}", "derivation": "\\dot{z}{(m_{s})} = e^{e^{m_{s}}} and e^{\\dot{z}{(m_{s})}} = e^{e^{e^{m_{s}}}} and e^{- \\dot{z}{(m_{s})}} = e^{- 2 \\dot{z}{(m_{s})}} e^{e^{e^{m_{s}}}} and e^{- 2 \\dot{z}{(m_{s})}} = e^{- 3 \\dot{z}{(m_{s})}} e^{e^{e^{m_{s}}}} and e^{- \\dot{z}{(m_{s})}} = e^{- 3 \\dot{z}{(m_{s})}} e^{2 e^{e^{m_{s}}}} and e^{2 \\dot{z}{(m_{s})}} + e^{- \\dot{z}{(m_{s})}} = e^{2 \\dot{z}{(m_{s})}} + e^{- 3 \\dot{z}{(m_{s})}} e^{2 e^{e^{m_{s}}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m_s', commutative=True)), exp(exp(Symbol('m_s', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\dot{z}')(Symbol('m_s', commutative=True))), exp(exp(exp(Symbol('m_s', commutative=True)))))"], [["divide", 2, "exp(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True))))"], "Equality(exp(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), exp(exp(exp(Symbol('m_s', commutative=True))))))"], [["times", 3, "exp(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m_s', commutative=True))))"], "Equality(exp(Mul(Integer(-1), Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(3), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), exp(exp(exp(Symbol('m_s', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(3), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), exp(Mul(Integer(2), exp(exp(Symbol('m_s', commutative=True)))))))"], [["add", 5, "exp(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True))))"], "Equality(Add(exp(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), exp(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('m_s', commutative=True))))), Add(exp(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(3), Function('\\\\dot{z}')(Symbol('m_s', commutative=True)))), exp(Mul(Integer(2), exp(exp(Symbol('m_s', commutative=True))))))))"]]}, {"prompt": "Given \\mathbf{v}{(\\rho,v)} = \\cos{(\\rho v)}, then obtain \\rho v \\frac{\\partial}{\\partial \\rho} \\mathbf{v}{(\\rho,v)} = - \\rho v^{2} \\sin{(\\rho v)}", "derivation": "\\mathbf{v}{(\\rho,v)} = \\cos{(\\rho v)} and \\frac{\\partial}{\\partial \\rho} \\mathbf{v}{(\\rho,v)} = \\frac{\\partial}{\\partial \\rho} \\cos{(\\rho v)} and \\rho v \\frac{\\partial}{\\partial \\rho} \\mathbf{v}{(\\rho,v)} = \\rho v \\frac{\\partial}{\\partial \\rho} \\cos{(\\rho v)} and \\rho v \\frac{\\partial}{\\partial \\rho} \\mathbf{v}{(\\rho,v)} = - \\rho v^{2} \\sin{(\\rho v)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True)), cos(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["times", 2, "Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True), Derivative(cos(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), Pow(Symbol('v', commutative=True), Integer(2)), sin(Mul(Symbol('\\\\rho', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given A{(\\lambda)} = \\log{(\\lambda)}, then obtain \\lambda (\\lambda + A{(\\lambda)}) + \\lambda + \\log{(\\lambda)} = \\lambda (\\lambda + A{(\\lambda)}) + \\frac{\\lambda (\\lambda + \\log{(\\lambda)})}{\\lambda + A{(\\lambda)}} + \\log{(\\lambda)}", "derivation": "A{(\\lambda)} = \\log{(\\lambda)} and \\lambda + A{(\\lambda)} = \\lambda + \\log{(\\lambda)} and \\lambda (\\lambda + A{(\\lambda)}) = \\lambda (\\lambda + \\log{(\\lambda)}) and \\lambda = \\frac{\\lambda (\\lambda + \\log{(\\lambda)})}{\\lambda + A{(\\lambda)}} and \\lambda (\\lambda + A{(\\lambda)}) + \\lambda + \\log{(\\lambda)} = \\lambda (\\lambda + A{(\\lambda)}) + \\frac{\\lambda (\\lambda + \\log{(\\lambda)})}{\\lambda + A{(\\lambda)}} + \\log{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))))"], [["times", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True)))), Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True)))))"], [["divide", 3, "Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True)))"], "Equality(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\lambda', commutative=True), Pow(Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True)))))"], [["add", 4, "Add(Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True)))), log(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True)))), Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Add(Mul(Symbol('\\\\lambda', commutative=True), Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True)))), Mul(Symbol('\\\\lambda', commutative=True), Pow(Add(Symbol('\\\\lambda', commutative=True), Function('A')(Symbol('\\\\lambda', commutative=True))), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True)))), log(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given S{(r_{0},J)} = J^{r_{0}}, then obtain (- J^{r_{0}} + S{(r_{0},J)}) S^{2}{(r_{0},J)} = J^{2 r_{0}} (- J^{r_{0}} + S{(r_{0},J)})", "derivation": "S{(r_{0},J)} = J^{r_{0}} and (- J^{r_{0}} + S{(r_{0},J)}) S{(r_{0},J)} = J^{r_{0}} (- J^{r_{0}} + S{(r_{0},J)}) and (- J^{r_{0}} + S{(r_{0},J)}) S^{2}{(r_{0},J)} = J^{r_{0}} (- J^{r_{0}} + S{(r_{0},J)}) S{(r_{0},J)} and J^{r_{0}} (- J^{r_{0}} + S{(r_{0},J)}) S{(r_{0},J)} = J^{2 r_{0}} (- J^{r_{0}} + S{(r_{0},J)}) and (- J^{r_{0}} + S{(r_{0},J)}) S^{2}{(r_{0},J)} = J^{2 r_{0}} (- J^{r_{0}} + S{(r_{0},J)})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)))))"], [["times", 2, "Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Pow(Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Integer(2))), Mul(Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Mul(Integer(2), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True))), Pow(Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)), Integer(2))), Mul(Pow(Symbol('J', commutative=True), Mul(Integer(2), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Symbol('r_0', commutative=True))), Function('S')(Symbol('r_0', commutative=True), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\pi,\\omega)} = - \\omega + \\pi, then obtain \\omega + \\operatorname{v_{z}}^{\\pi}{(\\pi,\\omega)} + 1 = \\omega + (- \\omega + \\pi)^{\\pi} + 1", "derivation": "\\operatorname{v_{z}}{(\\pi,\\omega)} = - \\omega + \\pi and \\operatorname{v_{z}}^{\\pi}{(\\pi,\\omega)} = (- \\omega + \\pi)^{\\pi} and \\operatorname{v_{z}}^{\\pi}{(\\pi,\\omega)} + 1 = (- \\omega + \\pi)^{\\pi} + 1 and \\omega + \\operatorname{v_{z}}^{\\pi}{(\\pi,\\omega)} + 1 = \\omega + (- \\omega + \\pi)^{\\pi} + 1", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Pow(Function('v_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1)))"], [["add", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Symbol('\\\\omega', commutative=True), Pow(Function('v_z')(Symbol('\\\\pi', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Integer(1)))"]]}, {"prompt": "Given z{(\\mathbf{D},A_{2})} = A_{2} + \\mathbf{D}, then obtain - 2 \\mathbf{D} + z{(\\mathbf{D},A_{2})} + 1 = A_{2} - \\mathbf{D} + 1", "derivation": "z{(\\mathbf{D},A_{2})} = A_{2} + \\mathbf{D} and - \\mathbf{D} + z{(\\mathbf{D},A_{2})} = A_{2} and - 2 \\mathbf{D} + z{(\\mathbf{D},A_{2})} = A_{2} - \\mathbf{D} and - 2 \\mathbf{D} + z{(\\mathbf{D},A_{2})} + 1 = A_{2} - \\mathbf{D} + 1", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True))), Symbol('A_2', commutative=True))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{D}', commutative=True)), Function('z')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('A_2', commutative=True)), Integer(1)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given f{(\\eta^{\\prime},\\theta_1)} = \\theta_1^{\\eta^{\\prime}} and \\hat{\\mathbf{x}}{(\\eta^{\\prime},\\theta_1)} = \\theta_1^{\\eta^{\\prime}}, then obtain \\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{x}}^{\\theta_1}{(\\eta^{\\prime},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\eta^{\\prime}})^{\\theta_1}", "derivation": "f{(\\eta^{\\prime},\\theta_1)} = \\theta_1^{\\eta^{\\prime}} and f^{\\theta_1}{(\\eta^{\\prime},\\theta_1)} = (\\theta_1^{\\eta^{\\prime}})^{\\theta_1} and \\hat{\\mathbf{x}}{(\\eta^{\\prime},\\theta_1)} = \\theta_1^{\\eta^{\\prime}} and \\frac{\\partial}{\\partial \\theta_1} f^{\\theta_1}{(\\eta^{\\prime},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\eta^{\\prime}})^{\\theta_1} and f{(\\eta^{\\prime},\\theta_1)} = \\hat{\\mathbf{x}}{(\\eta^{\\prime},\\theta_1)} and \\frac{\\partial}{\\partial \\theta_1} \\hat{\\mathbf{x}}^{\\theta_1}{(\\eta^{\\prime},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} (\\theta_1^{\\eta^{\\prime}})^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Pow(Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('f')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Derivative(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(L,\\hat{H}_l)} = L - \\hat{H}_l, then obtain \\frac{\\partial}{\\partial L} (2 \\mathbf{E}{(L,\\hat{H}_l)} + 2) = \\frac{\\partial}{\\partial L} (L - \\hat{H}_l + \\mathbf{E}{(L,\\hat{H}_l)} + 2)", "derivation": "\\mathbf{E}{(L,\\hat{H}_l)} = L - \\hat{H}_l and \\mathbf{E}{(L,\\hat{H}_l)} + 1 = L - \\hat{H}_l + 1 and L - \\hat{H}_l + \\mathbf{E}{(L,\\hat{H}_l)} + 2 = 2 L - 2 \\hat{H}_l + 2 and 2 \\mathbf{E}{(L,\\hat{H}_l)} + 2 = 2 L - 2 \\hat{H}_l + 2 and 2 \\mathbf{E}{(L,\\hat{H}_l)} + 2 = L - \\hat{H}_l + \\mathbf{E}{(L,\\hat{H}_l)} + 2 and \\frac{\\partial}{\\partial L} (2 \\mathbf{E}{(L,\\hat{H}_l)} + 2) = \\frac{\\partial}{\\partial L} (L - \\hat{H}_l + \\mathbf{E}{(L,\\hat{H}_l)} + 2)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1)))"], [["add", 2, "Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Integer(1))"], "Equality(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), Add(Mul(Integer(2), Symbol('L', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Add(Mul(Integer(2), Symbol('L', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)))"], [["differentiate", 5, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Integer(2)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('\\\\mathbf{E}')(Symbol('L', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(v_{2},\\Omega)} = \\log{(\\Omega + v_{2})}, then derive \\int \\hat{X}{(v_{2},\\Omega)} d\\Omega = \\Omega \\log{(\\Omega + v_{2})} - \\Omega + b + v_{2} \\log{(\\Omega + v_{2})}, then obtain \\frac{\\int \\hat{X}{(v_{2},\\Omega)} d\\Omega}{v_{2} \\log{(\\Omega + v_{2})}} = \\frac{\\Omega \\hat{X}{(v_{2},\\Omega)} - \\Omega + b + v_{2} \\hat{X}{(v_{2},\\Omega)}}{v_{2} \\log{(\\Omega + v_{2})}}", "derivation": "\\hat{X}{(v_{2},\\Omega)} = \\log{(\\Omega + v_{2})} and \\int \\hat{X}{(v_{2},\\Omega)} d\\Omega = \\int \\log{(\\Omega + v_{2})} d\\Omega and \\int \\hat{X}{(v_{2},\\Omega)} d\\Omega = \\Omega \\log{(\\Omega + v_{2})} - \\Omega + b + v_{2} \\log{(\\Omega + v_{2})} and \\int \\hat{X}{(v_{2},\\Omega)} d\\Omega = \\Omega \\hat{X}{(v_{2},\\Omega)} - \\Omega + b + v_{2} \\hat{X}{(v_{2},\\Omega)} and \\frac{\\int \\hat{X}{(v_{2},\\Omega)} d\\Omega}{v_{2} \\log{(\\Omega + v_{2})}} = \\frac{\\Omega \\hat{X}{(v_{2},\\Omega)} - \\Omega + b + v_{2} \\hat{X}{(v_{2},\\Omega)}}{v_{2} \\log{(\\Omega + v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('b', commutative=True), Mul(Symbol('v_2', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('b', commutative=True), Mul(Symbol('v_2', commutative=True), Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["divide", 4, "Mul(Symbol('v_2', commutative=True), log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))))"], "Equality(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Integral(Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('b', commutative=True), Mul(Symbol('v_2', commutative=True), Function('\\\\hat{X}')(Symbol('v_2', commutative=True), Symbol('\\\\Omega', commutative=True)))), Pow(log(Add(Symbol('\\\\Omega', commutative=True), Symbol('v_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given H{(\\hat{p})} = e^{\\hat{p}} and \\tilde{g}^*{(E,\\omega)} = \\log{(- E + \\omega)} and T{(E)} = - E, then obtain \\frac{\\partial^{2}}{\\partial \\omega\\partial \\hat{p}} (- H{(\\hat{p})} + \\tilde{g}^*{(E,\\omega)}) = \\frac{\\partial^{2}}{\\partial \\omega\\partial \\hat{p}} (- H{(\\hat{p})} + \\log{(\\omega + T{(E)})})", "derivation": "H{(\\hat{p})} = e^{\\hat{p}} and \\tilde{g}^*{(E,\\omega)} = \\log{(- E + \\omega)} and T{(E)} = - E and \\tilde{g}^*{(E,\\omega)} = \\log{(\\omega + T{(E)})} and \\tilde{g}^*{(E,\\omega)} - e^{\\hat{p}} = - e^{\\hat{p}} + \\log{(\\omega + T{(E)})} and - H{(\\hat{p})} + \\tilde{g}^*{(E,\\omega)} = - H{(\\hat{p})} + \\log{(\\omega + T{(E)})} and \\frac{\\partial}{\\partial \\hat{p}} (- H{(\\hat{p})} + \\tilde{g}^*{(E,\\omega)}) = \\frac{\\partial}{\\partial \\hat{p}} (- H{(\\hat{p})} + \\log{(\\omega + T{(E)})}) and \\frac{\\partial^{2}}{\\partial \\omega\\partial \\hat{p}} (- H{(\\hat{p})} + \\tilde{g}^*{(E,\\omega)}) = \\frac{\\partial^{2}}{\\partial \\omega\\partial \\hat{p}} (- H{(\\hat{p})} + \\log{(\\omega + T{(E)})})", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True)), log(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('E', commutative=True)), Mul(Integer(-1), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True)), log(Add(Symbol('\\\\omega', commutative=True), Function('T')(Symbol('E', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{p}', commutative=True))), log(Add(Symbol('\\\\omega', commutative=True), Function('T')(Symbol('E', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), log(Add(Symbol('\\\\omega', commutative=True), Function('T')(Symbol('E', commutative=True))))))"], [["differentiate", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), log(Add(Symbol('\\\\omega', commutative=True), Function('T')(Symbol('E', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["differentiate", 7, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('E', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('H')(Symbol('\\\\hat{p}', commutative=True))), log(Add(Symbol('\\\\omega', commutative=True), Function('T')(Symbol('E', commutative=True))))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})} and \\varepsilon{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})}, then obtain \\frac{d^{2}}{d \\mathbf{A}^{2}} \\omega^{\\mathbf{A}}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})}", "derivation": "\\omega{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})} and \\omega^{\\mathbf{A}}{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})}^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\omega^{\\mathbf{A}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\log{(\\sin{(\\mathbf{A})})}^{\\mathbf{A}} and \\varepsilon{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})} and \\frac{d}{d \\mathbf{A}} \\omega^{\\mathbf{A}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})} and \\frac{d^{2}}{d \\mathbf{A}^{2}} \\omega^{\\mathbf{A}}{(\\mathbf{A})} = \\frac{d^{2}}{d \\mathbf{A}^{2}} \\varepsilon^{\\mathbf{A}}{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{A}', commutative=True)), log(sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(log(sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(log(sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), log(sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\omega')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))), Derivative(Pow(Function('\\\\varepsilon')(Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(2))))"]]}, {"prompt": "Given t{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}}, then obtain \\sin{(V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} t{(f_{\\mathbf{p}},V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}})}", "derivation": "t{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} t{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}} and V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} t{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}} and \\sin{(V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} t{(f_{\\mathbf{p}},V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\frac{\\sin{(V_{\\mathbf{B}})}}{f_{\\mathbf{p}}})}", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Function('t')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["sin", 3], "Equality(sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Function('t')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))), sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})} = e^{\\sin{(f_{\\mathbf{p}})}}, then obtain \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} (f_{\\mathbf{p}} + \\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})}) = \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} (f_{\\mathbf{p}} + e^{\\sin{(f_{\\mathbf{p}})}})", "derivation": "\\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})} = e^{\\sin{(f_{\\mathbf{p}})}} and f_{\\mathbf{p}} + \\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + e^{\\sin{(f_{\\mathbf{p}})}} and \\frac{d}{d f_{\\mathbf{p}}} (f_{\\mathbf{p}} + \\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})}) = \\frac{d}{d f_{\\mathbf{p}}} (f_{\\mathbf{p}} + e^{\\sin{(f_{\\mathbf{p}})}}) and \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} (f_{\\mathbf{p}} + \\operatorname{g_{\\varepsilon}}{(f_{\\mathbf{p}})}) = \\frac{d^{2}}{d f_{\\mathbf{p}}^{2}} (f_{\\mathbf{p}} + e^{\\sin{(f_{\\mathbf{p}})}})", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["differentiate", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))), Derivative(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(sin(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{x})} = \\sin{(\\hat{x})} and c{(\\tilde{g},\\mathbf{H})} = \\mathbf{H} + \\tilde{g}, then obtain - \\mathbf{H} - \\tilde{g} + c{(\\tilde{g},\\mathbf{H})} = 0", "derivation": "\\operatorname{z^{*}}{(\\hat{x})} = \\sin{(\\hat{x})} and 1 = \\frac{\\sin{(\\hat{x})}}{\\operatorname{z^{*}}{(\\hat{x})}} and c{(\\tilde{g},\\mathbf{H})} = \\mathbf{H} + \\tilde{g} and \\int 1 d\\hat{x} = \\int \\frac{\\sin{(\\hat{x})}}{\\operatorname{z^{*}}{(\\hat{x})}} d\\hat{x} and c{(\\tilde{g},\\mathbf{H})} - \\int \\frac{\\sin{(\\hat{x})}}{\\operatorname{z^{*}}{(\\hat{x})}} d\\hat{x} = \\mathbf{H} + \\tilde{g} - \\int \\frac{\\sin{(\\hat{x})}}{\\operatorname{z^{*}}{(\\hat{x})}} d\\hat{x} and c{(\\tilde{g},\\mathbf{H})} - \\int 1 d\\hat{x} = \\mathbf{H} + \\tilde{g} - \\int 1 d\\hat{x} and - \\mathbf{H} - \\tilde{g} + c{(\\tilde{g},\\mathbf{H})} = 0", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 1, "Function('z^*')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))))"], ["get_premise", "Equality(Function('c')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Mul(Pow(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 3, "Integral(Mul(Pow(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Function('c')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Integral(Mul(Pow(Function('z^*')(Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('c')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True))))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True))))))"], [["minus", 6, "Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('c')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\mathbf{A}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})}, then derive \\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon} = J_{\\varepsilon} \\log{(J_{\\varepsilon})} - J_{\\varepsilon} + n_{1}, then obtain \\sin{((\\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon})^{n_{1}})} = \\sin{((J_{\\varepsilon} \\log{(J_{\\varepsilon})} - J_{\\varepsilon} + n_{1})^{n_{1}})}", "derivation": "\\mathbf{A}{(J_{\\varepsilon})} = \\log{(J_{\\varepsilon})} and \\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon} and \\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon} = J_{\\varepsilon} \\log{(J_{\\varepsilon})} - J_{\\varepsilon} + n_{1} and (\\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon})^{n_{1}} = (J_{\\varepsilon} \\log{(J_{\\varepsilon})} - J_{\\varepsilon} + n_{1})^{n_{1}} and \\sin{((\\int \\mathbf{A}{(J_{\\varepsilon})} dJ_{\\varepsilon})^{n_{1}})} = \\sin{((J_{\\varepsilon} \\log{(J_{\\varepsilon})} - J_{\\varepsilon} + n_{1})^{n_{1}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True)))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["sin", 4], "Equality(sin(Pow(Integral(Function('\\\\mathbf{A}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('n_1', commutative=True))), sin(Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), log(Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given J{(h,\\mathbf{B})} = \\frac{h}{\\mathbf{B}} and \\operatorname{f^{\\prime}}{(h,\\mathbf{B})} = \\frac{\\int \\frac{h}{\\mathbf{B}} dh}{- J{(h,\\mathbf{B})} + \\frac{h}{\\mathbf{B}}}, then obtain \\tilde{\\infty} \\int J{(h,\\mathbf{B})} dh = \\operatorname{f^{\\prime}}{(h,\\mathbf{B})}", "derivation": "J{(h,\\mathbf{B})} = \\frac{h}{\\mathbf{B}} and \\int J{(h,\\mathbf{B})} dh = \\int \\frac{h}{\\mathbf{B}} dh and \\frac{\\int J{(h,\\mathbf{B})} dh}{- J{(h,\\mathbf{B})} + \\frac{h}{\\mathbf{B}}} = \\frac{\\int \\frac{h}{\\mathbf{B}} dh}{- J{(h,\\mathbf{B})} + \\frac{h}{\\mathbf{B}}} and \\tilde{\\infty} \\int J{(h,\\mathbf{B})} dh = \\tilde{\\infty} \\int \\frac{h}{\\mathbf{B}} dh and \\operatorname{f^{\\prime}}{(h,\\mathbf{B})} = \\frac{\\int \\frac{h}{\\mathbf{B}} dh}{- J{(h,\\mathbf{B})} + \\frac{h}{\\mathbf{B}}} and \\operatorname{f^{\\prime}}{(h,\\mathbf{B})} = \\tilde{\\infty} \\int \\frac{h}{\\mathbf{B}} dh and \\tilde{\\infty} \\int J{(h,\\mathbf{B})} dh = \\operatorname{f^{\\prime}}{(h,\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], [["integrate", 1, "Symbol('h', commutative=True)"], "Equality(Integral(Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True))), Integer(-1)), Integral(Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(zoo, Integral(Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Mul(zoo, Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True))), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('f^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(zoo, Integral(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(zoo, Integral(Function('J')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('h', commutative=True)))), Function('f^{\\\\prime}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\theta_1)} = \\cos{(\\theta_1)}, then derive \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\theta_1 \\sin{(\\theta_1)} + f_{\\mathbf{p}} + \\cos{(\\theta_1)}, then obtain \\theta_1 \\mathbf{B}{(\\theta_1)} \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\theta_1 (\\theta_1 \\sin{(\\theta_1)} + f_{\\mathbf{p}} + \\mathbf{B}{(\\theta_1)}) \\mathbf{B}{(\\theta_1)}", "derivation": "\\mathbf{B}{(\\theta_1)} = \\cos{(\\theta_1)} and \\theta_1 \\mathbf{B}{(\\theta_1)} = \\theta_1 \\cos{(\\theta_1)} and \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\int \\theta_1 \\cos{(\\theta_1)} d\\theta_1 and \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\theta_1 \\sin{(\\theta_1)} + f_{\\mathbf{p}} + \\cos{(\\theta_1)} and \\theta_1 \\mathbf{B}{(\\theta_1)} \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\theta_1 (\\theta_1 \\sin{(\\theta_1)} + f_{\\mathbf{p}} + \\cos{(\\theta_1)}) \\mathbf{B}{(\\theta_1)} and \\theta_1 \\mathbf{B}{(\\theta_1)} \\int \\theta_1 \\mathbf{B}{(\\theta_1)} d\\theta_1 = \\theta_1 (\\theta_1 \\sin{(\\theta_1)} + f_{\\mathbf{p}} + \\mathbf{B}{(\\theta_1)}) \\mathbf{B}{(\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True)), cos(Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))))"], [["times", 4, "Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Add(Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Symbol('\\\\theta_1', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True)), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Symbol('\\\\theta_1', commutative=True), Add(Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\operatorname{F_{H}}{(\\sigma_p)} = 2 \\hat{x}{(\\sigma_p)}, then obtain \\operatorname{F_{H}}{(\\sigma_p)} + 1 = \\hat{x}{(\\sigma_p)} + \\frac{\\hat{x}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} + \\sin{(\\sigma_p)}", "derivation": "\\hat{x}{(\\sigma_p)} = \\sin{(\\sigma_p)} and \\frac{\\hat{x}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} = 1 and \\frac{\\hat{x}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} + \\sin{(\\sigma_p)} = \\sin{(\\sigma_p)} + 1 and 2 \\hat{x}{(\\sigma_p)} = \\hat{x}{(\\sigma_p)} + \\sin{(\\sigma_p)} and \\operatorname{F_{H}}{(\\sigma_p)} = 2 \\hat{x}{(\\sigma_p)} and \\operatorname{F_{H}}{(\\sigma_p)} = \\hat{x}{(\\sigma_p)} + \\sin{(\\sigma_p)} and \\operatorname{F_{H}}{(\\sigma_p)} + 1 = \\hat{x}{(\\sigma_p)} + \\sin{(\\sigma_p)} + 1 and \\operatorname{F_{H}}{(\\sigma_p)} + 1 = \\hat{x}{(\\sigma_p)} + \\frac{\\hat{x}{(\\sigma_p)}}{\\sin{(\\sigma_p)}} + \\sin{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "sin(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), sin(Symbol('\\\\sigma_p', commutative=True))), Add(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(1)))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True))), Add(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('F_H')(Symbol('\\\\sigma_p', commutative=True)), Add(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 6, "Integer(-1)"], "Equality(Add(Function('F_H')(Symbol('\\\\sigma_p', commutative=True)), Integer(1)), Add(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Function('F_H')(Symbol('\\\\sigma_p', commutative=True)), Integer(1)), Add(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Mul(Function('\\\\hat{x}')(Symbol('\\\\sigma_p', commutative=True)), Pow(sin(Symbol('\\\\sigma_p', commutative=True)), Integer(-1))), sin(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given f{(\\rho_b)} = e^{\\rho_b}, then obtain - 2 \\rho_b e^{\\rho_b} + f^{2}{(\\rho_b)} + f{(\\rho_b)} e^{\\rho_b} = - 2 \\rho_b e^{\\rho_b} + 2 f{(\\rho_b)} e^{\\rho_b}", "derivation": "f{(\\rho_b)} = e^{\\rho_b} and \\rho_b f{(\\rho_b)} = \\rho_b e^{\\rho_b} and f^{2}{(\\rho_b)} = f{(\\rho_b)} e^{\\rho_b} and - \\rho_b f{(\\rho_b)} + f^{2}{(\\rho_b)} = - \\rho_b f{(\\rho_b)} + f{(\\rho_b)} e^{\\rho_b} and - 2 \\rho_b f{(\\rho_b)} + f^{2}{(\\rho_b)} + f{(\\rho_b)} e^{\\rho_b} = - 2 \\rho_b f{(\\rho_b)} + 2 f{(\\rho_b)} e^{\\rho_b} and - 2 \\rho_b e^{\\rho_b} + f^{2}{(\\rho_b)} + f{(\\rho_b)} e^{\\rho_b} = - 2 \\rho_b e^{\\rho_b} + 2 f{(\\rho_b)} e^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "Function('f')(Symbol('\\\\rho_b', commutative=True))"], "Equality(Pow(Function('f')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 3, "Mul(Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Pow(Function('f')(Symbol('\\\\rho_b', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Mul(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Mul(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Pow(Function('f')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True), Function('f')(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(2), Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Pow(Function('f')(Symbol('\\\\rho_b', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\rho_b', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(2), Function('f')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given Z{(p,\\mathbf{p},r)} = \\mathbf{p} - p + r, then obtain (\\mathbf{p} - p + r)^{p} Z^{p}{(p,\\mathbf{p},r)} - \\int Z^{p}{(p,\\mathbf{p},r)} dr = (\\mathbf{p} - p + r)^{2 p} - \\int Z^{p}{(p,\\mathbf{p},r)} dr", "derivation": "Z{(p,\\mathbf{p},r)} = \\mathbf{p} - p + r and Z^{p}{(p,\\mathbf{p},r)} = (\\mathbf{p} - p + r)^{p} and (\\mathbf{p} - p + r)^{p} Z^{p}{(p,\\mathbf{p},r)} = (\\mathbf{p} - p + r)^{2 p} and (\\mathbf{p} - p + r)^{p} Z^{p}{(p,\\mathbf{p},r)} - \\int Z^{p}{(p,\\mathbf{p},r)} dr = (\\mathbf{p} - p + r)^{2 p} - \\int Z^{p}{(p,\\mathbf{p},r)} dr", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Symbol('p', commutative=True)))"], [["times", 2, "Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True))), Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))))"], [["minus", 3, "Integral(Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('r', commutative=True)))"], "Equality(Add(Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('r', commutative=True))))), Add(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('p', commutative=True)), Symbol('r', commutative=True)), Mul(Integer(2), Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('Z')(Symbol('p', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\omega{(\\hat{\\mathbf{x}},x)} = (e^{x})^{\\hat{\\mathbf{x}}}, then derive \\frac{\\partial}{\\partial x} \\omega{(\\hat{\\mathbf{x}},x)} = \\hat{\\mathbf{x}} (e^{x})^{\\hat{\\mathbf{x}}}, then obtain \\log{(\\frac{\\partial}{\\partial x} \\omega{(\\hat{\\mathbf{x}},x)})} = \\log{(\\hat{\\mathbf{x}} (e^{x})^{\\hat{\\mathbf{x}}})}", "derivation": "\\omega{(\\hat{\\mathbf{x}},x)} = (e^{x})^{\\hat{\\mathbf{x}}} and \\frac{\\partial}{\\partial x} \\omega{(\\hat{\\mathbf{x}},x)} = \\frac{\\partial}{\\partial x} (e^{x})^{\\hat{\\mathbf{x}}} and \\frac{\\partial}{\\partial x} \\omega{(\\hat{\\mathbf{x}},x)} = \\hat{\\mathbf{x}} (e^{x})^{\\hat{\\mathbf{x}}} and \\log{(\\frac{\\partial}{\\partial x} \\omega{(\\hat{\\mathbf{x}},x)})} = \\log{(\\hat{\\mathbf{x}} (e^{x})^{\\hat{\\mathbf{x}}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x', commutative=True)), Pow(exp(Symbol('x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(exp(Symbol('x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["log", 3], "Equality(log(Derivative(Function('\\\\omega')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), log(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Pow(exp(Symbol('x', commutative=True)), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given W{(n_{1})} = e^{n_{1}}, then obtain \\frac{d^{2}}{d n_{1}^{2}} (- e^{n_{1}} + \\frac{d}{d n_{1}} W{(n_{1})}) = \\frac{d^{2}}{d n_{1}^{2}} (- e^{n_{1}} + \\frac{d}{d n_{1}} e^{n_{1}})", "derivation": "W{(n_{1})} = e^{n_{1}} and \\frac{d}{d n_{1}} W{(n_{1})} = \\frac{d}{d n_{1}} e^{n_{1}} and - e^{n_{1}} + \\frac{d}{d n_{1}} W{(n_{1})} = - e^{n_{1}} + \\frac{d}{d n_{1}} e^{n_{1}} and \\frac{d}{d n_{1}} (- e^{n_{1}} + \\frac{d}{d n_{1}} W{(n_{1})}) = \\frac{d}{d n_{1}} (- e^{n_{1}} + \\frac{d}{d n_{1}} e^{n_{1}}) and \\frac{d^{2}}{d n_{1}^{2}} (- e^{n_{1}} + \\frac{d}{d n_{1}} W{(n_{1})}) = \\frac{d^{2}}{d n_{1}^{2}} (- e^{n_{1}} + \\frac{d}{d n_{1}} e^{n_{1}})", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('n_1', commutative=True)), exp(Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["minus", 2, "exp(Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(Function('W')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(Function('W')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(Function('W')(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), exp(Symbol('n_1', commutative=True))), Derivative(exp(Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Tuple(Symbol('n_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(i,q)} = \\cos{(i + q)}, then obtain \\frac{\\partial}{\\partial q} (\\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\operatorname{F_{x}}{(i,q)})}) = \\frac{\\partial}{\\partial q} (\\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\cos{(i + q)})})", "derivation": "\\operatorname{F_{x}}{(i,q)} = \\cos{(i + q)} and \\cos{(\\operatorname{F_{x}}{(i,q)})} = \\cos{(\\cos{(i + q)})} and \\frac{\\partial}{\\partial q} \\cos{(\\operatorname{F_{x}}{(i,q)})} = \\frac{\\partial}{\\partial q} \\cos{(\\cos{(i + q)})} and \\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\operatorname{F_{x}}{(i,q)})} = \\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\cos{(i + q)})} and \\frac{\\partial}{\\partial q} (\\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\operatorname{F_{x}}{(i,q)})}) = \\frac{\\partial}{\\partial q} (\\cos{(\\cos{(i + q)})} + \\frac{\\partial}{\\partial q} \\cos{(\\cos{(i + q)})})", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('i', commutative=True), Symbol('q', commutative=True)), cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True))))"], [["cos", 1], "Equality(cos(Function('F_x')(Symbol('i', commutative=True), Symbol('q', commutative=True))), cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(cos(Function('F_x')(Symbol('i', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 3, "cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True))))"], "Equality(Add(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Derivative(cos(Function('F_x')(Symbol('i', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Derivative(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Derivative(cos(Function('F_x')(Symbol('i', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Derivative(cos(cos(Add(Symbol('i', commutative=True), Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(p,h)} = \\int p^{h} dh and \\operatorname{E_{n}}{(p,h)} = \\int p^{h} dh, then obtain \\frac{\\frac{\\partial}{\\partial h} \\operatorname{C_{2}}{(p,h)}}{\\frac{\\partial}{\\partial h} \\operatorname{E_{n}}{(p,h)}} = 1", "derivation": "\\operatorname{C_{2}}{(p,h)} = \\int p^{h} dh and \\frac{\\partial}{\\partial h} \\operatorname{C_{2}}{(p,h)} = \\frac{\\partial}{\\partial h} \\int p^{h} dh and p \\frac{\\partial}{\\partial h} \\operatorname{C_{2}}{(p,h)} = p \\frac{\\partial}{\\partial h} \\int p^{h} dh and \\frac{\\frac{\\partial}{\\partial h} \\operatorname{C_{2}}{(p,h)}}{\\frac{\\partial}{\\partial h} \\int p^{h} dh} = 1 and \\operatorname{E_{n}}{(p,h)} = \\int p^{h} dh and \\frac{\\frac{\\partial}{\\partial h} \\operatorname{C_{2}}{(p,h)}}{\\frac{\\partial}{\\partial h} \\operatorname{E_{n}}{(p,h)}} = 1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 2, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Derivative(Function('C_2')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('p', commutative=True), Derivative(Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Symbol('p', commutative=True), Derivative(Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], "Equality(Mul(Derivative(Function('C_2')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Integral(Pow(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Derivative(Function('C_2')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Pow(Derivative(Function('E_n')(Symbol('p', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given I{(n)} = \\log{(\\sin{(n)})}, then obtain \\frac{(\\frac{I{(n)}}{n})^{n}}{F_{H}} = \\frac{(\\frac{\\log{(\\sin{(n)})}}{n})^{n}}{F_{H}}", "derivation": "I{(n)} = \\log{(\\sin{(n)})} and \\frac{I{(n)}}{n} = \\frac{\\log{(\\sin{(n)})}}{n} and (\\frac{I{(n)}}{n})^{n} = (\\frac{\\log{(\\sin{(n)})}}{n})^{n} and \\frac{(\\frac{I{(n)}}{n})^{n}}{F_{H}} = \\frac{(\\frac{\\log{(\\sin{(n)})}}{n})^{n}}{F_{H}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('n', commutative=True)), log(sin(Symbol('n', commutative=True))))"], [["divide", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), log(sin(Symbol('n', commutative=True)))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), log(sin(Symbol('n', commutative=True)))), Symbol('n', commutative=True)))"], [["divide", 3, "Symbol('F_H', commutative=True)"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('I')(Symbol('n', commutative=True))), Symbol('n', commutative=True))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), log(sin(Symbol('n', commutative=True)))), Symbol('n', commutative=True))))"]]}, {"prompt": "Given E{(\\Psi^{\\dagger},f)} = \\Psi^{\\dagger} f and \\operatorname{n_{2}}{(\\theta_1,U)} = U \\theta_1, then obtain U^{2} \\theta_1^{2} E{(\\Psi^{\\dagger},f)} = U^{2} \\Psi^{\\dagger} \\theta_1^{2} f", "derivation": "E{(\\Psi^{\\dagger},f)} = \\Psi^{\\dagger} f and \\operatorname{n_{2}}{(\\theta_1,U)} = U \\theta_1 and E{(\\Psi^{\\dagger},f)} \\operatorname{n_{2}}{(\\theta_1,U)} = \\Psi^{\\dagger} f \\operatorname{n_{2}}{(\\theta_1,U)} and U \\theta_1 E{(\\Psi^{\\dagger},f)} = U \\Psi^{\\dagger} \\theta_1 f and U^{2} \\theta_1^{2} E{(\\Psi^{\\dagger},f)} = U^{2} \\Psi^{\\dagger} \\theta_1^{2} f", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)))"], ["get_premise", "Equality(Function('n_2')(Symbol('\\\\theta_1', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Function('n_2')(Symbol('\\\\theta_1', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Function('E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)), Function('n_2')(Symbol('\\\\theta_1', commutative=True), Symbol('U', commutative=True))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True), Function('n_2')(Symbol('\\\\theta_1', commutative=True), Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True), Function('E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('U', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('f', commutative=True)))"], [["times", 4, "Mul(Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)), Function('E')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(2)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2)), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = \\frac{\\hat{p}_0}{F_{g}}, then derive \\frac{\\partial}{\\partial F_{g}} \\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = - \\frac{\\hat{p}_0}{F_{g}^{2}}, then obtain - \\frac{\\operatorname{m_{s}}{(F_{g},\\hat{p}_0)}}{F_{g}} = \\frac{\\partial}{\\partial F_{g}} \\frac{\\hat{p}_0}{F_{g}}", "derivation": "\\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = \\frac{\\hat{p}_0}{F_{g}} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = \\frac{\\partial}{\\partial F_{g}} \\frac{\\hat{p}_0}{F_{g}} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = - \\frac{\\hat{p}_0}{F_{g}^{2}} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{m_{s}}{(F_{g},\\hat{p}_0)} = - \\frac{\\operatorname{m_{s}}{(F_{g},\\hat{p}_0)}}{F_{g}} and - \\frac{\\operatorname{m_{s}}{(F_{g},\\hat{p}_0)}}{F_{g}} = \\frac{\\partial}{\\partial F_{g}} \\frac{\\hat{p}_0}{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-2)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('m_s')(Symbol('F_g', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"]]}, {"prompt": "Given p{(t)} = \\sin{(t)}, then derive \\int p{(t)} dt = \\theta - \\cos{(t)}, then obtain 2 \\theta - 2 \\cos{(t)} - \\int p{(t)} dt = \\theta - \\cos{(t)}", "derivation": "p{(t)} = \\sin{(t)} and \\int p{(t)} dt = \\int \\sin{(t)} dt and - \\int p{(t)} dt = - \\int \\sin{(t)} dt and - \\int p{(t)} dt + \\int \\sin{(t)} dt = 0 and \\int p{(t)} dt = \\theta - \\cos{(t)} and \\theta - \\cos{(t)} - \\int p{(t)} dt + \\int \\sin{(t)} dt = \\theta - \\cos{(t)} and \\int \\sin{(t)} dt = \\theta - \\cos{(t)} and 2 \\theta - 2 \\cos{(t)} - \\int p{(t)} dt = \\theta - \\cos{(t)}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Integer(-1), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["add", 3, "Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"], [["add", 4, "Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True))))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True))), Mul(Integer(-1), Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('t', commutative=True))), Mul(Integer(-1), Integral(Function('p')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(h,\\eta^{\\prime})} = \\frac{\\log{(\\eta^{\\prime})}}{h} and q{(h,\\eta^{\\prime})} = \\cos{((\\frac{\\log{(\\eta^{\\prime})}}{h})^{\\eta^{\\prime}})}, then obtain \\cos{(\\operatorname{g_{\\varepsilon}}^{\\eta^{\\prime}}{(h,\\eta^{\\prime})})} = q{(h,\\eta^{\\prime})}", "derivation": "\\operatorname{g_{\\varepsilon}}{(h,\\eta^{\\prime})} = \\frac{\\log{(\\eta^{\\prime})}}{h} and \\operatorname{g_{\\varepsilon}}^{\\eta^{\\prime}}{(h,\\eta^{\\prime})} = (\\frac{\\log{(\\eta^{\\prime})}}{h})^{\\eta^{\\prime}} and \\cos{(\\operatorname{g_{\\varepsilon}}^{\\eta^{\\prime}}{(h,\\eta^{\\prime})})} = \\cos{((\\frac{\\log{(\\eta^{\\prime})}}{h})^{\\eta^{\\prime}})} and q{(h,\\eta^{\\prime})} = \\cos{((\\frac{\\log{(\\eta^{\\prime})}}{h})^{\\eta^{\\prime}})} and \\cos{(\\operatorname{g_{\\varepsilon}}^{\\eta^{\\prime}}{(h,\\eta^{\\prime})})} = q{(h,\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["power", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('g_{\\\\varepsilon}')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), cos(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), cos(Pow(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), log(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(cos(Pow(Function('g_{\\\\varepsilon}')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Function('q')(Symbol('h', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(A_{2})} = \\log{(\\cos{(A_{2})})}, then obtain \\frac{d}{d A_{2}} (2 A_{2} + \\operatorname{F_{c}}{(A_{2})} + \\log{(\\cos{(A_{2})})}) = \\frac{d}{d A_{2}} (2 A_{2} + 2 \\log{(\\cos{(A_{2})})})", "derivation": "\\operatorname{F_{c}}{(A_{2})} = \\log{(\\cos{(A_{2})})} and A_{2} + \\operatorname{F_{c}}{(A_{2})} = A_{2} + \\log{(\\cos{(A_{2})})} and 2 A_{2} + \\operatorname{F_{c}}{(A_{2})} + \\log{(\\cos{(A_{2})})} = 2 A_{2} + 2 \\log{(\\cos{(A_{2})})} and \\frac{d}{d A_{2}} (2 A_{2} + \\operatorname{F_{c}}{(A_{2})} + \\log{(\\cos{(A_{2})})}) = \\frac{d}{d A_{2}} (2 A_{2} + 2 \\log{(\\cos{(A_{2})})})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True))))"], [["add", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Function('F_c')(Symbol('A_2', commutative=True))), Add(Symbol('A_2', commutative=True), log(cos(Symbol('A_2', commutative=True)))))"], [["add", 2, "Add(Symbol('A_2', commutative=True), log(cos(Symbol('A_2', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Function('F_c')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True)))), Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Mul(Integer(2), log(cos(Symbol('A_2', commutative=True))))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Function('F_c')(Symbol('A_2', commutative=True)), log(cos(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('A_2', commutative=True)), Mul(Integer(2), log(cos(Symbol('A_2', commutative=True))))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and v{(E,\\mathbf{H})} = E + \\mathbf{H}, then obtain v{(E,\\mathbf{H})} \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} = (E + \\mathbf{H}) \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})}", "derivation": "\\mathbf{A}{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\mathbf{A}{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} and v{(E,\\mathbf{H})} = E + \\mathbf{H} and v{(E,\\mathbf{H})} \\frac{d}{d \\mathbf{J}} \\mathbf{A}{(\\mathbf{J})} = (E + \\mathbf{H}) \\frac{d}{d \\mathbf{J}} \\mathbf{A}{(\\mathbf{J})} and v{(E,\\mathbf{H})} \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} = (E + \\mathbf{H}) \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}', commutative=True)), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('v')(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["times", 3, "Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))"], "Equality(Mul(Function('v')(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('v')(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Add(Symbol('E', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(r_{0})} = e^{\\cos{(r_{0})}}, then derive \\frac{d}{d r_{0}} \\omega{(r_{0})} = - e^{\\cos{(r_{0})}} \\sin{(r_{0})}, then obtain \\frac{d}{d r_{0}} \\omega{(r_{0})} = - \\omega{(r_{0})} \\sin{(r_{0})}", "derivation": "\\omega{(r_{0})} = e^{\\cos{(r_{0})}} and \\frac{d}{d r_{0}} \\omega{(r_{0})} = \\frac{d}{d r_{0}} e^{\\cos{(r_{0})}} and \\frac{d}{d r_{0}} \\omega{(r_{0})} = - e^{\\cos{(r_{0})}} \\sin{(r_{0})} and \\frac{d}{d r_{0}} \\omega{(r_{0})} = - \\omega{(r_{0})} \\sin{(r_{0})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\omega')(Symbol('r_0', commutative=True)), exp(cos(Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('r_0', commutative=True))), sin(Symbol('r_0', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\omega')(Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Mul(Integer(-1), Function('\\\\omega')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\rho,A_{1},F_{c})} = \\frac{A_{1}^{F_{c}}}{\\rho}, then obtain - \\frac{1}{\\rho} = - \\frac{A_{1}^{F_{c}}}{\\rho^{2} \\operatorname{A_{y}}{(\\rho,A_{1},F_{c})}}", "derivation": "\\operatorname{A_{y}}{(\\rho,A_{1},F_{c})} = \\frac{A_{1}^{F_{c}}}{\\rho} and \\frac{\\operatorname{A_{y}}{(\\rho,A_{1},F_{c})}}{F_{c}} = \\frac{A_{1}^{F_{c}}}{F_{c} \\rho} and \\frac{1}{\\rho} = \\frac{A_{1}^{F_{c}}}{\\rho^{2} \\operatorname{A_{y}}{(\\rho,A_{1},F_{c})}} and - \\frac{1}{\\rho} = - \\frac{A_{1}^{F_{c}}}{\\rho^{2} \\operatorname{A_{y}}{(\\rho,A_{1},F_{c})}}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('F_c', commutative=True)"], "Equality(Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True), Symbol('F_c', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('F_c', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["divide", 2, "Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True), Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)))"], "Equality(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Mul(Pow(Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Function('A_y')(Symbol('\\\\rho', commutative=True), Symbol('A_1', commutative=True), Symbol('F_c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{J}_f,\\omega)} = \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f}, then obtain (\\frac{\\tilde{g}{(\\mathbf{J}_f,\\omega)} - \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f}}{\\mathbf{J}_f \\tilde{g}{(\\mathbf{J}_f,\\omega)}})^{\\mathbf{J}_f} = 0^{\\mathbf{J}_f}", "derivation": "\\tilde{g}{(\\mathbf{J}_f,\\omega)} = \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f} and \\tilde{g}{(\\mathbf{J}_f,\\omega)} - \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f} = 0 and \\frac{\\tilde{g}{(\\mathbf{J}_f,\\omega)} - \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f}}{\\tilde{g}{(\\mathbf{J}_f,\\omega)}} = 0 and \\frac{\\tilde{g}{(\\mathbf{J}_f,\\omega)} - \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f}}{\\mathbf{J}_f \\tilde{g}{(\\mathbf{J}_f,\\omega)}} = 0 and (\\frac{\\tilde{g}{(\\mathbf{J}_f,\\omega)} - \\frac{\\cos{(\\omega)}}{\\mathbf{J}_f}}{\\mathbf{J}_f \\tilde{g}{(\\mathbf{J}_f,\\omega)}})^{\\mathbf{J}_f} = 0^{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Integer(0))"], [["divide", 2, "Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Integer(0))"], [["divide", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Integer(0))"], [["power", 4, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Add(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), cos(Symbol('\\\\omega', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given J{(\\sigma_x,\\chi)} = \\chi^{\\sigma_x}, then obtain (- \\chi^{\\sigma_x} + J{(\\sigma_x,\\chi)}) (\\chi^{\\sigma_x} + J^{\\chi}{(\\sigma_x,\\chi)}) = 0", "derivation": "J{(\\sigma_x,\\chi)} = \\chi^{\\sigma_x} and J^{\\chi}{(\\sigma_x,\\chi)} = (\\chi^{\\sigma_x})^{\\chi} and - (\\chi^{\\sigma_x})^{\\chi} + J{(\\sigma_x,\\chi)} = \\chi^{\\sigma_x} - (\\chi^{\\sigma_x})^{\\chi} and J{(\\sigma_x,\\chi)} - J^{\\chi}{(\\sigma_x,\\chi)} = \\chi^{\\sigma_x} - J^{\\chi}{(\\sigma_x,\\chi)} and - \\chi^{\\sigma_x} + J{(\\sigma_x,\\chi)} = 0 and (- \\chi^{\\sigma_x} + J{(\\sigma_x,\\chi)}) (\\chi^{\\sigma_x} + J^{\\chi}{(\\sigma_x,\\chi)}) = 0", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\chi', commutative=True))), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))))"], [["minus", 4, "Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(0))"], [["times", 5, "Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Pow(Symbol('\\\\chi', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Pow(Function('J')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\nabla{(G,V_{\\mathbf{B}})} = - G + \\log{(V_{\\mathbf{B}})} and J{(G,Z,v_{1},V_{\\mathbf{B}})} = (- G + \\log{(V_{\\mathbf{B}})}) \\operatorname{P_{g}}{(v_{1},Z)}, then obtain J{(G,Z,v_{1},V_{\\mathbf{B}})} = \\operatorname{P_{g}}{(v_{1},Z)} \\nabla{(G,V_{\\mathbf{B}})}", "derivation": "\\nabla{(G,V_{\\mathbf{B}})} = - G + \\log{(V_{\\mathbf{B}})} and \\operatorname{P_{g}}{(v_{1},Z)} \\nabla{(G,V_{\\mathbf{B}})} = (- G + \\log{(V_{\\mathbf{B}})}) \\operatorname{P_{g}}{(v_{1},Z)} and J{(G,Z,v_{1},V_{\\mathbf{B}})} = (- G + \\log{(V_{\\mathbf{B}})}) \\operatorname{P_{g}}{(v_{1},Z)} and J{(G,Z,v_{1},V_{\\mathbf{B}})} = \\operatorname{P_{g}}{(v_{1},Z)} \\nabla{(G,V_{\\mathbf{B}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["times", 1, "Function('P_g')(Symbol('v_1', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('v_1', commutative=True), Symbol('Z', commutative=True)), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('G', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('P_g')(Symbol('v_1', commutative=True), Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('G', commutative=True), Symbol('Z', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('G', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Function('P_g')(Symbol('v_1', commutative=True), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('J')(Symbol('G', commutative=True), Symbol('Z', commutative=True), Symbol('v_1', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Function('P_g')(Symbol('v_1', commutative=True), Symbol('Z', commutative=True)), Function('\\\\nabla')(Symbol('G', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{r},W)} = W \\mathbf{r}, then obtain (\\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW)^{2} = (W \\mathbf{r} + \\int W \\mathbf{r} dW) (\\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW)", "derivation": "\\hat{H}_l{(\\mathbf{r},W)} = W \\mathbf{r} and \\int \\hat{H}_l{(\\mathbf{r},W)} dW = \\int W \\mathbf{r} dW and \\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW = \\hat{H}_l{(\\mathbf{r},W)} + \\int W \\mathbf{r} dW and \\hat{H}_l{(\\mathbf{r},W)} + \\int W \\mathbf{r} dW = W \\mathbf{r} + \\int W \\mathbf{r} dW and \\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW = W \\mathbf{r} + \\int W \\mathbf{r} dW and (\\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW)^{2} = (W \\mathbf{r} + \\int W \\mathbf{r} dW) (\\hat{H}_l{(\\mathbf{r},W)} + \\int \\hat{H}_l{(\\mathbf{r},W)} dW)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["add", 2, "Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["add", 1, "Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["times", 5, "Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], "Equality(Pow(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Integer(2)), Mul(Add(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\omega{(x,z)} = z + e^{x} and C{(x,z)} = \\frac{\\partial}{\\partial z} e^{\\int (- z + \\omega{(x,z)} - e^{x}) dz}, then obtain C{(x,z)} = \\frac{d}{d z} 1", "derivation": "\\omega{(x,z)} = z + e^{x} and - z + \\omega{(x,z)} - e^{x} = 0 and \\int (- z + \\omega{(x,z)} - e^{x}) dz = \\int 0 dz and e^{\\int (- z + \\omega{(x,z)} - e^{x}) dz} = 1 and \\frac{\\partial}{\\partial z} e^{\\int (- z + \\omega{(x,z)} - e^{x}) dz} = \\frac{d}{d z} 1 and C{(x,z)} = \\frac{\\partial}{\\partial z} e^{\\int (- z + \\omega{(x,z)} - e^{x}) dz} and C{(x,z)} = \\frac{d}{d z} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Add(Symbol('z', commutative=True), exp(Symbol('x', commutative=True))))"], [["minus", 1, "Add(Symbol('z', commutative=True), exp(Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Tuple(Symbol('z', commutative=True))), Integral(Integer(0), Tuple(Symbol('z', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Tuple(Symbol('z', commutative=True)))), Integer(1))"], [["differentiate", 4, "Symbol('z', commutative=True)"], "Equality(Derivative(exp(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('C')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Derivative(exp(Integral(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\omega')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(Symbol('x', commutative=True)))), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('C')(Symbol('x', commutative=True), Symbol('z', commutative=True)), Derivative(Integer(1), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} e^{\\mathbf{D}}, then derive \\hat{\\mathbf{x}}{(\\mathbf{D})} = e^{\\mathbf{D}}, then obtain (\\frac{d}{d \\mathbf{D}} \\hat{\\mathbf{x}}{(\\mathbf{D})})^{\\mathbf{D}} = (e^{\\mathbf{D}})^{\\mathbf{D}}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} e^{\\mathbf{D}} and \\hat{\\mathbf{x}}{(\\mathbf{D})} = e^{\\mathbf{D}} and \\hat{\\mathbf{x}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\hat{\\mathbf{x}}{(\\mathbf{D})} and \\hat{\\mathbf{x}}^{\\mathbf{D}}{(\\mathbf{D})} = (e^{\\mathbf{D}})^{\\mathbf{D}} and (\\frac{d}{d \\mathbf{D}} \\hat{\\mathbf{x}}{(\\mathbf{D})})^{\\mathbf{D}} = (e^{\\mathbf{D}})^{\\mathbf{D}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), exp(Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"]]}, {"prompt": "Given Q{(J)} = e^{J}, then obtain (\\frac{d^{2}}{d J^{2}} Q^{J}{(J)})^{2} + \\frac{d}{d J} (e^{J})^{J} = \\frac{d}{d J} (e^{J})^{J} + (\\frac{d^{2}}{d J^{2}} (e^{J})^{J})^{2}", "derivation": "Q{(J)} = e^{J} and Q^{J}{(J)} = (e^{J})^{J} and \\frac{d}{d J} Q^{J}{(J)} = \\frac{d}{d J} (e^{J})^{J} and \\frac{d^{2}}{d J^{2}} Q^{J}{(J)} = \\frac{d^{2}}{d J^{2}} (e^{J})^{J} and (\\frac{d^{2}}{d J^{2}} Q^{J}{(J)})^{2} = (\\frac{d^{2}}{d J^{2}} (e^{J})^{J})^{2} and (\\frac{d^{2}}{d J^{2}} Q^{J}{(J)})^{2} + \\frac{d}{d J} (e^{J})^{J} = \\frac{d}{d J} (e^{J})^{J} + (\\frac{d^{2}}{d J^{2}} (e^{J})^{J})^{2}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Function('Q')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Function('Q')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))))"], [["power", 4, 2], "Equality(Pow(Derivative(Pow(Function('Q')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(2)))"], [["add", 5, "Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Pow(Derivative(Pow(Function('Q')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(2)), Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Pow(Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\Omega)} = e^{\\cos{(\\Omega)}}, then obtain U + \\operatorname{M_{E}}{(\\Omega)} = \\psi^* + e^{\\cos{(\\Omega)}}", "derivation": "\\operatorname{M_{E}}{(\\Omega)} = e^{\\cos{(\\Omega)}} and \\frac{d}{d \\Omega} \\operatorname{M_{E}}{(\\Omega)} = \\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} and \\int \\frac{d}{d \\Omega} \\operatorname{M_{E}}{(\\Omega)} d\\Omega = \\int \\frac{d}{d \\Omega} e^{\\cos{(\\Omega)}} d\\Omega and U + \\operatorname{M_{E}}{(\\Omega)} = \\psi^* + e^{\\cos{(\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\Omega', commutative=True)), exp(cos(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Function('M_E')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(exp(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('U', commutative=True), Function('M_E')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), exp(cos(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(f,\\omega)} = \\cos{(\\frac{f}{\\omega})}, then obtain - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega = - \\operatorname{F_{g}}^{\\omega}{(f,\\omega)} + \\cos^{\\omega}{(\\frac{f}{\\omega})} - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega", "derivation": "\\operatorname{F_{g}}{(f,\\omega)} = \\cos{(\\frac{f}{\\omega})} and \\operatorname{F_{g}}^{\\omega}{(f,\\omega)} = \\cos^{\\omega}{(\\frac{f}{\\omega})} and \\operatorname{F_{g}}^{\\omega}{(f,\\omega)} + \\frac{1}{\\omega} = \\cos^{\\omega}{(\\frac{f}{\\omega})} + \\frac{1}{\\omega} and \\operatorname{F_{g}}^{\\omega}{(f,\\omega)} - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega + \\frac{1}{\\omega} = \\cos^{\\omega}{(\\frac{f}{\\omega})} - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega + \\frac{1}{\\omega} and - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega = - \\operatorname{F_{g}}^{\\omega}{(f,\\omega)} + \\cos^{\\omega}{(\\frac{f}{\\omega})} - \\frac{\\partial}{\\partial f} \\int \\cos{(\\frac{f}{\\omega})} d\\omega", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\omega', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Add(Pow(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["minus", 3, "Derivative(Integral(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Derivative(Integral(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Add(Pow(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Derivative(Integral(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["minus", 4, "Add(Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], "Equality(Mul(Integer(-1), Derivative(Integral(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('F_g')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Pow(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Derivative(Integral(cos(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega}, then obtain (\\int \\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} d\\Omega + \\frac{1}{\\Omega})^{\\mathbf{H}} = (\\int \\frac{\\mathbf{H}}{\\Omega} d\\Omega + \\frac{1}{\\Omega})^{\\mathbf{H}}", "derivation": "\\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} = \\frac{\\mathbf{H}}{\\Omega} and \\int \\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} d\\Omega = \\int \\frac{\\mathbf{H}}{\\Omega} d\\Omega and \\int \\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} d\\Omega + \\frac{1}{\\Omega} = \\int \\frac{\\mathbf{H}}{\\Omega} d\\Omega + \\frac{1}{\\Omega} and (\\int \\Psi^{\\dagger}{(\\mathbf{H},\\Omega)} d\\Omega + \\frac{1}{\\Omega})^{\\mathbf{H}} = (\\int \\frac{\\mathbf{H}}{\\Omega} d\\Omega + \\frac{1}{\\Omega})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Add(Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Add(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Integral(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(F_{N})} = \\log{(F_{N})} and \\nabla{(F_{N})} = \\log{(F_{N})}, then obtain \\int (\\int 2 \\nabla{(F_{N})} dF_{N})^{2} dF_{N} = \\int (\\int (\\nabla{(F_{N})} + \\log{(F_{N})}) dF_{N})^{2} dF_{N}", "derivation": "\\operatorname{z^{*}}{(F_{N})} = \\log{(F_{N})} and 2 \\operatorname{z^{*}}{(F_{N})} = \\operatorname{z^{*}}{(F_{N})} + \\log{(F_{N})} and \\nabla{(F_{N})} = \\log{(F_{N})} and \\operatorname{z^{*}}{(F_{N})} = \\nabla{(F_{N})} and 2 \\nabla{(F_{N})} = \\nabla{(F_{N})} + \\log{(F_{N})} and \\int 2 \\nabla{(F_{N})} dF_{N} = \\int (\\nabla{(F_{N})} + \\log{(F_{N})}) dF_{N} and (\\int 2 \\nabla{(F_{N})} dF_{N})^{2} = (\\int (\\nabla{(F_{N})} + \\log{(F_{N})}) dF_{N})^{2} and \\int (\\int 2 \\nabla{(F_{N})} dF_{N})^{2} dF_{N} = \\int (\\int (\\nabla{(F_{N})} + \\log{(F_{N})}) dF_{N})^{2} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["add", 1, "Function('z^*')(Symbol('F_N', commutative=True))"], "Equality(Mul(Integer(2), Function('z^*')(Symbol('F_N', commutative=True))), Add(Function('z^*')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('z^*')(Symbol('F_N', commutative=True)), Function('\\\\nabla')(Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('F_N', commutative=True))), Add(Function('\\\\nabla')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))))"], [["integrate", 5, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\nabla')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Function('\\\\nabla')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["power", 6, 2], "Equality(Pow(Integral(Mul(Integer(2), Function('\\\\nabla')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(2)), Pow(Integral(Add(Function('\\\\nabla')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(2)))"], [["integrate", 7, "Symbol('F_N', commutative=True)"], "Equality(Integral(Pow(Integral(Mul(Integer(2), Function('\\\\nabla')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(2)), Tuple(Symbol('F_N', commutative=True))), Integral(Pow(Integral(Add(Function('\\\\nabla')(Symbol('F_N', commutative=True)), log(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integer(2)), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and m{(\\mathbf{B})} = 2 \\Psi{(\\mathbf{B})}, then obtain m{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} + \\sin{(\\mathbf{B})}", "derivation": "\\Psi{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and 2 \\Psi{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} + \\sin{(\\mathbf{B})} and m{(\\mathbf{B})} = 2 \\Psi{(\\mathbf{B})} and \\frac{m{(\\mathbf{B})}}{\\Psi{(\\mathbf{B})}} = 2 and m{(\\mathbf{B})} = \\Psi{(\\mathbf{B})} + \\sin{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["add", 1, "Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 3, "Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Function('m')(Symbol('\\\\mathbf{B}', commutative=True))), Integer(2))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('m')(Symbol('\\\\mathbf{B}', commutative=True)), Add(Function('\\\\Psi')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(W)} = \\log{(W)}, then obtain \\iint \\operatorname{J_{\\varepsilon}}^{W}{(W)} dW dW = \\iint \\log{(W)}^{W} dW dW", "derivation": "\\operatorname{J_{\\varepsilon}}{(W)} = \\log{(W)} and \\operatorname{J_{\\varepsilon}}^{W}{(W)} = \\log{(W)}^{W} and \\int \\operatorname{J_{\\varepsilon}}^{W}{(W)} dW = \\int \\log{(W)}^{W} dW and \\iint \\operatorname{J_{\\varepsilon}}^{W}{(W)} dW dW = \\iint \\log{(W)}^{W} dW dW", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["power", 1, "Symbol('W', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Pow(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(Pow(log(Symbol('W', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given L{(v_{2},W)} = \\sin{(W + v_{2})} and J{(v_{2},W)} = W + v_{2}, then obtain - n_{1} - \\sin{(W + v_{2})} + \\frac{\\partial}{\\partial v_{2}} J{(v_{2},W)} = - n_{1} - \\sin{(W + v_{2})} + \\frac{\\partial}{\\partial v_{2}} (W + v_{2})", "derivation": "L{(v_{2},W)} = \\sin{(W + v_{2})} and J{(v_{2},W)} = W + v_{2} and \\frac{\\partial}{\\partial v_{2}} J{(v_{2},W)} = \\frac{\\partial}{\\partial v_{2}} (W + v_{2}) and - n_{1} + \\frac{\\partial}{\\partial v_{2}} J{(v_{2},W)} = - n_{1} + \\frac{\\partial}{\\partial v_{2}} (W + v_{2}) and - n_{1} - L{(v_{2},W)} + \\frac{\\partial}{\\partial v_{2}} J{(v_{2},W)} = - n_{1} - L{(v_{2},W)} + \\frac{\\partial}{\\partial v_{2}} (W + v_{2}) and - n_{1} - \\sin{(W + v_{2})} + \\frac{\\partial}{\\partial v_{2}} J{(v_{2},W)} = - n_{1} - \\sin{(W + v_{2})} + \\frac{\\partial}{\\partial v_{2}} (W + v_{2})", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), sin(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('J')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Function('J')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Derivative(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["minus", 4, "Function('L')(Symbol('v_2', commutative=True), Symbol('W', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('v_2', commutative=True), Symbol('W', commutative=True))), Derivative(Function('J')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), Function('L')(Symbol('v_2', commutative=True), Symbol('W', commutative=True))), Derivative(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)))), Derivative(Function('J')(Symbol('v_2', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)))), Derivative(Add(Symbol('W', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(\\hbar)} = \\int e^{\\hbar} d\\hbar and \\dot{\\mathbf{r}}{(\\mathbf{g},\\hbar)} = 2 \\mathbf{g} + 2 e^{\\hbar}, then derive f{(\\hbar)} = B + e^{\\hbar}, then derive 2 \\mathbf{g} + 2 e^{\\hbar} = \\mathbf{g} + f{(\\hbar)} + e^{\\hbar}, then obtain e^{\\dot{\\mathbf{r}}{(\\mathbf{g},\\hbar)}} = e^{\\mathbf{g} + f{(\\hbar)} + e^{\\hbar}}", "derivation": "f{(\\hbar)} = \\int e^{\\hbar} d\\hbar and f{(\\hbar)} = B + e^{\\hbar} and f{(\\hbar)} + \\int e^{\\hbar} d\\hbar = B + e^{\\hbar} + \\int e^{\\hbar} d\\hbar and 2 \\int e^{\\hbar} d\\hbar = B + e^{\\hbar} + \\int e^{\\hbar} d\\hbar and 2 \\int e^{\\hbar} d\\hbar = f{(\\hbar)} + \\int e^{\\hbar} d\\hbar and 2 \\mathbf{g} + 2 e^{\\hbar} = \\mathbf{g} + f{(\\hbar)} + e^{\\hbar} and \\dot{\\mathbf{r}}{(\\mathbf{g},\\hbar)} = 2 \\mathbf{g} + 2 e^{\\hbar} and \\dot{\\mathbf{r}}{(\\mathbf{g},\\hbar)} = \\mathbf{g} + f{(\\hbar)} + e^{\\hbar} and e^{\\dot{\\mathbf{r}}{(\\mathbf{g},\\hbar)}} = e^{\\mathbf{g} + f{(\\hbar)} + e^{\\hbar}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('f')(Symbol('\\\\hbar', commutative=True)), Add(Symbol('B', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Function('f')(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('B', commutative=True), exp(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('B', commutative=True), exp(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Function('f')(Symbol('\\\\hbar', commutative=True)), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Function('f')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Function('f')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True))))"], [["exp", 8], "Equality(exp(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\hbar', commutative=True))), exp(Add(Symbol('\\\\mathbf{g}', commutative=True), Function('f')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\lambda,\\varepsilon_0)} = \\cos{(\\lambda - \\varepsilon_0)}, then obtain \\int \\mathbf{J}_P^{\\lambda}{(\\lambda,\\varepsilon_0)} d\\varepsilon_0 + 1 = \\int \\cos^{\\lambda}{(\\lambda - \\varepsilon_0)} d\\varepsilon_0 + 1", "derivation": "\\mathbf{J}_P{(\\lambda,\\varepsilon_0)} = \\cos{(\\lambda - \\varepsilon_0)} and \\mathbf{J}_P^{\\lambda}{(\\lambda,\\varepsilon_0)} = \\cos^{\\lambda}{(\\lambda - \\varepsilon_0)} and \\int \\mathbf{J}_P^{\\lambda}{(\\lambda,\\varepsilon_0)} d\\varepsilon_0 = \\int \\cos^{\\lambda}{(\\lambda - \\varepsilon_0)} d\\varepsilon_0 and \\int \\mathbf{J}_P^{\\lambda}{(\\lambda,\\varepsilon_0)} d\\varepsilon_0 + 1 = \\int \\cos^{\\lambda}{(\\lambda - \\varepsilon_0)} d\\varepsilon_0 + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integer(1)), Add(Integral(Pow(cos(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integer(1)))"]]}, {"prompt": "Given y{(\\hat{H})} = \\sin{(\\log{(\\hat{H})})} and \\hat{\\mathbf{r}}{(L)} = \\cos{(L)}, then obtain - \\hat{H} + \\hat{\\mathbf{r}}{(L)} - \\frac{1}{y{(\\hat{H})}} = - \\hat{H} + \\cos{(L)} - \\frac{1}{y{(\\hat{H})}}", "derivation": "y{(\\hat{H})} = \\sin{(\\log{(\\hat{H})})} and \\hat{\\mathbf{r}}{(L)} = \\cos{(L)} and \\hat{\\mathbf{r}}{(L)} - \\frac{1}{y{(\\hat{H})}} = \\cos{(L)} - \\frac{1}{y{(\\hat{H})}} and \\hat{\\mathbf{r}}{(L)} - \\frac{1}{\\sin{(\\log{(\\hat{H})})}} = \\cos{(L)} - \\frac{1}{\\sin{(\\log{(\\hat{H})})}} and - \\hat{H} + \\hat{\\mathbf{r}}{(L)} - \\frac{1}{\\sin{(\\log{(\\hat{H})})}} = - \\hat{H} + \\cos{(L)} - \\frac{1}{\\sin{(\\log{(\\hat{H})})}} and - \\hat{H} + \\hat{\\mathbf{r}}{(L)} - \\frac{1}{y{(\\hat{H})}} = - \\hat{H} + \\cos{(L)} - \\frac{1}{y{(\\hat{H})}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\hat{H}', commutative=True)), sin(log(Symbol('\\\\hat{H}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["minus", 2, "Pow(Function('y')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))), Add(cos(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)))), Add(cos(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)))))"], [["minus", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(sin(log(Symbol('\\\\hat{H}', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('L', commutative=True)), Mul(Integer(-1), Pow(Function('y')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} = \\sigma_x \\theta_1 and \\mathbf{S}{(\\sigma_x,\\theta_1)} = 2 \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)}, then obtain \\frac{\\partial}{\\partial \\sigma_x} (2 \\sigma_x \\theta_1 + \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)}) = \\frac{\\partial}{\\partial \\sigma_x} 3 \\sigma_x \\theta_1", "derivation": "\\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} = \\sigma_x \\theta_1 and 2 \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} = \\sigma_x \\theta_1 + \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} and \\mathbf{S}{(\\sigma_x,\\theta_1)} = 2 \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} and \\mathbf{S}{(\\sigma_x,\\theta_1)} = \\sigma_x \\theta_1 + \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} and \\mathbf{S}{(\\sigma_x,\\theta_1)} = 2 \\sigma_x \\theta_1 and \\sigma_x \\theta_1 + \\mathbf{S}{(\\sigma_x,\\theta_1)} = 3 \\sigma_x \\theta_1 and 2 \\sigma_x \\theta_1 + \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)} = 3 \\sigma_x \\theta_1 and \\frac{\\partial}{\\partial \\sigma_x} (2 \\sigma_x \\theta_1 + \\operatorname{C_{2}}{(\\sigma_x,\\theta_1)}) = \\frac{\\partial}{\\partial \\sigma_x} 3 \\sigma_x \\theta_1", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Integer(2), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["add", 5, "Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('\\\\mathbf{S}')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(3), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Integer(3), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 7, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Integer(3), Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{M},f_{E})} = \\frac{\\mathbf{M}}{f_{E}}, then obtain \\tilde{\\infty} \\int 0 df_{E} = \\tilde{\\infty} \\int - \\frac{\\frac{\\mathbf{M}}{f_{E}} - \\tilde{g}{(\\mathbf{M},f_{E})}}{\\tilde{g}{(\\mathbf{M},f_{E})}} df_{E}", "derivation": "\\tilde{g}{(\\mathbf{M},f_{E})} = \\frac{\\mathbf{M}}{f_{E}} and 0 = \\frac{\\mathbf{M}}{f_{E}} - \\tilde{g}{(\\mathbf{M},f_{E})} and 0 = - \\frac{\\frac{\\mathbf{M}}{f_{E}} - \\tilde{g}{(\\mathbf{M},f_{E})}}{\\tilde{g}{(\\mathbf{M},f_{E})}} and \\int 0 df_{E} = \\int - \\frac{\\frac{\\mathbf{M}}{f_{E}} - \\tilde{g}{(\\mathbf{M},f_{E})}}{\\tilde{g}{(\\mathbf{M},f_{E})}} df_{E} and \\tilde{\\infty} \\int 0 df_{E} = \\tilde{\\infty} \\int - \\frac{\\frac{\\mathbf{M}}{f_{E}} - \\tilde{g}{(\\mathbf{M},f_{E})}}{\\tilde{g}{(\\mathbf{M},f_{E})}} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["minus", 1, "Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('f_E', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Integer(-1), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Tuple(Symbol('f_E', commutative=True))))"], [["divide", 4, 0], "Equality(Mul(zoo, Integral(Integer(0), Tuple(Symbol('f_E', commutative=True)))), Mul(zoo, Integral(Mul(Integer(-1), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)))), Pow(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))), Tuple(Symbol('f_E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(t_{2},L)} = L + t_{2}, then obtain 0^{L} + L + t_{2} - \\mathbf{J}_P{(t_{2},L)} = 1", "derivation": "\\mathbf{J}_P{(t_{2},L)} = L + t_{2} and 0 = L + t_{2} - \\mathbf{J}_P{(t_{2},L)} and 0^{L} = (L + t_{2} - \\mathbf{J}_P{(t_{2},L)})^{L} and 0^{L} = 0^{L} + L + t_{2} - \\mathbf{J}_P{(t_{2},L)} and 0^{L} + L + t_{2} - \\mathbf{J}_P{(t_{2},L)} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('t_2', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(0), Add(Symbol('L', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True)))))"], [["power", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Integer(0), Symbol('L', commutative=True)), Pow(Add(Symbol('L', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True)))), Symbol('L', commutative=True)))"], [["add", 2, "Pow(Integer(0), Symbol('L', commutative=True))"], "Equality(Pow(Integer(0), Symbol('L', commutative=True)), Add(Pow(Integer(0), Symbol('L', commutative=True)), Symbol('L', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Pow(Integer(0), Symbol('L', commutative=True)), Symbol('L', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('t_2', commutative=True), Symbol('L', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\Psi_{nl}{(v_{z},s)} = s v_{z} and \\mathbf{F}{(v_{z},s)} = v_{z} + \\Psi_{nl}{(v_{z},s)}, then obtain (s v_{z} + v_{z}) \\mathbf{F}{(v_{z},s)} = (s v_{z} + v_{z})^{2}", "derivation": "\\Psi_{nl}{(v_{z},s)} = s v_{z} and v_{z} + \\Psi_{nl}{(v_{z},s)} = s v_{z} + v_{z} and (v_{z} + \\Psi_{nl}{(v_{z},s)}) (s v_{z} + v_{z}) = (s v_{z} + v_{z})^{2} and \\mathbf{F}{(v_{z},s)} = v_{z} + \\Psi_{nl}{(v_{z},s)} and (s v_{z} + v_{z}) \\mathbf{F}{(v_{z},s)} = (s v_{z} + v_{z})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)))"], [["add", 1, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["times", 2, "Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Mul(Add(Symbol('v_z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True))), Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))), Pow(Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True)), Add(Symbol('v_z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Function('\\\\mathbf{F}')(Symbol('v_z', commutative=True), Symbol('s', commutative=True))), Pow(Add(Mul(Symbol('s', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\bar{\\h}{(n,p,s)} = (s^{n})^{p}, then obtain \\frac{s + \\bar{\\h}^{p}{(n,p,s)}}{s} = \\frac{s + ((s^{n})^{p})^{p}}{s}", "derivation": "\\bar{\\h}{(n,p,s)} = (s^{n})^{p} and \\bar{\\h}^{p}{(n,p,s)} = ((s^{n})^{p})^{p} and s + \\bar{\\h}^{p}{(n,p,s)} = s + ((s^{n})^{p})^{p} and \\frac{s + \\bar{\\h}^{p}{(n,p,s)}}{s} = \\frac{s + ((s^{n})^{p})^{p}}{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True)), Pow(Pow(Symbol('s', commutative=True), Symbol('n', commutative=True)), Symbol('p', commutative=True)))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(Pow(Symbol('s', commutative=True), Symbol('n', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["add", 2, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Pow(Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True)), Symbol('p', commutative=True))), Add(Symbol('s', commutative=True), Pow(Pow(Pow(Symbol('s', commutative=True), Symbol('n', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))))"], [["divide", 3, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), Pow(Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('p', commutative=True), Symbol('s', commutative=True)), Symbol('p', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Add(Symbol('s', commutative=True), Pow(Pow(Pow(Symbol('s', commutative=True), Symbol('n', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(y)} = e^{\\cos{(y)}}, then obtain \\cos{(y - \\hat{H}{(y)})} = \\cos{(y - e^{\\frac{(\\hat{H}^{y}{(y)} - 1) \\cos{(y)}}{(e^{\\cos{(y)}})^{y} - 1}})}", "derivation": "\\hat{H}{(y)} = e^{\\cos{(y)}} and - y + \\hat{H}{(y)} = - y + e^{\\cos{(y)}} and \\hat{H}^{y}{(y)} = (e^{\\cos{(y)}})^{y} and \\hat{H}^{y}{(y)} - 1 = (e^{\\cos{(y)}})^{y} - 1 and \\frac{\\hat{H}^{y}{(y)} - 1}{(e^{\\cos{(y)}})^{y} - 1} = 1 and \\frac{(\\hat{H}^{y}{(y)} - 1) \\cos{(y)}}{(e^{\\cos{(y)}})^{y} - 1} = \\cos{(y)} and - y + \\hat{H}{(y)} = - y + e^{\\frac{(\\hat{H}^{y}{(y)} - 1) \\cos{(y)}}{(e^{\\cos{(y)}})^{y} - 1}} and \\cos{(y - \\hat{H}{(y)})} = \\cos{(y - e^{\\frac{(\\hat{H}^{y}{(y)} - 1) \\cos{(y)}}{(e^{\\cos{(y)}})^{y} - 1}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('y', commutative=True)), exp(cos(Symbol('y', commutative=True))))"], [["minus", 1, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{H}')(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), exp(cos(Symbol('y', commutative=True)))))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Integer(-1)), Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1)))"], [["divide", 4, "Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Integer(-1)), Pow(Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1)), Integer(-1))), Integer(1))"], [["times", 5, "cos(Symbol('y', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Integer(-1)), Pow(Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1)), Integer(-1)), cos(Symbol('y', commutative=True))), cos(Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Function('\\\\hat{H}')(Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), exp(Mul(Add(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Integer(-1)), Pow(Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1)), Integer(-1)), cos(Symbol('y', commutative=True))))))"], [["cos", 7], "Equality(cos(Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('y', commutative=True))))), cos(Add(Symbol('y', commutative=True), Mul(Integer(-1), exp(Mul(Add(Pow(Function('\\\\hat{H}')(Symbol('y', commutative=True)), Symbol('y', commutative=True)), Integer(-1)), Pow(Add(Pow(exp(cos(Symbol('y', commutative=True))), Symbol('y', commutative=True)), Integer(-1)), Integer(-1)), cos(Symbol('y', commutative=True))))))))"]]}, {"prompt": "Given \\mathbf{S}{(\\mathbf{P})} = \\int e^{\\mathbf{P}} d\\mathbf{P}, then obtain \\frac{d}{d \\mathbf{P}} \\iint \\mathbf{S}{(\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} = \\frac{d}{d \\mathbf{P}} \\iiint e^{\\mathbf{P}} d\\mathbf{P} d\\mathbf{P} d\\mathbf{P}", "derivation": "\\mathbf{S}{(\\mathbf{P})} = \\int e^{\\mathbf{P}} d\\mathbf{P} and \\int \\mathbf{S}{(\\mathbf{P})} d\\mathbf{P} = \\iint e^{\\mathbf{P}} d\\mathbf{P} d\\mathbf{P} and \\iint \\mathbf{S}{(\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} = \\iiint e^{\\mathbf{P}} d\\mathbf{P} d\\mathbf{P} d\\mathbf{P} and \\frac{d}{d \\mathbf{P}} \\iint \\mathbf{S}{(\\mathbf{P})} d\\mathbf{P} d\\mathbf{P} = \\frac{d}{d \\mathbf{P}} \\iiint e^{\\mathbf{P}} d\\mathbf{P} d\\mathbf{P} d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{S}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(y^{\\prime})} = y^{\\prime}, then obtain - 2 y^{\\prime} + 4 \\operatorname{F_{g}}{(y^{\\prime})} = 2 \\operatorname{F_{g}}{(y^{\\prime})}", "derivation": "\\operatorname{F_{g}}{(y^{\\prime})} = y^{\\prime} and - y^{\\prime} + \\operatorname{F_{g}}{(y^{\\prime})} = 0 and - 2 y^{\\prime} + 2 \\operatorname{F_{g}}{(y^{\\prime})} = - y^{\\prime} + \\operatorname{F_{g}}{(y^{\\prime})} and - 2 y^{\\prime} + 2 \\operatorname{F_{g}}{(y^{\\prime})} = 0 and - 2 y^{\\prime} + 4 \\operatorname{F_{g}}{(y^{\\prime})} = 2 \\operatorname{F_{g}}{(y^{\\prime})}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], [["minus", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True))), Integer(0))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)))), Integer(0))"], [["add", 4, "Mul(Integer(2), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(4), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(2), Function('F_g')(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(S)} = e^{S} and \\mathbf{B}{(S)} = S \\operatorname{v_{1}}{(S)}, then obtain \\mathbf{B}{(S)} \\operatorname{v_{1}}{(S)} = S \\operatorname{v_{1}}^{2}{(S)}", "derivation": "\\operatorname{v_{1}}{(S)} = e^{S} and S \\operatorname{v_{1}}{(S)} = S e^{S} and \\mathbf{B}{(S)} = S \\operatorname{v_{1}}{(S)} and \\mathbf{B}{(S)} = S e^{S} and \\mathbf{B}{(S)} \\operatorname{v_{1}}{(S)} = S \\operatorname{v_{1}}{(S)} e^{S} and \\mathbf{B}{(S)} \\operatorname{v_{1}}{(S)} = \\mathbf{B}{(S)} e^{S} and \\mathbf{B}{(S)} e^{S} = S \\operatorname{v_{1}}{(S)} e^{S} and \\mathbf{B}{(S)} \\operatorname{v_{1}}{(S)} = S \\operatorname{v_{1}}^{2}{(S)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["times", 1, "Symbol('S', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Function('v_1')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), exp(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Function('v_1')(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), exp(Symbol('S', commutative=True))))"], [["times", 4, "Function('v_1')(Symbol('S', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), Function('v_1')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), Function('v_1')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), Function('v_1')(Symbol('S', commutative=True))), Mul(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), Function('v_1')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('S', commutative=True)), Function('v_1')(Symbol('S', commutative=True))), Mul(Symbol('S', commutative=True), Pow(Function('v_1')(Symbol('S', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\mathbf{P}{(f^{\\prime},E_{n})} = \\frac{\\cos{(E_{n})}}{f^{\\prime}}, then obtain \\frac{\\partial}{\\partial E_{n}} \\int \\log{(\\frac{\\mathbf{P}{(f^{\\prime},E_{n})}}{\\cos{(E_{n})}})} dE_{n} = \\frac{\\partial}{\\partial E_{n}} \\int \\log{(\\frac{1}{f^{\\prime}})} dE_{n}", "derivation": "\\mathbf{P}{(f^{\\prime},E_{n})} = \\frac{\\cos{(E_{n})}}{f^{\\prime}} and \\frac{\\mathbf{P}{(f^{\\prime},E_{n})}}{\\cos{(E_{n})}} = \\frac{1}{f^{\\prime}} and \\log{(\\frac{\\mathbf{P}{(f^{\\prime},E_{n})}}{\\cos{(E_{n})}})} = \\log{(\\frac{1}{f^{\\prime}})} and \\int \\log{(\\frac{\\mathbf{P}{(f^{\\prime},E_{n})}}{\\cos{(E_{n})}})} dE_{n} = \\int \\log{(\\frac{1}{f^{\\prime}})} dE_{n} and \\frac{\\partial}{\\partial E_{n}} \\int \\log{(\\frac{\\mathbf{P}{(f^{\\prime},E_{n})}}{\\cos{(E_{n})}})} dE_{n} = \\frac{\\partial}{\\partial E_{n}} \\int \\log{(\\frac{1}{f^{\\prime}})} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), cos(Symbol('E_n', commutative=True))))"], [["divide", 1, "cos(Symbol('E_n', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Integer(-1))), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)))"], [["log", 2], "Equality(log(Mul(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Integer(-1)))), log(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(log(Mul(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Integer(-1)))), Tuple(Symbol('E_n', commutative=True))), Integral(log(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True))))"], [["differentiate", 4, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Integral(log(Mul(Function('\\\\mathbf{P}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('E_n', commutative=True)), Pow(cos(Symbol('E_n', commutative=True)), Integer(-1)))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Integral(log(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(F_{g})} = e^{F_{g}}, then obtain e^{\\eta^{\\prime}{(F_{g})} + \\int e^{F_{g}} dF_{g}} = e^{e^{F_{g}} + \\int e^{F_{g}} dF_{g}}", "derivation": "\\eta^{\\prime}{(F_{g})} = e^{F_{g}} and \\int \\eta^{\\prime}{(F_{g})} dF_{g} = \\int e^{F_{g}} dF_{g} and \\eta^{\\prime}{(F_{g})} + \\int \\eta^{\\prime}{(F_{g})} dF_{g} = e^{F_{g}} + \\int \\eta^{\\prime}{(F_{g})} dF_{g} and e^{\\eta^{\\prime}{(F_{g})} + \\int \\eta^{\\prime}{(F_{g})} dF_{g}} = e^{e^{F_{g}} + \\int \\eta^{\\prime}{(F_{g})} dF_{g}} and e^{\\eta^{\\prime}{(F_{g})} + \\int e^{F_{g}} dF_{g}} = e^{e^{F_{g}} + \\int e^{F_{g}} dF_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))"], [["add", 1, "Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))"], "Equality(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Add(exp(Symbol('F_g', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))), exp(Add(exp(Symbol('F_g', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Add(Function('\\\\eta^{\\\\prime}')(Symbol('F_g', commutative=True)), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))), exp(Add(exp(Symbol('F_g', commutative=True)), Integral(exp(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(G,\\hat{p}_0)} = G^{\\hat{p}_0} and l{(G,\\hat{p}_0)} = G^{\\hat{p}_0}, then obtain (G^{\\hat{p}_0})^{G} = l^{G}{(G,\\hat{p}_0)}", "derivation": "\\operatorname{a^{\\dagger}}{(G,\\hat{p}_0)} = G^{\\hat{p}_0} and \\operatorname{a^{\\dagger}}^{G}{(G,\\hat{p}_0)} = (G^{\\hat{p}_0})^{G} and l{(G,\\hat{p}_0)} = G^{\\hat{p}_0} and \\operatorname{a^{\\dagger}}^{G}{(G,\\hat{p}_0)} = l^{G}{(G,\\hat{p}_0)} and (G^{\\hat{p}_0})^{G} = l^{G}{(G,\\hat{p}_0)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["power", 1, "Symbol('G', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)), Pow(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)), Pow(Function('l')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Pow(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)), Pow(Function('l')(Symbol('G', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\sigma_p,\\hbar)} = \\hbar \\sigma_p, then derive (\\frac{\\partial}{\\partial \\hbar} \\operatorname{v_{z}}{(\\sigma_p,\\hbar)})^{\\sigma_p} = \\sigma_p^{\\sigma_p}, then obtain \\hbar \\sigma_p^{\\sigma_p} + b = \\int \\sigma_p^{\\sigma_p} d\\hbar", "derivation": "\\operatorname{v_{z}}{(\\sigma_p,\\hbar)} = \\hbar \\sigma_p and \\frac{\\partial}{\\partial \\hbar} \\operatorname{v_{z}}{(\\sigma_p,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\hbar \\sigma_p and (\\frac{\\partial}{\\partial \\hbar} \\operatorname{v_{z}}{(\\sigma_p,\\hbar)})^{\\sigma_p} = (\\frac{\\partial}{\\partial \\hbar} \\hbar \\sigma_p)^{\\sigma_p} and (\\frac{\\partial}{\\partial \\hbar} \\operatorname{v_{z}}{(\\sigma_p,\\hbar)})^{\\sigma_p} = \\sigma_p^{\\sigma_p} and (\\frac{\\partial}{\\partial \\hbar} \\hbar \\sigma_p)^{\\sigma_p} = \\sigma_p^{\\sigma_p} and \\int (\\frac{\\partial}{\\partial \\hbar} \\hbar \\sigma_p)^{\\sigma_p} d\\hbar = \\int \\sigma_p^{\\sigma_p} d\\hbar and \\hbar \\sigma_p^{\\sigma_p} + b = \\int \\sigma_p^{\\sigma_p} d\\hbar", "srepr_derivation": [["get_premise", "Equality(Function('v_z')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Derivative(Function('v_z')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('v_z')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Symbol('\\\\hbar', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Symbol('b', commutative=True)), Integral(Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(\\theta_2)} = \\cos{(e^{\\theta_2})}, then obtain \\frac{d}{d \\theta_2} (\\mathbf{s}{(\\theta_2)} + \\mathbf{s}^{\\theta_2}{(\\theta_2)}) = \\frac{d}{d \\theta_2} (\\mathbf{s}^{\\theta_2}{(\\theta_2)} + \\cos{(e^{\\theta_2})})", "derivation": "\\mathbf{s}{(\\theta_2)} = \\cos{(e^{\\theta_2})} and \\mathbf{s}^{\\theta_2}{(\\theta_2)} = \\cos^{\\theta_2}{(e^{\\theta_2})} and \\mathbf{s}{(\\theta_2)} + \\cos^{\\theta_2}{(e^{\\theta_2})} = \\cos{(e^{\\theta_2})} + \\cos^{\\theta_2}{(e^{\\theta_2})} and \\frac{d}{d \\theta_2} (\\mathbf{s}{(\\theta_2)} + \\cos^{\\theta_2}{(e^{\\theta_2})}) = \\frac{d}{d \\theta_2} (\\cos{(e^{\\theta_2})} + \\cos^{\\theta_2}{(e^{\\theta_2})}) and \\frac{d}{d \\theta_2} (\\mathbf{s}{(\\theta_2)} + \\mathbf{s}^{\\theta_2}{(\\theta_2)}) = \\frac{d}{d \\theta_2} (\\mathbf{s}^{\\theta_2}{(\\theta_2)} + \\cos{(e^{\\theta_2})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), cos(exp(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Add(cos(exp(Symbol('\\\\theta_2', commutative=True))), Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(cos(exp(Symbol('\\\\theta_2', commutative=True))), Pow(cos(exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Pow(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Pow(Function('\\\\mathbf{s}')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), cos(exp(Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given U{(C_{d},I,M_{E})} = \\frac{C_{d} + I}{M_{E}}, then obtain - M_{E} - h + 1 = - M_{E} - h + (\\frac{C_{d} + I}{M_{E} U{(C_{d},I,M_{E})}})^{M_{E}}", "derivation": "U{(C_{d},I,M_{E})} = \\frac{C_{d} + I}{M_{E}} and 1 = \\frac{C_{d} + I}{M_{E} U{(C_{d},I,M_{E})}} and 1 = (\\frac{C_{d} + I}{M_{E} U{(C_{d},I,M_{E})}})^{M_{E}} and 1 - \\int 1 dM_{E} = (\\frac{C_{d} + I}{M_{E} U{(C_{d},I,M_{E})}})^{M_{E}} - \\int 1 dM_{E} and - M_{E} - h + 1 = - M_{E} - h + (\\frac{C_{d} + I}{M_{E} U{(C_{d},I,M_{E})}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('I', commutative=True))))"], [["divide", 1, "Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('I', commutative=True)), Pow(Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('M_E', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('I', commutative=True)), Pow(Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Symbol('M_E', commutative=True)))"], [["minus", 3, "Integral(Integer(1), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('M_E', commutative=True))))), Add(Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('I', commutative=True)), Pow(Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Symbol('M_E', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('M_E', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Add(Symbol('C_d', commutative=True), Symbol('I', commutative=True)), Pow(Function('U')(Symbol('C_d', commutative=True), Symbol('I', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given S{(T,\\varphi)} = T + \\varphi and s{(T,\\varphi)} = \\int S{(T,\\varphi)} dT, then derive (T + \\varphi - S{(T,\\varphi)} + s{(T,\\varphi)})^{\\varphi} = (\\frac{T^{2}}{2} + T \\varphi + T + \\varphi + q - S{(T,\\varphi)})^{\\varphi}, then obtain s^{\\varphi}{(T,\\varphi)} = (\\frac{T^{2}}{2} + T \\varphi + q)^{\\varphi}", "derivation": "S{(T,\\varphi)} = T + \\varphi and \\int S{(T,\\varphi)} dT = \\int (T + \\varphi) dT and s{(T,\\varphi)} = \\int S{(T,\\varphi)} dT and T + \\varphi - S{(T,\\varphi)} + \\int S{(T,\\varphi)} dT = T + \\varphi - S{(T,\\varphi)} + \\int (T + \\varphi) dT and (T + \\varphi - S{(T,\\varphi)} + \\int S{(T,\\varphi)} dT)^{\\varphi} = (T + \\varphi - S{(T,\\varphi)} + \\int (T + \\varphi) dT)^{\\varphi} and (T + \\varphi - S{(T,\\varphi)} + s{(T,\\varphi)})^{\\varphi} = (T + \\varphi - S{(T,\\varphi)} + \\int (T + \\varphi) dT)^{\\varphi} and (T + \\varphi - S{(T,\\varphi)} + s{(T,\\varphi)})^{\\varphi} = (\\frac{T^{2}}{2} + T \\varphi + T + \\varphi + q - S{(T,\\varphi)})^{\\varphi} and s^{\\varphi}{(T,\\varphi)} = (\\frac{T^{2}}{2} + T \\varphi + q)^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Integral(Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["power", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Integral(Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('s')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('T', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('s')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('q', commutative=True), Mul(Integer(-1), Function('S')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Pow(Function('s')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('q', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given t{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then obtain \\int \\frac{t{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} d\\mathbf{S} = \\int \\frac{\\cos{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} d\\mathbf{S}", "derivation": "t{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and t{(\\mathbf{S})} \\cos{(\\mathbf{S})} = \\cos^{2}{(\\mathbf{S})} and \\frac{t{(\\mathbf{S})}}{- \\mathbf{S} + t{(\\mathbf{S})} \\cos{(\\mathbf{S})} + \\cos^{2}{(\\mathbf{S})}} = \\frac{\\cos{(\\mathbf{S})}}{- \\mathbf{S} + t{(\\mathbf{S})} \\cos{(\\mathbf{S})} + \\cos^{2}{(\\mathbf{S})}} and \\frac{t{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} = \\frac{\\cos{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} and \\int \\frac{t{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} d\\mathbf{S} = \\int \\frac{\\cos{(\\mathbf{S})}}{- \\mathbf{S} + 2 \\cos^{2}{(\\mathbf{S})}} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('t')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Function('t')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Function('t')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Integer(-1)), Function('t')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Function('t')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), Integer(-1)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))), Integer(-1)), Function('t')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))), Integer(-1)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))), Integer(-1)), Function('t')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))), Integer(-1)), cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(u)} = \\log{(u)}, then derive \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = Z + u \\log{(u)} - u, then obtain - \\sigma_{p}{(u)} - \\int A_{2} \\mathbf{P} dA_{2} + \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = Z + u \\sigma_{p}{(u)} - u - \\sigma_{p}{(u)} - \\int A_{2} \\mathbf{P} dA_{2}", "derivation": "\\sigma_{p}{(u)} = \\log{(u)} and \\int \\sigma_{p}{(u)} du = \\int \\log{(u)} du and \\frac{d}{d u} \\int \\sigma_{p}{(u)} du = \\frac{d}{d u} \\int \\log{(u)} du and \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = \\int \\frac{d}{d u} \\int \\log{(u)} du du and \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = Z + u \\log{(u)} - u and \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = Z + u \\sigma_{p}{(u)} - u and - \\sigma_{p}{(u)} - \\int A_{2} \\mathbf{P} dA_{2} + \\int \\frac{d}{d u} \\int \\sigma_{p}{(u)} du du = Z + u \\sigma_{p}{(u)} - u - \\sigma_{p}{(u)} - \\int A_{2} \\mathbf{P} dA_{2}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Derivative(Integral(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Derivative(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Add(Symbol('Z', commutative=True), Mul(Symbol('u', commutative=True), log(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Derivative(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Add(Symbol('Z', commutative=True), Mul(Symbol('u', commutative=True), Function('\\\\sigma_p')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["minus", 6, "Add(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('u', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Integral(Derivative(Integral(Function('\\\\sigma_p')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))), Add(Symbol('Z', commutative=True), Mul(Symbol('u', commutative=True), Function('\\\\sigma_p')(Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('u', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('A_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and E{(L)} = e^{L}, then obtain (- \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + E{(L)})^{\\hat{H}_l} = (- \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + e^{L})^{\\hat{H}_l}", "derivation": "\\hat{H}_{\\lambda}{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} = \\hat{H}_{\\lambda}{(\\hat{H}_l)} \\log{(\\hat{H}_l)} and E{(L)} = e^{L} and - \\hat{H}_l \\hat{H}_{\\lambda}{(\\hat{H}_l)} \\log{(\\hat{H}_l)} + E{(L)} = - \\hat{H}_l \\hat{H}_{\\lambda}{(\\hat{H}_l)} \\log{(\\hat{H}_l)} + e^{L} and - \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + E{(L)} = - \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + e^{L} and (- \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + E{(L)})^{\\hat{H}_l} = (- \\hat{H}_l \\hat{H}_{\\lambda}^{2}{(\\hat{H}_l)} + e^{L})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["times", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))))"], ["get_premise", "Equality(Function('E')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["minus", 3, "Mul(Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Function('E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), exp(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), Function('E')(Symbol('L', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), exp(Symbol('L', commutative=True))))"], [["power", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), Function('E')(Symbol('L', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(2))), exp(Symbol('L', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given v{(x^\\prime,b)} = b x^\\prime and \\operatorname{P_{g}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int v{(x^\\prime,b)} dx^\\prime)}, then obtain - x^\\prime + \\operatorname{P_{g}}{(x^\\prime,b)} - \\int v{(x^\\prime,b)} dx^\\prime = - x^\\prime + \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int b x^\\prime dx^\\prime)} - \\int v{(x^\\prime,b)} dx^\\prime", "derivation": "v{(x^\\prime,b)} = b x^\\prime and \\int v{(x^\\prime,b)} dx^\\prime = \\int b x^\\prime dx^\\prime and \\log{(\\int v{(x^\\prime,b)} dx^\\prime)} = \\log{(\\int b x^\\prime dx^\\prime)} and \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int v{(x^\\prime,b)} dx^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int b x^\\prime dx^\\prime)} and \\operatorname{P_{g}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int v{(x^\\prime,b)} dx^\\prime)} and \\operatorname{P_{g}}{(x^\\prime,b)} = \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int b x^\\prime dx^\\prime)} and - x^\\prime + \\operatorname{P_{g}}{(x^\\prime,b)} - \\int v{(x^\\prime,b)} dx^\\prime = - x^\\prime + \\frac{\\partial}{\\partial x^\\prime} \\log{(\\int b x^\\prime dx^\\prime)} - \\int v{(x^\\prime,b)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), log(Integral(Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(log(Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Integral(Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Derivative(log(Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('P_g')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Derivative(log(Integral(Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["minus", 6, "Add(Symbol('x^\\\\prime', commutative=True), Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('P_g')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))), Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Derivative(log(Integral(Mul(Symbol('b', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Function('v')(Symbol('x^\\\\prime', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))))"]]}, {"prompt": "Given A{(Z)} = e^{Z}, then obtain A{(Z)} = - A{(Z)} + 2 e^{Z}", "derivation": "A{(Z)} = e^{Z} and 2 A{(Z)} + e^{Z} = A{(Z)} + 2 e^{Z} and e^{Z} = - A{(Z)} + 2 e^{Z} and A{(Z)} = - A{(Z)} + 2 e^{Z}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["add", 1, "Add(Function('A')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('A')(Symbol('Z', commutative=True))), exp(Symbol('Z', commutative=True))), Add(Function('A')(Symbol('Z', commutative=True)), Mul(Integer(2), exp(Symbol('Z', commutative=True)))))"], [["minus", 2, "Mul(Integer(2), Function('A')(Symbol('Z', commutative=True)))"], "Equality(exp(Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Function('A')(Symbol('Z', commutative=True))), Mul(Integer(2), exp(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('A')(Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Function('A')(Symbol('Z', commutative=True))), Mul(Integer(2), exp(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given c{(x^\\prime)} = \\log{(x^\\prime)} and \\varphi^{*}{(x^\\prime)} = \\log{(x^\\prime)}, then obtain (2 \\frac{d}{d x^\\prime} c{(x^\\prime)} + \\frac{2}{x^\\prime})^{4} = \\frac{256}{(x^\\prime)^{4}}", "derivation": "c{(x^\\prime)} = \\log{(x^\\prime)} and c{(x^\\prime)} + \\log{(x^\\prime)} = 2 \\log{(x^\\prime)} and \\varphi^{*}{(x^\\prime)} = \\log{(x^\\prime)} and \\varphi^{*}{(x^\\prime)} + c{(x^\\prime)} = 2 \\varphi^{*}{(x^\\prime)} and 2 \\varphi^{*}{(x^\\prime)} + 2 c{(x^\\prime)} = 4 \\varphi^{*}{(x^\\prime)} and 2 c{(x^\\prime)} + 2 \\log{(x^\\prime)} = 4 \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} (2 c{(x^\\prime)} + 2 \\log{(x^\\prime)}) = \\frac{d}{d x^\\prime} 4 \\log{(x^\\prime)} and (\\frac{d}{d x^\\prime} (2 c{(x^\\prime)} + 2 \\log{(x^\\prime)}))^{4} = (\\frac{d}{d x^\\prime} 4 \\log{(x^\\prime)})^{4} and (2 \\frac{d}{d x^\\prime} c{(x^\\prime)} + \\frac{2}{x^\\prime})^{4} = \\frac{256}{(x^\\prime)^{4}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('c')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True)), Function('c')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 4, "Rational(1, 2)"], "Equality(Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Function('c')(Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(4), Function('\\\\varphi^*')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(2), Function('c')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(4), log(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('c')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Integer(4), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 7, 4], "Equality(Pow(Derivative(Add(Mul(Integer(2), Function('c')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(4)), Pow(Derivative(Mul(Integer(4), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(4)))"], [["evaluate_derivatives", 8], "Equality(Pow(Add(Mul(Integer(2), Derivative(Function('c')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))), Integer(4)), Mul(Integer(256), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-4))))"]]}, {"prompt": "Given \\ddot{x}{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then obtain (e^{\\int \\ddot{x}{(a^{\\dagger})} da^{\\dagger}})^{a^{\\dagger}} = (e^{\\hat{p}_0 - \\cos{(a^{\\dagger})}})^{a^{\\dagger}}", "derivation": "\\ddot{x}{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and \\int \\ddot{x}{(a^{\\dagger})} da^{\\dagger} = \\int \\sin{(a^{\\dagger})} da^{\\dagger} and e^{\\int \\ddot{x}{(a^{\\dagger})} da^{\\dagger}} = e^{\\int \\sin{(a^{\\dagger})} da^{\\dagger}} and (e^{\\int \\ddot{x}{(a^{\\dagger})} da^{\\dagger}})^{a^{\\dagger}} = (e^{\\int \\sin{(a^{\\dagger})} da^{\\dagger}})^{a^{\\dagger}} and (e^{\\int \\ddot{x}{(a^{\\dagger})} da^{\\dagger}})^{a^{\\dagger}} = (e^{\\hat{p}_0 - \\cos{(a^{\\dagger})}})^{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('\\\\ddot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), exp(Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["power", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(exp(Integral(Function('\\\\ddot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(Integral(sin(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(exp(Integral(Function('\\\\ddot{x}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Symbol('a^{\\\\dagger}', commutative=True)), Pow(exp(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True))))), Symbol('a^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mu_0)} = e^{\\mu_0}, then derive \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} = e^{\\mu_0}, then derive i + \\frac{d}{d \\mu_0} \\operatorname{F_{x}}{(\\mu_0)} = v_{1} + e^{\\mu_0}, then obtain i + \\frac{d}{d \\mu_0} \\operatorname{F_{x}}{(\\mu_0)} + \\int \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} d\\mu_0 = v_{1} + e^{\\mu_0} + \\int \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} d\\mu_0", "derivation": "\\operatorname{F_{x}}{(\\mu_0)} = e^{\\mu_0} and \\frac{d}{d \\mu_0} \\operatorname{F_{x}}{(\\mu_0)} = \\frac{d}{d \\mu_0} e^{\\mu_0} and \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} = \\frac{d^{2}}{d \\mu_0^{2}} e^{\\mu_0} and \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} = e^{\\mu_0} and \\int \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} d\\mu_0 = \\int e^{\\mu_0} d\\mu_0 and i + \\frac{d}{d \\mu_0} \\operatorname{F_{x}}{(\\mu_0)} = v_{1} + e^{\\mu_0} and i + \\frac{d}{d \\mu_0} \\operatorname{F_{x}}{(\\mu_0)} + \\int \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} d\\mu_0 = v_{1} + e^{\\mu_0} + \\int \\frac{d^{2}}{d \\mu_0^{2}} \\operatorname{F_{x}}{(\\mu_0)} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), exp(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), exp(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(exp(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('i', commutative=True), Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\mu_0', commutative=True))))"], [["add", 6, "Integral(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Symbol('i', commutative=True), Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integral(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Add(Symbol('v_1', commutative=True), exp(Symbol('\\\\mu_0', commutative=True)), Integral(Derivative(Function('F_x')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(2))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(c)} = \\sin{(c)}, then derive c \\frac{d}{d c} \\rho_{f}{(c)} + \\rho_{f}{(c)} = c \\cos{(c)} + \\sin{(c)}, then obtain \\frac{\\int (c \\frac{d}{d c} \\sin{(c)} + \\sin{(c)}) dc}{\\frac{d}{d \\Psi} U{(\\Psi)}} = \\frac{\\int (c \\cos{(c)} + \\sin{(c)}) dc}{\\frac{d}{d \\Psi} U{(\\Psi)}}", "derivation": "\\rho_{f}{(c)} = \\sin{(c)} and c \\rho_{f}{(c)} = c \\sin{(c)} and \\frac{d}{d c} c \\rho_{f}{(c)} = \\frac{d}{d c} c \\sin{(c)} and c \\frac{d}{d c} \\rho_{f}{(c)} + \\rho_{f}{(c)} = c \\cos{(c)} + \\sin{(c)} and c \\frac{d}{d c} \\sin{(c)} + \\sin{(c)} = c \\cos{(c)} + \\sin{(c)} and \\int (c \\frac{d}{d c} \\sin{(c)} + \\sin{(c)}) dc = \\int (c \\cos{(c)} + \\sin{(c)}) dc and \\frac{\\int (c \\frac{d}{d c} \\sin{(c)} + \\sin{(c)}) dc}{\\frac{d}{d \\Psi} U{(\\Psi)}} = \\frac{\\int (c \\cos{(c)} + \\sin{(c)}) dc}{\\frac{d}{d \\Psi} U{(\\Psi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('c', commutative=True)), sin(Symbol('c', commutative=True)))"], [["times", 1, "Symbol('c', commutative=True)"], "Equality(Mul(Symbol('c', commutative=True), Function('\\\\rho_f')(Symbol('c', commutative=True))), Mul(Symbol('c', commutative=True), sin(Symbol('c', commutative=True))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Symbol('c', commutative=True), Function('\\\\rho_f')(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Mul(Symbol('c', commutative=True), sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('c', commutative=True), Derivative(Function('\\\\rho_f')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Function('\\\\rho_f')(Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))), sin(Symbol('c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('c', commutative=True), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), sin(Symbol('c', commutative=True))), Add(Mul(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))), sin(Symbol('c', commutative=True))))"], [["integrate", 5, "Symbol('c', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('c', commutative=True), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))), Integral(Add(Mul(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))), sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True))))"], [["divide", 6, "Derivative(Function('U')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('U')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Mul(Symbol('c', commutative=True), Derivative(sin(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))), Mul(Pow(Derivative(Function('U')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Mul(Symbol('c', commutative=True), cos(Symbol('c', commutative=True))), sin(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given c{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and u{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} c{(L_{\\varepsilon})}, then obtain u^{L_{\\varepsilon}}{(L_{\\varepsilon})} = (\\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})})^{L_{\\varepsilon}}", "derivation": "c{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} c{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} and u{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} c{(L_{\\varepsilon})} and u^{L_{\\varepsilon}}{(L_{\\varepsilon})} = (\\frac{d}{d L_{\\varepsilon}} c{(L_{\\varepsilon})})^{L_{\\varepsilon}} and u^{L_{\\varepsilon}}{(L_{\\varepsilon})} = (\\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('u')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(Function('c')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('u')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Derivative(Function('c')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('u')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(z)} = \\cos{(z)} and \\operatorname{P_{g}}{(\\mathbf{F})} = \\mathbf{F}, then obtain - \\frac{- \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\operatorname{v_{2}}{(z)} dz}{\\mathbf{F}} = - \\frac{- \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\cos{(z)} dz}{\\mathbf{F}}", "derivation": "\\operatorname{v_{2}}{(z)} = \\cos{(z)} and \\int \\operatorname{v_{2}}{(z)} dz = \\int \\cos{(z)} dz and \\operatorname{P_{g}}{(\\mathbf{F})} = \\mathbf{F} and - \\mathbf{F} + \\int \\operatorname{v_{2}}{(z)} dz = - \\mathbf{F} + \\int \\cos{(z)} dz and - \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\operatorname{v_{2}}{(z)} dz = - \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\cos{(z)} dz and - \\frac{- \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\operatorname{v_{2}}{(z)} dz}{\\mathbf{F}} = - \\frac{- \\operatorname{P_{g}}{(\\mathbf{F})} + \\int \\cos{(z)} dz}{\\mathbf{F}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))"], [["minus", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Integral(Function('v_2')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Function('v_2')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["divide", 5, "Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Function('v_2')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given B{(C_{1},\\hbar)} = - C_{1} + \\hbar and \\operatorname{P_{g}}{(E_{x})} = \\log{(E_{x})}, then obtain - C_{1} - B{(C_{1},\\hbar)} - \\operatorname{P_{g}}{(E_{x})} = - \\hbar - \\operatorname{P_{g}}{(E_{x})}", "derivation": "B{(C_{1},\\hbar)} = - C_{1} + \\hbar and - B{(C_{1},\\hbar)} = C_{1} - \\hbar and - C_{1} - B{(C_{1},\\hbar)} = - \\hbar and \\operatorname{P_{g}}{(E_{x})} = \\log{(E_{x})} and - C_{1} - B{(C_{1},\\hbar)} - \\log{(E_{x})} = - \\hbar - \\log{(E_{x})} and - C_{1} - B{(C_{1},\\hbar)} - \\operatorname{P_{g}}{(E_{x})} = - \\hbar - \\operatorname{P_{g}}{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('C_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True)))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], ["get_premise", "Equality(Function('P_g')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["minus", 3, "log(Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), log(Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), log(Symbol('E_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Function('B')(Symbol('C_1', commutative=True), Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Function('P_g')(Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('P_g')(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given m{(k)} = \\log{(k)}, then obtain 1 = \\frac{- \\frac{1}{\\log{(k)}} + \\frac{1}{k \\log{(k)}}}{\\frac{\\frac{d}{d k} m{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}}", "derivation": "m{(k)} = \\log{(k)} and \\frac{d}{d k} m{(k)} = \\frac{d}{d k} \\log{(k)} and \\frac{\\frac{d}{d k} m{(k)}}{\\log{(k)}} = \\frac{\\frac{d}{d k} \\log{(k)}}{\\log{(k)}} and \\frac{\\frac{d}{d k} m{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} = \\frac{\\frac{d}{d k} \\log{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}} and 1 = \\frac{\\frac{\\frac{d}{d k} \\log{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}}{\\frac{\\frac{d}{d k} m{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}} and 1 = \\frac{- \\frac{1}{\\log{(k)}} + \\frac{1}{k \\log{(k)}}}{\\frac{\\frac{d}{d k} m{(k)}}{\\log{(k)}} - \\frac{1}{\\log{(k)}}}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 2, "log(Symbol('k', commutative=True))"], "Equality(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["minus", 3, "Pow(log(Symbol('k', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))), Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))))"], [["divide", 4, "Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1))))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))), Integer(-1)), Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(log(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1))))))"], [["evaluate_derivatives", 5], "Equality(Integer(1), Mul(Pow(Add(Mul(Pow(log(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('m')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1)))), Integer(-1)), Add(Mul(Integer(-1), Pow(log(Symbol('k', commutative=True)), Integer(-1))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Pow(log(Symbol('k', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given \\ddot{x}{(Q)} = \\cos{(Q)}, then derive \\int \\ddot{x}{(Q)} dQ = m + \\sin{(Q)}, then derive \\phi_1 + \\sin{(Q)} = m + \\sin{(Q)}, then obtain \\int (\\phi_1 + \\sin{(Q)}) d\\phi_1 = \\iint \\ddot{x}{(Q)} dQ d\\phi_1", "derivation": "\\ddot{x}{(Q)} = \\cos{(Q)} and \\int \\ddot{x}{(Q)} dQ = \\int \\cos{(Q)} dQ and \\int \\ddot{x}{(Q)} dQ = m + \\sin{(Q)} and \\int \\cos{(Q)} dQ = m + \\sin{(Q)} and \\phi_1 + \\sin{(Q)} = m + \\sin{(Q)} and \\int (\\phi_1 + \\sin{(Q)}) d\\phi_1 = \\int (m + \\sin{(Q)}) d\\phi_1 and \\int (\\phi_1 + \\sin{(Q)}) d\\phi_1 = \\iint \\ddot{x}{(Q)} dQ d\\phi_1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('m', commutative=True), sin(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('m', commutative=True), sin(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('Q', commutative=True))), Add(Symbol('m', commutative=True), sin(Symbol('Q', commutative=True))))"], [["integrate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Add(Symbol('m', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integral(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Function('\\\\ddot{x}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given z{(m_{s})} = \\sin{(m_{s})}, then derive (\\frac{d}{d m_{s}} z{(m_{s})})^{m_{s}} = \\cos^{m_{s}}{(m_{s})}, then obtain \\frac{d}{d m_{s}} (\\frac{d}{d m_{s}} \\sin{(m_{s})})^{m_{s}} = \\frac{d}{d m_{s}} \\cos^{m_{s}}{(m_{s})}", "derivation": "z{(m_{s})} = \\sin{(m_{s})} and \\frac{d}{d m_{s}} z{(m_{s})} = \\frac{d}{d m_{s}} \\sin{(m_{s})} and (\\frac{d}{d m_{s}} z{(m_{s})})^{m_{s}} = (\\frac{d}{d m_{s}} \\sin{(m_{s})})^{m_{s}} and (\\frac{d}{d m_{s}} z{(m_{s})})^{m_{s}} = \\cos^{m_{s}}{(m_{s})} and (\\frac{d}{d m_{s}} \\sin{(m_{s})})^{m_{s}} = \\cos^{m_{s}}{(m_{s})} and \\frac{d}{d m_{s}} (\\frac{d}{d m_{s}} \\sin{(m_{s})})^{m_{s}} = \\frac{d}{d m_{s}} \\cos^{m_{s}}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Derivative(Function('z')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('m_s', commutative=True)), Pow(Derivative(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('m_s', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('z')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('m_s', commutative=True)), Pow(cos(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('m_s', commutative=True)), Pow(cos(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["differentiate", 5, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Pow(Derivative(sin(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given u{(s)} = s, then derive (\\frac{d^{2}}{d s^{2}} u{(s)})^{2} = 0, then obtain (\\frac{d^{2}}{d u{(s)}^{2}} u{(s)})^{2} = 0", "derivation": "u{(s)} = s and \\frac{d}{d s} u{(s)} = \\frac{d}{d s} s and \\frac{d^{2}}{d s^{2}} u{(s)} = \\frac{d^{2}}{d s^{2}} s and (\\frac{d^{2}}{d s^{2}} u{(s)})^{2} = (\\frac{d^{2}}{d s^{2}} s)^{2} and (\\frac{d^{2}}{d s^{2}} u{(s)})^{2} = 0 and (\\frac{d^{2}}{d s^{2}} s)^{2} = 0 and (\\frac{d^{2}}{d u{(s)}^{2}} u{(s)})^{2} = 0", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Symbol('s', commutative=True), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Derivative(Symbol('s', commutative=True), Tuple(Symbol('s', commutative=True), Integer(2))))"], [["power", 3, 2], "Equality(Pow(Derivative(Function('u')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2)), Pow(Derivative(Symbol('s', commutative=True), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('u')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2)), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Derivative(Symbol('s', commutative=True), Tuple(Symbol('s', commutative=True), Integer(2))), Integer(2)), Integer(0))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Pow(Derivative(Function('u')(Symbol('s', commutative=True)), Tuple(Function('u')(Symbol('s', commutative=True)), Integer(2))), Integer(2)), Integer(0))"]]}, {"prompt": "Given a{(\\sigma_p,v_{y})} = v_{y}^{\\sigma_p}, then obtain v_{y} a{(\\sigma_p,v_{y})} + a{(\\sigma_p,v_{y})} = v_{y} a{(\\sigma_p,v_{y})} + v_{y}^{\\sigma_p}", "derivation": "a{(\\sigma_p,v_{y})} = v_{y}^{\\sigma_p} and v_{y} a{(\\sigma_p,v_{y})} = v_{y} v_{y}^{\\sigma_p} and v_{y} v_{y}^{\\sigma_p} + a{(\\sigma_p,v_{y})} = v_{y} v_{y}^{\\sigma_p} + v_{y}^{\\sigma_p} and v_{y} a{(\\sigma_p,v_{y})} + a{(\\sigma_p,v_{y})} = v_{y} a{(\\sigma_p,v_{y})} + v_{y}^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True)), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["times", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 1, "Mul(Symbol('v_y', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Symbol('v_y', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Symbol('v_y', commutative=True), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Symbol('v_y', commutative=True), Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True))), Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Symbol('v_y', commutative=True), Function('a')(Symbol('\\\\sigma_p', commutative=True), Symbol('v_y', commutative=True))), Pow(Symbol('v_y', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(m_{s},\\mathbf{f})} = (e^{\\mathbf{f}})^{m_{s}}, then obtain \\theta + \\int \\cos{(e^{\\Psi_{nl}{(m_{s},\\mathbf{f})}})} d\\mathbf{f} = \\theta + \\int \\cos{(e^{(e^{\\mathbf{f}})^{m_{s}}})} d\\mathbf{f}", "derivation": "\\Psi_{nl}{(m_{s},\\mathbf{f})} = (e^{\\mathbf{f}})^{m_{s}} and e^{\\Psi_{nl}{(m_{s},\\mathbf{f})}} = e^{(e^{\\mathbf{f}})^{m_{s}}} and \\cos{(e^{\\Psi_{nl}{(m_{s},\\mathbf{f})}})} = \\cos{(e^{(e^{\\mathbf{f}})^{m_{s}}})} and \\int \\cos{(e^{\\Psi_{nl}{(m_{s},\\mathbf{f})}})} d\\mathbf{f} = \\int \\cos{(e^{(e^{\\mathbf{f}})^{m_{s}}})} d\\mathbf{f} and \\theta + \\int \\cos{(e^{\\Psi_{nl}{(m_{s},\\mathbf{f})}})} d\\mathbf{f} = \\theta + \\int \\cos{(e^{(e^{\\mathbf{f}})^{m_{s}}})} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('m_s', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\Psi_{nl}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), exp(Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('m_s', commutative=True))))"], [["cos", 2], "Equality(cos(exp(Function('\\\\Psi_{nl}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), cos(exp(Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('m_s', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(cos(exp(Function('\\\\Psi_{nl}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(cos(exp(Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('m_s', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Integral(cos(exp(Function('\\\\Psi_{nl}')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Integral(cos(exp(Pow(exp(Symbol('\\\\mathbf{f}', commutative=True)), Symbol('m_s', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given J{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain - \\sigma_p + 2 J{(\\sigma_p)} - \\log{(\\sigma_p)} - 2 \\log{(\\sigma_p - J{(\\sigma_p)} + \\log{(\\sigma_p)})} = - \\sigma_p + J{(\\sigma_p)} - \\log{(\\sigma_p)} - \\log{(\\sigma_p - J{(\\sigma_p)} + \\log{(\\sigma_p)})}", "derivation": "J{(\\sigma_p)} = \\log{(\\sigma_p)} and J{(\\sigma_p)} - \\log{(\\sigma_p)} = 0 and - \\sigma_p + J{(\\sigma_p)} - \\log{(\\sigma_p)} = - \\sigma_p and - \\sigma_p + J{(\\sigma_p)} - 2 \\log{(\\sigma_p)} = - \\sigma_p - \\log{(\\sigma_p)} and - \\sigma_p + 2 J{(\\sigma_p)} - \\log{(\\sigma_p)} - 2 \\log{(\\sigma_p - J{(\\sigma_p)} + \\log{(\\sigma_p)})} = - \\sigma_p + J{(\\sigma_p)} - \\log{(\\sigma_p)} - \\log{(\\sigma_p - J{(\\sigma_p)} + \\log{(\\sigma_p)})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('J')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), Integer(0))"], [["minus", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('J')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"], [["add", 3, "Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('J')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Function('J')(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integer(2), log(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('J')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True)))))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('J')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), log(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('J')(Symbol('\\\\sigma_p', commutative=True))), log(Symbol('\\\\sigma_p', commutative=True)))))))"]]}, {"prompt": "Given \\omega{(A_{x})} = \\cos{(A_{x})}, then derive \\frac{d}{d A_{x}} \\omega{(A_{x})} = - \\sin{(A_{x})}, then obtain \\frac{d}{d A_{x}} \\cos{(A_{x})} = - \\sin{(A_{x})}", "derivation": "\\omega{(A_{x})} = \\cos{(A_{x})} and \\frac{d}{d A_{x}} \\omega{(A_{x})} = \\frac{d}{d A_{x}} \\cos{(A_{x})} and \\frac{d}{d A_{x}} \\omega{(A_{x})} = - \\sin{(A_{x})} and \\frac{d}{d A_{x}} \\cos{(A_{x})} = - \\sin{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('A_x', commutative=True)), cos(Symbol('A_x', commutative=True)))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(C_{2})} = \\sin{(C_{2})} and S{(C_{2})} = \\sin{(C_{2})}, then obtain \\frac{\\hat{H}_{\\lambda}{(C_{2})} + \\sin{(C_{2})}}{\\int \\sin{(C_{2})} dC_{2}} = \\frac{2 \\sin{(C_{2})}}{\\int \\sin{(C_{2})} dC_{2}}", "derivation": "\\hat{H}_{\\lambda}{(C_{2})} = \\sin{(C_{2})} and \\hat{H}_{\\lambda}{(C_{2})} + \\sin{(C_{2})} = 2 \\sin{(C_{2})} and S{(C_{2})} = \\sin{(C_{2})} and \\frac{\\hat{H}_{\\lambda}{(C_{2})} + \\sin{(C_{2})}}{\\int S{(C_{2})} dC_{2}} = \\frac{2 \\sin{(C_{2})}}{\\int S{(C_{2})} dC_{2}} and \\frac{\\hat{H}_{\\lambda}{(C_{2})} + \\sin{(C_{2})}}{\\int \\sin{(C_{2})} dC_{2}} = \\frac{2 \\sin{(C_{2})}}{\\int \\sin{(C_{2})} dC_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["add", 1, "sin(Symbol('C_2', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True))), Mul(Integer(2), sin(Symbol('C_2', commutative=True))))"], ["renaming_premise", "Equality(Function('S')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["divide", 2, "Integral(Function('S')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True))), Pow(Integral(Function('S')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))), Mul(Integer(2), sin(Symbol('C_2', commutative=True)), Pow(Integral(Function('S')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True))), Pow(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))), Mul(Integer(2), sin(Symbol('C_2', commutative=True)), Pow(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(Z,U)} = U^{Z}, then obtain Z + \\frac{\\partial}{\\partial U} (U^{Z} + \\operatorname{f_{E}}{(Z,U)})^{2} + 1 = Z + \\frac{\\partial}{\\partial U} 2 U^{Z} (U^{Z} + \\operatorname{f_{E}}{(Z,U)}) + 1", "derivation": "\\operatorname{f_{E}}{(Z,U)} = U^{Z} and U^{Z} + \\operatorname{f_{E}}{(Z,U)} = 2 U^{Z} and (U^{Z} + \\operatorname{f_{E}}{(Z,U)})^{2} = 2 U^{Z} (U^{Z} + \\operatorname{f_{E}}{(Z,U)}) and \\frac{\\partial}{\\partial U} (U^{Z} + \\operatorname{f_{E}}{(Z,U)})^{2} = \\frac{\\partial}{\\partial U} 2 U^{Z} (U^{Z} + \\operatorname{f_{E}}{(Z,U)}) and \\frac{\\partial}{\\partial U} (U^{Z} + \\operatorname{f_{E}}{(Z,U)})^{2} + 1 = \\frac{\\partial}{\\partial U} 2 U^{Z} (U^{Z} + \\operatorname{f_{E}}{(Z,U)}) + 1 and Z + \\frac{\\partial}{\\partial U} (U^{Z} + \\operatorname{f_{E}}{(Z,U)})^{2} + 1 = Z + \\frac{\\partial}{\\partial U} 2 U^{Z} (U^{Z} + \\operatorname{f_{E}}{(Z,U)}) + 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)))"], [["add", 1, "Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True))"], "Equality(Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True))))"], [["times", 2, "Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)))"], "Equality(Pow(Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Integer(2)), Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Pow(Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Integer(2)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["add", 4, 1], "Equality(Add(Derivative(Pow(Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Integer(2)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)))"], [["add", 5, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Derivative(Pow(Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True))), Integer(2)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)), Add(Symbol('Z', commutative=True), Derivative(Mul(Integer(2), Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Add(Pow(Symbol('U', commutative=True), Symbol('Z', commutative=True)), Function('f_E')(Symbol('Z', commutative=True), Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(C_{d})} = e^{e^{C_{d}}} and \\operatorname{A_{2}}{(C_{d})} = e^{C_{d}}, then obtain \\int \\operatorname{L_{\\varepsilon}}{(C_{d})} e^{e^{C_{d}}} dC_{d} = \\int e^{2 e^{C_{d}}} dC_{d}", "derivation": "\\operatorname{L_{\\varepsilon}}{(C_{d})} = e^{e^{C_{d}}} and \\operatorname{A_{2}}{(C_{d})} = e^{C_{d}} and \\operatorname{L_{\\varepsilon}}{(C_{d})} = e^{\\operatorname{A_{2}}{(C_{d})}} and \\operatorname{L_{\\varepsilon}}{(C_{d})} e^{e^{C_{d}}} = e^{\\operatorname{A_{2}}{(C_{d})}} e^{e^{C_{d}}} and \\int \\operatorname{L_{\\varepsilon}}{(C_{d})} e^{e^{C_{d}}} dC_{d} = \\int e^{\\operatorname{A_{2}}{(C_{d})}} e^{e^{C_{d}}} dC_{d} and \\int \\operatorname{L_{\\varepsilon}}{(C_{d})} e^{e^{C_{d}}} dC_{d} = \\int e^{2 e^{C_{d}}} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True))))"], ["renaming_premise", "Equality(Function('A_2')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(Function('A_2')(Symbol('C_d', commutative=True))))"], [["times", 3, "exp(exp(Symbol('C_d', commutative=True)))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True)))), Mul(exp(Function('A_2')(Symbol('C_d', commutative=True))), exp(exp(Symbol('C_d', commutative=True)))))"], [["integrate", 4, "Symbol('C_d', commutative=True)"], "Equality(Integral(Mul(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(exp(Function('A_2')(Symbol('C_d', commutative=True))), exp(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Mul(Function('L_{\\\\varepsilon}')(Symbol('C_d', commutative=True)), exp(exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))), Integral(exp(Mul(Integer(2), exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\tilde{g}^*)} = \\cos{(\\sin{(\\tilde{g}^*)})}, then obtain e^{\\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\sigma_{p}{(\\tilde{g}^*)}} = e^{\\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\cos{(\\sin{(\\tilde{g}^*)})}}", "derivation": "\\sigma_{p}{(\\tilde{g}^*)} = \\cos{(\\sin{(\\tilde{g}^*)})} and \\frac{d}{d \\tilde{g}^*} \\sigma_{p}{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} \\cos{(\\sin{(\\tilde{g}^*)})} and \\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\sigma_{p}{(\\tilde{g}^*)} = \\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\cos{(\\sin{(\\tilde{g}^*)})} and e^{\\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\sigma_{p}{(\\tilde{g}^*)}} = e^{\\frac{d^{2}}{d (\\tilde{g}^*)^{2}} \\cos{(\\sin{(\\tilde{g}^*)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\tilde{g}^*', commutative=True)), cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2))), Derivative(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2))))"], [["exp", 3], "Equality(exp(Derivative(Function('\\\\sigma_p')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2)))), exp(Derivative(cos(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\Omega,\\theta)} = \\Omega + \\theta, then derive \\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\Omega \\theta + \\sigma_x, then obtain \\frac{\\frac{\\Omega^{2}}{2} + \\Omega \\theta + \\sigma_x}{\\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega} = \\frac{\\int (\\Omega + \\theta) d\\Omega}{\\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega}", "derivation": "\\operatorname{C_{d}}{(\\Omega,\\theta)} = \\Omega + \\theta and \\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega = \\int (\\Omega + \\theta) d\\Omega and \\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega = \\frac{\\Omega^{2}}{2} + \\Omega \\theta + \\sigma_x and \\frac{\\Omega^{2}}{2} + \\Omega \\theta + \\sigma_x = \\int (\\Omega + \\theta) d\\Omega and \\frac{\\frac{\\Omega^{2}}{2} + \\Omega \\theta + \\sigma_x}{\\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega} = \\frac{\\int (\\Omega + \\theta) d\\Omega}{\\int \\operatorname{C_{d}}{(\\Omega,\\theta)} d\\Omega}", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 4, "Integral(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(2))), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Integral(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Mul(Integral(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Pow(Integral(Function('C_d')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(v_{x},\\mathbf{s})} = \\frac{\\partial}{\\partial v_{x}} \\mathbf{s} v_{x}, then derive \\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})} = 2 \\mathbf{s}, then obtain \\frac{d}{d v_{x}} 2 \\mathbf{s} + \\frac{\\partial}{\\partial v_{x}} (\\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})}) = 2 \\frac{d}{d v_{x}} 2 \\mathbf{s}", "derivation": "\\theta_{2}{(v_{x},\\mathbf{s})} = \\frac{\\partial}{\\partial v_{x}} \\mathbf{s} v_{x} and \\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})} = \\mathbf{s} + \\frac{\\partial}{\\partial v_{x}} \\mathbf{s} v_{x} and \\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})} = 2 \\mathbf{s} and \\frac{\\partial}{\\partial v_{x}} (\\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})}) = \\frac{d}{d v_{x}} 2 \\mathbf{s} and \\frac{d}{d v_{x}} 2 \\mathbf{s} + \\frac{\\partial}{\\partial v_{x}} (\\mathbf{s} + \\theta_{2}{(v_{x},\\mathbf{s})}) = 2 \\frac{d}{d v_{x}} 2 \\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["add", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 3, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\theta_2')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Integer(2), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given W{(\\theta_1,\\varphi^*,E_{\\lambda})} = \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*} and \\operatorname{E_{x}}{(\\theta_1,\\varphi^*,E_{\\lambda})} = - \\varphi^* + W{(\\theta_1,\\varphi^*,E_{\\lambda})}, then obtain \\frac{\\operatorname{E_{x}}{(\\theta_1,\\varphi^*,E_{\\lambda})}}{\\operatorname{A_{2}}{(r,M_{E})}} = \\frac{- \\varphi^* + \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*}}{\\operatorname{A_{2}}{(r,M_{E})}}", "derivation": "W{(\\theta_1,\\varphi^*,E_{\\lambda})} = \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*} and - \\varphi^* + W{(\\theta_1,\\varphi^*,E_{\\lambda})} = - \\varphi^* + \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*} and \\operatorname{E_{x}}{(\\theta_1,\\varphi^*,E_{\\lambda})} = - \\varphi^* + W{(\\theta_1,\\varphi^*,E_{\\lambda})} and \\operatorname{E_{x}}{(\\theta_1,\\varphi^*,E_{\\lambda})} = - \\varphi^* + \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*} and \\frac{\\operatorname{E_{x}}{(\\theta_1,\\varphi^*,E_{\\lambda})}}{\\operatorname{A_{2}}{(r,M_{E})}} = \\frac{- \\varphi^* + \\frac{E_{\\lambda} + \\theta_1}{\\varphi^*}}{\\operatorname{A_{2}}{(r,M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('W')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["divide", 4, "Function('A_2')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Pow(Function('A_2')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)), Function('E_x')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Pow(Function('A_2')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{H},A_{y})} = A_{y} \\mathbf{H}, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(A_{y} \\mathbf{H})} + \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(\\operatorname{n_{1}}{(\\mathbf{H},A_{y})})} = 2 \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(A_{y} \\mathbf{H})}", "derivation": "\\operatorname{n_{1}}{(\\mathbf{H},A_{y})} = A_{y} \\mathbf{H} and \\log{(\\operatorname{n_{1}}{(\\mathbf{H},A_{y})})} = \\log{(A_{y} \\mathbf{H})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(\\operatorname{n_{1}}{(\\mathbf{H},A_{y})})} = \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(A_{y} \\mathbf{H})} and \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(A_{y} \\mathbf{H})} + \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(\\operatorname{n_{1}}{(\\mathbf{H},A_{y})})} = 2 \\frac{\\partial}{\\partial \\mathbf{H}} \\log{(A_{y} \\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["log", 1], "Equality(log(Function('n_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_y', commutative=True))), log(Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(log(Function('n_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["add", 3, "Derivative(log(Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(log(Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Function('n_1')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_y', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(log(Mul(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(\\hat{p},n_{1})} = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}), then derive \\frac{l^{n_{1}}{(\\hat{p},n_{1})}}{\\hat{p} + n_{1}} = \\frac{1}{\\hat{p} + n_{1}}, then obtain \\frac{l^{2 n_{1}}{(\\hat{p},n_{1})}}{\\hat{p} + n_{1}} = \\frac{l^{n_{1}}{(\\hat{p},n_{1})} (\\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}))^{n_{1}}}{\\hat{p} + n_{1}}", "derivation": "l{(\\hat{p},n_{1})} = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}) and l^{n_{1}}{(\\hat{p},n_{1})} = (\\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}))^{n_{1}} and \\frac{l^{n_{1}}{(\\hat{p},n_{1})}}{\\hat{p} + n_{1}} = \\frac{(\\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}))^{n_{1}}}{\\hat{p} + n_{1}} and \\frac{l^{n_{1}}{(\\hat{p},n_{1})}}{\\hat{p} + n_{1}} = \\frac{1}{\\hat{p} + n_{1}} and \\frac{l^{2 n_{1}}{(\\hat{p},n_{1})}}{\\hat{p} + n_{1}} = \\frac{l^{n_{1}}{(\\hat{p},n_{1})} (\\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + n_{1}))^{n_{1}}}{\\hat{p} + n_{1}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('n_1', commutative=True)))"], [["divide", 2, "Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Pow(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Pow(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('n_1', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Pow(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True))), Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Pow(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True)))), Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Pow(Function('l')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}}, then derive \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{P_{e}}{(V_{\\mathbf{B}})} = 1, then obtain (\\frac{d}{d V_{\\mathbf{B}}} \\operatorname{P_{e}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = 1", "derivation": "\\operatorname{P_{e}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} and \\operatorname{P_{e}}{(V_{\\mathbf{B}})} - 1 = V_{\\mathbf{B}} - 1 and \\frac{d}{d V_{\\mathbf{B}}} (\\operatorname{P_{e}}{(V_{\\mathbf{B}})} - 1) = \\frac{d}{d V_{\\mathbf{B}}} (V_{\\mathbf{B}} - 1) and \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{P_{e}}{(V_{\\mathbf{B}})} = 1 and (\\frac{d}{d V_{\\mathbf{B}}} \\operatorname{P_{e}}{(V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('P_e')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_e')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Function('P_e')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('P_e')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Derivative(Function('P_e')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{p}{(L,M)} = L^{M}, then obtain \\int \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L \\mathbf{p}{(L,M)} dL)^{L} dL = \\int \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L L^{M} dL)^{L} dL", "derivation": "\\mathbf{p}{(L,M)} = L^{M} and L \\mathbf{p}{(L,M)} = L L^{M} and \\int L \\mathbf{p}{(L,M)} dL = \\int L L^{M} dL and (\\int L \\mathbf{p}{(L,M)} dL)^{L} = (\\int L L^{M} dL)^{L} and \\frac{\\partial}{\\partial M} (\\int L \\mathbf{p}{(L,M)} dL)^{L} = \\frac{\\partial}{\\partial M} (\\int L L^{M} dL)^{L} and \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L \\mathbf{p}{(L,M)} dL)^{L} = \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L L^{M} dL)^{L} and \\int \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L \\mathbf{p}{(L,M)} dL)^{L} dL = \\int \\frac{\\partial^{2}}{\\partial L\\partial M} (\\int L L^{M} dL)^{L} dL", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True)))"], [["times", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('L', commutative=True)"], "Equality(Integral(Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Pow(Integral(Mul(Symbol('L', commutative=True), Pow(Symbol('L', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('L', commutative=True))), Symbol('L', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(B,t)} = t^{B} and \\operatorname{V_{\\mathbf{B}}}{(B,t)} = t^{B} \\log{(t)}, then derive \\frac{\\partial}{\\partial B} \\mathbf{A}{(B,t)} = t^{B} \\log{(t)}, then obtain \\operatorname{V_{\\mathbf{B}}}{(B,t)} = \\frac{\\partial}{\\partial B} t^{B}", "derivation": "\\mathbf{A}{(B,t)} = t^{B} and \\frac{\\partial}{\\partial B} \\mathbf{A}{(B,t)} = \\frac{\\partial}{\\partial B} t^{B} and \\frac{\\partial}{\\partial B} \\mathbf{A}{(B,t)} = t^{B} \\log{(t)} and \\operatorname{V_{\\mathbf{B}}}{(B,t)} = t^{B} \\log{(t)} and t^{B} \\log{(t)} = \\frac{\\partial}{\\partial B} t^{B} and \\operatorname{V_{\\mathbf{B}}}{(B,t)} = \\frac{\\partial}{\\partial B} t^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('B', commutative=True), Symbol('t', commutative=True)), Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('B', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('B', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Mul(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), log(Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), log(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), log(Symbol('t', commutative=True))), Derivative(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('B', commutative=True), Symbol('t', commutative=True)), Derivative(Pow(Symbol('t', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(H)} = \\log{(\\log{(H)})}, then derive \\frac{d}{d H} \\operatorname{v_{z}}{(H)} = \\frac{1}{H \\log{(H)}}, then obtain \\int \\frac{1}{H \\log{(H)}} dH + \\int \\frac{d}{d H} \\operatorname{v_{z}}{(H)} dH = 2 \\int \\frac{1}{H \\log{(H)}} dH", "derivation": "\\operatorname{v_{z}}{(H)} = \\log{(\\log{(H)})} and \\frac{d}{d H} \\operatorname{v_{z}}{(H)} = \\frac{d}{d H} \\log{(\\log{(H)})} and \\frac{d}{d H} \\operatorname{v_{z}}{(H)} = \\frac{1}{H \\log{(H)}} and \\int \\frac{d}{d H} \\operatorname{v_{z}}{(H)} dH = \\int \\frac{1}{H \\log{(H)}} dH and \\int \\frac{1}{H \\log{(H)}} dH + \\int \\frac{d}{d H} \\operatorname{v_{z}}{(H)} dH = 2 \\int \\frac{1}{H \\log{(H)}} dH", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(log(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Function('v_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Tuple(Symbol('H', commutative=True))))"], [["add", 4, "Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Function('v_z')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True)))), Mul(Integer(2), Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(log(Symbol('H', commutative=True)), Integer(-1))), Tuple(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(q)} = \\cos{(\\log{(q)})}, then derive - \\log{(q)} + \\frac{d}{d q} \\hat{p}_0{(q)} = - \\log{(q)} - \\frac{\\sin{(\\log{(q)})}}{q}, then obtain \\frac{d}{d q} (- \\log{(q)} + \\frac{d}{d q} \\hat{p}_0{(q)}) = \\frac{d}{d q} (- \\log{(q)} - \\frac{\\sin{(\\log{(q)})}}{q})", "derivation": "\\hat{p}_0{(q)} = \\cos{(\\log{(q)})} and \\frac{d}{d q} \\hat{p}_0{(q)} = \\frac{d}{d q} \\cos{(\\log{(q)})} and - \\log{(q)} + \\frac{d}{d q} \\hat{p}_0{(q)} = - \\log{(q)} + \\frac{d}{d q} \\cos{(\\log{(q)})} and - \\log{(q)} + \\frac{d}{d q} \\hat{p}_0{(q)} = - \\log{(q)} - \\frac{\\sin{(\\log{(q)})}}{q} and \\frac{d}{d q} (- \\log{(q)} + \\frac{d}{d q} \\hat{p}_0{(q)}) = \\frac{d}{d q} (- \\log{(q)} - \\frac{\\sin{(\\log{(q)})}}{q})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('q', commutative=True)), cos(log(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 2, "log(Symbol('q', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Derivative(Function('\\\\hat{p}_0')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Derivative(cos(log(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Derivative(Function('\\\\hat{p}_0')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), sin(log(Symbol('q', commutative=True))))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Derivative(Function('\\\\hat{p}_0')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), sin(log(Symbol('q', commutative=True))))), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(v)} = \\int \\sin{(v)} dv, then obtain - v + 2 \\mathbf{B}{(v)} = - v + 2 \\int \\sin{(v)} dv", "derivation": "\\mathbf{B}{(v)} = \\int \\sin{(v)} dv and - v + \\mathbf{B}{(v)} = - v + \\int \\sin{(v)} dv and - v + 2 \\mathbf{B}{(v)} = - v + \\mathbf{B}{(v)} + \\int \\sin{(v)} dv and - v + 2 \\mathbf{B}{(v)} = - v + 2 \\int \\sin{(v)} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["minus", 1, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["add", 2, "Function('\\\\mathbf{B}')(Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Function('\\\\mathbf{B}')(Symbol('v', commutative=True)), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Integral(sin(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(p,q)} = p + q, then obtain \\frac{\\partial}{\\partial p} 2 \\hat{\\mathbf{x}}^{2}{(p,q)} = \\frac{\\partial}{\\partial p} (p + q + \\hat{\\mathbf{x}}{(p,q)}) \\hat{\\mathbf{x}}{(p,q)}", "derivation": "\\hat{\\mathbf{x}}{(p,q)} = p + q and 2 \\hat{\\mathbf{x}}{(p,q)} = p + q + \\hat{\\mathbf{x}}{(p,q)} and 2 \\hat{\\mathbf{x}}^{2}{(p,q)} = (p + q + \\hat{\\mathbf{x}}{(p,q)}) \\hat{\\mathbf{x}}{(p,q)} and \\frac{\\partial}{\\partial p} 2 \\hat{\\mathbf{x}}^{2}{(p,q)} = \\frac{\\partial}{\\partial p} (p + q + \\hat{\\mathbf{x}}{(p,q)}) \\hat{\\mathbf{x}}{(p,q)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True)), Add(Symbol('p', commutative=True), Symbol('q', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))), Add(Symbol('p', commutative=True), Symbol('q', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))))"], [["times", 2, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True)), Integer(2))), Mul(Add(Symbol('p', commutative=True), Symbol('q', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True)), Integer(2))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('p', commutative=True), Symbol('q', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('p', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(m_{s})} = \\sin{(m_{s})}, then obtain y{(m_{s})} + y^{m_{s}}{(m_{s})} = y^{m_{s}}{(m_{s})} + \\sin{(m_{s})}", "derivation": "y{(m_{s})} = \\sin{(m_{s})} and y^{m_{s}}{(m_{s})} = \\sin^{m_{s}}{(m_{s})} and y{(m_{s})} + \\sin^{m_{s}}{(m_{s})} = \\sin{(m_{s})} + \\sin^{m_{s}}{(m_{s})} and y{(m_{s})} + y^{m_{s}}{(m_{s})} = y^{m_{s}}{(m_{s})} + \\sin{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True)))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('y')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["add", 1, "Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], "Equality(Add(Function('y')(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Add(sin(Symbol('m_s', commutative=True)), Pow(sin(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('y')(Symbol('m_s', commutative=True)), Pow(Function('y')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))), Add(Pow(Function('y')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), sin(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\varphi^*)} = \\int \\cos{(\\varphi^*)} d\\varphi^*, then derive - \\varphi^* + \\hat{x}{(\\varphi^*)} = - \\varphi^* + t_{1} + \\sin{(\\varphi^*)}, then obtain 1 = \\frac{- \\varphi^* + t_{1} + \\sin{(\\varphi^*)}}{- \\varphi^* + \\hat{x}{(\\varphi^*)}}", "derivation": "\\hat{x}{(\\varphi^*)} = \\int \\cos{(\\varphi^*)} d\\varphi^* and - \\varphi^* + \\hat{x}{(\\varphi^*)} = - \\varphi^* + \\int \\cos{(\\varphi^*)} d\\varphi^* and - \\varphi^* + \\hat{x}{(\\varphi^*)} = - \\varphi^* + t_{1} + \\sin{(\\varphi^*)} and 1 = \\frac{- \\varphi^* + t_{1} + \\sin{(\\varphi^*)}}{- \\varphi^* + \\hat{x}{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True)), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Integral(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('t_1', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\hat{x}')(Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Symbol('t_1', commutative=True), sin(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given p{(z^{*},\\omega)} = \\omega z^{*}, then obtain (p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega = (\\omega z^{*} \\int \\omega z^{*} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega", "derivation": "p{(z^{*},\\omega)} = \\omega z^{*} and \\int p{(z^{*},\\omega)} d\\omega = \\int \\omega z^{*} d\\omega and p{(z^{*},\\omega)} \\int p{(z^{*},\\omega)} d\\omega = \\omega z^{*} \\int p{(z^{*},\\omega)} d\\omega and (p{(z^{*},\\omega)} \\int p{(z^{*},\\omega)} d\\omega)^{\\omega} = (\\omega z^{*} \\int p{(z^{*},\\omega)} d\\omega)^{\\omega} and (p{(z^{*},\\omega)} \\int p{(z^{*},\\omega)} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int p{(z^{*},\\omega)} d\\omega = (\\omega z^{*} \\int p{(z^{*},\\omega)} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int p{(z^{*},\\omega)} d\\omega and (p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega = (\\omega z^{*} \\int \\omega z^{*} d\\omega)^{\\omega} + p{(z^{*},\\omega)} \\int \\omega z^{*} d\\omega", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)))"], [["add", 4, "Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], "Equality(Add(Pow(Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Symbol('\\\\omega', commutative=True)), Mul(Function('p')(Symbol('z^*', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Symbol('\\\\omega', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(y^{\\prime})} = \\log{(y^{\\prime})}, then derive - \\varphi^* - \\frac{(y^{\\prime})^{2} \\log{(y^{\\prime})}}{2} + \\frac{(y^{\\prime})^{2}}{4} + \\int y^{\\prime} \\dot{y}{(y^{\\prime})} dy^{\\prime} = 0, then obtain - \\varphi^* - \\frac{(y^{\\prime})^{2} \\log{(y^{\\prime})}}{2} + \\frac{(y^{\\prime})^{2}}{4} + \\int y^{\\prime} \\log{(y^{\\prime})} dy^{\\prime} = 0", "derivation": "\\dot{y}{(y^{\\prime})} = \\log{(y^{\\prime})} and y^{\\prime} \\dot{y}{(y^{\\prime})} = y^{\\prime} \\log{(y^{\\prime})} and \\int y^{\\prime} \\dot{y}{(y^{\\prime})} dy^{\\prime} = \\int y^{\\prime} \\log{(y^{\\prime})} dy^{\\prime} and 0 = - \\int y^{\\prime} \\dot{y}{(y^{\\prime})} dy^{\\prime} + \\int y^{\\prime} \\log{(y^{\\prime})} dy^{\\prime} and \\int y^{\\prime} \\dot{y}{(y^{\\prime})} dy^{\\prime} - \\int y^{\\prime} \\log{(y^{\\prime})} dy^{\\prime} = 0 and - \\varphi^* - \\frac{(y^{\\prime})^{2} \\log{(y^{\\prime})}}{2} + \\frac{(y^{\\prime})^{2}}{4} + \\int y^{\\prime} \\dot{y}{(y^{\\prime})} dy^{\\prime} = 0 and - \\varphi^* - \\frac{(y^{\\prime})^{2} \\log{(y^{\\prime})}}{2} + \\frac{(y^{\\prime})^{2}}{4} + \\int y^{\\prime} \\log{(y^{\\prime})} dy^{\\prime} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True)))"], [["times", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 3, "Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Add(Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))), Integer(0))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2))), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), Function('\\\\dot{y}')(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2)), log(Symbol('y^{\\\\prime}', commutative=True))), Mul(Rational(1, 4), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(2))), Integral(Mul(Symbol('y^{\\\\prime}', commutative=True), log(Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(x^\\prime,z)} = \\log{(x^\\prime + z)}, then obtain \\cos{(\\operatorname{n_{2}}{(x^\\prime,z)})} + \\cos^{x^\\prime}{(\\operatorname{n_{2}}{(x^\\prime,z)})} = \\cos{(\\operatorname{n_{2}}{(x^\\prime,z)})} + \\cos^{x^\\prime}{(\\log{(x^\\prime + z)})}", "derivation": "\\operatorname{n_{2}}{(x^\\prime,z)} = \\log{(x^\\prime + z)} and \\cos{(\\operatorname{n_{2}}{(x^\\prime,z)})} = \\cos{(\\log{(x^\\prime + z)})} and \\cos^{x^\\prime}{(\\operatorname{n_{2}}{(x^\\prime,z)})} = \\cos^{x^\\prime}{(\\log{(x^\\prime + z)})} and \\cos{(\\operatorname{n_{2}}{(x^\\prime,z)})} + \\cos^{x^\\prime}{(\\operatorname{n_{2}}{(x^\\prime,z)})} = \\cos{(\\operatorname{n_{2}}{(x^\\prime,z)})} + \\cos^{x^\\prime}{(\\log{(x^\\prime + z)})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True)), log(Add(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))))"], [["cos", 1], "Equality(cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))), cos(log(Add(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True)))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Pow(cos(log(Add(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True)))), Symbol('x^\\\\prime', commutative=True)))"], [["add", 3, "cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True)))"], "Equality(Add(cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))), Pow(cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))), Symbol('x^\\\\prime', commutative=True))), Add(cos(Function('n_2')(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True))), Pow(cos(log(Add(Symbol('x^\\\\prime', commutative=True), Symbol('z', commutative=True)))), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\varphi)} = \\log{(\\varphi)}, then obtain \\frac{d}{d \\varphi} \\log{(\\varphi)}^{3} = \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}^{2}{(\\varphi)} \\log{(\\varphi)}", "derivation": "\\operatorname{y^{\\prime}}{(\\varphi)} = \\log{(\\varphi)} and \\operatorname{y^{\\prime}}{(\\varphi)} \\log{(\\varphi)} = \\log{(\\varphi)}^{2} and \\operatorname{y^{\\prime}}{(\\varphi)} \\log{(\\varphi)}^{2} = \\log{(\\varphi)}^{3} and \\operatorname{y^{\\prime}}^{2}{(\\varphi)} \\log{(\\varphi)} = \\log{(\\varphi)}^{3} and \\operatorname{y^{\\prime}}{(\\varphi)} \\log{(\\varphi)}^{2} = \\operatorname{y^{\\prime}}^{2}{(\\varphi)} \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}{(\\varphi)} \\log{(\\varphi)}^{2} = \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}^{2}{(\\varphi)} \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\log{(\\varphi)}^{3} = \\frac{d}{d \\varphi} \\operatorname{y^{\\prime}}^{2}{(\\varphi)} \\log{(\\varphi)}", "srepr_derivation": [["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "log(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True))), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2)))"], [["times", 1, "Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), log(Symbol('\\\\varphi', commutative=True))), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(3)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))), Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), log(Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Pow(log(Symbol('\\\\varphi', commutative=True)), Integer(3)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('y^{\\\\prime}')(Symbol('\\\\varphi', commutative=True)), Integer(2)), log(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(y)} = \\log{(y)}, then obtain - (y - \\frac{\\log{(y)}}{y}) \\mu{(y)} = - (y - \\mu{(y)} + \\log{(y)} - \\frac{\\log{(y)}}{y}) \\mu{(y)}", "derivation": "\\mu{(y)} = \\log{(y)} and \\frac{\\mu{(y)}}{y} = \\frac{\\log{(y)}}{y} and \\mu{(y)} - \\frac{\\log{(y)}}{y} = \\log{(y)} - \\frac{\\log{(y)}}{y} and \\mu{(y)} - \\frac{\\mu{(y)}}{y} = \\log{(y)} - \\frac{\\mu{(y)}}{y} and y + \\mu{(y)} - \\frac{\\mu{(y)}}{y} = y + \\log{(y)} - \\frac{\\mu{(y)}}{y} and y - \\frac{\\mu{(y)}}{y} = y - \\mu{(y)} + \\log{(y)} - \\frac{\\mu{(y)}}{y} and - (y - \\frac{\\mu{(y)}}{y}) \\mu{(y)} = - (y - \\mu{(y)} + \\log{(y)} - \\frac{\\mu{(y)}}{y}) \\mu{(y)} and - (y - \\frac{\\log{(y)}}{y}) \\mu{(y)} = - (y - \\mu{(y)} + \\log{(y)} - \\frac{\\log{(y)}}{y}) \\mu{(y)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True)))"], "Equality(Add(Function('\\\\mu')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True)))), Add(log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\mu')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))), Add(log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))))"], [["add", 4, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('\\\\mu')(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))))"], [["minus", 5, "Function('\\\\mu')(Symbol('y', commutative=True))"], "Equality(Add(Symbol('y', commutative=True), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))), Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('y', commutative=True))), log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))))"], [["times", 6, "Mul(Integer(-1), Function('\\\\mu')(Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))), Function('\\\\mu')(Symbol('y', commutative=True))), Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('y', commutative=True))), log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), Function('\\\\mu')(Symbol('y', commutative=True)))), Function('\\\\mu')(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True)))), Function('\\\\mu')(Symbol('y', commutative=True))), Mul(Integer(-1), Add(Symbol('y', commutative=True), Mul(Integer(-1), Function('\\\\mu')(Symbol('y', commutative=True))), log(Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)), log(Symbol('y', commutative=True)))), Function('\\\\mu')(Symbol('y', commutative=True))))"]]}, {"prompt": "Given s{(G,F_{H})} = \\cos{(F_{H} + G)}, then obtain \\iint \\frac{s{(G,F_{H})}}{\\cos{(F_{H} + G)}} dG dF_{H} = \\iint 1 dG dF_{H}", "derivation": "s{(G,F_{H})} = \\cos{(F_{H} + G)} and \\frac{s{(G,F_{H})}}{\\cos{(F_{H} + G)}} = 1 and \\int \\frac{s{(G,F_{H})}}{\\cos{(F_{H} + G)}} dG = \\int 1 dG and \\iint \\frac{s{(G,F_{H})}}{\\cos{(F_{H} + G)}} dG dF_{H} = \\iint 1 dG dF_{H}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('G', commutative=True), Symbol('F_H', commutative=True)), cos(Add(Symbol('F_H', commutative=True), Symbol('G', commutative=True))))"], [["divide", 1, "cos(Add(Symbol('F_H', commutative=True), Symbol('G', commutative=True)))"], "Equality(Mul(Function('s')(Symbol('G', commutative=True), Symbol('F_H', commutative=True)), Pow(cos(Add(Symbol('F_H', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Function('s')(Symbol('G', commutative=True), Symbol('F_H', commutative=True)), Pow(cos(Add(Symbol('F_H', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True))), Integral(Integer(1), Tuple(Symbol('G', commutative=True))))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Function('s')(Symbol('G', commutative=True), Symbol('F_H', commutative=True)), Pow(cos(Add(Symbol('F_H', commutative=True), Symbol('G', commutative=True))), Integer(-1))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(1), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(A)} = \\sin{(A)}, then obtain 3 \\phi_{1}{(A)} = 2 (A (\\phi_{1}{(A)} + \\sin{(A)}))^{A} (2 A \\phi_{1}{(A)})^{- A} \\phi_{1}{(A)} + \\phi_{1}{(A)}", "derivation": "\\phi_{1}{(A)} = \\sin{(A)} and 2 \\phi_{1}{(A)} = \\phi_{1}{(A)} + \\sin{(A)} and 2 A \\phi_{1}{(A)} = A (\\phi_{1}{(A)} + \\sin{(A)}) and (2 A \\phi_{1}{(A)})^{A} = (A (\\phi_{1}{(A)} + \\sin{(A)}))^{A} and 2 (2 A \\phi_{1}{(A)})^{A} \\phi_{1}{(A)} = 2 (A (\\phi_{1}{(A)} + \\sin{(A)}))^{A} \\phi_{1}{(A)} and 2 \\phi_{1}{(A)} = 2 (A (\\phi_{1}{(A)} + \\sin{(A)}))^{A} (2 A \\phi_{1}{(A)})^{- A} \\phi_{1}{(A)} and 3 \\phi_{1}{(A)} = 2 (A (\\phi_{1}{(A)} + \\sin{(A)}))^{A} (2 A \\phi_{1}{(A)})^{- A} \\phi_{1}{(A)} + \\phi_{1}{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["add", 1, "Function('\\\\phi_1')(Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('A', commutative=True))), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))))"], [["times", 2, "Symbol('A', commutative=True)"], "Equality(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Mul(Symbol('A', commutative=True), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))), Symbol('A', commutative=True)))"], [["times", 4, "Mul(Integer(2), Function('\\\\phi_1')(Symbol('A', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Function('\\\\phi_1')(Symbol('A', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('A', commutative=True), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Function('\\\\phi_1')(Symbol('A', commutative=True))))"], [["divide", 5, "Pow(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\phi_1')(Symbol('A', commutative=True))), Mul(Integer(2), Pow(Mul(Symbol('A', commutative=True), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Pow(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True))), Function('\\\\phi_1')(Symbol('A', commutative=True))))"], [["add", 6, "Function('\\\\phi_1')(Symbol('A', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\phi_1')(Symbol('A', commutative=True))), Add(Mul(Integer(2), Pow(Mul(Symbol('A', commutative=True), Add(Function('\\\\phi_1')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))), Symbol('A', commutative=True)), Pow(Mul(Integer(2), Symbol('A', commutative=True), Function('\\\\phi_1')(Symbol('A', commutative=True))), Mul(Integer(-1), Symbol('A', commutative=True))), Function('\\\\phi_1')(Symbol('A', commutative=True))), Function('\\\\phi_1')(Symbol('A', commutative=True))))"]]}, {"prompt": "Given z{(\\omega)} = \\sin{(\\omega)}, then derive - z{(\\omega)} + \\int z{(\\omega)} d\\omega = \\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)}, then obtain (\\int (\\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)}) d\\omega)^{\\hat{\\mathbf{x}}} = (\\int (- z{(\\omega)} + \\int \\sin{(\\omega)} d\\omega) d\\omega)^{\\hat{\\mathbf{x}}}", "derivation": "z{(\\omega)} = \\sin{(\\omega)} and \\int z{(\\omega)} d\\omega = \\int \\sin{(\\omega)} d\\omega and - z{(\\omega)} + \\int z{(\\omega)} d\\omega = - z{(\\omega)} + \\int \\sin{(\\omega)} d\\omega and - z{(\\omega)} + \\int z{(\\omega)} d\\omega = \\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)} and \\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)} = - z{(\\omega)} + \\int \\sin{(\\omega)} d\\omega and \\int (\\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)}) d\\omega = \\int (- z{(\\omega)} + \\int \\sin{(\\omega)} d\\omega) d\\omega and (\\int (\\hat{\\mathbf{x}} - z{(\\omega)} - \\cos{(\\omega)}) d\\omega)^{\\hat{\\mathbf{x}}} = (\\int (- z{(\\omega)} + \\int \\sin{(\\omega)} d\\omega) d\\omega)^{\\hat{\\mathbf{x}}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Function('z')(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(Function('z')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(Function('z')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["power", 6, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('z')(Symbol('\\\\omega', commutative=True))), Integral(sin(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(q)} = \\sin{(q)}, then obtain \\frac{2 \\operatorname{v_{x}}{(q)}}{\\sin{(q)}} - 2 - \\frac{2}{\\sin{(q)}} = - \\frac{2}{\\sin{(q)}}", "derivation": "\\operatorname{v_{x}}{(q)} = \\sin{(q)} and \\frac{\\operatorname{v_{x}}{(q)}}{\\sin{(q)}} = 1 and \\frac{\\operatorname{v_{x}}{(q)}}{\\sin{(q)}} - \\frac{1}{\\sin{(q)}} = 1 - \\frac{1}{\\sin{(q)}} and \\frac{\\operatorname{v_{x}}{(q)}}{\\sin{(q)}} - \\frac{2}{\\sin{(q)}} = 1 - \\frac{2}{\\sin{(q)}} and \\frac{\\operatorname{v_{x}}{(q)}}{\\sin{(q)}} - 1 - \\frac{2}{\\sin{(q)}} = - \\frac{2}{\\sin{(q)}} and \\frac{2 \\operatorname{v_{x}}{(q)}}{\\sin{(q)}} - 2 - \\frac{2}{\\sin{(q)}} = - \\frac{2}{\\sin{(q)}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["divide", 1, "sin(Symbol('q', commutative=True))"], "Equality(Mul(Function('v_x')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Pow(sin(Symbol('q', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('v_x')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))))"], [["minus", 3, "Pow(sin(Symbol('q', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('v_x')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))))"], [["minus", 4, 1], "Equality(Add(Mul(Function('v_x')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Integer(-1), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(2), Function('v_x')(Symbol('q', commutative=True)), Pow(sin(Symbol('q', commutative=True)), Integer(-1))), Integer(-2), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1)))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given r{(\\mathbf{E},\\varepsilon,A_{2})} = (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}}, then obtain (\\frac{A_{2}}{\\varepsilon} - (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}} + r{(\\mathbf{E},\\varepsilon,A_{2})})^{\\mathbf{E}} = (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}}", "derivation": "r{(\\mathbf{E},\\varepsilon,A_{2})} = (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}} and \\frac{A_{2}}{\\varepsilon} - (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}} + r{(\\mathbf{E},\\varepsilon,A_{2})} = \\frac{A_{2}}{\\varepsilon} and r{(\\mathbf{E},\\varepsilon,A_{2})} = (\\frac{A_{2}}{\\varepsilon} - (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}} + r{(\\mathbf{E},\\varepsilon,A_{2})})^{\\mathbf{E}} and (\\frac{A_{2}}{\\varepsilon} - (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}} + r{(\\mathbf{E},\\varepsilon,A_{2})})^{\\mathbf{E}} = (\\frac{A_{2}}{\\varepsilon})^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A_2', commutative=True)), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Add(Mul(Integer(-1), Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True))), Function('r')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('r')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A_2', commutative=True)), Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True))), Function('r')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A_2', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True))), Function('r')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('A_2', commutative=True))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Symbol('\\\\mathbf{E}', commutative=True)))"]]}, {"prompt": "Given c{(y)} = \\cos{(\\cos{(y)})}, then obtain 2 c^{2}{(y)} \\cos{(y)} = 2 c{(y)} \\cos{(y)} \\cos{(\\cos{(y)})}", "derivation": "c{(y)} = \\cos{(\\cos{(y)})} and c^{2}{(y)} = c{(y)} \\cos{(\\cos{(y)})} and 2 c^{2}{(y)} = c^{2}{(y)} + c{(y)} \\cos{(\\cos{(y)})} and 2 c^{2}{(y)} \\cos{(y)} = (c^{2}{(y)} + c{(y)} \\cos{(\\cos{(y)})}) \\cos{(y)} and c^{2}{(y)} \\cos{(y)} = c{(y)} \\cos{(y)} \\cos{(\\cos{(y)})} and 2 c{(y)} \\cos{(y)} \\cos{(\\cos{(y)})} = (c^{2}{(y)} + c{(y)} \\cos{(\\cos{(y)})}) \\cos{(y)} and 2 c^{2}{(y)} \\cos{(y)} = 2 c{(y)} \\cos{(y)} \\cos{(\\cos{(y)})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))"], [["times", 1, "Function('c')(Symbol('y', commutative=True))"], "Equality(Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True)))))"], [["add", 2, "Pow(Function('c')(Symbol('y', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('y', commutative=True)), Integer(2))), Add(Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))))"], [["times", 3, "cos(Symbol('y', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), cos(Symbol('y', commutative=True))), Mul(Add(Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))), cos(Symbol('y', commutative=True))))"], [["times", 2, "cos(Symbol('y', commutative=True))"], "Equality(Mul(Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), cos(Symbol('y', commutative=True))), Mul(Function('c')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Function('c')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True)))), Mul(Add(Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), Mul(Function('c')(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True))))), cos(Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Integer(2), Pow(Function('c')(Symbol('y', commutative=True)), Integer(2)), cos(Symbol('y', commutative=True))), Mul(Integer(2), Function('c')(Symbol('y', commutative=True)), cos(Symbol('y', commutative=True)), cos(cos(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(t,U)} = \\frac{U}{t} and \\rho_{f}{(U)} = U^{2}, then obtain \\frac{U \\ddot{x}{(t,U)}}{t} = \\frac{\\rho_{f}{(U)}}{t^{2}}", "derivation": "\\ddot{x}{(t,U)} = \\frac{U}{t} and \\frac{U \\ddot{x}{(t,U)}}{t} = \\frac{U^{2}}{t^{2}} and \\rho_{f}{(U)} = U^{2} and \\frac{U \\ddot{x}{(t,U)}}{t} = \\frac{\\rho_{f}{(U)}}{t^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('U', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1))))"], [["times", 1, "Mul(Symbol('U', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('U', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('U', commutative=True))), Mul(Pow(Symbol('U', commutative=True), Integer(2)), Pow(Symbol('t', commutative=True), Integer(-2))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('U', commutative=True), Pow(Symbol('t', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('t', commutative=True), Symbol('U', commutative=True))), Mul(Pow(Symbol('t', commutative=True), Integer(-2)), Function('\\\\rho_f')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\theta{(x^\\prime,H)} = - x^\\prime + \\log{(H)}, then obtain (- x^\\prime + \\theta{(x^\\prime,H)})^{2} = (- 2 x^\\prime + \\log{(H)})^{2}", "derivation": "\\theta{(x^\\prime,H)} = - x^\\prime + \\log{(H)} and - x^\\prime + \\theta{(x^\\prime,H)} = - 2 x^\\prime + \\log{(H)} and (- x^\\prime + \\theta{(x^\\prime,H)})^{2} = (- 2 x^\\prime + \\log{(H)}) (- x^\\prime + \\theta{(x^\\prime,H)}) and (- 2 x^\\prime + \\log{(H)}) (- x^\\prime + \\theta{(x^\\prime,H)}) = (- 2 x^\\prime + \\log{(H)})^{2} and (- x^\\prime + \\theta{(x^\\prime,H)})^{2} = (- 2 x^\\prime + \\log{(H)})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True))), Integer(2)), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True)))))"], [["times", 2, "Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True)))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Function('\\\\theta')(Symbol('x^\\\\prime', commutative=True), Symbol('H', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), log(Symbol('H', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\theta_{2}{(A_{x},v,U)} = \\frac{A_{x} - v}{U}, then derive \\cos{(\\int \\theta_{2}{(A_{x},v,U)} dv)} = \\cos{(\\frac{A_{x} v}{U} + \\Omega - \\frac{v^{2}}{2 U})}, then obtain \\cos{(\\frac{A_{x} v}{U} + z - \\frac{v^{2}}{2 U})} = \\cos{(\\frac{A_{x} v}{U} + \\Omega - \\frac{v^{2}}{2 U})}", "derivation": "\\theta_{2}{(A_{x},v,U)} = \\frac{A_{x} - v}{U} and \\int \\theta_{2}{(A_{x},v,U)} dv = \\int \\frac{A_{x} - v}{U} dv and \\cos{(\\int \\theta_{2}{(A_{x},v,U)} dv)} = \\cos{(\\int \\frac{A_{x} - v}{U} dv)} and \\cos{(\\int \\theta_{2}{(A_{x},v,U)} dv)} = \\cos{(\\frac{A_{x} v}{U} + \\Omega - \\frac{v^{2}}{2 U})} and \\cos{(\\int \\frac{A_{x} - v}{U} dv)} = \\cos{(\\frac{A_{x} v}{U} + \\Omega - \\frac{v^{2}}{2 U})} and \\cos{(\\frac{A_{x} v}{U} + z - \\frac{v^{2}}{2 U})} = \\cos{(\\frac{A_{x} v}{U} + \\Omega - \\frac{v^{2}}{2 U})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('A_x', commutative=True), Symbol('v', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('A_x', commutative=True), Symbol('v', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\theta_2')(Symbol('A_x', commutative=True), Symbol('v', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('v', commutative=True)))), cos(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(cos(Integral(Function('\\\\theta_2')(Symbol('A_x', commutative=True), Symbol('v', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('v', commutative=True)))), cos(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(cos(Integral(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Add(Symbol('A_x', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))), Tuple(Symbol('v', commutative=True)))), cos(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(2))))))"], [["evaluate_integrals", 5], "Equality(cos(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('z', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(2))))), cos(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\delta)} = e^{e^{\\delta}} and \\mathbf{P}{(\\delta)} = e^{e^{\\delta}}, then obtain \\mathbf{P}{(\\delta)} \\int \\mathbf{J}_P{(\\delta)} d\\delta = \\mathbf{P}{(\\delta)} \\int \\mathbf{P}{(\\delta)} d\\delta", "derivation": "\\mathbf{J}_P{(\\delta)} = e^{e^{\\delta}} and \\mathbf{P}{(\\delta)} = e^{e^{\\delta}} and \\mathbf{J}_P{(\\delta)} = \\mathbf{P}{(\\delta)} and \\int \\mathbf{J}_P{(\\delta)} d\\delta = \\int \\mathbf{P}{(\\delta)} d\\delta and \\mathbf{P}{(\\delta)} \\int \\mathbf{J}_P{(\\delta)} d\\delta = \\mathbf{P}{(\\delta)} \\int \\mathbf{P}{(\\delta)} d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\delta', commutative=True)), exp(exp(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)), exp(exp(Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\delta', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 4, "Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)), Integral(Function('\\\\mathbf{P}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\phi{(m_{s})} = \\int e^{m_{s}} dm_{s}, then derive \\phi{(m_{s})} = \\theta + e^{m_{s}}, then derive \\frac{A_{z} + e^{m_{s}}}{\\theta} = \\frac{\\theta + e^{m_{s}}}{\\theta}, then obtain \\frac{(A_{z} + e^{m_{s}}) e^{- m_{s}}}{\\theta} = \\frac{e^{- m_{s}} \\int e^{m_{s}} dm_{s}}{\\theta}", "derivation": "\\phi{(m_{s})} = \\int e^{m_{s}} dm_{s} and \\phi{(m_{s})} = \\theta + e^{m_{s}} and \\int e^{m_{s}} dm_{s} = \\theta + e^{m_{s}} and \\frac{\\int e^{m_{s}} dm_{s}}{\\theta} = \\frac{\\theta + e^{m_{s}}}{\\theta} and \\frac{A_{z} + e^{m_{s}}}{\\theta} = \\frac{\\theta + e^{m_{s}}}{\\theta} and \\frac{A_{z} + e^{m_{s}}}{\\theta} = \\frac{\\int e^{m_{s}} dm_{s}}{\\theta} and \\frac{(A_{z} + e^{m_{s}}) e^{- m_{s}}}{\\theta} = \\frac{e^{- m_{s}} \\int e^{m_{s}} dm_{s}}{\\theta}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('m_s', commutative=True)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\phi')(Symbol('m_s', commutative=True)), Add(Symbol('\\\\theta', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Add(Symbol('\\\\theta', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["divide", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\theta', commutative=True), exp(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), exp(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('\\\\theta', commutative=True), exp(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), exp(Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"], [["divide", 6, "exp(Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Add(Symbol('A_z', commutative=True), exp(Symbol('m_s', commutative=True))), exp(Mul(Integer(-1), Symbol('m_s', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), exp(Mul(Integer(-1), Symbol('m_s', commutative=True))), Integral(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(f)} = \\sin{(f)} and W{(f)} = - \\sin{(f)} + \\frac{\\sin{(f)}}{f}, then obtain - \\sin{(f)} + \\frac{\\mathbf{p}{(f)}}{f} = - \\mathbf{p}{(f)} + \\frac{\\mathbf{p}{(f)}}{f}", "derivation": "\\mathbf{p}{(f)} = \\sin{(f)} and \\frac{\\mathbf{p}{(f)}}{f} = \\frac{\\sin{(f)}}{f} and - \\sin{(f)} + \\frac{\\mathbf{p}{(f)}}{f} = - \\sin{(f)} + \\frac{\\sin{(f)}}{f} and W{(f)} = - \\sin{(f)} + \\frac{\\sin{(f)}}{f} and W{(f)} = - \\mathbf{p}{(f)} + \\frac{\\mathbf{p}{(f)}}{f} and - \\sin{(f)} + \\frac{\\mathbf{p}{(f)}}{f} = W{(f)} and - \\sin{(f)} + \\frac{\\mathbf{p}{(f)}}{f} = - \\mathbf{p}{(f)} + \\frac{\\mathbf{p}{(f)}}{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('f', commutative=True)), sin(Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), sin(Symbol('f', commutative=True))))"], [["minus", 2, "sin(Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), sin(Symbol('f', commutative=True)))))"], ["renaming_premise", "Equality(Function('W')(Symbol('f', commutative=True)), Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), sin(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('W')(Symbol('f', commutative=True)), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)))), Function('W')(Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{p}')(Symbol('f', commutative=True))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given G{(\\sigma_x,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\sigma_x), then obtain (\\frac{\\partial}{\\partial A_{1}} G{(\\sigma_x,A_{1})} + 1)^{\\sigma_x} = 1", "derivation": "G{(\\sigma_x,A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} + \\sigma_x) and \\frac{\\partial}{\\partial A_{1}} G{(\\sigma_x,A_{1})} = \\frac{\\partial^{2}}{\\partial A_{1}^{2}} (A_{1} + \\sigma_x) and \\frac{\\partial}{\\partial A_{1}} G{(\\sigma_x,A_{1})} + 1 = \\frac{\\partial^{2}}{\\partial A_{1}^{2}} (A_{1} + \\sigma_x) + 1 and (\\frac{\\partial}{\\partial A_{1}} G{(\\sigma_x,A_{1})} + 1)^{\\sigma_x} = (\\frac{\\partial^{2}}{\\partial A_{1}^{2}} (A_{1} + \\sigma_x) + 1)^{\\sigma_x} and (\\frac{\\partial}{\\partial A_{1}} G{(\\sigma_x,A_{1})} + 1)^{\\sigma_x} = 1", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_1', commutative=True)), Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('G')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))), Integer(1)))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Add(Derivative(Function('G')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)), Pow(Add(Derivative(Add(Symbol('A_1', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(2))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('G')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\sigma_x', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\dot{x}{(\\hat{x}_0)} = e^{\\hat{x}_0}, then derive L + e^{\\hat{x}_0} + \\int \\dot{x}{(\\hat{x}_0)} d\\hat{x}_0 = 2 L + 2 e^{\\hat{x}_0}, then obtain L + e^{\\hat{x}_0} + \\int e^{\\hat{x}_0} d\\hat{x}_0 = 2 L + 2 e^{\\hat{x}_0}", "derivation": "\\dot{x}{(\\hat{x}_0)} = e^{\\hat{x}_0} and \\int \\dot{x}{(\\hat{x}_0)} d\\hat{x}_0 = \\int e^{\\hat{x}_0} d\\hat{x}_0 and \\int \\dot{x}{(\\hat{x}_0)} d\\hat{x}_0 + \\int e^{\\hat{x}_0} d\\hat{x}_0 = 2 \\int e^{\\hat{x}_0} d\\hat{x}_0 and L + e^{\\hat{x}_0} + \\int \\dot{x}{(\\hat{x}_0)} d\\hat{x}_0 = 2 L + 2 e^{\\hat{x}_0} and L + e^{\\hat{x}_0} + \\int e^{\\hat{x}_0} d\\hat{x}_0 = 2 L + 2 e^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), exp(Symbol('\\\\hat{x}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(exp(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["add", 2, "Integral(exp(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Add(Integral(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(exp(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('L', commutative=True), exp(Symbol('\\\\hat{x}_0', commutative=True)), Integral(Function('\\\\dot{x}')(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Integer(2), Symbol('L', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('L', commutative=True), exp(Symbol('\\\\hat{x}_0', commutative=True)), Integral(exp(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)))), Add(Mul(Integer(2), Symbol('L', commutative=True)), Mul(Integer(2), exp(Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{g}{(G)} = e^{\\cos{(G)}}, then obtain G + (G + \\mathbf{g}{(G)}) \\mathbf{g}^{G}{(G)} + \\mathbf{g}{(G)} = G + (G + e^{\\cos{(G)}}) \\mathbf{g}^{G}{(G)} + \\mathbf{g}{(G)}", "derivation": "\\mathbf{g}{(G)} = e^{\\cos{(G)}} and G + \\mathbf{g}{(G)} = G + e^{\\cos{(G)}} and (G + \\mathbf{g}{(G)}) \\mathbf{g}^{G}{(G)} = (G + e^{\\cos{(G)}}) \\mathbf{g}^{G}{(G)} and G + (G + \\mathbf{g}{(G)}) \\mathbf{g}^{G}{(G)} + \\mathbf{g}{(G)} = G + (G + e^{\\cos{(G)}}) \\mathbf{g}^{G}{(G)} + \\mathbf{g}{(G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), exp(cos(Symbol('G', commutative=True))))"], [["add", 1, "Symbol('G', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Function('\\\\mathbf{g}')(Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), exp(cos(Symbol('G', commutative=True)))))"], [["times", 2, "Pow(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), Symbol('G', commutative=True))"], "Equality(Mul(Add(Symbol('G', commutative=True), Function('\\\\mathbf{g}')(Symbol('G', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Mul(Add(Symbol('G', commutative=True), exp(cos(Symbol('G', commutative=True)))), Pow(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), Symbol('G', commutative=True))))"], [["add", 3, "Add(Symbol('G', commutative=True), Function('\\\\mathbf{g}')(Symbol('G', commutative=True)))"], "Equality(Add(Symbol('G', commutative=True), Mul(Add(Symbol('G', commutative=True), Function('\\\\mathbf{g}')(Symbol('G', commutative=True))), Pow(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Function('\\\\mathbf{g}')(Symbol('G', commutative=True))), Add(Symbol('G', commutative=True), Mul(Add(Symbol('G', commutative=True), exp(cos(Symbol('G', commutative=True)))), Pow(Function('\\\\mathbf{g}')(Symbol('G', commutative=True)), Symbol('G', commutative=True))), Function('\\\\mathbf{g}')(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} = e^{E_{x} + \\mu} and \\operatorname{v_{y}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then obtain \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} - \\operatorname{v_{y}}{(f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}} = e^{E_{x} + \\mu}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} = e^{E_{x} + \\mu} and \\operatorname{v_{y}}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} + \\operatorname{v_{y}}{(f_{\\mathbf{v}})} = \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} + e^{f_{\\mathbf{v}}} and \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} = \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} - \\operatorname{v_{y}}{(f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}} and \\operatorname{f_{\\mathbf{p}}}{(\\mu,E_{x})} - \\operatorname{v_{y}}{(f_{\\mathbf{v}})} + e^{f_{\\mathbf{v}}} = e^{E_{x} + \\mu}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), exp(Add(Symbol('E_x', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["get_premise", "Equality(Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 2, "Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["minus", 3, "Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mu', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))), exp(Add(Symbol('E_x', commutative=True), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})}, then obtain (- \\sin{(E_{n})})^{E_{n}} + \\frac{d}{d E_{n}} \\mu_{0}^{E_{n}}{(E_{n})} = (- \\sin{(E_{n})})^{E_{n}} + \\frac{d}{d E_{n}} (\\frac{d}{d E_{n}} \\cos{(E_{n})})^{E_{n}}", "derivation": "\\mu_{0}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})} and \\mu_{0}^{E_{n}}{(E_{n})} = (\\frac{d}{d E_{n}} \\cos{(E_{n})})^{E_{n}} and \\frac{d}{d E_{n}} \\mu_{0}^{E_{n}}{(E_{n})} = \\frac{d}{d E_{n}} (\\frac{d}{d E_{n}} \\cos{(E_{n})})^{E_{n}} and (- \\sin{(E_{n})})^{E_{n}} + \\frac{d}{d E_{n}} \\mu_{0}^{E_{n}}{(E_{n})} = (- \\sin{(E_{n})})^{E_{n}} + \\frac{d}{d E_{n}} (\\frac{d}{d E_{n}} \\cos{(E_{n})})^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('E_n', commutative=True)), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["power", 1, "Symbol('E_n', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mu_0')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Pow(Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["add", 3, "Pow(Mul(Integer(-1), sin(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True))"], "Equality(Add(Pow(Mul(Integer(-1), sin(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Derivative(Pow(Function('\\\\mu_0')(Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Add(Pow(Mul(Integer(-1), sin(Symbol('E_n', commutative=True))), Symbol('E_n', commutative=True)), Derivative(Pow(Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(Z)} = e^{Z}, then obtain \\frac{(- Z + \\operatorname{v_{t}}{(Z)})^{2}}{(- Z + e^{Z})^{2}} = \\frac{- Z + \\operatorname{v_{t}}{(Z)}}{- Z + e^{Z}}", "derivation": "\\operatorname{v_{t}}{(Z)} = e^{Z} and - Z + \\operatorname{v_{t}}{(Z)} = - Z + e^{Z} and \\frac{- Z + \\operatorname{v_{t}}{(Z)}}{- Z + e^{Z}} = 1 and \\frac{(- Z + \\operatorname{v_{t}}{(Z)})^{2}}{(- Z + e^{Z})^{2}} = \\frac{- Z + \\operatorname{v_{t}}{(Z)}}{- Z + e^{Z}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], [["minus", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('Z', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))), Integer(-1))), Integer(1))"], [["times", 3, "Mul(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('Z', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('Z', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))), Integer(-2))), Mul(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('v_t')(Symbol('Z', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), exp(Symbol('Z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{J}_P,\\rho_b)} = e^{\\frac{\\rho_b}{\\mathbf{J}_P}}, then obtain (\\frac{\\partial}{\\partial \\rho_b} \\varepsilon{(\\mathbf{J}_P,\\rho_b)} e^{- \\frac{\\rho_b}{\\mathbf{J}_P}})^{\\rho_b} = (\\frac{d}{d \\rho_b} 1)^{\\rho_b}", "derivation": "\\varepsilon{(\\mathbf{J}_P,\\rho_b)} = e^{\\frac{\\rho_b}{\\mathbf{J}_P}} and \\varepsilon{(\\mathbf{J}_P,\\rho_b)} e^{- \\frac{\\rho_b}{\\mathbf{J}_P}} = 1 and \\frac{\\partial}{\\partial \\rho_b} \\varepsilon{(\\mathbf{J}_P,\\rho_b)} e^{- \\frac{\\rho_b}{\\mathbf{J}_P}} = \\frac{d}{d \\rho_b} 1 and (\\frac{\\partial}{\\partial \\rho_b} \\varepsilon{(\\mathbf{J}_P,\\rho_b)} e^{- \\frac{\\rho_b}{\\mathbf{J}_P}})^{\\rho_b} = (\\frac{d}{d \\rho_b} 1)^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True))))"], [["divide", 1, "exp(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('\\\\varepsilon')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\rho_b', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('\\\\rho_b', commutative=True)), Pow(Derivative(Integer(1), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\psi{(\\tilde{g})} = \\int e^{\\tilde{g}} d\\tilde{g}, then derive \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} (\\dot{z} + e^{\\tilde{g}}), then derive \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} = e^{\\tilde{g}}, then obtain L + \\psi{(\\tilde{g})} = \\psi{(\\tilde{g})}", "derivation": "\\psi{(\\tilde{g})} = \\int e^{\\tilde{g}} d\\tilde{g} and \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\int e^{\\tilde{g}} d\\tilde{g} and \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} (\\dot{z} + e^{\\tilde{g}}) and \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} = e^{\\tilde{g}} and \\int \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} d\\tilde{g} = \\int e^{\\tilde{g}} d\\tilde{g} and \\int \\frac{d}{d \\tilde{g}} \\psi{(\\tilde{g})} d\\tilde{g} = \\psi{(\\tilde{g})} and L + \\psi{(\\tilde{g})} = \\psi{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(exp(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Derivative(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('L', commutative=True), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True))), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True)))"]]}, {"prompt": "Given Q{(x^\\prime)} = x^\\prime and \\operatorname{F_{H}}{(x^\\prime)} = - Q{(x^\\prime)}, then obtain \\frac{- x^\\prime - \\int x^\\prime dx^\\prime}{\\int Q{(x^\\prime)} dx^\\prime} = \\frac{- Q{(x^\\prime)} - \\int x^\\prime dx^\\prime}{\\int Q{(x^\\prime)} dx^\\prime}", "derivation": "Q{(x^\\prime)} = x^\\prime and \\operatorname{F_{H}}{(x^\\prime)} = - Q{(x^\\prime)} and \\operatorname{F_{H}}{(x^\\prime)} = - x^\\prime and - x^\\prime = - Q{(x^\\prime)} and - x^\\prime - \\int x^\\prime dx^\\prime = - Q{(x^\\prime)} - \\int x^\\prime dx^\\prime and \\frac{- x^\\prime - \\int x^\\prime dx^\\prime}{\\int Q{(x^\\prime)} dx^\\prime} = \\frac{- Q{(x^\\prime)} - \\int x^\\prime dx^\\prime}{\\int Q{(x^\\prime)} dx^\\prime}", "srepr_derivation": [["renaming_premise", "Equality(Function('Q')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('F_H')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Function('Q')(Symbol('x^\\\\prime', commutative=True))))"], [["minus", 4, "Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))), Add(Mul(Integer(-1), Function('Q')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))))"], [["divide", 5, "Integral(Function('Q')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))), Pow(Integral(Function('Q')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(-1), Function('Q')(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(-1), Integral(Symbol('x^\\\\prime', commutative=True), Tuple(Symbol('x^\\\\prime', commutative=True))))), Pow(Integral(Function('Q')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{H}{(f_{E},v)} = \\frac{f_{E}}{v}, then obtain - \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} + \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} dv = - \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} + \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\frac{f_{E}}{v} dv", "derivation": "\\mathbf{H}{(f_{E},v)} = \\frac{f_{E}}{v} and \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} = \\frac{\\partial}{\\partial v} \\frac{f_{E}}{v} and \\int \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} dv = \\int \\frac{\\partial}{\\partial v} \\frac{f_{E}}{v} dv and \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} dv = \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\frac{f_{E}}{v} dv and - \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} + \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} dv = - \\frac{\\partial}{\\partial v} \\mathbf{H}{(f_{E},v)} + \\frac{\\partial}{\\partial v} \\int \\frac{\\partial}{\\partial v} \\frac{f_{E}}{v} dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Integral(Derivative(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["minus", 4, "Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Derivative(Integral(Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{H}')(Symbol('f_E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Derivative(Integral(Derivative(Mul(Symbol('f_E', commutative=True), Pow(Symbol('v', commutative=True), Integer(-1))), Tuple(Symbol('v', commutative=True), Integer(1))), Tuple(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(\\varepsilon)} = \\log{(e^{\\varepsilon})}, then obtain - (- \\log{(e^{\\varepsilon})})^{\\varepsilon} - \\log{(e^{\\varepsilon})} = - (- \\log{(e^{\\varepsilon})})^{\\varepsilon} - \\nabla{(\\varepsilon)}", "derivation": "\\nabla{(\\varepsilon)} = \\log{(e^{\\varepsilon})} and \\nabla{(\\varepsilon)} - \\log{(e^{\\varepsilon})} = 0 and - \\log{(e^{\\varepsilon})} = - \\nabla{(\\varepsilon)} and (- \\log{(e^{\\varepsilon})})^{\\varepsilon} = (- \\nabla{(\\varepsilon)})^{\\varepsilon} and - (- \\nabla{(\\varepsilon)})^{\\varepsilon} - \\log{(e^{\\varepsilon})} = - (- \\nabla{(\\varepsilon)})^{\\varepsilon} - \\nabla{(\\varepsilon)} and - (- \\log{(e^{\\varepsilon})})^{\\varepsilon} - \\log{(e^{\\varepsilon})} = - (- \\log{(e^{\\varepsilon})})^{\\varepsilon} - \\nabla{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), log(exp(Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 1, "log(exp(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True))))), Integer(0))"], [["minus", 2, "Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True)), Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 3, "Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True))))), Add(Mul(Integer(-1), Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Integer(-1), Pow(Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True))))), Add(Mul(Integer(-1), Pow(Mul(Integer(-1), log(exp(Symbol('\\\\varepsilon', commutative=True)))), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Function('\\\\nabla')(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\Psi^{\\dagger},\\hat{p}_0)} = \\cos{(\\Psi^{\\dagger} + \\hat{p}_0)} and Z{(\\chi,V)} = V \\chi, then obtain \\frac{Z{(\\chi,V)}}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} - \\frac{1}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} = \\frac{V \\chi}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} - \\frac{1}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}}", "derivation": "\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)} = \\cos{(\\Psi^{\\dagger} + \\hat{p}_0)} and Z{(\\chi,V)} = V \\chi and \\frac{Z{(\\chi,V)}}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} = \\frac{V \\chi}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} and \\frac{Z{(\\chi,V)}}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} - \\frac{1}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} = \\frac{V \\chi}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} - \\frac{1}{\\Omega{(\\Psi^{\\dagger},\\hat{p}_0)}} and \\frac{Z{(\\chi,V)}}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} - \\frac{1}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} = \\frac{V \\chi}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}} - \\frac{1}{\\cos{(\\Psi^{\\dagger} + \\hat{p}_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], ["get_premise", "Equality(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["divide", 2, "Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Mul(Symbol('V', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))))"], [["minus", 3, "Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))), Add(Mul(Symbol('V', commutative=True), Symbol('\\\\chi', commutative=True), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Function('Z')(Symbol('\\\\chi', commutative=True), Symbol('V', commutative=True)), Pow(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1)))), Add(Mul(Symbol('V', commutative=True), Symbol('\\\\chi', commutative=True), Pow(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(cos(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\Psi{(l)} = \\sin{(l)}, then obtain 3 \\Psi{(l)} = \\Psi{(l)} + 2 \\sin{(l)}", "derivation": "\\Psi{(l)} = \\sin{(l)} and 2 \\Psi{(l)} = \\Psi{(l)} + \\sin{(l)} and 3 \\Psi{(l)} = 2 \\Psi{(l)} + \\sin{(l)} and 3 \\Psi{(l)} = \\Psi{(l)} + 2 \\sin{(l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True)))"], [["add", 1, "Function('\\\\Psi')(Symbol('l', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi')(Symbol('l', commutative=True))), Add(Function('\\\\Psi')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('\\\\Psi')(Symbol('l', commutative=True)))"], "Equality(Mul(Integer(3), Function('\\\\Psi')(Symbol('l', commutative=True))), Add(Mul(Integer(2), Function('\\\\Psi')(Symbol('l', commutative=True))), sin(Symbol('l', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\Psi')(Symbol('l', commutative=True))), Add(Function('\\\\Psi')(Symbol('l', commutative=True)), Mul(Integer(2), sin(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given f{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\mathbf{s}{(\\varphi^*)} = \\cos{(\\varphi^*)}, then obtain \\frac{- \\varphi^* + f{(\\varphi^*)}}{f{(\\varphi^*)}} = \\frac{- \\varphi^* + \\mathbf{s}{(\\varphi^*)}}{f{(\\varphi^*)}}", "derivation": "f{(\\varphi^*)} = \\cos{(\\varphi^*)} and - \\varphi^* + f{(\\varphi^*)} = - \\varphi^* + \\cos{(\\varphi^*)} and \\mathbf{s}{(\\varphi^*)} = \\cos{(\\varphi^*)} and - \\varphi^* + f{(\\varphi^*)} = - \\varphi^* + \\mathbf{s}{(\\varphi^*)} and \\frac{- \\varphi^* + f{(\\varphi^*)}}{\\cos{(\\varphi^*)}} = \\frac{- \\varphi^* + \\mathbf{s}{(\\varphi^*)}}{\\cos{(\\varphi^*)}} and \\frac{- \\varphi^* + f{(\\varphi^*)}}{f{(\\varphi^*)}} = \\frac{- \\varphi^* + \\mathbf{s}{(\\varphi^*)}}{f{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('f')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('f')(Symbol('\\\\varphi^*', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 4, "cos(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('f')(Symbol('\\\\varphi^*', commutative=True))), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True))), Pow(cos(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('f')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('f')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('f')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(v_{t})} = e^{v_{t}}, then obtain - \\mathbf{A} + v_{t} + \\frac{d}{d v_{t}} (- 2 v_{t} + \\dot{y}{(v_{t})}) = - \\mathbf{A} + v_{t} + \\frac{d}{d v_{t}} (- 2 v_{t} + e^{v_{t}})", "derivation": "\\dot{y}{(v_{t})} = e^{v_{t}} and - v_{t} + \\dot{y}{(v_{t})} = - v_{t} + e^{v_{t}} and - 2 v_{t} + \\dot{y}{(v_{t})} = - 2 v_{t} + e^{v_{t}} and \\frac{d}{d v_{t}} (- 2 v_{t} + \\dot{y}{(v_{t})}) = \\frac{d}{d v_{t}} (- 2 v_{t} + e^{v_{t}}) and - \\mathbf{A} + \\frac{d}{d v_{t}} (- 2 v_{t} + \\dot{y}{(v_{t})}) = - \\mathbf{A} + \\frac{d}{d v_{t}} (- 2 v_{t} + e^{v_{t}}) and - \\mathbf{A} + v_{t} + \\frac{d}{d v_{t}} (- 2 v_{t} + \\dot{y}{(v_{t})}) = - \\mathbf{A} + v_{t} + \\frac{d}{d v_{t}} (- 2 v_{t} + e^{v_{t}})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["minus", 1, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('\\\\dot{y}')(Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))))"], [["minus", 2, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), Function('\\\\dot{y}')(Symbol('v_t', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), Function('\\\\dot{y}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), Function('\\\\dot{y}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"], [["add", 5, "Symbol('v_t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('v_t', commutative=True), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), Function('\\\\dot{y}')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('v_t', commutative=True), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{H}{(v_{2})} = \\sin{(v_{2})}, then obtain \\frac{d}{d v_{2}} (\\mathbf{H}{(v_{2})} - \\sin{(v_{2})} + \\int \\sin{(v_{2})} dv_{2}) = \\frac{d}{d v_{2}} \\int \\sin{(v_{2})} dv_{2}", "derivation": "\\mathbf{H}{(v_{2})} = \\sin{(v_{2})} and \\mathbf{H}{(v_{2})} - \\sin{(v_{2})} = 0 and \\int \\mathbf{H}{(v_{2})} dv_{2} = \\int \\sin{(v_{2})} dv_{2} and \\mathbf{H}{(v_{2})} - \\sin{(v_{2})} + \\int \\sin{(v_{2})} dv_{2} = \\int \\sin{(v_{2})} dv_{2} and \\frac{d}{d v_{2}} \\int \\mathbf{H}{(v_{2})} dv_{2} = \\frac{d}{d v_{2}} \\int \\sin{(v_{2})} dv_{2} and \\int \\mathbf{H}{(v_{2})} dv_{2} = \\mathbf{H}{(v_{2})} - \\sin{(v_{2})} + \\int \\sin{(v_{2})} dv_{2} and \\frac{d}{d v_{2}} (\\mathbf{H}{(v_{2})} - \\sin{(v_{2})} + \\int \\sin{(v_{2})} dv_{2}) = \\frac{d}{d v_{2}} \\int \\sin{(v_{2})} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), sin(Symbol('v_2', commutative=True)))"], [["minus", 1, "sin(Symbol('v_2', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), sin(Symbol('v_2', commutative=True)))), Integer(0))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["add", 2, "Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), sin(Symbol('v_2', commutative=True))), Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 3, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), sin(Symbol('v_2', commutative=True))), Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Add(Function('\\\\mathbf{H}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), sin(Symbol('v_2', commutative=True))), Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True)))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and q{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain (- \\dot{z}{(\\mathbf{J}_P)} + \\dot{z}^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (- \\dot{z}{(\\mathbf{J}_P)} + q^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P}", "derivation": "\\dot{z}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\dot{z}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = (e^{\\mathbf{J}_P})^{\\mathbf{J}_P} and - \\dot{z}{(\\mathbf{J}_P)} + \\dot{z}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = - \\dot{z}{(\\mathbf{J}_P)} + (e^{\\mathbf{J}_P})^{\\mathbf{J}_P} and q{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and - \\dot{z}{(\\mathbf{J}_P)} + \\dot{z}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = - \\dot{z}{(\\mathbf{J}_P)} + q^{\\mathbf{J}_P}{(\\mathbf{J}_P)} and (- \\dot{z}{(\\mathbf{J}_P)} + \\dot{z}^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (- \\dot{z}{(\\mathbf{J}_P)} + q^{\\mathbf{J}_P}{(\\mathbf{J}_P)})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 2, "Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('q')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Pow(Function('q')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(l,U)} = U^{l}, then obtain \\int (\\frac{\\partial}{\\partial l} \\frac{\\operatorname{E_{x}}^{2}{(l,U)}}{l})^{2} dl = \\int (\\frac{\\partial}{\\partial l} \\frac{U^{l} \\operatorname{E_{x}}{(l,U)}}{l})^{2} dl", "derivation": "\\operatorname{E_{x}}{(l,U)} = U^{l} and \\operatorname{E_{x}}^{2}{(l,U)} = U^{l} \\operatorname{E_{x}}{(l,U)} and \\frac{\\operatorname{E_{x}}^{2}{(l,U)}}{l} = \\frac{U^{l} \\operatorname{E_{x}}{(l,U)}}{l} and \\frac{\\partial}{\\partial l} \\frac{\\operatorname{E_{x}}^{2}{(l,U)}}{l} = \\frac{\\partial}{\\partial l} \\frac{U^{l} \\operatorname{E_{x}}{(l,U)}}{l} and (\\frac{\\partial}{\\partial l} \\frac{\\operatorname{E_{x}}^{2}{(l,U)}}{l})^{2} = (\\frac{\\partial}{\\partial l} \\frac{U^{l} \\operatorname{E_{x}}{(l,U)}}{l})^{2} and \\int (\\frac{\\partial}{\\partial l} \\frac{\\operatorname{E_{x}}^{2}{(l,U)}}{l})^{2} dl = \\int (\\frac{\\partial}{\\partial l} \\frac{U^{l} \\operatorname{E_{x}}{(l,U)}}{l})^{2} dl", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)))"], [["times", 1, "Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))"], "Equality(Pow(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Integer(2)), Mul(Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)), Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))))"], [["divide", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Integer(2))), Mul(Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Integer(2))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Integer(2))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2)))"], [["integrate", 5, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True)), Integer(2))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Derivative(Mul(Pow(Symbol('U', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Integer(-1)), Function('E_x')(Symbol('l', commutative=True), Symbol('U', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(v_{z})} = \\int e^{v_{z}} dv_{z}, then derive \\operatorname{f_{\\mathbf{p}}}{(v_{z})} = n_{1} + e^{v_{z}}, then derive e^{v_{z}} = \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z}, then obtain \\int e^{v_{z}} dv_{z} = n_{1} + \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(v_{z})} = \\int e^{v_{z}} dv_{z} and \\frac{d}{d v_{z}} \\operatorname{f_{\\mathbf{p}}}{(v_{z})} = \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z} and \\operatorname{f_{\\mathbf{p}}}{(v_{z})} = n_{1} + e^{v_{z}} and \\frac{\\partial}{\\partial v_{z}} (n_{1} + e^{v_{z}}) = \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z} and \\int e^{v_{z}} dv_{z} = n_{1} + e^{v_{z}} and e^{v_{z}} = \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z} and \\int e^{v_{z}} dv_{z} = n_{1} + \\frac{d}{d v_{z}} \\int e^{v_{z}} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True)), Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_integrals", 1], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('v_z', commutative=True)), Add(Symbol('n_1', commutative=True), exp(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('n_1', commutative=True), exp(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Symbol('n_1', commutative=True), exp(Symbol('v_z', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(exp(Symbol('v_z', commutative=True)), Derivative(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Add(Symbol('n_1', commutative=True), Derivative(Integral(exp(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(v_{y},H)} = H - v_{y}, then derive \\frac{\\partial}{\\partial v_{y}} \\Psi_{nl}{(v_{y},H)} = -1, then obtain \\frac{\\partial^{2}}{\\partial H\\partial v_{y}} (H - v_{y}) = \\frac{d}{d H} (-1)", "derivation": "\\Psi_{nl}{(v_{y},H)} = H - v_{y} and \\Psi_{nl}{(v_{y},H)} - 1 = H - v_{y} - 1 and \\frac{\\partial}{\\partial v_{y}} (\\Psi_{nl}{(v_{y},H)} - 1) = \\frac{\\partial}{\\partial v_{y}} (H - v_{y} - 1) and \\frac{\\partial}{\\partial v_{y}} \\Psi_{nl}{(v_{y},H)} = -1 and \\frac{\\partial^{2}}{\\partial H\\partial v_{y}} \\Psi_{nl}{(v_{y},H)} = \\frac{d}{d H} (-1) and \\frac{\\partial^{2}}{\\partial H\\partial v_{y}} (H - v_{y}) = \\frac{d}{d H} (-1)", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True), Symbol('H', commutative=True)), Integer(-1)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True)), Integer(-1)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))"], [["differentiate", 4, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('v_y', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\mathbf{J}_M{(\\sigma_x)} = \\log{(\\sigma_x)}^{2}, then obtain (\\operatorname{z^{*}}^{2}{(\\sigma_x)})^{\\sigma_x} = (\\operatorname{z^{*}}{(\\sigma_x)} \\log{(\\sigma_x)})^{\\sigma_x}", "derivation": "\\operatorname{z^{*}}{(\\sigma_x)} = \\log{(\\sigma_x)} and \\operatorname{z^{*}}{(\\sigma_x)} \\log{(\\sigma_x)} = \\log{(\\sigma_x)}^{2} and \\mathbf{J}_M{(\\sigma_x)} = \\log{(\\sigma_x)}^{2} and \\mathbf{J}_M{(\\sigma_x)} = \\operatorname{z^{*}}{(\\sigma_x)} \\log{(\\sigma_x)} and \\mathbf{J}_M^{\\sigma_x}{(\\sigma_x)} = (\\operatorname{z^{*}}{(\\sigma_x)} \\log{(\\sigma_x)})^{\\sigma_x} and \\mathbf{J}_M{(\\sigma_x)} = \\operatorname{z^{*}}^{2}{(\\sigma_x)} and \\mathbf{J}_M^{\\sigma_x}{(\\sigma_x)} = (\\operatorname{z^{*}}^{2}{(\\sigma_x)})^{\\sigma_x} and (\\operatorname{z^{*}}^{2}{(\\sigma_x)})^{\\sigma_x} = (\\operatorname{z^{*}}{(\\sigma_x)} \\log{(\\sigma_x)})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "log(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True))), Pow(log(Symbol('\\\\sigma_x', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_x', commutative=True)), Pow(log(Symbol('\\\\sigma_x', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_x', commutative=True)), Mul(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_x', commutative=True)), Pow(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)))"], [["power", 6, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Pow(Pow(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Function('z^*')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\theta{(v_{y},g)} = - g + v_{y}, then obtain (- g + v_{y})^{2} \\frac{\\partial}{\\partial v_{y}} (- g + v_{y}) \\frac{\\partial}{\\partial v_{y}} \\theta{(v_{y},g)} = (- g + v_{y})^{2} (\\frac{\\partial}{\\partial v_{y}} (- g + v_{y}))^{2}", "derivation": "\\theta{(v_{y},g)} = - g + v_{y} and \\frac{\\partial}{\\partial v_{y}} \\theta{(v_{y},g)} = \\frac{\\partial}{\\partial v_{y}} (- g + v_{y}) and (- g + v_{y}) \\frac{\\partial}{\\partial v_{y}} \\theta{(v_{y},g)} = (- g + v_{y}) \\frac{\\partial}{\\partial v_{y}} (- g + v_{y}) and (- g + v_{y})^{2} \\frac{\\partial}{\\partial v_{y}} (- g + v_{y}) \\frac{\\partial}{\\partial v_{y}} \\theta{(v_{y},g)} = (- g + v_{y})^{2} (\\frac{\\partial}{\\partial v_{y}} (- g + v_{y}))^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('v_y', commutative=True), Symbol('g', commutative=True)), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('v_y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Derivative(Function('\\\\theta')(Symbol('v_y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["times", 3, "Mul(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Integer(2)), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Function('\\\\theta')(Symbol('v_y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Integer(2)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\hat{x}{(\\pi)} = \\sin{(\\pi)}, then derive \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = \\cos{(\\pi)}, then derive \\Psi + \\hat{x}{(\\pi)} = f + \\sin{(\\pi)}, then obtain (\\Psi + \\hat{x}{(\\pi)}) \\cos{(\\pi)} = (f + \\sin{(\\pi)}) \\cos{(\\pi)}", "derivation": "\\hat{x}{(\\pi)} = \\sin{(\\pi)} and \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = \\frac{d}{d \\pi} \\sin{(\\pi)} and \\frac{d}{d \\pi} \\hat{x}{(\\pi)} = \\cos{(\\pi)} and \\cos{(\\pi)} = \\frac{d}{d \\pi} \\sin{(\\pi)} and \\int \\frac{d}{d \\pi} \\hat{x}{(\\pi)} d\\pi = \\int \\cos{(\\pi)} d\\pi and \\Psi + \\hat{x}{(\\pi)} = f + \\sin{(\\pi)} and (\\Psi + \\hat{x}{(\\pi)}) \\frac{d}{d \\pi} \\sin{(\\pi)} = (f + \\sin{(\\pi)}) \\frac{d}{d \\pi} \\sin{(\\pi)} and (\\Psi + \\hat{x}{(\\pi)}) \\cos{(\\pi)} = (f + \\sin{(\\pi)}) \\cos{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), cos(Symbol('\\\\pi', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\pi', commutative=True)), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(cos(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True))), Add(Symbol('f', commutative=True), sin(Symbol('\\\\pi', commutative=True))))"], [["times", 6, "Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))"], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True))), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Add(Symbol('f', commutative=True), sin(Symbol('\\\\pi', commutative=True))), Derivative(sin(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Mul(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))), Mul(Add(Symbol('f', commutative=True), sin(Symbol('\\\\pi', commutative=True))), cos(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(V)} = \\sin{(V)} and \\operatorname{E_{x}}{(P_{e})} = \\cos{(P_{e})}, then obtain \\frac{\\frac{d}{d P_{e}} \\operatorname{E_{x}}{(P_{e})}}{\\sin{(V)}} = \\frac{\\frac{d}{d P_{e}} \\cos{(P_{e})}}{\\sin{(V)}}", "derivation": "\\hat{\\mathbf{r}}{(V)} = \\sin{(V)} and \\operatorname{E_{x}}{(P_{e})} = \\cos{(P_{e})} and \\frac{d}{d P_{e}} \\operatorname{E_{x}}{(P_{e})} = \\frac{d}{d P_{e}} \\cos{(P_{e})} and \\frac{\\frac{d}{d P_{e}} \\operatorname{E_{x}}{(P_{e})}}{\\hat{\\mathbf{r}}{(V)}} = \\frac{\\frac{d}{d P_{e}} \\cos{(P_{e})}}{\\hat{\\mathbf{r}}{(V)}} and \\frac{\\frac{d}{d P_{e}} \\operatorname{E_{x}}{(P_{e})}}{\\sin{(V)}} = \\frac{\\frac{d}{d P_{e}} \\cos{(P_{e})}}{\\sin{(V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True)), sin(Symbol('V', commutative=True)))"], ["get_premise", "Equality(Function('E_x')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["differentiate", 2, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["divide", 3, "Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True)), Integer(-1)), Derivative(Function('E_x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V', commutative=True)), Integer(-1)), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), Derivative(Function('E_x')(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('V', commutative=True)), Integer(-1)), Derivative(cos(Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(Q)} = \\cos{(\\cos{(Q)})} and \\Psi^{\\dagger}{(Q)} = \\operatorname{f_{\\mathbf{p}}}{(Q)} \\cos{(Q)}, then obtain \\frac{\\partial}{\\partial Q} (- \\mathbb{I}{(A)} + \\operatorname{f_{\\mathbf{p}}}{(Q)} \\cos{(Q)}) = \\frac{\\partial}{\\partial Q} (- \\mathbb{I}{(A)} + \\cos{(Q)} \\cos{(\\cos{(Q)})})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(Q)} = \\cos{(\\cos{(Q)})} and \\Psi^{\\dagger}{(Q)} = \\operatorname{f_{\\mathbf{p}}}{(Q)} \\cos{(Q)} and \\Psi^{\\dagger}{(Q)} = \\cos{(Q)} \\cos{(\\cos{(Q)})} and \\Psi^{\\dagger}{(Q)} - \\mathbb{I}{(A)} = - \\mathbb{I}{(A)} + \\cos{(Q)} \\cos{(\\cos{(Q)})} and - \\mathbb{I}{(A)} + \\operatorname{f_{\\mathbf{p}}}{(Q)} \\cos{(Q)} = - \\mathbb{I}{(A)} + \\cos{(Q)} \\cos{(\\cos{(Q)})} and \\frac{\\partial}{\\partial Q} (- \\mathbb{I}{(A)} + \\operatorname{f_{\\mathbf{p}}}{(Q)} \\cos{(Q)}) = \\frac{\\partial}{\\partial Q} (- \\mathbb{I}{(A)} + \\cos{(Q)} \\cos{(\\cos{(Q)})})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('Q', commutative=True)), cos(cos(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Mul(cos(Symbol('Q', commutative=True)), cos(cos(Symbol('Q', commutative=True)))))"], [["minus", 3, "Function('\\\\mathbb{I}')(Symbol('A', commutative=True))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True))), Mul(cos(Symbol('Q', commutative=True)), cos(cos(Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True))), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True))), Mul(cos(Symbol('Q', commutative=True)), cos(cos(Symbol('Q', commutative=True))))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True))), Mul(Function('f_{\\\\mathbf{p}}')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A', commutative=True))), Mul(cos(Symbol('Q', commutative=True)), cos(cos(Symbol('Q', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\Psi)} = \\sin{(\\cos{(\\Psi)})}, then obtain \\varphi{(\\Psi)} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\varphi{(\\Psi)} = \\sin{(\\cos{(\\Psi)})} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\varphi{(\\Psi)}", "derivation": "\\varphi{(\\Psi)} = \\sin{(\\cos{(\\Psi)})} and \\varphi{(\\Psi)} - \\cos{(\\Psi)} = \\sin{(\\cos{(\\Psi)})} - \\cos{(\\Psi)} and \\frac{d}{d \\Psi} \\varphi{(\\Psi)} = \\frac{d}{d \\Psi} \\sin{(\\cos{(\\Psi)})} and \\varphi{(\\Psi)} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\sin{(\\cos{(\\Psi)})} = \\sin{(\\cos{(\\Psi)})} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\sin{(\\cos{(\\Psi)})} and \\varphi{(\\Psi)} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\varphi{(\\Psi)} = \\sin{(\\cos{(\\Psi)})} - \\cos{(\\Psi)} + \\frac{d}{d \\Psi} \\varphi{(\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), sin(cos(Symbol('\\\\Psi', commutative=True))))"], [["minus", 1, "cos(Symbol('\\\\Psi', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))), Add(sin(cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["add", 2, "Derivative(sin(cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Derivative(sin(cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(sin(cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Derivative(sin(cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Add(sin(cos(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Derivative(Function('\\\\varphi')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(\\psi,F_{N})} = F_{N}^{\\psi}, then obtain (F_{N}^{\\psi} + \\pi^{F_{N}}{(\\psi,F_{N})}) \\pi^{- F_{N}}{(\\psi,F_{N})} = (F_{N}^{\\psi} + (F_{N}^{\\psi})^{F_{N}}) \\pi^{- F_{N}}{(\\psi,F_{N})}", "derivation": "\\pi{(\\psi,F_{N})} = F_{N}^{\\psi} and \\pi^{F_{N}}{(\\psi,F_{N})} = (F_{N}^{\\psi})^{F_{N}} and F_{N}^{\\psi} + \\pi^{F_{N}}{(\\psi,F_{N})} = F_{N}^{\\psi} + (F_{N}^{\\psi})^{F_{N}} and (F_{N}^{\\psi} + \\pi^{F_{N}}{(\\psi,F_{N})}) \\pi^{- F_{N}}{(\\psi,F_{N})} = (F_{N}^{\\psi} + (F_{N}^{\\psi})^{F_{N}}) \\pi^{- F_{N}}{(\\psi,F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["power", 1, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('F_N', commutative=True)))"], [["add", 2, "Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True))"], "Equality(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('F_N', commutative=True))))"], [["divide", 3, "Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], "Equality(Mul(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))), Mul(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Pow(Pow(Symbol('F_N', commutative=True), Symbol('\\\\psi', commutative=True)), Symbol('F_N', commutative=True))), Pow(Function('\\\\pi')(Symbol('\\\\psi', commutative=True), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(A_{z},v_{y})} = A_{z}^{v_{y}}, then obtain (\\frac{\\partial}{\\partial v_{y}} (A_{z}^{v_{y}} + \\operatorname{F_{N}}{(A_{z},v_{y})}))^{A_{z}} = (\\frac{\\partial}{\\partial v_{y}} 2 A_{z}^{v_{y}})^{A_{z}}", "derivation": "\\operatorname{F_{N}}{(A_{z},v_{y})} = A_{z}^{v_{y}} and A_{z}^{v_{y}} + \\operatorname{F_{N}}{(A_{z},v_{y})} = 2 A_{z}^{v_{y}} and \\frac{\\partial}{\\partial v_{y}} (A_{z}^{v_{y}} + \\operatorname{F_{N}}{(A_{z},v_{y})}) = \\frac{\\partial}{\\partial v_{y}} 2 A_{z}^{v_{y}} and (\\frac{\\partial}{\\partial v_{y}} (A_{z}^{v_{y}} + \\operatorname{F_{N}}{(A_{z},v_{y})}))^{A_{z}} = (\\frac{\\partial}{\\partial v_{y}} 2 A_{z}^{v_{y}})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True)))"], [["add", 1, "Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Add(Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True)), Function('F_N')(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True)), Function('F_N')(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('A_z', commutative=True)"], "Equality(Pow(Derivative(Add(Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True)), Function('F_N')(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('A_z', commutative=True)), Pow(Derivative(Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\omega{(F_{x})} = \\log{(\\sin{(F_{x})})}, then obtain \\sin{(\\log{(e^{\\omega{(F_{x})}})}^{F_{x}})} = \\sin{(\\log{(\\sin{(F_{x})})}^{F_{x}})}", "derivation": "\\omega{(F_{x})} = \\log{(\\sin{(F_{x})})} and e^{\\omega{(F_{x})}} = \\sin{(F_{x})} and \\omega^{F_{x}}{(F_{x})} = \\log{(\\sin{(F_{x})})}^{F_{x}} and \\sin{(\\omega^{F_{x}}{(F_{x})})} = \\sin{(\\log{(\\sin{(F_{x})})}^{F_{x}})} and \\omega{(F_{x})} = \\log{(e^{\\omega{(F_{x})}})} and \\log{(e^{\\omega{(F_{x})}})}^{F_{x}} = \\log{(\\sin{(F_{x})})}^{F_{x}} and \\omega^{F_{x}}{(F_{x})} = \\log{(e^{\\omega{(F_{x})}})}^{F_{x}} and \\sin{(\\log{(e^{\\omega{(F_{x})}})}^{F_{x}})} = \\sin{(\\log{(\\sin{(F_{x})})}^{F_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('F_x', commutative=True)), log(sin(Symbol('F_x', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\omega')(Symbol('F_x', commutative=True))), sin(Symbol('F_x', commutative=True)))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\omega')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(log(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Function('\\\\omega')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), sin(Pow(log(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\omega')(Symbol('F_x', commutative=True)), log(exp(Function('\\\\omega')(Symbol('F_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(log(exp(Function('\\\\omega')(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)), Pow(log(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Function('\\\\omega')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(log(exp(Function('\\\\omega')(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 7], "Equality(sin(Pow(log(exp(Function('\\\\omega')(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True))), sin(Pow(log(sin(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\dot{x})} = \\cos{(\\dot{x})} and \\theta_{2}{(\\dot{x})} = - \\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})}, then obtain 1 - \\theta_{2}^{\\dot{x}}{(\\dot{x})} = 0", "derivation": "\\tilde{g}^*{(\\dot{x})} = \\cos{(\\dot{x})} and \\theta_{2}{(\\dot{x})} = - \\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})} and \\theta_{2}{(\\dot{x})} = 0 and \\theta_{2}^{\\dot{x}}{(\\dot{x})} = 0^{\\dot{x}} and (- \\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})})^{\\dot{x}} = 0^{\\dot{x}} and (- \\tilde{g}^*{(\\dot{x})} + \\cos{(\\dot{x})})^{\\dot{x}} + \\tilde{g}^*^{\\dot{x}}{(\\dot{x})} = 0^{\\dot{x}} + \\tilde{g}^*^{\\dot{x}}{(\\dot{x})} and 2 = \\theta_{2}^{\\dot{x}}{(\\dot{x})} + 1 and 1 - \\theta_{2}^{\\dot{x}}{(\\dot{x})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True))), cos(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Integer(0))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Integer(0), Symbol('\\\\dot{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True))), cos(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Integer(0), Symbol('\\\\dot{x}', commutative=True)))"], [["add", 5, "Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True))), cos(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\dot{x}', commutative=True)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Integer(2), Add(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Integer(1)))"], [["minus", 7, "Add(Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Integer(1))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\theta_2')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(C_{2},l)} = C_{2} l, then obtain (C_{2} l)^{C_{2}} \\operatorname{r_{0}}^{C_{2}}{(C_{2},l)} + ((C_{2} l)^{C_{2}})^{l} = (C_{2} l)^{2 C_{2}} + ((C_{2} l)^{C_{2}})^{l}", "derivation": "\\operatorname{r_{0}}{(C_{2},l)} = C_{2} l and \\operatorname{r_{0}}^{C_{2}}{(C_{2},l)} = (C_{2} l)^{C_{2}} and (C_{2} l)^{C_{2}} \\operatorname{r_{0}}^{C_{2}}{(C_{2},l)} = (C_{2} l)^{2 C_{2}} and (C_{2} l)^{C_{2}} \\operatorname{r_{0}}^{C_{2}}{(C_{2},l)} + ((C_{2} l)^{C_{2}})^{l} = (C_{2} l)^{2 C_{2}} + ((C_{2} l)^{C_{2}})^{l}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)))"], [["power", 1, "Symbol('C_2', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)))"], [["times", 2, "Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('r_0')(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True))), Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))))"], [["add", 3, "Pow(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Symbol('l', commutative=True))"], "Equality(Add(Mul(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('r_0')(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True))), Pow(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Symbol('l', commutative=True))), Add(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Mul(Integer(2), Symbol('C_2', commutative=True))), Pow(Pow(Mul(Symbol('C_2', commutative=True), Symbol('l', commutative=True)), Symbol('C_2', commutative=True)), Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(E_{n},\\Omega)} = E_{n} + \\Omega, then obtain (E_{n} + \\Omega + \\theta_{2}{(E_{n},\\Omega)})^{E_{n}} = (2 E_{n} + 2 \\Omega)^{E_{n}}", "derivation": "\\theta_{2}{(E_{n},\\Omega)} = E_{n} + \\Omega and E_{n} + \\Omega + \\theta_{2}{(E_{n},\\Omega)} = 2 E_{n} + 2 \\Omega and 2 \\theta_{2}{(E_{n},\\Omega)} = 2 E_{n} + 2 \\Omega and (2 \\theta_{2}{(E_{n},\\Omega)})^{E_{n}} = (2 E_{n} + 2 \\Omega)^{E_{n}} and E_{n} + \\Omega + \\theta_{2}{(E_{n},\\Omega)} = 2 \\theta_{2}{(E_{n},\\Omega)} and (E_{n} + \\Omega + \\theta_{2}{(E_{n},\\Omega)})^{E_{n}} = (2 E_{n} + 2 \\Omega)^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["add", 1, "Add(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('E_n', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Symbol('E_n', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Add(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True), Function('\\\\theta_2')(Symbol('E_n', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('E_n', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('E_n', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True))), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given z{(f_{E},I)} = \\log{(I)}^{f_{E}} and \\hat{x}_0{(f_{E},I)} = z{(f_{E},I)} - \\log{(I)}^{f_{E}}, then obtain \\int \\frac{1}{I} dI = \\int 0 dI", "derivation": "z{(f_{E},I)} = \\log{(I)}^{f_{E}} and \\hat{x}_0{(f_{E},I)} = z{(f_{E},I)} - \\log{(I)}^{f_{E}} and - \\frac{\\hat{x}_0{(f_{E},I)} z{(f_{E},I)} \\log{(I)}^{- f_{E}}}{I} = - \\frac{(z{(f_{E},I)} - \\log{(I)}^{f_{E}}) z{(f_{E},I)} \\log{(I)}^{- f_{E}}}{I} and - \\frac{\\hat{x}_0{(f_{E},I)}}{I} = 0 and \\frac{\\log{(I)}^{f_{E}}}{z{(f_{E},I)}} = 0 and \\frac{\\log{(I)}^{f_{E}}}{I z{(f_{E},I)}} = 0 and \\frac{1}{I} = 0 and \\int \\frac{1}{I} dI = \\int 0 dI", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Add(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Symbol('I', commutative=True), Pow(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True)))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Add(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True)))), Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('f_E', commutative=True), Symbol('I', commutative=True))), Integer(0))"], [["divide", 4, "Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], "Equality(Mul(Pow(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True))), Integer(0))"], [["divide", 5, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Pow(Function('z')(Symbol('f_E', commutative=True), Symbol('I', commutative=True)), Integer(-1)), Pow(log(Symbol('I', commutative=True)), Symbol('f_E', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Symbol('I', commutative=True), Integer(-1)), Integer(0))"], [["integrate", 7, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Symbol('I', commutative=True), Integer(-1)), Tuple(Symbol('I', commutative=True))), Integral(Integer(0), Tuple(Symbol('I', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\phi_1)} = \\log{(\\phi_1)} and \\mathbf{H}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)}, then obtain (\\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)})^{\\phi_1} = \\mathbf{H}^{\\phi_1}{(\\phi_1)}", "derivation": "\\Psi{(\\phi_1)} = \\log{(\\phi_1)} and \\Psi^{\\phi_1}{(\\phi_1)} = \\log{(\\phi_1)}^{\\phi_1} and \\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\log{(\\phi_1)}^{\\phi_1} and (\\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)})^{\\phi_1} = (\\frac{d}{d \\phi_1} \\log{(\\phi_1)}^{\\phi_1})^{\\phi_1} and \\mathbf{H}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)} and \\mathbf{H}^{\\phi_1}{(\\phi_1)} = (\\frac{d}{d \\phi_1} \\log{(\\phi_1)}^{\\phi_1})^{\\phi_1} and (\\frac{d}{d \\phi_1} \\Psi^{\\phi_1}{(\\phi_1)})^{\\phi_1} = \\mathbf{H}^{\\phi_1}{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\phi_1', commutative=True)), Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Pow(Derivative(Pow(Function('\\\\Psi')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given u{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then derive \\frac{d}{d V_{\\mathbf{B}}} u{(V_{\\mathbf{B}})} - \\frac{1}{V_{\\mathbf{B}}} = 0, then obtain (\\frac{d}{d V_{\\mathbf{B}}} u{(V_{\\mathbf{B}})} - \\frac{1}{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} = 0^{V_{\\mathbf{B}}}", "derivation": "u{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and u{(V_{\\mathbf{B}})} - \\log{(V_{\\mathbf{B}})} = 0 and \\frac{d}{d V_{\\mathbf{B}}} (u{(V_{\\mathbf{B}})} - \\log{(V_{\\mathbf{B}})}) = \\frac{d}{d V_{\\mathbf{B}}} 0 and \\frac{d}{d V_{\\mathbf{B}}} u{(V_{\\mathbf{B}})} - \\frac{1}{V_{\\mathbf{B}}} = 0 and (\\frac{d}{d V_{\\mathbf{B}}} u{(V_{\\mathbf{B}})} - \\frac{1}{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} = 0^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["minus", 1, "log(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Add(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))), Integer(0))"], [["power", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Add(Derivative(Function('u')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Integer(0), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(A)} = \\sin{(A)}, then derive \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\rho - \\cos{(A)}} = \\frac{\\sin{(A)}}{\\rho - \\cos{(A)}}, then obtain \\rho + 1 + \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\rho - \\cos{(A)}} = \\rho + 1 + \\frac{\\sin{(A)}}{\\rho - \\cos{(A)}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(A)} = \\sin{(A)} and \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\int \\sin{(A)} dA} = \\frac{\\sin{(A)}}{\\int \\sin{(A)} dA} and \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\rho - \\cos{(A)}} = \\frac{\\sin{(A)}}{\\rho - \\cos{(A)}} and \\rho + \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\rho - \\cos{(A)}} = \\rho + \\frac{\\sin{(A)}}{\\rho - \\cos{(A)}} and \\rho + 1 + \\frac{\\operatorname{g_{\\varepsilon}}{(A)}}{\\rho - \\cos{(A)}} = \\rho + 1 + \\frac{\\sin{(A)}}{\\rho - \\cos{(A)}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["divide", 1, "Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))"], "Equality(Mul(Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True)), Pow(Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1))), Mul(sin(Symbol('A', commutative=True)), Pow(Integral(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True))), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), sin(Symbol('A', commutative=True))))"], [["add", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), sin(Symbol('A', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Symbol('\\\\rho', commutative=True), Integer(1), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True)))), Add(Symbol('\\\\rho', commutative=True), Integer(1), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Integer(-1)), sin(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{\\varepsilon}, then obtain \\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})} - \\int 1 d\\varepsilon = \\Psi_{\\lambda}^{\\varepsilon} - \\int 1 d\\varepsilon", "derivation": "\\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{\\varepsilon} and 1 = \\frac{\\Psi_{\\lambda}^{\\varepsilon}}{\\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})}} and \\int 1 d\\varepsilon = \\int \\frac{\\Psi_{\\lambda}^{\\varepsilon}}{\\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})}} d\\varepsilon and \\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})} - \\int \\frac{\\Psi_{\\lambda}^{\\varepsilon}}{\\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})}} d\\varepsilon = \\Psi_{\\lambda}^{\\varepsilon} - \\int \\frac{\\Psi_{\\lambda}^{\\varepsilon}}{\\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})}} d\\varepsilon and \\bar{\\h}{(\\varepsilon,\\Psi_{\\lambda})} - \\int 1 d\\varepsilon = \\Psi_{\\lambda}^{\\varepsilon} - \\int 1 d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["minus", 1, "Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))), Add(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Integral(Mul(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\hbar')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True))))), Add(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\varepsilon', commutative=True))))))"]]}, {"prompt": "Given q{(\\pi,U)} = \\cos{(\\pi^{U})} and E{(\\pi,U)} = \\frac{\\partial}{\\partial \\pi} (- U + \\cos{(\\pi^{U})} - 1), then obtain \\frac{U \\frac{\\partial}{\\partial \\pi} (- U + \\cos{(\\pi^{U})} - 1)}{q{(\\pi,U)}} = \\frac{U E{(\\pi,U)}}{q{(\\pi,U)}}", "derivation": "q{(\\pi,U)} = \\cos{(\\pi^{U})} and - U + q{(\\pi,U)} = - U + \\cos{(\\pi^{U})} and - U + q{(\\pi,U)} - 1 = - U + \\cos{(\\pi^{U})} - 1 and \\frac{\\partial}{\\partial \\pi} (- U + q{(\\pi,U)} - 1) = \\frac{\\partial}{\\partial \\pi} (- U + \\cos{(\\pi^{U})} - 1) and E{(\\pi,U)} = \\frac{\\partial}{\\partial \\pi} (- U + \\cos{(\\pi^{U})} - 1) and \\frac{\\partial}{\\partial \\pi} (- U + q{(\\pi,U)} - 1) = E{(\\pi,U)} and \\frac{\\frac{\\partial}{\\partial \\pi} (- U + q{(\\pi,U)} - 1)}{q{(\\pi,U)}} = \\frac{E{(\\pi,U)}}{q{(\\pi,U)}} and \\frac{U \\frac{\\partial}{\\partial \\pi} (- U + q{(\\pi,U)} - 1)}{q{(\\pi,U)}} = \\frac{U E{(\\pi,U)}}{q{(\\pi,U)}} and \\frac{U \\frac{\\partial}{\\partial \\pi} (- U + \\cos{(\\pi^{U})} - 1)}{q{(\\pi,U)}} = \\frac{U E{(\\pi,U)}}{q{(\\pi,U)}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Function('E')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)))"], [["divide", 6, "Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Function('E')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1))))"], [["times", 7, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Symbol('U', commutative=True), Function('E')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Mul(Symbol('U', commutative=True), Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Pow(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Symbol('U', commutative=True), Function('E')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Pow(Function('q')(Symbol('\\\\pi', commutative=True), Symbol('U', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\ddot{x}{(\\delta,B)} = \\cos{(B \\delta)}, then obtain \\iint \\frac{\\partial}{\\partial \\delta} \\frac{\\ddot{x}{(\\delta,B)}}{\\cos{(B \\delta)}} dB dB = \\iint \\frac{d}{d \\delta} 1 dB dB", "derivation": "\\ddot{x}{(\\delta,B)} = \\cos{(B \\delta)} and \\frac{\\ddot{x}{(\\delta,B)}}{\\cos{(B \\delta)}} = 1 and \\frac{\\partial}{\\partial \\delta} \\frac{\\ddot{x}{(\\delta,B)}}{\\cos{(B \\delta)}} = \\frac{d}{d \\delta} 1 and \\int \\frac{\\partial}{\\partial \\delta} \\frac{\\ddot{x}{(\\delta,B)}}{\\cos{(B \\delta)}} dB = \\int \\frac{d}{d \\delta} 1 dB and \\iint \\frac{\\partial}{\\partial \\delta} \\frac{\\ddot{x}{(\\delta,B)}}{\\cos{(B \\delta)}} dB dB = \\iint \\frac{d}{d \\delta} 1 dB dB", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Pow(cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\ddot{x}')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Pow(cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('\\\\ddot{x}')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Pow(cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["integrate", 4, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('\\\\ddot{x}')(Symbol('\\\\delta', commutative=True), Symbol('B', commutative=True)), Pow(cos(Mul(Symbol('B', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})} and s{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})} and \\Omega{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})}, then obtain s{(F_{x},\\mathbf{v})} = \\Omega{(F_{x},\\mathbf{v})}", "derivation": "\\operatorname{A_{z}}{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})} and \\frac{\\operatorname{A_{z}}{(F_{x},\\mathbf{v})}}{\\sin{(F_{x} + \\mathbf{v})}} = 1 and s{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})} and \\Omega{(F_{x},\\mathbf{v})} = \\sin{(F_{x} + \\mathbf{v})} and s{(F_{x},\\mathbf{v})} = \\operatorname{A_{z}}{(F_{x},\\mathbf{v})} and \\frac{\\operatorname{A_{z}}{(F_{x},\\mathbf{v})}}{\\Omega{(F_{x},\\mathbf{v})}} = 1 and \\frac{s{(F_{x},\\mathbf{v})}}{\\Omega{(F_{x},\\mathbf{v})}} = 1 and s{(F_{x},\\mathbf{v})} = \\Omega{(F_{x},\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["divide", 1, "sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Function('A_z')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Pow(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('s')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('s')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('A_z')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Function('A_z')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('\\\\Omega')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Pow(Function('\\\\Omega')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Function('s')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Integer(1))"], [["divide", 7, "Pow(Function('\\\\Omega')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))"], "Equality(Function('s')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\Omega')(Symbol('F_x', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given f{(M)} = \\cos{(M)}, then obtain (\\frac{d}{d M} (- M + f{(M)}) - 1) \\int (f{(M)} + \\cos{(M)}) dM = (\\frac{d}{d M} (- M + \\cos{(M)}) - 1) \\int (f{(M)} + \\cos{(M)}) dM", "derivation": "f{(M)} = \\cos{(M)} and - M + f{(M)} = - M + \\cos{(M)} and \\frac{d}{d M} (- M + f{(M)}) = \\frac{d}{d M} (- M + \\cos{(M)}) and \\frac{d}{d M} (- M + f{(M)}) - 1 = \\frac{d}{d M} (- M + \\cos{(M)}) - 1 and (\\frac{d}{d M} (- M + f{(M)}) - 1) \\int (f{(M)} + \\cos{(M)}) dM = (\\frac{d}{d M} (- M + \\cos{(M)}) - 1) \\int (f{(M)} + \\cos{(M)}) dM", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True)))"], [["minus", 1, "Symbol('M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('f')(Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('f')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('f')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1)))"], [["times", 4, "Integral(Add(Function('f')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Add(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Function('f')(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Function('f')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))), Mul(Add(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Function('f')(Symbol('M', commutative=True)), cos(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given c{(\\mathbb{I},\\theta)} = \\frac{\\mathbb{I}}{\\theta}, then obtain \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta + \\int \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta d\\mathbb{I} = \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta + \\int \\theta \\int \\frac{\\mathbb{I}}{\\theta} d\\theta d\\mathbb{I}", "derivation": "c{(\\mathbb{I},\\theta)} = \\frac{\\mathbb{I}}{\\theta} and \\int c{(\\mathbb{I},\\theta)} d\\theta = \\int \\frac{\\mathbb{I}}{\\theta} d\\theta and \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta = \\theta \\int \\frac{\\mathbb{I}}{\\theta} d\\theta and \\int \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta d\\mathbb{I} = \\int \\theta \\int \\frac{\\mathbb{I}}{\\theta} d\\theta d\\mathbb{I} and \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta + \\int \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta d\\mathbb{I} = \\theta \\int c{(\\mathbb{I},\\theta)} d\\theta + \\int \\theta \\int \\frac{\\mathbb{I}}{\\theta} d\\theta d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 2, "Symbol('\\\\theta', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Symbol('\\\\theta', commutative=True), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Mul(Symbol('\\\\theta', commutative=True), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["add", 4, "Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Integral(Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), Add(Mul(Symbol('\\\\theta', commutative=True), Integral(Function('c')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Integral(Mul(Symbol('\\\\theta', commutative=True), Integral(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('\\\\theta', commutative=True), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\chi,U)} = \\sin^{\\chi}{(U)}, then obtain (\\delta^{U}{(\\chi,U)})^{U} \\delta{(\\chi,U)} \\delta^{U}{(\\chi,U)} = ((\\sin^{\\chi}{(U)})^{U})^{U} \\delta{(\\chi,U)} \\delta^{U}{(\\chi,U)}", "derivation": "\\delta{(\\chi,U)} = \\sin^{\\chi}{(U)} and \\delta^{U}{(\\chi,U)} = (\\sin^{\\chi}{(U)})^{U} and (\\delta^{U}{(\\chi,U)})^{U} = ((\\sin^{\\chi}{(U)})^{U})^{U} and \\delta{(\\chi,U)} \\delta^{U}{(\\chi,U)} = \\delta^{U}{(\\chi,U)} \\sin^{\\chi}{(U)} and (\\delta^{U}{(\\chi,U)})^{U} \\delta^{U}{(\\chi,U)} \\sin^{\\chi}{(U)} = ((\\sin^{\\chi}{(U)})^{U})^{U} \\delta^{U}{(\\chi,U)} \\sin^{\\chi}{(U)} and (\\delta^{U}{(\\chi,U)})^{U} \\delta{(\\chi,U)} \\delta^{U}{(\\chi,U)} = ((\\sin^{\\chi}{(U)})^{U})^{U} \\delta{(\\chi,U)} \\delta^{U}{(\\chi,U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('U', commutative=True)))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Pow(Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["times", 1, "Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["times", 3, "Mul(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Pow(Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True))), Mul(Pow(Pow(Pow(sin(Symbol('U', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Pow(Function('\\\\delta')(Symbol('\\\\chi', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True))))"]]}, {"prompt": "Given g{(\\hat{x})} = e^{\\hat{x}}, then derive \\frac{d}{d \\hat{x}} g{(\\hat{x})} = e^{\\hat{x}}, then obtain \\frac{d}{d \\hat{x}} g{(\\hat{x})} + 1 = g{(\\hat{x})} + 1", "derivation": "g{(\\hat{x})} = e^{\\hat{x}} and \\frac{d}{d \\hat{x}} g{(\\hat{x})} = \\frac{d}{d \\hat{x}} e^{\\hat{x}} and \\frac{d}{d \\hat{x}} g{(\\hat{x})} = e^{\\hat{x}} and \\frac{d}{d \\hat{x}} e^{\\hat{x}} = e^{\\hat{x}} and \\frac{d}{d \\hat{x}} e^{\\hat{x}} + 1 = e^{\\hat{x}} + 1 and \\frac{d}{d \\hat{x}} g{(\\hat{x})} + 1 = g{(\\hat{x})} + 1", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["add", 4, 1], "Equality(Add(Derivative(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('\\\\hat{x}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('g')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Integer(1)), Add(Function('g')(Symbol('\\\\hat{x}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given L{(\\hat{H}_l)} = \\log{(\\hat{H}_l)}, then obtain \\frac{d}{d \\hat{H}_l} ((\\hat{H}_l + L{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)}) = \\frac{d}{d \\hat{H}_l} ((\\hat{H}_l + \\log{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)})", "derivation": "L{(\\hat{H}_l)} = \\log{(\\hat{H}_l)} and \\hat{H}_l + L{(\\hat{H}_l)} = \\hat{H}_l + \\log{(\\hat{H}_l)} and (\\hat{H}_l + L{(\\hat{H}_l)})^{\\hat{H}_l} = (\\hat{H}_l + \\log{(\\hat{H}_l)})^{\\hat{H}_l} and (\\hat{H}_l + L{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)} = (\\hat{H}_l + \\log{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)} and \\frac{d}{d \\hat{H}_l} ((\\hat{H}_l + L{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)}) = \\frac{d}{d \\hat{H}_l} ((\\hat{H}_l + \\log{(\\hat{H}_l)})^{\\hat{H}_l} + \\log{(\\hat{H}_l)})", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('L')(Symbol('\\\\hat{H}_l', commutative=True))), Add(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('L')(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), log(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('L')(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Add(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), Function('L')(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Add(Pow(Add(Symbol('\\\\hat{H}_l', commutative=True), log(Symbol('\\\\hat{H}_l', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), log(Symbol('\\\\hat{H}_l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\hat{X},\\varepsilon_0)} = \\varepsilon_0^{\\hat{X}} and \\operatorname{M_{E}}{(\\delta,v_{t})} = \\frac{e^{\\delta}}{v_{t}}, then obtain \\operatorname{M_{E}}{(\\delta,v_{t})} + \\varphi{(\\hat{X},\\varepsilon_0)} = \\varphi{(\\hat{X},\\varepsilon_0)} + \\frac{e^{\\delta}}{v_{t}}", "derivation": "\\varphi{(\\hat{X},\\varepsilon_0)} = \\varepsilon_0^{\\hat{X}} and \\operatorname{M_{E}}{(\\delta,v_{t})} = \\frac{e^{\\delta}}{v_{t}} and \\varepsilon_0^{\\hat{X}} + \\operatorname{M_{E}}{(\\delta,v_{t})} = \\varepsilon_0^{\\hat{X}} + \\frac{e^{\\delta}}{v_{t}} and \\operatorname{M_{E}}{(\\delta,v_{t})} + \\varphi{(\\hat{X},\\varepsilon_0)} = \\varphi{(\\hat{X},\\varepsilon_0)} + \\frac{e^{\\delta}}{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], ["get_premise", "Equality(Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), exp(Symbol('\\\\delta', commutative=True))))"], [["add", 2, "Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True))), Add(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), exp(Symbol('\\\\delta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('M_E')(Symbol('\\\\delta', commutative=True), Symbol('v_t', commutative=True)), Function('\\\\varphi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))), Add(Function('\\\\varphi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), exp(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given m{(\\nabla)} = \\nabla, then derive \\int \\nabla m{(\\nabla)} d\\nabla = \\frac{\\nabla^{3}}{3} + y^{\\prime}, then obtain \\frac{d}{d y^{\\prime}} e^{\\iint \\nabla m{(\\nabla)} d\\nabla d\\nabla} = \\frac{\\partial}{\\partial y^{\\prime}} e^{\\int (\\frac{\\nabla^{3}}{3} + y^{\\prime}) d\\nabla}", "derivation": "m{(\\nabla)} = \\nabla and \\nabla m{(\\nabla)} = \\nabla^{2} and \\int \\nabla m{(\\nabla)} d\\nabla = \\int \\nabla^{2} d\\nabla and \\int \\nabla m{(\\nabla)} d\\nabla = \\frac{\\nabla^{3}}{3} + y^{\\prime} and \\iint \\nabla m{(\\nabla)} d\\nabla d\\nabla = \\int (\\frac{\\nabla^{3}}{3} + y^{\\prime}) d\\nabla and e^{\\iint \\nabla m{(\\nabla)} d\\nabla d\\nabla} = e^{\\int (\\frac{\\nabla^{3}}{3} + y^{\\prime}) d\\nabla} and \\frac{d}{d y^{\\prime}} e^{\\iint \\nabla m{(\\nabla)} d\\nabla d\\nabla} = \\frac{\\partial}{\\partial y^{\\prime}} e^{\\int (\\frac{\\nabla^{3}}{3} + y^{\\prime}) d\\nabla}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["times", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Pow(Symbol('\\\\nabla', commutative=True), Integer(2)))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Pow(Symbol('\\\\nabla', commutative=True), Integer(2)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Mul(Rational(1, 3), Pow(Symbol('\\\\nabla', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Mul(Rational(1, 3), Pow(Symbol('\\\\nabla', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["exp", 5], "Equality(exp(Integral(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), exp(Integral(Add(Mul(Rational(1, 3), Pow(Symbol('\\\\nabla', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["differentiate", 6, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(exp(Integral(Mul(Symbol('\\\\nabla', commutative=True), Function('m')(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Integral(Add(Mul(Rational(1, 3), Pow(Symbol('\\\\nabla', commutative=True), Integer(3))), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\hat{p},E_{x})} = E_{x} + \\hat{p}, then obtain - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{\\mathbf{D}^{3}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{4}} = - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{\\mathbf{D}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{2}}", "derivation": "\\mathbf{D}{(\\hat{p},E_{x})} = E_{x} + \\hat{p} and \\frac{\\mathbf{D}{(\\hat{p},E_{x})}}{E_{x} + \\hat{p}} = 1 and \\frac{\\mathbf{D}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{2}} = \\frac{1}{E_{x} + \\hat{p}} and - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{\\mathbf{D}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{2}} = - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{1}{E_{x} + \\hat{p}} and - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{\\mathbf{D}^{3}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{4}} = - \\mathbf{D}{(\\hat{p},E_{x})} + \\frac{\\mathbf{D}{(\\hat{p},E_{x})}}{(E_{x} + \\hat{p})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Integer(1))"], [["times", 2, "Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-2)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-1)))"], [["minus", 3, "Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-2)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-4)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)), Integer(3)))), Add(Mul(Integer(-1), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Add(Symbol('E_x', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Integer(-2)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{p}', commutative=True), Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given z{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\mathbf{E}{(y^{\\prime})} = \\cos{(\\sin{(y^{\\prime})})}, then obtain \\mathbf{E}{(y^{\\prime})} + 2 \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} = 2 \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} + \\cos{(\\sin{(y^{\\prime})})}", "derivation": "z{(\\Psi_{\\lambda})} = e^{\\Psi_{\\lambda}} and \\mathbf{E}{(y^{\\prime})} = \\cos{(\\sin{(y^{\\prime})})} and \\mathbf{E}{(y^{\\prime})} + \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} = \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} + \\cos{(\\sin{(y^{\\prime})})} and \\mathbf{E}{(y^{\\prime})} + \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} + \\log{(e^{\\Psi_{\\lambda}})}^{\\Psi_{\\lambda}} = \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} + \\log{(e^{\\Psi_{\\lambda}})}^{\\Psi_{\\lambda}} + \\cos{(\\sin{(y^{\\prime})})} and \\mathbf{E}{(y^{\\prime})} + 2 \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} = 2 \\log{(z{(\\Psi_{\\lambda})})}^{\\Psi_{\\lambda}} + \\cos{(\\sin{(y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('y^{\\\\prime}', commutative=True)), cos(sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 2, "Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 3, "Pow(log(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('y^{\\\\prime}', commutative=True)), Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(log(exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(2), Pow(log(Function('z')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), cos(sin(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\phi_2)} = \\sin{(\\phi_2)}, then obtain \\frac{d}{d \\phi_2} (\\frac{\\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}}) = \\frac{d}{d \\phi_2} (-1 + \\frac{2 \\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}})", "derivation": "\\operatorname{P_{e}}{(\\phi_2)} = \\sin{(\\phi_2)} and 1 = \\frac{\\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} and 1 + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}} = \\frac{\\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}} and \\frac{d}{d \\phi_2} (1 + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}}) = \\frac{d}{d \\phi_2} (\\frac{\\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}}) and \\frac{d}{d \\phi_2} (\\frac{\\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}}) = \\frac{d}{d \\phi_2} (-1 + \\frac{2 \\sin{(\\phi_2)}}{\\operatorname{P_{e}}{(\\phi_2)}} + \\frac{1}{\\operatorname{P_{e}}{(\\phi_2)}})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True)))"], [["divide", 1, "Function('P_e')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))))"], [["add", 2, "Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Add(Mul(Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Add(Integer(1), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Mul(Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Integer(-1), Mul(Integer(2), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), sin(Symbol('\\\\phi_2', commutative=True))), Pow(Function('P_e')(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(Z)} = \\sin{(Z)}, then obtain - Z + \\frac{d}{d Z} b^{Z}{(Z)} = - Z + \\frac{d}{d Z} \\sin^{Z}{(Z)}", "derivation": "b{(Z)} = \\sin{(Z)} and b^{Z}{(Z)} = \\sin^{Z}{(Z)} and \\frac{d}{d Z} b^{Z}{(Z)} = \\frac{d}{d Z} \\sin^{Z}{(Z)} and - Z + \\frac{d}{d Z} b^{Z}{(Z)} = - Z + \\frac{d}{d Z} \\sin^{Z}{(Z)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Symbol('Z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Derivative(Pow(Function('b')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Derivative(Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(C_{1})} = \\cos{(C_{1})}, then derive \\frac{d}{d C_{1}} s{(C_{1})} = - \\sin{(C_{1})}, then derive \\omega + \\cos{(C_{1})} = \\int - \\sin{(C_{1})} dC_{1}, then obtain (\\omega + s{(C_{1})}) \\int - \\sin{(C_{1})} dC_{1} = (\\int - \\sin{(C_{1})} dC_{1})^{2}", "derivation": "s{(C_{1})} = \\cos{(C_{1})} and \\frac{d}{d C_{1}} s{(C_{1})} = \\frac{d}{d C_{1}} \\cos{(C_{1})} and \\frac{d}{d C_{1}} s{(C_{1})} = - \\sin{(C_{1})} and \\frac{d}{d C_{1}} \\cos{(C_{1})} = - \\sin{(C_{1})} and \\int \\frac{d}{d C_{1}} \\cos{(C_{1})} dC_{1} = \\int - \\sin{(C_{1})} dC_{1} and \\omega + \\cos{(C_{1})} = \\int - \\sin{(C_{1})} dC_{1} and \\omega + s{(C_{1})} = \\int - \\sin{(C_{1})} dC_{1} and (\\omega + s{(C_{1})}) \\int - \\sin{(C_{1})} dC_{1} = (\\int - \\sin{(C_{1})} dC_{1})^{2}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('s')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_1', commutative=True))))"], [["integrate", 4, "Symbol('C_1', commutative=True)"], "Equality(Integral(Derivative(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\omega', commutative=True), Function('s')(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["times", 7, "Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Function('s')(Symbol('C_1', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))), Pow(Integral(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\theta_{2}{(U)} = \\cos{(U)} and \\mathbf{M}{(U)} = - U + \\theta_{2}{(U)}, then obtain e^{\\int (- U + \\theta_{2}{(U)})^{U} dU} = e^{\\int (- U + \\cos{(U)})^{U} dU}", "derivation": "\\theta_{2}{(U)} = \\cos{(U)} and - U + \\theta_{2}{(U)} = - U + \\cos{(U)} and \\mathbf{M}{(U)} = - U + \\theta_{2}{(U)} and (- U + \\theta_{2}{(U)})^{U} = (- U + \\cos{(U)})^{U} and \\mathbf{M}^{U}{(U)} = (- U + \\cos{(U)})^{U} and \\int \\mathbf{M}^{U}{(U)} dU = \\int (- U + \\cos{(U)})^{U} dU and \\int (- U + \\theta_{2}{(U)})^{U} dU = \\int (- U + \\cos{(U)})^{U} dU and e^{\\int (- U + \\theta_{2}{(U)})^{U} dU} = e^{\\int (- U + \\cos{(U)})^{U} dU}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\theta_2')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\theta_2')(Symbol('U', commutative=True))))"], [["power", 2, "Symbol('U', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\theta_2')(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\theta_2')(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["exp", 7], "Equality(exp(Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\theta_2')(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), exp(Integral(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), cos(Symbol('U', commutative=True))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{H},\\varphi)} = \\mathbf{H} + \\varphi, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} (\\mathbf{H} \\varphi + \\omega + \\frac{\\varphi^{2}}{2}) = \\frac{\\partial}{\\partial \\mathbf{H}} (\\frac{\\varphi^{2}}{2} + i + \\int \\mathbf{H} d\\varphi)", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{H},\\varphi)} = \\mathbf{H} + \\varphi and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{H},\\varphi)} d\\varphi = \\int (\\mathbf{H} + \\varphi) d\\varphi and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{H},\\varphi)} d\\varphi = \\int \\mathbf{H} d\\varphi + \\int \\varphi d\\varphi and \\int (\\mathbf{H} + \\varphi) d\\varphi = \\int \\mathbf{H} d\\varphi + \\int \\varphi d\\varphi and \\frac{\\partial}{\\partial \\mathbf{H}} \\int (\\mathbf{H} + \\varphi) d\\varphi = \\frac{\\partial}{\\partial \\mathbf{H}} (\\int \\mathbf{H} d\\varphi + \\int \\varphi d\\varphi) and \\frac{\\partial}{\\partial \\mathbf{H}} (\\mathbf{H} \\varphi + \\omega + \\frac{\\varphi^{2}}{2}) = \\frac{\\partial}{\\partial \\mathbf{H}} (\\frac{\\varphi^{2}}{2} + i + \\int \\mathbf{H} d\\varphi)", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Symbol('\\\\varphi', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2))), Symbol('i', commutative=True), Integral(Symbol('\\\\mathbf{H}', commutative=True), Tuple(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\mathbf{A})} = \\log{(\\cos{(\\mathbf{A})})}, then obtain \\frac{1}{\\operatorname{P_{g}}{(\\mathbf{A})}} = \\frac{1}{\\log{(\\cos{(\\mathbf{A})})}}", "derivation": "\\operatorname{P_{g}}{(\\mathbf{A})} = \\log{(\\cos{(\\mathbf{A})})} and \\log{(\\cos{(\\mathbf{A})})} = - \\operatorname{P_{g}}{(\\mathbf{A})} + 2 \\log{(\\cos{(\\mathbf{A})})} and \\operatorname{P_{g}}{(\\mathbf{A})} = - \\operatorname{P_{g}}{(\\mathbf{A})} + 2 \\log{(\\cos{(\\mathbf{A})})} and \\frac{\\log{(\\cos{(\\mathbf{A})})}}{\\operatorname{P_{g}}{(\\mathbf{A})}} = \\frac{- \\operatorname{P_{g}}{(\\mathbf{A})} + 2 \\log{(\\cos{(\\mathbf{A})})}}{\\operatorname{P_{g}}{(\\mathbf{A})}} and \\frac{\\log{(\\cos{(\\mathbf{A})})}}{\\operatorname{P_{g}}{(\\mathbf{A})}} = 1 and \\frac{1}{\\operatorname{P_{g}}{(\\mathbf{A})}} = \\frac{1}{\\log{(\\cos{(\\mathbf{A})})}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), log(cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 1, "Add(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), log(cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], "Equality(log(cos(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\mathbf{A}', commutative=True))))))"], [["divide", 2, "Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Pow(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), log(cos(Symbol('\\\\mathbf{A}', commutative=True))))), Pow(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), log(cos(Symbol('\\\\mathbf{A}', commutative=True)))), Integer(1))"], [["divide", 5, "log(cos(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Pow(Function('P_g')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Pow(log(cos(Symbol('\\\\mathbf{A}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbb{I}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*}, then derive \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = e^{\\varphi^*}, then derive - e^{\\varphi^*} + \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = 0, then obtain - \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} + \\frac{d^{3}}{d (\\varphi^*)^{3}} \\mathbb{I}{(\\varphi^*)} = 0", "derivation": "\\mathbb{I}{(\\varphi^*)} = \\frac{d}{d \\varphi^*} e^{\\varphi^*} and \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = \\frac{d^{2}}{d (\\varphi^*)^{2}} e^{\\varphi^*} and \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = e^{\\varphi^*} and - e^{\\varphi^*} + \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = - e^{\\varphi^*} + \\frac{d^{2}}{d (\\varphi^*)^{2}} e^{\\varphi^*} and - e^{\\varphi^*} + \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} = 0 and - e^{\\varphi^*} + \\frac{d^{2}}{d (\\varphi^*)^{2}} e^{\\varphi^*} = 0 and - \\frac{d}{d \\varphi^*} \\mathbb{I}{(\\varphi^*)} + \\frac{d^{3}}{d (\\varphi^*)^{3}} \\mathbb{I}{(\\varphi^*)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 2, "exp(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Add(Mul(Integer(-1), exp(Symbol('\\\\varphi^*', commutative=True))), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\varphi^*', commutative=True))), Derivative(exp(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(2)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Derivative(Function('\\\\mathbb{I}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(3)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{H}{(\\lambda)} = \\sin{(\\lambda)} and \\operatorname{r_{0}}{(\\lambda)} = \\mathbf{H}^{\\lambda}{(\\lambda)}, then obtain \\operatorname{r_{0}}{(\\lambda)} + \\frac{d}{d \\lambda} \\mathbf{H}^{\\lambda}{(\\lambda)} = \\operatorname{r_{0}}{(\\lambda)} + \\frac{d}{d \\lambda} \\operatorname{r_{0}}{(\\lambda)}", "derivation": "\\mathbf{H}{(\\lambda)} = \\sin{(\\lambda)} and \\mathbf{H}^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} and \\operatorname{r_{0}}{(\\lambda)} = \\mathbf{H}^{\\lambda}{(\\lambda)} and \\operatorname{r_{0}}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} and \\frac{d}{d \\lambda} \\mathbf{H}^{\\lambda}{(\\lambda)} = \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} and \\sin^{\\lambda}{(\\lambda)} + \\frac{d}{d \\lambda} \\mathbf{H}^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} + \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} and \\operatorname{r_{0}}{(\\lambda)} + \\frac{d}{d \\lambda} \\mathbf{H}^{\\lambda}{(\\lambda)} = \\operatorname{r_{0}}{(\\lambda)} + \\frac{d}{d \\lambda} \\operatorname{r_{0}}{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('r_0')(Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["add", 5, "Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('r_0')(Symbol('\\\\lambda', commutative=True)), Derivative(Pow(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Add(Function('r_0')(Symbol('\\\\lambda', commutative=True)), Derivative(Function('r_0')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(t_{1},\\omega)} = - \\omega + \\cos{(t_{1})}, then obtain - E_{n} + \\int (u{(t_{1},\\omega)} + 1) d\\omega = - E_{n} + \\int (- \\omega + \\cos{(t_{1})} + 1) d\\omega", "derivation": "u{(t_{1},\\omega)} = - \\omega + \\cos{(t_{1})} and u{(t_{1},\\omega)} + 1 = - \\omega + \\cos{(t_{1})} + 1 and \\int (u{(t_{1},\\omega)} + 1) d\\omega = \\int (- \\omega + \\cos{(t_{1})} + 1) d\\omega and - E_{n} + \\int (u{(t_{1},\\omega)} + 1) d\\omega = - E_{n} + \\int (- \\omega + \\cos{(t_{1})} + 1) d\\omega", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('t_1', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), cos(Symbol('t_1', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('u')(Symbol('t_1', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), cos(Symbol('t_1', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Add(Function('u')(Symbol('t_1', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), cos(Symbol('t_1', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Add(Function('u')(Symbol('t_1', commutative=True), Symbol('\\\\omega', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), cos(Symbol('t_1', commutative=True)), Integer(1)), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(F_{N},t)} = F_{N} - t, then derive \\frac{\\int \\operatorname{A_{1}}{(F_{N},t)} dt}{F_{N}} = \\frac{F_{N} t + P_{e} - \\frac{t^{2}}{2}}{F_{N}}, then obtain \\frac{F_{N} t + \\tilde{g}^* - \\frac{t^{2}}{2}}{F_{N}} = \\frac{F_{N} t + P_{e} - \\frac{t^{2}}{2}}{F_{N}}", "derivation": "\\operatorname{A_{1}}{(F_{N},t)} = F_{N} - t and \\int \\operatorname{A_{1}}{(F_{N},t)} dt = \\int (F_{N} - t) dt and \\frac{\\int \\operatorname{A_{1}}{(F_{N},t)} dt}{F_{N}} = \\frac{\\int (F_{N} - t) dt}{F_{N}} and \\frac{\\int \\operatorname{A_{1}}{(F_{N},t)} dt}{F_{N}} = \\frac{F_{N} t + P_{e} - \\frac{t^{2}}{2}}{F_{N}} and \\frac{\\int (F_{N} - t) dt}{F_{N}} = \\frac{F_{N} t + P_{e} - \\frac{t^{2}}{2}}{F_{N}} and \\frac{F_{N} t + \\tilde{g}^* - \\frac{t^{2}}{2}}{F_{N}} = \\frac{F_{N} t + P_{e} - \\frac{t^{2}}{2}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('A_1')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["divide", 2, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Function('A_1')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Function('A_1')(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('P_e', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('P_e', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"], [["evaluate_integrals", 5], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), Symbol('t', commutative=True)), Symbol('P_e', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))))))"]]}, {"prompt": "Given \\varphi^{*}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\delta{(P_{g})} = \\cos{(P_{g})}, then obtain 2 \\delta{(P_{g})} \\cos{(P_{g})} \\cos{(\\tilde{g})} = 2 \\cos^{2}{(P_{g})} \\cos{(\\tilde{g})}", "derivation": "\\varphi^{*}{(\\tilde{g})} = \\cos{(\\tilde{g})} and \\delta{(P_{g})} = \\cos{(P_{g})} and \\delta{(P_{g})} \\cos{(P_{g})} = \\cos^{2}{(P_{g})} and 2 \\delta{(P_{g})} \\varphi^{*}{(\\tilde{g})} \\cos{(P_{g})} = 2 \\varphi^{*}{(\\tilde{g})} \\cos^{2}{(P_{g})} and 2 \\delta{(P_{g})} \\cos{(P_{g})} \\cos{(\\tilde{g})} = 2 \\cos^{2}{(P_{g})} \\cos{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True)))"], [["times", 2, "cos(Symbol('P_g', commutative=True))"], "Equality(Mul(Function('\\\\delta')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True))), Pow(cos(Symbol('P_g', commutative=True)), Integer(2)))"], [["times", 3, "Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('P_g', commutative=True)), Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True)), cos(Symbol('P_g', commutative=True))), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True)), Pow(cos(Symbol('P_g', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Function('\\\\delta')(Symbol('P_g', commutative=True)), cos(Symbol('P_g', commutative=True)), cos(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), Pow(cos(Symbol('P_g', commutative=True)), Integer(2)), cos(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\hbar)} = e^{\\hbar}, then derive \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = J + e^{\\hbar}, then derive \\frac{d}{d \\hbar} \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = e^{\\hbar}, then derive \\frac{\\partial}{\\partial \\hbar} (t_{1} + e^{\\hbar}) = e^{\\hbar}, then obtain \\frac{d}{d \\hbar} \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = \\frac{\\partial}{\\partial \\hbar} (t_{1} + e^{\\hbar})", "derivation": "\\operatorname{v_{1}}{(\\hbar)} = e^{\\hbar} and \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = \\int e^{\\hbar} d\\hbar and \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = J + e^{\\hbar} and \\frac{d}{d \\hbar} \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = \\frac{\\partial}{\\partial \\hbar} (J + e^{\\hbar}) and \\frac{d}{d \\hbar} \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = e^{\\hbar} and \\frac{d}{d \\hbar} \\int e^{\\hbar} d\\hbar = e^{\\hbar} and \\frac{\\partial}{\\partial \\hbar} (t_{1} + e^{\\hbar}) = e^{\\hbar} and \\frac{d}{d \\hbar} \\int e^{\\hbar} d\\hbar = \\frac{\\partial}{\\partial \\hbar} (t_{1} + e^{\\hbar}) and \\frac{d}{d \\hbar} \\int \\operatorname{v_{1}}{(\\hbar)} d\\hbar = \\frac{\\partial}{\\partial \\hbar} (t_{1} + e^{\\hbar})", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('J', commutative=True), exp(Symbol('\\\\hbar', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Integral(Function('v_1')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('v_1')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), exp(Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), exp(Symbol('\\\\hbar', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Derivative(Add(Symbol('t_1', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), exp(Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Integral(exp(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 8, 2], "Equality(Derivative(Integral(Function('v_1')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), exp(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(s,\\Omega)} = \\Omega s, then derive \\frac{\\partial}{\\partial s} \\operatorname{P_{g}}{(s,\\Omega)} = \\Omega, then obtain \\Omega s + \\frac{\\partial}{\\partial s} \\Omega s = \\Omega s + \\Omega", "derivation": "\\operatorname{P_{g}}{(s,\\Omega)} = \\Omega s and \\frac{\\partial}{\\partial s} \\operatorname{P_{g}}{(s,\\Omega)} = \\frac{\\partial}{\\partial s} \\Omega s and \\frac{\\partial}{\\partial s} \\operatorname{P_{g}}{(s,\\Omega)} = \\Omega and \\Omega s + \\frac{\\partial}{\\partial s} \\operatorname{P_{g}}{(s,\\Omega)} = \\Omega s + \\Omega and \\Omega s + \\frac{\\partial}{\\partial s} \\Omega s = \\Omega s + \\Omega", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))"], [["add", 3, "Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Derivative(Function('P_g')(Symbol('s', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('s', commutative=True)), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})}, then obtain \\int 2 \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int 2 \\log{(\\Psi_{\\lambda})}^{2} d\\Psi_{\\lambda}", "derivation": "\\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})} = 2 \\log{(\\Psi_{\\lambda})} and (\\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}) \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} = (\\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}) \\log{(\\Psi_{\\lambda})} and 2 \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} = 2 \\log{(\\Psi_{\\lambda})}^{2} and \\int 2 \\hat{\\mathbf{x}}{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int 2 \\log{(\\Psi_{\\lambda})}^{2} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 1, "log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 1, "Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), Pow(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Mul(Integer(2), Pow(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(G,\\mu_0)} = \\sin{(\\frac{G}{\\mu_0})}, then obtain \\iint \\tilde{g}^*{(G,\\mu_0)} dG d\\mu_0 + 1 = \\iint \\sin{(\\frac{G}{\\mu_0})} dG d\\mu_0 + 1", "derivation": "\\tilde{g}^*{(G,\\mu_0)} = \\sin{(\\frac{G}{\\mu_0})} and \\int \\tilde{g}^*{(G,\\mu_0)} dG = \\int \\sin{(\\frac{G}{\\mu_0})} dG and \\iint \\tilde{g}^*{(G,\\mu_0)} dG d\\mu_0 = \\iint \\sin{(\\frac{G}{\\mu_0})} dG d\\mu_0 and \\iint \\tilde{g}^*{(G,\\mu_0)} dG d\\mu_0 + 1 = \\iint \\sin{(\\frac{G}{\\mu_0})} dG d\\mu_0 + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mu_0', commutative=True)), sin(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(sin(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))), Tuple(Symbol('G', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(sin(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\tilde{g}^*')(Symbol('G', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)), Add(Integral(sin(Mul(Symbol('G', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)))), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\delta)} = \\delta, then obtain \\frac{d}{d \\delta} (- \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta) = \\frac{d}{d \\delta} (- \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int 2 \\delta d\\delta)", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\delta)} = \\delta and \\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)} = 2 \\delta and \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta = \\int 2 \\delta d\\delta and - \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta = - \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int 2 \\delta d\\delta and - 2 \\delta + \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta = - 2 \\delta + \\int 2 \\delta d\\delta and \\frac{d}{d \\delta} (- 2 \\delta + \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta) = \\frac{d}{d \\delta} (- 2 \\delta + \\int 2 \\delta d\\delta) and \\frac{d}{d \\delta} (- \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int (\\delta + \\operatorname{f_{\\mathbf{p}}}{(\\delta)}) d\\delta) = \\frac{d}{d \\delta} (- \\delta - \\operatorname{f_{\\mathbf{p}}}{(\\delta)} + \\int 2 \\delta d\\delta)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Integral(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Integral(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Integral(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Integral(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Integral(Add(Symbol('\\\\delta', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(2), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(x)} = \\cos{(\\sin{(x)})}, then derive \\frac{d}{d x} \\hat{p}{(x)} = - \\sin{(\\sin{(x)})} \\cos{(x)}, then obtain (\\frac{d}{d x} \\cos{(\\sin{(x)})})^{2} = - \\sin{(\\sin{(x)})} \\cos{(x)} \\frac{d}{d x} \\cos{(\\sin{(x)})}", "derivation": "\\hat{p}{(x)} = \\cos{(\\sin{(x)})} and \\frac{d}{d x} \\hat{p}{(x)} = \\frac{d}{d x} \\cos{(\\sin{(x)})} and \\frac{d}{d x} \\hat{p}{(x)} = - \\sin{(\\sin{(x)})} \\cos{(x)} and \\frac{d}{d x} \\hat{p}{(x)} \\frac{d}{d x} \\cos{(\\sin{(x)})} = - \\sin{(\\sin{(x)})} \\cos{(x)} \\frac{d}{d x} \\cos{(\\sin{(x)})} and (\\frac{d}{d x} \\cos{(\\sin{(x)})})^{2} = - \\sin{(\\sin{(x)})} \\cos{(x)} \\frac{d}{d x} \\cos{(\\sin{(x)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('x', commutative=True)), cos(sin(Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), sin(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True))))"], [["times", 3, "Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\hat{p}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Integer(-1), sin(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True)), Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(sin(Symbol('x', commutative=True))), cos(Symbol('x', commutative=True)), Derivative(cos(sin(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(E_{\\lambda},z)} = - z + \\sin{(E_{\\lambda})}, then derive \\frac{\\partial}{\\partial z} a{(E_{\\lambda},z)} = -1, then obtain \\frac{\\partial^{- \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})}}{\\partial z^{- \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})}} a{(E_{\\lambda},z)} = \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})", "derivation": "a{(E_{\\lambda},z)} = - z + \\sin{(E_{\\lambda})} and \\frac{\\partial}{\\partial z} a{(E_{\\lambda},z)} = \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})}) and \\frac{\\partial}{\\partial z} a{(E_{\\lambda},z)} = -1 and \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})}) = -1 and \\frac{\\partial^{- \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})}}{\\partial z^{- \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})}} a{(E_{\\lambda},z)} = \\frac{\\partial}{\\partial z} (- z + \\sin{(E_{\\lambda})})", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(-1))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('a')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1)))))), Derivative(Add(Mul(Integer(-1), Symbol('z', commutative=True)), sin(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{p}{(F_{H})} = \\sin{(F_{H})} and \\operatorname{t_{2}}{(F_{H})} = \\sin{(F_{H})}, then obtain \\frac{1}{\\sin{(F_{H})}} = \\frac{1}{\\mathbf{p}{(F_{H})}}", "derivation": "\\mathbf{p}{(F_{H})} = \\sin{(F_{H})} and \\operatorname{t_{2}}{(F_{H})} = \\sin{(F_{H})} and \\frac{\\mathbf{p}{(F_{H})}}{\\operatorname{t_{2}}{(F_{H})}} = \\frac{\\sin{(F_{H})}}{\\operatorname{t_{2}}{(F_{H})}} and \\frac{1}{\\operatorname{t_{2}}{(F_{H})}} = \\frac{\\sin{(F_{H})}}{\\mathbf{p}{(F_{H})} \\operatorname{t_{2}}{(F_{H})}} and \\frac{1}{\\sin{(F_{H})}} = \\frac{1}{\\mathbf{p}{(F_{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('F_H', commutative=True)), sin(Symbol('F_H', commutative=True)))"], [["divide", 1, "Function('t_2')(Symbol('F_H', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{p}')(Symbol('F_H', commutative=True)), Pow(Function('t_2')(Symbol('F_H', commutative=True)), Integer(-1))), Mul(Pow(Function('t_2')(Symbol('F_H', commutative=True)), Integer(-1)), sin(Symbol('F_H', commutative=True))))"], [["divide", 3, "Function('\\\\mathbf{p}')(Symbol('F_H', commutative=True))"], "Equality(Pow(Function('t_2')(Symbol('F_H', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('t_2')(Symbol('F_H', commutative=True)), Integer(-1)), sin(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(sin(Symbol('F_H', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('F_H', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\mathbf{v})} = \\cos{(\\mathbf{v})}, then obtain \\cos{(\\sigma_x)} + 2 = \\cos{(\\sigma_x)} + 1 + \\frac{\\cos{(\\mathbf{v})}}{\\operatorname{E_{x}}{(\\mathbf{v})}}", "derivation": "\\operatorname{E_{x}}{(\\mathbf{v})} = \\cos{(\\mathbf{v})} and 1 = \\frac{\\cos{(\\mathbf{v})}}{\\operatorname{E_{x}}{(\\mathbf{v})}} and \\cos{(\\sigma_x)} + 1 = \\cos{(\\sigma_x)} + \\frac{\\cos{(\\mathbf{v})}}{\\operatorname{E_{x}}{(\\mathbf{v})}} and \\cos{(\\sigma_x)} + 2 = \\cos{(\\sigma_x)} + 1 + \\frac{\\cos{(\\mathbf{v})}}{\\operatorname{E_{x}}{(\\mathbf{v})}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\mathbf{v}', commutative=True)), cos(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "Function('E_x')(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('E_x')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["add", 2, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(1)), Add(cos(Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Function('E_x')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(2)), Add(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(1), Mul(Pow(Function('E_x')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(J)} = e^{J}, then derive - f + \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - e^{J} = - f - e^{J} + (e^{J})^{2 J}, then obtain - \\frac{(- f + \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - e^{J}) e^{- J}}{- f - e^{J} + (e^{J})^{2 J}} = - e^{- J}", "derivation": "\\varepsilon_{0}{(J)} = e^{J} and \\varepsilon_{0}^{J}{(J)} = (e^{J})^{J} and \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} = (e^{J})^{2 J} and \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - \\int e^{J} dJ = (e^{J})^{2 J} - \\int e^{J} dJ and - f + \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - e^{J} = - f - e^{J} + (e^{J})^{2 J} and - (- f + \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - e^{J}) e^{- J} = - (- f - e^{J} + (e^{J})^{2 J}) e^{- J} and - \\frac{(- f + \\varepsilon_{0}^{J}{(J)} (e^{J})^{J} - e^{J}) e^{- J}}{- f - e^{J} + (e^{J})^{2 J}} = - e^{- J}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))))"], [["minus", 3, "Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))), Add(Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))), Mul(Integer(-1), Integral(exp(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), exp(Symbol('J', commutative=True)))), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True)))))"], [["divide", 5, "Mul(Integer(-1), exp(Symbol('J', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), exp(Symbol('J', commutative=True)))), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True)))), exp(Mul(Integer(-1), Symbol('J', commutative=True)))))"], [["divide", 6, "Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True))), Mul(Integer(-1), exp(Symbol('J', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), exp(Symbol('J', commutative=True))), Pow(exp(Symbol('J', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True)))), Integer(-1)), exp(Mul(Integer(-1), Symbol('J', commutative=True)))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then derive \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\frac{1}{\\hat{H}_{\\lambda}}, then obtain \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} = \\frac{1}{\\hat{H}_{\\lambda}}", "derivation": "\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} and \\frac{d}{d \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\frac{1}{\\hat{H}_{\\lambda}} and \\frac{d}{d \\hat{H}_{\\lambda}} \\log{(\\hat{H}_{\\lambda})} = \\frac{1}{\\hat{H}_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\sigma_{p}{(\\Psi_{nl},\\nabla)} = \\Psi_{nl} + \\nabla, then derive \\frac{\\partial}{\\partial \\Psi_{nl}} \\sigma_{p}{(\\Psi_{nl},\\nabla)} = 1, then obtain e^{\\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi_{nl} + \\nabla)} = e", "derivation": "\\sigma_{p}{(\\Psi_{nl},\\nabla)} = \\Psi_{nl} + \\nabla and \\frac{\\partial}{\\partial \\Psi_{nl}} \\sigma_{p}{(\\Psi_{nl},\\nabla)} = \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi_{nl} + \\nabla) and \\frac{\\partial}{\\partial \\Psi_{nl}} \\sigma_{p}{(\\Psi_{nl},\\nabla)} = 1 and \\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi_{nl} + \\nabla) = 1 and e^{\\frac{\\partial}{\\partial \\Psi_{nl}} (\\Psi_{nl} + \\nabla)} = e", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(1))"], [["exp", 4], "Equality(exp(Derivative(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1)))), E)"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\chi,S)} = \\frac{S}{\\chi}, then obtain \\frac{S}{\\chi} + \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\operatorname{m_{s}}{(\\chi,S)} dS) = \\frac{S}{\\chi} + \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\frac{S}{\\chi} dS)", "derivation": "\\operatorname{m_{s}}{(\\chi,S)} = \\frac{S}{\\chi} and \\int \\operatorname{m_{s}}{(\\chi,S)} dS = \\int \\frac{S}{\\chi} dS and - \\operatorname{m_{s}}{(\\chi,S)} + \\int \\operatorname{m_{s}}{(\\chi,S)} dS = - \\operatorname{m_{s}}{(\\chi,S)} + \\int \\frac{S}{\\chi} dS and \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\operatorname{m_{s}}{(\\chi,S)} dS) = \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\frac{S}{\\chi} dS) and \\frac{S}{\\chi} + \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\operatorname{m_{s}}{(\\chi,S)} dS) = \\frac{S}{\\chi} + \\frac{\\partial}{\\partial \\chi} (- \\operatorname{m_{s}}{(\\chi,S)} + \\int \\frac{S}{\\chi} dS)", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True))))"], [["minus", 2, "Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["add", 4, "Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Derivative(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\chi', commutative=True), Symbol('S', commutative=True))), Integral(Mul(Symbol('S', commutative=True), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Tuple(Symbol('S', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given H{(\\pi,M_{E})} = M_{E}^{\\pi}, then obtain \\int \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} H{(\\pi,M_{E})})} d\\pi = \\int \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} M_{E}^{\\pi})} d\\pi", "derivation": "H{(\\pi,M_{E})} = M_{E}^{\\pi} and \\frac{\\partial}{\\partial \\pi} H{(\\pi,M_{E})} = \\frac{\\partial}{\\partial \\pi} M_{E}^{\\pi} and \\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} H{(\\pi,M_{E})} = \\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} M_{E}^{\\pi} and \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} H{(\\pi,M_{E})})} = \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} M_{E}^{\\pi})} and \\int \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} H{(\\pi,M_{E})})} d\\pi = \\int \\cos{(\\frac{\\partial^{2}}{\\partial M_{E}\\partial \\pi} M_{E}^{\\pi})} d\\pi", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Symbol('M_E', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Pow(Symbol('M_E', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), cos(Derivative(Pow(Symbol('M_E', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(cos(Derivative(Function('H')(Symbol('\\\\pi', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(cos(Derivative(Pow(Symbol('M_E', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given v{(\\varepsilon_0,T)} = - \\varepsilon_0 + \\sin{(T)}, then derive \\varepsilon_0 + \\int (v{(\\varepsilon_0,T)} + 1) dT = - T \\varepsilon_0 + T + \\varepsilon_0 + g^{\\prime}_{\\varepsilon} - \\cos{(T)}, then obtain \\hat{X} + \\varepsilon_0 + \\int (- \\varepsilon_0 + \\sin{(T)} + 1) dT = - T \\varepsilon_0 + T + \\hat{X} + \\varepsilon_0 + g^{\\prime}_{\\varepsilon} - \\cos{(T)}", "derivation": "v{(\\varepsilon_0,T)} = - \\varepsilon_0 + \\sin{(T)} and v{(\\varepsilon_0,T)} + 1 = - \\varepsilon_0 + \\sin{(T)} + 1 and \\int (v{(\\varepsilon_0,T)} + 1) dT = \\int (- \\varepsilon_0 + \\sin{(T)} + 1) dT and \\varepsilon_0 + \\int (v{(\\varepsilon_0,T)} + 1) dT = \\varepsilon_0 + \\int (- \\varepsilon_0 + \\sin{(T)} + 1) dT and \\varepsilon_0 + \\int (v{(\\varepsilon_0,T)} + 1) dT = - T \\varepsilon_0 + T + \\varepsilon_0 + g^{\\prime}_{\\varepsilon} - \\cos{(T)} and \\varepsilon_0 + \\int (- \\varepsilon_0 + \\sin{(T)} + 1) dT = - T \\varepsilon_0 + T + \\varepsilon_0 + g^{\\prime}_{\\varepsilon} - \\cos{(T)} and \\hat{X} + \\varepsilon_0 + \\int (- \\varepsilon_0 + \\sin{(T)} + 1) dT = - T \\varepsilon_0 + T + \\hat{X} + \\varepsilon_0 + g^{\\prime}_{\\varepsilon} - \\cos{(T)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True))))"], [["add", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True)))), Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Function('v')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('T', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('T', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["add", 6, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)), sin(Symbol('T', commutative=True)), Integer(1)), Tuple(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('T', commutative=True), Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given y{(\\mathbf{J}_P,F_{N})} = F_{N}^{\\mathbf{J}_P}, then obtain F_{N} + \\int (F_{N}^{\\mathbf{J}_P} + y{(\\mathbf{J}_P,F_{N})}) dF_{N} = F_{N} + \\int 2 F_{N}^{\\mathbf{J}_P} dF_{N}", "derivation": "y{(\\mathbf{J}_P,F_{N})} = F_{N}^{\\mathbf{J}_P} and F_{N}^{\\mathbf{J}_P} + y{(\\mathbf{J}_P,F_{N})} = 2 F_{N}^{\\mathbf{J}_P} and \\int (F_{N}^{\\mathbf{J}_P} + y{(\\mathbf{J}_P,F_{N})}) dF_{N} = \\int 2 F_{N}^{\\mathbf{J}_P} dF_{N} and F_{N} + \\int (F_{N}^{\\mathbf{J}_P} + y{(\\mathbf{J}_P,F_{N})}) dF_{N} = F_{N} + \\int 2 F_{N}^{\\mathbf{J}_P} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_N', commutative=True)), Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 1, "Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_N', commutative=True))), Mul(Integer(2), Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["add", 3, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Integral(Add(Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))), Add(Symbol('F_N', commutative=True), Integral(Mul(Integer(2), Pow(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\eta^{\\prime})} = - \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})}, then derive \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\eta^{\\prime})} = \\cos{(f_{\\mathbf{v}})}, then obtain e^{\\frac{\\partial}{\\partial f_{\\mathbf{v}}} (- \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})})} = e^{\\cos{(f_{\\mathbf{v}})}}", "derivation": "\\operatorname{n_{1}}{(f_{\\mathbf{v}},\\eta^{\\prime})} = - \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\eta^{\\prime})} = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (- \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})}) and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\eta^{\\prime})} = \\cos{(f_{\\mathbf{v}})} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (- \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})}) = \\cos{(f_{\\mathbf{v}})} and e^{\\frac{\\partial}{\\partial f_{\\mathbf{v}}} (- \\eta^{\\prime} + \\sin{(f_{\\mathbf{v}})})} = e^{\\cos{(f_{\\mathbf{v}})}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["exp", 4], "Equality(exp(Derivative(Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1)))), exp(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} = \\sin{(e^{\\hat{\\mathbf{r}}})}, then obtain (\\hat{\\mathbf{r}} \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})}) e^{- \\hat{\\mathbf{r}}} = (\\hat{\\mathbf{r}} \\sin{(e^{\\hat{\\mathbf{r}}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})}) e^{- \\hat{\\mathbf{r}}}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} = \\sin{(e^{\\hat{\\mathbf{r}}})} and \\hat{\\mathbf{r}} \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\sin{(e^{\\hat{\\mathbf{r}}})} and \\hat{\\mathbf{r}} \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\sin{(e^{\\hat{\\mathbf{r}}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} and (\\hat{\\mathbf{r}} \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})}) e^{- \\hat{\\mathbf{r}}} = (\\hat{\\mathbf{r}} \\sin{(e^{\\hat{\\mathbf{r}}})} + \\dot{\\mathbf{r}}{(\\hat{\\mathbf{r}})}) e^{- \\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["add", 2, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 3, "exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given H{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}, then obtain - \\mathbf{J}_f + \\int (- \\mathbf{J}_f + H^{\\mathbf{J}_f}{(\\mathbf{J}_f)}) d\\mathbf{J}_f = - \\mathbf{J}_f + \\int (- \\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f}) d\\mathbf{J}_f", "derivation": "H{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and H^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f} and - \\mathbf{J}_f + H^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = - \\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f} and \\int (- \\mathbf{J}_f + H^{\\mathbf{J}_f}{(\\mathbf{J}_f)}) d\\mathbf{J}_f = \\int (- \\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f}) d\\mathbf{J}_f and - \\mathbf{J}_f + \\int (- \\mathbf{J}_f + H^{\\mathbf{J}_f}{(\\mathbf{J}_f)}) d\\mathbf{J}_f = - \\mathbf{J}_f + \\int (- \\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f}) d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(z^{*})} = \\cos{(z^{*})} and n{(z^{*})} = \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}}, then obtain \\frac{d^{2}}{d (z^{*})^{2}} \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = \\frac{d^{2}}{d (z^{*})^{2}} 1", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(z^{*})} = \\cos{(z^{*})} and n{(z^{*})} = \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}} and \\frac{d}{d z^{*}} n{(z^{*})} = \\frac{d}{d z^{*}} \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}} and \\frac{d}{d z^{*}} n{(z^{*})} = \\frac{d}{d z^{*}} 1 and \\frac{d}{d z^{*}} \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = \\frac{d}{d z^{*}} 1 and \\frac{d^{2}}{d (z^{*})^{2}} \\frac{\\cos{(z^{*})}}{\\operatorname{f_{\\mathbf{p}}}{(z^{*})}} = \\frac{d^{2}}{d (z^{*})^{2}} 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('z^*', commutative=True)), Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True))))"], [["differentiate", 2, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('n')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('z^*', commutative=True)), Integer(-1)), cos(Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('z^*', commutative=True), Integer(2))))"]]}, {"prompt": "Given i{(f_{\\mathbf{v}},a^{\\dagger})} = a^{\\dagger} f_{\\mathbf{v}}, then obtain \\frac{\\partial^{2}}{\\partial f_{\\mathbf{v}}\\partial a^{\\dagger}} (- \\frac{a^{\\dagger} f_{\\mathbf{v}}}{i{(f_{\\mathbf{v}},a^{\\dagger})}} + 1) = \\frac{d^{2}}{d f_{\\mathbf{v}}d a^{\\dagger}} 0", "derivation": "i{(f_{\\mathbf{v}},a^{\\dagger})} = a^{\\dagger} f_{\\mathbf{v}} and 1 = \\frac{a^{\\dagger} f_{\\mathbf{v}}}{i{(f_{\\mathbf{v}},a^{\\dagger})}} and - \\frac{a^{\\dagger} f_{\\mathbf{v}}}{i{(f_{\\mathbf{v}},a^{\\dagger})}} + 1 = 0 and \\frac{\\partial}{\\partial a^{\\dagger}} (- \\frac{a^{\\dagger} f_{\\mathbf{v}}}{i{(f_{\\mathbf{v}},a^{\\dagger})}} + 1) = \\frac{d}{d a^{\\dagger}} 0 and \\frac{\\partial^{2}}{\\partial f_{\\mathbf{v}}\\partial a^{\\dagger}} (- \\frac{a^{\\dagger} f_{\\mathbf{v}}}{i{(f_{\\mathbf{v}},a^{\\dagger})}} + 1) = \\frac{d^{2}}{d f_{\\mathbf{v}}d a^{\\dagger}} 0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 1, "Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Integer(1), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))))"], [["minus", 2, "Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(1)), Integer(0))"], [["differentiate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Pow(Function('i')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\psi,J)} = J + e^{\\psi} and \\operatorname{P_{e}}{(v_{z},\\mathbf{s})} = e^{\\mathbf{s} + v_{z}}, then obtain ((\\psi + \\operatorname{x^{{\\}'}}{(\\psi,J)}) e^{\\mathbf{s} + v_{z}})^{\\psi} = ((J + \\psi + e^{\\psi}) e^{\\mathbf{s} + v_{z}})^{\\psi}", "derivation": "\\operatorname{x^{{\\}'}}{(\\psi,J)} = J + e^{\\psi} and \\psi + \\operatorname{x^{{\\}'}}{(\\psi,J)} = J + \\psi + e^{\\psi} and \\operatorname{P_{e}}{(v_{z},\\mathbf{s})} = e^{\\mathbf{s} + v_{z}} and (\\psi + \\operatorname{x^{{\\}'}}{(\\psi,J)}) \\operatorname{P_{e}}{(v_{z},\\mathbf{s})} = (J + \\psi + e^{\\psi}) \\operatorname{P_{e}}{(v_{z},\\mathbf{s})} and (\\psi + \\operatorname{x^{{\\}'}}{(\\psi,J)}) e^{\\mathbf{s} + v_{z}} = (J + \\psi + e^{\\psi}) e^{\\mathbf{s} + v_{z}} and ((\\psi + \\operatorname{x^{{\\}'}}{(\\psi,J)}) e^{\\mathbf{s} + v_{z}})^{\\psi} = ((J + \\psi + e^{\\psi}) e^{\\mathbf{s} + v_{z}})^{\\psi}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), exp(Symbol('\\\\psi', commutative=True))))"], [["add", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Add(Symbol('\\\\psi', commutative=True), Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('J', commutative=True))), Add(Symbol('J', commutative=True), Symbol('\\\\psi', commutative=True), exp(Symbol('\\\\psi', commutative=True))))"], ["get_premise", "Equality(Function('P_e')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), exp(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_z', commutative=True))))"], [["times", 2, "Function('P_e')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\psi', commutative=True), Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('J', commutative=True))), Function('P_e')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Add(Symbol('J', commutative=True), Symbol('\\\\psi', commutative=True), exp(Symbol('\\\\psi', commutative=True))), Function('P_e')(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('\\\\psi', commutative=True), Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('J', commutative=True))), exp(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_z', commutative=True)))), Mul(Add(Symbol('J', commutative=True), Symbol('\\\\psi', commutative=True), exp(Symbol('\\\\psi', commutative=True))), exp(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_z', commutative=True)))))"], [["power", 5, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\psi', commutative=True), Function('x^\\\\prime')(Symbol('\\\\psi', commutative=True), Symbol('J', commutative=True))), exp(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_z', commutative=True)))), Symbol('\\\\psi', commutative=True)), Pow(Mul(Add(Symbol('J', commutative=True), Symbol('\\\\psi', commutative=True), exp(Symbol('\\\\psi', commutative=True))), exp(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_z', commutative=True)))), Symbol('\\\\psi', commutative=True)))"]]}, {"prompt": "Given p{(A_{y})} = \\log{(\\cos{(A_{y})})} and \\mathbb{I}{(A_{y})} = \\cos{(A_{y})}, then obtain \\log{(\\cos{(A_{y})})} = \\log{(\\mathbb{I}{(A_{y})})}", "derivation": "p{(A_{y})} = \\log{(\\cos{(A_{y})})} and \\mathbb{I}{(A_{y})} = \\cos{(A_{y})} and p{(A_{y})} = \\log{(\\mathbb{I}{(A_{y})})} and \\log{(\\cos{(A_{y})})} = \\log{(\\mathbb{I}{(A_{y})})}", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('A_y', commutative=True)), log(cos(Symbol('A_y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('p')(Symbol('A_y', commutative=True)), log(Function('\\\\mathbb{I}')(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(log(cos(Symbol('A_y', commutative=True))), log(Function('\\\\mathbb{I}')(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given I{(C_{d},Z)} = \\frac{e^{C_{d}}}{Z}, then obtain (I{(C_{d},Z)} + 1) \\frac{\\partial}{\\partial Z} (- 2 Z + 2 I{(C_{d},Z)} + 2) = (I{(C_{d},Z)} + 1) \\frac{\\partial}{\\partial Z} (- 2 Z + I{(C_{d},Z)} + 2 + \\frac{e^{C_{d}}}{Z})", "derivation": "I{(C_{d},Z)} = \\frac{e^{C_{d}}}{Z} and I{(C_{d},Z)} + 1 = 1 + \\frac{e^{C_{d}}}{Z} and - Z + I{(C_{d},Z)} + 1 = - Z + 1 + \\frac{e^{C_{d}}}{Z} and - 2 Z + 2 I{(C_{d},Z)} + 2 = - 2 Z + I{(C_{d},Z)} + 2 + \\frac{e^{C_{d}}}{Z} and \\frac{\\partial}{\\partial Z} (- 2 Z + 2 I{(C_{d},Z)} + 2) = \\frac{\\partial}{\\partial Z} (- 2 Z + I{(C_{d},Z)} + 2 + \\frac{e^{C_{d}}}{Z}) and (I{(C_{d},Z)} + 1) \\frac{\\partial}{\\partial Z} (- 2 Z + 2 I{(C_{d},Z)} + 2) = (I{(C_{d},Z)} + 1) \\frac{\\partial}{\\partial Z} (- 2 Z + I{(C_{d},Z)} + 2 + \\frac{e^{C_{d}}}{Z})", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True)))))"], [["minus", 2, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Integer(1), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True))), Integer(2)), Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(2), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True)))))"], [["differentiate", 4, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True))), Integer(2)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(2), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["times", 5, "Add(Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1))"], "Equality(Mul(Add(Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1)), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True))), Integer(2)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Mul(Add(Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(1)), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('Z', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('Z', commutative=True)), Integer(2), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), exp(Symbol('C_d', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(x)} = \\sin{(x)}, then obtain \\int \\frac{\\nabla^{x}{(x)}}{x} dx - \\int \\sin^{x}{(x)} dx = \\int \\frac{\\sin^{x}{(x)}}{x} dx - \\int \\sin^{x}{(x)} dx", "derivation": "\\nabla{(x)} = \\sin{(x)} and \\nabla^{x}{(x)} = \\sin^{x}{(x)} and \\frac{\\nabla^{x}{(x)}}{x} = \\frac{\\sin^{x}{(x)}}{x} and \\int \\nabla^{x}{(x)} dx = \\int \\sin^{x}{(x)} dx and \\int \\frac{\\nabla^{x}{(x)}}{x} dx = \\int \\frac{\\sin^{x}{(x)}}{x} dx and \\int \\frac{\\nabla^{x}{(x)}}{x} dx - \\int \\nabla^{x}{(x)} dx = \\int \\frac{\\sin^{x}{(x)}}{x} dx - \\int \\nabla^{x}{(x)} dx and \\int \\frac{\\nabla^{x}{(x)}}{x} dx - \\int \\sin^{x}{(x)} dx = \\int \\frac{\\sin^{x}{(x)}}{x} dx - \\int \\sin^{x}{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["divide", 2, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["minus", 5, "Integral(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))), Add(Integral(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given c{(a^{\\dagger})} = \\cos{(\\sin{(a^{\\dagger})})}, then derive \\int \\frac{c{(a^{\\dagger})}}{\\cos{(\\sin{(a^{\\dagger})})}} da^{\\dagger} = a^{\\dagger} + f, then obtain \\frac{\\partial^{2}}{\\partial f^{2}} \\frac{\\log{(\\int 1 da^{\\dagger})}}{\\log{(a^{\\dagger} + f)}} = \\frac{d^{2}}{d f^{2}} 1", "derivation": "c{(a^{\\dagger})} = \\cos{(\\sin{(a^{\\dagger})})} and \\frac{c{(a^{\\dagger})}}{\\cos{(\\sin{(a^{\\dagger})})}} = 1 and \\int \\frac{c{(a^{\\dagger})}}{\\cos{(\\sin{(a^{\\dagger})})}} da^{\\dagger} = \\int 1 da^{\\dagger} and \\int \\frac{c{(a^{\\dagger})}}{\\cos{(\\sin{(a^{\\dagger})})}} da^{\\dagger} = a^{\\dagger} + f and \\int 1 da^{\\dagger} = a^{\\dagger} + f and \\log{(\\int 1 da^{\\dagger})} = \\log{(a^{\\dagger} + f)} and \\frac{\\log{(\\int 1 da^{\\dagger})}}{\\log{(a^{\\dagger} + f)}} = 1 and \\frac{\\partial}{\\partial f} \\frac{\\log{(\\int 1 da^{\\dagger})}}{\\log{(a^{\\dagger} + f)}} = \\frac{d}{d f} 1 and \\frac{\\partial^{2}}{\\partial f^{2}} \\frac{\\log{(\\int 1 da^{\\dagger})}}{\\log{(a^{\\dagger} + f)}} = \\frac{d^{2}}{d f^{2}} 1", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), cos(sin(Symbol('a^{\\\\dagger}', commutative=True))))"], [["divide", 1, "cos(sin(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('c')(Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(sin(Symbol('a^{\\\\dagger}', commutative=True))), Integer(-1))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)))"], [["log", 5], "Equality(log(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), log(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))))"], [["divide", 6, "log(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Pow(log(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), log(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Integer(1))"], [["differentiate", 7, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Pow(log(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), log(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 8, "Symbol('f', commutative=True)"], "Equality(Derivative(Mul(Pow(log(Add(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f', commutative=True))), Integer(-1)), log(Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))), Tuple(Symbol('f', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('f', commutative=True), Integer(2))))"]]}, {"prompt": "Given E{(z^{*},\\phi_2)} = \\int (\\phi_2 + z^{*}) d\\phi_2, then derive E{(z^{*},\\phi_2)} = \\mathbf{J}_P + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*}, then derive \\mathbf{J}_P + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} = E_{n} + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*}, then obtain E_{n} + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} = \\int (\\phi_2 + z^{*}) d\\phi_2", "derivation": "E{(z^{*},\\phi_2)} = \\int (\\phi_2 + z^{*}) d\\phi_2 and E{(z^{*},\\phi_2)} = \\mathbf{J}_P + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} and \\mathbf{J}_P + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} = \\int (\\phi_2 + z^{*}) d\\phi_2 and \\mathbf{J}_P + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} = E_{n} + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} and E_{n} + \\frac{\\phi_2^{2}}{2} + \\phi_2 z^{*} = \\int (\\phi_2 + z^{*}) d\\phi_2", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('z^*', commutative=True), Symbol('\\\\phi_2', commutative=True)), Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('E')(Symbol('z^*', commutative=True), Symbol('\\\\phi_2', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True))), Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('E_n', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Mul(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True))), Integral(Add(Symbol('\\\\phi_2', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(E_{n},f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}}}{E_{n}}, then derive \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\ddot{x}{(E_{n},f_{\\mathbf{p}})} = \\frac{1}{E_{n}}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{f_{\\mathbf{p}}}{E_{n}} = \\frac{1}{E_{n}}", "derivation": "\\ddot{x}{(E_{n},f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}}}{E_{n}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\ddot{x}{(E_{n},f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{f_{\\mathbf{p}}}{E_{n}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\ddot{x}{(E_{n},f_{\\mathbf{p}})} = \\frac{1}{E_{n}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\frac{f_{\\mathbf{p}}}{E_{n}} = \\frac{1}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Pow(Symbol('E_n', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Pow(Symbol('E_n', commutative=True), Integer(-1)))"]]}, {"prompt": "Given q{(b,c)} = \\frac{\\partial}{\\partial b} \\frac{b}{c}, then derive \\frac{\\frac{\\partial}{\\partial b} q{(b,c)}}{c} = 0, then obtain \\frac{d}{d c} - \\frac{1}{c} = \\frac{\\partial}{\\partial c} (- \\frac{\\frac{\\partial^{2}}{\\partial b^{2}} \\frac{b}{c}}{c} - \\frac{1}{c})", "derivation": "q{(b,c)} = \\frac{\\partial}{\\partial b} \\frac{b}{c} and \\frac{q{(b,c)}}{c} = \\frac{\\frac{\\partial}{\\partial b} \\frac{b}{c}}{c} and \\frac{\\partial}{\\partial b} \\frac{q{(b,c)}}{c} = \\frac{\\partial}{\\partial b} \\frac{\\frac{\\partial}{\\partial b} \\frac{b}{c}}{c} and \\frac{\\frac{\\partial}{\\partial b} q{(b,c)}}{c} = 0 and \\frac{\\frac{\\partial^{2}}{\\partial b^{2}} \\frac{b}{c}}{c} = 0 and 0 = - \\frac{\\frac{\\partial^{2}}{\\partial b^{2}} \\frac{b}{c}}{c} and - \\frac{1}{c} = - \\frac{\\frac{\\partial^{2}}{\\partial b^{2}} \\frac{b}{c}}{c} - \\frac{1}{c} and \\frac{d}{d c} - \\frac{1}{c} = \\frac{\\partial}{\\partial c} (- \\frac{\\frac{\\partial^{2}}{\\partial b^{2}} \\frac{b}{c}}{c} - \\frac{1}{c})", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 1, "Pow(Symbol('c', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('q')(Symbol('b', commutative=True), Symbol('c', commutative=True))), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Function('q')(Symbol('b', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Function('q')(Symbol('b', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2)))), Integer(0))"], [["minus", 5, "Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2))))"], "Equality(Integer(0), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2)))))"], [["minus", 6, "Pow(Symbol('c', commutative=True), Integer(-1))"], "Equality(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))))"], [["differentiate", 7, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Derivative(Mul(Symbol('b', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))), Tuple(Symbol('b', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(G)} = \\frac{d}{d G} e^{G}, then derive \\int \\operatorname{r_{0}}{(G)} dG = t + e^{G}, then obtain G ((t + e^{G})^{t} + \\int \\frac{d}{d G} e^{G} dG) = G (t + (t + e^{G})^{t} + e^{G})", "derivation": "\\operatorname{r_{0}}{(G)} = \\frac{d}{d G} e^{G} and \\int \\operatorname{r_{0}}{(G)} dG = \\int \\frac{d}{d G} e^{G} dG and \\int \\operatorname{r_{0}}{(G)} dG = t + e^{G} and (\\int \\operatorname{r_{0}}{(G)} dG)^{t} = (t + e^{G})^{t} and \\int \\frac{d}{d G} e^{G} dG = t + e^{G} and (\\int \\operatorname{r_{0}}{(G)} dG)^{t} + \\int \\frac{d}{d G} e^{G} dG = t + e^{G} + (\\int \\operatorname{r_{0}}{(G)} dG)^{t} and (t + e^{G})^{t} + \\int \\frac{d}{d G} e^{G} dG = t + (t + e^{G})^{t} + e^{G} and G ((t + e^{G})^{t} + \\int \\frac{d}{d G} e^{G} dG) = G (t + (t + e^{G})^{t} + e^{G})", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('G', commutative=True)), Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))))"], [["add", 5, "Pow(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('t', commutative=True))"], "Equality(Add(Pow(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('t', commutative=True)), Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True)), Pow(Integral(Function('r_0')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))), Symbol('t', commutative=True)), Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Add(Symbol('t', commutative=True), Pow(Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))), Symbol('t', commutative=True)), exp(Symbol('G', commutative=True))))"], [["times", 7, "Symbol('G', commutative=True)"], "Equality(Mul(Symbol('G', commutative=True), Add(Pow(Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))), Symbol('t', commutative=True)), Integral(Derivative(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))), Mul(Symbol('G', commutative=True), Add(Symbol('t', commutative=True), Pow(Add(Symbol('t', commutative=True), exp(Symbol('G', commutative=True))), Symbol('t', commutative=True)), exp(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)}, then derive \\varepsilon_{0}{(\\theta_2)} = \\frac{1}{\\theta_2}, then obtain \\frac{- \\theta_2 + \\varepsilon_{0}{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} = \\frac{- \\theta_2 + \\frac{1}{\\theta_2}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}}", "derivation": "\\varepsilon_{0}{(\\theta_2)} = \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and \\varepsilon_{0}{(\\theta_2)} = \\frac{1}{\\theta_2} and - \\theta_2 + \\varepsilon_{0}{(\\theta_2)} = - \\theta_2 + \\frac{d}{d \\theta_2} \\log{(\\theta_2)} and \\frac{d}{d \\theta_2} \\log{(\\theta_2)} = \\frac{1}{\\theta_2} and - \\theta_2 + \\varepsilon_{0}{(\\theta_2)} = - \\theta_2 + \\frac{1}{\\theta_2} and \\frac{- \\theta_2 + \\varepsilon_{0}{(\\theta_2)}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}} = \\frac{- \\theta_2 + \\frac{1}{\\theta_2}}{\\frac{d}{d \\theta_2} \\log{(\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))"], [["minus", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["divide", 5, "Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\theta_2', commutative=True))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Pow(Derivative(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\ddot{x})} = \\cos{(\\ddot{x})}, then obtain (- \\ddot{x} + \\cos{(\\ddot{x})})^{\\ddot{x}} \\mathbf{E}{(\\ddot{x})} = (- \\ddot{x} + \\cos{(\\ddot{x})})^{\\ddot{x}} \\cos{(\\ddot{x})}", "derivation": "\\mathbf{E}{(\\ddot{x})} = \\cos{(\\ddot{x})} and - \\ddot{x} + \\mathbf{E}{(\\ddot{x})} = - \\ddot{x} + \\cos{(\\ddot{x})} and (- \\ddot{x} + \\mathbf{E}{(\\ddot{x})})^{\\ddot{x}} \\mathbf{E}{(\\ddot{x})} = (- \\ddot{x} + \\mathbf{E}{(\\ddot{x})})^{\\ddot{x}} \\cos{(\\ddot{x})} and (- \\ddot{x} + \\cos{(\\ddot{x})})^{\\ddot{x}} \\mathbf{E}{(\\ddot{x})} = (- \\ddot{x} + \\cos{(\\ddot{x})})^{\\ddot{x}} \\cos{(\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True)))"], [["minus", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["times", 1, "Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(Q)} = \\sin{(\\cos{(Q)})}, then obtain \\mathbf{J}_P{(Q)} + \\frac{d^{2}}{d Q^{2}} \\sin{(\\cos{(Q)})} = \\mathbf{J}_P{(Q)} + \\frac{d^{2}}{d Q^{2}} (- \\mathbf{J}_P{(Q)} + 2 \\sin{(\\cos{(Q)})})", "derivation": "\\mathbf{J}_P{(Q)} = \\sin{(\\cos{(Q)})} and 0 = - \\mathbf{J}_P{(Q)} + \\sin{(\\cos{(Q)})} and \\sin{(\\cos{(Q)})} = - \\mathbf{J}_P{(Q)} + 2 \\sin{(\\cos{(Q)})} and \\frac{d}{d Q} \\sin{(\\cos{(Q)})} = \\frac{d}{d Q} (- \\mathbf{J}_P{(Q)} + 2 \\sin{(\\cos{(Q)})}) and \\frac{d^{2}}{d Q^{2}} \\sin{(\\cos{(Q)})} = \\frac{d^{2}}{d Q^{2}} (- \\mathbf{J}_P{(Q)} + 2 \\sin{(\\cos{(Q)})}) and \\mathbf{J}_P{(Q)} + \\frac{d^{2}}{d Q^{2}} \\sin{(\\cos{(Q)})} = \\mathbf{J}_P{(Q)} + \\frac{d^{2}}{d Q^{2}} (- \\mathbf{J}_P{(Q)} + 2 \\sin{(\\cos{(Q)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True)), sin(cos(Symbol('Q', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))), sin(cos(Symbol('Q', commutative=True)))))"], [["add", 2, "sin(cos(Symbol('Q', commutative=True)))"], "Equality(sin(cos(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))), Mul(Integer(2), sin(cos(Symbol('Q', commutative=True))))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(sin(cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))), Mul(Integer(2), sin(cos(Symbol('Q', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(sin(cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))), Mul(Integer(2), sin(cos(Symbol('Q', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(2))))"], [["minus", 5, "Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True)), Derivative(sin(cos(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(2)))), Add(Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('Q', commutative=True))), Mul(Integer(2), sin(cos(Symbol('Q', commutative=True))))), Tuple(Symbol('Q', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} = \\mu_0 + \\cos{(f_{\\mathbf{v}})}, then derive \\sin{(f_{\\mathbf{v}})} + \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} + 1 = 1, then obtain \\sin{(f_{\\mathbf{v}})} + \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\mu_0 + \\cos{(f_{\\mathbf{v}})}) + 1 = 1", "derivation": "\\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} = \\mu_0 + \\cos{(f_{\\mathbf{v}})} and \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} - \\cos{(f_{\\mathbf{v}})} = \\mu_0 and f_{\\mathbf{v}} + \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} - \\cos{(f_{\\mathbf{v}})} = \\mu_0 + f_{\\mathbf{v}} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (f_{\\mathbf{v}} + \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} - \\cos{(f_{\\mathbf{v}})}) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\mu_0 + f_{\\mathbf{v}}) and \\sin{(f_{\\mathbf{v}})} + \\frac{\\partial}{\\partial f_{\\mathbf{v}}} \\operatorname{n_{1}}{(f_{\\mathbf{v}},\\mu_0)} + 1 = 1 and \\sin{(f_{\\mathbf{v}})} + \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (\\mu_0 + \\cos{(f_{\\mathbf{v}})}) + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["minus", 1, "cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Symbol('\\\\mu_0', commutative=True))"], [["add", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mu_0', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Function('n_1')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(sin(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Add(Symbol('\\\\mu_0', commutative=True), cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\mathbf{S}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi, then derive \\psi^* + \\mathbf{S}{(\\Psi)} = \\Psi \\log{(\\Psi)} - \\Psi + \\theta_2, then obtain \\psi^* + \\int \\log{(\\Psi)} d\\Psi = \\Psi \\log{(\\Psi)} - \\Psi + \\theta_2", "derivation": "\\mathbf{S}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi and \\frac{d}{d \\Psi} \\mathbf{S}{(\\Psi)} = \\frac{d}{d \\Psi} \\int \\log{(\\Psi)} d\\Psi and \\int \\frac{d}{d \\Psi} \\mathbf{S}{(\\Psi)} d\\Psi = \\int \\frac{d}{d \\Psi} \\int \\log{(\\Psi)} d\\Psi d\\Psi and \\psi^* + \\mathbf{S}{(\\Psi)} = \\Psi \\log{(\\Psi)} - \\Psi + \\theta_2 and \\psi^* + \\int \\log{(\\Psi)} d\\Psi = \\Psi \\log{(\\Psi)} - \\Psi + \\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\Psi', commutative=True)), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Derivative(Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Function('\\\\mathbf{S}')(Symbol('\\\\Psi', commutative=True))), Add(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\psi^*', commutative=True), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))), Add(Mul(Symbol('\\\\Psi', commutative=True), log(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(G)} = \\cos{(G)}, then derive \\frac{d}{d G} \\theta_{1}{(G)} - 1 = - \\sin{(G)} - 1, then obtain (\\frac{d}{d G} \\cos{(G)} - 1)^{G} = (- \\sin{(G)} - 1)^{G}", "derivation": "\\theta_{1}{(G)} = \\cos{(G)} and - G + \\theta_{1}{(G)} = - G + \\cos{(G)} and \\frac{d}{d G} (- G + \\theta_{1}{(G)}) = \\frac{d}{d G} (- G + \\cos{(G)}) and \\frac{d}{d G} \\theta_{1}{(G)} - 1 = - \\sin{(G)} - 1 and \\frac{d}{d G} \\cos{(G)} - 1 = - \\sin{(G)} - 1 and (\\frac{d}{d G} \\cos{(G)} - 1)^{G} = (- \\sin{(G)} - 1)^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\theta_1')(Symbol('G', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))))"], [["differentiate", 2, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\theta_1')(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True)), cos(Symbol('G', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integer(-1)))"], [["power", 5, "Symbol('G', commutative=True)"], "Equality(Pow(Add(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(-1)), Symbol('G', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('G', commutative=True))), Integer(-1)), Symbol('G', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(A_{2})} = \\sin{(\\cos{(A_{2})})}, then obtain - \\log{(\\mathbf{P})} + \\cos{(A_{2})} \\frac{d}{d A_{2}} \\operatorname{V_{\\mathbf{B}}}{(A_{2})} = - \\log{(\\mathbf{P})} + \\cos{(A_{2})} \\frac{d}{d A_{2}} \\sin{(\\cos{(A_{2})})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(A_{2})} = \\sin{(\\cos{(A_{2})})} and \\frac{d}{d A_{2}} \\operatorname{V_{\\mathbf{B}}}{(A_{2})} = \\frac{d}{d A_{2}} \\sin{(\\cos{(A_{2})})} and \\cos{(A_{2})} \\frac{d}{d A_{2}} \\operatorname{V_{\\mathbf{B}}}{(A_{2})} = \\cos{(A_{2})} \\frac{d}{d A_{2}} \\sin{(\\cos{(A_{2})})} and - \\log{(\\mathbf{P})} + \\cos{(A_{2})} \\frac{d}{d A_{2}} \\operatorname{V_{\\mathbf{B}}}{(A_{2})} = - \\log{(\\mathbf{P})} + \\cos{(A_{2})} \\frac{d}{d A_{2}} \\sin{(\\cos{(A_{2})})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('A_2', commutative=True)), sin(cos(Symbol('A_2', commutative=True))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["times", 2, "cos(Symbol('A_2', commutative=True))"], "Equality(Mul(cos(Symbol('A_2', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(cos(Symbol('A_2', commutative=True)), Derivative(sin(cos(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["minus", 3, "log(Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(cos(Symbol('A_2', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{P}', commutative=True))), Mul(cos(Symbol('A_2', commutative=True)), Derivative(sin(cos(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\dot{y}{(\\hbar)} = \\log{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} = 2 \\dot{y}{(\\hbar)}, then obtain \\int \\operatorname{r_{0}}{(\\hbar)} d\\hbar = l - 2 \\int - \\operatorname{r_{0}}{(\\hbar)} d\\hbar - 2 \\int \\log{(\\hbar)} d\\hbar", "derivation": "\\dot{y}{(\\hbar)} = \\log{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} = 2 \\dot{y}{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} = 2 \\log{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} - \\log{(\\hbar)} = \\log{(\\hbar)} and \\operatorname{r_{0}}{(\\hbar)} = 2 \\operatorname{r_{0}}{(\\hbar)} - 2 \\log{(\\hbar)} and \\int \\operatorname{r_{0}}{(\\hbar)} d\\hbar = \\int (2 \\operatorname{r_{0}}{(\\hbar)} - 2 \\log{(\\hbar)}) d\\hbar and \\int \\operatorname{r_{0}}{(\\hbar)} d\\hbar = l - 2 \\int - \\operatorname{r_{0}}{(\\hbar)} d\\hbar - 2 \\int \\log{(\\hbar)} d\\hbar", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], ["renaming_premise", "Equality(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), log(Symbol('\\\\hbar', commutative=True))))"], [["minus", 3, "log(Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\hbar', commutative=True)))), log(Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(2), Function('r_0')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\hbar', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(2), Function('r_0')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Integer(2), log(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Integer(2), Add(Integral(Mul(Integer(-1), Function('r_0')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))))"]]}, {"prompt": "Given \\hat{H}{(f_{E},\\mathbf{J}_P)} = \\mathbf{J}_P^{f_{E}} and n{(f_{E},\\mathbf{J}_P)} = \\mathbf{J}_P^{f_{E}}, then obtain (\\mathbf{J}_P^{f_{E}})^{\\mathbf{J}_P} - \\hat{H}^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} = 0", "derivation": "\\hat{H}{(f_{E},\\mathbf{J}_P)} = \\mathbf{J}_P^{f_{E}} and \\hat{H}^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} = (\\mathbf{J}_P^{f_{E}})^{\\mathbf{J}_P} and n{(f_{E},\\mathbf{J}_P)} = \\mathbf{J}_P^{f_{E}} and \\hat{H}^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} = n^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} and (\\mathbf{J}_P^{f_{E}})^{\\mathbf{J}_P} = n^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} and (\\mathbf{J}_P^{f_{E}})^{\\mathbf{J}_P} - n^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} = 0 and (\\mathbf{J}_P^{f_{E}})^{\\mathbf{J}_P} - \\hat{H}^{\\mathbf{J}_P}{(f_{E},\\mathbf{J}_P)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('n')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('n')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Function('n')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 5, "Pow(Function('n')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Pow(Function('n')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Pow(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('f_E', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\hat{H}')(Symbol('f_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\delta,J)} = e^{\\frac{\\delta}{J}}, then obtain ((\\operatorname{v_{y}}{(\\delta,J)} + 1)^{J})^{J} = ((e^{\\frac{\\delta}{J}} + 1)^{J})^{J}", "derivation": "\\operatorname{v_{y}}{(\\delta,J)} = e^{\\frac{\\delta}{J}} and \\operatorname{v_{y}}{(\\delta,J)} + 1 = e^{\\frac{\\delta}{J}} + 1 and (\\operatorname{v_{y}}{(\\delta,J)} + 1)^{J} = (e^{\\frac{\\delta}{J}} + 1)^{J} and ((\\operatorname{v_{y}}{(\\delta,J)} + 1)^{J})^{J} = ((e^{\\frac{\\delta}{J}} + 1)^{J})^{J}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('J', commutative=True)), exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('J', commutative=True)), Integer(1)), Add(exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))), Integer(1)))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('J', commutative=True)), Integer(1)), Symbol('J', commutative=True)), Pow(Add(exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))), Integer(1)), Symbol('J', commutative=True)))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Add(Function('v_y')(Symbol('\\\\delta', commutative=True), Symbol('J', commutative=True)), Integer(1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Add(exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))), Integer(1)), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"]]}, {"prompt": "Given J{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(\\hat{x})}, then derive J{(\\hat{x})} = \\frac{1}{\\hat{x}}, then derive \\theta_1 + \\log{(\\hat{x})} = \\int J{(\\hat{x})} d\\hat{x}, then derive \\theta_1 + \\log{(\\hat{x})} = U + \\log{(\\hat{x})}, then obtain \\int (\\theta_1 + \\log{(\\hat{x})}) dU = \\int (U + \\log{(\\hat{x})}) dU", "derivation": "J{(\\hat{x})} = \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} and \\int J{(\\hat{x})} d\\hat{x} = \\int \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} d\\hat{x} and J{(\\hat{x})} = \\frac{1}{\\hat{x}} and \\int \\frac{1}{\\hat{x}} d\\hat{x} = \\int \\frac{d}{d \\hat{x}} \\log{(\\hat{x})} d\\hat{x} and \\int \\frac{1}{\\hat{x}} d\\hat{x} = \\int J{(\\hat{x})} d\\hat{x} and \\theta_1 + \\log{(\\hat{x})} = \\int J{(\\hat{x})} d\\hat{x} and \\theta_1 + \\log{(\\hat{x})} = \\int \\frac{1}{\\hat{x}} d\\hat{x} and \\theta_1 + \\log{(\\hat{x})} = U + \\log{(\\hat{x})} and \\int (\\theta_1 + \\log{(\\hat{x})}) dU = \\int (U + \\log{(\\hat{x})}) dU", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{x}', commutative=True)), Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('J')(Symbol('\\\\hat{x}', commutative=True)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(log(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Function('J')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Integral(Function('J')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Add(Symbol('U', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))))"], [["integrate", 8, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\theta_1', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('U', commutative=True), log(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})}, then obtain \\log{(\\log{(\\operatorname{P_{e}}{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}})} = \\log{(\\log{(\\sin{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}})}", "derivation": "\\operatorname{P_{e}}{(\\hat{\\mathbf{x}})} = \\sin{(\\hat{\\mathbf{x}})} and \\log{(\\operatorname{P_{e}}{(\\hat{\\mathbf{x}})})} = \\log{(\\sin{(\\hat{\\mathbf{x}})})} and \\log{(\\operatorname{P_{e}}{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}} = \\log{(\\sin{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}} and \\log{(\\log{(\\operatorname{P_{e}}{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}})} = \\log{(\\log{(\\sin{(\\hat{\\mathbf{x}})})}^{\\hat{\\mathbf{x}}})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["log", 1], "Equality(log(Function('P_e')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Pow(log(Function('P_e')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["log", 3], "Equality(log(Pow(log(Function('P_e')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Pow(log(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(v_{x})} = \\cos{(v_{x})}, then obtain (1 + \\frac{\\theta_{1}{(v_{x})}}{v_{x}^{2}})^{v_{x}} = (1 + \\frac{\\cos{(v_{x})}}{v_{x}^{2}})^{v_{x}}", "derivation": "\\theta_{1}{(v_{x})} = \\cos{(v_{x})} and \\frac{\\theta_{1}{(v_{x})}}{v_{x}} = \\frac{\\cos{(v_{x})}}{v_{x}} and \\frac{\\theta_{1}{(v_{x})}}{v_{x}^{2}} = \\frac{\\cos{(v_{x})}}{v_{x}^{2}} and 1 + \\frac{\\theta_{1}{(v_{x})}}{v_{x}^{2}} = 1 + \\frac{\\cos{(v_{x})}}{v_{x}^{2}} and (1 + \\frac{\\theta_{1}{(v_{x})}}{v_{x}^{2}})^{v_{x}} = (1 + \\frac{\\cos{(v_{x})}}{v_{x}^{2}})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('v_x', commutative=True)), cos(Symbol('v_x', commutative=True)))"], [["divide", 1, "Symbol('v_x', commutative=True)"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), Function('\\\\theta_1')(Symbol('v_x', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), cos(Symbol('v_x', commutative=True))))"], [["times", 2, "Pow(Symbol('v_x', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Function('\\\\theta_1')(Symbol('v_x', commutative=True))), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), cos(Symbol('v_x', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Function('\\\\theta_1')(Symbol('v_x', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), cos(Symbol('v_x', commutative=True)))))"], [["power", 4, "Symbol('v_x', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), Function('\\\\theta_1')(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)), Pow(Add(Integer(1), Mul(Pow(Symbol('v_x', commutative=True), Integer(-2)), cos(Symbol('v_x', commutative=True)))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(P_{e},x^\\prime)} = \\frac{P_{e}}{x^\\prime}, then derive \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(P_{e},x^\\prime)} = - \\frac{P_{e}}{(x^\\prime)^{2}}, then obtain \\frac{P_{e}}{x^\\prime} + g = \\int - \\frac{P_{e}}{(x^\\prime)^{2}} dx^\\prime", "derivation": "\\mathbf{p}{(P_{e},x^\\prime)} = \\frac{P_{e}}{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(P_{e},x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} \\frac{P_{e}}{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(P_{e},x^\\prime)} = - \\frac{P_{e}}{(x^\\prime)^{2}} and \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(P_{e},x^\\prime)} = - \\frac{\\mathbf{p}{(P_{e},x^\\prime)}}{x^\\prime} and \\int \\frac{\\partial}{\\partial x^\\prime} \\mathbf{p}{(P_{e},x^\\prime)} dx^\\prime = \\int - \\frac{\\mathbf{p}{(P_{e},x^\\prime)}}{x^\\prime} dx^\\prime and \\int \\frac{\\partial}{\\partial x^\\prime} \\frac{P_{e}}{x^\\prime} dx^\\prime = \\int - \\frac{P_{e}}{(x^\\prime)^{2}} dx^\\prime and \\frac{P_{e}}{x^\\prime} + g = \\int - \\frac{P_{e}}{(x^\\prime)^{2}} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('P_e', commutative=True), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Symbol('g', commutative=True)), Integral(Mul(Integer(-1), Symbol('P_e', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-2))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(C_{1},\\rho)} = \\frac{\\rho}{C_{1}}, then obtain (C_{1} + \\operatorname{P_{g}}{(C_{1},\\rho)} - 1)^{C_{1}} = (C_{1} - 1 + \\frac{\\rho}{C_{1}})^{C_{1}}", "derivation": "\\operatorname{P_{g}}{(C_{1},\\rho)} = \\frac{\\rho}{C_{1}} and \\operatorname{P_{g}}{(C_{1},\\rho)} - 1 = -1 + \\frac{\\rho}{C_{1}} and C_{1} + \\operatorname{P_{g}}{(C_{1},\\rho)} - 1 = C_{1} - 1 + \\frac{\\rho}{C_{1}} and (C_{1} + \\operatorname{P_{g}}{(C_{1},\\rho)} - 1)^{C_{1}} = (C_{1} - 1 + \\frac{\\rho}{C_{1}})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["add", 2, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Add(Symbol('C_1', commutative=True), Integer(-1), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Symbol('C_1', commutative=True), Function('P_g')(Symbol('C_1', commutative=True), Symbol('\\\\rho', commutative=True)), Integer(-1)), Symbol('C_1', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), Integer(-1), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\eta{(A)} = \\sin{(\\sin{(A)})}, then derive \\frac{d}{d A} \\eta{(A)} = \\cos{(A)} \\cos{(\\sin{(A)})}, then obtain \\frac{\\frac{d}{d A} \\eta{(A)}}{\\cos{(\\sin{(A)})}} = \\cos{(A)}", "derivation": "\\eta{(A)} = \\sin{(\\sin{(A)})} and \\frac{d}{d A} \\eta{(A)} = \\frac{d}{d A} \\sin{(\\sin{(A)})} and \\frac{d}{d A} \\eta{(A)} = \\cos{(A)} \\cos{(\\sin{(A)})} and \\frac{\\frac{d}{d A} \\eta{(A)}}{\\cos{(\\sin{(A)})}} = \\cos{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('A', commutative=True)), sin(sin(Symbol('A', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Mul(cos(Symbol('A', commutative=True)), cos(sin(Symbol('A', commutative=True)))))"], [["divide", 3, "cos(sin(Symbol('A', commutative=True)))"], "Equality(Mul(Pow(cos(sin(Symbol('A', commutative=True))), Integer(-1)), Derivative(Function('\\\\eta')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), cos(Symbol('A', commutative=True)))"]]}, {"prompt": "Given B{(L,\\mathbf{D})} = \\log{(L + \\mathbf{D})}, then obtain ((\\int (- \\mathbf{D} + B{(L,\\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}})^{L} = ((\\int (- \\mathbf{D} + \\log{(L + \\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}})^{L}", "derivation": "B{(L,\\mathbf{D})} = \\log{(L + \\mathbf{D})} and - \\mathbf{D} + B{(L,\\mathbf{D})} = - \\mathbf{D} + \\log{(L + \\mathbf{D})} and \\int (- \\mathbf{D} + B{(L,\\mathbf{D})}) d\\mathbf{D} = \\int (- \\mathbf{D} + \\log{(L + \\mathbf{D})}) d\\mathbf{D} and (\\int (- \\mathbf{D} + B{(L,\\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}} = (\\int (- \\mathbf{D} + \\log{(L + \\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}} and ((\\int (- \\mathbf{D} + B{(L,\\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}})^{L} = ((\\int (- \\mathbf{D} + \\log{(L + \\mathbf{D})}) d\\mathbf{D})^{\\mathbf{D}})^{L}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('B')(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('B')(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('B')(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 4, "Symbol('L', commutative=True)"], "Equality(Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('B')(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('L', commutative=True)), Pow(Pow(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('L', commutative=True)))"]]}, {"prompt": "Given p{(n_{2})} = \\sin{(n_{2})}, then derive \\int (- n_{2} + p{(n_{2})}) dn_{2} = T - \\frac{n_{2}^{2}}{2} - \\cos{(n_{2})}, then obtain T - \\frac{n_{2}^{2}}{2} - \\cos{(n_{2})} = \\int (- n_{2} + \\sin{(n_{2})}) dn_{2}", "derivation": "p{(n_{2})} = \\sin{(n_{2})} and - n_{2} + p{(n_{2})} = - n_{2} + \\sin{(n_{2})} and \\int (- n_{2} + p{(n_{2})}) dn_{2} = \\int (- n_{2} + \\sin{(n_{2})}) dn_{2} and \\int (- n_{2} + p{(n_{2})}) dn_{2} = T - \\frac{n_{2}^{2}}{2} - \\cos{(n_{2})} and T - \\frac{n_{2}^{2}}{2} - \\cos{(n_{2})} = \\int (- n_{2} + \\sin{(n_{2})}) dn_{2}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('n_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('p')(Symbol('n_2', commutative=True))), Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('p')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), Function('p')(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))), Add(Symbol('T', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('n_2', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('n_2', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('n_2', commutative=True)), sin(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(A)} = \\cos{(A)}, then derive \\int \\Psi_{nl}{(A)} dA = \\Omega + \\sin{(A)}, then obtain \\Omega + \\sin{(A)} = \\int \\cos{(A)} dA", "derivation": "\\Psi_{nl}{(A)} = \\cos{(A)} and \\int \\Psi_{nl}{(A)} dA = \\int \\cos{(A)} dA and \\int \\Psi_{nl}{(A)} dA = \\Omega + \\sin{(A)} and \\Omega + \\sin{(A)} = \\int \\cos{(A)} dA", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\Omega', commutative=True), sin(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\rho,\\mathbf{A})} = \\rho^{\\mathbf{A}}, then obtain \\sin{(\\cos{(\\theta^{\\rho}{(\\rho,\\mathbf{A})})})} = \\sin{(\\cos{((\\rho^{\\mathbf{A}})^{\\rho})})}", "derivation": "\\theta{(\\rho,\\mathbf{A})} = \\rho^{\\mathbf{A}} and \\theta^{\\rho}{(\\rho,\\mathbf{A})} = (\\rho^{\\mathbf{A}})^{\\rho} and \\cos{(\\theta^{\\rho}{(\\rho,\\mathbf{A})})} = \\cos{((\\rho^{\\mathbf{A}})^{\\rho})} and \\sin{(\\cos{(\\theta^{\\rho}{(\\rho,\\mathbf{A})})})} = \\sin{(\\cos{((\\rho^{\\mathbf{A}})^{\\rho})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('\\\\theta')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True))), cos(Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True))))"], [["sin", 3], "Equality(sin(cos(Pow(Function('\\\\theta')(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True)))), sin(cos(Pow(Pow(Symbol('\\\\rho', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(\\dot{y})} = e^{\\sin{(\\dot{y})}}, then obtain \\int \\sin{(\\int \\nabla{(\\dot{y})} d\\dot{y})} d\\dot{y} = \\int \\sin{(\\int e^{\\sin{(\\dot{y})}} d\\dot{y})} d\\dot{y}", "derivation": "\\nabla{(\\dot{y})} = e^{\\sin{(\\dot{y})}} and \\int \\nabla{(\\dot{y})} d\\dot{y} = \\int e^{\\sin{(\\dot{y})}} d\\dot{y} and \\sin{(\\int \\nabla{(\\dot{y})} d\\dot{y})} = \\sin{(\\int e^{\\sin{(\\dot{y})}} d\\dot{y})} and \\int \\sin{(\\int \\nabla{(\\dot{y})} d\\dot{y})} d\\dot{y} = \\int \\sin{(\\int e^{\\sin{(\\dot{y})}} d\\dot{y})} d\\dot{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), exp(sin(Symbol('\\\\dot{y}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(exp(sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), sin(Integral(exp(sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(sin(Integral(Function('\\\\nabla')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(sin(Integral(exp(sin(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(s,Q)} = \\frac{Q}{s}, then obtain s^{9} \\operatorname{v_{1}}^{6}{(s,Q)} = Q^{6} s^{3}", "derivation": "\\operatorname{v_{1}}{(s,Q)} = \\frac{Q}{s} and \\frac{s \\operatorname{v_{1}}{(s,Q)}}{Q} = 1 and s \\operatorname{v_{1}}{(s,Q)} = Q and s^{2} \\operatorname{v_{1}}{(s,Q)} = Q s and s^{3} \\operatorname{v_{1}}^{2}{(s,Q)} = Q s^{2} \\operatorname{v_{1}}{(s,Q)} and Q s^{2} \\operatorname{v_{1}}{(s,Q)} = Q^{2} s and s^{3} \\operatorname{v_{1}}^{2}{(s,Q)} = Q^{2} s and s^{9} \\operatorname{v_{1}}^{6}{(s,Q)} = Q^{6} s^{3}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["divide", 1, "Mul(Symbol('Q', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('s', commutative=True), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integer(1))"], [["divide", 2, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('s', commutative=True), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True))"], [["times", 3, "Symbol('s', commutative=True)"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(2)), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('Q', commutative=True), Symbol('s', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('s', commutative=True), Integer(2)), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(3)), Pow(Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Mul(Symbol('Q', commutative=True), Pow(Symbol('s', commutative=True), Integer(2)), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('Q', commutative=True), Pow(Symbol('s', commutative=True), Integer(2)), Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(3)), Pow(Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('s', commutative=True)))"], [["power", 7, 3], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(9)), Pow(Function('v_1')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Integer(6))), Mul(Pow(Symbol('Q', commutative=True), Integer(6)), Pow(Symbol('s', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\bar{\\h}{(P_{g})} = \\cos{(\\log{(P_{g})})} and \\mathbf{S}{(P_{g})} = \\frac{1}{\\cos{(\\log{(P_{g})})}}, then obtain \\frac{\\sin{(\\log{(P_{g})})}}{P_{g} \\cos^{2}{(\\log{(P_{g})})}} = - \\frac{\\frac{d}{d P_{g}} \\bar{\\h}{(P_{g})}}{\\bar{\\h}^{2}{(P_{g})}}", "derivation": "\\bar{\\h}{(P_{g})} = \\cos{(\\log{(P_{g})})} and \\mathbf{S}{(P_{g})} = \\frac{1}{\\cos{(\\log{(P_{g})})}} and \\mathbf{S}{(P_{g})} = \\frac{1}{\\bar{\\h}{(P_{g})}} and \\frac{1}{\\cos{(\\log{(P_{g})})}} = \\frac{1}{\\bar{\\h}{(P_{g})}} and \\frac{d}{d P_{g}} \\frac{1}{\\cos{(\\log{(P_{g})})}} = \\frac{d}{d P_{g}} \\frac{1}{\\bar{\\h}{(P_{g})}} and \\frac{\\sin{(\\log{(P_{g})})}}{P_{g} \\cos^{2}{(\\log{(P_{g})})}} = - \\frac{\\frac{d}{d P_{g}} \\bar{\\h}{(P_{g})}}{\\bar{\\h}^{2}{(P_{g})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('P_g', commutative=True)), cos(log(Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('P_g', commutative=True)), Pow(cos(log(Symbol('P_g', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('P_g', commutative=True)), Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(cos(log(Symbol('P_g', commutative=True))), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Pow(cos(log(Symbol('P_g', commutative=True))), Integer(-1)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True)), Integer(-1)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Pow(Symbol('P_g', commutative=True), Integer(-1)), sin(log(Symbol('P_g', commutative=True))), Pow(cos(log(Symbol('P_g', commutative=True))), Integer(-2))), Mul(Integer(-1), Pow(Function('\\\\hbar')(Symbol('P_g', commutative=True)), Integer(-2)), Derivative(Function('\\\\hbar')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(h)} = e^{e^{h}}, then obtain (h (\\pi{(h)} e^{- (\\pi{(h)} e^{- e^{h}})^{h} e^{h}})^{h} - e^{- e^{h}})^{h} = (h - e^{- e^{h}})^{h}", "derivation": "\\pi{(h)} = e^{e^{h}} and \\pi{(h)} e^{- e^{h}} = 1 and (\\pi{(h)} e^{- e^{h}})^{h} = 1 and - (\\pi{(h)} e^{- e^{h}})^{h} e^{h} = - e^{h} and (\\pi{(h)} e^{- (\\pi{(h)} e^{- e^{h}})^{h} e^{h}})^{h} = 1 and h (\\pi{(h)} e^{- (\\pi{(h)} e^{- e^{h}})^{h} e^{h}})^{h} = h and h (\\pi{(h)} e^{- (\\pi{(h)} e^{- e^{h}})^{h} e^{h}})^{h} - e^{- e^{h}} = h - e^{- e^{h}} and (h (\\pi{(h)} e^{- (\\pi{(h)} e^{- e^{h}})^{h} e^{h}})^{h} - e^{- e^{h}})^{h} = (h - e^{- e^{h}})^{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('h', commutative=True)), exp(exp(Symbol('h', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('h', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Integer(1))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), Integer(1))"], [["times", 3, "Mul(Integer(-1), exp(Symbol('h', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Mul(Integer(-1), exp(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), Integer(1))"], [["times", 5, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True))), Symbol('h', commutative=True))"], [["minus", 6, "exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))"], "Equality(Add(Mul(Symbol('h', commutative=True), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True)))))), Add(Symbol('h', commutative=True), Mul(Integer(-1), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True)))))))"], [["power", 7, "Symbol('h', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('h', commutative=True), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), Pow(Mul(Function('\\\\pi')(Symbol('h', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))), Symbol('h', commutative=True))), Mul(Integer(-1), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True)))))), Symbol('h', commutative=True)), Pow(Add(Symbol('h', commutative=True), Mul(Integer(-1), exp(Mul(Integer(-1), exp(Symbol('h', commutative=True)))))), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\rho{(B)} = \\int \\sin{(B)} dB, then derive - B + \\rho{(B)} = - B + \\hat{H}_l - \\cos{(B)}, then derive - B + c - \\cos{(B)} = - B + \\hat{H}_l - \\cos{(B)}, then obtain \\frac{- B + c - \\cos{(B)}}{B} = \\frac{- B + \\int \\sin{(B)} dB}{B}", "derivation": "\\rho{(B)} = \\int \\sin{(B)} dB and - B + \\rho{(B)} = - B + \\int \\sin{(B)} dB and - B + \\rho{(B)} = - B + \\hat{H}_l - \\cos{(B)} and - B + \\int \\sin{(B)} dB = - B + \\hat{H}_l - \\cos{(B)} and - B + c - \\cos{(B)} = - B + \\hat{H}_l - \\cos{(B)} and \\frac{- B + c - \\cos{(B)}}{B} = \\frac{- B + \\hat{H}_l - \\cos{(B)}}{B} and \\frac{- B + c - \\cos{(B)}}{B} = \\frac{- B + \\int \\sin{(B)} dB}{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["minus", 1, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\rho')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\rho')(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('c', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))))"], [["divide", 5, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('c', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True))))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('c', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True))))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(\\mu_0,\\mathbf{J}_f,z)} = (\\mathbf{J}_f - z)^{\\mu_0}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_f} \\int \\theta_{1}^{\\mu_0}{(\\mu_0,\\mathbf{J}_f,z)} dz = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\int ((\\mathbf{J}_f - z)^{\\mu_0})^{\\mu_0} dz", "derivation": "\\theta_{1}{(\\mu_0,\\mathbf{J}_f,z)} = (\\mathbf{J}_f - z)^{\\mu_0} and \\theta_{1}^{\\mu_0}{(\\mu_0,\\mathbf{J}_f,z)} = ((\\mathbf{J}_f - z)^{\\mu_0})^{\\mu_0} and \\int \\theta_{1}^{\\mu_0}{(\\mu_0,\\mathbf{J}_f,z)} dz = \\int ((\\mathbf{J}_f - z)^{\\mu_0})^{\\mu_0} dz and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\int \\theta_{1}^{\\mu_0}{(\\mu_0,\\mathbf{J}_f,z)} dz = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\int ((\\mathbf{J}_f - z)^{\\mu_0})^{\\mu_0} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integral(Pow(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(F_{g},\\nabla)} = F_{g}^{\\nabla}, then obtain ((n^{F_{g}}{(F_{g},\\nabla)})^{\\nabla})^{\\nabla} = (((F_{g}^{\\nabla})^{F_{g}})^{\\nabla})^{\\nabla}", "derivation": "n{(F_{g},\\nabla)} = F_{g}^{\\nabla} and n^{F_{g}}{(F_{g},\\nabla)} = (F_{g}^{\\nabla})^{F_{g}} and (n^{F_{g}}{(F_{g},\\nabla)})^{\\nabla} = ((F_{g}^{\\nabla})^{F_{g}})^{\\nabla} and ((n^{F_{g}}{(F_{g},\\nabla)})^{\\nabla})^{\\nabla} = (((F_{g}^{\\nabla})^{F_{g}})^{\\nabla})^{\\nabla}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('n')(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)), Pow(Pow(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)))"], [["power", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Pow(Function('n')(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Pow(Pow(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\nabla', commutative=True)))"], [["power", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Pow(Pow(Pow(Function('n')(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)), Pow(Pow(Pow(Pow(Symbol('F_g', commutative=True), Symbol('\\\\nabla', commutative=True)), Symbol('F_g', commutative=True)), Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True)))"]]}, {"prompt": "Given U{(\\mathbf{B})} = \\sin{(\\mathbf{B})}, then obtain \\frac{d}{d \\mathbf{B}} (\\int 1 d\\mathbf{B})^{\\mathbf{B}} = \\frac{d}{d \\mathbf{B}} (\\int \\frac{\\sin{(\\mathbf{B})}}{U{(\\mathbf{B})}} d\\mathbf{B})^{\\mathbf{B}}", "derivation": "U{(\\mathbf{B})} = \\sin{(\\mathbf{B})} and 1 = \\frac{\\sin{(\\mathbf{B})}}{U{(\\mathbf{B})}} and \\int 1 d\\mathbf{B} = \\int \\frac{\\sin{(\\mathbf{B})}}{U{(\\mathbf{B})}} d\\mathbf{B} and (\\int 1 d\\mathbf{B})^{\\mathbf{B}} = (\\int \\frac{\\sin{(\\mathbf{B})}}{U{(\\mathbf{B})}} d\\mathbf{B})^{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} (\\int 1 d\\mathbf{B})^{\\mathbf{B}} = \\frac{d}{d \\mathbf{B}} (\\int \\frac{\\sin{(\\mathbf{B})}}{U{(\\mathbf{B})}} d\\mathbf{B})^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{B}', commutative=True)), sin(Symbol('\\\\mathbf{B}', commutative=True)))"], [["divide", 1, "Function('U')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('U')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Mul(Pow(Function('U')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Integral(Mul(Pow(Function('U')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Pow(Integral(Mul(Pow(Function('U')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), sin(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given l{(\\phi)} = \\cos{(\\phi)}, then obtain \\phi (\\frac{l{(\\phi)}}{\\phi \\cos{(\\phi)}})^{\\phi} = \\phi (\\frac{1}{\\phi})^{\\phi}", "derivation": "l{(\\phi)} = \\cos{(\\phi)} and \\frac{l{(\\phi)}}{\\phi \\cos{(\\phi)}} = \\frac{1}{\\phi} and (\\frac{l{(\\phi)}}{\\phi \\cos{(\\phi)}})^{\\phi} = (\\frac{1}{\\phi})^{\\phi} and \\phi (\\frac{l{(\\phi)}}{\\phi \\cos{(\\phi)}})^{\\phi} = \\phi (\\frac{1}{\\phi})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\phi', commutative=True)), cos(Symbol('\\\\phi', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\phi', commutative=True), cos(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('l')(Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Integer(-1))), Pow(Symbol('\\\\phi', commutative=True), Integer(-1)))"], [["power", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('l')(Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('\\\\phi', commutative=True)), Pow(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))"], [["divide", 3, "Pow(Symbol('\\\\phi', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Pow(Mul(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Function('l')(Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Integer(-1))), Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), Pow(Pow(Symbol('\\\\phi', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(f,A_{z})} = f \\sin{(A_{z})}, then obtain \\frac{(\\int \\hat{X}{(f,A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)}}{\\sin^{2}{(I)}} = \\frac{(\\int f \\sin{(A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)}}{\\sin^{2}{(I)}}", "derivation": "\\hat{X}{(f,A_{z})} = f \\sin{(A_{z})} and \\int \\hat{X}{(f,A_{z})} df = \\int f \\sin{(A_{z})} df and \\int \\hat{X}{(f,A_{z})} df - \\frac{1}{I} = \\int f \\sin{(A_{z})} df - \\frac{1}{I} and (\\int \\hat{X}{(f,A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)} = (\\int f \\sin{(A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)} and \\frac{(\\int \\hat{X}{(f,A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)}}{\\sin^{2}{(I)}} = \\frac{(\\int f \\sin{(A_{z})} df - \\frac{1}{I}) \\operatorname{P_{e}}{(I)}}{\\sin^{2}{(I)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{X}')(Symbol('f', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('f', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('f', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('f', commutative=True), sin(Symbol('A_z', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["minus", 2, "Pow(Symbol('I', commutative=True), Integer(-1))"], "Equality(Add(Integral(Function('\\\\hat{X}')(Symbol('f', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))), Add(Integral(Mul(Symbol('f', commutative=True), sin(Symbol('A_z', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))))"], [["times", 3, "Function('P_e')(Symbol('I', commutative=True))"], "Equality(Mul(Add(Integral(Function('\\\\hat{X}')(Symbol('f', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))), Function('P_e')(Symbol('I', commutative=True))), Mul(Add(Integral(Mul(Symbol('f', commutative=True), sin(Symbol('A_z', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))), Function('P_e')(Symbol('I', commutative=True))))"], [["divide", 4, "Pow(sin(Symbol('I', commutative=True)), Integer(2))"], "Equality(Mul(Add(Integral(Function('\\\\hat{X}')(Symbol('f', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))), Function('P_e')(Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Integer(-2))), Mul(Add(Integral(Mul(Symbol('f', commutative=True), sin(Symbol('A_z', commutative=True))), Tuple(Symbol('f', commutative=True))), Mul(Integer(-1), Pow(Symbol('I', commutative=True), Integer(-1)))), Function('P_e')(Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(C)} = \\cos{(\\sin{(C)})}, then obtain \\frac{d^{3}}{d C^{3}} (\\operatorname{y^{\\prime}}{(C)} + \\sin{(C)}) = \\frac{d^{3}}{d C^{3}} (\\sin{(C)} + \\cos{(\\sin{(C)})})", "derivation": "\\operatorname{y^{\\prime}}{(C)} = \\cos{(\\sin{(C)})} and \\operatorname{y^{\\prime}}{(C)} + \\sin{(C)} = \\sin{(C)} + \\cos{(\\sin{(C)})} and \\frac{d}{d C} (\\operatorname{y^{\\prime}}{(C)} + \\sin{(C)}) = \\frac{d}{d C} (\\sin{(C)} + \\cos{(\\sin{(C)})}) and \\frac{d^{2}}{d C^{2}} (\\operatorname{y^{\\prime}}{(C)} + \\sin{(C)}) = \\frac{d^{2}}{d C^{2}} (\\sin{(C)} + \\cos{(\\sin{(C)})}) and \\frac{d^{3}}{d C^{3}} (\\operatorname{y^{\\prime}}{(C)} + \\sin{(C)}) = \\frac{d^{3}}{d C^{3}} (\\sin{(C)} + \\cos{(\\sin{(C)})})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True))))"], [["add", 1, "sin(Symbol('C', commutative=True))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Add(sin(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('y^{\\\\prime}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('y^{\\\\prime}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(Add(sin(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('y^{\\\\prime}')(Symbol('C', commutative=True)), sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(3))), Derivative(Add(sin(Symbol('C', commutative=True)), cos(sin(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(3))))"]]}, {"prompt": "Given \\phi_{2}{(p)} = \\log{(\\sin{(p)})} and \\mathbf{H}{(p)} = \\log{(\\sin{(p)})}^{p} and y{(\\mathbf{J}_P,\\chi)} = \\mathbf{J}_P \\cos{(\\chi)}, then obtain \\mathbf{H}^{p}{(p)} + \\frac{\\partial}{\\partial \\chi} y{(\\mathbf{J}_P,\\chi)} = \\mathbf{H}^{p}{(p)} + \\frac{\\partial}{\\partial \\chi} \\mathbf{J}_P \\cos{(\\chi)}", "derivation": "\\phi_{2}{(p)} = \\log{(\\sin{(p)})} and \\phi_{2}^{p}{(p)} = \\log{(\\sin{(p)})}^{p} and \\mathbf{H}{(p)} = \\log{(\\sin{(p)})}^{p} and y{(\\mathbf{J}_P,\\chi)} = \\mathbf{J}_P \\cos{(\\chi)} and \\frac{\\partial}{\\partial \\chi} y{(\\mathbf{J}_P,\\chi)} = \\frac{\\partial}{\\partial \\chi} \\mathbf{J}_P \\cos{(\\chi)} and \\phi_{2}^{p}{(p)} = \\mathbf{H}{(p)} and (\\phi_{2}^{p}{(p)})^{p} + \\frac{\\partial}{\\partial \\chi} y{(\\mathbf{J}_P,\\chi)} = (\\phi_{2}^{p}{(p)})^{p} + \\frac{\\partial}{\\partial \\chi} \\mathbf{J}_P \\cos{(\\chi)} and \\mathbf{H}^{p}{(p)} + \\frac{\\partial}{\\partial \\chi} y{(\\mathbf{J}_P,\\chi)} = \\mathbf{H}^{p}{(p)} + \\frac{\\partial}{\\partial \\chi} \\mathbf{J}_P \\cos{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('p', commutative=True)), log(sin(Symbol('p', commutative=True))))"], [["power", 1, "Symbol('p', commutative=True)"], "Equality(Pow(Function('\\\\phi_2')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(log(sin(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('p', commutative=True)), Pow(log(sin(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], ["get_premise", "Equality(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\phi_2')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Function('\\\\mathbf{H}')(Symbol('p', commutative=True)))"], [["add", 5, "Pow(Pow(Function('\\\\phi_2')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True))"], "Equality(Add(Pow(Pow(Function('\\\\phi_2')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Pow(Pow(Function('\\\\phi_2')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Pow(Function('\\\\mathbf{H}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Pow(Function('\\\\mathbf{H}')(Symbol('p', commutative=True)), Symbol('p', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\rho_f)} = \\cos{(\\rho_f)}, then derive \\int \\dot{x}{(\\rho_f)} d\\rho_f = \\mathbf{J}_f + \\sin{(\\rho_f)}, then obtain - \\mathbf{J}_f - \\sin{(\\rho_f)} + \\iint \\dot{x}{(\\rho_f)} d\\rho_f d\\mathbf{J}_f = - \\mathbf{J}_f - \\sin{(\\rho_f)} + \\int (\\mathbf{J}_f + \\sin{(\\rho_f)}) d\\mathbf{J}_f", "derivation": "\\dot{x}{(\\rho_f)} = \\cos{(\\rho_f)} and \\int \\dot{x}{(\\rho_f)} d\\rho_f = \\int \\cos{(\\rho_f)} d\\rho_f and \\int \\dot{x}{(\\rho_f)} d\\rho_f = \\mathbf{J}_f + \\sin{(\\rho_f)} and \\iint \\dot{x}{(\\rho_f)} d\\rho_f d\\mathbf{J}_f = \\int (\\mathbf{J}_f + \\sin{(\\rho_f)}) d\\mathbf{J}_f and - \\mathbf{J}_f - \\sin{(\\rho_f)} + \\iint \\dot{x}{(\\rho_f)} d\\rho_f d\\mathbf{J}_f = - \\mathbf{J}_f - \\sin{(\\rho_f)} + \\int (\\mathbf{J}_f + \\sin{(\\rho_f)}) d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(cos(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["minus", 4, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\rho_f', commutative=True))), Integral(Function('\\\\dot{x}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\rho_f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), sin(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)} = - n_{1} + \\sin{(U)}, then obtain \\int (- (- 2 n_{1} + \\sin{(U)})^{n_{1}} + (- n_{1} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)})^{n_{1}}) dU = \\int 0 dU", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(n_{1},U)} = - n_{1} + \\sin{(U)} and - n_{1} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)} = - 2 n_{1} + \\sin{(U)} and (- n_{1} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)})^{n_{1}} = (- 2 n_{1} + \\sin{(U)})^{n_{1}} and - (- 2 n_{1} + \\sin{(U)})^{n_{1}} + (- n_{1} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)})^{n_{1}} = 0 and \\int (- (- 2 n_{1} + \\sin{(U)})^{n_{1}} + (- n_{1} + \\operatorname{V_{\\mathbf{E}}}{(n_{1},U)})^{n_{1}}) dU = \\int 0 dU", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('U', commutative=True))), Symbol('n_1', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))), Symbol('n_1', commutative=True)))"], [["minus", 3, "Pow(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))), Symbol('n_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))), Symbol('n_1', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('U', commutative=True))), Symbol('n_1', commutative=True))), Integer(0))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('n_1', commutative=True)), sin(Symbol('U', commutative=True))), Symbol('n_1', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('n_1', commutative=True), Symbol('U', commutative=True))), Symbol('n_1', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Integer(0), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(m)} = \\sin{(m)}, then obtain \\int - \\sin{(m)} dm = \\int - \\hat{H}_l{(m)} dm", "derivation": "\\hat{H}_l{(m)} = \\sin{(m)} and \\hat{H}_l{(m)} - \\sin{(m)} = 0 and - \\sin{(m)} = - \\hat{H}_l{(m)} and \\int - \\sin{(m)} dm = \\int - \\hat{H}_l{(m)} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('m', commutative=True)), sin(Symbol('m', commutative=True)))"], [["minus", 1, "sin(Symbol('m', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('m', commutative=True)), Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Integer(0))"], [["minus", 2, "Function('\\\\hat{H}_l')(Symbol('m', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('m', commutative=True))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given E{(f^{*})} = \\int \\log{(f^{*})} df^{*}, then derive E{(f^{*})} = C_{2} + f^{*} \\log{(f^{*})} - f^{*}, then obtain (\\frac{\\int \\log{(f^{*})} df^{*}}{l})^{C_{2}} = (\\frac{E{(f^{*})}}{l})^{C_{2}}", "derivation": "E{(f^{*})} = \\int \\log{(f^{*})} df^{*} and E{(f^{*})} = C_{2} + f^{*} \\log{(f^{*})} - f^{*} and \\frac{E{(f^{*})}}{l} = \\frac{C_{2} + f^{*} \\log{(f^{*})} - f^{*}}{l} and (\\frac{E{(f^{*})}}{l})^{C_{2}} = (\\frac{C_{2} + f^{*} \\log{(f^{*})} - f^{*}}{l})^{C_{2}} and (\\frac{\\int \\log{(f^{*})} df^{*}}{l})^{C_{2}} = (\\frac{C_{2} + f^{*} \\log{(f^{*})} - f^{*}}{l})^{C_{2}} and (\\frac{\\int \\log{(f^{*})} df^{*}}{l})^{C_{2}} = (\\frac{E{(f^{*})}}{l})^{C_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('f^*', commutative=True)), Integral(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('E')(Symbol('f^*', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Symbol('f^*', commutative=True), log(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["divide", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E')(Symbol('f^*', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Symbol('f^*', commutative=True), log(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["power", 3, "Symbol('C_2', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E')(Symbol('f^*', commutative=True))), Symbol('C_2', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Symbol('f^*', commutative=True), log(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('C_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Integral(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Symbol('C_2', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Symbol('f^*', commutative=True), log(Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Symbol('C_2', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Integral(log(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Symbol('C_2', commutative=True)), Pow(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('E')(Symbol('f^*', commutative=True))), Symbol('C_2', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then obtain - \\log{(\\mathbf{J}_P)} \\frac{d}{d \\mathbf{J}_P} (\\psi^{*}{(\\mathbf{J}_P)} - 2 \\log{(\\mathbf{J}_P)}) = - \\log{(\\mathbf{J}_P)} \\frac{d}{d \\mathbf{J}_P} - \\log{(\\mathbf{J}_P)}", "derivation": "\\psi^{*}{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\psi^{*}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} = 0 and \\psi^{*}{(\\mathbf{J}_P)} - 2 \\log{(\\mathbf{J}_P)} = - \\log{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} (\\psi^{*}{(\\mathbf{J}_P)} - 2 \\log{(\\mathbf{J}_P)}) = \\frac{d}{d \\mathbf{J}_P} - \\log{(\\mathbf{J}_P)} and - \\log{(\\mathbf{J}_P)} \\frac{d}{d \\mathbf{J}_P} (\\psi^{*}{(\\mathbf{J}_P)} - 2 \\log{(\\mathbf{J}_P)}) = - \\log{(\\mathbf{J}_P)} \\frac{d}{d \\mathbf{J}_P} - \\log{(\\mathbf{J}_P)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Integer(0))"], [["minus", 2, "log(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Add(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["times", 4, "Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(Add(Function('\\\\psi^*')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True)), Derivative(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(k,t)} = \\frac{e^{t}}{k}, then obtain \\frac{\\partial}{\\partial t} \\int k \\operatorname{C_{2}}{(k,t)} e^{- t} dt = \\frac{d}{d t} \\int 1 dt", "derivation": "\\operatorname{C_{2}}{(k,t)} = \\frac{e^{t}}{k} and k \\operatorname{C_{2}}{(k,t)} e^{- t} = 1 and \\int k \\operatorname{C_{2}}{(k,t)} e^{- t} dt = \\int 1 dt and \\frac{\\partial}{\\partial t} \\int k \\operatorname{C_{2}}{(k,t)} e^{- t} dt = \\frac{d}{d t} \\int 1 dt", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('k', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), exp(Symbol('t', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('k', commutative=True), Integer(-1)), exp(Symbol('t', commutative=True)))"], "Equality(Mul(Symbol('k', commutative=True), Function('C_2')(Symbol('k', commutative=True), Symbol('t', commutative=True)), exp(Mul(Integer(-1), Symbol('t', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('t', commutative=True)"], "Equality(Integral(Mul(Symbol('k', commutative=True), Function('C_2')(Symbol('k', commutative=True), Symbol('t', commutative=True)), exp(Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Integral(Integer(1), Tuple(Symbol('t', commutative=True))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('k', commutative=True), Function('C_2')(Symbol('k', commutative=True), Symbol('t', commutative=True)), exp(Mul(Integer(-1), Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(\\varepsilon_0,\\mathbf{B})} = \\mathbf{B}^{\\varepsilon_0}, then obtain \\frac{\\partial}{\\partial \\varepsilon_0} (b^{\\varepsilon_0}{(\\varepsilon_0,\\mathbf{B})})^{\\varepsilon_0} = \\frac{\\partial}{\\partial \\varepsilon_0} ((\\mathbf{B}^{\\varepsilon_0})^{\\varepsilon_0})^{\\varepsilon_0}", "derivation": "b{(\\varepsilon_0,\\mathbf{B})} = \\mathbf{B}^{\\varepsilon_0} and b^{\\varepsilon_0}{(\\varepsilon_0,\\mathbf{B})} = (\\mathbf{B}^{\\varepsilon_0})^{\\varepsilon_0} and (b^{\\varepsilon_0}{(\\varepsilon_0,\\mathbf{B})})^{\\varepsilon_0} = ((\\mathbf{B}^{\\varepsilon_0})^{\\varepsilon_0})^{\\varepsilon_0} and \\frac{\\partial}{\\partial \\varepsilon_0} (b^{\\varepsilon_0}{(\\varepsilon_0,\\mathbf{B})})^{\\varepsilon_0} = \\frac{\\partial}{\\partial \\varepsilon_0} ((\\mathbf{B}^{\\varepsilon_0})^{\\varepsilon_0})^{\\varepsilon_0}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 2, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Pow(Function('b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('b')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))), Derivative(Pow(Pow(Pow(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(I,\\sigma_p)} = I + \\sigma_p, then derive - \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} - 1 = -2, then obtain (- \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} - 1)^{\\sigma_p} = (-2)^{\\sigma_p}", "derivation": "\\bar{\\h}{(I,\\sigma_p)} = I + \\sigma_p and \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} = \\frac{\\partial}{\\partial I} (I + \\sigma_p) and - \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} = - \\frac{\\partial}{\\partial I} (I + \\sigma_p) and - \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} - 1 = - \\frac{\\partial}{\\partial I} (I + \\sigma_p) - 1 and - \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} - 1 = -2 and (- \\frac{\\partial}{\\partial I} \\bar{\\h}{(I,\\sigma_p)} - 1)^{\\sigma_p} = (-2)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["minus", 3, 1], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Integer(-1)), Add(Mul(Integer(-1), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Integer(-1)), Integer(-2))"], [["power", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Integer(-1)), Symbol('\\\\sigma_p', commutative=True)), Pow(Integer(-2), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given Z{(A_{x})} = \\log{(A_{x})}, then obtain \\frac{d}{d A_{x}} (- \\log{(A_{x})} + \\frac{Z{(A_{x})}}{A_{x}}) = \\frac{d}{d A_{x}} (- \\log{(A_{x})} + \\frac{\\log{(A_{x})}}{A_{x}})", "derivation": "Z{(A_{x})} = \\log{(A_{x})} and \\frac{Z{(A_{x})}}{A_{x}} = \\frac{\\log{(A_{x})}}{A_{x}} and - \\log{(A_{x})} + \\frac{Z{(A_{x})}}{A_{x}} = - \\log{(A_{x})} + \\frac{\\log{(A_{x})}}{A_{x}} and \\frac{d}{d A_{x}} (- \\log{(A_{x})} + \\frac{Z{(A_{x})}}{A_{x}}) = \\frac{d}{d A_{x}} (- \\log{(A_{x})} + \\frac{\\log{(A_{x})}}{A_{x}})", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('A_x', commutative=True)), log(Symbol('A_x', commutative=True)))"], [["divide", 1, "Symbol('A_x', commutative=True)"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('Z')(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True))))"], [["minus", 2, "log(Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('Z')(Symbol('A_x', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True)))))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Function('Z')(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('A_x', commutative=True))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), log(Symbol('A_x', commutative=True)))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(b,k)} = - b + k, then derive \\frac{\\partial}{\\partial k} \\rho_{b}{(b,k)} = 1, then obtain \\frac{\\frac{\\partial^{2}}{\\partial b\\partial k} (- b + k)}{\\hat{H}_l{(b,k)}} = \\frac{\\frac{d}{d b} 1}{\\hat{H}_l{(b,k)}}", "derivation": "\\rho_{b}{(b,k)} = - b + k and \\frac{\\partial}{\\partial k} \\rho_{b}{(b,k)} = \\frac{\\partial}{\\partial k} (- b + k) and \\frac{\\partial}{\\partial k} \\rho_{b}{(b,k)} = 1 and \\frac{\\partial}{\\partial k} (- b + k) = 1 and \\frac{\\partial^{2}}{\\partial b\\partial k} (- b + k) = \\frac{d}{d b} 1 and \\frac{\\frac{\\partial^{2}}{\\partial b\\partial k} (- b + k)}{\\hat{H}_l{(b,k)}} = \\frac{\\frac{d}{d b} 1}{\\hat{H}_l{(b,k)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('b', commutative=True), Symbol('k', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('k', commutative=True)))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('b', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('b', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 5, "Function('\\\\hat{H}_l')(Symbol('b', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{H}_l')(Symbol('b', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\hat{H}_l')(Symbol('b', commutative=True), Symbol('k', commutative=True)), Integer(-1)), Derivative(Integer(1), Tuple(Symbol('b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given q{(Q)} = \\sin{(\\cos{(Q)})} and \\dot{y}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J},Q)} = \\dot{y}{(\\mathbf{J})} q{(Q)}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{y}{(\\mathbf{J})} q{(Q)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J},Q)}", "derivation": "q{(Q)} = \\sin{(\\cos{(Q)})} and \\dot{y}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\dot{y}{(\\mathbf{J})} \\sin{(\\cos{(Q)})} = \\sin{(\\mathbf{J})} \\sin{(\\cos{(Q)})} and \\dot{y}{(\\mathbf{J})} q{(Q)} = q{(Q)} \\sin{(\\mathbf{J})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J},Q)} = \\dot{y}{(\\mathbf{J})} q{(Q)} and \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{y}{(\\mathbf{J})} q{(Q)} = \\frac{\\partial}{\\partial \\mathbf{J}} q{(Q)} \\sin{(\\mathbf{J})} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J},Q)} = q{(Q)} \\sin{(\\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} \\dot{y}{(\\mathbf{J})} q{(Q)} = \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{J},Q)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('Q', commutative=True)), sin(cos(Symbol('Q', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 2, "sin(cos(Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), sin(cos(Symbol('Q', commutative=True)))), Mul(sin(Symbol('\\\\mathbf{J}', commutative=True)), sin(cos(Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), Function('q')(Symbol('Q', commutative=True))), Mul(Function('q')(Symbol('Q', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('Q', commutative=True)), Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), Function('q')(Symbol('Q', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), Function('q')(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Function('q')(Symbol('Q', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('Q', commutative=True)), Mul(Function('q')(Symbol('Q', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Mul(Function('\\\\dot{y}')(Symbol('\\\\mathbf{J}', commutative=True)), Function('q')(Symbol('Q', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(h)} = e^{e^{h}}, then obtain H^{2}{(h)} e^{4 h} = e^{4 h} e^{2 e^{h}}", "derivation": "H{(h)} = e^{e^{h}} and H{(h)} e^{h} = e^{h} e^{e^{h}} and H{(h)} e^{2 h} = e^{2 h} e^{e^{h}} and H^{2}{(h)} e^{4 h} = e^{4 h} e^{2 e^{h}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('h', commutative=True)), exp(exp(Symbol('h', commutative=True))))"], [["times", 1, "exp(Symbol('h', commutative=True))"], "Equality(Mul(Function('H')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Mul(exp(Symbol('h', commutative=True)), exp(exp(Symbol('h', commutative=True)))))"], [["times", 2, "exp(Symbol('h', commutative=True))"], "Equality(Mul(Function('H')(Symbol('h', commutative=True)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Mul(exp(Mul(Integer(2), Symbol('h', commutative=True))), exp(exp(Symbol('h', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Pow(Function('H')(Symbol('h', commutative=True)), Integer(2)), exp(Mul(Integer(4), Symbol('h', commutative=True)))), Mul(exp(Mul(Integer(4), Symbol('h', commutative=True))), exp(Mul(Integer(2), exp(Symbol('h', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(v_{1})} = e^{v_{1}}, then derive 1 = \\frac{\\hat{p}_0 + e^{v_{1}}}{\\int \\hat{p}{(v_{1})} dv_{1}}, then obtain 1 = \\frac{\\hat{p}_0 + \\hat{p}{(v_{1})}}{\\int \\hat{p}{(v_{1})} dv_{1}}", "derivation": "\\hat{p}{(v_{1})} = e^{v_{1}} and \\int \\hat{p}{(v_{1})} dv_{1} = \\int e^{v_{1}} dv_{1} and 1 = \\frac{\\int e^{v_{1}} dv_{1}}{\\int \\hat{p}{(v_{1})} dv_{1}} and 1 = \\frac{\\hat{p}_0 + e^{v_{1}}}{\\int \\hat{p}{(v_{1})} dv_{1}} and 1 = \\frac{\\hat{p}_0 + \\hat{p}{(v_{1})}}{\\int \\hat{p}{(v_{1})} dv_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), exp(Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integer(-1)), Integral(exp(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('v_1', commutative=True))), Pow(Integral(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(1), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Function('\\\\hat{p}')(Symbol('v_1', commutative=True))), Pow(Integral(Function('\\\\hat{p}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given A{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} and \\mathbb{I}{(f_{\\mathbf{v}},f)} = f + 2 f_{\\mathbf{v}} + A{(f_{\\mathbf{v}},f)}, then obtain \\mathbb{I}{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} + f + 2 f_{\\mathbf{v}}", "derivation": "A{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} and f + A{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} + f and f + f_{\\mathbf{v}} + A{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} + f + f_{\\mathbf{v}} and f + 2 f_{\\mathbf{v}} + A{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} + f + 2 f_{\\mathbf{v}} and \\mathbb{I}{(f_{\\mathbf{v}},f)} = f + 2 f_{\\mathbf{v}} + A{(f_{\\mathbf{v}},f)} and \\mathbb{I}{(f_{\\mathbf{v}},f)} = f \\sin{(f_{\\mathbf{v}})} + f + 2 f_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('f', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('A')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('f', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f', commutative=True)))"], [["add", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('A')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('f', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('A')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('f', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f', commutative=True), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('f', commutative=True), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('A')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\mathbb{I}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f', commutative=True)), Add(Mul(Symbol('f', commutative=True), sin(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('f', commutative=True), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} = \\frac{M_{E}}{\\dot{\\mathbf{r}}}, then obtain \\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} dM_{E} = - \\frac{M_{E}^{2}}{2 \\dot{\\mathbf{r}}^{2}} + V", "derivation": "\\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} = \\frac{M_{E}}{\\dot{\\mathbf{r}}} and \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} = \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\frac{M_{E}}{\\dot{\\mathbf{r}}} and \\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} dM_{E} = \\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\frac{M_{E}}{\\dot{\\mathbf{r}}} dM_{E} and \\int \\frac{\\partial}{\\partial \\dot{\\mathbf{r}}} \\mathbf{B}{(\\dot{\\mathbf{r}},M_{E})} dM_{E} = - \\frac{M_{E}^{2}}{2 \\dot{\\mathbf{r}}^{2}} + V", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))), Integral(Derivative(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('\\\\mathbf{B}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(1))), Tuple(Symbol('M_E', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2)), Pow(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Integer(-2))), Symbol('V', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(t_{1},q)} = \\frac{\\partial}{\\partial q} (q + t_{1}) and W{(t_{1},q)} = \\frac{\\partial}{\\partial q} (q + t_{1}), then derive W{(t_{1},q)} = 1, then obtain 1 = \\frac{1}{\\frac{\\partial}{\\partial q} (q + t_{1})}", "derivation": "\\mathbf{B}{(t_{1},q)} = \\frac{\\partial}{\\partial q} (q + t_{1}) and W{(t_{1},q)} = \\frac{\\partial}{\\partial q} (q + t_{1}) and W{(t_{1},q)} = 1 and \\frac{\\partial}{\\partial q} (q + t_{1}) = 1 and \\frac{\\frac{\\partial}{\\partial q} (q + t_{1})}{\\mathbf{B}{(t_{1},q)}} = \\frac{1}{\\mathbf{B}{(t_{1},q)}} and 1 = \\frac{1}{\\frac{\\partial}{\\partial q} (q + t_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Derivative(Add(Symbol('q', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('W')(Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Derivative(Add(Symbol('q', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Function('W')(Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('q', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1))"], [["divide", 4, "Function('\\\\mathbf{B}')(Symbol('t_1', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)), Derivative(Add(Symbol('q', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Pow(Function('\\\\mathbf{B}')(Symbol('t_1', commutative=True), Symbol('q', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Pow(Derivative(Add(Symbol('q', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)} = L_{\\varepsilon} \\theta_1, then obtain 1 = \\frac{L_{\\varepsilon} \\theta_1}{\\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)}}", "derivation": "\\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)} = L_{\\varepsilon} \\theta_1 and - L_{\\varepsilon} \\theta_1 + \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)} = 0 and - L_{\\varepsilon} \\theta_1 + 2 \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)} = \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)} and \\frac{\\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)}}{- L_{\\varepsilon} \\theta_1 + 2 \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)}} = \\frac{L_{\\varepsilon} \\theta_1}{- L_{\\varepsilon} \\theta_1 + 2 \\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)}} and 1 = \\frac{L_{\\varepsilon} \\theta_1}{\\operatorname{E_{n}}{(L_{\\varepsilon},\\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Integer(0))"], [["add", 2, "Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(2), Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(1), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True), Pow(Function('E_n')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given h{(\\omega,k)} = \\log{(- \\omega + k)}, then obtain (- \\omega + h^{k}{(\\omega,k)}) h{(\\omega,k)} = (- \\omega + \\log{(- \\omega + k)}^{k}) h{(\\omega,k)}", "derivation": "h{(\\omega,k)} = \\log{(- \\omega + k)} and h^{k}{(\\omega,k)} = \\log{(- \\omega + k)}^{k} and - \\omega + h^{k}{(\\omega,k)} = - \\omega + \\log{(- \\omega + k)}^{k} and (- \\omega + h^{k}{(\\omega,k)}) h{(\\omega,k)} = (- \\omega + \\log{(- \\omega + k)}^{k}) h{(\\omega,k)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True))))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True))))"], [["times", 3, "Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True)), Symbol('k', commutative=True))), Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Symbol('k', commutative=True))), Symbol('k', commutative=True))), Function('h')(Symbol('\\\\omega', commutative=True), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} = V_{\\mathbf{E}} \\theta_1, then obtain \\frac{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} - 1}{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)}} = \\frac{V_{\\mathbf{E}} \\theta_1 - 1}{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)}}", "derivation": "\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} = V_{\\mathbf{E}} \\theta_1 and 0 = V_{\\mathbf{E}} \\theta_1 - \\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} and -1 = V_{\\mathbf{E}} \\theta_1 - \\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} - 1 and \\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} - 1 = V_{\\mathbf{E}} \\theta_1 - 1 and \\frac{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)} - 1}{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)}} = \\frac{V_{\\mathbf{E}} \\theta_1 - 1}{\\operatorname{A_{1}}{(V_{\\mathbf{E}},\\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Integer(-1)))"], [["add", 3, "Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)))"], [["divide", 4, "Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Add(Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Pow(Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Pow(Function('A_1')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\theta{(T,\\varphi)} = - T + \\varphi, then obtain \\int (\\int \\theta{(T,\\varphi)} d\\varphi)^{T} d\\varphi = \\int (\\int (- T + \\varphi) d\\varphi)^{T} d\\varphi", "derivation": "\\theta{(T,\\varphi)} = - T + \\varphi and \\int \\theta{(T,\\varphi)} d\\varphi = \\int (- T + \\varphi) d\\varphi and (\\int \\theta{(T,\\varphi)} d\\varphi)^{T} = (\\int (- T + \\varphi) d\\varphi)^{T} and \\int (\\int \\theta{(T,\\varphi)} d\\varphi)^{T} d\\varphi = \\int (\\int (- T + \\varphi) d\\varphi)^{T} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Integral(Function('\\\\theta')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('T', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('T', commutative=True)))"], [["integrate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\theta')(Symbol('T', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Pow(Integral(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given a{(\\hat{x}_0,m)} = \\hat{x}_0 + m, then derive \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)} = 1, then obtain (- a^{m}{(\\hat{x}_0,m)} + \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)})^{m} = (1 - a^{m}{(\\hat{x}_0,m)})^{m}", "derivation": "a{(\\hat{x}_0,m)} = \\hat{x}_0 + m and \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)} = \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + m) and \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)} = 1 and \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + m) = 1 and - a^{m}{(\\hat{x}_0,m)} + \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + m) = 1 - a^{m}{(\\hat{x}_0,m)} and - a^{m}{(\\hat{x}_0,m)} + \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)} = 1 - a^{m}{(\\hat{x}_0,m)} and (- a^{m}{(\\hat{x}_0,m)} + \\frac{\\partial}{\\partial \\hat{x}_0} a{(\\hat{x}_0,m)})^{m} = (1 - a^{m}{(\\hat{x}_0,m)})^{m}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Derivative(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["power", 6, "Symbol('m', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Derivative(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Symbol('m', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Function('a')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(t,q)} = q t, then derive 2 \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - 1 = q + \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - 1, then obtain 2 \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - \\int \\frac{- t + \\tilde{g}{(t,q)}}{q t - t} dq - 1 = q + \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - \\int \\frac{- t + \\tilde{g}{(t,q)}}{q t - t} dq - 1", "derivation": "\\tilde{g}{(t,q)} = q t and - t + 2 \\tilde{g}{(t,q)} = q t - t + \\tilde{g}{(t,q)} and - t + 2 \\tilde{g}{(t,q)} + 1 = q t - t + \\tilde{g}{(t,q)} + 1 and \\frac{\\partial}{\\partial t} (- t + 2 \\tilde{g}{(t,q)} + 1) = \\frac{\\partial}{\\partial t} (q t - t + \\tilde{g}{(t,q)} + 1) and 2 \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - 1 = q + \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - 1 and 2 \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - \\int \\frac{- t + \\tilde{g}{(t,q)}}{q t - t} dq - 1 = q + \\frac{\\partial}{\\partial t} \\tilde{g}{(t,q)} - \\int \\frac{- t + \\tilde{g}{(t,q)}}{q t - t} dq - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)))), Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))))"], [["add", 2, 1], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Integer(1)), Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(-1)), Add(Symbol('q', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 5, "Integral(Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Pow(Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Integer(2), Derivative(Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Integer(-1), Integral(Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Pow(Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Integer(-1)), Add(Symbol('q', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Mul(Integer(-1), Integral(Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\tilde{g}')(Symbol('t', commutative=True), Symbol('q', commutative=True))), Pow(Add(Mul(Symbol('q', commutative=True), Symbol('t', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True))), Integer(-1))), Tuple(Symbol('q', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given W{(A,\\eta)} = \\frac{A}{\\eta}, then obtain (2 \\frac{d^{2}}{d \\etad A} 0)^{\\eta} = (\\frac{d^{2}}{d \\etad A} 0 + \\frac{\\partial^{2}}{\\partial \\eta\\partial A} (\\frac{A}{\\eta} - W{(A,\\eta)}))^{\\eta}", "derivation": "W{(A,\\eta)} = \\frac{A}{\\eta} and 0 = \\frac{A}{\\eta} - W{(A,\\eta)} and \\frac{d}{d A} 0 = \\frac{\\partial}{\\partial A} (\\frac{A}{\\eta} - W{(A,\\eta)}) and \\frac{d^{2}}{d \\etad A} 0 = \\frac{\\partial^{2}}{\\partial \\eta\\partial A} (\\frac{A}{\\eta} - W{(A,\\eta)}) and 2 \\frac{d^{2}}{d \\etad A} 0 = \\frac{d^{2}}{d \\etad A} 0 + \\frac{\\partial^{2}}{\\partial \\eta\\partial A} (\\frac{A}{\\eta} - W{(A,\\eta)}) and (2 \\frac{d^{2}}{d \\etad A} 0)^{\\eta} = (\\frac{d^{2}}{d \\etad A} 0 + \\frac{\\partial^{2}}{\\partial \\eta\\partial A} (\\frac{A}{\\eta} - W{(A,\\eta)}))^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))))"], [["minus", 1, "Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["add", 4, "Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Integer(2), Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Symbol('\\\\eta', commutative=True)), Pow(Add(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('\\\\eta', commutative=True), Integer(-1))), Mul(Integer(-1), Function('W')(Symbol('A', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\ddot{x}{(E_{n})} = \\sin{(E_{n})} and \\operatorname{v_{y}}{(E_{n})} = \\int - E_{n} dE_{n}, then obtain \\operatorname{v_{y}}{(E_{n})} = \\int (- E_{n} - \\ddot{x}{(E_{n})} + \\sin{(E_{n})}) dE_{n}", "derivation": "\\ddot{x}{(E_{n})} = \\sin{(E_{n})} and 0 = - \\ddot{x}{(E_{n})} + \\sin{(E_{n})} and - E_{n} = - E_{n} - \\ddot{x}{(E_{n})} + \\sin{(E_{n})} and \\int - E_{n} dE_{n} = \\int (- E_{n} - \\ddot{x}{(E_{n})} + \\sin{(E_{n})}) dE_{n} and \\operatorname{v_{y}}{(E_{n})} = \\int - E_{n} dE_{n} and \\operatorname{v_{y}}{(E_{n})} = \\int (- E_{n} - \\ddot{x}{(E_{n})} + \\sin{(E_{n})}) dE_{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["minus", 1, "Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))))"], [["minus", 2, "Symbol('E_n', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('E_n', commutative=True)), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))))"], [["integrate", 3, "Symbol('E_n', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('E_n', commutative=True)), Integral(Mul(Integer(-1), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('v_y')(Symbol('E_n', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('\\\\ddot{x}')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given M{(\\theta)} = \\cos{(\\log{(\\theta)})} and \\operatorname{a^{\\dagger}}{(\\theta)} = \\frac{d}{d \\theta} M{(\\theta)}, then obtain \\operatorname{a^{\\dagger}}{(\\theta)} = \\frac{d}{d \\theta} \\cos{(\\log{(\\theta)})}", "derivation": "M{(\\theta)} = \\cos{(\\log{(\\theta)})} and \\frac{d}{d \\theta} M{(\\theta)} = \\frac{d}{d \\theta} \\cos{(\\log{(\\theta)})} and \\operatorname{a^{\\dagger}}{(\\theta)} = \\frac{d}{d \\theta} M{(\\theta)} and \\operatorname{a^{\\dagger}}{(\\theta)} = \\frac{d}{d \\theta} \\cos{(\\log{(\\theta)})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\theta', commutative=True)), cos(log(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)), Derivative(Function('M')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True)), Derivative(cos(log(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})}, then derive \\int \\operatorname{F_{H}}{(E_{\\lambda})} dE_{\\lambda} = \\delta + \\sin{(E_{\\lambda})}, then obtain (\\int \\operatorname{F_{H}}{(E_{\\lambda})} dE_{\\lambda})^{\\delta} = (\\delta + \\sin{(E_{\\lambda})})^{\\delta}", "derivation": "\\operatorname{F_{H}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\int \\operatorname{F_{H}}{(E_{\\lambda})} dE_{\\lambda} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and \\int \\operatorname{F_{H}}{(E_{\\lambda})} dE_{\\lambda} = \\delta + \\sin{(E_{\\lambda})} and (\\int \\operatorname{F_{H}}{(E_{\\lambda})} dE_{\\lambda})^{\\delta} = (\\delta + \\sin{(E_{\\lambda})})^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('F_H')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_H')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\delta', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Function('F_H')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Add(Symbol('\\\\delta', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(f_{\\mathbf{v}},C)} = C - f_{\\mathbf{v}}, then derive \\int \\sigma_{x}{(f_{\\mathbf{v}},C)} dC = \\frac{C^{2}}{2} - C f_{\\mathbf{v}} + C_{2}, then obtain \\frac{\\int (C - f_{\\mathbf{v}}) dC}{\\sigma_{x}{(f_{\\mathbf{v}},C)}} = \\frac{\\frac{C^{2}}{2} - C f_{\\mathbf{v}} + C_{2}}{\\sigma_{x}{(f_{\\mathbf{v}},C)}}", "derivation": "\\sigma_{x}{(f_{\\mathbf{v}},C)} = C - f_{\\mathbf{v}} and \\int \\sigma_{x}{(f_{\\mathbf{v}},C)} dC = \\int (C - f_{\\mathbf{v}}) dC and \\int \\sigma_{x}{(f_{\\mathbf{v}},C)} dC = \\frac{C^{2}}{2} - C f_{\\mathbf{v}} + C_{2} and \\frac{\\int \\sigma_{x}{(f_{\\mathbf{v}},C)} dC}{\\sigma_{x}{(f_{\\mathbf{v}},C)}} = \\frac{\\frac{C^{2}}{2} - C f_{\\mathbf{v}} + C_{2}}{\\sigma_{x}{(f_{\\mathbf{v}},C)}} and \\frac{\\int (C - f_{\\mathbf{v}}) dC}{\\sigma_{x}{(f_{\\mathbf{v}},C)}} = \\frac{\\frac{C^{2}}{2} - C f_{\\mathbf{v}} + C_{2}}{\\sigma_{x}{(f_{\\mathbf{v}},C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('C', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('C_2', commutative=True)))"], [["divide", 3, "Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Integral(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('C', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Integer(-1)), Integral(Add(Symbol('C', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Tuple(Symbol('C', commutative=True)))), Mul(Add(Mul(Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('C', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('C_2', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('C', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\lambda{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain (\\frac{\\frac{d}{d \\mu_0} \\lambda^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}})^{\\mu_0} = (\\frac{\\frac{d}{d \\mu_0} \\cos^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}})^{\\mu_0}", "derivation": "\\lambda{(\\mu_0)} = \\cos{(\\mu_0)} and \\lambda^{\\mu_0}{(\\mu_0)} = \\cos^{\\mu_0}{(\\mu_0)} and \\frac{d}{d \\mu_0} \\lambda^{\\mu_0}{(\\mu_0)} = \\frac{d}{d \\mu_0} \\cos^{\\mu_0}{(\\mu_0)} and \\frac{\\frac{d}{d \\mu_0} \\lambda^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}} = \\frac{\\frac{d}{d \\mu_0} \\cos^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}} and (\\frac{\\frac{d}{d \\mu_0} \\lambda^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}})^{\\mu_0} = (\\frac{\\frac{d}{d \\mu_0} \\cos^{\\mu_0}{(\\mu_0)}}{2 \\cos{(\\mu_0)}})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(2), cos(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Mul(Rational(1, 2), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Derivative(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Rational(1, 2), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Derivative(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(F_{H})} = \\log{(F_{H})}, then obtain \\log{(F_{H})} \\sin{(\\int \\operatorname{f_{\\mathbf{v}}}{(F_{H})} dF_{H})} = \\log{(F_{H})} \\sin{(\\int \\log{(F_{H})} dF_{H})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(F_{H})} = \\log{(F_{H})} and \\int \\operatorname{f_{\\mathbf{v}}}{(F_{H})} dF_{H} = \\int \\log{(F_{H})} dF_{H} and \\sin{(\\int \\operatorname{f_{\\mathbf{v}}}{(F_{H})} dF_{H})} = \\sin{(\\int \\log{(F_{H})} dF_{H})} and \\log{(F_{H})} \\sin{(\\int \\operatorname{f_{\\mathbf{v}}}{(F_{H})} dF_{H})} = \\log{(F_{H})} \\sin{(\\int \\log{(F_{H})} dF_{H})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["integrate", 1, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), sin(Integral(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"], [["times", 3, "log(Symbol('F_H', commutative=True))"], "Equality(Mul(log(Symbol('F_H', commutative=True)), sin(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))), Mul(log(Symbol('F_H', commutative=True)), sin(Integral(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\delta{(\\mathbf{f})} = e^{e^{\\mathbf{f}}}, then obtain (- \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} \\delta{(\\mathbf{f})}) e^{- \\mathbf{f}} = (- \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} e^{e^{\\mathbf{f}}}) e^{- \\mathbf{f}}", "derivation": "\\delta{(\\mathbf{f})} = e^{e^{\\mathbf{f}}} and \\frac{d}{d \\mathbf{f}} \\delta{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} e^{e^{\\mathbf{f}}} and - \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} \\delta{(\\mathbf{f})} = - \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} e^{e^{\\mathbf{f}}} and (- \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} \\delta{(\\mathbf{f})}) e^{- \\mathbf{f}} = (- \\delta{(\\mathbf{f})} + e^{e^{\\mathbf{f}}} + \\frac{d}{d \\mathbf{f}} e^{e^{\\mathbf{f}}}) e^{- \\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True)), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["minus", 2, "Add(Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{f}', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Derivative(Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Derivative(exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))))"], [["divide", 3, "exp(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Derivative(Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\mathbf{f}', commutative=True))), exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Derivative(exp(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), exp(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(l)} = e^{e^{l}} and f{(l)} = \\frac{d}{d l} e^{e^{l}}, then derive \\frac{d}{d l} \\Psi^{\\dagger}{(l)} = e^{l} e^{e^{l}}, then obtain f{(l)} - e^{l} = e^{l} e^{e^{l}} - e^{l}", "derivation": "\\Psi^{\\dagger}{(l)} = e^{e^{l}} and \\frac{d}{d l} \\Psi^{\\dagger}{(l)} = \\frac{d}{d l} e^{e^{l}} and \\frac{d}{d l} \\Psi^{\\dagger}{(l)} = e^{l} e^{e^{l}} and \\frac{d}{d l} e^{e^{l}} = e^{l} e^{e^{l}} and f{(l)} = \\frac{d}{d l} e^{e^{l}} and - e^{l} + \\frac{d}{d l} e^{e^{l}} = e^{l} e^{e^{l}} - e^{l} and f{(l)} - e^{l} = e^{l} e^{e^{l}} - e^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True))))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(exp(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Mul(exp(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('f')(Symbol('l', commutative=True)), Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["minus", 4, "exp(Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('l', commutative=True))), Derivative(exp(exp(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Add(Mul(exp(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True)))), Mul(Integer(-1), exp(Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('f')(Symbol('l', commutative=True)), Mul(Integer(-1), exp(Symbol('l', commutative=True)))), Add(Mul(exp(Symbol('l', commutative=True)), exp(exp(Symbol('l', commutative=True)))), Mul(Integer(-1), exp(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(m)} = \\cos{(m)} and g{(m)} = \\frac{d}{d m} \\int \\hat{p}{(m)} dm, then derive g{(m)} = \\frac{\\partial}{\\partial m} (A_{x} + \\sin{(m)}), then obtain \\int g^{A_{x}}{(m)} dA_{x} = \\int (\\frac{\\partial}{\\partial m} (A_{x} + \\sin{(m)}))^{A_{x}} dA_{x}", "derivation": "\\hat{p}{(m)} = \\cos{(m)} and \\int \\hat{p}{(m)} dm = \\int \\cos{(m)} dm and \\frac{d}{d m} \\int \\hat{p}{(m)} dm = \\frac{d}{d m} \\int \\cos{(m)} dm and g{(m)} = \\frac{d}{d m} \\int \\hat{p}{(m)} dm and g{(m)} = \\frac{d}{d m} \\int \\cos{(m)} dm and g{(m)} = \\frac{\\partial}{\\partial m} (A_{x} + \\sin{(m)}) and g^{A_{x}}{(m)} = (\\frac{\\partial}{\\partial m} (A_{x} + \\sin{(m)}))^{A_{x}} and \\int g^{A_{x}}{(m)} dA_{x} = \\int (\\frac{\\partial}{\\partial m} (A_{x} + \\sin{(m)}))^{A_{x}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('m', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{p}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g')(Symbol('m', commutative=True)), Derivative(Integral(Function('\\\\hat{p}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('g')(Symbol('m', commutative=True)), Derivative(Integral(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Function('g')(Symbol('m', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 6, "Symbol('A_x', commutative=True)"], "Equality(Pow(Function('g')(Symbol('m', commutative=True)), Symbol('A_x', commutative=True)), Pow(Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('A_x', commutative=True)))"], [["integrate", 7, "Symbol('A_x', commutative=True)"], "Equality(Integral(Pow(Function('g')(Symbol('m', commutative=True)), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(Pow(Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{t_{2}}, then derive \\frac{\\partial}{\\partial t_{2}} \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{t_{2}} \\log{(e^{\\hat{\\mathbf{x}}})}, then obtain \\frac{\\partial}{\\partial t_{2}} (e^{\\hat{\\mathbf{x}}})^{t_{2}} = \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} \\log{(e^{\\hat{\\mathbf{x}}})}", "derivation": "\\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{t_{2}} and \\frac{\\partial}{\\partial t_{2}} \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = \\frac{\\partial}{\\partial t_{2}} (e^{\\hat{\\mathbf{x}}})^{t_{2}} and \\frac{\\partial}{\\partial t_{2}} \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = (e^{\\hat{\\mathbf{x}}})^{t_{2}} \\log{(e^{\\hat{\\mathbf{x}}})} and \\frac{\\partial}{\\partial t_{2}} \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} = \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} \\log{(e^{\\hat{\\mathbf{x}}})} and \\frac{\\partial}{\\partial t_{2}} (e^{\\hat{\\mathbf{x}}})^{t_{2}} = \\operatorname{v_{x}}{(t_{2},\\hat{\\mathbf{x}})} \\log{(e^{\\hat{\\mathbf{x}}})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('t_2', commutative=True)), log(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Pow(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Mul(Function('v_x')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(exp(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\rho,\\varepsilon)} = \\cos{(\\rho + \\varepsilon)}, then obtain (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)})^{4} = 2 (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)})^{3} \\cos{(\\rho + \\varepsilon)}", "derivation": "\\mathbf{s}{(\\rho,\\varepsilon)} = \\cos{(\\rho + \\varepsilon)} and \\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)} = 2 \\cos{(\\rho + \\varepsilon)} and (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)})^{2} = 2 (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)}) \\cos{(\\rho + \\varepsilon)} and (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)})^{4} = 2 (\\mathbf{s}{(\\rho,\\varepsilon)} + \\cos{(\\rho + \\varepsilon)})^{3} \\cos{(\\rho + \\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["add", 1, "cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 2, "Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], "Equality(Pow(Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Integer(2)), Mul(Integer(2), Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["times", 3, "Pow(Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Integer(2))"], "Equality(Pow(Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Integer(4)), Mul(Integer(2), Pow(Add(Function('\\\\mathbf{s}')(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), Integer(3)), cos(Add(Symbol('\\\\rho', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given s{(A_{z},U)} = A_{z} + U, then obtain \\frac{e^{s{(A_{z},U)}}}{2 s{(A_{z},U)}} = \\frac{e^{A_{z} + U}}{2 s{(A_{z},U)}}", "derivation": "s{(A_{z},U)} = A_{z} + U and 2 s{(A_{z},U)} = A_{z} + U + s{(A_{z},U)} and e^{s{(A_{z},U)}} = e^{A_{z} + U} and \\frac{e^{s{(A_{z},U)}}}{A_{z} + U + s{(A_{z},U)}} = \\frac{e^{A_{z} + U}}{A_{z} + U + s{(A_{z},U)}} and \\frac{e^{s{(A_{z},U)}}}{2 s{(A_{z},U)}} = \\frac{e^{A_{z} + U}}{2 s{(A_{z},U)}}", "srepr_derivation": [["get_premise", "Equality(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))"], [["add", 1, "Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))"], "Equality(Mul(Integer(2), Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))), Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True), Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))))"], [["exp", 1], "Equality(exp(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))), exp(Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True))))"], [["divide", 3, "Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True), Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True), Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))), Integer(-1)), exp(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))), Mul(Pow(Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True), Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True))), Integer(-1)), exp(Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Rational(1, 2), Pow(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)), Integer(-1)), exp(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))), Mul(Rational(1, 2), Pow(Function('s')(Symbol('A_z', commutative=True), Symbol('U', commutative=True)), Integer(-1)), exp(Add(Symbol('A_z', commutative=True), Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\Psi_{\\lambda},U)} = - U + \\Psi_{\\lambda}, then obtain (- \\frac{1}{U} - \\frac{- U + 2 \\Psi_{\\lambda}}{U^{2}}) \\log{((\\Psi_{\\lambda} + \\chi{(\\Psi_{\\lambda},U)})^{\\Psi_{\\lambda}})} = (- \\frac{1}{U} - \\frac{- U + 2 \\Psi_{\\lambda}}{U^{2}}) \\log{((- U + 2 \\Psi_{\\lambda})^{\\Psi_{\\lambda}})}", "derivation": "\\chi{(\\Psi_{\\lambda},U)} = - U + \\Psi_{\\lambda} and \\Psi_{\\lambda} + \\chi{(\\Psi_{\\lambda},U)} = - U + 2 \\Psi_{\\lambda} and (\\Psi_{\\lambda} + \\chi{(\\Psi_{\\lambda},U)})^{\\Psi_{\\lambda}} = (- U + 2 \\Psi_{\\lambda})^{\\Psi_{\\lambda}} and \\log{((\\Psi_{\\lambda} + \\chi{(\\Psi_{\\lambda},U)})^{\\Psi_{\\lambda}})} = \\log{((- U + 2 \\Psi_{\\lambda})^{\\Psi_{\\lambda}})} and (- \\frac{1}{U} - \\frac{- U + 2 \\Psi_{\\lambda}}{U^{2}}) \\log{((\\Psi_{\\lambda} + \\chi{(\\Psi_{\\lambda},U)})^{\\Psi_{\\lambda}})} = (- \\frac{1}{U} - \\frac{- U + 2 \\Psi_{\\lambda}}{U^{2}}) \\log{((- U + 2 \\Psi_{\\lambda})^{\\Psi_{\\lambda}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["add", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["power", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["log", 3], "Equality(log(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 4, "Add(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), log(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\chi')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('U', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Add(Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('U', commutative=True), Integer(-2)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), log(Pow(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\Omega,y)} = - \\Omega + y and \\mathbf{r}{(\\Omega,y)} = - \\Omega + y, then derive \\frac{\\partial}{\\partial \\Omega} \\operatorname{z^{*}}{(\\Omega,y)} = -1, then obtain 1 = - \\frac{\\partial}{\\partial \\Omega} \\mathbf{r}{(\\Omega,y)}", "derivation": "\\operatorname{z^{*}}{(\\Omega,y)} = - \\Omega + y and \\frac{\\partial}{\\partial \\Omega} \\operatorname{z^{*}}{(\\Omega,y)} = \\frac{\\partial}{\\partial \\Omega} (- \\Omega + y) and - \\Omega \\frac{\\partial}{\\partial \\Omega} \\operatorname{z^{*}}{(\\Omega,y)} = - \\Omega \\frac{\\partial}{\\partial \\Omega} (- \\Omega + y) and 1 = \\frac{\\frac{\\partial}{\\partial \\Omega} (- \\Omega + y)}{\\frac{\\partial}{\\partial \\Omega} \\operatorname{z^{*}}{(\\Omega,y)}} and \\frac{\\partial}{\\partial \\Omega} \\operatorname{z^{*}}{(\\Omega,y)} = -1 and 1 = - \\frac{\\partial}{\\partial \\Omega} (- \\Omega + y) and \\mathbf{r}{(\\Omega,y)} = - \\Omega + y and 1 = - \\frac{\\partial}{\\partial \\Omega} \\mathbf{r}{(\\Omega,y)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Derivative(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Derivative(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Pow(Derivative(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('z^*')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(1), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integer(1), Mul(Integer(-1), Derivative(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\pi,A)} = - A + \\log{(\\pi)}, then obtain - \\iint \\eta^{\\prime}{(\\pi,A)} d\\pi d\\pi = - \\iint (- A + \\log{(\\pi)}) d\\pi d\\pi", "derivation": "\\eta^{\\prime}{(\\pi,A)} = - A + \\log{(\\pi)} and \\int \\eta^{\\prime}{(\\pi,A)} d\\pi = \\int (- A + \\log{(\\pi)}) d\\pi and \\iint \\eta^{\\prime}{(\\pi,A)} d\\pi d\\pi = \\iint (- A + \\log{(\\pi)}) d\\pi d\\pi and - \\iint \\eta^{\\prime}{(\\pi,A)} d\\pi d\\pi = - \\iint (- A + \\log{(\\pi)}) d\\pi d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Add(Mul(Integer(-1), Symbol('A', commutative=True)), log(Symbol('\\\\pi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Mul(Integer(-1), Integral(Add(Mul(Integer(-1), Symbol('A', commutative=True)), log(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given q{(V)} = e^{\\sin{(V)}}, then obtain - \\cos{(V)} + \\frac{d}{d V} q{(V)} + 1 = e^{\\sin{(V)}} \\cos{(V)} - \\cos{(V)} + 1", "derivation": "q{(V)} = e^{\\sin{(V)}} and q{(V)} - \\sin{(V)} = e^{\\sin{(V)}} - \\sin{(V)} and V + q{(V)} - \\sin{(V)} = V + e^{\\sin{(V)}} - \\sin{(V)} and \\frac{d}{d V} (V + q{(V)} - \\sin{(V)}) = \\frac{d}{d V} (V + e^{\\sin{(V)}} - \\sin{(V)}) and - \\cos{(V)} + \\frac{d}{d V} q{(V)} + 1 = e^{\\sin{(V)}} \\cos{(V)} - \\cos{(V)} + 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('V', commutative=True)), exp(sin(Symbol('V', commutative=True))))"], [["minus", 1, "sin(Symbol('V', commutative=True))"], "Equality(Add(Function('q')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Add(exp(sin(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True)))))"], [["add", 2, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('q')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Add(Symbol('V', commutative=True), exp(sin(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True)))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), Function('q')(Symbol('V', commutative=True)), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Symbol('V', commutative=True), exp(sin(Symbol('V', commutative=True))), Mul(Integer(-1), sin(Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(-1), cos(Symbol('V', commutative=True))), Derivative(Function('q')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Integer(1)), Add(Mul(exp(sin(Symbol('V', commutative=True))), cos(Symbol('V', commutative=True))), Mul(Integer(-1), cos(Symbol('V', commutative=True))), Integer(1)))"]]}, {"prompt": "Given J{(\\chi)} = \\log{(\\chi)}, then obtain 2 \\log{(J{(\\chi)})}^{2} = (\\log{(J{(\\chi)})} + \\log{(\\log{(\\chi)})}) \\log{(J{(\\chi)})}", "derivation": "J{(\\chi)} = \\log{(\\chi)} and \\log{(J{(\\chi)})} = \\log{(\\log{(\\chi)})} and 2 \\log{(J{(\\chi)})} = \\log{(J{(\\chi)})} + \\log{(\\log{(\\chi)})} and 2 \\log{(J{(\\chi)})}^{2} = (\\log{(J{(\\chi)})} + \\log{(\\log{(\\chi)})}) \\log{(J{(\\chi)})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["log", 1], "Equality(log(Function('J')(Symbol('\\\\chi', commutative=True))), log(log(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "log(Function('J')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integer(2), log(Function('J')(Symbol('\\\\chi', commutative=True)))), Add(log(Function('J')(Symbol('\\\\chi', commutative=True))), log(log(Symbol('\\\\chi', commutative=True)))))"], [["times", 3, "log(Function('J')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integer(2), Pow(log(Function('J')(Symbol('\\\\chi', commutative=True))), Integer(2))), Mul(Add(log(Function('J')(Symbol('\\\\chi', commutative=True))), log(log(Symbol('\\\\chi', commutative=True)))), log(Function('J')(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\eta)} = \\sin{(\\eta)}, then derive \\int (\\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}})^{\\eta} d\\eta + \\frac{1}{x} = \\eta + \\phi_1 + \\frac{1}{x}, then obtain (\\int 1 d\\eta + \\frac{1}{x})^{\\phi_1} = (\\eta + \\phi_1 + \\frac{1}{x})^{\\phi_1}", "derivation": "\\dot{\\mathbf{r}}{(\\eta)} = \\sin{(\\eta)} and \\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}} = 1 and (\\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}})^{\\eta} = 1 and \\int (\\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}})^{\\eta} d\\eta = \\int 1 d\\eta and \\int (\\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}})^{\\eta} d\\eta + \\frac{1}{x} = \\int 1 d\\eta + \\frac{1}{x} and \\int (\\frac{\\dot{\\mathbf{r}}{(\\eta)}}{\\sin{(\\eta)}})^{\\eta} d\\eta + \\frac{1}{x} = \\eta + \\phi_1 + \\frac{1}{x} and \\int 1 d\\eta + \\frac{1}{x} = \\eta + \\phi_1 + \\frac{1}{x} and (\\int 1 d\\eta + \\frac{1}{x})^{\\phi_1} = (\\eta + \\phi_1 + \\frac{1}{x})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\eta', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(-1))), Symbol('\\\\eta', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(-1))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))))"], [["add", 4, "Pow(Symbol('x', commutative=True), Integer(-1))"], "Equality(Add(Integral(Pow(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(-1))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('x', commutative=True), Integer(-1))), Add(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["evaluate_integrals", 5], "Equality(Add(Integral(Pow(Mul(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\eta', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(-1))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('x', commutative=True), Integer(-1))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('x', commutative=True), Integer(-1))), Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["power", 7, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\phi_1', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(v_{1},b)} = b - v_{1} and \\mathbf{S}{(v_{1})} = v_{1}, then obtain (\\frac{\\partial}{\\partial b} (b - 2 v_{1}) \\mathbf{S}{(v_{1})})^{b} = (\\frac{\\partial}{\\partial b} v_{1} (b - 2 v_{1}))^{b}", "derivation": "\\operatorname{F_{x}}{(v_{1},b)} = b - v_{1} and - v_{1} + \\operatorname{F_{x}}{(v_{1},b)} = b - 2 v_{1} and \\mathbf{S}{(v_{1})} = v_{1} and (- v_{1} + \\operatorname{F_{x}}{(v_{1},b)}) \\mathbf{S}{(v_{1})} = v_{1} (- v_{1} + \\operatorname{F_{x}}{(v_{1},b)}) and (b - 2 v_{1}) \\mathbf{S}{(v_{1})} = v_{1} (b - 2 v_{1}) and \\frac{\\partial}{\\partial b} (b - 2 v_{1}) \\mathbf{S}{(v_{1})} = \\frac{\\partial}{\\partial b} v_{1} (b - 2 v_{1}) and (\\frac{\\partial}{\\partial b} (b - 2 v_{1}) \\mathbf{S}{(v_{1})})^{b} = (\\frac{\\partial}{\\partial b} v_{1} (b - 2 v_{1}))^{b}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('v_1', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('F_x')(Symbol('v_1', commutative=True), Symbol('b', commutative=True))), Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('F_x')(Symbol('v_1', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('F_x')(Symbol('v_1', commutative=True), Symbol('b', commutative=True))), Function('\\\\mathbf{S}')(Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('F_x')(Symbol('v_1', commutative=True), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))), Function('\\\\mathbf{S}')(Symbol('v_1', commutative=True))), Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)))))"], [["differentiate", 5, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))), Function('\\\\mathbf{S}')(Symbol('v_1', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["power", 6, "Symbol('b', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True))), Function('\\\\mathbf{S}')(Symbol('v_1', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True)), Pow(Derivative(Mul(Symbol('v_1', commutative=True), Add(Symbol('b', commutative=True), Mul(Integer(-1), Integer(2), Symbol('v_1', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(A_{1},\\lambda)} = e^{\\frac{A_{1}}{\\lambda}}, then obtain \\frac{\\partial}{\\partial A_{1}} \\Psi_{nl}{(A_{1},\\lambda)} - \\frac{\\partial}{\\partial A_{1}} e^{\\frac{A_{1}}{\\lambda}} - \\frac{1}{\\lambda} = - \\frac{1}{\\lambda}", "derivation": "\\Psi_{nl}{(A_{1},\\lambda)} = e^{\\frac{A_{1}}{\\lambda}} and \\frac{\\partial}{\\partial A_{1}} \\Psi_{nl}{(A_{1},\\lambda)} = \\frac{\\partial}{\\partial A_{1}} e^{\\frac{A_{1}}{\\lambda}} and \\frac{\\partial}{\\partial A_{1}} \\Psi_{nl}{(A_{1},\\lambda)} - \\frac{1}{\\lambda} = \\frac{\\partial}{\\partial A_{1}} e^{\\frac{A_{1}}{\\lambda}} - \\frac{1}{\\lambda} and \\frac{\\partial}{\\partial A_{1}} \\Psi_{nl}{(A_{1},\\lambda)} - \\frac{\\partial}{\\partial A_{1}} e^{\\frac{A_{1}}{\\lambda}} - \\frac{1}{\\lambda} = - \\frac{1}{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Add(Derivative(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))))"], [["minus", 3, "Derivative(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('A_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(z)} = \\log{(\\cos{(z)})}, then obtain \\int \\frac{d}{d z} \\int \\dot{y}{(z)} dz dz = \\int \\frac{d}{d z} \\int \\log{(\\cos{(z)})} dz dz", "derivation": "\\dot{y}{(z)} = \\log{(\\cos{(z)})} and \\int \\dot{y}{(z)} dz = \\int \\log{(\\cos{(z)})} dz and \\frac{d}{d z} \\int \\dot{y}{(z)} dz = \\frac{d}{d z} \\int \\log{(\\cos{(z)})} dz and \\int \\frac{d}{d z} \\int \\dot{y}{(z)} dz dz = \\int \\frac{d}{d z} \\int \\log{(\\cos{(z)})} dz dz", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('z', commutative=True)), log(cos(Symbol('z', commutative=True))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(log(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\dot{y}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integral(log(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('z', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\dot{y}')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))), Integral(Derivative(Integral(log(cos(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(f_{E},i)} = i + \\log{(f_{E})}, then obtain (\\frac{\\int 0 df_{E}}{\\log{(f_{E})}})^{f_{E}} = (\\frac{\\int (i + \\log{(f_{E})}) (i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})}) df_{E}}{\\log{(f_{E})}})^{f_{E}}", "derivation": "\\operatorname{E_{x}}{(f_{E},i)} = i + \\log{(f_{E})} and 0 = i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})} and 0 = (i + \\log{(f_{E})}) (i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})}) and \\int 0 df_{E} = \\int (i + \\log{(f_{E})}) (i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})}) df_{E} and \\frac{\\int 0 df_{E}}{\\log{(f_{E})}} = \\frac{\\int (i + \\log{(f_{E})}) (i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})}) df_{E}}{\\log{(f_{E})}} and (\\frac{\\int 0 df_{E}}{\\log{(f_{E})}})^{f_{E}} = (\\frac{\\int (i + \\log{(f_{E})}) (i - \\operatorname{E_{x}}{(f_{E},i)} + \\log{(f_{E})}) df_{E}}{\\log{(f_{E})}})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True))))"], [["minus", 1, "Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))"], "Equality(Integer(0), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))), log(Symbol('f_E', commutative=True))))"], [["times", 2, "Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True)))"], "Equality(Integer(0), Mul(Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))), log(Symbol('f_E', commutative=True)))))"], [["integrate", 3, "Symbol('f_E', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True))))"], [["divide", 4, "log(Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(log(Symbol('f_E', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('f_E', commutative=True)))), Mul(Pow(log(Symbol('f_E', commutative=True)), Integer(-1)), Integral(Mul(Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True)))))"], [["power", 5, "Symbol('f_E', commutative=True)"], "Equality(Pow(Mul(Pow(log(Symbol('f_E', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Pow(Mul(Pow(log(Symbol('f_E', commutative=True)), Integer(-1)), Integral(Mul(Add(Symbol('i', commutative=True), log(Symbol('f_E', commutative=True))), Add(Symbol('i', commutative=True), Mul(Integer(-1), Function('E_x')(Symbol('f_E', commutative=True), Symbol('i', commutative=True))), log(Symbol('f_E', commutative=True)))), Tuple(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given E{(Z)} = \\log{(Z)}, then derive (\\frac{Z \\frac{d}{d Z} E{(Z)}}{E{(Z)}} + \\log{(E{(Z)})}) E^{Z}{(Z)} = (\\log{(\\log{(Z)})} + \\frac{1}{\\log{(Z)}}) \\log{(Z)}^{Z}, then obtain (\\frac{Z \\frac{d}{d Z} E{(Z)}}{E{(Z)}} + \\log{(E{(Z)})}) \\log{(Z)}^{Z} = (\\log{(\\log{(Z)})} + \\frac{1}{\\log{(Z)}}) \\log{(Z)}^{Z}", "derivation": "E{(Z)} = \\log{(Z)} and E^{Z}{(Z)} = \\log{(Z)}^{Z} and \\frac{d}{d Z} E^{Z}{(Z)} = \\frac{d}{d Z} \\log{(Z)}^{Z} and (\\frac{Z \\frac{d}{d Z} E{(Z)}}{E{(Z)}} + \\log{(E{(Z)})}) E^{Z}{(Z)} = (\\log{(\\log{(Z)})} + \\frac{1}{\\log{(Z)}}) \\log{(Z)}^{Z} and (\\frac{Z \\frac{d}{d Z} E{(Z)}}{E{(Z)}} + \\log{(E{(Z)})}) \\log{(Z)}^{Z} = (\\log{(\\log{(Z)})} + \\frac{1}{\\log{(Z)}}) \\log{(Z)}^{Z}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('E')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Function('E')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('Z', commutative=True), Pow(Function('E')(Symbol('Z', commutative=True)), Integer(-1)), Derivative(Function('E')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), log(Function('E')(Symbol('Z', commutative=True)))), Pow(Function('E')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Add(log(log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('Z', commutative=True), Pow(Function('E')(Symbol('Z', commutative=True)), Integer(-1)), Derivative(Function('E')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), log(Function('E')(Symbol('Z', commutative=True)))), Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Mul(Add(log(log(Symbol('Z', commutative=True))), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Pow(log(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\hat{x})} = e^{\\hat{x}} and E{(\\hat{x})} = \\int \\omega{(\\hat{x})} d\\hat{x}, then obtain \\int (2 E{(\\hat{x})} + \\int \\omega{(\\hat{x})} d\\hat{x}) d\\hat{x} = \\int (2 E{(\\hat{x})} + \\int e^{\\hat{x}} d\\hat{x}) d\\hat{x}", "derivation": "\\omega{(\\hat{x})} = e^{\\hat{x}} and \\int \\omega{(\\hat{x})} d\\hat{x} = \\int e^{\\hat{x}} d\\hat{x} and E{(\\hat{x})} = \\int \\omega{(\\hat{x})} d\\hat{x} and E{(\\hat{x})} = \\int e^{\\hat{x}} d\\hat{x} and 3 E{(\\hat{x})} = 2 E{(\\hat{x})} + \\int e^{\\hat{x}} d\\hat{x} and 3 E{(\\hat{x})} = 2 E{(\\hat{x})} + \\int \\omega{(\\hat{x})} d\\hat{x} and 2 E{(\\hat{x})} + \\int \\omega{(\\hat{x})} d\\hat{x} = 2 E{(\\hat{x})} + \\int e^{\\hat{x}} d\\hat{x} and \\int (2 E{(\\hat{x})} + \\int \\omega{(\\hat{x})} d\\hat{x}) d\\hat{x} = \\int (2 E{(\\hat{x})} + \\int e^{\\hat{x}} d\\hat{x}) d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), exp(Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Integral(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["add", 4, "Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Integer(3), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Integer(3), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["integrate", 7, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(Function('\\\\omega')(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Mul(Integer(2), Function('E')(Symbol('\\\\hat{x}', commutative=True))), Integral(exp(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} = - \\mathbf{B} + e^{t_{1}} and \\operatorname{C_{2}}{(t_{1})} = t_{1}, then obtain \\int (\\mathbf{B} + \\operatorname{C_{2}}{(t_{1})}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} dt_{1} = \\int (\\mathbf{B} + t_{1}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} dt_{1}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{B},t_{1})} = - \\mathbf{B} + e^{t_{1}} and \\operatorname{C_{2}}{(t_{1})} = t_{1} and \\mathbf{B} + \\operatorname{C_{2}}{(t_{1})} = \\mathbf{B} + t_{1} and (- \\mathbf{B} + e^{t_{1}}) (\\mathbf{B} + \\operatorname{C_{2}}{(t_{1})}) = (- \\mathbf{B} + e^{t_{1}}) (\\mathbf{B} + t_{1}) and (\\mathbf{B} + \\operatorname{C_{2}}{(t_{1})}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} = (\\mathbf{B} + t_{1}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} and \\int (\\mathbf{B} + \\operatorname{C_{2}}{(t_{1})}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} dt_{1} = \\int (\\mathbf{B} + t_{1}) \\operatorname{F_{x}}{(\\mathbf{B},t_{1})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('C_2')(Symbol('t_1', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('t_1', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('t_1', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Function('C_2')(Symbol('t_1', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('t_1', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('C_2')(Symbol('t_1', commutative=True))), Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True))), Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True))))"], [["integrate", 5, "Symbol('t_1', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Function('C_2')(Symbol('t_1', commutative=True))), Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))), Integral(Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True)), Function('F_x')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\Psi,E_{x})} = \\frac{E_{x}}{\\Psi} and \\operatorname{z^{*}}{(\\Psi,E_{x})} = - \\mathbb{I}{(\\Psi,E_{x})}, then obtain \\operatorname{z^{*}}{(\\Psi,E_{x})} = - \\frac{E_{x}}{\\Psi}", "derivation": "\\mathbb{I}{(\\Psi,E_{x})} = \\frac{E_{x}}{\\Psi} and - \\mathbb{I}{(\\Psi,E_{x})} = - \\frac{E_{x}}{\\Psi} and \\operatorname{z^{*}}{(\\Psi,E_{x})} = - \\mathbb{I}{(\\Psi,E_{x})} and \\operatorname{z^{*}}{(\\Psi,E_{x})} = - \\frac{E_{x}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True), Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\Psi', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('\\\\Psi', commutative=True), Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('z^*')(Symbol('\\\\Psi', commutative=True), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\omega{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\mathbf{J}_f{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain \\cos{(2 \\mathbf{J}_f{(\\hat{H}_{\\lambda})} \\omega{(\\hat{H}_{\\lambda})})} = \\cos{(2 \\omega^{2}{(\\hat{H}_{\\lambda})})}", "derivation": "\\omega{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\mathbf{J}_f{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and (\\omega{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}) \\mathbf{J}_f{(\\hat{H}_{\\lambda})} = (\\omega{(\\hat{H}_{\\lambda})} + \\log{(\\hat{H}_{\\lambda})}) \\log{(\\hat{H}_{\\lambda})} and 2 \\mathbf{J}_f{(\\hat{H}_{\\lambda})} \\omega{(\\hat{H}_{\\lambda})} = 2 \\omega^{2}{(\\hat{H}_{\\lambda})} and \\cos{(2 \\mathbf{J}_f{(\\hat{H}_{\\lambda})} \\omega{(\\hat{H}_{\\lambda})})} = \\cos{(2 \\omega^{2}{(\\hat{H}_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["times", 2, "Add(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Add(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Add(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(2), Pow(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2))))"], [["cos", 4], "Equality(cos(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), cos(Mul(Integer(2), Pow(Function('\\\\omega')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given G{(\\mathbf{J}_M,u)} = \\mathbf{J}_M u, then obtain \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M u + 2 G{(\\mathbf{J}_M,u)}) = \\frac{\\partial}{\\partial \\mathbf{J}_M} 3 \\mathbf{J}_M u", "derivation": "G{(\\mathbf{J}_M,u)} = \\mathbf{J}_M u and \\mathbf{J}_M u + G{(\\mathbf{J}_M,u)} = 2 \\mathbf{J}_M u and 2 \\mathbf{J}_M u + G{(\\mathbf{J}_M,u)} = 3 \\mathbf{J}_M u and \\frac{\\partial}{\\partial \\mathbf{J}_M} (2 \\mathbf{J}_M u + G{(\\mathbf{J}_M,u)}) = \\frac{\\partial}{\\partial \\mathbf{J}_M} 3 \\mathbf{J}_M u and \\frac{\\partial}{\\partial \\mathbf{J}_M} (\\mathbf{J}_M u + 2 G{(\\mathbf{J}_M,u)}) = \\frac{\\partial}{\\partial \\mathbf{J}_M} 3 \\mathbf{J}_M u", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Function('G')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Function('G')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True))), Mul(Integer(3), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Function('G')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Integer(3), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Mul(Integer(2), Function('G')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Mul(Integer(3), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(F_{H})} = \\sin{(\\sin{(F_{H})})} and \\operatorname{A_{z}}{(F_{H})} = \\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H}, then obtain \\frac{\\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H}}{\\operatorname{A_{z}}{(F_{H})}} = 1", "derivation": "\\Omega{(F_{H})} = \\sin{(\\sin{(F_{H})})} and F_{H} \\Omega{(F_{H})} = F_{H} \\sin{(\\sin{(F_{H})})} and \\int F_{H} \\Omega{(F_{H})} dF_{H} = \\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H} and \\operatorname{A_{z}}{(F_{H})} = \\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H} and \\frac{\\int F_{H} \\Omega{(F_{H})} dF_{H}}{\\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H}} = 1 and \\frac{\\int F_{H} \\Omega{(F_{H})} dF_{H}}{\\operatorname{A_{z}}{(F_{H})}} = 1 and \\frac{\\int F_{H} \\sin{(\\sin{(F_{H})})} dF_{H}}{\\operatorname{A_{z}}{(F_{H})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('F_H', commutative=True)), sin(sin(Symbol('F_H', commutative=True))))"], [["times", 1, "Symbol('F_H', commutative=True)"], "Equality(Mul(Symbol('F_H', commutative=True), Function('\\\\Omega')(Symbol('F_H', commutative=True))), Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Symbol('F_H', commutative=True), Function('\\\\Omega')(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('F_H', commutative=True)), Integral(Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))))"], [["divide", 3, "Integral(Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Mul(Integral(Mul(Symbol('F_H', commutative=True), Function('\\\\Omega')(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Pow(Integral(Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Function('A_z')(Symbol('F_H', commutative=True)), Integer(-1)), Integral(Mul(Symbol('F_H', commutative=True), Function('\\\\Omega')(Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Function('A_z')(Symbol('F_H', commutative=True)), Integer(-1)), Integral(Mul(Symbol('F_H', commutative=True), sin(sin(Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(Z,\\dot{x})} = Z^{\\dot{x}}, then obtain Z^{\\dot{x}} \\dot{x} \\operatorname{E_{n}}{(Z,\\dot{x})} = Z^{2 \\dot{x}} \\dot{x}", "derivation": "\\operatorname{E_{n}}{(Z,\\dot{x})} = Z^{\\dot{x}} and \\dot{x} \\operatorname{E_{n}}{(Z,\\dot{x})} = Z^{\\dot{x}} \\dot{x} and \\dot{x} \\operatorname{E_{n}}^{2}{(Z,\\dot{x})} = Z^{\\dot{x}} \\dot{x} \\operatorname{E_{n}}{(Z,\\dot{x})} and Z^{\\dot{x}} \\dot{x} \\operatorname{E_{n}}{(Z,\\dot{x})} = Z^{2 \\dot{x}} \\dot{x}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\dot{x}', commutative=True), Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Integer(2))), Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True), Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True), Function('E_n')(Symbol('Z', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Mul(Integer(2), Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(f_{\\mathbf{p}})} = \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}}, then derive \\mathbf{H}{(f_{\\mathbf{p}})} = \\mu + e^{f_{\\mathbf{p}}}, then obtain \\frac{\\mathbf{H}{(f_{\\mathbf{p}})}}{\\mu + e^{f_{\\mathbf{p}}}} = 1", "derivation": "\\mathbf{H}{(f_{\\mathbf{p}})} = \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} and \\mathbf{H}{(f_{\\mathbf{p}})} = \\mu + e^{f_{\\mathbf{p}}} and \\frac{\\mathbf{H}{(f_{\\mathbf{p}})}}{\\mu + e^{f_{\\mathbf{p}}}} = \\frac{\\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}}}{\\mu + e^{f_{\\mathbf{p}}}} and \\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}} = \\mu + e^{f_{\\mathbf{p}}} and \\frac{\\mathbf{H}{(f_{\\mathbf{p}})}}{\\int e^{f_{\\mathbf{p}}} df_{\\mathbf{p}}} = 1 and \\frac{\\mathbf{H}{(f_{\\mathbf{p}})}}{\\mu + e^{f_{\\mathbf{p}}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1)), Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Integral(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1)), Function('\\\\mathbf{H}')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\hat{x}{(i,\\varepsilon)} = \\varepsilon^{i} and x{(i,\\varepsilon)} = \\int (\\frac{\\partial}{\\partial i} \\hat{x}{(i,\\varepsilon)})^{i} di, then obtain \\frac{\\partial}{\\partial \\varepsilon} x{(i,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} \\int (\\frac{\\partial}{\\partial i} \\varepsilon^{i})^{i} di", "derivation": "\\hat{x}{(i,\\varepsilon)} = \\varepsilon^{i} and \\frac{\\partial}{\\partial i} \\hat{x}{(i,\\varepsilon)} = \\frac{\\partial}{\\partial i} \\varepsilon^{i} and (\\frac{\\partial}{\\partial i} \\hat{x}{(i,\\varepsilon)})^{i} = (\\frac{\\partial}{\\partial i} \\varepsilon^{i})^{i} and \\int (\\frac{\\partial}{\\partial i} \\hat{x}{(i,\\varepsilon)})^{i} di = \\int (\\frac{\\partial}{\\partial i} \\varepsilon^{i})^{i} di and x{(i,\\varepsilon)} = \\int (\\frac{\\partial}{\\partial i} \\hat{x}{(i,\\varepsilon)})^{i} di and x{(i,\\varepsilon)} = \\int (\\frac{\\partial}{\\partial i} \\varepsilon^{i})^{i} di and \\frac{\\partial}{\\partial \\varepsilon} x{(i,\\varepsilon)} = \\frac{\\partial}{\\partial \\varepsilon} \\int (\\frac{\\partial}{\\partial i} \\varepsilon^{i})^{i} di", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{x}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\hat{x}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integral(Pow(Derivative(Function('\\\\hat{x}')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integral(Pow(Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('i', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(Integral(Pow(Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\theta)} = e^{\\cos{(\\theta)}}, then obtain 1 = - \\operatorname{A_{2}}{(\\theta)} e^{\\cos{(\\theta)}} + e^{2 \\cos{(\\theta)}} + 1", "derivation": "\\operatorname{A_{2}}{(\\theta)} = e^{\\cos{(\\theta)}} and \\operatorname{A_{2}}{(\\theta)} e^{\\cos{(\\theta)}} = e^{2 \\cos{(\\theta)}} and 0 = - \\operatorname{A_{2}}{(\\theta)} e^{\\cos{(\\theta)}} + e^{2 \\cos{(\\theta)}} and 1 = - \\operatorname{A_{2}}{(\\theta)} e^{\\cos{(\\theta)}} + e^{2 \\cos{(\\theta)}} + 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\theta', commutative=True)), exp(cos(Symbol('\\\\theta', commutative=True))))"], [["times", 1, "exp(cos(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Function('A_2')(Symbol('\\\\theta', commutative=True)), exp(cos(Symbol('\\\\theta', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))))"], [["minus", 2, "Mul(Function('A_2')(Symbol('\\\\theta', commutative=True)), exp(cos(Symbol('\\\\theta', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\theta', commutative=True)), exp(cos(Symbol('\\\\theta', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True))))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(1), Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\theta', commutative=True)), exp(cos(Symbol('\\\\theta', commutative=True)))), exp(Mul(Integer(2), cos(Symbol('\\\\theta', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\varphi{(T)} = \\sin{(T)}, then derive \\int \\varphi{(T)} dT = \\mathbf{v} - \\cos{(T)}, then obtain (\\mathbf{v} - \\cos{(T)}) \\cos{(T)} = \\cos{(T)} \\int \\sin{(T)} dT", "derivation": "\\varphi{(T)} = \\sin{(T)} and \\int \\varphi{(T)} dT = \\int \\sin{(T)} dT and \\int \\varphi{(T)} dT = \\mathbf{v} - \\cos{(T)} and \\cos{(T)} \\int \\varphi{(T)} dT = (\\mathbf{v} - \\cos{(T)}) \\cos{(T)} and \\int \\sin{(T)} dT = \\mathbf{v} - \\cos{(T)} and \\cos{(T)} \\int \\varphi{(T)} dT = \\cos{(T)} \\int \\sin{(T)} dT and (\\mathbf{v} - \\cos{(T)}) \\cos{(T)} = \\cos{(T)} \\int \\sin{(T)} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["times", 3, "cos(Symbol('T', commutative=True))"], "Equality(Mul(cos(Symbol('T', commutative=True)), Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(cos(Symbol('T', commutative=True)), Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(cos(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Integer(-1), cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True))), Mul(cos(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\rho_{b}{(\\chi)} = e^{e^{\\chi}}, then obtain (\\frac{d}{d \\chi} \\rho_{b}{(\\chi)})^{\\chi} = (e^{\\chi} e^{e^{\\chi}})^{\\chi}", "derivation": "\\rho_{b}{(\\chi)} = e^{e^{\\chi}} and \\frac{d}{d \\chi} \\rho_{b}{(\\chi)} = \\frac{d}{d \\chi} e^{e^{\\chi}} and (\\frac{d}{d \\chi} \\rho_{b}{(\\chi)})^{\\chi} = (\\frac{d}{d \\chi} e^{e^{\\chi}})^{\\chi} and (\\frac{d}{d \\chi} \\rho_{b}{(\\chi)})^{\\chi} = (e^{\\chi} e^{e^{\\chi}})^{\\chi}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\chi', commutative=True)), exp(exp(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(exp(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\rho_b')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Pow(Derivative(exp(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\rho_b')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Pow(Mul(exp(Symbol('\\\\chi', commutative=True)), exp(exp(Symbol('\\\\chi', commutative=True)))), Symbol('\\\\chi', commutative=True)))"]]}, {"prompt": "Given G{(n,\\phi_1)} = \\phi_1 + n, then obtain 2 G{(n,\\phi_1)} = \\phi_1 + n + G{(n,\\phi_1)}", "derivation": "G{(n,\\phi_1)} = \\phi_1 + n and \\phi_1 + n + G{(n,\\phi_1)} = 2 \\phi_1 + 2 n and \\phi_1 - n + G{(n,\\phi_1)} = 2 \\phi_1 and 2 G{(n,\\phi_1)} = 2 \\phi_1 + 2 n and 2 G{(n,\\phi_1)} = \\phi_1 + n + G{(n,\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True))"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True), Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('n', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Symbol('n', commutative=True))"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)), Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(2), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True), Function('G')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hbar)} = \\cos{(\\log{(\\hbar)})}, then obtain - \\cos{(\\log{(\\hbar)})} + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\hbar)}}{\\hbar} d\\hbar = \\psi^* + \\sin{(\\log{(\\hbar)})} - \\cos{(\\log{(\\hbar)})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hbar)} = \\cos{(\\log{(\\hbar)})} and \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\hbar)}}{\\hbar} = \\frac{\\cos{(\\log{(\\hbar)})}}{\\hbar} and \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\hbar)}}{\\hbar} d\\hbar = \\int \\frac{\\cos{(\\log{(\\hbar)})}}{\\hbar} d\\hbar and - \\cos{(\\log{(\\hbar)})} + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\hbar)}}{\\hbar} d\\hbar = - \\cos{(\\log{(\\hbar)})} + \\int \\frac{\\cos{(\\log{(\\hbar)})}}{\\hbar} d\\hbar and - \\cos{(\\log{(\\hbar)})} + \\int \\frac{\\operatorname{f_{\\mathbf{p}}}{(\\hbar)}}{\\hbar} d\\hbar = \\psi^* + \\sin{(\\log{(\\hbar)})} - \\cos{(\\log{(\\hbar)})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True)), cos(log(Symbol('\\\\hbar', commutative=True))))"], [["divide", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(log(Symbol('\\\\hbar', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(log(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 3, "cos(log(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(log(Symbol('\\\\hbar', commutative=True)))), Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), cos(log(Symbol('\\\\hbar', commutative=True)))), Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(log(Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), cos(log(Symbol('\\\\hbar', commutative=True)))), Integral(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Symbol('\\\\psi^*', commutative=True), sin(log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), cos(log(Symbol('\\\\hbar', commutative=True))))))"]]}, {"prompt": "Given y{(C_{1})} = \\cos{(\\cos{(C_{1})})}, then derive \\frac{d}{d C_{1}} y{(C_{1})} - 1 = \\sin{(C_{1})} \\sin{(\\cos{(C_{1})})} - 1, then obtain (\\frac{d}{d C_{1}} y{(C_{1})} - 1)^{C_{1}} = (\\sin{(C_{1})} \\sin{(\\cos{(C_{1})})} - 1)^{C_{1}}", "derivation": "y{(C_{1})} = \\cos{(\\cos{(C_{1})})} and \\frac{d}{d C_{1}} y{(C_{1})} = \\frac{d}{d C_{1}} \\cos{(\\cos{(C_{1})})} and \\frac{d}{d C_{1}} y{(C_{1})} - 1 = \\frac{d}{d C_{1}} \\cos{(\\cos{(C_{1})})} - 1 and \\frac{d}{d C_{1}} y{(C_{1})} - 1 = \\sin{(C_{1})} \\sin{(\\cos{(C_{1})})} - 1 and (\\frac{d}{d C_{1}} y{(C_{1})} - 1)^{C_{1}} = (\\sin{(C_{1})} \\sin{(\\cos{(C_{1})})} - 1)^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('C_1', commutative=True)), cos(cos(Symbol('C_1', commutative=True))))"], [["differentiate", 1, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('y')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('y')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1)), Add(Mul(sin(Symbol('C_1', commutative=True)), sin(cos(Symbol('C_1', commutative=True)))), Integer(-1)))"], [["power", 4, "Symbol('C_1', commutative=True)"], "Equality(Pow(Add(Derivative(Function('y')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Integer(-1)), Symbol('C_1', commutative=True)), Pow(Add(Mul(sin(Symbol('C_1', commutative=True)), sin(cos(Symbol('C_1', commutative=True)))), Integer(-1)), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given V{(\\rho,B,S)} = \\frac{B}{S} - \\rho, then derive g_{\\varepsilon} + V{(\\rho,B,S)} = \\frac{B}{S} + \\chi, then obtain \\log{(g_{\\varepsilon} + V{(\\rho,B,S)})} = \\log{(\\frac{B}{S} + \\chi)}", "derivation": "V{(\\rho,B,S)} = \\frac{B}{S} - \\rho and \\frac{\\partial}{\\partial B} V{(\\rho,B,S)} = \\frac{\\partial}{\\partial B} (\\frac{B}{S} - \\rho) and \\int \\frac{\\partial}{\\partial B} V{(\\rho,B,S)} dB = \\int \\frac{\\partial}{\\partial B} (\\frac{B}{S} - \\rho) dB and g_{\\varepsilon} + V{(\\rho,B,S)} = \\frac{B}{S} + \\chi and \\log{(g_{\\varepsilon} + V{(\\rho,B,S)})} = \\log{(\\frac{B}{S} + \\chi)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\rho', commutative=True), Symbol('B', commutative=True), Symbol('S', commutative=True)), Add(Mul(Symbol('B', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\rho', commutative=True), Symbol('B', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Derivative(Function('V')(Symbol('\\\\rho', commutative=True), Symbol('B', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))), Integral(Derivative(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('V')(Symbol('\\\\rho', commutative=True), Symbol('B', commutative=True), Symbol('S', commutative=True))), Add(Mul(Symbol('B', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('\\\\chi', commutative=True)))"], [["log", 4], "Equality(log(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('V')(Symbol('\\\\rho', commutative=True), Symbol('B', commutative=True), Symbol('S', commutative=True)))), log(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\pi,F_{c})} = \\frac{\\pi}{F_{c}}, then obtain \\int (- \\phi{(\\pi,F_{c})} - 1)^{\\pi} d\\pi = \\int (-1 - \\frac{\\pi}{F_{c}})^{\\pi} d\\pi", "derivation": "\\phi{(\\pi,F_{c})} = \\frac{\\pi}{F_{c}} and - \\phi{(\\pi,F_{c})} = - \\frac{\\pi}{F_{c}} and - \\phi{(\\pi,F_{c})} - 1 = -1 - \\frac{\\pi}{F_{c}} and (- \\phi{(\\pi,F_{c})} - 1)^{\\pi} = (-1 - \\frac{\\pi}{F_{c}})^{\\pi} and \\int (- \\phi{(\\pi,F_{c})} - 1)^{\\pi} d\\pi = \\int (-1 - \\frac{\\pi}{F_{c}})^{\\pi} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\pi', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\pi', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True)))"], [["add", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\pi', commutative=True), Symbol('F_c', commutative=True))), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True))))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\pi', commutative=True), Symbol('F_c', commutative=True))), Integer(-1)), Symbol('\\\\pi', commutative=True)), Pow(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\pi', commutative=True), Symbol('F_c', commutative=True))), Integer(-1)), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Pow(Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('F_c', commutative=True), Integer(-1)), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(n)} = \\sin{(e^{n})} and z{(n)} = e^{n}, then derive \\frac{d}{d n} \\mathbf{B}{(n)} = e^{n} \\cos{(e^{n})}, then obtain \\mathbf{B}{(n)} + \\frac{d}{d n} \\sin{(z{(n)})} = \\mathbf{B}{(n)} + \\frac{d}{d n} \\mathbf{B}{(n)}", "derivation": "\\mathbf{B}{(n)} = \\sin{(e^{n})} and \\frac{d}{d n} \\mathbf{B}{(n)} = \\frac{d}{d n} \\sin{(e^{n})} and \\frac{d}{d n} \\mathbf{B}{(n)} = e^{n} \\cos{(e^{n})} and z{(n)} = e^{n} and \\mathbf{B}{(n)} + \\frac{d}{d n} \\mathbf{B}{(n)} = \\mathbf{B}{(n)} + e^{n} \\cos{(e^{n})} and \\mathbf{B}{(n)} + \\frac{d}{d n} \\sin{(e^{n})} = \\mathbf{B}{(n)} + e^{n} \\cos{(e^{n})} and \\mathbf{B}{(n)} + \\frac{d}{d n} \\sin{(e^{n})} = \\mathbf{B}{(n)} + \\frac{d}{d n} \\mathbf{B}{(n)} and \\mathbf{B}{(n)} + \\frac{d}{d n} \\sin{(z{(n)})} = \\mathbf{B}{(n)} + \\frac{d}{d n} \\mathbf{B}{(n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), sin(exp(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Mul(exp(Symbol('n', commutative=True)), cos(exp(Symbol('n', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["add", 3, "Function('\\\\mathbf{B}')(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Mul(exp(Symbol('n', commutative=True)), cos(exp(Symbol('n', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(sin(exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Mul(exp(Symbol('n', commutative=True)), cos(exp(Symbol('n', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(sin(exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 7, 4], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(sin(Function('z')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Derivative(Function('\\\\mathbf{B}')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{b}{(A)} = \\sin{(A)} and \\lambda{(A)} = \\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}}, then obtain (\\frac{\\rho_{b}{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A} = (\\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A}", "derivation": "\\rho_{b}{(A)} = \\sin{(A)} and \\frac{\\rho_{b}{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}} = \\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}} and \\lambda{(A)} = \\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}} and \\lambda^{A}{(A)} = (\\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A} and \\lambda{(A)} = \\frac{\\rho_{b}{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}} and \\lambda^{A}{(A)} = (\\frac{\\rho_{b}{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A} and (\\frac{\\rho_{b}{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A} = (\\frac{\\sin{(A)}}{- \\rho_{b}{(A)} + \\sin{(A)}})^{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('A', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), sin(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('A', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), sin(Symbol('A', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\lambda')(Symbol('A', commutative=True)), Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('A', commutative=True))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), Function('\\\\rho_b')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A', commutative=True))), sin(Symbol('A', commutative=True))), Integer(-1)), sin(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\psi^{*}{(\\rho_b)} = \\cos{(\\rho_b)}, then obtain \\frac{\\psi^{*}^{\\rho_b}{(\\rho_b)}}{\\rho_b^{2}} = \\frac{\\cos^{\\rho_b}{(\\rho_b)}}{\\rho_b^{2}}", "derivation": "\\psi^{*}{(\\rho_b)} = \\cos{(\\rho_b)} and \\psi^{*}^{\\rho_b}{(\\rho_b)} = \\cos^{\\rho_b}{(\\rho_b)} and \\frac{\\psi^{*}^{\\rho_b}{(\\rho_b)}}{\\rho_b} = \\frac{\\cos^{\\rho_b}{(\\rho_b)}}{\\rho_b} and \\frac{\\psi^{*}^{\\rho_b}{(\\rho_b)}}{\\rho_b^{2}} = \\frac{\\cos^{\\rho_b}{(\\rho_b)}}{\\rho_b^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["divide", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(Function('\\\\psi^*')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["times", 3, "Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2)), Pow(Function('\\\\psi^*')(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2)), Pow(cos(Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{v})} = e^{\\mathbf{v}}, then obtain e^{\\frac{d^{2}}{d \\mathbf{v}^{2}} (\\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}})^{\\mathbf{v}}} = e^{\\frac{d^{2}}{d \\mathbf{v}^{2}} 1}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{v})} = e^{\\mathbf{v}} and \\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}} = 1 and (\\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}})^{\\mathbf{v}} = 1 and \\frac{d}{d \\mathbf{v}} (\\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}})^{\\mathbf{v}} = \\frac{d}{d \\mathbf{v}} 1 and \\frac{d^{2}}{d \\mathbf{v}^{2}} (\\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}})^{\\mathbf{v}} = \\frac{d^{2}}{d \\mathbf{v}^{2}} 1 and e^{\\frac{d^{2}}{d \\mathbf{v}^{2}} (\\operatorname{F_{c}}{(\\mathbf{v})} e^{- \\mathbf{v}})^{\\mathbf{v}}} = e^{\\frac{d^{2}}{d \\mathbf{v}^{2}} 1}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))), Integer(1))"], [["power", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Mul(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2))))"], [["exp", 5], "Equality(exp(Derivative(Pow(Mul(Function('F_c')(Symbol('\\\\mathbf{v}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2)))), exp(Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\varphi{(\\phi_1,A_{2},q)} = \\frac{\\phi_1 q}{A_{2}}, then obtain \\frac{\\partial^{2}}{\\partial q\\partial A_{2}} (- \\varphi{(\\phi_1,A_{2},q)} + \\frac{1}{A_{2}}) = \\frac{\\partial^{2}}{\\partial q\\partial A_{2}} (- \\frac{\\phi_1 q}{A_{2}} + \\frac{1}{A_{2}})", "derivation": "\\varphi{(\\phi_1,A_{2},q)} = \\frac{\\phi_1 q}{A_{2}} and \\varphi{(\\phi_1,A_{2},q)} - \\frac{1}{A_{2}} = \\frac{\\phi_1 q}{A_{2}} - \\frac{1}{A_{2}} and - \\varphi{(\\phi_1,A_{2},q)} + \\frac{1}{A_{2}} = - \\frac{\\phi_1 q}{A_{2}} + \\frac{1}{A_{2}} and \\frac{\\partial}{\\partial A_{2}} (- \\varphi{(\\phi_1,A_{2},q)} + \\frac{1}{A_{2}}) = \\frac{\\partial}{\\partial A_{2}} (- \\frac{\\phi_1 q}{A_{2}} + \\frac{1}{A_{2}}) and \\frac{\\partial^{2}}{\\partial q\\partial A_{2}} (- \\varphi{(\\phi_1,A_{2},q)} + \\frac{1}{A_{2}}) = \\frac{\\partial^{2}}{\\partial q\\partial A_{2}} (- \\frac{\\phi_1 q}{A_{2}} + \\frac{1}{A_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\phi_1', commutative=True), Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Symbol('q', commutative=True)))"], [["minus", 1, "Pow(Symbol('A_2', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\phi_1', commutative=True), Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\phi_1', commutative=True), Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Pow(Symbol('A_2', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('A_2', commutative=True), Integer(-1))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\phi_1', commutative=True), Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Pow(Symbol('A_2', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('A_2', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\phi_1', commutative=True), Symbol('A_2', commutative=True), Symbol('q', commutative=True))), Pow(Symbol('A_2', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Symbol('q', commutative=True)), Pow(Symbol('A_2', commutative=True), Integer(-1))), Tuple(Symbol('A_2', commutative=True), Integer(1)), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\sigma_{x}{(A_{z},z)} = z + \\sin{(A_{z})}, then obtain A_{z} (z + \\sin{(A_{z})})^{z} \\sigma_{x}^{- z}{(A_{z},z)} \\log{(\\sigma_{x}{(A_{z},z)})} = A_{z} (z + \\sin{(A_{z})})^{z} \\sigma_{x}^{- z}{(A_{z},z)} \\log{(z + \\sin{(A_{z})})}", "derivation": "\\sigma_{x}{(A_{z},z)} = z + \\sin{(A_{z})} and \\log{(\\sigma_{x}{(A_{z},z)})} = \\log{(z + \\sin{(A_{z})})} and (z + \\sin{(A_{z})})^{z} \\log{(\\sigma_{x}{(A_{z},z)})} = (z + \\sin{(A_{z})})^{z} \\log{(z + \\sin{(A_{z})})} and A_{z} (z + \\sin{(A_{z})})^{z} \\sigma_{x}^{- z}{(A_{z},z)} \\log{(\\sigma_{x}{(A_{z},z)})} = A_{z} (z + \\sin{(A_{z})})^{z} \\sigma_{x}^{- z}{(A_{z},z)} \\log{(z + \\sin{(A_{z})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)), Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True))), log(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True)))))"], [["times", 2, "Pow(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('z', commutative=True)), log(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)))), Mul(Pow(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('z', commutative=True)), log(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))))))"], [["divide", 3, "Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], "Equality(Mul(Symbol('A_z', commutative=True), Pow(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('z', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), log(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)))), Mul(Symbol('A_z', commutative=True), Pow(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))), Symbol('z', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('A_z', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Symbol('z', commutative=True))), log(Add(Symbol('z', commutative=True), sin(Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(M)} = \\frac{d}{d M} \\sin{(M)}, then obtain \\frac{\\operatorname{x^{{\\}'}}{(M)}}{M \\cos{(M)}} = \\frac{1}{M}", "derivation": "\\operatorname{x^{{\\}'}}{(M)} = \\frac{d}{d M} \\sin{(M)} and \\frac{\\operatorname{x^{{\\}'}}{(M)}}{M} = \\frac{\\frac{d}{d M} \\sin{(M)}}{M} and \\frac{\\operatorname{x^{{\\}'}}{(M)}}{M \\frac{d}{d M} \\sin{(M)}} = \\frac{1}{M} and \\frac{\\operatorname{x^{{\\}'}}{(M)}}{M \\cos{(M)}} = \\frac{1}{M}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('M', commutative=True)), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))))"], [["divide", 2, "Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('M', commutative=True)), Pow(Derivative(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Function('x^\\\\prime')(Symbol('M', commutative=True)), Pow(cos(Symbol('M', commutative=True)), Integer(-1))), Pow(Symbol('M', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_P{(J,k)} = \\frac{J}{k} and U{(J,k)} = \\frac{J}{k} and \\mathbf{s}{(J,k)} = k + \\mathbf{J}_P{(J,k)}, then obtain \\frac{\\partial}{\\partial J} \\mathbf{s}{(J,k)} = \\frac{\\partial}{\\partial J} (k + U{(J,k)})", "derivation": "\\mathbf{J}_P{(J,k)} = \\frac{J}{k} and k + \\mathbf{J}_P{(J,k)} = \\frac{J}{k} + k and \\frac{\\partial}{\\partial J} (k + \\mathbf{J}_P{(J,k)}) = \\frac{\\partial}{\\partial J} (\\frac{J}{k} + k) and U{(J,k)} = \\frac{J}{k} and \\frac{\\partial}{\\partial J} (k + \\mathbf{J}_P{(J,k)}) = \\frac{\\partial}{\\partial J} (k + U{(J,k)}) and \\mathbf{s}{(J,k)} = k + \\mathbf{J}_P{(J,k)} and \\mathbf{s}{(J,k)} = \\frac{J}{k} + k and \\frac{\\partial}{\\partial J} (k + \\mathbf{J}_P{(J,k)}) = \\frac{\\partial}{\\partial J} \\mathbf{s}{(J,k)} and \\frac{\\partial}{\\partial J} \\mathbf{s}{(J,k)} = \\frac{\\partial}{\\partial J} (k + U{(J,k)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('k', commutative=True)))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Add(Symbol('k', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('J', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('k', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('U')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Add(Symbol('k', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), Function('U')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Add(Symbol('k', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Function('\\\\mathbf{s}')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Add(Mul(Symbol('J', commutative=True), Pow(Symbol('k', commutative=True), Integer(-1))), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 7], "Equality(Derivative(Add(Symbol('k', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{s}')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 8], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('J', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('k', commutative=True), Function('U')(Symbol('J', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(v_{1},\\varphi)} = v_{1} \\log{(\\varphi)}, then obtain \\frac{d}{d v_{1}} \\int 0^{v_{1}} dv_{1} = \\frac{\\partial}{\\partial v_{1}} \\int (\\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} - 1)^{v_{1}} dv_{1}", "derivation": "\\mathbb{I}{(v_{1},\\varphi)} = v_{1} \\log{(\\varphi)} and 1 = \\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} and 0 = \\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} - 1 and 0^{v_{1}} = (\\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} - 1)^{v_{1}} and \\int 0^{v_{1}} dv_{1} = \\int (\\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} - 1)^{v_{1}} dv_{1} and \\frac{d}{d v_{1}} \\int 0^{v_{1}} dv_{1} = \\frac{\\partial}{\\partial v_{1}} \\int (\\frac{v_{1} \\log{(\\varphi)}}{\\mathbb{I}{(v_{1},\\varphi)}} - 1)^{v_{1}} dv_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('v_1', commutative=True), log(Symbol('\\\\varphi', commutative=True))))"], [["divide", 1, "Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(1), Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), log(Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)))"], [["power", 3, "Symbol('v_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('v_1', commutative=True)), Pow(Add(Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Symbol('v_1', commutative=True)))"], [["integrate", 4, "Symbol('v_1', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Pow(Add(Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integral(Pow(Integer(0), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Mul(Symbol('v_1', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('v_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), log(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(Z,c)} = Z - c, then obtain (c + \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)})^{Z} - \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)} = (c + \\frac{\\partial}{\\partial Z} (Z - c))^{Z} - \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)}", "derivation": "\\operatorname{t_{2}}{(Z,c)} = Z - c and \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)} = \\frac{\\partial}{\\partial Z} (Z - c) and c + \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)} = c + \\frac{\\partial}{\\partial Z} (Z - c) and (c + \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)})^{Z} = (c + \\frac{\\partial}{\\partial Z} (Z - c))^{Z} and (c + \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)})^{Z} - \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)} = (c + \\frac{\\partial}{\\partial Z} (Z - c))^{Z} - \\frac{\\partial}{\\partial Z} \\operatorname{t_{2}}{(Z,c)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Symbol('c', commutative=True), Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Add(Symbol('c', commutative=True), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Symbol('c', commutative=True), Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Symbol('Z', commutative=True)), Pow(Add(Symbol('c', commutative=True), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Symbol('Z', commutative=True)))"], [["minus", 4, "Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Add(Pow(Add(Symbol('c', commutative=True), Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), Symbol('Z', commutative=True)), Mul(Integer(-1), Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))), Add(Pow(Add(Symbol('c', commutative=True), Derivative(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), Symbol('Z', commutative=True)), Mul(Integer(-1), Derivative(Function('t_2')(Symbol('Z', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(a,\\ddot{x})} = \\ddot{x} + a, then obtain - \\ddot{x} + \\Psi^{\\dagger}{(a,\\ddot{x})} - \\frac{\\ddot{x} + a}{\\ddot{x}} = a - \\frac{\\ddot{x} + a}{\\ddot{x}}", "derivation": "\\Psi^{\\dagger}{(a,\\ddot{x})} = \\ddot{x} + a and \\frac{\\Psi^{\\dagger}{(a,\\ddot{x})}}{\\ddot{x}} = \\frac{\\ddot{x} + a}{\\ddot{x}} and \\Psi^{\\dagger}{(a,\\ddot{x})} - \\frac{\\Psi^{\\dagger}{(a,\\ddot{x})}}{\\ddot{x}} = \\ddot{x} + a - \\frac{\\Psi^{\\dagger}{(a,\\ddot{x})}}{\\ddot{x}} and \\Psi^{\\dagger}{(a,\\ddot{x})} - \\frac{\\ddot{x} + a}{\\ddot{x}} = \\ddot{x} + a - \\frac{\\ddot{x} + a}{\\ddot{x}} and - \\ddot{x} + \\Psi^{\\dagger}{(a,\\ddot{x})} - \\frac{\\ddot{x} + a}{\\ddot{x}} = a - \\frac{\\ddot{x} + a}{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True)))"], [["divide", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True)))), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True)))))"], [["minus", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('a', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True)))), Add(Symbol('a', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\psi)} = e^{\\psi}, then derive \\operatorname{C_{1}}{(\\psi)} - e^{\\psi} = 0, then obtain \\operatorname{C_{1}}{(\\psi)} - e^{\\psi} + \\frac{d}{d \\psi} \\operatorname{C_{1}}{(\\psi)} = \\frac{d}{d \\psi} \\operatorname{C_{1}}{(\\psi)}", "derivation": "\\operatorname{C_{1}}{(\\psi)} = e^{\\psi} and \\frac{d}{d \\psi} \\operatorname{C_{1}}{(\\psi)} = \\frac{d}{d \\psi} e^{\\psi} and \\operatorname{C_{1}}{(\\psi)} - \\frac{d}{d \\psi} e^{\\psi} = e^{\\psi} - \\frac{d}{d \\psi} e^{\\psi} and \\operatorname{C_{1}}{(\\psi)} - e^{\\psi} = 0 and \\operatorname{C_{1}}{(\\psi)} - e^{\\psi} + \\frac{d}{d \\psi} e^{\\psi} = \\frac{d}{d \\psi} e^{\\psi} and \\operatorname{C_{1}}{(\\psi)} - e^{\\psi} + \\frac{d}{d \\psi} \\operatorname{C_{1}}{(\\psi)} = \\frac{d}{d \\psi} \\operatorname{C_{1}}{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))"], "Equality(Add(Function('C_1')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('C_1')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True)))), Integer(0))"], [["minus", 4, "Mul(Integer(-1), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], "Equality(Add(Function('C_1')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Derivative(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Function('C_1')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), Derivative(Function('C_1')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Derivative(Function('C_1')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon_{0}{(h)} = \\cos{(h)}, then derive \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh = \\varepsilon + h, then obtain \\int (\\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh)^{2} d\\varepsilon = \\int (\\varepsilon + h) \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh d\\varepsilon", "derivation": "\\varepsilon_{0}{(h)} = \\cos{(h)} and \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} = 1 and \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh = \\int 1 dh and \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh = \\varepsilon + h and (\\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh)^{2} = (\\varepsilon + h) \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh and \\int (\\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh)^{2} d\\varepsilon = \\int (\\varepsilon + h) \\int \\frac{\\varepsilon_{0}{(h)}}{\\cos{(h)}} dh d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["divide", 1, "cos(Symbol('h', commutative=True))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True))), Integral(Integer(1), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)))"], [["times", 4, "Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True)))"], "Equality(Pow(Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Pow(Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('h', commutative=True)), Integral(Mul(Function('\\\\varepsilon_0')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(s)} = \\sin{(e^{s})}, then obtain 1 = ((\\frac{- e^{s} + \\sin{(e^{s})}}{\\ddot{x}{(s)} - e^{s}})^{s})^{s}", "derivation": "\\ddot{x}{(s)} = \\sin{(e^{s})} and \\ddot{x}{(s)} - e^{s} = - e^{s} + \\sin{(e^{s})} and s (\\ddot{x}{(s)} - e^{s}) = s (- e^{s} + \\sin{(e^{s})}) and 1 = \\frac{- e^{s} + \\sin{(e^{s})}}{\\ddot{x}{(s)} - e^{s}} and 1 = (\\frac{- e^{s} + \\sin{(e^{s})}}{\\ddot{x}{(s)} - e^{s}})^{s} and 1 = ((\\frac{- e^{s} + \\sin{(e^{s})}}{\\ddot{x}{(s)} - e^{s}})^{s})^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), sin(exp(Symbol('s', commutative=True))))"], [["minus", 1, "exp(Symbol('s', commutative=True))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True))), sin(exp(Symbol('s', commutative=True)))))"], [["times", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True))))), Mul(Symbol('s', commutative=True), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True))), sin(exp(Symbol('s', commutative=True))))))"], [["divide", 3, "Mul(Symbol('s', commutative=True), Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)))))"], "Equality(Integer(1), Mul(Pow(Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True))), sin(exp(Symbol('s', commutative=True))))))"], [["power", 4, "Symbol('s', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True))), sin(exp(Symbol('s', commutative=True))))), Symbol('s', commutative=True)))"], [["power", 5, "Symbol('s', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Pow(Add(Function('\\\\ddot{x}')(Symbol('s', commutative=True)), Mul(Integer(-1), exp(Symbol('s', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('s', commutative=True))), sin(exp(Symbol('s', commutative=True))))), Symbol('s', commutative=True)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(t_{2})} = e^{e^{t_{2}}}, then obtain \\sin{(\\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + e^{t_{2}}) dt_{2})} = \\sin{(\\int (e^{t_{2}} + e^{e^{t_{2}}}) dt_{2})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(t_{2})} = e^{e^{t_{2}}} and \\operatorname{V_{\\mathbf{B}}}{(t_{2})} + e^{t_{2}} = e^{t_{2}} + e^{e^{t_{2}}} and \\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + e^{t_{2}}) dt_{2} = \\int (e^{t_{2}} + e^{e^{t_{2}}}) dt_{2} and \\sin{(\\int (\\operatorname{V_{\\mathbf{B}}}{(t_{2})} + e^{t_{2}}) dt_{2})} = \\sin{(\\int (e^{t_{2}} + e^{e^{t_{2}}}) dt_{2})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), exp(exp(Symbol('t_2', commutative=True))))"], [["add", 1, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Add(exp(Symbol('t_2', commutative=True)), exp(exp(Symbol('t_2', commutative=True)))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Add(exp(Symbol('t_2', commutative=True)), exp(exp(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Add(Function('V_{\\\\mathbf{B}}')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True)))), sin(Integral(Add(exp(Symbol('t_2', commutative=True)), exp(exp(Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True)))))"]]}, {"prompt": "Given p{(v_{x},y,g)} = g + v_{x} + y, then derive \\int p{(v_{x},y,g)} dg = \\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y), then obtain \\frac{\\partial}{\\partial v_{x}} \\int (g + v_{x} + y) dg - 1 = \\frac{\\partial}{\\partial v_{x}} (\\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y)) - 1", "derivation": "p{(v_{x},y,g)} = g + v_{x} + y and \\int p{(v_{x},y,g)} dg = \\int (g + v_{x} + y) dg and \\int p{(v_{x},y,g)} dg = \\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y) and \\int (g + v_{x} + y) dg = \\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y) and \\frac{\\partial}{\\partial v_{x}} \\int (g + v_{x} + y) dg = \\frac{\\partial}{\\partial v_{x}} (\\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y)) and \\frac{\\partial}{\\partial v_{x}} \\int (g + v_{x} + y) dg - 1 = \\frac{\\partial}{\\partial v_{x}} (\\dot{z} + \\frac{g^{2}}{2} + g (v_{x} + y)) - 1", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('v_x', commutative=True), Symbol('y', commutative=True), Symbol('g', commutative=True)), Add(Symbol('g', commutative=True), Symbol('v_x', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('p')(Symbol('v_x', commutative=True), Symbol('y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('g', commutative=True), Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('p')(Symbol('v_x', commutative=True), Symbol('y', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('g', commutative=True), Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))))"], [["differentiate", 4, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('g', commutative=True), Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Derivative(Integral(Add(Symbol('g', commutative=True), Symbol('v_x', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('g', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g', commutative=True), Integer(2))), Mul(Symbol('g', commutative=True), Add(Symbol('v_x', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\omega{(v)} = \\sin{(v)} and \\Psi^{\\dagger}{(v)} = \\sin{(v)}, then obtain \\Psi^{\\dagger}{(v)} = \\frac{\\Psi^{\\dagger}^{2}{(v)}}{\\sin{(v)}}", "derivation": "\\omega{(v)} = \\sin{(v)} and \\Psi^{\\dagger}{(v)} = \\sin{(v)} and \\omega{(v)} = \\Psi^{\\dagger}{(v)} and \\Psi^{\\dagger}{(v)} = \\frac{\\Psi^{\\dagger}^{2}{(v)}}{\\omega{(v)}} and \\Psi^{\\dagger}{(v)} = \\frac{\\Psi^{\\dagger}^{2}{(v)}}{\\sin{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), sin(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\omega')(Symbol('v', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)))"], [["times", 3, "Mul(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), Pow(Function('\\\\omega')(Symbol('v', commutative=True)), Integer(-1)))"], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), Integer(2)), Pow(Function('\\\\omega')(Symbol('v', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('v', commutative=True)), Integer(2)), Pow(sin(Symbol('v', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(b)} = \\sin{(b)} and \\hat{p}{(b)} = \\delta{(b)} \\sin{(b)}, then obtain \\delta^{2}{(b)} = \\sin^{2}{(b)}", "derivation": "\\delta{(b)} = \\sin{(b)} and \\delta^{2}{(b)} = \\delta{(b)} \\sin{(b)} and \\hat{p}{(b)} = \\delta{(b)} \\sin{(b)} and \\hat{p}{(b)} = \\sin^{2}{(b)} and \\delta{(b)} \\sin{(b)} = \\sin^{2}{(b)} and \\delta^{2}{(b)} = \\sin^{2}{(b)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["times", 1, "Function('\\\\delta')(Symbol('b', commutative=True))"], "Equality(Pow(Function('\\\\delta')(Symbol('b', commutative=True)), Integer(2)), Mul(Function('\\\\delta')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('b', commutative=True)), Mul(Function('\\\\delta')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\hat{p}')(Symbol('b', commutative=True)), Pow(sin(Symbol('b', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\delta')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True))), Pow(sin(Symbol('b', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('\\\\delta')(Symbol('b', commutative=True)), Integer(2)), Pow(sin(Symbol('b', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\theta_{1}{(c)} = e^{c}, then derive \\frac{d}{d c} \\theta_{1}{(c)} = e^{c}, then obtain (e^{c})^{c} \\frac{d}{d c} \\theta_{1}{(c)} = \\frac{d}{d c} \\theta_{1}{(c)} (\\frac{d}{d c} e^{c})^{c}", "derivation": "\\theta_{1}{(c)} = e^{c} and \\frac{d}{d c} \\theta_{1}{(c)} = \\frac{d}{d c} e^{c} and (\\frac{d}{d c} \\theta_{1}{(c)})^{c} = (\\frac{d}{d c} e^{c})^{c} and \\frac{d}{d c} \\theta_{1}{(c)} = e^{c} and (e^{c})^{c} = (\\frac{d}{d c} e^{c})^{c} and (e^{c})^{c} \\frac{d}{d c} \\theta_{1}{(c)} = \\frac{d}{d c} \\theta_{1}{(c)} (\\frac{d}{d c} e^{c})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('c', commutative=True)), exp(Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)), Pow(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), exp(Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True)))"], [["times", 5, "Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Mul(Pow(exp(Symbol('c', commutative=True)), Symbol('c', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Mul(Derivative(Function('\\\\theta_1')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Pow(Derivative(exp(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(T,\\mathbf{D})} = T + \\mathbf{D}, then derive \\frac{\\partial}{\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = 0", "derivation": "\\operatorname{P_{g}}{(T,\\mathbf{D})} = T + \\mathbf{D} and \\frac{\\partial}{\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = \\frac{\\partial}{\\partial T} (T + \\mathbf{D}) and \\frac{\\partial}{\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = 1 and \\frac{\\partial}{\\partial T} (T + \\mathbf{D}) = 1 and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} (T + \\mathbf{D}) = \\frac{d}{d \\mathbf{D}} 1 and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} 1 and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} (T + \\mathbf{D}) and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial T} \\operatorname{P_{g}}{(T,\\mathbf{D})} = 0", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Derivative(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('P_g')(Symbol('T', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\varphi^{*}{(V)} = e^{V}, then obtain \\frac{d}{d V} \\int 1 dV = \\frac{d}{d V} \\int (\\frac{e^{V}}{\\varphi^{*}{(V)}})^{V} dV", "derivation": "\\varphi^{*}{(V)} = e^{V} and 1 = \\frac{e^{V}}{\\varphi^{*}{(V)}} and 1 = (\\frac{e^{V}}{\\varphi^{*}{(V)}})^{V} and \\int 1 dV = \\int (\\frac{e^{V}}{\\varphi^{*}{(V)}})^{V} dV and \\frac{d}{d V} \\int 1 dV = \\frac{d}{d V} \\int (\\frac{e^{V}}{\\varphi^{*}{(V)}})^{V} dV", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('V', commutative=True)), exp(Symbol('V', commutative=True)))"], [["divide", 1, "Function('\\\\varphi^*')(Symbol('V', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True)), Integer(-1)), exp(Symbol('V', commutative=True))))"], [["power", 2, "Symbol('V', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True)), Integer(-1)), exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)))"], [["integrate", 3, "Symbol('V', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Integral(Pow(Mul(Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True)), Integer(-1)), exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Pow(Function('\\\\varphi^*')(Symbol('V', commutative=True)), Integer(-1)), exp(Symbol('V', commutative=True))), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(F_{g},v_{z})} = \\log{(v_{z}^{F_{g}})}, then derive - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})} = - F_{g} + \\log{(v_{z})}, then obtain \\int (- F_{g} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})}) dF_{g} = \\int (- F_{g} + \\log{(v_{z})}) dF_{g}", "derivation": "\\operatorname{n_{2}}{(F_{g},v_{z})} = \\log{(v_{z}^{F_{g}})} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})} = \\frac{\\partial}{\\partial F_{g}} \\log{(v_{z}^{F_{g}})} and - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})} = - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\log{(v_{z}^{F_{g}})} and - F_{g} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})} = - F_{g} + \\log{(v_{z})} and \\int (- F_{g} + \\frac{\\partial}{\\partial F_{g}} \\operatorname{n_{2}}{(F_{g},v_{z})}) dF_{g} = \\int (- F_{g} + \\log{(v_{z})}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('F_g', commutative=True), Symbol('v_z', commutative=True)), log(Pow(Symbol('v_z', commutative=True), Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('F_g', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(log(Pow(Symbol('v_z', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('n_2')(Symbol('F_g', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(log(Pow(Symbol('v_z', commutative=True), Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('n_2')(Symbol('F_g', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('v_z', commutative=True))))"], [["integrate", 4, "Symbol('F_g', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Derivative(Function('n_2')(Symbol('F_g', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), log(Symbol('v_z', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(A,\\theta)} = \\frac{\\partial}{\\partial \\theta} (- A + \\theta), then derive \\operatorname{m_{s}}{(A,\\theta)} = 1, then derive \\frac{A^{2}}{2} + A + A_{y} = \\frac{A^{2}}{2} + A + \\hbar, then obtain \\frac{\\frac{A^{2}}{2} + A + A_{y}}{\\frac{A^{2}}{2} + A + \\hbar} = 1", "derivation": "\\operatorname{m_{s}}{(A,\\theta)} = \\frac{\\partial}{\\partial \\theta} (- A + \\theta) and \\operatorname{m_{s}}{(A,\\theta)} = 1 and A + \\operatorname{m_{s}}{(A,\\theta)} = A + 1 and A + \\frac{\\partial}{\\partial \\theta} (- A + \\theta) = A + 1 and \\int (A + \\frac{\\partial}{\\partial \\theta} (- A + \\theta)) dA = \\int (A + 1) dA and \\frac{A^{2}}{2} + A + A_{y} = \\frac{A^{2}}{2} + A + \\hbar and \\frac{\\frac{A^{2}}{2} + A + A_{y}}{\\frac{A^{2}}{2} + A + \\hbar} = 1", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('A', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('m_s')(Symbol('A', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1))"], [["minus", 2, "Mul(Integer(-1), Symbol('A', commutative=True))"], "Equality(Add(Symbol('A', commutative=True), Function('m_s')(Symbol('A', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('A', commutative=True), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('A', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Symbol('A', commutative=True), Integer(1)))"], [["integrate", 4, "Symbol('A', commutative=True)"], "Equality(Integral(Add(Symbol('A', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('A', commutative=True)), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('A', commutative=True))), Integral(Add(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('A', commutative=True), Symbol('A_y', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('A', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 6, "Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('A', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('A', commutative=True), Symbol('A_y', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('A', commutative=True), Integer(2))), Symbol('A', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{f}{(M_{E})} = \\cos{(\\cos{(M_{E})})}, then obtain ((M_{E} + \\mathbf{f}{(M_{E})}) (M_{E} + \\cos{(\\cos{(M_{E})})}))^{M_{E}} = ((M_{E} + \\cos{(\\cos{(M_{E})})})^{2})^{M_{E}}", "derivation": "\\mathbf{f}{(M_{E})} = \\cos{(\\cos{(M_{E})})} and M_{E} + \\mathbf{f}{(M_{E})} = M_{E} + \\cos{(\\cos{(M_{E})})} and (M_{E} + \\mathbf{f}{(M_{E})}) (M_{E} + \\cos{(\\cos{(M_{E})})}) = (M_{E} + \\cos{(\\cos{(M_{E})})})^{2} and ((M_{E} + \\mathbf{f}{(M_{E})}) (M_{E} + \\cos{(\\cos{(M_{E})})}))^{M_{E}} = ((M_{E} + \\cos{(\\cos{(M_{E})})})^{2})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('M_E', commutative=True)), cos(cos(Symbol('M_E', commutative=True))))"], [["add", 1, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Function('\\\\mathbf{f}')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True)))))"], [["times", 2, "Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True))))"], "Equality(Mul(Add(Symbol('M_E', commutative=True), Function('\\\\mathbf{f}')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True))))), Pow(Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True)))), Integer(2)))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('M_E', commutative=True), Function('\\\\mathbf{f}')(Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True))))), Symbol('M_E', commutative=True)), Pow(Pow(Add(Symbol('M_E', commutative=True), cos(cos(Symbol('M_E', commutative=True)))), Integer(2)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\theta{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})}, then obtain (\\Psi_{\\lambda} + \\int \\theta{(\\Psi_{\\lambda})} d\\Psi_{\\lambda})^{\\Psi_{\\lambda}} = (\\Psi_{\\lambda} + \\int \\log{(\\log{(\\Psi_{\\lambda})})} d\\Psi_{\\lambda})^{\\Psi_{\\lambda}}", "derivation": "\\theta{(\\Psi_{\\lambda})} = \\log{(\\log{(\\Psi_{\\lambda})})} and \\int \\theta{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\int \\log{(\\log{(\\Psi_{\\lambda})})} d\\Psi_{\\lambda} and \\Psi_{\\lambda} + \\int \\theta{(\\Psi_{\\lambda})} d\\Psi_{\\lambda} = \\Psi_{\\lambda} + \\int \\log{(\\log{(\\Psi_{\\lambda})})} d\\Psi_{\\lambda} and (\\Psi_{\\lambda} + \\int \\theta{(\\Psi_{\\lambda})} d\\Psi_{\\lambda})^{\\Psi_{\\lambda}} = (\\Psi_{\\lambda} + \\int \\log{(\\log{(\\Psi_{\\lambda})})} d\\Psi_{\\lambda})^{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["add", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["power", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('\\\\theta')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(log(log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(\\dot{x})} = \\cos{(\\dot{x})} and \\hat{p}{(\\dot{x})} = \\int \\dot{z}{(\\dot{x})} d\\dot{x}, then obtain 1 = \\frac{\\hat{p}{(\\dot{x})}}{\\int \\dot{z}{(\\dot{x})} d\\dot{x}}", "derivation": "\\dot{z}{(\\dot{x})} = \\cos{(\\dot{x})} and \\int \\dot{z}{(\\dot{x})} d\\dot{x} = \\int \\cos{(\\dot{x})} d\\dot{x} and \\hat{p}{(\\dot{x})} = \\int \\dot{z}{(\\dot{x})} d\\dot{x} and 1 = \\frac{\\int \\dot{z}{(\\dot{x})} d\\dot{x}}{\\hat{p}{(\\dot{x})}} and \\hat{p}{(\\dot{x})} = \\int \\cos{(\\dot{x})} d\\dot{x} and 1 = \\frac{\\int \\dot{z}{(\\dot{x})} d\\dot{x}}{\\int \\cos{(\\dot{x})} d\\dot{x}} and 1 = \\frac{\\hat{p}{(\\dot{x})}}{\\int \\cos{(\\dot{x})} d\\dot{x}} and 1 = \\frac{\\hat{p}{(\\dot{x})}}{\\int \\dot{z}{(\\dot{x})} d\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), cos(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True)), Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 3, "Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True)), Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(1), Mul(Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Pow(Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Integer(1), Mul(Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(cos(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Integer(1), Mul(Function('\\\\hat{p}')(Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(Function('\\\\dot{z}')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(A_{y})} = \\log{(A_{y})}, then derive \\int \\frac{Z \\operatorname{c_{0}}{(A_{y})}}{\\log{(A_{y})}} dZ = \\frac{Z^{2}}{2} + b, then obtain \\int Z dZ = \\frac{Z^{2}}{2} + b", "derivation": "\\operatorname{c_{0}}{(A_{y})} = \\log{(A_{y})} and \\frac{\\operatorname{c_{0}}{(A_{y})}}{\\log{(A_{y})}} = 1 and \\frac{Z \\operatorname{c_{0}}{(A_{y})}}{\\log{(A_{y})}} = Z and \\int \\frac{Z \\operatorname{c_{0}}{(A_{y})}}{\\log{(A_{y})}} dZ = \\int Z dZ and \\int \\frac{Z \\operatorname{c_{0}}{(A_{y})}}{\\log{(A_{y})}} dZ = \\frac{Z^{2}}{2} + b and \\int Z dZ = \\frac{Z^{2}}{2} + b", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('A_y', commutative=True)), log(Symbol('A_y', commutative=True)))"], [["divide", 1, "log(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('c_0')(Symbol('A_y', commutative=True)), Pow(log(Symbol('A_y', commutative=True)), Integer(-1))), Integer(1))"], [["divide", 2, "Pow(Symbol('Z', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('Z', commutative=True), Function('c_0')(Symbol('A_y', commutative=True)), Pow(log(Symbol('A_y', commutative=True)), Integer(-1))), Symbol('Z', commutative=True))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('c_0')(Symbol('A_y', commutative=True)), Pow(log(Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('Z', commutative=True))), Integral(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Symbol('Z', commutative=True), Function('c_0')(Symbol('A_y', commutative=True)), Pow(log(Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('Z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('Z', commutative=True), Integer(2))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\mathbf{g}{(A)} = \\sin{(A)}, then derive \\int \\mathbf{g}{(A)} \\sin{(A)} dA = \\frac{A}{2} + \\hat{H} - \\frac{\\sin{(A)} \\cos{(A)}}{2}, then obtain \\frac{A}{2} + x - \\frac{\\sin{(A)} \\cos{(A)}}{2} = \\frac{A}{2} + \\hat{H} - \\frac{\\mathbf{g}{(A)} \\cos{(A)}}{2}", "derivation": "\\mathbf{g}{(A)} = \\sin{(A)} and \\mathbf{g}{(A)} \\sin{(A)} = \\sin^{2}{(A)} and \\int \\mathbf{g}{(A)} \\sin{(A)} dA = \\int \\sin^{2}{(A)} dA and \\int \\mathbf{g}{(A)} \\sin{(A)} dA = \\frac{A}{2} + \\hat{H} - \\frac{\\sin{(A)} \\cos{(A)}}{2} and \\int \\sin^{2}{(A)} dA = \\frac{A}{2} + \\hat{H} - \\frac{\\sin{(A)} \\cos{(A)}}{2} and \\int \\sin^{2}{(A)} dA = \\frac{A}{2} + \\hat{H} - \\frac{\\mathbf{g}{(A)} \\cos{(A)}}{2} and \\frac{A}{2} + x - \\frac{\\sin{(A)} \\cos{(A)}}{2} = \\frac{A}{2} + \\hat{H} - \\frac{\\mathbf{g}{(A)} \\cos{(A)}}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["times", 1, "sin(Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Pow(sin(Symbol('A', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Integral(Pow(sin(Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Add(Mul(Rational(1, 2), Symbol('A', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(sin(Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True))), Add(Mul(Rational(1, 2), Symbol('A', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Pow(sin(Symbol('A', commutative=True)), Integer(2)), Tuple(Symbol('A', commutative=True))), Add(Mul(Rational(1, 2), Symbol('A', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Rational(1, 2), Symbol('A', commutative=True)), Symbol('x', commutative=True), Mul(Integer(-1), Rational(1, 2), sin(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))), Add(Mul(Rational(1, 2), Symbol('A', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), Function('\\\\mathbf{g}')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(i)} = \\sin{(i)}, then obtain \\rho_{f}{(i)} + \\log{(i \\rho_{f}{(i)})} = \\log{(i \\rho_{f}{(i)})} + \\sin{(i)}", "derivation": "\\rho_{f}{(i)} = \\sin{(i)} and i \\rho_{f}{(i)} = i \\sin{(i)} and \\log{(i \\rho_{f}{(i)})} = \\log{(i \\sin{(i)})} and \\rho_{f}{(i)} + \\log{(i \\sin{(i)})} = \\log{(i \\sin{(i)})} + \\sin{(i)} and \\rho_{f}{(i)} + \\log{(i \\rho_{f}{(i)})} = \\log{(i \\rho_{f}{(i)})} + \\sin{(i)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["times", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Function('\\\\rho_f')(Symbol('i', commutative=True))), Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], [["log", 2], "Equality(log(Mul(Symbol('i', commutative=True), Function('\\\\rho_f')(Symbol('i', commutative=True)))), log(Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))))"], [["add", 1, "log(Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))"], "Equality(Add(Function('\\\\rho_f')(Symbol('i', commutative=True)), log(Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True))))), Add(log(Mul(Symbol('i', commutative=True), sin(Symbol('i', commutative=True)))), sin(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\rho_f')(Symbol('i', commutative=True)), log(Mul(Symbol('i', commutative=True), Function('\\\\rho_f')(Symbol('i', commutative=True))))), Add(log(Mul(Symbol('i', commutative=True), Function('\\\\rho_f')(Symbol('i', commutative=True)))), sin(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\tilde{g})} = e^{\\tilde{g}}, then obtain \\varepsilon_{0}{(\\tilde{g})} - \\int 2 e^{\\tilde{g}} d\\tilde{g} = e^{\\tilde{g}} - \\int 2 e^{\\tilde{g}} d\\tilde{g}", "derivation": "\\varepsilon_{0}{(\\tilde{g})} = e^{\\tilde{g}} and \\varepsilon_{0}{(\\tilde{g})} + e^{\\tilde{g}} = 2 e^{\\tilde{g}} and \\int (\\varepsilon_{0}{(\\tilde{g})} + e^{\\tilde{g}}) d\\tilde{g} = \\int 2 e^{\\tilde{g}} d\\tilde{g} and \\varepsilon_{0}{(\\tilde{g})} - \\int (\\varepsilon_{0}{(\\tilde{g})} + e^{\\tilde{g}}) d\\tilde{g} = e^{\\tilde{g}} - \\int (\\varepsilon_{0}{(\\tilde{g})} + e^{\\tilde{g}}) d\\tilde{g} and \\varepsilon_{0}{(\\tilde{g})} - \\int 2 e^{\\tilde{g}} d\\tilde{g} = e^{\\tilde{g}} - \\int 2 e^{\\tilde{g}} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 1, "Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))), Add(exp(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integral(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))), Add(exp(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(-1), Integral(Mul(Integer(2), exp(Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon_{0}{(g)} = \\cos{(\\sin{(g)})} and \\mu_{0}{(g)} = \\sin{(g)}, then obtain \\iint (\\varepsilon_{0}{(g)} + \\sin{(g)}) dg dg = \\iint (\\sin{(g)} + \\cos{(\\mu_{0}{(g)})}) dg dg", "derivation": "\\varepsilon_{0}{(g)} = \\cos{(\\sin{(g)})} and \\mu_{0}{(g)} = \\sin{(g)} and \\varepsilon_{0}{(g)} = \\cos{(\\mu_{0}{(g)})} and \\varepsilon_{0}{(g)} + \\sin{(g)} = \\sin{(g)} + \\cos{(\\mu_{0}{(g)})} and \\int (\\varepsilon_{0}{(g)} + \\sin{(g)}) dg = \\int (\\sin{(g)} + \\cos{(\\mu_{0}{(g)})}) dg and \\iint (\\varepsilon_{0}{(g)} + \\sin{(g)}) dg dg = \\iint (\\sin{(g)} + \\cos{(\\mu_{0}{(g)})}) dg dg", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('g', commutative=True)), cos(sin(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varepsilon_0')(Symbol('g', commutative=True)), cos(Function('\\\\mu_0')(Symbol('g', commutative=True))))"], [["add", 3, "sin(Symbol('g', commutative=True))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True))), Add(sin(Symbol('g', commutative=True)), cos(Function('\\\\mu_0')(Symbol('g', commutative=True)))))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon_0')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Add(sin(Symbol('g', commutative=True)), cos(Function('\\\\mu_0')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["integrate", 5, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon_0')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Add(sin(Symbol('g', commutative=True)), cos(Function('\\\\mu_0')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given k{(\\hat{x}_0)} = \\cos{(\\sin{(\\hat{x}_0)})}, then obtain \\frac{d}{d \\hat{x}_0} (\\frac{k{(\\hat{x}_0)}}{\\hat{x}_0})^{\\hat{x}_0} = \\frac{d}{d \\hat{x}_0} (\\frac{\\cos{(\\sin{(\\hat{x}_0)})}}{\\hat{x}_0})^{\\hat{x}_0}", "derivation": "k{(\\hat{x}_0)} = \\cos{(\\sin{(\\hat{x}_0)})} and \\frac{k{(\\hat{x}_0)}}{\\hat{x}_0} = \\frac{\\cos{(\\sin{(\\hat{x}_0)})}}{\\hat{x}_0} and (\\frac{k{(\\hat{x}_0)}}{\\hat{x}_0})^{\\hat{x}_0} = (\\frac{\\cos{(\\sin{(\\hat{x}_0)})}}{\\hat{x}_0})^{\\hat{x}_0} and \\frac{d}{d \\hat{x}_0} (\\frac{k{(\\hat{x}_0)}}{\\hat{x}_0})^{\\hat{x}_0} = \\frac{d}{d \\hat{x}_0} (\\frac{\\cos{(\\sin{(\\hat{x}_0)})}}{\\hat{x}_0})^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True))))"], [["divide", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), cos(sin(Symbol('\\\\hat{x}_0', commutative=True)))), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} = (e^{\\mathbf{J}_f})^{G}, then obtain G (e^{\\mathbf{J}_f})^{G} - \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} = 0", "derivation": "\\operatorname{F_{H}}{(G,\\mathbf{J}_f)} = (e^{\\mathbf{J}_f})^{G} and \\mathbf{J}_f + \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} = \\mathbf{J}_f + (e^{\\mathbf{J}_f})^{G} and \\mathbf{J}_f + \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} + 1 = \\mathbf{J}_f + (e^{\\mathbf{J}_f})^{G} + 1 and \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} - (e^{\\mathbf{J}_f})^{G} + 1 = 1 and - \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} + (e^{\\mathbf{J}_f})^{G} - 1 = -1 and \\frac{\\partial}{\\partial \\mathbf{J}_f} (- \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} + (e^{\\mathbf{J}_f})^{G} - 1) = \\frac{d}{d \\mathbf{J}_f} (-1) and G (e^{\\mathbf{J}_f})^{G} - \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{F_{H}}{(G,\\mathbf{J}_f)} = 0", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True))))"], [["add", 2, 1], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(1)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True)), Integer(1)))"], [["minus", 3, "Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True)))"], "Equality(Add(Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True))), Integer(1)), Integer(1))"], [["divide", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True)), Integer(-1)), Integer(-1))"], [["differentiate", 5, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Mul(Symbol('G', commutative=True), Pow(exp(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('G', commutative=True))), Mul(Integer(-1), Derivative(Function('F_H')(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\mathbf{B}{(\\ddot{x},V)} = \\frac{\\cos{(\\ddot{x})}}{V} and \\operatorname{E_{n}}{(\\ddot{x},V)} = \\frac{\\cos{(\\ddot{x})}}{V}, then obtain e^{\\mathbf{B}{(\\ddot{x},V)}} = e^{\\operatorname{E_{n}}{(\\ddot{x},V)}}", "derivation": "\\mathbf{B}{(\\ddot{x},V)} = \\frac{\\cos{(\\ddot{x})}}{V} and \\operatorname{E_{n}}{(\\ddot{x},V)} = \\frac{\\cos{(\\ddot{x})}}{V} and e^{\\mathbf{B}{(\\ddot{x},V)}} = e^{\\frac{\\cos{(\\ddot{x})}}{V}} and e^{\\mathbf{B}{(\\ddot{x},V)}} = e^{\\operatorname{E_{n}}{(\\ddot{x},V)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{B}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True))), exp(Mul(Pow(Symbol('V', commutative=True), Integer(-1)), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(exp(Function('\\\\mathbf{B}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True))), exp(Function('E_n')(Symbol('\\\\ddot{x}', commutative=True), Symbol('V', commutative=True))))"]]}, {"prompt": "Given E{(\\hat{x}_0,\\hbar)} = \\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0, then obtain \\sin{(\\frac{E^{\\hat{x}_0}{(\\hat{x}_0,\\hbar)}}{\\omega})} = \\sin{(\\frac{(\\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0)^{\\hat{x}_0}}{\\omega})}", "derivation": "E{(\\hat{x}_0,\\hbar)} = \\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0 and E^{\\hat{x}_0}{(\\hat{x}_0,\\hbar)} = (\\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0)^{\\hat{x}_0} and \\frac{E^{\\hat{x}_0}{(\\hat{x}_0,\\hbar)}}{\\omega} = \\frac{(\\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0)^{\\hat{x}_0}}{\\omega} and \\sin{(\\frac{E^{\\hat{x}_0}{(\\hat{x}_0,\\hbar)}}{\\omega})} = \\sin{(\\frac{(\\int \\frac{\\hat{x}_0}{\\hbar} d\\hat{x}_0)^{\\hat{x}_0}}{\\omega})}", "srepr_derivation": [["get_premise", "Equality(Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["divide", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))))"], [["sin", 3], "Equality(sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Function('E')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)))), sin(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Pow(Integral(Mul(Symbol('\\\\hat{x}_0', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))))"]]}, {"prompt": "Given \\chi{(v_{x},g_{\\varepsilon})} = - g_{\\varepsilon} + \\log{(v_{x})}, then obtain \\frac{\\partial}{\\partial g_{\\varepsilon}} v_{x} (- g_{\\varepsilon} + \\chi{(v_{x},g_{\\varepsilon})}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} v_{x} (- 2 g_{\\varepsilon} + \\log{(v_{x})})", "derivation": "\\chi{(v_{x},g_{\\varepsilon})} = - g_{\\varepsilon} + \\log{(v_{x})} and - g_{\\varepsilon} + \\chi{(v_{x},g_{\\varepsilon})} = - 2 g_{\\varepsilon} + \\log{(v_{x})} and v_{x} (- g_{\\varepsilon} + \\chi{(v_{x},g_{\\varepsilon})}) = v_{x} (- 2 g_{\\varepsilon} + \\log{(v_{x})}) and \\frac{\\partial}{\\partial g_{\\varepsilon}} v_{x} (- g_{\\varepsilon} + \\chi{(v_{x},g_{\\varepsilon})}) = \\frac{\\partial}{\\partial g_{\\varepsilon}} v_{x} (- 2 g_{\\varepsilon} + \\log{(v_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('v_x', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('v_x', commutative=True))))"], [["times", 2, "Symbol('v_x', commutative=True)"], "Equality(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('v_x', commutative=True)))))"], [["differentiate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)), Function('\\\\chi')(Symbol('v_x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Symbol('v_x', commutative=True), Add(Mul(Integer(-1), Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('v_x', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(i,\\theta_2)} = \\theta_2 i and i{(\\hbar)} = \\cos{(\\hbar)}, then obtain \\theta_2 i + i^{\\hbar}{(\\hbar)} = \\theta_2 i + \\cos^{\\hbar}{(\\hbar)}", "derivation": "\\operatorname{v_{x}}{(i,\\theta_2)} = \\theta_2 i and i{(\\hbar)} = \\cos{(\\hbar)} and i^{\\hbar}{(\\hbar)} = \\cos^{\\hbar}{(\\hbar)} and i^{\\hbar}{(\\hbar)} + \\operatorname{v_{x}}{(i,\\theta_2)} = \\operatorname{v_{x}}{(i,\\theta_2)} + \\cos^{\\hbar}{(\\hbar)} and \\theta_2 i + i^{\\hbar}{(\\hbar)} = \\theta_2 i + \\cos^{\\hbar}{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\theta_2', commutative=True), Symbol('i', commutative=True)))"], ["get_premise", "Equality(Function('i')(Symbol('\\\\hbar', commutative=True)), cos(Symbol('\\\\hbar', commutative=True)))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('i')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["add", 3, "Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Pow(Function('i')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\theta_2', commutative=True))), Add(Function('v_x')(Symbol('i', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('i', commutative=True)), Pow(Function('i')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('i', commutative=True)), Pow(cos(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(F_{g},\\theta_1)} = F_{g} + \\theta_1 and \\mathbf{J}_f{(F_{g},\\theta_1)} = \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)}, then obtain (F_{g} + \\theta_1)^{F_{g}} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1 = \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1", "derivation": "\\mathbf{A}{(F_{g},\\theta_1)} = F_{g} + \\theta_1 and \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)} = (F_{g} + \\theta_1)^{F_{g}} and \\mathbf{J}_f{(F_{g},\\theta_1)} = \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)} and \\mathbf{J}_f{(F_{g},\\theta_1)} = (F_{g} + \\theta_1)^{F_{g}} and \\mathbf{J}_f{(F_{g},\\theta_1)} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1 = \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1 and (F_{g} + \\theta_1)^{F_{g}} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1 = \\mathbf{A}^{F_{g}}{(F_{g},\\theta_1)} - \\int (F_{g} + \\theta_1)^{F_{g}} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["power", 1, "Symbol('F_g', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{J}_f')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)))"], [["minus", 3, "Integral(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Integral(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), Add(Pow(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integral(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integral(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))), Add(Pow(Function('\\\\mathbf{A}')(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Mul(Integer(-1), Integral(Pow(Add(Symbol('F_g', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('F_g', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given h{(A_{1})} = \\sin{(A_{1})} and \\varepsilon_{0}{(A_{1})} = h{(A_{1})} + \\sin{(A_{1})}, then obtain 2 h{(A_{1})} = 2 \\sin{(A_{1})}", "derivation": "h{(A_{1})} = \\sin{(A_{1})} and h{(A_{1})} + \\sin{(A_{1})} = 2 \\sin{(A_{1})} and \\varepsilon_{0}{(A_{1})} = h{(A_{1})} + \\sin{(A_{1})} and \\varepsilon_{0}{(A_{1})} = 2 \\sin{(A_{1})} and \\varepsilon_{0}{(A_{1})} = 2 h{(A_{1})} and 2 h{(A_{1})} = 2 \\sin{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["add", 1, "sin(Symbol('A_1', commutative=True))"], "Equality(Add(Function('h')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))), Mul(Integer(2), sin(Symbol('A_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_1', commutative=True)), Add(Function('h')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\varepsilon_0')(Symbol('A_1', commutative=True)), Mul(Integer(2), sin(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\varepsilon_0')(Symbol('A_1', commutative=True)), Mul(Integer(2), Function('h')(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(2), Function('h')(Symbol('A_1', commutative=True))), Mul(Integer(2), sin(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given n{(\\mathbf{p},\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*} and \\theta{(\\mathbf{p},\\varphi^*)} = \\frac{\\mathbf{p}}{\\varphi^*}, then obtain \\theta{(\\mathbf{p},\\varphi^*)} \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*} = \\frac{\\mathbf{p} \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*}}{\\varphi^*}", "derivation": "n{(\\mathbf{p},\\varphi^*)} = \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*} and \\theta{(\\mathbf{p},\\varphi^*)} = \\frac{\\mathbf{p}}{\\varphi^*} and \\theta{(\\mathbf{p},\\varphi^*)} n{(\\mathbf{p},\\varphi^*)} = \\frac{\\mathbf{p} n{(\\mathbf{p},\\varphi^*)}}{\\varphi^*} and \\theta{(\\mathbf{p},\\varphi^*)} \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*} = \\frac{\\mathbf{p} \\frac{\\partial}{\\partial \\varphi^*} \\frac{\\mathbf{p}}{\\varphi^*}}{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))"], [["times", 2, "Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Function('n')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\theta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Omega{(F_{g})} = \\log{(F_{g})} and B{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)}, then obtain (\\Omega{(F_{g})} + \\cos{(\\hat{x}_0)}) t{(F_{g})} = (\\log{(F_{g})} + \\cos{(\\hat{x}_0)}) t{(F_{g})}", "derivation": "\\Omega{(F_{g})} = \\log{(F_{g})} and B{(\\hat{x}_0)} = \\cos{(\\hat{x}_0)} and B{(\\hat{x}_0)} + \\Omega{(F_{g})} = B{(\\hat{x}_0)} + \\log{(F_{g})} and \\Omega{(F_{g})} + \\cos{(\\hat{x}_0)} = \\log{(F_{g})} + \\cos{(\\hat{x}_0)} and (\\Omega{(F_{g})} + \\cos{(\\hat{x}_0)}) t{(F_{g})} = (\\log{(F_{g})} + \\cos{(\\hat{x}_0)}) t{(F_{g})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('F_g', commutative=True)), log(Symbol('F_g', commutative=True)))"], ["get_premise", "Equality(Function('B')(Symbol('\\\\hat{x}_0', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "Function('B')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Function('B')(Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\Omega')(Symbol('F_g', commutative=True))), Add(Function('B')(Symbol('\\\\hat{x}_0', commutative=True)), log(Symbol('F_g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\Omega')(Symbol('F_g', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))), Add(log(Symbol('F_g', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))))"], [["times", 4, "Function('t')(Symbol('F_g', commutative=True))"], "Equality(Mul(Add(Function('\\\\Omega')(Symbol('F_g', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))), Function('t')(Symbol('F_g', commutative=True))), Mul(Add(log(Symbol('F_g', commutative=True)), cos(Symbol('\\\\hat{x}_0', commutative=True))), Function('t')(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(m)} = e^{m}, then derive \\frac{d}{d m} \\dot{z}{(m)} = e^{m}, then obtain e^{m} + \\int \\frac{d^{2}}{d m^{2}} \\dot{z}{(m)} dm = e^{m} + \\int \\frac{d}{d m} \\dot{z}{(m)} dm", "derivation": "\\dot{z}{(m)} = e^{m} and \\frac{d}{d m} \\dot{z}{(m)} = \\frac{d}{d m} e^{m} and \\frac{d}{d m} \\dot{z}{(m)} = e^{m} and \\frac{d^{2}}{d m^{2}} \\dot{z}{(m)} = \\frac{d}{d m} e^{m} and \\frac{d^{2}}{d m^{2}} \\dot{z}{(m)} = \\frac{d}{d m} \\dot{z}{(m)} and \\int \\frac{d^{2}}{d m^{2}} \\dot{z}{(m)} dm = \\int \\frac{d}{d m} \\dot{z}{(m)} dm and e^{m} + \\int \\frac{d^{2}}{d m^{2}} \\dot{z}{(m)} dm = e^{m} + \\int \\frac{d}{d m} \\dot{z}{(m)} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True)), exp(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), exp(Symbol('m', commutative=True)))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Derivative(exp(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Tuple(Symbol('m', commutative=True))), Integral(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))))"], [["add", 6, "exp(Symbol('m', commutative=True))"], "Equality(Add(exp(Symbol('m', commutative=True)), Integral(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(2))), Tuple(Symbol('m', commutative=True)))), Add(exp(Symbol('m', commutative=True)), Integral(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\eta{(G)} = \\sin{(\\sin{(G)})} and \\operatorname{P_{g}}{(G)} = \\int \\eta^{2}{(G)} dG, then obtain \\int \\eta{(G)} \\sin{(\\sin{(G)})} dG = \\int \\sin^{2}{(\\sin{(G)})} dG", "derivation": "\\eta{(G)} = \\sin{(\\sin{(G)})} and \\eta^{2}{(G)} = \\eta{(G)} \\sin{(\\sin{(G)})} and \\int \\eta^{2}{(G)} dG = \\int \\eta{(G)} \\sin{(\\sin{(G)})} dG and \\operatorname{P_{g}}{(G)} = \\int \\eta^{2}{(G)} dG and \\operatorname{P_{g}}{(G)} = \\int \\eta{(G)} \\sin{(\\sin{(G)})} dG and \\operatorname{P_{g}}{(G)} = \\int \\sin^{2}{(\\sin{(G)})} dG and \\int \\eta{(G)} \\sin{(\\sin{(G)})} dG = \\int \\sin^{2}{(\\sin{(G)})} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True))))"], [["times", 1, "Function('\\\\eta')(Symbol('G', commutative=True))"], "Equality(Pow(Function('\\\\eta')(Symbol('G', commutative=True)), Integer(2)), Mul(Function('\\\\eta')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('G', commutative=True)), Integer(2)), Tuple(Symbol('G', commutative=True))), Integral(Mul(Function('\\\\eta')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('G', commutative=True)), Integral(Pow(Function('\\\\eta')(Symbol('G', commutative=True)), Integer(2)), Tuple(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('P_g')(Symbol('G', commutative=True)), Integral(Mul(Function('\\\\eta')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('P_g')(Symbol('G', commutative=True)), Integral(Pow(sin(sin(Symbol('G', commutative=True))), Integer(2)), Tuple(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Mul(Function('\\\\eta')(Symbol('G', commutative=True)), sin(sin(Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))), Integral(Pow(sin(sin(Symbol('G', commutative=True))), Integer(2)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}}, then derive \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{C_{2}}{(V_{\\mathbf{B}})} = 1, then obtain \\frac{\\Psi_{\\lambda} \\frac{d}{d V_{\\mathbf{B}}} V_{\\mathbf{B}}}{V_{\\mathbf{B}}} = \\frac{\\Psi_{\\lambda}}{V_{\\mathbf{B}}}", "derivation": "\\operatorname{C_{2}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} and \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{C_{2}}{(V_{\\mathbf{B}})} = \\frac{d}{d V_{\\mathbf{B}}} V_{\\mathbf{B}} and \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{C_{2}}{(V_{\\mathbf{B}})} = 1 and \\frac{\\Psi_{\\lambda} \\frac{d}{d V_{\\mathbf{B}}} \\operatorname{C_{2}}{(V_{\\mathbf{B}})}}{V_{\\mathbf{B}}} = \\frac{\\Psi_{\\lambda}}{V_{\\mathbf{B}}} and \\frac{\\Psi_{\\lambda} \\frac{d}{d V_{\\mathbf{B}}} V_{\\mathbf{B}}}{V_{\\mathbf{B}}} = \\frac{\\Psi_{\\lambda}}{V_{\\mathbf{B}}}", "srepr_derivation": [["renaming_premise", "Equality(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Symbol('V_{\\\\mathbf{B}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Derivative(Function('C_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Derivative(Symbol('V_{\\\\mathbf{B}}', commutative=True), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}{(F_{g},E_{x})} = E_{x}^{F_{g}}, then obtain - F_{g} \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{\\tilde{g}{(F_{g},E_{x})}}{E_{x}})} = - F_{g} \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{E_{x}^{F_{g}}}{E_{x}})}", "derivation": "\\tilde{g}{(F_{g},E_{x})} = E_{x}^{F_{g}} and \\frac{\\tilde{g}{(F_{g},E_{x})}}{E_{x}} = \\frac{E_{x}^{F_{g}}}{E_{x}} and \\cos{(\\frac{\\tilde{g}{(F_{g},E_{x})}}{E_{x}})} = \\cos{(\\frac{E_{x}^{F_{g}}}{E_{x}})} and \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{\\tilde{g}{(F_{g},E_{x})}}{E_{x}})} = \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{E_{x}^{F_{g}}}{E_{x}})} and - F_{g} \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{\\tilde{g}{(F_{g},E_{x})}}{E_{x}})} = - F_{g} \\frac{\\partial}{\\partial E_{x}} \\cos{(\\frac{E_{x}^{F_{g}}}{E_{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('F_g', commutative=True)))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('E_x', commutative=True), Symbol('F_g', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)))), cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('E_x', commutative=True), Symbol('F_g', commutative=True)))))"], [["differentiate", 3, "Symbol('E_x', commutative=True)"], "Equality(Derivative(cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('E_x', commutative=True), Symbol('F_g', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["times", 4, "Mul(Integer(-1), Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_g', commutative=True), Derivative(cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True), Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('F_g', commutative=True), Derivative(cos(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Pow(Symbol('E_x', commutative=True), Symbol('F_g', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(s)} = \\sin{(s)}, then derive \\int \\mathbf{s}{(s)} ds = \\mathbf{H} - \\cos{(s)}, then obtain (\\sin^{s}{(s)} + \\int \\mathbf{s}{(s)} ds) (\\mathbf{H} - \\cos{(s)} + 1) = (\\mathbf{H} + \\sin^{s}{(s)} - \\cos{(s)}) (\\mathbf{H} - \\cos{(s)} + 1)", "derivation": "\\mathbf{s}{(s)} = \\sin{(s)} and \\int \\mathbf{s}{(s)} ds = \\int \\sin{(s)} ds and \\int \\mathbf{s}{(s)} ds = \\mathbf{H} - \\cos{(s)} and \\int \\sin{(s)} ds = \\mathbf{H} - \\cos{(s)} and \\int \\sin{(s)} ds + 1 = \\mathbf{H} - \\cos{(s)} + 1 and \\sin^{s}{(s)} + \\int \\mathbf{s}{(s)} ds = \\mathbf{H} + \\sin^{s}{(s)} - \\cos{(s)} and (\\sin^{s}{(s)} + \\int \\mathbf{s}{(s)} ds) (\\int \\sin{(s)} ds + 1) = (\\int \\sin{(s)} ds + 1) (\\mathbf{H} + \\sin^{s}{(s)} - \\cos{(s)}) and (\\sin^{s}{(s)} + \\int \\mathbf{s}{(s)} ds) (\\mathbf{H} - \\cos{(s)} + 1) = (\\mathbf{H} + \\sin^{s}{(s)} - \\cos{(s)}) (\\mathbf{H} - \\cos{(s)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integer(1)))"], [["add", 3, "Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True)))))"], [["times", 6, "Add(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1))"], "Equality(Mul(Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1))), Mul(Add(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Add(Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Integral(Function('\\\\mathbf{s}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integer(1))), Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Pow(sin(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True))), Integer(1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(E_{n})} = \\frac{d}{d E_{n}} \\sin{(E_{n})}, then derive \\dot{\\mathbf{r}}{(E_{n})} = \\cos{(E_{n})}, then obtain - \\frac{d}{d E_{n}} \\cos{(E_{n})} = - \\frac{d^{2}}{d E_{n}^{2}} \\sin{(E_{n})}", "derivation": "\\dot{\\mathbf{r}}{(E_{n})} = \\frac{d}{d E_{n}} \\sin{(E_{n})} and \\dot{\\mathbf{r}}{(E_{n})} = \\cos{(E_{n})} and \\frac{d}{d E_{n}} \\dot{\\mathbf{r}}{(E_{n})} = \\frac{d}{d E_{n}} \\cos{(E_{n})} and \\frac{d}{d E_{n}} \\sin{(E_{n})} = \\cos{(E_{n})} and \\frac{d}{d E_{n}} \\dot{\\mathbf{r}}{(E_{n})} = \\frac{d^{2}}{d E_{n}^{2}} \\sin{(E_{n})} and - \\frac{d}{d E_{n}} \\dot{\\mathbf{r}}{(E_{n})} = - \\frac{d^{2}}{d E_{n}^{2}} \\sin{(E_{n})} and - \\frac{d}{d E_{n}} \\cos{(E_{n})} = - \\frac{d^{2}}{d E_{n}^{2}} \\sin{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('E_n', commutative=True)), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), cos(Symbol('E_n', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2))))"], [["times", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Integer(-1), Derivative(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(sin(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(2)))))"]]}, {"prompt": "Given z{(\\theta_1)} = \\log{(\\theta_1)} and \\mathbf{s}{(\\theta_1)} = \\log{(\\theta_1)} - \\log{(z{(\\theta_1)})}, then obtain (\\mathbf{s}{(\\theta_1)} - 1) e^{(\\log{(\\theta_1)} + \\log{(z{(\\theta_1)})})^{2}} = (z{(\\theta_1)} - \\log{(\\log{(\\theta_1)})} - 1) e^{(\\log{(\\theta_1)} + \\log{(z{(\\theta_1)})})^{2}}", "derivation": "z{(\\theta_1)} = \\log{(\\theta_1)} and \\log{(z{(\\theta_1)})} = \\log{(\\log{(\\theta_1)})} and z{(\\theta_1)} - \\log{(z{(\\theta_1)})} = \\log{(\\theta_1)} - \\log{(z{(\\theta_1)})} and \\mathbf{s}{(\\theta_1)} = \\log{(\\theta_1)} - \\log{(z{(\\theta_1)})} and \\mathbf{s}{(\\theta_1)} = z{(\\theta_1)} - \\log{(z{(\\theta_1)})} and \\mathbf{s}{(\\theta_1)} - 1 = z{(\\theta_1)} - \\log{(z{(\\theta_1)})} - 1 and \\mathbf{s}{(\\theta_1)} - 1 = z{(\\theta_1)} - \\log{(\\log{(\\theta_1)})} - 1 and (\\mathbf{s}{(\\theta_1)} - 1) e^{(\\log{(\\theta_1)} + \\log{(z{(\\theta_1)})})^{2}} = (z{(\\theta_1)} - \\log{(\\log{(\\theta_1)})} - 1) e^{(\\log{(\\theta_1)} + \\log{(z{(\\theta_1)})})^{2}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\theta_1', commutative=True)), log(Symbol('\\\\theta_1', commutative=True)))"], [["log", 1], "Equality(log(Function('z')(Symbol('\\\\theta_1', commutative=True))), log(log(Symbol('\\\\theta_1', commutative=True))))"], [["minus", 1, "log(Function('z')(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Add(Function('z')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Function('z')(Symbol('\\\\theta_1', commutative=True))))), Add(log(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Function('z')(Symbol('\\\\theta_1', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), Add(log(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Function('z')(Symbol('\\\\theta_1', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), Add(Function('z')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Function('z')(Symbol('\\\\theta_1', commutative=True))))))"], [["minus", 5, 1], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Add(Function('z')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(Function('z')(Symbol('\\\\theta_1', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Add(Function('z')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\theta_1', commutative=True)))), Integer(-1)))"], [["times", 7, "exp(Pow(Add(log(Symbol('\\\\theta_1', commutative=True)), log(Function('z')(Symbol('\\\\theta_1', commutative=True)))), Integer(2)))"], "Equality(Mul(Add(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Pow(Add(log(Symbol('\\\\theta_1', commutative=True)), log(Function('z')(Symbol('\\\\theta_1', commutative=True)))), Integer(2)))), Mul(Add(Function('z')(Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\theta_1', commutative=True)))), Integer(-1)), exp(Pow(Add(log(Symbol('\\\\theta_1', commutative=True)), log(Function('z')(Symbol('\\\\theta_1', commutative=True)))), Integer(2)))))"]]}, {"prompt": "Given \\Omega{(n_{2},\\mathbf{J}_f)} = \\mathbf{J}_f n_{2}, then obtain \\int (\\mathbf{J}_f + 1) d\\mathbf{J}_f = \\int (\\mathbf{J}_f + (\\frac{\\mathbf{J}_f n_{2}}{\\Omega{(n_{2},\\mathbf{J}_f)}})^{\\mathbf{J}_f}) d\\mathbf{J}_f", "derivation": "\\Omega{(n_{2},\\mathbf{J}_f)} = \\mathbf{J}_f n_{2} and 1 = \\frac{\\mathbf{J}_f n_{2}}{\\Omega{(n_{2},\\mathbf{J}_f)}} and 1 = (\\frac{\\mathbf{J}_f n_{2}}{\\Omega{(n_{2},\\mathbf{J}_f)}})^{\\mathbf{J}_f} and \\mathbf{J}_f + 1 = \\mathbf{J}_f + (\\frac{\\mathbf{J}_f n_{2}}{\\Omega{(n_{2},\\mathbf{J}_f)}})^{\\mathbf{J}_f} and \\int (\\mathbf{J}_f + 1) d\\mathbf{J}_f = \\int (\\mathbf{J}_f + (\\frac{\\mathbf{J}_f n_{2}}{\\Omega{(n_{2},\\mathbf{J}_f)}})^{\\mathbf{J}_f}) d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_2', commutative=True)))"], [["divide", 1, "Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_2', commutative=True), Pow(Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_2', commutative=True), Pow(Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_2', commutative=True), Pow(Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('n_2', commutative=True), Pow(Function('\\\\Omega')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})}, then obtain \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})}", "derivation": "\\operatorname{y^{\\prime}}{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} and \\frac{d}{d J_{\\varepsilon}} \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} and \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} and \\operatorname{y^{\\prime}}{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})} = \\cos{(J_{\\varepsilon})} \\frac{d}{d J_{\\varepsilon}} \\cos{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Mul(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(cos(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(A_{x},B,H)} = A_{x} B - H, then derive \\int \\varphi^{*}{(A_{x},B,H)} dH = A_{x} B H - \\frac{H^{2}}{2} + \\omega, then obtain \\hbar \\frac{\\partial}{\\partial A_{x}} \\int \\varphi^{*}{(A_{x},B,H)} dH = \\hbar \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} B - H) dH", "derivation": "\\varphi^{*}{(A_{x},B,H)} = A_{x} B - H and \\int \\varphi^{*}{(A_{x},B,H)} dH = \\int (A_{x} B - H) dH and \\int \\varphi^{*}{(A_{x},B,H)} dH = A_{x} B H - \\frac{H^{2}}{2} + \\omega and \\frac{\\partial}{\\partial A_{x}} \\int \\varphi^{*}{(A_{x},B,H)} dH = \\frac{\\partial}{\\partial A_{x}} (A_{x} B H - \\frac{H^{2}}{2} + \\omega) and \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} B - H) dH = \\frac{\\partial}{\\partial A_{x}} (A_{x} B H - \\frac{H^{2}}{2} + \\omega) and \\frac{\\partial}{\\partial A_{x}} \\int \\varphi^{*}{(A_{x},B,H)} dH = \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} B - H) dH and \\hbar \\frac{\\partial}{\\partial A_{x}} \\int \\varphi^{*}{(A_{x},B,H)} dH = \\hbar \\frac{\\partial}{\\partial A_{x}} \\int (A_{x} B - H) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\omega', commutative=True)))"], [["differentiate", 3, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["times", 6, "Symbol('\\\\hbar', commutative=True)"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Derivative(Integral(Function('\\\\varphi^*')(Symbol('A_x', commutative=True), Symbol('B', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))), Mul(Symbol('\\\\hbar', commutative=True), Derivative(Integral(Add(Mul(Symbol('A_x', commutative=True), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} = - \\eta^{\\prime} + n_{1}, then obtain - n_{1} + (\\eta^{\\prime} - n_{1} + \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} + 1)^{n_{1}} = 1 - n_{1}", "derivation": "\\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} = - \\eta^{\\prime} + n_{1} and \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} + 1 = - \\eta^{\\prime} + n_{1} + 1 and \\eta^{\\prime} - n_{1} + \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} + 1 = 1 and (\\eta^{\\prime} - n_{1} + \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} + 1)^{n_{1}} = 1 and - n_{1} + (\\eta^{\\prime} - n_{1} + \\operatorname{A_{2}}{(\\eta^{\\prime},n_{1})} + 1)^{n_{1}} = 1 - n_{1}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('n_1', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('A_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_1', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('n_1', commutative=True), Integer(1)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\eta^{\\\\prime}', commutative=True)), Symbol('n_1', commutative=True))"], "Equality(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('A_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_1', commutative=True)), Integer(1)), Integer(1))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('A_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_1', commutative=True)), Integer(1)), Symbol('n_1', commutative=True)), Integer(1))"], [["minus", 4, "Symbol('n_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n_1', commutative=True)), Pow(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Symbol('n_1', commutative=True)), Function('A_2')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('n_1', commutative=True)), Integer(1)), Symbol('n_1', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(k)} = \\sin{(k)} and \\mathbf{M}{(k)} = - e^{\\Psi_{\\lambda}{(k)}} \\sin{(k)} + e^{\\sin{(k)}}, then obtain (e^{\\Psi_{\\lambda}{(k)}} - e^{\\sin{(k)}} \\sin{(k)})^{k} = \\mathbf{M}^{k}{(k)}", "derivation": "\\Psi_{\\lambda}{(k)} = \\sin{(k)} and e^{\\Psi_{\\lambda}{(k)}} = e^{\\sin{(k)}} and e^{\\Psi_{\\lambda}{(k)}} \\sin{(k)} = e^{\\sin{(k)}} \\sin{(k)} and e^{\\Psi_{\\lambda}{(k)}} - e^{\\sin{(k)}} \\sin{(k)} = - e^{\\sin{(k)}} \\sin{(k)} + e^{\\sin{(k)}} and (e^{\\Psi_{\\lambda}{(k)}} - e^{\\sin{(k)}} \\sin{(k)})^{k} = (- e^{\\sin{(k)}} \\sin{(k)} + e^{\\sin{(k)}})^{k} and \\mathbf{M}{(k)} = - e^{\\Psi_{\\lambda}{(k)}} \\sin{(k)} + e^{\\sin{(k)}} and \\mathbf{M}{(k)} = - e^{\\sin{(k)}} \\sin{(k)} + e^{\\sin{(k)}} and (e^{\\Psi_{\\lambda}{(k)}} - e^{\\sin{(k)}} \\sin{(k)})^{k} = \\mathbf{M}^{k}{(k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), exp(sin(Symbol('k', commutative=True))))"], [["times", 2, "sin(Symbol('k', commutative=True))"], "Equality(Mul(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), Mul(exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))))"], [["minus", 2, "Mul(exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True)))"], "Equality(Add(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True)))), Add(Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), exp(sin(Symbol('k', commutative=True)))))"], [["power", 4, "Symbol('k', commutative=True)"], "Equality(Pow(Add(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), exp(sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Add(Mul(Integer(-1), exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), exp(sin(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Add(Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True))), exp(sin(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Pow(Add(exp(Function('\\\\Psi_{\\\\lambda}')(Symbol('k', commutative=True))), Mul(Integer(-1), exp(sin(Symbol('k', commutative=True))), sin(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Pow(Function('\\\\mathbf{M}')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\tilde{g}^*{(\\mathbf{J}_M,\\varphi^*)} = \\mathbf{J}_M + \\varphi^*, then obtain \\int (- \\frac{(\\mathbf{J}_M + \\varphi^*)^{\\mathbf{J}_M}}{\\mathbf{J}_M} + \\frac{\\tilde{g}^*^{\\mathbf{J}_M}{(\\mathbf{J}_M,\\varphi^*)}}{\\mathbf{J}_M}) d\\varphi^* = \\int 0 d\\varphi^*", "derivation": "\\tilde{g}^*{(\\mathbf{J}_M,\\varphi^*)} = \\mathbf{J}_M + \\varphi^* and \\tilde{g}^*^{\\mathbf{J}_M}{(\\mathbf{J}_M,\\varphi^*)} = (\\mathbf{J}_M + \\varphi^*)^{\\mathbf{J}_M} and \\frac{\\tilde{g}^*^{\\mathbf{J}_M}{(\\mathbf{J}_M,\\varphi^*)}}{\\mathbf{J}_M} = \\frac{(\\mathbf{J}_M + \\varphi^*)^{\\mathbf{J}_M}}{\\mathbf{J}_M} and - \\frac{(\\mathbf{J}_M + \\varphi^*)^{\\mathbf{J}_M}}{\\mathbf{J}_M} + \\frac{\\tilde{g}^*^{\\mathbf{J}_M}{(\\mathbf{J}_M,\\varphi^*)}}{\\mathbf{J}_M} = 0 and \\int (- \\frac{(\\mathbf{J}_M + \\varphi^*)^{\\mathbf{J}_M}}{\\mathbf{J}_M} + \\frac{\\tilde{g}^*^{\\mathbf{J}_M}{(\\mathbf{J}_M,\\varphi^*)}}{\\mathbf{J}_M}) d\\varphi^* = \\int 0 d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["divide", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))), Integer(0))"], [["integrate", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)), Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given S{(L_{\\varepsilon},f,\\mathbf{s})} = - L_{\\varepsilon} + \\mathbf{s} f, then obtain \\frac{\\mathbf{s} f \\int \\frac{S{(L_{\\varepsilon},f,\\mathbf{s})}}{\\mathbf{s} f} d\\mathbf{s}}{S{(L_{\\varepsilon},f,\\mathbf{s})}} = \\frac{\\mathbf{s} f \\int \\frac{- L_{\\varepsilon} + \\mathbf{s} f}{\\mathbf{s} f} d\\mathbf{s}}{S{(L_{\\varepsilon},f,\\mathbf{s})}}", "derivation": "S{(L_{\\varepsilon},f,\\mathbf{s})} = - L_{\\varepsilon} + \\mathbf{s} f and \\frac{S{(L_{\\varepsilon},f,\\mathbf{s})}}{\\mathbf{s} f} = \\frac{- L_{\\varepsilon} + \\mathbf{s} f}{\\mathbf{s} f} and \\int \\frac{S{(L_{\\varepsilon},f,\\mathbf{s})}}{\\mathbf{s} f} d\\mathbf{s} = \\int \\frac{- L_{\\varepsilon} + \\mathbf{s} f}{\\mathbf{s} f} d\\mathbf{s} and \\frac{\\mathbf{s} f \\int \\frac{S{(L_{\\varepsilon},f,\\mathbf{s})}}{\\mathbf{s} f} d\\mathbf{s}}{S{(L_{\\varepsilon},f,\\mathbf{s})}} = \\frac{\\mathbf{s} f \\int \\frac{- L_{\\varepsilon} + \\mathbf{s} f}{\\mathbf{s} f} d\\mathbf{s}}{S{(L_{\\varepsilon},f,\\mathbf{s})}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True), Pow(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True), Pow(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('f', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1)), Integral(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given H{(n_{1},\\varphi^*)} = \\frac{\\varphi^*}{n_{1}} + n_{1} and \\operatorname{t_{1}}{(n_{1},\\varphi^*)} = \\frac{\\varphi^*}{n_{1}} + n_{1}, then obtain ((\\frac{\\varphi^*}{n_{1}} + n_{1})^{n_{1}})^{n_{1}} = (\\operatorname{t_{1}}^{n_{1}}{(n_{1},\\varphi^*)})^{n_{1}}", "derivation": "H{(n_{1},\\varphi^*)} = \\frac{\\varphi^*}{n_{1}} + n_{1} and H^{n_{1}}{(n_{1},\\varphi^*)} = (\\frac{\\varphi^*}{n_{1}} + n_{1})^{n_{1}} and (H^{n_{1}}{(n_{1},\\varphi^*)})^{n_{1}} = ((\\frac{\\varphi^*}{n_{1}} + n_{1})^{n_{1}})^{n_{1}} and \\operatorname{t_{1}}{(n_{1},\\varphi^*)} = \\frac{\\varphi^*}{n_{1}} + n_{1} and (H^{n_{1}}{(n_{1},\\varphi^*)})^{n_{1}} = (\\operatorname{t_{1}}^{n_{1}}{(n_{1},\\varphi^*)})^{n_{1}} and ((\\frac{\\varphi^*}{n_{1}} + n_{1})^{n_{1}})^{n_{1}} = (\\operatorname{t_{1}}^{n_{1}}{(n_{1},\\varphi^*)})^{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Symbol('n_1', commutative=True)))"], [["power", 1, "Symbol('n_1', commutative=True)"], "Equality(Pow(Function('H')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('n_1', commutative=True)), Pow(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Pow(Function('H')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Pow(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('H')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Pow(Function('t_1')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)), Pow(Pow(Function('t_1')(Symbol('n_1', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('n_1', commutative=True)), Symbol('n_1', commutative=True)))"]]}, {"prompt": "Given z{(\\theta)} = \\cos{(e^{\\theta})}, then obtain z{(\\theta)} e^{z^{\\theta}{(\\theta)}} = e^{z^{\\theta}{(\\theta)}} \\cos{(e^{\\theta})}", "derivation": "z{(\\theta)} = \\cos{(e^{\\theta})} and z^{\\theta}{(\\theta)} = \\cos^{\\theta}{(e^{\\theta})} and z{(\\theta)} e^{\\cos^{\\theta}{(e^{\\theta})}} = e^{\\cos^{\\theta}{(e^{\\theta})}} \\cos{(e^{\\theta})} and z{(\\theta)} e^{z^{\\theta}{(\\theta)}} = e^{z^{\\theta}{(\\theta)}} \\cos{(e^{\\theta})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\theta', commutative=True)), cos(exp(Symbol('\\\\theta', commutative=True))))"], [["power", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('z')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(cos(exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "exp(Pow(cos(exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Function('z')(Symbol('\\\\theta', commutative=True)), exp(Pow(cos(exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))), Mul(exp(Pow(cos(exp(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('z')(Symbol('\\\\theta', commutative=True)), exp(Pow(Function('z')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))), Mul(exp(Pow(Function('z')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), cos(exp(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\tilde{g},Q)} = \\frac{\\tilde{g}}{Q}, then obtain \\int Q^{2} \\mathbf{J}_M^{2}{(\\tilde{g},Q)} d\\tilde{g} = \\int \\frac{Q^{4} \\mathbf{J}_M^{4}{(\\tilde{g},Q)}}{\\tilde{g}^{2}} d\\tilde{g}", "derivation": "\\mathbf{J}_M{(\\tilde{g},Q)} = \\frac{\\tilde{g}}{Q} and Q \\mathbf{J}_M{(\\tilde{g},Q)} = \\tilde{g} and \\frac{1}{Q \\mathbf{J}_M{(\\tilde{g},Q)}} = \\frac{\\tilde{g}}{Q^{2} \\mathbf{J}_M^{2}{(\\tilde{g},Q)}} and Q^{2} \\mathbf{J}_M^{2}{(\\tilde{g},Q)} = \\frac{Q^{4} \\mathbf{J}_M^{4}{(\\tilde{g},Q)}}{\\tilde{g}^{2}} and \\int Q^{2} \\mathbf{J}_M^{2}{(\\tilde{g},Q)} d\\tilde{g} = \\int \\frac{Q^{4} \\mathbf{J}_M^{4}{(\\tilde{g},Q)}}{\\tilde{g}^{2}} d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)))"], [["divide", 1, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\tilde{g}', commutative=True))"], [["divide", 2, "Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-2)), Symbol('\\\\tilde{g}', commutative=True), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(-2))))"], [["power", 3, "Integer(-2)"], "Equality(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Mul(Pow(Symbol('Q', commutative=True), Integer(4)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(4))))"], [["integrate", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(2)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(2))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(4)), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('Q', commutative=True)), Integer(4))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given t{(\\hat{H},\\nabla)} = \\nabla + \\log{(\\hat{H})}, then obtain \\hat{H} = \\frac{\\hat{H} (\\nabla + t{(\\hat{H},\\nabla)} + \\log{(\\hat{H})})}{2 t{(\\hat{H},\\nabla)}}", "derivation": "t{(\\hat{H},\\nabla)} = \\nabla + \\log{(\\hat{H})} and 2 t{(\\hat{H},\\nabla)} = \\nabla + t{(\\hat{H},\\nabla)} + \\log{(\\hat{H})} and 2 \\hat{H} t{(\\hat{H},\\nabla)} = \\hat{H} (\\nabla + t{(\\hat{H},\\nabla)} + \\log{(\\hat{H})}) and \\hat{H} = \\frac{\\hat{H} (\\nabla + t{(\\hat{H},\\nabla)} + \\log{(\\hat{H})})}{2 t{(\\hat{H},\\nabla)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('\\\\nabla', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))))"], [["add", 1, "Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Symbol('\\\\nabla', commutative=True), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))))"], [["divide", 3, "Mul(Integer(2), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], "Equality(Symbol('\\\\hat{H}', commutative=True), Mul(Rational(1, 2), Symbol('\\\\hat{H}', commutative=True), Add(Symbol('\\\\nabla', commutative=True), Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True))), Pow(Function('t')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(t)} = \\cos{(t)} and \\operatorname{v_{y}}{(t)} = \\cos{(t)}, then obtain \\sin{((\\operatorname{v_{y}}{(t)} - \\operatorname{x^{{\\}'}}{(t)})^{4})} = 0", "derivation": "\\operatorname{x^{{\\}'}}{(t)} = \\cos{(t)} and \\operatorname{v_{y}}{(t)} = \\cos{(t)} and \\operatorname{v_{y}}{(t)} - \\cos{(t)} = 0 and (\\operatorname{v_{y}}{(t)} - \\cos{(t)})^{2} = 0 and (\\operatorname{v_{y}}{(t)} - \\cos{(t)})^{4} = 0 and (\\operatorname{v_{y}}{(t)} - \\operatorname{x^{{\\}'}}{(t)})^{4} = 0 and \\sin{((\\operatorname{v_{y}}{(t)} - \\operatorname{x^{{\\}'}}{(t)})^{4})} = 0", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["add", 2, "Mul(Integer(-1), cos(Symbol('t', commutative=True)))"], "Equality(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Integer(0))"], [["times", 3, "Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True))))"], "Equality(Pow(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Integer(2)), Integer(0))"], [["times", 4, "Pow(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Integer(2))"], "Equality(Pow(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), cos(Symbol('t', commutative=True)))), Integer(4)), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('t', commutative=True)))), Integer(4)), Integer(0))"], [["sin", 6], "Equality(sin(Pow(Add(Function('v_y')(Symbol('t', commutative=True)), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('t', commutative=True)))), Integer(4))), Integer(0))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})} = \\frac{(f^{\\prime})^{f^{*}}}{i}, then obtain \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} (- (f^{\\prime})^{f^{*}} + \\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})}) = \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} (- (f^{\\prime})^{f^{*}} + \\frac{(f^{\\prime})^{f^{*}}}{i})", "derivation": "\\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})} = \\frac{(f^{\\prime})^{f^{*}}}{i} and - (f^{\\prime})^{f^{*}} + \\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})} = - (f^{\\prime})^{f^{*}} + \\frac{(f^{\\prime})^{f^{*}}}{i} and \\frac{\\partial}{\\partial f^{\\prime}} (- (f^{\\prime})^{f^{*}} + \\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})}) = \\frac{\\partial}{\\partial f^{\\prime}} (- (f^{\\prime})^{f^{*}} + \\frac{(f^{\\prime})^{f^{*}}}{i}) and \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} (- (f^{\\prime})^{f^{*}} + \\operatorname{y^{\\prime}}{(f^{\\prime},i,f^{*})}) = \\frac{\\partial^{2}}{\\partial (f^{\\prime})^{2}} (- (f^{\\prime})^{f^{*}} + \\frac{(f^{\\prime})^{f^{*}}}{i})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('i', commutative=True), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('i', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('i', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Function('y^{\\\\prime}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('i', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('i', commutative=True), Integer(-1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{P}{(z)} = \\cos{(z)} and H{(z)} = \\mathbf{P}^{z}{(z)}, then obtain H^{z}{(z)} = (\\cos^{z}{(z)})^{z}", "derivation": "\\mathbf{P}{(z)} = \\cos{(z)} and \\mathbf{P}^{z}{(z)} = \\cos^{z}{(z)} and (\\mathbf{P}^{z}{(z)})^{z} = (\\cos^{z}{(z)})^{z} and H{(z)} = \\mathbf{P}^{z}{(z)} and H^{z}{(z)} = (\\cos^{z}{(z)})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["power", 2, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('z', commutative=True)), Pow(Function('\\\\mathbf{P}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('H')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\mathbf{E})} = \\sin{(\\mathbf{E})}, then derive \\int \\operatorname{E_{\\lambda}}{(\\mathbf{E})} d\\mathbf{E} = v_{t} - \\cos{(\\mathbf{E})}, then obtain v_{t} - \\cos{(\\mathbf{E})} = \\int \\sin{(\\mathbf{E})} d\\mathbf{E}", "derivation": "\\operatorname{E_{\\lambda}}{(\\mathbf{E})} = \\sin{(\\mathbf{E})} and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{E})} d\\mathbf{E} = \\int \\sin{(\\mathbf{E})} d\\mathbf{E} and \\int \\operatorname{E_{\\lambda}}{(\\mathbf{E})} d\\mathbf{E} = v_{t} - \\cos{(\\mathbf{E})} and v_{t} - \\cos{(\\mathbf{E})} = \\int \\sin{(\\mathbf{E})} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), sin(Symbol('\\\\mathbf{E}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('v_t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('v_t', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Integral(sin(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})}, then derive \\frac{d}{d \\mathbf{p}} \\operatorname{n_{2}}{(\\mathbf{p})} = - \\sin{(\\mathbf{p})}, then obtain \\frac{d}{d \\mathbf{p}} \\cos{(\\mathbf{p})} = - \\sin{(\\mathbf{p})}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{p})} = \\cos{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\operatorname{n_{2}}{(\\mathbf{p})} = \\frac{d}{d \\mathbf{p}} \\cos{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\operatorname{n_{2}}{(\\mathbf{p})} = - \\sin{(\\mathbf{p})} and \\frac{d}{d \\mathbf{p}} \\cos{(\\mathbf{p})} = - \\sin{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True)), cos(Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given l{(B)} = \\int e^{B} dB, then derive l{(B)} = \\hat{H}_{\\lambda} + e^{B}, then obtain \\frac{d}{d \\hat{H}_{\\lambda}} \\iint e^{B} dB dB = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\int (\\hat{H}_{\\lambda} + e^{B}) dB", "derivation": "l{(B)} = \\int e^{B} dB and l{(B)} = \\hat{H}_{\\lambda} + e^{B} and \\int e^{B} dB = \\hat{H}_{\\lambda} + e^{B} and \\iint e^{B} dB dB = \\int (\\hat{H}_{\\lambda} + e^{B}) dB and \\frac{d}{d \\hat{H}_{\\lambda}} \\iint e^{B} dB dB = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\int (\\hat{H}_{\\lambda} + e^{B}) dB", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('B', commutative=True)), Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('l')(Symbol('B', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(Symbol('B', commutative=True))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Integral(exp(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), exp(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given k{(H)} = \\frac{d}{d H} \\log{(H)}, then derive k{(H)} = \\frac{1}{H}, then obtain \\int \\frac{d}{d H} 2 (\\frac{1}{H})^{H} dH = \\int \\frac{d}{d H} ((\\frac{1}{H})^{H} + (\\frac{d}{d H} \\log{(H)})^{H}) dH", "derivation": "k{(H)} = \\frac{d}{d H} \\log{(H)} and k^{H}{(H)} = (\\frac{d}{d H} \\log{(H)})^{H} and 2 k^{H}{(H)} = k^{H}{(H)} + (\\frac{d}{d H} \\log{(H)})^{H} and k{(H)} = \\frac{1}{H} and 2 (\\frac{1}{H})^{H} = (\\frac{1}{H})^{H} + (\\frac{d}{d H} \\log{(H)})^{H} and \\frac{d}{d H} 2 (\\frac{1}{H})^{H} = \\frac{d}{d H} ((\\frac{1}{H})^{H} + (\\frac{d}{d H} \\log{(H)})^{H}) and \\int \\frac{d}{d H} 2 (\\frac{1}{H})^{H} dH = \\int \\frac{d}{d H} ((\\frac{1}{H})^{H} + (\\frac{d}{d H} \\log{(H)})^{H}) dH", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('H', commutative=True)), Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('k')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)))"], [["add", 2, "Pow(Function('k')(Symbol('H', commutative=True)), Symbol('H', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('k')(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Add(Pow(Function('k')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('k')(Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(2), Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Add(Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))))"], [["differentiate", 5, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))), Integral(Derivative(Add(Pow(Pow(Symbol('H', commutative=True), Integer(-1)), Symbol('H', commutative=True)), Pow(Derivative(log(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given k{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)}, then obtain \\int \\frac{d}{d \\mathbf{J}_M} \\int k{(\\mathbf{J}_M)} d\\mathbf{J}_M d\\mathbf{J}_M = \\int \\frac{d}{d \\mathbf{J}_M} \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M d\\mathbf{J}_M", "derivation": "k{(\\mathbf{J}_M)} = \\sin{(\\mathbf{J}_M)} and \\int k{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\frac{d}{d \\mathbf{J}_M} \\int k{(\\mathbf{J}_M)} d\\mathbf{J}_M = \\frac{d}{d \\mathbf{J}_M} \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M and \\int \\frac{d}{d \\mathbf{J}_M} \\int k{(\\mathbf{J}_M)} d\\mathbf{J}_M d\\mathbf{J}_M = \\int \\frac{d}{d \\mathbf{J}_M} \\int \\sin{(\\mathbf{J}_M)} d\\mathbf{J}_M d\\mathbf{J}_M", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), sin(Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(Integral(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('k')(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Derivative(Integral(sin(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\phi,f^{*})} = \\phi f^{*}, then derive \\frac{\\partial}{\\partial f^{*}} \\operatorname{P_{e}}{(\\phi,f^{*})} = \\phi, then obtain \\phi = \\frac{\\partial}{\\partial f^{*}} \\phi f^{*}", "derivation": "\\operatorname{P_{e}}{(\\phi,f^{*})} = \\phi f^{*} and \\frac{\\partial}{\\partial f^{*}} \\operatorname{P_{e}}{(\\phi,f^{*})} = \\frac{\\partial}{\\partial f^{*}} \\phi f^{*} and \\frac{\\partial}{\\partial f^{*}} \\operatorname{P_{e}}{(\\phi,f^{*})} = \\phi and \\phi = \\frac{\\partial}{\\partial f^{*}} \\phi f^{*}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('P_e')(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Symbol('\\\\phi', commutative=True), Derivative(Mul(Symbol('\\\\phi', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})} = \\frac{P_{g} f^{*}}{s}, then obtain (\\frac{\\partial}{\\partial P_{g}} \\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})})^{f^{*}} = (\\frac{f^{*}}{s})^{f^{*}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})} = \\frac{P_{g} f^{*}}{s} and \\frac{\\partial}{\\partial P_{g}} \\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})} = \\frac{\\partial}{\\partial P_{g}} \\frac{P_{g} f^{*}}{s} and (\\frac{\\partial}{\\partial P_{g}} \\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})})^{f^{*}} = (\\frac{\\partial}{\\partial P_{g}} \\frac{P_{g} f^{*}}{s})^{f^{*}} and (\\frac{\\partial}{\\partial P_{g}} \\operatorname{V_{\\mathbf{E}}}{(f^{*},s,P_{g})})^{f^{*}} = (\\frac{f^{*}}{s})^{f^{*}}", "srepr_derivation": [["get_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True), Symbol('s', commutative=True), Symbol('P_g', commutative=True)), Mul(Symbol('P_g', commutative=True), Symbol('f^*', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('P_g', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True), Symbol('s', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_g', commutative=True), Symbol('f^*', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('f^*', commutative=True)"], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True), Symbol('s', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Derivative(Mul(Symbol('P_g', commutative=True), Symbol('f^*', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('f^*', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('f^*', commutative=True), Symbol('s', commutative=True), Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Symbol('f^*', commutative=True)), Pow(Mul(Symbol('f^*', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(Q)} = e^{Q}, then derive \\int \\mathbf{J}_M{(Q)} dQ = f^{*} + e^{Q}, then obtain \\int \\frac{d}{d Q} \\cos{(\\int \\mathbf{J}_M{(Q)} dQ - 1)} df^{*} = A_{z} + e^{Q} \\cos{(f^{*} + e^{Q} - 1)}", "derivation": "\\mathbf{J}_M{(Q)} = e^{Q} and \\int \\mathbf{J}_M{(Q)} dQ = \\int e^{Q} dQ and \\int \\mathbf{J}_M{(Q)} dQ = f^{*} + e^{Q} and \\int \\mathbf{J}_M{(Q)} dQ - 1 = f^{*} + e^{Q} - 1 and \\cos{(\\int \\mathbf{J}_M{(Q)} dQ - 1)} = \\cos{(f^{*} + e^{Q} - 1)} and \\frac{d}{d Q} \\cos{(\\int \\mathbf{J}_M{(Q)} dQ - 1)} = \\frac{\\partial}{\\partial Q} \\cos{(f^{*} + e^{Q} - 1)} and \\int \\frac{d}{d Q} \\cos{(\\int \\mathbf{J}_M{(Q)} dQ - 1)} df^{*} = \\int \\frac{\\partial}{\\partial Q} \\cos{(f^{*} + e^{Q} - 1)} df^{*} and \\int \\frac{d}{d Q} \\cos{(\\int \\mathbf{J}_M{(Q)} dQ - 1)} df^{*} = A_{z} + e^{Q} \\cos{(f^{*} + e^{Q} - 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(exp(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True))))"], [["minus", 3, 1], "Equality(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1)), Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True)), Integer(-1)))"], [["cos", 4], "Equality(cos(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))), cos(Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True)), Integer(-1))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(cos(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('f^*', commutative=True)"], "Equality(Integral(Derivative(cos(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))), Integral(Derivative(cos(Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True)), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 7], "Equality(Integral(Derivative(cos(Add(Integral(Function('\\\\mathbf{J}_M')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('A_z', commutative=True), Mul(exp(Symbol('Q', commutative=True)), cos(Add(Symbol('f^*', commutative=True), exp(Symbol('Q', commutative=True)), Integer(-1))))))"]]}, {"prompt": "Given a{(m_{s},F_{N})} = F_{N} + e^{m_{s}}, then derive \\int \\frac{a{(m_{s},F_{N})}}{F_{N}} dF_{N} = F_{N} + u + e^{m_{s}} \\log{(F_{N})}, then obtain F_{N} + u + e^{m_{s}} \\log{(F_{N})} = \\int \\frac{F_{N} + e^{m_{s}}}{F_{N}} dF_{N}", "derivation": "a{(m_{s},F_{N})} = F_{N} + e^{m_{s}} and \\frac{a{(m_{s},F_{N})}}{F_{N}} = \\frac{F_{N} + e^{m_{s}}}{F_{N}} and \\int \\frac{a{(m_{s},F_{N})}}{F_{N}} dF_{N} = \\int \\frac{F_{N} + e^{m_{s}}}{F_{N}} dF_{N} and \\int \\frac{a{(m_{s},F_{N})}}{F_{N}} dF_{N} = F_{N} + u + e^{m_{s}} \\log{(F_{N})} and F_{N} + u + e^{m_{s}} \\log{(F_{N})} = \\int \\frac{F_{N} + e^{m_{s}}}{F_{N}} dF_{N}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('m_s', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), exp(Symbol('m_s', commutative=True))))"], [["divide", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('a')(Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), exp(Symbol('m_s', commutative=True)))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('a')(Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), exp(Symbol('m_s', commutative=True)))), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('a')(Symbol('m_s', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), Symbol('u', commutative=True), Mul(exp(Symbol('m_s', commutative=True)), log(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('F_N', commutative=True), Symbol('u', commutative=True), Mul(exp(Symbol('m_s', commutative=True)), log(Symbol('F_N', commutative=True)))), Integral(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Symbol('F_N', commutative=True), exp(Symbol('m_s', commutative=True)))), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given T{(C)} = \\log{(C)} and \\dot{z}{(C)} = - \\frac{\\log{(C)}}{T{(C)}}, then obtain (\\dot{z}{(C)} + \\frac{1}{\\log{(C)}})^{C} = (-1 + \\frac{1}{\\log{(C)}})^{C}", "derivation": "T{(C)} = \\log{(C)} and 1 = \\frac{\\log{(C)}}{T{(C)}} and -1 = - \\frac{\\log{(C)}}{T{(C)}} and \\dot{z}{(C)} = - \\frac{\\log{(C)}}{T{(C)}} and \\dot{z}{(C)} + \\frac{1}{T{(C)}} = - \\frac{\\log{(C)}}{T{(C)}} + \\frac{1}{T{(C)}} and \\dot{z}{(C)} + \\frac{1}{\\log{(C)}} = -1 + \\frac{1}{\\log{(C)}} and \\dot{z}{(C)} + \\log{(C)}^{- \\frac{\\log{(C)}}{T{(C)}}} = \\log{(C)}^{- \\frac{\\log{(C)}}{T{(C)}}} - \\frac{\\log{(C)}}{T{(C)}} and (\\dot{z}{(C)} + \\log{(C)}^{- \\frac{\\log{(C)}}{T{(C)}}})^{C} = (\\log{(C)}^{- \\frac{\\log{(C)}}{T{(C)}}} - \\frac{\\log{(C)}}{T{(C)}})^{C} and (\\dot{z}{(C)} + \\frac{1}{\\log{(C)}})^{C} = (-1 + \\frac{1}{\\log{(C)}})^{C}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["divide", 1, "Function('T')(Symbol('C', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))))"], [["add", 4, "Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Add(Integer(-1), Pow(log(Symbol('C', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))))), Add(Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True)))), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True)))))"], [["power", 7, "Symbol('C', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True))))), Symbol('C', commutative=True)), Pow(Add(Pow(log(Symbol('C', commutative=True)), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True)))), Mul(Integer(-1), Pow(Function('T')(Symbol('C', commutative=True)), Integer(-1)), log(Symbol('C', commutative=True)))), Symbol('C', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Pow(Add(Function('\\\\dot{z}')(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Symbol('C', commutative=True)), Pow(Add(Integer(-1), Pow(log(Symbol('C', commutative=True)), Integer(-1))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\phi{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and f{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then obtain (\\phi{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (\\phi{(V_{\\mathbf{E}})} + f{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}}", "derivation": "\\phi{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and f{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\phi{(V_{\\mathbf{E}})} = f{(V_{\\mathbf{E}})} and \\phi{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} = f{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} and 2 \\phi{(V_{\\mathbf{E}})} = \\phi{(V_{\\mathbf{E}})} + f{(V_{\\mathbf{E}})} and 2 \\phi{(V_{\\mathbf{E}})} = \\phi{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} and \\phi{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})} = \\phi{(V_{\\mathbf{E}})} + f{(V_{\\mathbf{E}})} and (\\phi{(V_{\\mathbf{E}})} + \\sin{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}} = (\\phi{(V_{\\mathbf{E}})} + f{(V_{\\mathbf{E}})})^{V_{\\mathbf{E}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], ["renaming_premise", "Equality(Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 3, "sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Integer(2), Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["power", 7, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Add(Function('\\\\phi')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('f')(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(Z,A_{y})} = e^{A_{y} + Z}, then derive \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})} = e^{A_{y} + Z}, then obtain (e^{A_{y} + Z} + \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})})^{2} + \\frac{\\partial^{2}}{\\partial A_{y}^{2}} \\operatorname{A_{2}}{(Z,A_{y})} = 4 e^{2 A_{y} + 2 Z} + \\frac{\\partial^{2}}{\\partial A_{y}^{2}} \\operatorname{A_{2}}{(Z,A_{y})}", "derivation": "\\operatorname{A_{2}}{(Z,A_{y})} = e^{A_{y} + Z} and \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})} = \\frac{\\partial}{\\partial A_{y}} e^{A_{y} + Z} and \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})} = e^{A_{y} + Z} and e^{A_{y} + Z} + \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})} = 2 e^{A_{y} + Z} and (e^{A_{y} + Z} + \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})})^{2} = 4 e^{2 A_{y} + 2 Z} and (e^{A_{y} + Z} + \\frac{\\partial}{\\partial A_{y}} \\operatorname{A_{2}}{(Z,A_{y})})^{2} + \\frac{\\partial^{2}}{\\partial A_{y}^{2}} \\operatorname{A_{2}}{(Z,A_{y})} = 4 e^{2 A_{y} + 2 Z} + \\frac{\\partial^{2}}{\\partial A_{y}^{2}} \\operatorname{A_{2}}{(Z,A_{y})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))))"], [["add", 3, "exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Add(exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Mul(Integer(2), exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)))))"], [["power", 4, 2], "Equality(Pow(Add(exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Integer(2)), Mul(Integer(4), exp(Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))))"], [["add", 5, "Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(2)))"], "Equality(Add(Pow(Add(exp(Add(Symbol('A_y', commutative=True), Symbol('Z', commutative=True))), Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))), Integer(2)), Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(2)))), Add(Mul(Integer(4), exp(Add(Mul(Integer(2), Symbol('A_y', commutative=True)), Mul(Integer(2), Symbol('Z', commutative=True))))), Derivative(Function('A_2')(Symbol('Z', commutative=True), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\theta)} = \\log{(\\sin{(\\theta)})}, then derive \\frac{d}{d \\theta} \\int \\frac{\\operatorname{F_{g}}{(\\theta)}}{\\log{(\\sin{(\\theta)})}} d\\theta = \\frac{\\partial}{\\partial \\theta} (\\theta + s), then obtain \\frac{d}{d \\theta} \\int 1 d\\theta = 1", "derivation": "\\operatorname{F_{g}}{(\\theta)} = \\log{(\\sin{(\\theta)})} and \\operatorname{F_{g}}{(\\theta)} \\log{(\\sin{(\\theta)})} = \\log{(\\sin{(\\theta)})}^{2} and \\frac{\\operatorname{F_{g}}{(\\theta)}}{\\log{(\\sin{(\\theta)})}} = 1 and \\int \\frac{\\operatorname{F_{g}}{(\\theta)}}{\\log{(\\sin{(\\theta)})}} d\\theta = \\int 1 d\\theta and \\frac{d}{d \\theta} \\int \\frac{\\operatorname{F_{g}}{(\\theta)}}{\\log{(\\sin{(\\theta)})}} d\\theta = \\frac{d}{d \\theta} \\int 1 d\\theta and \\frac{d}{d \\theta} \\int \\frac{\\operatorname{F_{g}}{(\\theta)}}{\\log{(\\sin{(\\theta)})}} d\\theta = \\frac{\\partial}{\\partial \\theta} (\\theta + s) and \\frac{d}{d \\theta} \\int 1 d\\theta = \\frac{\\partial}{\\partial \\theta} (\\theta + s) and \\frac{d}{d \\theta} \\int 1 d\\theta = 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\theta', commutative=True)), log(sin(Symbol('\\\\theta', commutative=True))))"], [["times", 1, "log(sin(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Function('F_g')(Symbol('\\\\theta', commutative=True)), log(sin(Symbol('\\\\theta', commutative=True)))), Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(2)))"], [["divide", 2, "Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(2))"], "Equality(Mul(Function('F_g')(Symbol('\\\\theta', commutative=True)), Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Mul(Function('F_g')(Symbol('\\\\theta', commutative=True)), Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\theta', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Integral(Mul(Function('F_g')(Symbol('\\\\theta', commutative=True)), Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Integral(Mul(Function('F_g')(Symbol('\\\\theta', commutative=True)), Pow(log(sin(Symbol('\\\\theta', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\pi{(\\sigma_x,\\hbar)} = \\sin{(\\hbar \\sigma_x)} and t{(\\sigma_x,\\hbar)} = \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x, then obtain \\frac{\\partial}{\\partial \\sigma_x} \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x - 1 = \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,\\hbar)} - 1", "derivation": "\\pi{(\\sigma_x,\\hbar)} = \\sin{(\\hbar \\sigma_x)} and \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x = \\int \\sin{(\\hbar \\sigma_x)} d\\sigma_x and t{(\\sigma_x,\\hbar)} = \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x and t{(\\sigma_x,\\hbar)} = \\int \\sin{(\\hbar \\sigma_x)} d\\sigma_x and \\frac{\\partial}{\\partial \\sigma_x} \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x = \\frac{\\partial}{\\partial \\sigma_x} \\int \\sin{(\\hbar \\sigma_x)} d\\sigma_x and \\frac{\\partial}{\\partial \\sigma_x} \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x - 1 = \\frac{\\partial}{\\partial \\sigma_x} \\int \\sin{(\\hbar \\sigma_x)} d\\sigma_x - 1 and \\frac{\\partial}{\\partial \\sigma_x} \\int \\pi{(\\sigma_x,\\hbar)} d\\sigma_x - 1 = \\frac{\\partial}{\\partial \\sigma_x} t{(\\sigma_x,\\hbar)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["minus", 5, 1], "Equality(Add(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integral(sin(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Derivative(Integral(Function('\\\\pi')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Function('t')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(v_{z})} = \\cos{(v_{z})}, then obtain (\\frac{d}{d v_{z}} \\int \\operatorname{P_{g}}{(v_{z})} dv_{z}) \\int \\cos{(v_{z})} dv_{z} = (\\frac{d}{d v_{z}} \\int \\cos{(v_{z})} dv_{z}) \\int \\cos{(v_{z})} dv_{z}", "derivation": "\\operatorname{P_{g}}{(v_{z})} = \\cos{(v_{z})} and \\int \\operatorname{P_{g}}{(v_{z})} dv_{z} = \\int \\cos{(v_{z})} dv_{z} and \\frac{d}{d v_{z}} \\int \\operatorname{P_{g}}{(v_{z})} dv_{z} = \\frac{d}{d v_{z}} \\int \\cos{(v_{z})} dv_{z} and (\\frac{d}{d v_{z}} \\int \\operatorname{P_{g}}{(v_{z})} dv_{z}) \\int \\cos{(v_{z})} dv_{z} = (\\frac{d}{d v_{z}} \\int \\cos{(v_{z})} dv_{z}) \\int \\cos{(v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Integral(Function('P_g')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["times", 3, "Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('P_g')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Derivative(Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integral(cos(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(\\tilde{g},m)} = \\frac{\\partial}{\\partial \\tilde{g}} (- \\tilde{g} + m), then derive \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = (-1)^{\\tilde{g}}, then obtain \\frac{\\partial}{\\partial m} \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = \\frac{d}{d m} (-1)^{\\tilde{g}}", "derivation": "\\lambda{(\\tilde{g},m)} = \\frac{\\partial}{\\partial \\tilde{g}} (- \\tilde{g} + m) and \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = (\\frac{\\partial}{\\partial \\tilde{g}} (- \\tilde{g} + m))^{\\tilde{g}} and \\frac{\\partial}{\\partial m} \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = \\frac{\\partial}{\\partial m} (\\frac{\\partial}{\\partial \\tilde{g}} (- \\tilde{g} + m))^{\\tilde{g}} and \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = (-1)^{\\tilde{g}} and (\\frac{\\partial}{\\partial \\tilde{g}} (- \\tilde{g} + m))^{\\tilde{g}} = (-1)^{\\tilde{g}} and \\frac{\\partial}{\\partial m} \\lambda^{\\tilde{g}}{(\\tilde{g},m)} = \\frac{d}{d m} (-1)^{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 2, "Symbol('m', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Pow(Function('\\\\lambda')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(Pow(Function('\\\\lambda')(Symbol('\\\\tilde{g}', commutative=True), Symbol('m', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Pow(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(C_{2},\\hbar)} = \\log{(C_{2} \\hbar)}, then derive C_{2} + \\frac{\\partial}{\\partial \\hbar} A{(C_{2},\\hbar)} = C_{2} + \\frac{1}{\\hbar}, then obtain \\log{(C_{2} + \\frac{\\partial}{\\partial \\hbar} \\log{(C_{2} \\hbar)})} = \\log{(C_{2} + \\frac{1}{\\hbar})}", "derivation": "A{(C_{2},\\hbar)} = \\log{(C_{2} \\hbar)} and C_{2} \\hbar + A{(C_{2},\\hbar)} = C_{2} \\hbar + \\log{(C_{2} \\hbar)} and \\frac{\\partial}{\\partial \\hbar} (C_{2} \\hbar + A{(C_{2},\\hbar)}) = \\frac{\\partial}{\\partial \\hbar} (C_{2} \\hbar + \\log{(C_{2} \\hbar)}) and C_{2} + \\frac{\\partial}{\\partial \\hbar} A{(C_{2},\\hbar)} = C_{2} + \\frac{1}{\\hbar} and C_{2} + \\frac{\\partial}{\\partial \\hbar} \\log{(C_{2} \\hbar)} = C_{2} + \\frac{1}{\\hbar} and \\log{(C_{2} + \\frac{\\partial}{\\partial \\hbar} A{(C_{2},\\hbar)})} = \\log{(C_{2} + \\frac{1}{\\hbar})} and C_{2} + \\frac{\\partial}{\\partial \\hbar} \\log{(C_{2} \\hbar)} = C_{2} + \\frac{\\partial}{\\partial \\hbar} A{(C_{2},\\hbar)} and \\log{(C_{2} + \\frac{\\partial}{\\partial \\hbar} \\log{(C_{2} \\hbar)})} = \\log{(C_{2} + \\frac{1}{\\hbar})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["add", 1, "Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('C_2', commutative=True), Derivative(Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Symbol('C_2', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('C_2', commutative=True), Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Symbol('C_2', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["log", 4], "Equality(log(Add(Symbol('C_2', commutative=True), Derivative(Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), log(Add(Symbol('C_2', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('C_2', commutative=True), Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Add(Symbol('C_2', commutative=True), Derivative(Function('A')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(log(Add(Symbol('C_2', commutative=True), Derivative(log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), log(Add(Symbol('C_2', commutative=True), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\hat{x}_0{(n_{2},F_{g},B)} = B F_{g} n_{2}, then obtain (F_{g} - \\hat{x}_0{(n_{2},F_{g},B)})^{n_{2}} = (- B F_{g} n_{2} + F_{g})^{n_{2}}", "derivation": "\\hat{x}_0{(n_{2},F_{g},B)} = B F_{g} n_{2} and - F_{g} + \\hat{x}_0{(n_{2},F_{g},B)} = B F_{g} n_{2} - F_{g} and F_{g} - \\hat{x}_0{(n_{2},F_{g},B)} = - B F_{g} n_{2} + F_{g} and (F_{g} - \\hat{x}_0{(n_{2},F_{g},B)})^{n_{2}} = (- B F_{g} n_{2} + F_{g})^{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('n_2', commutative=True), Symbol('F_g', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Symbol('F_g', commutative=True), Symbol('n_2', commutative=True)))"], [["minus", 1, "Symbol('F_g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True)), Function('\\\\hat{x}_0')(Symbol('n_2', commutative=True), Symbol('F_g', commutative=True), Symbol('B', commutative=True))), Add(Mul(Symbol('B', commutative=True), Symbol('F_g', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), Symbol('F_g', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('n_2', commutative=True), Symbol('F_g', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('F_g', commutative=True), Symbol('n_2', commutative=True)), Symbol('F_g', commutative=True)))"], [["power", 3, "Symbol('n_2', commutative=True)"], "Equality(Pow(Add(Symbol('F_g', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('n_2', commutative=True), Symbol('F_g', commutative=True), Symbol('B', commutative=True)))), Symbol('n_2', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('F_g', commutative=True), Symbol('n_2', commutative=True)), Symbol('F_g', commutative=True)), Symbol('n_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(c_{0})} = c_{0}, then obtain ((\\operatorname{n_{2}}^{c_{0}}{(c_{0})})^{c_{0}})^{c_{0}} = ((c_{0}^{c_{0}})^{c_{0}})^{c_{0}}", "derivation": "\\operatorname{n_{2}}{(c_{0})} = c_{0} and \\operatorname{n_{2}}^{c_{0}}{(c_{0})} = c_{0}^{c_{0}} and (\\operatorname{n_{2}}^{c_{0}}{(c_{0})})^{c_{0}} = (c_{0}^{c_{0}})^{c_{0}} and ((\\operatorname{n_{2}}^{c_{0}}{(c_{0})})^{c_{0}})^{c_{0}} = ((c_{0}^{c_{0}})^{c_{0}})^{c_{0}}", "srepr_derivation": [["renaming_premise", "Equality(Function('n_2')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)))"], [["power", 2, "Symbol('c_0', commutative=True)"], "Equality(Pow(Pow(Function('n_2')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Pow(Pow(Function('n_2')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(Pow(Pow(Symbol('c_0', commutative=True), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(r,E_{x})} = r + \\sin{(E_{x})}, then obtain 2 r + 2 \\sin{(E_{x})} + (\\frac{\\partial}{\\partial E_{x}} (r + \\mathbf{p}{(r,E_{x})} + \\sin{(E_{x})}))^{r} = 2 r + 2 \\sin{(E_{x})} + (\\frac{\\partial}{\\partial E_{x}} (2 r + 2 \\sin{(E_{x})}))^{r}", "derivation": "\\mathbf{p}{(r,E_{x})} = r + \\sin{(E_{x})} and r + \\mathbf{p}{(r,E_{x})} + \\sin{(E_{x})} = 2 r + 2 \\sin{(E_{x})} and \\frac{\\partial}{\\partial E_{x}} (r + \\mathbf{p}{(r,E_{x})} + \\sin{(E_{x})}) = \\frac{\\partial}{\\partial E_{x}} (2 r + 2 \\sin{(E_{x})}) and (\\frac{\\partial}{\\partial E_{x}} (r + \\mathbf{p}{(r,E_{x})} + \\sin{(E_{x})}))^{r} = (\\frac{\\partial}{\\partial E_{x}} (2 r + 2 \\sin{(E_{x})}))^{r} and 2 r + 2 \\sin{(E_{x})} + (\\frac{\\partial}{\\partial E_{x}} (r + \\mathbf{p}{(r,E_{x})} + \\sin{(E_{x})}))^{r} = 2 r + 2 \\sin{(E_{x})} + (\\frac{\\partial}{\\partial E_{x}} (2 r + 2 \\sin{(E_{x})}))^{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('r', commutative=True), Symbol('E_x', commutative=True)), Add(Symbol('r', commutative=True), sin(Symbol('E_x', commutative=True))))"], [["add", 1, "Add(Symbol('r', commutative=True), sin(Symbol('E_x', commutative=True)))"], "Equality(Add(Symbol('r', commutative=True), Function('\\\\mathbf{p}')(Symbol('r', commutative=True), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True)))))"], [["differentiate", 2, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Add(Symbol('r', commutative=True), Function('\\\\mathbf{p}')(Symbol('r', commutative=True), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('r', commutative=True), Function('\\\\mathbf{p}')(Symbol('r', commutative=True), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('r', commutative=True)), Pow(Derivative(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('r', commutative=True)))"], [["add", 4, "Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True))))"], "Equality(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True))), Pow(Derivative(Add(Symbol('r', commutative=True), Function('\\\\mathbf{p}')(Symbol('r', commutative=True), Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('r', commutative=True))), Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True))), Pow(Derivative(Add(Mul(Integer(2), Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('E_x', commutative=True)))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} = \\Psi_{\\lambda}^{V_{\\mathbf{E}}}, then obtain \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\int \\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} d\\Psi_{\\lambda} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\int \\Psi_{\\lambda}^{V_{\\mathbf{E}}} d\\Psi_{\\lambda}", "derivation": "\\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} = \\Psi_{\\lambda}^{V_{\\mathbf{E}}} and \\int \\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} d\\Psi_{\\lambda} = \\int \\Psi_{\\lambda}^{V_{\\mathbf{E}}} d\\Psi_{\\lambda} and \\Psi_{\\lambda} \\int \\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} d\\Psi_{\\lambda} = \\Psi_{\\lambda} \\int \\Psi_{\\lambda}^{V_{\\mathbf{E}}} d\\Psi_{\\lambda} and \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\int \\operatorname{z^{*}}{(\\Psi_{\\lambda},V_{\\mathbf{E}})} d\\Psi_{\\lambda} = \\frac{\\partial}{\\partial \\Psi_{\\lambda}} \\Psi_{\\lambda} \\int \\Psi_{\\lambda}^{V_{\\mathbf{E}}} d\\Psi_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integral(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 2, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('z^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Function('z^*')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integral(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(u,v_{z})} = u + v_{z}, then obtain \\frac{- 2 u - 2 v_{z} + 6 \\operatorname{F_{x}}{(u,v_{z})}}{u + v_{z}} = \\frac{4 \\operatorname{F_{x}}{(u,v_{z})}}{u + v_{z}}", "derivation": "\\operatorname{F_{x}}{(u,v_{z})} = u + v_{z} and 2 \\operatorname{F_{x}}{(u,v_{z})} = u + v_{z} + \\operatorname{F_{x}}{(u,v_{z})} and - u - v_{z} + 2 \\operatorname{F_{x}}{(u,v_{z})} = \\operatorname{F_{x}}{(u,v_{z})} and - 2 u - 2 v_{z} + 4 \\operatorname{F_{x}}{(u,v_{z})} = 2 \\operatorname{F_{x}}{(u,v_{z})} and - 2 u - 2 v_{z} + 6 \\operatorname{F_{x}}{(u,v_{z})} = 4 \\operatorname{F_{x}}{(u,v_{z})} and \\frac{- 2 u - 2 v_{z} + 6 \\operatorname{F_{x}}{(u,v_{z})}}{u + v_{z}} = \\frac{4 \\operatorname{F_{x}}{(u,v_{z})}}{u + v_{z}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))"], [["add", 1, "Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Mul(Integer(2), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))), Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))))"], [["minus", 2, "Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), Symbol('v_z', commutative=True)), Mul(Integer(2), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True)), Mul(Integer(4), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))), Mul(Integer(2), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 4, "Mul(Integer(2), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True)), Mul(Integer(6), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True)))), Mul(Integer(4), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))))"], [["divide", 5, "Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('v_z', commutative=True)), Mul(Integer(6), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))))), Mul(Integer(4), Pow(Add(Symbol('u', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Function('F_x')(Symbol('u', commutative=True), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(q,\\nabla)} = e^{\\frac{\\nabla}{q}}, then derive \\int \\Psi_{\\lambda}{(q,\\nabla)} d\\nabla = \\psi^* + q e^{\\frac{\\nabla}{q}}, then obtain (\\int e^{\\frac{\\nabla}{q}} d\\nabla)^{2} = (\\psi^* + q \\Psi_{\\lambda}{(q,\\nabla)}) \\int e^{\\frac{\\nabla}{q}} d\\nabla", "derivation": "\\Psi_{\\lambda}{(q,\\nabla)} = e^{\\frac{\\nabla}{q}} and \\int \\Psi_{\\lambda}{(q,\\nabla)} d\\nabla = \\int e^{\\frac{\\nabla}{q}} d\\nabla and \\int \\Psi_{\\lambda}{(q,\\nabla)} d\\nabla = \\psi^* + q e^{\\frac{\\nabla}{q}} and (\\int \\Psi_{\\lambda}{(q,\\nabla)} d\\nabla)^{2} = (\\psi^* + q e^{\\frac{\\nabla}{q}}) \\int \\Psi_{\\lambda}{(q,\\nabla)} d\\nabla and (\\int e^{\\frac{\\nabla}{q}} d\\nabla)^{2} = (\\psi^* + q e^{\\frac{\\nabla}{q}}) \\int e^{\\frac{\\nabla}{q}} d\\nabla and (\\int e^{\\frac{\\nabla}{q}} d\\nabla)^{2} = (\\psi^* + q \\Psi_{\\lambda}{(q,\\nabla)}) \\int e^{\\frac{\\nabla}{q}} d\\nabla", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('q', commutative=True), exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))))))"], [["times", 3, "Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('q', commutative=True), exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))))), Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('q', commutative=True), exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))))), Integral(exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integral(exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('q', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('q', commutative=True), Symbol('\\\\nabla', commutative=True)))), Integral(exp(Mul(Symbol('\\\\nabla', commutative=True), Pow(Symbol('q', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\nabla', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\delta,F_{H})} = F_{H} + \\delta, then derive \\int \\frac{\\mathbf{B}{(\\delta,F_{H})}}{F_{H} + \\delta} d\\delta = \\delta + h, then obtain \\int 1 d\\delta = \\delta + h", "derivation": "\\mathbf{B}{(\\delta,F_{H})} = F_{H} + \\delta and \\frac{\\mathbf{B}{(\\delta,F_{H})}}{F_{H} + \\delta} = 1 and \\int \\frac{\\mathbf{B}{(\\delta,F_{H})}}{F_{H} + \\delta} d\\delta = \\int 1 d\\delta and \\int \\frac{\\mathbf{B}{(\\delta,F_{H})}}{F_{H} + \\delta} d\\delta = \\delta + h and \\int 1 d\\delta = \\delta + h", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["divide", 1, "Add(Symbol('F_H', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Add(Symbol('F_H', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Symbol('h', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Symbol('h', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{B},\\theta)} = \\mathbf{B} - \\theta, then obtain \\mathbf{J}_f{(\\mathbf{B},\\theta)} + 1 = \\frac{\\mathbf{B}}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} - \\frac{\\theta}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} + \\mathbf{J}_f{(\\mathbf{B},\\theta)}", "derivation": "\\mathbf{J}_f{(\\mathbf{B},\\theta)} = \\mathbf{B} - \\theta and 1 = \\frac{\\mathbf{B} - \\theta}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} and 1 = \\frac{\\mathbf{B}}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} - \\frac{\\theta}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} and 1 = \\frac{\\mathbf{B}}{\\mathbf{B} - \\theta} - \\frac{\\theta}{\\mathbf{B} - \\theta} and \\mathbf{B} - \\theta + 1 = \\mathbf{B} + \\frac{\\mathbf{B}}{\\mathbf{B} - \\theta} - \\theta - \\frac{\\theta}{\\mathbf{B} - \\theta} and \\mathbf{J}_f{(\\mathbf{B},\\theta)} + 1 = \\frac{\\mathbf{B}}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} - \\frac{\\theta}{\\mathbf{J}_f{(\\mathbf{B},\\theta)}} + \\mathbf{J}_f{(\\mathbf{B},\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))))"], [["expand", 2], "Equality(Integer(1), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)))))"], [["add", 4, "Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1))), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given J{(G,\\hat{H}_{\\lambda})} = G \\hat{H}_{\\lambda}, then obtain \\int (- \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} + \\frac{1}{G^{2}}) dG = \\int (- 2 \\hat{H}_{\\lambda} + \\frac{1}{G^{2}}) dG", "derivation": "J{(G,\\hat{H}_{\\lambda})} = G \\hat{H}_{\\lambda} and \\frac{J{(G,\\hat{H}_{\\lambda})}}{G} = \\hat{H}_{\\lambda} and \\hat{H}_{\\lambda} + \\frac{J{(G,\\hat{H}_{\\lambda})}}{G} = 2 \\hat{H}_{\\lambda} and \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} = \\hat{H}_{\\lambda} + \\frac{J{(G,\\hat{H}_{\\lambda})}}{G} and \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} = 2 \\hat{H}_{\\lambda} and - \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} = - 2 \\hat{H}_{\\lambda} and - \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} + \\frac{1}{G^{2}} = - 2 \\hat{H}_{\\lambda} + \\frac{1}{G^{2}} and \\int (- \\frac{2 J{(G,\\hat{H}_{\\lambda})}}{G} + \\frac{1}{G^{2}}) dG = \\int (- 2 \\hat{H}_{\\lambda} + \\frac{1}{G^{2}}) dG", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["divide", 1, "Symbol('G', commutative=True)"], "Equality(Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], [["add", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["add", 6, "Pow(Symbol('G', commutative=True), Integer(-2))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(Symbol('G', commutative=True), Integer(-2))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('G', commutative=True), Integer(-2))))"], [["integrate", 7, "Symbol('G', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('G', commutative=True), Integer(-1)), Function('J')(Symbol('G', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Pow(Symbol('G', commutative=True), Integer(-2))), Tuple(Symbol('G', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Symbol('G', commutative=True), Integer(-2))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})}, then derive \\sin{(\\varepsilon_{0}{(A_{2})})} = \\sin{(\\frac{1}{A_{2}})}, then obtain \\sin{(\\frac{d}{d A_{2}} \\log{(A_{2})})} = \\sin{(\\frac{1}{A_{2}})}", "derivation": "\\varepsilon_{0}{(A_{2})} = \\frac{d}{d A_{2}} \\log{(A_{2})} and \\sin{(\\varepsilon_{0}{(A_{2})})} = \\sin{(\\frac{d}{d A_{2}} \\log{(A_{2})})} and \\sin{(\\varepsilon_{0}{(A_{2})})} = \\sin{(\\frac{1}{A_{2}})} and \\sin{(\\frac{d}{d A_{2}} \\log{(A_{2})})} = \\sin{(\\frac{1}{A_{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('A_2', commutative=True)), Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["sin", 1], "Equality(sin(Function('\\\\varepsilon_0')(Symbol('A_2', commutative=True))), sin(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(sin(Function('\\\\varepsilon_0')(Symbol('A_2', commutative=True))), sin(Pow(Symbol('A_2', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(sin(Derivative(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), sin(Pow(Symbol('A_2', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})}, then derive (\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})})^{2} = \\sin^{2}{(A_{z})}, then obtain \\frac{\\sin^{2}{(A_{z})}}{\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})}} = \\frac{(\\frac{d^{2}}{d A_{z}^{2}} \\sin{(A_{z})})^{2}}{\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})}}", "derivation": "\\operatorname{F_{g}}{(A_{z})} = \\frac{d}{d A_{z}} \\sin{(A_{z})} and \\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})} = \\frac{d^{2}}{d A_{z}^{2}} \\sin{(A_{z})} and (\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})})^{2} = (\\frac{d^{2}}{d A_{z}^{2}} \\sin{(A_{z})})^{2} and (\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})})^{2} = \\sin^{2}{(A_{z})} and \\sin^{2}{(A_{z})} = (\\frac{d^{2}}{d A_{z}^{2}} \\sin{(A_{z})})^{2} and \\frac{\\sin^{2}{(A_{z})}}{\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})}} = \\frac{(\\frac{d^{2}}{d A_{z}^{2}} \\sin{(A_{z})})^{2}}{\\frac{d}{d A_{z}} \\operatorname{F_{g}}{(A_{z})}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('A_z', commutative=True)), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Pow(Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(2)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(2)), Pow(sin(Symbol('A_z', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(sin(Symbol('A_z', commutative=True)), Integer(2)), Pow(Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(2)))"], [["divide", 5, "Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Mul(Pow(sin(Symbol('A_z', commutative=True)), Integer(2)), Pow(Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Derivative(Function('F_g')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(2))), Integer(2))))"]]}, {"prompt": "Given b{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and \\hat{x}_0{(\\lambda)} = \\log{(\\lambda)}, then derive \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = M + \\mathbf{J}_f, then obtain - \\hat{x}_0{(\\lambda)} + \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = M + \\mathbf{J}_f - \\hat{x}_0{(\\lambda)}", "derivation": "b{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} = 1 and \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = \\int 1 d\\mathbf{J}_f and \\hat{x}_0{(\\lambda)} = \\log{(\\lambda)} and \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = M + \\mathbf{J}_f and - \\log{(\\lambda)} + \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = M + \\mathbf{J}_f - \\log{(\\lambda)} and - \\hat{x}_0{(\\lambda)} + \\int \\frac{b{(\\mathbf{J}_f)}}{\\log{(\\mathbf{J}_f)}} d\\mathbf{J}_f = M + \\mathbf{J}_f - \\hat{x}_0{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Mul(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["minus", 5, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), log(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Function('b')(Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Symbol('M', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(B)} = \\sin{(\\log{(B)})}, then obtain \\operatorname{A_{2}}{(B)} - \\frac{d}{d B} \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})} = \\sin{(\\log{(B)})} - \\frac{d}{d B} \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})}", "derivation": "\\operatorname{A_{2}}{(B)} = \\sin{(\\log{(B)})} and \\frac{\\operatorname{A_{2}}{(B)}}{B} = \\frac{\\sin{(\\log{(B)})}}{B} and \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})} = \\sin{(\\frac{\\sin{(\\log{(B)})}}{B})} and \\frac{d}{d B} \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})} = \\frac{d}{d B} \\sin{(\\frac{\\sin{(\\log{(B)})}}{B})} and \\operatorname{A_{2}}{(B)} - \\frac{d}{d B} \\sin{(\\frac{\\sin{(\\log{(B)})}}{B})} = \\sin{(\\log{(B)})} - \\frac{d}{d B} \\sin{(\\frac{\\sin{(\\log{(B)})}}{B})} and \\operatorname{A_{2}}{(B)} - \\frac{d}{d B} \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})} = \\sin{(\\log{(B)})} - \\frac{d}{d B} \\sin{(\\frac{\\operatorname{A_{2}}{(B)}}{B})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('B', commutative=True)), sin(log(Symbol('B', commutative=True))))"], [["divide", 1, "Symbol('B', commutative=True)"], "Equality(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('A_2')(Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True)))))"], [["sin", 2], "Equality(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('A_2')(Symbol('B', commutative=True)))), sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True))))))"], [["differentiate", 3, "Symbol('B', commutative=True)"], "Equality(Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('A_2')(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1)))"], "Equality(Add(Function('A_2')(Symbol('B', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1))))), Add(sin(log(Symbol('B', commutative=True))), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), sin(log(Symbol('B', commutative=True))))), Tuple(Symbol('B', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Function('A_2')(Symbol('B', commutative=True)), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('A_2')(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))), Add(sin(log(Symbol('B', commutative=True))), Mul(Integer(-1), Derivative(sin(Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Function('A_2')(Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(V,F_{N})} = e^{\\frac{V}{F_{N}}}, then obtain 2 \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,F_{N})} = \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,F_{N})} + \\frac{e^{\\frac{V}{F_{N}}}}{F_{N}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(V,F_{N})} = e^{\\frac{V}{F_{N}}} and F_{N} + \\operatorname{L_{\\varepsilon}}{(V,F_{N})} = F_{N} + e^{\\frac{V}{F_{N}}} and 2 F_{N} + 2 \\operatorname{L_{\\varepsilon}}{(V,F_{N})} = 2 F_{N} + \\operatorname{L_{\\varepsilon}}{(V,F_{N})} + e^{\\frac{V}{F_{N}}} and \\frac{\\partial}{\\partial V} (2 F_{N} + 2 \\operatorname{L_{\\varepsilon}}{(V,F_{N})}) = \\frac{\\partial}{\\partial V} (2 F_{N} + \\operatorname{L_{\\varepsilon}}{(V,F_{N})} + e^{\\frac{V}{F_{N}}}) and 2 \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,F_{N})} = \\frac{\\partial}{\\partial V} \\operatorname{L_{\\varepsilon}}{(V,F_{N})} + \\frac{e^{\\frac{V}{F_{N}}}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)), exp(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('V', commutative=True))))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), exp(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('V', commutative=True)))))"], [["add", 2, "Add(Symbol('F_N', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)))), Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)), exp(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('V', commutative=True)))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Mul(Integer(2), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('F_N', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)), exp(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('V', commutative=True)))), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Derivative(Function('L_{\\\\varepsilon}')(Symbol('V', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('V', commutative=True))))))"]]}, {"prompt": "Given y{(v_{1},l)} = - l + v_{1}, then obtain \\log{(- l (- l + v_{1}) y{(v_{1},l)})} = \\log{(- l y^{2}{(v_{1},l)})}", "derivation": "y{(v_{1},l)} = - l + v_{1} and l y{(v_{1},l)} = l (- l + v_{1}) and - y{(v_{1},l)} = l - v_{1} and - l (- l + v_{1}) y{(v_{1},l)} = l (- l + v_{1}) (l - v_{1}) and - l (- l + v_{1})^{2} = l (- l + v_{1}) (l - v_{1}) and - l (- l + v_{1})^{2} = - l (- l + v_{1}) y{(v_{1},l)} and \\log{(- l (- l + v_{1})^{2})} = \\log{(- l (- l + v_{1}) y{(v_{1},l)})} and \\log{(- l (- l + v_{1}) y{(v_{1},l)})} = \\log{(- l y^{2}{(v_{1},l)})}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('l', commutative=True))"], "Equality(Mul(Integer(-1), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["times", 3, "Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('l', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Integer(2))), Mul(Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('v_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Symbol('l', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True))))"], [["log", 6], "Equality(log(Mul(Integer(-1), Symbol('l', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Integer(2)))), log(Mul(Integer(-1), Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(log(Mul(Integer(-1), Symbol('l', commutative=True), Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('v_1', commutative=True)), Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True)))), log(Mul(Integer(-1), Symbol('l', commutative=True), Pow(Function('y')(Symbol('v_1', commutative=True), Symbol('l', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\dot{x}{(\\mu_0,\\dot{y})} = - \\dot{y} + \\mu_0, then derive \\int \\dot{x}{(\\mu_0,\\dot{y})} d\\mu_0 = - \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2}, then obtain (\\frac{\\partial}{\\partial \\mathbf{P}} \\int (- \\dot{y} + \\mu_0) d\\mu_0)^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\mathbf{P}} (- \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2}))^{\\mathbf{P}}", "derivation": "\\dot{x}{(\\mu_0,\\dot{y})} = - \\dot{y} + \\mu_0 and \\int \\dot{x}{(\\mu_0,\\dot{y})} d\\mu_0 = \\int (- \\dot{y} + \\mu_0) d\\mu_0 and \\int \\dot{x}{(\\mu_0,\\dot{y})} d\\mu_0 = - \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2} and \\int (- \\dot{y} + \\mu_0) d\\mu_0 = - \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2} and \\frac{\\partial}{\\partial \\mathbf{P}} \\int (- \\dot{y} + \\mu_0) d\\mu_0 = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2}) and (\\frac{\\partial}{\\partial \\mathbf{P}} \\int (- \\dot{y} + \\mu_0) d\\mu_0)^{\\mathbf{P}} = (\\frac{\\partial}{\\partial \\mathbf{P}} (- \\dot{y} \\mu_0 + \\mathbf{P} + \\frac{\\mu_0^{2}}{2}))^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu_0', commutative=True), Integer(2)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\dot{z})} = \\cos{(\\dot{z})}, then derive \\frac{d}{d \\dot{z}} \\eta^{\\prime}{(\\dot{z})} = - \\sin{(\\dot{z})}, then obtain \\psi + \\eta^{\\prime}{(\\dot{z})} = \\int - \\sin{(\\dot{z})} d\\dot{z}", "derivation": "\\eta^{\\prime}{(\\dot{z})} = \\cos{(\\dot{z})} and \\frac{d}{d \\dot{z}} \\eta^{\\prime}{(\\dot{z})} = \\frac{d}{d \\dot{z}} \\cos{(\\dot{z})} and \\frac{d}{d \\dot{z}} \\eta^{\\prime}{(\\dot{z})} = - \\sin{(\\dot{z})} and \\int \\frac{d}{d \\dot{z}} \\eta^{\\prime}{(\\dot{z})} d\\dot{z} = \\int - \\sin{(\\dot{z})} d\\dot{z} and \\psi + \\eta^{\\prime}{(\\dot{z})} = \\int - \\sin{(\\dot{z})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\psi', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\dot{z}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = a^{\\dagger} - t_{1}, then derive \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} \\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = 0, then obtain \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} (a^{\\dagger} - t_{1}) = 0", "derivation": "\\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = a^{\\dagger} - t_{1} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = \\frac{\\partial}{\\partial a^{\\dagger}} (a^{\\dagger} - t_{1}) and \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} \\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} (a^{\\dagger} - t_{1}) and \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} \\operatorname{x^{{\\}'}}{(a^{\\dagger},t_{1})} = 0 and \\frac{\\partial^{2}}{\\partial t_{1}\\partial a^{\\dagger}} (a^{\\dagger} - t_{1}) = 0", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('x^\\\\prime')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})} = \\frac{A_{1} \\mathbf{H}}{\\hat{p}}, then obtain \\int (\\frac{\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})}}{\\hat{p}})^{\\hat{p}} d\\mathbf{H} = \\int (\\frac{A_{1} \\mathbf{H}}{\\hat{p}^{2}})^{\\hat{p}} d\\mathbf{H}", "derivation": "\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})} = \\frac{A_{1} \\mathbf{H}}{\\hat{p}} and \\frac{\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})}}{\\hat{p}} = \\frac{A_{1} \\mathbf{H}}{\\hat{p}^{2}} and (\\frac{\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})}}{\\hat{p}})^{\\hat{p}} = (\\frac{A_{1} \\mathbf{H}}{\\hat{p}^{2}})^{\\hat{p}} and \\int (\\frac{\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H},A_{1})}}{\\hat{p}})^{\\hat{p}} d\\mathbf{H} = \\int (\\frac{A_{1} \\mathbf{H}}{\\hat{p}^{2}})^{\\hat{p}} d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Pow(Mul(Symbol('A_1', commutative=True), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\theta_2,W)} = W + \\theta_2, then obtain \\theta_2 (W + \\theta_2 + \\varphi^{*}{(\\theta_2,W)}) = 2 \\theta_2 \\varphi^{*}{(\\theta_2,W)}", "derivation": "\\varphi^{*}{(\\theta_2,W)} = W + \\theta_2 and W + \\theta_2 + \\varphi^{*}{(\\theta_2,W)} = 2 W + 2 \\theta_2 and \\theta_2 (W + \\theta_2 + \\varphi^{*}{(\\theta_2,W)}) = \\theta_2 (2 W + 2 \\theta_2) and 2 \\theta_2 \\varphi^{*}{(\\theta_2,W)} = \\theta_2 (2 W + 2 \\theta_2) and \\theta_2 (W + \\theta_2 + \\varphi^{*}{(\\theta_2,W)}) = 2 \\theta_2 \\varphi^{*}{(\\theta_2,W)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "Add(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))))"], [["times", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True)))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(2), Symbol('\\\\theta_2', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Add(Symbol('W', commutative=True), Symbol('\\\\theta_2', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True)))), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\theta_2', commutative=True), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\Psi^{\\dagger},\\phi)} = \\frac{\\phi}{\\Psi^{\\dagger}} and \\operatorname{m_{s}}{(\\Psi^{\\dagger},\\phi)} = \\mathbf{A}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\phi)}, then obtain \\phi + (\\frac{\\phi}{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} = \\phi + \\mathbf{A}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\phi)}", "derivation": "\\mathbf{A}{(\\Psi^{\\dagger},\\phi)} = \\frac{\\phi}{\\Psi^{\\dagger}} and \\operatorname{m_{s}}{(\\Psi^{\\dagger},\\phi)} = \\mathbf{A}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\phi)} and \\operatorname{m_{s}}{(\\Psi^{\\dagger},\\phi)} = (\\frac{\\phi}{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} and \\phi + \\operatorname{m_{s}}{(\\Psi^{\\dagger},\\phi)} = \\phi + \\mathbf{A}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\phi)} and \\phi + (\\frac{\\phi}{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} = \\phi + \\mathbf{A}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},\\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('m_s')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["add", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Symbol('\\\\phi', commutative=True), Function('m_s')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Symbol('\\\\phi', commutative=True), Pow(Mul(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Symbol('\\\\phi', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given i{(\\mathbf{f})} = \\sin{(e^{\\mathbf{f}})}, then obtain ((\\frac{i{(\\mathbf{f})}}{\\sin{(e^{\\mathbf{f}})}})^{\\mathbf{f}})^{\\mathbf{f}} = 1", "derivation": "i{(\\mathbf{f})} = \\sin{(e^{\\mathbf{f}})} and \\frac{i{(\\mathbf{f})}}{\\sin{(e^{\\mathbf{f}})}} = 1 and (\\frac{i{(\\mathbf{f})}}{\\sin{(e^{\\mathbf{f}})}})^{\\mathbf{f}} = 1 and ((\\frac{i{(\\mathbf{f})}}{\\sin{(e^{\\mathbf{f}})}})^{\\mathbf{f}})^{\\mathbf{f}} = 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{f}', commutative=True)), sin(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 1, "sin(exp(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Function('i')(Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Mul(Function('i')(Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{f}', commutative=True)), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Pow(Mul(Function('i')(Symbol('\\\\mathbf{f}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1))), Symbol('\\\\mathbf{f}', commutative=True)), Symbol('\\\\mathbf{f}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(E,t_{1})} = \\int E t_{1} dE, then obtain \\operatorname{f_{E}}^{3}{(E,t_{1})} = \\operatorname{f_{E}}^{2}{(E,t_{1})} \\int E t_{1} dE", "derivation": "\\operatorname{f_{E}}{(E,t_{1})} = \\int E t_{1} dE and \\operatorname{f_{E}}^{2}{(E,t_{1})} = \\operatorname{f_{E}}{(E,t_{1})} \\int E t_{1} dE and \\operatorname{f_{E}}^{2}{(E,t_{1})} \\int E t_{1} dE = \\operatorname{f_{E}}{(E,t_{1})} (\\int E t_{1} dE)^{2} and \\operatorname{f_{E}}^{3}{(E,t_{1})} = \\operatorname{f_{E}}^{2}{(E,t_{1})} \\int E t_{1} dE", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["times", 1, "Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integer(2)), Mul(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["times", 2, "Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True)))"], "Equality(Mul(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integer(2)), Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True)))), Mul(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Pow(Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integer(3)), Mul(Pow(Function('f_E')(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Integer(2)), Integral(Mul(Symbol('E', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(L)} = \\log{(\\cos{(L)})} and \\operatorname{a^{\\dagger}}{(L)} = \\int \\Psi_{nl}{(L)} dL, then obtain \\frac{\\int \\log{(\\cos{(L)})} dL}{L} = \\frac{\\int \\Psi_{nl}{(L)} dL}{L}", "derivation": "\\Psi_{nl}{(L)} = \\log{(\\cos{(L)})} and \\int \\Psi_{nl}{(L)} dL = \\int \\log{(\\cos{(L)})} dL and \\operatorname{a^{\\dagger}}{(L)} = \\int \\Psi_{nl}{(L)} dL and \\frac{\\operatorname{a^{\\dagger}}{(L)}}{L} = \\frac{\\int \\Psi_{nl}{(L)} dL}{L} and \\operatorname{a^{\\dagger}}{(L)} = \\int \\log{(\\cos{(L)})} dL and \\frac{\\int \\log{(\\cos{(L)})} dL}{L} = \\frac{\\int \\Psi_{nl}{(L)} dL}{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('L', commutative=True)), log(cos(Symbol('L', commutative=True))))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(log(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('L', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["divide", 3, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('L', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('L', commutative=True)), Integral(log(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(log(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given J{(r)} = \\cos{(r)}, then obtain \\frac{d}{d r} (J{(r)} + \\cos{(r)} + \\int (J{(r)} + \\cos{(r)})^{2} dr) = \\frac{\\partial}{\\partial r} (\\mathbf{E} + 2 r + J{(r)} + 2 \\sin{(r)} \\cos{(r)} + \\cos{(r)})", "derivation": "J{(r)} = \\cos{(r)} and J{(r)} + \\cos{(r)} = 2 \\cos{(r)} and (J{(r)} + \\cos{(r)})^{2} = 4 \\cos^{2}{(r)} and \\int (J{(r)} + \\cos{(r)})^{2} dr = \\int 4 \\cos^{2}{(r)} dr and J{(r)} + \\cos{(r)} + \\int (J{(r)} + \\cos{(r)})^{2} dr = J{(r)} + \\cos{(r)} + \\int 4 \\cos^{2}{(r)} dr and \\frac{d}{d r} (J{(r)} + \\cos{(r)} + \\int (J{(r)} + \\cos{(r)})^{2} dr) = \\frac{d}{d r} (J{(r)} + \\cos{(r)} + \\int 4 \\cos^{2}{(r)} dr) and \\frac{d}{d r} (J{(r)} + \\cos{(r)} + \\int (J{(r)} + \\cos{(r)})^{2} dr) = \\frac{\\partial}{\\partial r} (\\mathbf{E} + 2 r + J{(r)} + 2 \\sin{(r)} \\cos{(r)} + \\cos{(r)})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["add", 1, "cos(Symbol('r', commutative=True))"], "Equality(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Mul(Integer(2), cos(Symbol('r', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Integer(2)), Mul(Integer(4), Pow(cos(Symbol('r', commutative=True)), Integer(2))))"], [["integrate", 3, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Integer(2)), Tuple(Symbol('r', commutative=True))), Integral(Mul(Integer(4), Pow(cos(Symbol('r', commutative=True)), Integer(2))), Tuple(Symbol('r', commutative=True))))"], [["add", 4, "Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], "Equality(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)), Integral(Pow(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Integer(2)), Tuple(Symbol('r', commutative=True)))), Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)), Integral(Mul(Integer(4), Pow(cos(Symbol('r', commutative=True)), Integer(2))), Tuple(Symbol('r', commutative=True)))))"], [["differentiate", 5, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)), Integral(Pow(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Integer(2)), Tuple(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)), Integral(Mul(Integer(4), Pow(cos(Symbol('r', commutative=True)), Integer(2))), Tuple(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)), Integral(Pow(Add(Function('J')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), Integer(2)), Tuple(Symbol('r', commutative=True)))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(2), Symbol('r', commutative=True)), Function('J')(Symbol('r', commutative=True)), Mul(Integer(2), sin(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True))), cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\varepsilon,\\pi,\\psi)} = \\frac{\\pi}{\\psi \\varepsilon}, then derive \\frac{\\partial}{\\partial \\pi} I{(\\varepsilon,\\pi,\\psi)} = \\frac{1}{\\psi \\varepsilon}, then derive L_{\\varepsilon} + I{(\\varepsilon,\\pi,\\psi)} = \\int \\frac{1}{\\psi \\varepsilon} d\\pi, then obtain e^{\\int (L_{\\varepsilon} + I{(\\varepsilon,\\pi,\\psi)}) d\\pi} = e^{\\iint \\frac{1}{\\psi \\varepsilon} d\\pi d\\pi}", "derivation": "I{(\\varepsilon,\\pi,\\psi)} = \\frac{\\pi}{\\psi \\varepsilon} and \\frac{\\partial}{\\partial \\pi} I{(\\varepsilon,\\pi,\\psi)} = \\frac{\\partial}{\\partial \\pi} \\frac{\\pi}{\\psi \\varepsilon} and \\frac{\\partial}{\\partial \\pi} I{(\\varepsilon,\\pi,\\psi)} = \\frac{1}{\\psi \\varepsilon} and \\int \\frac{\\partial}{\\partial \\pi} I{(\\varepsilon,\\pi,\\psi)} d\\pi = \\int \\frac{1}{\\psi \\varepsilon} d\\pi and L_{\\varepsilon} + I{(\\varepsilon,\\pi,\\psi)} = \\int \\frac{1}{\\psi \\varepsilon} d\\pi and \\int (L_{\\varepsilon} + I{(\\varepsilon,\\pi,\\psi)}) d\\pi = \\iint \\frac{1}{\\psi \\varepsilon} d\\pi d\\pi and e^{\\int (L_{\\varepsilon} + I{(\\varepsilon,\\pi,\\psi)}) d\\pi} = e^{\\iint \\frac{1}{\\psi \\varepsilon} d\\pi d\\pi}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True)), Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\pi', commutative=True), Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["exp", 6], "Equality(exp(Integral(Add(Symbol('L_{\\\\varepsilon}', commutative=True), Function('I')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\pi', commutative=True), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)))), exp(Integral(Mul(Pow(Symbol('\\\\psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given y{(L,W)} = L + W, then obtain \\int ((- L)^{W} - y{(L,W)}) dW = \\int ((W - y{(L,W)})^{W} - y{(L,W)}) dW", "derivation": "y{(L,W)} = L + W and 0 = L + W - y{(L,W)} and - L = W - y{(L,W)} and (- L)^{W} = (W - y{(L,W)})^{W} and (- L)^{W} - y{(L,W)} = (W - y{(L,W)})^{W} - y{(L,W)} and \\int ((- L)^{W} - y{(L,W)}) dW = \\int ((W - y{(L,W)})^{W} - y{(L,W)}) dW", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)), Add(Symbol('L', commutative=True), Symbol('W', commutative=True)))"], [["minus", 1, "Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Symbol('L', commutative=True), Symbol('W', commutative=True), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))))"], [["minus", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('L', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('W', commutative=True)), Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Symbol('W', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Pow(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Add(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))))"], [["integrate", 5, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Pow(Mul(Integer(-1), Symbol('L', commutative=True)), Symbol('W', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Add(Pow(Add(Symbol('W', commutative=True), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), Function('y')(Symbol('L', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\Omega{(A_{2})} = A_{2}, then obtain - \\frac{\\Omega{(A_{2})} \\operatorname{v_{y}}{(\\mathbf{J}_P)}}{A_{2} t_{1}} = - \\frac{e^{\\mathbf{J}_P}}{t_{1}}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\Omega{(A_{2})} = A_{2} and \\frac{\\operatorname{v_{y}}{(\\mathbf{J}_P)}}{A_{2} t_{1}} = \\frac{e^{\\mathbf{J}_P}}{A_{2} t_{1}} and - \\frac{\\Omega{(A_{2})} \\operatorname{v_{y}}{(\\mathbf{J}_P)}}{A_{2} t_{1}} = - \\frac{\\Omega{(A_{2})} e^{\\mathbf{J}_P}}{A_{2} t_{1}} and \\Omega{(A_{2})} e^{\\mathbf{J}_P} = A_{2} e^{\\mathbf{J}_P} and - \\frac{\\Omega{(A_{2})} \\operatorname{v_{y}}{(\\mathbf{J}_P)}}{A_{2} t_{1}} = - \\frac{e^{\\mathbf{J}_P}}{t_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], [["divide", 1, "Mul(Symbol('A_2', commutative=True), Symbol('t_1', commutative=True))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('t_1', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Function('\\\\Omega')(Symbol('A_2', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('A_2', commutative=True)), Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('A_2', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 2, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('A_2', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('A_2', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Pow(Symbol('t_1', commutative=True), Integer(-1)), Function('\\\\Omega')(Symbol('A_2', commutative=True)), Function('v_y')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), Pow(Symbol('t_1', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given r{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}}, then obtain \\frac{r^{2}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{e^{2 \\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}}", "derivation": "r{(\\hat{\\mathbf{r}})} = e^{\\hat{\\mathbf{r}}} and \\frac{r{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{e^{\\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}} and \\frac{r{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}} = \\frac{e^{2 \\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}} and \\frac{r^{2}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{r{(\\hat{\\mathbf{r}})} e^{\\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}} and \\frac{r^{2}{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{e^{2 \\hat{\\mathbf{r}}}}{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), exp(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Pow(Function('r')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given k{(g)} = e^{g}, then derive - g \\int k{(g)} dg = - g (\\rho_f + e^{g}), then obtain e^{- g} \\int - g \\int k{(g)} dg dg = (\\eta^{\\prime} - \\int \\rho_f g dg - \\int g k{(g)} dg) e^{- g}", "derivation": "k{(g)} = e^{g} and \\int k{(g)} dg = \\int e^{g} dg and - g \\int k{(g)} dg = - g \\int e^{g} dg and - g \\int k{(g)} dg = - g (\\rho_f + e^{g}) and - g \\int k{(g)} dg = - g (\\rho_f + k{(g)}) and \\int - g \\int k{(g)} dg dg = \\int - g (\\rho_f + k{(g)}) dg and e^{- g} \\int - g \\int k{(g)} dg dg = e^{- g} \\int - g (\\rho_f + k{(g)}) dg and e^{- g} \\int - g \\int k{(g)} dg dg = (\\eta^{\\prime} - \\int \\rho_f g dg - \\int g k{(g)} dg) e^{- g}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('g', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True), Integral(exp(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), exp(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Symbol('g', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Function('k')(Symbol('g', commutative=True)))))"], [["integrate", 5, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('g', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Function('k')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["divide", 6, "exp(Symbol('g', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('g', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Function('k')(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Mul(exp(Mul(Integer(-1), Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('g', commutative=True), Integral(Function('k')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Mul(Add(Symbol('\\\\eta^{\\\\prime}', commutative=True), Mul(Integer(-1), Integral(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('g', commutative=True), Function('k')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))), exp(Mul(Integer(-1), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\mathbf{E}{(\\mathbf{H})} = - \\sin{(\\mathbf{H})}, then obtain - \\sin{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})}", "derivation": "\\mathbf{E}{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{E}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{E}{(\\mathbf{H})} = - \\sin{(\\mathbf{H})} and - \\sin{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(\\hat{H},\\lambda)} = \\lambda^{\\hat{H}} and \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} = \\lambda^{\\hat{H}}, then obtain \\frac{\\partial}{\\partial \\lambda} \\int \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} d\\hat{H} = \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\hat{H}} d\\hat{H}", "derivation": "\\mathbf{M}{(\\hat{H},\\lambda)} = \\lambda^{\\hat{H}} and \\int \\mathbf{M}{(\\hat{H},\\lambda)} d\\hat{H} = \\int \\lambda^{\\hat{H}} d\\hat{H} and \\frac{\\partial}{\\partial \\lambda} \\int \\mathbf{M}{(\\hat{H},\\lambda)} d\\hat{H} = \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\hat{H}} d\\hat{H} and \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} = \\lambda^{\\hat{H}} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} d\\hat{H} = \\int \\lambda^{\\hat{H}} d\\hat{H} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} d\\hat{H} = \\int \\mathbf{M}{(\\hat{H},\\lambda)} d\\hat{H} and \\frac{\\partial}{\\partial \\lambda} \\int \\operatorname{V_{\\mathbf{B}}}{(\\hat{H},\\lambda)} d\\hat{H} = \\frac{\\partial}{\\partial \\lambda} \\int \\lambda^{\\hat{H}} d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Derivative(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(A_{2})} = A_{2}, then derive F_{x} + \\frac{\\mathbf{B}^{2}{(A_{2})}}{2} = \\int A_{2} d\\mathbf{B}{(A_{2})}, then obtain e^{(\\frac{A_{2}^{2}}{2} + F_{x})^{F_{x}}} = e^{(\\int A_{2} dA_{2})^{F_{x}}}", "derivation": "\\mathbf{B}{(A_{2})} = A_{2} and \\int \\mathbf{B}{(A_{2})} dA_{2} = \\int A_{2} dA_{2} and \\int \\mathbf{B}{(A_{2})} d\\mathbf{B}{(A_{2})} = \\int A_{2} d\\mathbf{B}{(A_{2})} and F_{x} + \\frac{\\mathbf{B}^{2}{(A_{2})}}{2} = \\int A_{2} d\\mathbf{B}{(A_{2})} and \\frac{A_{2}^{2}}{2} + F_{x} = \\int A_{2} dA_{2} and (\\frac{A_{2}^{2}}{2} + F_{x})^{F_{x}} = (\\int A_{2} dA_{2})^{F_{x}} and e^{(\\frac{A_{2}^{2}}{2} + F_{x})^{F_{x}}} = e^{(\\int A_{2} dA_{2})^{F_{x}}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(Symbol('A_2', commutative=True), Tuple(Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)), Tuple(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)))), Integral(Symbol('A_2', commutative=True), Tuple(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_x', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)), Integer(2)))), Integral(Symbol('A_2', commutative=True), Tuple(Function('\\\\mathbf{B}')(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('F_x', commutative=True)), Integral(Symbol('A_2', commutative=True), Tuple(Symbol('A_2', commutative=True))))"], [["power", 5, "Symbol('F_x', commutative=True)"], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Integral(Symbol('A_2', commutative=True), Tuple(Symbol('A_2', commutative=True))), Symbol('F_x', commutative=True)))"], [["exp", 6], "Equality(exp(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('A_2', commutative=True), Integer(2))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), exp(Pow(Integral(Symbol('A_2', commutative=True), Tuple(Symbol('A_2', commutative=True))), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(F_{H})} = F_{H}, then obtain \\int (- F_{H} + \\operatorname{v_{y}}{(F_{H})})^{F_{H}} d\\operatorname{v_{y}}{(F_{H})} = \\int 0^{F_{H}} d\\operatorname{v_{y}}{(F_{H})}", "derivation": "\\operatorname{v_{y}}{(F_{H})} = F_{H} and - F_{H} + \\operatorname{v_{y}}{(F_{H})} = 0 and (- F_{H} + \\operatorname{v_{y}}{(F_{H})})^{F_{H}} = 0^{F_{H}} and \\int (- F_{H} + \\operatorname{v_{y}}{(F_{H})})^{F_{H}} dF_{H} = \\int 0^{F_{H}} dF_{H} and \\int (- F_{H} + \\operatorname{v_{y}}{(F_{H})})^{F_{H}} d\\operatorname{v_{y}}{(F_{H})} = \\int 0^{F_{H}} d\\operatorname{v_{y}}{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["minus", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('v_y')(Symbol('F_H', commutative=True))), Integer(0))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('v_y')(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Pow(Integer(0), Symbol('F_H', commutative=True)))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('v_y')(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Pow(Integer(0), Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('v_y')(Symbol('F_H', commutative=True))), Symbol('F_H', commutative=True)), Tuple(Function('v_y')(Symbol('F_H', commutative=True)))), Integral(Pow(Integer(0), Symbol('F_H', commutative=True)), Tuple(Function('v_y')(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given u{(h,\\psi^*)} = - \\psi^* + \\log{(h)}, then obtain \\psi^* u{(h,\\psi^*)} \\frac{\\partial}{\\partial h} u{(h,\\psi^*)} = \\psi^* (- \\psi^* + \\log{(h)}) \\frac{\\partial}{\\partial h} u{(h,\\psi^*)}", "derivation": "u{(h,\\psi^*)} = - \\psi^* + \\log{(h)} and \\frac{\\partial}{\\partial h} u{(h,\\psi^*)} = \\frac{\\partial}{\\partial h} (- \\psi^* + \\log{(h)}) and u{(h,\\psi^*)} \\frac{\\partial}{\\partial h} (- \\psi^* + \\log{(h)}) = (- \\psi^* + \\log{(h)}) \\frac{\\partial}{\\partial h} (- \\psi^* + \\log{(h)}) and \\psi^* u{(h,\\psi^*)} \\frac{\\partial}{\\partial h} (- \\psi^* + \\log{(h)}) = \\psi^* (- \\psi^* + \\log{(h)}) \\frac{\\partial}{\\partial h} (- \\psi^* + \\log{(h)}) and \\psi^* u{(h,\\psi^*)} \\frac{\\partial}{\\partial h} u{(h,\\psi^*)} = \\psi^* (- \\psi^* + \\log{(h)}) \\frac{\\partial}{\\partial h} u{(h,\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Mul(Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["times", 3, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('\\\\psi^*', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), log(Symbol('h', commutative=True))), Derivative(Function('u')(Symbol('h', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(T,x)} = T - x, then derive \\frac{\\partial}{\\partial x} \\pi{(T,x)} = -1, then obtain \\frac{\\partial}{\\partial x} - \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx = \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial x} (T - x) \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx", "derivation": "\\pi{(T,x)} = T - x and \\frac{\\partial}{\\partial x} \\pi{(T,x)} = \\frac{\\partial}{\\partial x} (T - x) and \\frac{\\partial}{\\partial x} \\pi{(T,x)} = -1 and -1 = \\frac{\\partial}{\\partial x} (T - x) and - \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx = \\frac{\\partial}{\\partial x} (T - x) \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx and \\frac{\\partial}{\\partial x} - \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx = \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial x} (T - x) \\int (2 T - 2 x + \\operatorname{A_{x}}{(T,x)} + 1) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["times", 4, "Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Integer(1)), Tuple(Symbol('x', commutative=True)))"], "Equality(Mul(Integer(-1), Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Integer(1)), Tuple(Symbol('x', commutative=True)))), Mul(Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Integer(1)), Tuple(Symbol('x', commutative=True)))))"], [["differentiate", 5, "Symbol('x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Integer(1)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Mul(Derivative(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Integral(Add(Mul(Integer(2), Symbol('T', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('A_x')(Symbol('T', commutative=True), Symbol('x', commutative=True)), Integer(1)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{E})} = \\int e^{\\mathbf{E}} d\\mathbf{E}, then derive \\operatorname{F_{H}}{(\\mathbf{E})} + e^{\\mathbf{E}} = \\mu_0 + 2 e^{\\mathbf{E}}, then obtain - \\mathbf{B} + 2 \\operatorname{F_{H}}{(\\mathbf{E})} = - 2 \\mathbf{B} + \\mu_0 + 2 \\operatorname{F_{H}}{(\\mathbf{E})}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{E})} = \\int e^{\\mathbf{E}} d\\mathbf{E} and \\operatorname{F_{H}}{(\\mathbf{E})} + e^{\\mathbf{E}} = e^{\\mathbf{E}} + \\int e^{\\mathbf{E}} d\\mathbf{E} and \\operatorname{F_{H}}{(\\mathbf{E})} + e^{\\mathbf{E}} = \\mu_0 + 2 e^{\\mathbf{E}} and \\operatorname{F_{H}}{(\\mathbf{E})} + e^{\\mathbf{E}} - \\int e^{\\mathbf{E}} d\\mathbf{E} = e^{\\mathbf{E}} and 2 \\operatorname{F_{H}}{(\\mathbf{E})} + e^{\\mathbf{E}} - \\int e^{\\mathbf{E}} d\\mathbf{E} = \\mu_0 + 2 \\operatorname{F_{H}}{(\\mathbf{E})} + 2 e^{\\mathbf{E}} - 2 \\int e^{\\mathbf{E}} d\\mathbf{E} and - \\mathbf{B} + 2 \\operatorname{F_{H}}{(\\mathbf{E})} = - 2 \\mathbf{B} + \\mu_0 + 2 \\operatorname{F_{H}}{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["add", 1, "exp(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))), Add(exp(Symbol('\\\\mathbf{E}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(2), exp(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["minus", 2, "Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True)))"], "Equality(Add(Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(2), Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True))), exp(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(2), Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Integer(2), Integral(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mu_0', commutative=True), Mul(Integer(2), Function('F_H')(Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given i{(\\mathbf{v},Q)} = \\frac{\\mathbf{v}}{Q}, then derive \\frac{\\partial}{\\partial \\mathbf{v}} i{(\\mathbf{v},Q)} = \\frac{1}{Q}, then obtain \\frac{1}{Q} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{Q}", "derivation": "i{(\\mathbf{v},Q)} = \\frac{\\mathbf{v}}{Q} and \\frac{\\partial}{\\partial \\mathbf{v}} i{(\\mathbf{v},Q)} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{Q} and \\frac{\\partial}{\\partial \\mathbf{v}} i{(\\mathbf{v},Q)} = \\frac{1}{Q} and \\frac{1}{Q} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v}}{Q}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Symbol('Q', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('Q', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(f^{\\prime})} = \\log{(f^{\\prime})}, then obtain 1 = (\\frac{- f^{\\prime} + \\log{(f^{\\prime})}}{- f^{\\prime} + \\varepsilon{(f^{\\prime})}})^{f^{\\prime}}", "derivation": "\\varepsilon{(f^{\\prime})} = \\log{(f^{\\prime})} and - f^{\\prime} + \\varepsilon{(f^{\\prime})} = - f^{\\prime} + \\log{(f^{\\prime})} and 1 = \\frac{- f^{\\prime} + \\log{(f^{\\prime})}}{- f^{\\prime} + \\varepsilon{(f^{\\prime})}} and 1 = (\\frac{- f^{\\prime} + \\log{(f^{\\prime})}}{- f^{\\prime} + \\varepsilon{(f^{\\prime})}})^{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\varepsilon')(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\varepsilon')(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\varepsilon')(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))))"], [["power", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\varepsilon')(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(L)} = e^{L} and a{(L)} = \\int 2 \\operatorname{A_{2}}{(L)} dL, then obtain (\\operatorname{A_{2}}{(L)} + e^{L}) a{(L)} \\int \\frac{d}{d L} \\operatorname{A_{2}}{(L)} dL = (\\operatorname{A_{2}}{(L)} + e^{L}) (\\int (\\operatorname{A_{2}}{(L)} + e^{L}) dL) \\int \\frac{d}{d L} \\operatorname{A_{2}}{(L)} dL", "derivation": "\\operatorname{A_{2}}{(L)} = e^{L} and 2 \\operatorname{A_{2}}{(L)} = \\operatorname{A_{2}}{(L)} + e^{L} and \\int 2 \\operatorname{A_{2}}{(L)} dL = \\int (\\operatorname{A_{2}}{(L)} + e^{L}) dL and a{(L)} = \\int 2 \\operatorname{A_{2}}{(L)} dL and (\\operatorname{A_{2}}{(L)} + e^{L}) a{(L)} = (\\operatorname{A_{2}}{(L)} + e^{L}) \\int 2 \\operatorname{A_{2}}{(L)} dL and (\\operatorname{A_{2}}{(L)} + e^{L}) a{(L)} = (\\operatorname{A_{2}}{(L)} + e^{L}) \\int (\\operatorname{A_{2}}{(L)} + e^{L}) dL and (\\operatorname{A_{2}}{(L)} + e^{L}) a{(L)} \\int \\frac{d}{d L} \\operatorname{A_{2}}{(L)} dL = (\\operatorname{A_{2}}{(L)} + e^{L}) (\\int (\\operatorname{A_{2}}{(L)} + e^{L}) dL) \\int \\frac{d}{d L} \\operatorname{A_{2}}{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], [["add", 1, "Function('A_2')(Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('A_2')(Symbol('L', commutative=True))), Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))))"], [["integrate", 2, "Symbol('L', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('A_2')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], ["renaming_premise", "Equality(Function('a')(Symbol('L', commutative=True)), Integral(Mul(Integer(2), Function('A_2')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))))"], [["times", 4, "Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True)))"], "Equality(Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Function('a')(Symbol('L', commutative=True))), Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Integral(Mul(Integer(2), Function('A_2')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Function('a')(Symbol('L', commutative=True))), Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Integral(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True)))))"], [["times", 6, "Integral(Derivative(Function('A_2')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True)))"], "Equality(Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Function('a')(Symbol('L', commutative=True)), Integral(Derivative(Function('A_2')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True)))), Mul(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Integral(Add(Function('A_2')(Symbol('L', commutative=True)), exp(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True))), Integral(Derivative(Function('A_2')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\pi,\\mathbf{r})} = \\mathbf{r} \\pi, then obtain \\frac{\\partial}{\\partial \\pi} (- (\\int \\mathbf{r} \\pi d\\mathbf{r})^{\\mathbf{r}} + (\\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}) = 0", "derivation": "\\rho{(\\pi,\\mathbf{r})} = \\mathbf{r} \\pi and \\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r} = \\int \\mathbf{r} \\pi d\\mathbf{r} and (\\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = (\\int \\mathbf{r} \\pi d\\mathbf{r})^{\\mathbf{r}} and - (\\int \\mathbf{r} \\pi d\\mathbf{r})^{\\mathbf{r}} + (\\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}} = 0 and \\frac{\\partial}{\\partial \\pi} (- (\\int \\mathbf{r} \\pi d\\mathbf{r})^{\\mathbf{r}} + (\\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}) = \\frac{d}{d \\pi} 0 and \\frac{\\partial}{\\partial \\pi} (- (\\int \\mathbf{r} \\pi d\\mathbf{r})^{\\mathbf{r}} + (\\int \\rho{(\\pi,\\mathbf{r})} d\\mathbf{r})^{\\mathbf{r}}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True)))"], [["minus", 3, "Pow(Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Integral(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Integral(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Integral(Mul(Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Integral(Function('\\\\rho')(Symbol('\\\\pi', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True))), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given y{(p,F_{c})} = F_{c} p, then derive \\frac{\\partial}{\\partial p} y{(p,F_{c})} + \\int \\mathbf{M}{(v_{1})} dv_{1} = F_{c} + \\int \\mathbf{M}{(v_{1})} dv_{1}, then obtain \\frac{\\partial}{\\partial p} F_{c} p + \\int \\mathbf{M}{(v_{1})} dv_{1} = F_{c} + \\int \\mathbf{M}{(v_{1})} dv_{1}", "derivation": "y{(p,F_{c})} = F_{c} p and \\frac{\\partial}{\\partial p} y{(p,F_{c})} = \\frac{\\partial}{\\partial p} F_{c} p and \\frac{\\partial}{\\partial p} y{(p,F_{c})} + \\int \\mathbf{M}{(v_{1})} dv_{1} = \\frac{\\partial}{\\partial p} F_{c} p + \\int \\mathbf{M}{(v_{1})} dv_{1} and \\frac{\\partial}{\\partial p} y{(p,F_{c})} + \\int \\mathbf{M}{(v_{1})} dv_{1} = F_{c} + \\int \\mathbf{M}{(v_{1})} dv_{1} and \\frac{\\partial}{\\partial p} F_{c} p + \\int \\mathbf{M}{(v_{1})} dv_{1} = F_{c} + \\int \\mathbf{M}{(v_{1})} dv_{1}", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('p', commutative=True), Symbol('F_c', commutative=True)), Mul(Symbol('F_c', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('p', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 2, "Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Add(Derivative(Function('y')(Symbol('p', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Derivative(Mul(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('y')(Symbol('p', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('F_c', commutative=True), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Mul(Symbol('F_c', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Add(Symbol('F_c', commutative=True), Integral(Function('\\\\mathbf{M}')(Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{P},\\mathbf{p})} = \\mathbf{P} + \\mathbf{p} and \\delta{(\\mathbf{P},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} \\sin{(\\mathbf{P} + \\mathbf{p})}, then obtain \\sin{(\\delta{(\\mathbf{P},\\mathbf{p})})} = \\sin{(\\frac{\\partial}{\\partial \\mathbf{p}} \\sin{(\\operatorname{C_{d}}{(\\mathbf{P},\\mathbf{p})})})}", "derivation": "\\operatorname{C_{d}}{(\\mathbf{P},\\mathbf{p})} = \\mathbf{P} + \\mathbf{p} and \\delta{(\\mathbf{P},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} \\sin{(\\mathbf{P} + \\mathbf{p})} and \\delta{(\\mathbf{P},\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} \\sin{(\\operatorname{C_{d}}{(\\mathbf{P},\\mathbf{p})})} and \\sin{(\\delta{(\\mathbf{P},\\mathbf{p})})} = \\sin{(\\frac{\\partial}{\\partial \\mathbf{p}} \\sin{(\\operatorname{C_{d}}{(\\mathbf{P},\\mathbf{p})})})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(sin(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(sin(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Function('\\\\delta')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), sin(Derivative(sin(Function('C_d')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given u{(\\theta_1)} = e^{\\theta_1}, then obtain \\frac{d}{d \\theta_1} 1 \\int 1 d\\theta_1 = (\\frac{d}{d \\theta_1} \\frac{\\int (\\theta_1 + e^{\\theta_1}) d\\theta_1}{\\int (\\theta_1 + u{(\\theta_1)}) d\\theta_1}) \\int 1 d\\theta_1", "derivation": "u{(\\theta_1)} = e^{\\theta_1} and \\theta_1 + u{(\\theta_1)} = \\theta_1 + e^{\\theta_1} and \\int (\\theta_1 + u{(\\theta_1)}) d\\theta_1 = \\int (\\theta_1 + e^{\\theta_1}) d\\theta_1 and 1 = \\frac{\\int (\\theta_1 + e^{\\theta_1}) d\\theta_1}{\\int (\\theta_1 + u{(\\theta_1)}) d\\theta_1} and \\frac{d}{d \\theta_1} 1 = \\frac{d}{d \\theta_1} \\frac{\\int (\\theta_1 + e^{\\theta_1}) d\\theta_1}{\\int (\\theta_1 + u{(\\theta_1)}) d\\theta_1} and \\frac{d}{d \\theta_1} 1 \\int 1 d\\theta_1 = (\\frac{d}{d \\theta_1} \\frac{\\int (\\theta_1 + e^{\\theta_1}) d\\theta_1}{\\int (\\theta_1 + u{(\\theta_1)}) d\\theta_1}) \\int 1 d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["add", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 3, "Integral(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Integral(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["times", 5, "Integral(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Derivative(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Derivative(Mul(Pow(Integral(Add(Symbol('\\\\theta_1', commutative=True), Function('u')(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integer(-1)), Integral(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Integral(Integer(1), Tuple(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(E,r_{0})} = E + r_{0}, then obtain \\log{(\\int \\operatorname{E_{n}}{(E,r_{0})} dr_{0})} = \\log{(M_{E} + \\frac{r_{0}^{2}}{2} + \\int E dr_{0})}", "derivation": "\\operatorname{E_{n}}{(E,r_{0})} = E + r_{0} and \\int \\operatorname{E_{n}}{(E,r_{0})} dr_{0} = \\int (E + r_{0}) dr_{0} and \\log{(\\int \\operatorname{E_{n}}{(E,r_{0})} dr_{0})} = \\log{(\\int (E + r_{0}) dr_{0})} and \\log{(\\int \\operatorname{E_{n}}{(E,r_{0})} dr_{0})} = \\log{(\\int E dr_{0} + \\int r_{0} dr_{0})} and \\log{(\\int \\operatorname{E_{n}}{(E,r_{0})} dr_{0})} = \\log{(M_{E} + \\frac{r_{0}^{2}}{2} + \\int E dr_{0})}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('E', commutative=True), Symbol('r_0', commutative=True)))"], [["integrate", 1, "Symbol('r_0', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Add(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('E_n')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), log(Integral(Add(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))))"], [["expand", 3], "Equality(log(Integral(Function('E_n')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), log(Add(Integral(Symbol('E', commutative=True), Tuple(Symbol('r_0', commutative=True))), Integral(Symbol('r_0', commutative=True), Tuple(Symbol('r_0', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(log(Integral(Function('E_n')(Symbol('E', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), log(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Pow(Symbol('r_0', commutative=True), Integer(2))), Integral(Symbol('E', commutative=True), Tuple(Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(A,\\varphi^*)} = \\frac{\\partial}{\\partial A} (A - \\varphi^*), then derive \\operatorname{A_{2}}{(A,\\varphi^*)} = 1, then obtain 0^{A} = (- \\operatorname{A_{2}}{(A,\\varphi^*)} + (\\frac{\\partial}{\\partial A} (A - \\varphi^*))^{2})^{A}", "derivation": "\\operatorname{A_{2}}{(A,\\varphi^*)} = \\frac{\\partial}{\\partial A} (A - \\varphi^*) and \\operatorname{A_{2}}{(A,\\varphi^*)} = 1 and \\frac{\\partial}{\\partial A} (A - \\varphi^*) = 1 and (\\frac{\\partial}{\\partial A} (A - \\varphi^*))^{2} = \\frac{\\partial}{\\partial A} (A - \\varphi^*) and 0 = - \\operatorname{A_{2}}{(A,\\varphi^*)} + \\frac{\\partial}{\\partial A} (A - \\varphi^*) and 0^{A} = (- \\operatorname{A_{2}}{(A,\\varphi^*)} + \\frac{\\partial}{\\partial A} (A - \\varphi^*))^{A} and 0^{A} = (- \\operatorname{A_{2}}{(A,\\varphi^*)} + (\\frac{\\partial}{\\partial A} (A - \\varphi^*))^{2})^{A}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(1))"], [["times", 3, "Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["minus", 1, "Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["power", 5, "Symbol('A', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Integer(0), Symbol('A', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A_2')(Symbol('A', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Pow(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})}, then obtain 4 \\mathbf{J}_M{(f_{\\mathbf{v}})} = 2 \\mathbf{J}_M{(f_{\\mathbf{v}})} + 2 \\log{(f_{\\mathbf{v}})}", "derivation": "\\mathbf{J}_M{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and 2 \\mathbf{J}_M{(f_{\\mathbf{v}})} = \\mathbf{J}_M{(f_{\\mathbf{v}})} + \\log{(f_{\\mathbf{v}})} and 3 \\mathbf{J}_M{(f_{\\mathbf{v}})} = 2 \\mathbf{J}_M{(f_{\\mathbf{v}})} + \\log{(f_{\\mathbf{v}})} and 3 \\mathbf{J}_M{(f_{\\mathbf{v}})} = \\mathbf{J}_M{(f_{\\mathbf{v}})} + 2 \\log{(f_{\\mathbf{v}})} and 4 \\mathbf{J}_M{(f_{\\mathbf{v}})} = 2 \\mathbf{J}_M{(f_{\\mathbf{v}})} + 2 \\log{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Integer(3), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Integer(2), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["add", 4, "Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Integer(4), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Mul(Integer(2), Function('\\\\mathbf{J}_M')(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Mul(Integer(2), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(B)} = \\cos{(B)}, then obtain \\operatorname{F_{g}}{(B)} \\int \\operatorname{F_{g}}{(B)} dB = \\cos{(B)} \\int \\operatorname{F_{g}}{(B)} dB", "derivation": "\\operatorname{F_{g}}{(B)} = \\cos{(B)} and \\int \\operatorname{F_{g}}{(B)} dB = \\int \\cos{(B)} dB and \\operatorname{F_{g}}{(B)} \\int \\cos{(B)} dB = \\cos{(B)} \\int \\cos{(B)} dB and \\operatorname{F_{g}}{(B)} \\int \\operatorname{F_{g}}{(B)} dB = \\cos{(B)} \\int \\operatorname{F_{g}}{(B)} dB", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('F_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["times", 1, "Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))"], "Equality(Mul(Function('F_g')(Symbol('B', commutative=True)), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(cos(Symbol('B', commutative=True)), Integral(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('F_g')(Symbol('B', commutative=True)), Integral(Function('F_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(cos(Symbol('B', commutative=True)), Integral(Function('F_g')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(L,J)} = J - L, then obtain J \\dot{\\mathbf{r}}{(L,J)} = J (J - L)", "derivation": "\\dot{\\mathbf{r}}{(L,J)} = J - L and - \\dot{\\mathbf{r}}{(L,J)} = - J + L and J \\dot{\\mathbf{r}}{(L,J)} = - J (- J + L) and J (J - L) = - J (- J + L) and J \\dot{\\mathbf{r}}{(L,J)} = J (J - L)", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L', commutative=True), Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('L', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L', commutative=True), Symbol('J', commutative=True))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('J', commutative=True))"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L', commutative=True), Symbol('J', commutative=True))), Mul(Integer(-1), Symbol('J', commutative=True), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('L', commutative=True)))), Mul(Integer(-1), Symbol('J', commutative=True), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('L', commutative=True), Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given S{(W,E_{\\lambda})} = E_{\\lambda} - W and \\dot{x}{(\\omega)} = \\log{(\\omega)}, then obtain (\\frac{d}{d \\omega} \\dot{x}{(\\omega)})^{\\omega} - 2 \\int 0 dW = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} - 2 \\int 0 dW", "derivation": "S{(W,E_{\\lambda})} = E_{\\lambda} - W and \\dot{x}{(\\omega)} = \\log{(\\omega)} and \\frac{d}{d \\omega} \\dot{x}{(\\omega)} = \\frac{d}{d \\omega} \\log{(\\omega)} and (\\frac{d}{d \\omega} \\dot{x}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} and (\\frac{d}{d \\omega} \\dot{x}{(\\omega)})^{\\omega} - \\int (E_{\\lambda} - W - S{(W,E_{\\lambda})}) dW = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} - \\int (E_{\\lambda} - W - S{(W,E_{\\lambda})}) dW and (\\frac{d}{d \\omega} \\dot{x}{(\\omega)})^{\\omega} - \\int 0 dW = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} - \\int 0 dW and (\\frac{d}{d \\omega} \\dot{x}{(\\omega)})^{\\omega} - 2 \\int 0 dW = (\\frac{d}{d \\omega} \\log{(\\omega)})^{\\omega} - 2 \\int 0 dW", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), log(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["minus", 4, "Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Pow(Derivative(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('W', commutative=True))))), Add(Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('W', commutative=True)), Mul(Integer(-1), Function('S')(Symbol('W', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('W', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Pow(Derivative(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('W', commutative=True))))), Add(Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('W', commutative=True))))))"], [["minus", 6, "Integral(Integer(0), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Pow(Derivative(Function('\\\\dot{x}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integer(2), Integral(Integer(0), Tuple(Symbol('W', commutative=True))))), Add(Pow(Derivative(log(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Integer(2), Integral(Integer(0), Tuple(Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\psi{(I,r_{0})} = \\log{(- I + r_{0})}, then obtain \\frac{\\partial}{\\partial r_{0}} (- \\log{(- I + r_{0})} + \\frac{\\psi{(I,r_{0})}}{r_{0}})^{I} = \\frac{\\partial}{\\partial r_{0}} (- \\log{(- I + r_{0})} + \\frac{\\log{(- I + r_{0})}}{r_{0}})^{I}", "derivation": "\\psi{(I,r_{0})} = \\log{(- I + r_{0})} and \\frac{\\psi{(I,r_{0})}}{r_{0}} = \\frac{\\log{(- I + r_{0})}}{r_{0}} and - \\log{(- I + r_{0})} + \\frac{\\psi{(I,r_{0})}}{r_{0}} = - \\log{(- I + r_{0})} + \\frac{\\log{(- I + r_{0})}}{r_{0}} and (- \\log{(- I + r_{0})} + \\frac{\\psi{(I,r_{0})}}{r_{0}})^{I} = (- \\log{(- I + r_{0})} + \\frac{\\log{(- I + r_{0})}}{r_{0}})^{I} and \\frac{\\partial}{\\partial r_{0}} (- \\log{(- I + r_{0})} + \\frac{\\psi{(I,r_{0})}}{r_{0}})^{I} = \\frac{\\partial}{\\partial r_{0}} (- \\log{(- I + r_{0})} + \\frac{\\log{(- I + r_{0})}}{r_{0}})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('I', commutative=True), Symbol('r_0', commutative=True)), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True))))"], [["divide", 1, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('r_0', commutative=True))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))))"], [["minus", 2, "log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('r_0', commutative=True)))), Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True))))))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('r_0', commutative=True)))), Symbol('I', commutative=True)), Pow(Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True))))), Symbol('I', commutative=True)))"], [["differentiate", 4, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('I', commutative=True), Symbol('r_0', commutative=True)))), Symbol('I', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Integer(-1), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), log(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('r_0', commutative=True))))), Symbol('I', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given s{(r_{0},S)} = - S + r_{0}, then obtain \\int (1 - (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S})^{2} dS = \\int 0 dS", "derivation": "s{(r_{0},S)} = - S + r_{0} and s^{2}{(r_{0},S)} = (- S + r_{0}) s{(r_{0},S)} and - (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)} = 0 and (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S} = 0^{S} and - 0^{S} + (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S} = 0 and (- 0^{S} + (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S})^{2} = 0 and (1 - (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S})^{2} = 0 and \\int (1 - (- (- S + r_{0}) s{(r_{0},S)} + s^{2}{(r_{0},S)})^{S})^{2} dS = \\int 0 dS", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)))"], [["times", 1, "Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))"], "Equality(Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))))"], [["minus", 2, "Mul(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Integer(0))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True)), Pow(Integer(0), Symbol('S', commutative=True)))"], [["minus", 4, "Pow(Integer(0), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('S', commutative=True))), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True))), Integer(0))"], [["times", 5, "Add(Mul(Integer(-1), Pow(Integer(0), Symbol('S', commutative=True))), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('S', commutative=True))), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True))), Integer(2)), Integer(0))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True)))), Integer(2)), Integer(0))"], [["integrate", 7, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Add(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('r_0', commutative=True)), Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True))), Pow(Function('s')(Symbol('r_0', commutative=True), Symbol('S', commutative=True)), Integer(2))), Symbol('S', commutative=True)))), Integer(2)), Tuple(Symbol('S', commutative=True))), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(f_{E},l)} = \\frac{f_{E}}{l}, then obtain \\int (\\tilde{g}^{f_{E}}{(f_{E},l)} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} df_{E} = \\int ((\\frac{f_{E}}{l})^{f_{E}} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} df_{E}", "derivation": "\\tilde{g}{(f_{E},l)} = \\frac{f_{E}}{l} and \\tilde{g}^{f_{E}}{(f_{E},l)} = (\\frac{f_{E}}{l})^{f_{E}} and \\tilde{g}^{f_{E}}{(f_{E},l)} - \\frac{1}{l} = (\\frac{f_{E}}{l})^{f_{E}} - \\frac{1}{l} and (\\tilde{g}^{f_{E}}{(f_{E},l)} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} = ((\\frac{f_{E}}{l})^{f_{E}} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} and \\int (\\tilde{g}^{f_{E}}{(f_{E},l)} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} df_{E} = \\int ((\\frac{f_{E}}{l})^{f_{E}} - \\frac{1}{l}) \\tilde{g}{(f_{E},l)} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('f_E', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True)), Symbol('f_E', commutative=True)), Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)))"], [["minus", 2, "Pow(Symbol('l', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Add(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))))"], [["times", 3, "Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Add(Pow(Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True))), Mul(Add(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True))))"], [["integrate", 4, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Add(Pow(Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True)), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Mul(Add(Pow(Mul(Symbol('f_E', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f_E', commutative=True)), Mul(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1)))), Function('\\\\tilde{g}')(Symbol('f_E', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given Q{(\\phi,I)} = I \\phi, then obtain (I \\phi)^{\\phi} = (I \\phi)^{\\phi} + \\iiint (I \\phi)^{\\phi} dI d\\phi d\\phi - \\iiint Q^{\\phi}{(\\phi,I)} dI d\\phi d\\phi", "derivation": "Q{(\\phi,I)} = I \\phi and Q^{\\phi}{(\\phi,I)} = (I \\phi)^{\\phi} and \\int Q^{\\phi}{(\\phi,I)} dI = \\int (I \\phi)^{\\phi} dI and \\iint Q^{\\phi}{(\\phi,I)} dI d\\phi = \\iint (I \\phi)^{\\phi} dI d\\phi and \\iiint Q^{\\phi}{(\\phi,I)} dI d\\phi d\\phi = \\iiint (I \\phi)^{\\phi} dI d\\phi d\\phi and 0 = \\iiint (I \\phi)^{\\phi} dI d\\phi d\\phi - \\iiint Q^{\\phi}{(\\phi,I)} dI d\\phi d\\phi and (I \\phi)^{\\phi} = (I \\phi)^{\\phi} + \\iiint (I \\phi)^{\\phi} dI d\\phi d\\phi - \\iiint Q^{\\phi}{(\\phi,I)} dI d\\phi d\\phi", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 3, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["minus", 5, "Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Integer(0), Add(Integral(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))))"], [["add", 6, "Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Add(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Integral(Pow(Mul(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('I', commutative=True)), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))))"]]}, {"prompt": "Given \\psi^{*}{(a,\\Omega)} = \\Omega + a, then derive \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = 0, then obtain \\Omega + \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = \\Omega", "derivation": "\\psi^{*}{(a,\\Omega)} = \\Omega + a and \\frac{\\partial}{\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + a) and \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} (\\Omega + a) and \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = 0 and \\Omega + \\frac{\\partial^{2}}{\\partial a\\partial \\Omega} \\psi^{*}{(a,\\Omega)} = \\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(0))"], [["add", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Derivative(Function('\\\\psi^*')(Symbol('a', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1)))), Symbol('\\\\Omega', commutative=True))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mu_0,q)} = - \\mu_0 + q, then obtain Q (- \\mu_0 + q)^{q} \\cos{(\\int \\operatorname{v_{z}}{(\\mu_0,q)} d\\mu_0)} = Q (- \\mu_0 + q)^{q} \\cos{(\\int (- \\mu_0 + q) d\\mu_0)}", "derivation": "\\operatorname{v_{z}}{(\\mu_0,q)} = - \\mu_0 + q and \\int \\operatorname{v_{z}}{(\\mu_0,q)} d\\mu_0 = \\int (- \\mu_0 + q) d\\mu_0 and \\cos{(\\int \\operatorname{v_{z}}{(\\mu_0,q)} d\\mu_0)} = \\cos{(\\int (- \\mu_0 + q) d\\mu_0)} and (- \\mu_0 + q)^{q} \\cos{(\\int \\operatorname{v_{z}}{(\\mu_0,q)} d\\mu_0)} = (- \\mu_0 + q)^{q} \\cos{(\\int (- \\mu_0 + q) d\\mu_0)} and Q (- \\mu_0 + q)^{q} \\cos{(\\int \\operatorname{v_{z}}{(\\mu_0,q)} d\\mu_0)} = Q (- \\mu_0 + q)^{q} \\cos{(\\int (- \\mu_0 + q) d\\mu_0)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mu_0', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('\\\\mu_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('v_z')(Symbol('\\\\mu_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), cos(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["times", 3, "Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Integral(Function('v_z')(Symbol('\\\\mu_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))))"], [["times", 4, "Symbol('Q', commutative=True)"], "Equality(Mul(Symbol('Q', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Integral(Function('v_z')(Symbol('\\\\mu_0', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))), Mul(Symbol('Q', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Integral(Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(C)} = \\log{(C)}, then obtain \\frac{\\eta^{\\prime}^{C}{(C)}}{C + \\log{(C)}} = \\frac{\\log{(C)}^{C}}{C + \\log{(C)}}", "derivation": "\\eta^{\\prime}{(C)} = \\log{(C)} and C + \\eta^{\\prime}{(C)} = C + \\log{(C)} and \\eta^{\\prime}^{C}{(C)} = \\log{(C)}^{C} and \\frac{\\eta^{\\prime}^{C}{(C)}}{C + \\eta^{\\prime}{(C)}} = \\frac{\\log{(C)}^{C}}{C + \\eta^{\\prime}{(C)}} and \\frac{\\eta^{\\prime}^{C}{(C)}}{C + \\log{(C)}} = \\frac{\\log{(C)}^{C}}{C + \\log{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["add", 1, "Symbol('C', commutative=True)"], "Equality(Add(Symbol('C', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True))), Add(Symbol('C', commutative=True), log(Symbol('C', commutative=True))))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["divide", 3, "Add(Symbol('C', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True))), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Pow(Add(Symbol('C', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True))), Integer(-1)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Integer(-1)), Pow(Function('\\\\eta^{\\\\prime}')(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Mul(Pow(Add(Symbol('C', commutative=True), log(Symbol('C', commutative=True))), Integer(-1)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(m)} = \\frac{d}{d m} \\cos{(m)}, then derive 1 = (\\frac{- \\sin{(m)} - 1}{\\operatorname{C_{1}}{(m)} - 1})^{m}, then obtain 1 = (((\\frac{- \\sin{(m)} - 1}{\\frac{d}{d m} \\cos{(m)} - 1})^{m})^{m})^{m}", "derivation": "\\operatorname{C_{1}}{(m)} = \\frac{d}{d m} \\cos{(m)} and \\operatorname{C_{1}}{(m)} - 1 = \\frac{d}{d m} \\cos{(m)} - 1 and 1 = \\frac{\\frac{d}{d m} \\cos{(m)} - 1}{\\operatorname{C_{1}}{(m)} - 1} and 1 = (\\frac{\\frac{d}{d m} \\cos{(m)} - 1}{\\operatorname{C_{1}}{(m)} - 1})^{m} and 1 = (\\frac{- \\sin{(m)} - 1}{\\operatorname{C_{1}}{(m)} - 1})^{m} and 1 = (\\frac{- \\sin{(m)} - 1}{\\frac{d}{d m} \\cos{(m)} - 1})^{m} and 1 = ((\\frac{- \\sin{(m)} - 1}{\\frac{d}{d m} \\cos{(m)} - 1})^{m})^{m} and 1 = (((\\frac{- \\sin{(m)} - 1}{\\frac{d}{d m} \\cos{(m)} - 1})^{m})^{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('m', commutative=True)), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('C_1')(Symbol('m', commutative=True)), Integer(-1)), Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)))"], [["divide", 2, "Add(Function('C_1')(Symbol('m', commutative=True)), Integer(-1))"], "Equality(Integer(1), Mul(Pow(Add(Function('C_1')(Symbol('m', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('C_1')(Symbol('m', commutative=True)), Integer(-1)), Integer(-1)), Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1))), Symbol('m', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('C_1')(Symbol('m', commutative=True)), Integer(-1)), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integer(-1))), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(1), Pow(Mul(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Integer(-1))), Symbol('m', commutative=True)))"], [["power", 6, "Symbol('m', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Integer(-1))), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["power", 7, "Symbol('m', commutative=True)"], "Equality(Integer(1), Pow(Pow(Pow(Mul(Add(Mul(Integer(-1), sin(Symbol('m', commutative=True))), Integer(-1)), Pow(Add(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(-1)), Integer(-1))), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\pi{(\\mathbf{M},v_{1})} = \\mathbf{M} \\cos{(v_{1})}, then obtain ((\\int (\\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}})^{\\mathbf{M}} d\\mathbf{M})^{\\mathbf{M}})^{v_{1}} = ((\\int 1 d\\mathbf{M})^{\\mathbf{M}})^{v_{1}}", "derivation": "\\pi{(\\mathbf{M},v_{1})} = \\mathbf{M} \\cos{(v_{1})} and \\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}} = 1 and (\\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}})^{\\mathbf{M}} = 1 and \\int (\\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}})^{\\mathbf{M}} d\\mathbf{M} = \\int 1 d\\mathbf{M} and (\\int (\\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}})^{\\mathbf{M}} d\\mathbf{M})^{\\mathbf{M}} = (\\int 1 d\\mathbf{M})^{\\mathbf{M}} and ((\\int (\\frac{\\pi{(\\mathbf{M},v_{1})}}{\\mathbf{M} \\cos{(v_{1})}})^{\\mathbf{M}} d\\mathbf{M})^{\\mathbf{M}})^{v_{1}} = ((\\int 1 d\\mathbf{M})^{\\mathbf{M}})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('v_1', commutative=True))))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{M}', commutative=True), cos(Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["power", 5, "Symbol('v_1', commutative=True)"], "Equality(Pow(Pow(Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('v_1', commutative=True)), Pow(cos(Symbol('v_1', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('v_1', commutative=True)), Pow(Pow(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('v_1', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(\\nabla)} = \\log{(\\nabla)} and \\hat{p}{(\\phi)} = e^{\\phi}, then obtain \\frac{\\int (\\mathbf{v}{(\\nabla)} + e^{\\phi} + 1) d\\phi}{\\hat{p}{(\\phi)}} = \\frac{\\int (e^{\\phi} + \\log{(\\nabla)} + 1) d\\phi}{\\hat{p}{(\\phi)}}", "derivation": "\\mathbf{v}{(\\nabla)} = \\log{(\\nabla)} and \\hat{p}{(\\phi)} = e^{\\phi} and \\hat{p}{(\\phi)} + \\mathbf{v}{(\\nabla)} = \\hat{p}{(\\phi)} + \\log{(\\nabla)} and \\hat{p}{(\\phi)} + \\mathbf{v}{(\\nabla)} + 1 = \\hat{p}{(\\phi)} + \\log{(\\nabla)} + 1 and \\int (\\hat{p}{(\\phi)} + \\mathbf{v}{(\\nabla)} + 1) d\\phi = \\int (\\hat{p}{(\\phi)} + \\log{(\\nabla)} + 1) d\\phi and \\int (\\mathbf{v}{(\\nabla)} + e^{\\phi} + 1) d\\phi = \\int (e^{\\phi} + \\log{(\\nabla)} + 1) d\\phi and \\frac{\\int (\\mathbf{v}{(\\nabla)} + e^{\\phi} + 1) d\\phi}{\\hat{p}{(\\phi)}} = \\frac{\\int (e^{\\phi} + \\log{(\\nabla)} + 1) d\\phi}{\\hat{p}{(\\phi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\nabla', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integer(1)), Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\nabla', commutative=True)), Integer(1)))"], [["integrate", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\nabla', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integral(Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\phi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(exp(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\nabla', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["divide", 6, "Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Add(Function('\\\\mathbf{v}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\phi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True)))), Mul(Pow(Function('\\\\hat{p}')(Symbol('\\\\phi', commutative=True)), Integer(-1)), Integral(Add(exp(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\nabla', commutative=True)), Integer(1)), Tuple(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given Q{(\\phi,W)} = \\log{(\\phi)}^{W}, then obtain \\frac{W Q{(\\phi,W)} \\log{(\\phi)}^{- W}}{- \\phi + Q{(\\phi,W)}} + \\operatorname{m_{s}}{(F_{c},M_{E})} = \\frac{W}{- \\phi + Q{(\\phi,W)}} + \\operatorname{m_{s}}{(F_{c},M_{E})}", "derivation": "Q{(\\phi,W)} = \\log{(\\phi)}^{W} and - \\phi + Q{(\\phi,W)} = - \\phi + \\log{(\\phi)}^{W} and W Q{(\\phi,W)} = W \\log{(\\phi)}^{W} and \\frac{W Q{(\\phi,W)} \\log{(\\phi)}^{- W}}{- \\phi + \\log{(\\phi)}^{W}} = \\frac{W}{- \\phi + \\log{(\\phi)}^{W}} and \\frac{W Q{(\\phi,W)} \\log{(\\phi)}^{- W}}{- \\phi + Q{(\\phi,W)}} = \\frac{W}{- \\phi + Q{(\\phi,W)}} and \\frac{W Q{(\\phi,W)} \\log{(\\phi)}^{- W}}{- \\phi + Q{(\\phi,W)}} + \\operatorname{m_{s}}{(F_{c},M_{E})} = \\frac{W}{- \\phi + Q{(\\phi,W)}} + \\operatorname{m_{s}}{(F_{c},M_{E})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True)))"], [["minus", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True))))"], [["times", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Symbol('W', commutative=True), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Mul(Symbol('W', commutative=True), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True))))"], [["divide", 3, "Mul(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True))), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True)))"], "Equality(Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True))), Integer(-1)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)))), Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Symbol('W', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Integer(-1)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)))), Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Integer(-1))))"], [["add", 5, "Function('m_s')(Symbol('F_c', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Integer(-1)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Symbol('W', commutative=True)))), Function('m_s')(Symbol('F_c', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Symbol('W', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Function('Q')(Symbol('\\\\phi', commutative=True), Symbol('W', commutative=True))), Integer(-1))), Function('m_s')(Symbol('F_c', commutative=True), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given c{(\\mathbf{J}_M,\\eta)} = \\frac{\\mathbf{J}_M}{\\eta}, then obtain (- \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(c{(\\mathbf{J}_M,\\eta)})})^{\\eta} = (- \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(\\frac{\\mathbf{J}_M}{\\eta})})^{\\eta}", "derivation": "c{(\\mathbf{J}_M,\\eta)} = \\frac{\\mathbf{J}_M}{\\eta} and \\sin{(c{(\\mathbf{J}_M,\\eta)})} = \\sin{(\\frac{\\mathbf{J}_M}{\\eta})} and \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(c{(\\mathbf{J}_M,\\eta)})} = \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(\\frac{\\mathbf{J}_M}{\\eta})} and - \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(c{(\\mathbf{J}_M,\\eta)})} = - \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(\\frac{\\mathbf{J}_M}{\\eta})} and (- \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(c{(\\mathbf{J}_M,\\eta)})})^{\\eta} = (- \\cos{(S)} \\frac{\\partial}{\\partial \\mathbf{J}_M} \\sin{(\\frac{\\mathbf{J}_M}{\\eta})})^{\\eta}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["sin", 1], "Equality(sin(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\eta', commutative=True))), sin(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Derivative(sin(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))), Derivative(sin(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(-1), cos(Symbol('S', commutative=True)))"], "Equality(Mul(Integer(-1), cos(Symbol('S', commutative=True)), Derivative(sin(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Mul(Integer(-1), cos(Symbol('S', commutative=True)), Derivative(sin(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Mul(Integer(-1), cos(Symbol('S', commutative=True)), Derivative(sin(Function('c')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Symbol('\\\\eta', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('S', commutative=True)), Derivative(sin(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(1)))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{J}_P)} = \\sin{(\\log{(\\mathbf{J}_P)})}, then obtain \\frac{(\\phi_{2}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} - \\sin{(\\log{(\\mathbf{J}_P)})})^{\\mathbf{J}_P}}{\\mathbf{J}_P} = \\frac{(- \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P}}{\\mathbf{J}_P}", "derivation": "\\phi_{2}{(\\mathbf{J}_P)} = \\sin{(\\log{(\\mathbf{J}_P)})} and \\phi_{2}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} = - \\log{(\\mathbf{J}_P)} + \\sin{(\\log{(\\mathbf{J}_P)})} and \\phi_{2}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} - \\sin{(\\log{(\\mathbf{J}_P)})} = - \\log{(\\mathbf{J}_P)} and (\\phi_{2}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} - \\sin{(\\log{(\\mathbf{J}_P)})})^{\\mathbf{J}_P} = (- \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P} and \\frac{(\\phi_{2}{(\\mathbf{J}_P)} - \\log{(\\mathbf{J}_P)} - \\sin{(\\log{(\\mathbf{J}_P)})})^{\\mathbf{J}_P}}{\\mathbf{J}_P} = \\frac{(- \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P}}{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["minus", 1, "log(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), sin(log(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["minus", 2, "sin(log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Add(Function('\\\\phi_2')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\mathbf{J}_P', commutative=True))))), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Function('\\\\phi_2')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\mathbf{J}_P', commutative=True))))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Pow(Add(Function('\\\\phi_2')(Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(-1), sin(log(Symbol('\\\\mathbf{J}_P', commutative=True))))), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), log(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(T)} = \\sin{(T)}, then derive \\frac{d^{2}}{d T^{2}} \\operatorname{V_{\\mathbf{E}}}{(T)} = - \\sin{(T)}, then obtain - \\frac{\\frac{d^{2}}{d T^{2}} \\operatorname{V_{\\mathbf{E}}}{(T)}}{\\operatorname{V_{\\mathbf{E}}}{(T)}} = \\frac{\\sin{(T)}}{\\operatorname{V_{\\mathbf{E}}}{(T)}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(T)} = \\sin{(T)} and \\frac{d}{d T} \\operatorname{V_{\\mathbf{E}}}{(T)} = \\frac{d}{d T} \\sin{(T)} and \\frac{d^{2}}{d T^{2}} \\operatorname{V_{\\mathbf{E}}}{(T)} = \\frac{d^{2}}{d T^{2}} \\sin{(T)} and \\frac{d^{2}}{d T^{2}} \\operatorname{V_{\\mathbf{E}}}{(T)} = - \\sin{(T)} and - \\frac{\\frac{d^{2}}{d T^{2}} \\operatorname{V_{\\mathbf{E}}}{(T)}}{\\operatorname{V_{\\mathbf{E}}}{(T)}} = \\frac{\\sin{(T)}}{\\operatorname{V_{\\mathbf{E}}}{(T)}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('T', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Integer(-1)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2)))), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('T', commutative=True)), Integer(-1)), sin(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(V)} = \\log{(V)}, then obtain \\operatorname{f_{\\mathbf{p}}}^{2}{(V)} + 2 \\operatorname{f_{\\mathbf{p}}}{(V)} \\log{(V)} + 2 \\log{(V)}^{2} = (\\operatorname{f_{\\mathbf{p}}}{(V)} + \\log{(V)})^{2} + \\log{(V)}^{2}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(V)} = \\log{(V)} and \\operatorname{f_{\\mathbf{p}}}{(V)} + \\log{(V)} = 2 \\log{(V)} and (\\operatorname{f_{\\mathbf{p}}}{(V)} + \\log{(V)})^{2} = 4 \\log{(V)}^{2} and (\\operatorname{f_{\\mathbf{p}}}{(V)} + \\log{(V)})^{2} + \\log{(V)}^{2} = 5 \\log{(V)}^{2} and \\operatorname{f_{\\mathbf{p}}}^{2}{(V)} + 2 \\operatorname{f_{\\mathbf{p}}}{(V)} \\log{(V)} + 2 \\log{(V)}^{2} = 5 \\log{(V)}^{2} and \\operatorname{f_{\\mathbf{p}}}^{2}{(V)} + 2 \\operatorname{f_{\\mathbf{p}}}{(V)} \\log{(V)} + 2 \\log{(V)}^{2} = (\\operatorname{f_{\\mathbf{p}}}{(V)} + \\log{(V)})^{2} + \\log{(V)}^{2}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True)))"], [["add", 1, "log(Symbol('V', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Mul(Integer(2), log(Symbol('V', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Integer(2)), Mul(Integer(4), Pow(log(Symbol('V', commutative=True)), Integer(2))))"], [["add", 3, "Pow(log(Symbol('V', commutative=True)), Integer(2))"], "Equality(Add(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Integer(2)), Pow(log(Symbol('V', commutative=True)), Integer(2))), Mul(Integer(5), Pow(log(Symbol('V', commutative=True)), Integer(2))))"], [["expand", 4], "Equality(Add(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), Integer(2)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Mul(Integer(2), Pow(log(Symbol('V', commutative=True)), Integer(2)))), Mul(Integer(5), Pow(log(Symbol('V', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), Integer(2)), Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Mul(Integer(2), Pow(log(Symbol('V', commutative=True)), Integer(2)))), Add(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('V', commutative=True)), log(Symbol('V', commutative=True))), Integer(2)), Pow(log(Symbol('V', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\nabla{(u)} = \\sin{(u)}, then derive \\int \\nabla{(u)} du = C_{1} - \\cos{(u)}, then derive G - \\cos{(u)} = C_{1} - \\cos{(u)}, then obtain G - u - \\cos{(u)} - \\frac{d}{d u} \\sin{(u)} = C_{1} - u - \\cos{(u)} - \\frac{d}{d u} \\sin{(u)}", "derivation": "\\nabla{(u)} = \\sin{(u)} and \\int \\nabla{(u)} du = \\int \\sin{(u)} du and \\int \\nabla{(u)} du = C_{1} - \\cos{(u)} and \\int \\sin{(u)} du = C_{1} - \\cos{(u)} and G - \\cos{(u)} = C_{1} - \\cos{(u)} and G - u - \\cos{(u)} = C_{1} - u - \\cos{(u)} and G - u - \\cos{(u)} - \\frac{d}{d u} \\sin{(u)} = C_{1} - u - \\cos{(u)} - \\frac{d}{d u} \\sin{(u)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\nabla')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"], [["minus", 5, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), cos(Symbol('u', commutative=True)))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), cos(Symbol('u', commutative=True)))))"], [["minus", 6, "Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), cos(Symbol('u', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))), Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), cos(Symbol('u', commutative=True))), Mul(Integer(-1), Derivative(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(x^\\prime)} = \\int e^{x^\\prime} dx^\\prime, then derive \\operatorname{c_{0}}{(x^\\prime)} = \\hbar + e^{x^\\prime}, then obtain (\\frac{\\int e^{x^\\prime} dx^\\prime}{\\operatorname{c_{0}}{(x^\\prime)}})^{x^\\prime} = (\\frac{\\hbar + e^{x^\\prime}}{\\operatorname{c_{0}}{(x^\\prime)}})^{x^\\prime}", "derivation": "\\operatorname{c_{0}}{(x^\\prime)} = \\int e^{x^\\prime} dx^\\prime and \\operatorname{c_{0}}{(x^\\prime)} = \\hbar + e^{x^\\prime} and \\operatorname{c_{0}}{(x^\\prime)} \\log{(W)} = (\\hbar + e^{x^\\prime}) \\log{(W)} and \\log{(W)} \\int e^{x^\\prime} dx^\\prime = (\\hbar + e^{x^\\prime}) \\log{(W)} and \\frac{\\int e^{x^\\prime} dx^\\prime}{\\operatorname{c_{0}}{(x^\\prime)}} = \\frac{\\hbar + e^{x^\\prime}}{\\operatorname{c_{0}}{(x^\\prime)}} and (\\frac{\\int e^{x^\\prime} dx^\\prime}{\\operatorname{c_{0}}{(x^\\prime)}})^{x^\\prime} = (\\frac{\\hbar + e^{x^\\prime}}{\\operatorname{c_{0}}{(x^\\prime)}})^{x^\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))))"], [["times", 2, "log(Symbol('W', commutative=True))"], "Equality(Mul(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('W', commutative=True))), Mul(Add(Symbol('\\\\hbar', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))), log(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(log(Symbol('W', commutative=True)), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Add(Symbol('\\\\hbar', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))), log(Symbol('W', commutative=True))))"], [["divide", 4, "Mul(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('W', commutative=True)))"], "Equality(Mul(Pow(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Add(Symbol('\\\\hbar', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))), Pow(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(Pow(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Integral(exp(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Add(Symbol('\\\\hbar', commutative=True), exp(Symbol('x^\\\\prime', commutative=True))), Pow(Function('c_0')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\lambda{(v_{y},\\Omega)} = \\log{(\\frac{v_{y}}{\\Omega})}, then obtain \\int \\lambda{(v_{y},\\Omega)} \\log{(\\frac{v_{y}}{\\Omega})} dv_{y} - \\frac{v_{y}}{\\Omega} = v_{y} \\log{(\\frac{v_{y}}{\\Omega})}^{2} - 2 v_{y} \\log{(\\frac{v_{y}}{\\Omega})} + 2 v_{y} + z^{*} - \\frac{v_{y}}{\\Omega}", "derivation": "\\lambda{(v_{y},\\Omega)} = \\log{(\\frac{v_{y}}{\\Omega})} and \\lambda{(v_{y},\\Omega)} \\log{(\\frac{v_{y}}{\\Omega})} = \\log{(\\frac{v_{y}}{\\Omega})}^{2} and \\int \\lambda{(v_{y},\\Omega)} \\log{(\\frac{v_{y}}{\\Omega})} dv_{y} = \\int \\log{(\\frac{v_{y}}{\\Omega})}^{2} dv_{y} and \\int \\lambda{(v_{y},\\Omega)} \\log{(\\frac{v_{y}}{\\Omega})} dv_{y} - \\frac{v_{y}}{\\Omega} = \\int \\log{(\\frac{v_{y}}{\\Omega})}^{2} dv_{y} - \\frac{v_{y}}{\\Omega} and \\int \\lambda{(v_{y},\\Omega)} \\log{(\\frac{v_{y}}{\\Omega})} dv_{y} - \\frac{v_{y}}{\\Omega} = v_{y} \\log{(\\frac{v_{y}}{\\Omega})}^{2} - 2 v_{y} \\log{(\\frac{v_{y}}{\\Omega})} + 2 v_{y} + z^{*} - \\frac{v_{y}}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))))"], [["times", 1, "log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))), Pow(log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))), Integral(Pow(log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Integer(2)), Tuple(Symbol('v_y', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))"], "Equality(Add(Integral(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Add(Integral(Pow(log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Integer(2)), Tuple(Symbol('v_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Integral(Mul(Function('\\\\lambda')(Symbol('v_y', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))), Tuple(Symbol('v_y', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Add(Mul(Symbol('v_y', commutative=True), Pow(log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))), Integer(2))), Mul(Integer(-1), Integer(2), Symbol('v_y', commutative=True), log(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))), Mul(Integer(2), Symbol('v_y', commutative=True)), Symbol('z^*', commutative=True), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Symbol('v_y', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(g_{\\varepsilon},\\Omega)} = g_{\\varepsilon} + e^{\\Omega}, then derive \\frac{\\partial}{\\partial \\Omega} \\varepsilon{(g_{\\varepsilon},\\Omega)} = e^{\\Omega}, then obtain \\frac{(\\frac{\\partial}{\\partial \\Omega} (g_{\\varepsilon} + e^{\\Omega}))^{\\Omega}}{g_{\\varepsilon}} = \\frac{(e^{\\Omega})^{\\Omega}}{g_{\\varepsilon}}", "derivation": "\\varepsilon{(g_{\\varepsilon},\\Omega)} = g_{\\varepsilon} + e^{\\Omega} and \\frac{\\partial}{\\partial \\Omega} \\varepsilon{(g_{\\varepsilon},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (g_{\\varepsilon} + e^{\\Omega}) and \\frac{\\partial}{\\partial \\Omega} \\varepsilon{(g_{\\varepsilon},\\Omega)} = e^{\\Omega} and (\\frac{\\partial}{\\partial \\Omega} \\varepsilon{(g_{\\varepsilon},\\Omega)})^{\\Omega} = (e^{\\Omega})^{\\Omega} and (\\frac{\\partial}{\\partial \\Omega} (g_{\\varepsilon} + e^{\\Omega}))^{\\Omega} = (e^{\\Omega})^{\\Omega} and \\frac{(\\frac{\\partial}{\\partial \\Omega} (g_{\\varepsilon} + e^{\\Omega}))^{\\Omega}}{g_{\\varepsilon}} = \\frac{(e^{\\Omega})^{\\Omega}}{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), exp(Symbol('\\\\Omega', commutative=True)))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\varepsilon')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["divide", 5, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Derivative(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(P_{e})} = e^{P_{e}}, then obtain 0 = (- 2 \\operatorname{F_{g}}{(P_{e})} + 2 e^{P_{e}})^{4}", "derivation": "\\operatorname{F_{g}}{(P_{e})} = e^{P_{e}} and 0 = - \\operatorname{F_{g}}{(P_{e})} + e^{P_{e}} and e^{P_{e}} = - \\operatorname{F_{g}}{(P_{e})} + 2 e^{P_{e}} and 0 = - 2 \\operatorname{F_{g}}{(P_{e})} + 2 e^{P_{e}} and 0 = (- 2 \\operatorname{F_{g}}{(P_{e})} + 2 e^{P_{e}})^{2} and 0 = (- 2 \\operatorname{F_{g}}{(P_{e})} + 2 e^{P_{e}})^{4}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["minus", 1, "Function('F_g')(Symbol('P_e', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_g')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), exp(Symbol('P_e', commutative=True)))"], "Equality(exp(Symbol('P_e', commutative=True)), Add(Mul(Integer(-1), Function('F_g')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('F_g')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True)))))"], [["power", 4, 2], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Integer(2), Function('F_g')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True)))), Integer(2)))"], [["power", 5, 2], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Integer(2), Function('F_g')(Symbol('P_e', commutative=True))), Mul(Integer(2), exp(Symbol('P_e', commutative=True)))), Integer(4)))"]]}, {"prompt": "Given k{(B)} = \\cos{(B)} and \\phi_{2}{(B)} = \\frac{d^{2}}{d B^{2}} \\cos{(B)}, then derive \\frac{d^{2}}{d B^{2}} k{(B)} = - \\cos{(B)}, then obtain - \\phi_{2}{(B)} \\cos{(B)} = \\cos^{2}{(B)}", "derivation": "k{(B)} = \\cos{(B)} and \\frac{d}{d B} k{(B)} = \\frac{d}{d B} \\cos{(B)} and \\frac{d^{2}}{d B^{2}} k{(B)} = \\frac{d^{2}}{d B^{2}} \\cos{(B)} and \\frac{d^{2}}{d B^{2}} k{(B)} = - \\cos{(B)} and \\frac{d^{2}}{d B^{2}} \\cos{(B)} = - \\cos{(B)} and \\phi_{2}{(B)} = \\frac{d^{2}}{d B^{2}} \\cos{(B)} and \\phi_{2}{(B)} \\frac{d^{2}}{d B^{2}} \\cos{(B)} = (\\frac{d^{2}}{d B^{2}} \\cos{(B)})^{2} and - \\phi_{2}{(B)} \\cos{(B)} = \\cos^{2}{(B)}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('k')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('B', commutative=True)), Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))))"], [["times", 6, "Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2)))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('B', commutative=True)), Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2)))), Pow(Derivative(cos(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(2))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Integer(-1), Function('\\\\phi_2')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\hat{H}{(g)} = \\cos{(g)}, then obtain 2 \\cos{(g)} \\frac{d}{d g} \\hat{H}{(g)} = 2 \\cos{(g)} \\frac{d}{d g} (- \\hat{H}{(g)} + 2 \\cos{(g)})", "derivation": "\\hat{H}{(g)} = \\cos{(g)} and \\cos{(g)} = - \\hat{H}{(g)} + 2 \\cos{(g)} and \\frac{d}{d g} \\hat{H}{(g)} = \\frac{d}{d g} \\cos{(g)} and \\frac{d}{d g} \\cos{(g)} = \\frac{d}{d g} (- \\hat{H}{(g)} + 2 \\cos{(g)}) and \\frac{d}{d g} \\hat{H}{(g)} = \\frac{d}{d g} (- \\hat{H}{(g)} + 2 \\cos{(g)}) and 2 \\cos{(g)} \\frac{d}{d g} \\hat{H}{(g)} = 2 \\cos{(g)} \\frac{d}{d g} (- \\hat{H}{(g)} + 2 \\cos{(g)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["minus", 1, "Add(Function('\\\\hat{H}')(Symbol('g', commutative=True)), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"], "Equality(cos(Symbol('g', commutative=True)), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('g', commutative=True))), Mul(Integer(2), cos(Symbol('g', commutative=True)))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('g', commutative=True))), Mul(Integer(2), cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('g', commutative=True))), Mul(Integer(2), cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["times", 5, "Mul(Integer(2), cos(Symbol('g', commutative=True)))"], "Equality(Mul(Integer(2), cos(Symbol('g', commutative=True)), Derivative(Function('\\\\hat{H}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('g', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('g', commutative=True))), Mul(Integer(2), cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\phi_2,h)} = \\log{(- \\phi_2 + h)}, then derive - \\frac{\\frac{\\partial}{\\partial h} \\dot{x}{(\\phi_2,h)}}{\\phi_2} = - \\frac{1}{\\phi_2 (- \\phi_2 + h)}, then obtain \\frac{\\frac{\\partial}{\\partial h} \\log{(- \\phi_2 + h)}}{\\phi_2^{2}} = \\frac{1}{\\phi_2^{2} (- \\phi_2 + h)}", "derivation": "\\dot{x}{(\\phi_2,h)} = \\log{(- \\phi_2 + h)} and \\frac{\\partial}{\\partial h} \\dot{x}{(\\phi_2,h)} = \\frac{\\partial}{\\partial h} \\log{(- \\phi_2 + h)} and - \\frac{\\frac{\\partial}{\\partial h} \\dot{x}{(\\phi_2,h)}}{\\phi_2} = - \\frac{\\frac{\\partial}{\\partial h} \\log{(- \\phi_2 + h)}}{\\phi_2} and - \\frac{\\frac{\\partial}{\\partial h} \\dot{x}{(\\phi_2,h)}}{\\phi_2} = - \\frac{1}{\\phi_2 (- \\phi_2 + h)} and - \\frac{\\frac{\\partial}{\\partial h} \\log{(- \\phi_2 + h)}}{\\phi_2} = - \\frac{1}{\\phi_2 (- \\phi_2 + h)} and \\frac{\\frac{\\partial}{\\partial h} \\log{(- \\phi_2 + h)}}{\\phi_2^{2}} = \\frac{1}{\\phi_2^{2} (- \\phi_2 + h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(Function('\\\\dot{x}')(Symbol('\\\\phi_2', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True)), Integer(-1))))"], [["divide", 5, "Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Derivative(log(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Symbol('h', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(W)} = \\log{(W)}, then derive \\frac{d}{d W} \\operatorname{r_{0}}{(W)} = \\frac{1}{W}, then obtain \\frac{(\\frac{d}{d W} \\operatorname{r_{0}}{(W)})^{W}}{\\frac{d}{d W} \\operatorname{r_{0}}{(W)}} = \\frac{(\\frac{1}{W})^{W}}{\\frac{d}{d W} \\operatorname{r_{0}}{(W)}}", "derivation": "\\operatorname{r_{0}}{(W)} = \\log{(W)} and \\frac{d}{d W} \\operatorname{r_{0}}{(W)} = \\frac{d}{d W} \\log{(W)} and \\frac{d}{d W} \\operatorname{r_{0}}{(W)} = \\frac{1}{W} and (\\frac{d}{d W} \\operatorname{r_{0}}{(W)})^{W} = (\\frac{1}{W})^{W} and \\frac{(\\frac{d}{d W} \\operatorname{r_{0}}{(W)})^{W}}{\\frac{d}{d W} \\log{(W)}} = \\frac{(\\frac{1}{W})^{W}}{\\frac{d}{d W} \\log{(W)}} and \\frac{(\\frac{d}{d W} \\operatorname{r_{0}}{(W)})^{W}}{\\frac{d}{d W} \\operatorname{r_{0}}{(W)}} = \\frac{(\\frac{1}{W})^{W}}{\\frac{d}{d W} \\operatorname{r_{0}}{(W)}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Pow(Symbol('W', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('W', commutative=True)))"], [["divide", 4, "Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('W', commutative=True)), Pow(Derivative(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))), Mul(Pow(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('W', commutative=True)), Pow(Derivative(Function('r_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given M{(n)} = e^{n}, then obtain (M{(n)} - 1)^{n} M{(n)} = (e^{n} - 1)^{n} M{(n)}", "derivation": "M{(n)} = e^{n} and M{(n)} - 1 = e^{n} - 1 and (M{(n)} - 1)^{n} = (e^{n} - 1)^{n} and (M{(n)} - 1)^{n} M{(n)} = (e^{n} - 1)^{n} M{(n)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('n', commutative=True)), exp(Symbol('n', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('M')(Symbol('n', commutative=True)), Integer(-1)), Add(exp(Symbol('n', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Function('M')(Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)), Pow(Add(exp(Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)))"], [["times", 3, "Function('M')(Symbol('n', commutative=True))"], "Equality(Mul(Pow(Add(Function('M')(Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)), Function('M')(Symbol('n', commutative=True))), Mul(Pow(Add(exp(Symbol('n', commutative=True)), Integer(-1)), Symbol('n', commutative=True)), Function('M')(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(L,F_{N},\\theta_2)} = \\frac{F_{N} \\theta_2}{L}, then derive 2 \\nabla + 2 \\theta_2 = \\nabla + \\theta_2 + \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2, then obtain \\nabla + \\theta_2 + \\int 1 d\\theta_2 = \\nabla + \\theta_2 + \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2", "derivation": "\\operatorname{v_{2}}{(L,F_{N},\\theta_2)} = \\frac{F_{N} \\theta_2}{L} and 1 = \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} and \\int 1 d\\theta_2 = \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2 and 2 \\int 1 d\\theta_2 = \\int 1 d\\theta_2 + \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2 and 2 \\nabla + 2 \\theta_2 = \\nabla + \\theta_2 + \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2 and 2 \\nabla + 2 \\theta_2 = \\nabla + \\theta_2 + \\int 1 d\\theta_2 and \\nabla + \\theta_2 + \\int 1 d\\theta_2 = \\nabla + \\theta_2 + \\int \\frac{F_{N} \\theta_2}{L \\operatorname{v_{2}}{(L,F_{N},\\theta_2)}} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(1), Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Pow(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Pow(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["add", 3, "Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Pow(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(2), Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True), Integral(Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Pow(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\theta_2', commutative=True)))), Add(Symbol('\\\\nabla', commutative=True), Symbol('\\\\theta_2', commutative=True), Integral(Mul(Symbol('F_N', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True), Pow(Function('v_2')(Symbol('L', commutative=True), Symbol('F_N', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(a)} = \\cos{(\\log{(a)})}, then obtain (\\frac{\\dot{y}{(a)} - \\log{(a)}}{- \\log{(a)} + \\cos{(\\log{(a)})}})^{a} = 1", "derivation": "\\dot{y}{(a)} = \\cos{(\\log{(a)})} and \\dot{y}{(a)} - \\log{(a)} = - \\log{(a)} + \\cos{(\\log{(a)})} and \\frac{\\dot{y}{(a)} - \\log{(a)}}{- \\log{(a)} + \\cos{(\\log{(a)})}} = 1 and (\\frac{\\dot{y}{(a)} - \\log{(a)}}{- \\log{(a)} + \\cos{(\\log{(a)})}})^{a} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True))))"], [["minus", 1, "log(Symbol('a', commutative=True))"], "Equality(Add(Function('\\\\dot{y}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('a', commutative=True))), cos(log(Symbol('a', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), log(Symbol('a', commutative=True))), cos(log(Symbol('a', commutative=True))))"], "Equality(Mul(Add(Function('\\\\dot{y}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Pow(Add(Mul(Integer(-1), log(Symbol('a', commutative=True))), cos(log(Symbol('a', commutative=True)))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('a', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\dot{y}')(Symbol('a', commutative=True)), Mul(Integer(-1), log(Symbol('a', commutative=True)))), Pow(Add(Mul(Integer(-1), log(Symbol('a', commutative=True))), cos(log(Symbol('a', commutative=True)))), Integer(-1))), Symbol('a', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} = \\frac{\\partial}{\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}), then obtain \\frac{\\partial}{\\partial \\hat{X}} \\int \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} d\\hat{X} = \\frac{\\partial}{\\partial \\hat{X}} \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}) d\\hat{X}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} = \\frac{\\partial}{\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}) and \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} = \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}) and \\int \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} d\\hat{X} = \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}) d\\hat{X} and \\frac{\\partial}{\\partial \\hat{X}} \\int \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{V_{\\mathbf{B}}}{(E_{\\lambda},\\hat{X})} d\\hat{X} = \\frac{\\partial}{\\partial \\hat{X}} \\int \\frac{\\partial^{2}}{\\partial E_{\\lambda}\\partial \\hat{X}} (E_{\\lambda} - \\hat{X}) d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})} = \\cos{(\\hat{x}^{\\mathbf{S}})}, then obtain (\\frac{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})}}{\\mathbf{S}})^{\\mathbf{S}} = (\\frac{\\frac{\\partial}{\\partial \\hat{x}} \\cos{(\\hat{x}^{\\mathbf{S}})}}{\\mathbf{S}})^{\\mathbf{S}}", "derivation": "\\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})} = \\cos{(\\hat{x}^{\\mathbf{S}})} and \\frac{\\partial}{\\partial \\hat{x}} \\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})} = \\frac{\\partial}{\\partial \\hat{x}} \\cos{(\\hat{x}^{\\mathbf{S}})} and \\frac{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})}}{\\mathbf{S}} = \\frac{\\frac{\\partial}{\\partial \\hat{x}} \\cos{(\\hat{x}^{\\mathbf{S}})}}{\\mathbf{S}} and (\\frac{\\frac{\\partial}{\\partial \\hat{x}} \\operatorname{A_{1}}{(\\hat{x},\\mathbf{S})}}{\\mathbf{S}})^{\\mathbf{S}} = (\\frac{\\frac{\\partial}{\\partial \\hat{x}} \\cos{(\\hat{x}^{\\mathbf{S}})}}{\\mathbf{S}})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), Derivative(cos(Pow(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given p{(x,g_{\\varepsilon},Q)} = - Q + g_{\\varepsilon} x and \\rho_{b}{(C_{2},\\hbar)} = C_{2}^{\\hbar}, then obtain ((Q + \\rho_{b}{(C_{2},\\hbar)} + p{(x,g_{\\varepsilon},Q)})^{\\hbar})^{Q} = ((g_{\\varepsilon} x + \\rho_{b}{(C_{2},\\hbar)})^{\\hbar})^{Q}", "derivation": "p{(x,g_{\\varepsilon},Q)} = - Q + g_{\\varepsilon} x and Q + p{(x,g_{\\varepsilon},Q)} = g_{\\varepsilon} x and \\rho_{b}{(C_{2},\\hbar)} = C_{2}^{\\hbar} and C_{2}^{\\hbar} + Q + p{(x,g_{\\varepsilon},Q)} = C_{2}^{\\hbar} + g_{\\varepsilon} x and (C_{2}^{\\hbar} + Q + p{(x,g_{\\varepsilon},Q)})^{\\hbar} = (C_{2}^{\\hbar} + g_{\\varepsilon} x)^{\\hbar} and (Q + \\rho_{b}{(C_{2},\\hbar)} + p{(x,g_{\\varepsilon},Q)})^{\\hbar} = (g_{\\varepsilon} x + \\rho_{b}{(C_{2},\\hbar)})^{\\hbar} and ((Q + \\rho_{b}{(C_{2},\\hbar)} + p{(x,g_{\\varepsilon},Q)})^{\\hbar})^{Q} = ((g_{\\varepsilon} x + \\rho_{b}{(C_{2},\\hbar)})^{\\hbar})^{Q}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Add(Symbol('Q', commutative=True), Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["add", 2, "Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('Q', commutative=True), Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Add(Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))))"], [["power", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('Q', commutative=True), Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Pow(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Symbol('Q', commutative=True), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["power", 6, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('Q', commutative=True), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('p')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Add(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Function('\\\\rho_b')(Symbol('C_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given s{(r,F_{H})} = F_{H} - r, then obtain - F_{H} \\iint - s{(r,F_{H})} dF_{H} dF_{H} = - F_{H} \\iint (F_{H} - r - 2 s{(r,F_{H})}) dF_{H} dF_{H}", "derivation": "s{(r,F_{H})} = F_{H} - r and F_{H} - r + s{(r,F_{H})} = 2 F_{H} - 2 r and 2 s{(r,F_{H})} = 2 F_{H} - 2 r and 2 s{(r,F_{H})} = F_{H} - r + s{(r,F_{H})} and 0 = F_{H} - r - s{(r,F_{H})} and - s{(r,F_{H})} = F_{H} - r - 2 s{(r,F_{H})} and \\int - s{(r,F_{H})} dF_{H} = \\int (F_{H} - r - 2 s{(r,F_{H})}) dF_{H} and \\iint - s{(r,F_{H})} dF_{H} dF_{H} = \\iint (F_{H} - r - 2 s{(r,F_{H})}) dF_{H} dF_{H} and - F_{H} \\iint - s{(r,F_{H})} dF_{H} dF_{H} = - F_{H} \\iint (F_{H} - r - 2 s{(r,F_{H})}) dF_{H} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["add", 1, "Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)))"], "Equality(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(2), Symbol('F_H', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Integer(0), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))))"], [["add", 5, "Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))))"], [["integrate", 6, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True))))"], [["integrate", 7, "Symbol('F_H', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["times", 8, "Mul(Integer(-1), Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Mul(Integer(-1), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))), Mul(Integer(-1), Symbol('F_H', commutative=True), Integral(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True)), Mul(Integer(-1), Integer(2), Function('s')(Symbol('r', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(C,M_{E})} = \\frac{C}{M_{E}}, then obtain \\frac{C^{2}}{M_{E}^{2}} = \\frac{C^{3}}{M_{E}^{3} \\operatorname{v_{y}}{(C,M_{E})}}", "derivation": "\\operatorname{v_{y}}{(C,M_{E})} = \\frac{C}{M_{E}} and \\frac{C \\operatorname{v_{y}}{(C,M_{E})}}{M_{E}} = \\frac{C^{2}}{M_{E}^{2}} and \\frac{C^{2} \\operatorname{v_{y}}^{2}{(C,M_{E})}}{M_{E}^{2}} = \\frac{C^{4}}{M_{E}^{4}} and \\frac{C \\operatorname{v_{y}}{(C,M_{E})}}{M_{E}} = \\frac{C^{3}}{M_{E}^{3} \\operatorname{v_{y}}{(C,M_{E})}} and \\frac{C^{2}}{M_{E}^{2}} = \\frac{C^{3}}{M_{E}^{3} \\operatorname{v_{y}}{(C,M_{E})}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1))))"], [["times", 1, "Mul(Symbol('C', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('C', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(2)), Pow(Symbol('M_E', commutative=True), Integer(-2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Pow(Symbol('M_E', commutative=True), Integer(-2)), Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True)), Integer(2))), Mul(Pow(Symbol('C', commutative=True), Integer(4)), Pow(Symbol('M_E', commutative=True), Integer(-4))))"], [["divide", 3, "Mul(Symbol('C', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Mul(Symbol('C', commutative=True), Pow(Symbol('M_E', commutative=True), Integer(-1)), Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(3)), Pow(Symbol('M_E', commutative=True), Integer(-3)), Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(2)), Pow(Symbol('M_E', commutative=True), Integer(-2))), Mul(Pow(Symbol('C', commutative=True), Integer(3)), Pow(Symbol('M_E', commutative=True), Integer(-3)), Pow(Function('v_y')(Symbol('C', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{F})} = \\log{(\\mathbf{F})}, then obtain 2 \\int (- \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})}) d\\mathbf{F} = \\int (- \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})}) d\\mathbf{F} + \\int (- \\mathbf{F} + \\log{(\\mathbf{F})}) d\\mathbf{F}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{F})} = \\log{(\\mathbf{F})} and - \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})} = - \\mathbf{F} + \\log{(\\mathbf{F})} and \\int (- \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})}) d\\mathbf{F} = \\int (- \\mathbf{F} + \\log{(\\mathbf{F})}) d\\mathbf{F} and 2 \\int (- \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})}) d\\mathbf{F} = \\int (- \\mathbf{F} + \\dot{\\mathbf{r}}{(\\mathbf{F})}) d\\mathbf{F} + \\int (- \\mathbf{F} + \\log{(\\mathbf{F})}) d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), log(Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given T{(\\varepsilon_0,\\nabla)} = \\nabla^{\\varepsilon_0}, then obtain \\frac{T{(\\varepsilon_0,\\nabla)}}{\\nabla^{\\varepsilon_0} + 1} = \\frac{\\nabla^{\\varepsilon_0}}{\\nabla^{\\varepsilon_0} + 1}", "derivation": "T{(\\varepsilon_0,\\nabla)} = \\nabla^{\\varepsilon_0} and 1 = \\frac{\\nabla^{\\varepsilon_0}}{T{(\\varepsilon_0,\\nabla)}} and \\nabla^{\\varepsilon_0} + 1 = \\nabla^{\\varepsilon_0} + \\frac{\\nabla^{\\varepsilon_0}}{T{(\\varepsilon_0,\\nabla)}} and \\frac{T{(\\varepsilon_0,\\nabla)}}{\\nabla^{\\varepsilon_0} + \\frac{\\nabla^{\\varepsilon_0}}{T{(\\varepsilon_0,\\nabla)}}} = \\frac{\\nabla^{\\varepsilon_0}}{\\nabla^{\\varepsilon_0} + \\frac{\\nabla^{\\varepsilon_0}}{T{(\\varepsilon_0,\\nabla)}}} and \\frac{T{(\\varepsilon_0,\\nabla)}}{\\nabla^{\\varepsilon_0} + 1} = \\frac{\\nabla^{\\varepsilon_0}}{\\nabla^{\\varepsilon_0} + 1}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["divide", 1, "Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["add", 2, "Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)))))"], [["divide", 1, "Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], "Equality(Mul(Pow(Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)))), Integer(-1)), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1)))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Integer(-1)), Function('T')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Add(Pow(Symbol('\\\\nabla', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(E,v)} = \\int E v dv and \\psi^{*}{(E,v)} = E v, then obtain \\mathbf{g}{(E,v)} - \\psi^{*}{(E,v)} = - \\psi^{*}{(E,v)} + \\int E v dv", "derivation": "\\mathbf{g}{(E,v)} = \\int E v dv and - E v + \\mathbf{g}{(E,v)} = - E v + \\int E v dv and \\psi^{*}{(E,v)} = E v and \\mathbf{g}{(E,v)} - \\psi^{*}{(E,v)} = - \\psi^{*}{(E,v)} + \\int E v dv", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["minus", 1, "Mul(Symbol('E', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('v', commutative=True)), Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True), Symbol('v', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('E', commutative=True), Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('E', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Function('\\\\psi^*')(Symbol('E', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('E', commutative=True), Symbol('v', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(m_{s})} = \\sin{(e^{m_{s}})}, then derive y + \\phi_{1}{(m_{s})} = f^{*} + \\sin{(e^{m_{s}})}, then obtain y + \\phi_{1}{(m_{s})} = f^{*} + \\phi_{1}{(m_{s})}", "derivation": "\\phi_{1}{(m_{s})} = \\sin{(e^{m_{s}})} and \\frac{d}{d m_{s}} \\phi_{1}{(m_{s})} = \\frac{d}{d m_{s}} \\sin{(e^{m_{s}})} and \\int \\frac{d}{d m_{s}} \\phi_{1}{(m_{s})} dm_{s} = \\int \\frac{d}{d m_{s}} \\sin{(e^{m_{s}})} dm_{s} and y + \\phi_{1}{(m_{s})} = f^{*} + \\sin{(e^{m_{s}})} and y + \\phi_{1}{(m_{s})} = f^{*} + \\phi_{1}{(m_{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), sin(exp(Symbol('m_s', commutative=True))))"], [["differentiate", 1, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('m_s', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))), Integral(Derivative(sin(exp(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('m_s', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('y', commutative=True), Function('\\\\phi_1')(Symbol('m_s', commutative=True))), Add(Symbol('f^*', commutative=True), sin(exp(Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('y', commutative=True), Function('\\\\phi_1')(Symbol('m_s', commutative=True))), Add(Symbol('f^*', commutative=True), Function('\\\\phi_1')(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\mathbf{J}_P{(\\mathbf{S})} = \\mathbf{S}, then obtain - \\mathbf{S} + 3 \\operatorname{P_{e}}{(\\mathbf{S})} = - \\mathbf{S} + 2 \\operatorname{P_{e}}{(\\mathbf{S})} + \\cos{(\\mathbf{S})}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and 2 \\operatorname{P_{e}}{(\\mathbf{S})} = \\operatorname{P_{e}}{(\\mathbf{S})} + \\cos{(\\mathbf{S})} and \\mathbf{J}_P{(\\mathbf{S})} = \\mathbf{S} and 3 \\operatorname{P_{e}}{(\\mathbf{S})} = 2 \\operatorname{P_{e}}{(\\mathbf{S})} + \\cos{(\\mathbf{S})} and 3 \\operatorname{P_{e}}{(\\mathbf{S})} - \\mathbf{J}_P{(\\mathbf{S})} = 2 \\operatorname{P_{e}}{(\\mathbf{S})} - \\mathbf{J}_P{(\\mathbf{S})} + \\cos{(\\mathbf{S})} and - \\mathbf{S} + 3 \\operatorname{P_{e}}{(\\mathbf{S})} = - \\mathbf{S} + 2 \\operatorname{P_{e}}{(\\mathbf{S})} + \\cos{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Integer(2), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], [["add", 2, "Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Integer(3), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(2), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["minus", 4, "Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Mul(Integer(3), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(2), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('\\\\mathbf{S}', commutative=True))), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(3), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(2), Function('P_e')(Symbol('\\\\mathbf{S}', commutative=True))), cos(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given G{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain (\\frac{d}{d x^\\prime} \\int 0 dx^\\prime)^{x^\\prime} = (\\frac{d}{d x^\\prime} \\int (- G^{x^\\prime}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)}) dx^\\prime)^{x^\\prime}", "derivation": "G{(x^\\prime)} = \\cos{(x^\\prime)} and G^{x^\\prime}{(x^\\prime)} = \\cos^{x^\\prime}{(x^\\prime)} and 0 = - G^{x^\\prime}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)} and \\int 0 dx^\\prime = \\int (- G^{x^\\prime}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)}) dx^\\prime and \\frac{d}{d x^\\prime} \\int 0 dx^\\prime = \\frac{d}{d x^\\prime} \\int (- G^{x^\\prime}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)}) dx^\\prime and (\\frac{d}{d x^\\prime} \\int 0 dx^\\prime)^{x^\\prime} = (\\frac{d}{d x^\\prime} \\int (- G^{x^\\prime}{(x^\\prime)} + \\cos^{x^\\prime}{(x^\\prime)}) dx^\\prime)^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 2, "Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["integrate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["power", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Derivative(Integral(Integer(0), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)), Pow(Derivative(Integral(Add(Mul(Integer(-1), Pow(Function('G')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Pow(cos(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given \\omega{(F_{H},r_{0})} = - F_{H} + r_{0}, then obtain 1 = - \\frac{F_{H} - r_{0}}{- F_{H} + r_{0}}", "derivation": "\\omega{(F_{H},r_{0})} = - F_{H} + r_{0} and - \\omega{(F_{H},r_{0})} = F_{H} - r_{0} and 1 = - \\frac{F_{H} - r_{0}}{\\omega{(F_{H},r_{0})}} and 1 = - \\frac{F_{H} - r_{0}}{- F_{H} + r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('r_0', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\omega')(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\omega')(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Integer(1), Mul(Integer(-1), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Pow(Function('\\\\omega')(Symbol('F_H', commutative=True), Symbol('r_0', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integer(1), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('r_0', commutative=True)), Integer(-1)), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}}, then derive \\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})} = - \\frac{E_{\\lambda}}{(\\eta^{\\prime})^{2}}, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} e^{- \\frac{E_{\\lambda}}{(\\eta^{\\prime})^{2}}} = \\frac{\\partial}{\\partial E_{\\lambda}} e^{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}}}", "derivation": "\\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}} and e^{\\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})}} = e^{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}}} and \\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})} = - \\frac{E_{\\lambda}}{(\\eta^{\\prime})^{2}} and \\frac{\\partial}{\\partial E_{\\lambda}} e^{\\mathbf{s}{(\\eta^{\\prime},E_{\\lambda})}} = \\frac{\\partial}{\\partial E_{\\lambda}} e^{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}}} and \\frac{\\partial}{\\partial E_{\\lambda}} e^{- \\frac{E_{\\lambda}}{(\\eta^{\\prime})^{2}}} = \\frac{\\partial}{\\partial E_{\\lambda}} e^{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\frac{E_{\\lambda}}{\\eta^{\\prime}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{s}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), exp(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(exp(Function('\\\\mathbf{s}')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(exp(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(exp(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-2)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(exp(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"]]}, {"prompt": "Given h{(\\rho_b)} = \\cos{(\\cos{(\\rho_b)})}, then obtain \\cos{(\\frac{d}{d \\rho_b} h{(\\rho_b)} \\cos{(\\rho_b)} \\cos{(\\cos{(\\rho_b)})})} = \\cos{(\\frac{d}{d \\rho_b} \\cos{(\\rho_b)} \\cos^{2}{(\\cos{(\\rho_b)})})}", "derivation": "h{(\\rho_b)} = \\cos{(\\cos{(\\rho_b)})} and h{(\\rho_b)} \\cos{(\\cos{(\\rho_b)})} = \\cos^{2}{(\\cos{(\\rho_b)})} and h{(\\rho_b)} \\cos{(\\rho_b)} \\cos{(\\cos{(\\rho_b)})} = \\cos{(\\rho_b)} \\cos^{2}{(\\cos{(\\rho_b)})} and \\frac{d}{d \\rho_b} h{(\\rho_b)} \\cos{(\\rho_b)} \\cos{(\\cos{(\\rho_b)})} = \\frac{d}{d \\rho_b} \\cos{(\\rho_b)} \\cos^{2}{(\\cos{(\\rho_b)})} and \\cos{(\\frac{d}{d \\rho_b} h{(\\rho_b)} \\cos{(\\rho_b)} \\cos{(\\cos{(\\rho_b)})})} = \\cos{(\\frac{d}{d \\rho_b} \\cos{(\\rho_b)} \\cos^{2}{(\\cos{(\\rho_b)})})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "cos(cos(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True)))), Pow(cos(cos(Symbol('\\\\rho_b', commutative=True))), Integer(2)))"], [["times", 2, "cos(Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Function('h')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True)))), Mul(cos(Symbol('\\\\rho_b', commutative=True)), Pow(cos(cos(Symbol('\\\\rho_b', commutative=True))), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Function('h')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('\\\\rho_b', commutative=True)), Pow(cos(cos(Symbol('\\\\rho_b', commutative=True))), Integer(2))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["cos", 4], "Equality(cos(Derivative(Mul(Function('h')(Symbol('\\\\rho_b', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True)), cos(cos(Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), cos(Derivative(Mul(cos(Symbol('\\\\rho_b', commutative=True)), Pow(cos(cos(Symbol('\\\\rho_b', commutative=True))), Integer(2))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"]]}, {"prompt": "Given f{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})}, then obtain f^{2}{(\\Psi_{\\lambda})} + f{(\\Psi_{\\lambda})} = f^{2}{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}", "derivation": "f{(\\Psi_{\\lambda})} = \\log{(\\Psi_{\\lambda})} and f^{2}{(\\Psi_{\\lambda})} = f{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} and f{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} + f{(\\Psi_{\\lambda})} = f{(\\Psi_{\\lambda})} \\log{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})} and f^{2}{(\\Psi_{\\lambda})} + f{(\\Psi_{\\lambda})} = f^{2}{(\\Psi_{\\lambda})} + \\log{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["times", 1, "Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Pow(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2)), Mul(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["add", 1, "Mul(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Pow(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2)), Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Pow(Function('f')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(n)} = n \\operatorname{A_{z}}{(n)}, then obtain \\mathbf{v}{(n)} + \\int n \\mathbf{v}{(n)} dn = \\mathbf{v}{(n)} + \\int n^{2} \\operatorname{A_{z}}{(n)} dn", "derivation": "\\mathbf{v}{(n)} = n \\operatorname{A_{z}}{(n)} and n \\mathbf{v}{(n)} = n^{2} \\operatorname{A_{z}}{(n)} and \\int n \\mathbf{v}{(n)} dn = \\int n^{2} \\operatorname{A_{z}}{(n)} dn and \\mathbf{v}{(n)} + \\int n \\mathbf{v}{(n)} dn = \\mathbf{v}{(n)} + \\int n^{2} \\operatorname{A_{z}}{(n)} dn", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Mul(Symbol('n', commutative=True), Function('A_z')(Symbol('n', commutative=True))))"], [["times", 1, "Symbol('n', commutative=True)"], "Equality(Mul(Symbol('n', commutative=True), Function('\\\\mathbf{v}')(Symbol('n', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(2)), Function('A_z')(Symbol('n', commutative=True))))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Mul(Symbol('n', commutative=True), Function('\\\\mathbf{v}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(2)), Function('A_z')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True))))"], [["add", 3, "Function('\\\\mathbf{v}')(Symbol('n', commutative=True))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Integral(Mul(Symbol('n', commutative=True), Function('\\\\mathbf{v}')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))), Add(Function('\\\\mathbf{v}')(Symbol('n', commutative=True)), Integral(Mul(Pow(Symbol('n', commutative=True), Integer(2)), Function('A_z')(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(k,\\theta_2)} = \\sin{(\\frac{k}{\\theta_2})}, then obtain \\frac{\\partial}{\\partial \\theta_2} e^{\\int \\operatorname{v_{2}}{(k,\\theta_2)} dk} = \\frac{\\partial}{\\partial \\theta_2} e^{\\int \\sin{(\\frac{k}{\\theta_2})} dk}", "derivation": "\\operatorname{v_{2}}{(k,\\theta_2)} = \\sin{(\\frac{k}{\\theta_2})} and \\int \\operatorname{v_{2}}{(k,\\theta_2)} dk = \\int \\sin{(\\frac{k}{\\theta_2})} dk and e^{\\int \\operatorname{v_{2}}{(k,\\theta_2)} dk} = e^{\\int \\sin{(\\frac{k}{\\theta_2})} dk} and \\frac{\\partial}{\\partial \\theta_2} e^{\\int \\operatorname{v_{2}}{(k,\\theta_2)} dk} = \\frac{\\partial}{\\partial \\theta_2} e^{\\int \\sin{(\\frac{k}{\\theta_2})} dk}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('v_2')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True)))), exp(Integral(sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(exp(Integral(Function('v_2')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(exp(Integral(sin(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\sigma_p,i)} = \\sigma_p i, then derive i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)} = 2 i, then obtain \\frac{\\partial}{\\partial i} (i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)}) = \\frac{d}{d i} 2 i", "derivation": "\\mathbf{J}{(\\sigma_p,i)} = \\sigma_p i and \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)} = \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p i and i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)} = i + \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p i and i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)} = 2 i and \\frac{\\partial}{\\partial i} (i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)}) = \\frac{\\partial}{\\partial i} (i + \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p i) and i + \\frac{\\partial}{\\partial \\sigma_p} \\sigma_p i = 2 i and \\frac{\\partial}{\\partial i} (i + \\frac{\\partial}{\\partial \\sigma_p} \\mathbf{J}{(\\sigma_p,i)}) = \\frac{d}{d i} 2 i", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["add", 2, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('i', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('i', commutative=True)))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Symbol('i', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('i', commutative=True), Derivative(Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Derivative(Add(Symbol('i', commutative=True), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\sigma_p', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(u)} = \\log{(\\cos{(u)})}, then obtain (\\int e^{((\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u})^{u}} du)^{u} = (\\int e du)^{u}", "derivation": "\\operatorname{v_{2}}{(u)} = \\log{(\\cos{(u)})} and \\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}} = 1 and (\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u} = 1 and ((\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u})^{u} = 1 and e^{((\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u})^{u}} = e and \\int e^{((\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u})^{u}} du = \\int e du and (\\int e^{((\\frac{\\operatorname{v_{2}}{(u)}}{\\log{(\\cos{(u)})}})^{u})^{u}} du)^{u} = (\\int e du)^{u}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('u', commutative=True)), log(cos(Symbol('u', commutative=True))))"], [["divide", 1, "log(cos(Symbol('u', commutative=True)))"], "Equality(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Symbol('u', commutative=True)), Integer(1))"], [["power", 3, "Symbol('u', commutative=True)"], "Equality(Pow(Pow(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Integer(1))"], [["exp", 4], "Equality(exp(Pow(Pow(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Symbol('u', commutative=True)), Symbol('u', commutative=True))), E)"], [["integrate", 5, "Symbol('u', commutative=True)"], "Equality(Integral(exp(Pow(Pow(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Integral(E, Tuple(Symbol('u', commutative=True))))"], [["power", 6, "Symbol('u', commutative=True)"], "Equality(Pow(Integral(exp(Pow(Pow(Mul(Function('v_2')(Symbol('u', commutative=True)), Pow(log(cos(Symbol('u', commutative=True))), Integer(-1))), Symbol('u', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)), Pow(Integral(E, Tuple(Symbol('u', commutative=True))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} = (\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} and \\hat{p}{(\\hat{x}_0,\\mathbf{B},p)} = \\int ((\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} - 1) dp, then obtain \\hat{p}{(\\hat{x}_0,\\mathbf{B},p)} = \\int (\\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} - 1) dp", "derivation": "\\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} = (\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} and \\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} - 1 = (\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} - 1 and \\int (\\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} - 1) dp = \\int ((\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} - 1) dp and \\hat{p}{(\\hat{x}_0,\\mathbf{B},p)} = \\int ((\\frac{\\mathbf{B}}{p})^{\\hat{x}_0} - 1) dp and \\hat{p}{(\\hat{x}_0,\\mathbf{B},p)} = \\int (\\operatorname{v_{2}}{(\\hat{x}_0,\\mathbf{B},p)} - 1) dp", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Add(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)))"], [["integrate", 2, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Function('v_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True))), Integral(Add(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Integral(Add(Pow(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('p', commutative=True), Integer(-1))), Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Integral(Add(Function('v_2')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\mathbf{B}', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given z{(t_{2},a)} = \\frac{t_{2}}{a} and p{(t_{2},a)} = a z^{a}{(t_{2},a)}, then obtain p{(t_{2},a)} = a (\\frac{t_{2}}{a})^{a}", "derivation": "z{(t_{2},a)} = \\frac{t_{2}}{a} and z^{a}{(t_{2},a)} = (\\frac{t_{2}}{a})^{a} and a z^{a}{(t_{2},a)} = a (\\frac{t_{2}}{a})^{a} and p{(t_{2},a)} = a z^{a}{(t_{2},a)} and p{(t_{2},a)} = a (\\frac{t_{2}}{a})^{a}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('z')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('a', commutative=True)))"], [["times", 2, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('z')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('a', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Function('z')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('p')(Symbol('t_2', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), Pow(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)), Symbol('a', commutative=True))))"]]}, {"prompt": "Given t{(A_{1})} = \\log{(\\log{(A_{1})})}, then obtain \\log{(\\int \\cos{(t{(A_{1})})} dA_{1})}^{A_{1}} = \\log{(\\int \\cos{(\\log{(\\log{(A_{1})})})} dA_{1})}^{A_{1}}", "derivation": "t{(A_{1})} = \\log{(\\log{(A_{1})})} and \\cos{(t{(A_{1})})} = \\cos{(\\log{(\\log{(A_{1})})})} and \\int \\cos{(t{(A_{1})})} dA_{1} = \\int \\cos{(\\log{(\\log{(A_{1})})})} dA_{1} and \\log{(\\int \\cos{(t{(A_{1})})} dA_{1})} = \\log{(\\int \\cos{(\\log{(\\log{(A_{1})})})} dA_{1})} and \\log{(\\int \\cos{(t{(A_{1})})} dA_{1})}^{A_{1}} = \\log{(\\int \\cos{(\\log{(\\log{(A_{1})})})} dA_{1})}^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('A_1', commutative=True)), log(log(Symbol('A_1', commutative=True))))"], [["cos", 1], "Equality(cos(Function('t')(Symbol('A_1', commutative=True))), cos(log(log(Symbol('A_1', commutative=True)))))"], [["integrate", 2, "Symbol('A_1', commutative=True)"], "Equality(Integral(cos(Function('t')(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(cos(log(log(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True))))"], [["log", 3], "Equality(log(Integral(cos(Function('t')(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), log(Integral(cos(log(log(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True)))))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(log(Integral(cos(Function('t')(Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Pow(log(Integral(cos(log(log(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(v_{t})} = \\log{(e^{v_{t}})} and \\rho_{f}{(v_{t})} = 2 \\hat{x}{(v_{t})}, then obtain 2 \\hat{x}{(v_{t})} \\operatorname{f^{*}}{(F_{x})} = \\rho_{f}{(v_{t})} \\operatorname{f^{*}}{(F_{x})}", "derivation": "\\hat{x}{(v_{t})} = \\log{(e^{v_{t}})} and 2 \\hat{x}{(v_{t})} = \\hat{x}{(v_{t})} + \\log{(e^{v_{t}})} and 2 \\hat{x}{(v_{t})} \\operatorname{f^{*}}{(F_{x})} = (\\hat{x}{(v_{t})} + \\log{(e^{v_{t}})}) \\operatorname{f^{*}}{(F_{x})} and \\rho_{f}{(v_{t})} = 2 \\hat{x}{(v_{t})} and \\rho_{f}{(v_{t})} \\operatorname{f^{*}}{(F_{x})} = (\\hat{x}{(v_{t})} + \\log{(e^{v_{t}})}) \\operatorname{f^{*}}{(F_{x})} and 2 \\hat{x}{(v_{t})} \\operatorname{f^{*}}{(F_{x})} = \\rho_{f}{(v_{t})} \\operatorname{f^{*}}{(F_{x})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(exp(Symbol('v_t', commutative=True))))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('v_t', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('v_t', commutative=True))), Add(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(exp(Symbol('v_t', commutative=True)))))"], [["times", 2, "Function('f^*')(Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), Function('f^*')(Symbol('F_x', commutative=True))), Mul(Add(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(exp(Symbol('v_t', commutative=True)))), Function('f^*')(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('v_t', commutative=True)), Mul(Integer(2), Function('\\\\hat{x}')(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Function('\\\\rho_f')(Symbol('v_t', commutative=True)), Function('f^*')(Symbol('F_x', commutative=True))), Mul(Add(Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), log(exp(Symbol('v_t', commutative=True)))), Function('f^*')(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('v_t', commutative=True)), Function('f^*')(Symbol('F_x', commutative=True))), Mul(Function('\\\\rho_f')(Symbol('v_t', commutative=True)), Function('f^*')(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(m,a)} = - m + \\cos{(a)} and A{(a)} = \\cos{(a)}, then obtain (- m + A{(a)}) (- m + \\cos{(a)})^{a} = (- m + A{(a)}) (- m + A{(a)})^{a}", "derivation": "\\operatorname{t_{1}}{(m,a)} = - m + \\cos{(a)} and \\operatorname{t_{1}}^{a}{(m,a)} = (- m + \\cos{(a)})^{a} and A{(a)} = \\cos{(a)} and (- m + \\cos{(a)}) \\operatorname{t_{1}}^{a}{(m,a)} = (- m + \\cos{(a)}) (- m + \\cos{(a)})^{a} and (- m + A{(a)}) \\operatorname{t_{1}}^{a}{(m,a)} = (- m + A{(a)}) (- m + A{(a)})^{a} and (- m + A{(a)}) (- m + \\cos{(a)})^{a} = (- m + A{(a)}) (- m + A{(a)})^{a}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('A')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))), Pow(Function('t_1')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))), Symbol('a', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Pow(Function('t_1')(Symbol('m', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), cos(Symbol('a', commutative=True))), Symbol('a', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Function('A')(Symbol('a', commutative=True))), Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(\\theta_2,\\mathbf{H})} = \\frac{\\mathbf{H}}{\\theta_2}, then obtain (1 + \\frac{\\dot{y}{(\\theta_2,\\mathbf{H})}}{\\mathbf{H}} - \\frac{1}{\\mathbf{H}})^{\\mathbf{H}} = (1 + \\frac{1}{\\theta_2} - \\frac{1}{\\mathbf{H}})^{\\mathbf{H}}", "derivation": "\\dot{y}{(\\theta_2,\\mathbf{H})} = \\frac{\\mathbf{H}}{\\theta_2} and \\frac{\\dot{y}{(\\theta_2,\\mathbf{H})}}{\\mathbf{H}} = \\frac{1}{\\theta_2} and \\frac{\\dot{y}{(\\theta_2,\\mathbf{H})}}{\\mathbf{H}} - \\frac{1}{\\mathbf{H}} = \\frac{1}{\\theta_2} - \\frac{1}{\\mathbf{H}} and 1 + \\frac{\\dot{y}{(\\theta_2,\\mathbf{H})}}{\\mathbf{H}} - \\frac{1}{\\mathbf{H}} = 1 + \\frac{1}{\\theta_2} - \\frac{1}{\\mathbf{H}} and (1 + \\frac{\\dot{y}{(\\theta_2,\\mathbf{H})}}{\\mathbf{H}} - \\frac{1}{\\mathbf{H}})^{\\mathbf{H}} = (1 + \\frac{1}{\\theta_2} - \\frac{1}{\\mathbf{H}})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["divide", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))"], [["minus", 2, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))), Add(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))), Add(Integer(1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))))"], [["power", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Add(Integer(1), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Integer(1), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\phi{(\\Psi)} = \\sin{(\\Psi)} and c{(\\Psi)} = \\int (\\Psi + \\phi{(\\Psi)}) d\\Psi, then derive \\cos{(c{(\\Psi)})} = \\cos{(\\frac{\\Psi^{2}}{2} + q - \\cos{(\\Psi)})}, then obtain \\cos{(\\frac{\\Psi^{2}}{2} + q - \\cos{(\\Psi)})} = \\cos{(\\int (\\Psi + \\sin{(\\Psi)}) d\\Psi)}", "derivation": "\\phi{(\\Psi)} = \\sin{(\\Psi)} and \\Psi + \\phi{(\\Psi)} = \\Psi + \\sin{(\\Psi)} and \\int (\\Psi + \\phi{(\\Psi)}) d\\Psi = \\int (\\Psi + \\sin{(\\Psi)}) d\\Psi and c{(\\Psi)} = \\int (\\Psi + \\phi{(\\Psi)}) d\\Psi and c{(\\Psi)} = \\int (\\Psi + \\sin{(\\Psi)}) d\\Psi and \\cos{(c{(\\Psi)})} = \\cos{(\\int (\\Psi + \\sin{(\\Psi)}) d\\Psi)} and \\cos{(c{(\\Psi)})} = \\cos{(\\frac{\\Psi^{2}}{2} + q - \\cos{(\\Psi)})} and \\cos{(\\frac{\\Psi^{2}}{2} + q - \\cos{(\\Psi)})} = \\cos{(\\int (\\Psi + \\sin{(\\Psi)}) d\\Psi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True)), sin(Symbol('\\\\Psi', commutative=True)))"], [["add", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\Psi', commutative=True)), Integral(Add(Symbol('\\\\Psi', commutative=True), Function('\\\\phi')(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('c')(Symbol('\\\\Psi', commutative=True)), Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["cos", 5], "Equality(cos(Function('c')(Symbol('\\\\Psi', commutative=True))), cos(Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(cos(Function('c')(Symbol('\\\\Psi', commutative=True))), cos(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(cos(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Symbol('q', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))))), cos(Integral(Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))"]]}, {"prompt": "Given t{(t_{2})} = \\log{(e^{t_{2}})}, then obtain - t_{2} (e^{t_{2}} + \\sin{(t^{2}{(t_{2})})}) = - t_{2} (e^{t_{2}} + \\sin{(t{(t_{2})} \\log{(e^{t_{2}})})})", "derivation": "t{(t_{2})} = \\log{(e^{t_{2}})} and t^{2}{(t_{2})} = t{(t_{2})} \\log{(e^{t_{2}})} and \\sin{(t^{2}{(t_{2})})} = \\sin{(t{(t_{2})} \\log{(e^{t_{2}})})} and e^{t_{2}} + \\sin{(t^{2}{(t_{2})})} = e^{t_{2}} + \\sin{(t{(t_{2})} \\log{(e^{t_{2}})})} and - t_{2} (e^{t_{2}} + \\sin{(t^{2}{(t_{2})})}) = - t_{2} (e^{t_{2}} + \\sin{(t{(t_{2})} \\log{(e^{t_{2}})})})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True))))"], [["times", 1, "Function('t')(Symbol('t_2', commutative=True))"], "Equality(Pow(Function('t')(Symbol('t_2', commutative=True)), Integer(2)), Mul(Function('t')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True)))))"], [["sin", 2], "Equality(sin(Pow(Function('t')(Symbol('t_2', commutative=True)), Integer(2))), sin(Mul(Function('t')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True))))))"], [["add", 3, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(exp(Symbol('t_2', commutative=True)), sin(Pow(Function('t')(Symbol('t_2', commutative=True)), Integer(2)))), Add(exp(Symbol('t_2', commutative=True)), sin(Mul(Function('t')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True)))))))"], [["times", 4, "Mul(Integer(-1), Symbol('t_2', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('t_2', commutative=True), Add(exp(Symbol('t_2', commutative=True)), sin(Pow(Function('t')(Symbol('t_2', commutative=True)), Integer(2))))), Mul(Integer(-1), Symbol('t_2', commutative=True), Add(exp(Symbol('t_2', commutative=True)), sin(Mul(Function('t')(Symbol('t_2', commutative=True)), log(exp(Symbol('t_2', commutative=True))))))))"]]}, {"prompt": "Given \\theta{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda}, then obtain 0^{E_{\\lambda}} = (\\hat{x} - \\theta{(E_{\\lambda})} + \\sin{(E_{\\lambda})})^{E_{\\lambda}}", "derivation": "\\theta{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and 0 = - \\theta{(E_{\\lambda})} + \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and 0^{E_{\\lambda}} = (- \\theta{(E_{\\lambda})} + \\int \\cos{(E_{\\lambda})} dE_{\\lambda})^{E_{\\lambda}} and 0^{E_{\\lambda}} = (\\hat{x} - \\theta{(E_{\\lambda})} + \\sin{(E_{\\lambda})})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["power", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integer(0), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('\\\\theta')(Symbol('E_{\\\\lambda}', commutative=True))), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\delta{(\\rho_b,L)} = \\log{(L + \\rho_b)}, then obtain (L - \\delta{(\\rho_b,L)} + \\sin{(\\delta{(\\rho_b,L)})}) \\sin{(\\delta{(\\rho_b,L)})} = (L - \\delta{(\\rho_b,L)} + \\sin{(\\delta{(\\rho_b,L)})}) \\sin{(\\log{(L + \\rho_b)})}", "derivation": "\\delta{(\\rho_b,L)} = \\log{(L + \\rho_b)} and \\sin{(\\delta{(\\rho_b,L)})} = \\sin{(\\log{(L + \\rho_b)})} and L - \\delta{(\\rho_b,L)} + \\sin{(\\delta{(\\rho_b,L)})} = L - \\delta{(\\rho_b,L)} + \\sin{(\\log{(L + \\rho_b)})} and (L - \\delta{(\\rho_b,L)} + \\sin{(\\log{(L + \\rho_b)})}) \\sin{(\\delta{(\\rho_b,L)})} = (L - \\delta{(\\rho_b,L)} + \\sin{(\\log{(L + \\rho_b)})}) \\sin{(\\log{(L + \\rho_b)})} and (L - \\delta{(\\rho_b,L)} + \\sin{(\\delta{(\\rho_b,L)})}) \\sin{(\\delta{(\\rho_b,L)})} = (L - \\delta{(\\rho_b,L)} + \\sin{(\\delta{(\\rho_b,L)})}) \\sin{(\\log{(L + \\rho_b)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)), log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["sin", 1], "Equality(sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))"], "Equality(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))), Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))))"], [["times", 2, "Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], "Equality(Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))), sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))), sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))), Mul(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True))), sin(Function('\\\\delta')(Symbol('\\\\rho_b', commutative=True), Symbol('L', commutative=True)))), sin(log(Add(Symbol('L', commutative=True), Symbol('\\\\rho_b', commutative=True))))))"]]}, {"prompt": "Given G{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})}, then derive G{(\\mathbf{v})} = \\frac{1}{\\mathbf{v}}, then obtain \\frac{G^{\\mathbf{v}}{(\\mathbf{v})}}{\\log{(\\mathbf{v})}} = \\frac{(\\frac{1}{\\mathbf{v}})^{\\mathbf{v}}}{\\log{(\\mathbf{v})}}", "derivation": "G{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})} and G{(\\mathbf{v})} = \\frac{1}{\\mathbf{v}} and G^{\\mathbf{v}}{(\\mathbf{v})} = (\\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})})^{\\mathbf{v}} and \\frac{G^{\\mathbf{v}}{(\\mathbf{v})}}{\\log{(\\mathbf{v})}} = \\frac{(\\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})})^{\\mathbf{v}}}{\\log{(\\mathbf{v})}} and \\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})} = \\frac{1}{\\mathbf{v}} and \\frac{G^{\\mathbf{v}}{(\\mathbf{v})}}{\\log{(\\mathbf{v})}} = \\frac{(\\frac{1}{\\mathbf{v}})^{\\mathbf{v}}}{\\log{(\\mathbf{v})}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('G')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"], [["power", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Function('G')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True)))"], [["divide", 3, "log(Symbol('\\\\mathbf{v}', commutative=True))"], "Equality(Mul(Pow(Function('G')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), Pow(Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Function('G')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))), Mul(Pow(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\varphi^{*}{(\\phi)} = \\log{(\\phi)}, then obtain (2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}) \\varphi^{*}{(\\phi)} + \\varphi^{*}{(\\phi)} + 2 \\log{(\\phi)} = (2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}) \\varphi^{*}{(\\phi)} + 2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}", "derivation": "\\varphi^{*}{(\\phi)} = \\log{(\\phi)} and \\varphi^{*}{(\\phi)} + \\log{(\\phi)} = 2 \\log{(\\phi)} and \\varphi^{*}{(\\phi)} + 2 \\log{(\\phi)} = 3 \\log{(\\phi)} and 2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)} = 3 \\log{(\\phi)} and \\varphi^{*}{(\\phi)} + 2 \\log{(\\phi)} = 2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)} and (2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}) \\varphi^{*}{(\\phi)} + \\varphi^{*}{(\\phi)} + 2 \\log{(\\phi)} = (2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}) \\varphi^{*}{(\\phi)} + 2 \\varphi^{*}{(\\phi)} + \\log{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["add", 1, "log(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\phi', commutative=True))))"], [["add", 1, "Mul(Integer(2), log(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\phi', commutative=True)))), Mul(Integer(3), log(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(3), log(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\phi', commutative=True)))), Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))))"], [["add", 5, "Mul(Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Mul(Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True)), Mul(Integer(2), log(Symbol('\\\\phi', commutative=True)))), Add(Mul(Add(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('\\\\phi', commutative=True))), log(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(v_{z},S,\\mathbf{H})} = (\\mathbf{H} + v_{z})^{S} and \\theta_{1}{(x^\\prime,S)} = S + x^\\prime, then obtain S + x^\\prime + (\\operatorname{A_{y}}^{\\mathbf{H}}{(v_{z},S,\\mathbf{H})})^{S} = S + x^\\prime + (((\\mathbf{H} + v_{z})^{S})^{\\mathbf{H}})^{S}", "derivation": "\\operatorname{A_{y}}{(v_{z},S,\\mathbf{H})} = (\\mathbf{H} + v_{z})^{S} and \\operatorname{A_{y}}^{\\mathbf{H}}{(v_{z},S,\\mathbf{H})} = ((\\mathbf{H} + v_{z})^{S})^{\\mathbf{H}} and \\theta_{1}{(x^\\prime,S)} = S + x^\\prime and (\\operatorname{A_{y}}^{\\mathbf{H}}{(v_{z},S,\\mathbf{H})})^{S} = (((\\mathbf{H} + v_{z})^{S})^{\\mathbf{H}})^{S} and (\\operatorname{A_{y}}^{\\mathbf{H}}{(v_{z},S,\\mathbf{H})})^{S} + \\theta_{1}{(x^\\prime,S)} = (((\\mathbf{H} + v_{z})^{S})^{\\mathbf{H}})^{S} + \\theta_{1}{(x^\\prime,S)} and S + x^\\prime + (\\operatorname{A_{y}}^{\\mathbf{H}}{(v_{z},S,\\mathbf{H})})^{S} = S + x^\\prime + (((\\mathbf{H} + v_{z})^{S})^{\\mathbf{H}})^{S}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('v_z', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_z', commutative=True)), Symbol('S', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_z', commutative=True)), Symbol('S', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\theta_1')(Symbol('x^\\\\prime', commutative=True), Symbol('S', commutative=True)), Add(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_z', commutative=True)), Symbol('S', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)))"], [["add", 4, "Function('\\\\theta_1')(Symbol('x^\\\\prime', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Pow(Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)), Function('\\\\theta_1')(Symbol('x^\\\\prime', commutative=True), Symbol('S', commutative=True))), Add(Pow(Pow(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_z', commutative=True)), Symbol('S', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True)), Function('\\\\theta_1')(Symbol('x^\\\\prime', commutative=True), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Pow(Pow(Function('A_y')(Symbol('v_z', commutative=True), Symbol('S', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True))), Add(Symbol('S', commutative=True), Symbol('x^\\\\prime', commutative=True), Pow(Pow(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('v_z', commutative=True)), Symbol('S', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(z)} = e^{z}, then obtain ((z \\mathbf{B}{(z)} e^{- z})^{z})^{z} = (z^{z})^{z}", "derivation": "\\mathbf{B}{(z)} = e^{z} and z \\mathbf{B}{(z)} = z e^{z} and z \\mathbf{B}{(z)} e^{- z} = z and (z \\mathbf{B}{(z)} e^{- z})^{z} = z^{z} and ((z \\mathbf{B}{(z)} e^{- z})^{z})^{z} = (z^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["times", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{B}')(Symbol('z', commutative=True))), Mul(Symbol('z', commutative=True), exp(Symbol('z', commutative=True))))"], [["divide", 2, "exp(Symbol('z', commutative=True))"], "Equality(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), exp(Mul(Integer(-1), Symbol('z', commutative=True)))), Symbol('z', commutative=True))"], [["power", 3, "Symbol('z', commutative=True)"], "Equality(Pow(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), exp(Mul(Integer(-1), Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('z', commutative=True), Function('\\\\mathbf{B}')(Symbol('z', commutative=True)), exp(Mul(Integer(-1), Symbol('z', commutative=True)))), Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(Pow(Symbol('z', commutative=True), Symbol('z', commutative=True)), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = (\\mathbf{J}_P a^{\\dagger})^{A_{1}}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = \\frac{A_{1} (\\mathbf{J}_P a^{\\dagger})^{A_{1}}}{a^{\\dagger}}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = \\frac{A_{1} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)}}{a^{\\dagger}}", "derivation": "\\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = (\\mathbf{J}_P a^{\\dagger})^{A_{1}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = \\frac{\\partial}{\\partial a^{\\dagger}} (\\mathbf{J}_P a^{\\dagger})^{A_{1}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = \\frac{A_{1} (\\mathbf{J}_P a^{\\dagger})^{A_{1}}}{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)} = \\frac{A_{1} \\operatorname{f^{\\prime}}{(a^{\\dagger},A_{1},\\mathbf{J}_P)}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('A_1', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('A_1', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Symbol('A_1', commutative=True), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('f^{\\\\prime}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}}, then derive \\mathbf{v}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then derive \\int \\mathbf{v}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\sigma_p + e^{f_{\\mathbf{v}}}, then obtain \\int \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{v}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\sigma_p + e^{f_{\\mathbf{v}}}", "derivation": "\\mathbf{v}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} and \\mathbf{v}{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and \\mathbf{v}{(f_{\\mathbf{v}})} = \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{v}{(f_{\\mathbf{v}})} and \\int \\mathbf{v}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int \\frac{d}{d f_{\\mathbf{v}}} e^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and \\int \\mathbf{v}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\sigma_p + e^{f_{\\mathbf{v}}} and \\int \\frac{d}{d f_{\\mathbf{v}}} \\mathbf{v}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\sigma_p + e^{f_{\\mathbf{v}}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Derivative(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Derivative(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Derivative(Function('\\\\mathbf{v}')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(C,A_{y})} = \\frac{A_{y}}{C}, then obtain (\\frac{\\partial}{\\partial C} (\\Psi_{nl}{(C,A_{y})} - \\frac{1}{C}))^{C} = (\\frac{\\partial}{\\partial C} (\\frac{A_{y}}{C} - \\frac{1}{C}))^{C}", "derivation": "\\Psi_{nl}{(C,A_{y})} = \\frac{A_{y}}{C} and \\Psi_{nl}{(C,A_{y})} - \\frac{1}{C} = \\frac{A_{y}}{C} - \\frac{1}{C} and \\frac{\\partial}{\\partial C} (\\Psi_{nl}{(C,A_{y})} - \\frac{1}{C}) = \\frac{\\partial}{\\partial C} (\\frac{A_{y}}{C} - \\frac{1}{C}) and (\\frac{\\partial}{\\partial C} (\\Psi_{nl}{(C,A_{y})} - \\frac{1}{C}))^{C} = (\\frac{\\partial}{\\partial C} (\\frac{A_{y}}{C} - \\frac{1}{C}))^{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('A_y', commutative=True)), Mul(Symbol('A_y', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1))))"], [["minus", 1, "Pow(Symbol('C', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('C', commutative=True), Symbol('A_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)), Pow(Derivative(Add(Mul(Symbol('A_y', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)))), Tuple(Symbol('C', commutative=True), Integer(1))), Symbol('C', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H}{g^{\\prime}_{\\varepsilon}} and m{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H \\varepsilon_{0}^{H}{(g^{\\prime}_{\\varepsilon},H)}}{g^{\\prime}_{\\varepsilon}}, then obtain m{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H (\\frac{H}{g^{\\prime}_{\\varepsilon}})^{H}}{g^{\\prime}_{\\varepsilon}}", "derivation": "\\varepsilon_{0}{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H}{g^{\\prime}_{\\varepsilon}} and \\varepsilon_{0}^{H}{(g^{\\prime}_{\\varepsilon},H)} = (\\frac{H}{g^{\\prime}_{\\varepsilon}})^{H} and \\frac{H \\varepsilon_{0}^{H}{(g^{\\prime}_{\\varepsilon},H)}}{g^{\\prime}_{\\varepsilon}} = \\frac{H (\\frac{H}{g^{\\prime}_{\\varepsilon}})^{H}}{g^{\\prime}_{\\varepsilon}} and m{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H \\varepsilon_{0}^{H}{(g^{\\prime}_{\\varepsilon},H)}}{g^{\\prime}_{\\varepsilon}} and m{(g^{\\prime}_{\\varepsilon},H)} = \\frac{H (\\frac{H}{g^{\\prime}_{\\varepsilon}})^{H}}{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))), Symbol('H', commutative=True)))"], [["times", 2, "Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))), Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))), Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\varepsilon_0')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('m')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Mul(Symbol('H', commutative=True), Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1))), Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(S,E_{n})} = E_{n}^{S}, then obtain (\\frac{\\partial}{\\partial S} \\mathbf{S}{(S,E_{n})} - 1)^{S} = (E_{n}^{S} \\log{(E_{n})} - 1)^{S}", "derivation": "\\mathbf{S}{(S,E_{n})} = E_{n}^{S} and - E_{n} + \\mathbf{S}{(S,E_{n})} = - E_{n} + E_{n}^{S} and \\frac{\\partial}{\\partial S} (- E_{n} + \\mathbf{S}{(S,E_{n})}) = \\frac{\\partial}{\\partial S} (- E_{n} + E_{n}^{S}) and \\frac{\\partial}{\\partial S} (- E_{n} + \\mathbf{S}{(S,E_{n})}) - 1 = \\frac{\\partial}{\\partial S} (- E_{n} + E_{n}^{S}) - 1 and (\\frac{\\partial}{\\partial S} (- E_{n} + \\mathbf{S}{(S,E_{n})}) - 1)^{S} = (\\frac{\\partial}{\\partial S} (- E_{n} + E_{n}^{S}) - 1)^{S} and (\\frac{\\partial}{\\partial S} \\mathbf{S}{(S,E_{n})} - 1)^{S} = (E_{n}^{S} \\log{(E_{n})} - 1)^{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True)))"], [["minus", 1, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)))"], [["power", 4, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)), Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Derivative(Function('\\\\mathbf{S}')(Symbol('S', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(-1)), Symbol('S', commutative=True)), Pow(Add(Mul(Pow(Symbol('E_n', commutative=True), Symbol('S', commutative=True)), log(Symbol('E_n', commutative=True))), Integer(-1)), Symbol('S', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(G)} = \\cos{(G)}, then derive (\\frac{d}{d G} \\operatorname{v_{y}}{(G)})^{2} = - \\sin{(G)} \\frac{d}{d G} \\operatorname{v_{y}}{(G)}, then obtain (\\frac{d}{d G} \\cos{(G)})^{2} = - \\sin{(G)} \\frac{d}{d G} \\cos{(G)}", "derivation": "\\operatorname{v_{y}}{(G)} = \\cos{(G)} and \\frac{d}{d G} \\operatorname{v_{y}}{(G)} = \\frac{d}{d G} \\cos{(G)} and (\\frac{d}{d G} \\operatorname{v_{y}}{(G)})^{2} = \\frac{d}{d G} \\operatorname{v_{y}}{(G)} \\frac{d}{d G} \\cos{(G)} and (\\frac{d}{d G} \\operatorname{v_{y}}{(G)})^{2} = - \\sin{(G)} \\frac{d}{d G} \\operatorname{v_{y}}{(G)} and (\\frac{d}{d G} \\cos{(G)})^{2} = - \\sin{(G)} \\frac{d}{d G} \\cos{(G)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Symbol('G', commutative=True)), Derivative(Function('v_y')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Symbol('G', commutative=True)), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{P}{(\\varphi,\\lambda)} = \\lambda \\varphi and \\mathbf{D}{(\\varphi,\\lambda)} = \\lambda \\varphi, then obtain \\log{(\\int \\mathbf{D}{(\\varphi,\\lambda)} d\\varphi)} - \\int \\lambda \\varphi d\\varphi = \\log{(\\int \\lambda \\varphi d\\varphi)} - \\int \\lambda \\varphi d\\varphi", "derivation": "\\mathbf{P}{(\\varphi,\\lambda)} = \\lambda \\varphi and \\int \\mathbf{P}{(\\varphi,\\lambda)} d\\varphi = \\int \\lambda \\varphi d\\varphi and \\log{(\\int \\mathbf{P}{(\\varphi,\\lambda)} d\\varphi)} = \\log{(\\int \\lambda \\varphi d\\varphi)} and \\mathbf{D}{(\\varphi,\\lambda)} = \\lambda \\varphi and \\mathbf{P}{(\\varphi,\\lambda)} = \\mathbf{D}{(\\varphi,\\lambda)} and \\log{(\\int \\mathbf{P}{(\\varphi,\\lambda)} d\\varphi)} - \\int \\lambda \\varphi d\\varphi = \\log{(\\int \\lambda \\varphi d\\varphi)} - \\int \\lambda \\varphi d\\varphi and \\log{(\\int \\mathbf{D}{(\\varphi,\\lambda)} d\\varphi)} - \\int \\lambda \\varphi d\\varphi = \\log{(\\int \\lambda \\varphi d\\varphi)} - \\int \\lambda \\varphi d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), log(Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["minus", 3, "Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(log(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))), Add(log(Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(log(Integral(Function('\\\\mathbf{D}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))), Add(log(Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))))"]]}, {"prompt": "Given \\phi_{1}{(\\Psi^{\\dagger},z)} = \\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z, then obtain (\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\phi_{1}{(\\Psi^{\\dagger},z)})}))^{z} = (\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z)}))^{z}", "derivation": "\\phi_{1}{(\\Psi^{\\dagger},z)} = \\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z and \\log{(\\phi_{1}{(\\Psi^{\\dagger},z)})} = \\log{(\\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z)} and - \\Psi^{\\dagger} z + \\log{(\\phi_{1}{(\\Psi^{\\dagger},z)})} = - \\Psi^{\\dagger} z + \\log{(\\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z)} and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\phi_{1}{(\\Psi^{\\dagger},z)})}) = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z)}) and (\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\phi_{1}{(\\Psi^{\\dagger},z)})}))^{z} = (\\frac{\\partial}{\\partial \\Psi^{\\dagger}} (- \\Psi^{\\dagger} z + \\log{(\\frac{\\partial}{\\partial z} \\Psi^{\\dagger} z)}))^{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('\\\\phi_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True))), log(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["minus", 2, "Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Function('\\\\phi_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))))"], [["differentiate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Function('\\\\phi_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Function('\\\\phi_1')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Symbol('z', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), log(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(c_{0})} = e^{c_{0}} and \\operatorname{y^{\\prime}}{(c_{0})} = c_{0}, then obtain (\\sigma_{x}{(c_{0})} - e^{c_{0}} + 1) \\operatorname{y^{\\prime}}{(c_{0})} + e^{c_{0}} = c_{0} + e^{c_{0}}", "derivation": "\\sigma_{x}{(c_{0})} = e^{c_{0}} and \\sigma_{x}{(c_{0})} - e^{c_{0}} = 0 and \\operatorname{y^{\\prime}}{(c_{0})} = c_{0} and \\sigma_{x}{(c_{0})} - e^{c_{0}} + 1 = 1 and (\\sigma_{x}{(c_{0})} - e^{c_{0}} + 1) \\operatorname{y^{\\prime}}{(c_{0})} = \\operatorname{y^{\\prime}}{(c_{0})} and \\operatorname{y^{\\prime}}{(c_{0})} + e^{c_{0}} = c_{0} + e^{c_{0}} and (\\sigma_{x}{(c_{0})} - e^{c_{0}} + 1) \\operatorname{y^{\\prime}}{(c_{0})} + e^{c_{0}} = c_{0} + e^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], [["minus", 1, "exp(Symbol('c_0', commutative=True))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))), Integer(0))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), Integer(1)), Integer(1))"], [["times", 4, "Function('y^{\\\\prime}')(Symbol('c_0', commutative=True))"], "Equality(Mul(Add(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), Integer(1)), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True))), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), exp(Symbol('c_0', commutative=True)))"], "Equality(Add(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Add(Function('\\\\sigma_x')(Symbol('c_0', commutative=True)), Mul(Integer(-1), exp(Symbol('c_0', commutative=True))), Integer(1)), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True))), exp(Symbol('c_0', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\mathbf{v})} = \\log{(\\cos{(\\mathbf{v})})}, then obtain \\chi^{2}{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{2} + \\chi{(\\mathbf{v})} = \\chi^{2}{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{2} + \\log{(\\cos{(\\mathbf{v})})}", "derivation": "\\chi{(\\mathbf{v})} = \\log{(\\cos{(\\mathbf{v})})} and \\chi{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})} = \\log{(\\cos{(\\mathbf{v})})}^{2} and \\chi^{2}{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{2} = \\chi{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{3} and \\chi{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{3} + \\chi{(\\mathbf{v})} = \\chi{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{3} + \\log{(\\cos{(\\mathbf{v})})} and \\chi^{2}{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{2} + \\chi{(\\mathbf{v})} = \\chi^{2}{(\\mathbf{v})} \\log{(\\cos{(\\mathbf{v})})}^{2} + \\log{(\\cos{(\\mathbf{v})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), log(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 1, "log(cos(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), log(cos(Symbol('\\\\mathbf{v}', commutative=True)))), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), log(cos(Symbol('\\\\mathbf{v}', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(2))), Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(3))))"], [["add", 1, "Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(3)))"], "Equality(Add(Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(3))), Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(3))), log(cos(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(2))), Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True))), Add(Mul(Pow(Function('\\\\chi')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)), Pow(log(cos(Symbol('\\\\mathbf{v}', commutative=True))), Integer(2))), log(cos(Symbol('\\\\mathbf{v}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(A,L)} = A L and \\operatorname{v_{x}}{(A,L)} = \\frac{\\partial}{\\partial A} \\operatorname{E_{x}}^{L}{(A,L)}, then obtain \\operatorname{v_{x}}{(A,L)} = \\frac{\\partial}{\\partial A} (A L)^{L}", "derivation": "\\operatorname{E_{x}}{(A,L)} = A L and \\operatorname{E_{x}}^{L}{(A,L)} = (A L)^{L} and \\frac{\\partial}{\\partial A} \\operatorname{E_{x}}^{L}{(A,L)} = \\frac{\\partial}{\\partial A} (A L)^{L} and \\operatorname{v_{x}}{(A,L)} = \\frac{\\partial}{\\partial A} \\operatorname{E_{x}}^{L}{(A,L)} and \\operatorname{v_{x}}{(A,L)} = \\frac{\\partial}{\\partial A} (A L)^{L}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Pow(Function('E_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Derivative(Pow(Function('E_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('v_x')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Derivative(Pow(Mul(Symbol('A', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\int (-1)^{J_{\\varepsilon}} dJ_{\\varepsilon} = \\int (- \\frac{e^{J_{\\varepsilon}}}{\\pi{(J_{\\varepsilon})}})^{J_{\\varepsilon}} dJ_{\\varepsilon}", "derivation": "\\pi{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and 1 = \\frac{e^{J_{\\varepsilon}}}{\\pi{(J_{\\varepsilon})}} and -1 = - \\frac{e^{J_{\\varepsilon}}}{\\pi{(J_{\\varepsilon})}} and (-1)^{J_{\\varepsilon}} = (- \\frac{e^{J_{\\varepsilon}}}{\\pi{(J_{\\varepsilon})}})^{J_{\\varepsilon}} and \\int (-1)^{J_{\\varepsilon}} dJ_{\\varepsilon} = \\int (- \\frac{e^{J_{\\varepsilon}}}{\\pi{(J_{\\varepsilon})}})^{J_{\\varepsilon}} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(k)} = \\sin{(\\sin{(k)})}, then obtain k + \\mathbf{H}{(k)} + \\int (k + \\mathbf{H}{(k)}) dk = k + \\mathbf{H}{(k)} + \\int (k + \\sin{(\\sin{(k)})}) dk", "derivation": "\\mathbf{H}{(k)} = \\sin{(\\sin{(k)})} and k + \\mathbf{H}{(k)} = k + \\sin{(\\sin{(k)})} and \\int (k + \\mathbf{H}{(k)}) dk = \\int (k + \\sin{(\\sin{(k)})}) dk and k + \\mathbf{H}{(k)} + \\int (k + \\mathbf{H}{(k)}) dk = k + \\mathbf{H}{(k)} + \\int (k + \\sin{(\\sin{(k)})}) dk", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('k', commutative=True)), sin(sin(Symbol('k', commutative=True))))"], [["add", 1, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True))), Add(Symbol('k', commutative=True), sin(sin(Symbol('k', commutative=True)))))"], [["integrate", 2, "Symbol('k', commutative=True)"], "Equality(Integral(Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Symbol('k', commutative=True), sin(sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True))))"], [["add", 3, "Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True)))"], "Equality(Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True)), Integral(Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Add(Symbol('k', commutative=True), Function('\\\\mathbf{H}')(Symbol('k', commutative=True)), Integral(Add(Symbol('k', commutative=True), sin(sin(Symbol('k', commutative=True)))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given \\theta{(\\varphi^*,h)} = e^{\\varphi^* + h} and \\bar{\\h}{(F_{N},\\mathbf{P},\\varphi^*,h)} = e^{- F_{N} + \\mathbf{P}} - \\frac{2 e^{\\varphi^* + h}}{\\varphi^* + h}, then obtain \\cos{(\\bar{\\h}{(F_{N},\\mathbf{P},\\varphi^*,h)})} = \\cos{(e^{- F_{N} + \\mathbf{P}} - \\frac{\\theta{(\\varphi^*,h)} + e^{\\varphi^* + h}}{\\varphi^* + h})}", "derivation": "\\theta{(\\varphi^*,h)} = e^{\\varphi^* + h} and \\theta{(\\varphi^*,h)} + e^{\\varphi^* + h} = 2 e^{\\varphi^* + h} and \\frac{\\theta{(\\varphi^*,h)} + e^{\\varphi^* + h}}{\\varphi^* + h} = \\frac{2 e^{\\varphi^* + h}}{\\varphi^* + h} and \\bar{\\h}{(F_{N},\\mathbf{P},\\varphi^*,h)} = e^{- F_{N} + \\mathbf{P}} - \\frac{2 e^{\\varphi^* + h}}{\\varphi^* + h} and \\bar{\\h}{(F_{N},\\mathbf{P},\\varphi^*,h)} = e^{- F_{N} + \\mathbf{P}} - \\frac{\\theta{(\\varphi^*,h)} + e^{\\varphi^* + h}}{\\varphi^* + h} and \\cos{(\\bar{\\h}{(F_{N},\\mathbf{P},\\varphi^*,h)})} = \\cos{(e^{- F_{N} + \\mathbf{P}} - \\frac{\\theta{(\\varphi^*,h)} + e^{\\varphi^* + h}}{\\varphi^* + h})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)))"], "Equality(Add(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)))), Mul(Integer(2), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)))))"], [["divide", 2, "Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Add(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))))), Mul(Integer(2), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Add(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Integer(-1)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\hbar')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Add(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Add(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)))))))"], [["cos", 5], "Equality(cos(Function('\\\\hbar')(Symbol('F_N', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))), cos(Add(exp(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), Integer(-1)), Add(Function('\\\\theta')(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True)), exp(Add(Symbol('\\\\varphi^*', commutative=True), Symbol('h', commutative=True))))))))"]]}, {"prompt": "Given Z{(\\hat{H})} = e^{\\hat{H}}, then derive \\frac{d}{d \\hat{H}} Z{(\\hat{H})} = e^{\\hat{H}}, then obtain \\frac{d}{d \\hat{H}} \\frac{e^{\\hat{H}}}{Z{(\\hat{H})}} = \\frac{d}{d \\hat{H}} \\frac{\\frac{d}{d \\hat{H}} e^{\\hat{H}}}{Z{(\\hat{H})}}", "derivation": "Z{(\\hat{H})} = e^{\\hat{H}} and \\frac{d}{d \\hat{H}} Z{(\\hat{H})} = \\frac{d}{d \\hat{H}} e^{\\hat{H}} and \\frac{d}{d \\hat{H}} Z{(\\hat{H})} = e^{\\hat{H}} and \\frac{\\frac{d}{d \\hat{H}} Z{(\\hat{H})}}{Z{(\\hat{H})}} = \\frac{\\frac{d}{d \\hat{H}} e^{\\hat{H}}}{Z{(\\hat{H})}} and \\frac{d}{d \\hat{H}} \\frac{\\frac{d}{d \\hat{H}} Z{(\\hat{H})}}{Z{(\\hat{H})}} = \\frac{d}{d \\hat{H}} \\frac{\\frac{d}{d \\hat{H}} e^{\\hat{H}}}{Z{(\\hat{H})}} and \\frac{d}{d \\hat{H}} \\frac{e^{\\hat{H}}}{Z{(\\hat{H})}} = \\frac{d}{d \\hat{H}} \\frac{\\frac{d}{d \\hat{H}} e^{\\hat{H}}}{Z{(\\hat{H})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 2, "Function('Z')(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), exp(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Z')(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(exp(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given t{(J_{\\varepsilon},y)} = \\cos{(\\frac{y}{J_{\\varepsilon}})}, then obtain (\\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} t{(J_{\\varepsilon},y)})^{J_{\\varepsilon}} = (\\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} \\cos{(\\frac{y}{J_{\\varepsilon}})})^{J_{\\varepsilon}}", "derivation": "t{(J_{\\varepsilon},y)} = \\cos{(\\frac{y}{J_{\\varepsilon}})} and \\frac{\\partial}{\\partial y} t{(J_{\\varepsilon},y)} = \\frac{\\partial}{\\partial y} \\cos{(\\frac{y}{J_{\\varepsilon}})} and \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} t{(J_{\\varepsilon},y)} = \\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} \\cos{(\\frac{y}{J_{\\varepsilon}})} and (\\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} t{(J_{\\varepsilon},y)})^{J_{\\varepsilon}} = (\\frac{\\partial^{2}}{\\partial J_{\\varepsilon}\\partial y} \\cos{(\\frac{y}{J_{\\varepsilon}})})^{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Derivative(Function('t')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Derivative(cos(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_f{(r)} = \\frac{d}{d r} \\cos{(r)}, then derive \\int \\mathbf{J}_f{(r)} dr = \\omega + \\cos{(r)}, then derive \\omega + \\cos{(r)} = t_{1} + \\cos{(r)}, then derive v_{x} + \\cos{(r)} = t_{1} + \\cos{(r)}, then obtain \\omega + \\cos{(r)} = v_{x} + \\cos{(r)}", "derivation": "\\mathbf{J}_f{(r)} = \\frac{d}{d r} \\cos{(r)} and \\int \\mathbf{J}_f{(r)} dr = \\int \\frac{d}{d r} \\cos{(r)} dr and \\int \\mathbf{J}_f{(r)} dr = \\omega + \\cos{(r)} and \\omega + \\cos{(r)} = \\int \\frac{d}{d r} \\cos{(r)} dr and \\omega + \\cos{(r)} = t_{1} + \\cos{(r)} and \\int \\frac{d}{d r} \\cos{(r)} dr = t_{1} + \\cos{(r)} and v_{x} + \\cos{(r)} = t_{1} + \\cos{(r)} and \\omega + \\cos{(r)} = v_{x} + \\cos{(r)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('r', commutative=True)), Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\omega', commutative=True), cos(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('r', commutative=True))), Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('r', commutative=True))), Add(Symbol('t_1', commutative=True), cos(Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Derivative(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Tuple(Symbol('r', commutative=True))), Add(Symbol('t_1', commutative=True), cos(Symbol('r', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('v_x', commutative=True), cos(Symbol('r', commutative=True))), Add(Symbol('t_1', commutative=True), cos(Symbol('r', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Symbol('\\\\omega', commutative=True), cos(Symbol('r', commutative=True))), Add(Symbol('v_x', commutative=True), cos(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(V,v_{y})} = V + v_{y}, then obtain (\\int \\frac{\\operatorname{C_{d}}{(V,v_{y})}}{V + v_{y}} dv_{y})^{v_{y}} = (\\int 1 dv_{y})^{v_{y}}", "derivation": "\\operatorname{C_{d}}{(V,v_{y})} = V + v_{y} and \\frac{\\operatorname{C_{d}}{(V,v_{y})}}{V + v_{y}} = 1 and \\int \\frac{\\operatorname{C_{d}}{(V,v_{y})}}{V + v_{y}} dv_{y} = \\int 1 dv_{y} and (\\int \\frac{\\operatorname{C_{d}}{(V,v_{y})}}{V + v_{y}} dv_{y})^{v_{y}} = (\\int 1 dv_{y})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)))"], [["divide", 1, "Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('C_d')(Symbol('V', commutative=True), Symbol('v_y', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('C_d')(Symbol('V', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Integral(Integer(1), Tuple(Symbol('v_y', commutative=True))))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Add(Symbol('V', commutative=True), Symbol('v_y', commutative=True)), Integer(-1)), Function('C_d')(Symbol('V', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(z,v_{z})} = \\sin{(\\frac{v_{z}}{z})}, then obtain (\\frac{v_{z}}{z} + \\frac{\\operatorname{C_{1}}{(z,v_{z})}}{z})^{v_{z}} = (\\frac{v_{z}}{z} + \\frac{\\sin{(\\frac{v_{z}}{z})}}{z})^{v_{z}}", "derivation": "\\operatorname{C_{1}}{(z,v_{z})} = \\sin{(\\frac{v_{z}}{z})} and \\frac{\\operatorname{C_{1}}{(z,v_{z})}}{z} = \\frac{\\sin{(\\frac{v_{z}}{z})}}{z} and \\frac{v_{z}}{z} + \\frac{\\operatorname{C_{1}}{(z,v_{z})}}{z} = \\frac{v_{z}}{z} + \\frac{\\sin{(\\frac{v_{z}}{z})}}{z} and (\\frac{v_{z}}{z} + \\frac{\\operatorname{C_{1}}{(z,v_{z})}}{z})^{v_{z}} = (\\frac{v_{z}}{z} + \\frac{\\sin{(\\frac{v_{z}}{z})}}{z})^{v_{z}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('z', commutative=True), Symbol('v_z', commutative=True)), sin(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))))"], [["divide", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C_1')(Symbol('z', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), sin(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))))))"], [["add", 2, "Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C_1')(Symbol('z', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), sin(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))))))"], [["power", 3, "Symbol('v_z', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('C_1')(Symbol('z', commutative=True), Symbol('v_z', commutative=True)))), Symbol('v_z', commutative=True)), Pow(Add(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), sin(Mul(Symbol('v_z', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)))))), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given A{(s,l)} = l - s and q{(s,l)} = l - s, then obtain - \\frac{\\partial}{\\partial s} A{(s,l)} = - \\frac{\\partial}{\\partial s} q{(s,l)}", "derivation": "A{(s,l)} = l - s and \\frac{\\partial}{\\partial s} A{(s,l)} = \\frac{\\partial}{\\partial s} (l - s) and q{(s,l)} = l - s and \\frac{\\partial}{\\partial s} A{(s,l)} = \\frac{\\partial}{\\partial s} q{(s,l)} and \\frac{\\partial}{\\partial s} (l - s) \\frac{\\partial}{\\partial s} A{(s,l)} = \\frac{\\partial}{\\partial s} (l - s) \\frac{\\partial}{\\partial s} q{(s,l)} and - \\frac{\\partial}{\\partial s} A{(s,l)} = - \\frac{\\partial}{\\partial s} q{(s,l)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('q')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('A')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('q')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["times", 4, "Derivative(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('A')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Derivative(Add(Symbol('l', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Function('q')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(-1), Derivative(Function('A')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('q')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(F_{H},S)} = \\log{(F_{H} S)} and \\rho{(F_{H})} = F_{H}, then obtain \\int (- F_{H} S + g{(F_{H},S)}) d\\rho{(F_{H})} = \\int (- F_{H} S + \\log{(F_{H} S)}) d\\rho{(F_{H})}", "derivation": "g{(F_{H},S)} = \\log{(F_{H} S)} and - F_{H} S + g{(F_{H},S)} = - F_{H} S + \\log{(F_{H} S)} and \\rho{(F_{H})} = F_{H} and \\int (- F_{H} S + g{(F_{H},S)}) dF_{H} = \\int (- F_{H} S + \\log{(F_{H} S)}) dF_{H} and \\int (- F_{H} S + g{(F_{H},S)}) d\\rho{(F_{H})} = \\int (- F_{H} S + \\log{(F_{H} S)}) d\\rho{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('F_H', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('S', commutative=True))))"], [["minus", 1, "Mul(Symbol('F_H', commutative=True), Symbol('S', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), Function('g')(Symbol('F_H', commutative=True), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('S', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), Function('g')(Symbol('F_H', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('S', commutative=True)))), Tuple(Symbol('F_H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), Function('g')(Symbol('F_H', commutative=True), Symbol('S', commutative=True))), Tuple(Function('\\\\rho')(Symbol('F_H', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True), Symbol('S', commutative=True)), log(Mul(Symbol('F_H', commutative=True), Symbol('S', commutative=True)))), Tuple(Function('\\\\rho')(Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\ddot{x},\\theta_2)} = \\frac{\\ddot{x}}{\\theta_2}, then obtain \\frac{\\partial}{\\partial \\ddot{x}} \\sin{(\\int \\mathbf{J}_P{(\\ddot{x},\\theta_2)} d\\ddot{x})} = \\frac{\\partial}{\\partial \\ddot{x}} \\sin{(\\int \\frac{\\ddot{x}}{\\theta_2} d\\ddot{x})}", "derivation": "\\mathbf{J}_P{(\\ddot{x},\\theta_2)} = \\frac{\\ddot{x}}{\\theta_2} and \\int \\mathbf{J}_P{(\\ddot{x},\\theta_2)} d\\ddot{x} = \\int \\frac{\\ddot{x}}{\\theta_2} d\\ddot{x} and \\sin{(\\int \\mathbf{J}_P{(\\ddot{x},\\theta_2)} d\\ddot{x})} = \\sin{(\\int \\frac{\\ddot{x}}{\\theta_2} d\\ddot{x})} and \\frac{\\partial}{\\partial \\ddot{x}} \\sin{(\\int \\mathbf{J}_P{(\\ddot{x},\\theta_2)} d\\ddot{x})} = \\frac{\\partial}{\\partial \\ddot{x}} \\sin{(\\int \\frac{\\ddot{x}}{\\theta_2} d\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["sin", 2], "Equality(sin(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), sin(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(sin(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(sin(Integral(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Tuple(Symbol('\\\\ddot{x}', commutative=True)))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(a,\\omega,f)} = \\frac{\\omega + f}{a} and \\mathbb{I}{(f,\\omega)} = \\omega + f, then obtain - (\\omega + f)^{2} + \\frac{\\partial}{\\partial a} \\omega \\mathbf{F}{(a,\\omega,f)} = - (\\omega + f)^{2} + \\frac{\\partial}{\\partial a} \\frac{\\omega \\mathbb{I}{(f,\\omega)}}{a}", "derivation": "\\mathbf{F}{(a,\\omega,f)} = \\frac{\\omega + f}{a} and \\omega \\mathbf{F}{(a,\\omega,f)} = \\frac{\\omega (\\omega + f)}{a} and \\mathbb{I}{(f,\\omega)} = \\omega + f and \\omega \\mathbf{F}{(a,\\omega,f)} = \\frac{\\omega \\mathbb{I}{(f,\\omega)}}{a} and \\frac{\\partial}{\\partial a} \\omega \\mathbf{F}{(a,\\omega,f)} = \\frac{\\partial}{\\partial a} \\frac{\\omega \\mathbb{I}{(f,\\omega)}}{a} and - (\\omega + f)^{2} + \\frac{\\partial}{\\partial a} \\omega \\mathbf{F}{(a,\\omega,f)} = - (\\omega + f)^{2} + \\frac{\\partial}{\\partial a} \\frac{\\omega \\mathbb{I}{(f,\\omega)}}{a}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["differentiate", 4, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 5, "Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True)), Integer(2))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('a', commutative=True), Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('f', commutative=True)), Integer(2))), Derivative(Mul(Symbol('\\\\omega', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('f', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given t{(\\theta_2)} = \\frac{d}{d \\theta_2} e^{\\theta_2}, then derive t{(\\theta_2)} = e^{\\theta_2}, then obtain t{(\\theta_2)} + \\frac{d}{d \\theta_2} e^{\\theta_2} = \\frac{d^{2}}{d \\theta_2^{2}} t{(\\theta_2)} + \\frac{d}{d \\theta_2} e^{\\theta_2}", "derivation": "t{(\\theta_2)} = \\frac{d}{d \\theta_2} e^{\\theta_2} and t{(\\theta_2)} = e^{\\theta_2} and t{(\\theta_2)} = \\frac{d}{d \\theta_2} t{(\\theta_2)} and \\frac{d}{d \\theta_2} t{(\\theta_2)} = e^{\\theta_2} and t{(\\theta_2)} = \\frac{d^{2}}{d \\theta_2^{2}} t{(\\theta_2)} and t{(\\theta_2)} + \\frac{d}{d \\theta_2} e^{\\theta_2} = \\frac{d^{2}}{d \\theta_2^{2}} t{(\\theta_2)} + \\frac{d}{d \\theta_2} e^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), exp(Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), Derivative(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), exp(Symbol('\\\\theta_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('t')(Symbol('\\\\theta_2', commutative=True)), Derivative(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))))"], [["add", 5, "Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))"], "Equality(Add(Function('t')(Symbol('\\\\theta_2', commutative=True)), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Add(Derivative(Function('t')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\dot{y},\\mathbf{H})} = e^{\\mathbf{H}^{\\dot{y}}}, then derive \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{z}}{(\\dot{y},\\mathbf{H})} = \\mathbf{H}^{\\dot{y}} e^{\\mathbf{H}^{\\dot{y}}} \\log{(\\mathbf{H})}, then obtain \\mathbf{H}^{\\dot{y}} e^{\\mathbf{H}^{\\dot{y}}} \\log{(\\mathbf{H})} = \\frac{\\partial}{\\partial \\dot{y}} e^{\\mathbf{H}^{\\dot{y}}}", "derivation": "\\operatorname{A_{z}}{(\\dot{y},\\mathbf{H})} = e^{\\mathbf{H}^{\\dot{y}}} and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{z}}{(\\dot{y},\\mathbf{H})} = \\frac{\\partial}{\\partial \\dot{y}} e^{\\mathbf{H}^{\\dot{y}}} and \\frac{\\partial}{\\partial \\dot{y}} \\operatorname{A_{z}}{(\\dot{y},\\mathbf{H})} = \\mathbf{H}^{\\dot{y}} e^{\\mathbf{H}^{\\dot{y}}} \\log{(\\mathbf{H})} and \\mathbf{H}^{\\dot{y}} e^{\\mathbf{H}^{\\dot{y}}} \\log{(\\mathbf{H})} = \\frac{\\partial}{\\partial \\dot{y}} e^{\\mathbf{H}^{\\dot{y}}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), exp(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), log(Symbol('\\\\mathbf{H}', commutative=True))), Derivative(exp(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given a{(\\omega)} = e^{\\omega}, then derive \\frac{d}{d \\omega} a{(\\omega)} = e^{\\omega}, then obtain \\frac{\\int \\frac{d}{d \\omega} a{(\\omega)} d\\omega}{\\log{(\\frac{d^{2}}{d \\omega^{2}} a{(\\omega)})}} = \\frac{\\int \\frac{d^{2}}{d \\omega^{2}} a{(\\omega)} d\\omega}{\\log{(\\frac{d^{2}}{d \\omega^{2}} a{(\\omega)})}}", "derivation": "a{(\\omega)} = e^{\\omega} and \\frac{d}{d \\omega} a{(\\omega)} = \\frac{d}{d \\omega} e^{\\omega} and \\frac{d}{d \\omega} a{(\\omega)} = e^{\\omega} and \\frac{d}{d \\omega} a{(\\omega)} = \\frac{d^{2}}{d \\omega^{2}} a{(\\omega)} and \\int \\frac{d}{d \\omega} a{(\\omega)} d\\omega = \\int \\frac{d^{2}}{d \\omega^{2}} a{(\\omega)} d\\omega and \\frac{\\int \\frac{d}{d \\omega} a{(\\omega)} d\\omega}{\\log{(\\frac{d^{2}}{d \\omega^{2}} a{(\\omega)})}} = \\frac{\\int \\frac{d^{2}}{d \\omega^{2}} a{(\\omega)} d\\omega}{\\log{(\\frac{d^{2}}{d \\omega^{2}} a{(\\omega)})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), exp(Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True))))"], [["divide", 5, "log(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], "Equality(Mul(Pow(log(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2)))), Integer(-1)), Integral(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(log(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2)))), Integer(-1)), Integral(Derivative(Function('a')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))), Tuple(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(g^{\\prime}_{\\varepsilon})} = \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon}, then derive C_{1} + \\operatorname{V_{\\mathbf{B}}}{(g^{\\prime}_{\\varepsilon})} - \\cos{(g^{\\prime}_{\\varepsilon})} = 2 C_{1} - 2 \\cos{(g^{\\prime}_{\\varepsilon})}, then obtain C_{1} - \\cos{(g^{\\prime}_{\\varepsilon})} + \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = 2 C_{1} - 2 \\cos{(g^{\\prime}_{\\varepsilon})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(g^{\\prime}_{\\varepsilon})} = \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and \\operatorname{V_{\\mathbf{B}}}{(g^{\\prime}_{\\varepsilon})} + \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = 2 \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} and C_{1} + \\operatorname{V_{\\mathbf{B}}}{(g^{\\prime}_{\\varepsilon})} - \\cos{(g^{\\prime}_{\\varepsilon})} = 2 C_{1} - 2 \\cos{(g^{\\prime}_{\\varepsilon})} and C_{1} - \\cos{(g^{\\prime}_{\\varepsilon})} + \\int \\sin{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = 2 C_{1} - 2 \\cos{(g^{\\prime}_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Symbol('C_1', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(2), Symbol('C_1', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given p{(r)} = e^{\\cos{(r)}}, then obtain \\log{((r + \\frac{p{(r)}}{r}) p{(r)})} = \\log{((r + \\frac{p{(r)}}{r}) e^{\\cos{(r)}})}", "derivation": "p{(r)} = e^{\\cos{(r)}} and \\frac{p{(r)}}{r} = \\frac{e^{\\cos{(r)}}}{r} and r + \\frac{p{(r)}}{r} = r + \\frac{e^{\\cos{(r)}}}{r} and (r + \\frac{e^{\\cos{(r)}}}{r}) p{(r)} = (r + \\frac{e^{\\cos{(r)}}}{r}) e^{\\cos{(r)}} and \\log{((r + \\frac{e^{\\cos{(r)}}}{r}) p{(r)})} = \\log{((r + \\frac{e^{\\cos{(r)}}}{r}) e^{\\cos{(r)}})} and \\log{((r + \\frac{p{(r)}}{r}) p{(r)})} = \\log{((r + \\frac{p{(r)}}{r}) e^{\\cos{(r)}})}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('r', commutative=True)), exp(cos(Symbol('r', commutative=True))))"], [["divide", 1, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('p')(Symbol('r', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True)))))"], [["add", 2, "Symbol('r', commutative=True)"], "Equality(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('p')(Symbol('r', commutative=True)))), Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True))))))"], [["times", 1, "Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True)))))"], "Equality(Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True))))), Function('p')(Symbol('r', commutative=True))), Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True))))), exp(cos(Symbol('r', commutative=True)))))"], [["log", 4], "Equality(log(Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True))))), Function('p')(Symbol('r', commutative=True)))), log(Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), exp(cos(Symbol('r', commutative=True))))), exp(cos(Symbol('r', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(log(Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('p')(Symbol('r', commutative=True)))), Function('p')(Symbol('r', commutative=True)))), log(Mul(Add(Symbol('r', commutative=True), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('p')(Symbol('r', commutative=True)))), exp(cos(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given \\hat{p}{(\\Omega,\\omega)} = \\sin{(\\Omega - \\omega)}, then obtain \\int (\\omega + (- \\omega + \\hat{p}{(\\Omega,\\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1) d\\Omega = \\int (\\omega + (- \\omega + \\sin{(\\Omega - \\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1) d\\Omega", "derivation": "\\hat{p}{(\\Omega,\\omega)} = \\sin{(\\Omega - \\omega)} and - \\omega + \\hat{p}{(\\Omega,\\omega)} = - \\omega + \\sin{(\\Omega - \\omega)} and (- \\omega + \\hat{p}{(\\Omega,\\omega)})^{\\Omega} = (- \\omega + \\sin{(\\Omega - \\omega)})^{\\Omega} and (- \\omega + \\hat{p}{(\\Omega,\\omega)})^{\\Omega} + 1 = (- \\omega + \\sin{(\\Omega - \\omega)})^{\\Omega} + 1 and \\omega + (- \\omega + \\hat{p}{(\\Omega,\\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1 = \\omega + (- \\omega + \\sin{(\\Omega - \\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1 and \\int (\\omega + (- \\omega + \\hat{p}{(\\Omega,\\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1) d\\Omega = \\int (\\omega + (- \\omega + \\sin{(\\Omega - \\omega)})^{\\Omega} - \\sin{(\\Omega - \\omega)} + 1) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], [["minus", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('\\\\Omega', commutative=True)))"], [["add", 3, 1], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Integer(1)), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('\\\\Omega', commutative=True)), Integer(1)))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], "Equality(Add(Symbol('\\\\omega', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integer(1)), Add(Symbol('\\\\omega', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integer(1)))"], [["integrate", 5, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\omega', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Symbol('\\\\omega', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given s{(q)} = \\cos{(q)}, then obtain s{(q)} + s^{q}{(q)} + \\frac{s^{q}{(q)} \\int (s{(q)} + s^{q}{(q)}) dq}{s{(q)}} = s^{q}{(q)} + \\cos{(q)} + \\frac{s^{q}{(q)} \\int (s{(q)} + s^{q}{(q)}) dq}{s{(q)}}", "derivation": "s{(q)} = \\cos{(q)} and s{(q)} + s^{q}{(q)} = s^{q}{(q)} + \\cos{(q)} and \\int (s{(q)} + s^{q}{(q)}) dq = \\int (s^{q}{(q)} + \\cos{(q)}) dq and s{(q)} + s^{q}{(q)} + \\frac{s^{q}{(q)} \\int (s^{q}{(q)} + \\cos{(q)}) dq}{s{(q)}} = s^{q}{(q)} + \\cos{(q)} + \\frac{s^{q}{(q)} \\int (s^{q}{(q)} + \\cos{(q)}) dq}{s{(q)}} and s{(q)} + s^{q}{(q)} + \\frac{s^{q}{(q)} \\int (s{(q)} + s^{q}{(q)}) dq}{s{(q)}} = s^{q}{(q)} + \\cos{(q)} + \\frac{s^{q}{(q)} \\int (s{(q)} + s^{q}{(q)}) dq}{s{(q)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["add", 1, "Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["add", 2, "Mul(Pow(Function('s')(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], "Equality(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Mul(Pow(Function('s')(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))), Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)), Mul(Pow(Function('s')(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Mul(Pow(Function('s')(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))), Add(Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)), Mul(Pow(Function('s')(Symbol('q', commutative=True)), Integer(-1)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Function('s')(Symbol('q', commutative=True)), Pow(Function('s')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))))"]]}, {"prompt": "Given m{(I)} = \\cos{(I)}, then obtain - \\mathbf{F}^{\\mathbf{J}_f}{(\\mathbf{J}_f)} + m^{I}{(I)} \\cos^{- I}{(I)} = 1 - \\mathbf{F}^{\\mathbf{J}_f}{(\\mathbf{J}_f)}", "derivation": "m{(I)} = \\cos{(I)} and m^{I}{(I)} = \\cos^{I}{(I)} and m^{I}{(I)} \\cos^{- I}{(I)} = 1 and - \\mathbf{F}^{\\mathbf{J}_f}{(\\mathbf{J}_f)} + m^{I}{(I)} \\cos^{- I}{(I)} = 1 - \\mathbf{F}^{\\mathbf{J}_f}{(\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('m')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], [["divide", 2, "Pow(cos(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Mul(Pow(Function('m')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)))), Integer(1))"], [["minus", 3, "Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Pow(Function('m')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\bar{\\h}{(b)} = \\cos{(b)}, then obtain - b + (\\int \\bar{\\h}^{b}{(b)} db)^{b} - \\int \\cos^{b}{(b)} db = - b - \\int \\cos^{b}{(b)} db + (\\int \\cos^{b}{(b)} db)^{b}", "derivation": "\\bar{\\h}{(b)} = \\cos{(b)} and \\bar{\\h}^{b}{(b)} = \\cos^{b}{(b)} and \\int \\bar{\\h}^{b}{(b)} db = \\int \\cos^{b}{(b)} db and (\\int \\bar{\\h}^{b}{(b)} db)^{b} = (\\int \\cos^{b}{(b)} db)^{b} and - b + (\\int \\bar{\\h}^{b}{(b)} db)^{b} - \\int \\cos^{b}{(b)} db = - b - \\int \\cos^{b}{(b)} db + (\\int \\cos^{b}{(b)} db)^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hbar')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["power", 3, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\hbar')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["minus", 4, "Add(Symbol('b', commutative=True), Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Pow(Integral(Pow(Function('\\\\hbar')(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Pow(Integral(Pow(cos(Symbol('b', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\phi_2)} = \\cos{(\\phi_2)}, then derive (\\frac{d}{d \\phi_2} \\chi{(\\phi_2)})^{2} = - \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)}, then obtain - \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)} + (\\frac{d}{d \\phi_2} \\chi{(\\phi_2)})^{2} = - 2 \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)}", "derivation": "\\chi{(\\phi_2)} = \\cos{(\\phi_2)} and \\frac{d}{d \\phi_2} \\chi{(\\phi_2)} = \\frac{d}{d \\phi_2} \\cos{(\\phi_2)} and (\\frac{d}{d \\phi_2} \\chi{(\\phi_2)})^{2} = \\frac{d}{d \\phi_2} \\chi{(\\phi_2)} \\frac{d}{d \\phi_2} \\cos{(\\phi_2)} and (\\frac{d}{d \\phi_2} \\chi{(\\phi_2)})^{2} = - \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)} and - \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)} + (\\frac{d}{d \\phi_2} \\chi{(\\phi_2)})^{2} = - 2 \\sin{(\\phi_2)} \\frac{d}{d \\phi_2} \\chi{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["add", 4, "Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(2))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\phi_2', commutative=True)), Derivative(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{1}{(\\hat{x},B)} = - B + \\hat{x}, then derive \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int \\phi_{1}{(\\hat{x},B)} d\\hat{x}) = \\frac{\\partial}{\\partial \\hat{x}} (- B \\hat{x} - B + \\frac{\\hat{x}^{2}}{2} + \\varphi), then obtain \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int (- B + \\hat{x}) d\\hat{x}) = \\frac{\\partial}{\\partial \\hat{x}} (- B \\hat{x} - B + \\frac{\\hat{x}^{2}}{2} + \\varphi)", "derivation": "\\phi_{1}{(\\hat{x},B)} = - B + \\hat{x} and \\int \\phi_{1}{(\\hat{x},B)} d\\hat{x} = \\int (- B + \\hat{x}) d\\hat{x} and - B + \\int \\phi_{1}{(\\hat{x},B)} d\\hat{x} = - B + \\int (- B + \\hat{x}) d\\hat{x} and \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int \\phi_{1}{(\\hat{x},B)} d\\hat{x}) = \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int (- B + \\hat{x}) d\\hat{x}) and \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int \\phi_{1}{(\\hat{x},B)} d\\hat{x}) = \\frac{\\partial}{\\partial \\hat{x}} (- B \\hat{x} - B + \\frac{\\hat{x}^{2}}{2} + \\varphi) and \\frac{\\partial}{\\partial \\hat{x}} (- B + \\int (- B + \\hat{x}) d\\hat{x}) = \\frac{\\partial}{\\partial \\hat{x}} (- B \\hat{x} - B + \\frac{\\hat{x}^{2}}{2} + \\varphi)", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Function('\\\\phi_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('B', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(n_{2},\\varphi^*)} = \\varphi^* n_{2} and t{(\\varphi^*)} = \\varphi^*, then obtain \\frac{t{(\\varphi^*)}}{C{(n_{2},\\varphi^*)}} = \\frac{1}{n_{2}}", "derivation": "C{(n_{2},\\varphi^*)} = \\varphi^* n_{2} and t{(\\varphi^*)} = \\varphi^* and \\frac{t{(\\varphi^*)}}{\\varphi^* n_{2}} = \\frac{1}{n_{2}} and \\frac{t{(\\varphi^*)}}{C{(n_{2},\\varphi^*)}} = \\frac{1}{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('n_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], [["divide", 2, "Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Symbol('n_2', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\varphi^*', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('C')(Symbol('n_2', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Function('t')(Symbol('\\\\varphi^*', commutative=True))), Pow(Symbol('n_2', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{A})} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A}, then derive \\operatorname{z^{*}}{(\\mathbf{A})} = \\Psi^{\\dagger} - \\cos{(\\mathbf{A})}, then obtain (\\sin{(\\mathbf{A})} \\int \\sin{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} = ((\\Psi^{\\dagger} - \\cos{(\\mathbf{A})}) \\sin{(\\mathbf{A})})^{\\mathbf{A}}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{A})} = \\int \\sin{(\\mathbf{A})} d\\mathbf{A} and \\operatorname{z^{*}}{(\\mathbf{A})} = \\Psi^{\\dagger} - \\cos{(\\mathbf{A})} and \\operatorname{z^{*}}{(\\mathbf{A})} \\sin{(\\mathbf{A})} = (\\Psi^{\\dagger} - \\cos{(\\mathbf{A})}) \\sin{(\\mathbf{A})} and (\\operatorname{z^{*}}{(\\mathbf{A})} \\sin{(\\mathbf{A})})^{\\mathbf{A}} = ((\\Psi^{\\dagger} - \\cos{(\\mathbf{A})}) \\sin{(\\mathbf{A})})^{\\mathbf{A}} and (\\sin{(\\mathbf{A})} \\int \\sin{(\\mathbf{A})} d\\mathbf{A})^{\\mathbf{A}} = ((\\Psi^{\\dagger} - \\cos{(\\mathbf{A})}) \\sin{(\\mathbf{A})})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('z^*')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["times", 2, "sin(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Function('z^*')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Mul(Function('z^*')(Symbol('\\\\mathbf{A}', commutative=True)), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Mul(sin(Symbol('\\\\mathbf{A}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{A}', commutative=True)))), sin(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given E{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain - (2 E{(a^{\\dagger})} + e^{a^{\\dagger}})^{2} + E{(a^{\\dagger})} = - (E{(a^{\\dagger})} + 2 e^{a^{\\dagger}}) (2 E{(a^{\\dagger})} + e^{a^{\\dagger}}) + E{(a^{\\dagger})}", "derivation": "E{(a^{\\dagger})} = e^{a^{\\dagger}} and E{(a^{\\dagger})} + e^{a^{\\dagger}} = 2 e^{a^{\\dagger}} and 2 E{(a^{\\dagger})} + e^{a^{\\dagger}} = E{(a^{\\dagger})} + 2 e^{a^{\\dagger}} and (2 E{(a^{\\dagger})} + e^{a^{\\dagger}})^{2} = (E{(a^{\\dagger})} + 2 e^{a^{\\dagger}}) (2 E{(a^{\\dagger})} + e^{a^{\\dagger}}) and (2 E{(a^{\\dagger})} + e^{a^{\\dagger}})^{2} - E{(a^{\\dagger})} = (E{(a^{\\dagger})} + 2 e^{a^{\\dagger}}) (2 E{(a^{\\dagger})} + e^{a^{\\dagger}}) - E{(a^{\\dagger})} and - (2 E{(a^{\\dagger})} + e^{a^{\\dagger}})^{2} + E{(a^{\\dagger})} = - (E{(a^{\\dagger})} + 2 e^{a^{\\dagger}}) (2 E{(a^{\\dagger})} + e^{a^{\\dagger}}) + E{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["add", 1, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(2), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 2, "Function('E')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Add(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["times", 3, "Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Pow(Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Integer(2)), Mul(Add(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["minus", 4, "Function('E')(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Integer(2)), Mul(Integer(-1), Function('E')(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Add(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Integer(-1), Function('E')(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["times", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True))), Integer(2))), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Add(Function('E')(Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(2), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(2), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))), exp(Symbol('a^{\\\\dagger}', commutative=True)))), Function('E')(Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given i{(\\theta_1,S)} = \\frac{\\sin{(S)}}{\\theta_1}, then obtain \\iint (i{(\\theta_1,S)} - \\frac{\\sin{(S)}}{\\theta_1}) dS dS = \\iint 0 dS dS", "derivation": "i{(\\theta_1,S)} = \\frac{\\sin{(S)}}{\\theta_1} and i{(\\theta_1,S)} - \\frac{\\sin{(S)}}{\\theta_1} = 0 and \\int (i{(\\theta_1,S)} - \\frac{\\sin{(S)}}{\\theta_1}) dS = \\int 0 dS and \\iint (i{(\\theta_1,S)} - \\frac{\\sin{(S)}}{\\theta_1}) dS dS = \\iint 0 dS dS", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('S', commutative=True)), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True))))"], [["minus", 1, "Mul(Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True)))"], "Equality(Add(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True))), Integral(Integer(0), Tuple(Symbol('S', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Function('i')(Symbol('\\\\theta_1', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\theta_1', commutative=True), Integer(-1)), sin(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Integer(0), Tuple(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\theta_1)} = \\theta_1, then derive \\int \\operatorname{v_{t}}{(\\theta_1)} d\\theta_1 = \\frac{\\theta_1^{2}}{2} + f, then obtain \\frac{\\theta_1^{2}}{2} + f = \\int \\theta_1 d\\theta_1", "derivation": "\\operatorname{v_{t}}{(\\theta_1)} = \\theta_1 and \\int \\operatorname{v_{t}}{(\\theta_1)} d\\theta_1 = \\int \\theta_1 d\\theta_1 and \\int \\operatorname{v_{t}}{(\\theta_1)} d\\theta_1 = \\frac{\\theta_1^{2}}{2} + f and \\frac{\\theta_1^{2}}{2} + f = \\int \\theta_1 d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\theta_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('v_t')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Symbol('\\\\theta_1', commutative=True), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_t')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\theta_1', commutative=True), Integer(2))), Symbol('f', commutative=True)), Integral(Symbol('\\\\theta_1', commutative=True), Tuple(Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\nabla)} = \\cos{(\\nabla)} and \\operatorname{J_{\\varepsilon}}{(\\nabla)} = \\cos{(\\nabla)} - 1, then obtain \\operatorname{v_{y}}{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla - 1 = \\cos{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla - 1", "derivation": "\\operatorname{v_{y}}{(\\nabla)} = \\cos{(\\nabla)} and \\operatorname{v_{y}}{(\\nabla)} - 1 = \\cos{(\\nabla)} - 1 and \\operatorname{J_{\\varepsilon}}{(\\nabla)} = \\cos{(\\nabla)} - 1 and \\operatorname{v_{y}}{(\\nabla)} - 1 = \\operatorname{J_{\\varepsilon}}{(\\nabla)} and \\operatorname{v_{y}}{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla - 1 = \\operatorname{J_{\\varepsilon}}{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla and \\operatorname{v_{y}}{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla - 1 = \\cos{(\\nabla)} + \\int \\log{(\\operatorname{v_{y}}{(\\nabla)} - 1)} d\\nabla - 1", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Add(cos(Symbol('\\\\nabla', commutative=True)), Integer(-1)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Add(cos(Symbol('\\\\nabla', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)))"], [["add", 4, "Integral(log(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integral(log(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(-1)), Add(Function('J_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Integral(log(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integral(log(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(-1)), Add(cos(Symbol('\\\\nabla', commutative=True)), Integral(log(Add(Function('v_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\nabla', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\lambda{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}}, then obtain (\\int \\lambda{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (A_{2} + e^{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}}", "derivation": "\\lambda{(\\Psi^{\\dagger})} = e^{\\Psi^{\\dagger}} and \\int \\lambda{(\\Psi^{\\dagger})} d\\Psi^{\\dagger} = \\int e^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} and (\\int \\lambda{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (\\int e^{\\Psi^{\\dagger}} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} and (\\int \\lambda{(\\Psi^{\\dagger})} d\\Psi^{\\dagger})^{\\Psi^{\\dagger}} = (A_{2} + e^{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Integral(exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\lambda')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Add(Symbol('A_2', commutative=True), exp(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(W)} = \\sin{(W)}, then derive \\int \\operatorname{m_{s}}{(W)} dW = V_{\\mathbf{E}} - \\cos{(W)}, then derive \\psi - \\cos{(W)} = V_{\\mathbf{E}} - \\cos{(W)}, then obtain \\int \\frac{\\partial}{\\partial W} (\\psi - \\cos{(W)}) d\\psi = \\int \\frac{\\partial}{\\partial W} (V_{\\mathbf{E}} - \\cos{(W)}) d\\psi", "derivation": "\\operatorname{m_{s}}{(W)} = \\sin{(W)} and \\int \\operatorname{m_{s}}{(W)} dW = \\int \\sin{(W)} dW and \\int \\operatorname{m_{s}}{(W)} dW = V_{\\mathbf{E}} - \\cos{(W)} and \\int \\sin{(W)} dW = V_{\\mathbf{E}} - \\cos{(W)} and \\psi - \\cos{(W)} = V_{\\mathbf{E}} - \\cos{(W)} and \\frac{\\partial}{\\partial W} (\\psi - \\cos{(W)}) = \\frac{\\partial}{\\partial W} (V_{\\mathbf{E}} - \\cos{(W)}) and \\int \\frac{\\partial}{\\partial W} (\\psi - \\cos{(W)}) d\\psi = \\int \\frac{\\partial}{\\partial W} (V_{\\mathbf{E}} - \\cos{(W)}) d\\psi", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m_s')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Derivative(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Mul(Integer(-1), cos(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Tuple(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\varphi{(C_{d})} = C_{d}, then derive 2 \\int \\varphi{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + Z + \\int \\varphi{(C_{d})} dC_{d}, then obtain Z + \\frac{\\varphi^{2}{(C_{d})}}{2} + 3 \\int C_{d} d\\varphi{(C_{d})} = 2 Z + \\varphi^{2}{(C_{d})} + 2 \\int C_{d} d\\varphi{(C_{d})}", "derivation": "\\varphi{(C_{d})} = C_{d} and \\int \\varphi{(C_{d})} dC_{d} = \\int C_{d} dC_{d} and 2 \\int \\varphi{(C_{d})} dC_{d} = \\int C_{d} dC_{d} + \\int \\varphi{(C_{d})} dC_{d} and 2 \\int \\varphi{(C_{d})} dC_{d} = \\frac{C_{d}^{2}}{2} + Z + \\int \\varphi{(C_{d})} dC_{d} and \\frac{C_{d}^{2}}{2} + Z + 3 \\int \\varphi{(C_{d})} dC_{d} = C_{d}^{2} + 2 Z + 2 \\int \\varphi{(C_{d})} dC_{d} and \\frac{C_{d}^{2}}{2} + Z + 3 \\int C_{d} dC_{d} = C_{d}^{2} + 2 Z + 2 \\int C_{d} dC_{d} and Z + \\frac{\\varphi^{2}{(C_{d})}}{2} + 3 \\int C_{d} d\\varphi{(C_{d})} = 2 Z + \\varphi^{2}{(C_{d})} + 2 \\int C_{d} d\\varphi{(C_{d})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))))"], [["add", 2, "Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Add(Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(2), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('Z', commutative=True), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True)))))"], [["add", 4, "Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('Z', commutative=True), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('Z', commutative=True), Mul(Integer(3), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))), Add(Pow(Symbol('C_d', commutative=True), Integer(2)), Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Integral(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('C_d', commutative=True), Integer(2))), Symbol('Z', commutative=True), Mul(Integer(3), Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))))), Add(Pow(Symbol('C_d', commutative=True), Integer(2)), Mul(Integer(2), Symbol('Z', commutative=True)), Mul(Integer(2), Integral(Symbol('C_d', commutative=True), Tuple(Symbol('C_d', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(2))), Mul(Integer(3), Integral(Symbol('C_d', commutative=True), Tuple(Function('\\\\varphi')(Symbol('C_d', commutative=True)))))), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Pow(Function('\\\\varphi')(Symbol('C_d', commutative=True)), Integer(2)), Mul(Integer(2), Integral(Symbol('C_d', commutative=True), Tuple(Function('\\\\varphi')(Symbol('C_d', commutative=True)))))))"]]}, {"prompt": "Given h{(t)} = \\log{(t)}, then derive (\\hat{x} + t \\log{(t)} - t) h{(t)} = (\\hat{x} + t \\log{(t)} - t) \\log{(t)}, then obtain - (\\hat{x} + t \\log{(t)} - t) h{(t)} = - (\\hat{x} + t \\log{(t)} - t) \\log{(t)}", "derivation": "h{(t)} = \\log{(t)} and \\int h{(t)} dt = \\int \\log{(t)} dt and h{(t)} \\int h{(t)} dt = \\log{(t)} \\int h{(t)} dt and h{(t)} \\int \\log{(t)} dt = \\log{(t)} \\int \\log{(t)} dt and (\\hat{x} + t \\log{(t)} - t) h{(t)} = (\\hat{x} + t \\log{(t)} - t) \\log{(t)} and - (\\hat{x} + t \\log{(t)} - t) h{(t)} = - (\\hat{x} + t \\log{(t)} - t) \\log{(t)}", "srepr_derivation": [["get_premise", "Equality(Function('h')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('h')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 1, "Integral(Function('h')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Function('h')(Symbol('t', commutative=True)), Integral(Function('h')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(log(Symbol('t', commutative=True)), Integral(Function('h')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('h')(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(log(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), Function('h')(Symbol('t', commutative=True))), Mul(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), log(Symbol('t', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), Function('h')(Symbol('t', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), log(Symbol('t', commutative=True))))"]]}, {"prompt": "Given p{(\\mathbf{p},s)} = e^{\\frac{\\mathbf{p}}{s}} and \\phi_{2}{(\\mathbf{p},s)} = s p{(\\mathbf{p},s)}, then obtain - \\frac{\\mathbf{p}}{s} + \\phi_{2}{(\\mathbf{p},s)} p{(\\mathbf{p},s)} = - \\frac{\\mathbf{p}}{s} + \\phi_{2}{(\\mathbf{p},s)} e^{\\frac{\\mathbf{p}}{s}}", "derivation": "p{(\\mathbf{p},s)} = e^{\\frac{\\mathbf{p}}{s}} and s p^{2}{(\\mathbf{p},s)} = s p{(\\mathbf{p},s)} e^{\\frac{\\mathbf{p}}{s}} and \\phi_{2}{(\\mathbf{p},s)} = s p{(\\mathbf{p},s)} and \\phi_{2}{(\\mathbf{p},s)} p{(\\mathbf{p},s)} = \\phi_{2}{(\\mathbf{p},s)} e^{\\frac{\\mathbf{p}}{s}} and - \\frac{\\mathbf{p}}{s} + \\phi_{2}{(\\mathbf{p},s)} p{(\\mathbf{p},s)} = - \\frac{\\mathbf{p}}{s} + \\phi_{2}{(\\mathbf{p},s)} e^{\\frac{\\mathbf{p}}{s}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["times", 1, "Mul(Symbol('s', commutative=True), Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Symbol('s', commutative=True), Pow(Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Symbol('s', commutative=True), Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True))), Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))))"], [["minus", 4, "Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), Function('p')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Function('\\\\phi_2')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('s', commutative=True)), exp(Mul(Symbol('\\\\mathbf{p}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(p)} = \\sin{(p)}, then obtain ((\\int \\frac{p \\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} dp)^{p})^{p} = ((\\int p dp)^{p})^{p}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(p)} = \\sin{(p)} and \\frac{\\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} = 1 and \\frac{p \\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} = p and \\int \\frac{p \\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} dp = \\int p dp and (\\int \\frac{p \\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} dp)^{p} = (\\int p dp)^{p} and ((\\int \\frac{p \\operatorname{V_{\\mathbf{B}}}{(p)}}{\\sin{(p)}} dp)^{p})^{p} = ((\\int p dp)^{p})^{p}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], [["divide", 1, "sin(Symbol('p', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))), Symbol('p', commutative=True))"], [["integrate", 3, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Symbol('p', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Integral(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True))))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('p', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["power", 5, "Symbol('p', commutative=True)"], "Equality(Pow(Pow(Integral(Mul(Symbol('p', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('p', commutative=True)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(Integral(Symbol('p', commutative=True), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"]]}, {"prompt": "Given z{(C_{2},t_{2},f_{E})} = (\\frac{C_{2}}{f_{E}})^{t_{2}}, then obtain \\int \\frac{C_{2} z{(C_{2},t_{2},f_{E})}}{f_{E}} dC_{2} + \\frac{1}{f_{E}} = \\int \\frac{C_{2} (\\frac{C_{2}}{f_{E}})^{t_{2}}}{f_{E}} dC_{2} + \\frac{1}{f_{E}}", "derivation": "z{(C_{2},t_{2},f_{E})} = (\\frac{C_{2}}{f_{E}})^{t_{2}} and \\frac{C_{2} z{(C_{2},t_{2},f_{E})}}{f_{E}} = \\frac{C_{2} (\\frac{C_{2}}{f_{E}})^{t_{2}}}{f_{E}} and \\int \\frac{C_{2} z{(C_{2},t_{2},f_{E})}}{f_{E}} dC_{2} = \\int \\frac{C_{2} (\\frac{C_{2}}{f_{E}})^{t_{2}}}{f_{E}} dC_{2} and \\int \\frac{C_{2} z{(C_{2},t_{2},f_{E})}}{f_{E}} dC_{2} + \\frac{1}{f_{E}} = \\int \\frac{C_{2} (\\frac{C_{2}}{f_{E}})^{t_{2}}}{f_{E}} dC_{2} + \\frac{1}{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('C_2', commutative=True), Symbol('t_2', commutative=True), Symbol('f_E', commutative=True)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Symbol('t_2', commutative=True)))"], [["times", 1, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('z')(Symbol('C_2', commutative=True), Symbol('t_2', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Symbol('t_2', commutative=True))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('z')(Symbol('C_2', commutative=True), Symbol('t_2', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Symbol('t_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["add", 3, "Pow(Symbol('f_E', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Function('z')(Symbol('C_2', commutative=True), Symbol('t_2', commutative=True), Symbol('f_E', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Pow(Symbol('f_E', commutative=True), Integer(-1))), Add(Integral(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1)), Pow(Mul(Symbol('C_2', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Symbol('t_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))), Pow(Symbol('f_E', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mu_{0}{(\\Psi^{\\dagger},\\sigma_x)} = \\Psi^{\\dagger} \\sigma_x, then obtain \\frac{\\partial}{\\partial \\sigma_x} \\int \\mu_{0}^{\\sigma_x}{(\\Psi^{\\dagger},\\sigma_x)} d\\Psi^{\\dagger} = \\frac{\\partial}{\\partial \\sigma_x} \\int (\\Psi^{\\dagger} \\sigma_x)^{\\sigma_x} d\\Psi^{\\dagger}", "derivation": "\\mu_{0}{(\\Psi^{\\dagger},\\sigma_x)} = \\Psi^{\\dagger} \\sigma_x and \\mu_{0}^{\\sigma_x}{(\\Psi^{\\dagger},\\sigma_x)} = (\\Psi^{\\dagger} \\sigma_x)^{\\sigma_x} and \\int \\mu_{0}^{\\sigma_x}{(\\Psi^{\\dagger},\\sigma_x)} d\\Psi^{\\dagger} = \\int (\\Psi^{\\dagger} \\sigma_x)^{\\sigma_x} d\\Psi^{\\dagger} and \\frac{\\partial}{\\partial \\sigma_x} \\int \\mu_{0}^{\\sigma_x}{(\\Psi^{\\dagger},\\sigma_x)} d\\Psi^{\\dagger} = \\frac{\\partial}{\\partial \\sigma_x} \\int (\\Psi^{\\dagger} \\sigma_x)^{\\sigma_x} d\\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Integral(Pow(Function('\\\\mu_0')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Integral(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(z)} = e^{z}, then obtain \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\Psi^{\\dagger} \\int \\operatorname{F_{N}}{(z)} dz = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\Psi^{\\dagger} \\int e^{z} dz", "derivation": "\\operatorname{F_{N}}{(z)} = e^{z} and \\int \\operatorname{F_{N}}{(z)} dz = \\int e^{z} dz and \\Psi^{\\dagger} \\int \\operatorname{F_{N}}{(z)} dz = \\Psi^{\\dagger} \\int e^{z} dz and \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\Psi^{\\dagger} \\int \\operatorname{F_{N}}{(z)} dz = \\frac{\\partial}{\\partial \\Psi^{\\dagger}} \\Psi^{\\dagger} \\int e^{z} dz", "srepr_derivation": [["get_premise", "Equality(Function('F_N')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(Function('F_N')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi{(M_{E},\\mathbf{J}_P)} = \\cos{(M_{E} \\mathbf{J}_P)}, then obtain - \\frac{1}{\\mathbf{J}_P} = - \\frac{\\cos{(M_{E} \\mathbf{J}_P)}}{\\mathbf{J}_P \\Psi{(M_{E},\\mathbf{J}_P)}}", "derivation": "\\Psi{(M_{E},\\mathbf{J}_P)} = \\cos{(M_{E} \\mathbf{J}_P)} and 1 = \\frac{\\cos{(M_{E} \\mathbf{J}_P)}}{\\Psi{(M_{E},\\mathbf{J}_P)}} and -1 = - \\frac{\\cos{(M_{E} \\mathbf{J}_P)}}{\\Psi{(M_{E},\\mathbf{J}_P)}} and - \\frac{1}{\\mathbf{J}_P} = - \\frac{\\cos{(M_{E} \\mathbf{J}_P)}}{\\mathbf{J}_P \\Psi{(M_{E},\\mathbf{J}_P)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["divide", 1, "Function('\\\\Psi')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\Psi')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), cos(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["divide", 2, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Function('\\\\Psi')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), cos(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["divide", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Pow(Function('\\\\Psi')(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), cos(Mul(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(x,y)} = \\log{(y)}^{x}, then obtain - 2 x + 3 \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} - \\log{(y)}^{x} = - 2 x + (- \\operatorname{E_{\\lambda}}{(x,y)} + \\log{(y)} + \\log{(y)}^{x})^{x} + 2 \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} - \\log{(y)}^{x}", "derivation": "\\operatorname{E_{\\lambda}}{(x,y)} = \\log{(y)}^{x} and - x + \\operatorname{E_{\\lambda}}{(x,y)} = - x + \\log{(y)}^{x} and - 2 x + 2 \\operatorname{E_{\\lambda}}{(x,y)} = - 2 x + \\operatorname{E_{\\lambda}}{(x,y)} + \\log{(y)}^{x} and - 2 x + 2 \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} = - 2 x + \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} + \\log{(y)}^{x} and \\log{(y)} = - \\operatorname{E_{\\lambda}}{(x,y)} + \\log{(y)} + \\log{(y)}^{x} and - 2 x + 3 \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} - \\log{(y)}^{x} = - 2 x + (- \\operatorname{E_{\\lambda}}{(x,y)} + \\log{(y)} + \\log{(y)}^{x})^{x} + 2 \\operatorname{E_{\\lambda}}{(x,y)} - \\log{(y)} - \\log{(y)}^{x}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True))))"], [["minus", 3, "log(Symbol('y', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), log(Symbol('y', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), log(Symbol('y', commutative=True))), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), log(Symbol('y', commutative=True))))"], "Equality(log(Symbol('y', commutative=True)), Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), log(Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Mul(Integer(3), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), log(Symbol('y', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), log(Symbol('y', commutative=True)), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('x', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), log(Symbol('y', commutative=True))), Mul(Integer(-1), Pow(log(Symbol('y', commutative=True)), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given H{(\\varepsilon,y^{\\prime},v_{z})} = \\varepsilon^{v_{z}} - y^{\\prime}, then derive \\hat{x}_0 + v_{z} = \\int \\frac{\\varepsilon^{v_{z}} - y^{\\prime}}{H{(\\varepsilon,y^{\\prime},v_{z})}} dv_{z}, then obtain \\hat{x}_0 + v_{z} = \\int 1 dv_{z}", "derivation": "H{(\\varepsilon,y^{\\prime},v_{z})} = \\varepsilon^{v_{z}} - y^{\\prime} and 1 = \\frac{\\varepsilon^{v_{z}} - y^{\\prime}}{H{(\\varepsilon,y^{\\prime},v_{z})}} and \\int 1 dv_{z} = \\int \\frac{\\varepsilon^{v_{z}} - y^{\\prime}}{H{(\\varepsilon,y^{\\prime},v_{z})}} dv_{z} and \\hat{x}_0 + v_{z} = \\int \\frac{\\varepsilon^{v_{z}} - y^{\\prime}}{H{(\\varepsilon,y^{\\prime},v_{z})}} dv_{z} and \\hat{x}_0 + v_{z} = \\int 1 dv_{z}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\varepsilon', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('v_z', commutative=True)), Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Function('H')(Symbol('\\\\varepsilon', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Integer(1), Mul(Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('H')(Symbol('\\\\varepsilon', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v_z', commutative=True))), Integral(Mul(Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('H')(Symbol('\\\\varepsilon', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Integral(Mul(Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Pow(Function('H')(Symbol('\\\\varepsilon', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('v_z', commutative=True)), Integer(-1))), Tuple(Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Integral(Integer(1), Tuple(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(J,M_{E})} = M_{E}^{J} and \\psi{(J,M_{E})} = M_{E}^{J}, then obtain \\psi{(J,M_{E})} + \\frac{\\tilde{g}{(J,M_{E})}}{J} = M_{E}^{J} + \\frac{\\tilde{g}{(J,M_{E})}}{J}", "derivation": "\\tilde{g}{(J,M_{E})} = M_{E}^{J} and \\frac{\\tilde{g}{(J,M_{E})}}{J} = \\frac{M_{E}^{J}}{J} and \\psi{(J,M_{E})} = M_{E}^{J} and \\psi{(J,M_{E})} + \\frac{M_{E}^{J}}{J} = M_{E}^{J} + \\frac{M_{E}^{J}}{J} and \\psi{(J,M_{E})} + \\frac{\\tilde{g}{(J,M_{E})}}{J} = M_{E}^{J} + \\frac{\\tilde{g}{(J,M_{E})}}{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)))"], [["divide", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)))"], [["add", 3, "Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)))"], "Equality(Add(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)))), Add(Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\psi')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)))), Add(Pow(Symbol('M_E', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('\\\\tilde{g}')(Symbol('J', commutative=True), Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(t,z)} = z^{t}, then obtain \\operatorname{C_{d}}{(t,z)} = (\\frac{z^{t}}{\\operatorname{C_{d}}{(t,z)}})^{t} \\operatorname{C_{d}}{(t,z)}", "derivation": "\\operatorname{C_{d}}{(t,z)} = z^{t} and 1 = \\frac{z^{t}}{\\operatorname{C_{d}}{(t,z)}} and 1 = (\\frac{z^{t}}{\\operatorname{C_{d}}{(t,z)}})^{t} and \\operatorname{C_{d}}{(t,z)} = (\\frac{z^{t}}{\\operatorname{C_{d}}{(t,z)}})^{t} \\operatorname{C_{d}}{(t,z)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('t', commutative=True)))"], [["divide", 1, "Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('z', commutative=True), Symbol('t', commutative=True)), Pow(Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Symbol('z', commutative=True), Symbol('t', commutative=True)), Pow(Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True)), Integer(-1))), Symbol('t', commutative=True)))"], [["times", 3, "Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True))"], "Equality(Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Mul(Pow(Symbol('z', commutative=True), Symbol('t', commutative=True)), Pow(Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True)), Integer(-1))), Symbol('t', commutative=True)), Function('C_d')(Symbol('t', commutative=True), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(v_{x},\\mathbf{p})} = v_{x}^{\\mathbf{p}}, then obtain - v_{x} + \\cos{(\\int \\dot{\\mathbf{r}}^{v_{x}}{(v_{x},\\mathbf{p})} d\\mathbf{p})} = - v_{x} + \\cos{(\\int (v_{x}^{\\mathbf{p}})^{v_{x}} d\\mathbf{p})}", "derivation": "\\dot{\\mathbf{r}}{(v_{x},\\mathbf{p})} = v_{x}^{\\mathbf{p}} and \\dot{\\mathbf{r}}^{v_{x}}{(v_{x},\\mathbf{p})} = (v_{x}^{\\mathbf{p}})^{v_{x}} and \\int \\dot{\\mathbf{r}}^{v_{x}}{(v_{x},\\mathbf{p})} d\\mathbf{p} = \\int (v_{x}^{\\mathbf{p}})^{v_{x}} d\\mathbf{p} and \\cos{(\\int \\dot{\\mathbf{r}}^{v_{x}}{(v_{x},\\mathbf{p})} d\\mathbf{p})} = \\cos{(\\int (v_{x}^{\\mathbf{p}})^{v_{x}} d\\mathbf{p})} and - v_{x} + \\cos{(\\int \\dot{\\mathbf{r}}^{v_{x}}{(v_{x},\\mathbf{p})} d\\mathbf{p})} = - v_{x} + \\cos{(\\int (v_{x}^{\\mathbf{p}})^{v_{x}} d\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Pow(Pow(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Pow(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), cos(Integral(Pow(Pow(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["minus", 4, "Symbol('v_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), cos(Integral(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), cos(Integral(Pow(Pow(Symbol('v_x', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{F}{(\\delta,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\delta})} and \\omega{(Q)} = \\log{(e^{Q})}, then obtain \\delta \\mathbf{F}{(\\delta,\\mathbf{f})} + \\log{(e^{Q})} + 1 = \\delta \\cos{(\\frac{\\mathbf{f}}{\\delta})} + \\log{(e^{Q})} + 1", "derivation": "\\mathbf{F}{(\\delta,\\mathbf{f})} = \\cos{(\\frac{\\mathbf{f}}{\\delta})} and \\delta \\mathbf{F}{(\\delta,\\mathbf{f})} = \\delta \\cos{(\\frac{\\mathbf{f}}{\\delta})} and \\delta \\mathbf{F}{(\\delta,\\mathbf{f})} + 1 = \\delta \\cos{(\\frac{\\mathbf{f}}{\\delta})} + 1 and \\omega{(Q)} = \\log{(e^{Q})} and \\delta \\mathbf{F}{(\\delta,\\mathbf{f})} + \\omega{(Q)} + 1 = \\delta \\cos{(\\frac{\\mathbf{f}}{\\delta})} + \\omega{(Q)} + 1 and \\delta \\mathbf{F}{(\\delta,\\mathbf{f})} + \\log{(e^{Q})} + 1 = \\delta \\cos{(\\frac{\\mathbf{f}}{\\delta})} + \\log{(e^{Q})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), cos(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), cos(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\delta', commutative=True), cos(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), Integer(1)))"], ["get_premise", "Equality(Function('\\\\omega')(Symbol('Q', commutative=True)), log(exp(Symbol('Q', commutative=True))))"], [["add", 3, "Function('\\\\omega')(Symbol('Q', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Function('\\\\omega')(Symbol('Q', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\delta', commutative=True), cos(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), Function('\\\\omega')(Symbol('Q', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Symbol('\\\\delta', commutative=True), Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), log(exp(Symbol('Q', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\delta', commutative=True), cos(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mathbf{f}', commutative=True)))), log(exp(Symbol('Q', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} = \\log{(J_{\\varepsilon} \\hat{p}_0)}, then obtain \\frac{(2 \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)})^{\\hat{p}_0}}{\\hat{p}_0} = \\frac{(\\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} + \\log{(J_{\\varepsilon} \\hat{p}_0)})^{\\hat{p}_0}}{\\hat{p}_0}", "derivation": "\\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} = \\log{(J_{\\varepsilon} \\hat{p}_0)} and 2 \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} = \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} + \\log{(J_{\\varepsilon} \\hat{p}_0)} and (2 \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)})^{\\hat{p}_0} = (\\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} + \\log{(J_{\\varepsilon} \\hat{p}_0)})^{\\hat{p}_0} and \\frac{(2 \\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)})^{\\hat{p}_0}}{\\hat{p}_0} = \\frac{(\\operatorname{t_{1}}{(J_{\\varepsilon},\\hat{p}_0)} + \\log{(J_{\\varepsilon} \\hat{p}_0)})^{\\hat{p}_0}}{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["add", 1, "Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Add(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True)))"], [["divide", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Mul(Integer(2), Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Add(Function('t_1')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), log(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\psi)} = \\log{(e^{\\psi})}, then obtain (\\sigma_{p}{(\\psi)} - \\log{(e^{\\psi})}) \\int (\\sigma_{p}{(\\psi)} - e^{\\psi}) d\\psi = 0", "derivation": "\\sigma_{p}{(\\psi)} = \\log{(e^{\\psi})} and \\sigma_{p}{(\\psi)} - e^{\\psi} = - e^{\\psi} + \\log{(e^{\\psi})} and \\int (\\sigma_{p}{(\\psi)} - e^{\\psi}) d\\psi = \\int (- e^{\\psi} + \\log{(e^{\\psi})}) d\\psi and \\sigma_{p}{(\\psi)} - \\log{(e^{\\psi})} = 0 and (\\sigma_{p}{(\\psi)} - \\log{(e^{\\psi})}) \\int (- e^{\\psi} + \\log{(e^{\\psi})}) d\\psi = 0 and (\\sigma_{p}{(\\psi)} - \\log{(e^{\\psi})}) \\int (\\sigma_{p}{(\\psi)} - e^{\\psi}) d\\psi = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), log(exp(Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\psi', commutative=True))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), log(exp(Symbol('\\\\psi', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), log(exp(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["minus", 1, "log(exp(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\psi', commutative=True))))), Integer(0))"], [["times", 4, "Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), log(exp(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Mul(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\psi', commutative=True))))), Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True))), log(exp(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), log(exp(Symbol('\\\\psi', commutative=True))))), Integral(Add(Function('\\\\sigma_p')(Symbol('\\\\psi', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\psi', commutative=True)))), Tuple(Symbol('\\\\psi', commutative=True)))), Integer(0))"]]}, {"prompt": "Given s{(z^{*},S)} = \\cos{(S z^{*})}, then obtain \\frac{\\partial}{\\partial S} \\int s{(z^{*},S)} dS + \\int s{(z^{*},S)} dz^{*} = \\frac{\\partial}{\\partial S} \\int \\cos{(S z^{*})} dS + \\int s{(z^{*},S)} dz^{*}", "derivation": "s{(z^{*},S)} = \\cos{(S z^{*})} and \\int s{(z^{*},S)} dS = \\int \\cos{(S z^{*})} dS and \\int s{(z^{*},S)} dz^{*} = \\int \\cos{(S z^{*})} dz^{*} and \\frac{\\partial}{\\partial S} \\int s{(z^{*},S)} dS = \\frac{\\partial}{\\partial S} \\int \\cos{(S z^{*})} dS and \\frac{\\partial}{\\partial S} \\int s{(z^{*},S)} dS + \\int \\cos{(S z^{*})} dz^{*} = \\frac{\\partial}{\\partial S} \\int \\cos{(S z^{*})} dS + \\int \\cos{(S z^{*})} dz^{*} and \\frac{\\partial}{\\partial S} \\int s{(z^{*},S)} dS + \\int s{(z^{*},S)} dz^{*} = \\frac{\\partial}{\\partial S} \\int \\cos{(S z^{*})} dS + \\int s{(z^{*},S)} dz^{*}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 4, "Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))"], "Equality(Add(Derivative(Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))), Add(Derivative(Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Derivative(Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('z^*', commutative=True)))), Add(Derivative(Integral(cos(Mul(Symbol('S', commutative=True), Symbol('z^*', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Integral(Function('s')(Symbol('z^*', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(z^{*})} = \\sin{(z^{*})}, then derive \\int \\operatorname{A_{y}}{(z^{*})} dz^{*} = I - \\cos{(z^{*})}, then obtain \\int \\sin{(z^{*})} dz^{*} = I - \\cos{(z^{*})}", "derivation": "\\operatorname{A_{y}}{(z^{*})} = \\sin{(z^{*})} and \\int \\operatorname{A_{y}}{(z^{*})} dz^{*} = \\int \\sin{(z^{*})} dz^{*} and \\int \\operatorname{A_{y}}{(z^{*})} dz^{*} = I - \\cos{(z^{*})} and \\int \\sin{(z^{*})} dz^{*} = I - \\cos{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('A_y')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_y')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('I', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\phi{(n_{2})} = \\log{(e^{n_{2}})}, then derive v_{2} + \\phi{(n_{2})} = n_{2} + v, then obtain - (n_{2} + v)^{v_{2}} + (v_{2} + \\phi{(n_{2})})^{v_{2}} = 0", "derivation": "\\phi{(n_{2})} = \\log{(e^{n_{2}})} and \\frac{d}{d n_{2}} \\phi{(n_{2})} = \\frac{d}{d n_{2}} \\log{(e^{n_{2}})} and \\int \\frac{d}{d n_{2}} \\phi{(n_{2})} dn_{2} = \\int \\frac{d}{d n_{2}} \\log{(e^{n_{2}})} dn_{2} and v_{2} + \\phi{(n_{2})} = n_{2} + v and (v_{2} + \\phi{(n_{2})})^{v_{2}} = (n_{2} + v)^{v_{2}} and - (n_{2} + v)^{v_{2}} + (v_{2} + \\phi{(n_{2})})^{v_{2}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('n_2', commutative=True)), log(exp(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('\\\\phi')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(log(exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('n_2', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))), Integral(Derivative(log(exp(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Tuple(Symbol('n_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True))), Add(Symbol('n_2', commutative=True), Symbol('v', commutative=True)))"], [["power", 4, "Symbol('v_2', commutative=True)"], "Equality(Pow(Add(Symbol('v_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True))), Symbol('v_2', commutative=True)), Pow(Add(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Symbol('v_2', commutative=True)))"], [["minus", 5, "Pow(Add(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('n_2', commutative=True), Symbol('v', commutative=True)), Symbol('v_2', commutative=True))), Pow(Add(Symbol('v_2', commutative=True), Function('\\\\phi')(Symbol('n_2', commutative=True))), Symbol('v_2', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} = \\log{(f_{\\mathbf{p}} x^\\prime)} and \\Psi^{\\dagger}{(f_{\\mathbf{p}},x^\\prime)} = \\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)}, then obtain \\sin{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)})} = \\sin{(\\Psi^{\\dagger}{(f_{\\mathbf{p}},x^\\prime)})}", "derivation": "\\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} = \\log{(f_{\\mathbf{p}} x^\\prime)} and \\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)} = 2 \\log{(f_{\\mathbf{p}} x^\\prime)} and \\sin{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)})} = \\sin{(2 \\log{(f_{\\mathbf{p}} x^\\prime)})} and \\Psi^{\\dagger}{(f_{\\mathbf{p}},x^\\prime)} = \\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)} and \\Psi^{\\dagger}{(f_{\\mathbf{p}},x^\\prime)} = 2 \\log{(f_{\\mathbf{p}} x^\\prime)} and \\sin{(\\operatorname{F_{x}}{(f_{\\mathbf{p}},x^\\prime)} + \\log{(f_{\\mathbf{p}} x^\\prime)})} = \\sin{(\\Psi^{\\dagger}{(f_{\\mathbf{p}},x^\\prime)})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], "Equality(Add(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(2), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["sin", 2], "Equality(sin(Add(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True))))), sin(Mul(Integer(2), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(sin(Add(Function('F_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True)), log(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True))))), sin(Function('\\\\Psi^{\\\\dagger}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\Psi{(\\hat{\\mathbf{r}})} = \\cos{(\\log{(\\hat{\\mathbf{r}})})}, then obtain 0 = \\frac{- 2 \\Psi{(\\hat{\\mathbf{r}})} + 2 \\cos{(\\log{(\\hat{\\mathbf{r}})})}}{\\log{(\\hat{\\mathbf{r}})}}", "derivation": "\\Psi{(\\hat{\\mathbf{r}})} = \\cos{(\\log{(\\hat{\\mathbf{r}})})} and 0 = - \\Psi{(\\hat{\\mathbf{r}})} + \\cos{(\\log{(\\hat{\\mathbf{r}})})} and - \\Psi{(\\hat{\\mathbf{r}})} = - 2 \\Psi{(\\hat{\\mathbf{r}})} + \\cos{(\\log{(\\hat{\\mathbf{r}})})} and 0 = \\frac{- \\Psi{(\\hat{\\mathbf{r}})} + \\cos{(\\log{(\\hat{\\mathbf{r}})})}}{\\log{(\\hat{\\mathbf{r}})}} and 0 = \\frac{- 2 \\Psi{(\\hat{\\mathbf{r}})} + 2 \\cos{(\\log{(\\hat{\\mathbf{r}})})}}{\\log{(\\hat{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["minus", 1, "Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), cos(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["minus", 2, "Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), cos(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["divide", 2, "log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), cos(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Pow(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integer(2), Function('\\\\Psi')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Integer(2), cos(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))), Pow(log(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given p{(m)} = e^{e^{m}} and \\dot{z}{(m)} = 2 m p{(m)} e^{- e^{m}}, then derive \\frac{d}{d m} \\dot{z}{(m)} = 2, then obtain m p^{3}{(m)} e^{- 3 e^{m}} \\frac{d}{d m} \\dot{z}{(m)} = m (p{(m)} + e^{e^{m}}) e^{- e^{m}}", "derivation": "p{(m)} = e^{e^{m}} and 2 p{(m)} = p{(m)} + e^{e^{m}} and 2 m p{(m)} e^{- e^{m}} = m (p{(m)} + e^{e^{m}}) e^{- e^{m}} and \\dot{z}{(m)} = 2 m p{(m)} e^{- e^{m}} and \\dot{z}{(m)} = 2 m and \\frac{d}{d m} \\dot{z}{(m)} = \\frac{d}{d m} 2 m and \\frac{d}{d m} \\dot{z}{(m)} = 2 and 2 m p{(m)} e^{- e^{m}} = 2 m and 2 m p^{3}{(m)} e^{- 3 e^{m}} = m (p{(m)} + e^{e^{m}}) e^{- e^{m}} and m p^{3}{(m)} e^{- 3 e^{m}} \\frac{d}{d m} \\dot{z}{(m)} = m (p{(m)} + e^{e^{m}}) e^{- e^{m}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('m', commutative=True)), exp(exp(Symbol('m', commutative=True))))"], [["add", 1, "Function('p')(Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Function('p')(Symbol('m', commutative=True))), Add(Function('p')(Symbol('m', commutative=True)), exp(exp(Symbol('m', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('m', commutative=True), Integer(-1)), exp(exp(Symbol('m', commutative=True))))"], "Equality(Mul(Integer(2), Symbol('m', commutative=True), Function('p')(Symbol('m', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))), Mul(Symbol('m', commutative=True), Add(Function('p')(Symbol('m', commutative=True)), exp(exp(Symbol('m', commutative=True)))), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Mul(Integer(2), Symbol('m', commutative=True), Function('p')(Symbol('m', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Mul(Integer(2), Symbol('m', commutative=True)))"], [["differentiate", 5, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(2))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Symbol('m', commutative=True), Function('p')(Symbol('m', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))), Mul(Integer(2), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 8], "Equality(Mul(Integer(2), Symbol('m', commutative=True), Pow(Function('p')(Symbol('m', commutative=True)), Integer(3)), exp(Mul(Integer(-1), Integer(3), exp(Symbol('m', commutative=True))))), Mul(Symbol('m', commutative=True), Add(Function('p')(Symbol('m', commutative=True)), exp(exp(Symbol('m', commutative=True)))), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))))"], [["substitute_RHS_for_LHS", 9, 7], "Equality(Mul(Symbol('m', commutative=True), Pow(Function('p')(Symbol('m', commutative=True)), Integer(3)), exp(Mul(Integer(-1), Integer(3), exp(Symbol('m', commutative=True)))), Derivative(Function('\\\\dot{z}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Symbol('m', commutative=True), Add(Function('p')(Symbol('m', commutative=True)), exp(exp(Symbol('m', commutative=True)))), exp(Mul(Integer(-1), exp(Symbol('m', commutative=True))))))"]]}, {"prompt": "Given C{(\\mathbf{E})} = e^{\\mathbf{E}} and \\mathbf{r}{(\\mathbf{E})} = e^{\\mathbf{E}}, then obtain e^{\\mathbf{E}} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} = \\mathbf{r}{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})}", "derivation": "C{(\\mathbf{E})} = e^{\\mathbf{E}} and \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} and C{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} = e^{\\mathbf{E}} + \\frac{d}{d \\mathbf{E}} e^{\\mathbf{E}} and C{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} = e^{\\mathbf{E}} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} and \\mathbf{r}{(\\mathbf{E})} = e^{\\mathbf{E}} and C{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} = \\mathbf{r}{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} and e^{\\mathbf{E}} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})} = \\mathbf{r}{(\\mathbf{E})} + \\frac{d}{d \\mathbf{E}} C{(\\mathbf{E})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Add(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(exp(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(exp(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), Add(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(Function('C')(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\nabla{(t_{2},\\hat{\\mathbf{r}})} = \\frac{t_{2}}{\\hat{\\mathbf{r}}}, then obtain e^{- \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}} \\frac{\\partial}{\\partial t_{2}} \\frac{\\nabla{(t_{2},\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = e^{- \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}} \\frac{\\partial}{\\partial t_{2}} \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}", "derivation": "\\nabla{(t_{2},\\hat{\\mathbf{r}})} = \\frac{t_{2}}{\\hat{\\mathbf{r}}} and \\frac{\\nabla{(t_{2},\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}} and \\frac{\\partial}{\\partial t_{2}} \\frac{\\nabla{(t_{2},\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{\\partial}{\\partial t_{2}} \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}} and e^{- \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}} \\frac{\\partial}{\\partial t_{2}} \\frac{\\nabla{(t_{2},\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = e^{- \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}} \\frac{\\partial}{\\partial t_{2}} \\frac{t_{2}}{\\hat{\\mathbf{r}}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Symbol('t_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True)))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["divide", 3, "exp(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True))), Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('\\\\nabla')(Symbol('t_2', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True))), Derivative(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(F_{c},\\mathbf{f})} = e^{\\frac{F_{c}}{\\mathbf{f}}}, then obtain \\frac{\\partial}{\\partial F_{c}} (a^{\\dagger} + M{(F_{c},\\mathbf{f})}) = \\frac{\\partial}{\\partial F_{c}} (E_{\\lambda} + e^{\\frac{F_{c}}{\\mathbf{f}}})", "derivation": "M{(F_{c},\\mathbf{f})} = e^{\\frac{F_{c}}{\\mathbf{f}}} and \\frac{\\partial}{\\partial \\mathbf{f}} M{(F_{c},\\mathbf{f})} = \\frac{\\partial}{\\partial \\mathbf{f}} e^{\\frac{F_{c}}{\\mathbf{f}}} and \\int \\frac{\\partial}{\\partial \\mathbf{f}} M{(F_{c},\\mathbf{f})} d\\mathbf{f} = \\int \\frac{\\partial}{\\partial \\mathbf{f}} e^{\\frac{F_{c}}{\\mathbf{f}}} d\\mathbf{f} and \\frac{\\partial}{\\partial F_{c}} \\int \\frac{\\partial}{\\partial \\mathbf{f}} M{(F_{c},\\mathbf{f})} d\\mathbf{f} = \\frac{\\partial}{\\partial F_{c}} \\int \\frac{\\partial}{\\partial \\mathbf{f}} e^{\\frac{F_{c}}{\\mathbf{f}}} d\\mathbf{f} and \\frac{\\partial}{\\partial F_{c}} (a^{\\dagger} + M{(F_{c},\\mathbf{f})}) = \\frac{\\partial}{\\partial F_{c}} (E_{\\lambda} + e^{\\frac{F_{c}}{\\mathbf{f}}})", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), exp(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Derivative(Function('M')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Derivative(exp(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 3, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('M')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Integral(Derivative(exp(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('M')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), exp(Mul(Symbol('F_c', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\sigma_x)} = \\log{(\\sigma_x)}, then obtain \\frac{d^{2}}{d \\sigma_x^{2}} (\\sigma_x C{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}) = \\frac{d^{2}}{d \\sigma_x^{2}} (\\sigma_x \\log{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)})", "derivation": "C{(\\sigma_x)} = \\log{(\\sigma_x)} and \\sigma_x C{(\\sigma_x)} = \\sigma_x \\log{(\\sigma_x)} and \\sigma_x C{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} = \\sigma_x \\log{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)} and \\frac{d}{d \\sigma_x} (\\sigma_x C{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}) = \\frac{d}{d \\sigma_x} (\\sigma_x \\log{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}) and \\frac{d^{2}}{d \\sigma_x^{2}} (\\sigma_x C{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)}) = \\frac{d^{2}}{d \\sigma_x^{2}} (\\sigma_x \\log{(\\sigma_x)} + \\frac{d}{d \\sigma_x} \\log{(\\sigma_x)})", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["times", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 2, "Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('\\\\sigma_x', commutative=True), log(Symbol('\\\\sigma_x', commutative=True))), Derivative(log(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(g)} = \\cos{(g)} and m{(g)} = - \\cos{(g)}, then obtain - \\operatorname{f_{\\mathbf{p}}}{(g)} = - \\cos{(g)}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(g)} = \\cos{(g)} and m{(g)} = - \\cos{(g)} and m{(g)} = - \\operatorname{f_{\\mathbf{p}}}{(g)} and - \\operatorname{f_{\\mathbf{p}}}{(g)} = - \\cos{(g)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('m')(Symbol('g', commutative=True)), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('g', commutative=True)), Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Function('f_{\\\\mathbf{p}}')(Symbol('g', commutative=True))), Mul(Integer(-1), cos(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\phi{(f_{\\mathbf{v}},\\lambda)} = \\lambda + f_{\\mathbf{v}} and \\Psi_{nl}{(f_{\\mathbf{v}},\\lambda)} = (\\lambda + f_{\\mathbf{v}}) \\phi{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)}, then obtain \\phi^{2}{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)} = \\Psi_{nl}{(f_{\\mathbf{v}},\\lambda)}", "derivation": "\\phi{(f_{\\mathbf{v}},\\lambda)} = \\lambda + f_{\\mathbf{v}} and \\phi^{2}{(f_{\\mathbf{v}},\\lambda)} = (\\lambda + f_{\\mathbf{v}}) \\phi{(f_{\\mathbf{v}},\\lambda)} and \\phi^{2}{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)} = (\\lambda + f_{\\mathbf{v}}) \\phi{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)} and \\Psi_{nl}{(f_{\\mathbf{v}},\\lambda)} = (\\lambda + f_{\\mathbf{v}}) \\phi{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)} and \\phi^{2}{(f_{\\mathbf{v}},\\lambda)} + \\phi{(f_{\\mathbf{v}},\\lambda)} = \\Psi_{nl}{(f_{\\mathbf{v}},\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 1, "Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Pow(Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["add", 2, "Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Pow(Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Add(Symbol('\\\\lambda', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(2)), Function('\\\\phi')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True))), Function('\\\\Psi_{nl}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\lambda', commutative=True)))"]]}, {"prompt": "Given B{(\\sigma_x,\\Omega,p)} = p (\\Omega + \\sigma_x) and \\operatorname{F_{x}}{(\\sigma_x,\\Omega,p)} = \\sigma_x B{(\\sigma_x,\\Omega,p)}, then obtain \\frac{\\partial}{\\partial p} \\operatorname{F_{x}}{(\\sigma_x,\\Omega,p)} = \\frac{\\partial}{\\partial p} \\sigma_x p (\\Omega + \\sigma_x)", "derivation": "B{(\\sigma_x,\\Omega,p)} = p (\\Omega + \\sigma_x) and \\sigma_x B{(\\sigma_x,\\Omega,p)} = \\sigma_x p (\\Omega + \\sigma_x) and \\operatorname{F_{x}}{(\\sigma_x,\\Omega,p)} = \\sigma_x B{(\\sigma_x,\\Omega,p)} and \\operatorname{F_{x}}{(\\sigma_x,\\Omega,p)} = \\sigma_x p (\\Omega + \\sigma_x) and \\frac{\\partial}{\\partial p} \\operatorname{F_{x}}{(\\sigma_x,\\Omega,p)} = \\frac{\\partial}{\\partial p} \\sigma_x p (\\Omega + \\sigma_x)", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["times", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Symbol('\\\\sigma_x', commutative=True), Function('B')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('p', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Function('B')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('F_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('p', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["differentiate", 4, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\Omega', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\sigma_x', commutative=True), Symbol('p', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\mathbf{H})} = \\mathbf{H}, then obtain ((\\operatorname{f^{*}}^{\\mathbf{H}}{(\\mathbf{H})})^{\\mathbf{H}})^{\\mathbf{H}} = ((\\mathbf{H}^{\\mathbf{H}})^{\\mathbf{H}})^{\\mathbf{H}}", "derivation": "\\operatorname{f^{*}}{(\\mathbf{H})} = \\mathbf{H} and \\operatorname{f^{*}}^{\\mathbf{H}}{(\\mathbf{H})} = \\mathbf{H}^{\\mathbf{H}} and (\\operatorname{f^{*}}^{\\mathbf{H}}{(\\mathbf{H})})^{\\mathbf{H}} = (\\mathbf{H}^{\\mathbf{H}})^{\\mathbf{H}} and ((\\operatorname{f^{*}}^{\\mathbf{H}}{(\\mathbf{H})})^{\\mathbf{H}})^{\\mathbf{H}} = ((\\mathbf{H}^{\\mathbf{H}})^{\\mathbf{H}})^{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('f^*')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Pow(Function('f^*')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Pow(Pow(Function('f^*')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Pow(Pow(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(c,m)} = c + m, then obtain \\operatorname{F_{x}}{(c,m)} \\int (c + m)^{c} dc = (c + m) \\int (c + m)^{c} dc", "derivation": "\\operatorname{F_{x}}{(c,m)} = c + m and \\operatorname{F_{x}}^{c}{(c,m)} = (c + m)^{c} and \\int \\operatorname{F_{x}}^{c}{(c,m)} dc = \\int (c + m)^{c} dc and \\operatorname{F_{x}}{(c,m)} \\int \\operatorname{F_{x}}^{c}{(c,m)} dc = (c + m) \\int \\operatorname{F_{x}}^{c}{(c,m)} dc and \\operatorname{F_{x}}{(c,m)} \\int (c + m)^{c} dc = (c + m) \\int (c + m)^{c} dc", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Pow(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["times", 1, "Integral(Pow(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Mul(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Integral(Pow(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Integral(Pow(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('F_x')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Integral(Pow(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Mul(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Integral(Pow(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\dot{x})} = \\log{(\\dot{x})}, then derive 1 = \\frac{\\dot{x} \\log{(\\dot{x})} - \\dot{x} + g}{\\int \\psi{(\\dot{x})} d\\dot{x}}, then obtain - c + \\int \\log{(\\dot{x})} d\\dot{x} = \\dot{x} \\log{(\\dot{x})} - \\dot{x} - c + g", "derivation": "\\psi{(\\dot{x})} = \\log{(\\dot{x})} and \\int \\psi{(\\dot{x})} d\\dot{x} = \\int \\log{(\\dot{x})} d\\dot{x} and 1 = \\frac{\\int \\log{(\\dot{x})} d\\dot{x}}{\\int \\psi{(\\dot{x})} d\\dot{x}} and 1 = \\frac{\\dot{x} \\log{(\\dot{x})} - \\dot{x} + g}{\\int \\psi{(\\dot{x})} d\\dot{x}} and \\int \\log{(\\dot{x})} d\\dot{x} = \\frac{(\\dot{x} \\log{(\\dot{x})} - \\dot{x} + g) \\int \\log{(\\dot{x})} d\\dot{x}}{\\int \\psi{(\\dot{x})} d\\dot{x}} and \\int \\log{(\\dot{x})} d\\dot{x} = \\dot{x} \\log{(\\dot{x})} - \\dot{x} + g and - c + \\int \\log{(\\dot{x})} d\\dot{x} = \\dot{x} \\log{(\\dot{x})} - \\dot{x} - c + g", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["divide", 2, "Integral(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(1), Mul(Add(Mul(Symbol('\\\\dot{x}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"], [["times", 4, "Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))"], "Equality(Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Mul(Add(Mul(Symbol('\\\\dot{x}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(Function('\\\\psi')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Symbol('g', commutative=True)))"], [["minus", 6, "Symbol('c', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Integral(log(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), log(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(x^\\prime)} = \\sin{(x^\\prime)}, then derive \\frac{d}{d x^\\prime} \\int \\operatorname{v_{2}}{(x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}), then obtain 2 \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) = \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) + \\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime", "derivation": "\\operatorname{v_{2}}{(x^\\prime)} = \\sin{(x^\\prime)} and \\int \\operatorname{v_{2}}{(x^\\prime)} dx^\\prime = \\int \\sin{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} \\int \\operatorname{v_{2}}{(x^\\prime)} dx^\\prime = \\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} \\int \\operatorname{v_{2}}{(x^\\prime)} dx^\\prime = \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) and \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) = \\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime and 2 \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) = \\frac{\\partial}{\\partial x^\\prime} (J - \\cos{(x^\\prime)}) + \\frac{d}{d x^\\prime} \\int \\sin{(x^\\prime)} dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), sin(Symbol('x^\\\\prime', commutative=True)))"], [["integrate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Integral(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('v_2')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Derivative(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["add", 5, "Derivative(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('J', commutative=True), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"]]}, {"prompt": "Given M{(J)} = \\log{(\\log{(J)})}, then derive \\int M{(J)} dJ = J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)}, then obtain \\cos^{\\mathbf{J}}{((\\int M{(J)} dJ)^{J})} = \\cos^{\\mathbf{J}}{((J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)})^{J})}", "derivation": "M{(J)} = \\log{(\\log{(J)})} and \\int M{(J)} dJ = \\int \\log{(\\log{(J)})} dJ and \\int M{(J)} dJ = J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)} and (\\int M{(J)} dJ)^{J} = (J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)})^{J} and \\cos{((\\int M{(J)} dJ)^{J})} = \\cos{((J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)})^{J})} and \\cos^{\\mathbf{J}}{((\\int M{(J)} dJ)^{J})} = \\cos^{\\mathbf{J}}{((J \\log{(\\log{(J)})} + \\mathbf{J} - \\operatorname{li}{(J)})^{J})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('J', commutative=True)), log(log(Symbol('J', commutative=True))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('M')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(log(log(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('M')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Add(Mul(Symbol('J', commutative=True), log(log(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), li(Symbol('J', commutative=True)))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Function('M')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Add(Mul(Symbol('J', commutative=True), log(log(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), li(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"], [["cos", 4], "Equality(cos(Pow(Integral(Function('M')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), cos(Pow(Add(Mul(Symbol('J', commutative=True), log(log(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), li(Symbol('J', commutative=True)))), Symbol('J', commutative=True))))"], [["power", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(cos(Pow(Integral(Function('M')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Pow(cos(Pow(Add(Mul(Symbol('J', commutative=True), log(log(Symbol('J', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), li(Symbol('J', commutative=True)))), Symbol('J', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\hat{X})} = \\frac{1}{\\hat{X}} and \\mathbf{D}{(\\hat{H}_{\\lambda},r_{0})} = \\sin{(\\hat{H}_{\\lambda} - r_{0})}, then obtain \\mathbf{D}{(\\hat{H}_{\\lambda},r_{0})} + \\operatorname{t_{2}}^{\\frac{1}{\\operatorname{t_{2}}{(\\hat{X})}}}{(\\hat{X})} = \\operatorname{t_{2}}^{\\frac{1}{\\operatorname{t_{2}}{(\\hat{X})}}}{(\\hat{X})} + \\sin{(\\hat{H}_{\\lambda} - r_{0})}", "derivation": "\\operatorname{t_{2}}{(\\hat{X})} = \\frac{1}{\\hat{X}} and \\mathbf{D}{(\\hat{H}_{\\lambda},r_{0})} = \\sin{(\\hat{H}_{\\lambda} - r_{0})} and (\\frac{1}{\\hat{X}})^{\\hat{X}} + \\mathbf{D}{(\\hat{H}_{\\lambda},r_{0})} = (\\frac{1}{\\hat{X}})^{\\hat{X}} + \\sin{(\\hat{H}_{\\lambda} - r_{0})} and \\mathbf{D}{(\\hat{H}_{\\lambda},r_{0})} + \\operatorname{t_{2}}^{\\frac{1}{\\operatorname{t_{2}}{(\\hat{X})}}}{(\\hat{X})} = \\operatorname{t_{2}}^{\\frac{1}{\\operatorname{t_{2}}{(\\hat{X})}}}{(\\hat{X})} + \\sin{(\\hat{H}_{\\lambda} - r_{0})}", "srepr_derivation": [["renaming_premise", "Equality(Function('t_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)))"], ["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))))"], [["add", 2, "Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True))), Add(Pow(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Symbol('\\\\hat{X}', commutative=True)), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Pow(Function('t_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(Function('t_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))), Add(Pow(Function('t_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(Function('t_2')(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), sin(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(f,\\delta)} = \\log{(\\delta + f)}, then obtain \\int (- 2 \\delta + \\operatorname{t_{2}}^{2 \\delta}{(f,\\delta)}) d\\delta = \\int (- 2 \\delta + \\operatorname{t_{2}}^{\\delta}{(f,\\delta)} \\log{(\\delta + f)}^{\\delta}) d\\delta", "derivation": "\\operatorname{t_{2}}{(f,\\delta)} = \\log{(\\delta + f)} and \\operatorname{t_{2}}^{\\delta}{(f,\\delta)} = \\log{(\\delta + f)}^{\\delta} and \\operatorname{t_{2}}^{2 \\delta}{(f,\\delta)} = \\operatorname{t_{2}}^{\\delta}{(f,\\delta)} \\log{(\\delta + f)}^{\\delta} and - 2 \\delta + \\operatorname{t_{2}}^{2 \\delta}{(f,\\delta)} = - 2 \\delta + \\operatorname{t_{2}}^{\\delta}{(f,\\delta)} \\log{(\\delta + f)}^{\\delta} and \\int (- 2 \\delta + \\operatorname{t_{2}}^{2 \\delta}{(f,\\delta)}) d\\delta = \\int (- 2 \\delta + \\operatorname{t_{2}}^{\\delta}{(f,\\delta)} \\log{(\\delta + f)}^{\\delta}) d\\delta", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), log(Add(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Add(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["times", 2, "Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Mul(Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Add(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\delta', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Add(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\delta', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True)), Mul(Pow(Function('t_2')(Symbol('f', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(log(Add(Symbol('\\\\delta', commutative=True), Symbol('f', commutative=True))), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\nabla)} = e^{\\nabla} and I{(\\nabla)} = \\frac{d}{d \\nabla} e^{\\nabla}, then obtain \\cos{(\\mathbf{r})} + \\int I{(\\nabla)} d\\nabla = \\psi + e^{\\nabla} + \\cos{(\\mathbf{r})}", "derivation": "\\sigma_{p}{(\\nabla)} = e^{\\nabla} and I{(\\nabla)} = \\frac{d}{d \\nabla} e^{\\nabla} and I{(\\nabla)} = \\frac{d}{d \\nabla} \\sigma_{p}{(\\nabla)} and \\int I{(\\nabla)} d\\nabla = \\int \\frac{d}{d \\nabla} \\sigma_{p}{(\\nabla)} d\\nabla and \\int I{(\\nabla)} d\\nabla = \\int \\frac{d}{d \\nabla} e^{\\nabla} d\\nabla and \\cos{(\\mathbf{r})} + \\int I{(\\nabla)} d\\nabla = \\cos{(\\mathbf{r})} + \\int \\frac{d}{d \\nabla} e^{\\nabla} d\\nabla and \\cos{(\\mathbf{r})} + \\int I{(\\nabla)} d\\nabla = \\psi + e^{\\nabla} + \\cos{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\nabla', commutative=True)), Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('I')(Symbol('\\\\nabla', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('I')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(Function('\\\\sigma_p')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Function('I')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 5, "cos(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integral(Function('I')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integral(Derivative(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(cos(Symbol('\\\\mathbf{r}', commutative=True)), Integral(Function('I')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('\\\\psi', commutative=True), exp(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain ((- \\varepsilon + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = ((- \\varepsilon + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon}", "derivation": "\\theta_{1}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and - \\varepsilon + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)} = - \\varepsilon + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and (- \\varepsilon + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)})^{\\varepsilon} = (- \\varepsilon + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon} and ((- \\varepsilon + \\frac{d}{d \\varepsilon} \\theta_{1}{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = ((- \\varepsilon + \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\delta{(\\hat{\\mathbf{x}},k)} = \\cos{(\\hat{\\mathbf{x}} k)}, then obtain 0 = \\hat{\\mathbf{x}} k (- \\int \\delta{(\\hat{\\mathbf{x}},k)} d\\hat{\\mathbf{x}} + \\int \\cos{(\\hat{\\mathbf{x}} k)} d\\hat{\\mathbf{x}}) \\cos{(\\hat{\\mathbf{x}} k)}", "derivation": "\\delta{(\\hat{\\mathbf{x}},k)} = \\cos{(\\hat{\\mathbf{x}} k)} and \\int \\delta{(\\hat{\\mathbf{x}},k)} d\\hat{\\mathbf{x}} = \\int \\cos{(\\hat{\\mathbf{x}} k)} d\\hat{\\mathbf{x}} and 0 = - \\int \\delta{(\\hat{\\mathbf{x}},k)} d\\hat{\\mathbf{x}} + \\int \\cos{(\\hat{\\mathbf{x}} k)} d\\hat{\\mathbf{x}} and 0 = \\hat{\\mathbf{x}} k (- \\int \\delta{(\\hat{\\mathbf{x}},k)} d\\hat{\\mathbf{x}} + \\int \\cos{(\\hat{\\mathbf{x}} k)} d\\hat{\\mathbf{x}}) \\cos{(\\hat{\\mathbf{x}} k)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)), cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integral(cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["times", 3, "Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True), cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True))))"], "Equality(Integer(0), Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True), Add(Mul(Integer(-1), Integral(Function('\\\\delta')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integral(cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), cos(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('k', commutative=True)))))"]]}, {"prompt": "Given Q{(H)} = \\sin{(\\log{(H)})} and \\phi_{1}{(H)} = Q{(H)} + \\sin{(\\log{(H)})}, then obtain \\phi_{1}{(H)} = - 2 Q{(H)} + 4 \\sin{(\\log{(H)})}", "derivation": "Q{(H)} = \\sin{(\\log{(H)})} and Q{(H)} + \\sin{(\\log{(H)})} = 2 \\sin{(\\log{(H)})} and \\phi_{1}{(H)} = Q{(H)} + \\sin{(\\log{(H)})} and \\phi_{1}{(H)} = 2 \\sin{(\\log{(H)})} and \\sin{(\\log{(H)})} = - Q{(H)} + 2 \\sin{(\\log{(H)})} and \\phi_{1}{(H)} = - 2 Q{(H)} + 4 \\sin{(\\log{(H)})}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('H', commutative=True)), sin(log(Symbol('H', commutative=True))))"], [["add", 1, "sin(log(Symbol('H', commutative=True)))"], "Equality(Add(Function('Q')(Symbol('H', commutative=True)), sin(log(Symbol('H', commutative=True)))), Mul(Integer(2), sin(log(Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('H', commutative=True)), Add(Function('Q')(Symbol('H', commutative=True)), sin(log(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\phi_1')(Symbol('H', commutative=True)), Mul(Integer(2), sin(log(Symbol('H', commutative=True)))))"], [["minus", 2, "Function('Q')(Symbol('H', commutative=True))"], "Equality(sin(log(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Function('Q')(Symbol('H', commutative=True))), Mul(Integer(2), sin(log(Symbol('H', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('\\\\phi_1')(Symbol('H', commutative=True)), Add(Mul(Integer(-1), Integer(2), Function('Q')(Symbol('H', commutative=True))), Mul(Integer(4), sin(log(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given z{(\\sigma_x)} = \\log{(\\sigma_x)}, then obtain \\log{(\\sigma_x)} \\int (z{(\\sigma_x)} - \\log{(\\sigma_x)}) d\\sigma_x = \\log{(\\sigma_x)} \\int 0 d\\sigma_x", "derivation": "z{(\\sigma_x)} = \\log{(\\sigma_x)} and z{(\\sigma_x)} - \\log{(\\sigma_x)} = 0 and \\int (z{(\\sigma_x)} - \\log{(\\sigma_x)}) d\\sigma_x = \\int 0 d\\sigma_x and \\log{(\\sigma_x)} \\int (z{(\\sigma_x)} - \\log{(\\sigma_x)}) d\\sigma_x = \\log{(\\sigma_x)} \\int 0 d\\sigma_x", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\sigma_x', commutative=True)), log(Symbol('\\\\sigma_x', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Add(Function('z')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Function('z')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 3, "log(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(log(Symbol('\\\\sigma_x', commutative=True)), Integral(Add(Function('z')(Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True)))), Mul(log(Symbol('\\\\sigma_x', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\sigma_x', commutative=True)))))"]]}, {"prompt": "Given T{(v)} = \\log{(\\log{(v)})} and x{(v)} = - \\log{(v)} + \\int T{(v)} dv, then derive - ((- \\log{(v)} + \\int T{(v)} dv)^{v}) \\log{(v)} = - (\\psi^* + v \\log{(\\log{(v)})} - \\log{(v)} - \\operatorname{li}{(v)})^{v} \\log{(v)}, then obtain - x^{v}{(v)} \\log{(v)} = - (\\psi^* + v \\log{(\\log{(v)})} - \\log{(v)} - \\operatorname{li}{(v)})^{v} \\log{(v)}", "derivation": "T{(v)} = \\log{(\\log{(v)})} and \\int T{(v)} dv = \\int \\log{(\\log{(v)})} dv and - \\log{(v)} + \\int T{(v)} dv = - \\log{(v)} + \\int \\log{(\\log{(v)})} dv and (- \\log{(v)} + \\int T{(v)} dv)^{v} = (- \\log{(v)} + \\int \\log{(\\log{(v)})} dv)^{v} and x{(v)} = - \\log{(v)} + \\int T{(v)} dv and - ((- \\log{(v)} + \\int T{(v)} dv)^{v}) \\log{(v)} = - ((- \\log{(v)} + \\int \\log{(\\log{(v)})} dv)^{v}) \\log{(v)} and - ((- \\log{(v)} + \\int T{(v)} dv)^{v}) \\log{(v)} = - (\\psi^* + v \\log{(\\log{(v)})} - \\log{(v)} - \\operatorname{li}{(v)})^{v} \\log{(v)} and - x^{v}{(v)} \\log{(v)} = - (\\psi^* + v \\log{(\\log{(v)})} - \\log{(v)} - \\operatorname{li}{(v)})^{v} \\log{(v)}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('v', commutative=True)), log(log(Symbol('v', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))))"], [["minus", 2, "log(Symbol('v', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('v', commutative=True)), Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), log(Symbol('v', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(log(log(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), log(Symbol('v', commutative=True))), Integral(Function('T')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True)))), Mul(Integer(-1), log(Symbol('v', commutative=True))), Mul(Integer(-1), li(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Mul(Integer(-1), Pow(Function('x')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Symbol('v', commutative=True), log(log(Symbol('v', commutative=True)))), Mul(Integer(-1), log(Symbol('v', commutative=True))), Mul(Integer(-1), li(Symbol('v', commutative=True)))), Symbol('v', commutative=True)), log(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(E_{x})} = \\cos{(\\log{(E_{x})})}, then obtain \\frac{\\sin{(\\hat{X}^{E_{x}}{(E_{x})})} - \\frac{1}{\\mathbf{v}}}{\\operatorname{P_{e}}{(E_{x})}} = \\frac{\\sin{(\\cos^{E_{x}}{(\\log{(E_{x})})})} - \\frac{1}{\\mathbf{v}}}{\\operatorname{P_{e}}{(E_{x})}}", "derivation": "\\hat{X}{(E_{x})} = \\cos{(\\log{(E_{x})})} and \\hat{X}^{E_{x}}{(E_{x})} = \\cos^{E_{x}}{(\\log{(E_{x})})} and \\sin{(\\hat{X}^{E_{x}}{(E_{x})})} = \\sin{(\\cos^{E_{x}}{(\\log{(E_{x})})})} and \\sin{(\\hat{X}^{E_{x}}{(E_{x})})} - \\frac{1}{\\mathbf{v}} = \\sin{(\\cos^{E_{x}}{(\\log{(E_{x})})})} - \\frac{1}{\\mathbf{v}} and \\frac{\\sin{(\\hat{X}^{E_{x}}{(E_{x})})} - \\frac{1}{\\mathbf{v}}}{\\operatorname{P_{e}}{(E_{x})}} = \\frac{\\sin{(\\cos^{E_{x}}{(\\log{(E_{x})})})} - \\frac{1}{\\mathbf{v}}}{\\operatorname{P_{e}}{(E_{x})}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), cos(log(Symbol('E_x', commutative=True))))"], [["power", 1, "Symbol('E_x', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True)), Pow(cos(log(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), sin(Pow(cos(log(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))))"], [["minus", 3, "Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))"], "Equality(Add(sin(Pow(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Add(sin(Pow(cos(log(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))))"], [["divide", 4, "Function('P_e')(Symbol('E_x', commutative=True))"], "Equality(Mul(Add(sin(Pow(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), Symbol('E_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Pow(Function('P_e')(Symbol('E_x', commutative=True)), Integer(-1))), Mul(Add(sin(Pow(cos(log(Symbol('E_x', commutative=True))), Symbol('E_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))), Pow(Function('P_e')(Symbol('E_x', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(v)} = \\sin{(e^{v})} and L{(\\mathbb{I},\\mathbf{J})} = \\mathbb{I} \\mathbf{J}, then obtain \\mathbb{I} \\mathbf{J} \\operatorname{y^{\\prime}}{(v)} = \\mathbb{I} \\mathbf{J} \\sin{(e^{v})}", "derivation": "\\operatorname{y^{\\prime}}{(v)} = \\sin{(e^{v})} and L{(\\mathbb{I},\\mathbf{J})} = \\mathbb{I} \\mathbf{J} and L{(\\mathbb{I},\\mathbf{J})} \\operatorname{y^{\\prime}}{(v)} = L{(\\mathbb{I},\\mathbf{J})} \\sin{(e^{v})} and \\mathbb{I} \\mathbf{J} \\operatorname{y^{\\prime}}{(v)} = \\mathbb{I} \\mathbf{J} \\sin{(e^{v})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('v', commutative=True)), sin(exp(Symbol('v', commutative=True))))"], ["get_premise", "Equality(Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], [["times", 1, "Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Function('y^{\\\\prime}')(Symbol('v', commutative=True))), Mul(Function('L')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(exp(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('y^{\\\\prime}')(Symbol('v', commutative=True))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), sin(exp(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(U)} = \\cos{(U)}, then obtain 0 = -1 + \\frac{\\cos{(\\int \\cos{(U)} dU)}}{\\cos{(\\int \\operatorname{m_{s}}{(U)} dU)}}", "derivation": "\\operatorname{m_{s}}{(U)} = \\cos{(U)} and \\int \\operatorname{m_{s}}{(U)} dU = \\int \\cos{(U)} dU and \\cos{(\\int \\operatorname{m_{s}}{(U)} dU)} = \\cos{(\\int \\cos{(U)} dU)} and 1 = \\frac{\\cos{(\\int \\cos{(U)} dU)}}{\\cos{(\\int \\operatorname{m_{s}}{(U)} dU)}} and 0 = -1 + \\frac{\\cos{(\\int \\cos{(U)} dU)}}{\\cos{(\\int \\operatorname{m_{s}}{(U)} dU)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('m_s')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), cos(Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["divide", 3, "cos(Integral(Function('m_s')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], "Equality(Integer(1), Mul(Pow(cos(Integral(Function('m_s')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integer(-1)), cos(Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))))"], [["add", 4, "Integer(-1)"], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(cos(Integral(Function('m_s')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Integer(-1)), cos(Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))))"]]}, {"prompt": "Given u{(f^{\\prime})} = \\cos{(f^{\\prime})}, then obtain \\frac{d}{d f^{\\prime}} \\frac{\\frac{d}{d f^{\\prime}} u{(f^{\\prime})}}{\\cos{(f^{\\prime})}} = \\frac{d}{d f^{\\prime}} \\frac{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}}{\\cos{(f^{\\prime})}}", "derivation": "u{(f^{\\prime})} = \\cos{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} u{(f^{\\prime})} = \\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})} and \\frac{\\frac{d}{d f^{\\prime}} u{(f^{\\prime})}}{\\cos{(f^{\\prime})}} = \\frac{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}}{\\cos{(f^{\\prime})}} and \\frac{d}{d f^{\\prime}} \\frac{\\frac{d}{d f^{\\prime}} u{(f^{\\prime})}}{\\cos{(f^{\\prime})}} = \\frac{d}{d f^{\\prime}} \\frac{\\frac{d}{d f^{\\prime}} \\cos{(f^{\\prime})}}{\\cos{(f^{\\prime})}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('f^{\\\\prime}', commutative=True)), cos(Symbol('f^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('u')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["divide", 2, "cos(Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(Function('u')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(Function('u')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(cos(Symbol('f^{\\\\prime}', commutative=True)), Integer(-1)), Derivative(cos(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_l{(t_{1})} = t_{1}, then obtain - \\frac{\\hat{H}_l{(t_{1})} - \\frac{d}{d t_{1}} t_{1}}{\\frac{d}{d t_{1}} t_{1}} = - \\frac{t_{1} - \\frac{d}{d t_{1}} t_{1}}{\\frac{d}{d t_{1}} t_{1}}", "derivation": "\\hat{H}_l{(t_{1})} = t_{1} and \\frac{d}{d t_{1}} \\hat{H}_l{(t_{1})} = \\frac{d}{d t_{1}} t_{1} and \\hat{H}_l{(t_{1})} - \\frac{d}{d t_{1}} \\hat{H}_l{(t_{1})} = t_{1} - \\frac{d}{d t_{1}} \\hat{H}_l{(t_{1})} and \\hat{H}_l{(t_{1})} - \\frac{d}{d t_{1}} t_{1} = t_{1} - \\frac{d}{d t_{1}} t_{1} and - \\frac{\\hat{H}_l{(t_{1})} - \\frac{d}{d t_{1}} t_{1}}{\\frac{d}{d t_{1}} t_{1}} = - \\frac{t_{1} - \\frac{d}{d t_{1}} t_{1}}{\\frac{d}{d t_{1}} t_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Derivative(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))))"], [["divide", 4, "Mul(Integer(-1), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\hat{H}_l')(Symbol('t_1', commutative=True)), Mul(Integer(-1), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Pow(Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Pow(Derivative(Symbol('t_1', commutative=True), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given S{(\\mu_0,\\theta_2)} = \\sin{(\\frac{\\mu_0}{\\theta_2})}, then obtain \\tilde{\\infty} \\iint (- \\sin{(S{(\\mu_0,\\theta_2)})} + \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})}) d\\mu_0 d\\theta_2 = \\tilde{\\infty} \\iint 0 d\\mu_0 d\\theta_2", "derivation": "S{(\\mu_0,\\theta_2)} = \\sin{(\\frac{\\mu_0}{\\theta_2})} and - S{(\\mu_0,\\theta_2)} = - \\sin{(\\frac{\\mu_0}{\\theta_2})} and - \\sin{(S{(\\mu_0,\\theta_2)})} = - \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})} and - \\sin{(S{(\\mu_0,\\theta_2)})} + \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})} = 0 and \\int (- \\sin{(S{(\\mu_0,\\theta_2)})} + \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})}) d\\mu_0 = \\int 0 d\\mu_0 and \\iint (- \\sin{(S{(\\mu_0,\\theta_2)})} + \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})}) d\\mu_0 d\\theta_2 = \\iint 0 d\\mu_0 d\\theta_2 and \\tilde{\\infty} \\iint (- \\sin{(S{(\\mu_0,\\theta_2)})} + \\sin{(\\sin{(\\frac{\\mu_0}{\\theta_2})})}) d\\mu_0 d\\theta_2 = \\tilde{\\infty} \\iint 0 d\\mu_0 d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))))"], [["minus", 3, "Mul(Integer(-1), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))))"], "Equality(Add(Mul(Integer(-1), sin(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))), Integer(0))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), sin(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["integrate", 5, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), sin(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))), Tuple(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["divide", 6, 0], "Equality(Mul(zoo, Integral(Add(Mul(Integer(-1), sin(Function('S')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)))), sin(sin(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)))))), Tuple(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(zoo, Integral(Integer(0), Tuple(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\nabla{(x)} = \\sin{(x)}, then obtain - y + (\\nabla{(x)} - \\nabla^{x}{(x)})^{2} = - y + (\\nabla{(x)} - \\nabla^{x}{(x)}) (- \\nabla^{x}{(x)} + \\sin{(x)})", "derivation": "\\nabla{(x)} = \\sin{(x)} and \\nabla^{x}{(x)} = \\sin^{x}{(x)} and \\nabla{(x)} - \\nabla^{x}{(x)} = - \\nabla^{x}{(x)} + \\sin{(x)} and \\nabla{(x)} - \\sin^{x}{(x)} = \\sin{(x)} - \\sin^{x}{(x)} and (\\nabla{(x)} - \\nabla^{x}{(x)}) (\\nabla{(x)} - \\sin^{x}{(x)}) = (\\nabla{(x)} - \\nabla^{x}{(x)}) (\\sin{(x)} - \\sin^{x}{(x)}) and (\\nabla{(x)} - \\nabla^{x}{(x)})^{2} = (\\nabla{(x)} - \\nabla^{x}{(x)}) (- \\nabla^{x}{(x)} + \\sin{(x)}) and - y + (\\nabla{(x)} - \\nabla^{x}{(x)})^{2} = - y + (\\nabla{(x)} - \\nabla^{x}{(x)}) (- \\nabla^{x}{(x)} + \\sin{(x)})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\nabla')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), sin(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True)))))"], [["times", 4, "Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))))"], "Equality(Mul(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))), Mul(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(sin(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(sin(Symbol('x', commutative=True)), Symbol('x', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Integer(2)), Mul(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), sin(Symbol('x', commutative=True)))))"], [["minus", 6, "Symbol('y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y', commutative=True)), Pow(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Integer(2))), Add(Mul(Integer(-1), Symbol('y', commutative=True)), Mul(Add(Function('\\\\nabla')(Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('x', commutative=True)), Symbol('x', commutative=True))), sin(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given Q{(c,f)} = c f, then derive 8 Q{(c,f)} (\\frac{\\partial}{\\partial c} Q{(c,f)})^{2} = 2 (f + \\frac{\\partial}{\\partial c} Q{(c,f)})^{2} Q{(c,f)}, then obtain 8 c f (\\frac{\\partial}{\\partial c} c f)^{2} = 2 c f (f + \\frac{\\partial}{\\partial c} c f)^{2}", "derivation": "Q{(c,f)} = c f and 2 Q{(c,f)} = c f + Q{(c,f)} and \\frac{\\partial}{\\partial c} 2 Q{(c,f)} = \\frac{\\partial}{\\partial c} (c f + Q{(c,f)}) and (\\frac{\\partial}{\\partial c} 2 Q{(c,f)})^{2} = (\\frac{\\partial}{\\partial c} (c f + Q{(c,f)}))^{2} and 2 Q{(c,f)} (\\frac{\\partial}{\\partial c} 2 Q{(c,f)})^{2} = 2 Q{(c,f)} (\\frac{\\partial}{\\partial c} (c f + Q{(c,f)}))^{2} and 8 Q{(c,f)} (\\frac{\\partial}{\\partial c} Q{(c,f)})^{2} = 2 (f + \\frac{\\partial}{\\partial c} Q{(c,f)})^{2} Q{(c,f)} and 8 c f (\\frac{\\partial}{\\partial c} c f)^{2} = 2 c f (f + \\frac{\\partial}{\\partial c} c f)^{2}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)))"], [["add", 1, "Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["power", 3, 2], "Equality(Pow(Derivative(Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2)))"], [["times", 4, "Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Pow(Derivative(Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2))), Mul(Integer(2), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Pow(Derivative(Add(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Mul(Integer(8), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Pow(Derivative(Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2))), Mul(Integer(2), Pow(Add(Symbol('f', commutative=True), Derivative(Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Integer(2)), Function('Q')(Symbol('c', commutative=True), Symbol('f', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Integer(8), Symbol('c', commutative=True), Symbol('f', commutative=True), Pow(Derivative(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(2))), Mul(Integer(2), Symbol('c', commutative=True), Symbol('f', commutative=True), Pow(Add(Symbol('f', commutative=True), Derivative(Mul(Symbol('c', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Integer(2))))"]]}, {"prompt": "Given c{(A_{z})} = \\sin{(A_{z})}, then derive \\int c{(A_{z})} dA_{z} = m_{s} - \\cos{(A_{z})}, then derive - x^\\prime + \\cos{(A_{z})} = - \\int c{(A_{z})} dA_{z}, then obtain - x^\\prime + \\cos{(A_{z})} = - \\int \\sin{(A_{z})} dA_{z}", "derivation": "c{(A_{z})} = \\sin{(A_{z})} and \\int c{(A_{z})} dA_{z} = \\int \\sin{(A_{z})} dA_{z} and \\int c{(A_{z})} dA_{z} = m_{s} - \\cos{(A_{z})} and - \\int c{(A_{z})} dA_{z} = - m_{s} + \\cos{(A_{z})} and - \\int \\sin{(A_{z})} dA_{z} = - m_{s} + \\cos{(A_{z})} and - \\int \\sin{(A_{z})} dA_{z} = - \\int c{(A_{z})} dA_{z} and - x^\\prime + \\cos{(A_{z})} = - \\int c{(A_{z})} dA_{z} and - x^\\prime + \\cos{(A_{z})} = - \\int \\sin{(A_{z})} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('A_z', commutative=True)), sin(Symbol('A_z', commutative=True)))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('c')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('c')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('A_z', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('c')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), cos(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True)), cos(Symbol('A_z', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integer(-1), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))), Mul(Integer(-1), Integral(Function('c')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), cos(Symbol('A_z', commutative=True))), Mul(Integer(-1), Integral(Function('c')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), cos(Symbol('A_z', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(A_{y},h)} = \\cos{(A_{y} h)}, then obtain \\frac{\\partial}{\\partial A_{y}} \\varphi{(A_{y},h)} + 1 = - h \\sin{(A_{y} h)} + 1", "derivation": "\\varphi{(A_{y},h)} = \\cos{(A_{y} h)} and \\frac{\\partial}{\\partial A_{y}} \\varphi{(A_{y},h)} = \\frac{\\partial}{\\partial A_{y}} \\cos{(A_{y} h)} and \\frac{\\partial}{\\partial A_{y}} \\varphi{(A_{y},h)} + 1 = \\frac{\\partial}{\\partial A_{y}} \\cos{(A_{y} h)} + 1 and \\frac{\\partial}{\\partial A_{y}} \\varphi{(A_{y},h)} + 1 = - h \\sin{(A_{y} h)} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('A_y', commutative=True), Symbol('h', commutative=True)), cos(Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\varphi')(Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('\\\\varphi')(Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)), Add(Derivative(cos(Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\varphi')(Symbol('A_y', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(1)), Add(Mul(Integer(-1), Symbol('h', commutative=True), sin(Mul(Symbol('A_y', commutative=True), Symbol('h', commutative=True)))), Integer(1)))"]]}, {"prompt": "Given \\Psi_{nl}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})}, then derive \\frac{d}{d V_{\\mathbf{E}}} \\Psi_{nl}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}, then obtain \\frac{d}{d V_{\\mathbf{E}}} \\sin{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}", "derivation": "\\Psi_{nl}{(V_{\\mathbf{E}})} = \\sin{(V_{\\mathbf{E}})} and \\frac{d}{d V_{\\mathbf{E}}} \\Psi_{nl}{(V_{\\mathbf{E}})} = \\frac{d}{d V_{\\mathbf{E}}} \\sin{(V_{\\mathbf{E}})} and \\frac{d}{d V_{\\mathbf{E}}} \\Psi_{nl}{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})} and \\frac{d}{d V_{\\mathbf{E}}} \\sin{(V_{\\mathbf{E}})} = \\cos{(V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"]]}, {"prompt": "Given y{(q,b)} = b q, then obtain \\frac{(2 b q + \\cos{(\\int b q db)} + \\cos{(\\int y{(q,b)} db)})^{2}}{(y{(q,b)} + \\cos{(\\int b q db)})^{2}} = \\frac{(2 b q + 2 \\cos{(\\int b q db)})^{2}}{(y{(q,b)} + \\cos{(\\int b q db)})^{2}}", "derivation": "y{(q,b)} = b q and \\int y{(q,b)} db = \\int b q db and \\cos{(\\int y{(q,b)} db)} = \\cos{(\\int b q db)} and b q + \\cos{(\\int y{(q,b)} db)} = b q + \\cos{(\\int b q db)} and 2 b q + \\cos{(\\int b q db)} + \\cos{(\\int y{(q,b)} db)} = 2 b q + 2 \\cos{(\\int b q db)} and \\frac{2 b q + \\cos{(\\int b q db)} + \\cos{(\\int y{(q,b)} db)}}{y{(q,b)} + \\cos{(\\int b q db)}} = \\frac{2 b q + 2 \\cos{(\\int b q db)}}{y{(q,b)} + \\cos{(\\int b q db)}} and \\frac{(2 b q + \\cos{(\\int b q db)} + \\cos{(\\int y{(q,b)} db)})^{2}}{(y{(q,b)} + \\cos{(\\int b q db)})^{2}} = \\frac{(2 b q + 2 \\cos{(\\int b q db)})^{2}}{(y{(q,b)} + \\cos{(\\int b q db)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["add", 3, "Mul(Symbol('b', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))), Add(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))))"], [["add", 4, "Add(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], "Equality(Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))), cos(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))), Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))))"], [["divide", 5, "Add(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], "Equality(Mul(Pow(Add(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(-1)), Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))), cos(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))), Mul(Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))), Pow(Add(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(-1))))"], [["power", 6, 2], "Equality(Mul(Pow(Add(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(-2)), Pow(Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))), cos(Integral(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(2))), Mul(Pow(Add(Mul(Integer(2), Symbol('b', commutative=True), Symbol('q', commutative=True)), Mul(Integer(2), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True)))))), Integer(2)), Pow(Add(Function('y')(Symbol('q', commutative=True), Symbol('b', commutative=True)), cos(Integral(Mul(Symbol('b', commutative=True), Symbol('q', commutative=True)), Tuple(Symbol('b', commutative=True))))), Integer(-2))))"]]}, {"prompt": "Given Z{(A_{2},A_{x})} = \\int (A_{2} + A_{x}) dA_{x}, then derive 1 = (\\frac{A_{2} A_{x} + \\frac{A_{x}^{2}}{2} + C_{2}}{Z{(A_{2},A_{x})}})^{A_{2}}, then obtain 1 = (\\frac{A_{2} A_{x} + \\frac{A_{x}^{2}}{2} + C_{2}}{\\int (A_{2} + A_{x}) dA_{x}})^{A_{2}}", "derivation": "Z{(A_{2},A_{x})} = \\int (A_{2} + A_{x}) dA_{x} and 1 = \\frac{\\int (A_{2} + A_{x}) dA_{x}}{Z{(A_{2},A_{x})}} and 1 = (\\frac{\\int (A_{2} + A_{x}) dA_{x}}{Z{(A_{2},A_{x})}})^{A_{2}} and 1 = (\\frac{A_{2} A_{x} + \\frac{A_{x}^{2}}{2} + C_{2}}{Z{(A_{2},A_{x})}})^{A_{2}} and 1 = (\\frac{A_{2} A_{x} + \\frac{A_{x}^{2}}{2} + C_{2}}{\\int (A_{2} + A_{x}) dA_{x}})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))))"], [["divide", 1, "Function('Z')(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Integral(Add(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('Z')(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Integer(-1)), Integral(Add(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Symbol('A_2', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Integer(1), Pow(Mul(Add(Mul(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Symbol('C_2', commutative=True)), Pow(Function('Z')(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Integer(-1))), Symbol('A_2', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Pow(Mul(Add(Mul(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('A_x', commutative=True), Integer(2))), Symbol('C_2', commutative=True)), Pow(Integral(Add(Symbol('A_2', commutative=True), Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integer(-1))), Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given b{(J_{\\varepsilon},y^{\\prime})} = \\frac{y^{\\prime}}{J_{\\varepsilon}} and \\mathbf{g}{(J_{\\varepsilon},y^{\\prime})} = J_{\\varepsilon} b{(J_{\\varepsilon},y^{\\prime})} + y^{\\prime}, then obtain 2 y^{\\prime} = J_{\\varepsilon} b{(J_{\\varepsilon},y^{\\prime})} + y^{\\prime}", "derivation": "b{(J_{\\varepsilon},y^{\\prime})} = \\frac{y^{\\prime}}{J_{\\varepsilon}} and \\mathbf{g}{(J_{\\varepsilon},y^{\\prime})} = J_{\\varepsilon} b{(J_{\\varepsilon},y^{\\prime})} + y^{\\prime} and \\mathbf{g}{(J_{\\varepsilon},y^{\\prime})} = 2 y^{\\prime} and 2 y^{\\prime} = J_{\\varepsilon} b{(J_{\\varepsilon},y^{\\prime})} + y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('b')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{g}')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Function('b')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},\\Psi_{\\lambda})} = \\frac{A_{2}}{\\Psi_{\\lambda}}, then obtain (\\frac{\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1}{\\frac{A_{2}}{\\Psi_{\\lambda}} + 1})^{A_{2}} = 1", "derivation": "\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} = \\frac{A_{2}}{\\Psi_{\\lambda}} and \\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1 = \\frac{A_{2}}{\\Psi_{\\lambda}} + 1 and \\Psi_{\\lambda} (\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1) = \\Psi_{\\lambda} (\\frac{A_{2}}{\\Psi_{\\lambda}} + 1) and - \\Psi_{\\lambda} (\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1) = - \\Psi_{\\lambda} (\\frac{A_{2}}{\\Psi_{\\lambda}} + 1) and \\frac{\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1}{\\frac{A_{2}}{\\Psi_{\\lambda}} + 1} = 1 and (\\frac{\\mathbf{S}{(A_{2},\\Psi_{\\lambda})} + 1}{\\frac{A_{2}}{\\Psi_{\\lambda}} + 1})^{A_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1)), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1)))"], [["divide", 2, "Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1))), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1))), Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1)))"], "Equality(Mul(Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1)), Integer(-1)), Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1))), Integer(1))"], [["power", 5, "Symbol('A_2', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(-1))), Integer(1)), Integer(-1)), Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(1))), Symbol('A_2', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\mathbf{F}{(\\hat{p},F_{x})} = \\sin{(F_{x} + \\hat{p})} and \\operatorname{C_{1}}{(\\hat{p},F_{x})} = \\sin{(F_{x} + \\hat{p})}, then obtain \\mathbf{F}^{F_{x}}{(\\hat{p},F_{x})} = \\operatorname{C_{1}}^{F_{x}}{(\\hat{p},F_{x})}", "derivation": "\\mathbf{F}{(\\hat{p},F_{x})} = \\sin{(F_{x} + \\hat{p})} and \\mathbf{F}^{F_{x}}{(\\hat{p},F_{x})} = \\sin^{F_{x}}{(F_{x} + \\hat{p})} and \\operatorname{C_{1}}{(\\hat{p},F_{x})} = \\sin{(F_{x} + \\hat{p})} and \\mathbf{F}^{F_{x}}{(\\hat{p},F_{x})} = \\operatorname{C_{1}}^{F_{x}}{(\\hat{p},F_{x})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["power", 1, "Symbol('F_x', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True))), Symbol('F_x', commutative=True)))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), sin(Add(Symbol('F_x', commutative=True), Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Function('C_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(i)} = \\log{(i)}, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(i)} di = \\rho + i \\log{(i)} - i, then obtain \\frac{\\partial}{\\partial i} (\\rho + i \\operatorname{f_{\\mathbf{p}}}{(i)} - i) = \\frac{d}{d i} \\int \\log{(i)} di", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(i)} = \\log{(i)} and \\int \\operatorname{f_{\\mathbf{p}}}{(i)} di = \\int \\log{(i)} di and \\frac{d}{d i} \\int \\operatorname{f_{\\mathbf{p}}}{(i)} di = \\frac{d}{d i} \\int \\log{(i)} di and \\int \\operatorname{f_{\\mathbf{p}}}{(i)} di = \\rho + i \\log{(i)} - i and \\int \\operatorname{f_{\\mathbf{p}}}{(i)} di = \\rho + i \\operatorname{f_{\\mathbf{p}}}{(i)} - i and \\frac{\\partial}{\\partial i} (\\rho + i \\operatorname{f_{\\mathbf{p}}}{(i)} - i) = \\frac{d}{d i} \\int \\log{(i)} di", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True)), log(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Symbol('i', commutative=True), log(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\rho', commutative=True), Mul(Symbol('i', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Derivative(Add(Symbol('\\\\rho', commutative=True), Mul(Symbol('i', commutative=True), Function('f_{\\\\mathbf{p}}')(Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given H{(f_{\\mathbf{v}})} = \\sin{(\\cos{(f_{\\mathbf{v}})})}, then obtain \\int - \\frac{H{(f_{\\mathbf{v}})}}{\\cos{(f_{\\mathbf{v}})}} df_{\\mathbf{v}} = \\int - \\frac{\\sin{(\\cos{(f_{\\mathbf{v}})})}}{\\cos{(f_{\\mathbf{v}})}} df_{\\mathbf{v}}", "derivation": "H{(f_{\\mathbf{v}})} = \\sin{(\\cos{(f_{\\mathbf{v}})})} and \\frac{H{(f_{\\mathbf{v}})}}{\\cos{(f_{\\mathbf{v}})}} = \\frac{\\sin{(\\cos{(f_{\\mathbf{v}})})}}{\\cos{(f_{\\mathbf{v}})}} and - \\frac{H{(f_{\\mathbf{v}})}}{\\cos{(f_{\\mathbf{v}})}} = - \\frac{\\sin{(\\cos{(f_{\\mathbf{v}})})}}{\\cos{(f_{\\mathbf{v}})}} and \\int - \\frac{H{(f_{\\mathbf{v}})}}{\\cos{(f_{\\mathbf{v}})}} df_{\\mathbf{v}} = \\int - \\frac{\\sin{(\\cos{(f_{\\mathbf{v}})})}}{\\cos{(f_{\\mathbf{v}})}} df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), sin(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["divide", 1, "cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Function('H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Mul(sin(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('H')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Mul(Integer(-1), sin(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(cos(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"]]}, {"prompt": "Given c{(q)} = \\int \\log{(q)} dq, then derive c{(q)} = \\hat{\\mathbf{r}} + q \\log{(q)} - q, then derive \\frac{d^{2}}{d q^{2}} \\int c{(q)} dq = \\log{(q)}, then obtain (\\hat{\\mathbf{r}} + q \\frac{d^{2}}{d q^{2}} \\int c{(q)} dq - q)^{\\hat{\\mathbf{r}}} = (\\int \\log{(q)} dq)^{\\hat{\\mathbf{r}}}", "derivation": "c{(q)} = \\int \\log{(q)} dq and \\int c{(q)} dq = \\iint \\log{(q)} dq dq and c{(q)} = \\hat{\\mathbf{r}} + q \\log{(q)} - q and \\frac{d}{d q} \\int c{(q)} dq = \\frac{d}{d q} \\iint \\log{(q)} dq dq and \\frac{d^{2}}{d q^{2}} \\int c{(q)} dq = \\frac{d^{2}}{d q^{2}} \\iint \\log{(q)} dq dq and \\frac{d^{2}}{d q^{2}} \\int c{(q)} dq = \\log{(q)} and \\hat{\\mathbf{r}} + q \\log{(q)} - q = \\int \\log{(q)} dq and (\\hat{\\mathbf{r}} + q \\log{(q)} - q)^{\\hat{\\mathbf{r}}} = (\\int \\log{(q)} dq)^{\\hat{\\mathbf{r}}} and (\\hat{\\mathbf{r}} + q \\frac{d^{2}}{d q^{2}} \\int c{(q)} dq - q)^{\\hat{\\mathbf{r}}} = (\\int \\log{(q)} dq)^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["get_premise", "Equality(Function('c')(Symbol('q', commutative=True)), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["integrate", 1, "Symbol('q', commutative=True)"], "Equality(Integral(Function('c')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('c')(Symbol('q', commutative=True)), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Integral(Function('c')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(2))), Derivative(Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(2))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Function('c')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(2))), log(Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["power", 7, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Symbol('q', commutative=True), log(Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('q', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Symbol('q', commutative=True), Derivative(Integral(Function('c')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(2)))), Mul(Integer(-1), Symbol('q', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Integral(log(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given W{(U)} = \\log{(U)}, then derive (- U + W{(U)}) (U \\log{(U)} - U + k) = (- U + \\log{(U)}) (U \\log{(U)} - U + k), then obtain (k + \\frac{(- U + W{(U)}) (U \\log{(U)} - U + k)}{k}) \\log{(U)} = (k + \\frac{(- U + \\log{(U)}) (U \\log{(U)} - U + k)}{k}) \\log{(U)}", "derivation": "W{(U)} = \\log{(U)} and - U + W{(U)} = - U + \\log{(U)} and (- U + W{(U)}) \\int \\log{(U)} dU = (- U + \\log{(U)}) \\int \\log{(U)} dU and (- U + W{(U)}) (U \\log{(U)} - U + k) = (- U + \\log{(U)}) (U \\log{(U)} - U + k) and \\frac{(- U + W{(U)}) (U \\log{(U)} - U + k)}{k} = \\frac{(- U + \\log{(U)}) (U \\log{(U)} - U + k)}{k} and k + \\frac{(- U + W{(U)}) (U \\log{(U)} - U + k)}{k} = k + \\frac{(- U + \\log{(U)}) (U \\log{(U)} - U + k)}{k} and (k + \\frac{(- U + W{(U)}) (U \\log{(U)} - U + k)}{k}) \\log{(U)} = (k + \\frac{(- U + \\log{(U)}) (U \\log{(U)} - U + k)}{k}) \\log{(U)}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('U', commutative=True)), log(Symbol('U', commutative=True)))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))))"], [["times", 2, "Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Integral(log(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True))))"], [["divide", 4, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True))))"], [["add", 5, "Symbol('k', commutative=True)"], "Equality(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True)))), Add(Symbol('k', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True)))))"], [["times", 6, "log(Symbol('U', commutative=True))"], "Equality(Mul(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('W')(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True)))), log(Symbol('U', commutative=True))), Mul(Add(Symbol('k', commutative=True), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), log(Symbol('U', commutative=True))), Mul(Integer(-1), Symbol('U', commutative=True)), Symbol('k', commutative=True)))), log(Symbol('U', commutative=True))))"]]}, {"prompt": "Given h{(\\dot{x})} = \\log{(\\dot{x})}, then derive \\frac{\\frac{d}{d \\dot{x}} h{(\\dot{x})}}{\\log{(\\dot{x})}} - \\frac{h{(\\dot{x})}}{\\dot{x} \\log{(\\dot{x})}^{2}} = 0, then obtain - \\dot{x} + \\frac{\\frac{d}{d \\dot{x}} h{(\\dot{x})}}{\\log{(\\dot{x})}} - \\frac{h{(\\dot{x})}}{\\dot{x} \\log{(\\dot{x})}^{2}} = - \\dot{x}", "derivation": "h{(\\dot{x})} = \\log{(\\dot{x})} and \\frac{h{(\\dot{x})}}{\\log{(\\dot{x})}} = 1 and \\frac{d}{d \\dot{x}} \\frac{h{(\\dot{x})}}{\\log{(\\dot{x})}} = \\frac{d}{d \\dot{x}} 1 and \\frac{\\frac{d}{d \\dot{x}} h{(\\dot{x})}}{\\log{(\\dot{x})}} - \\frac{h{(\\dot{x})}}{\\dot{x} \\log{(\\dot{x})}^{2}} = 0 and - \\dot{x} + \\frac{\\frac{d}{d \\dot{x}} h{(\\dot{x})}}{\\log{(\\dot{x})}} - \\frac{h{(\\dot{x})}}{\\dot{x} \\log{(\\dot{x})}^{2}} = - \\dot{x}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('h')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Derivative(Mul(Function('h')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), Derivative(Function('h')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-2)))), Integer(0))"], [["minus", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)), Mul(Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)), Derivative(Function('h')(Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-2)))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given \\hat{x}_0{(\\hat{x},\\tilde{g})} = \\frac{\\hat{x}}{\\tilde{g}}, then obtain \\frac{\\partial}{\\partial \\hat{x}} (- \\hat{x}_0{(\\hat{x},\\tilde{g})} - 1) = \\frac{\\partial}{\\partial \\hat{x}} (- \\frac{\\hat{x}}{\\tilde{g}} - 1)", "derivation": "\\hat{x}_0{(\\hat{x},\\tilde{g})} = \\frac{\\hat{x}}{\\tilde{g}} and \\hat{x}_0{(\\hat{x},\\tilde{g})} + 1 = \\frac{\\hat{x}}{\\tilde{g}} + 1 and - \\hat{x}_0{(\\hat{x},\\tilde{g})} - 1 = - \\frac{\\hat{x}}{\\tilde{g}} - 1 and \\frac{\\partial}{\\partial \\hat{x}} (- \\hat{x}_0{(\\hat{x},\\tilde{g})} - 1) = \\frac{\\partial}{\\partial \\hat{x}} (- \\frac{\\hat{x}}{\\tilde{g}} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{x}_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(1)))"], [["times", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1)))"], [["differentiate", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\hat{x}_0')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Integer(-1)), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(F_{g})} = e^{F_{g}}, then obtain F_{g} \\phi^{2}{(F_{g})} e^{F_{g}} = F_{g} e^{3 F_{g}}", "derivation": "\\phi{(F_{g})} = e^{F_{g}} and \\phi{(F_{g})} e^{F_{g}} = e^{2 F_{g}} and \\phi{(F_{g})} e^{2 F_{g}} = e^{3 F_{g}} and F_{g} \\phi{(F_{g})} e^{2 F_{g}} = F_{g} e^{3 F_{g}} and F_{g} \\phi^{2}{(F_{g})} e^{F_{g}} = F_{g} e^{3 F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True)))"], [["times", 1, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('F_g', commutative=True)), exp(Symbol('F_g', commutative=True))), exp(Mul(Integer(2), Symbol('F_g', commutative=True))))"], [["times", 2, "exp(Symbol('F_g', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('F_g', commutative=True)), exp(Mul(Integer(2), Symbol('F_g', commutative=True)))), exp(Mul(Integer(3), Symbol('F_g', commutative=True))))"], [["times", 3, "Symbol('F_g', commutative=True)"], "Equality(Mul(Symbol('F_g', commutative=True), Function('\\\\phi')(Symbol('F_g', commutative=True)), exp(Mul(Integer(2), Symbol('F_g', commutative=True)))), Mul(Symbol('F_g', commutative=True), exp(Mul(Integer(3), Symbol('F_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('F_g', commutative=True), Pow(Function('\\\\phi')(Symbol('F_g', commutative=True)), Integer(2)), exp(Symbol('F_g', commutative=True))), Mul(Symbol('F_g', commutative=True), exp(Mul(Integer(3), Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\omega)} = \\int e^{\\omega} d\\omega and \\tilde{g}^*{(\\omega)} = - \\omega + \\int e^{\\omega} d\\omega, then derive \\tilde{g}^*{(\\omega)} = L_{\\varepsilon} - \\omega + e^{\\omega}, then obtain \\frac{- \\omega - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + 2 \\int e^{\\omega} d\\omega}{\\omega + \\int e^{\\omega} d\\omega} = \\frac{L_{\\varepsilon} - \\omega + e^{\\omega}}{\\omega + \\int e^{\\omega} d\\omega}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\omega)} = \\int e^{\\omega} d\\omega and - \\omega = - \\omega - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + \\int e^{\\omega} d\\omega and \\tilde{g}^*{(\\omega)} = - \\omega + \\int e^{\\omega} d\\omega and \\tilde{g}^*{(\\omega)} = L_{\\varepsilon} - \\omega + e^{\\omega} and \\tilde{g}^*{(\\omega)} = - \\omega - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + 2 \\int e^{\\omega} d\\omega and - \\omega - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + 2 \\int e^{\\omega} d\\omega = L_{\\varepsilon} - \\omega + e^{\\omega} and \\frac{- \\omega - \\operatorname{V_{\\mathbf{B}}}{(\\omega)} + 2 \\int e^{\\omega} d\\omega}{\\omega + \\int e^{\\omega} d\\omega} = \\frac{L_{\\varepsilon} - \\omega + e^{\\omega}}{\\omega + \\int e^{\\omega} d\\omega}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\omega', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True))))"], [["divide", 6, "Add(Symbol('\\\\omega', commutative=True), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('\\\\omega', commutative=True), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))), Mul(Pow(Add(Symbol('\\\\omega', commutative=True), Integral(exp(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Integer(-1)), Add(Symbol('L_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), exp(Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\eta^{\\prime}{(g)} = \\cos{(\\cos{(g)})}, then obtain \\frac{((\\frac{\\int 0 dg}{\\cos{(g)}})^{g}) \\int 0 dg}{\\cos{(g)}} = \\frac{((\\frac{\\int (- \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})}) dg}{\\cos{(g)}})^{g}) \\int 0 dg}{\\cos{(g)}}", "derivation": "\\eta^{\\prime}{(g)} = \\cos{(\\cos{(g)})} and 0 = - \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})} and \\int 0 dg = \\int (- \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})}) dg and \\frac{\\int 0 dg}{\\cos{(g)}} = \\frac{\\int (- \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})}) dg}{\\cos{(g)}} and (\\frac{\\int 0 dg}{\\cos{(g)}})^{g} = (\\frac{\\int (- \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})}) dg}{\\cos{(g)}})^{g} and \\frac{((\\frac{\\int 0 dg}{\\cos{(g)}})^{g}) \\int 0 dg}{\\cos{(g)}} = \\frac{((\\frac{\\int (- \\eta^{\\prime}{(g)} + \\cos{(\\cos{(g)})}) dg}{\\cos{(g)}})^{g}) \\int 0 dg}{\\cos{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True)), cos(cos(Symbol('g', commutative=True))))"], [["minus", 1, "Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True))))"], [["divide", 3, "cos(Symbol('g', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))), Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)))"], [["times", 5, "Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True))))"], "Equality(Mul(Pow(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Mul(Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\eta^{\\\\prime}')(Symbol('g', commutative=True))), cos(cos(Symbol('g', commutative=True)))), Tuple(Symbol('g', commutative=True)))), Symbol('g', commutative=True)), Pow(cos(Symbol('g', commutative=True)), Integer(-1)), Integral(Integer(0), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(C_{2})} = \\cos{(\\log{(C_{2})})}, then derive 0 = - \\frac{d}{d C_{2}} \\mathbf{J}_P{(C_{2})} - \\frac{\\sin{(\\log{(C_{2})})}}{C_{2}}, then obtain \\frac{d}{d C_{2}} 0 = \\frac{d}{d C_{2}} (- \\frac{d}{d C_{2}} \\mathbf{J}_P{(C_{2})} - \\frac{\\sin{(\\log{(C_{2})})}}{C_{2}})", "derivation": "\\mathbf{J}_P{(C_{2})} = \\cos{(\\log{(C_{2})})} and 0 = - \\mathbf{J}_P{(C_{2})} + \\cos{(\\log{(C_{2})})} and \\frac{d}{d C_{2}} 0 = \\frac{d}{d C_{2}} (- \\mathbf{J}_P{(C_{2})} + \\cos{(\\log{(C_{2})})}) and 0 = - \\frac{d}{d C_{2}} \\mathbf{J}_P{(C_{2})} - \\frac{\\sin{(\\log{(C_{2})})}}{C_{2}} and \\frac{d}{d C_{2}} 0 = \\frac{d}{d C_{2}} (- \\frac{d}{d C_{2}} \\mathbf{J}_P{(C_{2})} - \\frac{\\sin{(\\log{(C_{2})})}}{C_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True)), cos(log(Symbol('C_2', commutative=True))))"], [["minus", 1, "Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True))), cos(log(Symbol('C_2', commutative=True)))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True))), cos(log(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(log(Symbol('C_2', commutative=True))))))"], [["differentiate", 4, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_P')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C_2', commutative=True), Integer(-1)), sin(log(Symbol('C_2', commutative=True))))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi^{*}{(\\lambda)} = \\log{(\\lambda)} and \\operatorname{E_{n}}{(m)} = \\cos{(e^{m})}, then obtain - (\\lambda \\varphi^{*}{(\\lambda)})^{\\lambda} + \\operatorname{E_{n}}{(m)} = - (\\lambda \\varphi^{*}{(\\lambda)})^{\\lambda} + \\cos{(e^{m})}", "derivation": "\\varphi^{*}{(\\lambda)} = \\log{(\\lambda)} and \\lambda \\varphi^{*}{(\\lambda)} = \\lambda \\log{(\\lambda)} and \\operatorname{E_{n}}{(m)} = \\cos{(e^{m})} and - (\\lambda \\log{(\\lambda)})^{\\lambda} + \\operatorname{E_{n}}{(m)} = - (\\lambda \\log{(\\lambda)})^{\\lambda} + \\cos{(e^{m})} and - (\\lambda \\varphi^{*}{(\\lambda)})^{\\lambda} + \\operatorname{E_{n}}{(m)} = - (\\lambda \\varphi^{*}{(\\lambda)})^{\\lambda} + \\cos{(e^{m})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))))"], ["get_premise", "Equality(Function('E_n')(Symbol('m', commutative=True)), cos(exp(Symbol('m', commutative=True))))"], [["minus", 3, "Pow(Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Function('E_n')(Symbol('m', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\lambda', commutative=True), log(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), cos(exp(Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), Function('E_n')(Symbol('m', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\lambda', commutative=True), Function('\\\\varphi^*')(Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))), cos(exp(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\theta)} = \\log{(\\theta)} and \\mathbf{M}{(\\theta)} = \\theta + \\operatorname{c_{0}}{(\\theta)} and \\mathbf{v}{(\\theta)} = (\\theta + \\log{(\\theta)})^{\\theta}, then obtain (\\theta + \\operatorname{c_{0}}{(\\theta)})^{\\theta} = \\mathbf{v}{(\\theta)}", "derivation": "\\operatorname{c_{0}}{(\\theta)} = \\log{(\\theta)} and \\theta + \\operatorname{c_{0}}{(\\theta)} = \\theta + \\log{(\\theta)} and \\mathbf{M}{(\\theta)} = \\theta + \\operatorname{c_{0}}{(\\theta)} and \\mathbf{M}^{\\theta}{(\\theta)} = (\\theta + \\operatorname{c_{0}}{(\\theta)})^{\\theta} and \\mathbf{M}{(\\theta)} = \\theta + \\log{(\\theta)} and \\mathbf{M}^{\\theta}{(\\theta)} = (\\theta + \\log{(\\theta)})^{\\theta} and \\mathbf{v}{(\\theta)} = (\\theta + \\log{(\\theta)})^{\\theta} and \\mathbf{M}^{\\theta}{(\\theta)} = \\mathbf{v}{(\\theta)} and (\\theta + \\operatorname{c_{0}}{(\\theta)})^{\\theta} = \\mathbf{v}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\theta', commutative=True)), log(Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('c_0')(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Function('c_0')(Symbol('\\\\theta', commutative=True))))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Function('c_0')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))))"], [["power", 5, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\theta', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), log(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Pow(Add(Symbol('\\\\theta', commutative=True), Function('c_0')(Symbol('\\\\theta', commutative=True))), Symbol('\\\\theta', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\psi{(k)} = \\log{(k)}, then derive \\mathbf{v} + \\frac{k^{2}}{2} = \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk, then obtain \\int k dk + \\frac{2 (\\mathbf{v} + \\frac{k^{2}}{2}) \\log{(k)}^{k}}{k^{2}} = \\int k dk + \\frac{2 \\log{(k)}^{k} \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk}{k^{2}}", "derivation": "\\psi{(k)} = \\log{(k)} and k \\psi{(k)} = k \\log{(k)} and k = \\frac{k \\log{(k)}}{\\psi{(k)}} and \\int k dk = \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk and \\mathbf{v} + \\frac{k^{2}}{2} = \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk and (\\mathbf{v} + \\frac{k^{2}}{2}) \\log{(k)}^{k} = \\log{(k)}^{k} \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk and \\frac{2 (\\mathbf{v} + \\frac{k^{2}}{2}) \\log{(k)}^{k}}{k^{2}} = \\frac{2 \\log{(k)}^{k} \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk}{k^{2}} and \\int k dk + \\frac{2 (\\mathbf{v} + \\frac{k^{2}}{2}) \\log{(k)}^{k}}{k^{2}} = \\int k dk + \\frac{2 \\log{(k)}^{k} \\int \\frac{k \\log{(k)}}{\\psi{(k)}} dk}{k^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('k', commutative=True)), log(Symbol('k', commutative=True)))"], [["times", 1, "Symbol('k', commutative=True)"], "Equality(Mul(Symbol('k', commutative=True), Function('\\\\psi')(Symbol('k', commutative=True))), Mul(Symbol('k', commutative=True), log(Symbol('k', commutative=True))))"], [["divide", 2, "Function('\\\\psi')(Symbol('k', commutative=True))"], "Equality(Symbol('k', commutative=True), Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))), Integral(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["times", 5, "Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["divide", 6, "Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))"], "Equality(Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(-2)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(-2)), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["add", 7, "Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True)))"], "Equality(Add(Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True))), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(-2)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('k', commutative=True), Integer(2)))), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Add(Integral(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True))), Mul(Integer(2), Pow(Symbol('k', commutative=True), Integer(-2)), Pow(log(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Integral(Mul(Symbol('k', commutative=True), Pow(Function('\\\\psi')(Symbol('k', commutative=True)), Integer(-1)), log(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))))"]]}, {"prompt": "Given q{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then derive \\int q{(\\mathbf{F})} d\\mathbf{F} = t_{1} + \\sin{(\\mathbf{F})}, then obtain (\\int q{(\\mathbf{F})} d\\mathbf{F})^{t_{1}} = (t_{1} + \\sin{(\\mathbf{F})})^{t_{1}}", "derivation": "q{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\int q{(\\mathbf{F})} d\\mathbf{F} = \\int \\cos{(\\mathbf{F})} d\\mathbf{F} and \\int q{(\\mathbf{F})} d\\mathbf{F} = t_{1} + \\sin{(\\mathbf{F})} and (\\int q{(\\mathbf{F})} d\\mathbf{F})^{t_{1}} = (t_{1} + \\sin{(\\mathbf{F})})^{t_{1}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('q')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 3, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integral(Function('q')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('t_1', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), sin(Symbol('\\\\mathbf{F}', commutative=True))), Symbol('t_1', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(C)} = \\cos{(C)} and \\operatorname{z^{*}}{(C)} = \\frac{\\cos{(C)}}{\\hat{H}{(C)}}, then obtain \\frac{d}{d C} 1 + \\frac{d}{d C} \\frac{\\cos{(C)}}{\\hat{H}{(C)}} = 2 \\frac{d}{d C} \\frac{\\cos{(C)}}{\\hat{H}{(C)}}", "derivation": "\\hat{H}{(C)} = \\cos{(C)} and 1 = \\frac{\\cos{(C)}}{\\hat{H}{(C)}} and \\operatorname{z^{*}}{(C)} = \\frac{\\cos{(C)}}{\\hat{H}{(C)}} and 1 = \\operatorname{z^{*}}{(C)} and \\frac{d}{d C} 1 = \\frac{d}{d C} \\operatorname{z^{*}}{(C)} and \\frac{d}{d C} 1 = \\frac{d}{d C} \\frac{\\cos{(C)}}{\\hat{H}{(C)}} and \\frac{d}{d C} 1 + \\frac{d}{d C} \\frac{\\cos{(C)}}{\\hat{H}{(C)}} = 2 \\frac{d}{d C} \\frac{\\cos{(C)}}{\\hat{H}{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["divide", 1, "Function('\\\\hat{H}')(Symbol('C', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('C', commutative=True)), Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Integer(1), Function('z^*')(Symbol('C', commutative=True)))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Function('z^*')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 6, "Derivative(Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Pow(Function('\\\\hat{H}')(Symbol('C', commutative=True)), Integer(-1)), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Omega{(f_{E})} = \\frac{d}{d f_{E}} \\cos{(f_{E})}, then derive \\frac{\\Omega{(f_{E})} + \\sin{(f_{E})}}{\\sin{(f_{E})}} = 0, then obtain (\\frac{\\sin{(f_{E})} + \\frac{d}{d f_{E}} \\cos{(f_{E})}}{\\sin{(f_{E})}} - 1)^{f_{E}} = (-1)^{f_{E}}", "derivation": "\\Omega{(f_{E})} = \\frac{d}{d f_{E}} \\cos{(f_{E})} and \\Omega{(f_{E})} - \\frac{d}{d f_{E}} \\cos{(f_{E})} = 0 and - \\frac{\\Omega{(f_{E})} - \\frac{d}{d f_{E}} \\cos{(f_{E})}}{\\frac{d}{d f_{E}} \\cos{(f_{E})}} = 0 and \\frac{\\Omega{(f_{E})} + \\sin{(f_{E})}}{\\sin{(f_{E})}} = 0 and \\frac{\\Omega{(f_{E})} + \\sin{(f_{E})}}{\\sin{(f_{E})}} - 1 = -1 and \\frac{\\sin{(f_{E})} + \\frac{d}{d f_{E}} \\cos{(f_{E})}}{\\sin{(f_{E})}} - 1 = -1 and (\\frac{\\sin{(f_{E})} + \\frac{d}{d f_{E}} \\cos{(f_{E})}}{\\sin{(f_{E})}} - 1)^{f_{E}} = (-1)^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('f_E', commutative=True)), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\Omega')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))), Integer(0))"], [["divide", 2, "Mul(Integer(-1), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Add(Function('\\\\Omega')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))), Pow(Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Function('\\\\Omega')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True))), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Integer(0))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Add(Function('\\\\Omega')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True))), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Integer(-1)), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Add(sin(Symbol('f_E', commutative=True)), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Integer(-1)), Integer(-1))"], [["power", 6, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Mul(Add(sin(Symbol('f_E', commutative=True)), Derivative(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Pow(sin(Symbol('f_E', commutative=True)), Integer(-1))), Integer(-1)), Symbol('f_E', commutative=True)), Pow(Integer(-1), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given c{(\\Omega)} = \\sin{(\\sin{(\\Omega)})}, then derive \\int \\frac{c{(\\Omega)}}{\\sin{(\\sin{(\\Omega)})}} d\\Omega = S + \\Omega, then obtain \\int 1 d\\Omega = S + \\Omega", "derivation": "c{(\\Omega)} = \\sin{(\\sin{(\\Omega)})} and \\frac{c{(\\Omega)}}{\\sin{(\\sin{(\\Omega)})}} = 1 and \\int \\frac{c{(\\Omega)}}{\\sin{(\\sin{(\\Omega)})}} d\\Omega = \\int 1 d\\Omega and \\int \\frac{c{(\\Omega)}}{\\sin{(\\sin{(\\Omega)})}} d\\Omega = S + \\Omega and \\int 1 d\\Omega = S + \\Omega", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Omega', commutative=True)), sin(sin(Symbol('\\\\Omega', commutative=True))))"], [["divide", 1, "sin(sin(Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Function('c')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Function('c')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('c')(Symbol('\\\\Omega', commutative=True)), Pow(sin(sin(Symbol('\\\\Omega', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Omega', commutative=True))), Add(Symbol('S', commutative=True), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\varphi{(q)} = \\sin{(q)}, then obtain \\frac{1}{\\sin{(q)} \\int \\varphi{(q)} \\sin{(q)} dq} = \\frac{1}{\\varphi{(q)} \\int \\varphi{(q)} \\sin{(q)} dq}", "derivation": "\\varphi{(q)} = \\sin{(q)} and \\varphi^{2}{(q)} = \\varphi{(q)} \\sin{(q)} and \\int \\varphi^{2}{(q)} dq = \\int \\varphi{(q)} \\sin{(q)} dq and \\varphi{(q)} \\int \\varphi^{2}{(q)} dq = \\sin{(q)} \\int \\varphi^{2}{(q)} dq and \\frac{1}{\\varphi{(q)}} = \\frac{\\sin{(q)}}{\\varphi^{2}{(q)}} and \\frac{1}{\\varphi{(q)} \\int \\varphi^{2}{(q)} dq} = \\frac{\\sin{(q)}}{\\varphi^{2}{(q)} \\int \\varphi^{2}{(q)} dq} and \\frac{1}{\\sin{(q)} \\int \\varphi^{2}{(q)} dq} = \\frac{1}{\\varphi{(q)} \\int \\varphi^{2}{(q)} dq} and \\frac{1}{\\sin{(q)} \\int \\varphi{(q)} \\sin{(q)} dq} = \\frac{1}{\\varphi{(q)} \\int \\varphi{(q)} \\sin{(q)} dq}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True)))"], [["times", 1, "Function('\\\\varphi')(Symbol('q', commutative=True))"], "Equality(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Mul(Function('\\\\varphi')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True))), Integral(Mul(Function('\\\\varphi')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["times", 1, "Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('q', commutative=True)), Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True)))), Mul(sin(Symbol('q', commutative=True)), Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True)))))"], [["divide", 1, "Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-2)), sin(Symbol('q', commutative=True))))"], [["divide", 5, "Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Mul(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-2)), sin(Symbol('q', commutative=True)), Pow(Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True))), Integer(-1))), Mul(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(sin(Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Mul(Function('\\\\varphi')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integer(-1))), Mul(Pow(Function('\\\\varphi')(Symbol('q', commutative=True)), Integer(-1)), Pow(Integral(Mul(Function('\\\\varphi')(Symbol('q', commutative=True)), sin(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(t_{1})} = e^{t_{1}} and n{(t_{1})} = 2 t_{1}, then obtain \\frac{d}{d t_{1}} (t_{1} + \\hat{\\mathbf{x}}^{2}{(t_{1})}) = \\frac{d}{d t_{1}} (t_{1} + e^{n{(t_{1})}})", "derivation": "\\hat{\\mathbf{x}}{(t_{1})} = e^{t_{1}} and \\hat{\\mathbf{x}}{(t_{1})} e^{t_{1}} = e^{2 t_{1}} and n{(t_{1})} = 2 t_{1} and \\hat{\\mathbf{x}}{(t_{1})} e^{t_{1}} = e^{n{(t_{1})}} and \\hat{\\mathbf{x}}^{2}{(t_{1})} = e^{n{(t_{1})}} and t_{1} + \\hat{\\mathbf{x}}^{2}{(t_{1})} = t_{1} + e^{n{(t_{1})}} and \\frac{d}{d t_{1}} (t_{1} + \\hat{\\mathbf{x}}^{2}{(t_{1})}) = \\frac{d}{d t_{1}} (t_{1} + e^{n{(t_{1})}})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True)))"], [["times", 1, "exp(Symbol('t_1', commutative=True))"], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True))), exp(Mul(Integer(2), Symbol('t_1', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('t_1', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), exp(Symbol('t_1', commutative=True))), exp(Function('n')(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), Integer(2)), exp(Function('n')(Symbol('t_1', commutative=True))))"], [["add", 5, "Symbol('t_1', commutative=True)"], "Equality(Add(Symbol('t_1', commutative=True), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), Integer(2))), Add(Symbol('t_1', commutative=True), exp(Function('n')(Symbol('t_1', commutative=True)))))"], [["differentiate", 6, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Symbol('t_1', commutative=True), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('t_1', commutative=True)), Integer(2))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), exp(Function('n')(Symbol('t_1', commutative=True)))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\ddot{x})} = e^{\\ddot{x}}, then obtain \\frac{2 \\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} - 1 + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}} = \\frac{\\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}}", "derivation": "\\hat{p}_0{(\\ddot{x})} = e^{\\ddot{x}} and \\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} = 1 and \\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} + e^{- \\ddot{x}} = 1 + e^{- \\ddot{x}} and \\frac{\\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}} = \\frac{1 + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}} and \\frac{2 \\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} - 1 + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}} = \\frac{\\hat{p}_0{(\\ddot{x})} e^{- \\ddot{x}} + e^{- \\ddot{x}}}{\\hat{p}_0{(\\ddot{x})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Symbol('\\\\ddot{x}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Integer(1))"], [["add", 2, "exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Add(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Add(Integer(1), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))))"], [["divide", 3, "Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Add(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Mul(Add(Integer(1), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))), Mul(Add(Mul(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)))), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\ddot{x}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{H}{(U)} = \\log{(e^{U})}, then obtain ((- U + \\mathbf{H}{(U)}) e^{- U})^{U} (- U + \\mathbf{H}{(U)}) = ((- U + \\log{(e^{U})}) e^{- U})^{U} (- U + \\mathbf{H}{(U)})", "derivation": "\\mathbf{H}{(U)} = \\log{(e^{U})} and - U + \\mathbf{H}{(U)} = - U + \\log{(e^{U})} and (- U + \\mathbf{H}{(U)}) e^{- U} = (- U + \\log{(e^{U})}) e^{- U} and ((- U + \\mathbf{H}{(U)}) e^{- U})^{U} = ((- U + \\log{(e^{U})}) e^{- U})^{U} and ((- U + \\mathbf{H}{(U)}) e^{- U})^{U} (- U + \\mathbf{H}{(U)}) = ((- U + \\log{(e^{U})}) e^{- U})^{U} (- U + \\mathbf{H}{(U)})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('U', commutative=True)), log(exp(Symbol('U', commutative=True))))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(exp(Symbol('U', commutative=True)))))"], [["divide", 2, "exp(Symbol('U', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(exp(Symbol('U', commutative=True)))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))))"], [["power", 3, "Symbol('U', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(exp(Symbol('U', commutative=True)))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Symbol('U', commutative=True)))"], [["times", 4, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True)))"], "Equality(Mul(Pow(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True)))), Mul(Pow(Mul(Add(Mul(Integer(-1), Symbol('U', commutative=True)), log(exp(Symbol('U', commutative=True)))), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Symbol('U', commutative=True)), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{H}')(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\sigma_x,V)} = \\sigma_x^{V} and \\mathbf{v}{(\\sigma_x,V)} = \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)}, then obtain 2 \\sigma_x + 2 \\mathbf{v}{(\\sigma_x,V)} = 2 \\sigma_x + \\mathbf{v}{(\\sigma_x,V)} + \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)}", "derivation": "\\operatorname{v_{z}}{(\\sigma_x,V)} = \\sigma_x^{V} and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)} = \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x^{V} and \\mathbf{v}{(\\sigma_x,V)} = \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)} and \\mathbf{v}{(\\sigma_x,V)} = \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x^{V} and \\sigma_x + \\mathbf{v}{(\\sigma_x,V)} = \\sigma_x + \\frac{\\partial}{\\partial \\sigma_x} \\sigma_x^{V} and \\sigma_x + \\mathbf{v}{(\\sigma_x,V)} = \\sigma_x + \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)} and 2 \\sigma_x + 2 \\mathbf{v}{(\\sigma_x,V)} = 2 \\sigma_x + \\mathbf{v}{(\\sigma_x,V)} + \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{v_{z}}{(\\sigma_x,V)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Derivative(Function('v_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Derivative(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), Derivative(Function('v_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"], [["add", 6, "Add(Symbol('\\\\sigma_x', commutative=True), Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\sigma_x', commutative=True)), Function('\\\\mathbf{v}')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Derivative(Function('v_z')(Symbol('\\\\sigma_x', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\omega,L)} = L - \\omega and V{(L,\\omega)} = \\operatorname{F_{c}}^{L}{(\\omega,L)}, then obtain \\frac{\\partial}{\\partial \\omega} (- \\omega + \\log{(V{(L,\\omega)})}) = \\frac{\\partial}{\\partial \\omega} (- \\omega + \\log{((L - \\omega)^{L})})", "derivation": "\\operatorname{F_{c}}{(\\omega,L)} = L - \\omega and \\operatorname{F_{c}}^{L}{(\\omega,L)} = (L - \\omega)^{L} and V{(L,\\omega)} = \\operatorname{F_{c}}^{L}{(\\omega,L)} and \\log{(\\operatorname{F_{c}}^{L}{(\\omega,L)})} = \\log{((L - \\omega)^{L})} and - \\omega + \\log{(\\operatorname{F_{c}}^{L}{(\\omega,L)})} = - \\omega + \\log{((L - \\omega)^{L})} and - \\omega + \\log{(V{(L,\\omega)})} = - \\omega + \\log{((L - \\omega)^{L})} and \\frac{\\partial}{\\partial \\omega} (- \\omega + \\log{(V{(L,\\omega)})}) = \\frac{\\partial}{\\partial \\omega} (- \\omega + \\log{((L - \\omega)^{L})})", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('L', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('V')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), log(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Symbol('L', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Pow(Function('F_c')(Symbol('\\\\omega', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Function('V')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Symbol('L', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Function('V')(Symbol('L', commutative=True), Symbol('\\\\omega', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), log(Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))), Symbol('L', commutative=True)))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})}, then obtain (\\mathbf{E}^{\\ddot{x}}{(\\ddot{x})})^{2 \\ddot{x}} = (\\mathbf{E}^{\\ddot{x}}{(\\ddot{x})})^{\\ddot{x}} (\\log{(\\sin{(\\ddot{x})})}^{\\ddot{x}})^{\\ddot{x}}", "derivation": "\\mathbf{E}{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})} and \\mathbf{E}^{\\ddot{x}}{(\\ddot{x})} = \\log{(\\sin{(\\ddot{x})})}^{\\ddot{x}} and (\\mathbf{E}^{\\ddot{x}}{(\\ddot{x})})^{\\ddot{x}} = (\\log{(\\sin{(\\ddot{x})})}^{\\ddot{x}})^{\\ddot{x}} and (\\mathbf{E}^{\\ddot{x}}{(\\ddot{x})})^{2 \\ddot{x}} = (\\mathbf{E}^{\\ddot{x}}{(\\ddot{x})})^{\\ddot{x}} (\\log{(\\sin{(\\ddot{x})})}^{\\ddot{x}})^{\\ddot{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), log(sin(Symbol('\\\\ddot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)))"], [["power", 2, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Pow(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)))"], [["times", 3, "Pow(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(2), Symbol('\\\\ddot{x}', commutative=True))), Mul(Pow(Pow(Function('\\\\mathbf{E}')(Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True)), Pow(Pow(log(sin(Symbol('\\\\ddot{x}', commutative=True))), Symbol('\\\\ddot{x}', commutative=True)), Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(U)} = e^{U} and \\operatorname{V_{\\mathbf{E}}}{(U)} = \\mathbf{A}{(U)} e^{U}, then obtain \\frac{\\operatorname{V_{\\mathbf{E}}}{(U)}}{\\mathbf{A}{(U)}} = e^{U}", "derivation": "\\mathbf{A}{(U)} = e^{U} and \\operatorname{V_{\\mathbf{E}}}{(U)} = \\mathbf{A}{(U)} e^{U} and \\operatorname{V_{\\mathbf{E}}}{(U)} = \\mathbf{A}^{2}{(U)} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(U)}}{\\mathbf{A}{(U)}} = \\mathbf{A}{(U)} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(U)}}{\\mathbf{A}{(U)}} = e^{U}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], ["renaming_premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True)), Mul(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), Integer(2)))"], [["divide", 3, "Function('\\\\mathbf{A}')(Symbol('U', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), Integer(-1))), Function('\\\\mathbf{A}')(Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('U', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('U', commutative=True)), Integer(-1))), exp(Symbol('U', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(A_{y},V)} = \\frac{e^{A_{y}}}{V} and \\sigma_{x}{(A_{y},V)} = \\frac{\\partial}{\\partial A_{y}} V \\hat{x}{(A_{y},V)}, then obtain 0 = \\frac{\\partial^{2}}{\\partial A_{y}^{2}} V \\hat{x}{(A_{y},V)} - \\frac{\\partial}{\\partial A_{y}} \\sigma_{x}{(A_{y},V)}", "derivation": "\\hat{x}{(A_{y},V)} = \\frac{e^{A_{y}}}{V} and V \\hat{x}{(A_{y},V)} = e^{A_{y}} and \\frac{\\partial}{\\partial A_{y}} V \\hat{x}{(A_{y},V)} = \\frac{d}{d A_{y}} e^{A_{y}} and \\sigma_{x}{(A_{y},V)} = \\frac{\\partial}{\\partial A_{y}} V \\hat{x}{(A_{y},V)} and \\sigma_{x}{(A_{y},V)} = \\frac{d}{d A_{y}} e^{A_{y}} and \\frac{\\partial}{\\partial A_{y}} \\sigma_{x}{(A_{y},V)} = \\frac{d^{2}}{d A_{y}^{2}} e^{A_{y}} and \\frac{\\partial}{\\partial A_{y}} \\sigma_{x}{(A_{y},V)} = \\frac{\\partial^{2}}{\\partial A_{y}^{2}} V \\hat{x}{(A_{y},V)} and 0 = \\frac{\\partial^{2}}{\\partial A_{y}^{2}} V \\hat{x}{(A_{y},V)} - \\frac{\\partial}{\\partial A_{y}} \\sigma_{x}{(A_{y},V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), exp(Symbol('A_y', commutative=True))))"], [["times", 1, "Symbol('V', commutative=True)"], "Equality(Mul(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True))), exp(Symbol('A_y', commutative=True)))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Derivative(Mul(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Derivative(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(2))))"], [["minus", 7, "Derivative(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('A_y', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(2))), Mul(Integer(-1), Derivative(Function('\\\\sigma_x')(Symbol('A_y', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} = \\cos{(I \\mathbf{r})}, then derive \\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} = - \\mathbf{r} \\sin{(I \\mathbf{r})}, then obtain \\frac{\\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} - 1}{\\frac{\\partial}{\\partial I} \\cos{(I \\mathbf{r})} - 1} = 1", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} = \\cos{(I \\mathbf{r})} and \\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} = \\frac{\\partial}{\\partial I} \\cos{(I \\mathbf{r})} and \\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} = - \\mathbf{r} \\sin{(I \\mathbf{r})} and \\frac{\\partial}{\\partial I} \\cos{(I \\mathbf{r})} = - \\mathbf{r} \\sin{(I \\mathbf{r})} and \\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} - 1 = - \\mathbf{r} \\sin{(I \\mathbf{r})} - 1 and \\frac{\\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} - 1}{- \\mathbf{r} \\sin{(I \\mathbf{r})} - 1} = 1 and \\frac{\\frac{\\partial}{\\partial I} \\operatorname{V_{\\mathbf{B}}}{(I,\\mathbf{r})} - 1}{\\frac{\\partial}{\\partial I} \\cos{(I \\mathbf{r})} - 1} = 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Integer(-1)))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Integer(-1))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))), Integer(-1)), Integer(-1)), Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Add(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Pow(Add(Derivative(cos(Mul(Symbol('I', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(-1)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\sigma_{p}{(\\hbar,b)} = b + \\cos{(\\hbar)}, then derive \\int \\sigma_{p}{(\\hbar,b)} db = \\mathbf{M} + \\frac{b^{2}}{2} + b \\cos{(\\hbar)}, then obtain b \\cos{(\\hbar)} + \\int \\sigma_{p}{(\\hbar,b)} db = \\mathbf{M} + \\frac{b^{2}}{2} + 2 b \\cos{(\\hbar)}", "derivation": "\\sigma_{p}{(\\hbar,b)} = b + \\cos{(\\hbar)} and \\int \\sigma_{p}{(\\hbar,b)} db = \\int (b + \\cos{(\\hbar)}) db and \\int \\sigma_{p}{(\\hbar,b)} db = \\mathbf{M} + \\frac{b^{2}}{2} + b \\cos{(\\hbar)} and b \\cos{(\\hbar)} + \\int \\sigma_{p}{(\\hbar,b)} db = \\mathbf{M} + \\frac{b^{2}}{2} + 2 b \\cos{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True), Symbol('b', commutative=True)), Add(Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Add(Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True)))))"], [["add", 3, "Mul(Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True))), Integral(Function('\\\\sigma_p')(Symbol('\\\\hbar', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('b', commutative=True), Integer(2))), Mul(Integer(2), Symbol('b', commutative=True), cos(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given L{(\\chi,x^\\prime)} = \\frac{\\chi}{x^\\prime}, then obtain \\int \\frac{(\\frac{\\int L{(\\chi,x^\\prime)} d\\chi}{\\int \\frac{\\chi}{x^\\prime} d\\chi})^{x^\\prime}}{\\int \\frac{\\chi}{x^\\prime} d\\chi} d\\chi = \\int \\frac{1}{\\int \\frac{\\chi}{x^\\prime} d\\chi} d\\chi", "derivation": "L{(\\chi,x^\\prime)} = \\frac{\\chi}{x^\\prime} and \\int L{(\\chi,x^\\prime)} d\\chi = \\int \\frac{\\chi}{x^\\prime} d\\chi and \\frac{\\int L{(\\chi,x^\\prime)} d\\chi}{\\int \\frac{\\chi}{x^\\prime} d\\chi} = 1 and (\\frac{\\int L{(\\chi,x^\\prime)} d\\chi}{\\int \\frac{\\chi}{x^\\prime} d\\chi})^{x^\\prime} = 1 and \\frac{(\\frac{\\int L{(\\chi,x^\\prime)} d\\chi}{\\int \\frac{\\chi}{x^\\prime} d\\chi})^{x^\\prime}}{\\int \\frac{\\chi}{x^\\prime} d\\chi} = \\frac{1}{\\int \\frac{\\chi}{x^\\prime} d\\chi} and \\int \\frac{(\\frac{\\int L{(\\chi,x^\\prime)} d\\chi}{\\int \\frac{\\chi}{x^\\prime} d\\chi})^{x^\\prime}}{\\int \\frac{\\chi}{x^\\prime} d\\chi} d\\chi = \\int \\frac{1}{\\int \\frac{\\chi}{x^\\prime} d\\chi} d\\chi", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["divide", 2, "Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Integer(1))"], [["divide", 4, "Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Mul(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))), Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)))"], [["integrate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Pow(Mul(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Integral(Function('L')(Symbol('\\\\chi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Symbol('x^\\\\prime', commutative=True)), Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Integral(Mul(Symbol('\\\\chi', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1))), Tuple(Symbol('\\\\chi', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}, then derive \\psi^{*}{(\\mathbf{E})} = - \\sin{(\\mathbf{E})}, then obtain - \\frac{\\psi^{*}{(\\mathbf{E})}}{\\sin{(\\mathbf{E})}} = 1", "derivation": "\\psi^{*}{(\\mathbf{E})} = \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} and \\frac{\\psi^{*}{(\\mathbf{E})}}{\\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})}} = 1 and \\psi^{*}{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} and \\frac{d}{d \\mathbf{E}} \\cos{(\\mathbf{E})} = - \\sin{(\\mathbf{E})} and - \\frac{\\psi^{*}{(\\mathbf{E})}}{\\sin{(\\mathbf{E})}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{E}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(n)} = \\cos{(e^{n})}, then obtain (\\Psi_{\\lambda}{(n)} e^{- n} - e^{n})^{n} = (- e^{n} + e^{- n} \\cos{(e^{n})})^{n}", "derivation": "\\Psi_{\\lambda}{(n)} = \\cos{(e^{n})} and \\Psi_{\\lambda}{(n)} e^{- n} = e^{- n} \\cos{(e^{n})} and \\Psi_{\\lambda}{(n)} e^{- n} - e^{n} = - e^{n} + e^{- n} \\cos{(e^{n})} and (\\Psi_{\\lambda}{(n)} e^{- n} - e^{n})^{n} = (- e^{n} + e^{- n} \\cos{(e^{n})})^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('n', commutative=True)), cos(exp(Symbol('n', commutative=True))))"], [["divide", 1, "exp(Symbol('n', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('n', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('n', commutative=True))), cos(exp(Symbol('n', commutative=True)))))"], [["minus", 2, "exp(Symbol('n', commutative=True))"], "Equality(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('n', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Mul(Integer(-1), exp(Symbol('n', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('n', commutative=True))), cos(exp(Symbol('n', commutative=True))))))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Mul(Function('\\\\Psi_{\\\\lambda}')(Symbol('n', commutative=True)), exp(Mul(Integer(-1), Symbol('n', commutative=True)))), Mul(Integer(-1), exp(Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), exp(Symbol('n', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('n', commutative=True))), cos(exp(Symbol('n', commutative=True))))), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C_{d},M)} = - \\sin{(C_{d} - M)}, then obtain \\frac{(Q \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d})^{M}}{\\frac{\\partial}{\\partial C_{d}} - Q \\int - \\sin{(C_{d} - M)} dC_{d}} = \\frac{(Q \\int - \\sin{(C_{d} - M)} dC_{d})^{M}}{\\frac{\\partial}{\\partial C_{d}} - Q \\int - \\sin{(C_{d} - M)} dC_{d}}", "derivation": "\\operatorname{M_{E}}{(C_{d},M)} = - \\sin{(C_{d} - M)} and \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d} = \\int - \\sin{(C_{d} - M)} dC_{d} and - Q \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d} = - Q \\int - \\sin{(C_{d} - M)} dC_{d} and Q \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d} = Q \\int - \\sin{(C_{d} - M)} dC_{d} and (Q \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d})^{M} = (Q \\int - \\sin{(C_{d} - M)} dC_{d})^{M} and \\frac{(Q \\int \\operatorname{M_{E}}{(C_{d},M)} dC_{d})^{M}}{\\frac{\\partial}{\\partial C_{d}} - Q \\int - \\sin{(C_{d} - M)} dC_{d}} = \\frac{(Q \\int - \\sin{(C_{d} - M)} dC_{d})^{M}}{\\frac{\\partial}{\\partial C_{d}} - Q \\int - \\sin{(C_{d} - M)} dC_{d}}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('Q', commutative=True), Integral(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Mul(Integer(-1), Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Symbol('Q', commutative=True), Integral(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Mul(Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))))"], [["power", 4, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Symbol('Q', commutative=True), Integral(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Symbol('M', commutative=True)), Pow(Mul(Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Symbol('M', commutative=True)))"], [["divide", 5, "Derivative(Mul(Integer(-1), Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Mul(Symbol('Q', commutative=True), Integral(Function('M_E')(Symbol('C_d', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_d', commutative=True)))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Integer(-1), Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Mul(Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Symbol('M', commutative=True)), Pow(Derivative(Mul(Integer(-1), Symbol('Q', commutative=True), Integral(Mul(Integer(-1), sin(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), Symbol('M', commutative=True))))), Tuple(Symbol('C_d', commutative=True)))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\delta{(T)} = \\log{(T)}, then derive \\frac{d^{2}}{d T^{2}} \\delta{(T)} = - \\frac{1}{T^{2}}, then obtain (\\frac{d^{2}}{d T^{2}} \\delta{(T)} + \\frac{1}{T^{2}})^{2} + 1 = 1", "derivation": "\\delta{(T)} = \\log{(T)} and \\frac{d}{d T} \\delta{(T)} = \\frac{d}{d T} \\log{(T)} and \\frac{d^{2}}{d T^{2}} \\delta{(T)} = \\frac{d^{2}}{d T^{2}} \\log{(T)} and \\frac{d^{2}}{d T^{2}} \\delta{(T)} = - \\frac{1}{T^{2}} and \\frac{d^{2}}{d T^{2}} \\delta{(T)} + \\frac{1}{T^{2}} = 0 and (\\frac{d^{2}}{d T^{2}} \\delta{(T)} + \\frac{1}{T^{2}})^{2} = 0 and (\\frac{d^{2}}{d T^{2}} \\delta{(T)} + \\frac{1}{T^{2}})^{2} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Derivative(log(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2))))"], [["minus", 4, "Mul(Integer(-1), Pow(Symbol('T', commutative=True), Integer(-2)))"], "Equality(Add(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Pow(Symbol('T', commutative=True), Integer(-2))), Integer(0))"], [["power", 5, 2], "Equality(Pow(Add(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Pow(Symbol('T', commutative=True), Integer(-2))), Integer(2)), Integer(0))"], [["add", 6, 1], "Equality(Add(Pow(Add(Derivative(Function('\\\\delta')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(2))), Pow(Symbol('T', commutative=True), Integer(-2))), Integer(2)), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\psi{(\\phi,F_{g},\\omega)} = F_{g} (\\omega + \\phi) and \\mathbf{B}{(\\phi,F_{g},\\omega)} = F_{g}^{2} (\\omega + \\phi)^{2}, then obtain F_{g}^{2} (\\omega + \\phi)^{2} - F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} = 0", "derivation": "\\psi{(\\phi,F_{g},\\omega)} = F_{g} (\\omega + \\phi) and F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} = F_{g}^{2} (\\omega + \\phi)^{2} and \\mathbf{B}{(\\phi,F_{g},\\omega)} = F_{g}^{2} (\\omega + \\phi)^{2} and - F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} + \\mathbf{B}{(\\phi,F_{g},\\omega)} = F_{g}^{2} (\\omega + \\phi)^{2} - F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} and - F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} + \\mathbf{B}{(\\phi,F_{g},\\omega)} = 0 and F_{g}^{2} (\\omega + \\phi)^{2} - F_{g} (\\omega + \\phi) \\psi{(\\phi,F_{g},\\omega)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["times", 1, "Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('F_g', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(2))))"], [["minus", 3, "Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Pow(Symbol('F_g', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Function('\\\\mathbf{B}')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Pow(Symbol('F_g', commutative=True), Integer(2)), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('F_g', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\phi', commutative=True)), Function('\\\\psi')(Symbol('\\\\phi', commutative=True), Symbol('F_g', commutative=True), Symbol('\\\\omega', commutative=True)))), Integer(0))"]]}, {"prompt": "Given h{(f^{\\prime},W,\\rho)} = W (\\rho - f^{\\prime}) and \\delta{(f^{\\prime},W,\\rho)} = \\int W (\\rho - f^{\\prime}) d\\rho, then obtain \\delta{(f^{\\prime},W,\\rho)} + h{(f^{\\prime},W,\\rho)} = h{(f^{\\prime},W,\\rho)} + \\int W (\\rho - f^{\\prime}) d\\rho", "derivation": "h{(f^{\\prime},W,\\rho)} = W (\\rho - f^{\\prime}) and \\int h{(f^{\\prime},W,\\rho)} d\\rho = \\int W (\\rho - f^{\\prime}) d\\rho and \\delta{(f^{\\prime},W,\\rho)} = \\int W (\\rho - f^{\\prime}) d\\rho and h{(f^{\\prime},W,\\rho)} + \\int h{(f^{\\prime},W,\\rho)} d\\rho = h{(f^{\\prime},W,\\rho)} + \\int W (\\rho - f^{\\prime}) d\\rho and \\delta{(f^{\\prime},W,\\rho)} = \\int h{(f^{\\prime},W,\\rho)} d\\rho and \\delta{(f^{\\prime},W,\\rho)} + h{(f^{\\prime},W,\\rho)} = h{(f^{\\prime},W,\\rho)} + \\int W (\\rho - f^{\\prime}) d\\rho", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('W', commutative=True), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Mul(Symbol('W', commutative=True), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["add", 2, "Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True)))), Add(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\delta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('\\\\delta')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Function('h')(Symbol('f^{\\\\prime}', commutative=True), Symbol('W', commutative=True), Symbol('\\\\rho', commutative=True)), Integral(Mul(Symbol('W', commutative=True), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(i)} = e^{\\sin{(i)}} and G{(i)} = \\frac{d}{d i} 2 e^{\\sin{(i)}}, then obtain (G{(i)} - \\sin{(i)})^{i} = (- \\sin{(i)} + \\frac{d}{d i} (\\operatorname{v_{y}}{(i)} + e^{\\sin{(i)}}))^{i}", "derivation": "\\operatorname{v_{y}}{(i)} = e^{\\sin{(i)}} and \\operatorname{v_{y}}{(i)} + e^{\\sin{(i)}} = 2 e^{\\sin{(i)}} and \\frac{d}{d i} (\\operatorname{v_{y}}{(i)} + e^{\\sin{(i)}}) = \\frac{d}{d i} 2 e^{\\sin{(i)}} and G{(i)} = \\frac{d}{d i} 2 e^{\\sin{(i)}} and G{(i)} - \\sin{(i)} = - \\sin{(i)} + \\frac{d}{d i} 2 e^{\\sin{(i)}} and G{(i)} - \\sin{(i)} = - \\sin{(i)} + \\frac{d}{d i} (\\operatorname{v_{y}}{(i)} + e^{\\sin{(i)}}) and (G{(i)} - \\sin{(i)})^{i} = (- \\sin{(i)} + \\frac{d}{d i} (\\operatorname{v_{y}}{(i)} + e^{\\sin{(i)}}))^{i}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True))))"], [["add", 1, "exp(sin(Symbol('i', commutative=True)))"], "Equality(Add(Function('v_y')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True)))), Mul(Integer(2), exp(sin(Symbol('i', commutative=True)))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('G')(Symbol('i', commutative=True)), Derivative(Mul(Integer(2), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["minus", 4, "sin(Symbol('i', commutative=True))"], "Equality(Add(Function('G')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('i', commutative=True))), Derivative(Mul(Integer(2), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('G')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('i', commutative=True))), Derivative(Add(Function('v_y')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Function('G')(Symbol('i', commutative=True)), Mul(Integer(-1), sin(Symbol('i', commutative=True)))), Symbol('i', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('i', commutative=True))), Derivative(Add(Function('v_y')(Symbol('i', commutative=True)), exp(sin(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1)))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given q{(\\psi,t_{2})} = \\psi - t_{2} and v{(\\psi,t_{2})} = \\int (\\psi - t_{2})^{t_{2}} dt_{2}, then obtain \\frac{v{(\\psi,t_{2})}}{\\int q^{t_{2}}{(\\psi,t_{2})} dt_{2}} = 1", "derivation": "q{(\\psi,t_{2})} = \\psi - t_{2} and q^{t_{2}}{(\\psi,t_{2})} = (\\psi - t_{2})^{t_{2}} and \\int q^{t_{2}}{(\\psi,t_{2})} dt_{2} = \\int (\\psi - t_{2})^{t_{2}} dt_{2} and v{(\\psi,t_{2})} = \\int (\\psi - t_{2})^{t_{2}} dt_{2} and \\frac{v{(\\psi,t_{2})}}{\\int (\\psi - t_{2})^{t_{2}} dt_{2}} = 1 and \\frac{v{(\\psi,t_{2})}}{\\int q^{t_{2}}{(\\psi,t_{2})} dt_{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Pow(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["divide", 4, "Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))"], "Equality(Mul(Function('v')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Pow(Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Function('v')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Pow(Integral(Pow(Function('q')(Symbol('\\\\psi', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\delta{(A_{x},\\mathbf{r})} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})}, then obtain \\delta{(A_{x},\\mathbf{r})} + \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + 1 + \\frac{1}{A_{x}} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + 1 + \\frac{1}{A_{x}}", "derivation": "\\delta{(A_{x},\\mathbf{r})} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})} and \\delta{(A_{x},\\mathbf{r})} + \\frac{1}{A_{x}} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + \\frac{1}{A_{x}} and \\delta{(A_{x},\\mathbf{r})} + 1 + \\frac{1}{A_{x}} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + 1 + \\frac{1}{A_{x}} and \\delta{(A_{x},\\mathbf{r})} + \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + 1 + \\frac{1}{A_{x}} = \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + \\frac{\\partial}{\\partial \\mathbf{r}} \\cos{(\\frac{\\mathbf{r}}{A_{x}})} + 1 + \\frac{1}{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 1, "Pow(Symbol('A_x', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\delta')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Symbol('A_x', commutative=True), Integer(-1))), Add(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Pow(Symbol('A_x', commutative=True), Integer(-1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\delta')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))), Add(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))))"], [["add", 3, "Derivative(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\delta')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))), Add(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Derivative(cos(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1), Pow(Symbol('A_x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given V{(U,a)} = \\frac{e^{U}}{a}, then derive \\frac{\\partial}{\\partial a} V{(U,a)} - e^{- U} = - e^{- U} - \\frac{e^{U}}{a^{2}}, then obtain \\frac{\\partial}{\\partial a} V{(U,a)} - e^{- U} = - e^{- U} - \\frac{V{(U,a)}}{a}", "derivation": "V{(U,a)} = \\frac{e^{U}}{a} and \\frac{\\partial}{\\partial a} V{(U,a)} = \\frac{\\partial}{\\partial a} \\frac{e^{U}}{a} and \\frac{\\partial}{\\partial a} V{(U,a)} - e^{- U} = \\frac{\\partial}{\\partial a} \\frac{e^{U}}{a} - e^{- U} and \\frac{\\partial}{\\partial a} V{(U,a)} - e^{- U} = - e^{- U} - \\frac{e^{U}}{a^{2}} and \\frac{\\partial}{\\partial a} \\frac{e^{U}}{a} - e^{- U} = - e^{- U} - \\frac{e^{U}}{a^{2}} and \\frac{\\partial}{\\partial a} V{(U,a)} - e^{- U} = - e^{- U} - \\frac{V{(U,a)}}{a}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('U', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["minus", 2, "exp(Mul(Integer(-1), Symbol('U', commutative=True)))"], "Equality(Add(Derivative(Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True))))), Add(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True))))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-2)), exp(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), exp(Symbol('U', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-2)), exp(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Derivative(Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True))))), Add(Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('U', commutative=True)))), Mul(Integer(-1), Pow(Symbol('a', commutative=True), Integer(-1)), Function('V')(Symbol('U', commutative=True), Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}}, then derive \\frac{d}{d \\eta^{\\prime}} \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}}, then obtain \\frac{d}{d \\eta^{\\prime}} \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} e^{\\eta^{\\prime}}", "derivation": "\\hat{\\mathbf{r}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}} and \\frac{d}{d \\eta^{\\prime}} \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} e^{\\eta^{\\prime}} and \\frac{d}{d \\eta^{\\prime}} \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = e^{\\eta^{\\prime}} and \\frac{d}{d \\eta^{\\prime}} e^{\\eta^{\\prime}} = e^{\\eta^{\\prime}} and \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} e^{\\eta^{\\prime}} = \\frac{d}{d \\eta^{\\prime}} e^{\\eta^{\\prime}} and \\frac{d}{d \\eta^{\\prime}} \\hat{\\mathbf{r}}{(\\eta^{\\prime})} = \\frac{d^{2}}{d (\\eta^{\\prime})^{2}} e^{\\eta^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))), Derivative(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\sigma_{x}{(E)} = \\int \\log{(E)} dE, then derive \\sigma_{x}{(E)} = E \\log{(E)} - E + t_{1}, then derive E \\log{(E)} - E + t_{1} = E \\log{(E)} - E + W, then obtain \\frac{\\partial}{\\partial E} (E \\log{(E)} - E + W) = \\frac{d}{d E} \\int \\log{(E)} dE", "derivation": "\\sigma_{x}{(E)} = \\int \\log{(E)} dE and \\sigma_{x}{(E)} = E \\log{(E)} - E + t_{1} and E \\log{(E)} - E + t_{1} = \\int \\log{(E)} dE and E \\log{(E)} - E + t_{1} = E \\log{(E)} - E + W and E \\log{(E)} - E + W = \\int \\log{(E)} dE and \\frac{\\partial}{\\partial E} (E \\log{(E)} - E + W) = \\frac{d}{d E} \\int \\log{(E)} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('E', commutative=True)), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\sigma_x')(Symbol('E', commutative=True)), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('t_1', commutative=True)), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('t_1', commutative=True)), Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('W', commutative=True)), Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["differentiate", 5, "Symbol('E', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('E', commutative=True), log(Symbol('E', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('W', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\eta)} = e^{\\eta}, then derive (\\hat{H} + e^{\\eta}) n{(\\eta)} = (\\hat{H} + e^{\\eta}) e^{\\eta}, then obtain - \\hat{H} + (\\hat{H} + e^{\\eta}) n{(\\eta)} - e^{\\eta} = - \\hat{H} + (\\hat{H} + e^{\\eta}) e^{\\eta} - e^{\\eta}", "derivation": "n{(\\eta)} = e^{\\eta} and n{(\\eta)} \\int e^{\\eta} d\\eta = e^{\\eta} \\int e^{\\eta} d\\eta and (\\hat{H} + e^{\\eta}) n{(\\eta)} = (\\hat{H} + e^{\\eta}) e^{\\eta} and - \\hat{H} + (\\hat{H} + e^{\\eta}) n{(\\eta)} - e^{\\eta} = - \\hat{H} + (\\hat{H} + e^{\\eta}) e^{\\eta} - e^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["times", 1, "Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Function('n')(Symbol('\\\\eta', commutative=True)), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(exp(Symbol('\\\\eta', commutative=True)), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Function('n')(Symbol('\\\\eta', commutative=True))), Mul(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), exp(Symbol('\\\\eta', commutative=True))))"], [["minus", 3, "Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), Function('n')(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('\\\\eta', commutative=True))), exp(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), exp(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(z)} = \\sin{(e^{z})}, then obtain \\frac{d}{d z} \\mathbf{f}^{z}{(z)} \\sin^{z}{(e^{z})} = \\frac{d}{d z} \\sin^{2 z}{(e^{z})}", "derivation": "\\mathbf{f}{(z)} = \\sin{(e^{z})} and \\mathbf{f}^{z}{(z)} = \\sin^{z}{(e^{z})} and \\mathbf{f}^{z}{(z)} \\sin^{z}{(e^{z})} = \\sin^{2 z}{(e^{z})} and \\frac{d}{d z} \\mathbf{f}^{z}{(z)} \\sin^{z}{(e^{z})} = \\frac{d}{d z} \\sin^{2 z}{(e^{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), sin(exp(Symbol('z', commutative=True))))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(sin(exp(Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["times", 2, "Pow(sin(exp(Symbol('z', commutative=True))), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(sin(exp(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Pow(sin(exp(Symbol('z', commutative=True))), Mul(Integer(2), Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\mathbf{f}')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(sin(exp(Symbol('z', commutative=True))), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(sin(exp(Symbol('z', commutative=True))), Mul(Integer(2), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = A_{x} + \\sin{(\\mathbf{s})}, then obtain \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{2}\\partial A_{x}} \\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{2}\\partial A_{x}} (A_{x} + \\sin{(\\mathbf{s})})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = A_{x} + \\sin{(\\mathbf{s})} and \\frac{\\partial}{\\partial A_{x}} \\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\sin{(\\mathbf{s})}) and \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial A_{x}} \\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}\\partial A_{x}} (A_{x} + \\sin{(\\mathbf{s})}) and \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{2}\\partial A_{x}} \\operatorname{V_{\\mathbf{B}}}{(A_{x},\\mathbf{s})} = \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{2}\\partial A_{x}} (A_{x} + \\sin{(\\mathbf{s})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('A_x', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('A_x', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Derivative(Add(Symbol('A_x', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('A_x', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\rho{(v_{y})} = \\log{(v_{y})}, then obtain - v_{y} \\rho{(v_{y})} \\log{(v_{y})} = - v_{y} \\log{(v_{y})}^{2}", "derivation": "\\rho{(v_{y})} = \\log{(v_{y})} and v_{y} \\rho{(v_{y})} = v_{y} \\log{(v_{y})} and v_{y} \\rho^{2}{(v_{y})} = v_{y} \\rho{(v_{y})} \\log{(v_{y})} and - v_{y} \\rho^{2}{(v_{y})} = - v_{y} \\rho{(v_{y})} \\log{(v_{y})} and - v_{y} \\rho{(v_{y})} \\log{(v_{y})} = - v_{y} \\log{(v_{y})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["times", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Function('\\\\rho')(Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), log(Symbol('v_y', commutative=True))))"], [["times", 2, "Function('\\\\rho')(Symbol('v_y', commutative=True))"], "Equality(Mul(Symbol('v_y', commutative=True), Pow(Function('\\\\rho')(Symbol('v_y', commutative=True)), Integer(2))), Mul(Symbol('v_y', commutative=True), Function('\\\\rho')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(Function('\\\\rho')(Symbol('v_y', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('v_y', commutative=True), Function('\\\\rho')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(-1), Symbol('v_y', commutative=True), Function('\\\\rho')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Mul(Integer(-1), Symbol('v_y', commutative=True), Pow(log(Symbol('v_y', commutative=True)), Integer(2))))"]]}, {"prompt": "Given W{(\\hat{p}_0,\\hat{X})} = \\hat{X}^{\\hat{p}_0}, then obtain \\sin{(\\int \\frac{\\cos{(\\hat{X} W{(\\hat{p}_0,\\hat{X})})}}{\\cos{(\\hat{X} \\hat{X}^{\\hat{p}_0})}} d\\hat{X})} = \\sin{(\\int 1 d\\hat{X})}", "derivation": "W{(\\hat{p}_0,\\hat{X})} = \\hat{X}^{\\hat{p}_0} and \\hat{X} W{(\\hat{p}_0,\\hat{X})} = \\hat{X} \\hat{X}^{\\hat{p}_0} and \\cos{(\\hat{X} W{(\\hat{p}_0,\\hat{X})})} = \\cos{(\\hat{X} \\hat{X}^{\\hat{p}_0})} and \\frac{\\cos{(\\hat{X} W{(\\hat{p}_0,\\hat{X})})}}{\\cos{(\\hat{X} \\hat{X}^{\\hat{p}_0})}} = 1 and \\int \\frac{\\cos{(\\hat{X} W{(\\hat{p}_0,\\hat{X})})}}{\\cos{(\\hat{X} \\hat{X}^{\\hat{p}_0})}} d\\hat{X} = \\int 1 d\\hat{X} and \\sin{(\\int \\frac{\\cos{(\\hat{X} W{(\\hat{p}_0,\\hat{X})})}}{\\cos{(\\hat{X} \\hat{X}^{\\hat{p}_0})}} d\\hat{X})} = \\sin{(\\int 1 d\\hat{X})}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Symbol('\\\\hat{X}', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["divide", 3, "cos(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], "Equality(Mul(Pow(cos(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Integer(-1)), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Integer(1))"], [["integrate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Mul(Pow(cos(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Integer(-1)), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["sin", 5], "Equality(sin(Integral(Mul(Pow(cos(Mul(Symbol('\\\\hat{X}', commutative=True), Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Integer(-1)), cos(Mul(Symbol('\\\\hat{X}', commutative=True), Function('W')(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), sin(Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"]]}, {"prompt": "Given E{(\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda})}, then obtain - E{(\\hat{H}_{\\lambda})} + \\int E{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = A_{z} - E{(\\hat{H}_{\\lambda})} + \\sin{(\\hat{H}_{\\lambda})}", "derivation": "E{(\\hat{H}_{\\lambda})} = \\cos{(\\hat{H}_{\\lambda})} and \\int E{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = \\int \\cos{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and - \\cos{(\\hat{H}_{\\lambda})} + \\int E{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = - \\cos{(\\hat{H}_{\\lambda})} + \\int \\cos{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and - E{(\\hat{H}_{\\lambda})} + \\int E{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = - E{(\\hat{H}_{\\lambda})} + \\int \\cos{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} and - E{(\\hat{H}_{\\lambda})} + \\int E{(\\hat{H}_{\\lambda})} d\\hat{H}_{\\lambda} = A_{z} - E{(\\hat{H}_{\\lambda})} + \\sin{(\\hat{H}_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 2, "cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Mul(Integer(-1), Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Function('E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), sin(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(c,I)} = \\frac{c}{I} and \\dot{z}{(c,I)} = c + \\iint \\frac{c}{I} dc dc, then obtain 1 = \\frac{c + \\iint \\frac{c}{I} dc dc}{\\dot{z}{(c,I)}}", "derivation": "\\operatorname{v_{z}}{(c,I)} = \\frac{c}{I} and \\int \\operatorname{v_{z}}{(c,I)} dc = \\int \\frac{c}{I} dc and \\iint \\operatorname{v_{z}}{(c,I)} dc dc = \\iint \\frac{c}{I} dc dc and c + \\iint \\operatorname{v_{z}}{(c,I)} dc dc = c + \\iint \\frac{c}{I} dc dc and \\dot{z}{(c,I)} = c + \\iint \\frac{c}{I} dc dc and \\dot{z}{(c,I)} = c + \\iint \\operatorname{v_{z}}{(c,I)} dc dc and 1 = \\frac{c + \\iint \\operatorname{v_{z}}{(c,I)} dc dc}{\\dot{z}{(c,I)}} and 1 = \\frac{c + \\iint \\frac{c}{I} dc dc}{\\dot{z}{(c,I)}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["integrate", 2, "Symbol('c', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["add", 3, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Integral(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Add(Symbol('c', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Add(Symbol('c', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\dot{z}')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Add(Symbol('c', commutative=True), Integral(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))))"], [["divide", 6, "Function('\\\\dot{z}')(Symbol('c', commutative=True), Symbol('I', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('c', commutative=True), Integral(Function('v_z')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Pow(Function('\\\\dot{z}')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Integer(1), Mul(Add(Symbol('c', commutative=True), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))), Pow(Function('\\\\dot{z}')(Symbol('c', commutative=True), Symbol('I', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given n{(\\mu_0,\\varphi^*)} = \\log{(\\mu_0 \\varphi^*)}, then obtain 2 n^{\\varphi^*}{(\\mu_0,\\varphi^*)} + \\log{(\\mu_0 \\varphi^*)} = n^{\\varphi^*}{(\\mu_0,\\varphi^*)} + \\log{(\\mu_0 \\varphi^*)} + \\log{(\\mu_0 \\varphi^*)}^{\\varphi^*}", "derivation": "n{(\\mu_0,\\varphi^*)} = \\log{(\\mu_0 \\varphi^*)} and n^{\\varphi^*}{(\\mu_0,\\varphi^*)} = \\log{(\\mu_0 \\varphi^*)}^{\\varphi^*} and 2 n^{\\varphi^*}{(\\mu_0,\\varphi^*)} = n^{\\varphi^*}{(\\mu_0,\\varphi^*)} + \\log{(\\mu_0 \\varphi^*)}^{\\varphi^*} and 2 n^{\\varphi^*}{(\\mu_0,\\varphi^*)} + \\log{(\\mu_0 \\varphi^*)} = n^{\\varphi^*}{(\\mu_0,\\varphi^*)} + \\log{(\\mu_0 \\varphi^*)} + \\log{(\\mu_0 \\varphi^*)}^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["power", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True)))"], [["add", 2, "Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))), Add(Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), Pow(log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))))"], [["add", 3, "log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Add(Mul(Integer(2), Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)))), Add(Pow(Function('n')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True)), log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Pow(log(Mul(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given q{(g,\\hat{p},t_{2})} = \\frac{\\hat{p}}{g t_{2}}, then obtain e^{g + (- \\hat{p} + q{(g,\\hat{p},t_{2})}) q{(g,\\hat{p},t_{2})}} = e^{g + (- \\hat{p} + \\frac{\\hat{p}}{g t_{2}}) q{(g,\\hat{p},t_{2})}}", "derivation": "q{(g,\\hat{p},t_{2})} = \\frac{\\hat{p}}{g t_{2}} and - \\hat{p} + q{(g,\\hat{p},t_{2})} = - \\hat{p} + \\frac{\\hat{p}}{g t_{2}} and (- \\hat{p} + q{(g,\\hat{p},t_{2})}) q{(g,\\hat{p},t_{2})} = (- \\hat{p} + \\frac{\\hat{p}}{g t_{2}}) q{(g,\\hat{p},t_{2})} and g + (- \\hat{p} + q{(g,\\hat{p},t_{2})}) q{(g,\\hat{p},t_{2})} = g + (- \\hat{p} + \\frac{\\hat{p}}{g t_{2}}) q{(g,\\hat{p},t_{2})} and e^{g + (- \\hat{p} + q{(g,\\hat{p},t_{2})}) q{(g,\\hat{p},t_{2})}} = e^{g + (- \\hat{p} + \\frac{\\hat{p}}{g t_{2}}) q{(g,\\hat{p},t_{2})}}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1)))))"], [["times", 2, "Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))))"], [["add", 3, "Symbol('g', commutative=True)"], "Equality(Add(Symbol('g', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True)))), Add(Symbol('g', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True)))))"], [["exp", 4], "Equality(exp(Add(Symbol('g', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))))), exp(Add(Symbol('g', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('t_2', commutative=True), Integer(-1)))), Function('q')(Symbol('g', commutative=True), Symbol('\\\\hat{p}', commutative=True), Symbol('t_2', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(E,I,V)} = \\frac{E - I}{V}, then obtain E - I = E - I + \\frac{- E + I}{V} + \\frac{E - I}{V}", "derivation": "\\hat{\\mathbf{r}}{(E,I,V)} = \\frac{E - I}{V} and - E + I + \\hat{\\mathbf{r}}{(E,I,V)} = - E + I + \\frac{E - I}{V} and E - I - \\hat{\\mathbf{r}}{(E,I,V)} = E - I - \\frac{E - I}{V} and E - I - \\hat{\\mathbf{r}}{(E,I,V)} = E - I + \\frac{- E + I}{V} and E - I - \\hat{\\mathbf{r}}{(E,I,V)} + \\frac{E - I}{V} = E - I + \\frac{- E + I}{V} + \\frac{E - I}{V} and E - I = E - I + \\hat{\\mathbf{r}}{(E,I,V)} + \\frac{- E + I}{V} and E - I = E - I + \\frac{- E + I}{V} + \\frac{E - I}{V}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["minus", 1, "Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True)))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True)))))"], [["minus", 4, "Mul(Integer(-1), Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))"], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('E', commutative=True), Symbol('I', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('I', commutative=True))), Mul(Pow(Symbol('V', commutative=True), Integer(-1)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(G)} = e^{G}, then derive \\int \\operatorname{P_{e}}{(G)} dG = \\theta_1 + e^{G}, then obtain \\int (\\int \\operatorname{P_{e}}{(G)} dG)^{G} dG = \\int (\\theta_1 + e^{G})^{G} dG", "derivation": "\\operatorname{P_{e}}{(G)} = e^{G} and \\int \\operatorname{P_{e}}{(G)} dG = \\int e^{G} dG and \\int \\operatorname{P_{e}}{(G)} dG = \\theta_1 + e^{G} and (\\int \\operatorname{P_{e}}{(G)} dG)^{G} = (\\theta_1 + e^{G})^{G} and \\int (\\int \\operatorname{P_{e}}{(G)} dG)^{G} dG = \\int (\\theta_1 + e^{G})^{G} dG", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(exp(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('G', commutative=True))))"], [["power", 3, "Symbol('G', commutative=True)"], "Equality(Pow(Integral(Function('P_e')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Pow(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('G', commutative=True))), Symbol('G', commutative=True)))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Pow(Integral(Function('P_e')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Pow(Add(Symbol('\\\\theta_1', commutative=True), exp(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given M{(q,\\mathbf{D})} = q^{\\mathbf{D}}, then obtain - \\frac{M^{\\mathbf{D}}{(q,\\mathbf{D})}}{M{(q,\\mathbf{D})}} = - \\frac{(q^{\\mathbf{D}})^{\\mathbf{D}}}{M{(q,\\mathbf{D})}}", "derivation": "M{(q,\\mathbf{D})} = q^{\\mathbf{D}} and M^{\\mathbf{D}}{(q,\\mathbf{D})} = (q^{\\mathbf{D}})^{\\mathbf{D}} and \\frac{M^{\\mathbf{D}}{(q,\\mathbf{D})}}{M{(q,\\mathbf{D})}} = \\frac{(q^{\\mathbf{D}})^{\\mathbf{D}}}{M{(q,\\mathbf{D})}} and - \\frac{M^{\\mathbf{D}}{(q,\\mathbf{D})}}{M{(q,\\mathbf{D})}} = - \\frac{(q^{\\mathbf{D}})^{\\mathbf{D}}}{M{(q,\\mathbf{D})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Pow(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["divide", 2, "Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Pow(Pow(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Pow(Pow(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('M')(Symbol('q', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\operatorname{y^{\\prime}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then obtain (\\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})})^{\\mathbf{S}} = (\\frac{d}{d \\mathbf{S}} \\operatorname{y^{\\prime}}{(\\mathbf{S})})^{\\mathbf{S}}", "derivation": "\\hat{\\mathbf{x}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\frac{d}{d \\mathbf{S}} \\hat{\\mathbf{x}}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})} and \\operatorname{y^{\\prime}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\frac{d}{d \\mathbf{S}} \\hat{\\mathbf{x}}{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} \\operatorname{y^{\\prime}}{(\\mathbf{S})} and (\\frac{d}{d \\mathbf{S}} \\hat{\\mathbf{x}}{(\\mathbf{S})})^{\\mathbf{S}} = (\\frac{d}{d \\mathbf{S}} \\operatorname{y^{\\prime}}{(\\mathbf{S})})^{\\mathbf{S}} and (\\frac{d}{d \\mathbf{S}} \\cos{(\\mathbf{S})})^{\\mathbf{S}} = (\\frac{d}{d \\mathbf{S}} \\operatorname{y^{\\prime}}{(\\mathbf{S})})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Derivative(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Derivative(Function('y^{\\\\prime}')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{S}{(F_{H})} = \\sin{(\\cos{(F_{H})})}, then obtain \\iint (\\mathbf{S}{(F_{H})} - \\sin{(\\cos{(F_{H})})}) dF_{H} dF_{H} = \\iint 0 dF_{H} dF_{H}", "derivation": "\\mathbf{S}{(F_{H})} = \\sin{(\\cos{(F_{H})})} and \\mathbf{S}{(F_{H})} - \\sin{(\\cos{(F_{H})})} = 0 and \\int (\\mathbf{S}{(F_{H})} - \\sin{(\\cos{(F_{H})})}) dF_{H} = \\int 0 dF_{H} and \\iint (\\mathbf{S}{(F_{H})} - \\sin{(\\cos{(F_{H})})}) dF_{H} dF_{H} = \\iint 0 dF_{H} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('F_H', commutative=True)), sin(cos(Symbol('F_H', commutative=True))))"], [["minus", 1, "sin(cos(Symbol('F_H', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('F_H', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{S}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('F_H', commutative=True))))), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))))"], [["integrate", 3, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{S}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('F_H', commutative=True))))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given b{(m,J)} = \\frac{m}{J}, then obtain \\frac{J (- b{(m,J)} + \\frac{m}{J})}{b{(m,J)}} + b{(m,J)} = \\frac{m}{J}", "derivation": "b{(m,J)} = \\frac{m}{J} and 0 = - b{(m,J)} + \\frac{m}{J} and 0 = J (- b{(m,J)} + \\frac{m}{J}) and 0 = \\frac{J (- b{(m,J)} + \\frac{m}{J})}{b{(m,J)}} and b{(m,J)} = \\frac{J (- b{(m,J)} + \\frac{m}{J})}{b{(m,J)}} + b{(m,J)} and \\frac{J (- b{(m,J)} + \\frac{m}{J})}{b{(m,J)}} + b{(m,J)} = \\frac{m}{J}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["minus", 1, "Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True))))"], [["times", 2, "Symbol('J', commutative=True)"], "Equality(Integer(0), Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True)))))"], [["divide", 3, "Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))"], "Equality(Integer(0), Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Integer(-1))))"], [["minus", 4, "Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)))"], "Equality(Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Add(Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Integer(-1))), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Add(Mul(Symbol('J', commutative=True), Add(Mul(Integer(-1), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True))), Pow(Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True)), Integer(-1))), Function('b')(Symbol('m', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(\\omega)} = \\omega, then obtain - \\omega + \\hat{H}_l^{\\omega}{(\\omega)} + \\frac{d}{d \\omega} (- \\omega + \\omega^{\\omega}) = - \\omega + \\omega^{\\omega} + \\frac{d}{d \\omega} (- \\omega + \\omega^{\\omega})", "derivation": "\\hat{H}_l{(\\omega)} = \\omega and \\hat{H}_l^{\\omega}{(\\omega)} = \\omega^{\\omega} and - \\omega + \\hat{H}_l^{\\omega}{(\\omega)} = - \\omega + \\omega^{\\omega} and \\frac{d}{d \\omega} (- \\omega + \\hat{H}_l^{\\omega}{(\\omega)}) = \\frac{d}{d \\omega} (- \\omega + \\omega^{\\omega}) and - \\omega + \\hat{H}_l^{\\omega}{(\\omega)} + \\frac{d}{d \\omega} (- \\omega + \\hat{H}_l^{\\omega}{(\\omega)}) = - \\omega + \\omega^{\\omega} + \\frac{d}{d \\omega} (- \\omega + \\hat{H}_l^{\\omega}{(\\omega)}) and - \\omega + \\hat{H}_l^{\\omega}{(\\omega)} + \\frac{d}{d \\omega} (- \\omega + \\omega^{\\omega}) = - \\omega + \\omega^{\\omega} + \\frac{d}{d \\omega} (- \\omega + \\omega^{\\omega})", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["minus", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Symbol('\\\\omega', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(b,r)} = \\frac{\\log{(r)}}{b}, then obtain \\log{(b^{3} \\operatorname{E_{x}}{(b,r)} \\log{(r)})} = \\log{(b^{2} \\log{(r)}^{2})}", "derivation": "\\operatorname{E_{x}}{(b,r)} = \\frac{\\log{(r)}}{b} and \\frac{\\operatorname{E_{x}}{(b,r)} \\log{(r)}}{b} = \\frac{\\log{(r)}^{2}}{b^{2}} and b^{2} \\operatorname{E_{x}}{(b,r)} = b \\log{(r)} and b^{3} \\operatorname{E_{x}}{(b,r)} \\log{(r)} = b^{2} \\log{(r)}^{2} and \\log{(b^{3} \\operatorname{E_{x}}{(b,r)} \\log{(r)})} = \\log{(b^{2} \\log{(r)}^{2})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('b', commutative=True), Symbol('r', commutative=True)), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('r', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('b', commutative=True), Integer(-1)), log(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('E_x')(Symbol('b', commutative=True), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Pow(log(Symbol('r', commutative=True)), Integer(2))))"], [["divide", 2, "Mul(Pow(Symbol('b', commutative=True), Integer(-3)), log(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Function('E_x')(Symbol('b', commutative=True), Symbol('r', commutative=True))), Mul(Symbol('b', commutative=True), log(Symbol('r', commutative=True))))"], [["times", 3, "Mul(Symbol('b', commutative=True), log(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(3)), Function('E_x')(Symbol('b', commutative=True), Symbol('r', commutative=True)), log(Symbol('r', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(log(Symbol('r', commutative=True)), Integer(2))))"], [["log", 4], "Equality(log(Mul(Pow(Symbol('b', commutative=True), Integer(3)), Function('E_x')(Symbol('b', commutative=True), Symbol('r', commutative=True)), log(Symbol('r', commutative=True)))), log(Mul(Pow(Symbol('b', commutative=True), Integer(2)), Pow(log(Symbol('r', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\rho_b,A_{z})} = A_{z} + \\rho_b, then obtain \\frac{\\partial^{2}}{\\partial A_{z}^{2}} (- A_{z} + \\operatorname{A_{1}}^{\\rho_b}{(\\rho_b,A_{z})}) = \\frac{\\partial^{2}}{\\partial A_{z}^{2}} (- A_{z} + (A_{z} + \\rho_b)^{\\rho_b})", "derivation": "\\operatorname{A_{1}}{(\\rho_b,A_{z})} = A_{z} + \\rho_b and \\operatorname{A_{1}}^{\\rho_b}{(\\rho_b,A_{z})} = (A_{z} + \\rho_b)^{\\rho_b} and - A_{z} + \\operatorname{A_{1}}^{\\rho_b}{(\\rho_b,A_{z})} = - A_{z} + (A_{z} + \\rho_b)^{\\rho_b} and \\frac{\\partial}{\\partial A_{z}} (- A_{z} + \\operatorname{A_{1}}^{\\rho_b}{(\\rho_b,A_{z})}) = \\frac{\\partial}{\\partial A_{z}} (- A_{z} + (A_{z} + \\rho_b)^{\\rho_b}) and \\frac{\\partial^{2}}{\\partial A_{z}^{2}} (- A_{z} + \\operatorname{A_{1}}^{\\rho_b}{(\\rho_b,A_{z})}) = \\frac{\\partial^{2}}{\\partial A_{z}^{2}} (- A_{z} + (A_{z} + \\rho_b)^{\\rho_b})", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["power", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\rho_b', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"], [["minus", 2, "Symbol('A_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Function('A_1')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 3, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Function('A_1')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Function('A_1')(Symbol('\\\\rho_b', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Add(Symbol('A_z', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mathbf{D}{(C_{d},A_{z})} = \\int C_{d}^{A_{z}} dA_{z} and \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} = \\int C_{d}^{A_{z}} dA_{z}, then obtain 0 = \\frac{\\partial}{\\partial A_{z}} \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} - \\frac{\\partial}{\\partial A_{z}} \\int C_{d}^{A_{z}} dA_{z}", "derivation": "\\mathbf{D}{(C_{d},A_{z})} = \\int C_{d}^{A_{z}} dA_{z} and \\frac{\\partial}{\\partial A_{z}} \\mathbf{D}{(C_{d},A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\int C_{d}^{A_{z}} dA_{z} and \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} = \\int C_{d}^{A_{z}} dA_{z} and \\frac{\\partial}{\\partial A_{z}} \\mathbf{D}{(C_{d},A_{z})} = \\frac{\\partial}{\\partial A_{z}} \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} and \\frac{\\partial}{\\partial A_{z}} \\int C_{d}^{A_{z}} dA_{z} = \\frac{\\partial}{\\partial A_{z}} \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} and 0 = \\frac{\\partial}{\\partial A_{z}} \\operatorname{x^{{\\}'}}{(C_{d},A_{z})} - \\frac{\\partial}{\\partial A_{z}} \\int C_{d}^{A_{z}} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Function('x^\\\\prime')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(Function('x^\\\\prime')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Function('x^\\\\prime')(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(Pow(Symbol('C_d', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('A_z', commutative=True), Integer(1))))))"]]}, {"prompt": "Given r{(v_{2},E_{n})} = v_{2}^{E_{n}} and \\mathbf{s}{(v_{2},E_{n})} = v_{2}^{E_{n}}, then obtain \\frac{\\mathbf{s}{(v_{2},E_{n})}}{\\phi{(\\chi,\\eta,\\delta)}} = \\frac{v_{2}^{E_{n}}}{\\phi{(\\chi,\\eta,\\delta)}}", "derivation": "r{(v_{2},E_{n})} = v_{2}^{E_{n}} and - r{(v_{2},E_{n})} = - v_{2}^{E_{n}} and - \\frac{r{(v_{2},E_{n})}}{\\phi{(\\chi,\\eta,\\delta)}} = - \\frac{v_{2}^{E_{n}}}{\\phi{(\\chi,\\eta,\\delta)}} and \\mathbf{s}{(v_{2},E_{n})} = v_{2}^{E_{n}} and \\mathbf{s}{(v_{2},E_{n})} = r{(v_{2},E_{n})} and \\frac{r{(v_{2},E_{n})}}{\\phi{(\\chi,\\eta,\\delta)}} = \\frac{v_{2}^{E_{n}}}{\\phi{(\\chi,\\eta,\\delta)}} and \\frac{\\mathbf{s}{(v_{2},E_{n})}}{\\phi{(\\chi,\\eta,\\delta)}} = \\frac{v_{2}^{E_{n}}}{\\phi{(\\chi,\\eta,\\delta)}}", "srepr_derivation": [["get_premise", "Equality(Function('r')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('r')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True))))"], [["divide", 2, "Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Function('r')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Function('r')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1)), Function('r')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True))), Mul(Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Function('\\\\mathbf{s}')(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))), Mul(Pow(Symbol('v_2', commutative=True), Symbol('E_n', commutative=True)), Pow(Function('\\\\phi')(Symbol('\\\\chi', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\tilde{g}{(a^{\\dagger},\\mathbf{g})} = \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g}, then obtain - \\mathbf{g} a^{\\dagger} + a^{\\dagger} (a^{\\dagger} + \\tilde{g}{(a^{\\dagger},\\mathbf{g})}) = - \\mathbf{g} a^{\\dagger} + a^{\\dagger} (a^{\\dagger} + \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g})", "derivation": "\\tilde{g}{(a^{\\dagger},\\mathbf{g})} = \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g} and a^{\\dagger} + \\tilde{g}{(a^{\\dagger},\\mathbf{g})} = a^{\\dagger} + \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g} and a^{\\dagger} (a^{\\dagger} + \\tilde{g}{(a^{\\dagger},\\mathbf{g})}) = a^{\\dagger} (a^{\\dagger} + \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g}) and - \\mathbf{g} a^{\\dagger} + a^{\\dagger} (a^{\\dagger} + \\tilde{g}{(a^{\\dagger},\\mathbf{g})}) = - \\mathbf{g} a^{\\dagger} + a^{\\dagger} (a^{\\dagger} + \\int \\mathbf{g} a^{\\dagger} d\\mathbf{g})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["add", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Symbol('a^{\\\\dagger}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))"], [["times", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))))"], [["minus", 3, "Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Function('\\\\tilde{g}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('a^{\\\\dagger}', commutative=True), Add(Symbol('a^{\\\\dagger}', commutative=True), Integral(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(v_{2})} = e^{v_{2}}, then obtain 0 = - e^{8 v_{2}} + \\frac{e^{16 v_{2}}}{\\operatorname{F_{g}}^{8}{(v_{2})}}", "derivation": "\\operatorname{F_{g}}{(v_{2})} = e^{v_{2}} and \\operatorname{F_{g}}{(v_{2})} e^{v_{2}} = e^{2 v_{2}} and e^{v_{2}} = \\frac{e^{2 v_{2}}}{\\operatorname{F_{g}}{(v_{2})}} and e^{2 v_{2}} = \\frac{e^{4 v_{2}}}{\\operatorname{F_{g}}^{2}{(v_{2})}} and e^{8 v_{2}} = \\frac{e^{16 v_{2}}}{\\operatorname{F_{g}}^{8}{(v_{2})}} and 0 = - e^{8 v_{2}} + \\frac{e^{16 v_{2}}}{\\operatorname{F_{g}}^{8}{(v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True)))"], [["times", 1, "exp(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('F_g')(Symbol('v_2', commutative=True)), exp(Symbol('v_2', commutative=True))), exp(Mul(Integer(2), Symbol('v_2', commutative=True))))"], [["divide", 2, "Function('F_g')(Symbol('v_2', commutative=True))"], "Equality(exp(Symbol('v_2', commutative=True)), Mul(Pow(Function('F_g')(Symbol('v_2', commutative=True)), Integer(-1)), exp(Mul(Integer(2), Symbol('v_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Mul(Integer(2), Symbol('v_2', commutative=True))), Mul(Pow(Function('F_g')(Symbol('v_2', commutative=True)), Integer(-2)), exp(Mul(Integer(4), Symbol('v_2', commutative=True)))))"], [["power", 4, 4], "Equality(exp(Mul(Integer(8), Symbol('v_2', commutative=True))), Mul(Pow(Function('F_g')(Symbol('v_2', commutative=True)), Integer(-8)), exp(Mul(Integer(16), Symbol('v_2', commutative=True)))))"], [["minus", 5, "exp(Mul(Integer(8), Symbol('v_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Mul(Integer(8), Symbol('v_2', commutative=True)))), Mul(Pow(Function('F_g')(Symbol('v_2', commutative=True)), Integer(-8)), exp(Mul(Integer(16), Symbol('v_2', commutative=True))))))"]]}, {"prompt": "Given y{(C_{d},c,\\phi_1)} = C_{d} c - \\phi_1 and \\operatorname{v_{z}}{(C_{d},c,\\phi_1)} = C_{d} c - \\phi_1, then obtain C_{d} \\operatorname{v_{z}}^{c}{(C_{d},c,\\phi_1)} + C_{d} y^{c}{(C_{d},c,\\phi_1)} - y^{c}{(C_{d},c,\\phi_1)} = 2 C_{d} \\operatorname{v_{z}}^{c}{(C_{d},c,\\phi_1)} - y^{c}{(C_{d},c,\\phi_1)}", "derivation": "y{(C_{d},c,\\phi_1)} = C_{d} c - \\phi_1 and y^{c}{(C_{d},c,\\phi_1)} = (C_{d} c - \\phi_1)^{c} and C_{d} y^{c}{(C_{d},c,\\phi_1)} = C_{d} (C_{d} c - \\phi_1)^{c} and C_{d} (C_{d} c - \\phi_1)^{c} + C_{d} y^{c}{(C_{d},c,\\phi_1)} - y^{c}{(C_{d},c,\\phi_1)} = 2 C_{d} (C_{d} c - \\phi_1)^{c} - y^{c}{(C_{d},c,\\phi_1)} and \\operatorname{v_{z}}{(C_{d},c,\\phi_1)} = C_{d} c - \\phi_1 and C_{d} \\operatorname{v_{z}}^{c}{(C_{d},c,\\phi_1)} + C_{d} y^{c}{(C_{d},c,\\phi_1)} - y^{c}{(C_{d},c,\\phi_1)} = 2 C_{d} \\operatorname{v_{z}}^{c}{(C_{d},c,\\phi_1)} - y^{c}{(C_{d},c,\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Symbol('c', commutative=True)))"], [["times", 2, "Symbol('C_d', commutative=True)"], "Equality(Mul(Symbol('C_d', commutative=True), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True))), Mul(Symbol('C_d', commutative=True), Pow(Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Symbol('c', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('C_d', commutative=True), Pow(Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Symbol('c', commutative=True))), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)))"], "Equality(Add(Mul(Symbol('C_d', commutative=True), Pow(Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Symbol('c', commutative=True))), Mul(Symbol('C_d', commutative=True), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)))), Add(Mul(Integer(2), Symbol('C_d', commutative=True), Pow(Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Symbol('C_d', commutative=True), Pow(Function('v_z')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True))), Mul(Symbol('C_d', commutative=True), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)))), Add(Mul(Integer(2), Symbol('C_d', commutative=True), Pow(Function('v_z')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True))), Mul(Integer(-1), Pow(Function('y')(Symbol('C_d', commutative=True), Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(c,m)} = \\cos{(c + m)}, then obtain - \\frac{I}{l} + (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\operatorname{v_{z}}^{m}{(c,m)} = - \\frac{I}{l} + (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\cos^{m}{(c + m)}", "derivation": "\\operatorname{v_{z}}{(c,m)} = \\cos{(c + m)} and \\operatorname{v_{z}}^{m}{(c,m)} = \\cos^{m}{(c + m)} and \\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)} = 2 \\cos{(c + m)} and 2 \\operatorname{v_{z}}^{m}{(c,m)} \\cos{(c + m)} = 2 \\cos{(c + m)} \\cos^{m}{(c + m)} and (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\operatorname{v_{z}}^{m}{(c,m)} = (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\cos^{m}{(c + m)} and - \\frac{I}{l} + (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\operatorname{v_{z}}^{m}{(c,m)} = - \\frac{I}{l} + (\\operatorname{v_{z}}{(c,m)} + \\cos{(c + m)}) \\cos^{m}{(c + m)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["add", 1, "cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))"], "Equality(Add(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))))"], [["times", 2, "Mul(Integer(2), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))), Pow(cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Pow(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Add(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Pow(cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["minus", 5, "Mul(Symbol('I', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Add(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Pow(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Add(Mul(Integer(-1), Symbol('I', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Mul(Add(Function('v_z')(Symbol('c', commutative=True), Symbol('m', commutative=True)), cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True)))), Pow(cos(Add(Symbol('c', commutative=True), Symbol('m', commutative=True))), Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{J},L)} = \\mathbf{J} + \\sin{(L)}, then obtain 0 = - \\sin{(L (2 \\mathbf{J} + 2 \\sin{(L)}) - L (\\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)}))}", "derivation": "\\mathbf{p}{(\\mathbf{J},L)} = \\mathbf{J} + \\sin{(L)} and 2 \\mathbf{p}{(\\mathbf{J},L)} = \\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)} and 2 L \\mathbf{p}{(\\mathbf{J},L)} = L (\\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)}) and 2 L (\\mathbf{J} + \\sin{(L)}) = L (2 \\mathbf{J} + 2 \\sin{(L)}) and 2 L \\mathbf{p}{(\\mathbf{J},L)} = L (2 \\mathbf{J} + 2 \\sin{(L)}) and 0 = L (\\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)}) - 2 L \\mathbf{p}{(\\mathbf{J},L)} and 0 = \\sin{(L (\\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)}) - 2 L \\mathbf{p}{(\\mathbf{J},L)})} and 0 = - \\sin{(L (2 \\mathbf{J} + 2 \\sin{(L)}) - L (\\mathbf{J} + \\mathbf{p}{(\\mathbf{J},L)} + \\sin{(L)}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('L', commutative=True))))"], [["add", 1, "Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Integer(2), Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Symbol('L', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('L', commutative=True)))), Mul(Symbol('L', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), sin(Symbol('L', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), sin(Symbol('L', commutative=True))))))"], [["minus", 3, "Mul(Integer(2), Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('L', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)))))"], [["sin", 6], "Equality(Integer(0), sin(Add(Mul(Symbol('L', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('L', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Integer(0), Mul(Integer(-1), sin(Add(Mul(Symbol('L', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(2), sin(Symbol('L', commutative=True))))), Mul(Integer(-1), Symbol('L', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('L', commutative=True)), sin(Symbol('L', commutative=True))))))))"]]}, {"prompt": "Given \\dot{x}{(m_{s})} = m_{s}, then derive v_{z} + \\frac{\\dot{x}^{2}{(m_{s})}}{2} = \\int m_{s} d\\dot{x}{(m_{s})}, then obtain \\frac{m_{s}^{2}}{2} + v_{z} = \\int m_{s} dm_{s}", "derivation": "\\dot{x}{(m_{s})} = m_{s} and \\int \\dot{x}{(m_{s})} dm_{s} = \\int m_{s} dm_{s} and \\int \\dot{x}{(m_{s})} d\\dot{x}{(m_{s})} = \\int m_{s} d\\dot{x}{(m_{s})} and v_{z} + \\frac{\\dot{x}^{2}{(m_{s})}}{2} = \\int m_{s} d\\dot{x}{(m_{s})} and \\frac{m_{s}^{2}}{2} + v_{z} = \\int m_{s} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Tuple(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)))), Integral(Symbol('m_s', commutative=True), Tuple(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('v_z', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)), Integer(2)))), Integral(Symbol('m_s', commutative=True), Tuple(Function('\\\\dot{x}')(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('m_s', commutative=True), Integer(2))), Symbol('v_z', commutative=True)), Integral(Symbol('m_s', commutative=True), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\varphi^*,M_{E})} = \\frac{M_{E}}{\\varphi^*} and \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} = (\\frac{M_{E}}{\\varphi^* \\operatorname{r_{0}}{(\\varphi^*,M_{E})}})^{\\varphi^*}, then obtain \\frac{\\int 1 d\\varphi^*}{\\int \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} d\\varphi^*} = 1", "derivation": "\\operatorname{r_{0}}{(\\varphi^*,M_{E})} = \\frac{M_{E}}{\\varphi^*} and 1 = \\frac{M_{E}}{\\varphi^* \\operatorname{r_{0}}{(\\varphi^*,M_{E})}} and 1 = (\\frac{M_{E}}{\\varphi^* \\operatorname{r_{0}}{(\\varphi^*,M_{E})}})^{\\varphi^*} and \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} = (\\frac{M_{E}}{\\varphi^* \\operatorname{r_{0}}{(\\varphi^*,M_{E})}})^{\\varphi^*} and 1 = \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} and \\int 1 d\\varphi^* = \\int \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} d\\varphi^* and \\frac{\\int 1 d\\varphi^*}{\\int \\operatorname{E_{\\lambda}}{(\\varphi^*,M_{E})} d\\varphi^*} = 1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1))))"], [["divide", 1, "Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Integer(1), Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Pow(Mul(Symbol('M_E', commutative=True), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(-1)), Pow(Function('r_0')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Integer(-1))), Symbol('\\\\varphi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(1), Function('E_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)))"], [["integrate", 5, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Function('E_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 6, "Integral(Function('E_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Pow(Integral(Function('E_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\hat{X}{(\\mathbf{s})} = \\sin{(\\mathbf{s})}, then obtain - \\cos{(\\mathbf{s})} = - \\frac{d}{d \\mathbf{s}} \\hat{X}{(\\mathbf{s})}", "derivation": "\\hat{X}{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and - \\hat{X}{(\\mathbf{s})} = - \\sin{(\\mathbf{s})} and - 2 \\hat{X}{(\\mathbf{s})} + \\sin{(\\mathbf{s})} = - \\hat{X}{(\\mathbf{s})} and - 2 \\hat{X}{(\\mathbf{s})} + \\sin{(\\mathbf{s})} = - \\sin{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} (- 2 \\hat{X}{(\\mathbf{s})} + \\sin{(\\mathbf{s})}) = \\frac{d}{d \\mathbf{s}} - \\hat{X}{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} - \\sin{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} - \\hat{X}{(\\mathbf{s})} and - \\cos{(\\mathbf{s})} = - \\frac{d}{d \\mathbf{s}} \\hat{X}{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 2, "Add(Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(-1), cos(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hat{X}')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\psi)} = e^{\\sin{(\\psi)}}, then obtain \\frac{\\int \\operatorname{J_{\\varepsilon}}{(\\psi)} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} - \\frac{\\int e^{\\sin{(\\psi)}} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} = 0", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\psi)} = e^{\\sin{(\\psi)}} and \\int \\operatorname{J_{\\varepsilon}}{(\\psi)} d\\psi = \\int e^{\\sin{(\\psi)}} d\\psi and \\frac{\\int \\operatorname{J_{\\varepsilon}}{(\\psi)} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} = \\frac{\\int e^{\\sin{(\\psi)}} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} and \\frac{\\int \\operatorname{J_{\\varepsilon}}{(\\psi)} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} - \\frac{\\int e^{\\sin{(\\psi)}} d\\psi}{\\operatorname{J_{\\varepsilon}}{(\\psi)}} = 0", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), exp(sin(Symbol('\\\\psi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(sin(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 2, "Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(-1)), Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True)))))"], [["minus", 3, "Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))"], "Equality(Add(Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(-1)), Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Integer(-1), Pow(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi', commutative=True)), Integer(-1)), Integral(exp(sin(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})}, then derive \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{B}}}{(F_{x})} = - \\cos{(F_{x})}, then obtain \\int \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{B}}}{(F_{x})} dF_{x} = \\int - \\cos{(F_{x})} dF_{x}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})} and \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{B}}}{(F_{x})} = \\frac{d^{2}}{d F_{x}^{2}} \\cos{(F_{x})} and \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{B}}}{(F_{x})} = - \\cos{(F_{x})} and \\int \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{B}}}{(F_{x})} dF_{x} = \\int - \\cos{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('F_x', commutative=True)), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('F_x', commutative=True))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given C{(g_{\\varepsilon})} = \\sin{(\\log{(g_{\\varepsilon})})}, then derive \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = M_{E} + \\frac{g_{\\varepsilon} \\sin{(\\log{(g_{\\varepsilon})})}}{2} - \\frac{g_{\\varepsilon} \\cos{(\\log{(g_{\\varepsilon})})}}{2}, then obtain \\frac{d}{d g_{\\varepsilon}} \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = \\frac{\\partial}{\\partial g_{\\varepsilon}} (M_{E} + \\frac{g_{\\varepsilon} \\sin{(\\log{(g_{\\varepsilon})})}}{2} - \\frac{g_{\\varepsilon} \\cos{(\\log{(g_{\\varepsilon})})}}{2})", "derivation": "C{(g_{\\varepsilon})} = \\sin{(\\log{(g_{\\varepsilon})})} and \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\sin{(\\log{(g_{\\varepsilon})})} dg_{\\varepsilon} and \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = M_{E} + \\frac{g_{\\varepsilon} \\sin{(\\log{(g_{\\varepsilon})})}}{2} - \\frac{g_{\\varepsilon} \\cos{(\\log{(g_{\\varepsilon})})}}{2} and \\frac{d}{d g_{\\varepsilon}} \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = \\frac{\\partial}{\\partial g_{\\varepsilon}} (M_{E} + \\frac{g_{\\varepsilon} \\sin{(\\log{(g_{\\varepsilon})})}}{2} - \\frac{g_{\\varepsilon} \\cos{(\\log{(g_{\\varepsilon})})}}{2})", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), sin(log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(sin(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Symbol('g_{\\\\varepsilon}', commutative=True), sin(log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('g_{\\\\varepsilon}', commutative=True), cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["differentiate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Add(Symbol('M_E', commutative=True), Mul(Rational(1, 2), Symbol('g_{\\\\varepsilon}', commutative=True), sin(log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('g_{\\\\varepsilon}', commutative=True), cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(A_{y})} = \\int \\log{(A_{y})} dA_{y} and \\mathbf{P}{(A_{y})} = (\\int \\log{(A_{y})} dA_{y})^{A_{y}}, then obtain \\frac{- p + \\mathbf{P}{(A_{y})}}{A_{x}} = \\frac{- p + n^{A_{y}}{(A_{y})}}{A_{x}}", "derivation": "n{(A_{y})} = \\int \\log{(A_{y})} dA_{y} and n^{A_{y}}{(A_{y})} = (\\int \\log{(A_{y})} dA_{y})^{A_{y}} and \\mathbf{P}{(A_{y})} = (\\int \\log{(A_{y})} dA_{y})^{A_{y}} and - p + \\mathbf{P}{(A_{y})} = - p + (\\int \\log{(A_{y})} dA_{y})^{A_{y}} and - p + \\mathbf{P}{(A_{y})} = - p + n^{A_{y}}{(A_{y})} and \\frac{- p + \\mathbf{P}{(A_{y})}}{A_{x}} = \\frac{- p + n^{A_{y}}{(A_{y})}}{A_{x}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('A_y', commutative=True)), Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["power", 1, "Symbol('A_y', commutative=True)"], "Equality(Pow(Function('n')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)), Pow(Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["minus", 3, "Symbol('p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Integral(log(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True))), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('n')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))))"], [["divide", 5, "Symbol('A_x', commutative=True)"], "Equality(Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Function('\\\\mathbf{P}')(Symbol('A_y', commutative=True)))), Mul(Pow(Symbol('A_x', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('p', commutative=True)), Pow(Function('n')(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\mathbf{E},\\mathbf{g})} = - \\mathbf{E} + \\mathbf{g}, then derive \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})} = -1, then obtain (\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})})^{2} = - \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})}", "derivation": "\\mathbf{p}{(\\mathbf{E},\\mathbf{g})} = - \\mathbf{E} + \\mathbf{g} and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} + \\mathbf{g}) and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})} = -1 and \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} + \\mathbf{g}) = -1 and (\\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} + \\mathbf{g}))^{2} = - \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} + \\mathbf{g}) and (\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})})^{2} = - \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{p}{(\\mathbf{E},\\mathbf{g})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(-1))"], [["times", 4, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(t,L)} = \\log{(\\frac{t}{L})} and \\sigma_{x}{(t,L)} = \\frac{t}{L}, then obtain \\log{(\\sigma_{x}{(t,L)})} = \\log{(\\frac{t}{L})}", "derivation": "\\varepsilon{(t,L)} = \\log{(\\frac{t}{L})} and \\sigma_{x}{(t,L)} = \\frac{t}{L} and \\varepsilon{(t,L)} = \\log{(\\sigma_{x}{(t,L)})} and \\log{(\\sigma_{x}{(t,L)})} = \\log{(\\frac{t}{L})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('t', commutative=True), Symbol('L', commutative=True)), log(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('t', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\varepsilon')(Symbol('t', commutative=True), Symbol('L', commutative=True)), log(Function('\\\\sigma_x')(Symbol('t', commutative=True), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(log(Function('\\\\sigma_x')(Symbol('t', commutative=True), Symbol('L', commutative=True))), log(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('t', commutative=True))))"]]}, {"prompt": "Given V{(E_{n})} = \\cos{(E_{n})}, then obtain \\frac{\\iint V{(E_{n})} dE_{n} dE_{n}}{C_{1} - \\tilde{g}} = \\frac{\\iint \\cos{(E_{n})} dE_{n} dE_{n}}{C_{1} - \\tilde{g}}", "derivation": "V{(E_{n})} = \\cos{(E_{n})} and \\int V{(E_{n})} dE_{n} = \\int \\cos{(E_{n})} dE_{n} and \\iint V{(E_{n})} dE_{n} dE_{n} = \\iint \\cos{(E_{n})} dE_{n} dE_{n} and \\frac{\\iint V{(E_{n})} dE_{n} dE_{n}}{C_{1} - \\tilde{g}} = \\frac{\\iint \\cos{(E_{n})} dE_{n} dE_{n}}{C_{1} - \\tilde{g}}", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('V')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["integrate", 2, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('V')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["divide", 3, "Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Integral(Function('V')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Mul(Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Integer(-1)), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given x{(\\eta^{\\prime},l)} = (\\eta^{\\prime})^{l}, then obtain \\int 0 d\\eta^{\\prime} = \\int ((\\eta^{\\prime})^{l} \\int x{(\\eta^{\\prime},l)} dl - x{(\\eta^{\\prime},l)} \\int x{(\\eta^{\\prime},l)} dl) d\\eta^{\\prime}", "derivation": "x{(\\eta^{\\prime},l)} = (\\eta^{\\prime})^{l} and \\int x{(\\eta^{\\prime},l)} dl = \\int (\\eta^{\\prime})^{l} dl and x{(\\eta^{\\prime},l)} \\int (\\eta^{\\prime})^{l} dl = (\\eta^{\\prime})^{l} \\int (\\eta^{\\prime})^{l} dl and 0 = (\\eta^{\\prime})^{l} \\int (\\eta^{\\prime})^{l} dl - x{(\\eta^{\\prime},l)} \\int (\\eta^{\\prime})^{l} dl and 0 = (\\eta^{\\prime})^{l} \\int x{(\\eta^{\\prime},l)} dl - x{(\\eta^{\\prime},l)} \\int x{(\\eta^{\\prime},l)} dl and \\int 0 d\\eta^{\\prime} = \\int ((\\eta^{\\prime})^{l} \\int x{(\\eta^{\\prime},l)} dl - x{(\\eta^{\\prime},l)} \\int x{(\\eta^{\\prime},l)} dl) d\\eta^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 1, "Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["minus", 3, "Mul(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Add(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))))"], [["integrate", 5, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Add(Mul(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Integer(-1), Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Integral(Function('x')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given h{(m_{s},\\varphi)} = \\cos{(\\varphi m_{s})} and \\hat{X}{(m_{s},\\varphi)} = \\varphi \\cos{(\\varphi m_{s})}, then obtain \\hat{X}{(m_{s},\\varphi)} \\cos{(\\varphi m_{s})} = \\varphi \\cos^{2}{(\\varphi m_{s})}", "derivation": "h{(m_{s},\\varphi)} = \\cos{(\\varphi m_{s})} and \\varphi h{(m_{s},\\varphi)} = \\varphi \\cos{(\\varphi m_{s})} and \\hat{X}{(m_{s},\\varphi)} = \\varphi \\cos{(\\varphi m_{s})} and \\varphi h{(m_{s},\\varphi)} = \\hat{X}{(m_{s},\\varphi)} and \\varphi h{(m_{s},\\varphi)} \\cos{(\\varphi m_{s})} = \\varphi \\cos^{2}{(\\varphi m_{s})} and \\hat{X}{(m_{s},\\varphi)} \\cos{(\\varphi m_{s})} = \\varphi \\cos^{2}{(\\varphi m_{s})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True))))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Symbol('\\\\varphi', commutative=True), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True))), Function('\\\\hat{X}')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["times", 2, "cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True)))"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('h')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True)))), Mul(Symbol('\\\\varphi', commutative=True), Pow(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Function('\\\\hat{X}')(Symbol('m_s', commutative=True), Symbol('\\\\varphi', commutative=True)), cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True)))), Mul(Symbol('\\\\varphi', commutative=True), Pow(cos(Mul(Symbol('\\\\varphi', commutative=True), Symbol('m_s', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(E)} = \\cos{(E)} and i{(E)} = \\cos{(E)} and \\operatorname{v_{1}}{(E)} = \\frac{1}{\\cos{(E)}}, then obtain \\operatorname{v_{1}}{(E)} = \\frac{1}{i{(E)}}", "derivation": "\\operatorname{F_{x}}{(E)} = \\cos{(E)} and i{(E)} = \\cos{(E)} and i{(E)} = \\operatorname{F_{x}}{(E)} and \\operatorname{v_{1}}{(E)} = \\frac{1}{\\cos{(E)}} and \\operatorname{v_{1}}{(E)} = \\frac{1}{\\operatorname{F_{x}}{(E)}} and \\operatorname{v_{1}}{(E)} = \\frac{1}{i{(E)}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('i')(Symbol('E', commutative=True)), cos(Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('i')(Symbol('E', commutative=True)), Function('F_x')(Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('v_1')(Symbol('E', commutative=True)), Pow(cos(Symbol('E', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('v_1')(Symbol('E', commutative=True)), Pow(Function('F_x')(Symbol('E', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Function('v_1')(Symbol('E', commutative=True)), Pow(Function('i')(Symbol('E', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(A_{2})} = \\sin{(A_{2})} and \\Psi_{nl}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi, then obtain \\Psi_{nl}{(\\Psi)} + \\int \\sin{(A_{2})} dA_{2} = \\int \\log{(\\Psi)} d\\Psi + \\int \\sin{(A_{2})} dA_{2}", "derivation": "\\operatorname{C_{1}}{(A_{2})} = \\sin{(A_{2})} and \\int \\operatorname{C_{1}}{(A_{2})} dA_{2} = \\int \\sin{(A_{2})} dA_{2} and \\Psi_{nl}{(\\Psi)} = \\int \\log{(\\Psi)} d\\Psi and \\Psi_{nl}{(\\Psi)} + \\int \\operatorname{C_{1}}{(A_{2})} dA_{2} = \\int \\operatorname{C_{1}}{(A_{2})} dA_{2} + \\int \\log{(\\Psi)} d\\Psi and \\Psi_{nl}{(\\Psi)} + \\int \\sin{(A_{2})} dA_{2} = \\int \\log{(\\Psi)} d\\Psi + \\int \\sin{(A_{2})} dA_{2}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["add", 3, "Integral(Function('C_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Integral(Function('C_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Integral(Function('C_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Add(Integral(log(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(C_{1})} = \\cos{(C_{1})}, then obtain (\\int 2 \\operatorname{n_{1}}{(C_{1})} \\cos{(C_{1})} dC_{1})^{2} = (\\int (\\operatorname{n_{1}}{(C_{1})} + \\cos{(C_{1})}) \\cos{(C_{1})} dC_{1})^{2}", "derivation": "\\operatorname{n_{1}}{(C_{1})} = \\cos{(C_{1})} and 2 \\operatorname{n_{1}}{(C_{1})} = \\operatorname{n_{1}}{(C_{1})} + \\cos{(C_{1})} and 2 \\operatorname{n_{1}}{(C_{1})} \\cos{(C_{1})} = (\\operatorname{n_{1}}{(C_{1})} + \\cos{(C_{1})}) \\cos{(C_{1})} and \\int 2 \\operatorname{n_{1}}{(C_{1})} \\cos{(C_{1})} dC_{1} = \\int (\\operatorname{n_{1}}{(C_{1})} + \\cos{(C_{1})}) \\cos{(C_{1})} dC_{1} and (\\int 2 \\operatorname{n_{1}}{(C_{1})} \\cos{(C_{1})} dC_{1})^{2} = (\\int (\\operatorname{n_{1}}{(C_{1})} + \\cos{(C_{1})}) \\cos{(C_{1})} dC_{1})^{2}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["add", 1, "Function('n_1')(Symbol('C_1', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('C_1', commutative=True))), Add(Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))))"], [["times", 2, "cos(Symbol('C_1', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), Mul(Add(Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), cos(Symbol('C_1', commutative=True))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Add(Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Integral(Mul(Integer(2), Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(2)), Pow(Integral(Mul(Add(Function('n_1')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True))), cos(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbf{E})} = e^{e^{\\mathbf{E}}}, then obtain \\frac{\\cos{(\\mathbf{E} - \\dot{\\mathbf{r}}{(\\mathbf{E})})}}{\\mathbf{E} - e^{e^{\\mathbf{E}}}} = \\frac{\\cos{(\\mathbf{E} - e^{e^{\\mathbf{E}}})}}{\\mathbf{E} - e^{e^{\\mathbf{E}}}}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbf{E})} = e^{e^{\\mathbf{E}}} and - \\mathbf{E} + \\dot{\\mathbf{r}}{(\\mathbf{E})} = - \\mathbf{E} + e^{e^{\\mathbf{E}}} and \\cos{(\\mathbf{E} - \\dot{\\mathbf{r}}{(\\mathbf{E})})} = \\cos{(\\mathbf{E} - e^{e^{\\mathbf{E}}})} and \\frac{\\cos{(\\mathbf{E} - \\dot{\\mathbf{r}}{(\\mathbf{E})})}}{\\mathbf{E} - e^{e^{\\mathbf{E}}}} = \\frac{\\cos{(\\mathbf{E} - e^{e^{\\mathbf{E}}})}}{\\mathbf{E} - e^{e^{\\mathbf{E}}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{E}', commutative=True)), exp(exp(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{E}', commutative=True))))), cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))))))"], [["divide", 3, "Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{E}', commutative=True)))))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{E}', commutative=True))))), Integer(-1)), cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbf{E}', commutative=True)))))), Mul(Pow(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{E}', commutative=True))))), Integer(-1)), cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), exp(exp(Symbol('\\\\mathbf{E}', commutative=True))))))))"]]}, {"prompt": "Given \\hat{H}{(t_{2},v)} = - v + e^{t_{2}}, then obtain \\frac{\\partial}{\\partial v} \\hat{H}{(t_{2},v)} = -1", "derivation": "\\hat{H}{(t_{2},v)} = - v + e^{t_{2}} and \\hat{H}{(t_{2},v)} - e^{t_{2}} = - v and \\frac{\\partial}{\\partial v} (\\hat{H}{(t_{2},v)} - e^{t_{2}}) = \\frac{d}{d v} - v and \\frac{\\partial}{\\partial v} \\hat{H}{(t_{2},v)} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('v', commutative=True)), Add(Mul(Integer(-1), Symbol('v', commutative=True)), exp(Symbol('t_2', commutative=True))))"], [["minus", 1, "exp(Symbol('t_2', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('t_2', commutative=True)))), Mul(Integer(-1), Symbol('v', commutative=True)))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), exp(Symbol('t_2', commutative=True)))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('t_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\psi{(\\tilde{g},\\mathbf{g})} = \\mathbf{g} + \\tilde{g}, then obtain 3 \\mathbf{g} + e^{\\mathbf{g} (2 \\mathbf{g} + 2 \\tilde{g} + \\psi{(\\tilde{g},\\mathbf{g})})} = 3 \\mathbf{g} + e^{\\mathbf{g} (3 \\mathbf{g} + 3 \\tilde{g})}", "derivation": "\\psi{(\\tilde{g},\\mathbf{g})} = \\mathbf{g} + \\tilde{g} and 2 \\mathbf{g} + 2 \\tilde{g} + \\psi{(\\tilde{g},\\mathbf{g})} = 3 \\mathbf{g} + 3 \\tilde{g} and \\mathbf{g} (2 \\mathbf{g} + 2 \\tilde{g} + \\psi{(\\tilde{g},\\mathbf{g})}) = \\mathbf{g} (3 \\mathbf{g} + 3 \\tilde{g}) and e^{\\mathbf{g} (2 \\mathbf{g} + 2 \\tilde{g} + \\psi{(\\tilde{g},\\mathbf{g})})} = e^{\\mathbf{g} (3 \\mathbf{g} + 3 \\tilde{g})} and 3 \\mathbf{g} + e^{\\mathbf{g} (2 \\mathbf{g} + 2 \\tilde{g} + \\psi{(\\tilde{g},\\mathbf{g})})} = 3 \\mathbf{g} + e^{\\mathbf{g} (3 \\mathbf{g} + 3 \\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Symbol('\\\\mathbf{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(3), Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(3), Symbol('\\\\tilde{g}', commutative=True)))))"], [["exp", 3], "Equality(exp(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))), exp(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(3), Symbol('\\\\tilde{g}', commutative=True))))))"], [["add", 4, "Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), exp(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(2), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)), Function('\\\\psi')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))))), Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), exp(Mul(Symbol('\\\\mathbf{g}', commutative=True), Add(Mul(Integer(3), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(3), Symbol('\\\\tilde{g}', commutative=True)))))))"]]}, {"prompt": "Given \\chi{(\\phi_2,F_{c})} = F_{c} - \\phi_2, then derive \\int \\chi{(\\phi_2,F_{c})} d\\phi_2 = F_{c} \\phi_2 - \\frac{\\phi_2^{2}}{2} + \\psi^*, then obtain \\int (F_{c} \\phi_2 - \\frac{\\phi_2^{2}}{2} + \\psi^*) dF_{c} = \\iint (F_{c} - \\phi_2) d\\phi_2 dF_{c}", "derivation": "\\chi{(\\phi_2,F_{c})} = F_{c} - \\phi_2 and \\int \\chi{(\\phi_2,F_{c})} d\\phi_2 = \\int (F_{c} - \\phi_2) d\\phi_2 and \\int \\chi{(\\phi_2,F_{c})} d\\phi_2 = F_{c} \\phi_2 - \\frac{\\phi_2^{2}}{2} + \\psi^* and F_{c} \\phi_2 - \\frac{\\phi_2^{2}}{2} + \\psi^* = \\int (F_{c} - \\phi_2) d\\phi_2 and \\int (F_{c} \\phi_2 - \\frac{\\phi_2^{2}}{2} + \\psi^*) dF_{c} = \\iint (F_{c} - \\phi_2) d\\phi_2 dF_{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\chi')(Symbol('\\\\phi_2', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('\\\\psi^*', commutative=True)), Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 4, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('F_c', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\phi_2', commutative=True), Integer(2))), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(E_{x})} = \\sin{(E_{x})}, then derive \\sin{(E_{x})} + \\int \\operatorname{v_{y}}{(E_{x})} dE_{x} = a + \\sin{(E_{x})} - \\cos{(E_{x})}, then obtain \\mathbf{s} + \\sin{(E_{x})} - \\cos{(E_{x})} = a + \\sin{(E_{x})} - \\cos{(E_{x})}", "derivation": "\\operatorname{v_{y}}{(E_{x})} = \\sin{(E_{x})} and \\int \\operatorname{v_{y}}{(E_{x})} dE_{x} = \\int \\sin{(E_{x})} dE_{x} and \\sin{(E_{x})} + \\int \\operatorname{v_{y}}{(E_{x})} dE_{x} = \\sin{(E_{x})} + \\int \\sin{(E_{x})} dE_{x} and \\sin{(E_{x})} + \\int \\operatorname{v_{y}}{(E_{x})} dE_{x} = a + \\sin{(E_{x})} - \\cos{(E_{x})} and \\sin{(E_{x})} + \\int \\sin{(E_{x})} dE_{x} = a + \\sin{(E_{x})} - \\cos{(E_{x})} and \\mathbf{s} + \\sin{(E_{x})} - \\cos{(E_{x})} = a + \\sin{(E_{x})} - \\cos{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('E_x', commutative=True)), sin(Symbol('E_x', commutative=True)))"], [["integrate", 1, "Symbol('E_x', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))), Integral(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["add", 2, "sin(Symbol('E_x', commutative=True))"], "Equality(Add(sin(Symbol('E_x', commutative=True)), Integral(Function('v_y')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(sin(Symbol('E_x', commutative=True)), Integral(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(sin(Symbol('E_x', commutative=True)), Integral(Function('v_y')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(Symbol('a', commutative=True), sin(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(sin(Symbol('E_x', commutative=True)), Integral(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True)))), Add(Symbol('a', commutative=True), sin(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Add(Symbol('a', commutative=True), sin(Symbol('E_x', commutative=True)), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(g,C_{1})} = - C_{1} + \\sin{(g)}, then obtain \\frac{\\partial^{2}}{\\partial g\\partial C_{1}} \\mathbf{P}{(g,C_{1})} = 0", "derivation": "\\mathbf{P}{(g,C_{1})} = - C_{1} + \\sin{(g)} and C_{1} + \\mathbf{P}{(g,C_{1})} = \\sin{(g)} and C_{1} + \\mathbf{P}{(g,C_{1})} + 1 = \\sin{(g)} + 1 and \\frac{\\partial}{\\partial C_{1}} (C_{1} + \\mathbf{P}{(g,C_{1})} + 1) = \\frac{d}{d C_{1}} (\\sin{(g)} + 1) and \\frac{\\partial^{2}}{\\partial g\\partial C_{1}} (C_{1} + \\mathbf{P}{(g,C_{1})} + 1) = \\frac{d^{2}}{d gd C_{1}} (\\sin{(g)} + 1) and \\frac{\\partial^{2}}{\\partial g\\partial C_{1}} \\mathbf{P}{(g,C_{1})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), sin(Symbol('g', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('C_1', commutative=True))"], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True))), sin(Symbol('g', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True)), Integer(1)), Add(sin(Symbol('g', commutative=True)), Integer(1)))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True)), Integer(1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('g', commutative=True)), Integer(1)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Symbol('C_1', commutative=True), Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True)), Integer(1)), Tuple(Symbol('C_1', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('g', commutative=True)), Integer(1)), Tuple(Symbol('C_1', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('g', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} = \\frac{\\hat{X} + \\varphi}{C_{1}}, then obtain \\iiint \\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} d\\varphi d\\hat{X} dC_{1} = \\iiint \\frac{\\hat{X} + \\varphi}{C_{1}} d\\varphi d\\hat{X} dC_{1}", "derivation": "\\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} = \\frac{\\hat{X} + \\varphi}{C_{1}} and \\int \\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} d\\varphi = \\int \\frac{\\hat{X} + \\varphi}{C_{1}} d\\varphi and \\iint \\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} d\\varphi d\\hat{X} = \\iint \\frac{\\hat{X} + \\varphi}{C_{1}} d\\varphi d\\hat{X} and \\iiint \\hat{\\mathbf{x}}{(\\varphi,C_{1},\\hat{X})} d\\varphi d\\hat{X} dC_{1} = \\iiint \\frac{\\hat{X} + \\varphi}{C_{1}} d\\varphi d\\hat{X} dC_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\varphi', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(x,v)} = v - x, then obtain \\frac{\\partial}{\\partial v} \\int (v - x) \\operatorname{A_{z}}{(x,v)} dx = \\frac{\\partial}{\\partial v} \\int (v - x)^{2} dx", "derivation": "\\operatorname{A_{z}}{(x,v)} = v - x and (v - x) \\operatorname{A_{z}}{(x,v)} = (v - x)^{2} and \\int (v - x) \\operatorname{A_{z}}{(x,v)} dx = \\int (v - x)^{2} dx and \\frac{\\partial}{\\partial v} \\int (v - x) \\operatorname{A_{z}}{(x,v)} dx = \\frac{\\partial}{\\partial v} \\int (v - x)^{2} dx", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('x', commutative=True), Symbol('v', commutative=True)), Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["times", 1, "Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)))"], "Equality(Mul(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Function('A_z')(Symbol('x', commutative=True), Symbol('v', commutative=True))), Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Function('A_z')(Symbol('x', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(2)), Tuple(Symbol('x', commutative=True))))"], [["differentiate", 3, "Symbol('v', commutative=True)"], "Equality(Derivative(Integral(Mul(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Function('A_z')(Symbol('x', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Integral(Pow(Add(Symbol('v', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))), Integer(2)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(g,q)} = g + q, then obtain (g + q)^{q} e^{- g - q} e^{\\operatorname{f^{\\prime}}{(g,q)}} = (g + q)^{q} e^{- g - q} e^{g + q}", "derivation": "\\operatorname{f^{\\prime}}{(g,q)} = g + q and \\operatorname{f^{\\prime}}^{q}{(g,q)} = (g + q)^{q} and e^{\\operatorname{f^{\\prime}}{(g,q)}} = e^{g + q} and \\operatorname{f^{\\prime}}^{q}{(g,q)} e^{\\operatorname{f^{\\prime}}{(g,q)}} = \\operatorname{f^{\\prime}}^{q}{(g,q)} e^{g + q} and (g + q)^{q} e^{\\operatorname{f^{\\prime}}{(g,q)}} = (g + q)^{q} e^{g + q} and (g + q)^{q} e^{- g - q} e^{\\operatorname{f^{\\prime}}{(g,q)}} = (g + q)^{q} e^{- g - q} e^{g + q}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)), Add(Symbol('g', commutative=True), Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["exp", 1], "Equality(exp(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True))), exp(Add(Symbol('g', commutative=True), Symbol('q', commutative=True))))"], [["times", 3, "Pow(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)))))"], [["divide", 5, "exp(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True)))), exp(Function('f^{\\\\prime}')(Symbol('g', commutative=True), Symbol('q', commutative=True)))), Mul(Pow(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True)))), exp(Add(Symbol('g', commutative=True), Symbol('q', commutative=True)))))"]]}, {"prompt": "Given i{(A_{2})} = \\log{(A_{2})}, then obtain \\frac{i{(A_{2})} \\int \\cos{(\\int i{(A_{2})} dA_{2})} dA_{2}}{A_{2}} = \\frac{i{(A_{2})} \\int \\cos{(\\int \\log{(A_{2})} dA_{2})} dA_{2}}{A_{2}}", "derivation": "i{(A_{2})} = \\log{(A_{2})} and \\int i{(A_{2})} dA_{2} = \\int \\log{(A_{2})} dA_{2} and \\cos{(\\int i{(A_{2})} dA_{2})} = \\cos{(\\int \\log{(A_{2})} dA_{2})} and \\int \\cos{(\\int i{(A_{2})} dA_{2})} dA_{2} = \\int \\cos{(\\int \\log{(A_{2})} dA_{2})} dA_{2} and \\frac{\\log{(A_{2})} \\int \\cos{(\\int i{(A_{2})} dA_{2})} dA_{2}}{A_{2}} = \\frac{\\log{(A_{2})} \\int \\cos{(\\int \\log{(A_{2})} dA_{2})} dA_{2}}{A_{2}} and \\frac{i{(A_{2})} \\int \\cos{(\\int i{(A_{2})} dA_{2})} dA_{2}}{A_{2}} = \\frac{i{(A_{2})} \\int \\cos{(\\int \\log{(A_{2})} dA_{2})} dA_{2}}{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('i')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('i')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), cos(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(cos(Integral(Function('i')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))), Integral(cos(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True))))"], [["times", 4, "Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), log(Symbol('A_2', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), log(Symbol('A_2', commutative=True)), Integral(cos(Integral(Function('i')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), log(Symbol('A_2', commutative=True)), Integral(cos(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('i')(Symbol('A_2', commutative=True)), Integral(cos(Integral(Function('i')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('i')(Symbol('A_2', commutative=True)), Integral(cos(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Tuple(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given f{(\\phi,L)} = L + \\sin{(\\phi)} and \\operatorname{M_{E}}{(\\phi,L)} = f^{\\phi}{(\\phi,L)}, then obtain 2 \\operatorname{M_{E}}{(\\phi,L)} f^{2 \\phi}{(\\phi,L)} = 2 f^{3 \\phi}{(\\phi,L)}", "derivation": "f{(\\phi,L)} = L + \\sin{(\\phi)} and f^{\\phi}{(\\phi,L)} = (L + \\sin{(\\phi)})^{\\phi} and \\operatorname{M_{E}}{(\\phi,L)} = f^{\\phi}{(\\phi,L)} and ((L + \\sin{(\\phi)})^{\\phi} + f^{\\phi}{(\\phi,L)}) \\operatorname{M_{E}}{(\\phi,L)} = ((L + \\sin{(\\phi)})^{\\phi} + f^{\\phi}{(\\phi,L)}) f^{\\phi}{(\\phi,L)} and 2 \\operatorname{M_{E}}{(\\phi,L)} f^{\\phi}{(\\phi,L)} = 2 f^{2 \\phi}{(\\phi,L)} and 2 \\operatorname{M_{E}}{(\\phi,L)} f^{2 \\phi}{(\\phi,L)} = 2 f^{3 \\phi}{(\\phi,L)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), sin(Symbol('\\\\phi', commutative=True))))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Add(Symbol('L', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["times", 3, "Add(Pow(Add(Symbol('L', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Add(Pow(Add(Symbol('L', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True))), Mul(Add(Pow(Add(Symbol('L', commutative=True), sin(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(2), Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Integer(2), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))))"], [["times", 5, "Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Integer(2), Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True)))), Mul(Integer(2), Pow(Function('f')(Symbol('\\\\phi', commutative=True), Symbol('L', commutative=True)), Mul(Integer(3), Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(A_{x},\\mu)} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mu), then derive \\lambda{(A_{x},\\mu)} = 1, then derive A_{x} + v_{t} = A_{x} + S, then obtain (A_{x} + v_{t}) \\frac{\\partial}{\\partial v_{t}} (A_{x} + v_{t}) = (A_{x} + v_{t}) \\frac{\\partial}{\\partial v_{t}} (A_{x} + S)", "derivation": "\\lambda{(A_{x},\\mu)} = \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mu) and \\lambda{(A_{x},\\mu)} = 1 and \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mu) = 1 and \\int \\frac{\\partial}{\\partial A_{x}} (A_{x} + \\mu) dA_{x} = \\int 1 dA_{x} and A_{x} + v_{t} = A_{x} + S and \\frac{\\partial}{\\partial v_{t}} (A_{x} + v_{t}) = \\frac{\\partial}{\\partial v_{t}} (A_{x} + S) and (A_{x} + v_{t}) \\frac{\\partial}{\\partial v_{t}} (A_{x} + v_{t}) = (A_{x} + v_{t}) \\frac{\\partial}{\\partial v_{t}} (A_{x} + S)", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\lambda')(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('A_x', commutative=True)"], "Equality(Integral(Derivative(Add(Symbol('A_x', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('A_x', commutative=True), Integer(1))), Tuple(Symbol('A_x', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('S', commutative=True)))"], [["differentiate", 5, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["times", 6, "Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))), Mul(Add(Symbol('A_x', commutative=True), Symbol('v_t', commutative=True)), Derivative(Add(Symbol('A_x', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{X}{(m_{s})} = e^{e^{m_{s}}} and \\mathbf{J}_P{(m_{s})} = e^{m_{s}}, then obtain e^{- \\mathbf{J}_P{(m_{s})}} e^{e^{m_{s}}} \\frac{d}{d m_{s}} \\hat{X}{(m_{s})} = e^{- \\mathbf{J}_P{(m_{s})}} e^{e^{m_{s}}} \\frac{d}{d m_{s}} e^{\\mathbf{J}_P{(m_{s})}}", "derivation": "\\hat{X}{(m_{s})} = e^{e^{m_{s}}} and \\mathbf{J}_P{(m_{s})} = e^{m_{s}} and \\hat{X}{(m_{s})} = e^{\\mathbf{J}_P{(m_{s})}} and \\frac{d}{d m_{s}} \\hat{X}{(m_{s})} = \\frac{d}{d m_{s}} e^{\\mathbf{J}_P{(m_{s})}} and e^{e^{m_{s}}} \\frac{d}{d m_{s}} \\hat{X}{(m_{s})} = e^{e^{m_{s}}} \\frac{d}{d m_{s}} e^{\\mathbf{J}_P{(m_{s})}} and e^{- \\mathbf{J}_P{(m_{s})}} e^{e^{m_{s}}} \\frac{d}{d m_{s}} \\hat{X}{(m_{s})} = e^{- \\mathbf{J}_P{(m_{s})}} e^{e^{m_{s}}} \\frac{d}{d m_{s}} e^{\\mathbf{J}_P{(m_{s})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('m_s', commutative=True)), exp(exp(Symbol('m_s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{X}')(Symbol('m_s', commutative=True)), exp(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True))))"], [["differentiate", 3, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(exp(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["times", 4, "exp(exp(Symbol('m_s', commutative=True)))"], "Equality(Mul(exp(exp(Symbol('m_s', commutative=True))), Derivative(Function('\\\\hat{X}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(exp(exp(Symbol('m_s', commutative=True))), Derivative(exp(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["divide", 5, "exp(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True)))), exp(exp(Symbol('m_s', commutative=True))), Derivative(Function('\\\\hat{X}')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(exp(Mul(Integer(-1), Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True)))), exp(exp(Symbol('m_s', commutative=True))), Derivative(exp(Function('\\\\mathbf{J}_P')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(z,\\mathbf{g})} = \\mathbf{g} z, then derive \\frac{\\partial}{\\partial z} a{(z,\\mathbf{g})} = \\mathbf{g}, then obtain \\mathbf{g} z + \\frac{\\partial}{\\partial z} \\mathbf{g} z = \\mathbf{g} z + \\mathbf{g}", "derivation": "a{(z,\\mathbf{g})} = \\mathbf{g} z and \\frac{\\partial}{\\partial z} a{(z,\\mathbf{g})} = \\frac{\\partial}{\\partial z} \\mathbf{g} z and \\frac{\\partial}{\\partial z} a{(z,\\mathbf{g})} = \\mathbf{g} and \\frac{\\partial}{\\partial z} \\mathbf{g} z = \\mathbf{g} and a{(z,\\mathbf{g})} + \\frac{\\partial}{\\partial z} \\mathbf{g} z = \\mathbf{g} + a{(z,\\mathbf{g})} and \\mathbf{g} z + \\frac{\\partial}{\\partial z} \\mathbf{g} z = \\mathbf{g} z + \\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Symbol('\\\\mathbf{g}', commutative=True))"], [["add", 4, "Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Add(Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Function('a')(Symbol('z', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{g}', commutative=True), Symbol('z', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(f_{E})} = \\sin{(f_{E})} and v{(f_{E})} = (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}}, then obtain (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}} v{(f_{E})} = 0^{f_{E}} v{(f_{E})}", "derivation": "\\operatorname{F_{H}}{(f_{E})} = \\sin{(f_{E})} and \\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})} = 0 and (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}} = 0^{f_{E}} and v{(f_{E})} = (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}} and v{(f_{E})} = 0^{f_{E}} and v^{2}{(f_{E})} = 0^{f_{E}} v{(f_{E})} and v^{2}{(f_{E})} = (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}} v{(f_{E})} and (\\operatorname{F_{H}}{(f_{E})} - \\sin{(f_{E})})^{f_{E}} v{(f_{E})} = 0^{f_{E}} v{(f_{E})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('f_E', commutative=True)), sin(Symbol('f_E', commutative=True)))"], [["minus", 1, "sin(Symbol('f_E', commutative=True))"], "Equality(Add(Function('F_H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Function('F_H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Pow(Integer(0), Symbol('f_E', commutative=True)))"], ["renaming_premise", "Equality(Function('v')(Symbol('f_E', commutative=True)), Pow(Add(Function('F_H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('v')(Symbol('f_E', commutative=True)), Pow(Integer(0), Symbol('f_E', commutative=True)))"], [["times", 5, "Function('v')(Symbol('f_E', commutative=True))"], "Equality(Pow(Function('v')(Symbol('f_E', commutative=True)), Integer(2)), Mul(Pow(Integer(0), Symbol('f_E', commutative=True)), Function('v')(Symbol('f_E', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Function('v')(Symbol('f_E', commutative=True)), Integer(2)), Mul(Pow(Add(Function('F_H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Function('v')(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Mul(Pow(Add(Function('F_H')(Symbol('f_E', commutative=True)), Mul(Integer(-1), sin(Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Function('v')(Symbol('f_E', commutative=True))), Mul(Pow(Integer(0), Symbol('f_E', commutative=True)), Function('v')(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{H},z,\\phi_1)} = - \\mathbf{H} + \\phi_1 - z, then obtain \\int (-1) dz = \\int - (- \\frac{\\mathbf{H} - \\phi_1 + z}{- \\mathbf{H} + \\phi_1 - z})^{z} dz", "derivation": "\\rho_{b}{(\\mathbf{H},z,\\phi_1)} = - \\mathbf{H} + \\phi_1 - z and - \\rho_{b}{(\\mathbf{H},z,\\phi_1)} = \\mathbf{H} - \\phi_1 + z and -1 = \\frac{\\mathbf{H} - \\phi_1 + z}{\\rho_{b}{(\\mathbf{H},z,\\phi_1)}} and 1 = - \\frac{\\mathbf{H} - \\phi_1 + z}{\\rho_{b}{(\\mathbf{H},z,\\phi_1)}} and 1 = (- \\frac{\\mathbf{H} - \\phi_1 + z}{\\rho_{b}{(\\mathbf{H},z,\\phi_1)}})^{z} and 1 = (- \\frac{\\mathbf{H} - \\phi_1 + z}{- \\mathbf{H} + \\phi_1 - z})^{z} and -1 = - (- \\frac{\\mathbf{H} - \\phi_1 + z}{- \\mathbf{H} + \\phi_1 - z})^{z} and \\int (-1) dz = \\int - (- \\frac{\\mathbf{H} - \\phi_1 + z}{- \\mathbf{H} + \\phi_1 - z})^{z} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True)))"], [["divide", 2, "Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(-1), Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))))"], [["times", 3, "Integer(-1)"], "Equality(Integer(1), Mul(Integer(-1), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Integer(1), Pow(Mul(Integer(-1), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True)), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Symbol('z', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True))), Symbol('z', commutative=True)))"], [["divide", 6, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True))), Symbol('z', commutative=True))))"], [["integrate", 7, "Symbol('z', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('z', commutative=True))), Integral(Mul(Integer(-1), Pow(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Symbol('z', commutative=True))), Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(z,\\mathbf{E},M)} = \\frac{\\mathbf{E} z}{M}, then derive \\frac{\\frac{\\partial}{\\partial z} \\operatorname{v_{1}}{(z,\\mathbf{E},M)}}{\\mathbf{E}} = \\frac{1}{M}, then obtain \\frac{\\frac{\\partial}{\\partial z} \\frac{\\mathbf{E} z}{M}}{\\mathbf{E}} = \\frac{1}{M}", "derivation": "\\operatorname{v_{1}}{(z,\\mathbf{E},M)} = \\frac{\\mathbf{E} z}{M} and \\frac{\\operatorname{v_{1}}{(z,\\mathbf{E},M)}}{\\mathbf{E}} = \\frac{z}{M} and \\frac{\\partial}{\\partial z} \\frac{\\operatorname{v_{1}}{(z,\\mathbf{E},M)}}{\\mathbf{E}} = \\frac{\\partial}{\\partial z} \\frac{z}{M} and \\frac{\\frac{\\partial}{\\partial z} \\operatorname{v_{1}}{(z,\\mathbf{E},M)}}{\\mathbf{E}} = \\frac{1}{M} and \\frac{\\frac{\\partial}{\\partial z} \\frac{\\mathbf{E} z}{M}}{\\mathbf{E}} = \\frac{1}{M}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('z', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('z', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('v_1')(Symbol('z', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Function('v_1')(Symbol('z', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Derivative(Function('v_1')(Symbol('z', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Pow(Symbol('M', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('M', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Pow(Symbol('M', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(v)} = e^{\\cos{(v)}}, then derive \\frac{d}{d v} \\operatorname{E_{\\lambda}}{(v)} = - e^{\\cos{(v)}} \\sin{(v)}, then obtain \\operatorname{E_{\\lambda}}{(v)} \\frac{d}{d v} e^{\\cos{(v)}} = - \\operatorname{E_{\\lambda}}{(v)} e^{\\cos{(v)}} \\sin{(v)}", "derivation": "\\operatorname{E_{\\lambda}}{(v)} = e^{\\cos{(v)}} and \\frac{d}{d v} \\operatorname{E_{\\lambda}}{(v)} = \\frac{d}{d v} e^{\\cos{(v)}} and \\frac{d}{d v} \\operatorname{E_{\\lambda}}{(v)} = - e^{\\cos{(v)}} \\sin{(v)} and \\operatorname{E_{\\lambda}}{(v)} \\frac{d}{d v} \\operatorname{E_{\\lambda}}{(v)} = - \\operatorname{E_{\\lambda}}{(v)} e^{\\cos{(v)}} \\sin{(v)} and \\operatorname{E_{\\lambda}}{(v)} \\frac{d}{d v} e^{\\cos{(v)}} = - \\operatorname{E_{\\lambda}}{(v)} e^{\\cos{(v)}} \\sin{(v)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), exp(cos(Symbol('v', commutative=True))))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('v', commutative=True))), sin(Symbol('v', commutative=True))))"], [["times", 3, "Function('E_{\\\\lambda}')(Symbol('v', commutative=True))"], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), Derivative(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), exp(cos(Symbol('v', commutative=True))), sin(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), Derivative(exp(cos(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('v', commutative=True)), exp(cos(Symbol('v', commutative=True))), sin(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\mu{(A_{z},\\mu)} = \\mu \\sin{(A_{z})} and B{(f^{\\prime},y)} = \\frac{y}{f^{\\prime}}, then obtain \\frac{B{(f^{\\prime},y)} \\mu{(A_{z},\\mu)}}{\\sin{(A_{z})}} - \\frac{y \\mu{(A_{z},\\mu)}}{f^{\\prime} \\sin{(A_{z})}} = 0", "derivation": "\\mu{(A_{z},\\mu)} = \\mu \\sin{(A_{z})} and \\frac{\\mu{(A_{z},\\mu)}}{\\sin{(A_{z})}} = \\mu and B{(f^{\\prime},y)} = \\frac{y}{f^{\\prime}} and \\mu B{(f^{\\prime},y)} = \\frac{\\mu y}{f^{\\prime}} and \\mu B{(f^{\\prime},y)} - \\frac{\\mu y}{f^{\\prime}} = 0 and \\frac{B{(f^{\\prime},y)} \\mu{(A_{z},\\mu)}}{\\sin{(A_{z})}} - \\frac{y \\mu{(A_{z},\\mu)}}{f^{\\prime} \\sin{(A_{z})}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('A_z', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), sin(Symbol('A_z', commutative=True))))"], [["divide", 1, "sin(Symbol('A_z', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('A_z', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Symbol('\\\\mu', commutative=True))"], ["get_premise", "Equality(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["times", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["minus", 4, "Mul(Symbol('\\\\mu', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Function('B')(Symbol('f^{\\\\prime}', commutative=True), Symbol('y', commutative=True)), Function('\\\\mu')(Symbol('A_z', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Symbol('y', commutative=True), Function('\\\\mu')(Symbol('A_z', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(sin(Symbol('A_z', commutative=True)), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\omega{(\\mathbf{S},\\mathbf{H})} = \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})}, then obtain \\omega{(\\mathbf{S},\\mathbf{H})} \\int \\omega{(\\mathbf{S},\\mathbf{H})} d\\mathbf{S} = \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} \\int \\omega{(\\mathbf{S},\\mathbf{H})} d\\mathbf{S}", "derivation": "\\omega{(\\mathbf{S},\\mathbf{H})} = \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} and \\int \\omega{(\\mathbf{S},\\mathbf{H})} d\\mathbf{S} = \\int \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} d\\mathbf{S} and \\omega{(\\mathbf{S},\\mathbf{H})} \\int \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} d\\mathbf{S} = \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} \\int \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} d\\mathbf{S} and \\omega{(\\mathbf{S},\\mathbf{H})} \\int \\omega{(\\mathbf{S},\\mathbf{H})} d\\mathbf{S} = \\log{(\\frac{\\mathbf{H}}{\\mathbf{S}})} \\int \\omega{(\\mathbf{S},\\mathbf{H})} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Integral(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integral(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(log(Mul(Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Integral(Function('\\\\omega')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given L{(\\dot{x},\\dot{z},\\hat{p}_0)} = \\frac{\\dot{z}^{\\hat{p}_0}}{\\dot{x}} and \\mathbf{E}{(\\hat{p}_0)} = 2 \\hat{p}_0, then obtain \\dot{z}^{\\hat{p}_0} L{(\\dot{x},\\dot{z},\\hat{p}_0)} = \\frac{\\dot{z}^{\\mathbf{E}{(\\hat{p}_0)}}}{\\dot{x}}", "derivation": "L{(\\dot{x},\\dot{z},\\hat{p}_0)} = \\frac{\\dot{z}^{\\hat{p}_0}}{\\dot{x}} and \\dot{z}^{\\hat{p}_0} L{(\\dot{x},\\dot{z},\\hat{p}_0)} = \\frac{\\dot{z}^{2 \\hat{p}_0}}{\\dot{x}} and \\mathbf{E}{(\\hat{p}_0)} = 2 \\hat{p}_0 and \\dot{z}^{\\hat{p}_0} L{(\\dot{x},\\dot{z},\\hat{p}_0)} = \\frac{\\dot{z}^{\\mathbf{E}{(\\hat{p}_0)}}}{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('L')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Function('L')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Symbol('\\\\dot{z}', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(u,v_{1})} = u v_{1}, then obtain \\frac{\\partial}{\\partial v_{1}} (- u v_{1} + 2 \\operatorname{E_{x}}{(u,v_{1})}) = \\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{x}}{(u,v_{1})}", "derivation": "\\operatorname{E_{x}}{(u,v_{1})} = u v_{1} and - u v_{1} + \\operatorname{E_{x}}{(u,v_{1})} = 0 and - u v_{1} + 2 \\operatorname{E_{x}}{(u,v_{1})} = \\operatorname{E_{x}}{(u,v_{1})} and \\frac{\\partial}{\\partial v_{1}} (- u v_{1} + 2 \\operatorname{E_{x}}{(u,v_{1})}) = \\frac{\\partial}{\\partial v_{1}} \\operatorname{E_{x}}{(u,v_{1})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Mul(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))"], [["minus", 1, "Mul(Symbol('u', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True))), Integer(0))"], [["add", 2, "Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))), Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 3, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Mul(Integer(2), Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Function('E_x')(Symbol('u', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi{(\\hat{H})} = \\log{(\\sin{(\\hat{H})})} and \\varepsilon_{0}{(E,f^{*})} = \\frac{f^{*}}{E}, then obtain \\int (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1) \\varepsilon_{0}{(E,f^{*})} dE = \\int \\frac{f^{*} (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1)}{E} dE", "derivation": "\\psi{(\\hat{H})} = \\log{(\\sin{(\\hat{H})})} and \\psi^{\\hat{H}}{(\\hat{H})} = \\log{(\\sin{(\\hat{H})})}^{\\hat{H}} and \\varepsilon_{0}{(E,f^{*})} = \\frac{f^{*}}{E} and (\\psi^{\\hat{H}}{(\\hat{H})} - 1) \\varepsilon_{0}{(E,f^{*})} = \\frac{f^{*} (\\psi^{\\hat{H}}{(\\hat{H})} - 1)}{E} and (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1) \\varepsilon_{0}{(E,f^{*})} = \\frac{f^{*} (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1)}{E} and \\int (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1) \\varepsilon_{0}{(E,f^{*})} dE = \\int \\frac{f^{*} (\\log{(\\sin{(\\hat{H})})}^{\\hat{H}} - 1)}{E} dE", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\hat{H}', commutative=True)), log(sin(Symbol('\\\\hat{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('E', commutative=True), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f^*', commutative=True)))"], [["times", 3, "Add(Pow(Function('\\\\psi')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))"], "Equality(Mul(Add(Pow(Function('\\\\psi')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('E', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f^*', commutative=True), Add(Pow(Function('\\\\psi')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Pow(log(sin(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('E', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f^*', commutative=True), Add(Pow(log(sin(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))))"], [["integrate", 5, "Symbol('E', commutative=True)"], "Equality(Integral(Mul(Add(Pow(log(sin(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Function('\\\\varepsilon_0')(Symbol('E', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), Symbol('f^*', commutative=True), Add(Pow(log(sin(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Tuple(Symbol('E', commutative=True))))"]]}, {"prompt": "Given t{(k,\\theta)} = \\log{(\\frac{k}{\\theta})}, then derive (\\int t{(k,\\theta)} d\\theta)^{k} = (S + \\theta \\log{(\\frac{k}{\\theta})} + \\theta)^{k}, then obtain h (S + \\theta \\log{(\\frac{k}{\\theta})} + \\theta)^{k} = h (\\int \\log{(\\frac{k}{\\theta})} d\\theta)^{k}", "derivation": "t{(k,\\theta)} = \\log{(\\frac{k}{\\theta})} and \\int t{(k,\\theta)} d\\theta = \\int \\log{(\\frac{k}{\\theta})} d\\theta and (\\int t{(k,\\theta)} d\\theta)^{k} = (\\int \\log{(\\frac{k}{\\theta})} d\\theta)^{k} and (\\int t{(k,\\theta)} d\\theta)^{k} = (S + \\theta \\log{(\\frac{k}{\\theta})} + \\theta)^{k} and (S + \\theta \\log{(\\frac{k}{\\theta})} + \\theta)^{k} = (\\int \\log{(\\frac{k}{\\theta})} d\\theta)^{k} and h (S + \\theta \\log{(\\frac{k}{\\theta})} + \\theta)^{k} = h (\\int \\log{(\\frac{k}{\\theta})} d\\theta)^{k}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Integral(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('k', commutative=True)), Pow(Integral(log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('k', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Symbol('S', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True)))), Symbol('\\\\theta', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Add(Symbol('S', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True)))), Symbol('\\\\theta', commutative=True)), Symbol('k', commutative=True)), Pow(Integral(log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('k', commutative=True)))"], [["divide", 5, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('h', commutative=True), Pow(Add(Symbol('S', commutative=True), Mul(Symbol('\\\\theta', commutative=True), log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True)))), Symbol('\\\\theta', commutative=True)), Symbol('k', commutative=True))), Mul(Symbol('h', commutative=True), Pow(Integral(log(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Symbol('k', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Symbol('k', commutative=True))))"]]}, {"prompt": "Given i{(C,F_{g})} = C F_{g}, then derive \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = C, then obtain i{(C,F_{g})} \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = C i{(C,F_{g})}", "derivation": "i{(C,F_{g})} = C F_{g} and \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = \\frac{\\partial}{\\partial F_{g}} C F_{g} and \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = C and C F_{g} \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = C^{2} F_{g} and C F_{g} \\frac{\\partial}{\\partial F_{g}} C F_{g} = C^{2} F_{g} and i{(C,F_{g})} \\frac{\\partial}{\\partial F_{g}} i{(C,F_{g})} = C i{(C,F_{g})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('C', commutative=True))"], [["times", 3, "Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True), Derivative(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True), Derivative(Mul(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(2)), Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Derivative(Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Symbol('C', commutative=True), Function('i')(Symbol('C', commutative=True), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given L{(\\mu,f)} = - \\mu + f, then obtain \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} L^{2}{(\\mu,f)}}{\\mu^{2}} = \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} (- \\mu + f) L{(\\mu,f)}}{\\mu^{2}}", "derivation": "L{(\\mu,f)} = - \\mu + f and \\frac{L{(\\mu,f)}}{\\mu} = \\frac{- \\mu + f}{\\mu} and L^{2}{(\\mu,f)} = (- \\mu + f) L{(\\mu,f)} and \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} (- \\mu + f) L{(\\mu,f)}}{\\mu^{2}} = \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} (- \\mu + f)^{2}}{\\mu^{2}} and \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} L^{2}{(\\mu,f)}}{\\mu^{2}} = \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} (- \\mu + f)^{2}}{\\mu^{2}} and \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} L^{2}{(\\mu,f)}}{\\mu^{2}} = \\frac{(\\frac{L{(\\mu,f)}}{\\mu})^{f} (- \\mu + f) L{(\\mu,f)}}{\\mu^{2}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True))))"], [["times", 1, "Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))"], "Equality(Pow(Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Pow(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu', commutative=True)), Symbol('f', commutative=True)), Function('L')(Symbol('\\\\mu', commutative=True), Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} = e^{t^{V_{\\mathbf{E}}}}, then obtain \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} + 2 \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt = \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} + \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt + \\int e^{t^{V_{\\mathbf{E}}}} dt", "derivation": "\\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} = e^{t^{V_{\\mathbf{E}}}} and \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt = \\int e^{t^{V_{\\mathbf{E}}}} dt and 2 \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt = \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt + \\int e^{t^{V_{\\mathbf{E}}}} dt and \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} + 2 \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt = \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} + \\int \\operatorname{m_{s}}{(t,V_{\\mathbf{E}})} dt + \\int e^{t^{V_{\\mathbf{E}}}} dt", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Pow(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Pow(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["add", 2, "Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Pow(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["add", 3, "Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Add(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Integer(2), Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(Function('m_s')(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(exp(Pow(Symbol('t', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Tuple(Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(c_{0},\\theta_2)} = (e^{c_{0}})^{\\theta_2}, then obtain \\operatorname{A_{z}}{(c_{0},\\theta_2)} + \\operatorname{A_{z}}^{\\theta_2}{(c_{0},\\theta_2)} + e^{c_{0}} = ((e^{c_{0}})^{\\theta_2})^{\\theta_2} + \\operatorname{A_{z}}{(c_{0},\\theta_2)} + e^{c_{0}}", "derivation": "\\operatorname{A_{z}}{(c_{0},\\theta_2)} = (e^{c_{0}})^{\\theta_2} and \\operatorname{A_{z}}{(c_{0},\\theta_2)} + e^{c_{0}} = e^{c_{0}} + (e^{c_{0}})^{\\theta_2} and \\operatorname{A_{z}}^{\\theta_2}{(c_{0},\\theta_2)} = ((e^{c_{0}})^{\\theta_2})^{\\theta_2} and \\operatorname{A_{z}}^{\\theta_2}{(c_{0},\\theta_2)} + e^{c_{0}} + (e^{c_{0}})^{\\theta_2} = ((e^{c_{0}})^{\\theta_2})^{\\theta_2} + e^{c_{0}} + (e^{c_{0}})^{\\theta_2} and \\operatorname{A_{z}}{(c_{0},\\theta_2)} + \\operatorname{A_{z}}^{\\theta_2}{(c_{0},\\theta_2)} + e^{c_{0}} = ((e^{c_{0}})^{\\theta_2})^{\\theta_2} + \\operatorname{A_{z}}{(c_{0},\\theta_2)} + e^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["add", 1, "exp(Symbol('c_0', commutative=True))"], "Equality(Add(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('c_0', commutative=True))), Add(exp(Symbol('c_0', commutative=True)), Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["add", 3, "Add(exp(Symbol('c_0', commutative=True)), Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Add(Pow(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('c_0', commutative=True)), Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Add(Pow(Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('c_0', commutative=True)), Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('c_0', commutative=True))), Add(Pow(Pow(exp(Symbol('c_0', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Function('A_z')(Symbol('c_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)} = \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}}, then obtain e^{L + \\frac{\\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)}}{V_{\\mathbf{B}}}} = e^{L + \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}^{2}}}", "derivation": "\\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)} = \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}} and \\frac{\\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)}}{V_{\\mathbf{B}}} = \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}^{2}} and L + \\frac{\\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)}}{V_{\\mathbf{B}}} = L + \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}^{2}} and e^{L + \\frac{\\theta{(\\mathbf{J}_P,V_{\\mathbf{B}},L)}}{V_{\\mathbf{B}}}} = e^{L + \\frac{L \\mathbf{J}_P}{V_{\\mathbf{B}}^{2}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["divide", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["add", 2, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('L', commutative=True)))), Add(Symbol('L', commutative=True), Mul(Symbol('L', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["exp", 3], "Equality(exp(Add(Symbol('L', commutative=True), Mul(Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('L', commutative=True))))), exp(Add(Symbol('L', commutative=True), Mul(Symbol('L', commutative=True), Pow(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(-2)), Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(c,\\mathbf{s})} = \\mathbf{s} + c, then derive \\frac{\\partial^{2}}{\\partial \\mathbf{s}^{2}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = 0, then obtain \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{3}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} 0", "derivation": "\\operatorname{v_{2}}{(c,\\mathbf{s})} = \\mathbf{s} + c and \\frac{\\partial}{\\partial \\mathbf{s}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + c) and \\frac{\\partial^{2}}{\\partial \\mathbf{s}^{2}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = \\frac{\\partial^{2}}{\\partial \\mathbf{s}^{2}} (\\mathbf{s} + c) and \\frac{\\partial^{2}}{\\partial \\mathbf{s}^{2}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = 0 and \\frac{\\partial^{3}}{\\partial \\mathbf{s}^{3}} \\operatorname{v_{2}}{(c,\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} 0", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('c', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('c', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('c', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('v_2')(Symbol('c', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(2))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('v_2')(Symbol('c', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(3))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(\\mathbf{S})} = \\int \\log{(\\mathbf{S})} d\\mathbf{S}, then derive \\frac{I{(\\mathbf{S})}}{\\log{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} + a^{\\dagger}}{\\log{(\\mathbf{S})}}, then obtain \\frac{I{(\\mathbf{S})}}{\\mathbf{r}{(\\Psi_{\\lambda})} \\log{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} + a^{\\dagger}}{\\mathbf{r}{(\\Psi_{\\lambda})} \\log{(\\mathbf{S})}}", "derivation": "I{(\\mathbf{S})} = \\int \\log{(\\mathbf{S})} d\\mathbf{S} and \\frac{I{(\\mathbf{S})}}{\\log{(\\mathbf{S})}} = \\frac{\\int \\log{(\\mathbf{S})} d\\mathbf{S}}{\\log{(\\mathbf{S})}} and \\frac{I{(\\mathbf{S})}}{\\log{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} + a^{\\dagger}}{\\log{(\\mathbf{S})}} and \\frac{I{(\\mathbf{S})}}{\\mathbf{r}{(\\Psi_{\\lambda})} \\log{(\\mathbf{S})}} = \\frac{\\mathbf{S} \\log{(\\mathbf{S})} - \\mathbf{S} + a^{\\dagger}}{\\mathbf{r}{(\\Psi_{\\lambda})} \\log{(\\mathbf{S})}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "log(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1)), Integral(log(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"], [["divide", 3, "Function('\\\\mathbf{r}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))), Mul(Add(Mul(Symbol('\\\\mathbf{S}', commutative=True), log(Symbol('\\\\mathbf{S}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1)), Pow(log(Symbol('\\\\mathbf{S}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = \\frac{\\cos{(C_{2})}}{\\mathbf{A}}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = - \\frac{\\cos{(C_{2})}}{\\mathbf{A}^{2}}, then obtain \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = - \\frac{\\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})}}{\\mathbf{A}}", "derivation": "\\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = \\frac{\\cos{(C_{2})}}{\\mathbf{A}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} \\frac{\\cos{(C_{2})}}{\\mathbf{A}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = - \\frac{\\cos{(C_{2})}}{\\mathbf{A}^{2}} and \\frac{\\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\cos{(C_{2})}}{\\mathbf{A}^{2}} and \\frac{\\partial}{\\partial \\mathbf{A}} \\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})} = - \\frac{\\operatorname{E_{\\lambda}}{(C_{2},\\mathbf{A})}}{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('C_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('C_2', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2)), cos(Symbol('C_2', commutative=True))))"], [["times", 1, "Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-2)), cos(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('E_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\Omega{(\\mathbf{J}_M,H)} = e^{H + \\mathbf{J}_M}, then obtain \\frac{\\Omega{(\\mathbf{J}_M,H)} \\Omega^{\\mathbf{J}_M}{(\\mathbf{J}_M,H)}}{\\sin{(c_{0} + q)}} = \\frac{\\Omega{(\\mathbf{J}_M,H)} (e^{H + \\mathbf{J}_M})^{\\mathbf{J}_M}}{\\sin{(c_{0} + q)}}", "derivation": "\\Omega{(\\mathbf{J}_M,H)} = e^{H + \\mathbf{J}_M} and \\Omega^{\\mathbf{J}_M}{(\\mathbf{J}_M,H)} = (e^{H + \\mathbf{J}_M})^{\\mathbf{J}_M} and \\Omega{(\\mathbf{J}_M,H)} \\Omega^{\\mathbf{J}_M}{(\\mathbf{J}_M,H)} = \\Omega{(\\mathbf{J}_M,H)} (e^{H + \\mathbf{J}_M})^{\\mathbf{J}_M} and \\frac{\\Omega{(\\mathbf{J}_M,H)} \\Omega^{\\mathbf{J}_M}{(\\mathbf{J}_M,H)}}{\\sin{(c_{0} + q)}} = \\frac{\\Omega{(\\mathbf{J}_M,H)} (e^{H + \\mathbf{J}_M})^{\\mathbf{J}_M}}{\\sin{(c_{0} + q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["times", 2, "Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))), Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["divide", 3, "sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True)))"], "Equality(Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Pow(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Integer(-1))), Mul(Function('\\\\Omega')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('H', commutative=True)), Pow(exp(Add(Symbol('H', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(\\hat{\\mathbf{r}},L)} = \\hat{\\mathbf{r}}^{L}, then obtain L \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} + \\dot{z}{(\\hat{\\mathbf{r}},L)} = L \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} + \\hat{\\mathbf{r}}^{L}", "derivation": "\\dot{z}{(\\hat{\\mathbf{r}},L)} = \\hat{\\mathbf{r}}^{L} and \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} = \\log{(\\hat{\\mathbf{r}}^{L})} and L \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} = L \\log{(\\hat{\\mathbf{r}}^{L})} and L \\log{(\\hat{\\mathbf{r}}^{L})} + \\dot{z}{(\\hat{\\mathbf{r}},L)} = L \\log{(\\hat{\\mathbf{r}}^{L})} + \\hat{\\mathbf{r}}^{L} and L \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} + \\dot{z}{(\\hat{\\mathbf{r}},L)} = L \\log{(\\dot{z}{(\\hat{\\mathbf{r}},L)})} + \\hat{\\mathbf{r}}^{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), log(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))))"], [["times", 2, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), log(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))), Mul(Symbol('L', commutative=True), log(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))))"], [["add", 1, "Mul(Symbol('L', commutative=True), log(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))))"], "Equality(Add(Mul(Symbol('L', commutative=True), log(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))), Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), log(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Symbol('L', commutative=True), log(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))), Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))), Add(Mul(Symbol('L', commutative=True), log(Function('\\\\dot{z}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True)))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\phi{(A_{x})} = \\log{(\\sin{(A_{x})})} and \\Psi_{nl}{(A_{x})} = \\log{(\\sin{(A_{x})})}, then obtain 0 = (\\Psi_{nl}{(A_{x})} - \\phi{(A_{x})}) \\Psi_{nl}{(A_{x})}", "derivation": "\\phi{(A_{x})} = \\log{(\\sin{(A_{x})})} and \\Psi_{nl}{(A_{x})} = \\log{(\\sin{(A_{x})})} and 0 = - \\phi{(A_{x})} + \\log{(\\sin{(A_{x})})} and 0 = (- \\phi{(A_{x})} + \\log{(\\sin{(A_{x})})}) \\Psi_{nl}{(A_{x})} and 0 = (\\Psi_{nl}{(A_{x})} - \\phi{(A_{x})}) \\Psi_{nl}{(A_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('A_x', commutative=True)), log(sin(Symbol('A_x', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A_x', commutative=True)), log(sin(Symbol('A_x', commutative=True))))"], [["minus", 1, "Function('\\\\phi')(Symbol('A_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('A_x', commutative=True))), log(sin(Symbol('A_x', commutative=True)))))"], [["times", 3, "Function('\\\\Psi_{nl}')(Symbol('A_x', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\phi')(Symbol('A_x', commutative=True))), log(sin(Symbol('A_x', commutative=True)))), Function('\\\\Psi_{nl}')(Symbol('A_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Mul(Add(Function('\\\\Psi_{nl}')(Symbol('A_x', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('A_x', commutative=True)))), Function('\\\\Psi_{nl}')(Symbol('A_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and G{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})}, then obtain G{(\\mathbf{M})} - \\sin{(\\mathbf{M})} = \\operatorname{F_{x}}{(\\mathbf{M})} - \\sin{(\\mathbf{M})}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and G{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and G{(\\mathbf{M})} = \\operatorname{F_{x}}{(\\mathbf{M})} and G{(\\mathbf{M})} + \\frac{d^{2}}{d \\mathbf{M}^{2}} \\sin{(\\mathbf{M})} = \\operatorname{F_{x}}{(\\mathbf{M})} + \\frac{d^{2}}{d \\mathbf{M}^{2}} \\sin{(\\mathbf{M})} and G{(\\mathbf{M})} - \\sin{(\\mathbf{M})} = \\operatorname{F_{x}}{(\\mathbf{M})} - \\sin{(\\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('G')(Symbol('\\\\mathbf{M}', commutative=True)), Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 3, "Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))"], "Equality(Add(Function('G')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))), Add(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 4], "Equality(Add(Function('G')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}^*{(t_{2},B)} = B + t_{2}, then derive \\int - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} dt_{2} = \\mathbf{E} + t_{2}, then obtain \\iint - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} dt_{2} dt_{2} = \\int (\\mathbf{E} + t_{2}) dt_{2}", "derivation": "\\tilde{g}^*{(t_{2},B)} = B + t_{2} and 0 = B + t_{2} - \\tilde{g}^*{(t_{2},B)} and - \\tilde{g}^*{(t_{2},B)} = B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)} and - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} = 1 and \\int - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} dt_{2} = \\int 1 dt_{2} and \\int - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} dt_{2} = \\mathbf{E} + t_{2} and \\iint - \\frac{\\tilde{g}^*{(t_{2},B)}}{B + t_{2} - 2 \\tilde{g}^*{(t_{2},B)}} dt_{2} dt_{2} = \\int (\\mathbf{E} + t_{2}) dt_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True)))"], [["minus", 1, "Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))))"], [["minus", 2, "Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))))"], [["divide", 3, "Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["integrate", 4, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t_2', commutative=True)))"], [["integrate", 6, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Add(Symbol('B', commutative=True), Symbol('t_2', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True)))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t_2', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(\\mathbf{B},\\eta)} = - \\eta + \\cos{(\\mathbf{B})}, then obtain \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\int \\mathbf{P}{(\\mathbf{B},\\eta)} d\\mathbf{B} d\\eta = \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- \\eta + \\cos{(\\mathbf{B})}) d\\mathbf{B} d\\eta", "derivation": "\\mathbf{P}{(\\mathbf{B},\\eta)} = - \\eta + \\cos{(\\mathbf{B})} and \\int \\mathbf{P}{(\\mathbf{B},\\eta)} d\\mathbf{B} = \\int (- \\eta + \\cos{(\\mathbf{B})}) d\\mathbf{B} and \\frac{\\partial}{\\partial \\mathbf{B}} \\int \\mathbf{P}{(\\mathbf{B},\\eta)} d\\mathbf{B} = \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- \\eta + \\cos{(\\mathbf{B})}) d\\mathbf{B} and \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\int \\mathbf{P}{(\\mathbf{B},\\eta)} d\\mathbf{B} d\\eta = \\int \\frac{\\partial}{\\partial \\mathbf{B}} \\int (- \\eta + \\cos{(\\mathbf{B})}) d\\mathbf{B} d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Derivative(Integral(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Derivative(Integral(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), cos(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\hat{H}{(b,m_{s})} = - b + m_{s}, then derive t_{1} + \\frac{\\partial}{\\partial m_{s}} \\hat{H}{(b,m_{s})} = \\theta, then obtain \\int (t_{1} + \\frac{\\partial}{\\partial m_{s}} \\hat{H}{(b,m_{s})}) db = \\int \\theta db", "derivation": "\\hat{H}{(b,m_{s})} = - b + m_{s} and \\hat{H}{(b,m_{s})} - 1 = - b + m_{s} - 1 and \\frac{\\partial}{\\partial b} (\\hat{H}{(b,m_{s})} - 1) = \\frac{\\partial}{\\partial b} (- b + m_{s} - 1) and \\frac{\\partial^{2}}{\\partial m_{s}\\partial b} (\\hat{H}{(b,m_{s})} - 1) = \\frac{\\partial^{2}}{\\partial m_{s}\\partial b} (- b + m_{s} - 1) and \\int \\frac{\\partial^{2}}{\\partial m_{s}\\partial b} (\\hat{H}{(b,m_{s})} - 1) db = \\int \\frac{\\partial^{2}}{\\partial m_{s}\\partial b} (- b + m_{s} - 1) db and t_{1} + \\frac{\\partial}{\\partial m_{s}} \\hat{H}{(b,m_{s})} = \\theta and \\int (t_{1} + \\frac{\\partial}{\\partial m_{s}} \\hat{H}{(b,m_{s})}) db = \\int \\theta db", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('m_s', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)))"], [["differentiate", 2, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Symbol('m_s', commutative=True), Integer(-1)), Tuple(Symbol('b', commutative=True), Integer(1)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('t_1', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Symbol('\\\\theta', commutative=True))"], [["integrate", 6, "Symbol('b', commutative=True)"], "Equality(Integral(Add(Symbol('t_1', commutative=True), Derivative(Function('\\\\hat{H}')(Symbol('b', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True))), Integral(Symbol('\\\\theta', commutative=True), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given S{(C,J,A)} = A + C + J, then obtain (\\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dC) \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dJ = (\\int 0 dJ) \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dC", "derivation": "S{(C,J,A)} = A + C + J and S^{C}{(C,J,A)} = (A + C + J)^{C} and - (A + C + J)^{C} + S^{C}{(C,J,A)} = 0 and \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dJ = \\int 0 dJ and \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dC = \\int 0 dC and (\\int 0 dC) \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dJ = (\\int 0 dC) \\int 0 dJ and (\\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dC) \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dJ = (\\int 0 dJ) \\int (- (A + C + J)^{C} + S^{C}{(C,J,A)}) dC", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True)), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True)))"], [["minus", 2, "Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('J', commutative=True))), Integral(Integer(0), Tuple(Symbol('J', commutative=True))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Integer(0), Tuple(Symbol('C', commutative=True))))"], [["times", 4, "Integral(Integer(0), Tuple(Symbol('C', commutative=True)))"], "Equality(Mul(Integral(Integer(0), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Integral(Integer(0), Tuple(Symbol('C', commutative=True))), Integral(Integer(0), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Mul(Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('J', commutative=True)))), Mul(Integral(Integer(0), Tuple(Symbol('J', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(Add(Symbol('A', commutative=True), Symbol('C', commutative=True), Symbol('J', commutative=True)), Symbol('C', commutative=True))), Pow(Function('S')(Symbol('C', commutative=True), Symbol('J', commutative=True), Symbol('A', commutative=True)), Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given M{(\\hat{x}_0)} = - \\frac{1}{\\hat{x}_0}, then obtain C_{1} (- \\varphi + M{(\\hat{x}_0)}) - \\varphi + \\mathbf{D}{(\\hat{x}_0,C_{1},\\varphi)} + \\frac{1}{\\hat{x}_0} = C_{1} (- \\varphi - \\frac{1}{\\hat{x}_0}) - \\varphi + \\mathbf{D}{(\\hat{x}_0,C_{1},\\varphi)} + \\frac{1}{\\hat{x}_0}", "derivation": "M{(\\hat{x}_0)} = - \\frac{1}{\\hat{x}_0} and - \\varphi + M{(\\hat{x}_0)} = - \\varphi - \\frac{1}{\\hat{x}_0} and C_{1} (- \\varphi + M{(\\hat{x}_0)}) = C_{1} (- \\varphi - \\frac{1}{\\hat{x}_0}) and C_{1} (- \\varphi + M{(\\hat{x}_0)}) - \\varphi - \\frac{1}{\\hat{x}_0} = C_{1} (- \\varphi - \\frac{1}{\\hat{x}_0}) - \\varphi - \\frac{1}{\\hat{x}_0} and C_{1} (- \\varphi + M{(\\hat{x}_0)}) - \\varphi + \\mathbf{D}{(\\hat{x}_0,C_{1},\\varphi)} + \\frac{1}{\\hat{x}_0} = C_{1} (- \\varphi - \\frac{1}{\\hat{x}_0}) - \\varphi + \\mathbf{D}{(\\hat{x}_0,C_{1},\\varphi)} + \\frac{1}{\\hat{x}_0}", "srepr_derivation": [["renaming_premise", "Equality(Function('M')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)))))"], [["times", 2, "Symbol('C_1', commutative=True)"], "Equality(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)))), Add(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)))))"], [["add", 4, "Add(Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('M')(Symbol('\\\\hat{x}_0', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))), Add(Mul(Symbol('C_1', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Function('\\\\mathbf{D}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('C_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given l{(\\mathbf{A})} = e^{\\mathbf{A}} and \\hat{p}_0{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A}, then derive \\int l{(\\mathbf{A})} d\\mathbf{A} = g_{\\varepsilon} + e^{\\mathbf{A}}, then obtain g_{\\varepsilon} + l{(\\mathbf{A})} = \\hat{p}_0{(\\mathbf{A})}", "derivation": "l{(\\mathbf{A})} = e^{\\mathbf{A}} and \\int l{(\\mathbf{A})} d\\mathbf{A} = \\int e^{\\mathbf{A}} d\\mathbf{A} and \\int l{(\\mathbf{A})} d\\mathbf{A} = g_{\\varepsilon} + e^{\\mathbf{A}} and \\hat{p}_0{(\\mathbf{A})} = \\int e^{\\mathbf{A}} d\\mathbf{A} and g_{\\varepsilon} + e^{\\mathbf{A}} = \\int e^{\\mathbf{A}} d\\mathbf{A} and g_{\\varepsilon} + e^{\\mathbf{A}} = \\hat{p}_0{(\\mathbf{A})} and g_{\\varepsilon} + l{(\\mathbf{A})} = \\hat{p}_0{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('l')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True)), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), exp(Symbol('\\\\mathbf{A}', commutative=True))), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('l')(Symbol('\\\\mathbf{A}', commutative=True))), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})} = \\sin{(\\hat{H}_l + \\mathbf{f})}, then obtain (\\frac{\\mathbf{f}}{\\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})}})^{\\mathbf{f}} = (\\frac{\\mathbf{f} \\sin{(\\hat{H}_l + \\mathbf{f})}}{\\operatorname{g_{\\varepsilon}}^{2}{(\\hat{H}_l,\\mathbf{f})}})^{\\mathbf{f}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})} = \\sin{(\\hat{H}_l + \\mathbf{f})} and \\mathbf{f} \\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})} = \\mathbf{f} \\sin{(\\hat{H}_l + \\mathbf{f})} and \\frac{\\mathbf{f}}{\\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})}} = \\frac{\\mathbf{f} \\sin{(\\hat{H}_l + \\mathbf{f})}}{\\operatorname{g_{\\varepsilon}}^{2}{(\\hat{H}_l,\\mathbf{f})}} and (\\frac{\\mathbf{f}}{\\operatorname{g_{\\varepsilon}}{(\\hat{H}_l,\\mathbf{f})}})^{\\mathbf{f}} = (\\frac{\\mathbf{f} \\sin{(\\hat{H}_l + \\mathbf{f})}}{\\operatorname{g_{\\varepsilon}}^{2}{(\\hat{H}_l,\\mathbf{f})}})^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Symbol('\\\\mathbf{f}', commutative=True), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["divide", 2, "Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(2))"], "Equality(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-2)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Function('g_{\\\\varepsilon}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-2)), sin(Add(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given x{(E_{x})} = \\frac{d}{d E_{x}} \\sin{(E_{x})}, then derive x{(E_{x})} = \\cos{(E_{x})}, then obtain 1 - \\frac{d}{d E_{x}} \\sin{(E_{x})} = - \\frac{d}{d E_{x}} \\sin{(E_{x})} + \\frac{\\frac{d}{d E_{x}} \\sin{(E_{x})}}{x{(E_{x})}}", "derivation": "x{(E_{x})} = \\frac{d}{d E_{x}} \\sin{(E_{x})} and x{(E_{x})} = \\cos{(E_{x})} and \\cos{(E_{x})} = \\frac{d}{d E_{x}} \\sin{(E_{x})} and 1 = \\frac{\\cos{(E_{x})}}{x{(E_{x})}} and 1 - \\cos{(E_{x})} = - \\cos{(E_{x})} + \\frac{\\cos{(E_{x})}}{x{(E_{x})}} and 1 - \\frac{d}{d E_{x}} \\sin{(E_{x})} = - \\frac{d}{d E_{x}} \\sin{(E_{x})} + \\frac{\\frac{d}{d E_{x}} \\sin{(E_{x})}}{x{(E_{x})}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('E_x', commutative=True)), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('x')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('E_x', commutative=True)), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["divide", 2, "Function('x')(Symbol('E_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('x')(Symbol('E_x', commutative=True)), Integer(-1)), cos(Symbol('E_x', commutative=True))))"], [["minus", 4, "cos(Symbol('E_x', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), cos(Symbol('E_x', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('E_x', commutative=True))), Mul(Pow(Function('x')(Symbol('E_x', commutative=True)), Integer(-1)), cos(Symbol('E_x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Pow(Function('x')(Symbol('E_x', commutative=True)), Integer(-1)), Derivative(sin(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{E}{(g,t)} = \\cos{(g t)} and \\operatorname{f_{\\mathbf{p}}}{(g,t)} = (\\int \\cos{(g t)} dg)^{2}, then obtain \\frac{(\\int \\mathbf{E}{(g,t)} dg) \\int \\cos{(g t)} dg}{g^{2}} = \\frac{\\operatorname{f_{\\mathbf{p}}}{(g,t)}}{g^{2}}", "derivation": "\\mathbf{E}{(g,t)} = \\cos{(g t)} and \\int \\mathbf{E}{(g,t)} dg = \\int \\cos{(g t)} dg and \\frac{\\int \\mathbf{E}{(g,t)} dg}{g} = \\frac{\\int \\cos{(g t)} dg}{g} and \\frac{(\\int \\mathbf{E}{(g,t)} dg) \\int \\cos{(g t)} dg}{g^{2}} = \\frac{(\\int \\cos{(g t)} dg)^{2}}{g^{2}} and \\operatorname{f_{\\mathbf{p}}}{(g,t)} = (\\int \\cos{(g t)} dg)^{2} and \\frac{(\\int \\mathbf{E}{(g,t)} dg) \\int \\cos{(g t)} dg}{g^{2}} = \\frac{\\operatorname{f_{\\mathbf{p}}}{(g,t)}}{g^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["divide", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(Function('\\\\mathbf{E}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["times", 3, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True))))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Integral(Function('\\\\mathbf{E}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Pow(Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True))), Integer(2))))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Pow(Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Integral(Function('\\\\mathbf{E}')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(cos(Mul(Symbol('g', commutative=True), Symbol('t', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Pow(Symbol('g', commutative=True), Integer(-2)), Function('f_{\\\\mathbf{p}}')(Symbol('g', commutative=True), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(A,\\mathbf{D})} = A + \\mathbf{D}, then obtain \\frac{d}{d A} 0 = \\frac{\\partial}{\\partial A} (\\mathbf{D} (A + \\mathbf{D}) - \\mathbf{D} \\operatorname{P_{e}}{(A,\\mathbf{D})})", "derivation": "\\operatorname{P_{e}}{(A,\\mathbf{D})} = A + \\mathbf{D} and \\mathbf{D} \\operatorname{P_{e}}{(A,\\mathbf{D})} = \\mathbf{D} (A + \\mathbf{D}) and 0 = \\mathbf{D} (A + \\mathbf{D}) - \\mathbf{D} \\operatorname{P_{e}}{(A,\\mathbf{D})} and \\frac{d}{d A} 0 = \\frac{\\partial}{\\partial A} (\\mathbf{D} (A + \\mathbf{D}) - \\mathbf{D} \\operatorname{P_{e}}{(A,\\mathbf{D})})", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{D}', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["differentiate", 3, "Symbol('A', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{D}', commutative=True), Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Function('P_e')(Symbol('A', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(U,Q)} = Q U, then derive e^{\\int (U + v{(U,Q)}) dQ} = e^{\\frac{Q^{2} U}{2} + Q U + t_{2}}, then obtain \\int e^{\\int (Q U + U) dQ} dt_{2} = \\int e^{\\int (U + v{(U,Q)}) dQ} dt_{2}", "derivation": "v{(U,Q)} = Q U and U + v{(U,Q)} = Q U + U and \\int (U + v{(U,Q)}) dQ = \\int (Q U + U) dQ and e^{\\int (U + v{(U,Q)}) dQ} = e^{\\int (Q U + U) dQ} and e^{\\int (U + v{(U,Q)}) dQ} = e^{\\frac{Q^{2} U}{2} + Q U + t_{2}} and \\int e^{\\int (U + v{(U,Q)}) dQ} dt_{2} = \\int e^{\\frac{Q^{2} U}{2} + Q U + t_{2}} dt_{2} and \\int e^{\\int (U + v{(U,Q)}) dQ} dt_{2} = \\int e^{\\frac{Q v{(U,Q)}}{2} + t_{2} + v{(U,Q)}} dt_{2} and \\int e^{\\int (Q U + U) dQ} dt_{2} = \\int e^{\\frac{Q v{(U,Q)}}{2} + t_{2} + v{(U,Q)}} dt_{2} and \\int e^{\\int (Q U + U) dQ} dt_{2} = \\int e^{\\int (U + v{(U,Q)}) dQ} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)))"], [["add", 1, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)))"], [["integrate", 2, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["exp", 3], "Equality(exp(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), exp(Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(exp(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), exp(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('U', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('t_2', commutative=True))))"], [["integrate", 5, "Symbol('t_2', commutative=True)"], "Equality(Integral(exp(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2)), Symbol('U', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integral(exp(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Add(Mul(Rational(1, 2), Symbol('Q', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Symbol('t_2', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Integral(exp(Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Add(Mul(Rational(1, 2), Symbol('Q', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Symbol('t_2', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Integral(exp(Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(exp(Integral(Add(Symbol('U', commutative=True), Function('v')(Symbol('U', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(g,\\nabla)} = e^{- \\nabla + g}, then derive \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = - e^{- \\nabla + g}, then obtain \\frac{\\partial}{\\partial \\nabla} - \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} e^{- \\nabla + g}", "derivation": "\\operatorname{y^{\\prime}}{(g,\\nabla)} = e^{- \\nabla + g} and \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} e^{- \\nabla + g} and \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = - e^{- \\nabla + g} and \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = - \\operatorname{y^{\\prime}}{(g,\\nabla)} and - \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = \\operatorname{y^{\\prime}}{(g,\\nabla)} and \\frac{\\partial}{\\partial \\nabla} - \\frac{\\partial}{\\partial \\nabla} \\operatorname{y^{\\prime}}{(g,\\nabla)} = \\frac{\\partial}{\\partial \\nabla} e^{- \\nabla + g}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Derivative(Mul(Integer(-1), Derivative(Function('y^{\\\\prime}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('g', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(v_{t})} = e^{v_{t}}, then derive \\int \\mu{(v_{t})} dv_{t} = r + e^{v_{t}}, then obtain \\int \\mu{(v_{t})} dv_{t} = r + \\mu{(v_{t})}", "derivation": "\\mu{(v_{t})} = e^{v_{t}} and \\int \\mu{(v_{t})} dv_{t} = \\int e^{v_{t}} dv_{t} and \\int \\mu{(v_{t})} dv_{t} = r + e^{v_{t}} and \\int \\mu{(v_{t})} dv_{t} = r + \\mu{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('v_t', commutative=True)), exp(Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\mu')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(exp(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mu')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('r', commutative=True), exp(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mu')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Add(Symbol('r', commutative=True), Function('\\\\mu')(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(B)} = \\cos{(B)} and \\operatorname{n_{1}}{(B)} = \\theta_{1}{(B)} + \\cos{(B)}, then obtain \\operatorname{n_{1}}^{2}{(B)} = 4 \\cos^{2}{(B)}", "derivation": "\\theta_{1}{(B)} = \\cos{(B)} and \\operatorname{n_{1}}{(B)} = \\theta_{1}{(B)} + \\cos{(B)} and \\operatorname{n_{1}}{(B)} = 2 \\cos{(B)} and \\operatorname{n_{1}}^{2}{(B)} = 4 \\cos^{2}{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('B', commutative=True)), Add(Function('\\\\theta_1')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('n_1')(Symbol('B', commutative=True)), Mul(Integer(2), cos(Symbol('B', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('n_1')(Symbol('B', commutative=True)), Integer(2)), Mul(Integer(4), Pow(cos(Symbol('B', commutative=True)), Integer(2))))"]]}, {"prompt": "Given v{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}, then obtain (\\mathbf{J}_f + v^{\\mathbf{J}_f}{(\\mathbf{J}_f)}) \\cos{(\\frac{d}{d \\mathbf{J}_f} v{(\\mathbf{J}_f)})} = (\\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f}) \\cos{(\\frac{d}{d \\mathbf{J}_f} v{(\\mathbf{J}_f)})}", "derivation": "v{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)} and v^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f} and \\mathbf{J}_f + v^{\\mathbf{J}_f}{(\\mathbf{J}_f)} = \\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f} and (\\mathbf{J}_f + v^{\\mathbf{J}_f}{(\\mathbf{J}_f)}) \\cos{(\\frac{d}{d \\mathbf{J}_f} v{(\\mathbf{J}_f)})} = (\\mathbf{J}_f + \\log{(\\mathbf{J}_f)}^{\\mathbf{J}_f}) \\cos{(\\frac{d}{d \\mathbf{J}_f} v{(\\mathbf{J}_f)})}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["times", 3, "cos(Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], "Equality(Mul(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), cos(Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))), Mul(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True))), cos(Derivative(Function('v')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{x}{(p)} = p, then obtain \\frac{d}{d p} (p + \\hat{x}^{2}{(p)}) = \\frac{d}{d p} (p \\hat{x}{(p)} + p)", "derivation": "\\hat{x}{(p)} = p and \\hat{x}^{2}{(p)} = p \\hat{x}{(p)} and p + \\hat{x}^{2}{(p)} = p \\hat{x}{(p)} + p and \\frac{d}{d p} (p + \\hat{x}^{2}{(p)}) = \\frac{d}{d p} (p \\hat{x}{(p)} + p)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Symbol('p', commutative=True))"], [["times", 1, "Function('\\\\hat{x}')(Symbol('p', commutative=True))"], "Equality(Pow(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Integer(2)), Mul(Symbol('p', commutative=True), Function('\\\\hat{x}')(Symbol('p', commutative=True))))"], [["add", 2, "Symbol('p', commutative=True)"], "Equality(Add(Symbol('p', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Integer(2))), Add(Mul(Symbol('p', commutative=True), Function('\\\\hat{x}')(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["differentiate", 3, "Symbol('p', commutative=True)"], "Equality(Derivative(Add(Symbol('p', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('p', commutative=True)), Integer(2))), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('p', commutative=True), Function('\\\\hat{x}')(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(r)} = e^{r}, then derive (\\operatorname{P_{g}}{(r)} e^{- r})^{r} (\\frac{r (- \\operatorname{P_{g}}{(r)} e^{- r} + e^{- r} \\frac{d}{d r} \\operatorname{P_{g}}{(r)}) e^{r}}{\\operatorname{P_{g}}{(r)}} + \\log{(\\operatorname{P_{g}}{(r)} e^{- r})}) = 0, then obtain \\int r (-1 + \\frac{\\frac{d}{d r} \\operatorname{P_{g}}{(r)}}{\\operatorname{P_{g}}{(r)}}) dr = \\int 0 dr", "derivation": "\\operatorname{P_{g}}{(r)} = e^{r} and \\operatorname{P_{g}}{(r)} e^{- r} = 1 and (\\operatorname{P_{g}}{(r)} e^{- r})^{r} = 1 and \\frac{d}{d r} (\\operatorname{P_{g}}{(r)} e^{- r})^{r} = \\frac{d}{d r} 1 and (\\operatorname{P_{g}}{(r)} e^{- r})^{r} (\\frac{r (- \\operatorname{P_{g}}{(r)} e^{- r} + e^{- r} \\frac{d}{d r} \\operatorname{P_{g}}{(r)}) e^{r}}{\\operatorname{P_{g}}{(r)}} + \\log{(\\operatorname{P_{g}}{(r)} e^{- r})}) = 0 and r (-1 + \\frac{\\frac{d}{d r} \\operatorname{P_{g}}{(r)}}{\\operatorname{P_{g}}{(r)}}) = 0 and \\int r (-1 + \\frac{\\frac{d}{d r} \\operatorname{P_{g}}{(r)}}{\\operatorname{P_{g}}{(r)}}) dr = \\int 0 dr", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["divide", 1, "exp(Symbol('r', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Integer(1))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Mul(Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Integer(1))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Mul(Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Mul(Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Symbol('r', commutative=True)), Add(Mul(Symbol('r', commutative=True), Add(Mul(Integer(-1), Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('r', commutative=True))), Derivative(Function('P_g')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))), Pow(Function('P_g')(Symbol('r', commutative=True)), Integer(-1)), exp(Symbol('r', commutative=True))), log(Mul(Function('P_g')(Symbol('r', commutative=True)), exp(Mul(Integer(-1), Symbol('r', commutative=True))))))), Integer(0))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Symbol('r', commutative=True), Add(Integer(-1), Mul(Pow(Function('P_g')(Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))), Integer(0))"], [["integrate", 6, "Symbol('r', commutative=True)"], "Equality(Integral(Mul(Symbol('r', commutative=True), Add(Integer(-1), Mul(Pow(Function('P_g')(Symbol('r', commutative=True)), Integer(-1)), Derivative(Function('P_g')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))))), Tuple(Symbol('r', commutative=True))), Integral(Integer(0), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given G{(f,g_{\\varepsilon})} = \\log{(f)}^{g_{\\varepsilon}}, then obtain \\iint 2 G{(f,g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint (G{(f,g_{\\varepsilon})} + \\log{(f)}^{g_{\\varepsilon}}) dg_{\\varepsilon} dg_{\\varepsilon}", "derivation": "G{(f,g_{\\varepsilon})} = \\log{(f)}^{g_{\\varepsilon}} and 2 G{(f,g_{\\varepsilon})} = G{(f,g_{\\varepsilon})} + \\log{(f)}^{g_{\\varepsilon}} and \\int 2 G{(f,g_{\\varepsilon})} dg_{\\varepsilon} = \\int (G{(f,g_{\\varepsilon})} + \\log{(f)}^{g_{\\varepsilon}}) dg_{\\varepsilon} and \\iint 2 G{(f,g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint (G{(f,g_{\\varepsilon})} + \\log{(f)}^{g_{\\varepsilon}}) dg_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Integer(2), Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Function('G')(Symbol('f', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('f', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\mu_0)} = \\log{(\\mu_0)} and \\eta^{\\prime}{(\\mu_0)} = \\frac{\\int \\lambda{(\\mu_0)} d\\mu_0}{\\mu_0}, then derive \\eta^{\\prime}{(\\mu_0)} = \\frac{\\mathbf{v} + \\mu_0 \\log{(\\mu_0)} - \\mu_0}{\\mu_0}, then obtain \\frac{d}{d \\mathbf{v}} \\frac{\\int \\lambda{(\\mu_0)} d\\mu_0}{\\mu_0} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v} + \\mu_0 \\lambda{(\\mu_0)} - \\mu_0}{\\mu_0}", "derivation": "\\lambda{(\\mu_0)} = \\log{(\\mu_0)} and \\int \\lambda{(\\mu_0)} d\\mu_0 = \\int \\log{(\\mu_0)} d\\mu_0 and \\eta^{\\prime}{(\\mu_0)} = \\frac{\\int \\lambda{(\\mu_0)} d\\mu_0}{\\mu_0} and \\eta^{\\prime}{(\\mu_0)} = \\frac{\\int \\log{(\\mu_0)} d\\mu_0}{\\mu_0} and \\eta^{\\prime}{(\\mu_0)} = \\frac{\\mathbf{v} + \\mu_0 \\log{(\\mu_0)} - \\mu_0}{\\mu_0} and \\eta^{\\prime}{(\\mu_0)} = \\frac{\\mathbf{v} + \\mu_0 \\lambda{(\\mu_0)} - \\mu_0}{\\mu_0} and \\frac{\\int \\lambda{(\\mu_0)} d\\mu_0}{\\mu_0} = \\frac{\\mathbf{v} + \\mu_0 \\lambda{(\\mu_0)} - \\mu_0}{\\mu_0} and \\frac{d}{d \\mathbf{v}} \\frac{\\int \\lambda{(\\mu_0)} d\\mu_0}{\\mu_0} = \\frac{\\partial}{\\partial \\mathbf{v}} \\frac{\\mathbf{v} + \\mu_0 \\lambda{(\\mu_0)} - \\mu_0}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), log(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))))"], [["differentiate", 7, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{v}', commutative=True), Mul(Symbol('\\\\mu_0', commutative=True), Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True))), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(A_{2})} = \\cos{(A_{2})}, then obtain \\operatorname{E_{\\lambda}}{(A_{2})} - \\cos^{A_{2}}{(A_{2})} = \\cos{(A_{2})} - \\cos^{A_{2}}{(A_{2})}", "derivation": "\\operatorname{E_{\\lambda}}{(A_{2})} = \\cos{(A_{2})} and \\operatorname{E_{\\lambda}}^{A_{2}}{(A_{2})} = \\cos^{A_{2}}{(A_{2})} and \\operatorname{E_{\\lambda}}{(A_{2})} - \\operatorname{E_{\\lambda}}^{A_{2}}{(A_{2})} = - \\operatorname{E_{\\lambda}}^{A_{2}}{(A_{2})} + \\cos{(A_{2})} and \\operatorname{E_{\\lambda}}{(A_{2})} - \\cos^{A_{2}}{(A_{2})} = \\cos{(A_{2})} - \\cos^{A_{2}}{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["power", 1, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["minus", 1, "Pow(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))), cos(Symbol('A_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))), Add(cos(Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given E{(B,y)} = B + \\cos{(y)}, then obtain \\iint (- B - \\cos{(y)} + 1) dy dy = \\iint (1 - E{(B,y)}) dy dy", "derivation": "E{(B,y)} = B + \\cos{(y)} and 0 = B - E{(B,y)} + \\cos{(y)} and 1 = B - E{(B,y)} + \\cos{(y)} + 1 and - B - \\cos{(y)} + 1 = 1 - E{(B,y)} and \\int (- B - \\cos{(y)} + 1) dy = \\int (1 - E{(B,y)}) dy and \\iint (- B - \\cos{(y)} + 1) dy dy = \\iint (1 - E{(B,y)}) dy dy", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True)), Add(Symbol('B', commutative=True), cos(Symbol('y', commutative=True))))"], [["minus", 1, "Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True))"], "Equality(Integer(0), Add(Symbol('B', commutative=True), Mul(Integer(-1), Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True))), cos(Symbol('y', commutative=True))))"], [["add", 2, 1], "Equality(Integer(1), Add(Symbol('B', commutative=True), Mul(Integer(-1), Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True))), cos(Symbol('y', commutative=True)), Integer(1)))"], [["minus", 3, "Add(Symbol('B', commutative=True), cos(Symbol('y', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True)))))"], [["integrate", 4, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True))))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Mul(Integer(-1), cos(Symbol('y', commutative=True))), Integer(1)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Function('E')(Symbol('B', commutative=True), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"]]}, {"prompt": "Given z{(I)} = \\sin{(\\sin{(I)})} and \\Psi_{nl}{(I)} = \\sin{(I)}, then obtain z^{I}{(I)} = \\sin^{I}{(\\Psi_{nl}{(I)})}", "derivation": "z{(I)} = \\sin{(\\sin{(I)})} and z^{I}{(I)} = \\sin^{I}{(\\sin{(I)})} and \\Psi_{nl}{(I)} = \\sin{(I)} and z^{I}{(I)} = \\sin^{I}{(\\Psi_{nl}{(I)})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('I', commutative=True)), sin(sin(Symbol('I', commutative=True))))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('z')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(sin(sin(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('z')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(sin(Function('\\\\Psi_{nl}')(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(A_{y})} = \\cos{(A_{y})}, then obtain \\frac{d}{d A_{y}} \\frac{\\mathbf{B}{(A_{y})}}{\\mathbf{B}{(A_{y})} + \\cos{(A_{y})}} = \\frac{d}{d A_{y}} \\frac{1}{2}", "derivation": "\\mathbf{B}{(A_{y})} = \\cos{(A_{y})} and \\mathbf{B}{(A_{y})} + \\cos{(A_{y})} = 2 \\cos{(A_{y})} and \\frac{\\mathbf{B}{(A_{y})}}{\\mathbf{B}{(A_{y})} + \\cos{(A_{y})}} = \\frac{\\cos{(A_{y})}}{\\mathbf{B}{(A_{y})} + \\cos{(A_{y})}} and \\frac{\\mathbf{B}{(A_{y})}}{2 \\cos{(A_{y})}} = \\frac{1}{2} and \\frac{\\mathbf{B}{(A_{y})}}{\\mathbf{B}{(A_{y})} + \\cos{(A_{y})}} = \\frac{1}{2} and \\frac{d}{d A_{y}} \\frac{\\mathbf{B}{(A_{y})}}{\\mathbf{B}{(A_{y})} + \\cos{(A_{y})}} = \\frac{d}{d A_{y}} \\frac{1}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["add", 1, "cos(Symbol('A_y', commutative=True))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Mul(Integer(2), cos(Symbol('A_y', commutative=True))))"], [["divide", 1, "Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], "Equality(Mul(Pow(Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True))), Mul(Pow(Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Integer(-1)), cos(Symbol('A_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Rational(1, 2), Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Rational(1, 2))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True))), Rational(1, 2))"], [["differentiate", 5, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True))), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Rational(1, 2), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(c,\\hat{H}_{\\lambda},r_{0})} = r_{0} (\\hat{H}_{\\lambda} - c), then derive \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(c,\\hat{H}_{\\lambda},r_{0})} = r_{0}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} r_{0} (\\hat{H}_{\\lambda} - c) = r_{0}", "derivation": "\\hat{\\mathbf{r}}{(c,\\hat{H}_{\\lambda},r_{0})} = r_{0} (\\hat{H}_{\\lambda} - c) and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(c,\\hat{H}_{\\lambda},r_{0})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} r_{0} (\\hat{H}_{\\lambda} - c) and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\hat{\\mathbf{r}}{(c,\\hat{H}_{\\lambda},r_{0})} = r_{0} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} r_{0} (\\hat{H}_{\\lambda} - c) = r_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('r_0', commutative=True), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Symbol('r_0', commutative=True))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})} = B - E_{\\lambda} + \\mathbb{I}, then derive \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})} = 0, then obtain \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (B - E_{\\lambda} + \\mathbb{I}) = 0", "derivation": "\\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})} = B - E_{\\lambda} + \\mathbb{I} and B + \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})} = 2 B - E_{\\lambda} + \\mathbb{I} and \\frac{\\partial}{\\partial \\mathbb{I}} (B + \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})}) = \\frac{\\partial}{\\partial \\mathbb{I}} (2 B - E_{\\lambda} + \\mathbb{I}) and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (B + \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})}) = \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (2 B - E_{\\lambda} + \\mathbb{I}) and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} \\hat{\\mathbf{r}}{(\\mathbb{I},B,E_{\\lambda})} = 0 and \\frac{\\partial^{2}}{\\partial \\mathbb{I}^{2}} (B - E_{\\lambda} + \\mathbb{I}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Symbol('B', commutative=True)"], "Equality(Add(Symbol('B', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Derivative(Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('B', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Add(Symbol('B', commutative=True), Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\mathbf{M}{(C)} = e^{C}, then obtain \\mathbf{M}^{3}{(C)} e^{C} = e^{4 C}", "derivation": "\\mathbf{M}{(C)} = e^{C} and \\mathbf{M}{(C)} e^{C} = e^{2 C} and \\mathbf{M}^{2}{(C)} e^{2 C} = e^{4 C} and \\mathbf{M}^{3}{(C)} e^{C} = \\mathbf{M}^{2}{(C)} e^{2 C} and \\mathbf{M}^{3}{(C)} e^{C} = e^{4 C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["times", 1, "exp(Symbol('C', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True))), exp(Mul(Integer(2), Symbol('C', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('C', commutative=True)))), exp(Mul(Integer(4), Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), Integer(3)), exp(Symbol('C', commutative=True))), Mul(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbf{M}')(Symbol('C', commutative=True)), Integer(3)), exp(Symbol('C', commutative=True))), exp(Mul(Integer(4), Symbol('C', commutative=True))))"]]}, {"prompt": "Given L{(\\Psi,\\mathbf{p})} = \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p}), then derive 1 = e^{1 - L{(\\Psi,\\mathbf{p})}}, then obtain 1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p}) = (1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p})) e^{1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p})}", "derivation": "L{(\\Psi,\\mathbf{p})} = \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p}) and 0 = - L{(\\Psi,\\mathbf{p})} + \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p}) and 1 = e^{- L{(\\Psi,\\mathbf{p})} + \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p})} and 1 = e^{1 - L{(\\Psi,\\mathbf{p})}} and 1 - L{(\\Psi,\\mathbf{p})} = (1 - L{(\\Psi,\\mathbf{p})}) e^{1 - L{(\\Psi,\\mathbf{p})}} and 1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p}) = (1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p})) e^{1 - \\frac{\\partial}{\\partial \\Psi} (\\Psi + \\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["minus", 1, "Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["exp", 2], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(1), exp(Add(Integer(1), Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))))"], [["times", 4, "Add(Integer(1), Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), Mul(Add(Integer(1), Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))), exp(Add(Integer(1), Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Mul(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), exp(Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\mathbf{p}, then obtain e^{- U} \\int \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} d\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = e^{- U} \\int \\mathbf{p} d\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = \\mathbf{p} and \\int \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} d\\mathbf{p} = \\int \\mathbf{p} d\\mathbf{p} and e^{- U} \\int \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} d\\mathbf{p} = e^{- U} \\int \\mathbf{p} d\\mathbf{p} and e^{- U} \\int \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} d\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})} = e^{- U} \\int \\mathbf{p} d\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["divide", 2, "exp(Symbol('U', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Symbol('\\\\mathbf{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(exp(Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True))))), Mul(exp(Mul(Integer(-1), Symbol('U', commutative=True))), Integral(Symbol('\\\\mathbf{p}', commutative=True), Tuple(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{p}', commutative=True))))))"]]}, {"prompt": "Given I{(\\rho_f,\\mathbf{J}_f)} = - \\mathbf{J}_f + \\rho_f, then obtain \\frac{(\\frac{I{(\\rho_f,\\mathbf{J}_f)}}{- \\mathbf{J}_f + \\rho_f})^{\\mathbf{J}_f}}{\\sin{(\\mathbf{J}_f - \\rho_f)}} = \\frac{1}{\\sin{(\\mathbf{J}_f - \\rho_f)}}", "derivation": "I{(\\rho_f,\\mathbf{J}_f)} = - \\mathbf{J}_f + \\rho_f and \\frac{I{(\\rho_f,\\mathbf{J}_f)}}{- \\mathbf{J}_f + \\rho_f} = 1 and (\\frac{I{(\\rho_f,\\mathbf{J}_f)}}{- \\mathbf{J}_f + \\rho_f})^{\\mathbf{J}_f} = 1 and \\frac{(\\frac{I{(\\rho_f,\\mathbf{J}_f)}}{- \\mathbf{J}_f + \\rho_f})^{\\mathbf{J}_f}}{\\sin{(\\mathbf{J}_f - \\rho_f)}} = \\frac{1}{\\sin{(\\mathbf{J}_f - \\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Function('I')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Function('I')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(1))"], [["divide", 3, "sin(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], "Equality(Mul(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Function('I')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(sin(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(-1))), Pow(sin(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given g{(A_{z},g^{\\prime}_{\\varepsilon})} = A_{z} + g^{\\prime}_{\\varepsilon}, then obtain g^{\\prime}_{\\varepsilon} (A_{z} + g^{\\prime}_{\\varepsilon}) = - g^{\\prime}_{\\varepsilon} (- A_{z} - g^{\\prime}_{\\varepsilon})", "derivation": "g{(A_{z},g^{\\prime}_{\\varepsilon})} = A_{z} + g^{\\prime}_{\\varepsilon} and - g{(A_{z},g^{\\prime}_{\\varepsilon})} = - A_{z} - g^{\\prime}_{\\varepsilon} and g^{\\prime}_{\\varepsilon} g{(A_{z},g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} (- A_{z} - g^{\\prime}_{\\varepsilon}) and g^{\\prime}_{\\varepsilon} (A_{z} + g^{\\prime}_{\\varepsilon}) = - g^{\\prime}_{\\varepsilon} (- A_{z} - g^{\\prime}_{\\varepsilon})", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('A_z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('A_z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('g')(Symbol('A_z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('g')(Symbol('A_z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Add(Symbol('A_z', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{B},f)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{f}{\\mathbf{B}}, then derive \\dot{y}{(\\mathbf{B},f)} = - \\frac{f}{\\mathbf{B}^{2}}, then obtain - \\frac{f}{\\mathbf{B}^{4}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{B}} \\frac{f}{\\mathbf{B}}}{\\mathbf{B}^{2}}", "derivation": "\\dot{y}{(\\mathbf{B},f)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{f}{\\mathbf{B}} and \\dot{y}{(\\mathbf{B},f)} = - \\frac{f}{\\mathbf{B}^{2}} and \\frac{\\dot{y}{(\\mathbf{B},f)}}{\\mathbf{B}^{2}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{B}} \\frac{f}{\\mathbf{B}}}{\\mathbf{B}^{2}} and - \\frac{f}{\\mathbf{B}^{4}} = \\frac{\\frac{\\partial}{\\partial \\mathbf{B}} \\frac{f}{\\mathbf{B}}}{\\mathbf{B}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Symbol('f', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('f', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Derivative(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-4)), Symbol('f', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-2)), Derivative(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given s{(v_{1},J)} = \\log{(v_{1})}^{J}, then obtain 2 J + \\frac{\\partial}{\\partial J} s{(v_{1},J)} \\log{(v_{1})}^{J} = 2 J + \\frac{\\partial}{\\partial J} \\log{(v_{1})}^{2 J}", "derivation": "s{(v_{1},J)} = \\log{(v_{1})}^{J} and s{(v_{1},J)} \\log{(v_{1})}^{J} = \\log{(v_{1})}^{2 J} and \\frac{\\partial}{\\partial J} s{(v_{1},J)} \\log{(v_{1})}^{J} = \\frac{\\partial}{\\partial J} \\log{(v_{1})}^{2 J} and 2 J + \\frac{\\partial}{\\partial J} s{(v_{1},J)} \\log{(v_{1})}^{J} = 2 J + \\frac{\\partial}{\\partial J} \\log{(v_{1})}^{2 J}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('J', commutative=True)))"], [["times", 1, "Pow(log(Symbol('v_1', commutative=True)), Symbol('J', commutative=True))"], "Equality(Mul(Function('s')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('J', commutative=True))), Pow(log(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Mul(Function('s')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(2), Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('J', commutative=True)), Derivative(Mul(Function('s')(Symbol('v_1', commutative=True), Symbol('J', commutative=True)), Pow(log(Symbol('v_1', commutative=True)), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Mul(Integer(2), Symbol('J', commutative=True)), Derivative(Pow(log(Symbol('v_1', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\psi^{*}{(A)} = \\sin{(e^{A})}, then obtain \\psi^{*}{(A)} - \\sin^{A}{(e^{A})} = \\sin{(e^{A})} - \\sin^{A}{(e^{A})}", "derivation": "\\psi^{*}{(A)} = \\sin{(e^{A})} and \\psi^{*}^{A}{(A)} = \\sin^{A}{(e^{A})} and \\psi^{*}{(A)} - \\psi^{*}^{A}{(A)} = - \\psi^{*}^{A}{(A)} + \\sin{(e^{A})} and \\psi^{*}{(A)} - \\sin^{A}{(e^{A})} = \\sin{(e^{A})} - \\sin^{A}{(e^{A})}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\psi^*')(Symbol('A', commutative=True)), sin(exp(Symbol('A', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(sin(exp(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["minus", 1, "Pow(Function('\\\\psi^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('A', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\psi^*')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), sin(exp(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\psi^*')(Symbol('A', commutative=True)), Mul(Integer(-1), Pow(sin(exp(Symbol('A', commutative=True))), Symbol('A', commutative=True)))), Add(sin(exp(Symbol('A', commutative=True))), Mul(Integer(-1), Pow(sin(exp(Symbol('A', commutative=True))), Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(F_{c},\\mathbf{A})} = F_{c} + \\mathbf{A}, then derive \\int \\operatorname{v_{1}}{(F_{c},\\mathbf{A})} dF_{c} = \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + f, then derive \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + f = \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + \\varepsilon_0, then obtain \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + \\varepsilon_0 = \\int (F_{c} + \\mathbf{A}) dF_{c}", "derivation": "\\operatorname{v_{1}}{(F_{c},\\mathbf{A})} = F_{c} + \\mathbf{A} and \\int \\operatorname{v_{1}}{(F_{c},\\mathbf{A})} dF_{c} = \\int (F_{c} + \\mathbf{A}) dF_{c} and \\int \\operatorname{v_{1}}{(F_{c},\\mathbf{A})} dF_{c} = \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + f and \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + f = \\int (F_{c} + \\mathbf{A}) dF_{c} and \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + f = \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + \\varepsilon_0 and \\frac{F_{c}^{2}}{2} + F_{c} \\mathbf{A} + \\varepsilon_0 = \\int (F_{c} + \\mathbf{A}) dF_{c}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('f', commutative=True)), Integral(Add(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('f', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('F_c', commutative=True), Integer(2))), Mul(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Add(Symbol('F_c', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given t{(\\mathbf{g})} = \\log{(\\mathbf{g})}, then obtain \\frac{t{(\\mathbf{g})}}{\\mathbf{g}} + \\frac{t^{2}{(\\mathbf{g})} \\log{(\\mathbf{g})}}{\\mathbf{g}^{2}} = \\frac{t{(\\mathbf{g})}}{\\mathbf{g}} + \\frac{t{(\\mathbf{g})} \\log{(\\mathbf{g})}^{2}}{\\mathbf{g}^{2}}", "derivation": "t{(\\mathbf{g})} = \\log{(\\mathbf{g})} and \\frac{t{(\\mathbf{g})}}{\\mathbf{g}} = \\frac{\\log{(\\mathbf{g})}}{\\mathbf{g}} and \\frac{t^{2}{(\\mathbf{g})}}{\\mathbf{g}^{2}} = \\frac{t{(\\mathbf{g})} \\log{(\\mathbf{g})}}{\\mathbf{g}^{2}} and \\frac{t^{2}{(\\mathbf{g})} \\log{(\\mathbf{g})}}{\\mathbf{g}^{2}} = \\frac{t{(\\mathbf{g})} \\log{(\\mathbf{g})}^{2}}{\\mathbf{g}^{2}} and \\frac{t{(\\mathbf{g})}}{\\mathbf{g}} + \\frac{t^{2}{(\\mathbf{g})} \\log{(\\mathbf{g})}}{\\mathbf{g}^{2}} = \\frac{t{(\\mathbf{g})}}{\\mathbf{g}} + \\frac{t{(\\mathbf{g})} \\log{(\\mathbf{g})}^{2}}{\\mathbf{g}^{2}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Pow(Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))))"], [["times", 3, "log(Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Pow(Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(log(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Pow(Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)), log(Symbol('\\\\mathbf{g}', commutative=True)))), Add(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-2)), Function('t')(Symbol('\\\\mathbf{g}', commutative=True)), Pow(log(Symbol('\\\\mathbf{g}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\hat{X}{(g)} = \\frac{d}{d g} \\log{(g)}, then derive \\frac{d}{d g} \\hat{X}{(g)} = - \\frac{1}{g^{2}}, then obtain - \\frac{1}{g^{2}} = \\frac{d^{2}}{d g^{2}} \\log{(g)}", "derivation": "\\hat{X}{(g)} = \\frac{d}{d g} \\log{(g)} and \\frac{d}{d g} \\hat{X}{(g)} = \\frac{d^{2}}{d g^{2}} \\log{(g)} and \\frac{d}{d g} \\hat{X}{(g)} = - \\frac{1}{g^{2}} and - \\frac{1}{g^{2}} = \\frac{d^{2}}{d g^{2}} \\log{(g)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('g', commutative=True)), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{X}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-2))), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\hat{H},m_{s})} = m_{s} \\log{(\\hat{H})} and \\phi{(\\hat{H})} = \\log{(\\hat{H})}, then obtain \\sin{(\\operatorname{P_{e}}{(\\hat{H},m_{s})})} = \\sin{(m_{s} \\phi{(\\hat{H})})}", "derivation": "\\operatorname{P_{e}}{(\\hat{H},m_{s})} = m_{s} \\log{(\\hat{H})} and \\sin{(\\operatorname{P_{e}}{(\\hat{H},m_{s})})} = \\sin{(m_{s} \\log{(\\hat{H})})} and \\phi{(\\hat{H})} = \\log{(\\hat{H})} and \\sin{(\\operatorname{P_{e}}{(\\hat{H},m_{s})})} = \\sin{(m_{s} \\phi{(\\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('m_s', commutative=True), log(Symbol('\\\\hat{H}', commutative=True))))"], [["sin", 1], "Equality(sin(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('m_s', commutative=True))), sin(Mul(Symbol('m_s', commutative=True), log(Symbol('\\\\hat{H}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)), log(Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(sin(Function('P_e')(Symbol('\\\\hat{H}', commutative=True), Symbol('m_s', commutative=True))), sin(Mul(Symbol('m_s', commutative=True), Function('\\\\phi')(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(E_{x})} = \\log{(E_{x})}, then obtain E_{x} - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = E_{x} - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\log{(E_{x})}", "derivation": "\\mathbf{J}_M{(E_{x})} = \\log{(E_{x})} and \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and - \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = - \\frac{d}{d E_{x}} \\log{(E_{x})} and - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\log{(E_{x})} and E_{x} - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\mathbf{J}_M{(E_{x})} = E_{x} - (- E_{x} \\log{(E_{x})}^{2} - E_{x}) \\frac{d}{d E_{x}} \\log{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(log(Symbol('E_x', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('E_x', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(log(Symbol('E_x', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('E_x', commutative=True))), Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(log(Symbol('E_x', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('E_x', commutative=True))), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1)))))"], [["add", 4, "Symbol('E_x', commutative=True)"], "Equality(Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(log(Symbol('E_x', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('E_x', commutative=True))), Derivative(Function('\\\\mathbf{J}_M')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))), Add(Symbol('E_x', commutative=True), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('E_x', commutative=True), Pow(log(Symbol('E_x', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('E_x', commutative=True))), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))))"]]}, {"prompt": "Given a{(b)} = \\sin{(b)}, then derive \\int a{(b)} db = \\mathbf{J}_M - \\cos{(b)}, then obtain (\\mathbf{J}_M - \\cos{(b)})^{b} = (\\int \\sin{(b)} db)^{b}", "derivation": "a{(b)} = \\sin{(b)} and \\int a{(b)} db = \\int \\sin{(b)} db and \\int a{(b)} db = \\mathbf{J}_M - \\cos{(b)} and (\\int a{(b)} db)^{b} = (\\int \\sin{(b)} db)^{b} and (\\mathbf{J}_M - \\cos{(b)})^{b} = (\\int \\sin{(b)} db)^{b}", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('b', commutative=True)), sin(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('a')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('b', commutative=True)))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(Function('a')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), cos(Symbol('b', commutative=True)))), Symbol('b', commutative=True)), Pow(Integral(sin(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\delta)} = \\sin{(\\log{(\\delta)})} and s{(\\delta)} = \\int \\sin{(\\log{(\\delta)})} d\\delta, then obtain s^{\\delta}{(\\delta)} \\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta = (\\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta) (\\int \\operatorname{z^{*}}{(\\delta)} d\\delta)^{\\delta}", "derivation": "\\operatorname{z^{*}}{(\\delta)} = \\sin{(\\log{(\\delta)})} and \\int \\operatorname{z^{*}}{(\\delta)} d\\delta = \\int \\sin{(\\log{(\\delta)})} d\\delta and s{(\\delta)} = \\int \\sin{(\\log{(\\delta)})} d\\delta and s^{\\delta}{(\\delta)} = (\\int \\sin{(\\log{(\\delta)})} d\\delta)^{\\delta} and s^{\\delta}{(\\delta)} \\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta = (\\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta) (\\int \\sin{(\\log{(\\delta)})} d\\delta)^{\\delta} and s^{\\delta}{(\\delta)} \\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta = (\\int \\frac{\\sin{(\\log{(\\delta)})}}{\\log{(\\delta)}} d\\delta) (\\int \\operatorname{z^{*}}{(\\delta)} d\\delta)^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\delta', commutative=True)), sin(log(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\delta', commutative=True)), Integral(sin(log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('s')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Integral(sin(log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["times", 4, "Integral(Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), sin(log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Function('s')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), sin(log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integral(Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), sin(log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Pow(Integral(sin(log(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('s')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Integral(Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), sin(log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Integral(Mul(Pow(log(Symbol('\\\\delta', commutative=True)), Integer(-1)), sin(log(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Pow(Integral(Function('z^*')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given H{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)}, then obtain (\\mathbf{J}_f + 1)^{\\mathbf{J}_f} = (\\mathbf{J}_f + \\frac{\\cos{(\\mathbf{J}_f)}}{H{(\\mathbf{J}_f)}})^{\\mathbf{J}_f}", "derivation": "H{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and 1 = \\frac{\\cos{(\\mathbf{J}_f)}}{H{(\\mathbf{J}_f)}} and \\mathbf{J}_f + 1 = \\mathbf{J}_f + \\frac{\\cos{(\\mathbf{J}_f)}}{H{(\\mathbf{J}_f)}} and (\\mathbf{J}_f + 1)^{\\mathbf{J}_f} = (\\mathbf{J}_f + \\frac{\\cos{(\\mathbf{J}_f)}}{H{(\\mathbf{J}_f)}})^{\\mathbf{J}_f}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["divide", 1, "Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Mul(Pow(Function('H')(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))), Symbol('\\\\mathbf{J}_f', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(I)} = \\frac{d}{d I} e^{I}, then derive \\bar{\\h}{(I)} = e^{I}, then obtain 0 = - \\frac{d}{d I} e^{I} + \\frac{d^{2}}{d I^{2}} e^{I}", "derivation": "\\bar{\\h}{(I)} = \\frac{d}{d I} e^{I} and \\bar{\\h}{(I)} = e^{I} and 0 = - \\bar{\\h}{(I)} + \\frac{d}{d I} e^{I} and 0 = - \\bar{\\h}{(I)} + \\frac{d}{d I} \\bar{\\h}{(I)} and \\bar{\\h}{(I)} = \\frac{d}{d I} \\bar{\\h}{(I)} and e^{I} = \\frac{d}{d I} e^{I} and 0 = - e^{I} + \\frac{d}{d I} e^{I} and 0 = - \\frac{d}{d I} e^{I} + \\frac{d^{2}}{d I^{2}} e^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('I', commutative=True)), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hbar')(Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('I', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('I', commutative=True))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('I', commutative=True))), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hbar')(Symbol('I', commutative=True)), Derivative(Function('\\\\hbar')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(exp(Symbol('I', commutative=True)), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))), Derivative(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2)))))"]]}, {"prompt": "Given r{(\\mu_0)} = \\log{(\\mu_0)}, then derive \\frac{d}{d \\mu_0} r{(\\mu_0)} + 1 = 1 + \\frac{1}{\\mu_0}, then obtain \\frac{d}{d \\mu_0} (\\frac{d}{d \\mu_0} r{(\\mu_0)} + 1) = \\frac{d}{d \\mu_0} (\\frac{d}{d \\mu_0} \\log{(\\mu_0)} + 1)", "derivation": "r{(\\mu_0)} = \\log{(\\mu_0)} and \\frac{d}{d \\mu_0} r{(\\mu_0)} = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} and \\frac{d}{d \\mu_0} r{(\\mu_0)} + 1 = \\frac{d}{d \\mu_0} \\log{(\\mu_0)} + 1 and \\frac{d}{d \\mu_0} r{(\\mu_0)} + 1 = 1 + \\frac{1}{\\mu_0} and \\frac{d}{d \\mu_0} \\log{(\\mu_0)} + 1 = 1 + \\frac{1}{\\mu_0} and \\frac{d}{d \\mu_0} (\\frac{d}{d \\mu_0} r{(\\mu_0)} + 1) = \\frac{d}{d \\mu_0} (1 + \\frac{1}{\\mu_0}) and \\frac{d}{d \\mu_0} (\\frac{d}{d \\mu_0} r{(\\mu_0)} + 1) = \\frac{d}{d \\mu_0} (\\frac{d}{d \\mu_0} \\log{(\\mu_0)} + 1)", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('r')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Add(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('r')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('r')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Integer(1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Derivative(Add(Derivative(Function('r')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Add(Derivative(log(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(v)} = e^{v}, then obtain \\frac{e^{- 2 v}}{(- \\mathbf{J}_f{(v)} - 2 e^{v})^{2}} = \\frac{e^{- 2 v}}{9 \\mathbf{J}_f^{2}{(v)}}", "derivation": "\\mathbf{J}_f{(v)} = e^{v} and 2 \\mathbf{J}_f{(v)} = \\mathbf{J}_f{(v)} + e^{v} and - e^{v} = - \\mathbf{J}_f{(v)} and - 2 \\mathbf{J}_f{(v)} - e^{v} = - 3 \\mathbf{J}_f{(v)} and (- 2 \\mathbf{J}_f{(v)} - e^{v}) e^{v} = - 3 \\mathbf{J}_f{(v)} e^{v} and \\frac{e^{- 2 v}}{(- 2 \\mathbf{J}_f{(v)} - e^{v})^{2}} = \\frac{e^{- 2 v}}{9 \\mathbf{J}_f^{2}{(v)}} and \\frac{e^{- 2 v}}{(- \\mathbf{J}_f{(v)} - 2 e^{v})^{2}} = \\frac{e^{- 2 v}}{9 \\mathbf{J}_f^{2}{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))), Add(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))"], [["minus", 1, "Add(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Symbol('v', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Mul(Integer(-1), Integer(3), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))))"], [["times", 4, "exp(Symbol('v', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), exp(Symbol('v', commutative=True))), Mul(Integer(-1), Integer(3), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True))))"], [["power", 5, "Integer(-2)"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))), Mul(Integer(-1), exp(Symbol('v', commutative=True)))), Integer(-2)), exp(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))), Mul(Rational(1, 9), Pow(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Integer(-2)), exp(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True))), Mul(Integer(-1), Integer(2), exp(Symbol('v', commutative=True)))), Integer(-2)), exp(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))), Mul(Rational(1, 9), Pow(Function('\\\\mathbf{J}_f')(Symbol('v', commutative=True)), Integer(-2)), exp(Mul(Integer(-1), Integer(2), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})}, then obtain \\cos{(1)} = \\cos{((\\frac{\\sin{(\\dot{\\mathbf{r}})}}{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}})^{\\dot{\\mathbf{r}}})}", "derivation": "\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})} = \\sin{(\\dot{\\mathbf{r}})} and 1 = \\frac{\\sin{(\\dot{\\mathbf{r}})}}{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}} and 1 = (\\frac{\\sin{(\\dot{\\mathbf{r}})}}{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}})^{\\dot{\\mathbf{r}}} and \\cos{(1)} = \\cos{((\\frac{\\sin{(\\dot{\\mathbf{r}})}}{\\operatorname{t_{2}}{(\\dot{\\mathbf{r}})}})^{\\dot{\\mathbf{r}}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["divide", 1, "Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["power", 2, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["cos", 3], "Equality(cos(Integer(1)), cos(Pow(Mul(Pow(Function('t_2')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Integer(-1)), sin(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given H{(\\omega)} = \\cos{(\\cos{(\\omega)})} and \\mu{(\\omega)} = \\cos{(\\omega)}, then obtain (H^{\\omega}{(\\omega)})^{\\omega} + \\cos{(\\omega)} = (\\cos^{\\omega}{(\\mu{(\\omega)})})^{\\omega} + \\cos{(\\omega)}", "derivation": "H{(\\omega)} = \\cos{(\\cos{(\\omega)})} and H^{\\omega}{(\\omega)} = \\cos^{\\omega}{(\\cos{(\\omega)})} and (H^{\\omega}{(\\omega)})^{\\omega} = (\\cos^{\\omega}{(\\cos{(\\omega)})})^{\\omega} and \\mu{(\\omega)} = \\cos{(\\omega)} and (H^{\\omega}{(\\omega)})^{\\omega} = (\\cos^{\\omega}{(\\mu{(\\omega)})})^{\\omega} and (H^{\\omega}{(\\omega)})^{\\omega} + \\cos{(\\omega)} = (\\cos^{\\omega}{(\\mu{(\\omega)})})^{\\omega} + \\cos{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\omega', commutative=True)), cos(cos(Symbol('\\\\omega', commutative=True))))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('H')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(cos(cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Pow(Function('H')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(cos(cos(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Pow(Function('H')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(Pow(cos(Function('\\\\mu')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["add", 5, "cos(Symbol('\\\\omega', commutative=True))"], "Equality(Add(Pow(Pow(Function('H')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True))), Add(Pow(Pow(cos(Function('\\\\mu')(Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), cos(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given G{(\\chi)} = e^{\\chi}, then obtain \\frac{d}{d \\chi} \\int G{(\\chi)} d\\chi = \\frac{\\partial}{\\partial \\chi} (z^{*} + e^{\\chi})", "derivation": "G{(\\chi)} = e^{\\chi} and \\int G{(\\chi)} d\\chi = \\int e^{\\chi} d\\chi and \\frac{d}{d \\chi} \\int G{(\\chi)} d\\chi = \\frac{d}{d \\chi} \\int e^{\\chi} d\\chi and \\frac{d}{d \\chi} \\int G{(\\chi)} d\\chi = \\frac{\\partial}{\\partial \\chi} (z^{*} + e^{\\chi})", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Integral(Function('G')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('G')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('z^*', commutative=True), exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(v_{x})} = \\log{(v_{x})}, then obtain \\psi^{*}^{v_{x}}{(v_{x})} - \\cos{(\\psi^{*}^{2 v_{x}}{(v_{x})})} = \\log{(v_{x})}^{v_{x}} - \\cos{(\\psi^{*}^{2 v_{x}}{(v_{x})})}", "derivation": "\\psi^{*}{(v_{x})} = \\log{(v_{x})} and \\psi^{*}^{v_{x}}{(v_{x})} = \\log{(v_{x})}^{v_{x}} and \\psi^{*}^{2 v_{x}}{(v_{x})} = \\psi^{*}^{v_{x}}{(v_{x})} \\log{(v_{x})}^{v_{x}} and \\psi^{*}^{v_{x}}{(v_{x})} - \\cos{(\\psi^{*}^{v_{x}}{(v_{x})} \\log{(v_{x})}^{v_{x}})} = \\log{(v_{x})}^{v_{x}} - \\cos{(\\psi^{*}^{v_{x}}{(v_{x})} \\log{(v_{x})}^{v_{x}})} and \\psi^{*}^{v_{x}}{(v_{x})} - \\cos{(\\psi^{*}^{2 v_{x}}{(v_{x})})} = \\log{(v_{x})}^{v_{x}} - \\cos{(\\psi^{*}^{2 v_{x}}{(v_{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), log(Symbol('v_x', commutative=True)))"], [["power", 1, "Symbol('v_x', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))"], [["times", 2, "Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))"], "Equality(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True))), Mul(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], [["minus", 2, "cos(Mul(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True))))"], "Equality(Add(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))))), Add(Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Mul(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True)))))), Add(Pow(log(Symbol('v_x', commutative=True)), Symbol('v_x', commutative=True)), Mul(Integer(-1), cos(Pow(Function('\\\\psi^*')(Symbol('v_x', commutative=True)), Mul(Integer(2), Symbol('v_x', commutative=True)))))))"]]}, {"prompt": "Given T{(l)} = \\cos{(l)}, then obtain \\frac{\\int (l + T{(l)}) dl}{T{(l)}} = \\frac{\\int (l + \\cos{(l)}) dl}{T{(l)}}", "derivation": "T{(l)} = \\cos{(l)} and l + T{(l)} = l + \\cos{(l)} and \\int (l + T{(l)}) dl = \\int (l + \\cos{(l)}) dl and \\frac{\\int (l + T{(l)}) dl}{T{(l)}} = \\frac{\\int (l + \\cos{(l)}) dl}{T{(l)}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["add", 1, "Symbol('l', commutative=True)"], "Equality(Add(Symbol('l', commutative=True), Function('T')(Symbol('l', commutative=True))), Add(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('l', commutative=True), Function('T')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["divide", 3, "Function('T')(Symbol('l', commutative=True))"], "Equality(Mul(Pow(Function('T')(Symbol('l', commutative=True)), Integer(-1)), Integral(Add(Symbol('l', commutative=True), Function('T')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Function('T')(Symbol('l', commutative=True)), Integer(-1)), Integral(Add(Symbol('l', commutative=True), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given Q{(\\delta,\\hat{X})} = - \\hat{X} + e^{\\delta}, then obtain 2 \\log{(Q{(\\delta,\\hat{X})} - e^{\\delta} - 1)} = \\log{(- \\hat{X} - 1)} + \\log{(Q{(\\delta,\\hat{X})} - e^{\\delta} - 1)}", "derivation": "Q{(\\delta,\\hat{X})} = - \\hat{X} + e^{\\delta} and Q{(\\delta,\\hat{X})} - 1 = - \\hat{X} + e^{\\delta} - 1 and Q{(\\delta,\\hat{X})} - e^{\\delta} - 1 = - \\hat{X} - 1 and \\log{(Q{(\\delta,\\hat{X})} - e^{\\delta} - 1)} = \\log{(- \\hat{X} - 1)} and 2 \\log{(Q{(\\delta,\\hat{X})} - e^{\\delta} - 1)} = \\log{(- \\hat{X} - 1)} + \\log{(Q{(\\delta,\\hat{X})} - e^{\\delta} - 1)}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), exp(Symbol('\\\\delta', commutative=True)), Integer(-1)))"], [["minus", 2, "exp(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))"], [["log", 3], "Equality(log(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Integer(-1))), log(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))))"], [["add", 4, "log(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(2), log(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Integer(-1)))), Add(log(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), log(Add(Function('Q')(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\delta', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\phi_2)} = \\sin{(\\phi_2)}, then derive \\frac{d}{d \\phi_2} \\operatorname{v_{z}}{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain \\frac{\\operatorname{v_{z}}{(\\phi_2)} - \\sin{(\\phi_2)}}{\\frac{d}{d \\phi_2} \\sin{(\\phi_2)}} + \\tilde{\\infty} \\cos^{\\phi_2}{(\\phi_2)} = \\tilde{\\infty} \\cos^{\\phi_2}{(\\phi_2)}", "derivation": "\\operatorname{v_{z}}{(\\phi_2)} = \\sin{(\\phi_2)} and \\frac{d}{d \\phi_2} \\operatorname{v_{z}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)} and \\frac{d}{d \\phi_2} \\operatorname{v_{z}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\cos{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)} and \\operatorname{v_{z}}{(\\phi_2)} - \\sin{(\\phi_2)} = 0 and \\frac{\\operatorname{v_{z}}{(\\phi_2)} - \\sin{(\\phi_2)}}{\\cos{(\\phi_2)}} = 0 and \\frac{\\operatorname{v_{z}}{(\\phi_2)} - \\sin{(\\phi_2)}}{\\frac{d}{d \\phi_2} \\sin{(\\phi_2)}} = 0 and \\frac{\\operatorname{v_{z}}{(\\phi_2)} - \\sin{(\\phi_2)}}{\\frac{d}{d \\phi_2} \\sin{(\\phi_2)}} + \\tilde{\\infty} \\cos^{\\phi_2}{(\\phi_2)} = \\tilde{\\infty} \\cos^{\\phi_2}{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), cos(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\phi_2', commutative=True)), Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 1, "sin(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)))), Integer(0))"], [["divide", 5, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Add(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)))), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Add(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)))), Pow(Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["add", 7, "Mul(zoo, Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Add(Function('v_z')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\phi_2', commutative=True)))), Pow(Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Integer(-1))), Mul(zoo, Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))), Mul(zoo, Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})} = \\phi_2 - \\tilde{g}, then derive \\phi_2 (\\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})} - 1) = - 2 \\phi_2, then obtain \\int \\phi_2 (\\frac{\\partial}{\\partial \\tilde{g}} (\\phi_2 - \\tilde{g}) - 1) d\\tilde{g} = \\int - 2 \\phi_2 d\\tilde{g}", "derivation": "\\operatorname{F_{H}}{(\\phi_2,\\tilde{g})} = \\phi_2 - \\tilde{g} and - \\tilde{g} + \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})} = \\phi_2 - 2 \\tilde{g} and \\phi_2 (- \\tilde{g} + \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})}) = \\phi_2 (\\phi_2 - 2 \\tilde{g}) and \\frac{\\partial}{\\partial \\tilde{g}} \\phi_2 (- \\tilde{g} + \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})}) = \\frac{\\partial}{\\partial \\tilde{g}} \\phi_2 (\\phi_2 - 2 \\tilde{g}) and \\phi_2 (\\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{H}}{(\\phi_2,\\tilde{g})} - 1) = - 2 \\phi_2 and \\phi_2 (\\frac{\\partial}{\\partial \\tilde{g}} (\\phi_2 - \\tilde{g}) - 1) = - 2 \\phi_2 and \\int \\phi_2 (\\frac{\\partial}{\\partial \\tilde{g}} (\\phi_2 - \\tilde{g}) - 1) d\\tilde{g} = \\int - 2 \\phi_2 d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))))"], [["minus", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('F_H')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}', commutative=True))))"], [["times", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('F_H')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True)), Function('F_H')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Add(Derivative(Function('F_H')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Symbol('\\\\phi_2', commutative=True), Add(Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 6, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi_2', commutative=True), Add(Derivative(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Mul(Integer(-1), Integer(2), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(r_{0})} = \\sin{(r_{0})}, then obtain \\operatorname{J_{\\varepsilon}}^{r_{0}}{(r_{0})} = \\frac{\\sin{(r_{0})} \\sin^{r_{0}}{(r_{0})}}{\\operatorname{J_{\\varepsilon}}{(r_{0})}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(r_{0})} = \\sin{(r_{0})} and \\operatorname{J_{\\varepsilon}}^{r_{0}}{(r_{0})} = \\sin^{r_{0}}{(r_{0})} and 1 = \\frac{\\sin{(r_{0})}}{\\operatorname{J_{\\varepsilon}}{(r_{0})}} and \\sin^{r_{0}}{(r_{0})} = \\frac{\\sin{(r_{0})} \\sin^{r_{0}}{(r_{0})}}{\\operatorname{J_{\\varepsilon}}{(r_{0})}} and \\operatorname{J_{\\varepsilon}}^{r_{0}}{(r_{0})} = \\frac{\\sin{(r_{0})} \\sin^{r_{0}}{(r_{0})}}{\\operatorname{J_{\\varepsilon}}{(r_{0})}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), sin(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(sin(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["divide", 1, "Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True))))"], [["times", 3, "Pow(sin(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Pow(sin(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True)), Pow(sin(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Mul(Pow(Function('J_{\\\\varepsilon}')(Symbol('r_0', commutative=True)), Integer(-1)), sin(Symbol('r_0', commutative=True)), Pow(sin(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbb{I},F_{g})} = \\frac{F_{g}}{\\mathbb{I}}, then obtain \\frac{1}{\\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})}} = \\frac{\\mathbb{I}}{2 F_{g}}", "derivation": "\\mathbf{J}{(\\mathbb{I},F_{g})} = \\frac{F_{g}}{\\mathbb{I}} and \\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})} = \\frac{2 F_{g}}{\\mathbb{I}} and (\\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})})^{2} = \\frac{4 F_{g}^{2}}{\\mathbb{I}^{2}} and \\frac{1}{\\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})}} = \\frac{2 F_{g}}{\\mathbb{I} (\\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})})^{2}} and \\frac{1}{\\frac{F_{g}}{\\mathbb{I}} + \\mathbf{J}{(\\mathbb{I},F_{g})}} = \\frac{\\mathbb{I}}{2 F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True)), Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["add", 1, "Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Mul(Integer(2), Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))))"], [["power", 2, 2], "Equality(Pow(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Integer(2)), Mul(Integer(4), Pow(Symbol('F_g', commutative=True), Integer(2)), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-2))))"], [["divide", 2, "Pow(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Integer(2))"], "Equality(Pow(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Integer(-1)), Mul(Integer(2), Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Pow(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Symbol('F_g', commutative=True), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1))), Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_g', commutative=True))), Integer(-1)), Mul(Rational(1, 2), Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given b{(v)} = e^{v}, then obtain 0 = - \\frac{z^{*} + e^{v}}{v} + \\frac{\\int b{(v)} dv}{v}", "derivation": "b{(v)} = e^{v} and \\int b{(v)} dv = \\int e^{v} dv and - \\frac{\\int b{(v)} dv}{v} = - \\frac{\\int e^{v} dv}{v} and 1 - \\frac{\\int b{(v)} dv}{v} = 1 - \\frac{\\int e^{v} dv}{v} and 0 = \\frac{\\int b{(v)} dv}{v} - \\frac{\\int e^{v} dv}{v} and 0 = - \\frac{z^{*} + e^{v}}{v} + \\frac{\\int b{(v)} dv}{v}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('v', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"], [["minus", 4, "Add(Integer(1), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], "Equality(Integer(0), Add(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('z^*', commutative=True), exp(Symbol('v', commutative=True)))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Function('b')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(A_{1})} = \\cos{(e^{A_{1}})} and \\dot{\\mathbf{r}}{(A_{1})} = \\mathbf{J}_P{(A_{1})} - e^{A_{1}}, then obtain \\frac{d}{d A_{1}} (\\dot{\\mathbf{r}}{(A_{1})} + \\cos{(e^{A_{1}})} - 1) = \\frac{d}{d A_{1}} (- e^{A_{1}} + 2 \\cos{(e^{A_{1}})} - 1)", "derivation": "\\mathbf{J}_P{(A_{1})} = \\cos{(e^{A_{1}})} and \\mathbf{J}_P{(A_{1})} - e^{A_{1}} = - e^{A_{1}} + \\cos{(e^{A_{1}})} and \\dot{\\mathbf{r}}{(A_{1})} = \\mathbf{J}_P{(A_{1})} - e^{A_{1}} and \\dot{\\mathbf{r}}{(A_{1})} = - e^{A_{1}} + \\cos{(e^{A_{1}})} and \\dot{\\mathbf{r}}{(A_{1})} + \\cos{(e^{A_{1}})} = - e^{A_{1}} + 2 \\cos{(e^{A_{1}})} and \\dot{\\mathbf{r}}{(A_{1})} + \\cos{(e^{A_{1}})} - 1 = - e^{A_{1}} + 2 \\cos{(e^{A_{1}})} - 1 and \\frac{d}{d A_{1}} (\\dot{\\mathbf{r}}{(A_{1})} + \\cos{(e^{A_{1}})} - 1) = \\frac{d}{d A_{1}} (- e^{A_{1}} + 2 \\cos{(e^{A_{1}})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))))"], [["minus", 1, "exp(Symbol('A_1', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), cos(exp(Symbol('A_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_1', commutative=True)), Add(Function('\\\\mathbf{J}_P')(Symbol('A_1', commutative=True)), Mul(Integer(-1), exp(Symbol('A_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), cos(exp(Symbol('A_1', commutative=True)))))"], [["add", 4, "cos(exp(Symbol('A_1', commutative=True)))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Mul(Integer(2), cos(exp(Symbol('A_1', commutative=True))))))"], [["add", 5, "Integer(-1)"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Mul(Integer(2), cos(exp(Symbol('A_1', commutative=True)))), Integer(-1)))"], [["differentiate", 6, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('A_1', commutative=True)), cos(exp(Symbol('A_1', commutative=True))), Integer(-1)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('A_1', commutative=True))), Mul(Integer(2), cos(exp(Symbol('A_1', commutative=True)))), Integer(-1)), Tuple(Symbol('A_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(C_{1},M)} = \\frac{e^{M}}{C_{1}}, then obtain - C_{1} \\rho_{b}{(C_{1},M)} e^{- M} + C_{1} e^{- M} \\frac{\\partial}{\\partial M} \\rho_{b}{(C_{1},M)} = 0", "derivation": "\\rho_{b}{(C_{1},M)} = \\frac{e^{M}}{C_{1}} and C_{1} \\rho_{b}{(C_{1},M)} e^{- M} = 1 and \\frac{\\partial}{\\partial M} C_{1} \\rho_{b}{(C_{1},M)} e^{- M} = \\frac{d}{d M} 1 and - C_{1} \\rho_{b}{(C_{1},M)} e^{- M} + C_{1} e^{- M} \\frac{\\partial}{\\partial M} \\rho_{b}{(C_{1},M)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), exp(Symbol('M', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), exp(Symbol('M', commutative=True)))"], "Equality(Mul(Symbol('C_1', commutative=True), Function('\\\\rho_b')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Integer(1))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Symbol('C_1', commutative=True), Function('\\\\rho_b')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True), Function('\\\\rho_b')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), exp(Mul(Integer(-1), Symbol('M', commutative=True)))), Mul(Symbol('C_1', commutative=True), exp(Mul(Integer(-1), Symbol('M', commutative=True))), Derivative(Function('\\\\rho_b')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given I{(C,i)} = - C + i, then obtain (\\int \\frac{\\partial}{\\partial C} \\frac{I{(C,i)}}{- C + i} dC)^{i} = (\\int 0 dC)^{i}", "derivation": "I{(C,i)} = - C + i and \\frac{I{(C,i)}}{- C + i} = 1 and \\frac{\\partial}{\\partial C} \\frac{I{(C,i)}}{- C + i} = \\frac{d}{d C} 1 and \\int \\frac{\\partial}{\\partial C} \\frac{I{(C,i)}}{- C + i} dC = \\int \\frac{d}{d C} 1 dC and (\\int \\frac{\\partial}{\\partial C} \\frac{I{(C,i)}}{- C + i} dC)^{i} = (\\int \\frac{d}{d C} 1 dC)^{i} and (\\int \\frac{\\partial}{\\partial C} \\frac{I{(C,i)}}{- C + i} dC)^{i} = (\\int 0 dC)^{i}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Integer(-1)), Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True))), Integer(1))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Integer(-1)), Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Integer(-1)), Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Integral(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Integer(-1)), Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(Derivative(Integer(1), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Symbol('i', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Integral(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('i', commutative=True)), Integer(-1)), Function('I')(Symbol('C', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('C', commutative=True))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})} = \\mathbb{I} - \\phi_2, then obtain \\int \\cos{(\\frac{\\partial}{\\partial \\phi_2} \\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})})} d\\mathbb{I} = \\int \\cos{(\\frac{\\partial}{\\partial \\phi_2} (\\mathbb{I} - \\phi_2))} d\\mathbb{I}", "derivation": "\\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})} = \\mathbb{I} - \\phi_2 and \\frac{\\partial}{\\partial \\phi_2} \\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})} = \\frac{\\partial}{\\partial \\phi_2} (\\mathbb{I} - \\phi_2) and \\cos{(\\frac{\\partial}{\\partial \\phi_2} \\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})})} = \\cos{(\\frac{\\partial}{\\partial \\phi_2} (\\mathbb{I} - \\phi_2))} and \\int \\cos{(\\frac{\\partial}{\\partial \\phi_2} \\operatorname{A_{x}}{(\\phi_2,\\mathbb{I})})} d\\mathbb{I} = \\int \\cos{(\\frac{\\partial}{\\partial \\phi_2} (\\mathbb{I} - \\phi_2))} d\\mathbb{I}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('A_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), cos(Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(cos(Derivative(Function('A_x')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(cos(Derivative(Add(Symbol('\\\\mathbb{I}', commutative=True), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\phi,x^\\prime)} = \\phi x^\\prime, then obtain - \\phi x^\\prime (\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)}) + \\sin{(\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)})} = - \\phi x^\\prime (\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)}) + \\sin{(2 \\phi x^\\prime)}", "derivation": "\\Psi_{nl}{(\\phi,x^\\prime)} = \\phi x^\\prime and \\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)} = 2 \\phi x^\\prime and \\phi x^\\prime (\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)}) = 2 \\phi^{2} (x^\\prime)^{2} and \\sin{(\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)})} = \\sin{(2 \\phi x^\\prime)} and - 2 \\phi^{2} (x^\\prime)^{2} + \\sin{(\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)})} = - 2 \\phi^{2} (x^\\prime)^{2} + \\sin{(2 \\phi x^\\prime)} and - \\phi x^\\prime (\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)}) + \\sin{(\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)})} = - \\phi x^\\prime (\\phi x^\\prime + \\Psi_{nl}{(\\phi,x^\\prime)}) + \\sin{(2 \\phi x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["times", 2, "Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Mul(Integer(2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))))"], [["sin", 2], "Equality(sin(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))), sin(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))), sin(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Add(Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)), Pow(Symbol('x^\\\\prime', commutative=True), Integer(2))), sin(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))), sin(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True), Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))), sin(Mul(Integer(2), Symbol('\\\\phi', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\mathbf{J}_M,E_{n})} = \\frac{\\mathbf{J}_M}{E_{n}}, then obtain (\\operatorname{A_{1}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{n})} + \\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} = ((\\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} + \\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M}", "derivation": "\\operatorname{A_{1}}{(\\mathbf{J}_M,E_{n})} = \\frac{\\mathbf{J}_M}{E_{n}} and \\operatorname{A_{1}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{n})} = (\\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} and \\operatorname{A_{1}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{n})} + \\frac{\\mathbf{J}_M}{E_{n}} = (\\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} + \\frac{\\mathbf{J}_M}{E_{n}} and (\\operatorname{A_{1}}^{\\mathbf{J}_M}{(\\mathbf{J}_M,E_{n})} + \\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} = ((\\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M} + \\frac{\\mathbf{J}_M}{E_{n}})^{\\mathbf{J}_M}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["add", 2, "Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))"], "Equality(Add(Pow(Function('A_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Add(Pow(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Add(Pow(Function('A_1')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('E_n', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Add(Pow(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True))), Symbol('\\\\mathbf{J}_M', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\mathbf{E}{(\\varphi^*)} = 2 \\cos{(\\varphi^*)}, then obtain \\frac{1}{\\mathbf{F}{(\\varphi^*)} + \\cos{(\\varphi^*)}} = \\frac{1}{\\mathbf{E}{(\\varphi^*)}}", "derivation": "\\mathbf{F}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\mathbf{F}{(\\varphi^*)} + \\cos{(\\varphi^*)} = 2 \\cos{(\\varphi^*)} and \\mathbf{E}{(\\varphi^*)} = 2 \\cos{(\\varphi^*)} and 1 = \\frac{2 \\cos{(\\varphi^*)}}{\\mathbf{E}{(\\varphi^*)}} and 1 = \\frac{\\mathbf{F}{(\\varphi^*)} + \\cos{(\\varphi^*)}}{\\mathbf{E}{(\\varphi^*)}} and \\frac{1}{\\mathbf{F}{(\\varphi^*)} + \\cos{(\\varphi^*)}} = \\frac{1}{\\mathbf{E}{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Function('\\\\mathbf{F}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\varphi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\varphi^*', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 3, "Function('\\\\mathbf{E}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(1), Mul(Integer(2), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(1), Mul(Add(Function('\\\\mathbf{F}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["divide", 5, "Add(Function('\\\\mathbf{F}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Pow(Add(Function('\\\\mathbf{F}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Pow(Function('\\\\mathbf{E}')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given v{(n_{1},\\mathbf{F})} = \\cos{(\\frac{\\mathbf{F}}{n_{1}})}, then obtain 2 v{(n_{1},\\mathbf{F})} - \\frac{v{(n_{1},\\mathbf{F})}}{n_{1}} = v{(n_{1},\\mathbf{F})} + \\cos{(\\frac{\\mathbf{F}}{n_{1}})} - \\frac{v{(n_{1},\\mathbf{F})}}{n_{1}}", "derivation": "v{(n_{1},\\mathbf{F})} = \\cos{(\\frac{\\mathbf{F}}{n_{1}})} and 2 v{(n_{1},\\mathbf{F})} = v{(n_{1},\\mathbf{F})} + \\cos{(\\frac{\\mathbf{F}}{n_{1}})} and \\frac{v{(n_{1},\\mathbf{F})}}{n_{1}} = \\frac{\\cos{(\\frac{\\mathbf{F}}{n_{1}})}}{n_{1}} and 2 v{(n_{1},\\mathbf{F})} - \\frac{\\cos{(\\frac{\\mathbf{F}}{n_{1}})}}{n_{1}} = v{(n_{1},\\mathbf{F})} + \\cos{(\\frac{\\mathbf{F}}{n_{1}})} - \\frac{\\cos{(\\frac{\\mathbf{F}}{n_{1}})}}{n_{1}} and 2 v{(n_{1},\\mathbf{F})} - \\frac{v{(n_{1},\\mathbf{F})}}{n_{1}} = v{(n_{1},\\mathbf{F})} + \\cos{(\\frac{\\mathbf{F}}{n_{1}})} - \\frac{v{(n_{1},\\mathbf{F})}}{n_{1}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"], [["add", 1, "Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Integer(2), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))))"], [["divide", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1))))))"], [["minus", 2, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))"], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))), Add(Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{F}', commutative=True), Pow(Symbol('n_1', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Integer(-1)), Function('v')(Symbol('n_1', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} = A_{1} - \\mathbf{H}, then obtain - \\frac{\\log{(\\frac{\\partial}{\\partial A_{1}} \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} - 1)}}{\\mathbf{H}} = - \\frac{\\log{(\\frac{\\partial}{\\partial A_{1}} (A_{1} - \\mathbf{H}) - 1)}}{\\mathbf{H}}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{H},A_{1})} = A_{1} - \\mathbf{H} and \\frac{\\partial}{\\partial A_{1}} \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} = \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\mathbf{H}) and \\frac{\\partial}{\\partial A_{1}} \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} - 1 = \\frac{\\partial}{\\partial A_{1}} (A_{1} - \\mathbf{H}) - 1 and \\log{(\\frac{\\partial}{\\partial A_{1}} \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} - 1)} = \\log{(\\frac{\\partial}{\\partial A_{1}} (A_{1} - \\mathbf{H}) - 1)} and - \\frac{\\log{(\\frac{\\partial}{\\partial A_{1}} \\operatorname{v_{x}}{(\\mathbf{H},A_{1})} - 1)}}{\\mathbf{H}} = - \\frac{\\log{(\\frac{\\partial}{\\partial A_{1}} (A_{1} - \\mathbf{H}) - 1)}}{\\mathbf{H}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)))"], [["log", 3], "Equality(log(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1))), log(Add(Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), log(Add(Derivative(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), log(Add(Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\nabla)} = e^{\\nabla}, then derive \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = v_{2} + e^{\\nabla}, then obtain - \\operatorname{L_{\\varepsilon}}{(\\nabla)} \\log{(c_{0})} + \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = v_{2} - \\operatorname{L_{\\varepsilon}}{(\\nabla)} \\log{(c_{0})} + \\operatorname{L_{\\varepsilon}}{(\\nabla)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\nabla)} = e^{\\nabla} and \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = \\int e^{\\nabla} d\\nabla and \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = v_{2} + e^{\\nabla} and \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = v_{2} + \\operatorname{L_{\\varepsilon}}{(\\nabla)} and - \\operatorname{L_{\\varepsilon}}{(\\nabla)} \\log{(c_{0})} + \\int \\operatorname{L_{\\varepsilon}}{(\\nabla)} d\\nabla = v_{2} - \\operatorname{L_{\\varepsilon}}{(\\nabla)} \\log{(c_{0})} + \\operatorname{L_{\\varepsilon}}{(\\nabla)}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), exp(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(exp(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('v_2', commutative=True), exp(Symbol('\\\\nabla', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('v_2', commutative=True), Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True))))"], [["minus", 4, "Mul(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True))), Integral(Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True))), Function('L_{\\\\varepsilon}')(Symbol('\\\\nabla', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(G,H)} = \\log{(G^{H})} and \\Omega{(G,H)} = - G - H + \\hat{\\mathbf{r}}{(G,H)}, then obtain \\frac{\\partial}{\\partial G} \\frac{\\Omega{(G,H)}}{- G - H + \\log{(G^{H})}} = \\frac{d}{d G} 1", "derivation": "\\hat{\\mathbf{r}}{(G,H)} = \\log{(G^{H})} and - G + \\hat{\\mathbf{r}}{(G,H)} = - G + \\log{(G^{H})} and - G - H + \\hat{\\mathbf{r}}{(G,H)} = - G - H + \\log{(G^{H})} and \\Omega{(G,H)} = - G - H + \\hat{\\mathbf{r}}{(G,H)} and \\Omega{(G,H)} = - G - H + \\log{(G^{H})} and \\frac{\\Omega{(G,H)}}{- G - H + \\log{(G^{H})}} = 1 and \\frac{\\partial}{\\partial G} \\frac{\\Omega{(G,H)}}{- G - H + \\log{(G^{H})}} = \\frac{d}{d G} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True)))))"], [["minus", 2, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Omega')(Symbol('G', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('G', commutative=True), Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\Omega')(Symbol('G', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True)))))"], [["divide", 5, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True)))), Integer(-1)), Function('\\\\Omega')(Symbol('G', commutative=True), Symbol('H', commutative=True))), Integer(1))"], [["differentiate", 6, "Symbol('G', commutative=True)"], "Equality(Derivative(Mul(Pow(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('G', commutative=True), Symbol('H', commutative=True)))), Integer(-1)), Function('\\\\Omega')(Symbol('G', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(\\hat{x},I)} = \\sin{(I \\hat{x})}, then obtain \\frac{(\\frac{\\partial}{\\partial I} M{(\\hat{x},I)})^{2}}{\\sin{(I \\hat{x})}} = \\frac{\\frac{\\partial}{\\partial I} M{(\\hat{x},I)} \\frac{\\partial}{\\partial I} \\sin{(I \\hat{x})}}{\\sin{(I \\hat{x})}}", "derivation": "M{(\\hat{x},I)} = \\sin{(I \\hat{x})} and \\frac{\\partial}{\\partial I} M{(\\hat{x},I)} = \\frac{\\partial}{\\partial I} \\sin{(I \\hat{x})} and (\\frac{\\partial}{\\partial I} M{(\\hat{x},I)})^{2} = \\frac{\\partial}{\\partial I} M{(\\hat{x},I)} \\frac{\\partial}{\\partial I} \\sin{(I \\hat{x})} and \\frac{(\\frac{\\partial}{\\partial I} M{(\\hat{x},I)})^{2}}{\\sin{(I \\hat{x})}} = \\frac{\\frac{\\partial}{\\partial I} M{(\\hat{x},I)} \\frac{\\partial}{\\partial I} \\sin{(I \\hat{x})}}{\\sin{(I \\hat{x})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["divide", 3, "sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Pow(sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Integer(-1)), Pow(Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(2))), Mul(Pow(sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Integer(-1)), Derivative(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('I', commutative=True), Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given J{(A)} = \\cos{(A)} and \\operatorname{E_{n}}{(A)} = \\int J{(A)} dA, then derive \\operatorname{E_{n}}{(A)} = g_{\\varepsilon} + \\sin{(A)}, then obtain (A + \\int \\cos{(A)} dA)^{g_{\\varepsilon}} = (A + g_{\\varepsilon} + \\sin{(A)})^{g_{\\varepsilon}}", "derivation": "J{(A)} = \\cos{(A)} and \\int J{(A)} dA = \\int \\cos{(A)} dA and \\operatorname{E_{n}}{(A)} = \\int J{(A)} dA and \\operatorname{E_{n}}{(A)} = \\int \\cos{(A)} dA and \\operatorname{E_{n}}{(A)} = g_{\\varepsilon} + \\sin{(A)} and \\int \\cos{(A)} dA = g_{\\varepsilon} + \\sin{(A)} and A + \\int \\cos{(A)} dA = A + g_{\\varepsilon} + \\sin{(A)} and (A + \\int \\cos{(A)} dA)^{g_{\\varepsilon}} = (A + g_{\\varepsilon} + \\sin{(A)})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('J')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('E_n')(Symbol('A', commutative=True)), Integral(Function('J')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_n')(Symbol('A', commutative=True)), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Function('E_n')(Symbol('A', commutative=True)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('A', commutative=True))))"], [["add", 6, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Add(Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('A', commutative=True))))"], [["power", 7, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Symbol('A', commutative=True), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('A', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('A', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{H}{(A_{z},\\hbar,\\varepsilon)} = \\hbar^{A_{z}} + \\varepsilon, then derive \\frac{\\hbar + k}{\\int \\frac{\\hbar^{A_{z}} + \\varepsilon}{\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)}} d\\hbar} = 1, then obtain \\frac{\\hbar + k}{\\int 1 d\\hbar} = 1", "derivation": "\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)} = \\hbar^{A_{z}} + \\varepsilon and 1 = \\frac{\\hbar^{A_{z}} + \\varepsilon}{\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)}} and \\int 1 d\\hbar = \\int \\frac{\\hbar^{A_{z}} + \\varepsilon}{\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)}} d\\hbar and \\frac{\\int 1 d\\hbar}{\\int \\frac{\\hbar^{A_{z}} + \\varepsilon}{\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)}} d\\hbar} = 1 and \\frac{\\hbar + k}{\\int \\frac{\\hbar^{A_{z}} + \\varepsilon}{\\mathbf{H}{(A_{z},\\hbar,\\varepsilon)}} d\\hbar} = 1 and \\frac{\\hbar + k}{\\int 1 d\\hbar} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Integer(1), Mul(Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Mul(Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["divide", 3, "Integral(Mul(Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True)))"], "Equality(Mul(Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))), Pow(Integral(Mul(Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 4], "Equality(Mul(Add(Symbol('\\\\hbar', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Mul(Add(Pow(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Function('\\\\mathbf{H}')(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\hbar', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Symbol('\\\\hbar', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\hbar', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)} = \\cos{(\\hat{H}_{\\lambda} \\phi_1)} and g{(\\hat{H}_{\\lambda},\\phi_1)} = - \\phi_1 + \\cos{(\\hat{H}_{\\lambda} \\phi_1)}, then obtain \\frac{g{(\\hat{H}_{\\lambda},\\phi_1)}}{\\phi_1} = \\frac{- \\phi_1 + \\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)}}{\\phi_1}", "derivation": "\\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)} = \\cos{(\\hat{H}_{\\lambda} \\phi_1)} and - \\phi_1 + \\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)} = - \\phi_1 + \\cos{(\\hat{H}_{\\lambda} \\phi_1)} and g{(\\hat{H}_{\\lambda},\\phi_1)} = - \\phi_1 + \\cos{(\\hat{H}_{\\lambda} \\phi_1)} and g{(\\hat{H}_{\\lambda},\\phi_1)} = - \\phi_1 + \\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)} and \\frac{g{(\\hat{H}_{\\lambda},\\phi_1)}}{\\phi_1} = \\frac{- \\phi_1 + \\operatorname{F_{x}}{(\\hat{H}_{\\lambda},\\phi_1)}}{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), cos(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('F_x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), cos(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('F_x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True))))"], [["divide", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Function('g')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True)), Function('F_x')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given H{(t_{2},\\phi_1)} = (e^{\\phi_1})^{t_{2}}, then obtain - t_{2} + \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + H{(t_{2},\\phi_1)}}{H{(t_{2},\\phi_1)}} = - t_{2} + \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + (e^{\\phi_1})^{t_{2}}}{H{(t_{2},\\phi_1)}}", "derivation": "H{(t_{2},\\phi_1)} = (e^{\\phi_1})^{t_{2}} and \\phi_1 + H{(t_{2},\\phi_1)} = \\phi_1 + (e^{\\phi_1})^{t_{2}} and \\frac{\\phi_1 + H{(t_{2},\\phi_1)}}{H{(t_{2},\\phi_1)}} = \\frac{\\phi_1 + (e^{\\phi_1})^{t_{2}}}{H{(t_{2},\\phi_1)}} and \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + H{(t_{2},\\phi_1)}}{H{(t_{2},\\phi_1)}} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + (e^{\\phi_1})^{t_{2}}}{H{(t_{2},\\phi_1)}} and - t_{2} + \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + H{(t_{2},\\phi_1)}}{H{(t_{2},\\phi_1)}} = - t_{2} + \\frac{\\partial}{\\partial t_{2}} \\frac{\\phi_1 + (e^{\\phi_1})^{t_{2}}}{H{(t_{2},\\phi_1)}}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True))))"], [["divide", 2, "Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\phi_1', commutative=True), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["minus", 4, "Symbol('t_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('t_2', commutative=True)), Derivative(Mul(Add(Symbol('\\\\phi_1', commutative=True), Pow(exp(Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True))), Pow(Function('H')(Symbol('t_2', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1))), Tuple(Symbol('t_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\chi)} = \\sin{(\\sin{(\\chi)})}, then obtain - 2 \\sin{(\\sin{(\\chi)})} + (\\int \\operatorname{J_{\\varepsilon}}{(\\chi)} d\\chi)^{\\chi} = - 2 \\sin{(\\sin{(\\chi)})} + (\\int \\sin{(\\sin{(\\chi)})} d\\chi)^{\\chi}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and \\int \\operatorname{J_{\\varepsilon}}{(\\chi)} d\\chi = \\int \\sin{(\\sin{(\\chi)})} d\\chi and (\\int \\operatorname{J_{\\varepsilon}}{(\\chi)} d\\chi)^{\\chi} = (\\int \\sin{(\\sin{(\\chi)})} d\\chi)^{\\chi} and - 2 \\sin{(\\sin{(\\chi)})} + (\\int \\operatorname{J_{\\varepsilon}}{(\\chi)} d\\chi)^{\\chi} = - 2 \\sin{(\\sin{(\\chi)})} + (\\int \\sin{(\\sin{(\\chi)})} d\\chi)^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\chi', commutative=True)), sin(sin(Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integral(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["minus", 3, "Mul(Integer(2), sin(sin(Symbol('\\\\chi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), sin(sin(Symbol('\\\\chi', commutative=True)))), Pow(Integral(Function('J_{\\\\varepsilon}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Integer(2), sin(sin(Symbol('\\\\chi', commutative=True)))), Pow(Integral(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(r_{0})} = e^{r_{0}}, then obtain \\operatorname{y^{\\prime}}^{9}{(r_{0})} = \\operatorname{y^{\\prime}}^{6}{(r_{0})} e^{3 r_{0}}", "derivation": "\\operatorname{y^{\\prime}}{(r_{0})} = e^{r_{0}} and \\operatorname{y^{\\prime}}^{2}{(r_{0})} = \\operatorname{y^{\\prime}}{(r_{0})} e^{r_{0}} and \\operatorname{y^{\\prime}}^{3}{(r_{0})} = \\operatorname{y^{\\prime}}^{2}{(r_{0})} e^{r_{0}} and \\operatorname{y^{\\prime}}^{9}{(r_{0})} = \\operatorname{y^{\\prime}}^{6}{(r_{0})} e^{3 r_{0}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], [["times", 1, "Function('y^{\\\\prime}')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), Integer(2)), Mul(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))))"], [["times", 2, "Function('y^{\\\\prime}')(Symbol('r_0', commutative=True))"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), Integer(3)), Mul(Pow(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), Integer(2)), exp(Symbol('r_0', commutative=True))))"], [["power", 3, 3], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), Integer(9)), Mul(Pow(Function('y^{\\\\prime}')(Symbol('r_0', commutative=True)), Integer(6)), exp(Mul(Integer(3), Symbol('r_0', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\hat{p})} = \\cos{(e^{\\hat{p}})}, then obtain \\hat{p} + 3 \\omega{(\\hat{p})} + \\cos{(e^{\\hat{p}})} = \\hat{p} + \\omega{(\\hat{p})} + 3 \\cos{(e^{\\hat{p}})}", "derivation": "\\omega{(\\hat{p})} = \\cos{(e^{\\hat{p}})} and \\omega{(\\hat{p})} + \\cos{(e^{\\hat{p}})} = 2 \\cos{(e^{\\hat{p}})} and 2 \\omega{(\\hat{p})} + 2 \\cos{(e^{\\hat{p}})} = \\omega{(\\hat{p})} + 3 \\cos{(e^{\\hat{p}})} and \\hat{p} + 2 \\omega{(\\hat{p})} + 2 \\cos{(e^{\\hat{p}})} = \\hat{p} + \\omega{(\\hat{p})} + 3 \\cos{(e^{\\hat{p}})} and \\hat{p} + 3 \\omega{(\\hat{p})} + \\cos{(e^{\\hat{p}})} = \\hat{p} + \\omega{(\\hat{p})} + 3 \\cos{(e^{\\hat{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), cos(exp(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 1, "cos(exp(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), cos(exp(Symbol('\\\\hat{p}', commutative=True)))), Mul(Integer(2), cos(exp(Symbol('\\\\hat{p}', commutative=True)))))"], [["add", 2, "Add(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), cos(exp(Symbol('\\\\hat{p}', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), cos(exp(Symbol('\\\\hat{p}', commutative=True))))), Add(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(3), cos(exp(Symbol('\\\\hat{p}', commutative=True))))))"], [["add", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True))), Mul(Integer(2), cos(exp(Symbol('\\\\hat{p}', commutative=True))))), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(3), cos(exp(Symbol('\\\\hat{p}', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Integer(3), Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True))), cos(exp(Symbol('\\\\hat{p}', commutative=True)))), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(3), cos(exp(Symbol('\\\\hat{p}', commutative=True))))))"]]}, {"prompt": "Given x{(\\hat{x}_0,v_{y})} = \\hat{x}_0 v_{y} and \\mathbf{J}_f{(\\hat{x}_0,v_{y})} = \\hat{x}_0 v_{y}, then derive \\frac{\\partial}{\\partial \\hat{x}_0} x{(\\hat{x}_0,v_{y})} = v_{y}, then obtain \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{x}_0 v_{y} = v_{y}", "derivation": "x{(\\hat{x}_0,v_{y})} = \\hat{x}_0 v_{y} and \\mathbf{J}_f{(\\hat{x}_0,v_{y})} = \\hat{x}_0 v_{y} and x{(\\hat{x}_0,v_{y})} = \\mathbf{J}_f{(\\hat{x}_0,v_{y})} and \\frac{\\partial}{\\partial \\hat{x}_0} x{(\\hat{x}_0,v_{y})} = \\frac{\\partial}{\\partial \\hat{x}_0} \\mathbf{J}_f{(\\hat{x}_0,v_{y})} and \\frac{\\partial}{\\partial \\hat{x}_0} x{(\\hat{x}_0,v_{y})} = \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{x}_0 v_{y} and \\frac{\\partial}{\\partial \\hat{x}_0} x{(\\hat{x}_0,v_{y})} = v_{y} and \\frac{\\partial}{\\partial \\hat{x}_0} \\hat{x}_0 v_{y} = v_{y}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('x')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Symbol('v_y', commutative=True))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Symbol('v_y', commutative=True))"]]}, {"prompt": "Given \\theta_{1}{(V_{\\mathbf{B}})} = \\int \\log{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}, then derive \\theta_{1}{(V_{\\mathbf{B}})} - \\log{(V_{\\mathbf{B}})} = G + V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} - V_{\\mathbf{B}} - \\log{(V_{\\mathbf{B}})}, then obtain G + V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} - V_{\\mathbf{B}} = \\int \\log{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}", "derivation": "\\theta_{1}{(V_{\\mathbf{B}})} = \\int \\log{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and \\theta_{1}{(V_{\\mathbf{B}})} - \\log{(V_{\\mathbf{B}})} = - \\log{(V_{\\mathbf{B}})} + \\int \\log{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and \\theta_{1}{(V_{\\mathbf{B}})} - \\log{(V_{\\mathbf{B}})} = G + V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} - V_{\\mathbf{B}} - \\log{(V_{\\mathbf{B}})} and \\theta_{1}{(V_{\\mathbf{B}})} = G + V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} - V_{\\mathbf{B}} and G + V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} - V_{\\mathbf{B}} = \\int \\log{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integral(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 1, "log(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Function('\\\\theta_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Function('\\\\theta_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Integer(-1), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], [["add", 3, "log(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Function('\\\\theta_1')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('G', commutative=True), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Add(Symbol('G', commutative=True), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given U{(t_{2},G)} = G t_{2} and G{(V_{\\mathbf{B}},n,v)} = \\frac{n^{V_{\\mathbf{B}}}}{v}, then obtain \\int (G{(V_{\\mathbf{B}},n,v)} + U{(t_{2},G)})^{t_{2}} dt_{2} = \\int (\\frac{n^{V_{\\mathbf{B}}}}{v} + U{(t_{2},G)})^{t_{2}} dt_{2}", "derivation": "U{(t_{2},G)} = G t_{2} and G{(V_{\\mathbf{B}},n,v)} = \\frac{n^{V_{\\mathbf{B}}}}{v} and G t_{2} + G{(V_{\\mathbf{B}},n,v)} = G t_{2} + \\frac{n^{V_{\\mathbf{B}}}}{v} and G{(V_{\\mathbf{B}},n,v)} + U{(t_{2},G)} = \\frac{n^{V_{\\mathbf{B}}}}{v} + U{(t_{2},G)} and (G{(V_{\\mathbf{B}},n,v)} + U{(t_{2},G)})^{t_{2}} = (\\frac{n^{V_{\\mathbf{B}}}}{v} + U{(t_{2},G)})^{t_{2}} and \\int (G{(V_{\\mathbf{B}},n,v)} + U{(t_{2},G)})^{t_{2}} dt_{2} = \\int (\\frac{n^{V_{\\mathbf{B}}}}{v} + U{(t_{2},G)})^{t_{2}} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('t_2', commutative=True)))"], ["get_premise", "Equality(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1))))"], [["add", 2, "Mul(Symbol('G', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Symbol('G', commutative=True), Symbol('t_2', commutative=True)), Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n', commutative=True), Symbol('v', commutative=True))), Add(Mul(Symbol('G', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))), Add(Mul(Pow(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1))), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))))"], [["power", 4, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))), Symbol('t_2', commutative=True)), Pow(Add(Mul(Pow(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1))), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))), Symbol('t_2', commutative=True)))"], [["integrate", 5, "Symbol('t_2', commutative=True)"], "Equality(Integral(Pow(Add(Function('G')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('n', commutative=True), Symbol('v', commutative=True)), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Pow(Add(Mul(Pow(Symbol('n', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Symbol('v', commutative=True), Integer(-1))), Function('U')(Symbol('t_2', commutative=True), Symbol('G', commutative=True))), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\phi{(u,\\mathbf{F})} = \\sin{(\\mathbf{F} - u)}, then obtain (\\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\phi{(u,\\mathbf{F})}}{\\phi{(u,\\mathbf{F})}})^{u} = (\\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\sin{(\\mathbf{F} - u)}}{\\phi{(u,\\mathbf{F})}})^{u}", "derivation": "\\phi{(u,\\mathbf{F})} = \\sin{(\\mathbf{F} - u)} and - u + \\phi{(u,\\mathbf{F})} = - u + \\sin{(\\mathbf{F} - u)} and \\frac{- u + \\phi{(u,\\mathbf{F})}}{\\phi{(u,\\mathbf{F})}} = \\frac{- u + \\sin{(\\mathbf{F} - u)}}{\\phi{(u,\\mathbf{F})}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\phi{(u,\\mathbf{F})}}{\\phi{(u,\\mathbf{F})}} = \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\sin{(\\mathbf{F} - u)}}{\\phi{(u,\\mathbf{F})}} and (\\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\phi{(u,\\mathbf{F})}}{\\phi{(u,\\mathbf{F})}})^{u} = (\\frac{\\partial}{\\partial \\mathbf{F}} \\frac{- u + \\sin{(\\mathbf{F} - u)}}{\\phi{(u,\\mathbf{F})}})^{u}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True)))))"], [["add", 1, "Mul(Integer(-1), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))))"], [["divide", 2, "Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('u', commutative=True)"], "Equality(Pow(Derivative(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('u', commutative=True)), Pow(Derivative(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), sin(Add(Symbol('\\\\mathbf{F}', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))), Pow(Function('\\\\phi')(Symbol('u', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given Q{(p)} = \\log{(e^{p})}, then obtain (\\int Q{(p)} dp + \\int \\log{(e^{p})} dp) \\log{(e^{p})} - \\int Q{(p)} dp - \\int \\log{(e^{p})} dp = 2 \\log{(e^{p})} \\int \\log{(e^{p})} dp - \\int Q{(p)} dp - \\int \\log{(e^{p})} dp", "derivation": "Q{(p)} = \\log{(e^{p})} and \\int Q{(p)} dp = \\int \\log{(e^{p})} dp and \\int Q{(p)} dp + \\int \\log{(e^{p})} dp = 2 \\int \\log{(e^{p})} dp and (\\int Q{(p)} dp + \\int \\log{(e^{p})} dp) \\log{(e^{p})} = 2 \\log{(e^{p})} \\int \\log{(e^{p})} dp and (\\int Q{(p)} dp + \\int \\log{(e^{p})} dp) \\log{(e^{p})} - \\int Q{(p)} dp - \\int \\log{(e^{p})} dp = 2 \\log{(e^{p})} \\int \\log{(e^{p})} dp - \\int Q{(p)} dp - \\int \\log{(e^{p})} dp", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('p', commutative=True)), log(exp(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["add", 2, "Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))"], "Equality(Add(Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Mul(Integer(2), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))))"], [["times", 3, "log(exp(Symbol('p', commutative=True)))"], "Equality(Mul(Add(Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), log(exp(Symbol('p', commutative=True)))), Mul(Integer(2), log(exp(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))))"], [["minus", 4, "Add(Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], "Equality(Add(Mul(Add(Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), log(exp(Symbol('p', commutative=True)))), Mul(Integer(-1), Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(-1), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))), Add(Mul(Integer(2), log(exp(Symbol('p', commutative=True))), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Mul(Integer(-1), Integral(Function('Q')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Integer(-1), Integral(log(exp(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(A_{z},m)} = \\frac{m}{A_{z}}, then obtain \\int (- \\nabla{(A_{z},m)})^{A_{z}} dm + \\frac{1}{A_{z}} = \\int (- \\frac{m}{A_{z}})^{A_{z}} dm + \\frac{1}{A_{z}}", "derivation": "\\nabla{(A_{z},m)} = \\frac{m}{A_{z}} and - \\nabla{(A_{z},m)} = - \\frac{m}{A_{z}} and (- \\nabla{(A_{z},m)})^{A_{z}} = (- \\frac{m}{A_{z}})^{A_{z}} and \\int (- \\nabla{(A_{z},m)})^{A_{z}} dm = \\int (- \\frac{m}{A_{z}})^{A_{z}} dm and \\int (- \\nabla{(A_{z},m)})^{A_{z}} dm + \\frac{1}{A_{z}} = \\int (- \\frac{m}{A_{z}})^{A_{z}} dm + \\frac{1}{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('A_z', commutative=True), Symbol('m', commutative=True)), Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_z', commutative=True), Symbol('m', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["power", 2, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_z', commutative=True), Symbol('m', commutative=True))), Symbol('A_z', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Symbol('A_z', commutative=True)))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_z', commutative=True), Symbol('m', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["add", 4, "Pow(Symbol('A_z', commutative=True), Integer(-1))"], "Equality(Add(Integral(Pow(Mul(Integer(-1), Function('\\\\nabla')(Symbol('A_z', commutative=True), Symbol('m', commutative=True))), Symbol('A_z', commutative=True)), Tuple(Symbol('m', commutative=True))), Pow(Symbol('A_z', commutative=True), Integer(-1))), Add(Integral(Pow(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Symbol('m', commutative=True)), Symbol('A_z', commutative=True)), Tuple(Symbol('m', commutative=True))), Pow(Symbol('A_z', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\dot{z}{(J)} = \\cos{(J)}, then derive \\frac{d}{d J} \\dot{z}{(J)} = - \\sin{(J)}, then derive \\frac{d^{2}}{d J^{2}} \\dot{z}{(J)} = - \\cos{(J)}, then obtain \\frac{d^{2}}{d J^{2}} \\cos{(J)} = \\frac{d}{d J} - \\sin{(J)}", "derivation": "\\dot{z}{(J)} = \\cos{(J)} and \\frac{d}{d J} \\dot{z}{(J)} = \\frac{d}{d J} \\cos{(J)} and \\frac{d}{d J} \\dot{z}{(J)} = - \\sin{(J)} and \\frac{d^{2}}{d J^{2}} \\dot{z}{(J)} = \\frac{d}{d J} - \\sin{(J)} and \\frac{d^{2}}{d J^{2}} \\dot{z}{(J)} = - \\cos{(J)} and \\frac{d^{2}}{d J^{2}} \\cos{(J)} = - \\cos{(J)} and - \\cos{(J)} = \\frac{d}{d J} - \\sin{(J)} and \\frac{d^{2}}{d J^{2}} \\cos{(J)} = \\frac{d}{d J} - \\sin{(J)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('J', commutative=True))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Integer(-1), cos(Symbol('J', commutative=True))), Derivative(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(\\tilde{g},M)} = M + \\tilde{g}, then derive \\frac{\\partial}{\\partial \\tilde{g}} n{(\\tilde{g},M)} = 1, then obtain - (M + \\tilde{g}) n{(\\tilde{g},M)} + \\frac{\\partial}{\\partial \\tilde{g}} (M + \\tilde{g}) = - (M + \\tilde{g}) n{(\\tilde{g},M)} + 1", "derivation": "n{(\\tilde{g},M)} = M + \\tilde{g} and \\frac{\\partial}{\\partial \\tilde{g}} n{(\\tilde{g},M)} = \\frac{\\partial}{\\partial \\tilde{g}} (M + \\tilde{g}) and \\frac{\\partial}{\\partial \\tilde{g}} n{(\\tilde{g},M)} = 1 and \\frac{\\partial}{\\partial \\tilde{g}} (M + \\tilde{g}) = 1 and - (M + \\tilde{g}) n{(\\tilde{g},M)} + \\frac{\\partial}{\\partial \\tilde{g}} (M + \\tilde{g}) = - (M + \\tilde{g}) n{(\\tilde{g},M)} + 1", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True)), Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1))"], [["minus", 4, "Mul(Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True))), Derivative(Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Add(Symbol('M', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Function('n')(Symbol('\\\\tilde{g}', commutative=True), Symbol('M', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(U,\\delta)} = e^{\\frac{\\delta}{U}}, then derive 1 = 0^{\\delta}, then obtain \\int 1 d\\delta = \\int 0^{\\delta} d\\delta", "derivation": "\\operatorname{f^{*}}{(U,\\delta)} = e^{\\frac{\\delta}{U}} and \\operatorname{f^{*}}{(U,\\delta)} - e^{\\frac{\\delta}{U}} = 0 and (\\operatorname{f^{*}}{(U,\\delta)} - e^{\\frac{\\delta}{U}})^{\\delta} = 0^{\\delta} and U (\\operatorname{f^{*}}{(U,\\delta)} - e^{\\frac{\\delta}{U}})^{\\delta} = 0^{\\delta} U and U = U (\\operatorname{f^{*}}{(U,\\delta)} - e^{\\frac{\\delta}{U}})^{\\delta} and U = 0^{\\delta} U and \\frac{d}{d U} U = \\frac{\\partial}{\\partial U} 0^{\\delta} U and 1 = 0^{\\delta} and \\int 1 d\\delta = \\int 0^{\\delta} d\\delta", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))"], [["minus", 1, "exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))), Symbol('\\\\delta', commutative=True)), Pow(Integer(0), Symbol('\\\\delta', commutative=True)))"], [["times", 3, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Pow(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))), Symbol('\\\\delta', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Symbol('U', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Symbol('U', commutative=True), Mul(Symbol('U', commutative=True), Pow(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), exp(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Symbol('U', commutative=True), Mul(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Symbol('U', commutative=True)))"], [["differentiate", 6, "Symbol('U', commutative=True)"], "Equality(Derivative(Symbol('U', commutative=True), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Mul(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(1), Pow(Integer(0), Symbol('\\\\delta', commutative=True)))"], [["integrate", 8, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Pow(Integer(0), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\Omega{(t_{1},A_{1})} = A_{1} \\sin{(t_{1})}, then obtain - \\frac{1}{\\Omega^{2}{(t_{1},A_{1})}} + \\frac{1}{A_{1}^{2} \\sin{(t_{1})}} = - \\frac{1}{\\Omega^{2}{(t_{1},A_{1})}} + \\frac{1}{A_{1} \\Omega{(t_{1},A_{1})}}", "derivation": "\\Omega{(t_{1},A_{1})} = A_{1} \\sin{(t_{1})} and A_{1} \\Omega{(t_{1},A_{1})} = A_{1}^{2} \\sin{(t_{1})} and 1 = \\frac{A_{1} \\sin{(t_{1})}}{\\Omega{(t_{1},A_{1})}} and \\frac{1}{A_{1} \\Omega{(t_{1},A_{1})}} = \\frac{\\sin{(t_{1})}}{\\Omega^{2}{(t_{1},A_{1})}} and \\frac{1}{A_{1}^{2} \\sin{(t_{1})}} = \\frac{\\sin{(t_{1})}}{\\Omega^{2}{(t_{1},A_{1})}} and \\frac{1}{A_{1}^{2} \\sin{(t_{1})}} = \\frac{1}{A_{1} \\Omega{(t_{1},A_{1})}} and - \\frac{1}{\\Omega^{2}{(t_{1},A_{1})}} + \\frac{1}{A_{1}^{2} \\sin{(t_{1})}} = - \\frac{1}{\\Omega^{2}{(t_{1},A_{1})}} + \\frac{1}{A_{1} \\Omega{(t_{1},A_{1})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), sin(Symbol('t_1', commutative=True))))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), sin(Symbol('t_1', commutative=True))))"], [["divide", 2, "Mul(Symbol('A_1', commutative=True), Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Integer(1), Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)), sin(Symbol('t_1', commutative=True))))"], [["divide", 3, "Mul(Symbol('A_1', commutative=True), Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-2)), sin(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(sin(Symbol('t_1', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-2)), sin(Symbol('t_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(sin(Symbol('t_1', commutative=True)), Integer(-1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-1))))"], [["minus", 6, "Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-2)), Pow(sin(Symbol('t_1', commutative=True)), Integer(-1)))), Add(Mul(Integer(-1), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-2))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Function('\\\\Omega')(Symbol('t_1', commutative=True), Symbol('A_1', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\varphi^*)} = \\varphi^*, then obtain \\varphi^* \\int 4 \\hat{H}_{\\lambda}^{2}{(\\varphi^*)} d\\varphi^* = \\varphi^* \\int (\\varphi^* + \\hat{H}_{\\lambda}{(\\varphi^*)})^{2} d\\varphi^*", "derivation": "\\hat{H}_{\\lambda}{(\\varphi^*)} = \\varphi^* and 2 \\hat{H}_{\\lambda}{(\\varphi^*)} = \\varphi^* + \\hat{H}_{\\lambda}{(\\varphi^*)} and 4 \\hat{H}_{\\lambda}^{2}{(\\varphi^*)} = (\\varphi^* + \\hat{H}_{\\lambda}{(\\varphi^*)})^{2} and \\int 4 \\hat{H}_{\\lambda}^{2}{(\\varphi^*)} d\\varphi^* = \\int (\\varphi^* + \\hat{H}_{\\lambda}{(\\varphi^*)})^{2} d\\varphi^* and \\varphi^* \\int 4 \\hat{H}_{\\lambda}^{2}{(\\varphi^*)} d\\varphi^* = \\varphi^* \\int (\\varphi^* + \\hat{H}_{\\lambda}{(\\varphi^*)})^{2} d\\varphi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True)), Symbol('\\\\varphi^*', commutative=True))"], [["add", 1, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Integer(2)))"], [["integrate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Pow(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 4, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Mul(Integer(4), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(Symbol('\\\\varphi^*', commutative=True), Integral(Pow(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\varphi^*', commutative=True))), Integer(2)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(\\phi_2)} = e^{\\phi_2}, then obtain \\int 0 d\\phi_2 = \\int \\frac{(- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) e^{\\phi_2}}{\\hat{p}{(\\phi_2)}} d\\phi_2", "derivation": "\\hat{p}{(\\phi_2)} = e^{\\phi_2} and 0 = - \\hat{p}{(\\phi_2)} + e^{\\phi_2} and (- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) \\hat{p}{(\\phi_2)} = (- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) e^{\\phi_2} and - \\hat{p}{(\\phi_2)} + e^{\\phi_2} = \\frac{(- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) e^{\\phi_2}}{\\hat{p}{(\\phi_2)}} and 0 = \\frac{(- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) e^{\\phi_2}}{\\hat{p}{(\\phi_2)}} and \\int 0 d\\phi_2 = \\int \\frac{(- \\hat{p}{(\\phi_2)} + e^{\\phi_2}) e^{\\phi_2}}{\\hat{p}{(\\phi_2)}} d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))"], [["minus", 1, "Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))))"], [["divide", 3, "Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), Mul(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True))), exp(Symbol('\\\\phi_2', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('\\\\phi_2', commutative=True)), Integer(-1)), exp(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\nabla)} = \\cos{(\\nabla)}, then obtain \\operatorname{A_{y}}{(\\nabla)} + 2 \\int 1 d\\nabla = \\cos{(\\nabla)} + 2 \\int 1 d\\nabla", "derivation": "\\operatorname{A_{y}}{(\\nabla)} = \\cos{(\\nabla)} and 1 = \\frac{\\cos{(\\nabla)}}{\\operatorname{A_{y}}{(\\nabla)}} and \\int 1 d\\nabla = \\int \\frac{\\cos{(\\nabla)}}{\\operatorname{A_{y}}{(\\nabla)}} d\\nabla and \\operatorname{A_{y}}{(\\nabla)} + \\int \\frac{\\cos{(\\nabla)}}{\\operatorname{A_{y}}{(\\nabla)}} d\\nabla = \\cos{(\\nabla)} + \\int \\frac{\\cos{(\\nabla)}}{\\operatorname{A_{y}}{(\\nabla)}} d\\nabla and \\operatorname{A_{y}}{(\\nabla)} + \\int 1 d\\nabla = \\cos{(\\nabla)} + \\int 1 d\\nabla and \\operatorname{A_{y}}{(\\nabla)} + 2 \\int 1 d\\nabla = \\cos{(\\nabla)} + 2 \\int 1 d\\nabla", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["divide", 1, "Function('A_y')(Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), cos(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 2, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Mul(Pow(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 1, "Integral(Mul(Pow(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integral(Mul(Pow(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(cos(Symbol('\\\\nabla', commutative=True)), Integral(Mul(Pow(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), cos(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)))), Add(cos(Symbol('\\\\nabla', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))))"], "Equality(Add(Function('A_y')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))))), Add(cos(Symbol('\\\\nabla', commutative=True)), Mul(Integer(2), Integral(Integer(1), Tuple(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\phi)} = \\sin{(\\phi)}, then derive (\\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)})^{\\phi} = \\cos^{\\phi}{(\\phi)}, then obtain \\cos^{\\phi}{(\\phi)} (\\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = \\cos^{2 \\phi}{(\\phi)}", "derivation": "\\hat{\\mathbf{x}}{(\\phi)} = \\sin{(\\phi)} and \\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and (\\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)})^{\\phi} = (\\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} and (\\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)})^{\\phi} = \\cos^{\\phi}{(\\phi)} and (\\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = \\cos^{\\phi}{(\\phi)} and (\\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)})^{\\phi} (\\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = \\cos^{\\phi}{(\\phi)} (\\frac{d}{d \\phi} \\hat{\\mathbf{x}}{(\\phi)})^{\\phi} and \\cos^{\\phi}{(\\phi)} (\\frac{d}{d \\phi} \\sin{(\\phi)})^{\\phi} = \\cos^{2 \\phi}{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)))"], [["times", 5, "Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))), Mul(Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(cos(Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Symbol('\\\\phi', commutative=True))), Pow(cos(Symbol('\\\\phi', commutative=True)), Mul(Integer(2), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\lambda{(u,\\mathbf{S})} = \\log{(\\mathbf{S} u)}, then derive \\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du = t_{1} + 2 u \\log{(\\mathbf{S} u)} - 2 u, then obtain ((\\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du)^{t_{1}})^{\\mathbf{S}} = ((t_{1} + 2 u \\log{(\\mathbf{S} u)} - 2 u)^{t_{1}})^{\\mathbf{S}}", "derivation": "\\lambda{(u,\\mathbf{S})} = \\log{(\\mathbf{S} u)} and \\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)} = 2 \\log{(\\mathbf{S} u)} and \\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du = \\int 2 \\log{(\\mathbf{S} u)} du and \\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du = t_{1} + 2 u \\log{(\\mathbf{S} u)} - 2 u and (\\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du)^{t_{1}} = (t_{1} + 2 u \\log{(\\mathbf{S} u)} - 2 u)^{t_{1}} and ((\\int (\\lambda{(u,\\mathbf{S})} + \\log{(\\mathbf{S} u)}) du)^{t_{1}})^{\\mathbf{S}} = ((t_{1} + 2 u \\log{(\\mathbf{S} u)} - 2 u)^{t_{1}})^{\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True))))"], [["add", 1, "log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))"], "Equality(Add(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Mul(Integer(2), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))))"], [["integrate", 2, "Symbol('u', commutative=True)"], "Equality(Integral(Add(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(2), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))))"], [["power", 4, "Symbol('t_1', commutative=True)"], "Equality(Pow(Integral(Add(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('t_1', commutative=True)), Pow(Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))), Symbol('t_1', commutative=True)))"], [["power", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Pow(Integral(Add(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True))), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Pow(Add(Symbol('t_1', commutative=True), Mul(Integer(2), Symbol('u', commutative=True), log(Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('u', commutative=True)))), Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))), Symbol('t_1', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)} = \\psi^* \\cos{(\\hat{p})}, then obtain \\frac{\\partial}{\\partial \\hat{p}} (- \\psi^* \\cos{(\\hat{p})} + \\frac{\\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)}}{\\hat{p}}) = \\frac{\\partial}{\\partial \\hat{p}} (- \\psi^* \\cos{(\\hat{p})} + \\frac{\\psi^* \\cos{(\\hat{p})}}{\\hat{p}})", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)} = \\psi^* \\cos{(\\hat{p})} and \\frac{\\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)}}{\\hat{p}} = \\frac{\\psi^* \\cos{(\\hat{p})}}{\\hat{p}} and - \\psi^* \\cos{(\\hat{p})} + \\frac{\\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)}}{\\hat{p}} = - \\psi^* \\cos{(\\hat{p})} + \\frac{\\psi^* \\cos{(\\hat{p})}}{\\hat{p}} and \\frac{\\partial}{\\partial \\hat{p}} (- \\psi^* \\cos{(\\hat{p})} + \\frac{\\operatorname{J_{\\varepsilon}}{(\\hat{p},\\psi^*)}}{\\hat{p}}) = \\frac{\\partial}{\\partial \\hat{p}} (- \\psi^* \\cos{(\\hat{p})} + \\frac{\\psi^* \\cos{(\\hat{p})}}{\\hat{p}})", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True), cos(Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(q,\\mathbf{J}_M,c)} = - \\mathbf{J}_M + c + q and \\operatorname{n_{2}}{(q,\\mathbf{J}_M,c)} = \\operatorname{F_{x}}^{c}{(q,\\mathbf{J}_M,c)}, then obtain ((- \\mathbf{J}_M + c + q)^{c})^{c} = \\operatorname{n_{2}}^{c}{(q,\\mathbf{J}_M,c)}", "derivation": "\\operatorname{F_{x}}{(q,\\mathbf{J}_M,c)} = - \\mathbf{J}_M + c + q and \\operatorname{F_{x}}^{c}{(q,\\mathbf{J}_M,c)} = (- \\mathbf{J}_M + c + q)^{c} and (\\operatorname{F_{x}}^{c}{(q,\\mathbf{J}_M,c)})^{c} = ((- \\mathbf{J}_M + c + q)^{c})^{c} and \\operatorname{n_{2}}{(q,\\mathbf{J}_M,c)} = \\operatorname{F_{x}}^{c}{(q,\\mathbf{J}_M,c)} and \\operatorname{n_{2}}{(q,\\mathbf{J}_M,c)} = (- \\mathbf{J}_M + c + q)^{c} and (\\operatorname{F_{x}}^{c}{(q,\\mathbf{J}_M,c)})^{c} = \\operatorname{n_{2}}^{c}{(q,\\mathbf{J}_M,c)} and ((- \\mathbf{J}_M + c + q)^{c})^{c} = \\operatorname{n_{2}}^{c}{(q,\\mathbf{J}_M,c)}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('c', commutative=True), Symbol('q', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('c', commutative=True)))"], [["power", 2, "Symbol('c', commutative=True)"], "Equality(Pow(Pow(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Pow(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('n_2')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Pow(Pow(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('n_2')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('c', commutative=True), Symbol('q', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Function('n_2')(Symbol('q', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(E,F_{g})} = E e^{F_{g}}, then derive \\frac{\\partial}{\\partial E} \\mathbf{J}{(E,F_{g})} = e^{F_{g}}, then obtain \\frac{\\partial}{\\partial E} E e^{F_{g}} + \\frac{\\frac{\\partial}{\\partial E} E e^{F_{g}}}{F_{g}} = e^{F_{g}} + \\frac{\\frac{\\partial}{\\partial E} E e^{F_{g}}}{F_{g}}", "derivation": "\\mathbf{J}{(E,F_{g})} = E e^{F_{g}} and \\frac{\\partial}{\\partial E} \\mathbf{J}{(E,F_{g})} = \\frac{\\partial}{\\partial E} E e^{F_{g}} and \\frac{\\partial}{\\partial E} \\mathbf{J}{(E,F_{g})} = e^{F_{g}} and \\frac{\\partial}{\\partial E} E e^{F_{g}} = e^{F_{g}} and \\frac{\\partial}{\\partial E} E e^{F_{g}} + \\frac{\\frac{\\partial}{\\partial E} E e^{F_{g}}}{F_{g}} = e^{F_{g}} + \\frac{\\frac{\\partial}{\\partial E} E e^{F_{g}}}{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('E', commutative=True), Symbol('F_g', commutative=True)), Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('E', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('E', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), exp(Symbol('F_g', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), exp(Symbol('F_g', commutative=True)))"], [["add", 4, "Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], "Equality(Add(Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))), Add(exp(Symbol('F_g', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(Mul(Symbol('E', commutative=True), exp(Symbol('F_g', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))))"]]}, {"prompt": "Given m{(\\mu_0,\\theta_2)} = \\mu_0 - \\theta_2 and \\operatorname{t_{1}}{(\\mu_0,\\theta_2)} = \\mu_0 - \\theta_2, then obtain - \\frac{\\mathbf{H} (\\mu_0 - \\theta_2)}{\\mu_0 \\theta_2} = - \\frac{\\mathbf{H} \\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\mu_0 \\theta_2}", "derivation": "m{(\\mu_0,\\theta_2)} = \\mu_0 - \\theta_2 and \\frac{m{(\\mu_0,\\theta_2)}}{\\theta_2} = \\frac{\\mu_0 - \\theta_2}{\\theta_2} and \\operatorname{t_{1}}{(\\mu_0,\\theta_2)} = \\mu_0 - \\theta_2 and \\frac{m{(\\mu_0,\\theta_2)}}{\\theta_2} = \\frac{\\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\theta_2} and \\frac{\\mu_0 - \\theta_2}{\\theta_2} = \\frac{\\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\theta_2} and \\frac{\\mu_0 - \\theta_2}{\\mu_0 \\theta_2} = \\frac{\\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\mu_0 \\theta_2} and - \\frac{\\mu_0 - \\theta_2}{\\mu_0 \\theta_2} = - \\frac{\\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\mu_0 \\theta_2} and - \\frac{\\mathbf{H} (\\mu_0 - \\theta_2)}{\\mu_0 \\theta_2} = - \\frac{\\mathbf{H} \\operatorname{t_{1}}{(\\mu_0,\\theta_2)}}{\\mu_0 \\theta_2}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('m')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('m')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 5, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 6, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["times", 7, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True), Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('t_1')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(A_{y})} = \\cos{(A_{y})}, then derive \\sin{(\\frac{\\operatorname{F_{x}}{(A_{y})} \\sin{(A_{y})}}{\\cos^{2}{(A_{y})}} + \\frac{\\frac{d}{d A_{y}} \\operatorname{F_{x}}{(A_{y})}}{\\cos{(A_{y})}})} = 0, then obtain \\sin{(\\frac{\\sin{(A_{y})}}{\\cos{(A_{y})}} + \\frac{\\frac{d}{d A_{y}} \\cos{(A_{y})}}{\\cos{(A_{y})}})} = 0", "derivation": "\\operatorname{F_{x}}{(A_{y})} = \\cos{(A_{y})} and \\frac{\\operatorname{F_{x}}{(A_{y})}}{\\cos{(A_{y})}} = 1 and \\frac{d}{d A_{y}} \\frac{\\operatorname{F_{x}}{(A_{y})}}{\\cos{(A_{y})}} = \\frac{d}{d A_{y}} 1 and \\sin{(\\frac{d}{d A_{y}} \\frac{\\operatorname{F_{x}}{(A_{y})}}{\\cos{(A_{y})}})} = \\sin{(\\frac{d}{d A_{y}} 1)} and \\sin{(\\frac{\\operatorname{F_{x}}{(A_{y})} \\sin{(A_{y})}}{\\cos^{2}{(A_{y})}} + \\frac{\\frac{d}{d A_{y}} \\operatorname{F_{x}}{(A_{y})}}{\\cos{(A_{y})}})} = 0 and \\sin{(\\frac{\\sin{(A_{y})}}{\\cos{(A_{y})}} + \\frac{\\frac{d}{d A_{y}} \\cos{(A_{y})}}{\\cos{(A_{y})}})} = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('A_y', commutative=True)), cos(Symbol('A_y', commutative=True)))"], [["divide", 1, "cos(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('F_x')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Function('F_x')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Mul(Function('F_x')(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Tuple(Symbol('A_y', commutative=True), Integer(1)))), sin(Derivative(Integer(1), Tuple(Symbol('A_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(sin(Add(Mul(Function('F_x')(Symbol('A_y', commutative=True)), sin(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-2))), Mul(Pow(cos(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(Function('F_x')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(sin(Add(Mul(sin(Symbol('A_y', commutative=True)), Pow(cos(Symbol('A_y', commutative=True)), Integer(-1))), Mul(Pow(cos(Symbol('A_y', commutative=True)), Integer(-1)), Derivative(cos(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1)))))), Integer(0))"]]}, {"prompt": "Given \\varphi^{*}{(v_{2})} = \\cos{(v_{2})}, then obtain \\int (-1 + \\frac{\\frac{d}{d v_{2}} \\varphi^{*}{(v_{2})}}{\\cos{(v_{2})}}) dv_{2} = \\int (-1 + \\frac{\\frac{d}{d v_{2}} \\cos{(v_{2})}}{\\cos{(v_{2})}}) dv_{2}", "derivation": "\\varphi^{*}{(v_{2})} = \\cos{(v_{2})} and \\frac{d}{d v_{2}} \\varphi^{*}{(v_{2})} = \\frac{d}{d v_{2}} \\cos{(v_{2})} and \\frac{\\frac{d}{d v_{2}} \\varphi^{*}{(v_{2})}}{\\cos{(v_{2})}} = \\frac{\\frac{d}{d v_{2}} \\cos{(v_{2})}}{\\cos{(v_{2})}} and -1 + \\frac{\\frac{d}{d v_{2}} \\varphi^{*}{(v_{2})}}{\\cos{(v_{2})}} = -1 + \\frac{\\frac{d}{d v_{2}} \\cos{(v_{2})}}{\\cos{(v_{2})}} and \\int (-1 + \\frac{\\frac{d}{d v_{2}} \\varphi^{*}{(v_{2})}}{\\cos{(v_{2})}}) dv_{2} = \\int (-1 + \\frac{\\frac{d}{d v_{2}} \\cos{(v_{2})}}{\\cos{(v_{2})}}) dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["divide", 2, "cos(Symbol('v_2', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(Function('\\\\varphi^*')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["minus", 3, 1], "Equality(Add(Integer(-1), Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(Function('\\\\varphi^*')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Add(Integer(-1), Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Add(Integer(-1), Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(Function('\\\\varphi^*')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Integer(-1), Mul(Pow(cos(Symbol('v_2', commutative=True)), Integer(-1)), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given t{(\\hat{x},t_{1})} = \\hat{x} + t_{1}, then derive \\frac{\\partial}{\\partial t_{1}} t{(\\hat{x},t_{1})} + 2 = 3, then obtain (\\frac{\\partial}{\\partial t_{1}} (\\hat{x} + t_{1}) + 2) \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1}) = 3 \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1})", "derivation": "t{(\\hat{x},t_{1})} = \\hat{x} + t_{1} and t_{1} + t{(\\hat{x},t_{1})} = \\hat{x} + 2 t_{1} and 2 t_{1} + t{(\\hat{x},t_{1})} = \\hat{x} + 3 t_{1} and \\frac{\\partial}{\\partial t_{1}} (2 t_{1} + t{(\\hat{x},t_{1})}) = \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1}) and \\frac{\\partial}{\\partial t_{1}} t{(\\hat{x},t_{1})} + 2 = 3 and (\\frac{\\partial}{\\partial t_{1}} t{(\\hat{x},t_{1})} + 2) \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1}) = 3 \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1}) and (\\frac{\\partial}{\\partial t_{1}} (\\hat{x} + t_{1}) + 2) \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1}) = 3 \\frac{\\partial}{\\partial t_{1}} (\\hat{x} + 3 t_{1})", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('t_1', commutative=True))"], "Equality(Add(Symbol('t_1', commutative=True), Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(2), Symbol('t_1', commutative=True))))"], [["add", 2, "Symbol('t_1', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('t_1', commutative=True)), Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Symbol('t_1', commutative=True)), Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(2)), Integer(3))"], [["times", 5, "Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Mul(Add(Derivative(Function('t')(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(2)), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Integer(3), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Add(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Integer(2)), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Mul(Integer(3), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(3), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varepsilon{(b)} = \\log{(e^{b})}, then obtain \\int \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\varepsilon{(b)}}{b^{2}} db = \\int \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\log{(e^{b})}}{b^{2}} db", "derivation": "\\varepsilon{(b)} = \\log{(e^{b})} and \\frac{d}{d b} \\varepsilon{(b)} = \\frac{d}{d b} \\log{(e^{b})} and \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\varepsilon{(b)}}{b} = \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\log{(e^{b})}}{b} and \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\varepsilon{(b)}}{b^{2}} = \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\log{(e^{b})}}{b^{2}} and \\int \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\varepsilon{(b)}}{b^{2}} db = \\int \\frac{\\varepsilon{(b)} \\frac{d}{d b} \\log{(e^{b})}}{b^{2}} db", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('b', commutative=True)), log(exp(Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(log(exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["times", 2, "Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('b', commutative=True)))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(log(exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["divide", 3, "Symbol('b', commutative=True)"], "Equality(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(log(exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True))), Integral(Mul(Pow(Symbol('b', commutative=True), Integer(-2)), Function('\\\\varepsilon')(Symbol('b', commutative=True)), Derivative(log(exp(Symbol('b', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1)))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(B,\\mathbf{s})} = \\mathbf{s}^{B}, then obtain \\sin{(\\mathbf{s}^{B} \\log{(\\mathbf{s})} + \\frac{\\partial}{\\partial B} \\operatorname{n_{2}}{(B,\\mathbf{s})})} = \\sin{(2 \\mathbf{s}^{B} \\log{(\\mathbf{s})})}", "derivation": "\\operatorname{n_{2}}{(B,\\mathbf{s})} = \\mathbf{s}^{B} and \\mathbf{s}^{B} + \\operatorname{n_{2}}{(B,\\mathbf{s})} = 2 \\mathbf{s}^{B} and \\frac{\\partial}{\\partial B} (\\mathbf{s}^{B} + \\operatorname{n_{2}}{(B,\\mathbf{s})}) = \\frac{\\partial}{\\partial B} 2 \\mathbf{s}^{B} and \\sin{(\\frac{\\partial}{\\partial B} (\\mathbf{s}^{B} + \\operatorname{n_{2}}{(B,\\mathbf{s})}))} = \\sin{(\\frac{\\partial}{\\partial B} 2 \\mathbf{s}^{B})} and \\sin{(\\mathbf{s}^{B} \\log{(\\mathbf{s})} + \\frac{\\partial}{\\partial B} \\operatorname{n_{2}}{(B,\\mathbf{s})})} = \\sin{(2 \\mathbf{s}^{B} \\log{(\\mathbf{s})})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('B', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)))"], [["add", 1, "Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)), Function('n_2')(Symbol('B', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)), Function('n_2')(Symbol('B', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Add(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)), Function('n_2')(Symbol('B', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))), sin(Derivative(Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(sin(Add(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)), log(Symbol('\\\\mathbf{s}', commutative=True))), Derivative(Function('n_2')(Symbol('B', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))), sin(Mul(Integer(2), Pow(Symbol('\\\\mathbf{s}', commutative=True), Symbol('B', commutative=True)), log(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(z,\\hat{p}_0)} = \\hat{p}_0 - z, then obtain \\int 1 d\\hat{p}_0 = \\int \\frac{2 \\hat{p}_0 - 2 z}{2 (\\hat{p}_0 - z)} d\\hat{p}_0", "derivation": "\\operatorname{a^{\\dagger}}{(z,\\hat{p}_0)} = \\hat{p}_0 - z and \\hat{p}_0 - z + \\operatorname{a^{\\dagger}}{(z,\\hat{p}_0)} = 2 \\hat{p}_0 - 2 z and 1 = \\frac{2 \\hat{p}_0 - 2 z}{\\hat{p}_0 - z + \\operatorname{a^{\\dagger}}{(z,\\hat{p}_0)}} and 1 = \\frac{2 \\hat{p}_0 - 2 z}{2 \\operatorname{a^{\\dagger}}{(z,\\hat{p}_0)}} and 1 = \\frac{2 \\hat{p}_0 - 2 z}{2 (\\hat{p}_0 - z)} and \\int 1 d\\hat{p}_0 = \\int \\frac{2 \\hat{p}_0 - 2 z}{2 (\\hat{p}_0 - z)} d\\hat{p}_0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))"], "Equality(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))"], "Equality(Integer(1), Mul(Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)), Function('a^{\\\\dagger}')(Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Rational(1, 2), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))), Pow(Function('a^{\\\\dagger}')(Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Rational(1, 2), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)))))"], [["integrate", 5, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Mul(Rational(1, 2), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(S)} = \\cos{(S)}, then derive F_{g} + \\log{(S)} = \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS, then obtain \\cos{(S)} \\frac{\\partial}{\\partial S} (F_{g} + \\log{(S)}) = \\cos{(S)} \\frac{d}{d S} \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS", "derivation": "\\operatorname{v_{z}}{(S)} = \\cos{(S)} and 1 = \\frac{\\cos{(S)}}{\\operatorname{v_{z}}{(S)}} and \\frac{1}{S} = \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} and \\int \\frac{1}{S} dS = \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS and F_{g} + \\log{(S)} = \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS and \\frac{\\partial}{\\partial S} (F_{g} + \\log{(S)}) = \\frac{d}{d S} \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS and \\cos{(S)} \\frac{\\partial}{\\partial S} (F_{g} + \\log{(S)}) = \\cos{(S)} \\frac{d}{d S} \\int \\frac{\\cos{(S)}}{S \\operatorname{v_{z}}{(S)}} dS", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True)))"], [["divide", 1, "Function('v_z')(Symbol('S', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))))"], [["divide", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Symbol('S', commutative=True), Integer(-1)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Pow(Symbol('S', commutative=True), Integer(-1)), Tuple(Symbol('S', commutative=True))), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('F_g', commutative=True), log(Symbol('S', commutative=True))), Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["differentiate", 5, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Symbol('F_g', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 6, "cos(Symbol('S', commutative=True))"], "Equality(Mul(cos(Symbol('S', commutative=True)), Derivative(Add(Symbol('F_g', commutative=True), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Mul(cos(Symbol('S', commutative=True)), Derivative(Integral(Mul(Pow(Symbol('S', commutative=True), Integer(-1)), Pow(Function('v_z')(Symbol('S', commutative=True)), Integer(-1)), cos(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{x}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})}, then obtain \\dot{x}{(\\hat{\\mathbf{x}})} \\int 0 d\\hat{\\mathbf{x}} = \\dot{x}{(\\hat{\\mathbf{x}})} \\int (- \\dot{x}{(\\hat{\\mathbf{x}})} + \\log{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}}", "derivation": "\\dot{x}{(\\hat{\\mathbf{x}})} = \\log{(\\hat{\\mathbf{x}})} and 0 = - \\dot{x}{(\\hat{\\mathbf{x}})} + \\log{(\\hat{\\mathbf{x}})} and \\int 0 d\\hat{\\mathbf{x}} = \\int (- \\dot{x}{(\\hat{\\mathbf{x}})} + \\log{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}} and \\dot{x}{(\\hat{\\mathbf{x}})} \\int 0 d\\hat{\\mathbf{x}} = \\dot{x}{(\\hat{\\mathbf{x}})} \\int (- \\dot{x}{(\\hat{\\mathbf{x}})} + \\log{(\\hat{\\mathbf{x}})}) d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))"], [["minus", 1, "Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["times", 3, "Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))"], "Equality(Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integral(Integer(0), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Mul(Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Integral(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"]]}, {"prompt": "Given \\theta{(u,r,E_{n})} = E_{n} u^{r}, then obtain (\\int (u^{r} + \\theta{(u,r,E_{n})}) dr)^{E_{n}} = (\\int (E_{n} u^{r} + u^{r}) dr)^{E_{n}}", "derivation": "\\theta{(u,r,E_{n})} = E_{n} u^{r} and u^{r} + \\theta{(u,r,E_{n})} = E_{n} u^{r} + u^{r} and \\int (u^{r} + \\theta{(u,r,E_{n})}) dr = \\int (E_{n} u^{r} + u^{r}) dr and (\\int (u^{r} + \\theta{(u,r,E_{n})}) dr)^{E_{n}} = (\\int (E_{n} u^{r} + u^{r}) dr)^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('u', commutative=True), Symbol('r', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))))"], [["add", 1, "Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Pow(Symbol('u', commutative=True), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True), Symbol('r', commutative=True), Symbol('E_n', commutative=True))), Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Add(Pow(Symbol('u', commutative=True), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True), Symbol('r', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(Integral(Add(Pow(Symbol('u', commutative=True), Symbol('r', commutative=True)), Function('\\\\theta')(Symbol('u', commutative=True), Symbol('r', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('r', commutative=True))), Symbol('E_n', commutative=True)), Pow(Integral(Add(Mul(Symbol('E_n', commutative=True), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))), Pow(Symbol('u', commutative=True), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})}, then obtain (\\int \\frac{\\operatorname{m_{s}}{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}} = (\\int \\frac{\\cos{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}}", "derivation": "\\operatorname{m_{s}}{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\frac{\\operatorname{m_{s}}{(\\mathbf{A})}}{\\mathbf{A}} = \\frac{\\cos{(\\mathbf{A})}}{\\mathbf{A}} and \\int \\frac{\\operatorname{m_{s}}{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A} = \\int \\frac{\\cos{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A} and (\\int \\frac{\\operatorname{m_{s}}{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}} = (\\int \\frac{\\cos{(\\mathbf{A})}}{\\mathbf{A}} d\\mathbf{A})^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["divide", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), Function('m_s')(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given y{(c,x,\\delta)} = \\delta x - c, then derive \\int x y{(c,x,\\delta)} dx = \\frac{\\delta x^{3}}{3} + \\hat{x}_0 - \\frac{c x^{2}}{2}, then obtain \\frac{\\delta x^{3}}{3} + \\hat{x}_0 - \\frac{c x^{2}}{2} + \\int x (\\delta x - c) dx = 2 \\int x (\\delta x - c) dx", "derivation": "y{(c,x,\\delta)} = \\delta x - c and x y{(c,x,\\delta)} = x (\\delta x - c) and \\int x y{(c,x,\\delta)} dx = \\int x (\\delta x - c) dx and \\int x (\\delta x - c) dx + \\int x y{(c,x,\\delta)} dx = 2 \\int x (\\delta x - c) dx and \\int x y{(c,x,\\delta)} dx = \\frac{\\delta x^{3}}{3} + \\hat{x}_0 - \\frac{c x^{2}}{2} and \\frac{\\delta x^{3}}{3} + \\hat{x}_0 - \\frac{c x^{2}}{2} + \\int x (\\delta x - c) dx = 2 \\int x (\\delta x - c) dx", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('y')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('y')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["add", 3, "Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True)))"], "Equality(Add(Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), Function('y')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('x', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('y')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('x', commutative=True))), Add(Mul(Rational(1, 3), Symbol('\\\\delta', commutative=True), Pow(Symbol('x', commutative=True), Integer(3))), Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Symbol('c', commutative=True), Pow(Symbol('x', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Rational(1, 3), Symbol('\\\\delta', commutative=True), Pow(Symbol('x', commutative=True), Integer(3))), Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), Rational(1, 2), Symbol('c', commutative=True), Pow(Symbol('x', commutative=True), Integer(2))), Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True)))), Mul(Integer(2), Integral(Mul(Symbol('x', commutative=True), Add(Mul(Symbol('\\\\delta', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Symbol('c', commutative=True)))), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given Z{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})}, then obtain 4 Z{(\\theta_2)} \\cos{(\\log{(\\theta_2)})} = (Z{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})})^{2}", "derivation": "Z{(\\theta_2)} = \\cos{(\\log{(\\theta_2)})} and Z^{2}{(\\theta_2)} = Z{(\\theta_2)} \\cos{(\\log{(\\theta_2)})} and 2 Z{(\\theta_2)} = Z{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})} and 4 Z^{2}{(\\theta_2)} = (Z{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})})^{2} and 4 Z{(\\theta_2)} \\cos{(\\log{(\\theta_2)})} = (Z{(\\theta_2)} + \\cos{(\\log{(\\theta_2)})})^{2}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True))))"], [["times", 1, "Function('Z')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Pow(Function('Z')(Symbol('\\\\theta_2', commutative=True)), Integer(2)), Mul(Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True)))))"], [["add", 1, "Function('Z')(Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Integer(2), Function('Z')(Symbol('\\\\theta_2', commutative=True))), Add(Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True)))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('Z')(Symbol('\\\\theta_2', commutative=True)), Integer(2))), Pow(Add(Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True)))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Integer(4), Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True)))), Pow(Add(Function('Z')(Symbol('\\\\theta_2', commutative=True)), cos(log(Symbol('\\\\theta_2', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\mathbf{P}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain \\mathbf{P}{(J_{\\varepsilon})} \\theta_{2}{(\\mathbf{J}_f)} + \\int \\theta_{2}{(\\mathbf{J}_f)} \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\theta_{2}{(\\mathbf{J}_f)} e^{J_{\\varepsilon}} + \\int \\theta_{2}{(\\mathbf{J}_f)} \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f", "derivation": "\\theta_{2}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\mathbf{P}{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\mathbf{P}{(J_{\\varepsilon})} \\cos{(\\mathbf{J}_f)} = e^{J_{\\varepsilon}} \\cos{(\\mathbf{J}_f)} and \\mathbf{P}{(J_{\\varepsilon})} \\theta_{2}{(\\mathbf{J}_f)} = \\theta_{2}{(\\mathbf{J}_f)} e^{J_{\\varepsilon}} and \\mathbf{P}{(J_{\\varepsilon})} \\theta_{2}{(\\mathbf{J}_f)} + \\int \\theta_{2}{(\\mathbf{J}_f)} \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f = \\theta_{2}{(\\mathbf{J}_f)} e^{J_{\\varepsilon}} + \\int \\theta_{2}{(\\mathbf{J}_f)} \\cos{(\\mathbf{J}_f)} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["times", 2, "cos(Symbol('\\\\mathbf{J}_f', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{P}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 4, "Integral(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Mul(Function('\\\\mathbf{P}')(Symbol('J_{\\\\varepsilon}', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))), Add(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\sigma_x)} = \\sigma_x, then derive \\frac{d}{d \\sigma_x} \\operatorname{E_{n}}{(\\sigma_x)} = 1, then obtain \\frac{d}{d \\sigma_x} \\sigma_x = 1", "derivation": "\\operatorname{E_{n}}{(\\sigma_x)} = \\sigma_x and \\frac{d}{d \\sigma_x} \\operatorname{E_{n}}{(\\sigma_x)} = \\frac{d}{d \\sigma_x} \\sigma_x and \\frac{d}{d \\sigma_x} \\operatorname{E_{n}}{(\\sigma_x)} = 1 and \\frac{d}{d \\sigma_x} \\sigma_x = 1", "srepr_derivation": [["renaming_premise", "Equality(Function('E_n')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Symbol('\\\\sigma_x', commutative=True), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('\\\\sigma_x', commutative=True), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(t)} = \\sin{(t)}, then derive - \\cos{(t)} + \\frac{d}{d t} \\operatorname{F_{g}}{(t)} = 0, then obtain - \\frac{- \\cos{(t)} + \\frac{d}{d t} \\operatorname{F_{g}}{(t)}}{\\cos{(t)}} = 0", "derivation": "\\operatorname{F_{g}}{(t)} = \\sin{(t)} and \\operatorname{F_{g}}{(t)} - \\sin{(t)} = 0 and \\frac{d}{d t} (\\operatorname{F_{g}}{(t)} - \\sin{(t)}) = \\frac{d}{d t} 0 and - \\cos{(t)} + \\frac{d}{d t} \\operatorname{F_{g}}{(t)} = 0 and - \\frac{- \\cos{(t)} + \\frac{d}{d t} \\operatorname{F_{g}}{(t)}}{\\cos{(t)}} = 0", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["minus", 1, "sin(Symbol('t', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Function('F_g')(Symbol('t', commutative=True)), Mul(Integer(-1), sin(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(Function('F_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Integer(0))"], [["divide", 4, "Mul(Integer(-1), cos(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), cos(Symbol('t', commutative=True))), Derivative(Function('F_g')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Pow(cos(Symbol('t', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\hat{p}{(\\tilde{g},\\varepsilon,v)} = \\frac{\\varepsilon v}{\\tilde{g}}, then derive \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\hat{p}{(\\tilde{g},\\varepsilon,v)} = - \\frac{\\varepsilon^{2} v}{\\tilde{g}^{2}}, then obtain \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\varepsilon v}{\\tilde{g}} = - \\frac{\\varepsilon^{2} v}{\\tilde{g}^{2}}", "derivation": "\\hat{p}{(\\tilde{g},\\varepsilon,v)} = \\frac{\\varepsilon v}{\\tilde{g}} and \\frac{\\partial}{\\partial \\tilde{g}} \\hat{p}{(\\tilde{g},\\varepsilon,v)} = \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\varepsilon v}{\\tilde{g}} and \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\hat{p}{(\\tilde{g},\\varepsilon,v)} = \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\varepsilon v}{\\tilde{g}} and \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\hat{p}{(\\tilde{g},\\varepsilon,v)} = - \\frac{\\varepsilon^{2} v}{\\tilde{g}^{2}} and \\varepsilon \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\varepsilon v}{\\tilde{g}} = - \\frac{\\varepsilon^{2} v}{\\tilde{g}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["times", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-2)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Symbol('v', commutative=True)))"]]}, {"prompt": "Given t{(m)} = \\cos{(m)} and \\operatorname{F_{c}}{(m)} = m + \\cos{(m)}, then obtain (m + \\cos{(m)})^{m} + \\operatorname{F_{c}}^{m}{(m)} = 2 (m + \\cos{(m)})^{m}", "derivation": "t{(m)} = \\cos{(m)} and m + t{(m)} = m + \\cos{(m)} and \\operatorname{F_{c}}{(m)} = m + \\cos{(m)} and \\operatorname{F_{c}}^{m}{(m)} = (m + \\cos{(m)})^{m} and \\operatorname{F_{c}}^{m}{(m)} = (m + t{(m)})^{m} and (m + t{(m)})^{m} + \\operatorname{F_{c}}^{m}{(m)} = 2 (m + t{(m)})^{m} and (m + \\cos{(m)})^{m} + \\operatorname{F_{c}}^{m}{(m)} = 2 (m + \\cos{(m)})^{m}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["add", 1, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('t')(Symbol('m', commutative=True))), Add(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Function('F_c')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Add(Symbol('m', commutative=True), Function('t')(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["add", 5, "Pow(Add(Symbol('m', commutative=True), Function('t')(Symbol('m', commutative=True))), Symbol('m', commutative=True))"], "Equality(Add(Pow(Add(Symbol('m', commutative=True), Function('t')(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Function('F_c')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('m', commutative=True), Function('t')(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Pow(Add(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Function('F_c')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Integer(2), Pow(Add(Symbol('m', commutative=True), cos(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(h)} = \\sin{(h)}, then derive \\int \\frac{\\ddot{x}{(h)}}{h} dh + \\frac{1}{h} = F_{N} + \\operatorname{Si}{(h)} + \\frac{1}{h}, then obtain \\int \\frac{\\sin{(h)}}{h} dh + \\frac{1}{h} = F_{N} + \\operatorname{Si}{(h)} + \\frac{1}{h}", "derivation": "\\ddot{x}{(h)} = \\sin{(h)} and \\frac{\\ddot{x}{(h)}}{h} = \\frac{\\sin{(h)}}{h} and \\int \\frac{\\ddot{x}{(h)}}{h} dh = \\int \\frac{\\sin{(h)}}{h} dh and \\int \\frac{\\ddot{x}{(h)}}{h} dh + \\frac{1}{h} = \\int \\frac{\\sin{(h)}}{h} dh + \\frac{1}{h} and \\int \\frac{\\ddot{x}{(h)}}{h} dh + \\frac{1}{h} = F_{N} + \\operatorname{Si}{(h)} + \\frac{1}{h} and \\int \\frac{\\sin{(h)}}{h} dh + \\frac{1}{h} = F_{N} + \\operatorname{Si}{(h)} + \\frac{1}{h}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["add", 3, "Pow(Symbol('h', commutative=True), Integer(-1))"], "Equality(Add(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Pow(Symbol('h', commutative=True), Integer(-1))), Add(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Add(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\ddot{x}')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Pow(Symbol('h', commutative=True), Integer(-1))), Add(Symbol('F_N', commutative=True), Si(Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Pow(Symbol('h', commutative=True), Integer(-1))), Add(Symbol('F_N', commutative=True), Si(Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(q,\\hat{X})} = \\hat{X} + q, then obtain ((\\hat{X} + q) \\operatorname{F_{x}}{(q,\\hat{X})})^{2 q} ((\\hat{X} + q)^{2})^{2 q} = ((\\hat{X} + q)^{2})^{4 q}", "derivation": "\\operatorname{F_{x}}{(q,\\hat{X})} = \\hat{X} + q and (\\hat{X} + q) \\operatorname{F_{x}}{(q,\\hat{X})} = (\\hat{X} + q)^{2} and ((\\hat{X} + q) \\operatorname{F_{x}}{(q,\\hat{X})})^{q} = ((\\hat{X} + q)^{2})^{q} and ((\\hat{X} + q) \\operatorname{F_{x}}{(q,\\hat{X})})^{q} ((\\hat{X} + q)^{2})^{q} = ((\\hat{X} + q)^{2})^{2 q} and ((\\hat{X} + q) \\operatorname{F_{x}}{(q,\\hat{X})})^{2 q} ((\\hat{X} + q)^{2})^{2 q} = ((\\hat{X} + q)^{2})^{4 q}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Symbol('q', commutative=True)))"], [["times", 3, "Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Symbol('q', commutative=True))"], "Equality(Mul(Pow(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Symbol('q', commutative=True)), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Symbol('q', commutative=True))), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('q', commutative=True))))"], [["power", 4, 2], "Equality(Mul(Pow(Mul(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Function('F_x')(Symbol('q', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Integer(2), Symbol('q', commutative=True))), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Mul(Integer(2), Symbol('q', commutative=True)))), Pow(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('q', commutative=True)), Integer(2)), Mul(Integer(4), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\rho_{f}{(\\mathbf{J}_M,M)} = e^{\\frac{M}{\\mathbf{J}_M}}, then obtain \\mathbf{A} \\int \\dot{x}{(\\mathbf{A})} d\\mathbf{A} + \\rho_{f}{(\\mathbf{J}_M,M)} = \\mathbf{A} \\int \\log{(\\mathbf{A})} d\\mathbf{A} + \\rho_{f}{(\\mathbf{J}_M,M)}", "derivation": "\\dot{x}{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\int \\dot{x}{(\\mathbf{A})} d\\mathbf{A} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\rho_{f}{(\\mathbf{J}_M,M)} = e^{\\frac{M}{\\mathbf{J}_M}} and \\mathbf{A} \\int \\dot{x}{(\\mathbf{A})} d\\mathbf{A} = \\mathbf{A} \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\mathbf{A} \\int \\dot{x}{(\\mathbf{A})} d\\mathbf{A} + e^{\\frac{M}{\\mathbf{J}_M}} = \\mathbf{A} \\int \\log{(\\mathbf{A})} d\\mathbf{A} + e^{\\frac{M}{\\mathbf{J}_M}} and \\mathbf{A} \\int \\dot{x}{(\\mathbf{A})} d\\mathbf{A} + \\rho_{f}{(\\mathbf{J}_M,M)} = \\mathbf{A} \\int \\log{(\\mathbf{A})} d\\mathbf{A} + \\rho_{f}{(\\mathbf{J}_M,M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M', commutative=True)), exp(Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)))))"], [["times", 2, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["minus", 4, "Mul(Integer(-1), exp(Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1)))))"], "Equality(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), exp(Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), exp(Mul(Symbol('M', commutative=True), Pow(Symbol('\\\\mathbf{J}_M', commutative=True), Integer(-1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(Function('\\\\dot{x}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M', commutative=True))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Function('\\\\rho_f')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('M', commutative=True))))"]]}, {"prompt": "Given t{(Z)} = \\log{(\\log{(Z)})} and y{(Z)} = t^{Z}{(Z)}, then obtain \\sin{(\\frac{d}{d Z} (- Z + y{(Z)}))} = \\sin{(\\frac{d}{d Z} (- Z + \\log{(\\log{(Z)})}^{Z}))}", "derivation": "t{(Z)} = \\log{(\\log{(Z)})} and t^{Z}{(Z)} = \\log{(\\log{(Z)})}^{Z} and y{(Z)} = t^{Z}{(Z)} and y{(Z)} = \\log{(\\log{(Z)})}^{Z} and - Z + y{(Z)} = - Z + \\log{(\\log{(Z)})}^{Z} and \\frac{d}{d Z} (- Z + y{(Z)}) = \\frac{d}{d Z} (- Z + \\log{(\\log{(Z)})}^{Z}) and \\sin{(\\frac{d}{d Z} (- Z + y{(Z)}))} = \\sin{(\\frac{d}{d Z} (- Z + \\log{(\\log{(Z)})}^{Z}))}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('Z', commutative=True)), log(log(Symbol('Z', commutative=True))))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('t')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(log(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('Z', commutative=True)), Pow(Function('t')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('y')(Symbol('Z', commutative=True)), Pow(log(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["minus", 4, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('y')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Pow(log(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('y')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Pow(log(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["sin", 6], "Equality(sin(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('y')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))), sin(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Pow(log(log(Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi{(\\hat{x})} = \\sin{(e^{\\hat{x}})}, then obtain \\cos^{\\hat{x}}{(\\varphi{(\\hat{x})} - e^{\\hat{x}})} = \\cos^{\\hat{x}}{(e^{\\hat{x}} - \\sin{(e^{\\hat{x}})})}", "derivation": "\\varphi{(\\hat{x})} = \\sin{(e^{\\hat{x}})} and \\varphi{(\\hat{x})} - e^{\\hat{x}} = - e^{\\hat{x}} + \\sin{(e^{\\hat{x}})} and \\cos{(\\varphi{(\\hat{x})} - e^{\\hat{x}})} = \\cos{(e^{\\hat{x}} - \\sin{(e^{\\hat{x}})})} and \\cos^{\\hat{x}}{(\\varphi{(\\hat{x})} - e^{\\hat{x}})} = \\cos^{\\hat{x}}{(e^{\\hat{x}} - \\sin{(e^{\\hat{x}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True)), sin(exp(Symbol('\\\\hat{x}', commutative=True))))"], [["minus", 1, "exp(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))), sin(exp(Symbol('\\\\hat{x}', commutative=True)))))"], [["cos", 2], "Equality(cos(Add(Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))))), cos(Add(exp(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('\\\\hat{x}', commutative=True)))))))"], [["power", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(cos(Add(Function('\\\\varphi')(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{x}', commutative=True))))), Symbol('\\\\hat{x}', commutative=True)), Pow(cos(Add(exp(Symbol('\\\\hat{x}', commutative=True)), Mul(Integer(-1), sin(exp(Symbol('\\\\hat{x}', commutative=True)))))), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(G)} = \\cos{(e^{G})} and \\mathbf{p}{(\\chi)} = e^{\\chi}, then obtain \\sigma_{p}{(G)} e^{G} + \\int \\mathbf{p}^{\\chi}{(\\chi)} d\\chi = e^{G} \\cos{(e^{G})} + \\int \\mathbf{p}^{\\chi}{(\\chi)} d\\chi", "derivation": "\\sigma_{p}{(G)} = \\cos{(e^{G})} and \\sigma_{p}{(G)} e^{G} = e^{G} \\cos{(e^{G})} and \\mathbf{p}{(\\chi)} = e^{\\chi} and \\mathbf{p}^{\\chi}{(\\chi)} = (e^{\\chi})^{\\chi} and \\int \\mathbf{p}^{\\chi}{(\\chi)} d\\chi = \\int (e^{\\chi})^{\\chi} d\\chi and \\sigma_{p}{(G)} e^{G} + \\int (e^{\\chi})^{\\chi} d\\chi = e^{G} \\cos{(e^{G})} + \\int (e^{\\chi})^{\\chi} d\\chi and \\sigma_{p}{(G)} e^{G} + \\int \\mathbf{p}^{\\chi}{(\\chi)} d\\chi = e^{G} \\cos{(e^{G})} + \\int \\mathbf{p}^{\\chi}{(\\chi)} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True))))"], [["times", 1, "exp(Symbol('G', commutative=True))"], "Equality(Mul(Function('\\\\sigma_p')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))), Mul(exp(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))))"], ["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), exp(Symbol('\\\\chi', commutative=True)))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Integral(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))), Integral(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(exp(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))), Integral(Pow(exp(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Function('\\\\sigma_p')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True))), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(exp(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True)))), Integral(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\dot{z},S)} = \\cos{(S \\dot{z})} and C{(\\dot{z},S)} = \\int \\operatorname{r_{0}}{(\\dot{z},S)} d\\dot{z}, then obtain C{(\\dot{z},S)} = \\int \\cos{(S \\dot{z})} d\\dot{z}", "derivation": "\\operatorname{r_{0}}{(\\dot{z},S)} = \\cos{(S \\dot{z})} and \\int \\operatorname{r_{0}}{(\\dot{z},S)} d\\dot{z} = \\int \\cos{(S \\dot{z})} d\\dot{z} and C{(\\dot{z},S)} = \\int \\operatorname{r_{0}}{(\\dot{z},S)} d\\dot{z} and C{(\\dot{z},S)} = \\int \\cos{(S \\dot{z})} d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('S', commutative=True)), cos(Mul(Symbol('S', commutative=True), Symbol('\\\\dot{z}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\dot{z}', commutative=True), Symbol('S', commutative=True)), Integral(Function('r_0')(Symbol('\\\\dot{z}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('C')(Symbol('\\\\dot{z}', commutative=True), Symbol('S', commutative=True)), Integral(cos(Mul(Symbol('S', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given z{(A_{y})} = e^{A_{y}}, then derive 0 = z^{*} + e^{A_{y}} - \\int z{(A_{y})} dA_{y}, then obtain \\int 0 dz^{*} = \\int (z^{*} + z{(A_{y})} - \\int e^{A_{y}} dA_{y}) dz^{*}", "derivation": "z{(A_{y})} = e^{A_{y}} and \\int z{(A_{y})} dA_{y} = \\int e^{A_{y}} dA_{y} and z{(A_{y})} + \\int z{(A_{y})} dA_{y} = z{(A_{y})} + \\int e^{A_{y}} dA_{y} and 0 = - \\int z{(A_{y})} dA_{y} + \\int e^{A_{y}} dA_{y} and 0 = z^{*} + e^{A_{y}} - \\int z{(A_{y})} dA_{y} and 0 = z^{*} + e^{A_{y}} - \\int e^{A_{y}} dA_{y} and 0 = z^{*} + z{(A_{y})} - \\int e^{A_{y}} dA_{y} and \\int 0 dz^{*} = \\int (z^{*} + z{(A_{y})} - \\int e^{A_{y}} dA_{y}) dz^{*}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["integrate", 1, "Symbol('A_y', commutative=True)"], "Equality(Integral(Function('z')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], [["add", 2, "Function('z')(Symbol('A_y', commutative=True))"], "Equality(Add(Function('z')(Symbol('A_y', commutative=True)), Integral(Function('z')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Add(Function('z')(Symbol('A_y', commutative=True)), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"], [["minus", 3, "Add(Function('z')(Symbol('A_y', commutative=True)), Integral(Function('z')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('z')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Integer(0), Add(Symbol('z^*', commutative=True), exp(Symbol('A_y', commutative=True)), Mul(Integer(-1), Integral(Function('z')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Symbol('z^*', commutative=True), exp(Symbol('A_y', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Add(Symbol('z^*', commutative=True), Function('z')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))))"], [["integrate", 7, "Symbol('z^*', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('z^*', commutative=True))), Integral(Add(Symbol('z^*', commutative=True), Function('z')(Symbol('A_y', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True))))), Tuple(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given l{(\\pi,f^{*})} = \\pi^{f^{*}}, then obtain \\frac{\\partial}{\\partial \\pi} (f^{*} + \\pi^{- 2 f^{*}} l^{2}{(\\pi,f^{*})}) = \\frac{d}{d \\pi} (f^{*} + 1)", "derivation": "l{(\\pi,f^{*})} = \\pi^{f^{*}} and \\pi^{- f^{*}} l{(\\pi,f^{*})} = 1 and f^{*} + \\pi^{- f^{*}} l{(\\pi,f^{*})} = f^{*} + 1 and \\pi^{- 2 f^{*}} l{(\\pi,f^{*})} = \\pi^{- f^{*}} and f^{*} + \\pi^{- 2 f^{*}} l^{2}{(\\pi,f^{*})} = f^{*} + 1 and \\frac{\\partial}{\\partial \\pi} (f^{*} + \\pi^{- 2 f^{*}} l^{2}{(\\pi,f^{*})}) = \\frac{d}{d \\pi} (f^{*} + 1)", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True)), Pow(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True)))"], [["divide", 1, "Pow(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True))), Integer(1))"], [["add", 2, "Symbol('f^*', commutative=True)"], "Equality(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True)))), Add(Symbol('f^*', commutative=True), Integer(1)))"], [["divide", 2, "Pow(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True))), Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True))), Pow(Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True)), Integer(2)))), Add(Symbol('f^*', commutative=True), Integer(1)))"], [["differentiate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Add(Symbol('f^*', commutative=True), Mul(Pow(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True))), Pow(Function('l')(Symbol('\\\\pi', commutative=True), Symbol('f^*', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), Integer(1)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(\\lambda,\\hat{p}_0)} = \\hat{p}_0 \\lambda and \\varphi{(\\lambda,\\hat{p}_0)} = \\frac{1}{S{(\\lambda,\\hat{p}_0)}}, then obtain (\\int \\varphi{(\\lambda,\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\int \\frac{1}{\\hat{p}_0 \\lambda} d\\hat{p}_0)^{\\hat{p}_0}", "derivation": "S{(\\lambda,\\hat{p}_0)} = \\hat{p}_0 \\lambda and \\varphi{(\\lambda,\\hat{p}_0)} = \\frac{1}{S{(\\lambda,\\hat{p}_0)}} and \\varphi{(\\lambda,\\hat{p}_0)} = \\frac{1}{\\hat{p}_0 \\lambda} and \\int \\varphi{(\\lambda,\\hat{p}_0)} d\\hat{p}_0 = \\int \\frac{1}{S{(\\lambda,\\hat{p}_0)}} d\\hat{p}_0 and \\int \\frac{1}{\\hat{p}_0 \\lambda} d\\hat{p}_0 = \\int \\frac{1}{S{(\\lambda,\\hat{p}_0)}} d\\hat{p}_0 and \\int \\varphi{(\\lambda,\\hat{p}_0)} d\\hat{p}_0 = \\int \\frac{1}{\\hat{p}_0 \\lambda} d\\hat{p}_0 and (\\int \\varphi{(\\lambda,\\hat{p}_0)} d\\hat{p}_0)^{\\hat{p}_0} = (\\int \\frac{1}{\\hat{p}_0 \\lambda} d\\hat{p}_0)^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Function('S')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Pow(Function('S')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Pow(Function('S')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["power", 6, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Integral(Function('\\\\varphi')(Symbol('\\\\lambda', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\hat{p}_0', commutative=True), Integer(-1)), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\hat{X}{(\\psi,v_{2})} = \\psi \\log{(v_{2})}, then derive \\int \\hat{X}{(\\psi,v_{2})} dv_{2} = C_{1} + \\psi v_{2} \\log{(v_{2})} - \\psi v_{2}, then obtain 1 = \\frac{\\int \\psi \\log{(v_{2})} dv_{2}}{C_{1} - \\psi v_{2} + v_{2} \\hat{X}{(\\psi,v_{2})}}", "derivation": "\\hat{X}{(\\psi,v_{2})} = \\psi \\log{(v_{2})} and \\int \\hat{X}{(\\psi,v_{2})} dv_{2} = \\int \\psi \\log{(v_{2})} dv_{2} and \\int \\hat{X}{(\\psi,v_{2})} dv_{2} = C_{1} + \\psi v_{2} \\log{(v_{2})} - \\psi v_{2} and C_{1} + \\psi v_{2} \\log{(v_{2})} - \\psi v_{2} = \\int \\psi \\log{(v_{2})} dv_{2} and \\frac{C_{1} + \\psi v_{2} \\log{(v_{2})} - \\psi v_{2}}{\\int \\psi \\log{(v_{2})} dv_{2}} = 1 and \\frac{C_{1} - \\psi v_{2} + v_{2} \\hat{X}{(\\psi,v_{2})}}{\\int \\psi \\log{(v_{2})} dv_{2}} = 1 and 1 = \\frac{\\int \\psi \\log{(v_{2})} dv_{2}}{C_{1} - \\psi v_{2} + v_{2} \\hat{X}{(\\psi,v_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True), log(Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True), log(Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True))), Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["divide", 4, "Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True)))"], "Equality(Mul(Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True), log(Symbol('v_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True))), Pow(Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('v_2', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)))), Pow(Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1))), Integer(1))"], [["divide", 6, "Mul(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('v_2', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)))), Pow(Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)), Mul(Symbol('v_2', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\psi', commutative=True), Symbol('v_2', commutative=True)))), Integer(-1)), Integral(Mul(Symbol('\\\\psi', commutative=True), log(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(J)} = e^{J}, then obtain J + e^{J} = \\frac{(J + e^{J}) e^{J}}{\\operatorname{A_{x}}{(J)}}", "derivation": "\\operatorname{A_{x}}{(J)} = e^{J} and \\operatorname{A_{x}}{(J)} e^{J} = e^{2 J} and 1 = \\frac{e^{J}}{\\operatorname{A_{x}}{(J)}} and J + e^{J} = \\frac{(J + e^{J}) e^{J}}{\\operatorname{A_{x}}{(J)}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["times", 1, "exp(Symbol('J', commutative=True))"], "Equality(Mul(Function('A_x')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True))), exp(Mul(Integer(2), Symbol('J', commutative=True))))"], [["divide", 2, "Mul(Function('A_x')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('A_x')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))))"], [["times", 3, "Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True)))"], "Equality(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Mul(Add(Symbol('J', commutative=True), exp(Symbol('J', commutative=True))), Pow(Function('A_x')(Symbol('J', commutative=True)), Integer(-1)), exp(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\hat{p})} = e^{\\hat{p}} and \\operatorname{P_{e}}{(f,G)} = - G + f, then derive \\int \\omega{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} + e^{\\hat{p}}, then derive M + e^{\\hat{p}} = V_{\\mathbf{B}} + e^{\\hat{p}}, then obtain \\operatorname{P_{e}}{(f,G)} + \\int (M + e^{\\hat{p}}) d\\hat{p} = \\operatorname{P_{e}}{(f,G)} + \\int (V_{\\mathbf{B}} + e^{\\hat{p}}) d\\hat{p}", "derivation": "\\omega{(\\hat{p})} = e^{\\hat{p}} and \\int \\omega{(\\hat{p})} d\\hat{p} = \\int e^{\\hat{p}} d\\hat{p} and \\int \\omega{(\\hat{p})} d\\hat{p} = V_{\\mathbf{B}} + e^{\\hat{p}} and \\int e^{\\hat{p}} d\\hat{p} = V_{\\mathbf{B}} + e^{\\hat{p}} and \\operatorname{P_{e}}{(f,G)} = - G + f and M + e^{\\hat{p}} = V_{\\mathbf{B}} + e^{\\hat{p}} and \\int (M + e^{\\hat{p}}) d\\hat{p} = \\int (V_{\\mathbf{B}} + e^{\\hat{p}}) d\\hat{p} and - G + f + \\int (M + e^{\\hat{p}}) d\\hat{p} = - G + f + \\int (V_{\\mathbf{B}} + e^{\\hat{p}}) d\\hat{p} and \\operatorname{P_{e}}{(f,G)} + \\int (M + e^{\\hat{p}}) d\\hat{p} = \\operatorname{P_{e}}{(f,G)} + \\int (V_{\\mathbf{B}} + e^{\\hat{p}}) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], ["get_premise", "Equality(Function('P_e')(Symbol('f', commutative=True), Symbol('G', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('f', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('M', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["integrate", 6, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Symbol('M', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True))))"], [["add", 7, "Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('f', commutative=True), Integral(Add(Symbol('M', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Symbol('f', commutative=True), Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Add(Function('P_e')(Symbol('f', commutative=True), Symbol('G', commutative=True)), Integral(Add(Symbol('M', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))), Add(Function('P_e')(Symbol('f', commutative=True), Symbol('G', commutative=True)), Integral(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), exp(Symbol('\\\\hat{p}', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\hat{x})} = \\hat{x}, then obtain \\hat{x} (\\hat{x} - \\operatorname{F_{N}}{(\\hat{x})})^{\\hat{x}} = \\hat{x}", "derivation": "\\operatorname{F_{N}}{(\\hat{x})} = \\hat{x} and 0 = \\hat{x} - \\operatorname{F_{N}}{(\\hat{x})} and 0^{\\hat{x}} = (\\hat{x} - \\operatorname{F_{N}}{(\\hat{x})})^{\\hat{x}} and 0^{\\hat{x}} \\hat{x} \\operatorname{F_{N}}{(\\hat{x})} = \\hat{x} (\\hat{x} - \\operatorname{F_{N}}{(\\hat{x})})^{\\hat{x}} \\operatorname{F_{N}}{(\\hat{x})} and 0^{\\hat{x}} \\hat{x} = \\hat{x} (\\hat{x} - \\operatorname{F_{N}}{(\\hat{x})})^{\\hat{x}} and \\hat{x} (\\hat{x} - \\operatorname{F_{N}}{(\\hat{x})})^{\\hat{x}} = \\hat{x}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True))"], [["minus", 1, "Function('F_N')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\hat{x}', commutative=True)), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)))"], [["times", 3, "Mul(Symbol('\\\\hat{x}', commutative=True), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True), Function('F_N')(Symbol('\\\\hat{x}', commutative=True))), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True)), Function('F_N')(Symbol('\\\\hat{x}', commutative=True))))"], [["divide", 4, "Function('F_N')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Function('F_N')(Symbol('\\\\hat{x}', commutative=True)))), Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\rho_f)} = \\cos{(\\rho_f)}, then obtain \\cos{(\\rho_f)} + \\frac{\\cos{(\\rho_f)}}{\\dot{\\mathbf{r}}^{2}{(\\rho_f)}} = \\cos{(\\rho_f)} + \\frac{\\cos^{3}{(\\rho_f)}}{\\dot{\\mathbf{r}}^{4}{(\\rho_f)}}", "derivation": "\\dot{\\mathbf{r}}{(\\rho_f)} = \\cos{(\\rho_f)} and \\rho_f \\dot{\\mathbf{r}}{(\\rho_f)} = \\rho_f \\cos{(\\rho_f)} and 1 = \\frac{\\cos{(\\rho_f)}}{\\dot{\\mathbf{r}}{(\\rho_f)}} and \\frac{1}{\\dot{\\mathbf{r}}{(\\rho_f)}} = \\frac{\\cos{(\\rho_f)}}{\\dot{\\mathbf{r}}^{2}{(\\rho_f)}} and \\cos{(\\rho_f)} + \\frac{1}{\\dot{\\mathbf{r}}{(\\rho_f)}} = \\cos{(\\rho_f)} + \\frac{\\cos{(\\rho_f)}}{\\dot{\\mathbf{r}}^{2}{(\\rho_f)}} and \\cos{(\\rho_f)} + \\frac{\\cos{(\\rho_f)}}{\\dot{\\mathbf{r}}^{2}{(\\rho_f)}} = \\cos{(\\rho_f)} + \\frac{\\cos^{3}{(\\rho_f)}}{\\dot{\\mathbf{r}}^{4}{(\\rho_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), cos(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\rho_f', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho_f', commutative=True))))"], [["divide", 3, "Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True))"], "Equality(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-2)), cos(Symbol('\\\\rho_f', commutative=True))))"], [["add", 4, "cos(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(cos(Symbol('\\\\rho_f', commutative=True)), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-1))), Add(cos(Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-2)), cos(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(cos(Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-2)), cos(Symbol('\\\\rho_f', commutative=True)))), Add(cos(Symbol('\\\\rho_f', commutative=True)), Mul(Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_f', commutative=True)), Integer(-4)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(3)))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\Omega)} = \\sin{(\\Omega)}, then derive (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = \\cos^{\\Omega}{(\\Omega)}, then obtain \\operatorname{v_{1}}{(\\Omega)} (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = \\operatorname{v_{1}}{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega}", "derivation": "\\tilde{g}^*{(\\Omega)} = \\sin{(\\Omega)} and \\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)} = \\frac{d}{d \\Omega} \\sin{(\\Omega)} and (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega} and (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = \\cos^{\\Omega}{(\\Omega)} and \\operatorname{v_{1}}{(\\Omega)} (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = \\operatorname{v_{1}}{(\\Omega)} \\cos^{\\Omega}{(\\Omega)} and \\cos^{\\Omega}{(\\Omega)} = (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega} and \\operatorname{v_{1}}{(\\Omega)} (\\frac{d}{d \\Omega} \\tilde{g}^*{(\\Omega)})^{\\Omega} = \\operatorname{v_{1}}{(\\Omega)} (\\frac{d}{d \\Omega} \\sin{(\\Omega)})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)), Pow(cos(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["times", 4, "Function('v_1')(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Mul(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Pow(cos(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(cos(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))), Mul(Function('v_1')(Symbol('\\\\Omega', commutative=True)), Pow(Derivative(sin(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given U{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})}, then derive U{(\\mathbf{H})} = \\cos{(\\mathbf{H})}, then obtain (\\mathbf{H} + \\cos{(\\mathbf{H})})^{3} = (\\mathbf{H} + \\cos{(\\mathbf{H})})^{2} (\\mathbf{H} + \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})})", "derivation": "U{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} and U{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\cos{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} and \\mathbf{H} + \\cos{(\\mathbf{H})} = \\mathbf{H} + \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})} and (\\mathbf{H} + \\cos{(\\mathbf{H})})^{2} = (\\mathbf{H} + \\cos{(\\mathbf{H})}) (\\mathbf{H} + \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})}) and (\\mathbf{H} + \\cos{(\\mathbf{H})})^{3} = (\\mathbf{H} + \\cos{(\\mathbf{H})})^{2} (\\mathbf{H} + \\frac{d}{d \\mathbf{H}} \\sin{(\\mathbf{H})})", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\mathbf{H}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('U')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('\\\\mathbf{H}', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["add", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"], [["times", 4, "Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))))"], [["times", 5, "Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(3)), Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2)), Add(Symbol('\\\\mathbf{H}', commutative=True), Derivative(sin(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta_2, then obtain (\\frac{d}{d \\theta_2} 0)^{\\mathbf{J}_P} = (\\frac{\\partial}{\\partial \\theta_2} (\\mathbf{J}_P \\theta_2 - \\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)}))^{\\mathbf{J}_P}", "derivation": "\\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)} = \\mathbf{J}_P \\theta_2 and 0 = \\mathbf{J}_P \\theta_2 - \\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)} and \\frac{d}{d \\theta_2} 0 = \\frac{\\partial}{\\partial \\theta_2} (\\mathbf{J}_P \\theta_2 - \\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)}) and (\\frac{d}{d \\theta_2} 0)^{\\mathbf{J}_P} = (\\frac{\\partial}{\\partial \\theta_2} (\\mathbf{J}_P \\theta_2 - \\operatorname{A_{y}}{(\\theta_2,\\mathbf{J}_P)}))^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["minus", 1, "Function('A_y')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Derivative(Add(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('\\\\theta_2', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given s{(\\mathbf{M},F_{g})} = \\frac{\\mathbf{M}}{F_{g}} and \\hat{\\mathbf{x}}{(\\mathbf{M},F_{g})} = \\frac{\\partial}{\\partial F_{g}} s{(\\mathbf{M},F_{g})}, then obtain \\frac{\\hat{\\mathbf{x}}{(\\mathbf{M},F_{g})}}{\\mathbf{M}} = \\frac{\\frac{\\partial}{\\partial F_{g}} \\frac{\\mathbf{M}}{F_{g}}}{\\mathbf{M}}", "derivation": "s{(\\mathbf{M},F_{g})} = \\frac{\\mathbf{M}}{F_{g}} and \\frac{\\partial}{\\partial F_{g}} s{(\\mathbf{M},F_{g})} = \\frac{\\partial}{\\partial F_{g}} \\frac{\\mathbf{M}}{F_{g}} and \\frac{\\frac{\\partial}{\\partial F_{g}} s{(\\mathbf{M},F_{g})}}{\\mathbf{M}} = \\frac{\\frac{\\partial}{\\partial F_{g}} \\frac{\\mathbf{M}}{F_{g}}}{\\mathbf{M}} and \\hat{\\mathbf{x}}{(\\mathbf{M},F_{g})} = \\frac{\\partial}{\\partial F_{g}} s{(\\mathbf{M},F_{g})} and \\frac{\\hat{\\mathbf{x}}{(\\mathbf{M},F_{g})}}{\\mathbf{M}} = \\frac{\\frac{\\partial}{\\partial F_{g}} \\frac{\\mathbf{M}}{F_{g}}}{\\mathbf{M}}", "srepr_derivation": [["get_premise", "Equality(Function('s')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Function('s')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True)), Derivative(Function('s')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('F_g', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(E)} = \\log{(e^{E})}, then obtain - \\log{(e^{E})} + \\int (\\operatorname{P_{g}}{(E)} - \\log{(e^{E})}) dE = - \\log{(e^{E})} + \\int 0 dE", "derivation": "\\operatorname{P_{g}}{(E)} = \\log{(e^{E})} and \\operatorname{P_{g}}{(E)} - \\log{(e^{E})} = 0 and \\int (\\operatorname{P_{g}}{(E)} - \\log{(e^{E})}) dE = \\int 0 dE and - \\log{(e^{E})} + \\int (\\operatorname{P_{g}}{(E)} - \\log{(e^{E})}) dE = - \\log{(e^{E})} + \\int 0 dE", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('E', commutative=True)), log(exp(Symbol('E', commutative=True))))"], [["minus", 1, "log(exp(Symbol('E', commutative=True)))"], "Equality(Add(Function('P_g')(Symbol('E', commutative=True)), Mul(Integer(-1), log(exp(Symbol('E', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Function('P_g')(Symbol('E', commutative=True)), Mul(Integer(-1), log(exp(Symbol('E', commutative=True))))), Tuple(Symbol('E', commutative=True))), Integral(Integer(0), Tuple(Symbol('E', commutative=True))))"], [["minus", 3, "log(exp(Symbol('E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), log(exp(Symbol('E', commutative=True)))), Integral(Add(Function('P_g')(Symbol('E', commutative=True)), Mul(Integer(-1), log(exp(Symbol('E', commutative=True))))), Tuple(Symbol('E', commutative=True)))), Add(Mul(Integer(-1), log(exp(Symbol('E', commutative=True)))), Integral(Integer(0), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = \\mathbf{f} - i + u, then derive - i \\frac{\\partial}{\\partial u} \\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = - i, then obtain \\int - i \\frac{\\partial}{\\partial u} (\\mathbf{f} - i + u) di = \\int - i di", "derivation": "\\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = \\mathbf{f} - i + u and - i \\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = - i (\\mathbf{f} - i + u) and \\frac{\\partial}{\\partial u} - i \\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = \\frac{\\partial}{\\partial u} - i (\\mathbf{f} - i + u) and - i \\frac{\\partial}{\\partial u} \\operatorname{a^{\\dagger}}{(u,i,\\mathbf{f})} = - i and - i \\frac{\\partial}{\\partial u} (\\mathbf{f} - i + u) = - i and \\int - i \\frac{\\partial}{\\partial u} (\\mathbf{f} - i + u) di = \\int - i di", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('u', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('u', commutative=True)))"], [["times", 1, "Mul(Integer(-1), Symbol('i', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Function('a^{\\\\dagger}')(Symbol('u', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('u', commutative=True))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('i', commutative=True), Function('a^{\\\\dagger}')(Symbol('u', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('i', commutative=True), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('u', commutative=True), Symbol('i', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('i', commutative=True), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["integrate", 5, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('i', commutative=True), Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('i', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Tuple(Symbol('i', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\phi_1)} = e^{e^{\\phi_1}}, then derive - \\mathbf{g} - \\operatorname{Ei}{(2 e^{\\phi_1})} + \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1 = 0, then obtain (- \\mathbf{g} - \\operatorname{Ei}{(2 e^{\\phi_1})} + \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1)^{\\mathbf{g}} = 0^{\\mathbf{g}}", "derivation": "\\operatorname{v_{y}}{(\\phi_1)} = e^{e^{\\phi_1}} and \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} = e^{2 e^{\\phi_1}} and \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1 = \\int e^{2 e^{\\phi_1}} d\\phi_1 and \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1 - \\int e^{2 e^{\\phi_1}} d\\phi_1 = 0 and - \\mathbf{g} - \\operatorname{Ei}{(2 e^{\\phi_1})} + \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1 = 0 and (- \\mathbf{g} - \\operatorname{Ei}{(2 e^{\\phi_1})} + \\int \\operatorname{v_{y}}{(\\phi_1)} e^{e^{\\phi_1}} d\\phi_1)^{\\mathbf{g}} = 0^{\\mathbf{g}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True))))"], [["times", 1, "exp(exp(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True)))), exp(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(exp(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["minus", 3, "Integral(exp(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Integral(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))), Mul(Integer(-1), Integral(exp(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True))))), Integer(0))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Ei(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True))))), Integral(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Integer(0))"], [["power", 5, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Ei(Mul(Integer(2), exp(Symbol('\\\\phi_1', commutative=True))))), Integral(Mul(Function('v_y')(Symbol('\\\\phi_1', commutative=True)), exp(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{g}', commutative=True)))"]]}, {"prompt": "Given \\rho{(F_{x})} = \\sin{(F_{x})}, then obtain F_{x} \\rho{(F_{x})} + \\sin{(F_{x} \\sin{(F_{x})})} = F_{x} \\sin{(F_{x})} + \\sin{(F_{x} \\sin{(F_{x})})}", "derivation": "\\rho{(F_{x})} = \\sin{(F_{x})} and F_{x} \\rho{(F_{x})} = F_{x} \\sin{(F_{x})} and \\sin{(F_{x} \\rho{(F_{x})})} = \\sin{(F_{x} \\sin{(F_{x})})} and F_{x} \\rho{(F_{x})} + \\sin{(F_{x} \\rho{(F_{x})})} = F_{x} \\sin{(F_{x})} + \\sin{(F_{x} \\rho{(F_{x})})} and F_{x} \\rho{(F_{x})} + \\sin{(F_{x} \\sin{(F_{x})})} = F_{x} \\sin{(F_{x})} + \\sin{(F_{x} \\sin{(F_{x})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('F_x', commutative=True)), sin(Symbol('F_x', commutative=True)))"], [["times", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))), Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True)))), sin(Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True)))))"], [["add", 2, "sin(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))))"], "Equality(Add(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))), sin(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))))), Add(Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True))), sin(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('F_x', commutative=True), Function('\\\\rho')(Symbol('F_x', commutative=True))), sin(Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True))))), Add(Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True))), sin(Mul(Symbol('F_x', commutative=True), sin(Symbol('F_x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(U,\\Omega)} = U \\Omega, then obtain \\frac{\\frac{\\partial}{\\partial U} (- U + \\operatorname{v_{t}}^{U}{(U,\\Omega)})}{U \\Omega} = \\frac{\\frac{\\partial}{\\partial U} (- U + (U \\Omega)^{U})}{U \\Omega}", "derivation": "\\operatorname{v_{t}}{(U,\\Omega)} = U \\Omega and \\operatorname{v_{t}}^{U}{(U,\\Omega)} = (U \\Omega)^{U} and - U + \\operatorname{v_{t}}^{U}{(U,\\Omega)} = - U + (U \\Omega)^{U} and \\frac{\\partial}{\\partial U} (- U + \\operatorname{v_{t}}^{U}{(U,\\Omega)}) = \\frac{\\partial}{\\partial U} (- U + (U \\Omega)^{U}) and \\frac{\\frac{\\partial}{\\partial U} (- U + \\operatorname{v_{t}}^{U}{(U,\\Omega)})}{U \\Omega} = \\frac{\\frac{\\partial}{\\partial U} (- U + (U \\Omega)^{U})}{U \\Omega}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('v_t')(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True)))"], [["minus", 2, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Function('v_t')(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('U', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Function('v_t')(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Function('v_t')(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Mul(Pow(Symbol('U', commutative=True), Integer(-1)), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\pi{(t,\\hat{p}_0)} = \\hat{p}_0 + t, then obtain \\frac{- \\hat{p}_0 + 2 (\\hat{p}_0 + t)^{2}}{2 t} = \\frac{- \\hat{p}_0 + (\\hat{p}_0 + t) (2 \\hat{p}_0 + 2 t)}{2 t}", "derivation": "\\pi{(t,\\hat{p}_0)} = \\hat{p}_0 + t and 2 \\pi{(t,\\hat{p}_0)} = \\hat{p}_0 + t + \\pi{(t,\\hat{p}_0)} and 2 (\\hat{p}_0 + t) \\pi{(t,\\hat{p}_0)} = (\\hat{p}_0 + t) (\\hat{p}_0 + t + \\pi{(t,\\hat{p}_0)}) and - \\hat{p}_0 + 2 (\\hat{p}_0 + t) \\pi{(t,\\hat{p}_0)} = - \\hat{p}_0 + (\\hat{p}_0 + t) (\\hat{p}_0 + t + \\pi{(t,\\hat{p}_0)}) and - \\hat{p}_0 + 2 (\\hat{p}_0 + t)^{2} = - \\hat{p}_0 + (\\hat{p}_0 + t) (2 \\hat{p}_0 + 2 t) and \\frac{- \\hat{p}_0 + 2 (\\hat{p}_0 + t)^{2}}{2 t} = \\frac{- \\hat{p}_0 + (\\hat{p}_0 + t) (2 \\hat{p}_0 + 2 t)}{2 t}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)))"], [["add", 1, "Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))))"], [["minus", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True), Function('\\\\pi')(Symbol('t', commutative=True), Symbol('\\\\hat{p}_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Symbol('t', commutative=True))))))"], [["divide", 5, "Mul(Integer(2), Symbol('t', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Pow(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Integer(2))))), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('t', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(2), Symbol('t', commutative=True)))))))"]]}, {"prompt": "Given \\ddot{x}{(C,U)} = \\frac{C}{U}, then obtain \\frac{\\int \\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)} dC}{\\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U}} = \\frac{\\int \\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U} dC}{\\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U}}", "derivation": "\\ddot{x}{(C,U)} = \\frac{C}{U} and \\ddot{x}^{U}{(C,U)} = (\\frac{C}{U})^{U} and \\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)} = \\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U} and \\int \\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)} dC = \\int \\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U} dC and \\frac{\\int \\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)} dC}{\\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)}} = \\frac{\\int \\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U} dC}{\\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)}} and \\frac{\\int \\frac{\\partial}{\\partial U} \\ddot{x}^{U}{(C,U)} dC}{\\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U}} = \\frac{\\int \\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U} dC}{\\frac{\\partial}{\\partial U} (\\frac{C}{U})^{U}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('U', commutative=True)"], "Equality(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)))"], [["differentiate", 2, "Symbol('U', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))))"], [["divide", 4, "Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Pow(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('U', commutative=True)), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)), Integral(Derivative(Pow(Mul(Symbol('C', commutative=True), Pow(Symbol('U', commutative=True), Integer(-1))), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(F_{g})} = \\sin{(\\cos{(F_{g})})}, then obtain \\int \\frac{\\int \\operatorname{y^{\\prime}}{(F_{g})} dF_{g}}{\\cos{(F_{g})}} dF_{g} = \\int \\frac{\\int \\sin{(\\cos{(F_{g})})} dF_{g}}{\\cos{(F_{g})}} dF_{g}", "derivation": "\\operatorname{y^{\\prime}}{(F_{g})} = \\sin{(\\cos{(F_{g})})} and \\int \\operatorname{y^{\\prime}}{(F_{g})} dF_{g} = \\int \\sin{(\\cos{(F_{g})})} dF_{g} and \\frac{\\int \\operatorname{y^{\\prime}}{(F_{g})} dF_{g}}{\\cos{(F_{g})}} = \\frac{\\int \\sin{(\\cos{(F_{g})})} dF_{g}}{\\cos{(F_{g})}} and \\int \\frac{\\int \\operatorname{y^{\\prime}}{(F_{g})} dF_{g}}{\\cos{(F_{g})}} dF_{g} = \\int \\frac{\\int \\sin{(\\cos{(F_{g})})} dF_{g}}{\\cos{(F_{g})}} dF_{g}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True))))"], [["integrate", 1, "Symbol('F_g', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True))), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True))))"], [["divide", 2, "cos(Symbol('F_g', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('F_g', commutative=True)), Integer(-1)), Integral(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Mul(Pow(cos(Symbol('F_g', commutative=True)), Integer(-1)), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Mul(Pow(cos(Symbol('F_g', commutative=True)), Integer(-1)), Integral(Function('y^{\\\\prime}')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))), Integral(Mul(Pow(cos(Symbol('F_g', commutative=True)), Integer(-1)), Integral(sin(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(F_{N})} = \\sin{(\\log{(F_{N})})}, then derive \\int (\\mathbf{H}{(F_{N})} + \\sin{(\\log{(F_{N})})}) dF_{N} = F_{N} \\sin{(\\log{(F_{N})})} - F_{N} \\cos{(\\log{(F_{N})})} + i, then obtain \\iint 2 \\sin{(\\log{(F_{N})})} dF_{N} dF_{N} = \\int (F_{N} \\sin{(\\log{(F_{N})})} - F_{N} \\cos{(\\log{(F_{N})})} + i) dF_{N}", "derivation": "\\mathbf{H}{(F_{N})} = \\sin{(\\log{(F_{N})})} and \\mathbf{H}{(F_{N})} + \\sin{(\\log{(F_{N})})} = 2 \\sin{(\\log{(F_{N})})} and \\int (\\mathbf{H}{(F_{N})} + \\sin{(\\log{(F_{N})})}) dF_{N} = \\int 2 \\sin{(\\log{(F_{N})})} dF_{N} and \\int (\\mathbf{H}{(F_{N})} + \\sin{(\\log{(F_{N})})}) dF_{N} = F_{N} \\sin{(\\log{(F_{N})})} - F_{N} \\cos{(\\log{(F_{N})})} + i and \\int 2 \\sin{(\\log{(F_{N})})} dF_{N} = F_{N} \\sin{(\\log{(F_{N})})} - F_{N} \\cos{(\\log{(F_{N})})} + i and \\iint 2 \\sin{(\\log{(F_{N})})} dF_{N} dF_{N} = \\int (F_{N} \\sin{(\\log{(F_{N})})} - F_{N} \\cos{(\\log{(F_{N})})} + i) dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), sin(log(Symbol('F_N', commutative=True))))"], [["add", 1, "sin(log(Symbol('F_N', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), sin(log(Symbol('F_N', commutative=True)))), Mul(Integer(2), sin(log(Symbol('F_N', commutative=True)))))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), sin(log(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Integer(2), sin(log(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('F_N', commutative=True)), sin(log(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), sin(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('F_N', commutative=True), cos(log(Symbol('F_N', commutative=True)))), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Mul(Integer(2), sin(log(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True))), Add(Mul(Symbol('F_N', commutative=True), sin(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('F_N', commutative=True), cos(log(Symbol('F_N', commutative=True)))), Symbol('i', commutative=True)))"], [["integrate", 5, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Integer(2), sin(log(Symbol('F_N', commutative=True)))), Tuple(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Mul(Symbol('F_N', commutative=True), sin(log(Symbol('F_N', commutative=True)))), Mul(Integer(-1), Symbol('F_N', commutative=True), cos(log(Symbol('F_N', commutative=True)))), Symbol('i', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(b,C_{1})} = - b + \\log{(C_{1})}, then obtain - \\frac{b^{2} \\mathbb{I}^{b}{(b,C_{1})} \\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(b,C_{1})}}{\\mathbb{I}{(b,C_{1})}} = - \\frac{b^{2} (- b + \\log{(C_{1})})^{b}}{C_{1} (- b + \\log{(C_{1})})}", "derivation": "\\mathbb{I}{(b,C_{1})} = - b + \\log{(C_{1})} and \\mathbb{I}^{b}{(b,C_{1})} = (- b + \\log{(C_{1})})^{b} and - b \\mathbb{I}^{b}{(b,C_{1})} = - b (- b + \\log{(C_{1})})^{b} and \\frac{\\partial}{\\partial C_{1}} - b \\mathbb{I}^{b}{(b,C_{1})} = \\frac{\\partial}{\\partial C_{1}} - b (- b + \\log{(C_{1})})^{b} and - \\frac{b^{2} \\mathbb{I}^{b}{(b,C_{1})} \\frac{\\partial}{\\partial C_{1}} \\mathbb{I}{(b,C_{1})}}{\\mathbb{I}{(b,C_{1})}} = - \\frac{b^{2} (- b + \\log{(C_{1})})^{b}}{C_{1} (- b + \\log{(C_{1})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Symbol('b', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))), Symbol('b', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('b', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('b', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Symbol('b', commutative=True))), Mul(Integer(-1), Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))), Symbol('b', commutative=True))))"], [["differentiate", 3, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('b', commutative=True), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Symbol('b', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('b', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))), Symbol('b', commutative=True))), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Symbol('b', commutative=True), Integer(2)), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Symbol('b', commutative=True)), Derivative(Function('\\\\mathbb{I}')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Symbol('C_1', commutative=True))), Symbol('b', commutative=True))))"]]}, {"prompt": "Given B{(f^{\\prime})} = \\sin{(\\sin{(f^{\\prime})})}, then obtain - \\frac{f^{\\prime} + B{(f^{\\prime})}}{(f^{\\prime} + \\sin{(\\sin{(f^{\\prime})})}) \\sin{(\\sin{(f^{\\prime})})}} = - \\frac{1}{\\sin{(\\sin{(f^{\\prime})})}}", "derivation": "B{(f^{\\prime})} = \\sin{(\\sin{(f^{\\prime})})} and f^{\\prime} + B{(f^{\\prime})} = f^{\\prime} + \\sin{(\\sin{(f^{\\prime})})} and \\frac{f^{\\prime} + B{(f^{\\prime})}}{f^{\\prime} + \\sin{(\\sin{(f^{\\prime})})}} = 1 and - \\frac{f^{\\prime} + B{(f^{\\prime})}}{(f^{\\prime} + \\sin{(\\sin{(f^{\\prime})})}) \\sin{(\\sin{(f^{\\prime})})}} = - \\frac{1}{\\sin{(\\sin{(f^{\\prime})})}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('f^{\\\\prime}', commutative=True)), sin(sin(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Symbol('f^{\\\\prime}', commutative=True), Function('B')(Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('f^{\\\\prime}', commutative=True), sin(sin(Symbol('f^{\\\\prime}', commutative=True)))))"], [["divide", 2, "Add(Symbol('f^{\\\\prime}', commutative=True), sin(sin(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Mul(Add(Symbol('f^{\\\\prime}', commutative=True), Function('B')(Symbol('f^{\\\\prime}', commutative=True))), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), sin(sin(Symbol('f^{\\\\prime}', commutative=True)))), Integer(-1))), Integer(1))"], [["divide", 3, "Mul(Integer(-1), sin(sin(Symbol('f^{\\\\prime}', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Symbol('f^{\\\\prime}', commutative=True), Function('B')(Symbol('f^{\\\\prime}', commutative=True))), Pow(Add(Symbol('f^{\\\\prime}', commutative=True), sin(sin(Symbol('f^{\\\\prime}', commutative=True)))), Integer(-1)), Pow(sin(sin(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))), Mul(Integer(-1), Pow(sin(sin(Symbol('f^{\\\\prime}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(i)} = e^{i}, then derive (\\frac{d}{d i} \\operatorname{E_{n}}{(i)})^{2} = e^{i} \\frac{d}{d i} \\operatorname{E_{n}}{(i)}, then obtain \\operatorname{E_{n}}{(i)} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} + 1 = e^{i} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} + 1", "derivation": "\\operatorname{E_{n}}{(i)} = e^{i} and \\frac{d}{d i} \\operatorname{E_{n}}{(i)} = \\frac{d}{d i} e^{i} and (\\frac{d}{d i} \\operatorname{E_{n}}{(i)})^{2} = \\frac{d}{d i} \\operatorname{E_{n}}{(i)} \\frac{d}{d i} e^{i} and (\\frac{d}{d i} \\operatorname{E_{n}}{(i)})^{2} = e^{i} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} and (\\frac{d}{d i} \\operatorname{E_{n}}{(i)})^{2} = \\operatorname{E_{n}}{(i)} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} and \\operatorname{E_{n}}{(i)} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} = e^{i} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} and \\operatorname{E_{n}}{(i)} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} + 1 = e^{i} \\frac{d}{d i} \\operatorname{E_{n}}{(i)} + 1", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('i', commutative=True)), exp(Symbol('i', commutative=True)))"], [["differentiate", 1, "Symbol('i', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(exp(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(2)), Mul(exp(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Pow(Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(2)), Mul(Function('E_n')(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Function('E_n')(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Mul(exp(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))))"], [["add", 6, 1], "Equality(Add(Mul(Function('E_n')(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Integer(1)), Add(Mul(exp(Symbol('i', commutative=True)), Derivative(Function('E_n')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1)))), Integer(1)))"]]}, {"prompt": "Given E{(M_{E})} = e^{M_{E}} and \\theta{(\\mu_0,L)} = e^{L + \\mu_0}, then obtain \\theta{(\\mu_0,L)} = e^{L + \\mu_0 - E{(M_{E})} + e^{M_{E}}}", "derivation": "E{(M_{E})} = e^{M_{E}} and 0 = - E{(M_{E})} + e^{M_{E}} and \\theta{(\\mu_0,L)} = e^{L + \\mu_0} and L + \\mu_0 = L + \\mu_0 - E{(M_{E})} + e^{M_{E}} and \\theta{(\\mu_0,L)} = e^{L + \\mu_0 - E{(M_{E})} + e^{M_{E}}}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["minus", 1, "Function('E')(Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('E')(Symbol('M_E', commutative=True))), exp(Symbol('M_E', commutative=True))))"], ["get_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mu_0', commutative=True), Symbol('L', commutative=True)), exp(Add(Symbol('L', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["add", 2, "Add(Symbol('L', commutative=True), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Symbol('L', commutative=True), Symbol('\\\\mu_0', commutative=True)), Add(Symbol('L', commutative=True), Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Function('E')(Symbol('M_E', commutative=True))), exp(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Function('\\\\theta')(Symbol('\\\\mu_0', commutative=True), Symbol('L', commutative=True)), exp(Add(Symbol('L', commutative=True), Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Function('E')(Symbol('M_E', commutative=True))), exp(Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(M_{E})} = \\cos{(M_{E})} and \\operatorname{f_{\\mathbf{p}}}{(M_{E})} = - \\sin{(M_{E})}, then derive \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})} = - \\sin{(M_{E})}, then obtain (\\operatorname{f_{\\mathbf{p}}}{(M_{E})} + \\sin{(M_{E})})^{M_{E}} = 0^{M_{E}}", "derivation": "\\operatorname{F_{x}}{(M_{E})} = \\cos{(M_{E})} and \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})} = \\frac{d}{d M_{E}} \\cos{(M_{E})} and \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})} = - \\sin{(M_{E})} and \\sin{(M_{E})} + \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})} = 0 and \\operatorname{f_{\\mathbf{p}}}{(M_{E})} = - \\sin{(M_{E})} and \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})} = \\operatorname{f_{\\mathbf{p}}}{(M_{E})} and (\\sin{(M_{E})} + \\frac{d}{d M_{E}} \\operatorname{F_{x}}{(M_{E})})^{M_{E}} = 0^{M_{E}} and (\\operatorname{f_{\\mathbf{p}}}{(M_{E})} + \\sin{(M_{E})})^{M_{E}} = 0^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(cos(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('M_E', commutative=True))))"], [["add", 3, "sin(Symbol('M_E', commutative=True))"], "Equality(Add(sin(Symbol('M_E', commutative=True)), Derivative(Function('F_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Integer(0))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), sin(Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Derivative(Function('F_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Function('f_{\\\\mathbf{p}}')(Symbol('M_E', commutative=True)))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Add(sin(Symbol('M_E', commutative=True)), Derivative(Function('F_x')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Symbol('M_E', commutative=True)), Pow(Integer(0), Symbol('M_E', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Add(Function('f_{\\\\mathbf{p}}')(Symbol('M_E', commutative=True)), sin(Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True)), Pow(Integer(0), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})}, then derive \\rho{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}}, then obtain \\frac{d}{d \\mathbf{F}} \\frac{1}{\\mathbf{F}} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\mathbf{F})}", "derivation": "\\rho{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\log{(\\mathbf{F})} and \\rho{(\\mathbf{F})} = \\frac{1}{\\mathbf{F}} and \\frac{d}{d \\mathbf{F}} \\rho{(\\mathbf{F})} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\frac{1}{\\mathbf{F}} = \\frac{d^{2}}{d \\mathbf{F}^{2}} \\log{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('\\\\rho')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(k)} = \\frac{d}{d k} \\cos{(k)}, then derive \\operatorname{t_{2}}{(k)} = - \\sin{(k)}, then obtain (\\frac{d}{d k} \\cos{(k)} + \\int - \\sin{(k)} dk) \\sin^{k}{(\\int \\operatorname{t_{2}}{(k)} dk)} = (\\frac{d}{d k} \\cos{(k)} + \\int - \\sin{(k)} dk) \\sin^{k}{(\\int - \\sin{(k)} dk)}", "derivation": "\\operatorname{t_{2}}{(k)} = \\frac{d}{d k} \\cos{(k)} and \\operatorname{t_{2}}{(k)} = - \\sin{(k)} and \\int \\operatorname{t_{2}}{(k)} dk = \\int \\frac{d}{d k} \\cos{(k)} dk and \\sin{(\\int \\operatorname{t_{2}}{(k)} dk)} = \\sin{(\\int \\frac{d}{d k} \\cos{(k)} dk)} and \\sin{(\\int - \\sin{(k)} dk)} = \\sin{(\\int \\frac{d}{d k} \\cos{(k)} dk)} and \\sin{(\\int \\operatorname{t_{2}}{(k)} dk)} = \\sin{(\\int - \\sin{(k)} dk)} and \\sin^{k}{(\\int \\operatorname{t_{2}}{(k)} dk)} = \\sin^{k}{(\\int - \\sin{(k)} dk)} and (\\frac{d}{d k} \\cos{(k)} + \\int - \\sin{(k)} dk) \\sin^{k}{(\\int \\operatorname{t_{2}}{(k)} dk)} = (\\frac{d}{d k} \\cos{(k)} + \\int - \\sin{(k)} dk) \\sin^{k}{(\\int - \\sin{(k)} dk)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('k', commutative=True)), Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('t_2')(Symbol('k', commutative=True)), Mul(Integer(-1), sin(Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Function('t_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), sin(Integral(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(sin(Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), sin(Integral(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(sin(Integral(Function('t_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), sin(Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(sin(Integral(Function('t_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Pow(sin(Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["times", 7, "Add(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True))))"], "Equality(Mul(Add(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Pow(sin(Integral(Function('t_2')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Symbol('k', commutative=True))), Mul(Add(Derivative(cos(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Pow(sin(Integral(Mul(Integer(-1), sin(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True)))), Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(C_{2},\\mathbf{P})} = C_{2} + e^{\\mathbf{P}}, then obtain - C_{2} + \\Psi_{\\lambda}{(C_{2},\\mathbf{P})} - e^{\\mathbf{P}} = 0", "derivation": "\\Psi_{\\lambda}{(C_{2},\\mathbf{P})} = C_{2} + e^{\\mathbf{P}} and - \\mathbf{P} + \\Psi_{\\lambda}{(C_{2},\\mathbf{P})} = C_{2} - \\mathbf{P} + e^{\\mathbf{P}} and - 2 \\mathbf{P} + \\Psi_{\\lambda}{(C_{2},\\mathbf{P})} = C_{2} - 2 \\mathbf{P} + e^{\\mathbf{P}} and - C_{2} + \\Psi_{\\lambda}{(C_{2},\\mathbf{P})} - e^{\\mathbf{P}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('C_2', commutative=True), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], [["minus", 3, "Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C_2', commutative=True)), Function('\\\\Psi_{\\\\lambda}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\mathbf{P}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(k)} = \\sin{(e^{k})}, then obtain \\frac{d}{d k} (\\frac{\\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} - e^{k} - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}}) = \\frac{d}{d k} (- e^{k} + 1 - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}})", "derivation": "\\operatorname{y^{\\prime}}{(k)} = \\sin{(e^{k})} and \\frac{\\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} = 1 and \\frac{\\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} - e^{k} = 1 - e^{k} and \\frac{\\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} - e^{k} - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} = - e^{k} + 1 - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} and \\frac{d}{d k} (\\frac{\\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}} - e^{k} - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}}) = \\frac{d}{d k} (- e^{k} + 1 - \\frac{\\frac{d}{d k} \\operatorname{y^{\\prime}}{(k)}}{\\sin{(e^{k})}})", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), sin(exp(Symbol('k', commutative=True))))"], [["divide", 1, "sin(exp(Symbol('k', commutative=True)))"], "Equality(Mul(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "exp(Symbol('k', commutative=True))"], "Equality(Add(Mul(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1))), Mul(Integer(-1), exp(Symbol('k', commutative=True)))), Add(Integer(1), Mul(Integer(-1), exp(Symbol('k', commutative=True)))))"], [["minus", 3, "Mul(Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Add(Mul(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1))), Mul(Integer(-1), exp(Symbol('k', commutative=True))), Mul(Integer(-1), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Add(Mul(Integer(-1), exp(Symbol('k', commutative=True))), Integer(1), Mul(Integer(-1), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Add(Mul(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1))), Mul(Integer(-1), exp(Symbol('k', commutative=True))), Mul(Integer(-1), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('k', commutative=True))), Integer(1), Mul(Integer(-1), Pow(sin(exp(Symbol('k', commutative=True))), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(A)} = \\cos{(A)}, then derive \\frac{\\cos{(A)} \\int \\frac{\\pi{(A)}}{\\cos{(A)}} dA}{\\pi{(A)}} = \\frac{(A + V) \\cos{(A)}}{\\pi{(A)}}, then obtain \\frac{(A + V) \\cos{(A)}}{\\pi{(A)}} = \\frac{\\cos{(A)} \\int 1 dA}{\\pi{(A)}}", "derivation": "\\pi{(A)} = \\cos{(A)} and \\frac{\\pi{(A)}}{\\cos{(A)}} = 1 and \\int \\frac{\\pi{(A)}}{\\cos{(A)}} dA = \\int 1 dA and \\frac{\\cos{(A)} \\int \\frac{\\pi{(A)}}{\\cos{(A)}} dA}{\\pi{(A)}} = \\frac{\\cos{(A)} \\int 1 dA}{\\pi{(A)}} and \\frac{\\cos{(A)} \\int \\frac{\\pi{(A)}}{\\cos{(A)}} dA}{\\pi{(A)}} = \\frac{(A + V) \\cos{(A)}}{\\pi{(A)}} and \\int \\frac{\\pi{(A)}}{\\cos{(A)}} dA = A + V and \\frac{(A + V) \\cos{(A)}}{\\pi{(A)}} = \\frac{\\cos{(A)} \\int 1 dA}{\\pi{(A)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["divide", 1, "cos(Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Tuple(Symbol('A', commutative=True))), Integral(Integer(1), Tuple(Symbol('A', commutative=True))))"], [["divide", 3, "Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True)), Integral(Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Tuple(Symbol('A', commutative=True)))), Mul(Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True)), Integral(Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Tuple(Symbol('A', commutative=True)))), Mul(Add(Symbol('A', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Integral(Mul(Function('\\\\pi')(Symbol('A', commutative=True)), Pow(cos(Symbol('A', commutative=True)), Integer(-1))), Tuple(Symbol('A', commutative=True))), Add(Symbol('A', commutative=True), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Mul(Add(Symbol('A', commutative=True), Symbol('V', commutative=True)), Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True))), Mul(Pow(Function('\\\\pi')(Symbol('A', commutative=True)), Integer(-1)), cos(Symbol('A', commutative=True)), Integral(Integer(1), Tuple(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(W)} = \\log{(W)}, then obtain - (\\int \\operatorname{c_{0}}{(W)} dW) \\int \\log{(W)} dW = - \\operatorname{c_{0}}{(W)} + \\log{(W)} - (\\int \\operatorname{c_{0}}{(W)} dW) \\int \\log{(W)} dW", "derivation": "\\operatorname{c_{0}}{(W)} = \\log{(W)} and \\int \\operatorname{c_{0}}{(W)} dW = \\int \\log{(W)} dW and 0 = - \\operatorname{c_{0}}{(W)} + \\log{(W)} and (\\int \\operatorname{c_{0}}{(W)} dW)^{2} = (\\int \\operatorname{c_{0}}{(W)} dW) \\int \\log{(W)} dW and - (\\int \\operatorname{c_{0}}{(W)} dW)^{2} = - \\operatorname{c_{0}}{(W)} + \\log{(W)} - (\\int \\operatorname{c_{0}}{(W)} dW)^{2} and - (\\int \\operatorname{c_{0}}{(W)} dW) \\int \\log{(W)} dW = - \\operatorname{c_{0}}{(W)} + \\log{(W)} - (\\int \\operatorname{c_{0}}{(W)} dW) \\int \\log{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('W', commutative=True)), log(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["minus", 1, "Function('c_0')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('c_0')(Symbol('W', commutative=True))), log(Symbol('W', commutative=True))))"], [["times", 2, "Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Pow(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)), Mul(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["minus", 3, "Pow(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2))"], "Equality(Mul(Integer(-1), Pow(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2))), Add(Mul(Integer(-1), Function('c_0')(Symbol('W', commutative=True))), log(Symbol('W', commutative=True)), Mul(Integer(-1), Pow(Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integer(2)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('c_0')(Symbol('W', commutative=True))), log(Symbol('W', commutative=True)), Mul(Integer(-1), Integral(Function('c_0')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A_{1})} = \\sin{(A_{1})}, then obtain \\frac{d}{d A_{1}} \\operatorname{v_{y}}{(A_{1})} = 2 \\cos{(A_{1})} - \\frac{d}{d A_{1}} \\operatorname{v_{y}}{(A_{1})}", "derivation": "\\operatorname{v_{y}}{(A_{1})} = \\sin{(A_{1})} and 0 = - \\operatorname{v_{y}}{(A_{1})} + \\sin{(A_{1})} and \\sin{(A_{1})} = - \\operatorname{v_{y}}{(A_{1})} + 2 \\sin{(A_{1})} and \\operatorname{v_{y}}{(A_{1})} = - \\operatorname{v_{y}}{(A_{1})} + 2 \\sin{(A_{1})} and \\frac{d}{d A_{1}} \\operatorname{v_{y}}{(A_{1})} = \\frac{d}{d A_{1}} (- \\operatorname{v_{y}}{(A_{1})} + 2 \\sin{(A_{1})}) and \\frac{d}{d A_{1}} \\operatorname{v_{y}}{(A_{1})} = 2 \\cos{(A_{1})} - \\frac{d}{d A_{1}} \\operatorname{v_{y}}{(A_{1})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A_1', commutative=True)), sin(Symbol('A_1', commutative=True)))"], [["minus", 1, "Function('v_y')(Symbol('A_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_1', commutative=True))), sin(Symbol('A_1', commutative=True))))"], [["add", 2, "sin(Symbol('A_1', commutative=True))"], "Equality(sin(Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_1', commutative=True))), Mul(Integer(2), sin(Symbol('A_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Function('v_y')(Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_1', commutative=True))), Mul(Integer(2), sin(Symbol('A_1', commutative=True)))))"], [["differentiate", 4, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('v_y')(Symbol('A_1', commutative=True))), Mul(Integer(2), sin(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('v_y')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))), Add(Mul(Integer(2), cos(Symbol('A_1', commutative=True))), Mul(Integer(-1), Derivative(Function('v_y')(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{E})} = \\log{(\\mathbf{E})}, then obtain \\frac{d}{d \\mathbf{E}} \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\rho_{b}{(\\mathbf{E})})})} = \\frac{d}{d \\mathbf{E}} \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\log{(\\mathbf{E})})})}", "derivation": "\\rho_{b}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and - \\mathbf{E} + \\rho_{b}{(\\mathbf{E})} = - \\mathbf{E} + \\log{(\\mathbf{E})} and \\cos{(\\mathbf{E} - \\rho_{b}{(\\mathbf{E})})} = \\cos{(\\mathbf{E} - \\log{(\\mathbf{E})})} and \\cos^{\\mathbf{E}}{(\\mathbf{E} - \\rho_{b}{(\\mathbf{E})})} = \\cos^{\\mathbf{E}}{(\\mathbf{E} - \\log{(\\mathbf{E})})} and \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\rho_{b}{(\\mathbf{E})})})} = \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\log{(\\mathbf{E})})})} and \\frac{d}{d \\mathbf{E}} \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\rho_{b}{(\\mathbf{E})})})} = \\frac{d}{d \\mathbf{E}} \\log{(\\cos^{\\mathbf{E}}{(\\mathbf{E} - \\log{(\\mathbf{E})})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True))))), cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))))))"], [["power", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True)))"], [["log", 4], "Equality(log(Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True))), log(Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(log(Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\rho_b')(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(log(Pow(cos(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), log(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\pi{(\\theta,\\hat{X})} = - \\hat{X} + \\theta, then obtain 0 = - \\frac{\\theta (- \\hat{X} + \\theta)^{\\theta} (- \\hat{X} + \\theta - \\pi{(\\theta,\\hat{X})})}{- \\hat{X} + \\theta} + ((- \\hat{X} + \\theta)^{\\theta} + 1) (- \\frac{\\partial}{\\partial \\hat{X}} \\pi{(\\theta,\\hat{X})} - 1)", "derivation": "\\pi{(\\theta,\\hat{X})} = - \\hat{X} + \\theta and 2 \\pi{(\\theta,\\hat{X})} = - \\hat{X} + \\theta + \\pi{(\\theta,\\hat{X})} and 0 = - \\hat{X} + \\theta - \\pi{(\\theta,\\hat{X})} and 0 = ((- \\hat{X} + \\theta)^{\\theta} + 1) (- \\hat{X} + \\theta - \\pi{(\\theta,\\hat{X})}) and \\frac{d}{d \\hat{X}} 0 = \\frac{\\partial}{\\partial \\hat{X}} ((- \\hat{X} + \\theta)^{\\theta} + 1) (- \\hat{X} + \\theta - \\pi{(\\theta,\\hat{X})}) and 0 = - \\frac{\\theta (- \\hat{X} + \\theta)^{\\theta} (- \\hat{X} + \\theta - \\pi{(\\theta,\\hat{X})})}{- \\hat{X} + \\theta} + ((- \\hat{X} + \\theta)^{\\theta} + 1) (- \\frac{\\partial}{\\partial \\hat{X}} \\pi{(\\theta,\\hat{X})} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["add", 1, "Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 2, "Mul(Integer(2), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)))))"], [["times", 3, "Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integer(1))"], "Equality(Integer(0), Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))), Derivative(Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Mul(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Derivative(Function('\\\\pi')(Symbol('\\\\theta', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True), Integer(1)))), Integer(-1)))))"]]}, {"prompt": "Given G{(x)} = \\log{(\\cos{(x)})}, then obtain (- 2 G{(x)} + 2 \\log{(\\cos{(x)})}) \\int (\\frac{d}{d x} 0 - \\frac{d}{d x} (- G{(x)} + \\log{(\\cos{(x)})})) dx = (- 2 G{(x)} + 2 \\log{(\\cos{(x)})}) \\int 0 dx", "derivation": "G{(x)} = \\log{(\\cos{(x)})} and 0 = - G{(x)} + \\log{(\\cos{(x)})} and - G{(x)} + \\log{(\\cos{(x)})} = - 2 G{(x)} + 2 \\log{(\\cos{(x)})} and 0 = - 2 G{(x)} + 2 \\log{(\\cos{(x)})} and \\frac{d}{d x} 0 = \\frac{d}{d x} (- 2 G{(x)} + 2 \\log{(\\cos{(x)})}) and \\frac{d}{d x} 0 = \\frac{d}{d x} (- G{(x)} + \\log{(\\cos{(x)})}) and \\frac{d}{d x} 0 - \\frac{d}{d x} (- G{(x)} + \\log{(\\cos{(x)})}) = 0 and \\int (\\frac{d}{d x} 0 - \\frac{d}{d x} (- G{(x)} + \\log{(\\cos{(x)})})) dx = \\int 0 dx and (- 2 G{(x)} + 2 \\log{(\\cos{(x)})}) \\int (\\frac{d}{d x} 0 - \\frac{d}{d x} (- G{(x)} + \\log{(\\cos{(x)})})) dx = (- 2 G{(x)} + 2 \\log{(\\cos{(x)})}) \\int 0 dx", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('x', commutative=True)), log(cos(Symbol('x', commutative=True))))"], [["minus", 1, "Function('G')(Symbol('x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True))))))"], [["differentiate", 4, "Symbol('x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True))))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["minus", 6, "Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))), Integer(0))"], [["integrate", 7, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))), Tuple(Symbol('x', commutative=True))), Integral(Integer(0), Tuple(Symbol('x', commutative=True))))"], [["times", 8, "Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True))))), Integral(Add(Derivative(Integer(0), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('x', commutative=True))), log(cos(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True), Integer(1))))), Tuple(Symbol('x', commutative=True)))), Mul(Add(Mul(Integer(-1), Integer(2), Function('G')(Symbol('x', commutative=True))), Mul(Integer(2), log(cos(Symbol('x', commutative=True))))), Integral(Integer(0), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\psi^{*}{(n_{1},\\chi)} = n_{1}^{\\chi}, then obtain \\frac{- \\psi^{*}{(n_{1},\\chi)} + \\int \\psi^{*}{(n_{1},\\chi)} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}} = \\frac{- \\psi^{*}{(n_{1},\\chi)} + \\int n_{1}^{\\chi} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}}", "derivation": "\\psi^{*}{(n_{1},\\chi)} = n_{1}^{\\chi} and \\int \\psi^{*}{(n_{1},\\chi)} d\\chi = \\int n_{1}^{\\chi} d\\chi and - n_{1}^{\\chi} + \\int \\psi^{*}{(n_{1},\\chi)} d\\chi = - n_{1}^{\\chi} + \\int n_{1}^{\\chi} d\\chi and \\frac{- n_{1}^{\\chi} + \\int \\psi^{*}{(n_{1},\\chi)} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}} = \\frac{- n_{1}^{\\chi} + \\int n_{1}^{\\chi} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}} and \\frac{- \\psi^{*}{(n_{1},\\chi)} + \\int \\psi^{*}{(n_{1},\\chi)} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}} = \\frac{- \\psi^{*}{(n_{1},\\chi)} + \\int n_{1}^{\\chi} d\\chi}{- \\chi + \\psi^{*}{(n_{1},\\chi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["minus", 2, "Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\psi^*')(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True))), Integral(Pow(Symbol('n_1', commutative=True), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(C,k)} = C k, then obtain \\frac{C k}{\\int C k dk} + \\frac{\\operatorname{v_{y}}{(C,k)}}{\\int C k dk} + 1 = \\frac{2 C k}{\\int C k dk} + 1", "derivation": "\\operatorname{v_{y}}{(C,k)} = C k and \\int \\operatorname{v_{y}}{(C,k)} dk = \\int C k dk and \\frac{\\operatorname{v_{y}}{(C,k)}}{\\int C k dk} = \\frac{C k}{\\int C k dk} and \\frac{\\operatorname{v_{y}}{(C,k)}}{\\int \\operatorname{v_{y}}{(C,k)} dk} = \\frac{C k}{\\int \\operatorname{v_{y}}{(C,k)} dk} and \\frac{C k}{\\int C k dk} + \\frac{\\operatorname{v_{y}}{(C,k)}}{\\int \\operatorname{v_{y}}{(C,k)} dk} + 1 = \\frac{C k}{\\int \\operatorname{v_{y}}{(C,k)} dk} + \\frac{C k}{\\int C k dk} + 1 and \\frac{C k}{\\int C k dk} + \\frac{\\operatorname{v_{y}}{(C,k)}}{\\int C k dk} + 1 = \\frac{2 C k}{\\int C k dk} + 1", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["divide", 1, "Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))"], "Equality(Mul(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))))"], [["add", 4, "Add(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Integer(1))"], "Equality(Add(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Mul(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Integer(1)), Add(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Mul(Function('v_y')(Symbol('C', commutative=True), Symbol('k', commutative=True)), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Integer(1)), Add(Mul(Integer(2), Symbol('C', commutative=True), Symbol('k', commutative=True), Pow(Integral(Mul(Symbol('C', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integer(-1))), Integer(1)))"]]}, {"prompt": "Given t{(\\chi,\\varepsilon)} = \\sin{(\\chi \\varepsilon)} and \\mathbf{E}{(\\chi,\\varepsilon)} = \\chi \\varepsilon, then obtain \\mathbf{E}{(\\chi,\\varepsilon)} t{(\\chi,\\varepsilon)} = \\mathbf{E}{(\\chi,\\varepsilon)} \\sin{(\\mathbf{E}{(\\chi,\\varepsilon)})}", "derivation": "t{(\\chi,\\varepsilon)} = \\sin{(\\chi \\varepsilon)} and \\mathbf{E}{(\\chi,\\varepsilon)} = \\chi \\varepsilon and \\mathbf{E}{(\\chi,\\varepsilon)} t{(\\chi,\\varepsilon)} = \\mathbf{E}{(\\chi,\\varepsilon)} \\sin{(\\chi \\varepsilon)} and \\mathbf{E}{(\\chi,\\varepsilon)} t{(\\chi,\\varepsilon)} = \\mathbf{E}{(\\chi,\\varepsilon)} \\sin{(\\mathbf{E}{(\\chi,\\varepsilon)})}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('t')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('t')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given c{(F_{c},\\theta_2)} = e^{F_{c}^{\\theta_2}}, then derive F_{c} + G = \\int \\frac{e^{F_{c}^{\\theta_2}}}{c{(F_{c},\\theta_2)}} dF_{c}, then obtain F_{c} + G - c{(F_{c},\\theta_2)} = - c{(F_{c},\\theta_2)} + \\int 1 dF_{c}", "derivation": "c{(F_{c},\\theta_2)} = e^{F_{c}^{\\theta_2}} and 1 = \\frac{e^{F_{c}^{\\theta_2}}}{c{(F_{c},\\theta_2)}} and \\int 1 dF_{c} = \\int \\frac{e^{F_{c}^{\\theta_2}}}{c{(F_{c},\\theta_2)}} dF_{c} and F_{c} + G = \\int \\frac{e^{F_{c}^{\\theta_2}}}{c{(F_{c},\\theta_2)}} dF_{c} and F_{c} + G = \\int 1 dF_{c} and F_{c} + G - c{(F_{c},\\theta_2)} = - c{(F_{c},\\theta_2)} + \\int 1 dF_{c}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Pow(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 1, "Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), exp(Pow(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)))))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_c', commutative=True))), Integral(Mul(Pow(Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), exp(Pow(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('F_c', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('F_c', commutative=True), Symbol('G', commutative=True)), Integral(Mul(Pow(Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), exp(Pow(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Tuple(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('F_c', commutative=True), Symbol('G', commutative=True)), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True))))"], [["minus", 5, "Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Symbol('F_c', commutative=True), Symbol('G', commutative=True), Mul(Integer(-1), Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), Function('c')(Symbol('F_c', commutative=True), Symbol('\\\\theta_2', commutative=True))), Integral(Integer(1), Tuple(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(F_{c},\\delta)} = - \\delta + e^{F_{c}}, then derive \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},\\delta)} = e^{F_{c}}, then obtain \\operatorname{t_{1}}{(F_{c},\\delta)} = - \\delta + \\frac{\\partial}{\\partial F_{c}} (- \\delta + e^{F_{c}})", "derivation": "\\operatorname{t_{1}}{(F_{c},\\delta)} = - \\delta + e^{F_{c}} and \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},\\delta)} = \\frac{\\partial}{\\partial F_{c}} (- \\delta + e^{F_{c}}) and \\frac{\\partial}{\\partial F_{c}} \\operatorname{t_{1}}{(F_{c},\\delta)} = e^{F_{c}} and e^{F_{c}} = \\frac{\\partial}{\\partial F_{c}} (- \\delta + e^{F_{c}}) and \\operatorname{t_{1}}{(F_{c},\\delta)} = - \\delta + \\frac{\\partial}{\\partial F_{c}} (- \\delta + e^{F_{c}})", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('F_c', commutative=True))))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), exp(Symbol('F_c', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Symbol('F_c', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 1, 4], "Equality(Function('t_1')(Symbol('F_c', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), exp(Symbol('F_c', commutative=True))), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given L{(\\varepsilon)} = \\cos{(\\varepsilon)}, then derive \\frac{d}{d \\varepsilon} L{(\\varepsilon)} = - \\sin{(\\varepsilon)}, then obtain \\frac{d}{d \\varepsilon} - \\sin{(\\varepsilon)} = \\frac{d^{2}}{d \\varepsilon^{2}} \\cos{(\\varepsilon)}", "derivation": "L{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} L{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} L{(\\varepsilon)} = - \\sin{(\\varepsilon)} and - \\sin{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} - \\sin{(\\varepsilon)} = \\frac{d^{2}}{d \\varepsilon^{2}} \\cos{(\\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('L')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"]]}, {"prompt": "Given r{(A_{2})} = \\int \\sin{(A_{2})} dA_{2}, then derive 4 r^{2}{(A_{2})} = (p + r{(A_{2})} - \\cos{(A_{2})})^{2}, then obtain (p + r{(A_{2})} - \\cos{(A_{2})})^{2} - \\cos{(A_{2})} = 2 (r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2}) r{(A_{2})} - \\cos{(A_{2})}", "derivation": "r{(A_{2})} = \\int \\sin{(A_{2})} dA_{2} and 2 r{(A_{2})} = r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2} and 4 r^{2}{(A_{2})} = (r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2})^{2} and 4 r^{2}{(A_{2})} = (p + r{(A_{2})} - \\cos{(A_{2})})^{2} and 4 r^{2}{(A_{2})} = 2 (r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2}) r{(A_{2})} and 4 r^{2}{(A_{2})} - \\cos{(A_{2})} = 2 (r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2}) r{(A_{2})} - \\cos{(A_{2})} and (p + r{(A_{2})} - \\cos{(A_{2})})^{2} - \\cos{(A_{2})} = 2 (r{(A_{2})} + \\int \\sin{(A_{2})} dA_{2}) r{(A_{2})} - \\cos{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["add", 1, "Function('r')(Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(2), Function('r')(Symbol('A_2', commutative=True))), Add(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('r')(Symbol('A_2', commutative=True)), Integer(2))), Pow(Add(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Integer(2)))"], [["evaluate_integrals", 3], "Equality(Mul(Integer(4), Pow(Function('r')(Symbol('A_2', commutative=True)), Integer(2))), Pow(Add(Symbol('p', commutative=True), Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Integer(2)))"], [["times", 2, "Mul(Integer(2), Function('r')(Symbol('A_2', commutative=True)))"], "Equality(Mul(Integer(4), Pow(Function('r')(Symbol('A_2', commutative=True)), Integer(2))), Mul(Integer(2), Add(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Function('r')(Symbol('A_2', commutative=True))))"], [["add", 5, "Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))"], "Equality(Add(Mul(Integer(4), Pow(Function('r')(Symbol('A_2', commutative=True)), Integer(2))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Add(Mul(Integer(2), Add(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Function('r')(Symbol('A_2', commutative=True))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Pow(Add(Symbol('p', commutative=True), Function('r')(Symbol('A_2', commutative=True)), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Integer(2)), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))), Add(Mul(Integer(2), Add(Function('r')(Symbol('A_2', commutative=True)), Integral(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True)))), Function('r')(Symbol('A_2', commutative=True))), Mul(Integer(-1), cos(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(J)} = \\sin{(e^{J})}, then obtain (\\frac{d}{d J} (\\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ)^{J})^{J} = (\\frac{\\partial}{\\partial J} (\\tilde{g}^* + \\operatorname{Si}{(e^{J})})^{J})^{J}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(J)} = \\sin{(e^{J})} and \\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ = \\int \\sin{(e^{J})} dJ and (\\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ)^{J} = (\\int \\sin{(e^{J})} dJ)^{J} and \\frac{d}{d J} (\\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ)^{J} = \\frac{d}{d J} (\\int \\sin{(e^{J})} dJ)^{J} and (\\frac{d}{d J} (\\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ)^{J})^{J} = (\\frac{d}{d J} (\\int \\sin{(e^{J})} dJ)^{J})^{J} and (\\frac{d}{d J} (\\int \\operatorname{f_{\\mathbf{v}}}{(J)} dJ)^{J})^{J} = (\\frac{\\partial}{\\partial J} (\\tilde{g}^* + \\operatorname{Si}{(e^{J})})^{J})^{J}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), sin(exp(Symbol('J', commutative=True))))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(sin(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(sin(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(Pow(Integral(sin(exp(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Pow(Derivative(Pow(Integral(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(Pow(Add(Symbol('\\\\tilde{g}^*', commutative=True), Si(exp(Symbol('J', commutative=True)))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})}, then obtain (\\int (- g^{\\prime}_{\\varepsilon} + \\sigma_{x}{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} = (\\int (- g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}}", "derivation": "\\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and - g^{\\prime}_{\\varepsilon} + \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = - g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})} and \\int (- g^{\\prime}_{\\varepsilon} + \\sigma_{x}{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon} = \\int (- g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon} and (\\int (- g^{\\prime}_{\\varepsilon} + \\sigma_{x}{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}} = (\\int (- g^{\\prime}_{\\varepsilon} + \\log{(g^{\\prime}_{\\varepsilon})}) dg^{\\prime}_{\\varepsilon})^{g^{\\prime}_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["minus", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\psi{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\operatorname{f_{E}}{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}}, then obtain \\operatorname{f_{E}}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}} = 0", "derivation": "\\psi{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\psi{(L_{\\varepsilon})} - \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} = 0 and \\operatorname{f_{E}}{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\psi{(L_{\\varepsilon})} = \\operatorname{f_{E}}{(L_{\\varepsilon})} and \\operatorname{f_{E}}{(L_{\\varepsilon})} - \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} = 0 and \\operatorname{f_{E}}{(L_{\\varepsilon})} - e^{L_{\\varepsilon}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\psi')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))), Integer(0))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\psi')(Symbol('L_{\\\\varepsilon}', commutative=True)), Function('f_E')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Function('f_E')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 5], "Equality(Add(Function('f_E')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\dot{x}{(c_{0})} = \\sin{(c_{0})} and \\mathbf{M}{(c_{0})} = \\dot{x}^{3}{(c_{0})} \\sin{(c_{0})} and C{(c_{0})} = \\sin{(c_{0})}, then obtain C^{12}{(c_{0})} - C^{3}{(c_{0})} \\dot{x}^{9}{(c_{0})} = 0", "derivation": "\\dot{x}{(c_{0})} = \\sin{(c_{0})} and \\mathbf{M}{(c_{0})} = \\dot{x}^{3}{(c_{0})} \\sin{(c_{0})} and \\mathbf{M}^{3}{(c_{0})} = \\dot{x}^{9}{(c_{0})} \\sin^{3}{(c_{0})} and \\mathbf{M}^{3}{(c_{0})} = \\sin^{12}{(c_{0})} and C{(c_{0})} = \\sin{(c_{0})} and \\sin^{12}{(c_{0})} = \\dot{x}^{9}{(c_{0})} \\sin^{3}{(c_{0})} and C^{12}{(c_{0})} = C^{3}{(c_{0})} \\dot{x}^{9}{(c_{0})} and C^{12}{(c_{0})} + \\mathbf{M}{(c_{0})} = C^{3}{(c_{0})} \\dot{x}^{9}{(c_{0})} + \\mathbf{M}{(c_{0})} and C^{12}{(c_{0})} - C^{3}{(c_{0})} \\dot{x}^{9}{(c_{0})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True)), Mul(Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(3)), sin(Symbol('c_0', commutative=True))))"], [["power", 2, 3], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True)), Integer(3)), Mul(Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9)), Pow(sin(Symbol('c_0', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True)), Integer(3)), Pow(sin(Symbol('c_0', commutative=True)), Integer(12)))"], ["renaming_premise", "Equality(Function('C')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(sin(Symbol('c_0', commutative=True)), Integer(12)), Mul(Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9)), Pow(sin(Symbol('c_0', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(12)), Mul(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(3)), Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9))))"], [["add", 7, "Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True))"], "Equality(Add(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(12)), Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True))), Add(Mul(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(3)), Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9))), Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True))))"], [["minus", 8, "Add(Mul(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(3)), Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9))), Function('\\\\mathbf{M}')(Symbol('c_0', commutative=True)))"], "Equality(Add(Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(12)), Mul(Integer(-1), Pow(Function('C')(Symbol('c_0', commutative=True)), Integer(3)), Pow(Function('\\\\dot{x}')(Symbol('c_0', commutative=True)), Integer(9)))), Integer(0))"]]}, {"prompt": "Given m{(A)} = \\sin{(\\cos{(A)})} and \\operatorname{n_{1}}{(A)} = 2 \\sin^{A}{(\\cos{(A)})}, then obtain \\frac{d}{d A} - 2 m^{A}{(A)} = \\frac{d}{d A} - \\operatorname{n_{1}}{(A)}", "derivation": "m{(A)} = \\sin{(\\cos{(A)})} and m^{A}{(A)} = \\sin^{A}{(\\cos{(A)})} and m^{A}{(A)} + \\sin^{A}{(\\cos{(A)})} = 2 \\sin^{A}{(\\cos{(A)})} and \\operatorname{n_{1}}{(A)} = 2 \\sin^{A}{(\\cos{(A)})} and m^{A}{(A)} + \\sin^{A}{(\\cos{(A)})} = \\operatorname{n_{1}}{(A)} and - m^{A}{(A)} - \\sin^{A}{(\\cos{(A)})} = - \\operatorname{n_{1}}{(A)} and - 2 \\sin^{A}{(\\cos{(A)})} = - \\operatorname{n_{1}}{(A)} and - 2 m^{A}{(A)} = - \\operatorname{n_{1}}{(A)} and \\frac{d}{d A} - 2 m^{A}{(A)} = \\frac{d}{d A} - \\operatorname{n_{1}}{(A)}", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('A', commutative=True)), sin(cos(Symbol('A', commutative=True))))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["add", 2, "Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))"], "Equality(Add(Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))), Mul(Integer(2), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('A', commutative=True)), Mul(Integer(2), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))), Function('n_1')(Symbol('A', commutative=True)))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Integer(-1), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True)))), Mul(Integer(-1), Function('n_1')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Integer(-1), Integer(2), Pow(sin(cos(Symbol('A', commutative=True))), Symbol('A', commutative=True))), Mul(Integer(-1), Function('n_1')(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Mul(Integer(-1), Integer(2), Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Mul(Integer(-1), Function('n_1')(Symbol('A', commutative=True))))"], [["differentiate", 8, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integer(2), Pow(Function('m')(Symbol('A', commutative=True)), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('n_1')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"]]}, {"prompt": "Given m{(I,\\phi)} = \\log{(\\frac{\\phi}{I})}, then derive \\int m{(I,\\phi)} d\\phi = \\Psi^{\\dagger} + \\phi \\log{(\\frac{\\phi}{I})} - \\phi, then obtain \\int \\log{(\\frac{\\phi}{I})} d\\phi = \\Psi^{\\dagger} + \\phi \\log{(\\frac{\\phi}{I})} - \\phi", "derivation": "m{(I,\\phi)} = \\log{(\\frac{\\phi}{I})} and \\int m{(I,\\phi)} d\\phi = \\int \\log{(\\frac{\\phi}{I})} d\\phi and \\int m{(I,\\phi)} d\\phi = \\Psi^{\\dagger} + \\phi \\log{(\\frac{\\phi}{I})} - \\phi and \\int \\log{(\\frac{\\phi}{I})} d\\phi = \\Psi^{\\dagger} + \\phi \\log{(\\frac{\\phi}{I})} - \\phi", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), log(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('m')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(log(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('m')(Symbol('I', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(\\hbar)} = \\log{(\\hbar)}, then derive \\int \\mathbf{B}{(\\hbar)} d\\hbar = M + \\hbar \\log{(\\hbar)} - \\hbar, then obtain - C - \\hbar \\mathbf{B}{(\\hbar)} + \\hbar + (\\int \\mathbf{B}{(\\hbar)} d\\hbar)^{\\hbar} = - C - \\hbar \\mathbf{B}{(\\hbar)} + \\hbar + (M + \\hbar \\log{(\\hbar)} - \\hbar)^{\\hbar}", "derivation": "\\mathbf{B}{(\\hbar)} = \\log{(\\hbar)} and \\int \\mathbf{B}{(\\hbar)} d\\hbar = \\int \\log{(\\hbar)} d\\hbar and \\int \\mathbf{B}{(\\hbar)} d\\hbar = M + \\hbar \\log{(\\hbar)} - \\hbar and (\\int \\mathbf{B}{(\\hbar)} d\\hbar)^{\\hbar} = (\\int \\log{(\\hbar)} d\\hbar)^{\\hbar} and \\int \\log{(\\hbar)} d\\hbar = M + \\hbar \\log{(\\hbar)} - \\hbar and (\\int \\mathbf{B}{(\\hbar)} d\\hbar)^{\\hbar} = (M + \\hbar \\log{(\\hbar)} - \\hbar)^{\\hbar} and - C - \\hbar \\mathbf{B}{(\\hbar)} + \\hbar + (\\int \\mathbf{B}{(\\hbar)} d\\hbar)^{\\hbar} = - C - \\hbar \\mathbf{B}{(\\hbar)} + \\hbar + (M + \\hbar \\log{(\\hbar)} - \\hbar)^{\\hbar}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('M', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["minus", 6, "Add(Symbol('C', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True), Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{B}')(Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True), Pow(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\hbar', commutative=True), log(Symbol('\\\\hbar', commutative=True))), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{E})} = e^{\\mathbf{E}} and \\operatorname{c_{0}}{(\\mathbf{E})} = \\frac{1}{\\operatorname{n_{2}}{(\\mathbf{E})}} and v{(\\mathbf{E})} = e^{- \\mathbf{E}}, then obtain v{(\\mathbf{E})} = \\frac{1}{\\operatorname{n_{2}}{(\\mathbf{E})}}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{E})} = e^{\\mathbf{E}} and \\operatorname{c_{0}}{(\\mathbf{E})} = \\frac{1}{\\operatorname{n_{2}}{(\\mathbf{E})}} and \\operatorname{c_{0}}{(\\mathbf{E})} = e^{- \\mathbf{E}} and v{(\\mathbf{E})} = e^{- \\mathbf{E}} and e^{- \\mathbf{E}} = \\frac{1}{\\operatorname{n_{2}}{(\\mathbf{E})}} and v{(\\mathbf{E})} = \\frac{1}{\\operatorname{n_{2}}{(\\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Symbol('\\\\mathbf{E}', commutative=True)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('c_0')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], ["renaming_premise", "Equality(Function('v')(Symbol('\\\\mathbf{E}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(exp(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))), Pow(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Function('v')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(Function('n_2')(Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\sigma_{p}{(g,\\mu)} = \\mu g and \\operatorname{a^{\\dagger}}{(g,\\mu)} = - \\sigma_{p}{(g,\\mu)}, then obtain \\iint - \\sigma_{p}{(g,\\mu)} dg dg = \\iint - \\mu g dg dg", "derivation": "\\sigma_{p}{(g,\\mu)} = \\mu g and \\operatorname{a^{\\dagger}}{(g,\\mu)} = - \\sigma_{p}{(g,\\mu)} and \\operatorname{a^{\\dagger}}{(g,\\mu)} = - \\mu g and \\int \\operatorname{a^{\\dagger}}{(g,\\mu)} dg = \\int - \\mu g dg and \\int - \\sigma_{p}{(g,\\mu)} dg = \\int - \\mu g dg and \\iint - \\sigma_{p}{(g,\\mu)} dg dg = \\iint - \\mu g dg dg", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["integrate", 5, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(\\mathbf{s})} = \\sin{(\\cos{(\\mathbf{s})})}, then obtain (e^{\\frac{d}{d \\mathbf{s}} \\psi^{*}{(\\mathbf{s})}})^{\\mathbf{s}} = (e^{\\frac{d}{d \\mathbf{s}} \\sin{(\\cos{(\\mathbf{s})})}})^{\\mathbf{s}}", "derivation": "\\psi^{*}{(\\mathbf{s})} = \\sin{(\\cos{(\\mathbf{s})})} and \\frac{d}{d \\mathbf{s}} \\psi^{*}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\sin{(\\cos{(\\mathbf{s})})} and e^{\\frac{d}{d \\mathbf{s}} \\psi^{*}{(\\mathbf{s})}} = e^{\\frac{d}{d \\mathbf{s}} \\sin{(\\cos{(\\mathbf{s})})}} and (e^{\\frac{d}{d \\mathbf{s}} \\psi^{*}{(\\mathbf{s})}})^{\\mathbf{s}} = (e^{\\frac{d}{d \\mathbf{s}} \\sin{(\\cos{(\\mathbf{s})})}})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True)), sin(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), exp(Derivative(sin(cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(exp(Derivative(Function('\\\\psi^*')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{s}', commutative=True)), Pow(exp(Derivative(sin(cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\phi{(y^{\\prime},V_{\\mathbf{E}})} = - y^{\\prime} + e^{V_{\\mathbf{E}}}, then obtain (\\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\phi{(y^{\\prime},V_{\\mathbf{E}})} e^{- V_{\\mathbf{E}}})^{y^{\\prime}} = (\\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- y^{\\prime} + e^{V_{\\mathbf{E}}}) e^{- V_{\\mathbf{E}}})^{y^{\\prime}}", "derivation": "\\phi{(y^{\\prime},V_{\\mathbf{E}})} = - y^{\\prime} + e^{V_{\\mathbf{E}}} and \\phi{(y^{\\prime},V_{\\mathbf{E}})} e^{- V_{\\mathbf{E}}} = (- y^{\\prime} + e^{V_{\\mathbf{E}}}) e^{- V_{\\mathbf{E}}} and \\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\phi{(y^{\\prime},V_{\\mathbf{E}})} e^{- V_{\\mathbf{E}}} = \\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- y^{\\prime} + e^{V_{\\mathbf{E}}}) e^{- V_{\\mathbf{E}}} and (\\frac{\\partial}{\\partial V_{\\mathbf{E}}} \\phi{(y^{\\prime},V_{\\mathbf{E}})} e^{- V_{\\mathbf{E}}})^{y^{\\prime}} = (\\frac{\\partial}{\\partial V_{\\mathbf{E}}} (- y^{\\prime} + e^{V_{\\mathbf{E}}}) e^{- V_{\\mathbf{E}}})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["divide", 1, "exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Function('\\\\phi')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["differentiate", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\phi')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Mul(Function('\\\\phi')(Symbol('y^{\\\\prime}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(Mul(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('V_{\\\\mathbf{E}}', commutative=True))), exp(Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{v}{(H)} = \\sin{(\\sin{(H)})} and \\hat{H}_{\\lambda}{(H)} = \\sin{(\\sin{(H)})}, then obtain (\\frac{\\mathbf{v}^{2}{(H)}}{H})^{H} - \\hat{H}_{\\lambda}{(H)} = (\\frac{\\hat{H}_{\\lambda}{(H)} \\mathbf{v}{(H)}}{H})^{H} - \\hat{H}_{\\lambda}{(H)}", "derivation": "\\mathbf{v}{(H)} = \\sin{(\\sin{(H)})} and \\mathbf{v}^{2}{(H)} = \\mathbf{v}{(H)} \\sin{(\\sin{(H)})} and \\hat{H}_{\\lambda}{(H)} = \\sin{(\\sin{(H)})} and \\mathbf{v}^{2}{(H)} = \\hat{H}_{\\lambda}{(H)} \\mathbf{v}{(H)} and \\frac{\\mathbf{v}^{2}{(H)}}{H} = \\frac{\\hat{H}_{\\lambda}{(H)} \\mathbf{v}{(H)}}{H} and (\\frac{\\mathbf{v}^{2}{(H)}}{H})^{H} = (\\frac{\\hat{H}_{\\lambda}{(H)} \\mathbf{v}{(H)}}{H})^{H} and (\\frac{\\mathbf{v}^{2}{(H)}}{H})^{H} - \\hat{H}_{\\lambda}{(H)} = (\\frac{\\hat{H}_{\\lambda}{(H)} \\mathbf{v}{(H)}}{H})^{H} - \\hat{H}_{\\lambda}{(H)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{v}')(Symbol('H', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)), sin(sin(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), Integer(2)), Mul(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)), Function('\\\\mathbf{v}')(Symbol('H', commutative=True))))"], [["divide", 4, "Symbol('H', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), Integer(2))), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)), Function('\\\\mathbf{v}')(Symbol('H', commutative=True))))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), Integer(2))), Symbol('H', commutative=True)), Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)), Function('\\\\mathbf{v}')(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["minus", 6, "Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{v}')(Symbol('H', commutative=True)), Integer(2))), Symbol('H', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)))), Add(Pow(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)), Function('\\\\mathbf{v}')(Symbol('H', commutative=True))), Symbol('H', commutative=True)), Mul(Integer(-1), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(m)} = \\log{(e^{m})}, then obtain F_{g} + 2 m - \\operatorname{n_{1}}{(m)} = \\dot{y} + 4 m - 3 \\operatorname{n_{1}}{(m)}", "derivation": "\\operatorname{n_{1}}{(m)} = \\log{(e^{m})} and \\operatorname{n_{1}}{(m)} + \\log{(e^{m})} = 2 \\log{(e^{m})} and \\log{(e^{m})} = - \\operatorname{n_{1}}{(m)} + 2 \\log{(e^{m})} and \\frac{d}{d m} \\log{(e^{m})} = \\frac{d}{d m} (- \\operatorname{n_{1}}{(m)} + 2 \\log{(e^{m})}) and \\frac{d}{d m} (- \\operatorname{n_{1}}{(m)} + 2 \\log{(e^{m})}) = \\frac{d}{d m} (- 3 \\operatorname{n_{1}}{(m)} + 4 \\log{(e^{m})}) and \\int \\frac{d}{d m} (- \\operatorname{n_{1}}{(m)} + 2 \\log{(e^{m})}) dm = \\int \\frac{d}{d m} (- 3 \\operatorname{n_{1}}{(m)} + 4 \\log{(e^{m})}) dm and F_{g} + 2 m - \\operatorname{n_{1}}{(m)} = \\dot{y} + 4 m - 3 \\operatorname{n_{1}}{(m)}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True))))"], [["add", 1, "log(exp(Symbol('m', commutative=True)))"], "Equality(Add(Function('n_1')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True)))), Mul(Integer(2), log(exp(Symbol('m', commutative=True)))))"], [["minus", 2, "Function('n_1')(Symbol('m', commutative=True))"], "Equality(log(exp(Symbol('m', commutative=True))), Add(Mul(Integer(-1), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(2), log(exp(Symbol('m', commutative=True))))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(log(exp(Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(2), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Add(Mul(Integer(-1), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(2), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(3), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(4), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(2), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Integer(3), Function('n_1')(Symbol('m', commutative=True))), Mul(Integer(4), log(exp(Symbol('m', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))), Tuple(Symbol('m', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('F_g', commutative=True), Mul(Integer(2), Symbol('m', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('m', commutative=True)))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(4), Symbol('m', commutative=True)), Mul(Integer(-1), Integer(3), Function('n_1')(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\delta{(x)} = \\sin{(\\cos{(x)})}, then obtain - \\frac{d}{d x} \\sin{(\\cos{(x)})} + \\int (\\delta{(x)} + \\frac{d}{d x} \\delta{(x)} + 1) dx = - \\frac{d}{d x} \\sin{(\\cos{(x)})} + \\int (\\sin{(\\cos{(x)})} + \\frac{d}{d x} \\delta{(x)} + 1) dx", "derivation": "\\delta{(x)} = \\sin{(\\cos{(x)})} and \\delta{(x)} + \\frac{d}{d x} \\delta{(x)} = \\sin{(\\cos{(x)})} + \\frac{d}{d x} \\delta{(x)} and \\delta{(x)} + \\frac{d}{d x} \\delta{(x)} + 1 = \\sin{(\\cos{(x)})} + \\frac{d}{d x} \\delta{(x)} + 1 and \\int (\\delta{(x)} + \\frac{d}{d x} \\delta{(x)} + 1) dx = \\int (\\sin{(\\cos{(x)})} + \\frac{d}{d x} \\delta{(x)} + 1) dx and - \\frac{d}{d x} \\sin{(\\cos{(x)})} + \\int (\\delta{(x)} + \\frac{d}{d x} \\delta{(x)} + 1) dx = - \\frac{d}{d x} \\sin{(\\cos{(x)})} + \\int (\\sin{(\\cos{(x)})} + \\frac{d}{d x} \\delta{(x)} + 1) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('x', commutative=True)), sin(cos(Symbol('x', commutative=True))))"], [["add", 1, "Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\delta')(Symbol('x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Add(sin(cos(Symbol('x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["add", 2, 1], "Equality(Add(Function('\\\\delta')(Symbol('x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Add(sin(cos(Symbol('x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Function('\\\\delta')(Symbol('x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True))), Integral(Add(sin(cos(Symbol('x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True))))"], [["minus", 4, "Derivative(sin(cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(sin(cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Integral(Add(Function('\\\\delta')(Symbol('x', commutative=True)), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Derivative(sin(cos(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), Integral(Add(sin(cos(Symbol('x', commutative=True))), Derivative(Function('\\\\delta')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon_{0}{(n,y^{\\prime},\\Omega)} = \\frac{\\Omega^{y^{\\prime}}}{n}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\varepsilon_{0}{(n,y^{\\prime},\\Omega)} = \\frac{\\Omega^{y^{\\prime}} \\log{(\\Omega)}}{n}, then obtain \\frac{\\partial^{2}}{\\partial n\\partial y^{\\prime}} \\frac{\\Omega^{y^{\\prime}}}{n} = \\frac{\\partial}{\\partial n} \\frac{\\Omega^{y^{\\prime}} \\log{(\\Omega)}}{n}", "derivation": "\\varepsilon_{0}{(n,y^{\\prime},\\Omega)} = \\frac{\\Omega^{y^{\\prime}}}{n} and \\frac{\\partial}{\\partial y^{\\prime}} \\varepsilon_{0}{(n,y^{\\prime},\\Omega)} = \\frac{\\partial}{\\partial y^{\\prime}} \\frac{\\Omega^{y^{\\prime}}}{n} and \\frac{\\partial}{\\partial y^{\\prime}} \\varepsilon_{0}{(n,y^{\\prime},\\Omega)} = \\frac{\\Omega^{y^{\\prime}} \\log{(\\Omega)}}{n} and \\frac{\\partial}{\\partial y^{\\prime}} \\frac{\\Omega^{y^{\\prime}}}{n} = \\frac{\\Omega^{y^{\\prime}} \\log{(\\Omega)}}{n} and \\frac{\\partial^{2}}{\\partial n\\partial y^{\\prime}} \\frac{\\Omega^{y^{\\prime}}}{n} = \\frac{\\partial}{\\partial n} \\frac{\\Omega^{y^{\\prime}} \\log{(\\Omega)}}{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('n', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('n', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('n', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)), log(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)), log(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 4, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('n', commutative=True), Integer(-1)), log(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(c)} = \\frac{d}{d c} \\log{(c)}, then derive \\frac{d^{2}}{d c^{2}} T{(c)} = \\frac{2}{c^{3}}, then obtain \\frac{d}{d c} \\int \\frac{d^{2}}{d c^{2}} T{(c)} dc = \\frac{d}{d c} \\int \\frac{2}{c^{3}} dc", "derivation": "T{(c)} = \\frac{d}{d c} \\log{(c)} and \\frac{d}{d c} T{(c)} = \\frac{d^{2}}{d c^{2}} \\log{(c)} and \\frac{d^{2}}{d c^{2}} T{(c)} = \\frac{d^{3}}{d c^{3}} \\log{(c)} and \\frac{d^{2}}{d c^{2}} T{(c)} = \\frac{2}{c^{3}} and \\int \\frac{d^{2}}{d c^{2}} T{(c)} dc = \\int \\frac{2}{c^{3}} dc and \\frac{d}{d c} \\int \\frac{d^{2}}{d c^{2}} T{(c)} dc = \\frac{d}{d c} \\int \\frac{2}{c^{3}} dc", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('c', commutative=True)), Derivative(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Derivative(log(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(3))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('T')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-3))))"], [["integrate", 4, "Symbol('c', commutative=True)"], "Equality(Integral(Derivative(Function('T')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Tuple(Symbol('c', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-3))), Tuple(Symbol('c', commutative=True))))"], [["differentiate", 5, "Symbol('c', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('T')(Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(2))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), Pow(Symbol('c', commutative=True), Integer(-3))), Tuple(Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} = \\frac{\\lambda + b}{\\sigma_x}, then obtain \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} + \\int \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} d\\lambda = \\int \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} d\\lambda + \\frac{\\lambda + b}{\\sigma_x}", "derivation": "\\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} = \\frac{\\lambda + b}{\\sigma_x} and \\int \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} d\\lambda = \\int \\frac{\\lambda + b}{\\sigma_x} d\\lambda and \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} + \\int \\frac{\\lambda + b}{\\sigma_x} d\\lambda = \\int \\frac{\\lambda + b}{\\sigma_x} d\\lambda + \\frac{\\lambda + b}{\\sigma_x} and \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} + \\int \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} d\\lambda = \\int \\operatorname{x^{{\\}'}}{(\\sigma_x,b,\\lambda)} d\\lambda + \\frac{\\lambda + b}{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True))))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Integral(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Integral(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Integral(Function('x^\\\\prime')(Symbol('\\\\sigma_x', commutative=True), Symbol('b', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Symbol('b', commutative=True)))))"]]}, {"prompt": "Given J{(C,\\mathbf{s})} = \\log{(C)}^{\\mathbf{s}} and \\ddot{x}{(C,\\mathbf{s})} = \\cos{(J{(C,\\mathbf{s})})}, then obtain \\cos{(J{(C,\\mathbf{s})})} + 2 = \\ddot{x}{(C,\\mathbf{s})} + 2", "derivation": "J{(C,\\mathbf{s})} = \\log{(C)}^{\\mathbf{s}} and \\cos{(J{(C,\\mathbf{s})})} = \\cos{(\\log{(C)}^{\\mathbf{s}})} and \\ddot{x}{(C,\\mathbf{s})} = \\cos{(J{(C,\\mathbf{s})})} and \\cos{(J{(C,\\mathbf{s})})} + 1 = \\cos{(\\log{(C)}^{\\mathbf{s}})} + 1 and \\ddot{x}{(C,\\mathbf{s})} = \\cos{(\\log{(C)}^{\\mathbf{s}})} and \\cos{(J{(C,\\mathbf{s})})} + 2 = \\cos{(\\log{(C)}^{\\mathbf{s}})} + 2 and \\cos{(J{(C,\\mathbf{s})})} + 2 = \\ddot{x}{(C,\\mathbf{s})} + 2", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), cos(Pow(log(Symbol('C', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), cos(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(cos(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)), Add(cos(Pow(log(Symbol('C', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), cos(Pow(log(Symbol('C', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 4, 1], "Equality(Add(cos(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(2)), Add(cos(Pow(log(Symbol('C', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(cos(Function('J')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Integer(2)), Add(Function('\\\\ddot{x}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(v_{x},\\dot{z})} = - \\sin{(\\dot{z} - v_{x})}, then obtain \\int \\frac{\\partial}{\\partial v_{x}} - \\frac{\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})}}{\\sin{(\\dot{z} - v_{x})}} d\\dot{z} = \\int 0 d\\dot{z}", "derivation": "\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})} = - \\sin{(\\dot{z} - v_{x})} and - \\frac{\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})}}{\\sin{(\\dot{z} - v_{x})}} = 1 and \\frac{\\partial}{\\partial v_{x}} - \\frac{\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})}}{\\sin{(\\dot{z} - v_{x})}} = \\frac{d}{d v_{x}} 1 and \\int \\frac{\\partial}{\\partial v_{x}} - \\frac{\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})}}{\\sin{(\\dot{z} - v_{x})}} d\\dot{z} = \\int \\frac{d}{d v_{x}} 1 d\\dot{z} and \\int \\frac{\\partial}{\\partial v_{x}} - \\frac{\\operatorname{f^{\\prime}}{(v_{x},\\dot{z})}}{\\sin{(\\dot{z} - v_{x})}} d\\dot{z} = \\int 0 d\\dot{z}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))))))"], [["divide", 1, "Mul(Integer(-1), sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))))"], "Equality(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Integer(-1))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Integer(-1))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Derivative(Integer(1), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integral(Derivative(Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('v_x', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Pow(sin(Add(Symbol('\\\\dot{z}', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))), Integer(-1))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Tuple(Symbol('\\\\dot{z}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\dot{z}', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(\\psi^*)} = e^{\\psi^*}, then derive e^{\\psi^*} + \\frac{d}{d \\psi^*} \\dot{z}{(\\psi^*)} = 2 e^{\\psi^*}, then obtain \\int (\\dot{z}{(\\psi^*)} + \\frac{d}{d \\psi^*} \\dot{z}{(\\psi^*)}) d\\psi^* = \\int 2 \\dot{z}{(\\psi^*)} d\\psi^*", "derivation": "\\dot{z}{(\\psi^*)} = e^{\\psi^*} and \\dot{z}{(\\psi^*)} + e^{\\psi^*} = 2 e^{\\psi^*} and \\frac{d}{d \\psi^*} (\\dot{z}{(\\psi^*)} + e^{\\psi^*}) = \\frac{d}{d \\psi^*} 2 e^{\\psi^*} and e^{\\psi^*} + \\frac{d}{d \\psi^*} \\dot{z}{(\\psi^*)} = 2 e^{\\psi^*} and \\dot{z}{(\\psi^*)} + \\frac{d}{d \\psi^*} \\dot{z}{(\\psi^*)} = 2 \\dot{z}{(\\psi^*)} and \\int (\\dot{z}{(\\psi^*)} + \\frac{d}{d \\psi^*} \\dot{z}{(\\psi^*)}) d\\psi^* = \\int 2 \\dot{z}{(\\psi^*)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Add(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(exp(Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 5, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(2), Function('\\\\dot{z}')(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(v_{x})} = e^{v_{x}} and B{(v_{x})} = e^{v_{x}}, then obtain ((B^{2}{(v_{x})})^{- v_{x}} B{(v_{x})} e^{v_{x}})^{v_{x}} = ((B^{2}{(v_{x})})^{- v_{x}} B^{2}{(v_{x})})^{v_{x}}", "derivation": "\\theta_{1}{(v_{x})} = e^{v_{x}} and B{(v_{x})} = e^{v_{x}} and B{(v_{x})} \\theta_{1}{(v_{x})} = B{(v_{x})} e^{v_{x}} and \\theta_{1}{(v_{x})} e^{v_{x}} = e^{2 v_{x}} and B{(v_{x})} \\theta_{1}{(v_{x})} = B^{2}{(v_{x})} and B{(v_{x})} e^{v_{x}} = B^{2}{(v_{x})} and (B^{2}{(v_{x})})^{- v_{x}} B{(v_{x})} e^{v_{x}} = (B^{2}{(v_{x})})^{- v_{x}} B^{2}{(v_{x})} and ((B^{2}{(v_{x})})^{- v_{x}} B{(v_{x})} e^{v_{x}})^{v_{x}} = ((B^{2}{(v_{x})})^{- v_{x}} B^{2}{(v_{x})})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True)))"], [["times", 1, "Function('B')(Symbol('v_x', commutative=True))"], "Equality(Mul(Function('B')(Symbol('v_x', commutative=True)), Function('\\\\theta_1')(Symbol('v_x', commutative=True))), Mul(Function('B')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\theta_1')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), exp(Mul(Integer(2), Symbol('v_x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Function('B')(Symbol('v_x', commutative=True)), Function('\\\\theta_1')(Symbol('v_x', commutative=True))), Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Function('B')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)))"], [["divide", 6, "Pow(Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)), Symbol('v_x', commutative=True))"], "Equality(Mul(Pow(Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Function('B')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), Mul(Pow(Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2))))"], [["power", 7, "Symbol('v_x', commutative=True)"], "Equality(Pow(Mul(Pow(Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Function('B')(Symbol('v_x', commutative=True)), exp(Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Pow(Mul(Pow(Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2)), Mul(Integer(-1), Symbol('v_x', commutative=True))), Pow(Function('B')(Symbol('v_x', commutative=True)), Integer(2))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\pi,A_{z})} = A_{z}^{\\pi}, then obtain (- A_{z} + \\operatorname{A_{1}}{(\\pi,A_{z})})^{A_{z}} = (- A_{z} + 2 A_{z}^{\\pi} - \\operatorname{A_{1}}{(\\pi,A_{z})})^{A_{z}}", "derivation": "\\operatorname{A_{1}}{(\\pi,A_{z})} = A_{z}^{\\pi} and - A_{z} + \\operatorname{A_{1}}{(\\pi,A_{z})} = - A_{z} + A_{z}^{\\pi} and - A_{z} + 2 \\operatorname{A_{1}}{(\\pi,A_{z})} = - A_{z} + A_{z}^{\\pi} + \\operatorname{A_{1}}{(\\pi,A_{z})} and - A_{z} = - A_{z} + A_{z}^{\\pi} - \\operatorname{A_{1}}{(\\pi,A_{z})} and - A_{z} + A_{z}^{\\pi} = - A_{z} + 2 A_{z}^{\\pi} - \\operatorname{A_{1}}{(\\pi,A_{z})} and (- A_{z} + A_{z}^{\\pi})^{A_{z}} = (- A_{z} + 2 A_{z}^{\\pi} - \\operatorname{A_{1}}{(\\pi,A_{z})})^{A_{z}} and (- A_{z} + \\operatorname{A_{1}}{(\\pi,A_{z})})^{A_{z}} = (- A_{z} + 2 A_{z}^{\\pi} - \\operatorname{A_{1}}{(\\pi,A_{z})})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "Symbol('A_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))))"], [["add", 2, "Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(2), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True))))"], [["minus", 3, "Mul(Integer(2), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('A_z', commutative=True)), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))), Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_z', commutative=True)), Mul(Integer(2), Pow(Symbol('A_z', commutative=True), Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Function('A_1')(Symbol('\\\\pi', commutative=True), Symbol('A_z', commutative=True)))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(u)} = \\cos{(u)}, then derive \\int \\operatorname{f_{E}}{(u)} du = \\mathbf{s} + \\sin{(u)}, then obtain (\\int \\cos{(u)} du)^{2} = (\\mathbf{s} + \\sin{(u)}) \\int \\cos{(u)} du", "derivation": "\\operatorname{f_{E}}{(u)} = \\cos{(u)} and \\int \\operatorname{f_{E}}{(u)} du = \\int \\cos{(u)} du and \\int \\operatorname{f_{E}}{(u)} du = \\mathbf{s} + \\sin{(u)} and \\int \\cos{(u)} du = \\mathbf{s} + \\sin{(u)} and (\\int \\cos{(u)} du)^{2} = (\\mathbf{s} + \\sin{(u)}) \\int \\cos{(u)} du", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('u', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('u', commutative=True))))"], [["times", 4, "Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))"], "Equality(Pow(Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(2)), Mul(Add(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('u', commutative=True))), Integral(cos(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given A{(q,\\mathbf{S})} = - \\sin{(\\mathbf{S} - q)}, then obtain - q \\log{(z^{*})} + A{(q,\\mathbf{S})} \\log{(z^{*})} \\sin{(\\mathbf{S} - q)} + \\log{(z^{*})}^{2} = - q \\log{(z^{*})} - A^{2}{(q,\\mathbf{S})} \\log{(z^{*})} + \\log{(z^{*})}^{2}", "derivation": "A{(q,\\mathbf{S})} = - \\sin{(\\mathbf{S} - q)} and A{(q,\\mathbf{S})} \\sin{(\\mathbf{S} - q)} = - \\sin^{2}{(\\mathbf{S} - q)} and A{(q,\\mathbf{S})} \\sin{(\\mathbf{S} - q)} = - A^{2}{(q,\\mathbf{S})} and - q + A{(q,\\mathbf{S})} \\sin{(\\mathbf{S} - q)} + \\log{(z^{*})} = - q - A^{2}{(q,\\mathbf{S})} + \\log{(z^{*})} and (- q + A{(q,\\mathbf{S})} \\sin{(\\mathbf{S} - q)} + \\log{(z^{*})}) \\log{(z^{*})} = (- q - A^{2}{(q,\\mathbf{S})} + \\log{(z^{*})}) \\log{(z^{*})} and - q \\log{(z^{*})} + A{(q,\\mathbf{S})} \\log{(z^{*})} \\sin{(\\mathbf{S} - q)} + \\log{(z^{*})}^{2} = - q \\log{(z^{*})} - A^{2}{(q,\\mathbf{S})} \\log{(z^{*})} + \\log{(z^{*})}^{2}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))))"], [["times", 1, "sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], "Equality(Mul(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Mul(Integer(-1), Pow(sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Mul(Integer(-1), Pow(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('q', commutative=True)), log(Symbol('z^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), log(Symbol('z^*', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), log(Symbol('z^*', commutative=True))))"], [["times", 4, "log(Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), log(Symbol('z^*', commutative=True))), log(Symbol('z^*', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))), log(Symbol('z^*', commutative=True))), log(Symbol('z^*', commutative=True))))"], [["expand", 5], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True), log(Symbol('z^*', commutative=True))), Mul(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Symbol('z^*', commutative=True)), sin(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))), Pow(log(Symbol('z^*', commutative=True)), Integer(2))), Add(Mul(Integer(-1), Symbol('q', commutative=True), log(Symbol('z^*', commutative=True))), Mul(Integer(-1), Pow(Function('A')(Symbol('q', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), log(Symbol('z^*', commutative=True))), Pow(log(Symbol('z^*', commutative=True)), Integer(2))))"]]}, {"prompt": "Given G{(W)} = \\cos{(W)}, then obtain E_{x} r + (W + u)^{W} = E_{x} r + (\\int (- G{(W)} + \\cos{(W)} + 1) dW)^{W}", "derivation": "G{(W)} = \\cos{(W)} and 0 = - G{(W)} + \\cos{(W)} and 1 = - G{(W)} + \\cos{(W)} + 1 and \\int 1 dW = \\int (- G{(W)} + \\cos{(W)} + 1) dW and (\\int 1 dW)^{W} = (\\int (- G{(W)} + \\cos{(W)} + 1) dW)^{W} and E_{x} r + (\\int 1 dW)^{W} = E_{x} r + (\\int (- G{(W)} + \\cos{(W)} + 1) dW)^{W} and E_{x} r + (W + u)^{W} = E_{x} r + (\\int (- G{(W)} + \\cos{(W)} + 1) dW)^{W}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True)))"], [["minus", 1, "Function('G')(Symbol('W', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True))))"], [["add", 2, 1], "Equality(Integer(1), Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Integer(1)))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Integral(Integer(1), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["add", 5, "Mul(Symbol('E_x', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True))), Add(Mul(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Pow(Add(Symbol('W', commutative=True), Symbol('u', commutative=True)), Symbol('W', commutative=True))), Add(Mul(Symbol('E_x', commutative=True), Symbol('r', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Function('G')(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Integer(1)), Tuple(Symbol('W', commutative=True))), Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(z,G)} = z^{G}, then derive \\frac{\\partial}{\\partial G} \\bar{\\h}{(z,G)} = z^{G} \\log{(z)}, then obtain z^{G} \\log{(z)} = \\frac{\\partial}{\\partial G} z^{G}", "derivation": "\\bar{\\h}{(z,G)} = z^{G} and \\frac{\\partial}{\\partial G} \\bar{\\h}{(z,G)} = \\frac{\\partial}{\\partial G} z^{G} and \\frac{\\partial}{\\partial G} \\bar{\\h}{(z,G)} = z^{G} \\log{(z)} and z^{G} \\log{(z)} = \\frac{\\partial}{\\partial G} z^{G}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('\\\\hbar')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Pow(Symbol('z', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hbar')(Symbol('z', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Mul(Pow(Symbol('z', commutative=True), Symbol('G', commutative=True)), log(Symbol('z', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('z', commutative=True), Symbol('G', commutative=True)), log(Symbol('z', commutative=True))), Derivative(Pow(Symbol('z', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{b}{(\\hat{H}_l)} = \\log{(\\sin{(\\hat{H}_l)})}, then obtain \\int \\rho_{b}{(\\hat{H}_l)} d\\hat{H}_l + 1 = (\\frac{\\log{(\\sin{(\\hat{H}_l)})}}{\\rho_{b}{(\\hat{H}_l)}})^{\\hat{H}_l} + \\int \\rho_{b}{(\\hat{H}_l)} d\\hat{H}_l", "derivation": "\\rho_{b}{(\\hat{H}_l)} = \\log{(\\sin{(\\hat{H}_l)})} and 1 = \\frac{\\log{(\\sin{(\\hat{H}_l)})}}{\\rho_{b}{(\\hat{H}_l)}} and 1 = (\\frac{\\log{(\\sin{(\\hat{H}_l)})}}{\\rho_{b}{(\\hat{H}_l)}})^{\\hat{H}_l} and \\int \\rho_{b}{(\\hat{H}_l)} d\\hat{H}_l + 1 = (\\frac{\\log{(\\sin{(\\hat{H}_l)})}}{\\rho_{b}{(\\hat{H}_l)}})^{\\hat{H}_l} + \\int \\rho_{b}{(\\hat{H}_l)} d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), log(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["divide", 1, "Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["add", 3, "Integral(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Add(Integral(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integer(1)), Add(Pow(Mul(Pow(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Integer(-1)), log(sin(Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given \\chi{(A)} = \\log{(A)} and \\mu_{0}{(A)} = \\frac{d}{d A} \\chi{(A)}, then obtain \\frac{(\\frac{d}{d A} \\chi{(A)})^{2}}{(\\frac{d}{d A} \\log{(A)})^{2}} = 1", "derivation": "\\chi{(A)} = \\log{(A)} and \\frac{d}{d A} \\chi{(A)} = \\frac{d}{d A} \\log{(A)} and \\mu_{0}{(A)} = \\frac{d}{d A} \\chi{(A)} and \\mu_{0}{(A)} = \\frac{d}{d A} \\log{(A)} and \\mu_{0}{(A)} \\frac{d}{d A} \\chi{(A)} = \\mu_{0}{(A)} \\frac{d}{d A} \\log{(A)} and (\\frac{d}{d A} \\chi{(A)})^{2} = \\frac{d}{d A} \\chi{(A)} \\frac{d}{d A} \\log{(A)} and (\\frac{d}{d A} \\chi{(A)})^{4} = (\\frac{d}{d A} \\chi{(A)})^{2} (\\frac{d}{d A} \\log{(A)})^{2} and \\frac{(\\frac{d}{d A} \\chi{(A)})^{2}}{\\mu_{0}{(A)} \\frac{d}{d A} \\log{(A)}} = \\frac{\\frac{d}{d A} \\log{(A)}}{\\mu_{0}{(A)}} and \\frac{(\\frac{d}{d A} \\chi{(A)})^{2}}{(\\frac{d}{d A} \\log{(A)})^{2}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('A', commutative=True)), log(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('A', commutative=True)), Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mu_0')(Symbol('A', commutative=True)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 2, "Function('\\\\mu_0')(Symbol('A', commutative=True))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('A', commutative=True)), Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Function('\\\\mu_0')(Symbol('A', commutative=True)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["power", 6, 2], "Equality(Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(4)), Mul(Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2))))"], [["divide", 7, "Mul(Function('\\\\mu_0')(Symbol('A', commutative=True)), Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Function('\\\\mu_0')(Symbol('A', commutative=True)), Integer(-1)), Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Function('\\\\mu_0')(Symbol('A', commutative=True)), Integer(-1)), Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Mul(Pow(Derivative(Function('\\\\chi')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(log(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(-2))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\mathbf{M})} = \\sin{(\\mathbf{M})}, then derive \\int \\operatorname{F_{x}}{(\\mathbf{M})} d\\mathbf{M} = C_{d} - \\cos{(\\mathbf{M})}, then obtain 1 = \\frac{C_{d} - \\cos{(\\mathbf{M})}}{\\int \\operatorname{F_{x}}{(\\mathbf{M})} d\\mathbf{M}}", "derivation": "\\operatorname{F_{x}}{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\int \\operatorname{F_{x}}{(\\mathbf{M})} d\\mathbf{M} = \\int \\sin{(\\mathbf{M})} d\\mathbf{M} and \\int \\operatorname{F_{x}}{(\\mathbf{M})} d\\mathbf{M} = C_{d} - \\cos{(\\mathbf{M})} and \\int \\sin{(\\mathbf{M})} d\\mathbf{M} = C_{d} - \\cos{(\\mathbf{M})} and 1 = \\frac{C_{d} - \\cos{(\\mathbf{M})}}{\\int \\sin{(\\mathbf{M})} d\\mathbf{M}} and 1 = \\frac{C_{d} - \\cos{(\\mathbf{M})}}{\\int \\operatorname{F_{x}}{(\\mathbf{M})} d\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["divide", 4, "Integral(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Pow(Integral(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integer(1), Mul(Add(Symbol('C_d', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Pow(Integral(Function('F_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(V)} = \\cos{(V)}, then derive \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\cos{(V)}, then obtain \\mathbf{J}_f{(V)} \\cos{(V)} \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\mathbf{J}_f^{3}{(V)}", "derivation": "\\mathbf{J}_f{(V)} = \\cos{(V)} and \\mathbf{J}_f^{2}{(V)} = \\mathbf{J}_f{(V)} \\cos{(V)} and \\frac{d}{d V} \\mathbf{J}_f{(V)} = \\frac{d}{d V} \\cos{(V)} and \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = \\frac{d^{2}}{d V^{2}} \\cos{(V)} and \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\cos{(V)} and \\mathbf{J}_f{(V)} \\cos{(V)} \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\mathbf{J}_f{(V)} \\cos^{2}{(V)} and \\mathbf{J}_f^{2}{(V)} \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\mathbf{J}_f^{3}{(V)} and \\mathbf{J}_f{(V)} \\cos{(V)} \\frac{d^{2}}{d V^{2}} \\mathbf{J}_f{(V)} = - \\mathbf{J}_f^{3}{(V)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True))))"], [["differentiate", 1, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('V', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))), Derivative(cos(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('V', commutative=True))))"], [["times", 5, "Mul(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2)))), Mul(Integer(-1), Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Pow(cos(Symbol('V', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Integer(2)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Integer(3))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), cos(Symbol('V', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(2)))), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_f')(Symbol('V', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\varphi^{*}{(t)} = \\log{(t)}, then obtain 0 = (\\int \\varphi^{*}{(t)} dt + \\int \\log{(t)} dt) \\varphi^{*}{(t)} - 2 \\varphi^{*}{(t)} \\int \\varphi^{*}{(t)} dt", "derivation": "\\varphi^{*}{(t)} = \\log{(t)} and \\int \\varphi^{*}{(t)} dt = \\int \\log{(t)} dt and 2 \\int \\varphi^{*}{(t)} dt = \\int \\varphi^{*}{(t)} dt + \\int \\log{(t)} dt and 2 \\varphi^{*}{(t)} \\int \\varphi^{*}{(t)} dt = (\\int \\varphi^{*}{(t)} dt + \\int \\log{(t)} dt) \\varphi^{*}{(t)} and - t + 2 \\varphi^{*}{(t)} \\int \\varphi^{*}{(t)} dt = - t + (\\int \\varphi^{*}{(t)} dt + \\int \\log{(t)} dt) \\varphi^{*}{(t)} and 0 = (\\int \\varphi^{*}{(t)} dt + \\int \\log{(t)} dt) \\varphi^{*}{(t)} - 2 \\varphi^{*}{(t)} \\int \\varphi^{*}{(t)} dt", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["add", 2, "Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Add(Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["times", 3, "Function('\\\\varphi^*')(Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Add(Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Function('\\\\varphi^*')(Symbol('t', commutative=True))))"], [["minus", 4, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Add(Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Function('\\\\varphi^*')(Symbol('t', commutative=True)))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('t', commutative=True)), Mul(Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], "Equality(Integer(0), Add(Mul(Add(Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True)))), Function('\\\\varphi^*')(Symbol('t', commutative=True))), Mul(Integer(-1), Integer(2), Function('\\\\varphi^*')(Symbol('t', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\hat{x}{(\\nabla)} = \\log{(\\nabla)}, then obtain \\hat{x}{(\\nabla)} - \\int \\log{(\\nabla)} d\\nabla - 1 = \\log{(\\nabla)} - \\int \\log{(\\nabla)} d\\nabla - 1", "derivation": "\\hat{x}{(\\nabla)} = \\log{(\\nabla)} and \\int \\hat{x}{(\\nabla)} d\\nabla = \\int \\log{(\\nabla)} d\\nabla and \\hat{x}{(\\nabla)} - 1 = \\log{(\\nabla)} - 1 and \\hat{x}{(\\nabla)} - \\int \\hat{x}{(\\nabla)} d\\nabla - 1 = \\log{(\\nabla)} - \\int \\hat{x}{(\\nabla)} d\\nabla - 1 and \\hat{x}{(\\nabla)} - \\int \\log{(\\nabla)} d\\nabla - 1 = \\log{(\\nabla)} - \\int \\log{(\\nabla)} d\\nabla - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), log(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Integer(-1)), Add(log(Symbol('\\\\nabla', commutative=True)), Integer(-1)))"], [["minus", 3, "Integral(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(log(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Integer(-1)), Add(log(Symbol('\\\\nabla', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given B{(\\mathbf{p})} = e^{\\mathbf{p}}, then obtain B{(\\mathbf{p})} - B^{2 \\mathbf{p}}{(\\mathbf{p})} = B{(\\mathbf{p})} - B^{\\mathbf{p}}{(\\mathbf{p})} (e^{\\mathbf{p}})^{\\mathbf{p}}", "derivation": "B{(\\mathbf{p})} = e^{\\mathbf{p}} and B^{\\mathbf{p}}{(\\mathbf{p})} = (e^{\\mathbf{p}})^{\\mathbf{p}} and B^{2 \\mathbf{p}}{(\\mathbf{p})} = B^{\\mathbf{p}}{(\\mathbf{p})} (e^{\\mathbf{p}})^{\\mathbf{p}} and - B{(\\mathbf{p})} + B^{2 \\mathbf{p}}{(\\mathbf{p})} = - B{(\\mathbf{p})} + B^{\\mathbf{p}}{(\\mathbf{p})} (e^{\\mathbf{p}})^{\\mathbf{p}} and B{(\\mathbf{p})} - B^{2 \\mathbf{p}}{(\\mathbf{p})} = B{(\\mathbf{p})} - B^{\\mathbf{p}}{(\\mathbf{p})} (e^{\\mathbf{p}})^{\\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('\\\\mathbf{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 2, "Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True))))"], [["minus", 3, "Function('B')(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('B')(Symbol('\\\\mathbf{p}', commutative=True))), Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True)))), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True))))), Add(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Pow(Function('B')(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Pow(exp(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))))"]]}, {"prompt": "Given \\pi{(E_{\\lambda},Q)} = E_{\\lambda} Q and \\hat{x}_0{(E_{\\lambda},Q)} = \\frac{\\pi{(E_{\\lambda},Q)}}{Q}, then obtain \\frac{\\pi{(E_{\\lambda},Q)}}{E_{\\lambda} Q^{2} \\int E_{\\lambda} Q dE_{\\lambda}} = \\frac{1}{Q \\int E_{\\lambda} Q dE_{\\lambda}}", "derivation": "\\pi{(E_{\\lambda},Q)} = E_{\\lambda} Q and \\hat{x}_0{(E_{\\lambda},Q)} = \\frac{\\pi{(E_{\\lambda},Q)}}{Q} and \\hat{x}_0{(E_{\\lambda},Q)} = E_{\\lambda} and \\frac{\\hat{x}_0{(E_{\\lambda},Q)}}{\\pi{(E_{\\lambda},Q)}} = \\frac{E_{\\lambda}}{\\pi{(E_{\\lambda},Q)}} and \\frac{\\hat{x}_0{(E_{\\lambda},Q)}}{E_{\\lambda} Q} = \\frac{1}{Q} and \\frac{\\hat{x}_0{(E_{\\lambda},Q)}}{E_{\\lambda} Q \\int E_{\\lambda} Q dE_{\\lambda}} = \\frac{1}{Q \\int E_{\\lambda} Q dE_{\\lambda}} and \\frac{\\pi{(E_{\\lambda},Q)}}{E_{\\lambda} Q^{2} \\int E_{\\lambda} Q dE_{\\lambda}} = \\frac{1}{Q \\int E_{\\lambda} Q dE_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["divide", 3, "Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Pow(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True))), Pow(Symbol('Q', commutative=True), Integer(-1)))"], [["divide", 5, "Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Pow(Symbol('Q', commutative=True), Integer(-2)), Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Pow(Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(T)} = \\cos{(T)}, then obtain \\sin^{3}{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)} = (\\sin{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)}) \\sin^{2}{(\\int \\log{(2 \\cos{(T)})} dT)}", "derivation": "\\operatorname{v_{x}}{(T)} = \\cos{(T)} and \\operatorname{v_{x}}{(T)} + \\cos{(T)} = 2 \\cos{(T)} and \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} = \\log{(2 \\cos{(T)})} and \\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT = \\int \\log{(2 \\cos{(T)})} dT and \\sin{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)} = \\sin{(\\int \\log{(2 \\cos{(T)})} dT)} and \\sin^{2}{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)} = \\sin^{2}{(\\int \\log{(2 \\cos{(T)})} dT)} and \\sin^{3}{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)} = (\\sin{(\\int \\log{(\\operatorname{v_{x}}{(T)} + \\cos{(T)})} dT)}) \\sin^{2}{(\\int \\log{(2 \\cos{(T)})} dT)}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))"], [["add", 1, "cos(Symbol('T', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True))), Mul(Integer(2), cos(Symbol('T', commutative=True))))"], [["log", 2], "Equality(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), log(Mul(Integer(2), cos(Symbol('T', commutative=True)))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))), Integral(log(Mul(Integer(2), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], [["sin", 4], "Equality(sin(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), sin(Integral(log(Mul(Integer(2), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))))"], [["power", 5, 2], "Equality(Pow(sin(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Integer(2)), Pow(sin(Integral(log(Mul(Integer(2), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Integer(2)))"], [["times", 6, "sin(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True))))"], "Equality(Pow(sin(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Integer(3)), Mul(sin(Integral(log(Add(Function('v_x')(Symbol('T', commutative=True)), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Pow(sin(Integral(log(Mul(Integer(2), cos(Symbol('T', commutative=True)))), Tuple(Symbol('T', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(b,C_{1})} = b^{C_{1}}, then obtain \\frac{C_{1} b^{C_{1}}}{b} + \\frac{\\partial}{\\partial b} \\operatorname{f_{E}}{(b,C_{1})} + 1 = \\frac{C_{1} b^{C_{1}}}{b} + \\frac{\\partial}{\\partial b} b^{C_{1}} + 1", "derivation": "\\operatorname{f_{E}}{(b,C_{1})} = b^{C_{1}} and \\frac{\\partial}{\\partial b} \\operatorname{f_{E}}{(b,C_{1})} = \\frac{\\partial}{\\partial b} b^{C_{1}} and \\frac{\\partial}{\\partial b} \\operatorname{f_{E}}{(b,C_{1})} + 1 = \\frac{\\partial}{\\partial b} b^{C_{1}} + 1 and \\frac{C_{1} b^{C_{1}}}{b} + \\frac{\\partial}{\\partial b} \\operatorname{f_{E}}{(b,C_{1})} + 1 = \\frac{C_{1} b^{C_{1}}}{b} + \\frac{\\partial}{\\partial b} b^{C_{1}} + 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('f_E')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"], [["add", 3, "Mul(Symbol('C_1', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True)))"], "Equality(Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True))), Derivative(Function('f_E')(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)), Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True))), Derivative(Pow(Symbol('b', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\psi{(A_{2},\\theta_2)} = \\frac{A_{2}}{\\theta_2}, then derive \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\psi{(A_{2},\\theta_2)}}{\\psi{(A_{2},\\theta_2)}} = - \\frac{A_{2} (\\frac{A_{2}}{\\theta_2})^{A_{2}}}{\\theta_2}, then obtain \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\psi{(A_{2},\\theta_2)}}{\\psi{(A_{2},\\theta_2)}} = - \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)}}{\\theta_2}", "derivation": "\\psi{(A_{2},\\theta_2)} = \\frac{A_{2}}{\\theta_2} and \\psi^{A_{2}}{(A_{2},\\theta_2)} = (\\frac{A_{2}}{\\theta_2})^{A_{2}} and \\frac{\\partial}{\\partial \\theta_2} \\psi^{A_{2}}{(A_{2},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\frac{A_{2}}{\\theta_2})^{A_{2}} and \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\psi{(A_{2},\\theta_2)}}{\\psi{(A_{2},\\theta_2)}} = - \\frac{A_{2} (\\frac{A_{2}}{\\theta_2})^{A_{2}}}{\\theta_2} and \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)} \\frac{\\partial}{\\partial \\theta_2} \\psi{(A_{2},\\theta_2)}}{\\psi{(A_{2},\\theta_2)}} = - \\frac{A_{2} \\psi^{A_{2}}{(A_{2},\\theta_2)}}{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Symbol('A_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Symbol('A_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('A_2', commutative=True), Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1))), Symbol('A_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('A_2', commutative=True), Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('A_2', commutative=True), Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Pow(Function('\\\\psi')(Symbol('A_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('A_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(v_{2},\\psi^*)} = e^{v_{2}^{\\psi^*}} and G{(v_{2},\\psi^*)} = v_{2}^{\\psi^*}, then obtain \\mathbf{D}^{v_{2}}{(v_{2},\\psi^*)} - e^{G{(v_{2},\\psi^*)}} = - e^{G{(v_{2},\\psi^*)}} + (e^{G{(v_{2},\\psi^*)}})^{v_{2}}", "derivation": "\\mathbf{D}{(v_{2},\\psi^*)} = e^{v_{2}^{\\psi^*}} and \\mathbf{D}^{v_{2}}{(v_{2},\\psi^*)} = (e^{v_{2}^{\\psi^*}})^{v_{2}} and \\mathbf{D}^{v_{2}}{(v_{2},\\psi^*)} - e^{v_{2}^{\\psi^*}} = - e^{v_{2}^{\\psi^*}} + (e^{v_{2}^{\\psi^*}})^{v_{2}} and G{(v_{2},\\psi^*)} = v_{2}^{\\psi^*} and \\mathbf{D}^{v_{2}}{(v_{2},\\psi^*)} - e^{G{(v_{2},\\psi^*)}} = - e^{G{(v_{2},\\psi^*)}} + (e^{G{(v_{2},\\psi^*)}})^{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True)), Pow(exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('v_2', commutative=True)))"], [["minus", 2, "exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))))), Add(Mul(Integer(-1), exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Pow(exp(Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('G')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('\\\\mathbf{D}')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), exp(Function('G')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))))), Add(Mul(Integer(-1), exp(Function('G')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True)))), Pow(exp(Function('G')(Symbol('v_2', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\mu{(\\Omega)} = \\sin{(\\Omega)}, then obtain \\frac{\\mu^{2 \\Omega}{(\\Omega)}}{\\Omega} = \\frac{\\mu^{\\Omega}{(\\Omega)} \\sin^{\\Omega}{(\\Omega)}}{\\Omega}", "derivation": "\\mu{(\\Omega)} = \\sin{(\\Omega)} and \\mu^{\\Omega}{(\\Omega)} = \\sin^{\\Omega}{(\\Omega)} and \\frac{\\mu^{\\Omega}{(\\Omega)}}{\\Omega} = \\frac{\\sin^{\\Omega}{(\\Omega)}}{\\Omega} and \\frac{\\mu^{\\Omega}{(\\Omega)} \\sin^{\\Omega}{(\\Omega)}}{\\Omega} = \\frac{\\sin^{2 \\Omega}{(\\Omega)}}{\\Omega} and \\frac{\\mu^{2 \\Omega}{(\\Omega)}}{\\Omega} = \\frac{\\mu^{\\Omega}{(\\Omega)} \\sin^{\\Omega}{(\\Omega)}}{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), sin(Symbol('\\\\Omega', commutative=True)))"], [["power", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], [["divide", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"], [["times", 2, "Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True)), Pow(sin(Symbol('\\\\Omega', commutative=True)), Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\rho{(E)} = \\log{(E)}, then obtain \\frac{(\\frac{\\rho{(E)}}{\\log{(E)}})^{E} \\rho{(E)}}{\\log{(E)}^{2}} + \\frac{\\rho{(E)}}{\\log{(E)}} = \\frac{\\rho{(E)}}{\\log{(E)}} + \\frac{\\rho{(E)}}{\\log{(E)}^{2}}", "derivation": "\\rho{(E)} = \\log{(E)} and \\frac{\\rho{(E)}}{\\log{(E)}} = 1 and (\\frac{\\rho{(E)}}{\\log{(E)}})^{E} = 1 and \\frac{(\\frac{\\rho{(E)}}{\\log{(E)}})^{E}}{\\log{(E)}} = \\frac{1}{\\log{(E)}} and \\frac{(\\frac{\\rho{(E)}}{\\log{(E)}})^{E} \\rho{(E)}}{\\log{(E)}^{2}} = \\frac{\\rho{(E)}}{\\log{(E)}^{2}} and \\frac{(\\frac{\\rho{(E)}}{\\log{(E)}})^{E} \\rho{(E)}}{\\log{(E)}^{2}} + \\frac{\\rho{(E)}}{\\log{(E)}} = \\frac{\\rho{(E)}}{\\log{(E)}} + \\frac{\\rho{(E)}}{\\log{(E)}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('E', commutative=True)), log(Symbol('E', commutative=True)))"], [["divide", 1, "log(Symbol('E', commutative=True))"], "Equality(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Integer(1))"], [["times", 3, "Pow(log(Symbol('E', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Pow(log(Symbol('E', commutative=True)), Integer(-1)))"], [["times", 4, "Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-2))), Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-2))))"], [["add", 5, "Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Pow(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Symbol('E', commutative=True)), Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-2))), Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1)))), Add(Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-1))), Mul(Function('\\\\rho')(Symbol('E', commutative=True)), Pow(log(Symbol('E', commutative=True)), Integer(-2)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(g)} = \\cos{(g)}, then obtain (\\int (\\frac{d}{d g} \\dot{\\mathbf{r}}{(g)} - 1) dg)^{g} = (\\int (\\frac{d}{d g} \\cos{(g)} - 1) dg)^{g}", "derivation": "\\dot{\\mathbf{r}}{(g)} = \\cos{(g)} and \\frac{d}{d g} \\dot{\\mathbf{r}}{(g)} = \\frac{d}{d g} \\cos{(g)} and \\frac{d}{d g} \\dot{\\mathbf{r}}{(g)} - 1 = \\frac{d}{d g} \\cos{(g)} - 1 and \\int (\\frac{d}{d g} \\dot{\\mathbf{r}}{(g)} - 1) dg = \\int (\\frac{d}{d g} \\cos{(g)} - 1) dg and (\\int (\\frac{d}{d g} \\dot{\\mathbf{r}}{(g)} - 1) dg)^{g} = (\\int (\\frac{d}{d g} \\cos{(g)} - 1) dg)^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('g', commutative=True))), Integral(Add(Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('g', commutative=True))))"], [["power", 4, "Symbol('g', commutative=True)"], "Equality(Pow(Integral(Add(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Integral(Add(Derivative(cos(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\delta)} = \\cos{(\\cos{(\\delta)})}, then obtain 1 = \\frac{\\frac{d}{d \\delta} \\int \\cos{(\\cos{(\\delta)})} d\\delta}{\\frac{d}{d \\delta} \\int \\mathbf{F}{(\\delta)} d\\delta}", "derivation": "\\mathbf{F}{(\\delta)} = \\cos{(\\cos{(\\delta)})} and \\int \\mathbf{F}{(\\delta)} d\\delta = \\int \\cos{(\\cos{(\\delta)})} d\\delta and \\frac{d}{d \\delta} \\int \\mathbf{F}{(\\delta)} d\\delta = \\frac{d}{d \\delta} \\int \\cos{(\\cos{(\\delta)})} d\\delta and 1 = \\frac{\\frac{d}{d \\delta} \\int \\cos{(\\cos{(\\delta)})} d\\delta}{\\frac{d}{d \\delta} \\int \\mathbf{F}{(\\delta)} d\\delta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), cos(cos(Symbol('\\\\delta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(cos(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Integral(cos(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(-1)), Derivative(Integral(cos(cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given b{(E,\\rho)} = \\rho \\sin{(E)}, then derive \\int (- E + b{(E,\\rho)}) dE = - \\frac{E^{2}}{2} - \\rho \\cos{(E)} + c_{0}, then obtain (\\int (- E + b{(E,\\rho)}) dE)^{\\rho} = (\\int (- E + \\rho \\sin{(E)}) dE)^{\\rho}", "derivation": "b{(E,\\rho)} = \\rho \\sin{(E)} and - E + b{(E,\\rho)} = - E + \\rho \\sin{(E)} and \\int (- E + b{(E,\\rho)}) dE = \\int (- E + \\rho \\sin{(E)}) dE and \\int (- E + b{(E,\\rho)}) dE = - \\frac{E^{2}}{2} - \\rho \\cos{(E)} + c_{0} and (\\int (- E + b{(E,\\rho)}) dE)^{\\rho} = (- \\frac{E^{2}}{2} - \\rho \\cos{(E)} + c_{0})^{\\rho} and (\\int (- E + \\rho \\sin{(E)}) dE)^{\\rho} = (- \\frac{E^{2}}{2} - \\rho \\cos{(E)} + c_{0})^{\\rho} and (\\int (- E + b{(E,\\rho)}) dE)^{\\rho} = (\\int (- E + \\rho \\sin{(E)}) dE)^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('E', commutative=True))))"], [["minus", 1, "Symbol('E', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('E', commutative=True)))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), cos(Symbol('E', commutative=True))), Symbol('c_0', commutative=True)))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), cos(Symbol('E', commutative=True))), Symbol('c_0', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True), cos(Symbol('E', commutative=True))), Symbol('c_0', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Function('b')(Symbol('E', commutative=True), Symbol('\\\\rho', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('E', commutative=True)))), Tuple(Symbol('E', commutative=True))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}_M{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})}, then obtain \\varphi + \\mathbf{J}_M{(L_{\\varepsilon})} = T + \\sin{(L_{\\varepsilon})}", "derivation": "\\mathbf{J}_M{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} \\mathbf{J}_M{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} and \\int \\frac{d}{d L_{\\varepsilon}} \\mathbf{J}_M{(L_{\\varepsilon})} dL_{\\varepsilon} = \\int \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} dL_{\\varepsilon} and \\varphi + \\mathbf{J}_M{(L_{\\varepsilon})} = T + \\sin{(L_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{J}_M')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))), Integral(Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Symbol('T', commutative=True), sin(Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given h{(\\varepsilon_0,b,G)} = G + \\frac{b}{\\varepsilon_0}, then obtain \\frac{\\iint h{(\\varepsilon_0,b,G)} dG d\\varepsilon_0}{\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG} = \\frac{\\int (\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG) d\\varepsilon_0}{\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG}", "derivation": "h{(\\varepsilon_0,b,G)} = G + \\frac{b}{\\varepsilon_0} and \\int h{(\\varepsilon_0,b,G)} dG = \\int (G + \\frac{b}{\\varepsilon_0}) dG and \\int h{(\\varepsilon_0,b,G)} dG = \\int G dG + \\int \\frac{b}{\\varepsilon_0} dG and \\iint h{(\\varepsilon_0,b,G)} dG d\\varepsilon_0 = \\int (\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG) d\\varepsilon_0 and \\frac{\\iint h{(\\varepsilon_0,b,G)} dG d\\varepsilon_0}{\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG} = \\frac{\\int (\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG) d\\varepsilon_0}{\\int G dG + \\int \\frac{b}{\\varepsilon_0} dG}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('b', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True))))"], [["integrate", 1, "Symbol('G', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('b', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Integral(Add(Symbol('G', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('h')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('b', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True))), Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Function('h')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('b', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"], [["divide", 4, "Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True))))"], "Equality(Mul(Pow(Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True)))), Integer(-1)), Integral(Function('h')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('b', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True)), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))), Mul(Pow(Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True)))), Integer(-1)), Integral(Add(Integral(Symbol('G', commutative=True), Tuple(Symbol('G', commutative=True))), Integral(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('b', commutative=True)), Tuple(Symbol('G', commutative=True)))), Tuple(Symbol('\\\\varepsilon_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\Omega,v_{z})} = \\Omega v_{z}, then obtain - \\Omega v_{z} - \\operatorname{f^{*}}{(\\Omega,v_{z})} + \\int \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{*}}{(\\Omega,v_{z})} d\\Omega = - \\Omega v_{z} - \\operatorname{f^{*}}{(\\Omega,v_{z})} + \\int \\frac{\\partial}{\\partial \\Omega} \\Omega v_{z} d\\Omega", "derivation": "\\operatorname{f^{*}}{(\\Omega,v_{z})} = \\Omega v_{z} and \\Omega v_{z} + \\operatorname{f^{*}}{(\\Omega,v_{z})} = 2 \\Omega v_{z} and \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{*}}{(\\Omega,v_{z})} = \\frac{\\partial}{\\partial \\Omega} \\Omega v_{z} and \\int \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{*}}{(\\Omega,v_{z})} d\\Omega = \\int \\frac{\\partial}{\\partial \\Omega} \\Omega v_{z} d\\Omega and - 2 \\Omega v_{z} + \\int \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{*}}{(\\Omega,v_{z})} d\\Omega = - 2 \\Omega v_{z} + \\int \\frac{\\partial}{\\partial \\Omega} \\Omega v_{z} d\\Omega and - \\Omega v_{z} - \\operatorname{f^{*}}{(\\Omega,v_{z})} + \\int \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{*}}{(\\Omega,v_{z})} d\\Omega = - \\Omega v_{z} - \\operatorname{f^{*}}{(\\Omega,v_{z})} + \\int \\frac{\\partial}{\\partial \\Omega} \\Omega v_{z} d\\Omega", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Integral(Derivative(Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Integral(Derivative(Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Mul(Integer(-1), Function('f^*')(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{S}{(q,U,\\mathbf{J}_f)} = (U q)^{\\mathbf{J}_f}, then obtain \\frac{\\partial}{\\partial q} (- \\frac{\\partial}{\\partial \\mathbf{J}_f} ((U q)^{\\mathbf{J}_f})^{q} + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{S}^{q}{(q,U,\\mathbf{J}_f)}) = \\frac{d}{d q} 0", "derivation": "\\mathbf{S}{(q,U,\\mathbf{J}_f)} = (U q)^{\\mathbf{J}_f} and \\mathbf{S}^{q}{(q,U,\\mathbf{J}_f)} = ((U q)^{\\mathbf{J}_f})^{q} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{S}^{q}{(q,U,\\mathbf{J}_f)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} ((U q)^{\\mathbf{J}_f})^{q} and - \\frac{\\partial}{\\partial \\mathbf{J}_f} ((U q)^{\\mathbf{J}_f})^{q} + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{S}^{q}{(q,U,\\mathbf{J}_f)} = 0 and \\frac{\\partial}{\\partial q} (- \\frac{\\partial}{\\partial \\mathbf{J}_f} ((U q)^{\\mathbf{J}_f})^{q} + \\frac{\\partial}{\\partial \\mathbf{J}_f} \\mathbf{S}^{q}{(q,U,\\mathbf{J}_f)}) = \\frac{d}{d q} 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('q', commutative=True), Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{S}')(Symbol('q', commutative=True), Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Pow(Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('q', commutative=True), Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Pow(Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Pow(Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('q', commutative=True), Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Integer(0))"], [["differentiate", 4, "Symbol('q', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Derivative(Pow(Pow(Mul(Symbol('U', commutative=True), Symbol('q', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Derivative(Pow(Function('\\\\mathbf{S}')(Symbol('q', commutative=True), Symbol('U', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} = - \\mathbf{B} + \\log{(C_{2})}, then obtain - \\frac{- \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} - \\log{(C_{2})}}{\\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})}} = - \\frac{\\mathbf{B} - 2 \\log{(C_{2})}}{\\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} = - \\mathbf{B} + \\log{(C_{2})} and - \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} = \\mathbf{B} - \\log{(C_{2})} and - \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} - \\log{(C_{2})} = \\mathbf{B} - 2 \\log{(C_{2})} and - (- \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} - \\log{(C_{2})}) \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} = - (\\mathbf{B} - 2 \\log{(C_{2})}) \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} and - \\frac{- \\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})} - \\log{(C_{2})}}{\\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})}} = - \\frac{\\mathbf{B} - 2 \\log{(C_{2})}}{\\operatorname{J_{\\varepsilon}}{(C_{2},\\mathbf{B})}}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), log(Symbol('C_2', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), log(Symbol('C_2', commutative=True)))))"], [["minus", 2, "log(Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), log(Symbol('C_2', commutative=True)))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), log(Symbol('C_2', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), log(Symbol('C_2', commutative=True)))), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), log(Symbol('C_2', commutative=True)))), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))))"], [["divide", 4, "Pow(Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(2))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), log(Symbol('C_2', commutative=True)))), Pow(Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), log(Symbol('C_2', commutative=True)))), Pow(Function('J_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} = v_{z}^{\\hat{x}_0}, then obtain \\iiint \\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} d\\hat{x}_0 dv_{z} d\\hat{x}_0 = \\iiint v_{z}^{\\hat{x}_0} d\\hat{x}_0 dv_{z} d\\hat{x}_0", "derivation": "\\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} = v_{z}^{\\hat{x}_0} and \\int \\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} d\\hat{x}_0 = \\int v_{z}^{\\hat{x}_0} d\\hat{x}_0 and \\iint \\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} d\\hat{x}_0 dv_{z} = \\iint v_{z}^{\\hat{x}_0} d\\hat{x}_0 dv_{z} and \\iiint \\operatorname{y^{\\prime}}{(\\hat{x}_0,v_{z})} d\\hat{x}_0 dv_{z} d\\hat{x}_0 = \\iiint v_{z}^{\\hat{x}_0} d\\hat{x}_0 dv_{z} d\\hat{x}_0", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('v_z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Pow(Symbol('v_z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Pow(Symbol('v_z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["integrate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Integral(Function('y^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))), Integral(Pow(Symbol('v_z', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('v_z', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True))))"]]}, {"prompt": "Given E{(\\lambda)} = e^{\\lambda} and \\mathbf{f}{(\\lambda)} = E^{\\lambda}{(\\lambda)}, then obtain \\frac{\\mathbf{f}^{2}{(\\lambda)} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda}{e^{\\lambda} + \\int E{(\\lambda)} d\\lambda} = \\frac{\\mathbf{f}{(\\lambda)} (e^{\\lambda})^{\\lambda} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda}{e^{\\lambda} + \\int E{(\\lambda)} d\\lambda}", "derivation": "E{(\\lambda)} = e^{\\lambda} and E^{\\lambda}{(\\lambda)} = (e^{\\lambda})^{\\lambda} and \\mathbf{f}{(\\lambda)} = E^{\\lambda}{(\\lambda)} and \\mathbf{f}{(\\lambda)} = (e^{\\lambda})^{\\lambda} and \\mathbf{f}^{2}{(\\lambda)} = \\mathbf{f}{(\\lambda)} (e^{\\lambda})^{\\lambda} and \\mathbf{f}^{2}{(\\lambda)} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda = \\mathbf{f}{(\\lambda)} (e^{\\lambda})^{\\lambda} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda and \\frac{\\mathbf{f}^{2}{(\\lambda)} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda}{e^{\\lambda} + \\int E{(\\lambda)} d\\lambda} = \\frac{\\mathbf{f}{(\\lambda)} (e^{\\lambda})^{\\lambda} + e^{\\lambda} + \\int e^{\\lambda} d\\lambda}{e^{\\lambda} + \\int E{(\\lambda)} d\\lambda}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Pow(Function('E')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["times", 4, "Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))))"], [["add", 5, "Add(exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], "Equality(Add(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Integer(2)), exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Add(Mul(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["divide", 6, "Add(exp(Symbol('\\\\lambda', commutative=True)), Integral(Function('E')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], "Equality(Mul(Pow(Add(exp(Symbol('\\\\lambda', commutative=True)), Integral(Function('E')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Integer(-1)), Add(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Integer(2)), exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Mul(Pow(Add(exp(Symbol('\\\\lambda', commutative=True)), Integral(Function('E')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Integer(-1)), Add(Mul(Function('\\\\mathbf{f}')(Symbol('\\\\lambda', commutative=True)), Pow(exp(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True)), Integral(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given s{(\\ddot{x},\\delta)} = \\ddot{x} - \\delta and \\mathbf{s}{(\\ddot{x},\\delta)} = - \\frac{\\ddot{x} - \\delta}{\\delta}, then obtain 1 - \\frac{s{(\\ddot{x},\\delta)}}{\\delta} = 1 - \\frac{\\ddot{x} - \\delta}{\\delta}", "derivation": "s{(\\ddot{x},\\delta)} = \\ddot{x} - \\delta and - \\frac{s{(\\ddot{x},\\delta)}}{\\delta} = - \\frac{\\ddot{x} - \\delta}{\\delta} and \\mathbf{s}{(\\ddot{x},\\delta)} = - \\frac{\\ddot{x} - \\delta}{\\delta} and \\mathbf{s}{(\\ddot{x},\\delta)} + 1 = 1 - \\frac{\\ddot{x} - \\delta}{\\delta} and \\mathbf{s}{(\\ddot{x},\\delta)} = - \\frac{s{(\\ddot{x},\\delta)}}{\\delta} and 1 - \\frac{s{(\\ddot{x},\\delta)}}{\\delta} = 1 - \\frac{\\ddot{x} - \\delta}{\\delta}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\delta', commutative=True)))), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Add(Symbol('\\\\ddot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))))))"]]}, {"prompt": "Given x{(\\mathbf{E})} = \\cos{(\\mathbf{E})}, then obtain \\iint (x{(\\mathbf{E})} - \\cos{(\\mathbf{E})}) d\\mathbf{E} d\\mathbf{E} = \\iint 0 d\\mathbf{E} d\\mathbf{E}", "derivation": "x{(\\mathbf{E})} = \\cos{(\\mathbf{E})} and x{(\\mathbf{E})} - \\cos{(\\mathbf{E})} = 0 and \\int (x{(\\mathbf{E})} - \\cos{(\\mathbf{E})}) d\\mathbf{E} = \\int 0 d\\mathbf{E} and \\iint (x{(\\mathbf{E})} - \\cos{(\\mathbf{E})}) d\\mathbf{E} d\\mathbf{E} = \\iint 0 d\\mathbf{E} d\\mathbf{E}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{E}', commutative=True)), cos(Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Add(Function('x')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Function('x')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Integral(Add(Function('x')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{E}', commutative=True)))), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} = \\Psi_{nl}^{\\dot{y}} and \\operatorname{C_{2}}{(\\dot{y},\\Psi_{nl})} = \\Psi_{nl}^{\\dot{y}}, then obtain - \\Psi_{nl} + \\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} + 1 = - \\Psi_{nl} + \\operatorname{C_{2}}{(\\dot{y},\\Psi_{nl})} + 1", "derivation": "\\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} = \\Psi_{nl}^{\\dot{y}} and - \\Psi_{nl} + \\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} = - \\Psi_{nl} + \\Psi_{nl}^{\\dot{y}} and \\operatorname{C_{2}}{(\\dot{y},\\Psi_{nl})} = \\Psi_{nl}^{\\dot{y}} and - \\Psi_{nl} + \\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} + 1 = - \\Psi_{nl} + \\Psi_{nl}^{\\dot{y}} + 1 and - \\Psi_{nl} + \\mathbf{J}_M{(\\dot{y},\\Psi_{nl})} + 1 = - \\Psi_{nl} + \\operatorname{C_{2}}{(\\dot{y},\\Psi_{nl})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["minus", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Function('C_2')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given V{(A_{x},\\varphi)} = \\varphi^{A_{x}}, then obtain \\varphi + \\varphi^{A_{x}} V{(A_{x},\\varphi)} - \\varphi^{A_{x}} + V{(A_{x},\\varphi)} = \\varphi + \\varphi^{2 A_{x}}", "derivation": "V{(A_{x},\\varphi)} = \\varphi^{A_{x}} and \\varphi^{A_{x}} V{(A_{x},\\varphi)} = \\varphi^{2 A_{x}} and \\varphi^{A_{x}} V{(A_{x},\\varphi)} - \\varphi^{A_{x}} = \\varphi^{2 A_{x}} - \\varphi^{A_{x}} and \\varphi + V{(A_{x},\\varphi)} = \\varphi + \\varphi^{A_{x}} and \\varphi + \\varphi^{2 A_{x}} - \\varphi^{A_{x}} + V{(A_{x},\\varphi)} = \\varphi + \\varphi^{2 A_{x}} and \\varphi + \\varphi^{A_{x}} V{(A_{x},\\varphi)} - \\varphi^{A_{x}} + V{(A_{x},\\varphi)} = \\varphi + \\varphi^{2 A_{x}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True))))"], [["minus", 2, "Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))"], "Equality(Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)))), Add(Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)))))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))))"], [["add", 4, "Add(Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))))"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True)), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Symbol('A_x', commutative=True))), Function('V')(Symbol('A_x', commutative=True), Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Mul(Integer(2), Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(v_{2})} = \\cos{(v_{2})}, then obtain \\frac{\\operatorname{v_{1}}{(v_{2})} - \\cos{(v_{2})}}{\\frac{d}{d v_{2}} \\cos{(v_{2})}} = 0", "derivation": "\\operatorname{v_{1}}{(v_{2})} = \\cos{(v_{2})} and \\frac{d}{d v_{2}} \\operatorname{v_{1}}{(v_{2})} = \\frac{d}{d v_{2}} \\cos{(v_{2})} and \\operatorname{v_{1}}{(v_{2})} - \\cos{(v_{2})} = 0 and \\frac{\\operatorname{v_{1}}{(v_{2})} - \\cos{(v_{2})}}{\\frac{d}{d v_{2}} \\operatorname{v_{1}}{(v_{2})}} = 0 and \\frac{\\operatorname{v_{1}}{(v_{2})} - \\cos{(v_{2})}}{\\frac{d}{d v_{2}} \\cos{(v_{2})}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["minus", 1, "cos(Symbol('v_2', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Symbol('v_2', commutative=True)))), Integer(0))"], [["divide", 3, "Derivative(Function('v_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('v_1')(Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Symbol('v_2', commutative=True)))), Pow(Derivative(Function('v_1')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Function('v_1')(Symbol('v_2', commutative=True)), Mul(Integer(-1), cos(Symbol('v_2', commutative=True)))), Pow(Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\varepsilon{(n_{1},F_{g})} = F_{g} + n_{1} and x{(n_{1},F_{g})} = \\varepsilon^{2}{(n_{1},F_{g})}, then obtain x{(n_{1},F_{g})} = (F_{g} + n_{1}) \\varepsilon{(n_{1},F_{g})}", "derivation": "\\varepsilon{(n_{1},F_{g})} = F_{g} + n_{1} and \\varepsilon^{2}{(n_{1},F_{g})} = (F_{g} + n_{1}) \\varepsilon{(n_{1},F_{g})} and x{(n_{1},F_{g})} = \\varepsilon^{2}{(n_{1},F_{g})} and x{(n_{1},F_{g})} = (F_{g} + n_{1}) \\varepsilon{(n_{1},F_{g})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('n_1', commutative=True)))"], [["times", 1, "Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Integer(2)), Mul(Add(Symbol('F_g', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True))))"], ["renaming_premise", "Equality(Function('x')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Pow(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('x')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Mul(Add(Symbol('F_g', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(r)} = \\cos{(r)}, then derive \\int \\operatorname{f^{*}}{(r)} dr = \\Psi_{nl} + \\sin{(r)}, then obtain \\frac{(\\int \\cos{(r)} dr)^{r}}{x + \\sin{(r)}} = \\frac{(\\int \\operatorname{f^{*}}{(r)} dr)^{r}}{x + \\sin{(r)}}", "derivation": "\\operatorname{f^{*}}{(r)} = \\cos{(r)} and \\int \\operatorname{f^{*}}{(r)} dr = \\int \\cos{(r)} dr and \\int \\operatorname{f^{*}}{(r)} dr = \\Psi_{nl} + \\sin{(r)} and (\\int \\operatorname{f^{*}}{(r)} dr)^{r} = (\\Psi_{nl} + \\sin{(r)})^{r} and (\\int \\cos{(r)} dr)^{r} = (\\Psi_{nl} + \\sin{(r)})^{r} and (\\int \\cos{(r)} dr)^{r} = (\\int \\operatorname{f^{*}}{(r)} dr)^{r} and \\frac{(\\int \\cos{(r)} dr)^{r}}{x + \\sin{(r)}} = \\frac{(\\int \\operatorname{f^{*}}{(r)} dr)^{r}}{x + \\sin{(r)}}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('r', commutative=True)), cos(Symbol('r', commutative=True)))"], [["integrate", 1, "Symbol('r', commutative=True)"], "Equality(Integral(Function('f^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('r', commutative=True))))"], [["power", 3, "Symbol('r', commutative=True)"], "Equality(Pow(Integral(Function('f^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)), Pow(Integral(Function('f^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["divide", 6, "Add(Symbol('x', commutative=True), sin(Symbol('r', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('x', commutative=True), sin(Symbol('r', commutative=True))), Integer(-1)), Pow(Integral(cos(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Mul(Pow(Add(Symbol('x', commutative=True), sin(Symbol('r', commutative=True))), Integer(-1)), Pow(Integral(Function('f^*')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\nabla{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi}, then derive \\nabla{(\\pi)} = e^{\\pi}, then derive \\frac{d}{d \\pi} \\nabla{(\\pi)} = e^{\\pi}, then obtain \\cos{(\\frac{d}{d \\pi} \\nabla{(\\pi)})} = \\cos{(e^{\\pi})}", "derivation": "\\nabla{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} and \\nabla{(\\pi)} = e^{\\pi} and \\frac{d}{d \\pi} \\nabla{(\\pi)} = \\frac{d}{d \\pi} e^{\\pi} and \\frac{d}{d \\pi} \\nabla{(\\pi)} = e^{\\pi} and \\cos{(\\frac{d}{d \\pi} \\nabla{(\\pi)})} = \\cos{(e^{\\pi})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), exp(Symbol('\\\\pi', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), exp(Symbol('\\\\pi', commutative=True)))"], [["cos", 4], "Equality(cos(Derivative(Function('\\\\nabla')(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), cos(exp(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{M},x)} = - x + \\log{(\\mathbf{M})}, then derive \\int \\operatorname{z^{*}}{(\\mathbf{M},x)} d\\mathbf{M} = \\mathbf{M} (- x - 1) + \\mathbf{M} \\log{(\\mathbf{M})} + g_{\\varepsilon}, then obtain \\iint (- x + \\log{(\\mathbf{M})}) d\\mathbf{M} dg_{\\varepsilon} = \\int (\\mathbf{M} (- x - 1) + \\mathbf{M} \\log{(\\mathbf{M})} + g_{\\varepsilon}) dg_{\\varepsilon}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{M},x)} = - x + \\log{(\\mathbf{M})} and \\int \\operatorname{z^{*}}{(\\mathbf{M},x)} d\\mathbf{M} = \\int (- x + \\log{(\\mathbf{M})}) d\\mathbf{M} and \\int \\operatorname{z^{*}}{(\\mathbf{M},x)} d\\mathbf{M} = \\mathbf{M} (- x - 1) + \\mathbf{M} \\log{(\\mathbf{M})} + g_{\\varepsilon} and \\int (- x + \\log{(\\mathbf{M})}) d\\mathbf{M} = \\mathbf{M} (- x - 1) + \\mathbf{M} \\log{(\\mathbf{M})} + g_{\\varepsilon} and \\iint (- x + \\log{(\\mathbf{M})}) d\\mathbf{M} dg_{\\varepsilon} = \\int (\\mathbf{M} (- x - 1) + \\mathbf{M} \\log{(\\mathbf{M})} + g_{\\varepsilon}) dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('z^*')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('x', commutative=True)), log(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integer(-1))), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given T{(V_{\\mathbf{E}},\\mathbf{v})} = \\frac{\\mathbf{v}}{V_{\\mathbf{E}}}, then obtain (\\int - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} d\\mathbf{v}) \\int - T{(V_{\\mathbf{E}},\\mathbf{v})} d\\mathbf{v} = (\\int - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} d\\mathbf{v})^{2}", "derivation": "T{(V_{\\mathbf{E}},\\mathbf{v})} = \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} and - T{(V_{\\mathbf{E}},\\mathbf{v})} = - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} and \\int - T{(V_{\\mathbf{E}},\\mathbf{v})} d\\mathbf{v} = \\int - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} d\\mathbf{v} and (\\int - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} d\\mathbf{v}) \\int - T{(V_{\\mathbf{E}},\\mathbf{v})} d\\mathbf{v} = (\\int - \\frac{\\mathbf{v}}{V_{\\mathbf{E}}} d\\mathbf{v})^{2}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["times", 3, "Integral(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Mul(Integral(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Mul(Integer(-1), Function('T')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True)))), Pow(Integral(Mul(Integer(-1), Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\pi{(A_{1})} = \\int \\log{(A_{1})} dA_{1}, then obtain (\\pi{(A_{1})} \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} - 1 = ((\\int \\log{(A_{1})} dA_{1}) \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} - 1", "derivation": "\\pi{(A_{1})} = \\int \\log{(A_{1})} dA_{1} and \\pi{(A_{1})} \\iint \\log{(A_{1})} dA_{1} dA_{1} = (\\int \\log{(A_{1})} dA_{1}) \\iint \\log{(A_{1})} dA_{1} dA_{1} and (\\pi{(A_{1})} \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} = ((\\int \\log{(A_{1})} dA_{1}) \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} and (\\pi{(A_{1})} \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} - 1 = ((\\int \\log{(A_{1})} dA_{1}) \\iint \\log{(A_{1})} dA_{1} dA_{1})^{A_{1}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('A_1', commutative=True)), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))))"], [["times", 1, "Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))"], "Equality(Mul(Function('\\\\pi')(Symbol('A_1', commutative=True)), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Mul(Function('\\\\pi')(Symbol('A_1', commutative=True)), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Pow(Mul(Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Mul(Function('\\\\pi')(Symbol('A_1', commutative=True)), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(-1)), Add(Pow(Mul(Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True))), Integral(log(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given W{(v_{1},\\mathbf{p},r_{0})} = \\mathbf{p} r_{0} v_{1}, then derive \\frac{\\frac{\\partial}{\\partial v_{1}} W{(v_{1},\\mathbf{p},r_{0})}}{\\mathbf{p} r_{0} v_{1}} = \\frac{1}{v_{1}}, then obtain \\frac{\\frac{\\partial}{\\partial v_{1}} \\mathbf{p} r_{0} v_{1}}{\\mathbf{p} r_{0} v_{1}} = \\frac{1}{v_{1}}", "derivation": "W{(v_{1},\\mathbf{p},r_{0})} = \\mathbf{p} r_{0} v_{1} and \\frac{\\partial}{\\partial v_{1}} W{(v_{1},\\mathbf{p},r_{0})} = \\frac{\\partial}{\\partial v_{1}} \\mathbf{p} r_{0} v_{1} and \\frac{\\frac{\\partial}{\\partial v_{1}} W{(v_{1},\\mathbf{p},r_{0})}}{\\mathbf{p} r_{0} v_{1}} = \\frac{\\frac{\\partial}{\\partial v_{1}} \\mathbf{p} r_{0} v_{1}}{\\mathbf{p} r_{0} v_{1}} and \\frac{\\frac{\\partial}{\\partial v_{1}} W{(v_{1},\\mathbf{p},r_{0})}}{\\mathbf{p} r_{0} v_{1}} = \\frac{1}{v_{1}} and \\frac{\\frac{\\partial}{\\partial v_{1}} \\mathbf{p} r_{0} v_{1}}{\\mathbf{p} r_{0} v_{1}} = \\frac{1}{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True), Symbol('v_1', commutative=True)))"], [["differentiate", 1, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Function('W')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Symbol('v_1', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), Pow(Symbol('r_0', commutative=True), Integer(-1)), Pow(Symbol('v_1', commutative=True), Integer(-1)), Derivative(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('r_0', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1)))), Pow(Symbol('v_1', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\delta{(m,\\mathbf{s})} = \\log{(\\mathbf{s} m)}, then derive \\frac{\\partial}{\\partial m} \\delta{(m,\\mathbf{s})} = \\frac{1}{m}, then obtain \\frac{\\frac{\\partial}{\\partial m} \\delta{(m,\\mathbf{s})}}{\\mathbf{s} m} = \\frac{1}{\\mathbf{s} m^{2}}", "derivation": "\\delta{(m,\\mathbf{s})} = \\log{(\\mathbf{s} m)} and \\frac{\\partial}{\\partial m} \\delta{(m,\\mathbf{s})} = \\frac{\\partial}{\\partial m} \\log{(\\mathbf{s} m)} and \\frac{\\partial}{\\partial m} \\delta{(m,\\mathbf{s})} = \\frac{1}{m} and \\frac{\\partial}{\\partial m} \\log{(\\mathbf{s} m)} = \\frac{1}{m} and \\frac{\\frac{\\partial}{\\partial m} \\log{(\\mathbf{s} m)}}{\\mathbf{s} m} = \\frac{1}{\\mathbf{s} m^{2}} and \\frac{\\frac{\\partial}{\\partial m} \\delta{(m,\\mathbf{s})}}{\\mathbf{s} m} = \\frac{1}{\\mathbf{s} m^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), log(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["divide", 4, "Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Derivative(log(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-1)), Derivative(Function('\\\\delta')(Symbol('m', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Pow(Symbol('m', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(B)} = \\sin{(B)} and \\mathbf{p}{(B,G)} = (G - \\cos{(B)}) \\operatorname{v_{x}}{(B)}, then derive (G - \\cos{(B)}) \\operatorname{v_{x}}{(B)} = (G - \\cos{(B)}) \\sin{(B)}, then obtain \\cos{((G - \\cos{(B)}) \\sin{(B)})} = \\cos{((G - \\cos{(B)}) \\operatorname{v_{x}}{(B)})}", "derivation": "\\operatorname{v_{x}}{(B)} = \\sin{(B)} and \\operatorname{v_{x}}{(B)} \\int \\sin{(B)} dB = \\sin{(B)} \\int \\sin{(B)} dB and (G - \\cos{(B)}) \\operatorname{v_{x}}{(B)} = (G - \\cos{(B)}) \\sin{(B)} and \\mathbf{p}{(B,G)} = (G - \\cos{(B)}) \\operatorname{v_{x}}{(B)} and \\mathbf{p}{(B,G)} = (G - \\cos{(B)}) \\sin{(B)} and \\cos{(\\mathbf{p}{(B,G)})} = \\cos{((G - \\cos{(B)}) \\operatorname{v_{x}}{(B)})} and \\cos{((G - \\cos{(B)}) \\sin{(B)})} = \\cos{((G - \\cos{(B)}) \\operatorname{v_{x}}{(B)})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["times", 1, "Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))"], "Equality(Mul(Function('v_x')(Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))), Mul(sin(Symbol('B', commutative=True)), Integral(sin(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('v_x')(Symbol('B', commutative=True))), Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), sin(Symbol('B', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('B', commutative=True), Symbol('G', commutative=True)), Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('v_x')(Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\mathbf{p}')(Symbol('B', commutative=True), Symbol('G', commutative=True)), Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), sin(Symbol('B', commutative=True))))"], [["cos", 4], "Equality(cos(Function('\\\\mathbf{p}')(Symbol('B', commutative=True), Symbol('G', commutative=True))), cos(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('v_x')(Symbol('B', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(cos(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), sin(Symbol('B', commutative=True)))), cos(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('v_x')(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(Z)} = \\cos{(\\log{(Z)})}, then derive \\cos{(\\frac{d}{d Z} \\hat{H}_l{(Z)})} = \\cos{(\\frac{\\sin{(\\log{(Z)})}}{Z})}, then obtain (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{d}{d Z} \\cos{(\\log{(Z)})})} = (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{\\sin{(\\log{(Z)})}}{Z})}", "derivation": "\\hat{H}_l{(Z)} = \\cos{(\\log{(Z)})} and \\frac{d}{d Z} \\hat{H}_l{(Z)} = \\frac{d}{d Z} \\cos{(\\log{(Z)})} and \\cos{(\\frac{d}{d Z} \\hat{H}_l{(Z)})} = \\cos{(\\frac{d}{d Z} \\cos{(\\log{(Z)})})} and \\cos{(\\frac{d}{d Z} \\hat{H}_l{(Z)})} = \\cos{(\\frac{\\sin{(\\log{(Z)})}}{Z})} and (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{d}{d Z} \\hat{H}_l{(Z)})} = (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{\\sin{(\\log{(Z)})}}{Z})} and (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{d}{d Z} \\cos{(\\log{(Z)})})} = (\\hat{H}_l{(Z)} + \\cos{(\\log{(Z)})}) \\cos{(\\frac{\\sin{(\\log{(Z)})}}{Z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True))))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), cos(Derivative(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))), cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True))))))"], [["times", 4, "Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True))))"], "Equality(Mul(Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True)))), cos(Derivative(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))), Mul(Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True)))), cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True)))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True)))), cos(Derivative(cos(log(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))), Mul(Add(Function('\\\\hat{H}_l')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True)))), cos(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\cos{(\\mathbf{B} \\mathbf{J})}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{B}\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\frac{\\partial^{2}}{\\partial \\mathbf{B}\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\cos{(\\mathbf{B} \\mathbf{J})}", "derivation": "\\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\cos{(\\mathbf{B} \\mathbf{J})} and \\mathbf{B} \\mathbf{J} \\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\mathbf{B} \\mathbf{J} \\cos{(\\mathbf{B} \\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\cos{(\\mathbf{B} \\mathbf{J})} and \\frac{\\partial^{2}}{\\partial \\mathbf{B}\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\mathbf{J}_f{(\\mathbf{B},\\mathbf{J})} = \\frac{\\partial^{2}}{\\partial \\mathbf{B}\\partial \\mathbf{J}} \\mathbf{B} \\mathbf{J} \\cos{(\\mathbf{B} \\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(P_{e})} = e^{\\cos{(P_{e})}}, then obtain \\operatorname{E_{x}}{(P_{e})} + e^{\\cos{(P_{e})}} - (e^{\\cos{(P_{e})}})^{P_{e}} = 2 e^{\\cos{(P_{e})}} - (e^{\\cos{(P_{e})}})^{P_{e}}", "derivation": "\\operatorname{E_{x}}{(P_{e})} = e^{\\cos{(P_{e})}} and \\operatorname{E_{x}}^{P_{e}}{(P_{e})} = (e^{\\cos{(P_{e})}})^{P_{e}} and \\operatorname{E_{x}}{(P_{e})} + e^{\\cos{(P_{e})}} = 2 e^{\\cos{(P_{e})}} and \\operatorname{E_{x}}{(P_{e})} - \\operatorname{E_{x}}^{P_{e}}{(P_{e})} + e^{\\cos{(P_{e})}} = - \\operatorname{E_{x}}^{P_{e}}{(P_{e})} + 2 e^{\\cos{(P_{e})}} and \\operatorname{E_{x}}{(P_{e})} + e^{\\cos{(P_{e})}} - (e^{\\cos{(P_{e})}})^{P_{e}} = 2 e^{\\cos{(P_{e})}} - (e^{\\cos{(P_{e})}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('P_e', commutative=True)), exp(cos(Symbol('P_e', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(exp(cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["add", 1, "exp(cos(Symbol('P_e', commutative=True)))"], "Equality(Add(Function('E_x')(Symbol('P_e', commutative=True)), exp(cos(Symbol('P_e', commutative=True)))), Mul(Integer(2), exp(cos(Symbol('P_e', commutative=True)))))"], [["minus", 3, "Pow(Function('E_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], "Equality(Add(Function('E_x')(Symbol('P_e', commutative=True)), Mul(Integer(-1), Pow(Function('E_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), exp(cos(Symbol('P_e', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('E_x')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))), Mul(Integer(2), exp(cos(Symbol('P_e', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Function('E_x')(Symbol('P_e', commutative=True)), exp(cos(Symbol('P_e', commutative=True))), Mul(Integer(-1), Pow(exp(cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))), Add(Mul(Integer(2), exp(cos(Symbol('P_e', commutative=True)))), Mul(Integer(-1), Pow(exp(cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(C_{2},\\theta_1)} = \\cos{(\\frac{\\theta_1}{C_{2}})}, then obtain \\int (\\operatorname{t_{2}}{(C_{2},\\theta_1)} + \\frac{1}{C_{2}}) dC_{2} = C_{2} \\cos{(\\frac{\\theta_1}{C_{2}})} + \\mu + \\theta_1 \\operatorname{Si}{(\\frac{\\theta_1}{C_{2}})} + \\log{(C_{2})}", "derivation": "\\operatorname{t_{2}}{(C_{2},\\theta_1)} = \\cos{(\\frac{\\theta_1}{C_{2}})} and \\operatorname{t_{2}}{(C_{2},\\theta_1)} + \\frac{1}{C_{2}} = \\cos{(\\frac{\\theta_1}{C_{2}})} + \\frac{1}{C_{2}} and \\int (\\operatorname{t_{2}}{(C_{2},\\theta_1)} + \\frac{1}{C_{2}}) dC_{2} = \\int (\\cos{(\\frac{\\theta_1}{C_{2}})} + \\frac{1}{C_{2}}) dC_{2} and \\int (\\operatorname{t_{2}}{(C_{2},\\theta_1)} + \\frac{1}{C_{2}}) dC_{2} = C_{2} \\cos{(\\frac{\\theta_1}{C_{2}})} + \\mu + \\theta_1 \\operatorname{Si}{(\\frac{\\theta_1}{C_{2}})} + \\log{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('C_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))))"], [["add", 1, "Pow(Symbol('C_2', commutative=True), Integer(-1))"], "Equality(Add(Function('t_2')(Symbol('C_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('C_2', commutative=True), Integer(-1))), Add(cos(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Pow(Symbol('C_2', commutative=True), Integer(-1))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Add(Function('t_2')(Symbol('C_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('C_2', commutative=True), Integer(-1))), Tuple(Symbol('C_2', commutative=True))), Integral(Add(cos(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Pow(Symbol('C_2', commutative=True), Integer(-1))), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Function('t_2')(Symbol('C_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Symbol('C_2', commutative=True), Integer(-1))), Tuple(Symbol('C_2', commutative=True))), Add(Mul(Symbol('C_2', commutative=True), cos(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))), Symbol('\\\\mu', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Si(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))), log(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(\\theta_2)} = \\sin{(\\theta_2)}, then derive \\int \\frac{\\rho_{f}{(\\theta_2)}}{\\theta_2} d\\theta_2 = M_{E} + \\operatorname{Si}{(\\theta_2)}, then obtain M_{E} + \\operatorname{Si}{(\\theta_2)} = \\int \\frac{\\sin{(\\theta_2)}}{\\theta_2} d\\theta_2", "derivation": "\\rho_{f}{(\\theta_2)} = \\sin{(\\theta_2)} and \\frac{\\rho_{f}{(\\theta_2)}}{\\theta_2} = \\frac{\\sin{(\\theta_2)}}{\\theta_2} and \\int \\frac{\\rho_{f}{(\\theta_2)}}{\\theta_2} d\\theta_2 = \\int \\frac{\\sin{(\\theta_2)}}{\\theta_2} d\\theta_2 and \\int \\frac{\\rho_{f}{(\\theta_2)}}{\\theta_2} d\\theta_2 = M_{E} + \\operatorname{Si}{(\\theta_2)} and M_{E} + \\operatorname{Si}{(\\theta_2)} = \\int \\frac{\\sin{(\\theta_2)}}{\\theta_2} d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\theta_2', commutative=True)), sin(Symbol('\\\\theta_2', commutative=True)))"], [["divide", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), Function('\\\\rho_f')(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('M_E', commutative=True), Si(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('M_E', commutative=True), Si(Symbol('\\\\theta_2', commutative=True))), Integral(Mul(Pow(Symbol('\\\\theta_2', commutative=True), Integer(-1)), sin(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\hbar)} = \\log{(\\hbar)}, then derive \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} = \\frac{1}{\\hbar}, then obtain \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} - \\frac{1}{\\hbar} = 0", "derivation": "\\operatorname{r_{0}}{(\\hbar)} = \\log{(\\hbar)} and \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} = \\frac{d}{d \\hbar} \\log{(\\hbar)} and \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} = \\frac{1}{\\hbar} and \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} - \\frac{d}{d \\hbar} \\log{(\\hbar)} = - \\frac{d}{d \\hbar} \\log{(\\hbar)} + \\frac{1}{\\hbar} and \\frac{d}{d \\hbar} \\operatorname{r_{0}}{(\\hbar)} - \\frac{1}{\\hbar} = 0", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))"], [["minus", 3, "Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(log(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('r_0')(Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\Psi{(m,Z)} = Z^{m} and W{(\\mu)} = e^{\\cos{(\\mu)}}, then obtain e^{(- Z^{m} + \\Psi{(m,Z)})^{Z}} + \\sin{(W{(\\mu)})} = e^{(- Z^{m} + \\Psi{(m,Z)})^{Z}} + \\sin{(e^{\\cos{(\\mu)}})}", "derivation": "\\Psi{(m,Z)} = Z^{m} and - Z^{m} + \\Psi{(m,Z)} = 0 and W{(\\mu)} = e^{\\cos{(\\mu)}} and (- Z^{m} + \\Psi{(m,Z)})^{Z} = 0^{Z} and \\sin{(W{(\\mu)})} = \\sin{(e^{\\cos{(\\mu)}})} and e^{0^{Z}} + \\sin{(W{(\\mu)})} = e^{0^{Z}} + \\sin{(e^{\\cos{(\\mu)}})} and e^{(- Z^{m} + \\Psi{(m,Z)})^{Z}} + \\sin{(W{(\\mu)})} = e^{(- Z^{m} + \\Psi{(m,Z)})^{Z}} + \\sin{(e^{\\cos{(\\mu)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True)))"], [["minus", 1, "Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True))), Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Integer(0))"], ["get_premise", "Equality(Function('W')(Symbol('\\\\mu', commutative=True)), exp(cos(Symbol('\\\\mu', commutative=True))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True))), Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Integer(0), Symbol('Z', commutative=True)))"], [["sin", 3], "Equality(sin(Function('W')(Symbol('\\\\mu', commutative=True))), sin(exp(cos(Symbol('\\\\mu', commutative=True)))))"], [["add", 5, "exp(Pow(Integer(0), Symbol('Z', commutative=True)))"], "Equality(Add(exp(Pow(Integer(0), Symbol('Z', commutative=True))), sin(Function('W')(Symbol('\\\\mu', commutative=True)))), Add(exp(Pow(Integer(0), Symbol('Z', commutative=True))), sin(exp(cos(Symbol('\\\\mu', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(exp(Pow(Add(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True))), Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), sin(Function('W')(Symbol('\\\\mu', commutative=True)))), Add(exp(Pow(Add(Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Symbol('m', commutative=True))), Function('\\\\Psi')(Symbol('m', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True))), sin(exp(cos(Symbol('\\\\mu', commutative=True))))))"]]}, {"prompt": "Given \\ddot{x}{(\\hat{H},F_{g})} = \\sin{(F_{g} \\hat{H})}, then obtain (\\frac{\\partial}{\\partial F_{g}} \\ddot{x}{(\\hat{H},F_{g})})^{\\hat{H}} = (\\hat{H} \\cos{(F_{g} \\hat{H})})^{\\hat{H}}", "derivation": "\\ddot{x}{(\\hat{H},F_{g})} = \\sin{(F_{g} \\hat{H})} and \\frac{\\partial}{\\partial F_{g}} \\ddot{x}{(\\hat{H},F_{g})} = \\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} \\hat{H})} and (\\frac{\\partial}{\\partial F_{g}} \\ddot{x}{(\\hat{H},F_{g})})^{\\hat{H}} = (\\frac{\\partial}{\\partial F_{g}} \\sin{(F_{g} \\hat{H})})^{\\hat{H}} and (\\frac{\\partial}{\\partial F_{g}} \\ddot{x}{(\\hat{H},F_{g})})^{\\hat{H}} = (\\hat{H} \\cos{(F_{g} \\hat{H})})^{\\hat{H}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('F_g', commutative=True)), sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)), Pow(Derivative(sin(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\hat{H}', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('\\\\hat{H}', commutative=True)), Pow(Mul(Symbol('\\\\hat{H}', commutative=True), cos(Mul(Symbol('F_g', commutative=True), Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given z{(C,m_{s})} = e^{C^{m_{s}}} and \\operatorname{a^{\\dagger}}{(C,m_{s})} = - C + e^{C^{m_{s}}} and b{(C)} = - C, then obtain \\operatorname{a^{\\dagger}}{(C,m_{s})} = b{(C)} + z{(C,m_{s})}", "derivation": "z{(C,m_{s})} = e^{C^{m_{s}}} and - C + z{(C,m_{s})} = - C + e^{C^{m_{s}}} and \\operatorname{a^{\\dagger}}{(C,m_{s})} = - C + e^{C^{m_{s}}} and b{(C)} = - C and \\operatorname{a^{\\dagger}}{(C,m_{s})} = - C + z{(C,m_{s})} and \\operatorname{a^{\\dagger}}{(C,m_{s})} = b{(C)} + z{(C,m_{s})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), exp(Pow(Symbol('C', commutative=True), Symbol('m_s', commutative=True))))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), exp(Pow(Symbol('C', commutative=True), Symbol('m_s', commutative=True)))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), exp(Pow(Symbol('C', commutative=True), Symbol('m_s', commutative=True)))))"], ["renaming_premise", "Equality(Function('b')(Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('a^{\\\\dagger}')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('m_s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('a^{\\\\dagger}')(Symbol('C', commutative=True), Symbol('m_s', commutative=True)), Add(Function('b')(Symbol('C', commutative=True)), Function('z')(Symbol('C', commutative=True), Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(v_{2},v)} = v v_{2}, then obtain v (v v_{2}^{2} - v_{2} + \\int \\operatorname{C_{d}}{(v_{2},v)} dv) = v (v v_{2}^{2} - v_{2} + \\int v v_{2} dv)", "derivation": "\\operatorname{C_{d}}{(v_{2},v)} = v v_{2} and v_{2} \\operatorname{C_{d}}{(v_{2},v)} = v v_{2}^{2} and v_{2} \\operatorname{C_{d}}{(v_{2},v)} - v_{2} = v v_{2}^{2} - v_{2} and \\int \\operatorname{C_{d}}{(v_{2},v)} dv = \\int v v_{2} dv and v_{2} \\operatorname{C_{d}}{(v_{2},v)} - v_{2} + \\int \\operatorname{C_{d}}{(v_{2},v)} dv = v_{2} \\operatorname{C_{d}}{(v_{2},v)} - v_{2} + \\int v v_{2} dv and v v_{2}^{2} - v_{2} + \\int \\operatorname{C_{d}}{(v_{2},v)} dv = v v_{2}^{2} - v_{2} + \\int v v_{2} dv and v (v v_{2}^{2} - v_{2} + \\int \\operatorname{C_{d}}{(v_{2},v)} dv) = v (v v_{2}^{2} - v_{2} + \\int v v_{2} dv)", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), Symbol('v_2', commutative=True)))"], [["times", 1, "Symbol('v_2', commutative=True)"], "Equality(Mul(Symbol('v_2', commutative=True), Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))))"], [["minus", 2, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Symbol('v_2', commutative=True), Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True))), Add(Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Mul(Symbol('v', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["add", 4, "Add(Mul(Symbol('v_2', commutative=True), Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True)))"], "Equality(Add(Mul(Symbol('v_2', commutative=True), Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Symbol('v_2', commutative=True), Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Mul(Symbol('v', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Add(Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Mul(Symbol('v', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["times", 6, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Add(Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Function('C_d')(Symbol('v_2', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))), Mul(Symbol('v', commutative=True), Add(Mul(Symbol('v', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('v_2', commutative=True)), Integral(Mul(Symbol('v', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given k{(\\mu)} = \\log{(\\mu)}, then obtain \\frac{d}{d \\mu} k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{k{(\\mu)}}{\\mu}) = \\frac{d}{d \\mu} k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{\\log{(\\mu)}}{\\mu})", "derivation": "k{(\\mu)} = \\log{(\\mu)} and \\frac{k{(\\mu)}}{\\mu} = \\frac{\\log{(\\mu)}}{\\mu} and - k{(\\mu)} + \\frac{k{(\\mu)}}{\\mu} = - k{(\\mu)} + \\frac{\\log{(\\mu)}}{\\mu} and \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{k{(\\mu)}}{\\mu}) = \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{\\log{(\\mu)}}{\\mu}) and k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{k{(\\mu)}}{\\mu}) = k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{\\log{(\\mu)}}{\\mu}) and \\frac{d}{d \\mu} k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{k{(\\mu)}}{\\mu}) = \\frac{d}{d \\mu} k{(\\mu)} \\frac{d}{d \\mu} (- k{(\\mu)} + \\frac{\\log{(\\mu)}}{\\mu})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["divide", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Function('k')(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\mu', commutative=True)))), Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["times", 4, "Function('k')(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('k')(Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Function('k')(Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["differentiate", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Function('k')(Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Function('k')(Symbol('\\\\mu', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('k')(Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), log(Symbol('\\\\mu', commutative=True)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\nabla{(C_{1},Z)} = \\int (- C_{1} + Z) dZ, then derive \\frac{\\partial^{2}}{\\partial Z^{2}} \\frac{\\nabla{(C_{1},Z)}}{\\int (- C_{1} + Z) dZ} = 0, then obtain \\frac{d^{2}}{d Z^{2}} 1 + 1 = 1", "derivation": "\\nabla{(C_{1},Z)} = \\int (- C_{1} + Z) dZ and \\frac{\\nabla{(C_{1},Z)}}{\\int (- C_{1} + Z) dZ} = 1 and \\frac{\\partial}{\\partial Z} \\frac{\\nabla{(C_{1},Z)}}{\\int (- C_{1} + Z) dZ} = \\frac{d}{d Z} 1 and \\frac{\\partial^{2}}{\\partial Z^{2}} \\frac{\\nabla{(C_{1},Z)}}{\\int (- C_{1} + Z) dZ} = \\frac{d^{2}}{d Z^{2}} 1 and \\frac{\\partial^{2}}{\\partial Z^{2}} \\frac{\\nabla{(C_{1},Z)}}{\\int (- C_{1} + Z) dZ} = 0 and \\frac{d^{2}}{d Z^{2}} 1 = 0 and \\frac{d^{2}}{d Z^{2}} 1 + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["divide", 1, "Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))"], "Equality(Mul(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Tuple(Symbol('Z', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Mul(Function('\\\\nabla')(Symbol('C_1', commutative=True), Symbol('Z', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integer(-1))), Tuple(Symbol('Z', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2))), Integer(0))"], [["add", 6, 1], "Equality(Add(Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(2))), Integer(1)), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(t,u)} = t - u, then obtain \\frac{\\partial^{2}}{\\partial t\\partial u} \\operatorname{v_{y}}^{u}{(t,u)} = \\frac{\\partial^{2}}{\\partial t\\partial u} (t - u)^{u}", "derivation": "\\operatorname{v_{y}}{(t,u)} = t - u and \\operatorname{v_{y}}^{u}{(t,u)} = (t - u)^{u} and \\frac{\\partial}{\\partial u} \\operatorname{v_{y}}^{u}{(t,u)} = \\frac{\\partial}{\\partial u} (t - u)^{u} and \\frac{\\partial^{2}}{\\partial t\\partial u} \\operatorname{v_{y}}^{u}{(t,u)} = \\frac{\\partial^{2}}{\\partial t\\partial u} (t - u)^{u}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('u', commutative=True)))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Pow(Function('v_y')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('t', commutative=True)"], "Equality(Derivative(Pow(Function('v_y')(Symbol('t', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('t', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\psi^{*}{(A_{y})} = e^{A_{y}} and \\theta_{2}{(A_{y})} = e^{A_{y}}, then obtain 1 = ((\\frac{\\theta_{2}{(A_{y})}}{\\psi^{*}{(A_{y})}})^{A_{y}})^{A_{y}}", "derivation": "\\psi^{*}{(A_{y})} = e^{A_{y}} and 1 = \\frac{e^{A_{y}}}{\\psi^{*}{(A_{y})}} and 1 = (\\frac{e^{A_{y}}}{\\psi^{*}{(A_{y})}})^{A_{y}} and \\theta_{2}{(A_{y})} = e^{A_{y}} and 1 = ((\\frac{e^{A_{y}}}{\\psi^{*}{(A_{y})}})^{A_{y}})^{A_{y}} and 1 = ((\\frac{\\theta_{2}{(A_{y})}}{\\psi^{*}{(A_{y})}})^{A_{y}})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["divide", 1, "Function('\\\\psi^*')(Symbol('A_y', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\psi^*')(Symbol('A_y', commutative=True)), Integer(-1)), exp(Symbol('A_y', commutative=True))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\psi^*')(Symbol('A_y', commutative=True)), Integer(-1)), exp(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta_2')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["power", 3, "Symbol('A_y', commutative=True)"], "Equality(Integer(1), Pow(Pow(Mul(Pow(Function('\\\\psi^*')(Symbol('A_y', commutative=True)), Integer(-1)), exp(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(1), Pow(Pow(Mul(Pow(Function('\\\\psi^*')(Symbol('A_y', commutative=True)), Integer(-1)), Function('\\\\theta_2')(Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"]]}, {"prompt": "Given I{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})}, then obtain - \\frac{d}{d \\varepsilon} I{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} I{(\\varepsilon)})^{\\varepsilon} = - \\frac{d}{d \\varepsilon} I{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\log{(\\cos{(\\varepsilon)})})^{\\varepsilon}", "derivation": "I{(\\varepsilon)} = \\log{(\\cos{(\\varepsilon)})} and \\frac{d}{d \\varepsilon} I{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\log{(\\cos{(\\varepsilon)})} and (\\frac{d}{d \\varepsilon} I{(\\varepsilon)})^{\\varepsilon} = (\\frac{d}{d \\varepsilon} \\log{(\\cos{(\\varepsilon)})})^{\\varepsilon} and - \\frac{d}{d \\varepsilon} I{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} I{(\\varepsilon)})^{\\varepsilon} = - \\frac{d}{d \\varepsilon} I{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\log{(\\cos{(\\varepsilon)})})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\varepsilon', commutative=True)), log(cos(Symbol('\\\\varepsilon', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 3, "Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Pow(Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Derivative(Function('I')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1)))), Pow(Derivative(log(cos(Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\varphi)} = \\log{(\\varphi)}, then derive \\int \\operatorname{v_{y}}{(\\varphi)} d\\varphi = C_{2} + \\varphi \\log{(\\varphi)} - \\varphi, then obtain \\int (C_{2} + \\varphi \\operatorname{v_{y}}{(\\varphi)} - \\varphi) d\\varphi = \\iint \\log{(\\varphi)} d\\varphi d\\varphi", "derivation": "\\operatorname{v_{y}}{(\\varphi)} = \\log{(\\varphi)} and \\int \\operatorname{v_{y}}{(\\varphi)} d\\varphi = \\int \\log{(\\varphi)} d\\varphi and \\int \\operatorname{v_{y}}{(\\varphi)} d\\varphi = C_{2} + \\varphi \\log{(\\varphi)} - \\varphi and \\int \\operatorname{v_{y}}{(\\varphi)} d\\varphi = C_{2} + \\varphi \\operatorname{v_{y}}{(\\varphi)} - \\varphi and C_{2} + \\varphi \\operatorname{v_{y}}{(\\varphi)} - \\varphi = \\int \\log{(\\varphi)} d\\varphi and \\int (C_{2} + \\varphi \\operatorname{v_{y}}{(\\varphi)} - \\varphi) d\\varphi = \\iint \\log{(\\varphi)} d\\varphi d\\varphi", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), log(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v_y')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Function('v_y')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Function('v_y')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Symbol('C_2', commutative=True), Mul(Symbol('\\\\varphi', commutative=True), Function('v_y')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(J,f^{*})} = J f^{*}, then obtain (\\int \\frac{\\int \\mathbf{J}_f{(J,f^{*})} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*})^{J} = (\\int \\frac{\\int J f^{*} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*})^{J}", "derivation": "\\mathbf{J}_f{(J,f^{*})} = J f^{*} and \\int \\mathbf{J}_f{(J,f^{*})} dJ = \\int J f^{*} dJ and \\frac{\\int \\mathbf{J}_f{(J,f^{*})} dJ}{\\mathbf{J}_f{(J,f^{*})}} = \\frac{\\int J f^{*} dJ}{\\mathbf{J}_f{(J,f^{*})}} and \\int \\frac{\\int \\mathbf{J}_f{(J,f^{*})} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*} = \\int \\frac{\\int J f^{*} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*} and (\\int \\frac{\\int \\mathbf{J}_f{(J,f^{*})} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*})^{J} = (\\int \\frac{\\int J f^{*} dJ}{\\mathbf{J}_f{(J,f^{*})}} df^{*})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["integrate", 3, "Symbol('f^*', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^*', commutative=True))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(Mul(Pow(Function('\\\\mathbf{J}_f')(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Integer(-1)), Integral(Mul(Symbol('J', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('J', commutative=True)))), Tuple(Symbol('f^*', commutative=True))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(A_{z})} = e^{\\cos{(A_{z})}}, then obtain (A_{z} + \\sin{(e^{\\cos{(A_{z})}})}) \\sin{(\\operatorname{A_{x}}{(A_{z})})} = (A_{z} + \\sin{(e^{\\cos{(A_{z})}})}) \\sin{(e^{\\cos{(A_{z})}})}", "derivation": "\\operatorname{A_{x}}{(A_{z})} = e^{\\cos{(A_{z})}} and \\sin{(\\operatorname{A_{x}}{(A_{z})})} = \\sin{(e^{\\cos{(A_{z})}})} and A_{z} + \\sin{(\\operatorname{A_{x}}{(A_{z})})} = A_{z} + \\sin{(e^{\\cos{(A_{z})}})} and (A_{z} + \\sin{(\\operatorname{A_{x}}{(A_{z})})}) \\sin{(\\operatorname{A_{x}}{(A_{z})})} = (A_{z} + \\sin{(\\operatorname{A_{x}}{(A_{z})})}) \\sin{(e^{\\cos{(A_{z})}})} and (A_{z} + \\sin{(e^{\\cos{(A_{z})}})}) \\sin{(\\operatorname{A_{x}}{(A_{z})})} = (A_{z} + \\sin{(e^{\\cos{(A_{z})}})}) \\sin{(e^{\\cos{(A_{z})}})}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('A_z', commutative=True)), exp(cos(Symbol('A_z', commutative=True))))"], [["sin", 1], "Equality(sin(Function('A_x')(Symbol('A_z', commutative=True))), sin(exp(cos(Symbol('A_z', commutative=True)))))"], [["add", 2, "Symbol('A_z', commutative=True)"], "Equality(Add(Symbol('A_z', commutative=True), sin(Function('A_x')(Symbol('A_z', commutative=True)))), Add(Symbol('A_z', commutative=True), sin(exp(cos(Symbol('A_z', commutative=True))))))"], [["times", 2, "Add(Symbol('A_z', commutative=True), sin(Function('A_x')(Symbol('A_z', commutative=True))))"], "Equality(Mul(Add(Symbol('A_z', commutative=True), sin(Function('A_x')(Symbol('A_z', commutative=True)))), sin(Function('A_x')(Symbol('A_z', commutative=True)))), Mul(Add(Symbol('A_z', commutative=True), sin(Function('A_x')(Symbol('A_z', commutative=True)))), sin(exp(cos(Symbol('A_z', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Symbol('A_z', commutative=True), sin(exp(cos(Symbol('A_z', commutative=True))))), sin(Function('A_x')(Symbol('A_z', commutative=True)))), Mul(Add(Symbol('A_z', commutative=True), sin(exp(cos(Symbol('A_z', commutative=True))))), sin(exp(cos(Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given \\ddot{x}{(M_{E})} = \\log{(M_{E})} and H{(M_{E})} = \\log{(M_{E})}, then derive \\int H{(M_{E})} dM_{E} = A_{2} + M_{E} \\log{(M_{E})} - M_{E}, then obtain \\int \\ddot{x}{(M_{E})} dM_{E} = A_{2} + M_{E} \\log{(M_{E})} - M_{E}", "derivation": "\\ddot{x}{(M_{E})} = \\log{(M_{E})} and H{(M_{E})} = \\log{(M_{E})} and \\int \\ddot{x}{(M_{E})} dM_{E} = \\int \\log{(M_{E})} dM_{E} and \\int H{(M_{E})} dM_{E} = \\int \\log{(M_{E})} dM_{E} and \\int H{(M_{E})} dM_{E} = A_{2} + M_{E} \\log{(M_{E})} - M_{E} and \\int \\log{(M_{E})} dM_{E} = A_{2} + M_{E} \\log{(M_{E})} - M_{E} and \\int \\ddot{x}{(M_{E})} dM_{E} = A_{2} + M_{E} \\log{(M_{E})} - M_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], ["renaming_premise", "Equality(Function('H')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('H')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Function('H')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integral(Function('\\\\ddot{x}')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Symbol('M_E', commutative=True), log(Symbol('M_E', commutative=True))), Mul(Integer(-1), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(L_{\\varepsilon},W)} = \\sin{(L_{\\varepsilon}^{W})} and q{(\\lambda,l)} = \\lambda - l, then obtain \\frac{\\operatorname{v_{x}}{(L_{\\varepsilon},W)}}{\\lambda - l} - \\frac{1}{\\lambda - l} = \\frac{\\sin{(L_{\\varepsilon}^{W})}}{\\lambda - l} - \\frac{1}{\\lambda - l}", "derivation": "\\operatorname{v_{x}}{(L_{\\varepsilon},W)} = \\sin{(L_{\\varepsilon}^{W})} and q{(\\lambda,l)} = \\lambda - l and \\frac{\\operatorname{v_{x}}{(L_{\\varepsilon},W)}}{q{(\\lambda,l)}} = \\frac{\\sin{(L_{\\varepsilon}^{W})}}{q{(\\lambda,l)}} and \\frac{\\operatorname{v_{x}}{(L_{\\varepsilon},W)}}{q{(\\lambda,l)}} - \\frac{1}{q{(\\lambda,l)}} = \\frac{\\sin{(L_{\\varepsilon}^{W})}}{q{(\\lambda,l)}} - \\frac{1}{q{(\\lambda,l)}} and \\frac{\\operatorname{v_{x}}{(L_{\\varepsilon},W)}}{\\lambda - l} - \\frac{1}{\\lambda - l} = \\frac{\\sin{(L_{\\varepsilon}^{W})}}{\\lambda - l} - \\frac{1}{\\lambda - l}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), sin(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))))"], ["get_premise", "Equality(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["divide", 1, "Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Function('v_x')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)), sin(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)))))"], [["minus", 3, "Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)), Function('v_x')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)))), Add(Mul(Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)), sin(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)))), Mul(Integer(-1), Pow(Function('q')(Symbol('\\\\lambda', commutative=True), Symbol('l', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), Function('v_x')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)))), Add(Mul(Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)), sin(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)))), Mul(Integer(-1), Pow(Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\Psi_{\\lambda})} = e^{\\cos{(\\Psi_{\\lambda})}} and q{(\\Psi_{\\lambda})} = - \\frac{- 2 \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 2 e^{\\cos{(\\Psi_{\\lambda})}}}{\\operatorname{A_{2}}{(\\Psi_{\\lambda})}}, then obtain 0 = - (- 4 \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 4 e^{\\cos{(\\Psi_{\\lambda})}}) q{(\\Psi_{\\lambda})}", "derivation": "\\operatorname{A_{2}}{(\\Psi_{\\lambda})} = e^{\\cos{(\\Psi_{\\lambda})}} and 0 = - \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + e^{\\cos{(\\Psi_{\\lambda})}} and e^{\\cos{(\\Psi_{\\lambda})}} = - \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 2 e^{\\cos{(\\Psi_{\\lambda})}} and q{(\\Psi_{\\lambda})} = - \\frac{- 2 \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 2 e^{\\cos{(\\Psi_{\\lambda})}}}{\\operatorname{A_{2}}{(\\Psi_{\\lambda})}} and q{(\\Psi_{\\lambda})} = 0 and 0 = - q{(\\Psi_{\\lambda})} and 0 = - (- 2 \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 2 e^{\\cos{(\\Psi_{\\lambda})}}) q{(\\Psi_{\\lambda})} and 0 = - (- 4 \\operatorname{A_{2}}{(\\Psi_{\\lambda})} + 4 e^{\\cos{(\\Psi_{\\lambda})}}) q{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["minus", 1, "Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["add", 2, "exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))))"], ["renaming_premise", "Equality(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Pow(Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(0))"], [["minus", 5, "Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["times", 6, "Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Integer(2), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(2), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Integer(4), Function('A_2')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Mul(Integer(4), exp(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))), Function('q')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given n{(\\Psi)} = e^{\\Psi}, then obtain \\frac{n{(\\Psi)}}{v_{x} + e^{\\Psi}} = \\frac{e^{\\Psi}}{v_{x} + e^{\\Psi}}", "derivation": "n{(\\Psi)} = e^{\\Psi} and \\int n{(\\Psi)} d\\Psi = \\int e^{\\Psi} d\\Psi and \\frac{n{(\\Psi)}}{\\int n{(\\Psi)} d\\Psi} = \\frac{e^{\\Psi}}{\\int n{(\\Psi)} d\\Psi} and \\frac{n{(\\Psi)}}{\\int e^{\\Psi} d\\Psi} = \\frac{e^{\\Psi}}{\\int e^{\\Psi} d\\Psi} and \\frac{n{(\\Psi)}}{v_{x} + e^{\\Psi}} = \\frac{e^{\\Psi}}{v_{x} + e^{\\Psi}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('n')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["divide", 1, "Integral(Function('n')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True)))"], "Equality(Mul(Function('n')(Symbol('\\\\Psi', commutative=True)), Pow(Integral(Function('n')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(exp(Symbol('\\\\Psi', commutative=True)), Pow(Integral(Function('n')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('n')(Symbol('\\\\Psi', commutative=True)), Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))), Mul(exp(Symbol('\\\\Psi', commutative=True)), Pow(Integral(exp(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integer(-1))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Add(Symbol('v_x', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Integer(-1)), Function('n')(Symbol('\\\\Psi', commutative=True))), Mul(Pow(Add(Symbol('v_x', commutative=True), exp(Symbol('\\\\Psi', commutative=True))), Integer(-1)), exp(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\mathbf{A})} = e^{\\mathbf{A}} and g{(\\mathbf{A})} = e^{\\mathbf{A}}, then obtain - \\mathbf{A} - g{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + \\int \\frac{d}{d \\mathbf{A}} \\lambda{(\\mathbf{A})} d\\mathbf{A} = - \\mathbf{A} - g{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + \\int \\frac{d}{d \\mathbf{A}} g{(\\mathbf{A})} d\\mathbf{A}", "derivation": "\\lambda{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\lambda{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and g{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\lambda{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} g{(\\mathbf{A})} and \\int \\frac{d}{d \\mathbf{A}} \\lambda{(\\mathbf{A})} d\\mathbf{A} = \\int \\frac{d}{d \\mathbf{A}} g{(\\mathbf{A})} d\\mathbf{A} and - \\mathbf{A} - g{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + \\int \\frac{d}{d \\mathbf{A}} \\lambda{(\\mathbf{A})} d\\mathbf{A} = - \\mathbf{A} - g{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} + \\int \\frac{d}{d \\mathbf{A}} g{(\\mathbf{A})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Function('g')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Derivative(Function('g')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 5, "Add(Symbol('\\\\mathbf{A}', commutative=True), Function('g')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integral(Derivative(Function('\\\\lambda')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Function('g')(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Integral(Derivative(Function('g')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(W)} = \\log{(\\cos{(W)})}, then obtain (\\int (- W + \\int \\hat{\\mathbf{x}}{(W)} dW) dW) \\int \\hat{\\mathbf{x}}{(W)} dW = (\\int (- W + \\int \\log{(\\cos{(W)})} dW) dW) \\int \\hat{\\mathbf{x}}{(W)} dW", "derivation": "\\hat{\\mathbf{x}}{(W)} = \\log{(\\cos{(W)})} and \\int \\hat{\\mathbf{x}}{(W)} dW = \\int \\log{(\\cos{(W)})} dW and - W + \\int \\hat{\\mathbf{x}}{(W)} dW = - W + \\int \\log{(\\cos{(W)})} dW and \\int (- W + \\int \\hat{\\mathbf{x}}{(W)} dW) dW = \\int (- W + \\int \\log{(\\cos{(W)})} dW) dW and (\\int (- W + \\int \\hat{\\mathbf{x}}{(W)} dW) dW) \\int \\hat{\\mathbf{x}}{(W)} dW = (\\int (- W + \\int \\log{(\\cos{(W)})} dW) dW) \\int \\hat{\\mathbf{x}}{(W)} dW", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), log(cos(Symbol('W', commutative=True))))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(log(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["minus", 2, "Symbol('W', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(log(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))))"], [["integrate", 3, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(log(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))))"], [["times", 4, "Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Mul(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integral(Add(Mul(Integer(-1), Symbol('W', commutative=True)), Integral(log(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\lambda{(c_{0})} = \\cos{(c_{0})}, then obtain \\frac{d}{d c_{0}} \\frac{\\int \\lambda{(c_{0})} dc_{0}}{\\int \\cos{(c_{0})} dc_{0}} = \\frac{d}{d c_{0}} 1", "derivation": "\\lambda{(c_{0})} = \\cos{(c_{0})} and \\int \\lambda{(c_{0})} dc_{0} = \\int \\cos{(c_{0})} dc_{0} and \\tilde{\\infty}^{c_{0}} \\int \\lambda{(c_{0})} dc_{0} = \\tilde{\\infty}^{c_{0}} \\int \\cos{(c_{0})} dc_{0} and \\frac{\\int \\lambda{(c_{0})} dc_{0}}{\\int \\cos{(c_{0})} dc_{0}} = 1 and \\frac{d}{d c_{0}} \\frac{\\int \\lambda{(c_{0})} dc_{0}}{\\int \\cos{(c_{0})} dc_{0}} = \\frac{d}{d c_{0}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["divide", 2, "Pow(Integer(0), Symbol('c_0', commutative=True))"], "Equality(Mul(Pow(zoo, Symbol('c_0', commutative=True)), Integral(Function('\\\\lambda')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Pow(zoo, Symbol('c_0', commutative=True)), Integral(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["divide", 3, "Mul(Pow(zoo, Symbol('c_0', commutative=True)), Integral(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], "Equality(Mul(Integral(Function('\\\\lambda')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Pow(Integral(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 4, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Mul(Integral(Function('\\\\lambda')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Pow(Integral(cos(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integer(-1))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi_{nl}{(c_{0})} = e^{c_{0}}, then derive \\int \\Psi_{nl}{(c_{0})} dc_{0} = v_{1} + e^{c_{0}}, then derive v_{1} + e^{c_{0}} = \\varepsilon_0 + e^{c_{0}}, then obtain \\int \\Psi_{nl}{(c_{0})} dc_{0} = \\varepsilon_0 + e^{c_{0}}", "derivation": "\\Psi_{nl}{(c_{0})} = e^{c_{0}} and \\int \\Psi_{nl}{(c_{0})} dc_{0} = \\int e^{c_{0}} dc_{0} and \\int \\Psi_{nl}{(c_{0})} dc_{0} = v_{1} + e^{c_{0}} and v_{1} + e^{c_{0}} = \\int e^{c_{0}} dc_{0} and v_{1} + e^{c_{0}} = \\varepsilon_0 + e^{c_{0}} and \\int \\Psi_{nl}{(c_{0})} dc_{0} = \\varepsilon_0 + e^{c_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), exp(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Symbol('v_1', commutative=True), exp(Symbol('c_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('v_1', commutative=True), exp(Symbol('c_0', commutative=True))), Integral(exp(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('v_1', commutative=True), exp(Symbol('c_0', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('c_0', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Symbol('\\\\varepsilon_0', commutative=True), exp(Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given \\Omega{(l)} = \\cos{(l)}, then derive \\int \\Omega{(l)} dl = m + \\sin{(l)}, then derive L + m + \\sin{(l)} + \\int \\Omega{(l)} dl = L + 2 m + 2 \\sin{(l)}, then obtain \\frac{L + m + \\sin{(l)} + \\int \\Omega{(l)} dl}{L} = \\frac{L + 2 m + 2 \\sin{(l)}}{L}", "derivation": "\\Omega{(l)} = \\cos{(l)} and \\int \\Omega{(l)} dl = \\int \\cos{(l)} dl and \\int \\Omega{(l)} dl = m + \\sin{(l)} and m + \\int \\Omega{(l)} dl + \\int \\cos{(l)} dl = 2 m + \\sin{(l)} + \\int \\cos{(l)} dl and L + m + \\sin{(l)} + \\int \\Omega{(l)} dl = L + 2 m + 2 \\sin{(l)} and \\frac{L + m + \\sin{(l)} + \\int \\Omega{(l)} dl}{L} = \\frac{L + 2 m + 2 \\sin{(l)}}{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('m', commutative=True), sin(Symbol('l', commutative=True))))"], [["add", 3, "Add(Symbol('m', commutative=True), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], "Equality(Add(Symbol('m', commutative=True), Integral(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(2), Symbol('m', commutative=True)), sin(Symbol('l', commutative=True)), Integral(cos(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('L', commutative=True), Symbol('m', commutative=True), sin(Symbol('l', commutative=True)), Integral(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Symbol('L', commutative=True), Mul(Integer(2), Symbol('m', commutative=True)), Mul(Integer(2), sin(Symbol('l', commutative=True)))))"], [["divide", 5, "Symbol('L', commutative=True)"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Symbol('m', commutative=True), sin(Symbol('l', commutative=True)), Integral(Function('\\\\Omega')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Symbol('L', commutative=True), Mul(Integer(2), Symbol('m', commutative=True)), Mul(Integer(2), sin(Symbol('l', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})} and Q{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain \\frac{Q{(\\mu_0)}}{\\sin{(\\cos{(\\mu_0)})}} = \\frac{\\cos{(\\mu_0)}}{\\sin{(\\cos{(\\mu_0)})}}", "derivation": "\\operatorname{F_{c}}{(\\mu_0)} = \\sin{(\\cos{(\\mu_0)})} and Q{(\\mu_0)} = \\cos{(\\mu_0)} and \\frac{Q{(\\mu_0)}}{\\operatorname{F_{c}}{(\\mu_0)}} = \\frac{\\cos{(\\mu_0)}}{\\operatorname{F_{c}}{(\\mu_0)}} and \\frac{Q{(\\mu_0)}}{\\sin{(\\cos{(\\mu_0)})}} = \\frac{\\cos{(\\mu_0)}}{\\sin{(\\cos{(\\mu_0)})}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mu_0', commutative=True)), sin(cos(Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["divide", 2, "Function('F_c')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Function('F_c')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Function('Q')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Function('F_c')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('Q')(Symbol('\\\\mu_0', commutative=True)), Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Integer(-1))), Mul(Pow(sin(cos(Symbol('\\\\mu_0', commutative=True))), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(t_{2},B,\\nabla)} = B + \\nabla t_{2}, then obtain 2 \\mathbf{J}{(t_{2},B,\\nabla)} = 2 B + 2 \\nabla t_{2}", "derivation": "\\mathbf{J}{(t_{2},B,\\nabla)} = B + \\nabla t_{2} and 1 = \\frac{B + \\nabla t_{2}}{\\mathbf{J}{(t_{2},B,\\nabla)}} and 2 \\mathbf{J}{(t_{2},B,\\nabla)} = B + \\nabla t_{2} + \\mathbf{J}{(t_{2},B,\\nabla)} and B + \\nabla t_{2} + \\mathbf{J}{(t_{2},B,\\nabla)} = \\frac{(B + \\nabla t_{2}) (B + \\nabla t_{2} + \\mathbf{J}{(t_{2},B,\\nabla)})}{\\mathbf{J}{(t_{2},B,\\nabla)}} and 2 \\mathbf{J}{(t_{2},B,\\nabla)} = 2 B + 2 \\nabla t_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True))))"], [["divide", 1, "Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True))), Pow(Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))))"], [["times", 2, "Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)))"], "Equality(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))), Mul(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True)), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))), Pow(Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('t_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(2), Symbol('B', commutative=True)), Mul(Integer(2), Symbol('\\\\nabla', commutative=True), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(g)} = e^{g}, then obtain 3 + \\frac{3 e^{g}}{\\operatorname{C_{d}}{(g)}} = (1 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}}) (2 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}})", "derivation": "\\operatorname{C_{d}}{(g)} = e^{g} and 1 = \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}} and 2 = 1 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}} and 3 = 2 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}} and 3 + \\frac{3 e^{g}}{\\operatorname{C_{d}}{(g)}} = (1 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}}) (2 + \\frac{e^{g}}{\\operatorname{C_{d}}{(g)}})", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('g', commutative=True)), exp(Symbol('g', commutative=True)))"], [["divide", 1, "Function('C_d')(Symbol('g', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Integer(3), Add(Integer(2), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True)))))"], [["times", 4, "Add(Integer(1), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True))))"], "Equality(Add(Integer(3), Mul(Integer(3), Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True)))), Mul(Add(Integer(1), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True)))), Add(Integer(2), Mul(Pow(Function('C_d')(Symbol('g', commutative=True)), Integer(-1)), exp(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon{(v_{2})} = \\cos{(v_{2})}, then obtain \\varepsilon^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\cos{(v_{2})} = \\cos^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\cos{(v_{2})}", "derivation": "\\varepsilon{(v_{2})} = \\cos{(v_{2})} and \\frac{d}{d v_{2}} \\varepsilon{(v_{2})} = \\frac{d}{d v_{2}} \\cos{(v_{2})} and \\varepsilon^{v_{2}}{(v_{2})} = \\cos^{v_{2}}{(v_{2})} and \\varepsilon^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\varepsilon{(v_{2})} = \\cos^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\varepsilon{(v_{2})} and \\varepsilon^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\cos{(v_{2})} = \\cos^{v_{2}}{(v_{2})} - \\frac{d}{d v_{2}} \\cos{(v_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["power", 1, "Symbol('v_2', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Pow(cos(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)))"], [["minus", 3, "Derivative(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Add(Pow(cos(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('\\\\varepsilon')(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))), Add(Pow(cos(Symbol('v_2', commutative=True)), Symbol('v_2', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given g{(\\tilde{g},f_{E})} = \\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}}, then obtain \\frac{d}{d f_{E}} 1 = \\frac{\\partial}{\\partial f_{E}} (\\frac{\\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}}}{g{(\\tilde{g},f_{E})}})^{f_{E}}", "derivation": "g{(\\tilde{g},f_{E})} = \\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}} and 1 = \\frac{\\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}}}{g{(\\tilde{g},f_{E})}} and 1 = (\\frac{\\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}}}{g{(\\tilde{g},f_{E})}})^{f_{E}} and \\frac{d}{d f_{E}} 1 = \\frac{\\partial}{\\partial f_{E}} (\\frac{\\frac{\\partial}{\\partial \\tilde{g}} f_{E}^{\\tilde{g}}}{g{(\\tilde{g},f_{E})}})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_E', commutative=True)), Derivative(Pow(Symbol('f_E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["divide", 1, "Function('g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('f_E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('f_E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Symbol('f_E', commutative=True)))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Function('g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('f_E', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('f_E', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = \\Omega \\cos{(\\hat{\\mathbf{x}})}, then derive \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{x}}\\partial \\Omega} \\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = - \\sin{(\\hat{\\mathbf{x}})}, then obtain \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{x}}\\partial \\Omega} \\Omega \\cos{(\\hat{\\mathbf{x}})} = - \\sin{(\\hat{\\mathbf{x}})}", "derivation": "\\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = \\Omega \\cos{(\\hat{\\mathbf{x}})} and \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = \\frac{\\partial}{\\partial \\hat{\\mathbf{x}}} \\Omega \\cos{(\\hat{\\mathbf{x}})} and \\frac{\\partial^{2}}{\\partial \\Omega\\partial \\hat{\\mathbf{x}}} \\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = \\frac{\\partial^{2}}{\\partial \\Omega\\partial \\hat{\\mathbf{x}}} \\Omega \\cos{(\\hat{\\mathbf{x}})} and \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{x}}\\partial \\Omega} \\mathbf{S}{(\\hat{\\mathbf{x}},\\Omega)} = - \\sin{(\\hat{\\mathbf{x}})} and \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{x}}\\partial \\Omega} \\Omega \\cos{(\\hat{\\mathbf{x}})} = - \\sin{(\\hat{\\mathbf{x}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given g{(\\delta)} = e^{\\delta}, then obtain - \\frac{\\partial}{\\partial P_{g}} P_{g} h + (\\iint (g^{2}{(\\delta)} - g{(\\delta)}) d\\delta d\\delta)^{\\delta} = - \\frac{\\partial}{\\partial P_{g}} P_{g} h + (\\iint (g{(\\delta)} e^{\\delta} - g{(\\delta)}) d\\delta d\\delta)^{\\delta}", "derivation": "g{(\\delta)} = e^{\\delta} and g^{2}{(\\delta)} = g{(\\delta)} e^{\\delta} and g^{2}{(\\delta)} - g{(\\delta)} = g{(\\delta)} e^{\\delta} - g{(\\delta)} and \\int (g^{2}{(\\delta)} - g{(\\delta)}) d\\delta = \\int (g{(\\delta)} e^{\\delta} - g{(\\delta)}) d\\delta and \\iint (g^{2}{(\\delta)} - g{(\\delta)}) d\\delta d\\delta = \\iint (g{(\\delta)} e^{\\delta} - g{(\\delta)}) d\\delta d\\delta and (\\iint (g^{2}{(\\delta)} - g{(\\delta)}) d\\delta d\\delta)^{\\delta} = (\\iint (g{(\\delta)} e^{\\delta} - g{(\\delta)}) d\\delta d\\delta)^{\\delta} and - \\frac{\\partial}{\\partial P_{g}} P_{g} h + (\\iint (g^{2}{(\\delta)} - g{(\\delta)}) d\\delta d\\delta)^{\\delta} = - \\frac{\\partial}{\\partial P_{g}} P_{g} h + (\\iint (g{(\\delta)} e^{\\delta} - g{(\\delta)}) d\\delta d\\delta)^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True)))"], [["times", 1, "Function('g')(Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Function('g')(Symbol('\\\\delta', commutative=True))"], "Equality(Add(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Add(Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Add(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 5, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Add(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(Add(Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["minus", 6, "Derivative(Mul(Symbol('P_g', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('P_g', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Pow(Integral(Add(Pow(Function('g')(Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))), Add(Mul(Integer(-1), Derivative(Mul(Symbol('P_g', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Pow(Integral(Add(Mul(Function('g')(Symbol('\\\\delta', commutative=True)), exp(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Function('g')(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given U{(k,l)} = k + l, then obtain \\int 4 (\\int U^{l}{(k,l)} dl)^{2} dk = \\int 2 (\\int (k + l)^{l} dl + \\int U^{l}{(k,l)} dl) \\int U^{l}{(k,l)} dl dk", "derivation": "U{(k,l)} = k + l and U^{l}{(k,l)} = (k + l)^{l} and \\int U^{l}{(k,l)} dl = \\int (k + l)^{l} dl and 2 \\int U^{l}{(k,l)} dl = \\int (k + l)^{l} dl + \\int U^{l}{(k,l)} dl and 4 (\\int U^{l}{(k,l)} dl)^{2} = 2 (\\int (k + l)^{l} dl + \\int U^{l}{(k,l)} dl) \\int U^{l}{(k,l)} dl and \\int 4 (\\int U^{l}{(k,l)} dl)^{2} dk = \\int 2 (\\int (k + l)^{l} dl + \\int U^{l}{(k,l)} dl) \\int U^{l}{(k,l)} dl dk", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Add(Symbol('k', commutative=True), Symbol('l', commutative=True)))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Add(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Add(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["add", 3, "Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Integral(Pow(Add(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["times", 4, "Mul(Integer(2), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], "Equality(Mul(Integer(4), Pow(Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(2))), Mul(Integer(2), Add(Integral(Pow(Add(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["integrate", 5, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Integer(4), Pow(Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(2))), Tuple(Symbol('k', commutative=True))), Integral(Mul(Integer(2), Add(Integral(Pow(Add(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Integral(Pow(Function('U')(Symbol('k', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('k', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(V,\\theta)} = \\theta e^{V} and \\operatorname{x^{{\\}'}}{(V,\\theta)} = \\theta e^{V}, then obtain \\int (\\theta e^{V} + \\operatorname{x^{{\\}'}}{(V,\\theta)}) d\\theta = \\int (\\theta e^{V} + \\operatorname{C_{2}}{(V,\\theta)}) d\\theta", "derivation": "\\operatorname{C_{2}}{(V,\\theta)} = \\theta e^{V} and \\operatorname{x^{{\\}'}}{(V,\\theta)} = \\theta e^{V} and \\operatorname{x^{{\\}'}}{(V,\\theta)} = \\operatorname{C_{2}}{(V,\\theta)} and \\theta e^{V} + \\operatorname{x^{{\\}'}}{(V,\\theta)} = \\theta e^{V} + \\operatorname{C_{2}}{(V,\\theta)} and \\int (\\theta e^{V} + \\operatorname{x^{{\\}'}}{(V,\\theta)}) d\\theta = \\int (\\theta e^{V} + \\operatorname{C_{2}}{(V,\\theta)}) d\\theta", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('x^\\\\prime')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True)), Function('C_2')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))), Function('x^\\\\prime')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))), Function('C_2')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True))))"], [["integrate", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))), Function('x^\\\\prime')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Mul(Symbol('\\\\theta', commutative=True), exp(Symbol('V', commutative=True))), Function('C_2')(Symbol('V', commutative=True), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"]]}, {"prompt": "Given a{(C_{d},v_{t},i)} = C_{d}^{v_{t}} - i and \\operatorname{F_{N}}{(C_{d},v_{t},i)} = C_{d}^{v_{t}} - i, then obtain \\frac{\\partial}{\\partial i} \\frac{\\operatorname{F_{N}}{(C_{d},v_{t},i)}}{C_{d}} = \\frac{\\partial}{\\partial i} \\frac{a{(C_{d},v_{t},i)}}{C_{d}}", "derivation": "a{(C_{d},v_{t},i)} = C_{d}^{v_{t}} - i and \\operatorname{F_{N}}{(C_{d},v_{t},i)} = C_{d}^{v_{t}} - i and \\frac{\\operatorname{F_{N}}{(C_{d},v_{t},i)}}{C_{d}} = \\frac{C_{d}^{v_{t}} - i}{C_{d}} and \\frac{\\partial}{\\partial i} \\frac{\\operatorname{F_{N}}{(C_{d},v_{t},i)}}{C_{d}} = \\frac{\\partial}{\\partial i} \\frac{C_{d}^{v_{t}} - i}{C_{d}} and \\frac{\\partial}{\\partial i} \\frac{\\operatorname{F_{N}}{(C_{d},v_{t},i)}}{C_{d}} = \\frac{\\partial}{\\partial i} \\frac{a{(C_{d},v_{t},i)}}{C_{d}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True)), Add(Pow(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True)), Add(Pow(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True))))"], [["divide", 2, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('F_N')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Pow(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('F_N')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Pow(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('F_N')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Function('a')(Symbol('C_d', commutative=True), Symbol('v_t', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\chi)}, then derive \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} = - \\sin{(\\chi)}, then obtain \\operatorname{C_{d}}{(\\chi)} \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} + \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} = \\operatorname{C_{d}}{(\\chi)} \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} - \\sin{(\\chi)}", "derivation": "\\operatorname{C_{d}}{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\chi)} and \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} = \\frac{d^{2}}{d \\chi^{2}} \\sin{(\\chi)} and \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} = - \\sin{(\\chi)} and \\frac{d^{2}}{d \\chi^{2}} \\sin{(\\chi)} = - \\sin{(\\chi)} and \\operatorname{C_{d}}{(\\chi)} \\frac{d^{2}}{d \\chi^{2}} \\sin{(\\chi)} + \\frac{d^{2}}{d \\chi^{2}} \\sin{(\\chi)} = \\operatorname{C_{d}}{(\\chi)} \\frac{d^{2}}{d \\chi^{2}} \\sin{(\\chi)} - \\sin{(\\chi)} and \\operatorname{C_{d}}{(\\chi)} \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} + \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} = \\operatorname{C_{d}}{(\\chi)} \\frac{d}{d \\chi} \\operatorname{C_{d}}{(\\chi)} - \\sin{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))))"], [["add", 4, "Mul(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))))"], "Equality(Add(Mul(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))), Add(Mul(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Derivative(Function('C_d')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Add(Mul(Function('C_d')(Symbol('\\\\chi', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f \\psi^*}{\\phi_1} and \\mathbf{g}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\int \\frac{\\mathbf{J}_f \\psi^*}{\\phi_1} d\\psi^*, then obtain \\mathbf{g}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\int \\Psi_{nl}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} d\\psi^*", "derivation": "\\Psi_{nl}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f \\psi^*}{\\phi_1} and \\int \\Psi_{nl}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} d\\psi^* = \\int \\frac{\\mathbf{J}_f \\psi^*}{\\phi_1} d\\psi^* and \\mathbf{g}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\int \\frac{\\mathbf{J}_f \\psi^*}{\\phi_1} d\\psi^* and \\mathbf{g}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} = \\int \\Psi_{nl}{(\\phi_1,\\psi^*,\\mathbf{J}_f)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\psi^*', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given u{(L)} = \\sin{(L)} and \\tilde{g}{(L)} = - \\sin{(L)}, then obtain 2 \\tilde{g}{(L)} - 2 u{(L)} = \\tilde{g}{(L)} - 3 u{(L)}", "derivation": "u{(L)} = \\sin{(L)} and \\tilde{g}{(L)} = - \\sin{(L)} and \\tilde{g}{(L)} = - u{(L)} and \\tilde{g}{(L)} - u{(L)} = - 2 u{(L)} and 2 \\tilde{g}{(L)} - 2 u{(L)} = \\tilde{g}{(L)} - 3 u{(L)}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('L', commutative=True)), sin(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Mul(Integer(-1), sin(Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('L', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('u')(Symbol('L', commutative=True)))"], "Equality(Add(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('L', commutative=True)))), Mul(Integer(-1), Integer(2), Function('u')(Symbol('L', commutative=True))))"], [["add", 4, "Add(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Mul(Integer(-1), Function('u')(Symbol('L', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('L', commutative=True))), Mul(Integer(-1), Integer(2), Function('u')(Symbol('L', commutative=True)))), Add(Function('\\\\tilde{g}')(Symbol('L', commutative=True)), Mul(Integer(-1), Integer(3), Function('u')(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\chi{(J_{\\varepsilon},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}), then obtain \\chi{(J_{\\varepsilon},y^{\\prime})} (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{2} = (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{3}", "derivation": "\\chi{(J_{\\varepsilon},y^{\\prime})} = \\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}) and \\chi{(J_{\\varepsilon},y^{\\prime})} \\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}) = (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{2} and \\chi^{2}{(J_{\\varepsilon},y^{\\prime})} \\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}) = \\chi{(J_{\\varepsilon},y^{\\prime})} (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{2} and \\chi{(J_{\\varepsilon},y^{\\prime})} (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{2} = (\\frac{\\partial}{\\partial y^{\\prime}} (- J_{\\varepsilon} + y^{\\prime}))^{3}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Integer(2)))"], [["times", 2, "Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(2)), Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1)))), Mul(Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\chi')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Integer(3)))"]]}, {"prompt": "Given \\hat{p}{(f_{\\mathbf{p}},a)} = f_{\\mathbf{p}} + \\sin{(a)}, then obtain f_{\\mathbf{p}} + \\sin{(a)} = \\frac{(f_{\\mathbf{p}} + \\sin{(a)})^{2}}{\\hat{p}{(f_{\\mathbf{p}},a)}}", "derivation": "\\hat{p}{(f_{\\mathbf{p}},a)} = f_{\\mathbf{p}} + \\sin{(a)} and \\hat{p}{(f_{\\mathbf{p}},a)} \\sin{(a)} = (f_{\\mathbf{p}} + \\sin{(a)}) \\sin{(a)} and \\hat{p}^{2}{(f_{\\mathbf{p}},a)} \\sin{(a)} = (f_{\\mathbf{p}} + \\sin{(a)}) \\hat{p}{(f_{\\mathbf{p}},a)} \\sin{(a)} and (f_{\\mathbf{p}} + \\sin{(a)}) \\hat{p}{(f_{\\mathbf{p}},a)} \\sin{(a)} = (f_{\\mathbf{p}} + \\sin{(a)})^{2} \\sin{(a)} and f_{\\mathbf{p}} + \\sin{(a)} = \\frac{(f_{\\mathbf{p}} + \\sin{(a)})^{2}}{\\hat{p}{(f_{\\mathbf{p}},a)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))))"], [["times", 1, "sin(Symbol('a', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), sin(Symbol('a', commutative=True))))"], [["times", 2, "Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True))"], "Equality(Mul(Pow(Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), Integer(2)), sin(Symbol('a', commutative=True))), Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True))), Mul(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(2)), sin(Symbol('a', commutative=True))))"], [["divide", 4, "Mul(Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), Mul(Pow(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), sin(Symbol('a', commutative=True))), Integer(2)), Pow(Function('\\\\hat{p}')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('a', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\chi{(b,\\mathbf{H})} = \\mathbf{H} + \\cos{(b)} and \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})} = \\mathbf{J} + \\sin{(\\hat{H})}, then obtain b \\chi{(b,\\mathbf{H})} \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})} = b (\\mathbf{H} + \\cos{(b)}) \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})}", "derivation": "\\chi{(b,\\mathbf{H})} = \\mathbf{H} + \\cos{(b)} and b \\chi{(b,\\mathbf{H})} = b (\\mathbf{H} + \\cos{(b)}) and \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})} = \\mathbf{J} + \\sin{(\\hat{H})} and b (\\mathbf{J} + \\sin{(\\hat{H})}) \\chi{(b,\\mathbf{H})} = b (\\mathbf{H} + \\cos{(b)}) (\\mathbf{J} + \\sin{(\\hat{H})}) and b \\chi{(b,\\mathbf{H})} \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})} = b (\\mathbf{H} + \\cos{(b)}) \\operatorname{r_{0}}{(\\hat{H},\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('b', commutative=True))))"], [["times", 1, "Symbol('b', commutative=True)"], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\chi')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('b', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('b', commutative=True)))))"], ["get_premise", "Equality(Function('r_0')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 2, "Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)))"], "Equality(Mul(Symbol('b', commutative=True), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True))), Function('\\\\chi')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Symbol('b', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('b', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), sin(Symbol('\\\\hat{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Symbol('b', commutative=True), Function('\\\\chi')(Symbol('b', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('r_0')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('b', commutative=True), Add(Symbol('\\\\mathbf{H}', commutative=True), cos(Symbol('b', commutative=True))), Function('r_0')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(Z,\\phi_1)} = \\frac{Z}{\\phi_1}, then obtain \\frac{\\phi_1 (\\sigma_{p}^{\\phi_1}{(Z,\\phi_1)})^{Z} \\sigma_{p}{(Z,\\phi_1)}}{Z} = \\frac{\\phi_1 ((\\frac{Z}{\\phi_1})^{\\phi_1})^{Z} \\sigma_{p}{(Z,\\phi_1)}}{Z}", "derivation": "\\sigma_{p}{(Z,\\phi_1)} = \\frac{Z}{\\phi_1} and \\sigma_{p}^{\\phi_1}{(Z,\\phi_1)} = (\\frac{Z}{\\phi_1})^{\\phi_1} and (\\sigma_{p}^{\\phi_1}{(Z,\\phi_1)})^{Z} = ((\\frac{Z}{\\phi_1})^{\\phi_1})^{Z} and \\frac{\\phi_1 (\\sigma_{p}^{\\phi_1}{(Z,\\phi_1)})^{Z} \\sigma_{p}{(Z,\\phi_1)}}{Z} = \\frac{\\phi_1 ((\\frac{Z}{\\phi_1})^{\\phi_1})^{Z} \\sigma_{p}{(Z,\\phi_1)}}{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Symbol('\\\\phi_1', commutative=True)))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Pow(Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('Z', commutative=True)), Pow(Pow(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Symbol('\\\\phi_1', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 3, "Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Pow(Pow(Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\phi_1', commutative=True), Pow(Pow(Mul(Symbol('Z', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Symbol('\\\\phi_1', commutative=True)), Symbol('Z', commutative=True)), Function('\\\\sigma_p')(Symbol('Z', commutative=True), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\hat{H},s)} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} + s), then derive \\int \\tilde{g}^*{(\\hat{H},s)} ds = \\mathbf{H} + s, then obtain \\hat{H} + s + \\int \\tilde{g}^*{(\\hat{H},s)} ds + 1 = \\hat{H} + \\mathbf{H} + 2 s + 1", "derivation": "\\tilde{g}^*{(\\hat{H},s)} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} + s) and \\int \\tilde{g}^*{(\\hat{H},s)} ds = \\int \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} + s) ds and \\int \\tilde{g}^*{(\\hat{H},s)} ds = \\mathbf{H} + s and s + \\int \\tilde{g}^*{(\\hat{H},s)} ds = \\mathbf{H} + 2 s and s + \\int \\tilde{g}^*{(\\hat{H},s)} ds + 1 = \\mathbf{H} + 2 s + 1 and \\hat{H} + s + \\int \\tilde{g}^*{(\\hat{H},s)} ds + 1 = \\hat{H} + \\mathbf{H} + 2 s + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('s', commutative=True)))"], [["add", 3, "Symbol('s', commutative=True)"], "Equality(Add(Symbol('s', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Symbol('s', commutative=True))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Symbol('s', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1)), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Symbol('s', commutative=True)), Integer(1)))"], [["add", 5, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True), Integral(Function('\\\\tilde{g}^*')(Symbol('\\\\hat{H}', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(1)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(2), Symbol('s', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\psi^{*}{(E_{x})} = \\int \\cos{(E_{x})} dE_{x} and \\operatorname{F_{x}}{(E_{x})} = \\int \\cos{(E_{x})} dE_{x}, then derive \\operatorname{F_{x}}{(E_{x})} = \\mathbf{D} + \\sin{(E_{x})}, then obtain \\frac{d}{d \\mathbf{D}} \\psi^{*}{(E_{x})} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + \\sin{(E_{x})})", "derivation": "\\psi^{*}{(E_{x})} = \\int \\cos{(E_{x})} dE_{x} and \\operatorname{F_{x}}{(E_{x})} = \\int \\cos{(E_{x})} dE_{x} and \\operatorname{F_{x}}{(E_{x})} = \\mathbf{D} + \\sin{(E_{x})} and \\mathbf{D} + \\sin{(E_{x})} = \\int \\cos{(E_{x})} dE_{x} and \\psi^{*}{(E_{x})} = \\mathbf{D} + \\sin{(E_{x})} and \\frac{d}{d \\mathbf{D}} \\psi^{*}{(E_{x})} = \\frac{\\partial}{\\partial \\mathbf{D}} (\\mathbf{D} + \\sin{(E_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('E_x', commutative=True)), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Function('F_x')(Symbol('E_x', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('E_x', commutative=True))), Integral(cos(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\psi^*')(Symbol('E_x', commutative=True)), Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('E_x', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\psi^*')(Symbol('E_x', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), sin(Symbol('E_x', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(f_{E})} = \\log{(f_{E})}, then obtain \\frac{d}{d f_{E}} \\hat{p}{(f_{E})} - \\int \\log{(\\log{(f_{E})})} df_{E} = \\frac{d}{d f_{E}} \\log{(f_{E})} - \\int \\log{(\\log{(f_{E})})} df_{E}", "derivation": "\\hat{p}{(f_{E})} = \\log{(f_{E})} and \\log{(\\hat{p}{(f_{E})})} = \\log{(\\log{(f_{E})})} and \\int \\log{(\\hat{p}{(f_{E})})} df_{E} = \\int \\log{(\\log{(f_{E})})} df_{E} and \\frac{d}{d f_{E}} \\hat{p}{(f_{E})} = \\frac{d}{d f_{E}} \\log{(f_{E})} and \\frac{d}{d f_{E}} \\hat{p}{(f_{E})} - \\int \\log{(\\hat{p}{(f_{E})})} df_{E} = \\frac{d}{d f_{E}} \\log{(f_{E})} - \\int \\log{(\\hat{p}{(f_{E})})} df_{E} and \\frac{d}{d f_{E}} \\hat{p}{(f_{E})} - \\int \\log{(\\log{(f_{E})})} df_{E} = \\frac{d}{d f_{E}} \\log{(f_{E})} - \\int \\log{(\\log{(f_{E})})} df_{E}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('f_E', commutative=True)), log(Symbol('f_E', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\hat{p}')(Symbol('f_E', commutative=True))), log(log(Symbol('f_E', commutative=True))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(log(Function('\\\\hat{p}')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(log(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["minus", 4, "Integral(log(Function('\\\\hat{p}')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True)))"], "Equality(Add(Derivative(Function('\\\\hat{p}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Integral(log(Function('\\\\hat{p}')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))), Add(Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Integral(log(Function('\\\\hat{p}')(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Derivative(Function('\\\\hat{p}')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Integral(log(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))), Add(Derivative(log(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), Integral(log(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given l{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} \\theta and \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} V_{\\mathbf{E}} \\theta, then derive \\frac{\\partial}{\\partial \\theta} l{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}}, then obtain (\\dot{\\mathbf{r}} + z^{*}) \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} (\\dot{\\mathbf{r}} + z^{*})", "derivation": "l{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} \\theta and \\frac{\\partial}{\\partial \\theta} l{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} V_{\\mathbf{E}} \\theta and \\frac{\\partial}{\\partial \\theta} l{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} and \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} V_{\\mathbf{E}} \\theta and \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = \\frac{\\partial}{\\partial \\theta} l{(\\theta,V_{\\mathbf{E}})} and \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} and (\\dot{\\mathbf{r}} + z^{*}) \\mathbf{J}{(\\theta,V_{\\mathbf{E}})} = V_{\\mathbf{E}} (\\dot{\\mathbf{r}} + z^{*})", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('l')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('l')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Derivative(Function('l')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], [["times", 6, "Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z^*', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\theta', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\rho_f)} = \\sin{(\\rho_f)}, then derive \\int \\operatorname{g_{\\varepsilon}}{(\\rho_f)} d\\rho_f = \\hat{x}_0 - \\cos{(\\rho_f)}, then obtain (\\int \\operatorname{g_{\\varepsilon}}{(\\rho_f)} d\\rho_f)^{\\rho_f} = (\\int \\sin{(\\rho_f)} d\\rho_f)^{\\rho_f}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\rho_f)} = \\sin{(\\rho_f)} and \\int \\operatorname{g_{\\varepsilon}}{(\\rho_f)} d\\rho_f = \\int \\sin{(\\rho_f)} d\\rho_f and \\int \\operatorname{g_{\\varepsilon}}{(\\rho_f)} d\\rho_f = \\hat{x}_0 - \\cos{(\\rho_f)} and \\hat{x}_0 - \\cos{(\\rho_f)} = \\int \\sin{(\\rho_f)} d\\rho_f and (\\hat{x}_0 - \\cos{(\\rho_f)})^{\\rho_f} = (\\int \\sin{(\\rho_f)} d\\rho_f)^{\\rho_f} and (\\int \\operatorname{g_{\\varepsilon}}{(\\rho_f)} d\\rho_f)^{\\rho_f} = (\\int \\sin{(\\rho_f)} d\\rho_f)^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\rho_f', commutative=True)), sin(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))), Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{x}_0', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\rho_f', commutative=True)))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integral(Function('g_{\\\\varepsilon}')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(sin(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{H}_l,\\delta)} = \\delta^{\\hat{H}_l}, then obtain (\\delta^{- \\hat{H}_l} \\theta_{1}^{\\delta}{(\\hat{H}_l,\\delta)})^{\\hat{H}_l} = (\\delta^{- \\hat{H}_l} (\\delta^{\\hat{H}_l})^{\\delta})^{\\hat{H}_l}", "derivation": "\\theta_{1}{(\\hat{H}_l,\\delta)} = \\delta^{\\hat{H}_l} and \\theta_{1}^{\\delta}{(\\hat{H}_l,\\delta)} = (\\delta^{\\hat{H}_l})^{\\delta} and \\delta^{- \\hat{H}_l} \\theta_{1}^{\\delta}{(\\hat{H}_l,\\delta)} = \\delta^{- \\hat{H}_l} (\\delta^{\\hat{H}_l})^{\\delta} and (\\delta^{- \\hat{H}_l} \\theta_{1}^{\\delta}{(\\hat{H}_l,\\delta)})^{\\hat{H}_l} = (\\delta^{- \\hat{H}_l} (\\delta^{\\hat{H}_l})^{\\delta})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["divide", 2, "Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Mul(Pow(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True))), Pow(Pow(Symbol('\\\\delta', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\delta', commutative=True))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given G{(\\dot{y},\\Omega)} = \\cos{(\\Omega + \\dot{y})}, then obtain - \\sin{(\\Omega + \\dot{y})} + \\frac{\\partial}{\\partial \\Omega} G{(\\dot{y},\\Omega)} = - 2 \\sin{(\\Omega + \\dot{y})}", "derivation": "G{(\\dot{y},\\Omega)} = \\cos{(\\Omega + \\dot{y})} and \\frac{\\partial}{\\partial \\Omega} G{(\\dot{y},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\cos{(\\Omega + \\dot{y})} and \\frac{\\partial}{\\partial \\Omega} G{(\\dot{y},\\Omega)} + \\frac{\\partial}{\\partial \\Omega} \\cos{(\\Omega + \\dot{y})} = 2 \\frac{\\partial}{\\partial \\Omega} \\cos{(\\Omega + \\dot{y})} and - \\sin{(\\Omega + \\dot{y})} + \\frac{\\partial}{\\partial \\Omega} G{(\\dot{y},\\Omega)} = - 2 \\sin{(\\Omega + \\dot{y})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Omega', commutative=True)), cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["add", 2, "Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('G')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(cos(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True)))), Derivative(Function('G')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(F_{x})} = \\cos{(F_{x})}, then derive \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = \\sigma_x + \\sin{(F_{x})}, then derive v_{x} + \\sin{(F_{x})} = \\sigma_x + \\sin{(F_{x})}, then obtain v_{x} + \\sin{(F_{x})} - \\cos{(F_{x})} = - \\cos{(F_{x})} + \\int \\cos{(F_{x})} dF_{x}", "derivation": "\\operatorname{x^{{\\}'}}{(F_{x})} = \\cos{(F_{x})} and \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = \\int \\cos{(F_{x})} dF_{x} and \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} = \\sigma_x + \\sin{(F_{x})} and \\int \\cos{(F_{x})} dF_{x} = \\sigma_x + \\sin{(F_{x})} and v_{x} + \\sin{(F_{x})} = \\sigma_x + \\sin{(F_{x})} and v_{x} + \\sin{(F_{x})} = \\int \\operatorname{x^{{\\}'}}{(F_{x})} dF_{x} and v_{x} + \\sin{(F_{x})} = \\int \\cos{(F_{x})} dF_{x} and v_{x} + \\sin{(F_{x})} - \\cos{(F_{x})} = - \\cos{(F_{x})} + \\int \\cos{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('v_x', commutative=True), sin(Symbol('F_x', commutative=True))), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('F_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Symbol('v_x', commutative=True), sin(Symbol('F_x', commutative=True))), Integral(Function('x^\\\\prime')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Symbol('v_x', commutative=True), sin(Symbol('F_x', commutative=True))), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["add", 7, "Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))"], "Equality(Add(Symbol('v_x', commutative=True), sin(Symbol('F_x', commutative=True)), Mul(Integer(-1), cos(Symbol('F_x', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('F_x', commutative=True))), Integral(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{p}{(P_{e},y^{\\prime})} = \\frac{P_{e}}{y^{\\prime}} and b{(P_{e})} = P_{e}, then obtain P_{e} b{(P_{e})} + \\frac{y^{\\prime}}{P_{e}} = P_{e}^{2} + \\frac{y^{\\prime}}{P_{e}}", "derivation": "\\sigma_{p}{(P_{e},y^{\\prime})} = \\frac{P_{e}}{y^{\\prime}} and b{(P_{e})} = P_{e} and P_{e} b{(P_{e})} = P_{e}^{2} and P_{e} b{(P_{e})} + \\frac{1}{\\sigma_{p}{(P_{e},y^{\\prime})}} = P_{e}^{2} + \\frac{1}{\\sigma_{p}{(P_{e},y^{\\prime})}} and P_{e} b{(P_{e})} + \\frac{y^{\\prime}}{P_{e}} = P_{e}^{2} + \\frac{y^{\\prime}}{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('P_e', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('b')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True))"], [["times", 2, "Symbol('P_e', commutative=True)"], "Equality(Mul(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Pow(Symbol('P_e', commutative=True), Integer(2)))"], [["add", 3, "Pow(Function('\\\\sigma_p')(Symbol('P_e', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Pow(Function('\\\\sigma_p')(Symbol('P_e', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))), Add(Pow(Symbol('P_e', commutative=True), Integer(2)), Pow(Function('\\\\sigma_p')(Symbol('P_e', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('P_e', commutative=True), Function('b')(Symbol('P_e', commutative=True))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))), Add(Pow(Symbol('P_e', commutative=True), Integer(2)), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\chi,B)} = B + \\cos{(\\chi)}, then obtain \\frac{\\partial}{\\partial B} \\varepsilon_{0}{(\\chi,B)} + (\\frac{\\partial}{\\partial \\chi} \\varepsilon_{0}^{B}{(\\chi,B)})^{\\chi} = \\frac{\\partial}{\\partial B} (B + \\cos{(\\chi)}) + (\\frac{\\partial}{\\partial \\chi} \\varepsilon_{0}^{B}{(\\chi,B)})^{\\chi}", "derivation": "\\varepsilon_{0}{(\\chi,B)} = B + \\cos{(\\chi)} and \\frac{\\partial}{\\partial B} \\varepsilon_{0}{(\\chi,B)} = \\frac{\\partial}{\\partial B} (B + \\cos{(\\chi)}) and \\varepsilon_{0}^{B}{(\\chi,B)} = (B + \\cos{(\\chi)})^{B} and (\\frac{\\partial}{\\partial \\chi} (B + \\cos{(\\chi)})^{B})^{\\chi} + \\frac{\\partial}{\\partial B} \\varepsilon_{0}{(\\chi,B)} = \\frac{\\partial}{\\partial B} (B + \\cos{(\\chi)}) + (\\frac{\\partial}{\\partial \\chi} (B + \\cos{(\\chi)})^{B})^{\\chi} and \\frac{\\partial}{\\partial B} \\varepsilon_{0}{(\\chi,B)} + (\\frac{\\partial}{\\partial \\chi} \\varepsilon_{0}^{B}{(\\chi,B)})^{\\chi} = \\frac{\\partial}{\\partial B} (B + \\cos{(\\chi)}) + (\\frac{\\partial}{\\partial \\chi} \\varepsilon_{0}^{B}{(\\chi,B)})^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 1, "Symbol('B', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Symbol('B', commutative=True)))"], [["add", 2, "Pow(Derivative(Pow(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True))"], "Equality(Add(Pow(Derivative(Pow(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True)), Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1)))), Add(Derivative(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Pow(Derivative(Pow(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Derivative(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Pow(Derivative(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True))), Add(Derivative(Add(Symbol('B', commutative=True), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Pow(Derivative(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\chi', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(B)} = \\cos{(\\sin{(B)})}, then obtain \\hat{\\mathbf{x}}{(B)} (\\int \\hat{\\mathbf{x}}{(B)} dB)^{B} = \\hat{\\mathbf{x}}{(B)} (\\int \\cos{(\\sin{(B)})} dB)^{B}", "derivation": "\\hat{\\mathbf{x}}{(B)} = \\cos{(\\sin{(B)})} and \\int \\hat{\\mathbf{x}}{(B)} dB = \\int \\cos{(\\sin{(B)})} dB and (\\int \\hat{\\mathbf{x}}{(B)} dB)^{B} = (\\int \\cos{(\\sin{(B)})} dB)^{B} and \\cos{(\\sin{(B)})} (\\int \\hat{\\mathbf{x}}{(B)} dB)^{B} = \\cos{(\\sin{(B)})} (\\int \\cos{(\\sin{(B)})} dB)^{B} and \\hat{\\mathbf{x}}{(B)} (\\int \\hat{\\mathbf{x}}{(B)} dB)^{B} = \\hat{\\mathbf{x}}{(B)} (\\int \\cos{(\\sin{(B)})} dB)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), cos(sin(Symbol('B', commutative=True))))"], [["integrate", 1, "Symbol('B', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Integral(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True)))"], [["times", 3, "cos(sin(Symbol('B', commutative=True)))"], "Equality(Mul(cos(sin(Symbol('B', commutative=True))), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(cos(sin(Symbol('B', commutative=True))), Pow(Integral(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Pow(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))), Mul(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('B', commutative=True)), Pow(Integral(cos(sin(Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True))), Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(F_{x},Q)} = F_{x} + Q and \\operatorname{v_{z}}{(J,F_{N})} = F_{N} + J, then obtain - F_{N} - F_{x} - J - Q + \\hat{\\mathbf{x}}{(F_{x},Q)} + \\operatorname{v_{z}}{(J,F_{N})} = 0", "derivation": "\\hat{\\mathbf{x}}{(F_{x},Q)} = F_{x} + Q and 2 \\hat{\\mathbf{x}}{(F_{x},Q)} = F_{x} + Q + \\hat{\\mathbf{x}}{(F_{x},Q)} and \\operatorname{v_{z}}{(J,F_{N})} = F_{N} + J and - F_{N} - J + \\operatorname{v_{z}}{(J,F_{N})} = 0 and - F_{x} - Q + \\hat{\\mathbf{x}}{(F_{x},Q)} = 0 and - F_{N} - F_{x} - J - Q + \\hat{\\mathbf{x}}{(F_{x},Q)} + \\operatorname{v_{z}}{(J,F_{N})} = - F_{x} - Q + \\hat{\\mathbf{x}}{(F_{x},Q)} and - F_{N} - F_{x} - J - Q + \\hat{\\mathbf{x}}{(F_{x},Q)} + \\operatorname{v_{z}}{(J,F_{N})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)))"], [["add", 1, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True))), Add(Symbol('F_x', commutative=True), Symbol('Q', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True))))"], ["get_premise", "Equality(Function('v_z')(Symbol('J', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('J', commutative=True)))"], [["minus", 3, "Add(Symbol('F_N', commutative=True), Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Function('v_z')(Symbol('J', commutative=True), Symbol('F_N', commutative=True))), Integer(0))"], [["minus", 2, "Add(Symbol('F_x', commutative=True), Symbol('Q', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True))), Integer(0))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)), Function('v_z')(Symbol('J', commutative=True), Symbol('F_N', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('F_x', commutative=True)), Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('Q', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('F_x', commutative=True), Symbol('Q', commutative=True)), Function('v_z')(Symbol('J', commutative=True), Symbol('F_N', commutative=True))), Integer(0))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(H)} = H, then obtain (H + \\operatorname{C_{1}}{(H)}) \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)} = 2 H \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)}", "derivation": "\\operatorname{C_{1}}{(H)} = H and H + \\operatorname{C_{1}}{(H)} = 2 H and 2 \\operatorname{C_{1}}^{2}{(H)} = 2 H \\operatorname{C_{1}}{(H)} and 2 \\operatorname{C_{1}}^{2}{(H)} + \\operatorname{C_{1}}{(H)} = 2 H \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)} and 2 \\operatorname{C_{1}}^{2}{(H)} + \\operatorname{C_{1}}{(H)} = (H + \\operatorname{C_{1}}{(H)}) \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)} and (H + \\operatorname{C_{1}}{(H)}) \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)} = 2 H \\operatorname{C_{1}}{(H)} + \\operatorname{C_{1}}{(H)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('H', commutative=True)), Symbol('H', commutative=True))"], [["add", 1, "Symbol('H', commutative=True)"], "Equality(Add(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Mul(Integer(2), Symbol('H', commutative=True)))"], [["times", 1, "Mul(Integer(2), Function('C_1')(Symbol('H', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('C_1')(Symbol('H', commutative=True)), Integer(2))), Mul(Integer(2), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))))"], [["add", 3, "Function('C_1')(Symbol('H', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Function('C_1')(Symbol('H', commutative=True)), Integer(2))), Function('C_1')(Symbol('H', commutative=True))), Add(Mul(Integer(2), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(2), Pow(Function('C_1')(Symbol('H', commutative=True)), Integer(2))), Function('C_1')(Symbol('H', commutative=True))), Add(Mul(Add(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Add(Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))), Add(Mul(Integer(2), Symbol('H', commutative=True), Function('C_1')(Symbol('H', commutative=True))), Function('C_1')(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} = \\mathbf{J}_M - g_{\\varepsilon}, then obtain - \\mathbf{J}_M + \\iint \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} d\\mathbf{J}_M dg_{\\varepsilon} = - \\mathbf{J}_M + \\iint (\\mathbf{J}_M - g_{\\varepsilon}) d\\mathbf{J}_M dg_{\\varepsilon}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} = \\mathbf{J}_M - g_{\\varepsilon} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} d\\mathbf{J}_M = \\int (\\mathbf{J}_M - g_{\\varepsilon}) d\\mathbf{J}_M and \\iint \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} d\\mathbf{J}_M dg_{\\varepsilon} = \\iint (\\mathbf{J}_M - g_{\\varepsilon}) d\\mathbf{J}_M dg_{\\varepsilon} and - \\mathbf{J}_M + \\iint \\operatorname{f_{\\mathbf{p}}}{(\\mathbf{J}_M,g_{\\varepsilon})} d\\mathbf{J}_M dg_{\\varepsilon} = - \\mathbf{J}_M + \\iint (\\mathbf{J}_M - g_{\\varepsilon}) d\\mathbf{J}_M dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Integral(Add(Symbol('\\\\mathbf{J}_M', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\operatorname{n_{2}}{(V_{\\mathbf{B}})}, then obtain \\sin{(\\operatorname{v_{x}}{(V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})}", "derivation": "\\operatorname{n_{2}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and V_{\\mathbf{B}} \\operatorname{n_{2}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})} and \\sin{(V_{\\mathbf{B}} \\operatorname{n_{2}}{(V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})} and \\operatorname{v_{x}}{(V_{\\mathbf{B}})} = V_{\\mathbf{B}} \\operatorname{n_{2}}{(V_{\\mathbf{B}})} and \\sin{(\\operatorname{v_{x}}{(V_{\\mathbf{B}})})} = \\sin{(V_{\\mathbf{B}} \\log{(V_{\\mathbf{B}})})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('n_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["sin", 2], "Equality(sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('n_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('n_2')(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Function('v_x')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), sin(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(C_{2})} = \\log{(C_{2})}, then obtain 2 C_{2} - \\frac{\\operatorname{v_{z}}{(C_{2})}}{\\log{(C_{2})}} = 2 C_{2} - 1", "derivation": "\\operatorname{v_{z}}{(C_{2})} = \\log{(C_{2})} and \\frac{\\operatorname{v_{z}}{(C_{2})}}{\\log{(C_{2})}} = 1 and - \\frac{\\operatorname{v_{z}}{(C_{2})}}{\\log{(C_{2})}} = -1 and C_{2} - \\frac{\\operatorname{v_{z}}{(C_{2})}}{\\log{(C_{2})}} = C_{2} - 1 and 2 C_{2} - \\frac{\\operatorname{v_{z}}{(C_{2})}}{\\log{(C_{2})}} = 2 C_{2} - 1", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["divide", 1, "log(Symbol('C_2', commutative=True))"], "Equality(Mul(Function('v_z')(Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Integer(-1))), Integer(1))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_z')(Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Integer(-1))), Integer(-1))"], [["add", 3, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Function('v_z')(Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Integer(-1)))), Add(Symbol('C_2', commutative=True), Integer(-1)))"], [["add", 4, "Symbol('C_2', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Mul(Integer(-1), Function('v_z')(Symbol('C_2', commutative=True)), Pow(log(Symbol('C_2', commutative=True)), Integer(-1)))), Add(Mul(Integer(2), Symbol('C_2', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\tilde{g}, then obtain ((\\hat{\\mathbf{r}} \\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})})^{\\tilde{g}})^{\\hat{\\mathbf{r}}} = ((\\hat{\\mathbf{r}}^{2} \\tilde{g})^{\\tilde{g}})^{\\hat{\\mathbf{r}}}", "derivation": "\\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\tilde{g} and \\hat{\\mathbf{r}} \\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}}^{2} \\tilde{g} and (\\hat{\\mathbf{r}} \\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})})^{\\tilde{g}} = (\\hat{\\mathbf{r}}^{2} \\tilde{g})^{\\tilde{g}} and ((\\hat{\\mathbf{r}} \\mathbf{J}_P{(\\tilde{g},\\hat{\\mathbf{r}})})^{\\tilde{g}})^{\\hat{\\mathbf{r}}} = ((\\hat{\\mathbf{r}}^{2} \\tilde{g})^{\\tilde{g}})^{\\hat{\\mathbf{r}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 2, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Pow(Pow(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\mathbf{J}_P')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Pow(Pow(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(2)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(v_{y},\\omega)} = \\omega + v_{y}, then obtain \\frac{\\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y}) \\operatorname{v_{x}}{(v_{y},\\omega)} d\\omega}{\\omega} = \\frac{\\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y})^{2} d\\omega}{\\omega}", "derivation": "\\operatorname{v_{x}}{(v_{y},\\omega)} = \\omega + v_{y} and (\\omega + v_{y}) \\operatorname{v_{x}}{(v_{y},\\omega)} = (\\omega + v_{y})^{2} and \\int (\\omega + v_{y}) \\operatorname{v_{x}}{(v_{y},\\omega)} d\\omega = \\int (\\omega + v_{y})^{2} d\\omega and \\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y}) \\operatorname{v_{x}}{(v_{y},\\omega)} d\\omega = \\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y})^{2} d\\omega and \\frac{\\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y}) \\operatorname{v_{x}}{(v_{y},\\omega)} d\\omega}{\\omega} = \\frac{\\operatorname{v_{x}}{(v_{y},\\omega)} + \\int (\\omega + v_{y})^{2} d\\omega}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Integer(2)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["add", 3, "Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True)))), Add(Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Integer(2)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["divide", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Mul(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Function('v_x')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Integral(Pow(Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)), Integer(2)), Tuple(Symbol('\\\\omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)}, then obtain \\frac{\\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\operatorname{A_{1}}{(\\hat{x}_0)}}{\\hat{x}_0 \\sin{(\\hat{x}_0)}} = \\frac{\\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\sin{(\\hat{x}_0)}}{\\hat{x}_0 \\sin{(\\hat{x}_0)}}", "derivation": "\\operatorname{A_{1}}{(\\hat{x}_0)} = \\sin{(\\hat{x}_0)} and \\hat{x}_0 \\operatorname{A_{1}}{(\\hat{x}_0)} = \\hat{x}_0 \\sin{(\\hat{x}_0)} and \\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\operatorname{A_{1}}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\sin{(\\hat{x}_0)} and \\frac{\\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\operatorname{A_{1}}{(\\hat{x}_0)}}{\\hat{x}_0 \\sin{(\\hat{x}_0)}} = \\frac{\\frac{d}{d \\hat{x}_0} \\hat{x}_0 \\sin{(\\hat{x}_0)}}{\\hat{x}_0 \\sin{(\\hat{x}_0)}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True)), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], [["times", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('A_1')(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{x}_0', commutative=True)), Integer(-1)), Derivative(Mul(Symbol('\\\\hat{x}_0', commutative=True), sin(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\dot{y}{(C_{d})} = \\cos{(C_{d})}, then obtain \\cos^{4}{(C_{d})} = \\frac{\\cos^{8}{(C_{d})}}{\\dot{y}^{4}{(C_{d})}}", "derivation": "\\dot{y}{(C_{d})} = \\cos{(C_{d})} and \\dot{y}{(C_{d})} \\cos{(C_{d})} = \\cos^{2}{(C_{d})} and \\cos{(C_{d})} = \\frac{\\cos^{2}{(C_{d})}}{\\dot{y}{(C_{d})}} and \\cos^{2}{(C_{d})} = \\frac{\\cos^{4}{(C_{d})}}{\\dot{y}^{2}{(C_{d})}} and \\cos^{4}{(C_{d})} = \\frac{\\cos^{8}{(C_{d})}}{\\dot{y}^{4}{(C_{d})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True)))"], [["times", 1, "cos(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), cos(Symbol('C_d', commutative=True))), Pow(cos(Symbol('C_d', commutative=True)), Integer(2)))"], [["divide", 2, "Function('\\\\dot{y}')(Symbol('C_d', commutative=True))"], "Equality(cos(Symbol('C_d', commutative=True)), Mul(Pow(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), Integer(-1)), Pow(cos(Symbol('C_d', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(cos(Symbol('C_d', commutative=True)), Integer(2)), Mul(Pow(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), Integer(-2)), Pow(cos(Symbol('C_d', commutative=True)), Integer(4))))"], [["power", 4, 2], "Equality(Pow(cos(Symbol('C_d', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\dot{y}')(Symbol('C_d', commutative=True)), Integer(-4)), Pow(cos(Symbol('C_d', commutative=True)), Integer(8))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(V_{\\mathbf{E}})} = \\sin{(\\sin{(V_{\\mathbf{E}})})} and \\operatorname{g_{\\varepsilon}}{(A,F_{c},V)} = \\frac{F_{c}^{V}}{A}, then obtain \\frac{V_{\\mathbf{E}} \\operatorname{g_{\\varepsilon}}{(A,F_{c},V)}}{\\sin{(\\sin{(V_{\\mathbf{E}})})}} = \\frac{F_{c}^{V} V_{\\mathbf{E}}}{A \\sin{(\\sin{(V_{\\mathbf{E}})})}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(V_{\\mathbf{E}})} = \\sin{(\\sin{(V_{\\mathbf{E}})})} and \\operatorname{g_{\\varepsilon}}{(A,F_{c},V)} = \\frac{F_{c}^{V}}{A} and \\frac{V_{\\mathbf{E}} \\operatorname{g_{\\varepsilon}}{(A,F_{c},V)}}{\\operatorname{V_{\\mathbf{B}}}{(V_{\\mathbf{E}})}} = \\frac{F_{c}^{V} V_{\\mathbf{E}}}{A \\operatorname{V_{\\mathbf{B}}}{(V_{\\mathbf{E}})}} and \\frac{V_{\\mathbf{E}} \\operatorname{g_{\\varepsilon}}{(A,F_{c},V)}}{\\sin{(\\sin{(V_{\\mathbf{E}})})}} = \\frac{F_{c}^{V} V_{\\mathbf{E}}}{A \\sin{(\\sin{(V_{\\mathbf{E}})})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), sin(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], ["get_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True), Symbol('F_c', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('V', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('V_{\\\\mathbf{E}}', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True), Symbol('F_c', commutative=True), Symbol('V', commutative=True))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('V', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('g_{\\\\varepsilon}')(Symbol('A', commutative=True), Symbol('F_c', commutative=True), Symbol('V', commutative=True)), Pow(sin(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(-1))), Mul(Pow(Symbol('A', commutative=True), Integer(-1)), Pow(Symbol('F_c', commutative=True), Symbol('V', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True), Pow(sin(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(f^{\\prime})} = \\int \\log{(f^{\\prime})} df^{\\prime}, then derive \\operatorname{C_{1}}{(f^{\\prime})} = G + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}, then derive G + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} = \\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}, then obtain - G - f^{\\prime} \\operatorname{C_{1}}{(f^{\\prime})} - f^{\\prime} \\log{(f^{\\prime})} + f^{\\prime} = - G - f^{\\prime} (\\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) - f^{\\prime} \\log{(f^{\\prime})} + f^{\\prime}", "derivation": "\\operatorname{C_{1}}{(f^{\\prime})} = \\int \\log{(f^{\\prime})} df^{\\prime} and \\operatorname{C_{1}}{(f^{\\prime})} = G + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} and G + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} = \\int \\log{(f^{\\prime})} df^{\\prime} and G + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} = \\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} and \\operatorname{C_{1}}{(f^{\\prime})} = \\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime} and - f^{\\prime} \\operatorname{C_{1}}{(f^{\\prime})} = - f^{\\prime} (\\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) and - G - f^{\\prime} \\operatorname{C_{1}}{(f^{\\prime})} - f^{\\prime} \\log{(f^{\\prime})} + f^{\\prime} = - G - f^{\\prime} (\\pi + f^{\\prime} \\log{(f^{\\prime})} - f^{\\prime}) - f^{\\prime} \\log{(f^{\\prime})} + f^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Integral(log(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('C_1')(Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))"], [["times", 5, "Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 6, "Add(Symbol('G', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Function('C_1')(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True), log(Symbol('f^{\\\\prime}', commutative=True))), Symbol('f^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(s,x^\\prime)} = s x^\\prime, then obtain \\int \\frac{\\partial}{\\partial s} (\\bar{\\h}{(s,x^\\prime)} + \\bar{\\h}^{s}{(s,x^\\prime)}) ds = \\int \\frac{\\partial}{\\partial s} (s x^\\prime + \\bar{\\h}^{s}{(s,x^\\prime)}) ds", "derivation": "\\bar{\\h}{(s,x^\\prime)} = s x^\\prime and \\bar{\\h}^{s}{(s,x^\\prime)} = (s x^\\prime)^{s} and (s x^\\prime)^{s} + \\bar{\\h}{(s,x^\\prime)} = s x^\\prime + (s x^\\prime)^{s} and \\bar{\\h}{(s,x^\\prime)} + \\bar{\\h}^{s}{(s,x^\\prime)} = s x^\\prime + \\bar{\\h}^{s}{(s,x^\\prime)} and \\frac{\\partial}{\\partial s} (\\bar{\\h}{(s,x^\\prime)} + \\bar{\\h}^{s}{(s,x^\\prime)}) = \\frac{\\partial}{\\partial s} (s x^\\prime + \\bar{\\h}^{s}{(s,x^\\prime)}) and \\int \\frac{\\partial}{\\partial s} (\\bar{\\h}{(s,x^\\prime)} + \\bar{\\h}^{s}{(s,x^\\prime)}) ds = \\int \\frac{\\partial}{\\partial s} (s x^\\prime + \\bar{\\h}^{s}{(s,x^\\prime)}) ds", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Pow(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)))"], [["add", 1, "Pow(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True)), Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True))), Add(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))), Add(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["integrate", 5, "Symbol('s', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))), Integral(Derivative(Add(Mul(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\hbar')(Symbol('s', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Tuple(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(J,h)} = \\sin{(J + h)}, then obtain \\hat{H}_l{(J,h)} - \\sin{(J + h)} = 0", "derivation": "\\hat{H}_l{(J,h)} = \\sin{(J + h)} and J + h + \\hat{H}_l{(J,h)} = J + h + \\sin{(J + h)} and \\frac{\\hat{H}_l{(J,h)}}{J + h + \\hat{H}_l{(J,h)}} = \\frac{\\sin{(J + h)}}{J + h + \\hat{H}_l{(J,h)}} and \\frac{(J + h + \\sin{(J + h)}) \\hat{H}_l{(J,h)}}{J + h + \\hat{H}_l{(J,h)}} = \\frac{(J + h + \\sin{(J + h)}) \\sin{(J + h)}}{J + h + \\hat{H}_l{(J,h)}} and \\frac{(J + h + \\sin{(J + h)}) \\hat{H}_l{(J,h)}}{J + h + \\hat{H}_l{(J,h)}} - \\frac{(J + h + \\sin{(J + h)}) \\sin{(J + h)}}{J + h + \\hat{H}_l{(J,h)}} = 0 and \\hat{H}_l{(J,h)} - \\sin{(J + h)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True)), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Add(Symbol('J', commutative=True), Symbol('h', commutative=True))"], "Equality(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))))"], [["divide", 1, "Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))))"], [["times", 3, "Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))))"], [["minus", 4, "Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True))))"], "Equality(Add(Mul(Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('J', commutative=True), Symbol('h', commutative=True), Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True))), Integer(-1)), Add(Symbol('J', commutative=True), Symbol('h', commutative=True), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True)))), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('J', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('J', commutative=True), Symbol('h', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P \\mu)}, then derive \\frac{\\partial}{\\partial \\mu} \\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(\\mathbf{J}_P \\mu)}, then obtain - \\mathbf{J}_P \\sin{(\\mathbf{J}_P \\mu)} + \\mu = \\mu + \\frac{\\partial}{\\partial \\mu} \\cos{(\\mathbf{J}_P \\mu)}", "derivation": "\\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = \\cos{(\\mathbf{J}_P \\mu)} and \\frac{\\partial}{\\partial \\mu} \\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mu} \\cos{(\\mathbf{J}_P \\mu)} and \\mu + \\frac{\\partial}{\\partial \\mu} \\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = \\mu + \\frac{\\partial}{\\partial \\mu} \\cos{(\\mathbf{J}_P \\mu)} and \\frac{\\partial}{\\partial \\mu} \\operatorname{a^{\\dagger}}{(\\mu,\\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(\\mathbf{J}_P \\mu)} and - \\mathbf{J}_P \\sin{(\\mathbf{J}_P \\mu)} + \\mu = \\mu + \\frac{\\partial}{\\partial \\mu} \\cos{(\\mathbf{J}_P \\mu)}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Add(Symbol('\\\\mu', commutative=True), Derivative(cos(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Derivative(cos(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\phi,Z)} = Z + \\phi and \\hat{\\mathbf{r}}{(\\phi,Z)} = Z + \\phi, then obtain ((Z + \\Psi_{nl}{(\\phi,Z)}) \\Psi_{nl}{(\\phi,Z)})^{\\phi} = ((Z + \\Psi_{nl}{(\\phi,Z)}) \\hat{\\mathbf{r}}{(\\phi,Z)})^{\\phi}", "derivation": "\\Psi_{nl}{(\\phi,Z)} = Z + \\phi and Z + \\Psi_{nl}{(\\phi,Z)} = 2 Z + \\phi and \\hat{\\mathbf{r}}{(\\phi,Z)} = Z + \\phi and \\Psi_{nl}{(\\phi,Z)} = \\hat{\\mathbf{r}}{(\\phi,Z)} and (2 Z + \\phi) \\Psi_{nl}{(\\phi,Z)} = (2 Z + \\phi) \\hat{\\mathbf{r}}{(\\phi,Z)} and ((2 Z + \\phi) \\Psi_{nl}{(\\phi,Z)})^{\\phi} = ((2 Z + \\phi) \\hat{\\mathbf{r}}{(\\phi,Z)})^{\\phi} and ((Z + \\Psi_{nl}{(\\phi,Z)}) \\Psi_{nl}{(\\phi,Z)})^{\\phi} = ((Z + \\Psi_{nl}{(\\phi,Z)}) \\hat{\\mathbf{r}}{(\\phi,Z)})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True)))"], [["times", 4, "Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Mul(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))))"], [["power", 5, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Mul(Add(Mul(Integer(2), Symbol('Z', commutative=True)), Symbol('\\\\phi', commutative=True)), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\phi', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Mul(Add(Symbol('Z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\phi', commutative=True)), Pow(Mul(Add(Symbol('Z', commutative=True), Function('\\\\Psi_{nl}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True), Symbol('Z', commutative=True))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\rho{(M)} = \\sin{(M)}, then obtain \\frac{d^{3}}{d M^{3}} \\int \\rho{(M)} dM = \\frac{d^{3}}{d M^{3}} \\int \\sin{(M)} dM", "derivation": "\\rho{(M)} = \\sin{(M)} and \\int \\rho{(M)} dM = \\int \\sin{(M)} dM and \\frac{d}{d M} \\int \\rho{(M)} dM = \\frac{d}{d M} \\int \\sin{(M)} dM and \\frac{d^{2}}{d M^{2}} \\int \\rho{(M)} dM = \\frac{d^{2}}{d M^{2}} \\int \\sin{(M)} dM and \\frac{d^{3}}{d M^{3}} \\int \\rho{(M)} dM = \\frac{d^{3}}{d M^{3}} \\int \\sin{(M)} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('M', commutative=True)), sin(Symbol('M', commutative=True)))"], [["integrate", 1, "Symbol('M', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(2))), Derivative(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(2))))"], [["differentiate", 4, "Symbol('M', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\rho')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(3))), Derivative(Integral(sin(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(3))))"]]}, {"prompt": "Given J{(v_{t})} = \\sin{(v_{t})}, then obtain \\frac{d}{d v_{t}} (J{(v_{t})} \\sin{(v_{t})} + J{(v_{t})}) = \\frac{d}{d v_{t}} (J{(v_{t})} \\sin{(v_{t})} + \\sin{(v_{t})})", "derivation": "J{(v_{t})} = \\sin{(v_{t})} and J^{2}{(v_{t})} = J{(v_{t})} \\sin{(v_{t})} and J^{2}{(v_{t})} + J{(v_{t})} = J^{2}{(v_{t})} + \\sin{(v_{t})} and \\frac{d}{d v_{t}} (J^{2}{(v_{t})} + J{(v_{t})}) = \\frac{d}{d v_{t}} (J^{2}{(v_{t})} + \\sin{(v_{t})}) and \\frac{d}{d v_{t}} (J{(v_{t})} \\sin{(v_{t})} + J{(v_{t})}) = \\frac{d}{d v_{t}} (J{(v_{t})} \\sin{(v_{t})} + \\sin{(v_{t})})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True)))"], [["times", 1, "Function('J')(Symbol('v_t', commutative=True))"], "Equality(Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2)), Mul(Function('J')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True))))"], [["add", 1, "Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2))"], "Equality(Add(Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2)), Function('J')(Symbol('v_t', commutative=True))), Add(Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2)), sin(Symbol('v_t', commutative=True))))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2)), Function('J')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Pow(Function('J')(Symbol('v_t', commutative=True)), Integer(2)), sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Mul(Function('J')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True))), Function('J')(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Add(Mul(Function('J')(Symbol('v_t', commutative=True)), sin(Symbol('v_t', commutative=True))), sin(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(i,\\mathbf{A})} = - \\mathbf{A} + i and \\rho{(\\mathbf{A})} = - \\mathbf{A}, then obtain (- \\mathbf{A} (- \\mathbf{A} + i))^{\\mathbf{A}} = (- \\mathbf{A} (i + \\rho{(\\mathbf{A})}))^{\\mathbf{A}}", "derivation": "C{(i,\\mathbf{A})} = - \\mathbf{A} + i and \\rho{(\\mathbf{A})} = - \\mathbf{A} and C{(i,\\mathbf{A})} = i + \\rho{(\\mathbf{A})} and - \\mathbf{A} + i = i + \\rho{(\\mathbf{A})} and - \\mathbf{A} (- \\mathbf{A} + i) = - \\mathbf{A} (i + \\rho{(\\mathbf{A})}) and (- \\mathbf{A} (- \\mathbf{A} + i))^{\\mathbf{A}} = (- \\mathbf{A} (i + \\rho{(\\mathbf{A})}))^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('i', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('i', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C')(Symbol('i', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('i', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('i', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["power", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('i', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('i', commutative=True), Function('\\\\rho')(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(v_{1})} = \\sin{(v_{1})}, then obtain (\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} - \\sin{(v_{1})} = (\\sin^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} - \\sin{(v_{1})}", "derivation": "\\operatorname{L_{\\varepsilon}}{(v_{1})} = \\sin{(v_{1})} and \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} = \\sin^{v_{1}}{(v_{1})} and (\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})})^{v_{1}} = (\\sin^{v_{1}}{(v_{1})})^{v_{1}} and (\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} = (\\sin^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} and (\\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} - \\sin{(v_{1})} = (\\sin^{v_{1}}{(v_{1})})^{v_{1}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(v_{1})} - \\sin{(v_{1})}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), sin(Symbol('v_1', commutative=True)))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["power", 2, "Symbol('v_1', commutative=True)"], "Equality(Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)))"], [["divide", 3, "Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))"], "Equality(Mul(Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Pow(Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))))"], [["minus", 4, "sin(Symbol('v_1', commutative=True))"], "Equality(Add(Mul(Pow(Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), sin(Symbol('v_1', commutative=True)))), Add(Mul(Pow(Pow(sin(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('v_1', commutative=True)), Symbol('v_1', commutative=True))), Mul(Integer(-1), sin(Symbol('v_1', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\sigma_x)} = e^{\\sigma_x}, then obtain \\frac{d}{d \\sigma_x} (- \\sigma_x + \\frac{\\psi{(\\sigma_x)}}{\\sigma_x^{4}}) = \\frac{d}{d \\sigma_x} (- \\sigma_x + \\frac{e^{\\sigma_x}}{\\sigma_x^{4}})", "derivation": "\\psi{(\\sigma_x)} = e^{\\sigma_x} and \\frac{\\psi{(\\sigma_x)}}{\\sigma_x} = \\frac{e^{\\sigma_x}}{\\sigma_x} and \\frac{\\psi{(\\sigma_x)}}{\\sigma_x^{2}} = \\frac{e^{\\sigma_x}}{\\sigma_x^{2}} and \\frac{\\psi{(\\sigma_x)}}{\\sigma_x^{4}} = \\frac{e^{\\sigma_x}}{\\sigma_x^{4}} and - \\sigma_x + \\frac{\\psi{(\\sigma_x)}}{\\sigma_x^{4}} = - \\sigma_x + \\frac{e^{\\sigma_x}}{\\sigma_x^{4}} and \\frac{d}{d \\sigma_x} (- \\sigma_x + \\frac{\\psi{(\\sigma_x)}}{\\sigma_x^{4}}) = \\frac{d}{d \\sigma_x} (- \\sigma_x + \\frac{e^{\\sigma_x}}{\\sigma_x^{4}})", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True)), exp(Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["divide", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-2)), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-2)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["times", 3, "Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-2))"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), exp(Symbol('\\\\sigma_x', commutative=True))))"], [["minus", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), exp(Symbol('\\\\sigma_x', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), Function('\\\\psi')(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-4)), exp(Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(J)} = \\cos{(J)}, then obtain \\int (\\cos^{2}{(\\hat{X}{(J)})} - \\cos{(\\hat{X}{(J)})} \\cos{(\\cos{(J)})}) dJ = \\int 0 dJ", "derivation": "\\hat{X}{(J)} = \\cos{(J)} and \\cos{(\\hat{X}{(J)})} = \\cos{(\\cos{(J)})} and \\cos^{2}{(\\hat{X}{(J)})} = \\cos{(\\hat{X}{(J)})} \\cos{(\\cos{(J)})} and \\cos^{2}{(\\hat{X}{(J)})} - \\cos{(\\hat{X}{(J)})} \\cos{(\\cos{(J)})} = 0 and \\int (\\cos^{2}{(\\hat{X}{(J)})} - \\cos{(\\hat{X}{(J)})} \\cos{(\\cos{(J)})}) dJ = \\int 0 dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), cos(cos(Symbol('J', commutative=True))))"], [["times", 2, "cos(Function('\\\\hat{X}')(Symbol('J', commutative=True)))"], "Equality(Pow(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), Integer(2)), Mul(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), cos(cos(Symbol('J', commutative=True)))))"], [["minus", 3, "Mul(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), cos(cos(Symbol('J', commutative=True))))"], "Equality(Add(Pow(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), Integer(2)), Mul(Integer(-1), cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), cos(cos(Symbol('J', commutative=True))))), Integer(0))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Add(Pow(cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), Integer(2)), Mul(Integer(-1), cos(Function('\\\\hat{X}')(Symbol('J', commutative=True))), cos(cos(Symbol('J', commutative=True))))), Tuple(Symbol('J', commutative=True))), Integral(Integer(0), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(v)} = \\cos{(v)}, then obtain \\frac{\\mathbb{I}^{8}{(v)}}{\\cos^{6}{(v)}} = \\frac{\\mathbb{I}^{4}{(v)}}{\\cos^{2}{(v)}}", "derivation": "\\mathbb{I}{(v)} = \\cos{(v)} and \\mathbb{I}^{2}{(v)} = \\mathbb{I}{(v)} \\cos{(v)} and \\frac{\\mathbb{I}^{2}{(v)}}{\\cos{(v)}} = \\mathbb{I}{(v)} and \\frac{\\mathbb{I}^{4}{(v)}}{\\cos^{2}{(v)}} = \\mathbb{I}^{2}{(v)} and \\frac{\\mathbb{I}^{8}{(v)}}{\\cos^{6}{(v)}} = \\frac{\\mathbb{I}^{4}{(v)}}{\\cos^{2}{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["times", 1, "Function('\\\\mathbb{I}')(Symbol('v', commutative=True))"], "Equality(Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(2)), Mul(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True))))"], [["divide", 2, "cos(Symbol('v', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(2)), Pow(cos(Symbol('v', commutative=True)), Integer(-1))), Function('\\\\mathbb{I}')(Symbol('v', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(4)), Pow(cos(Symbol('v', commutative=True)), Integer(-2))), Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(8)), Pow(cos(Symbol('v', commutative=True)), Integer(-6))), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('v', commutative=True)), Integer(4)), Pow(cos(Symbol('v', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\rho_{b}{(\\phi,\\mathbf{v})} = \\phi + \\log{(\\mathbf{v})}, then obtain \\rho_{b}{(\\phi,\\mathbf{v})} + \\int \\rho_{b}{(\\phi,\\mathbf{v})} d\\phi = \\mu + \\frac{\\phi^{2}}{2} + \\phi \\log{(\\mathbf{v})} + \\rho_{b}{(\\phi,\\mathbf{v})}", "derivation": "\\rho_{b}{(\\phi,\\mathbf{v})} = \\phi + \\log{(\\mathbf{v})} and \\int \\rho_{b}{(\\phi,\\mathbf{v})} d\\phi = \\int (\\phi + \\log{(\\mathbf{v})}) d\\phi and \\phi + \\log{(\\mathbf{v})} + \\int \\rho_{b}{(\\phi,\\mathbf{v})} d\\phi = \\phi + \\log{(\\mathbf{v})} + \\int (\\phi + \\log{(\\mathbf{v})}) d\\phi and \\rho_{b}{(\\phi,\\mathbf{v})} + \\int \\rho_{b}{(\\phi,\\mathbf{v})} d\\phi = \\rho_{b}{(\\phi,\\mathbf{v})} + \\int (\\phi + \\log{(\\mathbf{v})}) d\\phi and \\rho_{b}{(\\phi,\\mathbf{v})} + \\int \\rho_{b}{(\\phi,\\mathbf{v})} d\\phi = \\mu + \\frac{\\phi^{2}}{2} + \\phi \\log{(\\mathbf{v})} + \\rho_{b}{(\\phi,\\mathbf{v})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["add", 2, "Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True)))"], "Equality(Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True)), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Add(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Integral(Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Symbol('\\\\mu', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\mathbf{v}', commutative=True))), Function('\\\\rho_b')(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))))"]]}, {"prompt": "Given C{(v_{y})} = \\log{(v_{y})}, then derive \\frac{d}{d v_{y}} C{(v_{y})} = \\frac{1}{v_{y}}, then obtain ((\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}} - 1)^{v_{y}} = ((\\frac{1}{v_{y}})^{v_{y}} - 1)^{v_{y}}", "derivation": "C{(v_{y})} = \\log{(v_{y})} and \\frac{d}{d v_{y}} C{(v_{y})} = \\frac{d}{d v_{y}} \\log{(v_{y})} and \\frac{d}{d v_{y}} C{(v_{y})} = \\frac{1}{v_{y}} and (\\frac{d}{d v_{y}} C{(v_{y})})^{v_{y}} = (\\frac{1}{v_{y}})^{v_{y}} and (\\frac{d}{d v_{y}} C{(v_{y})})^{v_{y}} - 1 = (\\frac{1}{v_{y}})^{v_{y}} - 1 and (\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}} - 1 = (\\frac{1}{v_{y}})^{v_{y}} - 1 and ((\\frac{d}{d v_{y}} \\log{(v_{y})})^{v_{y}} - 1)^{v_{y}} = ((\\frac{1}{v_{y}})^{v_{y}} - 1)^{v_{y}}", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Pow(Symbol('v_y', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Derivative(Function('C')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)))"], [["minus", 4, 1], "Equality(Add(Pow(Derivative(Function('C')(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Integer(-1)), Add(Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Pow(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Integer(-1)), Add(Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)), Integer(-1)))"], [["power", 6, "Symbol('v_y', commutative=True)"], "Equality(Pow(Add(Pow(Derivative(log(Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Integer(-1)), Symbol('v_y', commutative=True)), Pow(Add(Pow(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('v_y', commutative=True)), Integer(-1)), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(c_{0},a)} = \\frac{\\log{(c_{0})}}{a}, then derive \\frac{\\partial}{\\partial c_{0}} \\operatorname{E_{x}}{(c_{0},a)} = \\frac{1}{a c_{0}}, then obtain \\frac{\\partial}{\\partial c_{0}} \\frac{\\log{(c_{0})}}{a} = \\frac{1}{a c_{0}}", "derivation": "\\operatorname{E_{x}}{(c_{0},a)} = \\frac{\\log{(c_{0})}}{a} and \\operatorname{E_{x}}{(c_{0},a)} + 1 = 1 + \\frac{\\log{(c_{0})}}{a} and \\frac{\\partial}{\\partial c_{0}} (\\operatorname{E_{x}}{(c_{0},a)} + 1) = \\frac{\\partial}{\\partial c_{0}} (1 + \\frac{\\log{(c_{0})}}{a}) and \\frac{\\partial}{\\partial c_{0}} \\operatorname{E_{x}}{(c_{0},a)} = \\frac{1}{a c_{0}} and \\frac{\\partial}{\\partial c_{0}} \\frac{\\log{(c_{0})}}{a} = \\frac{1}{a c_{0}}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('c_0', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('c_0', commutative=True)))))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Function('E_x')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Integer(1)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('E_x')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Symbol('c_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})} = A_{z} \\hat{x} and \\dot{y}{(v)} = \\cos{(v)}, then obtain - \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})} \\cos{(v)}}{A_{z} \\hat{x}} = - \\cos{(v)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})} = A_{z} \\hat{x} and \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})}}{A_{z} \\hat{x}} = 1 and \\dot{y}{(v)} = \\cos{(v)} and \\frac{\\dot{y}{(v)} \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})}}{A_{z} \\hat{x}} = \\dot{y}{(v)} and - \\frac{\\dot{y}{(v)} \\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})}}{A_{z} \\hat{x}} = - \\dot{y}{(v)} and - \\frac{\\operatorname{f_{\\mathbf{v}}}{(\\hat{x},A_{z})} \\cos{(v)}}{A_{z} \\hat{x}} = - \\cos{(v)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 1, "Mul(Symbol('A_z', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_z', commutative=True))), Integer(1))"], ["get_premise", "Equality(Function('\\\\dot{y}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["times", 2, "Function('\\\\dot{y}')(Symbol('v', commutative=True))"], "Equality(Mul(Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('v', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_z', commutative=True))), Function('\\\\dot{y}')(Symbol('v', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('\\\\dot{y}')(Symbol('v', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('v', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('A_z', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\hat{x}', commutative=True), Symbol('A_z', commutative=True)), cos(Symbol('v', commutative=True))), Mul(Integer(-1), cos(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(\\theta_1)} = e^{\\theta_1}, then obtain \\frac{d}{d \\theta_1} \\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}} = \\frac{d}{d \\theta_1} \\frac{(\\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}})^{\\theta_1} e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}}", "derivation": "\\hat{p}_0{(\\theta_1)} = e^{\\theta_1} and 1 = \\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}} and 1 = (\\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}})^{\\theta_1} and \\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}} = \\frac{(\\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}})^{\\theta_1} e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}} and \\frac{d}{d \\theta_1} \\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}} = \\frac{d}{d \\theta_1} \\frac{(\\frac{e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}})^{\\theta_1} e^{\\theta_1}}{\\hat{p}_0{(\\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["divide", 1, "Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"], [["times", 3, "Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), exp(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}}, then derive \\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})} = \\log{(e^{J_{\\varepsilon}})}, then obtain \\frac{\\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})}^{J_{\\varepsilon}}}{\\log{(\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})}} = \\frac{\\log{(e^{J_{\\varepsilon}})}^{J_{\\varepsilon}}}{\\log{(\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})}}", "derivation": "\\operatorname{P_{e}}{(J_{\\varepsilon})} = \\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}} and \\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})} = \\log{(\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})} and \\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})} = \\log{(e^{J_{\\varepsilon}})} and \\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})}^{J_{\\varepsilon}} = \\log{(e^{J_{\\varepsilon}})}^{J_{\\varepsilon}} and \\frac{\\log{(\\operatorname{P_{e}}{(J_{\\varepsilon})})}^{J_{\\varepsilon}}}{\\log{(\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})}} = \\frac{\\log{(e^{J_{\\varepsilon}})}^{J_{\\varepsilon}}}{\\log{(\\frac{d}{d J_{\\varepsilon}} e^{J_{\\varepsilon}})}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('J_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('P_e')(Symbol('J_{\\\\varepsilon}', commutative=True))), log(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(log(Function('P_e')(Symbol('J_{\\\\varepsilon}', commutative=True))), log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["power", 3, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(log(Function('P_e')(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 4, "log(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], "Equality(Mul(Pow(log(Function('P_e')(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(log(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(-1))), Mul(Pow(log(exp(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(log(Derivative(exp(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F},M)} = - M + \\mathbf{F}, then obtain - (- M + \\mathbf{F})^{\\mathbf{F}} + \\iint \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbf{F}}{(\\mathbf{F},M)} dM d\\mathbf{F} = - (- M + \\mathbf{F})^{\\mathbf{F}} + \\iint (- M + \\mathbf{F})^{\\mathbf{F}} dM d\\mathbf{F}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{F},M)} = - M + \\mathbf{F} and \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbf{F}}{(\\mathbf{F},M)} = (- M + \\mathbf{F})^{\\mathbf{F}} and \\int \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbf{F}}{(\\mathbf{F},M)} dM = \\int (- M + \\mathbf{F})^{\\mathbf{F}} dM and \\iint \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbf{F}}{(\\mathbf{F},M)} dM d\\mathbf{F} = \\iint (- M + \\mathbf{F})^{\\mathbf{F}} dM d\\mathbf{F} and - (- M + \\mathbf{F})^{\\mathbf{F}} + \\iint \\operatorname{g^{\\prime}_{\\varepsilon}}^{\\mathbf{F}}{(\\mathbf{F},M)} dM d\\mathbf{F} = - (- M + \\mathbf{F})^{\\mathbf{F}} + \\iint (- M + \\mathbf{F})^{\\mathbf{F}} dM d\\mathbf{F}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 4, "Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Pow(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('M', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} = (\\mathbf{s} - \\psi)^{b} and \\mathbb{I}{(\\psi,b,\\mathbf{s})} = \\frac{1}{\\int (\\mathbf{s} - \\psi)^{b} db}, then obtain \\frac{1}{\\int \\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} db} = \\frac{1}{\\int (\\mathbf{s} - \\psi)^{b} db}", "derivation": "\\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} = (\\mathbf{s} - \\psi)^{b} and \\int \\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} db = \\int (\\mathbf{s} - \\psi)^{b} db and \\mathbb{I}{(\\psi,b,\\mathbf{s})} = \\frac{1}{\\int (\\mathbf{s} - \\psi)^{b} db} and \\mathbb{I}{(\\psi,b,\\mathbf{s})} = \\frac{1}{\\int \\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} db} and \\frac{1}{\\int \\operatorname{C_{2}}{(\\psi,b,\\mathbf{s})} db} = \\frac{1}{\\int (\\mathbf{s} - \\psi)^{b} db}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Integral(Function('C_2')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Integral(Function('C_2')(Symbol('\\\\psi', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1)), Pow(Integral(Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Symbol('\\\\psi', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given J{(P_{e})} = e^{P_{e}}, then obtain ((- J^{2}{(P_{e})} + J{(P_{e})} e^{P_{e}} + e^{P_{e}}) J{(P_{e})})^{P_{e}} = (J{(P_{e})} e^{P_{e}})^{P_{e}}", "derivation": "J{(P_{e})} = e^{P_{e}} and J^{2}{(P_{e})} = J{(P_{e})} e^{P_{e}} and J^{2}{(P_{e})} + e^{P_{e}} = J{(P_{e})} e^{P_{e}} + e^{P_{e}} and e^{P_{e}} = - J^{2}{(P_{e})} + J{(P_{e})} e^{P_{e}} + e^{P_{e}} and (J^{2}{(P_{e})})^{P_{e}} = (J{(P_{e})} e^{P_{e}})^{P_{e}} and J^{2}{(P_{e})} = (- J^{2}{(P_{e})} + J{(P_{e})} e^{P_{e}} + e^{P_{e}}) J{(P_{e})} and ((- J^{2}{(P_{e})} + J{(P_{e})} e^{P_{e}} + e^{P_{e}}) J{(P_{e})})^{P_{e}} = (J{(P_{e})} e^{P_{e}})^{P_{e}}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True)))"], [["times", 1, "Function('J')(Symbol('P_e', commutative=True))"], "Equality(Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))))"], [["add", 2, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2)), exp(Symbol('P_e', commutative=True))), Add(Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))))"], [["minus", 3, "Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2))"], "Equality(exp(Symbol('P_e', commutative=True)), Add(Mul(Integer(-1), Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2))), Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2)), Symbol('P_e', commutative=True)), Pow(Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2))), Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))), Function('J')(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Mul(Add(Mul(Integer(-1), Pow(Function('J')(Symbol('P_e', commutative=True)), Integer(2))), Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True))), Function('J')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Pow(Mul(Function('J')(Symbol('P_e', commutative=True)), exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"]]}, {"prompt": "Given z{(n_{1})} = \\cos{(e^{n_{1}})} and \\tilde{g}^*{(x,g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} - x, then obtain \\tilde{g}^*{(x,g^{\\prime}_{\\varepsilon})} + 2 z{(n_{1})} = g^{\\prime}_{\\varepsilon} - x + 2 z{(n_{1})}", "derivation": "z{(n_{1})} = \\cos{(e^{n_{1}})} and \\tilde{g}^*{(x,g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} - x and \\tilde{g}^*{(x,g^{\\prime}_{\\varepsilon})} + z{(n_{1})} + \\cos{(e^{n_{1}})} = g^{\\prime}_{\\varepsilon} - x + z{(n_{1})} + \\cos{(e^{n_{1}})} and \\tilde{g}^*{(x,g^{\\prime}_{\\varepsilon})} + 2 z{(n_{1})} = g^{\\prime}_{\\varepsilon} - x + 2 z{(n_{1})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["add", 2, "Add(Function('z')(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True))))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Function('z')(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True)))), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Function('z')(Symbol('n_1', commutative=True)), cos(exp(Symbol('n_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('x', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Function('z')(Symbol('n_1', commutative=True)))), Add(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(2), Function('z')(Symbol('n_1', commutative=True)))))"]]}, {"prompt": "Given g{(\\mathbf{J})} = \\cos{(\\mathbf{J})}, then derive \\frac{d}{d \\mathbf{J}} g{(\\mathbf{J})} = - \\sin{(\\mathbf{J})}, then obtain \\frac{\\sigma_x + \\cos{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\int - \\sin{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}}", "derivation": "g{(\\mathbf{J})} = \\cos{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} g{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} g{(\\mathbf{J})} = - \\sin{(\\mathbf{J})} and \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} = - \\sin{(\\mathbf{J})} and \\int \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} d\\mathbf{J} = \\int - \\sin{(\\mathbf{J})} d\\mathbf{J} and \\frac{\\int \\frac{d}{d \\mathbf{J}} \\cos{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}} = \\frac{\\int - \\sin{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}} and \\frac{\\sigma_x + \\cos{(\\mathbf{J})}}{\\mathbf{J}} = \\frac{\\int - \\sin{(\\mathbf{J})} d\\mathbf{J}}{\\mathbf{J}}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\mathbf{J}', commutative=True)), cos(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["divide", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Integral(Derivative(cos(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Add(Symbol('\\\\sigma_x', commutative=True), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain \\int \\frac{2 \\operatorname{n_{1}}{(\\mathbf{S})}}{\\operatorname{n_{1}}{(\\mathbf{S})} + e^{\\mathbf{S}}} d\\mathbf{S} = \\mathbf{S} + m", "derivation": "\\operatorname{n_{1}}{(\\mathbf{S})} = e^{\\mathbf{S}} and 2 \\operatorname{n_{1}}{(\\mathbf{S})} = \\operatorname{n_{1}}{(\\mathbf{S})} + e^{\\mathbf{S}} and \\frac{2 \\operatorname{n_{1}}{(\\mathbf{S})}}{\\operatorname{n_{1}}{(\\mathbf{S})} + e^{\\mathbf{S}}} = 1 and \\int \\frac{2 \\operatorname{n_{1}}{(\\mathbf{S})}}{\\operatorname{n_{1}}{(\\mathbf{S})} + e^{\\mathbf{S}}} d\\mathbf{S} = \\int 1 d\\mathbf{S} and \\int \\frac{2 \\operatorname{n_{1}}{(\\mathbf{S})}}{\\operatorname{n_{1}}{(\\mathbf{S})} + e^{\\mathbf{S}}} d\\mathbf{S} = \\mathbf{S} + m", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["add", 1, "Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 2, "Add(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True))), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Integer(2), Pow(Add(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Integer(2), Pow(Add(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1)), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('m', commutative=True)))"]]}, {"prompt": "Given A{(I,c)} = I + c, then obtain \\frac{2 A{(I,c)}}{2 I + 2 c} = 1", "derivation": "A{(I,c)} = I + c and 2 A{(I,c)} = I + c + A{(I,c)} and \\frac{2 A{(I,c)}}{I + c + A{(I,c)}} = 1 and \\frac{2 (I + c)}{2 I + 2 c} = 1 and \\frac{2 A{(I,c)}}{2 I + 2 c} = 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True)), Add(Symbol('I', commutative=True), Symbol('c', commutative=True)))"], [["add", 1, "Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Add(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))))"], [["divide", 2, "Add(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Symbol('I', commutative=True), Symbol('c', commutative=True), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Integer(-1)), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Add(Symbol('I', commutative=True), Symbol('c', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(2), Symbol('c', commutative=True))), Integer(-1)), Function('A')(Symbol('I', commutative=True), Symbol('c', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\bar{\\h}{(\\hat{H}_l)} = e^{\\sin{(\\hat{H}_l)}} and \\Psi_{nl}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain e^{\\Psi_{nl}{(\\hat{H}_l)} e^{\\Psi_{nl}{(\\hat{H}_l)}}} = e^{e^{\\Psi_{nl}{(\\hat{H}_l)}} \\sin{(\\hat{H}_l)}}", "derivation": "\\bar{\\h}{(\\hat{H}_l)} = e^{\\sin{(\\hat{H}_l)}} and \\Psi_{nl}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\bar{\\h}{(\\hat{H}_l)} = e^{\\Psi_{nl}{(\\hat{H}_l)}} and \\Psi_{nl}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}} = e^{\\sin{(\\hat{H}_l)}} \\sin{(\\hat{H}_l)} and e^{\\Psi_{nl}{(\\hat{H}_l)} e^{\\sin{(\\hat{H}_l)}}} = e^{e^{\\sin{(\\hat{H}_l)}} \\sin{(\\hat{H}_l)}} and e^{\\Psi_{nl}{(\\hat{H}_l)}} = e^{\\sin{(\\hat{H}_l)}} and e^{\\Psi_{nl}{(\\hat{H}_l)} e^{\\Psi_{nl}{(\\hat{H}_l)}}} = e^{e^{\\Psi_{nl}{(\\hat{H}_l)}} \\sin{(\\hat{H}_l)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hbar')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True))))"], [["times", 2, "exp(sin(Symbol('\\\\hat{H}_l', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True)))), Mul(exp(sin(Symbol('\\\\hat{H}_l', commutative=True))), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["exp", 4], "Equality(exp(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True)), exp(sin(Symbol('\\\\hat{H}_l', commutative=True))))), exp(Mul(exp(sin(Symbol('\\\\hat{H}_l', commutative=True))), sin(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(exp(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True))), exp(sin(Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(exp(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True)), exp(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True))))), exp(Mul(exp(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{H}_l', commutative=True))), sin(Symbol('\\\\hat{H}_l', commutative=True)))))"]]}, {"prompt": "Given v{(C,W)} = C W and \\Psi{(C,W)} = C W, then obtain \\int v{(C,W)} dC = \\int \\Psi{(C,W)} dC", "derivation": "v{(C,W)} = C W and \\Psi{(C,W)} = C W and v{(C,W)} = \\Psi{(C,W)} and \\int v{(C,W)} dC = \\int \\Psi{(C,W)} dC", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('C', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('W', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('v')(Symbol('C', commutative=True), Symbol('W', commutative=True)), Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('W', commutative=True)))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Function('v')(Symbol('C', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Function('\\\\Psi')(Symbol('C', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\mathbf{J},G)} = \\cos{(G + \\mathbf{J})}, then obtain \\cos{(G + \\mathbf{J})} \\int G d\\mathbf{J} = \\cos{(G + \\mathbf{J})} \\int (G + \\operatorname{F_{c}}{(\\mathbf{J},G)} - \\cos{(G + \\mathbf{J})}) d\\mathbf{J}", "derivation": "\\operatorname{F_{c}}{(\\mathbf{J},G)} = \\cos{(G + \\mathbf{J})} and - G - \\mathbf{J} = - G - \\mathbf{J} - \\operatorname{F_{c}}{(\\mathbf{J},G)} + \\cos{(G + \\mathbf{J})} and G + \\mathbf{J} = G + \\mathbf{J} + \\operatorname{F_{c}}{(\\mathbf{J},G)} - \\cos{(G + \\mathbf{J})} and G = G + \\operatorname{F_{c}}{(\\mathbf{J},G)} - \\cos{(G + \\mathbf{J})} and \\int G d\\mathbf{J} = \\int (G + \\operatorname{F_{c}}{(\\mathbf{J},G)} - \\cos{(G + \\mathbf{J})}) d\\mathbf{J} and \\cos{(G + \\mathbf{J})} \\int G d\\mathbf{J} = \\cos{(G + \\mathbf{J})} \\int (G + \\operatorname{F_{c}}{(\\mathbf{J},G)} - \\cos{(G + \\mathbf{J})}) d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 1, "Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True))), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Symbol('G', commutative=True), Add(Symbol('G', commutative=True), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Symbol('G', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Symbol('G', commutative=True), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["times", 5, "cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integral(Symbol('G', commutative=True), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))), Mul(cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Symbol('G', commutative=True), Function('F_c')(Symbol('\\\\mathbf{J}', commutative=True), Symbol('G', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))), Tuple(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given Q{(v)} = e^{v}, then derive \\int Q{(v)} dv = A + e^{v}, then obtain \\frac{1}{\\int e^{v} dv} = \\frac{A + e^{v}}{(\\int Q{(v)} dv) \\int e^{v} dv}", "derivation": "Q{(v)} = e^{v} and \\int Q{(v)} dv = \\int e^{v} dv and \\int Q{(v)} dv = A + e^{v} and \\frac{\\int Q{(v)} dv}{\\int e^{v} dv} = \\frac{A + e^{v}}{\\int e^{v} dv} and \\frac{1}{\\int e^{v} dv} = \\frac{A + e^{v}}{(\\int Q{(v)} dv) \\int e^{v} dv}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["integrate", 1, "Symbol('v', commutative=True)"], "Equality(Integral(Function('Q')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Q')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Add(Symbol('A', commutative=True), exp(Symbol('v', commutative=True))))"], [["divide", 3, "Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Mul(Integral(Function('Q')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Pow(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))), Mul(Add(Symbol('A', commutative=True), exp(Symbol('v', commutative=True))), Pow(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))))"], [["divide", 4, "Integral(Function('Q')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Pow(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1)), Mul(Add(Symbol('A', commutative=True), exp(Symbol('v', commutative=True))), Pow(Integral(Function('Q')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1)), Pow(Integral(exp(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}{(P_{e})} = \\cos{(P_{e})}, then obtain P_{e} + \\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}} = P_{e} (\\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}})^{P_{e}} + \\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}}", "derivation": "\\hat{p}{(P_{e})} = \\cos{(P_{e})} and 1 = \\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}} and 1 = (\\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}})^{P_{e}} and P_{e} = P_{e} (\\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}})^{P_{e}} and P_{e} + \\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}} = P_{e} (\\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}})^{P_{e}} + \\frac{\\cos{(P_{e})}}{\\hat{p}{(P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), cos(Symbol('P_e', commutative=True)))"], [["divide", 1, "Function('\\\\hat{p}')(Symbol('P_e', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["times", 3, "Symbol('P_e', commutative=True)"], "Equality(Symbol('P_e', commutative=True), Mul(Symbol('P_e', commutative=True), Pow(Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))))"], [["add", 4, "Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))"], "Equality(Add(Symbol('P_e', commutative=True), Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))), Add(Mul(Symbol('P_e', commutative=True), Pow(Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))), Mul(Pow(Function('\\\\hat{p}')(Symbol('P_e', commutative=True)), Integer(-1)), cos(Symbol('P_e', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(H)} = \\frac{d}{d H} \\cos{(H)}, then derive \\varepsilon{(H)} = - \\sin{(H)}, then derive \\frac{d}{d H} \\varepsilon{(H)} - 1 = - \\cos{(H)} - 1, then obtain \\frac{d}{d H} - \\sin{(H)} - 1 = - \\cos{(H)} - 1", "derivation": "\\varepsilon{(H)} = \\frac{d}{d H} \\cos{(H)} and - H + \\varepsilon{(H)} = - H + \\frac{d}{d H} \\cos{(H)} and - H + \\varepsilon{(H)} + 1 = - H + \\frac{d}{d H} \\cos{(H)} + 1 and \\varepsilon{(H)} = - \\sin{(H)} and \\frac{d}{d H} (- H + \\varepsilon{(H)} + 1) = \\frac{d}{d H} (- H + \\frac{d}{d H} \\cos{(H)} + 1) and \\frac{d}{d H} \\varepsilon{(H)} - 1 = - \\cos{(H)} - 1 and \\frac{d}{d H} - \\sin{(H)} - 1 = - \\cos{(H)} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\varepsilon')(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\varepsilon')(Symbol('H', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(1)))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\varepsilon')(Symbol('H', commutative=True)), Mul(Integer(-1), sin(Symbol('H', commutative=True))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('\\\\varepsilon')(Symbol('H', commutative=True)), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Derivative(cos(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('\\\\varepsilon')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('H', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Derivative(Mul(Integer(-1), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), cos(Symbol('H', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\pi)} = \\log{(e^{\\pi})}, then obtain \\varepsilon_{0}^{\\pi}{(\\pi)} - \\log{(e^{\\pi})}^{\\pi} = 0", "derivation": "\\varepsilon_{0}{(\\pi)} = \\log{(e^{\\pi})} and \\varepsilon_{0}^{\\pi}{(\\pi)} = \\log{(e^{\\pi})}^{\\pi} and \\varepsilon_{0}{(\\pi)} + \\varepsilon_{0}^{\\pi}{(\\pi)} - \\log{(e^{\\pi})}^{\\pi} = \\varepsilon_{0}{(\\pi)} and \\varepsilon_{0}^{\\pi}{(\\pi)} - \\log{(e^{\\pi})}^{\\pi} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)), log(exp(Symbol('\\\\pi', commutative=True))))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True))), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))), Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Pow(Function('\\\\varepsilon_0')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), Pow(log(exp(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))), Integer(0))"]]}, {"prompt": "Given Z{(\\rho)} = \\cos{(\\rho)} and \\psi{(\\rho)} = \\cos{(\\rho)}, then obtain ((\\psi^{\\rho}{(\\rho)})^{\\rho})^{\\rho} = ((\\cos^{\\rho}{(\\rho)})^{\\rho})^{\\rho}", "derivation": "Z{(\\rho)} = \\cos{(\\rho)} and Z^{\\rho}{(\\rho)} = \\cos^{\\rho}{(\\rho)} and \\psi{(\\rho)} = \\cos{(\\rho)} and (Z^{\\rho}{(\\rho)})^{\\rho} = (\\cos^{\\rho}{(\\rho)})^{\\rho} and Z^{\\rho}{(\\rho)} = \\psi^{\\rho}{(\\rho)} and (\\psi^{\\rho}{(\\rho)})^{\\rho} = (\\cos^{\\rho}{(\\rho)})^{\\rho} and ((\\psi^{\\rho}{(\\rho)})^{\\rho})^{\\rho} = ((\\cos^{\\rho}{(\\rho)})^{\\rho})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["power", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('Z')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\psi')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Function('Z')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('Z')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\psi')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Function('\\\\psi')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["power", 6, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\psi')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Pow(Pow(cos(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given g{(\\pi,n_{1})} = \\cos^{n_{1}}{(\\pi)} and \\mathbf{J}{(\\pi,n_{1})} = \\cos^{n_{1}}{(\\pi)}, then obtain g^{\\pi}{(\\pi,n_{1})} = (\\cos^{n_{1}}{(\\pi)})^{\\pi}", "derivation": "g{(\\pi,n_{1})} = \\cos^{n_{1}}{(\\pi)} and \\mathbf{J}{(\\pi,n_{1})} = \\cos^{n_{1}}{(\\pi)} and g{(\\pi,n_{1})} = \\mathbf{J}{(\\pi,n_{1})} and g^{\\pi}{(\\pi,n_{1})} = \\mathbf{J}^{\\pi}{(\\pi,n_{1})} and g^{\\pi}{(\\pi,n_{1})} = (\\cos^{n_{1}}{(\\pi)})^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('n_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('n_1', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('g')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)))"], [["power", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('g')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('g')(Symbol('\\\\pi', commutative=True), Symbol('n_1', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Pow(cos(Symbol('\\\\pi', commutative=True)), Symbol('n_1', commutative=True)), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} = \\frac{\\tilde{g}}{\\phi_2}, then obtain \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\phi_2} = \\frac{\\tilde{g} \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\phi_2}}{\\phi_2}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} = \\frac{\\tilde{g}}{\\phi_2} and \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} = \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\phi_2} and \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} = \\frac{\\tilde{g} \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})}}{\\phi_2} and \\operatorname{V_{\\mathbf{E}}}{(\\phi_2,\\tilde{g})} \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\phi_2} = \\frac{\\tilde{g} \\frac{\\partial}{\\partial \\tilde{g}} \\frac{\\tilde{g}}{\\phi_2}}{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{B}{(z,\\mathbf{A})} = \\sin{(\\mathbf{A} - z)}, then derive \\int \\mathbf{B}{(z,\\mathbf{A})} dz = \\Psi_{nl} + \\cos{(\\mathbf{A} - z)}, then obtain \\Psi_{nl} + \\cos{(\\mathbf{A} - z)} - \\int \\mathbf{B}{(z,\\mathbf{A})} dz = f_{\\mathbf{p}} + \\cos{(\\mathbf{A} - z)} - \\int \\mathbf{B}{(z,\\mathbf{A})} dz", "derivation": "\\mathbf{B}{(z,\\mathbf{A})} = \\sin{(\\mathbf{A} - z)} and \\int \\mathbf{B}{(z,\\mathbf{A})} dz = \\int \\sin{(\\mathbf{A} - z)} dz and \\int \\mathbf{B}{(z,\\mathbf{A})} dz - \\int \\sin{(\\mathbf{A} - z)} dz = 0 and \\int \\mathbf{B}{(z,\\mathbf{A})} dz = \\Psi_{nl} + \\cos{(\\mathbf{A} - z)} and \\Psi_{nl} + \\cos{(\\mathbf{A} - z)} - \\int \\sin{(\\mathbf{A} - z)} dz = 0 and \\Psi_{nl} + \\cos{(\\mathbf{A} - z)} - \\int \\mathbf{B}{(z,\\mathbf{A})} dz = - \\int \\mathbf{B}{(z,\\mathbf{A})} dz + \\int \\sin{(\\mathbf{A} - z)} dz and \\Psi_{nl} + \\cos{(\\mathbf{A} - z)} - \\int \\mathbf{B}{(z,\\mathbf{A})} dz = f_{\\mathbf{p}} + \\cos{(\\mathbf{A} - z)} - \\int \\mathbf{B}{(z,\\mathbf{A})} dz", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))"], [["minus", 2, "Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))), Integer(0))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Integer(-1), Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True))))), Integer(0))"], [["minus", 5, "Add(Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))), Mul(Integer(-1), Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True)))))"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True)))), Integral(sin(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), cos(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), cos(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\mathbf{B}')(Symbol('z', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('z', commutative=True))))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\hat{H},u)} = \\hat{H} + e^{u}, then obtain \\frac{(\\frac{\\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}})^{- \\hat{H}} \\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}} = 1", "derivation": "\\Psi_{\\lambda}{(\\hat{H},u)} = \\hat{H} + e^{u} and \\frac{\\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}} = 1 and (\\frac{\\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}})^{\\hat{H}} = 1 and (\\frac{\\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}})^{\\hat{H}} (\\hat{H} + e^{u}) = \\hat{H} + e^{u} and \\frac{(\\frac{\\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}})^{- \\hat{H}} \\Psi_{\\lambda}{(\\hat{H},u)}}{\\hat{H} + e^{u}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Integer(1))"], [["divide", 3, "Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Pow(Mul(Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True))), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Pow(Add(Symbol('\\\\hat{H}', commutative=True), exp(Symbol('u', commutative=True))), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\hat{H}', commutative=True), Symbol('u', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(v_{y},p)} = p v_{y}, then derive \\frac{\\partial}{\\partial v_{y}} \\operatorname{t_{1}}{(v_{y},p)} = p, then obtain p^{3} v_{y} \\frac{\\partial}{\\partial v_{y}} p v_{y} = p^{4} v_{y}", "derivation": "\\operatorname{t_{1}}{(v_{y},p)} = p v_{y} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{t_{1}}{(v_{y},p)} = \\frac{\\partial}{\\partial v_{y}} p v_{y} and \\frac{\\partial}{\\partial v_{y}} \\operatorname{t_{1}}{(v_{y},p)} = p and p^{2} \\frac{\\partial}{\\partial v_{y}} \\operatorname{t_{1}}{(v_{y},p)} = p^{3} and p^{3} v_{y} \\frac{\\partial}{\\partial v_{y}} \\operatorname{t_{1}}{(v_{y},p)} = p^{4} v_{y} and p^{3} v_{y} \\frac{\\partial}{\\partial v_{y}} p v_{y} = p^{4} v_{y}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('p', commutative=True))"], [["times", 3, "Pow(Symbol('p', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(2)), Derivative(Function('t_1')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Pow(Symbol('p', commutative=True), Integer(3)))"], [["times", 4, "Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(3)), Symbol('v_y', commutative=True), Derivative(Function('t_1')(Symbol('v_y', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Pow(Symbol('p', commutative=True), Integer(4)), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(3)), Symbol('v_y', commutative=True), Derivative(Mul(Symbol('p', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Pow(Symbol('p', commutative=True), Integer(4)), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given p{(E_{x},V)} = E_{x} \\cos{(V)}, then obtain \\frac{\\partial}{\\partial V} - \\frac{2 p{(E_{x},V)}}{E_{x} \\sin{(V)}} = \\frac{d}{d V} - \\frac{2 \\cos{(V)}}{\\sin{(V)}}", "derivation": "p{(E_{x},V)} = E_{x} \\cos{(V)} and E_{x} \\cos{(V)} + p{(E_{x},V)} = 2 E_{x} \\cos{(V)} and - \\frac{E_{x} \\cos{(V)} + p{(E_{x},V)}}{E_{x} \\sin{(V)}} = - \\frac{2 \\cos{(V)}}{\\sin{(V)}} and - \\frac{2 p{(E_{x},V)}}{E_{x} \\sin{(V)}} = - \\frac{2 \\cos{(V)}}{\\sin{(V)}} and \\frac{\\partial}{\\partial V} - \\frac{2 p{(E_{x},V)}}{E_{x} \\sin{(V)}} = \\frac{d}{d V} - \\frac{2 \\cos{(V)}}{\\sin{(V)}}", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('E_x', commutative=True), cos(Symbol('V', commutative=True))))"], [["add", 1, "Mul(Symbol('E_x', commutative=True), cos(Symbol('V', commutative=True)))"], "Equality(Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('V', commutative=True))), Function('p')(Symbol('E_x', commutative=True), Symbol('V', commutative=True))), Mul(Integer(2), Symbol('E_x', commutative=True), cos(Symbol('V', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Symbol('E_x', commutative=True), sin(Symbol('V', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('E_x', commutative=True), Integer(-1)), Add(Mul(Symbol('E_x', commutative=True), cos(Symbol('V', commutative=True))), Function('p')(Symbol('E_x', commutative=True), Symbol('V', commutative=True))), Pow(sin(Symbol('V', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('V', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Integer(-1), Integer(2), Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('p')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Integer(-1))), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('V', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))))"], [["differentiate", 4, "Symbol('V', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Integer(2), Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('p')(Symbol('E_x', commutative=True), Symbol('V', commutative=True)), Pow(sin(Symbol('V', commutative=True)), Integer(-1))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Integer(2), Pow(sin(Symbol('V', commutative=True)), Integer(-1)), cos(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(n,r_{0})} = r_{0} + \\log{(n)}, then obtain A{(n,r_{0})} - A^{r_{0}}{(n,r_{0})} = r_{0} - A^{r_{0}}{(n,r_{0})} + \\log{(n)}", "derivation": "A{(n,r_{0})} = r_{0} + \\log{(n)} and A^{r_{0}}{(n,r_{0})} = (r_{0} + \\log{(n)})^{r_{0}} and - (r_{0} + \\log{(n)})^{r_{0}} + A{(n,r_{0})} = r_{0} - (r_{0} + \\log{(n)})^{r_{0}} + \\log{(n)} and A{(n,r_{0})} - A^{r_{0}}{(n,r_{0})} = r_{0} - A^{r_{0}}{(n,r_{0})} + \\log{(n)}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('r_0', commutative=True), log(Symbol('n', commutative=True))))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Add(Symbol('r_0', commutative=True), log(Symbol('n', commutative=True))), Symbol('r_0', commutative=True)))"], [["minus", 1, "Pow(Add(Symbol('r_0', commutative=True), log(Symbol('n', commutative=True))), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('r_0', commutative=True), log(Symbol('n', commutative=True))), Symbol('r_0', commutative=True))), Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('r_0', commutative=True), log(Symbol('n', commutative=True))), Symbol('r_0', commutative=True))), log(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Pow(Function('A')(Symbol('n', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True))), log(Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\psi^*)} = e^{\\psi^*}, then derive \\int \\hat{X}{(\\psi^*)} d\\psi^* = l + e^{\\psi^*}, then obtain e^{\\psi^*} + \\int e^{\\psi^*} d\\psi^* = e^{\\psi^*} + \\int \\hat{X}{(\\psi^*)} d\\psi^*", "derivation": "\\hat{X}{(\\psi^*)} = e^{\\psi^*} and \\int \\hat{X}{(\\psi^*)} d\\psi^* = \\int e^{\\psi^*} d\\psi^* and \\int \\hat{X}{(\\psi^*)} d\\psi^* = l + e^{\\psi^*} and l + e^{\\psi^*} = \\int e^{\\psi^*} d\\psi^* and l + 2 e^{\\psi^*} = e^{\\psi^*} + \\int e^{\\psi^*} d\\psi^* and l + 2 e^{\\psi^*} = e^{\\psi^*} + \\int \\hat{X}{(\\psi^*)} d\\psi^* and e^{\\psi^*} + \\int e^{\\psi^*} d\\psi^* = e^{\\psi^*} + \\int \\hat{X}{(\\psi^*)} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{X}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('l', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('l', commutative=True), exp(Symbol('\\\\psi^*', commutative=True))), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 4, "exp(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True)))), Add(exp(Symbol('\\\\psi^*', commutative=True)), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('l', commutative=True), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True)))), Add(exp(Symbol('\\\\psi^*', commutative=True)), Integral(Function('\\\\hat{X}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(exp(Symbol('\\\\psi^*', commutative=True)), Integral(exp(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(exp(Symbol('\\\\psi^*', commutative=True)), Integral(Function('\\\\hat{X}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given C{(E_{x},M)} = E_{x} + M, then obtain \\frac{\\partial}{\\partial M} (C{(E_{x},M)} - C^{M}{(E_{x},M)}) (E_{x} + M - C^{M}{(E_{x},M)}) = \\frac{\\partial}{\\partial M} (E_{x} + M - C^{M}{(E_{x},M)})^{2}", "derivation": "C{(E_{x},M)} = E_{x} + M and C^{M}{(E_{x},M)} = (E_{x} + M)^{M} and - (E_{x} + M)^{M} + C{(E_{x},M)} = E_{x} + M - (E_{x} + M)^{M} and (- (E_{x} + M)^{M} + C{(E_{x},M)}) (E_{x} + M - (E_{x} + M)^{M}) = (E_{x} + M - (E_{x} + M)^{M})^{2} and (C{(E_{x},M)} - C^{M}{(E_{x},M)}) (E_{x} + M - C^{M}{(E_{x},M)}) = (E_{x} + M - C^{M}{(E_{x},M)})^{2} and \\frac{\\partial}{\\partial M} (C{(E_{x},M)} - C^{M}{(E_{x},M)}) (E_{x} + M - C^{M}{(E_{x},M)}) = \\frac{\\partial}{\\partial M} (E_{x} + M - C^{M}{(E_{x},M)})^{2}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)))"], [["power", 1, "Symbol('M', commutative=True)"], "Equality(Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))"], [["minus", 1, "Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))))"], [["times", 3, "Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))), Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True))), Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))))), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))))), Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Integer(2)))"], [["differentiate", 5, "Symbol('M', commutative=True)"], "Equality(Derivative(Mul(Add(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True))))), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('E_x', commutative=True), Symbol('M', commutative=True), Mul(Integer(-1), Pow(Function('C')(Symbol('E_x', commutative=True), Symbol('M', commutative=True)), Symbol('M', commutative=True)))), Integer(2)), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mathbf{S},\\mathbf{P},\\ddot{x})} = \\ddot{x} + \\mathbf{P} - \\mathbf{S} and \\mathbf{A}{(\\mathbf{S})} = \\mathbf{S}, then obtain \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} - \\mathbf{P} + \\mathbf{S} + \\mathbf{A}{(\\mathbf{S})}) = \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} - \\mathbf{P} + 2 \\mathbf{S})", "derivation": "\\operatorname{n_{1}}{(\\mathbf{S},\\mathbf{P},\\ddot{x})} = \\ddot{x} + \\mathbf{P} - \\mathbf{S} and \\mathbf{A}{(\\mathbf{S})} = \\mathbf{S} and \\mathbf{A}{(\\mathbf{S})} - \\operatorname{n_{1}}{(\\mathbf{S},\\mathbf{P},\\ddot{x})} = \\mathbf{S} - \\operatorname{n_{1}}{(\\mathbf{S},\\mathbf{P},\\ddot{x})} and - \\ddot{x} - \\mathbf{P} + \\mathbf{S} + \\mathbf{A}{(\\mathbf{S})} = - \\ddot{x} - \\mathbf{P} + 2 \\mathbf{S} and \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} - \\mathbf{P} + \\mathbf{S} + \\mathbf{A}{(\\mathbf{S})}) = \\frac{\\partial}{\\partial \\ddot{x}} (- \\ddot{x} - \\mathbf{P} + 2 \\mathbf{S})", "srepr_derivation": [["get_premise", "Equality(Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Add(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], [["minus", 2, "Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\ddot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(g,\\ddot{x})} = \\cos{(\\ddot{x}^{g})}, then obtain \\frac{1}{2} = \\frac{\\cos{(\\ddot{x}^{g})}}{\\operatorname{C_{1}}{(g,\\ddot{x})} + \\cos{(\\ddot{x}^{g})}}", "derivation": "\\operatorname{C_{1}}{(g,\\ddot{x})} = \\cos{(\\ddot{x}^{g})} and 2 \\operatorname{C_{1}}{(g,\\ddot{x})} = \\operatorname{C_{1}}{(g,\\ddot{x})} + \\cos{(\\ddot{x}^{g})} and \\frac{1}{2} = \\frac{\\cos{(\\ddot{x}^{g})}}{2 \\operatorname{C_{1}}{(g,\\ddot{x})}} and \\frac{1}{2} = \\frac{\\cos{(\\ddot{x}^{g})}}{\\operatorname{C_{1}}{(g,\\ddot{x})} + \\cos{(\\ddot{x}^{g})}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), cos(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('g', commutative=True))))"], [["add", 1, "Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True))"], "Equality(Mul(Integer(2), Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True))), Add(Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), cos(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('g', commutative=True)))))"], [["divide", 1, "Mul(Integer(2), Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True)))"], "Equality(Rational(1, 2), Mul(Rational(1, 2), Pow(Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), Integer(-1)), cos(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('g', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Rational(1, 2), Mul(Pow(Add(Function('C_1')(Symbol('g', commutative=True), Symbol('\\\\ddot{x}', commutative=True)), cos(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('g', commutative=True)))), Integer(-1)), cos(Pow(Symbol('\\\\ddot{x}', commutative=True), Symbol('g', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\hat{X})} = \\sin{(\\hat{X})}, then obtain ((- \\theta_{2}{(\\hat{X})} + \\frac{\\theta_{2}{(\\hat{X})}}{\\sin{(\\hat{X})}})^{\\hat{X}})^{\\hat{X}} = ((1 - \\theta_{2}{(\\hat{X})})^{\\hat{X}})^{\\hat{X}}", "derivation": "\\theta_{2}{(\\hat{X})} = \\sin{(\\hat{X})} and \\frac{\\theta_{2}{(\\hat{X})}}{\\sin{(\\hat{X})}} = 1 and - \\theta_{2}{(\\hat{X})} + \\frac{\\theta_{2}{(\\hat{X})}}{\\sin{(\\hat{X})}} = 1 - \\theta_{2}{(\\hat{X})} and (- \\theta_{2}{(\\hat{X})} + \\frac{\\theta_{2}{(\\hat{X})}}{\\sin{(\\hat{X})}})^{\\hat{X}} = (1 - \\theta_{2}{(\\hat{X})})^{\\hat{X}} and ((- \\theta_{2}{(\\hat{X})} + \\frac{\\theta_{2}{(\\hat{X})}}{\\sin{(\\hat{X})}})^{\\hat{X}})^{\\hat{X}} = ((1 - \\theta_{2}{(\\hat{X})})^{\\hat{X}})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), sin(Symbol('\\\\hat{X}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))), Add(Integer(1), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)))))"], [["power", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"], [["power", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)), Pow(sin(Symbol('\\\\hat{X}', commutative=True)), Integer(-1)))), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Pow(Add(Integer(1), Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(s)} = \\cos{(e^{s})}, then derive \\int \\operatorname{A_{z}}{(s)} ds = \\hat{p}_0 + \\operatorname{Ci}{(e^{s})}, then obtain \\frac{(\\hat{p}_0 + \\operatorname{Ci}{(e^{s})}) \\cos{(e^{s})}}{\\int \\cos{(e^{s})} ds} = \\cos{(e^{s})}", "derivation": "\\operatorname{A_{z}}{(s)} = \\cos{(e^{s})} and \\int \\operatorname{A_{z}}{(s)} ds = \\int \\cos{(e^{s})} ds and \\frac{\\int \\operatorname{A_{z}}{(s)} ds}{\\int \\cos{(e^{s})} ds} = 1 and \\frac{\\cos{(e^{s})} \\int \\operatorname{A_{z}}{(s)} ds}{\\int \\cos{(e^{s})} ds} = \\cos{(e^{s})} and \\int \\operatorname{A_{z}}{(s)} ds = \\hat{p}_0 + \\operatorname{Ci}{(e^{s})} and \\frac{(\\hat{p}_0 + \\operatorname{Ci}{(e^{s})}) \\cos{(e^{s})}}{\\int \\cos{(e^{s})} ds} = \\cos{(e^{s})}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('s', commutative=True)), cos(exp(Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["divide", 2, "Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))"], "Equality(Mul(Integral(Function('A_z')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Pow(Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integer(-1))), Integer(1))"], [["times", 3, "cos(exp(Symbol('s', commutative=True)))"], "Equality(Mul(cos(exp(Symbol('s', commutative=True))), Integral(Function('A_z')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Pow(Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integer(-1))), cos(exp(Symbol('s', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_z')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Add(Symbol('\\\\hat{p}_0', commutative=True), Ci(exp(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Add(Symbol('\\\\hat{p}_0', commutative=True), Ci(exp(Symbol('s', commutative=True)))), cos(exp(Symbol('s', commutative=True))), Pow(Integral(cos(exp(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integer(-1))), cos(exp(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} = \\frac{A_{x}}{\\hat{\\mathbf{r}}}, then obtain \\frac{A_{x}}{\\hat{\\mathbf{r}}} + \\hat{\\mathbf{r}} \\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}^{2}} = A_{x} + \\frac{A_{x}}{\\hat{\\mathbf{r}}} + \\frac{1}{\\hat{\\mathbf{r}}^{2}}", "derivation": "\\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} = \\frac{A_{x}}{\\hat{\\mathbf{r}}} and \\frac{\\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}}} = \\frac{A_{x}}{\\hat{\\mathbf{r}}^{2}} and \\hat{\\mathbf{r}} \\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} = A_{x} and \\hat{\\mathbf{r}} \\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}^{2}} = A_{x} + \\frac{1}{\\hat{\\mathbf{r}}^{2}} and \\frac{A_{x}}{\\hat{\\mathbf{r}}} + \\hat{\\mathbf{r}} \\operatorname{F_{H}}{(A_{x},\\hat{\\mathbf{r}})} + \\frac{1}{\\hat{\\mathbf{r}}^{2}} = A_{x} + \\frac{A_{x}}{\\hat{\\mathbf{r}}} + \\frac{1}{\\hat{\\mathbf{r}}^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)), Function('F_H')(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))))"], [["divide", 2, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('F_H')(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Symbol('A_x', commutative=True))"], [["add", 3, "Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))"], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('F_H')(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))), Add(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))))"], [["add", 4, "Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('F_H')(Symbol('A_x', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))), Add(Symbol('A_x', commutative=True), Mul(Symbol('A_x', commutative=True), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-1))), Pow(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given z{(M_{E})} = \\log{(M_{E})}, then obtain \\frac{\\int z{(M_{E})} dM_{E}}{\\iint z{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\log{(M_{E})} dM_{E}}{\\iint z{(M_{E})} dM_{E} dM_{E}}", "derivation": "z{(M_{E})} = \\log{(M_{E})} and \\int z{(M_{E})} dM_{E} = \\int \\log{(M_{E})} dM_{E} and \\iint z{(M_{E})} dM_{E} dM_{E} = \\iint \\log{(M_{E})} dM_{E} dM_{E} and \\frac{\\int z{(M_{E})} dM_{E}}{\\iint \\log{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\log{(M_{E})} dM_{E}}{\\iint \\log{(M_{E})} dM_{E} dM_{E}} and \\frac{\\int z{(M_{E})} dM_{E}}{\\iint z{(M_{E})} dM_{E} dM_{E}} = \\frac{\\int \\log{(M_{E})} dM_{E}}{\\iint z{(M_{E})} dM_{E} dM_{E}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('M_E', commutative=True)), log(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["integrate", 2, "Symbol('M_E', commutative=True)"], "Equality(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))))"], [["divide", 2, "Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)))"], "Equality(Mul(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))), Mul(Integral(log(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Pow(Integral(Function('z')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}_0{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})}, then obtain \\int \\frac{d}{d x^\\prime} \\hat{p}_0{(x^\\prime)} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime dx^\\prime = \\int \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime dx^\\prime", "derivation": "\\hat{p}_0{(x^\\prime)} = \\log{(\\sin{(x^\\prime)})} and \\hat{p}_0{(x^\\prime)} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime = \\log{(\\sin{(x^\\prime)})} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime and \\frac{d}{d x^\\prime} \\hat{p}_0{(x^\\prime)} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime = \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime and \\int \\frac{d}{d x^\\prime} \\hat{p}_0{(x^\\prime)} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime dx^\\prime = \\int \\frac{d}{d x^\\prime} \\log{(\\sin{(x^\\prime)})} \\int \\hat{p}_0{(x^\\prime)} dx^\\prime dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), log(sin(Symbol('x^\\\\prime', commutative=True))))"], [["times", 1, "Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(log(sin(Symbol('x^\\\\prime', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(log(sin(Symbol('x^\\\\prime', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(Mul(log(sin(Symbol('x^\\\\prime', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(t_{2})} = e^{t_{2}}, then obtain \\frac{d}{d t_{2}} \\operatorname{t_{1}}{(t_{2})} + 1 = e^{t_{2}} + 1", "derivation": "\\operatorname{t_{1}}{(t_{2})} = e^{t_{2}} and t_{2} + \\operatorname{t_{1}}{(t_{2})} = t_{2} + e^{t_{2}} and \\frac{d}{d t_{2}} (t_{2} + \\operatorname{t_{1}}{(t_{2})}) = \\frac{d}{d t_{2}} (t_{2} + e^{t_{2}}) and \\frac{d}{d t_{2}} \\operatorname{t_{1}}{(t_{2})} + 1 = e^{t_{2}} + 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('t_2', commutative=True)"], "Equality(Add(Symbol('t_2', commutative=True), Function('t_1')(Symbol('t_2', commutative=True))), Add(Symbol('t_2', commutative=True), exp(Symbol('t_2', commutative=True))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Symbol('t_2', commutative=True), Function('t_1')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Symbol('t_2', commutative=True), exp(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t_1')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(1)), Add(exp(Symbol('t_2', commutative=True)), Integer(1)))"]]}, {"prompt": "Given y{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)}, then obtain (\\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\frac{1}{\\mathbf{J}_P})^{\\mathbf{J}_P}", "derivation": "y{(\\mathbf{J}_P)} = \\log{(\\mathbf{J}_P)} and \\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)} = \\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)} and (\\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\frac{d}{d \\mathbf{J}_P} \\log{(\\mathbf{J}_P)})^{\\mathbf{J}_P} and (\\frac{d}{d \\mathbf{J}_P} y{(\\mathbf{J}_P)})^{\\mathbf{J}_P} = (\\frac{1}{\\mathbf{J}_P})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), log(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Derivative(log(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given s{(y)} = e^{y}, then obtain (- s{(y)} + \\int s{(y)} dy) \\frac{d}{d y} (s{(y)} - e^{y}) = (- s{(y)} + \\int s{(y)} dy) \\frac{d}{d y} 0", "derivation": "s{(y)} = e^{y} and s{(y)} - e^{y} = 0 and \\frac{d}{d y} (s{(y)} - e^{y}) = \\frac{d}{d y} 0 and (- s{(y)} + \\int s{(y)} dy) \\frac{d}{d y} (s{(y)} - e^{y}) = (- s{(y)} + \\int s{(y)} dy) \\frac{d}{d y} 0", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('y', commutative=True)), exp(Symbol('y', commutative=True)))"], [["minus", 1, "exp(Symbol('y', commutative=True))"], "Equality(Add(Function('s')(Symbol('y', commutative=True)), Mul(Integer(-1), exp(Symbol('y', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Add(Function('s')(Symbol('y', commutative=True)), Mul(Integer(-1), exp(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["times", 3, "Add(Mul(Integer(-1), Function('s')(Symbol('y', commutative=True))), Integral(Function('s')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Function('s')(Symbol('y', commutative=True))), Integral(Function('s')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Derivative(Add(Function('s')(Symbol('y', commutative=True)), Mul(Integer(-1), exp(Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1)))), Mul(Add(Mul(Integer(-1), Function('s')(Symbol('y', commutative=True))), Integral(Function('s')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True)))), Derivative(Integer(0), Tuple(Symbol('y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(F_{N})} = e^{F_{N}} and l{(v_{t},v_{z})} = \\frac{v_{z}}{v_{t}}, then derive \\int \\lambda{(F_{N})} dF_{N} - \\frac{v_{z}}{v_{t}} = \\hat{X} + e^{F_{N}} - \\frac{v_{z}}{v_{t}}, then obtain \\int \\lambda{(F_{N})} dF_{N} - \\frac{v_{z}}{v_{t}} = \\hat{X} + \\lambda{(F_{N})} - \\frac{v_{z}}{v_{t}}", "derivation": "\\lambda{(F_{N})} = e^{F_{N}} and \\int \\lambda{(F_{N})} dF_{N} = \\int e^{F_{N}} dF_{N} and l{(v_{t},v_{z})} = \\frac{v_{z}}{v_{t}} and - l{(v_{t},v_{z})} + \\int \\lambda{(F_{N})} dF_{N} = - l{(v_{t},v_{z})} + \\int e^{F_{N}} dF_{N} and \\int \\lambda{(F_{N})} dF_{N} - \\frac{v_{z}}{v_{t}} = \\int e^{F_{N}} dF_{N} - \\frac{v_{z}}{v_{t}} and \\int \\lambda{(F_{N})} dF_{N} - \\frac{v_{z}}{v_{t}} = \\hat{X} + e^{F_{N}} - \\frac{v_{z}}{v_{t}} and \\int \\lambda{(F_{N})} dF_{N} - \\frac{v_{z}}{v_{t}} = \\hat{X} + \\lambda{(F_{N})} - \\frac{v_{z}}{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], ["get_premise", "Equality(Function('l')(Symbol('v_t', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["minus", 2, "Function('l')(Symbol('v_t', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('l')(Symbol('v_t', commutative=True), Symbol('v_z', commutative=True))), Integral(Function('\\\\lambda')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Integer(-1), Function('l')(Symbol('v_t', commutative=True), Symbol('v_z', commutative=True))), Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Integral(Function('\\\\lambda')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))), Add(Integral(exp(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Integral(Function('\\\\lambda')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), exp(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Integral(Function('\\\\lambda')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Function('\\\\lambda')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_t', commutative=True), Integer(-1)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(F_{H},I,r_{0})} = - F_{H} + I + r_{0}, then obtain (\\iint \\sigma_{p}{(F_{H},I,r_{0})} dI dF_{H})^{I} = (\\iint (- F_{H} + I + r_{0}) dI dF_{H})^{I}", "derivation": "\\sigma_{p}{(F_{H},I,r_{0})} = - F_{H} + I + r_{0} and \\int \\sigma_{p}{(F_{H},I,r_{0})} dI = \\int (- F_{H} + I + r_{0}) dI and \\iint \\sigma_{p}{(F_{H},I,r_{0})} dI dF_{H} = \\iint (- F_{H} + I + r_{0}) dI dF_{H} and (\\iint \\sigma_{p}{(F_{H},I,r_{0})} dI dF_{H})^{I} = (\\iint (- F_{H} + I + r_{0}) dI dF_{H})^{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('F_H', commutative=True), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('I', commutative=True), Symbol('r_0', commutative=True)))"], [["integrate", 1, "Symbol('I', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('F_H', commutative=True), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('F_H', commutative=True), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Integral(Function('\\\\sigma_p')(Symbol('F_H', commutative=True), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('I', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('I', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('I', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Symbol('I', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\hat{H},n_{1})} = \\hat{H} n_{1}, then obtain \\int \\frac{\\partial}{\\partial \\hat{H}} (\\operatorname{E_{n}}{(\\hat{H},n_{1})} - 1)^{n_{1}} dn_{1} = \\int \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} n_{1} - 1)^{n_{1}} dn_{1}", "derivation": "\\operatorname{E_{n}}{(\\hat{H},n_{1})} = \\hat{H} n_{1} and \\operatorname{E_{n}}{(\\hat{H},n_{1})} - 1 = \\hat{H} n_{1} - 1 and (\\operatorname{E_{n}}{(\\hat{H},n_{1})} - 1)^{n_{1}} = (\\hat{H} n_{1} - 1)^{n_{1}} and \\frac{\\partial}{\\partial \\hat{H}} (\\operatorname{E_{n}}{(\\hat{H},n_{1})} - 1)^{n_{1}} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} n_{1} - 1)^{n_{1}} and \\int \\frac{\\partial}{\\partial \\hat{H}} (\\operatorname{E_{n}}{(\\hat{H},n_{1})} - 1)^{n_{1}} dn_{1} = \\int \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} n_{1} - 1)^{n_{1}} dn_{1}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('E_n')(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)))"], [["power", 2, "Symbol('n_1', commutative=True)"], "Equality(Pow(Add(Function('E_n')(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Pow(Add(Function('E_n')(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('n_1', commutative=True)"], "Equality(Integral(Derivative(Pow(Add(Function('E_n')(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('n_1', commutative=True))), Integral(Derivative(Pow(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('n_1', commutative=True)), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Tuple(Symbol('n_1', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(v_{x},h)} = - v_{x} + \\sin{(h)}, then derive a + \\dot{y}{(v_{x},h)} = \\hat{\\mathbf{x}} + \\sin{(h)}, then obtain a - v_{x} + \\sin{(h)} = a + \\dot{y}{(v_{x},h)}", "derivation": "\\dot{y}{(v_{x},h)} = - v_{x} + \\sin{(h)} and \\frac{\\partial}{\\partial h} \\dot{y}{(v_{x},h)} = \\frac{\\partial}{\\partial h} (- v_{x} + \\sin{(h)}) and \\int \\frac{\\partial}{\\partial h} \\dot{y}{(v_{x},h)} dh = \\int \\frac{\\partial}{\\partial h} (- v_{x} + \\sin{(h)}) dh and a + \\dot{y}{(v_{x},h)} = \\hat{\\mathbf{x}} + \\sin{(h)} and a - v_{x} + \\sin{(h)} = \\hat{\\mathbf{x}} + \\sin{(h)} and a - v_{x} + \\sin{(h)} = a + \\dot{y}{(v_{x},h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('h', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('a', commutative=True), Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('h', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('h', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), sin(Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('a', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), sin(Symbol('h', commutative=True))), Add(Symbol('a', commutative=True), Function('\\\\dot{y}')(Symbol('v_x', commutative=True), Symbol('h', commutative=True))))"]]}, {"prompt": "Given i{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\mathbf{P}{(\\mathbf{H})} = \\frac{1}{\\cos{(\\mathbf{H})}}, then obtain \\frac{d}{d \\mathbf{H}} (\\mathbf{P}{(\\mathbf{H})} + \\frac{1}{i{(\\mathbf{H})}}) = \\frac{d}{d \\mathbf{H}} \\frac{2}{i{(\\mathbf{H})}}", "derivation": "i{(\\mathbf{H})} = \\cos{(\\mathbf{H})} and \\mathbf{P}{(\\mathbf{H})} = \\frac{1}{\\cos{(\\mathbf{H})}} and \\mathbf{P}{(\\mathbf{H})} = \\frac{1}{i{(\\mathbf{H})}} and \\mathbf{P}{(\\mathbf{H})} + \\frac{1}{i{(\\mathbf{H})}} = \\frac{2}{i{(\\mathbf{H})}} and \\frac{d}{d \\mathbf{H}} (\\mathbf{P}{(\\mathbf{H})} + \\frac{1}{i{(\\mathbf{H})}}) = \\frac{d}{d \\mathbf{H}} \\frac{2}{i{(\\mathbf{H})}}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], [["add", 3, "Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{P}')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Function('i')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(t_{1})} = \\sin{(t_{1})} and \\phi{(t_{1})} = \\sin{(t_{1})} and \\mathbf{J}_M{(t_{1})} = \\operatorname{C_{d}}^{t_{1}}{(t_{1})} \\int 0 dt_{1}, then obtain \\mathbf{J}_M{(t_{1})} = \\sin^{t_{1}}{(t_{1})} \\int 0 dt_{1}", "derivation": "\\operatorname{C_{d}}{(t_{1})} = \\sin{(t_{1})} and \\phi{(t_{1})} = \\sin{(t_{1})} and \\phi^{t_{1}}{(t_{1})} = \\sin^{t_{1}}{(t_{1})} and \\phi^{t_{1}}{(t_{1})} \\int 0 dt_{1} = \\sin^{t_{1}}{(t_{1})} \\int 0 dt_{1} and \\phi^{t_{1}}{(t_{1})} \\int 0 dt_{1} = \\operatorname{C_{d}}^{t_{1}}{(t_{1})} \\int 0 dt_{1} and \\mathbf{J}_M{(t_{1})} = \\operatorname{C_{d}}^{t_{1}}{(t_{1})} \\int 0 dt_{1} and \\sin^{t_{1}}{(t_{1})} \\int 0 dt_{1} = \\operatorname{C_{d}}^{t_{1}}{(t_{1})} \\int 0 dt_{1} and \\mathbf{J}_M{(t_{1})} = \\sin^{t_{1}}{(t_{1})} \\int 0 dt_{1}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], [["power", 2, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(sin(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["times", 3, "Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(sin(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('\\\\phi')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Function('C_d')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Mul(Pow(Function('C_d')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(sin(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Function('C_d')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Function('\\\\mathbf{J}_M')(Symbol('t_1', commutative=True)), Mul(Pow(sin(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Integral(Integer(0), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{E},F_{H})} = F_{H} + \\mathbf{E}, then derive \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})} = 1, then obtain e^{e^{\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})}}} = e^{e^{\\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E})}}", "derivation": "\\mathbf{D}{(\\mathbf{E},F_{H})} = F_{H} + \\mathbf{E} and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})} = \\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E}) and \\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})} = 1 and \\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E}) = 1 and e^{\\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E})} = e and e^{\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})}} = e and e^{\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})}} = e^{\\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E})} and e^{e^{\\frac{\\partial}{\\partial \\mathbf{E}} \\mathbf{D}{(\\mathbf{E},F_{H})}}} = e^{e^{\\frac{\\partial}{\\partial \\mathbf{E}} (F_{H} + \\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["exp", 4], "Equality(exp(Derivative(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), E)"], [["substitute_RHS_for_LHS", 5, 1], "Equality(exp(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), E)"], [["substitute_RHS_for_LHS", 6, 5], "Equality(exp(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))), exp(Derivative(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1)))))"], [["exp", 7], "Equality(exp(exp(Derivative(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))), exp(exp(Derivative(Add(Symbol('F_H', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))))"]]}, {"prompt": "Given l{(L_{\\varepsilon})} = \\log{(\\log{(L_{\\varepsilon})})} and \\operatorname{F_{x}}{(L_{\\varepsilon})} = \\log{(\\log{(L_{\\varepsilon})})}, then obtain \\operatorname{F_{x}}^{L_{\\varepsilon}}{(L_{\\varepsilon})} + \\int \\log{(\\log{(L_{\\varepsilon})})} dL_{\\varepsilon} = l^{L_{\\varepsilon}}{(L_{\\varepsilon})} + \\int \\log{(\\log{(L_{\\varepsilon})})} dL_{\\varepsilon}", "derivation": "l{(L_{\\varepsilon})} = \\log{(\\log{(L_{\\varepsilon})})} and \\operatorname{F_{x}}{(L_{\\varepsilon})} = \\log{(\\log{(L_{\\varepsilon})})} and \\operatorname{F_{x}}{(L_{\\varepsilon})} = l{(L_{\\varepsilon})} and \\operatorname{F_{x}}^{L_{\\varepsilon}}{(L_{\\varepsilon})} = l^{L_{\\varepsilon}}{(L_{\\varepsilon})} and \\operatorname{F_{x}}^{L_{\\varepsilon}}{(L_{\\varepsilon})} + \\int \\log{(\\log{(L_{\\varepsilon})})} dL_{\\varepsilon} = l^{L_{\\varepsilon}}{(L_{\\varepsilon})} + \\int \\log{(\\log{(L_{\\varepsilon})})} dL_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(log(Symbol('L_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), log(log(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Function('l')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Function('l')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["add", 4, "Integral(log(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Pow(Function('F_x')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(log(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))), Add(Pow(Function('l')(Symbol('L_{\\\\varepsilon}', commutative=True)), Symbol('L_{\\\\varepsilon}', commutative=True)), Integral(log(log(Symbol('L_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\varepsilon,L)} = \\sin{(\\frac{\\varepsilon}{L})} and \\hat{x}_0{(\\varepsilon,L)} = \\operatorname{x^{{\\}'}}{(\\varepsilon,L)} + \\sin{(\\frac{\\varepsilon}{L})}, then obtain \\int \\hat{x}_0^{4}{(\\varepsilon,L)} d\\varepsilon = \\int 16 \\operatorname{x^{{\\}'}}^{4}{(\\varepsilon,L)} d\\varepsilon", "derivation": "\\operatorname{x^{{\\}'}}{(\\varepsilon,L)} = \\sin{(\\frac{\\varepsilon}{L})} and \\hat{x}_0{(\\varepsilon,L)} = \\operatorname{x^{{\\}'}}{(\\varepsilon,L)} + \\sin{(\\frac{\\varepsilon}{L})} and \\hat{x}_0{(\\varepsilon,L)} = 2 \\sin{(\\frac{\\varepsilon}{L})} and \\hat{x}_0{(\\varepsilon,L)} = 2 \\operatorname{x^{{\\}'}}{(\\varepsilon,L)} and \\hat{x}_0^{2}{(\\varepsilon,L)} = 4 \\operatorname{x^{{\\}'}}^{2}{(\\varepsilon,L)} and \\hat{x}_0^{4}{(\\varepsilon,L)} = 16 \\operatorname{x^{{\\}'}}^{4}{(\\varepsilon,L)} and \\int \\hat{x}_0^{4}{(\\varepsilon,L)} d\\varepsilon = \\int 16 \\operatorname{x^{{\\}'}}^{4}{(\\varepsilon,L)} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), sin(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Add(Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), sin(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Mul(Integer(2), sin(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Mul(Integer(2), Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(2)), Mul(Integer(4), Pow(Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(2))))"], [["power", 5, 2], "Equality(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(4)), Mul(Integer(16), Pow(Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(4))))"], [["integrate", 6, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(4)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Mul(Integer(16), Pow(Function('x^\\\\prime')(Symbol('\\\\varepsilon', commutative=True), Symbol('L', commutative=True)), Integer(4))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\phi_2,t)} = \\log{(\\phi_2 + t)}, then obtain (- t + \\Psi^{\\dagger}{(\\phi_2,t)})^{t} - 1 = (- t + \\log{(\\phi_2 + t)})^{t} - 1", "derivation": "\\Psi^{\\dagger}{(\\phi_2,t)} = \\log{(\\phi_2 + t)} and - t + \\Psi^{\\dagger}{(\\phi_2,t)} = - t + \\log{(\\phi_2 + t)} and (- t + \\Psi^{\\dagger}{(\\phi_2,t)})^{t} = (- t + \\log{(\\phi_2 + t)})^{t} and (- t + \\Psi^{\\dagger}{(\\phi_2,t)})^{t} - 1 = (- t + \\log{(\\phi_2 + t)})^{t} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)), log(Add(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True))))"], [["minus", 1, "Symbol('t', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True))), Add(Mul(Integer(-1), Symbol('t', commutative=True)), log(Add(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)))))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), log(Add(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True))), Symbol('t', commutative=True)), Integer(-1)), Add(Pow(Add(Mul(Integer(-1), Symbol('t', commutative=True)), log(Add(Symbol('\\\\phi_2', commutative=True), Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(n_{1},F_{g})} = \\frac{n_{1}}{F_{g}}, then derive \\frac{\\partial}{\\partial n_{1}} \\operatorname{n_{2}}{(n_{1},F_{g})} = \\frac{1}{F_{g}}, then obtain \\frac{2}{F_{g}} = \\frac{\\partial}{\\partial n_{1}} \\frac{n_{1}}{F_{g}} + \\frac{1}{F_{g}}", "derivation": "\\operatorname{n_{2}}{(n_{1},F_{g})} = \\frac{n_{1}}{F_{g}} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{n_{2}}{(n_{1},F_{g})} = \\frac{\\partial}{\\partial n_{1}} \\frac{n_{1}}{F_{g}} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{n_{2}}{(n_{1},F_{g})} = \\frac{1}{F_{g}} and \\frac{\\partial}{\\partial n_{1}} \\operatorname{n_{2}}{(n_{1},F_{g})} + \\frac{1}{F_{g}} = \\frac{\\partial}{\\partial n_{1}} \\frac{n_{1}}{F_{g}} + \\frac{1}{F_{g}} and \\frac{2}{F_{g}} = \\frac{\\partial}{\\partial n_{1}} \\frac{n_{1}}{F_{g}} + \\frac{1}{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)))"], [["differentiate", 1, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1)))"], [["add", 2, "Pow(Symbol('F_g', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('n_2')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1))), Add(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Pow(Symbol('F_g', commutative=True), Integer(-1))), Add(Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))), Pow(Symbol('F_g', commutative=True), Integer(-1))))"]]}, {"prompt": "Given s{(g,\\mathbf{F})} = \\frac{e^{\\mathbf{F}}}{g}, then derive \\frac{\\partial}{\\partial g} s{(g,\\mathbf{F})} - 1 = -1 - \\frac{e^{\\mathbf{F}}}{g^{2}}, then obtain \\frac{\\partial}{\\partial g} s{(g,\\mathbf{F})} - 1 = -1 - \\frac{s{(g,\\mathbf{F})}}{g}", "derivation": "s{(g,\\mathbf{F})} = \\frac{e^{\\mathbf{F}}}{g} and - g + s{(g,\\mathbf{F})} = - g + \\frac{e^{\\mathbf{F}}}{g} and \\frac{\\partial}{\\partial g} (- g + s{(g,\\mathbf{F})}) = \\frac{\\partial}{\\partial g} (- g + \\frac{e^{\\mathbf{F}}}{g}) and \\frac{\\partial}{\\partial g} s{(g,\\mathbf{F})} - 1 = -1 - \\frac{e^{\\mathbf{F}}}{g^{2}} and \\frac{\\partial}{\\partial g} s{(g,\\mathbf{F})} - 1 = -1 - \\frac{s{(g,\\mathbf{F})}}{g}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), exp(Symbol('\\\\mathbf{F}', commutative=True)))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-2)), exp(Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)), Function('s')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given n{(\\psi^*)} = \\cos{(\\psi^*)} and \\mu{(\\psi^*)} = \\frac{1}{\\cos{(\\psi^*)}}, then obtain \\frac{\\mu{(\\psi^*)}}{\\cos{(\\psi^*)}} = \\frac{1}{n{(\\psi^*)} \\cos{(\\psi^*)}}", "derivation": "n{(\\psi^*)} = \\cos{(\\psi^*)} and \\mu{(\\psi^*)} = \\frac{1}{\\cos{(\\psi^*)}} and \\mu{(\\psi^*)} = \\frac{1}{n{(\\psi^*)}} and \\frac{\\mu{(\\psi^*)}}{n{(\\psi^*)}} = \\frac{1}{n{(\\psi^*)} \\cos{(\\psi^*)}} and \\frac{\\mu{(\\psi^*)}}{\\cos{(\\psi^*)}} = \\frac{1}{\\cos^{2}{(\\psi^*)}} and \\frac{1}{n{(\\psi^*)} \\cos{(\\psi^*)}} = \\frac{1}{\\cos^{2}{(\\psi^*)}} and \\frac{\\mu{(\\psi^*)}}{\\cos{(\\psi^*)}} = \\frac{1}{n{(\\psi^*)} \\cos{(\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\psi^*', commutative=True)), cos(Symbol('\\\\psi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Pow(Function('n')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)))"], [["divide", 2, "Function('n')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Pow(Function('n')(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Mul(Pow(Function('n')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-2)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Function('n')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-2)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Function('\\\\mu')(Symbol('\\\\psi^*', commutative=True)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))), Mul(Pow(Function('n')(Symbol('\\\\psi^*', commutative=True)), Integer(-1)), Pow(cos(Symbol('\\\\psi^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\Omega)} = \\cos{(\\cos{(\\Omega)})}, then derive \\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)} = \\sin{(\\Omega)} \\sin{(\\cos{(\\Omega)})}, then obtain e^{\\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)}} = e^{\\sin{(\\Omega)} \\sin{(\\cos{(\\Omega)})}}", "derivation": "\\operatorname{x^{{\\}'}}{(\\Omega)} = \\cos{(\\cos{(\\Omega)})} and \\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)} = \\frac{d}{d \\Omega} \\cos{(\\cos{(\\Omega)})} and e^{\\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)}} = e^{\\frac{d}{d \\Omega} \\cos{(\\cos{(\\Omega)})}} and \\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)} = \\sin{(\\Omega)} \\sin{(\\cos{(\\Omega)})} and \\frac{d}{d \\Omega} \\cos{(\\cos{(\\Omega)})} = \\sin{(\\Omega)} \\sin{(\\cos{(\\Omega)})} and e^{\\frac{d}{d \\Omega} \\operatorname{x^{{\\}'}}{(\\Omega)}} = e^{\\sin{(\\Omega)} \\sin{(\\cos{(\\Omega)})}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\Omega', commutative=True)), cos(cos(Symbol('\\\\Omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["exp", 2], "Equality(exp(Derivative(Function('x^\\\\prime')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), exp(Derivative(cos(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(sin(Symbol('\\\\Omega', commutative=True)), sin(cos(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(cos(cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Mul(sin(Symbol('\\\\Omega', commutative=True)), sin(cos(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(exp(Derivative(Function('x^\\\\prime')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), exp(Mul(sin(Symbol('\\\\Omega', commutative=True)), sin(cos(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(P_{g})} = P_{g}, then obtain \\int (\\operatorname{F_{x}}{(P_{g})} - \\sin{(P_{g})}) d\\operatorname{F_{x}}{(P_{g})} = \\int (P_{g} - \\sin{(P_{g})}) d\\operatorname{F_{x}}{(P_{g})}", "derivation": "\\operatorname{F_{x}}{(P_{g})} = P_{g} and \\operatorname{F_{x}}{(P_{g})} - \\sin{(P_{g})} = P_{g} - \\sin{(P_{g})} and \\int (\\operatorname{F_{x}}{(P_{g})} - \\sin{(P_{g})}) dP_{g} = \\int (P_{g} - \\sin{(P_{g})}) dP_{g} and \\int (\\operatorname{F_{x}}{(P_{g})} - \\sin{(P_{g})}) d\\operatorname{F_{x}}{(P_{g})} = \\int (P_{g} - \\sin{(P_{g})}) d\\operatorname{F_{x}}{(P_{g})}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_x')(Symbol('P_g', commutative=True)), Symbol('P_g', commutative=True))"], [["minus", 1, "sin(Symbol('P_g', commutative=True))"], "Equality(Add(Function('F_x')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Add(Symbol('P_g', commutative=True), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))))"], [["integrate", 2, "Symbol('P_g', commutative=True)"], "Equality(Integral(Add(Function('F_x')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Add(Function('F_x')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Tuple(Function('F_x')(Symbol('P_g', commutative=True)))), Integral(Add(Symbol('P_g', commutative=True), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Tuple(Function('F_x')(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mu_0)} = \\sin{(\\mu_0)} and G{(\\mu_0)} = \\frac{e^{\\sin{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}}, then obtain - \\frac{e^{\\sin{(\\mu_0)}}}{\\sin{(\\mu_0)}} + \\frac{e^{\\Psi^{\\dagger}{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} = 0", "derivation": "\\Psi^{\\dagger}{(\\mu_0)} = \\sin{(\\mu_0)} and e^{\\Psi^{\\dagger}{(\\mu_0)}} = e^{\\sin{(\\mu_0)}} and \\frac{e^{\\Psi^{\\dagger}{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} = \\frac{e^{\\sin{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} and G{(\\mu_0)} = \\frac{e^{\\sin{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} and G{(\\mu_0)} - \\frac{e^{\\sin{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} = 0 and G{(\\mu_0)} - \\frac{e^{\\sin{(\\mu_0)}}}{\\sin{(\\mu_0)}} = 0 and - \\frac{e^{\\sin{(\\mu_0)}}}{\\sin{(\\mu_0)}} + \\frac{e^{\\sin{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} = 0 and - \\frac{e^{\\sin{(\\mu_0)}}}{\\sin{(\\mu_0)}} + \\frac{e^{\\Psi^{\\dagger}{(\\mu_0)}}}{\\Psi^{\\dagger}{(\\mu_0)}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True))), exp(sin(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 2, "Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"], ["renaming_premise", "Equality(Function('G')(Symbol('\\\\mu_0', commutative=True)), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 4, "Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Function('G')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\mu_0', commutative=True))))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('G')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), exp(sin(Symbol('\\\\mu_0', commutative=True))), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Integer(-1), exp(sin(Symbol('\\\\mu_0', commutative=True))), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(sin(Symbol('\\\\mu_0', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(-1), exp(sin(Symbol('\\\\mu_0', commutative=True))), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), exp(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True))))), Integer(0))"]]}, {"prompt": "Given z{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\mathbf{B}{(g^{\\prime}_{\\varepsilon})} = \\frac{z{(g^{\\prime}_{\\varepsilon})}}{\\log{(g^{\\prime}_{\\varepsilon})}}, then obtain \\mathbf{B}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} - 1 = 0", "derivation": "z{(g^{\\prime}_{\\varepsilon})} = \\log{(g^{\\prime}_{\\varepsilon})} and \\frac{z{(g^{\\prime}_{\\varepsilon})}}{\\log{(g^{\\prime}_{\\varepsilon})}} = 1 and \\mathbf{B}{(g^{\\prime}_{\\varepsilon})} = \\frac{z{(g^{\\prime}_{\\varepsilon})}}{\\log{(g^{\\prime}_{\\varepsilon})}} and (\\frac{z{(g^{\\prime}_{\\varepsilon})}}{\\log{(g^{\\prime}_{\\varepsilon})}})^{g^{\\prime}_{\\varepsilon}} = 1 and 0 = 1 - (\\frac{z{(g^{\\prime}_{\\varepsilon})}}{\\log{(g^{\\prime}_{\\varepsilon})}})^{g^{\\prime}_{\\varepsilon}} and 0 = 1 - \\mathbf{B}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} and \\mathbf{B}^{g^{\\prime}_{\\varepsilon}}{(g^{\\prime}_{\\varepsilon})} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Mul(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))"], [["minus", 4, "Pow(Mul(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Mul(Function('z')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Pow(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["minus", 6, "Add(Integer(1), Mul(Integer(-1), Pow(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], "Equality(Add(Pow(Function('\\\\mathbf{B}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Integer(0))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{H},c)} = \\frac{e^{c}}{\\hat{H}}, then obtain 2 \\hat{H}^{2} c \\operatorname{a^{\\dagger}}{(\\hat{H},c)} e^{- c} - c = \\hat{H}^{2} c (\\operatorname{a^{\\dagger}}{(\\hat{H},c)} + \\frac{e^{c}}{\\hat{H}}) e^{- c} - c", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{H},c)} = \\frac{e^{c}}{\\hat{H}} and 2 \\operatorname{a^{\\dagger}}{(\\hat{H},c)} = \\operatorname{a^{\\dagger}}{(\\hat{H},c)} + \\frac{e^{c}}{\\hat{H}} and 2 \\hat{H} c \\operatorname{a^{\\dagger}}{(\\hat{H},c)} e^{- c} = \\hat{H} c (\\operatorname{a^{\\dagger}}{(\\hat{H},c)} + \\frac{e^{c}}{\\hat{H}}) e^{- c} and 2 \\hat{H}^{2} c \\operatorname{a^{\\dagger}}{(\\hat{H},c)} e^{- c} = \\hat{H}^{2} c (\\operatorname{a^{\\dagger}}{(\\hat{H},c)} + \\frac{e^{c}}{\\hat{H}}) e^{- c} and 2 \\hat{H}^{2} c \\operatorname{a^{\\dagger}}{(\\hat{H},c)} e^{- c} - c = \\hat{H}^{2} c (\\operatorname{a^{\\dagger}}{(\\hat{H},c)} + \\frac{e^{c}}{\\hat{H}}) e^{- c} - c", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True))))"], [["add", 1, "Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True))), Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Pow(Symbol('c', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True), Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True)))), exp(Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["divide", 3, "Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('c', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('c', commutative=True), Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True)))), exp(Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('c', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('c', commutative=True), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Integer(-1), Symbol('c', commutative=True))), Add(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(2)), Symbol('c', commutative=True), Add(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}', commutative=True), Symbol('c', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), exp(Symbol('c', commutative=True)))), exp(Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Integer(-1), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\Omega{(s)} = \\frac{d}{d s} e^{s}, then derive \\Omega{(s)} = e^{s}, then derive \\Omega^{s}{(s)} = (e^{s})^{s}, then obtain \\frac{\\Omega^{- s}{(s)} (\\frac{d}{d s} \\Omega{(s)})^{s}}{\\cos{(\\cos{(\\rho_f)})}} = \\frac{1}{\\cos{(\\cos{(\\rho_f)})}}", "derivation": "\\Omega{(s)} = \\frac{d}{d s} e^{s} and \\Omega{(s)} = e^{s} and \\Omega^{s}{(s)} = (\\frac{d}{d s} e^{s})^{s} and \\Omega^{s}{(s)} = (e^{s})^{s} and (\\frac{d}{d s} e^{s})^{s} = (e^{s})^{s} and (\\frac{d}{d s} \\Omega{(s)})^{s} = \\Omega^{s}{(s)} and \\frac{\\Omega^{- s}{(s)} (\\frac{d}{d s} \\Omega{(s)})^{s}}{\\cos{(\\cos{(\\rho_f)})}} = \\frac{1}{\\cos{(\\cos{(\\rho_f)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('s', commutative=True)), Derivative(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Omega')(Symbol('s', commutative=True)), exp(Symbol('s', commutative=True)))"], [["power", 1, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\Omega')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Derivative(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Function('\\\\Omega')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(exp(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(exp(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(exp(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Derivative(Function('\\\\Omega')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Function('\\\\Omega')(Symbol('s', commutative=True)), Symbol('s', commutative=True)))"], [["divide", 6, "Mul(Pow(Function('\\\\Omega')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), cos(cos(Symbol('\\\\rho_f', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\Omega')(Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Pow(cos(cos(Symbol('\\\\rho_f', commutative=True))), Integer(-1)), Pow(Derivative(Function('\\\\Omega')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True))), Pow(cos(cos(Symbol('\\\\rho_f', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{M}{(S)} = \\log{(S)} and i{(S)} = \\int (- S + \\mathbf{M}{(S)}) dS, then obtain 1 = \\frac{- \\frac{S^{2}}{2} + S \\log{(S)} - S + \\hat{x}_0}{i{(S)}}", "derivation": "\\mathbf{M}{(S)} = \\log{(S)} and - S + \\mathbf{M}{(S)} = - S + \\log{(S)} and \\int (- S + \\mathbf{M}{(S)}) dS = \\int (- S + \\log{(S)}) dS and i{(S)} = \\int (- S + \\mathbf{M}{(S)}) dS and 1 = \\frac{\\int (- S + \\log{(S)}) dS}{\\int (- S + \\mathbf{M}{(S)}) dS} and 1 = \\frac{\\int (- S + \\log{(S)}) dS}{i{(S)}} and 1 = \\frac{- \\frac{S^{2}}{2} + S \\log{(S)} - S + \\hat{x}_0}{i{(S)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["minus", 1, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\mathbf{M}')(Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('S', commutative=True))))"], [["integrate", 2, "Symbol('S', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\mathbf{M}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('S', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\mathbf{M}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["divide", 3, "Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\mathbf{M}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('\\\\mathbf{M}')(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(1), Mul(Pow(Function('i')(Symbol('S', commutative=True)), Integer(-1)), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True)))))"], [["evaluate_integrals", 6], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('S', commutative=True), Integer(2))), Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Function('i')(Symbol('S', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\delta{(u,\\varphi)} = \\varphi - u, then derive \\frac{\\partial}{\\partial u} \\delta{(u,\\varphi)} = -1, then obtain \\frac{\\partial}{\\partial u} (\\varphi - u) + \\int J^{\\rho} dJ = \\int J^{\\rho} dJ - 1", "derivation": "\\delta{(u,\\varphi)} = \\varphi - u and \\frac{\\partial}{\\partial u} \\delta{(u,\\varphi)} = \\frac{\\partial}{\\partial u} (\\varphi - u) and \\frac{\\partial}{\\partial u} \\delta{(u,\\varphi)} = -1 and \\frac{\\partial}{\\partial u} (\\varphi - u) = -1 and \\frac{\\partial}{\\partial u} (\\varphi - u) + \\int J^{\\rho} dJ = \\int J^{\\rho} dJ - 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\delta')(Symbol('u', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('u', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('u', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1))"], [["add", 4, "Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Add(Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('J', commutative=True)))), Add(Integral(Pow(Symbol('J', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('J', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(E_{x})} = \\cos{(E_{x})}, then obtain 2 \\operatorname{E_{\\lambda}}{(E_{x})} + \\cos{(E_{x})} = 3 \\cos{(E_{x})}", "derivation": "\\operatorname{E_{\\lambda}}{(E_{x})} = \\cos{(E_{x})} and \\operatorname{E_{\\lambda}}{(E_{x})} + \\cos{(E_{x})} = 2 \\cos{(E_{x})} and \\operatorname{E_{\\lambda}}{(E_{x})} + 2 \\cos{(E_{x})} = 3 \\cos{(E_{x})} and 2 \\operatorname{E_{\\lambda}}{(E_{x})} + \\cos{(E_{x})} = 3 \\cos{(E_{x})}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True)))"], [["add", 1, "cos(Symbol('E_x', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('E_x', commutative=True)), cos(Symbol('E_x', commutative=True))), Mul(Integer(2), cos(Symbol('E_x', commutative=True))))"], [["add", 1, "Mul(Integer(2), cos(Symbol('E_x', commutative=True)))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('E_x', commutative=True)), Mul(Integer(2), cos(Symbol('E_x', commutative=True)))), Mul(Integer(3), cos(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('E_x', commutative=True))), cos(Symbol('E_x', commutative=True))), Mul(Integer(3), cos(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\operatorname{f_{E}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})}, then obtain \\iint \\operatorname{f_{E}}{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "derivation": "\\hat{\\mathbf{r}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\operatorname{f_{E}}{(g_{\\varepsilon})} = \\cos{(g_{\\varepsilon})} and \\int \\hat{\\mathbf{r}}{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} and \\hat{\\mathbf{r}}{(g_{\\varepsilon})} = \\operatorname{f_{E}}{(g_{\\varepsilon})} and \\int \\operatorname{f_{E}}{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} and \\iint \\operatorname{f_{E}}{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon} = \\iint \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('f_E')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Function('f_E')(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integral(Function('f_E')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 5, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given h{(c_{0})} = \\sin{(c_{0})}, then derive - \\frac{h{(c_{0})} \\cos{(c_{0})}}{\\sin^{2}{(c_{0})}} + \\frac{\\frac{d}{d c_{0}} h{(c_{0})}}{\\sin{(c_{0})}} = 0, then obtain - \\frac{\\cos{(c_{0})}}{\\sin{(c_{0})}} + \\frac{\\frac{d}{d c_{0}} \\sin{(c_{0})}}{\\sin{(c_{0})}} = 0", "derivation": "h{(c_{0})} = \\sin{(c_{0})} and \\frac{h{(c_{0})}}{\\sin{(c_{0})}} = 1 and \\frac{d}{d c_{0}} \\frac{h{(c_{0})}}{\\sin{(c_{0})}} = \\frac{d}{d c_{0}} 1 and - \\frac{h{(c_{0})} \\cos{(c_{0})}}{\\sin^{2}{(c_{0})}} + \\frac{\\frac{d}{d c_{0}} h{(c_{0})}}{\\sin{(c_{0})}} = 0 and - \\frac{\\cos{(c_{0})}}{\\sin{(c_{0})}} + \\frac{\\frac{d}{d c_{0}} \\sin{(c_{0})}}{\\sin{(c_{0})}} = 0", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["divide", 1, "sin(Symbol('c_0', commutative=True))"], "Equality(Mul(Function('h')(Symbol('c_0', commutative=True)), Pow(sin(Symbol('c_0', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Mul(Function('h')(Symbol('c_0', commutative=True)), Pow(sin(Symbol('c_0', commutative=True)), Integer(-1))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('h')(Symbol('c_0', commutative=True)), Pow(sin(Symbol('c_0', commutative=True)), Integer(-2)), cos(Symbol('c_0', commutative=True))), Mul(Pow(sin(Symbol('c_0', commutative=True)), Integer(-1)), Derivative(Function('h')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(sin(Symbol('c_0', commutative=True)), Integer(-1)), cos(Symbol('c_0', commutative=True))), Mul(Pow(sin(Symbol('c_0', commutative=True)), Integer(-1)), Derivative(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(L_{\\varepsilon},h)} = L_{\\varepsilon} h, then obtain - \\sin{(\\frac{\\operatorname{F_{g}}{(L_{\\varepsilon},h)}}{L_{\\varepsilon} h})} = - \\sin{(1)}", "derivation": "\\operatorname{F_{g}}{(L_{\\varepsilon},h)} = L_{\\varepsilon} h and \\frac{\\operatorname{F_{g}}{(L_{\\varepsilon},h)}}{L_{\\varepsilon}} = h and \\frac{\\operatorname{F_{g}}{(L_{\\varepsilon},h)}}{L_{\\varepsilon} h} = 1 and - \\frac{\\operatorname{F_{g}}{(L_{\\varepsilon},h)}}{L_{\\varepsilon} h} = -1 and - \\sin{(\\frac{\\operatorname{F_{g}}{(L_{\\varepsilon},h)}}{L_{\\varepsilon} h})} = - \\sin{(1)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True)))"], [["divide", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True))), Symbol('h', commutative=True))"], [["divide", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True))), Integer(1))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True))), Integer(-1))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Function('F_g')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('h', commutative=True))))), Mul(Integer(-1), sin(Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(v_{y},\\omega)} = \\omega + v_{y} and i{(v_{z},\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{v_{z}}, then obtain (\\omega \\mathbf{F}{(v_{y},\\omega)})^{\\omega} + e^{\\Psi_{\\lambda}^{v_{z}}} = (\\omega (\\omega + v_{y}))^{\\omega} + e^{\\Psi_{\\lambda}^{v_{z}}}", "derivation": "\\mathbf{F}{(v_{y},\\omega)} = \\omega + v_{y} and i{(v_{z},\\Psi_{\\lambda})} = \\Psi_{\\lambda}^{v_{z}} and e^{i{(v_{z},\\Psi_{\\lambda})}} = e^{\\Psi_{\\lambda}^{v_{z}}} and \\omega \\mathbf{F}{(v_{y},\\omega)} = \\omega (\\omega + v_{y}) and (\\omega \\mathbf{F}{(v_{y},\\omega)})^{\\omega} = (\\omega (\\omega + v_{y}))^{\\omega} and (\\omega \\mathbf{F}{(v_{y},\\omega)})^{\\omega} + e^{i{(v_{z},\\Psi_{\\lambda})}} = (\\omega (\\omega + v_{y}))^{\\omega} + e^{i{(v_{z},\\Psi_{\\lambda})}} and (\\omega \\mathbf{F}{(v_{y},\\omega)})^{\\omega} + e^{\\Psi_{\\lambda}^{v_{z}}} = (\\omega (\\omega + v_{y}))^{\\omega} + e^{\\Psi_{\\lambda}^{v_{z}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True)))"], ["get_premise", "Equality(Function('i')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)))"], [["exp", 2], "Equality(exp(Function('i')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), exp(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True))))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))))"], [["power", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))), Symbol('\\\\omega', commutative=True)))"], [["add", 5, "exp(Function('i')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), exp(Function('i')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))), Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))), Symbol('\\\\omega', commutative=True)), exp(Function('i')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True)), exp(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)))), Add(Pow(Mul(Symbol('\\\\omega', commutative=True), Add(Symbol('\\\\omega', commutative=True), Symbol('v_y', commutative=True))), Symbol('\\\\omega', commutative=True)), exp(Pow(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\pi{(\\rho_f)} = \\cos{(\\rho_f)} and \\hat{X}{(\\rho_f)} = \\frac{1}{\\rho_f}, then obtain 0 = (- \\hat{X}{(\\rho_f)} + \\frac{1}{\\rho_f}) \\pi{(\\rho_f)}", "derivation": "\\pi{(\\rho_f)} = \\cos{(\\rho_f)} and \\hat{X}{(\\rho_f)} = \\frac{1}{\\rho_f} and \\hat{X}{(\\rho_f)} - \\frac{1}{\\rho_f} = 0 and \\hat{X}{(\\rho_f)} + \\cos{(\\rho_f)} - \\frac{1}{\\rho_f} = \\cos{(\\rho_f)} and - \\pi{(\\rho_f)} + \\cos{(\\rho_f)} = - \\hat{X}{(\\rho_f)} - \\pi{(\\rho_f)} + \\cos{(\\rho_f)} + \\frac{1}{\\rho_f} and 0 = - \\hat{X}{(\\rho_f)} + \\frac{1}{\\rho_f} and 0 = (- \\hat{X}{(\\rho_f)} + \\frac{1}{\\rho_f}) \\pi{(\\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))"], [["minus", 2, "Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))), Integer(0))"], [["add", 3, "cos(Symbol('\\\\rho_f', commutative=True))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1)))), cos(Symbol('\\\\rho_f', commutative=True)))"], [["minus", 4, "Add(Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True)), Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True))), cos(Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True))), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))))"], [["divide", 6, "Pow(Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True)), Integer(-1))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\rho_f', commutative=True))), Pow(Symbol('\\\\rho_f', commutative=True), Integer(-1))), Function('\\\\pi')(Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(x,\\phi)} = \\cos{(\\phi + x)}, then obtain x + \\operatorname{r_{0}}{(x,\\phi)} \\operatorname{r_{0}}^{x}{(x,\\phi)} = x + \\operatorname{r_{0}}^{x}{(x,\\phi)} \\cos{(\\phi + x)}", "derivation": "\\operatorname{r_{0}}{(x,\\phi)} = \\cos{(\\phi + x)} and \\operatorname{r_{0}}^{x}{(x,\\phi)} = \\cos^{x}{(\\phi + x)} and \\operatorname{r_{0}}{(x,\\phi)} \\cos^{x}{(\\phi + x)} = \\cos{(\\phi + x)} \\cos^{x}{(\\phi + x)} and \\operatorname{r_{0}}{(x,\\phi)} \\operatorname{r_{0}}^{x}{(x,\\phi)} = \\operatorname{r_{0}}^{x}{(x,\\phi)} \\cos{(\\phi + x)} and x + \\operatorname{r_{0}}{(x,\\phi)} \\operatorname{r_{0}}^{x}{(x,\\phi)} = x + \\operatorname{r_{0}}^{x}{(x,\\phi)} \\cos{(\\phi + x)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('x', commutative=True)), Pow(cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["times", 1, "Pow(cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True))"], "Equality(Mul(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True))), Mul(cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))), Pow(cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('x', commutative=True))), Mul(Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('x', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True)))))"], [["add", 4, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Mul(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('x', commutative=True)))), Add(Symbol('x', commutative=True), Mul(Pow(Function('r_0')(Symbol('x', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('x', commutative=True)), cos(Add(Symbol('\\\\phi', commutative=True), Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\varepsilon_{0}{(m_{s})} = \\sin{(\\sin{(m_{s})})} and \\mathbf{H}{(m_{s})} = - \\varepsilon_{0}{(m_{s})}, then obtain - \\sin{(\\sin{(m_{s})})} = \\varepsilon_{0}{(m_{s})} - 2 \\sin{(\\sin{(m_{s})})}", "derivation": "\\varepsilon_{0}{(m_{s})} = \\sin{(\\sin{(m_{s})})} and \\mathbf{H}{(m_{s})} = - \\varepsilon_{0}{(m_{s})} and \\mathbf{H}{(m_{s})} = - \\sin{(\\sin{(m_{s})})} and - \\varepsilon_{0}{(m_{s})} = - \\sin{(\\sin{(m_{s})})} and - \\sin{(\\sin{(m_{s})})} = \\varepsilon_{0}{(m_{s})} - 2 \\sin{(\\sin{(m_{s})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)), sin(sin(Symbol('m_s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('m_s', commutative=True)), Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{H}')(Symbol('m_s', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('m_s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True))), Mul(Integer(-1), sin(sin(Symbol('m_s', commutative=True)))))"], [["minus", 4, "Add(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True))), sin(sin(Symbol('m_s', commutative=True))))"], "Equality(Mul(Integer(-1), sin(sin(Symbol('m_s', commutative=True)))), Add(Function('\\\\varepsilon_0')(Symbol('m_s', commutative=True)), Mul(Integer(-1), Integer(2), sin(sin(Symbol('m_s', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{J}_P{(F_{g},v)} = F_{g} + v, then obtain ((\\frac{1}{v})^{v})^{v} (\\frac{1}{v})^{v} = (\\frac{1}{v})^{v} ((\\frac{F_{g} + v}{v \\mathbf{J}_P{(F_{g},v)}})^{v})^{v}", "derivation": "\\mathbf{J}_P{(F_{g},v)} = F_{g} + v and \\frac{\\mathbf{J}_P{(F_{g},v)}}{v} = \\frac{F_{g} + v}{v} and \\frac{1}{v} = \\frac{F_{g} + v}{v \\mathbf{J}_P{(F_{g},v)}} and (\\frac{1}{v})^{v} = (\\frac{F_{g} + v}{v \\mathbf{J}_P{(F_{g},v)}})^{v} and ((\\frac{1}{v})^{v})^{v} = ((\\frac{F_{g} + v}{v \\mathbf{J}_P{(F_{g},v)}})^{v})^{v} and ((\\frac{1}{v})^{v})^{v} (\\frac{1}{v})^{v} = (\\frac{1}{v})^{v} ((\\frac{F_{g} + v}{v \\mathbf{J}_P{(F_{g},v)}})^{v})^{v}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True)))"], [["divide", 1, "Symbol('v', commutative=True)"], "Equality(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True))))"], [["divide", 2, "Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True))"], "Equality(Pow(Symbol('v', commutative=True), Integer(-1)), Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('v', commutative=True)"], "Equality(Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Symbol('v', commutative=True)))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["times", 5, "Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True))), Mul(Pow(Pow(Symbol('v', commutative=True), Integer(-1)), Symbol('v', commutative=True)), Pow(Pow(Mul(Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Pow(Function('\\\\mathbf{J}_P')(Symbol('F_g', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Symbol('v', commutative=True)), Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(J,\\theta)} = J + \\theta and \\theta{(f^{\\prime})} = f^{\\prime}, then derive 0 = - (\\frac{\\partial}{\\partial J} \\operatorname{n_{1}}{(J,\\theta)} - 1) \\frac{d}{d f^{\\prime}} \\theta{(f^{\\prime})}, then obtain 0 = - (\\frac{\\partial}{\\partial J} (J + \\theta) - 1) \\frac{d}{d f^{\\prime}} \\theta{(f^{\\prime})}", "derivation": "\\operatorname{n_{1}}{(J,\\theta)} = J + \\theta and 0 = J + \\theta - \\operatorname{n_{1}}{(J,\\theta)} and \\theta{(f^{\\prime})} = f^{\\prime} and 0 = f^{\\prime} (J + \\theta - \\operatorname{n_{1}}{(J,\\theta)}) and 0 = (J + \\theta - \\operatorname{n_{1}}{(J,\\theta)}) \\theta{(f^{\\prime})} and \\frac{d}{d f^{\\prime}} 0 = \\frac{\\partial}{\\partial f^{\\prime}} (J + \\theta - \\operatorname{n_{1}}{(J,\\theta)}) \\theta{(f^{\\prime})} and \\frac{d^{2}}{d Jd f^{\\prime}} 0 = \\frac{\\partial^{2}}{\\partial J\\partial f^{\\prime}} (J + \\theta - \\operatorname{n_{1}}{(J,\\theta)}) \\theta{(f^{\\prime})} and 0 = - (\\frac{\\partial}{\\partial J} \\operatorname{n_{1}}{(J,\\theta)} - 1) \\frac{d}{d f^{\\prime}} \\theta{(f^{\\prime})} and 0 = - (\\frac{\\partial}{\\partial J} (J + \\theta) - 1) \\frac{d}{d f^{\\prime}} \\theta{(f^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["minus", 1, "Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Integer(0), Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["times", 2, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('f^{\\\\prime}', commutative=True), Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(0), Mul(Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)))), Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True))))"], [["differentiate", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)))), Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('J', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)))), Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Mul(Integer(-1), Add(Derivative(Function('n_1')(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 8, 1], "Equality(Integer(0), Mul(Integer(-1), Add(Derivative(Add(Symbol('J', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\mathbf{P},M_{E})} = M_{E} + \\mathbf{P}, then obtain - M_{E} - \\mathbf{P} + 2 S^{2}{(\\mathbf{P},M_{E})} = - M_{E} - \\mathbf{P} + 2 (M_{E} + \\mathbf{P}) S{(\\mathbf{P},M_{E})}", "derivation": "S{(\\mathbf{P},M_{E})} = M_{E} + \\mathbf{P} and 2 S{(\\mathbf{P},M_{E})} = M_{E} + \\mathbf{P} + S{(\\mathbf{P},M_{E})} and (M_{E} + \\mathbf{P} + S{(\\mathbf{P},M_{E})}) S{(\\mathbf{P},M_{E})} = (M_{E} + \\mathbf{P}) (M_{E} + \\mathbf{P} + S{(\\mathbf{P},M_{E})}) and - M_{E} - \\mathbf{P} + (M_{E} + \\mathbf{P} + S{(\\mathbf{P},M_{E})}) S{(\\mathbf{P},M_{E})} = - M_{E} - \\mathbf{P} + (M_{E} + \\mathbf{P}) (M_{E} + \\mathbf{P} + S{(\\mathbf{P},M_{E})}) and - M_{E} - \\mathbf{P} + 2 S^{2}{(\\mathbf{P},M_{E})} = - M_{E} - \\mathbf{P} + 2 (M_{E} + \\mathbf{P}) S{(\\mathbf{P},M_{E})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 1, "Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Mul(Integer(2), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))))"], [["times", 1, "Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)))"], "Equality(Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))), Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)))))"], [["minus", 3, "Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Pow(Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(2), Add(Symbol('M_E', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('M_E', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} = Z + \\hat{H}_{\\lambda} c, then obtain (\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} + 1)^{c} = (\\hat{H}_{\\lambda} (Z + \\hat{H}_{\\lambda} c) + 1)^{c}", "derivation": "\\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} = Z + \\hat{H}_{\\lambda} c and \\hat{H}_{\\lambda} \\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} = \\hat{H}_{\\lambda} (Z + \\hat{H}_{\\lambda} c) and \\hat{H}_{\\lambda} \\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} + 1 = \\hat{H}_{\\lambda} (Z + \\hat{H}_{\\lambda} c) + 1 and (\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(c,\\hat{H}_{\\lambda},Z)} + 1)^{c} = (\\hat{H}_{\\lambda} (Z + \\hat{H}_{\\lambda} c) + 1)^{c}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Add(Symbol('Z', commutative=True), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Symbol('Z', commutative=True), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True))), Integer(1)), Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Symbol('Z', commutative=True), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)))), Integer(1)))"], [["power", 3, "Symbol('c', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True))), Integer(1)), Symbol('c', commutative=True)), Pow(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Add(Symbol('Z', commutative=True), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)))), Integer(1)), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\dot{z}{(g)} = \\log{(g)}, then derive \\int \\dot{z}{(g)} dg = \\hbar + g \\log{(g)} - g, then derive \\hat{\\mathbf{x}} + g \\log{(g)} - g = \\hbar + g \\log{(g)} - g, then obtain (\\hat{\\mathbf{x}} + g \\log{(g)} - g)^{\\hbar} = (\\hbar + g \\log{(g)} - g)^{\\hbar}", "derivation": "\\dot{z}{(g)} = \\log{(g)} and \\int \\dot{z}{(g)} dg = \\int \\log{(g)} dg and \\int \\dot{z}{(g)} dg = \\hbar + g \\log{(g)} - g and \\int \\log{(g)} dg = \\hbar + g \\log{(g)} - g and \\hat{\\mathbf{x}} + g \\log{(g)} - g = \\hbar + g \\log{(g)} - g and (\\hat{\\mathbf{x}} + g \\log{(g)} - g)^{\\hbar} = (\\hbar + g \\log{(g)} - g)^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["integrate", 1, "Symbol('g', commutative=True)"], "Equality(Integral(Function('\\\\dot{z}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{z}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))))"], [["power", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Mul(Integer(-1), Symbol('g', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(C,\\mathbf{E})} = - C + \\mathbf{E} and \\psi^{*}{(C,\\mathbf{E})} = - \\frac{\\mathbf{E} - \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}}, then obtain \\psi^{*}{(C,\\mathbf{E})} = \\frac{- \\mathbf{E} + \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}}", "derivation": "\\hat{H}{(C,\\mathbf{E})} = - C + \\mathbf{E} and - \\mathbf{E} + \\hat{H}{(C,\\mathbf{E})} = - C and \\frac{- \\mathbf{E} + \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}} = - \\frac{C}{\\mathbf{E}} and \\frac{- \\mathbf{E} + \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}} = - \\frac{\\mathbf{E} - \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}} and \\psi^{*}{(C,\\mathbf{E})} = - \\frac{\\mathbf{E} - \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}} and \\psi^{*}{(C,\\mathbf{E})} = \\frac{- \\mathbf{E} + \\hat{H}{(C,\\mathbf{E})}}{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)))"], [["divide", 2, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Integer(-1), Symbol('C', commutative=True), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('\\\\psi^*')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{E}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('\\\\hat{H}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(b,J_{\\varepsilon})} = b e^{J_{\\varepsilon}} and n{(J_{\\varepsilon})} = e^{J_{\\varepsilon}}, then obtain - J_{\\varepsilon} + \\frac{\\mathbf{A}{(b,J_{\\varepsilon})}}{J_{\\varepsilon}} = - J_{\\varepsilon} + \\frac{b n{(J_{\\varepsilon})}}{J_{\\varepsilon}}", "derivation": "\\mathbf{A}{(b,J_{\\varepsilon})} = b e^{J_{\\varepsilon}} and n{(J_{\\varepsilon})} = e^{J_{\\varepsilon}} and \\frac{\\mathbf{A}{(b,J_{\\varepsilon})}}{J_{\\varepsilon}} = \\frac{b e^{J_{\\varepsilon}}}{J_{\\varepsilon}} and - J_{\\varepsilon} + \\frac{\\mathbf{A}{(b,J_{\\varepsilon})}}{J_{\\varepsilon}} = - J_{\\varepsilon} + \\frac{b e^{J_{\\varepsilon}}}{J_{\\varepsilon}} and - J_{\\varepsilon} + \\frac{\\mathbf{A}{(b,J_{\\varepsilon})}}{J_{\\varepsilon}} = - J_{\\varepsilon} + \\frac{b n{(J_{\\varepsilon})}}{J_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('b', commutative=True), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('n')(Symbol('J_{\\\\varepsilon}', commutative=True)), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True), exp(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True), exp(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('b', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('b', commutative=True), Function('n')(Symbol('J_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given h{(t_{1})} = \\sin{(t_{1})}, then obtain h^{3}{(t_{1})} \\sin{(t_{1})} = h^{2}{(t_{1})} \\sin^{2}{(t_{1})}", "derivation": "h{(t_{1})} = \\sin{(t_{1})} and h{(t_{1})} \\sin{(t_{1})} = \\sin^{2}{(t_{1})} and h^{2}{(t_{1})} \\sin^{2}{(t_{1})} = \\sin^{4}{(t_{1})} and h^{3}{(t_{1})} \\sin{(t_{1})} = h^{2}{(t_{1})} \\sin^{2}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True)))"], [["times", 1, "sin(Symbol('t_1', commutative=True))"], "Equality(Mul(Function('h')(Symbol('t_1', commutative=True)), sin(Symbol('t_1', commutative=True))), Pow(sin(Symbol('t_1', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('h')(Symbol('t_1', commutative=True)), Integer(2)), Pow(sin(Symbol('t_1', commutative=True)), Integer(2))), Pow(sin(Symbol('t_1', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('h')(Symbol('t_1', commutative=True)), Integer(3)), sin(Symbol('t_1', commutative=True))), Mul(Pow(Function('h')(Symbol('t_1', commutative=True)), Integer(2)), Pow(sin(Symbol('t_1', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\varepsilon_{0}{(F_{c})} = \\sin{(F_{c})}, then obtain (\\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\varepsilon_{0}{(F_{c})})})^{F_{c}} = (\\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\sin{(F_{c})})})^{F_{c}}", "derivation": "\\varepsilon_{0}{(F_{c})} = \\sin{(F_{c})} and \\cos{(\\varepsilon_{0}{(F_{c})})} = \\cos{(\\sin{(F_{c})})} and \\cos^{F_{c}}{(\\varepsilon_{0}{(F_{c})})} = \\cos^{F_{c}}{(\\sin{(F_{c})})} and \\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\varepsilon_{0}{(F_{c})})} = \\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\sin{(F_{c})})} and (\\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\varepsilon_{0}{(F_{c})})})^{F_{c}} = (\\sin^{- F_{c}}{(F_{c})} \\cos^{F_{c}}{(\\sin{(F_{c})})})^{F_{c}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('F_c', commutative=True)), sin(Symbol('F_c', commutative=True)))"], [["cos", 1], "Equality(cos(Function('\\\\varepsilon_0')(Symbol('F_c', commutative=True))), cos(sin(Symbol('F_c', commutative=True))))"], [["power", 2, "Symbol('F_c', commutative=True)"], "Equality(Pow(cos(Function('\\\\varepsilon_0')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(cos(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"], [["divide", 3, "Pow(sin(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(cos(Function('\\\\varepsilon_0')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Mul(Pow(sin(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(cos(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))))"], [["power", 4, "Symbol('F_c', commutative=True)"], "Equality(Pow(Mul(Pow(sin(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(cos(Function('\\\\varepsilon_0')(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)), Pow(Mul(Pow(sin(Symbol('F_c', commutative=True)), Mul(Integer(-1), Symbol('F_c', commutative=True))), Pow(cos(sin(Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True))), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(P_{e},\\mathbf{f})} = \\frac{P_{e}}{\\mathbf{f}}, then obtain (\\int \\frac{\\mu_{0}{(P_{e},\\mathbf{f})} - 1}{\\mathbf{f}} d\\mathbf{f})^{\\mathbf{f}} = (\\int \\frac{\\frac{P_{e}}{\\mathbf{f}} - 1}{\\mathbf{f}} d\\mathbf{f})^{\\mathbf{f}}", "derivation": "\\mu_{0}{(P_{e},\\mathbf{f})} = \\frac{P_{e}}{\\mathbf{f}} and \\mu_{0}{(P_{e},\\mathbf{f})} - 1 = \\frac{P_{e}}{\\mathbf{f}} - 1 and \\frac{\\mu_{0}{(P_{e},\\mathbf{f})} - 1}{\\mathbf{f}} = \\frac{\\frac{P_{e}}{\\mathbf{f}} - 1}{\\mathbf{f}} and \\int \\frac{\\mu_{0}{(P_{e},\\mathbf{f})} - 1}{\\mathbf{f}} d\\mathbf{f} = \\int \\frac{\\frac{P_{e}}{\\mathbf{f}} - 1}{\\mathbf{f}} d\\mathbf{f} and (\\int \\frac{\\mu_{0}{(P_{e},\\mathbf{f})} - 1}{\\mathbf{f}} d\\mathbf{f})^{\\mathbf{f}} = (\\int \\frac{\\frac{P_{e}}{\\mathbf{f}} - 1}{\\mathbf{f}} d\\mathbf{f})^{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mu_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1)), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Integer(-1)))"], [["divide", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Function('\\\\mu_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Function('\\\\mu_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Integer(-1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["power", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Function('\\\\mu_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Integral(Mul(Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), Add(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))), Integer(-1))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True)))"]]}, {"prompt": "Given \\chi{(\\mu_0)} = e^{\\sin{(\\mu_0)}}, then obtain \\chi{(\\mu_0)} + e^{\\sin{(\\mu_0)}} \\sin{(\\mu_0)} = e^{\\sin{(\\mu_0)}} \\sin{(\\mu_0)} + e^{\\sin{(\\mu_0)}}", "derivation": "\\chi{(\\mu_0)} = e^{\\sin{(\\mu_0)}} and \\chi{(\\mu_0)} \\sin{(\\mu_0)} = e^{\\sin{(\\mu_0)}} \\sin{(\\mu_0)} and \\chi{(\\mu_0)} \\sin{(\\mu_0)} + \\chi{(\\mu_0)} = \\chi{(\\mu_0)} \\sin{(\\mu_0)} + e^{\\sin{(\\mu_0)}} and \\chi{(\\mu_0)} + e^{\\sin{(\\mu_0)}} \\sin{(\\mu_0)} = e^{\\sin{(\\mu_0)}} \\sin{(\\mu_0)} + e^{\\sin{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), exp(sin(Symbol('\\\\mu_0', commutative=True))))"], [["times", 1, "sin(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Mul(exp(sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))))"], [["add", 1, "Mul(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\chi')(Symbol('\\\\mu_0', commutative=True)), Mul(exp(sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(exp(sin(Symbol('\\\\mu_0', commutative=True))), sin(Symbol('\\\\mu_0', commutative=True))), exp(sin(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\theta{(b,M)} = \\sin^{M}{(b)}, then derive 0 = \\frac{\\log{(\\sin{(b)})} \\sin^{M}{(b)}}{\\theta{(b,M)}} - \\frac{\\sin^{M}{(b)} \\frac{\\partial}{\\partial M} \\theta{(b,M)}}{\\theta^{2}{(b,M)}}, then obtain 0 = - \\frac{(\\log{(\\sin{(b)})} - \\frac{\\frac{\\partial}{\\partial M} \\theta{(b,M)}}{\\theta{(b,M)}}) \\theta{(b,M)}}{\\frac{\\partial}{\\partial M} \\theta{(b,M)}}", "derivation": "\\theta{(b,M)} = \\sin^{M}{(b)} and 1 = \\frac{\\sin^{M}{(b)}}{\\theta{(b,M)}} and \\frac{d}{d M} 1 = \\frac{\\partial}{\\partial M} \\frac{\\sin^{M}{(b)}}{\\theta{(b,M)}} and 0 = \\frac{\\log{(\\sin{(b)})} \\sin^{M}{(b)}}{\\theta{(b,M)}} - \\frac{\\sin^{M}{(b)} \\frac{\\partial}{\\partial M} \\theta{(b,M)}}{\\theta^{2}{(b,M)}} and 0 = \\log{(\\sin{(b)})} - \\frac{\\frac{\\partial}{\\partial M} \\theta{(b,M)}}{\\theta{(b,M)}} and 0 = - \\frac{(\\log{(\\sin{(b)})} - \\frac{\\frac{\\partial}{\\partial M} \\theta{(b,M)}}{\\theta{(b,M)}}) \\theta{(b,M)}}{\\frac{\\partial}{\\partial M} \\theta{(b,M)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Pow(sin(Symbol('b', commutative=True)), Symbol('M', commutative=True)))"], [["divide", 1, "Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(sin(Symbol('b', commutative=True)), Symbol('M', commutative=True))))"], [["differentiate", 2, "Symbol('M', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(sin(Symbol('b', commutative=True)), Symbol('M', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), log(sin(Symbol('b', commutative=True))), Pow(sin(Symbol('b', commutative=True)), Symbol('M', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-2)), Pow(sin(Symbol('b', commutative=True)), Symbol('M', commutative=True)), Derivative(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(0), Add(log(sin(Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))))"], [["divide", 5, "Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], "Equality(Integer(0), Mul(Integer(-1), Add(log(sin(Symbol('b', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Derivative(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))), Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Pow(Derivative(Function('\\\\theta')(Symbol('b', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{M}{(Q)} = \\sin{(\\log{(Q)})} and \\Psi{(J_{\\varepsilon})} = \\log{(\\log{(J_{\\varepsilon})})}, then obtain \\iint \\frac{Q \\Psi{(J_{\\varepsilon})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} dQ = \\iint \\frac{Q \\log{(\\log{(J_{\\varepsilon})})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} dQ", "derivation": "\\mathbf{M}{(Q)} = \\sin{(\\log{(Q)})} and \\Psi{(J_{\\varepsilon})} = \\log{(\\log{(J_{\\varepsilon})})} and \\frac{Q \\Psi{(J_{\\varepsilon})}}{\\mathbf{M}{(Q)}} = \\frac{Q \\log{(\\log{(J_{\\varepsilon})})}}{\\mathbf{M}{(Q)}} and \\frac{Q \\Psi{(J_{\\varepsilon})}}{\\sin{(\\log{(Q)})}} = \\frac{Q \\log{(\\log{(J_{\\varepsilon})})}}{\\sin{(\\log{(Q)})}} and \\int \\frac{Q \\Psi{(J_{\\varepsilon})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} = \\int \\frac{Q \\log{(\\log{(J_{\\varepsilon})})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} and \\iint \\frac{Q \\Psi{(J_{\\varepsilon})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} dQ = \\iint \\frac{Q \\log{(\\log{(J_{\\varepsilon})})}}{\\sin{(\\log{(Q)})}} dJ_{\\varepsilon} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('Q', commutative=True)), sin(log(Symbol('Q', commutative=True))))"], ["get_premise", "Equality(Function('\\\\Psi')(Symbol('J_{\\\\varepsilon}', commutative=True)), log(log(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('Q', commutative=True)))"], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\Psi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Function('\\\\mathbf{M}')(Symbol('Q', commutative=True)), Integer(-1))), Mul(Symbol('Q', commutative=True), Pow(Function('\\\\mathbf{M}')(Symbol('Q', commutative=True)), Integer(-1)), log(log(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('Q', commutative=True), Function('\\\\Psi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))), Mul(Symbol('Q', commutative=True), log(log(Symbol('J_{\\\\varepsilon}', commutative=True))), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\Psi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), log(log(Symbol('J_{\\\\varepsilon}', commutative=True))), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Mul(Symbol('Q', commutative=True), Function('\\\\Psi')(Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Symbol('Q', commutative=True), log(log(Symbol('J_{\\\\varepsilon}', commutative=True))), Pow(sin(log(Symbol('Q', commutative=True))), Integer(-1))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given q{(\\hat{p},n_{2})} = \\hat{p} + \\log{(n_{2})}, then obtain \\frac{\\partial}{\\partial n_{2}} (\\hat{p} + \\log{(n_{2})}) + \\frac{\\partial}{\\partial \\hat{p}} q{(\\hat{p},n_{2})} - 1 = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\log{(n_{2})}) + \\frac{\\partial}{\\partial n_{2}} (\\hat{p} + \\log{(n_{2})}) - 1", "derivation": "q{(\\hat{p},n_{2})} = \\hat{p} + \\log{(n_{2})} and \\frac{\\partial}{\\partial \\hat{p}} q{(\\hat{p},n_{2})} = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\log{(n_{2})}) and \\frac{\\partial}{\\partial \\hat{p}} q{(\\hat{p},n_{2})} - 1 = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\log{(n_{2})}) - 1 and \\frac{\\partial}{\\partial n_{2}} (\\hat{p} + \\log{(n_{2})}) + \\frac{\\partial}{\\partial \\hat{p}} q{(\\hat{p},n_{2})} - 1 = \\frac{\\partial}{\\partial \\hat{p}} (\\hat{p} + \\log{(n_{2})}) + \\frac{\\partial}{\\partial n_{2}} (\\hat{p} + \\log{(n_{2})}) - 1", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('q')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)))"], [["add", 3, "Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Function('q')(Symbol('\\\\hat{p}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{p}', commutative=True), log(Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\mu_{0}{(x,p)} = \\sin{(p^{x})} and y{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and S{(x,\\mathbf{J}_P,p)} = y{(\\mathbf{J}_P)} \\sin{(p^{x})}, then obtain S{(x,\\mathbf{J}_P,p)} = e^{\\mathbf{J}_P} \\sin{(p^{x})}", "derivation": "\\mu_{0}{(x,p)} = \\sin{(p^{x})} and y{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and \\mu_{0}{(x,p)} y{(\\mathbf{J}_P)} = \\mu_{0}{(x,p)} e^{\\mathbf{J}_P} and y{(\\mathbf{J}_P)} \\sin{(p^{x})} = e^{\\mathbf{J}_P} \\sin{(p^{x})} and S{(x,\\mathbf{J}_P,p)} = y{(\\mathbf{J}_P)} \\sin{(p^{x})} and S{(x,\\mathbf{J}_P,p)} = e^{\\mathbf{J}_P} \\sin{(p^{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('p', commutative=True)), sin(Pow(Symbol('p', commutative=True), Symbol('x', commutative=True))))"], ["get_premise", "Equality(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["times", 2, "Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('p', commutative=True)), Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Function('\\\\mu_0')(Symbol('x', commutative=True), Symbol('p', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Pow(Symbol('p', commutative=True), Symbol('x', commutative=True)))), Mul(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Pow(Symbol('p', commutative=True), Symbol('x', commutative=True)))))"], ["renaming_premise", "Equality(Function('S')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)), Mul(Function('y')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Pow(Symbol('p', commutative=True), Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('S')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('p', commutative=True)), Mul(exp(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Pow(Symbol('p', commutative=True), Symbol('x', commutative=True)))))"]]}, {"prompt": "Given k{(\\hat{x},f^{*})} = \\hat{x} - f^{*}, then derive e^{\\int k{(\\hat{x},f^{*})} d\\hat{x}} = e^{\\frac{\\hat{x}^{2}}{2} - \\hat{x} f^{*} + k}, then obtain e^{\\frac{\\hat{x}^{2}}{2} - \\hat{x} f^{*} + k} = e^{\\int (\\hat{x} - f^{*}) d\\hat{x}}", "derivation": "k{(\\hat{x},f^{*})} = \\hat{x} - f^{*} and \\int k{(\\hat{x},f^{*})} d\\hat{x} = \\int (\\hat{x} - f^{*}) d\\hat{x} and e^{\\int k{(\\hat{x},f^{*})} d\\hat{x}} = e^{\\int (\\hat{x} - f^{*}) d\\hat{x}} and e^{\\int k{(\\hat{x},f^{*})} d\\hat{x}} = e^{\\frac{\\hat{x}^{2}}{2} - \\hat{x} f^{*} + k} and e^{\\frac{\\hat{x}^{2}}{2} - \\hat{x} f^{*} + k} = e^{\\int (\\hat{x} - f^{*}) d\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["exp", 2], "Equality(exp(Integral(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), exp(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(exp(Integral(Function('k')(Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Symbol('k', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(exp(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True), Symbol('f^*', commutative=True)), Symbol('k', commutative=True))), exp(Integral(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given W{(\\sigma_p)} = \\sin{(\\sigma_p)}, then obtain \\int (\\sigma_p - W{(\\sigma_p)}) d\\sigma_p = \\int (\\sigma_p - \\sin{(\\sigma_p)}) d\\sigma_p", "derivation": "W{(\\sigma_p)} = \\sin{(\\sigma_p)} and - \\sigma_p + W{(\\sigma_p)} = - \\sigma_p + \\sin{(\\sigma_p)} and \\sigma_p - W{(\\sigma_p)} = \\sigma_p - \\sin{(\\sigma_p)} and \\int (\\sigma_p - W{(\\sigma_p)}) d\\sigma_p = \\int (\\sigma_p - \\sin{(\\sigma_p)}) d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('W')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), sin(Symbol('\\\\sigma_p', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('W')(Symbol('\\\\sigma_p', commutative=True)))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\sigma_p', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('W')(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), sin(Symbol('\\\\sigma_p', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given h{(\\pi,H)} = H \\pi, then obtain \\int \\frac{h{(\\pi,H)}}{H \\pi^{2}} d\\pi + 1 + \\frac{1}{\\pi} = \\int \\frac{1}{\\pi} d\\pi + 1 + \\frac{1}{\\pi}", "derivation": "h{(\\pi,H)} = H \\pi and \\frac{h{(\\pi,H)}}{H \\pi} = 1 and \\frac{h{(\\pi,H)}}{H \\pi^{2}} = \\frac{1}{\\pi} and 1 + \\frac{h{(\\pi,H)}}{H \\pi^{2}} = 1 + \\frac{1}{\\pi} and \\int \\frac{h{(\\pi,H)}}{H \\pi^{2}} d\\pi = \\int \\frac{1}{\\pi} d\\pi and \\int \\frac{h{(\\pi,H)}}{H \\pi^{2}} d\\pi + 1 + \\frac{h^{3}{(\\pi,H)}}{H^{3} \\pi^{4}} = \\int \\frac{1}{\\pi} d\\pi + 1 + \\frac{h^{3}{(\\pi,H)}}{H^{3} \\pi^{4}} and \\int \\frac{h{(\\pi,H)}}{H \\pi^{2}} d\\pi + 1 + \\frac{h{(\\pi,H)}}{H \\pi^{2}} = \\int \\frac{1}{\\pi} d\\pi + 1 + \\frac{h{(\\pi,H)}}{H \\pi^{2}} and \\int \\frac{h{(\\pi,H)}}{H \\pi^{2}} d\\pi + 1 + \\frac{1}{\\pi} = \\int \\frac{1}{\\pi} d\\pi + 1 + \\frac{1}{\\pi}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Mul(Symbol('H', commutative=True), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Integer(1))"], [["divide", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Integer(-1)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)))), Add(Integer(1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["add", 5, "Add(Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-3)), Pow(Symbol('\\\\pi', commutative=True), Integer(-4)), Pow(Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Integer(3))))"], "Equality(Add(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-3)), Pow(Symbol('\\\\pi', commutative=True), Integer(-4)), Pow(Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Integer(3)))), Add(Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-3)), Pow(Symbol('\\\\pi', commutative=True), Integer(-4)), Pow(Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)), Integer(3)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)))), Add(Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 4], "Equality(Add(Integral(Mul(Pow(Symbol('H', commutative=True), Integer(-1)), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)), Function('h')(Symbol('\\\\pi', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Add(Integral(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Tuple(Symbol('\\\\pi', commutative=True))), Integer(1), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"]]}, {"prompt": "Given b{(\\lambda)} = \\sin{(\\lambda)}, then obtain \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} b^{\\lambda}{(\\lambda)} - \\frac{d}{d \\lambda} b{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} - \\frac{d}{d \\lambda} b{(\\lambda)}", "derivation": "b{(\\lambda)} = \\sin{(\\lambda)} and b^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} and \\frac{d}{d \\lambda} b^{\\lambda}{(\\lambda)} = \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} and \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} b^{\\lambda}{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} and \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} b^{\\lambda}{(\\lambda)} - \\frac{d}{d \\lambda} b{(\\lambda)} = \\sin^{\\lambda}{(\\lambda)} \\frac{d}{d \\lambda} \\sin^{\\lambda}{(\\lambda)} - \\frac{d}{d \\lambda} b{(\\lambda)}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Pow(Function('b')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["times", 3, "Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(Function('b')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))))"], [["minus", 4, "Derivative(Function('b')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(Function('b')(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('b')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Add(Mul(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Derivative(Pow(sin(Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Function('b')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))))"]]}, {"prompt": "Given i{(Z,Q)} = Q Z and \\varphi^{*}{(Z,Q)} = (\\iint i^{Z}{(Z,Q)} dZ dQ)^{Q}, then obtain \\frac{\\partial}{\\partial Z} \\varphi^{*}{(Z,Q)} = \\frac{\\partial}{\\partial Z} (\\iint (Q Z)^{Z} dZ dQ)^{Q}", "derivation": "i{(Z,Q)} = Q Z and i^{Z}{(Z,Q)} = (Q Z)^{Z} and \\int i^{Z}{(Z,Q)} dZ = \\int (Q Z)^{Z} dZ and \\iint i^{Z}{(Z,Q)} dZ dQ = \\iint (Q Z)^{Z} dZ dQ and (\\iint i^{Z}{(Z,Q)} dZ dQ)^{Q} = (\\iint (Q Z)^{Z} dZ dQ)^{Q} and \\frac{\\partial}{\\partial Z} (\\iint i^{Z}{(Z,Q)} dZ dQ)^{Q} = \\frac{\\partial}{\\partial Z} (\\iint (Q Z)^{Z} dZ dQ)^{Q} and \\varphi^{*}{(Z,Q)} = (\\iint i^{Z}{(Z,Q)} dZ dQ)^{Q} and \\frac{\\partial}{\\partial Z} \\varphi^{*}{(Z,Q)} = \\frac{\\partial}{\\partial Z} (\\iint (Q Z)^{Z} dZ dQ)^{Q}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["integrate", 2, "Symbol('Z', commutative=True)"], "Equality(Integral(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))), Integral(Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Integral(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Integral(Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Integral(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Pow(Integral(Pow(Function('i')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('Z', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Integral(Pow(Mul(Symbol('Q', commutative=True), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)), Tuple(Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\phi,Q)} = \\frac{\\phi}{Q}, then obtain Q \\operatorname{M_{E}}{(\\phi,Q)} \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q}) = \\phi \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q})", "derivation": "\\operatorname{M_{E}}{(\\phi,Q)} = \\frac{\\phi}{Q} and \\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q} = \\frac{\\phi}{Q} + \\frac{1}{Q} and \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q}) = \\frac{\\partial}{\\partial Q} (\\frac{\\phi}{Q} + \\frac{1}{Q}) and \\operatorname{M_{E}}{(\\phi,Q)} \\frac{\\partial}{\\partial Q} (\\frac{\\phi}{Q} + \\frac{1}{Q}) = \\frac{\\phi \\frac{\\partial}{\\partial Q} (\\frac{\\phi}{Q} + \\frac{1}{Q})}{Q} and \\operatorname{M_{E}}{(\\phi,Q)} \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q}) = \\frac{\\phi \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q})}{Q} and Q \\operatorname{M_{E}}{(\\phi,Q)} \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q}) = \\phi \\frac{\\partial}{\\partial Q} (\\operatorname{M_{E}}{(\\phi,Q)} + \\frac{1}{Q})", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)))"], [["add", 1, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))"], "Equality(Mul(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Derivative(Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Derivative(Add(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Derivative(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\phi', commutative=True), Derivative(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["divide", 5, "Pow(Symbol('Q', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('Q', commutative=True), Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Derivative(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Mul(Symbol('\\\\phi', commutative=True), Derivative(Add(Function('M_E')(Symbol('\\\\phi', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('Q', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given p{(\\hat{H},\\lambda)} = \\hat{H} \\lambda, then derive \\frac{\\partial}{\\partial \\hat{H}} p{(\\hat{H},\\lambda)} = \\lambda, then obtain \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} \\lambda \\frac{\\partial}{\\partial \\hat{H}} (p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})}) = \\lambda \\frac{\\partial}{\\partial \\hat{H}} (p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})})", "derivation": "p{(\\hat{H},\\lambda)} = \\hat{H} \\lambda and p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})} = \\hat{H} \\lambda + \\cos^{\\varphi}{(f_{E})} and \\frac{\\partial}{\\partial \\hat{H}} (p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})}) = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} \\lambda + \\cos^{\\varphi}{(f_{E})}) and \\frac{\\partial}{\\partial \\hat{H}} p{(\\hat{H},\\lambda)} = \\lambda and \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} \\lambda = \\lambda and \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} \\lambda \\frac{\\partial}{\\partial \\hat{H}} (p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})}) = \\lambda \\frac{\\partial}{\\partial \\hat{H}} (p{(\\hat{H},\\lambda)} + \\cos^{\\varphi}{(f_{E})})", "srepr_derivation": [["get_premise", "Equality(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Symbol('\\\\lambda', commutative=True))"], [["times", 5, "Derivative(Add(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Add(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Symbol('\\\\lambda', commutative=True), Derivative(Add(Function('p')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(cos(Symbol('f_E', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})} = J_{\\varepsilon} \\varepsilon and \\delta{(\\hbar,n)} = \\sin{(\\hbar + n)}, then obtain (\\delta{(\\hbar,n)} + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})})^{\\hbar} = (J_{\\varepsilon} \\varepsilon + \\delta{(\\hbar,n)})^{\\hbar}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})} = J_{\\varepsilon} \\varepsilon and \\delta{(\\hbar,n)} = \\sin{(\\hbar + n)} and \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})} + \\sin{(\\hbar + n)} = J_{\\varepsilon} \\varepsilon + \\sin{(\\hbar + n)} and (\\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})} + \\sin{(\\hbar + n)})^{\\hbar} = (J_{\\varepsilon} \\varepsilon + \\sin{(\\hbar + n)})^{\\hbar} and (\\delta{(\\hbar,n)} + \\operatorname{f_{\\mathbf{v}}}{(\\varepsilon,J_{\\varepsilon})})^{\\hbar} = (J_{\\varepsilon} \\varepsilon + \\delta{(\\hbar,n)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], ["get_premise", "Equality(Function('\\\\delta')(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))))"], [["add", 1, "sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))))"], [["power", 3, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), sin(Add(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)))), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Add(Function('\\\\delta')(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('\\\\varepsilon', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Function('\\\\delta')(Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\hbar', commutative=True)))"]]}, {"prompt": "Given \\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})} = \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}}, then obtain \\int (\\mathbf{E} + \\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})}) d\\sigma_x - 1 = \\int (\\mathbf{E} + \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}}) d\\sigma_x - 1", "derivation": "\\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})} = \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}} and \\mathbf{E} + \\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})} = \\mathbf{E} + \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}} and \\int (\\mathbf{E} + \\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})}) d\\sigma_x = \\int (\\mathbf{E} + \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}}) d\\sigma_x and \\int (\\mathbf{E} + \\mu_{0}{(\\sigma_x,\\mathbf{E},F_{H})}) d\\sigma_x - 1 = \\int (\\mathbf{E} + \\frac{\\mathbf{E}^{\\sigma_x}}{F_{H}}) d\\sigma_x - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\sigma_x', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mu_0')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{E}', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)), Add(Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\sigma_x', commutative=True)))), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})} = F_{N} f_{\\mathbf{v}}, then derive \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})} = f_{\\mathbf{v}}, then obtain (\\frac{\\partial}{\\partial F_{N}} F_{N} f_{\\mathbf{v}})^{f_{\\mathbf{v}}} = f_{\\mathbf{v}}^{f_{\\mathbf{v}}}", "derivation": "\\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})} = F_{N} f_{\\mathbf{v}} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})} = \\frac{\\partial}{\\partial F_{N}} F_{N} f_{\\mathbf{v}} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})} = f_{\\mathbf{v}} and (\\frac{\\partial}{\\partial F_{N}} \\operatorname{A_{z}}{(f_{\\mathbf{v}},F_{N})})^{f_{\\mathbf{v}}} = f_{\\mathbf{v}}^{f_{\\mathbf{v}}} and (\\frac{\\partial}{\\partial F_{N}} F_{N} f_{\\mathbf{v}})^{f_{\\mathbf{v}}} = f_{\\mathbf{v}}^{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('F_N', commutative=True)), Mul(Symbol('F_N', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_N', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], [["power", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Derivative(Function('A_z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Mul(Symbol('F_N', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\rho)} = \\cos{(\\log{(\\rho)})} and c{(\\rho)} = \\cos{(\\log{(\\rho)})}, then obtain c^{\\rho}{(\\rho)} \\operatorname{t_{2}}^{- 2 \\rho}{(\\rho)} = \\operatorname{t_{2}}^{- \\rho}{(\\rho)}", "derivation": "\\operatorname{t_{2}}{(\\rho)} = \\cos{(\\log{(\\rho)})} and c{(\\rho)} = \\cos{(\\log{(\\rho)})} and c^{\\rho}{(\\rho)} = \\cos^{\\rho}{(\\log{(\\rho)})} and c^{\\rho}{(\\rho)} \\cos^{- \\rho}{(\\log{(\\rho)})} = 1 and c^{\\rho}{(\\rho)} \\cos^{- 2 \\rho}{(\\log{(\\rho)})} = \\cos^{- \\rho}{(\\log{(\\rho)})} and c^{\\rho}{(\\rho)} \\operatorname{t_{2}}^{- 2 \\rho}{(\\rho)} = \\operatorname{t_{2}}^{- \\rho}{(\\rho)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True))))"], ["renaming_premise", "Equality(Function('c')(Symbol('\\\\rho', commutative=True)), cos(log(Symbol('\\\\rho', commutative=True))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('c')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(log(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["divide", 3, "Pow(cos(log(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(Function('c')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(log(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True)))), Integer(1))"], [["divide", 4, "Pow(cos(log(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Pow(Function('c')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(cos(log(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\rho', commutative=True)))), Pow(cos(log(Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('c')(Symbol('\\\\rho', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Function('t_2')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\rho', commutative=True)))), Pow(Function('t_2')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\lambda)} = \\cos{(\\lambda)}, then derive \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda = \\hat{\\mathbf{r}} + \\sin{(\\lambda)}, then obtain (\\sin{(\\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda)}) \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda = \\sin{(\\hat{\\mathbf{r}} + \\sin{(\\lambda)})} \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda", "derivation": "\\operatorname{A_{2}}{(\\lambda)} = \\cos{(\\lambda)} and \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda = \\int \\cos{(\\lambda)} d\\lambda and \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda = \\hat{\\mathbf{r}} + \\sin{(\\lambda)} and \\sin{(\\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda)} = \\sin{(\\hat{\\mathbf{r}} + \\sin{(\\lambda)})} and (\\sin{(\\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda)}) \\int \\cos{(\\lambda)} d\\lambda = \\sin{(\\hat{\\mathbf{r}} + \\sin{(\\lambda)})} \\int \\cos{(\\lambda)} d\\lambda and (\\sin{(\\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda)}) \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda = \\sin{(\\hat{\\mathbf{r}} + \\sin{(\\lambda)})} \\int \\operatorname{A_{2}}{(\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\lambda', commutative=True)), cos(Symbol('\\\\lambda', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\lambda', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), sin(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))))"], [["times", 4, "Integral(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(sin(Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Integral(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(sin(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))), Integral(cos(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(sin(Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))), Mul(sin(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), sin(Symbol('\\\\lambda', commutative=True)))), Integral(Function('A_2')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\psi,\\Omega)} = \\Omega \\psi, then derive \\frac{\\partial}{\\partial \\Omega} \\varphi^{*}{(\\psi,\\Omega)} = \\psi, then derive \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\varphi^{*}{(\\psi,\\Omega)} = 0, then obtain \\frac{d}{d \\Omega} \\psi = 0", "derivation": "\\varphi^{*}{(\\psi,\\Omega)} = \\Omega \\psi and \\frac{\\partial}{\\partial \\Omega} \\varphi^{*}{(\\psi,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\Omega \\psi and \\frac{\\partial}{\\partial \\Omega} \\varphi^{*}{(\\psi,\\Omega)} = \\psi and \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\varphi^{*}{(\\psi,\\Omega)} = \\frac{d}{d \\Omega} \\psi and \\frac{\\partial^{2}}{\\partial \\Omega^{2}} \\varphi^{*}{(\\psi,\\Omega)} = 0 and \\frac{d}{d \\Omega} \\psi = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Symbol('\\\\psi', commutative=True))"], [["differentiate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\psi', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(2))), Integer(0))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Symbol('\\\\psi', commutative=True), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\psi^{*}{(\\theta_1)} = e^{\\theta_1} and \\pi{(\\theta_1)} = \\frac{\\int \\psi^{*}{(\\theta_1)} d\\theta_1}{\\psi^{*}{(\\theta_1)}}, then obtain 0 = - \\pi{(\\theta_1)} + e^{- \\theta_1} \\int e^{\\theta_1} d\\theta_1", "derivation": "\\psi^{*}{(\\theta_1)} = e^{\\theta_1} and \\int \\psi^{*}{(\\theta_1)} d\\theta_1 = \\int e^{\\theta_1} d\\theta_1 and e^{- \\theta_1} \\int \\psi^{*}{(\\theta_1)} d\\theta_1 = e^{- \\theta_1} \\int e^{\\theta_1} d\\theta_1 and \\frac{\\int \\psi^{*}{(\\theta_1)} d\\theta_1}{\\psi^{*}{(\\theta_1)}} = \\frac{\\int e^{\\theta_1} d\\theta_1}{\\psi^{*}{(\\theta_1)}} and \\pi{(\\theta_1)} = \\frac{\\int \\psi^{*}{(\\theta_1)} d\\theta_1}{\\psi^{*}{(\\theta_1)}} and \\pi{(\\theta_1)} = \\frac{\\int e^{\\theta_1} d\\theta_1}{\\psi^{*}{(\\theta_1)}} and 0 = - \\pi{(\\theta_1)} + \\frac{\\int e^{\\theta_1} d\\theta_1}{\\psi^{*}{(\\theta_1)}} and 0 = - \\pi{(\\theta_1)} + e^{- \\theta_1} \\int e^{\\theta_1} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), exp(Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["divide", 2, "exp(Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(exp(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integral(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))), Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integral(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\pi')(Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True)))))"], [["minus", 6, "Function('\\\\pi')(Symbol('\\\\theta_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Function('\\\\psi^*')(Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\pi')(Symbol('\\\\theta_1', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True))), Integral(exp(Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{p}{(s,\\rho,x^\\prime)} = \\rho - s + x^\\prime and \\operatorname{F_{H}}{(s,\\rho,x^\\prime)} = \\rho - s + x^\\prime, then obtain \\frac{\\operatorname{F_{H}}^{\\rho}{(s,\\rho,x^\\prime)}}{\\mathbf{p}{(s,\\rho,x^\\prime)}} = \\frac{\\mathbf{p}^{\\rho}{(s,\\rho,x^\\prime)}}{\\mathbf{p}{(s,\\rho,x^\\prime)}}", "derivation": "\\mathbf{p}{(s,\\rho,x^\\prime)} = \\rho - s + x^\\prime and \\operatorname{F_{H}}{(s,\\rho,x^\\prime)} = \\rho - s + x^\\prime and \\operatorname{F_{H}}^{\\rho}{(s,\\rho,x^\\prime)} = (\\rho - s + x^\\prime)^{\\rho} and \\operatorname{F_{H}}^{\\rho}{(s,\\rho,x^\\prime)} = \\mathbf{p}^{\\rho}{(s,\\rho,x^\\prime)} and \\frac{\\operatorname{F_{H}}^{\\rho}{(s,\\rho,x^\\prime)}}{\\rho - s + x^\\prime} = \\frac{\\mathbf{p}^{\\rho}{(s,\\rho,x^\\prime)}}{\\rho - s + x^\\prime} and \\frac{\\operatorname{F_{H}}^{\\rho}{(s,\\rho,x^\\prime)}}{\\mathbf{p}{(s,\\rho,x^\\prime)}} = \\frac{\\mathbf{p}^{\\rho}{(s,\\rho,x^\\prime)}}{\\mathbf{p}{(s,\\rho,x^\\prime)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Function('F_H')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Pow(Function('F_H')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True)))"], [["divide", 4, "Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Pow(Function('F_H')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True))), Mul(Pow(Add(Symbol('\\\\rho', commutative=True), Mul(Integer(-1), Symbol('s', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('F_H')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(-1))), Mul(Pow(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{p}')(Symbol('s', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('\\\\rho', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(q,k)} = \\log{(- k + q)}, then obtain - \\frac{k - q + \\frac{q + \\operatorname{t_{2}}{(q,k)}}{k}}{k} = - \\frac{k - q + \\frac{q + \\log{(- k + q)}}{k}}{k}", "derivation": "\\operatorname{t_{2}}{(q,k)} = \\log{(- k + q)} and q + \\operatorname{t_{2}}{(q,k)} = q + \\log{(- k + q)} and \\frac{q + \\operatorname{t_{2}}{(q,k)}}{k} = \\frac{q + \\log{(- k + q)}}{k} and k - q + \\frac{q + \\operatorname{t_{2}}{(q,k)}}{k} = k - q + \\frac{q + \\log{(- k + q)}}{k} and - \\frac{k - q + \\frac{q + \\operatorname{t_{2}}{(q,k)}}{k}}{k} = - \\frac{k - q + \\frac{q + \\log{(- k + q)}}{k}}{k}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('q', commutative=True), Symbol('k', commutative=True)), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))))"], [["add", 1, "Symbol('q', commutative=True)"], "Equality(Add(Symbol('q', commutative=True), Function('t_2')(Symbol('q', commutative=True), Symbol('k', commutative=True))), Add(Symbol('q', commutative=True), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True)))))"], [["divide", 2, "Symbol('k', commutative=True)"], "Equality(Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), Function('t_2')(Symbol('q', commutative=True), Symbol('k', commutative=True)))), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), Function('t_2')(Symbol('q', commutative=True), Symbol('k', commutative=True))))), Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True)))))))"], [["divide", 4, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), Function('t_2')(Symbol('q', commutative=True), Symbol('k', commutative=True)))))), Mul(Integer(-1), Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('k', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Pow(Symbol('k', commutative=True), Integer(-1)), Add(Symbol('q', commutative=True), log(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Symbol('q', commutative=True))))))))"]]}, {"prompt": "Given s{(\\rho_f,\\theta_1)} = e^{- \\rho_f + \\theta_1}, then obtain \\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1} + \\frac{\\frac{\\partial}{\\partial \\rho_f} s{(\\rho_f,\\theta_1)}}{s{(\\rho_f,\\theta_1)}} = \\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1} + \\frac{\\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1}}{s{(\\rho_f,\\theta_1)}}", "derivation": "s{(\\rho_f,\\theta_1)} = e^{- \\rho_f + \\theta_1} and \\frac{\\partial}{\\partial \\rho_f} s{(\\rho_f,\\theta_1)} = \\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1} and \\frac{\\frac{\\partial}{\\partial \\rho_f} s{(\\rho_f,\\theta_1)}}{s{(\\rho_f,\\theta_1)}} = \\frac{\\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1}}{s{(\\rho_f,\\theta_1)}} and \\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1} + \\frac{\\frac{\\partial}{\\partial \\rho_f} s{(\\rho_f,\\theta_1)}}{s{(\\rho_f,\\theta_1)}} = \\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1} + \\frac{\\frac{\\partial}{\\partial \\rho_f} e^{- \\rho_f + \\theta_1}}{s{(\\rho_f,\\theta_1)}}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["divide", 2, "Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Mul(Pow(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Derivative(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))), Mul(Pow(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1)))))"], [["minus", 3, "Mul(Integer(-1), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], "Equality(Add(Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Pow(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Derivative(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))), Add(Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Mul(Pow(Function('s')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Derivative(exp(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\eta{(I,f^{\\prime})} = - I + f^{\\prime}, then obtain \\frac{- v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(I - \\eta{(I,f^{\\prime})})}}{f^{\\prime}} = \\frac{- v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(2 I - f^{\\prime})}}{f^{\\prime}}", "derivation": "\\eta{(I,f^{\\prime})} = - I + f^{\\prime} and - I + \\eta{(I,f^{\\prime})} = - 2 I + f^{\\prime} and - \\sin{(I - \\eta{(I,f^{\\prime})})} = - \\sin{(2 I - f^{\\prime})} and - \\eta{(I,f^{\\prime})} - \\sin{(I - \\eta{(I,f^{\\prime})})} = - \\eta{(I,f^{\\prime})} - \\sin{(2 I - f^{\\prime})} and - v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(I - \\eta{(I,f^{\\prime})})} = - v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(2 I - f^{\\prime})} and \\frac{- v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(I - \\eta{(I,f^{\\prime})})}}{f^{\\prime}} = \\frac{- v_{2} - \\eta{(I,f^{\\prime})} - \\sin{(2 I - f^{\\prime})}}{f^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('I', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('I', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["sin", 2], "Equality(Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))))"], [["minus", 3, "Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))))), Add(Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))))"], [["minus", 4, "Symbol('v_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))))))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)))))))"], [["divide", 5, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Symbol('I', commutative=True), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))))))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\eta')(Symbol('I', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Mul(Integer(-1), sin(Add(Mul(Integer(2), Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True))))))))"]]}, {"prompt": "Given h{(A_{z},k)} = \\cos{(A_{z} + k)}, then obtain \\frac{\\frac{\\partial}{\\partial k} \\int h{(A_{z},k)} dA_{z}}{\\int h{(A_{z},k)} dA_{z}} = \\frac{\\frac{\\partial}{\\partial k} \\int \\cos{(A_{z} + k)} dA_{z}}{\\int h{(A_{z},k)} dA_{z}}", "derivation": "h{(A_{z},k)} = \\cos{(A_{z} + k)} and \\int h{(A_{z},k)} dA_{z} = \\int \\cos{(A_{z} + k)} dA_{z} and \\frac{\\partial}{\\partial k} \\int h{(A_{z},k)} dA_{z} = \\frac{\\partial}{\\partial k} \\int \\cos{(A_{z} + k)} dA_{z} and \\frac{\\frac{\\partial}{\\partial k} \\int h{(A_{z},k)} dA_{z}}{\\int h{(A_{z},k)} dA_{z}} = \\frac{\\frac{\\partial}{\\partial k} \\int \\cos{(A_{z} + k)} dA_{z}}{\\int h{(A_{z},k)} dA_{z}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), cos(Add(Symbol('A_z', commutative=True), Symbol('k', commutative=True))))"], [["integrate", 1, "Symbol('A_z', commutative=True)"], "Equality(Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integral(cos(Add(Symbol('A_z', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('A_z', commutative=True))))"], [["differentiate", 2, "Symbol('k', commutative=True)"], "Equality(Derivative(Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Integral(cos(Add(Symbol('A_z', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 3, "Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True)))"], "Equality(Mul(Derivative(Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Pow(Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integer(-1))), Mul(Derivative(Integral(cos(Add(Symbol('A_z', commutative=True), Symbol('k', commutative=True))), Tuple(Symbol('A_z', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Pow(Integral(Function('h')(Symbol('A_z', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('A_z', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given y{(\\rho_b,M)} = M \\rho_b, then obtain (M^{2} \\rho_b^{2})^{M} + (M \\rho_b y{(\\rho_b,M)})^{M} - \\lambda{(\\mu_0,E)} = 2 (M^{2} \\rho_b^{2})^{M} - \\lambda{(\\mu_0,E)}", "derivation": "y{(\\rho_b,M)} = M \\rho_b and M \\rho_b y{(\\rho_b,M)} = M^{2} \\rho_b^{2} and (M \\rho_b y{(\\rho_b,M)})^{M} = (M^{2} \\rho_b^{2})^{M} and (M^{2} \\rho_b^{2})^{M} + (M \\rho_b y{(\\rho_b,M)})^{M} = 2 (M^{2} \\rho_b^{2})^{M} and (M^{2} \\rho_b^{2})^{M} + (M \\rho_b y{(\\rho_b,M)})^{M} - \\lambda{(\\mu_0,E)} = 2 (M^{2} \\rho_b^{2})^{M} - \\lambda{(\\mu_0,E)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["times", 1, "Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('M', commutative=True))), Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))))"], [["power", 2, "Symbol('M', commutative=True)"], "Equality(Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True)))"], [["add", 3, "Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True))), Mul(Integer(2), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True))))"], [["minus", 4, "Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True), Symbol('E', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True)), Pow(Mul(Symbol('M', commutative=True), Symbol('\\\\rho_b', commutative=True), Function('y')(Symbol('\\\\rho_b', commutative=True), Symbol('M', commutative=True))), Symbol('M', commutative=True)), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True), Symbol('E', commutative=True)))), Add(Mul(Integer(2), Pow(Mul(Pow(Symbol('M', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Symbol('M', commutative=True))), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mu_0', commutative=True), Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\phi_2)} = \\sin{(\\phi_2)}, then derive \\frac{d}{d \\phi_2} \\mathbf{p}{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain \\cos{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)}", "derivation": "\\mathbf{p}{(\\phi_2)} = \\sin{(\\phi_2)} and \\frac{d}{d \\phi_2} \\mathbf{p}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)} and \\frac{d}{d \\phi_2} \\mathbf{p}{(\\phi_2)} = \\cos{(\\phi_2)} and \\cos{(\\phi_2)} = \\frac{d}{d \\phi_2} \\sin{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), sin(Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), cos(Symbol('\\\\phi_2', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(cos(Symbol('\\\\phi_2', commutative=True)), Derivative(sin(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\sin{(L_{\\varepsilon} x)}, then obtain \\operatorname{t_{2}}^{x}{(L_{\\varepsilon},x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\operatorname{t_{2}}^{x}{(L_{\\varepsilon},x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\sin{(L_{\\varepsilon} x)}", "derivation": "\\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\sin{(L_{\\varepsilon} x)} and \\operatorname{t_{2}}^{x}{(L_{\\varepsilon},x)} = \\sin^{x}{(L_{\\varepsilon} x)} and \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\frac{\\partial}{\\partial L_{\\varepsilon}} \\sin{(L_{\\varepsilon} x)} and \\sin^{x}{(L_{\\varepsilon} x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\sin^{x}{(L_{\\varepsilon} x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\sin{(L_{\\varepsilon} x)} and \\operatorname{t_{2}}^{x}{(L_{\\varepsilon},x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\operatorname{t_{2}}{(L_{\\varepsilon},x)} = \\operatorname{t_{2}}^{x}{(L_{\\varepsilon},x)} + \\frac{\\partial}{\\partial L_{\\varepsilon}} \\sin{(L_{\\varepsilon} x)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))))"], [["power", 1, "Symbol('x', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Pow(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["add", 3, "Pow(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True))"], "Equality(Add(Pow(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Derivative(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Pow(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))), Add(Pow(Function('t_2')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Symbol('x', commutative=True)), Derivative(sin(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given V{(A)} = \\sin{(A)}, then derive \\cos{(A)} + \\frac{d}{d A} V{(A)} = 2 \\cos{(A)}, then obtain \\cos{(A)} + \\frac{d}{d A} V{(A)} = \\cos{(A)} + \\frac{d}{d A} \\sin{(A)}", "derivation": "V{(A)} = \\sin{(A)} and V{(A)} + \\sin{(A)} = 2 \\sin{(A)} and \\frac{d}{d A} (V{(A)} + \\sin{(A)}) = \\frac{d}{d A} 2 \\sin{(A)} and \\cos{(A)} + \\frac{d}{d A} V{(A)} = 2 \\cos{(A)} and \\cos{(A)} + \\frac{d}{d A} \\sin{(A)} = 2 \\cos{(A)} and \\cos{(A)} + \\frac{d}{d A} V{(A)} = \\cos{(A)} + \\frac{d}{d A} \\sin{(A)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["add", 1, "sin(Symbol('A', commutative=True))"], "Equality(Add(Function('V')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Mul(Integer(2), sin(Symbol('A', commutative=True))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Function('V')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Integer(2), sin(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(Function('V')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Mul(Integer(2), cos(Symbol('A', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(cos(Symbol('A', commutative=True)), Derivative(Function('V')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(cos(Symbol('A', commutative=True)), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\rho_f,T)} = T - \\rho_f, then obtain \\operatorname{E_{x}}{(\\rho_f,T)} + \\frac{\\partial}{\\partial T} \\operatorname{E_{x}}^{T}{(\\rho_f,T)} - 1 = T - \\rho_f + \\frac{\\partial}{\\partial T} \\operatorname{E_{x}}^{T}{(\\rho_f,T)} - 1", "derivation": "\\operatorname{E_{x}}{(\\rho_f,T)} = T - \\rho_f and \\operatorname{E_{x}}{(\\rho_f,T)} - 1 = T - \\rho_f - 1 and \\operatorname{E_{x}}^{T}{(\\rho_f,T)} = (T - \\rho_f)^{T} and \\operatorname{E_{x}}{(\\rho_f,T)} + \\frac{\\partial}{\\partial T} (T - \\rho_f)^{T} - 1 = T - \\rho_f + \\frac{\\partial}{\\partial T} (T - \\rho_f)^{T} - 1 and \\operatorname{E_{x}}{(\\rho_f,T)} + \\frac{\\partial}{\\partial T} \\operatorname{E_{x}}^{T}{(\\rho_f,T)} - 1 = T - \\rho_f + \\frac{\\partial}{\\partial T} \\operatorname{E_{x}}^{T}{(\\rho_f,T)} - 1", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Integer(-1)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('T', commutative=True)))"], [["add", 2, "Derivative(Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))"], "Equality(Add(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Derivative(Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Derivative(Pow(Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Derivative(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)), Add(Symbol('T', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Derivative(Pow(Function('E_x')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\theta_{2}{(a,Z,p)} = (p^{a})^{Z}, then obtain \\frac{\\partial}{\\partial Z} (p^{a})^{- Z} \\theta_{2}{(a,Z,p)} = \\frac{d}{d Z} 1", "derivation": "\\theta_{2}{(a,Z,p)} = (p^{a})^{Z} and p \\theta_{2}{(a,Z,p)} = p (p^{a})^{Z} and (p^{a})^{- Z} \\theta_{2}{(a,Z,p)} = 1 and \\frac{\\partial}{\\partial Z} (p^{a})^{- Z} \\theta_{2}{(a,Z,p)} = \\frac{d}{d Z} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('a', commutative=True), Symbol('Z', commutative=True), Symbol('p', commutative=True)), Pow(Pow(Symbol('p', commutative=True), Symbol('a', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\theta_2')(Symbol('a', commutative=True), Symbol('Z', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Pow(Pow(Symbol('p', commutative=True), Symbol('a', commutative=True)), Symbol('Z', commutative=True))))"], [["divide", 2, "Mul(Symbol('p', commutative=True), Pow(Pow(Symbol('p', commutative=True), Symbol('a', commutative=True)), Symbol('Z', commutative=True)))"], "Equality(Mul(Pow(Pow(Symbol('p', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('\\\\theta_2')(Symbol('a', commutative=True), Symbol('Z', commutative=True), Symbol('p', commutative=True))), Integer(1))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Pow(Pow(Symbol('p', commutative=True), Symbol('a', commutative=True)), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('\\\\theta_2')(Symbol('a', commutative=True), Symbol('Z', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(M)} = \\int e^{M} dM, then obtain \\frac{r{(M)} \\int e^{M} dM}{\\int r{(M)} \\int e^{M} dM dM} = \\frac{(\\int e^{M} dM)^{2}}{\\int r{(M)} \\int e^{M} dM dM}", "derivation": "r{(M)} = \\int e^{M} dM and r{(M)} \\int e^{M} dM = (\\int e^{M} dM)^{2} and \\int r{(M)} \\int e^{M} dM dM = \\int (\\int e^{M} dM)^{2} dM and \\frac{r{(M)} \\int e^{M} dM}{\\int (\\int e^{M} dM)^{2} dM} = \\frac{(\\int e^{M} dM)^{2}}{\\int (\\int e^{M} dM)^{2} dM} and \\frac{r{(M)} \\int e^{M} dM}{\\int r{(M)} \\int e^{M} dM dM} = \\frac{(\\int e^{M} dM)^{2}}{\\int r{(M)} \\int e^{M} dM dM}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["times", 1, "Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integral(Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)), Tuple(Symbol('M', commutative=True))))"], [["divide", 2, "Integral(Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)), Tuple(Symbol('M', commutative=True)))"], "Equality(Mul(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Pow(Integral(Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)), Tuple(Symbol('M', commutative=True))), Integer(-1))), Mul(Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)), Pow(Integral(Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2)), Tuple(Symbol('M', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('r')(Symbol('M', commutative=True)), Pow(Integral(Mul(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integer(-1)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Mul(Pow(Integral(Mul(Function('r')(Symbol('M', commutative=True)), Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Integer(-1)), Pow(Integral(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{F}{(v_{y})} = \\log{(v_{y})}, then obtain (\\frac{d}{d v_{y}} 2 \\mathbf{F}{(v_{y})})^{v_{y}} = (\\frac{d}{d v_{y}} (\\mathbf{F}{(v_{y})} + \\log{(v_{y})}))^{v_{y}}", "derivation": "\\mathbf{F}{(v_{y})} = \\log{(v_{y})} and 2 \\mathbf{F}{(v_{y})} = \\mathbf{F}{(v_{y})} + \\log{(v_{y})} and \\frac{d}{d v_{y}} 2 \\mathbf{F}{(v_{y})} = \\frac{d}{d v_{y}} (\\mathbf{F}{(v_{y})} + \\log{(v_{y})}) and (\\frac{d}{d v_{y}} 2 \\mathbf{F}{(v_{y})})^{v_{y}} = (\\frac{d}{d v_{y}} (\\mathbf{F}{(v_{y})} + \\log{(v_{y})}))^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True))), Add(Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))))"], [["differentiate", 2, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Add(Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('v_y', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(2), Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)), Pow(Derivative(Add(Function('\\\\mathbf{F}')(Symbol('v_y', commutative=True)), log(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\varphi{(J,P_{e})} = J P_{e}, then obtain \\frac{d}{d P_{e}} 1 = \\frac{\\partial}{\\partial P_{e}} \\frac{J P_{e}}{\\varphi{(J,P_{e})}}", "derivation": "\\varphi{(J,P_{e})} = J P_{e} and \\int \\varphi{(J,P_{e})} dJ = \\int J P_{e} dJ and \\varphi{(J,P_{e})} \\int \\varphi{(J,P_{e})} dJ = J P_{e} \\int \\varphi{(J,P_{e})} dJ and (J P_{e} + \\varphi{(J,P_{e})}) \\varphi{(J,P_{e})} \\int \\varphi{(J,P_{e})} dJ = J P_{e} (J P_{e} + \\varphi{(J,P_{e})}) \\int \\varphi{(J,P_{e})} dJ and (J P_{e} + \\varphi{(J,P_{e})}) \\varphi{(J,P_{e})} \\int J P_{e} dJ = J P_{e} (J P_{e} + \\varphi{(J,P_{e})}) \\int J P_{e} dJ and 1 = \\frac{J P_{e}}{\\varphi{(J,P_{e})}} and \\frac{d}{d P_{e}} 1 = \\frac{\\partial}{\\partial P_{e}} \\frac{J P_{e}}{\\varphi{(J,P_{e})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)))"], [["integrate", 1, "Symbol('J', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["times", 1, "Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True), Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["times", 3, "Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)))"], "Equality(Mul(Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True))), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True), Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True))), Integral(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True))), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integral(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))), Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True), Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True)))))"], [["divide", 5, "Mul(Add(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True))), Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integral(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('J', commutative=True))))"], "Equality(Integer(1), Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True), Pow(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))))"], [["differentiate", 6, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Symbol('J', commutative=True), Symbol('P_e', commutative=True), Pow(Function('\\\\varphi')(Symbol('J', commutative=True), Symbol('P_e', commutative=True)), Integer(-1))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = \\Omega + \\cos{(\\hat{x}_0)}, then derive - \\Omega + \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = 1 - \\Omega, then obtain \\frac{d}{d \\hat{x}_0} 0 = \\frac{\\partial}{\\partial \\hat{x}_0} (1 - \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)})", "derivation": "\\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = \\Omega + \\cos{(\\hat{x}_0)} and \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\cos{(\\hat{x}_0)}) and - \\Omega + \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = - \\Omega + \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\cos{(\\hat{x}_0)}) and - \\Omega + \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} = 1 - \\Omega and - \\Omega + \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\cos{(\\hat{x}_0)}) = 1 - \\Omega and 0 = 1 - \\frac{\\partial}{\\partial \\Omega} (\\Omega + \\cos{(\\hat{x}_0)}) and 0 = 1 - \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)} and \\frac{d}{d \\hat{x}_0} 0 = \\frac{\\partial}{\\partial \\hat{x}_0} (1 - \\frac{\\partial}{\\partial \\Omega} \\operatorname{f^{\\prime}}{(\\hat{x}_0,\\Omega)})", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))"], [["minus", 5, "Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Derivative(Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\hat{x}_0', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))))"], [["differentiate", 7, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\dot{y})} = e^{\\dot{y}}, then obtain \\operatorname{P_{e}}^{2}{(\\dot{y})} e^{\\dot{y}} = \\operatorname{P_{e}}{(\\dot{y})} e^{2 \\dot{y}}", "derivation": "\\operatorname{P_{e}}{(\\dot{y})} = e^{\\dot{y}} and \\operatorname{P_{e}}^{2}{(\\dot{y})} = \\operatorname{P_{e}}{(\\dot{y})} e^{\\dot{y}} and \\operatorname{P_{e}}^{3}{(\\dot{y})} = \\operatorname{P_{e}}^{2}{(\\dot{y})} e^{\\dot{y}} and \\operatorname{P_{e}}^{3}{(\\dot{y})} = \\operatorname{P_{e}}{(\\dot{y})} e^{2 \\dot{y}} and \\operatorname{P_{e}}^{2}{(\\dot{y})} e^{\\dot{y}} = \\operatorname{P_{e}}{(\\dot{y})} e^{2 \\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["times", 1, "Function('P_e')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), Mul(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["times", 2, "Function('P_e')(Symbol('\\\\dot{y}', commutative=True))"], "Equality(Pow(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), Integer(3)), Mul(Pow(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), exp(Symbol('\\\\dot{y}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), Integer(3)), Mul(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), Integer(2)), exp(Symbol('\\\\dot{y}', commutative=True))), Mul(Function('P_e')(Symbol('\\\\dot{y}', commutative=True)), exp(Mul(Integer(2), Symbol('\\\\dot{y}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(u,\\mathbf{g})} = \\frac{\\mathbf{g}}{u} and A{(u)} = \\frac{1}{u}, then obtain u \\sin{(\\operatorname{r_{0}}^{u}{(u,\\mathbf{g})})} = u \\sin{((\\frac{\\mathbf{g}}{u})^{u})}", "derivation": "\\operatorname{r_{0}}{(u,\\mathbf{g})} = \\frac{\\mathbf{g}}{u} and A{(u)} = \\frac{1}{u} and \\operatorname{r_{0}}^{u}{(u,\\mathbf{g})} = (\\frac{\\mathbf{g}}{u})^{u} and \\sin{(\\operatorname{r_{0}}^{u}{(u,\\mathbf{g})})} = \\sin{((\\frac{\\mathbf{g}}{u})^{u})} and \\frac{\\sin{(\\operatorname{r_{0}}^{u}{(u,\\mathbf{g})})}}{A{(u)}} = \\frac{\\sin{((\\frac{\\mathbf{g}}{u})^{u})}}{A{(u)}} and u \\sin{(\\operatorname{r_{0}}^{u}{(u,\\mathbf{g})})} = u \\sin{((\\frac{\\mathbf{g}}{u})^{u})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('u', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], ["renaming_premise", "Equality(Function('A')(Symbol('u', commutative=True)), Pow(Symbol('u', commutative=True), Integer(-1)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('r_0')(Symbol('u', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('u', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('u', commutative=True)))"], [["sin", 3], "Equality(sin(Pow(Function('r_0')(Symbol('u', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('u', commutative=True))), sin(Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('u', commutative=True))))"], [["divide", 4, "Function('A')(Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('A')(Symbol('u', commutative=True)), Integer(-1)), sin(Pow(Function('r_0')(Symbol('u', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('u', commutative=True)))), Mul(Pow(Function('A')(Symbol('u', commutative=True)), Integer(-1)), sin(Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('u', commutative=True), sin(Pow(Function('r_0')(Symbol('u', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('u', commutative=True)))), Mul(Symbol('u', commutative=True), sin(Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} = \\hat{\\mathbf{r}} \\cos{(h)}, then obtain \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} \\int \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} d\\hat{\\mathbf{r}} = \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} \\int \\hat{\\mathbf{r}} \\cos{(h)} d\\hat{\\mathbf{r}}", "derivation": "\\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} = \\hat{\\mathbf{r}} \\cos{(h)} and \\int \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} d\\hat{\\mathbf{r}} = \\int \\hat{\\mathbf{r}} \\cos{(h)} d\\hat{\\mathbf{r}} and \\hat{\\mathbf{r}} \\cos{(h)} \\int \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} d\\hat{\\mathbf{r}} = \\hat{\\mathbf{r}} \\cos{(h)} \\int \\hat{\\mathbf{r}} \\cos{(h)} d\\hat{\\mathbf{r}} and \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} \\int \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} d\\hat{\\mathbf{r}} = \\operatorname{n_{2}}{(\\hat{\\mathbf{r}},h)} \\int \\hat{\\mathbf{r}} \\cos{(h)} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["times", 2, "Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True)))"], "Equality(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True)), Integral(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True)), Integral(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Integral(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))), Mul(Function('n_2')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('h', commutative=True)), Integral(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), cos(Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(t_{2},M_{E},f)} = \\frac{M_{E} + f}{t_{2}}, then obtain \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)})^{2} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\partial}{\\partial M_{E}} \\frac{M_{E} + f}{t_{2}} \\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)}", "derivation": "\\hat{\\mathbf{r}}{(t_{2},M_{E},f)} = \\frac{M_{E} + f}{t_{2}} and \\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)} = \\frac{\\partial}{\\partial M_{E}} \\frac{M_{E} + f}{t_{2}} and (\\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)})^{2} = \\frac{\\partial}{\\partial M_{E}} \\frac{M_{E} + f}{t_{2}} \\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)} and \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)})^{2} = \\frac{\\partial}{\\partial t_{2}} \\frac{\\partial}{\\partial M_{E}} \\frac{M_{E} + f}{t_{2}} \\frac{\\partial}{\\partial M_{E}} \\hat{\\mathbf{r}}{(t_{2},M_{E},f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('M_E', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('t_2', commutative=True), Symbol('M_E', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\sigma_x,Q)} = \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}, then derive \\operatorname{F_{g}}{(\\sigma_x,Q)} = \\frac{1}{\\sigma_x}, then obtain \\frac{a{(\\sigma_x)}}{\\sigma_x (\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x})} = \\frac{a{(\\sigma_x)} \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}}{\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}}", "derivation": "\\operatorname{F_{g}}{(\\sigma_x,Q)} = \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x} and \\operatorname{F_{g}}{(\\sigma_x,Q)} = \\frac{1}{\\sigma_x} and \\frac{1}{\\sigma_x} = \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x} and \\frac{1}{\\sigma_x (\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x})} = \\frac{\\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}}{\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}} and \\frac{a{(\\sigma_x)}}{\\sigma_x (\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x})} = \\frac{a{(\\sigma_x)} \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}}{\\operatorname{F_{g}}{(\\sigma_x,Q)} + \\frac{\\partial}{\\partial Q} \\frac{Q}{\\sigma_x}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["divide", 3, "Add(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Add(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(-1))), Mul(Pow(Add(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(-1)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"], [["times", 4, "Function('a')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Pow(Add(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(-1)), Function('a')(Symbol('\\\\sigma_x', commutative=True))), Mul(Pow(Add(Function('F_g')(Symbol('\\\\sigma_x', commutative=True), Symbol('Q', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))), Integer(-1)), Function('a')(Symbol('\\\\sigma_x', commutative=True)), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_P{(q,A_{y})} = \\frac{q}{A_{y}}, then obtain - A_{y} + \\int \\mathbf{J}_P^{q}{(q,A_{y})} dq = - A_{y} + \\int (\\frac{q}{A_{y}})^{q} dq", "derivation": "\\mathbf{J}_P{(q,A_{y})} = \\frac{q}{A_{y}} and \\mathbf{J}_P^{q}{(q,A_{y})} = (\\frac{q}{A_{y}})^{q} and \\int \\mathbf{J}_P^{q}{(q,A_{y})} dq = \\int (\\frac{q}{A_{y}})^{q} dq and - A_{y} + \\int \\mathbf{J}_P^{q}{(q,A_{y})} dq = - A_{y} + \\int (\\frac{q}{A_{y}})^{q} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('A_y', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('A_y', commutative=True)), Symbol('q', commutative=True)), Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('A_y', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))), Integral(Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True))))"], [["minus", 3, "Symbol('A_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Integral(Pow(Function('\\\\mathbf{J}_P')(Symbol('q', commutative=True), Symbol('A_y', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_y', commutative=True)), Integral(Pow(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(t_{2})} = \\log{(t_{2})}, then derive \\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})} = \\frac{1}{t_{2}}, then obtain \\frac{\\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})}}{t_{2}^{2}} = \\frac{1}{t_{2}^{3}}", "derivation": "\\Psi_{nl}{(t_{2})} = \\log{(t_{2})} and \\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})} = \\frac{d}{d t_{2}} \\log{(t_{2})} and \\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})} = \\frac{1}{t_{2}} and \\frac{1}{t_{2}} = \\frac{d}{d t_{2}} \\log{(t_{2})} and \\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})} \\frac{d}{d t_{2}} \\log{(t_{2})} = (\\frac{d}{d t_{2}} \\log{(t_{2})})^{2} and \\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})} (\\frac{d}{d t_{2}} \\log{(t_{2})})^{2} = (\\frac{d}{d t_{2}} \\log{(t_{2})})^{3} and \\frac{\\frac{d}{d t_{2}} \\Psi_{nl}{(t_{2})}}{t_{2}^{2}} = \\frac{1}{t_{2}^{3}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True)))"], [["differentiate", 1, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Pow(Symbol('t_2', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('t_2', commutative=True), Integer(-1)), Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["times", 2, "Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Pow(Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(2)))"], [["times", 5, "Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(2))), Pow(Derivative(log(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(3)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-2)), Derivative(Function('\\\\Psi_{nl}')(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1)))), Pow(Symbol('t_2', commutative=True), Integer(-3)))"]]}, {"prompt": "Given S{(I,g_{\\varepsilon})} = e^{I^{g_{\\varepsilon}}}, then obtain \\frac{\\partial}{\\partial I} I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + S{(I,g_{\\varepsilon})}) = \\frac{\\partial}{\\partial I} I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + e^{I^{g_{\\varepsilon}}})", "derivation": "S{(I,g_{\\varepsilon})} = e^{I^{g_{\\varepsilon}}} and - I + S{(I,g_{\\varepsilon})} = - I + e^{I^{g_{\\varepsilon}}} and \\frac{\\partial}{\\partial I} (- I + S{(I,g_{\\varepsilon})}) = \\frac{\\partial}{\\partial I} (- I + e^{I^{g_{\\varepsilon}}}) and I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + S{(I,g_{\\varepsilon})}) = I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + e^{I^{g_{\\varepsilon}}}) and \\frac{\\partial}{\\partial I} I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + S{(I,g_{\\varepsilon})}) = \\frac{\\partial}{\\partial I} I^{g_{\\varepsilon}} \\frac{\\partial}{\\partial I} (- I + e^{I^{g_{\\varepsilon}}})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), exp(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 3, "Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('I', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('S')(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('I', commutative=True)), exp(Pow(Symbol('I', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('I', commutative=True), Integer(1)))), Tuple(Symbol('I', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\sigma_p)} = e^{\\sigma_p} and \\mathbf{g}{(\\sigma_p)} = \\sigma_p + e^{\\sigma_p}, then obtain 2 \\mathbf{A}{(\\sigma_p)} = \\mathbf{A}{(\\sigma_p)} + e^{\\sigma_p}", "derivation": "\\mathbf{A}{(\\sigma_p)} = e^{\\sigma_p} and \\sigma_p + \\mathbf{A}{(\\sigma_p)} = \\sigma_p + e^{\\sigma_p} and \\mathbf{g}{(\\sigma_p)} = \\sigma_p + e^{\\sigma_p} and \\sigma_p + \\mathbf{A}{(\\sigma_p)} + \\mathbf{g}{(\\sigma_p)} = \\sigma_p + \\mathbf{g}{(\\sigma_p)} + e^{\\sigma_p} and \\sigma_p + \\mathbf{A}{(\\sigma_p)} = \\mathbf{g}{(\\sigma_p)} and 2 \\sigma_p + 2 \\mathbf{A}{(\\sigma_p)} = 2 \\sigma_p + \\mathbf{A}{(\\sigma_p)} + e^{\\sigma_p} and 2 \\mathbf{A}{(\\sigma_p)} = \\mathbf{A}{(\\sigma_p)} + e^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\sigma_p', commutative=True)), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{g}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{g}')(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{g}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Function('\\\\mathbf{g}')(Symbol('\\\\sigma_p', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 6, "Mul(Integer(2), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True))), Add(Function('\\\\mathbf{A}')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given A{(\\mathbf{f})} = \\log{(\\mathbf{f})}, then obtain \\mathbf{f} + 2 A{(\\mathbf{f})} = \\mathbf{f} + 2 \\log{(\\mathbf{f})}", "derivation": "A{(\\mathbf{f})} = \\log{(\\mathbf{f})} and \\mathbf{f} + A{(\\mathbf{f})} = \\mathbf{f} + \\log{(\\mathbf{f})} and \\mathbf{f} + 2 A{(\\mathbf{f})} = \\mathbf{f} + A{(\\mathbf{f})} + \\log{(\\mathbf{f})} and \\mathbf{f} + 2 A{(\\mathbf{f})} = \\mathbf{f} + 2 \\log{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["add", 1, "Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A')(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(2), Function('A')(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Function('A')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(2), Function('A')(Symbol('\\\\mathbf{f}', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(2), log(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\varepsilon_0,\\mu_0)} = \\varepsilon_0^{\\mu_0}, then obtain e^{\\frac{\\varphi^{*}{(\\varepsilon_0,\\mu_0)}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}}} = e^{\\frac{\\varepsilon_0^{\\mu_0}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}}}", "derivation": "\\varphi^{*}{(\\varepsilon_0,\\mu_0)} = \\varepsilon_0^{\\mu_0} and \\frac{\\partial}{\\partial \\mu_0} \\varphi^{*}{(\\varepsilon_0,\\mu_0)} = \\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0} and \\frac{\\varphi^{*}{(\\varepsilon_0,\\mu_0)}}{\\frac{\\partial}{\\partial \\mu_0} \\varphi^{*}{(\\varepsilon_0,\\mu_0)}} = \\frac{\\varepsilon_0^{\\mu_0}}{\\frac{\\partial}{\\partial \\mu_0} \\varphi^{*}{(\\varepsilon_0,\\mu_0)}} and \\frac{\\varphi^{*}{(\\varepsilon_0,\\mu_0)}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}} = \\frac{\\varepsilon_0^{\\mu_0}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}} and e^{\\frac{\\varphi^{*}{(\\varepsilon_0,\\mu_0)}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}}} = e^{\\frac{\\varepsilon_0^{\\mu_0}}{\\frac{\\partial}{\\partial \\mu_0} \\varepsilon_0^{\\mu_0}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1))))"], [["exp", 4], "Equality(exp(Mul(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)))), exp(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Derivative(Pow(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(k,\\theta_2)} = \\theta_2 + k and t{(k,\\theta_2)} = \\theta_2 + k, then obtain \\frac{\\partial}{\\partial k} \\operatorname{F_{N}}{(k,\\theta_2)} \\frac{\\partial}{\\partial k} t{(k,\\theta_2)} = (\\frac{\\partial}{\\partial k} t{(k,\\theta_2)})^{2}", "derivation": "\\operatorname{F_{N}}{(k,\\theta_2)} = \\theta_2 + k and t{(k,\\theta_2)} = \\theta_2 + k and \\operatorname{F_{N}}{(k,\\theta_2)} = t{(k,\\theta_2)} and \\frac{\\partial}{\\partial k} \\operatorname{F_{N}}{(k,\\theta_2)} = \\frac{\\partial}{\\partial k} t{(k,\\theta_2)} and \\frac{\\partial}{\\partial k} (\\theta_2 + k) = \\frac{\\partial}{\\partial k} t{(k,\\theta_2)} and \\frac{\\partial}{\\partial k} \\operatorname{F_{N}}{(k,\\theta_2)} = \\frac{\\partial}{\\partial k} (\\theta_2 + k) and \\frac{\\partial}{\\partial k} (\\theta_2 + k) \\frac{\\partial}{\\partial k} \\operatorname{F_{N}}{(k,\\theta_2)} = (\\frac{\\partial}{\\partial k} (\\theta_2 + k))^{2} and \\frac{\\partial}{\\partial k} \\operatorname{F_{N}}{(k,\\theta_2)} \\frac{\\partial}{\\partial k} t{(k,\\theta_2)} = (\\frac{\\partial}{\\partial k} t{(k,\\theta_2)})^{2}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)))"], ["renaming_premise", "Equality(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["times", 6, "Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('\\\\theta_2', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Derivative(Function('F_N')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Pow(Derivative(Function('t')(Symbol('k', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\psi{(z,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + z and T{(z,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + z, then obtain \\log{(2 \\psi{(z,V_{\\mathbf{B}})})} = \\log{(T{(z,V_{\\mathbf{B}})} + \\psi{(z,V_{\\mathbf{B}})})}", "derivation": "\\psi{(z,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + z and 2 \\psi{(z,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + z + \\psi{(z,V_{\\mathbf{B}})} and \\log{(2 \\psi{(z,V_{\\mathbf{B}})})} = \\log{(V_{\\mathbf{B}} + z + \\psi{(z,V_{\\mathbf{B}})})} and T{(z,V_{\\mathbf{B}})} = V_{\\mathbf{B}} + z and \\log{(2 \\psi{(z,V_{\\mathbf{B}})})} = \\log{(T{(z,V_{\\mathbf{B}})} + \\psi{(z,V_{\\mathbf{B}})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)))"], [["add", 1, "Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["log", 2], "Equality(log(Mul(Integer(2), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), log(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"], ["renaming_premise", "Equality(Function('T')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(log(Mul(Integer(2), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))), log(Add(Function('T')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\psi')(Symbol('z', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\phi,M_{E})} = M_{E}^{\\phi} and u{(\\phi,M_{E})} = M_{E}^{\\phi}, then obtain M_{E} + \\operatorname{n_{1}}^{M_{E}}{(\\phi,M_{E})} = M_{E} + (M_{E}^{- \\phi} \\operatorname{n_{1}}{(\\phi,M_{E})} u{(\\phi,M_{E})})^{M_{E}}", "derivation": "\\operatorname{n_{1}}{(\\phi,M_{E})} = M_{E}^{\\phi} and u{(\\phi,M_{E})} = M_{E}^{\\phi} and M_{E}^{- \\phi} \\operatorname{n_{1}}{(\\phi,M_{E})} u{(\\phi,M_{E})} = \\operatorname{n_{1}}{(\\phi,M_{E})} and \\operatorname{n_{1}}^{M_{E}}{(\\phi,M_{E})} = (M_{E}^{\\phi})^{M_{E}} and M_{E}^{- \\phi} \\operatorname{n_{1}}{(\\phi,M_{E})} u{(\\phi,M_{E})} = M_{E}^{\\phi} and M_{E} + \\operatorname{n_{1}}^{M_{E}}{(\\phi,M_{E})} = M_{E} + (M_{E}^{\\phi})^{M_{E}} and M_{E} + \\operatorname{n_{1}}^{M_{E}}{(\\phi,M_{E})} = M_{E} + (M_{E}^{- \\phi} \\operatorname{n_{1}}{(\\phi,M_{E})} u{(\\phi,M_{E})})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["divide", 2, "Mul(Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Function('u')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True))), Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)))"], [["power", 1, "Symbol('M_E', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Mul(Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Function('u')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True))), Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)))"], [["add", 4, "Symbol('M_E', commutative=True)"], "Equality(Add(Symbol('M_E', commutative=True), Pow(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Pow(Pow(Symbol('M_E', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Symbol('M_E', commutative=True), Pow(Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True))), Add(Symbol('M_E', commutative=True), Pow(Mul(Pow(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Function('n_1')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True)), Function('u')(Symbol('\\\\phi', commutative=True), Symbol('M_E', commutative=True))), Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(\\theta)} = \\sin{(\\theta)}, then derive \\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)} = \\cos{(\\theta)}, then obtain (\\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)})^{\\theta} = (\\frac{d}{d \\theta} \\sin{(\\theta)})^{\\theta}", "derivation": "\\operatorname{F_{x}}{(\\theta)} = \\sin{(\\theta)} and \\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\theta)} and \\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)} = \\cos{(\\theta)} and (\\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)})^{\\theta} = \\cos^{\\theta}{(\\theta)} and (\\frac{d}{d \\theta} \\sin{(\\theta)})^{\\theta} = \\cos^{\\theta}{(\\theta)} and (\\frac{d}{d \\theta} \\operatorname{F_{x}}{(\\theta)})^{\\theta} = (\\frac{d}{d \\theta} \\sin{(\\theta)})^{\\theta}", "srepr_derivation": [["renaming_premise", "Equality(Function('F_x')(Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), cos(Symbol('\\\\theta', commutative=True)))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Derivative(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Function('F_x')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True)), Pow(Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(v_{t},n_{2})} = n_{2} v_{t}, then obtain \\operatorname{A_{1}}{(v_{t},n_{2})} \\operatorname{A_{1}}^{v_{t}}{(v_{t},n_{2})} = n_{2} v_{t} \\operatorname{A_{1}}^{v_{t}}{(v_{t},n_{2})}", "derivation": "\\operatorname{A_{1}}{(v_{t},n_{2})} = n_{2} v_{t} and \\operatorname{A_{1}}^{v_{t}}{(v_{t},n_{2})} = (n_{2} v_{t})^{v_{t}} and (n_{2} v_{t})^{v_{t}} \\operatorname{A_{1}}{(v_{t},n_{2})} = n_{2} v_{t} (n_{2} v_{t})^{v_{t}} and \\operatorname{A_{1}}{(v_{t},n_{2})} \\operatorname{A_{1}}^{v_{t}}{(v_{t},n_{2})} = n_{2} v_{t} \\operatorname{A_{1}}^{v_{t}}{(v_{t},n_{2})}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True)))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True)), Symbol('v_t', commutative=True)), Pow(Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)))"], [["times", 1, "Pow(Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True))), Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True), Pow(Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True)), Pow(Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True)), Symbol('v_t', commutative=True))), Mul(Symbol('n_2', commutative=True), Symbol('v_t', commutative=True), Pow(Function('A_1')(Symbol('v_t', commutative=True), Symbol('n_2', commutative=True)), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\phi_{1}{(\\mathbf{H})} = \\mathbf{H}, then obtain \\frac{0^{\\mathbf{H}} (1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}})^{\\mathbf{H}}}{\\mathbf{H}^{2}} = \\frac{0^{\\mathbf{H}}}{\\mathbf{H}^{2}}", "derivation": "\\phi_{1}{(\\mathbf{H})} = \\mathbf{H} and \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}} = 1 and 0 = 1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}} and 0^{\\mathbf{H}} = (1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}})^{\\mathbf{H}} and \\frac{0^{\\mathbf{H}}}{\\mathbf{H}^{2}} = \\frac{(1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}})^{\\mathbf{H}}}{\\mathbf{H}^{2}} and \\frac{(1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}})^{\\mathbf{H}}}{\\mathbf{H}^{2}} = \\frac{1}{\\mathbf{H}^{2}} and \\frac{0^{\\mathbf{H}}}{\\mathbf{H}^{2}} = \\frac{1}{\\mathbf{H}^{2}} and \\frac{0^{\\mathbf{H}} (1 - \\frac{\\phi_{1}{(\\mathbf{H})}}{\\mathbf{H}})^{\\mathbf{H}}}{\\mathbf{H}^{2}} = \\frac{0^{\\mathbf{H}}}{\\mathbf{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], [["divide", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True))), Integer(1))"], [["minus", 2, "Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 4, "Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('\\\\mathbf{H}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('\\\\mathbf{H}', commutative=True)))), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Pow(Integer(0), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\varepsilon^{V_{\\mathbf{B}}}, then derive \\varepsilon^{V_{\\mathbf{B}}} + \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\varepsilon^{V_{\\mathbf{B}}} \\log{(\\varepsilon)} + \\varepsilon^{V_{\\mathbf{B}}}, then obtain \\varepsilon + \\varepsilon^{V_{\\mathbf{B}}} + \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\varepsilon + \\varepsilon^{V_{\\mathbf{B}}} \\log{(\\varepsilon)} + \\varepsilon^{V_{\\mathbf{B}}}", "derivation": "\\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\varepsilon^{V_{\\mathbf{B}}} and \\varepsilon^{V_{\\mathbf{B}}} + \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\varepsilon^{V_{\\mathbf{B}}} + \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\varepsilon^{V_{\\mathbf{B}}} and \\varepsilon^{V_{\\mathbf{B}}} + \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\varepsilon^{V_{\\mathbf{B}}} \\log{(\\varepsilon)} + \\varepsilon^{V_{\\mathbf{B}}} and \\varepsilon + \\varepsilon^{V_{\\mathbf{B}}} + \\mu_{0}{(\\varepsilon,V_{\\mathbf{B}})} = \\varepsilon + \\varepsilon^{V_{\\mathbf{B}}} \\log{(\\varepsilon)} + \\varepsilon^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["add", 1, "Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Derivative(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["add", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Function('\\\\mu_0')(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Symbol('\\\\varepsilon', commutative=True), Mul(Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True))), Pow(Symbol('\\\\varepsilon', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\mu{(a)} = \\log{(a)}, then obtain \\int \\sin{(\\frac{1}{a \\log{(a)}})} da = \\int \\sin{(\\frac{\\log{(a)}}{a \\mu^{2}{(a)}})} da", "derivation": "\\mu{(a)} = \\log{(a)} and a \\mu{(a)} = a \\log{(a)} and \\frac{1}{a} = \\frac{\\log{(a)}}{a \\mu{(a)}} and \\frac{1}{a \\mu{(a)}} = \\frac{\\log{(a)}}{a \\mu^{2}{(a)}} and \\frac{1}{a \\log{(a)}} = \\frac{1}{a \\mu{(a)}} and \\frac{1}{a \\log{(a)}} = \\frac{\\log{(a)}}{a \\mu^{2}{(a)}} and \\sin{(\\frac{1}{a \\log{(a)}})} = \\sin{(\\frac{\\log{(a)}}{a \\mu^{2}{(a)}})} and \\int \\sin{(\\frac{1}{a \\log{(a)}})} da = \\int \\sin{(\\frac{\\log{(a)}}{a \\mu^{2}{(a)}})} da", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('a', commutative=True)), log(Symbol('a', commutative=True)))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('\\\\mu')(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), log(Symbol('a', commutative=True))))"], [["divide", 1, "Mul(Symbol('a', commutative=True), Function('\\\\mu')(Symbol('a', commutative=True)))"], "Equality(Pow(Symbol('a', commutative=True), Integer(-1)), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-1)), log(Symbol('a', commutative=True))))"], [["divide", 3, "Function('\\\\mu')(Symbol('a', commutative=True))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-2)), log(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(log(Symbol('a', commutative=True)), Integer(-1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(log(Symbol('a', commutative=True)), Integer(-1))), Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-2)), log(Symbol('a', commutative=True))))"], [["sin", 6], "Equality(sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(log(Symbol('a', commutative=True)), Integer(-1)))), sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-2)), log(Symbol('a', commutative=True)))))"], [["integrate", 7, "Symbol('a', commutative=True)"], "Equality(Integral(sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(log(Symbol('a', commutative=True)), Integer(-1)))), Tuple(Symbol('a', commutative=True))), Integral(sin(Mul(Pow(Symbol('a', commutative=True), Integer(-1)), Pow(Function('\\\\mu')(Symbol('a', commutative=True)), Integer(-2)), log(Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(P_{g},m)} = - P_{g} + m, then obtain \\operatorname{z^{*}}{(P_{g},m)} \\int \\operatorname{z^{*}}{(P_{g},m)} dP_{g} = (- P_{g} + m) \\int \\operatorname{z^{*}}{(P_{g},m)} dP_{g}", "derivation": "\\operatorname{z^{*}}{(P_{g},m)} = - P_{g} + m and \\int \\operatorname{z^{*}}{(P_{g},m)} dP_{g} = \\int (- P_{g} + m) dP_{g} and \\operatorname{z^{*}}{(P_{g},m)} \\int (- P_{g} + m) dP_{g} = (- P_{g} + m) \\int (- P_{g} + m) dP_{g} and \\operatorname{z^{*}}{(P_{g},m)} \\int \\operatorname{z^{*}}{(P_{g},m)} dP_{g} = (- P_{g} + m) \\int \\operatorname{z^{*}}{(P_{g},m)} dP_{g}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["times", 1, "Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True)))"], "Equality(Mul(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Integral(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('m', commutative=True)), Integral(Function('z^*')(Symbol('P_g', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given c{(E,h)} = \\cos{(E + h)}, then obtain (\\frac{\\partial}{\\partial h} c{(E,h)})^{2} = - \\sin{(E + h)} \\frac{\\partial}{\\partial h} c{(E,h)}", "derivation": "c{(E,h)} = \\cos{(E + h)} and \\frac{\\partial}{\\partial h} c{(E,h)} = \\frac{\\partial}{\\partial h} \\cos{(E + h)} and (\\frac{\\partial}{\\partial h} c{(E,h)})^{2} = \\frac{\\partial}{\\partial h} c{(E,h)} \\frac{\\partial}{\\partial h} \\cos{(E + h)} and (\\frac{\\partial}{\\partial h} c{(E,h)})^{2} = - \\sin{(E + h)} \\frac{\\partial}{\\partial h} c{(E,h)}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('E', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(cos(Add(Symbol('E', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)), Mul(Integer(-1), sin(Add(Symbol('E', commutative=True), Symbol('h', commutative=True))), Derivative(Function('c')(Symbol('E', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{2}{(J)} = \\log{(\\cos{(J)})} and \\operatorname{z^{*}}{(J)} = \\int \\frac{d}{d J} \\log{(\\cos{(J)})} dJ, then obtain \\operatorname{z^{*}}{(J)} \\frac{d}{d J} \\log{(\\cos{(J)})} = \\frac{d}{d J} \\log{(\\cos{(J)})} \\int \\frac{d}{d J} \\phi_{2}{(J)} dJ", "derivation": "\\phi_{2}{(J)} = \\log{(\\cos{(J)})} and \\frac{d}{d J} \\phi_{2}{(J)} = \\frac{d}{d J} \\log{(\\cos{(J)})} and \\int \\frac{d}{d J} \\phi_{2}{(J)} dJ = \\int \\frac{d}{d J} \\log{(\\cos{(J)})} dJ and \\operatorname{z^{*}}{(J)} = \\int \\frac{d}{d J} \\log{(\\cos{(J)})} dJ and \\operatorname{z^{*}}{(J)} \\frac{d}{d J} \\phi_{2}{(J)} = \\frac{d}{d J} \\phi_{2}{(J)} \\int \\frac{d}{d J} \\log{(\\cos{(J)})} dJ and \\operatorname{z^{*}}{(J)} \\frac{d}{d J} \\log{(\\cos{(J)})} = \\frac{d}{d J} \\log{(\\cos{(J)})} \\int \\frac{d}{d J} \\log{(\\cos{(J)})} dJ and \\operatorname{z^{*}}{(J)} \\frac{d}{d J} \\log{(\\cos{(J)})} = \\frac{d}{d J} \\log{(\\cos{(J)})} \\int \\frac{d}{d J} \\phi_{2}{(J)} dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('J', commutative=True)), log(cos(Symbol('J', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('J', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Integral(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], ["renaming_premise", "Equality(Function('z^*')(Symbol('J', commutative=True)), Integral(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["times", 4, "Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Mul(Function('z^*')(Symbol('J', commutative=True)), Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Integral(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Function('z^*')(Symbol('J', commutative=True)), Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integral(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Function('z^*')(Symbol('J', commutative=True)), Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Derivative(log(cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Integral(Derivative(Function('\\\\phi_2')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(J)} = \\cos{(J)} and \\Psi_{nl}{(n)} = \\log{(n)}, then obtain \\frac{\\cos{(\\Psi_{nl}{(n)} + 4 \\varepsilon{(J)} - 4 \\cos{(J)})}}{2 \\varepsilon{(J)} - 2 \\cos{(J)}} = \\frac{\\cos{(4 \\varepsilon{(J)} + \\log{(n)} - 4 \\cos{(J)})}}{2 \\varepsilon{(J)} - 2 \\cos{(J)}}", "derivation": "\\varepsilon{(J)} = \\cos{(J)} and \\varepsilon{(J)} - \\cos{(J)} = 0 and 2 \\varepsilon{(J)} - \\cos{(J)} = \\varepsilon{(J)} and \\Psi_{nl}{(n)} = \\log{(n)} and \\Psi_{nl}{(n)} + 2 \\varepsilon{(J)} - 2 \\cos{(J)} = 2 \\varepsilon{(J)} + \\log{(n)} - 2 \\cos{(J)} and \\Psi_{nl}{(n)} + 4 \\varepsilon{(J)} - 4 \\cos{(J)} = 4 \\varepsilon{(J)} + \\log{(n)} - 4 \\cos{(J)} and \\cos{(\\Psi_{nl}{(n)} + 4 \\varepsilon{(J)} - 4 \\cos{(J)})} = \\cos{(4 \\varepsilon{(J)} + \\log{(n)} - 4 \\cos{(J)})} and \\frac{\\cos{(\\Psi_{nl}{(n)} + 4 \\varepsilon{(J)} - 4 \\cos{(J)})}}{2 \\varepsilon{(J)} - 2 \\cos{(J)}} = \\frac{\\cos{(4 \\varepsilon{(J)} + \\log{(n)} - 4 \\cos{(J)})}}{2 \\varepsilon{(J)} - 2 \\cos{(J)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["minus", 1, "cos(Symbol('J', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('J', commutative=True)), Mul(Integer(-1), cos(Symbol('J', commutative=True)))), Integer(0))"], [["add", 2, "Function('\\\\varepsilon')(Symbol('J', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), cos(Symbol('J', commutative=True)))), Function('\\\\varepsilon')(Symbol('J', commutative=True)))"], ["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["add", 4, "Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True))))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('n', commutative=True)), Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True)))), Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), log(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('n', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True)))), Add(Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), log(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True)))))"], [["cos", 6], "Equality(cos(Add(Function('\\\\Psi_{nl}')(Symbol('n', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True))))), cos(Add(Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), log(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True))))))"], [["divide", 7, "Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True)))), Integer(-1)), cos(Add(Function('\\\\Psi_{nl}')(Symbol('n', commutative=True)), Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True)))))), Mul(Pow(Add(Mul(Integer(2), Function('\\\\varepsilon')(Symbol('J', commutative=True))), Mul(Integer(-1), Integer(2), cos(Symbol('J', commutative=True)))), Integer(-1)), cos(Add(Mul(Integer(4), Function('\\\\varepsilon')(Symbol('J', commutative=True))), log(Symbol('n', commutative=True)), Mul(Integer(-1), Integer(4), cos(Symbol('J', commutative=True)))))))"]]}, {"prompt": "Given y{(E)} = e^{E}, then derive \\frac{d}{d E} y{(E)} = e^{E}, then obtain (\\frac{d^{2}}{d E^{2}} e^{E})^{E} = (e^{E})^{E}", "derivation": "y{(E)} = e^{E} and \\frac{d}{d E} y{(E)} = \\frac{d}{d E} e^{E} and \\frac{d}{d E} y{(E)} = e^{E} and (\\frac{d}{d E} y{(E)})^{E} = (e^{E})^{E} and (\\frac{d}{d E} e^{E})^{E} = (e^{E})^{E} and \\frac{d}{d E} y{(E)} = \\frac{d^{2}}{d E^{2}} y{(E)} and (\\frac{d}{d E} y{(E)})^{E} = (\\frac{d^{2}}{d E^{2}} y{(E)})^{E} and (\\frac{d}{d E} e^{E})^{E} = (\\frac{d^{2}}{d E^{2}} e^{E})^{E} and (\\frac{d^{2}}{d E^{2}} e^{E})^{E} = (e^{E})^{E}", "srepr_derivation": [["get_premise", "Equality(Function('y')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), exp(Symbol('E', commutative=True)))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))))"], [["power", 6, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(Function('y')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 8], "Equality(Pow(Derivative(exp(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(2))), Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(\\varepsilon_0,\\dot{\\mathbf{r}},f^{*})} = \\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0}, then derive \\frac{\\partial}{\\partial f^{*}} \\mathbf{s}{(\\varepsilon_0,\\dot{\\mathbf{r}},f^{*})} = \\frac{1}{\\varepsilon_0}, then obtain \\operatorname{F_{g}}{(\\Psi,f_{\\mathbf{p}})} + \\frac{1}{\\varepsilon_0} = \\operatorname{F_{g}}{(\\Psi,f_{\\mathbf{p}})} + \\frac{\\partial}{\\partial f^{*}} (\\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0})", "derivation": "\\mathbf{s}{(\\varepsilon_0,\\dot{\\mathbf{r}},f^{*})} = \\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0} and \\frac{\\partial}{\\partial f^{*}} \\mathbf{s}{(\\varepsilon_0,\\dot{\\mathbf{r}},f^{*})} = \\frac{\\partial}{\\partial f^{*}} (\\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0}) and \\frac{\\partial}{\\partial f^{*}} \\mathbf{s}{(\\varepsilon_0,\\dot{\\mathbf{r}},f^{*})} = \\frac{1}{\\varepsilon_0} and \\frac{1}{\\varepsilon_0} = \\frac{\\partial}{\\partial f^{*}} (\\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0}) and \\operatorname{F_{g}}{(\\Psi,f_{\\mathbf{p}})} + \\frac{1}{\\varepsilon_0} = \\operatorname{F_{g}}{(\\Psi,f_{\\mathbf{p}})} + \\frac{\\partial}{\\partial f^{*}} (\\dot{\\mathbf{r}} + \\frac{f^{*}}{\\varepsilon_0})", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["add", 4, "Function('F_g')(Symbol('\\\\Psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], "Equality(Add(Function('F_g')(Symbol('\\\\Psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1))), Add(Function('F_g')(Symbol('\\\\Psi', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True), Integer(1)))))"]]}, {"prompt": "Given h{(I)} = \\cos{(\\cos{(I)})}, then obtain \\frac{\\int \\frac{h{(I)}}{I} dI}{\\int \\frac{\\cos{(\\cos{(I)})}}{I} dI} = 1", "derivation": "h{(I)} = \\cos{(\\cos{(I)})} and \\frac{h{(I)}}{I} = \\frac{\\cos{(\\cos{(I)})}}{I} and \\int \\frac{h{(I)}}{I} dI = \\int \\frac{\\cos{(\\cos{(I)})}}{I} dI and \\frac{\\int \\frac{h{(I)}}{I} dI}{\\int \\frac{\\cos{(\\cos{(I)})}}{I} dI} = 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('I', commutative=True)), cos(cos(Symbol('I', commutative=True))))"], [["divide", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('h')(Symbol('I', commutative=True))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(cos(Symbol('I', commutative=True)))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('h')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(cos(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))))"], [["divide", 3, "Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(cos(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Function('h')(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True))), Pow(Integral(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), cos(cos(Symbol('I', commutative=True)))), Tuple(Symbol('I', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(B)} = \\cos{(\\cos{(B)})} and \\dot{z}{(Z)} = \\sin{(e^{Z})}, then obtain - \\cos^{2}{(\\cos{(B)})} + \\int (- Z + \\dot{z}{(Z)}) dZ = - \\cos^{2}{(\\cos{(B)})} + \\int (- Z + \\sin{(e^{Z})}) dZ", "derivation": "\\dot{\\mathbf{r}}{(B)} = \\cos{(\\cos{(B)})} and \\dot{z}{(Z)} = \\sin{(e^{Z})} and - Z + \\dot{z}{(Z)} = - Z + \\sin{(e^{Z})} and \\int (- Z + \\dot{z}{(Z)}) dZ = \\int (- Z + \\sin{(e^{Z})}) dZ and - \\dot{\\mathbf{r}}^{2}{(B)} + \\int (- Z + \\dot{z}{(Z)}) dZ = - \\dot{\\mathbf{r}}^{2}{(B)} + \\int (- Z + \\sin{(e^{Z})}) dZ and - \\cos^{2}{(\\cos{(B)})} + \\int (- Z + \\dot{z}{(Z)}) dZ = - \\cos^{2}{(\\cos{(B)})} + \\int (- Z + \\sin{(e^{Z})}) dZ", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('B', commutative=True)), cos(cos(Symbol('B', commutative=True))))"], ["get_premise", "Equality(Function('\\\\dot{z}')(Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True))))"], [["minus", 2, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\dot{z}')(Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True)))))"], [["integrate", 3, "Symbol('Z', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\dot{z}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True))))"], [["minus", 4, "Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('B', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('B', commutative=True)), Integer(2))), Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\dot{z}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('B', commutative=True)), Integer(2))), Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Pow(cos(cos(Symbol('B', commutative=True))), Integer(2))), Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\dot{z}')(Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Add(Mul(Integer(-1), Pow(cos(cos(Symbol('B', commutative=True))), Integer(2))), Integral(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), sin(exp(Symbol('Z', commutative=True)))), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} = A_{x} (- V_{\\mathbf{B}} + v_{z}), then obtain \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} dv_{z} = - A_{x} v_{z} + F_{c}", "derivation": "\\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} = A_{x} (- V_{\\mathbf{B}} + v_{z}) and \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} = \\frac{\\partial}{\\partial V_{\\mathbf{B}}} A_{x} (- V_{\\mathbf{B}} + v_{z}) and \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} dv_{z} = \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} A_{x} (- V_{\\mathbf{B}} + v_{z}) dv_{z} and \\int \\frac{\\partial}{\\partial V_{\\mathbf{B}}} \\operatorname{F_{c}}{(A_{x},V_{\\mathbf{B}},v_{z})} dv_{z} = - A_{x} v_{z} + F_{c}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('A_x', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('A_x', commutative=True), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('v_z', commutative=True))))"], [["differentiate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('A_x', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_x', commutative=True), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('v_z', commutative=True)"], "Equality(Integral(Derivative(Function('F_c')(Symbol('A_x', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True))), Integral(Derivative(Mul(Symbol('A_x', commutative=True), Add(Mul(Integer(-1), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('v_z', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('F_c')(Symbol('A_x', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True), Integer(1))), Tuple(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('A_x', commutative=True), Symbol('v_z', commutative=True)), Symbol('F_c', commutative=True)))"]]}, {"prompt": "Given \\mu{(\\Psi,E)} = \\frac{E}{\\Psi}, then obtain \\frac{\\frac{\\partial^{3}}{\\partial \\Psi^{2}\\partial E} \\Psi \\mu{(\\Psi,E)}}{\\frac{d^{3}}{d \\Psi^{2}d E} E} = 1", "derivation": "\\mu{(\\Psi,E)} = \\frac{E}{\\Psi} and \\Psi \\mu{(\\Psi,E)} = E and \\frac{\\partial}{\\partial E} \\Psi \\mu{(\\Psi,E)} = \\frac{d}{d E} E and \\frac{\\partial^{2}}{\\partial \\Psi\\partial E} \\Psi \\mu{(\\Psi,E)} = \\frac{d^{2}}{d \\Psid E} E and \\frac{\\partial^{3}}{\\partial \\Psi^{2}\\partial E} \\Psi \\mu{(\\Psi,E)} = \\frac{d^{3}}{d \\Psi^{2}d E} E and \\frac{\\frac{\\partial^{3}}{\\partial \\Psi^{2}\\partial E} \\Psi \\mu{(\\Psi,E)}}{\\frac{d^{3}}{d \\Psi^{2}d E} E} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True))), Symbol('E', commutative=True))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["differentiate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2))), Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2))))"], [["divide", 5, "Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2)))"], "Equality(Mul(Pow(Derivative(Symbol('E', commutative=True), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2))), Integer(-1)), Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\mu')(Symbol('\\\\Psi', commutative=True), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(2)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(l,f)} = \\frac{l}{f}, then derive \\frac{\\partial}{\\partial l} \\operatorname{C_{1}}{(l,f)} = \\frac{1}{f}, then obtain \\frac{\\frac{\\partial}{\\partial l} \\operatorname{C_{1}}{(l,f)}}{- E_{x} - \\nabla + \\frac{l}{f}} = \\frac{1}{f (- E_{x} - \\nabla + \\frac{l}{f})}", "derivation": "\\operatorname{C_{1}}{(l,f)} = \\frac{l}{f} and - E_{x} - \\nabla + \\operatorname{C_{1}}{(l,f)} = - E_{x} - \\nabla + \\frac{l}{f} and \\frac{\\partial}{\\partial l} (- E_{x} - \\nabla + \\operatorname{C_{1}}{(l,f)}) = \\frac{\\partial}{\\partial l} (- E_{x} - \\nabla + \\frac{l}{f}) and \\frac{\\partial}{\\partial l} \\operatorname{C_{1}}{(l,f)} = \\frac{1}{f} and \\frac{\\frac{\\partial}{\\partial l} \\operatorname{C_{1}}{(l,f)}}{- E_{x} - \\nabla + \\frac{l}{f}} = \\frac{1}{f (- E_{x} - \\nabla + \\frac{l}{f})}", "srepr_derivation": [["get_premise", "Equality(Function('C_1')(Symbol('l', commutative=True), Symbol('f', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], [["minus", 1, "Add(Symbol('E_x', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('C_1')(Symbol('l', commutative=True), Symbol('f', commutative=True))), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Function('C_1')(Symbol('l', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('C_1')(Symbol('l', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Pow(Symbol('f', commutative=True), Integer(-1)))"], [["divide", 4, "Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Integer(-1)), Derivative(Function('C_1')(Symbol('l', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('l', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(\\eta)} = \\log{(e^{\\eta})}, then derive \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} = 1, then obtain \\int (\\int \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} d\\eta)^{\\eta} d\\eta = \\int (\\int 1 d\\eta)^{\\eta} d\\eta", "derivation": "\\operatorname{x^{{\\}'}}{(\\eta)} = \\log{(e^{\\eta})} and \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} = \\frac{d}{d \\eta} \\log{(e^{\\eta})} and \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} = 1 and \\int \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} d\\eta = \\int 1 d\\eta and (\\int \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} d\\eta)^{\\eta} = (\\int 1 d\\eta)^{\\eta} and \\int (\\int \\frac{d}{d \\eta} \\operatorname{x^{{\\}'}}{(\\eta)} d\\eta)^{\\eta} d\\eta = \\int (\\int 1 d\\eta)^{\\eta} d\\eta", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), log(exp(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Derivative(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))))"], [["power", 4, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Pow(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["integrate", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Pow(Integral(Derivative(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Pow(Integral(Integer(1), Tuple(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(G,\\Psi_{nl})} = \\sin{(G \\Psi_{nl})} and \\Psi{(G,\\Psi_{nl})} = \\sin{(G \\Psi_{nl})}, then obtain \\frac{\\Psi{(G,\\Psi_{nl})}}{\\sin{(G \\Psi_{nl})}} = 1", "derivation": "\\operatorname{t_{1}}{(G,\\Psi_{nl})} = \\sin{(G \\Psi_{nl})} and \\Psi{(G,\\Psi_{nl})} = \\sin{(G \\Psi_{nl})} and \\frac{\\Psi{(G,\\Psi_{nl})}}{\\operatorname{t_{1}}{(G,\\Psi_{nl})}} = \\frac{\\sin{(G \\Psi_{nl})}}{\\operatorname{t_{1}}{(G,\\Psi_{nl})}} and \\frac{\\Psi{(G,\\Psi_{nl})}}{\\sin{(G \\Psi_{nl})}} = 1", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Mul(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), sin(Mul(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))))"], [["divide", 2, "Function('t_1')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Mul(Function('\\\\Psi')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Function('t_1')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1))), Mul(Pow(Function('t_1')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Integer(-1)), sin(Mul(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('\\\\Psi')(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(sin(Mul(Symbol('G', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\hat{X}{(k,n_{2},S)} = - S + k + n_{2}, then derive 2 \\frac{\\partial}{\\partial k} \\hat{X}{(k,n_{2},S)} = 2, then obtain \\frac{\\int 2 \\hat{X}{(k,n_{2},S)} dk}{2} = \\frac{\\int (- 2 S + 2 k + 2 n_{2}) dk}{2}", "derivation": "\\hat{X}{(k,n_{2},S)} = - S + k + n_{2} and - S + k + n_{2} + \\hat{X}{(k,n_{2},S)} = - 2 S + 2 k + 2 n_{2} and 2 \\hat{X}{(k,n_{2},S)} = - 2 S + 2 k + 2 n_{2} and \\frac{\\partial}{\\partial k} 2 \\hat{X}{(k,n_{2},S)} = \\frac{\\partial}{\\partial k} (- 2 S + 2 k + 2 n_{2}) and \\int 2 \\hat{X}{(k,n_{2},S)} dk = \\int (- 2 S + 2 k + 2 n_{2}) dk and 2 \\frac{\\partial}{\\partial k} \\hat{X}{(k,n_{2},S)} = 2 and \\frac{\\int 2 \\hat{X}{(k,n_{2},S)} dk}{2 \\frac{\\partial}{\\partial k} \\hat{X}{(k,n_{2},S)}} = \\frac{\\int (- 2 S + 2 k + 2 n_{2}) dk}{2 \\frac{\\partial}{\\partial k} \\hat{X}{(k,n_{2},S)}} and \\frac{\\int 2 \\hat{X}{(k,n_{2},S)} dk}{2} = \\frac{\\int (- 2 S + 2 k + 2 n_{2}) dk}{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('k', commutative=True), Symbol('n_2', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('k', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('k', commutative=True), Symbol('n_2', commutative=True), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('k', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('k', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))), Tuple(Symbol('k', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Integer(2))"], [["divide", 5, "Mul(Integer(2), Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], "Equality(Mul(Rational(1, 2), Pow(Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Rational(1, 2), Pow(Derivative(Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))), Tuple(Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Mul(Rational(1, 2), Integral(Mul(Integer(2), Function('\\\\hat{X}')(Symbol('k', commutative=True), Symbol('n_2', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('k', commutative=True)))), Mul(Rational(1, 2), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('S', commutative=True)), Mul(Integer(2), Symbol('k', commutative=True)), Mul(Integer(2), Symbol('n_2', commutative=True))), Tuple(Symbol('k', commutative=True)))))"]]}, {"prompt": "Given Z{(\\phi_2)} = \\sin{(\\cos{(\\phi_2)})} and \\operatorname{v_{t}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\frac{(\\sin{(\\cos{(\\phi_2)})} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2}, then obtain \\operatorname{v_{t}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\frac{(Z{(\\phi_2)} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2}", "derivation": "Z{(\\phi_2)} = \\sin{(\\cos{(\\phi_2)})} and Z{(\\phi_2)} \\cos{(\\phi_2)} = \\sin{(\\cos{(\\phi_2)})} \\cos{(\\phi_2)} and (Z{(\\phi_2)} \\cos{(\\phi_2)})^{\\phi_2} = (\\sin{(\\cos{(\\phi_2)})} \\cos{(\\phi_2)})^{\\phi_2} and \\frac{(Z{(\\phi_2)} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2} = \\frac{(\\sin{(\\cos{(\\phi_2)})} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2} and \\operatorname{v_{t}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\frac{(\\sin{(\\cos{(\\phi_2)})} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2} and \\operatorname{v_{t}}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\frac{(Z{(\\phi_2)} \\cos{(\\phi_2)})^{\\phi_2}}{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\phi_2', commutative=True)), sin(cos(Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(sin(cos(Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Mul(Function('Z')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), Pow(Mul(sin(cos(Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["times", 3, "Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Mul(Function('Z')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))), Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Mul(sin(cos(Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))))"], ["renaming_premise", "Equality(Function('v_t')(Symbol('\\\\phi_2', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Mul(sin(cos(Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('v_t')(Symbol('\\\\phi_2', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\phi_2', commutative=True), Integer(-1)), Pow(Mul(Function('Z')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(E)} = \\frac{d}{d E} \\cos{(E)}, then derive \\hat{x}_0{(E)} = - \\sin{(E)}, then derive \\frac{d}{d E} \\hat{x}_0{(E)} = - \\cos{(E)}, then obtain \\frac{d}{d E} \\hat{x}_0{(E)} + (\\frac{d}{d E} \\hat{x}_0{(E)})^{E} + \\frac{d}{d E} \\cos{(E)} = (- \\cos{(E)})^{E} + \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{d}{d E} \\cos{(E)}", "derivation": "\\hat{x}_0{(E)} = \\frac{d}{d E} \\cos{(E)} and \\hat{x}_0{(E)} = - \\sin{(E)} and \\frac{d}{d E} \\hat{x}_0{(E)} = \\frac{d}{d E} - \\sin{(E)} and \\frac{d}{d E} \\hat{x}_0{(E)} = - \\cos{(E)} and (\\frac{d}{d E} \\hat{x}_0{(E)})^{E} = (- \\cos{(E)})^{E} and \\frac{d}{d E} \\hat{x}_0{(E)} + (\\frac{d}{d E} \\hat{x}_0{(E)})^{E} + \\frac{d}{d E} \\cos{(E)} = (- \\cos{(E)})^{E} + \\frac{d}{d E} \\hat{x}_0{(E)} + \\frac{d}{d E} \\cos{(E)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Mul(Integer(-1), sin(Symbol('E', commutative=True))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Mul(Integer(-1), cos(Symbol('E', commutative=True))))"], [["power", 4, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Mul(Integer(-1), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"], [["add", 5, "Add(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))))"], "Equality(Add(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Add(Pow(Mul(Integer(-1), cos(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Derivative(Function('\\\\hat{x}_0')(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(cos(Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\phi_1)} = \\phi_1 and \\mathbf{S}{(\\Omega,\\phi_1)} = \\int \\frac{\\Omega}{\\phi_1} d\\phi_1, then obtain (\\int \\frac{\\Omega}{\\phi_1} d\\phi_1) \\int \\mathbf{J}_M{(\\phi_1)} d\\phi_1 = (\\int \\phi_1 d\\phi_1) \\int \\frac{\\Omega}{\\phi_1} d\\phi_1", "derivation": "\\mathbf{J}_M{(\\phi_1)} = \\phi_1 and \\int \\mathbf{J}_M{(\\phi_1)} d\\phi_1 = \\int \\phi_1 d\\phi_1 and \\mathbf{S}{(\\Omega,\\phi_1)} = \\int \\frac{\\Omega}{\\phi_1} d\\phi_1 and \\mathbf{S}{(\\Omega,\\phi_1)} \\int \\mathbf{J}_M{(\\phi_1)} d\\phi_1 = \\mathbf{S}{(\\Omega,\\phi_1)} \\int \\phi_1 d\\phi_1 and (\\int \\frac{\\Omega}{\\phi_1} d\\phi_1) \\int \\mathbf{J}_M{(\\phi_1)} d\\phi_1 = (\\int \\phi_1 d\\phi_1) \\int \\frac{\\Omega}{\\phi_1} d\\phi_1", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\phi_1', commutative=True))"], [["integrate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True))))"], [["times", 2, "Function('\\\\mathbf{S}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{S}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Function('\\\\mathbf{S}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integral(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True)))), Mul(Integral(Symbol('\\\\phi_1', commutative=True), Tuple(Symbol('\\\\phi_1', commutative=True))), Integral(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\phi_1', commutative=True), Integer(-1))), Tuple(Symbol('\\\\phi_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(z)} = \\cos{(z)} and \\varepsilon{(z)} = \\cos^{z}{(z)}, then derive (- \\frac{z \\sin{(z)}}{\\cos{(z)}} + \\log{(\\cos{(z)})}) \\cos^{z}{(z)} = \\frac{d}{d z} \\varepsilon{(z)}, then obtain (- \\frac{z \\sin{(z)}}{\\cos{(z)}} + \\log{(\\cos{(z)})}) \\cos^{z}{(z)} = \\frac{d}{d z} \\cos^{z}{(z)}", "derivation": "\\operatorname{E_{x}}{(z)} = \\cos{(z)} and \\operatorname{E_{x}}^{z}{(z)} = \\cos^{z}{(z)} and \\frac{d}{d z} \\operatorname{E_{x}}^{z}{(z)} = \\frac{d}{d z} \\cos^{z}{(z)} and \\varepsilon{(z)} = \\cos^{z}{(z)} and \\frac{d}{d z} \\operatorname{E_{x}}^{z}{(z)} = \\frac{d}{d z} \\varepsilon{(z)} and \\frac{d}{d z} \\cos^{z}{(z)} = \\frac{d}{d z} \\varepsilon{(z)} and (- \\frac{z \\sin{(z)}}{\\cos{(z)}} + \\log{(\\cos{(z)})}) \\cos^{z}{(z)} = \\frac{d}{d z} \\varepsilon{(z)} and (- \\frac{z \\sin{(z)}}{\\cos{(z)}} + \\log{(\\cos{(z)})}) \\cos^{z}{(z)} = \\frac{d}{d z} \\cos^{z}{(z)}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('E_x')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Pow(Function('E_x')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('E_x')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True), sin(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), log(cos(Symbol('z', commutative=True)))), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Derivative(Function('\\\\varepsilon')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Mul(Add(Mul(Integer(-1), Symbol('z', commutative=True), sin(Symbol('z', commutative=True)), Pow(cos(Symbol('z', commutative=True)), Integer(-1))), log(cos(Symbol('z', commutative=True)))), Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Derivative(Pow(cos(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\sigma_p,M,Q)} = M + Q + \\sigma_p, then derive y^{\\prime} = \\int (M + Q + \\sigma_p - \\operatorname{r_{0}}{(\\sigma_p,M,Q)})^{Q} dQ, then obtain y^{\\prime} = \\int 0^{Q} dQ", "derivation": "\\operatorname{r_{0}}{(\\sigma_p,M,Q)} = M + Q + \\sigma_p and 0 = M + Q + \\sigma_p - \\operatorname{r_{0}}{(\\sigma_p,M,Q)} and 0^{Q} = (M + Q + \\sigma_p - \\operatorname{r_{0}}{(\\sigma_p,M,Q)})^{Q} and \\int 0^{Q} dQ = \\int (M + Q + \\sigma_p - \\operatorname{r_{0}}{(\\sigma_p,M,Q)})^{Q} dQ and y^{\\prime} = \\int (M + Q + \\sigma_p - \\operatorname{r_{0}}{(\\sigma_p,M,Q)})^{Q} dQ and y^{\\prime} = \\int 0^{Q} dQ", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('M', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 1, "Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(0), Add(Symbol('M', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True)))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Integer(0), Symbol('Q', commutative=True)), Pow(Add(Symbol('M', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)))"], [["integrate", 3, "Symbol('Q', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Add(Symbol('M', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Symbol('y^{\\\\prime}', commutative=True), Integral(Pow(Add(Symbol('M', commutative=True), Symbol('Q', commutative=True), Symbol('\\\\sigma_p', commutative=True), Mul(Integer(-1), Function('r_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('M', commutative=True), Symbol('Q', commutative=True)))), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Symbol('y^{\\\\prime}', commutative=True), Integral(Pow(Integer(0), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given M{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}}, then obtain M{(f_{\\mathbf{v}})} + \\int M^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = M{(f_{\\mathbf{v}})} + \\int (e^{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} df_{\\mathbf{v}}", "derivation": "M{(f_{\\mathbf{v}})} = e^{f_{\\mathbf{v}}} and M^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} = (e^{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} and \\int M^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = \\int (e^{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and e^{f_{\\mathbf{v}}} + \\int M^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = e^{f_{\\mathbf{v}}} + \\int (e^{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and M{(f_{\\mathbf{v}})} + \\int M^{f_{\\mathbf{v}}}{(f_{\\mathbf{v}})} df_{\\mathbf{v}} = M{(f_{\\mathbf{v}})} + \\int (e^{f_{\\mathbf{v}}})^{f_{\\mathbf{v}}} df_{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["power", 1, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 2, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Integral(Pow(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))), Integral(Pow(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 3, "exp(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Add(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Pow(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Pow(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Pow(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Function('M')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integral(Pow(exp(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(c,x)} = \\sin^{x}{(c)}, then derive \\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)} = \\log{(\\sin{(c)})} \\sin^{x}{(c)}, then obtain \\frac{\\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)}}{\\sin{(c)}} = \\frac{\\operatorname{f_{E}}{(c,x)} \\log{(\\sin{(c)})}}{\\sin{(c)}}", "derivation": "\\operatorname{f_{E}}{(c,x)} = \\sin^{x}{(c)} and \\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)} = \\frac{\\partial}{\\partial x} \\sin^{x}{(c)} and \\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)} = \\log{(\\sin{(c)})} \\sin^{x}{(c)} and \\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)} = \\operatorname{f_{E}}{(c,x)} \\log{(\\sin{(c)})} and \\frac{\\frac{\\partial}{\\partial x} \\operatorname{f_{E}}{(c,x)}}{\\sin{(c)}} = \\frac{\\operatorname{f_{E}}{(c,x)} \\log{(\\sin{(c)})}}{\\sin{(c)}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), Pow(sin(Symbol('c', commutative=True)), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('c', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(log(sin(Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), log(sin(Symbol('c', commutative=True)))))"], [["divide", 4, "sin(Symbol('c', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('c', commutative=True)), Integer(-1)), Derivative(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), Mul(Function('f_E')(Symbol('c', commutative=True), Symbol('x', commutative=True)), log(sin(Symbol('c', commutative=True))), Pow(sin(Symbol('c', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\eta{(f_{E})} = \\log{(\\cos{(f_{E})})} and \\rho{(M_{E})} = \\cos{(M_{E})}, then obtain 1 - \\int \\eta{(f_{E})} df_{E} = e^{- \\rho{(M_{E})} + \\cos{(M_{E})}} - \\int \\eta{(f_{E})} df_{E}", "derivation": "\\eta{(f_{E})} = \\log{(\\cos{(f_{E})})} and \\rho{(M_{E})} = \\cos{(M_{E})} and \\int \\eta{(f_{E})} df_{E} = \\int \\log{(\\cos{(f_{E})})} df_{E} and 0 = - \\rho{(M_{E})} + \\cos{(M_{E})} and 1 = e^{- \\rho{(M_{E})} + \\cos{(M_{E})}} and 1 - \\int \\log{(\\cos{(f_{E})})} df_{E} = e^{- \\rho{(M_{E})} + \\cos{(M_{E})}} - \\int \\log{(\\cos{(f_{E})})} df_{E} and 1 - \\int \\eta{(f_{E})} df_{E} = e^{- \\rho{(M_{E})} + \\cos{(M_{E})}} - \\int \\eta{(f_{E})} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('f_E', commutative=True)), log(cos(Symbol('f_E', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(log(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["minus", 2, "Function('\\\\rho')(Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M_E', commutative=True))), cos(Symbol('M_E', commutative=True))))"], [["exp", 4], "Equality(Integer(1), exp(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M_E', commutative=True))), cos(Symbol('M_E', commutative=True)))))"], [["minus", 5, "Integral(log(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(log(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))), Add(exp(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M_E', commutative=True))), cos(Symbol('M_E', commutative=True)))), Mul(Integer(-1), Integral(log(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Function('\\\\eta')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))), Add(exp(Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('M_E', commutative=True))), cos(Symbol('M_E', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\eta')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given \\theta{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})}, then obtain 3 \\theta^{2}{(f_{\\mathbf{v}})} + \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} = \\theta^{2}{(f_{\\mathbf{v}})} + 3 \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})}", "derivation": "\\theta{(f_{\\mathbf{v}})} = \\log{(f_{\\mathbf{v}})} and \\theta^{2}{(f_{\\mathbf{v}})} = \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} and 2 \\theta^{2}{(f_{\\mathbf{v}})} = \\theta^{2}{(f_{\\mathbf{v}})} + \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} and 3 \\theta^{2}{(f_{\\mathbf{v}})} + \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} = 2 \\theta^{2}{(f_{\\mathbf{v}})} + 2 \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} and 3 \\theta^{2}{(f_{\\mathbf{v}})} + \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})} = \\theta^{2}{(f_{\\mathbf{v}})} + 3 \\theta{(f_{\\mathbf{v}})} \\log{(f_{\\mathbf{v}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["times", 1, "Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Mul(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 2, "Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))), Add(Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Mul(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["add", 3, "Add(Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Mul(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], "Equality(Add(Mul(Integer(3), Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))), Mul(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Mul(Integer(2), Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))), Mul(Integer(2), Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(3), Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2))), Mul(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Add(Pow(Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(2)), Mul(Integer(3), Function('\\\\theta')(Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"]]}, {"prompt": "Given \\varphi^{*}{(F_{H})} = \\log{(F_{H})} and \\eta{(F_{H})} = - \\sin{(\\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} - \\frac{d}{d F_{H}} \\log{(F_{H})})}, then obtain 0 = \\eta{(F_{H})} - \\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} + \\frac{d}{d F_{H}} \\log{(F_{H})}", "derivation": "\\varphi^{*}{(F_{H})} = \\log{(F_{H})} and \\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} = \\frac{d}{d F_{H}} \\log{(F_{H})} and 0 = - \\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} + \\frac{d}{d F_{H}} \\log{(F_{H})} and \\eta{(F_{H})} = - \\sin{(\\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} - \\frac{d}{d F_{H}} \\log{(F_{H})})} and \\eta{(F_{H})} = 0 and \\eta{(F_{H})} + \\frac{d}{d F_{H}} \\log{(F_{H})} = \\frac{d}{d F_{H}} \\log{(F_{H})} and 0 = \\eta{(F_{H})} - \\frac{d}{d F_{H}} \\varphi^{*}{(F_{H})} + \\frac{d}{d F_{H}} \\log{(F_{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["differentiate", 1, "Symbol('F_H', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('F_H', commutative=True)), Mul(Integer(-1), sin(Add(Derivative(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\eta')(Symbol('F_H', commutative=True)), Integer(0))"], [["minus", 5, "Mul(Integer(-1), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], "Equality(Add(Function('\\\\eta')(Symbol('F_H', commutative=True)), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Integer(0), Add(Function('\\\\eta')(Symbol('F_H', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\varphi^*')(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))), Derivative(log(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}^*{(t,\\hat{\\mathbf{r}})} = - t + \\cos{(\\hat{\\mathbf{r}})}, then obtain \\frac{\\tilde{g}^*{(t,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - t + \\cos{(\\hat{\\mathbf{r}})}} = \\frac{- t + \\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - t + \\cos{(\\hat{\\mathbf{r}})}}", "derivation": "\\tilde{g}^*{(t,\\hat{\\mathbf{r}})} = - t + \\cos{(\\hat{\\mathbf{r}})} and \\hat{\\mathbf{r}} + \\tilde{g}^*{(t,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} - t + \\cos{(\\hat{\\mathbf{r}})} and \\frac{\\tilde{g}^*{(t,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + \\tilde{g}^*{(t,\\hat{\\mathbf{r}})}} = \\frac{- t + \\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} + \\tilde{g}^*{(t,\\hat{\\mathbf{r}})}} and \\frac{\\tilde{g}^*{(t,\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - t + \\cos{(\\hat{\\mathbf{r}})}} = \\frac{- t + \\cos{(\\hat{\\mathbf{r}})}}{\\hat{\\mathbf{r}} - t + \\cos{(\\hat{\\mathbf{r}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["divide", 1, "Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1)), Function('\\\\tilde{g}^*')(Symbol('t', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Pow(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True)), cos(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(y,S)} = \\frac{\\partial}{\\partial S} (S + y), then derive \\operatorname{F_{N}}^{3}{(y,S)} = \\operatorname{F_{N}}^{2}{(y,S)}, then obtain e^{\\operatorname{F_{N}}^{3}{(y,S)}} = e^{\\operatorname{F_{N}}^{3}{(y,S)} (\\frac{\\partial}{\\partial S} (S + y))^{2}}", "derivation": "\\operatorname{F_{N}}{(y,S)} = \\frac{\\partial}{\\partial S} (S + y) and \\operatorname{F_{N}}^{2}{(y,S)} = \\operatorname{F_{N}}{(y,S)} \\frac{\\partial}{\\partial S} (S + y) and \\operatorname{F_{N}}^{3}{(y,S)} = \\operatorname{F_{N}}^{2}{(y,S)} \\frac{\\partial}{\\partial S} (S + y) and \\operatorname{F_{N}}^{3}{(y,S)} = \\operatorname{F_{N}}^{2}{(y,S)} and \\operatorname{F_{N}}^{3}{(y,S)} = \\operatorname{F_{N}}^{3}{(y,S)} \\frac{\\partial}{\\partial S} (S + y) and \\operatorname{F_{N}}^{3}{(y,S)} \\frac{\\partial}{\\partial S} (S + y) = \\operatorname{F_{N}}^{2}{(y,S)} and \\operatorname{F_{N}}^{3}{(y,S)} = \\operatorname{F_{N}}^{3}{(y,S)} (\\frac{\\partial}{\\partial S} (S + y))^{2} and e^{\\operatorname{F_{N}}^{3}{(y,S)}} = e^{\\operatorname{F_{N}}^{3}{(y,S)} (\\frac{\\partial}{\\partial S} (S + y))^{2}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["times", 1, "Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(2)), Mul(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["times", 2, "Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True))"], "Equality(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Mul(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(2)), Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Mul(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Mul(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Pow(Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(2))))"], [["exp", 7], "Equality(exp(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3))), exp(Mul(Pow(Function('F_N')(Symbol('y', commutative=True), Symbol('S', commutative=True)), Integer(3)), Pow(Derivative(Add(Symbol('S', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(2)))))"]]}, {"prompt": "Given \\lambda{(r,S)} = - S + \\log{(r)}, then derive \\frac{\\partial}{\\partial r} \\lambda{(r,S)} = \\frac{1}{r}, then obtain \\frac{\\frac{\\partial}{\\partial r} \\lambda{(r,S)}}{r} = \\frac{1}{r^{2}}", "derivation": "\\lambda{(r,S)} = - S + \\log{(r)} and \\frac{\\partial}{\\partial r} \\lambda{(r,S)} = \\frac{\\partial}{\\partial r} (- S + \\log{(r)}) and \\frac{\\partial}{\\partial r} \\lambda{(r,S)} = \\frac{1}{r} and \\frac{\\partial}{\\partial r} (- S + \\log{(r)}) \\frac{\\partial}{\\partial r} \\lambda{(r,S)} = \\frac{\\frac{\\partial}{\\partial r} (- S + \\log{(r)})}{r} and (\\frac{\\partial}{\\partial r} (- S + \\log{(r)}))^{2} = \\frac{\\frac{\\partial}{\\partial r} (- S + \\log{(r)})}{r} and \\frac{\\partial}{\\partial r} (- S + \\log{(r)}) \\frac{\\partial}{\\partial r} \\lambda{(r,S)} = (\\frac{\\partial}{\\partial r} (- S + \\log{(r)}))^{2} and \\frac{\\frac{\\partial}{\\partial r} \\lambda{(r,S)}}{r} = \\frac{1}{r^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Pow(Symbol('r', commutative=True), Integer(-1)))"], [["times", 3, "Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(2)), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('S', commutative=True)), log(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(2)))"], [["evaluate_derivatives", 6], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Derivative(Function('\\\\lambda')(Symbol('r', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Pow(Symbol('r', commutative=True), Integer(-2)))"]]}, {"prompt": "Given \\phi{(F_{N})} = \\log{(\\sin{(F_{N})})} and m{(F_{N})} = F_{N} + \\phi{(F_{N})} + \\log{(\\sin{(F_{N})})}^{F_{N}}, then obtain m{(F_{N})} = F_{N} + \\log{(\\sin{(F_{N})})} + \\log{(\\sin{(F_{N})})}^{F_{N}}", "derivation": "\\phi{(F_{N})} = \\log{(\\sin{(F_{N})})} and F_{N} + \\phi{(F_{N})} = F_{N} + \\log{(\\sin{(F_{N})})} and F_{N} + \\phi{(F_{N})} + \\log{(\\sin{(F_{N})})}^{F_{N}} = F_{N} + \\log{(\\sin{(F_{N})})} + \\log{(\\sin{(F_{N})})}^{F_{N}} and m{(F_{N})} = F_{N} + \\phi{(F_{N})} + \\log{(\\sin{(F_{N})})}^{F_{N}} and m{(F_{N})} = F_{N} + \\log{(\\sin{(F_{N})})} + \\log{(\\sin{(F_{N})})}^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('F_N', commutative=True)), log(sin(Symbol('F_N', commutative=True))))"], [["add", 1, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Function('\\\\phi')(Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), log(sin(Symbol('F_N', commutative=True)))))"], [["add", 2, "Pow(log(sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))"], "Equality(Add(Symbol('F_N', commutative=True), Function('\\\\phi')(Symbol('F_N', commutative=True)), Pow(log(sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))), Add(Symbol('F_N', commutative=True), log(sin(Symbol('F_N', commutative=True))), Pow(log(sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Function('\\\\phi')(Symbol('F_N', commutative=True)), Pow(log(sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('m')(Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), log(sin(Symbol('F_N', commutative=True))), Pow(log(sin(Symbol('F_N', commutative=True))), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given m{(A,v_{y})} = A v_{y}, then derive \\frac{\\partial}{\\partial A} m{(A,v_{y})} = v_{y}, then obtain m{(A,\\frac{\\partial}{\\partial A} A v_{y})} (\\frac{\\partial}{\\partial A} m{(A,v_{y})})^{A} = v_{y}^{A} m{(A,\\frac{\\partial}{\\partial A} A v_{y})}", "derivation": "m{(A,v_{y})} = A v_{y} and \\frac{\\partial}{\\partial A} m{(A,v_{y})} = \\frac{\\partial}{\\partial A} A v_{y} and \\frac{\\partial}{\\partial A} m{(A,v_{y})} = v_{y} and \\frac{\\partial}{\\partial A} A v_{y} = v_{y} and (\\frac{\\partial}{\\partial A} m{(A,v_{y})})^{A} = (\\frac{\\partial}{\\partial A} A v_{y})^{A} and (\\frac{\\partial}{\\partial A} m{(A,v_{y})})^{A} = v_{y}^{A} and m{(A,\\frac{\\partial}{\\partial A} A v_{y})} (\\frac{\\partial}{\\partial A} m{(A,v_{y})})^{A} = v_{y}^{A} m{(A,\\frac{\\partial}{\\partial A} A v_{y})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('v_y', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('v_y', commutative=True))"], [["power", 2, "Symbol('A', commutative=True)"], "Equality(Pow(Derivative(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True)), Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Derivative(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True)), Pow(Symbol('v_y', commutative=True), Symbol('A', commutative=True)))"], [["times", 6, "Function('m')(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], "Equality(Mul(Function('m')(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Pow(Derivative(Function('m')(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Symbol('A', commutative=True))), Mul(Pow(Symbol('v_y', commutative=True), Symbol('A', commutative=True)), Function('m')(Symbol('A', commutative=True), Derivative(Mul(Symbol('A', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"]]}, {"prompt": "Given I{(\\psi,y^{\\prime})} = \\psi + y^{\\prime}, then obtain \\frac{(\\psi + I{(\\psi,y^{\\prime})}) \\int I^{y^{\\prime}}{(\\psi,y^{\\prime})} d\\psi}{\\psi + y^{\\prime}} = \\frac{(\\psi + I{(\\psi,y^{\\prime})}) \\int (\\psi + y^{\\prime})^{y^{\\prime}} d\\psi}{\\psi + y^{\\prime}}", "derivation": "I{(\\psi,y^{\\prime})} = \\psi + y^{\\prime} and I^{y^{\\prime}}{(\\psi,y^{\\prime})} = (\\psi + y^{\\prime})^{y^{\\prime}} and \\int I^{y^{\\prime}}{(\\psi,y^{\\prime})} d\\psi = \\int (\\psi + y^{\\prime})^{y^{\\prime}} d\\psi and \\frac{(\\psi + I{(\\psi,y^{\\prime})}) \\int I^{y^{\\prime}}{(\\psi,y^{\\prime})} d\\psi}{\\psi + y^{\\prime}} = \\frac{(\\psi + I{(\\psi,y^{\\prime})}) \\int (\\psi + y^{\\prime})^{y^{\\prime}} d\\psi}{\\psi + y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Pow(Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["divide", 3, "Mul(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Pow(Add(Symbol('\\\\psi', commutative=True), Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Pow(Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))), Mul(Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Symbol('\\\\psi', commutative=True), Function('I')(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Integral(Pow(Add(Symbol('\\\\psi', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given u{(\\chi,F_{H})} = \\frac{e^{\\chi}}{F_{H}} and W{(\\chi)} = \\chi, then obtain \\frac{l^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} e^{\\chi} \\frac{d}{d \\chi} W{(\\chi)}}{F_{H}} = \\frac{l^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} e^{\\chi} \\frac{d}{d \\chi} \\chi}{F_{H}}", "derivation": "u{(\\chi,F_{H})} = \\frac{e^{\\chi}}{F_{H}} and W{(\\chi)} = \\chi and \\frac{d}{d \\chi} W{(\\chi)} = \\frac{d}{d \\chi} \\chi and u{(\\chi,F_{H})} \\frac{d}{d \\chi} W{(\\chi)} = u{(\\chi,F_{H})} \\frac{d}{d \\chi} \\chi and \\frac{e^{\\chi} \\frac{d}{d \\chi} W{(\\chi)}}{F_{H}} = \\frac{e^{\\chi} \\frac{d}{d \\chi} \\chi}{F_{H}} and \\frac{l^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} e^{\\chi} \\frac{d}{d \\chi} W{(\\chi)}}{F_{H}} = \\frac{l^{V_{\\mathbf{E}}}{(V_{\\mathbf{E}})} e^{\\chi} \\frac{d}{d \\chi} \\chi}{F_{H}}", "srepr_derivation": [["get_premise", "Equality(Function('u')(Symbol('\\\\chi', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), exp(Symbol('\\\\chi', commutative=True))))"], ["renaming_premise", "Equality(Function('W')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('W')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["times", 3, "Function('u')(Symbol('\\\\chi', commutative=True), Symbol('F_H', commutative=True))"], "Equality(Mul(Function('u')(Symbol('\\\\chi', commutative=True), Symbol('F_H', commutative=True)), Derivative(Function('W')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Function('u')(Symbol('\\\\chi', commutative=True), Symbol('F_H', commutative=True)), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), exp(Symbol('\\\\chi', commutative=True)), Derivative(Function('W')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), exp(Symbol('\\\\chi', commutative=True)), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["times", 5, "Pow(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('\\\\chi', commutative=True)), Derivative(Function('W')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Function('l')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), exp(Symbol('\\\\chi', commutative=True)), Derivative(Symbol('\\\\chi', commutative=True), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Z{(\\rho_f,V)} = - V + \\log{(\\rho_f)}, then derive \\int Z{(\\rho_f,V)} dV = - \\frac{V^{2}}{2} + V \\log{(\\rho_f)} + q, then obtain \\int Z{(\\rho_f,V)} dV = - \\frac{V^{2}}{2} + V (V + Z{(\\rho_f,V)}) + q", "derivation": "Z{(\\rho_f,V)} = - V + \\log{(\\rho_f)} and V + Z{(\\rho_f,V)} = \\log{(\\rho_f)} and \\int Z{(\\rho_f,V)} dV = \\int (- V + \\log{(\\rho_f)}) dV and \\int Z{(\\rho_f,V)} dV = - \\frac{V^{2}}{2} + V \\log{(\\rho_f)} + q and \\int Z{(\\rho_f,V)} dV = - \\frac{V^{2}}{2} + V (V + Z{(\\rho_f,V)}) + q", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('V', commutative=True))"], "Equality(Add(Symbol('V', commutative=True), Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True))), log(Symbol('\\\\rho_f', commutative=True)))"], [["integrate", 1, "Symbol('V', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('V', commutative=True)), log(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Symbol('V', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Symbol('q', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('V', commutative=True), Integer(2))), Mul(Symbol('V', commutative=True), Add(Symbol('V', commutative=True), Function('Z')(Symbol('\\\\rho_f', commutative=True), Symbol('V', commutative=True)))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(Q)} = e^{e^{Q}}, then obtain (e^{(F_{H} + \\log{(J)}) (\\operatorname{f^{\\prime}}{(Q)} + \\log{(J)}) \\lambda{(J,F_{H})}})^{J} = (e^{(F_{H} + \\log{(J)}) (e^{e^{Q}} + \\log{(J)}) \\lambda{(J,F_{H})}})^{J}", "derivation": "\\operatorname{f^{\\prime}}{(Q)} = e^{e^{Q}} and \\operatorname{f^{\\prime}}{(Q)} + \\log{(J)} = e^{e^{Q}} + \\log{(J)} and (F_{H} + \\log{(J)}) (\\operatorname{f^{\\prime}}{(Q)} + \\log{(J)}) \\lambda{(J,F_{H})} = (F_{H} + \\log{(J)}) (e^{e^{Q}} + \\log{(J)}) \\lambda{(J,F_{H})} and e^{(F_{H} + \\log{(J)}) (\\operatorname{f^{\\prime}}{(Q)} + \\log{(J)}) \\lambda{(J,F_{H})}} = e^{(F_{H} + \\log{(J)}) (e^{e^{Q}} + \\log{(J)}) \\lambda{(J,F_{H})}} and (e^{(F_{H} + \\log{(J)}) (\\operatorname{f^{\\prime}}{(Q)} + \\log{(J)}) \\lambda{(J,F_{H})}})^{J} = (e^{(F_{H} + \\log{(J)}) (e^{e^{Q}} + \\log{(J)}) \\lambda{(J,F_{H})}})^{J}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('Q', commutative=True)), exp(exp(Symbol('Q', commutative=True))))"], [["add", 1, "log(Symbol('J', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('J', commutative=True))), Add(exp(exp(Symbol('Q', commutative=True))), log(Symbol('J', commutative=True))))"], [["times", 2, "Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))"], "Equality(Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(Function('f^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))), Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(exp(exp(Symbol('Q', commutative=True))), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True))))"], [["exp", 3], "Equality(exp(Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(Function('f^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), exp(Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(exp(exp(Symbol('Q', commutative=True))), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(exp(Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(Function('f^{\\\\prime}')(Symbol('Q', commutative=True)), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Symbol('J', commutative=True)), Pow(exp(Mul(Add(Symbol('F_H', commutative=True), log(Symbol('J', commutative=True))), Add(exp(exp(Symbol('Q', commutative=True))), log(Symbol('J', commutative=True))), Function('\\\\lambda')(Symbol('J', commutative=True), Symbol('F_H', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(C_{2},v)} = C_{2} - v, then obtain \\int \\frac{\\operatorname{g_{\\varepsilon}}{(C_{2},v)}}{C_{2} - v} dv = \\int 1 dv", "derivation": "\\operatorname{g_{\\varepsilon}}{(C_{2},v)} = C_{2} - v and \\frac{\\operatorname{g_{\\varepsilon}}{(C_{2},v)}}{C_{2}} = \\frac{C_{2} - v}{C_{2}} and \\frac{\\operatorname{g_{\\varepsilon}}{(C_{2},v)}}{C_{2} - v} = 1 and \\int \\frac{\\operatorname{g_{\\varepsilon}}{(C_{2},v)}}{C_{2} - v} dv = \\int 1 dv", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('v', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('v', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)))))"], [["divide", 2, "Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('v', commutative=True))), Integer(1))"], [["integrate", 3, "Symbol('v', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('C_2', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))), Integer(-1)), Function('g_{\\\\varepsilon}')(Symbol('C_2', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True))), Integral(Integer(1), Tuple(Symbol('v', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(v_{1},\\mathbf{s})} = \\mathbf{s} v_{1} and \\ddot{x}{(T)} = e^{\\cos{(T)}}, then obtain M{(A)} + \\ddot{x}{(T)} \\int \\mathbf{s} v_{1} dv_{1} - \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1} = M{(A)} + e^{\\cos{(T)}} \\int \\mathbf{s} v_{1} dv_{1} - \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1}", "derivation": "\\sigma_{p}{(v_{1},\\mathbf{s})} = \\mathbf{s} v_{1} and \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1} = \\int \\mathbf{s} v_{1} dv_{1} and \\ddot{x}{(T)} = e^{\\cos{(T)}} and \\ddot{x}{(T)} \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1} = e^{\\cos{(T)}} \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1} and \\ddot{x}{(T)} \\int \\mathbf{s} v_{1} dv_{1} = e^{\\cos{(T)}} \\int \\mathbf{s} v_{1} dv_{1} and M{(A)} + \\ddot{x}{(T)} \\int \\mathbf{s} v_{1} dv_{1} - \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1} = M{(A)} + e^{\\cos{(T)}} \\int \\mathbf{s} v_{1} dv_{1} - \\int \\sigma_{p}{(v_{1},\\mathbf{s})} dv_{1}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)))"], [["integrate", 1, "Symbol('v_1', commutative=True)"], "Equality(Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], ["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), exp(cos(Symbol('T', commutative=True))))"], [["times", 3, "Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True)))"], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(exp(cos(Symbol('T', commutative=True))), Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(exp(cos(Symbol('T', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))))"], [["minus", 5, "Add(Mul(Integer(-1), Function('M')(Symbol('A', commutative=True))), Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True))))"], "Equality(Add(Function('M')(Symbol('A', commutative=True)), Mul(Function('\\\\ddot{x}')(Symbol('T', commutative=True)), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True))))), Add(Function('M')(Symbol('A', commutative=True)), Mul(exp(cos(Symbol('T', commutative=True))), Integral(Mul(Symbol('\\\\mathbf{s}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True)))), Mul(Integer(-1), Integral(Function('\\\\sigma_p')(Symbol('v_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('v_1', commutative=True))))))"]]}, {"prompt": "Given M{(c_{0},\\phi)} = \\frac{\\phi}{c_{0}}, then derive \\eta + M{(c_{0},\\phi)} = \\lambda + \\frac{\\phi}{c_{0}}, then obtain \\eta + M{(c_{0},\\phi)} = \\lambda + M{(c_{0},\\phi)}", "derivation": "M{(c_{0},\\phi)} = \\frac{\\phi}{c_{0}} and \\frac{\\partial}{\\partial c_{0}} M{(c_{0},\\phi)} = \\frac{\\partial}{\\partial c_{0}} \\frac{\\phi}{c_{0}} and \\int \\frac{\\partial}{\\partial c_{0}} M{(c_{0},\\phi)} dc_{0} = \\int \\frac{\\partial}{\\partial c_{0}} \\frac{\\phi}{c_{0}} dc_{0} and \\eta + M{(c_{0},\\phi)} = \\lambda + \\frac{\\phi}{c_{0}} and \\eta + M{(c_{0},\\phi)} = \\lambda + M{(c_{0},\\phi)}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('c_0', commutative=True)"], "Equality(Integral(Derivative(Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Tuple(Symbol('c_0', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), Function('M')(Symbol('c_0', commutative=True), Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\lambda{(u,f_{\\mathbf{p}})} = \\frac{\\cos{(u)}}{f_{\\mathbf{p}}}, then obtain - \\frac{\\lambda{(u,f_{\\mathbf{p}})} \\sin{(u)}}{f_{\\mathbf{p}}} = - \\frac{\\sin{(u)} \\cos{(u)}}{f_{\\mathbf{p}}^{2}}", "derivation": "\\lambda{(u,f_{\\mathbf{p}})} = \\frac{\\cos{(u)}}{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial u} \\lambda{(u,f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial u} \\frac{\\cos{(u)}}{f_{\\mathbf{p}}} and \\lambda{(u,f_{\\mathbf{p}})} \\frac{\\partial}{\\partial u} \\lambda{(u,f_{\\mathbf{p}})} = \\frac{\\cos{(u)} \\frac{\\partial}{\\partial u} \\lambda{(u,f_{\\mathbf{p}})}}{f_{\\mathbf{p}}} and \\lambda{(u,f_{\\mathbf{p}})} \\frac{\\partial}{\\partial u} \\frac{\\cos{(u)}}{f_{\\mathbf{p}}} = \\frac{\\cos{(u)} \\frac{\\partial}{\\partial u} \\frac{\\cos{(u)}}{f_{\\mathbf{p}}}}{f_{\\mathbf{p}}} and - \\frac{\\lambda{(u,f_{\\mathbf{p}})} \\sin{(u)}}{f_{\\mathbf{p}}} = - \\frac{\\sin{(u)} \\cos{(u)}}{f_{\\mathbf{p}}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 1, "Derivative(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True)), Derivative(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True)), Derivative(Mul(Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), cos(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)), Function('\\\\lambda')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), sin(Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-2)), sin(Symbol('u', commutative=True)), cos(Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\mathbb{I},z)} = z^{\\mathbb{I}} and \\operatorname{t_{1}}{(\\mathbb{I},z)} = z^{\\mathbb{I}}, then obtain \\frac{- z + z^{\\mathbb{I}}}{\\mathbb{I}} = \\frac{- z + \\operatorname{t_{1}}{(\\mathbb{I},z)}}{\\mathbb{I}}", "derivation": "\\operatorname{r_{0}}{(\\mathbb{I},z)} = z^{\\mathbb{I}} and - z + \\operatorname{r_{0}}{(\\mathbb{I},z)} = - z + z^{\\mathbb{I}} and \\frac{- z + \\operatorname{r_{0}}{(\\mathbb{I},z)}}{\\mathbb{I}} = \\frac{- z + z^{\\mathbb{I}}}{\\mathbb{I}} and \\operatorname{t_{1}}{(\\mathbb{I},z)} = z^{\\mathbb{I}} and \\frac{- z + \\operatorname{r_{0}}{(\\mathbb{I},z)}}{\\mathbb{I}} = \\frac{- z + \\operatorname{t_{1}}{(\\mathbb{I},z)}}{\\mathbb{I}} and \\frac{- z + z^{\\mathbb{I}}}{\\mathbb{I}} = \\frac{- z + \\operatorname{t_{1}}{(\\mathbb{I},z)}}{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "Symbol('z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('r_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True))), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('r_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('r_0')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('t_1')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)} = E_{\\lambda} + \\Omega, then obtain (\\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)}}{E_{\\lambda}})^{\\Omega} = (\\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} (E_{\\lambda} + \\Omega)}{E_{\\lambda}})^{\\Omega}", "derivation": "\\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)} = E_{\\lambda} + \\Omega and \\frac{\\partial}{\\partial \\Omega} \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)} = \\frac{\\partial}{\\partial \\Omega} (E_{\\lambda} + \\Omega) and E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)} = E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} (E_{\\lambda} + \\Omega) and \\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)}}{E_{\\lambda}} = \\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} (E_{\\lambda} + \\Omega)}{E_{\\lambda}} and (\\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} \\operatorname{a^{\\dagger}}{(E_{\\lambda},\\Omega)}}{E_{\\lambda}})^{\\Omega} = (\\frac{E_{\\lambda} + \\frac{\\partial}{\\partial \\Omega} (E_{\\lambda} + \\Omega)}{E_{\\lambda}})^{\\Omega}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["add", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))))"], [["divide", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))))"], [["power", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Function('a^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Symbol('\\\\Omega', commutative=True)), Pow(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Add(Symbol('E_{\\\\lambda}', commutative=True), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Symbol('\\\\Omega', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{p})} = \\sin{(\\hat{p})} and \\phi_{1}{(\\hat{p})} = \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})}, then obtain \\operatorname{v_{2}}^{2}{(\\hat{p})} - \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})} = \\phi_{1}{(\\hat{p})} - \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})}", "derivation": "\\operatorname{v_{2}}{(\\hat{p})} = \\sin{(\\hat{p})} and \\operatorname{v_{2}}^{2}{(\\hat{p})} = \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})} and \\phi_{1}{(\\hat{p})} = \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})} and \\operatorname{v_{2}}^{2}{(\\hat{p})} = \\phi_{1}{(\\hat{p})} and \\operatorname{v_{2}}^{2}{(\\hat{p})} - \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})} = \\phi_{1}{(\\hat{p})} - \\operatorname{v_{2}}{(\\hat{p})} \\sin{(\\hat{p})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], [["times", 1, "Function('v_2')(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Pow(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), Integer(2)), Mul(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\hat{p}', commutative=True)), Mul(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), Integer(2)), Function('\\\\phi_1')(Symbol('\\\\hat{p}', commutative=True)))"], [["minus", 4, "Mul(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Pow(Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), Integer(2)), Mul(Integer(-1), Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))), Add(Function('\\\\phi_1')(Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Function('v_2')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\delta)} = \\sin{(\\delta)} and u{(l,\\mathbf{A})} = \\frac{\\mathbf{A}}{l}, then obtain (\\tilde{\\infty} u{(l,\\mathbf{A})})^{l} = (\\frac{\\tilde{\\infty} \\mathbf{A}}{l})^{l}", "derivation": "\\operatorname{a^{\\dagger}}{(\\delta)} = \\sin{(\\delta)} and u{(l,\\mathbf{A})} = \\frac{\\mathbf{A}}{l} and \\frac{u{(l,\\mathbf{A})}}{- \\operatorname{a^{\\dagger}}{(\\delta)} + \\sin{(\\delta)}} = \\frac{\\mathbf{A}}{l (- \\operatorname{a^{\\dagger}}{(\\delta)} + \\sin{(\\delta)})} and \\tilde{\\infty} u{(l,\\mathbf{A})} = \\frac{\\tilde{\\infty} \\mathbf{A}}{l} and (\\tilde{\\infty} u{(l,\\mathbf{A})})^{l} = (\\frac{\\tilde{\\infty} \\mathbf{A}}{l})^{l}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('u')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["divide", 2, "Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\delta', commutative=True))), sin(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\delta', commutative=True))), sin(Symbol('\\\\delta', commutative=True))), Integer(-1)), Function('u')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Function('a^{\\\\dagger}')(Symbol('\\\\delta', commutative=True))), sin(Symbol('\\\\delta', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(zoo, Function('u')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Mul(zoo, Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Mul(zoo, Function('u')(Symbol('l', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('l', commutative=True)), Pow(Mul(zoo, Symbol('\\\\mathbf{A}', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\varepsilon)} = \\log{(\\varepsilon)}, then obtain \\varepsilon \\int \\log{(\\varepsilon)} d\\varepsilon + \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon = \\varepsilon \\int \\log{(\\varepsilon)} d\\varepsilon + \\int \\log{(\\varepsilon)} d\\varepsilon", "derivation": "\\operatorname{A_{x}}{(\\varepsilon)} = \\log{(\\varepsilon)} and \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon = \\int \\log{(\\varepsilon)} d\\varepsilon and \\varepsilon \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon = \\varepsilon \\int \\log{(\\varepsilon)} d\\varepsilon and \\varepsilon \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon + \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon = \\varepsilon \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon + \\int \\log{(\\varepsilon)} d\\varepsilon and \\varepsilon \\int \\log{(\\varepsilon)} d\\varepsilon + \\int \\operatorname{A_{x}}{(\\varepsilon)} d\\varepsilon = \\varepsilon \\int \\log{(\\varepsilon)} d\\varepsilon + \\int \\log{(\\varepsilon)} d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), log(Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Mul(Symbol('\\\\varepsilon', commutative=True), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 2, "Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], "Equality(Add(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(Function('A_x')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Add(Mul(Symbol('\\\\varepsilon', commutative=True), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))), Integral(log(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\sigma_p)} = \\log{(\\sigma_p)}, then obtain \\iint \\frac{\\operatorname{C_{d}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p d\\sigma_p = \\iint \\frac{\\log{(\\sigma_p)}}{\\sigma_p} d\\sigma_p d\\sigma_p", "derivation": "\\operatorname{C_{d}}{(\\sigma_p)} = \\log{(\\sigma_p)} and \\frac{\\operatorname{C_{d}}{(\\sigma_p)}}{\\sigma_p} = \\frac{\\log{(\\sigma_p)}}{\\sigma_p} and \\int \\frac{\\operatorname{C_{d}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p = \\int \\frac{\\log{(\\sigma_p)}}{\\sigma_p} d\\sigma_p and \\iint \\frac{\\operatorname{C_{d}}{(\\sigma_p)}}{\\sigma_p} d\\sigma_p d\\sigma_p = \\iint \\frac{\\log{(\\sigma_p)}}{\\sigma_p} d\\sigma_p d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\sigma_p', commutative=True)), log(Symbol('\\\\sigma_p', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 3, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('C_d')(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), log(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(m,z,a)} = (a^{z})^{m}, then obtain \\int e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} \\mathbf{A}{(m,z,a)}}{z}} dm = \\int e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} (a^{z})^{m}}{z}} dm", "derivation": "\\mathbf{A}{(m,z,a)} = (a^{z})^{m} and a^{z} \\mathbf{A}{(m,z,a)} = a^{z} (a^{z})^{m} and \\frac{a^{z} \\mathbf{A}{(m,z,a)}}{z} = \\frac{a^{z} (a^{z})^{m}}{z} and \\frac{\\partial}{\\partial m} \\frac{a^{z} \\mathbf{A}{(m,z,a)}}{z} = \\frac{\\partial}{\\partial m} \\frac{a^{z} (a^{z})^{m}}{z} and e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} \\mathbf{A}{(m,z,a)}}{z}} = e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} (a^{z})^{m}}{z}} and \\int e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} \\mathbf{A}{(m,z,a)}}{z}} dm = \\int e^{\\frac{\\partial}{\\partial m} \\frac{a^{z} (a^{z})^{m}}{z}} dm", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True)))"], [["times", 1, "Pow(Symbol('a', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True))))"], [["divide", 2, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True))), Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["exp", 4], "Equality(exp(Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), exp(Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(exp(Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('m', commutative=True), Symbol('z', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))), Integral(exp(Derivative(Mul(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Pow(Symbol('a', commutative=True), Symbol('z', commutative=True)), Symbol('m', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1)))), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\Psi_{nl}{(A_{y})} = \\sin{(\\cos{(A_{y})})}, then obtain \\sigma_x + 2 \\Psi_{nl}{(A_{y})} = V_{\\mathbf{E}} + \\Psi_{nl}{(A_{y})} + \\sin{(\\cos{(A_{y})})}", "derivation": "\\Psi_{nl}{(A_{y})} = \\sin{(\\cos{(A_{y})})} and 2 \\Psi_{nl}{(A_{y})} = \\Psi_{nl}{(A_{y})} + \\sin{(\\cos{(A_{y})})} and \\frac{d}{d A_{y}} 2 \\Psi_{nl}{(A_{y})} = \\frac{d}{d A_{y}} (\\Psi_{nl}{(A_{y})} + \\sin{(\\cos{(A_{y})})}) and \\int \\frac{d}{d A_{y}} 2 \\Psi_{nl}{(A_{y})} dA_{y} = \\int \\frac{d}{d A_{y}} (\\Psi_{nl}{(A_{y})} + \\sin{(\\cos{(A_{y})})}) dA_{y} and \\sigma_x + 2 \\Psi_{nl}{(A_{y})} = V_{\\mathbf{E}} + \\Psi_{nl}{(A_{y})} + \\sin{(\\cos{(A_{y})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True))))"], [["add", 1, "Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True))), Add(Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))))"], [["differentiate", 2, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('A_y', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('A_y', commutative=True))), Integral(Derivative(Add(Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Tuple(Symbol('A_y', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(2), Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('\\\\Psi_{nl}')(Symbol('A_y', commutative=True)), sin(cos(Symbol('A_y', commutative=True)))))"]]}, {"prompt": "Given k{(u)} = \\log{(u)}, then obtain u \\frac{d}{d u} k{(u)} = 1", "derivation": "k{(u)} = \\log{(u)} and \\frac{d}{d u} k{(u)} = \\frac{d}{d u} \\log{(u)} and \\frac{\\frac{d}{d u} k{(u)}}{\\frac{d}{d u} \\log{(u)}} = 1 and u \\frac{d}{d u} k{(u)} = 1", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('u', commutative=True)), log(Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('k')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Pow(Derivative(log(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('u', commutative=True), Derivative(Function('k')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\eta)} = \\cos{(\\cos{(\\eta)})}, then obtain \\operatorname{V_{\\mathbf{B}}}{(\\eta)} \\int \\cos^{\\eta}{(\\cos{(\\eta)})} d\\eta = \\cos{(\\cos{(\\eta)})} \\int \\cos^{\\eta}{(\\cos{(\\eta)})} d\\eta", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\eta)} = \\cos{(\\cos{(\\eta)})} and \\operatorname{V_{\\mathbf{B}}}^{\\eta}{(\\eta)} = \\cos^{\\eta}{(\\cos{(\\eta)})} and \\int \\operatorname{V_{\\mathbf{B}}}^{\\eta}{(\\eta)} d\\eta = \\int \\cos^{\\eta}{(\\cos{(\\eta)})} d\\eta and \\operatorname{V_{\\mathbf{B}}}{(\\eta)} \\int \\operatorname{V_{\\mathbf{B}}}^{\\eta}{(\\eta)} d\\eta = \\cos{(\\cos{(\\eta)})} \\int \\operatorname{V_{\\mathbf{B}}}^{\\eta}{(\\eta)} d\\eta and \\operatorname{V_{\\mathbf{B}}}{(\\eta)} \\int \\cos^{\\eta}{(\\cos{(\\eta)})} d\\eta = \\cos{(\\cos{(\\eta)})} \\int \\cos^{\\eta}{(\\cos{(\\eta)})} d\\eta", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), cos(cos(Symbol('\\\\eta', commutative=True))))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(cos(cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["integrate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Pow(cos(cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["times", 1, "Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(cos(cos(Symbol('\\\\eta', commutative=True))), Integral(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\eta', commutative=True)), Integral(Pow(cos(cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Mul(cos(cos(Symbol('\\\\eta', commutative=True))), Integral(Pow(cos(cos(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given k{(t,\\mathbf{S})} = \\mathbf{S} + t and G{(t,\\mathbf{S})} = \\mathbf{S} + t, then obtain \\frac{\\partial}{\\partial t} \\int k{(t,\\mathbf{S})} dt = \\frac{\\partial}{\\partial t} (J_{\\varepsilon} + \\mathbf{S} t + \\frac{t^{2}}{2})", "derivation": "k{(t,\\mathbf{S})} = \\mathbf{S} + t and G{(t,\\mathbf{S})} = \\mathbf{S} + t and G{(t,\\mathbf{S})} = k{(t,\\mathbf{S})} and \\int G{(t,\\mathbf{S})} dt = \\int k{(t,\\mathbf{S})} dt and \\int G{(t,\\mathbf{S})} dt = \\int (\\mathbf{S} + t) dt and \\frac{\\partial}{\\partial t} \\int G{(t,\\mathbf{S})} dt = \\frac{\\partial}{\\partial t} \\int (\\mathbf{S} + t) dt and \\frac{\\partial}{\\partial t} \\int k{(t,\\mathbf{S})} dt = \\frac{\\partial}{\\partial t} \\int (\\mathbf{S} + t) dt and \\frac{\\partial}{\\partial t} \\int k{(t,\\mathbf{S})} dt = \\frac{\\partial}{\\partial t} (J_{\\varepsilon} + \\mathbf{S} t + \\frac{t^{2}}{2})", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('G')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('G')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Function('k')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(Function('G')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Function('k')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Function('G')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["differentiate", 5, "Symbol('t', commutative=True)"], "Equality(Derivative(Integral(Function('G')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Integral(Function('k')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["evaluate_integrals", 7], "Equality(Derivative(Integral(Function('k')(Symbol('t', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Tuple(Symbol('t', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\theta_2)} = \\log{(\\theta_2)}, then derive \\int \\operatorname{f_{\\mathbf{p}}}{(\\theta_2)} d\\theta_2 = C_{d} + \\theta_2 \\log{(\\theta_2)} - \\theta_2, then obtain B + \\theta_2 \\log{(\\theta_2)} - \\theta_2 = C_{d} + \\theta_2 \\log{(\\theta_2)} - \\theta_2", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\theta_2)} = \\log{(\\theta_2)} and \\int \\operatorname{f_{\\mathbf{p}}}{(\\theta_2)} d\\theta_2 = \\int \\log{(\\theta_2)} d\\theta_2 and \\int \\operatorname{f_{\\mathbf{p}}}{(\\theta_2)} d\\theta_2 = C_{d} + \\theta_2 \\log{(\\theta_2)} - \\theta_2 and \\int \\log{(\\theta_2)} d\\theta_2 = C_{d} + \\theta_2 \\log{(\\theta_2)} - \\theta_2 and B + \\theta_2 \\log{(\\theta_2)} - \\theta_2 = C_{d} + \\theta_2 \\log{(\\theta_2)} - \\theta_2", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_2', commutative=True)), log(Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Add(Symbol('C_d', commutative=True), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given C{(\\dot{\\mathbf{r}},E_{\\lambda})} = E_{\\lambda} \\dot{\\mathbf{r}}, then obtain \\log{(\\cos{(C^{E_{\\lambda}}{(\\dot{\\mathbf{r}},E_{\\lambda})})})} = \\log{(\\cos{((E_{\\lambda} \\dot{\\mathbf{r}})^{E_{\\lambda}})})}", "derivation": "C{(\\dot{\\mathbf{r}},E_{\\lambda})} = E_{\\lambda} \\dot{\\mathbf{r}} and C^{E_{\\lambda}}{(\\dot{\\mathbf{r}},E_{\\lambda})} = (E_{\\lambda} \\dot{\\mathbf{r}})^{E_{\\lambda}} and \\cos{(C^{E_{\\lambda}}{(\\dot{\\mathbf{r}},E_{\\lambda})})} = \\cos{((E_{\\lambda} \\dot{\\mathbf{r}})^{E_{\\lambda}})} and \\log{(\\cos{(C^{E_{\\lambda}}{(\\dot{\\mathbf{r}},E_{\\lambda})})})} = \\log{(\\cos{((E_{\\lambda} \\dot{\\mathbf{r}})^{E_{\\lambda}})})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('C')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))"], [["cos", 2], "Equality(cos(Pow(Function('C')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))), cos(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))))"], [["log", 3], "Equality(log(cos(Pow(Function('C')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))), log(cos(Pow(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(c_{0},H)} = H - c_{0}, then derive (\\frac{\\partial}{\\partial H} \\mathbf{D}{(c_{0},H)})^{c_{0}} = 1, then obtain \\int (\\frac{\\partial}{\\partial H} \\mathbf{D}{(c_{0},H)})^{c_{0}} dH = \\int 1 dH", "derivation": "\\mathbf{D}{(c_{0},H)} = H - c_{0} and t + \\mathbf{D}{(c_{0},H)} = H - c_{0} + t and \\frac{\\partial}{\\partial H} (t + \\mathbf{D}{(c_{0},H)}) = \\frac{\\partial}{\\partial H} (H - c_{0} + t) and (\\frac{\\partial}{\\partial H} (t + \\mathbf{D}{(c_{0},H)}))^{c_{0}} = (\\frac{\\partial}{\\partial H} (H - c_{0} + t))^{c_{0}} and (\\frac{\\partial}{\\partial H} \\mathbf{D}{(c_{0},H)})^{c_{0}} = 1 and \\int (\\frac{\\partial}{\\partial H} \\mathbf{D}{(c_{0},H)})^{c_{0}} dH = \\int 1 dH", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["add", 1, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('t', commutative=True)))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Add(Symbol('t', commutative=True), Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["power", 3, "Symbol('c_0', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('t', commutative=True), Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('c_0', commutative=True)), Pow(Derivative(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('t', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('c_0', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('c_0', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('H', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\mathbf{D}')(Symbol('c_0', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('c_0', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Integer(1), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\ddot{x},n_{2})} = e^{\\ddot{x} n_{2}}, then obtain \\operatorname{m_{s}}{(\\ddot{x},n_{2})} - e^{\\ddot{x} n_{2}} + \\int \\operatorname{m_{s}}{(\\ddot{x},n_{2})} dn_{2} = \\operatorname{m_{s}}{(\\ddot{x},n_{2})} - e^{\\ddot{x} n_{2}} + \\int e^{\\ddot{x} n_{2}} dn_{2}", "derivation": "\\operatorname{m_{s}}{(\\ddot{x},n_{2})} = e^{\\ddot{x} n_{2}} and \\int \\operatorname{m_{s}}{(\\ddot{x},n_{2})} dn_{2} = \\int e^{\\ddot{x} n_{2}} dn_{2} and - e^{\\ddot{x} n_{2}} + \\int \\operatorname{m_{s}}{(\\ddot{x},n_{2})} dn_{2} = - e^{\\ddot{x} n_{2}} + \\int e^{\\ddot{x} n_{2}} dn_{2} and \\operatorname{m_{s}}{(\\ddot{x},n_{2})} - e^{\\ddot{x} n_{2}} + \\int \\operatorname{m_{s}}{(\\ddot{x},n_{2})} dn_{2} = \\operatorname{m_{s}}{(\\ddot{x},n_{2})} - e^{\\ddot{x} n_{2}} + \\int e^{\\ddot{x} n_{2}} dn_{2}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))))"], [["integrate", 1, "Symbol('n_2', commutative=True)"], "Equality(Integral(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True))), Integral(exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True))))"], [["minus", 2, "exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)))), Integral(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Mul(Integer(-1), exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)))), Integral(exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))))"], [["add", 3, "Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)))), Integral(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True)))), Add(Function('m_s')(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(-1), exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True)))), Integral(exp(Mul(Symbol('\\\\ddot{x}', commutative=True), Symbol('n_2', commutative=True))), Tuple(Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(L_{\\varepsilon},v_{y})} = L_{\\varepsilon} v_{y}, then obtain \\frac{\\cos{(\\mu_{0}{(L_{\\varepsilon},v_{y})})}}{L_{\\varepsilon} v_{y} - v_{y}} = \\frac{\\cos{(L_{\\varepsilon} v_{y})}}{L_{\\varepsilon} v_{y} - v_{y}}", "derivation": "\\mu_{0}{(L_{\\varepsilon},v_{y})} = L_{\\varepsilon} v_{y} and - v_{y} + \\mu_{0}{(L_{\\varepsilon},v_{y})} = L_{\\varepsilon} v_{y} - v_{y} and \\cos{(\\mu_{0}{(L_{\\varepsilon},v_{y})})} = \\cos{(L_{\\varepsilon} v_{y})} and \\frac{\\cos{(\\mu_{0}{(L_{\\varepsilon},v_{y})})}}{- v_{y} + \\mu_{0}{(L_{\\varepsilon},v_{y})}} = \\frac{\\cos{(L_{\\varepsilon} v_{y})}}{- v_{y} + \\mu_{0}{(L_{\\varepsilon},v_{y})}} and \\frac{\\cos{(\\mu_{0}{(L_{\\varepsilon},v_{y})})}}{L_{\\varepsilon} v_{y} - v_{y}} = \\frac{\\cos{(L_{\\varepsilon} v_{y})}}{L_{\\varepsilon} v_{y} - v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))"], [["minus", 1, "Symbol('v_y', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True))), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True))), Integer(-1)), cos(Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True))), Integer(-1)), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))), Integer(-1)), cos(Function('\\\\mu_0')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))), Mul(Pow(Add(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Symbol('v_y', commutative=True))), Integer(-1)), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(h)} = e^{h} and \\operatorname{a^{\\dagger}}{(h)} = e^{h}, then obtain \\cos{(h - e^{2 h} + (e^{h})^{h})} = \\cos{(h - \\operatorname{M_{E}}{(h)} e^{h} + (e^{h})^{h})}", "derivation": "\\operatorname{M_{E}}{(h)} = e^{h} and \\operatorname{a^{\\dagger}}{(h)} = e^{h} and \\operatorname{M_{E}}{(h)} = \\operatorname{a^{\\dagger}}{(h)} and \\operatorname{M_{E}}{(h)} e^{h} = \\operatorname{a^{\\dagger}}{(h)} e^{h} and - h + \\operatorname{M_{E}}{(h)} e^{h} = - h + \\operatorname{a^{\\dagger}}{(h)} e^{h} and - h + e^{2 h} = - h + \\operatorname{a^{\\dagger}}{(h)} e^{h} and - h + e^{2 h} = - h + \\operatorname{M_{E}}{(h)} e^{h} and - h + e^{2 h} - (e^{h})^{h} = - h + \\operatorname{M_{E}}{(h)} e^{h} - (e^{h})^{h} and \\cos{(h - e^{2 h} + (e^{h})^{h})} = \\cos{(h - \\operatorname{M_{E}}{(h)} e^{h} + (e^{h})^{h})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('M_E')(Symbol('h', commutative=True)), Function('a^{\\\\dagger}')(Symbol('h', commutative=True)))"], [["times", 3, "exp(Symbol('h', commutative=True))"], "Equality(Mul(Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))))"], [["minus", 4, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Function('a^{\\\\dagger}')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True)))))"], [["minus", 7, "Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), exp(Mul(Integer(2), Symbol('h', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), Mul(Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Mul(Integer(-1), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))))"], [["cos", 8], "Equality(cos(Add(Symbol('h', commutative=True), Mul(Integer(-1), exp(Mul(Integer(2), Symbol('h', commutative=True)))), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))), cos(Add(Symbol('h', commutative=True), Mul(Integer(-1), Function('M_E')(Symbol('h', commutative=True)), exp(Symbol('h', commutative=True))), Pow(exp(Symbol('h', commutative=True)), Symbol('h', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})} = \\sin{(E_{\\lambda} - \\mathbf{H})}, then obtain \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})} - 1 = \\cos{(E_{\\lambda} - \\mathbf{H})} - 1", "derivation": "\\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})} = \\sin{(E_{\\lambda} - \\mathbf{H})} and - E_{\\lambda} + \\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})} = - E_{\\lambda} + \\sin{(E_{\\lambda} - \\mathbf{H})} and \\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + \\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})}) = \\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + \\sin{(E_{\\lambda} - \\mathbf{H})}) and \\frac{\\partial}{\\partial E_{\\lambda}} \\operatorname{L_{\\varepsilon}}{(E_{\\lambda},\\mathbf{H})} - 1 = \\cos{(E_{\\lambda} - \\mathbf{H})} - 1", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))))"], [["minus", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), sin(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True))))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('L_{\\\\varepsilon}')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1)), Add(cos(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{H}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\eta{(\\mu)} = e^{\\mu} and \\sigma_{p}{(\\mu)} = \\frac{d}{d \\mu} e^{\\mu}, then obtain 0 = \\sigma_{p}{(\\mu)} - \\frac{d}{d \\mu} \\eta{(\\mu)}", "derivation": "\\eta{(\\mu)} = e^{\\mu} and \\frac{d}{d \\mu} \\eta{(\\mu)} = \\frac{d}{d \\mu} e^{\\mu} and \\sigma_{p}{(\\mu)} = \\frac{d}{d \\mu} e^{\\mu} and \\frac{d}{d \\mu} \\eta{(\\mu)} = \\sigma_{p}{(\\mu)} and 0 = \\sigma_{p}{(\\mu)} - \\frac{d}{d \\mu} \\eta{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mu', commutative=True)), exp(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mu', commutative=True)), Derivative(exp(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Function('\\\\sigma_p')(Symbol('\\\\mu', commutative=True)))"], [["minus", 4, "Derivative(Function('\\\\eta')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Function('\\\\sigma_p')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\eta')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\mathbf{S},v_{y})} = \\int (\\mathbf{S} - v_{y}) d\\mathbf{S} and \\mathbf{f}{(\\mathbf{S},v_{y})} = \\frac{\\operatorname{v_{1}}{(\\mathbf{S},v_{y})}}{\\int (\\mathbf{S} - v_{y}) d\\mathbf{S}}, then obtain \\int \\frac{\\operatorname{v_{1}}{(\\mathbf{S},v_{y})}}{\\int (\\mathbf{S} - v_{y}) d\\mathbf{S}} d\\mathbf{S} = \\int 1 d\\mathbf{S}", "derivation": "\\operatorname{v_{1}}{(\\mathbf{S},v_{y})} = \\int (\\mathbf{S} - v_{y}) d\\mathbf{S} and - \\operatorname{v_{1}}{(\\mathbf{S},v_{y})} = - \\int (\\mathbf{S} - v_{y}) d\\mathbf{S} and \\frac{\\operatorname{v_{1}}{(\\mathbf{S},v_{y})}}{\\int (\\mathbf{S} - v_{y}) d\\mathbf{S}} = 1 and \\mathbf{f}{(\\mathbf{S},v_{y})} = \\frac{\\operatorname{v_{1}}{(\\mathbf{S},v_{y})}}{\\int (\\mathbf{S} - v_{y}) d\\mathbf{S}} and \\mathbf{f}{(\\mathbf{S},v_{y})} = 1 and \\int \\mathbf{f}{(\\mathbf{S},v_{y})} d\\mathbf{S} = \\int 1 d\\mathbf{S} and \\int \\frac{\\operatorname{v_{1}}{(\\mathbf{S},v_{y})}}{\\int (\\mathbf{S} - v_{y}) d\\mathbf{S}} d\\mathbf{S} = \\int 1 d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["divide", 2, "Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], "Equality(Mul(Function('v_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Mul(Function('v_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Integer(1))"], [["integrate", 5, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integral(Mul(Function('v_1')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(y^{\\prime})} = \\cos{(y^{\\prime})}, then obtain \\frac{- \\phi^{\\mathbf{M}} + \\operatorname{C_{2}}{(y^{\\prime})} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})}}{\\mathbf{g}} = \\frac{- \\phi^{\\mathbf{M}} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})} + \\cos{(y^{\\prime})}}{\\mathbf{g}}", "derivation": "\\operatorname{C_{2}}{(y^{\\prime})} = \\cos{(y^{\\prime})} and - \\phi^{\\mathbf{M}} + \\operatorname{C_{2}}{(y^{\\prime})} = - \\phi^{\\mathbf{M}} + \\cos{(y^{\\prime})} and - \\phi^{\\mathbf{M}} + \\operatorname{C_{2}}{(y^{\\prime})} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})} = - \\phi^{\\mathbf{M}} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})} + \\cos{(y^{\\prime})} and \\frac{- \\phi^{\\mathbf{M}} + \\operatorname{C_{2}}{(y^{\\prime})} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})}}{\\mathbf{g}} = \\frac{- \\phi^{\\mathbf{M}} + \\hat{p}_0{(\\mathbf{g})} + \\log{(\\mathbf{g})} + \\cos{(y^{\\prime})}}{\\mathbf{g}}", "srepr_derivation": [["get_premise", "Equality(Function('C_2')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))"], [["minus", 1, "Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('C_2')(Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["add", 2, "Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('C_2')(Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True))), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('C_2')(Symbol('y^{\\\\prime}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('\\\\phi', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{g}', commutative=True)), log(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given g{(x^\\prime,Z)} = e^{Z + x^\\prime} and W{(x^\\prime,Z)} = 2 g{(x^\\prime,Z)}, then obtain g{(x^\\prime,Z)} + e^{Z + x^\\prime} + \\int 2 g{(x^\\prime,Z)} dx^\\prime = g{(x^\\prime,Z)} + e^{Z + x^\\prime} + \\int (g{(x^\\prime,Z)} + e^{Z + x^\\prime}) dx^\\prime", "derivation": "g{(x^\\prime,Z)} = e^{Z + x^\\prime} and 2 g{(x^\\prime,Z)} = g{(x^\\prime,Z)} + e^{Z + x^\\prime} and \\int 2 g{(x^\\prime,Z)} dx^\\prime = \\int (g{(x^\\prime,Z)} + e^{Z + x^\\prime}) dx^\\prime and 2 g{(x^\\prime,Z)} + \\int 2 g{(x^\\prime,Z)} dx^\\prime = 2 g{(x^\\prime,Z)} + \\int (g{(x^\\prime,Z)} + e^{Z + x^\\prime}) dx^\\prime and W{(x^\\prime,Z)} = 2 g{(x^\\prime,Z)} and W{(x^\\prime,Z)} = g{(x^\\prime,Z)} + e^{Z + x^\\prime} and W{(x^\\prime,Z)} + \\int 2 g{(x^\\prime,Z)} dx^\\prime = W{(x^\\prime,Z)} + \\int (g{(x^\\prime,Z)} + e^{Z + x^\\prime}) dx^\\prime and g{(x^\\prime,Z)} + e^{Z + x^\\prime} + \\int 2 g{(x^\\prime,Z)} dx^\\prime = g{(x^\\prime,Z)} + e^{Z + x^\\prime} + \\int (g{(x^\\prime,Z)} + e^{Z + x^\\prime}) dx^\\prime", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True))))"], [["add", 1, "Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["add", 3, "Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Integral(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Integral(Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], ["renaming_premise", "Equality(Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Integral(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integral(Mul(Integer(2), Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True))), Integral(Add(Function('g')(Symbol('x^\\\\prime', commutative=True), Symbol('Z', commutative=True)), exp(Add(Symbol('Z', commutative=True), Symbol('x^\\\\prime', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\rho_b,B)} = B \\sin{(\\rho_b)} and M{(\\rho_b,B)} = B \\sin{(\\rho_b)} - \\sigma_{x}{(\\rho_b,B)}, then obtain 0^{B} = 1", "derivation": "\\sigma_{x}{(\\rho_b,B)} = B \\sin{(\\rho_b)} and 0 = B \\sin{(\\rho_b)} - \\sigma_{x}{(\\rho_b,B)} and M{(\\rho_b,B)} = B \\sin{(\\rho_b)} - \\sigma_{x}{(\\rho_b,B)} and 0^{B} = (B \\sin{(\\rho_b)} - \\sigma_{x}{(\\rho_b,B)})^{B} and 0^{B} = M^{B}{(\\rho_b,B)} and M^{B}{(\\rho_b,B)} = 1 and 0^{B} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))))"], [["minus", 1, "Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('B', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)))))"], ["renaming_premise", "Equality(Function('M')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)), Add(Mul(Symbol('B', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Integer(0), Symbol('B', commutative=True)), Pow(Add(Mul(Symbol('B', commutative=True), sin(Symbol('\\\\rho_b', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)))), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Integer(0), Symbol('B', commutative=True)), Pow(Function('M')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('M')(Symbol('\\\\rho_b', commutative=True), Symbol('B', commutative=True)), Symbol('B', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Pow(Integer(0), Symbol('B', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\rho_{b}{(J,p)} = J p and \\operatorname{P_{g}}{(J,p)} = J p + \\rho_{b}{(J,p)}, then obtain \\iiint ((J p + \\rho_{b}{(J,p)})^{J})^{J} dp dJ dJ = \\iiint ((2 J p)^{J})^{J} dp dJ dJ", "derivation": "\\rho_{b}{(J,p)} = J p and \\operatorname{P_{g}}{(J,p)} = J p + \\rho_{b}{(J,p)} and \\operatorname{P_{g}}{(J,p)} = 2 J p and \\operatorname{P_{g}}^{J}{(J,p)} = (2 J p)^{J} and (\\operatorname{P_{g}}^{J}{(J,p)})^{J} = ((2 J p)^{J})^{J} and \\int (\\operatorname{P_{g}}^{J}{(J,p)})^{J} dp = \\int ((2 J p)^{J})^{J} dp and \\iint (\\operatorname{P_{g}}^{J}{(J,p)})^{J} dp dJ = \\iint ((2 J p)^{J})^{J} dp dJ and \\iiint (\\operatorname{P_{g}}^{J}{(J,p)})^{J} dp dJ dJ = \\iiint ((2 J p)^{J})^{J} dp dJ dJ and \\iiint ((J p + \\rho_{b}{(J,p)})^{J})^{J} dp dJ dJ = \\iiint ((2 J p)^{J})^{J} dp dJ dJ", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('p', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('J', commutative=True), Symbol('p', commutative=True)), Function('\\\\rho_b')(Symbol('J', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)))"], [["power", 4, "Symbol('J', commutative=True)"], "Equality(Pow(Pow(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["integrate", 5, "Symbol('p', commutative=True)"], "Equality(Integral(Pow(Pow(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["integrate", 6, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Pow(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["integrate", 7, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Pow(Function('P_g')(Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 2], "Equality(Integral(Pow(Pow(Add(Mul(Symbol('J', commutative=True), Symbol('p', commutative=True)), Function('\\\\rho_b')(Symbol('J', commutative=True), Symbol('p', commutative=True))), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Pow(Mul(Integer(2), Symbol('J', commutative=True), Symbol('p', commutative=True)), Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(J,\\lambda,\\mathbf{g})} = \\frac{\\lambda - \\mathbf{g}}{J}, then obtain (- \\mathbf{E}{(J,\\lambda,\\mathbf{g})} + \\frac{1}{J})^{J} = (\\frac{- \\lambda + \\mathbf{g}}{J} + \\frac{1}{J})^{J}", "derivation": "\\mathbf{E}{(J,\\lambda,\\mathbf{g})} = \\frac{\\lambda - \\mathbf{g}}{J} and - \\mathbf{E}{(J,\\lambda,\\mathbf{g})} = - \\frac{\\lambda - \\mathbf{g}}{J} and - \\mathbf{E}{(J,\\lambda,\\mathbf{g})} + \\frac{1}{J} = - \\frac{\\lambda - \\mathbf{g}}{J} + \\frac{1}{J} and (- \\mathbf{E}{(J,\\lambda,\\mathbf{g})} + \\frac{1}{J})^{J} = (- \\frac{\\lambda - \\mathbf{g}}{J} + \\frac{1}{J})^{J} and - \\mathbf{E}{(J,\\lambda,\\mathbf{g})} = \\frac{- \\lambda + \\mathbf{g}}{J} and \\frac{- \\lambda + \\mathbf{g}}{J} = - \\frac{\\lambda - \\mathbf{g}}{J} and (- \\mathbf{E}{(J,\\lambda,\\mathbf{g})} + \\frac{1}{J})^{J} = (\\frac{- \\lambda + \\mathbf{g}}{J} + \\frac{1}{J})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["add", 2, "Pow(Symbol('J', commutative=True), Integer(-1))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Symbol('J', commutative=True)), Pow(Add(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))), Pow(Symbol('J', commutative=True), Integer(-1))), Symbol('J', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Add(Symbol('\\\\lambda', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('J', commutative=True), Symbol('\\\\lambda', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Symbol('J', commutative=True)), Pow(Add(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))), Pow(Symbol('J', commutative=True), Integer(-1))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} = \\rho (- \\dot{\\mathbf{r}} + \\sigma_p) and \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}}, then obtain V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} + \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = \\rho \\sigma_p + \\rho \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} + \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}", "derivation": "V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} = \\rho (- \\dot{\\mathbf{r}} + \\sigma_p) and V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} \\rho + \\rho \\sigma_p and - \\dot{\\mathbf{r}} + V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} \\rho - \\dot{\\mathbf{r}} + \\rho \\sigma_p and \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} and V{(\\sigma_p,\\rho,\\dot{\\mathbf{r}})} + \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} = \\rho \\sigma_p + \\rho \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})} + \\hat{\\mathbf{x}}{(\\dot{\\mathbf{r}})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\sigma_p', commutative=True))))"], [["expand", 1], "Equality(Function('V')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('V')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('V')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\rho', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(A_{1})} = e^{\\sin{(A_{1})}}, then obtain \\int 0^{A_{1}} (- \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}})^{- A_{1}} dA_{1} = \\int 0^{A_{1}} \\tilde{\\infty}^{A_{1}} dA_{1}", "derivation": "\\rho_{b}{(A_{1})} = e^{\\sin{(A_{1})}} and 0 = - \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}} and 0^{A_{1}} = (- \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}})^{A_{1}} and 0^{A_{1}} (- \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}})^{- A_{1}} = 1 and \\int 0^{A_{1}} (- \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}})^{- A_{1}} dA_{1} = \\int 1 dA_{1} and \\int 0^{A_{1}} \\tilde{\\infty}^{A_{1}} dA_{1} = \\int 1 dA_{1} and \\int 0^{A_{1}} (- \\rho_{b}{(A_{1})} + e^{\\sin{(A_{1})}})^{- A_{1}} dA_{1} = \\int 0^{A_{1}} \\tilde{\\infty}^{A_{1}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('A_1', commutative=True)), exp(sin(Symbol('A_1', commutative=True))))"], [["minus", 1, "Function('\\\\rho_b')(Symbol('A_1', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))))"], [["power", 2, "Symbol('A_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)))"], [["divide", 3, "Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))), Mul(Integer(-1), Symbol('A_1', commutative=True)))), Integer(1))"], [["integrate", 4, "Symbol('A_1', commutative=True)"], "Equality(Integral(Mul(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))), Mul(Integer(-1), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Mul(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(zoo, Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Integral(Mul(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\rho_b')(Symbol('A_1', commutative=True))), exp(sin(Symbol('A_1', commutative=True)))), Mul(Integer(-1), Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True))), Integral(Mul(Pow(Integer(0), Symbol('A_1', commutative=True)), Pow(zoo, Symbol('A_1', commutative=True))), Tuple(Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\eta)} = e^{\\eta}, then obtain - \\operatorname{t_{1}}{(\\eta)} + \\int \\operatorname{t_{1}}{(\\eta)} d\\eta = - \\operatorname{t_{1}}{(\\eta)} + \\int e^{\\eta} d\\eta", "derivation": "\\operatorname{t_{1}}{(\\eta)} = e^{\\eta} and \\int \\operatorname{t_{1}}{(\\eta)} d\\eta = \\int e^{\\eta} d\\eta and - e^{\\eta} + \\int \\operatorname{t_{1}}{(\\eta)} d\\eta = - e^{\\eta} + \\int e^{\\eta} d\\eta and - \\operatorname{t_{1}}{(\\eta)} + \\int \\operatorname{t_{1}}{(\\eta)} d\\eta = - \\operatorname{t_{1}}{(\\eta)} + \\int e^{\\eta} d\\eta", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"], [["minus", 2, "exp(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\eta', commutative=True))), Integral(Function('t_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\eta', commutative=True))), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('t_1')(Symbol('\\\\eta', commutative=True))), Integral(Function('t_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Function('t_1')(Symbol('\\\\eta', commutative=True))), Integral(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\hat{x},v_{2})} = \\sin{(\\frac{\\hat{x}}{v_{2}})}, then obtain \\frac{\\partial}{\\partial \\hat{x}} (2 \\mathbf{B}{(\\hat{x},v_{2})} - \\frac{1}{v_{2}}) = \\frac{\\partial}{\\partial \\hat{x}} (2 \\sin{(\\frac{\\hat{x}}{v_{2}})} - \\frac{1}{v_{2}})", "derivation": "\\mathbf{B}{(\\hat{x},v_{2})} = \\sin{(\\frac{\\hat{x}}{v_{2}})} and \\mathbf{B}{(\\hat{x},v_{2})} - \\frac{1}{v_{2}} = \\sin{(\\frac{\\hat{x}}{v_{2}})} - \\frac{1}{v_{2}} and 2 \\mathbf{B}{(\\hat{x},v_{2})} - \\frac{1}{v_{2}} = \\mathbf{B}{(\\hat{x},v_{2})} + \\sin{(\\frac{\\hat{x}}{v_{2}})} - \\frac{1}{v_{2}} and 2 \\mathbf{B}{(\\hat{x},v_{2})} - \\frac{1}{v_{2}} = 2 \\sin{(\\frac{\\hat{x}}{v_{2}})} - \\frac{1}{v_{2}} and \\frac{\\partial}{\\partial \\hat{x}} (2 \\mathbf{B}{(\\hat{x},v_{2})} - \\frac{1}{v_{2}}) = \\frac{\\partial}{\\partial \\hat{x}} (2 \\sin{(\\frac{\\hat{x}}{v_{2}})} - \\frac{1}{v_{2}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)))))"], [["minus", 1, "Pow(Symbol('v_2', commutative=True), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Add(sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))))"], [["add", 2, "Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Add(Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Add(Mul(Integer(2), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))))"], [["differentiate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\hat{x}', commutative=True), Symbol('v_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Symbol('v_2', commutative=True), Integer(-1)))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(\\rho)} = \\cos{(e^{\\rho})} and \\operatorname{C_{1}}{(\\rho)} = e^{\\rho}, then obtain (- \\rho + \\hat{X}{(\\rho)})^{\\rho} = (- \\rho + \\cos{(\\operatorname{C_{1}}{(\\rho)})})^{\\rho}", "derivation": "\\hat{X}{(\\rho)} = \\cos{(e^{\\rho})} and - \\rho + \\hat{X}{(\\rho)} = - \\rho + \\cos{(e^{\\rho})} and (- \\rho + \\hat{X}{(\\rho)})^{\\rho} = (- \\rho + \\cos{(e^{\\rho})})^{\\rho} and \\operatorname{C_{1}}{(\\rho)} = e^{\\rho} and (- \\rho + \\hat{X}{(\\rho)})^{\\rho} = (- \\rho + \\cos{(\\operatorname{C_{1}}{(\\rho)})})^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True)), cos(exp(Symbol('\\\\rho', commutative=True))))"], [["minus", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(exp(Symbol('\\\\rho', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(exp(Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\rho', commutative=True)), exp(Symbol('\\\\rho', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho', commutative=True)), cos(Function('C_1')(Symbol('\\\\rho', commutative=True)))), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given u{(n_{2},c)} = - c + n_{2} and E{(\\rho_f,f_{E})} = \\rho_f + f_{E}, then obtain 1 = \\frac{\\rho_f + f_{E}}{E{(\\rho_f,f_{E})}}", "derivation": "u{(n_{2},c)} = - c + n_{2} and E{(\\rho_f,f_{E})} = \\rho_f + f_{E} and \\frac{E{(\\rho_f,f_{E})}}{u{(n_{2},c)}} = \\frac{\\rho_f + f_{E}}{u{(n_{2},c)}} and \\frac{E{(\\rho_f,f_{E})}}{- c + n_{2}} = \\frac{\\rho_f + f_{E}}{- c + n_{2}} and 1 = \\frac{\\rho_f + f_{E}}{E{(\\rho_f,f_{E})}}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('n_2', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('n_2', commutative=True)))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)))"], [["divide", 2, "Function('u')(Symbol('n_2', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('u')(Symbol('n_2', commutative=True), Symbol('c', commutative=True)), Integer(-1))), Mul(Add(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('u')(Symbol('n_2', commutative=True), Symbol('c', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('n_2', commutative=True)), Integer(-1)), Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True))), Mul(Add(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('n_2', commutative=True)), Integer(-1))))"], [["divide", 4, "Mul(Pow(Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('n_2', commutative=True)), Integer(-1)), Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Pow(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('f_E', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given I{(l,v_{y})} = v_{y} + e^{l} and \\operatorname{E_{x}}{(l,v_{y})} = v_{y} I{(l,v_{y})}, then obtain \\operatorname{E_{x}}{(l,v_{y})} e^{- l} = v_{y} (v_{y} + e^{l}) e^{- l}", "derivation": "I{(l,v_{y})} = v_{y} + e^{l} and v_{y} I{(l,v_{y})} = v_{y} (v_{y} + e^{l}) and \\operatorname{E_{x}}{(l,v_{y})} = v_{y} I{(l,v_{y})} and \\operatorname{E_{x}}{(l,v_{y})} = v_{y} (v_{y} + e^{l}) and \\operatorname{E_{x}}{(l,v_{y})} e^{- l} = v_{y} (v_{y} + e^{l}) e^{- l}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('v_y', commutative=True), exp(Symbol('l', commutative=True))))"], [["times", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Function('I')(Symbol('l', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Add(Symbol('v_y', commutative=True), exp(Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('v_y', commutative=True), Function('I')(Symbol('l', commutative=True), Symbol('v_y', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_x')(Symbol('l', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('v_y', commutative=True), Add(Symbol('v_y', commutative=True), exp(Symbol('l', commutative=True)))))"], [["divide", 4, "exp(Symbol('l', commutative=True))"], "Equality(Mul(Function('E_x')(Symbol('l', commutative=True), Symbol('v_y', commutative=True)), exp(Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Symbol('v_y', commutative=True), Add(Symbol('v_y', commutative=True), exp(Symbol('l', commutative=True))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\eta{(v_{2})} = \\int \\cos{(v_{2})} dv_{2}, then derive \\eta{(v_{2})} = \\phi_1 + \\sin{(v_{2})}, then obtain (\\Psi_{\\lambda} + \\sin{(v_{2})})^{\\phi_1} = (\\phi_1 + \\sin{(v_{2})})^{\\phi_1}", "derivation": "\\eta{(v_{2})} = \\int \\cos{(v_{2})} dv_{2} and \\eta{(v_{2})} = \\phi_1 + \\sin{(v_{2})} and \\eta^{\\phi_1}{(v_{2})} = (\\phi_1 + \\sin{(v_{2})})^{\\phi_1} and (\\int \\cos{(v_{2})} dv_{2})^{\\phi_1} = (\\phi_1 + \\sin{(v_{2})})^{\\phi_1} and (\\Psi_{\\lambda} + \\sin{(v_{2})})^{\\phi_1} = (\\phi_1 + \\sin{(v_{2})})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('v_2', commutative=True)), Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\eta')(Symbol('v_2', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('v_2', commutative=True))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('v_2', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('v_2', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(cos(Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('v_2', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Symbol('v_2', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('v_2', commutative=True))), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\varphi)} = \\log{(\\varphi)}, then derive \\frac{d}{d \\varphi} \\Psi_{nl}{(\\varphi)} - \\frac{1}{\\varphi} = 0, then obtain \\frac{d}{d \\varphi} \\log{(\\varphi)} - \\frac{1}{\\varphi} = 0", "derivation": "\\Psi_{nl}{(\\varphi)} = \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\Psi_{nl}{(\\varphi)} = \\frac{d}{d \\varphi} \\log{(\\varphi)} and \\frac{d}{d \\varphi} \\Psi_{nl}{(\\varphi)} - \\frac{d}{d \\varphi} \\log{(\\varphi)} = 0 and \\frac{d}{d \\varphi} \\Psi_{nl}{(\\varphi)} - \\frac{1}{\\varphi} = 0 and \\frac{d}{d \\varphi} \\log{(\\varphi)} - \\frac{1}{\\varphi} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given S{(v_{2},\\theta_2)} = \\cos{(\\theta_2^{v_{2}})} and m{(v_{2},\\theta_2)} = \\frac{1}{\\cos{(\\theta_2^{v_{2}})}}, then obtain \\frac{\\partial^{2}}{\\partial v_{2}\\partial \\theta_2} m{(v_{2},\\theta_2)} = \\frac{\\partial^{2}}{\\partial v_{2}\\partial \\theta_2} \\frac{1}{S{(v_{2},\\theta_2)}}", "derivation": "S{(v_{2},\\theta_2)} = \\cos{(\\theta_2^{v_{2}})} and m{(v_{2},\\theta_2)} = \\frac{1}{\\cos{(\\theta_2^{v_{2}})}} and m{(v_{2},\\theta_2)} = \\frac{1}{S{(v_{2},\\theta_2)}} and \\frac{1}{\\cos{(\\theta_2^{v_{2}})}} = \\frac{1}{S{(v_{2},\\theta_2)}} and \\frac{\\partial}{\\partial \\theta_2} \\frac{1}{\\cos{(\\theta_2^{v_{2}})}} = \\frac{\\partial}{\\partial \\theta_2} \\frac{1}{S{(v_{2},\\theta_2)}} and \\frac{\\partial}{\\partial \\theta_2} m{(v_{2},\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} \\frac{1}{S{(v_{2},\\theta_2)}} and \\frac{\\partial^{2}}{\\partial v_{2}\\partial \\theta_2} m{(v_{2},\\theta_2)} = \\frac{\\partial^{2}}{\\partial v_{2}\\partial \\theta_2} \\frac{1}{S{(v_{2},\\theta_2)}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), cos(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))))"], ["renaming_premise", "Equality(Function('m')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(cos(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('m')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(cos(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Pow(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Pow(cos(Pow(Symbol('\\\\theta_2', commutative=True), Symbol('v_2', commutative=True))), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Derivative(Function('m')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Pow(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Pow(Function('S')(Symbol('v_2', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\rho_f,B)} = - B + \\rho_f, then obtain (\\iint - \\frac{\\phi{(\\rho_f,B)}}{B} d\\rho_f dB)^{\\rho_f} = (\\iint - \\frac{- B + \\rho_f}{B} d\\rho_f dB)^{\\rho_f}", "derivation": "\\phi{(\\rho_f,B)} = - B + \\rho_f and - \\frac{\\phi{(\\rho_f,B)}}{B} = - \\frac{- B + \\rho_f}{B} and \\int - \\frac{\\phi{(\\rho_f,B)}}{B} d\\rho_f = \\int - \\frac{- B + \\rho_f}{B} d\\rho_f and \\iint - \\frac{\\phi{(\\rho_f,B)}}{B} d\\rho_f dB = \\iint - \\frac{- B + \\rho_f}{B} d\\rho_f dB and (\\iint - \\frac{\\phi{(\\rho_f,B)}}{B} d\\rho_f dB)^{\\rho_f} = (\\iint - \\frac{- B + \\rho_f}{B} d\\rho_f dB)^{\\rho_f}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\rho_f', commutative=True), Symbol('B', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('B', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\rho_f', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\rho_f', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 3, "Symbol('B', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\rho_f', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('\\\\phi')(Symbol('\\\\rho_f', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\rho{(\\phi_2,v_{2},\\lambda)} = v_{2} (- \\lambda + \\phi_2) and \\mathbf{H}{(\\lambda)} = - \\lambda, then obtain - v_{2} (- \\lambda + \\phi_2) - v_{2} (\\phi_2 + \\mathbf{H}{(\\lambda)}) + 2 \\rho{(\\phi_2,v_{2},\\lambda)} = 0", "derivation": "\\rho{(\\phi_2,v_{2},\\lambda)} = v_{2} (- \\lambda + \\phi_2) and - v_{2} (- \\lambda + \\phi_2) + \\rho{(\\phi_2,v_{2},\\lambda)} = 0 and \\mathbf{H}{(\\lambda)} = - \\lambda and - v_{2} (\\phi_2 + \\mathbf{H}{(\\lambda)}) + \\rho{(\\phi_2,v_{2},\\lambda)} = 0 and - v_{2} (- \\lambda + \\phi_2) - v_{2} (\\phi_2 + \\mathbf{H}{(\\lambda)}) + 2 \\rho{(\\phi_2,v_{2},\\lambda)} = - v_{2} (- \\lambda + \\phi_2) + \\rho{(\\phi_2,v_{2},\\lambda)} and - v_{2} (- \\lambda + \\phi_2) - v_{2} (\\phi_2 + \\mathbf{H}{(\\lambda)}) + 2 \\rho{(\\phi_2,v_{2},\\lambda)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["minus", 1, "Mul(Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(0))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)))), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Integer(0))"], [["add", 4, "Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Symbol('v_2', commutative=True), Add(Symbol('\\\\phi_2', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(2), Function('\\\\rho')(Symbol('\\\\phi_2', commutative=True), Symbol('v_2', commutative=True), Symbol('\\\\lambda', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\theta_{2}{(\\varepsilon_0,\\chi)} = \\chi \\varepsilon_0, then obtain \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} + \\int \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} d\\chi = \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} + \\int (\\chi \\varepsilon_0)^{\\varepsilon_0} d\\chi", "derivation": "\\theta_{2}{(\\varepsilon_0,\\chi)} = \\chi \\varepsilon_0 and \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} = (\\chi \\varepsilon_0)^{\\varepsilon_0} and \\int \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} d\\chi = \\int (\\chi \\varepsilon_0)^{\\varepsilon_0} d\\chi and (\\chi \\varepsilon_0)^{\\varepsilon_0} + \\int \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} d\\chi = (\\chi \\varepsilon_0)^{\\varepsilon_0} + \\int (\\chi \\varepsilon_0)^{\\varepsilon_0} d\\chi and \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} + \\int \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} d\\chi = \\theta_{2}^{\\varepsilon_0}{(\\varepsilon_0,\\chi)} + \\int (\\chi \\varepsilon_0)^{\\varepsilon_0} d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 3, "Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Pow(Function('\\\\theta_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Integral(Pow(Mul(Symbol('\\\\chi', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(E)} = e^{E}, then obtain (\\int \\log{(\\mu_{0}{(E)})} dE)^{E} = (\\int \\log{(e^{E})} dE)^{E}", "derivation": "\\mu_{0}{(E)} = e^{E} and \\log{(\\mu_{0}{(E)})} = \\log{(e^{E})} and \\int \\log{(\\mu_{0}{(E)})} dE = \\int \\log{(e^{E})} dE and (\\int \\log{(\\mu_{0}{(E)})} dE)^{E} = (\\int \\log{(e^{E})} dE)^{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mu_0')(Symbol('E', commutative=True))), log(exp(Symbol('E', commutative=True))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(log(Function('\\\\mu_0')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Integral(log(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Integral(log(Function('\\\\mu_0')(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)), Pow(Integral(log(exp(Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\hat{p}{(r,t_{2})} = t_{2}^{r}, then derive \\frac{t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\frac{\\partial}{\\partial r} \\hat{p}{(r,t_{2})}}{\\hat{p}{(r,t_{2})}} = t_{2} (t_{2}^{r})^{t_{2}} \\log{(t_{2})}, then obtain \\frac{t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\frac{\\partial}{\\partial r} \\hat{p}{(r,t_{2})}}{\\hat{p}{(r,t_{2})}} = t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\log{(t_{2})}", "derivation": "\\hat{p}{(r,t_{2})} = t_{2}^{r} and \\hat{p}^{t_{2}}{(r,t_{2})} = (t_{2}^{r})^{t_{2}} and \\frac{\\partial}{\\partial r} \\hat{p}^{t_{2}}{(r,t_{2})} = \\frac{\\partial}{\\partial r} (t_{2}^{r})^{t_{2}} and \\frac{t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\frac{\\partial}{\\partial r} \\hat{p}{(r,t_{2})}}{\\hat{p}{(r,t_{2})}} = t_{2} (t_{2}^{r})^{t_{2}} \\log{(t_{2})} and \\frac{t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\frac{\\partial}{\\partial r} \\hat{p}{(r,t_{2})}}{\\hat{p}{(r,t_{2})}} = t_{2} \\hat{p}^{t_{2}}{(r,t_{2})} \\log{(t_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('r', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(Pow(Symbol('t_2', commutative=True), Symbol('r', commutative=True)), Symbol('t_2', commutative=True)))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('t_2', commutative=True), Symbol('r', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('t_2', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('t_2', commutative=True), Pow(Pow(Symbol('t_2', commutative=True), Symbol('r', commutative=True)), Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('t_2', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Derivative(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Mul(Symbol('t_2', commutative=True), Pow(Function('\\\\hat{p}')(Symbol('r', commutative=True), Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), log(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(v_{y})} = \\sin{(v_{y})}, then derive \\int \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} dv_{y} = c + \\operatorname{Si}{(v_{y})}, then obtain (\\int \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} dv_{y})^{c} = (c + \\operatorname{Si}{(v_{y})})^{c}", "derivation": "\\mathbf{B}{(v_{y})} = \\sin{(v_{y})} and \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} = \\frac{\\sin{(v_{y})}}{v_{y}} and \\int \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} dv_{y} = \\int \\frac{\\sin{(v_{y})}}{v_{y}} dv_{y} and \\int \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} dv_{y} = c + \\operatorname{Si}{(v_{y})} and (\\int \\frac{\\mathbf{B}{(v_{y})}}{v_{y}} dv_{y})^{c} = (c + \\operatorname{Si}{(v_{y})})^{c}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True)), sin(Symbol('v_y', commutative=True)))"], [["divide", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True))), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True))))"], [["integrate", 2, "Symbol('v_y', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Integral(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), sin(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Add(Symbol('c', commutative=True), Si(Symbol('v_y', commutative=True))))"], [["power", 4, "Symbol('c', commutative=True)"], "Equality(Pow(Integral(Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Symbol('c', commutative=True)), Pow(Add(Symbol('c', commutative=True), Si(Symbol('v_y', commutative=True))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(z^{*})} = \\cos{(z^{*})}, then derive \\frac{d}{d z^{*}} \\operatorname{t_{1}}{(z^{*})} = - \\sin{(z^{*})}, then obtain z^{*} + \\frac{d}{d z^{*}} \\cos{(z^{*})} = z^{*} - \\sin{(z^{*})}", "derivation": "\\operatorname{t_{1}}{(z^{*})} = \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{t_{1}}{(z^{*})} = \\frac{d}{d z^{*}} \\cos{(z^{*})} and \\frac{d}{d z^{*}} \\operatorname{t_{1}}{(z^{*})} = - \\sin{(z^{*})} and \\frac{d}{d z^{*}} \\cos{(z^{*})} = - \\sin{(z^{*})} and z^{*} + \\frac{d}{d z^{*}} \\cos{(z^{*})} = z^{*} - \\sin{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t_1')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('z^*', commutative=True))))"], [["add", 4, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Derivative(cos(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Add(Symbol('z^*', commutative=True), Mul(Integer(-1), sin(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = \\frac{\\cos{(\\mathbf{s})}}{a^{\\dagger}}, then derive \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = - \\frac{\\cos{(\\mathbf{s})}}{(a^{\\dagger})^{2}}, then obtain \\frac{\\partial}{\\partial a^{\\dagger}} \\frac{\\cos{(\\mathbf{s})}}{a^{\\dagger}} = - \\frac{\\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})}}{a^{\\dagger}}", "derivation": "\\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = \\frac{\\cos{(\\mathbf{s})}}{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = \\frac{\\partial}{\\partial a^{\\dagger}} \\frac{\\cos{(\\mathbf{s})}}{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = - \\frac{\\cos{(\\mathbf{s})}}{(a^{\\dagger})^{2}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})} = - \\frac{\\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})}}{a^{\\dagger}} and \\frac{\\partial}{\\partial a^{\\dagger}} \\frac{\\cos{(\\mathbf{s})}}{a^{\\dagger}} = - \\frac{\\operatorname{A_{2}}{(a^{\\dagger},\\mathbf{s})}}{a^{\\dagger}}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-2)), cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Mul(Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('a^{\\\\dagger}', commutative=True), Integer(-1)), Function('A_2')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\rho_b,H)} = e^{H + \\rho_b}, then obtain \\frac{\\mathbf{F}{(\\rho_b,H)} e^{H + \\rho_b}}{\\frac{\\partial}{\\partial H} (H + \\rho_b + e^{H + \\rho_b})} = \\frac{\\mathbf{F}^{2}{(\\rho_b,H)}}{\\frac{\\partial}{\\partial H} (H + \\rho_b + e^{H + \\rho_b})}", "derivation": "\\mathbf{F}{(\\rho_b,H)} = e^{H + \\rho_b} and \\mathbf{F}{(\\rho_b,H)} e^{H + \\rho_b} = e^{2 H + 2 \\rho_b} and \\mathbf{F}^{2}{(\\rho_b,H)} = e^{2 H + 2 \\rho_b} and \\mathbf{F}{(\\rho_b,H)} e^{H + \\rho_b} = \\mathbf{F}^{2}{(\\rho_b,H)} and \\frac{\\mathbf{F}{(\\rho_b,H)} e^{H + \\rho_b}}{\\frac{\\partial}{\\partial H} (H + \\rho_b + e^{H + \\rho_b})} = \\frac{\\mathbf{F}^{2}{(\\rho_b,H)}}{\\frac{\\partial}{\\partial H} (H + \\rho_b + e^{H + \\rho_b})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), exp(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), Integer(2)), exp(Add(Mul(Integer(2), Symbol('H', commutative=True)), Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), Integer(2)))"], [["divide", 4, "Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True))), Pow(Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\rho_b', commutative=True), Symbol('H', commutative=True)), Integer(2)), Pow(Derivative(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True), exp(Add(Symbol('H', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})} = - \\dot{z} + \\hat{H}_l, then derive (\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})} - 1)^{\\hat{H}_l} = 0^{\\hat{H}_l}, then obtain (\\frac{\\partial}{\\partial \\hat{H}_l} (- \\dot{z} + \\hat{H}_l) - 1)^{\\hat{H}_l} = 0^{\\hat{H}_l}", "derivation": "\\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})} = - \\dot{z} + \\hat{H}_l and - \\hat{H}_l + \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})} = - \\dot{z} and \\frac{\\partial}{\\partial \\hat{H}_l} (- \\hat{H}_l + \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})}) = \\frac{d}{d \\hat{H}_l} - \\dot{z} and (\\frac{\\partial}{\\partial \\hat{H}_l} (- \\hat{H}_l + \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})}))^{\\hat{H}_l} = (\\frac{d}{d \\hat{H}_l} - \\dot{z})^{\\hat{H}_l} and (\\frac{\\partial}{\\partial \\hat{H}_l} \\operatorname{a^{\\dagger}}{(\\hat{H}_l,\\dot{z})} - 1)^{\\hat{H}_l} = 0^{\\hat{H}_l} and (\\frac{\\partial}{\\partial \\hat{H}_l} (- \\dot{z} + \\hat{H}_l) - 1)^{\\hat{H}_l} = 0^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Derivative(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Symbol('\\\\hat{H}_l', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\dot{z}', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Integer(0), Symbol('\\\\hat{H}_l', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Integer(0), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})} = (A_{1} - v_{x})^{f_{\\mathbf{v}}}, then obtain \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (A_{1} - v_{x} - \\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})}) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (A_{1} - v_{x} - (A_{1} - v_{x})^{f_{\\mathbf{v}}})", "derivation": "\\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})} = (A_{1} - v_{x})^{f_{\\mathbf{v}}} and - A_{1} + v_{x} + \\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})} = - A_{1} + v_{x} + (A_{1} - v_{x})^{f_{\\mathbf{v}}} and A_{1} - v_{x} - \\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})} = A_{1} - v_{x} - (A_{1} - v_{x})^{f_{\\mathbf{v}}} and \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (A_{1} - v_{x} - \\mathbf{M}{(f_{\\mathbf{v}},v_{x},A_{1})}) = \\frac{\\partial}{\\partial f_{\\mathbf{v}}} (A_{1} - v_{x} - (A_{1} - v_{x})^{f_{\\mathbf{v}}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["minus", 1, "Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('v_x', commutative=True), Function('\\\\mathbf{M}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Symbol('v_x', commutative=True), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True), Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["differentiate", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('v_x', commutative=True), Symbol('A_1', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))), Derivative(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('v_x', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{J}_M{(C_{2},v_{z})} = \\frac{v_{z}}{C_{2}} and \\delta{(C_{2},v_{z})} = \\iint \\mathbf{J}_M{(C_{2},v_{z})} dC_{2} dC_{2}, then obtain \\delta{(C_{2},v_{z})} = \\iint \\frac{v_{z}}{C_{2}} dC_{2} dC_{2}", "derivation": "\\mathbf{J}_M{(C_{2},v_{z})} = \\frac{v_{z}}{C_{2}} and \\int \\mathbf{J}_M{(C_{2},v_{z})} dC_{2} = \\int \\frac{v_{z}}{C_{2}} dC_{2} and \\iint \\mathbf{J}_M{(C_{2},v_{z})} dC_{2} dC_{2} = \\iint \\frac{v_{z}}{C_{2}} dC_{2} dC_{2} and \\delta{(C_{2},v_{z})} = \\iint \\mathbf{J}_M{(C_{2},v_{z})} dC_{2} dC_{2} and \\delta{(C_{2},v_{z})} = \\iint \\frac{v_{z}}{C_{2}} dC_{2} dC_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["integrate", 2, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\delta')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Integral(Function('\\\\mathbf{J}_M')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\delta')(Symbol('C_2', commutative=True), Symbol('v_z', commutative=True)), Integral(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(b)} = e^{b}, then obtain - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} \\sigma_{x}{(b)} + \\int e^{b} db = - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} e^{b} + \\int e^{b} db", "derivation": "\\sigma_{x}{(b)} = e^{b} and \\frac{d}{d b} \\sigma_{x}{(b)} = \\frac{d}{d b} e^{b} and \\int \\sigma_{x}{(b)} db = \\int e^{b} db and - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} \\sigma_{x}{(b)} = - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} e^{b} and - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} \\sigma_{x}{(b)} + \\int \\sigma_{x}{(b)} db = - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} e^{b} + \\int \\sigma_{x}{(b)} db and - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} \\sigma_{x}{(b)} + \\int e^{b} db = - \\cos{(\\frac{d}{d b} e^{b})} + \\frac{d}{d b} e^{b} + \\int e^{b} db", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('b', commutative=True)), exp(Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('b', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["minus", 2, "cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))), Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1)))))"], [["add", 4, "Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integral(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(Function('\\\\sigma_x')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), cos(Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))), Derivative(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integral(exp(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})} = \\log{(\\hat{X} \\mathbf{f})}, then obtain \\frac{\\log{(\\hat{X} \\mathbf{f} \\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})})}}{\\hat{X} \\mathbf{f}} = \\frac{\\log{(\\hat{X} \\mathbf{f} \\log{(\\hat{X} \\mathbf{f})})}}{\\hat{X} \\mathbf{f}}", "derivation": "\\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})} = \\log{(\\hat{X} \\mathbf{f})} and \\hat{X} \\mathbf{f} \\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})} = \\hat{X} \\mathbf{f} \\log{(\\hat{X} \\mathbf{f})} and \\log{(\\hat{X} \\mathbf{f} \\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})})} = \\log{(\\hat{X} \\mathbf{f} \\log{(\\hat{X} \\mathbf{f})})} and \\frac{\\log{(\\hat{X} \\mathbf{f} \\operatorname{F_{H}}{(\\mathbf{f},\\hat{X})})}}{\\hat{X} \\mathbf{f}} = \\frac{\\log{(\\hat{X} \\mathbf{f} \\log{(\\hat{X} \\mathbf{f})})}}{\\hat{X} \\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))), Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))"], [["log", 2], "Equality(log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))))"], [["divide", 3, "Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), Function('F_H')(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Mul(Pow(Symbol('\\\\hat{X}', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True), log(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\pi)} = \\sin{(\\pi)}, then derive \\mathbf{F} + \\pi = \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi, then obtain \\sin{(\\pi)} \\frac{d}{d \\mathbf{F}} \\int 1 d\\pi = \\sin{(\\pi)} \\frac{d}{d \\mathbf{F}} \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi", "derivation": "\\mathbf{A}{(\\pi)} = \\sin{(\\pi)} and 1 = \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} and \\int 1 d\\pi = \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi and \\mathbf{F} + \\pi = \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi and \\int 1 d\\pi = \\mathbf{F} + \\pi and \\frac{d}{d \\mathbf{F}} \\int 1 d\\pi = \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} + \\pi) and \\frac{d}{d \\mathbf{F}} \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi = \\frac{\\partial}{\\partial \\mathbf{F}} (\\mathbf{F} + \\pi) and \\frac{d}{d \\mathbf{F}} \\int 1 d\\pi = \\frac{d}{d \\mathbf{F}} \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi and \\sin{(\\pi)} \\frac{d}{d \\mathbf{F}} \\int 1 d\\pi = \\sin{(\\pi)} \\frac{d}{d \\mathbf{F}} \\int \\frac{\\sin{(\\pi)}}{\\mathbf{A}{(\\pi)}} d\\pi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\pi', commutative=True)), Integral(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Integral(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["times", 8, "sin(Symbol('\\\\pi', commutative=True))"], "Equality(Mul(sin(Symbol('\\\\pi', commutative=True)), Derivative(Integral(Integer(1), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(sin(Symbol('\\\\pi', commutative=True)), Derivative(Integral(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\chi,\\phi)} = \\cos{(\\chi - \\phi)}, then obtain \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} \\sigma_{x}^{\\phi}{(\\chi,\\phi)})^{\\chi} = \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} \\cos^{\\phi}{(\\chi - \\phi)})^{\\chi}", "derivation": "\\sigma_{x}{(\\chi,\\phi)} = \\cos{(\\chi - \\phi)} and \\sigma_{x}^{\\phi}{(\\chi,\\phi)} = \\cos^{\\phi}{(\\chi - \\phi)} and - \\mathbf{E} \\sigma_{x}^{\\phi}{(\\chi,\\phi)} = - \\mathbf{E} \\cos^{\\phi}{(\\chi - \\phi)} and (- \\mathbf{E} \\sigma_{x}^{\\phi}{(\\chi,\\phi)})^{\\chi} = (- \\mathbf{E} \\cos^{\\phi}{(\\chi - \\phi)})^{\\chi} and \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} \\sigma_{x}^{\\phi}{(\\chi,\\phi)})^{\\chi} = \\frac{\\partial}{\\partial \\mathbf{E}} (- \\mathbf{E} \\cos^{\\phi}{(\\chi - \\phi)})^{\\chi}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))))"], [["power", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True)), Pow(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Function('\\\\sigma_x')(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True))))"], [["power", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Function('\\\\sigma_x')(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(Function('\\\\sigma_x')(Symbol('\\\\chi', commutative=True), Symbol('\\\\phi', commutative=True)), Symbol('\\\\phi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True), Pow(cos(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\theta,F_{H})} = \\theta^{F_{H}}, then obtain (\\frac{- F_{H} + C{(\\theta,F_{H})}}{F_{H}})^{\\theta} = (\\frac{- F_{H} + \\theta^{F_{H}}}{F_{H}})^{\\theta}", "derivation": "C{(\\theta,F_{H})} = \\theta^{F_{H}} and - F_{H} + C{(\\theta,F_{H})} = - F_{H} + \\theta^{F_{H}} and \\frac{- F_{H} + C{(\\theta,F_{H})}}{F_{H}} = \\frac{- F_{H} + \\theta^{F_{H}}}{F_{H}} and (\\frac{- F_{H} + C{(\\theta,F_{H})}}{F_{H}})^{\\theta} = (\\frac{- F_{H} + \\theta^{F_{H}}}{F_{H}})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))"], [["minus", 1, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('C')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True))))"], [["divide", 2, "Symbol('F_H', commutative=True)"], "Equality(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('C')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))))"], [["power", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('C')(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Pow(Symbol('\\\\theta', commutative=True), Symbol('F_H', commutative=True)))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given h{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain (- 0^{\\mathbf{J}} h{(\\mathbf{J})} + e^{\\mathbf{J}})^{\\mathbf{J}} = 1", "derivation": "h{(\\mathbf{J})} = e^{\\mathbf{J}} and 0 = - h{(\\mathbf{J})} + e^{\\mathbf{J}} and 0^{\\mathbf{J}} = (- h{(\\mathbf{J})} + e^{\\mathbf{J}})^{\\mathbf{J}} and 0^{\\mathbf{J}} e^{- \\mathbf{J}} = (- h{(\\mathbf{J})} + e^{\\mathbf{J}})^{\\mathbf{J}} e^{- \\mathbf{J}} and (- h{(\\mathbf{J})} + e^{\\mathbf{J}})^{\\mathbf{J}} = 1 and 0^{\\mathbf{J}} = 1 and 0^{\\mathbf{J}} e^{\\mathbf{J}} = e^{\\mathbf{J}} and 0^{\\mathbf{J}} h{(\\mathbf{J})} = h{(\\mathbf{J})} and (- 0^{\\mathbf{J}} h{(\\mathbf{J})} + e^{\\mathbf{J}})^{\\mathbf{J}} = 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["minus", 1, "Function('h')(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 3, "exp(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Pow(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1))"], [["times", 6, "exp(Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), Function('h')(Symbol('\\\\mathbf{J}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 8], "Equality(Pow(Add(Mul(Integer(-1), Pow(Integer(0), Symbol('\\\\mathbf{J}', commutative=True)), Function('h')(Symbol('\\\\mathbf{J}', commutative=True))), exp(Symbol('\\\\mathbf{J}', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Integer(1))"]]}, {"prompt": "Given C{(C_{2})} = \\cos{(C_{2})} and \\mathbf{A}{(C_{2})} = \\cos{(C_{2})}, then derive \\frac{d}{d C_{2}} C{(C_{2})} = - \\sin{(C_{2})}, then obtain \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = - \\sin{(C_{2})}", "derivation": "C{(C_{2})} = \\cos{(C_{2})} and \\frac{d}{d C_{2}} C{(C_{2})} = \\frac{d}{d C_{2}} \\cos{(C_{2})} and \\mathbf{A}{(C_{2})} = \\cos{(C_{2})} and \\frac{d}{d C_{2}} C{(C_{2})} = - \\sin{(C_{2})} and \\frac{d}{d C_{2}} \\cos{(C_{2})} = - \\sin{(C_{2})} and \\frac{d}{d C_{2}} \\mathbf{A}{(C_{2})} = - \\sin{(C_{2})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), cos(Symbol('C_2', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(cos(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('C_2', commutative=True))))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbf{r})} = e^{\\mathbf{r}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{r})} = (\\frac{d}{d \\mathbf{r}} \\phi_{2}{(\\mathbf{r})} - 1)^{\\mathbf{r}}, then obtain \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{r})} = (\\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} - 1)^{\\mathbf{r}}", "derivation": "\\phi_{2}{(\\mathbf{r})} = e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\phi_{2}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} and \\frac{d}{d \\mathbf{r}} \\phi_{2}{(\\mathbf{r})} - 1 = \\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} - 1 and (\\frac{d}{d \\mathbf{r}} \\phi_{2}{(\\mathbf{r})} - 1)^{\\mathbf{r}} = (\\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} - 1)^{\\mathbf{r}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{r})} = (\\frac{d}{d \\mathbf{r}} \\phi_{2}{(\\mathbf{r})} - 1)^{\\mathbf{r}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{r})} = (\\frac{d}{d \\mathbf{r}} e^{\\mathbf{r}} - 1)^{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbf{r}', commutative=True)), exp(Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)))"], [["power", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Pow(Add(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(Add(Derivative(exp(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(-1)), Symbol('\\\\mathbf{r}', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(c,\\phi_1)} = \\phi_1 - c, then derive \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} + \\frac{\\partial}{\\partial c} \\phi_{2}{(c,\\phi_1)} = \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} - 1, then obtain \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} + \\frac{\\partial}{\\partial c} (\\phi_1 - c) = \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} - 1", "derivation": "\\phi_{2}{(c,\\phi_1)} = \\phi_1 - c and - (\\phi_1 - c)^{\\phi_1} + \\phi_{2}{(c,\\phi_1)} = \\phi_1 - c - (\\phi_1 - c)^{\\phi_1} and \\frac{\\partial}{\\partial c} (- (\\phi_1 - c)^{\\phi_1} + \\phi_{2}{(c,\\phi_1)}) = \\frac{\\partial}{\\partial c} (\\phi_1 - c - (\\phi_1 - c)^{\\phi_1}) and \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} + \\frac{\\partial}{\\partial c} \\phi_{2}{(c,\\phi_1)} = \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} - 1 and \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} + \\frac{\\partial}{\\partial c} (\\phi_1 - c) = \\frac{\\phi_1 (\\phi_1 - c)^{\\phi_1}}{\\phi_1 - c} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))))"], [["minus", 1, "Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Function('\\\\phi_2')(Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True)))))"], [["differentiate", 2, "Symbol('c', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Function('\\\\phi_2')(Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True)), Mul(Integer(-1), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Derivative(Function('\\\\phi_2')(Symbol('c', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Derivative(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\phi_1', commutative=True), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Integer(-1)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('c', commutative=True))), Symbol('\\\\phi_1', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\hat{x},U,\\eta)} = U + \\eta + \\hat{x} and \\operatorname{v_{y}}{(\\hat{x},\\eta,U)} = \\frac{(U + \\eta + \\hat{x})^{3}}{\\Psi_{nl}^{2}{(\\hat{x},U,\\eta)}}, then obtain \\operatorname{v_{y}}{(\\hat{x},\\eta,U)} = U + \\eta + \\hat{x}", "derivation": "\\Psi_{nl}{(\\hat{x},U,\\eta)} = U + \\eta + \\hat{x} and U + \\eta + \\hat{x} = \\frac{(U + \\eta + \\hat{x})^{2}}{\\Psi_{nl}{(\\hat{x},U,\\eta)}} and \\Psi_{nl}{(\\hat{x},U,\\eta)} = \\frac{(U + \\eta + \\hat{x})^{2}}{\\Psi_{nl}{(\\hat{x},U,\\eta)}} and \\operatorname{v_{y}}{(\\hat{x},\\eta,U)} = \\frac{(U + \\eta + \\hat{x})^{3}}{\\Psi_{nl}^{2}{(\\hat{x},U,\\eta)}} and \\operatorname{v_{y}}{(\\hat{x},\\eta,U)} = \\frac{\\Psi_{nl}^{2}{(\\hat{x},U,\\eta)}}{U + \\eta + \\hat{x}} and \\operatorname{v_{y}}{(\\hat{x},\\eta,U)} = U + \\eta + \\hat{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 1, "Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(3)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('v_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Mul(Pow(Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Integer(-1)), Pow(Function('\\\\Psi_{nl}')(Symbol('\\\\hat{x}', commutative=True), Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('v_y')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('U', commutative=True)), Add(Symbol('U', commutative=True), Symbol('\\\\eta', commutative=True), Symbol('\\\\hat{x}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and B{(\\mathbf{E})} = \\log{(\\mathbf{E})}^{\\mathbf{E}}, then obtain (\\mathbf{B}^{\\mathbf{E}}{(\\mathbf{E})})^{\\mathbf{E}} = (\\log{(\\mathbf{E})}^{\\mathbf{E}})^{\\mathbf{E}}", "derivation": "\\mathbf{B}{(\\mathbf{E})} = \\log{(\\mathbf{E})} and \\mathbf{B}^{\\mathbf{E}}{(\\mathbf{E})} = \\log{(\\mathbf{E})}^{\\mathbf{E}} and B{(\\mathbf{E})} = \\log{(\\mathbf{E})}^{\\mathbf{E}} and \\mathbf{B}^{\\mathbf{E}}{(\\mathbf{E})} = B{(\\mathbf{E})} and B^{\\mathbf{E}}{(\\mathbf{E})} = (\\log{(\\mathbf{E})}^{\\mathbf{E}})^{\\mathbf{E}} and (\\mathbf{B}^{\\mathbf{E}}{(\\mathbf{E})})^{\\mathbf{E}} = (\\log{(\\mathbf{E})}^{\\mathbf{E}})^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), log(Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\mathbf{E}', commutative=True)), Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Function('B')(Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Function('B')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Pow(log(Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"]]}, {"prompt": "Given a{(C_{2})} = e^{\\cos{(C_{2})}} and x{(C_{2})} = \\int a{(C_{2})} dC_{2}, then obtain 0 = \\frac{- \\int a{(C_{2})} dC_{2} + \\int e^{\\cos{(C_{2})}} dC_{2}}{C_{2}}", "derivation": "a{(C_{2})} = e^{\\cos{(C_{2})}} and \\int a{(C_{2})} dC_{2} = \\int e^{\\cos{(C_{2})}} dC_{2} and C_{2} + \\int a{(C_{2})} dC_{2} = C_{2} + \\int e^{\\cos{(C_{2})}} dC_{2} and x{(C_{2})} = \\int a{(C_{2})} dC_{2} and C_{2} + x{(C_{2})} = C_{2} + \\int e^{\\cos{(C_{2})}} dC_{2} and 0 = - x{(C_{2})} + \\int e^{\\cos{(C_{2})}} dC_{2} and 0 = - \\int a{(C_{2})} dC_{2} + \\int e^{\\cos{(C_{2})}} dC_{2} and 0 = \\frac{- \\int a{(C_{2})} dC_{2} + \\int e^{\\cos{(C_{2})}} dC_{2}}{C_{2}}", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('C_2', commutative=True)), exp(cos(Symbol('C_2', commutative=True))))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('a')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))))"], [["add", 2, "Symbol('C_2', commutative=True)"], "Equality(Add(Symbol('C_2', commutative=True), Integral(Function('a')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Add(Symbol('C_2', commutative=True), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('x')(Symbol('C_2', commutative=True)), Integral(Function('a')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('C_2', commutative=True), Function('x')(Symbol('C_2', commutative=True))), Add(Symbol('C_2', commutative=True), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["minus", 5, "Add(Symbol('C_2', commutative=True), Function('x')(Symbol('C_2', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('x')(Symbol('C_2', commutative=True))), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('a')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True)))))"], [["divide", 7, "Symbol('C_2', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Integral(Function('a')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))), Integral(exp(cos(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\mu_0)} = \\sin{(\\mu_0)}, then derive \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 = \\rho_f - \\cos{(\\mu_0)}, then obtain - \\rho_f + \\cos{(\\mu_0)} + \\int \\sin{(\\mu_0)} d\\mu_0 + 1 = - \\rho_f + \\cos{(\\mu_0)} + \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 + 1", "derivation": "\\operatorname{a^{\\dagger}}{(\\mu_0)} = \\sin{(\\mu_0)} and \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 = \\int \\sin{(\\mu_0)} d\\mu_0 and \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 = \\rho_f - \\cos{(\\mu_0)} and \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 + 1 = \\rho_f - \\cos{(\\mu_0)} + 1 and \\int \\sin{(\\mu_0)} d\\mu_0 + 1 = \\rho_f - \\cos{(\\mu_0)} + 1 and \\int \\sin{(\\mu_0)} d\\mu_0 + 1 = \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 + 1 and - \\rho_f + \\cos{(\\mu_0)} + \\int \\sin{(\\mu_0)} d\\mu_0 + 1 = - \\rho_f + \\cos{(\\mu_0)} + \\int \\operatorname{a^{\\dagger}}{(\\mu_0)} d\\mu_0 + 1", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True)))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)), Add(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)))"], [["minus", 6, "Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu_0', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)), Integral(sin(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integer(1)))"]]}, {"prompt": "Given M{(l)} = e^{l}, then obtain l (1 + \\frac{M{(l)}}{l}) e^{- l} = l (1 + \\frac{e^{l}}{l}) e^{- l}", "derivation": "M{(l)} = e^{l} and \\frac{M{(l)}}{l} = \\frac{e^{l}}{l} and 1 + \\frac{M{(l)}}{l} = 1 + \\frac{e^{l}}{l} and l (1 + \\frac{M{(l)}}{l}) e^{- l} = l (1 + \\frac{e^{l}}{l}) e^{- l}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('l', commutative=True)), exp(Symbol('l', commutative=True)))"], [["divide", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('M')(Symbol('l', commutative=True))), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('M')(Symbol('l', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('l', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True)))"], "Equality(Mul(Symbol('l', commutative=True), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), Function('M')(Symbol('l', commutative=True)))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Symbol('l', commutative=True), Add(Integer(1), Mul(Pow(Symbol('l', commutative=True), Integer(-1)), exp(Symbol('l', commutative=True)))), exp(Mul(Integer(-1), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given u{(\\Psi^{\\dagger},J)} = \\cos{(J - \\Psi^{\\dagger})}, then obtain \\frac{u^{2}{(\\Psi^{\\dagger},J)}}{J} = \\frac{u{(\\Psi^{\\dagger},J)} \\cos{(J - \\Psi^{\\dagger})}}{J}", "derivation": "u{(\\Psi^{\\dagger},J)} = \\cos{(J - \\Psi^{\\dagger})} and \\frac{u{(\\Psi^{\\dagger},J)}}{J} = \\frac{\\cos{(J - \\Psi^{\\dagger})}}{J} and \\frac{u{(\\Psi^{\\dagger},J)} \\cos{(J - \\Psi^{\\dagger})}}{J} = \\frac{\\cos^{2}{(J - \\Psi^{\\dagger})}}{J} and \\frac{u^{2}{(\\Psi^{\\dagger},J)}}{J} = \\frac{u{(\\Psi^{\\dagger},J)} \\cos{(J - \\Psi^{\\dagger})}}{J}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["divide", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"], [["times", 2, "cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Pow(Function('u')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), Integer(2))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Function('u')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('J', commutative=True)), cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"]]}, {"prompt": "Given r{(Z,\\omega)} = \\sin{(Z - \\omega)}, then obtain (Z + \\sin{(Z - \\omega)}) \\int - \\omega (e^{r{(Z,\\omega)}})^{Z} dZ = (Z + \\sin{(Z - \\omega)}) \\int - \\omega (e^{\\sin{(Z - \\omega)}})^{Z} dZ", "derivation": "r{(Z,\\omega)} = \\sin{(Z - \\omega)} and Z + r{(Z,\\omega)} = Z + \\sin{(Z - \\omega)} and e^{r{(Z,\\omega)}} = e^{\\sin{(Z - \\omega)}} and (e^{r{(Z,\\omega)}})^{Z} = (e^{\\sin{(Z - \\omega)}})^{Z} and - \\omega (e^{r{(Z,\\omega)}})^{Z} = - \\omega (e^{\\sin{(Z - \\omega)}})^{Z} and \\int - \\omega (e^{r{(Z,\\omega)}})^{Z} dZ = \\int - \\omega (e^{\\sin{(Z - \\omega)}})^{Z} dZ and (Z + r{(Z,\\omega)}) \\int - \\omega (e^{r{(Z,\\omega)}})^{Z} dZ = (Z + r{(Z,\\omega)}) \\int - \\omega (e^{\\sin{(Z - \\omega)}})^{Z} dZ and (Z + \\sin{(Z - \\omega)}) \\int - \\omega (e^{r{(Z,\\omega)}})^{Z} dZ = (Z + \\sin{(Z - \\omega)}) \\int - \\omega (e^{\\sin{(Z - \\omega)}})^{Z} dZ", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True)))))"], [["add", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Symbol('Z', commutative=True), Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Symbol('Z', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))))"], [["exp", 1], "Equality(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))))"], [["power", 3, "Symbol('Z', commutative=True)"], "Equality(Pow(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('Z', commutative=True)), Pow(exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('Z', commutative=True)))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('Z', commutative=True))), Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('Z', commutative=True))))"], [["integrate", 5, "Symbol('Z', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True))))"], [["times", 6, "Add(Symbol('Z', commutative=True), Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Add(Symbol('Z', commutative=True), Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Mul(Add(Symbol('Z', commutative=True), Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Add(Symbol('Z', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(Function('r')(Symbol('Z', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))), Mul(Add(Symbol('Z', commutative=True), sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Integral(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Pow(exp(sin(Add(Symbol('Z', commutative=True), Mul(Integer(-1), Symbol('\\\\omega', commutative=True))))), Symbol('Z', commutative=True))), Tuple(Symbol('Z', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(v_{z})} = \\sin{(v_{z})}, then obtain \\frac{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} + \\frac{d}{d v_{z}} \\sin{(v_{z})}}{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} \\frac{d}{d v_{z}} \\sin{(v_{z})}} = \\frac{2}{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})}}", "derivation": "\\mathbf{J}_f{(v_{z})} = \\sin{(v_{z})} and \\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} = \\frac{d}{d v_{z}} \\sin{(v_{z})} and \\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} + \\frac{d}{d v_{z}} \\sin{(v_{z})} = 2 \\frac{d}{d v_{z}} \\sin{(v_{z})} and \\frac{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} + \\frac{d}{d v_{z}} \\sin{(v_{z})}}{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})} \\frac{d}{d v_{z}} \\sin{(v_{z})}} = \\frac{2}{\\frac{d}{d v_{z}} \\mathbf{J}_f{(v_{z})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), sin(Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["add", 2, "Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], "Equality(Mul(Add(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(-1)), Pow(Derivative(sin(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(2), Pow(Derivative(Function('\\\\mathbf{J}_f')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\rho_{f}{(\\mathbf{F})} = \\sin{(\\mathbf{F})}, then derive \\frac{d}{d \\mathbf{F}} \\rho_{f}{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then obtain \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} = \\cos{(\\mathbf{F})}", "derivation": "\\rho_{f}{(\\mathbf{F})} = \\sin{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\rho_{f}{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\rho_{f}{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} \\sin{(\\mathbf{F})} = \\cos{(\\mathbf{F})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{F}', commutative=True)))"]]}, {"prompt": "Given L{(\\mathbf{P})} = \\mathbf{P}, then obtain 1 = \\frac{\\mathbf{P}^{\\mathbf{P}} + z^{*}}{z^{*} + L^{\\mathbf{P}}{(\\mathbf{P})}}", "derivation": "L{(\\mathbf{P})} = \\mathbf{P} and L^{\\mathbf{P}}{(\\mathbf{P})} = \\mathbf{P}^{\\mathbf{P}} and z^{*} + L^{\\mathbf{P}}{(\\mathbf{P})} = \\mathbf{P}^{\\mathbf{P}} + z^{*} and 1 = \\frac{\\mathbf{P}^{\\mathbf{P}} + z^{*}}{z^{*} + L^{\\mathbf{P}}{(\\mathbf{P})}}", "srepr_derivation": [["renaming_premise", "Equality(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], [["add", 2, "Symbol('z^*', commutative=True)"], "Equality(Add(Symbol('z^*', commutative=True), Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Add(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('z^*', commutative=True)))"], [["divide", 3, "Add(Symbol('z^*', commutative=True), Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Integer(1), Mul(Add(Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('z^*', commutative=True)), Pow(Add(Symbol('z^*', commutative=True), Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(x,f^{*})} = f^{*} + x, then derive (\\frac{\\partial}{\\partial f^{*}} \\operatorname{v_{t}}{(x,f^{*})} - 1)^{x} = 0^{x}, then obtain (\\frac{\\partial}{\\partial f^{*}} (f^{*} + x) - 1)^{x} = 0^{x}", "derivation": "\\operatorname{v_{t}}{(x,f^{*})} = f^{*} + x and \\frac{\\partial}{\\partial f^{*}} \\operatorname{v_{t}}{(x,f^{*})} = \\frac{\\partial}{\\partial f^{*}} (f^{*} + x) and - \\frac{\\partial}{\\partial f^{*}} (f^{*} + x) + \\frac{\\partial}{\\partial f^{*}} \\operatorname{v_{t}}{(x,f^{*})} = 0 and (- \\frac{\\partial}{\\partial f^{*}} (f^{*} + x) + \\frac{\\partial}{\\partial f^{*}} \\operatorname{v_{t}}{(x,f^{*})})^{x} = 0^{x} and (\\frac{\\partial}{\\partial f^{*}} \\operatorname{v_{t}}{(x,f^{*})} - 1)^{x} = 0^{x} and (\\frac{\\partial}{\\partial f^{*}} (f^{*} + x) - 1)^{x} = 0^{x}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('x', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('x', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Derivative(Function('v_t')(Symbol('x', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Integer(0))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Derivative(Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Derivative(Function('v_t')(Symbol('x', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1)))), Symbol('x', commutative=True)), Pow(Integer(0), Symbol('x', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Add(Derivative(Function('v_t')(Symbol('x', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-1)), Symbol('x', commutative=True)), Pow(Integer(0), Symbol('x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Add(Derivative(Add(Symbol('f^*', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Integer(-1)), Symbol('x', commutative=True)), Pow(Integer(0), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} = \\frac{\\Psi_{\\lambda}}{f_{E}}, then obtain \\frac{\\partial}{\\partial f_{E}} \\int 2 \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} df_{E} = \\frac{\\partial}{\\partial f_{E}} \\int (\\frac{\\Psi_{\\lambda}}{f_{E}} + \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})}) df_{E}", "derivation": "\\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} = \\frac{\\Psi_{\\lambda}}{f_{E}} and 2 \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} = \\frac{\\Psi_{\\lambda}}{f_{E}} + \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} and \\int 2 \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} df_{E} = \\int (\\frac{\\Psi_{\\lambda}}{f_{E}} + \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})}) df_{E} and \\frac{\\partial}{\\partial f_{E}} \\int 2 \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})} df_{E} = \\frac{\\partial}{\\partial f_{E}} \\int (\\frac{\\Psi_{\\lambda}}{f_{E}} + \\operatorname{a^{\\dagger}}{(f_{E},\\Psi_{\\lambda})}) df_{E}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))))"], [["add", 1, "Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["differentiate", 3, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Pow(Symbol('f_E', commutative=True), Integer(-1))), Function('a^{\\\\dagger}')(Symbol('f_E', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(m,S)} = \\cos^{m}{(S)}, then obtain \\int \\dot{y}{(m,S)} \\dot{y}^{S}{(m,S)} dS = \\int (\\cos^{m}{(S)})^{S} \\dot{y}{(m,S)} dS", "derivation": "\\dot{y}{(m,S)} = \\cos^{m}{(S)} and \\dot{y}^{S}{(m,S)} = (\\cos^{m}{(S)})^{S} and \\dot{y}{(m,S)} \\dot{y}^{S}{(m,S)} = (\\cos^{m}{(S)})^{S} \\dot{y}{(m,S)} and \\int \\dot{y}{(m,S)} \\dot{y}^{S}{(m,S)} dS = \\int (\\cos^{m}{(S)})^{S} \\dot{y}{(m,S)} dS", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Pow(cos(Symbol('S', commutative=True)), Symbol('m', commutative=True)))"], [["power", 1, "Symbol('S', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(cos(Symbol('S', commutative=True)), Symbol('m', commutative=True)), Symbol('S', commutative=True)))"], [["times", 2, "Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Symbol('S', commutative=True))), Mul(Pow(Pow(cos(Symbol('S', commutative=True)), Symbol('m', commutative=True)), Symbol('S', commutative=True)), Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True))))"], [["integrate", 3, "Symbol('S', commutative=True)"], "Equality(Integral(Mul(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Pow(Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True)), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Integral(Mul(Pow(Pow(cos(Symbol('S', commutative=True)), Symbol('m', commutative=True)), Symbol('S', commutative=True)), Function('\\\\dot{y}')(Symbol('m', commutative=True), Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\pi{(E_{\\lambda})} = e^{E_{\\lambda}} and B{(E_{\\lambda})} = E_{\\lambda}, then obtain (B^{E_{\\lambda}}{(E_{\\lambda})} - e^{E_{\\lambda}})^{E_{\\lambda}} = (E_{\\lambda}^{E_{\\lambda}} - e^{E_{\\lambda}})^{E_{\\lambda}}", "derivation": "\\pi{(E_{\\lambda})} = e^{E_{\\lambda}} and B{(E_{\\lambda})} = E_{\\lambda} and B^{E_{\\lambda}}{(E_{\\lambda})} = E_{\\lambda}^{E_{\\lambda}} and B^{E_{\\lambda}}{(E_{\\lambda})} - \\pi{(E_{\\lambda})} = E_{\\lambda}^{E_{\\lambda}} - \\pi{(E_{\\lambda})} and B^{E_{\\lambda}}{(E_{\\lambda})} - e^{E_{\\lambda}} = E_{\\lambda}^{E_{\\lambda}} - e^{E_{\\lambda}} and (B^{E_{\\lambda}}{(E_{\\lambda})} - e^{E_{\\lambda}})^{E_{\\lambda}} = (E_{\\lambda}^{E_{\\lambda}} - e^{E_{\\lambda}})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('B')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True))"], [["power", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('B')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)))"], "Equality(Add(Pow(Function('B')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)))), Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Function('\\\\pi')(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Function('B')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))), Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["power", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Pow(Function('B')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Pow(Symbol('E_{\\\\lambda}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), exp(Symbol('E_{\\\\lambda}', commutative=True)))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(G)} = \\cos{(\\sin{(G)})} and \\varepsilon{(\\sigma_p)} = e^{\\sigma_p}, then obtain \\varepsilon{(\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sin{(G)})}) = \\varepsilon{(\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} (e^{\\sigma_p} - \\cos{(\\sin{(G)})})", "derivation": "\\mathbf{r}{(G)} = \\cos{(\\sin{(G)})} and \\varepsilon{(\\sigma_p)} = e^{\\sigma_p} and - \\mathbf{r}{(G)} + \\varepsilon{(\\sigma_p)} = - \\mathbf{r}{(G)} + e^{\\sigma_p} and \\varepsilon{(\\sigma_p)} - \\cos{(\\sin{(G)})} = e^{\\sigma_p} - \\cos{(\\sin{(G)})} and \\frac{\\partial}{\\partial \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sin{(G)})}) = \\frac{\\partial}{\\partial \\sigma_p} (e^{\\sigma_p} - \\cos{(\\sin{(G)})}) and \\varepsilon{(\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} (\\varepsilon{(\\sigma_p)} - \\cos{(\\sin{(G)})}) = \\varepsilon{(\\sigma_p)} + \\frac{\\partial}{\\partial \\sigma_p} (e^{\\sigma_p} - \\cos{(\\sin{(G)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('G', commutative=True)), cos(sin(Symbol('G', commutative=True))))"], ["get_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 2, "Function('\\\\mathbf{r}')(Symbol('G', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('G', commutative=True))), Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('G', commutative=True))), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))), Add(exp(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))))"], [["differentiate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"], [["add", 5, "Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))), Add(Function('\\\\varepsilon')(Symbol('\\\\sigma_p', commutative=True)), Derivative(Add(exp(Symbol('\\\\sigma_p', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('G', commutative=True))))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\rho_{f}{(\\chi)} = \\sin{(\\chi)}, then obtain \\frac{d}{d \\chi} (\\rho_{f}^{\\chi}{(\\chi)})^{\\chi} (\\sin^{\\chi}{(\\chi)})^{\\chi} = \\frac{d}{d \\chi} (\\sin^{\\chi}{(\\chi)})^{2 \\chi}", "derivation": "\\rho_{f}{(\\chi)} = \\sin{(\\chi)} and \\rho_{f}^{\\chi}{(\\chi)} = \\sin^{\\chi}{(\\chi)} and (\\rho_{f}^{\\chi}{(\\chi)})^{\\chi} = (\\sin^{\\chi}{(\\chi)})^{\\chi} and (\\rho_{f}^{\\chi}{(\\chi)})^{\\chi} (\\sin^{\\chi}{(\\chi)})^{\\chi} = (\\sin^{\\chi}{(\\chi)})^{2 \\chi} and \\frac{d}{d \\chi} (\\rho_{f}^{\\chi}{(\\chi)})^{\\chi} (\\sin^{\\chi}{(\\chi)})^{\\chi} = \\frac{d}{d \\chi} (\\sin^{\\chi}{(\\chi)})^{2 \\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["power", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["times", 3, "Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Mul(Pow(Pow(Function('\\\\rho_f')(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True)), Mul(Integer(2), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given x{(H)} = \\cos{(e^{H})}, then obtain \\frac{d}{d H} (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + 2 x{(H)} - \\cos{(e^{H})}) dH = \\frac{d}{d H} (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + x{(H)}) dH", "derivation": "x{(H)} = \\cos{(e^{H})} and - H + x{(H)} = - H + \\cos{(e^{H})} and - H + x{(H)} - \\cos{(e^{H})} = - H and - H + 2 x{(H)} - \\cos{(e^{H})} = - H + x{(H)} and \\int (- H + 2 x{(H)} - \\cos{(e^{H})}) dH = \\int (- H + x{(H)}) dH and (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + 2 x{(H)} - \\cos{(e^{H})}) dH = (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + x{(H)}) dH and \\frac{d}{d H} (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + 2 x{(H)} - \\cos{(e^{H})}) dH = \\frac{d}{d H} (- H + x{(H)} - \\cos{(e^{H})}) \\int (- H + x{(H)}) dH", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True))))"], [["minus", 1, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), cos(exp(Symbol('H', commutative=True)))))"], [["minus", 2, "cos(exp(Symbol('H', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Mul(Integer(-1), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True))))"], [["integrate", 4, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))))"], [["differentiate", 6, "Symbol('H', commutative=True)"], "Equality(Derivative(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), Function('x')(Symbol('H', commutative=True))), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('H', commutative=True))))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Function('x')(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}{(\\theta_1,\\lambda)} = \\lambda \\theta_1 and \\operatorname{t_{1}}{(\\theta_1,\\lambda)} = \\frac{\\lambda \\theta_1}{\\hat{x}{(\\theta_1,\\lambda)}}, then obtain 1 - \\int \\hat{x}{(\\theta_1,\\lambda)} d\\lambda = \\operatorname{t_{1}}{(\\theta_1,\\lambda)} - \\int \\hat{x}{(\\theta_1,\\lambda)} d\\lambda", "derivation": "\\hat{x}{(\\theta_1,\\lambda)} = \\lambda \\theta_1 and \\int \\hat{x}{(\\theta_1,\\lambda)} d\\lambda = \\int \\lambda \\theta_1 d\\lambda and 1 = \\frac{\\lambda \\theta_1}{\\hat{x}{(\\theta_1,\\lambda)}} and 1 - \\int \\lambda \\theta_1 d\\lambda = \\frac{\\lambda \\theta_1}{\\hat{x}{(\\theta_1,\\lambda)}} - \\int \\lambda \\theta_1 d\\lambda and \\operatorname{t_{1}}{(\\theta_1,\\lambda)} = \\frac{\\lambda \\theta_1}{\\hat{x}{(\\theta_1,\\lambda)}} and 1 - \\int \\lambda \\theta_1 d\\lambda = \\operatorname{t_{1}}{(\\theta_1,\\lambda)} - \\int \\lambda \\theta_1 d\\lambda and 1 - \\int \\hat{x}{(\\theta_1,\\lambda)} d\\lambda = \\operatorname{t_{1}}{(\\theta_1,\\lambda)} - \\int \\hat{x}{(\\theta_1,\\lambda)} d\\lambda", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["integrate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))))"], [["minus", 3, "Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"], ["renaming_premise", "Equality(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('\\\\lambda', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Integer(1), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))), Add(Function('t_1')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\hat{x}')(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} = \\log{(\\log{(\\hat{\\mathbf{x}})})}, then derive \\int \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} \\log{(\\log{(\\hat{\\mathbf{x}})})} + l - \\operatorname{li}{(\\hat{\\mathbf{x}})}, then obtain \\hat{\\mathbf{x}} \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} + l - \\operatorname{li}{(\\hat{\\mathbf{x}})} = \\int \\log{(\\log{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}}", "derivation": "\\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} = \\log{(\\log{(\\hat{\\mathbf{x}})})} and \\int \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\int \\log{(\\log{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}} and \\int \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} \\log{(\\log{(\\hat{\\mathbf{x}})})} + l - \\operatorname{li}{(\\hat{\\mathbf{x}})} and \\int \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} d\\hat{\\mathbf{x}} = \\hat{\\mathbf{x}} \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} + l - \\operatorname{li}{(\\hat{\\mathbf{x}})} and \\hat{\\mathbf{x}} \\operatorname{C_{1}}{(\\hat{\\mathbf{x}})} + l - \\operatorname{li}{(\\hat{\\mathbf{x}})} = \\int \\log{(\\log{(\\hat{\\mathbf{x}})})} d\\hat{\\mathbf{x}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), log(log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Integral(Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Integral(log(log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Symbol('l', commutative=True), Mul(Integer(-1), li(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('l', commutative=True), Mul(Integer(-1), li(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Add(Mul(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Function('C_1')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Symbol('l', commutative=True), Mul(Integer(-1), li(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)))), Integral(log(log(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(\\hat{\\mathbf{r}},\\theta_2)} = e^{- \\hat{\\mathbf{r}} + \\theta_2}, then obtain 1 = \\mathbb{I}{(\\hat{\\mathbf{r}},\\theta_2)} e^{\\hat{\\mathbf{r}} - \\theta_2}", "derivation": "\\mathbb{I}{(\\hat{\\mathbf{r}},\\theta_2)} = e^{- \\hat{\\mathbf{r}} + \\theta_2} and 1 = \\frac{e^{- \\hat{\\mathbf{r}} + \\theta_2}}{\\mathbb{I}{(\\hat{\\mathbf{r}},\\theta_2)}} and 1 = e^{- \\hat{\\mathbf{r}} + \\theta_2} e^{\\hat{\\mathbf{r}} - \\theta_2} and 1 = \\mathbb{I}{(\\hat{\\mathbf{r}},\\theta_2)} e^{\\hat{\\mathbf{r}} - \\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["divide", 1, "Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), exp(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\theta_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integer(1), Mul(exp(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\theta_2', commutative=True))), exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(1), Mul(Function('\\\\mathbb{I}')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\theta_2', commutative=True)), exp(Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))))))"]]}, {"prompt": "Given h{(\\rho_b,\\Omega)} = \\frac{\\Omega}{\\rho_b}, then derive \\frac{\\partial}{\\partial \\rho_b} h{(\\rho_b,\\Omega)} = - \\frac{\\Omega}{\\rho_b^{2}}, then obtain \\frac{\\partial}{\\partial \\rho_b} \\frac{\\Omega}{\\rho_b} = - \\frac{\\Omega}{\\rho_b^{2}}", "derivation": "h{(\\rho_b,\\Omega)} = \\frac{\\Omega}{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} h{(\\rho_b,\\Omega)} = \\frac{\\partial}{\\partial \\rho_b} \\frac{\\Omega}{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} h{(\\rho_b,\\Omega)} = - \\frac{\\Omega}{\\rho_b^{2}} and \\frac{\\partial}{\\partial \\rho_b} h{(\\rho_b,\\Omega)} = - \\frac{h{(\\rho_b,\\Omega)}}{\\rho_b} and \\frac{\\partial}{\\partial \\rho_b} \\frac{\\Omega}{\\rho_b} = - \\frac{\\Omega}{\\rho_b^{2}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Function('h')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('h')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('h')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Function('h')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Symbol('\\\\rho_b', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\phi)} = e^{\\phi} and \\operatorname{F_{N}}{(\\phi)} = - \\dot{\\mathbf{r}}{(\\phi)}, then obtain 0 = \\operatorname{F_{N}}{(\\phi)} - \\dot{\\mathbf{r}}{(\\phi)} + 2 e^{\\phi}", "derivation": "\\dot{\\mathbf{r}}{(\\phi)} = e^{\\phi} and \\dot{\\mathbf{r}}{(\\phi)} - \\frac{d}{d \\phi} e^{\\phi} = e^{\\phi} - \\frac{d}{d \\phi} e^{\\phi} and 0 = - \\dot{\\mathbf{r}}{(\\phi)} + e^{\\phi} and \\operatorname{F_{N}}{(\\phi)} = - \\dot{\\mathbf{r}}{(\\phi)} and \\operatorname{F_{N}}{(\\phi)} = \\operatorname{F_{N}}{(\\phi)} - \\dot{\\mathbf{r}}{(\\phi)} + e^{\\phi} and 0 = \\operatorname{F_{N}}{(\\phi)} + e^{\\phi} and 0 = \\operatorname{F_{N}}{(\\phi)} - \\dot{\\mathbf{r}}{(\\phi)} + 2 e^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["minus", 1, "Derivative(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))))"], [["minus", 2, "Add(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True))))"], [["add", 3, "Function('F_N')(Symbol('\\\\phi', commutative=True))"], "Equality(Function('F_N')(Symbol('\\\\phi', commutative=True)), Add(Function('F_N')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True))), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integer(0), Add(Function('F_N')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integer(0), Add(Function('F_N')(Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\phi', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\phi', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\sigma_x)} = \\sin{(\\sigma_x)}, then derive \\int \\Psi_{\\lambda}{(\\sigma_x)} d\\sigma_x = v - \\cos{(\\sigma_x)}, then obtain 1 = \\frac{v - \\cos{(\\sigma_x)}}{\\int \\Psi_{\\lambda}{(\\sigma_x)} d\\sigma_x}", "derivation": "\\Psi_{\\lambda}{(\\sigma_x)} = \\sin{(\\sigma_x)} and \\int \\Psi_{\\lambda}{(\\sigma_x)} d\\sigma_x = \\int \\sin{(\\sigma_x)} d\\sigma_x and \\int \\Psi_{\\lambda}{(\\sigma_x)} d\\sigma_x = v - \\cos{(\\sigma_x)} and 1 = \\frac{v - \\cos{(\\sigma_x)}}{\\int \\Psi_{\\lambda}{(\\sigma_x)} d\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), sin(Symbol('\\\\sigma_x', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integral(sin(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('v', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\sigma_x', commutative=True)))), Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\mathbf{f})} = \\log{(\\mathbf{f})}, then derive \\frac{d}{d \\mathbf{f}} \\operatorname{v_{z}}{(\\mathbf{f})} - \\frac{2}{\\mathbf{f}} = - \\frac{1}{\\mathbf{f}}, then obtain \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - \\frac{2}{\\mathbf{f}} = - \\frac{1}{\\mathbf{f}}", "derivation": "\\operatorname{v_{z}}{(\\mathbf{f})} = \\log{(\\mathbf{f})} and \\operatorname{v_{z}}{(\\mathbf{f})} - \\log{(\\mathbf{f})} = 0 and \\operatorname{v_{z}}{(\\mathbf{f})} - 2 \\log{(\\mathbf{f})} = - \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} (\\operatorname{v_{z}}{(\\mathbf{f})} - 2 \\log{(\\mathbf{f})}) = \\frac{d}{d \\mathbf{f}} - \\log{(\\mathbf{f})} and \\frac{d}{d \\mathbf{f}} \\operatorname{v_{z}}{(\\mathbf{f})} - \\frac{2}{\\mathbf{f}} = - \\frac{1}{\\mathbf{f}} and \\frac{d}{d \\mathbf{f}} \\log{(\\mathbf{f})} - \\frac{2}{\\mathbf{f}} = - \\frac{1}{\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\mathbf{f}', commutative=True)), log(Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "log(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), log(Symbol('\\\\mathbf{f}', commutative=True)))), Integer(0))"], [["minus", 2, "log(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Integer(-1), log(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Add(Function('v_z')(Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Integer(2), log(Symbol('\\\\mathbf{f}', commutative=True)))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), log(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('v_z')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Derivative(log(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{f}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(E_{n})} = \\cos{(\\sin{(E_{n})})}, then obtain \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} + \\cos{(\\sin{(E_{n})})} = \\sin{(E_{n})} + 2 \\cos{(\\sin{(E_{n})})}", "derivation": "\\operatorname{P_{e}}{(E_{n})} = \\cos{(\\sin{(E_{n})})} and \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} = \\sin{(E_{n})} + \\cos{(\\sin{(E_{n})})} and 2 \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} = \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} + \\cos{(\\sin{(E_{n})})} and 2 \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} = \\sin{(E_{n})} + 2 \\cos{(\\sin{(E_{n})})} and \\operatorname{P_{e}}{(E_{n})} + \\sin{(E_{n})} + \\cos{(\\sin{(E_{n})})} = \\sin{(E_{n})} + 2 \\cos{(\\sin{(E_{n})})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True))))"], [["add", 1, "sin(Symbol('E_n', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True))), Add(sin(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))))"], [["add", 1, "Add(Function('P_e')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('P_e')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Add(Function('P_e')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('P_e')(Symbol('E_n', commutative=True))), sin(Symbol('E_n', commutative=True))), Add(sin(Symbol('E_n', commutative=True)), Mul(Integer(2), cos(sin(Symbol('E_n', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('P_e')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)), cos(sin(Symbol('E_n', commutative=True)))), Add(sin(Symbol('E_n', commutative=True)), Mul(Integer(2), cos(sin(Symbol('E_n', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(f^{*})} = \\cos{(e^{f^{*}})}, then derive \\int \\operatorname{P_{g}}{(f^{*})} df^{*} = \\rho_b + \\operatorname{Ci}{(e^{f^{*}})}, then derive T + \\operatorname{Ci}{(e^{f^{*}})} = \\rho_b + \\operatorname{Ci}{(e^{f^{*}})}, then obtain \\iint \\operatorname{P_{g}}{(f^{*})} df^{*} dT = \\int (T + \\operatorname{Ci}{(e^{f^{*}})}) dT", "derivation": "\\operatorname{P_{g}}{(f^{*})} = \\cos{(e^{f^{*}})} and \\int \\operatorname{P_{g}}{(f^{*})} df^{*} = \\int \\cos{(e^{f^{*}})} df^{*} and \\int \\operatorname{P_{g}}{(f^{*})} df^{*} = \\rho_b + \\operatorname{Ci}{(e^{f^{*}})} and \\int \\cos{(e^{f^{*}})} df^{*} = \\rho_b + \\operatorname{Ci}{(e^{f^{*}})} and T + \\operatorname{Ci}{(e^{f^{*}})} = \\rho_b + \\operatorname{Ci}{(e^{f^{*}})} and \\int \\operatorname{P_{g}}{(f^{*})} df^{*} = T + \\operatorname{Ci}{(e^{f^{*}})} and \\iint \\operatorname{P_{g}}{(f^{*})} df^{*} dT = \\int (T + \\operatorname{Ci}{(e^{f^{*}})}) dT", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('f^*', commutative=True)), cos(exp(Symbol('f^*', commutative=True))))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(cos(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_g')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(exp(Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\rho_b', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('T', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))), Add(Symbol('\\\\rho_b', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Integral(Function('P_g')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('T', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))))"], [["integrate", 6, "Symbol('T', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), Ci(exp(Symbol('f^*', commutative=True)))), Tuple(Symbol('T', commutative=True))))"]]}, {"prompt": "Given r{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})}, then derive r{(\\mathbf{s})} = \\frac{1}{\\mathbf{s}}, then derive \\int r{(\\mathbf{s})} d\\mathbf{s} = A + \\log{(\\mathbf{s})}, then obtain \\int \\frac{1}{\\mathbf{s}} d\\mathbf{s} = A + \\log{(\\mathbf{s})}", "derivation": "r{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})} and \\int r{(\\mathbf{s})} d\\mathbf{s} = \\int \\frac{d}{d \\mathbf{s}} \\log{(\\mathbf{s})} d\\mathbf{s} and r{(\\mathbf{s})} = \\frac{1}{\\mathbf{s}} and \\int r{(\\mathbf{s})} d\\mathbf{s} = A + \\log{(\\mathbf{s})} and \\int \\frac{1}{\\mathbf{s}} d\\mathbf{s} = A + \\log{(\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('r')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Derivative(log(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_derivatives", 1], "Equality(Function('r')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('A', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('A', commutative=True), log(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given t{(A_{1},Q)} = \\frac{Q}{A_{1}} and \\operatorname{y^{\\prime}}{(A_{1},Q)} = \\frac{Q}{A_{1}} and \\mathbb{I}{(Q,A_{1})} = t{(A_{1},Q)} + 1, then obtain \\frac{Q \\mathbb{I}{(Q,A_{1})}}{A_{1}} = \\frac{Q (1 + \\frac{Q}{A_{1}})}{A_{1}}", "derivation": "t{(A_{1},Q)} = \\frac{Q}{A_{1}} and t{(A_{1},Q)} + 1 = 1 + \\frac{Q}{A_{1}} and \\operatorname{y^{\\prime}}{(A_{1},Q)} = \\frac{Q}{A_{1}} and t{(A_{1},Q)} + 1 = \\operatorname{y^{\\prime}}{(A_{1},Q)} + 1 and \\mathbb{I}{(Q,A_{1})} = t{(A_{1},Q)} + 1 and \\frac{\\mathbb{I}{(Q,A_{1})}}{A_{1}} = \\frac{t{(A_{1},Q)} + 1}{A_{1}} and \\frac{Q (t{(A_{1},Q)} + 1)}{A_{1}} = \\frac{Q (\\operatorname{y^{\\prime}}{(A_{1},Q)} + 1)}{A_{1}} and \\frac{Q (t{(A_{1},Q)} + 1)}{A_{1}} = \\frac{Q (1 + \\frac{Q}{A_{1}})}{A_{1}} and \\frac{Q \\mathbb{I}{(Q,A_{1})}}{A_{1}} = \\frac{Q (1 + \\frac{Q}{A_{1}})}{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Add(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1)), Add(Function('y^{\\\\prime}')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True)), Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1)))"], [["divide", 5, "Symbol('A_1', commutative=True)"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1))))"], [["times", 4, "Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True))"], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Add(Function('y^{\\\\prime}')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Add(Function('t')(Symbol('A_1', commutative=True), Symbol('Q', commutative=True)), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Add(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Function('\\\\mathbb{I}')(Symbol('Q', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True), Add(Integer(1), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given B{(u,\\mathbf{E})} = \\log{(\\frac{\\mathbf{E}}{u})}, then derive \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} = - \\frac{1}{u}, then obtain \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} \\int - \\frac{1}{u} du = - \\frac{\\int - \\frac{1}{u} du}{u}", "derivation": "B{(u,\\mathbf{E})} = \\log{(\\frac{\\mathbf{E}}{u})} and \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} = \\frac{\\partial}{\\partial u} \\log{(\\frac{\\mathbf{E}}{u})} and \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} = - \\frac{1}{u} and \\int \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} du = \\int - \\frac{1}{u} du and \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} \\int \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} du = - \\frac{\\int \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} du}{u} and \\frac{\\partial}{\\partial u} B{(u,\\mathbf{E})} \\int - \\frac{1}{u} du = - \\frac{\\int - \\frac{1}{u} du}{u}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), log(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('\\\\mathbf{E}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1)))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["integrate", 3, "Symbol('u', commutative=True)"], "Equality(Integral(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))))"], [["times", 3, "Integral(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))"], "Equality(Mul(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Derivative(Function('B')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integral(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)), Integral(Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A,\\mathbf{H})} = \\int (A + \\mathbf{H}) d\\mathbf{H}, then obtain \\frac{\\partial^{2}}{\\partial A^{2}} \\operatorname{v_{y}}{(A,\\mathbf{H})} = \\frac{\\partial^{2}}{\\partial A^{2}} (A \\mathbf{H} + \\frac{\\mathbf{H}^{2}}{2} + \\phi_2)", "derivation": "\\operatorname{v_{y}}{(A,\\mathbf{H})} = \\int (A + \\mathbf{H}) d\\mathbf{H} and \\frac{\\partial}{\\partial A} \\operatorname{v_{y}}{(A,\\mathbf{H})} = \\frac{\\partial}{\\partial A} \\int (A + \\mathbf{H}) d\\mathbf{H} and \\frac{\\partial^{2}}{\\partial A^{2}} \\operatorname{v_{y}}{(A,\\mathbf{H})} = \\frac{\\partial^{2}}{\\partial A^{2}} \\int (A + \\mathbf{H}) d\\mathbf{H} and \\frac{\\partial^{2}}{\\partial A^{2}} \\operatorname{v_{y}}{(A,\\mathbf{H})} = \\frac{\\partial^{2}}{\\partial A^{2}} (A \\mathbf{H} + \\frac{\\mathbf{H}^{2}}{2} + \\phi_2)", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(Integral(Add(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(2))))"], [["evaluate_integrals", 3], "Equality(Derivative(Function('v_y')(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(2))))"]]}, {"prompt": "Given Z{(\\dot{y})} = \\cos{(\\dot{y})} and \\mathbf{J}_f{(f^{\\prime})} = \\sin{(\\cos{(f^{\\prime})})}, then obtain \\mathbf{J}_f{(f^{\\prime})} + 1 + \\frac{1}{\\cos{(\\dot{y})}} = \\sin{(\\cos{(f^{\\prime})})} + 1 + \\frac{1}{\\cos{(\\dot{y})}}", "derivation": "Z{(\\dot{y})} = \\cos{(\\dot{y})} and \\mathbf{J}_f{(f^{\\prime})} = \\sin{(\\cos{(f^{\\prime})})} and \\mathbf{J}_f{(f^{\\prime})} + \\frac{1}{Z{(\\dot{y})}} = \\sin{(\\cos{(f^{\\prime})})} + \\frac{1}{Z{(\\dot{y})}} and \\mathbf{J}_f{(f^{\\prime})} + \\frac{\\cos{(\\dot{y})}}{Z{(\\dot{y})}} + \\frac{1}{Z{(\\dot{y})}} = \\sin{(\\cos{(f^{\\prime})})} + \\frac{\\cos{(\\dot{y})}}{Z{(\\dot{y})}} + \\frac{1}{Z{(\\dot{y})}} and \\mathbf{J}_f{(f^{\\prime})} + 1 + \\frac{1}{Z{(\\dot{y})}} = \\sin{(\\cos{(f^{\\prime})})} + 1 + \\frac{1}{Z{(\\dot{y})}} and \\mathbf{J}_f{(f^{\\prime})} + 1 + \\frac{1}{\\cos{(\\dot{y})}} = \\sin{(\\cos{(f^{\\prime})})} + 1 + \\frac{1}{\\cos{(\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), cos(Symbol('\\\\dot{y}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), sin(cos(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Add(sin(cos(Symbol('f^{\\\\prime}', commutative=True))), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["add", 3, "Mul(Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{y}', commutative=True))), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Add(sin(cos(Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1)), cos(Symbol('\\\\dot{y}', commutative=True))), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Integer(1), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Add(sin(cos(Symbol('f^{\\\\prime}', commutative=True))), Integer(1), Pow(Function('Z')(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f^{\\\\prime}', commutative=True)), Integer(1), Pow(cos(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))), Add(sin(cos(Symbol('f^{\\\\prime}', commutative=True))), Integer(1), Pow(cos(Symbol('\\\\dot{y}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{S}{(A_{2},\\hat{H}_l)} = \\cos{(\\hat{H}_l^{A_{2}})}, then obtain \\mathbf{S}{(A_{2},\\hat{H}_l)} - 2 \\cos{(\\hat{H}_l^{A_{2}})} = - \\mathbf{S}{(A_{2},\\hat{H}_l)}", "derivation": "\\mathbf{S}{(A_{2},\\hat{H}_l)} = \\cos{(\\hat{H}_l^{A_{2}})} and \\mathbf{S}{(A_{2},\\hat{H}_l)} + \\cos{(\\hat{H}_l^{A_{2}})} = 2 \\cos{(\\hat{H}_l^{A_{2}})} and \\cos{(\\hat{H}_l^{A_{2}})} = - \\mathbf{S}{(A_{2},\\hat{H}_l)} + 2 \\cos{(\\hat{H}_l^{A_{2}})} and - \\cos{(\\hat{H}_l^{A_{2}})} = - \\mathbf{S}{(A_{2},\\hat{H}_l)} and \\mathbf{S}{(A_{2},\\hat{H}_l)} - 2 \\cos{(\\hat{H}_l^{A_{2}})} = - \\mathbf{S}{(A_{2},\\hat{H}_l)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True))))"], [["add", 1, "cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True)))), Mul(Integer(2), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True)))))"], [["minus", 2, "Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(2), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True))))))"], [["minus", 3, "Mul(Integer(2), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True))))"], "Equality(Mul(Integer(-1), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True)))), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Integer(2), cos(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('A_2', commutative=True))))), Mul(Integer(-1), Function('\\\\mathbf{S}')(Symbol('A_2', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\lambda)} = \\log{(\\lambda)}, then obtain \\log{(\\lambda)}^{2} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\operatorname{A_{x}}{(\\lambda)} \\log{(\\lambda)} = \\log{(\\lambda)}^{2} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\log{(\\lambda)}^{2}", "derivation": "\\operatorname{A_{x}}{(\\lambda)} = \\log{(\\lambda)} and \\operatorname{A_{x}}{(\\lambda)} \\log{(\\lambda)} = \\log{(\\lambda)}^{2} and \\mathbf{r} \\operatorname{A_{x}}{(\\lambda)} \\log{(\\lambda)} = \\mathbf{r} \\log{(\\lambda)}^{2} and \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\operatorname{A_{x}}{(\\lambda)} \\log{(\\lambda)} = \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\log{(\\lambda)}^{2} and \\log{(\\lambda)}^{2} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\operatorname{A_{x}}{(\\lambda)} \\log{(\\lambda)} = \\log{(\\lambda)}^{2} \\frac{\\partial}{\\partial \\mathbf{r}} \\mathbf{r} \\log{(\\lambda)}^{2}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True)))"], [["times", 1, "log(Symbol('\\\\lambda', commutative=True))"], "Equality(Mul(Function('A_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2)))"], [["times", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('A_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('A_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["times", 4, "Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))"], "Equality(Mul(Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2)), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Function('A_x')(Symbol('\\\\lambda', commutative=True)), log(Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))), Mul(Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2)), Derivative(Mul(Symbol('\\\\mathbf{r}', commutative=True), Pow(log(Symbol('\\\\lambda', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(C)} = \\cos{(C)}, then obtain \\frac{\\operatorname{v_{t}}^{C}{(C)} \\int \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} dC}{C} = \\frac{\\operatorname{v_{t}}^{C}{(C)} \\int \\frac{e^{\\cos{(C)}}}{C} dC}{C}", "derivation": "\\operatorname{v_{t}}{(C)} = \\cos{(C)} and e^{\\operatorname{v_{t}}{(C)}} = e^{\\cos{(C)}} and \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} = \\frac{e^{\\cos{(C)}}}{C} and \\int \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} dC = \\int \\frac{e^{\\cos{(C)}}}{C} dC and \\cos^{C}{(C)} \\int \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} dC = \\cos^{C}{(C)} \\int \\frac{e^{\\cos{(C)}}}{C} dC and \\frac{\\cos^{C}{(C)} \\int \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} dC}{C} = \\frac{\\cos^{C}{(C)} \\int \\frac{e^{\\cos{(C)}}}{C} dC}{C} and \\frac{\\operatorname{v_{t}}^{C}{(C)} \\int \\frac{e^{\\operatorname{v_{t}}{(C)}}}{C} dC}{C} = \\frac{\\operatorname{v_{t}}^{C}{(C)} \\int \\frac{e^{\\cos{(C)}}}{C} dC}{C}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["exp", 1], "Equality(exp(Function('v_t')(Symbol('C', commutative=True))), exp(cos(Symbol('C', commutative=True))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Function('v_t')(Symbol('C', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Function('v_t')(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["times", 4, "Pow(cos(Symbol('C', commutative=True)), Symbol('C', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Function('v_t')(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(cos(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))))"], [["times", 5, "Pow(Symbol('C', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(cos(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Function('v_t')(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(cos(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Function('v_t')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(Function('v_t')(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Function('v_t')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Integral(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), exp(cos(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{D}{(s,A)} = A s and \\hat{\\mathbf{x}}{(s,A)} = A s, then obtain \\frac{\\partial^{2}}{\\partial s\\partial A} (A - \\hat{\\mathbf{x}}^{A}{(s,A)} + \\mathbf{D}^{A}{(s,A)}) = \\frac{d^{2}}{d sd A} A", "derivation": "\\mathbf{D}{(s,A)} = A s and \\mathbf{D}^{A}{(s,A)} = (A s)^{A} and - (A s)^{A} + \\mathbf{D}^{A}{(s,A)} = 0 and A - (A s)^{A} + \\mathbf{D}^{A}{(s,A)} = A and \\frac{\\partial}{\\partial A} (A - (A s)^{A} + \\mathbf{D}^{A}{(s,A)}) = \\frac{d}{d A} A and \\hat{\\mathbf{x}}{(s,A)} = A s and \\frac{\\partial^{2}}{\\partial s\\partial A} (A - (A s)^{A} + \\mathbf{D}^{A}{(s,A)}) = \\frac{d^{2}}{d sd A} A and \\frac{\\partial^{2}}{\\partial s\\partial A} (A - \\hat{\\mathbf{x}}^{A}{(s,A)} + \\mathbf{D}^{A}{(s,A)}) = \\frac{d^{2}}{d sd A} A", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)))"], [["power", 1, "Symbol('A', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True)))"], [["minus", 2, "Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Integer(0))"], [["add", 3, "Symbol('A', commutative=True)"], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Symbol('A', commutative=True))"], [["differentiate", 4, "Symbol('A', commutative=True)"], "Equality(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)))"], [["differentiate", 5, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Pow(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Symbol('A', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 6], "Equality(Derivative(Add(Symbol('A', commutative=True), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Pow(Function('\\\\mathbf{D}')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Symbol('A', commutative=True), Tuple(Symbol('A', commutative=True), Integer(1)), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho{(A_{2})} = \\sin{(A_{2})} and \\bar{\\h}{(A_{2})} = \\sin{(A_{2})}, then obtain - \\bar{\\h}{(A_{2})} + \\rho{(A_{2})} + \\frac{d}{d A_{2}} - \\bar{\\h}{(A_{2})} = - \\bar{\\h}{(A_{2})} + \\rho{(A_{2})} + \\frac{d}{d A_{2}} - \\rho{(A_{2})}", "derivation": "\\rho{(A_{2})} = \\sin{(A_{2})} and 0 = - \\rho{(A_{2})} + \\sin{(A_{2})} and - \\sin{(A_{2})} = - \\rho{(A_{2})} and \\frac{d}{d A_{2}} - \\sin{(A_{2})} = \\frac{d}{d A_{2}} - \\rho{(A_{2})} and \\bar{\\h}{(A_{2})} = \\sin{(A_{2})} and \\rho{(A_{2})} - \\sin{(A_{2})} + \\frac{d}{d A_{2}} - \\sin{(A_{2})} = \\rho{(A_{2})} - \\sin{(A_{2})} + \\frac{d}{d A_{2}} - \\rho{(A_{2})} and - \\bar{\\h}{(A_{2})} + \\rho{(A_{2})} + \\frac{d}{d A_{2}} - \\bar{\\h}{(A_{2})} = - \\bar{\\h}{(A_{2})} + \\rho{(A_{2})} + \\frac{d}{d A_{2}} - \\rho{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["minus", 1, "Function('\\\\rho')(Symbol('A_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))), sin(Symbol('A_2', commutative=True))))"], [["minus", 2, "sin(Symbol('A_2', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["minus", 4, "Add(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))), sin(Symbol('A_2', commutative=True)))"], "Equality(Add(Function('\\\\rho')(Symbol('A_2', commutative=True)), Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Derivative(Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Function('\\\\rho')(Symbol('A_2', commutative=True)), Mul(Integer(-1), sin(Symbol('A_2', commutative=True))), Derivative(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_2', commutative=True))), Function('\\\\rho')(Symbol('A_2', commutative=True)), Derivative(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('A_2', commutative=True))), Function('\\\\rho')(Symbol('A_2', commutative=True)), Derivative(Mul(Integer(-1), Function('\\\\rho')(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given v{(J_{\\varepsilon})} = \\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon} and \\varepsilon{(J_{\\varepsilon})} = \\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon}, then obtain \\int v^{J_{\\varepsilon}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (\\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon})^{J_{\\varepsilon}} dJ_{\\varepsilon}", "derivation": "v{(J_{\\varepsilon})} = \\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon} and \\varepsilon{(J_{\\varepsilon})} = \\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon} and \\varepsilon{(J_{\\varepsilon})} = v{(J_{\\varepsilon})} and \\varepsilon^{J_{\\varepsilon}}{(J_{\\varepsilon})} = (\\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon})^{J_{\\varepsilon}} and \\int \\varepsilon^{J_{\\varepsilon}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (\\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon})^{J_{\\varepsilon}} dJ_{\\varepsilon} and \\int v^{J_{\\varepsilon}}{(J_{\\varepsilon})} dJ_{\\varepsilon} = \\int (\\int \\log{(J_{\\varepsilon})} dJ_{\\varepsilon})^{J_{\\varepsilon}} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Function('v')(Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["power", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Function('\\\\varepsilon')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Pow(Function('v')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Integral(log(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(U)} = \\sin{(U)}, then obtain - 4 U + 3 \\mathbf{B}{(U)} + \\sin{(U)} = - 4 U + 2 \\mathbf{B}{(U)} + 2 \\sin{(U)}", "derivation": "\\mathbf{B}{(U)} = \\sin{(U)} and 2 \\mathbf{B}{(U)} = \\mathbf{B}{(U)} + \\sin{(U)} and - U + 2 \\mathbf{B}{(U)} = - U + \\mathbf{B}{(U)} + \\sin{(U)} and - 2 U + 2 \\mathbf{B}{(U)} = - 2 U + \\mathbf{B}{(U)} + \\sin{(U)} and - 4 U + 3 \\mathbf{B}{(U)} + \\sin{(U)} = - 4 U + 2 \\mathbf{B}{(U)} + 2 \\sin{(U)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{B}')(Symbol('U', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('U', commutative=True))), Add(Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))))"], [["minus", 2, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True))))"], [["add", 4, "Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Function('\\\\mathbf{B}')(Symbol('U', commutative=True)), sin(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integer(4), Symbol('U', commutative=True)), Mul(Integer(3), Function('\\\\mathbf{B}')(Symbol('U', commutative=True))), sin(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Integer(4), Symbol('U', commutative=True)), Mul(Integer(2), Function('\\\\mathbf{B}')(Symbol('U', commutative=True))), Mul(Integer(2), sin(Symbol('U', commutative=True)))))"]]}, {"prompt": "Given v{(l)} = \\log{(l)}, then derive \\int v{(l)} dl = \\pi + l \\log{(l)} - l, then obtain (\\pi + l \\log{(l)} - l) \\iint (\\pi + l \\log{(l)} - l + \\int v{(l)} dl) d\\pi dl = (\\pi + l \\log{(l)} - l) \\iint (2 \\pi + l v{(l)} + l \\log{(l)} - 2 l) d\\pi dl", "derivation": "v{(l)} = \\log{(l)} and \\int v{(l)} dl = \\int \\log{(l)} dl and \\int v{(l)} dl = \\pi + l \\log{(l)} - l and \\int v{(l)} dl = \\pi + l v{(l)} - l and \\pi + l \\log{(l)} - l + \\int v{(l)} dl = 2 \\pi + l v{(l)} + l \\log{(l)} - 2 l and \\int (\\pi + l \\log{(l)} - l + \\int v{(l)} dl) d\\pi = \\int (2 \\pi + l v{(l)} + l \\log{(l)} - 2 l) d\\pi and \\iint (\\pi + l \\log{(l)} - l + \\int v{(l)} dl) d\\pi dl = \\iint (2 \\pi + l v{(l)} + l \\log{(l)} - 2 l) d\\pi dl and (\\pi + l \\log{(l)} - l) \\iint (\\pi + l \\log{(l)} - l + \\int v{(l)} dl) d\\pi dl = (\\pi + l \\log{(l)} - l) \\iint (2 \\pi + l v{(l)} + l \\log{(l)} - 2 l) d\\pi dl", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), Function('v')(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True))))"], [["add", 4, "Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Symbol('l', commutative=True), Function('v')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Symbol('l', commutative=True), Function('v')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 6, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Symbol('l', commutative=True), Function('v')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 7, "Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)), Integral(Function('v')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('l', commutative=True)))), Mul(Add(Symbol('\\\\pi', commutative=True), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Symbol('l', commutative=True), Function('v')(Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), log(Symbol('l', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given k{(I)} = \\sin{(\\cos{(I)})}, then obtain \\frac{d^{3}}{d I^{3}} k{(I)} = \\frac{d^{3}}{d I^{3}} \\sin{(\\cos{(I)})}", "derivation": "k{(I)} = \\sin{(\\cos{(I)})} and \\frac{d}{d I} k{(I)} = \\frac{d}{d I} \\sin{(\\cos{(I)})} and \\frac{d^{2}}{d I^{2}} k{(I)} = \\frac{d^{2}}{d I^{2}} \\sin{(\\cos{(I)})} and \\frac{d^{3}}{d I^{3}} k{(I)} = \\frac{d^{3}}{d I^{3}} \\sin{(\\cos{(I)})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('I', commutative=True)), sin(cos(Symbol('I', commutative=True))))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(2))), Derivative(sin(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(2))))"], [["differentiate", 3, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(3))), Derivative(sin(cos(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(3))))"]]}, {"prompt": "Given a{(\\tilde{g}^*,Z)} = \\log{(Z \\tilde{g}^*)}, then obtain \\frac{\\partial}{\\partial Z} \\log{(\\int a{(\\tilde{g}^*,Z)} d\\tilde{g}^*)} = \\frac{\\partial}{\\partial Z} \\log{(\\int \\log{(e^{a{(\\tilde{g}^*,Z)}})} d\\tilde{g}^*)}", "derivation": "a{(\\tilde{g}^*,Z)} = \\log{(Z \\tilde{g}^*)} and e^{a{(\\tilde{g}^*,Z)}} = Z \\tilde{g}^* and a{(\\tilde{g}^*,Z)} = \\log{(e^{a{(\\tilde{g}^*,Z)}})} and \\int a{(\\tilde{g}^*,Z)} d\\tilde{g}^* = \\int \\log{(e^{a{(\\tilde{g}^*,Z)}})} d\\tilde{g}^* and \\log{(\\int a{(\\tilde{g}^*,Z)} d\\tilde{g}^*)} = \\log{(\\int \\log{(e^{a{(\\tilde{g}^*,Z)}})} d\\tilde{g}^*)} and \\frac{\\partial}{\\partial Z} \\log{(\\int a{(\\tilde{g}^*,Z)} d\\tilde{g}^*)} = \\frac{\\partial}{\\partial Z} \\log{(\\int \\log{(e^{a{(\\tilde{g}^*,Z)}})} d\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)), log(Mul(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["exp", 1], "Equality(exp(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('Z', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)), log(exp(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)))))"], [["integrate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(log(exp(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["log", 4], "Equality(log(Integral(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), log(Integral(log(exp(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(log(Integral(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(log(Integral(log(exp(Function('a')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('Z', commutative=True)))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(P_{g})} = \\log{(P_{g})}, then obtain 0 = (\\frac{\\log{(P_{g})}}{\\hat{X}{(P_{g})}})^{P_{g}} - 1", "derivation": "\\hat{X}{(P_{g})} = \\log{(P_{g})} and 1 = \\frac{\\log{(P_{g})}}{\\hat{X}{(P_{g})}} and 1 = (\\frac{\\log{(P_{g})}}{\\hat{X}{(P_{g})}})^{P_{g}} and 0 = (\\frac{\\log{(P_{g})}}{\\hat{X}{(P_{g})}})^{P_{g}} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('P_g', commutative=True)), log(Symbol('P_g', commutative=True)))"], [["divide", 1, "Function('\\\\hat{X}')(Symbol('P_g', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('P_g', commutative=True)), Integer(-1)), log(Symbol('P_g', commutative=True))))"], [["power", 2, "Symbol('P_g', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('\\\\hat{X}')(Symbol('P_g', commutative=True)), Integer(-1)), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Integer(0), Add(Pow(Mul(Pow(Function('\\\\hat{X}')(Symbol('P_g', commutative=True)), Integer(-1)), log(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(I)} = \\cos{(I)}, then obtain (- I + \\operatorname{A_{x}}{(I)})^{I} - \\cos{(I)} = (- I + \\cos{(I)})^{I} - \\cos{(I)}", "derivation": "\\operatorname{A_{x}}{(I)} = \\cos{(I)} and - I + \\operatorname{A_{x}}{(I)} = - I + \\cos{(I)} and (- I + \\operatorname{A_{x}}{(I)})^{I} = (- I + \\cos{(I)})^{I} and (- I + \\operatorname{A_{x}}{(I)})^{I} - \\cos{(I)} = (- I + \\cos{(I)})^{I} - \\cos{(I)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["minus", 1, "Symbol('I', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('A_x')(Symbol('I', commutative=True))), Add(Mul(Integer(-1), Symbol('I', commutative=True)), cos(Symbol('I', commutative=True))))"], [["power", 2, "Symbol('I', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('A_x')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), cos(Symbol('I', commutative=True))), Symbol('I', commutative=True)))"], [["minus", 3, "cos(Symbol('I', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), Function('A_x')(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Symbol('I', commutative=True)), cos(Symbol('I', commutative=True))), Symbol('I', commutative=True)), Mul(Integer(-1), cos(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then derive \\frac{d}{d L_{\\varepsilon}} \\hat{x}_0{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain \\hat{x}_0{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}}", "derivation": "\\hat{x}_0{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\hat{x}_0{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\hat{x}_0{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} e^{L_{\\varepsilon}} = e^{L_{\\varepsilon}} and \\frac{d}{d L_{\\varepsilon}} \\hat{x}_0{(L_{\\varepsilon})} = \\hat{x}_0{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} \\hat{x}_0{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}} and \\hat{x}_0{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} e^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\hat{x}_0')(Symbol('L_{\\\\varepsilon}', commutative=True)), Derivative(exp(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\theta_{2}{(A)} = \\cos{(A)}, then derive \\int \\theta_{2}{(A)} dA = A_{1} + \\sin{(A)}, then obtain A_{1} (A_{1} + \\sin{(A)}) = A_{1} (\\sigma_x + \\sin{(A)})", "derivation": "\\theta_{2}{(A)} = \\cos{(A)} and \\int \\theta_{2}{(A)} dA = \\int \\cos{(A)} dA and \\int \\theta_{2}{(A)} dA = A_{1} + \\sin{(A)} and A_{1} \\int \\theta_{2}{(A)} dA = A_{1} \\int \\cos{(A)} dA and A_{1} (A_{1} + \\sin{(A)}) = A_{1} \\int \\cos{(A)} dA and A_{1} (A_{1} + \\sin{(A)}) = A_{1} (\\sigma_x + \\sin{(A)})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('A_1', commutative=True), sin(Symbol('A', commutative=True))))"], [["times", 2, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Integral(Function('\\\\theta_2')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))), Mul(Symbol('A_1', commutative=True), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), sin(Symbol('A', commutative=True)))), Mul(Symbol('A_1', commutative=True), Integral(cos(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), sin(Symbol('A', commutative=True)))), Mul(Symbol('A_1', commutative=True), Add(Symbol('\\\\sigma_x', commutative=True), sin(Symbol('A', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(v)} = \\cos{(v)}, then derive \\frac{d}{d v} \\operatorname{E_{x}}{(v)} = - \\sin{(v)}, then obtain \\log{(\\frac{d}{d v} \\operatorname{E_{x}}{(v)})} = \\log{(- \\sin{(v)})}", "derivation": "\\operatorname{E_{x}}{(v)} = \\cos{(v)} and \\frac{d}{d v} \\operatorname{E_{x}}{(v)} = \\frac{d}{d v} \\cos{(v)} and \\frac{d}{d v} \\operatorname{E_{x}}{(v)} = - \\sin{(v)} and \\log{(\\frac{d}{d v} \\operatorname{E_{x}}{(v)})} = \\log{(- \\sin{(v)})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(cos(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('v', commutative=True))))"], [["log", 3], "Equality(log(Derivative(Function('E_x')(Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), log(Mul(Integer(-1), sin(Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(a^{\\dagger},\\psi^*)} = \\psi^* + a^{\\dagger} and \\operatorname{v_{z}}{(a^{\\dagger})} = \\int 1 da^{\\dagger}, then obtain - (\\Psi_{\\lambda} y)^{M} + \\operatorname{v_{z}}{(a^{\\dagger})} = - (\\Psi_{\\lambda} y)^{M} + \\int \\frac{\\dot{z}{(a^{\\dagger},\\psi^*)}}{\\psi^* + a^{\\dagger}} da^{\\dagger}", "derivation": "\\dot{z}{(a^{\\dagger},\\psi^*)} = \\psi^* + a^{\\dagger} and \\frac{\\dot{z}{(a^{\\dagger},\\psi^*)}}{\\psi^* + a^{\\dagger}} = 1 and \\int \\frac{\\dot{z}{(a^{\\dagger},\\psi^*)}}{\\psi^* + a^{\\dagger}} da^{\\dagger} = \\int 1 da^{\\dagger} and \\operatorname{v_{z}}{(a^{\\dagger})} = \\int 1 da^{\\dagger} and - (\\Psi_{\\lambda} y)^{M} + \\operatorname{v_{z}}{(a^{\\dagger})} = - (\\Psi_{\\lambda} y)^{M} + \\int 1 da^{\\dagger} and - (\\Psi_{\\lambda} y)^{M} + \\operatorname{v_{z}}{(a^{\\dagger})} = - (\\Psi_{\\lambda} y)^{M} + \\int \\frac{\\dot{z}{(a^{\\dagger},\\psi^*)}}{\\psi^* + a^{\\dagger}} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\psi^*', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\psi^*', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\psi^*', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('v_z')(Symbol('a^{\\\\dagger}', commutative=True)), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 4, "Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('M', commutative=True))), Function('v_z')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('M', commutative=True))), Integral(Integer(1), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('M', commutative=True))), Function('v_z')(Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Integer(-1), Pow(Mul(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('y', commutative=True)), Symbol('M', commutative=True))), Integral(Mul(Pow(Add(Symbol('\\\\psi^*', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Integer(-1)), Function('\\\\dot{z}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(l,\\Omega)} = \\Omega + l, then obtain \\frac{\\partial}{\\partial l} \\ddot{x}{(l,\\Omega)} - 2 = -1", "derivation": "\\ddot{x}{(l,\\Omega)} = \\Omega + l and - \\Omega - l + \\ddot{x}{(l,\\Omega)} = 0 and - \\Omega - 2 l + \\ddot{x}{(l,\\Omega)} = - l and \\frac{\\partial}{\\partial l} (- \\Omega - 2 l + \\ddot{x}{(l,\\Omega)}) = \\frac{d}{d l} - l and \\frac{\\partial}{\\partial l} \\ddot{x}{(l,\\Omega)} - 2 = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\Omega', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)), Function('\\\\ddot{x}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Integer(0))"], [["add", 2, "Mul(Integer(-1), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True)), Function('\\\\ddot{x}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('l', commutative=True)))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('l', commutative=True)), Function('\\\\ddot{x}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\ddot{x}')(Symbol('l', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-2)), Integer(-1))"]]}, {"prompt": "Given \\mathbf{B}{(u)} = e^{u} and h{(u)} = e^{- u}, then obtain \\cos{(\\frac{\\mathbf{B}{(u)} h{(u)}}{\\mathbf{B}{(u)} e^{- u} + e^{- u}})} = \\cos{(\\frac{1}{\\mathbf{B}{(u)} e^{- u} + e^{- u}})}", "derivation": "\\mathbf{B}{(u)} = e^{u} and \\mathbf{B}{(u)} e^{- u} = 1 and h{(u)} = e^{- u} and \\mathbf{B}{(u)} h{(u)} = 1 and \\frac{\\mathbf{B}{(u)} h{(u)}}{\\mathbf{B}{(u)} e^{- u} + e^{- u}} = \\frac{1}{\\mathbf{B}{(u)} e^{- u} + e^{- u}} and \\cos{(\\frac{\\mathbf{B}{(u)} h{(u)}}{\\mathbf{B}{(u)} e^{- u} + e^{- u}})} = \\cos{(\\frac{1}{\\mathbf{B}{(u)} e^{- u} + e^{- u}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["divide", 1, "exp(Symbol('u', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('h')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), Function('h')(Symbol('u', commutative=True))), Integer(1))"], [["divide", 4, "Add(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Mul(Integer(-1), Symbol('u', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), Function('h')(Symbol('u', commutative=True))), Pow(Add(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(-1)))"], [["cos", 5], "Equality(cos(Mul(Pow(Add(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(-1)), Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), Function('h')(Symbol('u', commutative=True)))), cos(Pow(Add(Mul(Function('\\\\mathbf{B}')(Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), exp(Mul(Integer(-1), Symbol('u', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(G)} = \\log{(G)} and \\operatorname{c_{0}}{(G)} = \\operatorname{A_{1}}{(G)} \\log{(G)}, then obtain \\operatorname{A_{1}}^{2}{(G)} = \\log{(G)}^{2}", "derivation": "\\operatorname{A_{1}}{(G)} = \\log{(G)} and \\operatorname{A_{1}}{(G)} \\log{(G)} = \\log{(G)}^{2} and \\operatorname{c_{0}}{(G)} = \\operatorname{A_{1}}{(G)} \\log{(G)} and \\operatorname{c_{0}}{(G)} = \\log{(G)}^{2} and \\operatorname{c_{0}}{(G)} = \\operatorname{A_{1}}^{2}{(G)} and \\operatorname{A_{1}}^{2}{(G)} = \\log{(G)}^{2}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True)))"], [["times", 1, "log(Symbol('G', commutative=True))"], "Equality(Mul(Function('A_1')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True))), Pow(log(Symbol('G', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('c_0')(Symbol('G', commutative=True)), Mul(Function('A_1')(Symbol('G', commutative=True)), log(Symbol('G', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('c_0')(Symbol('G', commutative=True)), Pow(log(Symbol('G', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('c_0')(Symbol('G', commutative=True)), Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Function('A_1')(Symbol('G', commutative=True)), Integer(2)), Pow(log(Symbol('G', commutative=True)), Integer(2)))"]]}, {"prompt": "Given \\hat{p}_0{(\\sigma_p,x)} = \\sigma_p \\sin{(x)}, then obtain - \\hat{p}_0{(\\sigma_p,x)} = - \\hat{p}_0{(\\sigma_p,x)} + \\iint \\sigma_p \\sin{(x)} d\\sigma_p dx - \\iint \\hat{p}_0{(\\sigma_p,x)} d\\sigma_p dx", "derivation": "\\hat{p}_0{(\\sigma_p,x)} = \\sigma_p \\sin{(x)} and \\int \\hat{p}_0{(\\sigma_p,x)} d\\sigma_p = \\int \\sigma_p \\sin{(x)} d\\sigma_p and \\iint \\hat{p}_0{(\\sigma_p,x)} d\\sigma_p dx = \\iint \\sigma_p \\sin{(x)} d\\sigma_p dx and - \\hat{p}_0{(\\sigma_p,x)} + \\iint \\hat{p}_0{(\\sigma_p,x)} d\\sigma_p dx = - \\hat{p}_0{(\\sigma_p,x)} + \\iint \\sigma_p \\sin{(x)} d\\sigma_p dx and - \\hat{p}_0{(\\sigma_p,x)} = - \\hat{p}_0{(\\sigma_p,x)} + \\iint \\sigma_p \\sin{(x)} d\\sigma_p dx - \\iint \\hat{p}_0{(\\sigma_p,x)} d\\sigma_p dx", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["minus", 3, "Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True))), Integral(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True)))))"], [["minus", 4, "Integral(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True))), Integral(Mul(Symbol('\\\\sigma_p', commutative=True), sin(Symbol('x', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{p}_0')(Symbol('\\\\sigma_p', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(P_{e},\\omega)} = \\frac{\\partial}{\\partial \\omega} P_{e}^{\\omega}, then derive \\operatorname{C_{1}}{(P_{e},\\omega)} = P_{e}^{\\omega} \\log{(P_{e})}, then derive \\frac{\\partial}{\\partial \\omega} \\operatorname{C_{1}}{(P_{e},\\omega)} = P_{e}^{\\omega} \\log{(P_{e})}^{2}, then obtain \\frac{\\partial}{\\partial \\omega} \\operatorname{C_{1}}{(P_{e},\\omega)} = \\log{(P_{e})} \\frac{\\partial}{\\partial \\omega} P_{e}^{\\omega}", "derivation": "\\operatorname{C_{1}}{(P_{e},\\omega)} = \\frac{\\partial}{\\partial \\omega} P_{e}^{\\omega} and \\operatorname{C_{1}}{(P_{e},\\omega)} = P_{e}^{\\omega} \\log{(P_{e})} and \\frac{\\partial}{\\partial \\omega} P_{e}^{\\omega} = P_{e}^{\\omega} \\log{(P_{e})} and \\frac{\\partial}{\\partial \\omega} \\operatorname{C_{1}}{(P_{e},\\omega)} = \\frac{\\partial^{2}}{\\partial \\omega^{2}} P_{e}^{\\omega} and \\frac{\\partial}{\\partial \\omega} \\operatorname{C_{1}}{(P_{e},\\omega)} = P_{e}^{\\omega} \\log{(P_{e})}^{2} and \\frac{\\partial}{\\partial \\omega} \\operatorname{C_{1}}{(P_{e},\\omega)} = \\log{(P_{e})} \\frac{\\partial}{\\partial \\omega} P_{e}^{\\omega}", "srepr_derivation": [["get_premise", "Equality(Function('C_1')(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('C_1')(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), log(Symbol('P_e', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), log(Symbol('P_e', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(2))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('C_1')(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(log(Symbol('P_e', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('C_1')(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Mul(log(Symbol('P_e', commutative=True)), Derivative(Pow(Symbol('P_e', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(\\rho_f,T)} = \\frac{e^{\\rho_f}}{T}, then obtain - \\rho_f + \\int \\varphi^{*}{(\\rho_f,T)} d\\rho_f - 1 = - \\rho_f + \\int \\frac{e^{\\rho_f}}{T} d\\rho_f - 1", "derivation": "\\varphi^{*}{(\\rho_f,T)} = \\frac{e^{\\rho_f}}{T} and \\int \\varphi^{*}{(\\rho_f,T)} d\\rho_f = \\int \\frac{e^{\\rho_f}}{T} d\\rho_f and - \\rho_f + \\int \\varphi^{*}{(\\rho_f,T)} d\\rho_f = - \\rho_f + \\int \\frac{e^{\\rho_f}}{T} d\\rho_f and - \\rho_f + \\int \\varphi^{*}{(\\rho_f,T)} d\\rho_f - 1 = - \\rho_f + \\int \\frac{e^{\\rho_f}}{T} d\\rho_f - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["minus", 2, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Integral(Function('\\\\varphi^*')(Symbol('\\\\rho_f', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Integral(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given i{(C_{d})} = \\sin{(C_{d})}, then derive - \\frac{i{(C_{d})} \\cos{(C_{d})}}{\\sin^{2}{(C_{d})}} + \\frac{\\frac{d}{d C_{d}} i{(C_{d})}}{\\sin{(C_{d})}} = 0, then obtain - \\frac{\\cos{(C_{d})}}{i{(C_{d})}} + \\frac{\\frac{d}{d C_{d}} i{(C_{d})}}{i{(C_{d})}} = 0", "derivation": "i{(C_{d})} = \\sin{(C_{d})} and \\frac{i{(C_{d})}}{\\sin{(C_{d})}} = 1 and \\frac{d}{d C_{d}} \\frac{i{(C_{d})}}{\\sin{(C_{d})}} = \\frac{d}{d C_{d}} 1 and - \\frac{i{(C_{d})} \\cos{(C_{d})}}{\\sin^{2}{(C_{d})}} + \\frac{\\frac{d}{d C_{d}} i{(C_{d})}}{\\sin{(C_{d})}} = 0 and - \\frac{\\cos{(C_{d})}}{i{(C_{d})}} + \\frac{\\frac{d}{d C_{d}} i{(C_{d})}}{i{(C_{d})}} = 0", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["divide", 1, "sin(Symbol('C_d', commutative=True))"], "Equality(Mul(Function('i')(Symbol('C_d', commutative=True)), Pow(sin(Symbol('C_d', commutative=True)), Integer(-1))), Integer(1))"], [["differentiate", 2, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Mul(Function('i')(Symbol('C_d', commutative=True)), Pow(sin(Symbol('C_d', commutative=True)), Integer(-1))), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('i')(Symbol('C_d', commutative=True)), Pow(sin(Symbol('C_d', commutative=True)), Integer(-2)), cos(Symbol('C_d', commutative=True))), Mul(Pow(sin(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(Function('i')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Pow(Function('i')(Symbol('C_d', commutative=True)), Integer(-1)), cos(Symbol('C_d', commutative=True))), Mul(Pow(Function('i')(Symbol('C_d', commutative=True)), Integer(-1)), Derivative(Function('i')(Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given f{(\\hat{H})} = \\int \\sin{(\\hat{H})} d\\hat{H}, then derive \\frac{f{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} = \\frac{n - \\cos{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}}, then obtain \\frac{n - \\cos{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} = \\frac{\\int \\sin{(\\hat{H})} d\\hat{H}}{g_{\\varepsilon} - \\cos{(\\hat{H})}}", "derivation": "f{(\\hat{H})} = \\int \\sin{(\\hat{H})} d\\hat{H} and \\hat{H} f{(\\hat{H})} = \\hat{H} \\int \\sin{(\\hat{H})} d\\hat{H} and \\frac{f{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} = \\frac{\\int \\sin{(\\hat{H})} d\\hat{H}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} and \\frac{f{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} = \\frac{n - \\cos{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} and \\frac{n - \\cos{(\\hat{H})}}{g_{\\varepsilon} - \\cos{(\\hat{H})}} = \\frac{\\int \\sin{(\\hat{H})} d\\hat{H}}{g_{\\varepsilon} - \\cos{(\\hat{H})}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\hat{H}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('f')(Symbol('\\\\hat{H}', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["divide", 2, "Mul(Symbol('\\\\hat{H}', commutative=True), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))))"], "Equality(Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Function('f')(Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Function('f')(Symbol('\\\\hat{H}', commutative=True))), Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Add(Symbol('n', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True))))), Mul(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{H}', commutative=True)))), Integer(-1)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}{(r_{0})} = \\log{(\\log{(r_{0})})}, then obtain (- \\hat{p}{(r_{0})} + \\log{(\\log{(r_{0})})})^{r_{0}} + \\hat{p}{(r_{0})} = \\hat{p}{(r_{0})} + 1", "derivation": "\\hat{p}{(r_{0})} = \\log{(\\log{(r_{0})})} and 0 = - \\hat{p}{(r_{0})} + \\log{(\\log{(r_{0})})} and 0^{r_{0}} = (- \\hat{p}{(r_{0})} + \\log{(\\log{(r_{0})})})^{r_{0}} and 0^{r_{0}} + \\hat{p}{(r_{0})} = (- \\hat{p}{(r_{0})} + \\log{(\\log{(r_{0})})})^{r_{0}} + \\hat{p}{(r_{0})} and (- \\hat{p}{(r_{0})} + \\log{(\\log{(r_{0})})})^{r_{0}} + \\hat{p}{(r_{0})} = \\hat{p}{(r_{0})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('r_0', commutative=True)), log(log(Symbol('r_0', commutative=True))))"], [["minus", 1, "Function('\\\\hat{p}')(Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), log(log(Symbol('r_0', commutative=True)))))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Integer(0), Symbol('r_0', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), log(log(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)))"], [["minus", 3, "Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('r_0', commutative=True)))"], "Equality(Add(Pow(Integer(0), Symbol('r_0', commutative=True)), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), log(log(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), log(log(Symbol('r_0', commutative=True)))), Symbol('r_0', commutative=True)), Function('\\\\hat{p}')(Symbol('r_0', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('r_0', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)} = \\frac{\\sin{(\\mathbb{I})}}{J}, then derive (\\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)})^{J} = (- \\frac{\\sin{(\\mathbb{I})}}{J^{2}})^{J}, then obtain \\cos{((\\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)})^{J})} = \\cos{((- \\frac{\\sin{(\\mathbb{I})}}{J^{2}})^{J})}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)} = \\frac{\\sin{(\\mathbb{I})}}{J} and \\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)} = \\frac{\\partial}{\\partial J} \\frac{\\sin{(\\mathbb{I})}}{J} and (\\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)})^{J} = (\\frac{\\partial}{\\partial J} \\frac{\\sin{(\\mathbb{I})}}{J})^{J} and (\\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)})^{J} = (- \\frac{\\sin{(\\mathbb{I})}}{J^{2}})^{J} and \\cos{((\\frac{\\partial}{\\partial J} \\operatorname{f_{\\mathbf{v}}}{(\\mathbb{I},J)})^{J})} = \\cos{((- \\frac{\\sin{(\\mathbb{I})}}{J^{2}})^{J})}", "srepr_derivation": [["get_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), sin(Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2)), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('J', commutative=True)))"], [["cos", 4], "Equality(cos(Pow(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True))), cos(Pow(Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2)), sin(Symbol('\\\\mathbb{I}', commutative=True))), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C_{2},L)} = \\frac{C_{2}}{L}, then obtain \\frac{C_{2} (\\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{1}{L})}{L \\Psi^{\\dagger}{(C_{2},L)}} = \\frac{C_{2} (\\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{C_{2}}{L^{2} \\Psi^{\\dagger}{(C_{2},L)}})}{L \\Psi^{\\dagger}{(C_{2},L)}}", "derivation": "\\Psi^{\\dagger}{(C_{2},L)} = \\frac{C_{2}}{L} and 1 = \\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} and \\frac{1}{L} = \\frac{C_{2}}{L^{2} \\Psi^{\\dagger}{(C_{2},L)}} and \\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{1}{L} = \\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{C_{2}}{L^{2} \\Psi^{\\dagger}{(C_{2},L)}} and \\frac{C_{2} (\\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{1}{L})}{L \\Psi^{\\dagger}{(C_{2},L)}} = \\frac{C_{2} (\\frac{C_{2}}{L \\Psi^{\\dagger}{(C_{2},L)}} + \\frac{C_{2}}{L^{2} \\Psi^{\\dagger}{(C_{2},L)}})}{L \\Psi^{\\dagger}{(C_{2},L)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1))))"], [["divide", 1, "Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True))"], "Equality(Integer(1), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["divide", 2, "Symbol('L', commutative=True)"], "Equality(Pow(Symbol('L', commutative=True), Integer(-1)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-2)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["add", 3, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Pow(Symbol('L', commutative=True), Integer(-1))), Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-2)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))))"], [["times", 4, "Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))"], "Equality(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Pow(Symbol('L', commutative=True), Integer(-1))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L', commutative=True), Integer(-2)), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1)))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('C_2', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi{(\\hat{X},\\phi_1)} = \\phi_1^{\\hat{X}}, then obtain e^{(\\phi_1^{\\hat{X}} - \\phi{(\\hat{X},\\phi_1)})^{\\phi_1}} = e", "derivation": "\\phi{(\\hat{X},\\phi_1)} = \\phi_1^{\\hat{X}} and 0 = \\phi_1^{\\hat{X}} - \\phi{(\\hat{X},\\phi_1)} and 0^{\\phi_1} = (\\phi_1^{\\hat{X}} - \\phi{(\\hat{X},\\phi_1)})^{\\phi_1} and e^{0^{\\phi_1}} = e^{(\\phi_1^{\\hat{X}} - \\phi{(\\hat{X},\\phi_1)})^{\\phi_1}} and e^{(\\phi_1^{\\hat{X}} - \\phi{(\\hat{X},\\phi_1)})^{\\phi_1}} = e", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["minus", 1, "Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)))))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Integer(0), Symbol('\\\\phi_1', commutative=True))), exp(Pow(Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(exp(Pow(Add(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\phi_1', commutative=True)))), Symbol('\\\\phi_1', commutative=True))), E)"]]}, {"prompt": "Given \\theta_{2}{(v_{t},H)} = H^{v_{t}}, then obtain - H + \\log{(\\theta_{2}{(v_{t},H)})} - \\int \\theta_{2}{(v_{t},H)} dH = - H + \\log{(H^{v_{t}})} - \\int \\theta_{2}{(v_{t},H)} dH", "derivation": "\\theta_{2}{(v_{t},H)} = H^{v_{t}} and \\int \\theta_{2}{(v_{t},H)} dH = \\int H^{v_{t}} dH and \\log{(\\theta_{2}{(v_{t},H)})} = \\log{(H^{v_{t}})} and \\log{(\\theta_{2}{(v_{t},H)})} - \\int H^{v_{t}} dH = \\log{(H^{v_{t}})} - \\int H^{v_{t}} dH and - H + \\log{(\\theta_{2}{(v_{t},H)})} - \\int H^{v_{t}} dH = - H + \\log{(H^{v_{t}})} - \\int H^{v_{t}} dH and - H + \\log{(\\theta_{2}{(v_{t},H)})} - \\int \\theta_{2}{(v_{t},H)} dH = - H + \\log{(H^{v_{t}})} - \\int \\theta_{2}{(v_{t},H)} dH", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True))), log(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True))))"], [["minus", 3, "Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(log(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(log(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["minus", 4, "Symbol('H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), log(Pow(Symbol('H', commutative=True), Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_2')(Symbol('v_t', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"]]}, {"prompt": "Given A{(s,y)} = \\cos{(\\frac{s}{y})} and \\operatorname{J_{\\varepsilon}}{(s,y)} = \\cos{(\\frac{s}{y})}, then obtain \\frac{s \\operatorname{J_{\\varepsilon}}{(s,y)}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})} = \\frac{s \\cos{(\\frac{s}{y})}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})}", "derivation": "A{(s,y)} = \\cos{(\\frac{s}{y})} and \\operatorname{J_{\\varepsilon}}{(s,y)} = \\cos{(\\frac{s}{y})} and \\operatorname{J_{\\varepsilon}}{(s,y)} = A{(s,y)} and \\frac{s \\operatorname{J_{\\varepsilon}}{(s,y)}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})} = \\frac{s A{(s,y)}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})} and \\frac{s \\operatorname{J_{\\varepsilon}}{(s,y)}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})} = \\frac{s \\cos{(\\frac{s}{y})}}{y (\\cos{(\\frac{s}{y})} - \\frac{1}{y})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('s', commutative=True), Symbol('y', commutative=True)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('s', commutative=True), Symbol('y', commutative=True)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('s', commutative=True), Symbol('y', commutative=True)), Function('A')(Symbol('s', commutative=True), Symbol('y', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('s', commutative=True), Integer(-1)), Symbol('y', commutative=True), Add(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)))))"], "Equality(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Add(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)))), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('s', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Add(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)))), Integer(-1)), Function('A')(Symbol('s', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Add(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)))), Integer(-1)), Function('J_{\\\\varepsilon}')(Symbol('s', commutative=True), Symbol('y', commutative=True))), Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Add(cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)))), Mul(Integer(-1), Pow(Symbol('y', commutative=True), Integer(-1)))), Integer(-1)), cos(Mul(Symbol('s', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(k)} = k, then obtain (\\frac{\\frac{d}{d k} \\operatorname{A_{1}}{(k)}}{\\operatorname{A_{1}}{(k)}})^{k} = (\\frac{\\frac{d}{d k} k}{\\operatorname{A_{1}}{(k)}})^{k}", "derivation": "\\operatorname{A_{1}}{(k)} = k and \\frac{d}{d k} \\operatorname{A_{1}}{(k)} = \\frac{d}{d k} k and \\frac{\\frac{d}{d k} \\operatorname{A_{1}}{(k)}}{\\operatorname{A_{1}}{(k)}} = \\frac{\\frac{d}{d k} k}{\\operatorname{A_{1}}{(k)}} and (\\frac{\\frac{d}{d k} \\operatorname{A_{1}}{(k)}}{\\operatorname{A_{1}}{(k)}})^{k} = (\\frac{\\frac{d}{d k} k}{\\operatorname{A_{1}}{(k)}})^{k}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('k', commutative=True)), Symbol('k', commutative=True))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["divide", 2, "Function('A_1')(Symbol('k', commutative=True))"], "Equality(Mul(Pow(Function('A_1')(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Mul(Pow(Function('A_1')(Symbol('k', commutative=True)), Integer(-1)), Derivative(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Mul(Pow(Function('A_1')(Symbol('k', commutative=True)), Integer(-1)), Derivative(Function('A_1')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1)))), Symbol('k', commutative=True)), Pow(Mul(Pow(Function('A_1')(Symbol('k', commutative=True)), Integer(-1)), Derivative(Symbol('k', commutative=True), Tuple(Symbol('k', commutative=True), Integer(1)))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given a{(C,u)} = \\frac{C}{u} and k{(C,u)} = \\int \\frac{C}{u} du, then obtain \\frac{(\\int \\frac{C}{u} du) \\int a{(C,u)} du}{C} = \\frac{(\\int a{(C,u)} du)^{2}}{C}", "derivation": "a{(C,u)} = \\frac{C}{u} and \\int a{(C,u)} du = \\int \\frac{C}{u} du and k{(C,u)} = \\int \\frac{C}{u} du and \\frac{k{(C,u)}}{C} = \\frac{\\int \\frac{C}{u} du}{C} and k{(C,u)} = \\int a{(C,u)} du and \\frac{k{(C,u)}}{C} = \\frac{\\int a{(C,u)} du}{C} and \\frac{\\int \\frac{C}{u} du}{C} = \\frac{\\int a{(C,u)} du}{C} and \\frac{k{(C,u)} \\int \\frac{C}{u} du}{C} = \\frac{k{(C,u)} \\int a{(C,u)} du}{C} and \\frac{(\\int \\frac{C}{u} du) \\int a{(C,u)} du}{C} = \\frac{(\\int a{(C,u)} du)^{2}}{C}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))))"], [["divide", 3, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["times", 7, "Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Function('k')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 5], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Integral(Mul(Symbol('C', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Tuple(Symbol('u', commutative=True))), Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Integral(Function('a')(Symbol('C', commutative=True), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\phi_2)} = \\phi_2 and \\omega{(\\phi_2)} = \\phi_2^{\\phi_2}, then obtain - \\phi_2^{\\phi_2} - \\int \\omega{(\\phi_2)} d\\phi_2 = - \\operatorname{v_{y}}^{\\phi_2}{(\\phi_2)} - \\int \\omega{(\\phi_2)} d\\phi_2", "derivation": "\\operatorname{v_{y}}{(\\phi_2)} = \\phi_2 and \\operatorname{v_{y}}^{\\phi_2}{(\\phi_2)} = \\phi_2^{\\phi_2} and \\omega{(\\phi_2)} = \\phi_2^{\\phi_2} and - \\omega{(\\phi_2)} = - \\phi_2^{\\phi_2} and - \\omega{(\\phi_2)} = - \\operatorname{v_{y}}^{\\phi_2}{(\\phi_2)} and - \\phi_2^{\\phi_2} = - \\operatorname{v_{y}}^{\\phi_2}{(\\phi_2)} and - \\phi_2^{\\phi_2} - \\int \\omega{(\\phi_2)} d\\phi_2 = - \\operatorname{v_{y}}^{\\phi_2}{(\\phi_2)} - \\int \\omega{(\\phi_2)} d\\phi_2", "srepr_derivation": [["renaming_premise", "Equality(Function('v_y')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True)), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Integer(-1), Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Function('v_y')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Pow(Function('v_y')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))))"], [["minus", 6, "Integral(Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))), Add(Mul(Integer(-1), Pow(Function('v_y')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\omega')(Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(q)} = \\log{(q)}, then obtain e^{- \\frac{\\operatorname{v_{y}}{(q)} - \\log{(q)}}{q \\operatorname{v_{y}}{(q)}}} = 1", "derivation": "\\operatorname{v_{y}}{(q)} = \\log{(q)} and \\operatorname{v_{y}}{(q)} - \\log{(q)} = 0 and \\operatorname{v_{y}}{(q)} - \\log{(q)} - 1 = -1 and (\\operatorname{v_{y}}{(q)} - \\log{(q)} - 1) \\operatorname{v_{y}}{(q)} = - \\operatorname{v_{y}}{(q)} and \\frac{\\operatorname{v_{y}}{(q)} - \\log{(q)}}{q} = 0 and \\frac{\\operatorname{v_{y}}{(q)} - \\log{(q)}}{q (\\operatorname{v_{y}}{(q)} - \\log{(q)} - 1) \\operatorname{v_{y}}{(q)}} = 0 and - \\frac{\\operatorname{v_{y}}{(q)} - \\log{(q)}}{q \\operatorname{v_{y}}{(q)}} = 0 and e^{- \\frac{\\operatorname{v_{y}}{(q)} - \\log{(q)}}{q \\operatorname{v_{y}}{(q)}}} = 1", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('q', commutative=True)), log(Symbol('q', commutative=True)))"], [["minus", 1, "log(Symbol('q', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True)))), Integer(0))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True))), Integer(-1)), Integer(-1))"], [["times", 3, "Function('v_y')(Symbol('q', commutative=True))"], "Equality(Mul(Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True))), Integer(-1)), Function('v_y')(Symbol('q', commutative=True))), Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True))))"], [["divide", 2, "Symbol('q', commutative=True)"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True))))), Integer(0))"], [["divide", 5, "Mul(Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True))), Integer(-1)), Function('v_y')(Symbol('q', commutative=True)))"], "Equality(Mul(Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True)))), Pow(Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True))), Integer(-1)), Integer(-1)), Pow(Function('v_y')(Symbol('q', commutative=True)), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True)))), Pow(Function('v_y')(Symbol('q', commutative=True)), Integer(-1))), Integer(0))"], [["exp", 7], "Equality(exp(Mul(Integer(-1), Pow(Symbol('q', commutative=True), Integer(-1)), Add(Function('v_y')(Symbol('q', commutative=True)), Mul(Integer(-1), log(Symbol('q', commutative=True)))), Pow(Function('v_y')(Symbol('q', commutative=True)), Integer(-1)))), Integer(1))"]]}, {"prompt": "Given \\Psi_{nl}{(\\Psi,P_{e})} = \\frac{P_{e}}{\\Psi}, then obtain \\frac{\\partial}{\\partial P_{e}} \\Psi_{nl}{(\\Psi,P_{e})} \\int \\Psi_{nl}{(\\Psi,P_{e})} dP_{e} = \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\int \\Psi_{nl}{(\\Psi,P_{e})} dP_{e}}{\\Psi}", "derivation": "\\Psi_{nl}{(\\Psi,P_{e})} = \\frac{P_{e}}{\\Psi} and \\int \\Psi_{nl}{(\\Psi,P_{e})} dP_{e} = \\int \\frac{P_{e}}{\\Psi} dP_{e} and \\Psi_{nl}{(\\Psi,P_{e})} \\int \\frac{P_{e}}{\\Psi} dP_{e} = \\frac{P_{e} \\int \\frac{P_{e}}{\\Psi} dP_{e}}{\\Psi} and \\frac{\\partial}{\\partial P_{e}} \\Psi_{nl}{(\\Psi,P_{e})} \\int \\frac{P_{e}}{\\Psi} dP_{e} = \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\int \\frac{P_{e}}{\\Psi} dP_{e}}{\\Psi} and \\frac{\\partial}{\\partial P_{e}} \\Psi_{nl}{(\\Psi,P_{e})} \\int \\Psi_{nl}{(\\Psi,P_{e})} dP_{e} = \\frac{\\partial}{\\partial P_{e}} \\frac{P_{e} \\int \\Psi_{nl}{(\\Psi,P_{e})} dP_{e}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('P_e', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True))), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True))))"], [["times", 1, "Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))), Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))))"], [["differentiate", 3, "Symbol('P_e', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1))), Tuple(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Mul(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True), Integer(1))), Derivative(Mul(Symbol('P_e', commutative=True), Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\Psi', commutative=True), Symbol('P_e', commutative=True)), Tuple(Symbol('P_e', commutative=True)))), Tuple(Symbol('P_e', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\delta,S)} = S \\delta and \\operatorname{v_{2}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})}, then derive \\operatorname{v_{2}}{(\\mathbf{D})} = - \\sin{(\\mathbf{D})}, then obtain \\operatorname{v_{2}}{(\\mathbf{D})} - 1 = - \\sin{(\\mathbf{D})} - 1", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\delta,S)} = S \\delta and \\operatorname{v_{2}}{(\\mathbf{D})} = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} and - \\frac{S \\delta}{\\operatorname{f_{\\mathbf{p}}}{(\\delta,S)}} + \\operatorname{v_{2}}{(\\mathbf{D})} = - \\frac{S \\delta}{\\operatorname{f_{\\mathbf{p}}}{(\\delta,S)}} + \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} and \\operatorname{v_{2}}{(\\mathbf{D})} - 1 = \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} - 1 and \\operatorname{v_{2}}{(\\mathbf{D})} = - \\sin{(\\mathbf{D})} and \\frac{d}{d \\mathbf{D}} \\cos{(\\mathbf{D})} = - \\sin{(\\mathbf{D})} and \\operatorname{v_{2}}{(\\mathbf{D})} - 1 = - \\sin{(\\mathbf{D})} - 1", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True)))"], ["get_premise", "Equality(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["minus", 2, "Mul(Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), Integer(-1))), Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('\\\\delta', commutative=True), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\delta', commutative=True), Symbol('S', commutative=True)), Integer(-1))), Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Add(Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(cos(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Add(Function('v_2')(Symbol('\\\\mathbf{D}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{D}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\hat{H}{(F_{x},\\delta)} = F_{x} + \\delta, then obtain \\frac{\\partial}{\\partial F_{x}} \\int \\hat{H}{(F_{x},\\delta)} dF_{x} = \\frac{\\partial}{\\partial F_{x}} (\\frac{F_{x}^{2}}{2} + F_{x} \\delta + s)", "derivation": "\\hat{H}{(F_{x},\\delta)} = F_{x} + \\delta and \\int \\hat{H}{(F_{x},\\delta)} dF_{x} = \\int (F_{x} + \\delta) dF_{x} and \\frac{\\partial}{\\partial F_{x}} \\int \\hat{H}{(F_{x},\\delta)} dF_{x} = \\frac{\\partial}{\\partial F_{x}} \\int (F_{x} + \\delta) dF_{x} and \\frac{\\partial}{\\partial F_{x}} \\int \\hat{H}{(F_{x},\\delta)} dF_{x} = \\frac{\\partial}{\\partial F_{x}} (\\frac{F_{x}^{2}}{2} + F_{x} \\delta + s)", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}')(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_x', commutative=True))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hat{H}')(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\hat{H}')(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Rational(1, 2), Pow(Symbol('F_x', commutative=True), Integer(2))), Mul(Symbol('F_x', commutative=True), Symbol('\\\\delta', commutative=True)), Symbol('s', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(\\rho_b,f_{\\mathbf{v}})} = - \\rho_b + f_{\\mathbf{v}}, then derive \\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b = - \\frac{\\rho_b^{2}}{2} + \\rho_b f_{\\mathbf{v}} + \\varepsilon, then obtain \\log{(\\frac{\\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b}{\\rho_b})} = \\log{(\\frac{- \\frac{\\rho_b^{2}}{2} + \\rho_b f_{\\mathbf{v}} + \\varepsilon}{\\rho_b})}", "derivation": "Z{(\\rho_b,f_{\\mathbf{v}})} = - \\rho_b + f_{\\mathbf{v}} and \\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b = \\int (- \\rho_b + f_{\\mathbf{v}}) d\\rho_b and \\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b = - \\frac{\\rho_b^{2}}{2} + \\rho_b f_{\\mathbf{v}} + \\varepsilon and \\frac{\\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b}{\\rho_b} = \\frac{- \\frac{\\rho_b^{2}}{2} + \\rho_b f_{\\mathbf{v}} + \\varepsilon}{\\rho_b} and \\log{(\\frac{\\int Z{(\\rho_b,f_{\\mathbf{v}})} d\\rho_b}{\\rho_b})} = \\log{(\\frac{- \\frac{\\rho_b^{2}}{2} + \\rho_b f_{\\mathbf{v}} + \\varepsilon}{\\rho_b})}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('Z')(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["divide", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"], [["log", 4], "Equality(log(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Integral(Function('Z')(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))), log(Mul(Pow(Symbol('\\\\rho_b', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\rho_b', commutative=True), Integer(2))), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(W,l)} = e^{W - l} and \\hat{p}{(W,l)} = e^{W - l} and \\dot{z}{(W,l)} = (e^{W - l})^{l}, then obtain \\dot{z}{(W,l)} = \\hat{p}^{l}{(W,l)}", "derivation": "\\operatorname{J_{\\varepsilon}}{(W,l)} = e^{W - l} and \\hat{p}{(W,l)} = e^{W - l} and \\operatorname{J_{\\varepsilon}}^{l}{(W,l)} = (e^{W - l})^{l} and \\dot{z}{(W,l)} = (e^{W - l})^{l} and \\operatorname{J_{\\varepsilon}}^{l}{(W,l)} = \\hat{p}^{l}{(W,l)} and \\hat{p}^{l}{(W,l)} = (e^{W - l})^{l} and \\dot{z}{(W,l)} = \\hat{p}^{l}{(W,l)}", "srepr_derivation": [["get_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["power", 1, "Symbol('l', commutative=True)"], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Symbol('l', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Pow(exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Pow(Function('J_{\\\\varepsilon}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Function('\\\\hat{p}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)), Pow(exp(Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('l', commutative=True)))), Symbol('l', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Function('\\\\dot{z}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('W', commutative=True), Symbol('l', commutative=True)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\mathbf{r})} = \\log{(\\mathbf{r})}, then obtain \\frac{\\operatorname{A_{2}}{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = 1", "derivation": "\\operatorname{A_{2}}{(\\mathbf{r})} = \\log{(\\mathbf{r})} and \\operatorname{A_{2}}{(\\mathbf{r})} \\log{(\\mathbf{r})} = \\log{(\\mathbf{r})}^{2} and \\operatorname{A_{2}}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}^{2} = \\log{(\\mathbf{r})}^{4} and \\operatorname{A_{2}}^{3}{(\\mathbf{r})} \\log{(\\mathbf{r})} = \\operatorname{A_{2}}^{2}{(\\mathbf{r})} \\log{(\\mathbf{r})}^{2} and \\frac{\\operatorname{A_{2}}{(\\mathbf{r})}}{\\log{(\\mathbf{r})}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True)))"], [["times", 1, "log(Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Mul(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), log(Symbol('\\\\mathbf{r}', commutative=True))), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(3)), log(Symbol('\\\\mathbf{r}', commutative=True))), Mul(Pow(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2))))"], [["divide", 4, "Mul(Pow(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(2)))"], "Equality(Mul(Function('A_2')(Symbol('\\\\mathbf{r}', commutative=True)), Pow(log(Symbol('\\\\mathbf{r}', commutative=True)), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\mathbf{P}{(V_{\\mathbf{B}})} = e^{V_{\\mathbf{B}}} and \\hat{\\mathbf{x}}{(v_{2})} = \\cos{(v_{2})}, then obtain \\int (\\hat{\\mathbf{x}}{(v_{2})} - 1) dv_{2} = \\int (\\cos{(v_{2})} - 1) dv_{2}", "derivation": "\\mathbf{P}{(V_{\\mathbf{B}})} = e^{V_{\\mathbf{B}}} and \\hat{\\mathbf{x}}{(v_{2})} = \\cos{(v_{2})} and \\hat{\\mathbf{x}}{(v_{2})} - \\mathbf{P}{(V_{\\mathbf{B}})} + e^{V_{\\mathbf{B}}} = - \\mathbf{P}{(V_{\\mathbf{B}})} + e^{V_{\\mathbf{B}}} + \\cos{(v_{2})} and \\hat{\\mathbf{x}}{(v_{2})} - \\mathbf{P}{(V_{\\mathbf{B}})} + e^{V_{\\mathbf{B}}} - 1 = - \\mathbf{P}{(V_{\\mathbf{B}})} + e^{V_{\\mathbf{B}}} + \\cos{(v_{2})} - 1 and \\hat{\\mathbf{x}}{(v_{2})} - 1 = \\cos{(v_{2})} - 1 and \\int (\\hat{\\mathbf{x}}{(v_{2})} - 1) dv_{2} = \\int (\\cos{(v_{2})} - 1) dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_2', commutative=True)), cos(Symbol('v_2', commutative=True)))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('v_2', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_2', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{P}')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), exp(Symbol('V_{\\\\mathbf{B}}', commutative=True)), cos(Symbol('v_2', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_2', commutative=True)), Integer(-1)), Add(cos(Symbol('v_2', commutative=True)), Integer(-1)))"], [["integrate", 5, "Symbol('v_2', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('v_2', commutative=True)), Integer(-1)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(cos(Symbol('v_2', commutative=True)), Integer(-1)), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})} = \\mathbb{I} f^{*}, then derive \\int (\\sigma_p + \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})}) df^{*} = C_{2} + \\frac{\\mathbb{I} (f^{*})^{2}}{2} + \\sigma_p f^{*}, then obtain (C_{2} + \\frac{\\mathbb{I} (f^{*})^{2}}{2} + \\sigma_p f^{*})^{\\mathbb{I}} = (\\int (\\mathbb{I} f^{*} + \\sigma_p) df^{*})^{\\mathbb{I}}", "derivation": "\\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})} = \\mathbb{I} f^{*} and \\sigma_p + \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})} = \\mathbb{I} f^{*} + \\sigma_p and \\int (\\sigma_p + \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})}) df^{*} = \\int (\\mathbb{I} f^{*} + \\sigma_p) df^{*} and \\int (\\sigma_p + \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})}) df^{*} = C_{2} + \\frac{\\mathbb{I} (f^{*})^{2}}{2} + \\sigma_p f^{*} and (\\int (\\sigma_p + \\dot{\\mathbf{r}}{(\\mathbb{I},f^{*})}) df^{*})^{\\mathbb{I}} = (\\int (\\mathbb{I} f^{*} + \\sigma_p) df^{*})^{\\mathbb{I}} and (C_{2} + \\frac{\\mathbb{I} (f^{*})^{2}}{2} + \\sigma_p f^{*})^{\\mathbb{I}} = (\\int (\\mathbb{I} f^{*} + \\sigma_p) df^{*})^{\\mathbb{I}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)))"], [["add", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f^*', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\sigma_p', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('f^*', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Symbol('C_2', commutative=True), Mul(Rational(1, 2), Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(2))), Mul(Symbol('\\\\sigma_p', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{F}{(\\chi)} = \\cos{(\\chi)}, then derive \\frac{d}{d \\chi} \\mathbf{F}{(\\chi)} = - \\sin{(\\chi)}, then derive \\frac{d^{2}}{d \\chi^{2}} \\mathbf{F}{(\\chi)} = - \\cos{(\\chi)}, then obtain \\frac{d^{3}}{d \\chi^{3}} \\cos{(\\chi)} = \\frac{d}{d \\chi} - \\cos{(\\chi)}", "derivation": "\\mathbf{F}{(\\chi)} = \\cos{(\\chi)} and \\frac{d}{d \\chi} \\mathbf{F}{(\\chi)} = \\frac{d}{d \\chi} \\cos{(\\chi)} and \\frac{d}{d \\chi} \\mathbf{F}{(\\chi)} = - \\sin{(\\chi)} and \\frac{d^{2}}{d \\chi^{2}} \\mathbf{F}{(\\chi)} = \\frac{d}{d \\chi} - \\sin{(\\chi)} and \\frac{d^{2}}{d \\chi^{2}} \\mathbf{F}{(\\chi)} = - \\cos{(\\chi)} and \\frac{d^{2}}{d \\chi^{2}} \\cos{(\\chi)} = - \\cos{(\\chi)} and \\frac{d^{3}}{d \\chi^{3}} \\cos{(\\chi)} = \\frac{d}{d \\chi} - \\cos{(\\chi)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(cos(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(3))), Derivative(Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(i)} = \\sin{(i)}, then derive \\int \\operatorname{P_{g}}{(i)} di = A - \\cos{(i)}, then obtain (\\int \\operatorname{P_{g}}{(i)} di)^{i} = (A - \\cos{(i)})^{i}", "derivation": "\\operatorname{P_{g}}{(i)} = \\sin{(i)} and \\int \\operatorname{P_{g}}{(i)} di = \\int \\sin{(i)} di and \\int \\operatorname{P_{g}}{(i)} di = A - \\cos{(i)} and (\\int \\operatorname{P_{g}}{(i)} di)^{i} = (\\int \\sin{(i)} di)^{i} and A - \\cos{(i)} = \\int \\sin{(i)} di and (\\int \\operatorname{P_{g}}{(i)} di)^{i} = (A - \\cos{(i)})^{i}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_g')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('i', commutative=True)))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Integral(Function('P_g')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('i', commutative=True)))), Integral(sin(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Integral(Function('P_g')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Symbol('A', commutative=True), Mul(Integer(-1), cos(Symbol('i', commutative=True)))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(h)} = \\cos{(h)}, then derive (\\frac{h \\frac{d}{d h} \\mathbf{J}{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)} = (- \\frac{h \\sin{(h)}}{\\cos{(h)}} + \\log{(\\cos{(h)})}) \\cos^{h}{(h)}, then obtain (\\frac{h \\frac{d}{d h} \\mathbf{J}{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)} = (- \\frac{h \\sin{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)}", "derivation": "\\mathbf{J}{(h)} = \\cos{(h)} and \\mathbf{J}^{h}{(h)} = \\cos^{h}{(h)} and \\frac{d}{d h} \\mathbf{J}^{h}{(h)} = \\frac{d}{d h} \\cos^{h}{(h)} and (\\frac{h \\frac{d}{d h} \\mathbf{J}{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)} = (- \\frac{h \\sin{(h)}}{\\cos{(h)}} + \\log{(\\cos{(h)})}) \\cos^{h}{(h)} and (\\frac{h \\frac{d}{d h} \\mathbf{J}{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)} = (- \\frac{h \\sin{(h)}}{\\mathbf{J}{(h)}} + \\log{(\\mathbf{J}{(h)})}) \\mathbf{J}^{h}{(h)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["power", 1, "Symbol('h', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('h', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), log(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)))), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True), sin(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), log(cos(Symbol('h', commutative=True)))), Pow(cos(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Symbol('h', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), log(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)))), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('h', commutative=True), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Integer(-1)), sin(Symbol('h', commutative=True))), log(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)))), Pow(Function('\\\\mathbf{J}')(Symbol('h', commutative=True)), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\cos{(\\dot{z})}, then derive \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} \\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = 0, then obtain \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} (- \\hat{\\mathbf{r}} + \\cos{(\\dot{z})}) = 0", "derivation": "\\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\cos{(\\dot{z})} and \\frac{\\partial}{\\partial \\dot{z}} \\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\dot{z}} (- \\hat{\\mathbf{r}} + \\cos{(\\dot{z})}) and \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} \\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} (- \\hat{\\mathbf{r}} + \\cos{(\\dot{z})}) and \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} \\mathbf{v}{(\\dot{z},\\hat{\\mathbf{r}})} = 0 and \\frac{\\partial^{2}}{\\partial \\hat{\\mathbf{r}}\\partial \\dot{z}} (- \\hat{\\mathbf{r}} + \\cos{(\\dot{z})}) = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\dot{z}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), cos(Symbol('\\\\dot{z}', commutative=True))), Tuple(Symbol('\\\\dot{z}', commutative=True), Integer(1)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Integer(0))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(L)} = \\log{(L)}, then obtain - \\cos{(\\operatorname{E_{n}}{(L)})} + \\int \\operatorname{E_{n}}{(L)} dL = - \\cos{(\\operatorname{E_{n}}{(L)})} + \\int \\log{(L)} dL", "derivation": "\\operatorname{E_{n}}{(L)} = \\log{(L)} and \\int \\operatorname{E_{n}}{(L)} dL = \\int \\log{(L)} dL and \\cos{(\\operatorname{E_{n}}{(L)})} = \\cos{(\\log{(L)})} and - \\cos{(\\log{(L)})} + \\int \\operatorname{E_{n}}{(L)} dL = - \\cos{(\\log{(L)})} + \\int \\log{(L)} dL and - \\cos{(\\operatorname{E_{n}}{(L)})} + \\int \\operatorname{E_{n}}{(L)} dL = - \\cos{(\\operatorname{E_{n}}{(L)})} + \\int \\log{(L)} dL", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('L', commutative=True)), log(Symbol('L', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["cos", 1], "Equality(cos(Function('E_n')(Symbol('L', commutative=True))), cos(log(Symbol('L', commutative=True))))"], [["minus", 2, "cos(log(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(log(Symbol('L', commutative=True)))), Integral(Function('E_n')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), cos(log(Symbol('L', commutative=True)))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(-1), cos(Function('E_n')(Symbol('L', commutative=True)))), Integral(Function('E_n')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Mul(Integer(-1), cos(Function('E_n')(Symbol('L', commutative=True)))), Integral(log(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given a{(C,t_{1},F_{H})} = (C + F_{H})^{t_{1}}, then obtain \\frac{\\partial}{\\partial t_{1}} a{(C,t_{1},F_{H})} = (C + F_{H})^{t_{1}} \\log{(C + F_{H})}", "derivation": "a{(C,t_{1},F_{H})} = (C + F_{H})^{t_{1}} and - C + a{(C,t_{1},F_{H})} = - C + (C + F_{H})^{t_{1}} and - C - F_{H} + a{(C,t_{1},F_{H})} = - C - F_{H} + (C + F_{H})^{t_{1}} and \\frac{\\partial}{\\partial t_{1}} (- C - F_{H} + a{(C,t_{1},F_{H})}) = \\frac{\\partial}{\\partial t_{1}} (- C - F_{H} + (C + F_{H})^{t_{1}}) and \\frac{\\partial}{\\partial t_{1}} a{(C,t_{1},F_{H})} = (C + F_{H})^{t_{1}} \\log{(C + F_{H})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C', commutative=True), Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('a')(Symbol('C', commutative=True), Symbol('t_1', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Symbol('t_1', commutative=True))))"], [["minus", 2, "Symbol('F_H', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('a')(Symbol('C', commutative=True), Symbol('t_1', commutative=True), Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('F_H', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Symbol('t_1', commutative=True))))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('F_H', commutative=True)), Function('a')(Symbol('C', commutative=True), Symbol('t_1', commutative=True), Symbol('F_H', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Mul(Integer(-1), Symbol('F_H', commutative=True)), Pow(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('a')(Symbol('C', commutative=True), Symbol('t_1', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Pow(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)), Symbol('t_1', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('F_H', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(g,\\mathbf{F})} = e^{g^{\\mathbf{F}}}, then obtain \\frac{\\rho_{f}{(g,\\mathbf{F})}}{- g + \\rho_{f}{(g,\\mathbf{F})}} = \\frac{e^{g^{\\mathbf{F}}}}{- g + \\rho_{f}{(g,\\mathbf{F})}}", "derivation": "\\rho_{f}{(g,\\mathbf{F})} = e^{g^{\\mathbf{F}}} and - g + \\rho_{f}{(g,\\mathbf{F})} = - g + e^{g^{\\mathbf{F}}} and \\frac{\\rho_{f}{(g,\\mathbf{F})}}{- g + e^{g^{\\mathbf{F}}}} = \\frac{e^{g^{\\mathbf{F}}}}{- g + e^{g^{\\mathbf{F}}}} and \\frac{\\rho_{f}{(g,\\mathbf{F})}}{- g + \\rho_{f}{(g,\\mathbf{F})}} = \\frac{e^{g^{\\mathbf{F}}}}{- g + \\rho_{f}{(g,\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\rho_f')(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1)), exp(Pow(Symbol('g', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(u,\\mathbf{f})} = \\frac{\\mathbf{f}}{u}, then obtain \\iiint u \\tilde{g}{(u,\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} d\\mathbf{f} = \\iiint \\mathbf{f} d\\mathbf{f} d\\mathbf{f} d\\mathbf{f}", "derivation": "\\tilde{g}{(u,\\mathbf{f})} = \\frac{\\mathbf{f}}{u} and u \\tilde{g}{(u,\\mathbf{f})} = \\mathbf{f} and \\int u \\tilde{g}{(u,\\mathbf{f})} d\\mathbf{f} = \\int \\mathbf{f} d\\mathbf{f} and \\iint u \\tilde{g}{(u,\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} = \\iint \\mathbf{f} d\\mathbf{f} d\\mathbf{f} and \\iiint u \\tilde{g}{(u,\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} d\\mathbf{f} = \\iiint \\mathbf{f} d\\mathbf{f} d\\mathbf{f} d\\mathbf{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Symbol('\\\\mathbf{f}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["times", 1, "Symbol('u', commutative=True)"], "Equality(Mul(Symbol('u', commutative=True), Function('\\\\tilde{g}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Symbol('\\\\mathbf{f}', commutative=True))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Symbol('u', commutative=True), Function('\\\\tilde{g}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Symbol('\\\\mathbf{f}', commutative=True), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Symbol('u', commutative=True), Function('\\\\tilde{g}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Symbol('\\\\mathbf{f}', commutative=True), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 4, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Mul(Symbol('u', commutative=True), Function('\\\\tilde{g}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(Symbol('\\\\mathbf{f}', commutative=True), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given U{(v_{2},\\mu)} = v_{2}^{\\mu} and \\bar{\\h}{(v_{2},\\mu)} = 2 v_{2}^{\\mu}, then obtain ((\\mu + v_{2}^{\\mu} + U{(v_{2},\\mu)} + \\bar{\\h}{(v_{2},\\mu)})^{\\mu})^{\\mu} = ((\\mu + 2 \\bar{\\h}{(v_{2},\\mu)})^{\\mu})^{\\mu}", "derivation": "U{(v_{2},\\mu)} = v_{2}^{\\mu} and v_{2}^{\\mu} + U{(v_{2},\\mu)} = 2 v_{2}^{\\mu} and \\bar{\\h}{(v_{2},\\mu)} = 2 v_{2}^{\\mu} and v_{2}^{\\mu} + U{(v_{2},\\mu)} = \\bar{\\h}{(v_{2},\\mu)} and v_{2}^{\\mu} + U{(v_{2},\\mu)} + \\bar{\\h}{(v_{2},\\mu)} = 2 \\bar{\\h}{(v_{2},\\mu)} and \\mu + v_{2}^{\\mu} + U{(v_{2},\\mu)} + \\bar{\\h}{(v_{2},\\mu)} = \\mu + 2 \\bar{\\h}{(v_{2},\\mu)} and (\\mu + v_{2}^{\\mu} + U{(v_{2},\\mu)} + \\bar{\\h}{(v_{2},\\mu)})^{\\mu} = (\\mu + 2 \\bar{\\h}{(v_{2},\\mu)})^{\\mu} and ((\\mu + v_{2}^{\\mu} + U{(v_{2},\\mu)} + \\bar{\\h}{(v_{2},\\mu)})^{\\mu})^{\\mu} = ((\\mu + 2 \\bar{\\h}{(v_{2},\\mu)})^{\\mu})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 1, "Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Integer(2), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["add", 4, "Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))))"], [["add", 5, "Symbol('\\\\mu', commutative=True)"], "Equality(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(2), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)))))"], [["power", 6, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(2), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], [["power", 7, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Add(Symbol('\\\\mu', commutative=True), Pow(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('U')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(2), Function('\\\\hbar')(Symbol('v_2', commutative=True), Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given r{(\\theta,\\mathbf{v})} = \\mathbf{v} + \\theta, then derive \\frac{\\partial}{\\partial \\theta} r{(\\theta,\\mathbf{v})} = 1, then obtain \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta) = 1", "derivation": "r{(\\theta,\\mathbf{v})} = \\mathbf{v} + \\theta and \\frac{\\partial}{\\partial \\theta} r{(\\theta,\\mathbf{v})} = \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta) and r{(\\theta,\\mathbf{v})} + \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta) = \\mathbf{v} + \\theta + \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta) and \\frac{\\partial}{\\partial \\theta} (r{(\\theta,\\mathbf{v})} + \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta)) = \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta + \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta)) and \\frac{\\partial}{\\partial \\theta} r{(\\theta,\\mathbf{v})} = 1 and \\frac{\\partial}{\\partial \\theta} (\\mathbf{v} + \\theta) = 1", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('r')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))"], "Equality(Add(Function('r')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True), Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('r')(Symbol('\\\\theta', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given l{(\\tilde{g},\\mathbf{f})} = \\mathbf{f} + \\tilde{g}, then obtain \\mathbf{f} + \\tilde{g} + l{(\\tilde{g},\\mathbf{f})} = 2 l{(\\tilde{g},\\mathbf{f})}", "derivation": "l{(\\tilde{g},\\mathbf{f})} = \\mathbf{f} + \\tilde{g} and \\mathbf{f} + \\tilde{g} + l{(\\tilde{g},\\mathbf{f})} = 2 \\mathbf{f} + 2 \\tilde{g} and 2 l{(\\tilde{g},\\mathbf{f})} = 2 \\mathbf{f} + 2 \\tilde{g} and \\mathbf{f} + \\tilde{g} + l{(\\tilde{g},\\mathbf{f})} = 2 l{(\\tilde{g},\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Symbol('\\\\tilde{g}', commutative=True), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(2), Function('l')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{S}{(c_{0},\\hat{H})} = - \\hat{H} + c_{0}, then derive \\int \\mathbf{S}{(c_{0},\\hat{H})} dc_{0} = - \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2}, then obtain (- \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2}) \\int \\mathbf{S}{(c_{0},\\hat{H})} dc_{0} = (- \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2})^{2}", "derivation": "\\mathbf{S}{(c_{0},\\hat{H})} = - \\hat{H} + c_{0} and \\int \\mathbf{S}{(c_{0},\\hat{H})} dc_{0} = \\int (- \\hat{H} + c_{0}) dc_{0} and \\int \\mathbf{S}{(c_{0},\\hat{H})} dc_{0} = - \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2} and (- \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2}) \\int \\mathbf{S}{(c_{0},\\hat{H})} dc_{0} = (- \\hat{H} c_{0} + \\omega + \\frac{c_{0}^{2}}{2})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{S}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{S}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))))"], [["times", 3, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))), Integral(Function('\\\\mathbf{S}')(Symbol('c_0', commutative=True), Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('c_0', commutative=True)), Symbol('\\\\omega', commutative=True), Mul(Rational(1, 2), Pow(Symbol('c_0', commutative=True), Integer(2)))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(x)} = \\cos{(x)}, then obtain \\frac{\\operatorname{a^{\\dagger}}{(x)} - \\cos{(x)} + 1}{x} - \\frac{1}{x} = 0", "derivation": "\\operatorname{a^{\\dagger}}{(x)} = \\cos{(x)} and 2 \\operatorname{a^{\\dagger}}{(x)} = \\operatorname{a^{\\dagger}}{(x)} + \\cos{(x)} and \\operatorname{a^{\\dagger}}{(x)} - \\cos{(x)} = 0 and \\operatorname{a^{\\dagger}}{(x)} - \\cos{(x)} + 1 = 1 and \\frac{\\operatorname{a^{\\dagger}}{(x)} - \\cos{(x)} + 1}{x} = \\frac{1}{x} and \\frac{\\operatorname{a^{\\dagger}}{(x)} - \\cos{(x)} + 1}{x} - \\frac{1}{x} = 0", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], [["add", 1, "Function('a^{\\\\dagger}')(Symbol('x', commutative=True))"], "Equality(Mul(Integer(2), Function('a^{\\\\dagger}')(Symbol('x', commutative=True))), Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))))"], [["minus", 2, "Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True)))"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Integer(0))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True))), Integer(1)), Integer(1))"], [["divide", 4, "Symbol('x', commutative=True)"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True))), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["minus", 5, "Pow(Symbol('x', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('a^{\\\\dagger}')(Symbol('x', commutative=True)), Mul(Integer(-1), cos(Symbol('x', commutative=True))), Integer(1))), Mul(Integer(-1), Pow(Symbol('x', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given W{(\\mu)} = \\sin{(\\mu)}, then derive \\int W{(\\mu)} d\\mu = F_{c} - \\cos{(\\mu)}, then obtain (\\int W{(\\mu)} d\\mu)^{\\mu} = (F_{c} - \\cos{(\\mu)})^{\\mu}", "derivation": "W{(\\mu)} = \\sin{(\\mu)} and \\int W{(\\mu)} d\\mu = \\int \\sin{(\\mu)} d\\mu and (\\int W{(\\mu)} d\\mu)^{\\mu} = (\\int \\sin{(\\mu)} d\\mu)^{\\mu} and \\int W{(\\mu)} d\\mu = F_{c} - \\cos{(\\mu)} and \\int \\sin{(\\mu)} d\\mu = F_{c} - \\cos{(\\mu)} and (\\int W{(\\mu)} d\\mu)^{\\mu} = (F_{c} - \\cos{(\\mu)})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integral(Function('W')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Integral(Function('W')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(sin(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Integral(Function('W')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given A{(\\mathbf{J})} = e^{\\mathbf{J}}, then obtain (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)^{- \\mathbf{J}} \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} A{(\\mathbf{J})} + 1) = (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)^{- \\mathbf{J}} \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)", "derivation": "A{(\\mathbf{J})} = e^{\\mathbf{J}} and \\frac{d}{d \\mathbf{J}} A{(\\mathbf{J})} = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} and \\frac{d}{d \\mathbf{J}} A{(\\mathbf{J})} + 1 = \\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1 and \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} A{(\\mathbf{J})} + 1) = \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1) and (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)^{- \\mathbf{J}} \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} A{(\\mathbf{J})} + 1) = (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)^{- \\mathbf{J}} \\frac{d}{d \\mathbf{J}} (\\frac{d}{d \\mathbf{J}} e^{\\mathbf{J}} + 1)", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{J}', commutative=True)), exp(Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('A')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Add(Derivative(Function('A')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["divide", 4, "Pow(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Derivative(Add(Derivative(Function('A')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))), Mul(Pow(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Derivative(Add(Derivative(exp(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\lambda{(z^{*})} = \\sin{(z^{*})} and \\eta{(C_{2},z^{*})} = C_{2} - \\cos{(z^{*})}, then derive \\int \\lambda{(z^{*})} dz^{*} = C_{2} - \\cos{(z^{*})}, then obtain \\int \\sin{(z^{*})} dz^{*} = \\eta{(C_{2},z^{*})}", "derivation": "\\lambda{(z^{*})} = \\sin{(z^{*})} and \\int \\lambda{(z^{*})} dz^{*} = \\int \\sin{(z^{*})} dz^{*} and \\int \\lambda{(z^{*})} dz^{*} = C_{2} - \\cos{(z^{*})} and \\eta{(C_{2},z^{*})} = C_{2} - \\cos{(z^{*})} and \\int \\lambda{(z^{*})} dz^{*} = \\eta{(C_{2},z^{*})} and \\int \\sin{(z^{*})} dz^{*} = \\eta{(C_{2},z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('z^*', commutative=True)), sin(Symbol('z^*', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('C_2', commutative=True), Symbol('z^*', commutative=True)), Add(Symbol('C_2', commutative=True), Mul(Integer(-1), cos(Symbol('z^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('\\\\lambda')(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Function('\\\\eta')(Symbol('C_2', commutative=True), Symbol('z^*', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(sin(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Function('\\\\eta')(Symbol('C_2', commutative=True), Symbol('z^*', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(E_{n})} = \\log{(E_{n})}, then obtain \\phi_{2}{(E_{n})} \\log{(E_{n})} + \\phi_{2}{(E_{n})} + \\log{(E_{n})} = \\phi_{2}{(E_{n})} \\log{(E_{n})} + 2 \\log{(E_{n})}", "derivation": "\\phi_{2}{(E_{n})} = \\log{(E_{n})} and \\phi_{2}{(E_{n})} + \\log{(E_{n})} = 2 \\log{(E_{n})} and \\phi_{2}{(E_{n})} \\log{(E_{n})} = \\log{(E_{n})}^{2} and \\phi_{2}{(E_{n})} + \\log{(E_{n})}^{2} + \\log{(E_{n})} = \\log{(E_{n})}^{2} + 2 \\log{(E_{n})} and \\phi_{2}{(E_{n})} \\log{(E_{n})} + \\phi_{2}{(E_{n})} + \\log{(E_{n})} = \\phi_{2}{(E_{n})} \\log{(E_{n})} + 2 \\log{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True)))"], [["add", 1, "log(Symbol('E_n', commutative=True))"], "Equality(Add(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True))))"], [["times", 1, "log(Symbol('E_n', commutative=True))"], "Equality(Mul(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Pow(log(Symbol('E_n', commutative=True)), Integer(2)))"], [["add", 2, "Pow(log(Symbol('E_n', commutative=True)), Integer(2))"], "Equality(Add(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), Pow(log(Symbol('E_n', commutative=True)), Integer(2)), log(Symbol('E_n', commutative=True))), Add(Pow(log(Symbol('E_n', commutative=True)), Integer(2)), Mul(Integer(2), log(Symbol('E_n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Add(Mul(Function('\\\\phi_2')(Symbol('E_n', commutative=True)), log(Symbol('E_n', commutative=True))), Mul(Integer(2), log(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given Z{(J_{\\varepsilon},E_{n})} = E_{n} J_{\\varepsilon}, then obtain ((- \\int Z{(J_{\\varepsilon},E_{n})} dJ_{\\varepsilon})^{E_{n}})^{E_{n}} = ((- \\int E_{n} J_{\\varepsilon} dJ_{\\varepsilon})^{E_{n}})^{E_{n}}", "derivation": "Z{(J_{\\varepsilon},E_{n})} = E_{n} J_{\\varepsilon} and \\int Z{(J_{\\varepsilon},E_{n})} dJ_{\\varepsilon} = \\int E_{n} J_{\\varepsilon} dJ_{\\varepsilon} and - \\int Z{(J_{\\varepsilon},E_{n})} dJ_{\\varepsilon} = - \\int E_{n} J_{\\varepsilon} dJ_{\\varepsilon} and (- \\int Z{(J_{\\varepsilon},E_{n})} dJ_{\\varepsilon})^{E_{n}} = (- \\int E_{n} J_{\\varepsilon} dJ_{\\varepsilon})^{E_{n}} and ((- \\int Z{(J_{\\varepsilon},E_{n})} dJ_{\\varepsilon})^{E_{n}})^{E_{n}} = ((- \\int E_{n} J_{\\varepsilon} dJ_{\\varepsilon})^{E_{n}})^{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('Z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Mul(Integer(-1), Integral(Mul(Symbol('E_n', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))))"], [["power", 3, "Symbol('E_n', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Integral(Function('Z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Symbol('E_n', commutative=True)), Pow(Mul(Integer(-1), Integral(Mul(Symbol('E_n', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Symbol('E_n', commutative=True)))"], [["power", 4, "Symbol('E_n', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(-1), Integral(Function('Z')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)), Pow(Pow(Mul(Integer(-1), Integral(Mul(Symbol('E_n', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)))), Symbol('E_n', commutative=True)), Symbol('E_n', commutative=True)))"]]}, {"prompt": "Given \\varphi{(S)} = \\log{(S)}, then derive \\frac{d}{d S} 0 = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S \\log{(S)} - S + \\mathbf{A}) - \\frac{d}{d S} \\int \\varphi{(S)} dS), then obtain \\frac{d}{d S} 0 = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S \\log{(S)} - S + \\mathbf{A}) - \\frac{d}{d S} \\int \\log{(S)} dS)", "derivation": "\\varphi{(S)} = \\log{(S)} and \\int \\varphi{(S)} dS = \\int \\log{(S)} dS and \\frac{d}{d S} \\int \\varphi{(S)} dS = \\frac{d}{d S} \\int \\log{(S)} dS and 0 = - \\frac{d}{d S} \\int \\varphi{(S)} dS + \\frac{d}{d S} \\int \\log{(S)} dS and \\frac{d}{d S} 0 = \\frac{d}{d S} (- \\frac{d}{d S} \\int \\varphi{(S)} dS + \\frac{d}{d S} \\int \\log{(S)} dS) and \\frac{d}{d S} 0 = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S \\log{(S)} - S + \\mathbf{A}) - \\frac{d}{d S} \\int \\varphi{(S)} dS) and \\frac{d}{d S} 0 = \\frac{\\partial}{\\partial S} (\\frac{\\partial}{\\partial S} (S \\log{(S)} - S + \\mathbf{A}) - \\frac{d}{d S} \\int \\log{(S)} dS)", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('S', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Derivative(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Derivative(Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_integrals", 5], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(Function('\\\\varphi')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Add(Derivative(Add(Mul(Symbol('S', commutative=True), log(Symbol('S', commutative=True))), Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta{(\\theta_2)} = \\log{(\\cos{(\\theta_2)})}, then obtain \\log{(\\cos{(\\theta_2)})}^{- \\theta_2} \\iint \\eta^{\\theta_2}{(\\theta_2)} d\\theta_2 d\\theta_2 = \\log{(\\cos{(\\theta_2)})}^{- \\theta_2} \\iint \\log{(\\cos{(\\theta_2)})}^{\\theta_2} d\\theta_2 d\\theta_2", "derivation": "\\eta{(\\theta_2)} = \\log{(\\cos{(\\theta_2)})} and \\eta^{\\theta_2}{(\\theta_2)} = \\log{(\\cos{(\\theta_2)})}^{\\theta_2} and \\int \\eta^{\\theta_2}{(\\theta_2)} d\\theta_2 = \\int \\log{(\\cos{(\\theta_2)})}^{\\theta_2} d\\theta_2 and \\iint \\eta^{\\theta_2}{(\\theta_2)} d\\theta_2 d\\theta_2 = \\iint \\log{(\\cos{(\\theta_2)})}^{\\theta_2} d\\theta_2 d\\theta_2 and \\log{(\\cos{(\\theta_2)})}^{- \\theta_2} \\iint \\eta^{\\theta_2}{(\\theta_2)} d\\theta_2 d\\theta_2 = \\log{(\\cos{(\\theta_2)})}^{- \\theta_2} \\iint \\log{(\\cos{(\\theta_2)})}^{\\theta_2} d\\theta_2 d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\theta_2', commutative=True)), log(cos(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 2, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Pow(Function('\\\\eta')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["times", 4, "Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True)))"], "Equality(Mul(Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Integral(Pow(Function('\\\\eta')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))), Mul(Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta_2', commutative=True))), Integral(Pow(log(cos(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and T{(\\Psi_{\\lambda})} = \\Psi_{\\lambda}, then obtain \\Psi_{\\lambda} + \\mathbf{J}{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(\\Psi_{\\lambda})}", "derivation": "\\mathbf{J}{(\\Psi_{\\lambda})} = \\cos{(\\Psi_{\\lambda})} and T{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} and T{(\\Psi_{\\lambda})} + \\mathbf{J}{(\\Psi_{\\lambda})} = T{(\\Psi_{\\lambda})} + \\cos{(\\Psi_{\\lambda})} and \\Psi_{\\lambda} + \\mathbf{J}{(\\Psi_{\\lambda})} = \\Psi_{\\lambda} + \\cos{(\\Psi_{\\lambda})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], [["add", 1, "Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))"], "Equality(Add(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Function('\\\\mathbf{J}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Function('T')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Function('\\\\mathbf{J}')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{P}{(B,E_{n})} = e^{- B + E_{n}}, then obtain - (- \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} + \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}})^{B} = -1", "derivation": "\\mathbf{P}{(B,E_{n})} = e^{- B + E_{n}} and \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} = \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}} and 0 = - \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} + \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}} and 0^{B} = (- \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} + \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}})^{B} and - 0^{B} = - (- \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} + \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}})^{B} and - (- \\frac{\\partial}{\\partial E_{n}} \\mathbf{P}{(B,E_{n})} + \\frac{\\partial}{\\partial E_{n}} e^{- B + E_{n}})^{B} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('B', commutative=True)"], "Equality(Pow(Integer(0), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('B', commutative=True)))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Integer(0), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('B', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Derivative(Function('\\\\mathbf{P}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Derivative(exp(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Symbol('B', commutative=True))), Integer(-1))"]]}, {"prompt": "Given \\ddot{x}{(C_{2},\\lambda)} = \\log{(C_{2} \\lambda)}, then obtain \\frac{\\ddot{x}{(C_{2},\\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} = \\frac{\\log{(e^{\\ddot{x}{(C_{2},\\lambda)}})}}{\\operatorname{C_{d}}{(A_{y},Z)}}", "derivation": "\\ddot{x}{(C_{2},\\lambda)} = \\log{(C_{2} \\lambda)} and - \\frac{\\ddot{x}{(C_{2},\\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} = - \\frac{\\log{(C_{2} \\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} and e^{\\ddot{x}{(C_{2},\\lambda)}} = C_{2} \\lambda and \\frac{\\ddot{x}{(C_{2},\\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} = \\frac{\\log{(C_{2} \\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} and \\frac{\\ddot{x}{(C_{2},\\lambda)}}{\\operatorname{C_{d}}{(A_{y},Z)}} = \\frac{\\log{(e^{\\ddot{x}{(C_{2},\\lambda)}})}}{\\operatorname{C_{d}}{(A_{y},Z)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["exp", 1], "Equality(exp(Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), log(Mul(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Function('C_d')(Symbol('A_y', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), log(exp(Function('\\\\ddot{x}')(Symbol('C_2', commutative=True), Symbol('\\\\lambda', commutative=True))))))"]]}, {"prompt": "Given \\rho{(\\eta)} = \\cos{(\\log{(\\eta)})} and \\hat{H}_{\\lambda}{(\\eta)} = \\cos{(\\log{(\\eta)})}, then obtain \\int \\hat{H}_{\\lambda}{(\\eta)} d\\eta = \\int \\rho{(\\eta)} d\\eta", "derivation": "\\rho{(\\eta)} = \\cos{(\\log{(\\eta)})} and \\int \\rho{(\\eta)} d\\eta = \\int \\cos{(\\log{(\\eta)})} d\\eta and \\hat{H}_{\\lambda}{(\\eta)} = \\cos{(\\log{(\\eta)})} and \\int \\hat{H}_{\\lambda}{(\\eta)} d\\eta = \\int \\cos{(\\log{(\\eta)})} d\\eta and \\int \\hat{H}_{\\lambda}{(\\eta)} d\\eta = \\int \\rho{(\\eta)} d\\eta", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), cos(log(Symbol('\\\\eta', commutative=True))))"], [["integrate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\eta', commutative=True)), cos(log(Symbol('\\\\eta', commutative=True))))"], [["integrate", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(cos(log(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))), Integral(Function('\\\\rho')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(\\sigma_p,h)} = \\sigma_p^{h}, then obtain \\int \\frac{\\partial}{\\partial h} (\\frac{\\mathbf{M}{(\\sigma_p,h)}}{\\sigma_p})^{\\sigma_p} d\\sigma_p = \\int \\frac{\\partial}{\\partial h} (\\frac{\\sigma_p^{h}}{\\sigma_p})^{\\sigma_p} d\\sigma_p", "derivation": "\\mathbf{M}{(\\sigma_p,h)} = \\sigma_p^{h} and \\frac{\\mathbf{M}{(\\sigma_p,h)}}{\\sigma_p} = \\frac{\\sigma_p^{h}}{\\sigma_p} and (\\frac{\\mathbf{M}{(\\sigma_p,h)}}{\\sigma_p})^{\\sigma_p} = (\\frac{\\sigma_p^{h}}{\\sigma_p})^{\\sigma_p} and \\frac{\\partial}{\\partial h} (\\frac{\\mathbf{M}{(\\sigma_p,h)}}{\\sigma_p})^{\\sigma_p} = \\frac{\\partial}{\\partial h} (\\frac{\\sigma_p^{h}}{\\sigma_p})^{\\sigma_p} and \\int \\frac{\\partial}{\\partial h} (\\frac{\\mathbf{M}{(\\sigma_p,h)}}{\\sigma_p})^{\\sigma_p} d\\sigma_p = \\int \\frac{\\partial}{\\partial h} (\\frac{\\sigma_p^{h}}{\\sigma_p})^{\\sigma_p} d\\sigma_p", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True)))"], [["divide", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Derivative(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Function('\\\\mathbf{M}')(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Derivative(Pow(Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Symbol('h', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(A_{x},Q)} = e^{A_{x} + Q}, then derive \\int \\mathbf{p}{(A_{x},Q)} dA_{x} = c_{0} + e^{A_{x} + Q}, then obtain \\frac{c_{0} + \\mathbf{p}{(A_{x},Q)} + e^{A_{x} + Q}}{\\mathbf{p}{(A_{x},Q)} + \\int e^{A_{x} + Q} dA_{x}} = 1", "derivation": "\\mathbf{p}{(A_{x},Q)} = e^{A_{x} + Q} and \\int \\mathbf{p}{(A_{x},Q)} dA_{x} = \\int e^{A_{x} + Q} dA_{x} and \\mathbf{p}{(A_{x},Q)} + \\int \\mathbf{p}{(A_{x},Q)} dA_{x} = \\mathbf{p}{(A_{x},Q)} + \\int e^{A_{x} + Q} dA_{x} and \\int \\mathbf{p}{(A_{x},Q)} dA_{x} = c_{0} + e^{A_{x} + Q} and \\frac{\\mathbf{p}{(A_{x},Q)} + \\int \\mathbf{p}{(A_{x},Q)} dA_{x}}{\\mathbf{p}{(A_{x},Q)} + \\int e^{A_{x} + Q} dA_{x}} = 1 and \\frac{c_{0} + \\mathbf{p}{(A_{x},Q)} + e^{A_{x} + Q}}{\\mathbf{p}{(A_{x},Q)} + \\int e^{A_{x} + Q} dA_{x}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["add", 2, "Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('A_x', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Add(Symbol('c_0', commutative=True), exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)))))"], [["divide", 3, "Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], "Equality(Mul(Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Pow(Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('A_x', commutative=True)))), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), Integral(exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('A_x', commutative=True)))), Integer(-1)), Add(Symbol('c_0', commutative=True), Function('\\\\mathbf{p}')(Symbol('A_x', commutative=True), Symbol('Q', commutative=True)), exp(Add(Symbol('A_x', commutative=True), Symbol('Q', commutative=True))))), Integer(1))"]]}, {"prompt": "Given x{(t)} = \\log{(t)}, then obtain (\\frac{d}{d t} 2 x{(t)})^{t} = (\\frac{d}{d t} (x{(t)} + \\log{(t)}))^{t}", "derivation": "x{(t)} = \\log{(t)} and 2 x{(t)} = x{(t)} + \\log{(t)} and \\frac{d}{d t} 2 x{(t)} = \\frac{d}{d t} (x{(t)} + \\log{(t)}) and (\\frac{d}{d t} 2 x{(t)})^{t} = (\\frac{d}{d t} (x{(t)} + \\log{(t)}))^{t}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True)))"], [["add", 1, "Function('x')(Symbol('t', commutative=True))"], "Equality(Mul(Integer(2), Function('x')(Symbol('t', commutative=True))), Add(Function('x')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('x')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Function('x')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(2), Function('x')(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)), Pow(Derivative(Add(Function('x')(Symbol('t', commutative=True)), log(Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given I{(P_{g},t_{1})} = \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}), then derive I{(P_{g},t_{1})} + 1 = 2, then obtain (I{(P_{g},t_{1})} + \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}))^{2} (\\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) + 1) = 2 (I{(P_{g},t_{1})} + \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}))^{2}", "derivation": "I{(P_{g},t_{1})} = \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) and I{(P_{g},t_{1})} + \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) = 2 \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) and I{(P_{g},t_{1})} + 1 = 2 and \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) + 1 = 2 and (I{(P_{g},t_{1})} + \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}))^{2} (\\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}) + 1) = 2 (I{(P_{g},t_{1})} + \\frac{\\partial}{\\partial P_{g}} (P_{g} + t_{1}))^{2}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))"], "Equality(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Integer(1)), Integer(2))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["times", 4, "Pow(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(2))"], "Equality(Mul(Pow(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(2)), Add(Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))), Integer(1))), Mul(Integer(2), Pow(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Derivative(Add(Symbol('P_g', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(f)} = \\log{(\\sin{(f)})}, then obtain (\\frac{d}{d f} \\operatorname{C_{1}}{(f)})^{f} = (\\frac{\\cos{(f)}}{\\sin{(f)}})^{f}", "derivation": "\\operatorname{C_{1}}{(f)} = \\log{(\\sin{(f)})} and \\frac{d}{d f} \\operatorname{C_{1}}{(f)} = \\frac{d}{d f} \\log{(\\sin{(f)})} and (\\frac{d}{d f} \\operatorname{C_{1}}{(f)})^{f} = (\\frac{d}{d f} \\log{(\\sin{(f)})})^{f} and (\\frac{d}{d f} \\operatorname{C_{1}}{(f)})^{f} = (\\frac{\\cos{(f)}}{\\sin{(f)}})^{f}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('f', commutative=True)), log(sin(Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 2, "Symbol('f', commutative=True)"], "Equality(Pow(Derivative(Function('C_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Derivative(log(sin(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('C_1')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Mul(Pow(sin(Symbol('f', commutative=True)), Integer(-1)), cos(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})}, then obtain \\sigma_{x}{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sigma_{x}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sigma_{x}{(\\eta^{\\prime})}", "derivation": "\\sigma_{x}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} and \\frac{d}{d \\eta^{\\prime}} \\sigma_{x}{(\\eta^{\\prime})} = \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} and \\sigma_{x}{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sin{(\\eta^{\\prime})} and \\sigma_{x}{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sigma_{x}{(\\eta^{\\prime})} = \\sin{(\\eta^{\\prime})} + \\frac{d}{d \\eta^{\\prime}} \\sigma_{x}{(\\eta^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 1, "Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Add(sin(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Derivative(Function('\\\\sigma_x')(Symbol('\\\\eta^{\\\\prime}', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\hat{H})} = \\sin{(\\hat{H})} and \\dot{\\mathbf{r}}{(\\hat{H})} = \\hat{H}, then obtain - \\hat{H} + \\int \\operatorname{E_{x}}{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})} = - \\hat{H} + \\int \\sin{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})}", "derivation": "\\operatorname{E_{x}}{(\\hat{H})} = \\sin{(\\hat{H})} and \\dot{\\mathbf{r}}{(\\hat{H})} = \\hat{H} and \\int \\operatorname{E_{x}}{(\\hat{H})} d\\hat{H} = \\int \\sin{(\\hat{H})} d\\hat{H} and \\int \\operatorname{E_{x}}{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})} = \\int \\sin{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})} and - \\hat{H} + \\int \\operatorname{E_{x}}{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})} = - \\hat{H} + \\int \\sin{(\\hat{H})} d\\dot{\\mathbf{r}}{(\\hat{H})}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\hat{H}', commutative=True)), sin(Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], [["integrate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Integral(Function('E_x')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)))), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Integral(Function('E_x')(Symbol('\\\\hat{H}', commutative=True)), Tuple(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Integral(sin(Symbol('\\\\hat{H}', commutative=True)), Tuple(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{P}{(z^{*},a,\\mathbf{v})} = (\\frac{z^{*}}{\\mathbf{v}})^{a}, then obtain \\frac{\\partial}{\\partial \\mathbf{v}} (\\int \\mathbf{P}{(z^{*},a,\\mathbf{v})} dz^{*} + 1) = \\frac{\\partial}{\\partial \\mathbf{v}} (\\int (\\frac{z^{*}}{\\mathbf{v}})^{a} dz^{*} + 1)", "derivation": "\\mathbf{P}{(z^{*},a,\\mathbf{v})} = (\\frac{z^{*}}{\\mathbf{v}})^{a} and \\int \\mathbf{P}{(z^{*},a,\\mathbf{v})} dz^{*} = \\int (\\frac{z^{*}}{\\mathbf{v}})^{a} dz^{*} and \\int \\mathbf{P}{(z^{*},a,\\mathbf{v})} dz^{*} + 1 = \\int (\\frac{z^{*}}{\\mathbf{v}})^{a} dz^{*} + 1 and \\frac{\\partial}{\\partial \\mathbf{v}} (\\int \\mathbf{P}{(z^{*},a,\\mathbf{v})} dz^{*} + 1) = \\frac{\\partial}{\\partial \\mathbf{v}} (\\int (\\frac{z^{*}}{\\mathbf{v}})^{a} dz^{*} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('z^*', commutative=True), Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('z^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{P}')(Symbol('z^*', commutative=True), Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('z^*', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('\\\\mathbf{P}')(Symbol('z^*', commutative=True), Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(1)), Add(Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(1)))"], [["differentiate", 3, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Add(Integral(Function('\\\\mathbf{P}')(Symbol('z^*', commutative=True), Symbol('a', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(1)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(Add(Integral(Pow(Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('z^*', commutative=True))), Integer(1)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + e^{f_{\\mathbf{p}}} and E{(\\hat{p},f_{\\mathbf{p}})} = 2 \\hat{p} + e^{f_{\\mathbf{p}}}, then obtain E{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + \\mathbf{A}{(\\hat{p},f_{\\mathbf{p}})}", "derivation": "\\mathbf{A}{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + e^{f_{\\mathbf{p}}} and \\hat{p} + \\mathbf{A}{(\\hat{p},f_{\\mathbf{p}})} = 2 \\hat{p} + e^{f_{\\mathbf{p}}} and E{(\\hat{p},f_{\\mathbf{p}})} = 2 \\hat{p} + e^{f_{\\mathbf{p}}} and E{(\\hat{p},f_{\\mathbf{p}})} = \\hat{p} + \\mathbf{A}{(\\hat{p},f_{\\mathbf{p}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["add", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], ["renaming_premise", "Equality(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Mul(Integer(2), Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('E')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Function('\\\\mathbf{A}')(Symbol('\\\\hat{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)} = \\sin{(u)}, then obtain (\\frac{- 2 u - 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)}}{\\sin{(u)}})^{u} = (\\frac{- u - \\sin{(u)}}{\\sin{(u)}} - \\frac{u + \\sin{(u)}}{\\sin{(u)}})^{u}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(u)} = \\sin{(u)} and u + \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)} = u + \\sin{(u)} and - \\frac{u + \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)}}{\\sin{(u)}} = - \\frac{u + \\sin{(u)}}{\\sin{(u)}} and - \\frac{2 (u + \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)})}{\\sin{(u)}} = - \\frac{u + \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)}}{\\sin{(u)}} - \\frac{u + \\sin{(u)}}{\\sin{(u)}} and \\frac{- 2 u - 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)}}{\\sin{(u)}} = \\frac{- u - \\sin{(u)}}{\\sin{(u)}} - \\frac{u + \\sin{(u)}}{\\sin{(u)}} and (\\frac{- 2 u - 2 \\operatorname{g^{\\prime}_{\\varepsilon}}{(u)}}{\\sin{(u)}})^{u} = (\\frac{- u - \\sin{(u)}}{\\sin{(u)}} - \\frac{u + \\sin{(u)}}{\\sin{(u)}})^{u}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["add", 1, "Symbol('u', commutative=True)"], "Equality(Add(Symbol('u', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True))), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), sin(Symbol('u', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Symbol('u', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))))"], [["add", 3, "Mul(Integer(-1), Add(Symbol('u', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1)))"], "Equality(Mul(Integer(-1), Integer(2), Add(Symbol('u', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Add(Mul(Integer(-1), Add(Symbol('u', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True)))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Add(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1)))))"], [["power", 5, "Symbol('u', commutative=True)"], "Equality(Pow(Mul(Add(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True)), Mul(Integer(-1), Integer(2), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True)))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Symbol('u', commutative=True)), Pow(Add(Mul(Add(Mul(Integer(-1), Symbol('u', commutative=True)), Mul(Integer(-1), sin(Symbol('u', commutative=True)))), Pow(sin(Symbol('u', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Symbol('u', commutative=True), sin(Symbol('u', commutative=True))), Pow(sin(Symbol('u', commutative=True)), Integer(-1)))), Symbol('u', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(x)} = \\int \\log{(x)} dx, then derive \\hat{\\mathbf{x}}{(x)} = A + x \\log{(x)} - x, then derive A + x \\log{(x)} - x = \\dot{\\mathbf{r}} + x \\log{(x)} - x, then obtain \\sin{(F_{H})} \\int \\hat{\\mathbf{x}}{(x)} dx = \\sin{(F_{H})} \\int (\\dot{\\mathbf{r}} + x \\log{(x)} - x) dx", "derivation": "\\hat{\\mathbf{x}}{(x)} = \\int \\log{(x)} dx and \\hat{\\mathbf{x}}{(x)} = A + x \\log{(x)} - x and A + x \\log{(x)} - x = \\int \\log{(x)} dx and A + x \\log{(x)} - x = \\dot{\\mathbf{r}} + x \\log{(x)} - x and \\hat{\\mathbf{x}}{(x)} = \\dot{\\mathbf{r}} + x \\log{(x)} - x and \\int \\hat{\\mathbf{x}}{(x)} dx = \\int (\\dot{\\mathbf{r}} + x \\log{(x)} - x) dx and \\sin{(F_{H})} \\int \\hat{\\mathbf{x}}{(x)} dx = \\sin{(F_{H})} \\int (\\dot{\\mathbf{r}} + x \\log{(x)} - x) dx", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True)), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True)), Add(Symbol('A', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Add(Symbol('A', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), Integral(log(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))))"], [["integrate", 5, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["times", 6, "sin(Symbol('F_H', commutative=True))"], "Equality(Mul(sin(Symbol('F_H', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(sin(Symbol('F_H', commutative=True)), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Symbol('x', commutative=True), log(Symbol('x', commutative=True))), Mul(Integer(-1), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain \\int (\\operatorname{E_{x}}{(\\phi_2)} - \\cos^{2}{(\\phi_2)}) d\\phi_2 = \\int (- \\cos^{2}{(\\phi_2)} + \\cos{(\\phi_2)}) d\\phi_2", "derivation": "\\operatorname{E_{x}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\operatorname{E_{x}}{(\\phi_2)} \\cos{(\\phi_2)} = \\cos^{2}{(\\phi_2)} and - \\operatorname{E_{x}}{(\\phi_2)} \\cos{(\\phi_2)} + \\operatorname{E_{x}}{(\\phi_2)} = - \\operatorname{E_{x}}{(\\phi_2)} \\cos{(\\phi_2)} + \\cos{(\\phi_2)} and \\operatorname{E_{x}}{(\\phi_2)} - \\cos^{2}{(\\phi_2)} = - \\cos^{2}{(\\phi_2)} + \\cos{(\\phi_2)} and \\int (\\operatorname{E_{x}}{(\\phi_2)} - \\cos^{2}{(\\phi_2)}) d\\phi_2 = \\int (- \\cos^{2}{(\\phi_2)} + \\cos{(\\phi_2)}) d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["times", 1, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('E_x')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(2)))"], [["minus", 1, "Mul(Function('E_x')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Function('E_x')(Symbol('\\\\phi_2', commutative=True))), Add(Mul(Integer(-1), Function('E_x')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), cos(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('E_x')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(2)))), Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(2))), cos(Symbol('\\\\phi_2', commutative=True))))"], [["integrate", 4, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Add(Function('E_x')(Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(2)))), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Integer(2))), cos(Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(f,E_{n})} = E_{n} e^{f}, then obtain E_{n} (\\int \\operatorname{C_{d}}{(f,E_{n})} e^{2 f} df + \\int - e^{2 f} df) + \\hat{H}_l = \\frac{E_{n}^{2} e^{3 f}}{3} - \\frac{E_{n} e^{2 f}}{2} + \\hat{H}_l", "derivation": "\\operatorname{C_{d}}{(f,E_{n})} = E_{n} e^{f} and \\operatorname{C_{d}}{(f,E_{n})} e^{f} = E_{n} e^{2 f} and \\operatorname{C_{d}}{(f,E_{n})} e^{f} - e^{f} = E_{n} e^{2 f} - e^{f} and E_{n} (\\operatorname{C_{d}}{(f,E_{n})} e^{f} - e^{f}) e^{f} = E_{n} (E_{n} e^{2 f} - e^{f}) e^{f} and \\int E_{n} (\\operatorname{C_{d}}{(f,E_{n})} e^{f} - e^{f}) e^{f} df = \\int E_{n} (E_{n} e^{2 f} - e^{f}) e^{f} df and E_{n} (\\int \\operatorname{C_{d}}{(f,E_{n})} e^{2 f} df + \\int - e^{2 f} df) + \\hat{H}_l = \\frac{E_{n}^{2} e^{3 f}}{3} - \\frac{E_{n} e^{2 f}}{2} + \\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), Mul(Symbol('E_n', commutative=True), exp(Symbol('f', commutative=True))))"], [["times", 1, "exp(Symbol('f', commutative=True))"], "Equality(Mul(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), exp(Symbol('f', commutative=True))), Mul(Symbol('E_n', commutative=True), exp(Mul(Integer(2), Symbol('f', commutative=True)))))"], [["minus", 2, "exp(Symbol('f', commutative=True))"], "Equality(Add(Mul(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), exp(Symbol('f', commutative=True))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))), Add(Mul(Symbol('E_n', commutative=True), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))))"], [["times", 3, "Mul(Symbol('E_n', commutative=True), exp(Symbol('f', commutative=True)))"], "Equality(Mul(Symbol('E_n', commutative=True), Add(Mul(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), exp(Symbol('f', commutative=True))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))), exp(Symbol('f', commutative=True))), Mul(Symbol('E_n', commutative=True), Add(Mul(Symbol('E_n', commutative=True), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))), exp(Symbol('f', commutative=True))))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Symbol('E_n', commutative=True), Add(Mul(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), exp(Symbol('f', commutative=True))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))), exp(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))), Integral(Mul(Symbol('E_n', commutative=True), Add(Mul(Symbol('E_n', commutative=True), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Mul(Integer(-1), exp(Symbol('f', commutative=True)))), exp(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('E_n', commutative=True), Add(Integral(Mul(Function('C_d')(Symbol('f', commutative=True), Symbol('E_n', commutative=True)), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Integral(Mul(Integer(-1), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))))), Symbol('\\\\hat{H}_l', commutative=True)), Add(Mul(Rational(1, 3), Pow(Symbol('E_n', commutative=True), Integer(2)), exp(Mul(Integer(3), Symbol('f', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('E_n', commutative=True), exp(Mul(Integer(2), Symbol('f', commutative=True)))), Symbol('\\\\hat{H}_l', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\sigma_{p}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})}, then obtain \\frac{\\log{(V_{\\mathbf{B}})}}{\\sigma_{p}{(V_{\\mathbf{B}})}} = 1", "derivation": "\\operatorname{r_{0}}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\frac{\\operatorname{r_{0}}{(V_{\\mathbf{B}})}}{\\log{(V_{\\mathbf{B}})}} = 1 and \\sigma_{p}{(V_{\\mathbf{B}})} = \\log{(V_{\\mathbf{B}})} and \\frac{\\operatorname{r_{0}}{(V_{\\mathbf{B}})}}{\\sigma_{p}{(V_{\\mathbf{B}})}} = 1 and \\frac{\\log{(V_{\\mathbf{B}})}}{\\sigma_{p}{(V_{\\mathbf{B}})}} = 1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["divide", 1, "log(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Mul(Function('r_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), Function('r_0')(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Function('\\\\sigma_p')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(-1)), log(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\chi{(A_{2},z)} = \\frac{z}{A_{2}}, then obtain (\\int (- A_{2} - \\chi{(A_{2},z)}) dA_{2})^{z} = (\\int (- A_{2} - \\frac{z}{A_{2}}) dA_{2})^{z}", "derivation": "\\chi{(A_{2},z)} = \\frac{z}{A_{2}} and A_{2} + \\chi{(A_{2},z)} = A_{2} + \\frac{z}{A_{2}} and - A_{2} - \\chi{(A_{2},z)} = - A_{2} - \\frac{z}{A_{2}} and \\int (- A_{2} - \\chi{(A_{2},z)}) dA_{2} = \\int (- A_{2} - \\frac{z}{A_{2}}) dA_{2} and (\\int (- A_{2} - \\chi{(A_{2},z)}) dA_{2})^{z} = (\\int (- A_{2} - \\frac{z}{A_{2}}) dA_{2})^{z}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('z', commutative=True)))"], [["add", 1, "Symbol('A_2', commutative=True)"], "Equality(Add(Symbol('A_2', commutative=True), Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('z', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)))), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('z', commutative=True))))"], [["integrate", 3, "Symbol('A_2', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('A_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Tuple(Symbol('A_2', commutative=True))))"], [["power", 4, "Symbol('z', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('z', commutative=True)))), Tuple(Symbol('A_2', commutative=True))), Symbol('z', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('z', commutative=True))), Tuple(Symbol('A_2', commutative=True))), Symbol('z', commutative=True)))"]]}, {"prompt": "Given b{(v_{y})} = \\sin{(\\log{(v_{y})})}, then obtain 0 = (- 2 b^{v_{y}}{(v_{y})} + 2 \\sin^{v_{y}}{(\\log{(v_{y})})}) \\sin^{- v_{y}}{(\\log{(v_{y})})}", "derivation": "b{(v_{y})} = \\sin{(\\log{(v_{y})})} and b^{v_{y}}{(v_{y})} = \\sin^{v_{y}}{(\\log{(v_{y})})} and 0 = - b^{v_{y}}{(v_{y})} + \\sin^{v_{y}}{(\\log{(v_{y})})} and - b^{v_{y}}{(v_{y})} = - 2 b^{v_{y}}{(v_{y})} + \\sin^{v_{y}}{(\\log{(v_{y})})} and 0 = - 2 b^{v_{y}}{(v_{y})} + 2 \\sin^{v_{y}}{(\\log{(v_{y})})} and 0 = (- 2 b^{v_{y}}{(v_{y})} + 2 \\sin^{v_{y}}{(\\log{(v_{y})})}) \\sin^{- v_{y}}{(\\log{(v_{y})})}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('v_y', commutative=True)), sin(log(Symbol('v_y', commutative=True))))"], [["power", 1, "Symbol('v_y', commutative=True)"], "Equality(Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)), Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["minus", 2, "Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))))"], [["minus", 3, "Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Add(Mul(Integer(-1), Integer(2), Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Mul(Integer(2), Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))))"], [["divide", 5, "Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Integer(2), Pow(Function('b')(Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), Mul(Integer(2), Pow(sin(log(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))), Pow(sin(log(Symbol('v_y', commutative=True))), Mul(Integer(-1), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given H{(\\varphi)} = \\cos{(\\varphi)} and C{(\\varphi)} = \\frac{\\varphi + \\cos{(\\varphi)}}{\\varphi + H{(\\varphi)}}, then obtain (\\frac{e^{C^{\\varphi}{(\\varphi)}}}{\\varphi})^{\\varphi} = (\\frac{e}{\\varphi})^{\\varphi}", "derivation": "H{(\\varphi)} = \\cos{(\\varphi)} and \\varphi + H{(\\varphi)} = \\varphi + \\cos{(\\varphi)} and C{(\\varphi)} = \\frac{\\varphi + \\cos{(\\varphi)}}{\\varphi + H{(\\varphi)}} and C^{\\varphi}{(\\varphi)} = (\\frac{\\varphi + \\cos{(\\varphi)}}{\\varphi + H{(\\varphi)}})^{\\varphi} and e^{C^{\\varphi}{(\\varphi)}} = e^{(\\frac{\\varphi + \\cos{(\\varphi)}}{\\varphi + H{(\\varphi)}})^{\\varphi}} and \\frac{e^{C^{\\varphi}{(\\varphi)}}}{\\varphi} = \\frac{e^{(\\frac{\\varphi + \\cos{(\\varphi)}}{\\varphi + H{(\\varphi)}})^{\\varphi}}}{\\varphi} and \\frac{e^{C^{\\varphi}{(\\varphi)}}}{\\varphi} = \\frac{e}{\\varphi} and (\\frac{e^{C^{\\varphi}{(\\varphi)}}}{\\varphi})^{\\varphi} = (\\frac{e}{\\varphi})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('H')(Symbol('\\\\varphi', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\varphi', commutative=True))))"], ["renaming_premise", "Equality(Function('C')(Symbol('\\\\varphi', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('H')(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\varphi', commutative=True)))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('C')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('H')(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))"], [["exp", 4], "Equality(exp(Pow(Function('C')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), exp(Pow(Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('H')(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True))))"], [["divide", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), exp(Pow(Function('C')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), exp(Pow(Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Function('H')(Symbol('\\\\varphi', commutative=True))), Integer(-1)), Add(Symbol('\\\\varphi', commutative=True), cos(Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), exp(Pow(Function('C')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))), Mul(E, Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))))"], [["power", 7, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), exp(Pow(Function('C')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Pow(Mul(E, Pow(Symbol('\\\\varphi', commutative=True), Integer(-1))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given B{(\\varphi)} = e^{\\varphi} and Q{(\\varphi)} = - \\varphi, then obtain B^{2}{(\\varphi)} e^{\\varphi} e^{2 Q{(\\varphi)}} + e^{3 \\varphi} e^{Q{(\\varphi)}} = B^{2}{(\\varphi)} e^{\\varphi} e^{2 Q{(\\varphi)}} + e^{2 \\varphi}", "derivation": "B{(\\varphi)} = e^{\\varphi} and B{(\\varphi)} e^{- \\varphi} = 1 and Q{(\\varphi)} = - \\varphi and B{(\\varphi)} e^{Q{(\\varphi)}} = 1 and B{(\\varphi)} e^{\\varphi} e^{Q{(\\varphi)}} = e^{\\varphi} and B^{2}{(\\varphi)} e^{Q{(\\varphi)}} = B{(\\varphi)} and B^{3}{(\\varphi)} e^{Q{(\\varphi)}} = B^{2}{(\\varphi)} and e^{3 \\varphi} e^{Q{(\\varphi)}} = e^{2 \\varphi} and B^{2}{(\\varphi)} e^{\\varphi} e^{2 Q{(\\varphi)}} + e^{3 \\varphi} e^{Q{(\\varphi)}} = B^{2}{(\\varphi)} e^{\\varphi} e^{2 Q{(\\varphi)}} + e^{2 \\varphi}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('B')(Symbol('\\\\varphi', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Integer(1))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('B')(Symbol('\\\\varphi', commutative=True)), exp(Function('Q')(Symbol('\\\\varphi', commutative=True)))), Integer(1))"], [["divide", 4, "exp(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Function('B')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)), exp(Function('Q')(Symbol('\\\\varphi', commutative=True)))), exp(Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(2)), exp(Function('Q')(Symbol('\\\\varphi', commutative=True)))), Function('B')(Symbol('\\\\varphi', commutative=True)))"], [["times", 6, "Function('B')(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(3)), exp(Function('Q')(Symbol('\\\\varphi', commutative=True)))), Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(exp(Mul(Integer(3), Symbol('\\\\varphi', commutative=True))), exp(Function('Q')(Symbol('\\\\varphi', commutative=True)))), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True))))"], [["add", 8, "Mul(Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(2)), exp(Symbol('\\\\varphi', commutative=True)), exp(Mul(Integer(2), Function('Q')(Symbol('\\\\varphi', commutative=True)))))"], "Equality(Add(Mul(Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(2)), exp(Symbol('\\\\varphi', commutative=True)), exp(Mul(Integer(2), Function('Q')(Symbol('\\\\varphi', commutative=True))))), Mul(exp(Mul(Integer(3), Symbol('\\\\varphi', commutative=True))), exp(Function('Q')(Symbol('\\\\varphi', commutative=True))))), Add(Mul(Pow(Function('B')(Symbol('\\\\varphi', commutative=True)), Integer(2)), exp(Symbol('\\\\varphi', commutative=True)), exp(Mul(Integer(2), Function('Q')(Symbol('\\\\varphi', commutative=True))))), exp(Mul(Integer(2), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given \\Psi{(t,E_{x})} = E_{x} t, then obtain \\frac{\\int 1 dE_{x}}{E_{x} t + \\log{(\\Psi{(t,E_{x})})}} = \\frac{\\int \\frac{\\log{(E_{x} t)}}{\\log{(\\Psi{(t,E_{x})})}} dE_{x}}{E_{x} t + \\log{(\\Psi{(t,E_{x})})}}", "derivation": "\\Psi{(t,E_{x})} = E_{x} t and \\log{(\\Psi{(t,E_{x})})} = \\log{(E_{x} t)} and 1 = \\frac{\\log{(E_{x} t)}}{\\log{(\\Psi{(t,E_{x})})}} and \\int 1 dE_{x} = \\int \\frac{\\log{(E_{x} t)}}{\\log{(\\Psi{(t,E_{x})})}} dE_{x} and \\frac{\\int 1 dE_{x}}{E_{x} t + \\log{(\\Psi{(t,E_{x})})}} = \\frac{\\int \\frac{\\log{(E_{x} t)}}{\\log{(\\Psi{(t,E_{x})})}} dE_{x}}{E_{x} t + \\log{(\\Psi{(t,E_{x})})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True)), Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True))), log(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True))))"], [["divide", 2, "log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True)))"], "Equality(Integer(1), Mul(log(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True))), Pow(log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True))), Integer(-1))))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('E_x', commutative=True))), Integral(Mul(log(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True))), Pow(log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True))), Integer(-1))), Tuple(Symbol('E_x', commutative=True))))"], [["divide", 4, "Add(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True)), log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True)), log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True)))), Integer(-1)), Integral(Integer(1), Tuple(Symbol('E_x', commutative=True)))), Mul(Pow(Add(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True)), log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True)))), Integer(-1)), Integral(Mul(log(Mul(Symbol('E_x', commutative=True), Symbol('t', commutative=True))), Pow(log(Function('\\\\Psi')(Symbol('t', commutative=True), Symbol('E_x', commutative=True))), Integer(-1))), Tuple(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given J{(x,g_{\\varepsilon})} = \\frac{g_{\\varepsilon}}{x}, then derive \\frac{\\partial}{\\partial g_{\\varepsilon}} J{(x,g_{\\varepsilon})} = \\frac{1}{x}, then obtain (\\frac{\\partial}{\\partial g_{\\varepsilon}} J{(x,g_{\\varepsilon})})^{x} = (\\frac{1}{x})^{x}", "derivation": "J{(x,g_{\\varepsilon})} = \\frac{g_{\\varepsilon}}{x} and \\frac{\\partial}{\\partial g_{\\varepsilon}} J{(x,g_{\\varepsilon})} = \\frac{\\partial}{\\partial g_{\\varepsilon}} \\frac{g_{\\varepsilon}}{x} and \\frac{\\partial}{\\partial g_{\\varepsilon}} J{(x,g_{\\varepsilon})} = \\frac{1}{x} and (\\frac{\\partial}{\\partial g_{\\varepsilon}} J{(x,g_{\\varepsilon})})^{x} = (\\frac{1}{x})^{x}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('J')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Pow(Symbol('x', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Derivative(Function('J')(Symbol('x', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(1))), Symbol('x', commutative=True)), Pow(Pow(Symbol('x', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"]]}, {"prompt": "Given A{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\dot{x}{(\\mathbf{M})} = \\sin{(\\mathbf{M})}, then obtain \\frac{\\frac{d}{d \\mathbf{M}} \\dot{x}{(\\mathbf{M})}}{\\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})}} = 1", "derivation": "A{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} A{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})} and \\frac{\\frac{d}{d \\mathbf{M}} A{(\\mathbf{M})}}{\\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})}} = 1 and \\dot{x}{(\\mathbf{M})} = \\sin{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} A{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\dot{x}{(\\mathbf{M})} and \\frac{\\frac{d}{d \\mathbf{M}} \\dot{x}{(\\mathbf{M})}}{\\frac{d}{d \\mathbf{M}} \\sin{(\\mathbf{M})}} = 1", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Pow(Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('A')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Derivative(Function('\\\\dot{x}')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Pow(Derivative(sin(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given c{(E_{x},i,f)} = \\frac{E_{x}}{f i}, then obtain (\\frac{\\partial}{\\partial f} (2 c{(E_{x},i,f)})^{i})^{i} = (\\frac{\\partial}{\\partial f} (\\frac{E_{x}}{f i} + c{(E_{x},i,f)})^{i})^{i}", "derivation": "c{(E_{x},i,f)} = \\frac{E_{x}}{f i} and 2 c{(E_{x},i,f)} = \\frac{E_{x}}{f i} + c{(E_{x},i,f)} and (2 c{(E_{x},i,f)})^{i} = (\\frac{E_{x}}{f i} + c{(E_{x},i,f)})^{i} and \\frac{\\partial}{\\partial f} (2 c{(E_{x},i,f)})^{i} = \\frac{\\partial}{\\partial f} (\\frac{E_{x}}{f i} + c{(E_{x},i,f)})^{i} and (\\frac{\\partial}{\\partial f} (2 c{(E_{x},i,f)})^{i})^{i} = (\\frac{\\partial}{\\partial f} (\\frac{E_{x}}{f i} + c{(E_{x},i,f)})^{i})^{i}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('E_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(-1))))"], [["add", 1, "Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))"], "Equality(Mul(Integer(2), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(-1))), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)), Pow(Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(-1))), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)))"], [["differentiate", 3, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Mul(Integer(2), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(-1))), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["power", 4, "Symbol('i', commutative=True)"], "Equality(Pow(Derivative(Pow(Mul(Integer(2), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('i', commutative=True)), Pow(Derivative(Pow(Add(Mul(Symbol('E_x', commutative=True), Pow(Symbol('f', commutative=True), Integer(-1)), Pow(Symbol('i', commutative=True), Integer(-1))), Function('c')(Symbol('E_x', commutative=True), Symbol('i', commutative=True), Symbol('f', commutative=True))), Symbol('i', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('i', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(s)} = \\sin{(\\sin{(s)})}, then derive \\frac{d}{d s} \\operatorname{C_{1}}{(s)} = \\cos{(s)} \\cos{(\\sin{(s)})}, then obtain \\frac{d}{d s} \\sin{(\\sin{(s)})} = \\cos{(s)} \\cos{(\\sin{(s)})}", "derivation": "\\operatorname{C_{1}}{(s)} = \\sin{(\\sin{(s)})} and \\frac{d}{d s} \\operatorname{C_{1}}{(s)} = \\frac{d}{d s} \\sin{(\\sin{(s)})} and \\frac{d}{d s} \\operatorname{C_{1}}{(s)} = \\cos{(s)} \\cos{(\\sin{(s)})} and \\frac{d}{d s} \\sin{(\\sin{(s)})} = \\cos{(s)} \\cos{(\\sin{(s)})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('s', commutative=True)), sin(sin(Symbol('s', commutative=True))))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(cos(Symbol('s', commutative=True)), cos(sin(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(sin(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True), Integer(1))), Mul(cos(Symbol('s', commutative=True)), cos(sin(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(\\hat{p})} = e^{\\hat{p}}, then derive \\frac{d}{d \\hat{p}} \\mathbf{p}{(\\hat{p})} = e^{\\hat{p}}, then obtain (\\mathbf{p}^{\\hat{p}}{(\\hat{p})})^{\\hat{p}} = ((\\frac{d}{d \\hat{p}} e^{\\hat{p}})^{\\hat{p}})^{\\hat{p}}", "derivation": "\\mathbf{p}{(\\hat{p})} = e^{\\hat{p}} and \\mathbf{p}^{\\hat{p}}{(\\hat{p})} = (e^{\\hat{p}})^{\\hat{p}} and (\\mathbf{p}^{\\hat{p}}{(\\hat{p})})^{\\hat{p}} = ((e^{\\hat{p}})^{\\hat{p}})^{\\hat{p}} and \\frac{d}{d \\hat{p}} \\mathbf{p}{(\\hat{p})} = \\frac{d}{d \\hat{p}} e^{\\hat{p}} and \\frac{d}{d \\hat{p}} \\mathbf{p}{(\\hat{p})} = e^{\\hat{p}} and \\frac{d}{d \\hat{p}} e^{\\hat{p}} = e^{\\hat{p}} and (\\mathbf{p}^{\\hat{p}}{(\\hat{p})})^{\\hat{p}} = ((\\frac{d}{d \\hat{p}} e^{\\hat{p}})^{\\hat{p}})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(exp(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Pow(exp(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Pow(Pow(Function('\\\\mathbf{p}')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Pow(Derivative(exp(Symbol('\\\\hat{p}', commutative=True)), Tuple(Symbol('\\\\hat{p}', commutative=True), Integer(1))), Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})}, then obtain (\\int (\\operatorname{t_{2}}{(g_{\\varepsilon})} - \\log{(e^{g_{\\varepsilon}})}) e^{g_{\\varepsilon}} dg_{\\varepsilon})^{g_{\\varepsilon}} = (\\int 0 dg_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "\\operatorname{t_{2}}{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})} and \\operatorname{t_{2}}{(g_{\\varepsilon})} - \\log{(e^{g_{\\varepsilon}})} = 0 and (\\operatorname{t_{2}}{(g_{\\varepsilon})} - \\log{(e^{g_{\\varepsilon}})}) e^{g_{\\varepsilon}} = 0 and \\int (\\operatorname{t_{2}}{(g_{\\varepsilon})} - \\log{(e^{g_{\\varepsilon}})}) e^{g_{\\varepsilon}} dg_{\\varepsilon} = \\int 0 dg_{\\varepsilon} and (\\int (\\operatorname{t_{2}}{(g_{\\varepsilon})} - \\log{(e^{g_{\\varepsilon}})}) e^{g_{\\varepsilon}} dg_{\\varepsilon})^{g_{\\varepsilon}} = (\\int 0 dg_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 1, "log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))), Integer(0))"], [["times", 2, "exp(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Add(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(0))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Add(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 4, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Integral(Mul(Add(Function('t_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))), exp(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\mathbf{s}{(A_{2})} = \\frac{d}{d A_{2}} \\cos{(A_{2})}, then derive \\mathbf{s}{(A_{2})} + 1 = 1 - \\sin{(A_{2})}, then obtain \\frac{1 - \\sin{(A_{2})}}{\\frac{d}{d A_{2}} \\cos{(A_{2})}} = \\frac{\\frac{d}{d A_{2}} \\cos{(A_{2})} + 1}{\\frac{d}{d A_{2}} \\cos{(A_{2})}}", "derivation": "\\mathbf{s}{(A_{2})} = \\frac{d}{d A_{2}} \\cos{(A_{2})} and \\mathbf{s}{(A_{2})} + 1 = \\frac{d}{d A_{2}} \\cos{(A_{2})} + 1 and \\frac{\\mathbf{s}{(A_{2})} + 1}{\\frac{d}{d A_{2}} \\cos{(A_{2})}} = \\frac{\\frac{d}{d A_{2}} \\cos{(A_{2})} + 1}{\\frac{d}{d A_{2}} \\cos{(A_{2})}} and \\mathbf{s}{(A_{2})} + 1 = 1 - \\sin{(A_{2})} and \\frac{1 - \\sin{(A_{2})}}{\\frac{d}{d A_{2}} \\cos{(A_{2})}} = \\frac{\\frac{d}{d A_{2}} \\cos{(A_{2})} + 1}{\\frac{d}{d A_{2}} \\cos{(A_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('A_2', commutative=True)), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('A_2', commutative=True)), Integer(1)), Add(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(1)))"], [["divide", 2, "Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\mathbf{s}')(Symbol('A_2', commutative=True)), Integer(1)), Pow(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(1)), Pow(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{s}')(Symbol('A_2', commutative=True)), Integer(1)), Add(Integer(1), Mul(Integer(-1), sin(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), sin(Symbol('A_2', commutative=True)))), Pow(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(1)), Pow(Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given a{(x,\\eta)} = - \\eta + \\sin{(x)}, then obtain \\log{(\\frac{\\partial}{\\partial x} a{(x,\\eta)})} = \\log{(\\cos{(x)})}", "derivation": "a{(x,\\eta)} = - \\eta + \\sin{(x)} and \\eta + a{(x,\\eta)} = \\sin{(x)} and \\frac{\\partial}{\\partial x} (\\eta + a{(x,\\eta)}) = \\frac{d}{d x} \\sin{(x)} and \\log{(\\frac{\\partial}{\\partial x} (\\eta + a{(x,\\eta)}))} = \\log{(\\frac{d}{d x} \\sin{(x)})} and \\log{(\\frac{\\partial}{\\partial x} a{(x,\\eta)})} = \\log{(\\cos{(x)})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), sin(Symbol('x', commutative=True))))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Symbol('\\\\eta', commutative=True), Function('a')(Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), sin(Symbol('x', commutative=True)))"], [["differentiate", 2, "Symbol('x', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\eta', commutative=True), Function('a')(Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["log", 3], "Equality(log(Derivative(Add(Symbol('\\\\eta', commutative=True), Function('a')(Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('x', commutative=True), Integer(1)))), log(Derivative(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(log(Derivative(Function('a')(Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1)))), log(cos(Symbol('x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\theta_2)} = \\cos{(\\theta_2)} and \\dot{x}{(\\theta_2)} = - \\cos{(\\theta_2)}, then obtain - \\operatorname{F_{g}}{(\\theta_2)} + \\operatorname{F_{g}}^{\\theta_2}{(\\theta_2)} = - \\operatorname{F_{g}}{(\\theta_2)} + \\cos^{\\theta_2}{(\\theta_2)}", "derivation": "\\operatorname{F_{g}}{(\\theta_2)} = \\cos{(\\theta_2)} and \\operatorname{F_{g}}^{\\theta_2}{(\\theta_2)} = \\cos^{\\theta_2}{(\\theta_2)} and \\operatorname{F_{g}}^{\\theta_2}{(\\theta_2)} - \\cos{(\\theta_2)} = - \\cos{(\\theta_2)} + \\cos^{\\theta_2}{(\\theta_2)} and \\dot{x}{(\\theta_2)} = - \\cos{(\\theta_2)} and \\dot{x}{(\\theta_2)} = - \\operatorname{F_{g}}{(\\theta_2)} and \\operatorname{F_{g}}^{\\theta_2}{(\\theta_2)} + \\dot{x}{(\\theta_2)} = \\dot{x}{(\\theta_2)} + \\cos^{\\theta_2}{(\\theta_2)} and - \\operatorname{F_{g}}{(\\theta_2)} + \\operatorname{F_{g}}^{\\theta_2}{(\\theta_2)} = - \\operatorname{F_{g}}{(\\theta_2)} + \\cos^{\\theta_2}{(\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\theta_2', commutative=True)), cos(Symbol('\\\\theta_2', commutative=True)))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('F_g')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)))"], [["minus", 2, "cos(Symbol('\\\\theta_2', commutative=True))"], "Equality(Add(Pow(Function('F_g')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True)), Mul(Integer(-1), Function('F_g')(Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Pow(Function('F_g')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True))), Add(Function('\\\\dot{x}')(Symbol('\\\\theta_2', commutative=True)), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\theta_2', commutative=True))), Pow(Function('F_g')(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))), Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\theta_2', commutative=True))), Pow(cos(Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(t_{1})} = \\int e^{t_{1}} dt_{1}, then derive \\operatorname{J_{\\varepsilon}}{(t_{1})} = \\dot{y} + e^{t_{1}}, then obtain - \\mathbf{p} + u + e^{t_{1}} = \\dot{y} - \\mathbf{p} + e^{t_{1}}", "derivation": "\\operatorname{J_{\\varepsilon}}{(t_{1})} = \\int e^{t_{1}} dt_{1} and \\operatorname{J_{\\varepsilon}}{(t_{1})} = \\dot{y} + e^{t_{1}} and - \\mathbf{p} + \\operatorname{J_{\\varepsilon}}{(t_{1})} = \\dot{y} - \\mathbf{p} + e^{t_{1}} and - \\mathbf{p} + \\int e^{t_{1}} dt_{1} = \\dot{y} - \\mathbf{p} + e^{t_{1}} and - \\mathbf{p} + u + e^{t_{1}} = \\dot{y} - \\mathbf{p} + e^{t_{1}}", "srepr_derivation": [["get_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True)), Integral(exp(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), exp(Symbol('t_1', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('t_1', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('t_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Integral(exp(Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('t_1', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Symbol('u', commutative=True), exp(Symbol('t_1', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), exp(Symbol('t_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})} = \\frac{\\log{(m_{s})}}{\\hat{H}_{\\lambda}}, then obtain \\frac{\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})}}{\\log{(m_{s})}} + 1 + \\frac{1}{\\log{(m_{s})}} = 2 + \\frac{1}{\\log{(m_{s})}}", "derivation": "\\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})} = \\frac{\\log{(m_{s})}}{\\hat{H}_{\\lambda}} and \\frac{\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})}}{\\log{(m_{s})}} = 1 and \\frac{\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})}}{\\log{(m_{s})}} + \\frac{1}{\\log{(m_{s})}} = 1 + \\frac{1}{\\log{(m_{s})}} and \\frac{\\hat{H}_{\\lambda} \\operatorname{t_{2}}{(m_{s},\\hat{H}_{\\lambda})}}{\\log{(m_{s})}} + 1 + \\frac{1}{\\log{(m_{s})}} = 2 + \\frac{1}{\\log{(m_{s})}}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('m_s', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), log(Symbol('m_s', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(-1)), log(Symbol('m_s', commutative=True)))"], "Equality(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('m_s', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Pow(log(Symbol('m_s', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('m_s', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))), Add(Integer(1), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('t_2')(Symbol('m_s', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))), Integer(1), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))), Add(Integer(2), Pow(log(Symbol('m_s', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{x}_0{(P_{e},\\mathbf{J})} = \\mathbf{J} + \\cos{(P_{e})}, then derive \\int (P_{e} + \\hat{x}_0{(P_{e},\\mathbf{J})} - 1) d\\mathbf{J} = G + \\frac{\\mathbf{J}^{2}}{2} + \\mathbf{J} (P_{e} + \\cos{(P_{e})} - 1), then obtain G + \\frac{\\mathbf{J}^{2}}{2} + \\mathbf{J} (P_{e} + \\cos{(P_{e})} - 1) = \\int (P_{e} + \\mathbf{J} + \\cos{(P_{e})} - 1) d\\mathbf{J}", "derivation": "\\hat{x}_0{(P_{e},\\mathbf{J})} = \\mathbf{J} + \\cos{(P_{e})} and P_{e} + \\hat{x}_0{(P_{e},\\mathbf{J})} = P_{e} + \\mathbf{J} + \\cos{(P_{e})} and P_{e} + \\hat{x}_0{(P_{e},\\mathbf{J})} - 1 = P_{e} + \\mathbf{J} + \\cos{(P_{e})} - 1 and \\int (P_{e} + \\hat{x}_0{(P_{e},\\mathbf{J})} - 1) d\\mathbf{J} = \\int (P_{e} + \\mathbf{J} + \\cos{(P_{e})} - 1) d\\mathbf{J} and \\int (P_{e} + \\hat{x}_0{(P_{e},\\mathbf{J})} - 1) d\\mathbf{J} = G + \\frac{\\mathbf{J}^{2}}{2} + \\mathbf{J} (P_{e} + \\cos{(P_{e})} - 1) and G + \\frac{\\mathbf{J}^{2}}{2} + \\mathbf{J} (P_{e} + \\cos{(P_{e})} - 1) = \\int (P_{e} + \\mathbf{J} + \\cos{(P_{e})} - 1) d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["add", 1, "Symbol('P_e', commutative=True)"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('P_e', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('P_e', commutative=True)), Integer(-1)))"], [["integrate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Add(Symbol('P_e', commutative=True), Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('P_e', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Add(Symbol('P_e', commutative=True), Function('\\\\hat{x}_0')(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('G', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('G', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Add(Symbol('P_e', commutative=True), cos(Symbol('P_e', commutative=True)), Integer(-1)))), Integral(Add(Symbol('P_e', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), cos(Symbol('P_e', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\hbar,v_{x})} = \\hbar + v_{x}, then obtain \\int (\\int (\\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})}) d\\hbar)^{v_{x}} d\\hbar = \\chi + \\int (\\hbar^{2} + \\hbar v_{x})^{v_{x}} d\\hbar", "derivation": "\\operatorname{f^{\\prime}}{(\\hbar,v_{x})} = \\hbar + v_{x} and \\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})} = 2 \\hbar + v_{x} and \\int (\\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})}) d\\hbar = \\int (2 \\hbar + v_{x}) d\\hbar and (\\int (\\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})}) d\\hbar)^{v_{x}} = (\\int (2 \\hbar + v_{x}) d\\hbar)^{v_{x}} and \\int (\\int (\\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})}) d\\hbar)^{v_{x}} d\\hbar = \\int (\\int (2 \\hbar + v_{x}) d\\hbar)^{v_{x}} d\\hbar and \\int (\\int (\\hbar + \\operatorname{f^{\\prime}}{(\\hbar,v_{x})}) d\\hbar)^{v_{x}} d\\hbar = \\chi + \\int (\\hbar^{2} + \\hbar v_{x})^{v_{x}} d\\hbar", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True)))"], [["add", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Symbol('v_x', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\hbar', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["power", 3, "Symbol('v_x', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('\\\\hbar', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('v_x', commutative=True)), Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('v_x', commutative=True)))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Pow(Integral(Add(Symbol('\\\\hbar', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Pow(Integral(Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Pow(Integral(Add(Symbol('\\\\hbar', commutative=True), Function('f^{\\\\prime}')(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True))), Add(Symbol('\\\\chi', commutative=True), Integral(Pow(Add(Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('v_x', commutative=True))), Symbol('v_x', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{F}{(F_{N})} = e^{F_{N}}, then obtain \\int (F_{N} \\mathbf{F}{(F_{N})} + e^{F_{N}} - 1) dF_{N} = \\int (F_{N} e^{F_{N}} + e^{F_{N}} - 1) dF_{N}", "derivation": "\\mathbf{F}{(F_{N})} = e^{F_{N}} and F_{N} \\mathbf{F}{(F_{N})} = F_{N} e^{F_{N}} and F_{N} \\mathbf{F}{(F_{N})} - 1 = F_{N} e^{F_{N}} - 1 and F_{N} \\mathbf{F}{(F_{N})} + e^{F_{N}} - 1 = F_{N} e^{F_{N}} + e^{F_{N}} - 1 and \\int (F_{N} \\mathbf{F}{(F_{N})} + e^{F_{N}} - 1) dF_{N} = \\int (F_{N} e^{F_{N}} + e^{F_{N}} - 1) dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('F_N', commutative=True)), exp(Symbol('F_N', commutative=True)))"], [["times", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_N', commutative=True))), Mul(Symbol('F_N', commutative=True), exp(Symbol('F_N', commutative=True))))"], [["minus", 2, 1], "Equality(Add(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_N', commutative=True))), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), exp(Symbol('F_N', commutative=True))), Integer(-1)))"], [["add", 3, "exp(Symbol('F_N', commutative=True))"], "Equality(Add(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True)), Integer(-1)), Add(Mul(Symbol('F_N', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True)), Integer(-1)))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('F_N', commutative=True), Function('\\\\mathbf{F}')(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True))), Integral(Add(Mul(Symbol('F_N', commutative=True), exp(Symbol('F_N', commutative=True))), exp(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = (f^{*})^{f_{\\mathbf{p}}}, then derive \\frac{\\partial}{\\partial f^{*}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = \\frac{(f^{*})^{f_{\\mathbf{p}}} f_{\\mathbf{p}}}{f^{*}}, then obtain \\frac{\\partial}{\\partial f^{*}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})}}{f^{*}}", "derivation": "\\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = (f^{*})^{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial f^{*}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = \\frac{\\partial}{\\partial f^{*}} (f^{*})^{f_{\\mathbf{p}}} and \\frac{\\partial}{\\partial f^{*}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = \\frac{(f^{*})^{f_{\\mathbf{p}}} f_{\\mathbf{p}}}{f^{*}} and \\frac{\\partial}{\\partial f^{*}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})} = \\frac{f_{\\mathbf{p}} \\mathbf{M}{(f^{*},f_{\\mathbf{p}})}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Derivative(Pow(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f^*', commutative=True), Integer(1))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('\\\\mathbf{M}')(Symbol('f^*', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"]]}, {"prompt": "Given Q{(r,\\rho)} = \\frac{\\partial}{\\partial \\rho} \\rho^{r}, then derive Q{(r,\\rho)} = \\frac{\\rho^{r} r}{\\rho}, then derive \\frac{\\partial}{\\partial \\rho} Q{(r,\\rho)} = \\frac{\\rho^{r} r^{2}}{\\rho^{2}} - \\frac{\\rho^{r} r}{\\rho^{2}}, then obtain \\frac{\\partial}{\\partial \\rho} \\frac{\\rho^{r} r}{\\rho} = \\frac{\\rho^{r} r^{2}}{\\rho^{2}} - \\frac{\\rho^{r} r}{\\rho^{2}}", "derivation": "Q{(r,\\rho)} = \\frac{\\partial}{\\partial \\rho} \\rho^{r} and Q{(r,\\rho)} = \\frac{\\rho^{r} r}{\\rho} and \\frac{\\partial}{\\partial \\rho} Q{(r,\\rho)} = \\frac{\\partial}{\\partial \\rho} \\frac{\\rho^{r} r}{\\rho} and \\frac{\\partial}{\\partial \\rho} Q{(r,\\rho)} = \\frac{\\rho^{r} r^{2}}{\\rho^{2}} - \\frac{\\rho^{r} r}{\\rho^{2}} and \\frac{\\partial}{\\partial \\rho} \\frac{\\rho^{r} r}{\\rho} = \\frac{\\rho^{r} r^{2}}{\\rho^{2}} - \\frac{\\rho^{r} r}{\\rho^{2}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Derivative(Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('Q')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('Q')(Symbol('r', commutative=True), Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Add(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Pow(Symbol('r', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Pow(Symbol('\\\\rho', commutative=True), Symbol('r', commutative=True)), Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(I)} = \\sin{(I)}, then obtain \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} + \\sin^{2}{(I)} + \\sin{(I)} = \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} - \\operatorname{A_{x}}{(I)} + \\sin^{2}{(I)} + 2 \\sin{(I)}", "derivation": "\\operatorname{A_{x}}{(I)} = \\sin{(I)} and I + \\operatorname{A_{x}}{(I)} = I + \\sin{(I)} and \\sin{(I)} = - \\operatorname{A_{x}}{(I)} + 2 \\sin{(I)} and \\frac{I + \\sin{(I)}}{\\operatorname{A_{x}}{(I)}} + \\sin{(I)} = \\frac{I + \\sin{(I)}}{\\operatorname{A_{x}}{(I)}} - \\operatorname{A_{x}}{(I)} + 2 \\sin{(I)} and \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} + \\sin{(I)} = \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} - \\operatorname{A_{x}}{(I)} + 2 \\sin{(I)} and \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} + \\sin^{2}{(I)} + \\sin{(I)} = \\frac{I + \\operatorname{A_{x}}{(I)}}{\\operatorname{A_{x}}{(I)}} - \\operatorname{A_{x}}{(I)} + \\sin^{2}{(I)} + 2 \\sin{(I)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["add", 1, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Function('A_x')(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Function('A_x')(Symbol('I', commutative=True))), sin(Symbol('I', commutative=True)))"], "Equality(sin(Symbol('I', commutative=True)), Add(Mul(Integer(-1), Function('A_x')(Symbol('I', commutative=True))), Mul(Integer(2), sin(Symbol('I', commutative=True)))))"], [["minus", 3, "Mul(Integer(-1), Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), sin(Symbol('I', commutative=True))), Add(Mul(Add(Symbol('I', commutative=True), sin(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('A_x')(Symbol('I', commutative=True))), Mul(Integer(2), sin(Symbol('I', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Add(Symbol('I', commutative=True), Function('A_x')(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), sin(Symbol('I', commutative=True))), Add(Mul(Add(Symbol('I', commutative=True), Function('A_x')(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('A_x')(Symbol('I', commutative=True))), Mul(Integer(2), sin(Symbol('I', commutative=True)))))"], [["minus", 5, "Mul(Integer(-1), Pow(sin(Symbol('I', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Add(Symbol('I', commutative=True), Function('A_x')(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), Pow(sin(Symbol('I', commutative=True)), Integer(2)), sin(Symbol('I', commutative=True))), Add(Mul(Add(Symbol('I', commutative=True), Function('A_x')(Symbol('I', commutative=True))), Pow(Function('A_x')(Symbol('I', commutative=True)), Integer(-1))), Mul(Integer(-1), Function('A_x')(Symbol('I', commutative=True))), Pow(sin(Symbol('I', commutative=True)), Integer(2)), Mul(Integer(2), sin(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\dot{x},f)} = \\dot{x} f, then obtain \\frac{f}{\\dot{x} (\\dot{x} f + f)} = \\frac{f^{2}}{(\\dot{x} f + f) \\operatorname{a^{\\dagger}}{(\\dot{x},f)}}", "derivation": "\\operatorname{a^{\\dagger}}{(\\dot{x},f)} = \\dot{x} f and \\frac{\\operatorname{a^{\\dagger}}{(\\dot{x},f)}}{\\dot{x} f + f} = \\frac{\\dot{x} f}{\\dot{x} f + f} and \\frac{(\\dot{x} f + f) \\operatorname{a^{\\dagger}}{(\\dot{x},f)}}{\\dot{x} f} = \\dot{x} f + f and \\frac{\\dot{x} f}{\\dot{x} f + f} = \\frac{\\dot{x}^{2} f^{2}}{(\\dot{x} f + f) \\operatorname{a^{\\dagger}}{(\\dot{x},f)}} and \\frac{f}{\\dot{x} (\\dot{x} f + f)} = \\frac{f^{2}}{(\\dot{x} f + f) \\operatorname{a^{\\dagger}}{(\\dot{x},f)}}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)))"], [["divide", 1, "Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1))))"], [["divide", 1, "Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Pow(Symbol('f', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True))), Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)), Pow(Symbol('f', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"], [["divide", 4, "Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2))"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Symbol('f', commutative=True), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1))), Mul(Pow(Symbol('f', commutative=True), Integer(2)), Pow(Add(Mul(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Integer(-1)), Pow(Function('a^{\\\\dagger}')(Symbol('\\\\dot{x}', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(b,J)} = \\frac{J}{b} and x{(b,J)} = \\frac{\\partial}{\\partial b} (\\frac{J}{b})^{b}, then obtain (\\int x{(b,J)} dJ)^{J} = (\\int \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}^{b}{(b,J)} dJ)^{J}", "derivation": "\\operatorname{t_{1}}{(b,J)} = \\frac{J}{b} and \\operatorname{t_{1}}^{b}{(b,J)} = (\\frac{J}{b})^{b} and x{(b,J)} = \\frac{\\partial}{\\partial b} (\\frac{J}{b})^{b} and x{(b,J)} = \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}^{b}{(b,J)} and \\int x{(b,J)} dJ = \\int \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}^{b}{(b,J)} dJ and (\\int x{(b,J)} dJ)^{J} = (\\int \\frac{\\partial}{\\partial b} \\operatorname{t_{1}}^{b}{(b,J)} dJ)^{J}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('b', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Symbol('b', commutative=True)), Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('x')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Derivative(Pow(Mul(Symbol('J', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('x')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Derivative(Pow(Function('t_1')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('J', commutative=True)"], "Equality(Integral(Function('x')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Derivative(Pow(Function('t_1')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))))"], [["power", 5, "Symbol('J', commutative=True)"], "Equality(Pow(Integral(Function('x')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Integral(Derivative(Pow(Function('t_1')(Symbol('b', commutative=True), Symbol('J', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(k,v)} = k + v, then obtain (- \\operatorname{f_{E}}{(k,v)} + \\int 1 dv) \\sin{(y^{\\prime})} = (- \\operatorname{f_{E}}{(k,v)} + \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv) \\sin{(y^{\\prime})}", "derivation": "\\operatorname{f_{E}}{(k,v)} = k + v and 1 = \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} and \\int 1 dv = \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv and - \\operatorname{f_{E}}{(k,v)} = - k - v and - \\operatorname{f_{E}}{(k,v)} + \\int 1 dv = - \\operatorname{f_{E}}{(k,v)} + \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv and - k - v + \\int 1 dv = - k - v + \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv and (- k - v + \\int 1 dv) \\sin{(y^{\\prime})} = (- k - v + \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv) \\sin{(y^{\\prime})} and (- \\operatorname{f_{E}}{(k,v)} + \\int 1 dv) \\sin{(y^{\\prime})} = (- \\operatorname{f_{E}}{(k,v)} + \\int \\frac{k + v}{\\operatorname{f_{E}}{(k,v)}} dv) \\sin{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Add(Symbol('k', commutative=True), Symbol('v', commutative=True)))"], [["divide", 1, "Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('v', commutative=True))), Integral(Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Integral(Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Integral(Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)))))"], [["times", 6, "sin(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), sin(Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('v', commutative=True)), Integral(Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)))), sin(Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Integral(Integer(1), Tuple(Symbol('v', commutative=True)))), sin(Symbol('y^{\\\\prime}', commutative=True))), Mul(Add(Mul(Integer(-1), Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True))), Integral(Mul(Add(Symbol('k', commutative=True), Symbol('v', commutative=True)), Pow(Function('f_E')(Symbol('k', commutative=True), Symbol('v', commutative=True)), Integer(-1))), Tuple(Symbol('v', commutative=True)))), sin(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given t{(\\tilde{g}^*,y^{\\prime})} = - \\tilde{g}^* + y^{\\prime}, then obtain \\frac{\\int 2 t{(\\tilde{g}^*,y^{\\prime})} d\\tilde{g}^*}{\\dot{x}} = \\frac{\\int (- 2 \\tilde{g}^* + 2 y^{\\prime}) d\\tilde{g}^*}{\\dot{x}}", "derivation": "t{(\\tilde{g}^*,y^{\\prime})} = - \\tilde{g}^* + y^{\\prime} and - \\tilde{g}^* + y^{\\prime} + t{(\\tilde{g}^*,y^{\\prime})} = - 2 \\tilde{g}^* + 2 y^{\\prime} and 2 t{(\\tilde{g}^*,y^{\\prime})} = - 2 \\tilde{g}^* + 2 y^{\\prime} and \\int 2 t{(\\tilde{g}^*,y^{\\prime})} d\\tilde{g}^* = \\int (- 2 \\tilde{g}^* + 2 y^{\\prime}) d\\tilde{g}^* and \\frac{\\int 2 t{(\\tilde{g}^*,y^{\\prime})} d\\tilde{g}^*}{\\dot{x}} = \\frac{\\int (- 2 \\tilde{g}^* + 2 y^{\\prime}) d\\tilde{g}^*}{\\dot{x}}", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), Symbol('y^{\\\\prime}', commutative=True), Function('t')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('t')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('t')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["divide", 4, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Integral(Mul(Integer(2), Function('t')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(2), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given k{(A_{z},B)} = \\int \\frac{A_{z}}{B} dB, then obtain ((k{(A_{z},B)} - 1) k^{A_{z}}{(A_{z},B)})^{A_{z}} = ((\\int \\frac{A_{z}}{B} dB - 1) k^{A_{z}}{(A_{z},B)})^{A_{z}}", "derivation": "k{(A_{z},B)} = \\int \\frac{A_{z}}{B} dB and k^{A_{z}}{(A_{z},B)} = (\\int \\frac{A_{z}}{B} dB)^{A_{z}} and k{(A_{z},B)} - 1 = \\int \\frac{A_{z}}{B} dB - 1 and (k{(A_{z},B)} - 1) (\\int \\frac{A_{z}}{B} dB)^{A_{z}} = (\\int \\frac{A_{z}}{B} dB - 1) (\\int \\frac{A_{z}}{B} dB)^{A_{z}} and (k{(A_{z},B)} - 1) k^{A_{z}}{(A_{z},B)} = (\\int \\frac{A_{z}}{B} dB - 1) k^{A_{z}}{(A_{z},B)} and ((k{(A_{z},B)} - 1) k^{A_{z}}{(A_{z},B)})^{A_{z}} = ((\\int \\frac{A_{z}}{B} dB - 1) k^{A_{z}}{(A_{z},B)})^{A_{z}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))))"], [["power", 1, "Symbol('A_z', commutative=True)"], "Equality(Pow(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Symbol('A_z', commutative=True)), Pow(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Symbol('A_z', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Add(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Integer(-1)))"], [["times", 3, "Pow(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Symbol('A_z', commutative=True))"], "Equality(Mul(Add(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Symbol('A_z', commutative=True))), Mul(Add(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Pow(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Symbol('A_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Symbol('A_z', commutative=True))), Mul(Add(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Pow(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Symbol('A_z', commutative=True))))"], [["power", 5, "Symbol('A_z', commutative=True)"], "Equality(Pow(Mul(Add(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Integer(-1)), Pow(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)), Pow(Mul(Add(Integral(Mul(Symbol('A_z', commutative=True), Pow(Symbol('B', commutative=True), Integer(-1))), Tuple(Symbol('B', commutative=True))), Integer(-1)), Pow(Function('k')(Symbol('A_z', commutative=True), Symbol('B', commutative=True)), Symbol('A_z', commutative=True))), Symbol('A_z', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} = \\theta_1 (H - \\mathbf{F}), then obtain (- \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} - 1)^{H} = (- \\theta_1 (H - \\mathbf{F}) - 1)^{H}", "derivation": "\\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} = \\theta_1 (H - \\mathbf{F}) and \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} = H \\theta_1 - \\mathbf{F} \\theta_1 and - \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} = - H \\theta_1 + \\mathbf{F} \\theta_1 and - \\theta_1 (H - \\mathbf{F}) = - H \\theta_1 + \\mathbf{F} \\theta_1 and - \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} - 1 = - H \\theta_1 + \\mathbf{F} \\theta_1 - 1 and (- \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} - 1)^{H} = (- H \\theta_1 + \\mathbf{F} \\theta_1 - 1)^{H} and (- \\operatorname{C_{1}}{(\\mathbf{F},\\theta_1,H)} - 1)^{H} = (- \\theta_1 (H - \\mathbf{F}) - 1)^{H}", "srepr_derivation": [["get_premise", "Equality(Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["expand", 1], "Equality(Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True)), Add(Mul(Symbol('H', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)))"], [["power", 5, "Symbol('H', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('H', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Integer(-1)), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Pow(Add(Mul(Integer(-1), Function('C_1')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('H', commutative=True))), Integer(-1)), Symbol('H', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)))), Integer(-1)), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(A_{2},Q)} = A_{2} + Q and \\operatorname{C_{2}}{(A_{2},Q)} = \\int \\operatorname{t_{1}}{(A_{2},Q)} dQ, then obtain Q + \\int (- Q + \\operatorname{C_{2}}{(A_{2},Q)}) dQ = Q + \\int (- Q + \\int (A_{2} + Q) dQ) dQ", "derivation": "\\operatorname{t_{1}}{(A_{2},Q)} = A_{2} + Q and \\int \\operatorname{t_{1}}{(A_{2},Q)} dQ = \\int (A_{2} + Q) dQ and \\operatorname{C_{2}}{(A_{2},Q)} = \\int \\operatorname{t_{1}}{(A_{2},Q)} dQ and \\operatorname{C_{2}}{(A_{2},Q)} = \\int (A_{2} + Q) dQ and - Q + \\operatorname{C_{2}}{(A_{2},Q)} = - Q + \\int (A_{2} + Q) dQ and \\int (- Q + \\operatorname{C_{2}}{(A_{2},Q)}) dQ = \\int (- Q + \\int (A_{2} + Q) dQ) dQ and Q + \\int (- Q + \\operatorname{C_{2}}{(A_{2},Q)}) dQ = Q + \\int (- Q + \\int (A_{2} + Q) dQ) dQ", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Integral(Function('t_1')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('C_2')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["minus", 4, "Symbol('Q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('C_2')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))))"], [["integrate", 5, "Symbol('Q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('C_2')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True))))"], [["add", 6, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Function('C_2')(Symbol('A_2', commutative=True), Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True)))), Add(Symbol('Q', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Integral(Add(Symbol('A_2', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(C)} = \\log{(C)}, then obtain \\frac{d}{d C} (\\operatorname{M_{E}}^{C}{(C)} \\log{(C)})^{C} = \\frac{d}{d C} (\\log{(C)} \\log{(C)}^{C})^{C}", "derivation": "\\operatorname{M_{E}}{(C)} = \\log{(C)} and \\operatorname{M_{E}}^{C}{(C)} = \\log{(C)}^{C} and \\operatorname{M_{E}}^{C}{(C)} \\log{(C)} = \\log{(C)} \\log{(C)}^{C} and (\\operatorname{M_{E}}^{C}{(C)} \\log{(C)})^{C} = (\\log{(C)} \\log{(C)}^{C})^{C} and \\frac{d}{d C} (\\operatorname{M_{E}}^{C}{(C)} \\log{(C)})^{C} = \\frac{d}{d C} (\\log{(C)} \\log{(C)}^{C})^{C}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["power", 1, "Symbol('C', commutative=True)"], "Equality(Pow(Function('M_E')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True)))"], [["times", 2, "log(Symbol('C', commutative=True))"], "Equality(Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Mul(log(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))))"], [["power", 3, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(log(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Symbol('C', commutative=True)))"], [["differentiate", 4, "Symbol('C', commutative=True)"], "Equality(Derivative(Pow(Mul(Pow(Function('M_E')(Symbol('C', commutative=True)), Symbol('C', commutative=True)), log(Symbol('C', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(Mul(log(Symbol('C', commutative=True)), Pow(log(Symbol('C', commutative=True)), Symbol('C', commutative=True))), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})}, then obtain \\iint (\\operatorname{v_{y}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})}) dg_{\\varepsilon} dg_{\\varepsilon} = \\iint 2 \\log{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "derivation": "\\operatorname{v_{y}}{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} and \\operatorname{v_{y}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})} = 2 \\log{(g_{\\varepsilon})} and \\int (\\operatorname{v_{y}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})}) dg_{\\varepsilon} = \\int 2 \\log{(g_{\\varepsilon})} dg_{\\varepsilon} and \\iint (\\operatorname{v_{y}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})}) dg_{\\varepsilon} dg_{\\varepsilon} = \\iint 2 \\log{(g_{\\varepsilon})} dg_{\\varepsilon} dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "log(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(2), log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(2), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Function('v_y')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Integer(2), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given t{(t_{1},r,f)} = f + r + t_{1}, then obtain \\frac{- t_{1} (f + r + t_{1}) + t{(t_{1},r,f)}}{t_{1} t{(t_{1},r,f)}} = \\frac{f + r - t_{1} (f + r + t_{1}) + t_{1}}{t_{1} t{(t_{1},r,f)}}", "derivation": "t{(t_{1},r,f)} = f + r + t_{1} and t_{1} t{(t_{1},r,f)} = t_{1} (f + r + t_{1}) and - t_{1} t{(t_{1},r,f)} + t{(t_{1},r,f)} = f + r - t_{1} t{(t_{1},r,f)} + t_{1} and - t_{1} (f + r + t_{1}) + t{(t_{1},r,f)} = f + r - t_{1} (f + r + t_{1}) + t_{1} and \\frac{- t_{1} (f + r + t_{1}) + t{(t_{1},r,f)}}{t_{1} t{(t_{1},r,f)}} = \\frac{f + r - t_{1} (f + r + t_{1}) + t_{1}}{t_{1} t{(t_{1},r,f)}}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True)), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True)))"], [["times", 1, "Symbol('t_1', commutative=True)"], "Equality(Mul(Symbol('t_1', commutative=True), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Mul(Symbol('t_1', commutative=True), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True))))"], [["minus", 1, "Mul(Symbol('t_1', commutative=True), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Symbol('t_1', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True))), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)))"], [["divide", 4, "Mul(Symbol('t_1', commutative=True), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True)))"], "Equality(Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('t_1', commutative=True), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True))), Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True))), Pow(Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(-1))), Mul(Pow(Symbol('t_1', commutative=True), Integer(-1)), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True), Add(Symbol('f', commutative=True), Symbol('r', commutative=True), Symbol('t_1', commutative=True))), Symbol('t_1', commutative=True)), Pow(Function('t')(Symbol('t_1', commutative=True), Symbol('r', commutative=True), Symbol('f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given f{(f_{\\mathbf{v}},l)} = \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}}, then obtain 4 l^{2} f^{2}{(f_{\\mathbf{v}},l)} = l^{2} (f{(f_{\\mathbf{v}},l)} + \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}})^{2}", "derivation": "f{(f_{\\mathbf{v}},l)} = \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and 2 f{(f_{\\mathbf{v}},l)} = f{(f_{\\mathbf{v}},l)} + \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}} and 2 l f{(f_{\\mathbf{v}},l)} = l (f{(f_{\\mathbf{v}},l)} + \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}}) and 4 l^{2} f^{2}{(f_{\\mathbf{v}},l)} = l^{2} (f{(f_{\\mathbf{v}},l)} + \\int l^{f_{\\mathbf{v}}} df_{\\mathbf{v}})^{2}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["add", 1, "Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Integer(2), Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True))), Add(Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))))"], [["times", 2, "Symbol('l', commutative=True)"], "Equality(Mul(Integer(2), Symbol('l', commutative=True), Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True))))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Symbol('l', commutative=True), Integer(2)), Pow(Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integer(2))), Mul(Pow(Symbol('l', commutative=True), Integer(2)), Pow(Add(Function('f')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('l', commutative=True)), Integral(Pow(Symbol('l', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{v}}', commutative=True)))), Integer(2))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)} = \\psi^{\\hat{H}_l}, then obtain (- \\psi)^{\\psi} (\\psi^{\\hat{H}_l})^{\\hat{H}_l} = (- \\psi + \\psi^{\\hat{H}_l} - \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)})^{\\psi} (\\psi^{\\hat{H}_l})^{\\hat{H}_l}", "derivation": "\\Psi_{\\lambda}{(\\psi,\\hat{H}_l)} = \\psi^{\\hat{H}_l} and 0 = \\psi^{\\hat{H}_l} - \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)} and - \\psi = - \\psi + \\psi^{\\hat{H}_l} - \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)} and (- \\psi)^{\\psi} = (- \\psi + \\psi^{\\hat{H}_l} - \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)})^{\\psi} and (- \\psi)^{\\psi} (\\psi^{\\hat{H}_l})^{\\hat{H}_l} = (- \\psi + \\psi^{\\hat{H}_l} - \\Psi_{\\lambda}{(\\psi,\\hat{H}_l)})^{\\psi} (\\psi^{\\hat{H}_l})^{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Integer(0), Add(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["minus", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))))"], [["power", 3, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\psi', commutative=True)))"], [["times", 4, "Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('\\\\psi', commutative=True)), Pow(Pow(Symbol('\\\\psi', commutative=True), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}{(F_{g})} = \\cos{(F_{g})}, then obtain \\int \\frac{d}{d F_{g}} 2 \\tilde{g}{(F_{g})} dF_{g} = \\int \\frac{d}{d F_{g}} (\\tilde{g}{(F_{g})} + \\cos{(F_{g})}) dF_{g}", "derivation": "\\tilde{g}{(F_{g})} = \\cos{(F_{g})} and 2 \\tilde{g}{(F_{g})} = \\tilde{g}{(F_{g})} + \\cos{(F_{g})} and \\frac{d}{d F_{g}} 2 \\tilde{g}{(F_{g})} = \\frac{d}{d F_{g}} (\\tilde{g}{(F_{g})} + \\cos{(F_{g})}) and \\int \\frac{d}{d F_{g}} 2 \\tilde{g}{(F_{g})} dF_{g} = \\int \\frac{d}{d F_{g}} (\\tilde{g}{(F_{g})} + \\cos{(F_{g})}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True)))"], [["add", 1, "Function('\\\\tilde{g}')(Symbol('F_g', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True))), Add(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Add(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('F_g', commutative=True)"], "Equality(Integral(Derivative(Mul(Integer(2), Function('\\\\tilde{g}')(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))), Integral(Derivative(Add(Function('\\\\tilde{g}')(Symbol('F_g', commutative=True)), cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given f{(\\phi_2,J)} = J \\phi_2, then obtain - \\frac{\\partial}{\\partial J} (J \\phi_2 - \\phi_2) + \\int f{(\\phi_2,J)} d\\phi_2 = - \\frac{\\partial}{\\partial J} (J \\phi_2 - \\phi_2) + \\int J \\phi_2 d\\phi_2", "derivation": "f{(\\phi_2,J)} = J \\phi_2 and \\int f{(\\phi_2,J)} d\\phi_2 = \\int J \\phi_2 d\\phi_2 and - \\phi_2 + f{(\\phi_2,J)} = J \\phi_2 - \\phi_2 and - \\frac{\\partial}{\\partial J} (- \\phi_2 + f{(\\phi_2,J)}) + \\int f{(\\phi_2,J)} d\\phi_2 = - \\frac{\\partial}{\\partial J} (- \\phi_2 + f{(\\phi_2,J)}) + \\int J \\phi_2 d\\phi_2 and - \\frac{\\partial}{\\partial J} (J \\phi_2 - \\phi_2) + \\int f{(\\phi_2,J)} d\\phi_2 = - \\frac{\\partial}{\\partial J} (J \\phi_2 - \\phi_2) + \\int J \\phi_2 d\\phi_2", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True)), Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True))))"], [["minus", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True))), Add(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))))"], [["minus", 2, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Derivative(Add(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Function('f')(Symbol('\\\\phi_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Derivative(Add(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1)))), Integral(Mul(Symbol('J', commutative=True), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True)))))"]]}, {"prompt": "Given J{(\\phi)} = e^{\\phi}, then derive \\int J{(\\phi)} d\\phi = f_{\\mathbf{v}} + e^{\\phi}, then obtain \\int J{(\\phi)} d\\phi = f_{\\mathbf{v}} + J{(\\phi)}", "derivation": "J{(\\phi)} = e^{\\phi} and \\int J{(\\phi)} d\\phi = \\int e^{\\phi} d\\phi and \\int J{(\\phi)} d\\phi = f_{\\mathbf{v}} + e^{\\phi} and \\int J{(\\phi)} d\\phi = f_{\\mathbf{v}} + J{(\\phi)}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\phi', commutative=True)), exp(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(exp(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), exp(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('J')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('J')(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\omega{(\\hat{H},\\theta_1)} = \\hat{H} + \\theta_1, then obtain \\int \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} (- \\hat{H} - \\theta_1 + \\omega{(\\hat{H},\\theta_1)}) d\\hat{H} = \\int \\frac{d^{2}}{d \\hat{H}^{2}} 0 d\\hat{H}", "derivation": "\\omega{(\\hat{H},\\theta_1)} = \\hat{H} + \\theta_1 and - \\hat{H} - \\theta_1 + \\omega{(\\hat{H},\\theta_1)} = 0 and \\frac{\\partial}{\\partial \\hat{H}} (- \\hat{H} - \\theta_1 + \\omega{(\\hat{H},\\theta_1)}) = \\frac{d}{d \\hat{H}} 0 and \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} (- \\hat{H} - \\theta_1 + \\omega{(\\hat{H},\\theta_1)}) = \\frac{d^{2}}{d \\hat{H}^{2}} 0 and \\int \\frac{\\partial^{2}}{\\partial \\hat{H}^{2}} (- \\hat{H} - \\theta_1 + \\omega{(\\hat{H},\\theta_1)}) d\\hat{H} = \\int \\frac{d^{2}}{d \\hat{H}^{2}} 0 d\\hat{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\omega')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{H}', commutative=True))), Integral(Derivative(Integer(0), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(2))), Tuple(Symbol('\\\\hat{H}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(v_{t},\\psi^*)} = \\psi^* - v_{t}, then obtain - 2 (\\psi^* - v_{t})^{v_{t}} + 2 \\operatorname{F_{N}}^{v_{t}}{(v_{t},\\psi^*)} = 0", "derivation": "\\operatorname{F_{N}}{(v_{t},\\psi^*)} = \\psi^* - v_{t} and \\operatorname{F_{N}}^{v_{t}}{(v_{t},\\psi^*)} = (\\psi^* - v_{t})^{v_{t}} and - (\\psi^* - v_{t})^{v_{t}} + \\operatorname{F_{N}}^{v_{t}}{(v_{t},\\psi^*)} = 0 and - 2 (\\psi^* - v_{t})^{v_{t}} + \\operatorname{F_{N}}^{v_{t}}{(v_{t},\\psi^*)} = - (\\psi^* - v_{t})^{v_{t}} and - 2 (\\psi^* - v_{t})^{v_{t}} + 2 \\operatorname{F_{N}}^{v_{t}}{(v_{t},\\psi^*)} = 0", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('v_t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["power", 1, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('F_N')(Symbol('v_t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_t', commutative=True)), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["minus", 2, "Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))), Pow(Function('F_N')(Symbol('v_t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_t', commutative=True))), Integer(0))"], [["minus", 3, "Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))), Pow(Function('F_N')(Symbol('v_t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_t', commutative=True))), Mul(Integer(-1), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Pow(Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))), Mul(Integer(2), Pow(Function('F_N')(Symbol('v_t', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('v_t', commutative=True)))), Integer(0))"]]}, {"prompt": "Given r{(\\psi)} = \\cos{(\\psi)}, then derive 2 \\frac{d}{d \\psi} r{(\\psi)} = - \\sin{(\\psi)} + \\frac{d}{d \\psi} r{(\\psi)}, then obtain \\sin{(\\psi (r{(\\psi)} + \\cos{(\\psi)}) - 1)} + 2 \\frac{d}{d \\psi} \\cos{(\\psi)} = - \\sin{(\\psi)} + \\sin{(\\psi (r{(\\psi)} + \\cos{(\\psi)}) - 1)} + \\frac{d}{d \\psi} \\cos{(\\psi)}", "derivation": "r{(\\psi)} = \\cos{(\\psi)} and 2 r{(\\psi)} = r{(\\psi)} + \\cos{(\\psi)} and \\frac{d}{d \\psi} 2 r{(\\psi)} = \\frac{d}{d \\psi} (r{(\\psi)} + \\cos{(\\psi)}) and 2 \\frac{d}{d \\psi} r{(\\psi)} = - \\sin{(\\psi)} + \\frac{d}{d \\psi} r{(\\psi)} and 2 \\frac{d}{d \\psi} \\cos{(\\psi)} = - \\sin{(\\psi)} + \\frac{d}{d \\psi} \\cos{(\\psi)} and \\sin{(\\psi (r{(\\psi)} + \\cos{(\\psi)}) - 1)} + 2 \\frac{d}{d \\psi} \\cos{(\\psi)} = - \\sin{(\\psi)} + \\sin{(\\psi (r{(\\psi)} + \\cos{(\\psi)}) - 1)} + \\frac{d}{d \\psi} \\cos{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))"], [["add", 1, "Function('r')(Symbol('\\\\psi', commutative=True))"], "Equality(Mul(Integer(2), Function('r')(Symbol('\\\\psi', commutative=True))), Add(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('r')(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Add(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True))), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))), Derivative(Function('r')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["add", 5, "sin(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))), Integer(-1)))"], "Equality(Add(sin(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))), Integer(-1))), Mul(Integer(2), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(Symbol('\\\\psi', commutative=True))), sin(Add(Mul(Symbol('\\\\psi', commutative=True), Add(Function('r')(Symbol('\\\\psi', commutative=True)), cos(Symbol('\\\\psi', commutative=True)))), Integer(-1))), Derivative(cos(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\theta{(M,A_{1})} = - A_{1} + \\log{(M)}, then derive (\\int A_{1} \\theta{(M,A_{1})} dM)^{A_{1}} = (A_{1} M \\log{(M)} + M (- A_{1}^{2} - A_{1}) + f)^{A_{1}}, then obtain - \\frac{(\\int A_{1} \\theta{(M,A_{1})} dM)^{A_{1}}}{A_{1} (- A_{1} + \\log{(M)})} = - \\frac{(A_{1} M \\log{(M)} + M (- A_{1}^{2} - A_{1}) + f)^{A_{1}}}{A_{1} (- A_{1} + \\log{(M)})}", "derivation": "\\theta{(M,A_{1})} = - A_{1} + \\log{(M)} and A_{1} \\theta{(M,A_{1})} = A_{1} (- A_{1} + \\log{(M)}) and \\int A_{1} \\theta{(M,A_{1})} dM = \\int A_{1} (- A_{1} + \\log{(M)}) dM and (\\int A_{1} \\theta{(M,A_{1})} dM)^{A_{1}} = (\\int A_{1} (- A_{1} + \\log{(M)}) dM)^{A_{1}} and (\\int A_{1} \\theta{(M,A_{1})} dM)^{A_{1}} = (A_{1} M \\log{(M)} + M (- A_{1}^{2} - A_{1}) + f)^{A_{1}} and - \\frac{(\\int A_{1} \\theta{(M,A_{1})} dM)^{A_{1}}}{A_{1} (- A_{1} + \\log{(M)})} = - \\frac{(A_{1} M \\log{(M)} + M (- A_{1}^{2} - A_{1}) + f)^{A_{1}}}{A_{1} (- A_{1} + \\log{(M)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True)), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True))))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True)))))"], [["integrate", 2, "Symbol('M', commutative=True)"], "Equality(Integral(Mul(Symbol('A_1', commutative=True), Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('M', commutative=True))), Integral(Mul(Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Integral(Mul(Symbol('A_1', commutative=True), Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('M', commutative=True))), Symbol('A_1', commutative=True)), Pow(Integral(Mul(Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True)))), Tuple(Symbol('M', commutative=True))), Symbol('A_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Integral(Mul(Symbol('A_1', commutative=True), Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('M', commutative=True))), Symbol('A_1', commutative=True)), Pow(Add(Mul(Symbol('A_1', commutative=True), Symbol('M', commutative=True), log(Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_1', commutative=True)))), Symbol('f', commutative=True)), Symbol('A_1', commutative=True)))"], [["divide", 5, "Mul(Integer(-1), Symbol('A_1', commutative=True), Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True))), Integer(-1)), Pow(Integral(Mul(Symbol('A_1', commutative=True), Function('\\\\theta')(Symbol('M', commutative=True), Symbol('A_1', commutative=True))), Tuple(Symbol('M', commutative=True))), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), log(Symbol('M', commutative=True))), Integer(-1)), Pow(Add(Mul(Symbol('A_1', commutative=True), Symbol('M', commutative=True), log(Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), Add(Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('A_1', commutative=True)))), Symbol('f', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given v{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)} and \\mathbf{P}{(\\tilde{g}^*)} = v{(\\tilde{g}^*)} \\log{(\\tilde{g}^*)}, then obtain \\mathbf{P}{(\\tilde{g}^*)} - 1 = v{(\\tilde{g}^*)} \\log{(\\tilde{g}^*)} - 1", "derivation": "v{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)} and v{(\\tilde{g}^*)} \\log{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)}^{2} and \\mathbf{P}{(\\tilde{g}^*)} = v{(\\tilde{g}^*)} \\log{(\\tilde{g}^*)} and \\mathbf{P}{(\\tilde{g}^*)} = v^{2}{(\\tilde{g}^*)} and \\mathbf{P}{(\\tilde{g}^*)} = \\log{(\\tilde{g}^*)}^{2} and \\mathbf{P}{(\\tilde{g}^*)} - 1 = \\log{(\\tilde{g}^*)}^{2} - 1 and \\mathbf{P}{(\\tilde{g}^*)} - 1 = v{(\\tilde{g}^*)} \\log{(\\tilde{g}^*)} - 1", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 1, "log(Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Function('v')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))), Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Function('v')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Function('v')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}^*', commutative=True)), Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)))"], [["add", 5, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Add(Pow(log(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(2)), Integer(-1)))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Function('\\\\mathbf{P}')(Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Add(Mul(Function('v')(Symbol('\\\\tilde{g}^*', commutative=True)), log(Symbol('\\\\tilde{g}^*', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\hat{X}{(E_{x})} = \\sin{(\\cos{(E_{x})})} and A{(E_{x})} = \\frac{\\hat{X}{(E_{x})}}{E_{x}}, then obtain A^{2}{(E_{x})} = \\frac{A{(E_{x})} \\sin{(\\cos{(E_{x})})}}{E_{x}}", "derivation": "\\hat{X}{(E_{x})} = \\sin{(\\cos{(E_{x})})} and \\frac{\\hat{X}{(E_{x})}}{E_{x}} = \\frac{\\sin{(\\cos{(E_{x})})}}{E_{x}} and A{(E_{x})} = \\frac{\\hat{X}{(E_{x})}}{E_{x}} and A^{2}{(E_{x})} = \\frac{A{(E_{x})} \\hat{X}{(E_{x})}}{E_{x}} and A^{2}{(E_{x})} = \\frac{A{(E_{x})} \\sin{(\\cos{(E_{x})})}}{E_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('E_x', commutative=True)), sin(cos(Symbol('E_x', commutative=True))))"], [["divide", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E_x', commutative=True))), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), sin(cos(Symbol('E_x', commutative=True)))))"], ["renaming_premise", "Equality(Function('A')(Symbol('E_x', commutative=True)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('\\\\hat{X}')(Symbol('E_x', commutative=True))))"], [["times", 3, "Function('A')(Symbol('E_x', commutative=True))"], "Equality(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('A')(Symbol('E_x', commutative=True)), Function('\\\\hat{X}')(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Function('A')(Symbol('E_x', commutative=True)), Integer(2)), Mul(Pow(Symbol('E_x', commutative=True), Integer(-1)), Function('A')(Symbol('E_x', commutative=True)), sin(cos(Symbol('E_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(C,F_{N})} = \\log{(C^{F_{N}})} and \\hat{X}{(C,F_{N})} = \\log{(C^{F_{N}})}, then derive F_{N} + \\int \\hat{X}{(C,F_{N})} dF_{N} = \\frac{F_{N}^{2} \\log{(C)}}{2} + F_{N} + \\mu_0, then obtain F_{N} + \\int \\operatorname{V_{\\mathbf{B}}}{(C,F_{N})} dF_{N} = \\frac{F_{N}^{2} \\log{(C)}}{2} + F_{N} + \\mu_0", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(C,F_{N})} = \\log{(C^{F_{N}})} and \\hat{X}{(C,F_{N})} = \\log{(C^{F_{N}})} and \\hat{X}{(C,F_{N})} = \\operatorname{V_{\\mathbf{B}}}{(C,F_{N})} and \\int \\hat{X}{(C,F_{N})} dF_{N} = \\int \\log{(C^{F_{N}})} dF_{N} and F_{N} + \\int \\hat{X}{(C,F_{N})} dF_{N} = F_{N} + \\int \\log{(C^{F_{N}})} dF_{N} and F_{N} + \\int \\hat{X}{(C,F_{N})} dF_{N} = \\frac{F_{N}^{2} \\log{(C)}}{2} + F_{N} + \\mu_0 and F_{N} + \\int \\operatorname{V_{\\mathbf{B}}}{(C,F_{N})} dF_{N} = \\frac{F_{N}^{2} \\log{(C)}}{2} + F_{N} + \\mu_0", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), log(Pow(Symbol('C', commutative=True), Symbol('F_N', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), log(Pow(Symbol('C', commutative=True), Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)))"], [["integrate", 2, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(log(Pow(Symbol('C', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["add", 4, "Symbol('F_N', commutative=True)"], "Equality(Add(Symbol('F_N', commutative=True), Integral(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Symbol('F_N', commutative=True), Integral(log(Pow(Symbol('C', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_N', commutative=True), Integral(Function('\\\\hat{X}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2)), log(Symbol('C', commutative=True))), Symbol('F_N', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('F_N', commutative=True), Integral(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2)), log(Symbol('C', commutative=True))), Symbol('F_N', commutative=True), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\eta^{\\prime}{(q,i)} = \\cos^{q}{(i)}, then derive \\frac{\\partial}{\\partial i} \\eta^{\\prime}{(q,i)} - 1 = - \\frac{q \\sin{(i)} \\cos^{q}{(i)}}{\\cos{(i)}} - 1, then obtain \\frac{q \\eta^{\\prime}{(q,i)} \\sin{(i)}}{\\cos{(i)}} + \\frac{\\partial}{\\partial i} \\eta^{\\prime}{(q,i)} - 1 = -1", "derivation": "\\eta^{\\prime}{(q,i)} = \\cos^{q}{(i)} and - i + \\eta^{\\prime}{(q,i)} = - i + \\cos^{q}{(i)} and \\frac{\\partial}{\\partial i} (- i + \\eta^{\\prime}{(q,i)}) = \\frac{\\partial}{\\partial i} (- i + \\cos^{q}{(i)}) and \\frac{\\partial}{\\partial i} \\eta^{\\prime}{(q,i)} - 1 = - \\frac{q \\sin{(i)} \\cos^{q}{(i)}}{\\cos{(i)}} - 1 and \\frac{q \\eta^{\\prime}{(q,i)} \\sin{(i)}}{\\cos{(i)}} + \\frac{\\partial}{\\partial i} \\eta^{\\prime}{(q,i)} - 1 = \\frac{q \\eta^{\\prime}{(q,i)} \\sin{(i)}}{\\cos{(i)}} - \\frac{q \\sin{(i)} \\cos^{q}{(i)}}{\\cos{(i)}} - 1 and \\frac{q \\eta^{\\prime}{(q,i)} \\sin{(i)}}{\\cos{(i)}} + \\frac{\\partial}{\\partial i} \\eta^{\\prime}{(q,i)} - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('i', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True))), Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Symbol('q', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('q', commutative=True), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1)), Pow(cos(Symbol('i', commutative=True)), Symbol('q', commutative=True))), Integer(-1)))"], [["minus", 4, "Mul(Integer(-1), Symbol('q', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Symbol('q', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Symbol('q', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1)), Pow(cos(Symbol('i', commutative=True)), Symbol('q', commutative=True))), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Symbol('q', commutative=True), Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), sin(Symbol('i', commutative=True)), Pow(cos(Symbol('i', commutative=True)), Integer(-1))), Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('q', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True), Integer(1))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given k{(V_{\\mathbf{B}},y)} = y^{V_{\\mathbf{B}}}, then derive \\frac{\\partial}{\\partial y} k{(V_{\\mathbf{B}},y)} = \\frac{V_{\\mathbf{B}} y^{V_{\\mathbf{B}}}}{y}, then obtain \\frac{V_{\\mathbf{B}} k{(V_{\\mathbf{B}},y)}}{y} - \\frac{\\partial}{\\partial y} y^{V_{\\mathbf{B}}} = 0", "derivation": "k{(V_{\\mathbf{B}},y)} = y^{V_{\\mathbf{B}}} and \\frac{\\partial}{\\partial y} k{(V_{\\mathbf{B}},y)} = \\frac{\\partial}{\\partial y} y^{V_{\\mathbf{B}}} and \\frac{k{(V_{\\mathbf{B}},y)}}{y} = \\frac{y^{V_{\\mathbf{B}}}}{y} and \\frac{\\partial}{\\partial y} k{(V_{\\mathbf{B}},y)} = \\frac{V_{\\mathbf{B}} y^{V_{\\mathbf{B}}}}{y} and - \\frac{\\partial}{\\partial y} y^{V_{\\mathbf{B}}} + \\frac{\\partial}{\\partial y} k{(V_{\\mathbf{B}},y)} = 0 and \\frac{\\partial}{\\partial y} k{(V_{\\mathbf{B}},y)} = \\frac{V_{\\mathbf{B}} k{(V_{\\mathbf{B}},y)}}{y} and \\frac{V_{\\mathbf{B}} k{(V_{\\mathbf{B}},y)}}{y} - \\frac{\\partial}{\\partial y} y^{V_{\\mathbf{B}}} = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 1, "Symbol('y', commutative=True)"], "Equality(Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))), Mul(Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 2, "Derivative(Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Derivative(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('y', commutative=True), Integer(-1)), Function('k')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Derivative(Pow(Symbol('y', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given h{(\\mathbf{B})} = e^{\\mathbf{B}} and \\operatorname{v_{x}}{(V,T)} = T + e^{V}, then obtain h^{- \\mathbf{B}}{(\\mathbf{B})} \\int \\operatorname{v_{x}}{(V,T)} dV + h^{- \\mathbf{B}}{(\\mathbf{B})} = h^{- \\mathbf{B}}{(\\mathbf{B})} \\int (T + e^{V}) dV + h^{- \\mathbf{B}}{(\\mathbf{B})}", "derivation": "h{(\\mathbf{B})} = e^{\\mathbf{B}} and \\operatorname{v_{x}}{(V,T)} = T + e^{V} and \\int \\operatorname{v_{x}}{(V,T)} dV = \\int (T + e^{V}) dV and (e^{\\mathbf{B}})^{- \\mathbf{B}} \\int \\operatorname{v_{x}}{(V,T)} dV = (e^{\\mathbf{B}})^{- \\mathbf{B}} \\int (T + e^{V}) dV and h^{- \\mathbf{B}}{(\\mathbf{B})} \\int \\operatorname{v_{x}}{(V,T)} dV = h^{- \\mathbf{B}}{(\\mathbf{B})} \\int (T + e^{V}) dV and h^{- \\mathbf{B}}{(\\mathbf{B})} \\int \\operatorname{v_{x}}{(V,T)} dV + h^{- \\mathbf{B}}{(\\mathbf{B})} = h^{- \\mathbf{B}}{(\\mathbf{B})} \\int (T + e^{V}) dV + h^{- \\mathbf{B}}{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), exp(Symbol('\\\\mathbf{B}', commutative=True)))"], ["get_premise", "Equality(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), exp(Symbol('V', commutative=True))))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True))))"], [["divide", 3, "Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Pow(exp(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))))"], [["add", 5, "Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Add(Mul(Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Function('v_x')(Symbol('V', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('V', commutative=True)))), Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))), Add(Mul(Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True))), Integral(Add(Symbol('T', commutative=True), exp(Symbol('V', commutative=True))), Tuple(Symbol('V', commutative=True)))), Pow(Function('h')(Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\dot{x}{(q,\\lambda)} = q^{\\lambda}, then obtain e^{(\\dot{x}{(q,\\lambda)} + \\dot{x}^{q}{(q,\\lambda)})^{q}} = e^{(q^{\\lambda} + \\dot{x}^{q}{(q,\\lambda)})^{q}}", "derivation": "\\dot{x}{(q,\\lambda)} = q^{\\lambda} and \\dot{x}{(q,\\lambda)} + \\dot{x}^{q}{(q,\\lambda)} = q^{\\lambda} + \\dot{x}^{q}{(q,\\lambda)} and (\\dot{x}{(q,\\lambda)} + \\dot{x}^{q}{(q,\\lambda)})^{q} = (q^{\\lambda} + \\dot{x}^{q}{(q,\\lambda)})^{q} and e^{(\\dot{x}{(q,\\lambda)} + \\dot{x}^{q}{(q,\\lambda)})^{q}} = e^{(q^{\\lambda} + \\dot{x}^{q}{(q,\\lambda)})^{q}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["add", 1, "Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Add(Pow(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))))"], [["power", 2, "Symbol('q', commutative=True)"], "Equality(Pow(Add(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Add(Pow(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Add(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True))), exp(Pow(Add(Pow(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Pow(Function('\\\\dot{x}')(Symbol('q', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(F_{H})} = \\cos{(F_{H})}, then obtain ((\\hat{p}_0^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} - ((\\cos^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} = 0", "derivation": "\\hat{p}_0{(F_{H})} = \\cos{(F_{H})} and \\hat{p}_0^{F_{H}}{(F_{H})} = \\cos^{F_{H}}{(F_{H})} and (\\hat{p}_0^{F_{H}}{(F_{H})})^{F_{H}} = (\\cos^{F_{H}}{(F_{H})})^{F_{H}} and ((\\hat{p}_0^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} = ((\\cos^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} and ((\\hat{p}_0^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} - ((\\cos^{F_{H}}{(F_{H})})^{F_{H}})^{F_{H}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('F_H', commutative=True)), cos(Symbol('F_H', commutative=True)))"], [["power", 1, "Symbol('F_H', commutative=True)"], "Equality(Pow(Function('\\\\hat{p}_0')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["power", 2, "Symbol('F_H', commutative=True)"], "Equality(Pow(Pow(Function('\\\\hat{p}_0')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["power", 3, "Symbol('F_H', commutative=True)"], "Equality(Pow(Pow(Pow(Function('\\\\hat{p}_0')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Pow(Pow(Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))"], [["minus", 4, "Pow(Pow(Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True))"], "Equality(Add(Pow(Pow(Pow(Function('\\\\hat{p}_0')(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Mul(Integer(-1), Pow(Pow(Pow(cos(Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)), Symbol('F_H', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{M}{(\\varphi^*)} = \\sin{(\\varphi^*)}, then obtain \\frac{\\mathbf{M}^{2}{(\\varphi^*)}}{- (- \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)}) \\mathbf{M}^{2}{(\\varphi^*)} + \\mathbf{M}^{2}{(\\varphi^*)}} = 1", "derivation": "\\mathbf{M}{(\\varphi^*)} = \\sin{(\\varphi^*)} and 0 = - \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)} and 0 = - (- \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)}) \\mathbf{M}{(\\varphi^*)} and 0 = - (- \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)}) \\mathbf{M}^{2}{(\\varphi^*)} and \\mathbf{M}^{2}{(\\varphi^*)} = - (- \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)}) \\mathbf{M}^{2}{(\\varphi^*)} + \\mathbf{M}^{2}{(\\varphi^*)} and \\frac{\\mathbf{M}^{2}{(\\varphi^*)}}{- (- \\mathbf{M}{(\\varphi^*)} + \\sin{(\\varphi^*)}) \\mathbf{M}^{2}{(\\varphi^*)} + \\mathbf{M}^{2}{(\\varphi^*)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 3, "Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Integer(0), Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["add", 4, "Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)), Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["divide", 5, "Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True))), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Integer(-1)), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Integer(1))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(t_{1},A)} = \\sin^{t_{1}}{(A)} and \\mathbf{g}{(t_{1},A)} = \\sin^{t_{1}}{(A)}, then obtain \\mathbf{g}^{2}{(t_{1},A)} \\sin^{- t_{1}}{(A)} = \\operatorname{A_{1}}{(t_{1},A)}", "derivation": "\\operatorname{A_{1}}{(t_{1},A)} = \\sin^{t_{1}}{(A)} and \\mathbf{g}{(t_{1},A)} = \\sin^{t_{1}}{(A)} and \\mathbf{g}{(t_{1},A)} = \\operatorname{A_{1}}{(t_{1},A)} and \\mathbf{g}^{2}{(t_{1},A)} = \\mathbf{g}{(t_{1},A)} \\sin^{t_{1}}{(A)} and \\mathbf{g}^{2}{(t_{1},A)} \\sin^{- t_{1}}{(A)} = \\mathbf{g}{(t_{1},A)} and \\mathbf{g}^{2}{(t_{1},A)} \\sin^{- t_{1}}{(A)} = \\operatorname{A_{1}}{(t_{1},A)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Pow(sin(Symbol('A', commutative=True)), Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Pow(sin(Symbol('A', commutative=True)), Symbol('t_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Function('A_1')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)))"], [["times", 2, "Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Pow(sin(Symbol('A', commutative=True)), Symbol('t_1', commutative=True))))"], [["divide", 4, "Pow(sin(Symbol('A', commutative=True)), Symbol('t_1', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Integer(2)), Pow(sin(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)), Integer(2)), Pow(sin(Symbol('A', commutative=True)), Mul(Integer(-1), Symbol('t_1', commutative=True)))), Function('A_1')(Symbol('t_1', commutative=True), Symbol('A', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = \\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime), then derive \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} + 1 = 2, then derive \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = 1, then obtain (\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1) \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = \\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1", "derivation": "\\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = \\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) and \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} + 1 = \\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1 and \\frac{\\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} + 1}{\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime)} = \\frac{\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1}{\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime)} and \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} + 1 = 2 and \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} - (\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime))^{\\delta} + 1 = 2 - (\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime))^{\\delta} and \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = 1 and (\\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1) \\operatorname{f^{\\prime}}{(x^\\prime,\\delta)} = \\frac{\\partial}{\\partial x^\\prime} (- \\delta + x^\\prime) + 1", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1)), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)))"], [["divide", 2, "Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 3], "Equality(Add(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1)), Integer(2))"], [["minus", 4, "Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(-1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True))), Integer(1)), Add(Integer(2), Mul(Integer(-1), Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Symbol('\\\\delta', commutative=True)))))"], [["evaluate_derivatives", 5], "Equality(Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1))"], [["times", 6, "Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1))"], "Equality(Mul(Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)), Function('f^{\\\\prime}')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Derivative(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given V{(S)} = \\log{(S)} and \\hat{H}_l{(S)} = (V{(S)} + \\log{(S)})^{S}, then obtain (V{(S)} + \\log{(S)})^{S} + ((2 \\log{(S)})^{S})^{S} = (V{(S)} + \\log{(S)})^{S} + \\hat{H}_l^{S}{(S)}", "derivation": "V{(S)} = \\log{(S)} and V{(S)} + \\log{(S)} = 2 \\log{(S)} and (V{(S)} + \\log{(S)})^{S} = (2 \\log{(S)})^{S} and ((V{(S)} + \\log{(S)})^{S})^{S} = ((2 \\log{(S)})^{S})^{S} and (V{(S)} + \\log{(S)})^{S} + ((V{(S)} + \\log{(S)})^{S})^{S} = (V{(S)} + \\log{(S)})^{S} + ((2 \\log{(S)})^{S})^{S} and \\hat{H}_l{(S)} = (V{(S)} + \\log{(S)})^{S} and \\hat{H}_l{(S)} = (2 \\log{(S)})^{S} and (V{(S)} + \\log{(S)})^{S} + ((V{(S)} + \\log{(S)})^{S})^{S} = (V{(S)} + \\log{(S)})^{S} + \\hat{H}_l^{S}{(S)} and (V{(S)} + \\log{(S)})^{S} + ((2 \\log{(S)})^{S})^{S} = (V{(S)} + \\log{(S)})^{S} + \\hat{H}_l^{S}{(S)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["add", 1, "log(Symbol('S', commutative=True))"], "Equality(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Mul(Integer(2), log(Symbol('S', commutative=True))))"], [["power", 2, "Symbol('S', commutative=True)"], "Equality(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Mul(Integer(2), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True)), Pow(Pow(Mul(Integer(2), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True)))"], [["add", 4, "Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True))"], "Equality(Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True))), Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Pow(Mul(Integer(2), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('S', commutative=True)), Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Function('\\\\hat{H}_l')(Symbol('S', commutative=True)), Pow(Mul(Integer(2), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 7], "Equality(Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True))), Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 4], "Equality(Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Pow(Mul(Integer(2), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Symbol('S', commutative=True))), Add(Pow(Add(Function('V')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True))), Symbol('S', commutative=True)), Pow(Function('\\\\hat{H}_l')(Symbol('S', commutative=True)), Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(g)} = \\cos{(g)} and \\operatorname{F_{H}}{(g)} = \\frac{g \\cos{(g)} \\frac{d}{d g} \\frac{g \\cos{(g)}}{\\operatorname{t_{2}}{(g)}}}{\\operatorname{t_{2}}{(g)}}, then obtain \\sin{(\\operatorname{F_{H}}{(g)})} = \\sin{(g \\frac{d}{d g} g)}", "derivation": "\\operatorname{t_{2}}{(g)} = \\cos{(g)} and g \\operatorname{t_{2}}{(g)} = g \\cos{(g)} and g = \\frac{g \\cos{(g)}}{\\operatorname{t_{2}}{(g)}} and \\operatorname{F_{H}}{(g)} = \\frac{g \\cos{(g)} \\frac{d}{d g} \\frac{g \\cos{(g)}}{\\operatorname{t_{2}}{(g)}}}{\\operatorname{t_{2}}{(g)}} and \\sin{(\\operatorname{F_{H}}{(g)})} = \\sin{(\\frac{g \\cos{(g)} \\frac{d}{d g} \\frac{g \\cos{(g)}}{\\operatorname{t_{2}}{(g)}}}{\\operatorname{t_{2}}{(g)}})} and \\sin{(\\operatorname{F_{H}}{(g)})} = \\sin{(g \\frac{d}{d g} g)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('t_2')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), cos(Symbol('g', commutative=True))))"], [["divide", 2, "Function('t_2')(Symbol('g', commutative=True))"], "Equality(Symbol('g', commutative=True), Mul(Symbol('g', commutative=True), Pow(Function('t_2')(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('F_H')(Symbol('g', commutative=True)), Mul(Symbol('g', commutative=True), Pow(Function('t_2')(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Pow(Function('t_2')(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["sin", 4], "Equality(sin(Function('F_H')(Symbol('g', commutative=True))), sin(Mul(Symbol('g', commutative=True), Pow(Function('t_2')(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True)), Derivative(Mul(Symbol('g', commutative=True), Pow(Function('t_2')(Symbol('g', commutative=True)), Integer(-1)), cos(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(sin(Function('F_H')(Symbol('g', commutative=True))), sin(Mul(Symbol('g', commutative=True), Derivative(Symbol('g', commutative=True), Tuple(Symbol('g', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(C,b)} = \\log{(C + b)}, then derive \\frac{\\partial}{\\partial C} \\operatorname{C_{1}}{(C,b)} = \\frac{1}{C + b}, then obtain \\frac{\\int \\frac{\\partial}{\\partial C} \\log{(C + b)} dC}{\\log{(C + b)}} = \\frac{\\int \\frac{1}{C + b} dC}{\\log{(C + b)}}", "derivation": "\\operatorname{C_{1}}{(C,b)} = \\log{(C + b)} and \\frac{\\partial}{\\partial C} \\operatorname{C_{1}}{(C,b)} = \\frac{\\partial}{\\partial C} \\log{(C + b)} and \\frac{\\partial}{\\partial C} \\operatorname{C_{1}}{(C,b)} = \\frac{1}{C + b} and \\frac{\\partial}{\\partial C} \\log{(C + b)} = \\frac{1}{C + b} and \\int \\frac{\\partial}{\\partial C} \\log{(C + b)} dC = \\int \\frac{1}{C + b} dC and \\frac{\\int \\frac{\\partial}{\\partial C} \\log{(C + b)} dC}{\\log{(C + b)}} = \\frac{\\int \\frac{1}{C + b} dC}{\\log{(C + b)}}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)), log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('C', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Add(Symbol('C', commutative=True), Symbol('b', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Pow(Add(Symbol('C', commutative=True), Symbol('b', commutative=True)), Integer(-1)))"], [["integrate", 4, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Pow(Add(Symbol('C', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True))))"], [["divide", 5, "log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True)))"], "Equality(Mul(Pow(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Integral(Derivative(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Pow(log(Add(Symbol('C', commutative=True), Symbol('b', commutative=True))), Integer(-1)), Integral(Pow(Add(Symbol('C', commutative=True), Symbol('b', commutative=True)), Integer(-1)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(B)} = \\cos{(B)}, then obtain (-1)^{B} = (- \\Psi_{nl}{(B)} \\cos{(B)} + \\cos^{2}{(B)} - 1)^{B}", "derivation": "\\Psi_{nl}{(B)} = \\cos{(B)} and \\Psi_{nl}{(B)} \\cos{(B)} = \\cos^{2}{(B)} and 0 = - \\Psi_{nl}{(B)} \\cos{(B)} + \\cos^{2}{(B)} and -1 = - \\Psi_{nl}{(B)} \\cos{(B)} + \\cos^{2}{(B)} - 1 and (-1)^{B} = (- \\Psi_{nl}{(B)} \\cos{(B)} + \\cos^{2}{(B)} - 1)^{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["times", 1, "cos(Symbol('B', commutative=True))"], "Equality(Mul(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(2)))"], [["minus", 2, "Mul(Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(2))))"], [["add", 3, "Integer(-1)"], "Equality(Integer(-1), Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(2)), Integer(-1)))"], [["power", 4, "Symbol('B', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\Psi_{nl}')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True))), Pow(cos(Symbol('B', commutative=True)), Integer(2)), Integer(-1)), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - \\mathbb{I}, then derive \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}^{2}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = 0", "derivation": "\\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} - \\mathbb{I} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} (\\hat{H}_{\\lambda} - \\mathbb{I}) and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = 1 and \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}^{2}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = \\frac{d}{d \\hat{H}_{\\lambda}} 1 and \\frac{\\partial^{2}}{\\partial \\hat{H}_{\\lambda}^{2}} \\phi_{2}{(\\mathbb{I},\\hat{H}_{\\lambda})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('\\\\phi_2')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(2))), Integer(0))"]]}, {"prompt": "Given \\nabla{(g,c)} = - c + g and \\operatorname{f^{*}}{(g,c)} = g + 3 \\nabla{(g,c)}, then obtain \\operatorname{f^{*}}{(g,c)} = c + 4 \\nabla{(g,c)}", "derivation": "\\nabla{(g,c)} = - c + g and 2 \\nabla{(g,c)} = - c + g + \\nabla{(g,c)} and c + 4 \\nabla{(g,c)} = g + 3 \\nabla{(g,c)} and \\operatorname{f^{*}}{(g,c)} = g + 3 \\nabla{(g,c)} and \\operatorname{f^{*}}{(g,c)} = c + 4 \\nabla{(g,c)}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True)), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('g', commutative=True)))"], [["add", 1, "Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True))), Add(Mul(Integer(-1), Symbol('c', commutative=True)), Symbol('g', commutative=True), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True))))"], [["add", 2, "Add(Symbol('c', commutative=True), Mul(Integer(2), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True))))"], "Equality(Add(Symbol('c', commutative=True), Mul(Integer(4), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True)))), Add(Symbol('g', commutative=True), Mul(Integer(3), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True)))))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('g', commutative=True), Symbol('c', commutative=True)), Add(Symbol('g', commutative=True), Mul(Integer(3), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('f^*')(Symbol('g', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Mul(Integer(4), Function('\\\\nabla')(Symbol('g', commutative=True), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\cos{(\\sin{(\\varepsilon)})}, then obtain \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} \\cos{(\\sin{(\\varepsilon)})} - \\cos^{2}{(\\sin{(\\varepsilon)})} + \\cos{(\\sin{(\\varepsilon)})}", "derivation": "\\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\cos{(\\sin{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} \\cos{(\\sin{(\\varepsilon)})} = \\cos^{2}{(\\sin{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} \\cos{(\\sin{(\\varepsilon)})} + \\cos{(\\sin{(\\varepsilon)})} = \\cos^{2}{(\\sin{(\\varepsilon)})} + \\cos{(\\sin{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} \\cos{(\\sin{(\\varepsilon)})} - \\cos^{2}{(\\sin{(\\varepsilon)})} + \\cos{(\\sin{(\\varepsilon)})} = \\cos{(\\sin{(\\varepsilon)})} and \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} = \\operatorname{J_{\\varepsilon}}{(\\varepsilon)} \\cos{(\\sin{(\\varepsilon)})} - \\cos^{2}{(\\sin{(\\varepsilon)})} + \\cos{(\\sin{(\\varepsilon)})}", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), cos(sin(Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "cos(sin(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), Pow(cos(sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2)))"], [["add", 2, "cos(sin(Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), Add(Pow(cos(sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2)), cos(sin(Symbol('\\\\varepsilon', commutative=True)))))"], [["minus", 3, "Pow(cos(sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2))"], "Equality(Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Pow(cos(sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2))), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), cos(sin(Symbol('\\\\varepsilon', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), Add(Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\varepsilon', commutative=True)), cos(sin(Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Pow(cos(sin(Symbol('\\\\varepsilon', commutative=True))), Integer(2))), cos(sin(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(u,f_{\\mathbf{v}})} = \\log{(u^{f_{\\mathbf{v}}})}, then obtain - \\frac{u^{- f_{\\mathbf{v}}} \\operatorname{g_{\\varepsilon}}^{u}{(u,f_{\\mathbf{v}})}}{f_{\\mathbf{v}}} = - \\frac{u^{- f_{\\mathbf{v}}} \\log{(u^{f_{\\mathbf{v}}})}^{u}}{f_{\\mathbf{v}}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(u,f_{\\mathbf{v}})} = \\log{(u^{f_{\\mathbf{v}}})} and \\operatorname{g_{\\varepsilon}}^{u}{(u,f_{\\mathbf{v}})} = \\log{(u^{f_{\\mathbf{v}}})}^{u} and u^{- f_{\\mathbf{v}}} \\operatorname{g_{\\varepsilon}}^{u}{(u,f_{\\mathbf{v}})} = u^{- f_{\\mathbf{v}}} \\log{(u^{f_{\\mathbf{v}}})}^{u} and - \\frac{u^{- f_{\\mathbf{v}}} \\operatorname{g_{\\varepsilon}}^{u}{(u,f_{\\mathbf{v}})}}{f_{\\mathbf{v}}} = - \\frac{u^{- f_{\\mathbf{v}}} \\log{(u^{f_{\\mathbf{v}}})}^{u}}{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), log(Pow(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('u', commutative=True)), Pow(log(Pow(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('u', commutative=True)))"], [["divide", 2, "Pow(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('u', commutative=True))), Mul(Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(log(Pow(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('u', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(Function('g_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Symbol('u', commutative=True))), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Pow(Symbol('u', commutative=True), Mul(Integer(-1), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Pow(log(Pow(Symbol('u', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True))), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\hat{H},\\mathbf{r},V)} = - V + \\hat{H} + \\mathbf{r}, then obtain - V + \\int \\operatorname{f_{E}}^{V}{(\\hat{H},\\mathbf{r},V)} dV = - V + \\int (- V + \\hat{H} + \\mathbf{r})^{V} dV", "derivation": "\\operatorname{f_{E}}{(\\hat{H},\\mathbf{r},V)} = - V + \\hat{H} + \\mathbf{r} and \\operatorname{f_{E}}^{V}{(\\hat{H},\\mathbf{r},V)} = (- V + \\hat{H} + \\mathbf{r})^{V} and \\int \\operatorname{f_{E}}^{V}{(\\hat{H},\\mathbf{r},V)} dV = \\int (- V + \\hat{H} + \\mathbf{r})^{V} dV and - V + \\int \\operatorname{f_{E}}^{V}{(\\hat{H},\\mathbf{r},V)} dV = - V + \\int (- V + \\hat{H} + \\mathbf{r})^{V} dV", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)))"], [["power", 1, "Symbol('V', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V', commutative=True)))"], [["integrate", 2, "Symbol('V', commutative=True)"], "Equality(Integral(Pow(Function('f_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Symbol('V', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Pow(Function('f_E')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('V', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Add(Mul(Integer(-1), Symbol('V', commutative=True)), Integral(Pow(Add(Mul(Integer(-1), Symbol('V', commutative=True)), Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(L)} = e^{\\cos{(L)}}, then obtain 2 \\operatorname{v_{1}}{(L)} - \\frac{d}{d L} e^{\\cos{(L)}} = \\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}} - \\frac{d}{d L} e^{\\cos{(L)}}", "derivation": "\\operatorname{v_{1}}{(L)} = e^{\\cos{(L)}} and \\frac{d}{d L} \\operatorname{v_{1}}{(L)} = \\frac{d}{d L} e^{\\cos{(L)}} and \\operatorname{v_{1}}{(L)} - \\frac{d}{d L} \\operatorname{v_{1}}{(L)} = e^{\\cos{(L)}} - \\frac{d}{d L} \\operatorname{v_{1}}{(L)} and \\operatorname{v_{1}}{(L)} - \\frac{d}{d L} e^{\\cos{(L)}} = e^{\\cos{(L)}} - \\frac{d}{d L} e^{\\cos{(L)}} and 2 \\operatorname{v_{1}}{(L)} - \\frac{d}{d L} e^{\\cos{(L)}} = \\operatorname{v_{1}}{(L)} + e^{\\cos{(L)}} - \\frac{d}{d L} e^{\\cos{(L)}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('v_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(Function('v_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))"], "Equality(Add(Function('v_1')(Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(exp(cos(Symbol('L', commutative=True))), Mul(Integer(-1), Derivative(Function('v_1')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('v_1')(Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(exp(cos(Symbol('L', commutative=True))), Mul(Integer(-1), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))))"], [["minus", 4, "Mul(Integer(-1), Function('v_1')(Symbol('L', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('v_1')(Symbol('L', commutative=True))), Mul(Integer(-1), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))), Add(Function('v_1')(Symbol('L', commutative=True)), exp(cos(Symbol('L', commutative=True))), Mul(Integer(-1), Derivative(exp(cos(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))))"]]}, {"prompt": "Given u{(\\tilde{g},r_{0})} = \\tilde{g} - r_{0}, then derive \\int (\\tilde{g} - r_{0}) u{(\\tilde{g},r_{0})} dr_{0} = \\tilde{g}^{2} r_{0} - \\tilde{g} r_{0}^{2} + h + \\frac{r_{0}^{3}}{3}, then obtain \\int (\\tilde{g} - r_{0})^{2} dr_{0} = \\tilde{g}^{2} r_{0} - \\tilde{g} r_{0}^{2} + h + \\frac{r_{0}^{3}}{3}", "derivation": "u{(\\tilde{g},r_{0})} = \\tilde{g} - r_{0} and (\\tilde{g} - r_{0}) u{(\\tilde{g},r_{0})} = (\\tilde{g} - r_{0})^{2} and \\int (\\tilde{g} - r_{0}) u{(\\tilde{g},r_{0})} dr_{0} = \\int (\\tilde{g} - r_{0})^{2} dr_{0} and \\int (\\tilde{g} - r_{0}) u{(\\tilde{g},r_{0})} dr_{0} = \\tilde{g}^{2} r_{0} - \\tilde{g} r_{0}^{2} + h + \\frac{r_{0}^{3}}{3} and \\int (\\tilde{g} - r_{0})^{2} dr_{0} = \\tilde{g}^{2} r_{0} - \\tilde{g} r_{0}^{2} + h + \\frac{r_{0}^{3}}{3}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\tilde{g}', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Function('u')(Symbol('\\\\tilde{g}', commutative=True), Symbol('r_0', commutative=True))), Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(2)))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Function('u')(Symbol('\\\\tilde{g}', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Integral(Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(2)), Tuple(Symbol('r_0', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Function('u')(Symbol('\\\\tilde{g}', commutative=True), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True))), Add(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(2))), Symbol('h', commutative=True), Mul(Rational(1, 3), Pow(Symbol('r_0', commutative=True), Integer(3)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Pow(Add(Symbol('\\\\tilde{g}', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Integer(2)), Tuple(Symbol('r_0', commutative=True))), Add(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(2)), Symbol('r_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(2))), Symbol('h', commutative=True), Mul(Rational(1, 3), Pow(Symbol('r_0', commutative=True), Integer(3)))))"]]}, {"prompt": "Given \\mathbb{I}{(E_{n})} = \\sin{(E_{n})} and C{(E_{n})} = \\sin{(E_{n})}, then obtain \\frac{\\mathbb{I}{(E_{n})}}{E_{n}} = \\frac{C{(E_{n})}}{E_{n}}", "derivation": "\\mathbb{I}{(E_{n})} = \\sin{(E_{n})} and C{(E_{n})} = \\sin{(E_{n})} and \\frac{\\mathbb{I}{(E_{n})}}{E_{n}} = \\frac{\\sin{(E_{n})}}{E_{n}} and \\frac{C{(E_{n})}}{E_{n}} = \\frac{\\sin{(E_{n})}}{E_{n}} and \\frac{\\mathbb{I}{(E_{n})}}{E_{n}} = \\frac{C{(E_{n})}}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], ["renaming_premise", "Equality(Function('C')(Symbol('E_n', commutative=True)), sin(Symbol('E_n', commutative=True)))"], [["divide", 1, "Symbol('E_n', commutative=True)"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), sin(Symbol('E_n', commutative=True))))"], [["times", 2, "Pow(Symbol('E_n', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('C')(Symbol('E_n', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), sin(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('\\\\mathbb{I}')(Symbol('E_n', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Function('C')(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\rho_f,r_{0})} = \\rho_f r_{0}, then obtain - \\rho_f r_{0} - \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} + \\mathbf{J}_M{(\\rho_f,r_{0})} - 1 = - \\rho_f r_{0} \\mathbf{J}_M{(\\rho_f,r_{0})} - 1", "derivation": "\\mathbf{J}_M{(\\rho_f,r_{0})} = \\rho_f r_{0} and 0 = \\rho_f r_{0} - \\mathbf{J}_M{(\\rho_f,r_{0})} and - \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} = - \\rho_f r_{0} \\mathbf{J}_M{(\\rho_f,r_{0})} and \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} = \\rho_f r_{0} + \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} - \\mathbf{J}_M{(\\rho_f,r_{0})} and - \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} - 1 = - \\rho_f r_{0} \\mathbf{J}_M{(\\rho_f,r_{0})} - 1 and - \\rho_f r_{0} - \\mathbf{J}_M^{2}{(\\rho_f,r_{0})} + \\mathbf{J}_M{(\\rho_f,r_{0})} - 1 = - \\rho_f r_{0} \\mathbf{J}_M{(\\rho_f,r_{0})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)))))"], [["times", 1, "Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 2, "Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Add(Mul(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True), Function('\\\\mathbf{J}_M')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given s{(k)} = \\sin{(k)}, then derive \\int s{(k)} dk = \\mathbf{J} - \\cos{(k)}, then obtain \\cos{(k)} + \\int \\sin{(k)} dk = \\mathbf{J}", "derivation": "s{(k)} = \\sin{(k)} and \\int s{(k)} dk = \\int \\sin{(k)} dk and \\int s{(k)} dk = \\mathbf{J} - \\cos{(k)} and \\cos{(k)} + \\int s{(k)} dk = \\mathbf{J} and \\cos{(k)} + \\int \\sin{(k)} dk = \\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["integrate", 1, "Symbol('k', commutative=True)"], "Equality(Integral(Function('s')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('s')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))))"], [["add", 3, "cos(Symbol('k', commutative=True))"], "Equality(Add(cos(Symbol('k', commutative=True)), Integral(Function('s')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(cos(Symbol('k', commutative=True)), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True)))), Symbol('\\\\mathbf{J}', commutative=True))"]]}, {"prompt": "Given m{(U)} = \\cos{(U)}, then derive \\frac{U \\cos{(U)} \\int m{(U)} dU}{2} = \\frac{U (v_{x} + \\sin{(U)}) \\cos{(U)}}{2}, then obtain \\frac{U \\cos{(U)} \\int \\cos{(U)} dU}{2} = \\frac{U (v_{x} + \\sin{(U)}) \\cos{(U)}}{2}", "derivation": "m{(U)} = \\cos{(U)} and \\int m{(U)} dU = \\int \\cos{(U)} dU and \\frac{\\int m{(U)} dU}{\\cos{(U)}} = \\frac{\\int \\cos{(U)} dU}{\\cos{(U)}} and \\frac{U \\int m{(U)} dU}{\\cos{(U)}} = \\frac{U \\int \\cos{(U)} dU}{\\cos{(U)}} and \\frac{U \\int m{(U)} dU}{2 \\cos^{2}{(U)}} = \\frac{U \\int \\cos{(U)} dU}{2 \\cos^{2}{(U)}} and \\frac{U \\int m{(U)} dU}{2 \\cos{(U)}} = \\frac{U \\int \\cos{(U)} dU}{2 \\cos{(U)}} and \\frac{U \\cos{(U)} \\int m{(U)} dU}{2} = \\frac{U \\cos{(U)} \\int \\cos{(U)} dU}{2} and \\frac{U \\cos{(U)} \\int m{(U)} dU}{2} = \\frac{U (v_{x} + \\sin{(U)}) \\cos{(U)}}{2} and \\frac{U \\cos{(U)} \\int \\cos{(U)} dU}{2} = \\frac{U (v_{x} + \\sin{(U)}) \\cos{(U)}}{2}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('U', commutative=True)), cos(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["divide", 2, "cos(Symbol('U', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["times", 3, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["divide", 4, "Mul(Integer(2), cos(Symbol('U', commutative=True)))"], "Equality(Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-2)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-2)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["times", 5, "cos(Symbol('U', commutative=True))"], "Equality(Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Rational(1, 2), Symbol('U', commutative=True), Pow(cos(Symbol('U', commutative=True)), Integer(-1)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["divide", 6, "Pow(cos(Symbol('U', commutative=True)), Integer(-2))"], "Equality(Mul(Rational(1, 2), Symbol('U', commutative=True), cos(Symbol('U', commutative=True)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Rational(1, 2), Symbol('U', commutative=True), cos(Symbol('U', commutative=True)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))))"], [["evaluate_integrals", 7], "Equality(Mul(Rational(1, 2), Symbol('U', commutative=True), cos(Symbol('U', commutative=True)), Integral(Function('m')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Rational(1, 2), Symbol('U', commutative=True), Add(Symbol('v_x', commutative=True), sin(Symbol('U', commutative=True))), cos(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Mul(Rational(1, 2), Symbol('U', commutative=True), cos(Symbol('U', commutative=True)), Integral(cos(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Mul(Rational(1, 2), Symbol('U', commutative=True), Add(Symbol('v_x', commutative=True), sin(Symbol('U', commutative=True))), cos(Symbol('U', commutative=True))))"]]}, {"prompt": "Given k{(\\mathbf{f})} = \\cos{(e^{\\mathbf{f}})}, then derive \\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})} = - e^{\\mathbf{f}} \\sin{(e^{\\mathbf{f}})}, then obtain \\cos{(\\cos{(\\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})})})} = \\cos{(\\cos{(e^{\\mathbf{f}} \\sin{(e^{\\mathbf{f}})})})}", "derivation": "k{(\\mathbf{f})} = \\cos{(e^{\\mathbf{f}})} and \\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})} = \\frac{d}{d \\mathbf{f}} \\cos{(e^{\\mathbf{f}})} and \\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})} = - e^{\\mathbf{f}} \\sin{(e^{\\mathbf{f}})} and \\cos{(\\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})})} = \\cos{(e^{\\mathbf{f}} \\sin{(e^{\\mathbf{f}})})} and \\cos{(\\cos{(\\frac{d}{d \\mathbf{f}} k{(\\mathbf{f})})})} = \\cos{(\\cos{(e^{\\mathbf{f}} \\sin{(e^{\\mathbf{f}})})})}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('\\\\mathbf{f}', commutative=True)), cos(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Derivative(cos(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))), Mul(Integer(-1), exp(Symbol('\\\\mathbf{f}', commutative=True)), sin(exp(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["cos", 3], "Equality(cos(Derivative(Function('k')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1)))), cos(Mul(exp(Symbol('\\\\mathbf{f}', commutative=True)), sin(exp(Symbol('\\\\mathbf{f}', commutative=True))))))"], [["cos", 4], "Equality(cos(cos(Derivative(Function('k')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True), Integer(1))))), cos(cos(Mul(exp(Symbol('\\\\mathbf{f}', commutative=True)), sin(exp(Symbol('\\\\mathbf{f}', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)}, then obtain (2 \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - 2 \\sin{(\\hat{H}_l)})^{2} = (\\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)}) (2 \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - 2 \\sin{(\\hat{H}_l)})", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} = \\sin{(\\hat{H}_l)} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} = 0 and 2 \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - 2 \\sin{(\\hat{H}_l)} = \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)} and (2 \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - 2 \\sin{(\\hat{H}_l)})^{2} = (\\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - \\sin{(\\hat{H}_l)}) (2 \\operatorname{f_{\\mathbf{p}}}{(\\hat{H}_l)} - 2 \\sin{(\\hat{H}_l)})", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True)), sin(Symbol('\\\\hat{H}_l', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\hat{H}_l', commutative=True))"], "Equality(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(0))"], [["add", 2, "Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))))"], [["times", 3, "Add(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True))))"], "Equality(Pow(Add(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Integer(2)), Mul(Add(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\hat{H}_l', commutative=True)))), Add(Mul(Integer(2), Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}_l', commutative=True))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\hat{H}_l', commutative=True))))))"]]}, {"prompt": "Given p{(h,\\omega)} = e^{\\omega h} and E{(\\varphi,E_{n})} = \\frac{\\varphi}{E_{n}}, then obtain \\frac{\\partial}{\\partial \\varphi} - E{(\\varphi,E_{n})} e^{- \\omega h} = \\frac{\\partial}{\\partial \\varphi} - \\frac{\\varphi e^{- \\omega h}}{E_{n}}", "derivation": "p{(h,\\omega)} = e^{\\omega h} and E{(\\varphi,E_{n})} = \\frac{\\varphi}{E_{n}} and E{(\\varphi,E_{n})} - p{(h,\\omega)} = - p{(h,\\omega)} + \\frac{\\varphi}{E_{n}} and E{(\\varphi,E_{n})} - p{(h,\\omega)} + e^{\\omega h} = - p{(h,\\omega)} + e^{\\omega h} + \\frac{\\varphi}{E_{n}} and - (E{(\\varphi,E_{n})} - p{(h,\\omega)} + e^{\\omega h}) e^{- \\omega h} = - (- p{(h,\\omega)} + e^{\\omega h} + \\frac{\\varphi}{E_{n}}) e^{- \\omega h} and - E{(\\varphi,E_{n})} e^{- \\omega h} = - \\frac{\\varphi e^{- \\omega h}}{E_{n}} and \\frac{\\partial}{\\partial \\varphi} - E{(\\varphi,E_{n})} e^{- \\omega h} = \\frac{\\partial}{\\partial \\varphi} - \\frac{\\varphi e^{- \\omega h}}{E_{n}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))))"], ["get_premise", "Equality(Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["minus", 2, "Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Add(Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))"], [["minus", 3, "Mul(Integer(-1), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))))"], "Equality(Add(Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))))"], "Equality(Mul(Integer(-1), Add(Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Mul(Integer(-1), Add(Mul(Integer(-1), Function('p')(Symbol('h', commutative=True), Symbol('\\\\omega', commutative=True))), exp(Mul(Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True))), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('E')(Symbol('\\\\varphi', commutative=True), Symbol('E_n', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\omega', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given I{(a^{\\dagger})} = \\sin{(a^{\\dagger})}, then obtain (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} \\int (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger} = (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} \\int (\\sin^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger}", "derivation": "I{(a^{\\dagger})} = \\sin{(a^{\\dagger})} and I^{a^{\\dagger}}{(a^{\\dagger})} = \\sin^{a^{\\dagger}}{(a^{\\dagger})} and (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} = (\\sin^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} and \\int (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger} = \\int (\\sin^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger} and (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} \\int (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger} = (I^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} \\int (\\sin^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), sin(Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 4, "Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(Pow(Function('I')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Pow(Pow(sin(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(\\mathbf{M},\\mathbf{S})} = \\mathbf{S} \\sin{(\\mathbf{M})}, then obtain (\\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})})^{\\mathbf{M}} - \\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})} = (\\mathbf{S} \\mathbf{r}{(\\mathbf{M},\\mathbf{S})} \\sin{(\\mathbf{M})})^{\\mathbf{M}} - \\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})}", "derivation": "\\mathbf{r}{(\\mathbf{M},\\mathbf{S})} = \\mathbf{S} \\sin{(\\mathbf{M})} and \\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})} = \\mathbf{S} \\mathbf{r}{(\\mathbf{M},\\mathbf{S})} \\sin{(\\mathbf{M})} and (\\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})})^{\\mathbf{M}} = (\\mathbf{S} \\mathbf{r}{(\\mathbf{M},\\mathbf{S})} \\sin{(\\mathbf{M})})^{\\mathbf{M}} and (\\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})})^{\\mathbf{M}} - \\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})} = (\\mathbf{S} \\mathbf{r}{(\\mathbf{M},\\mathbf{S})} \\sin{(\\mathbf{M})})^{\\mathbf{M}} - \\mathbf{r}^{2}{(\\mathbf{M},\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["times", 1, "Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 3, "Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2))"], "Equality(Add(Pow(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))), Add(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{M}', commutative=True))), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(t,y)} = \\frac{y}{t} and \\nabla{(t,y)} = - t^{2} \\operatorname{M_{E}}^{2}{(t,y)}, then obtain 1 = (- t^{2} \\operatorname{M_{E}}^{2}{(t,y)})^{t} \\nabla^{- t}{(t,y)}", "derivation": "\\operatorname{M_{E}}{(t,y)} = \\frac{y}{t} and t \\operatorname{M_{E}}{(t,y)} = y and - t \\operatorname{M_{E}}{(t,y)} = - y and - t^{2} \\operatorname{M_{E}}^{2}{(t,y)} = - t y \\operatorname{M_{E}}{(t,y)} and (- t^{2} \\operatorname{M_{E}}^{2}{(t,y)})^{t} = (- t y \\operatorname{M_{E}}{(t,y)})^{t} and \\nabla{(t,y)} = - t^{2} \\operatorname{M_{E}}^{2}{(t,y)} and \\nabla^{t}{(t,y)} = (- t y \\operatorname{M_{E}}{(t,y)})^{t} and 1 = (- t y \\operatorname{M_{E}}{(t,y)})^{t} \\nabla^{- t}{(t,y)} and 1 = (- t^{2} \\operatorname{M_{E}}^{2}{(t,y)})^{t} \\nabla^{- t}{(t,y)}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('t', commutative=True), Integer(-1)), Symbol('y', commutative=True)))"], [["divide", 1, "Pow(Symbol('t', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('t', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))), Symbol('y', commutative=True))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Symbol('t', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True)))"], [["times", 3, "Mul(Symbol('t', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(2)), Pow(Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('t', commutative=True), Symbol('y', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))))"], [["power", 4, "Symbol('t', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(2)), Pow(Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Integer(2))), Symbol('t', commutative=True)), Pow(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('y', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(2)), Pow(Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Pow(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Symbol('t', commutative=True)), Pow(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('y', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))), Symbol('t', commutative=True)))"], [["divide", 7, "Pow(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Symbol('t', commutative=True))"], "Equality(Integer(1), Mul(Pow(Mul(Integer(-1), Symbol('t', commutative=True), Symbol('y', commutative=True), Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True))), Symbol('t', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 5], "Equality(Integer(1), Mul(Pow(Mul(Integer(-1), Pow(Symbol('t', commutative=True), Integer(2)), Pow(Function('M_E')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Integer(2))), Symbol('t', commutative=True)), Pow(Function('\\\\nabla')(Symbol('t', commutative=True), Symbol('y', commutative=True)), Mul(Integer(-1), Symbol('t', commutative=True)))))"]]}, {"prompt": "Given \\theta{(g,M_{E})} = \\int (M_{E} - g) dM_{E}, then derive \\theta{(g,M_{E})} = \\frac{M_{E}^{2}}{2} - M_{E} g + r, then obtain \\frac{\\partial}{\\partial r} ((\\frac{M_{E}^{2}}{2} - M_{E} g + r)^{g})^{g} = \\frac{\\partial}{\\partial r} ((\\int (M_{E} - g) dM_{E})^{g})^{g}", "derivation": "\\theta{(g,M_{E})} = \\int (M_{E} - g) dM_{E} and \\theta^{g}{(g,M_{E})} = (\\int (M_{E} - g) dM_{E})^{g} and \\theta{(g,M_{E})} = \\frac{M_{E}^{2}}{2} - M_{E} g + r and \\frac{M_{E}^{2}}{2} - M_{E} g + r = \\int (M_{E} - g) dM_{E} and (\\theta^{g}{(g,M_{E})})^{g} = ((\\int (M_{E} - g) dM_{E})^{g})^{g} and \\theta^{g}{(g,M_{E})} = (\\frac{M_{E}^{2}}{2} - M_{E} g + r)^{g} and ((\\frac{M_{E}^{2}}{2} - M_{E} g + r)^{g})^{g} = ((\\int (M_{E} - g) dM_{E})^{g})^{g} and \\frac{\\partial}{\\partial r} ((\\frac{M_{E}^{2}}{2} - M_{E} g + r)^{g})^{g} = \\frac{\\partial}{\\partial r} ((\\int (M_{E} - g) dM_{E})^{g})^{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('g', commutative=True), Symbol('M_E', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('\\\\theta')(Symbol('g', commutative=True), Symbol('M_E', commutative=True)), Symbol('g', commutative=True)), Pow(Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('g', commutative=True)))"], [["evaluate_integrals", 1], "Equality(Function('\\\\theta')(Symbol('g', commutative=True), Symbol('M_E', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('g', commutative=True)), Symbol('r', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('g', commutative=True)), Symbol('r', commutative=True)), Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Pow(Function('\\\\theta')(Symbol('g', commutative=True), Symbol('M_E', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Pow(Function('\\\\theta')(Symbol('g', commutative=True), Symbol('M_E', commutative=True)), Symbol('g', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('g', commutative=True)), Symbol('r', commutative=True)), Symbol('g', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('g', commutative=True)), Symbol('r', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)))"], [["differentiate", 7, "Symbol('r', commutative=True)"], "Equality(Derivative(Pow(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('M_E', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('M_E', commutative=True), Symbol('g', commutative=True)), Symbol('r', commutative=True)), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Pow(Pow(Integral(Add(Symbol('M_E', commutative=True), Mul(Integer(-1), Symbol('g', commutative=True))), Tuple(Symbol('M_E', commutative=True))), Symbol('g', commutative=True)), Symbol('g', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(b)} = \\cos{(b)}, then obtain \\frac{d}{d b} C{(b)} (\\int C{(b)} db) \\int \\cos{(b)} db = \\frac{d}{d b} \\cos{(b)} (\\int \\cos{(b)} db)^{2}", "derivation": "C{(b)} = \\cos{(b)} and \\int C{(b)} db = \\int \\cos{(b)} db and C{(b)} \\int C{(b)} db = \\cos{(b)} \\int C{(b)} db and \\cos{(b)} (\\int C{(b)} db) \\int \\cos{(b)} db = \\cos{(b)} (\\int \\cos{(b)} db)^{2} and C{(b)} (\\int C{(b)} db) \\int \\cos{(b)} db = \\cos{(b)} (\\int \\cos{(b)} db)^{2} and \\frac{d}{d b} C{(b)} (\\int C{(b)} db) \\int \\cos{(b)} db = \\frac{d}{d b} \\cos{(b)} (\\int \\cos{(b)} db)^{2}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["integrate", 1, "Symbol('b', commutative=True)"], "Equality(Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], [["times", 1, "Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))"], "Equality(Mul(Function('C')(Symbol('b', commutative=True)), Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Mul(cos(Symbol('b', commutative=True)), Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))))"], [["times", 2, "Mul(cos(Symbol('b', commutative=True)), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))))"], "Equality(Mul(cos(Symbol('b', commutative=True)), Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Mul(cos(Symbol('b', commutative=True)), Pow(Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('C')(Symbol('b', commutative=True)), Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Mul(cos(Symbol('b', commutative=True)), Pow(Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(2))))"], [["differentiate", 5, "Symbol('b', commutative=True)"], "Equality(Derivative(Mul(Function('C')(Symbol('b', commutative=True)), Integral(Function('C')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Mul(cos(Symbol('b', commutative=True)), Pow(Integral(cos(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integer(2))), Tuple(Symbol('b', commutative=True), Integer(1))))"]]}, {"prompt": "Given L{(\\mathbf{P})} = \\log{(\\mathbf{P})}, then obtain (\\frac{d}{d \\mathbf{P}} ((L^{\\mathbf{P}}{(\\mathbf{P})})^{\\mathbf{P}} - (\\log{(\\mathbf{P})}^{\\mathbf{P}})^{\\mathbf{P}}))^{\\mathbf{P}} = (\\frac{d}{d \\mathbf{P}} 0)^{\\mathbf{P}}", "derivation": "L{(\\mathbf{P})} = \\log{(\\mathbf{P})} and L^{\\mathbf{P}}{(\\mathbf{P})} = \\log{(\\mathbf{P})}^{\\mathbf{P}} and (L^{\\mathbf{P}}{(\\mathbf{P})})^{\\mathbf{P}} = (\\log{(\\mathbf{P})}^{\\mathbf{P}})^{\\mathbf{P}} and (L^{\\mathbf{P}}{(\\mathbf{P})})^{\\mathbf{P}} - (\\log{(\\mathbf{P})}^{\\mathbf{P}})^{\\mathbf{P}} = 0 and \\frac{d}{d \\mathbf{P}} ((L^{\\mathbf{P}}{(\\mathbf{P})})^{\\mathbf{P}} - (\\log{(\\mathbf{P})}^{\\mathbf{P}})^{\\mathbf{P}}) = \\frac{d}{d \\mathbf{P}} 0 and (\\frac{d}{d \\mathbf{P}} ((L^{\\mathbf{P}}{(\\mathbf{P})})^{\\mathbf{P}} - (\\log{(\\mathbf{P})}^{\\mathbf{P}})^{\\mathbf{P}}))^{\\mathbf{P}} = (\\frac{d}{d \\mathbf{P}} 0)^{\\mathbf{P}}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), log(Symbol('\\\\mathbf{P}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["minus", 3, "Pow(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Add(Pow(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))), Integer(0))"], [["differentiate", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Add(Pow(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Add(Pow(Pow(Function('L')(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Mul(Integer(-1), Pow(Pow(log(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Integer(0), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"]]}, {"prompt": "Given S{(L_{\\varepsilon},\\dot{x})} = L_{\\varepsilon} \\dot{x}, then obtain (- \\cos{(L_{\\varepsilon} \\dot{x})} + \\cos{(S{(L_{\\varepsilon},\\dot{x})})})^{L_{\\varepsilon}} = 0^{L_{\\varepsilon}}", "derivation": "S{(L_{\\varepsilon},\\dot{x})} = L_{\\varepsilon} \\dot{x} and \\cos{(S{(L_{\\varepsilon},\\dot{x})})} = \\cos{(L_{\\varepsilon} \\dot{x})} and - L_{\\varepsilon} + \\cos{(S{(L_{\\varepsilon},\\dot{x})})} = - L_{\\varepsilon} + \\cos{(L_{\\varepsilon} \\dot{x})} and - \\cos{(L_{\\varepsilon} \\dot{x})} + \\cos{(S{(L_{\\varepsilon},\\dot{x})})} = 0 and (- \\cos{(L_{\\varepsilon} \\dot{x})} + \\cos{(S{(L_{\\varepsilon},\\dot{x})})})^{L_{\\varepsilon}} = 0^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), cos(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Integer(0))"], [["power", 4, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), cos(Mul(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), cos(Function('S')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Integer(0), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(f)} = \\cos{(f)}, then obtain (\\Psi_{\\lambda}{(f)} - \\cos{(f + \\cos{(f)})})^{f} = (\\cos{(f)} - \\cos{(f + \\cos{(f)})})^{f}", "derivation": "\\Psi_{\\lambda}{(f)} = \\cos{(f)} and f + \\Psi_{\\lambda}{(f)} = f + \\cos{(f)} and \\cos{(f + \\Psi_{\\lambda}{(f)})} = \\cos{(f + \\cos{(f)})} and \\Psi_{\\lambda}{(f)} - \\cos{(f + \\cos{(f)})} = \\cos{(f)} - \\cos{(f + \\cos{(f)})} and \\Psi_{\\lambda}{(f)} - \\cos{(f + \\Psi_{\\lambda}{(f)})} = \\cos{(f)} - \\cos{(f + \\Psi_{\\lambda}{(f)})} and (\\Psi_{\\lambda}{(f)} - \\cos{(f + \\Psi_{\\lambda}{(f)})})^{f} = (\\cos{(f)} - \\cos{(f + \\Psi_{\\lambda}{(f)})})^{f} and (\\Psi_{\\lambda}{(f)} - \\cos{(f + \\cos{(f)})})^{f} = (\\cos{(f)} - \\cos{(f + \\cos{(f)})})^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["add", 1, "Symbol('f', commutative=True)"], "Equality(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True))), Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))), cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)))))"], [["minus", 1, "cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True))))"], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)))))), Add(cos(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))))), Add(cos(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))))))"], [["power", 5, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))))), Symbol('f', commutative=True)), Pow(Add(cos(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)))))), Symbol('f', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Add(Function('\\\\Psi_{\\\\lambda}')(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)))))), Symbol('f', commutative=True)), Pow(Add(cos(Symbol('f', commutative=True)), Mul(Integer(-1), cos(Add(Symbol('f', commutative=True), cos(Symbol('f', commutative=True)))))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given \\mathbf{J}{(\\psi^*)} = \\log{(\\psi^*)} and \\operatorname{F_{c}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)}, then obtain \\operatorname{F_{c}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\mathbf{J}{(\\psi^*)}", "derivation": "\\mathbf{J}{(\\psi^*)} = \\log{(\\psi^*)} and \\frac{d}{d \\psi^*} \\mathbf{J}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and \\operatorname{F_{c}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\log{(\\psi^*)} and \\operatorname{F_{c}}{(\\psi^*)} = \\frac{d}{d \\psi^*} \\mathbf{J}{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('F_c')(Symbol('\\\\psi^*', commutative=True)), Derivative(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('F_c')(Symbol('\\\\psi^*', commutative=True)), Derivative(Function('\\\\mathbf{J}')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given A{(\\mathbf{g})} = \\sin{(\\sin{(\\mathbf{g})})}, then obtain \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\iint A{(\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} = \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\iint \\sin{(\\sin{(\\mathbf{g})})} d\\mathbf{g} d\\mathbf{g}", "derivation": "A{(\\mathbf{g})} = \\sin{(\\sin{(\\mathbf{g})})} and \\int A{(\\mathbf{g})} d\\mathbf{g} = \\int \\sin{(\\sin{(\\mathbf{g})})} d\\mathbf{g} and \\iint A{(\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} = \\iint \\sin{(\\sin{(\\mathbf{g})})} d\\mathbf{g} d\\mathbf{g} and \\frac{d}{d \\mathbf{g}} \\iint A{(\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} = \\frac{d}{d \\mathbf{g}} \\iint \\sin{(\\sin{(\\mathbf{g})})} d\\mathbf{g} d\\mathbf{g} and \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\iint A{(\\mathbf{g})} d\\mathbf{g} d\\mathbf{g} = \\mathbf{g} + \\frac{d}{d \\mathbf{g}} \\iint \\sin{(\\sin{(\\mathbf{g})})} d\\mathbf{g} d\\mathbf{g}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{g}', commutative=True)), sin(sin(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Integral(Function('A')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Integral(sin(sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Integral(Function('A')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Integral(sin(sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["add", 4, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Integral(Function('A')(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))), Add(Symbol('\\\\mathbf{g}', commutative=True), Derivative(Integral(sin(sin(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(n_{1},F_{g})} = \\frac{e^{F_{g}}}{n_{1}} and \\lambda{(F_{g})} = - F_{g}, then obtain \\frac{n_{1}^{2} \\operatorname{z^{*}}^{2}{(n_{1},F_{g})} e^{2 \\lambda{(F_{g})}}}{F_{g}^{2}} = \\frac{n_{1} \\operatorname{z^{*}}{(n_{1},F_{g})} e^{\\lambda{(F_{g})}}}{F_{g}^{2}}", "derivation": "\\operatorname{z^{*}}{(n_{1},F_{g})} = \\frac{e^{F_{g}}}{n_{1}} and \\frac{\\operatorname{z^{*}}{(n_{1},F_{g})}}{F_{g}} = \\frac{e^{F_{g}}}{F_{g} n_{1}} and \\frac{n_{1} \\operatorname{z^{*}}{(n_{1},F_{g})} e^{- F_{g}}}{F_{g}} = \\frac{1}{F_{g}} and \\lambda{(F_{g})} = - F_{g} and \\frac{n_{1} \\operatorname{z^{*}}{(n_{1},F_{g})} e^{\\lambda{(F_{g})}}}{F_{g}} = \\frac{1}{F_{g}} and \\frac{n_{1}^{2} \\operatorname{z^{*}}^{2}{(n_{1},F_{g})} e^{2 \\lambda{(F_{g})}}}{F_{g}^{2}} = \\frac{n_{1} \\operatorname{z^{*}}{(n_{1},F_{g})} e^{\\lambda{(F_{g})}}}{F_{g}^{2}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Symbol('F_g', commutative=True))))"], [["divide", 1, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Symbol('F_g', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('n_1', commutative=True), Integer(-1)), exp(Symbol('F_g', commutative=True)))"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), exp(Mul(Integer(-1), Symbol('F_g', commutative=True)))), Pow(Symbol('F_g', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('F_g', commutative=True)), Mul(Integer(-1), Symbol('F_g', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), exp(Function('\\\\lambda')(Symbol('F_g', commutative=True)))), Pow(Symbol('F_g', commutative=True), Integer(-1)))"], [["times", 5, "Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('n_1', commutative=True), Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), exp(Function('\\\\lambda')(Symbol('F_g', commutative=True))))"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-2)), Pow(Symbol('n_1', commutative=True), Integer(2)), Pow(Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), Integer(2)), exp(Mul(Integer(2), Function('\\\\lambda')(Symbol('F_g', commutative=True))))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-2)), Symbol('n_1', commutative=True), Function('z^*')(Symbol('n_1', commutative=True), Symbol('F_g', commutative=True)), exp(Function('\\\\lambda')(Symbol('F_g', commutative=True)))))"]]}, {"prompt": "Given M{(u,\\mathbf{E})} = \\mathbf{E} u and \\operatorname{F_{N}}{(u,\\mathbf{E})} = \\mathbf{E} u + M{(u,\\mathbf{E})}, then obtain 2 M{(u,\\mathbf{E})} = 2 \\mathbf{E} u", "derivation": "M{(u,\\mathbf{E})} = \\mathbf{E} u and 2 M{(u,\\mathbf{E})} = \\mathbf{E} u + M{(u,\\mathbf{E})} and \\operatorname{F_{N}}{(u,\\mathbf{E})} = \\mathbf{E} u + M{(u,\\mathbf{E})} and \\operatorname{F_{N}}{(u,\\mathbf{E})} = 2 M{(u,\\mathbf{E})} and \\operatorname{F_{N}}{(u,\\mathbf{E})} = 2 \\mathbf{E} u and \\mathbf{E} u + M{(u,\\mathbf{E})} = 2 \\mathbf{E} u and 2 M{(u,\\mathbf{E})} = 2 \\mathbf{E} u", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)))"], [["add", 1, "Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Mul(Integer(2), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Function('F_N')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 6], "Equality(Mul(Integer(2), Function('M')(Symbol('u', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Integer(2), Symbol('\\\\mathbf{E}', commutative=True), Symbol('u', commutative=True)))"]]}, {"prompt": "Given x{(E,J_{\\varepsilon})} = E^{J_{\\varepsilon}} and i{(E,J_{\\varepsilon})} = \\iint E^{J_{\\varepsilon}} dJ_{\\varepsilon} dJ_{\\varepsilon}, then obtain \\iint x{(E,J_{\\varepsilon})} dJ_{\\varepsilon} dJ_{\\varepsilon} = i{(E,J_{\\varepsilon})}", "derivation": "x{(E,J_{\\varepsilon})} = E^{J_{\\varepsilon}} and \\int x{(E,J_{\\varepsilon})} dJ_{\\varepsilon} = \\int E^{J_{\\varepsilon}} dJ_{\\varepsilon} and \\iint x{(E,J_{\\varepsilon})} dJ_{\\varepsilon} dJ_{\\varepsilon} = \\iint E^{J_{\\varepsilon}} dJ_{\\varepsilon} dJ_{\\varepsilon} and i{(E,J_{\\varepsilon})} = \\iint E^{J_{\\varepsilon}} dJ_{\\varepsilon} dJ_{\\varepsilon} and \\iint x{(E,J_{\\varepsilon})} dJ_{\\varepsilon} dJ_{\\varepsilon} = i{(E,J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Pow(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('x')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('x')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Pow(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('i')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Integral(Pow(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Function('x')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Function('i')(Symbol('E', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given H{(v_{2})} = \\log{(v_{2})}, then obtain (H{(v_{2})} \\log{(v_{2})}^{3})^{v_{2}} \\log{(v_{2})}^{4} = (\\log{(v_{2})}^{4})^{v_{2}} \\log{(v_{2})}^{4}", "derivation": "H{(v_{2})} = \\log{(v_{2})} and H{(v_{2})} \\log{(v_{2})} = \\log{(v_{2})}^{2} and H{(v_{2})} \\log{(v_{2})}^{3} = \\log{(v_{2})}^{4} and (H{(v_{2})} \\log{(v_{2})}^{3})^{v_{2}} = (\\log{(v_{2})}^{4})^{v_{2}} and (H{(v_{2})} \\log{(v_{2})}^{3})^{v_{2}} \\log{(v_{2})}^{4} = (\\log{(v_{2})}^{4})^{v_{2}} \\log{(v_{2})}^{4}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True)))"], [["times", 1, "log(Symbol('v_2', commutative=True))"], "Equality(Mul(Function('H')(Symbol('v_2', commutative=True)), log(Symbol('v_2', commutative=True))), Pow(log(Symbol('v_2', commutative=True)), Integer(2)))"], [["times", 2, "Pow(log(Symbol('v_2', commutative=True)), Integer(2))"], "Equality(Mul(Function('H')(Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(3))), Pow(log(Symbol('v_2', commutative=True)), Integer(4)))"], [["power", 3, "Symbol('v_2', commutative=True)"], "Equality(Pow(Mul(Function('H')(Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(3))), Symbol('v_2', commutative=True)), Pow(Pow(log(Symbol('v_2', commutative=True)), Integer(4)), Symbol('v_2', commutative=True)))"], [["times", 4, "Pow(log(Symbol('v_2', commutative=True)), Integer(4))"], "Equality(Mul(Pow(Mul(Function('H')(Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(3))), Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(4))), Mul(Pow(Pow(log(Symbol('v_2', commutative=True)), Integer(4)), Symbol('v_2', commutative=True)), Pow(log(Symbol('v_2', commutative=True)), Integer(4))))"]]}, {"prompt": "Given \\varphi^{*}{(\\tilde{g},n)} = \\tilde{g} \\sin{(n)} and \\hat{p}{(\\tilde{g},n)} = \\frac{\\tilde{g} \\sin{(n)}}{\\varphi^{*}{(\\tilde{g},n)}}, then obtain \\int (\\frac{\\tilde{g} \\sin{(n)}}{\\varphi^{*}{(\\tilde{g},n)}} - n) d\\tilde{g} = \\int (1 - n) d\\tilde{g}", "derivation": "\\varphi^{*}{(\\tilde{g},n)} = \\tilde{g} \\sin{(n)} and \\hat{p}{(\\tilde{g},n)} = \\frac{\\tilde{g} \\sin{(n)}}{\\varphi^{*}{(\\tilde{g},n)}} and \\hat{p}{(\\tilde{g},n)} = 1 and - n + \\hat{p}{(\\tilde{g},n)} = 1 - n and \\frac{\\tilde{g} \\sin{(n)}}{\\varphi^{*}{(\\tilde{g},n)}} - n = 1 - n and \\int (\\frac{\\tilde{g} \\sin{(n)}}{\\varphi^{*}{(\\tilde{g},n)}} - n) d\\tilde{g} = \\int (1 - n) d\\tilde{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), sin(Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Integer(-1)), sin(Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Integer(1))"], [["minus", 3, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\hat{p}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Integer(-1)), sin(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))), Add(Integer(1), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["integrate", 5, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('\\\\tilde{g}', commutative=True), Pow(Function('\\\\varphi^*')(Symbol('\\\\tilde{g}', commutative=True), Symbol('n', commutative=True)), Integer(-1)), sin(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))), Integral(Add(Integer(1), Mul(Integer(-1), Symbol('n', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(E_{\\lambda},\\theta)} = E_{\\lambda} - \\theta and \\tilde{g}^*{(\\Psi_{nl})} = \\cos{(\\sin{(\\Psi_{nl})})}, then obtain \\tilde{g}^*{(\\Psi_{nl})} + \\frac{1}{(- \\theta + \\hat{p}_0{(E_{\\lambda},\\theta)})^{2}} = \\cos{(\\sin{(\\Psi_{nl})})} + \\frac{1}{(- \\theta + \\hat{p}_0{(E_{\\lambda},\\theta)})^{2}}", "derivation": "\\hat{p}_0{(E_{\\lambda},\\theta)} = E_{\\lambda} - \\theta and - \\theta + \\hat{p}_0{(E_{\\lambda},\\theta)} = E_{\\lambda} - 2 \\theta and \\tilde{g}^*{(\\Psi_{nl})} = \\cos{(\\sin{(\\Psi_{nl})})} and \\tilde{g}^*{(\\Psi_{nl})} + \\frac{1}{(E_{\\lambda} - 2 \\theta)^{2}} = \\cos{(\\sin{(\\Psi_{nl})})} + \\frac{1}{(E_{\\lambda} - 2 \\theta)^{2}} and \\tilde{g}^*{(\\Psi_{nl})} + \\frac{1}{(- \\theta + \\hat{p}_0{(E_{\\lambda},\\theta)})^{2}} = \\cos{(\\sin{(\\Psi_{nl})})} + \\frac{1}{(- \\theta + \\hat{p}_0{(E_{\\lambda},\\theta)})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi_{nl}', commutative=True)), cos(sin(Symbol('\\\\Psi_{nl}', commutative=True))))"], [["add", 3, "Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True))), Integer(-2))"], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True))), Integer(-2))), Add(cos(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\theta', commutative=True))), Integer(-2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi_{nl}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-2))), Add(cos(sin(Symbol('\\\\Psi_{nl}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('\\\\hat{p}_0')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('\\\\theta', commutative=True))), Integer(-2))))"]]}, {"prompt": "Given \\mu{(\\sigma_x,f)} = - \\sigma_x + f, then obtain \\mu{(\\sigma_x,f)} + 2 \\mu^{f}{(\\sigma_x,f)} = 2 (- \\sigma_x + f)^{f} + \\mu{(\\sigma_x,f)}", "derivation": "\\mu{(\\sigma_x,f)} = - \\sigma_x + f and \\mu^{f}{(\\sigma_x,f)} = (- \\sigma_x + f)^{f} and \\mu{(\\sigma_x,f)} + \\mu^{f}{(\\sigma_x,f)} = (- \\sigma_x + f)^{f} + \\mu{(\\sigma_x,f)} and (- \\sigma_x + f)^{f} + \\mu{(\\sigma_x,f)} + \\mu^{f}{(\\sigma_x,f)} = 2 (- \\sigma_x + f)^{f} + \\mu{(\\sigma_x,f)} and \\mu{(\\sigma_x,f)} + 2 \\mu^{f}{(\\sigma_x,f)} = 2 (- \\sigma_x + f)^{f} + \\mu{(\\sigma_x,f)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)))"], [["add", 2, "Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))"], "Equality(Add(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Pow(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))))"], [["add", 3, "Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True))"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Pow(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Add(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Mul(Integer(2), Pow(Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)))), Add(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True)), Symbol('f', commutative=True)), Symbol('f', commutative=True))), Function('\\\\mu')(Symbol('\\\\sigma_x', commutative=True), Symbol('f', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(l)} = \\cos{(l)}, then obtain (l \\frac{d}{d l} \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl - \\int 2 \\cos{(l)} dl)^{2} = (l \\frac{d}{d l} \\int 2 \\cos{(l)} dl - \\int 2 \\cos{(l)} dl)^{2}", "derivation": "\\mathbf{H}{(l)} = \\cos{(l)} and \\mathbf{H}{(l)} + \\cos{(l)} = 2 \\cos{(l)} and \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl = \\int 2 \\cos{(l)} dl and \\frac{d}{d l} \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl = \\frac{d}{d l} \\int 2 \\cos{(l)} dl and l \\frac{d}{d l} \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl = l \\frac{d}{d l} \\int 2 \\cos{(l)} dl and l \\frac{d}{d l} \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl - \\int 2 \\cos{(l)} dl = l \\frac{d}{d l} \\int 2 \\cos{(l)} dl - \\int 2 \\cos{(l)} dl and (l \\frac{d}{d l} \\int (\\mathbf{H}{(l)} + \\cos{(l)}) dl - \\int 2 \\cos{(l)} dl)^{2} = (l \\frac{d}{d l} \\int 2 \\cos{(l)} dl - \\int 2 \\cos{(l)} dl)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True)))"], [["add", 1, "cos(Symbol('l', commutative=True))"], "Equality(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Mul(Integer(2), cos(Symbol('l', commutative=True))))"], [["integrate", 2, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["times", 4, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Symbol('l', commutative=True), Derivative(Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["minus", 5, "Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True)))"], "Equality(Add(Mul(Symbol('l', commutative=True), Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))), Add(Mul(Symbol('l', commutative=True), Derivative(Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))))"], [["power", 6, 2], "Equality(Pow(Add(Mul(Symbol('l', commutative=True), Derivative(Integral(Add(Function('\\\\mathbf{H}')(Symbol('l', commutative=True)), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))), Integer(2)), Pow(Add(Mul(Symbol('l', commutative=True), Derivative(Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Integer(-1), Integral(Mul(Integer(2), cos(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True))))), Integer(2)))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbb{I})} = \\sin{(e^{\\mathbb{I}})}, then obtain \\frac{(\\frac{\\mathbf{J}{(\\mathbb{I})}}{\\sin{(e^{\\mathbb{I}})}} + 1) \\sin{(e^{\\mathbb{I}})}}{\\mathbf{J}{(\\mathbb{I})}} = \\frac{2 \\sin{(e^{\\mathbb{I}})}}{\\mathbf{J}{(\\mathbb{I})}}", "derivation": "\\mathbf{J}{(\\mathbb{I})} = \\sin{(e^{\\mathbb{I}})} and \\frac{\\mathbf{J}{(\\mathbb{I})}}{\\sin{(e^{\\mathbb{I}})}} = 1 and \\frac{\\mathbf{J}{(\\mathbb{I})}}{\\sin{(e^{\\mathbb{I}})}} + 1 = 2 and \\frac{(\\frac{\\mathbf{J}{(\\mathbb{I})}}{\\sin{(e^{\\mathbb{I}})}} + 1) \\sin{(e^{\\mathbb{I}})}}{\\mathbf{J}{(\\mathbb{I})}} = \\frac{2 \\sin{(e^{\\mathbb{I}})}}{\\mathbf{J}{(\\mathbb{I})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), sin(exp(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 1, "sin(exp(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Integer(1))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Integer(1)), Integer(2))"], [["divide", 3, "Mul(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"], "Equality(Mul(Add(Mul(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(sin(exp(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Integer(1)), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), sin(exp(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Integer(2), Pow(Function('\\\\mathbf{J}')(Symbol('\\\\mathbb{I}', commutative=True)), Integer(-1)), sin(exp(Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given T{(f)} = \\int e^{f} df, then obtain \\frac{d}{d f} \\int \\frac{T{(f)} + \\int e^{f} df}{f} df = \\frac{d}{d f} \\int \\frac{2 \\int e^{f} df}{f} df", "derivation": "T{(f)} = \\int e^{f} df and T{(f)} + \\int e^{f} df = 2 \\int e^{f} df and \\frac{T{(f)} + \\int e^{f} df}{f} = \\frac{2 \\int e^{f} df}{f} and \\int \\frac{T{(f)} + \\int e^{f} df}{f} df = \\int \\frac{2 \\int e^{f} df}{f} df and \\frac{d}{d f} \\int \\frac{T{(f)} + \\int e^{f} df}{f} df = \\frac{d}{d f} \\int \\frac{2 \\int e^{f} df}{f} df", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["add", 1, "Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))"], "Equality(Add(Function('T')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Mul(Integer(2), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["divide", 2, "Symbol('f', commutative=True)"], "Equality(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Function('T')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Mul(Integer(2), Pow(Symbol('f', commutative=True), Integer(-1)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))))"], [["integrate", 3, "Symbol('f', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Function('T')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True))), Integral(Mul(Integer(2), Pow(Symbol('f', commutative=True), Integer(-1)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))))"], [["differentiate", 4, "Symbol('f', commutative=True)"], "Equality(Derivative(Integral(Mul(Pow(Symbol('f', commutative=True), Integer(-1)), Add(Function('T')(Symbol('f', commutative=True)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(2), Pow(Symbol('f', commutative=True), Integer(-1)), Integral(exp(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True)))), Tuple(Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"]]}, {"prompt": "Given z{(\\mathbf{E},\\theta)} = \\mathbf{E} - \\theta, then obtain \\theta^{2} z{(\\mathbf{E},\\theta)} \\int (\\mathbf{E} - \\theta) d\\theta = \\theta^{2} (\\mathbf{E} - \\theta) \\int (\\mathbf{E} - \\theta) d\\theta", "derivation": "z{(\\mathbf{E},\\theta)} = \\mathbf{E} - \\theta and - \\theta z{(\\mathbf{E},\\theta)} = - \\theta (\\mathbf{E} - \\theta) and \\int z{(\\mathbf{E},\\theta)} d\\theta = \\int (\\mathbf{E} - \\theta) d\\theta and - \\theta z{(\\mathbf{E},\\theta)} \\int z{(\\mathbf{E},\\theta)} d\\theta = - \\theta (\\mathbf{E} - \\theta) \\int z{(\\mathbf{E},\\theta)} d\\theta and \\theta^{2} z{(\\mathbf{E},\\theta)} \\int z{(\\mathbf{E},\\theta)} d\\theta = \\theta^{2} (\\mathbf{E} - \\theta) \\int z{(\\mathbf{E},\\theta)} d\\theta and \\theta^{2} z{(\\mathbf{E},\\theta)} \\int (\\mathbf{E} - \\theta) d\\theta = \\theta^{2} (\\mathbf{E} - \\theta) \\int (\\mathbf{E} - \\theta) d\\theta", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True)))))"], [["integrate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True))))"], [["times", 2, "Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Integer(-1), Symbol('\\\\theta', commutative=True), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["times", 4, "Mul(Integer(-1), Symbol('\\\\theta', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Function('z')(Symbol('\\\\mathbf{E}', commutative=True), Symbol('\\\\theta', commutative=True)), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Integral(Add(Symbol('\\\\mathbf{E}', commutative=True), Mul(Integer(-1), Symbol('\\\\theta', commutative=True))), Tuple(Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given H{(W)} = \\sin{(W)} and \\dot{x}{(W)} = \\sin{(W)}, then derive \\frac{d}{d W} \\dot{x}{(W)} = \\cos{(W)}, then obtain (W + H{(W)} + \\cos{(W)}) \\cos{(W)} \\frac{d}{d W} \\sin{(W)} = (W + H{(W)} + \\frac{d}{d W} H{(W)}) \\cos{(W)} \\frac{d}{d W} \\sin{(W)}", "derivation": "H{(W)} = \\sin{(W)} and \\dot{x}{(W)} = \\sin{(W)} and \\frac{d}{d W} \\dot{x}{(W)} = \\frac{d}{d W} \\sin{(W)} and \\frac{d}{d W} \\dot{x}{(W)} = \\cos{(W)} and \\cos{(W)} = \\frac{d}{d W} \\sin{(W)} and H{(W)} + \\cos{(W)} = H{(W)} + \\frac{d}{d W} \\sin{(W)} and H{(W)} + \\cos{(W)} = H{(W)} + \\frac{d}{d W} H{(W)} and W + H{(W)} + \\cos{(W)} = W + H{(W)} + \\frac{d}{d W} H{(W)} and (W + H{(W)} + \\cos{(W)}) \\cos{(W)} \\frac{d}{d W} \\sin{(W)} = (W + H{(W)} + \\frac{d}{d W} H{(W)}) \\cos{(W)} \\frac{d}{d W} \\sin{(W)}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{x}')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True)))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\dot{x}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), cos(Symbol('W', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(cos(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["add", 5, "Function('H')(Symbol('W', commutative=True))"], "Equality(Add(Function('H')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Add(Function('H')(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Function('H')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Add(Function('H')(Symbol('W', commutative=True)), Derivative(Function('H')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["add", 7, "Symbol('W', commutative=True)"], "Equality(Add(Symbol('W', commutative=True), Function('H')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), Add(Symbol('W', commutative=True), Function('H')(Symbol('W', commutative=True)), Derivative(Function('H')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["times", 8, "Mul(cos(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], "Equality(Mul(Add(Symbol('W', commutative=True), Function('H')(Symbol('W', commutative=True)), cos(Symbol('W', commutative=True))), cos(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Add(Symbol('W', commutative=True), Function('H')(Symbol('W', commutative=True)), Derivative(Function('H')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), cos(Symbol('W', commutative=True)), Derivative(sin(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{v})} = \\cos{(e^{\\mathbf{v}})}, then derive \\int \\operatorname{f_{E}}{(\\mathbf{v})} d\\mathbf{v} = t_{2} + \\operatorname{Ci}{(e^{\\mathbf{v}})}, then obtain \\int \\cos{(e^{\\mathbf{v}})} d\\mathbf{v} + 1 = t_{2} + \\operatorname{Ci}{(e^{\\mathbf{v}})} + 1", "derivation": "\\operatorname{f_{E}}{(\\mathbf{v})} = \\cos{(e^{\\mathbf{v}})} and \\int \\operatorname{f_{E}}{(\\mathbf{v})} d\\mathbf{v} = \\int \\cos{(e^{\\mathbf{v}})} d\\mathbf{v} and \\int \\operatorname{f_{E}}{(\\mathbf{v})} d\\mathbf{v} = t_{2} + \\operatorname{Ci}{(e^{\\mathbf{v}})} and \\int \\cos{(e^{\\mathbf{v}})} d\\mathbf{v} = t_{2} + \\operatorname{Ci}{(e^{\\mathbf{v}})} and \\int \\cos{(e^{\\mathbf{v}})} d\\mathbf{v} + 1 = t_{2} + \\operatorname{Ci}{(e^{\\mathbf{v}})} + 1", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{v}', commutative=True)), cos(exp(Symbol('\\\\mathbf{v}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(cos(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('t_2', commutative=True), Ci(exp(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Add(Symbol('t_2', commutative=True), Ci(exp(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Add(Integral(cos(exp(Symbol('\\\\mathbf{v}', commutative=True))), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integer(1)), Add(Symbol('t_2', commutative=True), Ci(exp(Symbol('\\\\mathbf{v}', commutative=True))), Integer(1)))"]]}, {"prompt": "Given a{(s,F_{x})} = F_{x}^{s}, then obtain 0 = - \\frac{F_{x}^{s} \\frac{\\partial}{\\partial F_{x}} a{(s,F_{x})}}{a^{2}{(s,F_{x})}} + \\frac{F_{x}^{s} s}{F_{x} a{(s,F_{x})}}", "derivation": "a{(s,F_{x})} = F_{x}^{s} and 1 = \\frac{F_{x}^{s}}{a{(s,F_{x})}} and \\frac{d}{d F_{x}} 1 = \\frac{\\partial}{\\partial F_{x}} \\frac{F_{x}^{s}}{a{(s,F_{x})}} and 0 = - \\frac{F_{x}^{s} \\frac{\\partial}{\\partial F_{x}} a{(s,F_{x})}}{a^{2}{(s,F_{x})}} + \\frac{F_{x}^{s} s}{F_{x} a{(s,F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Pow(Symbol('F_x', commutative=True), Symbol('s', commutative=True)))"], [["divide", 1, "Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('F_x', commutative=True), Symbol('s', commutative=True)), Pow(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_x', commutative=True), Symbol('s', commutative=True)), Pow(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Integer(-1))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Symbol('F_x', commutative=True), Symbol('s', commutative=True)), Pow(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Integer(-2)), Derivative(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(Symbol('F_x', commutative=True), Symbol('s', commutative=True)), Symbol('s', commutative=True), Pow(Function('a')(Symbol('s', commutative=True), Symbol('F_x', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\varphi^*)} = \\cos{(\\varphi^*)}, then derive - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\operatorname{g_{\\varepsilon}}{(\\varphi^*)} = - 2 \\sin{(\\varphi^*)}, then obtain - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = - 2 \\sin{(\\varphi^*)}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\varphi^*)} = \\cos{(\\varphi^*)} and \\operatorname{g_{\\varepsilon}}{(\\varphi^*)} + \\cos{(\\varphi^*)} = 2 \\cos{(\\varphi^*)} and \\frac{d}{d \\varphi^*} (\\operatorname{g_{\\varepsilon}}{(\\varphi^*)} + \\cos{(\\varphi^*)}) = \\frac{d}{d \\varphi^*} 2 \\cos{(\\varphi^*)} and - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\operatorname{g_{\\varepsilon}}{(\\varphi^*)} = - 2 \\sin{(\\varphi^*)} and - \\sin{(\\varphi^*)} + \\frac{d}{d \\varphi^*} \\cos{(\\varphi^*)} = - 2 \\sin{(\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "cos(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\varphi^*', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), cos(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Integer(2), cos(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\varphi^*', commutative=True))), Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\varphi^*', commutative=True))), Derivative(cos(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1)))), Mul(Integer(-1), Integer(2), sin(Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(C_{2})} = \\sin{(C_{2})}, then derive \\int \\theta_{1}{(C_{2})} dC_{2} = \\mu - \\cos{(C_{2})}, then obtain 1 = \\frac{\\int \\theta_{1}{(C_{2})} dC_{2}}{\\mu - \\cos{(C_{2})}}", "derivation": "\\theta_{1}{(C_{2})} = \\sin{(C_{2})} and \\int \\theta_{1}{(C_{2})} dC_{2} = \\int \\sin{(C_{2})} dC_{2} and \\int \\theta_{1}{(C_{2})} dC_{2} = \\mu - \\cos{(C_{2})} and 1 = \\frac{\\mu - \\cos{(C_{2})}}{\\int \\theta_{1}{(C_{2})} dC_{2}} and 1 = \\frac{\\mu - \\cos{(C_{2})}}{\\int \\sin{(C_{2})} dC_{2}} and 1 = \\frac{\\int \\theta_{1}{(C_{2})} dC_{2}}{\\int \\sin{(C_{2})} dC_{2}} and \\int \\sin{(C_{2})} dC_{2} = \\mu - \\cos{(C_{2})} and 1 = \\frac{\\int \\theta_{1}{(C_{2})} dC_{2}}{\\mu - \\cos{(C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["integrate", 1, "Symbol('C_2', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))))"], [["divide", 3, "Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))"], "Equality(Integer(1), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Pow(Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integer(1), Mul(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Pow(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Integer(1), Mul(Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Pow(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), cos(Symbol('C_2', commutative=True)))), Integer(-1)), Integral(Function('\\\\theta_1')(Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{s}{(\\hbar,f^{*})} = \\frac{e^{f^{*}}}{\\hbar} and \\hat{x}_0{(\\hbar)} = \\hbar, then obtain \\frac{(\\hbar \\mathbf{s}{(\\hbar,f^{*})} + \\hbar) e^{f^{*}}}{\\hbar} = \\frac{(\\hbar \\mathbf{s}{(\\hbar,f^{*})} + \\hbar) e^{f^{*}}}{\\hat{x}_0{(\\hbar)}}", "derivation": "\\mathbf{s}{(\\hbar,f^{*})} = \\frac{e^{f^{*}}}{\\hbar} and \\hat{x}_0{(\\hbar)} = \\hbar and \\hat{x}_0{(\\hbar)} \\mathbf{s}{(\\hbar,f^{*})} = \\hbar \\mathbf{s}{(\\hbar,f^{*})} and \\frac{\\hat{x}_0{(\\hbar)} e^{f^{*}}}{\\hbar} = e^{f^{*}} and \\frac{e^{f^{*}}}{\\hbar} = \\frac{e^{f^{*}}}{\\hat{x}_0{(\\hbar)}} and \\frac{(\\hbar \\mathbf{s}{(\\hbar,f^{*})} + \\hbar) e^{f^{*}}}{\\hbar} = \\frac{(\\hbar \\mathbf{s}{(\\hbar,f^{*})} + \\hbar) e^{f^{*}}}{\\hat{x}_0{(\\hbar)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), exp(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], [["times", 2, "Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))), Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True)), exp(Symbol('f^*', commutative=True))), exp(Symbol('f^*', commutative=True)))"], [["divide", 4, "Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), exp(Symbol('f^*', commutative=True))), Mul(Pow(Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True)), Integer(-1)), exp(Symbol('f^*', commutative=True))))"], [["times", 5, "Add(Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\hbar', commutative=True)), exp(Symbol('f^*', commutative=True))), Mul(Add(Mul(Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{s}')(Symbol('\\\\hbar', commutative=True), Symbol('f^*', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Function('\\\\hat{x}_0')(Symbol('\\\\hbar', commutative=True)), Integer(-1)), exp(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(W)} = \\int \\cos{(W)} dW, then derive \\frac{\\partial}{\\partial W} (\\pi + \\operatorname{A_{1}}{(W)} + \\sin{(W)}) = \\frac{\\partial}{\\partial W} (2 \\pi + 2 \\sin{(W)}), then obtain \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} (\\pi + \\operatorname{A_{1}}{(W)} + \\sin{(W)}))^{2} = \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} (2 \\pi + 2 \\sin{(W)}))^{2}", "derivation": "\\operatorname{A_{1}}{(W)} = \\int \\cos{(W)} dW and \\operatorname{A_{1}}{(W)} + \\int \\cos{(W)} dW = 2 \\int \\cos{(W)} dW and \\frac{d}{d W} (\\operatorname{A_{1}}{(W)} + \\int \\cos{(W)} dW) = \\frac{d}{d W} 2 \\int \\cos{(W)} dW and \\frac{\\partial}{\\partial W} (\\pi + \\operatorname{A_{1}}{(W)} + \\sin{(W)}) = \\frac{\\partial}{\\partial W} (2 \\pi + 2 \\sin{(W)}) and (\\frac{\\partial}{\\partial W} (\\pi + \\operatorname{A_{1}}{(W)} + \\sin{(W)}))^{2} = (\\frac{\\partial}{\\partial W} (2 \\pi + 2 \\sin{(W)}))^{2} and \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} (\\pi + \\operatorname{A_{1}}{(W)} + \\sin{(W)}))^{2} = \\frac{\\partial}{\\partial W} (\\frac{\\partial}{\\partial W} (2 \\pi + 2 \\sin{(W)}))^{2}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('W', commutative=True)), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["add", 1, "Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))"], "Equality(Add(Function('A_1')(Symbol('W', commutative=True)), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Add(Function('A_1')(Symbol('W', commutative=True)), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Integral(cos(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Add(Symbol('\\\\pi', commutative=True), Function('A_1')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), sin(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(Add(Symbol('\\\\pi', commutative=True), Function('A_1')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), sin(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Symbol('\\\\pi', commutative=True), Function('A_1')(Symbol('W', commutative=True)), sin(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Integer(2), Symbol('\\\\pi', commutative=True)), Mul(Integer(2), sin(Symbol('W', commutative=True)))), Tuple(Symbol('W', commutative=True), Integer(1))), Integer(2)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(y)} = \\log{(y)}, then derive \\Psi^{\\dagger} + \\operatorname{M_{E}}{(y)} = f_{\\mathbf{v}} + \\log{(y)}, then obtain (\\Psi^{\\dagger} + \\log{(y)})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} + \\log{(y)})^{f_{\\mathbf{v}}}", "derivation": "\\operatorname{M_{E}}{(y)} = \\log{(y)} and \\frac{d}{d y} \\operatorname{M_{E}}{(y)} = \\frac{d}{d y} \\log{(y)} and \\int \\frac{d}{d y} \\operatorname{M_{E}}{(y)} dy = \\int \\frac{d}{d y} \\log{(y)} dy and \\Psi^{\\dagger} + \\operatorname{M_{E}}{(y)} = f_{\\mathbf{v}} + \\log{(y)} and \\Psi^{\\dagger} + \\log{(y)} = f_{\\mathbf{v}} + \\log{(y)} and (\\Psi^{\\dagger} + \\log{(y)})^{f_{\\mathbf{v}}} = (f_{\\mathbf{v}} + \\log{(y)})^{f_{\\mathbf{v}}}", "srepr_derivation": [["get_premise", "Equality(Function('M_E')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('M_E')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Derivative(Function('M_E')(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))), Integral(Derivative(log(Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Function('M_E')(Symbol('y', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), log(Symbol('y', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('y', commutative=True))), Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), log(Symbol('y', commutative=True))))"], [["power", 5, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), log(Symbol('y', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Pow(Add(Symbol('f_{\\\\mathbf{v}}', commutative=True), log(Symbol('y', commutative=True))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(M)} = e^{M}, then derive \\frac{d}{d M} \\rho_{f}{(M)} = e^{M}, then derive S + \\rho_{f}{(M)} = \\int \\rho_{f}{(M)} dM, then obtain S + \\frac{d}{d M} \\rho_{f}{(M)} = \\int \\frac{d}{d M} \\rho_{f}{(M)} dM", "derivation": "\\rho_{f}{(M)} = e^{M} and \\frac{d}{d M} \\rho_{f}{(M)} = \\frac{d}{d M} e^{M} and \\frac{d}{d M} \\rho_{f}{(M)} = e^{M} and \\frac{d}{d M} \\rho_{f}{(M)} = \\rho_{f}{(M)} and \\int \\frac{d}{d M} \\rho_{f}{(M)} dM = \\int \\rho_{f}{(M)} dM and S + \\rho_{f}{(M)} = \\int \\rho_{f}{(M)} dM and S + \\frac{d}{d M} \\rho_{f}{(M)} = \\int \\frac{d}{d M} \\rho_{f}{(M)} dM", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('M', commutative=True)), exp(Symbol('M', commutative=True)))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(exp(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), exp(Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Function('\\\\rho_f')(Symbol('M', commutative=True)))"], [["integrate", 4, "Symbol('M', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))), Integral(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('S', commutative=True), Function('\\\\rho_f')(Symbol('M', commutative=True))), Integral(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Symbol('S', commutative=True), Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), Integral(Derivative(Function('\\\\rho_f')(Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(y^{\\prime},M_{E},x^\\prime)} = \\frac{- M_{E} + y^{\\prime}}{x^\\prime}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\sigma_{p}{(y^{\\prime},M_{E},x^\\prime)} = \\frac{1}{x^\\prime}, then obtain (\\frac{1}{x^\\prime})^{M_{E}} = (\\frac{\\partial}{\\partial y^{\\prime}} \\frac{- M_{E} + y^{\\prime}}{x^\\prime})^{M_{E}}", "derivation": "\\sigma_{p}{(y^{\\prime},M_{E},x^\\prime)} = \\frac{- M_{E} + y^{\\prime}}{x^\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\sigma_{p}{(y^{\\prime},M_{E},x^\\prime)} = \\frac{\\partial}{\\partial y^{\\prime}} \\frac{- M_{E} + y^{\\prime}}{x^\\prime} and \\frac{\\partial}{\\partial y^{\\prime}} \\sigma_{p}{(y^{\\prime},M_{E},x^\\prime)} = \\frac{1}{x^\\prime} and \\frac{1}{x^\\prime} = \\frac{\\partial}{\\partial y^{\\prime}} \\frac{- M_{E} + y^{\\prime}}{x^\\prime} and (\\frac{1}{x^\\prime})^{M_{E}} = (\\frac{\\partial}{\\partial y^{\\prime}} \\frac{- M_{E} + y^{\\prime}}{x^\\prime})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('y^{\\\\prime}', commutative=True), Symbol('M_E', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Symbol('M_E', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('x^\\\\prime', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('M_E', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\lambda{(i)} = \\cos{(i)}, then obtain \\int (\\int (\\lambda{(i)} + \\cos{(i)}) di) \\int 2 \\lambda{(i)} di di = \\int (\\int (\\lambda{(i)} + \\cos{(i)}) di)^{2} di", "derivation": "\\lambda{(i)} = \\cos{(i)} and 2 \\lambda{(i)} = \\lambda{(i)} + \\cos{(i)} and \\int 2 \\lambda{(i)} di = \\int (\\lambda{(i)} + \\cos{(i)}) di and (\\int (\\lambda{(i)} + \\cos{(i)}) di) \\int 2 \\lambda{(i)} di = (\\int (\\lambda{(i)} + \\cos{(i)}) di)^{2} and \\int (\\int (\\lambda{(i)} + \\cos{(i)}) di) \\int 2 \\lambda{(i)} di di = \\int (\\int (\\lambda{(i)} + \\cos{(i)}) di)^{2} di", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True)))"], [["add", 1, "Function('\\\\lambda')(Symbol('i', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\lambda')(Symbol('i', commutative=True))), Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))))"], [["integrate", 2, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\lambda')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["times", 3, "Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Mul(Integer(2), Function('\\\\lambda')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Pow(Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integer(2)))"], [["integrate", 4, "Symbol('i', commutative=True)"], "Equality(Integral(Mul(Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integral(Mul(Integer(2), Function('\\\\lambda')(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))), Tuple(Symbol('i', commutative=True))), Integral(Pow(Integral(Add(Function('\\\\lambda')(Symbol('i', commutative=True)), cos(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))), Integer(2)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given f{(c_{0},v)} = c_{0} v, then obtain f{(c_{0},v)} \\int f^{v}{(c_{0},v)} dc_{0} - f{(c_{0},v)} = f{(c_{0},v)} \\int (c_{0} v)^{v} dc_{0} - f{(c_{0},v)}", "derivation": "f{(c_{0},v)} = c_{0} v and f^{v}{(c_{0},v)} = (c_{0} v)^{v} and \\int f^{v}{(c_{0},v)} dc_{0} = \\int (c_{0} v)^{v} dc_{0} and c_{0} v \\int f^{v}{(c_{0},v)} dc_{0} = c_{0} v \\int (c_{0} v)^{v} dc_{0} and f{(c_{0},v)} \\int f^{v}{(c_{0},v)} dc_{0} = f{(c_{0},v)} \\int (c_{0} v)^{v} dc_{0} and f{(c_{0},v)} \\int f^{v}{(c_{0},v)} dc_{0} - f{(c_{0},v)} = f{(c_{0},v)} \\int (c_{0} v)^{v} dc_{0} - f{(c_{0},v)}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["integrate", 2, "Symbol('c_0', commutative=True)"], "Equality(Integral(Pow(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(Pow(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["times", 3, "Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True), Integral(Pow(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True), Integral(Pow(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["minus", 5, "Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True))"], "Equality(Add(Mul(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Integer(-1), Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))), Add(Mul(Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Mul(Symbol('c_0', commutative=True), Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Mul(Integer(-1), Function('f')(Symbol('c_0', commutative=True), Symbol('v', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(U,\\delta)} = \\frac{\\partial}{\\partial U} (U - \\delta), then derive \\theta_{1}{(U,\\delta)} = 1, then obtain \\int \\sin{(\\theta_{1}{(U,\\delta)})} dU = \\int \\sin{(\\frac{\\partial}{\\partial U} (U - \\delta))} dU", "derivation": "\\theta_{1}{(U,\\delta)} = \\frac{\\partial}{\\partial U} (U - \\delta) and \\theta_{1}{(U,\\delta)} = 1 and \\sin{(\\theta_{1}{(U,\\delta)})} = \\sin{(1)} and \\sin{(\\frac{\\partial}{\\partial U} (U - \\delta))} = \\sin{(1)} and \\sin{(\\theta_{1}{(U,\\delta)})} = \\sin{(\\frac{\\partial}{\\partial U} (U - \\delta))} and \\int \\sin{(\\theta_{1}{(U,\\delta)})} dU = \\int \\sin{(\\frac{\\partial}{\\partial U} (U - \\delta))} dU", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(1))"], [["sin", 2], "Equality(sin(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))), sin(Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(sin(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), sin(Integer(1)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(sin(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))), sin(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(sin(Function('\\\\theta_1')(Symbol('U', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(sin(Derivative(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('U', commutative=True), Integer(1)))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given U{(\\hat{x},\\theta_1,\\theta)} = \\frac{\\theta \\theta_1}{\\hat{x}} and p{(\\hat{x},\\theta,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} U{(\\hat{x},\\theta_1,\\theta)}, then obtain p{(\\hat{x},\\theta,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\frac{\\theta \\theta_1}{\\hat{x}}", "derivation": "U{(\\hat{x},\\theta_1,\\theta)} = \\frac{\\theta \\theta_1}{\\hat{x}} and \\frac{\\partial}{\\partial \\theta_1} U{(\\hat{x},\\theta_1,\\theta)} = \\frac{\\partial}{\\partial \\theta_1} \\frac{\\theta \\theta_1}{\\hat{x}} and p{(\\hat{x},\\theta,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} U{(\\hat{x},\\theta_1,\\theta)} and p{(\\hat{x},\\theta,\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\frac{\\theta \\theta_1}{\\hat{x}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Function('U')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('p')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_1', commutative=True)), Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\theta', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}}, then obtain \\log{(\\tilde{g})} + \\cos{(f_{\\mathbf{p}} + b{(f_{\\mathbf{p}})})} = \\log{(\\tilde{g})} + \\cos{(f_{\\mathbf{p}} + e^{f_{\\mathbf{p}}})}", "derivation": "b{(f_{\\mathbf{p}})} = e^{f_{\\mathbf{p}}} and f_{\\mathbf{p}} + b{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + e^{f_{\\mathbf{p}}} and \\cos{(f_{\\mathbf{p}} + b{(f_{\\mathbf{p}})})} = \\cos{(f_{\\mathbf{p}} + e^{f_{\\mathbf{p}}})} and \\log{(\\tilde{g})} + \\cos{(f_{\\mathbf{p}} + b{(f_{\\mathbf{p}})})} = \\log{(\\tilde{g})} + \\cos{(f_{\\mathbf{p}} + e^{f_{\\mathbf{p}}})}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["cos", 2], "Equality(cos(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True)))), cos(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True)))))"], [["add", 3, "log(Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Add(log(Symbol('\\\\tilde{g}', commutative=True)), cos(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Function('b')(Symbol('f_{\\\\mathbf{p}}', commutative=True))))), Add(log(Symbol('\\\\tilde{g}', commutative=True)), cos(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{p}{(b)} = \\cos{(b)} and \\theta{(b)} = \\cos{(b)} and \\mu_{0}{(b)} = \\frac{d}{d b} (e^{\\cos{(b)}})^{b}, then obtain \\frac{(e^{\\theta{(b)}})^{b}}{\\frac{d}{d b} (e^{\\cos{(b)}})^{b}} = \\frac{(e^{\\cos{(b)}})^{b}}{\\frac{d}{d b} (e^{\\cos{(b)}})^{b}}", "derivation": "\\mathbf{p}{(b)} = \\cos{(b)} and e^{\\mathbf{p}{(b)}} = e^{\\cos{(b)}} and (e^{\\mathbf{p}{(b)}})^{b} = (e^{\\cos{(b)}})^{b} and \\theta{(b)} = \\cos{(b)} and \\mathbf{p}{(b)} = \\theta{(b)} and (e^{\\theta{(b)}})^{b} = (e^{\\cos{(b)}})^{b} and \\mu_{0}{(b)} = \\frac{d}{d b} (e^{\\cos{(b)}})^{b} and \\frac{(e^{\\theta{(b)}})^{b}}{\\mu_{0}{(b)}} = \\frac{(e^{\\cos{(b)}})^{b}}{\\mu_{0}{(b)}} and \\frac{(e^{\\theta{(b)}})^{b}}{\\frac{d}{d b} (e^{\\cos{(b)}})^{b}} = \\frac{(e^{\\cos{(b)}})^{b}}{\\frac{d}{d b} (e^{\\cos{(b)}})^{b}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), exp(cos(Symbol('b', commutative=True))))"], [["power", 2, "Symbol('b', commutative=True)"], "Equality(Pow(exp(Function('\\\\mathbf{p}')(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('b', commutative=True)), cos(Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\mathbf{p}')(Symbol('b', commutative=True)), Function('\\\\theta')(Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(exp(Function('\\\\theta')(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('b', commutative=True)), Derivative(Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["divide", 6, "Function('\\\\mu_0')(Symbol('b', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mu_0')(Symbol('b', commutative=True)), Integer(-1)), Pow(exp(Function('\\\\theta')(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Mul(Pow(Function('\\\\mu_0')(Symbol('b', commutative=True)), Integer(-1)), Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Mul(Pow(exp(Function('\\\\theta')(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Derivative(Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1))), Mul(Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Derivative(Pow(exp(cos(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given T{(x^\\prime,\\theta_1)} = (e^{\\theta_1})^{x^\\prime} and \\mathbf{s}{(\\theta_1,x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} T{(x^\\prime,\\theta_1)}, then derive \\frac{\\partial}{\\partial x^\\prime} T{(x^\\prime,\\theta_1)} = (e^{\\theta_1})^{x^\\prime} \\log{(e^{\\theta_1})}, then obtain \\mathbf{s}{(\\theta_1,x^\\prime)} = T{(x^\\prime,\\theta_1)} \\log{(e^{\\theta_1})}", "derivation": "T{(x^\\prime,\\theta_1)} = (e^{\\theta_1})^{x^\\prime} and T{(x^\\prime,\\theta_1)} - e^{\\theta_1} = - e^{\\theta_1} + (e^{\\theta_1})^{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} (T{(x^\\prime,\\theta_1)} - e^{\\theta_1}) = \\frac{\\partial}{\\partial x^\\prime} (- e^{\\theta_1} + (e^{\\theta_1})^{x^\\prime}) and \\frac{\\partial}{\\partial x^\\prime} T{(x^\\prime,\\theta_1)} = (e^{\\theta_1})^{x^\\prime} \\log{(e^{\\theta_1})} and \\frac{\\partial}{\\partial x^\\prime} T{(x^\\prime,\\theta_1)} = T{(x^\\prime,\\theta_1)} \\log{(e^{\\theta_1})} and \\mathbf{s}{(\\theta_1,x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} T{(x^\\prime,\\theta_1)} and \\mathbf{s}{(\\theta_1,x^\\prime)} = T{(x^\\prime,\\theta_1)} \\log{(e^{\\theta_1})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\theta_1', commutative=True))"], "Equality(Add(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True))), Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), exp(Symbol('\\\\theta_1', commutative=True))), Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Pow(exp(Symbol('\\\\theta_1', commutative=True)), Symbol('x^\\\\prime', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Mul(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Derivative(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\theta_1', commutative=True), Symbol('x^\\\\prime', commutative=True)), Mul(Function('T')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\theta_1', commutative=True)), log(exp(Symbol('\\\\theta_1', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(A)} = \\sin{(A)}, then derive \\phi_1 + \\Omega{(A)} = v_{1} + \\sin{(A)}, then obtain \\phi_1 + \\sin{(A)} = v_{1} + \\sin{(A)}", "derivation": "\\Omega{(A)} = \\sin{(A)} and \\frac{d}{d A} \\Omega{(A)} = \\frac{d}{d A} \\sin{(A)} and \\int \\frac{d}{d A} \\Omega{(A)} dA = \\int \\frac{d}{d A} \\sin{(A)} dA and \\phi_1 + \\Omega{(A)} = v_{1} + \\sin{(A)} and \\phi_1 + \\sin{(A)} = v_{1} + \\sin{(A)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('A', commutative=True)), sin(Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('A', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Omega')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))), Integral(Derivative(sin(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\phi_1', commutative=True), Function('\\\\Omega')(Symbol('A', commutative=True))), Add(Symbol('v_1', commutative=True), sin(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('\\\\phi_1', commutative=True), sin(Symbol('A', commutative=True))), Add(Symbol('v_1', commutative=True), sin(Symbol('A', commutative=True))))"]]}, {"prompt": "Given \\theta_{2}{(g_{\\varepsilon},E_{n})} = \\cos{(E_{n} - g_{\\varepsilon})}, then obtain \\int (- E_{n} + \\int \\theta_{2}{(g_{\\varepsilon},E_{n})} dg_{\\varepsilon}) dg_{\\varepsilon} = \\int (- E_{n} + \\int \\cos{(E_{n} - g_{\\varepsilon})} dg_{\\varepsilon}) dg_{\\varepsilon}", "derivation": "\\theta_{2}{(g_{\\varepsilon},E_{n})} = \\cos{(E_{n} - g_{\\varepsilon})} and \\int \\theta_{2}{(g_{\\varepsilon},E_{n})} dg_{\\varepsilon} = \\int \\cos{(E_{n} - g_{\\varepsilon})} dg_{\\varepsilon} and - E_{n} + \\int \\theta_{2}{(g_{\\varepsilon},E_{n})} dg_{\\varepsilon} = - E_{n} + \\int \\cos{(E_{n} - g_{\\varepsilon})} dg_{\\varepsilon} and \\int (- E_{n} + \\int \\theta_{2}{(g_{\\varepsilon},E_{n})} dg_{\\varepsilon}) dg_{\\varepsilon} = \\int (- E_{n} + \\int \\cos{(E_{n} - g_{\\varepsilon})} dg_{\\varepsilon}) dg_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Symbol('E_n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 3, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_n', commutative=True)), Integral(cos(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given g{(\\sigma_p,\\tilde{g}^*,W)} = W^{\\tilde{g}^*} - \\sigma_p, then obtain \\frac{d}{d \\sigma_p} 1 = \\frac{\\partial}{\\partial \\sigma_p} (2 - \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p})", "derivation": "g{(\\sigma_p,\\tilde{g}^*,W)} = W^{\\tilde{g}^*} - \\sigma_p and \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p} = 1 and -1 + \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p} = 0 and 0 = 1 - \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p} and 1 = 2 - \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p} and \\frac{d}{d \\sigma_p} 1 = \\frac{\\partial}{\\partial \\sigma_p} (2 - \\frac{g{(\\sigma_p,\\tilde{g}^*,W)}}{W^{\\tilde{g}^*} - \\sigma_p})", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True)), Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))"], [["divide", 1, "Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Integer(-1), Mul(Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True)))), Integer(0))"], [["minus", 3, "Add(Integer(-1), Mul(Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True))))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True)))))"], [["minus", 4, "Integer(-1)"], "Equality(Integer(1), Add(Integer(2), Mul(Integer(-1), Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True)))))"], [["differentiate", 5, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))), Derivative(Add(Integer(2), Mul(Integer(-1), Pow(Add(Pow(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Function('g')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True)))), Tuple(Symbol('\\\\sigma_p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(y^{\\prime},T)} = \\sin^{T}{(y^{\\prime})}, then derive (\\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{H}{(y^{\\prime},T)})^{y^{\\prime}} = (\\frac{T \\sin^{T}{(y^{\\prime})} \\cos{(y^{\\prime})}}{\\sin{(y^{\\prime})}})^{y^{\\prime}}, then obtain (\\frac{\\partial}{\\partial y^{\\prime}} \\sin^{T}{(y^{\\prime})})^{y^{\\prime}} = (\\frac{T \\sin^{T}{(y^{\\prime})} \\cos{(y^{\\prime})}}{\\sin{(y^{\\prime})}})^{y^{\\prime}}", "derivation": "\\mathbf{H}{(y^{\\prime},T)} = \\sin^{T}{(y^{\\prime})} and \\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{H}{(y^{\\prime},T)} = \\frac{\\partial}{\\partial y^{\\prime}} \\sin^{T}{(y^{\\prime})} and (\\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{H}{(y^{\\prime},T)})^{y^{\\prime}} = (\\frac{\\partial}{\\partial y^{\\prime}} \\sin^{T}{(y^{\\prime})})^{y^{\\prime}} and (\\frac{\\partial}{\\partial y^{\\prime}} \\mathbf{H}{(y^{\\prime},T)})^{y^{\\prime}} = (\\frac{T \\sin^{T}{(y^{\\prime})} \\cos{(y^{\\prime})}}{\\sin{(y^{\\prime})}})^{y^{\\prime}} and (\\frac{\\partial}{\\partial y^{\\prime}} \\sin^{T}{(y^{\\prime})})^{y^{\\prime}} = (\\frac{T \\sin^{T}{(y^{\\prime})} \\cos{(y^{\\prime})}}{\\sin{(y^{\\prime})}})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["power", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Derivative(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('\\\\mathbf{H}')(Symbol('y^{\\\\prime}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Derivative(Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Symbol('T', commutative=True), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('T', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given A{(\\mu,\\varphi)} = \\frac{\\varphi}{\\mu}, then derive \\frac{(\\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)})^{\\mu}}{\\varphi} = \\frac{(- \\frac{\\varphi}{\\mu^{2}})^{\\mu}}{\\varphi}, then obtain \\frac{(- \\frac{A{(\\mu,\\varphi)}}{\\mu})^{\\mu}}{\\varphi} = \\frac{(- \\frac{\\varphi}{\\mu^{2}})^{\\mu}}{\\varphi}", "derivation": "A{(\\mu,\\varphi)} = \\frac{\\varphi}{\\mu} and \\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)} = \\frac{\\partial}{\\partial \\mu} \\frac{\\varphi}{\\mu} and (\\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)})^{\\mu} = (\\frac{\\partial}{\\partial \\mu} \\frac{\\varphi}{\\mu})^{\\mu} and \\frac{(\\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)})^{\\mu}}{\\varphi} = \\frac{(\\frac{\\partial}{\\partial \\mu} \\frac{\\varphi}{\\mu})^{\\mu}}{\\varphi} and \\frac{(\\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)})^{\\mu}}{\\varphi} = \\frac{(- \\frac{\\varphi}{\\mu^{2}})^{\\mu}}{\\varphi} and \\frac{(\\frac{\\partial}{\\partial \\mu} A{(\\mu,\\varphi)})^{\\mu}}{\\varphi} = \\frac{(- \\frac{A{(\\mu,\\varphi)}}{\\mu})^{\\mu}}{\\varphi} and \\frac{(- \\frac{A{(\\mu,\\varphi)}}{\\mu})^{\\mu}}{\\varphi} = \\frac{(- \\frac{\\varphi}{\\mu^{2}})^{\\mu}}{\\varphi}", "srepr_derivation": [["get_premise", "Equality(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Derivative(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True)))"], [["divide", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Derivative(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Derivative(Mul(Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Derivative(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Derivative(Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Function('A')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mu', commutative=True), Integer(-2)), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given W{(t,\\chi)} = \\log{(t)}^{\\chi}, then obtain - \\frac{\\frac{\\partial}{\\partial t} (W{(t,\\chi)} - \\log{(t)})}{\\log{(t)}} = - \\frac{\\frac{\\partial}{\\partial t} (- \\log{(t)} + \\log{(t)}^{\\chi})}{\\log{(t)}}", "derivation": "W{(t,\\chi)} = \\log{(t)}^{\\chi} and W{(t,\\chi)} - \\log{(t)} = - \\log{(t)} + \\log{(t)}^{\\chi} and \\frac{\\partial}{\\partial t} (W{(t,\\chi)} - \\log{(t)}) = \\frac{\\partial}{\\partial t} (- \\log{(t)} + \\log{(t)}^{\\chi}) and - \\frac{\\frac{\\partial}{\\partial t} (W{(t,\\chi)} - \\log{(t)})}{\\log{(t)}} = - \\frac{\\frac{\\partial}{\\partial t} (- \\log{(t)} + \\log{(t)}^{\\chi})}{\\log{(t)}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('t', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(log(Symbol('t', commutative=True)), Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "log(Symbol('t', commutative=True))"], "Equality(Add(Function('W')(Symbol('t', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('t', commutative=True)"], "Equality(Derivative(Add(Function('W')(Symbol('t', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["divide", 3, "Mul(Integer(-1), log(Symbol('t', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(log(Symbol('t', commutative=True)), Integer(-1)), Derivative(Add(Function('W')(Symbol('t', commutative=True), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), log(Symbol('t', commutative=True)))), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(log(Symbol('t', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), log(Symbol('t', commutative=True))), Pow(log(Symbol('t', commutative=True)), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\Psi{(\\hat{x}_0)} = \\hat{x}_0 and S{(J)} = \\log{(e^{J})}, then obtain \\hat{x}_0 (S{(J)} + \\cos{(\\mathbf{M})}) = \\hat{x}_0 (\\log{(e^{J})} + \\cos{(\\mathbf{M})})", "derivation": "\\operatorname{F_{N}}{(\\mathbf{M})} = \\cos{(\\mathbf{M})} and \\Psi{(\\hat{x}_0)} = \\hat{x}_0 and S{(J)} = \\log{(e^{J})} and S{(J)} + \\cos{(\\mathbf{M})} = \\log{(e^{J})} + \\cos{(\\mathbf{M})} and \\operatorname{F_{N}}{(\\mathbf{M})} + S{(J)} = \\operatorname{F_{N}}{(\\mathbf{M})} + \\log{(e^{J})} and (\\operatorname{F_{N}}{(\\mathbf{M})} + S{(J)}) \\Psi{(\\hat{x}_0)} = (\\operatorname{F_{N}}{(\\mathbf{M})} + \\log{(e^{J})}) \\Psi{(\\hat{x}_0)} and \\hat{x}_0 (\\operatorname{F_{N}}{(\\mathbf{M})} + S{(J)}) = \\hat{x}_0 (\\operatorname{F_{N}}{(\\mathbf{M})} + \\log{(e^{J})}) and \\hat{x}_0 (S{(J)} + \\cos{(\\mathbf{M})}) = \\hat{x}_0 (\\log{(e^{J})} + \\cos{(\\mathbf{M})})", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], ["get_premise", "Equality(Function('S')(Symbol('J', commutative=True)), log(exp(Symbol('J', commutative=True))))"], [["add", 3, "cos(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True))), Add(log(exp(Symbol('J', commutative=True))), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), Function('S')(Symbol('J', commutative=True))), Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), log(exp(Symbol('J', commutative=True)))))"], [["times", 5, "Function('\\\\Psi')(Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Mul(Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), Function('S')(Symbol('J', commutative=True))), Function('\\\\Psi')(Symbol('\\\\hat{x}_0', commutative=True))), Mul(Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), log(exp(Symbol('J', commutative=True)))), Function('\\\\Psi')(Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), Function('S')(Symbol('J', commutative=True)))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Function('F_N')(Symbol('\\\\mathbf{M}', commutative=True)), log(exp(Symbol('J', commutative=True))))))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Function('S')(Symbol('J', commutative=True)), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(log(exp(Symbol('J', commutative=True))), cos(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given v{(t_{1},\\mathbf{s})} = - \\mathbf{s} + t_{1} and \\lambda{(\\mathbf{s})} = \\mathbf{s}, then obtain \\frac{\\partial}{\\partial \\mathbf{s}} v{(t_{1},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (- \\mathbf{s} + t_{1})", "derivation": "v{(t_{1},\\mathbf{s})} = - \\mathbf{s} + t_{1} and \\mathbf{s} + v{(t_{1},\\mathbf{s})} = t_{1} and \\lambda{(\\mathbf{s})} = \\mathbf{s} and \\mathbf{s} - \\lambda{(\\mathbf{s})} + v{(t_{1},\\mathbf{s})} = t_{1} - \\lambda{(\\mathbf{s})} and \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} - \\lambda{(\\mathbf{s})} + v{(t_{1},\\mathbf{s})}) = \\frac{\\partial}{\\partial \\mathbf{s}} (t_{1} - \\lambda{(\\mathbf{s})}) and \\frac{\\partial}{\\partial \\mathbf{s}} v{(t_{1},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (- \\mathbf{s} + t_{1})", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('t_1', commutative=True)))"], [["minus", 1, "Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Function('v')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('t_1', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True))"], [["minus", 2, "Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Function('v')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True))), Function('v')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Symbol('t_1', commutative=True), Mul(Integer(-1), Function('\\\\lambda')(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Function('v')(Symbol('t_1', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\rho_{f}{(v_{z})} = \\cos{(\\cos{(v_{z})})} and \\mathbf{p}{(v_{z})} = \\cos{(v_{z})} + \\cos^{v_{z}}{(\\cos{(v_{z})})}, then obtain \\mathbf{p}{(v_{z})} = \\rho_{f}^{v_{z}}{(v_{z})} + \\cos{(v_{z})}", "derivation": "\\rho_{f}{(v_{z})} = \\cos{(\\cos{(v_{z})})} and \\rho_{f}^{v_{z}}{(v_{z})} = \\cos^{v_{z}}{(\\cos{(v_{z})})} and \\rho_{f}^{v_{z}}{(v_{z})} + \\cos{(v_{z})} = \\cos{(v_{z})} + \\cos^{v_{z}}{(\\cos{(v_{z})})} and \\mathbf{p}{(v_{z})} = \\cos{(v_{z})} + \\cos^{v_{z}}{(\\cos{(v_{z})})} and \\mathbf{p}{(v_{z})} = \\rho_{f}^{v_{z}}{(v_{z})} + \\cos{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('v_z', commutative=True)), cos(cos(Symbol('v_z', commutative=True))))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(cos(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], [["add", 2, "cos(Symbol('v_z', commutative=True))"], "Equality(Add(Pow(Function('\\\\rho_f')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True))), Add(cos(Symbol('v_z', commutative=True)), Pow(cos(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('v_z', commutative=True)), Add(cos(Symbol('v_z', commutative=True)), Pow(cos(cos(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{p}')(Symbol('v_z', commutative=True)), Add(Pow(Function('\\\\rho_f')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), cos(Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{F},\\omega)} = \\mathbf{F} \\omega, then obtain \\int \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} d\\omega - \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} = \\int 1 d\\omega - \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega}", "derivation": "\\mathbf{J}{(\\mathbf{F},\\omega)} = \\mathbf{F} \\omega and \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} = 1 and \\int \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} d\\omega = \\int 1 d\\omega and \\int \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} d\\omega - \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega} = \\int 1 d\\omega - \\frac{\\mathbf{J}{(\\mathbf{F},\\omega)}}{\\mathbf{F} \\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["divide", 1, "Mul(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\omega', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Integral(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('\\\\omega', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\sigma_p,V_{\\mathbf{B}},Z)} = \\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p and \\hat{\\mathbf{x}}{(\\sigma_p,V_{\\mathbf{B}},Z)} = \\frac{\\partial}{\\partial Z} \\int \\Psi_{nl}{(\\sigma_p,V_{\\mathbf{B}},Z)} dV_{\\mathbf{B}}, then obtain \\hat{\\mathbf{x}}^{\\sigma_p}{(\\sigma_p,V_{\\mathbf{B}},Z)} = (\\frac{\\partial}{\\partial Z} \\int (\\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p) dV_{\\mathbf{B}})^{\\sigma_p}", "derivation": "\\Psi_{nl}{(\\sigma_p,V_{\\mathbf{B}},Z)} = \\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p and \\int \\Psi_{nl}{(\\sigma_p,V_{\\mathbf{B}},Z)} dV_{\\mathbf{B}} = \\int (\\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p) dV_{\\mathbf{B}} and \\hat{\\mathbf{x}}{(\\sigma_p,V_{\\mathbf{B}},Z)} = \\frac{\\partial}{\\partial Z} \\int \\Psi_{nl}{(\\sigma_p,V_{\\mathbf{B}},Z)} dV_{\\mathbf{B}} and \\hat{\\mathbf{x}}{(\\sigma_p,V_{\\mathbf{B}},Z)} = \\frac{\\partial}{\\partial Z} \\int (\\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p) dV_{\\mathbf{B}} and \\hat{\\mathbf{x}}^{\\sigma_p}{(\\sigma_p,V_{\\mathbf{B}},Z)} = (\\frac{\\partial}{\\partial Z} \\int (\\frac{V_{\\mathbf{B}}}{Z} - \\sigma_p) dV_{\\mathbf{B}})^{\\sigma_p}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Derivative(Integral(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\sigma_p', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('Z', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Pow(Derivative(Integral(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('Z', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} = \\cos{(\\frac{\\rho}{\\mathbf{B}})}, then obtain 0 = (- \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} + \\cos{(\\frac{\\rho}{\\mathbf{B}})}) (\\cos{(\\frac{\\rho}{\\mathbf{B}})} - \\frac{1}{\\mathbf{B}})", "derivation": "\\operatorname{m_{s}}{(\\mathbf{B},\\rho)} = \\cos{(\\frac{\\rho}{\\mathbf{B}})} and \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} - \\frac{1}{\\mathbf{B}} = \\cos{(\\frac{\\rho}{\\mathbf{B}})} - \\frac{1}{\\mathbf{B}} and 0 = - \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} + \\cos{(\\frac{\\rho}{\\mathbf{B}})} and 0 = (- \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} + \\cos{(\\frac{\\rho}{\\mathbf{B}})}) (\\operatorname{m_{s}}{(\\mathbf{B},\\rho)} - \\frac{1}{\\mathbf{B}}) and 0 = (- \\operatorname{m_{s}}{(\\mathbf{B},\\rho)} + \\cos{(\\frac{\\rho}{\\mathbf{B}})}) (\\cos{(\\frac{\\rho}{\\mathbf{B}})} - \\frac{1}{\\mathbf{B}})", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True)), cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))"], "Equality(Add(Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))), Add(cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))))"], [["add", 1, "Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))))"], [["times", 3, "Add(Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))), Add(Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('m_s')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('\\\\rho', commutative=True))), cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True)))), Add(cos(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Symbol('\\\\rho', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(\\rho,r_{0})} = \\frac{\\rho}{r_{0}}, then derive \\frac{\\partial}{\\partial \\rho} \\operatorname{E_{n}}{(\\rho,r_{0})} = \\frac{1}{r_{0}}, then obtain \\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{r_{0}} - 1 = -1 + \\frac{1}{r_{0}}", "derivation": "\\operatorname{E_{n}}{(\\rho,r_{0})} = \\frac{\\rho}{r_{0}} and \\frac{\\partial}{\\partial \\rho} \\operatorname{E_{n}}{(\\rho,r_{0})} = \\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{r_{0}} and \\frac{\\partial}{\\partial \\rho} \\operatorname{E_{n}}{(\\rho,r_{0})} = \\frac{1}{r_{0}} and \\frac{\\partial}{\\partial \\rho} \\operatorname{E_{n}}{(\\rho,r_{0})} - 1 = -1 + \\frac{1}{r_{0}} and \\frac{\\partial}{\\partial \\rho} \\frac{\\rho}{r_{0}} - 1 = -1 + \\frac{1}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('E_n')(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('E_n')(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Pow(Symbol('r_0', commutative=True), Integer(-1)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('E_n')(Symbol('\\\\rho', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Derivative(Mul(Symbol('\\\\rho', commutative=True), Pow(Symbol('r_0', commutative=True), Integer(-1))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('r_0', commutative=True), Integer(-1))))"]]}, {"prompt": "Given A{(q,G)} = G - q, then obtain \\frac{A^{q}{(q,G)}}{(G - q) A{(q,G)}} = \\frac{(G - q)^{q}}{(G - q) A{(q,G)}}", "derivation": "A{(q,G)} = G - q and (G - q) A{(q,G)} = (G - q)^{2} and A^{q}{(q,G)} = (G - q)^{q} and \\frac{A^{q}{(q,G)}}{(G - q)^{2}} = \\frac{(G - q)^{q}}{(G - q)^{2}} and \\frac{A^{q}{(q,G)}}{(G - q) A{(q,G)}} = \\frac{(G - q)^{q}}{(G - q) A{(q,G)}}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["times", 1, "Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True)))"], "Equality(Mul(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True))), Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(2)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('q', commutative=True)), Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["divide", 3, "Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(2))"], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(-2)), Pow(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('q', commutative=True))), Mul(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(-2)), Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(-1)), Pow(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Integer(-1)), Pow(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('q', commutative=True))), Mul(Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Integer(-1)), Pow(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('q', commutative=True))), Symbol('q', commutative=True)), Pow(Function('A')(Symbol('q', commutative=True), Symbol('G', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(F_{x})} = \\log{(\\log{(F_{x})})}, then derive \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} = \\frac{1}{F_{x} \\log{(F_{x})}}, then obtain 0 = - \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} + \\frac{1}{F_{x} \\log{(F_{x})}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(F_{x})} = \\log{(\\log{(F_{x})})} and \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} = \\frac{d}{d F_{x}} \\log{(\\log{(F_{x})})} and \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} = \\frac{1}{F_{x} \\log{(F_{x})}} and \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} - \\frac{d}{d F_{x}} \\log{(\\log{(F_{x})})} = - \\frac{d}{d F_{x}} \\log{(\\log{(F_{x})})} + \\frac{1}{F_{x} \\log{(F_{x})}} and 0 = - \\frac{d}{d F_{x}} \\operatorname{V_{\\mathbf{E}}}{(F_{x})} + \\frac{1}{F_{x} \\log{(F_{x})}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('F_x', commutative=True)), log(log(Symbol('F_x', commutative=True))))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(log(log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1))))"], [["minus", 3, "Derivative(log(log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(log(log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Derivative(log(log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), Pow(log(Symbol('F_x', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\nabla)} = \\cos{(\\nabla)}, then derive \\int \\operatorname{v_{1}}{(\\nabla)} d\\nabla = x + \\sin{(\\nabla)}, then obtain 0 = \\frac{\\partial^{2}}{\\partial x\\partial \\nabla} \\int (x + \\sin{(\\nabla)}) d\\nabla", "derivation": "\\operatorname{v_{1}}{(\\nabla)} = \\cos{(\\nabla)} and \\int \\operatorname{v_{1}}{(\\nabla)} d\\nabla = \\int \\cos{(\\nabla)} d\\nabla and \\int \\operatorname{v_{1}}{(\\nabla)} d\\nabla = x + \\sin{(\\nabla)} and \\int \\cos{(\\nabla)} d\\nabla = x + \\sin{(\\nabla)} and \\iint \\cos{(\\nabla)} d\\nabla d\\nabla = \\int (x + \\sin{(\\nabla)}) d\\nabla and \\frac{d}{d \\nabla} \\iint \\cos{(\\nabla)} d\\nabla d\\nabla = \\frac{\\partial}{\\partial \\nabla} \\int (x + \\sin{(\\nabla)}) d\\nabla and \\frac{d^{2}}{d xd \\nabla} \\iint \\cos{(\\nabla)} d\\nabla d\\nabla = \\frac{\\partial^{2}}{\\partial x\\partial \\nabla} \\int (x + \\sin{(\\nabla)}) d\\nabla and 0 = \\frac{\\partial^{2}}{\\partial x\\partial \\nabla} \\int (x + \\sin{(\\nabla)}) d\\nabla", "srepr_derivation": [["get_premise", "Equality(Function('v_1')(Symbol('\\\\nabla', commutative=True)), cos(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_1')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))))"], [["integrate", 4, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\nabla', commutative=True)"], "Equality(Derivative(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1))))"], [["differentiate", 6, "Symbol('x', commutative=True)"], "Equality(Derivative(Integral(cos(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(0), Derivative(Integral(Add(Symbol('x', commutative=True), sin(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True))), Tuple(Symbol('\\\\nabla', commutative=True), Integer(1)), Tuple(Symbol('x', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(z,i)} = \\sin{(i + z)} and \\operatorname{A_{x}}{(z,i)} = i + z + \\frac{\\sin{(i + z)}}{i}, then obtain \\frac{\\int (\\operatorname{A_{x}}{(z,i)} - \\frac{\\sin{(i + z)}}{i}) dz}{i + z + \\frac{\\sin{(i + z)}}{i} - \\frac{1}{i}} = \\frac{\\int (i + z) dz}{i + z + \\frac{\\sin{(i + z)}}{i} - \\frac{1}{i}}", "derivation": "v{(z,i)} = \\sin{(i + z)} and \\frac{v{(z,i)}}{i} = \\frac{\\sin{(i + z)}}{i} and \\operatorname{A_{x}}{(z,i)} = i + z + \\frac{\\sin{(i + z)}}{i} and \\operatorname{A_{x}}{(z,i)} - \\frac{\\sin{(i + z)}}{i} = i + z and \\int (\\operatorname{A_{x}}{(z,i)} - \\frac{\\sin{(i + z)}}{i}) dz = \\int (i + z) dz and \\frac{\\int (\\operatorname{A_{x}}{(z,i)} - \\frac{\\sin{(i + z)}}{i}) dz}{i + z + \\frac{v{(z,i)}}{i} - \\frac{1}{i}} = \\frac{\\int (i + z) dz}{i + z + \\frac{v{(z,i)}}{i} - \\frac{1}{i}} and \\frac{\\int (\\operatorname{A_{x}}{(z,i)} - \\frac{\\sin{(i + z)}}{i}) dz}{i + z + \\frac{\\sin{(i + z)}}{i} - \\frac{1}{i}} = \\frac{\\int (i + z) dz}{i + z + \\frac{\\sin{(i + z)}}{i} - \\frac{1}{i}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('z', commutative=True), Symbol('i', commutative=True)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))"], [["divide", 1, "Symbol('i', commutative=True)"], "Equality(Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('v')(Symbol('z', commutative=True), Symbol('i', commutative=True))), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('z', commutative=True), Symbol('i', commutative=True)), Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))))"], [["minus", 3, "Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))"], "Equality(Add(Function('A_x')(Symbol('z', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))), Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 4, "Symbol('z', commutative=True)"], "Equality(Integral(Add(Function('A_x')(Symbol('z', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))), Tuple(Symbol('z', commutative=True))), Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["divide", 5, "Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('v')(Symbol('z', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1))))"], "Equality(Mul(Pow(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('v')(Symbol('z', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(-1)), Integral(Add(Function('A_x')(Symbol('z', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))), Tuple(Symbol('z', commutative=True)))), Mul(Pow(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), Function('v')(Symbol('z', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(-1)), Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Mul(Pow(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(-1)), Integral(Add(Function('A_x')(Symbol('z', commutative=True), Symbol('i', commutative=True)), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True))))), Tuple(Symbol('z', commutative=True)))), Mul(Pow(Add(Symbol('i', commutative=True), Symbol('z', commutative=True), Mul(Pow(Symbol('i', commutative=True), Integer(-1)), sin(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)))), Mul(Integer(-1), Pow(Symbol('i', commutative=True), Integer(-1)))), Integer(-1)), Integral(Add(Symbol('i', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\ddot{x})} = \\sin{(\\ddot{x})}, then derive \\int \\hat{x}_0{(\\ddot{x})} d\\ddot{x} = G - \\cos{(\\ddot{x})}, then derive y^{\\prime} - \\cos{(\\ddot{x})} = G - \\cos{(\\ddot{x})}, then obtain - y^{\\prime} + \\cos{(\\ddot{x})} = - G + \\cos{(\\ddot{x})}", "derivation": "\\hat{x}_0{(\\ddot{x})} = \\sin{(\\ddot{x})} and \\int \\hat{x}_0{(\\ddot{x})} d\\ddot{x} = \\int \\sin{(\\ddot{x})} d\\ddot{x} and \\int \\hat{x}_0{(\\ddot{x})} d\\ddot{x} = G - \\cos{(\\ddot{x})} and \\int \\sin{(\\ddot{x})} d\\ddot{x} = G - \\cos{(\\ddot{x})} and y^{\\prime} - \\cos{(\\ddot{x})} = G - \\cos{(\\ddot{x})} and - y^{\\prime} + \\cos{(\\ddot{x})} = - G + \\cos{(\\ddot{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\ddot{x}', commutative=True)), sin(Symbol('\\\\ddot{x}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True))), Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('y^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))), Add(Symbol('G', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\ddot{x}', commutative=True)))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))), Add(Mul(Integer(-1), Symbol('G', commutative=True)), cos(Symbol('\\\\ddot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(k)} = \\int \\sin{(k)} dk, then derive \\operatorname{A_{y}}^{k}{(k)} = (\\delta - \\cos{(k)})^{k}, then obtain (((\\delta - \\cos{(k)})^{k})^{k})^{k} = ((\\operatorname{A_{y}}^{k}{(k)})^{k})^{k}", "derivation": "\\operatorname{A_{y}}{(k)} = \\int \\sin{(k)} dk and \\operatorname{A_{y}}^{k}{(k)} = (\\int \\sin{(k)} dk)^{k} and \\operatorname{A_{y}}^{k}{(k)} = (\\delta - \\cos{(k)})^{k} and (\\operatorname{A_{y}}^{k}{(k)})^{k} = ((\\int \\sin{(k)} dk)^{k})^{k} and ((\\delta - \\cos{(k)})^{k})^{k} = ((\\int \\sin{(k)} dk)^{k})^{k} and ((\\delta - \\cos{(k)})^{k})^{k} = (\\operatorname{A_{y}}^{k}{(k)})^{k} and (((\\delta - \\cos{(k)})^{k})^{k})^{k} = ((\\operatorname{A_{y}}^{k}{(k)})^{k})^{k}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('k', commutative=True)), Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('A_y')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Function('A_y')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Integral(sin(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Function('A_y')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["power", 6, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), cos(Symbol('k', commutative=True)))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Pow(Function('A_y')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"]]}, {"prompt": "Given \\Omega{(q)} = \\log{(e^{q})}, then derive 1 = 2 - \\frac{d}{d q} \\Omega{(q)}, then obtain \\Omega{(q)} + \\log{(e^{q})} - \\frac{d}{d q} 1 = 2 \\log{(e^{q})} - \\frac{d}{d q} 1", "derivation": "\\Omega{(q)} = \\log{(e^{q})} and \\frac{d}{d q} \\Omega{(q)} = \\frac{d}{d q} \\log{(e^{q})} and \\frac{d}{d q} \\Omega{(q)} + 1 = \\frac{d}{d q} \\log{(e^{q})} + 1 and 1 = - \\frac{d}{d q} \\Omega{(q)} + \\frac{d}{d q} \\log{(e^{q})} + 1 and \\Omega{(q)} + \\log{(e^{q})} = 2 \\log{(e^{q})} and 1 = 2 - \\frac{d}{d q} \\Omega{(q)} and \\frac{d}{d q} 1 = \\frac{d}{d q} (2 - \\frac{d}{d q} \\Omega{(q)}) and \\Omega{(q)} + \\log{(e^{q})} - \\frac{d}{d q} (2 - \\frac{d}{d q} \\Omega{(q)}) = 2 \\log{(e^{q})} - \\frac{d}{d q} (2 - \\frac{d}{d q} \\Omega{(q)}) and \\Omega{(q)} + \\log{(e^{q})} - \\frac{d}{d q} 1 = 2 \\log{(e^{q})} - \\frac{d}{d q} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('q', commutative=True)), log(exp(Symbol('q', commutative=True))))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(log(exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["add", 2, 1], "Equality(Add(Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1)), Add(Derivative(log(exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1)))"], [["minus", 3, "Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Integer(1), Add(Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), Derivative(log(exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(1)))"], [["add", 1, "log(exp(Symbol('q', commutative=True)))"], "Equality(Add(Function('\\\\Omega')(Symbol('q', commutative=True)), log(exp(Symbol('q', commutative=True)))), Mul(Integer(2), log(exp(Symbol('q', commutative=True)))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["differentiate", 6, "Symbol('q', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["minus", 5, "Derivative(Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\Omega')(Symbol('q', commutative=True)), log(exp(Symbol('q', commutative=True))), Mul(Integer(-1), Derivative(Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True), Integer(1))))), Add(Mul(Integer(2), log(exp(Symbol('q', commutative=True)))), Mul(Integer(-1), Derivative(Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\Omega')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Add(Function('\\\\Omega')(Symbol('q', commutative=True)), log(exp(Symbol('q', commutative=True))), Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))))), Add(Mul(Integer(2), log(exp(Symbol('q', commutative=True)))), Mul(Integer(-1), Derivative(Integer(1), Tuple(Symbol('q', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(m)} = \\int \\log{(m)} dm and \\sigma_{x}{(m)} = \\frac{\\operatorname{t_{1}}{(m)}}{m}, then derive (\\frac{\\operatorname{t_{1}}{(m)}}{m})^{m} = (\\frac{g + m \\log{(m)} - m}{m})^{m}, then obtain \\sigma_{x}^{m}{(m)} = (\\frac{g + m \\log{(m)} - m}{m})^{m}", "derivation": "\\operatorname{t_{1}}{(m)} = \\int \\log{(m)} dm and \\frac{\\operatorname{t_{1}}{(m)}}{m} = \\frac{\\int \\log{(m)} dm}{m} and \\sigma_{x}{(m)} = \\frac{\\operatorname{t_{1}}{(m)}}{m} and (\\frac{\\operatorname{t_{1}}{(m)}}{m})^{m} = (\\frac{\\int \\log{(m)} dm}{m})^{m} and (\\frac{\\operatorname{t_{1}}{(m)}}{m})^{m} = (\\frac{g + m \\log{(m)} - m}{m})^{m} and \\sigma_{x}^{m}{(m)} = (\\frac{g + m \\log{(m)} - m}{m})^{m}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('m', commutative=True)), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["divide", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('t_1')(Symbol('m', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('m', commutative=True)), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('t_1')(Symbol('m', commutative=True))))"], [["power", 2, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('t_1')(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Integral(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('t_1')(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('g', commutative=True), Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Function('\\\\sigma_x')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('g', commutative=True), Mul(Symbol('m', commutative=True), log(Symbol('m', commutative=True))), Mul(Integer(-1), Symbol('m', commutative=True)))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given p{(\\psi^*)} = e^{\\psi^*}, then obtain p{(\\psi^*)} + 3 e^{\\psi^*} = 2 p{(\\psi^*)} + 2 e^{\\psi^*}", "derivation": "p{(\\psi^*)} = e^{\\psi^*} and 2 p{(\\psi^*)} = p{(\\psi^*)} + e^{\\psi^*} and 3 p{(\\psi^*)} + e^{\\psi^*} = 2 p{(\\psi^*)} + 2 e^{\\psi^*} and 3 p{(\\psi^*)} + e^{\\psi^*} = p{(\\psi^*)} + 3 e^{\\psi^*} and p{(\\psi^*)} + 3 e^{\\psi^*} = 2 p{(\\psi^*)} + 2 e^{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "Function('p')(Symbol('\\\\psi^*', commutative=True))"], "Equality(Mul(Integer(2), Function('p')(Symbol('\\\\psi^*', commutative=True))), Add(Function('p')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True))))"], [["add", 2, "Add(Function('p')(Symbol('\\\\psi^*', commutative=True)), exp(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Mul(Integer(3), Function('p')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Add(Mul(Integer(2), Function('p')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(3), Function('p')(Symbol('\\\\psi^*', commutative=True))), exp(Symbol('\\\\psi^*', commutative=True))), Add(Function('p')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Function('p')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(3), exp(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(2), Function('p')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(s,B)} = e^{s^{B}}, then derive \\frac{\\partial}{\\partial B} \\operatorname{v_{x}}{(s,B)} + 1 = s^{B} e^{s^{B}} \\log{(s)} + 1, then obtain \\frac{\\partial}{\\partial B} \\operatorname{v_{x}}{(s,B)} + 1 = s^{B} \\operatorname{v_{x}}{(s,B)} \\log{(s)} + 1", "derivation": "\\operatorname{v_{x}}{(s,B)} = e^{s^{B}} and B + \\operatorname{v_{x}}{(s,B)} = B + e^{s^{B}} and \\frac{\\partial}{\\partial B} (B + \\operatorname{v_{x}}{(s,B)}) = \\frac{\\partial}{\\partial B} (B + e^{s^{B}}) and \\frac{\\partial}{\\partial B} \\operatorname{v_{x}}{(s,B)} + 1 = s^{B} e^{s^{B}} \\log{(s)} + 1 and \\frac{\\partial}{\\partial B} \\operatorname{v_{x}}{(s,B)} + 1 = s^{B} \\operatorname{v_{x}}{(s,B)} \\log{(s)} + 1", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True)), exp(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True))))"], [["add", 1, "Symbol('B', commutative=True)"], "Equality(Add(Symbol('B', commutative=True), Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Add(Symbol('B', commutative=True), exp(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True)))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Add(Symbol('B', commutative=True), Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Add(Symbol('B', commutative=True), exp(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True)))), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(1)), Add(Mul(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True)), exp(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True))), log(Symbol('s', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Derivative(Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Integer(1)), Add(Mul(Pow(Symbol('s', commutative=True), Symbol('B', commutative=True)), Function('v_x')(Symbol('s', commutative=True), Symbol('B', commutative=True)), log(Symbol('s', commutative=True))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{B}{(\\mu)} = \\mu, then derive \\frac{d}{d \\mu} (\\int \\mathbf{B}{(\\mu)} d\\mu)^{\\mu} = \\frac{\\partial}{\\partial \\mu} (\\delta + \\frac{\\mu^{2}}{2})^{\\mu}, then obtain \\frac{d}{d \\mu} (2 \\mu - 2 \\mathbf{B}{(\\mu)} + \\int \\mu d\\mu)^{\\mu} = \\frac{\\partial}{\\partial \\mu} (\\delta + \\frac{\\mu^{2}}{2})^{\\mu}", "derivation": "\\mathbf{B}{(\\mu)} = \\mu and \\int \\mathbf{B}{(\\mu)} d\\mu = \\int \\mu d\\mu and \\int \\mu d\\mu = \\mu - \\mathbf{B}{(\\mu)} + \\int \\mu d\\mu and (\\int \\mathbf{B}{(\\mu)} d\\mu)^{\\mu} = (\\int \\mu d\\mu)^{\\mu} and \\int \\mathbf{B}{(\\mu)} d\\mu = \\mu - \\mathbf{B}{(\\mu)} + \\int \\mu d\\mu and \\frac{d}{d \\mu} (\\int \\mathbf{B}{(\\mu)} d\\mu)^{\\mu} = \\frac{d}{d \\mu} (\\int \\mu d\\mu)^{\\mu} and \\frac{d}{d \\mu} (\\int \\mathbf{B}{(\\mu)} d\\mu)^{\\mu} = \\frac{\\partial}{\\partial \\mu} (\\delta + \\frac{\\mu^{2}}{2})^{\\mu} and \\int \\mathbf{B}{(\\mu)} d\\mu = 2 \\mu - 2 \\mathbf{B}{(\\mu)} + \\int \\mu d\\mu and \\frac{d}{d \\mu} (2 \\mu - 2 \\mathbf{B}{(\\mu)} + \\int \\mu d\\mu)^{\\mu} = \\frac{\\partial}{\\partial \\mu} (\\delta + \\frac{\\mu^{2}}{2})^{\\mu}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))))"], [["minus", 1, "Add(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))))"], "Equality(Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_integrals", 6], "Equality(Derivative(Pow(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2)))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 8], "Equality(Derivative(Pow(Add(Mul(Integer(2), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integer(2), Function('\\\\mathbf{B}')(Symbol('\\\\mu', commutative=True))), Integral(Symbol('\\\\mu', commutative=True), Tuple(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\delta', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mu', commutative=True), Integer(2)))), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(A,H)} = \\cos{(A H)} and L{(A,H)} = A H, then obtain \\int A H \\cos{(L{(A,H)})} dH = \\int A H \\cos{(A H)} dH", "derivation": "\\operatorname{P_{g}}{(A,H)} = \\cos{(A H)} and L{(A,H)} = A H and \\operatorname{P_{g}}{(A,H)} = \\cos{(L{(A,H)})} and A H \\operatorname{P_{g}}{(A,H)} = A H \\cos{(L{(A,H)})} and A H \\operatorname{P_{g}}{(A,H)} = A H \\cos{(A H)} and A H \\cos{(L{(A,H)})} = A H \\cos{(A H)} and \\int A H \\cos{(L{(A,H)})} dH = \\int A H \\cos{(A H)} dH", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('A', commutative=True), Symbol('H', commutative=True)), cos(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('A', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('H', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('P_g')(Symbol('A', commutative=True), Symbol('H', commutative=True)), cos(Function('L')(Symbol('A', commutative=True), Symbol('H', commutative=True))))"], [["times", 3, "Mul(Symbol('A', commutative=True), Symbol('H', commutative=True))"], "Equality(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), Function('P_g')(Symbol('A', commutative=True), Symbol('H', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Function('L')(Symbol('A', commutative=True), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), Function('P_g')(Symbol('A', commutative=True), Symbol('H', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Function('L')(Symbol('A', commutative=True), Symbol('H', commutative=True)))), Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True)))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Function('L')(Symbol('A', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))), Integral(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True), cos(Mul(Symbol('A', commutative=True), Symbol('H', commutative=True)))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given G{(f_{E},\\eta)} = e^{\\eta f_{E}} and \\mathbf{A}{(\\eta)} = \\eta, then obtain \\frac{\\eta e^{- \\eta f_{E}} \\sin{(\\mathbf{A}{(\\eta)})}}{\\eta \\sin{(\\eta)} - \\eta \\sin{(\\mathbf{A}{(\\eta)})}} = \\frac{\\eta e^{- \\eta f_{E}} \\sin{(\\eta)}}{\\eta \\sin{(\\eta)} - \\eta \\sin{(\\mathbf{A}{(\\eta)})}}", "derivation": "G{(f_{E},\\eta)} = e^{\\eta f_{E}} and \\mathbf{A}{(\\eta)} = \\eta and \\sin{(\\mathbf{A}{(\\eta)})} = \\sin{(\\eta)} and \\frac{\\eta \\sin{(\\mathbf{A}{(\\eta)})}}{G{(f_{E},\\eta)}} = \\frac{\\eta \\sin{(\\eta)}}{G{(f_{E},\\eta)}} and \\eta e^{- \\eta f_{E}} \\sin{(\\mathbf{A}{(\\eta)})} = \\eta e^{- \\eta f_{E}} \\sin{(\\eta)} and \\frac{\\eta e^{- \\eta f_{E}} \\sin{(\\mathbf{A}{(\\eta)})}}{\\eta \\sin{(\\eta)} - \\eta \\sin{(\\mathbf{A}{(\\eta)})}} = \\frac{\\eta e^{- \\eta f_{E}} \\sin{(\\eta)}}{\\eta \\sin{(\\eta)} - \\eta \\sin{(\\mathbf{A}{(\\eta)})}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('f_E', commutative=True), Symbol('\\\\eta', commutative=True)), exp(Mul(Symbol('\\\\eta', commutative=True), Symbol('f_E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["sin", 2], "Equality(sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True))), sin(Symbol('\\\\eta', commutative=True)))"], [["divide", 3, "Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Function('G')(Symbol('f_E', commutative=True), Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Pow(Function('G')(Symbol('f_E', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True)))), Mul(Symbol('\\\\eta', commutative=True), Pow(Function('G')(Symbol('f_E', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(-1)), sin(Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Symbol('\\\\eta', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Symbol('f_E', commutative=True))), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True)))), Mul(Symbol('\\\\eta', commutative=True), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Symbol('f_E', commutative=True))), sin(Symbol('\\\\eta', commutative=True))))"], [["divide", 5, "Add(Mul(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True)))))"], "Equality(Mul(Symbol('\\\\eta', commutative=True), Pow(Add(Mul(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True))))), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Symbol('f_E', commutative=True))), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True)))), Mul(Symbol('\\\\eta', commutative=True), Pow(Add(Mul(Symbol('\\\\eta', commutative=True), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), sin(Function('\\\\mathbf{A}')(Symbol('\\\\eta', commutative=True))))), Integer(-1)), exp(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Symbol('f_E', commutative=True))), sin(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given v{(n,\\Omega)} = \\frac{\\Omega}{n}, then obtain - \\frac{\\Omega}{\\frac{d}{d n} \\Omega} + n v{(n,\\Omega)} = \\Omega - \\frac{\\Omega}{\\frac{d}{d n} \\Omega}", "derivation": "v{(n,\\Omega)} = \\frac{\\Omega}{n} and n v{(n,\\Omega)} = \\Omega and \\frac{\\partial}{\\partial n} n v{(n,\\Omega)} = \\frac{d}{d n} \\Omega and \\frac{n v{(n,\\Omega)}}{\\frac{\\partial}{\\partial n} n v{(n,\\Omega)}} = \\frac{\\Omega}{\\frac{\\partial}{\\partial n} n v{(n,\\Omega)}} and n v{(n,\\Omega)} - \\frac{n v{(n,\\Omega)}}{\\frac{\\partial}{\\partial n} n v{(n,\\Omega)}} = \\Omega - \\frac{n v{(n,\\Omega)}}{\\frac{\\partial}{\\partial n} n v{(n,\\Omega)}} and n v{(n,\\Omega)} - \\frac{n v{(n,\\Omega)}}{\\frac{d}{d n} \\Omega} = \\Omega - \\frac{n v{(n,\\Omega)}}{\\frac{d}{d n} \\Omega} and \\frac{n v{(n,\\Omega)}}{\\frac{d}{d n} \\Omega} = \\frac{\\Omega}{\\frac{d}{d n} \\Omega} and - \\frac{\\Omega}{\\frac{d}{d n} \\Omega} + n v{(n,\\Omega)} = \\Omega - \\frac{\\Omega}{\\frac{d}{d n} \\Omega}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["divide", 1, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1)))"], "Equality(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))))"], [["minus", 2, "Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('\\\\Omega', commutative=True), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('n', commutative=True), Function('v')(Symbol('n', commutative=True), Symbol('\\\\Omega', commutative=True)))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Derivative(Symbol('\\\\Omega', commutative=True), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})} = A_{1} \\mathbf{J}_f and \\mathbf{J}_M{(A_{1},\\mathbf{J}_f)} = A_{1}^{2} \\mathbf{J}_f, then obtain \\int \\mathbf{J}_M^{A_{1}}{(A_{1},\\mathbf{J}_f)} d\\mathbf{J}_f = \\int (A_{1} \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})})^{A_{1}} d\\mathbf{J}_f", "derivation": "\\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})} = A_{1} \\mathbf{J}_f and A_{1} \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})} = A_{1}^{2} \\mathbf{J}_f and \\mathbf{J}_M{(A_{1},\\mathbf{J}_f)} = A_{1}^{2} \\mathbf{J}_f and \\mathbf{J}_M{(A_{1},\\mathbf{J}_f)} = A_{1} \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})} and \\mathbf{J}_M^{A_{1}}{(A_{1},\\mathbf{J}_f)} = (A_{1} \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})})^{A_{1}} and \\int \\mathbf{J}_M^{A_{1}}{(A_{1},\\mathbf{J}_f)} d\\mathbf{J}_f = \\int (A_{1} \\operatorname{C_{d}}{(\\mathbf{J}_f,A_{1})})^{A_{1}} d\\mathbf{J}_f", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(2)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\mathbf{J}_M')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('A_1', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))))"], [["power", 4, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}_M')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('A_1', commutative=True)), Pow(Mul(Symbol('A_1', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)))"], [["integrate", 5, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{J}_M')(Symbol('A_1', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Pow(Mul(Symbol('A_1', commutative=True), Function('C_d')(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\hat{H},n)} = - \\hat{H} + n, then obtain \\hat{H} \\operatorname{A_{2}}{(\\hat{H},n)} + \\cos{(\\hat{H} - n)} = \\hat{H} (- \\hat{H} + n) + \\cos{(\\hat{H} - n)}", "derivation": "\\operatorname{A_{2}}{(\\hat{H},n)} = - \\hat{H} + n and \\cos{(\\operatorname{A_{2}}{(\\hat{H},n)})} = \\cos{(\\hat{H} - n)} and \\hat{H} \\operatorname{A_{2}}{(\\hat{H},n)} = \\hat{H} (- \\hat{H} + n) and \\hat{H} \\operatorname{A_{2}}{(\\hat{H},n)} + \\cos{(\\operatorname{A_{2}}{(\\hat{H},n)})} = \\hat{H} (- \\hat{H} + n) + \\cos{(\\operatorname{A_{2}}{(\\hat{H},n)})} and \\hat{H} \\operatorname{A_{2}}{(\\hat{H},n)} + \\cos{(\\hat{H} - n)} = \\hat{H} (- \\hat{H} + n) + \\cos{(\\hat{H} - n)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)))"], [["cos", 1], "Equality(cos(Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), cos(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True)))))"], [["times", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{H}', commutative=True), Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True))))"], [["add", 3, "cos(Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), cos(Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True))), cos(Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Symbol('\\\\hat{H}', commutative=True), Function('A_2')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True))), cos(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))), Add(Mul(Symbol('\\\\hat{H}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True))), cos(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}{(P_{e})} = e^{e^{P_{e}}} and \\rho_{b}{(m,\\theta_1)} = \\cos{(\\theta_1 m)}, then obtain \\frac{\\partial}{\\partial m} (P_{e} - \\hat{H}{(P_{e})} + \\rho_{b}{(m,\\theta_1)}) = \\frac{\\partial}{\\partial m} (P_{e} - \\hat{H}{(P_{e})} + \\cos{(\\theta_1 m)})", "derivation": "\\hat{H}{(P_{e})} = e^{e^{P_{e}}} and \\rho_{b}{(m,\\theta_1)} = \\cos{(\\theta_1 m)} and P_{e} + \\rho_{b}{(m,\\theta_1)} - e^{e^{P_{e}}} = P_{e} - e^{e^{P_{e}}} + \\cos{(\\theta_1 m)} and \\frac{\\partial}{\\partial m} (P_{e} + \\rho_{b}{(m,\\theta_1)} - e^{e^{P_{e}}}) = \\frac{\\partial}{\\partial m} (P_{e} - e^{e^{P_{e}}} + \\cos{(\\theta_1 m)}) and \\frac{\\partial}{\\partial m} (P_{e} - \\hat{H}{(P_{e})} + \\rho_{b}{(m,\\theta_1)}) = \\frac{\\partial}{\\partial m} (P_{e} - \\hat{H}{(P_{e})} + \\cos{(\\theta_1 m)})", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('P_e', commutative=True)), exp(exp(Symbol('P_e', commutative=True))))"], ["get_premise", "Equality(Function('\\\\rho_b')(Symbol('m', commutative=True), Symbol('\\\\theta_1', commutative=True)), cos(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('m', commutative=True))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('P_e', commutative=True)), exp(exp(Symbol('P_e', commutative=True))))"], "Equality(Add(Symbol('P_e', commutative=True), Function('\\\\rho_b')(Symbol('m', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('P_e', commutative=True))))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), exp(exp(Symbol('P_e', commutative=True)))), cos(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('m', commutative=True)))))"], [["differentiate", 3, "Symbol('m', commutative=True)"], "Equality(Derivative(Add(Symbol('P_e', commutative=True), Function('\\\\rho_b')(Symbol('m', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('P_e', commutative=True))))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), exp(exp(Symbol('P_e', commutative=True)))), cos(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('P_e', commutative=True))), Function('\\\\rho_b')(Symbol('m', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('P_e', commutative=True))), cos(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('m', commutative=True)))), Tuple(Symbol('m', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu_{0}{(g_{\\varepsilon},E_{n})} = g_{\\varepsilon}^{E_{n}} and s{(g_{\\varepsilon},E_{n})} = - g_{\\varepsilon}^{E_{n}}, then obtain e^{s{(g_{\\varepsilon},E_{n})} - 1} = e^{- \\mu_{0}{(g_{\\varepsilon},E_{n})} - 1}", "derivation": "\\mu_{0}{(g_{\\varepsilon},E_{n})} = g_{\\varepsilon}^{E_{n}} and s{(g_{\\varepsilon},E_{n})} = - g_{\\varepsilon}^{E_{n}} and s{(g_{\\varepsilon},E_{n})} - 1 = - g_{\\varepsilon}^{E_{n}} - 1 and e^{s{(g_{\\varepsilon},E_{n})} - 1} = e^{- g_{\\varepsilon}^{E_{n}} - 1} and e^{s{(g_{\\varepsilon},E_{n})} - 1} = e^{- \\mu_{0}{(g_{\\varepsilon},E_{n})} - 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)))"], ["renaming_premise", "Equality(Function('s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Function('s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True))), Integer(-1)))"], [["exp", 3], "Equality(exp(Add(Function('s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Integer(-1))), exp(Add(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(exp(Add(Function('s')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True)), Integer(-1))), exp(Add(Mul(Integer(-1), Function('\\\\mu_0')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('E_n', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given J{(k,y^{\\prime})} = \\frac{\\partial}{\\partial k} (y^{\\prime})^{k}, then derive J{(k,y^{\\prime})} - 1 = (y^{\\prime})^{k} \\log{(y^{\\prime})} - 1, then obtain \\frac{\\partial}{\\partial k} (y^{\\prime})^{k} - 1 = (y^{\\prime})^{k} \\log{(y^{\\prime})} - 1", "derivation": "J{(k,y^{\\prime})} = \\frac{\\partial}{\\partial k} (y^{\\prime})^{k} and J{(k,y^{\\prime})} - 1 = \\frac{\\partial}{\\partial k} (y^{\\prime})^{k} - 1 and J{(k,y^{\\prime})} - 1 = (y^{\\prime})^{k} \\log{(y^{\\prime})} - 1 and \\frac{\\partial}{\\partial k} (y^{\\prime})^{k} - 1 = (y^{\\prime})^{k} \\log{(y^{\\prime})} - 1", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('J')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Add(Function('J')(Symbol('k', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Add(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('k', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Derivative(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('k', commutative=True)), log(Symbol('y^{\\\\prime}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(\\mu_0,\\sigma_p,\\theta_2)} = - \\sigma_p + \\frac{\\theta_2}{\\mu_0} and \\hat{H}{(\\mu_0,\\sigma_p,\\theta_2)} = - \\sigma_p + \\frac{\\theta_2}{\\mu_0}, then obtain \\hat{H}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)} = \\operatorname{m_{s}}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)}", "derivation": "\\operatorname{m_{s}}{(\\mu_0,\\sigma_p,\\theta_2)} = - \\sigma_p + \\frac{\\theta_2}{\\mu_0} and \\hat{H}{(\\mu_0,\\sigma_p,\\theta_2)} = - \\sigma_p + \\frac{\\theta_2}{\\mu_0} and \\operatorname{m_{s}}{(\\mu_0,\\sigma_p,\\theta_2)} = \\hat{H}{(\\mu_0,\\sigma_p,\\theta_2)} and \\hat{H}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)} = (- \\sigma_p + \\frac{\\theta_2}{\\mu_0})^{\\mu_0} and \\operatorname{m_{s}}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)} = (- \\sigma_p + \\frac{\\theta_2}{\\mu_0})^{\\mu_0} and \\hat{H}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)} = \\operatorname{m_{s}}^{\\mu_0}{(\\mu_0,\\sigma_p,\\theta_2)}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Function('\\\\hat{H}')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})} = A_{2} - \\mathbf{D} + \\mathbf{M}, then obtain - \\frac{- \\mathbf{M} + \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}}{\\mathbf{D} \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}} = - \\frac{A_{2} - \\mathbf{D}}{\\mathbf{D} \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}}", "derivation": "\\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})} = A_{2} - \\mathbf{D} + \\mathbf{M} and - \\mathbf{M} + \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})} = A_{2} - \\mathbf{D} and \\frac{- \\mathbf{M} + \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}}{\\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}} = \\frac{A_{2} - \\mathbf{D}}{\\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}} and - \\frac{- \\mathbf{M} + \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}}{\\mathbf{D} \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}} = - \\frac{A_{2} - \\mathbf{D}}{\\mathbf{D} \\phi_{2}{(A_{2},\\mathbf{D},\\mathbf{M})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))))"], [["divide", 2, "Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Mul(Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"], [["divide", 3, "Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True))), Pow(Function('\\\\phi_2')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{M},\\varphi)} = \\mathbf{M} + \\varphi, then derive \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\mathbf{M} \\varphi + \\varepsilon + \\frac{\\varphi^{2}}{2}, then derive \\frac{\\partial}{\\partial \\varphi} \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\mathbf{M} + \\varphi, then obtain \\varphi \\mathbf{E}{(\\mathbf{M},\\varphi)} = \\varphi \\frac{\\partial}{\\partial \\varphi} \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi", "derivation": "\\mathbf{E}{(\\mathbf{M},\\varphi)} = \\mathbf{M} + \\varphi and \\varphi \\mathbf{E}{(\\mathbf{M},\\varphi)} = \\varphi (\\mathbf{M} + \\varphi) and \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\int (\\mathbf{M} + \\varphi) d\\varphi and \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\mathbf{M} \\varphi + \\varepsilon + \\frac{\\varphi^{2}}{2} and \\frac{\\partial}{\\partial \\varphi} \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\frac{\\partial}{\\partial \\varphi} (\\mathbf{M} \\varphi + \\varepsilon + \\frac{\\varphi^{2}}{2}) and \\frac{\\partial}{\\partial \\varphi} \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi = \\mathbf{M} + \\varphi and \\varphi \\mathbf{E}{(\\mathbf{M},\\varphi)} = \\varphi \\frac{\\partial}{\\partial \\varphi} \\int \\mathbf{E}{(\\mathbf{M},\\varphi)} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2)))))"], [["differentiate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\varphi', commutative=True), Integer(2)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Add(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), Derivative(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{p}{(Q)} = e^{Q}, then obtain (\\hat{p}{(Q)} + \\frac{2 \\hat{p}{(Q)}}{\\hat{p}{(Q)} + e^{Q}})^{2} = (\\hat{p}{(Q)} + 1)^{2}", "derivation": "\\hat{p}{(Q)} = e^{Q} and 2 \\hat{p}{(Q)} = \\hat{p}{(Q)} + e^{Q} and \\frac{2 \\hat{p}{(Q)}}{\\hat{p}{(Q)} + e^{Q}} = 1 and \\hat{p}{(Q)} + \\frac{2 \\hat{p}{(Q)}}{\\hat{p}{(Q)} + e^{Q}} = \\hat{p}{(Q)} + 1 and (\\hat{p}{(Q)} + \\frac{2 \\hat{p}{(Q)}}{\\hat{p}{(Q)} + e^{Q}})^{2} = (\\hat{p}{(Q)} + 1)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], [["add", 1, "Function('\\\\hat{p}')(Symbol('Q', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}')(Symbol('Q', commutative=True))), Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))))"], [["divide", 2, "Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Integer(-1)), Function('\\\\hat{p}')(Symbol('Q', commutative=True))), Integer(1))"], [["add", 3, "Function('\\\\hat{p}')(Symbol('Q', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), Mul(Integer(2), Pow(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Integer(-1)), Function('\\\\hat{p}')(Symbol('Q', commutative=True)))), Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), Integer(1)))"], [["power", 4, 2], "Equality(Pow(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), Mul(Integer(2), Pow(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), exp(Symbol('Q', commutative=True))), Integer(-1)), Function('\\\\hat{p}')(Symbol('Q', commutative=True)))), Integer(2)), Pow(Add(Function('\\\\hat{p}')(Symbol('Q', commutative=True)), Integer(1)), Integer(2)))"]]}, {"prompt": "Given x{(t,x^\\prime)} = e^{- t + x^\\prime}, then obtain \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} x^{x^\\prime}{(t,x^\\prime)} = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (e^{- t + x^\\prime})^{x^\\prime}", "derivation": "x{(t,x^\\prime)} = e^{- t + x^\\prime} and x^{x^\\prime}{(t,x^\\prime)} = (e^{- t + x^\\prime})^{x^\\prime} and \\frac{\\partial}{\\partial x^\\prime} x^{x^\\prime}{(t,x^\\prime)} = \\frac{\\partial}{\\partial x^\\prime} (e^{- t + x^\\prime})^{x^\\prime} and \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} x^{x^\\prime}{(t,x^\\prime)} = \\frac{\\partial^{2}}{\\partial (x^\\prime)^{2}} (e^{- t + x^\\prime})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('x')(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(exp(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Function('x')(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(exp(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Function('x')(Symbol('t', commutative=True), Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Pow(exp(Add(Mul(Integer(-1), Symbol('t', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(x^\\prime)} = \\cos{(x^\\prime)}, then obtain (- \\operatorname{C_{d}}{(x^\\prime)} + \\cos{(x^\\prime)})^{x^\\prime} - \\cos{(x^\\prime)} = 1 - \\cos{(x^\\prime)}", "derivation": "\\operatorname{C_{d}}{(x^\\prime)} = \\cos{(x^\\prime)} and 0 = - \\operatorname{C_{d}}{(x^\\prime)} + \\cos{(x^\\prime)} and 0^{x^\\prime} = (- \\operatorname{C_{d}}{(x^\\prime)} + \\cos{(x^\\prime)})^{x^\\prime} and 0^{x^\\prime} - \\cos{(x^\\prime)} = (- \\operatorname{C_{d}}{(x^\\prime)} + \\cos{(x^\\prime)})^{x^\\prime} - \\cos{(x^\\prime)} and (- \\operatorname{C_{d}}{(x^\\prime)} + \\cos{(x^\\prime)})^{x^\\prime} - \\cos{(x^\\prime)} = 1 - \\cos{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('x^\\\\prime', commutative=True)), cos(Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "Function('C_d')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_d')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))))"], [["power", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)))"], [["minus", 3, "cos(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Add(Mul(Integer(-1), Function('C_d')(Symbol('x^\\\\prime', commutative=True))), cos(Symbol('x^\\\\prime', commutative=True))), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))), Add(Integer(1), Mul(Integer(-1), cos(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\mu{(\\phi)} = \\log{(\\phi)}, then obtain \\int \\phi \\mu{(\\phi)} d\\phi = \\frac{\\phi^{2} \\log{(\\phi)}}{2} - \\frac{\\phi^{2}}{4} + c", "derivation": "\\mu{(\\phi)} = \\log{(\\phi)} and \\phi \\mu{(\\phi)} = \\phi \\log{(\\phi)} and \\int \\phi \\mu{(\\phi)} d\\phi = \\int \\phi \\log{(\\phi)} d\\phi and \\int \\phi \\mu{(\\phi)} d\\phi = \\frac{\\phi^{2} \\log{(\\phi)}}{2} - \\frac{\\phi^{2}}{4} + c", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\mu')(Symbol('\\\\phi', commutative=True))), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))))"], [["integrate", 2, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\mu')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('\\\\phi', commutative=True), Function('\\\\mu')(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\phi', commutative=True), Integer(2)), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('\\\\phi', commutative=True), Integer(2))), Symbol('c', commutative=True)))"]]}, {"prompt": "Given h{(a)} = \\cos{(a)}, then obtain a h{(a)} \\cos^{2}{(a)} + a \\cos{(a)} = a \\cos^{3}{(a)} + a \\cos{(a)}", "derivation": "h{(a)} = \\cos{(a)} and a h{(a)} = a \\cos{(a)} and a h^{3}{(a)} = a h^{2}{(a)} \\cos{(a)} and a h^{2}{(a)} \\cos{(a)} = a h{(a)} \\cos^{2}{(a)} and a h^{2}{(a)} \\cos{(a)} + a \\cos{(a)} = a h{(a)} \\cos^{2}{(a)} + a \\cos{(a)} and a h{(a)} \\cos^{2}{(a)} + a \\cos{(a)} = a \\cos^{3}{(a)} + a \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('h')(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True))))"], [["times", 2, "Pow(Function('h')(Symbol('a', commutative=True)), Integer(2))"], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('h')(Symbol('a', commutative=True)), Integer(3))), Mul(Symbol('a', commutative=True), Pow(Function('h')(Symbol('a', commutative=True)), Integer(2)), cos(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('a', commutative=True), Pow(Function('h')(Symbol('a', commutative=True)), Integer(2)), cos(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), Function('h')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(2))))"], [["add", 4, "Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True)))"], "Equality(Add(Mul(Symbol('a', commutative=True), Pow(Function('h')(Symbol('a', commutative=True)), Integer(2)), cos(Symbol('a', commutative=True))), Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True)))), Add(Mul(Symbol('a', commutative=True), Function('h')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(2))), Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Symbol('a', commutative=True), Function('h')(Symbol('a', commutative=True)), Pow(cos(Symbol('a', commutative=True)), Integer(2))), Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True)))), Add(Mul(Symbol('a', commutative=True), Pow(cos(Symbol('a', commutative=True)), Integer(3))), Mul(Symbol('a', commutative=True), cos(Symbol('a', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)} = (Z + \\mathbf{B})^{W}, then obtain (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)})^{W} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} (Z + \\mathbf{B})^{W})^{W}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)} = (Z + \\mathbf{B})^{W} and \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)} = \\mathbf{B} (Z + \\mathbf{B})^{W} and \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} (Z + \\mathbf{B})^{W} and (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\mathbf{B},W,Z)})^{W} = (\\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} (Z + \\mathbf{B})^{W})^{W}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('Z', commutative=True)), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('W', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('Z', commutative=True))), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('W', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["power", 3, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('W', commutative=True), Symbol('Z', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('Z', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Symbol('W', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(f,A_{1})} = \\frac{f}{A_{1}}, then obtain A_{1} \\phi_{1}^{2}{(f,A_{1})} - \\phi_{1}{(f,A_{1})} = f \\phi_{1}{(f,A_{1})} - \\phi_{1}{(f,A_{1})}", "derivation": "\\phi_{1}{(f,A_{1})} = \\frac{f}{A_{1}} and \\phi_{1}^{2}{(f,A_{1})} = \\frac{f \\phi_{1}{(f,A_{1})}}{A_{1}} and A_{1} \\phi_{1}^{2}{(f,A_{1})} = f \\phi_{1}{(f,A_{1})} and A_{1} \\phi_{1}^{2}{(f,A_{1})} - \\frac{f}{A_{1}} = f \\phi_{1}{(f,A_{1})} - \\frac{f}{A_{1}} and A_{1} \\phi_{1}^{2}{(f,A_{1})} - \\phi_{1}{(f,A_{1})} = f \\phi_{1}{(f,A_{1})} - \\phi_{1}{(f,A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f', commutative=True)))"], [["times", 1, "Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Pow(Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Integer(2)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f', commutative=True), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True))))"], [["divide", 2, "Pow(Symbol('A_1', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Symbol('f', commutative=True), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True))))"], [["minus", 3, "Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f', commutative=True))"], "Equality(Add(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f', commutative=True))), Add(Mul(Symbol('f', commutative=True), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Symbol('A_1', commutative=True), Pow(Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Symbol('f', commutative=True), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True))), Mul(Integer(-1), Function('\\\\phi_1')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{P}{(a^{\\dagger},v_{y})} = \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y}, then derive v_{y} \\mathbf{P}{(a^{\\dagger},v_{y})} = a^{\\dagger} v_{y}, then obtain v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} = v_{y} \\frac{\\partial}{\\partial v_{y}} v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y}", "derivation": "\\mathbf{P}{(a^{\\dagger},v_{y})} = \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} and v_{y} \\mathbf{P}{(a^{\\dagger},v_{y})} = v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} and v_{y} \\mathbf{P}{(a^{\\dagger},v_{y})} = a^{\\dagger} v_{y} and a^{\\dagger} v_{y} = v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} and v_{y} \\mathbf{P}{(a^{\\dagger},v_{y})} = v_{y} \\frac{\\partial}{\\partial v_{y}} v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} and v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y} = v_{y} \\frac{\\partial}{\\partial v_{y}} v_{y} \\frac{\\partial}{\\partial v_{y}} a^{\\dagger} v_{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["times", 1, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Symbol('v_y', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Mul(Symbol('v_y', commutative=True), Function('\\\\mathbf{P}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True))), Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('v_y', commutative=True), Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))), Tuple(Symbol('v_y', commutative=True), Integer(1)))))"]]}, {"prompt": "Given i{(u,\\mathbf{B})} = \\frac{\\mathbf{B}}{u}, then obtain 1 - \\frac{1}{\\mathbf{B} + u} = \\frac{\\mathbf{B}}{\\mathbf{B} + u} + \\frac{1}{i{(u,\\mathbf{B})} + 1} - \\frac{1}{\\mathbf{B} + u}", "derivation": "i{(u,\\mathbf{B})} = \\frac{\\mathbf{B}}{u} and i{(u,\\mathbf{B})} + 1 = \\frac{\\mathbf{B}}{u} + 1 and \\frac{i{(u,\\mathbf{B})} + 1}{u} = \\frac{\\frac{\\mathbf{B}}{u} + 1}{u} and 1 = \\frac{\\frac{\\mathbf{B}}{u} + 1}{i{(u,\\mathbf{B})} + 1} and 1 = \\frac{\\mathbf{B}}{u i{(u,\\mathbf{B})} + u} + \\frac{1}{i{(u,\\mathbf{B})} + 1} and 1 = \\frac{\\mathbf{B}}{\\mathbf{B} + u} + \\frac{1}{\\frac{\\mathbf{B}}{u} + 1} and 1 = \\frac{\\mathbf{B}}{\\mathbf{B} + u} + \\frac{1}{i{(u,\\mathbf{B})} + 1} and 1 - \\frac{1}{\\mathbf{B} + u} = \\frac{\\mathbf{B}}{\\mathbf{B} + u} + \\frac{1}{i{(u,\\mathbf{B})} + 1} - \\frac{1}{\\mathbf{B} + u}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Integer(1)))"], [["times", 2, "Pow(Symbol('u', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1))), Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Integer(1))))"], [["divide", 3, "Mul(Pow(Symbol('u', commutative=True), Integer(-1)), Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)))"], "Equality(Integer(1), Mul(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Integer(1)), Pow(Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)), Integer(-1))))"], [["expand", 4], "Equality(Integer(1), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Mul(Symbol('u', commutative=True), Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True))), Symbol('u', commutative=True)), Integer(-1))), Pow(Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integer(1), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Pow(Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Integer(1)), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Integer(1), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Pow(Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)), Integer(-1))))"], [["minus", 7, "Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1))), Pow(Add(Function('i')(Symbol('u', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1)), Integer(-1)), Mul(Integer(-1), Pow(Add(Symbol('\\\\mathbf{B}', commutative=True), Symbol('u', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(P_{e})} = \\log{(e^{P_{e}})} and G{(\\pi,v_{t})} = \\pi + v_{t}, then obtain (G{(\\pi,v_{t})} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}}) (\\pi + v_{t} + e^{P_{e}}) = (\\pi + v_{t} + e^{P_{e}}) (\\pi + v_{t} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}})", "derivation": "\\operatorname{E_{\\lambda}}{(P_{e})} = \\log{(e^{P_{e}})} and G{(\\pi,v_{t})} = \\pi + v_{t} and e^{\\operatorname{E_{\\lambda}}{(P_{e})}} = e^{P_{e}} and G{(\\pi,v_{t})} + e^{P_{e}} = \\pi + v_{t} + e^{P_{e}} and G{(\\pi,v_{t})} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}} = \\pi + v_{t} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}} and (G{(\\pi,v_{t})} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}}) (\\pi + v_{t} + e^{P_{e}}) = (\\pi + v_{t} + e^{P_{e}}) (\\pi + v_{t} + e^{\\operatorname{E_{\\lambda}}{(P_{e})}})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True)), log(exp(Symbol('P_e', commutative=True))))"], ["get_premise", "Equality(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True)))"], [["exp", 1], "Equality(exp(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True))), exp(Symbol('P_e', commutative=True)))"], [["add", 2, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True)), exp(Symbol('P_e', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Symbol('P_e', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True)), exp(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True)))))"], [["times", 5, "Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Symbol('P_e', commutative=True)))"], "Equality(Mul(Add(Function('G')(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True)), exp(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True)))), Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Symbol('P_e', commutative=True)))), Mul(Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Symbol('P_e', commutative=True))), Add(Symbol('\\\\pi', commutative=True), Symbol('v_t', commutative=True), exp(Function('E_{\\\\lambda}')(Symbol('P_e', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(\\mu,B)} = \\mu^{B} and \\operatorname{g_{\\varepsilon}}{(\\mu,B)} = \\mu^{B}, then obtain \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\hat{H}_{\\lambda}{(\\mu,B)} = \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\operatorname{g_{\\varepsilon}}{(\\mu,B)}", "derivation": "\\hat{H}_{\\lambda}{(\\mu,B)} = \\mu^{B} and \\frac{\\partial}{\\partial \\mu} \\hat{H}_{\\lambda}{(\\mu,B)} = \\frac{\\partial}{\\partial \\mu} \\mu^{B} and \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\hat{H}_{\\lambda}{(\\mu,B)} = \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\mu^{B} and \\operatorname{g_{\\varepsilon}}{(\\mu,B)} = \\mu^{B} and \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\hat{H}_{\\lambda}{(\\mu,B)} = \\frac{\\partial^{2}}{\\partial B\\partial \\mu} \\operatorname{g_{\\varepsilon}}{(\\mu,B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('B', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Pow(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Function('g_{\\\\varepsilon}')(Symbol('\\\\mu', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(\\mathbf{B},y)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} y, then derive \\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)} = 1, then obtain 2 y \\cos{((\\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)})^{\\mathbf{B}})} = 2 y \\cos{(1)}", "derivation": "\\dot{y}{(\\mathbf{B},y)} = \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} y and \\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)} = \\frac{\\partial^{2}}{\\partial y\\partial \\mathbf{B}} \\mathbf{B} y and \\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)} = 1 and (\\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)})^{\\mathbf{B}} = 1 and \\cos{((\\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)})^{\\mathbf{B}})} = \\cos{(1)} and (y + \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} y) \\cos{((\\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)})^{\\mathbf{B}})} = (y + \\frac{\\partial}{\\partial \\mathbf{B}} \\mathbf{B} y) \\cos{(1)} and 2 y \\cos{((\\frac{\\partial}{\\partial y} \\dot{y}{(\\mathbf{B},y)})^{\\mathbf{B}})} = 2 y \\cos{(1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)), Integer(1))"], [["cos", 4], "Equality(cos(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True))), cos(Integer(1)))"], [["times", 5, "Add(Symbol('y', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], "Equality(Mul(Add(Symbol('y', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), cos(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Add(Symbol('y', commutative=True), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), cos(Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Integer(2), Symbol('y', commutative=True), cos(Pow(Derivative(Function('\\\\dot{y}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\mathbf{B}', commutative=True)))), Mul(Integer(2), Symbol('y', commutative=True), cos(Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})} and B{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})}, then derive 0 = - \\operatorname{z^{*}}{(\\mathbf{B})} - \\sin{(\\mathbf{B})}, then obtain 0 = - B{(\\mathbf{B})} - \\sin{(\\mathbf{B})}", "derivation": "\\operatorname{z^{*}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})} and B{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})} and 0 = - \\operatorname{z^{*}}{(\\mathbf{B})} + \\frac{d}{d \\mathbf{B}} \\cos{(\\mathbf{B})} and 0 = - \\operatorname{z^{*}}{(\\mathbf{B})} - \\sin{(\\mathbf{B})} and \\operatorname{z^{*}}{(\\mathbf{B})} = B{(\\mathbf{B})} and 0 = - B{(\\mathbf{B})} - \\sin{(\\mathbf{B})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('B')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["minus", 1, "Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('z^*')(Symbol('\\\\mathbf{B}', commutative=True)), Function('B')(Symbol('\\\\mathbf{B}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Integer(0), Add(Mul(Integer(-1), Function('B')(Symbol('\\\\mathbf{B}', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mu,\\varphi)} = \\mu + \\varphi, then derive \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = \\mu + \\sigma_x, then obtain (\\mu + \\varphi) \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = (\\mu + \\sigma_x) (\\mu + \\varphi)", "derivation": "\\operatorname{F_{N}}{(\\mu,\\varphi)} = \\mu + \\varphi and \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} = 1 and \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = \\int 1 d\\mu and \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = \\mu + \\sigma_x and (\\mu + \\varphi) \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = (\\mu + \\varphi) \\int 1 d\\mu and \\int 1 d\\mu = \\mu + \\sigma_x and (\\mu + \\varphi) \\int \\frac{\\operatorname{F_{N}}{(\\mu,\\varphi)}}{\\mu + \\varphi} d\\mu = (\\mu + \\sigma_x) (\\mu + \\varphi)", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["times", 3, "Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\mu', commutative=True))), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integral(Mul(Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Function('F_N')(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True)))), Mul(Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Add(Symbol('\\\\mu', commutative=True), Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\eta)} = \\log{(\\eta)} and \\hat{X}{(\\eta)} = \\eta, then obtain F_{H} - \\eta \\operatorname{f_{E}}{(\\eta)} + \\eta \\log{(\\eta)} - \\eta + \\hat{X}{(\\eta)} = F_{H} - \\eta \\operatorname{f_{E}}{(\\eta)} + \\eta \\log{(\\eta)}", "derivation": "\\operatorname{f_{E}}{(\\eta)} = \\log{(\\eta)} and \\hat{X}{(\\eta)} = \\eta and \\hat{X}{(\\eta)} + \\int \\operatorname{f_{E}}{(\\eta)} d\\eta = \\eta + \\int \\operatorname{f_{E}}{(\\eta)} d\\eta and \\hat{X}{(\\eta)} + \\int \\log{(\\eta)} d\\eta = \\eta + \\int \\log{(\\eta)} d\\eta and - \\eta \\operatorname{f_{E}}{(\\eta)} + \\hat{X}{(\\eta)} + \\int \\log{(\\eta)} d\\eta = - \\eta \\operatorname{f_{E}}{(\\eta)} + \\eta + \\int \\log{(\\eta)} d\\eta and F_{H} - \\eta \\operatorname{f_{E}}{(\\eta)} + \\eta \\log{(\\eta)} - \\eta + \\hat{X}{(\\eta)} = F_{H} - \\eta \\operatorname{f_{E}}{(\\eta)} + \\eta \\log{(\\eta)}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\eta', commutative=True)), log(Symbol('\\\\eta', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["add", 2, "Integral(Function('f_E')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True)), Integral(Function('f_E')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), Integral(Function('f_E')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True)), Integral(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Symbol('\\\\eta', commutative=True), Integral(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["minus", 4, "Mul(Symbol('\\\\eta', commutative=True), Function('f_E')(Symbol('\\\\eta', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Function('f_E')(Symbol('\\\\eta', commutative=True))), Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True)), Integral(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Function('f_E')(Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True), Integral(log(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Function('f_E')(Symbol('\\\\eta', commutative=True))), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\eta', commutative=True))), Add(Symbol('F_H', commutative=True), Mul(Integer(-1), Symbol('\\\\eta', commutative=True), Function('f_E')(Symbol('\\\\eta', commutative=True))), Mul(Symbol('\\\\eta', commutative=True), log(Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(h)} = \\log{(h)}, then derive h \\frac{d}{d h} \\operatorname{A_{z}}{(h)} = 1, then obtain h \\frac{d}{d h} \\log{(h)} = 1", "derivation": "\\operatorname{A_{z}}{(h)} = \\log{(h)} and \\frac{d}{d h} \\operatorname{A_{z}}{(h)} = \\frac{d}{d h} \\log{(h)} and h \\frac{d}{d h} \\operatorname{A_{z}}{(h)} = h \\frac{d}{d h} \\log{(h)} and h \\frac{d}{d h} \\operatorname{A_{z}}{(h)} = 1 and h \\frac{d}{d h} \\log{(h)} = 1", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('h', commutative=True)), log(Symbol('h', commutative=True)))"], [["differentiate", 1, "Symbol('h', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 2, "Symbol('h', commutative=True)"], "Equality(Mul(Symbol('h', commutative=True), Derivative(Function('A_z')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Symbol('h', commutative=True), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('h', commutative=True), Derivative(Function('A_z')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('h', commutative=True), Derivative(log(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\hat{H}_l,Q)} = - Q + \\hat{H}_l, then obtain \\frac{(- \\frac{- Q + \\hat{H}_l}{Q})^{Q}}{\\hat{H}_l} = \\frac{(\\frac{Q - \\hat{H}_l}{Q})^{Q}}{\\hat{H}_l}", "derivation": "\\operatorname{f_{E}}{(\\hat{H}_l,Q)} = - Q + \\hat{H}_l and - \\frac{\\operatorname{f_{E}}{(\\hat{H}_l,Q)}}{Q} = - \\frac{- Q + \\hat{H}_l}{Q} and (- \\frac{\\operatorname{f_{E}}{(\\hat{H}_l,Q)}}{Q})^{Q} = (- \\frac{- Q + \\hat{H}_l}{Q})^{Q} and (- \\frac{- Q + \\hat{H}_l}{Q})^{Q} = (\\frac{Q - \\hat{H}_l}{Q})^{Q} and (- \\frac{\\operatorname{f_{E}}{(\\hat{H}_l,Q)}}{Q})^{Q} = (\\frac{Q - \\hat{H}_l}{Q})^{Q} and \\frac{(- \\frac{\\operatorname{f_{E}}{(\\hat{H}_l,Q)}}{Q})^{Q}}{\\hat{H}_l} = \\frac{(\\frac{Q - \\hat{H}_l}{Q})^{Q}}{\\hat{H}_l} and \\frac{(- \\frac{- Q + \\hat{H}_l}{Q})^{Q}}{\\hat{H}_l} = \\frac{(\\frac{Q - \\hat{H}_l}{Q})^{Q}}{\\hat{H}_l}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True)))"], [["divide", 1, "Mul(Integer(-1), Symbol('Q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))))"], [["power", 2, "Symbol('Q', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('Q', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('Q', commutative=True)))"], [["divide", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('Q', commutative=True))), Symbol('Q', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\hat{H}_l', commutative=True))), Symbol('Q', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Pow(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Add(Symbol('Q', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)))), Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = \\theta_2 - \\tilde{g}^*, then derive \\frac{\\partial}{\\partial \\theta_2} \\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = 1, then obtain \\frac{\\partial^{2}}{\\partial \\tilde{g}^*\\partial \\theta_2} (\\theta_2 - \\tilde{g}^*) = \\frac{d}{d \\tilde{g}^*} 1", "derivation": "\\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = \\theta_2 - \\tilde{g}^* and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = \\frac{\\partial}{\\partial \\theta_2} (\\theta_2 - \\tilde{g}^*) and \\frac{\\partial}{\\partial \\theta_2} \\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = 1 and \\frac{\\partial^{2}}{\\partial \\tilde{g}^*\\partial \\theta_2} \\operatorname{n_{2}}{(\\tilde{g}^*,\\theta_2)} = \\frac{d}{d \\tilde{g}^*} 1 and \\frac{\\partial^{2}}{\\partial \\tilde{g}^*\\partial \\theta_2} (\\theta_2 - \\tilde{g}^*) = \\frac{d}{d \\tilde{g}^*} 1", "srepr_derivation": [["get_premise", "Equality(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1))), Integer(1))"], [["differentiate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('n_2')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Symbol('\\\\theta_2', commutative=True), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True), Integer(1)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\mu,\\delta)} = \\frac{\\mu}{\\delta} and \\tilde{g}^*{(v_{t})} = \\log{(\\sin{(v_{t})})}, then obtain \\tilde{g}^*^{v_{t}}{(v_{t})} + \\operatorname{z^{*}}{(\\mu,\\delta)} = \\operatorname{z^{*}}{(\\mu,\\delta)} + \\log{(\\sin{(v_{t})})}^{v_{t}}", "derivation": "\\operatorname{z^{*}}{(\\mu,\\delta)} = \\frac{\\mu}{\\delta} and \\delta \\operatorname{z^{*}}{(\\mu,\\delta)} = \\mu and \\tilde{g}^*{(v_{t})} = \\log{(\\sin{(v_{t})})} and \\tilde{g}^*^{v_{t}}{(v_{t})} = \\log{(\\sin{(v_{t})})}^{v_{t}} and \\tilde{g}^*^{v_{t}}{(v_{t})} + \\frac{\\mu}{\\delta} = \\log{(\\sin{(v_{t})})}^{v_{t}} + \\frac{\\mu}{\\delta} and \\tilde{g}^*^{v_{t}}{(v_{t})} + \\operatorname{z^{*}}{(\\mu,\\delta)} = \\operatorname{z^{*}}{(\\mu,\\delta)} + \\log{(\\sin{(v_{t})})}^{v_{t}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('z^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True))), Symbol('\\\\mu', commutative=True))"], ["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), log(sin(Symbol('v_t', commutative=True))))"], [["power", 3, "Symbol('v_t', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Pow(log(sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)))"], [["add", 4, "Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))"], "Equality(Add(Pow(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))), Add(Pow(log(sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True)), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(-1)), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Pow(Function('\\\\tilde{g}^*')(Symbol('v_t', commutative=True)), Symbol('v_t', commutative=True)), Function('z^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Function('z^*')(Symbol('\\\\mu', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(log(sin(Symbol('v_t', commutative=True))), Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(z)} = \\int e^{z} dz, then obtain (\\frac{d}{d z} \\int e^{z} dz) \\int \\hat{H}_l{(z)} dz = (\\frac{d}{d z} \\int e^{z} dz) \\iint e^{z} dz dz", "derivation": "\\hat{H}_l{(z)} = \\int e^{z} dz and \\frac{d}{d z} \\hat{H}_l{(z)} = \\frac{d}{d z} \\int e^{z} dz and \\int \\hat{H}_l{(z)} dz = \\iint e^{z} dz dz and \\frac{d}{d z} \\hat{H}_l{(z)} \\int \\hat{H}_l{(z)} dz = \\frac{d}{d z} \\hat{H}_l{(z)} \\iint e^{z} dz dz and (\\frac{d}{d z} \\int e^{z} dz) \\int \\hat{H}_l{(z)} dz = (\\frac{d}{d z} \\int e^{z} dz) \\iint e^{z} dz dz", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["integrate", 1, "Symbol('z', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["times", 3, "Derivative(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integral(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Derivative(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Derivative(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Integral(Function('\\\\hat{H}_l')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))), Mul(Derivative(Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Integral(exp(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(B)} = \\sin{(B)} and \\operatorname{y^{\\prime}}{(c_{0},B)} = 2 (c_{0} - \\cos{(B)}) \\hat{p}_0{(B)}, then obtain \\frac{\\operatorname{y^{\\prime}}{(c_{0},B)}}{2 B (c_{0} - \\cos{(B)}) \\sin{(B)}} = \\frac{1}{B}", "derivation": "\\hat{p}_0{(B)} = \\sin{(B)} and \\operatorname{y^{\\prime}}{(c_{0},B)} = 2 (c_{0} - \\cos{(B)}) \\hat{p}_0{(B)} and \\frac{\\operatorname{y^{\\prime}}{(c_{0},B)}}{2 (c_{0} - \\cos{(B)}) \\hat{p}_0{(B)}} = 1 and \\frac{\\operatorname{y^{\\prime}}{(c_{0},B)}}{2 B (c_{0} - \\cos{(B)}) \\hat{p}_0{(B)}} = \\frac{1}{B} and \\frac{\\operatorname{y^{\\prime}}{(c_{0},B)}}{2 B (c_{0} - \\cos{(B)}) \\sin{(B)}} = \\frac{1}{B}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('B', commutative=True)), Mul(Integer(2), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('B', commutative=True))))"], [["divide", 2, "Mul(Integer(2), Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Function('\\\\hat{p}_0')(Symbol('B', commutative=True)))"], "Equality(Mul(Rational(1, 2), Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('B', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('B', commutative=True))), Integer(1))"], [["divide", 3, "Symbol('B', commutative=True)"], "Equality(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Integer(-1)), Pow(Function('\\\\hat{p}_0')(Symbol('B', commutative=True)), Integer(-1)), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('B', commutative=True))), Pow(Symbol('B', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Rational(1, 2), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Add(Symbol('c_0', commutative=True), Mul(Integer(-1), cos(Symbol('B', commutative=True)))), Integer(-1)), Function('y^{\\\\prime}')(Symbol('c_0', commutative=True), Symbol('B', commutative=True)), Pow(sin(Symbol('B', commutative=True)), Integer(-1))), Pow(Symbol('B', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mu_{0}{(Q,\\phi_1)} = \\phi_1^{Q} and \\bar{\\h}{(\\phi_1,Q)} = \\mu_{0}^{Q}{(Q,\\phi_1)}, then obtain \\bar{\\h}^{\\phi_1}{(\\phi_1,Q)} = (\\mu_{0}^{Q}{(Q,\\phi_1)})^{\\phi_1}", "derivation": "\\mu_{0}{(Q,\\phi_1)} = \\phi_1^{Q} and \\mu_{0}^{Q}{(Q,\\phi_1)} = (\\phi_1^{Q})^{Q} and \\bar{\\h}{(\\phi_1,Q)} = \\mu_{0}^{Q}{(Q,\\phi_1)} and \\bar{\\h}{(\\phi_1,Q)} = (\\phi_1^{Q})^{Q} and \\bar{\\h}^{\\phi_1}{(\\phi_1,Q)} = ((\\phi_1^{Q})^{Q})^{\\phi_1} and \\bar{\\h}^{\\phi_1}{(\\phi_1,Q)} = (\\mu_{0}^{Q}{(Q,\\phi_1)})^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)))"], [["power", 1, "Symbol('Q', commutative=True)"], "Equality(Pow(Function('\\\\mu_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\mu_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hbar')(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Pow(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Pow(Pow(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\phi_1', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Pow(Pow(Function('\\\\mu_0')(Symbol('Q', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('Q', commutative=True)), Symbol('\\\\phi_1', commutative=True)))"]]}, {"prompt": "Given \\phi{(J)} = \\sin{(J)}, then obtain \\frac{d}{d J} (J \\phi{(J)})^{J} = \\frac{d}{d J} (J \\sin{(J)})^{J}", "derivation": "\\phi{(J)} = \\sin{(J)} and J \\phi{(J)} = J \\sin{(J)} and (J \\phi{(J)})^{J} = (J \\sin{(J)})^{J} and \\frac{d}{d J} (J \\phi{(J)})^{J} = \\frac{d}{d J} (J \\sin{(J)})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('J', commutative=True)), sin(Symbol('J', commutative=True)))"], [["times", 1, "Symbol('J', commutative=True)"], "Equality(Mul(Symbol('J', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True))), Mul(Symbol('J', commutative=True), sin(Symbol('J', commutative=True))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Mul(Symbol('J', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Pow(Mul(Symbol('J', commutative=True), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('J', commutative=True), Function('\\\\phi')(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('J', commutative=True), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{E}{(l,p)} = l + p, then derive \\int \\mathbf{E}{(l,p)} dl = \\rho_f + \\frac{l^{2}}{2} + l p, then obtain \\frac{\\int (\\rho_f + \\frac{l^{2}}{2} + l p) dl}{\\frac{\\partial}{\\partial l} (l + p)} = \\frac{\\iint (l + p) dl dl}{\\frac{\\partial}{\\partial l} (l + p)}", "derivation": "\\mathbf{E}{(l,p)} = l + p and \\int \\mathbf{E}{(l,p)} dl = \\int (l + p) dl and \\int \\mathbf{E}{(l,p)} dl = \\rho_f + \\frac{l^{2}}{2} + l p and \\rho_f + \\frac{l^{2}}{2} + l p = \\int (l + p) dl and \\int (\\rho_f + \\frac{l^{2}}{2} + l p) dl = \\iint (l + p) dl dl and \\frac{\\int (\\rho_f + \\frac{l^{2}}{2} + l p) dl}{\\frac{\\partial}{\\partial l} \\mathbf{E}{(l,p)}} = \\frac{\\iint (l + p) dl dl}{\\frac{\\partial}{\\partial l} \\mathbf{E}{(l,p)}} and \\frac{\\int (\\rho_f + \\frac{l^{2}}{2} + l p) dl}{\\frac{\\partial}{\\partial l} (l + p)} = \\frac{\\iint (l + p) dl dl}{\\frac{\\partial}{\\partial l} (l + p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Add(Symbol('l', commutative=True), Symbol('p', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\rho_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('p', commutative=True))), Integral(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('l', commutative=True))), Integral(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["divide", 5, "Derivative(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Derivative(Function('\\\\mathbf{E}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Pow(Derivative(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Symbol('\\\\rho_f', commutative=True), Mul(Rational(1, 2), Pow(Symbol('l', commutative=True), Integer(2))), Mul(Symbol('l', commutative=True), Symbol('p', commutative=True))), Tuple(Symbol('l', commutative=True)))), Mul(Pow(Derivative(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Integral(Add(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given V{(v_{2},y^{\\prime})} = v_{2} - y^{\\prime}, then obtain - \\frac{V{(v_{2},y^{\\prime})} \\int - \\frac{V{(v_{2},y^{\\prime})}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}} = - \\frac{V{(v_{2},y^{\\prime})} \\int - \\frac{v_{2} - y^{\\prime}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}}", "derivation": "V{(v_{2},y^{\\prime})} = v_{2} - y^{\\prime} and - \\frac{V{(v_{2},y^{\\prime})}}{y^{\\prime}} = - \\frac{v_{2} - y^{\\prime}}{y^{\\prime}} and \\int - \\frac{V{(v_{2},y^{\\prime})}}{y^{\\prime}} dy^{\\prime} = \\int - \\frac{v_{2} - y^{\\prime}}{y^{\\prime}} dy^{\\prime} and - \\frac{(v_{2} - y^{\\prime}) \\int - \\frac{V{(v_{2},y^{\\prime})}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}} = - \\frac{(v_{2} - y^{\\prime}) \\int - \\frac{v_{2} - y^{\\prime}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}} and - \\frac{V{(v_{2},y^{\\prime})} \\int - \\frac{V{(v_{2},y^{\\prime})}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}} = - \\frac{V{(v_{2},y^{\\prime})} \\int - \\frac{v_{2} - y^{\\prime}}{y^{\\prime}} dy^{\\prime}}{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))))"], [["integrate", 2, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))))"], "Equality(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('V')(Symbol('v_2', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integral(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))), Tuple(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{A}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)}, then obtain 0 = (- \\mathbf{A}{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)}) (- \\hat{x}_0 + \\mathbf{A}{(\\hat{x}_0)} - \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} + 1)", "derivation": "\\mathbf{A}{(\\hat{x}_0)} = \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} and \\mathbf{A}{(\\hat{x}_0)} - \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} = 0 and \\mathbf{A}{(\\hat{x}_0)} - \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} + 1 = 1 and - \\hat{x}_0 + \\mathbf{A}{(\\hat{x}_0)} - \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} + 1 = 1 - \\hat{x}_0 and 0 = - \\mathbf{A}{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} and 0 = (- \\mathbf{A}{(\\hat{x}_0)} + \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)}) (- \\hat{x}_0 + \\mathbf{A}{(\\hat{x}_0)} - \\frac{d}{d \\hat{x}_0} \\cos{(\\hat{x}_0)} + 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))), Integer(0))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(1)), Integer(1))"], [["minus", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(1))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(1))"], "Equality(Integer(0), Mul(Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True))), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\mathbf{A}')(Symbol('\\\\hat{x}_0', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('\\\\hat{x}_0', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1)))), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{S},v_{y})} = \\mathbf{S} - v_{y}, then obtain (\\int \\hat{p}_0{(\\mathbf{S},v_{y})} dv_{y})^{v_{y}} = (M + \\mathbf{S} v_{y} - \\frac{v_{y}^{2}}{2})^{v_{y}}", "derivation": "\\hat{p}_0{(\\mathbf{S},v_{y})} = \\mathbf{S} - v_{y} and \\int \\hat{p}_0{(\\mathbf{S},v_{y})} dv_{y} = \\int (\\mathbf{S} - v_{y}) dv_{y} and (\\int \\hat{p}_0{(\\mathbf{S},v_{y})} dv_{y})^{v_{y}} = (\\int (\\mathbf{S} - v_{y}) dv_{y})^{v_{y}} and (\\int \\hat{p}_0{(\\mathbf{S},v_{y})} dv_{y})^{v_{y}} = (M + \\mathbf{S} v_{y} - \\frac{v_{y}^{2}}{2})^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))))"], [["integrate", 1, "Symbol('v_y', commutative=True)"], "Equality(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), Mul(Integer(-1), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True))), Symbol('v_y', commutative=True)), Pow(Add(Symbol('M', commutative=True), Mul(Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v_y', commutative=True), Integer(2)))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(f^{*},y,\\mathbf{D})} = (f^{*} y)^{\\mathbf{D}} and \\mathbf{B}{(f^{*},y)} = f^{*} y, then obtain \\iint (f^{*} y)^{\\mathbf{D}} d\\mathbf{D} df^{*} = \\iint \\mathbf{B}^{\\mathbf{D}}{(f^{*},y)} d\\mathbf{D} df^{*}", "derivation": "\\operatorname{C_{d}}{(f^{*},y,\\mathbf{D})} = (f^{*} y)^{\\mathbf{D}} and \\mathbf{B}{(f^{*},y)} = f^{*} y and \\operatorname{C_{d}}{(f^{*},y,\\mathbf{D})} = \\mathbf{B}^{\\mathbf{D}}{(f^{*},y)} and (f^{*} y)^{\\mathbf{D}} = \\mathbf{B}^{\\mathbf{D}}{(f^{*},y)} and \\int (f^{*} y)^{\\mathbf{D}} d\\mathbf{D} = \\int \\mathbf{B}^{\\mathbf{D}}{(f^{*},y)} d\\mathbf{D} and \\iint (f^{*} y)^{\\mathbf{D}} d\\mathbf{D} df^{*} = \\iint \\mathbf{B}^{\\mathbf{D}}{(f^{*},y)} d\\mathbf{D} df^{*}", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('f^*', commutative=True), Symbol('y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Mul(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{B}')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Mul(Symbol('f^*', commutative=True), Symbol('y', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C_d')(Symbol('f^*', commutative=True), Symbol('y', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\mathbf{B}')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Mul(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Function('\\\\mathbf{B}')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Pow(Function('\\\\mathbf{B}')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["integrate", 5, "Symbol('f^*', commutative=True)"], "Equality(Integral(Pow(Mul(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Pow(Function('\\\\mathbf{B}')(Symbol('f^*', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(v_{1},a)} = \\log{(a^{v_{1}})} and \\mathbf{J}{(v_{1},a)} = \\log{(a^{v_{1}})}, then obtain \\sigma_{x}^{a}{(v_{1},a)} = \\mathbf{J}^{a}{(v_{1},a)}", "derivation": "\\sigma_{x}{(v_{1},a)} = \\log{(a^{v_{1}})} and \\mathbf{J}{(v_{1},a)} = \\log{(a^{v_{1}})} and \\mathbf{J}^{a}{(v_{1},a)} = \\log{(a^{v_{1}})}^{a} and \\sigma_{x}{(v_{1},a)} = \\mathbf{J}{(v_{1},a)} and \\sigma_{x}^{a}{(v_{1},a)} = \\log{(a^{v_{1}})}^{a} and \\sigma_{x}^{a}{(v_{1},a)} = \\mathbf{J}^{a}{(v_{1},a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), log(Pow(Symbol('a', commutative=True), Symbol('v_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), log(Pow(Symbol('a', commutative=True), Symbol('v_1', commutative=True))))"], [["power", 2, "Symbol('a', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(log(Pow(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\sigma_x')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), Function('\\\\mathbf{J}')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\sigma_x')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(log(Pow(Symbol('a', commutative=True), Symbol('v_1', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Function('\\\\sigma_x')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('v_1', commutative=True), Symbol('a', commutative=True)), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\omega{(F_{x})} = e^{\\cos{(F_{x})}}, then obtain (\\int e^{\\cos{(F_{x})}} \\int \\omega{(F_{x})} dF_{x} dF_{x}) \\int \\omega{(F_{x})} dF_{x} = (\\int e^{\\cos{(F_{x})}} \\int e^{\\cos{(F_{x})}} dF_{x} dF_{x}) \\int \\omega{(F_{x})} dF_{x}", "derivation": "\\omega{(F_{x})} = e^{\\cos{(F_{x})}} and \\int \\omega{(F_{x})} dF_{x} = \\int e^{\\cos{(F_{x})}} dF_{x} and e^{\\cos{(F_{x})}} \\int \\omega{(F_{x})} dF_{x} = e^{\\cos{(F_{x})}} \\int e^{\\cos{(F_{x})}} dF_{x} and \\int e^{\\cos{(F_{x})}} \\int \\omega{(F_{x})} dF_{x} dF_{x} = \\int e^{\\cos{(F_{x})}} \\int e^{\\cos{(F_{x})}} dF_{x} dF_{x} and (\\int e^{\\cos{(F_{x})}} \\int \\omega{(F_{x})} dF_{x} dF_{x}) \\int \\omega{(F_{x})} dF_{x} = (\\int e^{\\cos{(F_{x})}} \\int e^{\\cos{(F_{x})}} dF_{x} dF_{x}) \\int \\omega{(F_{x})} dF_{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('F_x', commutative=True)), exp(cos(Symbol('F_x', commutative=True))))"], [["integrate", 1, "Symbol('F_x', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True))), Integral(exp(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True))))"], [["times", 2, "exp(cos(Symbol('F_x', commutative=True)))"], "Equality(Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(exp(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(exp(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))"], [["times", 4, "Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))"], "Equality(Mul(Integral(Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))), Mul(Integral(Mul(exp(cos(Symbol('F_x', commutative=True))), Integral(exp(cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Integral(Function('\\\\omega')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f^{*})} = \\sin{(f^{*})}, then derive \\int \\operatorname{E_{n}}{(f^{*})} df^{*} = \\Omega - \\cos{(f^{*})}, then obtain \\Omega - \\cos{(f^{*})} = \\int \\sin{(f^{*})} df^{*}", "derivation": "\\operatorname{E_{n}}{(f^{*})} = \\sin{(f^{*})} and \\int \\operatorname{E_{n}}{(f^{*})} df^{*} = \\int \\sin{(f^{*})} df^{*} and \\frac{\\int \\operatorname{E_{n}}{(f^{*})} df^{*}}{p{(f^{*})}} = \\frac{\\int \\sin{(f^{*})} df^{*}}{p{(f^{*})}} and \\int \\operatorname{E_{n}}{(f^{*})} df^{*} = \\Omega - \\cos{(f^{*})} and \\frac{\\Omega - \\cos{(f^{*})}}{p{(f^{*})}} = \\frac{\\int \\sin{(f^{*})} df^{*}}{p{(f^{*})}} and \\Omega - \\cos{(f^{*})} = \\int \\sin{(f^{*})} df^{*}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f^*', commutative=True)), sin(Symbol('f^*', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('E_n')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["divide", 2, "Function('p')(Symbol('f^*', commutative=True))"], "Equality(Mul(Pow(Function('p')(Symbol('f^*', commutative=True)), Integer(-1)), Integral(Function('E_n')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Mul(Pow(Function('p')(Symbol('f^*', commutative=True)), Integer(-1)), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_n')(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('f^*', commutative=True)))), Pow(Function('p')(Symbol('f^*', commutative=True)), Integer(-1))), Mul(Pow(Function('p')(Symbol('f^*', commutative=True)), Integer(-1)), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["times", 5, "Function('p')(Symbol('f^*', commutative=True))"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('f^*', commutative=True)))), Integral(sin(Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given m{(M_{E})} = \\cos{(M_{E})} and \\Psi^{\\dagger}{(M_{E})} = - (2 m{(M_{E})} - \\cos{(M_{E})}) \\cos{(M_{E})}, then obtain - \\cos^{2}{(M_{E})} = - m{(M_{E})} \\cos{(M_{E})}", "derivation": "m{(M_{E})} = \\cos{(M_{E})} and m{(M_{E})} - \\cos{(M_{E})} = 0 and 2 m{(M_{E})} - \\cos{(M_{E})} = m{(M_{E})} and - (2 m{(M_{E})} - \\cos{(M_{E})}) \\cos{(M_{E})} = - m{(M_{E})} \\cos{(M_{E})} and \\Psi^{\\dagger}{(M_{E})} = - (2 m{(M_{E})} - \\cos{(M_{E})}) \\cos{(M_{E})} and \\Psi^{\\dagger}{(M_{E})} = - m{(M_{E})} \\cos{(M_{E})} and \\Psi^{\\dagger}{(M_{E})} = - \\cos^{2}{(M_{E})} and - \\cos^{2}{(M_{E})} = - m{(M_{E})} \\cos{(M_{E})}", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True)))"], [["minus", 1, "cos(Symbol('M_E', commutative=True))"], "Equality(Add(Function('m')(Symbol('M_E', commutative=True)), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))), Integer(0))"], [["add", 2, "Function('m')(Symbol('M_E', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('m')(Symbol('M_E', commutative=True))), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))), Function('m')(Symbol('M_E', commutative=True)))"], [["times", 3, "Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(2), Function('m')(Symbol('M_E', commutative=True))), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))), cos(Symbol('M_E', commutative=True))), Mul(Integer(-1), Function('m')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Add(Mul(Integer(2), Function('m')(Symbol('M_E', commutative=True))), Mul(Integer(-1), cos(Symbol('M_E', commutative=True)))), cos(Symbol('M_E', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Function('m')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('M_E', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('M_E', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Mul(Integer(-1), Pow(cos(Symbol('M_E', commutative=True)), Integer(2))), Mul(Integer(-1), Function('m')(Symbol('M_E', commutative=True)), cos(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given \\theta{(p,L)} = L^{p}, then obtain \\int L^{2 p} (L^{p} \\theta{(p,L)})^{p} dp = \\int L^{2 p} (L^{2 p})^{p} dp", "derivation": "\\theta{(p,L)} = L^{p} and L^{p} \\theta{(p,L)} = L^{2 p} and (L^{p} \\theta{(p,L)})^{p} = (L^{2 p})^{p} and L^{2 p} (L^{p} \\theta{(p,L)})^{p} = L^{2 p} (L^{2 p})^{p} and \\int L^{2 p} (L^{p} \\theta{(p,L)})^{p} dp = \\int L^{2 p} (L^{2 p})^{p} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('p', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Pow(Symbol('L', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Symbol('p', commutative=True)), Function('\\\\theta')(Symbol('p', commutative=True), Symbol('L', commutative=True))), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('L', commutative=True), Symbol('p', commutative=True)), Function('\\\\theta')(Symbol('p', commutative=True), Symbol('L', commutative=True))), Symbol('p', commutative=True)), Pow(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["times", 3, "Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Pow(Mul(Pow(Symbol('L', commutative=True), Symbol('p', commutative=True)), Function('\\\\theta')(Symbol('p', commutative=True), Symbol('L', commutative=True))), Symbol('p', commutative=True))), Mul(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Pow(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Symbol('p', commutative=True))))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Pow(Mul(Pow(Symbol('L', commutative=True), Symbol('p', commutative=True)), Function('\\\\theta')(Symbol('p', commutative=True), Symbol('L', commutative=True))), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Integral(Mul(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Pow(Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('p', commutative=True))), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(\\psi^*,\\omega)} = \\psi^* \\sin{(\\omega)} and \\Psi{(\\psi^*,\\omega)} = \\psi^* \\frac{\\partial}{\\partial \\psi^*} \\psi^* \\sin{(\\omega)}, then derive \\psi^* \\frac{\\partial}{\\partial \\psi^*} \\operatorname{f^{*}}{(\\psi^*,\\omega)} + \\operatorname{f^{*}}{(\\psi^*,\\omega)} = 2 \\psi^* \\sin{(\\omega)}, then obtain \\psi^* \\sin{(\\omega)} + \\Psi{(\\psi^*,\\omega)} = 2 \\psi^* \\sin{(\\omega)}", "derivation": "\\operatorname{f^{*}}{(\\psi^*,\\omega)} = \\psi^* \\sin{(\\omega)} and \\psi^* \\operatorname{f^{*}}{(\\psi^*,\\omega)} = (\\psi^*)^{2} \\sin{(\\omega)} and \\frac{\\partial}{\\partial \\psi^*} \\psi^* \\operatorname{f^{*}}{(\\psi^*,\\omega)} = \\frac{\\partial}{\\partial \\psi^*} (\\psi^*)^{2} \\sin{(\\omega)} and \\psi^* \\frac{\\partial}{\\partial \\psi^*} \\operatorname{f^{*}}{(\\psi^*,\\omega)} + \\operatorname{f^{*}}{(\\psi^*,\\omega)} = 2 \\psi^* \\sin{(\\omega)} and \\psi^* \\sin{(\\omega)} + \\psi^* \\frac{\\partial}{\\partial \\psi^*} \\psi^* \\sin{(\\omega)} = 2 \\psi^* \\sin{(\\omega)} and \\Psi{(\\psi^*,\\omega)} = \\psi^* \\frac{\\partial}{\\partial \\psi^*} \\psi^* \\sin{(\\omega)} and \\psi^* \\sin{(\\omega)} + \\Psi{(\\psi^*,\\omega)} = 2 \\psi^* \\sin{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], [["times", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\psi^*', commutative=True), Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), sin(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\psi^*', commutative=True), Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\psi^*', commutative=True), Integer(2)), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\psi^*', commutative=True), Derivative(Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))), Function('f^*')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1))))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Symbol('\\\\psi^*', commutative=True), Derivative(Mul(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Mul(Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))), Function('\\\\Psi')(Symbol('\\\\psi^*', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Integer(2), Symbol('\\\\psi^*', commutative=True), sin(Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\pi{(\\Psi_{nl})} = e^{\\Psi_{nl}}, then obtain \\frac{d}{d \\Psi_{nl}} (\\pi{(\\Psi_{nl})} - \\pi^{\\Psi_{nl}}{(\\Psi_{nl})}) = \\frac{d}{d \\Psi_{nl}} (- \\pi^{\\Psi_{nl}}{(\\Psi_{nl})} + e^{\\Psi_{nl}})", "derivation": "\\pi{(\\Psi_{nl})} = e^{\\Psi_{nl}} and \\pi^{\\Psi_{nl}}{(\\Psi_{nl})} = (e^{\\Psi_{nl}})^{\\Psi_{nl}} and \\pi{(\\Psi_{nl})} - (e^{\\Psi_{nl}})^{\\Psi_{nl}} = e^{\\Psi_{nl}} - (e^{\\Psi_{nl}})^{\\Psi_{nl}} and \\frac{d}{d \\Psi_{nl}} (\\pi{(\\Psi_{nl})} - (e^{\\Psi_{nl}})^{\\Psi_{nl}}) = \\frac{d}{d \\Psi_{nl}} (e^{\\Psi_{nl}} - (e^{\\Psi_{nl}})^{\\Psi_{nl}}) and \\frac{d}{d \\Psi_{nl}} (\\pi{(\\Psi_{nl})} - \\pi^{\\Psi_{nl}}{(\\Psi_{nl})}) = \\frac{d}{d \\Psi_{nl}} (- \\pi^{\\Psi_{nl}}{(\\Psi_{nl})} + e^{\\Psi_{nl}})", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), exp(Symbol('\\\\Psi_{nl}', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)), Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))"], [["minus", 1, "Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))"], "Equality(Add(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))), Add(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Add(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True)))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('\\\\pi')(Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('\\\\Psi_{nl}', commutative=True))), exp(Symbol('\\\\Psi_{nl}', commutative=True))), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(t,\\phi)} = \\phi + t, then derive \\int \\operatorname{r_{0}}{(t,\\phi)} dt = \\hat{X} + \\phi t + \\frac{t^{2}}{2}, then derive \\phi t + \\frac{t^{2}}{2} + v_{z} = \\hat{X} + \\phi t + \\frac{t^{2}}{2}, then obtain \\int (\\phi t + \\frac{t^{2}}{2} + v_{z})^{\\hat{X}} d\\phi = \\int (\\hat{X} + \\phi t + \\frac{t^{2}}{2})^{\\hat{X}} d\\phi", "derivation": "\\operatorname{r_{0}}{(t,\\phi)} = \\phi + t and \\int \\operatorname{r_{0}}{(t,\\phi)} dt = \\int (\\phi + t) dt and \\int \\operatorname{r_{0}}{(t,\\phi)} dt = \\hat{X} + \\phi t + \\frac{t^{2}}{2} and \\int (\\phi + t) dt = \\hat{X} + \\phi t + \\frac{t^{2}}{2} and \\phi t + \\frac{t^{2}}{2} + v_{z} = \\hat{X} + \\phi t + \\frac{t^{2}}{2} and (\\phi t + \\frac{t^{2}}{2} + v_{z})^{\\hat{X}} = (\\hat{X} + \\phi t + \\frac{t^{2}}{2})^{\\hat{X}} and \\int (\\phi t + \\frac{t^{2}}{2} + v_{z})^{\\hat{X}} d\\phi = \\int (\\hat{X} + \\phi t + \\frac{t^{2}}{2})^{\\hat{X}} d\\phi", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('t', commutative=True), Symbol('\\\\phi', commutative=True)), Add(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('r_0')(Symbol('t', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('r_0')(Symbol('t', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('v_z', commutative=True)), Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))))"], [["power", 5, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('v_z', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('v_z', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(Pow(Add(Symbol('\\\\hat{X}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), Symbol('t', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('t', commutative=True), Integer(2)))), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})} = W - \\mathbf{F} + \\tilde{g}^*, then obtain 1 = \\frac{W + \\tilde{g}^*}{\\mathbf{F} + \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})}}", "derivation": "\\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})} = W - \\mathbf{F} + \\tilde{g}^* and \\mathbf{F} + \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})} = W + \\tilde{g}^* and (\\mathbf{F} + \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})})^{2} = (W + \\tilde{g}^*) (\\mathbf{F} + \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})}) and 1 = \\frac{W + \\tilde{g}^*}{\\mathbf{F} + \\hat{X}{(\\tilde{g}^*,W,\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Add(Symbol('W', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{F}', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["times", 2, "Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], "Equality(Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(2)), Mul(Add(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["divide", 3, "Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(2))"], "Equality(Integer(1), Mul(Add(Symbol('W', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Function('\\\\hat{X}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('W', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{M},A_{2})} = e^{\\mathbf{M}^{A_{2}}} and \\mathbf{F}{(\\mathbf{M},A_{2})} = \\mathbf{M}^{A_{2}}, then obtain \\frac{A_{2} e^{\\mathbf{M}^{A_{2}}}}{\\mathbf{M}} = \\frac{A_{2} e^{\\mathbf{F}{(\\mathbf{M},A_{2})}}}{\\mathbf{M}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\mathbf{M},A_{2})} = e^{\\mathbf{M}^{A_{2}}} and A_{2} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{M},A_{2})} = A_{2} e^{\\mathbf{M}^{A_{2}}} and \\frac{A_{2} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{M},A_{2})}}{\\mathbf{M}} = \\frac{A_{2} e^{\\mathbf{M}^{A_{2}}}}{\\mathbf{M}} and \\mathbf{F}{(\\mathbf{M},A_{2})} = \\mathbf{M}^{A_{2}} and \\frac{A_{2} \\operatorname{V_{\\mathbf{B}}}{(\\mathbf{M},A_{2})}}{\\mathbf{M}} = \\frac{A_{2} e^{\\mathbf{F}{(\\mathbf{M},A_{2})}}}{\\mathbf{M}} and \\frac{A_{2} e^{\\mathbf{M}^{A_{2}}}}{\\mathbf{M}} = \\frac{A_{2} e^{\\mathbf{F}{(\\mathbf{M},A_{2})}}}{\\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)), exp(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True))))"], [["times", 1, "Symbol('A_2', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), exp(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))))"], [["divide", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), exp(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), exp(Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))), Mul(Symbol('A_2', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(-1)), exp(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given a{(z,m)} = \\int \\frac{z}{m} dz, then obtain \\iint (\\frac{\\partial}{\\partial m} a{(z,m)} + 1) dm dm = \\iint (\\frac{\\partial}{\\partial m} \\int \\frac{z}{m} dz + 1) dm dm", "derivation": "a{(z,m)} = \\int \\frac{z}{m} dz and \\frac{\\partial}{\\partial m} a{(z,m)} = \\frac{\\partial}{\\partial m} \\int \\frac{z}{m} dz and \\frac{\\partial}{\\partial m} a{(z,m)} + 1 = \\frac{\\partial}{\\partial m} \\int \\frac{z}{m} dz + 1 and \\int (\\frac{\\partial}{\\partial m} a{(z,m)} + 1) dm = \\int (\\frac{\\partial}{\\partial m} \\int \\frac{z}{m} dz + 1) dm and \\iint (\\frac{\\partial}{\\partial m} a{(z,m)} + 1) dm dm = \\iint (\\frac{\\partial}{\\partial m} \\int \\frac{z}{m} dz + 1) dm dm", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('z', commutative=True), Symbol('m', commutative=True)), Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('m', commutative=True))), Integral(Add(Derivative(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('m', commutative=True))))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Derivative(Function('a')(Symbol('z', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Derivative(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True))), Tuple(Symbol('m', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\Psi{(E_{x},\\psi^*)} = \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x}}{\\psi^*} and \\mu_{0}{(\\psi^*)} = (\\frac{1}{\\psi^*})^{\\psi^*}, then derive \\Psi^{\\psi^*}{(E_{x},\\psi^*)} - \\frac{1}{\\psi^*} = (\\frac{1}{\\psi^*})^{\\psi^*} - \\frac{1}{\\psi^*}, then obtain \\Psi^{\\psi^*}{(E_{x},\\psi^*)} - \\frac{1}{\\psi^*} = \\mu_{0}{(\\psi^*)} - \\frac{1}{\\psi^*}", "derivation": "\\Psi{(E_{x},\\psi^*)} = \\frac{\\partial}{\\partial E_{x}} \\frac{E_{x}}{\\psi^*} and \\Psi^{\\psi^*}{(E_{x},\\psi^*)} = (\\frac{\\partial}{\\partial E_{x}} \\frac{E_{x}}{\\psi^*})^{\\psi^*} and \\Psi^{\\psi^*}{(E_{x},\\psi^*)} - \\frac{1}{\\psi^*} = (\\frac{\\partial}{\\partial E_{x}} \\frac{E_{x}}{\\psi^*})^{\\psi^*} - \\frac{1}{\\psi^*} and \\Psi^{\\psi^*}{(E_{x},\\psi^*)} - \\frac{1}{\\psi^*} = (\\frac{1}{\\psi^*})^{\\psi^*} - \\frac{1}{\\psi^*} and \\mu_{0}{(\\psi^*)} = (\\frac{1}{\\psi^*})^{\\psi^*} and \\Psi^{\\psi^*}{(E_{x},\\psi^*)} - \\frac{1}{\\psi^*} = \\mu_{0}{(\\psi^*)} - \\frac{1}{\\psi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('E_x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Derivative(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["power", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Function('\\\\Psi')(Symbol('E_x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Pow(Derivative(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('\\\\psi^*', commutative=True)))"], [["minus", 2, "Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))"], "Equality(Add(Pow(Function('\\\\Psi')(Symbol('E_x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Add(Pow(Derivative(Mul(Symbol('E_x', commutative=True), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1))), Tuple(Symbol('E_x', commutative=True), Integer(1))), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Pow(Function('\\\\Psi')(Symbol('E_x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Add(Pow(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True)), Pow(Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)), Symbol('\\\\psi^*', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Pow(Function('\\\\Psi')(Symbol('E_x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))), Add(Function('\\\\mu_0')(Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\psi^*', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)} = \\eta - \\varphi, then derive 1 = \\tilde{\\infty} (1 - \\frac{\\partial}{\\partial \\eta} \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}), then obtain 0 = \\tilde{\\infty} \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\eta} \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}", "derivation": "\\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)} = \\eta - \\varphi and 0 = \\eta - \\varphi - \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)} and \\frac{d}{d \\eta} 0 = \\frac{\\partial}{\\partial \\eta} (\\eta - \\varphi - \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}) and 1 = \\frac{\\frac{\\partial}{\\partial \\eta} (\\eta - \\varphi - \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)})}{\\frac{d}{d \\eta} 0} and 1 = \\tilde{\\infty} (1 - \\frac{\\partial}{\\partial \\eta} \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}) and \\frac{d}{d \\varphi} 1 = \\frac{\\partial}{\\partial \\varphi} \\tilde{\\infty} (1 - \\frac{\\partial}{\\partial \\eta} \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}) and 0 = \\tilde{\\infty} \\frac{\\partial^{2}}{\\partial \\varphi\\partial \\eta} \\operatorname{L_{\\varepsilon}}{(\\varphi,\\eta)}", "srepr_derivation": [["get_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["minus", 1, "Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Integer(0), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(-1)), Derivative(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Integer(1), Mul(zoo, Add(Integer(1), Mul(Integer(-1), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))))"], [["differentiate", 5, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Mul(zoo, Add(Integer(1), Mul(Integer(-1), Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Mul(zoo, Derivative(Function('L_{\\\\varepsilon}')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))))"]]}, {"prompt": "Given x{(\\mathbb{I},A)} = A \\mathbb{I} and i{(U)} = \\log{(\\cos{(U)})}, then obtain A \\mathbb{I} \\cos{(U)} = (A \\mathbb{I} + \\int i{(U)} dU - \\int \\log{(\\cos{(U)})} dU) \\cos{(U)}", "derivation": "x{(\\mathbb{I},A)} = A \\mathbb{I} and i{(U)} = \\log{(\\cos{(U)})} and x{(\\mathbb{I},A)} \\cos{(U)} = A \\mathbb{I} \\cos{(U)} and \\int i{(U)} dU = \\int \\log{(\\cos{(U)})} dU and A \\mathbb{I} + \\int i{(U)} dU = A \\mathbb{I} + \\int \\log{(\\cos{(U)})} dU and A \\mathbb{I} + \\int i{(U)} dU - \\int \\log{(\\cos{(U)})} dU = A \\mathbb{I} and x{(\\mathbb{I},A)} \\cos{(U)} = (A \\mathbb{I} + \\int i{(U)} dU - \\int \\log{(\\cos{(U)})} dU) \\cos{(U)} and A \\mathbb{I} \\cos{(U)} = (A \\mathbb{I} + \\int i{(U)} dU - \\int \\log{(\\cos{(U)})} dU) \\cos{(U)}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], ["get_premise", "Equality(Function('i')(Symbol('U', commutative=True)), log(cos(Symbol('U', commutative=True))))"], [["times", 1, "cos(Symbol('U', commutative=True))"], "Equality(Mul(Function('x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A', commutative=True)), cos(Symbol('U', commutative=True))), Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('U', commutative=True))))"], [["integrate", 2, "Symbol('U', commutative=True)"], "Equality(Integral(Function('i')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["add", 4, "Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('i')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)))), Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"], [["minus", 5, "Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))"], "Equality(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('i')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(-1), Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Mul(Function('x')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('A', commutative=True)), cos(Symbol('U', commutative=True))), Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('i')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(-1), Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), cos(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), cos(Symbol('U', commutative=True))), Mul(Add(Mul(Symbol('A', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integral(Function('i')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(-1), Integral(log(cos(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), cos(Symbol('U', commutative=True))))"]]}, {"prompt": "Given x{(\\mathbf{M},\\eta)} = \\cos{(\\eta + \\mathbf{M})}, then obtain - 2 \\mathbf{M} + x{(\\mathbf{M},\\eta)} + \\cos{(\\eta + \\mathbf{M})} = - 2 \\mathbf{M} + 2 \\cos{(\\eta + \\mathbf{M})}", "derivation": "x{(\\mathbf{M},\\eta)} = \\cos{(\\eta + \\mathbf{M})} and - \\eta - \\mathbf{M} + x{(\\mathbf{M},\\eta)} = - \\eta - \\mathbf{M} + \\cos{(\\eta + \\mathbf{M})} and - \\mathbf{M} + x{(\\mathbf{M},\\eta)} = - \\mathbf{M} + \\cos{(\\eta + \\mathbf{M})} and - 2 \\mathbf{M} + x{(\\mathbf{M},\\eta)} + \\cos{(\\eta + \\mathbf{M})} = - 2 \\mathbf{M} + 2 \\cos{(\\eta + \\mathbf{M})}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('x')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["minus", 2, "Mul(Integer(-1), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('x')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Function('x')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\eta', commutative=True)), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), cos(Add(Symbol('\\\\eta', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))))))"]]}, {"prompt": "Given \\omega{(b,\\delta)} = \\int b^{\\delta} d\\delta, then obtain (e^{(\\int \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} db)^{b}})^{b} = (e^{(\\int \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} db)^{b}})^{b}", "derivation": "\\omega{(b,\\delta)} = \\int b^{\\delta} d\\delta and b^{- \\delta} \\omega{(b,\\delta)} = b^{- \\delta} \\int b^{\\delta} d\\delta and \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} = \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} and \\int \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} db = \\int \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} db and (\\int \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} db)^{b} = (\\int \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} db)^{b} and e^{(\\int \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} db)^{b}} = e^{(\\int \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} db)^{b}} and (e^{(\\int \\cos{(b^{- \\delta} \\omega{(b,\\delta)})} db)^{b}})^{b} = (e^{(\\int \\cos{(b^{- \\delta} \\int b^{\\delta} d\\delta)} db)^{b}})^{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["divide", 1, "Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["cos", 2], "Equality(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)))), cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"], [["integrate", 3, "Symbol('b', commutative=True)"], "Equality(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('b', commutative=True))))"], [["power", 4, "Symbol('b', commutative=True)"], "Equality(Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True)))"], [["exp", 5], "Equality(exp(Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), exp(Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))))"], [["power", 6, "Symbol('b', commutative=True)"], "Equality(Pow(exp(Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Function('\\\\omega')(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Symbol('b', commutative=True)), Pow(exp(Pow(Integral(cos(Mul(Pow(Symbol('b', commutative=True), Mul(Integer(-1), Symbol('\\\\delta', commutative=True))), Integral(Pow(Symbol('b', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Tuple(Symbol('b', commutative=True))), Symbol('b', commutative=True))), Symbol('b', commutative=True)))"]]}, {"prompt": "Given n{(E_{x},S)} = \\frac{\\cos{(E_{x})}}{S}, then obtain \\log{(\\frac{n{(E_{x},S)}}{\\cos{(E_{x})}})}^{S} = \\log{(\\frac{1}{S})}^{S}", "derivation": "n{(E_{x},S)} = \\frac{\\cos{(E_{x})}}{S} and \\frac{n{(E_{x},S)}}{\\cos{(E_{x})}} = \\frac{1}{S} and \\log{(\\frac{n{(E_{x},S)}}{\\cos{(E_{x})}})} = \\log{(\\frac{1}{S})} and \\log{(\\frac{n{(E_{x},S)}}{\\cos{(E_{x})}})}^{S} = \\log{(\\frac{1}{S})}^{S}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Mul(Pow(Symbol('S', commutative=True), Integer(-1)), cos(Symbol('E_x', commutative=True))))"], [["divide", 1, "cos(Symbol('E_x', commutative=True))"], "Equality(Mul(Function('n')(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Pow(cos(Symbol('E_x', commutative=True)), Integer(-1))), Pow(Symbol('S', commutative=True), Integer(-1)))"], [["log", 2], "Equality(log(Mul(Function('n')(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Pow(cos(Symbol('E_x', commutative=True)), Integer(-1)))), log(Pow(Symbol('S', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('S', commutative=True)"], "Equality(Pow(log(Mul(Function('n')(Symbol('E_x', commutative=True), Symbol('S', commutative=True)), Pow(cos(Symbol('E_x', commutative=True)), Integer(-1)))), Symbol('S', commutative=True)), Pow(log(Pow(Symbol('S', commutative=True), Integer(-1))), Symbol('S', commutative=True)))"]]}, {"prompt": "Given q{(s)} = \\cos{(s)} and \\mathbf{p}{(s)} = \\frac{q{(s)} + \\cos{(s)}}{q{(s)}}, then obtain \\mathbf{p}{(s)} = \\frac{2 \\cos{(s)}}{q{(s)}}", "derivation": "q{(s)} = \\cos{(s)} and q{(s)} + \\cos{(s)} = 2 \\cos{(s)} and \\frac{q{(s)} + \\cos{(s)}}{q{(s)}} = \\frac{2 \\cos{(s)}}{q{(s)}} and \\mathbf{p}{(s)} = \\frac{q{(s)} + \\cos{(s)}}{q{(s)}} and \\mathbf{p}{(s)} = \\frac{2 \\cos{(s)}}{q{(s)}}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True)))"], [["add", 1, "cos(Symbol('s', commutative=True))"], "Equality(Add(Function('q')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))), Mul(Integer(2), cos(Symbol('s', commutative=True))))"], [["divide", 2, "Function('q')(Symbol('s', commutative=True))"], "Equality(Mul(Add(Function('q')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))), Pow(Function('q')(Symbol('s', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('q')(Symbol('s', commutative=True)), Integer(-1)), cos(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('s', commutative=True)), Mul(Add(Function('q')(Symbol('s', commutative=True)), cos(Symbol('s', commutative=True))), Pow(Function('q')(Symbol('s', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{p}')(Symbol('s', commutative=True)), Mul(Integer(2), Pow(Function('q')(Symbol('s', commutative=True)), Integer(-1)), cos(Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\chi{(P_{e})} = e^{e^{P_{e}}}, then obtain \\sin{(\\chi^{P_{e}}{(P_{e})} e^{- P_{e}} + e^{P_{e}})} = \\sin{(e^{P_{e}} + e^{- P_{e}} (e^{e^{P_{e}}})^{P_{e}})}", "derivation": "\\chi{(P_{e})} = e^{e^{P_{e}}} and \\chi^{P_{e}}{(P_{e})} = (e^{e^{P_{e}}})^{P_{e}} and \\chi^{P_{e}}{(P_{e})} e^{- P_{e}} = e^{- P_{e}} (e^{e^{P_{e}}})^{P_{e}} and \\chi^{P_{e}}{(P_{e})} e^{- P_{e}} + e^{P_{e}} = e^{P_{e}} + e^{- P_{e}} (e^{e^{P_{e}}})^{P_{e}} and \\sin{(\\chi^{P_{e}}{(P_{e})} e^{- P_{e}} + e^{P_{e}})} = \\sin{(e^{P_{e}} + e^{- P_{e}} (e^{e^{P_{e}}})^{P_{e}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('P_e', commutative=True)), exp(exp(Symbol('P_e', commutative=True))))"], [["power", 1, "Symbol('P_e', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), Pow(exp(exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["divide", 2, "exp(Symbol('P_e', commutative=True))"], "Equality(Mul(Pow(Function('\\\\chi')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))))"], [["add", 3, "exp(Symbol('P_e', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\chi')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), exp(Symbol('P_e', commutative=True))), Add(exp(Symbol('P_e', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))))"], [["sin", 4], "Equality(sin(Add(Mul(Pow(Function('\\\\chi')(Symbol('P_e', commutative=True)), Symbol('P_e', commutative=True)), exp(Mul(Integer(-1), Symbol('P_e', commutative=True)))), exp(Symbol('P_e', commutative=True)))), sin(Add(exp(Symbol('P_e', commutative=True)), Mul(exp(Mul(Integer(-1), Symbol('P_e', commutative=True))), Pow(exp(exp(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(v_{x})} = \\sin{(v_{x})}, then obtain \\log{(\\int \\operatorname{n_{1}}{(v_{x})} dv_{x})} + \\int \\operatorname{n_{1}}{(v_{x})} dv_{x} = \\log{(\\int \\operatorname{n_{1}}{(v_{x})} dv_{x})} + \\int \\sin{(v_{x})} dv_{x}", "derivation": "\\operatorname{n_{1}}{(v_{x})} = \\sin{(v_{x})} and \\int \\operatorname{n_{1}}{(v_{x})} dv_{x} = \\int \\sin{(v_{x})} dv_{x} and \\log{(\\int \\operatorname{n_{1}}{(v_{x})} dv_{x})} = \\log{(\\int \\sin{(v_{x})} dv_{x})} and \\log{(\\int \\sin{(v_{x})} dv_{x})} + \\int \\operatorname{n_{1}}{(v_{x})} dv_{x} = \\log{(\\int \\sin{(v_{x})} dv_{x})} + \\int \\sin{(v_{x})} dv_{x} and \\log{(\\int \\operatorname{n_{1}}{(v_{x})} dv_{x})} + \\int \\operatorname{n_{1}}{(v_{x})} dv_{x} = \\log{(\\int \\operatorname{n_{1}}{(v_{x})} dv_{x})} + \\int \\sin{(v_{x})} dv_{x}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["integrate", 1, "Symbol('v_x', commutative=True)"], "Equality(Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), log(Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["add", 2, "log(Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], "Equality(Add(log(Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Add(log(Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(log(Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Add(log(Integral(Function('n_1')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))), Integral(sin(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}{(g)} = \\cos{(g)}, then obtain (\\frac{d}{d g} (\\hat{H}{(g)} + \\cos{(g)})^{g})^{2} = (\\frac{d}{d g} (2 \\cos{(g)})^{g})^{2}", "derivation": "\\hat{H}{(g)} = \\cos{(g)} and \\hat{H}{(g)} + \\cos{(g)} = 2 \\cos{(g)} and (\\hat{H}{(g)} + \\cos{(g)})^{g} = (2 \\cos{(g)})^{g} and \\frac{d}{d g} (\\hat{H}{(g)} + \\cos{(g)})^{g} = \\frac{d}{d g} (2 \\cos{(g)})^{g} and (\\frac{d}{d g} (\\hat{H}{(g)} + \\cos{(g)})^{g})^{2} = (\\frac{d}{d g} (2 \\cos{(g)})^{g})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True)))"], [["add", 1, "cos(Symbol('g', commutative=True))"], "Equality(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Mul(Integer(2), cos(Symbol('g', commutative=True))))"], [["power", 2, "Symbol('g', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Pow(Mul(Integer(2), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)))"], [["differentiate", 3, "Symbol('g', commutative=True)"], "Equality(Derivative(Pow(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(2), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(Pow(Add(Function('\\\\hat{H}')(Symbol('g', commutative=True)), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Pow(Mul(Integer(2), cos(Symbol('g', commutative=True))), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\mathbb{I}{(v_{z})} = \\log{(v_{z})} and a{(v_{z})} = - v_{z} + \\log{(v_{z})}, then obtain (- v_{z} + \\mathbb{I}{(v_{z})})^{v_{z}} = a^{v_{z}}{(v_{z})}", "derivation": "\\mathbb{I}{(v_{z})} = \\log{(v_{z})} and - v_{z} + \\mathbb{I}{(v_{z})} = - v_{z} + \\log{(v_{z})} and (- v_{z} + \\mathbb{I}{(v_{z})})^{v_{z}} = (- v_{z} + \\log{(v_{z})})^{v_{z}} and a{(v_{z})} = - v_{z} + \\log{(v_{z})} and (- v_{z} + \\mathbb{I}{(v_{z})})^{v_{z}} = a^{v_{z}}{(v_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True)))"], [["minus", 1, "Symbol('v_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\mathbb{I}')(Symbol('v_z', commutative=True))), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True))))"], [["power", 2, "Symbol('v_z', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\mathbb{I}')(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('a')(Symbol('v_z', commutative=True)), Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), log(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Function('\\\\mathbb{I}')(Symbol('v_z', commutative=True))), Symbol('v_z', commutative=True)), Pow(Function('a')(Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given r{(\\mathbb{I},\\Omega)} = \\Omega + \\mathbb{I}, then obtain (r^{2}{(\\mathbb{I},\\Omega)})^{\\mathbb{I}} = ((\\Omega + \\mathbb{I})^{2})^{\\mathbb{I}}", "derivation": "r{(\\mathbb{I},\\Omega)} = \\Omega + \\mathbb{I} and (\\Omega + \\mathbb{I}) r{(\\mathbb{I},\\Omega)} = (\\Omega + \\mathbb{I})^{2} and r^{2}{(\\mathbb{I},\\Omega)} = (\\Omega + \\mathbb{I}) r{(\\mathbb{I},\\Omega)} and r^{2}{(\\mathbb{I},\\Omega)} = (\\Omega + \\mathbb{I})^{2} and (r^{2}{(\\mathbb{I},\\Omega)})^{\\mathbb{I}} = ((\\Omega + \\mathbb{I})^{2})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["times", 1, "Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))"], [["times", 1, "Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Pow(Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)))"], [["power", 4, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Pow(Function('r')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(2)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Pow(Add(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Integer(2)), Symbol('\\\\mathbb{I}', commutative=True)))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain ((\\int \\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} = ((\\int \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "derivation": "\\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\int \\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\int \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and (\\int \\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}} = (\\int \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}} and ((\\int \\hat{\\mathbf{r}}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}} = ((\\int \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}})^{V_{\\mathbf{B}}})^{V_{\\mathbf{B}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["integrate", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["power", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Integral(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Pow(Integral(sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(q,J)} = \\log{(J q)} and \\mu_{0}{(q,J)} = \\frac{1}{\\operatorname{v_{2}}{(q,J)}} and c{(\\rho_b,g_{\\varepsilon})} = \\rho_b g_{\\varepsilon}, then obtain c{(\\rho_b,g_{\\varepsilon})} - \\frac{1}{\\operatorname{v_{2}}{(q,J)}} = \\rho_b g_{\\varepsilon} - \\frac{1}{\\operatorname{v_{2}}{(q,J)}}", "derivation": "\\operatorname{v_{2}}{(q,J)} = \\log{(J q)} and \\mu_{0}{(q,J)} = \\frac{1}{\\operatorname{v_{2}}{(q,J)}} and \\mu_{0}{(q,J)} = \\frac{1}{\\log{(J q)}} and c{(\\rho_b,g_{\\varepsilon})} = \\rho_b g_{\\varepsilon} and c{(\\rho_b,g_{\\varepsilon})} - \\frac{1}{\\log{(J q)}} = \\rho_b g_{\\varepsilon} - \\frac{1}{\\log{(J q)}} and \\frac{1}{\\log{(J q)}} = \\frac{1}{\\operatorname{v_{2}}{(q,J)}} and c{(\\rho_b,g_{\\varepsilon})} - \\frac{1}{\\operatorname{v_{2}}{(q,J)}} = \\rho_b g_{\\varepsilon} - \\frac{1}{\\operatorname{v_{2}}{(q,J)}}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('q', commutative=True), Symbol('J', commutative=True)), log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu_0')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Pow(Function('v_2')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mu_0')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Pow(log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))), Integer(-1)))"], ["get_premise", "Equality(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 4, "Pow(log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))), Integer(-1))"], "Equality(Add(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))), Integer(-1)))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Pow(log(Mul(Symbol('J', commutative=True), Symbol('q', commutative=True))), Integer(-1)), Pow(Function('v_2')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Function('c')(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Function('v_2')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Integer(-1)))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Function('v_2')(Symbol('q', commutative=True), Symbol('J', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(a^{\\dagger})} = \\log{(a^{\\dagger})}, then obtain e^{(\\operatorname{v_{y}}^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}}} = e^{(\\log{(a^{\\dagger})}^{a^{\\dagger}})^{a^{\\dagger}}}", "derivation": "\\operatorname{v_{y}}{(a^{\\dagger})} = \\log{(a^{\\dagger})} and \\operatorname{v_{y}}^{a^{\\dagger}}{(a^{\\dagger})} = \\log{(a^{\\dagger})}^{a^{\\dagger}} and (\\operatorname{v_{y}}^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}} = (\\log{(a^{\\dagger})}^{a^{\\dagger}})^{a^{\\dagger}} and e^{(\\operatorname{v_{y}}^{a^{\\dagger}}{(a^{\\dagger})})^{a^{\\dagger}}} = e^{(\\log{(a^{\\dagger})}^{a^{\\dagger}})^{a^{\\dagger}}}", "srepr_derivation": [["get_premise", "Equality(Function('v_y')(Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Pow(Function('v_y')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["exp", 3], "Equality(exp(Pow(Pow(Function('v_y')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), exp(Pow(Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})} = \\hat{H}_{\\lambda}^{V_{\\mathbf{E}}} \\mathbf{H}, then obtain \\frac{\\hat{H}_{\\lambda}^{- V_{\\mathbf{E}}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\partial}{\\partial \\mathbf{H}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}", "derivation": "\\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})} = \\hat{H}_{\\lambda}^{V_{\\mathbf{E}}} \\mathbf{H} and 1 = \\frac{\\hat{H}_{\\lambda}^{V_{\\mathbf{E}}} \\mathbf{H}}{\\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}} and \\frac{\\hat{H}_{\\lambda}^{- V_{\\mathbf{E}}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}}{\\mathbf{H}} = 1 and \\frac{\\hat{H}_{\\lambda}^{- V_{\\mathbf{E}}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})} \\frac{\\partial}{\\partial \\mathbf{H}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}}{\\mathbf{H}} = \\frac{\\partial}{\\partial \\mathbf{H}} \\hat{x}{(V_{\\mathbf{E}},\\hat{H}_{\\lambda},\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 1, "Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))))"], [["divide", 2, "Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Pow(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Integer(1))"], [["times", 3, "Derivative(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))), Derivative(Function('\\\\hat{x}')(Symbol('V_{\\\\mathbf{E}}', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})}, then obtain (\\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\mu{(L_{\\varepsilon})})^{L_{\\varepsilon}} = (- \\sin{(L_{\\varepsilon})})^{L_{\\varepsilon}}", "derivation": "\\mu{(L_{\\varepsilon})} = \\sin{(L_{\\varepsilon})} and \\frac{d}{d L_{\\varepsilon}} \\mu{(L_{\\varepsilon})} = \\frac{d}{d L_{\\varepsilon}} \\sin{(L_{\\varepsilon})} and \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\mu{(L_{\\varepsilon})} = \\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})} and (\\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\mu{(L_{\\varepsilon})})^{L_{\\varepsilon}} = (\\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\sin{(L_{\\varepsilon})})^{L_{\\varepsilon}} and (\\frac{d^{2}}{d L_{\\varepsilon}^{2}} \\mu{(L_{\\varepsilon})})^{L_{\\varepsilon}} = (- \\sin{(L_{\\varepsilon})})^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('L_{\\\\varepsilon}', commutative=True)), sin(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["differentiate", 1, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Function('\\\\mu')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))))"], [["power", 3, "Symbol('L_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mu')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Derivative(sin(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["evaluate_derivatives", 4], "Equality(Pow(Derivative(Function('\\\\mu')(Symbol('L_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(2))), Symbol('L_{\\\\varepsilon}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('L_{\\\\varepsilon}', commutative=True))), Symbol('L_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon{(F_{g},\\mathbf{D})} = \\frac{\\mathbf{D}}{F_{g}}, then obtain \\mathbf{D} \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\varepsilon{(F_{g},\\mathbf{D})} d\\mathbf{D} = \\mathbf{D} \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\frac{\\mathbf{D}}{F_{g}} d\\mathbf{D}", "derivation": "\\varepsilon{(F_{g},\\mathbf{D})} = \\frac{\\mathbf{D}}{F_{g}} and \\int \\varepsilon{(F_{g},\\mathbf{D})} d\\mathbf{D} = \\int \\frac{\\mathbf{D}}{F_{g}} d\\mathbf{D} and \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\varepsilon{(F_{g},\\mathbf{D})} d\\mathbf{D} = \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\frac{\\mathbf{D}}{F_{g}} d\\mathbf{D} and \\mathbf{D} \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\varepsilon{(F_{g},\\mathbf{D})} d\\mathbf{D} = \\mathbf{D} \\frac{\\partial}{\\partial \\mathbf{D}} \\int \\frac{\\mathbf{D}}{F_{g}} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Integral(Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["times", 3, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Integral(Function('\\\\varepsilon')(Symbol('F_g', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{D}', commutative=True), Derivative(Integral(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\phi_{2}{(T)} = \\log{(T)}, then obtain - T + (- T + \\log{(T)}) \\phi_{2}{(T)} = - T + (- T + \\log{(T)}) \\log{(T)}", "derivation": "\\phi_{2}{(T)} = \\log{(T)} and - T + \\phi_{2}{(T)} = - T + \\log{(T)} and (- T + \\phi_{2}{(T)}) \\phi_{2}{(T)} = (- T + \\phi_{2}{(T)}) \\log{(T)} and - T + (- T + \\phi_{2}{(T)}) \\phi_{2}{(T)} = - T + (- T + \\phi_{2}{(T)}) \\log{(T)} and - T + (- T + \\log{(T)}) \\phi_{2}{(T)} = - T + (- T + \\log{(T)}) \\log{(T)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('T', commutative=True)), log(Symbol('T', commutative=True)))"], [["minus", 1, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), log(Symbol('T', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True))), Function('\\\\phi_2')(Symbol('T', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True))))"], [["minus", 3, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True))), Function('\\\\phi_2')(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Function('\\\\phi_2')(Symbol('T', commutative=True))), log(Symbol('T', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), log(Symbol('T', commutative=True))), Function('\\\\phi_2')(Symbol('T', commutative=True)))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Mul(Add(Mul(Integer(-1), Symbol('T', commutative=True)), log(Symbol('T', commutative=True))), log(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{x})} = \\int \\cos{(v_{x})} dv_{x}, then derive 0 = \\frac{\\partial}{\\partial v_{x}} (\\mathbf{f} + \\sin{(v_{x})}) - \\frac{d}{d v_{x}} \\operatorname{F_{H}}{(v_{x})}, then obtain \\int 0 dv_{x} = n_{2} - \\operatorname{F_{H}}{(v_{x})} + \\sin{(v_{x})}", "derivation": "\\operatorname{F_{H}}{(v_{x})} = \\int \\cos{(v_{x})} dv_{x} and \\frac{d}{d v_{x}} \\operatorname{F_{H}}{(v_{x})} = \\frac{d}{d v_{x}} \\int \\cos{(v_{x})} dv_{x} and 0 = - \\frac{d}{d v_{x}} \\operatorname{F_{H}}{(v_{x})} + \\frac{d}{d v_{x}} \\int \\cos{(v_{x})} dv_{x} and 0 = \\frac{\\partial}{\\partial v_{x}} (\\mathbf{f} + \\sin{(v_{x})}) - \\frac{d}{d v_{x}} \\operatorname{F_{H}}{(v_{x})} and \\int 0 dv_{x} = \\int (\\frac{\\partial}{\\partial v_{x}} (\\mathbf{f} + \\sin{(v_{x})}) - \\frac{d}{d v_{x}} \\operatorname{F_{H}}{(v_{x})}) dv_{x} and \\int 0 dv_{x} = n_{2} - \\operatorname{F_{H}}{(v_{x})} + \\sin{(v_{x})}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_x', commutative=True)), Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('F_H')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Function('F_H')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Derivative(Integral(cos(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('F_H')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))))"], [["integrate", 4, "Symbol('v_x', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Derivative(Add(Symbol('\\\\mathbf{f}', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('F_H')(Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))))), Tuple(Symbol('v_x', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Integer(0), Tuple(Symbol('v_x', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), Function('F_H')(Symbol('v_x', commutative=True))), sin(Symbol('v_x', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(E_{x})} = \\log{(E_{x})}, then derive \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} = \\frac{1}{E_{x}}, then obtain \\int \\frac{d}{d E_{x}} \\log{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} dE_{x}", "derivation": "\\operatorname{C_{d}}{(E_{x})} = \\log{(E_{x})} and \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} = \\frac{d}{d E_{x}} \\log{(E_{x})} and \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} = \\frac{1}{E_{x}} and \\int \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} dE_{x} = \\int \\frac{1}{E_{x}} dE_{x} and \\int \\frac{d}{d E_{x}} \\log{(E_{x})} dE_{x} = \\int \\frac{1}{E_{x}} dE_{x} and \\int \\frac{d}{d E_{x}} \\log{(E_{x})} dE_{x} = \\int \\frac{d}{d E_{x}} \\operatorname{C_{d}}{(E_{x})} dE_{x}", "srepr_derivation": [["get_premise", "Equality(Function('C_d')(Symbol('E_x', commutative=True)), log(Symbol('E_x', commutative=True)))"], [["differentiate", 1, "Symbol('E_x', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_d')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Pow(Symbol('E_x', commutative=True), Integer(-1)))"], [["integrate", 3, "Symbol('E_x', commutative=True)"], "Equality(Integral(Derivative(Function('C_d')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(Symbol('E_x', commutative=True), Integer(-1)), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Pow(Symbol('E_x', commutative=True), Integer(-1)), Tuple(Symbol('E_x', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integral(Derivative(log(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))), Integral(Derivative(Function('C_d')(Symbol('E_x', commutative=True)), Tuple(Symbol('E_x', commutative=True), Integer(1))), Tuple(Symbol('E_x', commutative=True))))"]]}, {"prompt": "Given L{(x,\\mathbf{J}_P)} = \\mathbf{J}_P + x, then obtain ((\\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)})^{2})^{\\mathbf{J}_P} = (\\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + x) \\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)})^{\\mathbf{J}_P}", "derivation": "L{(x,\\mathbf{J}_P)} = \\mathbf{J}_P + x and \\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + x) and (\\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)})^{2} = \\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + x) \\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)} and ((\\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)})^{2})^{\\mathbf{J}_P} = (\\frac{\\partial}{\\partial \\mathbf{J}_P} (\\mathbf{J}_P + x) \\frac{\\partial}{\\partial \\mathbf{J}_P} L{(x,\\mathbf{J}_P)})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))))"], [["times", 2, "Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))"], "Equality(Pow(Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(2)), Mul(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Integer(2)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Mul(Derivative(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1))), Derivative(Function('L')(Symbol('x', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True), Integer(1)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\phi_{1}{(T)} = \\sin{(T)}, then derive \\sin{(V_{\\mathbf{B}} + \\phi_{1}{(T)})} = \\sin{(l + \\sin{(T)})}, then obtain \\sin{(V_{\\mathbf{B}} + \\phi_{1}{(T)})} = \\sin{(l + \\phi_{1}{(T)})}", "derivation": "\\phi_{1}{(T)} = \\sin{(T)} and \\frac{d}{d T} \\phi_{1}{(T)} = \\frac{d}{d T} \\sin{(T)} and \\int \\frac{d}{d T} \\phi_{1}{(T)} dT = \\int \\frac{d}{d T} \\sin{(T)} dT and \\sin{(\\int \\frac{d}{d T} \\phi_{1}{(T)} dT)} = \\sin{(\\int \\frac{d}{d T} \\sin{(T)} dT)} and \\sin{(V_{\\mathbf{B}} + \\phi_{1}{(T)})} = \\sin{(l + \\sin{(T)})} and \\sin{(V_{\\mathbf{B}} + \\phi_{1}{(T)})} = \\sin{(l + \\phi_{1}{(T)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('T', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\phi_1')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))), Integral(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True))))"], [["sin", 3], "Equality(sin(Integral(Derivative(Function('\\\\phi_1')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True)))), sin(Integral(Derivative(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\phi_1')(Symbol('T', commutative=True)))), sin(Add(Symbol('l', commutative=True), sin(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(sin(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Function('\\\\phi_1')(Symbol('T', commutative=True)))), sin(Add(Symbol('l', commutative=True), Function('\\\\phi_1')(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given V{(x^\\prime)} = \\log{(x^\\prime)}, then obtain \\frac{d^{2}}{d (x^\\prime)^{2}} (V{(x^\\prime)} + \\log{(x^\\prime)}) = \\frac{d^{2}}{d (x^\\prime)^{2}} 2 \\log{(x^\\prime)}", "derivation": "V{(x^\\prime)} = \\log{(x^\\prime)} and V{(x^\\prime)} + \\log{(x^\\prime)} = 2 \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} (V{(x^\\prime)} + \\log{(x^\\prime)}) = \\frac{d}{d x^\\prime} 2 \\log{(x^\\prime)} and \\frac{d^{2}}{d (x^\\prime)^{2}} (V{(x^\\prime)} + \\log{(x^\\prime)}) = \\frac{d^{2}}{d (x^\\prime)^{2}} 2 \\log{(x^\\prime)}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('V')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Function('V')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Add(Function('V')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))), Derivative(Mul(Integer(2), log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(2))))"]]}, {"prompt": "Given v{(\\pi)} = \\cos{(\\pi)}, then obtain \\iint ((v{(\\pi)} - \\cos{(\\pi)})^{\\pi} + \\cos{(\\pi)}) d\\pi d\\pi = \\iint (0^{\\pi} + \\cos{(\\pi)}) d\\pi d\\pi", "derivation": "v{(\\pi)} = \\cos{(\\pi)} and v{(\\pi)} - \\cos{(\\pi)} = 0 and (v{(\\pi)} - \\cos{(\\pi)})^{\\pi} = 0^{\\pi} and (v{(\\pi)} - \\cos{(\\pi)})^{\\pi} + \\cos{(\\pi)} = 0^{\\pi} + \\cos{(\\pi)} and \\int ((v{(\\pi)} - \\cos{(\\pi)})^{\\pi} + \\cos{(\\pi)}) d\\pi = \\int (0^{\\pi} + \\cos{(\\pi)}) d\\pi and \\iint ((v{(\\pi)} - \\cos{(\\pi)})^{\\pi} + \\cos{(\\pi)}) d\\pi d\\pi = \\iint (0^{\\pi} + \\cos{(\\pi)}) d\\pi d\\pi", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Function('v')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Add(Function('v')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), Pow(Integer(0), Symbol('\\\\pi', commutative=True)))"], [["add", 3, "cos(Symbol('\\\\pi', commutative=True))"], "Equality(Add(Pow(Add(Function('v')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Pow(Add(Function('v')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Pow(Integer(0), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Add(Pow(Add(Function('v')(Symbol('\\\\pi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\pi', commutative=True)))), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Add(Pow(Integer(0), Symbol('\\\\pi', commutative=True)), cos(Symbol('\\\\pi', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(x)} = \\sin{(x)}, then derive \\int \\mathbf{F}{(x)} dx = n_{2} - \\cos{(x)}, then obtain 1 = \\frac{d}{d n_{2}} \\int \\mathbf{F}{(x)} dx", "derivation": "\\mathbf{F}{(x)} = \\sin{(x)} and \\int \\mathbf{F}{(x)} dx = \\int \\sin{(x)} dx and \\int \\mathbf{F}{(x)} dx = n_{2} - \\cos{(x)} and \\int \\sin{(x)} dx = n_{2} - \\cos{(x)} and \\frac{d}{d n_{2}} \\int \\sin{(x)} dx = \\frac{\\partial}{\\partial n_{2}} (n_{2} - \\cos{(x)}) and \\frac{d}{d n_{2}} \\int \\sin{(x)} dx = \\frac{d}{d n_{2}} \\int \\mathbf{F}{(x)} dx and \\frac{\\partial}{\\partial n_{2}} (n_{2} - \\cos{(x)}) = \\frac{d}{d n_{2}} \\int \\mathbf{F}{(x)} dx and 1 = \\frac{d}{d n_{2}} \\int \\mathbf{F}{(x)} dx", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), sin(Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))))"], [["differentiate", 4, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Integral(sin(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Add(Symbol('n_2', commutative=True), Mul(Integer(-1), cos(Symbol('x', commutative=True)))), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Integer(1), Derivative(Integral(Function('\\\\mathbf{F}')(Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\varphi^*,p)} = p^{\\varphi^*}, then obtain \\varphi^* \\mathbf{r}{(\\varphi^*,p)} + \\int p^{\\varphi^*} dp - \\int \\mathbf{r}{(\\varphi^*,p)} dp = \\varphi^* p^{\\varphi^*}", "derivation": "\\mathbf{r}{(\\varphi^*,p)} = p^{\\varphi^*} and \\int \\mathbf{r}{(\\varphi^*,p)} dp = \\int p^{\\varphi^*} dp and \\varphi^* \\mathbf{r}{(\\varphi^*,p)} = \\varphi^* p^{\\varphi^*} and 0 = \\int p^{\\varphi^*} dp - \\int \\mathbf{r}{(\\varphi^*,p)} dp and \\varphi^* \\mathbf{r}{(\\varphi^*,p)} = \\varphi^* \\mathbf{r}{(\\varphi^*,p)} + \\int p^{\\varphi^*} dp - \\int \\mathbf{r}{(\\varphi^*,p)} dp and \\varphi^* \\mathbf{r}{(\\varphi^*,p)} + \\int p^{\\varphi^*} dp - \\int \\mathbf{r}{(\\varphi^*,p)} dp = \\varphi^* p^{\\varphi^*}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True)))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["times", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Integer(0), Add(Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"], [["add", 4, "Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True))), Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Add(Mul(Symbol('\\\\varphi^*', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True))), Integral(Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mathbf{r}')(Symbol('\\\\varphi^*', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))), Mul(Symbol('\\\\varphi^*', commutative=True), Pow(Symbol('p', commutative=True), Symbol('\\\\varphi^*', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(A,f^{*})} = \\frac{A}{f^{*}} and \\phi_{1}{(f^{*},A)} = \\frac{2 \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}}, then obtain \\frac{A}{f^{*}} + \\phi_{1}{(f^{*},A)} + \\operatorname{n_{1}}{(A,f^{*})} = \\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})} + \\frac{\\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}}", "derivation": "\\operatorname{n_{1}}{(A,f^{*})} = \\frac{A}{f^{*}} and 2 \\operatorname{n_{1}}{(A,f^{*})} = \\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})} and \\frac{2 \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}} = \\frac{\\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}} and \\phi_{1}{(f^{*},A)} = \\frac{2 \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}} and \\phi_{1}{(f^{*},A)} = \\frac{\\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}} and \\frac{A}{f^{*}} + \\phi_{1}{(f^{*},A)} + \\operatorname{n_{1}}{(A,f^{*})} = \\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})} + \\frac{\\frac{A}{f^{*}} + \\operatorname{n_{1}}{(A,f^{*})}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))))"], [["add", 1, "Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(2), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))))"], [["times", 2, "Pow(Symbol('f^*', commutative=True), Integer(-1))"], "Equality(Mul(Integer(2), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\phi_1')(Symbol('f^*', commutative=True), Symbol('A', commutative=True)), Mul(Integer(2), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('\\\\phi_1')(Symbol('f^*', commutative=True), Symbol('A', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True)))))"], [["add", 5, "Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True)))"], "Equality(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('\\\\phi_1')(Symbol('f^*', commutative=True), Symbol('A', commutative=True)), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True)), Mul(Pow(Symbol('f^*', commutative=True), Integer(-1)), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('f^*', commutative=True), Integer(-1))), Function('n_1')(Symbol('A', commutative=True), Symbol('f^*', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(m,W)} = \\frac{m}{W}, then obtain \\cos{(\\frac{\\frac{\\partial}{\\partial W} \\operatorname{m_{s}}{(m,W)}}{W} - \\frac{\\operatorname{m_{s}}{(m,W)}}{W^{2}})} = \\cos{(\\frac{2 m}{W^{3}})}", "derivation": "\\operatorname{m_{s}}{(m,W)} = \\frac{m}{W} and \\frac{\\operatorname{m_{s}}{(m,W)}}{W} = \\frac{m}{W^{2}} and \\frac{\\partial}{\\partial W} \\frac{\\operatorname{m_{s}}{(m,W)}}{W} = \\frac{\\partial}{\\partial W} \\frac{m}{W^{2}} and \\cos{(\\frac{\\partial}{\\partial W} \\frac{\\operatorname{m_{s}}{(m,W)}}{W})} = \\cos{(\\frac{\\partial}{\\partial W} \\frac{m}{W^{2}})} and \\cos{(\\frac{\\frac{\\partial}{\\partial W} \\operatorname{m_{s}}{(m,W)}}{W} - \\frac{\\operatorname{m_{s}}{(m,W)}}{W^{2}})} = \\cos{(\\frac{2 m}{W^{3}})}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["divide", 1, "Symbol('W', commutative=True)"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-2)), Symbol('m', commutative=True)))"], [["differentiate", 2, "Symbol('W', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-2)), Symbol('m', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["cos", 3], "Equality(cos(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1)))), cos(Derivative(Mul(Pow(Symbol('W', commutative=True), Integer(-2)), Symbol('m', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(cos(Add(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Derivative(Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('W', commutative=True), Integer(-2)), Function('m_s')(Symbol('m', commutative=True), Symbol('W', commutative=True))))), cos(Mul(Integer(2), Pow(Symbol('W', commutative=True), Integer(-3)), Symbol('m', commutative=True))))"]]}, {"prompt": "Given t{(n_{1},L)} = \\sin{(L n_{1})}, then derive \\frac{\\partial}{\\partial L} t{(n_{1},L)} = n_{1} \\cos{(L n_{1})}, then obtain n_{1} z{(\\sigma_p,G)} \\cos{(L n_{1})} = z{(\\sigma_p,G)} \\frac{\\partial}{\\partial L} \\sin{(L n_{1})}", "derivation": "t{(n_{1},L)} = \\sin{(L n_{1})} and \\frac{\\partial}{\\partial L} t{(n_{1},L)} = \\frac{\\partial}{\\partial L} \\sin{(L n_{1})} and \\frac{\\partial}{\\partial L} t{(n_{1},L)} = n_{1} \\cos{(L n_{1})} and z{(\\sigma_p,G)} \\frac{\\partial}{\\partial L} t{(n_{1},L)} = z{(\\sigma_p,G)} \\frac{\\partial}{\\partial L} \\sin{(L n_{1})} and n_{1} z{(\\sigma_p,G)} \\cos{(L n_{1})} = z{(\\sigma_p,G)} \\frac{\\partial}{\\partial L} \\sin{(L n_{1})}", "srepr_derivation": [["get_premise", "Equality(Function('t')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), sin(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))))"], [["differentiate", 1, "Symbol('L', commutative=True)"], "Equality(Derivative(Function('t')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('t')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Mul(Symbol('n_1', commutative=True), cos(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True)))))"], [["divide", 2, "Pow(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Integer(-1))"], "Equality(Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Derivative(Function('t')(Symbol('n_1', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Derivative(sin(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('n_1', commutative=True), Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), cos(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True)))), Mul(Function('z')(Symbol('\\\\sigma_p', commutative=True), Symbol('G', commutative=True)), Derivative(sin(Mul(Symbol('L', commutative=True), Symbol('n_1', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(a,G)} = G a, then obtain \\int (a + \\sin{(\\operatorname{t_{1}}{(a,G)})}) da = \\int (a + \\sin{(G a)}) da", "derivation": "\\operatorname{t_{1}}{(a,G)} = G a and \\sin{(\\operatorname{t_{1}}{(a,G)})} = \\sin{(G a)} and a + \\sin{(\\operatorname{t_{1}}{(a,G)})} = a + \\sin{(G a)} and \\int (a + \\sin{(\\operatorname{t_{1}}{(a,G)})}) da = \\int (a + \\sin{(G a)}) da", "srepr_derivation": [["get_premise", "Equality(Function('t_1')(Symbol('a', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Symbol('a', commutative=True)))"], [["sin", 1], "Equality(sin(Function('t_1')(Symbol('a', commutative=True), Symbol('G', commutative=True))), sin(Mul(Symbol('G', commutative=True), Symbol('a', commutative=True))))"], [["add", 2, "Symbol('a', commutative=True)"], "Equality(Add(Symbol('a', commutative=True), sin(Function('t_1')(Symbol('a', commutative=True), Symbol('G', commutative=True)))), Add(Symbol('a', commutative=True), sin(Mul(Symbol('G', commutative=True), Symbol('a', commutative=True)))))"], [["integrate", 3, "Symbol('a', commutative=True)"], "Equality(Integral(Add(Symbol('a', commutative=True), sin(Function('t_1')(Symbol('a', commutative=True), Symbol('G', commutative=True)))), Tuple(Symbol('a', commutative=True))), Integral(Add(Symbol('a', commutative=True), sin(Mul(Symbol('G', commutative=True), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_P{(E_{x},n_{2})} = - E_{x} + e^{n_{2}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = \\sin{(\\sin{(\\pi)})}, then obtain \\mathbf{J}_P{(E_{x},n_{2})} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} \\sin^{- \\pi}{(\\sin{(\\pi)})} = \\mathbf{J}_P{(E_{x},n_{2})} \\sin{(\\sin{(\\pi)})} \\sin^{- \\pi}{(\\sin{(\\pi)})}", "derivation": "\\mathbf{J}_P{(E_{x},n_{2})} = - E_{x} + e^{n_{2}} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} = \\sin{(\\sin{(\\pi)})} and (- E_{x} + e^{n_{2}}) \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} \\sin^{- \\pi}{(\\sin{(\\pi)})} = (- E_{x} + e^{n_{2}}) \\sin{(\\sin{(\\pi)})} \\sin^{- \\pi}{(\\sin{(\\pi)})} and \\mathbf{J}_P{(E_{x},n_{2})} \\operatorname{g^{\\prime}_{\\varepsilon}}{(\\pi)} \\sin^{- \\pi}{(\\sin{(\\pi)})} = \\mathbf{J}_P{(E_{x},n_{2})} \\sin{(\\sin{(\\pi)})} \\sin^{- \\pi}{(\\sin{(\\pi)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), exp(Symbol('n_2', commutative=True))))"], ["get_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True))))"], [["divide", 2, "Mul(Pow(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), exp(Symbol('n_2', commutative=True))), Integer(-1)), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), exp(Symbol('n_2', commutative=True))), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('E_x', commutative=True)), exp(Symbol('n_2', commutative=True))), sin(sin(Symbol('\\\\pi', commutative=True))), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('\\\\pi', commutative=True)), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))), Mul(Function('\\\\mathbf{J}_P')(Symbol('E_x', commutative=True), Symbol('n_2', commutative=True)), sin(sin(Symbol('\\\\pi', commutative=True))), Pow(sin(sin(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Symbol('\\\\pi', commutative=True)))))"]]}, {"prompt": "Given \\varepsilon{(h,\\mathbf{D})} = \\log{(\\mathbf{D} + h)}, then derive \\frac{\\partial}{\\partial \\mathbf{D}} \\varepsilon{(h,\\mathbf{D})} = \\frac{1}{\\mathbf{D} + h}, then obtain e^{\\frac{\\partial}{\\partial \\mathbf{D}} \\varepsilon{(h,\\mathbf{D})}} = e^{\\frac{1}{\\mathbf{D} + h}}", "derivation": "\\varepsilon{(h,\\mathbf{D})} = \\log{(\\mathbf{D} + h)} and h + \\varepsilon{(h,\\mathbf{D})} = h + \\log{(\\mathbf{D} + h)} and \\frac{\\partial}{\\partial \\mathbf{D}} (h + \\varepsilon{(h,\\mathbf{D})}) = \\frac{\\partial}{\\partial \\mathbf{D}} (h + \\log{(\\mathbf{D} + h)}) and \\frac{\\partial}{\\partial \\mathbf{D}} \\varepsilon{(h,\\mathbf{D})} = \\frac{1}{\\mathbf{D} + h} and e^{\\frac{\\partial}{\\partial \\mathbf{D}} \\varepsilon{(h,\\mathbf{D})}} = e^{\\frac{1}{\\mathbf{D} + h}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('h', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('h', commutative=True))))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('\\\\varepsilon')(Symbol('h', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Add(Symbol('h', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('h', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Add(Symbol('h', commutative=True), Function('\\\\varepsilon')(Symbol('h', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Add(Symbol('h', commutative=True), log(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('h', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('h', commutative=True)), Integer(-1)))"], [["exp", 4], "Equality(exp(Derivative(Function('\\\\varepsilon')(Symbol('h', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))), exp(Pow(Add(Symbol('\\\\mathbf{D}', commutative=True), Symbol('h', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given q{(\\hat{H},A_{1})} = A_{1} - \\hat{H}, then obtain A_{1} q^{2}{(\\hat{H},A_{1})} = A_{1} (A_{1} - \\hat{H}) q{(\\hat{H},A_{1})}", "derivation": "q{(\\hat{H},A_{1})} = A_{1} - \\hat{H} and A_{1} q{(\\hat{H},A_{1})} = A_{1} (A_{1} - \\hat{H}) and A_{1} q{(\\hat{H},A_{1})} = A_{1}^{2} - A_{1} \\hat{H} and A_{1} q^{2}{(\\hat{H},A_{1})} = (A_{1}^{2} - A_{1} \\hat{H}) q{(\\hat{H},A_{1})} and A_{1} (A_{1} - \\hat{H}) = A_{1}^{2} - A_{1} \\hat{H} and A_{1} q^{2}{(\\hat{H},A_{1})} = A_{1} (A_{1} - \\hat{H}) q{(\\hat{H},A_{1})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))))"], [["times", 1, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True))), Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))))"], [["expand", 2], "Equality(Mul(Symbol('A_1', commutative=True), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True))), Add(Pow(Symbol('A_1', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["times", 3, "Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Add(Pow(Symbol('A_1', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))), Add(Pow(Symbol('A_1', commutative=True), Integer(2)), Mul(Integer(-1), Symbol('A_1', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('A_1', commutative=True), Pow(Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True)), Integer(2))), Mul(Symbol('A_1', commutative=True), Add(Symbol('A_1', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True))), Function('q')(Symbol('\\\\hat{H}', commutative=True), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(g,\\nabla)} = \\nabla + g, then obtain \\int 0 dg = \\int (\\nabla + g - \\mathbf{g}{(g,\\nabla)}) dg", "derivation": "\\mathbf{g}{(g,\\nabla)} = \\nabla + g and - g + \\mathbf{g}{(g,\\nabla)} = \\nabla and 0 = \\nabla + g - \\mathbf{g}{(g,\\nabla)} and \\int 0 dg = \\int (\\nabla + g - \\mathbf{g}{(g,\\nabla)}) dg", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Symbol('\\\\nabla', commutative=True), Symbol('g', commutative=True)))"], [["minus", 1, "Symbol('g', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True))), Symbol('\\\\nabla', commutative=True))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('g', commutative=True)), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\nabla', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)))))"], [["integrate", 3, "Symbol('g', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('g', commutative=True))), Integral(Add(Symbol('\\\\nabla', commutative=True), Symbol('g', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('g', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('g', commutative=True))))"]]}, {"prompt": "Given \\dot{y}{(k,r_{0})} = \\cos^{r_{0}}{(k)}, then obtain ((- k + \\dot{y}{(k,r_{0})})^{k})^{r_{0}} = ((- k + \\cos^{r_{0}}{(k)})^{k})^{r_{0}}", "derivation": "\\dot{y}{(k,r_{0})} = \\cos^{r_{0}}{(k)} and - k + \\dot{y}{(k,r_{0})} = - k + \\cos^{r_{0}}{(k)} and (- k + \\dot{y}{(k,r_{0})})^{k} = (- k + \\cos^{r_{0}}{(k)})^{k} and ((- k + \\dot{y}{(k,r_{0})})^{k})^{r_{0}} = ((- k + \\cos^{r_{0}}{(k)})^{k})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('r_0', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Symbol('k', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('r_0', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('r_0', commutative=True))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('r_0', commutative=True))), Symbol('k', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('r_0', commutative=True))), Symbol('k', commutative=True)))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Function('\\\\dot{y}')(Symbol('k', commutative=True), Symbol('r_0', commutative=True))), Symbol('k', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('k', commutative=True)), Pow(cos(Symbol('k', commutative=True)), Symbol('r_0', commutative=True))), Symbol('k', commutative=True)), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\chi{(\\eta^{\\prime},t)} = \\eta^{\\prime} t, then obtain 1 = \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} t + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)}}{2 \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)}}", "derivation": "\\chi{(\\eta^{\\prime},t)} = \\eta^{\\prime} t and \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} t and 2 \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)} = \\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} t + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)} and 1 = \\frac{\\frac{\\partial}{\\partial \\eta^{\\prime}} \\eta^{\\prime} t + \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)}}{2 \\frac{\\partial}{\\partial \\eta^{\\prime}} \\chi{(\\eta^{\\prime},t)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Integer(2), Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))))"], "Equality(Integer(1), Mul(Rational(1, 2), Add(Derivative(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1)))), Pow(Derivative(Function('\\\\chi')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('t', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\chi{(\\varepsilon,f)} = \\varepsilon f, then obtain \\sigma_x + 2 \\int \\chi{(\\varepsilon,f)} \\frac{\\partial}{\\partial f} \\chi{(\\varepsilon,f)} d\\varepsilon = t_{1} + \\int \\varepsilon (f \\frac{\\partial}{\\partial f} \\chi{(\\varepsilon,f)} + \\chi{(\\varepsilon,f)}) d\\varepsilon", "derivation": "\\chi{(\\varepsilon,f)} = \\varepsilon f and \\chi^{2}{(\\varepsilon,f)} = \\varepsilon f \\chi{(\\varepsilon,f)} and \\frac{\\partial}{\\partial f} \\chi^{2}{(\\varepsilon,f)} = \\frac{\\partial}{\\partial f} \\varepsilon f \\chi{(\\varepsilon,f)} and \\int \\frac{\\partial}{\\partial f} \\chi^{2}{(\\varepsilon,f)} d\\varepsilon = \\int \\frac{\\partial}{\\partial f} \\varepsilon f \\chi{(\\varepsilon,f)} d\\varepsilon and \\sigma_x + 2 \\int \\chi{(\\varepsilon,f)} \\frac{\\partial}{\\partial f} \\chi{(\\varepsilon,f)} d\\varepsilon = t_{1} + \\int \\varepsilon (f \\frac{\\partial}{\\partial f} \\chi{(\\varepsilon,f)} + \\chi{(\\varepsilon,f)}) d\\varepsilon", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)))"], [["times", 1, "Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Integer(2)), Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True), Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True))))"], [["differentiate", 2, "Symbol('f', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Integer(2)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True), Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Integral(Derivative(Pow(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Integer(2)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True), Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('\\\\varepsilon', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(2), Integral(Mul(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Derivative(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varepsilon', commutative=True))))), Add(Symbol('t_1', commutative=True), Integral(Mul(Symbol('\\\\varepsilon', commutative=True), Add(Mul(Symbol('f', commutative=True), Derivative(Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Function('\\\\chi')(Symbol('\\\\varepsilon', commutative=True), Symbol('f', commutative=True)))), Tuple(Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(z)} = \\sin{(z)} and \\operatorname{v_{y}}{(z)} = \\int \\operatorname{C_{2}}{(z)} \\sin{(z)} dz, then obtain \\frac{d}{d z} (\\operatorname{v_{y}}{(z)} + \\sin{(z)}) = \\frac{\\partial}{\\partial z} (x + \\frac{z}{2} - \\frac{\\sin{(z)} \\cos{(z)}}{2} + \\sin{(z)})", "derivation": "\\operatorname{C_{2}}{(z)} = \\sin{(z)} and \\operatorname{C_{2}}{(z)} \\sin{(z)} = \\sin^{2}{(z)} and \\int \\operatorname{C_{2}}{(z)} \\sin{(z)} dz = \\int \\sin^{2}{(z)} dz and \\sin{(z)} + \\int \\operatorname{C_{2}}{(z)} \\sin{(z)} dz = \\sin{(z)} + \\int \\sin^{2}{(z)} dz and \\operatorname{v_{y}}{(z)} = \\int \\operatorname{C_{2}}{(z)} \\sin{(z)} dz and \\operatorname{v_{y}}{(z)} + \\sin{(z)} = \\sin{(z)} + \\int \\sin^{2}{(z)} dz and \\frac{d}{d z} (\\operatorname{v_{y}}{(z)} + \\sin{(z)}) = \\frac{d}{d z} (\\sin{(z)} + \\int \\sin^{2}{(z)} dz) and \\frac{d}{d z} (\\operatorname{v_{y}}{(z)} + \\sin{(z)}) = \\frac{\\partial}{\\partial z} (x + \\frac{z}{2} - \\frac{\\sin{(z)} \\cos{(z)}}{2} + \\sin{(z)})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True)))"], [["times", 1, "sin(Symbol('z', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Pow(sin(Symbol('z', commutative=True)), Integer(2)))"], [["integrate", 2, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Function('C_2')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))), Integral(Pow(sin(Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True))))"], [["add", 3, "sin(Symbol('z', commutative=True))"], "Equality(Add(sin(Symbol('z', commutative=True)), Integral(Mul(Function('C_2')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True)))), Add(sin(Symbol('z', commutative=True)), Integral(Pow(sin(Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True)))))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('z', commutative=True)), Integral(Mul(Function('C_2')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Function('v_y')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Add(sin(Symbol('z', commutative=True)), Integral(Pow(sin(Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True)))))"], [["differentiate", 6, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(sin(Symbol('z', commutative=True)), Integral(Pow(sin(Symbol('z', commutative=True)), Integer(2)), Tuple(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_integrals", 7], "Equality(Derivative(Add(Function('v_y')(Symbol('z', commutative=True)), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Symbol('x', commutative=True), Mul(Rational(1, 2), Symbol('z', commutative=True)), Mul(Integer(-1), Rational(1, 2), sin(Symbol('z', commutative=True)), cos(Symbol('z', commutative=True))), sin(Symbol('z', commutative=True))), Tuple(Symbol('z', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\sigma_x,A_{z})} = - \\sin{(A_{z} - \\sigma_x)} and \\operatorname{F_{N}}{(\\sigma_x,A_{z})} = A_{z} - \\sigma_x, then obtain e^{- \\sigma_x C{(\\sigma_x,A_{z})}} = e^{\\sigma_x \\sin{(\\operatorname{F_{N}}{(\\sigma_x,A_{z})})}}", "derivation": "C{(\\sigma_x,A_{z})} = - \\sin{(A_{z} - \\sigma_x)} and \\operatorname{F_{N}}{(\\sigma_x,A_{z})} = A_{z} - \\sigma_x and C{(\\sigma_x,A_{z})} = - \\sin{(\\operatorname{F_{N}}{(\\sigma_x,A_{z})})} and - \\sigma_x C{(\\sigma_x,A_{z})} = \\sigma_x \\sin{(\\operatorname{F_{N}}{(\\sigma_x,A_{z})})} and e^{- \\sigma_x C{(\\sigma_x,A_{z})}} = e^{\\sigma_x \\sin{(\\operatorname{F_{N}}{(\\sigma_x,A_{z})})}}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))))"], ["renaming_premise", "Equality(Function('F_N')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)), Add(Symbol('A_z', commutative=True), Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('C')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)), Mul(Integer(-1), sin(Function('F_N')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)))))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True))), Mul(Symbol('\\\\sigma_x', commutative=True), sin(Function('F_N')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)))))"], [["exp", 4], "Equality(exp(Mul(Integer(-1), Symbol('\\\\sigma_x', commutative=True), Function('C')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True)))), exp(Mul(Symbol('\\\\sigma_x', commutative=True), sin(Function('F_N')(Symbol('\\\\sigma_x', commutative=True), Symbol('A_z', commutative=True))))))"]]}, {"prompt": "Given \\theta{(t_{2},\\theta_1)} = \\theta_1 + t_{2}, then obtain \\frac{\\partial}{\\partial t_{2}} ((- \\theta_1 + \\theta{(t_{2},\\theta_1)})^{t_{2}} - 1) = \\frac{d}{d t_{2}} (t_{2}^{t_{2}} - 1)", "derivation": "\\theta{(t_{2},\\theta_1)} = \\theta_1 + t_{2} and - \\theta_1 + \\theta{(t_{2},\\theta_1)} = t_{2} and (- \\theta_1 + \\theta{(t_{2},\\theta_1)})^{t_{2}} = t_{2}^{t_{2}} and (- \\theta_1 + \\theta{(t_{2},\\theta_1)})^{t_{2}} - 1 = t_{2}^{t_{2}} - 1 and \\frac{\\partial}{\\partial t_{2}} ((- \\theta_1 + \\theta{(t_{2},\\theta_1)})^{t_{2}} - 1) = \\frac{d}{d t_{2}} (t_{2}^{t_{2}} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True)), Add(Symbol('\\\\theta_1', commutative=True), Symbol('t_2', commutative=True)))"], [["minus", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\theta')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('t_2', commutative=True))"], [["power", 2, "Symbol('t_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\theta')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('t_2', commutative=True)), Pow(Symbol('t_2', commutative=True), Symbol('t_2', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\theta')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('t_2', commutative=True)), Integer(-1)), Add(Pow(Symbol('t_2', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\theta_1', commutative=True)), Function('\\\\theta')(Symbol('t_2', commutative=True), Symbol('\\\\theta_1', commutative=True))), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('t_2', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(\\mathbf{p},f_{\\mathbf{p}})} = \\mathbf{p} f_{\\mathbf{p}} and \\mathbf{v}{(z^{*})} = \\log{(z^{*})}, then obtain \\frac{\\mathbf{v}{(z^{*})}}{\\mathbf{p} f_{\\mathbf{p}} + \\mathbf{p}} = \\frac{\\log{(z^{*})}}{\\mathbf{p} f_{\\mathbf{p}} + \\mathbf{p}}", "derivation": "\\omega{(\\mathbf{p},f_{\\mathbf{p}})} = \\mathbf{p} f_{\\mathbf{p}} and \\mathbf{p} + \\omega{(\\mathbf{p},f_{\\mathbf{p}})} = \\mathbf{p} f_{\\mathbf{p}} + \\mathbf{p} and \\mathbf{v}{(z^{*})} = \\log{(z^{*})} and \\frac{\\mathbf{v}{(z^{*})}}{\\mathbf{p} + \\omega{(\\mathbf{p},f_{\\mathbf{p}})}} = \\frac{\\log{(z^{*})}}{\\mathbf{p} + \\omega{(\\mathbf{p},f_{\\mathbf{p}})}} and \\frac{\\mathbf{v}{(z^{*})}}{\\mathbf{p} f_{\\mathbf{p}} + \\mathbf{p}} = \\frac{\\log{(z^{*})}}{\\mathbf{p} f_{\\mathbf{p}} + \\mathbf{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)))"], [["divide", 3, "Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True))), Mul(Pow(Add(Symbol('\\\\mathbf{p}', commutative=True), Function('\\\\omega')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Integer(-1)), log(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('z^*', commutative=True))), Mul(Pow(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('\\\\mathbf{p}', commutative=True)), Integer(-1)), log(Symbol('z^*', commutative=True))))"]]}, {"prompt": "Given \\chi{(A_{2},\\mathbf{H})} = A_{2} + \\mathbf{H} and A{(A_{2},\\mathbf{H})} = A_{2} + \\mathbf{H}, then obtain \\chi^{\\mathbf{H}}{(A_{2},\\mathbf{H})} = A^{\\mathbf{H}}{(A_{2},\\mathbf{H})}", "derivation": "\\chi{(A_{2},\\mathbf{H})} = A_{2} + \\mathbf{H} and A{(A_{2},\\mathbf{H})} = A_{2} + \\mathbf{H} and \\chi{(A_{2},\\mathbf{H})} = A{(A_{2},\\mathbf{H})} and \\chi^{\\mathbf{H}}{(A_{2},\\mathbf{H})} = A^{\\mathbf{H}}{(A_{2},\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('A')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('A')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], [["power", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)), Pow(Function('A')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True)))"]]}, {"prompt": "Given i{(\\mathbf{P},\\mathbf{g})} = \\log{(\\mathbf{P} + \\mathbf{g})}, then obtain (\\mathbf{P} + \\mathbf{g}) \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} i{(\\mathbf{P},\\mathbf{g})} = (\\mathbf{P} + \\mathbf{g}) \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} \\log{(\\mathbf{P} + \\mathbf{g})}", "derivation": "i{(\\mathbf{P},\\mathbf{g})} = \\log{(\\mathbf{P} + \\mathbf{g})} and \\frac{\\partial}{\\partial \\mathbf{g}} i{(\\mathbf{P},\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} \\log{(\\mathbf{P} + \\mathbf{g})} and \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} i{(\\mathbf{P},\\mathbf{g})} = \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} \\log{(\\mathbf{P} + \\mathbf{g})} and (\\mathbf{P} + \\mathbf{g}) \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} i{(\\mathbf{P},\\mathbf{g})} = (\\mathbf{P} + \\mathbf{g}) \\frac{\\partial^{2}}{\\partial \\mathbf{g}^{2}} \\log{(\\mathbf{P} + \\mathbf{g})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), log(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))), Derivative(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2))))"], [["times", 3, "Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(Function('i')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)))), Mul(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Derivative(log(Add(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(2)))))"]]}, {"prompt": "Given W{(\\Psi^{\\dagger},t)} = \\Psi^{\\dagger} t, then obtain \\int (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} + 1 = (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} + \\int (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger}", "derivation": "W{(\\Psi^{\\dagger},t)} = \\Psi^{\\dagger} t and 1 = \\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}} and 1 = (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} and \\int 1 d\\Psi^{\\dagger} = \\int (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} and \\int 1 d\\Psi^{\\dagger} + 1 = (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} + \\int 1 d\\Psi^{\\dagger} and \\int (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger} + 1 = (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} + \\int (\\frac{\\Psi^{\\dagger} t}{W{(\\Psi^{\\dagger},t)}})^{\\Psi^{\\dagger}} d\\Psi^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)))"], [["divide", 1, "Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True))"], "Equality(Integer(1), Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))))"], [["power", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integer(1), Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["add", 3, "Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], "Equality(Add(Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Add(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Integer(1), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Integral(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integer(1)), Add(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Integral(Pow(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True), Pow(Function('W')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('t', commutative=True)), Integer(-1))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given M{(\\mathbf{P})} = e^{\\cos{(\\mathbf{P})}}, then derive \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = - e^{\\cos{(\\mathbf{P})}} \\sin{(\\mathbf{P})}, then obtain - \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = - \\frac{d}{d \\mathbf{P}} e^{\\cos{(\\mathbf{P})}}", "derivation": "M{(\\mathbf{P})} = e^{\\cos{(\\mathbf{P})}} and \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = \\frac{d}{d \\mathbf{P}} e^{\\cos{(\\mathbf{P})}} and \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = - e^{\\cos{(\\mathbf{P})}} \\sin{(\\mathbf{P})} and - \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = e^{\\cos{(\\mathbf{P})}} \\sin{(\\mathbf{P})} and - \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = M{(\\mathbf{P})} \\sin{(\\mathbf{P})} and - \\frac{d}{d \\mathbf{P}} e^{\\cos{(\\mathbf{P})}} = e^{\\cos{(\\mathbf{P})}} \\sin{(\\mathbf{P})} and - \\frac{d}{d \\mathbf{P}} M{(\\mathbf{P})} = - \\frac{d}{d \\mathbf{P}} e^{\\cos{(\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), exp(cos(Symbol('\\\\mathbf{P}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Derivative(exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Mul(Integer(-1), Derivative(Function('M')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(exp(cos(Symbol('\\\\mathbf{P}', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\tilde{g}{(v_{t})} = \\cos{(v_{t})}, then obtain \\tilde{g}^{4}{(v_{t})} = \\tilde{g}{(v_{t})} \\cos^{3}{(v_{t})}", "derivation": "\\tilde{g}{(v_{t})} = \\cos{(v_{t})} and \\tilde{g}^{2}{(v_{t})} = \\tilde{g}{(v_{t})} \\cos{(v_{t})} and \\tilde{g}^{4}{(v_{t})} = \\tilde{g}^{2}{(v_{t})} \\cos^{2}{(v_{t})} and \\tilde{g}^{2}{(v_{t})} \\cos^{2}{(v_{t})} = \\tilde{g}{(v_{t})} \\cos^{3}{(v_{t})} and \\tilde{g}^{4}{(v_{t})} = \\tilde{g}{(v_{t})} \\cos^{3}{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["times", 1, "Function('\\\\tilde{g}')(Symbol('v_t', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Integer(2)), Mul(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Integer(2)), Pow(cos(Symbol('v_t', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Integer(2)), Pow(cos(Symbol('v_t', commutative=True)), Integer(2))), Mul(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Integer(3))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Integer(4)), Mul(Function('\\\\tilde{g}')(Symbol('v_t', commutative=True)), Pow(cos(Symbol('v_t', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} = \\cos{(\\cos{(\\hat{p}_0)})} and \\hat{x}{(\\hat{p}_0)} = \\cos{(\\cos{(\\hat{p}_0)})}, then obtain f \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} = f \\cos{(\\cos{(\\hat{p}_0)})} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)}", "derivation": "\\operatorname{E_{\\lambda}}{(\\hat{p}_0)} = \\cos{(\\cos{(\\hat{p}_0)})} and \\hat{x}{(\\hat{p}_0)} = \\cos{(\\cos{(\\hat{p}_0)})} and \\hat{x}{(\\hat{p}_0)} = \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} and f \\hat{x}{(\\hat{p}_0)} = f \\cos{(\\cos{(\\hat{p}_0)})} and f \\hat{x}{(\\hat{p}_0)} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} = f \\cos{(\\cos{(\\hat{p}_0)})} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} and f \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)} = f \\cos{(\\cos{(\\hat{p}_0)})} - \\operatorname{E_{\\lambda}}{(\\hat{p}_0)}", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)), cos(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), cos(cos(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True)), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)))"], [["times", 2, "Symbol('f', commutative=True)"], "Equality(Mul(Symbol('f', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Symbol('f', commutative=True), cos(cos(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["minus", 4, "Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Mul(Symbol('f', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Symbol('f', commutative=True), cos(cos(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Symbol('f', commutative=True), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)))), Add(Mul(Symbol('f', commutative=True), cos(cos(Symbol('\\\\hat{p}_0', commutative=True)))), Mul(Integer(-1), Function('E_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(S)} = \\log{(S)}, then obtain \\frac{d}{d S} - (\\operatorname{P_{g}}{(S)} - \\log{(S)}) \\log{(S)} = \\frac{d}{d S} 0", "derivation": "\\operatorname{P_{g}}{(S)} = \\log{(S)} and \\operatorname{P_{g}}{(S)} - \\log{(S)} = 0 and - (\\operatorname{P_{g}}{(S)} - \\log{(S)}) \\log{(S)} = 0 and \\frac{d}{d S} - (\\operatorname{P_{g}}{(S)} - \\log{(S)}) \\log{(S)} = \\frac{d}{d S} 0", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], [["minus", 1, "log(Symbol('S', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('S', commutative=True)), Mul(Integer(-1), log(Symbol('S', commutative=True)))), Integer(0))"], [["times", 2, "Mul(Integer(-1), log(Symbol('S', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Function('P_g')(Symbol('S', commutative=True)), Mul(Integer(-1), log(Symbol('S', commutative=True)))), log(Symbol('S', commutative=True))), Integer(0))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Add(Function('P_g')(Symbol('S', commutative=True)), Mul(Integer(-1), log(Symbol('S', commutative=True)))), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))))"]]}, {"prompt": "Given C{(\\hbar,\\phi_1)} = \\hbar \\phi_1, then derive \\hbar \\phi_1 \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\hbar^{2} \\phi_1, then obtain C{(\\hbar,\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\hbar C{(\\hbar,\\phi_1)}", "derivation": "C{(\\hbar,\\phi_1)} = \\hbar \\phi_1 and \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} \\hbar \\phi_1 and \\hbar \\phi_1 \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\hbar \\phi_1 \\frac{\\partial}{\\partial \\phi_1} \\hbar \\phi_1 and \\hbar \\phi_1 \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\hbar^{2} \\phi_1 and C{(\\hbar,\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} C{(\\hbar,\\phi_1)} = \\hbar C{(\\hbar,\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["times", 2, "Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True), Derivative(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True), Derivative(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True), Derivative(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(2)), Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Derivative(Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Symbol('\\\\hbar', commutative=True), Function('C')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(g,\\mathbf{J})} = \\sin^{\\mathbf{J}}{(g)} and \\phi{(g,\\mathbf{J})} = (\\sin^{\\mathbf{J}}{(g)})^{g}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} g \\phi{(g,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} g (\\sin^{\\mathbf{J}}{(g)})^{g}", "derivation": "\\operatorname{t_{1}}{(g,\\mathbf{J})} = \\sin^{\\mathbf{J}}{(g)} and \\operatorname{t_{1}}^{g}{(g,\\mathbf{J})} = (\\sin^{\\mathbf{J}}{(g)})^{g} and \\phi{(g,\\mathbf{J})} = (\\sin^{\\mathbf{J}}{(g)})^{g} and g \\phi{(g,\\mathbf{J})} = g (\\sin^{\\mathbf{J}}{(g)})^{g} and g \\phi{(g,\\mathbf{J})} = g \\operatorname{t_{1}}^{g}{(g,\\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} g \\phi{(g,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} g \\operatorname{t_{1}}^{g}{(g,\\mathbf{J})} and \\frac{\\partial}{\\partial \\mathbf{J}} g \\phi{(g,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} g (\\sin^{\\mathbf{J}}{(g)})^{g}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(sin(Symbol('g', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 1, "Symbol('g', commutative=True)"], "Equality(Pow(Function('t_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True)), Pow(Pow(sin(Symbol('g', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(sin(Symbol('g', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True)))"], [["times", 3, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('g', commutative=True), Pow(Pow(sin(Symbol('g', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('g', commutative=True), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Symbol('g', commutative=True), Pow(Function('t_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), Pow(Function('t_1')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\phi')(Symbol('g', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), Pow(Pow(sin(Symbol('g', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('g', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given W{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} and g{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}}, then obtain - a^{\\dagger} + g{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}} - a^{\\dagger}", "derivation": "W{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} and W^{a^{\\dagger}}{(a^{\\dagger},\\mu)} = (\\mu a^{\\dagger})^{a^{\\dagger}} and \\mu a^{\\dagger} W^{a^{\\dagger}}{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}} and g{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}} and g{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} W^{a^{\\dagger}}{(a^{\\dagger},\\mu)} and \\mu a^{\\dagger} W^{a^{\\dagger}}{(a^{\\dagger},\\mu)} - a^{\\dagger} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}} - a^{\\dagger} and - a^{\\dagger} + g{(a^{\\dagger},\\mu)} = \\mu a^{\\dagger} (\\mu a^{\\dagger})^{a^{\\dagger}} - a^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 2, "Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], ["renaming_premise", "Equality(Function('g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 3, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Function('W')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Function('g')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\mu', commutative=True))), Add(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Pow(Mul(Symbol('\\\\mu', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(\\Omega)} = e^{\\Omega}, then obtain \\operatorname{r_{0}}{(\\Omega)} + (e^{- \\Omega})^{\\Omega} - e^{- \\Omega} = (e^{- \\Omega})^{\\Omega} + e^{\\Omega} - e^{- \\Omega}", "derivation": "\\operatorname{r_{0}}{(\\Omega)} = e^{\\Omega} and \\operatorname{r_{0}}{(\\Omega)} e^{- \\Omega} = 1 and e^{- \\Omega} = \\frac{1}{\\operatorname{r_{0}}{(\\Omega)}} and \\operatorname{r_{0}}{(\\Omega)} - e^{- \\Omega} = e^{\\Omega} - e^{- \\Omega} and (e^{- \\Omega})^{\\Omega} = (\\frac{1}{\\operatorname{r_{0}}{(\\Omega)}})^{\\Omega} and (\\frac{1}{\\operatorname{r_{0}}{(\\Omega)}})^{\\Omega} + \\operatorname{r_{0}}{(\\Omega)} - e^{- \\Omega} = (\\frac{1}{\\operatorname{r_{0}}{(\\Omega)}})^{\\Omega} + e^{\\Omega} - e^{- \\Omega} and \\operatorname{r_{0}}{(\\Omega)} + (e^{- \\Omega})^{\\Omega} - e^{- \\Omega} = (e^{- \\Omega})^{\\Omega} + e^{\\Omega} - e^{- \\Omega}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Function('r_0')(Symbol('\\\\Omega', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))), Integer(1))"], [["divide", 2, "Function('r_0')(Symbol('\\\\Omega', commutative=True))"], "Equality(exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Pow(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Integer(-1)))"], [["minus", 1, "exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)))"], "Equality(Add(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Add(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Pow(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Symbol('\\\\Omega', commutative=True)))"], [["add", 4, "Pow(Pow(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Symbol('\\\\Omega', commutative=True))"], "Equality(Add(Pow(Pow(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Symbol('\\\\Omega', commutative=True)), Function('r_0')(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Add(Pow(Pow(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Integer(-1)), Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Function('r_0')(Symbol('\\\\Omega', commutative=True)), Pow(exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))), Add(Pow(exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given C{(\\hat{p})} = \\hat{p} and \\mathbf{J}_f{(\\hat{p})} = \\frac{\\hat{p}}{\\sin{(e^{\\hat{p}})}}, then obtain (\\frac{\\hat{p}}{\\sin{(e^{\\hat{p}})}})^{\\hat{p}} = \\mathbf{J}_f^{\\hat{p}}{(\\hat{p})}", "derivation": "C{(\\hat{p})} = \\hat{p} and \\frac{C{(\\hat{p})}}{\\sin{(e^{\\hat{p}})}} = \\frac{\\hat{p}}{\\sin{(e^{\\hat{p}})}} and \\mathbf{J}_f{(\\hat{p})} = \\frac{\\hat{p}}{\\sin{(e^{\\hat{p}})}} and \\frac{C{(\\hat{p})}}{\\sin{(e^{\\hat{p}})}} = \\mathbf{J}_f{(\\hat{p})} and (\\frac{C{(\\hat{p})}}{\\sin{(e^{\\hat{p}})}})^{\\hat{p}} = \\mathbf{J}_f^{\\hat{p}}{(\\hat{p})} and (\\frac{\\hat{p}}{\\sin{(e^{\\hat{p}})}})^{\\hat{p}} = \\mathbf{J}_f^{\\hat{p}}{(\\hat{p})}", "srepr_derivation": [["renaming_premise", "Equality(Function('C')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], [["divide", 1, "sin(exp(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Mul(Function('C')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True)), Mul(Symbol('\\\\hat{p}', commutative=True), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Function('C')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True)))"], [["power", 4, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Mul(Function('C')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Symbol('\\\\hat{p}', commutative=True), Pow(sin(exp(Symbol('\\\\hat{p}', commutative=True))), Integer(-1))), Symbol('\\\\hat{p}', commutative=True)), Pow(Function('\\\\mathbf{J}_f')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)} = (\\phi_1 - t_{2})^{F_{x}}, then obtain \\frac{t_{2} \\operatorname{c_{0}}^{t_{2}}{(t_{2},F_{x},\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} \\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)}}{\\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)}} = \\frac{F_{x} t_{2} ((\\phi_1 - t_{2})^{F_{x}})^{t_{2}}}{\\phi_1 - t_{2}}", "derivation": "\\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)} = (\\phi_1 - t_{2})^{F_{x}} and \\operatorname{c_{0}}^{t_{2}}{(t_{2},F_{x},\\phi_1)} = ((\\phi_1 - t_{2})^{F_{x}})^{t_{2}} and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{c_{0}}^{t_{2}}{(t_{2},F_{x},\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} ((\\phi_1 - t_{2})^{F_{x}})^{t_{2}} and \\frac{t_{2} \\operatorname{c_{0}}^{t_{2}}{(t_{2},F_{x},\\phi_1)} \\frac{\\partial}{\\partial \\phi_1} \\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)}}{\\operatorname{c_{0}}{(t_{2},F_{x},\\phi_1)}} = \\frac{F_{x} t_{2} ((\\phi_1 - t_{2})^{F_{x}})^{t_{2}}}{\\phi_1 - t_{2}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('F_x', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True)), Pow(Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('F_x', commutative=True)), Symbol('t_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('F_x', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('t_2', commutative=True), Pow(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Integer(-1)), Pow(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('t_2', commutative=True)), Derivative(Function('c_0')(Symbol('t_2', commutative=True), Symbol('F_x', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1)))), Mul(Symbol('F_x', commutative=True), Symbol('t_2', commutative=True), Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Integer(-1)), Pow(Pow(Add(Symbol('\\\\phi_1', commutative=True), Mul(Integer(-1), Symbol('t_2', commutative=True))), Symbol('F_x', commutative=True)), Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\phi_1)} = \\log{(\\phi_1)} and \\lambda{(\\phi_1)} = \\log{(\\phi_1)}, then obtain \\frac{d}{d \\phi_1} \\log{(\\phi_1)} = \\frac{d}{d \\phi_1} \\lambda{(\\phi_1)}", "derivation": "\\varphi^{*}{(\\phi_1)} = \\log{(\\phi_1)} and \\frac{d}{d \\phi_1} \\varphi^{*}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\log{(\\phi_1)} and \\lambda{(\\phi_1)} = \\log{(\\phi_1)} and \\frac{d}{d \\phi_1} \\varphi^{*}{(\\phi_1)} = \\frac{d}{d \\phi_1} \\lambda{(\\phi_1)} and \\frac{d}{d \\phi_1} \\log{(\\phi_1)} = \\frac{d}{d \\phi_1} \\lambda{(\\phi_1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True)), log(Symbol('\\\\phi_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('\\\\varphi^*')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(log(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Function('\\\\lambda')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(P_{g})} = \\sin{(P_{g})}, then derive (- \\cos{(P_{g})} + \\frac{d}{d P_{g}} T{(P_{g})}) e^{T{(P_{g})} - \\sin{(P_{g})}} = 0, then obtain - 2 \\cos{(P_{g})} + 2 \\frac{d}{d P_{g}} T{(P_{g})} = 0", "derivation": "T{(P_{g})} = \\sin{(P_{g})} and T{(P_{g})} - \\sin{(P_{g})} = 0 and 2 T{(P_{g})} - \\sin{(P_{g})} = T{(P_{g})} and e^{T{(P_{g})} - \\sin{(P_{g})}} = 1 and \\frac{d}{d P_{g}} e^{T{(P_{g})} - \\sin{(P_{g})}} = \\frac{d}{d P_{g}} 1 and (- \\cos{(P_{g})} + \\frac{d}{d P_{g}} T{(P_{g})}) e^{T{(P_{g})} - \\sin{(P_{g})}} = 0 and - \\cos{(P_{g})} + \\frac{d}{d P_{g}} T{(P_{g})} = 0 and - \\cos{(P_{g})} + \\frac{d}{d P_{g}} (2 T{(P_{g})} - \\sin{(P_{g})}) = 0 and - 2 \\cos{(P_{g})} + 2 \\frac{d}{d P_{g}} T{(P_{g})} = 0", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('P_g', commutative=True)), sin(Symbol('P_g', commutative=True)))"], [["minus", 1, "sin(Symbol('P_g', commutative=True))"], "Equality(Add(Function('T')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Integer(0))"], [["add", 2, "Function('T')(Symbol('P_g', commutative=True))"], "Equality(Add(Mul(Integer(2), Function('T')(Symbol('P_g', commutative=True))), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Function('T')(Symbol('P_g', commutative=True)))"], [["exp", 2], "Equality(exp(Add(Function('T')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True))))), Integer(1))"], [["differentiate", 4, "Symbol('P_g', commutative=True)"], "Equality(Derivative(exp(Add(Function('T')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True))))), Tuple(Symbol('P_g', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('P_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Mul(Add(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Derivative(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), exp(Add(Function('T')(Symbol('P_g', commutative=True)), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Derivative(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(0))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Add(Mul(Integer(-1), cos(Symbol('P_g', commutative=True))), Derivative(Add(Mul(Integer(2), Function('T')(Symbol('P_g', commutative=True))), Mul(Integer(-1), sin(Symbol('P_g', commutative=True)))), Tuple(Symbol('P_g', commutative=True), Integer(1)))), Integer(0))"], [["evaluate_derivatives", 8], "Equality(Add(Mul(Integer(-1), Integer(2), cos(Symbol('P_g', commutative=True))), Mul(Integer(2), Derivative(Function('T')(Symbol('P_g', commutative=True)), Tuple(Symbol('P_g', commutative=True), Integer(1))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = \\frac{\\hat{H}_l}{l} + f^{*} and u{(\\hat{H}_l,f^{*},l)} = l (\\frac{\\hat{H}_l}{l} + f^{*}), then obtain \\frac{\\partial}{\\partial \\hat{H}_l} l^{2} \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = \\frac{\\partial}{\\partial \\hat{H}_l} l u{(\\hat{H}_l,f^{*},l)}", "derivation": "\\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = \\frac{\\hat{H}_l}{l} + f^{*} and l \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = l (\\frac{\\hat{H}_l}{l} + f^{*}) and u{(\\hat{H}_l,f^{*},l)} = l (\\frac{\\hat{H}_l}{l} + f^{*}) and l \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = u{(\\hat{H}_l,f^{*},l)} and l^{2} \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = l u{(\\hat{H}_l,f^{*},l)} and \\frac{\\partial}{\\partial \\hat{H}_l} l^{2} \\operatorname{v_{t}}{(\\hat{H}_l,f^{*},l)} = \\frac{\\partial}{\\partial \\hat{H}_l} l u{(\\hat{H}_l,f^{*},l)}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f^*', commutative=True)))"], [["times", 1, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('l', commutative=True), Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True)), Mul(Symbol('l', commutative=True), Add(Mul(Symbol('\\\\hat{H}_l', commutative=True), Pow(Symbol('l', commutative=True), Integer(-1))), Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('l', commutative=True), Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))), Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True)))"], [["divide", 4, "Pow(Symbol('l', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))), Mul(Symbol('l', commutative=True), Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('l', commutative=True), Integer(2)), Function('v_t')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))), Derivative(Mul(Symbol('l', commutative=True), Function('u')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('f^*', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('\\\\hat{H}_l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(C,\\mathbf{F})} = \\frac{\\mathbf{F}}{C} and \\mathbf{A}{(\\hat{H},a^{\\dagger})} = \\hat{H} + a^{\\dagger}, then obtain \\mathbf{A}{(\\hat{H},a^{\\dagger})} - \\frac{\\mathbf{F}}{C} = \\hat{H} + a^{\\dagger} - \\frac{\\mathbf{F}}{C}", "derivation": "\\operatorname{y^{\\prime}}{(C,\\mathbf{F})} = \\frac{\\mathbf{F}}{C} and \\mathbf{A}{(\\hat{H},a^{\\dagger})} = \\hat{H} + a^{\\dagger} and \\mathbf{A}{(\\hat{H},a^{\\dagger})} - \\operatorname{y^{\\prime}}{(C,\\mathbf{F})} = \\hat{H} + a^{\\dagger} - \\operatorname{y^{\\prime}}{(C,\\mathbf{F})} and \\mathbf{A}{(\\hat{H},a^{\\dagger})} - \\frac{\\mathbf{F}}{C} = \\hat{H} + a^{\\dagger} - \\frac{\\mathbf{F}}{C}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], ["get_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)))"], [["minus", 2, "Function('y^{\\\\prime}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Function('y^{\\\\prime}')(Symbol('C', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))), Add(Symbol('\\\\hat{H}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(q,\\dot{y})} = \\dot{y} + q, then obtain (\\dot{y} (\\dot{y} + q))^{- \\dot{y}} = \\frac{(\\dot{y} (\\dot{y} + q))^{- \\dot{y}} \\frac{\\partial}{\\partial q} \\dot{y} (\\dot{y} + q)}{\\frac{\\partial}{\\partial q} \\dot{y} \\operatorname{E_{n}}{(q,\\dot{y})}}", "derivation": "\\operatorname{E_{n}}{(q,\\dot{y})} = \\dot{y} + q and \\dot{y} \\operatorname{E_{n}}{(q,\\dot{y})} = \\dot{y} (\\dot{y} + q) and \\frac{\\partial}{\\partial q} \\dot{y} \\operatorname{E_{n}}{(q,\\dot{y})} = \\frac{\\partial}{\\partial q} \\dot{y} (\\dot{y} + q) and 1 = \\frac{\\frac{\\partial}{\\partial q} \\dot{y} (\\dot{y} + q)}{\\frac{\\partial}{\\partial q} \\dot{y} \\operatorname{E_{n}}{(q,\\dot{y})}} and (\\dot{y} (\\dot{y} + q))^{- \\dot{y}} = \\frac{(\\dot{y} (\\dot{y} + q))^{- \\dot{y}} \\frac{\\partial}{\\partial q} \\dot{y} (\\dot{y} + q)}{\\frac{\\partial}{\\partial q} \\dot{y} \\operatorname{E_{n}}{(q,\\dot{y})}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True)))"], [["times", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Mul(Symbol('\\\\dot{y}', commutative=True), Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Pow(Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))))"], [["divide", 4, "Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Mul(Pow(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))), Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Pow(Derivative(Mul(Symbol('\\\\dot{y}', commutative=True), Function('E_n')(Symbol('q', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)}, then derive \\operatorname{C_{2}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)}, then obtain \\frac{d}{d \\mathbf{J}_f} (\\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)}) = \\frac{d}{d \\mathbf{J}_f} (\\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\cos{(\\mathbf{J}_f)})", "derivation": "\\operatorname{C_{2}}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)} and \\operatorname{C_{2}}{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)} = \\cos{(\\mathbf{J}_f)} and \\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)} = \\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\cos{(\\mathbf{J}_f)} and \\frac{d}{d \\mathbf{J}_f} (\\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\frac{d}{d \\mathbf{J}_f} \\sin{(\\mathbf{J}_f)}) = \\frac{d}{d \\mathbf{J}_f} (\\mathbf{J}_f \\operatorname{C_{2}}{(\\mathbf{J}_f)} + \\cos{(\\mathbf{J}_f)})", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True)), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), cos(Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["add", 3, "Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), Derivative(sin(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Function('C_2')(Symbol('\\\\mathbf{J}_f', commutative=True))), cos(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(A_{y},\\mathbf{J})} = \\frac{\\mathbf{J}}{A_{y}}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{r_{0}}{(A_{y},\\mathbf{J})} d\\mathbf{J} = \\frac{\\partial}{\\partial \\mathbf{J}} \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J}}{A_{y}} d\\mathbf{J}", "derivation": "\\operatorname{r_{0}}{(A_{y},\\mathbf{J})} = \\frac{\\mathbf{J}}{A_{y}} and \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{r_{0}}{(A_{y},\\mathbf{J})} = \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J}}{A_{y}} and \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{r_{0}}{(A_{y},\\mathbf{J})} d\\mathbf{J} = \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J}}{A_{y}} d\\mathbf{J} and \\frac{\\partial}{\\partial \\mathbf{J}} \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\operatorname{r_{0}}{(A_{y},\\mathbf{J})} d\\mathbf{J} = \\frac{\\partial}{\\partial \\mathbf{J}} \\int \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{\\mathbf{J}}{A_{y}} d\\mathbf{J}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Derivative(Function('r_0')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Integral(Derivative(Function('r_0')(Symbol('A_y', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(Derivative(Mul(Pow(Symbol('A_y', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{F}{(\\mathbf{J}_M,L)} = L \\mathbf{J}_M, then derive \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} \\frac{\\partial}{\\partial L} \\mathbf{F}{(\\mathbf{J}_M,L)}}{\\mathbf{F}{(\\mathbf{J}_M,L)}} = \\frac{\\mathbf{J}_M (L \\mathbf{J}_M)^{\\mathbf{J}_M}}{L}, then obtain \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} \\frac{\\partial}{\\partial L} \\mathbf{F}{(\\mathbf{J}_M,L)}}{\\mathbf{F}{(\\mathbf{J}_M,L)}} = \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)}}{L}", "derivation": "\\mathbf{F}{(\\mathbf{J}_M,L)} = L \\mathbf{J}_M and \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} = (L \\mathbf{J}_M)^{\\mathbf{J}_M} and \\frac{\\partial}{\\partial L} \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} = \\frac{\\partial}{\\partial L} (L \\mathbf{J}_M)^{\\mathbf{J}_M} and \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} \\frac{\\partial}{\\partial L} \\mathbf{F}{(\\mathbf{J}_M,L)}}{\\mathbf{F}{(\\mathbf{J}_M,L)}} = \\frac{\\mathbf{J}_M (L \\mathbf{J}_M)^{\\mathbf{J}_M}}{L} and \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)} \\frac{\\partial}{\\partial L} \\mathbf{F}{(\\mathbf{J}_M,L)}}{\\mathbf{F}{(\\mathbf{J}_M,L)}} = \\frac{\\mathbf{J}_M \\mathbf{F}^{\\mathbf{J}_M}{(\\mathbf{J}_M,L)}}{L}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Pow(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Mul(Symbol('L', commutative=True), Symbol('\\\\mathbf{J}_M', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True)), Derivative(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('\\\\mathbf{J}_M', commutative=True), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\mathbf{J}_M', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(\\sigma_p)} = \\cos{(\\cos{(\\sigma_p)})}, then obtain - \\dot{x}{(\\sigma_p)} + (\\int \\dot{x}{(\\sigma_p)} d\\sigma_p)^{\\sigma_p} = - \\dot{x}{(\\sigma_p)} + (\\int \\cos{(\\cos{(\\sigma_p)})} d\\sigma_p)^{\\sigma_p}", "derivation": "\\dot{x}{(\\sigma_p)} = \\cos{(\\cos{(\\sigma_p)})} and \\int \\dot{x}{(\\sigma_p)} d\\sigma_p = \\int \\cos{(\\cos{(\\sigma_p)})} d\\sigma_p and (\\int \\dot{x}{(\\sigma_p)} d\\sigma_p)^{\\sigma_p} = (\\int \\cos{(\\cos{(\\sigma_p)})} d\\sigma_p)^{\\sigma_p} and - \\dot{x}{(\\sigma_p)} + (\\int \\dot{x}{(\\sigma_p)} d\\sigma_p)^{\\sigma_p} = - \\dot{x}{(\\sigma_p)} + (\\int \\cos{(\\cos{(\\sigma_p)})} d\\sigma_p)^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True)), cos(cos(Symbol('\\\\sigma_p', commutative=True))))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(cos(cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)), Pow(Integral(cos(cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True)))"], [["minus", 3, "Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))), Add(Mul(Integer(-1), Function('\\\\dot{x}')(Symbol('\\\\sigma_p', commutative=True))), Pow(Integral(cos(cos(Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\sigma_p', commutative=True))), Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(F_{N})} = \\frac{d}{d F_{N}} \\sin{(F_{N})}, then derive \\mathbf{g}{(F_{N})} - \\cos{(F_{N})} = 0, then obtain \\int (\\mathbf{g}{(F_{N})} - \\cos{(F_{N})}) \\frac{d}{d F_{N}} \\sin{(F_{N})} dF_{N} = \\int 0 dF_{N}", "derivation": "\\mathbf{g}{(F_{N})} = \\frac{d}{d F_{N}} \\sin{(F_{N})} and \\mathbf{g}{(F_{N})} - \\frac{d}{d F_{N}} \\sin{(F_{N})} = 0 and \\mathbf{g}{(F_{N})} - \\cos{(F_{N})} = 0 and (\\mathbf{g}{(F_{N})} - \\cos{(F_{N})}) \\frac{d}{d F_{N}} \\sin{(F_{N})} = 0 and \\int (\\mathbf{g}{(F_{N})} - \\cos{(F_{N})}) \\frac{d}{d F_{N}} \\sin{(F_{N})} dF_{N} = \\int 0 dF_{N}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('F_N', commutative=True)), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["minus", 1, "Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Add(Function('\\\\mathbf{g}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Integer(0))"], [["times", 3, "Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))"], "Equality(Mul(Add(Function('\\\\mathbf{g}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Mul(Add(Function('\\\\mathbf{g}')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Tuple(Symbol('F_N', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\mathbb{I})} = e^{\\mathbb{I}}, then obtain - \\mathbb{I} + (\\eta^{\\prime}^{\\mathbb{I}}{(\\mathbb{I})})^{\\mathbb{I}} = - \\mathbb{I} + ((e^{\\mathbb{I}})^{\\mathbb{I}})^{\\mathbb{I}}", "derivation": "\\eta^{\\prime}{(\\mathbb{I})} = e^{\\mathbb{I}} and \\eta^{\\prime}^{\\mathbb{I}}{(\\mathbb{I})} = (e^{\\mathbb{I}})^{\\mathbb{I}} and (\\eta^{\\prime}^{\\mathbb{I}}{(\\mathbb{I})})^{\\mathbb{I}} = ((e^{\\mathbb{I}})^{\\mathbb{I}})^{\\mathbb{I}} and - \\mathbb{I} + (\\eta^{\\prime}^{\\mathbb{I}}{(\\mathbb{I})})^{\\mathbb{I}} = - \\mathbb{I} + ((e^{\\mathbb{I}})^{\\mathbb{I}})^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True)), exp(Symbol('\\\\mathbb{I}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 3, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Pow(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Pow(exp(Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\mathbb{I}', commutative=True))))"]]}, {"prompt": "Given a{(A_{y})} = e^{A_{y}}, then obtain \\int a^{2}{(A_{y})} e^{- 2 A_{y}} dA_{y} = \\int 1 dA_{y}", "derivation": "a{(A_{y})} = e^{A_{y}} and a{(A_{y})} (e^{A_{y}})^{- A_{y}} = e^{A_{y}} (e^{A_{y}})^{- A_{y}} and a{(A_{y})} e^{- A_{y}} = 1 and a{(A_{y})} e^{- 2 A_{y}} = e^{- A_{y}} and a^{2}{(A_{y})} e^{- 2 A_{y}} = 1 and \\int a^{2}{(A_{y})} e^{- 2 A_{y}} dA_{y} = \\int 1 dA_{y}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["divide", 1, "Pow(exp(Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True))"], "Equality(Mul(Function('a')(Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))), Mul(exp(Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True)))))"], [["divide", 2, "Mul(exp(Symbol('A_y', commutative=True)), Pow(exp(Symbol('A_y', commutative=True)), Mul(Integer(-1), Symbol('A_y', commutative=True))))"], "Equality(Mul(Function('a')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('A_y', commutative=True)))), Integer(1))"], [["times", 3, "exp(Mul(Integer(-1), Symbol('A_y', commutative=True)))"], "Equality(Mul(Function('a')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)))), exp(Mul(Integer(-1), Symbol('A_y', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('a')(Symbol('A_y', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)))), Integer(1))"], [["integrate", 5, "Symbol('A_y', commutative=True)"], "Equality(Integral(Mul(Pow(Function('a')(Symbol('A_y', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(2), Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True))), Integral(Integer(1), Tuple(Symbol('A_y', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbb{I})} = \\int \\log{(\\mathbb{I})} d\\mathbb{I}, then derive \\frac{\\mathbf{A}{(\\mathbb{I})}}{\\hat{p} + \\mathbb{I} \\log{(\\mathbb{I})} - \\mathbb{I}} = 1, then obtain \\frac{1}{\\hat{p} + \\mathbb{I} \\log{(\\mathbb{I})} - \\mathbb{I}} = \\frac{1}{\\int \\log{(\\mathbb{I})} d\\mathbb{I}}", "derivation": "\\mathbf{A}{(\\mathbb{I})} = \\int \\log{(\\mathbb{I})} d\\mathbb{I} and \\frac{\\mathbf{A}{(\\mathbb{I})}}{\\int \\log{(\\mathbb{I})} d\\mathbb{I}} = 1 and \\frac{\\mathbf{A}{(\\mathbb{I})}}{\\hat{p} + \\mathbb{I} \\log{(\\mathbb{I})} - \\mathbb{I}} = 1 and \\frac{\\mathbf{A}{(\\mathbb{I})}}{(\\hat{p} + \\mathbb{I} \\log{(\\mathbb{I})} - \\mathbb{I}) \\int \\log{(\\mathbb{I})} d\\mathbb{I}} = \\frac{1}{\\int \\log{(\\mathbb{I})} d\\mathbb{I}} and \\frac{1}{\\hat{p} + \\mathbb{I} \\log{(\\mathbb{I})} - \\mathbb{I}} = \\frac{1}{\\int \\log{(\\mathbb{I})} d\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbb{I}', commutative=True)), Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["divide", 1, "Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{A}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Integer(1))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\mathbb{I}', commutative=True))), Integer(1))"], [["times", 3, "Pow(Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1))), Pow(Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Add(Symbol('\\\\hat{p}', commutative=True), Mul(Symbol('\\\\mathbb{I}', commutative=True), log(Symbol('\\\\mathbb{I}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)), Pow(Integral(log(Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} = \\frac{e^{F_{x}}}{\\mathbf{p}}, then obtain \\hat{H} = \\int (- \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} + \\frac{e^{F_{x}}}{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p}", "derivation": "\\operatorname{A_{z}}{(\\mathbf{p},F_{x})} = \\frac{e^{F_{x}}}{\\mathbf{p}} and 0 = - \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} + \\frac{e^{F_{x}}}{\\mathbf{p}} and 0^{\\mathbf{p}} = (- \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} + \\frac{e^{F_{x}}}{\\mathbf{p}})^{\\mathbf{p}} and \\int 0^{\\mathbf{p}} d\\mathbf{p} = \\int (- \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} + \\frac{e^{F_{x}}}{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p} and \\hat{H} = \\int (- \\operatorname{A_{z}}{(\\mathbf{p},F_{x})} + \\frac{e^{F_{x}}}{\\mathbf{p}})^{\\mathbf{p}} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True))))"], [["minus", 1, "Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{p}', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Pow(Integer(0), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Symbol('\\\\hat{H}', commutative=True), Integral(Pow(Add(Mul(Integer(-1), Function('A_z')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('F_x', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{p}', commutative=True), Integer(-1)), exp(Symbol('F_x', commutative=True)))), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(g)} = \\log{(g)}, then derive g \\frac{d}{d g} \\mathbf{v}{(g)} + \\mathbf{v}{(g)} = \\log{(g)} + 1, then obtain g \\frac{d}{d g} \\mathbf{v}{(g)} + \\mathbf{v}{(g)} + \\frac{1}{\\frac{d}{d g} g \\mathbf{v}{(g)}} = \\log{(g)} + 1 + \\frac{1}{\\frac{d}{d g} g \\mathbf{v}{(g)}}", "derivation": "\\mathbf{v}{(g)} = \\log{(g)} and g \\mathbf{v}{(g)} = g \\log{(g)} and \\frac{d}{d g} g \\mathbf{v}{(g)} = \\frac{d}{d g} g \\log{(g)} and g \\frac{d}{d g} \\mathbf{v}{(g)} + \\mathbf{v}{(g)} = \\log{(g)} + 1 and g \\frac{d}{d g} \\mathbf{v}{(g)} + \\mathbf{v}{(g)} + \\frac{1}{\\frac{d}{d g} g \\mathbf{v}{(g)}} = \\log{(g)} + 1 + \\frac{1}{\\frac{d}{d g} g \\mathbf{v}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["times", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Mul(Symbol('g', commutative=True), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('g', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Add(log(Symbol('g', commutative=True)), Integer(1)))"], [["add", 4, "Pow(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))"], "Equality(Add(Mul(Symbol('g', commutative=True), Derivative(Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Function('\\\\mathbf{v}')(Symbol('g', commutative=True)), Pow(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))), Add(log(Symbol('g', commutative=True)), Integer(1), Pow(Derivative(Mul(Symbol('g', commutative=True), Function('\\\\mathbf{v}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(g_{\\varepsilon})} = \\cos{(\\log{(g_{\\varepsilon})})}, then obtain \\frac{\\operatorname{y^{\\prime}}^{g_{\\varepsilon}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})}}{n_{2}} = \\frac{\\log{(g_{\\varepsilon})} + \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})}}{n_{2}}", "derivation": "\\operatorname{y^{\\prime}}{(g_{\\varepsilon})} = \\cos{(\\log{(g_{\\varepsilon})})} and \\operatorname{y^{\\prime}}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})} and \\operatorname{y^{\\prime}}^{g_{\\varepsilon}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})} = \\log{(g_{\\varepsilon})} + \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})} and \\frac{\\operatorname{y^{\\prime}}^{g_{\\varepsilon}}{(g_{\\varepsilon})} + \\log{(g_{\\varepsilon})}}{n_{2}} = \\frac{\\log{(g_{\\varepsilon})} + \\cos^{g_{\\varepsilon}}{(\\log{(g_{\\varepsilon})})}}{n_{2}}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('y^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 2, "log(Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Pow(Function('y^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["divide", 3, "Symbol('n_2', commutative=True)"], "Equality(Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(Pow(Function('y^{\\\\prime}')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), log(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Pow(Symbol('n_2', commutative=True), Integer(-1)), Add(log(Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(cos(log(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(q)} = \\cos{(q)}, then obtain \\cos{(q)} \\cos{(\\frac{d}{d q} \\hat{H}_{\\lambda}{(q)})} - 1 = \\cos{(q)} \\cos{(\\sin{(q)})} - 1", "derivation": "\\hat{H}_{\\lambda}{(q)} = \\cos{(q)} and \\frac{d}{d q} \\hat{H}_{\\lambda}{(q)} = \\frac{d}{d q} \\cos{(q)} and \\cos{(\\frac{d}{d q} \\hat{H}_{\\lambda}{(q)})} = \\cos{(\\frac{d}{d q} \\cos{(q)})} and \\cos{(q)} \\cos{(\\frac{d}{d q} \\hat{H}_{\\lambda}{(q)})} = \\cos{(q)} \\cos{(\\frac{d}{d q} \\cos{(q)})} and \\cos{(q)} \\cos{(\\frac{d}{d q} \\hat{H}_{\\lambda}{(q)})} - 1 = \\cos{(q)} \\cos{(\\frac{d}{d q} \\cos{(q)})} - 1 and \\cos{(q)} \\cos{(\\frac{d}{d q} \\hat{H}_{\\lambda}{(q)})} - 1 = \\cos{(q)} \\cos{(\\sin{(q)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["differentiate", 1, "Symbol('q', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))), cos(Derivative(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1)))))"], [["times", 3, "cos(Symbol('q', commutative=True))"], "Equality(Mul(cos(Symbol('q', commutative=True)), cos(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Mul(cos(Symbol('q', commutative=True)), cos(Derivative(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["minus", 4, 1], "Equality(Add(Mul(cos(Symbol('q', commutative=True)), cos(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Integer(-1)), Add(Mul(cos(Symbol('q', commutative=True)), cos(Derivative(cos(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Integer(-1)))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(cos(Symbol('q', commutative=True)), cos(Derivative(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Integer(-1)), Add(Mul(cos(Symbol('q', commutative=True)), cos(sin(Symbol('q', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\varepsilon_0,\\chi)} = \\chi - \\varepsilon_0 and l{(\\varepsilon_0,\\chi)} = (\\chi - \\varepsilon_0)^{2}, then obtain \\int (- \\mathbf{J}_M \\nabla + l{(\\varepsilon_0,\\chi)}) d\\varepsilon_0 = \\int (- \\mathbf{J}_M \\nabla + (\\chi - \\varepsilon_0)^{2}) d\\varepsilon_0", "derivation": "\\mathbf{J}_M{(\\varepsilon_0,\\chi)} = \\chi - \\varepsilon_0 and (\\chi - \\varepsilon_0) \\mathbf{J}_M{(\\varepsilon_0,\\chi)} = (\\chi - \\varepsilon_0)^{2} and l{(\\varepsilon_0,\\chi)} = (\\chi - \\varepsilon_0)^{2} and (\\chi - \\varepsilon_0) \\mathbf{J}_M{(\\varepsilon_0,\\chi)} = l{(\\varepsilon_0,\\chi)} and - \\mathbf{J}_M \\nabla + (\\chi - \\varepsilon_0) \\mathbf{J}_M{(\\varepsilon_0,\\chi)} = - \\mathbf{J}_M \\nabla + (\\chi - \\varepsilon_0)^{2} and - \\mathbf{J}_M \\nabla + l{(\\varepsilon_0,\\chi)} = - \\mathbf{J}_M \\nabla + (\\chi - \\varepsilon_0)^{2} and \\int (- \\mathbf{J}_M \\nabla + l{(\\varepsilon_0,\\chi)}) d\\varepsilon_0 = \\int (- \\mathbf{J}_M \\nabla + (\\chi - \\varepsilon_0)^{2}) d\\varepsilon_0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True))), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2)))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True))), Function('l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Mul(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Function('\\\\mathbf{J}_M')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2))))"], [["integrate", 6, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Function('l')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\nabla', commutative=True)), Pow(Add(Symbol('\\\\chi', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon_0', commutative=True))), Integer(2))), Tuple(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given y{(m)} = \\frac{d}{d m} \\cos{(m)}, then derive \\log{(y{(m)})} = \\log{(- \\sin{(m)})}, then obtain y^{2}{(m)} \\log{(- \\sin{(m)})}^{m} = y^{2}{(m)} \\log{(\\frac{d}{d m} \\cos{(m)})}^{m}", "derivation": "y{(m)} = \\frac{d}{d m} \\cos{(m)} and \\log{(y{(m)})} = \\log{(\\frac{d}{d m} \\cos{(m)})} and \\log{(y{(m)})} = \\log{(- \\sin{(m)})} and \\log{(- \\sin{(m)})} = \\log{(\\frac{d}{d m} \\cos{(m)})} and \\log{(- \\sin{(m)})}^{m} = \\log{(\\frac{d}{d m} \\cos{(m)})}^{m} and y{(m)} \\log{(- \\sin{(m)})}^{m} = y{(m)} \\log{(\\frac{d}{d m} \\cos{(m)})}^{m} and y^{2}{(m)} \\log{(- \\sin{(m)})}^{m} = y^{2}{(m)} \\log{(\\frac{d}{d m} \\cos{(m)})}^{m}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('m', commutative=True)), Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('y')(Symbol('m', commutative=True))), log(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(log(Function('y')(Symbol('m', commutative=True))), log(Mul(Integer(-1), sin(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(log(Mul(Integer(-1), sin(Symbol('m', commutative=True)))), log(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(log(Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True)), Pow(log(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True)))"], [["times", 5, "Function('y')(Symbol('m', commutative=True))"], "Equality(Mul(Function('y')(Symbol('m', commutative=True)), Pow(log(Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True))), Mul(Function('y')(Symbol('m', commutative=True)), Pow(log(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True))))"], [["times", 6, "Function('y')(Symbol('m', commutative=True))"], "Equality(Mul(Pow(Function('y')(Symbol('m', commutative=True)), Integer(2)), Pow(log(Mul(Integer(-1), sin(Symbol('m', commutative=True)))), Symbol('m', commutative=True))), Mul(Pow(Function('y')(Symbol('m', commutative=True)), Integer(2)), Pow(log(Derivative(cos(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1)))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(t_{1},v)} = v \\sin{(t_{1})}, then derive \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} = v \\cos{(t_{1})}, then obtain - v + 2 \\frac{\\partial}{\\partial t_{1}} v \\sin{(t_{1})} = v \\cos{(t_{1})} - v + \\frac{\\partial}{\\partial t_{1}} v \\sin{(t_{1})}", "derivation": "\\operatorname{C_{1}}{(t_{1},v)} = v \\sin{(t_{1})} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} = \\frac{\\partial}{\\partial t_{1}} v \\sin{(t_{1})} and \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} = v \\cos{(t_{1})} and 2 \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} = v \\cos{(t_{1})} + \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} and - v + 2 \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} = v \\cos{(t_{1})} - v + \\frac{\\partial}{\\partial t_{1}} \\operatorname{C_{1}}{(t_{1},v)} and - v + 2 \\frac{\\partial}{\\partial t_{1}} v \\sin{(t_{1})} = v \\cos{(t_{1})} - v + \\frac{\\partial}{\\partial t_{1}} v \\sin{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), sin(Symbol('t_1', commutative=True))))"], [["differentiate", 1, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Mul(Symbol('v', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Mul(Symbol('v', commutative=True), cos(Symbol('t_1', commutative=True))))"], [["add", 3, "Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))), Add(Mul(Symbol('v', commutative=True), cos(Symbol('t_1', commutative=True))), Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["minus", 4, "Symbol('v', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(Mul(Symbol('v', commutative=True), cos(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True)), Derivative(Function('C_1')(Symbol('t_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('v', commutative=True)), Mul(Integer(2), Derivative(Mul(Symbol('v', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1))))), Add(Mul(Symbol('v', commutative=True), cos(Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('v', commutative=True)), Derivative(Mul(Symbol('v', commutative=True), sin(Symbol('t_1', commutative=True))), Tuple(Symbol('t_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(F_{N},v_{1},\\mathbf{v})} = \\mathbf{v} v_{1}^{F_{N}}, then obtain (\\mathbf{v} v_{1}^{F_{N}})^{v_{1}} \\operatorname{L_{\\varepsilon}}{(F_{N},v_{1},\\mathbf{v})} = \\mathbf{v} v_{1}^{F_{N}} (\\mathbf{v} v_{1}^{F_{N}})^{v_{1}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(F_{N},v_{1},\\mathbf{v})} = \\mathbf{v} v_{1}^{F_{N}} and \\operatorname{L_{\\varepsilon}}^{v_{1}}{(F_{N},v_{1},\\mathbf{v})} = (\\mathbf{v} v_{1}^{F_{N}})^{v_{1}} and \\operatorname{L_{\\varepsilon}}{(F_{N},v_{1},\\mathbf{v})} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(F_{N},v_{1},\\mathbf{v})} = \\mathbf{v} v_{1}^{F_{N}} \\operatorname{L_{\\varepsilon}}^{v_{1}}{(F_{N},v_{1},\\mathbf{v})} and (\\mathbf{v} v_{1}^{F_{N}})^{v_{1}} \\operatorname{L_{\\varepsilon}}{(F_{N},v_{1},\\mathbf{v})} = \\mathbf{v} v_{1}^{F_{N}} (\\mathbf{v} v_{1}^{F_{N}})^{v_{1}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True))))"], [["power", 1, "Symbol('v_1', commutative=True)"], "Equality(Pow(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('v_1', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True))), Symbol('v_1', commutative=True)))"], [["times", 1, "Pow(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('v_1', commutative=True))"], "Equality(Mul(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('v_1', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True)), Pow(Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Symbol('v_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True))), Symbol('v_1', commutative=True)), Function('L_{\\\\varepsilon}')(Symbol('F_N', commutative=True), Symbol('v_1', commutative=True), Symbol('\\\\mathbf{v}', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Symbol('v_1', commutative=True), Symbol('F_N', commutative=True))), Symbol('v_1', commutative=True))))"]]}, {"prompt": "Given E{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}, then obtain E{(\\hat{x})} \\int E^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\log{(\\sin{(\\hat{x})})} \\int E^{\\hat{x}}{(\\hat{x})} d\\hat{x}", "derivation": "E{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})} and E^{\\hat{x}}{(\\hat{x})} = \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} and \\int E^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\int \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} d\\hat{x} and E{(\\hat{x})} \\int \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} d\\hat{x} = \\log{(\\sin{(\\hat{x})})} \\int \\log{(\\sin{(\\hat{x})})}^{\\hat{x}} d\\hat{x} and E{(\\hat{x})} \\int E^{\\hat{x}}{(\\hat{x})} d\\hat{x} = \\log{(\\sin{(\\hat{x})})} \\int E^{\\hat{x}}{(\\hat{x})} d\\hat{x}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{x}', commutative=True)), log(sin(Symbol('\\\\hat{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Pow(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Pow(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 1, "Integral(Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Mul(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Integral(Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(log(sin(Symbol('\\\\hat{x}', commutative=True))), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Integral(Pow(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))), Mul(log(sin(Symbol('\\\\hat{x}', commutative=True))), Integral(Pow(Function('E')(Symbol('\\\\hat{x}', commutative=True)), Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(C_{d},p)} = C_{d} + p, then obtain \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial C_{d}} (p \\hat{H}_{\\lambda}{(C_{d},p)})^{C_{d}})^{p} = \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial C_{d}} (p (C_{d} + p))^{C_{d}})^{p}", "derivation": "\\hat{H}_{\\lambda}{(C_{d},p)} = C_{d} + p and p \\hat{H}_{\\lambda}{(C_{d},p)} = p (C_{d} + p) and (p \\hat{H}_{\\lambda}{(C_{d},p)})^{C_{d}} = (p (C_{d} + p))^{C_{d}} and \\frac{\\partial}{\\partial C_{d}} (p \\hat{H}_{\\lambda}{(C_{d},p)})^{C_{d}} = \\frac{\\partial}{\\partial C_{d}} (p (C_{d} + p))^{C_{d}} and (\\frac{\\partial}{\\partial C_{d}} (p \\hat{H}_{\\lambda}{(C_{d},p)})^{C_{d}})^{p} = (\\frac{\\partial}{\\partial C_{d}} (p (C_{d} + p))^{C_{d}})^{p} and \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial C_{d}} (p \\hat{H}_{\\lambda}{(C_{d},p)})^{C_{d}})^{p} = \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial C_{d}} (p (C_{d} + p))^{C_{d}})^{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('p', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True))))"], [["power", 2, "Symbol('C_d', commutative=True)"], "Equality(Pow(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Pow(Mul(Symbol('p', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)))"], [["differentiate", 3, "Symbol('C_d', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('p', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))))"], [["power", 4, "Symbol('p', commutative=True)"], "Equality(Pow(Derivative(Pow(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('p', commutative=True)), Pow(Derivative(Pow(Mul(Symbol('p', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('p', commutative=True)))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Derivative(Pow(Mul(Symbol('p', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Derivative(Pow(Mul(Symbol('p', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('p', commutative=True))), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True), Integer(1))), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(q)} = e^{q}, then derive \\int (- q + \\omega{(q)}) dq = \\mathbf{H} - \\frac{q^{2}}{2} + e^{q}, then obtain (\\mathbf{H} - \\frac{q^{2}}{2} + e^{q})^{2} \\omega^{2}{(q)} = \\omega^{2}{(q)} (\\int (- q + e^{q}) dq)^{2}", "derivation": "\\omega{(q)} = e^{q} and - q + \\omega{(q)} = - q + e^{q} and \\int (- q + \\omega{(q)}) dq = \\int (- q + e^{q}) dq and e^{q} \\int (- q + \\omega{(q)}) dq = e^{q} \\int (- q + e^{q}) dq and \\int (- q + \\omega{(q)}) dq = \\mathbf{H} - \\frac{q^{2}}{2} + e^{q} and \\omega{(q)} \\int (- q + \\omega{(q)}) dq = \\omega{(q)} \\int (- q + e^{q}) dq and (\\mathbf{H} - \\frac{q^{2}}{2} + e^{q}) \\omega{(q)} = \\omega{(q)} \\int (- q + e^{q}) dq and (\\mathbf{H} - \\frac{q^{2}}{2} + e^{q})^{2} \\omega^{2}{(q)} = \\omega^{2}{(q)} (\\int (- q + e^{q}) dq)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["minus", 1, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))))"], [["integrate", 2, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))))"], [["times", 3, "exp(Symbol('q', commutative=True))"], "Equality(Mul(exp(Symbol('q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Mul(exp(Symbol('q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), exp(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('\\\\omega')(Symbol('q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Function('\\\\omega')(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))), Mul(Function('\\\\omega')(Symbol('q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), exp(Symbol('q', commutative=True))), Function('\\\\omega')(Symbol('q', commutative=True))), Mul(Function('\\\\omega')(Symbol('q', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True)))))"], [["power", 7, 2], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('q', commutative=True), Integer(2))), exp(Symbol('q', commutative=True))), Integer(2)), Pow(Function('\\\\omega')(Symbol('q', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\omega')(Symbol('q', commutative=True)), Integer(2)), Pow(Integral(Add(Mul(Integer(-1), Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\phi)} = \\log{(\\phi)}, then derive \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\phi \\log{(\\phi)} - \\phi + \\rho_f, then derive \\mathbf{f} + \\phi \\log{(\\phi)} - \\phi = \\phi \\log{(\\phi)} - \\phi + \\rho_f, then obtain - \\phi \\log{(\\phi)} + \\phi - a^{\\dagger} + \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\mathbf{f} - a^{\\dagger}", "derivation": "\\operatorname{a^{\\dagger}}{(\\phi)} = \\log{(\\phi)} and \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\int \\log{(\\phi)} d\\phi and \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\phi \\log{(\\phi)} - \\phi + \\rho_f and \\int \\log{(\\phi)} d\\phi = \\phi \\log{(\\phi)} - \\phi + \\rho_f and \\mathbf{f} + \\phi \\log{(\\phi)} - \\phi = \\phi \\log{(\\phi)} - \\phi + \\rho_f and \\mathbf{f} + \\phi \\log{(\\phi)} - \\phi = \\int \\log{(\\phi)} d\\phi and \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\mathbf{f} + \\phi \\log{(\\phi)} - \\phi and \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi - \\int \\log{(\\phi)} d\\phi = \\mathbf{f} + \\phi \\log{(\\phi)} - \\phi - \\int \\log{(\\phi)} d\\phi and - \\phi \\log{(\\phi)} + \\phi - a^{\\dagger} + \\int \\operatorname{a^{\\dagger}}{(\\phi)} d\\phi = \\mathbf{f} - a^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Add(Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True))))"], [["minus", 7, "Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))"], "Equality(Add(Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Symbol('\\\\phi', commutative=True)), Mul(Integer(-1), Integral(log(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True))))))"], [["evaluate_integrals", 8], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi', commutative=True), log(Symbol('\\\\phi', commutative=True))), Symbol('\\\\phi', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('a^{\\\\dagger}')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True)))), Add(Symbol('\\\\mathbf{f}', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)} = \\cos{(U \\hat{H})}, then obtain - \\cos^{\\hat{H}}{(U \\hat{H})} + \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)})} = - \\cos^{\\hat{H}}{(U \\hat{H})} + \\cos{(\\cos{(U \\hat{H})})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)} = \\cos{(U \\hat{H})} and \\operatorname{f_{\\mathbf{p}}}^{\\hat{H}}{(\\hat{H},U)} = \\cos^{\\hat{H}}{(U \\hat{H})} and \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)})} = \\cos{(\\cos{(U \\hat{H})})} and - \\operatorname{f_{\\mathbf{p}}}^{\\hat{H}}{(\\hat{H},U)} + \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)})} = - \\operatorname{f_{\\mathbf{p}}}^{\\hat{H}}{(\\hat{H},U)} + \\cos{(\\cos{(U \\hat{H})})} and - \\cos^{\\hat{H}}{(U \\hat{H})} + \\cos{(\\operatorname{f_{\\mathbf{p}}}{(\\hat{H},U)})} = - \\cos^{\\hat{H}}{(U \\hat{H})} + \\cos{(\\cos{(U \\hat{H})})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)), cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["cos", 1], "Equality(cos(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True))), cos(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True)))))"], [["minus", 3, "Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), cos(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\hat{H}', commutative=True))), cos(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), cos(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Pow(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True))), cos(cos(Mul(Symbol('U', commutative=True), Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(F_{N},\\Omega)} = - \\Omega + \\cos{(F_{N})}, then obtain \\int \\frac{\\partial}{\\partial F_{N}} (\\operatorname{n_{1}}{(F_{N},\\Omega)} - 1) d\\Omega = \\int \\frac{\\partial}{\\partial F_{N}} (- \\Omega + \\cos{(F_{N})} - 1) d\\Omega", "derivation": "\\operatorname{n_{1}}{(F_{N},\\Omega)} = - \\Omega + \\cos{(F_{N})} and \\operatorname{n_{1}}{(F_{N},\\Omega)} - 1 = - \\Omega + \\cos{(F_{N})} - 1 and \\frac{\\partial}{\\partial F_{N}} (\\operatorname{n_{1}}{(F_{N},\\Omega)} - 1) = \\frac{\\partial}{\\partial F_{N}} (- \\Omega + \\cos{(F_{N})} - 1) and \\int \\frac{\\partial}{\\partial F_{N}} (\\operatorname{n_{1}}{(F_{N},\\Omega)} - 1) d\\Omega = \\int \\frac{\\partial}{\\partial F_{N}} (- \\Omega + \\cos{(F_{N})} - 1) d\\Omega", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), cos(Symbol('F_N', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('n_1')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), cos(Symbol('F_N', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Add(Function('n_1')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), cos(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Derivative(Add(Function('n_1')(Symbol('F_N', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True)), cos(Symbol('F_N', commutative=True)), Integer(-1)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{B}{(G)} = \\frac{d}{d G} \\sin{(G)}, then derive \\mathbf{B}{(G)} = \\cos{(G)}, then obtain \\frac{d}{d G} \\mathbf{B}{(G)} = \\frac{d}{d G} \\cos{(G)}", "derivation": "\\mathbf{B}{(G)} = \\frac{d}{d G} \\sin{(G)} and \\mathbf{B}{(G)} = \\cos{(G)} and \\cos{(G)} = \\frac{d}{d G} \\sin{(G)} and \\frac{d}{d G} \\cos{(G)} = \\frac{d^{2}}{d G^{2}} \\sin{(G)} and \\frac{d}{d G} \\mathbf{B}{(G)} = \\frac{d^{2}}{d G^{2}} \\sin{(G)} and \\frac{d}{d G} \\mathbf{B}{(G)} = \\frac{d}{d G} \\cos{(G)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('G', commutative=True)), Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\mathbf{B}')(Symbol('G', commutative=True)), cos(Symbol('G', commutative=True)))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(cos(Symbol('G', commutative=True)), Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('G', commutative=True)"], "Equality(Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(sin(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('\\\\mathbf{B}')(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(cos(Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta_{2}{(F_{x})} = \\sin{(\\log{(F_{x})})}, then derive \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} = \\frac{\\cos{(\\log{(F_{x})})}}{F_{x}}, then obtain (\\int \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} dF_{x})^{F_{x}} = (\\int \\frac{\\cos{(\\log{(F_{x})})}}{F_{x}} dF_{x})^{F_{x}}", "derivation": "\\theta_{2}{(F_{x})} = \\sin{(\\log{(F_{x})})} and \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} = \\frac{d}{d F_{x}} \\sin{(\\log{(F_{x})})} and \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} = \\frac{\\cos{(\\log{(F_{x})})}}{F_{x}} and \\int \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} dF_{x} = \\int \\frac{\\cos{(\\log{(F_{x})})}}{F_{x}} dF_{x} and (\\int \\frac{d}{d F_{x}} \\theta_{2}{(F_{x})} dF_{x})^{F_{x}} = (\\int \\frac{\\cos{(\\log{(F_{x})})}}{F_{x}} dF_{x})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('F_x', commutative=True)), sin(log(Symbol('F_x', commutative=True))))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\theta_2')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(sin(log(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_2')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(log(Symbol('F_x', commutative=True)))))"], [["integrate", 3, "Symbol('F_x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\theta_2')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Tuple(Symbol('F_x', commutative=True))), Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(log(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))))"], [["power", 4, "Symbol('F_x', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\theta_2')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Integral(Mul(Pow(Symbol('F_x', commutative=True), Integer(-1)), cos(log(Symbol('F_x', commutative=True)))), Tuple(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)))"]]}, {"prompt": "Given a{(m,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\theta^{m}, then derive m (- \\theta + a{(m,\\theta)}) = m (- \\theta + \\frac{\\theta^{m} m}{\\theta}), then obtain (m (- \\theta + \\frac{\\partial}{\\partial \\theta} \\theta^{m}))^{\\theta} = (m (- \\theta + \\frac{\\theta^{m} m}{\\theta}))^{\\theta}", "derivation": "a{(m,\\theta)} = \\frac{\\partial}{\\partial \\theta} \\theta^{m} and - \\theta + a{(m,\\theta)} = - \\theta + \\frac{\\partial}{\\partial \\theta} \\theta^{m} and m (- \\theta + a{(m,\\theta)}) = m (- \\theta + \\frac{\\partial}{\\partial \\theta} \\theta^{m}) and m (- \\theta + a{(m,\\theta)}) = m (- \\theta + \\frac{\\theta^{m} m}{\\theta}) and (m (- \\theta + a{(m,\\theta)}))^{\\theta} = (m (- \\theta + \\frac{\\theta^{m} m}{\\theta}))^{\\theta} and (m (- \\theta + \\frac{\\partial}{\\partial \\theta} \\theta^{m}))^{\\theta} = (m (- \\theta + \\frac{\\theta^{m} m}{\\theta}))^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('m', commutative=True), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["minus", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('m', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))))"], [["times", 2, "Symbol('m', commutative=True)"], "Equality(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('m', commutative=True), Symbol('\\\\theta', commutative=True)))), Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('m', commutative=True), Symbol('\\\\theta', commutative=True)))), Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))))"], [["power", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Function('a')(Symbol('m', commutative=True), Symbol('\\\\theta', commutative=True)))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Derivative(Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Symbol('m', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} = (\\mathbf{g}^{L})^{\\varphi} and \\operatorname{f_{E}}{(\\mu_0)} = \\cos{(\\mu_0)}, then obtain \\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} + \\frac{\\operatorname{f_{E}}{(\\mu_0)}}{\\mu_0} = \\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} + \\frac{\\cos{(\\mu_0)}}{\\mu_0}", "derivation": "\\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} = (\\mathbf{g}^{L})^{\\varphi} and \\operatorname{f_{E}}{(\\mu_0)} = \\cos{(\\mu_0)} and \\frac{\\operatorname{f_{E}}{(\\mu_0)}}{\\mu_0} = \\frac{\\cos{(\\mu_0)}}{\\mu_0} and (\\mathbf{g}^{L})^{\\varphi} + \\frac{\\operatorname{f_{E}}{(\\mu_0)}}{\\mu_0} = (\\mathbf{g}^{L})^{\\varphi} + \\frac{\\cos{(\\mu_0)}}{\\mu_0} and \\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} + \\frac{\\operatorname{f_{E}}{(\\mu_0)}}{\\mu_0} = \\operatorname{v_{2}}{(\\varphi,\\mathbf{g},L)} + \\frac{\\cos{(\\mu_0)}}{\\mu_0}", "srepr_derivation": [["get_premise", "Equality(Function('v_2')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], ["get_premise", "Equality(Function('f_E')(Symbol('\\\\mu_0', commutative=True)), cos(Symbol('\\\\mu_0', commutative=True)))"], [["divide", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True))))"], [["add", 3, "Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True))"], "Equality(Add(Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\mu_0', commutative=True)))), Add(Pow(Pow(Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Symbol('\\\\varphi', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('v_2')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Function('f_E')(Symbol('\\\\mu_0', commutative=True)))), Add(Function('v_2')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mathbf{g}', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), cos(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\nabla)} = \\sin{(\\nabla)}, then derive \\int \\operatorname{f_{E}}{(\\nabla)} d\\nabla = \\Omega - \\cos{(\\nabla)}, then obtain \\sin{(\\int \\operatorname{f_{E}}{(\\nabla)} d\\nabla)} = \\sin{(\\Omega - \\cos{(\\nabla)})}", "derivation": "\\operatorname{f_{E}}{(\\nabla)} = \\sin{(\\nabla)} and \\int \\operatorname{f_{E}}{(\\nabla)} d\\nabla = \\int \\sin{(\\nabla)} d\\nabla and \\int \\operatorname{f_{E}}{(\\nabla)} d\\nabla = \\Omega - \\cos{(\\nabla)} and \\sin{(\\int \\operatorname{f_{E}}{(\\nabla)} d\\nabla)} = \\sin{(\\Omega - \\cos{(\\nabla)})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\nabla', commutative=True)), sin(Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\nabla', commutative=True)"], "Equality(Integral(Function('f_E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Integral(sin(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('f_E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True)))))"], [["sin", 3], "Equality(sin(Integral(Function('f_E')(Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\nabla', commutative=True)))), sin(Add(Symbol('\\\\Omega', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\nabla', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} = W + f_{\\mathbf{v}}, then obtain \\frac{3 \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)}}{f_{\\mathbf{v}}} = \\frac{2 W + 2 f_{\\mathbf{v}} + \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)}}{f_{\\mathbf{v}}}", "derivation": "\\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} = W + f_{\\mathbf{v}} and W + f_{\\mathbf{v}} + 2 \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} = 2 W + 2 f_{\\mathbf{v}} + \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} and 3 \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} = 2 W + 2 f_{\\mathbf{v}} + \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)} and \\frac{3 \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)}}{f_{\\mathbf{v}}} = \\frac{2 W + 2 f_{\\mathbf{v}} + \\operatorname{E_{n}}{(f_{\\mathbf{v}},W)}}{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True)), Add(Symbol('W', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["add", 1, "Add(Symbol('W', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True)))"], "Equality(Add(Symbol('W', commutative=True), Symbol('f_{\\\\mathbf{v}}', commutative=True), Mul(Integer(2), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(3), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True))))"], [["divide", 3, "Symbol('f_{\\\\mathbf{v}}', commutative=True)"], "Equality(Mul(Integer(3), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)), Add(Mul(Integer(2), Symbol('W', commutative=True)), Mul(Integer(2), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Function('E_n')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} = e^{\\phi_1}, then obtain - J - \\mathbf{r} - \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1}}{A_{2}^{2}} = - J - \\mathbf{r} - \\frac{e^{2 \\phi_1}}{A_{2}^{2}}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} = e^{\\phi_1} and \\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1} = e^{2 \\phi_1} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1}}{A_{2}} = \\frac{e^{2 \\phi_1}}{A_{2}} and \\frac{\\partial}{\\partial A_{2}} \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1}}{A_{2}} = \\frac{\\partial}{\\partial A_{2}} \\frac{e^{2 \\phi_1}}{A_{2}} and - J - \\mathbf{r} + \\frac{\\partial}{\\partial A_{2}} \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1}}{A_{2}} = - J - \\mathbf{r} + \\frac{\\partial}{\\partial A_{2}} \\frac{e^{2 \\phi_1}}{A_{2}} and - J - \\mathbf{r} - \\frac{\\operatorname{V_{\\mathbf{E}}}{(\\phi_1)} e^{\\phi_1}}{A_{2}^{2}} = - J - \\mathbf{r} - \\frac{e^{2 \\phi_1}}{A_{2}^{2}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))"], [["divide", 2, "Symbol('A_2', commutative=True)"], "Equality(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["minus", 4, "Add(Symbol('J', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-2)), Function('V_{\\\\mathbf{E}}')(Symbol('\\\\phi_1', commutative=True)), exp(Symbol('\\\\phi_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(-1), Pow(Symbol('A_2', commutative=True), Integer(-2)), exp(Mul(Integer(2), Symbol('\\\\phi_1', commutative=True))))))"]]}, {"prompt": "Given a{(f_{E})} = \\sin{(\\log{(f_{E})})}, then derive \\int a{(f_{E})} df_{E} = A_{1} + \\frac{f_{E} \\sin{(\\log{(f_{E})})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2}, then obtain A_{1} + \\frac{f_{E} a{(f_{E})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2} = A_{1} + \\frac{f_{E} \\sin{(\\log{(f_{E})})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2}", "derivation": "a{(f_{E})} = \\sin{(\\log{(f_{E})})} and \\int a{(f_{E})} df_{E} = \\int \\sin{(\\log{(f_{E})})} df_{E} and \\int a{(f_{E})} df_{E} = A_{1} + \\frac{f_{E} \\sin{(\\log{(f_{E})})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2} and \\int a{(f_{E})} df_{E} = A_{1} + \\frac{f_{E} a{(f_{E})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2} and A_{1} + \\frac{f_{E} a{(f_{E})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2} = A_{1} + \\frac{f_{E} \\sin{(\\log{(f_{E})})}}{2} - \\frac{f_{E} \\cos{(\\log{(f_{E})})}}{2}", "srepr_derivation": [["get_premise", "Equality(Function('a')(Symbol('f_E', commutative=True)), sin(log(Symbol('f_E', commutative=True))))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('a')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(sin(log(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('a')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Symbol('f_E', commutative=True), sin(log(Symbol('f_E', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('f_E', commutative=True), cos(log(Symbol('f_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('a')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Symbol('f_E', commutative=True), Function('a')(Symbol('f_E', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('f_E', commutative=True), cos(log(Symbol('f_E', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Symbol('f_E', commutative=True), Function('a')(Symbol('f_E', commutative=True))), Mul(Integer(-1), Rational(1, 2), Symbol('f_E', commutative=True), cos(log(Symbol('f_E', commutative=True))))), Add(Symbol('A_1', commutative=True), Mul(Rational(1, 2), Symbol('f_E', commutative=True), sin(log(Symbol('f_E', commutative=True)))), Mul(Integer(-1), Rational(1, 2), Symbol('f_E', commutative=True), cos(log(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given \\varphi{(k)} = \\cos{(e^{k})} and \\dot{y}{(k)} = \\varphi^{2}{(k)}, then obtain \\varphi{(k)} \\varphi^{k}{(k)} - \\cos^{2}{(e^{k})} = \\varphi{(k)} \\cos^{k}{(e^{k})} - \\cos^{2}{(e^{k})}", "derivation": "\\varphi{(k)} = \\cos{(e^{k})} and \\varphi^{k}{(k)} = \\cos^{k}{(e^{k})} and \\varphi{(k)} \\varphi^{k}{(k)} = \\varphi{(k)} \\cos^{k}{(e^{k})} and - \\varphi^{2}{(k)} + \\varphi{(k)} \\varphi^{k}{(k)} = - \\varphi^{2}{(k)} + \\varphi{(k)} \\cos^{k}{(e^{k})} and \\dot{y}{(k)} = \\varphi^{2}{(k)} and - \\dot{y}{(k)} + \\varphi{(k)} \\varphi^{k}{(k)} = - \\dot{y}{(k)} + \\varphi{(k)} \\cos^{k}{(e^{k})} and \\dot{y}{(k)} = \\cos^{2}{(e^{k})} and \\varphi{(k)} \\varphi^{k}{(k)} - \\cos^{2}{(e^{k})} = \\varphi{(k)} \\cos^{k}{(e^{k})} - \\cos^{2}{(e^{k})}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('k', commutative=True)), cos(exp(Symbol('k', commutative=True))))"], [["power", 1, "Symbol('k', commutative=True)"], "Equality(Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))"], [["times", 2, "Function('\\\\varphi')(Symbol('k', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True))))"], [["minus", 3, "Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Integer(2))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Integer(2))), Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Integer(2))), Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('k', commutative=True))), Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Symbol('k', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\dot{y}')(Symbol('k', commutative=True))), Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Function('\\\\dot{y}')(Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 6, 7], "Equality(Add(Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(Function('\\\\varphi')(Symbol('k', commutative=True)), Symbol('k', commutative=True))), Mul(Integer(-1), Pow(cos(exp(Symbol('k', commutative=True))), Integer(2)))), Add(Mul(Function('\\\\varphi')(Symbol('k', commutative=True)), Pow(cos(exp(Symbol('k', commutative=True))), Symbol('k', commutative=True))), Mul(Integer(-1), Pow(cos(exp(Symbol('k', commutative=True))), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(f_{\\mathbf{p}},G,\\theta_2)} = (\\theta_2 f_{\\mathbf{p}})^{G} and g{(f_{\\mathbf{p}})} = f_{\\mathbf{p}}, then obtain \\int (\\theta_2 f_{\\mathbf{p}})^{- G} g{(f_{\\mathbf{p}})} dG = \\int f_{\\mathbf{p}} (\\theta_2 f_{\\mathbf{p}})^{- G} dG", "derivation": "\\operatorname{P_{g}}{(f_{\\mathbf{p}},G,\\theta_2)} = (\\theta_2 f_{\\mathbf{p}})^{G} and g{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} and \\frac{g{(f_{\\mathbf{p}})}}{\\operatorname{P_{g}}{(f_{\\mathbf{p}},G,\\theta_2)}} = \\frac{f_{\\mathbf{p}}}{\\operatorname{P_{g}}{(f_{\\mathbf{p}},G,\\theta_2)}} and (\\theta_2 f_{\\mathbf{p}})^{- G} g{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} (\\theta_2 f_{\\mathbf{p}})^{- G} and \\int (\\theta_2 f_{\\mathbf{p}})^{- G} g{(f_{\\mathbf{p}})} dG = \\int f_{\\mathbf{p}} (\\theta_2 f_{\\mathbf{p}})^{- G} dG", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('G', commutative=True)))"], ["renaming_premise", "Equality(Function('g')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))"], [["divide", 2, "Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True))"], "Equality(Mul(Pow(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1)), Function('g')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('P_g')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('G', commutative=True), Symbol('\\\\theta_2', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Pow(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))), Function('g')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))))"], [["integrate", 4, "Symbol('G', commutative=True)"], "Equality(Integral(Mul(Pow(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True))), Function('g')(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(Mul(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Mul(Symbol('\\\\theta_2', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Symbol('G', commutative=True)))), Tuple(Symbol('G', commutative=True))))"]]}, {"prompt": "Given p{(\\varphi)} = e^{\\varphi}, then obtain (p{(\\varphi)} + 1) e^{- \\varphi} = (e^{\\varphi} + 1) e^{- \\varphi}", "derivation": "p{(\\varphi)} = e^{\\varphi} and \\varphi p{(\\varphi)} = \\varphi e^{\\varphi} and p{(\\varphi)} + 1 = e^{\\varphi} + 1 and - \\varphi p{(\\varphi)} + \\varphi e^{\\varphi} + p{(\\varphi)} + 1 = - \\varphi p{(\\varphi)} + \\varphi e^{\\varphi} + e^{\\varphi} + 1 and \\frac{- \\varphi p{(\\varphi)} + \\varphi e^{\\varphi} + p{(\\varphi)} + 1}{\\frac{d}{d \\varphi} (e^{\\varphi} + 1)} = \\frac{- \\varphi p{(\\varphi)} + \\varphi e^{\\varphi} + e^{\\varphi} + 1}{\\frac{d}{d \\varphi} (e^{\\varphi} + 1)} and \\frac{p{(\\varphi)} + 1}{\\frac{d}{d \\varphi} (e^{\\varphi} + 1)} = \\frac{e^{\\varphi} + 1}{\\frac{d}{d \\varphi} (e^{\\varphi} + 1)} and (p{(\\varphi)} + 1) e^{- \\varphi} = (e^{\\varphi} + 1) e^{- \\varphi}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\varphi', commutative=True)), exp(Symbol('\\\\varphi', commutative=True)))"], [["times", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Mul(Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('p')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)))"], [["minus", 3, "Add(Mul(Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))), Function('p')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))), exp(Symbol('\\\\varphi', commutative=True)), Integer(1)))"], [["divide", 4, "Derivative(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))), Function('p')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Pow(Derivative(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True), Function('p')(Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\varphi', commutative=True), exp(Symbol('\\\\varphi', commutative=True))), exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Pow(Derivative(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Add(Function('p')(Symbol('\\\\varphi', commutative=True)), Integer(1)), Pow(Derivative(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1))), Mul(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Pow(Derivative(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 6], "Equality(Mul(Add(Function('p')(Symbol('\\\\varphi', commutative=True)), Integer(1)), exp(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Mul(Add(exp(Symbol('\\\\varphi', commutative=True)), Integer(1)), exp(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))))"]]}, {"prompt": "Given s{(\\lambda)} = \\sin{(\\lambda)}, then obtain s{(\\lambda)} - \\sin{(\\lambda)} - \\frac{s{(\\lambda)}}{\\lambda} = - \\frac{s{(\\lambda)}}{\\lambda}", "derivation": "s{(\\lambda)} = \\sin{(\\lambda)} and \\frac{s{(\\lambda)}}{\\lambda} = \\frac{\\sin{(\\lambda)}}{\\lambda} and s{(\\lambda)} + \\frac{\\sin{(\\lambda)}}{\\lambda} = \\sin{(\\lambda)} + \\frac{\\sin{(\\lambda)}}{\\lambda} and \\lambda + s{(\\lambda)} + \\frac{\\sin{(\\lambda)}}{\\lambda} = \\lambda + \\sin{(\\lambda)} + \\frac{\\sin{(\\lambda)}}{\\lambda} and \\lambda + s{(\\lambda)} = \\lambda + \\sin{(\\lambda)} and s{(\\lambda)} - \\sin{(\\lambda)} - \\frac{\\sin{(\\lambda)}}{\\lambda} = - \\frac{\\sin{(\\lambda)}}{\\lambda} and s{(\\lambda)} - \\sin{(\\lambda)} - \\frac{s{(\\lambda)}}{\\lambda} = - \\frac{s{(\\lambda)}}{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\lambda', commutative=True)), sin(Symbol('\\\\lambda', commutative=True)))"], [["divide", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True))))"], [["add", 1, "Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Function('s')(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))), Add(sin(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))))"], [["add", 3, "Symbol('\\\\lambda', commutative=True)"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('s')(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))))"], [["minus", 4, "Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))"], "Equality(Add(Symbol('\\\\lambda', commutative=True), Function('s')(Symbol('\\\\lambda', commutative=True))), Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\lambda', commutative=True))))"], [["minus", 5, "Add(Symbol('\\\\lambda', commutative=True), sin(Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True))))"], "Equality(Add(Function('s')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), sin(Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Add(Function('s')(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\lambda', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\lambda', commutative=True)))), Mul(Integer(-1), Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('s')(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(q)} = \\cos{(q)} and \\hat{\\mathbf{r}}{(q)} = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)}, then obtain q + \\hat{\\mathbf{r}}{(q)} = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)} + q", "derivation": "\\operatorname{A_{2}}{(q)} = \\cos{(q)} and q \\operatorname{A_{2}}{(q)} = q \\cos{(q)} and 2 q \\operatorname{A_{2}}{(q)} = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)} and 2 q \\operatorname{A_{2}}{(q)} + q = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)} + q and \\hat{\\mathbf{r}}{(q)} = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)} and \\hat{\\mathbf{r}}{(q)} = 2 q \\operatorname{A_{2}}{(q)} and q + \\hat{\\mathbf{r}}{(q)} = q \\operatorname{A_{2}}{(q)} + q \\cos{(q)} + q", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True))))"], [["add", 2, "Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Add(Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True)))))"], [["add", 3, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Symbol('q', commutative=True)), Add(Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('q', commutative=True)), Add(Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(Add(Symbol('q', commutative=True), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('q', commutative=True))), Add(Mul(Symbol('q', commutative=True), Function('A_2')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), cos(Symbol('q', commutative=True))), Symbol('q', commutative=True)))"]]}, {"prompt": "Given \\theta_{1}{(F_{N},\\varphi,Z)} = - F_{N} + Z + \\varphi, then obtain \\log{(\\int 1 dF_{N})} = \\log{(\\int - \\frac{F_{N} - Z - \\varphi}{- F_{N} + Z + \\varphi} dF_{N})}", "derivation": "\\theta_{1}{(F_{N},\\varphi,Z)} = - F_{N} + Z + \\varphi and - \\theta_{1}{(F_{N},\\varphi,Z)} = F_{N} - Z - \\varphi and 1 = - \\frac{F_{N} - Z - \\varphi}{\\theta_{1}{(F_{N},\\varphi,Z)}} and \\int 1 dF_{N} = \\int - \\frac{F_{N} - Z - \\varphi}{\\theta_{1}{(F_{N},\\varphi,Z)}} dF_{N} and \\int 1 dF_{N} = \\int - \\frac{F_{N} - Z - \\varphi}{- F_{N} + Z + \\varphi} dF_{N} and \\log{(\\int 1 dF_{N})} = \\log{(\\int - \\frac{F_{N} - Z - \\varphi}{- F_{N} + Z + \\varphi} dF_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('F_N', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Z', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta_1')(Symbol('F_N', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))))"], [["divide", 2, "Mul(Integer(-1), Function('\\\\theta_1')(Symbol('F_N', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)))"], "Equality(Integer(1), Mul(Integer(-1), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Integer(-1))))"], [["integrate", 3, "Symbol('F_N', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Integer(-1), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True))), Pow(Function('\\\\theta_1')(Symbol('F_N', commutative=True), Symbol('\\\\varphi', commutative=True), Symbol('Z', commutative=True)), Integer(-1))), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(Integer(1), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Z', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('F_N', commutative=True))))"], [["log", 5], "Equality(log(Integral(Integer(1), Tuple(Symbol('F_N', commutative=True)))), log(Integral(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Symbol('Z', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Add(Symbol('F_N', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given n{(\\mu_0)} = - \\mu_0, then obtain - 2 \\mu_0 + \\tilde{g}^* = - \\mu_0 + \\tilde{g}^* + n{(\\mu_0)}", "derivation": "n{(\\mu_0)} = - \\mu_0 and \\tilde{g}^* + n{(\\mu_0)} = - \\mu_0 + \\tilde{g}^* and \\tilde{g}^* + 2 n{(\\mu_0)} = - \\mu_0 + \\tilde{g}^* + n{(\\mu_0)} and \\tilde{g}^* + 2 n{(\\mu_0)} = - 2 \\mu_0 + \\tilde{g}^* and - 2 \\mu_0 + \\tilde{g}^* = - \\mu_0 + \\tilde{g}^* + n{(\\mu_0)}", "srepr_derivation": [["renaming_premise", "Equality(Function('n')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)))"], [["add", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Function('n')(Symbol('\\\\mu_0', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["add", 2, "Function('n')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(2), Function('n')(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), Function('n')(Symbol('\\\\mu_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Integer(2), Function('n')(Symbol('\\\\mu_0', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True), Function('n')(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(\\mathbf{M})} = \\mathbf{M}, then derive \\int \\sigma_{x}{(\\mathbf{M})} d\\mathbf{M} = F_{c} + \\frac{\\mathbf{M}^{2}}{2}, then obtain \\iint \\sigma_{x}{(\\mathbf{M})} d\\sigma_{x}{(\\mathbf{M})} dF_{c} = \\int (F_{c} + \\frac{\\sigma_{x}^{2}{(\\mathbf{M})}}{2}) dF_{c}", "derivation": "\\sigma_{x}{(\\mathbf{M})} = \\mathbf{M} and \\int \\sigma_{x}{(\\mathbf{M})} d\\mathbf{M} = \\int \\mathbf{M} d\\mathbf{M} and \\int \\sigma_{x}{(\\mathbf{M})} d\\mathbf{M} = F_{c} + \\frac{\\mathbf{M}^{2}}{2} and \\iint \\sigma_{x}{(\\mathbf{M})} d\\mathbf{M} dF_{c} = \\int (F_{c} + \\frac{\\mathbf{M}^{2}}{2}) dF_{c} and \\iint \\sigma_{x}{(\\mathbf{M})} d\\sigma_{x}{(\\mathbf{M})} dF_{c} = \\int (F_{c} + \\frac{\\sigma_{x}^{2}{(\\mathbf{M})}}{2}) dF_{c}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Symbol('\\\\mathbf{M}', commutative=True), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))))"], [["integrate", 3, "Symbol('F_c', commutative=True)"], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{M}', commutative=True), Integer(2)))), Tuple(Symbol('F_c', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\sigma_x')(Symbol('\\\\mathbf{M}', commutative=True)), Integer(2)))), Tuple(Symbol('F_c', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(m)} = \\log{(m)}, then derive \\frac{d}{d m} \\ddot{x}{(m)} = \\frac{1}{m}, then obtain (\\int (\\frac{d}{d m} \\ddot{x}{(m)})^{m} dm)^{m} = (\\int (\\frac{d}{d m} \\log{(m)})^{m} dm)^{m}", "derivation": "\\ddot{x}{(m)} = \\log{(m)} and \\frac{d}{d m} \\ddot{x}{(m)} = \\frac{d}{d m} \\log{(m)} and \\frac{d}{d m} \\ddot{x}{(m)} = \\frac{1}{m} and (\\frac{d}{d m} \\ddot{x}{(m)})^{m} = (\\frac{1}{m})^{m} and (\\frac{d}{d m} \\log{(m)})^{m} = (\\frac{1}{m})^{m} and (\\frac{d}{d m} \\ddot{x}{(m)})^{m} = (\\frac{d}{d m} \\log{(m)})^{m} and \\int (\\frac{d}{d m} \\ddot{x}{(m)})^{m} dm = \\int (\\frac{d}{d m} \\log{(m)})^{m} dm and (\\int (\\frac{d}{d m} \\ddot{x}{(m)})^{m} dm)^{m} = (\\int (\\frac{d}{d m} \\log{(m)})^{m} dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), log(Symbol('m', commutative=True)))"], [["differentiate", 1, "Symbol('m', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Pow(Symbol('m', commutative=True), Integer(-1)))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(Pow(Symbol('m', commutative=True), Integer(-1)), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Pow(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)))"], [["integrate", 6, "Symbol('m', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Pow(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["power", 7, "Symbol('m', commutative=True)"], "Equality(Pow(Integral(Pow(Derivative(Function('\\\\ddot{x}')(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(Pow(Derivative(log(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given I{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})} and \\operatorname{f^{\\prime}}{(\\mu_0)} = \\sin{(\\mu_0)} and \\operatorname{A_{x}}{(\\mu_0)} = I{(\\mu_0)} - \\sin{(\\sin{(\\mu_0)})}, then obtain (0^{\\mu_0})^{\\mu_0} ((I{(\\mu_0)} - \\sin{(\\operatorname{f^{\\prime}}{(\\mu_0)})})^{\\mu_0})^{\\mu_0} = (0^{\\mu_0})^{2 \\mu_0}", "derivation": "I{(\\mu_0)} = \\sin{(\\sin{(\\mu_0)})} and \\operatorname{f^{\\prime}}{(\\mu_0)} = \\sin{(\\mu_0)} and \\operatorname{A_{x}}{(\\mu_0)} = I{(\\mu_0)} - \\sin{(\\sin{(\\mu_0)})} and \\operatorname{A_{x}}{(\\mu_0)} = 0 and \\operatorname{A_{x}}^{\\mu_0}{(\\mu_0)} = 0^{\\mu_0} and (I{(\\mu_0)} - \\sin{(\\sin{(\\mu_0)})})^{\\mu_0} = 0^{\\mu_0} and ((I{(\\mu_0)} - \\sin{(\\sin{(\\mu_0)})})^{\\mu_0})^{\\mu_0} = (0^{\\mu_0})^{\\mu_0} and ((I{(\\mu_0)} - \\sin{(\\operatorname{f^{\\prime}}{(\\mu_0)})})^{\\mu_0})^{\\mu_0} = (0^{\\mu_0})^{\\mu_0} and (0^{\\mu_0})^{\\mu_0} ((I{(\\mu_0)} - \\sin{(\\operatorname{f^{\\prime}}{(\\mu_0)})})^{\\mu_0})^{\\mu_0} = (0^{\\mu_0})^{2 \\mu_0}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mu_0', commutative=True)), sin(sin(Symbol('\\\\mu_0', commutative=True))))"], ["renaming_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('A_x')(Symbol('\\\\mu_0', commutative=True)), Add(Function('I')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('\\\\mu_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('A_x')(Symbol('\\\\mu_0', commutative=True)), Integer(0))"], [["power", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('A_x')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Add(Function('I')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)))"], [["power", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Pow(Add(Function('I')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(sin(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["substitute_RHS_for_LHS", 7, 2], "Equality(Pow(Pow(Add(Function('I')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], [["times", 8, "Pow(Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Add(Function('I')(Symbol('\\\\mu_0', commutative=True)), Mul(Integer(-1), sin(Function('f^{\\\\prime}')(Symbol('\\\\mu_0', commutative=True))))), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Pow(Pow(Integer(0), Symbol('\\\\mu_0', commutative=True)), Mul(Integer(2), Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\dot{z},\\theta_2)} = - \\dot{z} + \\log{(\\theta_2)}, then derive \\int \\hat{\\mathbf{r}}{(\\dot{z},\\theta_2)} d\\theta_2 = \\theta_2 (- \\dot{z} - 1) + \\theta_2 \\log{(\\theta_2)} + v_{y}, then obtain \\theta_2 (- \\dot{z} - 1) + \\theta_2 \\log{(\\theta_2)} + v_{y} = \\int (- \\dot{z} + \\log{(\\theta_2)}) d\\theta_2", "derivation": "\\hat{\\mathbf{r}}{(\\dot{z},\\theta_2)} = - \\dot{z} + \\log{(\\theta_2)} and \\int \\hat{\\mathbf{r}}{(\\dot{z},\\theta_2)} d\\theta_2 = \\int (- \\dot{z} + \\log{(\\theta_2)}) d\\theta_2 and \\int \\hat{\\mathbf{r}}{(\\dot{z},\\theta_2)} d\\theta_2 = \\theta_2 (- \\dot{z} - 1) + \\theta_2 \\log{(\\theta_2)} + v_{y} and \\theta_2 (- \\dot{z} - 1) + \\theta_2 \\log{(\\theta_2)} + v_{y} = \\int (- \\dot{z} + \\log{(\\theta_2)}) d\\theta_2", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))))"], [["integrate", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('\\\\theta_2', commutative=True))), Add(Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Symbol('\\\\theta_2', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\theta_2', commutative=True), log(Symbol('\\\\theta_2', commutative=True))), Symbol('v_y', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('\\\\dot{z}', commutative=True)), log(Symbol('\\\\theta_2', commutative=True))), Tuple(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\theta{(J,F_{x})} = \\frac{F_{x}}{J}, then obtain \\frac{\\partial}{\\partial J} (\\frac{F_{x}}{J} - 2 \\theta{(J,F_{x})}) = \\frac{\\partial}{\\partial J} - \\frac{F_{x}}{J}", "derivation": "\\theta{(J,F_{x})} = \\frac{F_{x}}{J} and 0 = \\frac{F_{x}}{J} - \\theta{(J,F_{x})} and - \\theta{(J,F_{x})} = \\frac{F_{x}}{J} - 2 \\theta{(J,F_{x})} and - \\theta{(J,F_{x})} = - \\frac{F_{x}}{J} and \\frac{F_{x}}{J} - 2 \\theta{(J,F_{x})} = - \\frac{F_{x}}{J} and \\frac{\\partial}{\\partial J} (\\frac{F_{x}}{J} - 2 \\theta{(J,F_{x})}) = \\frac{\\partial}{\\partial J} - \\frac{F_{x}}{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["minus", 1, "Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))), Mul(Integer(-1), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)))))"], [["add", 2, "Mul(Integer(-1), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True))), Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)))), Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))))"], [["differentiate", 5, "Symbol('J', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))), Mul(Integer(-1), Integer(2), Function('\\\\theta')(Symbol('J', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('F_x', commutative=True), Pow(Symbol('J', commutative=True), Integer(-1))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given E{(z)} = e^{z}, then obtain (\\frac{z \\frac{d}{d z} E{(z)}}{E{(z)}} + \\log{(E{(z)})}) E^{z}{(z)} = (z + \\log{(e^{z})}) (e^{z})^{z}", "derivation": "E{(z)} = e^{z} and E^{z}{(z)} = (e^{z})^{z} and \\frac{d}{d z} E^{z}{(z)} = \\frac{d}{d z} (e^{z})^{z} and (\\frac{z \\frac{d}{d z} E{(z)}}{E{(z)}} + \\log{(E{(z)})}) E^{z}{(z)} = (z + \\log{(e^{z})}) (e^{z})^{z}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('z', commutative=True)), exp(Symbol('z', commutative=True)))"], [["power", 1, "Symbol('z', commutative=True)"], "Equality(Pow(Function('E')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Pow(exp(Symbol('z', commutative=True)), Symbol('z', commutative=True)))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Pow(Function('E')(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('z', commutative=True)), Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Add(Mul(Symbol('z', commutative=True), Pow(Function('E')(Symbol('z', commutative=True)), Integer(-1)), Derivative(Function('E')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), log(Function('E')(Symbol('z', commutative=True)))), Pow(Function('E')(Symbol('z', commutative=True)), Symbol('z', commutative=True))), Mul(Add(Symbol('z', commutative=True), log(exp(Symbol('z', commutative=True)))), Pow(exp(Symbol('z', commutative=True)), Symbol('z', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(y^{\\prime},G)} = \\frac{G}{y^{\\prime}}, then obtain (\\frac{G \\sigma_{x}{(y^{\\prime},G)}}{(y^{\\prime})^{2}})^{y^{\\prime}} = (\\frac{G^{2}}{(y^{\\prime})^{3}})^{y^{\\prime}}", "derivation": "\\sigma_{x}{(y^{\\prime},G)} = \\frac{G}{y^{\\prime}} and \\frac{\\sigma_{x}{(y^{\\prime},G)}}{y^{\\prime}} = \\frac{G}{(y^{\\prime})^{2}} and \\frac{\\sigma_{x}^{2}{(y^{\\prime},G)}}{y^{\\prime}} = \\frac{G \\sigma_{x}{(y^{\\prime},G)}}{(y^{\\prime})^{2}} and \\frac{G \\sigma_{x}{(y^{\\prime},G)}}{(y^{\\prime})^{2}} = \\frac{G^{2}}{(y^{\\prime})^{3}} and (\\frac{G \\sigma_{x}{(y^{\\prime},G)}}{(y^{\\prime})^{2}})^{y^{\\prime}} = (\\frac{G^{2}}{(y^{\\prime})^{3}})^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('G', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))))"], [["times", 1, "Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True))), Mul(Symbol('G', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-2))))"], [["times", 2, "Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True))"], "Equality(Mul(Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-1)), Pow(Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True)), Integer(2))), Mul(Symbol('G', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-2)), Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Symbol('G', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-2)), Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-3))))"], [["power", 4, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Mul(Symbol('G', commutative=True), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-2)), Function('\\\\sigma_x')(Symbol('y^{\\\\prime}', commutative=True), Symbol('G', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)), Pow(Mul(Pow(Symbol('G', commutative=True), Integer(2)), Pow(Symbol('y^{\\\\prime}', commutative=True), Integer(-3))), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given G{(n_{2},\\theta)} = \\log{(\\theta n_{2})}, then obtain 0 = -1 + \\frac{\\log{(\\theta n_{2})}}{G{(n_{2},\\theta)}}", "derivation": "G{(n_{2},\\theta)} = \\log{(\\theta n_{2})} and \\theta n_{2} G{(n_{2},\\theta)} = \\theta n_{2} \\log{(\\theta n_{2})} and 1 = \\frac{\\log{(\\theta n_{2})}}{G{(n_{2},\\theta)}} and 0 = -1 + \\frac{\\log{(\\theta n_{2})}}{G{(n_{2},\\theta)}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('n_2', commutative=True), Symbol('\\\\theta', commutative=True)), log(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True), Function('G')(Symbol('n_2', commutative=True), Symbol('\\\\theta', commutative=True))), Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True), log(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))))"], [["divide", 2, "Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True), Function('G')(Symbol('n_2', commutative=True), Symbol('\\\\theta', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('G')(Symbol('n_2', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)), log(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True)))))"], [["minus", 3, 1], "Equality(Integer(0), Add(Integer(-1), Mul(Pow(Function('G')(Symbol('n_2', commutative=True), Symbol('\\\\theta', commutative=True)), Integer(-1)), log(Mul(Symbol('\\\\theta', commutative=True), Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(q,\\varphi)} = \\sin{(\\frac{q}{\\varphi})}, then obtain - \\operatorname{f^{*}}{(q,\\varphi)} \\sin{(\\frac{q}{\\varphi})} = - \\sin^{2}{(\\frac{q}{\\varphi})}", "derivation": "\\operatorname{f^{*}}{(q,\\varphi)} = \\sin{(\\frac{q}{\\varphi})} and \\operatorname{f^{*}}{(q,\\varphi)} \\sin{(\\frac{q}{\\varphi})} = \\sin^{2}{(\\frac{q}{\\varphi})} and - \\sin{(\\frac{q}{\\varphi})} = - \\frac{\\sin^{2}{(\\frac{q}{\\varphi})}}{\\operatorname{f^{*}}{(q,\\varphi)}} and - \\operatorname{f^{*}}{(q,\\varphi)} \\sin{(\\frac{q}{\\varphi})} = - \\sin^{2}{(\\frac{q}{\\varphi})}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True))))"], [["times", 1, "sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True)))"], "Equality(Mul(Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True)))), Pow(sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(2)))"], [["divide", 2, "Mul(Integer(-1), Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)))"], "Equality(Mul(Integer(-1), sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True)))), Mul(Integer(-1), Pow(Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), Integer(-1)), Pow(sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(2))))"], [["times", 3, "Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Integer(-1), Function('f^*')(Symbol('q', commutative=True), Symbol('\\\\varphi', commutative=True)), sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True)))), Mul(Integer(-1), Pow(sin(Mul(Pow(Symbol('\\\\varphi', commutative=True), Integer(-1)), Symbol('q', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\mathbf{v}{(\\omega)} = \\sin{(e^{\\omega})}, then obtain - \\mathbf{v}{(\\omega)} + (\\frac{d}{d \\omega} \\mathbf{v}{(\\omega)})^{\\omega} = - \\mathbf{v}{(\\omega)} + (\\frac{d}{d \\omega} \\sin{(e^{\\omega})})^{\\omega}", "derivation": "\\mathbf{v}{(\\omega)} = \\sin{(e^{\\omega})} and \\frac{d}{d \\omega} \\mathbf{v}{(\\omega)} = \\frac{d}{d \\omega} \\sin{(e^{\\omega})} and (\\frac{d}{d \\omega} \\mathbf{v}{(\\omega)})^{\\omega} = (\\frac{d}{d \\omega} \\sin{(e^{\\omega})})^{\\omega} and - \\mathbf{v}{(\\omega)} + (\\frac{d}{d \\omega} \\mathbf{v}{(\\omega)})^{\\omega} = - \\mathbf{v}{(\\omega)} + (\\frac{d}{d \\omega} \\sin{(e^{\\omega})})^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), sin(exp(Symbol('\\\\omega', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(sin(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)), Pow(Derivative(sin(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True))), Pow(Derivative(Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Function('\\\\mathbf{v}')(Symbol('\\\\omega', commutative=True))), Pow(Derivative(sin(exp(Symbol('\\\\omega', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{s}{(F_{H})} = \\log{(F_{H})}, then obtain 0 = F_{H} (- 2 \\mathbf{s}{(F_{H})} + 2 \\log{(F_{H})})", "derivation": "\\mathbf{s}{(F_{H})} = \\log{(F_{H})} and 0 = - \\mathbf{s}{(F_{H})} + \\log{(F_{H})} and 0 = F_{H} (- \\mathbf{s}{(F_{H})} + \\log{(F_{H})}) and - \\mathbf{s}{(F_{H})} = - 2 \\mathbf{s}{(F_{H})} + \\log{(F_{H})} and 0 = F_{H} (- 2 \\mathbf{s}{(F_{H})} + 2 \\log{(F_{H})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True)), log(Symbol('F_H', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True))))"], [["times", 2, "Symbol('F_H', commutative=True)"], "Equality(Integer(0), Mul(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True)))))"], [["minus", 2, "Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))), log(Symbol('F_H', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(0), Mul(Symbol('F_H', commutative=True), Add(Mul(Integer(-1), Integer(2), Function('\\\\mathbf{s}')(Symbol('F_H', commutative=True))), Mul(Integer(2), log(Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{x}{(Q)} = \\sin{(\\cos{(Q)})}, then obtain \\frac{\\log{(\\sigma_{x}{(Q)} - \\sin{(\\cos{(Q)})} \\cos{(Q)})}}{\\cos{(Q)}} = \\frac{\\log{(- \\sin{(\\cos{(Q)})} \\cos{(Q)} + \\sin{(\\cos{(Q)})})}}{\\cos{(Q)}}", "derivation": "\\sigma_{x}{(Q)} = \\sin{(\\cos{(Q)})} and \\sigma_{x}{(Q)} \\cos{(Q)} = \\sin{(\\cos{(Q)})} \\cos{(Q)} and - \\sigma_{x}{(Q)} \\cos{(Q)} + \\sigma_{x}{(Q)} = - \\sigma_{x}{(Q)} \\cos{(Q)} + \\sin{(\\cos{(Q)})} and \\log{(- \\sigma_{x}{(Q)} \\cos{(Q)} + \\sigma_{x}{(Q)})} = \\log{(- \\sigma_{x}{(Q)} \\cos{(Q)} + \\sin{(\\cos{(Q)})})} and \\log{(\\sigma_{x}{(Q)} - \\sin{(\\cos{(Q)})} \\cos{(Q)})} = \\log{(- \\sin{(\\cos{(Q)})} \\cos{(Q)} + \\sin{(\\cos{(Q)})})} and \\frac{\\log{(\\sigma_{x}{(Q)} - \\sin{(\\cos{(Q)})} \\cos{(Q)})}}{\\cos{(Q)}} = \\frac{\\log{(- \\sin{(\\cos{(Q)})} \\cos{(Q)} + \\sin{(\\cos{(Q)})})}}{\\cos{(Q)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), sin(cos(Symbol('Q', commutative=True))))"], [["times", 1, "cos(Symbol('Q', commutative=True))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Mul(sin(cos(Symbol('Q', commutative=True))), cos(Symbol('Q', commutative=True))))"], [["minus", 1, "Mul(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Function('\\\\sigma_x')(Symbol('Q', commutative=True))), Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), sin(cos(Symbol('Q', commutative=True)))))"], [["log", 3], "Equality(log(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), Function('\\\\sigma_x')(Symbol('Q', commutative=True)))), log(Add(Mul(Integer(-1), Function('\\\\sigma_x')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True))), sin(cos(Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(log(Add(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('Q', commutative=True))), cos(Symbol('Q', commutative=True))))), log(Add(Mul(Integer(-1), sin(cos(Symbol('Q', commutative=True))), cos(Symbol('Q', commutative=True))), sin(cos(Symbol('Q', commutative=True))))))"], [["divide", 5, "cos(Symbol('Q', commutative=True))"], "Equality(Mul(log(Add(Function('\\\\sigma_x')(Symbol('Q', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('Q', commutative=True))), cos(Symbol('Q', commutative=True))))), Pow(cos(Symbol('Q', commutative=True)), Integer(-1))), Mul(log(Add(Mul(Integer(-1), sin(cos(Symbol('Q', commutative=True))), cos(Symbol('Q', commutative=True))), sin(cos(Symbol('Q', commutative=True))))), Pow(cos(Symbol('Q', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(c_{0},F_{N})} = \\log{(\\frac{F_{N}}{c_{0}})}, then derive (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{F_{N}} = (\\frac{1}{F_{N}})^{F_{N}}, then obtain (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{2 F_{N}} = (\\frac{1}{F_{N}})^{F_{N}} (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{F_{N}}", "derivation": "\\operatorname{F_{c}}{(c_{0},F_{N})} = \\log{(\\frac{F_{N}}{c_{0}})} and \\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})} = \\frac{\\partial}{\\partial F_{N}} \\log{(\\frac{F_{N}}{c_{0}})} and (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{F_{N}} = (\\frac{\\partial}{\\partial F_{N}} \\log{(\\frac{F_{N}}{c_{0}})})^{F_{N}} and (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{F_{N}} = (\\frac{1}{F_{N}})^{F_{N}} and (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{2 F_{N}} = (\\frac{1}{F_{N}})^{F_{N}} (\\frac{\\partial}{\\partial F_{N}} \\operatorname{F_{c}}{(c_{0},F_{N})})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), log(Mul(Symbol('F_N', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(log(Mul(Symbol('F_N', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["power", 2, "Symbol('F_N', commutative=True)"], "Equality(Pow(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Pow(Derivative(log(Mul(Symbol('F_N', commutative=True), Pow(Symbol('c_0', commutative=True), Integer(-1)))), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Pow(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('F_N', commutative=True)))"], [["times", 4, "Pow(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))"], "Equality(Pow(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Integer(2), Symbol('F_N', commutative=True))), Mul(Pow(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('F_N', commutative=True)), Pow(Derivative(Function('F_c')(Symbol('c_0', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(E,V)} = \\frac{\\partial}{\\partial V} (E + V), then obtain (V + \\operatorname{f_{E}}{(E,V)})^{2} = (V + 1) (V + \\operatorname{f_{E}}{(E,V)})", "derivation": "\\operatorname{f_{E}}{(E,V)} = \\frac{\\partial}{\\partial V} (E + V) and V + \\operatorname{f_{E}}{(E,V)} = V + \\frac{\\partial}{\\partial V} (E + V) and (V + \\operatorname{f_{E}}{(E,V)})^{2} = (V + \\operatorname{f_{E}}{(E,V)}) (V + \\frac{\\partial}{\\partial V} (E + V)) and (V + \\operatorname{f_{E}}{(E,V)})^{2} = (V + 1) (V + \\operatorname{f_{E}}{(E,V)})", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True)), Derivative(Add(Symbol('E', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["add", 1, "Symbol('V', commutative=True)"], "Equality(Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True))), Add(Symbol('V', commutative=True), Derivative(Add(Symbol('E', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"], [["times", 2, "Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True)))"], "Equality(Pow(Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True))), Integer(2)), Mul(Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True))), Add(Symbol('V', commutative=True), Derivative(Add(Symbol('E', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Pow(Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True))), Integer(2)), Mul(Add(Symbol('V', commutative=True), Integer(1)), Add(Symbol('V', commutative=True), Function('f_E')(Symbol('E', commutative=True), Symbol('V', commutative=True)))))"]]}, {"prompt": "Given G{(\\mu_0)} = \\sin{(\\mu_0)}, then obtain \\frac{G^{2}{(\\mu_0)} \\int \\frac{d}{d \\mu_0} G{(\\mu_0)} \\sin{(\\mu_0)} d\\mu_0}{\\sin{(\\mu_0)}} = \\frac{G^{2}{(\\mu_0)} \\int \\frac{d}{d \\mu_0} \\sin^{2}{(\\mu_0)} d\\mu_0}{\\sin{(\\mu_0)}}", "derivation": "G{(\\mu_0)} = \\sin{(\\mu_0)} and G{(\\mu_0)} \\sin{(\\mu_0)} = \\sin^{2}{(\\mu_0)} and \\frac{d}{d \\mu_0} G{(\\mu_0)} \\sin{(\\mu_0)} = \\frac{d}{d \\mu_0} \\sin^{2}{(\\mu_0)} and \\int \\frac{d}{d \\mu_0} G{(\\mu_0)} \\sin{(\\mu_0)} d\\mu_0 = \\int \\frac{d}{d \\mu_0} \\sin^{2}{(\\mu_0)} d\\mu_0 and \\frac{G^{2}{(\\mu_0)} \\int \\frac{d}{d \\mu_0} G{(\\mu_0)} \\sin{(\\mu_0)} d\\mu_0}{\\sin{(\\mu_0)}} = \\frac{G^{2}{(\\mu_0)} \\int \\frac{d}{d \\mu_0} \\sin^{2}{(\\mu_0)} d\\mu_0}{\\sin{(\\mu_0)}}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Derivative(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Derivative(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Derivative(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 4, "Mul(Pow(Function('G')(Symbol('\\\\mu_0', commutative=True)), Integer(-2)), sin(Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Pow(Function('G')(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Derivative(Mul(Function('G')(Symbol('\\\\mu_0', commutative=True)), sin(Symbol('\\\\mu_0', commutative=True))), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Function('G')(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(-1)), Integral(Derivative(Pow(sin(Symbol('\\\\mu_0', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True), Integer(1))), Tuple(Symbol('\\\\mu_0', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(c,x,\\eta)} = \\frac{c x}{\\eta}, then obtain (c + \\operatorname{m_{s}}{(c,x,\\eta)}) (\\operatorname{m_{s}}{(c,x,\\eta)} - \\frac{1}{\\eta})^{- c} = (c + \\frac{c x}{\\eta}) (\\operatorname{m_{s}}{(c,x,\\eta)} - \\frac{1}{\\eta})^{- c}", "derivation": "\\operatorname{m_{s}}{(c,x,\\eta)} = \\frac{c x}{\\eta} and c + \\operatorname{m_{s}}{(c,x,\\eta)} = c + \\frac{c x}{\\eta} and \\operatorname{m_{s}}{(c,x,\\eta)} - \\frac{1}{\\eta} = \\frac{c x}{\\eta} - \\frac{1}{\\eta} and (c + \\operatorname{m_{s}}{(c,x,\\eta)}) (\\frac{c x}{\\eta} - \\frac{1}{\\eta})^{- c} = (c + \\frac{c x}{\\eta}) (\\frac{c x}{\\eta} - \\frac{1}{\\eta})^{- c} and (c + \\operatorname{m_{s}}{(c,x,\\eta)}) (\\operatorname{m_{s}}{(c,x,\\eta)} - \\frac{1}{\\eta})^{- c} = (c + \\frac{c x}{\\eta}) (\\operatorname{m_{s}}{(c,x,\\eta)} - \\frac{1}{\\eta})^{- c}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('c', commutative=True)"], "Equality(Add(Symbol('c', commutative=True), Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), Add(Symbol('c', commutative=True), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True))))"], [["minus", 1, "Pow(Symbol('\\\\eta', commutative=True), Integer(-1))"], "Equality(Add(Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))))"], [["divide", 2, "Pow(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Symbol('c', commutative=True))"], "Equality(Mul(Add(Symbol('c', commutative=True), Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), Pow(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True))), Pow(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Mul(Integer(-1), Symbol('c', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Add(Symbol('c', commutative=True), Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True))), Pow(Add(Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Mul(Integer(-1), Symbol('c', commutative=True)))), Mul(Add(Symbol('c', commutative=True), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), Symbol('c', commutative=True), Symbol('x', commutative=True))), Pow(Add(Function('m_s')(Symbol('c', commutative=True), Symbol('x', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\eta', commutative=True), Integer(-1)))), Mul(Integer(-1), Symbol('c', commutative=True)))))"]]}, {"prompt": "Given x{(\\tilde{g}^*,f_{E})} = \\sin{((\\tilde{g}^*)^{f_{E}})}, then obtain \\frac{\\partial}{\\partial \\tilde{g}^*} (x{(\\tilde{g}^*,f_{E})} + \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}} = \\frac{\\partial}{\\partial \\tilde{g}^*} (2 \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}}", "derivation": "x{(\\tilde{g}^*,f_{E})} = \\sin{((\\tilde{g}^*)^{f_{E}})} and x{(\\tilde{g}^*,f_{E})} + \\sin{((\\tilde{g}^*)^{f_{E}})} = 2 \\sin{((\\tilde{g}^*)^{f_{E}})} and (x{(\\tilde{g}^*,f_{E})} + \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}} = (2 \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}} and \\frac{\\partial}{\\partial \\tilde{g}^*} (x{(\\tilde{g}^*,f_{E})} + \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}} = \\frac{\\partial}{\\partial \\tilde{g}^*} (2 \\sin{((\\tilde{g}^*)^{f_{E}})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True))))"], [["add", 1, "sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))"], "Equality(Add(Function('x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))), Mul(Integer(2), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))))"], [["power", 2, "Symbol('f_E', commutative=True)"], "Equality(Pow(Add(Function('x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Pow(Mul(Integer(2), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Pow(Add(Function('x')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(2), sin(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('f_E', commutative=True)))), Symbol('f_E', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(s,f^{*})} = f^{*} + s and \\operatorname{A_{z}}{(s,f^{*})} = f^{*} + s and \\ddot{x}{(s,f^{*})} = \\int M{(s,f^{*})} df^{*}, then obtain \\ddot{x}{(s,f^{*})} = \\int (f^{*} + s) df^{*}", "derivation": "M{(s,f^{*})} = f^{*} + s and \\int M{(s,f^{*})} df^{*} = \\int (f^{*} + s) df^{*} and \\operatorname{A_{z}}{(s,f^{*})} = f^{*} + s and \\operatorname{A_{z}}{(s,f^{*})} = M{(s,f^{*})} and \\ddot{x}{(s,f^{*})} = \\int M{(s,f^{*})} df^{*} and \\int \\operatorname{A_{z}}{(s,f^{*})} df^{*} = \\int (f^{*} + s) df^{*} and \\int M{(s,f^{*})} df^{*} = \\int \\operatorname{A_{z}}{(s,f^{*})} df^{*} and \\ddot{x}{(s,f^{*})} = \\int \\operatorname{A_{z}}{(s,f^{*})} df^{*} and \\ddot{x}{(s,f^{*})} = \\int (f^{*} + s) df^{*}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('f^*', commutative=True)"], "Equality(Integral(Function('M')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], ["renaming_premise", "Equality(Function('A_z')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('f^*', commutative=True), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('A_z')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Function('M')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\ddot{x}')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Integral(Function('M')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integral(Function('A_z')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Add(Symbol('f^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 6], "Equality(Integral(Function('M')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Function('A_z')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Function('\\\\ddot{x}')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Integral(Function('A_z')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 8], "Equality(Function('\\\\ddot{x}')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Integral(Add(Symbol('f^*', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(E,v_{2})} = E - v_{2}, then obtain \\int (\\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} + \\frac{\\partial}{\\partial v_{2}} \\int \\mathbb{I}{(E,v_{2})} dv_{2}) dv_{2} = \\int 2 \\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} dv_{2}", "derivation": "\\mathbb{I}{(E,v_{2})} = E - v_{2} and \\int \\mathbb{I}{(E,v_{2})} dv_{2} = \\int (E - v_{2}) dv_{2} and \\frac{\\partial}{\\partial v_{2}} \\int \\mathbb{I}{(E,v_{2})} dv_{2} = \\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} and \\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} + \\frac{\\partial}{\\partial v_{2}} \\int \\mathbb{I}{(E,v_{2})} dv_{2} = 2 \\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} and \\int (\\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} + \\frac{\\partial}{\\partial v_{2}} \\int \\mathbb{I}{(E,v_{2})} dv_{2}) dv_{2} = \\int 2 \\frac{\\partial}{\\partial v_{2}} \\int (E - v_{2}) dv_{2} dv_{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))))"], [["integrate", 1, "Symbol('v_2', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))))"], [["differentiate", 2, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('v_2', commutative=True)"], "Equality(Integral(Add(Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('E', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('v_2', commutative=True))), Integral(Mul(Integer(2), Derivative(Integral(Add(Symbol('E', commutative=True), Mul(Integer(-1), Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True))), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('v_2', commutative=True))))"]]}, {"prompt": "Given \\chi{(\\dot{y})} = \\cos{(\\cos{(\\dot{y})})}, then derive \\frac{d}{d \\dot{y}} \\chi{(\\dot{y})} - 1 = \\sin{(\\dot{y})} \\sin{(\\cos{(\\dot{y})})} - 1, then obtain \\frac{d}{d \\dot{y}} \\cos{(\\cos{(\\dot{y})})} - 1 = \\sin{(\\dot{y})} \\sin{(\\cos{(\\dot{y})})} - 1", "derivation": "\\chi{(\\dot{y})} = \\cos{(\\cos{(\\dot{y})})} and - \\dot{y} + \\chi{(\\dot{y})} = - \\dot{y} + \\cos{(\\cos{(\\dot{y})})} and \\frac{d}{d \\dot{y}} (- \\dot{y} + \\chi{(\\dot{y})}) = \\frac{d}{d \\dot{y}} (- \\dot{y} + \\cos{(\\cos{(\\dot{y})})}) and \\frac{d}{d \\dot{y}} \\chi{(\\dot{y})} - 1 = \\sin{(\\dot{y})} \\sin{(\\cos{(\\dot{y})})} - 1 and \\frac{d}{d \\dot{y}} \\cos{(\\cos{(\\dot{y})})} - 1 = \\sin{(\\dot{y})} \\sin{(\\cos{(\\dot{y})})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('\\\\dot{y}', commutative=True)), cos(cos(Symbol('\\\\dot{y}', commutative=True))))"], [["minus", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\chi')(Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), cos(cos(Symbol('\\\\dot{y}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\chi')(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), cos(cos(Symbol('\\\\dot{y}', commutative=True)))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\chi')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(sin(Symbol('\\\\dot{y}', commutative=True)), sin(cos(Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(cos(cos(Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(-1)), Add(Mul(sin(Symbol('\\\\dot{y}', commutative=True)), sin(cos(Symbol('\\\\dot{y}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})}, then obtain 2 e^{\\mathbf{B}} \\frac{d}{d \\mathbf{B}} \\operatorname{n_{2}}{(\\mathbf{B})} = (e^{\\mathbf{B}} \\cos{(e^{\\mathbf{B}})} + \\frac{d}{d \\mathbf{B}} \\operatorname{n_{2}}{(\\mathbf{B})}) e^{\\mathbf{B}}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{B})} = \\sin{(e^{\\mathbf{B}})} and 2 \\operatorname{n_{2}}{(\\mathbf{B})} = \\operatorname{n_{2}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})} and \\frac{d}{d \\mathbf{B}} 2 \\operatorname{n_{2}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} (\\operatorname{n_{2}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})}) and e^{\\mathbf{B}} \\frac{d}{d \\mathbf{B}} 2 \\operatorname{n_{2}}{(\\mathbf{B})} = e^{\\mathbf{B}} \\frac{d}{d \\mathbf{B}} (\\operatorname{n_{2}}{(\\mathbf{B})} + \\sin{(e^{\\mathbf{B}})}) and 2 e^{\\mathbf{B}} \\frac{d}{d \\mathbf{B}} \\operatorname{n_{2}}{(\\mathbf{B})} = (e^{\\mathbf{B}} \\cos{(e^{\\mathbf{B}})} + \\frac{d}{d \\mathbf{B}} \\operatorname{n_{2}}{(\\mathbf{B})}) e^{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True))))"], [["add", 1, "Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["times", 3, "exp(Symbol('\\\\mathbf{B}', commutative=True))"], "Equality(Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Mul(Integer(2), Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Add(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), sin(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), exp(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Add(Mul(exp(Symbol('\\\\mathbf{B}', commutative=True)), cos(exp(Symbol('\\\\mathbf{B}', commutative=True)))), Derivative(Function('n_2')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), exp(Symbol('\\\\mathbf{B}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(A)} = e^{e^{A}}, then derive \\int \\operatorname{A_{z}}{(A)} dA = P_{e} + \\operatorname{Ei}{(e^{A})}, then obtain \\frac{\\partial}{\\partial A} (P_{e} + \\operatorname{Ei}{(e^{A})}) \\frac{d}{d A} \\int \\operatorname{A_{z}}{(A)} dA = (\\frac{\\partial}{\\partial A} (P_{e} + \\operatorname{Ei}{(e^{A})}))^{2}", "derivation": "\\operatorname{A_{z}}{(A)} = e^{e^{A}} and \\int \\operatorname{A_{z}}{(A)} dA = \\int e^{e^{A}} dA and \\int \\operatorname{A_{z}}{(A)} dA = P_{e} + \\operatorname{Ei}{(e^{A})} and \\int e^{e^{A}} dA = P_{e} + \\operatorname{Ei}{(e^{A})} and \\frac{d}{d A} \\int \\operatorname{A_{z}}{(A)} dA = \\frac{d}{d A} \\int e^{e^{A}} dA and (\\frac{d}{d A} \\int \\operatorname{A_{z}}{(A)} dA) \\frac{d}{d A} \\int e^{e^{A}} dA = (\\frac{d}{d A} \\int e^{e^{A}} dA)^{2} and \\frac{\\partial}{\\partial A} (P_{e} + \\operatorname{Ei}{(e^{A})}) \\frac{d}{d A} \\int \\operatorname{A_{z}}{(A)} dA = (\\frac{\\partial}{\\partial A} (P_{e} + \\operatorname{Ei}{(e^{A})}))^{2}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('A', commutative=True)), exp(exp(Symbol('A', commutative=True))))"], [["integrate", 1, "Symbol('A', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('A_z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Add(Symbol('P_e', commutative=True), Ei(exp(Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Add(Symbol('P_e', commutative=True), Ei(exp(Symbol('A', commutative=True)))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Integral(Function('A_z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["times", 5, "Derivative(Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Integral(Function('A_z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))), Pow(Derivative(Integral(exp(exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Derivative(Add(Symbol('P_e', commutative=True), Ei(exp(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Integral(Function('A_z')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('P_e', commutative=True), Ei(exp(Symbol('A', commutative=True)))), Tuple(Symbol('A', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(z)} = \\log{(z)} and f{(\\hat{x}_0,h)} = \\hat{x}_0 + h, then obtain \\frac{\\partial}{\\partial \\hat{x}_0} f{(\\hat{x}_0,h)} = 1", "derivation": "\\operatorname{P_{e}}{(z)} = \\log{(z)} and f{(\\hat{x}_0,h)} = \\hat{x}_0 + h and \\operatorname{P_{e}}{(z)} + f{(\\hat{x}_0,h)} = \\hat{x}_0 + h + \\operatorname{P_{e}}{(z)} and \\frac{\\partial}{\\partial \\hat{x}_0} (\\operatorname{P_{e}}{(z)} + f{(\\hat{x}_0,h)}) = \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + h + \\operatorname{P_{e}}{(z)}) and \\frac{\\partial}{\\partial \\hat{x}_0} (f{(\\hat{x}_0,h)} + \\log{(z)}) = \\frac{\\partial}{\\partial \\hat{x}_0} (\\hat{x}_0 + h + \\log{(z)}) and \\frac{\\partial}{\\partial \\hat{x}_0} f{(\\hat{x}_0,h)} = 1", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], ["get_premise", "Equality(Function('f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)))"], [["add", 2, "Function('P_e')(Symbol('z', commutative=True))"], "Equality(Add(Function('P_e')(Symbol('z', commutative=True)), Function('f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True))), Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True), Function('P_e')(Symbol('z', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Derivative(Add(Function('P_e')(Symbol('z', commutative=True)), Function('f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True), Function('P_e')(Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Function('f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), log(Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True), log(Symbol('z', commutative=True))), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('f')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\hat{x}_0', commutative=True), Integer(1))), Integer(1))"]]}, {"prompt": "Given r{(g_{\\varepsilon},\\Omega)} = (e^{\\Omega})^{g_{\\varepsilon}} and \\mathbf{v}{(g_{\\varepsilon},\\Omega)} = (e^{\\Omega})^{2 g_{\\varepsilon}}, then obtain \\cos{(\\mathbf{v}{(g_{\\varepsilon},\\Omega)} + r{(g_{\\varepsilon},\\Omega)} (e^{\\Omega})^{g_{\\varepsilon}})} = \\cos{(2 \\mathbf{v}{(g_{\\varepsilon},\\Omega)})}", "derivation": "r{(g_{\\varepsilon},\\Omega)} = (e^{\\Omega})^{g_{\\varepsilon}} and r{(g_{\\varepsilon},\\Omega)} (e^{\\Omega})^{g_{\\varepsilon}} = (e^{\\Omega})^{2 g_{\\varepsilon}} and r{(g_{\\varepsilon},\\Omega)} (e^{\\Omega})^{g_{\\varepsilon}} + (e^{\\Omega})^{2 g_{\\varepsilon}} = 2 (e^{\\Omega})^{2 g_{\\varepsilon}} and \\cos{(r{(g_{\\varepsilon},\\Omega)} (e^{\\Omega})^{g_{\\varepsilon}} + (e^{\\Omega})^{2 g_{\\varepsilon}})} = \\cos{(2 (e^{\\Omega})^{2 g_{\\varepsilon}})} and \\mathbf{v}{(g_{\\varepsilon},\\Omega)} = (e^{\\Omega})^{2 g_{\\varepsilon}} and \\cos{(\\mathbf{v}{(g_{\\varepsilon},\\Omega)} + r{(g_{\\varepsilon},\\Omega)} (e^{\\Omega})^{g_{\\varepsilon}})} = \\cos{(2 \\mathbf{v}{(g_{\\varepsilon},\\Omega)})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Function('r')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 2, "Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Add(Mul(Function('r')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Integer(2), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["cos", 3], "Equality(cos(Add(Mul(Function('r')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))), cos(Mul(Integer(2), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(cos(Add(Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Function('r')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)), Pow(exp(Symbol('\\\\Omega', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))))), cos(Mul(Integer(2), Function('\\\\mathbf{v}')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(k,\\lambda)} = - k + e^{\\lambda}, then obtain - k (- k + e^{\\lambda}) + (- k + e^{\\lambda}) e^{\\lambda} + e^{\\lambda} = k^{2} - 2 k e^{\\lambda} + e^{2 \\lambda} + e^{\\lambda}", "derivation": "\\operatorname{f^{*}}{(k,\\lambda)} = - k + e^{\\lambda} and (- k + e^{\\lambda} + 1) \\operatorname{f^{*}}{(k,\\lambda)} = (- k + e^{\\lambda}) (- k + e^{\\lambda} + 1) and k + (- k + e^{\\lambda} + 1) \\operatorname{f^{*}}{(k,\\lambda)} = k + (- k + e^{\\lambda}) (- k + e^{\\lambda} + 1) and - k \\operatorname{f^{*}}{(k,\\lambda)} + k + \\operatorname{f^{*}}{(k,\\lambda)} e^{\\lambda} + \\operatorname{f^{*}}{(k,\\lambda)} = k^{2} - 2 k e^{\\lambda} + e^{2 \\lambda} + e^{\\lambda} and - k (- k + e^{\\lambda}) + (- k + e^{\\lambda}) e^{\\lambda} + e^{\\lambda} = k^{2} - 2 k e^{\\lambda} + e^{2 \\lambda} + e^{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True)), Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)), Integer(1))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)), Integer(1)), Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)), Integer(1))))"], [["minus", 2, "Mul(Integer(-1), Symbol('k', commutative=True))"], "Equality(Add(Symbol('k', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)), Integer(1)), Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True)))), Add(Symbol('k', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)), Integer(1)))))"], [["expand", 3], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True), Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('k', commutative=True), Mul(Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Function('f^*')(Symbol('k', commutative=True), Symbol('\\\\lambda', commutative=True))), Add(Pow(Symbol('k', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('k', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('k', commutative=True), Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('k', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))), Add(Pow(Symbol('k', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('k', commutative=True), exp(Symbol('\\\\lambda', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\lambda', commutative=True))), exp(Symbol('\\\\lambda', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\tilde{g},\\mathbf{B})} = \\frac{\\mathbf{B}}{\\tilde{g}}, then derive \\frac{\\partial}{\\partial \\mathbf{B}} \\hat{x}{(\\tilde{g},\\mathbf{B})} = \\frac{1}{\\tilde{g}}, then obtain \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\tilde{g}} = \\frac{1}{\\tilde{g}}", "derivation": "\\hat{x}{(\\tilde{g},\\mathbf{B})} = \\frac{\\mathbf{B}}{\\tilde{g}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\hat{x}{(\\tilde{g},\\mathbf{B})} = \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\tilde{g}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\hat{x}{(\\tilde{g},\\mathbf{B})} = \\frac{1}{\\tilde{g}} and \\frac{\\partial}{\\partial \\mathbf{B}} \\frac{\\mathbf{B}}{\\tilde{g}} = \\frac{1}{\\tilde{g}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Symbol('\\\\mathbf{B}', commutative=True), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\tilde{g}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{f}{(\\Psi,\\hat{x}_0,f^{*})} = \\Psi + \\hat{x}_0 + f^{*}, then obtain \\log{(\\Psi (\\Psi + \\mathbf{f}^{\\Psi}{(\\Psi,\\hat{x}_0,f^{*})}))} = \\log{(\\Psi (\\Psi + (\\Psi + \\hat{x}_0 + f^{*})^{\\Psi}))}", "derivation": "\\mathbf{f}{(\\Psi,\\hat{x}_0,f^{*})} = \\Psi + \\hat{x}_0 + f^{*} and \\mathbf{f}^{\\Psi}{(\\Psi,\\hat{x}_0,f^{*})} = (\\Psi + \\hat{x}_0 + f^{*})^{\\Psi} and \\Psi + \\mathbf{f}^{\\Psi}{(\\Psi,\\hat{x}_0,f^{*})} = \\Psi + (\\Psi + \\hat{x}_0 + f^{*})^{\\Psi} and \\Psi (\\Psi + \\mathbf{f}^{\\Psi}{(\\Psi,\\hat{x}_0,f^{*})}) = \\Psi (\\Psi + (\\Psi + \\hat{x}_0 + f^{*})^{\\Psi}) and \\log{(\\Psi (\\Psi + \\mathbf{f}^{\\Psi}{(\\Psi,\\hat{x}_0,f^{*})}))} = \\log{(\\Psi (\\Psi + (\\Psi + \\hat{x}_0 + f^{*})^{\\Psi}))}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True)))"], [["add", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Add(Symbol('\\\\Psi', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True))))"], [["times", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True)))), Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True)))))"], [["log", 4], "Equality(log(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Pow(Function('\\\\mathbf{f}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True))))), log(Mul(Symbol('\\\\Psi', commutative=True), Add(Symbol('\\\\Psi', commutative=True), Pow(Add(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hat{x}_0', commutative=True), Symbol('f^*', commutative=True)), Symbol('\\\\Psi', commutative=True))))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(n_{2},\\varphi)} = \\varphi^{n_{2}}, then obtain ((\\varphi^{n_{2}} + n_{2} \\dot{\\mathbf{r}}{(n_{2},\\varphi)})^{\\varphi})^{\\varphi} = ((\\varphi^{n_{2}} n_{2} + \\varphi^{n_{2}})^{\\varphi})^{\\varphi}", "derivation": "\\dot{\\mathbf{r}}{(n_{2},\\varphi)} = \\varphi^{n_{2}} and n_{2} \\dot{\\mathbf{r}}{(n_{2},\\varphi)} = \\varphi^{n_{2}} n_{2} and \\varphi^{n_{2}} + n_{2} \\dot{\\mathbf{r}}{(n_{2},\\varphi)} = \\varphi^{n_{2}} n_{2} + \\varphi^{n_{2}} and (\\varphi^{n_{2}} + n_{2} \\dot{\\mathbf{r}}{(n_{2},\\varphi)})^{\\varphi} = (\\varphi^{n_{2}} n_{2} + \\varphi^{n_{2}})^{\\varphi} and ((\\varphi^{n_{2}} + n_{2} \\dot{\\mathbf{r}}{(n_{2},\\varphi)})^{\\varphi})^{\\varphi} = ((\\varphi^{n_{2}} n_{2} + \\varphi^{n_{2}})^{\\varphi})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)))"], [["times", 1, "Symbol('n_2', commutative=True)"], "Equality(Mul(Symbol('n_2', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)))"], [["add", 2, "Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)))), Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True))))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True))), Symbol('\\\\varphi', commutative=True)))"], [["power", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Add(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('n_2', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('n_2', commutative=True), Symbol('\\\\varphi', commutative=True)))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Add(Mul(Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('n_2', commutative=True))), Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(x,y^{\\prime},S)} = (x + y^{\\prime})^{S}, then derive \\frac{\\partial}{\\partial x} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{S (x + y^{\\prime})^{S}}{x + y^{\\prime}}, then derive \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{S (x + y^{\\prime})^{S}}{x + y^{\\prime}}, then obtain \\frac{\\partial}{\\partial x} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{\\partial}{\\partial y^{\\prime}} (x + y^{\\prime})^{S}", "derivation": "\\operatorname{A_{z}}{(x,y^{\\prime},S)} = (x + y^{\\prime})^{S} and \\frac{\\partial}{\\partial x} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{\\partial}{\\partial x} (x + y^{\\prime})^{S} and \\frac{\\partial}{\\partial x} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{S (x + y^{\\prime})^{S}}{x + y^{\\prime}} and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{\\partial}{\\partial y^{\\prime}} (x + y^{\\prime})^{S} and \\frac{\\partial}{\\partial y^{\\prime}} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{S (x + y^{\\prime})^{S}}{x + y^{\\prime}} and \\frac{\\partial}{\\partial y^{\\prime}} (x + y^{\\prime})^{S} = \\frac{S (x + y^{\\prime})^{S}}{x + y^{\\prime}} and \\frac{\\partial}{\\partial x} \\operatorname{A_{z}}{(x,y^{\\prime},S)} = \\frac{\\partial}{\\partial y^{\\prime}} (x + y^{\\prime})^{S}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True)))"], [["differentiate", 1, "Symbol('x', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Mul(Symbol('S', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True))))"], [["differentiate", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Derivative(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('S', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Derivative(Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Mul(Symbol('S', commutative=True), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Integer(-1)), Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Derivative(Function('A_z')(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('x', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Symbol('S', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(P_{g},n,v_{z})} = \\frac{P_{g} v_{z}}{n}, then obtain \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{v_{z}} + \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{n} + \\frac{1}{n} = \\frac{P_{g}}{n} + \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{n} + \\frac{1}{n}", "derivation": "\\operatorname{F_{H}}{(P_{g},n,v_{z})} = \\frac{P_{g} v_{z}}{n} and \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{v_{z}} = \\frac{P_{g}}{n} and \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{v_{z}} + \\frac{1}{n} = \\frac{P_{g}}{n} + \\frac{1}{n} and \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{v_{z}} + \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{n} + \\frac{1}{n} = \\frac{P_{g}}{n} + \\frac{\\operatorname{F_{H}}{(P_{g},n,v_{z})}}{n} + \\frac{1}{n}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["divide", 1, "Symbol('v_z', commutative=True)"], "Equality(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["add", 2, "Pow(Symbol('n', commutative=True), Integer(-1))"], "Equality(Add(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["add", 3, "Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('v_z', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))), Add(Mul(Symbol('P_g', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Function('F_H')(Symbol('P_g', commutative=True), Symbol('n', commutative=True), Symbol('v_z', commutative=True))), Pow(Symbol('n', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_f{(C,l)} = \\log{(l)}^{C} and V{(C,l)} = \\int \\mathbf{J}_f{(C,l)} dC, then obtain \\frac{\\frac{\\partial}{\\partial C} \\mathbf{J}_f{(C,l)} \\frac{\\partial}{\\partial l} \\int \\log{(l)}^{C} dC}{C} = \\frac{\\frac{\\partial}{\\partial C} \\log{(l)}^{C} \\frac{\\partial}{\\partial l} \\int \\log{(l)}^{C} dC}{C}", "derivation": "\\mathbf{J}_f{(C,l)} = \\log{(l)}^{C} and \\frac{\\partial}{\\partial C} \\mathbf{J}_f{(C,l)} = \\frac{\\partial}{\\partial C} \\log{(l)}^{C} and V{(C,l)} = \\int \\mathbf{J}_f{(C,l)} dC and V{(C,l)} = \\int \\log{(l)}^{C} dC and \\frac{\\partial}{\\partial l} V{(C,l)} = \\frac{\\partial}{\\partial l} \\int \\log{(l)}^{C} dC and \\frac{\\frac{\\partial}{\\partial C} \\mathbf{J}_f{(C,l)}}{C} = \\frac{\\frac{\\partial}{\\partial C} \\log{(l)}^{C}}{C} and \\frac{\\frac{\\partial}{\\partial l} V{(C,l)} \\frac{\\partial}{\\partial C} \\mathbf{J}_f{(C,l)}}{C} = \\frac{\\frac{\\partial}{\\partial l} V{(C,l)} \\frac{\\partial}{\\partial C} \\log{(l)}^{C}}{C} and \\frac{\\frac{\\partial}{\\partial C} \\mathbf{J}_f{(C,l)} \\frac{\\partial}{\\partial l} \\int \\log{(l)}^{C} dC}{C} = \\frac{\\frac{\\partial}{\\partial C} \\log{(l)}^{C} \\frac{\\partial}{\\partial l} \\int \\log{(l)}^{C} dC}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integral(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Integral(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 4, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Integral(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('C', commutative=True)"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["times", 6, "Derivative(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('V')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('C', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Derivative(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Integral(Pow(log(Symbol('l', commutative=True)), Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\eta,M_{E})} = e^{M_{E} \\eta}, then obtain \\frac{e^{\\operatorname{v_{2}}^{\\eta}{(\\eta,M_{E})}}}{\\eta} = \\frac{e^{(e^{M_{E} \\eta})^{\\eta}}}{\\eta}", "derivation": "\\operatorname{v_{2}}{(\\eta,M_{E})} = e^{M_{E} \\eta} and \\operatorname{v_{2}}^{\\eta}{(\\eta,M_{E})} = (e^{M_{E} \\eta})^{\\eta} and e^{\\operatorname{v_{2}}^{\\eta}{(\\eta,M_{E})}} = e^{(e^{M_{E} \\eta})^{\\eta}} and \\frac{e^{\\operatorname{v_{2}}^{\\eta}{(\\eta,M_{E})}}}{\\eta} = \\frac{e^{(e^{M_{E} \\eta})^{\\eta}}}{\\eta}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('v_2')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(exp(Mul(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('v_2')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\eta', commutative=True))), exp(Pow(exp(Mul(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True))))"], [["divide", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Pow(Function('v_2')(Symbol('\\\\eta', commutative=True), Symbol('M_E', commutative=True)), Symbol('\\\\eta', commutative=True)))), Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(-1)), exp(Pow(exp(Mul(Symbol('M_E', commutative=True), Symbol('\\\\eta', commutative=True))), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\sigma_{x}{(\\dot{x},Z)} = \\log{(\\frac{\\dot{x}}{Z})}, then obtain \\frac{\\partial}{\\partial Z} (\\dot{x} + \\sigma_{x}{(\\dot{x},Z)})^{Z} = \\frac{\\partial}{\\partial Z} (\\dot{x} + \\log{(\\frac{\\dot{x}}{Z})})^{Z}", "derivation": "\\sigma_{x}{(\\dot{x},Z)} = \\log{(\\frac{\\dot{x}}{Z})} and \\dot{x} + \\sigma_{x}{(\\dot{x},Z)} = \\dot{x} + \\log{(\\frac{\\dot{x}}{Z})} and (\\dot{x} + \\sigma_{x}{(\\dot{x},Z)})^{Z} = (\\dot{x} + \\log{(\\frac{\\dot{x}}{Z})})^{Z} and \\frac{\\partial}{\\partial Z} (\\dot{x} + \\sigma_{x}{(\\dot{x},Z)})^{Z} = \\frac{\\partial}{\\partial Z} (\\dot{x} + \\log{(\\frac{\\dot{x}}{Z})})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True)), log(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True))))"], [["add", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), log(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))))"], [["power", 2, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Pow(Add(Symbol('\\\\dot{x}', commutative=True), log(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))), Symbol('Z', commutative=True)))"], [["differentiate", 3, "Symbol('Z', commutative=True)"], "Equality(Derivative(Pow(Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\sigma_x')(Symbol('\\\\dot{x}', commutative=True), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\dot{x}', commutative=True), log(Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), Symbol('\\\\dot{x}', commutative=True)))), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\omega{(n)} = \\int \\log{(n)} dn, then derive \\frac{d}{d n} (\\omega{(n)} - 1) = \\frac{\\partial}{\\partial n} (E_{\\lambda} + n \\log{(n)} - n - 1), then obtain \\frac{d}{d n} (\\int \\log{(n)} dn - 1) + 1 = \\frac{\\partial}{\\partial n} (E_{\\lambda} + n \\log{(n)} - n - 1) + 1", "derivation": "\\omega{(n)} = \\int \\log{(n)} dn and \\omega{(n)} - 1 = \\int \\log{(n)} dn - 1 and \\frac{d}{d n} (\\omega{(n)} - 1) = \\frac{d}{d n} (\\int \\log{(n)} dn - 1) and \\frac{d}{d n} (\\omega{(n)} - 1) = \\frac{\\partial}{\\partial n} (E_{\\lambda} + n \\log{(n)} - n - 1) and \\frac{d}{d n} (\\int \\log{(n)} dn - 1) = \\frac{\\partial}{\\partial n} (E_{\\lambda} + n \\log{(n)} - n - 1) and \\frac{d}{d n} (\\int \\log{(n)} dn - 1) + 1 = \\frac{\\partial}{\\partial n} (E_{\\lambda} + n \\log{(n)} - n - 1) + 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\omega')(Symbol('n', commutative=True)), Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\omega')(Symbol('n', commutative=True)), Integer(-1)), Add(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)))"], [["differentiate", 2, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Function('\\\\omega')(Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Add(Function('\\\\omega')(Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Add(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Derivative(Add(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)), Add(Derivative(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(S)} = \\log{(S)} and T{(S)} = \\operatorname{J_{\\varepsilon}}{(S)} - \\log{(S)}, then derive \\frac{d}{d S} T{(S)} = 0, then obtain \\frac{d}{d S} T{(S)} + 1 = 1", "derivation": "\\operatorname{J_{\\varepsilon}}{(S)} = \\log{(S)} and T{(S)} = \\operatorname{J_{\\varepsilon}}{(S)} - \\log{(S)} and T{(S)} = 0 and \\frac{d}{d S} T{(S)} = \\frac{d}{d S} 0 and \\frac{d}{d S} T{(S)} = 0 and \\frac{d}{d S} T{(S)} + 1 = 1", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True)), log(Symbol('S', commutative=True)))"], ["renaming_premise", "Equality(Function('T')(Symbol('S', commutative=True)), Add(Function('J_{\\\\varepsilon}')(Symbol('S', commutative=True)), Mul(Integer(-1), log(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('T')(Symbol('S', commutative=True)), Integer(0))"], [["differentiate", 3, "Symbol('S', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Function('T')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(0))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Derivative(Function('T')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))), Integer(1)), Integer(1))"]]}, {"prompt": "Given M{(\\hat{x},x^\\prime,\\mathbf{g})} = \\frac{\\mathbf{g} x^\\prime}{\\hat{x}}, then obtain \\frac{\\partial}{\\partial f_{E}} \\frac{\\mathbf{g} a^{\\dagger} f_{E} x^\\prime}{\\hat{x}} = \\frac{\\partial}{\\partial f_{E}} \\frac{\\mathbf{g}^{3} a^{\\dagger} f_{E} (x^\\prime)^{3}}{\\hat{x}^{3} M^{2}{(\\hat{x},x^\\prime,\\mathbf{g})}}", "derivation": "M{(\\hat{x},x^\\prime,\\mathbf{g})} = \\frac{\\mathbf{g} x^\\prime}{\\hat{x}} and a^{\\dagger} f_{E} M{(\\hat{x},x^\\prime,\\mathbf{g})} = \\frac{\\mathbf{g} a^{\\dagger} f_{E} x^\\prime}{\\hat{x}} and a^{\\dagger} f_{E} = \\frac{\\mathbf{g} a^{\\dagger} f_{E} x^\\prime}{\\hat{x} M{(\\hat{x},x^\\prime,\\mathbf{g})}} and \\frac{\\partial}{\\partial f_{E}} a^{\\dagger} f_{E} M{(\\hat{x},x^\\prime,\\mathbf{g})} = \\frac{\\partial}{\\partial f_{E}} \\frac{\\mathbf{g} a^{\\dagger} f_{E} x^\\prime}{\\hat{x}} and \\frac{\\partial}{\\partial f_{E}} \\frac{\\mathbf{g} a^{\\dagger} f_{E} x^\\prime}{\\hat{x}} = \\frac{\\partial}{\\partial f_{E}} \\frac{\\mathbf{g}^{3} a^{\\dagger} f_{E} (x^\\prime)^{3}}{\\hat{x}^{3} M^{2}{(\\hat{x},x^\\prime,\\mathbf{g})}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["times", 1, "Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True))"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["divide", 2, "Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Symbol('x^\\\\prime', commutative=True), Pow(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-1))))"], [["differentiate", 2, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Mul(Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-3)), Pow(Symbol('\\\\mathbf{g}', commutative=True), Integer(3)), Symbol('a^{\\\\dagger}', commutative=True), Symbol('f_E', commutative=True), Pow(Symbol('x^\\\\prime', commutative=True), Integer(3)), Pow(Function('M')(Symbol('\\\\hat{x}', commutative=True), Symbol('x^\\\\prime', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Integer(-2))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{g}{(A_{2},\\mathbf{F})} = \\frac{\\mathbf{F}}{A_{2}}, then derive g_{\\varepsilon} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1 = \\dot{x} + 1 + \\frac{\\mathbf{F}}{A_{2}}, then obtain g_{\\varepsilon} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1 = \\dot{x} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1", "derivation": "\\mathbf{g}{(A_{2},\\mathbf{F})} = \\frac{\\mathbf{F}}{A_{2}} and \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{g}{(A_{2},\\mathbf{F})} = \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\mathbf{F}}{A_{2}} and \\int \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{g}{(A_{2},\\mathbf{F})} d\\mathbf{F} = \\int \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\mathbf{F}}{A_{2}} d\\mathbf{F} and \\int \\frac{\\partial}{\\partial \\mathbf{F}} \\mathbf{g}{(A_{2},\\mathbf{F})} d\\mathbf{F} + 1 = \\int \\frac{\\partial}{\\partial \\mathbf{F}} \\frac{\\mathbf{F}}{A_{2}} d\\mathbf{F} + 1 and g_{\\varepsilon} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1 = \\dot{x} + 1 + \\frac{\\mathbf{F}}{A_{2}} and g_{\\varepsilon} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1 = \\dot{x} + \\mathbf{g}{(A_{2},\\mathbf{F})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integral(Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Integral(Derivative(Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(1)), Add(Integral(Derivative(Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{F}', commutative=True))), Integer(1)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(1)), Add(Symbol('\\\\dot{x}', commutative=True), Integer(1), Mul(Pow(Symbol('A_2', commutative=True), Integer(-1)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(1)), Add(Symbol('\\\\dot{x}', commutative=True), Function('\\\\mathbf{g}')(Symbol('A_2', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(U,v)} = U - v, then derive - U - \\frac{\\int (\\operatorname{f^{*}}{(U,v)} + 1) dv}{v} = - U - \\frac{H - \\frac{v^{2}}{2} + v (U + 1)}{v}, then obtain - U - \\frac{H - \\frac{v^{2}}{2} + v (U + 1)}{v} = - U - \\frac{\\int (U - v + 1) dv}{v}", "derivation": "\\operatorname{f^{*}}{(U,v)} = U - v and \\operatorname{f^{*}}{(U,v)} + 1 = U - v + 1 and \\int (\\operatorname{f^{*}}{(U,v)} + 1) dv = \\int (U - v + 1) dv and - \\frac{\\int (\\operatorname{f^{*}}{(U,v)} + 1) dv}{v} = - \\frac{\\int (U - v + 1) dv}{v} and - U - \\frac{\\int (\\operatorname{f^{*}}{(U,v)} + 1) dv}{v} = - U - \\frac{\\int (U - v + 1) dv}{v} and - U - \\frac{\\int (\\operatorname{f^{*}}{(U,v)} + 1) dv}{v} = - U - \\frac{H - \\frac{v^{2}}{2} + v (U + 1)}{v} and - U - \\frac{H - \\frac{v^{2}}{2} + v (U + 1)}{v} = - U - \\frac{\\int (U - v + 1) dv}{v}", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Integer(1)), Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(1)))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))))"], [["divide", 3, "Mul(Integer(-1), Symbol('v', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True)))), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('U', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))))))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Function('f^*')(Symbol('U', commutative=True), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Symbol('v', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('v', commutative=True), Integer(2))), Mul(Symbol('v', commutative=True), Add(Symbol('U', commutative=True), Integer(1)))))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), Mul(Integer(-1), Pow(Symbol('v', commutative=True), Integer(-1)), Integral(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('v', commutative=True)), Integer(1)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(n_{1})} = \\log{(n_{1})}, then obtain (\\frac{n_{1}}{n_{1} + \\operatorname{v_{t}}{(n_{1})}} + \\frac{\\log{(n_{1})}}{n_{1} + \\operatorname{v_{t}}{(n_{1})}})^{n_{1}} = 1", "derivation": "\\operatorname{v_{t}}{(n_{1})} = \\log{(n_{1})} and n_{1} + \\operatorname{v_{t}}{(n_{1})} = n_{1} + \\log{(n_{1})} and \\frac{n_{1} + \\operatorname{v_{t}}{(n_{1})}}{n_{1} + \\log{(n_{1})}} = 1 and (\\frac{n_{1} + \\operatorname{v_{t}}{(n_{1})}}{n_{1} + \\log{(n_{1})}})^{n_{1}} = 1 and (\\frac{n_{1}}{n_{1} + \\log{(n_{1})}} + \\frac{\\operatorname{v_{t}}{(n_{1})}}{n_{1} + \\log{(n_{1})}})^{n_{1}} = 1 and (\\frac{n_{1}}{n_{1} + \\log{(n_{1})}} + \\frac{\\log{(n_{1})}}{n_{1} + \\log{(n_{1})}})^{n_{1}} = 1 and (\\frac{n_{1}}{n_{1} + \\operatorname{v_{t}}{(n_{1})}} + \\frac{\\log{(n_{1})}}{n_{1} + \\operatorname{v_{t}}{(n_{1})}})^{n_{1}} = 1", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('n_1', commutative=True)), log(Symbol('n_1', commutative=True)))"], [["add", 1, "Symbol('n_1', commutative=True)"], "Equality(Add(Symbol('n_1', commutative=True), Function('v_t')(Symbol('n_1', commutative=True))), Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))))"], [["divide", 2, "Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True)))"], "Equality(Mul(Add(Symbol('n_1', commutative=True), Function('v_t')(Symbol('n_1', commutative=True))), Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('n_1', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('n_1', commutative=True), Function('v_t')(Symbol('n_1', commutative=True))), Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1))), Symbol('n_1', commutative=True)), Integer(1))"], [["expand", 4], "Equality(Pow(Add(Mul(Symbol('n_1', commutative=True), Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1)), Function('v_t')(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Mul(Symbol('n_1', commutative=True), Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('n_1', commutative=True), log(Symbol('n_1', commutative=True))), Integer(-1)), log(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Integer(1))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Pow(Add(Mul(Symbol('n_1', commutative=True), Pow(Add(Symbol('n_1', commutative=True), Function('v_t')(Symbol('n_1', commutative=True))), Integer(-1))), Mul(Pow(Add(Symbol('n_1', commutative=True), Function('v_t')(Symbol('n_1', commutative=True))), Integer(-1)), log(Symbol('n_1', commutative=True)))), Symbol('n_1', commutative=True)), Integer(1))"]]}, {"prompt": "Given Z{(F_{H})} = e^{F_{H}}, then obtain \\iint \\frac{d}{d F_{H}} Z^{2}{(F_{H})} e^{- 4 F_{H}} dF_{H} dF_{H} = \\iint \\frac{d}{d F_{H}} e^{- 2 F_{H}} dF_{H} dF_{H}", "derivation": "Z{(F_{H})} = e^{F_{H}} and Z{(F_{H})} e^{- F_{H}} = 1 and Z{(F_{H})} e^{- 2 F_{H}} = e^{- F_{H}} and e^{- 2 F_{H}} = \\frac{e^{- F_{H}}}{Z{(F_{H})}} and \\frac{d}{d F_{H}} e^{- 2 F_{H}} = \\frac{d}{d F_{H}} \\frac{e^{- F_{H}}}{Z{(F_{H})}} and \\frac{d}{d F_{H}} Z^{2}{(F_{H})} e^{- 4 F_{H}} = \\frac{d}{d F_{H}} e^{- 2 F_{H}} and \\int \\frac{d}{d F_{H}} Z^{2}{(F_{H})} e^{- 4 F_{H}} dF_{H} = \\int \\frac{d}{d F_{H}} e^{- 2 F_{H}} dF_{H} and \\iint \\frac{d}{d F_{H}} Z^{2}{(F_{H})} e^{- 4 F_{H}} dF_{H} dF_{H} = \\iint \\frac{d}{d F_{H}} e^{- 2 F_{H}} dF_{H} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('F_H', commutative=True)), exp(Symbol('F_H', commutative=True)))"], [["divide", 1, "exp(Symbol('F_H', commutative=True))"], "Equality(Mul(Function('Z')(Symbol('F_H', commutative=True)), exp(Mul(Integer(-1), Symbol('F_H', commutative=True)))), Integer(1))"], [["times", 2, "exp(Mul(Integer(-1), Symbol('F_H', commutative=True)))"], "Equality(Mul(Function('Z')(Symbol('F_H', commutative=True)), exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True)))), exp(Mul(Integer(-1), Symbol('F_H', commutative=True))))"], [["divide", 3, "Function('Z')(Symbol('F_H', commutative=True))"], "Equality(exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('F_H', commutative=True)))))"], [["differentiate", 4, "Symbol('F_H', commutative=True)"], "Equality(Derivative(exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(-1)), exp(Mul(Integer(-1), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(4), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))))"], [["integrate", 6, "Symbol('F_H', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(4), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))), Integral(Derivative(exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True))))"], [["integrate", 7, "Symbol('F_H', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Function('Z')(Symbol('F_H', commutative=True)), Integer(2)), exp(Mul(Integer(-1), Integer(4), Symbol('F_H', commutative=True)))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))), Integral(Derivative(exp(Mul(Integer(-1), Integer(2), Symbol('F_H', commutative=True))), Tuple(Symbol('F_H', commutative=True), Integer(1))), Tuple(Symbol('F_H', commutative=True)), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\rho_b)} = e^{\\sin{(\\rho_b)}}, then obtain (\\rho_b \\mathbf{H}{(\\rho_b)})^{\\rho_b} - (\\rho_b e^{\\sin{(\\rho_b)}})^{\\rho_b} = 0", "derivation": "\\mathbf{H}{(\\rho_b)} = e^{\\sin{(\\rho_b)}} and \\rho_b \\mathbf{H}{(\\rho_b)} = \\rho_b e^{\\sin{(\\rho_b)}} and (\\rho_b \\mathbf{H}{(\\rho_b)})^{\\rho_b} = (\\rho_b e^{\\sin{(\\rho_b)}})^{\\rho_b} and (\\rho_b \\mathbf{H}{(\\rho_b)})^{\\rho_b} - (\\rho_b e^{\\sin{(\\rho_b)}})^{\\rho_b} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True)), exp(sin(Symbol('\\\\rho_b', commutative=True))))"], [["times", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), exp(sin(Symbol('\\\\rho_b', commutative=True)))))"], [["power", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Pow(Mul(Symbol('\\\\rho_b', commutative=True), exp(sin(Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True)))"], [["minus", 3, "Pow(Mul(Symbol('\\\\rho_b', commutative=True), exp(sin(Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Pow(Mul(Symbol('\\\\rho_b', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\rho_b', commutative=True))), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), Pow(Mul(Symbol('\\\\rho_b', commutative=True), exp(sin(Symbol('\\\\rho_b', commutative=True)))), Symbol('\\\\rho_b', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\mathbf{r}{(k,i)} = \\int i^{k} di, then obtain - \\frac{\\mathbf{r}{(k,i)}}{\\int i^{k} di} = -1", "derivation": "\\mathbf{r}{(k,i)} = \\int i^{k} di and \\frac{\\mathbf{r}{(k,i)}}{\\int i^{k} di} = 1 and \\frac{\\mathbf{r}^{2}{(k,i)}}{(\\int i^{k} di)^{2}} = \\frac{\\mathbf{r}{(k,i)}}{\\int i^{k} di} and \\frac{\\mathbf{r}^{2}{(k,i)}}{(\\int i^{k} di)^{2}} = 1 and - \\frac{\\mathbf{r}^{2}{(k,i)}}{(\\int i^{k} di)^{2}} = -1 and - \\frac{\\mathbf{r}{(k,i)}}{\\int i^{k} di} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["divide", 1, "Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Mul(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1)))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Integer(2)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-2))), Mul(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Integer(2)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-2))), Integer(1))"], [["divide", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Integer(2)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-2))), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('k', commutative=True), Symbol('i', commutative=True)), Pow(Integral(Pow(Symbol('i', commutative=True), Symbol('k', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\sigma_x,r_{0})} = \\sigma_x + r_{0}, then obtain \\frac{\\partial}{\\partial \\sigma_x} (\\frac{\\partial}{\\partial \\sigma_x} \\operatorname{C_{2}}{(\\sigma_x,r_{0})})^{\\sigma_x} = \\frac{\\partial}{\\partial \\sigma_x} (\\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + r_{0}))^{\\sigma_x}", "derivation": "\\operatorname{C_{2}}{(\\sigma_x,r_{0})} = \\sigma_x + r_{0} and \\frac{\\partial}{\\partial \\sigma_x} \\operatorname{C_{2}}{(\\sigma_x,r_{0})} = \\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + r_{0}) and (\\frac{\\partial}{\\partial \\sigma_x} \\operatorname{C_{2}}{(\\sigma_x,r_{0})})^{\\sigma_x} = (\\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + r_{0}))^{\\sigma_x} and \\frac{\\partial}{\\partial \\sigma_x} (\\frac{\\partial}{\\partial \\sigma_x} \\operatorname{C_{2}}{(\\sigma_x,r_{0})})^{\\sigma_x} = \\frac{\\partial}{\\partial \\sigma_x} (\\frac{\\partial}{\\partial \\sigma_x} (\\sigma_x + r_{0}))^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Derivative(Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)), Pow(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Pow(Derivative(Function('C_2')(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(U)} = \\sin{(e^{U})}, then obtain \\int (\\int - 2 U dU + 2 \\int \\sin{(e^{U})} dU) dU = \\int (\\int - 2 U dU + \\int \\phi{(U)} dU + \\int \\sin{(e^{U})} dU) dU", "derivation": "\\phi{(U)} = \\sin{(e^{U})} and - U + \\phi{(U)} = - U + \\sin{(e^{U})} and - 2 U + \\phi{(U)} + \\sin{(e^{U})} = - 2 U + 2 \\sin{(e^{U})} and \\int (- 2 U + \\phi{(U)} + \\sin{(e^{U})}) dU = \\int (- 2 U + 2 \\sin{(e^{U})}) dU and \\int - 2 U dU + \\int \\phi{(U)} dU + \\int \\sin{(e^{U})} dU = \\int - 2 U dU + \\int 2 \\sin{(e^{U})} dU and \\int - 2 U dU + 2 \\int \\sin{(e^{U})} dU = \\int - 2 U dU + \\int 2 \\sin{(e^{U})} dU and \\int - 2 U dU + 2 \\int \\sin{(e^{U})} dU = \\int - 2 U dU + \\int \\phi{(U)} dU + \\int \\sin{(e^{U})} dU and \\int (\\int - 2 U dU + 2 \\int \\sin{(e^{U})} dU) dU = \\int (\\int - 2 U dU + \\int \\phi{(U)} dU + \\int \\sin{(e^{U})} dU) dU", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True))))"], [["minus", 1, "Symbol('U', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('U', commutative=True)), Function('\\\\phi')(Symbol('U', commutative=True))), Add(Mul(Integer(-1), Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Function('\\\\phi')(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), sin(exp(Symbol('U', commutative=True))))))"], [["integrate", 3, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Function('\\\\phi')(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Mul(Integer(2), sin(exp(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))))"], [["expand", 4], "Equality(Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\phi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))), Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Integer(2), sin(exp(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(2), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Integer(2), sin(exp(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(2), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\phi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))))"], [["integrate", 7, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Mul(Integer(2), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True))))), Tuple(Symbol('U', commutative=True))), Integral(Add(Integral(Mul(Integer(-1), Integer(2), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('\\\\phi')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(sin(exp(Symbol('U', commutative=True))), Tuple(Symbol('U', commutative=True)))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(b,v_{1})} = e^{b + v_{1}}, then derive 0 = - \\varepsilon{(b,v_{1})} e^{- b - v_{1}} + e^{- b - v_{1}} \\frac{\\partial}{\\partial v_{1}} \\varepsilon{(b,v_{1})}, then obtain 0 = - e^{- b - v_{1}} e^{b + v_{1}} + e^{- b - v_{1}} \\frac{\\partial}{\\partial v_{1}} e^{b + v_{1}}", "derivation": "\\varepsilon{(b,v_{1})} = e^{b + v_{1}} and 1 = \\frac{e^{b + v_{1}}}{\\varepsilon{(b,v_{1})}} and \\frac{d}{d v_{1}} 1 = \\frac{\\partial}{\\partial v_{1}} \\frac{e^{b + v_{1}}}{\\varepsilon{(b,v_{1})}} and \\frac{d}{d v_{1}} 1 = \\frac{\\partial}{\\partial v_{1}} e^{- b - v_{1}} e^{b + v_{1}} and \\frac{d}{d v_{1}} 1 = \\frac{\\partial}{\\partial v_{1}} \\varepsilon{(b,v_{1})} e^{- b - v_{1}} and 0 = - \\varepsilon{(b,v_{1})} e^{- b - v_{1}} + e^{- b - v_{1}} \\frac{\\partial}{\\partial v_{1}} \\varepsilon{(b,v_{1})} and 0 = - e^{- b - v_{1}} e^{b + v_{1}} + e^{- b - v_{1}} \\frac{\\partial}{\\partial v_{1}} e^{b + v_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True))))"], [["divide", 1, "Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True)))))"], [["differentiate", 2, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Integer(-1)), exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Integer(1), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))), exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True)))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integer(1), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Tuple(Symbol('v_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True))))), Mul(exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Derivative(Function('\\\\varepsilon')(Symbol('b', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(0), Add(Mul(Integer(-1), exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))), exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True)))), Mul(exp(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Mul(Integer(-1), Symbol('v_1', commutative=True)))), Derivative(exp(Add(Symbol('b', commutative=True), Symbol('v_1', commutative=True))), Tuple(Symbol('v_1', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\theta_1)} = \\sin{(\\theta_1)}, then obtain \\frac{d}{d \\theta_1} \\int \\theta_1 \\operatorname{A_{1}}^{2}{(\\theta_1)} d\\theta_1 = \\frac{d}{d \\theta_1} \\int \\theta_1 \\sin^{2}{(\\theta_1)} d\\theta_1", "derivation": "\\operatorname{A_{1}}{(\\theta_1)} = \\sin{(\\theta_1)} and \\theta_1 \\operatorname{A_{1}}{(\\theta_1)} = \\theta_1 \\sin{(\\theta_1)} and \\theta_1 \\operatorname{A_{1}}{(\\theta_1)} \\sin{(\\theta_1)} = \\theta_1 \\sin^{2}{(\\theta_1)} and \\theta_1 \\operatorname{A_{1}}^{2}{(\\theta_1)} = \\theta_1 \\operatorname{A_{1}}{(\\theta_1)} \\sin{(\\theta_1)} and \\theta_1 \\operatorname{A_{1}}^{2}{(\\theta_1)} = \\theta_1 \\sin^{2}{(\\theta_1)} and \\int \\theta_1 \\operatorname{A_{1}}^{2}{(\\theta_1)} d\\theta_1 = \\int \\theta_1 \\sin^{2}{(\\theta_1)} d\\theta_1 and \\frac{d}{d \\theta_1} \\int \\theta_1 \\operatorname{A_{1}}^{2}{(\\theta_1)} d\\theta_1 = \\frac{d}{d \\theta_1} \\int \\theta_1 \\sin^{2}{(\\theta_1)} d\\theta_1", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True)))"], [["times", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('A_1')(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\theta_1', commutative=True), sin(Symbol('\\\\theta_1', commutative=True)))"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Function('A_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('A_1')(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Mul(Symbol('\\\\theta_1', commutative=True), Function('A_1')(Symbol('\\\\theta_1', commutative=True)), sin(Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('A_1')(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Mul(Symbol('\\\\theta_1', commutative=True), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(2))))"], [["integrate", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('A_1')(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["differentiate", 6, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(Function('A_1')(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('\\\\theta_1', commutative=True), Pow(sin(Symbol('\\\\theta_1', commutative=True)), Integer(2))), Tuple(Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(c_{0},\\nabla)} = - \\nabla + \\log{(c_{0})}, then obtain \\frac{\\partial}{\\partial c_{0}} (c_{0} + 2 \\dot{y}{(c_{0},\\nabla)}) = \\frac{\\partial}{\\partial c_{0}} (- 2 \\nabla + c_{0} + 2 \\log{(c_{0})})", "derivation": "\\dot{y}{(c_{0},\\nabla)} = - \\nabla + \\log{(c_{0})} and c_{0} + \\dot{y}{(c_{0},\\nabla)} = - \\nabla + c_{0} + \\log{(c_{0})} and c_{0} + 2 \\dot{y}{(c_{0},\\nabla)} = - \\nabla + c_{0} + \\dot{y}{(c_{0},\\nabla)} + \\log{(c_{0})} and \\frac{\\partial}{\\partial c_{0}} (c_{0} + 2 \\dot{y}{(c_{0},\\nabla)}) = \\frac{\\partial}{\\partial c_{0}} (- \\nabla + c_{0} + \\dot{y}{(c_{0},\\nabla)} + \\log{(c_{0})}) and \\frac{\\partial}{\\partial c_{0}} (c_{0} + 2 \\dot{y}{(c_{0},\\nabla)}) = \\frac{\\partial}{\\partial c_{0}} (- 2 \\nabla + c_{0} + 2 \\log{(c_{0})})", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["add", 1, "Symbol('c_0', commutative=True)"], "Equality(Add(Symbol('c_0', commutative=True), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('c_0', commutative=True), log(Symbol('c_0', commutative=True))))"], [["add", 2, "Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Symbol('c_0', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('c_0', commutative=True), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True))))"], [["differentiate", 3, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Add(Symbol('c_0', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\nabla', commutative=True)), Symbol('c_0', commutative=True), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Symbol('c_0', commutative=True), Mul(Integer(2), Function('\\\\dot{y}')(Symbol('c_0', commutative=True), Symbol('\\\\nabla', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\nabla', commutative=True)), Symbol('c_0', commutative=True), Mul(Integer(2), log(Symbol('c_0', commutative=True)))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(E_{n})} = e^{E_{n}} and \\operatorname{L_{\\varepsilon}}{(E_{n})} = \\int e^{E_{n}} dE_{n}, then derive \\int S{(E_{n})} dE_{n} = B + e^{E_{n}}, then obtain \\operatorname{L_{\\varepsilon}}{(E_{n})} = B + S{(E_{n})}", "derivation": "S{(E_{n})} = e^{E_{n}} and \\int S{(E_{n})} dE_{n} = \\int e^{E_{n}} dE_{n} and \\operatorname{L_{\\varepsilon}}{(E_{n})} = \\int e^{E_{n}} dE_{n} and \\int S{(E_{n})} dE_{n} = B + e^{E_{n}} and \\int S{(E_{n})} dE_{n} = B + S{(E_{n})} and B + S{(E_{n})} = \\int e^{E_{n}} dE_{n} and \\operatorname{L_{\\varepsilon}}{(E_{n})} = B + S{(E_{n})}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('E_n', commutative=True)), exp(Symbol('E_n', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('S')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True)), Integral(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('S')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Add(Symbol('B', commutative=True), exp(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Function('S')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Add(Symbol('B', commutative=True), Function('S')(Symbol('E_n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Add(Symbol('B', commutative=True), Function('S')(Symbol('E_n', commutative=True))), Integral(exp(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 6], "Equality(Function('L_{\\\\varepsilon}')(Symbol('E_n', commutative=True)), Add(Symbol('B', commutative=True), Function('S')(Symbol('E_n', commutative=True))))"]]}, {"prompt": "Given l{(C_{1})} = \\sin{(C_{1})}, then obtain \\int (l{(C_{1})} + \\frac{l{(C_{1})}}{\\sin{(C_{1})}}) dC_{1} = \\int (l{(C_{1})} + 1) dC_{1}", "derivation": "l{(C_{1})} = \\sin{(C_{1})} and \\frac{l{(C_{1})}}{\\sin{(C_{1})}} = 1 and l{(C_{1})} + \\frac{l{(C_{1})}}{\\sin{(C_{1})}} = l{(C_{1})} + 1 and \\int (l{(C_{1})} + \\frac{l{(C_{1})}}{\\sin{(C_{1})}}) dC_{1} = \\int (l{(C_{1})} + 1) dC_{1}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('C_1', commutative=True)), sin(Symbol('C_1', commutative=True)))"], [["divide", 1, "sin(Symbol('C_1', commutative=True))"], "Equality(Mul(Function('l')(Symbol('C_1', commutative=True)), Pow(sin(Symbol('C_1', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Function('l')(Symbol('C_1', commutative=True))"], "Equality(Add(Function('l')(Symbol('C_1', commutative=True)), Mul(Function('l')(Symbol('C_1', commutative=True)), Pow(sin(Symbol('C_1', commutative=True)), Integer(-1)))), Add(Function('l')(Symbol('C_1', commutative=True)), Integer(1)))"], [["integrate", 3, "Symbol('C_1', commutative=True)"], "Equality(Integral(Add(Function('l')(Symbol('C_1', commutative=True)), Mul(Function('l')(Symbol('C_1', commutative=True)), Pow(sin(Symbol('C_1', commutative=True)), Integer(-1)))), Tuple(Symbol('C_1', commutative=True))), Integral(Add(Function('l')(Symbol('C_1', commutative=True)), Integer(1)), Tuple(Symbol('C_1', commutative=True))))"]]}, {"prompt": "Given g{(\\dot{y})} = e^{\\dot{y}}, then obtain \\frac{d}{d \\dot{y}} 0 = \\frac{d}{d \\dot{y}} (- g^{\\dot{y}}{(\\dot{y})} + (e^{\\dot{y}})^{\\dot{y}})", "derivation": "g{(\\dot{y})} = e^{\\dot{y}} and g^{\\dot{y}}{(\\dot{y})} = (e^{\\dot{y}})^{\\dot{y}} and 0 = - g^{\\dot{y}}{(\\dot{y})} + (e^{\\dot{y}})^{\\dot{y}} and \\frac{d}{d \\dot{y}} 0 = \\frac{d}{d \\dot{y}} (- g^{\\dot{y}}{(\\dot{y})} + (e^{\\dot{y}})^{\\dot{y}})", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["power", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Pow(Function('g')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True)))"], [["minus", 2, "Pow(Function('g')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Pow(Function('g')(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Pow(exp(Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\dot{y}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mu{(x^\\prime)} = \\log{(\\log{(x^\\prime)})} and \\operatorname{C_{1}}{(x^\\prime)} = \\log{(\\log{(x^\\prime)})}, then obtain (\\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})})^{x^\\prime} = (\\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\mu{(x^\\prime)})^{x^\\prime}", "derivation": "\\mu{(x^\\prime)} = \\log{(\\log{(x^\\prime)})} and \\operatorname{C_{1}}{(x^\\prime)} = \\log{(\\log{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\operatorname{C_{1}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})} and \\frac{d}{d x^\\prime} \\operatorname{C_{1}}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\mu{(x^\\prime)} and \\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\operatorname{C_{1}}{(x^\\prime)} = \\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\mu{(x^\\prime)} and \\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})} = \\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\mu{(x^\\prime)} and (\\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\log{(\\log{(x^\\prime)})})^{x^\\prime} = (\\log{(x^\\prime)} \\frac{d}{d x^\\prime} \\mu{(x^\\prime)})^{x^\\prime}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mu')(Symbol('x^\\\\prime', commutative=True)), log(log(Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('x^\\\\prime', commutative=True)), log(log(Symbol('x^\\\\prime', commutative=True))))"], [["differentiate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('C_1')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Function('\\\\mu')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["times", 4, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('C_1')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mu')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mu')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))))"], [["power", 6, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(log(log(Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Symbol('x^\\\\prime', commutative=True)), Pow(Mul(log(Symbol('x^\\\\prime', commutative=True)), Derivative(Function('\\\\mu')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1)))), Symbol('x^\\\\prime', commutative=True)))"]]}, {"prompt": "Given q{(\\mathbf{A},\\hat{X})} = \\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A}, then obtain \\int \\sin{(q^{\\hat{X}}{(\\mathbf{A},\\hat{X})})} d\\mathbf{A} = \\int \\sin{((\\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A})^{\\hat{X}})} d\\mathbf{A}", "derivation": "q{(\\mathbf{A},\\hat{X})} = \\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A} and q^{\\hat{X}}{(\\mathbf{A},\\hat{X})} = (\\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A})^{\\hat{X}} and \\sin{(q^{\\hat{X}}{(\\mathbf{A},\\hat{X})})} = \\sin{((\\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A})^{\\hat{X}})} and \\int \\sin{(q^{\\hat{X}}{(\\mathbf{A},\\hat{X})})} d\\mathbf{A} = \\int \\sin{((\\int (\\hat{X} + \\mathbf{A}) d\\mathbf{A})^{\\hat{X}})} d\\mathbf{A}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Integral(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(Integral(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\hat{X}', commutative=True)))"], [["sin", 2], "Equality(sin(Pow(Function('q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), sin(Pow(Integral(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(sin(Pow(Function('q')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(sin(Pow(Integral(Add(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\hat{X}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{f}{(\\Psi^{\\dagger},h)} = h^{\\Psi^{\\dagger}}, then obtain \\cos{(\\int \\frac{\\mathbf{f}{(\\Psi^{\\dagger},h)}}{h} d\\Psi^{\\dagger})} = \\cos{(\\int \\frac{h^{\\Psi^{\\dagger}}}{h} d\\Psi^{\\dagger})}", "derivation": "\\mathbf{f}{(\\Psi^{\\dagger},h)} = h^{\\Psi^{\\dagger}} and \\frac{\\mathbf{f}{(\\Psi^{\\dagger},h)}}{h} = \\frac{h^{\\Psi^{\\dagger}}}{h} and \\int \\frac{\\mathbf{f}{(\\Psi^{\\dagger},h)}}{h} d\\Psi^{\\dagger} = \\int \\frac{h^{\\Psi^{\\dagger}}}{h} d\\Psi^{\\dagger} and \\cos{(\\int \\frac{\\mathbf{f}{(\\Psi^{\\dagger},h)}}{h} d\\Psi^{\\dagger})} = \\cos{(\\int \\frac{h^{\\Psi^{\\dagger}}}{h} d\\Psi^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('\\\\mathbf{f}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))), cos(Integral(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\delta{(L)} = e^{e^{L}}, then obtain 1 = \\frac{4 e^{e^{L}}}{\\delta{(L)} + 3 e^{e^{L}}}", "derivation": "\\delta{(L)} = e^{e^{L}} and \\delta{(L)} + e^{e^{L}} = 2 e^{e^{L}} and \\delta{(L)} + 3 e^{e^{L}} = 4 e^{e^{L}} and 1 = \\frac{4 e^{e^{L}}}{\\delta{(L)} + 3 e^{e^{L}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('L', commutative=True)), exp(exp(Symbol('L', commutative=True))))"], [["add", 1, "exp(exp(Symbol('L', commutative=True)))"], "Equality(Add(Function('\\\\delta')(Symbol('L', commutative=True)), exp(exp(Symbol('L', commutative=True)))), Mul(Integer(2), exp(exp(Symbol('L', commutative=True)))))"], [["add", 2, "Mul(Integer(2), exp(exp(Symbol('L', commutative=True))))"], "Equality(Add(Function('\\\\delta')(Symbol('L', commutative=True)), Mul(Integer(3), exp(exp(Symbol('L', commutative=True))))), Mul(Integer(4), exp(exp(Symbol('L', commutative=True)))))"], [["divide", 3, "Add(Function('\\\\delta')(Symbol('L', commutative=True)), Mul(Integer(3), exp(exp(Symbol('L', commutative=True)))))"], "Equality(Integer(1), Mul(Integer(4), Pow(Add(Function('\\\\delta')(Symbol('L', commutative=True)), Mul(Integer(3), exp(exp(Symbol('L', commutative=True))))), Integer(-1)), exp(exp(Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{H}{(r_{0})} = \\log{(r_{0})}, then obtain \\frac{2 \\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0}}{r_{0}} = \\frac{\\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0}}{r_{0}} + \\frac{\\int \\log{(r_{0})}^{r_{0}} dr_{0}}{r_{0}}", "derivation": "\\mathbf{H}{(r_{0})} = \\log{(r_{0})} and \\mathbf{H}^{r_{0}}{(r_{0})} = \\log{(r_{0})}^{r_{0}} and \\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0} = \\int \\log{(r_{0})}^{r_{0}} dr_{0} and \\frac{\\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0}}{r_{0}} = \\frac{\\int \\log{(r_{0})}^{r_{0}} dr_{0}}{r_{0}} and \\frac{2 \\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0}}{r_{0}} = \\frac{\\int \\mathbf{H}^{r_{0}}{(r_{0})} dr_{0}}{r_{0}} + \\frac{\\int \\log{(r_{0})}^{r_{0}} dr_{0}}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), log(Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["integrate", 2, "Symbol('r_0', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))), Integral(Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], [["divide", 3, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))))"], [["add", 4, "Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))"], "Equality(Mul(Integer(2), Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Add(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(Function('\\\\mathbf{H}')(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True)))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Integral(Pow(log(Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True))))))"]]}, {"prompt": "Given \\psi{(\\theta)} = \\cos{(\\theta)} and i{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)}, then obtain 2 \\theta + i{(\\mathbf{J}_P)} + \\cos{(\\theta)} - \\cos^{\\theta}{(\\theta)} = 2 \\theta + \\sin{(\\mathbf{J}_P)} + \\cos{(\\theta)} - \\cos^{\\theta}{(\\theta)}", "derivation": "\\psi{(\\theta)} = \\cos{(\\theta)} and i{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and \\theta - \\psi^{\\theta}{(\\theta)} + i{(\\mathbf{J}_P)} + \\cos{(\\theta)} = \\theta - \\psi^{\\theta}{(\\theta)} + \\sin{(\\mathbf{J}_P)} + \\cos{(\\theta)} and 2 \\theta - \\psi^{\\theta}{(\\theta)} + i{(\\mathbf{J}_P)} + \\cos{(\\theta)} = 2 \\theta - \\psi^{\\theta}{(\\theta)} + \\sin{(\\mathbf{J}_P)} + \\cos{(\\theta)} and 2 \\theta + i{(\\mathbf{J}_P)} + \\cos{(\\theta)} - \\cos^{\\theta}{(\\theta)} = 2 \\theta + \\sin{(\\mathbf{J}_P)} + \\cos{(\\theta)} - \\cos^{\\theta}{(\\theta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), cos(Symbol('\\\\theta', commutative=True)))"], ["get_premise", "Equality(Function('i')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), Pow(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\theta', commutative=True))))"], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Function('i')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), sin(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["add", 3, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), Function('i')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\psi')(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True))), sin(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), Function('i')(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)), cos(Symbol('\\\\theta', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('\\\\theta', commutative=True)), Symbol('\\\\theta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\hbar,h)} = \\frac{\\cos{(h)}}{\\hbar}, then obtain \\int (\\frac{1}{h})^{h} dh = \\int (\\frac{\\cos{(h)}}{\\hbar h \\operatorname{C_{2}}{(\\hbar,h)}})^{h} dh", "derivation": "\\operatorname{C_{2}}{(\\hbar,h)} = \\frac{\\cos{(h)}}{\\hbar} and \\frac{\\operatorname{C_{2}}{(\\hbar,h)}}{h} = \\frac{\\cos{(h)}}{\\hbar h} and \\frac{1}{h} = \\frac{\\cos{(h)}}{\\hbar h \\operatorname{C_{2}}{(\\hbar,h)}} and (\\frac{1}{h})^{h} = (\\frac{\\cos{(h)}}{\\hbar h \\operatorname{C_{2}}{(\\hbar,h)}})^{h} and \\int (\\frac{1}{h})^{h} dh = \\int (\\frac{\\cos{(h)}}{\\hbar h \\operatorname{C_{2}}{(\\hbar,h)}})^{h} dh", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), cos(Symbol('h', commutative=True))))"], [["divide", 1, "Symbol('h', commutative=True)"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), cos(Symbol('h', commutative=True))))"], [["divide", 2, "Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True))"], "Equality(Pow(Symbol('h', commutative=True), Integer(-1)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True)), Integer(-1)), cos(Symbol('h', commutative=True))))"], [["power", 3, "Symbol('h', commutative=True)"], "Equality(Pow(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True)), Integer(-1)), cos(Symbol('h', commutative=True))), Symbol('h', commutative=True)))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Pow(Pow(Symbol('h', commutative=True), Integer(-1)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Function('C_2')(Symbol('\\\\hbar', commutative=True), Symbol('h', commutative=True)), Integer(-1)), cos(Symbol('h', commutative=True))), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given G{(\\mathbf{S},a^{\\dagger})} = \\cos^{a^{\\dagger}}{(\\mathbf{S})}, then obtain - G{(\\mathbf{S},a^{\\dagger})} + \\iint G{(\\mathbf{S},a^{\\dagger})} d\\mathbf{S} da^{\\dagger} = - G{(\\mathbf{S},a^{\\dagger})} + \\iint \\cos^{a^{\\dagger}}{(\\mathbf{S})} d\\mathbf{S} da^{\\dagger}", "derivation": "G{(\\mathbf{S},a^{\\dagger})} = \\cos^{a^{\\dagger}}{(\\mathbf{S})} and \\int G{(\\mathbf{S},a^{\\dagger})} d\\mathbf{S} = \\int \\cos^{a^{\\dagger}}{(\\mathbf{S})} d\\mathbf{S} and \\iint G{(\\mathbf{S},a^{\\dagger})} d\\mathbf{S} da^{\\dagger} = \\iint \\cos^{a^{\\dagger}}{(\\mathbf{S})} d\\mathbf{S} da^{\\dagger} and - G{(\\mathbf{S},a^{\\dagger})} + \\iint G{(\\mathbf{S},a^{\\dagger})} d\\mathbf{S} da^{\\dagger} = - G{(\\mathbf{S},a^{\\dagger})} + \\iint \\cos^{a^{\\dagger}}{(\\mathbf{S})} d\\mathbf{S} da^{\\dagger}", "srepr_derivation": [["get_premise", "Equality(Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["minus", 3, "Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integral(Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Add(Mul(Integer(-1), Function('G')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Integral(Pow(cos(Symbol('\\\\mathbf{S}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} = \\frac{\\dot{y} + \\psi}{\\hat{x}_0}, then obtain \\hat{x}_0^{2} \\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} - 1 = \\hat{x}_0 (\\dot{y} + \\psi) - 1", "derivation": "\\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} = \\frac{\\dot{y} + \\psi}{\\hat{x}_0} and \\hat{x}_0 \\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} = \\dot{y} + \\psi and \\hat{x}_0^{2} \\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} = \\hat{x}_0 (\\dot{y} + \\psi) and \\hat{x}_0^{2} \\mu_{0}{(\\hat{x}_0,\\psi,\\dot{y})} - 1 = \\hat{x}_0 (\\dot{y} + \\psi) - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["times", 1, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Mul(Symbol('\\\\hat{x}_0', commutative=True), Function('\\\\mu_0')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["divide", 2, "Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\psi', commutative=True))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Mul(Pow(Symbol('\\\\hat{x}_0', commutative=True), Integer(2)), Function('\\\\mu_0')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('\\\\psi', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-1)), Add(Mul(Symbol('\\\\hat{x}_0', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\psi', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given b{(Z)} = \\cos{(\\log{(Z)})} and \\hat{X}{(Z)} = 0^{Z} (\\frac{d}{d Z} b{(Z)} + \\frac{\\sin{(\\log{(Z)})}}{Z})^{Z}, then derive \\frac{d}{d Z} b{(Z)} + \\frac{\\sin{(\\log{(Z)})}}{Z} = 0, then obtain \\hat{X}{(Z)} = 0^{Z}", "derivation": "b{(Z)} = \\cos{(\\log{(Z)})} and b{(Z)} - \\cos{(\\log{(Z)})} = 0 and \\frac{d}{d Z} (b{(Z)} - \\cos{(\\log{(Z)})}) = \\frac{d}{d Z} 0 and \\frac{d}{d Z} b{(Z)} + \\frac{\\sin{(\\log{(Z)})}}{Z} = 0 and (\\frac{d}{d Z} b{(Z)} + \\frac{\\sin{(\\log{(Z)})}}{Z})^{Z} = 0^{Z} and \\hat{X}{(Z)} = 0^{Z} (\\frac{d}{d Z} b{(Z)} + \\frac{\\sin{(\\log{(Z)})}}{Z})^{Z} and \\hat{X}{(Z)} = 0^{Z}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('Z', commutative=True)), cos(log(Symbol('Z', commutative=True))))"], [["minus", 1, "cos(log(Symbol('Z', commutative=True)))"], "Equality(Add(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(log(Symbol('Z', commutative=True))))), Integer(0))"], [["differentiate", 2, "Symbol('Z', commutative=True)"], "Equality(Derivative(Add(Function('b')(Symbol('Z', commutative=True)), Mul(Integer(-1), cos(log(Symbol('Z', commutative=True))))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('b')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True))))), Integer(0))"], [["power", 4, "Symbol('Z', commutative=True)"], "Equality(Pow(Add(Derivative(Function('b')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True))))), Symbol('Z', commutative=True)), Pow(Integer(0), Symbol('Z', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('Z', commutative=True)), Mul(Pow(Integer(0), Symbol('Z', commutative=True)), Pow(Add(Derivative(Function('b')(Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), sin(log(Symbol('Z', commutative=True))))), Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Function('\\\\hat{X}')(Symbol('Z', commutative=True)), Pow(Integer(0), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given J{(E_{n})} = \\cos{(E_{n})}, then obtain 2 \\int J{(E_{n})} dE_{n} + \\int \\cos{(E_{n})} dE_{n} = 3 \\int \\cos{(E_{n})} dE_{n}", "derivation": "J{(E_{n})} = \\cos{(E_{n})} and \\int J{(E_{n})} dE_{n} = \\int \\cos{(E_{n})} dE_{n} and \\int J{(E_{n})} dE_{n} + \\int \\cos{(E_{n})} dE_{n} = 2 \\int \\cos{(E_{n})} dE_{n} and \\int J{(E_{n})} dE_{n} + 2 \\int \\cos{(E_{n})} dE_{n} = 3 \\int \\cos{(E_{n})} dE_{n} and 2 \\int J{(E_{n})} dE_{n} + \\int \\cos{(E_{n})} dE_{n} = 3 \\int \\cos{(E_{n})} dE_{n}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('E_n', commutative=True)), cos(Symbol('E_n', commutative=True)))"], [["integrate", 1, "Symbol('E_n', commutative=True)"], "Equality(Integral(Function('J')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))"], [["add", 2, "Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Add(Integral(Function('J')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Mul(Integer(2), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"], [["add", 3, "Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))"], "Equality(Add(Integral(Function('J')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))), Mul(Integer(2), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True))))), Mul(Integer(3), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Integer(2), Integral(Function('J')(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))), Mul(Integer(3), Integral(cos(Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(A_{x},r_{0})} = r_{0}^{A_{x}} and Q{(A_{x},r_{0})} = \\hat{\\mathbf{r}}^{A_{x}}{(A_{x},r_{0})}, then obtain - r_{0} + \\frac{Q{(A_{x},r_{0})} + 1}{r_{0}} = - r_{0} + \\frac{(r_{0}^{A_{x}})^{A_{x}} + 1}{r_{0}}", "derivation": "\\hat{\\mathbf{r}}{(A_{x},r_{0})} = r_{0}^{A_{x}} and Q{(A_{x},r_{0})} = \\hat{\\mathbf{r}}^{A_{x}}{(A_{x},r_{0})} and Q{(A_{x},r_{0})} + 1 = \\hat{\\mathbf{r}}^{A_{x}}{(A_{x},r_{0})} + 1 and Q{(A_{x},r_{0})} + 1 = (r_{0}^{A_{x}})^{A_{x}} + 1 and \\frac{Q{(A_{x},r_{0})} + 1}{r_{0}} = \\frac{(r_{0}^{A_{x}})^{A_{x}} + 1}{r_{0}} and - r_{0} + \\frac{Q{(A_{x},r_{0})} + 1}{r_{0}} = - r_{0} + \\frac{(r_{0}^{A_{x}})^{A_{x}} + 1}{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Pow(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)))"], ["renaming_premise", "Equality(Function('Q')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Symbol('A_x', commutative=True)))"], [["add", 2, 1], "Equality(Add(Function('Q')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Integer(1)), Add(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Symbol('A_x', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('Q')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Integer(1)), Add(Pow(Pow(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Integer(1)))"], [["divide", 4, "Symbol('r_0', commutative=True)"], "Equality(Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Function('Q')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Integer(1))), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Pow(Pow(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Integer(1))))"], [["minus", 5, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Function('Q')(Symbol('A_x', commutative=True), Symbol('r_0', commutative=True)), Integer(1)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Mul(Pow(Symbol('r_0', commutative=True), Integer(-1)), Add(Pow(Pow(Symbol('r_0', commutative=True), Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True)), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(\\varphi^*)} = e^{\\varphi^*}, then derive \\int \\frac{\\varphi^* + \\operatorname{F_{c}}{(\\varphi^*)}}{\\varphi^* + e^{\\varphi^*}} d\\varphi^* = \\varphi^* + g_{\\varepsilon}, then obtain \\varepsilon + \\varphi^* = \\varphi^* + g_{\\varepsilon}", "derivation": "\\operatorname{F_{c}}{(\\varphi^*)} = e^{\\varphi^*} and \\varphi^* + \\operatorname{F_{c}}{(\\varphi^*)} = \\varphi^* + e^{\\varphi^*} and \\frac{\\varphi^* + \\operatorname{F_{c}}{(\\varphi^*)}}{\\varphi^* + e^{\\varphi^*}} = 1 and \\int \\frac{\\varphi^* + \\operatorname{F_{c}}{(\\varphi^*)}}{\\varphi^* + e^{\\varphi^*}} d\\varphi^* = \\int 1 d\\varphi^* and \\int \\frac{\\varphi^* + \\operatorname{F_{c}}{(\\varphi^*)}}{\\varphi^* + e^{\\varphi^*}} d\\varphi^* = \\varphi^* + g_{\\varepsilon} and \\int 1 d\\varphi^* = \\varphi^* + g_{\\varepsilon} and \\varepsilon + \\varphi^* = \\varphi^* + g_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('\\\\varphi^*', commutative=True)), exp(Symbol('\\\\varphi^*', commutative=True)))"], [["add", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('F_c')(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Function('F_c')(Symbol('\\\\varphi^*', commutative=True))), Pow(Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))), Integer(-1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Function('F_c')(Symbol('\\\\varphi^*', commutative=True))), Pow(Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Integral(Mul(Add(Symbol('\\\\varphi^*', commutative=True), Function('F_c')(Symbol('\\\\varphi^*', commutative=True))), Pow(Add(Symbol('\\\\varphi^*', commutative=True), exp(Symbol('\\\\varphi^*', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Integer(1), Tuple(Symbol('\\\\varphi^*', commutative=True))), Add(Symbol('\\\\varphi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Add(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Add(Symbol('\\\\varphi^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(\\phi)} = \\sin{(\\phi)}, then obtain \\int \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\varepsilon_{0}{(\\phi)} \\sin^{2}{(\\eta)} \\sin{(\\phi)} d\\Omega = \\int \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\sin^{2}{(\\eta)} \\sin^{2}{(\\phi)} d\\Omega", "derivation": "\\varepsilon_{0}{(\\phi)} = \\sin{(\\phi)} and - \\varepsilon_{0}{(\\phi)} \\sin{(\\eta)} = - \\sin{(\\eta)} \\sin{(\\phi)} and \\varepsilon_{0}{(\\phi)} \\sin^{2}{(\\eta)} \\sin{(\\phi)} = \\sin^{2}{(\\eta)} \\sin^{2}{(\\phi)} and \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\varepsilon_{0}{(\\phi)} \\sin^{2}{(\\eta)} \\sin{(\\phi)} = \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\sin^{2}{(\\eta)} \\sin^{2}{(\\phi)} and \\int \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\varepsilon_{0}{(\\phi)} \\sin^{2}{(\\eta)} \\sin{(\\phi)} d\\Omega = \\int \\mathbf{g}^{\\chi}{(\\chi,\\Omega)} \\sin^{2}{(\\eta)} \\sin^{2}{(\\phi)} d\\Omega", "srepr_derivation": [["get_premise", "Equality(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(-1), Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\phi', commutative=True))))"], [["times", 2, "Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], "Equality(Mul(Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), sin(Symbol('\\\\phi', commutative=True))), Mul(Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(2))))"], [["times", 3, "Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\chi', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), sin(Symbol('\\\\phi', commutative=True))), Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\chi', commutative=True)), Function('\\\\varepsilon_0')(Symbol('\\\\phi', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), sin(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Mul(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\Omega', commutative=True)), Symbol('\\\\chi', commutative=True)), Pow(sin(Symbol('\\\\eta', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\phi', commutative=True)), Integer(2))), Tuple(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\lambda{(\\sigma_p,L)} = L + \\sigma_p, then obtain 0^{\\sigma_p} = (\\frac{\\int (L + \\sigma_p) d\\sigma_p - \\int \\lambda{(\\sigma_p,L)} d\\sigma_p}{L \\sigma_p})^{\\sigma_p}", "derivation": "\\lambda{(\\sigma_p,L)} = L + \\sigma_p and \\int \\lambda{(\\sigma_p,L)} d\\sigma_p = \\int (L + \\sigma_p) d\\sigma_p and 0 = \\int (L + \\sigma_p) d\\sigma_p - \\int \\lambda{(\\sigma_p,L)} d\\sigma_p and 0 = \\frac{\\int (L + \\sigma_p) d\\sigma_p - \\int \\lambda{(\\sigma_p,L)} d\\sigma_p}{L \\sigma_p} and 0^{\\sigma_p} = (\\frac{\\int (L + \\sigma_p) d\\sigma_p - \\int \\lambda{(\\sigma_p,L)} d\\sigma_p}{L \\sigma_p})^{\\sigma_p}", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)))"], [["integrate", 1, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Integral(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Integral(Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Integer(0), Add(Integral(Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))))))"], [["divide", 3, "Mul(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True))"], "Equality(Integer(0), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))))))"], [["power", 4, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\sigma_p', commutative=True)), Pow(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Integral(Add(Symbol('L', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\lambda')(Symbol('\\\\sigma_p', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('\\\\sigma_p', commutative=True)))))), Symbol('\\\\sigma_p', commutative=True)))"]]}, {"prompt": "Given \\nabla{(\\dot{x},\\delta)} = \\delta - \\dot{x}, then obtain \\iint - \\dot{x} \\nabla{(\\dot{x},\\delta)} d\\delta d\\delta = \\int (\\int \\dot{x}^{2} d\\delta + \\int - \\delta \\dot{x} d\\delta) d\\delta", "derivation": "\\nabla{(\\dot{x},\\delta)} = \\delta - \\dot{x} and - \\dot{x} \\nabla{(\\dot{x},\\delta)} = - \\dot{x} (\\delta - \\dot{x}) and \\int - \\dot{x} \\nabla{(\\dot{x},\\delta)} d\\delta = \\int - \\dot{x} (\\delta - \\dot{x}) d\\delta and \\int - \\dot{x} \\nabla{(\\dot{x},\\delta)} d\\delta = \\int \\dot{x}^{2} d\\delta + \\int - \\delta \\dot{x} d\\delta and \\iint - \\dot{x} \\nabla{(\\dot{x},\\delta)} d\\delta d\\delta = \\int (\\int \\dot{x}^{2} d\\delta + \\int - \\delta \\dot{x} d\\delta) d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["expand", 3], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Add(Integral(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True), Function('\\\\nabla')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Integral(Pow(Symbol('\\\\dot{x}', commutative=True), Integer(2)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\mathbf{D}{(\\mathbf{P})} = \\sin{(\\mathbf{P})}, then obtain \\mathbf{P} + \\log{(\\mathbf{D}{(\\mathbf{P})})} + \\log{(\\sin{(\\mathbf{P})})} = \\mathbf{P} + 2 \\log{(\\sin{(\\mathbf{P})})}", "derivation": "\\mathbf{D}{(\\mathbf{P})} = \\sin{(\\mathbf{P})} and \\log{(\\mathbf{D}{(\\mathbf{P})})} = \\log{(\\sin{(\\mathbf{P})})} and \\log{(\\mathbf{D}{(\\mathbf{P})})} + \\log{(\\sin{(\\mathbf{P})})} = 2 \\log{(\\sin{(\\mathbf{P})})} and \\mathbf{P} + \\log{(\\mathbf{D}{(\\mathbf{P})})} + \\log{(\\sin{(\\mathbf{P})})} = \\mathbf{P} + 2 \\log{(\\sin{(\\mathbf{P})})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True)), sin(Symbol('\\\\mathbf{P}', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), log(sin(Symbol('\\\\mathbf{P}', commutative=True))))"], [["add", 2, "log(sin(Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Add(log(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), log(sin(Symbol('\\\\mathbf{P}', commutative=True)))), Mul(Integer(2), log(sin(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{P}', commutative=True), log(Function('\\\\mathbf{D}')(Symbol('\\\\mathbf{P}', commutative=True))), log(sin(Symbol('\\\\mathbf{P}', commutative=True)))), Add(Symbol('\\\\mathbf{P}', commutative=True), Mul(Integer(2), log(sin(Symbol('\\\\mathbf{P}', commutative=True))))))"]]}, {"prompt": "Given \\psi{(f)} = \\frac{d}{d f} \\cos{(f)}, then derive - \\frac{\\psi{(f)}}{\\sin{(f)}} = 1, then obtain - \\frac{\\frac{d}{d f} \\cos{(f)}}{\\sin{(f)}} = 1", "derivation": "\\psi{(f)} = \\frac{d}{d f} \\cos{(f)} and \\frac{\\psi{(f)}}{\\frac{d}{d f} \\cos{(f)}} = 1 and - \\frac{\\psi{(f)}}{\\sin{(f)}} = 1 and - \\frac{\\frac{d}{d f} \\cos{(f)}}{\\sin{(f)}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('f', commutative=True)), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\psi')(Symbol('f', commutative=True)), Pow(Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(-1), Function('\\\\psi')(Symbol('f', commutative=True)), Pow(sin(Symbol('f', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(sin(Symbol('f', commutative=True)), Integer(-1)), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{x^{{\\}'}}{(v_{x},v_{z})} = v_{x} v_{z}, then obtain (\\int (v_{z} - \\operatorname{x^{{\\}'}}{(v_{x},v_{z})}) dv_{z})^{v_{x}} = (\\int (- v_{x} v_{z} + v_{z}) dv_{z})^{v_{x}}", "derivation": "\\operatorname{x^{{\\}'}}{(v_{x},v_{z})} = v_{x} v_{z} and - \\operatorname{x^{{\\}'}}{(v_{x},v_{z})} = - v_{x} v_{z} and v_{z} - \\operatorname{x^{{\\}'}}{(v_{x},v_{z})} = - v_{x} v_{z} + v_{z} and \\int (v_{z} - \\operatorname{x^{{\\}'}}{(v_{x},v_{z})}) dv_{z} = \\int (- v_{x} v_{z} + v_{z}) dv_{z} and (\\int (v_{z} - \\operatorname{x^{{\\}'}}{(v_{x},v_{z})}) dv_{z})^{v_{x}} = (\\int (- v_{x} v_{z} + v_{z}) dv_{z})^{v_{x}}", "srepr_derivation": [["premise", "Equality(Function('x^\\\\prime')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('x^\\\\prime')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(-1), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))"], [["add", 2, "Symbol('v_z', commutative=True)"], "Equality(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))), Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["integrate", 3, "Symbol('v_z', commutative=True)"], "Equality(Integral(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["power", 4, "Symbol('v_x', commutative=True)"], "Equality(Pow(Integral(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Function('x^\\\\prime')(Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)))), Tuple(Symbol('v_z', commutative=True))), Symbol('v_x', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('v_x', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Symbol('v_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(U)} = \\sin{(e^{U})}, then obtain (U \\operatorname{A_{x}}{(U)} + e^{U}) \\operatorname{A_{x}}{(U)} = (U \\sin{(e^{U})} + e^{U}) \\operatorname{A_{x}}{(U)}", "derivation": "\\operatorname{A_{x}}{(U)} = \\sin{(e^{U})} and U \\operatorname{A_{x}}{(U)} = U \\sin{(e^{U})} and U \\operatorname{A_{x}}{(U)} + e^{U} = U \\sin{(e^{U})} + e^{U} and (U \\operatorname{A_{x}}{(U)} + e^{U}) \\operatorname{A_{x}}{(U)} = (U \\sin{(e^{U})} + e^{U}) \\operatorname{A_{x}}{(U)}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('U', commutative=True)), sin(exp(Symbol('U', commutative=True))))"], [["times", 1, "Symbol('U', commutative=True)"], "Equality(Mul(Symbol('U', commutative=True), Function('A_x')(Symbol('U', commutative=True))), Mul(Symbol('U', commutative=True), sin(exp(Symbol('U', commutative=True)))))"], [["add", 2, "exp(Symbol('U', commutative=True))"], "Equality(Add(Mul(Symbol('U', commutative=True), Function('A_x')(Symbol('U', commutative=True))), exp(Symbol('U', commutative=True))), Add(Mul(Symbol('U', commutative=True), sin(exp(Symbol('U', commutative=True)))), exp(Symbol('U', commutative=True))))"], [["times", 3, "Function('A_x')(Symbol('U', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('U', commutative=True), Function('A_x')(Symbol('U', commutative=True))), exp(Symbol('U', commutative=True))), Function('A_x')(Symbol('U', commutative=True))), Mul(Add(Mul(Symbol('U', commutative=True), sin(exp(Symbol('U', commutative=True)))), exp(Symbol('U', commutative=True))), Function('A_x')(Symbol('U', commutative=True))))"]]}, {"prompt": "Given l{(\\phi_1)} = \\log{(e^{\\phi_1})}, then derive \\frac{d}{d \\phi_1} l{(\\phi_1)} + 1 = 2, then obtain 2^{- \\phi_1} (\\frac{d}{d \\phi_1} l{(\\phi_1)} + 1)^{\\phi_1} = 1", "derivation": "l{(\\phi_1)} = \\log{(e^{\\phi_1})} and l{(\\phi_1)} + \\log{(e^{\\phi_1})} = 2 \\log{(e^{\\phi_1})} and \\frac{d}{d \\phi_1} (l{(\\phi_1)} + \\log{(e^{\\phi_1})}) = \\frac{d}{d \\phi_1} 2 \\log{(e^{\\phi_1})} and \\frac{d}{d \\phi_1} l{(\\phi_1)} + 1 = 2 and (\\frac{d}{d \\phi_1} l{(\\phi_1)} + 1)^{\\phi_1} = 2^{\\phi_1} and 2^{- \\phi_1} (\\frac{d}{d \\phi_1} l{(\\phi_1)} + 1)^{\\phi_1} = 1", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\phi_1', commutative=True)), log(exp(Symbol('\\\\phi_1', commutative=True))))"], [["add", 1, "log(exp(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Function('l')(Symbol('\\\\phi_1', commutative=True)), log(exp(Symbol('\\\\phi_1', commutative=True)))), Mul(Integer(2), log(exp(Symbol('\\\\phi_1', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Add(Function('l')(Symbol('\\\\phi_1', commutative=True)), log(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(exp(Symbol('\\\\phi_1', commutative=True)))), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('l')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1)), Integer(2))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Add(Derivative(Function('l')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\phi_1', commutative=True)), Pow(Integer(2), Symbol('\\\\phi_1', commutative=True)))"], [["divide", 5, "Pow(Integer(2), Symbol('\\\\phi_1', commutative=True))"], "Equality(Mul(Pow(Integer(2), Mul(Integer(-1), Symbol('\\\\phi_1', commutative=True))), Pow(Add(Derivative(Function('l')(Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1)), Symbol('\\\\phi_1', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\bar{\\h}{(i)} = \\cos{(e^{i})}, then obtain \\int (\\bar{\\h}{(i)} - \\cos{(e^{i})})^{i} di = \\int 0^{i} di", "derivation": "\\bar{\\h}{(i)} = \\cos{(e^{i})} and \\bar{\\h}{(i)} - \\cos{(e^{i})} = 0 and (\\bar{\\h}{(i)} - \\cos{(e^{i})})^{i} = 0^{i} and \\int (\\bar{\\h}{(i)} - \\cos{(e^{i})})^{i} di = \\int 0^{i} di", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('i', commutative=True)), cos(exp(Symbol('i', commutative=True))))"], [["minus", 1, "cos(exp(Symbol('i', commutative=True)))"], "Equality(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('i', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('i', commutative=True)"], "Equality(Pow(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), Pow(Integer(0), Symbol('i', commutative=True)))"], [["integrate", 3, "Symbol('i', commutative=True)"], "Equality(Integral(Pow(Add(Function('\\\\hbar')(Symbol('i', commutative=True)), Mul(Integer(-1), cos(exp(Symbol('i', commutative=True))))), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Pow(Integer(0), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\chi)} = \\sin{(\\log{(\\chi)})}, then obtain 0^{\\chi} - \\chi = 2 \\cdot 0^{\\chi} - \\chi - 1", "derivation": "\\mathbf{E}{(\\chi)} = \\sin{(\\log{(\\chi)})} and \\mathbf{E}{(\\chi)} - \\sin{(\\log{(\\chi)})} = 0 and (\\mathbf{E}{(\\chi)} - \\sin{(\\log{(\\chi)})})^{\\chi} = 0^{\\chi} and - \\chi + (\\mathbf{E}{(\\chi)} - \\sin{(\\log{(\\chi)})})^{\\chi} = 0^{\\chi} - \\chi and 1 - \\chi = - \\chi + (\\mathbf{E}{(\\chi)} - \\sin{(\\log{(\\chi)})})^{\\chi} and 1 - \\chi = 0^{\\chi} - \\chi and 0^{\\chi} - \\chi = 0^{\\chi} - \\chi + (\\mathbf{E}{(\\chi)} - \\sin{(\\log{(\\chi)})})^{\\chi} - 1 and 0^{\\chi} - \\chi = 2 \\cdot 0^{\\chi} - \\chi - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), sin(log(Symbol('\\\\chi', commutative=True))))"], [["minus", 1, "sin(log(Symbol('\\\\chi', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Integer(0))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Symbol('\\\\chi', commutative=True)), Pow(Integer(0), Symbol('\\\\chi', commutative=True)))"], [["minus", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Add(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Symbol('\\\\chi', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Add(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Pow(Add(Function('\\\\mathbf{E}')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), sin(log(Symbol('\\\\chi', commutative=True))))), Symbol('\\\\chi', commutative=True)), Integer(-1)))"], [["substitute_RHS_for_LHS", 7, 1], "Equality(Add(Pow(Integer(0), Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(2), Pow(Integer(0), Symbol('\\\\chi', commutative=True))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given A{(\\mathbf{M},k,V_{\\mathbf{B}})} = V_{\\mathbf{B}} k - \\mathbf{M}, then obtain \\int (A^{V_{\\mathbf{B}}}{(\\mathbf{M},k,V_{\\mathbf{B}})})^{\\mathbf{M}} d\\mathbf{M} = \\int ((V_{\\mathbf{B}} k - \\mathbf{M})^{V_{\\mathbf{B}}})^{\\mathbf{M}} d\\mathbf{M}", "derivation": "A{(\\mathbf{M},k,V_{\\mathbf{B}})} = V_{\\mathbf{B}} k - \\mathbf{M} and A^{V_{\\mathbf{B}}}{(\\mathbf{M},k,V_{\\mathbf{B}})} = (V_{\\mathbf{B}} k - \\mathbf{M})^{V_{\\mathbf{B}}} and (A^{V_{\\mathbf{B}}}{(\\mathbf{M},k,V_{\\mathbf{B}})})^{\\mathbf{M}} = ((V_{\\mathbf{B}} k - \\mathbf{M})^{V_{\\mathbf{B}}})^{\\mathbf{M}} and \\int (A^{V_{\\mathbf{B}}}{(\\mathbf{M},k,V_{\\mathbf{B}})})^{\\mathbf{M}} d\\mathbf{M} = \\int ((V_{\\mathbf{B}} k - \\mathbf{M})^{V_{\\mathbf{B}}})^{\\mathbf{M}} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 1, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["power", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Pow(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Pow(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Pow(Pow(Function('A')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('k', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Pow(Pow(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('k', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given I{(\\mathbf{F},y)} = \\mathbf{F} + y and \\operatorname{P_{g}}{(\\mathbf{F},y)} = \\frac{(\\mathbf{F} + y)^{\\mathbf{F}}}{\\mathbf{F} (\\mathbf{F} + y)}, then obtain \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{P_{g}}^{y}{(\\mathbf{F},y)} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\frac{I^{\\mathbf{F}}{(\\mathbf{F},y)}}{\\mathbf{F} (\\mathbf{F} + y)})^{y}", "derivation": "I{(\\mathbf{F},y)} = \\mathbf{F} + y and I^{\\mathbf{F}}{(\\mathbf{F},y)} = (\\mathbf{F} + y)^{\\mathbf{F}} and \\operatorname{P_{g}}{(\\mathbf{F},y)} = \\frac{(\\mathbf{F} + y)^{\\mathbf{F}}}{\\mathbf{F} (\\mathbf{F} + y)} and \\operatorname{P_{g}}{(\\mathbf{F},y)} = \\frac{I^{\\mathbf{F}}{(\\mathbf{F},y)}}{\\mathbf{F} (\\mathbf{F} + y)} and \\operatorname{P_{g}}^{y}{(\\mathbf{F},y)} = (\\frac{I^{\\mathbf{F}}{(\\mathbf{F},y)}}{\\mathbf{F} (\\mathbf{F} + y)})^{y} and \\frac{\\partial}{\\partial \\mathbf{F}} \\operatorname{P_{g}}^{y}{(\\mathbf{F},y)} = \\frac{\\partial}{\\partial \\mathbf{F}} (\\frac{I^{\\mathbf{F}}{(\\mathbf{F},y)}}{\\mathbf{F} (\\mathbf{F} + y)})^{y}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))))"], [["power", 4, "Symbol('y', commutative=True)"], "Equality(Pow(Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('y', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Pow(Function('P_g')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(Pow(Mul(Pow(Symbol('\\\\mathbf{F}', commutative=True), Integer(-1)), Pow(Add(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Pow(Function('I')(Symbol('\\\\mathbf{F}', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\mathbf{F}', commutative=True))), Symbol('y', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"]]}, {"prompt": "Given b{(C,n)} = \\frac{C}{n} and \\mathbf{f}{(C,n)} = b{(C,n)} + 1, then obtain \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int (b{(C,n)} + 1) dn) = \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int (\\frac{C}{n} + 1) dn)", "derivation": "b{(C,n)} = \\frac{C}{n} and b{(C,n)} + 1 = \\frac{C}{n} + 1 and \\mathbf{f}{(C,n)} = b{(C,n)} + 1 and \\mathbf{f}{(C,n)} = \\frac{C}{n} + 1 and \\int \\mathbf{f}{(C,n)} dn = \\int (\\frac{C}{n} + 1) dn and \\frac{C}{n} + \\int \\mathbf{f}{(C,n)} dn = \\frac{C}{n} + \\int (\\frac{C}{n} + 1) dn and \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int \\mathbf{f}{(C,n)} dn) = \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int (\\frac{C}{n} + 1) dn) and \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int (b{(C,n)} + 1) dn) = \\frac{\\partial}{\\partial n} (\\frac{C}{n} + \\int (\\frac{C}{n} + 1) dn)", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('b')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{f}')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Add(Function('b')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Integer(1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{f}')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)))"], [["integrate", 4, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('n', commutative=True))))"], [["add", 5, "Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Function('\\\\mathbf{f}')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('n', commutative=True)))))"], [["differentiate", 6, "Symbol('n', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Function('\\\\mathbf{f}')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Derivative(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Add(Function('b')(Symbol('C', commutative=True), Symbol('n', commutative=True)), Integer(1)), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integral(Add(Mul(Symbol('C', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('n', commutative=True)))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}{(\\hat{x}_0)} = \\cos{(\\log{(\\hat{x}_0)})} and \\hat{p}{(E_{x},f)} = E_{x}^{f}, then obtain (\\hat{p}{(E_{x},f)} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})})^{\\hat{x}_0} = (E_{x}^{f} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})})^{\\hat{x}_0}", "derivation": "\\hat{p}{(\\hat{x}_0)} = \\cos{(\\log{(\\hat{x}_0)})} and \\hat{p}{(E_{x},f)} = E_{x}^{f} and \\hat{p}^{\\hat{x}_0}{(\\hat{x}_0)} + \\hat{p}{(E_{x},f)} = E_{x}^{f} + \\hat{p}^{\\hat{x}_0}{(\\hat{x}_0)} and \\hat{p}{(E_{x},f)} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})} = E_{x}^{f} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})} and (\\hat{p}{(E_{x},f)} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})})^{\\hat{x}_0} = (E_{x}^{f} + \\cos^{\\hat{x}_0}{(\\log{(\\hat{x}_0)})})^{\\hat{x}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True)), cos(log(Symbol('\\\\hat{x}_0', commutative=True))))"], ["get_premise", "Equality(Function('\\\\hat{p}')(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(Symbol('E_x', commutative=True), Symbol('f', commutative=True)))"], [["add", 2, "Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True)), Function('\\\\hat{p}')(Symbol('E_x', commutative=True), Symbol('f', commutative=True))), Add(Pow(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(Function('\\\\hat{p}')(Symbol('\\\\hat{x}_0', commutative=True)), Symbol('\\\\hat{x}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('\\\\hat{p}')(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Add(Pow(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))))"], [["power", 4, "Symbol('\\\\hat{x}_0', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{p}')(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)), Pow(Add(Pow(Symbol('E_x', commutative=True), Symbol('f', commutative=True)), Pow(cos(log(Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True))), Symbol('\\\\hat{x}_0', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)}, then derive 0^{\\psi} = (- \\operatorname{F_{g}}{(\\psi)} + \\cos{(\\psi)})^{\\psi}, then obtain 0^{\\psi - \\operatorname{F_{g}}{(\\psi)} + \\cos{(\\psi)}} = 1", "derivation": "\\operatorname{F_{g}}{(\\psi)} = \\frac{d}{d \\psi} \\sin{(\\psi)} and 0 = - \\operatorname{F_{g}}{(\\psi)} + \\frac{d}{d \\psi} \\sin{(\\psi)} and 0^{\\psi} = (- \\operatorname{F_{g}}{(\\psi)} + \\frac{d}{d \\psi} \\sin{(\\psi)})^{\\psi} and 0^{\\psi} = (- \\operatorname{F_{g}}{(\\psi)} + \\cos{(\\psi)})^{\\psi} and (- \\operatorname{F_{g}}{(\\psi)} + \\frac{d}{d \\psi} \\sin{(\\psi)})^{\\psi} = 1 and \\psi = \\psi - \\operatorname{F_{g}}{(\\psi)} + \\frac{d}{d \\psi} \\sin{(\\psi)} and 0^{\\psi} = 1 and 0^{\\psi - \\operatorname{F_{g}}{(\\psi)} + \\frac{d}{d \\psi} \\sin{(\\psi)}} = 1 and 0^{\\psi - \\operatorname{F_{g}}{(\\psi)} + \\cos{(\\psi)}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\psi', commutative=True)), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["minus", 1, "Function('F_g')(Symbol('\\\\psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["power", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Symbol('\\\\psi', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), cos(Symbol('\\\\psi', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Add(Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))), Symbol('\\\\psi', commutative=True)), Integer(1))"], [["add", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Symbol('\\\\psi', commutative=True), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Integer(0), Symbol('\\\\psi', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Integer(0), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), Derivative(sin(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))), Integer(1))"], [["evaluate_derivatives", 8], "Equality(Pow(Integer(0), Add(Symbol('\\\\psi', commutative=True), Mul(Integer(-1), Function('F_g')(Symbol('\\\\psi', commutative=True))), cos(Symbol('\\\\psi', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(G)} = e^{G}, then obtain \\frac{d}{d G} (- G \\operatorname{t_{2}}{(G)} + G) = \\frac{d}{d G} (G (- 2 \\operatorname{t_{2}}{(G)} + e^{G}) + G)", "derivation": "\\operatorname{t_{2}}{(G)} = e^{G} and - \\operatorname{t_{2}}{(G)} = - 2 \\operatorname{t_{2}}{(G)} + e^{G} and - G \\operatorname{t_{2}}{(G)} = G (- 2 \\operatorname{t_{2}}{(G)} + e^{G}) and - G \\operatorname{t_{2}}{(G)} + G = G (- 2 \\operatorname{t_{2}}{(G)} + e^{G}) + G and \\frac{d}{d G} (- G \\operatorname{t_{2}}{(G)} + G) = \\frac{d}{d G} (G (- 2 \\operatorname{t_{2}}{(G)} + e^{G}) + G)", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["minus", 1, "Mul(Integer(2), Function('t_2')(Symbol('G', commutative=True)))"], "Equality(Mul(Integer(-1), Function('t_2')(Symbol('G', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('t_2')(Symbol('G', commutative=True))), exp(Symbol('G', commutative=True))))"], [["times", 2, "Symbol('G', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('G', commutative=True), Function('t_2')(Symbol('G', commutative=True))), Mul(Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Function('t_2')(Symbol('G', commutative=True))), exp(Symbol('G', commutative=True)))))"], [["add", 3, "Symbol('G', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('G', commutative=True), Function('t_2')(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Add(Mul(Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Function('t_2')(Symbol('G', commutative=True))), exp(Symbol('G', commutative=True)))), Symbol('G', commutative=True)))"], [["differentiate", 4, "Symbol('G', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('G', commutative=True), Function('t_2')(Symbol('G', commutative=True))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('G', commutative=True), Add(Mul(Integer(-1), Integer(2), Function('t_2')(Symbol('G', commutative=True))), exp(Symbol('G', commutative=True)))), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"]]}, {"prompt": "Given r{(c_{0})} = \\cos{(c_{0})}, then obtain - \\cos{(c_{0})} - \\cos^{c_{0}}{(c_{0})} + \\cos{(r{(c_{0})})} = - \\cos{(c_{0})} - \\cos^{c_{0}}{(c_{0})} + \\cos{(\\cos{(c_{0})})}", "derivation": "r{(c_{0})} = \\cos{(c_{0})} and r^{c_{0}}{(c_{0})} = \\cos^{c_{0}}{(c_{0})} and \\cos{(r{(c_{0})})} = \\cos{(\\cos{(c_{0})})} and - r^{c_{0}}{(c_{0})} + \\cos{(r{(c_{0})})} = - r^{c_{0}}{(c_{0})} + \\cos{(\\cos{(c_{0})})} and - \\cos^{c_{0}}{(c_{0})} + \\cos{(r{(c_{0})})} = - \\cos^{c_{0}}{(c_{0})} + \\cos{(\\cos{(c_{0})})} and - \\cos{(c_{0})} - \\cos^{c_{0}}{(c_{0})} + \\cos{(r{(c_{0})})} = - \\cos{(c_{0})} - \\cos^{c_{0}}{(c_{0})} + \\cos{(\\cos{(c_{0})})}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('c_0', commutative=True)), cos(Symbol('c_0', commutative=True)))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('r')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], [["cos", 1], "Equality(cos(Function('r')(Symbol('c_0', commutative=True))), cos(cos(Symbol('c_0', commutative=True))))"], [["add", 3, "Mul(Integer(-1), Pow(Function('r')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Function('r')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(Function('r')(Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('r')(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(cos(Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(Function('r')(Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(cos(Symbol('c_0', commutative=True)))))"], [["minus", 5, "cos(Symbol('c_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(Function('r')(Symbol('c_0', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('c_0', commutative=True)), Symbol('c_0', commutative=True))), cos(cos(Symbol('c_0', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(b,m)} = b^{m} and \\mathbf{J}_f{(b,m)} = b^{m}, then obtain \\frac{\\partial}{\\partial b} \\hat{\\mathbf{x}}{(b,m)} = \\frac{b^{m} m}{b}", "derivation": "\\hat{\\mathbf{x}}{(b,m)} = b^{m} and \\mathbf{J}_f{(b,m)} = b^{m} and \\hat{\\mathbf{x}}{(b,m)} = \\mathbf{J}_f{(b,m)} and m + \\hat{\\mathbf{x}}{(b,m)} = m + \\mathbf{J}_f{(b,m)} and \\frac{\\partial}{\\partial b} (m + \\hat{\\mathbf{x}}{(b,m)}) = \\frac{\\partial}{\\partial b} (m + \\mathbf{J}_f{(b,m)}) and b^{m} + m = m + \\mathbf{J}_f{(b,m)} and \\frac{\\partial}{\\partial b} (m + \\hat{\\mathbf{x}}{(b,m)}) = \\frac{\\partial}{\\partial b} (b^{m} + m) and \\frac{\\partial}{\\partial b} \\hat{\\mathbf{x}}{(b,m)} = \\frac{b^{m} m}{b}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Pow(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('m', commutative=True)))"], [["add", 3, "Symbol('m', commutative=True)"], "Equality(Add(Symbol('m', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True))), Add(Symbol('m', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('m', commutative=True))))"], [["differentiate", 4, "Symbol('b', commutative=True)"], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Symbol('m', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Pow(Symbol('b', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('b', commutative=True), Symbol('m', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Add(Symbol('m', commutative=True), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True))), Tuple(Symbol('b', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('b', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 7], "Equality(Derivative(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('b', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Mul(Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Symbol('b', commutative=True), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\hat{p},t_{1})} = \\hat{p}^{t_{1}}, then obtain \\operatorname{P_{e}}{(\\hat{p},t_{1})} \\int \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} dt_{1} = \\hat{p}^{t_{1}} \\int \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} dt_{1}", "derivation": "\\operatorname{P_{e}}{(\\hat{p},t_{1})} = \\hat{p}^{t_{1}} and \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} = (\\hat{p}^{t_{1}})^{t_{1}} and \\int \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} dt_{1} = \\int (\\hat{p}^{t_{1}})^{t_{1}} dt_{1} and \\operatorname{P_{e}}{(\\hat{p},t_{1})} \\int (\\hat{p}^{t_{1}})^{t_{1}} dt_{1} = \\hat{p}^{t_{1}} \\int (\\hat{p}^{t_{1}})^{t_{1}} dt_{1} and \\operatorname{P_{e}}{(\\hat{p},t_{1})} \\int \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} dt_{1} = \\hat{p}^{t_{1}} \\int \\operatorname{P_{e}}^{t_{1}}{(\\hat{p},t_{1})} dt_{1}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Pow(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["times", 1, "Integral(Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))"], "Equality(Mul(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Integral(Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Integral(Pow(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Integral(Pow(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Integral(Pow(Function('P_e')(Symbol('\\\\hat{p}', commutative=True), Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(\\rho_b,A_{1})} = \\varphi{(\\rho_b,A_{1})} + \\cos{(A_{1})}, then obtain A_{1} \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\operatorname{A_{z}}{(\\rho_b,A_{1})} + 2 \\cos{(A_{1})}) = A_{1} \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\varphi{(\\rho_b,A_{1})} + 3 \\cos{(A_{1})})", "derivation": "\\operatorname{A_{z}}{(\\rho_b,A_{1})} = \\varphi{(\\rho_b,A_{1})} + \\cos{(A_{1})} and - \\rho_b + \\operatorname{A_{z}}{(\\rho_b,A_{1})} + 2 \\cos{(A_{1})} = - \\rho_b + \\varphi{(\\rho_b,A_{1})} + 3 \\cos{(A_{1})} and \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\operatorname{A_{z}}{(\\rho_b,A_{1})} + 2 \\cos{(A_{1})}) = \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\varphi{(\\rho_b,A_{1})} + 3 \\cos{(A_{1})}) and A_{1} \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\operatorname{A_{z}}{(\\rho_b,A_{1})} + 2 \\cos{(A_{1})}) = A_{1} \\frac{\\partial}{\\partial A_{1}} (- \\rho_b + \\varphi{(\\rho_b,A_{1})} + 3 \\cos{(A_{1})})", "srepr_derivation": [["renaming_premise", "Equality(Function('A_z')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Add(Function('\\\\varphi')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), cos(Symbol('A_1', commutative=True))))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(2), cos(Symbol('A_1', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_z')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(2), cos(Symbol('A_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\varphi')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(3), cos(Symbol('A_1', commutative=True)))))"], [["differentiate", 2, "Symbol('A_1', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_z')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(2), cos(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\varphi')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(3), cos(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1))))"], [["times", 3, "Symbol('A_1', commutative=True)"], "Equality(Mul(Symbol('A_1', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('A_z')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(2), cos(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))), Mul(Symbol('A_1', commutative=True), Derivative(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), Function('\\\\varphi')(Symbol('\\\\rho_b', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(3), cos(Symbol('A_1', commutative=True)))), Tuple(Symbol('A_1', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(C,p)} = - C + p, then obtain \\int ((C Q{(C,p)})^{C} Q{(C,p)} - (C Q{(C,p)})^{- C}) dC = \\int ((C Q{(C,p)})^{C} (- C + p) - (C Q{(C,p)})^{- C}) dC", "derivation": "Q{(C,p)} = - C + p and C Q{(C,p)} = C (- C + p) and (C Q{(C,p)})^{C} = (C (- C + p))^{C} and (C (- C + p))^{C} Q{(C,p)} = (C (- C + p))^{C} (- C + p) and (C Q{(C,p)})^{C} Q{(C,p)} = (C Q{(C,p)})^{C} (- C + p) and (C Q{(C,p)})^{C} Q{(C,p)} - (C Q{(C,p)})^{- C} = (C Q{(C,p)})^{C} (- C + p) - (C Q{(C,p)})^{- C} and \\int ((C Q{(C,p)})^{C} Q{(C,p)} - (C Q{(C,p)})^{- C}) dC = \\int ((C Q{(C,p)})^{C} (- C + p) - (C Q{(C,p)})^{- C}) dC", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('C', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))))"], [["power", 2, "Symbol('C', commutative=True)"], "Equality(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Pow(Mul(Symbol('C', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Symbol('C', commutative=True)))"], [["times", 1, "Pow(Mul(Symbol('C', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Mul(Symbol('C', commutative=True), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))))"], [["minus", 5, "Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True)))"], "Equality(Add(Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True))))), Add(Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True))))))"], [["integrate", 6, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True))))), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Symbol('C', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('p', commutative=True))), Mul(Integer(-1), Pow(Mul(Symbol('C', commutative=True), Function('Q')(Symbol('C', commutative=True), Symbol('p', commutative=True))), Mul(Integer(-1), Symbol('C', commutative=True))))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\varphi{(T)} = \\sin{(T)}, then obtain (\\varphi{(T)} - \\sin{(T)}) \\varphi{(T)} + \\sin{(T)} + \\int \\varphi{(T)} dT = \\sin{(T)} + \\int \\varphi{(T)} dT", "derivation": "\\varphi{(T)} = \\sin{(T)} and \\varphi{(T)} - \\sin{(T)} = 0 and (\\varphi{(T)} - \\sin{(T)}) \\varphi{(T)} = 0 and \\int \\varphi{(T)} dT = \\int \\sin{(T)} dT and \\sin{(T)} + \\int \\varphi{(T)} dT = \\sin{(T)} + \\int \\sin{(T)} dT and (\\varphi{(T)} - \\sin{(T)}) \\varphi{(T)} + \\sin{(T)} + \\int \\sin{(T)} dT = \\sin{(T)} + \\int \\sin{(T)} dT and (\\varphi{(T)} - \\sin{(T)}) \\varphi{(T)} + \\sin{(T)} + \\int \\varphi{(T)} dT = \\sin{(T)} + \\int \\varphi{(T)} dT", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('T', commutative=True)), sin(Symbol('T', commutative=True)))"], [["minus", 1, "sin(Symbol('T', commutative=True))"], "Equality(Add(Function('\\\\varphi')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Integer(0))"], [["times", 2, "Function('\\\\varphi')(Symbol('T', commutative=True))"], "Equality(Mul(Add(Function('\\\\varphi')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Function('\\\\varphi')(Symbol('T', commutative=True))), Integer(0))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["minus", 4, "Mul(Integer(-1), sin(Symbol('T', commutative=True)))"], "Equality(Add(sin(Symbol('T', commutative=True)), Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(sin(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["add", 3, "Add(sin(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))))"], "Equality(Add(Mul(Add(Function('\\\\varphi')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Function('\\\\varphi')(Symbol('T', commutative=True))), sin(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(sin(Symbol('T', commutative=True)), Integral(sin(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Add(Function('\\\\varphi')(Symbol('T', commutative=True)), Mul(Integer(-1), sin(Symbol('T', commutative=True)))), Function('\\\\varphi')(Symbol('T', commutative=True))), sin(Symbol('T', commutative=True)), Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Add(sin(Symbol('T', commutative=True)), Integral(Function('\\\\varphi')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given T{(v_{x})} = \\sin{(v_{x})}, then obtain \\frac{- T{(v_{x})} + \\int (v_{x} + T{(v_{x})}) dv_{x}}{- T{(v_{x})} + \\int (v_{x} + \\sin{(v_{x})}) dv_{x}} = 1", "derivation": "T{(v_{x})} = \\sin{(v_{x})} and v_{x} + T{(v_{x})} = v_{x} + \\sin{(v_{x})} and \\int (v_{x} + T{(v_{x})}) dv_{x} = \\int (v_{x} + \\sin{(v_{x})}) dv_{x} and - \\sin{(v_{x})} + \\int (v_{x} + T{(v_{x})}) dv_{x} = - \\sin{(v_{x})} + \\int (v_{x} + \\sin{(v_{x})}) dv_{x} and - T{(v_{x})} + \\int (v_{x} + T{(v_{x})}) dv_{x} = - T{(v_{x})} + \\int (v_{x} + \\sin{(v_{x})}) dv_{x} and \\frac{- T{(v_{x})} + \\int (v_{x} + T{(v_{x})}) dv_{x}}{- T{(v_{x})} + \\int (v_{x} + \\sin{(v_{x})}) dv_{x}} = 1", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('v_x', commutative=True)), sin(Symbol('v_x', commutative=True)))"], [["add", 1, "Symbol('v_x', commutative=True)"], "Equality(Add(Symbol('v_x', commutative=True), Function('T')(Symbol('v_x', commutative=True))), Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))))"], [["integrate", 2, "Symbol('v_x', commutative=True)"], "Equality(Integral(Add(Symbol('v_x', commutative=True), Function('T')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"], [["minus", 3, "sin(Symbol('v_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), Function('T')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), Function('T')(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), Function('T')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Add(Mul(Integer(-1), Function('T')(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))))"], [["divide", 5, "Add(Mul(Integer(-1), Function('T')(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Function('T')(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), Function('T')(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Pow(Add(Mul(Integer(-1), Function('T')(Symbol('v_x', commutative=True))), Integral(Add(Symbol('v_x', commutative=True), sin(Symbol('v_x', commutative=True))), Tuple(Symbol('v_x', commutative=True)))), Integer(-1))), Integer(1))"]]}, {"prompt": "Given \\omega{(x)} = \\cos{(\\cos{(x)})}, then obtain \\operatorname{v_{z}}{(\\dot{x})} + \\frac{(\\omega{(x)} + \\cos{(x)}) \\omega{(x)}}{x} = \\operatorname{v_{z}}{(\\dot{x})} + \\frac{(\\omega{(x)} + \\cos{(x)}) \\cos{(\\cos{(x)})}}{x}", "derivation": "\\omega{(x)} = \\cos{(\\cos{(x)})} and \\omega{(x)} + \\cos{(x)} = \\cos{(x)} + \\cos{(\\cos{(x)})} and \\frac{(\\cos{(x)} + \\cos{(\\cos{(x)})}) \\omega{(x)}}{x} = \\frac{(\\cos{(x)} + \\cos{(\\cos{(x)})}) \\cos{(\\cos{(x)})}}{x} and \\frac{(\\omega{(x)} + \\cos{(x)}) \\omega{(x)}}{x} = \\frac{(\\omega{(x)} + \\cos{(x)}) \\cos{(\\cos{(x)})}}{x} and \\operatorname{v_{z}}{(\\dot{x})} + \\frac{(\\omega{(x)} + \\cos{(x)}) \\omega{(x)}}{x} = \\operatorname{v_{z}}{(\\dot{x})} + \\frac{(\\omega{(x)} + \\cos{(x)}) \\cos{(\\cos{(x)})}}{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True))))"], [["add", 1, "cos(Symbol('x', commutative=True))"], "Equality(Add(Function('\\\\omega')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Add(cos(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True)))))"], [["times", 1, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(cos(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True)))))"], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(cos(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True)))), Function('\\\\omega')(Symbol('x', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(cos(Symbol('x', commutative=True)), cos(cos(Symbol('x', commutative=True)))), cos(cos(Symbol('x', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Function('\\\\omega')(Symbol('x', commutative=True))), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), cos(cos(Symbol('x', commutative=True)))))"], [["add", 4, "Function('v_z')(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Function('v_z')(Symbol('\\\\dot{x}', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), Function('\\\\omega')(Symbol('x', commutative=True)))), Add(Function('v_z')(Symbol('\\\\dot{x}', commutative=True)), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Function('\\\\omega')(Symbol('x', commutative=True)), cos(Symbol('x', commutative=True))), cos(cos(Symbol('x', commutative=True))))))"]]}, {"prompt": "Given \\sigma_{x}{(a)} = \\sin{(a)}, then derive \\frac{d}{d a} \\sigma_{x}{(a)} = \\cos{(a)}, then obtain \\frac{d^{2}}{d a^{2}} \\sin{(a)} = \\frac{d}{d a} \\cos{(a)}", "derivation": "\\sigma_{x}{(a)} = \\sin{(a)} and \\frac{d}{d a} \\sigma_{x}{(a)} = \\frac{d}{d a} \\sin{(a)} and \\frac{d}{d a} \\sigma_{x}{(a)} = \\cos{(a)} and \\frac{d^{2}}{d a^{2}} \\sigma_{x}{(a)} = \\frac{d}{d a} \\cos{(a)} and \\frac{d^{2}}{d a^{2}} \\sin{(a)} = \\frac{d}{d a} \\cos{(a)}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('a', commutative=True)), sin(Symbol('a', commutative=True)))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), cos(Symbol('a', commutative=True)))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_x')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(sin(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(2))), Derivative(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(y,v_{z})} = \\frac{\\partial}{\\partial y} (- v_{z} + y), then obtain e^{1 - \\frac{\\frac{\\partial}{\\partial y} (- v_{z} + y)}{\\operatorname{V_{\\mathbf{E}}}{(y,v_{z})}}} = 1", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(y,v_{z})} = \\frac{\\partial}{\\partial y} (- v_{z} + y) and 1 = \\frac{\\frac{\\partial}{\\partial y} (- v_{z} + y)}{\\operatorname{V_{\\mathbf{E}}}{(y,v_{z})}} and 1 - \\frac{\\frac{\\partial}{\\partial y} (- v_{z} + y)}{\\operatorname{V_{\\mathbf{E}}}{(y,v_{z})}} = 0 and e^{1 - \\frac{\\frac{\\partial}{\\partial y} (- v_{z} + y)}{\\operatorname{V_{\\mathbf{E}}}{(y,v_{z})}}} = 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["divide", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))"], [["minus", 2, "Mul(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))))), Integer(0))"], [["exp", 3], "Equality(exp(Add(Integer(1), Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{E}}')(Symbol('y', commutative=True), Symbol('v_z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1)))))), Integer(1))"]]}, {"prompt": "Given H{(\\mathbb{I},F_{x})} = F_{x} \\mathbb{I} and Q{(F_{x})} = F_{x}, then derive 0 = F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},F_{x})}, then obtain \\int 0 dQ{(F_{x})} = \\int (F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I}) dQ{(F_{x})}", "derivation": "H{(\\mathbb{I},F_{x})} = F_{x} \\mathbb{I} and \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},F_{x})} = \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I} and 0 = \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I} - \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},F_{x})} and 0 = F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} H{(\\mathbb{I},F_{x})} and 0 = F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I} and \\int 0 dF_{x} = \\int (F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I}) dF_{x} and Q{(F_{x})} = F_{x} and \\int 0 dQ{(F_{x})} = \\int (F_{x} - \\frac{\\partial}{\\partial \\mathbb{I}} F_{x} \\mathbb{I}) dQ{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_x', commutative=True)), Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Derivative(Function('H')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('F_x', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))))"], [["integrate", 5, "Symbol('F_x', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('F_x', commutative=True))), Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))), Tuple(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], [["substitute_RHS_for_LHS", 6, 7], "Equality(Integral(Integer(0), Tuple(Function('Q')(Symbol('F_x', commutative=True)))), Integral(Add(Symbol('F_x', commutative=True), Mul(Integer(-1), Derivative(Mul(Symbol('F_x', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))), Tuple(Function('Q')(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbb{I},E_{n})} = E_{n}^{\\mathbb{I}}, then obtain \\frac{\\partial^{2}}{\\partial E_{n}^{2}} (E_{n}^{\\mathbb{I}} + \\mathbf{J}_f{(\\mathbb{I},E_{n})}) = \\frac{\\partial^{2}}{\\partial E_{n}^{2}} 2 E_{n}^{\\mathbb{I}}", "derivation": "\\mathbf{J}_f{(\\mathbb{I},E_{n})} = E_{n}^{\\mathbb{I}} and E_{n}^{\\mathbb{I}} + \\mathbf{J}_f{(\\mathbb{I},E_{n})} = 2 E_{n}^{\\mathbb{I}} and \\frac{\\partial}{\\partial E_{n}} (E_{n}^{\\mathbb{I}} + \\mathbf{J}_f{(\\mathbb{I},E_{n})}) = \\frac{\\partial}{\\partial E_{n}} 2 E_{n}^{\\mathbb{I}} and \\frac{\\partial^{2}}{\\partial E_{n}^{2}} (E_{n}^{\\mathbb{I}} + \\mathbf{J}_f{(\\mathbb{I},E_{n})}) = \\frac{\\partial^{2}}{\\partial E_{n}^{2}} 2 E_{n}^{\\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E_n', commutative=True)), Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)))"], [["add", 1, "Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E_n', commutative=True))), Mul(Integer(2), Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))))"], [["differentiate", 2, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Add(Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('E_n', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(2))), Derivative(Mul(Integer(2), Pow(Symbol('E_n', commutative=True), Symbol('\\\\mathbb{I}', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\mu_{0}{(\\mathbf{r})} = \\cos{(\\mathbf{r})}, then obtain \\frac{d^{2}}{d \\mathbf{r}^{2}} \\mu_{0}{(\\mathbf{r})} \\frac{d^{2}}{d \\mathbf{r}^{2}} \\cos{(\\mathbf{r})} = (\\frac{d^{2}}{d \\mathbf{r}^{2}} \\cos{(\\mathbf{r})})^{2}", "derivation": "\\mu_{0}{(\\mathbf{r})} = \\cos{(\\mathbf{r})} and \\frac{d}{d \\mathbf{r}} \\mu_{0}{(\\mathbf{r})} = \\frac{d}{d \\mathbf{r}} \\cos{(\\mathbf{r})} and \\frac{d^{2}}{d \\mathbf{r}^{2}} \\mu_{0}{(\\mathbf{r})} = \\frac{d^{2}}{d \\mathbf{r}^{2}} \\cos{(\\mathbf{r})} and \\frac{d^{2}}{d \\mathbf{r}^{2}} \\mu_{0}{(\\mathbf{r})} \\frac{d^{2}}{d \\mathbf{r}^{2}} \\cos{(\\mathbf{r})} = (\\frac{d^{2}}{d \\mathbf{r}^{2}} \\cos{(\\mathbf{r})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\mathbf{r}', commutative=True)), cos(Symbol('\\\\mathbf{r}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))))"], [["times", 3, "Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)))"], "Equality(Mul(Derivative(Function('\\\\mu_0')(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2)))), Pow(Derivative(cos(Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(2))), Integer(2)))"]]}, {"prompt": "Given \\dot{y}{(v_{2},f^{\\prime})} = f^{\\prime} + v_{2}, then derive \\int \\dot{y}{(v_{2},f^{\\prime})} df^{\\prime} = \\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q, then obtain (\\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q)^{2} = (\\int \\dot{y}{(v_{2},f^{\\prime})} df^{\\prime})^{2}", "derivation": "\\dot{y}{(v_{2},f^{\\prime})} = f^{\\prime} + v_{2} and \\int \\dot{y}{(v_{2},f^{\\prime})} df^{\\prime} = \\int (f^{\\prime} + v_{2}) df^{\\prime} and \\int \\dot{y}{(v_{2},f^{\\prime})} df^{\\prime} = \\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q and \\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q = \\int (f^{\\prime} + v_{2}) df^{\\prime} and (\\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q)^{2} = (\\int (f^{\\prime} + v_{2}) df^{\\prime})^{2} and (\\frac{(f^{\\prime})^{2}}{2} + f^{\\prime} v_{2} + q)^{2} = (\\int \\dot{y}{(v_{2},f^{\\prime})} df^{\\prime})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('v_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('v_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('v_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Symbol('q', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Symbol('q', commutative=True)), Integral(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Symbol('q', commutative=True)), Integer(2)), Pow(Integral(Add(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('f^{\\\\prime}', commutative=True), Integer(2))), Mul(Symbol('f^{\\\\prime}', commutative=True), Symbol('v_2', commutative=True)), Symbol('q', commutative=True)), Integer(2)), Pow(Integral(Function('\\\\dot{y}')(Symbol('v_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(t)} = \\sin{(t)} and \\bar{\\h}{(\\hat{x}_0,t,a^{\\dagger})} = ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\operatorname{t_{2}}{(t)})})^{t}, then obtain \\bar{\\h}{(\\hat{x}_0,t,a^{\\dagger})} = ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\sin{(t)})})^{t}", "derivation": "\\operatorname{t_{2}}{(t)} = \\sin{(t)} and \\cos{(\\operatorname{t_{2}}{(t)})} = \\cos{(\\sin{(t)})} and (a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\operatorname{t_{2}}{(t)})} = (a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\sin{(t)})} and ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\operatorname{t_{2}}{(t)})})^{t} = ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\sin{(t)})})^{t} and \\bar{\\h}{(\\hat{x}_0,t,a^{\\dagger})} = ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\operatorname{t_{2}}{(t)})})^{t} and \\bar{\\h}{(\\hat{x}_0,t,a^{\\dagger})} = ((a^{\\dagger})^{\\hat{x}_0} + \\cos{(\\sin{(t)})})^{t}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["cos", 1], "Equality(cos(Function('t_2')(Symbol('t', commutative=True))), cos(sin(Symbol('t', commutative=True))))"], [["add", 2, "Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True))"], "Equality(Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(Function('t_2')(Symbol('t', commutative=True)))), Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('t', commutative=True)))))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(Function('t_2')(Symbol('t', commutative=True)))), Symbol('t', commutative=True)), Pow(Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('t', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(Function('t_2')(Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\hbar')(Symbol('\\\\hat{x}_0', commutative=True), Symbol('t', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Pow(Add(Pow(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\hat{x}_0', commutative=True)), cos(sin(Symbol('t', commutative=True)))), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\hat{x}{(\\mathbb{I},Q)} = \\frac{\\mathbb{I}}{Q}, then derive \\frac{\\partial}{\\partial \\mathbb{I}} \\hat{x}{(\\mathbb{I},Q)} = \\frac{1}{Q}, then obtain \\frac{\\partial}{\\partial Q} \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Q} dQ = \\frac{d}{d Q} \\int \\frac{1}{Q} dQ", "derivation": "\\hat{x}{(\\mathbb{I},Q)} = \\frac{\\mathbb{I}}{Q} and \\frac{\\partial}{\\partial \\mathbb{I}} \\hat{x}{(\\mathbb{I},Q)} = \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Q} and \\frac{\\partial}{\\partial \\mathbb{I}} \\hat{x}{(\\mathbb{I},Q)} = \\frac{1}{Q} and \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Q} = \\frac{1}{Q} and \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Q} dQ = \\int \\frac{1}{Q} dQ and \\frac{\\partial}{\\partial Q} \\int \\frac{\\partial}{\\partial \\mathbb{I}} \\frac{\\mathbb{I}}{Q} dQ = \\frac{d}{d Q} \\int \\frac{1}{Q} dQ", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Pow(Symbol('Q', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Pow(Symbol('Q', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('Q', commutative=True)"], "Equality(Integral(Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Integral(Pow(Symbol('Q', commutative=True), Integer(-1)), Tuple(Symbol('Q', commutative=True))))"], [["differentiate", 5, "Symbol('Q', commutative=True)"], "Equality(Derivative(Integral(Derivative(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Integral(Pow(Symbol('Q', commutative=True), Integer(-1)), Tuple(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(y,n)} = - \\sin{(n - y)} and \\mathbf{g}{(y,n)} = - \\sin{(n - y)}, then obtain - (\\int \\varphi{(y,n)} dy) (\\iint \\varphi{(y,n)} dy dn)^{y} = - (\\int \\varphi{(y,n)} dy) (\\iint - \\sin{(n - y)} dy dn)^{y}", "derivation": "\\varphi{(y,n)} = - \\sin{(n - y)} and \\mathbf{g}{(y,n)} = - \\sin{(n - y)} and \\int \\varphi{(y,n)} dy = \\int - \\sin{(n - y)} dy and \\iint \\varphi{(y,n)} dy dn = \\iint - \\sin{(n - y)} dy dn and \\mathbf{g}{(y,n)} = \\varphi{(y,n)} and \\iint \\mathbf{g}{(y,n)} dy dn = \\iint - \\sin{(n - y)} dy dn and (\\iint \\mathbf{g}{(y,n)} dy dn)^{y} = (\\iint - \\sin{(n - y)} dy dn)^{y} and (\\iint \\varphi{(y,n)} dy dn)^{y} = (\\iint - \\sin{(n - y)} dy dn)^{y} and - (\\int \\varphi{(y,n)} dy) (\\iint \\varphi{(y,n)} dy dn)^{y} = - (\\int \\varphi{(y,n)} dy) (\\iint - \\sin{(n - y)} dy dn)^{y}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True))))"], [["integrate", 3, "Symbol('n', commutative=True)"], "Equality(Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["power", 6, "Symbol('y', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{g}')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True)), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True)))"], [["times", 8, "Mul(Integer(-1), Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True))))"], "Equality(Mul(Integer(-1), Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True))), Pow(Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\varphi')(Symbol('y', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('y', commutative=True))), Pow(Integral(Mul(Integer(-1), sin(Add(Symbol('n', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Tuple(Symbol('y', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('y', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\hbar,Q)} = Q \\hbar, then obtain - \\operatorname{A_{x}}{(\\hbar,Q)} + \\int (Q \\hbar + 3 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar = - \\operatorname{A_{x}}{(\\hbar,Q)} + \\int (2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar", "derivation": "\\operatorname{A_{x}}{(\\hbar,Q)} = Q \\hbar and Q \\hbar + \\operatorname{A_{x}}{(\\hbar,Q)} = 2 Q \\hbar and 2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)} = 4 Q \\hbar and Q \\hbar + 3 \\operatorname{A_{x}}{(\\hbar,Q)} = 2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)} and \\int (Q \\hbar + 3 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar = \\int (2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar and - Q \\hbar + \\int (Q \\hbar + 3 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar = - Q \\hbar + \\int (2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar and - \\operatorname{A_{x}}{(\\hbar,Q)} + \\int (Q \\hbar + 3 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar = - \\operatorname{A_{x}}{(\\hbar,Q)} + \\int (2 Q \\hbar + 2 \\operatorname{A_{x}}{(\\hbar,Q)}) d\\hbar", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["add", 1, "Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True))), Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["divide", 2, "Rational(1, 2)"], "Equality(Add(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Mul(Integer(4), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(3), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Add(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(3), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True))))"], [["minus", 5, "Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(3), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(3), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True)))), Add(Mul(Integer(-1), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Mul(Integer(2), Symbol('Q', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Function('A_x')(Symbol('\\\\hbar', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\int (\\mathbf{H} + \\frac{\\operatorname{v_{x}}{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}}) d\\mathbf{H} = \\int (\\mathbf{H} + 1) d\\mathbf{H}", "derivation": "\\operatorname{v_{x}}{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and \\frac{\\operatorname{v_{x}}{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}} = 1 and \\mathbf{H} + \\frac{\\operatorname{v_{x}}{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}} = \\mathbf{H} + 1 and \\int (\\mathbf{H} + \\frac{\\operatorname{v_{x}}{(\\mathbf{H})}}{\\sin{(\\mathbf{H})}}) d\\mathbf{H} = \\int (\\mathbf{H} + 1) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))), Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Mul(Function('v_x')(Symbol('\\\\mathbf{H}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(C_{d})} = e^{C_{d}} and C{(C_{d})} = C_{d}^{2} \\tilde{g}^*^{2 C_{d}}{(C_{d})} (e^{C_{d}})^{2 C_{d}}, then obtain C{(C_{d})} = C_{d}^{2} \\tilde{g}^*^{4 C_{d}}{(C_{d})}", "derivation": "\\tilde{g}^*{(C_{d})} = e^{C_{d}} and \\tilde{g}^*^{C_{d}}{(C_{d})} = (e^{C_{d}})^{C_{d}} and \\tilde{g}^*^{2 C_{d}}{(C_{d})} = \\tilde{g}^*^{C_{d}}{(C_{d})} (e^{C_{d}})^{C_{d}} and \\tilde{g}^*^{4 C_{d}}{(C_{d})} = \\tilde{g}^*^{2 C_{d}}{(C_{d})} (e^{C_{d}})^{2 C_{d}} and C{(C_{d})} = C_{d}^{2} \\tilde{g}^*^{2 C_{d}}{(C_{d})} (e^{C_{d}})^{2 C_{d}} and C{(C_{d})} = C_{d}^{2} \\tilde{g}^*^{4 C_{d}}{(C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), exp(Symbol('C_d', commutative=True)))"], [["power", 1, "Symbol('C_d', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(exp(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)))"], [["times", 2, "Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('C_d', commutative=True))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True)), Pow(exp(Symbol('C_d', commutative=True)), Symbol('C_d', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Mul(Integer(4), Symbol('C_d', commutative=True))), Mul(Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('C_d', commutative=True))), Pow(exp(Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('C_d', commutative=True)))))"], ["renaming_premise", "Equality(Function('C')(Symbol('C_d', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(2)), Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('C_d', commutative=True))), Pow(exp(Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('C_d', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Function('C')(Symbol('C_d', commutative=True)), Mul(Pow(Symbol('C_d', commutative=True), Integer(2)), Pow(Function('\\\\tilde{g}^*')(Symbol('C_d', commutative=True)), Mul(Integer(4), Symbol('C_d', commutative=True)))))"]]}, {"prompt": "Given T{(\\mu_0,B)} = e^{B + \\mu_0} and \\tilde{g}{(g,J)} = \\frac{g}{J}, then obtain T{(\\mu_0,B)} \\tilde{g}{(g,J)} + e^{B + \\mu_0} = e^{B + \\mu_0} + \\frac{g T{(\\mu_0,B)}}{J}", "derivation": "T{(\\mu_0,B)} = e^{B + \\mu_0} and \\tilde{g}{(g,J)} = \\frac{g}{J} and \\tilde{g}{(g,J)} e^{B + \\mu_0} = \\frac{g e^{B + \\mu_0}}{J} and T{(\\mu_0,B)} \\tilde{g}{(g,J)} = \\frac{g T{(\\mu_0,B)}}{J} and T{(\\mu_0,B)} \\tilde{g}{(g,J)} + e^{B + \\mu_0} = e^{B + \\mu_0} + \\frac{g T{(\\mu_0,B)}}{J}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('B', commutative=True)), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], ["get_premise", "Equality(Function('\\\\tilde{g}')(Symbol('g', commutative=True), Symbol('J', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('g', commutative=True)))"], [["times", 2, "exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Mul(Function('\\\\tilde{g}')(Symbol('g', commutative=True), Symbol('J', commutative=True)), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('g', commutative=True), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('B', commutative=True)), Function('\\\\tilde{g}')(Symbol('g', commutative=True), Symbol('J', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('g', commutative=True), Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('B', commutative=True))))"], [["add", 4, "exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], "Equality(Add(Mul(Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('B', commutative=True)), Function('\\\\tilde{g}')(Symbol('g', commutative=True), Symbol('J', commutative=True))), exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True)))), Add(exp(Add(Symbol('B', commutative=True), Symbol('\\\\mu_0', commutative=True))), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('g', commutative=True), Function('T')(Symbol('\\\\mu_0', commutative=True), Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(C,M)} = \\frac{C}{M}, then derive \\int \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} dC = - \\frac{C^{2}}{2 M^{2}} + \\mathbf{f}, then obtain (\\frac{\\partial}{\\partial M} \\frac{C}{M})^{- C} \\int \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} dC = (- \\frac{C^{2}}{2 M^{2}} + \\mathbf{f}) (\\frac{\\partial}{\\partial M} \\frac{C}{M})^{- C}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(C,M)} = \\frac{C}{M} and \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} = \\frac{\\partial}{\\partial M} \\frac{C}{M} and \\int \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} dC = \\int \\frac{\\partial}{\\partial M} \\frac{C}{M} dC and \\int \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} dC = - \\frac{C^{2}}{2 M^{2}} + \\mathbf{f} and (\\frac{\\partial}{\\partial M} \\frac{C}{M})^{- C} \\int \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(C,M)} dC = (- \\frac{C^{2}}{2 M^{2}} + \\mathbf{f}) (\\frac{\\partial}{\\partial M} \\frac{C}{M})^{- C}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Integral(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2)), Pow(Symbol('M', commutative=True), Integer(-2))), Symbol('\\\\mathbf{f}', commutative=True)))"], [["divide", 4, "Pow(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True), Integer(1))), Symbol('C', commutative=True))"], "Equality(Mul(Pow(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True))), Integral(Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('C', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Tuple(Symbol('C', commutative=True)))), Mul(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2)), Pow(Symbol('M', commutative=True), Integer(-2))), Symbol('\\\\mathbf{f}', commutative=True)), Pow(Derivative(Mul(Symbol('C', commutative=True), Pow(Symbol('M', commutative=True), Integer(-1))), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\hat{X}{(\\Psi^{\\dagger},U)} = - \\sin{(U - \\Psi^{\\dagger})}, then obtain \\int (- \\Psi^{\\dagger} \\hat{X}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},U)})^{\\Psi^{\\dagger}} dU = \\int (- \\Psi^{\\dagger} (- \\sin{(U - \\Psi^{\\dagger})})^{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} dU", "derivation": "\\hat{X}{(\\Psi^{\\dagger},U)} = - \\sin{(U - \\Psi^{\\dagger})} and \\hat{X}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},U)} = (- \\sin{(U - \\Psi^{\\dagger})})^{\\Psi^{\\dagger}} and - \\Psi^{\\dagger} \\hat{X}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},U)} = - \\Psi^{\\dagger} (- \\sin{(U - \\Psi^{\\dagger})})^{\\Psi^{\\dagger}} and (- \\Psi^{\\dagger} \\hat{X}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},U)})^{\\Psi^{\\dagger}} = (- \\Psi^{\\dagger} (- \\sin{(U - \\Psi^{\\dagger})})^{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} and \\int (- \\Psi^{\\dagger} \\hat{X}^{\\Psi^{\\dagger}}{(\\Psi^{\\dagger},U)})^{\\Psi^{\\dagger}} dU = \\int (- \\Psi^{\\dagger} (- \\sin{(U - \\Psi^{\\dagger})})^{\\Psi^{\\dagger}})^{\\Psi^{\\dagger}} dU", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))))"], [["power", 1, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('\\\\hat{X}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Function('\\\\hat{X}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('U', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Pow(Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Pow(Mul(Integer(-1), sin(Add(Symbol('U', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\mathbf{p},V)} = V \\mathbf{p}, then derive V + \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{C_{d}}{(\\mathbf{p},V)} = 2 V, then obtain \\mathbf{p} \\frac{\\partial}{\\partial \\mathbf{p}} 2 V \\mathbf{p} + \\frac{\\partial}{\\partial V} (V + \\frac{\\partial}{\\partial \\mathbf{p}} V \\mathbf{p}) = \\mathbf{p} \\frac{\\partial}{\\partial \\mathbf{p}} 2 V \\mathbf{p} + \\frac{d}{d V} 2 V", "derivation": "\\operatorname{C_{d}}{(\\mathbf{p},V)} = V \\mathbf{p} and V \\mathbf{p} + \\operatorname{C_{d}}{(\\mathbf{p},V)} = 2 V \\mathbf{p} and \\frac{\\partial}{\\partial \\mathbf{p}} (V \\mathbf{p} + \\operatorname{C_{d}}{(\\mathbf{p},V)}) = \\frac{\\partial}{\\partial \\mathbf{p}} 2 V \\mathbf{p} and V + \\frac{\\partial}{\\partial \\mathbf{p}} \\operatorname{C_{d}}{(\\mathbf{p},V)} = 2 V and V + \\frac{\\partial}{\\partial \\mathbf{p}} V \\mathbf{p} = 2 V and \\frac{\\partial}{\\partial V} (V + \\frac{\\partial}{\\partial \\mathbf{p}} V \\mathbf{p}) = \\frac{d}{d V} 2 V and \\mathbf{p} \\frac{\\partial}{\\partial \\mathbf{p}} 2 V \\mathbf{p} + \\frac{\\partial}{\\partial V} (V + \\frac{\\partial}{\\partial \\mathbf{p}} V \\mathbf{p}) = \\mathbf{p} \\frac{\\partial}{\\partial \\mathbf{p}} 2 V \\mathbf{p} + \\frac{d}{d V} 2 V", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)), Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["add", 1, "Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True))), Mul(Integer(2), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Add(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('V', commutative=True), Derivative(Function('C_d')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('V', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Mul(Integer(2), Symbol('V', commutative=True)))"], [["differentiate", 5, "Symbol('V', commutative=True)"], "Equality(Derivative(Add(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Tuple(Symbol('V', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1))))"], [["add", 6, "Mul(Symbol('\\\\mathbf{p}', commutative=True), Derivative(Mul(Integer(2), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], "Equality(Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Derivative(Mul(Integer(2), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Derivative(Add(Symbol('V', commutative=True), Derivative(Mul(Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Tuple(Symbol('V', commutative=True), Integer(1)))), Add(Mul(Symbol('\\\\mathbf{p}', commutative=True), Derivative(Mul(Integer(2), Symbol('V', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1)))), Derivative(Mul(Integer(2), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\varphi^{*}{(p)} = e^{\\sin{(p)}}, then obtain - \\varphi^{*}{(p)} + \\frac{d}{d p} ((\\int \\varphi^{*}{(p)} dp)^{p})^{p} + \\int \\varphi^{*}{(p)} dp = - \\varphi^{*}{(p)} + \\frac{d}{d p} ((\\int e^{\\sin{(p)}} dp)^{p})^{p} + \\int \\varphi^{*}{(p)} dp", "derivation": "\\varphi^{*}{(p)} = e^{\\sin{(p)}} and \\int \\varphi^{*}{(p)} dp = \\int e^{\\sin{(p)}} dp and (\\int \\varphi^{*}{(p)} dp)^{p} = (\\int e^{\\sin{(p)}} dp)^{p} and ((\\int \\varphi^{*}{(p)} dp)^{p})^{p} = ((\\int e^{\\sin{(p)}} dp)^{p})^{p} and \\frac{d}{d p} ((\\int \\varphi^{*}{(p)} dp)^{p})^{p} = \\frac{d}{d p} ((\\int e^{\\sin{(p)}} dp)^{p})^{p} and - \\varphi^{*}{(p)} + \\frac{d}{d p} ((\\int \\varphi^{*}{(p)} dp)^{p})^{p} + \\int \\varphi^{*}{(p)} dp = - \\varphi^{*}{(p)} + \\frac{d}{d p} ((\\int e^{\\sin{(p)}} dp)^{p})^{p} + \\int \\varphi^{*}{(p)} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('p', commutative=True)), exp(sin(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(exp(sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["power", 2, "Symbol('p', commutative=True)"], "Equality(Pow(Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Pow(Integral(exp(sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)))"], [["power", 3, "Symbol('p', commutative=True)"], "Equality(Pow(Pow(Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Pow(Pow(Integral(exp(sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)))"], [["differentiate", 4, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Pow(Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Pow(Pow(Integral(exp(sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["add", 5, "Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('p', commutative=True))), Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('p', commutative=True))), Derivative(Pow(Pow(Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varphi^*')(Symbol('p', commutative=True))), Derivative(Pow(Pow(Integral(exp(sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))), Symbol('p', commutative=True)), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Integral(Function('\\\\varphi^*')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)}, then obtain (\\frac{d}{d \\hat{p}_0} (\\mathbf{E}{(\\hat{p}_0)} + 1))^{\\hat{p}_0} = (\\frac{d}{d \\hat{p}_0} (\\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} + 1))^{\\hat{p}_0}", "derivation": "\\mathbf{E}{(\\hat{p}_0)} = \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} and \\mathbf{E}{(\\hat{p}_0)} + 1 = \\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} + 1 and \\frac{d}{d \\hat{p}_0} (\\mathbf{E}{(\\hat{p}_0)} + 1) = \\frac{d}{d \\hat{p}_0} (\\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} + 1) and (\\frac{d}{d \\hat{p}_0} (\\mathbf{E}{(\\hat{p}_0)} + 1))^{\\hat{p}_0} = (\\frac{d}{d \\hat{p}_0} (\\frac{d}{d \\hat{p}_0} \\cos{(\\hat{p}_0)} + 1))^{\\hat{p}_0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)), Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)), Add(Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integer(1)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Add(Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Pow(Derivative(Add(Function('\\\\mathbf{E}')(Symbol('\\\\hat{p}_0', commutative=True)), Integer(1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)), Pow(Derivative(Add(Derivative(cos(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Symbol('\\\\hat{p}_0', commutative=True)))"]]}, {"prompt": "Given \\omega{(\\mathbf{M})} = e^{\\mathbf{M}}, then derive \\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M} = \\Psi_{nl} + \\mathbf{M}, then obtain (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M}) (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M})^{- \\mathbf{M}} = (\\int 1 d\\mathbf{M}) (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M})^{- \\mathbf{M}}", "derivation": "\\omega{(\\mathbf{M})} = e^{\\mathbf{M}} and \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} = 1 and \\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M} = \\int 1 d\\mathbf{M} and \\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M} = \\Psi_{nl} + \\mathbf{M} and (\\Psi_{nl} + \\mathbf{M})^{- \\mathbf{M}} \\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M} = (\\Psi_{nl} + \\mathbf{M})^{- \\mathbf{M}} \\int 1 d\\mathbf{M} and (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M}) (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M})^{- \\mathbf{M}} = (\\int 1 d\\mathbf{M}) (\\int \\omega{(\\mathbf{M})} e^{- \\mathbf{M}} d\\mathbf{M})^{- \\mathbf{M}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 1, "exp(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Integer(1))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["divide", 3, "Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Pow(Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Mul(Integral(Integer(1), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Pow(Integral(Mul(Function('\\\\omega')(Symbol('\\\\mathbf{M}', commutative=True)), exp(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\hat{x}_0{(\\dot{y})} = e^{\\dot{y}}, then derive \\frac{d}{d \\dot{y}} \\hat{x}_0{(\\dot{y})} = e^{\\dot{y}}, then obtain \\hat{x}_0{(\\dot{y})} = \\frac{d}{d \\dot{y}} e^{\\dot{y}}", "derivation": "\\hat{x}_0{(\\dot{y})} = e^{\\dot{y}} and \\frac{d}{d \\dot{y}} \\hat{x}_0{(\\dot{y})} = \\frac{d}{d \\dot{y}} e^{\\dot{y}} and \\frac{d}{d \\dot{y}} \\hat{x}_0{(\\dot{y})} = e^{\\dot{y}} and \\frac{d}{d \\dot{y}} e^{\\dot{y}} = e^{\\dot{y}} and \\hat{x}_0{(\\dot{y})} = \\frac{d}{d \\dot{y}} e^{\\dot{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\dot{y}', commutative=True)), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), exp(Symbol('\\\\dot{y}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 4], "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\dot{y}', commutative=True)), Derivative(exp(Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(\\varphi^*)} = \\sin{(\\sin{(\\varphi^*)})}, then obtain 0 = - \\frac{V{(\\varphi^*)} + \\sin{(\\varphi^*)}}{V{(\\varphi^*)}} + \\frac{\\sin{(\\varphi^*)} + \\sin{(\\sin{(\\varphi^*)})}}{V{(\\varphi^*)}}", "derivation": "V{(\\varphi^*)} = \\sin{(\\sin{(\\varphi^*)})} and V{(\\varphi^*)} + \\sin{(\\varphi^*)} = \\sin{(\\varphi^*)} + \\sin{(\\sin{(\\varphi^*)})} and \\frac{V{(\\varphi^*)} + \\sin{(\\varphi^*)}}{V{(\\varphi^*)}} = \\frac{\\sin{(\\varphi^*)} + \\sin{(\\sin{(\\varphi^*)})}}{V{(\\varphi^*)}} and \\frac{V{(\\varphi^*)} + \\sin{(\\varphi^*)}}{V{(\\varphi^*)}} + \\frac{1}{V{(\\varphi^*)}} = \\frac{\\sin{(\\varphi^*)} + \\sin{(\\sin{(\\varphi^*)})}}{V{(\\varphi^*)}} + \\frac{1}{V{(\\varphi^*)}} and 0 = - \\frac{V{(\\varphi^*)} + \\sin{(\\varphi^*)}}{V{(\\varphi^*)}} + \\frac{\\sin{(\\varphi^*)} + \\sin{(\\sin{(\\varphi^*)})}}{V{(\\varphi^*)}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(sin(Symbol('\\\\varphi^*', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Add(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Add(sin(Symbol('\\\\varphi^*', commutative=True)), sin(sin(Symbol('\\\\varphi^*', commutative=True)))))"], [["divide", 2, "Function('V')(Symbol('\\\\varphi^*', commutative=True))"], "Equality(Mul(Add(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Add(sin(Symbol('\\\\varphi^*', commutative=True)), sin(sin(Symbol('\\\\varphi^*', commutative=True)))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["add", 3, "Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Add(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Add(Mul(Add(sin(Symbol('\\\\varphi^*', commutative=True)), sin(sin(Symbol('\\\\varphi^*', commutative=True)))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))))"], [["minus", 4, "Add(Mul(Add(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Add(Function('V')(Symbol('\\\\varphi^*', commutative=True)), sin(Symbol('\\\\varphi^*', commutative=True))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1))), Mul(Add(sin(Symbol('\\\\varphi^*', commutative=True)), sin(sin(Symbol('\\\\varphi^*', commutative=True)))), Pow(Function('V')(Symbol('\\\\varphi^*', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given I{(x^\\prime,n)} = n + x^\\prime, then obtain \\frac{- n - 2 x^\\prime + 2 I{(x^\\prime,n)}}{n} = 1", "derivation": "I{(x^\\prime,n)} = n + x^\\prime and n + x^\\prime + I{(x^\\prime,n)} = 2 n + 2 x^\\prime and 2 I{(x^\\prime,n)} = 2 n + 2 x^\\prime and - n - x^\\prime + 2 I{(x^\\prime,n)} = n + x^\\prime and - n - 2 x^\\prime + 2 I{(x^\\prime,n)} = n and \\frac{- n - 2 x^\\prime + 2 I{(x^\\prime,n)}}{n} = 1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('n', commutative=True)), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Mul(Integer(2), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(2), Symbol('n', commutative=True)), Mul(Integer(2), Symbol('x^\\\\prime', commutative=True))))"], [["minus", 3, "Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)))), Add(Symbol('n', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 4, "Mul(Integer(-1), Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True)))), Symbol('n', commutative=True))"], [["divide", 5, "Symbol('n', commutative=True)"], "Equality(Mul(Pow(Symbol('n', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('x^\\\\prime', commutative=True)), Mul(Integer(2), Function('I')(Symbol('x^\\\\prime', commutative=True), Symbol('n', commutative=True))))), Integer(1))"]]}, {"prompt": "Given E{(l)} = \\sin{(\\sin{(l)})}, then obtain \\frac{d}{d l} (E{(l)} + \\sin{(l)}) E{(l)} = \\frac{d}{d l} (\\sin{(l)} + \\sin{(\\sin{(l)})}) E{(l)}", "derivation": "E{(l)} = \\sin{(\\sin{(l)})} and E{(l)} + \\sin{(l)} = \\sin{(l)} + \\sin{(\\sin{(l)})} and (E{(l)} + \\sin{(l)}) E{(l)} = (\\sin{(l)} + \\sin{(\\sin{(l)})}) E{(l)} and \\frac{d}{d l} (E{(l)} + \\sin{(l)}) E{(l)} = \\frac{d}{d l} (\\sin{(l)} + \\sin{(\\sin{(l)})}) E{(l)}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('l', commutative=True)), sin(sin(Symbol('l', commutative=True))))"], [["add", 1, "sin(Symbol('l', commutative=True))"], "Equality(Add(Function('E')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Add(sin(Symbol('l', commutative=True)), sin(sin(Symbol('l', commutative=True)))))"], [["times", 2, "Function('E')(Symbol('l', commutative=True))"], "Equality(Mul(Add(Function('E')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Function('E')(Symbol('l', commutative=True))), Mul(Add(sin(Symbol('l', commutative=True)), sin(sin(Symbol('l', commutative=True)))), Function('E')(Symbol('l', commutative=True))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Mul(Add(Function('E')(Symbol('l', commutative=True)), sin(Symbol('l', commutative=True))), Function('E')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Add(sin(Symbol('l', commutative=True)), sin(sin(Symbol('l', commutative=True)))), Function('E')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(t)} = \\sin{(t)}, then obtain \\int \\log{(\\frac{d}{d t} \\Psi^{\\dagger}{(t)})} dt = \\int \\log{(\\frac{d}{d t} \\sin{(t)})} dt", "derivation": "\\Psi^{\\dagger}{(t)} = \\sin{(t)} and \\frac{d}{d t} \\Psi^{\\dagger}{(t)} = \\frac{d}{d t} \\sin{(t)} and \\log{(\\frac{d}{d t} \\Psi^{\\dagger}{(t)})} = \\log{(\\frac{d}{d t} \\sin{(t)})} and \\int \\log{(\\frac{d}{d t} \\Psi^{\\dagger}{(t)})} dt = \\int \\log{(\\frac{d}{d t} \\sin{(t)})} dt", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('t', commutative=True)), sin(Symbol('t', commutative=True)))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), log(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["integrate", 3, "Symbol('t', commutative=True)"], "Equality(Integral(log(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True))), Integral(log(Derivative(sin(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Tuple(Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\rho_b, then obtain - \\hat{\\mathbf{r}} + \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\rho_b", "derivation": "\\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = \\hat{\\mathbf{r}} \\rho_b and \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{\\mathbf{r}} \\rho_b and - \\frac{\\partial}{\\partial \\rho_b} \\hat{\\mathbf{r}} \\rho_b + \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\hat{\\mathbf{r}} \\rho_b - \\frac{\\partial}{\\partial \\rho_b} \\hat{\\mathbf{r}} \\rho_b and - \\hat{\\mathbf{r}} + \\frac{\\partial}{\\partial \\hat{\\mathbf{r}}} \\varepsilon{(\\rho_b,\\hat{\\mathbf{r}})} = - \\hat{\\mathbf{r}} + \\rho_b", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Derivative(Function('\\\\varepsilon')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Derivative(Function('\\\\varepsilon')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Symbol('\\\\rho_b', commutative=True)))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = C_{d} + F_{c} - c_{0}, then obtain 3 \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = 2 C_{d} + 2 F_{c} - 2 c_{0} + \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})}", "derivation": "\\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = C_{d} + F_{c} - c_{0} and 2 \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = C_{d} + F_{c} - c_{0} + \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} and 3 \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = C_{d} + F_{c} - c_{0} + 2 \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} and 3 \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})} = 2 C_{d} + 2 F_{c} - 2 c_{0} + \\operatorname{y^{\\prime}}{(C_{d},F_{c},c_{0})}", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True))))"], [["add", 1, "Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))"], "Equality(Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))))"], [["add", 1, "Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True)))"], "Equality(Mul(Integer(3), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))), Add(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('c_0', commutative=True)), Mul(Integer(2), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(3), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))), Add(Mul(Integer(2), Symbol('C_d', commutative=True)), Mul(Integer(2), Symbol('F_c', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Function('y^{\\\\prime}')(Symbol('C_d', commutative=True), Symbol('F_c', commutative=True), Symbol('c_0', commutative=True))))"]]}, {"prompt": "Given T{(\\hat{p})} = e^{\\hat{p}} and \\eta{(\\hat{p})} = \\frac{e^{\\hat{p}}}{\\hat{p}}, then obtain (\\frac{T{(\\hat{p})}}{\\hat{p}})^{\\hat{p}} = (\\frac{e^{\\hat{p}}}{\\hat{p}})^{\\hat{p}}", "derivation": "T{(\\hat{p})} = e^{\\hat{p}} and \\frac{T{(\\hat{p})}}{\\hat{p}} = \\frac{e^{\\hat{p}}}{\\hat{p}} and \\eta{(\\hat{p})} = \\frac{e^{\\hat{p}}}{\\hat{p}} and \\eta^{\\hat{p}}{(\\hat{p})} = (\\frac{e^{\\hat{p}}}{\\hat{p}})^{\\hat{p}} and \\frac{T{(\\hat{p})}}{\\hat{p}} = \\eta{(\\hat{p})} and (\\frac{T{(\\hat{p})}}{\\hat{p}})^{\\hat{p}} = (\\frac{e^{\\hat{p}}}{\\hat{p}})^{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\hat{p}', commutative=True)), exp(Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{p}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\eta')(Symbol('\\\\hat{p}', commutative=True)), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{p}', commutative=True))))"], [["power", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('\\\\eta')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\hat{p}', commutative=True))), Function('\\\\eta')(Symbol('\\\\hat{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('T')(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), exp(Symbol('\\\\hat{p}', commutative=True))), Symbol('\\\\hat{p}', commutative=True)))"]]}, {"prompt": "Given \\Omega{(C)} = e^{C}, then obtain C \\Omega{(C)} \\frac{d}{d C} e^{C} \\int \\Omega{(C)} dC = C \\Omega{(C)} \\frac{d}{d C} e^{C} \\int e^{C} dC", "derivation": "\\Omega{(C)} = e^{C} and C \\Omega{(C)} = C e^{C} and \\int \\Omega{(C)} dC = \\int e^{C} dC and \\frac{d}{d C} \\Omega{(C)} = \\frac{d}{d C} e^{C} and C e^{C} \\frac{d}{d C} \\Omega{(C)} = C e^{C} \\frac{d}{d C} e^{C} and C e^{C} \\frac{d}{d C} \\Omega{(C)} \\int \\Omega{(C)} dC = C e^{C} \\frac{d}{d C} \\Omega{(C)} \\int e^{C} dC and C \\Omega{(C)} \\frac{d}{d C} \\Omega{(C)} = C \\Omega{(C)} \\frac{d}{d C} e^{C} and C \\Omega{(C)} \\frac{d}{d C} \\Omega{(C)} \\int \\Omega{(C)} dC = C \\Omega{(C)} \\frac{d}{d C} \\Omega{(C)} \\int e^{C} dC and C \\Omega{(C)} \\frac{d}{d C} e^{C} \\int \\Omega{(C)} dC = C \\Omega{(C)} \\frac{d}{d C} e^{C} \\int e^{C} dC", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Omega')(Symbol('C', commutative=True)), exp(Symbol('C', commutative=True)))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True))))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 4, "Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)))"], "Equality(Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["times", 3, "Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], "Equality(Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), exp(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 7], "Equality(Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(Function('\\\\Omega')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Mul(Symbol('C', commutative=True), Function('\\\\Omega')(Symbol('C', commutative=True)), Derivative(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Integral(exp(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(f_{E})} = e^{\\cos{(f_{E})}}, then derive \\frac{d}{d f_{E}} \\operatorname{A_{x}}{(f_{E})} = - e^{\\cos{(f_{E})}} \\sin{(f_{E})}, then obtain (\\frac{d}{d f_{E}} e^{\\cos{(f_{E})}})^{f_{E}} = (\\frac{d}{d f_{E}} \\operatorname{A_{x}}{(f_{E})})^{f_{E}}", "derivation": "\\operatorname{A_{x}}{(f_{E})} = e^{\\cos{(f_{E})}} and \\frac{d}{d f_{E}} \\operatorname{A_{x}}{(f_{E})} = \\frac{d}{d f_{E}} e^{\\cos{(f_{E})}} and \\frac{d}{d f_{E}} \\operatorname{A_{x}}{(f_{E})} = - e^{\\cos{(f_{E})}} \\sin{(f_{E})} and \\frac{d}{d f_{E}} e^{\\cos{(f_{E})}} = - e^{\\cos{(f_{E})}} \\sin{(f_{E})} and (\\frac{d}{d f_{E}} e^{\\cos{(f_{E})}})^{f_{E}} = (- e^{\\cos{(f_{E})}} \\sin{(f_{E})})^{f_{E}} and (\\frac{d}{d f_{E}} e^{\\cos{(f_{E})}})^{f_{E}} = (\\frac{d}{d f_{E}} \\operatorname{A_{x}}{(f_{E})})^{f_{E}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('f_E', commutative=True)), exp(cos(Symbol('f_E', commutative=True))))"], [["differentiate", 1, "Symbol('f_E', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('A_x')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('f_E', commutative=True))), sin(Symbol('f_E', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('f_E', commutative=True))), sin(Symbol('f_E', commutative=True))))"], [["power", 4, "Symbol('f_E', commutative=True)"], "Equality(Pow(Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('f_E', commutative=True)), Pow(Mul(Integer(-1), exp(cos(Symbol('f_E', commutative=True))), sin(Symbol('f_E', commutative=True))), Symbol('f_E', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Derivative(exp(cos(Symbol('f_E', commutative=True))), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('f_E', commutative=True)), Pow(Derivative(Function('A_x')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Symbol('f_E', commutative=True)))"]]}, {"prompt": "Given k{(A_{2})} = \\sin{(\\sin{(A_{2})})}, then derive \\frac{d}{d A_{2}} k{(A_{2})} = \\cos{(A_{2})} \\cos{(\\sin{(A_{2})})}, then obtain (\\frac{d}{d A_{2}} \\sin{(\\sin{(A_{2})})})^{A_{2}} = (\\cos{(A_{2})} \\cos{(\\sin{(A_{2})})})^{A_{2}}", "derivation": "k{(A_{2})} = \\sin{(\\sin{(A_{2})})} and \\frac{d}{d A_{2}} k{(A_{2})} = \\frac{d}{d A_{2}} \\sin{(\\sin{(A_{2})})} and \\frac{d}{d A_{2}} k{(A_{2})} = \\cos{(A_{2})} \\cos{(\\sin{(A_{2})})} and \\frac{d}{d A_{2}} \\sin{(\\sin{(A_{2})})} = \\cos{(A_{2})} \\cos{(\\sin{(A_{2})})} and (\\frac{d}{d A_{2}} \\sin{(\\sin{(A_{2})})})^{A_{2}} = (\\cos{(A_{2})} \\cos{(\\sin{(A_{2})})})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('A_2', commutative=True)), sin(sin(Symbol('A_2', commutative=True))))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('k')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('k')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(cos(Symbol('A_2', commutative=True)), cos(sin(Symbol('A_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Mul(cos(Symbol('A_2', commutative=True)), cos(sin(Symbol('A_2', commutative=True)))))"], [["power", 4, "Symbol('A_2', commutative=True)"], "Equality(Pow(Derivative(sin(sin(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('A_2', commutative=True)), Pow(Mul(cos(Symbol('A_2', commutative=True)), cos(sin(Symbol('A_2', commutative=True)))), Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(C)} = \\int \\cos{(C)} dC and \\mathbf{S}{(C)} = \\operatorname{V_{\\mathbf{B}}}{(C)} + \\cos{(C)}, then obtain \\frac{d}{d C} \\mathbf{S}{(C)} = \\frac{d}{d C} (\\cos{(C)} + \\int \\cos{(C)} dC)", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(C)} = \\int \\cos{(C)} dC and \\operatorname{V_{\\mathbf{B}}}{(C)} + \\cos{(C)} = \\cos{(C)} + \\int \\cos{(C)} dC and \\frac{d}{d C} (\\operatorname{V_{\\mathbf{B}}}{(C)} + \\cos{(C)}) = \\frac{d}{d C} (\\cos{(C)} + \\int \\cos{(C)} dC) and \\mathbf{S}{(C)} = \\operatorname{V_{\\mathbf{B}}}{(C)} + \\cos{(C)} and \\frac{d}{d C} \\mathbf{S}{(C)} = \\frac{d}{d C} (\\cos{(C)} + \\int \\cos{(C)} dC)", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["add", 1, "cos(Symbol('C', commutative=True))"], "Equality(Add(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True))), Add(cos(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Add(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('C', commutative=True)), Add(Function('V_{\\\\mathbf{B}}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Function('\\\\mathbf{S}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"]]}, {"prompt": "Given c{(\\Psi_{\\lambda})} = \\log{(\\cos{(\\Psi_{\\lambda})})}, then derive \\frac{d}{d \\Psi_{\\lambda}} c{(\\Psi_{\\lambda})} = - \\frac{\\sin{(\\Psi_{\\lambda})}}{\\cos{(\\Psi_{\\lambda})}}, then obtain (\\frac{d}{d \\Psi_{\\lambda}} \\log{(\\cos{(\\Psi_{\\lambda})})})^{\\Psi_{\\lambda}} = (- \\frac{\\sin{(\\Psi_{\\lambda})}}{\\cos{(\\Psi_{\\lambda})}})^{\\Psi_{\\lambda}}", "derivation": "c{(\\Psi_{\\lambda})} = \\log{(\\cos{(\\Psi_{\\lambda})})} and \\frac{d}{d \\Psi_{\\lambda}} c{(\\Psi_{\\lambda})} = \\frac{d}{d \\Psi_{\\lambda}} \\log{(\\cos{(\\Psi_{\\lambda})})} and \\frac{d}{d \\Psi_{\\lambda}} c{(\\Psi_{\\lambda})} = - \\frac{\\sin{(\\Psi_{\\lambda})}}{\\cos{(\\Psi_{\\lambda})}} and (\\frac{d}{d \\Psi_{\\lambda}} c{(\\Psi_{\\lambda})})^{\\Psi_{\\lambda}} = (- \\frac{\\sin{(\\Psi_{\\lambda})}}{\\cos{(\\Psi_{\\lambda})}})^{\\Psi_{\\lambda}} and (\\frac{d}{d \\Psi_{\\lambda}} \\log{(\\cos{(\\Psi_{\\lambda})})})^{\\Psi_{\\lambda}} = (- \\frac{\\sin{(\\Psi_{\\lambda})}}{\\cos{(\\Psi_{\\lambda})}})^{\\Psi_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), log(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Derivative(log(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('\\\\Psi_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Function('c')(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Pow(Derivative(log(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Tuple(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Integer(1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Pow(cos(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Integer(-1))), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = \\Psi_{\\lambda} + v_{z}, then derive \\frac{\\partial}{\\partial v_{z}} \\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = 1, then obtain v_{z} - \\frac{\\partial}{\\partial v_{z}} (\\Psi_{\\lambda} + v_{z}) = v_{z} - 1", "derivation": "\\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = \\Psi_{\\lambda} + v_{z} and \\frac{\\partial}{\\partial v_{z}} \\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial v_{z}} (\\Psi_{\\lambda} + v_{z}) and \\frac{\\partial}{\\partial v_{z}} \\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = 1 and - v_{z} + \\frac{\\partial}{\\partial v_{z}} \\operatorname{r_{0}}{(v_{z},\\Psi_{\\lambda})} = 1 - v_{z} and - v_{z} + \\frac{\\partial}{\\partial v_{z}} (\\Psi_{\\lambda} + v_{z}) = 1 - v_{z} and v_{z} - \\frac{\\partial}{\\partial v_{z}} (\\Psi_{\\lambda} + v_{z}) = v_{z} - 1", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('r_0')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))), Integer(1))"], [["minus", 3, "Symbol('v_z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Derivative(Function('r_0')(Symbol('v_z', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_z', commutative=True)), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1)))), Add(Integer(1), Mul(Integer(-1), Symbol('v_z', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Add(Symbol('v_z', commutative=True), Mul(Integer(-1), Derivative(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True), Integer(1))))), Add(Symbol('v_z', commutative=True), Integer(-1)))"]]}, {"prompt": "Given A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})}, then obtain - v_{1} + A{(v_{1},a^{\\dagger})} \\log{(a^{\\dagger})} + A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})} - v_{1} + A{(v_{1},a^{\\dagger})} \\log{(a^{\\dagger})}", "derivation": "A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})} and A{(v_{1},a^{\\dagger})} \\log{(a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})}^{2} and - v_{1} + A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})} - v_{1} and v_{1} \\log{(a^{\\dagger})}^{2} - v_{1} + A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})}^{2} + v_{1} \\log{(a^{\\dagger})} - v_{1} and - v_{1} + A{(v_{1},a^{\\dagger})} \\log{(a^{\\dagger})} + A{(v_{1},a^{\\dagger})} = v_{1} \\log{(a^{\\dagger})} - v_{1} + A{(v_{1},a^{\\dagger})} \\log{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), Mul(Symbol('v_1', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "log(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('v_1', commutative=True), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))))"], [["minus", 1, "Symbol('v_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('v_1', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["add", 3, "Mul(Symbol('v_1', commutative=True), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Symbol('v_1', commutative=True), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('v_1', commutative=True)), Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('v_1', commutative=True), Pow(log(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))), Mul(Symbol('v_1', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True))), Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True))), Add(Mul(Symbol('v_1', commutative=True), log(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Integer(-1), Symbol('v_1', commutative=True)), Mul(Function('A')(Symbol('v_1', commutative=True), Symbol('a^{\\\\dagger}', commutative=True)), log(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given M{(\\mathbf{p},\\mathbf{r})} = \\mathbf{p} + \\cos{(\\mathbf{r})} and \\tilde{g}{(\\mathbf{p},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} M{(\\mathbf{p},\\mathbf{r})} + 1, then derive \\frac{\\partial}{\\partial \\mathbf{r}} M{(\\mathbf{p},\\mathbf{r})} + 1 = 1 - \\sin{(\\mathbf{r})}, then obtain \\tilde{g}{(\\mathbf{p},\\mathbf{r})} = 1 - \\sin{(\\mathbf{r})}", "derivation": "M{(\\mathbf{p},\\mathbf{r})} = \\mathbf{p} + \\cos{(\\mathbf{r})} and \\mathbf{r} + M{(\\mathbf{p},\\mathbf{r})} = \\mathbf{p} + \\mathbf{r} + \\cos{(\\mathbf{r})} and \\frac{\\partial}{\\partial \\mathbf{r}} (\\mathbf{r} + M{(\\mathbf{p},\\mathbf{r})}) = \\frac{\\partial}{\\partial \\mathbf{r}} (\\mathbf{p} + \\mathbf{r} + \\cos{(\\mathbf{r})}) and \\frac{\\partial}{\\partial \\mathbf{r}} M{(\\mathbf{p},\\mathbf{r})} + 1 = 1 - \\sin{(\\mathbf{r})} and \\tilde{g}{(\\mathbf{p},\\mathbf{r})} = \\frac{\\partial}{\\partial \\mathbf{r}} M{(\\mathbf{p},\\mathbf{r})} + 1 and \\tilde{g}{(\\mathbf{p},\\mathbf{r})} = 1 - \\sin{(\\mathbf{r})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["add", 1, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{r}', commutative=True), Function('M')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), cos(Symbol('\\\\mathbf{r}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{r}', commutative=True), Function('M')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), cos(Symbol('\\\\mathbf{r}', commutative=True))), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('M')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{r}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Derivative(Function('M')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Tuple(Symbol('\\\\mathbf{r}', commutative=True), Integer(1))), Integer(1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('\\\\mathbf{r}', commutative=True)), Add(Integer(1), Mul(Integer(-1), sin(Symbol('\\\\mathbf{r}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(C_{1},U,\\theta_1)} = \\frac{\\theta_1}{C_{1} U} and \\hat{x}_0{(C_{1},U,\\theta_1)} = e^{\\frac{\\theta_1}{C_{1} U}}, then obtain (\\int e^{\\operatorname{F_{c}}{(C_{1},U,\\theta_1)}} d\\theta_1)^{\\theta_1} = (\\int e^{\\frac{\\theta_1}{C_{1} U}} d\\theta_1)^{\\theta_1}", "derivation": "\\operatorname{F_{c}}{(C_{1},U,\\theta_1)} = \\frac{\\theta_1}{C_{1} U} and e^{\\operatorname{F_{c}}{(C_{1},U,\\theta_1)}} = e^{\\frac{\\theta_1}{C_{1} U}} and \\hat{x}_0{(C_{1},U,\\theta_1)} = e^{\\frac{\\theta_1}{C_{1} U}} and \\int \\hat{x}_0{(C_{1},U,\\theta_1)} d\\theta_1 = \\int e^{\\frac{\\theta_1}{C_{1} U}} d\\theta_1 and (\\int \\hat{x}_0{(C_{1},U,\\theta_1)} d\\theta_1)^{\\theta_1} = (\\int e^{\\frac{\\theta_1}{C_{1} U}} d\\theta_1)^{\\theta_1} and \\hat{x}_0{(C_{1},U,\\theta_1)} = e^{\\operatorname{F_{c}}{(C_{1},U,\\theta_1)}} and (\\int e^{\\operatorname{F_{c}}{(C_{1},U,\\theta_1)}} d\\theta_1)^{\\theta_1} = (\\int e^{\\frac{\\theta_1}{C_{1} U}} d\\theta_1)^{\\theta_1}", "srepr_derivation": [["get_premise", "Equality(Function('F_c')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)), Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True)))"], [["exp", 1], "Equality(exp(Function('F_c')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True))), exp(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}_0')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))))"], [["integrate", 3, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Function('\\\\hat{x}_0')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(exp(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["power", 4, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Integral(Function('\\\\hat{x}_0')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Integral(exp(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\hat{x}_0')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True)), exp(Function('F_c')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Pow(Integral(exp(Function('F_c')(Symbol('C_1', commutative=True), Symbol('U', commutative=True), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)), Pow(Integral(exp(Mul(Pow(Symbol('C_1', commutative=True), Integer(-1)), Pow(Symbol('U', commutative=True), Integer(-1)), Symbol('\\\\theta_1', commutative=True))), Tuple(Symbol('\\\\theta_1', commutative=True))), Symbol('\\\\theta_1', commutative=True)))"]]}, {"prompt": "Given U{(\\sigma_x,n_{2},\\Omega)} = \\frac{n_{2}}{\\Omega \\sigma_x}, then obtain \\Omega + \\frac{\\int (U{(\\sigma_x,n_{2},\\Omega)} - 1) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}} = \\Omega + \\frac{\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}}", "derivation": "U{(\\sigma_x,n_{2},\\Omega)} = \\frac{n_{2}}{\\Omega \\sigma_x} and U{(\\sigma_x,n_{2},\\Omega)} - 1 = -1 + \\frac{n_{2}}{\\Omega \\sigma_x} and \\int (U{(\\sigma_x,n_{2},\\Omega)} - 1) d\\Omega = \\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega and \\frac{\\int (U{(\\sigma_x,n_{2},\\Omega)} - 1) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}} = \\frac{\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}} and \\Omega + \\frac{\\int (U{(\\sigma_x,n_{2},\\Omega)} - 1) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}} = \\Omega + \\frac{\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega}{\\cos{(\\int (-1 + \\frac{n_{2}}{\\Omega \\sigma_x}) d\\Omega)}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\sigma_x', commutative=True), Symbol('n_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('U')(Symbol('\\\\sigma_x', commutative=True), Symbol('n_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Add(Function('U')(Symbol('\\\\sigma_x', commutative=True), Symbol('n_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["divide", 3, "cos(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))"], "Equality(Mul(Pow(cos(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Integral(Add(Function('U')(Symbol('\\\\sigma_x', commutative=True), Symbol('n_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(cos(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["add", 4, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Mul(Pow(cos(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Integral(Add(Function('U')(Symbol('\\\\sigma_x', commutative=True), Symbol('n_2', commutative=True), Symbol('\\\\Omega', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\Omega', commutative=True))))), Add(Symbol('\\\\Omega', commutative=True), Mul(Pow(cos(Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True)))), Integer(-1)), Integral(Add(Integer(-1), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Pow(Symbol('\\\\sigma_x', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(E)} = e^{\\sin{(E)}} and \\Psi_{\\lambda}{(E)} = e^{2 \\sin{(E)}}, then obtain \\Psi_{\\lambda}{(E)} = \\operatorname{m_{s}}{(E)} e^{\\sin{(E)}}", "derivation": "\\operatorname{m_{s}}{(E)} = e^{\\sin{(E)}} and \\operatorname{m_{s}}{(E)} e^{\\sin{(E)}} = e^{2 \\sin{(E)}} and \\Psi_{\\lambda}{(E)} = e^{2 \\sin{(E)}} and \\Psi_{\\lambda}{(E)} = \\operatorname{m_{s}}{(E)} e^{\\sin{(E)}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('E', commutative=True)), exp(sin(Symbol('E', commutative=True))))"], [["times", 1, "exp(sin(Symbol('E', commutative=True)))"], "Equality(Mul(Function('m_s')(Symbol('E', commutative=True)), exp(sin(Symbol('E', commutative=True)))), exp(Mul(Integer(2), sin(Symbol('E', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('E', commutative=True)), exp(Mul(Integer(2), sin(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('E', commutative=True)), Mul(Function('m_s')(Symbol('E', commutative=True)), exp(sin(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\varphi{(Q)} = \\log{(\\log{(Q)})}, then obtain \\frac{d}{d Q} \\frac{\\varphi{(Q)}}{\\log{(\\log{(Q)})}^{2}} = \\frac{d}{d Q} \\frac{1}{\\log{(\\log{(Q)})}}", "derivation": "\\varphi{(Q)} = \\log{(\\log{(Q)})} and \\frac{\\varphi{(Q)}}{\\log{(\\log{(Q)})}} = 1 and \\frac{\\varphi{(Q)}}{\\log{(\\log{(Q)})}^{2}} = \\frac{1}{\\log{(\\log{(Q)})}} and \\frac{d}{d Q} \\frac{\\varphi{(Q)}}{\\log{(\\log{(Q)})}^{2}} = \\frac{d}{d Q} \\frac{1}{\\log{(\\log{(Q)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('Q', commutative=True)), log(log(Symbol('Q', commutative=True))))"], [["divide", 1, "log(log(Symbol('Q', commutative=True)))"], "Equality(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(log(log(Symbol('Q', commutative=True))), Integer(-1))), Integer(1))"], [["times", 2, "Pow(log(log(Symbol('Q', commutative=True))), Integer(-1))"], "Equality(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(log(log(Symbol('Q', commutative=True))), Integer(-2))), Pow(log(log(Symbol('Q', commutative=True))), Integer(-1)))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Mul(Function('\\\\varphi')(Symbol('Q', commutative=True)), Pow(log(log(Symbol('Q', commutative=True))), Integer(-2))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Pow(log(log(Symbol('Q', commutative=True))), Integer(-1)), Tuple(Symbol('Q', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{r}{(\\Omega,\\mathbf{H})} = \\Omega \\cos{(\\mathbf{H})} and k{(\\Omega)} = \\Omega^{2}, then obtain - \\mathbf{r}{(\\Omega,\\mathbf{H})} k{(\\Omega)} \\cos{(\\mathbf{H})} = - \\Omega^{2} \\mathbf{r}{(\\Omega,\\mathbf{H})} \\cos{(\\mathbf{H})}", "derivation": "\\mathbf{r}{(\\Omega,\\mathbf{H})} = \\Omega \\cos{(\\mathbf{H})} and \\frac{\\mathbf{r}^{2}{(\\Omega,\\mathbf{H})}}{\\cos{(\\mathbf{H})}} = \\Omega \\mathbf{r}{(\\Omega,\\mathbf{H})} and - \\Omega \\mathbf{r}^{2}{(\\Omega,\\mathbf{H})} = - \\Omega^{2} \\mathbf{r}{(\\Omega,\\mathbf{H})} \\cos{(\\mathbf{H})} and k{(\\Omega)} = \\Omega^{2} and - \\Omega \\mathbf{r}^{2}{(\\Omega,\\mathbf{H})} = - \\mathbf{r}{(\\Omega,\\mathbf{H})} k{(\\Omega)} \\cos{(\\mathbf{H})} and - \\mathbf{r}{(\\Omega,\\mathbf{H})} k{(\\Omega)} \\cos{(\\mathbf{H})} = - \\Omega^{2} \\mathbf{r}{(\\Omega,\\mathbf{H})} \\cos{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 1, "Mul(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2)), Pow(cos(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1))), Mul(Symbol('\\\\Omega', commutative=True), Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), cos(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\Omega', commutative=True)), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Integer(-1), Symbol('\\\\Omega', commutative=True), Pow(Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('k')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('k')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Function('\\\\mathbf{r}')(Symbol('\\\\Omega', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), cos(Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given I{(u,\\Psi_{nl})} = - \\Psi_{nl} + u, then derive \\frac{\\partial}{\\partial \\Psi_{nl}} I{(u,\\Psi_{nl})} = -1, then obtain \\frac{\\partial}{\\partial \\Psi_{nl}} (- \\Psi_{nl} + u) = -1", "derivation": "I{(u,\\Psi_{nl})} = - \\Psi_{nl} + u and \\frac{\\partial}{\\partial \\Psi_{nl}} I{(u,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\Psi_{nl}} (- \\Psi_{nl} + u) and \\frac{\\partial}{\\partial \\Psi_{nl}} I{(u,\\Psi_{nl})} = -1 and \\frac{\\partial}{\\partial \\Psi_{nl}} (- \\Psi_{nl} + u) = -1", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('u', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('u', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\Psi_{nl}', commutative=True)"], "Equality(Derivative(Function('I')(Symbol('u', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('I')(Symbol('u', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('\\\\Psi_{nl}', commutative=True), Integer(1))), Integer(-1))"]]}, {"prompt": "Given \\mathbf{M}{(\\hbar,A_{z})} = A_{z} \\hbar, then obtain 2 \\mathbf{M}^{2}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} = 2 A_{z} \\hbar \\mathbf{M}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})}", "derivation": "\\mathbf{M}{(\\hbar,A_{z})} = A_{z} \\hbar and \\mathbf{M}^{2}{(\\hbar,A_{z})} = A_{z} \\hbar \\mathbf{M}{(\\hbar,A_{z})} and \\mathbf{M}^{2}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} = A_{z} \\hbar \\mathbf{M}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} and 2 \\mathbf{M}^{2}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} = A_{z} \\hbar \\mathbf{M}{(\\hbar,A_{z})} + \\mathbf{M}^{2}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} and 2 \\mathbf{M}^{2}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})} = 2 A_{z} \\hbar \\mathbf{M}{(\\hbar,A_{z})} - \\mathbf{M}{(\\hbar,A_{z})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))"], "Equality(Add(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))))"], [["add", 2, "Add(Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))))"], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2)), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)), Integer(2))), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))), Add(Mul(Integer(2), Symbol('A_z', commutative=True), Symbol('\\\\hbar', commutative=True), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{M}')(Symbol('\\\\hbar', commutative=True), Symbol('A_z', commutative=True)))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(\\delta)} = \\sin{(\\delta)} and l{(\\delta)} = \\sin{(\\delta)}, then obtain \\hat{\\mathbf{x}}^{2 \\delta}{(\\delta)} - \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} \\sin^{\\delta}{(\\delta)} = 0", "derivation": "\\hat{\\mathbf{x}}{(\\delta)} = \\sin{(\\delta)} and \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} = \\sin^{\\delta}{(\\delta)} and \\hat{\\mathbf{x}}^{2 \\delta}{(\\delta)} = \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} \\sin^{\\delta}{(\\delta)} and l{(\\delta)} = \\sin{(\\delta)} and \\hat{\\mathbf{x}}^{2 \\delta}{(\\delta)} = \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} l^{\\delta}{(\\delta)} and \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} l^{\\delta}{(\\delta)} = \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} \\sin^{\\delta}{(\\delta)} and \\hat{\\mathbf{x}}^{2 \\delta}{(\\delta)} - \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} l^{\\delta}{(\\delta)} = 0 and \\hat{\\mathbf{x}}^{2 \\delta}{(\\delta)} - \\hat{\\mathbf{x}}^{\\delta}{(\\delta)} \\sin^{\\delta}{(\\delta)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["power", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], [["times", 2, "Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], ["renaming_premise", "Equality(Function('l')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Function('l')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Function('l')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))), Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True))))"], [["minus", 5, "Mul(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Function('l')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))"], "Equality(Add(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(Function('l')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Integer(0))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Add(Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Pow(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)), Pow(sin(Symbol('\\\\delta', commutative=True)), Symbol('\\\\delta', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})} = \\frac{B n_{2}}{\\mathbf{v}} and \\mathbf{J}_P{(n_{2})} = n_{2}, then obtain e^{n_{2} + \\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})}} = e^{\\frac{B n_{2}}{\\mathbf{v}} + n_{2}}", "derivation": "\\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})} = \\frac{B n_{2}}{\\mathbf{v}} and \\mathbf{J}_P{(n_{2})} = n_{2} and \\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})} + \\mathbf{J}_P{(n_{2})} = \\frac{B n_{2}}{\\mathbf{v}} + \\mathbf{J}_P{(n_{2})} and e^{\\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})} + \\mathbf{J}_P{(n_{2})}} = e^{\\frac{B n_{2}}{\\mathbf{v}} + \\mathbf{J}_P{(n_{2})}} and e^{n_{2} + \\operatorname{E_{\\lambda}}{(n_{2},B,\\mathbf{v})}} = e^{\\frac{B n_{2}}{\\mathbf{v}} + n_{2}}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))"], [["add", 1, "Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True))), Add(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True))))"], [["exp", 3], "Equality(exp(Add(Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True)))), exp(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Function('\\\\mathbf{J}_P')(Symbol('n_2', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(exp(Add(Symbol('n_2', commutative=True), Function('E_{\\\\lambda}')(Symbol('n_2', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{v}', commutative=True)))), exp(Add(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)), Symbol('n_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(C)} = e^{e^{C}} and \\mathbf{v}{(C)} = \\int C e^{e^{C}} dC, then obtain \\mathbf{v}{(C)} - \\int C \\operatorname{F_{x}}{(C)} dC = 0", "derivation": "\\operatorname{F_{x}}{(C)} = e^{e^{C}} and C \\operatorname{F_{x}}{(C)} = C e^{e^{C}} and \\int C \\operatorname{F_{x}}{(C)} dC = \\int C e^{e^{C}} dC and \\mathbf{v}{(C)} = \\int C e^{e^{C}} dC and \\mathbf{v}{(C)} = \\int C \\operatorname{F_{x}}{(C)} dC and \\mathbf{v}{(C)} - \\int C \\operatorname{F_{x}}{(C)} dC = 0", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('C', commutative=True)), exp(exp(Symbol('C', commutative=True))))"], [["times", 1, "Symbol('C', commutative=True)"], "Equality(Mul(Symbol('C', commutative=True), Function('F_x')(Symbol('C', commutative=True))), Mul(Symbol('C', commutative=True), exp(exp(Symbol('C', commutative=True)))))"], [["integrate", 2, "Symbol('C', commutative=True)"], "Equality(Integral(Mul(Symbol('C', commutative=True), Function('F_x')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Symbol('C', commutative=True), exp(exp(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{v}')(Symbol('C', commutative=True)), Integral(Mul(Symbol('C', commutative=True), exp(exp(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\mathbf{v}')(Symbol('C', commutative=True)), Integral(Mul(Symbol('C', commutative=True), Function('F_x')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"], [["minus", 5, "Integral(Mul(Symbol('C', commutative=True), Function('F_x')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('C', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('C', commutative=True), Function('F_x')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))), Integer(0))"]]}, {"prompt": "Given h{(\\mathbf{r},a)} = \\mathbf{r} + a, then obtain \\frac{e^{h{(\\mathbf{r},a)}}}{\\frac{\\partial}{\\partial a} e^{h{(\\mathbf{r},a)}}} = \\frac{e^{\\mathbf{r} + a}}{\\frac{\\partial}{\\partial a} e^{h{(\\mathbf{r},a)}}}", "derivation": "h{(\\mathbf{r},a)} = \\mathbf{r} + a and e^{h{(\\mathbf{r},a)}} = e^{\\mathbf{r} + a} and \\frac{\\partial}{\\partial a} e^{h{(\\mathbf{r},a)}} = \\frac{\\partial}{\\partial a} e^{\\mathbf{r} + a} and \\frac{e^{h{(\\mathbf{r},a)}}}{\\frac{\\partial}{\\partial a} e^{\\mathbf{r} + a}} = \\frac{e^{\\mathbf{r} + a}}{\\frac{\\partial}{\\partial a} e^{\\mathbf{r} + a}} and \\frac{e^{h{(\\mathbf{r},a)}}}{\\frac{\\partial}{\\partial a} e^{h{(\\mathbf{r},a)}}} = \\frac{e^{\\mathbf{r} + a}}{\\frac{\\partial}{\\partial a} e^{h{(\\mathbf{r},a)}}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True)))"], [["exp", 1], "Equality(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))"], "Equality(Mul(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Pow(Derivative(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))), Mul(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Pow(Derivative(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Pow(Derivative(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))), Mul(exp(Add(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Pow(Derivative(exp(Function('h')(Symbol('\\\\mathbf{r}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\chi)} = \\sin{(\\chi)}, then obtain - \\mathbf{f}{(A)} + \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} - 1 = - \\mathbf{f}{(A)} + \\frac{d}{d \\chi} \\sin{(\\chi)} - 1", "derivation": "\\operatorname{f^{\\prime}}{(\\chi)} = \\sin{(\\chi)} and \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\chi)} and \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} - 1 = \\frac{d}{d \\chi} \\sin{(\\chi)} - 1 and - \\mathbf{f}{(A)} + \\frac{d}{d \\chi} \\operatorname{f^{\\prime}}{(\\chi)} - 1 = - \\mathbf{f}{(A)} + \\frac{d}{d \\chi} \\sin{(\\chi)} - 1", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), sin(Symbol('\\\\chi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)))"], [["minus", 3, "Function('\\\\mathbf{f}')(Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('A', commutative=True))), Derivative(Function('f^{\\\\prime}')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), Function('\\\\mathbf{f}')(Symbol('A', commutative=True))), Derivative(sin(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(J)} = e^{J}, then obtain \\operatorname{f_{\\mathbf{v}}}{(J)} + \\frac{d}{d J} \\operatorname{f_{\\mathbf{v}}}^{J}{(J)} = e^{J} + \\frac{d}{d J} \\operatorname{f_{\\mathbf{v}}}^{J}{(J)}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(J)} = e^{J} and \\operatorname{f_{\\mathbf{v}}}^{J}{(J)} = (e^{J})^{J} and \\operatorname{f_{\\mathbf{v}}}{(J)} + \\frac{d}{d J} (e^{J})^{J} = e^{J} + \\frac{d}{d J} (e^{J})^{J} and \\operatorname{f_{\\mathbf{v}}}{(J)} + \\frac{d}{d J} \\operatorname{f_{\\mathbf{v}}}^{J}{(J)} = e^{J} + \\frac{d}{d J} \\operatorname{f_{\\mathbf{v}}}^{J}{(J)}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), exp(Symbol('J', commutative=True)))"], [["power", 1, "Symbol('J', commutative=True)"], "Equality(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["add", 1, "Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(exp(Symbol('J', commutative=True)), Derivative(Pow(exp(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Derivative(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(exp(Symbol('J', commutative=True)), Derivative(Pow(Function('f_{\\\\mathbf{v}}')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(a,\\mathbf{H})} = \\mathbf{H} + a, then derive \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = F_{g} + \\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} a, then derive \\frac{\\partial}{\\partial a} \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = \\mathbf{H}, then obtain \\frac{\\partial}{\\partial a} (F_{g} + \\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} a) = \\mathbf{H}", "derivation": "\\mathbf{J}_f{(a,\\mathbf{H})} = \\mathbf{H} + a and \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = \\int (\\mathbf{H} + a) d\\mathbf{H} and \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = F_{g} + \\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} a and \\frac{\\partial}{\\partial a} \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = \\frac{\\partial}{\\partial a} (F_{g} + \\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} a) and \\frac{\\partial}{\\partial a} \\int \\mathbf{J}_f{(a,\\mathbf{H})} d\\mathbf{H} = \\mathbf{H} and \\frac{\\partial}{\\partial a} (F_{g} + \\frac{\\mathbf{H}^{2}}{2} + \\mathbf{H} a) = \\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('a', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('a', commutative=True))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\mathbf{J}_f')(Symbol('a', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Add(Symbol('F_g', commutative=True), Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('\\\\mathbf{H}', commutative=True))"]]}, {"prompt": "Given q{(\\hbar,\\phi_1)} = \\hbar + \\phi_1 and f{(\\hbar,\\phi_1)} = (\\hbar + \\phi_1)^{\\hbar}, then obtain \\frac{\\frac{\\partial}{\\partial \\hbar} q^{\\hbar}{(\\hbar,\\phi_1)}}{\\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\sin{(\\hat{\\mathbf{x}})})}} = \\frac{\\frac{\\partial}{\\partial \\hbar} f{(\\hbar,\\phi_1)}}{\\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\sin{(\\hat{\\mathbf{x}})})}}", "derivation": "q{(\\hbar,\\phi_1)} = \\hbar + \\phi_1 and q^{\\hbar}{(\\hbar,\\phi_1)} = (\\hbar + \\phi_1)^{\\hbar} and \\frac{\\partial}{\\partial \\hbar} q^{\\hbar}{(\\hbar,\\phi_1)} = \\frac{\\partial}{\\partial \\hbar} (\\hbar + \\phi_1)^{\\hbar} and f{(\\hbar,\\phi_1)} = (\\hbar + \\phi_1)^{\\hbar} and \\frac{\\partial}{\\partial \\hbar} q^{\\hbar}{(\\hbar,\\phi_1)} = \\frac{\\partial}{\\partial \\hbar} f{(\\hbar,\\phi_1)} and \\frac{\\frac{\\partial}{\\partial \\hbar} q^{\\hbar}{(\\hbar,\\phi_1)}}{\\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\sin{(\\hat{\\mathbf{x}})})}} = \\frac{\\frac{\\partial}{\\partial \\hbar} f{(\\hbar,\\phi_1)}}{\\frac{d}{d \\hat{\\mathbf{x}}} \\cos{(\\sin{(\\hat{\\mathbf{x}})})}}", "srepr_derivation": [["get_premise", "Equality(Function('q')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('q')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Pow(Function('q')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('f')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(Add(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Pow(Function('q')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Function('f')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["divide", 5, "Derivative(cos(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Pow(Function('q')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Pow(Derivative(cos(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Integer(-1))), Mul(Derivative(Function('f')(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Pow(Derivative(cos(sin(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True))), Tuple(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}{(\\mathbf{s})} = \\cos{(\\cos{(\\mathbf{s})})}, then obtain \\frac{d}{d \\mathbf{s}} (\\mathbf{J}{(\\mathbf{s})} - 2 \\cos{(\\mathbf{s})} + \\cos{(\\cos{(\\mathbf{s})})}) = \\frac{d}{d \\mathbf{s}} (- 2 \\cos{(\\mathbf{s})} + 2 \\cos{(\\cos{(\\mathbf{s})})})", "derivation": "\\mathbf{J}{(\\mathbf{s})} = \\cos{(\\cos{(\\mathbf{s})})} and \\mathbf{J}{(\\mathbf{s})} - \\cos{(\\mathbf{s})} = - \\cos{(\\mathbf{s})} + \\cos{(\\cos{(\\mathbf{s})})} and \\mathbf{J}{(\\mathbf{s})} - 2 \\cos{(\\mathbf{s})} + \\cos{(\\cos{(\\mathbf{s})})} = - 2 \\cos{(\\mathbf{s})} + 2 \\cos{(\\cos{(\\mathbf{s})})} and \\frac{d}{d \\mathbf{s}} (\\mathbf{J}{(\\mathbf{s})} - 2 \\cos{(\\mathbf{s})} + \\cos{(\\cos{(\\mathbf{s})})}) = \\frac{d}{d \\mathbf{s}} (- 2 \\cos{(\\mathbf{s})} + 2 \\cos{(\\cos{(\\mathbf{s})})})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{s}', commutative=True)), cos(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 1, "cos(Symbol('\\\\mathbf{s}', commutative=True))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{s}', commutative=True))), cos(cos(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), cos(Symbol('\\\\mathbf{s}', commutative=True))), cos(cos(Symbol('\\\\mathbf{s}', commutative=True))))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True))), cos(cos(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('\\\\mathbf{s}', commutative=True))))))"], [["differentiate", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{J}')(Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True))), cos(cos(Symbol('\\\\mathbf{s}', commutative=True)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Integer(2), cos(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(2), cos(cos(Symbol('\\\\mathbf{s}', commutative=True))))), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(Q)} = \\log{(\\sin{(Q)})}, then obtain \\frac{d^{2}}{d Q^{2}} (Q + \\tilde{g}{(Q)}) = \\frac{d^{2}}{d Q^{2}} (Q + \\log{(\\sin{(Q)})})", "derivation": "\\tilde{g}{(Q)} = \\log{(\\sin{(Q)})} and Q + \\tilde{g}{(Q)} = Q + \\log{(\\sin{(Q)})} and \\frac{d}{d Q} (Q + \\tilde{g}{(Q)}) = \\frac{d}{d Q} (Q + \\log{(\\sin{(Q)})}) and \\frac{d^{2}}{d Q^{2}} (Q + \\tilde{g}{(Q)}) = \\frac{d^{2}}{d Q^{2}} (Q + \\log{(\\sin{(Q)})})", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('Q', commutative=True)), log(sin(Symbol('Q', commutative=True))))"], [["add", 1, "Symbol('Q', commutative=True)"], "Equality(Add(Symbol('Q', commutative=True), Function('\\\\tilde{g}')(Symbol('Q', commutative=True))), Add(Symbol('Q', commutative=True), log(sin(Symbol('Q', commutative=True)))))"], [["differentiate", 2, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Symbol('Q', commutative=True), Function('\\\\tilde{g}')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Add(Symbol('Q', commutative=True), log(sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('Q', commutative=True)"], "Equality(Derivative(Add(Symbol('Q', commutative=True), Function('\\\\tilde{g}')(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(2))), Derivative(Add(Symbol('Q', commutative=True), log(sin(Symbol('Q', commutative=True)))), Tuple(Symbol('Q', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\dot{x}, then obtain - \\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} \\dot{x}", "derivation": "\\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\dot{x} and - \\dot{\\mathbf{r}} + \\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = \\dot{\\mathbf{r}} \\dot{x} - \\dot{\\mathbf{r}} and \\dot{\\mathbf{r}} - \\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} \\dot{x} + \\dot{\\mathbf{r}} and - \\hat{X}{(\\dot{x},\\dot{\\mathbf{r}})} = - \\dot{\\mathbf{r}} \\dot{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"], [["minus", 1, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Add(Mul(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"], [["divide", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\hat{X}')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))"]]}, {"prompt": "Given z{(v_{y},\\hat{X})} = \\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}, then obtain \\int v_{y} \\int \\frac{z{(v_{y},\\hat{X})}}{\\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}} d\\hat{X} d\\hat{X} = \\int v_{y} \\int 1 d\\hat{X} d\\hat{X}", "derivation": "z{(v_{y},\\hat{X})} = \\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y} and \\frac{z{(v_{y},\\hat{X})}}{\\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}} = 1 and \\int \\frac{z{(v_{y},\\hat{X})}}{\\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}} d\\hat{X} = \\int 1 d\\hat{X} and v_{y} \\int \\frac{z{(v_{y},\\hat{X})}}{\\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}} d\\hat{X} = v_{y} \\int 1 d\\hat{X} and \\int v_{y} \\int \\frac{z{(v_{y},\\hat{X})}}{\\frac{\\partial}{\\partial v_{y}} \\hat{X} v_{y}} d\\hat{X} d\\hat{X} = \\int v_{y} \\int 1 d\\hat{X} d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('v_y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"], [["divide", 1, "Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1)))"], "Equality(Mul(Function('z')(Symbol('v_y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Mul(Function('z')(Symbol('v_y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["times", 3, "Symbol('v_y', commutative=True)"], "Equality(Mul(Symbol('v_y', commutative=True), Integral(Mul(Function('z')(Symbol('v_y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Mul(Symbol('v_y', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Mul(Symbol('v_y', commutative=True), Integral(Mul(Function('z')(Symbol('v_y', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\hat{X}', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))), Integer(-1))), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Mul(Symbol('v_y', commutative=True), Integral(Integer(1), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(T,p)} = T^{p}, then derive \\log{(\\frac{\\frac{\\partial}{\\partial T} \\operatorname{r_{0}}{(T,p)}}{\\operatorname{r_{0}}{(T,p)}})} = \\log{(\\frac{T^{p} p}{T \\operatorname{r_{0}}{(T,p)}})}, then obtain \\log{(\\frac{T^{p} p}{T \\operatorname{r_{0}}{(T,p)}})} = \\log{(\\frac{\\frac{\\partial}{\\partial T} T^{p}}{\\operatorname{r_{0}}{(T,p)}})}", "derivation": "\\operatorname{r_{0}}{(T,p)} = T^{p} and \\frac{\\partial}{\\partial T} \\operatorname{r_{0}}{(T,p)} = \\frac{\\partial}{\\partial T} T^{p} and \\frac{\\frac{\\partial}{\\partial T} \\operatorname{r_{0}}{(T,p)}}{\\operatorname{r_{0}}{(T,p)}} = \\frac{\\frac{\\partial}{\\partial T} T^{p}}{\\operatorname{r_{0}}{(T,p)}} and \\log{(\\frac{\\frac{\\partial}{\\partial T} \\operatorname{r_{0}}{(T,p)}}{\\operatorname{r_{0}}{(T,p)}})} = \\log{(\\frac{\\frac{\\partial}{\\partial T} T^{p}}{\\operatorname{r_{0}}{(T,p)}})} and \\log{(\\frac{\\frac{\\partial}{\\partial T} \\operatorname{r_{0}}{(T,p)}}{\\operatorname{r_{0}}{(T,p)}})} = \\log{(\\frac{T^{p} p}{T \\operatorname{r_{0}}{(T,p)}})} and \\log{(\\frac{T^{p} p}{T \\operatorname{r_{0}}{(T,p)}})} = \\log{(\\frac{\\frac{\\partial}{\\partial T} T^{p}}{\\operatorname{r_{0}}{(T,p)}})}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["divide", 2, "Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True))"], "Equality(Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))), Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1)))))"], [["log", 3], "Equality(log(Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), log(Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(log(Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))), log(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True), Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(log(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Symbol('p', commutative=True), Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)))), log(Mul(Pow(Function('r_0')(Symbol('T', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('T', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\tilde{g}^*{(r_{0},L)} = \\frac{r_{0}}{L}, then obtain (\\frac{r_{0} + \\tilde{g}^*{(r_{0},L)}}{\\tilde{g}^*{(r_{0},L)}})^{r_{0}} = (\\frac{r_{0} + \\frac{r_{0}}{L}}{\\tilde{g}^*{(r_{0},L)}})^{r_{0}}", "derivation": "\\tilde{g}^*{(r_{0},L)} = \\frac{r_{0}}{L} and r_{0} + \\tilde{g}^*{(r_{0},L)} = r_{0} + \\frac{r_{0}}{L} and \\frac{r_{0} + \\tilde{g}^*{(r_{0},L)}}{\\tilde{g}^*{(r_{0},L)}} = \\frac{r_{0} + \\frac{r_{0}}{L}}{\\tilde{g}^*{(r_{0},L)}} and (\\frac{r_{0} + \\tilde{g}^*{(r_{0},L)}}{\\tilde{g}^*{(r_{0},L)}})^{r_{0}} = (\\frac{r_{0} + \\frac{r_{0}}{L}}{\\tilde{g}^*{(r_{0},L)}})^{r_{0}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('r_0', commutative=True)))"], [["add", 1, "Symbol('r_0', commutative=True)"], "Equality(Add(Symbol('r_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True))), Add(Symbol('r_0', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))))"], [["divide", 2, "Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Add(Symbol('r_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True))), Pow(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Mul(Add(Symbol('r_0', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Pow(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True)), Integer(-1))))"], [["power", 3, "Symbol('r_0', commutative=True)"], "Equality(Pow(Mul(Add(Symbol('r_0', commutative=True), Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True))), Pow(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)), Pow(Mul(Add(Symbol('r_0', commutative=True), Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Symbol('r_0', commutative=True))), Pow(Function('\\\\tilde{g}^*')(Symbol('r_0', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Symbol('r_0', commutative=True)))"]]}, {"prompt": "Given \\Omega{(z^{*},g_{\\varepsilon})} = g_{\\varepsilon} z^{*}, then obtain (\\int z^{*} \\Omega{(z^{*},g_{\\varepsilon})} dg_{\\varepsilon})^{2} = (\\int g_{\\varepsilon} (z^{*})^{2} dg_{\\varepsilon})^{2}", "derivation": "\\Omega{(z^{*},g_{\\varepsilon})} = g_{\\varepsilon} z^{*} and z^{*} \\Omega{(z^{*},g_{\\varepsilon})} = g_{\\varepsilon} (z^{*})^{2} and \\int z^{*} \\Omega{(z^{*},g_{\\varepsilon})} dg_{\\varepsilon} = \\int g_{\\varepsilon} (z^{*})^{2} dg_{\\varepsilon} and (\\int z^{*} \\Omega{(z^{*},g_{\\varepsilon})} dg_{\\varepsilon})^{2} = (\\int g_{\\varepsilon} (z^{*})^{2} dg_{\\varepsilon})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('z^*', commutative=True)))"], [["times", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(2))))"], [["integrate", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(2))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Mul(Symbol('z^*', commutative=True), Function('\\\\Omega')(Symbol('z^*', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2)), Pow(Integral(Mul(Symbol('g_{\\\\varepsilon}', commutative=True), Pow(Symbol('z^*', commutative=True), Integer(2))), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\psi{(\\Omega)} = \\log{(\\Omega)} and I{(\\Omega)} = \\log{(\\Omega)}, then obtain \\psi^{2}{(\\Omega)} = \\psi{(\\Omega)} \\log{(\\Omega)}", "derivation": "\\psi{(\\Omega)} = \\log{(\\Omega)} and I{(\\Omega)} = \\log{(\\Omega)} and \\psi{(\\Omega)} = I{(\\Omega)} and I^{2}{(\\Omega)} = I{(\\Omega)} \\log{(\\Omega)} and \\psi^{2}{(\\Omega)} = \\psi{(\\Omega)} \\log{(\\Omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], ["renaming_premise", "Equality(Function('I')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)), Function('I')(Symbol('\\\\Omega', commutative=True)))"], [["times", 2, "Function('I')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('I')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('I')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('\\\\psi')(Symbol('\\\\Omega', commutative=True)), log(Symbol('\\\\Omega', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(f)} = f, then derive Z + \\frac{\\mathbf{A}^{2}{(f)}}{2} = \\int f d\\mathbf{A}{(f)}, then obtain Z + \\frac{f \\mathbf{A}{(f)}}{2} = \\int f d\\mathbf{A}{(f)}", "derivation": "\\mathbf{A}{(f)} = f and \\mathbf{A}^{2}{(f)} = f \\mathbf{A}{(f)} and \\int \\mathbf{A}{(f)} df = \\int f df and \\int \\mathbf{A}{(f)} d\\mathbf{A}{(f)} = \\int f d\\mathbf{A}{(f)} and Z + \\frac{\\mathbf{A}^{2}{(f)}}{2} = \\int f d\\mathbf{A}{(f)} and Z + \\frac{f \\mathbf{A}{(f)}}{2} = \\int f d\\mathbf{A}{(f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)), Symbol('f', commutative=True))"], [["times", 1, "Function('\\\\mathbf{A}')(Symbol('f', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)), Integer(2)), Mul(Symbol('f', commutative=True), Function('\\\\mathbf{A}')(Symbol('f', commutative=True))))"], [["integrate", 1, "Symbol('f', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Symbol('f', commutative=True), Tuple(Symbol('f', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)), Tuple(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)))), Integral(Symbol('f', commutative=True), Tuple(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)), Integer(2)))), Integral(Symbol('f', commutative=True), Tuple(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('Z', commutative=True), Mul(Rational(1, 2), Symbol('f', commutative=True), Function('\\\\mathbf{A}')(Symbol('f', commutative=True)))), Integral(Symbol('f', commutative=True), Tuple(Function('\\\\mathbf{A}')(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\phi_2)} = \\cos{(\\phi_2)}, then obtain - \\cos^{\\phi_2}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2)} = - \\cos^{\\phi_2}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)}", "derivation": "\\operatorname{t_{2}}{(\\phi_2)} = \\cos{(\\phi_2)} and \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2)} = \\cos^{\\phi_2}{(\\phi_2)} and \\frac{d}{d \\phi_2} \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2)} = \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)} and - \\cos^{\\phi_2}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\operatorname{t_{2}}^{\\phi_2}{(\\phi_2)} = - \\cos^{\\phi_2}{(\\phi_2)} + \\frac{d}{d \\phi_2} \\cos^{\\phi_2}{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 3, "Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Derivative(Pow(Function('t_2')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True))), Derivative(Pow(cos(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(A_{2},A)} = A^{A_{2}}, then obtain (\\frac{\\partial}{\\partial A_{2}} 4 C{(A_{2},A)})^{A} = (\\frac{\\partial}{\\partial A_{2}} (A^{A_{2}} + 3 C{(A_{2},A)}))^{A}", "derivation": "C{(A_{2},A)} = A^{A_{2}} and 2 C{(A_{2},A)} = A^{A_{2}} + C{(A_{2},A)} and 4 C{(A_{2},A)} = A^{A_{2}} + 3 C{(A_{2},A)} and \\frac{\\partial}{\\partial A_{2}} 4 C{(A_{2},A)} = \\frac{\\partial}{\\partial A_{2}} (A^{A_{2}} + 3 C{(A_{2},A)}) and (\\frac{\\partial}{\\partial A_{2}} 4 C{(A_{2},A)})^{A} = (\\frac{\\partial}{\\partial A_{2}} (A^{A_{2}} + 3 C{(A_{2},A)}))^{A}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('A_2', commutative=True)))"], [["add", 1, "Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Add(Pow(Symbol('A', commutative=True), Symbol('A_2', commutative=True)), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))))"], [["add", 2, "Mul(Integer(2), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)))"], "Equality(Mul(Integer(4), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Add(Pow(Symbol('A', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(3), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)))))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Mul(Integer(4), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Add(Pow(Symbol('A', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(3), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["power", 4, "Symbol('A', commutative=True)"], "Equality(Pow(Derivative(Mul(Integer(4), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('A', commutative=True)), Pow(Derivative(Add(Pow(Symbol('A', commutative=True), Symbol('A_2', commutative=True)), Mul(Integer(3), Function('C')(Symbol('A_2', commutative=True), Symbol('A', commutative=True)))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('A', commutative=True)))"]]}, {"prompt": "Given T{(\\phi_2)} = \\cos{(e^{\\phi_2})} and C{(\\mathbf{J}_M,\\theta)} = \\mathbf{J}_M + \\theta, then obtain (\\theta + T{(\\phi_2)} e^{\\phi_2})^{\\phi_2} = (\\theta + e^{\\phi_2} \\cos{(e^{\\phi_2})})^{\\phi_2}", "derivation": "T{(\\phi_2)} = \\cos{(e^{\\phi_2})} and C{(\\mathbf{J}_M,\\theta)} = \\mathbf{J}_M + \\theta and T{(\\phi_2)} e^{\\phi_2} = e^{\\phi_2} \\cos{(e^{\\phi_2})} and - \\mathbf{J}_M + T{(\\phi_2)} e^{\\phi_2} = - \\mathbf{J}_M + e^{\\phi_2} \\cos{(e^{\\phi_2})} and - \\mathbf{J}_M + C{(\\mathbf{J}_M,\\theta)} + T{(\\phi_2)} e^{\\phi_2} = - \\mathbf{J}_M + C{(\\mathbf{J}_M,\\theta)} + e^{\\phi_2} \\cos{(e^{\\phi_2})} and \\theta + T{(\\phi_2)} e^{\\phi_2} = \\theta + e^{\\phi_2} \\cos{(e^{\\phi_2})} and (\\theta + T{(\\phi_2)} e^{\\phi_2})^{\\phi_2} = (\\theta + e^{\\phi_2} \\cos{(e^{\\phi_2})})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))"], ["get_premise", "Equality(Function('C')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta', commutative=True)), Add(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Function('T')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True))), Mul(exp(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True)))))"], [["minus", 3, "Symbol('\\\\mathbf{J}_M', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(Function('T')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Mul(exp(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))))"], [["add", 4, "Function('C')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('C')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(Function('T')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_M', commutative=True)), Function('C')(Symbol('\\\\mathbf{J}_M', commutative=True), Symbol('\\\\theta', commutative=True)), Mul(exp(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\theta', commutative=True), Mul(Function('T')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(exp(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))))"], [["power", 6, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\theta', commutative=True), Mul(Function('T')(Symbol('\\\\phi_2', commutative=True)), exp(Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Symbol('\\\\theta', commutative=True), Mul(exp(Symbol('\\\\phi_2', commutative=True)), cos(exp(Symbol('\\\\phi_2', commutative=True))))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given \\dot{y}{(f)} = \\cos{(f)}, then derive \\frac{d}{d f} \\dot{y}{(f)} = - \\sin{(f)}, then obtain (\\cos{(m_{s})} + \\frac{d}{d f} \\dot{y}{(f)})^{m_{s}} - \\operatorname{A_{y}}{(f)} = (- \\sin{(f)} + \\cos{(m_{s})})^{m_{s}} - \\operatorname{A_{y}}{(f)}", "derivation": "\\dot{y}{(f)} = \\cos{(f)} and \\frac{d}{d f} \\dot{y}{(f)} = \\frac{d}{d f} \\cos{(f)} and \\frac{d}{d f} \\dot{y}{(f)} = - \\sin{(f)} and \\cos{(m_{s})} + \\frac{d}{d f} \\dot{y}{(f)} = - \\sin{(f)} + \\cos{(m_{s})} and (\\cos{(m_{s})} + \\frac{d}{d f} \\dot{y}{(f)})^{m_{s}} = (- \\sin{(f)} + \\cos{(m_{s})})^{m_{s}} and (\\cos{(m_{s})} + \\frac{d}{d f} \\dot{y}{(f)})^{m_{s}} - \\operatorname{A_{y}}{(f)} = (- \\sin{(f)} + \\cos{(m_{s})})^{m_{s}} - \\operatorname{A_{y}}{(f)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{y}')(Symbol('f', commutative=True)), cos(Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(cos(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{y}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('f', commutative=True))))"], [["add", 3, "cos(Symbol('m_s', commutative=True))"], "Equality(Add(cos(Symbol('m_s', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), cos(Symbol('m_s', commutative=True))))"], [["power", 4, "Symbol('m_s', commutative=True)"], "Equality(Pow(Add(cos(Symbol('m_s', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Pow(Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), cos(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["minus", 5, "Function('A_y')(Symbol('f', commutative=True))"], "Equality(Add(Pow(Add(cos(Symbol('m_s', commutative=True)), Derivative(Function('\\\\dot{y}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Symbol('m_s', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('f', commutative=True)))), Add(Pow(Add(Mul(Integer(-1), sin(Symbol('f', commutative=True))), cos(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Mul(Integer(-1), Function('A_y')(Symbol('f', commutative=True)))))"]]}, {"prompt": "Given i{(g)} = \\sin{(g)}, then derive A_{x} + \\operatorname{Si}{(g)} = \\int \\frac{\\sin^{2}{(g)}}{g i{(g)}} dg, then obtain A_{x} - \\sin{(g)} + \\operatorname{Si}{(g)} = - \\sin{(g)} + \\int \\frac{\\sin^{2}{(g)}}{g i{(g)}} dg", "derivation": "i{(g)} = \\sin{(g)} and \\frac{i{(g)}}{g} = \\frac{\\sin{(g)}}{g} and 1 = \\frac{\\sin{(g)}}{i{(g)}} and \\frac{\\sin{(g)}}{g} = \\frac{\\sin^{2}{(g)}}{g i{(g)}} and \\int \\frac{\\sin{(g)}}{g} dg = \\int \\frac{\\sin^{2}{(g)}}{g i{(g)}} dg and A_{x} + \\operatorname{Si}{(g)} = \\int \\frac{\\sin^{2}{(g)}}{g i{(g)}} dg and A_{x} - \\sin{(g)} + \\operatorname{Si}{(g)} = - \\sin{(g)} + \\int \\frac{\\sin^{2}{(g)}}{g i{(g)}} dg", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('g', commutative=True)), sin(Symbol('g', commutative=True)))"], [["divide", 1, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('i')(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), sin(Symbol('g', commutative=True))))"], [["divide", 2, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Function('i')(Symbol('g', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Function('i')(Symbol('g', commutative=True)), Integer(-1)), sin(Symbol('g', commutative=True))))"], [["times", 3, "Mul(Pow(Symbol('g', commutative=True), Integer(-1)), sin(Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), sin(Symbol('g', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('i')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))))"], [["integrate", 4, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), sin(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('i')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('A_x', commutative=True), Si(Symbol('g', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('i')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True))))"], [["minus", 6, "sin(Symbol('g', commutative=True))"], "Equality(Add(Symbol('A_x', commutative=True), Mul(Integer(-1), sin(Symbol('g', commutative=True))), Si(Symbol('g', commutative=True))), Add(Mul(Integer(-1), sin(Symbol('g', commutative=True))), Integral(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('i')(Symbol('g', commutative=True)), Integer(-1)), Pow(sin(Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given M{(s,A)} = A s, then obtain (\\frac{\\partial}{\\partial s} M{(s,A)})^{s} = A^{s}", "derivation": "M{(s,A)} = A s and \\frac{\\partial}{\\partial s} M{(s,A)} = \\frac{\\partial}{\\partial s} A s and (\\frac{\\partial}{\\partial s} M{(s,A)})^{s} = (\\frac{\\partial}{\\partial s} A s)^{s} and (\\frac{\\partial}{\\partial s} M{(s,A)})^{s} = A^{s}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)))"], [["differentiate", 1, "Symbol('s', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["power", 2, "Symbol('s', commutative=True)"], "Equality(Pow(Derivative(Function('M')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Derivative(Mul(Symbol('A', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('M')(Symbol('s', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('s', commutative=True), Integer(1))), Symbol('s', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\rho_{f}{(h)} = \\sin{(\\sin{(h)})} and \\varphi^{*}{(h)} = \\sin{(h)}, then obtain \\rho_{f}{(h)} + \\frac{d}{d h} \\varphi^{*}{(h)} = \\sin{(\\varphi^{*}{(h)})} + \\frac{d}{d h} \\varphi^{*}{(h)}", "derivation": "\\rho_{f}{(h)} = \\sin{(\\sin{(h)})} and \\varphi^{*}{(h)} = \\sin{(h)} and \\rho_{f}{(h)} + \\frac{d}{d h} \\sin{(h)} = \\sin{(\\sin{(h)})} + \\frac{d}{d h} \\sin{(h)} and \\rho_{f}{(h)} + \\frac{d}{d h} \\varphi^{*}{(h)} = \\sin{(\\varphi^{*}{(h)})} + \\frac{d}{d h} \\varphi^{*}{(h)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho_f')(Symbol('h', commutative=True)), sin(sin(Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["add", 1, "Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\rho_f')(Symbol('h', commutative=True)), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(sin(sin(Symbol('h', commutative=True))), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\rho_f')(Symbol('h', commutative=True)), Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Add(sin(Function('\\\\varphi^*')(Symbol('h', commutative=True))), Derivative(Function('\\\\varphi^*')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(h)} = \\frac{d}{d h} \\sin{(h)}, then obtain 2 (- \\sin{(h)} + \\frac{d}{d h} \\operatorname{v_{y}}{(h)}) \\cos{(h)} = 2 \\cos{(h)} \\frac{d}{d h} 2 \\cos{(h)}", "derivation": "\\operatorname{v_{y}}{(h)} = \\frac{d}{d h} \\sin{(h)} and \\operatorname{v_{y}}{(h)} + \\frac{d}{d h} \\sin{(h)} = 2 \\frac{d}{d h} \\sin{(h)} and \\frac{d}{d h} (\\operatorname{v_{y}}{(h)} + \\frac{d}{d h} \\sin{(h)}) = \\frac{d}{d h} 2 \\frac{d}{d h} \\sin{(h)} and 2 \\frac{d}{d h} (\\operatorname{v_{y}}{(h)} + \\frac{d}{d h} \\sin{(h)}) \\frac{d}{d h} \\sin{(h)} = 2 \\frac{d}{d h} \\sin{(h)} \\frac{d}{d h} 2 \\frac{d}{d h} \\sin{(h)} and 2 (- \\sin{(h)} + \\frac{d}{d h} \\operatorname{v_{y}}{(h)}) \\cos{(h)} = 2 \\cos{(h)} \\frac{d}{d h} 2 \\cos{(h)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('h', commutative=True)), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["add", 1, "Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))"], "Equality(Add(Function('v_y')(Symbol('h', commutative=True)), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('h', commutative=True)"], "Equality(Derivative(Add(Function('v_y')(Symbol('h', commutative=True)), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["times", 3, "Mul(Integer(2), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))))"], "Equality(Mul(Integer(2), Derivative(Add(Function('v_y')(Symbol('h', commutative=True)), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), Tuple(Symbol('h', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), sin(Symbol('h', commutative=True))), Derivative(Function('v_y')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1)))), cos(Symbol('h', commutative=True))), Mul(Integer(2), cos(Symbol('h', commutative=True)), Derivative(Mul(Integer(2), cos(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(J_{\\varepsilon},\\mathbf{F})} = J_{\\varepsilon} \\mathbf{F} and h{(J_{\\varepsilon},\\mathbf{F})} = J_{\\varepsilon} \\mathbf{F}, then obtain \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int \\operatorname{z^{*}}{(J_{\\varepsilon},\\mathbf{F})} dJ_{\\varepsilon} = \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int J_{\\varepsilon} \\mathbf{F} dJ_{\\varepsilon}", "derivation": "\\operatorname{z^{*}}{(J_{\\varepsilon},\\mathbf{F})} = J_{\\varepsilon} \\mathbf{F} and h{(J_{\\varepsilon},\\mathbf{F})} = J_{\\varepsilon} \\mathbf{F} and h{(J_{\\varepsilon},\\mathbf{F})} = \\operatorname{z^{*}}{(J_{\\varepsilon},\\mathbf{F})} and \\int h{(J_{\\varepsilon},\\mathbf{F})} dJ_{\\varepsilon} = \\int J_{\\varepsilon} \\mathbf{F} dJ_{\\varepsilon} and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int h{(J_{\\varepsilon},\\mathbf{F})} dJ_{\\varepsilon} = \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int J_{\\varepsilon} \\mathbf{F} dJ_{\\varepsilon} and \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int \\operatorname{z^{*}}{(J_{\\varepsilon},\\mathbf{F})} dJ_{\\varepsilon} = \\frac{\\partial}{\\partial J_{\\varepsilon}} \\int J_{\\varepsilon} \\mathbf{F} dJ_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], ["renaming_premise", "Equality(Function('h')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('h')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Function('z^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)))"], [["integrate", 2, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('h')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["differentiate", 4, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Derivative(Integral(Function('h')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Integral(Function('z^*')(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))), Derivative(Integral(Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(n_{2})} = e^{n_{2}}, then derive \\frac{d}{d n_{2}} V{(n_{2})} = e^{n_{2}}, then obtain \\log{(- V{(n_{2})} + \\frac{d}{d n_{2}} V{(n_{2})})} = \\log{(- 2 V{(n_{2})} + 2 \\frac{d}{d n_{2}} V{(n_{2})})}", "derivation": "V{(n_{2})} = e^{n_{2}} and 0 = - V{(n_{2})} + e^{n_{2}} and \\frac{d}{d n_{2}} V{(n_{2})} = \\frac{d}{d n_{2}} e^{n_{2}} and \\frac{d}{d n_{2}} V{(n_{2})} = e^{n_{2}} and 0 = - V{(n_{2})} + \\frac{d}{d n_{2}} V{(n_{2})} and - V{(n_{2})} + \\frac{d}{d n_{2}} V{(n_{2})} = - 2 V{(n_{2})} + 2 \\frac{d}{d n_{2}} V{(n_{2})} and \\log{(- V{(n_{2})} + \\frac{d}{d n_{2}} V{(n_{2})})} = \\log{(- 2 V{(n_{2})} + 2 \\frac{d}{d n_{2}} V{(n_{2})})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('n_2', commutative=True)), exp(Symbol('n_2', commutative=True)))"], [["minus", 1, "Function('V')(Symbol('n_2', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V')(Symbol('n_2', commutative=True))), exp(Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(exp(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), exp(Symbol('n_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Integer(0), Add(Mul(Integer(-1), Function('V')(Symbol('n_2', commutative=True))), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))"], [["add", 5, "Add(Mul(Integer(-1), Function('V')(Symbol('n_2', commutative=True))), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('n_2', commutative=True))), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Integer(2), Function('V')(Symbol('n_2', commutative=True))), Mul(Integer(2), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))))"], [["log", 6], "Equality(log(Add(Mul(Integer(-1), Function('V')(Symbol('n_2', commutative=True))), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))))), log(Add(Mul(Integer(-1), Integer(2), Function('V')(Symbol('n_2', commutative=True))), Mul(Integer(2), Derivative(Function('V')(Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given T{(\\hat{H})} = e^{\\hat{H}}, then obtain \\cos{(T^{\\hat{H}}{(\\hat{H})} + e^{\\hat{H}})} = \\cos{(e^{\\hat{H}} + (e^{\\hat{H}})^{\\hat{H}})}", "derivation": "T{(\\hat{H})} = e^{\\hat{H}} and T^{\\hat{H}}{(\\hat{H})} = (e^{\\hat{H}})^{\\hat{H}} and T^{\\hat{H}}{(\\hat{H})} + e^{\\hat{H}} = e^{\\hat{H}} + (e^{\\hat{H}})^{\\hat{H}} and \\cos{(T^{\\hat{H}}{(\\hat{H})} + e^{\\hat{H}})} = \\cos{(e^{\\hat{H}} + (e^{\\hat{H}})^{\\hat{H}})}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], [["add", 2, "exp(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Add(Pow(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True))), Add(exp(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True))))"], [["cos", 3], "Equality(cos(Add(Pow(Function('T')(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), exp(Symbol('\\\\hat{H}', commutative=True)))), cos(Add(exp(Symbol('\\\\hat{H}', commutative=True)), Pow(exp(Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))))"]]}, {"prompt": "Given C{(\\mathbf{P},\\varphi)} = (e^{\\mathbf{P}})^{\\varphi}, then derive \\frac{\\partial}{\\partial \\varphi} C{(\\mathbf{P},\\varphi)} = (e^{\\mathbf{P}})^{\\varphi} \\log{(e^{\\mathbf{P}})}, then obtain \\frac{\\partial}{\\partial \\varphi} (e^{\\mathbf{P}})^{\\varphi} = (e^{\\mathbf{P}})^{\\varphi} \\log{(e^{\\mathbf{P}})}", "derivation": "C{(\\mathbf{P},\\varphi)} = (e^{\\mathbf{P}})^{\\varphi} and \\frac{\\partial}{\\partial \\varphi} C{(\\mathbf{P},\\varphi)} = \\frac{\\partial}{\\partial \\varphi} (e^{\\mathbf{P}})^{\\varphi} and \\frac{\\partial}{\\partial \\varphi} C{(\\mathbf{P},\\varphi)} = (e^{\\mathbf{P}})^{\\varphi} \\log{(e^{\\mathbf{P}})} and \\frac{\\partial}{\\partial \\varphi} (e^{\\mathbf{P}})^{\\varphi} = (e^{\\mathbf{P}})^{\\varphi} \\log{(e^{\\mathbf{P}})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(exp(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Pow(exp(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Pow(exp(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Pow(exp(Symbol('\\\\mathbf{P}', commutative=True)), Symbol('\\\\varphi', commutative=True)), log(exp(Symbol('\\\\mathbf{P}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(m,\\rho)} = \\sin{(m^{\\rho})}, then obtain (- \\operatorname{n_{2}}^{m}{(m,\\rho)} + \\sin^{m}{(m^{\\rho})})^{\\rho} = 0^{\\rho}", "derivation": "\\operatorname{n_{2}}{(m,\\rho)} = \\sin{(m^{\\rho})} and \\operatorname{n_{2}}^{m}{(m,\\rho)} = \\sin^{m}{(m^{\\rho})} and \\operatorname{n_{2}}^{m}{(m,\\rho)} - \\sin^{m}{(m^{\\rho})} = 0 and - \\operatorname{n_{2}}^{m}{(m,\\rho)} + \\sin^{m}{(m^{\\rho})} = 0 and (- \\operatorname{n_{2}}^{m}{(m,\\rho)} + \\sin^{m}{(m^{\\rho})})^{\\rho} = 0^{\\rho}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True)), sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('n_2')(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('m', commutative=True)), Pow(sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('m', commutative=True)))"], [["minus", 2, "Pow(sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('m', commutative=True))"], "Equality(Add(Pow(Function('n_2')(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), Pow(sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('m', commutative=True)))), Integer(0))"], [["times", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Pow(Function('n_2')(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('m', commutative=True))), Pow(sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('m', commutative=True))), Integer(0))"], [["power", 4, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Pow(Function('n_2')(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True)), Symbol('m', commutative=True))), Pow(sin(Pow(Symbol('m', commutative=True), Symbol('\\\\rho', commutative=True))), Symbol('m', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integer(0), Symbol('\\\\rho', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{P},\\eta)} = \\eta e^{\\mathbf{P}} and \\operatorname{C_{d}}{(\\mathbf{J})} = \\log{(e^{\\mathbf{J}})}, then obtain - \\eta e^{\\mathbf{P}} - \\mathbf{J} + \\operatorname{C_{d}}{(\\mathbf{J})} = - \\eta e^{\\mathbf{P}} - \\mathbf{J} + \\log{(e^{\\mathbf{J}})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{P},\\eta)} = \\eta e^{\\mathbf{P}} and \\operatorname{C_{d}}{(\\mathbf{J})} = \\log{(e^{\\mathbf{J}})} and - \\mathbf{J} + \\operatorname{C_{d}}{(\\mathbf{J})} = - \\mathbf{J} + \\log{(e^{\\mathbf{J}})} and - \\mathbf{J} + \\operatorname{C_{d}}{(\\mathbf{J})} - \\operatorname{t_{2}}{(\\mathbf{P},\\eta)} = - \\mathbf{J} - \\operatorname{t_{2}}{(\\mathbf{P},\\eta)} + \\log{(e^{\\mathbf{J}})} and - \\eta e^{\\mathbf{P}} - \\mathbf{J} + \\operatorname{C_{d}}{(\\mathbf{J})} = - \\eta e^{\\mathbf{P}} - \\mathbf{J} + \\log{(e^{\\mathbf{J}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), exp(Symbol('\\\\mathbf{P}', commutative=True))))"], ["get_premise", "Equality(Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), log(exp(Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), log(exp(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["minus", 3, "Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\eta', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\eta', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Integer(-1), Function('t_2')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('\\\\eta', commutative=True))), log(exp(Symbol('\\\\mathbf{J}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), exp(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('C_d')(Symbol('\\\\mathbf{J}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True), exp(Symbol('\\\\mathbf{P}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), log(exp(Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})} = \\sin^{\\mathbf{E}}{(k)} and l{(k)} = \\sin{(k)}, then obtain \\frac{d}{d k} 1 = \\frac{\\partial}{\\partial k} \\frac{l^{\\mathbf{E}}{(k)}}{\\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})}}", "derivation": "\\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})} = \\sin^{\\mathbf{E}}{(k)} and l{(k)} = \\sin{(k)} and 1 = \\frac{\\sin^{\\mathbf{E}}{(k)}}{\\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})}} and \\frac{d}{d k} 1 = \\frac{\\partial}{\\partial k} \\frac{\\sin^{\\mathbf{E}}{(k)}}{\\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})}} and \\frac{d}{d k} 1 = \\frac{\\partial}{\\partial k} \\frac{l^{\\mathbf{E}}{(k)}}{\\operatorname{L_{\\varepsilon}}{(k,\\mathbf{E})}}", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(sin(Symbol('k', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"], ["renaming_premise", "Equality(Function('l')(Symbol('k', commutative=True)), sin(Symbol('k', commutative=True)))"], [["divide", 1, "Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Pow(sin(Symbol('k', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))))"], [["differentiate", 3, "Symbol('k', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Pow(sin(Symbol('k', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Derivative(Integer(1), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('L_{\\\\varepsilon}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Integer(-1)), Pow(Function('l')(Symbol('k', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(A_{x},u)} = A_{x} + u, then obtain (A_{x} + u) (- A_{x} - u + \\operatorname{A_{2}}{(A_{x},u)})^{A_{x}} = 0^{A_{x}} (A_{x} + u)", "derivation": "\\operatorname{A_{2}}{(A_{x},u)} = A_{x} + u and - A_{x} - u + \\operatorname{A_{2}}{(A_{x},u)} = 0 and (- A_{x} - u + \\operatorname{A_{2}}{(A_{x},u)})^{A_{x}} = 0^{A_{x}} and (A_{x} + u) (- A_{x} - u + \\operatorname{A_{2}}{(A_{x},u)})^{A_{x}} = 0^{A_{x}} (A_{x} + u)", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)))"], [["minus", 1, "Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A_2')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Integer(0))"], [["power", 2, "Symbol('A_x', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A_2')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Symbol('A_x', commutative=True)), Pow(Integer(0), Symbol('A_x', commutative=True)))"], [["times", 3, "Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True))"], "Equality(Mul(Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('u', commutative=True)), Function('A_2')(Symbol('A_x', commutative=True), Symbol('u', commutative=True))), Symbol('A_x', commutative=True))), Mul(Pow(Integer(0), Symbol('A_x', commutative=True)), Add(Symbol('A_x', commutative=True), Symbol('u', commutative=True))))"]]}, {"prompt": "Given \\dot{z}{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})}, then obtain e^{\\dot{z}{(g^{\\prime}_{\\varepsilon})}} e^{- \\sin{(g^{\\prime}_{\\varepsilon})}} - 1 = 0", "derivation": "\\dot{z}{(g^{\\prime}_{\\varepsilon})} = \\sin{(g^{\\prime}_{\\varepsilon})} and e^{\\dot{z}{(g^{\\prime}_{\\varepsilon})}} = e^{\\sin{(g^{\\prime}_{\\varepsilon})}} and e^{\\dot{z}{(g^{\\prime}_{\\varepsilon})}} e^{- \\sin{(g^{\\prime}_{\\varepsilon})}} = 1 and e^{\\dot{z}{(g^{\\prime}_{\\varepsilon})}} e^{- \\sin{(g^{\\prime}_{\\varepsilon})}} + \\frac{\\dot{z}^{2}{(g^{\\prime}_{\\varepsilon})}}{g^{\\prime}_{\\varepsilon}} = 1 + \\frac{\\dot{z}^{2}{(g^{\\prime}_{\\varepsilon})}}{g^{\\prime}_{\\varepsilon}} and e^{\\dot{z}{(g^{\\prime}_{\\varepsilon})}} e^{- \\sin{(g^{\\prime}_{\\varepsilon})}} - 1 = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), exp(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["divide", 2, "exp(sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(exp(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Integer(1))"], [["add", 3, "Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)))"], "Equality(Add(Mul(exp(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)))), Add(Integer(1), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2)))))"], [["minus", 4, "Add(Integer(1), Mul(Pow(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Integer(-1)), Pow(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(2))))"], "Equality(Add(Mul(exp(Function('\\\\dot{z}')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))), Integer(-1)), Integer(0))"]]}, {"prompt": "Given T{(b,i,M_{E})} = b + i^{M_{E}}, then derive - \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)}}{2} = - (\\frac{b}{2} + \\frac{i^{M_{E}}}{2}) e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)}, then obtain \\int - \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)}}{2} db = \\int - (\\frac{b}{2} + \\frac{i^{M_{E}}}{2}) e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)} db", "derivation": "T{(b,i,M_{E})} = b + i^{M_{E}} and \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}}}{2} = \\frac{(b + i^{M_{E}}) e^{- \\sin{(\\sigma_x)}}}{2} and \\frac{\\partial}{\\partial \\sigma_x} \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}}}{2} = \\frac{\\partial}{\\partial \\sigma_x} \\frac{(b + i^{M_{E}}) e^{- \\sin{(\\sigma_x)}}}{2} and - \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)}}{2} = - (\\frac{b}{2} + \\frac{i^{M_{E}}}{2}) e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)} and \\int - \\frac{T{(b,i,M_{E})} e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)}}{2} db = \\int - (\\frac{b}{2} + \\frac{i^{M_{E}}}{2}) e^{- \\sin{(\\sigma_x)}} \\cos{(\\sigma_x)} db", "srepr_derivation": [["get_premise", "Equality(Function('T')(Symbol('b', commutative=True), Symbol('i', commutative=True), Symbol('M_E', commutative=True)), Add(Symbol('b', commutative=True), Pow(Symbol('i', commutative=True), Symbol('M_E', commutative=True))))"], [["divide", 1, "Mul(Integer(2), exp(sin(Symbol('\\\\sigma_x', commutative=True))))"], "Equality(Mul(Rational(1, 2), Function('T')(Symbol('b', commutative=True), Symbol('i', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))), Mul(Rational(1, 2), Add(Symbol('b', commutative=True), Pow(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))))"], [["differentiate", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Mul(Rational(1, 2), Function('T')(Symbol('b', commutative=True), Symbol('i', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Rational(1, 2), Add(Symbol('b', commutative=True), Pow(Symbol('i', commutative=True), Symbol('M_E', commutative=True))), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True))))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Rational(1, 2), Function('T')(Symbol('b', commutative=True), Symbol('i', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True)))), cos(Symbol('\\\\sigma_x', commutative=True))), Mul(Integer(-1), Add(Mul(Rational(1, 2), Symbol('b', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Symbol('M_E', commutative=True)))), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True)))), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Rational(1, 2), Function('T')(Symbol('b', commutative=True), Symbol('i', commutative=True), Symbol('M_E', commutative=True)), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True)))), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('b', commutative=True))), Integral(Mul(Integer(-1), Add(Mul(Rational(1, 2), Symbol('b', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Symbol('M_E', commutative=True)))), exp(Mul(Integer(-1), sin(Symbol('\\\\sigma_x', commutative=True)))), cos(Symbol('\\\\sigma_x', commutative=True))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mu)} = \\sin{(\\mu)} and \\hat{X}{(\\mu)} = \\mu \\sin{(\\mu)}, then obtain \\mu \\operatorname{P_{e}}{(\\mu)} + \\hat{X}{(\\mu)} = 2 \\hat{X}{(\\mu)}", "derivation": "\\operatorname{P_{e}}{(\\mu)} = \\sin{(\\mu)} and \\mu \\operatorname{P_{e}}{(\\mu)} = \\mu \\sin{(\\mu)} and \\mu \\operatorname{P_{e}}{(\\mu)} + \\mu \\sin{(\\mu)} = 2 \\mu \\sin{(\\mu)} and \\hat{X}{(\\mu)} = \\mu \\sin{(\\mu)} and \\mu \\operatorname{P_{e}}{(\\mu)} + \\hat{X}{(\\mu)} = 2 \\hat{X}{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('\\\\mu', commutative=True)), sin(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('P_e')(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\mu', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Function('P_e')(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\mu', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{X}')(Symbol('\\\\mu', commutative=True)), Mul(Symbol('\\\\mu', commutative=True), sin(Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Symbol('\\\\mu', commutative=True), Function('P_e')(Symbol('\\\\mu', commutative=True))), Function('\\\\hat{X}')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Function('\\\\hat{X}')(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{r_{0}}{(x)} = e^{x} and \\operatorname{J_{\\varepsilon}}{(\\psi^*)} = \\psi^*, then derive \\int x \\operatorname{r_{0}}{(x)} dx = \\psi^* + (x - 1) e^{x}, then obtain \\int x \\operatorname{r_{0}}{(x)} dx = (x - 1) e^{x} + \\operatorname{J_{\\varepsilon}}{(\\psi^*)}", "derivation": "\\operatorname{r_{0}}{(x)} = e^{x} and x \\operatorname{r_{0}}{(x)} = x e^{x} and \\int x \\operatorname{r_{0}}{(x)} dx = \\int x e^{x} dx and \\int x \\operatorname{r_{0}}{(x)} dx = \\psi^* + (x - 1) e^{x} and \\operatorname{J_{\\varepsilon}}{(\\psi^*)} = \\psi^* and \\int x \\operatorname{r_{0}}{(x)} dx = (x - 1) e^{x} + \\operatorname{J_{\\varepsilon}}{(\\psi^*)}", "srepr_derivation": [["premise", "Equality(Function('r_0')(Symbol('x', commutative=True)), exp(Symbol('x', commutative=True)))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('r_0')(Symbol('x', commutative=True))), Mul(Symbol('x', commutative=True), exp(Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('r_0')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('x', commutative=True), exp(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('r_0')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Add(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('x', commutative=True)))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True)), Symbol('\\\\psi^*', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integral(Mul(Symbol('x', commutative=True), Function('r_0')(Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Add(Mul(Add(Symbol('x', commutative=True), Integer(-1)), exp(Symbol('x', commutative=True))), Function('J_{\\\\varepsilon}')(Symbol('\\\\psi^*', commutative=True))))"]]}, {"prompt": "Given v{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\theta)}, then derive \\frac{\\frac{d}{d \\theta} v{(\\theta)}}{\\theta \\cos{(\\theta)}} = - \\frac{\\sin{(\\theta)}}{\\theta \\cos{(\\theta)}}, then obtain (\\frac{\\frac{d^{2}}{d \\theta^{2}} \\sin{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta} = (- \\frac{\\sin{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta}", "derivation": "v{(\\theta)} = \\frac{d}{d \\theta} \\sin{(\\theta)} and \\frac{d}{d \\theta} v{(\\theta)} = \\frac{d^{2}}{d \\theta^{2}} \\sin{(\\theta)} and \\frac{\\frac{d}{d \\theta} v{(\\theta)}}{\\theta \\cos{(\\theta)}} = \\frac{\\frac{d^{2}}{d \\theta^{2}} \\sin{(\\theta)}}{\\theta \\cos{(\\theta)}} and \\frac{\\frac{d}{d \\theta} v{(\\theta)}}{\\theta \\cos{(\\theta)}} = - \\frac{\\sin{(\\theta)}}{\\theta \\cos{(\\theta)}} and (\\frac{\\frac{d}{d \\theta} v{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta} = (- \\frac{\\sin{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta} and (\\frac{\\frac{d^{2}}{d \\theta^{2}} \\sin{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta} = (- \\frac{\\sin{(\\theta)}}{\\theta \\cos{(\\theta)}})^{\\theta}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\theta', commutative=True)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(2))))"], [["divide", 2, "Mul(Symbol('\\\\theta', commutative=True), cos(Symbol('\\\\theta', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(2)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), sin(Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('\\\\theta', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(1)))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), sin(Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1))), Symbol('\\\\theta', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Mul(Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1)), Derivative(sin(Symbol('\\\\theta', commutative=True)), Tuple(Symbol('\\\\theta', commutative=True), Integer(2)))), Symbol('\\\\theta', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\theta', commutative=True), Integer(-1)), sin(Symbol('\\\\theta', commutative=True)), Pow(cos(Symbol('\\\\theta', commutative=True)), Integer(-1))), Symbol('\\\\theta', commutative=True)))"]]}, {"prompt": "Given h{(F_{x},Z)} = \\int (F_{x} + Z) dZ and \\dot{z}{(c)} = \\frac{1}{c}, then obtain \\delta + \\dot{z}{(c)} + \\frac{F_{x} + \\int (F_{x} + Z) dZ}{c} = \\delta + \\frac{F_{x} + \\int (F_{x} + Z) dZ}{c} + \\frac{1}{c}", "derivation": "h{(F_{x},Z)} = \\int (F_{x} + Z) dZ and F_{x} + h{(F_{x},Z)} = F_{x} + \\int (F_{x} + Z) dZ and \\dot{z}{(c)} = \\frac{1}{c} and \\delta + \\dot{z}{(c)} = \\delta + \\frac{1}{c} and \\delta + \\dot{z}{(c)} + \\frac{F_{x} + h{(F_{x},Z)}}{c} = \\delta + \\frac{F_{x} + h{(F_{x},Z)}}{c} + \\frac{1}{c} and \\delta + \\dot{z}{(c)} + \\frac{F_{x} + \\int (F_{x} + Z) dZ}{c} = \\delta + \\frac{F_{x} + \\int (F_{x} + Z) dZ}{c} + \\frac{1}{c}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)), Integral(Add(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))"], [["add", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Symbol('F_x', commutative=True), Function('h')(Symbol('F_x', commutative=True), Symbol('Z', commutative=True))), Add(Symbol('F_x', commutative=True), Integral(Add(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('c', commutative=True)), Pow(Symbol('c', commutative=True), Integer(-1)))"], [["add", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('c', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["minus", 4, "Mul(Integer(-1), Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Function('h')(Symbol('F_x', commutative=True), Symbol('Z', commutative=True))))"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Function('h')(Symbol('F_x', commutative=True), Symbol('Z', commutative=True))))), Add(Symbol('\\\\delta', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Function('h')(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)))), Pow(Symbol('c', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\dot{z}')(Symbol('c', commutative=True)), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Integral(Add(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True)))))), Add(Symbol('\\\\delta', commutative=True), Mul(Pow(Symbol('c', commutative=True), Integer(-1)), Add(Symbol('F_x', commutative=True), Integral(Add(Symbol('F_x', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True))))), Pow(Symbol('c', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} = \\delta - a^{\\dagger}, then obtain \\operatorname{z^{*}}^{2}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} = (\\delta - a^{\\dagger}) \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(a^{\\dagger},\\delta)}", "derivation": "\\operatorname{z^{*}}{(a^{\\dagger},\\delta)} = \\delta - a^{\\dagger} and \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} = \\frac{\\partial}{\\partial \\delta} (\\delta - a^{\\dagger}) and \\operatorname{z^{*}}^{2}{(a^{\\dagger},\\delta)} = (\\delta - a^{\\dagger}) \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} and \\operatorname{z^{*}}^{2}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} (\\delta - a^{\\dagger}) = (\\delta - a^{\\dagger}) \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} (\\delta - a^{\\dagger}) and \\operatorname{z^{*}}^{2}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} = (\\delta - a^{\\dagger}) \\operatorname{z^{*}}{(a^{\\dagger},\\delta)} - \\frac{\\partial}{\\partial \\delta} \\operatorname{z^{*}}{(a^{\\dagger},\\delta)}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["times", 1, "Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True))"], "Equality(Pow(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["minus", 3, "Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Add(Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Derivative(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Pow(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2)), Mul(Integer(-1), Derivative(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))), Add(Mul(Add(Symbol('\\\\delta', commutative=True), Mul(Integer(-1), Symbol('a^{\\\\dagger}', commutative=True))), Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Derivative(Function('z^*')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))))"]]}, {"prompt": "Given f{(C_{1})} = e^{e^{C_{1}}}, then obtain (f{(C_{1})} + e^{e^{C_{1}}}) (f{(C_{1})} + 2 e^{e^{C_{1}}}) = 3 (f{(C_{1})} + e^{e^{C_{1}}}) e^{e^{C_{1}}}", "derivation": "f{(C_{1})} = e^{e^{C_{1}}} and f{(C_{1})} + e^{e^{C_{1}}} = 2 e^{e^{C_{1}}} and f{(C_{1})} + 2 e^{e^{C_{1}}} = 3 e^{e^{C_{1}}} and 2 f{(C_{1})} + e^{e^{C_{1}}} = 3 e^{e^{C_{1}}} and (f{(C_{1})} + e^{e^{C_{1}}}) (2 f{(C_{1})} + e^{e^{C_{1}}}) = 3 (f{(C_{1})} + e^{e^{C_{1}}}) e^{e^{C_{1}}} and f{(C_{1})} + 2 e^{e^{C_{1}}} = 2 f{(C_{1})} + e^{e^{C_{1}}} and (f{(C_{1})} + e^{e^{C_{1}}}) (f{(C_{1})} + 2 e^{e^{C_{1}}}) = 3 (f{(C_{1})} + e^{e^{C_{1}}}) e^{e^{C_{1}}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True))))"], [["add", 1, "exp(exp(Symbol('C_1', commutative=True)))"], "Equality(Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True)))))"], [["add", 2, "exp(exp(Symbol('C_1', commutative=True)))"], "Equality(Add(Function('f')(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True))))), Mul(Integer(3), exp(exp(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('f')(Symbol('C_1', commutative=True))), exp(exp(Symbol('C_1', commutative=True)))), Mul(Integer(3), exp(exp(Symbol('C_1', commutative=True)))))"], [["times", 4, "Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True))))"], "Equality(Mul(Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))), Add(Mul(Integer(2), Function('f')(Symbol('C_1', commutative=True))), exp(exp(Symbol('C_1', commutative=True))))), Mul(Integer(3), Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))), exp(exp(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('f')(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True))))), Add(Mul(Integer(2), Function('f')(Symbol('C_1', commutative=True))), exp(exp(Symbol('C_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))), Add(Function('f')(Symbol('C_1', commutative=True)), Mul(Integer(2), exp(exp(Symbol('C_1', commutative=True)))))), Mul(Integer(3), Add(Function('f')(Symbol('C_1', commutative=True)), exp(exp(Symbol('C_1', commutative=True)))), exp(exp(Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{p})} = \\sin{(\\mathbf{p})}, then obtain \\theta_{2}^{6}{(\\mathbf{p})} \\sin^{2}{(\\mathbf{p})} = \\theta_{2}^{5}{(\\mathbf{p})} \\sin^{3}{(\\mathbf{p})}", "derivation": "\\theta_{2}{(\\mathbf{p})} = \\sin{(\\mathbf{p})} and \\theta_{2}{(\\mathbf{p})} \\sin{(\\mathbf{p})} = \\sin^{2}{(\\mathbf{p})} and \\theta_{2}^{2}{(\\mathbf{p})} \\sin^{2}{(\\mathbf{p})} = \\sin^{4}{(\\mathbf{p})} and \\theta_{2}^{3}{(\\mathbf{p})} \\sin{(\\mathbf{p})} = \\theta_{2}^{2}{(\\mathbf{p})} \\sin^{2}{(\\mathbf{p})} and \\theta_{2}^{6}{(\\mathbf{p})} \\sin^{2}{(\\mathbf{p})} = \\theta_{2}^{5}{(\\mathbf{p})} \\sin^{3}{(\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], [["times", 1, "sin(Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), sin(Symbol('\\\\mathbf{p}', commutative=True))), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)))"], [["power", 2, 2], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(4)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{p}', commutative=True))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))))"], [["times", 4, "Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(3)), sin(Symbol('\\\\mathbf{p}', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(6)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(2))), Mul(Pow(Function('\\\\theta_2')(Symbol('\\\\mathbf{p}', commutative=True)), Integer(5)), Pow(sin(Symbol('\\\\mathbf{p}', commutative=True)), Integer(3))))"]]}, {"prompt": "Given \\mathbf{s}{(\\mathbf{B},z)} = \\mathbf{B} - z, then obtain \\frac{- \\mathbf{B} + 2 z}{\\mathbf{B}} + \\frac{- z + \\mathbf{s}{(\\mathbf{B},z)}}{\\mathbf{B}} = 0", "derivation": "\\mathbf{s}{(\\mathbf{B},z)} = \\mathbf{B} - z and - z + \\mathbf{s}{(\\mathbf{B},z)} = \\mathbf{B} - 2 z and \\frac{- z + \\mathbf{s}{(\\mathbf{B},z)}}{\\mathbf{B}} = \\frac{\\mathbf{B} - 2 z}{\\mathbf{B}} and - \\frac{\\mathbf{B} - 2 z}{\\mathbf{B}} + \\frac{- z + \\mathbf{s}{(\\mathbf{B},z)}}{\\mathbf{B}} = 0 and \\frac{- \\mathbf{B} + 2 z}{\\mathbf{B}} + \\frac{- z + \\mathbf{s}{(\\mathbf{B},z)}}{\\mathbf{B}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('z', commutative=True)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Symbol('z', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('z', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('z', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))))"], [["divide", 2, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)))))"], [["minus", 3, "Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Integer(-1), Integer(2), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('z', commutative=True))))), Integer(0))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True)))), Mul(Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('z', commutative=True)), Function('\\\\mathbf{s}')(Symbol('\\\\mathbf{B}', commutative=True), Symbol('z', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}{(r_{0})} = e^{r_{0}} and \\operatorname{F_{g}}{(r_{0})} = e^{r_{0}}, then obtain \\frac{2 \\operatorname{F_{g}}{(r_{0})}}{\\cos{(e^{r_{0}})}} = \\frac{\\operatorname{F_{g}}{(r_{0})} + e^{r_{0}}}{\\cos{(e^{r_{0}})}}", "derivation": "\\tilde{g}{(r_{0})} = e^{r_{0}} and \\operatorname{F_{g}}{(r_{0})} = e^{r_{0}} and 2 \\operatorname{F_{g}}{(r_{0})} = \\operatorname{F_{g}}{(r_{0})} + e^{r_{0}} and \\frac{2 \\operatorname{F_{g}}{(r_{0})}}{\\cos{(\\tilde{g}{(r_{0})})}} = \\frac{\\operatorname{F_{g}}{(r_{0})} + e^{r_{0}}}{\\cos{(\\tilde{g}{(r_{0})})}} and \\frac{2 \\operatorname{F_{g}}{(r_{0})}}{\\cos{(e^{r_{0}})}} = \\frac{\\operatorname{F_{g}}{(r_{0})} + e^{r_{0}}}{\\cos{(e^{r_{0}})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True)))"], [["add", 2, "Function('F_g')(Symbol('r_0', commutative=True))"], "Equality(Mul(Integer(2), Function('F_g')(Symbol('r_0', commutative=True))), Add(Function('F_g')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))))"], [["divide", 3, "cos(Function('\\\\tilde{g}')(Symbol('r_0', commutative=True)))"], "Equality(Mul(Integer(2), Function('F_g')(Symbol('r_0', commutative=True)), Pow(cos(Function('\\\\tilde{g}')(Symbol('r_0', commutative=True))), Integer(-1))), Mul(Add(Function('F_g')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))), Pow(cos(Function('\\\\tilde{g}')(Symbol('r_0', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Function('F_g')(Symbol('r_0', commutative=True)), Pow(cos(exp(Symbol('r_0', commutative=True))), Integer(-1))), Mul(Add(Function('F_g')(Symbol('r_0', commutative=True)), exp(Symbol('r_0', commutative=True))), Pow(cos(exp(Symbol('r_0', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\Psi_{nl}{(\\psi^*)} = \\sin{(\\psi^*)}, then obtain (\\int (\\Psi_{nl}{(\\psi^*)} + \\sin{(\\psi^*)}) d\\psi^*)^{2} = (\\int 2 \\sin{(\\psi^*)} d\\psi^*)^{2}", "derivation": "\\Psi_{nl}{(\\psi^*)} = \\sin{(\\psi^*)} and \\Psi_{nl}{(\\psi^*)} + \\sin{(\\psi^*)} = 2 \\sin{(\\psi^*)} and \\int (\\Psi_{nl}{(\\psi^*)} + \\sin{(\\psi^*)}) d\\psi^* = \\int 2 \\sin{(\\psi^*)} d\\psi^* and (\\int (\\Psi_{nl}{(\\psi^*)} + \\sin{(\\psi^*)}) d\\psi^*)^{2} = (\\int 2 \\sin{(\\psi^*)} d\\psi^*)^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True)))"], [["add", 1, "sin(Symbol('\\\\psi^*', commutative=True))"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Integral(Add(Function('\\\\Psi_{nl}')(Symbol('\\\\psi^*', commutative=True)), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(2)), Pow(Integral(Mul(Integer(2), sin(Symbol('\\\\psi^*', commutative=True))), Tuple(Symbol('\\\\psi^*', commutative=True))), Integer(2)))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(m_{s})} = \\frac{d}{d m_{s}} e^{m_{s}}, then derive \\operatorname{c_{0}}{(m_{s})} = e^{m_{s}}, then obtain e^{\\frac{\\partial}{\\partial a} a e^{m_{s}}} = e^{\\frac{\\partial}{\\partial a} a \\frac{d}{d m_{s}} e^{m_{s}}}", "derivation": "\\operatorname{c_{0}}{(m_{s})} = \\frac{d}{d m_{s}} e^{m_{s}} and a \\operatorname{c_{0}}{(m_{s})} = a \\frac{d}{d m_{s}} e^{m_{s}} and \\frac{\\partial}{\\partial a} a \\operatorname{c_{0}}{(m_{s})} = \\frac{\\partial}{\\partial a} a \\frac{d}{d m_{s}} e^{m_{s}} and \\operatorname{c_{0}}{(m_{s})} = e^{m_{s}} and \\frac{\\partial}{\\partial a} a \\operatorname{c_{0}}{(m_{s})} = \\frac{\\partial}{\\partial a} a \\frac{d}{d m_{s}} \\operatorname{c_{0}}{(m_{s})} and \\frac{\\partial}{\\partial a} a e^{m_{s}} = \\frac{\\partial}{\\partial a} a \\frac{d}{d m_{s}} e^{m_{s}} and e^{\\frac{\\partial}{\\partial a} a e^{m_{s}}} = e^{\\frac{\\partial}{\\partial a} a \\frac{d}{d m_{s}} e^{m_{s}}}", "srepr_derivation": [["get_premise", "Equality(Function('c_0')(Symbol('m_s', commutative=True)), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["times", 1, "Symbol('a', commutative=True)"], "Equality(Mul(Symbol('a', commutative=True), Function('c_0')(Symbol('m_s', commutative=True))), Mul(Symbol('a', commutative=True), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('a', commutative=True)"], "Equality(Derivative(Mul(Symbol('a', commutative=True), Function('c_0')(Symbol('m_s', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('c_0')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Mul(Symbol('a', commutative=True), Function('c_0')(Symbol('m_s', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Derivative(Function('c_0')(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Derivative(Mul(Symbol('a', commutative=True), exp(Symbol('m_s', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Mul(Symbol('a', commutative=True), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["exp", 6], "Equality(exp(Derivative(Mul(Symbol('a', commutative=True), exp(Symbol('m_s', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))), exp(Derivative(Mul(Symbol('a', commutative=True), Derivative(exp(Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given r{(C_{2})} = \\log{(C_{2})}, then obtain (\\frac{d}{d C_{2}} C_{2} r{(C_{2})})^{C_{2}} = (\\frac{d}{d C_{2}} C_{2} \\log{(C_{2})})^{C_{2}}", "derivation": "r{(C_{2})} = \\log{(C_{2})} and C_{2} r{(C_{2})} = C_{2} \\log{(C_{2})} and \\frac{d}{d C_{2}} C_{2} r{(C_{2})} = \\frac{d}{d C_{2}} C_{2} \\log{(C_{2})} and (\\frac{d}{d C_{2}} C_{2} r{(C_{2})})^{C_{2}} = (\\frac{d}{d C_{2}} C_{2} \\log{(C_{2})})^{C_{2}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C_2', commutative=True)), log(Symbol('C_2', commutative=True)))"], [["times", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Function('r')(Symbol('C_2', commutative=True))), Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Symbol('C_2', commutative=True), Function('r')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["power", 3, "Symbol('C_2', commutative=True)"], "Equality(Pow(Derivative(Mul(Symbol('C_2', commutative=True), Function('r')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)), Pow(Derivative(Mul(Symbol('C_2', commutative=True), log(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Symbol('C_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(T)} = e^{T}, then obtain 0 = (B{(\\psi)} + \\operatorname{F_{x}}{(T)}) (- \\operatorname{F_{x}}{(T)} + e^{T})^{2}", "derivation": "\\operatorname{F_{x}}{(T)} = e^{T} and 0 = - \\operatorname{F_{x}}{(T)} + e^{T} and 0 = (- \\operatorname{F_{x}}{(T)} + e^{T})^{2} and 0 = (B{(\\psi)} + \\operatorname{F_{x}}{(T)}) (- \\operatorname{F_{x}}{(T)} + e^{T})^{2}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["minus", 1, "Function('F_x')(Symbol('T', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('F_x')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Function('F_x')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True)))"], "Equality(Integer(0), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True))), Integer(2)))"], [["times", 3, "Add(Function('B')(Symbol('\\\\psi', commutative=True)), Function('F_x')(Symbol('T', commutative=True)))"], "Equality(Integer(0), Mul(Add(Function('B')(Symbol('\\\\psi', commutative=True)), Function('F_x')(Symbol('T', commutative=True))), Pow(Add(Mul(Integer(-1), Function('F_x')(Symbol('T', commutative=True))), exp(Symbol('T', commutative=True))), Integer(2))))"]]}, {"prompt": "Given S{(\\hat{H}_l,\\mathbf{P})} = - \\hat{H}_l + \\mathbf{P}, then obtain 2 \\mathbf{P} (- \\hat{H}_l + \\mathbf{P}) = \\frac{\\mathbf{P} ((- \\hat{H}_l + \\mathbf{P})^{2} + (- \\hat{H}_l + \\mathbf{P}) S{(\\hat{H}_l,\\mathbf{P})})}{S{(\\hat{H}_l,\\mathbf{P})}}", "derivation": "S{(\\hat{H}_l,\\mathbf{P})} = - \\hat{H}_l + \\mathbf{P} and (- \\hat{H}_l + \\mathbf{P}) S{(\\hat{H}_l,\\mathbf{P})} = (- \\hat{H}_l + \\mathbf{P})^{2} and 2 (- \\hat{H}_l + \\mathbf{P}) S{(\\hat{H}_l,\\mathbf{P})} = (- \\hat{H}_l + \\mathbf{P})^{2} + (- \\hat{H}_l + \\mathbf{P}) S{(\\hat{H}_l,\\mathbf{P})} and 2 \\mathbf{P} (- \\hat{H}_l + \\mathbf{P}) = \\frac{\\mathbf{P} ((- \\hat{H}_l + \\mathbf{P})^{2} + (- \\hat{H}_l + \\mathbf{P}) S{(\\hat{H}_l,\\mathbf{P})})}{S{(\\hat{H}_l,\\mathbf{P})}}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)))"], [["times", 1, "Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)))"], [["add", 2, "Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True))), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\mathbf{P}', commutative=True), Integer(-1)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))"], "Equality(Mul(Integer(2), Symbol('\\\\mathbf{P}', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True))), Mul(Symbol('\\\\mathbf{P}', commutative=True), Add(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Integer(2)), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_l', commutative=True)), Symbol('\\\\mathbf{P}', commutative=True)), Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)))), Pow(Function('S')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{P}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{D}{(\\psi,E_{\\lambda},Q)} = - E_{\\lambda} + \\psi^{Q}, then obtain (\\int (- m + \\mathbf{D}^{\\psi}{(\\psi,E_{\\lambda},Q)}) dm)^{m} = (\\int (- m + (- E_{\\lambda} + \\psi^{Q})^{\\psi}) dm)^{m}", "derivation": "\\mathbf{D}{(\\psi,E_{\\lambda},Q)} = - E_{\\lambda} + \\psi^{Q} and \\mathbf{D}^{\\psi}{(\\psi,E_{\\lambda},Q)} = (- E_{\\lambda} + \\psi^{Q})^{\\psi} and - m + \\mathbf{D}^{\\psi}{(\\psi,E_{\\lambda},Q)} = - m + (- E_{\\lambda} + \\psi^{Q})^{\\psi} and \\int (- m + \\mathbf{D}^{\\psi}{(\\psi,E_{\\lambda},Q)}) dm = \\int (- m + (- E_{\\lambda} + \\psi^{Q})^{\\psi}) dm and (\\int (- m + \\mathbf{D}^{\\psi}{(\\psi,E_{\\lambda},Q)}) dm)^{m} = (\\int (- m + (- E_{\\lambda} + \\psi^{Q})^{\\psi}) dm)^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))))"], [["power", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\psi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\psi', commutative=True)))"], [["add", 2, "Mul(Integer(-1), Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\psi', commutative=True))), Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\psi', commutative=True))))"], [["integrate", 3, "Symbol('m', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('m', commutative=True))))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('Q', commutative=True)), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Symbol('\\\\psi', commutative=True), Symbol('Q', commutative=True))), Symbol('\\\\psi', commutative=True))), Tuple(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given G{(s)} = \\sin{(s)}, then obtain \\frac{d}{d s} (- G{(s)} + \\frac{d}{d s} (G{(s)} - \\sin{(s)})) = \\frac{d}{d s} (- G{(s)} + \\frac{d}{d s} 0)", "derivation": "G{(s)} = \\sin{(s)} and G{(s)} - \\sin{(s)} = 0 and \\frac{d}{d s} (G{(s)} - \\sin{(s)}) = \\frac{d}{d s} 0 and - G{(s)} + \\frac{d}{d s} (G{(s)} - \\sin{(s)}) = - G{(s)} + \\frac{d}{d s} 0 and \\frac{d}{d s} (- G{(s)} + \\frac{d}{d s} (G{(s)} - \\sin{(s)})) = \\frac{d}{d s} (- G{(s)} + \\frac{d}{d s} 0)", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["minus", 1, "sin(Symbol('s', commutative=True))"], "Equality(Add(Function('G')(Symbol('s', commutative=True)), Mul(Integer(-1), sin(Symbol('s', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Function('G')(Symbol('s', commutative=True)), Mul(Integer(-1), sin(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('s', commutative=True), Integer(1))))"], [["add", 3, "Mul(Integer(-1), Function('G')(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('G')(Symbol('s', commutative=True))), Derivative(Add(Function('G')(Symbol('s', commutative=True)), Mul(Integer(-1), sin(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('G')(Symbol('s', commutative=True))), Derivative(Integer(0), Tuple(Symbol('s', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('s', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('s', commutative=True))), Derivative(Add(Function('G')(Symbol('s', commutative=True)), Mul(Integer(-1), sin(Symbol('s', commutative=True)))), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('G')(Symbol('s', commutative=True))), Derivative(Integer(0), Tuple(Symbol('s', commutative=True), Integer(1)))), Tuple(Symbol('s', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(F_{N},T)} = T^{F_{N}}, then derive \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = T^{F_{N}} \\log{(T)}, then obtain - \\log{(T)} + \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = v{(F_{N},T)} \\log{(T)} - \\log{(T)}", "derivation": "v{(F_{N},T)} = T^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = \\frac{\\partial}{\\partial F_{N}} T^{F_{N}} and \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = T^{F_{N}} \\log{(T)} and - \\log{(T)} + \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = T^{F_{N}} \\log{(T)} - \\log{(T)} and - \\log{(T)} + \\frac{\\partial}{\\partial F_{N}} v{(F_{N},T)} = v{(F_{N},T)} \\log{(T)} - \\log{(T)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('T', commutative=True), Symbol('F_N', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Mul(Pow(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), log(Symbol('T', commutative=True))))"], [["minus", 3, "log(Symbol('T', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Derivative(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('T', commutative=True), Symbol('F_N', commutative=True)), log(Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Mul(Integer(-1), log(Symbol('T', commutative=True))), Derivative(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1)))), Add(Mul(Function('v')(Symbol('F_N', commutative=True), Symbol('T', commutative=True)), log(Symbol('T', commutative=True))), Mul(Integer(-1), log(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(g,g_{\\varepsilon})} = \\frac{\\log{(g)}}{g_{\\varepsilon}} and \\hat{H}_l{(g,g_{\\varepsilon})} = - \\frac{\\log{(g)}}{g_{\\varepsilon}}, then obtain \\hat{H}_l{(g,g_{\\varepsilon})} \\int - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} dg = - \\frac{\\log{(g)} \\int - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} dg}{g_{\\varepsilon}}", "derivation": "\\Psi_{\\lambda}{(g,g_{\\varepsilon})} = \\frac{\\log{(g)}}{g_{\\varepsilon}} and - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} = - \\frac{\\log{(g)}}{g_{\\varepsilon}} and \\hat{H}_l{(g,g_{\\varepsilon})} = - \\frac{\\log{(g)}}{g_{\\varepsilon}} and \\int - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} dg = \\int - \\frac{\\log{(g)}}{g_{\\varepsilon}} dg and \\hat{H}_l{(g,g_{\\varepsilon})} \\int - \\frac{\\log{(g)}}{g_{\\varepsilon}} dg = - \\frac{\\log{(g)} \\int - \\frac{\\log{(g)}}{g_{\\varepsilon}} dg}{g_{\\varepsilon}} and \\hat{H}_l{(g,g_{\\varepsilon})} \\int - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} dg = - \\frac{\\log{(g)} \\int - \\Psi_{\\lambda}{(g,g_{\\varepsilon})} dg}{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))))"], [["integrate", 2, "Symbol('g', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g', commutative=True))), Integral(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True))))"], [["times", 3, "Integral(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)), Integral(Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g', commutative=True)))), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('g', commutative=True)), Integral(Mul(Integer(-1), Function('\\\\Psi_{\\\\lambda}')(Symbol('g', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g', commutative=True)))))"]]}, {"prompt": "Given y{(\\sigma_x,\\mathbf{A})} = \\sin{(\\sigma_x^{\\mathbf{A}})}, then obtain (\\frac{\\partial}{\\partial \\mathbf{A}} (y^{\\mathbf{A}}{(\\sigma_x,\\mathbf{A})})^{\\sigma_x})^{\\sigma_x} = (\\frac{\\partial}{\\partial \\mathbf{A}} (\\sin^{\\mathbf{A}}{(\\sigma_x^{\\mathbf{A}})})^{\\sigma_x})^{\\sigma_x}", "derivation": "y{(\\sigma_x,\\mathbf{A})} = \\sin{(\\sigma_x^{\\mathbf{A}})} and y^{\\mathbf{A}}{(\\sigma_x,\\mathbf{A})} = \\sin^{\\mathbf{A}}{(\\sigma_x^{\\mathbf{A}})} and (y^{\\mathbf{A}}{(\\sigma_x,\\mathbf{A})})^{\\sigma_x} = (\\sin^{\\mathbf{A}}{(\\sigma_x^{\\mathbf{A}})})^{\\sigma_x} and \\frac{\\partial}{\\partial \\mathbf{A}} (y^{\\mathbf{A}}{(\\sigma_x,\\mathbf{A})})^{\\sigma_x} = \\frac{\\partial}{\\partial \\mathbf{A}} (\\sin^{\\mathbf{A}}{(\\sigma_x^{\\mathbf{A}})})^{\\sigma_x} and (\\frac{\\partial}{\\partial \\mathbf{A}} (y^{\\mathbf{A}}{(\\sigma_x,\\mathbf{A})})^{\\sigma_x})^{\\sigma_x} = (\\frac{\\partial}{\\partial \\mathbf{A}} (\\sin^{\\mathbf{A}}{(\\sigma_x^{\\mathbf{A}})})^{\\sigma_x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Function('y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Pow(Function('y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Pow(Pow(Function('y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Pow(Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Derivative(Pow(Pow(Function('y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)), Pow(Derivative(Pow(Pow(sin(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given T{(\\varphi)} = \\cos{(\\varphi)}, then derive \\frac{d}{d \\varphi} T{(\\varphi)} = - \\sin{(\\varphi)}, then obtain \\frac{d}{d \\varphi} (\\varphi - 2 T{(\\varphi)} + \\frac{d}{d \\varphi} T{(\\varphi)}) = \\frac{d}{d \\varphi} (\\varphi - 2 T{(\\varphi)} - \\sin{(\\varphi)})", "derivation": "T{(\\varphi)} = \\cos{(\\varphi)} and \\frac{d}{d \\varphi} T{(\\varphi)} = \\frac{d}{d \\varphi} \\cos{(\\varphi)} and \\frac{d}{d \\varphi} T{(\\varphi)} = - \\sin{(\\varphi)} and \\frac{d}{d \\varphi} \\cos{(\\varphi)} = - \\sin{(\\varphi)} and \\varphi - 2 T{(\\varphi)} + \\frac{d}{d \\varphi} \\cos{(\\varphi)} = \\varphi - 2 T{(\\varphi)} - \\sin{(\\varphi)} and \\varphi - 2 T{(\\varphi)} + \\frac{d}{d \\varphi} T{(\\varphi)} = \\varphi - 2 T{(\\varphi)} - \\sin{(\\varphi)} and \\frac{d}{d \\varphi} (\\varphi - 2 T{(\\varphi)} + \\frac{d}{d \\varphi} T{(\\varphi)}) = \\frac{d}{d \\varphi} (\\varphi - 2 T{(\\varphi)} - \\sin{(\\varphi)})", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\varphi', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('T')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('T')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True))))"], [["minus", 4, "Add(Mul(Integer(-1), Symbol('\\\\varphi', commutative=True)), Mul(Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))))"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Derivative(cos(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Derivative(Function('T')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True)))))"], [["differentiate", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Derivative(Function('T')(Symbol('\\\\varphi', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\varphi', commutative=True), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\varphi', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(u)} = \\sin{(u)}, then derive 0 = \\frac{\\partial}{\\partial u} (M - \\cos{(u)}) - \\frac{d}{d u} \\int \\operatorname{v_{1}}{(u)} du, then obtain 0 = \\sin{(u)} - \\frac{d}{d u} \\int \\sin{(u)} du", "derivation": "\\operatorname{v_{1}}{(u)} = \\sin{(u)} and \\int \\operatorname{v_{1}}{(u)} du = \\int \\sin{(u)} du and \\frac{d}{d u} \\int \\operatorname{v_{1}}{(u)} du = \\frac{d}{d u} \\int \\sin{(u)} du and 0 = - \\frac{d}{d u} \\int \\operatorname{v_{1}}{(u)} du + \\frac{d}{d u} \\int \\sin{(u)} du and 0 = \\frac{\\partial}{\\partial u} (M - \\cos{(u)}) - \\frac{d}{d u} \\int \\operatorname{v_{1}}{(u)} du and 0 = \\frac{\\partial}{\\partial u} (M - \\cos{(u)}) - \\frac{d}{d u} \\int \\sin{(u)} du and 0 = \\sin{(u)} - \\frac{d}{d u} \\int \\sin{(u)} du", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('u', commutative=True)), sin(Symbol('u', commutative=True)))"], [["integrate", 1, "Symbol('u', commutative=True)"], "Equality(Integral(Function('v_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))))"], [["differentiate", 2, "Symbol('u', commutative=True)"], "Equality(Derivative(Integral(Function('v_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["minus", 3, "Derivative(Integral(Function('v_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Derivative(Integral(Function('v_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))), Derivative(Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1)))))"], [["evaluate_integrals", 4], "Equality(Integer(0), Add(Derivative(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(Function('v_1')(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Integer(0), Add(Derivative(Add(Symbol('M', commutative=True), Mul(Integer(-1), cos(Symbol('u', commutative=True)))), Tuple(Symbol('u', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 6], "Equality(Integer(0), Add(sin(Symbol('u', commutative=True)), Mul(Integer(-1), Derivative(Integral(sin(Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True))), Tuple(Symbol('u', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\hat{p}_0{(\\mathbf{M})} = \\int \\log{(\\mathbf{M})} d\\mathbf{M}, then derive 2 \\hat{p}_0{(\\mathbf{M})} = B + \\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + \\hat{p}_0{(\\mathbf{M})}, then obtain \\frac{d}{d B} 4 \\hat{p}_0^{2}{(\\mathbf{M})} = \\frac{\\partial}{\\partial B} (B + \\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + \\hat{p}_0{(\\mathbf{M})})^{2}", "derivation": "\\hat{p}_0{(\\mathbf{M})} = \\int \\log{(\\mathbf{M})} d\\mathbf{M} and 2 \\hat{p}_0{(\\mathbf{M})} = \\hat{p}_0{(\\mathbf{M})} + \\int \\log{(\\mathbf{M})} d\\mathbf{M} and 2 \\hat{p}_0{(\\mathbf{M})} = B + \\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + \\hat{p}_0{(\\mathbf{M})} and 4 \\hat{p}_0^{2}{(\\mathbf{M})} = (B + \\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + \\hat{p}_0{(\\mathbf{M})})^{2} and \\frac{d}{d B} 4 \\hat{p}_0^{2}{(\\mathbf{M})} = \\frac{\\partial}{\\partial B} (B + \\mathbf{M} \\log{(\\mathbf{M})} - \\mathbf{M} + \\hat{p}_0{(\\mathbf{M})})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True)), Integral(log(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["add", 1, "Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True)), Integral(log(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))), Add(Symbol('B', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))))"], [["power", 3, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Pow(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2)))"], [["differentiate", 4, "Symbol('B', commutative=True)"], "Equality(Derivative(Mul(Integer(4), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True)), Integer(2))), Tuple(Symbol('B', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('B', commutative=True), Mul(Symbol('\\\\mathbf{M}', commutative=True), log(Symbol('\\\\mathbf{M}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Function('\\\\hat{p}_0')(Symbol('\\\\mathbf{M}', commutative=True))), Integer(2)), Tuple(Symbol('B', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(s)} = \\sin{(s)} and \\psi^{*}{(s)} = \\int \\sin{(s)} ds, then derive (f^{\\prime} - \\cos{(s)})^{s} = (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s}, then obtain (f^{\\prime} - \\cos{(s)})^{s} - \\sin{(s)} = - \\sin{(s)} + (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s}", "derivation": "\\hat{H}_{\\lambda}{(s)} = \\sin{(s)} and \\int \\hat{H}_{\\lambda}{(s)} ds = \\int \\sin{(s)} ds and \\psi^{*}{(s)} = \\int \\sin{(s)} ds and \\psi^{*}^{s}{(s)} = (\\int \\sin{(s)} ds)^{s} and \\psi^{*}^{s}{(s)} = (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s} and (\\int \\sin{(s)} ds)^{s} = (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s} and (f^{\\prime} - \\cos{(s)})^{s} = (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s} and (f^{\\prime} - \\cos{(s)})^{s} - \\sin{(s)} = - \\sin{(s)} + (\\int \\hat{H}_{\\lambda}{(s)} ds)^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\psi^*')(Symbol('s', commutative=True)), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["power", 3, "Symbol('s', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('\\\\psi^*')(Symbol('s', commutative=True)), Symbol('s', commutative=True)), Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Pow(Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)), Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["minus", 7, "sin(Symbol('s', commutative=True))"], "Equality(Add(Pow(Add(Symbol('f^{\\\\prime}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Symbol('s', commutative=True)), Mul(Integer(-1), sin(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('s', commutative=True))), Pow(Integral(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\hbar,F_{N})} = F_{N} + \\hbar, then obtain \\frac{\\frac{\\partial}{\\partial \\hbar} \\rho_{b}{(\\hbar,F_{N})} - 1}{F_{N} + \\hbar} = 0", "derivation": "\\rho_{b}{(\\hbar,F_{N})} = F_{N} + \\hbar and - F_{N} - \\hbar + \\rho_{b}{(\\hbar,F_{N})} = 0 and \\frac{\\partial}{\\partial \\hbar} (- F_{N} - \\hbar + \\rho_{b}{(\\hbar,F_{N})}) = \\frac{d}{d \\hbar} 0 and \\frac{\\frac{\\partial}{\\partial \\hbar} (- F_{N} - \\hbar + \\rho_{b}{(\\hbar,F_{N})})}{F_{N} + \\hbar} = \\frac{\\frac{d}{d \\hbar} 0}{F_{N} + \\hbar} and \\frac{\\frac{\\partial}{\\partial \\hbar} \\rho_{b}{(\\hbar,F_{N})} - 1}{F_{N} + \\hbar} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\hbar', commutative=True), Symbol('F_N', commutative=True)), Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hbar', commutative=True), Symbol('F_N', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hbar', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["divide", 3, "Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Derivative(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\rho_b')(Symbol('\\\\hbar', commutative=True), Symbol('F_N', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(Symbol('F_N', commutative=True), Symbol('\\\\hbar', commutative=True)), Integer(-1)), Add(Derivative(Function('\\\\rho_b')(Symbol('\\\\hbar', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Integer(-1))), Integer(0))"]]}, {"prompt": "Given y{(A,L)} = L + \\cos{(A)}, then obtain \\frac{\\partial}{\\partial L} (y{(A,L)} - \\cos{(A)} + \\int L dL)^{L} = \\frac{d}{d L} (L + \\int L dL)^{L}", "derivation": "y{(A,L)} = L + \\cos{(A)} and y{(A,L)} - \\cos{(A)} = L and y{(A,L)} - \\cos{(A)} + \\int L dL = L + \\int L dL and (y{(A,L)} - \\cos{(A)} + \\int L dL)^{L} = (L + \\int L dL)^{L} and \\frac{\\partial}{\\partial L} (y{(A,L)} - \\cos{(A)} + \\int L dL)^{L} = \\frac{d}{d L} (L + \\int L dL)^{L}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Add(Symbol('L', commutative=True), cos(Symbol('A', commutative=True))))"], [["minus", 1, "cos(Symbol('A', commutative=True))"], "Equality(Add(Function('y')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('A', commutative=True)))), Symbol('L', commutative=True))"], [["add", 2, "Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))"], "Equality(Add(Function('y')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('A', commutative=True))), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))), Add(Symbol('L', commutative=True), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))))"], [["power", 3, "Symbol('L', commutative=True)"], "Equality(Pow(Add(Function('y')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('A', commutative=True))), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))), Symbol('L', commutative=True)), Pow(Add(Symbol('L', commutative=True), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))), Symbol('L', commutative=True)))"], [["differentiate", 4, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Add(Function('y')(Symbol('A', commutative=True), Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('A', commutative=True))), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('L', commutative=True), Integral(Symbol('L', commutative=True), Tuple(Symbol('L', commutative=True)))), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(r)} = \\sin{(r)}, then obtain \\int \\cos{((\\Omega^{2}{(r)})^{r})} dr = \\int \\cos{((\\Omega{(r)} \\sin{(r)})^{r})} dr", "derivation": "\\Omega{(r)} = \\sin{(r)} and \\Omega^{2}{(r)} = \\Omega{(r)} \\sin{(r)} and (\\Omega^{2}{(r)})^{r} = (\\Omega{(r)} \\sin{(r)})^{r} and \\cos{((\\Omega^{2}{(r)})^{r})} = \\cos{((\\Omega{(r)} \\sin{(r)})^{r})} and \\int \\cos{((\\Omega^{2}{(r)})^{r})} dr = \\int \\cos{((\\Omega{(r)} \\sin{(r)})^{r})} dr", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True)))"], [["times", 1, "Function('\\\\Omega')(Symbol('r', commutative=True))"], "Equality(Pow(Function('\\\\Omega')(Symbol('r', commutative=True)), Integer(2)), Mul(Function('\\\\Omega')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))))"], [["power", 2, "Symbol('r', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Omega')(Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True)), Pow(Mul(Function('\\\\Omega')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["cos", 3], "Equality(cos(Pow(Pow(Function('\\\\Omega')(Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True))), cos(Pow(Mul(Function('\\\\Omega')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))))"], [["integrate", 4, "Symbol('r', commutative=True)"], "Equality(Integral(cos(Pow(Pow(Function('\\\\Omega')(Symbol('r', commutative=True)), Integer(2)), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))), Integral(cos(Pow(Mul(Function('\\\\Omega')(Symbol('r', commutative=True)), sin(Symbol('r', commutative=True))), Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\rho_b)} = e^{\\rho_b}, then derive \\int \\operatorname{E_{x}}{(\\rho_b)} d\\rho_b = \\phi_1 + e^{\\rho_b}, then derive q + e^{\\rho_b} = \\phi_1 + e^{\\rho_b}, then obtain e^{q + e^{\\rho_b}} \\int e^{\\rho_b} d\\rho_b = (e^{\\int e^{\\rho_b} d\\rho_b}) \\int e^{\\rho_b} d\\rho_b", "derivation": "\\operatorname{E_{x}}{(\\rho_b)} = e^{\\rho_b} and \\int \\operatorname{E_{x}}{(\\rho_b)} d\\rho_b = \\int e^{\\rho_b} d\\rho_b and \\int \\operatorname{E_{x}}{(\\rho_b)} d\\rho_b = \\phi_1 + e^{\\rho_b} and \\int e^{\\rho_b} d\\rho_b = \\phi_1 + e^{\\rho_b} and q + e^{\\rho_b} = \\phi_1 + e^{\\rho_b} and e^{q + e^{\\rho_b}} = e^{\\phi_1 + e^{\\rho_b}} and e^{q + e^{\\rho_b}} = e^{\\int e^{\\rho_b} d\\rho_b} and e^{q + e^{\\rho_b}} \\int e^{\\rho_b} d\\rho_b = (e^{\\int e^{\\rho_b} d\\rho_b}) \\int e^{\\rho_b} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('\\\\rho_b', commutative=True)), exp(Symbol('\\\\rho_b', commutative=True)))"], [["integrate", 1, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('E_x')(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('q', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["exp", 5], "Equality(exp(Add(Symbol('q', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))), exp(Add(Symbol('\\\\phi_1', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(exp(Add(Symbol('q', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))), exp(Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"], [["times", 7, "Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))"], "Equality(Mul(exp(Add(Symbol('q', commutative=True), exp(Symbol('\\\\rho_b', commutative=True)))), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Mul(exp(Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))), Integral(exp(Symbol('\\\\rho_b', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\dot{z}{(g,\\mathbf{H})} = - (- g + \\log{(\\mathbf{H})})^{g} and \\phi{(g)} = - g, then obtain - (- g + \\log{(\\mathbf{H})})^{g} = - (\\phi{(g)} + \\log{(\\mathbf{H})})^{- \\phi{(g)}}", "derivation": "\\dot{z}{(g,\\mathbf{H})} = - (- g + \\log{(\\mathbf{H})})^{g} and \\phi{(g)} = - g and \\dot{z}{(g,\\mathbf{H})} = - (\\phi{(g)} + \\log{(\\mathbf{H})})^{- \\phi{(g)}} and - (- g + \\log{(\\mathbf{H})})^{g} = - (\\phi{(g)} + \\log{(\\mathbf{H})})^{- \\phi{(g)}}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('g', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\phi')(Symbol('g', commutative=True)), Mul(Integer(-1), Symbol('g', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\dot{z}')(Symbol('g', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), Pow(Add(Function('\\\\phi')(Symbol('g', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Function('\\\\phi')(Symbol('g', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('g', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Symbol('g', commutative=True))), Mul(Integer(-1), Pow(Add(Function('\\\\phi')(Symbol('g', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Function('\\\\phi')(Symbol('g', commutative=True))))))"]]}, {"prompt": "Given C{(\\rho_f)} = \\cos{(e^{\\rho_f})}, then derive \\int C{(\\rho_f)} d\\rho_f = M + \\operatorname{Ci}{(e^{\\rho_f})}, then obtain \\cos{(e^{\\rho_f})} \\int C{(\\rho_f)} d\\rho_f = (M + \\operatorname{Ci}{(e^{\\rho_f})}) \\cos{(e^{\\rho_f})}", "derivation": "C{(\\rho_f)} = \\cos{(e^{\\rho_f})} and \\int C{(\\rho_f)} d\\rho_f = \\int \\cos{(e^{\\rho_f})} d\\rho_f and \\int C{(\\rho_f)} d\\rho_f = M + \\operatorname{Ci}{(e^{\\rho_f})} and \\cos{(e^{\\rho_f})} \\int C{(\\rho_f)} d\\rho_f = (M + \\operatorname{Ci}{(e^{\\rho_f})}) \\cos{(e^{\\rho_f})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\rho_f', commutative=True)), cos(exp(Symbol('\\\\rho_f', commutative=True))))"], [["integrate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Function('C')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(cos(exp(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Add(Symbol('M', commutative=True), Ci(exp(Symbol('\\\\rho_f', commutative=True)))))"], [["times", 3, "cos(exp(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(cos(exp(Symbol('\\\\rho_f', commutative=True))), Integral(Function('C')(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True)))), Mul(Add(Symbol('M', commutative=True), Ci(exp(Symbol('\\\\rho_f', commutative=True)))), cos(exp(Symbol('\\\\rho_f', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{r}{(g)} = \\log{(g)}, then derive \\frac{\\frac{d}{d g} \\mathbf{r}{(g)}}{\\mathbf{r}{(g)}} = \\frac{1}{g \\log{(g)}}, then obtain \\frac{\\frac{d}{d g} \\mathbf{r}{(g)}}{\\mathbf{r}{(g)}} = \\frac{1}{g \\mathbf{r}{(g)}}", "derivation": "\\mathbf{r}{(g)} = \\log{(g)} and \\log{(\\mathbf{r}{(g)})} = \\log{(\\log{(g)})} and \\frac{d}{d g} \\log{(\\mathbf{r}{(g)})} = \\frac{d}{d g} \\log{(\\log{(g)})} and \\frac{\\frac{d}{d g} \\mathbf{r}{(g)}}{\\mathbf{r}{(g)}} = \\frac{1}{g \\log{(g)}} and \\frac{\\frac{d}{d g} \\log{(g)}}{\\log{(g)}} = \\frac{1}{g \\log{(g)}} and \\frac{\\frac{d}{d g} \\mathbf{r}{(g)}}{\\mathbf{r}{(g)}} = \\frac{1}{g \\mathbf{r}{(g)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), log(Symbol('g', commutative=True)))"], [["log", 1], "Equality(log(Function('\\\\mathbf{r}')(Symbol('g', commutative=True))), log(log(Symbol('g', commutative=True))))"], [["differentiate", 2, "Symbol('g', commutative=True)"], "Equality(Derivative(log(Function('\\\\mathbf{r}')(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(log(log(Symbol('g', commutative=True))), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(log(Symbol('g', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(log(Symbol('g', commutative=True)), Integer(-1)), Derivative(log(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(log(Symbol('g', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Pow(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('\\\\mathbf{r}')(Symbol('g', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{p}_0{(C)} = \\sin{(\\sin{(C)})}, then obtain \\frac{\\hat{p}_0{(C)} + \\frac{d}{d C} \\hat{p}_0{(C)}}{\\sin{(\\sin{(C)})}} = \\frac{\\sin{(\\sin{(C)})} + \\frac{d}{d C} \\hat{p}_0{(C)}}{\\sin{(\\sin{(C)})}}", "derivation": "\\hat{p}_0{(C)} = \\sin{(\\sin{(C)})} and \\frac{d}{d C} \\hat{p}_0{(C)} = \\frac{d}{d C} \\sin{(\\sin{(C)})} and \\hat{p}_0{(C)} + \\frac{d}{d C} \\sin{(\\sin{(C)})} = \\sin{(\\sin{(C)})} + \\frac{d}{d C} \\sin{(\\sin{(C)})} and \\frac{\\hat{p}_0{(C)} + \\frac{d}{d C} \\sin{(\\sin{(C)})}}{\\sin{(\\sin{(C)})}} = \\frac{\\sin{(\\sin{(C)})} + \\frac{d}{d C} \\sin{(\\sin{(C)})}}{\\sin{(\\sin{(C)})}} and \\frac{\\hat{p}_0{(C)} + \\frac{d}{d C} \\hat{p}_0{(C)}}{\\sin{(\\sin{(C)})}} = \\frac{\\sin{(\\sin{(C)})} + \\frac{d}{d C} \\hat{p}_0{(C)}}{\\sin{(\\sin{(C)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), sin(sin(Symbol('C', commutative=True))))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["add", 1, "Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))"], "Equality(Add(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Add(sin(sin(Symbol('C', commutative=True))), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["divide", 3, "sin(sin(Symbol('C', commutative=True)))"], "Equality(Mul(Add(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(sin(sin(Symbol('C', commutative=True))), Integer(-1))), Mul(Add(sin(sin(Symbol('C', commutative=True))), Derivative(sin(sin(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(sin(sin(Symbol('C', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Add(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Derivative(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(sin(sin(Symbol('C', commutative=True))), Integer(-1))), Mul(Add(sin(sin(Symbol('C', commutative=True))), Derivative(Function('\\\\hat{p}_0')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(sin(sin(Symbol('C', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\psi{(E_{\\lambda},Z)} = - E_{\\lambda} + Z, then derive \\frac{\\partial}{\\partial E_{\\lambda}} \\psi{(E_{\\lambda},Z)} = -1, then obtain (- (-1)^{E_{\\lambda}} + (\\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z))^{E_{\\lambda}}) \\frac{\\partial}{\\partial E_{\\lambda}} \\psi{(E_{\\lambda},Z)} = 0", "derivation": "\\psi{(E_{\\lambda},Z)} = - E_{\\lambda} + Z and \\frac{\\partial}{\\partial E_{\\lambda}} \\psi{(E_{\\lambda},Z)} = \\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z) and \\frac{\\partial}{\\partial E_{\\lambda}} \\psi{(E_{\\lambda},Z)} = -1 and \\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z) = -1 and (\\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z))^{E_{\\lambda}} = (-1)^{E_{\\lambda}} and - (-1)^{E_{\\lambda}} + (\\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z))^{E_{\\lambda}} = 0 and (- (-1)^{E_{\\lambda}} + (\\frac{\\partial}{\\partial E_{\\lambda}} (- E_{\\lambda} + Z))^{E_{\\lambda}}) \\frac{\\partial}{\\partial E_{\\lambda}} \\psi{(E_{\\lambda},Z)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('\\\\psi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\psi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Integer(-1))"], [["power", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 5, "Pow(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True))), Integer(0))"], [["times", 6, "Derivative(Function('\\\\psi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))"], "Equality(Mul(Add(Mul(Integer(-1), Pow(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True))), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True))), Derivative(Function('\\\\psi')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given U{(T)} = e^{T}, then obtain \\frac{T}{4 U{(T)}} = \\frac{T e^{T}}{(U{(T)} + e^{T})^{2}}", "derivation": "U{(T)} = e^{T} and 2 U{(T)} = U{(T)} + e^{T} and 4 U^{2}{(T)} = (U{(T)} + e^{T})^{2} and T U{(T)} = T e^{T} and \\frac{T}{4 U{(T)}} = \\frac{T e^{T}}{4 U^{2}{(T)}} and \\frac{T}{4 U{(T)}} = \\frac{T e^{T}}{(U{(T)} + e^{T})^{2}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["add", 1, "Function('U')(Symbol('T', commutative=True))"], "Equality(Mul(Integer(2), Function('U')(Symbol('T', commutative=True))), Add(Function('U')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('U')(Symbol('T', commutative=True)), Integer(2))), Pow(Add(Function('U')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integer(2)))"], [["times", 1, "Symbol('T', commutative=True)"], "Equality(Mul(Symbol('T', commutative=True), Function('U')(Symbol('T', commutative=True))), Mul(Symbol('T', commutative=True), exp(Symbol('T', commutative=True))))"], [["divide", 4, "Mul(Integer(4), Pow(Function('U')(Symbol('T', commutative=True)), Integer(2)))"], "Equality(Mul(Rational(1, 4), Symbol('T', commutative=True), Pow(Function('U')(Symbol('T', commutative=True)), Integer(-1))), Mul(Rational(1, 4), Symbol('T', commutative=True), Pow(Function('U')(Symbol('T', commutative=True)), Integer(-2)), exp(Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Rational(1, 4), Symbol('T', commutative=True), Pow(Function('U')(Symbol('T', commutative=True)), Integer(-1))), Mul(Symbol('T', commutative=True), Pow(Add(Function('U')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True))), Integer(-2)), exp(Symbol('T', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(U)} = e^{U}, then derive \\int \\mathbb{I}{(U)} dU = \\mathbf{E} + e^{U}, then derive \\frac{d}{d \\mathbf{E}} \\int \\mathbb{I}{(U)} dU = 1, then obtain (\\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + \\mathbb{I}{(U)}))^{\\mathbf{E}} = 1", "derivation": "\\mathbb{I}{(U)} = e^{U} and \\int \\mathbb{I}{(U)} dU = \\int e^{U} dU and \\int \\mathbb{I}{(U)} dU = \\mathbf{E} + e^{U} and \\frac{d}{d \\mathbf{E}} \\int \\mathbb{I}{(U)} dU = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + e^{U}) and \\int \\mathbb{I}{(U)} dU = \\mathbf{E} + \\mathbb{I}{(U)} and \\frac{d}{d \\mathbf{E}} \\int \\mathbb{I}{(U)} dU = \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + \\mathbb{I}{(U)}) and \\frac{d}{d \\mathbf{E}} \\int \\mathbb{I}{(U)} dU = 1 and \\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + \\mathbb{I}{(U)}) = 1 and (\\frac{\\partial}{\\partial \\mathbf{E}} (\\mathbf{E} + \\mathbb{I}{(U)}))^{\\mathbf{E}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), exp(Symbol('U', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(exp(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('U', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), exp(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbb{I}')(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbb{I}')(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Derivative(Integral(Function('\\\\mathbb{I}')(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbb{I}')(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Integer(1))"], [["power", 8, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\mathbf{E}', commutative=True), Function('\\\\mathbb{I}')(Symbol('U', commutative=True))), Tuple(Symbol('\\\\mathbf{E}', commutative=True), Integer(1))), Symbol('\\\\mathbf{E}', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(I,v_{z})} = I v_{z}, then derive x + \\operatorname{a^{\\dagger}}{(I,v_{z})} = I v_{z} + \\mathbf{s}, then obtain (\\int (I v_{z} + x) dx)^{x} = (\\int (I v_{z} + \\mathbf{s}) dx)^{x}", "derivation": "\\operatorname{a^{\\dagger}}{(I,v_{z})} = I v_{z} and \\frac{\\partial}{\\partial I} \\operatorname{a^{\\dagger}}{(I,v_{z})} = \\frac{\\partial}{\\partial I} I v_{z} and \\int \\frac{\\partial}{\\partial I} \\operatorname{a^{\\dagger}}{(I,v_{z})} dI = \\int \\frac{\\partial}{\\partial I} I v_{z} dI and x + \\operatorname{a^{\\dagger}}{(I,v_{z})} = I v_{z} + \\mathbf{s} and \\int (x + \\operatorname{a^{\\dagger}}{(I,v_{z})}) dx = \\int (I v_{z} + \\mathbf{s}) dx and \\int (I v_{z} + x) dx = \\int (I v_{z} + \\mathbf{s}) dx and (\\int (I v_{z} + x) dx)^{x} = (\\int (I v_{z} + \\mathbf{s}) dx)^{x}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Derivative(Function('a^{\\\\dagger}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('x', commutative=True), Function('a^{\\\\dagger}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 4, "Symbol('x', commutative=True)"], "Equality(Integral(Add(Symbol('x', commutative=True), Function('a^{\\\\dagger}')(Symbol('I', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('x', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('x', commutative=True))))"], [["power", 6, "Symbol('x', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integral(Add(Mul(Symbol('I', commutative=True), Symbol('v_z', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('x', commutative=True))), Symbol('x', commutative=True)))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} = \\log{(\\Psi_{nl} + \\mathbf{D})}, then obtain \\Psi_{nl} + \\mathbf{D} - \\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} - 1 = \\Psi_{nl} + \\mathbf{D} - \\log{(\\Psi_{nl} + \\mathbf{D})} - 1", "derivation": "\\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} = \\log{(\\Psi_{nl} + \\mathbf{D})} and - \\Psi_{nl} - \\mathbf{D} + \\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} = - \\Psi_{nl} - \\mathbf{D} + \\log{(\\Psi_{nl} + \\mathbf{D})} and \\Psi_{nl} + \\mathbf{D} - \\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} = \\Psi_{nl} + \\mathbf{D} - \\log{(\\Psi_{nl} + \\mathbf{D})} and \\Psi_{nl} + \\mathbf{D} - \\Psi^{\\dagger}{(\\mathbf{D},\\Psi_{nl})} - 1 = \\Psi_{nl} + \\mathbf{D} - \\log{(\\Psi_{nl} + \\mathbf{D})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True)), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)))), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True))), Integer(-1)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True), Mul(Integer(-1), log(Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}}, then derive \\operatorname{V_{\\mathbf{B}}}{(y^{\\prime})} = e^{y^{\\prime}}, then obtain r_{0} + e^{y^{\\prime}} = \\sigma_p + e^{y^{\\prime}}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(y^{\\prime})} = \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} and \\operatorname{V_{\\mathbf{B}}}{(y^{\\prime})} = e^{y^{\\prime}} and \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} = e^{y^{\\prime}} and \\int \\frac{d}{d y^{\\prime}} e^{y^{\\prime}} dy^{\\prime} = \\int e^{y^{\\prime}} dy^{\\prime} and r_{0} + e^{y^{\\prime}} = \\sigma_p + e^{y^{\\prime}}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('y^{\\\\prime}', commutative=True)), Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('y^{\\\\prime}', commutative=True)), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), exp(Symbol('y^{\\\\prime}', commutative=True)))"], [["integrate", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Integral(Derivative(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True), Integer(1))), Tuple(Symbol('y^{\\\\prime}', commutative=True))), Integral(exp(Symbol('y^{\\\\prime}', commutative=True)), Tuple(Symbol('y^{\\\\prime}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('r_0', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), exp(Symbol('y^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{F}{(\\chi)} = \\log{(\\chi)}, then obtain \\frac{2 \\mathbf{F}^{2}{(\\chi)}}{\\cos{(\\log{(\\chi)})}} = \\frac{\\mathbf{F}^{2}{(\\chi)} + \\mathbf{F}{(\\chi)} \\log{(\\chi)}}{\\cos{(\\log{(\\chi)})}}", "derivation": "\\mathbf{F}{(\\chi)} = \\log{(\\chi)} and \\mathbf{F}^{2}{(\\chi)} = \\mathbf{F}{(\\chi)} \\log{(\\chi)} and \\cos{(\\mathbf{F}{(\\chi)})} = \\cos{(\\log{(\\chi)})} and 2 \\mathbf{F}^{2}{(\\chi)} = \\mathbf{F}^{2}{(\\chi)} + \\mathbf{F}{(\\chi)} \\log{(\\chi)} and \\frac{2 \\mathbf{F}^{2}{(\\chi)}}{\\cos{(\\mathbf{F}{(\\chi)})}} = \\frac{\\mathbf{F}^{2}{(\\chi)} + \\mathbf{F}{(\\chi)} \\log{(\\chi)}}{\\cos{(\\mathbf{F}{(\\chi)})}} and \\frac{2 \\mathbf{F}^{2}{(\\chi)}}{\\cos{(\\log{(\\chi)})}} = \\frac{\\mathbf{F}^{2}{(\\chi)} + \\mathbf{F}{(\\chi)} \\log{(\\chi)}}{\\cos{(\\log{(\\chi)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["cos", 1], "Equality(cos(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True))), cos(log(Symbol('\\\\chi', commutative=True))))"], [["add", 2, "Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2))), Add(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))))"], [["divide", 4, "cos(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Pow(cos(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True))), Integer(-1))), Mul(Add(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))), Pow(cos(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True))), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(2), Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Pow(cos(log(Symbol('\\\\chi', commutative=True))), Integer(-1))), Mul(Add(Pow(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{F}')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))), Pow(cos(log(Symbol('\\\\chi', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(A_{2},\\rho_b)} = \\cos{(A_{2} + \\rho_b)}, then obtain \\cos{(\\frac{\\hat{H}_{\\lambda}{(A_{2},\\rho_b)} + \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}})} = \\cos{(\\frac{2 \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}})}", "derivation": "\\hat{H}_{\\lambda}{(A_{2},\\rho_b)} = \\cos{(A_{2} + \\rho_b)} and \\hat{H}_{\\lambda}{(A_{2},\\rho_b)} + \\cos{(A_{2} + \\rho_b)} = 2 \\cos{(A_{2} + \\rho_b)} and \\frac{\\hat{H}_{\\lambda}{(A_{2},\\rho_b)} + \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}} = \\frac{2 \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}} and \\cos{(\\frac{\\hat{H}_{\\lambda}{(A_{2},\\rho_b)} + \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}})} = \\cos{(\\frac{2 \\cos{(A_{2} + \\rho_b)}}{- \\rho_b + \\cos{(A_{2} + \\rho_b)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], [["add", 1, "cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], "Equality(Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Mul(Integer(2), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True))))), Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(-1)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))))"], [["cos", 3], "Equality(cos(Mul(Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))))), cos(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\rho_b', commutative=True)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True)))), Integer(-1)), cos(Add(Symbol('A_2', commutative=True), Symbol('\\\\rho_b', commutative=True))))))"]]}, {"prompt": "Given \\nabla{(y^{\\prime})} = y^{\\prime}, then obtain - \\nabla^{y^{\\prime}}{(y^{\\prime})} - \\sin^{y^{\\prime}}{(y^{\\prime})} = - (y^{\\prime})^{y^{\\prime}} - \\sin^{y^{\\prime}}{(y^{\\prime})}", "derivation": "\\nabla{(y^{\\prime})} = y^{\\prime} and \\nabla^{y^{\\prime}}{(y^{\\prime})} = (y^{\\prime})^{y^{\\prime}} and \\sin{(\\nabla{(y^{\\prime})})} = \\sin{(y^{\\prime})} and - \\nabla^{y^{\\prime}}{(y^{\\prime})} = - (y^{\\prime})^{y^{\\prime}} and - \\nabla^{y^{\\prime}}{(y^{\\prime})} - \\sin^{y^{\\prime}}{(\\nabla{(y^{\\prime})})} = - (y^{\\prime})^{y^{\\prime}} - \\sin^{y^{\\prime}}{(\\nabla{(y^{\\prime})})} and - \\nabla^{y^{\\prime}}{(y^{\\prime})} - \\sin^{y^{\\prime}}{(y^{\\prime})} = - (y^{\\prime})^{y^{\\prime}} - \\sin^{y^{\\prime}}{(y^{\\prime})}", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))"], [["power", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True))), sin(Symbol('y^{\\\\prime}', commutative=True)))"], [["divide", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 4, "Pow(sin(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(sin(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(sin(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True))), Symbol('y^{\\\\prime}', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\nabla')(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('y^{\\\\prime}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Pow(sin(Symbol('y^{\\\\prime}', commutative=True)), Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}{(q,M)} = - M + q, then obtain 0 = - 4 (- M + q)^{2} + 4 \\mathbf{J}^{2}{(q,M)}", "derivation": "\\mathbf{J}{(q,M)} = - M + q and 2 \\mathbf{J}{(q,M)} = - M + q + \\mathbf{J}{(q,M)} and 4 \\mathbf{J}^{2}{(q,M)} = (- M + q + \\mathbf{J}{(q,M)})^{2} and 4 (- M + q)^{2} = (- 2 M + 2 q)^{2} and 4 \\mathbf{J}^{2}{(q,M)} = (- 2 M + 2 q)^{2} and (- 2 M + 2 q)^{2} = (- M + q + \\mathbf{J}{(q,M)})^{2} and 4 (- M + q)^{2} = (- M + q + \\mathbf{J}{(q,M)})^{2} and 0 = - 4 (- M + q)^{2} + (- M + q + \\mathbf{J}{(q,M)})^{2} and 0 = - 4 (- M + q)^{2} + 4 \\mathbf{J}^{2}{(q,M)}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)))"], [["add", 1, "Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))), Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(4), Pow(Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 5], "Equality(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('M', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))), Integer(2)))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))), Integer(2)))"], [["minus", 7, "Mul(Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)), Integer(2)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)), Integer(2))), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True), Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 8, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(4), Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Symbol('q', commutative=True)), Integer(2))), Mul(Integer(4), Pow(Function('\\\\mathbf{J}')(Symbol('q', commutative=True), Symbol('M', commutative=True)), Integer(2)))))"]]}, {"prompt": "Given v{(\\rho)} = \\sin{(\\sin{(\\rho)})}, then derive \\hat{H}_{\\lambda} + v{(\\rho)} = v + \\sin{(\\sin{(\\rho)})}, then obtain (\\hat{H}_{\\lambda} + \\sin{(\\sin{(\\rho)})})^{v} = (v + \\sin{(\\sin{(\\rho)})})^{v}", "derivation": "v{(\\rho)} = \\sin{(\\sin{(\\rho)})} and \\frac{d}{d \\rho} v{(\\rho)} = \\frac{d}{d \\rho} \\sin{(\\sin{(\\rho)})} and \\int \\frac{d}{d \\rho} v{(\\rho)} d\\rho = \\int \\frac{d}{d \\rho} \\sin{(\\sin{(\\rho)})} d\\rho and \\hat{H}_{\\lambda} + v{(\\rho)} = v + \\sin{(\\sin{(\\rho)})} and (\\hat{H}_{\\lambda} + v{(\\rho)})^{v} = (v + \\sin{(\\sin{(\\rho)})})^{v} and (\\hat{H}_{\\lambda} + \\sin{(\\sin{(\\rho)})})^{v} = (v + \\sin{(\\sin{(\\rho)})})^{v}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\rho', commutative=True)), sin(sin(Symbol('\\\\rho', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Derivative(Function('v')(Symbol('\\\\rho', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Derivative(sin(sin(Symbol('\\\\rho', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('v')(Symbol('\\\\rho', commutative=True))), Add(Symbol('v', commutative=True), sin(sin(Symbol('\\\\rho', commutative=True)))))"], [["power", 4, "Symbol('v', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Function('v')(Symbol('\\\\rho', commutative=True))), Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), sin(sin(Symbol('\\\\rho', commutative=True)))), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Pow(Add(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), sin(sin(Symbol('\\\\rho', commutative=True)))), Symbol('v', commutative=True)), Pow(Add(Symbol('v', commutative=True), sin(sin(Symbol('\\\\rho', commutative=True)))), Symbol('v', commutative=True)))"]]}, {"prompt": "Given I{(E_{\\lambda},f)} = E_{\\lambda} - f, then obtain - 3 f - 3 I{(E_{\\lambda},f)} = - E_{\\lambda} - 2 f - 2 I{(E_{\\lambda},f)}", "derivation": "I{(E_{\\lambda},f)} = E_{\\lambda} - f and f + 2 I{(E_{\\lambda},f)} = E_{\\lambda} + I{(E_{\\lambda},f)} and 2 f + 3 I{(E_{\\lambda},f)} = E_{\\lambda} + f + 2 I{(E_{\\lambda},f)} and - 2 f - 3 I{(E_{\\lambda},f)} = - E_{\\lambda} - f - 2 I{(E_{\\lambda},f)} and - 3 f - 3 I{(E_{\\lambda},f)} = - E_{\\lambda} - 2 f - 2 I{(E_{\\lambda},f)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)), Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Integer(-1), Symbol('f', commutative=True))))"], [["add", 1, "Add(Symbol('f', commutative=True), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Symbol('f', commutative=True), Mul(Integer(2), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True))))"], [["add", 2, "Add(Symbol('f', commutative=True), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('f', commutative=True)), Mul(Integer(3), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True), Mul(Integer(2), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(3), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))))"], [["add", 4, "Mul(Integer(-1), Symbol('f', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(3), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(3), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f', commutative=True)), Mul(Integer(-1), Integer(2), Function('I')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})} = \\hat{H}_{\\lambda} c + r_{0}, then obtain \\int (- \\hat{H}_{\\lambda} c - r_{0} + \\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})})^{r_{0}} d\\hat{H}_{\\lambda} = \\int 0^{r_{0}} d\\hat{H}_{\\lambda}", "derivation": "\\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})} = \\hat{H}_{\\lambda} c + r_{0} and - \\hat{H}_{\\lambda} c - r_{0} + \\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})} = 0 and (- \\hat{H}_{\\lambda} c - r_{0} + \\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})})^{r_{0}} = 0^{r_{0}} and \\int (- \\hat{H}_{\\lambda} c - r_{0} + \\Psi^{\\dagger}{(c,\\hat{H}_{\\lambda},r_{0})})^{r_{0}} d\\hat{H}_{\\lambda} = \\int 0^{r_{0}} d\\hat{H}_{\\lambda}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True)), Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)), Symbol('r_0', commutative=True)))"], [["minus", 1, "Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)), Symbol('r_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True))), Integer(0))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Pow(Integer(0), Symbol('r_0', commutative=True)))"], [["integrate", 3, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('c', commutative=True)), Mul(Integer(-1), Symbol('r_0', commutative=True)), Function('\\\\Psi^{\\\\dagger}')(Symbol('c', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('r_0', commutative=True))), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(Pow(Integer(0), Symbol('r_0', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} = E_{\\lambda} (\\mathbf{s} + i), then obtain \\mathbf{s} + i \\int \\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} d\\mathbf{s} = \\mathbf{s} + i \\int E_{\\lambda} (\\mathbf{s} + i) d\\mathbf{s}", "derivation": "\\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} = E_{\\lambda} (\\mathbf{s} + i) and \\int \\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} d\\mathbf{s} = \\int E_{\\lambda} (\\mathbf{s} + i) d\\mathbf{s} and i \\int \\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} d\\mathbf{s} = i \\int E_{\\lambda} (\\mathbf{s} + i) d\\mathbf{s} and \\mathbf{s} + i \\int \\rho_{b}{(\\mathbf{s},E_{\\lambda},i)} d\\mathbf{s} = \\mathbf{s} + i \\int E_{\\lambda} (\\mathbf{s} + i) d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["times", 2, "Symbol('i', commutative=True)"], "Equality(Mul(Symbol('i', commutative=True), Integral(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Mul(Symbol('i', commutative=True), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"], [["add", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Symbol('i', commutative=True), Integral(Function('\\\\rho_b')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))), Add(Symbol('\\\\mathbf{s}', commutative=True), Mul(Symbol('i', commutative=True), Integral(Mul(Symbol('E_{\\\\lambda}', commutative=True), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{C_{1}}{(\\mu)} = \\cos{(\\mu)}, then derive \\frac{d^{2}}{d \\mu^{2}} \\operatorname{C_{1}}{(\\mu)} = - \\cos{(\\mu)}, then obtain \\operatorname{A_{1}}{(f_{E})} \\frac{d^{2}}{d \\mu^{2}} \\cos{(\\mu)} = - \\operatorname{A_{1}}{(f_{E})} \\cos{(\\mu)}", "derivation": "\\operatorname{C_{1}}{(\\mu)} = \\cos{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{C_{1}}{(\\mu)} = \\frac{d}{d \\mu} \\cos{(\\mu)} and \\frac{d^{2}}{d \\mu^{2}} \\operatorname{C_{1}}{(\\mu)} = \\frac{d^{2}}{d \\mu^{2}} \\cos{(\\mu)} and \\frac{d^{2}}{d \\mu^{2}} \\operatorname{C_{1}}{(\\mu)} = - \\cos{(\\mu)} and \\frac{d^{2}}{d \\mu^{2}} \\cos{(\\mu)} = - \\cos{(\\mu)} and \\operatorname{A_{1}}{(f_{E})} \\frac{d^{2}}{d \\mu^{2}} \\cos{(\\mu)} = - \\operatorname{A_{1}}{(f_{E})} \\cos{(\\mu)}", "srepr_derivation": [["premise", "Equality(Function('C_1')(Symbol('\\\\mu', commutative=True)), cos(Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('C_1')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('C_1')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('\\\\mu', commutative=True))))"], [["times", 5, "Function('A_1')(Symbol('f_E', commutative=True))"], "Equality(Mul(Function('A_1')(Symbol('f_E', commutative=True)), Derivative(cos(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(2)))), Mul(Integer(-1), Function('A_1')(Symbol('f_E', commutative=True)), cos(Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(l,p)} = \\log{(- l + p)}, then derive \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\frac{1}{- l + p}, then obtain \\hat{x}{(l,p)} + \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\hat{x}{(l,p)} + \\frac{1}{- l + p}", "derivation": "\\hat{x}{(l,p)} = \\log{(- l + p)} and \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\frac{\\partial}{\\partial p} \\log{(- l + p)} and \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\frac{1}{- l + p} and \\log{(- l + p)} + \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\log{(- l + p)} + \\frac{1}{- l + p} and \\hat{x}{(l,p)} + \\frac{\\partial}{\\partial p} \\hat{x}{(l,p)} = \\hat{x}{(l,p)} + \\frac{1}{- l + p}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), log(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True))))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(log(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True)), Integer(-1)))"], [["add", 3, "log(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True)))"], "Equality(Add(log(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True))), Derivative(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(log(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True))), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Derivative(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))), Add(Function('\\\\hat{x}')(Symbol('l', commutative=True), Symbol('p', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Symbol('p', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{g}{(\\chi,\\lambda)} = \\log{(\\chi + \\lambda)} and \\operatorname{C_{1}}{(\\chi,\\lambda)} = \\lambda \\log{(\\chi + \\lambda)}^{\\lambda}, then obtain \\iint \\lambda \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} d\\chi d\\chi = \\iint \\operatorname{C_{1}}{(\\chi,\\lambda)} d\\chi d\\chi", "derivation": "\\mathbf{g}{(\\chi,\\lambda)} = \\log{(\\chi + \\lambda)} and \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} = \\log{(\\chi + \\lambda)}^{\\lambda} and \\lambda \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} = \\lambda \\log{(\\chi + \\lambda)}^{\\lambda} and \\operatorname{C_{1}}{(\\chi,\\lambda)} = \\lambda \\log{(\\chi + \\lambda)}^{\\lambda} and \\lambda \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} = \\operatorname{C_{1}}{(\\chi,\\lambda)} and \\int \\lambda \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} d\\chi = \\int \\operatorname{C_{1}}{(\\chi,\\lambda)} d\\chi and \\iint \\lambda \\mathbf{g}^{\\lambda}{(\\chi,\\lambda)} d\\chi d\\chi = \\iint \\operatorname{C_{1}}{(\\chi,\\lambda)} d\\chi d\\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True))))"], [["power", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True)), Pow(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)))"], [["times", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Mul(Symbol('\\\\lambda', commutative=True), Pow(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], ["renaming_premise", "Equality(Function('C_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Symbol('\\\\lambda', commutative=True), Pow(log(Add(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Function('C_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)))"], [["integrate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Function('C_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["integrate", 6, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Mul(Symbol('\\\\lambda', commutative=True), Pow(Function('\\\\mathbf{g}')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Symbol('\\\\lambda', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Function('C_1')(Symbol('\\\\chi', commutative=True), Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P}, then obtain (\\mathbf{J}_P + e^{\\mathbf{J}_P})^{\\mathbf{J}_P} = (\\mathbf{J}_P - \\mathbf{A}{(\\mathbf{J}_P)} + 2 e^{\\mathbf{J}_P})^{\\mathbf{J}_P}", "derivation": "\\mathbf{A}{(\\mathbf{J}_P)} = e^{\\mathbf{J}_P} and 0 = - \\mathbf{A}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} and \\mathbf{J}_P = \\mathbf{J}_P - \\mathbf{A}{(\\mathbf{J}_P)} + e^{\\mathbf{J}_P} and \\mathbf{J}_P + e^{\\mathbf{J}_P} = \\mathbf{J}_P - \\mathbf{A}{(\\mathbf{J}_P)} + 2 e^{\\mathbf{J}_P} and (\\mathbf{J}_P + e^{\\mathbf{J}_P})^{\\mathbf{J}_P} = (\\mathbf{J}_P - \\mathbf{A}{(\\mathbf{J}_P)} + 2 e^{\\mathbf{J}_P})^{\\mathbf{J}_P}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True)), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Symbol('\\\\mathbf{J}_P', commutative=True), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True))), exp(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["add", 3, "exp(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["power", 4, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), exp(Symbol('\\\\mathbf{J}_P', commutative=True))), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\mathbf{J}_P', commutative=True)))), Symbol('\\\\mathbf{J}_P', commutative=True)))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\varepsilon)} = \\cos{(\\varepsilon)}, then obtain (\\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\dot{\\mathbf{r}}{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = (\\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon}", "derivation": "\\dot{\\mathbf{r}}{(\\varepsilon)} = \\cos{(\\varepsilon)} and \\frac{d}{d \\varepsilon} \\dot{\\mathbf{r}}{(\\varepsilon)} = \\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)} and (\\frac{d}{d \\varepsilon} \\dot{\\mathbf{r}}{(\\varepsilon)})^{\\varepsilon} = (\\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon} and \\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\dot{\\mathbf{r}}{(\\varepsilon)})^{\\varepsilon} = \\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon} and (\\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\dot{\\mathbf{r}}{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon} = (\\cos{(\\varepsilon)} + (\\frac{d}{d \\varepsilon} \\cos{(\\varepsilon)})^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True)), cos(Symbol('\\\\varepsilon', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True)))"], [["add", 3, "cos(Symbol('\\\\varepsilon', commutative=True))"], "Equality(Add(cos(Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))), Add(cos(Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))))"], [["power", 4, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Add(cos(Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(cos(Symbol('\\\\varepsilon', commutative=True)), Pow(Derivative(cos(Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\varepsilon', commutative=True), Integer(1))), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"]]}, {"prompt": "Given \\delta{(E_{\\lambda})} = e^{E_{\\lambda}}, then obtain (\\frac{d}{d E_{\\lambda}} \\frac{\\int 0 dE_{\\lambda}}{E_{\\lambda}})^{E_{\\lambda}} = (\\frac{d}{d E_{\\lambda}} \\frac{\\int (- \\delta{(E_{\\lambda})} + e^{E_{\\lambda}}) dE_{\\lambda}}{E_{\\lambda}})^{E_{\\lambda}}", "derivation": "\\delta{(E_{\\lambda})} = e^{E_{\\lambda}} and 0 = - \\delta{(E_{\\lambda})} + e^{E_{\\lambda}} and \\int 0 dE_{\\lambda} = \\int (- \\delta{(E_{\\lambda})} + e^{E_{\\lambda}}) dE_{\\lambda} and \\frac{\\int 0 dE_{\\lambda}}{E_{\\lambda}} = \\frac{\\int (- \\delta{(E_{\\lambda})} + e^{E_{\\lambda}}) dE_{\\lambda}}{E_{\\lambda}} and \\frac{d}{d E_{\\lambda}} \\frac{\\int 0 dE_{\\lambda}}{E_{\\lambda}} = \\frac{d}{d E_{\\lambda}} \\frac{\\int (- \\delta{(E_{\\lambda})} + e^{E_{\\lambda}}) dE_{\\lambda}}{E_{\\lambda}} and (\\frac{d}{d E_{\\lambda}} \\frac{\\int 0 dE_{\\lambda}}{E_{\\lambda}})^{E_{\\lambda}} = (\\frac{d}{d E_{\\lambda}} \\frac{\\int (- \\delta{(E_{\\lambda})} + e^{E_{\\lambda}}) dE_{\\lambda}}{E_{\\lambda}})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True)), exp(Symbol('E_{\\\\lambda}', commutative=True)))"], [["minus", 1, "Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))))"], [["integrate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["divide", 3, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))))"], [["differentiate", 4, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["power", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Integer(0), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Derivative(Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), Integral(Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('E_{\\\\lambda}', commutative=True))), exp(Symbol('E_{\\\\lambda}', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True)))), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given z{(I)} = \\sin{(I)} and \\operatorname{P_{g}}{(I)} = \\log{(z{(I)})}, then obtain \\operatorname{P_{g}}{(I)} z^{I}{(I)} = z^{I}{(I)} \\log{(\\sin{(I)})}", "derivation": "z{(I)} = \\sin{(I)} and z^{I}{(I)} = \\sin^{I}{(I)} and \\operatorname{P_{g}}{(I)} = \\log{(z{(I)})} and \\operatorname{P_{g}}{(I)} \\sin^{I}{(I)} = \\log{(z{(I)})} \\sin^{I}{(I)} and \\operatorname{P_{g}}{(I)} \\sin^{I}{(I)} = \\log{(\\sin{(I)})} \\sin^{I}{(I)} and \\operatorname{P_{g}}{(I)} z^{I}{(I)} = z^{I}{(I)} \\log{(\\sin{(I)})}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('I', commutative=True)), sin(Symbol('I', commutative=True)))"], [["power", 1, "Symbol('I', commutative=True)"], "Equality(Pow(Function('z')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True)))"], ["renaming_premise", "Equality(Function('P_g')(Symbol('I', commutative=True)), log(Function('z')(Symbol('I', commutative=True))))"], [["times", 3, "Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Mul(Function('P_g')(Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Mul(log(Function('z')(Symbol('I', commutative=True))), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Function('P_g')(Symbol('I', commutative=True)), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Mul(log(sin(Symbol('I', commutative=True))), Pow(sin(Symbol('I', commutative=True)), Symbol('I', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Function('P_g')(Symbol('I', commutative=True)), Pow(Function('z')(Symbol('I', commutative=True)), Symbol('I', commutative=True))), Mul(Pow(Function('z')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), log(sin(Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{p},\\dot{y})} = - \\dot{y} + \\hat{p}, then obtain - \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})} + \\frac{1}{(- \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})})^{2}} = - \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})} + \\frac{1}{(- 2 \\dot{y} + \\hat{p})^{2}}", "derivation": "\\theta_{1}{(\\hat{p},\\dot{y})} = - \\dot{y} + \\hat{p} and - \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})} = - 2 \\dot{y} + \\hat{p} and \\frac{1}{(- \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})})^{2}} = \\frac{1}{(- 2 \\dot{y} + \\hat{p})^{2}} and - \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})} + \\frac{1}{(- \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})})^{2}} = - \\dot{y} + \\theta_{1}{(\\hat{p},\\dot{y})} + \\frac{1}{(- 2 \\dot{y} + \\hat{p})^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))"], [["power", 2, "Integer(-2)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-2)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-2)))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True))), Integer(-2))), Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Function('\\\\theta_1')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Pow(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\dot{y}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(t_{2},F_{N})} = \\frac{\\partial}{\\partial t_{2}} (F_{N} + t_{2}), then derive \\operatorname{V_{\\mathbf{E}}}{(t_{2},F_{N})} = 1, then obtain - F_{N} - t_{2} + \\frac{\\partial^{2}}{\\partial t_{2}^{2}} (F_{N} + t_{2}) - 1 = - F_{N} - t_{2} + \\frac{d}{d t_{2}} 1 - 1", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(t_{2},F_{N})} = \\frac{\\partial}{\\partial t_{2}} (F_{N} + t_{2}) and \\operatorname{V_{\\mathbf{E}}}{(t_{2},F_{N})} = 1 and \\frac{\\partial}{\\partial t_{2}} \\operatorname{V_{\\mathbf{E}}}{(t_{2},F_{N})} = \\frac{d}{d t_{2}} 1 and \\frac{\\partial^{2}}{\\partial t_{2}^{2}} (F_{N} + t_{2}) = \\frac{d}{d t_{2}} 1 and - F_{N} - t_{2} + \\frac{\\partial^{2}}{\\partial t_{2}^{2}} (F_{N} + t_{2}) - 1 = - F_{N} - t_{2} + \\frac{d}{d t_{2}} 1 - 1", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('t_2', commutative=True), Symbol('F_N', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('t_2', commutative=True), Symbol('F_N', commutative=True)), Integer(1))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('t_2', commutative=True), Symbol('F_N', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('F_N', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(2))), Derivative(Integer(1), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["minus", 4, "Add(Symbol('F_N', commutative=True), Symbol('t_2', commutative=True), Integer(1))"], "Equality(Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True)), Derivative(Add(Symbol('F_N', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True), Integer(2))), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_N', commutative=True)), Mul(Integer(-1), Symbol('t_2', commutative=True)), Derivative(Integer(1), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\pi{(t_{1},B,\\mathbf{J})} = - B + \\mathbf{J} - t_{1}, then obtain 1 + (- B + \\mathbf{J} - t_{1})^{- \\mathbf{J}} \\pi^{\\mathbf{J}}{(t_{1},B,\\mathbf{J})} = 2", "derivation": "\\pi{(t_{1},B,\\mathbf{J})} = - B + \\mathbf{J} - t_{1} and \\pi^{\\mathbf{J}}{(t_{1},B,\\mathbf{J})} = (- B + \\mathbf{J} - t_{1})^{\\mathbf{J}} and (- B + \\mathbf{J} - t_{1})^{- \\mathbf{J}} \\pi^{\\mathbf{J}}{(t_{1},B,\\mathbf{J})} = 1 and 1 + (- B + \\mathbf{J} - t_{1})^{- \\mathbf{J}} \\pi^{\\mathbf{J}}{(t_{1},B,\\mathbf{J})} = 2", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('t_1', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('t_1', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["divide", 2, "Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Function('\\\\pi')(Symbol('t_1', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))), Integer(1))"], [["add", 3, 1], "Equality(Add(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True), Mul(Integer(-1), Symbol('t_1', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))), Pow(Function('\\\\pi')(Symbol('t_1', commutative=True), Symbol('B', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))), Integer(2))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} \\rho_b, then obtain \\int (\\hat{H}_{\\lambda} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} d\\rho_b = \\int (2 \\hat{H}_{\\lambda} \\rho_b)^{\\hat{H}_{\\lambda}} d\\rho_b", "derivation": "\\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})} = \\hat{H}_{\\lambda} \\rho_b and \\hat{H}_{\\lambda} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})} = 2 \\hat{H}_{\\lambda} \\rho_b and (\\hat{H}_{\\lambda} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} = (2 \\hat{H}_{\\lambda} \\rho_b)^{\\hat{H}_{\\lambda}} and \\int (\\hat{H}_{\\lambda} \\rho_b + \\dot{\\mathbf{r}}{(\\rho_b,\\hat{H}_{\\lambda})})^{\\hat{H}_{\\lambda}} d\\rho_b = \\int (2 \\hat{H}_{\\lambda} \\rho_b)^{\\hat{H}_{\\lambda}} d\\rho_b", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["add", 1, "Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)))"], [["power", 2, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))), Integral(Pow(Mul(Integer(2), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('\\\\rho_b', commutative=True)), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True))))"]]}, {"prompt": "Given \\eta^{\\prime}{(m_{s},i)} = \\frac{i}{m_{s}} and \\operatorname{t_{2}}{(m_{s},i)} = \\int \\frac{i}{m_{s}} di, then obtain - \\frac{\\operatorname{t_{2}}{(m_{s},i)}}{\\int \\eta^{\\prime}{(m_{s},i)} di} = -1", "derivation": "\\eta^{\\prime}{(m_{s},i)} = \\frac{i}{m_{s}} and \\int \\eta^{\\prime}{(m_{s},i)} di = \\int \\frac{i}{m_{s}} di and \\operatorname{t_{2}}{(m_{s},i)} = \\int \\frac{i}{m_{s}} di and \\operatorname{t_{2}}{(m_{s},i)} = \\int \\eta^{\\prime}{(m_{s},i)} di and - \\frac{\\operatorname{t_{2}}{(m_{s},i)}}{\\int \\eta^{\\prime}{(m_{s},i)} di} = -1", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Mul(Symbol('i', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Mul(Symbol('i', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('i', commutative=True))))"], ["renaming_premise", "Equality(Function('t_2')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Integral(Mul(Symbol('i', commutative=True), Pow(Symbol('m_s', commutative=True), Integer(-1))), Tuple(Symbol('i', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('t_2')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["divide", 4, "Mul(Integer(-1), Integral(Function('\\\\eta^{\\\\prime}')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], "Equality(Mul(Integer(-1), Function('t_2')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Pow(Integral(Function('\\\\eta^{\\\\prime}')(Symbol('m_s', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integer(-1))), Integer(-1))"]]}, {"prompt": "Given T{(F_{N})} = F_{N}, then derive \\tilde{g}^* + \\frac{T^{2}{(F_{N})}}{2} = \\int F_{N} dT{(F_{N})}, then obtain \\int \\frac{F_{N}^{2}}{2} dF_{N} + \\int \\tilde{g}^* dF_{N} = \\iint F_{N} dT{(F_{N})} dF_{N}", "derivation": "T{(F_{N})} = F_{N} and \\int T{(F_{N})} dF_{N} = \\int F_{N} dF_{N} and \\int T{(F_{N})} dT{(F_{N})} = \\int F_{N} dT{(F_{N})} and \\tilde{g}^* + \\frac{T^{2}{(F_{N})}}{2} = \\int F_{N} dT{(F_{N})} and \\int (\\tilde{g}^* + \\frac{T^{2}{(F_{N})}}{2}) dF_{N} = \\iint F_{N} dT{(F_{N})} dF_{N} and \\int \\tilde{g}^* dF_{N} + \\int \\frac{T^{2}{(F_{N})}}{2} dF_{N} = \\iint F_{N} dT{(F_{N})} dF_{N} and \\int \\frac{F_{N}^{2}}{2} dF_{N} + \\int \\tilde{g}^* dF_{N} = \\iint F_{N} dT{(F_{N})} dF_{N}", "srepr_derivation": [["renaming_premise", "Equality(Function('T')(Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('T')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(Symbol('F_N', commutative=True), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('T')(Symbol('F_N', commutative=True)), Tuple(Function('T')(Symbol('F_N', commutative=True)))), Integral(Symbol('F_N', commutative=True), Tuple(Function('T')(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(Function('T')(Symbol('F_N', commutative=True)), Integer(2)))), Integral(Symbol('F_N', commutative=True), Tuple(Function('T')(Symbol('F_N', commutative=True)))))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\tilde{g}^*', commutative=True), Mul(Rational(1, 2), Pow(Function('T')(Symbol('F_N', commutative=True)), Integer(2)))), Tuple(Symbol('F_N', commutative=True))), Integral(Symbol('F_N', commutative=True), Tuple(Function('T')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["expand", 5], "Equality(Add(Integral(Symbol('\\\\tilde{g}^*', commutative=True), Tuple(Symbol('F_N', commutative=True))), Integral(Mul(Rational(1, 2), Pow(Function('T')(Symbol('F_N', commutative=True)), Integer(2))), Tuple(Symbol('F_N', commutative=True)))), Integral(Symbol('F_N', commutative=True), Tuple(Function('T')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Integral(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Tuple(Symbol('F_N', commutative=True))), Integral(Symbol('\\\\tilde{g}^*', commutative=True), Tuple(Symbol('F_N', commutative=True)))), Integral(Symbol('F_N', commutative=True), Tuple(Function('T')(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(C_{2},h)} = h^{C_{2}} and \\lambda{(m,V,\\psi^*)} = V - \\psi^* - m, then obtain \\operatorname{A_{1}}{(C_{2},h)} \\lambda{(m,V,\\psi^*)} = (V - \\psi^* - m) \\operatorname{A_{1}}{(C_{2},h)}", "derivation": "\\operatorname{A_{1}}{(C_{2},h)} = h^{C_{2}} and \\lambda{(m,V,\\psi^*)} = V - \\psi^* - m and h^{C_{2}} \\lambda{(m,V,\\psi^*)} = h^{C_{2}} (V - \\psi^* - m) and \\operatorname{A_{1}}{(C_{2},h)} \\lambda{(m,V,\\psi^*)} = (V - \\psi^* - m) \\operatorname{A_{1}}{(C_{2},h)}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('C_2', commutative=True), Symbol('h', commutative=True)), Pow(Symbol('h', commutative=True), Symbol('C_2', commutative=True)))"], ["get_premise", "Equality(Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))))"], [["times", 2, "Pow(Symbol('h', commutative=True), Symbol('C_2', commutative=True))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Pow(Symbol('h', commutative=True), Symbol('C_2', commutative=True)), Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('A_1')(Symbol('C_2', commutative=True), Symbol('h', commutative=True)), Function('\\\\lambda')(Symbol('m', commutative=True), Symbol('V', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Add(Symbol('V', commutative=True), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), Mul(Integer(-1), Symbol('m', commutative=True))), Function('A_1')(Symbol('C_2', commutative=True), Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(A_{y})} = e^{A_{y}}, then obtain \\frac{\\operatorname{A_{1}}{(A_{y})} e^{- A_{y}}}{A_{y} + e^{A_{y}}} = \\frac{1}{A_{y} + e^{A_{y}}}", "derivation": "\\operatorname{A_{1}}{(A_{y})} = e^{A_{y}} and A_{y} + \\operatorname{A_{1}}{(A_{y})} = A_{y} + e^{A_{y}} and \\operatorname{A_{1}}{(A_{y})} e^{- A_{y}} = 1 and \\frac{\\operatorname{A_{1}}{(A_{y})} e^{- A_{y}}}{A_{y} + \\operatorname{A_{1}}{(A_{y})}} = \\frac{1}{A_{y} + \\operatorname{A_{1}}{(A_{y})}} and \\frac{\\operatorname{A_{1}}{(A_{y})} e^{- A_{y}}}{A_{y} + e^{A_{y}}} = \\frac{1}{A_{y} + e^{A_{y}}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('A_y', commutative=True)), exp(Symbol('A_y', commutative=True)))"], [["add", 1, "Symbol('A_y', commutative=True)"], "Equality(Add(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True))), Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True))))"], [["divide", 1, "exp(Symbol('A_y', commutative=True))"], "Equality(Mul(Function('A_1')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('A_y', commutative=True)))), Integer(1))"], [["divide", 3, "Add(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True)))"], "Equality(Mul(Pow(Add(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True))), Integer(-1)), Function('A_1')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('A_y', commutative=True)))), Pow(Add(Symbol('A_y', commutative=True), Function('A_1')(Symbol('A_y', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True))), Integer(-1)), Function('A_1')(Symbol('A_y', commutative=True)), exp(Mul(Integer(-1), Symbol('A_y', commutative=True)))), Pow(Add(Symbol('A_y', commutative=True), exp(Symbol('A_y', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(s,l)} = l^{s}, then obtain - s - \\hat{\\mathbf{x}}{(s,l)} + \\frac{\\partial}{\\partial l} (l^{s} - s) + \\int (- s + \\hat{\\mathbf{x}}{(s,l)}) ds = - s - \\hat{\\mathbf{x}}{(s,l)} + \\frac{\\partial}{\\partial l} (l^{s} - s) + \\int (l^{s} - s) ds", "derivation": "\\hat{\\mathbf{x}}{(s,l)} = l^{s} and - s + \\hat{\\mathbf{x}}{(s,l)} = l^{s} - s and \\int (- s + \\hat{\\mathbf{x}}{(s,l)}) ds = \\int (l^{s} - s) ds and - s + \\int (- s + \\hat{\\mathbf{x}}{(s,l)}) ds = - s + \\int (l^{s} - s) ds and - s - \\hat{\\mathbf{x}}{(s,l)} + \\int (- s + \\hat{\\mathbf{x}}{(s,l)}) ds = - s - \\hat{\\mathbf{x}}{(s,l)} + \\int (l^{s} - s) ds and - s - \\hat{\\mathbf{x}}{(s,l)} + \\frac{\\partial}{\\partial l} (l^{s} - s) + \\int (- s + \\hat{\\mathbf{x}}{(s,l)}) ds = - s - \\hat{\\mathbf{x}}{(s,l)} + \\frac{\\partial}{\\partial l} (l^{s} - s) + \\int (l^{s} - s) ds", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True)), Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)))"], [["minus", 1, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["minus", 3, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Integral(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["minus", 4, "Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Integral(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["add", 5, "Derivative(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Derivative(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Add(Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Symbol('s', commutative=True)), Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('s', commutative=True), Symbol('l', commutative=True))), Derivative(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Integral(Add(Pow(Symbol('l', commutative=True), Symbol('s', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\rho_{f}{(\\phi_2)} = e^{\\cos{(\\phi_2)}} and \\mathbb{I}{(\\phi_2)} = (e^{\\cos{(\\phi_2)}})^{\\phi_2}, then obtain \\mathbb{I}{(\\phi_2)} \\cos{(\\phi_2)} + (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)} = 2 (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)}", "derivation": "\\rho_{f}{(\\phi_2)} = e^{\\cos{(\\phi_2)}} and \\rho_{f}^{\\phi_2}{(\\phi_2)} = (e^{\\cos{(\\phi_2)}})^{\\phi_2} and \\rho_{f}^{\\phi_2}{(\\phi_2)} \\cos{(\\phi_2)} = (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)} and \\mathbb{I}{(\\phi_2)} = (e^{\\cos{(\\phi_2)}})^{\\phi_2} and \\rho_{f}^{\\phi_2}{(\\phi_2)} \\cos{(\\phi_2)} + (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)} = 2 (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)} and \\rho_{f}^{\\phi_2}{(\\phi_2)} \\cos{(\\phi_2)} = \\mathbb{I}{(\\phi_2)} \\cos{(\\phi_2)} and \\mathbb{I}{(\\phi_2)} \\cos{(\\phi_2)} + (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)} = 2 (e^{\\cos{(\\phi_2)}})^{\\phi_2} \\cos{(\\phi_2)}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True)), exp(cos(Symbol('\\\\phi_2', commutative=True))))"], [["power", 1, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["times", 2, "cos(Symbol('\\\\phi_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True)), Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)))"], [["add", 3, "Mul(Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))"], "Equality(Add(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(2), Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Pow(Function('\\\\rho_f')(Symbol('\\\\phi_2', commutative=True)), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Add(Mul(Function('\\\\mathbb{I}')(Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))), Mul(Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(2), Pow(exp(cos(Symbol('\\\\phi_2', commutative=True))), Symbol('\\\\phi_2', commutative=True)), cos(Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\omega{(C_{1})} = \\cos{(C_{1})}, then derive \\log{(\\int \\omega{(C_{1})} dC_{1})} = \\log{(C_{d} + \\sin{(C_{1})})}, then obtain \\int \\log{(\\int \\cos{(C_{1})} dC_{1})} dC_{d} = \\int \\log{(C_{d} + \\sin{(C_{1})})} dC_{d}", "derivation": "\\omega{(C_{1})} = \\cos{(C_{1})} and \\int \\omega{(C_{1})} dC_{1} = \\int \\cos{(C_{1})} dC_{1} and \\log{(\\int \\omega{(C_{1})} dC_{1})} = \\log{(\\int \\cos{(C_{1})} dC_{1})} and \\log{(\\int \\omega{(C_{1})} dC_{1})} = \\log{(C_{d} + \\sin{(C_{1})})} and \\log{(\\int \\cos{(C_{1})} dC_{1})} = \\log{(C_{d} + \\sin{(C_{1})})} and \\int \\log{(\\int \\cos{(C_{1})} dC_{1})} dC_{d} = \\int \\log{(C_{d} + \\sin{(C_{1})})} dC_{d}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["log", 2], "Equality(log(Integral(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), log(Integral(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(log(Integral(Function('\\\\omega')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), log(Add(Symbol('C_d', commutative=True), sin(Symbol('C_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(log(Integral(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), log(Add(Symbol('C_d', commutative=True), sin(Symbol('C_1', commutative=True)))))"], [["integrate", 5, "Symbol('C_d', commutative=True)"], "Equality(Integral(log(Integral(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_d', commutative=True))), Integral(log(Add(Symbol('C_d', commutative=True), sin(Symbol('C_1', commutative=True)))), Tuple(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given V{(V,\\Omega)} = V + \\cos{(\\Omega)} and \\hat{x}{(\\Omega)} = \\cos{(\\Omega)}, then derive \\frac{T^{2}}{2} + T (V + \\hat{x}{(\\Omega)}) + z^{*} = \\frac{T^{2}}{2} + T (V + \\cos{(\\Omega)}) + \\hat{X}, then obtain T^{2} + T (V + \\hat{x}{(\\Omega)}) + T (V + \\cos{(\\Omega)}) + \\dot{z} + z^{*} = T^{2} + 2 T (V + \\cos{(\\Omega)}) + \\dot{z} + \\hat{X}", "derivation": "V{(V,\\Omega)} = V + \\cos{(\\Omega)} and \\hat{x}{(\\Omega)} = \\cos{(\\Omega)} and V{(V,\\Omega)} = V + \\hat{x}{(\\Omega)} and V + \\hat{x}{(\\Omega)} = V + \\cos{(\\Omega)} and T + V + \\hat{x}{(\\Omega)} = T + V + \\cos{(\\Omega)} and \\int (T + V + \\hat{x}{(\\Omega)}) dT = \\int (T + V + \\cos{(\\Omega)}) dT and \\frac{T^{2}}{2} + T (V + \\hat{x}{(\\Omega)}) + z^{*} = \\frac{T^{2}}{2} + T (V + \\cos{(\\Omega)}) + \\hat{X} and \\frac{T^{2}}{2} + T (V + \\hat{x}{(\\Omega)}) + z^{*} + \\int (T + V + \\cos{(\\Omega)}) dT = \\frac{T^{2}}{2} + T (V + \\cos{(\\Omega)}) + \\hat{X} + \\int (T + V + \\cos{(\\Omega)}) dT and T^{2} + T (V + \\hat{x}{(\\Omega)}) + T (V + \\cos{(\\Omega)}) + \\dot{z} + z^{*} = T^{2} + 2 T (V + \\cos{(\\Omega)}) + \\dot{z} + \\hat{X}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('V', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True)), cos(Symbol('\\\\Omega', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('V')(Symbol('V', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], [["add", 4, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True))), Add(Symbol('T', commutative=True), Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 5, "Symbol('T', commutative=True)"], "Equality(Integral(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True)))), Symbol('z^*', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"], [["add", 7, "Integral(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))"], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True)))), Symbol('z^*', commutative=True), Integral(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))), Add(Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\hat{X}', commutative=True), Integral(Add(Symbol('T', commutative=True), Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["evaluate_integrals", 8], "Equality(Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('T', commutative=True), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\dot{z}', commutative=True), Symbol('z^*', commutative=True)), Add(Pow(Symbol('T', commutative=True), Integer(2)), Mul(Integer(2), Symbol('T', commutative=True), Add(Symbol('V', commutative=True), cos(Symbol('\\\\Omega', commutative=True)))), Symbol('\\\\dot{z}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(C_{1})} = \\cos{(C_{1})}, then derive \\int \\Psi_{\\lambda}{(C_{1})} dC_{1} = J + \\sin{(C_{1})}, then obtain \\frac{d}{d J} \\int \\Psi_{\\lambda}{(C_{1})} dC_{1} = \\frac{\\partial}{\\partial J} (J + \\sin{(C_{1})})", "derivation": "\\Psi_{\\lambda}{(C_{1})} = \\cos{(C_{1})} and \\int \\Psi_{\\lambda}{(C_{1})} dC_{1} = \\int \\cos{(C_{1})} dC_{1} and \\int \\Psi_{\\lambda}{(C_{1})} dC_{1} = J + \\sin{(C_{1})} and \\frac{d}{d J} \\int \\Psi_{\\lambda}{(C_{1})} dC_{1} = \\frac{\\partial}{\\partial J} (J + \\sin{(C_{1})})", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(cos(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Add(Symbol('J', commutative=True), sin(Symbol('C_1', commutative=True))))"], [["differentiate", 3, "Symbol('J', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Add(Symbol('J', commutative=True), sin(Symbol('C_1', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(M)} = \\log{(M)} and \\operatorname{L_{\\varepsilon}}{(M)} = Z^{2}{(M)}, then obtain \\log{(M)}^{2} = Z{(M)} \\log{(M)}", "derivation": "Z{(M)} = \\log{(M)} and Z^{2}{(M)} = Z{(M)} \\log{(M)} and \\operatorname{L_{\\varepsilon}}{(M)} = Z^{2}{(M)} and \\operatorname{L_{\\varepsilon}}{(M)} = \\log{(M)}^{2} and \\operatorname{L_{\\varepsilon}}{(M)} = Z{(M)} \\log{(M)} and \\log{(M)}^{2} = Z{(M)} \\log{(M)}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True)))"], [["times", 1, "Function('Z')(Symbol('M', commutative=True))"], "Equality(Pow(Function('Z')(Symbol('M', commutative=True)), Integer(2)), Mul(Function('Z')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(Function('Z')(Symbol('M', commutative=True)), Integer(2)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('L_{\\\\varepsilon}')(Symbol('M', commutative=True)), Pow(log(Symbol('M', commutative=True)), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('L_{\\\\varepsilon}')(Symbol('M', commutative=True)), Mul(Function('Z')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(log(Symbol('M', commutative=True)), Integer(2)), Mul(Function('Z')(Symbol('M', commutative=True)), log(Symbol('M', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}}, then derive \\frac{\\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{e^{\\sin{(\\tilde{g}^*)}} \\cos{(\\tilde{g}^*)}}{\\tilde{g}^*}, then obtain \\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} \\cos{(\\tilde{g}^*)}", "derivation": "\\hat{x}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} and \\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)} = \\frac{d}{d \\tilde{g}^*} e^{\\sin{(\\tilde{g}^*)}} and \\frac{\\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{\\frac{d}{d \\tilde{g}^*} e^{\\sin{(\\tilde{g}^*)}}}{\\tilde{g}^*} and \\frac{\\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)}}{\\tilde{g}^*} = \\frac{e^{\\sin{(\\tilde{g}^*)}} \\cos{(\\tilde{g}^*)}}{\\tilde{g}^*} and \\frac{d}{d \\tilde{g}^*} \\hat{x}{(\\tilde{g}^*)} = e^{\\sin{(\\tilde{g}^*)}} \\cos{(\\tilde{g}^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}^*', commutative=True)), exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))))"], [["divide", 2, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Derivative(exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)), exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), cos(Symbol('\\\\tilde{g}^*', commutative=True))))"], [["divide", 4, "Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))"], "Equality(Derivative(Function('\\\\hat{x}')(Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('\\\\tilde{g}^*', commutative=True), Integer(1))), Mul(exp(sin(Symbol('\\\\tilde{g}^*', commutative=True))), cos(Symbol('\\\\tilde{g}^*', commutative=True))))"]]}, {"prompt": "Given A{(g,v_{y})} = \\frac{\\partial}{\\partial g} g^{v_{y}}, then derive \\log{(A{(g,v_{y})})} = \\log{(\\frac{g^{v_{y}} v_{y}}{g})}, then obtain g \\log{(A{(g,v_{y})})} = g \\log{(\\frac{g^{v_{y}} v_{y}}{g})}", "derivation": "A{(g,v_{y})} = \\frac{\\partial}{\\partial g} g^{v_{y}} and \\log{(A{(g,v_{y})})} = \\log{(\\frac{\\partial}{\\partial g} g^{v_{y}})} and \\log{(A{(g,v_{y})})} = \\log{(\\frac{g^{v_{y}} v_{y}}{g})} and g \\log{(A{(g,v_{y})})} = g \\log{(\\frac{\\partial}{\\partial g} g^{v_{y}})} and \\log{(\\frac{g^{v_{y}} v_{y}}{g})} = \\log{(\\frac{\\partial}{\\partial g} g^{v_{y}})} and g \\log{(A{(g,v_{y})})} = g \\log{(\\frac{g^{v_{y}} v_{y}}{g})}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Derivative(Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["log", 1], "Equality(log(Function('A')(Symbol('g', commutative=True), Symbol('v_y', commutative=True))), log(Derivative(Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(log(Function('A')(Symbol('g', commutative=True), Symbol('v_y', commutative=True))), log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))))"], [["divide", 2, "Pow(Symbol('g', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('g', commutative=True), log(Function('A')(Symbol('g', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('g', commutative=True), log(Derivative(Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True))), log(Derivative(Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Symbol('g', commutative=True), log(Function('A')(Symbol('g', commutative=True), Symbol('v_y', commutative=True)))), Mul(Symbol('g', commutative=True), log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Symbol('g', commutative=True), Symbol('v_y', commutative=True)), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(r,V_{\\mathbf{E}})} = r^{V_{\\mathbf{E}}} and y{(r,V_{\\mathbf{E}})} = r^{V_{\\mathbf{E}}}, then obtain V_{\\mathbf{E}} + y{(r,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\operatorname{P_{e}}{(r,V_{\\mathbf{E}})}", "derivation": "\\operatorname{P_{e}}{(r,V_{\\mathbf{E}})} = r^{V_{\\mathbf{E}}} and y{(r,V_{\\mathbf{E}})} = r^{V_{\\mathbf{E}}} and y{(r,V_{\\mathbf{E}})} = \\operatorname{P_{e}}{(r,V_{\\mathbf{E}})} and V_{\\mathbf{E}} + y{(r,V_{\\mathbf{E}})} = V_{\\mathbf{E}} + \\operatorname{P_{e}}{(r,V_{\\mathbf{E}})}", "srepr_derivation": [["premise", "Equality(Function('P_e')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('y')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Function('P_e')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["add", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('y')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('V_{\\\\mathbf{E}}', commutative=True), Function('P_e')(Symbol('r', commutative=True), Symbol('V_{\\\\mathbf{E}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(A_{y},\\hat{H}_{\\lambda})} = A_{y} \\hat{H}_{\\lambda}, then derive \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(A_{y},\\hat{H}_{\\lambda})} = A_{y}, then obtain \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} A_{y} \\hat{H}_{\\lambda} = A_{y}", "derivation": "\\operatorname{g_{\\varepsilon}}{(A_{y},\\hat{H}_{\\lambda})} = A_{y} \\hat{H}_{\\lambda} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(A_{y},\\hat{H}_{\\lambda})} = \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} A_{y} \\hat{H}_{\\lambda} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} \\operatorname{g_{\\varepsilon}}{(A_{y},\\hat{H}_{\\lambda})} = A_{y} and \\frac{\\partial}{\\partial \\hat{H}_{\\lambda}} A_{y} \\hat{H}_{\\lambda} = A_{y}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('g_{\\\\varepsilon}')(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Symbol('A_y', commutative=True))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('A_y', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Integer(1))), Symbol('A_y', commutative=True))"]]}, {"prompt": "Given \\mathbf{E}{(\\mu)} = \\log{(\\log{(\\mu)})}, then obtain (- \\log{(\\mu)} + (\\int \\mathbf{E}{(\\mu)} d\\mu)^{\\mu})^{\\mu} = (- \\log{(\\mu)} + (\\int \\log{(\\log{(\\mu)})} d\\mu)^{\\mu})^{\\mu}", "derivation": "\\mathbf{E}{(\\mu)} = \\log{(\\log{(\\mu)})} and \\int \\mathbf{E}{(\\mu)} d\\mu = \\int \\log{(\\log{(\\mu)})} d\\mu and (\\int \\mathbf{E}{(\\mu)} d\\mu)^{\\mu} = (\\int \\log{(\\log{(\\mu)})} d\\mu)^{\\mu} and - \\log{(\\mu)} + (\\int \\mathbf{E}{(\\mu)} d\\mu)^{\\mu} = - \\log{(\\mu)} + (\\int \\log{(\\log{(\\mu)})} d\\mu)^{\\mu} and (- \\log{(\\mu)} + (\\int \\mathbf{E}{(\\mu)} d\\mu)^{\\mu})^{\\mu} = (- \\log{(\\mu)} + (\\int \\log{(\\log{(\\mu)})} d\\mu)^{\\mu})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True)), log(log(Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Integral(log(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Integral(log(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["minus", 3, "log(Symbol('\\\\mu', commutative=True))"], "Equality(Add(Mul(Integer(-1), log(Symbol('\\\\mu', commutative=True))), Pow(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Add(Mul(Integer(-1), log(Symbol('\\\\mu', commutative=True))), Pow(Integral(log(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), log(Symbol('\\\\mu', commutative=True))), Pow(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Integer(-1), log(Symbol('\\\\mu', commutative=True))), Pow(Integral(log(log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(l)} = \\log{(l)}, then obtain \\log{(\\frac{d}{d l} \\operatorname{C_{2}}{(l)})} = \\log{(\\frac{1}{l})}", "derivation": "\\operatorname{C_{2}}{(l)} = \\log{(l)} and \\frac{d}{d l} \\operatorname{C_{2}}{(l)} = \\frac{d}{d l} \\log{(l)} and \\log{(\\frac{d}{d l} \\operatorname{C_{2}}{(l)})} = \\log{(\\frac{d}{d l} \\log{(l)})} and \\log{(\\frac{d}{d l} \\operatorname{C_{2}}{(l)})} = \\log{(\\frac{1}{l})}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["differentiate", 1, "Symbol('l', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["log", 2], "Equality(log(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), log(Derivative(log(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(log(Derivative(Function('C_2')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1)))), log(Pow(Symbol('l', commutative=True), Integer(-1))))"]]}, {"prompt": "Given M{(g,F_{c})} = F_{c}^{g} and \\lambda{(g,F_{c})} = \\frac{\\partial}{\\partial g} M{(g,F_{c})}, then obtain F_{c} + \\lambda{(g,F_{c})} = F_{c} + \\frac{\\partial}{\\partial g} F_{c}^{g}", "derivation": "M{(g,F_{c})} = F_{c}^{g} and \\frac{\\partial}{\\partial g} M{(g,F_{c})} = \\frac{\\partial}{\\partial g} F_{c}^{g} and F_{c} + \\frac{\\partial}{\\partial g} M{(g,F_{c})} = F_{c} + \\frac{\\partial}{\\partial g} F_{c}^{g} and \\lambda{(g,F_{c})} = \\frac{\\partial}{\\partial g} M{(g,F_{c})} and F_{c} + \\lambda{(g,F_{c})} = F_{c} + \\frac{\\partial}{\\partial g} F_{c}^{g}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('g', commutative=True), Symbol('F_c', commutative=True)), Pow(Symbol('F_c', commutative=True), Symbol('g', commutative=True)))"], [["differentiate", 1, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('M')(Symbol('g', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["add", 2, "Symbol('F_c', commutative=True)"], "Equality(Add(Symbol('F_c', commutative=True), Derivative(Function('M')(Symbol('g', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))), Add(Symbol('F_c', commutative=True), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('g', commutative=True), Symbol('F_c', commutative=True)), Derivative(Function('M')(Symbol('g', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('F_c', commutative=True), Function('\\\\lambda')(Symbol('g', commutative=True), Symbol('F_c', commutative=True))), Add(Symbol('F_c', commutative=True), Derivative(Pow(Symbol('F_c', commutative=True), Symbol('g', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1)))))"]]}, {"prompt": "Given S{(\\hbar)} = \\log{(\\hbar)}, then obtain 2 S^{\\hbar}{(\\hbar)} = - S{(\\hbar)} + 2 S^{\\hbar}{(\\hbar)} + \\log{(\\hbar)}", "derivation": "S{(\\hbar)} = \\log{(\\hbar)} and 0 = - S{(\\hbar)} + \\log{(\\hbar)} and S^{\\hbar}{(\\hbar)} = \\log{(\\hbar)}^{\\hbar} and \\log{(\\hbar)}^{\\hbar} = - S{(\\hbar)} + \\log{(\\hbar)} + \\log{(\\hbar)}^{\\hbar} and 2 \\log{(\\hbar)}^{\\hbar} = - S{(\\hbar)} + \\log{(\\hbar)} + 2 \\log{(\\hbar)}^{\\hbar} and 2 S^{\\hbar}{(\\hbar)} = - S{(\\hbar)} + 2 S^{\\hbar}{(\\hbar)} + \\log{(\\hbar)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hbar', commutative=True)), log(Symbol('\\\\hbar', commutative=True)))"], [["minus", 1, "Function('S')(Symbol('\\\\hbar', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\hbar', commutative=True))), log(Symbol('\\\\hbar', commutative=True))))"], [["power", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Function('S')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["add", 2, "Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\hbar', commutative=True))), log(Symbol('\\\\hbar', commutative=True)), Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))))"], [["add", 4, "Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))"], "Equality(Mul(Integer(2), Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\hbar', commutative=True))), log(Symbol('\\\\hbar', commutative=True)), Mul(Integer(2), Pow(log(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Integer(2), Pow(Function('S')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Function('S')(Symbol('\\\\hbar', commutative=True))), Mul(Integer(2), Pow(Function('S')(Symbol('\\\\hbar', commutative=True)), Symbol('\\\\hbar', commutative=True))), log(Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given t{(l)} = \\log{(l)}, then derive \\frac{d}{d l} t{(l)} - 1 = -1 + \\frac{1}{l}, then obtain (\\frac{d}{d l} t{(l)} - 1) t^{- l}{(l)} = (-1 + \\frac{1}{l}) t^{- l}{(l)}", "derivation": "t{(l)} = \\log{(l)} and - l + t{(l)} = - l + \\log{(l)} and \\frac{d}{d l} (- l + t{(l)}) = \\frac{d}{d l} (- l + \\log{(l)}) and \\frac{d}{d l} t{(l)} - 1 = -1 + \\frac{1}{l} and (\\frac{d}{d l} t{(l)} - 1) \\log{(l)}^{- l} = (-1 + \\frac{1}{l}) \\log{(l)}^{- l} and (\\frac{d}{d l} t{(l)} - 1) t^{- l}{(l)} = (-1 + \\frac{1}{l}) t^{- l}{(l)}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('l', commutative=True)), log(Symbol('l', commutative=True)))"], [["minus", 1, "Symbol('l', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('t')(Symbol('l', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), log(Symbol('l', commutative=True))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Function('t')(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('l', commutative=True)), log(Symbol('l', commutative=True))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('t')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Add(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1))))"], [["divide", 4, "Pow(log(Symbol('l', commutative=True)), Symbol('l', commutative=True))"], "Equality(Mul(Add(Derivative(Function('t')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Pow(log(Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Add(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1))), Pow(log(Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Add(Derivative(Function('t')(Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True), Integer(1))), Integer(-1)), Pow(Function('t')(Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))), Mul(Add(Integer(-1), Pow(Symbol('l', commutative=True), Integer(-1))), Pow(Function('t')(Symbol('l', commutative=True)), Mul(Integer(-1), Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(\\delta,Q)} = \\frac{\\delta}{Q}, then obtain \\operatorname{P_{g}}{(\\delta,Q)} + \\int \\frac{\\delta}{Q} dQ = \\int \\frac{\\delta}{Q} dQ + \\frac{\\delta}{Q}", "derivation": "\\operatorname{P_{g}}{(\\delta,Q)} = \\frac{\\delta}{Q} and \\operatorname{P_{g}}{(\\delta,Q)} - \\frac{\\delta}{Q} = 0 and \\int \\operatorname{P_{g}}{(\\delta,Q)} dQ = \\int \\frac{\\delta}{Q} dQ and \\operatorname{P_{g}}{(\\delta,Q)} + \\int \\operatorname{P_{g}}{(\\delta,Q)} dQ - \\frac{\\delta}{Q} = \\int \\operatorname{P_{g}}{(\\delta,Q)} dQ and \\operatorname{P_{g}}{(\\delta,Q)} + \\int \\operatorname{P_{g}}{(\\delta,Q)} dQ = \\int \\operatorname{P_{g}}{(\\delta,Q)} dQ + \\frac{\\delta}{Q} and \\operatorname{P_{g}}{(\\delta,Q)} + \\int \\frac{\\delta}{Q} dQ = \\int \\frac{\\delta}{Q} dQ + \\frac{\\delta}{Q}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)))"], [["minus", 1, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))), Integer(0))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["add", 2, "Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))"], "Equality(Add(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Mul(Integer(-1), Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))), Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"], [["add", 4, "Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))"], "Equality(Add(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Integral(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Add(Function('P_g')(Symbol('\\\\delta', commutative=True), Symbol('Q', commutative=True)), Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('Q', commutative=True)))), Add(Integral(Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('Q', commutative=True))), Mul(Pow(Symbol('Q', commutative=True), Integer(-1)), Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f^{*}}{(C_{d},T)} = C_{d} + T and \\Psi_{nl}{(C_{d},T)} = T (C_{d} + T), then obtain \\int - h \\int \\Psi_{nl}{(C_{d},T)} dT dh = \\int - h \\int T \\operatorname{f^{*}}{(C_{d},T)} dT dh", "derivation": "\\operatorname{f^{*}}{(C_{d},T)} = C_{d} + T and \\Psi_{nl}{(C_{d},T)} = T (C_{d} + T) and \\Psi_{nl}{(C_{d},T)} = T \\operatorname{f^{*}}{(C_{d},T)} and \\int \\Psi_{nl}{(C_{d},T)} dT = \\int T \\operatorname{f^{*}}{(C_{d},T)} dT and - h \\int \\Psi_{nl}{(C_{d},T)} dT = - h \\int T \\operatorname{f^{*}}{(C_{d},T)} dT and \\int - h \\int \\Psi_{nl}{(C_{d},T)} dT dh = \\int - h \\int T \\operatorname{f^{*}}{(C_{d},T)} dT dh", "srepr_derivation": [["premise", "Equality(Function('f^*')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Add(Symbol('C_d', commutative=True), Symbol('T', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Mul(Symbol('T', commutative=True), Function('f^*')(Symbol('C_d', commutative=True), Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('T', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Mul(Symbol('T', commutative=True), Function('f^*')(Symbol('C_d', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Symbol('h', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('h', commutative=True), Integral(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Integer(-1), Symbol('h', commutative=True), Integral(Mul(Symbol('T', commutative=True), Function('f^*')(Symbol('C_d', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))))"], [["integrate", 5, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Symbol('h', commutative=True), Integral(Function('\\\\Psi_{nl}')(Symbol('C_d', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('h', commutative=True))), Integral(Mul(Integer(-1), Symbol('h', commutative=True), Integral(Mul(Symbol('T', commutative=True), Function('f^*')(Symbol('C_d', commutative=True), Symbol('T', commutative=True))), Tuple(Symbol('T', commutative=True)))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(a,\\hat{p}_0)} = \\hat{p}_0 e^{a}, then obtain g e^{\\Psi_{\\lambda}} \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} \\operatorname{f_{E}}^{a}{(a,\\hat{p}_0)} = g e^{\\Psi_{\\lambda}} \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} (\\hat{p}_0 e^{a})^{a}", "derivation": "\\operatorname{f_{E}}{(a,\\hat{p}_0)} = \\hat{p}_0 e^{a} and \\operatorname{f_{E}}^{a}{(a,\\hat{p}_0)} = (\\hat{p}_0 e^{a})^{a} and \\frac{\\partial}{\\partial \\hat{p}_0} \\operatorname{f_{E}}^{a}{(a,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\hat{p}_0} (\\hat{p}_0 e^{a})^{a} and \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} \\operatorname{f_{E}}^{a}{(a,\\hat{p}_0)} = \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} (\\hat{p}_0 e^{a})^{a} and g e^{\\Psi_{\\lambda}} \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} \\operatorname{f_{E}}^{a}{(a,\\hat{p}_0)} = g e^{\\Psi_{\\lambda}} \\frac{\\partial^{2}}{\\partial a\\partial \\hat{p}_0} (\\hat{p}_0 e^{a})^{a}", "srepr_derivation": [["get_premise", "Equality(Function('f_E')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('a', commutative=True))))"], [["power", 1, "Symbol('a', commutative=True)"], "Equality(Pow(Function('f_E')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('a', commutative=True)), Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Pow(Function('f_E')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('a', commutative=True)"], "Equality(Derivative(Pow(Function('f_E')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["times", 4, "Mul(Symbol('g', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Symbol('g', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Derivative(Pow(Function('f_E')(Symbol('a', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Symbol('g', commutative=True), exp(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Derivative(Pow(Mul(Symbol('\\\\hat{p}_0', commutative=True), exp(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)), Tuple(Symbol('a', commutative=True), Integer(1)))))"]]}, {"prompt": "Given l{(r,M)} = M r, then derive - M + \\frac{\\partial}{\\partial r} l{(r,M)} = 0, then obtain \\frac{\\partial}{\\partial M} (- M + \\frac{\\partial}{\\partial r} M r + 1) = \\frac{d}{d M} 1", "derivation": "l{(r,M)} = M r and - M r + l{(r,M)} = 0 and \\frac{\\partial}{\\partial r} (- M r + l{(r,M)}) = \\frac{d}{d r} 0 and - M + \\frac{\\partial}{\\partial r} l{(r,M)} = 0 and - M + \\frac{\\partial}{\\partial r} M r = 0 and - M + \\frac{\\partial}{\\partial r} M r + 1 = 1 and \\frac{\\partial}{\\partial M} (- M + \\frac{\\partial}{\\partial r} M r + 1) = \\frac{d}{d M} 1", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Mul(Symbol('M', commutative=True), Symbol('r', commutative=True)))"], [["minus", 1, "Mul(Symbol('M', commutative=True), Symbol('r', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('r', commutative=True)), Function('l')(Symbol('r', commutative=True), Symbol('M', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('r', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True), Symbol('r', commutative=True)), Function('l')(Symbol('r', commutative=True), Symbol('M', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Derivative(Function('l')(Symbol('r', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Derivative(Mul(Symbol('M', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1)))), Integer(0))"], [["add", 5, 1], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Derivative(Mul(Symbol('M', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Integer(1))"], [["differentiate", 6, "Symbol('M', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('M', commutative=True)), Derivative(Mul(Symbol('M', commutative=True), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(1)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(u,\\mathbf{S})} = \\frac{\\mathbf{S}}{u}, then obtain ((\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} - \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})})^{\\mathbf{S}} \\dot{y}{(u,\\mathbf{S})} = \\dot{y}{(u,\\mathbf{S})}", "derivation": "\\dot{y}{(u,\\mathbf{S})} = \\frac{\\mathbf{S}}{u} and \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})} = (\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} and 0 = (\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} - \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})} and 0^{\\mathbf{S}} = ((\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} - \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})})^{\\mathbf{S}} and 0^{\\mathbf{S}} \\dot{y}{(u,\\mathbf{S})} = ((\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} - \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})})^{\\mathbf{S}} \\dot{y}{(u,\\mathbf{S})} and ((\\frac{\\mathbf{S}}{u})^{\\mathbf{S}} - \\dot{y}^{\\mathbf{S}}{(u,\\mathbf{S})})^{\\mathbf{S}} \\dot{y}{(u,\\mathbf{S})} = \\dot{y}{(u,\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))))"], [["power", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["minus", 2, "Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Integer(0), Add(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["power", 3, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 4, "Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Pow(Integer(0), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Mul(Pow(Add(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Pow(Add(Pow(Mul(Symbol('\\\\mathbf{S}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('\\\\mathbf{S}', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))), Symbol('\\\\mathbf{S}', commutative=True)), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Function('\\\\dot{y}')(Symbol('u', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\eta)} = \\eta, then derive \\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)} = 1, then obtain \\frac{\\partial}{\\partial \\eta} (\\eta^{2} \\log{(\\Psi_{\\lambda})} + (\\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)})^{\\eta}) = \\frac{\\partial}{\\partial \\eta} (\\eta^{2} \\log{(\\Psi_{\\lambda})} + 1)", "derivation": "\\Psi_{nl}{(\\eta)} = \\eta and \\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)} = \\frac{d}{d \\eta} \\eta and \\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)} = 1 and (\\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)})^{\\eta} = 1 and \\eta^{2} \\log{(\\Psi_{\\lambda})} + (\\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)})^{\\eta} = \\eta^{2} \\log{(\\Psi_{\\lambda})} + 1 and \\frac{\\partial}{\\partial \\eta} (\\eta^{2} \\log{(\\Psi_{\\lambda})} + (\\frac{d}{d \\eta} \\Psi_{nl}{(\\eta)})^{\\eta}) = \\frac{\\partial}{\\partial \\eta} (\\eta^{2} \\log{(\\Psi_{\\lambda})} + 1)", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Symbol('\\\\eta', commutative=True), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\eta', commutative=True)), Integer(1))"], [["add", 4, "Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\eta', commutative=True))), Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)))"], [["differentiate", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Pow(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('\\\\eta', commutative=True), Integer(2)), log(Symbol('\\\\Psi_{\\\\lambda}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{g^{\\prime}_{\\varepsilon}}{(u,b,\\mathbf{J})} = (\\mathbf{J}^{u})^{b} and z{(u,b,\\mathbf{J})} = (\\mathbf{J}^{u})^{b}, then obtain - \\frac{\\partial}{\\partial u} (\\mathbf{J}^{u})^{b} \\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})} = - (\\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})})^{2}", "derivation": "\\operatorname{g^{\\prime}_{\\varepsilon}}{(u,b,\\mathbf{J})} = (\\mathbf{J}^{u})^{b} and \\frac{\\partial}{\\partial u} \\operatorname{g^{\\prime}_{\\varepsilon}}{(u,b,\\mathbf{J})} = \\frac{\\partial}{\\partial u} (\\mathbf{J}^{u})^{b} and z{(u,b,\\mathbf{J})} = (\\mathbf{J}^{u})^{b} and \\frac{\\partial}{\\partial u} \\operatorname{g^{\\prime}_{\\varepsilon}}{(u,b,\\mathbf{J})} = \\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})} and - \\frac{\\partial}{\\partial u} \\operatorname{g^{\\prime}_{\\varepsilon}}{(u,b,\\mathbf{J})} \\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})} = - (\\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})})^{2} and - \\frac{\\partial}{\\partial u} (\\mathbf{J}^{u})^{b} \\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})} = - (\\frac{\\partial}{\\partial u} z{(u,b,\\mathbf{J})})^{2}", "srepr_derivation": [["premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Symbol('b', commutative=True)))"], [["differentiate", 1, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Symbol('b', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], [["times", 4, "Mul(Integer(-1), Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"], "Equality(Mul(Integer(-1), Derivative(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(2))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Derivative(Pow(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('u', commutative=True)), Symbol('b', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Derivative(Function('z')(Symbol('u', commutative=True), Symbol('b', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Integer(2))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda}, then derive \\Psi^{\\dagger}^{E_{\\lambda}}{(E_{\\lambda})} = (\\varphi + \\sin{(E_{\\lambda})})^{E_{\\lambda}}, then obtain (\\Psi^{\\dagger}^{E_{\\lambda}}{(E_{\\lambda})})^{\\varphi} = ((\\varphi + \\sin{(E_{\\lambda})})^{E_{\\lambda}})^{\\varphi}", "derivation": "\\Psi^{\\dagger}{(E_{\\lambda})} = \\int \\cos{(E_{\\lambda})} dE_{\\lambda} and \\Psi^{\\dagger}^{E_{\\lambda}}{(E_{\\lambda})} = (\\int \\cos{(E_{\\lambda})} dE_{\\lambda})^{E_{\\lambda}} and \\Psi^{\\dagger}^{E_{\\lambda}}{(E_{\\lambda})} = (\\varphi + \\sin{(E_{\\lambda})})^{E_{\\lambda}} and (\\Psi^{\\dagger}^{E_{\\lambda}}{(E_{\\lambda})})^{\\varphi} = ((\\varphi + \\sin{(E_{\\lambda})})^{E_{\\lambda}})^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True)), Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Integral(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["evaluate_integrals", 2], "Equality(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"], [["power", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('E_{\\\\lambda}', commutative=True)), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(Pow(Add(Symbol('\\\\varphi', commutative=True), sin(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(\\Psi)} = \\cos{(\\Psi)}, then derive \\int 0 d\\Psi = P_{e} - 2 \\int \\hat{H}{(\\Psi)} d\\Psi - 2 \\int - \\cos{(\\Psi)} d\\Psi, then obtain \\int 0 d\\Psi = P_{e} - 2 \\int - \\cos{(\\Psi)} d\\Psi - 2 \\int \\cos{(\\Psi)} d\\Psi", "derivation": "\\hat{H}{(\\Psi)} = \\cos{(\\Psi)} and 0 = - \\hat{H}{(\\Psi)} + \\cos{(\\Psi)} and - \\hat{H}{(\\Psi)} + \\cos{(\\Psi)} = - 2 \\hat{H}{(\\Psi)} + 2 \\cos{(\\Psi)} and 0 = - 2 \\hat{H}{(\\Psi)} + 2 \\cos{(\\Psi)} and \\int 0 d\\Psi = \\int (- 2 \\hat{H}{(\\Psi)} + 2 \\cos{(\\Psi)}) d\\Psi and \\int 0 d\\Psi = P_{e} - 2 \\int \\hat{H}{(\\Psi)} d\\Psi - 2 \\int - \\cos{(\\Psi)} d\\Psi and \\int 0 d\\Psi = P_{e} - 2 \\int - \\cos{(\\Psi)} d\\Psi - 2 \\int \\cos{(\\Psi)} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True)), cos(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))))"], [["add", 2, "Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), cos(Symbol('\\\\Psi', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi', commutative=True)))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\Psi', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Integer(2), Add(Integral(Function('\\\\hat{H}')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True))), Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Integer(2), Integral(Mul(Integer(-1), cos(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True)))), Mul(Integer(-1), Integer(2), Integral(cos(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))))))"]]}, {"prompt": "Given \\tilde{g}^*{(A_{x},n_{2},f^{\\prime})} = A_{x} n_{2} + f^{\\prime}, then obtain e^{n_{2} + \\tilde{g}^*^{n_{2}}{(A_{x},n_{2},f^{\\prime})}} = e^{n_{2} + (A_{x} n_{2} + f^{\\prime})^{n_{2}}}", "derivation": "\\tilde{g}^*{(A_{x},n_{2},f^{\\prime})} = A_{x} n_{2} + f^{\\prime} and \\tilde{g}^*^{n_{2}}{(A_{x},n_{2},f^{\\prime})} = (A_{x} n_{2} + f^{\\prime})^{n_{2}} and n_{2} + \\tilde{g}^*^{n_{2}}{(A_{x},n_{2},f^{\\prime})} = n_{2} + (A_{x} n_{2} + f^{\\prime})^{n_{2}} and e^{n_{2} + \\tilde{g}^*^{n_{2}}{(A_{x},n_{2},f^{\\prime})}} = e^{n_{2} + (A_{x} n_{2} + f^{\\prime})^{n_{2}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)))"], [["power", 1, "Symbol('n_2', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True)), Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True)))"], [["add", 2, "Symbol('n_2', commutative=True)"], "Equality(Add(Symbol('n_2', commutative=True), Pow(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True))), Add(Symbol('n_2', commutative=True), Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True))))"], [["exp", 3], "Equality(exp(Add(Symbol('n_2', commutative=True), Pow(Function('\\\\tilde{g}^*')(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True)))), exp(Add(Symbol('n_2', commutative=True), Pow(Add(Mul(Symbol('A_x', commutative=True), Symbol('n_2', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Symbol('n_2', commutative=True)))))"]]}, {"prompt": "Given B{(\\eta)} = e^{\\eta}, then derive \\frac{d}{d \\eta} B{(\\eta)} = e^{\\eta}, then obtain e^{\\eta} + \\frac{d}{d \\eta} B{(\\eta)} = B{(\\eta)} + e^{\\eta}", "derivation": "B{(\\eta)} = e^{\\eta} and \\frac{d}{d \\eta} B{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\frac{d}{d \\eta} B{(\\eta)} = e^{\\eta} and \\frac{d}{d \\eta} B{(\\eta)} = B{(\\eta)} and \\frac{d}{d \\eta} B{(\\eta)} + \\frac{d}{d \\eta} e^{\\eta} = B{(\\eta)} + \\frac{d}{d \\eta} e^{\\eta} and e^{\\eta} + \\frac{d}{d \\eta} B{(\\eta)} = B{(\\eta)} + e^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Function('B')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('B')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), exp(Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('B')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Function('B')(Symbol('\\\\eta', commutative=True)))"], [["add", 4, "Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('B')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Function('B')(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(Function('B')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(Function('B')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(f_{E},y)} = \\frac{\\partial}{\\partial f_{E}} f_{E} y and \\hat{H}_l{(f_{E},y)} = \\frac{\\partial}{\\partial f_{E}} f_{E} y, then obtain \\int (\\hat{H}_l{(f_{E},y)} - 1) df_{E} = \\int (\\frac{\\partial}{\\partial f_{E}} f_{E} y - 1) df_{E}", "derivation": "\\hat{H}_{\\lambda}{(f_{E},y)} = \\frac{\\partial}{\\partial f_{E}} f_{E} y and \\hat{H}_l{(f_{E},y)} = \\frac{\\partial}{\\partial f_{E}} f_{E} y and \\hat{H}_l{(f_{E},y)} = \\hat{H}_{\\lambda}{(f_{E},y)} and \\hat{H}_l{(f_{E},y)} - 1 = \\hat{H}_{\\lambda}{(f_{E},y)} - 1 and \\hat{H}_l{(f_{E},y)} - 1 = \\frac{\\partial}{\\partial f_{E}} f_{E} y - 1 and \\int (\\hat{H}_l{(f_{E},y)} - 1) df_{E} = \\int (\\frac{\\partial}{\\partial f_{E}} f_{E} y - 1) df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Derivative(Mul(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Derivative(Mul(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\hat{H}_l')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)))"], [["add", 3, "Integer(-1)"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Add(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Add(Derivative(Mul(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 5, "Symbol('f_E', commutative=True)"], "Equality(Integral(Add(Function('\\\\hat{H}_l')(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Integer(-1)), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Derivative(Mul(Symbol('f_E', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('f_E', commutative=True))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(n)} = \\int e^{n} dn, then derive \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\frac{\\partial}{\\partial n} (v_{t} + e^{n}), then obtain \\frac{\\partial}{\\partial n} (v_{t} + e^{n}) = \\frac{\\partial}{\\partial n} (t_{2} + e^{n})", "derivation": "\\operatorname{m_{s}}{(n)} = \\int e^{n} dn and \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\frac{d}{d n} \\int e^{n} dn and \\frac{d}{d n} \\operatorname{m_{s}}{(n)} = \\frac{\\partial}{\\partial n} (v_{t} + e^{n}) and \\frac{\\partial}{\\partial n} (v_{t} + e^{n}) = \\frac{d}{d n} \\int e^{n} dn and \\frac{\\partial}{\\partial n} (v_{t} + e^{n}) = \\frac{\\partial}{\\partial n} (t_{2} + e^{n})", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('n', commutative=True)), Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('m_s')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('v_t', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('v_t', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('v_t', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Add(Symbol('t_2', commutative=True), exp(Symbol('n', commutative=True))), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{M}{(A,\\mathbf{J}_f)} = e^{A^{\\mathbf{J}_f}} and U{(A,\\mathbf{J}_f)} = A^{\\mathbf{J}_f}, then obtain \\mathbf{M}{(A,\\mathbf{J}_f)} + e^{A^{\\mathbf{J}_f}} = 2 \\mathbf{M}{(A,\\mathbf{J}_f)}", "derivation": "\\mathbf{M}{(A,\\mathbf{J}_f)} = e^{A^{\\mathbf{J}_f}} and \\mathbf{M}{(A,\\mathbf{J}_f)} + e^{A^{\\mathbf{J}_f}} = 2 e^{A^{\\mathbf{J}_f}} and U{(A,\\mathbf{J}_f)} = A^{\\mathbf{J}_f} and \\mathbf{M}{(A,\\mathbf{J}_f)} = e^{U{(A,\\mathbf{J}_f)}} and \\mathbf{M}{(A,\\mathbf{J}_f)} + e^{U{(A,\\mathbf{J}_f)}} = 2 e^{U{(A,\\mathbf{J}_f)}} and e^{A^{\\mathbf{J}_f}} + e^{U{(A,\\mathbf{J}_f)}} = 2 e^{U{(A,\\mathbf{J}_f)}} and \\mathbf{M}{(A,\\mathbf{J}_f)} + e^{A^{\\mathbf{J}_f}} = 2 \\mathbf{M}{(A,\\mathbf{J}_f)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["add", 1, "exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], ["renaming_premise", "Equality(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), exp(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), exp(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), exp(Function('U')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Add(Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), exp(Pow(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))), Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('A', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))))"]]}, {"prompt": "Given \\ddot{x}{(c,\\omega)} = \\omega + c and \\operatorname{n_{2}}{(c,\\omega)} = \\omega + c + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1, then obtain \\frac{\\partial}{\\partial c} (\\omega + c) = \\frac{\\partial}{\\partial c} (\\omega + c) + \\frac{\\ddot{x}{(c,\\omega)} - \\operatorname{n_{2}}{(c,\\omega)} + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1}{\\omega}", "derivation": "\\ddot{x}{(c,\\omega)} = \\omega + c and \\operatorname{n_{2}}{(c,\\omega)} = \\omega + c + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1 and 0 = \\omega + c - \\operatorname{n_{2}}{(c,\\omega)} + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1 and 0 = \\ddot{x}{(c,\\omega)} - \\operatorname{n_{2}}{(c,\\omega)} + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1 and 0 = \\frac{\\ddot{x}{(c,\\omega)} - \\operatorname{n_{2}}{(c,\\omega)} + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1}{\\omega} and \\frac{\\partial}{\\partial c} (\\omega + c) = \\frac{\\partial}{\\partial c} (\\omega + c) + \\frac{\\ddot{x}{(c,\\omega)} - \\operatorname{n_{2}}{(c,\\omega)} + \\frac{\\partial}{\\partial c} \\ddot{x}{(c,\\omega)} + 1}{\\omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)))"], ["renaming_premise", "Equality(Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Derivative(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(1)))"], [["minus", 2, "Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True), Mul(Integer(-1), Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(1)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integer(0), Add(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(1)))"], [["divide", 4, "Symbol('\\\\omega', commutative=True)"], "Equality(Integer(0), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(1))))"], [["add", 5, "Derivative(Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1)))"], "Equality(Derivative(Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Add(Derivative(Add(Symbol('\\\\omega', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Add(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Integer(-1), Function('n_2')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True))), Derivative(Function('\\\\ddot{x}')(Symbol('c', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('c', commutative=True), Integer(1))), Integer(1)))))"]]}, {"prompt": "Given b{(a)} = \\int e^{a} da, then derive \\frac{d}{d a} b{(a)} = \\frac{\\partial}{\\partial a} (\\tilde{g}^* + e^{a}), then derive \\frac{\\partial}{\\partial a} (\\tilde{g}^* + e^{a}) = \\frac{\\partial}{\\partial a} (\\mathbf{D} + e^{a}), then obtain (\\frac{d}{d a} b{(a)})^{a} = (\\frac{\\partial}{\\partial a} (\\mathbf{D} + e^{a}))^{a}", "derivation": "b{(a)} = \\int e^{a} da and \\frac{d}{d a} b{(a)} = \\frac{d}{d a} \\int e^{a} da and \\frac{d}{d a} b{(a)} = \\frac{\\partial}{\\partial a} (\\tilde{g}^* + e^{a}) and \\frac{\\partial}{\\partial a} (\\tilde{g}^* + e^{a}) = \\frac{d}{d a} \\int e^{a} da and \\frac{\\partial}{\\partial a} (\\tilde{g}^* + e^{a}) = \\frac{\\partial}{\\partial a} (\\mathbf{D} + e^{a}) and \\frac{d}{d a} b{(a)} = \\frac{\\partial}{\\partial a} (\\mathbf{D} + e^{a}) and (\\frac{d}{d a} b{(a)})^{a} = (\\frac{\\partial}{\\partial a} (\\mathbf{D} + e^{a}))^{a}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('a', commutative=True)), Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_integrals", 2], "Equality(Derivative(Function('b')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Integral(exp(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_integrals", 4], "Equality(Derivative(Add(Symbol('\\\\tilde{g}^*', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Derivative(Function('b')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["power", 6, "Symbol('a', commutative=True)"], "Equality(Pow(Derivative(Function('b')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)), Pow(Derivative(Add(Symbol('\\\\mathbf{D}', commutative=True), exp(Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(f^{*},M)} = \\cos{(M f^{*})}, then derive \\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)} = - f^{*} \\sin{(M f^{*})}, then obtain M f^{*} \\cos{(\\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)})} = M f^{*} \\cos{(f^{*} \\sin{(M f^{*})})}", "derivation": "\\operatorname{v_{x}}{(f^{*},M)} = \\cos{(M f^{*})} and \\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)} = \\frac{\\partial}{\\partial M} \\cos{(M f^{*})} and \\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)} = - f^{*} \\sin{(M f^{*})} and \\cos{(\\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)})} = \\cos{(f^{*} \\sin{(M f^{*})})} and M f^{*} \\cos{(\\frac{\\partial}{\\partial M} \\operatorname{v_{x}}{(f^{*},M)})} = M f^{*} \\cos{(f^{*} \\sin{(M f^{*})})}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('f^*', commutative=True), Symbol('M', commutative=True)), cos(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True))))"], [["differentiate", 1, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(cos(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True))), Tuple(Symbol('M', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('f^*', commutative=True), sin(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True)))))"], [["cos", 3], "Equality(cos(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1)))), cos(Mul(Symbol('f^*', commutative=True), sin(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True))))))"], [["times", 4, "Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True), cos(Derivative(Function('v_x')(Symbol('f^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))), Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True), cos(Mul(Symbol('f^*', commutative=True), sin(Mul(Symbol('M', commutative=True), Symbol('f^*', commutative=True)))))))"]]}, {"prompt": "Given S{(\\sigma_p,\\omega)} = \\sin{(\\omega + \\sigma_p)}, then derive \\sigma_p + \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)} = \\sigma_p + \\cos{(\\omega + \\sigma_p)}, then obtain (\\sigma_p + \\cos{(\\omega + \\sigma_p)})^{2} = (\\sigma_p + \\cos{(\\omega + \\sigma_p)}) (\\sigma_p + \\frac{\\partial}{\\partial \\omega} \\sin{(\\omega + \\sigma_p)})", "derivation": "S{(\\sigma_p,\\omega)} = \\sin{(\\omega + \\sigma_p)} and \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)} = \\frac{\\partial}{\\partial \\omega} \\sin{(\\omega + \\sigma_p)} and \\sigma_p + \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)} = \\sigma_p + \\frac{\\partial}{\\partial \\omega} \\sin{(\\omega + \\sigma_p)} and \\sigma_p + \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)} = \\sigma_p + \\cos{(\\omega + \\sigma_p)} and (\\sigma_p + \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)})^{2} = (\\sigma_p + \\frac{\\partial}{\\partial \\omega} S{(\\sigma_p,\\omega)}) (\\sigma_p + \\frac{\\partial}{\\partial \\omega} \\sin{(\\omega + \\sigma_p)}) and (\\sigma_p + \\cos{(\\omega + \\sigma_p)})^{2} = (\\sigma_p + \\cos{(\\omega + \\sigma_p)}) (\\sigma_p + \\frac{\\partial}{\\partial \\omega} \\sin{(\\omega + \\sigma_p)})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), sin(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], [["add", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Symbol('\\\\sigma_p', commutative=True), Derivative(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True)))))"], [["times", 3, "Add(Symbol('\\\\sigma_p', commutative=True), Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))"], "Equality(Pow(Add(Symbol('\\\\sigma_p', commutative=True), Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Integer(2)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Derivative(Function('S')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True), Integer(1)))), Add(Symbol('\\\\sigma_p', commutative=True), Derivative(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Add(Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Integer(2)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), cos(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True)))), Add(Symbol('\\\\sigma_p', commutative=True), Derivative(sin(Add(Symbol('\\\\omega', commutative=True), Symbol('\\\\sigma_p', commutative=True))), Tuple(Symbol('\\\\omega', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\psi^{*}{(\\hat{p})} = \\sin{(\\hat{p})}, then obtain \\int (\\frac{\\psi^{*}{(\\hat{p})}}{\\sin{(\\hat{p})}} + \\cos{(\\sin{(\\hat{p})})}) d\\hat{p} = \\int (\\cos{(\\sin{(\\hat{p})})} + 1) d\\hat{p}", "derivation": "\\psi^{*}{(\\hat{p})} = \\sin{(\\hat{p})} and \\frac{\\psi^{*}{(\\hat{p})}}{\\sin{(\\hat{p})}} = 1 and \\frac{\\psi^{*}{(\\hat{p})}}{\\sin{(\\hat{p})}} + \\cos{(\\sin{(\\hat{p})})} = \\cos{(\\sin{(\\hat{p})})} + 1 and \\int (\\frac{\\psi^{*}{(\\hat{p})}}{\\sin{(\\hat{p})}} + \\cos{(\\sin{(\\hat{p})})}) d\\hat{p} = \\int (\\cos{(\\sin{(\\hat{p})})} + 1) d\\hat{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\hat{p}', commutative=True)))"], [["divide", 1, "sin(Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Function('\\\\psi^*')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(Symbol('\\\\hat{p}', commutative=True)), Integer(-1))), Integer(1))"], [["add", 2, "cos(sin(Symbol('\\\\hat{p}', commutative=True)))"], "Equality(Add(Mul(Function('\\\\psi^*')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(Symbol('\\\\hat{p}', commutative=True)), Integer(-1))), cos(sin(Symbol('\\\\hat{p}', commutative=True)))), Add(cos(sin(Symbol('\\\\hat{p}', commutative=True))), Integer(1)))"], [["integrate", 3, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integral(Add(Mul(Function('\\\\psi^*')(Symbol('\\\\hat{p}', commutative=True)), Pow(sin(Symbol('\\\\hat{p}', commutative=True)), Integer(-1))), cos(sin(Symbol('\\\\hat{p}', commutative=True)))), Tuple(Symbol('\\\\hat{p}', commutative=True))), Integral(Add(cos(sin(Symbol('\\\\hat{p}', commutative=True))), Integer(1)), Tuple(Symbol('\\\\hat{p}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(v_{y},T)} = T^{v_{y}}, then obtain (\\psi^{*}^{T}{(v_{y},T)})^{3 v_{y}} = ((T^{v_{y}})^{T})^{v_{y}} (\\psi^{*}^{T}{(v_{y},T)})^{2 v_{y}}", "derivation": "\\psi^{*}{(v_{y},T)} = T^{v_{y}} and \\psi^{*}^{T}{(v_{y},T)} = (T^{v_{y}})^{T} and (\\psi^{*}^{T}{(v_{y},T)})^{v_{y}} = ((T^{v_{y}})^{T})^{v_{y}} and (\\psi^{*}^{T}{(v_{y},T)})^{2 v_{y}} = ((T^{v_{y}})^{T})^{v_{y}} (\\psi^{*}^{T}{(v_{y},T)})^{v_{y}} and (\\psi^{*}^{T}{(v_{y},T)})^{3 v_{y}} = ((T^{v_{y}})^{T})^{v_{y}} (\\psi^{*}^{T}{(v_{y},T)})^{2 v_{y}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Pow(Symbol('T', commutative=True), Symbol('v_y', commutative=True)))"], [["power", 1, "Symbol('T', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Pow(Pow(Symbol('T', commutative=True), Symbol('v_y', commutative=True)), Symbol('T', commutative=True)))"], [["power", 2, "Symbol('v_y', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True)), Pow(Pow(Pow(Symbol('T', commutative=True), Symbol('v_y', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True)))"], [["times", 3, "Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True))), Mul(Pow(Pow(Pow(Symbol('T', commutative=True), Symbol('v_y', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True)), Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True))))"], [["times", 4, "Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True))"], "Equality(Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(3), Symbol('v_y', commutative=True))), Mul(Pow(Pow(Pow(Symbol('T', commutative=True), Symbol('v_y', commutative=True)), Symbol('T', commutative=True)), Symbol('v_y', commutative=True)), Pow(Pow(Function('\\\\psi^*')(Symbol('v_y', commutative=True), Symbol('T', commutative=True)), Symbol('T', commutative=True)), Mul(Integer(2), Symbol('v_y', commutative=True)))))"]]}, {"prompt": "Given C{(g_{\\varepsilon})} = g_{\\varepsilon}, then derive \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = B + \\frac{g_{\\varepsilon}^{2}}{2}, then obtain B + \\frac{g_{\\varepsilon}^{2}}{2} - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}} = \\int g_{\\varepsilon} dg_{\\varepsilon} - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}}", "derivation": "C{(g_{\\varepsilon})} = g_{\\varepsilon} and \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int g_{\\varepsilon} dg_{\\varepsilon} and - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}} + \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = \\int g_{\\varepsilon} dg_{\\varepsilon} - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}} and \\int C{(g_{\\varepsilon})} dg_{\\varepsilon} = B + \\frac{g_{\\varepsilon}^{2}}{2} and B + \\frac{g_{\\varepsilon}^{2}}{2} - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}} = \\int g_{\\varepsilon} dg_{\\varepsilon} - (\\int g_{\\varepsilon} dg_{\\varepsilon})^{g_{\\varepsilon}}", "srepr_derivation": [["renaming_premise", "Equality(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))"], [["integrate", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Pow(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))), Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('C')(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Symbol('B', commutative=True), Mul(Rational(1, 2), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(2))), Mul(Integer(-1), Pow(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), Pow(Integral(Symbol('g_{\\\\varepsilon}', commutative=True), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\omega{(a,F_{H})} = \\cos{(\\frac{a}{F_{H}})} and \\hat{H}_l{(a,F_{H})} = \\frac{a}{F_{H}}, then derive \\frac{\\partial}{\\partial a} \\omega{(a,F_{H})} = - \\frac{\\sin{(\\frac{a}{F_{H}})}}{F_{H}}, then obtain \\int - \\sin{(\\hat{H}_l{(a,F_{H})})} da = \\int - \\sin{(\\frac{a}{F_{H}})} da", "derivation": "\\omega{(a,F_{H})} = \\cos{(\\frac{a}{F_{H}})} and \\frac{\\partial}{\\partial a} \\omega{(a,F_{H})} = \\frac{\\partial}{\\partial a} \\cos{(\\frac{a}{F_{H}})} and \\frac{\\partial}{\\partial a} \\omega{(a,F_{H})} = - \\frac{\\sin{(\\frac{a}{F_{H}})}}{F_{H}} and \\hat{H}_l{(a,F_{H})} = \\frac{a}{F_{H}} and F_{H} \\frac{\\partial}{\\partial a} \\omega{(a,F_{H})} = - \\sin{(\\frac{a}{F_{H}})} and F_{H} \\frac{\\partial}{\\partial a} \\omega{(a,F_{H})} = - \\sin{(\\hat{H}_l{(a,F_{H})})} and - \\sin{(\\hat{H}_l{(a,F_{H})})} = - \\sin{(\\frac{a}{F_{H}})} and \\int - \\sin{(\\hat{H}_l{(a,F_{H})})} da = \\int - \\sin{(\\frac{a}{F_{H}})} da", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(cos(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True)))"], [["divide", 3, "Pow(Symbol('F_H', commutative=True), Integer(-1))"], "Equality(Mul(Symbol('F_H', commutative=True), Derivative(Function('\\\\omega')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Mul(Symbol('F_H', commutative=True), Derivative(Function('\\\\omega')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Integer(-1), sin(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)))), Mul(Integer(-1), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True)))))"], [["integrate", 7, "Symbol('a', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Function('\\\\hat{H}_l')(Symbol('a', commutative=True), Symbol('F_H', commutative=True)))), Tuple(Symbol('a', commutative=True))), Integral(Mul(Integer(-1), sin(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('a', commutative=True)))), Tuple(Symbol('a', commutative=True))))"]]}, {"prompt": "Given \\rho{(\\mu)} = \\log{(e^{\\mu})}, then obtain (\\mu \\rho^{\\mu}{(\\mu)})^{\\mu} = (\\mu \\log{(e^{\\mu})}^{\\mu})^{\\mu}", "derivation": "\\rho{(\\mu)} = \\log{(e^{\\mu})} and \\rho^{\\mu}{(\\mu)} = \\log{(e^{\\mu})}^{\\mu} and \\mu \\rho^{\\mu}{(\\mu)} = \\mu \\log{(e^{\\mu})}^{\\mu} and (\\mu \\rho^{\\mu}{(\\mu)})^{\\mu} = (\\mu \\log{(e^{\\mu})}^{\\mu})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\mu', commutative=True)), log(exp(Symbol('\\\\mu', commutative=True))))"], [["power", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"], [["times", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mu', commutative=True), Pow(Function('\\\\rho')(Symbol('\\\\mu', commutative=True)), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Symbol('\\\\mu', commutative=True), Pow(log(exp(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(z^{*})} = \\cos{(z^{*})} and \\varphi{(z^{*})} = - z^{*} + \\cos{(z^{*} \\operatorname{F_{N}}{(z^{*})})}, then obtain 1 = \\frac{- z^{*} + \\cos{(z^{*} \\cos{(z^{*})})}}{\\varphi{(z^{*})}}", "derivation": "\\operatorname{F_{N}}{(z^{*})} = \\cos{(z^{*})} and z^{*} \\operatorname{F_{N}}{(z^{*})} = z^{*} \\cos{(z^{*})} and \\cos{(z^{*} \\operatorname{F_{N}}{(z^{*})})} = \\cos{(z^{*} \\cos{(z^{*})})} and - z^{*} + \\cos{(z^{*} \\operatorname{F_{N}}{(z^{*})})} = - z^{*} + \\cos{(z^{*} \\cos{(z^{*})})} and \\varphi{(z^{*})} = - z^{*} + \\cos{(z^{*} \\operatorname{F_{N}}{(z^{*})})} and \\varphi{(z^{*})} = - z^{*} + \\cos{(z^{*} \\cos{(z^{*})})} and 1 = \\frac{- z^{*} + \\cos{(z^{*} \\cos{(z^{*})})}}{\\varphi{(z^{*})}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('z^*', commutative=True)), cos(Symbol('z^*', commutative=True)))"], [["times", 1, "Symbol('z^*', commutative=True)"], "Equality(Mul(Symbol('z^*', commutative=True), Function('F_N')(Symbol('z^*', commutative=True))), Mul(Symbol('z^*', commutative=True), cos(Symbol('z^*', commutative=True))))"], [["cos", 2], "Equality(cos(Mul(Symbol('z^*', commutative=True), Function('F_N')(Symbol('z^*', commutative=True)))), cos(Mul(Symbol('z^*', commutative=True), cos(Symbol('z^*', commutative=True)))))"], [["minus", 3, "Symbol('z^*', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), cos(Mul(Symbol('z^*', commutative=True), Function('F_N')(Symbol('z^*', commutative=True))))), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), cos(Mul(Symbol('z^*', commutative=True), cos(Symbol('z^*', commutative=True))))))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), cos(Mul(Symbol('z^*', commutative=True), Function('F_N')(Symbol('z^*', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\varphi')(Symbol('z^*', commutative=True)), Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), cos(Mul(Symbol('z^*', commutative=True), cos(Symbol('z^*', commutative=True))))))"], [["divide", 6, "Function('\\\\varphi')(Symbol('z^*', commutative=True))"], "Equality(Integer(1), Mul(Add(Mul(Integer(-1), Symbol('z^*', commutative=True)), cos(Mul(Symbol('z^*', commutative=True), cos(Symbol('z^*', commutative=True))))), Pow(Function('\\\\varphi')(Symbol('z^*', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(\\chi)} = \\sin{(\\sin{(\\chi)})}, then obtain \\frac{\\cos{(\\frac{d}{d \\chi} \\operatorname{E_{x}}{(\\chi)})}}{h v} = \\frac{\\cos{(\\cos{(\\chi)} \\cos{(\\sin{(\\chi)})})}}{h v}", "derivation": "\\operatorname{E_{x}}{(\\chi)} = \\sin{(\\sin{(\\chi)})} and \\frac{d}{d \\chi} \\operatorname{E_{x}}{(\\chi)} = \\frac{d}{d \\chi} \\sin{(\\sin{(\\chi)})} and \\cos{(\\frac{d}{d \\chi} \\operatorname{E_{x}}{(\\chi)})} = \\cos{(\\frac{d}{d \\chi} \\sin{(\\sin{(\\chi)})})} and \\frac{\\cos{(\\frac{d}{d \\chi} \\operatorname{E_{x}}{(\\chi)})}}{h v} = \\frac{\\cos{(\\frac{d}{d \\chi} \\sin{(\\sin{(\\chi)})})}}{h v} and \\frac{\\cos{(\\frac{d}{d \\chi} \\operatorname{E_{x}}{(\\chi)})}}{h v} = \\frac{\\cos{(\\cos{(\\chi)} \\cos{(\\sin{(\\chi)})})}}{h v}", "srepr_derivation": [["get_premise", "Equality(Function('E_x')(Symbol('\\\\chi', commutative=True)), sin(sin(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Function('E_x')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('E_x')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), cos(Derivative(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))))"], [["divide", 3, "Mul(Symbol('h', commutative=True), Symbol('v', commutative=True))"], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), cos(Derivative(Function('E_x')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), cos(Derivative(sin(sin(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), cos(Derivative(Function('E_x')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))), Mul(Pow(Symbol('h', commutative=True), Integer(-1)), Pow(Symbol('v', commutative=True), Integer(-1)), cos(Mul(cos(Symbol('\\\\chi', commutative=True)), cos(sin(Symbol('\\\\chi', commutative=True)))))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\Psi^{\\dagger},v_{z})} = (\\Psi^{\\dagger})^{v_{z}}, then obtain (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{v_{z}} + (\\tilde{g}^*^{v_{z}}{(\\Psi^{\\dagger},v_{z})})^{\\Psi^{\\dagger}} = (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{\\Psi^{\\dagger}} + (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{v_{z}}", "derivation": "\\tilde{g}^*{(\\Psi^{\\dagger},v_{z})} = (\\Psi^{\\dagger})^{v_{z}} and \\tilde{g}^*^{v_{z}}{(\\Psi^{\\dagger},v_{z})} = ((\\Psi^{\\dagger})^{v_{z}})^{v_{z}} and (\\tilde{g}^*^{v_{z}}{(\\Psi^{\\dagger},v_{z})})^{\\Psi^{\\dagger}} = (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{\\Psi^{\\dagger}} and (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{v_{z}} + (\\tilde{g}^*^{v_{z}}{(\\Psi^{\\dagger},v_{z})})^{\\Psi^{\\dagger}} = (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{\\Psi^{\\dagger}} + (((\\Psi^{\\dagger})^{v_{z}})^{v_{z}})^{v_{z}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)))"], [["power", 1, "Symbol('v_z', commutative=True)"], "Equality(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)))"], [["power", 2, "Symbol('\\\\Psi^{\\\\dagger}', commutative=True)"], "Equality(Pow(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["add", 3, "Pow(Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Pow(Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Pow(Pow(Function('\\\\tilde{g}^*')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True))), Add(Pow(Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Pow(Pow(Pow(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True)), Symbol('v_z', commutative=True))))"]]}, {"prompt": "Given E{(\\mathbf{A},h)} = \\mathbf{A} h, then derive h + E{(\\mathbf{A},h)} = h (\\mathbf{A} + 1), then obtain \\mathbf{A} h + h = h (\\mathbf{A} + 1)", "derivation": "E{(\\mathbf{A},h)} = \\mathbf{A} h and h + E{(\\mathbf{A},h)} = \\mathbf{A} h + h and \\int (h + E{(\\mathbf{A},h)}) dh = \\int (\\mathbf{A} h + h) dh and \\iint (h + E{(\\mathbf{A},h)}) dh d\\mathbf{A} = \\iint (\\mathbf{A} h + h) dh d\\mathbf{A} and \\frac{\\partial}{\\partial h} \\iint (h + E{(\\mathbf{A},h)}) dh d\\mathbf{A} = \\frac{\\partial}{\\partial h} \\iint (\\mathbf{A} h + h) dh d\\mathbf{A} and \\frac{\\partial^{2}}{\\partial \\mathbf{A}\\partial h} \\iint (h + E{(\\mathbf{A},h)}) dh d\\mathbf{A} = \\frac{\\partial^{2}}{\\partial \\mathbf{A}\\partial h} \\iint (\\mathbf{A} h + h) dh d\\mathbf{A} and h + E{(\\mathbf{A},h)} = h (\\mathbf{A} + 1) and \\mathbf{A} h + h = h (\\mathbf{A} + 1)", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)))"], [["add", 1, "Symbol('h', commutative=True)"], "Equality(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 4, "Symbol('h', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Integral(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('h', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 6], "Equality(Add(Symbol('h', commutative=True), Function('E')(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True))), Mul(Symbol('h', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Add(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('h', commutative=True)), Symbol('h', commutative=True)), Mul(Symbol('h', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"]]}, {"prompt": "Given Z{(f_{\\mathbf{v}},\\dot{y})} = \\frac{\\dot{y}}{f_{\\mathbf{v}}}, then obtain \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int Z{(f_{\\mathbf{v}},\\dot{y})} d\\dot{y} + \\frac{1}{f_{\\mathbf{v}}} = \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int \\frac{\\dot{y}}{f_{\\mathbf{v}}} d\\dot{y} + \\frac{1}{f_{\\mathbf{v}}}", "derivation": "Z{(f_{\\mathbf{v}},\\dot{y})} = \\frac{\\dot{y}}{f_{\\mathbf{v}}} and \\int Z{(f_{\\mathbf{v}},\\dot{y})} d\\dot{y} = \\int \\frac{\\dot{y}}{f_{\\mathbf{v}}} d\\dot{y} and \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int Z{(f_{\\mathbf{v}},\\dot{y})} d\\dot{y} = \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int \\frac{\\dot{y}}{f_{\\mathbf{v}}} d\\dot{y} and \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int Z{(f_{\\mathbf{v}},\\dot{y})} d\\dot{y} + \\frac{1}{f_{\\mathbf{v}}} = \\frac{\\dot{y}}{f_{\\mathbf{v}}} + \\int \\frac{\\dot{y}}{f_{\\mathbf{v}}} d\\dot{y} + \\frac{1}{f_{\\mathbf{v}}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Integral(Function('Z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{y}', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Integral(Function('Z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True)))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{y}', commutative=True)))))"], [["add", 3, "Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))"], "Equality(Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Integral(Function('Z')(Symbol('f_{\\\\mathbf{v}}', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True))), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Add(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Integral(Mul(Symbol('\\\\dot{y}', commutative=True), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))), Tuple(Symbol('\\\\dot{y}', commutative=True))), Pow(Symbol('f_{\\\\mathbf{v}}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\Psi{(A_{y},M)} = A_{y} + M and \\operatorname{f_{\\mathbf{v}}}{(A_{y},M)} = A_{y} + M, then obtain \\frac{\\partial}{\\partial M} \\Psi{(A_{y},M)} = \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(A_{y},M)}", "derivation": "\\Psi{(A_{y},M)} = A_{y} + M and \\operatorname{f_{\\mathbf{v}}}{(A_{y},M)} = A_{y} + M and \\Psi{(A_{y},M)} = \\operatorname{f_{\\mathbf{v}}}{(A_{y},M)} and \\frac{\\partial}{\\partial M} \\Psi{(A_{y},M)} = \\frac{\\partial}{\\partial M} \\operatorname{f_{\\mathbf{v}}}{(A_{y},M)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))"], ["renaming_premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Add(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\Psi')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)))"], [["differentiate", 3, "Symbol('M', commutative=True)"], "Equality(Derivative(Function('\\\\Psi')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))), Derivative(Function('f_{\\\\mathbf{v}}')(Symbol('A_y', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('M', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\lambda{(\\Psi,q)} = e^{- \\Psi + q}, then obtain \\int (\\frac{\\partial}{\\partial \\Psi} \\Psi \\lambda{(\\Psi,q)} + \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q}) d\\Psi = \\int 2 \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q} d\\Psi", "derivation": "\\lambda{(\\Psi,q)} = e^{- \\Psi + q} and \\Psi \\lambda{(\\Psi,q)} = \\Psi e^{- \\Psi + q} and \\frac{\\partial}{\\partial \\Psi} \\Psi \\lambda{(\\Psi,q)} = \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q} and \\frac{\\partial}{\\partial \\Psi} \\Psi \\lambda{(\\Psi,q)} + \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q} = 2 \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q} and \\int (\\frac{\\partial}{\\partial \\Psi} \\Psi \\lambda{(\\Psi,q)} + \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q}) d\\Psi = \\int 2 \\frac{\\partial}{\\partial \\Psi} \\Psi e^{- \\Psi + q} d\\Psi", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('q', commutative=True)), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True))))"], [["times", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('q', commutative=True))), Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["add", 3, "Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Mul(Integer(2), Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], [["integrate", 4, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Add(Derivative(Mul(Symbol('\\\\Psi', commutative=True), Function('\\\\lambda')(Symbol('\\\\Psi', commutative=True), Symbol('q', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Mul(Integer(2), Derivative(Mul(Symbol('\\\\Psi', commutative=True), exp(Add(Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Symbol('q', commutative=True)))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))), Tuple(Symbol('\\\\Psi', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})}, then derive \\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})} = - \\cos{(E_{\\lambda})}, then obtain (\\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})})^{E_{\\lambda}} = (- \\operatorname{F_{N}}{(E_{\\lambda})})^{E_{\\lambda}}", "derivation": "\\operatorname{F_{N}}{(E_{\\lambda})} = \\cos{(E_{\\lambda})} and \\frac{d}{d E_{\\lambda}} \\operatorname{F_{N}}{(E_{\\lambda})} = \\frac{d}{d E_{\\lambda}} \\cos{(E_{\\lambda})} and \\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})} = \\frac{d^{2}}{d E_{\\lambda}^{2}} \\cos{(E_{\\lambda})} and \\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})} = - \\cos{(E_{\\lambda})} and \\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})} = - \\operatorname{F_{N}}{(E_{\\lambda})} and (\\frac{d^{2}}{d E_{\\lambda}^{2}} \\operatorname{F_{N}}{(E_{\\lambda})})^{E_{\\lambda}} = (- \\operatorname{F_{N}}{(E_{\\lambda})})^{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('E_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))), Derivative(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Derivative(cos(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Mul(Integer(-1), cos(Symbol('E_{\\\\lambda}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Mul(Integer(-1), Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 5, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Pow(Derivative(Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Symbol('E_{\\\\lambda}', commutative=True)), Pow(Mul(Integer(-1), Function('F_N')(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('E_{\\\\lambda}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(z^{*},M)} = M + z^{*}, then derive \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{(M + z^{*})^{2}} = \\frac{1}{(M + z^{*})^{2}}, then obtain -1 + \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{(M + z^{*})^{2}} = -1 + \\frac{1}{(M + z^{*})^{2}}", "derivation": "\\operatorname{A_{x}}{(z^{*},M)} = M + z^{*} and \\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)} = \\frac{\\partial}{\\partial z^{*}} (M + z^{*}) and \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{M + z^{*}} = \\frac{\\frac{\\partial}{\\partial z^{*}} (M + z^{*})}{M + z^{*}} and \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{(M + z^{*})^{2}} = \\frac{\\frac{\\partial}{\\partial z^{*}} (M + z^{*})}{(M + z^{*})^{2}} and \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{(M + z^{*})^{2}} = \\frac{1}{(M + z^{*})^{2}} and -1 + \\frac{\\frac{\\partial}{\\partial z^{*}} \\operatorname{A_{x}}{(z^{*},M)}}{(M + z^{*})^{2}} = -1 + \\frac{1}{(M + z^{*})^{2}}", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)))"], [["differentiate", 1, "Symbol('z^*', commutative=True)"], "Equality(Derivative(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Derivative(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["divide", 2, "Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-1)), Derivative(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-1)), Derivative(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["times", 3, "Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2)), Derivative(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2)), Derivative(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2)), Derivative(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1)))), Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2)))"], [["minus", 5, 1], "Equality(Add(Integer(-1), Mul(Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2)), Derivative(Function('A_x')(Symbol('z^*', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))), Add(Integer(-1), Pow(Add(Symbol('M', commutative=True), Symbol('z^*', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given C{(T)} = e^{T}, then obtain \\frac{0^{T} - T}{T} = \\frac{2 \\cdot 0^{T} - T - 1}{T}", "derivation": "C{(T)} = e^{T} and C{(T)} - e^{T} = 0 and (C{(T)} - e^{T})^{T} = 0^{T} and - T + (C{(T)} - e^{T})^{T} = 0^{T} - T and 1 - T = - T + (C{(T)} - e^{T})^{T} and 1 - T = 0^{T} - T and - T + (C{(T)} - e^{T})^{T} = 0^{T} - T + (C{(T)} - e^{T})^{T} - 1 and 0^{T} - T = 2 \\cdot 0^{T} - T - 1 and \\frac{0^{T} - T}{T} = \\frac{2 \\cdot 0^{T} - T - 1}{T}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('T', commutative=True)), exp(Symbol('T', commutative=True)))"], [["minus", 1, "exp(Symbol('T', commutative=True))"], "Equality(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('T', commutative=True)"], "Equality(Pow(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Pow(Integer(0), Symbol('T', commutative=True)))"], [["minus", 3, "Symbol('T', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Add(Integer(1), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True))), Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)), Pow(Add(Function('C')(Symbol('T', commutative=True)), Mul(Integer(-1), exp(Symbol('T', commutative=True)))), Symbol('T', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 7, 1], "Equality(Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True))), Add(Mul(Integer(2), Pow(Integer(0), Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Integer(-1)))"], [["divide", 8, "Symbol('T', commutative=True)"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Pow(Integer(0), Symbol('T', commutative=True)), Mul(Integer(-1), Symbol('T', commutative=True)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Mul(Integer(2), Pow(Integer(0), Symbol('T', commutative=True))), Mul(Integer(-1), Symbol('T', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\cos{(\\cos{(\\hat{H}_{\\lambda})})} and T{(\\hat{H}_{\\lambda})} = \\cos{(\\cos{(\\hat{H}_{\\lambda})})}, then obtain q + (q^{G})^{f_{\\mathbf{v}}} + \\frac{\\cos{(\\cos{(\\hat{H}_{\\lambda})})}}{\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})}} = q + (q^{G})^{f_{\\mathbf{v}}} + 1", "derivation": "\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})} = \\cos{(\\cos{(\\hat{H}_{\\lambda})})} and T{(\\hat{H}_{\\lambda})} = \\cos{(\\cos{(\\hat{H}_{\\lambda})})} and \\frac{T{(\\hat{H}_{\\lambda})}}{\\cos{(\\cos{(\\hat{H}_{\\lambda})})}} = 1 and \\frac{T{(\\hat{H}_{\\lambda})}}{\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})}} = 1 and \\frac{\\cos{(\\cos{(\\hat{H}_{\\lambda})})}}{\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})}} = 1 and q + (q^{G})^{f_{\\mathbf{v}}} + \\frac{\\cos{(\\cos{(\\hat{H}_{\\lambda})})}}{\\hat{\\mathbf{r}}{(\\hat{H}_{\\lambda})}} = q + (q^{G})^{f_{\\mathbf{v}}} + 1", "srepr_derivation": [["get_premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["divide", 2, "cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Mul(Function('T')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integer(-1))), Integer(1))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('T')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Integer(1))"], [["add", 5, "Add(Symbol('q', commutative=True), Pow(Pow(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], "Equality(Add(Symbol('q', commutative=True), Pow(Pow(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Mul(Pow(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Integer(-1)), cos(cos(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Add(Symbol('q', commutative=True), Pow(Pow(Symbol('q', commutative=True), Symbol('G', commutative=True)), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\hat{H}{(W)} = \\sin{(\\cos{(W)})}, then obtain (- \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\hat{H}{(W)})^{W})^{W} = ((- \\sin{(W)} \\cos{(\\cos{(W)})})^{W} - \\cos{(\\cos{(W)})})^{W}", "derivation": "\\hat{H}{(W)} = \\sin{(\\cos{(W)})} and \\frac{d}{d W} \\hat{H}{(W)} = \\frac{d}{d W} \\sin{(\\cos{(W)})} and (\\frac{d}{d W} \\hat{H}{(W)})^{W} = (\\frac{d}{d W} \\sin{(\\cos{(W)})})^{W} and - \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\hat{H}{(W)})^{W} = - \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\sin{(\\cos{(W)})})^{W} and (- \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\hat{H}{(W)})^{W})^{W} = (- \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\sin{(\\cos{(W)})})^{W})^{W} and (- \\cos{(\\cos{(W)})} + (\\frac{d}{d W} \\hat{H}{(W)})^{W})^{W} = ((- \\sin{(W)} \\cos{(\\cos{(W)})})^{W} - \\cos{(\\cos{(W)})})^{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('W', commutative=True)), sin(cos(Symbol('W', commutative=True))))"], [["differentiate", 1, "Symbol('W', commutative=True)"], "Equality(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(sin(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["power", 2, "Symbol('W', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)), Pow(Derivative(sin(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True)))"], [["minus", 3, "cos(cos(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(-1), cos(cos(Symbol('W', commutative=True)))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))), Add(Mul(Integer(-1), cos(cos(Symbol('W', commutative=True)))), Pow(Derivative(sin(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))))"], [["power", 4, "Symbol('W', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), cos(cos(Symbol('W', commutative=True)))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Mul(Integer(-1), cos(cos(Symbol('W', commutative=True)))), Pow(Derivative(sin(cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))), Symbol('W', commutative=True)))"], [["evaluate_derivatives", 5], "Equality(Pow(Add(Mul(Integer(-1), cos(cos(Symbol('W', commutative=True)))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Symbol('W', commutative=True))), Symbol('W', commutative=True)), Pow(Add(Pow(Mul(Integer(-1), sin(Symbol('W', commutative=True)), cos(cos(Symbol('W', commutative=True)))), Symbol('W', commutative=True)), Mul(Integer(-1), cos(cos(Symbol('W', commutative=True))))), Symbol('W', commutative=True)))"]]}, {"prompt": "Given V{(F_{g},f_{E})} = \\frac{f_{E}}{F_{g}}, then derive \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} V{(F_{g},f_{E})}}{F_{g}} = - \\frac{f_{E}^{2}}{F_{g}^{3}}, then obtain f_{E} + \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} \\frac{f_{E}}{F_{g}}}{F_{g}} = f_{E} - \\frac{f_{E}^{2}}{F_{g}^{3}}", "derivation": "V{(F_{g},f_{E})} = \\frac{f_{E}}{F_{g}} and \\frac{\\partial}{\\partial F_{g}} V{(F_{g},f_{E})} = \\frac{\\partial}{\\partial F_{g}} \\frac{f_{E}}{F_{g}} and \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} V{(F_{g},f_{E})}}{F_{g}} = \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} \\frac{f_{E}}{F_{g}}}{F_{g}} and \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} V{(F_{g},f_{E})}}{F_{g}} = - \\frac{f_{E}^{2}}{F_{g}^{3}} and \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} \\frac{f_{E}}{F_{g}}}{F_{g}} = - \\frac{f_{E}^{2}}{F_{g}^{3}} and f_{E} + \\frac{f_{E} \\frac{\\partial}{\\partial F_{g}} \\frac{f_{E}}{F_{g}}}{F_{g}} = f_{E} - \\frac{f_{E}^{2}}{F_{g}^{3}}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('F_g', commutative=True), Symbol('f_E', commutative=True)), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('F_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["times", 2, "Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True))"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Derivative(Function('V')(Symbol('F_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Derivative(Function('V')(Symbol('F_g', commutative=True), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-3)), Pow(Symbol('f_E', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-3)), Pow(Symbol('f_E', commutative=True), Integer(2))))"], [["add", 5, "Symbol('f_E', commutative=True)"], "Equality(Add(Symbol('f_E', commutative=True), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True), Derivative(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Symbol('f_E', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Pow(Symbol('F_g', commutative=True), Integer(-3)), Pow(Symbol('f_E', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(f^{\\prime})} = f^{\\prime}, then obtain \\int \\operatorname{v_{y}}^{2 f^{\\prime}}{(f^{\\prime})} df^{\\prime} = \\int (f^{\\prime})^{f^{\\prime}} \\operatorname{v_{y}}^{f^{\\prime}}{(f^{\\prime})} df^{\\prime}", "derivation": "\\operatorname{v_{y}}{(f^{\\prime})} = f^{\\prime} and \\operatorname{v_{y}}^{f^{\\prime}}{(f^{\\prime})} = (f^{\\prime})^{f^{\\prime}} and \\operatorname{v_{y}}^{2 f^{\\prime}}{(f^{\\prime})} = (f^{\\prime})^{f^{\\prime}} \\operatorname{v_{y}}^{f^{\\prime}}{(f^{\\prime})} and \\int \\operatorname{v_{y}}^{2 f^{\\prime}}{(f^{\\prime})} df^{\\prime} = \\int (f^{\\prime})^{f^{\\prime}} \\operatorname{v_{y}}^{f^{\\prime}}{(f^{\\prime})} df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], [["power", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["times", 2, "Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Mul(Pow(Symbol('f^{\\\\prime}', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Pow(Function('v_y')(Symbol('f^{\\\\prime}', commutative=True)), Symbol('f^{\\\\prime}', commutative=True))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given A{(s,Q)} = \\int (Q + s) ds, then obtain (((- A{(s,Q)} + \\int A{(s,Q)} ds)^{Q})^{Q})^{Q} = (((- A{(s,Q)} + \\iint (Q + s) ds ds)^{Q})^{Q})^{Q}", "derivation": "A{(s,Q)} = \\int (Q + s) ds and \\int A{(s,Q)} ds = \\iint (Q + s) ds ds and - \\int (Q + s) ds + \\int A{(s,Q)} ds = - \\int (Q + s) ds + \\iint (Q + s) ds ds and - A{(s,Q)} + \\int A{(s,Q)} ds = - A{(s,Q)} + \\iint (Q + s) ds ds and (- A{(s,Q)} + \\int A{(s,Q)} ds)^{Q} = (- A{(s,Q)} + \\iint (Q + s) ds ds)^{Q} and ((- A{(s,Q)} + \\int A{(s,Q)} ds)^{Q})^{Q} = ((- A{(s,Q)} + \\iint (Q + s) ds ds)^{Q})^{Q} and (((- A{(s,Q)} + \\int A{(s,Q)} ds)^{Q})^{Q})^{Q} = (((- A{(s,Q)} + \\iint (Q + s) ds ds)^{Q})^{Q})^{Q}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["minus", 2, "Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True)))), Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["power", 4, "Symbol('Q', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)), Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)))"], [["power", 5, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"], [["power", 6, "Symbol('Q', commutative=True)"], "Equality(Pow(Pow(Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Pow(Pow(Pow(Add(Mul(Integer(-1), Function('A')(Symbol('s', commutative=True), Symbol('Q', commutative=True))), Integral(Add(Symbol('Q', commutative=True), Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)), Symbol('Q', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(\\tilde{g}^*,v)} = \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}, then obtain \\frac{\\partial}{\\partial v} (v + \\mathbf{p}{(\\tilde{g}^*,v)}) \\frac{\\partial}{\\partial v} (v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}) = (\\frac{\\partial}{\\partial v} (v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}))^{2}", "derivation": "\\mathbf{p}{(\\tilde{g}^*,v)} = \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*} and v + \\mathbf{p}{(\\tilde{g}^*,v)} = v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*} and \\frac{\\partial}{\\partial v} (v + \\mathbf{p}{(\\tilde{g}^*,v)}) = \\frac{\\partial}{\\partial v} (v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}) and \\frac{\\partial}{\\partial v} (v + \\mathbf{p}{(\\tilde{g}^*,v)}) \\frac{\\partial}{\\partial v} (v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}) = (\\frac{\\partial}{\\partial v} (v + \\frac{\\partial}{\\partial v} v^{\\tilde{g}^*}))^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('v', commutative=True)), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["add", 1, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('v', commutative=True))), Add(Symbol('v', commutative=True), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Add(Symbol('v', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["times", 3, "Derivative(Add(Symbol('v', commutative=True), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Mul(Derivative(Add(Symbol('v', commutative=True), Function('\\\\mathbf{p}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Add(Symbol('v', commutative=True), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1)))), Pow(Derivative(Add(Symbol('v', commutative=True), Derivative(Pow(Symbol('v', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1)))), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given C{(\\chi)} = \\log{(\\chi)}, then derive \\frac{d}{d \\chi} C{(\\chi)} + \\frac{1}{\\chi} = \\frac{2}{\\chi}, then obtain \\frac{d}{d \\chi} \\log{(\\chi)} - \\frac{d}{d \\chi} 2 \\log{(\\chi)} + \\frac{1}{\\chi} = - \\frac{d}{d \\chi} 2 \\log{(\\chi)} + \\frac{2}{\\chi}", "derivation": "C{(\\chi)} = \\log{(\\chi)} and C{(\\chi)} + \\log{(\\chi)} = 2 \\log{(\\chi)} and \\frac{d}{d \\chi} (C{(\\chi)} + \\log{(\\chi)}) = \\frac{d}{d \\chi} 2 \\log{(\\chi)} and \\frac{d}{d \\chi} C{(\\chi)} + \\frac{1}{\\chi} = \\frac{2}{\\chi} and \\frac{d}{d \\chi} \\log{(\\chi)} + \\frac{1}{\\chi} = \\frac{2}{\\chi} and - \\frac{d}{d \\chi} (C{(\\chi)} + \\log{(\\chi)}) + \\frac{d}{d \\chi} \\log{(\\chi)} + \\frac{1}{\\chi} = - \\frac{d}{d \\chi} (C{(\\chi)} + \\log{(\\chi)}) + \\frac{2}{\\chi} and \\frac{d}{d \\chi} \\log{(\\chi)} - \\frac{d}{d \\chi} 2 \\log{(\\chi)} + \\frac{1}{\\chi} = - \\frac{d}{d \\chi} 2 \\log{(\\chi)} + \\frac{2}{\\chi}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True)))"], [["add", 1, "log(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Mul(Integer(2), log(Symbol('\\\\chi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('C')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))))"], [["minus", 5, "Derivative(Add(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Derivative(Add(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Derivative(Add(Function('C')(Symbol('\\\\chi', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Derivative(log(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Mul(Integer(2), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Pow(Symbol('\\\\chi', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Derivative(Mul(Integer(2), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\chi', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{v})} = \\log{(\\mathbf{v})}, then derive \\frac{\\log{(\\mathbf{v})} \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})}}{\\theta_{1}{(\\mathbf{v})}} = \\frac{\\log{(\\mathbf{v})}}{\\mathbf{v} \\theta_{1}{(\\mathbf{v})}}, then obtain \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})} = \\frac{1}{\\mathbf{v}}", "derivation": "\\theta_{1}{(\\mathbf{v})} = \\log{(\\mathbf{v})} and \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})} = \\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})} and \\frac{\\log{(\\mathbf{v})} \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})}}{\\theta_{1}{(\\mathbf{v})}} = \\frac{\\log{(\\mathbf{v})} \\frac{d}{d \\mathbf{v}} \\log{(\\mathbf{v})}}{\\theta_{1}{(\\mathbf{v})}} and \\frac{\\log{(\\mathbf{v})} \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})}}{\\theta_{1}{(\\mathbf{v})}} = \\frac{\\log{(\\mathbf{v})}}{\\mathbf{v} \\theta_{1}{(\\mathbf{v})}} and \\frac{d}{d \\mathbf{v}} \\theta_{1}{(\\mathbf{v})} = \\frac{1}{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), log(Symbol('\\\\mathbf{v}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))))"], [["divide", 2, "Mul(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Pow(log(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)))"], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1)))), Mul(Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)), Pow(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(-1)), log(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given I{(\\psi^*,f)} = - \\psi^* + \\sin{(f)}, then obtain I{(\\psi^*,f)} + \\int (- \\psi^* + \\sin{(f)})^{f} d\\psi^* = - \\psi^* + \\sin{(f)} + \\int (- \\psi^* + \\sin{(f)})^{f} d\\psi^*", "derivation": "I{(\\psi^*,f)} = - \\psi^* + \\sin{(f)} and I^{f}{(\\psi^*,f)} = (- \\psi^* + \\sin{(f)})^{f} and \\int I^{f}{(\\psi^*,f)} d\\psi^* = \\int (- \\psi^* + \\sin{(f)})^{f} d\\psi^* and I{(\\psi^*,f)} + \\int I^{f}{(\\psi^*,f)} d\\psi^* = - \\psi^* + \\sin{(f)} + \\int I^{f}{(\\psi^*,f)} d\\psi^* and I{(\\psi^*,f)} + \\int (- \\psi^* + \\sin{(f)})^{f} d\\psi^* = - \\psi^* + \\sin{(f)} + \\int (- \\psi^* + \\sin{(f)})^{f} d\\psi^*", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True))))"], [["power", 1, "Symbol('f', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"], [["integrate", 2, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Pow(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["add", 1, "Integral(Pow(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integral(Pow(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True)), Integral(Pow(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('I')(Symbol('\\\\psi^*', commutative=True), Symbol('f', commutative=True)), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True)), Integral(Pow(Add(Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True)), sin(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(F_{N},p)} = \\frac{\\sin{(p)}}{F_{N}}, then obtain -2 = - \\frac{2 \\sin{(p)}}{F_{N} \\phi_{1}{(F_{N},p)}}", "derivation": "\\phi_{1}{(F_{N},p)} = \\frac{\\sin{(p)}}{F_{N}} and \\frac{\\phi_{1}{(F_{N},p)} \\sin{(p)}}{F_{N}} = \\frac{\\sin^{2}{(p)}}{F_{N}^{2}} and \\frac{2 \\phi_{1}{(F_{N},p)} \\sin{(p)}}{F_{N}} = \\frac{\\phi_{1}{(F_{N},p)} \\sin{(p)}}{F_{N}} + \\frac{\\sin^{2}{(p)}}{F_{N}^{2}} and -2 = - \\frac{F_{N} (\\frac{\\phi_{1}{(F_{N},p)} \\sin{(p)}}{F_{N}} + \\frac{\\sin^{2}{(p)}}{F_{N}^{2}})}{\\phi_{1}{(F_{N},p)} \\sin{(p)}} and -2 = - \\frac{2 \\sin{(p)}}{F_{N} \\phi_{1}{(F_{N},p)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), sin(Symbol('p', commutative=True)))"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-2)), Pow(sin(Symbol('p', commutative=True)), Integer(2))))"], [["minus", 2, "Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Add(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-2)), Pow(sin(Symbol('p', commutative=True)), Integer(2)))))"], [["divide", 3, "Mul(Integer(-1), Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True)))"], "Equality(Integer(-2), Mul(Integer(-1), Symbol('F_N', commutative=True), Add(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), sin(Symbol('p', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-2)), Pow(sin(Symbol('p', commutative=True)), Integer(2)))), Pow(Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), Integer(-1)), Pow(sin(Symbol('p', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(-2), Mul(Integer(-1), Integer(2), Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Function('\\\\phi_1')(Symbol('F_N', commutative=True), Symbol('p', commutative=True)), Integer(-1)), sin(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\theta{(k)} = \\log{(e^{k})}, then obtain ((\\frac{d}{d k} \\theta{(k)})^{k})^{k} - \\frac{1}{\\theta{(k)}} = ((\\frac{d}{d k} \\log{(e^{k})})^{k})^{k} - \\frac{1}{\\theta{(k)}}", "derivation": "\\theta{(k)} = \\log{(e^{k})} and \\frac{d}{d k} \\theta{(k)} = \\frac{d}{d k} \\log{(e^{k})} and (\\frac{d}{d k} \\theta{(k)})^{k} = (\\frac{d}{d k} \\log{(e^{k})})^{k} and ((\\frac{d}{d k} \\theta{(k)})^{k})^{k} = ((\\frac{d}{d k} \\log{(e^{k})})^{k})^{k} and ((\\frac{d}{d k} \\theta{(k)})^{k})^{k} - \\frac{1}{\\theta{(k)}} = ((\\frac{d}{d k} \\log{(e^{k})})^{k})^{k} - \\frac{1}{\\theta{(k)}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\theta')(Symbol('k', commutative=True)), log(exp(Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('k', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(log(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))))"], [["power", 2, "Symbol('k', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\theta')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Pow(Derivative(log(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('\\\\theta')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Pow(Pow(Derivative(log(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)))"], [["minus", 4, "Pow(Function('\\\\theta')(Symbol('k', commutative=True)), Integer(-1))"], "Equality(Add(Pow(Pow(Derivative(Function('\\\\theta')(Symbol('k', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('k', commutative=True)), Integer(-1)))), Add(Pow(Pow(Derivative(log(exp(Symbol('k', commutative=True))), Tuple(Symbol('k', commutative=True), Integer(1))), Symbol('k', commutative=True)), Symbol('k', commutative=True)), Mul(Integer(-1), Pow(Function('\\\\theta')(Symbol('k', commutative=True)), Integer(-1)))))"]]}, {"prompt": "Given v{(\\hat{H},v_{x})} = \\frac{\\sin{(\\hat{H})}}{v_{x}}, then derive \\frac{\\partial}{\\partial v_{x}} v{(\\hat{H},v_{x})} = - \\frac{\\sin{(\\hat{H})}}{v_{x}^{2}}, then obtain \\frac{\\frac{\\partial}{\\partial v_{x}} v{(\\hat{H},v_{x})}}{\\sin{(\\hat{H})}} = - \\frac{1}{v_{x}^{2}}", "derivation": "v{(\\hat{H},v_{x})} = \\frac{\\sin{(\\hat{H})}}{v_{x}} and \\frac{\\partial}{\\partial v_{x}} v{(\\hat{H},v_{x})} = \\frac{\\partial}{\\partial v_{x}} \\frac{\\sin{(\\hat{H})}}{v_{x}} and \\frac{\\partial}{\\partial v_{x}} v{(\\hat{H},v_{x})} = - \\frac{\\sin{(\\hat{H})}}{v_{x}^{2}} and \\frac{\\frac{\\partial}{\\partial v_{x}} v{(\\hat{H},v_{x})}}{\\sin{(\\hat{H})}} = - \\frac{1}{v_{x}^{2}}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\hat{H}', commutative=True), Symbol('v_x', commutative=True)), Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["differentiate", 1, "Symbol('v_x', commutative=True)"], "Equality(Derivative(Function('v')(Symbol('\\\\hat{H}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_x', commutative=True), Integer(-1)), sin(Symbol('\\\\hat{H}', commutative=True))), Tuple(Symbol('v_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v')(Symbol('\\\\hat{H}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-2)), sin(Symbol('\\\\hat{H}', commutative=True))))"], [["divide", 3, "sin(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Pow(sin(Symbol('\\\\hat{H}', commutative=True)), Integer(-1)), Derivative(Function('v')(Symbol('\\\\hat{H}', commutative=True), Symbol('v_x', commutative=True)), Tuple(Symbol('v_x', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('v_x', commutative=True), Integer(-2))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = e^{a^{\\dagger}}, then derive \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} e^{a^{\\dagger}} + e^{a^{\\dagger}} \\frac{d}{d a^{\\dagger}} \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = 2 e^{2 a^{\\dagger}}, then obtain \\operatorname{V_{\\mathbf{B}}}^{2}{(a^{\\dagger})} + \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} \\frac{d}{d a^{\\dagger}} \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = 2 \\operatorname{V_{\\mathbf{B}}}^{2}{(a^{\\dagger})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = e^{a^{\\dagger}} and \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} e^{a^{\\dagger}} = e^{2 a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} e^{a^{\\dagger}} = \\frac{d}{d a^{\\dagger}} e^{2 a^{\\dagger}} and \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} e^{a^{\\dagger}} + e^{a^{\\dagger}} \\frac{d}{d a^{\\dagger}} \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = 2 e^{2 a^{\\dagger}} and \\operatorname{V_{\\mathbf{B}}}^{2}{(a^{\\dagger})} + \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} \\frac{d}{d a^{\\dagger}} \\operatorname{V_{\\mathbf{B}}}{(a^{\\dagger})} = 2 \\operatorname{V_{\\mathbf{B}}}^{2}{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 1, "exp(Symbol('a^{\\\\dagger}', commutative=True))"], "Equality(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))))"], [["differentiate", 2, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True))), Mul(exp(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Mul(Integer(2), exp(Mul(Integer(2), Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2)), Mul(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))), Mul(Integer(2), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('a^{\\\\dagger}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(\\hat{x},B)} = \\frac{\\sin{(B)}}{\\hat{x}}, then obtain - \\frac{\\sin{(B)}}{\\hat{x}} - \\frac{\\hat{x} \\operatorname{v_{2}}{(\\hat{x},B)}}{B} = - \\frac{\\sin{(B)}}{\\hat{x}} - \\frac{\\sin{(B)}}{B}", "derivation": "\\operatorname{v_{2}}{(\\hat{x},B)} = \\frac{\\sin{(B)}}{\\hat{x}} and - \\operatorname{v_{2}}{(\\hat{x},B)} = - \\frac{\\sin{(B)}}{\\hat{x}} and - \\frac{\\operatorname{v_{2}}{(\\hat{x},B)}}{B} = - \\frac{\\sin{(B)}}{B \\hat{x}} and - \\frac{\\hat{x} \\operatorname{v_{2}}{(\\hat{x},B)}}{B} = - \\frac{\\sin{(B)}}{B} and - \\frac{\\sin{(B)}}{\\hat{x}} - \\frac{\\hat{x} \\operatorname{v_{2}}{(\\hat{x},B)}}{B} = - \\frac{\\sin{(B)}}{\\hat{x}} - \\frac{\\sin{(B)}}{B}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))))"], [["divide", 2, "Symbol('B', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Function('v_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))))"], [["times", 3, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True), Function('v_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))))"], [["add", 4, "Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('\\\\hat{x}', commutative=True), Function('v_2')(Symbol('\\\\hat{x}', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('\\\\hat{x}', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True))), Mul(Integer(-1), Pow(Symbol('B', commutative=True), Integer(-1)), sin(Symbol('B', commutative=True)))))"]]}, {"prompt": "Given \\rho{(b)} = b and \\mathbf{H}{(b)} = \\frac{d}{d b} \\rho{(b)}, then obtain - b + \\rho{(b)} + \\int \\frac{d}{d b} b db = - b + \\rho{(b)} + \\int \\frac{d}{d b} \\rho{(b)} db", "derivation": "\\rho{(b)} = b and \\mathbf{H}{(b)} = \\frac{d}{d b} \\rho{(b)} and \\int \\mathbf{H}{(b)} db = \\int \\frac{d}{d b} \\rho{(b)} db and \\mathbf{H}{(b)} = \\frac{d}{d b} b and \\rho{(b)} + \\int \\mathbf{H}{(b)} db = \\rho{(b)} + \\int \\frac{d}{d b} \\rho{(b)} db and \\rho{(b)} + \\int \\frac{d}{d b} b db = \\rho{(b)} + \\int \\frac{d}{d b} \\rho{(b)} db and - b + \\rho{(b)} + \\int \\frac{d}{d b} b db = - b + \\rho{(b)} + \\int \\frac{d}{d b} \\rho{(b)} db", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('b', commutative=True)), Symbol('b', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('b', commutative=True)), Derivative(Function('\\\\rho')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('b', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{H}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True))), Integral(Derivative(Function('\\\\rho')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\mathbf{H}')(Symbol('b', commutative=True)), Derivative(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\rho')(Symbol('b', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True)))), Add(Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Derivative(Function('\\\\rho')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Derivative(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))), Add(Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Derivative(Function('\\\\rho')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))))"], [["minus", 6, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Derivative(Symbol('b', commutative=True), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('\\\\rho')(Symbol('b', commutative=True)), Integral(Derivative(Function('\\\\rho')(Symbol('b', commutative=True)), Tuple(Symbol('b', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(q,r,m_{s})} = m_{s} r + q, then derive - \\frac{\\partial}{\\partial m_{s}} \\operatorname{v_{y}}{(q,r,m_{s})} = - r, then obtain - q \\frac{\\partial}{\\partial m_{s}} \\operatorname{v_{y}}{(q,r,m_{s})} - \\operatorname{v_{y}}{(q,r,m_{s})} + \\frac{\\partial}{\\partial m_{s}} - \\operatorname{v_{y}}{(q,r,m_{s})} = - q r - \\operatorname{v_{y}}{(q,r,m_{s})} + \\frac{\\partial}{\\partial m_{s}} - \\operatorname{v_{y}}{(q,r,m_{s})}", "derivation": "\\operatorname{v_{y}}{(q,r,m_{s})} = m_{s} r + q and - \\operatorname{v_{y}}{(q,r,m_{s})} = - m_{s} r - q and \\frac{\\partial}{\\partial m_{s}} - \\operatorname{v_{y}}{(q,r,m_{s})} = \\frac{\\partial}{\\partial m_{s}} (- m_{s} r - q) and - \\frac{\\partial}{\\partial m_{s}} \\operatorname{v_{y}}{(q,r,m_{s})} = - r and - q \\frac{\\partial}{\\partial m_{s}} \\operatorname{v_{y}}{(q,r,m_{s})} = - q r and - q \\frac{\\partial}{\\partial m_{s}} \\operatorname{v_{y}}{(q,r,m_{s})} - \\operatorname{v_{y}}{(q,r,m_{s})} + \\frac{\\partial}{\\partial m_{s}} - \\operatorname{v_{y}}{(q,r,m_{s})} = - q r - \\operatorname{v_{y}}{(q,r,m_{s})} + \\frac{\\partial}{\\partial m_{s}} - \\operatorname{v_{y}}{(q,r,m_{s})}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True)), Add(Mul(Symbol('m_s', commutative=True), Symbol('r', commutative=True)), Symbol('q', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Add(Mul(Integer(-1), Symbol('m_s', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('m_s', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Symbol('q', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(-1), Derivative(Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('r', commutative=True)))"], [["times", 4, "Symbol('q', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Derivative(Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('q', commutative=True), Symbol('r', commutative=True)))"], [["add", 5, "Add(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Derivative(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True), Derivative(Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Derivative(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('q', commutative=True), Symbol('r', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Derivative(Mul(Integer(-1), Function('v_y')(Symbol('q', commutative=True), Symbol('r', commutative=True), Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mu_{0}{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and \\operatorname{F_{g}}{(\\hat{\\mathbf{r}})} = \\int \\mu_{0}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}, then obtain \\operatorname{F_{g}}{(\\hat{\\mathbf{r}})} = \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}", "derivation": "\\mu_{0}{(\\hat{\\mathbf{r}})} = \\sin{(\\hat{\\mathbf{r}})} and \\int \\mu_{0}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} = \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} and \\operatorname{F_{g}}{(\\hat{\\mathbf{r}})} = \\int \\mu_{0}{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}} and \\operatorname{F_{g}}{(\\hat{\\mathbf{r}})} = \\int \\sin{(\\hat{\\mathbf{r}})} d\\hat{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))), Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(Function('\\\\mu_0')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('F_g')(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Integral(sin(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True)), Tuple(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(\\mathbf{H})} = \\log{(\\mathbf{H})}, then derive \\frac{d}{d \\mathbf{H}} \\mathbf{A}{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}}, then obtain \\frac{\\frac{d^{2}}{d \\mathbf{H}^{2}} \\log{(\\mathbf{H})}}{\\mathbf{A}{(\\mathbf{H})}} = \\frac{\\frac{d}{d \\mathbf{H}} \\frac{1}{\\mathbf{H}}}{\\mathbf{A}{(\\mathbf{H})}}", "derivation": "\\mathbf{A}{(\\mathbf{H})} = \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{A}{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} \\mathbf{A}{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{d}{d \\mathbf{H}} \\log{(\\mathbf{H})} = \\frac{1}{\\mathbf{H}} and \\frac{d^{2}}{d \\mathbf{H}^{2}} \\log{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} \\frac{1}{\\mathbf{H}} and \\frac{\\frac{d^{2}}{d \\mathbf{H}^{2}} \\log{(\\mathbf{H})}}{\\mathbf{A}{(\\mathbf{H})}} = \\frac{\\frac{d}{d \\mathbf{H}} \\frac{1}{\\mathbf{H}}}{\\mathbf{A}{(\\mathbf{H})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True)), log(Symbol('\\\\mathbf{H}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2))), Derivative(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"], [["divide", 5, "Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)))), Mul(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\mathbf{H}', commutative=True)), Integer(-1)), Derivative(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{s}{(t_{2})} = \\cos{(\\sin{(t_{2})})} and \\operatorname{J_{\\varepsilon}}{(t_{2})} = \\sin{(t_{2})} and \\tilde{g}{(t_{2})} = \\cos{(\\operatorname{J_{\\varepsilon}}{(t_{2})})}, then obtain \\frac{d}{d t_{2}} \\frac{\\tilde{g}{(t_{2})}}{\\mathbf{s}{(t_{2})}} = \\frac{d}{d t_{2}} 1", "derivation": "\\mathbf{s}{(t_{2})} = \\cos{(\\sin{(t_{2})})} and \\operatorname{J_{\\varepsilon}}{(t_{2})} = \\sin{(t_{2})} and \\mathbf{s}{(t_{2})} = \\cos{(\\operatorname{J_{\\varepsilon}}{(t_{2})})} and \\tilde{g}{(t_{2})} = \\cos{(\\operatorname{J_{\\varepsilon}}{(t_{2})})} and \\tilde{g}{(t_{2})} = \\mathbf{s}{(t_{2})} and \\frac{\\tilde{g}{(t_{2})}}{\\mathbf{s}{(t_{2})}} = 1 and \\frac{d}{d t_{2}} \\frac{\\tilde{g}{(t_{2})}}{\\mathbf{s}{(t_{2})}} = \\frac{d}{d t_{2}} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True)), cos(sin(Symbol('t_2', commutative=True))))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('t_2', commutative=True)), sin(Symbol('t_2', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True)), cos(Function('J_{\\\\varepsilon}')(Symbol('t_2', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('t_2', commutative=True)), cos(Function('J_{\\\\varepsilon}')(Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('\\\\tilde{g}')(Symbol('t_2', commutative=True)), Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True)))"], [["divide", 5, "Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('t_2', commutative=True))), Integer(1))"], [["differentiate", 6, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Function('\\\\mathbf{s}')(Symbol('t_2', commutative=True)), Integer(-1)), Function('\\\\tilde{g}')(Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{J})} = \\sin{(\\mathbf{J})}, then derive \\frac{d}{d \\mathbf{J}} \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\frac{\\partial}{\\partial \\mathbf{J}} (\\sigma_x - \\cos{(\\mathbf{J})}), then obtain \\frac{d}{d \\mathbf{J}} \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\sin{(\\mathbf{J})}", "derivation": "\\bar{\\h}{(\\mathbf{J})} = \\sin{(\\mathbf{J})} and \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\int \\sin{(\\mathbf{J})} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\frac{d}{d \\mathbf{J}} \\int \\sin{(\\mathbf{J})} d\\mathbf{J} and \\frac{d}{d \\mathbf{J}} \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\frac{\\partial}{\\partial \\mathbf{J}} (\\sigma_x - \\cos{(\\mathbf{J})}) and \\frac{d}{d \\mathbf{J}} \\int \\bar{\\h}{(\\mathbf{J})} d\\mathbf{J} = \\sin{(\\mathbf{J})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{J}', commutative=True)), sin(Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Integral(sin(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integral(sin(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_integrals", 3], "Equality(Derivative(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\sigma_x', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{J}', commutative=True)))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Derivative(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mathbf{J}', commutative=True))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), sin(Symbol('\\\\mathbf{J}', commutative=True)))"]]}, {"prompt": "Given \\phi_{2}{(l,\\phi_1)} = - l + \\sin{(\\phi_1)}, then obtain (- l + (\\frac{\\phi_{2}{(l,\\phi_1)}}{- l + \\sin{(\\phi_1)}})^{\\phi_1} + \\sin{(\\phi_1)})^{l} = (- l + \\sin{(\\phi_1)} + 1)^{l}", "derivation": "\\phi_{2}{(l,\\phi_1)} = - l + \\sin{(\\phi_1)} and \\frac{\\phi_{2}{(l,\\phi_1)}}{- l + \\sin{(\\phi_1)}} = 1 and (\\frac{\\phi_{2}{(l,\\phi_1)}}{- l + \\sin{(\\phi_1)}})^{\\phi_1} = 1 and - l + (\\frac{\\phi_{2}{(l,\\phi_1)}}{- l + \\sin{(\\phi_1)}})^{\\phi_1} + \\sin{(\\phi_1)} = - l + \\sin{(\\phi_1)} + 1 and (- l + (\\frac{\\phi_{2}{(l,\\phi_1)}}{- l + \\sin{(\\phi_1)}})^{\\phi_1} + \\sin{(\\phi_1)})^{l} = (- l + \\sin{(\\phi_1)} + 1)^{l}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_2')(Symbol('l', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))))"], [["divide", 1, "Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Function('\\\\phi_2')(Symbol('l', commutative=True), Symbol('\\\\phi_1', commutative=True))), Integer(1))"], [["power", 2, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Function('\\\\phi_2')(Symbol('l', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), Integer(1))"], [["add", 3, "Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Function('\\\\phi_2')(Symbol('l', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)), Integer(1)))"], [["power", 4, "Symbol('l', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Integer(-1)), Function('\\\\phi_2')(Symbol('l', commutative=True), Symbol('\\\\phi_1', commutative=True))), Symbol('\\\\phi_1', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True))), Symbol('l', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('l', commutative=True)), sin(Symbol('\\\\phi_1', commutative=True)), Integer(1)), Symbol('l', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(m_{s})} = m_{s}, then obtain \\int (m_{s} + (2 \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})}) dm_{s} = \\int (m_{s} + (m_{s} + \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})}) dm_{s}", "derivation": "\\mathbf{M}{(m_{s})} = m_{s} and 2 \\mathbf{M}{(m_{s})} = m_{s} + \\mathbf{M}{(m_{s})} and (2 \\mathbf{M}{(m_{s})})^{m_{s}} = (m_{s} + \\mathbf{M}{(m_{s})})^{m_{s}} and m_{s} + (2 \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})} = m_{s} + (m_{s} + \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})} and \\int (m_{s} + (2 \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})}) dm_{s} = \\int (m_{s} + (m_{s} + \\mathbf{M}{(m_{s})})^{m_{s}} + \\mathbf{M}{(m_{s})}) dm_{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], [["add", 1, "Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Pow(Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["add", 3, "Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True)))"], "Equality(Add(Symbol('m_s', commutative=True), Pow(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Add(Symbol('m_s', commutative=True), Pow(Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))))"], [["integrate", 4, "Symbol('m_s', commutative=True)"], "Equality(Integral(Add(Symbol('m_s', commutative=True), Pow(Mul(Integer(2), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))), Integral(Add(Symbol('m_s', commutative=True), Pow(Add(Symbol('m_s', commutative=True), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)), Function('\\\\mathbf{M}')(Symbol('m_s', commutative=True))), Tuple(Symbol('m_s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mathbf{f})} = \\sin{(\\mathbf{f})}, then obtain - \\mathbf{f} + \\operatorname{f_{E}}{(\\mathbf{f})} = - \\mathbf{f} - \\operatorname{f_{E}}{(\\mathbf{f})} + 2 \\sin{(\\mathbf{f})}", "derivation": "\\operatorname{f_{E}}{(\\mathbf{f})} = \\sin{(\\mathbf{f})} and - \\mathbf{f} + \\operatorname{f_{E}}{(\\mathbf{f})} = - \\mathbf{f} + \\sin{(\\mathbf{f})} and - \\mathbf{f} = - \\mathbf{f} - \\operatorname{f_{E}}{(\\mathbf{f})} + \\sin{(\\mathbf{f})} and - \\mathbf{f} + \\sin{(\\mathbf{f})} = - \\mathbf{f} - \\operatorname{f_{E}}{(\\mathbf{f})} + 2 \\sin{(\\mathbf{f})} and - \\mathbf{f} + \\operatorname{f_{E}}{(\\mathbf{f})} = - \\mathbf{f} - \\operatorname{f_{E}}{(\\mathbf{f})} + 2 \\sin{(\\mathbf{f})}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["minus", 2, "Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))), sin(Symbol('\\\\mathbf{f}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), sin(Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{f}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Mul(Integer(-1), Function('f_E')(Symbol('\\\\mathbf{f}', commutative=True))), Mul(Integer(2), sin(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given Q{(F_{x})} = \\cos{(\\sin{(F_{x})})}, then obtain (2 Q{(F_{x})})^{- F_{x}} ((2 Q{(F_{x})})^{F_{x}})^{F_{x}} = (2 Q{(F_{x})})^{- F_{x}} ((Q{(F_{x})} + \\cos{(\\sin{(F_{x})})})^{F_{x}})^{F_{x}}", "derivation": "Q{(F_{x})} = \\cos{(\\sin{(F_{x})})} and 2 Q{(F_{x})} = Q{(F_{x})} + \\cos{(\\sin{(F_{x})})} and (2 Q{(F_{x})})^{F_{x}} = (Q{(F_{x})} + \\cos{(\\sin{(F_{x})})})^{F_{x}} and ((2 Q{(F_{x})})^{F_{x}})^{F_{x}} = ((Q{(F_{x})} + \\cos{(\\sin{(F_{x})})})^{F_{x}})^{F_{x}} and (2 Q{(F_{x})})^{- F_{x}} ((2 Q{(F_{x})})^{F_{x}})^{F_{x}} = (2 Q{(F_{x})})^{- F_{x}} ((Q{(F_{x})} + \\cos{(\\sin{(F_{x})})})^{F_{x}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('F_x', commutative=True)), cos(sin(Symbol('F_x', commutative=True))))"], [["add", 1, "Function('Q')(Symbol('F_x', commutative=True))"], "Equality(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Add(Function('Q')(Symbol('F_x', commutative=True)), cos(sin(Symbol('F_x', commutative=True)))))"], [["power", 2, "Symbol('F_x', commutative=True)"], "Equality(Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Pow(Add(Function('Q')(Symbol('F_x', commutative=True)), cos(sin(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Pow(Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)), Pow(Pow(Add(Function('Q')(Symbol('F_x', commutative=True)), cos(sin(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True)))"], [["divide", 4, "Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True))"], "Equality(Mul(Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Pow(Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Pow(Mul(Integer(2), Function('Q')(Symbol('F_x', commutative=True))), Mul(Integer(-1), Symbol('F_x', commutative=True))), Pow(Pow(Add(Function('Q')(Symbol('F_x', commutative=True)), cos(sin(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given y{(A_{x})} = e^{e^{A_{x}}}, then obtain y^{- A_{x}}{(A_{x})} e^{- A_{x}} (e^{e^{A_{x}}})^{A_{x}} \\int y{(A_{x})} dA_{x} = y^{- A_{x}}{(A_{x})} e^{- A_{x}} (e^{e^{A_{x}}})^{A_{x}} \\int e^{e^{A_{x}}} dA_{x}", "derivation": "y{(A_{x})} = e^{e^{A_{x}}} and \\int y{(A_{x})} dA_{x} = \\int e^{e^{A_{x}}} dA_{x} and y^{- A_{x}}{(A_{x})} \\int y{(A_{x})} dA_{x} = y^{- A_{x}}{(A_{x})} \\int e^{e^{A_{x}}} dA_{x} and y^{- A_{x}}{(A_{x})} e^{- A_{x}} (e^{e^{A_{x}}})^{A_{x}} \\int y{(A_{x})} dA_{x} = y^{- A_{x}}{(A_{x})} e^{- A_{x}} (e^{e^{A_{x}}})^{A_{x}} \\int e^{e^{A_{x}}} dA_{x}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('A_x', commutative=True)), exp(exp(Symbol('A_x', commutative=True))))"], [["integrate", 1, "Symbol('A_x', commutative=True)"], "Equality(Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True))), Integral(exp(exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True))))"], [["divide", 2, "Pow(Function('y')(Symbol('A_x', commutative=True)), Symbol('A_x', commutative=True))"], "Equality(Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('A_x', commutative=True))), Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('A_x', commutative=True))), Integral(exp(exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True)))))"], [["times", 3, "Mul(exp(Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(exp(exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)))"], "Equality(Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('A_x', commutative=True))), exp(Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(exp(exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Integral(Function('y')(Symbol('A_x', commutative=True)), Tuple(Symbol('A_x', commutative=True)))), Mul(Pow(Function('y')(Symbol('A_x', commutative=True)), Mul(Integer(-1), Symbol('A_x', commutative=True))), exp(Mul(Integer(-1), Symbol('A_x', commutative=True))), Pow(exp(exp(Symbol('A_x', commutative=True))), Symbol('A_x', commutative=True)), Integral(exp(exp(Symbol('A_x', commutative=True))), Tuple(Symbol('A_x', commutative=True)))))"]]}, {"prompt": "Given \\pi{(A_{2})} = \\log{(A_{2})}, then derive \\int \\pi{(A_{2})} dA_{2} = A_{2} \\log{(A_{2})} - A_{2} + \\mathbf{J}_f, then obtain \\frac{d^{2}}{d A_{2}^{2}} \\int \\pi{(A_{2})} dA_{2} = \\frac{\\partial^{2}}{\\partial A_{2}^{2}} (A_{2} \\log{(A_{2})} - A_{2} + \\mathbf{J}_f)", "derivation": "\\pi{(A_{2})} = \\log{(A_{2})} and \\int \\pi{(A_{2})} dA_{2} = \\int \\log{(A_{2})} dA_{2} and \\int \\pi{(A_{2})} dA_{2} = A_{2} \\log{(A_{2})} - A_{2} + \\mathbf{J}_f and \\frac{d}{d A_{2}} \\int \\pi{(A_{2})} dA_{2} = \\frac{d}{d A_{2}} \\int \\log{(A_{2})} dA_{2} and \\int \\log{(A_{2})} dA_{2} = A_{2} \\log{(A_{2})} - A_{2} + \\mathbf{J}_f and \\frac{d^{2}}{d A_{2}^{2}} \\int \\pi{(A_{2})} dA_{2} = \\frac{d^{2}}{d A_{2}^{2}} \\int \\log{(A_{2})} dA_{2} and \\frac{d^{2}}{d A_{2}^{2}} \\int \\pi{(A_{2})} dA_{2} = \\frac{\\partial^{2}}{\\partial A_{2}^{2}} (A_{2} \\log{(A_{2})} - A_{2} + \\mathbf{J}_f)", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('A_2', commutative=True)), log(Symbol('A_2', commutative=True)))"], [["integrate", 1, "Symbol('A_2', commutative=True)"], "Equality(Integral(Function('\\\\pi')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\pi')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), log(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 2, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Add(Mul(Symbol('A_2', commutative=True), log(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)))"], [["differentiate", 4, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(2))), Derivative(Integral(log(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Derivative(Integral(Function('\\\\pi')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True))), Tuple(Symbol('A_2', commutative=True), Integer(2))), Derivative(Add(Mul(Symbol('A_2', commutative=True), log(Symbol('A_2', commutative=True))), Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(q)} = e^{e^{q}}, then obtain - q (- q + 3 \\operatorname{P_{g}}{(q)} e^{- e^{q}} - 2) = - q (- q + \\operatorname{P_{g}}{(q)} e^{- e^{q}})", "derivation": "\\operatorname{P_{g}}{(q)} = e^{e^{q}} and \\operatorname{P_{g}}{(q)} e^{- e^{q}} = 1 and - q + \\operatorname{P_{g}}{(q)} e^{- e^{q}} = 1 - q and - q (- q + \\operatorname{P_{g}}{(q)} e^{- e^{q}}) = - q (1 - q) and - q (- q + 2 \\operatorname{P_{g}}{(q)} e^{- e^{q}} - 1) = - q (- q + \\operatorname{P_{g}}{(q)} e^{- e^{q}}) and - q (- q + 2 \\operatorname{P_{g}}{(q)} e^{- e^{q}} - 1) = - q (1 - q) and - q (- q + 3 \\operatorname{P_{g}}{(q)} e^{- e^{q}} - 2) = - q (- q + \\operatorname{P_{g}}{(q)} e^{- e^{q}})", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('q', commutative=True)), exp(exp(Symbol('q', commutative=True))))"], [["divide", 1, "exp(exp(Symbol('q', commutative=True)))"], "Equality(Mul(Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))), Integer(1))"], [["minus", 2, "Symbol('q', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True)))))), Add(Integer(1), Mul(Integer(-1), Symbol('q', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))))), Mul(Integer(-1), Symbol('q', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(2), Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))), Integer(-1))), Mul(Integer(-1), Symbol('q', commutative=True), Add(Integer(1), Mul(Integer(-1), Symbol('q', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Integer(3), Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))), Integer(-2))), Mul(Integer(-1), Symbol('q', commutative=True), Add(Mul(Integer(-1), Symbol('q', commutative=True)), Mul(Function('P_g')(Symbol('q', commutative=True)), exp(Mul(Integer(-1), exp(Symbol('q', commutative=True))))))))"]]}, {"prompt": "Given \\lambda{(\\psi,x,f)} = x (\\psi + f), then derive \\frac{\\partial^{2}}{\\partial f\\partial \\psi} \\lambda{(\\psi,x,f)} = 0, then obtain \\iint \\frac{\\partial^{2}}{\\partial f\\partial \\psi} \\lambda{(\\psi,x,f)} dx df = \\iint 0 dx df", "derivation": "\\lambda{(\\psi,x,f)} = x (\\psi + f) and \\frac{\\partial}{\\partial f} \\lambda{(\\psi,x,f)} = \\frac{\\partial}{\\partial f} x (\\psi + f) and \\frac{\\partial^{2}}{\\partial \\psi\\partial f} \\lambda{(\\psi,x,f)} = \\frac{\\partial^{2}}{\\partial \\psi\\partial f} x (\\psi + f) and \\frac{\\partial^{2}}{\\partial f\\partial \\psi} \\lambda{(\\psi,x,f)} = 0 and \\int \\frac{\\partial^{2}}{\\partial f\\partial \\psi} \\lambda{(\\psi,x,f)} dx = \\int 0 dx and \\iint \\frac{\\partial^{2}}{\\partial f\\partial \\psi} \\lambda{(\\psi,x,f)} dx df = \\iint 0 dx df", "srepr_derivation": [["premise", "Equality(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Mul(Symbol('x', commutative=True), Add(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True))))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Mul(Symbol('x', commutative=True), Add(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\psi', commutative=True)"], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))), Derivative(Mul(Symbol('x', commutative=True), Add(Symbol('\\\\psi', commutative=True), Symbol('f', commutative=True))), Tuple(Symbol('f', commutative=True), Integer(1)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Derivative(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1))), Integer(0))"], [["integrate", 4, "Symbol('x', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True))), Integral(Integer(0), Tuple(Symbol('x', commutative=True))))"], [["integrate", 5, "Symbol('f', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\lambda')(Symbol('\\\\psi', commutative=True), Symbol('x', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('f', commutative=True), Integer(1))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Integer(0), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('f', commutative=True))))"]]}, {"prompt": "Given C{(F_{x})} = \\cos{(F_{x})} and H{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})}, then derive \\frac{d}{d F_{x}} C{(F_{x})} = - \\sin{(F_{x})}, then obtain H{(F_{x})} = - \\sin{(F_{x})}", "derivation": "C{(F_{x})} = \\cos{(F_{x})} and \\frac{d}{d F_{x}} C{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})} and \\frac{d}{d F_{x}} C{(F_{x})} = - \\sin{(F_{x})} and H{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})} and H{(F_{x})} = \\frac{d}{d F_{x}} C{(F_{x})} and H{(F_{x})} = - \\sin{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True))))"], ["renaming_premise", "Equality(Function('H')(Symbol('F_x', commutative=True)), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Function('H')(Symbol('F_x', commutative=True)), Derivative(Function('C')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('H')(Symbol('F_x', commutative=True)), Mul(Integer(-1), sin(Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\nabla{(l,S)} = S l, then obtain S l^{2} (\\int \\nabla{(l,S)} dl)^{2} = S l^{2} (\\int S l dl) \\int \\nabla{(l,S)} dl", "derivation": "\\nabla{(l,S)} = S l and \\int \\nabla{(l,S)} dl = \\int S l dl and (\\int \\nabla{(l,S)} dl)^{2} = (\\int S l dl) \\int \\nabla{(l,S)} dl and S l (\\int \\nabla{(l,S)} dl)^{2} = S l (\\int S l dl) \\int \\nabla{(l,S)} dl and S l^{2} (\\int \\nabla{(l,S)} dl)^{2} = S l^{2} (\\int S l dl) \\int \\nabla{(l,S)} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Symbol('l', commutative=True)))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Symbol('S', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))))"], [["times", 2, "Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True)))"], "Equality(Pow(Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(2)), Mul(Integral(Mul(Symbol('S', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["times", 3, "Mul(Symbol('S', commutative=True), Symbol('l', commutative=True))"], "Equality(Mul(Symbol('S', commutative=True), Symbol('l', commutative=True), Pow(Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(2))), Mul(Symbol('S', commutative=True), Symbol('l', commutative=True), Integral(Mul(Symbol('S', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["times", 4, "Symbol('l', commutative=True)"], "Equality(Mul(Symbol('S', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)), Pow(Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(2))), Mul(Symbol('S', commutative=True), Pow(Symbol('l', commutative=True), Integer(2)), Integral(Mul(Symbol('S', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Function('\\\\nabla')(Symbol('l', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('l', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(\\eta)} = e^{\\sin{(\\eta)}} and \\operatorname{x^{{\\}'}}{(\\eta)} = \\mathbf{v}{(\\eta)} - \\sin{(\\eta)}, then obtain (e^{\\sin{(\\eta)}} - \\sin{(\\eta)})^{\\eta} = (\\mathbf{v}{(\\eta)} - \\sin{(\\eta)})^{\\eta}", "derivation": "\\mathbf{v}{(\\eta)} = e^{\\sin{(\\eta)}} and \\mathbf{v}{(\\eta)} - \\sin{(\\eta)} = e^{\\sin{(\\eta)}} - \\sin{(\\eta)} and \\operatorname{x^{{\\}'}}{(\\eta)} = \\mathbf{v}{(\\eta)} - \\sin{(\\eta)} and \\operatorname{x^{{\\}'}}^{\\eta}{(\\eta)} = (\\mathbf{v}{(\\eta)} - \\sin{(\\eta)})^{\\eta} and \\operatorname{x^{{\\}'}}{(\\eta)} = e^{\\sin{(\\eta)}} - \\sin{(\\eta)} and \\operatorname{x^{{\\}'}}^{\\eta}{(\\eta)} = (e^{\\sin{(\\eta)}} - \\sin{(\\eta)})^{\\eta} and (e^{\\sin{(\\eta)}} - \\sin{(\\eta)})^{\\eta} = (\\mathbf{v}{(\\eta)} - \\sin{(\\eta)})^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('\\\\eta', commutative=True)), exp(sin(Symbol('\\\\eta', commutative=True))))"], [["minus", 1, "sin(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Add(exp(sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Add(Function('\\\\mathbf{v}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], [["power", 3, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Add(Function('\\\\mathbf{v}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Add(exp(sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))))"], [["power", 5, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('x^\\\\prime')(Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Add(exp(sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Add(exp(sin(Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)), Pow(Add(Function('\\\\mathbf{v}')(Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\eta', commutative=True)))), Symbol('\\\\eta', commutative=True)))"]]}, {"prompt": "Given \\mathbf{p}{(F_{N},L)} = e^{F_{N}^{L}}, then obtain (\\mathbf{p}{(F_{N},L)} + \\frac{e^{F_{N}^{L}}}{F_{N}})^{F_{N}} = (e^{F_{N}^{L}} + \\frac{e^{F_{N}^{L}}}{F_{N}})^{F_{N}}", "derivation": "\\mathbf{p}{(F_{N},L)} = e^{F_{N}^{L}} and \\frac{\\mathbf{p}{(F_{N},L)}}{F_{N}} = \\frac{e^{F_{N}^{L}}}{F_{N}} and \\mathbf{p}{(F_{N},L)} + \\frac{\\mathbf{p}{(F_{N},L)}}{F_{N}} = e^{F_{N}^{L}} + \\frac{\\mathbf{p}{(F_{N},L)}}{F_{N}} and (\\mathbf{p}{(F_{N},L)} + \\frac{\\mathbf{p}{(F_{N},L)}}{F_{N}})^{F_{N}} = (e^{F_{N}^{L}} + \\frac{\\mathbf{p}{(F_{N},L)}}{F_{N}})^{F_{N}} and (\\mathbf{p}{(F_{N},L)} + \\frac{e^{F_{N}^{L}}}{F_{N}})^{F_{N}} = (e^{F_{N}^{L}} + \\frac{e^{F_{N}^{L}}}{F_{N}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)), exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))))"], [["divide", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))))"], [["add", 1, "Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))), Add(exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Add(Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))), Symbol('F_N', commutative=True)), Pow(Add(exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)))), Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Add(Function('\\\\mathbf{p}')(Symbol('F_N', commutative=True), Symbol('L', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))))), Symbol('F_N', commutative=True)), Pow(Add(exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), exp(Pow(Symbol('F_N', commutative=True), Symbol('L', commutative=True))))), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = - \\sin{(\\mathbf{J}_P - \\mathbf{S})} and H{(\\varepsilon_0,\\rho_f)} = \\rho_f + \\varepsilon_0, then obtain 2 (\\rho_f + \\varepsilon_0) \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = (\\rho_f + \\varepsilon_0) (\\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} - \\sin{(\\mathbf{J}_P - \\mathbf{S})})", "derivation": "\\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = - \\sin{(\\mathbf{J}_P - \\mathbf{S})} and 2 \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} - \\sin{(\\mathbf{J}_P - \\mathbf{S})} and H{(\\varepsilon_0,\\rho_f)} = \\rho_f + \\varepsilon_0 and 2 H{(\\varepsilon_0,\\rho_f)} \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = (\\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} - \\sin{(\\mathbf{J}_P - \\mathbf{S})}) H{(\\varepsilon_0,\\rho_f)} and 2 (\\rho_f + \\varepsilon_0) \\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} = (\\rho_f + \\varepsilon_0) (\\theta_{2}{(\\mathbf{S},\\mathbf{J}_P)} - \\sin{(\\mathbf{J}_P - \\mathbf{S})})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))))"], [["minus", 1, "Mul(Integer(-1), Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))))"], ["get_premise", "Equality(Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 2, "Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Integer(2), Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\rho_f', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Add(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True)))))), Function('H')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Integer(2), Add(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Add(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\varepsilon_0', commutative=True)), Add(Function('\\\\theta_2')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mathbf{J}_P', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\mathbf{J}_P', commutative=True), Mul(Integer(-1), Symbol('\\\\mathbf{S}', commutative=True))))))))"]]}, {"prompt": "Given \\hat{H}{(U,\\varphi^*)} = \\frac{\\partial}{\\partial U} U \\varphi^*, then obtain \\int (U + 2 \\hat{H}{(U,\\varphi^*)} - 1) dU = \\int (U + 2 \\frac{\\partial}{\\partial U} U \\varphi^* - 1) dU", "derivation": "\\hat{H}{(U,\\varphi^*)} = \\frac{\\partial}{\\partial U} U \\varphi^* and \\hat{H}{(U,\\varphi^*)} - 1 = \\frac{\\partial}{\\partial U} U \\varphi^* - 1 and 2 \\hat{H}{(U,\\varphi^*)} - 1 = \\hat{H}{(U,\\varphi^*)} + \\frac{\\partial}{\\partial U} U \\varphi^* - 1 and 2 \\hat{H}{(U,\\varphi^*)} - 1 = 2 \\frac{\\partial}{\\partial U} U \\varphi^* - 1 and U + 2 \\hat{H}{(U,\\varphi^*)} - 1 = U + 2 \\frac{\\partial}{\\partial U} U \\varphi^* - 1 and \\int (U + 2 \\hat{H}{(U,\\varphi^*)} - 1) dU = \\int (U + 2 \\frac{\\partial}{\\partial U} U \\varphi^* - 1) dU", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["minus", 1, 1], "Equality(Add(Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(-1)), Add(Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)))"], [["add", 1, "Add(Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(-1))"], "Equality(Add(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Add(Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Add(Mul(Integer(2), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integer(-1)))"], [["add", 4, "Symbol('U', commutative=True)"], "Equality(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Add(Symbol('U', commutative=True), Mul(Integer(2), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integer(-1)))"], [["integrate", 5, "Symbol('U', commutative=True)"], "Equality(Integral(Add(Symbol('U', commutative=True), Mul(Integer(2), Function('\\\\hat{H}')(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Integer(-1)), Tuple(Symbol('U', commutative=True))), Integral(Add(Symbol('U', commutative=True), Mul(Integer(2), Derivative(Mul(Symbol('U', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1)))), Integer(-1)), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given J{(\\mathbf{H})} = \\sin{(\\mathbf{H})}, then obtain \\frac{d}{d \\mathbf{H}} - \\sin{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} - J{(\\mathbf{H})}", "derivation": "J{(\\mathbf{H})} = \\sin{(\\mathbf{H})} and J{(\\mathbf{H})} - \\sin{(\\mathbf{H})} = 0 and - \\sin{(\\mathbf{H})} = - J{(\\mathbf{H})} and \\frac{d}{d \\mathbf{H}} - \\sin{(\\mathbf{H})} = \\frac{d}{d \\mathbf{H}} - J{(\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], [["minus", 1, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('J')(Symbol('\\\\mathbf{H}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Integer(0))"], [["minus", 2, "Function('J')(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))), Mul(Integer(-1), Function('J')(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), sin(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Function('J')(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(C,\\varepsilon,x)} = - x + \\frac{\\varepsilon}{C}, then obtain - C + (\\operatorname{A_{1}}^{\\varepsilon}{(C,\\varepsilon,x)})^{\\varepsilon} = - C + ((- x + \\frac{\\varepsilon}{C})^{\\varepsilon})^{\\varepsilon}", "derivation": "\\operatorname{A_{1}}{(C,\\varepsilon,x)} = - x + \\frac{\\varepsilon}{C} and \\operatorname{A_{1}}^{\\varepsilon}{(C,\\varepsilon,x)} = (- x + \\frac{\\varepsilon}{C})^{\\varepsilon} and (\\operatorname{A_{1}}^{\\varepsilon}{(C,\\varepsilon,x)})^{\\varepsilon} = ((- x + \\frac{\\varepsilon}{C})^{\\varepsilon})^{\\varepsilon} and - C + (\\operatorname{A_{1}}^{\\varepsilon}{(C,\\varepsilon,x)})^{\\varepsilon} = - C + ((- x + \\frac{\\varepsilon}{C})^{\\varepsilon})^{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('C', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Function('A_1')(Symbol('C', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)))"], [["power", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Pow(Pow(Function('A_1')(Symbol('C', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["minus", 3, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Pow(Function('A_1')(Symbol('C', commutative=True), Symbol('\\\\varepsilon', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('C', commutative=True), Integer(-1)), Symbol('\\\\varepsilon', commutative=True))), Symbol('\\\\varepsilon', commutative=True)), Symbol('\\\\varepsilon', commutative=True))))"]]}, {"prompt": "Given \\phi{(G,y)} = (e^{y})^{G}, then obtain ((e^{y})^{G})^{y} \\int \\log{(\\phi^{y}{(G,y)})} dG = ((e^{y})^{G})^{y} \\int \\log{(((e^{y})^{G})^{y})} dG", "derivation": "\\phi{(G,y)} = (e^{y})^{G} and \\phi^{y}{(G,y)} = ((e^{y})^{G})^{y} and \\log{(\\phi^{y}{(G,y)})} = \\log{(((e^{y})^{G})^{y})} and \\int \\log{(\\phi^{y}{(G,y)})} dG = \\int \\log{(((e^{y})^{G})^{y})} dG and ((e^{y})^{G})^{y} \\int \\log{(\\phi^{y}{(G,y)})} dG = ((e^{y})^{G})^{y} \\int \\log{(((e^{y})^{G})^{y})} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)))"], [["power", 1, "Symbol('y', commutative=True)"], "Equality(Pow(Function('\\\\phi')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True)), Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True)))"], [["log", 2], "Equality(log(Pow(Function('\\\\phi')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), log(Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True))))"], [["integrate", 3, "Symbol('G', commutative=True)"], "Equality(Integral(log(Pow(Function('\\\\phi')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('G', commutative=True))), Integral(log(Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('G', commutative=True))))"], [["times", 4, "Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True))"], "Equality(Mul(Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True)), Integral(log(Pow(Function('\\\\phi')(Symbol('G', commutative=True), Symbol('y', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('G', commutative=True)))), Mul(Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True)), Integral(log(Pow(Pow(exp(Symbol('y', commutative=True)), Symbol('G', commutative=True)), Symbol('y', commutative=True))), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given W{(\\mathbf{D})} = \\int \\sin{(\\mathbf{D})} d\\mathbf{D}, then derive W{(\\mathbf{D})} = a^{\\dagger} - \\cos{(\\mathbf{D})}, then obtain W^{\\mathbf{D}}{(\\mathbf{D})} - \\int \\sin{(\\mathbf{D})} d\\mathbf{D} = (a^{\\dagger} - \\cos{(\\mathbf{D})})^{\\mathbf{D}} - \\int \\sin{(\\mathbf{D})} d\\mathbf{D}", "derivation": "W{(\\mathbf{D})} = \\int \\sin{(\\mathbf{D})} d\\mathbf{D} and W{(\\mathbf{D})} = a^{\\dagger} - \\cos{(\\mathbf{D})} and W^{\\mathbf{D}}{(\\mathbf{D})} = (a^{\\dagger} - \\cos{(\\mathbf{D})})^{\\mathbf{D}} and W^{\\mathbf{D}}{(\\mathbf{D})} - \\int \\sin{(\\mathbf{D})} d\\mathbf{D} = (a^{\\dagger} - \\cos{(\\mathbf{D})})^{\\mathbf{D}} - \\int \\sin{(\\mathbf{D})} d\\mathbf{D}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{D}', commutative=True)), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('W')(Symbol('\\\\mathbf{D}', commutative=True)), Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))))"], [["power", 2, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Pow(Function('W')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)))"], [["minus", 3, "Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True)))"], "Equality(Add(Pow(Function('W')(Symbol('\\\\mathbf{D}', commutative=True)), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))), Add(Pow(Add(Symbol('a^{\\\\dagger}', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\mathbf{D}', commutative=True)))), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Integer(-1), Integral(sin(Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\pi,n)} = \\sin{(\\pi + n)} and z{(\\hat{H}_l,\\mathbf{E})} = \\frac{\\mathbf{E}}{\\hat{H}_l}, then obtain \\int (z{(\\hat{H}_l,\\mathbf{E})} \\sin^{\\pi}{(\\pi + n)})^{\\pi} d\\hat{H}_l = \\int (\\frac{\\mathbf{E} \\sin^{\\pi}{(\\pi + n)}}{\\hat{H}_l})^{\\pi} d\\hat{H}_l", "derivation": "\\operatorname{f^{\\prime}}{(\\pi,n)} = \\sin{(\\pi + n)} and z{(\\hat{H}_l,\\mathbf{E})} = \\frac{\\mathbf{E}}{\\hat{H}_l} and \\operatorname{f^{\\prime}}^{\\pi}{(\\pi,n)} z{(\\hat{H}_l,\\mathbf{E})} = \\frac{\\mathbf{E} \\operatorname{f^{\\prime}}^{\\pi}{(\\pi,n)}}{\\hat{H}_l} and z{(\\hat{H}_l,\\mathbf{E})} \\sin^{\\pi}{(\\pi + n)} = \\frac{\\mathbf{E} \\sin^{\\pi}{(\\pi + n)}}{\\hat{H}_l} and (z{(\\hat{H}_l,\\mathbf{E})} \\sin^{\\pi}{(\\pi + n)})^{\\pi} = (\\frac{\\mathbf{E} \\sin^{\\pi}{(\\pi + n)}}{\\hat{H}_l})^{\\pi} and \\int (z{(\\hat{H}_l,\\mathbf{E})} \\sin^{\\pi}{(\\pi + n)})^{\\pi} d\\hat{H}_l = \\int (\\frac{\\mathbf{E} \\sin^{\\pi}{(\\pi + n)}}{\\hat{H}_l})^{\\pi} d\\hat{H}_l", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True)))"], [["times", 2, "Pow(Function('f^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\pi', commutative=True))"], "Equality(Mul(Pow(Function('f^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\pi', commutative=True)), Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{E}', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(Function('f^{\\\\prime}')(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))))"], [["power", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Mul(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"], [["integrate", 5, "Symbol('\\\\hat{H}_l', commutative=True)"], "Equality(Integral(Pow(Mul(Function('z')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('\\\\mathbf{E}', commutative=True)), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))), Integral(Pow(Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Integer(-1)), Symbol('\\\\mathbf{E}', commutative=True), Pow(sin(Add(Symbol('\\\\pi', commutative=True), Symbol('n', commutative=True))), Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\hat{H}_l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(a^{\\dagger})} = \\sin{(\\cos{(a^{\\dagger})})}, then obtain \\operatorname{v_{x}}^{a^{\\dagger}}{(a^{\\dagger})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = \\sin^{a^{\\dagger}}{(\\cos{(a^{\\dagger})})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger}", "derivation": "\\operatorname{v_{x}}{(a^{\\dagger})} = \\sin{(\\cos{(a^{\\dagger})})} and \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = \\int \\sin{(\\cos{(a^{\\dagger})})} da^{\\dagger} and \\operatorname{v_{x}}^{a^{\\dagger}}{(a^{\\dagger})} = \\sin^{a^{\\dagger}}{(\\cos{(a^{\\dagger})})} and \\operatorname{v_{x}}^{a^{\\dagger}}{(a^{\\dagger})} \\int \\sin{(\\cos{(a^{\\dagger})})} da^{\\dagger} = \\sin^{a^{\\dagger}}{(\\cos{(a^{\\dagger})})} \\int \\sin{(\\cos{(a^{\\dagger})})} da^{\\dagger} and \\operatorname{v_{x}}^{a^{\\dagger}}{(a^{\\dagger})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger} = \\sin^{a^{\\dagger}}{(\\cos{(a^{\\dagger})})} \\int \\operatorname{v_{x}}{(a^{\\dagger})} da^{\\dagger}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), sin(cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["integrate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True))), Integral(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True))))"], [["power", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Pow(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Pow(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)))"], [["times", 3, "Integral(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))"], "Equality(Mul(Pow(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Integral(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Mul(Pow(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))), Mul(Pow(sin(cos(Symbol('a^{\\\\dagger}', commutative=True))), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Function('v_x')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{F_{c}}{(i,C_{1},F_{c})} = (C_{1}^{i})^{F_{c}}, then obtain \\frac{\\partial}{\\partial i} (i + \\operatorname{F_{c}}^{C_{1}}{(i,C_{1},F_{c})}) = \\frac{\\partial}{\\partial i} (i + ((C_{1}^{i})^{F_{c}})^{C_{1}})", "derivation": "\\operatorname{F_{c}}{(i,C_{1},F_{c})} = (C_{1}^{i})^{F_{c}} and \\operatorname{F_{c}}^{C_{1}}{(i,C_{1},F_{c})} = ((C_{1}^{i})^{F_{c}})^{C_{1}} and i + \\operatorname{F_{c}}^{C_{1}}{(i,C_{1},F_{c})} = i + ((C_{1}^{i})^{F_{c}})^{C_{1}} and \\frac{\\partial}{\\partial i} (i + \\operatorname{F_{c}}^{C_{1}}{(i,C_{1},F_{c})}) = \\frac{\\partial}{\\partial i} (i + ((C_{1}^{i})^{F_{c}})^{C_{1}})", "srepr_derivation": [["premise", "Equality(Function('F_c')(Symbol('i', commutative=True), Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Pow(Pow(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Symbol('F_c', commutative=True)))"], [["power", 1, "Symbol('C_1', commutative=True)"], "Equality(Pow(Function('F_c')(Symbol('i', commutative=True), Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True)), Pow(Pow(Pow(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True)))"], [["add", 2, "Symbol('i', commutative=True)"], "Equality(Add(Symbol('i', commutative=True), Pow(Function('F_c')(Symbol('i', commutative=True), Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True))), Add(Symbol('i', commutative=True), Pow(Pow(Pow(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True))))"], [["differentiate", 3, "Symbol('i', commutative=True)"], "Equality(Derivative(Add(Symbol('i', commutative=True), Pow(Function('F_c')(Symbol('i', commutative=True), Symbol('C_1', commutative=True), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Add(Symbol('i', commutative=True), Pow(Pow(Pow(Symbol('C_1', commutative=True), Symbol('i', commutative=True)), Symbol('F_c', commutative=True)), Symbol('C_1', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(q)} = \\cos{(q)} and \\mathbf{p}{(q)} = - \\cos^{q}{(q)}, then obtain - \\frac{\\cos^{q}{(q)}}{\\mathbf{p}{(q)}} = 1", "derivation": "\\operatorname{A_{2}}{(q)} = \\cos{(q)} and \\operatorname{A_{2}}^{q}{(q)} = \\cos^{q}{(q)} and \\mathbf{p}{(q)} = - \\cos^{q}{(q)} and \\mathbf{p}{(q)} = - \\operatorname{A_{2}}^{q}{(q)} and - \\operatorname{A_{2}}^{q}{(q)} = - \\cos^{q}{(q)} and - \\frac{\\operatorname{A_{2}}^{q}{(q)}}{\\mathbf{p}{(q)}} = - \\frac{\\cos^{q}{(q)}}{\\mathbf{p}{(q)}} and - \\frac{\\operatorname{A_{2}}^{q}{(q)}}{\\mathbf{p}{(q)}} = 1 and - \\frac{\\cos^{q}{(q)}}{\\mathbf{p}{(q)}} = 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('q', commutative=True)), cos(Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Mul(Integer(-1), Pow(Function('A_2')(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Function('A_2')(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["divide", 5, "Function('\\\\mathbf{p}')(Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Function('A_2')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Integer(-1)), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Mul(Integer(-1), Pow(Function('A_2')(Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Integer(-1))), Integer(1))"], [["substitute_LHS_for_RHS", 7, 2], "Equality(Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('q', commutative=True)), Integer(-1)), Pow(cos(Symbol('q', commutative=True)), Symbol('q', commutative=True))), Integer(1))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})} = \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}}, then obtain (g_{\\varepsilon} + \\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})})^{g_{\\varepsilon}} - \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}} = (g_{\\varepsilon} + \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}})^{g_{\\varepsilon}} - \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}}", "derivation": "\\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})} = \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}} and g_{\\varepsilon} + \\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})} = g_{\\varepsilon} + \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}} and (g_{\\varepsilon} + \\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})})^{g_{\\varepsilon}} = (g_{\\varepsilon} + \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}})^{g_{\\varepsilon}} and (g_{\\varepsilon} + \\operatorname{F_{g}}{(g_{\\varepsilon},\\mathbf{A})})^{g_{\\varepsilon}} - \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}} = (g_{\\varepsilon} + \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}})^{g_{\\varepsilon}} - \\frac{\\log{(\\mathbf{A})}}{g_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_g')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["power", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_g')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["minus", 3, "Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('F_g')(Symbol('g_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Pow(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))), Symbol('g_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), log(Symbol('\\\\mathbf{A}', commutative=True)))))"]]}, {"prompt": "Given \\chi{(\\tilde{g})} = \\tilde{g}, then obtain \\frac{d}{d \\tilde{g}} \\chi^{2 \\tilde{g}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\tilde{g}^{\\tilde{g}} \\chi^{\\tilde{g}}{(\\tilde{g})}", "derivation": "\\chi{(\\tilde{g})} = \\tilde{g} and \\chi^{\\tilde{g}}{(\\tilde{g})} = \\tilde{g}^{\\tilde{g}} and \\chi^{2 \\tilde{g}}{(\\tilde{g})} = \\tilde{g}^{\\tilde{g}} \\chi^{\\tilde{g}}{(\\tilde{g})} and \\frac{d}{d \\tilde{g}} \\chi^{2 \\tilde{g}}{(\\tilde{g})} = \\frac{d}{d \\tilde{g}} \\tilde{g}^{\\tilde{g}} \\chi^{\\tilde{g}}{(\\tilde{g})}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))"], [["power", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["times", 2, "Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Function('\\\\chi')(Symbol('\\\\tilde{g}', commutative=True)), Symbol('\\\\tilde{g}', commutative=True))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given G{(v_{2},Z)} = v_{2}^{Z}, then derive \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = \\frac{Z v_{2}^{Z}}{v_{2}}, then obtain \\frac{\\partial}{\\partial Z} v_{2} v_{2}^{- Z} \\frac{\\partial}{\\partial v_{2}} v_{2}^{Z} = \\frac{d}{d Z} Z", "derivation": "G{(v_{2},Z)} = v_{2}^{Z} and \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = \\frac{\\partial}{\\partial v_{2}} v_{2}^{Z} and \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = \\frac{Z v_{2}^{Z}}{v_{2}} and v_{2} v_{2}^{- Z} G{(v_{2},Z)} \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = Z G{(v_{2},Z)} and v_{2} v_{2}^{- Z} \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = Z and \\frac{\\partial}{\\partial Z} v_{2} v_{2}^{- Z} \\frac{\\partial}{\\partial v_{2}} G{(v_{2},Z)} = \\frac{d}{d Z} Z and \\frac{\\partial}{\\partial Z} v_{2} v_{2}^{- Z} \\frac{\\partial}{\\partial v_{2}} v_{2}^{Z} = \\frac{d}{d Z} Z", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('v_2', commutative=True)"], "Equality(Derivative(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Derivative(Pow(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1))), Mul(Symbol('Z', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Symbol('Z', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), Pow(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Pow(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Integer(-1)))"], "Equality(Mul(Symbol('v_2', commutative=True), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Derivative(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Mul(Symbol('Z', commutative=True), Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True))))"], [["times", 4, "Pow(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Integer(-1))"], "Equality(Mul(Symbol('v_2', commutative=True), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Symbol('Z', commutative=True))"], [["differentiate", 5, "Symbol('Z', commutative=True)"], "Equality(Derivative(Mul(Symbol('v_2', commutative=True), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Function('G')(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Mul(Symbol('v_2', commutative=True), Pow(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('Z', commutative=True))), Derivative(Pow(Symbol('v_2', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_2', commutative=True), Integer(1)))), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Symbol('Z', commutative=True), Tuple(Symbol('Z', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{A})} = e^{\\mathbf{A}}, then obtain \\frac{\\partial}{\\partial f^{\\prime}} E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} \\tilde{g}{(\\mathbf{A})} = \\frac{\\partial}{\\partial f^{\\prime}} E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}", "derivation": "\\tilde{g}{(\\mathbf{A})} = e^{\\mathbf{A}} and \\frac{d}{d \\mathbf{A}} \\tilde{g}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} \\tilde{g}{(\\mathbf{A})} = E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}} and \\frac{\\partial}{\\partial f^{\\prime}} E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} \\tilde{g}{(\\mathbf{A})} = \\frac{\\partial}{\\partial f^{\\prime}} E_{\\lambda} f^{\\prime} \\frac{d}{d \\mathbf{A}} e^{\\mathbf{A}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{A}', commutative=True)), exp(Symbol('\\\\mathbf{A}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["times", 2, "Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^{\\\\prime}', commutative=True))"], "Equality(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["differentiate", 3, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))), Derivative(Mul(Symbol('E_{\\\\lambda}', commutative=True), Symbol('f^{\\\\prime}', commutative=True), Derivative(exp(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))), Tuple(Symbol('f^{\\\\prime}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{x}}{(t_{1})} = \\cos{(t_{1})}, then obtain \\frac{d^{2}}{d t_{1}^{2}} \\operatorname{F_{x}}^{t_{1}}{(t_{1})} = \\frac{d^{2}}{d t_{1}^{2}} \\cos^{t_{1}}{(t_{1})}", "derivation": "\\operatorname{F_{x}}{(t_{1})} = \\cos{(t_{1})} and \\operatorname{F_{x}}^{t_{1}}{(t_{1})} = \\cos^{t_{1}}{(t_{1})} and \\frac{d}{d t_{1}} \\operatorname{F_{x}}^{t_{1}}{(t_{1})} = \\frac{d}{d t_{1}} \\cos^{t_{1}}{(t_{1})} and \\frac{d^{2}}{d t_{1}^{2}} \\operatorname{F_{x}}^{t_{1}}{(t_{1})} = \\frac{d^{2}}{d t_{1}^{2}} \\cos^{t_{1}}{(t_{1})}", "srepr_derivation": [["premise", "Equality(Function('F_x')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], [["power", 1, "Symbol('t_1', commutative=True)"], "Equality(Pow(Function('F_x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Pow(cos(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)))"], [["differentiate", 2, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Pow(Function('F_x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('t_1', commutative=True)"], "Equality(Derivative(Pow(Function('F_x')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(2))), Derivative(Pow(cos(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True)), Tuple(Symbol('t_1', commutative=True), Integer(2))))"]]}, {"prompt": "Given y{(v_{y},S)} = - v_{y} + \\log{(S)}, then obtain (\\int (- v_{y} + \\log{(S)}) dS)^{v_{y}} = (\\int - v_{y} dS + \\int \\log{(S)} dS)^{v_{y}}", "derivation": "y{(v_{y},S)} = - v_{y} + \\log{(S)} and \\int y{(v_{y},S)} dS = \\int (- v_{y} + \\log{(S)}) dS and \\int y{(v_{y},S)} dS = \\int - v_{y} dS + \\int \\log{(S)} dS and \\int (- v_{y} + \\log{(S)}) dS = \\int - v_{y} dS + \\int \\log{(S)} dS and (\\int (- v_{y} + \\log{(S)}) dS)^{v_{y}} = (\\int - v_{y} dS + \\int \\log{(S)} dS)^{v_{y}}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('v_y', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), log(Symbol('S', commutative=True))))"], [["integrate", 1, "Symbol('S', commutative=True)"], "Equality(Integral(Function('y')(Symbol('v_y', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))))"], [["expand", 2], "Equality(Integral(Function('y')(Symbol('v_y', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True))), Add(Integral(Mul(Integer(-1), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Integral(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Add(Integral(Mul(Integer(-1), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))))"], [["power", 4, "Symbol('v_y', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('v_y', commutative=True)), log(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True))), Symbol('v_y', commutative=True)), Pow(Add(Integral(Mul(Integer(-1), Symbol('v_y', commutative=True)), Tuple(Symbol('S', commutative=True))), Integral(log(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True)))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\hat{p}_0{(E_{x})} = e^{E_{x}} and V{(C_{d})} = \\sin{(C_{d})}, then obtain 2 E + 2 E_{x} e^{E_{x}} - 2 G + V{(C_{d})} = 2 E + 2 E_{x} e^{E_{x}} - 2 G + \\sin{(C_{d})}", "derivation": "\\hat{p}_0{(E_{x})} = e^{E_{x}} and E_{x} \\hat{p}_0{(E_{x})} = E_{x} e^{E_{x}} and V{(C_{d})} = \\sin{(C_{d})} and E + E_{x} \\hat{p}_0{(E_{x})} - G + V{(C_{d})} = E + E_{x} \\hat{p}_0{(E_{x})} - G + \\sin{(C_{d})} and E + E_{x} e^{E_{x}} - G + V{(C_{d})} = E + E_{x} e^{E_{x}} - G + \\sin{(C_{d})} and 2 E + E_{x} \\hat{p}_0{(E_{x})} + E_{x} e^{E_{x}} - 2 G + V{(C_{d})} = 2 E + E_{x} \\hat{p}_0{(E_{x})} + E_{x} e^{E_{x}} - 2 G + \\sin{(C_{d})} and 2 E + 2 E_{x} e^{E_{x}} - 2 G + V{(C_{d})} = 2 E + 2 E_{x} e^{E_{x}} - 2 G + \\sin{(C_{d})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True)), exp(Symbol('E_x', commutative=True)))"], [["times", 1, "Symbol('E_x', commutative=True)"], "Equality(Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))))"], ["get_premise", "Equality(Function('V')(Symbol('C_d', commutative=True)), sin(Symbol('C_d', commutative=True)))"], [["add", 3, "Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)))"], "Equality(Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Function('V')(Symbol('C_d', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), sin(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), Function('V')(Symbol('C_d', commutative=True))), Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)), sin(Symbol('C_d', commutative=True))))"], [["add", 5, "Add(Symbol('E', commutative=True), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Integer(-1), Symbol('G', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Function('V')(Symbol('C_d', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Symbol('E_x', commutative=True), Function('\\\\hat{p}_0')(Symbol('E_x', commutative=True))), Mul(Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), sin(Symbol('C_d', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 2], "Equality(Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(2), Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), Function('V')(Symbol('C_d', commutative=True))), Add(Mul(Integer(2), Symbol('E', commutative=True)), Mul(Integer(2), Symbol('E_x', commutative=True), exp(Symbol('E_x', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('G', commutative=True)), sin(Symbol('C_d', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\chi,F_{g})} = \\chi^{F_{g}}, then derive \\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{y}}{(\\chi,F_{g})} = \\chi^{F_{g}} \\log{(\\chi)}, then obtain \\frac{\\partial}{\\partial F_{g}} \\chi^{F_{g}} \\log{(\\chi)} = \\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\chi^{F_{g}}", "derivation": "\\operatorname{v_{y}}{(\\chi,F_{g})} = \\chi^{F_{g}} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{y}}{(\\chi,F_{g})} = \\frac{\\partial}{\\partial F_{g}} \\chi^{F_{g}} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{y}}{(\\chi,F_{g})} = \\chi^{F_{g}} \\log{(\\chi)} and \\chi^{F_{g}} \\log{(\\chi)} = \\frac{\\partial}{\\partial F_{g}} \\chi^{F_{g}} and \\operatorname{v_{y}}{(\\chi,F_{g})} \\log{(\\chi)} = \\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{y}}{(\\chi,F_{g})} and \\frac{\\partial}{\\partial F_{g}} \\operatorname{v_{y}}{(\\chi,F_{g})} \\log{(\\chi)} = \\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\operatorname{v_{y}}{(\\chi,F_{g})} and \\frac{\\partial}{\\partial F_{g}} \\chi^{F_{g}} \\log{(\\chi)} = \\frac{\\partial^{2}}{\\partial F_{g}^{2}} \\chi^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('\\\\chi', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Mul(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Function('v_y')(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Derivative(Mul(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), log(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\chi', commutative=True), Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{x}}{(\\psi)} = e^{\\psi} and z{(\\psi)} = \\int \\operatorname{A_{x}}{(\\psi)} d\\psi, then obtain z^{2}{(\\psi)} = z{(\\psi)} \\int e^{\\psi} d\\psi", "derivation": "\\operatorname{A_{x}}{(\\psi)} = e^{\\psi} and \\int \\operatorname{A_{x}}{(\\psi)} d\\psi = \\int e^{\\psi} d\\psi and (\\int \\operatorname{A_{x}}{(\\psi)} d\\psi)^{2} = (\\int \\operatorname{A_{x}}{(\\psi)} d\\psi) \\int e^{\\psi} d\\psi and z{(\\psi)} = \\int \\operatorname{A_{x}}{(\\psi)} d\\psi and z^{2}{(\\psi)} = z{(\\psi)} \\int e^{\\psi} d\\psi", "srepr_derivation": [["premise", "Equality(Function('A_x')(Symbol('\\\\psi', commutative=True)), exp(Symbol('\\\\psi', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi', commutative=True)"], "Equality(Integral(Function('A_x')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["times", 2, "Integral(Function('A_x')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))"], "Equality(Pow(Integral(Function('A_x')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integer(2)), Mul(Integral(Function('A_x')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))), Integral(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"], ["renaming_premise", "Equality(Function('z')(Symbol('\\\\psi', commutative=True)), Integral(Function('A_x')(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('z')(Symbol('\\\\psi', commutative=True)), Integer(2)), Mul(Function('z')(Symbol('\\\\psi', commutative=True)), Integral(exp(Symbol('\\\\psi', commutative=True)), Tuple(Symbol('\\\\psi', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{P_{e}}{(\\mathbf{A})} = \\log{(\\mathbf{A})}, then derive \\int \\operatorname{P_{e}}{(\\mathbf{A})} d\\mathbf{A} = C_{1} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A}, then obtain \\frac{\\ddot{x} \\int \\operatorname{P_{e}}{(\\mathbf{A})} d\\mathbf{A}}{\\operatorname{P_{e}}{(\\mathbf{A})}} = \\frac{\\ddot{x} (C_{1} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A})}{\\operatorname{P_{e}}{(\\mathbf{A})}}", "derivation": "\\operatorname{P_{e}}{(\\mathbf{A})} = \\log{(\\mathbf{A})} and \\int \\operatorname{P_{e}}{(\\mathbf{A})} d\\mathbf{A} = \\int \\log{(\\mathbf{A})} d\\mathbf{A} and \\int \\operatorname{P_{e}}{(\\mathbf{A})} d\\mathbf{A} = C_{1} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A} and \\frac{\\ddot{x} \\int \\operatorname{P_{e}}{(\\mathbf{A})} d\\mathbf{A}}{\\operatorname{P_{e}}{(\\mathbf{A})}} = \\frac{\\ddot{x} (C_{1} + \\mathbf{A} \\log{(\\mathbf{A})} - \\mathbf{A})}{\\operatorname{P_{e}}{(\\mathbf{A})}}", "srepr_derivation": [["get_premise", "Equality(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), log(Symbol('\\\\mathbf{A}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(log(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Mul(Symbol('\\\\ddot{x}', commutative=True), Pow(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1)), Integral(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True)))), Mul(Symbol('\\\\ddot{x}', commutative=True), Add(Symbol('C_1', commutative=True), Mul(Symbol('\\\\mathbf{A}', commutative=True), log(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True))), Pow(Function('P_e')(Symbol('\\\\mathbf{A}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\rho_{b}{(I,Z)} = I Z, then derive \\frac{\\partial}{\\partial Z} \\rho_{b}{(I,Z)} = I, then obtain 1 = \\frac{\\frac{\\partial}{\\partial Z} I Z}{I}", "derivation": "\\rho_{b}{(I,Z)} = I Z and \\frac{\\partial}{\\partial Z} \\rho_{b}{(I,Z)} = \\frac{\\partial}{\\partial Z} I Z and 1 = \\frac{\\frac{\\partial}{\\partial Z} I Z}{\\frac{\\partial}{\\partial Z} \\rho_{b}{(I,Z)}} and \\frac{\\partial}{\\partial Z} \\rho_{b}{(I,Z)} = I and 1 = \\frac{\\frac{\\partial}{\\partial Z} I Z}{I}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Mul(Symbol('I', commutative=True), Symbol('Z', commutative=True)))"], [["differentiate", 1, "Symbol('Z', commutative=True)"], "Equality(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Derivative(Mul(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))))"], [["divide", 2, "Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Derivative(Mul(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Pow(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Integer(-1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\rho_b')(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1))), Symbol('I', commutative=True))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Integer(1), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), Derivative(Mul(Symbol('I', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('Z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{m_{s}}{(z^{*},F_{N})} = \\int (F_{N} + z^{*}) dF_{N}, then derive \\operatorname{m_{s}}{(z^{*},F_{N})} = \\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\eta^{\\prime}, then derive \\frac{\\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\eta^{\\prime}}{F_{N}} = \\frac{\\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\mathbf{g}}{F_{N}}, then obtain \\frac{F_{N}}{2} + z^{*} + \\frac{\\eta^{\\prime}}{F_{N}} = \\frac{F_{N}}{2} + z^{*} + \\frac{\\mathbf{g}}{F_{N}}", "derivation": "\\operatorname{m_{s}}{(z^{*},F_{N})} = \\int (F_{N} + z^{*}) dF_{N} and \\frac{\\operatorname{m_{s}}{(z^{*},F_{N})}}{F_{N}} = \\frac{\\int (F_{N} + z^{*}) dF_{N}}{F_{N}} and \\operatorname{m_{s}}{(z^{*},F_{N})} = \\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\eta^{\\prime} and \\frac{\\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\eta^{\\prime}}{F_{N}} = \\frac{\\int (F_{N} + z^{*}) dF_{N}}{F_{N}} and \\frac{\\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\eta^{\\prime}}{F_{N}} = \\frac{\\frac{F_{N}^{2}}{2} + F_{N} z^{*} + \\mathbf{g}}{F_{N}} and \\frac{F_{N}}{2} + z^{*} + \\frac{\\eta^{\\prime}}{F_{N}} = \\frac{F_{N}}{2} + z^{*} + \\frac{\\mathbf{g}}{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('m_s')(Symbol('z^*', commutative=True), Symbol('F_N', commutative=True)), Integral(Add(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["divide", 1, "Symbol('F_N', commutative=True)"], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Function('m_s')(Symbol('z^*', commutative=True), Symbol('F_N', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 1], "Equality(Function('m_s')(Symbol('z^*', commutative=True), Symbol('F_N', commutative=True)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Integral(Add(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('F_N', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Add(Mul(Rational(1, 2), Pow(Symbol('F_N', commutative=True), Integer(2))), Mul(Symbol('F_N', commutative=True), Symbol('z^*', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True))))"], [["expand", 5], "Equality(Add(Mul(Rational(1, 2), Symbol('F_N', commutative=True)), Symbol('z^*', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\eta^{\\\\prime}', commutative=True))), Add(Mul(Rational(1, 2), Symbol('F_N', commutative=True)), Symbol('z^*', commutative=True), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Symbol('\\\\mathbf{g}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{A}{(p)} = \\cos{(p)} and \\operatorname{A_{z}}{(a,S)} = \\frac{S}{a}, then obtain - S + \\operatorname{A_{z}}{(a,S)} - \\frac{d}{d p} \\cos{(p)} = - S + \\frac{S}{a} - \\frac{d}{d p} \\cos{(p)}", "derivation": "\\mathbf{A}{(p)} = \\cos{(p)} and \\frac{d}{d p} \\mathbf{A}{(p)} = \\frac{d}{d p} \\cos{(p)} and \\operatorname{A_{z}}{(a,S)} = \\frac{S}{a} and - S + \\operatorname{A_{z}}{(a,S)} = - S + \\frac{S}{a} and - S + \\operatorname{A_{z}}{(a,S)} - \\frac{d}{d p} \\mathbf{A}{(p)} = - S + \\frac{S}{a} - \\frac{d}{d p} \\mathbf{A}{(p)} and - S + \\operatorname{A_{z}}{(a,S)} - \\frac{d}{d p} \\cos{(p)} = - S + \\frac{S}{a} - \\frac{d}{d p} \\cos{(p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('p', commutative=True)), cos(Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('p', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{A}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))"], ["get_premise", "Equality(Function('A_z')(Symbol('a', commutative=True), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))))"], [["minus", 3, "Symbol('S', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('A_z')(Symbol('a', commutative=True), Symbol('S', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1)))))"], [["minus", 4, "Derivative(Function('\\\\mathbf{A}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1)))"], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('A_z')(Symbol('a', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{A}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(Function('\\\\mathbf{A}')(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Function('A_z')(Symbol('a', commutative=True), Symbol('S', commutative=True)), Mul(Integer(-1), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Mul(Symbol('S', commutative=True), Pow(Symbol('a', commutative=True), Integer(-1))), Mul(Integer(-1), Derivative(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(m_{s},\\mathbf{A})} = \\log{(\\mathbf{A} m_{s})}, then obtain ((\\operatorname{v_{1}}^{m_{s}}{(m_{s},\\mathbf{A})} \\log{(\\mathbf{A} m_{s})}^{- m_{s}})^{\\mathbf{A}})^{m_{s}} = 1", "derivation": "\\operatorname{v_{1}}{(m_{s},\\mathbf{A})} = \\log{(\\mathbf{A} m_{s})} and \\operatorname{v_{1}}^{m_{s}}{(m_{s},\\mathbf{A})} = \\log{(\\mathbf{A} m_{s})}^{m_{s}} and \\operatorname{v_{1}}^{m_{s}}{(m_{s},\\mathbf{A})} \\log{(\\mathbf{A} m_{s})}^{- m_{s}} = 1 and (\\operatorname{v_{1}}^{m_{s}}{(m_{s},\\mathbf{A})} \\log{(\\mathbf{A} m_{s})}^{- m_{s}})^{\\mathbf{A}} = 1 and ((\\operatorname{v_{1}}^{m_{s}}{(m_{s},\\mathbf{A})} \\log{(\\mathbf{A} m_{s})}^{- m_{s}})^{\\mathbf{A}})^{m_{s}} = 1", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('v_1')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('m_s', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True)))"], [["divide", 2, "Pow(log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))), Symbol('m_s', commutative=True))"], "Equality(Mul(Pow(Function('v_1')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('m_s', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Pow(Mul(Pow(Function('v_1')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('m_s', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1))"], [["power", 4, "Symbol('m_s', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Function('v_1')(Symbol('m_s', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('m_s', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbf{A}', commutative=True), Symbol('m_s', commutative=True))), Mul(Integer(-1), Symbol('m_s', commutative=True)))), Symbol('\\\\mathbf{A}', commutative=True)), Symbol('m_s', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\hat{x}_0{(\\pi,z^{*})} = - \\sin{(\\pi - z^{*})}, then derive \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} = - \\cos{(\\pi - z^{*})}, then obtain (\\iint \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} dz^{*} d\\pi)^{\\pi} = (\\iint - \\cos{(\\pi - z^{*})} dz^{*} d\\pi)^{\\pi}", "derivation": "\\hat{x}_0{(\\pi,z^{*})} = - \\sin{(\\pi - z^{*})} and \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} = \\frac{\\partial}{\\partial \\pi} - \\sin{(\\pi - z^{*})} and \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} = - \\cos{(\\pi - z^{*})} and \\int \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} dz^{*} = \\int - \\cos{(\\pi - z^{*})} dz^{*} and \\iint \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} dz^{*} d\\pi = \\iint - \\cos{(\\pi - z^{*})} dz^{*} d\\pi and (\\iint \\frac{\\partial}{\\partial \\pi} \\hat{x}_0{(\\pi,z^{*})} dz^{*} d\\pi)^{\\pi} = (\\iint - \\cos{(\\pi - z^{*})} dz^{*} d\\pi)^{\\pi}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Mul(Integer(-1), sin(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), sin(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))))"], [["integrate", 3, "Symbol('z^*', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('z^*', commutative=True))), Integral(Mul(Integer(-1), cos(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('z^*', commutative=True))))"], [["integrate", 4, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(Mul(Integer(-1), cos(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))))"], [["power", 5, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Integral(Derivative(Function('\\\\hat{x}_0')(Symbol('\\\\pi', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)), Pow(Integral(Mul(Integer(-1), cos(Add(Symbol('\\\\pi', commutative=True), Mul(Integer(-1), Symbol('z^*', commutative=True))))), Tuple(Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Symbol('\\\\pi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mu_0,c)} = \\cos^{c}{(\\mu_0)} and p{(\\mu_0,c)} = \\operatorname{t_{2}}^{c}{(\\mu_0,c)}, then obtain \\mu_0 \\operatorname{t_{2}}^{c}{(\\mu_0,c)} + \\operatorname{m_{s}}{(\\mu_0,c)} = \\mu_0 p{(\\mu_0,c)} + \\operatorname{m_{s}}{(\\mu_0,c)}", "derivation": "\\operatorname{t_{2}}{(\\mu_0,c)} = \\cos^{c}{(\\mu_0)} and \\operatorname{t_{2}}^{c}{(\\mu_0,c)} = (\\cos^{c}{(\\mu_0)})^{c} and \\mu_0 \\operatorname{t_{2}}^{c}{(\\mu_0,c)} = \\mu_0 (\\cos^{c}{(\\mu_0)})^{c} and p{(\\mu_0,c)} = \\operatorname{t_{2}}^{c}{(\\mu_0,c)} and p{(\\mu_0,c)} = (\\cos^{c}{(\\mu_0)})^{c} and \\mu_0 \\operatorname{t_{2}}^{c}{(\\mu_0,c)} = \\mu_0 p{(\\mu_0,c)} and \\mu_0 \\operatorname{t_{2}}^{c}{(\\mu_0,c)} + \\operatorname{m_{s}}{(\\mu_0,c)} = \\mu_0 p{(\\mu_0,c)} + \\operatorname{m_{s}}{(\\mu_0,c)}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('c', commutative=True)))"], [["power", 1, "Symbol('c', commutative=True)"], "Equality(Pow(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)), Pow(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["times", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Pow(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Pow(Pow(cos(Symbol('\\\\mu_0', commutative=True)), Symbol('c', commutative=True)), Symbol('c', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 5], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True))))"], [["add", 6, "Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True))"], "Equality(Add(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Function('t_2')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True)), Symbol('c', commutative=True))), Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True))), Add(Mul(Symbol('\\\\mu_0', commutative=True), Function('p')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True))), Function('m_s')(Symbol('\\\\mu_0', commutative=True), Symbol('c', commutative=True))))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(\\theta,\\delta)} = - \\theta + \\sin{(\\delta)}, then obtain (- \\delta + \\frac{\\partial}{\\partial \\delta} \\operatorname{a^{\\dagger}}{(\\theta,\\delta)})^{\\delta} = (- \\delta + \\frac{\\partial}{\\partial \\delta} (- \\theta + \\sin{(\\delta)}))^{\\delta}", "derivation": "\\operatorname{a^{\\dagger}}{(\\theta,\\delta)} = - \\theta + \\sin{(\\delta)} and \\frac{\\partial}{\\partial \\delta} \\operatorname{a^{\\dagger}}{(\\theta,\\delta)} = \\frac{\\partial}{\\partial \\delta} (- \\theta + \\sin{(\\delta)}) and - \\delta + \\frac{\\partial}{\\partial \\delta} \\operatorname{a^{\\dagger}}{(\\theta,\\delta)} = - \\delta + \\frac{\\partial}{\\partial \\delta} (- \\theta + \\sin{(\\delta)}) and (- \\delta + \\frac{\\partial}{\\partial \\delta} \\operatorname{a^{\\dagger}}{(\\theta,\\delta)})^{\\delta} = (- \\delta + \\frac{\\partial}{\\partial \\delta} (- \\theta + \\sin{(\\delta)}))^{\\delta}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["minus", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Function('a^{\\\\dagger}')(Symbol('\\\\theta', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\theta', commutative=True)), sin(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1)))), Symbol('\\\\delta', commutative=True)))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\pi)} = \\sin{(\\pi)} and \\operatorname{m_{s}}{(\\pi)} = \\pi, then obtain \\pi \\operatorname{m_{s}}{(\\pi)} - \\operatorname{C_{2}}^{4}{(\\pi)} = \\pi^{2} - \\operatorname{C_{2}}^{4}{(\\pi)}", "derivation": "\\operatorname{C_{2}}{(\\pi)} = \\sin{(\\pi)} and \\operatorname{m_{s}}{(\\pi)} = \\pi and \\pi \\operatorname{m_{s}}{(\\pi)} = \\pi^{2} and \\operatorname{C_{2}}^{2}{(\\pi)} = \\operatorname{C_{2}}{(\\pi)} \\sin{(\\pi)} and \\operatorname{C_{2}}^{4}{(\\pi)} = \\operatorname{C_{2}}^{2}{(\\pi)} \\sin^{2}{(\\pi)} and \\pi \\operatorname{m_{s}}{(\\pi)} - \\operatorname{C_{2}}^{2}{(\\pi)} \\sin^{2}{(\\pi)} = \\pi^{2} - \\operatorname{C_{2}}^{2}{(\\pi)} \\sin^{2}{(\\pi)} and \\pi \\operatorname{m_{s}}{(\\pi)} - \\operatorname{C_{2}}^{4}{(\\pi)} = \\pi^{2} - \\operatorname{C_{2}}^{4}{(\\pi)}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], ["renaming_premise", "Equality(Function('m_s')(Symbol('\\\\pi', commutative=True)), Symbol('\\\\pi', commutative=True))"], [["times", 2, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Function('m_s')(Symbol('\\\\pi', commutative=True))), Pow(Symbol('\\\\pi', commutative=True), Integer(2)))"], [["times", 1, "Function('C_2')(Symbol('\\\\pi', commutative=True))"], "Equality(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Mul(Function('C_2')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True))))"], [["power", 4, 2], "Equality(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(4)), Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(2))))"], [["minus", 3, "Mul(Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('m_s')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(2)))), Add(Pow(Symbol('\\\\pi', commutative=True), Integer(2)), Mul(Integer(-1), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(2)), Pow(sin(Symbol('\\\\pi', commutative=True)), Integer(2)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Add(Mul(Symbol('\\\\pi', commutative=True), Function('m_s')(Symbol('\\\\pi', commutative=True))), Mul(Integer(-1), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(4)))), Add(Pow(Symbol('\\\\pi', commutative=True), Integer(2)), Mul(Integer(-1), Pow(Function('C_2')(Symbol('\\\\pi', commutative=True)), Integer(4)))))"]]}, {"prompt": "Given \\Psi_{nl}{(i)} = - i, then obtain ((\\frac{d}{d i} \\int \\Psi_{nl}{(i)} di)^{\\Psi_{nl}{(i)}}) \\int \\Psi_{nl}{(i)} di = ((\\frac{d}{d i} \\int \\Psi_{nl}{(i)} di)^{\\Psi_{nl}{(i)}}) \\int - i di", "derivation": "\\Psi_{nl}{(i)} = - i and \\int \\Psi_{nl}{(i)} di = \\int - i di and \\frac{d}{d i} \\int \\Psi_{nl}{(i)} di = \\frac{d}{d i} \\int - i di and ((\\frac{d}{d i} \\int - i di)^{- i}) \\int \\Psi_{nl}{(i)} di = ((\\frac{d}{d i} \\int - i di)^{- i}) \\int - i di and ((\\frac{d}{d i} \\int - i di)^{\\Psi_{nl}{(i)}}) \\int \\Psi_{nl}{(i)} di = ((\\frac{d}{d i} \\int - i di)^{\\Psi_{nl}{(i)}}) \\int - i di and ((\\frac{d}{d i} \\int \\Psi_{nl}{(i)} di)^{\\Psi_{nl}{(i)}}) \\int \\Psi_{nl}{(i)} di = ((\\frac{d}{d i} \\int \\Psi_{nl}{(i)} di)^{\\Psi_{nl}{(i)}}) \\int - i di", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Mul(Integer(-1), Symbol('i', commutative=True)))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))))"], [["differentiate", 2, "Symbol('i', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))))"], [["divide", 2, "Pow(Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Symbol('i', commutative=True))"], "Equality(Mul(Pow(Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('i', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Mul(Pow(Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Mul(Integer(-1), Symbol('i', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Pow(Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Mul(Pow(Derivative(Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))), Mul(Pow(Derivative(Integral(Function('\\\\Psi_{nl}')(Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True), Integer(1))), Function('\\\\Psi_{nl}')(Symbol('i', commutative=True))), Integral(Mul(Integer(-1), Symbol('i', commutative=True)), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given l{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})}, then obtain e^{\\frac{\\sin{(l{(g_{\\varepsilon})})}}{g_{\\varepsilon}}} = e^{\\frac{\\sin{(\\log{(e^{g_{\\varepsilon}})})}}{g_{\\varepsilon}}}", "derivation": "l{(g_{\\varepsilon})} = \\log{(e^{g_{\\varepsilon}})} and \\sin{(l{(g_{\\varepsilon})})} = \\sin{(\\log{(e^{g_{\\varepsilon}})})} and \\frac{\\sin{(l{(g_{\\varepsilon})})}}{g_{\\varepsilon}} = \\frac{\\sin{(\\log{(e^{g_{\\varepsilon}})})}}{g_{\\varepsilon}} and e^{\\frac{\\sin{(l{(g_{\\varepsilon})})}}{g_{\\varepsilon}}} = e^{\\frac{\\sin{(\\log{(e^{g_{\\varepsilon}})})}}{g_{\\varepsilon}}}", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True)), log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["sin", 1], "Equality(sin(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True))), sin(log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["divide", 2, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True)))), Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(log(exp(Symbol('g_{\\\\varepsilon}', commutative=True))))))"], [["exp", 3], "Equality(exp(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(Function('l')(Symbol('g_{\\\\varepsilon}', commutative=True))))), exp(Mul(Pow(Symbol('g_{\\\\varepsilon}', commutative=True), Integer(-1)), sin(log(exp(Symbol('g_{\\\\varepsilon}', commutative=True)))))))"]]}, {"prompt": "Given \\hat{x}{(\\Psi^{\\dagger},n_{1})} = e^{\\Psi^{\\dagger} n_{1}}, then obtain \\frac{\\partial}{\\partial n_{1}} 2 \\hat{x}{(\\Psi^{\\dagger},n_{1})} e^{\\Psi^{\\dagger} n_{1}} = \\frac{\\partial}{\\partial n_{1}} (\\hat{x}{(\\Psi^{\\dagger},n_{1})} + e^{\\Psi^{\\dagger} n_{1}}) e^{\\Psi^{\\dagger} n_{1}}", "derivation": "\\hat{x}{(\\Psi^{\\dagger},n_{1})} = e^{\\Psi^{\\dagger} n_{1}} and 2 \\hat{x}{(\\Psi^{\\dagger},n_{1})} = \\hat{x}{(\\Psi^{\\dagger},n_{1})} + e^{\\Psi^{\\dagger} n_{1}} and 2 \\hat{x}{(\\Psi^{\\dagger},n_{1})} e^{\\Psi^{\\dagger} n_{1}} = (\\hat{x}{(\\Psi^{\\dagger},n_{1})} + e^{\\Psi^{\\dagger} n_{1}}) e^{\\Psi^{\\dagger} n_{1}} and \\frac{\\partial}{\\partial n_{1}} 2 \\hat{x}{(\\Psi^{\\dagger},n_{1})} e^{\\Psi^{\\dagger} n_{1}} = \\frac{\\partial}{\\partial n_{1}} (\\hat{x}{(\\Psi^{\\dagger},n_{1})} + e^{\\Psi^{\\dagger} n_{1}}) e^{\\Psi^{\\dagger} n_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True))))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True))), Add(Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))))"], [["times", 2, "exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))), Mul(Add(Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))))"], [["differentiate", 3, "Symbol('n_1', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))), Derivative(Mul(Add(Function('\\\\hat{x}')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))), exp(Mul(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('n_1', commutative=True)))), Tuple(Symbol('n_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varepsilon{(\\mathbf{P},F_{H},h)} = \\frac{\\mathbf{P}^{h}}{F_{H}} and t{(\\mathbf{P},h)} = - \\mathbf{P}^{h}, then obtain \\int (\\varepsilon{(\\mathbf{P},F_{H},h)} + \\frac{2 t{(\\mathbf{P},h)}}{F_{H}}) dF_{H} = \\int \\frac{t{(\\mathbf{P},h)}}{F_{H}} dF_{H}", "derivation": "\\varepsilon{(\\mathbf{P},F_{H},h)} = \\frac{\\mathbf{P}^{h}}{F_{H}} and \\varepsilon{(\\mathbf{P},F_{H},h)} - \\frac{2 \\mathbf{P}^{h}}{F_{H}} = - \\frac{\\mathbf{P}^{h}}{F_{H}} and t{(\\mathbf{P},h)} = - \\mathbf{P}^{h} and \\varepsilon{(\\mathbf{P},F_{H},h)} + \\frac{2 t{(\\mathbf{P},h)}}{F_{H}} = \\frac{t{(\\mathbf{P},h)}}{F_{H}} and \\int (\\varepsilon{(\\mathbf{P},F_{H},h)} + \\frac{2 t{(\\mathbf{P},h)}}{F_{H}}) dF_{H} = \\int \\frac{t{(\\mathbf{P},h)}}{F_{H}} dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('F_H', commutative=True), Symbol('h', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], [["minus", 1, "Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))"], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('F_H', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))), Mul(Integer(-1), Pow(Symbol('F_H', commutative=True), Integer(-1)), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], ["renaming_premise", "Equality(Function('t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('F_H', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], [["integrate", 4, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Function('\\\\varepsilon')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('F_H', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)))), Tuple(Symbol('F_H', commutative=True))), Integral(Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Function('t')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('F_H', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(s)} = \\log{(s)}, then obtain \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds + \\int - \\mathbf{J}_M{(s)} ds + \\int \\log{(s)} ds = 2 \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds", "derivation": "\\mathbf{J}_M{(s)} = \\log{(s)} and 0 = - \\mathbf{J}_M{(s)} + \\log{(s)} and \\int 0 ds = \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds and \\int 0 ds = \\int - \\mathbf{J}_M{(s)} ds + \\int \\log{(s)} ds and \\int 0 ds + \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds = 2 \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds and \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds + \\int - \\mathbf{J}_M{(s)} ds + \\int \\log{(s)} ds = 2 \\int (- \\mathbf{J}_M{(s)} + \\log{(s)}) ds", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True)), log(Symbol('s', commutative=True)))"], [["minus", 1, "Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('s', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))))"], [["expand", 3], "Equality(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Add(Integral(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["add", 3, "Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('s', commutative=True))), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))), Mul(Integer(2), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True))), Integral(log(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Integer(2), Integral(Add(Mul(Integer(-1), Function('\\\\mathbf{J}_M')(Symbol('s', commutative=True))), log(Symbol('s', commutative=True))), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(f,p)} = f + \\sin{(p)}, then obtain (\\int (- \\hat{H}_l{(f,p)} + \\int \\hat{H}_l{(f,p)} dp)^{f} df)^{f} = (\\int (- \\hat{H}_l{(f,p)} + \\int (f + \\sin{(p)}) dp)^{f} df)^{f}", "derivation": "\\hat{H}_l{(f,p)} = f + \\sin{(p)} and \\int \\hat{H}_l{(f,p)} dp = \\int (f + \\sin{(p)}) dp and - \\hat{H}_l{(f,p)} + \\int \\hat{H}_l{(f,p)} dp = - \\hat{H}_l{(f,p)} + \\int (f + \\sin{(p)}) dp and (- \\hat{H}_l{(f,p)} + \\int \\hat{H}_l{(f,p)} dp)^{f} = (- \\hat{H}_l{(f,p)} + \\int (f + \\sin{(p)}) dp)^{f} and \\int (- \\hat{H}_l{(f,p)} + \\int \\hat{H}_l{(f,p)} dp)^{f} df = \\int (- \\hat{H}_l{(f,p)} + \\int (f + \\sin{(p)}) dp)^{f} df and (\\int (- \\hat{H}_l{(f,p)} + \\int \\hat{H}_l{(f,p)} dp)^{f} df)^{f} = (\\int (- \\hat{H}_l{(f,p)} + \\int (f + \\sin{(p)}) dp)^{f} df)^{f}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))))"], [["integrate", 1, "Symbol('p', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integral(Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True))))"], [["minus", 2, "Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))))"], [["power", 3, "Symbol('f', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)))"], [["integrate", 4, "Symbol('f', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))))"], [["power", 5, "Symbol('f', commutative=True)"], "Equality(Pow(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Integral(Pow(Add(Mul(Integer(-1), Function('\\\\hat{H}_l')(Symbol('f', commutative=True), Symbol('p', commutative=True))), Integral(Add(Symbol('f', commutative=True), sin(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)))), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given A{(C,g^{\\prime}_{\\varepsilon})} = C + g^{\\prime}_{\\varepsilon}, then obtain \\iint (\\frac{A{(C,g^{\\prime}_{\\varepsilon})}}{C + g^{\\prime}_{\\varepsilon}})^{g^{\\prime}_{\\varepsilon}} dC dg^{\\prime}_{\\varepsilon} = \\iint 1 dC dg^{\\prime}_{\\varepsilon}", "derivation": "A{(C,g^{\\prime}_{\\varepsilon})} = C + g^{\\prime}_{\\varepsilon} and \\frac{A{(C,g^{\\prime}_{\\varepsilon})}}{C + g^{\\prime}_{\\varepsilon}} = 1 and (\\frac{A{(C,g^{\\prime}_{\\varepsilon})}}{C + g^{\\prime}_{\\varepsilon}})^{g^{\\prime}_{\\varepsilon}} = 1 and \\int (\\frac{A{(C,g^{\\prime}_{\\varepsilon})}}{C + g^{\\prime}_{\\varepsilon}})^{g^{\\prime}_{\\varepsilon}} dC = \\int 1 dC and \\iint (\\frac{A{(C,g^{\\prime}_{\\varepsilon})}}{C + g^{\\prime}_{\\varepsilon}})^{g^{\\prime}_{\\varepsilon}} dC dg^{\\prime}_{\\varepsilon} = \\iint 1 dC dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))"], [["divide", 1, "Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('A')(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integer(1))"], [["power", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('A')(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(1))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('A')(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Integer(1), Tuple(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Pow(Mul(Pow(Add(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Integer(-1)), Function('A')(Symbol('C', commutative=True), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Integer(1), Tuple(Symbol('C', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\pi{(u)} = e^{u}, then obtain \\pi^{2 u}{(u)} e^{- 2 u} (e^{u})^{2 u} = e^{- 2 u} (e^{u})^{4 u}", "derivation": "\\pi{(u)} = e^{u} and \\pi^{u}{(u)} = (e^{u})^{u} and \\pi^{u}{(u)} (e^{u})^{u} = (e^{u})^{2 u} and \\pi^{u}{(u)} e^{- u} (e^{u})^{u} = e^{- u} (e^{u})^{2 u} and \\pi^{2 u}{(u)} e^{- 2 u} (e^{u})^{2 u} = e^{- 2 u} (e^{u})^{4 u}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('u', commutative=True)), exp(Symbol('u', commutative=True)))"], [["power", 1, "Symbol('u', commutative=True)"], "Equality(Pow(Function('\\\\pi')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["times", 2, "Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))))"], [["divide", 3, "exp(Symbol('u', commutative=True))"], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('u', commutative=True)), Symbol('u', commutative=True)), exp(Mul(Integer(-1), Symbol('u', commutative=True))), Pow(exp(Symbol('u', commutative=True)), Symbol('u', commutative=True))), Mul(exp(Mul(Integer(-1), Symbol('u', commutative=True))), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)))))"], [["power", 4, 2], "Equality(Mul(Pow(Function('\\\\pi')(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True))), exp(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(2), Symbol('u', commutative=True)))), Mul(exp(Mul(Integer(-1), Integer(2), Symbol('u', commutative=True))), Pow(exp(Symbol('u', commutative=True)), Mul(Integer(4), Symbol('u', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = - \\hat{p} + \\sin{(\\mathbf{H})}, then obtain 2 \\operatorname{A_{y}}^{2}{(\\hat{p},\\mathbf{H})} = (- 2 \\hat{p} + 2 \\sin{(\\mathbf{H})}) \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})}", "derivation": "\\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = - \\hat{p} + \\sin{(\\mathbf{H})} and 2 \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = - \\hat{p} + \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} + \\sin{(\\mathbf{H})} and 2 (- \\hat{p} + \\sin{(\\mathbf{H})}) \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} = (- \\hat{p} + \\sin{(\\mathbf{H})}) (- \\hat{p} + \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})} + \\sin{(\\mathbf{H})}) and 2 (- \\hat{p} + \\sin{(\\mathbf{H})})^{2} = (- 2 \\hat{p} + 2 \\sin{(\\mathbf{H})}) (- \\hat{p} + \\sin{(\\mathbf{H})}) and 2 \\operatorname{A_{y}}^{2}{(\\hat{p},\\mathbf{H})} = (- 2 \\hat{p} + 2 \\sin{(\\mathbf{H})}) \\operatorname{A_{y}}{(\\hat{p},\\mathbf{H})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Mul(Integer(2), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["times", 2, "Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Integer(2), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(2), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Integer(2))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Pow(Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Integer(2))), Mul(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Function('A_y')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"]]}, {"prompt": "Given q{(\\varphi^*,A_{1})} = \\frac{\\varphi^*}{A_{1}} and \\operatorname{C_{d}}{(A_{1})} = A_{1}, then obtain \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{C_{d}}{(A_{1})} \\int \\frac{\\varphi^*}{A_{1}} dA_{1} = \\frac{\\partial}{\\partial \\varphi^*} A_{1} \\int \\frac{\\varphi^*}{A_{1}} dA_{1}", "derivation": "q{(\\varphi^*,A_{1})} = \\frac{\\varphi^*}{A_{1}} and \\operatorname{C_{d}}{(A_{1})} = A_{1} and \\operatorname{C_{d}}{(A_{1})} \\int q{(\\varphi^*,A_{1})} dA_{1} = A_{1} \\int q{(\\varphi^*,A_{1})} dA_{1} and \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{C_{d}}{(A_{1})} \\int q{(\\varphi^*,A_{1})} dA_{1} = \\frac{\\partial}{\\partial \\varphi^*} A_{1} \\int q{(\\varphi^*,A_{1})} dA_{1} and \\frac{\\partial}{\\partial \\varphi^*} \\operatorname{C_{d}}{(A_{1})} \\int \\frac{\\varphi^*}{A_{1}} dA_{1} = \\frac{\\partial}{\\partial \\varphi^*} A_{1} \\int \\frac{\\varphi^*}{A_{1}} dA_{1}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)))"], ["renaming_premise", "Equality(Function('C_d')(Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))"], [["times", 2, "Integral(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))"], "Equality(Mul(Function('C_d')(Symbol('A_1', commutative=True)), Integral(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Mul(Symbol('A_1', commutative=True), Integral(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Derivative(Mul(Function('C_d')(Symbol('A_1', commutative=True)), Integral(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_1', commutative=True), Integral(Function('q')(Symbol('\\\\varphi^*', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(Mul(Function('C_d')(Symbol('A_1', commutative=True)), Integral(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))), Derivative(Mul(Symbol('A_1', commutative=True), Integral(Mul(Pow(Symbol('A_1', commutative=True), Integer(-1)), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('A_1', commutative=True)))), Tuple(Symbol('\\\\varphi^*', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(x,v_{y})} = \\frac{x}{v_{y}}, then obtain \\cos^{v_{y}}{((\\operatorname{z^{*}}{(x,v_{y})} - \\frac{x}{v_{y}})^{x})} = \\cos^{v_{y}}{(0^{x})}", "derivation": "\\operatorname{z^{*}}{(x,v_{y})} = \\frac{x}{v_{y}} and x + \\operatorname{z^{*}}{(x,v_{y})} = x + \\frac{x}{v_{y}} and \\operatorname{z^{*}}{(x,v_{y})} - \\frac{x}{v_{y}} = 0 and (\\operatorname{z^{*}}{(x,v_{y})} - \\frac{x}{v_{y}})^{x} = 0^{x} and \\cos{((\\operatorname{z^{*}}{(x,v_{y})} - \\frac{x}{v_{y}})^{x})} = \\cos{(0^{x})} and \\cos^{v_{y}}{((\\operatorname{z^{*}}{(x,v_{y})} - \\frac{x}{v_{y}})^{x})} = \\cos^{v_{y}}{(0^{x})}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True)), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], [["add", 1, "Symbol('x', commutative=True)"], "Equality(Add(Symbol('x', commutative=True), Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True))), Add(Symbol('x', commutative=True), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True))))"], [["minus", 2, "Add(Symbol('x', commutative=True), Mul(Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True)))"], "Equality(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Integer(0))"], [["power", 3, "Symbol('x', commutative=True)"], "Equality(Pow(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Symbol('x', commutative=True)), Pow(Integer(0), Symbol('x', commutative=True)))"], [["cos", 4], "Equality(cos(Pow(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Symbol('x', commutative=True))), cos(Pow(Integer(0), Symbol('x', commutative=True))))"], [["power", 5, "Symbol('v_y', commutative=True)"], "Equality(Pow(cos(Pow(Add(Function('z^*')(Symbol('x', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_y', commutative=True), Integer(-1)), Symbol('x', commutative=True))), Symbol('x', commutative=True))), Symbol('v_y', commutative=True)), Pow(cos(Pow(Integer(0), Symbol('x', commutative=True))), Symbol('v_y', commutative=True)))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})} = \\tilde{g} + g_{\\varepsilon}, then derive 0 = 1 - \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})}, then obtain \\frac{d}{d \\tilde{g}} 0 = \\frac{\\partial}{\\partial \\tilde{g}} (1 - \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})})", "derivation": "\\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})} = \\tilde{g} + g_{\\varepsilon} and \\tilde{g} + g_{\\varepsilon} + \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})} = 2 \\tilde{g} + 2 g_{\\varepsilon} and 0 = \\tilde{g} + g_{\\varepsilon} - \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})} and \\frac{d}{d \\tilde{g}} 0 = \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + g_{\\varepsilon} - \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})}) and 0 = 1 - \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})} and \\frac{d}{d \\tilde{g}} 0 = \\frac{\\partial}{\\partial \\tilde{g}} (1 - \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{t_{1}}{(\\tilde{g},g_{\\varepsilon})})", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\tilde{g}', commutative=True)), Mul(Integer(2), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 2, "Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))"], "Equality(Integer(0), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), Mul(Integer(-1), Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Derivative(Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))))"], [["differentiate", 5, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Derivative(Function('t_1')(Symbol('\\\\tilde{g}', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(\\hat{H},g,E_{n})} = \\frac{E_{n} + g}{\\hat{H}} and Z{(g,\\hat{H},E_{n})} = g + \\frac{\\partial}{\\partial \\hat{H}} S{(\\hat{H},g,E_{n})}, then obtain \\frac{\\partial}{\\partial g} Z{(g,\\hat{H},E_{n})} = \\frac{\\partial}{\\partial g} (g + \\frac{\\partial}{\\partial \\hat{H}} \\frac{E_{n} + g}{\\hat{H}})", "derivation": "S{(\\hat{H},g,E_{n})} = \\frac{E_{n} + g}{\\hat{H}} and \\frac{\\partial}{\\partial \\hat{H}} S{(\\hat{H},g,E_{n})} = \\frac{\\partial}{\\partial \\hat{H}} \\frac{E_{n} + g}{\\hat{H}} and Z{(g,\\hat{H},E_{n})} = g + \\frac{\\partial}{\\partial \\hat{H}} S{(\\hat{H},g,E_{n})} and Z{(g,\\hat{H},E_{n})} = g + \\frac{\\partial}{\\partial \\hat{H}} \\frac{E_{n} + g}{\\hat{H}} and \\frac{\\partial}{\\partial g} Z{(g,\\hat{H},E_{n})} = \\frac{\\partial}{\\partial g} (g + \\frac{\\partial}{\\partial \\hat{H}} \\frac{E_{n} + g}{\\hat{H}})", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('\\\\hat{H}', commutative=True), Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Symbol('g', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('\\\\hat{H}', commutative=True), Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('Z')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('E_n', commutative=True)), Add(Symbol('g', commutative=True), Derivative(Function('S')(Symbol('\\\\hat{H}', commutative=True), Symbol('g', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Function('Z')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('E_n', commutative=True)), Add(Symbol('g', commutative=True), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["differentiate", 4, "Symbol('g', commutative=True)"], "Equality(Derivative(Function('Z')(Symbol('g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('g', commutative=True), Integer(1))), Derivative(Add(Symbol('g', commutative=True), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Add(Symbol('E_n', commutative=True), Symbol('g', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Tuple(Symbol('g', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\hat{H},a)} = a \\cos{(\\hat{H})} and \\dot{z}{(\\hat{H})} = - \\hat{H}, then obtain ((\\dot{z}{(\\hat{H})} + \\eta^{\\prime}{(\\hat{H},a)})^{\\hat{H}})^{\\hat{H}} = ((a \\cos{(\\hat{H})} + \\dot{z}{(\\hat{H})})^{\\hat{H}})^{\\hat{H}}", "derivation": "\\eta^{\\prime}{(\\hat{H},a)} = a \\cos{(\\hat{H})} and - \\hat{H} + \\eta^{\\prime}{(\\hat{H},a)} = - \\hat{H} + a \\cos{(\\hat{H})} and (- \\hat{H} + \\eta^{\\prime}{(\\hat{H},a)})^{\\hat{H}} = (- \\hat{H} + a \\cos{(\\hat{H})})^{\\hat{H}} and ((- \\hat{H} + \\eta^{\\prime}{(\\hat{H},a)})^{\\hat{H}})^{\\hat{H}} = ((- \\hat{H} + a \\cos{(\\hat{H})})^{\\hat{H}})^{\\hat{H}} and \\dot{z}{(\\hat{H})} = - \\hat{H} and ((\\dot{z}{(\\hat{H})} + \\eta^{\\prime}{(\\hat{H},a)})^{\\hat{H}})^{\\hat{H}} = ((a \\cos{(\\hat{H})} + \\dot{z}{(\\hat{H})})^{\\hat{H}})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{H}', commutative=True))))"], [["minus", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{H}', commutative=True)))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)))"], [["power", 3, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{H}', commutative=True)))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True)), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Pow(Add(Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True)), Function('\\\\eta^{\\\\prime}')(Symbol('\\\\hat{H}', commutative=True), Symbol('a', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)), Pow(Pow(Add(Mul(Symbol('a', commutative=True), cos(Symbol('\\\\hat{H}', commutative=True))), Function('\\\\dot{z}')(Symbol('\\\\hat{H}', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given z{(t_{1})} = \\cos{(t_{1})} and \\varphi{(t_{1})} = t_{1}, then obtain - \\frac{- \\varphi{(t_{1})} - \\cos^{2}{(t_{1})}}{\\cos^{2}{(t_{1})}} = - \\frac{- t_{1} - \\cos^{2}{(t_{1})}}{\\cos^{2}{(t_{1})}}", "derivation": "z{(t_{1})} = \\cos{(t_{1})} and \\varphi{(t_{1})} = t_{1} and \\varphi{(t_{1})} + z{(t_{1})} \\cos{(t_{1})} = t_{1} + z{(t_{1})} \\cos{(t_{1})} and - \\varphi{(t_{1})} - z{(t_{1})} \\cos{(t_{1})} = - t_{1} - z{(t_{1})} \\cos{(t_{1})} and - \\frac{- \\varphi{(t_{1})} - z{(t_{1})} \\cos{(t_{1})}}{z{(t_{1})} \\cos{(t_{1})}} = - \\frac{- t_{1} - z{(t_{1})} \\cos{(t_{1})}}{z{(t_{1})} \\cos{(t_{1})}} and - \\frac{- \\varphi{(t_{1})} - \\cos^{2}{(t_{1})}}{\\cos^{2}{(t_{1})}} = - \\frac{- t_{1} - \\cos^{2}{(t_{1})}}{\\cos^{2}{(t_{1})}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\varphi')(Symbol('t_1', commutative=True)), Symbol('t_1', commutative=True))"], [["add", 2, "Mul(Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], "Equality(Add(Function('\\\\varphi')(Symbol('t_1', commutative=True)), Mul(Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))), Add(Symbol('t_1', commutative=True), Mul(Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))))"], [["divide", 3, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('t_1', commutative=True))), Mul(Integer(-1), Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))))"], [["divide", 4, "Mul(Integer(-1), Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))"], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('t_1', commutative=True))), Mul(Integer(-1), Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))), Pow(Function('z')(Symbol('t_1', commutative=True)), Integer(-1)), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Function('z')(Symbol('t_1', commutative=True)), cos(Symbol('t_1', commutative=True)))), Pow(Function('z')(Symbol('t_1', commutative=True)), Integer(-1)), Pow(cos(Symbol('t_1', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Mul(Integer(-1), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('t_1', commutative=True))), Mul(Integer(-1), Pow(cos(Symbol('t_1', commutative=True)), Integer(2)))), Pow(cos(Symbol('t_1', commutative=True)), Integer(-2))), Mul(Integer(-1), Add(Mul(Integer(-1), Symbol('t_1', commutative=True)), Mul(Integer(-1), Pow(cos(Symbol('t_1', commutative=True)), Integer(2)))), Pow(cos(Symbol('t_1', commutative=True)), Integer(-2))))"]]}, {"prompt": "Given v{(q)} = e^{q} and \\mathbf{M}{(q)} = \\frac{d}{d q} v^{2}{(q)}, then derive \\mathbf{M}{(q)} = v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)}, then obtain (v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)}) \\mathbf{M}{(q)} v{(q)} e^{q} = (v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)}) (e^{2 q} + e^{q} \\frac{d}{d q} e^{q}) v{(q)} e^{q}", "derivation": "v{(q)} = e^{q} and v^{2}{(q)} = v{(q)} e^{q} and \\frac{d}{d q} v^{2}{(q)} = \\frac{d}{d q} v{(q)} e^{q} and \\mathbf{M}{(q)} = \\frac{d}{d q} v^{2}{(q)} and \\mathbf{M}{(q)} = \\frac{d}{d q} v{(q)} e^{q} and \\mathbf{M}{(q)} = v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)} and \\mathbf{M}{(q)} = e^{2 q} + e^{q} \\frac{d}{d q} e^{q} and (v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)}) \\mathbf{M}{(q)} v{(q)} e^{q} = (v{(q)} e^{q} + e^{q} \\frac{d}{d q} v{(q)}) (e^{2 q} + e^{q} \\frac{d}{d q} e^{q}) v{(q)} e^{q}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], [["times", 1, "Function('v')(Symbol('q', commutative=True))"], "Equality(Pow(Function('v')(Symbol('q', commutative=True)), Integer(2)), Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))))"], [["differentiate", 2, "Symbol('q', commutative=True)"], "Equality(Derivative(Pow(Function('v')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True), Integer(1))), Derivative(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('q', commutative=True)), Derivative(Pow(Function('v')(Symbol('q', commutative=True)), Integer(2)), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{M}')(Symbol('q', commutative=True)), Derivative(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Tuple(Symbol('q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Function('\\\\mathbf{M}')(Symbol('q', commutative=True)), Add(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(Function('v')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Function('\\\\mathbf{M}')(Symbol('q', commutative=True)), Add(exp(Mul(Integer(2), Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))))"], [["times", 7, "Mul(Add(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(Function('v')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True)))"], "Equality(Mul(Add(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(Function('v')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Function('\\\\mathbf{M}')(Symbol('q', commutative=True)), Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(Add(Mul(Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(Function('v')(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Add(exp(Mul(Integer(2), Symbol('q', commutative=True))), Mul(exp(Symbol('q', commutative=True)), Derivative(exp(Symbol('q', commutative=True)), Tuple(Symbol('q', commutative=True), Integer(1))))), Function('v')(Symbol('q', commutative=True)), exp(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\dot{x}{(x^\\prime)} = \\log{(x^\\prime)}, then obtain \\frac{d}{d x^\\prime} 1 = \\frac{d}{d x^\\prime} (\\frac{\\dot{x}{(x^\\prime)} - \\log{(x^\\prime)}}{\\dot{x}{(x^\\prime)} + \\log{(x^\\prime)}})^{x^\\prime}", "derivation": "\\dot{x}{(x^\\prime)} = \\log{(x^\\prime)} and \\dot{x}{(x^\\prime)} - \\log{(x^\\prime)} = 0 and \\frac{\\dot{x}{(x^\\prime)} - \\log{(x^\\prime)}}{\\dot{x}{(x^\\prime)} + \\log{(x^\\prime)}} = 0 and (\\frac{\\dot{x}{(x^\\prime)} - \\log{(x^\\prime)}}{\\dot{x}{(x^\\prime)} + \\log{(x^\\prime)}})^{x^\\prime} = 0^{x^\\prime} and \\frac{d}{d x^\\prime} (\\frac{\\dot{x}{(x^\\prime)} - \\log{(x^\\prime)}}{\\dot{x}{(x^\\prime)} + \\log{(x^\\prime)}})^{x^\\prime} = \\frac{d}{d x^\\prime} 0^{x^\\prime} and \\frac{d}{d x^\\prime} 1 = \\frac{d}{d x^\\prime} (\\frac{\\dot{x}{(x^\\prime)} - \\log{(x^\\prime)}}{\\dot{x}{(x^\\prime)} + \\log{(x^\\prime)}})^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["minus", 1, "log(Symbol('x^\\\\prime', commutative=True))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Integer(0))"], [["divide", 2, "Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], "Equality(Mul(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Pow(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Integer(0))"], [["power", 3, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Mul(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Pow(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Symbol('x^\\\\prime', commutative=True)), Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 4, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Pow(Mul(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Pow(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Integer(1), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(Pow(Mul(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), Mul(Integer(-1), log(Symbol('x^\\\\prime', commutative=True)))), Pow(Add(Function('\\\\dot{x}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True))), Integer(-1))), Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(n)} = \\log{(n)}, then derive \\int y{(n)} dn = \\varphi + n \\log{(n)} - n, then obtain (\\int y{(n)} dn)^{\\varphi} = (\\varphi + n y{(n)} - n)^{\\varphi}", "derivation": "y{(n)} = \\log{(n)} and \\int y{(n)} dn = \\int \\log{(n)} dn and \\int y{(n)} dn = \\varphi + n \\log{(n)} - n and \\int \\log{(n)} dn = \\varphi + n \\log{(n)} - n and \\int \\log{(n)} dn = \\varphi + n y{(n)} - n and \\int y{(n)} dn = \\varphi + n y{(n)} - n and (\\int y{(n)} dn)^{\\varphi} = (\\varphi + n y{(n)} - n)^{\\varphi}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('n', commutative=True)), log(Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('y')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('y')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('n', commutative=True), log(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(log(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('n', commutative=True), Function('y')(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Integral(Function('y')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('n', commutative=True), Function('y')(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))))"], [["power", 6, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Integral(Function('y')(Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Symbol('\\\\varphi', commutative=True)), Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Symbol('n', commutative=True), Function('y')(Symbol('n', commutative=True))), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given \\omega{(I)} = I, then derive \\frac{d}{d I} \\omega{(I)} = 1, then obtain (- \\mathbf{r} + 2 \\int \\frac{d}{d I} I dI) \\int 1 dI - 2 \\int \\frac{d}{d I} I d\\omega{(I)} = (- \\mathbf{r} + \\int 1 dI + \\int \\frac{d}{d I} I dI) \\int 1 dI - 2 \\int \\frac{d}{d I} I d\\omega{(I)}", "derivation": "\\omega{(I)} = I and \\frac{d}{d I} \\omega{(I)} = \\frac{d}{d I} I and \\frac{d}{d I} \\omega{(I)} = 1 and \\frac{d}{d I} I = 1 and \\int \\frac{d}{d I} I dI = \\int 1 dI and 2 \\int \\frac{d}{d I} I dI = \\int 1 dI + \\int \\frac{d}{d I} I dI and - \\mathbf{r} + 2 \\int \\frac{d}{d I} I dI = - \\mathbf{r} + \\int 1 dI + \\int \\frac{d}{d I} I dI and (- \\mathbf{r} + 2 \\int \\frac{d}{d I} I dI) \\int 1 dI = (- \\mathbf{r} + \\int 1 dI + \\int \\frac{d}{d I} I dI) \\int 1 dI and (- \\mathbf{r} + 2 \\int \\frac{d}{d I} I dI) \\int 1 dI - 2 \\int \\frac{d}{d I} I d\\omega{(I)} = (- \\mathbf{r} + \\int 1 dI + \\int \\frac{d}{d I} I dI) \\int 1 dI - 2 \\int \\frac{d}{d I} I d\\omega{(I)}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], [["differentiate", 1, "Symbol('I', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\omega')(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Integer(1))"], [["integrate", 4, "Symbol('I', commutative=True)"], "Equality(Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))), Integral(Integer(1), Tuple(Symbol('I', commutative=True))))"], [["add", 5, "Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))), Add(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))))"], [["minus", 6, "Symbol('\\\\mathbf{r}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))))"], [["times", 7, "Integral(Integer(1), Tuple(Symbol('I', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))), Integral(Integer(1), Tuple(Symbol('I', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))), Integral(Integer(1), Tuple(Symbol('I', commutative=True)))))"], [["minus", 8, "Mul(Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Function('\\\\omega')(Symbol('I', commutative=True)))))"], "Equality(Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Mul(Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True))))), Integral(Integer(1), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Function('\\\\omega')(Symbol('I', commutative=True)))))), Add(Mul(Add(Mul(Integer(-1), Symbol('\\\\mathbf{r}', commutative=True)), Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Symbol('I', commutative=True)))), Integral(Integer(1), Tuple(Symbol('I', commutative=True)))), Mul(Integer(-1), Integer(2), Integral(Derivative(Symbol('I', commutative=True), Tuple(Symbol('I', commutative=True), Integer(1))), Tuple(Function('\\\\omega')(Symbol('I', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(f_{E},n,\\mathbf{J})} = (f_{E} - n)^{\\mathbf{J}}, then obtain \\int ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} dn + \\int \\operatorname{v_{z}}^{\\mathbf{J}}{(f_{E},n,\\mathbf{J})} dn = 2 \\int ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} dn", "derivation": "\\operatorname{v_{z}}{(f_{E},n,\\mathbf{J})} = (f_{E} - n)^{\\mathbf{J}} and \\operatorname{v_{z}}^{\\mathbf{J}}{(f_{E},n,\\mathbf{J})} = ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} and \\int \\operatorname{v_{z}}^{\\mathbf{J}}{(f_{E},n,\\mathbf{J})} dn = \\int ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} dn and \\int ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} dn + \\int \\operatorname{v_{z}}^{\\mathbf{J}}{(f_{E},n,\\mathbf{J})} dn = 2 \\int ((f_{E} - n)^{\\mathbf{J}})^{\\mathbf{J}} dn", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('f_E', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('v_z')(Symbol('f_E', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["integrate", 2, "Symbol('n', commutative=True)"], "Equality(Integral(Pow(Function('v_z')(Symbol('f_E', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["add", 3, "Integral(Pow(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True)))"], "Equality(Add(Integral(Pow(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Pow(Function('v_z')(Symbol('f_E', commutative=True), Symbol('n', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True)))), Mul(Integer(2), Integral(Pow(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('n', commutative=True))), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('n', commutative=True)))))"]]}, {"prompt": "Given p{(E,x)} = \\frac{E}{x}, then obtain p{(E,x)} + \\iint p{(E,x)} dx dE = p{(E,x)} + \\iint \\frac{E}{x} dx dE", "derivation": "p{(E,x)} = \\frac{E}{x} and \\int p{(E,x)} dx = \\int \\frac{E}{x} dx and \\iint p{(E,x)} dx dE = \\iint \\frac{E}{x} dx dE and \\frac{E}{x} + \\iint p{(E,x)} dx dE = \\frac{E}{x} + \\iint \\frac{E}{x} dx dE and p{(E,x)} + \\iint p{(E,x)} dx dE = p{(E,x)} + \\iint \\frac{E}{x} dx dE", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True))))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["add", 3, "Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Integral(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Integral(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True)))), Add(Function('p')(Symbol('E', commutative=True), Symbol('x', commutative=True)), Integral(Mul(Symbol('E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('x', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(x^\\prime)} = \\log{(x^\\prime)}, then derive n_{1} + \\Psi^{\\dagger}{(x^\\prime)} = v + \\log{(x^\\prime)}, then obtain \\frac{n_{1} + \\Psi^{\\dagger}{(x^\\prime)}}{\\Psi^{\\dagger}{(x^\\prime)}} = \\frac{v + \\log{(x^\\prime)}}{\\Psi^{\\dagger}{(x^\\prime)}}", "derivation": "\\Psi^{\\dagger}{(x^\\prime)} = \\log{(x^\\prime)} and \\frac{d}{d x^\\prime} \\Psi^{\\dagger}{(x^\\prime)} = \\frac{d}{d x^\\prime} \\log{(x^\\prime)} and \\int \\frac{d}{d x^\\prime} \\Psi^{\\dagger}{(x^\\prime)} dx^\\prime = \\int \\frac{d}{d x^\\prime} \\log{(x^\\prime)} dx^\\prime and n_{1} + \\Psi^{\\dagger}{(x^\\prime)} = v + \\log{(x^\\prime)} and \\frac{n_{1} + \\Psi^{\\dagger}{(x^\\prime)}}{\\Psi^{\\dagger}{(x^\\prime)}} = \\frac{v + \\log{(x^\\prime)}}{\\Psi^{\\dagger}{(x^\\prime)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), log(Symbol('x^\\\\prime', commutative=True)))"], [["differentiate", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Derivative(log(Symbol('x^\\\\prime', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True), Integer(1))), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('n_1', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True))), Add(Symbol('v', commutative=True), log(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 4, "Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Add(Symbol('n_1', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))), Mul(Add(Symbol('v', commutative=True), log(Symbol('x^\\\\prime', commutative=True))), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('x^\\\\prime', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(C_{2})} = e^{\\cos{(C_{2})}}, then obtain \\frac{d}{d C_{2}} \\frac{\\operatorname{y^{\\prime}}{(C_{2})}}{C_{2}} - 1 = \\frac{d}{d C_{2}} \\frac{e^{\\cos{(C_{2})}}}{C_{2}} - 1", "derivation": "\\operatorname{y^{\\prime}}{(C_{2})} = e^{\\cos{(C_{2})}} and \\frac{\\operatorname{y^{\\prime}}{(C_{2})}}{C_{2}} = \\frac{e^{\\cos{(C_{2})}}}{C_{2}} and \\frac{d}{d C_{2}} \\frac{\\operatorname{y^{\\prime}}{(C_{2})}}{C_{2}} = \\frac{d}{d C_{2}} \\frac{e^{\\cos{(C_{2})}}}{C_{2}} and \\frac{d}{d C_{2}} \\frac{\\operatorname{y^{\\prime}}{(C_{2})}}{C_{2}} - 1 = \\frac{d}{d C_{2}} \\frac{e^{\\cos{(C_{2})}}}{C_{2}} - 1", "srepr_derivation": [["premise", "Equality(Function('y^{\\\\prime}')(Symbol('C_2', commutative=True)), exp(cos(Symbol('C_2', commutative=True))))"], [["divide", 1, "Symbol('C_2', commutative=True)"], "Equality(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('C_2', commutative=True))), Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), exp(cos(Symbol('C_2', commutative=True)))))"], [["differentiate", 2, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), exp(cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), Function('y^{\\\\prime}')(Symbol('C_2', commutative=True))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Pow(Symbol('C_2', commutative=True), Integer(-1)), exp(cos(Symbol('C_2', commutative=True)))), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\chi{(y,\\phi_1)} = \\phi_1 + y, then derive (\\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy)^{\\phi_1} = (\\frac{\\partial}{\\partial y} (\\phi_1 y + \\varepsilon + \\frac{y^{2}}{2}))^{\\phi_1}, then obtain \\frac{\\partial}{\\partial \\phi_1} (\\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy)^{\\phi_1} = \\frac{\\partial}{\\partial \\phi_1} (\\frac{\\partial}{\\partial y} (\\phi_1 y + \\varepsilon + \\frac{y^{2}}{2}))^{\\phi_1}", "derivation": "\\chi{(y,\\phi_1)} = \\phi_1 + y and \\int \\chi{(y,\\phi_1)} dy = \\int (\\phi_1 + y) dy and \\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy = \\frac{\\partial}{\\partial y} \\int (\\phi_1 + y) dy and (\\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy)^{\\phi_1} = (\\frac{\\partial}{\\partial y} \\int (\\phi_1 + y) dy)^{\\phi_1} and (\\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy)^{\\phi_1} = (\\frac{\\partial}{\\partial y} (\\phi_1 y + \\varepsilon + \\frac{y^{2}}{2}))^{\\phi_1} and \\frac{\\partial}{\\partial \\phi_1} (\\frac{\\partial}{\\partial y} \\int \\chi{(y,\\phi_1)} dy)^{\\phi_1} = \\frac{\\partial}{\\partial \\phi_1} (\\frac{\\partial}{\\partial y} (\\phi_1 y + \\varepsilon + \\frac{y^{2}}{2}))^{\\phi_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)))"], [["integrate", 1, "Symbol('y', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))))"], [["differentiate", 2, "Symbol('y', commutative=True)"], "Equality(Derivative(Integral(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["power", 3, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Integral(Add(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], [["evaluate_integrals", 4], "Equality(Pow(Derivative(Integral(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Pow(Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)))"], [["differentiate", 5, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Pow(Derivative(Integral(Function('\\\\chi')(Symbol('y', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Pow(Derivative(Add(Mul(Symbol('\\\\phi_1', commutative=True), Symbol('y', commutative=True)), Symbol('\\\\varepsilon', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2)))), Tuple(Symbol('y', commutative=True), Integer(1))), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(x,\\nabla)} = \\sin{(\\nabla^{x})}, then obtain \\frac{\\int - \\int \\operatorname{A_{z}}{(x,\\nabla)} dx dx}{\\int \\sin{(\\nabla^{x})} dx} = \\frac{\\int - \\int \\sin{(\\nabla^{x})} dx dx}{\\int \\sin{(\\nabla^{x})} dx}", "derivation": "\\operatorname{A_{z}}{(x,\\nabla)} = \\sin{(\\nabla^{x})} and \\int \\operatorname{A_{z}}{(x,\\nabla)} dx = \\int \\sin{(\\nabla^{x})} dx and - \\int \\operatorname{A_{z}}{(x,\\nabla)} dx = - \\int \\sin{(\\nabla^{x})} dx and \\int - \\int \\operatorname{A_{z}}{(x,\\nabla)} dx dx = \\int - \\int \\sin{(\\nabla^{x})} dx dx and \\frac{\\int - \\int \\operatorname{A_{z}}{(x,\\nabla)} dx dx}{\\int \\operatorname{A_{z}}{(x,\\nabla)} dx} = \\frac{\\int - \\int \\sin{(\\nabla^{x})} dx dx}{\\int \\operatorname{A_{z}}{(x,\\nabla)} dx} and \\frac{\\int - \\int \\operatorname{A_{z}}{(x,\\nabla)} dx dx}{\\int \\sin{(\\nabla^{x})} dx} = \\frac{\\int - \\int \\sin{(\\nabla^{x})} dx dx}{\\int \\sin{(\\nabla^{x})} dx}", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))))"], [["integrate", 1, "Symbol('x', commutative=True)"], "Equality(Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True))), Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True)))), Mul(Integer(-1), Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))))"], [["integrate", 3, "Symbol('x', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))), Integral(Mul(Integer(-1), Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True))))"], [["divide", 4, "Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True)))), Mul(Pow(Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Integral(Function('A_z')(Symbol('x', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True)))), Mul(Pow(Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True))), Integer(-1)), Integral(Mul(Integer(-1), Integral(sin(Pow(Symbol('\\\\nabla', commutative=True), Symbol('x', commutative=True))), Tuple(Symbol('x', commutative=True)))), Tuple(Symbol('x', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(M_{E},n)} = \\sin{(\\frac{n}{M_{E}})}, then obtain (((- n + \\phi_{1}{(M_{E},n)})^{n})^{M_{E}})^{M_{E}} = (((- n + \\sin{(\\frac{n}{M_{E}})})^{n})^{M_{E}})^{M_{E}}", "derivation": "\\phi_{1}{(M_{E},n)} = \\sin{(\\frac{n}{M_{E}})} and - n + \\phi_{1}{(M_{E},n)} = - n + \\sin{(\\frac{n}{M_{E}})} and (- n + \\phi_{1}{(M_{E},n)})^{n} = (- n + \\sin{(\\frac{n}{M_{E}})})^{n} and ((- n + \\phi_{1}{(M_{E},n)})^{n})^{M_{E}} = ((- n + \\sin{(\\frac{n}{M_{E}})})^{n})^{M_{E}} and (((- n + \\phi_{1}{(M_{E},n)})^{n})^{M_{E}})^{M_{E}} = (((- n + \\sin{(\\frac{n}{M_{E}})})^{n})^{M_{E}})^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('M_E', commutative=True), Symbol('n', commutative=True)), sin(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('n', commutative=True))))"], [["minus", 1, "Symbol('n', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\phi_1')(Symbol('M_E', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), sin(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('n', commutative=True)))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\phi_1')(Symbol('M_E', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), sin(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)))"], [["power", 3, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\phi_1')(Symbol('M_E', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), sin(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Symbol('M_E', commutative=True)))"], [["power", 4, "Symbol('M_E', commutative=True)"], "Equality(Pow(Pow(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\phi_1')(Symbol('M_E', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)), Pow(Pow(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), sin(Mul(Pow(Symbol('M_E', commutative=True), Integer(-1)), Symbol('n', commutative=True)))), Symbol('n', commutative=True)), Symbol('M_E', commutative=True)), Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\eta{(C_{d},i)} = \\log{(\\frac{C_{d}}{i})}, then derive \\int \\eta{(C_{d},i)} dC_{d} = C_{d} \\log{(\\frac{C_{d}}{i})} - C_{d} + \\varphi, then obtain C_{d} \\log{(\\frac{C_{d}}{i})} - C_{d} + b = C_{d} \\eta{(C_{d},i)} - C_{d} + \\varphi", "derivation": "\\eta{(C_{d},i)} = \\log{(\\frac{C_{d}}{i})} and \\int \\eta{(C_{d},i)} dC_{d} = \\int \\log{(\\frac{C_{d}}{i})} dC_{d} and \\int \\eta{(C_{d},i)} dC_{d} = C_{d} \\log{(\\frac{C_{d}}{i})} - C_{d} + \\varphi and \\int \\eta{(C_{d},i)} dC_{d} = C_{d} \\eta{(C_{d},i)} - C_{d} + \\varphi and \\int \\log{(\\frac{C_{d}}{i})} dC_{d} = C_{d} \\eta{(C_{d},i)} - C_{d} + \\varphi and C_{d} \\log{(\\frac{C_{d}}{i})} - C_{d} + b = C_{d} \\eta{(C_{d},i)} - C_{d} + \\varphi", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), log(Mul(Symbol('C_d', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(log(Mul(Symbol('C_d', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))), Tuple(Symbol('C_d', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), log(Mul(Symbol('C_d', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Mul(Symbol('C_d', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1)))), Tuple(Symbol('C_d', commutative=True))), Add(Mul(Symbol('C_d', commutative=True), Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["evaluate_integrals", 5], "Equality(Add(Mul(Symbol('C_d', commutative=True), log(Mul(Symbol('C_d', commutative=True), Pow(Symbol('i', commutative=True), Integer(-1))))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('b', commutative=True)), Add(Mul(Symbol('C_d', commutative=True), Function('\\\\eta')(Symbol('C_d', commutative=True), Symbol('i', commutative=True))), Mul(Integer(-1), Symbol('C_d', commutative=True)), Symbol('\\\\varphi', commutative=True)))"]]}, {"prompt": "Given I{(f^{\\prime})} = e^{f^{\\prime}}, then derive I{(f^{\\prime})} + \\log{(\\psi)} + \\int I{(f^{\\prime})} df^{\\prime} = f + I{(f^{\\prime})} + e^{f^{\\prime}} + \\log{(\\psi)}, then obtain I{(f^{\\prime})} + \\log{(\\psi)} + \\int e^{f^{\\prime}} df^{\\prime} = f + I{(f^{\\prime})} + e^{f^{\\prime}} + \\log{(\\psi)}", "derivation": "I{(f^{\\prime})} = e^{f^{\\prime}} and \\int I{(f^{\\prime})} df^{\\prime} = \\int e^{f^{\\prime}} df^{\\prime} and I{(f^{\\prime})} + \\log{(\\psi)} + \\int I{(f^{\\prime})} df^{\\prime} = I{(f^{\\prime})} + \\log{(\\psi)} + \\int e^{f^{\\prime}} df^{\\prime} and I{(f^{\\prime})} + \\log{(\\psi)} + \\int I{(f^{\\prime})} df^{\\prime} = f + I{(f^{\\prime})} + e^{f^{\\prime}} + \\log{(\\psi)} and I{(f^{\\prime})} + \\log{(\\psi)} + \\int e^{f^{\\prime}} df^{\\prime} = f + I{(f^{\\prime})} + e^{f^{\\prime}} + \\log{(\\psi)}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, "Add(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True)))"], "Equality(Add(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True)), Integral(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Add(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True)), Integral(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True)), Integral(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Add(Symbol('f', commutative=True), Function('I')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Function('I')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True)), Integral(exp(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), Add(Symbol('f', commutative=True), Function('I')(Symbol('f^{\\\\prime}', commutative=True)), exp(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('\\\\psi', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(\\chi,F_{c})} = \\log{(F_{c} - \\chi)}, then derive \\int \\Psi^{\\dagger}{(\\chi,F_{c})} d\\chi = - F_{c} \\log{(- F_{c} + \\chi)} + M_{E} + \\chi \\log{(F_{c} - \\chi)} - \\chi, then obtain \\int \\Psi^{\\dagger}{(\\chi,F_{c})} d\\chi = - F_{c} \\log{(- F_{c} + \\chi)} + M_{E} + \\chi \\Psi^{\\dagger}{(\\chi,F_{c})} - \\chi", "derivation": "\\Psi^{\\dagger}{(\\chi,F_{c})} = \\log{(F_{c} - \\chi)} and \\chi \\Psi^{\\dagger}{(\\chi,F_{c})} = \\chi \\log{(F_{c} - \\chi)} and \\int \\Psi^{\\dagger}{(\\chi,F_{c})} d\\chi = \\int \\log{(F_{c} - \\chi)} d\\chi and \\int \\Psi^{\\dagger}{(\\chi,F_{c})} d\\chi = - F_{c} \\log{(- F_{c} + \\chi)} + M_{E} + \\chi \\log{(F_{c} - \\chi)} - \\chi and \\int \\Psi^{\\dagger}{(\\chi,F_{c})} d\\chi = - F_{c} \\log{(- F_{c} + \\chi)} + M_{E} + \\chi \\Psi^{\\dagger}{(\\chi,F_{c})} - \\chi", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True)), log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))))"], [["times", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Mul(Symbol('\\\\chi', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True))), Mul(Symbol('\\\\chi', commutative=True), log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), log(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\chi', commutative=True)))), Symbol('M_E', commutative=True), Mul(Symbol('\\\\chi', commutative=True), log(Add(Symbol('F_c', commutative=True), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Add(Mul(Integer(-1), Symbol('F_c', commutative=True), log(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\chi', commutative=True)))), Symbol('M_E', commutative=True), Mul(Symbol('\\\\chi', commutative=True), Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\chi', commutative=True), Symbol('F_c', commutative=True))), Mul(Integer(-1), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given W{(\\mathbf{f})} = \\sin{(e^{\\mathbf{f}})}, then obtain \\frac{\\iint W{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}}{\\int W{(\\mathbf{f})} d\\mathbf{f}} = \\frac{\\iint \\sin{(e^{\\mathbf{f}})} d\\mathbf{f} d\\mathbf{f}}{\\int W{(\\mathbf{f})} d\\mathbf{f}}", "derivation": "W{(\\mathbf{f})} = \\sin{(e^{\\mathbf{f}})} and \\int W{(\\mathbf{f})} d\\mathbf{f} = \\int \\sin{(e^{\\mathbf{f}})} d\\mathbf{f} and \\iint W{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f} = \\iint \\sin{(e^{\\mathbf{f}})} d\\mathbf{f} d\\mathbf{f} and \\frac{\\iint W{(\\mathbf{f})} d\\mathbf{f} d\\mathbf{f}}{\\int W{(\\mathbf{f})} d\\mathbf{f}} = \\frac{\\iint \\sin{(e^{\\mathbf{f}})} d\\mathbf{f} d\\mathbf{f}}{\\int W{(\\mathbf{f})} d\\mathbf{f}}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), sin(exp(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{f}', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integral(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))))"], [["divide", 3, "Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))), Mul(Pow(Integral(Function('W')(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True))), Integer(-1)), Integral(sin(exp(Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\mathbf{f}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(c_{0},a)} = c_{0} \\log{(a)}, then obtain \\operatorname{A_{2}}^{c_{0}}{(c_{0},a)} + \\frac{\\partial}{\\partial c_{0}} c_{0} \\log{(a)} = (c_{0} \\log{(a)})^{c_{0}} + \\frac{\\partial}{\\partial c_{0}} c_{0} \\log{(a)}", "derivation": "\\operatorname{A_{2}}{(c_{0},a)} = c_{0} \\log{(a)} and \\frac{\\partial}{\\partial c_{0}} \\operatorname{A_{2}}{(c_{0},a)} = \\frac{\\partial}{\\partial c_{0}} c_{0} \\log{(a)} and \\operatorname{A_{2}}^{c_{0}}{(c_{0},a)} = (c_{0} \\log{(a)})^{c_{0}} and \\operatorname{A_{2}}^{c_{0}}{(c_{0},a)} + \\frac{\\partial}{\\partial c_{0}} \\operatorname{A_{2}}{(c_{0},a)} = (c_{0} \\log{(a)})^{c_{0}} + \\frac{\\partial}{\\partial c_{0}} \\operatorname{A_{2}}{(c_{0},a)} and \\operatorname{A_{2}}^{c_{0}}{(c_{0},a)} + \\frac{\\partial}{\\partial c_{0}} c_{0} \\log{(a)} = (c_{0} \\log{(a)})^{c_{0}} + \\frac{\\partial}{\\partial c_{0}} c_{0} \\log{(a)}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["power", 1, "Symbol('c_0', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Symbol('c_0', commutative=True)), Pow(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Symbol('c_0', commutative=True)))"], [["add", 3, "Derivative(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))"], "Equality(Add(Pow(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Symbol('c_0', commutative=True)), Derivative(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Pow(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Symbol('c_0', commutative=True)), Derivative(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Add(Pow(Function('A_2')(Symbol('c_0', commutative=True), Symbol('a', commutative=True)), Symbol('c_0', commutative=True)), Derivative(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)))), Add(Pow(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Symbol('c_0', commutative=True)), Derivative(Mul(Symbol('c_0', commutative=True), log(Symbol('a', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1)))))"]]}, {"prompt": "Given C{(U,f^{\\prime})} = U f^{\\prime} and g{(U,f^{\\prime})} = \\int U f^{\\prime} dU, then obtain - \\operatorname{L_{\\varepsilon}}{(\\rho_b,t_{2})} + e^{\\iiint U f^{\\prime} dU dU df^{\\prime}} = - \\operatorname{L_{\\varepsilon}}{(\\rho_b,t_{2})} + e^{\\iiint C{(U,f^{\\prime})} dU dU df^{\\prime}}", "derivation": "C{(U,f^{\\prime})} = U f^{\\prime} and \\int C{(U,f^{\\prime})} dU = \\int U f^{\\prime} dU and g{(U,f^{\\prime})} = \\int U f^{\\prime} dU and g{(U,f^{\\prime})} = \\int C{(U,f^{\\prime})} dU and \\int g{(U,f^{\\prime})} dU = \\iint C{(U,f^{\\prime})} dU dU and \\iint U f^{\\prime} dU dU = \\iint C{(U,f^{\\prime})} dU dU and \\iiint U f^{\\prime} dU dU df^{\\prime} = \\iiint C{(U,f^{\\prime})} dU dU df^{\\prime} and e^{\\iiint U f^{\\prime} dU dU df^{\\prime}} = e^{\\iiint C{(U,f^{\\prime})} dU dU df^{\\prime}} and - \\operatorname{L_{\\varepsilon}}{(\\rho_b,t_{2})} + e^{\\iiint U f^{\\prime} dU dU df^{\\prime}} = - \\operatorname{L_{\\varepsilon}}{(\\rho_b,t_{2})} + e^{\\iiint C{(U,f^{\\prime})} dU dU df^{\\prime}}", "srepr_derivation": [["get_premise", "Equality(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('g')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Function('g')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 4, "Symbol('U', commutative=True)"], "Equality(Integral(Function('g')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True))))"], [["integrate", 6, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["exp", 7], "Equality(exp(Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))), exp(Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)))))"], [["minus", 8, "Function('L_{\\\\varepsilon}')(Symbol('\\\\rho_b', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\rho_b', commutative=True), Symbol('t_2', commutative=True))), exp(Integral(Mul(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))), Add(Mul(Integer(-1), Function('L_{\\\\varepsilon}')(Symbol('\\\\rho_b', commutative=True), Symbol('t_2', commutative=True))), exp(Integral(Function('C')(Symbol('U', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})} = \\frac{\\mathbb{I}}{u} + t_{2} and z{(u,\\mathbb{I},t_{2})} = - \\frac{\\mathbb{I}}{u} - t_{2} + \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})}, then obtain \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})} - \\frac{1}{u} = 0", "derivation": "\\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})} = \\frac{\\mathbb{I}}{u} + t_{2} and z{(u,\\mathbb{I},t_{2})} = - \\frac{\\mathbb{I}}{u} - t_{2} + \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})} and \\frac{\\partial}{\\partial \\mathbb{I}} z{(u,\\mathbb{I},t_{2})} = \\frac{\\partial}{\\partial \\mathbb{I}} (- \\frac{\\mathbb{I}}{u} - t_{2} + \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})}) and \\frac{\\partial}{\\partial \\mathbb{I}} z{(u,\\mathbb{I},t_{2})} = \\frac{d}{d \\mathbb{I}} 0 and \\frac{\\partial}{\\partial \\mathbb{I}} (- \\frac{\\mathbb{I}}{u} - t_{2} + \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})}) = \\frac{d}{d \\mathbb{I}} 0 and \\frac{\\partial}{\\partial \\mathbb{I}} \\operatorname{f_{E}}{(u,\\mathbb{I},t_{2})} - \\frac{1}{u} = 0", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Symbol('t_2', commutative=True)))"], ["renaming_premise", "Equality(Function('z')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f_E')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Derivative(Function('z')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f_E')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Function('z')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True), Pow(Symbol('u', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('t_2', commutative=True)), Function('f_E')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Derivative(Function('f_E')(Symbol('u', commutative=True), Symbol('\\\\mathbb{I}', commutative=True), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('u', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\sin^{\\theta_1}{(v_{1})}, then derive W + \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\mathbf{r} + \\sin^{\\theta_1}{(v_{1})}, then obtain W + \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\mathbf{r} + \\operatorname{v_{y}}{(v_{1},\\theta_1)}", "derivation": "\\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\sin^{\\theta_1}{(v_{1})} and \\frac{\\partial}{\\partial \\theta_1} \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\frac{\\partial}{\\partial \\theta_1} \\sin^{\\theta_1}{(v_{1})} and \\int \\frac{\\partial}{\\partial \\theta_1} \\operatorname{v_{y}}{(v_{1},\\theta_1)} d\\theta_1 = \\int \\frac{\\partial}{\\partial \\theta_1} \\sin^{\\theta_1}{(v_{1})} d\\theta_1 and W + \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\mathbf{r} + \\sin^{\\theta_1}{(v_{1})} and W + \\operatorname{v_{y}}{(v_{1},\\theta_1)} = \\mathbf{r} + \\operatorname{v_{y}}{(v_{1},\\theta_1)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(sin(Symbol('v_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Derivative(Pow(sin(Symbol('v_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Integral(Derivative(Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))), Integral(Derivative(Pow(sin(Symbol('v_1', commutative=True)), Symbol('\\\\theta_1', commutative=True)), Tuple(Symbol('\\\\theta_1', commutative=True), Integer(1))), Tuple(Symbol('\\\\theta_1', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('W', commutative=True), Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Pow(sin(Symbol('v_1', commutative=True)), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Symbol('W', commutative=True), Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True))), Add(Symbol('\\\\mathbf{r}', commutative=True), Function('v_y')(Symbol('v_1', commutative=True), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given b{(m,h)} = h + m, then obtain ((\\frac{h + m}{m})^{h})^{m} \\iint \\frac{b{(m,h)}}{m} dm dm = ((\\frac{h + m}{m})^{h})^{m} \\iint (\\frac{h}{m} + 1) dm dm", "derivation": "b{(m,h)} = h + m and \\frac{b{(m,h)}}{m} = \\frac{h + m}{m} and (\\frac{b{(m,h)}}{m})^{h} = (\\frac{h + m}{m})^{h} and \\frac{b{(m,h)}}{m} = \\frac{h}{m} + 1 and \\int \\frac{b{(m,h)}}{m} dm = \\int (\\frac{h}{m} + 1) dm and \\iint \\frac{b{(m,h)}}{m} dm dm = \\iint (\\frac{h}{m} + 1) dm dm and ((\\frac{b{(m,h)}}{m})^{h})^{m} \\iint \\frac{b{(m,h)}}{m} dm dm = ((\\frac{b{(m,h)}}{m})^{h})^{m} \\iint (\\frac{h}{m} + 1) dm dm and ((\\frac{h + m}{m})^{h})^{m} \\iint \\frac{b{(m,h)}}{m} dm dm = ((\\frac{h + m}{m})^{h})^{m} \\iint (\\frac{h}{m} + 1) dm dm", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True)), Add(Symbol('h', commutative=True), Symbol('m', commutative=True)))"], [["divide", 1, "Symbol('m', commutative=True)"], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Symbol('m', commutative=True))))"], [["power", 2, "Symbol('h', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Symbol('h', commutative=True)), Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Symbol('m', commutative=True))), Symbol('h', commutative=True)))"], [["expand", 2], "Equality(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Add(Mul(Symbol('h', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(1)))"], [["integrate", 4, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('m', commutative=True))))"], [["integrate", 5, "Symbol('m', commutative=True)"], "Equality(Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))), Integral(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True))))"], [["times", 6, "Pow(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Symbol('h', commutative=True)), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Symbol('h', commutative=True)), Symbol('m', commutative=True)), Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Symbol('h', commutative=True)), Symbol('m', commutative=True)), Integral(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"], [["substitute_LHS_for_RHS", 7, 3], "Equality(Mul(Pow(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Symbol('m', commutative=True))), Symbol('h', commutative=True)), Symbol('m', commutative=True)), Integral(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Function('b')(Symbol('m', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))), Mul(Pow(Pow(Mul(Pow(Symbol('m', commutative=True), Integer(-1)), Add(Symbol('h', commutative=True), Symbol('m', commutative=True))), Symbol('h', commutative=True)), Symbol('m', commutative=True)), Integral(Add(Mul(Symbol('h', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Integer(1)), Tuple(Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\phi_2,r_{0})} = \\log{(\\phi_2)}^{r_{0}}, then obtain (\\psi^{r_{0}}{(\\phi_2,r_{0})})^{r_{0}} \\log{(\\chi)}^{\\chi} = ((\\log{(\\phi_2)}^{r_{0}})^{r_{0}})^{r_{0}} \\log{(\\chi)}^{\\chi}", "derivation": "\\psi{(\\phi_2,r_{0})} = \\log{(\\phi_2)}^{r_{0}} and \\psi^{r_{0}}{(\\phi_2,r_{0})} = (\\log{(\\phi_2)}^{r_{0}})^{r_{0}} and (\\psi^{r_{0}}{(\\phi_2,r_{0})})^{r_{0}} = ((\\log{(\\phi_2)}^{r_{0}})^{r_{0}})^{r_{0}} and (\\psi^{r_{0}}{(\\phi_2,r_{0})})^{r_{0}} \\log{(\\chi)}^{\\chi} = ((\\log{(\\phi_2)}^{r_{0}})^{r_{0}})^{r_{0}} \\log{(\\chi)}^{\\chi}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True)), Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('r_0', commutative=True)))"], [["power", 1, "Symbol('r_0', commutative=True)"], "Equality(Pow(Function('\\\\psi')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["power", 2, "Symbol('r_0', commutative=True)"], "Equality(Pow(Pow(Function('\\\\psi')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(Pow(Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)))"], [["times", 3, "Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))"], "Equality(Mul(Pow(Pow(Function('\\\\psi')(Symbol('\\\\phi_2', commutative=True), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))), Mul(Pow(Pow(Pow(log(Symbol('\\\\phi_2', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Symbol('r_0', commutative=True)), Pow(log(Symbol('\\\\chi', commutative=True)), Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_f{(t,T)} = t + \\sin{(T)}, then obtain \\frac{(t + \\sin{(T)})^{2} \\frac{\\partial}{\\partial t} \\mathbf{J}_f{(t,T)}}{T} = \\frac{(t + \\sin{(T)})^{2} \\frac{\\partial}{\\partial t} (t + \\sin{(T)})}{T}", "derivation": "\\mathbf{J}_f{(t,T)} = t + \\sin{(T)} and \\frac{\\partial}{\\partial t} \\mathbf{J}_f{(t,T)} = \\frac{\\partial}{\\partial t} (t + \\sin{(T)}) and (t + \\sin{(T)}) \\mathbf{J}_f{(t,T)} = (t + \\sin{(T)})^{2} and \\frac{(t + \\sin{(T)}) \\mathbf{J}_f{(t,T)} \\frac{\\partial}{\\partial t} \\mathbf{J}_f{(t,T)}}{T} = \\frac{(t + \\sin{(T)}) \\mathbf{J}_f{(t,T)} \\frac{\\partial}{\\partial t} (t + \\sin{(T)})}{T} and \\frac{(t + \\sin{(T)})^{2} \\frac{\\partial}{\\partial t} \\mathbf{J}_f{(t,T)}}{T} = \\frac{(t + \\sin{(T)})^{2} \\frac{\\partial}{\\partial t} (t + \\sin{(T)})}{T}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))))"], [["differentiate", 1, "Symbol('t', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1))), Derivative(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1))))"], [["times", 1, "Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True)))"], "Equality(Mul(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True))), Pow(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)))"], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Derivative(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Integer(2)), Derivative(Function('\\\\mathbf{J}_f')(Symbol('t', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('t', commutative=True), Integer(1)))), Mul(Pow(Symbol('T', commutative=True), Integer(-1)), Pow(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Integer(2)), Derivative(Add(Symbol('t', commutative=True), sin(Symbol('T', commutative=True))), Tuple(Symbol('t', commutative=True), Integer(1)))))"]]}, {"prompt": "Given g{(A_{y},\\rho_f)} = e^{A_{y} + \\rho_f}, then obtain \\int (g{(A_{y},\\rho_f)} + e^{A_{y} + \\rho_f}) dA_{y} - \\int 2 e^{A_{y} + \\rho_f} dA_{y} = 0", "derivation": "g{(A_{y},\\rho_f)} = e^{A_{y} + \\rho_f} and g{(A_{y},\\rho_f)} + e^{A_{y} + \\rho_f} = 2 e^{A_{y} + \\rho_f} and \\int (g{(A_{y},\\rho_f)} + e^{A_{y} + \\rho_f}) dA_{y} = \\int 2 e^{A_{y} + \\rho_f} dA_{y} and \\int (g{(A_{y},\\rho_f)} + e^{A_{y} + \\rho_f}) dA_{y} - \\int 2 e^{A_{y} + \\rho_f} dA_{y} = 0", "srepr_derivation": [["get_premise", "Equality(Function('g')(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["add", 1, "exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Function('g')(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Mul(Integer(2), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))))"], [["integrate", 2, "Symbol('A_y', commutative=True)"], "Equality(Integral(Add(Function('g')(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('A_y', commutative=True))), Integral(Mul(Integer(2), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('A_y', commutative=True))))"], [["minus", 3, "Integral(Mul(Integer(2), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('A_y', commutative=True)))"], "Equality(Add(Integral(Add(Function('g')(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('A_y', commutative=True))), Mul(Integer(-1), Integral(Mul(Integer(2), exp(Add(Symbol('A_y', commutative=True), Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('A_y', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\tilde{g}{(r)} = \\cos{(\\cos{(r)})} and c{(r)} = \\frac{d}{d r} \\cos{(\\cos{(r)})}, then obtain \\frac{c{(r)}}{r \\frac{d}{d r} \\tilde{g}{(r)}} = \\frac{1}{r}", "derivation": "\\tilde{g}{(r)} = \\cos{(\\cos{(r)})} and \\frac{d}{d r} \\tilde{g}{(r)} = \\frac{d}{d r} \\cos{(\\cos{(r)})} and c{(r)} = \\frac{d}{d r} \\cos{(\\cos{(r)})} and \\frac{c{(r)}}{r} = \\frac{\\frac{d}{d r} \\cos{(\\cos{(r)})}}{r} and \\frac{c{(r)}}{r \\frac{d}{d r} \\cos{(\\cos{(r)})}} = \\frac{1}{r} and \\frac{c{(r)}}{r \\frac{d}{d r} \\tilde{g}{(r)}} = \\frac{1}{r}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('r', commutative=True)), cos(cos(Symbol('r', commutative=True))))"], [["differentiate", 1, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('c')(Symbol('r', commutative=True)), Derivative(cos(cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["divide", 3, "Symbol('r', commutative=True)"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('r', commutative=True))), Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Derivative(cos(cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))))"], [["divide", 4, "Derivative(cos(cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('r', commutative=True)), Pow(Derivative(cos(cos(Symbol('r', commutative=True))), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))), Pow(Symbol('r', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('r', commutative=True), Integer(-1)), Function('c')(Symbol('r', commutative=True)), Pow(Derivative(Function('\\\\tilde{g}')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Integer(-1))), Pow(Symbol('r', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{L_{\\varepsilon}}{(c,v)} = v^{c} and \\mathbf{A}{(c,v)} = \\int \\operatorname{L_{\\varepsilon}}{(c,v)} dc, then obtain \\mathbf{A}{(c,v)} - \\int v^{c} dc = 0", "derivation": "\\operatorname{L_{\\varepsilon}}{(c,v)} = v^{c} and \\int \\operatorname{L_{\\varepsilon}}{(c,v)} dc = \\int v^{c} dc and \\mathbf{A}{(c,v)} = \\int \\operatorname{L_{\\varepsilon}}{(c,v)} dc and \\mathbf{A}{(c,v)} = \\int v^{c} dc and \\mathbf{A}{(c,v)} - \\int v^{c} dc = 0", "srepr_derivation": [["premise", "Equality(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Pow(Symbol('v', commutative=True), Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('c', commutative=True)"], "Equality(Integral(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('c', commutative=True))), Integral(Pow(Symbol('v', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Integral(Function('L_{\\\\varepsilon}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Integral(Pow(Symbol('v', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))"], [["minus", 4, "Integral(Pow(Symbol('v', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{A}')(Symbol('c', commutative=True), Symbol('v', commutative=True)), Mul(Integer(-1), Integral(Pow(Symbol('v', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('c', commutative=True))))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(i)} = \\sin{(\\log{(i)})}, then obtain - \\sin{(\\operatorname{V_{\\mathbf{B}}}^{2}{(i)} - \\operatorname{V_{\\mathbf{B}}}{(i)})} = - \\sin{(\\operatorname{V_{\\mathbf{B}}}^{2}{(i)} - \\sin{(\\log{(i)})})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(i)} = \\sin{(\\log{(i)})} and \\operatorname{V_{\\mathbf{B}}}^{2}{(i)} = \\operatorname{V_{\\mathbf{B}}}{(i)} \\sin{(\\log{(i)})} and - \\operatorname{V_{\\mathbf{B}}}{(i)} \\sin{(\\log{(i)})} + \\operatorname{V_{\\mathbf{B}}}{(i)} = - \\operatorname{V_{\\mathbf{B}}}{(i)} \\sin{(\\log{(i)})} + \\sin{(\\log{(i)})} and - \\operatorname{V_{\\mathbf{B}}}^{2}{(i)} + \\operatorname{V_{\\mathbf{B}}}{(i)} = - \\operatorname{V_{\\mathbf{B}}}^{2}{(i)} + \\sin{(\\log{(i)})} and - \\sin{(\\operatorname{V_{\\mathbf{B}}}^{2}{(i)} - \\operatorname{V_{\\mathbf{B}}}{(i)})} = - \\sin{(\\operatorname{V_{\\mathbf{B}}}^{2}{(i)} - \\sin{(\\log{(i)})})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], [["times", 1, "Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True))"], "Equality(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Integer(2)), Mul(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True)))))"], [["minus", 1, "Mul(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True)))), Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True))), Add(Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), sin(log(Symbol('i', commutative=True)))), sin(log(Symbol('i', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Integer(2))), Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True))), Add(Mul(Integer(-1), Pow(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Integer(2))), sin(log(Symbol('i', commutative=True)))))"], [["sin", 4], "Equality(Mul(Integer(-1), sin(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Integer(2)), Mul(Integer(-1), Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)))))), Mul(Integer(-1), sin(Add(Pow(Function('V_{\\\\mathbf{B}}')(Symbol('i', commutative=True)), Integer(2)), Mul(Integer(-1), sin(log(Symbol('i', commutative=True))))))))"]]}, {"prompt": "Given m{(\\dot{y},\\mathbf{f})} = \\dot{y} + \\mathbf{f}, then derive \\frac{\\partial}{\\partial \\dot{y}} m{(\\dot{y},\\mathbf{f})} - 1 = 0, then obtain - \\dot{y} (\\dot{y} + \\mathbf{f}) + \\frac{\\partial}{\\partial \\dot{y}} m{(\\dot{y},\\mathbf{f})} - 1 = - \\dot{y} (\\dot{y} + \\mathbf{f})", "derivation": "m{(\\dot{y},\\mathbf{f})} = \\dot{y} + \\mathbf{f} and - \\dot{y} - \\mathbf{f} + m{(\\dot{y},\\mathbf{f})} = 0 and \\frac{\\partial}{\\partial \\dot{y}} (- \\dot{y} - \\mathbf{f} + m{(\\dot{y},\\mathbf{f})}) = \\frac{d}{d \\dot{y}} 0 and \\frac{\\partial}{\\partial \\dot{y}} m{(\\dot{y},\\mathbf{f})} - 1 = 0 and - \\dot{y} (\\dot{y} + \\mathbf{f}) + \\frac{\\partial}{\\partial \\dot{y}} m{(\\dot{y},\\mathbf{f})} - 1 = - \\dot{y} (\\dot{y} + \\mathbf{f})", "srepr_derivation": [["premise", "Equality(Function('m')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], [["minus", 1, "Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('m')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\dot{y}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{f}', commutative=True)), Function('m')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('m')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(-1)), Integer(0))"], [["minus", 4, "Mul(Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))), Derivative(Function('m')(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True)), Tuple(Symbol('\\\\dot{y}', commutative=True), Integer(1))), Integer(-1)), Mul(Integer(-1), Symbol('\\\\dot{y}', commutative=True), Add(Symbol('\\\\dot{y}', commutative=True), Symbol('\\\\mathbf{f}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(h,\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f + h)} and \\mathbb{I}{(h,\\mathbf{J}_f)} = \\mathbf{J}_f + h, then obtain - h + \\log{(\\mathbf{J}_f + h)} = - h + \\log{(\\mathbb{I}{(h,\\mathbf{J}_f)})}", "derivation": "\\operatorname{v_{2}}{(h,\\mathbf{J}_f)} = \\log{(\\mathbf{J}_f + h)} and - h + \\operatorname{v_{2}}{(h,\\mathbf{J}_f)} = - h + \\log{(\\mathbf{J}_f + h)} and \\mathbb{I}{(h,\\mathbf{J}_f)} = \\mathbf{J}_f + h and - h + \\operatorname{v_{2}}{(h,\\mathbf{J}_f)} = - h + \\log{(\\mathbb{I}{(h,\\mathbf{J}_f)})} and - h + \\log{(\\mathbf{J}_f + h)} = - h + \\log{(\\mathbb{I}{(h,\\mathbf{J}_f)})}", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('h', commutative=True))))"], [["minus", 1, "Symbol('h', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('h', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('h', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), Function('v_2')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(Add(Symbol('\\\\mathbf{J}_f', commutative=True), Symbol('h', commutative=True)))), Add(Mul(Integer(-1), Symbol('h', commutative=True)), log(Function('\\\\mathbb{I}')(Symbol('h', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} = \\frac{\\delta \\mathbf{r}}{\\sigma_p}, then obtain - \\delta \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta = (- \\delta + \\int \\frac{\\delta \\mathbf{r}}{\\sigma_p} d\\delta - \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta) \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta", "derivation": "\\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} = \\frac{\\delta \\mathbf{r}}{\\sigma_p} and \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta = \\int \\frac{\\delta \\mathbf{r}}{\\sigma_p} d\\delta and 0 = \\int \\frac{\\delta \\mathbf{r}}{\\sigma_p} d\\delta - \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta and - \\delta = - \\delta + \\int \\frac{\\delta \\mathbf{r}}{\\sigma_p} d\\delta - \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta and - \\delta \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta = (- \\delta + \\int \\frac{\\delta \\mathbf{r}}{\\sigma_p} d\\delta - \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta) \\int \\hat{H}_l{(\\sigma_p,\\mathbf{r},\\delta)} d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"], [["minus", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))))"], [["times", 4, "Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True)), Integral(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Integral(Function('\\\\hat{H}_l')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\mathbf{r}', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})} = E_{n} + \\cos{(\\lambda)}, then obtain \\cos{(I + \\frac{e^{E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})}}}{I})} = \\cos{(I + \\frac{e^{E_{n} (E_{n} + \\cos{(\\lambda)})}}{I})}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})} = E_{n} + \\cos{(\\lambda)} and E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})} = E_{n} (E_{n} + \\cos{(\\lambda)}) and e^{E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})}} = e^{E_{n} (E_{n} + \\cos{(\\lambda)})} and \\frac{e^{E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})}}}{I} = \\frac{e^{E_{n} (E_{n} + \\cos{(\\lambda)})}}{I} and I + \\frac{e^{E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})}}}{I} = I + \\frac{e^{E_{n} (E_{n} + \\cos{(\\lambda)})}}{I} and \\cos{(I + \\frac{e^{E_{n} \\operatorname{V_{\\mathbf{B}}}{(\\lambda,E_{n})}}}{I})} = \\cos{(I + \\frac{e^{E_{n} (E_{n} + \\cos{(\\lambda)})}}{I})}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True)), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True))))"], [["times", 1, "Symbol('E_n', commutative=True)"], "Equality(Mul(Symbol('E_n', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True))), Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True)))))"], [["exp", 2], "Equality(exp(Mul(Symbol('E_n', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True)))), exp(Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True))))))"], [["divide", 3, "Symbol('I', commutative=True)"], "Equality(Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True))))), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True)))))))"], [["add", 4, "Symbol('I', commutative=True)"], "Equality(Add(Symbol('I', commutative=True), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True)))))), Add(Symbol('I', commutative=True), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True))))))))"], [["cos", 5], "Equality(cos(Add(Symbol('I', commutative=True), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('\\\\lambda', commutative=True), Symbol('E_n', commutative=True))))))), cos(Add(Symbol('I', commutative=True), Mul(Pow(Symbol('I', commutative=True), Integer(-1)), exp(Mul(Symbol('E_n', commutative=True), Add(Symbol('E_n', commutative=True), cos(Symbol('\\\\lambda', commutative=True)))))))))"]]}, {"prompt": "Given a{(z^{*})} = \\frac{d}{d z^{*}} \\log{(z^{*})}, then derive a^{z^{*}}{(z^{*})} + \\log{(z^{*})} = (\\frac{1}{z^{*}})^{z^{*}} + \\log{(z^{*})}, then obtain \\log{(z^{*})} + (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} + 1 = (\\frac{1}{z^{*}})^{z^{*}} + \\log{(z^{*})} + 1", "derivation": "a{(z^{*})} = \\frac{d}{d z^{*}} \\log{(z^{*})} and a^{z^{*}}{(z^{*})} = (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} and a^{z^{*}}{(z^{*})} + \\log{(z^{*})} = \\log{(z^{*})} + (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} and a^{z^{*}}{(z^{*})} + \\log{(z^{*})} = (\\frac{1}{z^{*}})^{z^{*}} + \\log{(z^{*})} and \\log{(z^{*})} + (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} = (\\frac{1}{z^{*}})^{z^{*}} + \\log{(z^{*})} and \\log{(z^{*})} + (\\frac{d}{d z^{*}} \\log{(z^{*})})^{z^{*}} + 1 = (\\frac{1}{z^{*}})^{z^{*}} + \\log{(z^{*})} + 1", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('z^*', commutative=True)), Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))))"], [["power", 1, "Symbol('z^*', commutative=True)"], "Equality(Pow(Function('a')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)))"], [["add", 2, "log(Symbol('z^*', commutative=True))"], "Equality(Add(Pow(Function('a')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True))), Add(log(Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Add(Pow(Function('a')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True))), Add(Pow(Pow(Symbol('z^*', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(log(Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True))), Add(Pow(Pow(Symbol('z^*', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True))))"], [["add", 5, 1], "Equality(Add(log(Symbol('z^*', commutative=True)), Pow(Derivative(log(Symbol('z^*', commutative=True)), Tuple(Symbol('z^*', commutative=True), Integer(1))), Symbol('z^*', commutative=True)), Integer(1)), Add(Pow(Pow(Symbol('z^*', commutative=True), Integer(-1)), Symbol('z^*', commutative=True)), log(Symbol('z^*', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\mathbf{D}{(\\psi,F_{c})} = F_{c} + \\psi and \\mu{(F_{c})} = F_{c}, then obtain \\int (\\mathbf{D}{(\\psi,F_{c})} + 1) d\\mu{(F_{c})} = \\int (F_{c} + \\psi + 1) d\\mu{(F_{c})}", "derivation": "\\mathbf{D}{(\\psi,F_{c})} = F_{c} + \\psi and \\mathbf{D}{(\\psi,F_{c})} + 1 = F_{c} + \\psi + 1 and \\int (\\mathbf{D}{(\\psi,F_{c})} + 1) dF_{c} = \\int (F_{c} + \\psi + 1) dF_{c} and \\mu{(F_{c})} = F_{c} and \\int (\\mathbf{D}{(\\psi,F_{c})} + 1) d\\mu{(F_{c})} = \\int (F_{c} + \\psi + 1) d\\mu{(F_{c})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Add(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True)))"], [["add", 1, 1], "Equality(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Add(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True), Integer(1)))"], [["integrate", 2, "Symbol('F_c', commutative=True)"], "Equality(Integral(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Tuple(Symbol('F_c', commutative=True))), Integral(Add(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Symbol('F_c', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mu')(Symbol('F_c', commutative=True)), Symbol('F_c', commutative=True))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Integral(Add(Function('\\\\mathbf{D}')(Symbol('\\\\psi', commutative=True), Symbol('F_c', commutative=True)), Integer(1)), Tuple(Function('\\\\mu')(Symbol('F_c', commutative=True)))), Integral(Add(Symbol('F_c', commutative=True), Symbol('\\\\psi', commutative=True), Integer(1)), Tuple(Function('\\\\mu')(Symbol('F_c', commutative=True)))))"]]}, {"prompt": "Given \\delta{(\\hat{H},P_{g},W)} = (\\frac{W}{\\hat{H}})^{P_{g}} and I{(P_{g},\\hat{H},W)} = \\frac{\\hat{H} \\delta{(\\hat{H},P_{g},W)}}{W}, then obtain I{(P_{g},\\hat{H},W)} - \\delta{(\\hat{H},P_{g},W)} = - \\delta{(\\hat{H},P_{g},W)} + \\frac{\\hat{H} (\\frac{W}{\\hat{H}})^{P_{g}}}{W}", "derivation": "\\delta{(\\hat{H},P_{g},W)} = (\\frac{W}{\\hat{H}})^{P_{g}} and \\frac{\\hat{H} \\delta{(\\hat{H},P_{g},W)}}{W} = \\frac{\\hat{H} (\\frac{W}{\\hat{H}})^{P_{g}}}{W} and I{(P_{g},\\hat{H},W)} = \\frac{\\hat{H} \\delta{(\\hat{H},P_{g},W)}}{W} and I{(P_{g},\\hat{H},W)} = \\frac{\\hat{H} (\\frac{W}{\\hat{H}})^{P_{g}}}{W} and I{(P_{g},\\hat{H},W)} - \\delta{(\\hat{H},P_{g},W)} = - \\delta{(\\hat{H},P_{g},W)} + \\frac{\\hat{H} (\\frac{W}{\\hat{H}})^{P_{g}}}{W}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True)), Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Symbol('P_g', commutative=True)))"], [["divide", 1, "Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)))"], "Equality(Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Symbol('P_g', commutative=True))))"], ["renaming_premise", "Equality(Function('I')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('I')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('W', commutative=True)), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Symbol('P_g', commutative=True))))"], [["minus", 4, "Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True))"], "Equality(Add(Function('I')(Symbol('P_g', commutative=True), Symbol('\\\\hat{H}', commutative=True), Symbol('W', commutative=True)), Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\delta')(Symbol('\\\\hat{H}', commutative=True), Symbol('P_g', commutative=True), Symbol('W', commutative=True))), Mul(Pow(Symbol('W', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Pow(Mul(Symbol('W', commutative=True), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1))), Symbol('P_g', commutative=True)))))"]]}, {"prompt": "Given h{(\\sigma_p,F_{N})} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{\\sigma_p}, then derive h{(\\sigma_p,F_{N})} = \\frac{F_{N}^{\\sigma_p} \\sigma_p}{F_{N}}, then obtain (\\frac{\\partial}{\\partial F_{N}} F_{N}^{\\sigma_p})^{F_{N}} = (\\frac{F_{N}^{\\sigma_p} \\sigma_p}{F_{N}})^{F_{N}}", "derivation": "h{(\\sigma_p,F_{N})} = \\frac{\\partial}{\\partial F_{N}} F_{N}^{\\sigma_p} and h{(\\sigma_p,F_{N})} = \\frac{F_{N}^{\\sigma_p} \\sigma_p}{F_{N}} and h^{F_{N}}{(\\sigma_p,F_{N})} = (\\frac{F_{N}^{\\sigma_p} \\sigma_p}{F_{N}})^{F_{N}} and (\\frac{\\partial}{\\partial F_{N}} F_{N}^{\\sigma_p})^{F_{N}} = (\\frac{F_{N}^{\\sigma_p} \\sigma_p}{F_{N}})^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('\\\\sigma_p', commutative=True), Symbol('F_N', commutative=True)), Derivative(Pow(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('h')(Symbol('\\\\sigma_p', commutative=True), Symbol('F_N', commutative=True)), Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)))"], [["power", 2, "Symbol('F_N', commutative=True)"], "Equality(Pow(Function('h')(Symbol('\\\\sigma_p', commutative=True), Symbol('F_N', commutative=True)), Symbol('F_N', commutative=True)), Pow(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Derivative(Pow(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Symbol('F_N', commutative=True)), Pow(Mul(Pow(Symbol('F_N', commutative=True), Integer(-1)), Pow(Symbol('F_N', commutative=True), Symbol('\\\\sigma_p', commutative=True)), Symbol('\\\\sigma_p', commutative=True)), Symbol('F_N', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(k,F_{g})} = e^{F_{g} k}, then obtain \\frac{d}{d k} (\\frac{d}{d F_{g}} 1)^{F_{g}} = \\frac{\\partial}{\\partial k} (\\frac{\\partial}{\\partial F_{g}} \\frac{e^{F_{g} k}}{\\operatorname{A_{1}}{(k,F_{g})}})^{F_{g}}", "derivation": "\\operatorname{A_{1}}{(k,F_{g})} = e^{F_{g} k} and 1 = \\frac{e^{F_{g} k}}{\\operatorname{A_{1}}{(k,F_{g})}} and \\frac{d}{d F_{g}} 1 = \\frac{\\partial}{\\partial F_{g}} \\frac{e^{F_{g} k}}{\\operatorname{A_{1}}{(k,F_{g})}} and (\\frac{d}{d F_{g}} 1)^{F_{g}} = (\\frac{\\partial}{\\partial F_{g}} \\frac{e^{F_{g} k}}{\\operatorname{A_{1}}{(k,F_{g})}})^{F_{g}} and \\frac{d}{d k} (\\frac{d}{d F_{g}} 1)^{F_{g}} = \\frac{\\partial}{\\partial k} (\\frac{\\partial}{\\partial F_{g}} \\frac{e^{F_{g} k}}{\\operatorname{A_{1}}{(k,F_{g})}})^{F_{g}}", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True)), exp(Mul(Symbol('F_g', commutative=True), Symbol('k', commutative=True))))"], [["divide", 1, "Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True)), Integer(-1)), exp(Mul(Symbol('F_g', commutative=True), Symbol('k', commutative=True)))))"], [["differentiate", 2, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True)), Integer(-1)), exp(Mul(Symbol('F_g', commutative=True), Symbol('k', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["power", 3, "Symbol('F_g', commutative=True)"], "Equality(Pow(Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Pow(Derivative(Mul(Pow(Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True)), Integer(-1)), exp(Mul(Symbol('F_g', commutative=True), Symbol('k', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)))"], [["differentiate", 4, "Symbol('k', commutative=True)"], "Equality(Derivative(Pow(Derivative(Integer(1), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))), Derivative(Pow(Derivative(Mul(Pow(Function('A_1')(Symbol('k', commutative=True), Symbol('F_g', commutative=True)), Integer(-1)), exp(Mul(Symbol('F_g', commutative=True), Symbol('k', commutative=True)))), Tuple(Symbol('F_g', commutative=True), Integer(1))), Symbol('F_g', commutative=True)), Tuple(Symbol('k', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{z}{(k,\\mathbf{p})} = e^{\\mathbf{p} - k}, then derive \\frac{\\partial}{\\partial \\mathbf{p}} \\dot{z}{(k,\\mathbf{p})} = e^{\\mathbf{p} - k}, then obtain \\frac{\\partial}{\\partial \\mathbf{p}} \\dot{z}{(k,\\mathbf{p})} = \\dot{z}{(k,\\mathbf{p})}", "derivation": "\\dot{z}{(k,\\mathbf{p})} = e^{\\mathbf{p} - k} and \\frac{\\partial}{\\partial \\mathbf{p}} \\dot{z}{(k,\\mathbf{p})} = \\frac{\\partial}{\\partial \\mathbf{p}} e^{\\mathbf{p} - k} and \\frac{\\partial}{\\partial \\mathbf{p}} \\dot{z}{(k,\\mathbf{p})} = e^{\\mathbf{p} - k} and \\frac{\\partial}{\\partial \\mathbf{p}} e^{\\mathbf{p} - k} = e^{\\mathbf{p} - k} and \\frac{\\partial}{\\partial \\mathbf{p}} \\dot{z}{(k,\\mathbf{p})} = \\dot{z}{(k,\\mathbf{p})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), exp(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["differentiate", 1, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Derivative(exp(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(exp(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), exp(Add(Symbol('\\\\mathbf{p}', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)), Tuple(Symbol('\\\\mathbf{p}', commutative=True), Integer(1))), Function('\\\\dot{z}')(Symbol('k', commutative=True), Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} = \\sin{(\\cos{(\\mathbf{E})})}, then obtain ((\\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} - \\sin{(\\cos{(\\mathbf{E})})})^{\\mathbf{E}})^{\\mathbf{E}} = (0^{\\mathbf{E}})^{\\mathbf{E}}", "derivation": "\\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} = \\sin{(\\cos{(\\mathbf{E})})} and - \\mathbf{E} + \\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} = - \\mathbf{E} + \\sin{(\\cos{(\\mathbf{E})})} and \\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} - \\sin{(\\cos{(\\mathbf{E})})} = 0 and (\\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} - \\sin{(\\cos{(\\mathbf{E})})})^{\\mathbf{E}} = 0^{\\mathbf{E}} and ((\\operatorname{g_{\\varepsilon}}{(\\mathbf{E})} - \\sin{(\\cos{(\\mathbf{E})})})^{\\mathbf{E}})^{\\mathbf{E}} = (0^{\\mathbf{E}})^{\\mathbf{E}}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True)), sin(cos(Symbol('\\\\mathbf{E}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), sin(cos(Symbol('\\\\mathbf{E}', commutative=True)))))"], [["minus", 2, "Add(Mul(Integer(-1), Symbol('\\\\mathbf{E}', commutative=True)), sin(cos(Symbol('\\\\mathbf{E}', commutative=True))))"], "Equality(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\mathbf{E}', commutative=True))))), Integer(0))"], [["power", 3, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)))"], [["power", 4, "Symbol('\\\\mathbf{E}', commutative=True)"], "Equality(Pow(Pow(Add(Function('g_{\\\\varepsilon}')(Symbol('\\\\mathbf{E}', commutative=True)), Mul(Integer(-1), sin(cos(Symbol('\\\\mathbf{E}', commutative=True))))), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{E}', commutative=True)), Symbol('\\\\mathbf{E}', commutative=True)))"]]}, {"prompt": "Given S{(E,\\rho_f)} = \\cos{(E \\rho_f)}, then obtain - S^{3}{(E,\\rho_f)} = - S{(E,\\rho_f)} \\cos^{2}{(E \\rho_f)}", "derivation": "S{(E,\\rho_f)} = \\cos{(E \\rho_f)} and S^{2}{(E,\\rho_f)} = S{(E,\\rho_f)} \\cos{(E \\rho_f)} and - S^{3}{(E,\\rho_f)} = - S^{2}{(E,\\rho_f)} \\cos{(E \\rho_f)} and - S^{3}{(E,\\rho_f)} = - S{(E,\\rho_f)} \\cos^{2}{(E \\rho_f)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True))"], "Equality(Pow(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2)), Mul(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))))"], [["times", 2, "Mul(Integer(-1), Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(3))), Mul(Integer(-1), Pow(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(2)), cos(Mul(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Pow(Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), Integer(3))), Mul(Integer(-1), Function('S')(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True)), Pow(cos(Mul(Symbol('E', commutative=True), Symbol('\\\\rho_f', commutative=True))), Integer(2))))"]]}, {"prompt": "Given \\delta{(E)} = e^{E}, then obtain \\cos{(\\int \\delta^{E}{(E)} dE)} = \\cos{(\\int (e^{E})^{E} dE)}", "derivation": "\\delta{(E)} = e^{E} and \\delta^{E}{(E)} = (e^{E})^{E} and \\int \\delta^{E}{(E)} dE = \\int (e^{E})^{E} dE and \\cos{(\\int \\delta^{E}{(E)} dE)} = \\cos{(\\int (e^{E})^{E} dE)}", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('E', commutative=True)), exp(Symbol('E', commutative=True)))"], [["power", 1, "Symbol('E', commutative=True)"], "Equality(Pow(Function('\\\\delta')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)))"], [["integrate", 2, "Symbol('E', commutative=True)"], "Equality(Integral(Pow(Function('\\\\delta')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))), Integral(Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True))))"], [["cos", 3], "Equality(cos(Integral(Pow(Function('\\\\delta')(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))), cos(Integral(Pow(exp(Symbol('E', commutative=True)), Symbol('E', commutative=True)), Tuple(Symbol('E', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(B,E_{n})} = \\frac{\\partial}{\\partial E_{n}} (B + E_{n}), then derive \\frac{\\mathbf{v}{(B,E_{n})}}{B + E_{n}} = \\frac{1}{B + E_{n}}, then obtain \\frac{\\partial}{\\partial E_{n}} \\frac{B \\frac{\\partial}{\\partial E_{n}} (B + E_{n})}{B + E_{n}} = \\frac{\\partial}{\\partial E_{n}} \\frac{B}{B + E_{n}}", "derivation": "\\mathbf{v}{(B,E_{n})} = \\frac{\\partial}{\\partial E_{n}} (B + E_{n}) and \\frac{\\mathbf{v}{(B,E_{n})}}{B + E_{n}} = \\frac{\\frac{\\partial}{\\partial E_{n}} (B + E_{n})}{B + E_{n}} and \\frac{\\mathbf{v}{(B,E_{n})}}{B + E_{n}} = \\frac{1}{B + E_{n}} and \\frac{\\frac{\\partial}{\\partial E_{n}} (B + E_{n})}{B + E_{n}} = \\frac{1}{B + E_{n}} and \\frac{B \\frac{\\partial}{\\partial E_{n}} (B + E_{n})}{B + E_{n}} = \\frac{B}{B + E_{n}} and \\frac{\\partial}{\\partial E_{n}} \\frac{B \\frac{\\partial}{\\partial E_{n}} (B + E_{n})}{B + E_{n}} = \\frac{\\partial}{\\partial E_{n}} \\frac{B}{B + E_{n}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Derivative(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["divide", 1, "Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True))), Mul(Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Derivative(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Function('\\\\mathbf{v}')(Symbol('B', commutative=True), Symbol('E_n', commutative=True))), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Derivative(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)))"], [["times", 4, "Symbol('B', commutative=True)"], "Equality(Mul(Symbol('B', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Derivative(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Mul(Symbol('B', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1))))"], [["differentiate", 5, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Mul(Symbol('B', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1)), Derivative(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1)))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Pow(Add(Symbol('B', commutative=True), Symbol('E_n', commutative=True)), Integer(-1))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"]]}, {"prompt": "Given n{(n_{1},i)} = \\frac{\\partial}{\\partial n_{1}} i^{n_{1}}, then derive n_{1} n{(n_{1},i)} + n{(n_{1},i)} = i^{n_{1}} \\log{(i)} + n_{1} n{(n_{1},i)}, then obtain n_{1} n{(n_{1},i)} + \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} = i^{n_{1}} \\log{(i)} + n_{1} n{(n_{1},i)}", "derivation": "n{(n_{1},i)} = \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} and n_{1} n{(n_{1},i)} = n_{1} \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} and n_{1} n{(n_{1},i)} + n{(n_{1},i)} = n_{1} n{(n_{1},i)} + \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} and n_{1} n{(n_{1},i)} + n{(n_{1},i)} = i^{n_{1}} \\log{(i)} + n_{1} n{(n_{1},i)} and n_{1} \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} + \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} = i^{n_{1}} \\log{(i)} + n_{1} \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} and n_{1} n{(n_{1},i)} + \\frac{\\partial}{\\partial n_{1}} i^{n_{1}} = i^{n_{1}} \\log{(i)} + n_{1} n{(n_{1},i)}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True)), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))"], [["times", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Mul(Symbol('n_1', commutative=True), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["add", 1, "Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True)))"], "Equality(Add(Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Add(Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), log(Symbol('i', commutative=True))), Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('n_1', commutative=True), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), log(Symbol('i', commutative=True))), Mul(Symbol('n_1', commutative=True), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True))), Derivative(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), Tuple(Symbol('n_1', commutative=True), Integer(1)))), Add(Mul(Pow(Symbol('i', commutative=True), Symbol('n_1', commutative=True)), log(Symbol('i', commutative=True))), Mul(Symbol('n_1', commutative=True), Function('n')(Symbol('n_1', commutative=True), Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(C)} = \\cos{(C)}, then obtain \\int (\\operatorname{A_{1}}{(C)} + 3 \\cos{(C)}) dC = \\int 4 \\cos{(C)} dC", "derivation": "\\operatorname{A_{1}}{(C)} = \\cos{(C)} and \\operatorname{A_{1}}{(C)} + \\cos{(C)} = 2 \\cos{(C)} and \\operatorname{A_{1}}{(C)} + 3 \\cos{(C)} = 4 \\cos{(C)} and \\int (\\operatorname{A_{1}}{(C)} + 3 \\cos{(C)}) dC = \\int 4 \\cos{(C)} dC", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["add", 1, "cos(Symbol('C', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True))), Mul(Integer(2), cos(Symbol('C', commutative=True))))"], [["add", 2, "Mul(Integer(2), cos(Symbol('C', commutative=True)))"], "Equality(Add(Function('A_1')(Symbol('C', commutative=True)), Mul(Integer(3), cos(Symbol('C', commutative=True)))), Mul(Integer(4), cos(Symbol('C', commutative=True))))"], [["integrate", 3, "Symbol('C', commutative=True)"], "Equality(Integral(Add(Function('A_1')(Symbol('C', commutative=True)), Mul(Integer(3), cos(Symbol('C', commutative=True)))), Tuple(Symbol('C', commutative=True))), Integral(Mul(Integer(4), cos(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\varphi^{*}{(\\hat{x},n_{2})} = \\hat{x} n_{2}, then obtain \\frac{\\varphi^{*}{(\\hat{x},n_{2})} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})}}{\\frac{\\partial}{\\partial n_{2}} \\varphi^{*}{(\\hat{x},n_{2})}} = \\frac{\\hat{x} n_{2} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})}}{\\frac{\\partial}{\\partial n_{2}} \\varphi^{*}{(\\hat{x},n_{2})}}", "derivation": "\\varphi^{*}{(\\hat{x},n_{2})} = \\hat{x} n_{2} and \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})} = \\sin{(\\hat{x} n_{2})} and \\varphi^{*}{(\\hat{x},n_{2})} \\sin{(\\hat{x} n_{2})} = \\hat{x} n_{2} \\sin{(\\hat{x} n_{2})} and \\varphi^{*}{(\\hat{x},n_{2})} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})} = \\hat{x} n_{2} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})} and \\frac{\\varphi^{*}{(\\hat{x},n_{2})} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})}}{\\frac{\\partial}{\\partial n_{2}} \\varphi^{*}{(\\hat{x},n_{2})}} = \\frac{\\hat{x} n_{2} \\sin{(\\varphi^{*}{(\\hat{x},n_{2})})}}{\\frac{\\partial}{\\partial n_{2}} \\varphi^{*}{(\\hat{x},n_{2})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))"], [["sin", 1], "Equality(sin(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True))), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True))))"], [["times", 1, "sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True), sin(Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), sin(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True), sin(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)))))"], [["divide", 4, "Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1)))"], "Equality(Mul(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), sin(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True))), Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))), Mul(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True), sin(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True))), Pow(Derivative(Function('\\\\varphi^*')(Symbol('\\\\hat{x}', commutative=True), Symbol('n_2', commutative=True)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\eta^{\\prime},Z)} = (\\eta^{\\prime})^{Z}, then obtain \\frac{\\eta^{\\prime} \\int \\mathbf{J}_P{(\\eta^{\\prime},Z)} d\\eta^{\\prime}}{\\mathbf{J}_P{(\\eta^{\\prime},Z)}} = \\frac{\\eta^{\\prime} \\int (\\eta^{\\prime})^{Z} d\\eta^{\\prime}}{\\mathbf{J}_P{(\\eta^{\\prime},Z)}}", "derivation": "\\mathbf{J}_P{(\\eta^{\\prime},Z)} = (\\eta^{\\prime})^{Z} and \\int \\mathbf{J}_P{(\\eta^{\\prime},Z)} d\\eta^{\\prime} = \\int (\\eta^{\\prime})^{Z} d\\eta^{\\prime} and \\eta^{\\prime} \\int \\mathbf{J}_P{(\\eta^{\\prime},Z)} d\\eta^{\\prime} = \\eta^{\\prime} \\int (\\eta^{\\prime})^{Z} d\\eta^{\\prime} and \\frac{\\eta^{\\prime} \\int \\mathbf{J}_P{(\\eta^{\\prime},Z)} d\\eta^{\\prime}}{\\mathbf{J}_P{(\\eta^{\\prime},Z)}} = \\frac{\\eta^{\\prime} \\int (\\eta^{\\prime})^{Z} d\\eta^{\\prime}}{\\mathbf{J}_P{(\\eta^{\\prime},Z)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)))"], [["integrate", 1, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True))))"], [["times", 2, "Symbol('\\\\eta^{\\\\prime}', commutative=True)"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"], [["divide", 3, "Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True))"], "Equality(Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Integral(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))), Mul(Symbol('\\\\eta^{\\\\prime}', commutative=True), Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Integer(-1)), Integral(Pow(Symbol('\\\\eta^{\\\\prime}', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('\\\\eta^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given T{(B)} = e^{B} and q{(B)} = \\frac{d}{d B} (2 T{(B)} + e^{B})^{B}, then obtain q^{B}{(B)} = (\\frac{d}{d B} (T{(B)} + 2 e^{B})^{B})^{B}", "derivation": "T{(B)} = e^{B} and 2 T{(B)} + e^{B} = T{(B)} + 2 e^{B} and (2 T{(B)} + e^{B})^{B} = (T{(B)} + 2 e^{B})^{B} and q{(B)} = \\frac{d}{d B} (2 T{(B)} + e^{B})^{B} and q{(B)} = \\frac{d}{d B} (T{(B)} + 2 e^{B})^{B} and q^{B}{(B)} = (\\frac{d}{d B} (T{(B)} + 2 e^{B})^{B})^{B}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], [["add", 1, "Add(Function('T')(Symbol('B', commutative=True)), exp(Symbol('B', commutative=True)))"], "Equality(Add(Mul(Integer(2), Function('T')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Add(Function('T')(Symbol('B', commutative=True)), Mul(Integer(2), exp(Symbol('B', commutative=True)))))"], [["power", 2, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('T')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Function('T')(Symbol('B', commutative=True)), Mul(Integer(2), exp(Symbol('B', commutative=True)))), Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('q')(Symbol('B', commutative=True)), Derivative(Pow(Add(Mul(Integer(2), Function('T')(Symbol('B', commutative=True))), exp(Symbol('B', commutative=True))), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('q')(Symbol('B', commutative=True)), Derivative(Pow(Add(Function('T')(Symbol('B', commutative=True)), Mul(Integer(2), exp(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))))"], [["power", 5, "Symbol('B', commutative=True)"], "Equality(Pow(Function('q')(Symbol('B', commutative=True)), Symbol('B', commutative=True)), Pow(Derivative(Pow(Add(Function('T')(Symbol('B', commutative=True)), Mul(Integer(2), exp(Symbol('B', commutative=True)))), Symbol('B', commutative=True)), Tuple(Symbol('B', commutative=True), Integer(1))), Symbol('B', commutative=True)))"]]}, {"prompt": "Given \\sigma_{p}{(s,B)} = \\frac{s}{B}, then obtain ((s \\sigma_{p}{(s,B)} - \\sigma_{p}{(s,B)} + \\frac{s}{B})^{B})^{s} = ((- \\sigma_{p}{(s,B)} + \\frac{s^{2}}{B} + \\frac{s}{B})^{B})^{s}", "derivation": "\\sigma_{p}{(s,B)} = \\frac{s}{B} and s \\sigma_{p}{(s,B)} = \\frac{s^{2}}{B} and s \\sigma_{p}{(s,B)} - \\sigma_{p}{(s,B)} = - \\sigma_{p}{(s,B)} + \\frac{s^{2}}{B} and s \\sigma_{p}{(s,B)} - \\sigma_{p}{(s,B)} + \\frac{s}{B} = - \\sigma_{p}{(s,B)} + \\frac{s^{2}}{B} + \\frac{s}{B} and (s \\sigma_{p}{(s,B)} - \\sigma_{p}{(s,B)} + \\frac{s}{B})^{B} = (- \\sigma_{p}{(s,B)} + \\frac{s^{2}}{B} + \\frac{s}{B})^{B} and ((s \\sigma_{p}{(s,B)} - \\sigma_{p}{(s,B)} + \\frac{s}{B})^{B})^{s} = ((- \\sigma_{p}{(s,B)} + \\frac{s^{2}}{B} + \\frac{s}{B})^{B})^{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True)), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True)))"], [["times", 1, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(2))))"], [["minus", 2, "Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))"], "Equality(Add(Mul(Symbol('s', commutative=True), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(2)))))"], [["add", 3, "Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))"], "Equality(Add(Mul(Symbol('s', commutative=True), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))))"], [["power", 4, "Symbol('B', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('s', commutative=True), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Symbol('B', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Symbol('B', commutative=True)))"], [["power", 5, "Symbol('s', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Symbol('s', commutative=True), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Symbol('B', commutative=True)), Symbol('s', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\sigma_p')(Symbol('s', commutative=True), Symbol('B', commutative=True))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Pow(Symbol('s', commutative=True), Integer(2))), Mul(Pow(Symbol('B', commutative=True), Integer(-1)), Symbol('s', commutative=True))), Symbol('B', commutative=True)), Symbol('s', commutative=True)))"]]}, {"prompt": "Given \\mathbf{r}{(n_{1},C_{1})} = n_{1} \\log{(C_{1})} and V{(p,\\phi_1)} = \\log{(\\phi_1)}^{p}, then obtain \\int (V{(p,\\phi_1)} - \\int \\mathbf{r}{(n_{1},C_{1})} dC_{1}) dp = \\int (\\log{(\\phi_1)}^{p} - \\int \\mathbf{r}{(n_{1},C_{1})} dC_{1}) dp", "derivation": "\\mathbf{r}{(n_{1},C_{1})} = n_{1} \\log{(C_{1})} and \\int \\mathbf{r}{(n_{1},C_{1})} dC_{1} = \\int n_{1} \\log{(C_{1})} dC_{1} and V{(p,\\phi_1)} = \\log{(\\phi_1)}^{p} and V{(p,\\phi_1)} - \\int n_{1} \\log{(C_{1})} dC_{1} = \\log{(\\phi_1)}^{p} - \\int n_{1} \\log{(C_{1})} dC_{1} and \\int (V{(p,\\phi_1)} - \\int n_{1} \\log{(C_{1})} dC_{1}) dp = \\int (\\log{(\\phi_1)}^{p} - \\int n_{1} \\log{(C_{1})} dC_{1}) dp and \\int (V{(p,\\phi_1)} - \\int \\mathbf{r}{(n_{1},C_{1})} dC_{1}) dp = \\int (\\log{(\\phi_1)}^{p} - \\int \\mathbf{r}{(n_{1},C_{1})} dC_{1}) dp", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('n_1', commutative=True), Symbol('C_1', commutative=True)), Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))))"], [["integrate", 1, "Symbol('C_1', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{r}')(Symbol('n_1', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))"], ["get_premise", "Equality(Function('V')(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('p', commutative=True)))"], [["minus", 3, "Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Add(Function('V')(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))), Add(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Add(Function('V')(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))), Tuple(Symbol('p', commutative=True))), Integral(Add(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Mul(Symbol('n_1', commutative=True), log(Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))))), Tuple(Symbol('p', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Integral(Add(Function('V')(Symbol('p', commutative=True), Symbol('\\\\phi_1', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{r}')(Symbol('n_1', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))), Tuple(Symbol('p', commutative=True))), Integral(Add(Pow(log(Symbol('\\\\phi_1', commutative=True)), Symbol('p', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{r}')(Symbol('n_1', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_1', commutative=True))))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\hat{\\mathbf{x}}{(H)} = \\sin{(\\log{(H)})}, then obtain - \\log{(H)} = - \\hat{\\mathbf{x}}{(H)} - \\log{(H)} + \\sin{(\\log{(H)})}", "derivation": "\\hat{\\mathbf{x}}{(H)} = \\sin{(\\log{(H)})} and \\int \\hat{\\mathbf{x}}{(H)} dH = \\int \\sin{(\\log{(H)})} dH and \\hat{\\mathbf{x}}{(H)} + \\int \\sin{(\\log{(H)})} dH = \\sin{(\\log{(H)})} + \\int \\sin{(\\log{(H)})} dH and \\hat{\\mathbf{x}}{(H)} + \\int \\hat{\\mathbf{x}}{(H)} dH = \\sin{(\\log{(H)})} + \\int \\hat{\\mathbf{x}}{(H)} dH and 0 = - \\hat{\\mathbf{x}}{(H)} + \\sin{(\\log{(H)})} and - \\log{(H)} = - \\hat{\\mathbf{x}}{(H)} - \\log{(H)} + \\sin{(\\log{(H)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), sin(log(Symbol('H', commutative=True))))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(sin(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True))))"], [["add", 1, "Integral(sin(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))"], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Integral(sin(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))), Add(sin(log(Symbol('H', commutative=True))), Integral(sin(log(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))), Add(sin(log(Symbol('H', commutative=True))), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))))"], [["minus", 4, "Add(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Integral(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True))), sin(log(Symbol('H', commutative=True)))))"], [["minus", 5, "log(Symbol('H', commutative=True))"], "Equality(Mul(Integer(-1), log(Symbol('H', commutative=True))), Add(Mul(Integer(-1), Function('\\\\hat{\\\\mathbf{x}}')(Symbol('H', commutative=True))), Mul(Integer(-1), log(Symbol('H', commutative=True))), sin(log(Symbol('H', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\cos{(\\hbar^{\\phi_2})}, then obtain \\phi_2 - \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\phi_2 - \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\cos{(\\hbar^{\\phi_2})}", "derivation": "\\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\cos{(\\hbar^{\\phi_2})} and \\frac{\\partial}{\\partial \\hbar} \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\frac{\\partial}{\\partial \\hbar} \\cos{(\\hbar^{\\phi_2})} and \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\cos{(\\hbar^{\\phi_2})} and - \\phi_2 + \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = - \\phi_2 + \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\cos{(\\hbar^{\\phi_2})} and \\phi_2 - \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\operatorname{C_{d}}{(\\phi_2,\\hbar)} = \\phi_2 - \\frac{\\partial^{2}}{\\partial \\phi_2\\partial \\hbar} \\cos{(\\hbar^{\\phi_2})}", "srepr_derivation": [["premise", "Equality(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), cos(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))), Derivative(cos(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))"], [["minus", 3, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('\\\\phi_2', commutative=True)), Derivative(cos(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1)))))"], [["times", 4, "Integer(-1)"], "Equality(Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Derivative(Function('C_d')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))), Add(Symbol('\\\\phi_2', commutative=True), Mul(Integer(-1), Derivative(cos(Pow(Symbol('\\\\hbar', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)), Tuple(Symbol('\\\\phi_2', commutative=True), Integer(1))))))"]]}, {"prompt": "Given b{(\\hat{x},\\mathbf{A})} = \\hat{x} + \\cos{(\\mathbf{A})}, then derive \\frac{\\partial}{\\partial \\mathbf{A}} b{(\\hat{x},\\mathbf{A})} = - \\sin{(\\mathbf{A})}, then obtain \\frac{\\partial}{\\partial \\hat{x}} \\int - \\sin{(\\mathbf{A})} d\\hat{x} = \\frac{\\partial}{\\partial \\hat{x}} \\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\hat{x} + \\cos{(\\mathbf{A})}) d\\hat{x}", "derivation": "b{(\\hat{x},\\mathbf{A})} = \\hat{x} + \\cos{(\\mathbf{A})} and \\frac{\\partial}{\\partial \\mathbf{A}} b{(\\hat{x},\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} (\\hat{x} + \\cos{(\\mathbf{A})}) and \\frac{\\partial}{\\partial \\mathbf{A}} b{(\\hat{x},\\mathbf{A})} = - \\sin{(\\mathbf{A})} and - \\sin{(\\mathbf{A})} = \\frac{\\partial}{\\partial \\mathbf{A}} (\\hat{x} + \\cos{(\\mathbf{A})}) and \\int - \\sin{(\\mathbf{A})} d\\hat{x} = \\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\hat{x} + \\cos{(\\mathbf{A})}) d\\hat{x} and \\frac{\\partial}{\\partial \\hat{x}} \\int - \\sin{(\\mathbf{A})} d\\hat{x} = \\frac{\\partial}{\\partial \\hat{x}} \\int \\frac{\\partial}{\\partial \\mathbf{A}} (\\hat{x} + \\cos{(\\mathbf{A})}) d\\hat{x}", "srepr_derivation": [["get_premise", "Equality(Function('b')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Derivative(Function('b')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('b')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["differentiate", 5, "Symbol('\\\\hat{x}', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))), Derivative(Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))), Tuple(Symbol('\\\\hat{x}', commutative=True))), Tuple(Symbol('\\\\hat{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(C_{1},G)} = C_{1} G, then obtain (\\frac{\\partial}{\\partial G} \\operatorname{C_{2}}{(C_{1},G)})^{C_{1}} = C_{1}^{C_{1}}", "derivation": "\\operatorname{C_{2}}{(C_{1},G)} = C_{1} G and \\frac{\\partial}{\\partial G} \\operatorname{C_{2}}{(C_{1},G)} = \\frac{\\partial}{\\partial G} C_{1} G and (\\frac{\\partial}{\\partial G} \\operatorname{C_{2}}{(C_{1},G)})^{C_{1}} = (\\frac{\\partial}{\\partial G} C_{1} G)^{C_{1}} and (\\frac{\\partial}{\\partial G} \\operatorname{C_{2}}{(C_{1},G)})^{C_{1}} = C_{1}^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('G', commutative=True)))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('C_2')(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["power", 2, "Symbol('C_1', commutative=True)"], "Equality(Pow(Derivative(Function('C_2')(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Derivative(Mul(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('C_2')(Symbol('C_1', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Symbol('C_1', commutative=True), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given M{(A_{y},Q)} = \\log{(Q)}^{A_{y}}, then obtain \\frac{\\partial}{\\partial A_{y}} (\\cos^{A_{y}}{(M{(A_{y},Q)})})^{A_{y}} = \\frac{\\partial}{\\partial A_{y}} (\\cos^{A_{y}}{(\\log{(Q)}^{A_{y}})})^{A_{y}}", "derivation": "M{(A_{y},Q)} = \\log{(Q)}^{A_{y}} and \\cos{(M{(A_{y},Q)})} = \\cos{(\\log{(Q)}^{A_{y}})} and \\cos^{A_{y}}{(M{(A_{y},Q)})} = \\cos^{A_{y}}{(\\log{(Q)}^{A_{y}})} and (\\cos^{A_{y}}{(M{(A_{y},Q)})})^{A_{y}} = (\\cos^{A_{y}}{(\\log{(Q)}^{A_{y}})})^{A_{y}} and \\frac{\\partial}{\\partial A_{y}} (\\cos^{A_{y}}{(M{(A_{y},Q)})})^{A_{y}} = \\frac{\\partial}{\\partial A_{y}} (\\cos^{A_{y}}{(\\log{(Q)}^{A_{y}})})^{A_{y}}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True)), Pow(log(Symbol('Q', commutative=True)), Symbol('A_y', commutative=True)))"], [["cos", 1], "Equality(cos(Function('M')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), cos(Pow(log(Symbol('Q', commutative=True)), Symbol('A_y', commutative=True))))"], [["power", 2, "Symbol('A_y', commutative=True)"], "Equality(Pow(cos(Function('M')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), Symbol('A_y', commutative=True)), Pow(cos(Pow(log(Symbol('Q', commutative=True)), Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)))"], [["power", 3, "Symbol('A_y', commutative=True)"], "Equality(Pow(Pow(cos(Function('M')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Pow(Pow(cos(Pow(log(Symbol('Q', commutative=True)), Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)))"], [["differentiate", 4, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Pow(Pow(cos(Function('M')(Symbol('A_y', commutative=True), Symbol('Q', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(Pow(Pow(cos(Pow(log(Symbol('Q', commutative=True)), Symbol('A_y', commutative=True))), Symbol('A_y', commutative=True)), Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{X}{(s,f^{\\prime})} = \\frac{f^{\\prime}}{s}, then obtain - f^{\\prime} + \\frac{f^{\\prime}}{s} - s + \\hat{X}{(s,f^{\\prime})} = - f^{\\prime} + \\frac{2 f^{\\prime}}{s} - s", "derivation": "\\hat{X}{(s,f^{\\prime})} = \\frac{f^{\\prime}}{s} and - f^{\\prime} + \\hat{X}{(s,f^{\\prime})} = - f^{\\prime} + \\frac{f^{\\prime}}{s} and - f^{\\prime} - s + \\hat{X}{(s,f^{\\prime})} = - f^{\\prime} + \\frac{f^{\\prime}}{s} - s and - f^{\\prime} + \\frac{f^{\\prime}}{s} - s + \\hat{X}{(s,f^{\\prime})} = - f^{\\prime} + \\frac{2 f^{\\prime}}{s} - s", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('s', commutative=True), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))))"], [["minus", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('\\\\hat{X}')(Symbol('s', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))))"], [["minus", 2, "Symbol('s', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{X}')(Symbol('s', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True))))"], [["add", 3, "Mul(Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True)), Function('\\\\hat{X}')(Symbol('s', commutative=True), Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Symbol('f^{\\\\prime}', commutative=True), Pow(Symbol('s', commutative=True), Integer(-1))), Mul(Integer(-1), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\hat{X}{(P_{g},C_{1},p)} = C_{1} P_{g} p and \\sigma_{x}{(P_{g},C_{1},p)} = C_{1} P_{g}^{2} p, then obtain C_{1} + P_{g} \\hat{X}{(P_{g},C_{1},p)} = C_{1} + \\sigma_{x}{(P_{g},C_{1},p)}", "derivation": "\\hat{X}{(P_{g},C_{1},p)} = C_{1} P_{g} p and P_{g} \\hat{X}{(P_{g},C_{1},p)} = C_{1} P_{g}^{2} p and C_{1} + P_{g} \\hat{X}{(P_{g},C_{1},p)} = C_{1} P_{g}^{2} p + C_{1} and \\sigma_{x}{(P_{g},C_{1},p)} = C_{1} P_{g}^{2} p and C_{1} + P_{g} \\hat{X}{(P_{g},C_{1},p)} = C_{1} + \\sigma_{x}{(P_{g},C_{1},p)}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('C_1', commutative=True), Symbol('P_g', commutative=True), Symbol('p', commutative=True)))"], [["times", 1, "Symbol('P_g', commutative=True)"], "Equality(Mul(Symbol('P_g', commutative=True), Function('\\\\hat{X}')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True))), Mul(Symbol('C_1', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(2)), Symbol('p', commutative=True)))"], [["add", 2, "Symbol('C_1', commutative=True)"], "Equality(Add(Symbol('C_1', commutative=True), Mul(Symbol('P_g', commutative=True), Function('\\\\hat{X}')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True)))), Add(Mul(Symbol('C_1', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(2)), Symbol('p', commutative=True)), Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\sigma_x')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('C_1', commutative=True), Pow(Symbol('P_g', commutative=True), Integer(2)), Symbol('p', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Symbol('C_1', commutative=True), Mul(Symbol('P_g', commutative=True), Function('\\\\hat{X}')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True)))), Add(Symbol('C_1', commutative=True), Function('\\\\sigma_x')(Symbol('P_g', commutative=True), Symbol('C_1', commutative=True), Symbol('p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\tilde{g},H)} = H + \\tilde{g}, then derive \\int \\mathbf{J}_M{(\\tilde{g},H)} dH = A_{x} + \\frac{H^{2}}{2} + H \\tilde{g}, then derive \\frac{H^{2}}{2} + H \\tilde{g} + a^{\\dagger} = A_{x} + \\frac{H^{2}}{2} + H \\tilde{g}, then obtain \\frac{H^{2}}{2} + H \\tilde{g} + a^{\\dagger} = \\int (H + \\tilde{g}) dH", "derivation": "\\mathbf{J}_M{(\\tilde{g},H)} = H + \\tilde{g} and \\int \\mathbf{J}_M{(\\tilde{g},H)} dH = \\int (H + \\tilde{g}) dH and \\int \\mathbf{J}_M{(\\tilde{g},H)} dH = A_{x} + \\frac{H^{2}}{2} + H \\tilde{g} and \\int (H + \\tilde{g}) dH = A_{x} + \\frac{H^{2}}{2} + H \\tilde{g} and \\frac{H^{2}}{2} + H \\tilde{g} + a^{\\dagger} = A_{x} + \\frac{H^{2}}{2} + H \\tilde{g} and \\frac{H^{2}}{2} + H \\tilde{g} + a^{\\dagger} = \\int (H + \\tilde{g}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{J}_M')(Symbol('\\\\tilde{g}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('H', commutative=True))), Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('H', commutative=True), Integer(2))), Mul(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Symbol('a^{\\\\dagger}', commutative=True)), Integral(Add(Symbol('H', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given a{(C)} = \\int \\cos{(C)} dC, then derive - C + a{(C)} = - C + W + \\sin{(C)}, then obtain (- C + \\int \\cos{(C)} dC) \\int (- C + \\int \\cos{(C)} dC) dW = (- C + \\int \\cos{(C)} dC) \\int (- C + W + \\sin{(C)}) dW", "derivation": "a{(C)} = \\int \\cos{(C)} dC and - C + a{(C)} = - C + \\int \\cos{(C)} dC and - C + a{(C)} = - C + W + \\sin{(C)} and - C + \\int \\cos{(C)} dC = - C + W + \\sin{(C)} and \\int (- C + \\int \\cos{(C)} dC) dW = \\int (- C + W + \\sin{(C)}) dW and (- C + \\int \\cos{(C)} dC) \\int (- C + \\int \\cos{(C)} dC) dW = (- C + \\int \\cos{(C)} dC) \\int (- C + W + \\sin{(C)}) dW", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["minus", 1, "Symbol('C', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('a')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Function('a')(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('W', commutative=True), sin(Symbol('C', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('W', commutative=True), sin(Symbol('C', commutative=True))))"], [["integrate", 4, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('W', commutative=True), sin(Symbol('C', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["times", 5, "Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Tuple(Symbol('W', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Integral(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('W', commutative=True), sin(Symbol('C', commutative=True))), Tuple(Symbol('W', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(\\mathbf{H},H)} = - H + \\sin{(\\mathbf{H})}, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} \\int (\\operatorname{n_{2}}{(\\mathbf{H},H)} + \\sin{(\\mathbf{H})}) d\\mathbf{H} = \\frac{\\partial}{\\partial \\mathbf{H}} \\int (- H + 2 \\sin{(\\mathbf{H})}) d\\mathbf{H}", "derivation": "\\operatorname{n_{2}}{(\\mathbf{H},H)} = - H + \\sin{(\\mathbf{H})} and \\operatorname{n_{2}}{(\\mathbf{H},H)} + \\sin{(\\mathbf{H})} = - H + 2 \\sin{(\\mathbf{H})} and \\int (\\operatorname{n_{2}}{(\\mathbf{H},H)} + \\sin{(\\mathbf{H})}) d\\mathbf{H} = \\int (- H + 2 \\sin{(\\mathbf{H})}) d\\mathbf{H} and \\frac{\\partial}{\\partial \\mathbf{H}} \\int (\\operatorname{n_{2}}{(\\mathbf{H},H)} + \\sin{(\\mathbf{H})}) d\\mathbf{H} = \\frac{\\partial}{\\partial \\mathbf{H}} \\int (- H + 2 \\sin{(\\mathbf{H})}) d\\mathbf{H}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))))"], [["add", 1, "sin(Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('H', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{H}', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Add(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('H', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Integral(Add(Function('n_2')(Symbol('\\\\mathbf{H}', commutative=True), Symbol('H', commutative=True)), sin(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Mul(Integer(2), sin(Symbol('\\\\mathbf{H}', commutative=True)))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given v{(g_{\\varepsilon})} = \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon}, then derive v{(g_{\\varepsilon})} = \\Psi + \\sin{(g_{\\varepsilon})}, then derive g_{\\varepsilon} + v{(g_{\\varepsilon})} - 1 = \\phi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1, then obtain \\cos{(\\Psi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1)} = \\cos{(\\phi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1)}", "derivation": "v{(g_{\\varepsilon})} = \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} and v{(g_{\\varepsilon})} = \\Psi + \\sin{(g_{\\varepsilon})} and g_{\\varepsilon} + v{(g_{\\varepsilon})} = g_{\\varepsilon} + \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} and g_{\\varepsilon} + v{(g_{\\varepsilon})} - 1 = g_{\\varepsilon} + \\int \\cos{(g_{\\varepsilon})} dg_{\\varepsilon} - 1 and g_{\\varepsilon} + v{(g_{\\varepsilon})} - 1 = \\phi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1 and \\Psi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1 = \\phi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1 and \\cos{(\\Psi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1)} = \\cos{(\\phi + g_{\\varepsilon} + \\sin{(g_{\\varepsilon})} - 1)}", "srepr_derivation": [["get_premise", "Equality(Function('v')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('v')(Symbol('g_{\\\\varepsilon}', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["add", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v')(Symbol('g_{\\\\varepsilon}', commutative=True))), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True)))))"], [["add", 3, "Integer(-1)"], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Symbol('g_{\\\\varepsilon}', commutative=True), Integral(cos(Symbol('g_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g_{\\\\varepsilon}', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 4], "Equality(Add(Symbol('g_{\\\\varepsilon}', commutative=True), Function('v')(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Add(Symbol('\\\\Psi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Symbol('\\\\phi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1)))"], [["cos", 6], "Equality(cos(Add(Symbol('\\\\Psi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))), cos(Add(Symbol('\\\\phi', commutative=True), Symbol('g_{\\\\varepsilon}', commutative=True), sin(Symbol('g_{\\\\varepsilon}', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\phi_{2}{(W)} = e^{W}, then derive \\int \\phi_{2}{(W)} dW = \\hat{\\mathbf{r}} + e^{W}, then obtain \\int e^{W} dW = \\hat{\\mathbf{r}} + e^{W}", "derivation": "\\phi_{2}{(W)} = e^{W} and \\int \\phi_{2}{(W)} dW = \\int e^{W} dW and \\int \\phi_{2}{(W)} dW = \\hat{\\mathbf{r}} + e^{W} and \\int \\phi_{2}{(W)} dW = \\hat{\\mathbf{r}} + \\phi_{2}{(W)} and \\int e^{W} dW = \\hat{\\mathbf{r}} + e^{W}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\phi_2')(Symbol('W', commutative=True)), exp(Symbol('W', commutative=True)))"], [["integrate", 1, "Symbol('W', commutative=True)"], "Equality(Integral(Function('\\\\phi_2')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\phi_2')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('W', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('\\\\phi_2')(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), Function('\\\\phi_2')(Symbol('W', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Integral(exp(Symbol('W', commutative=True)), Tuple(Symbol('W', commutative=True))), Add(Symbol('\\\\hat{\\\\mathbf{r}}', commutative=True), exp(Symbol('W', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(t_{2})} = e^{t_{2}}, then obtain \\cos{(\\iint (\\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}}) dt_{2} dt_{2})} = 1", "derivation": "\\operatorname{A_{2}}{(t_{2})} = e^{t_{2}} and \\operatorname{A_{2}}^{t_{2}}{(t_{2})} = (e^{t_{2}})^{t_{2}} and \\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}} = 0 and \\int (\\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}}) dt_{2} = \\int 0 dt_{2} and \\iint (\\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}}) dt_{2} dt_{2} = \\iint 0 dt_{2} dt_{2} and - \\iint (\\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}}) dt_{2} dt_{2} = - \\iint 0 dt_{2} dt_{2} and \\cos{(\\iint (\\operatorname{A_{2}}^{t_{2}}{(t_{2})} - (e^{t_{2}})^{t_{2}}) dt_{2} dt_{2})} = 1", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('t_2', commutative=True)), exp(Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('t_2', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))"], [["minus", 2, "Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True))"], "Equality(Add(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))), Integer(0))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Add(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('t_2', commutative=True))))"], [["integrate", 4, "Symbol('t_2', commutative=True)"], "Equality(Integral(Add(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))), Integral(Integer(0), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True))))"], [["divide", 5, "Integer(-1)"], "Equality(Mul(Integer(-1), Integral(Add(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Mul(Integer(-1), Integral(Integer(0), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))))"], [["cos", 6], "Equality(cos(Integral(Add(Pow(Function('A_2')(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)), Mul(Integer(-1), Pow(exp(Symbol('t_2', commutative=True)), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True)), Tuple(Symbol('t_2', commutative=True)))), Integer(1))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(C)} = \\log{(C)}, then obtain (C + \\frac{d}{d C} \\log{(\\operatorname{v_{1}}{(C)})}) \\operatorname{v_{1}}^{4}{(C)} = (C + \\frac{d}{d C} \\log{(\\operatorname{v_{1}}{(C)})}) \\operatorname{v_{1}}^{3}{(C)} \\log{(C)}", "derivation": "\\operatorname{v_{1}}{(C)} = \\log{(C)} and \\log{(\\operatorname{v_{1}}{(C)})} = \\log{(\\log{(C)})} and \\frac{d}{d C} \\log{(\\operatorname{v_{1}}{(C)})} = \\frac{d}{d C} \\log{(\\log{(C)})} and - \\operatorname{v_{1}}^{2}{(C)} = - \\operatorname{v_{1}}{(C)} \\log{(C)} and (C + \\frac{d}{d C} \\log{(\\log{(C)})}) \\operatorname{v_{1}}^{4}{(C)} = (C + \\frac{d}{d C} \\log{(\\log{(C)})}) \\operatorname{v_{1}}^{3}{(C)} \\log{(C)} and (C + \\frac{d}{d C} \\log{(\\operatorname{v_{1}}{(C)})}) \\operatorname{v_{1}}^{4}{(C)} = (C + \\frac{d}{d C} \\log{(\\operatorname{v_{1}}{(C)})}) \\operatorname{v_{1}}^{3}{(C)} \\log{(C)}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True)))"], [["log", 1], "Equality(log(Function('v_1')(Symbol('C', commutative=True))), log(log(Symbol('C', commutative=True))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(log(Function('v_1')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["times", 1, "Mul(Integer(-1), Function('v_1')(Symbol('C', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(2))), Mul(Integer(-1), Function('v_1')(Symbol('C', commutative=True)), log(Symbol('C', commutative=True))))"], [["times", 4, "Mul(Integer(-1), Add(Symbol('C', commutative=True), Derivative(log(log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(2)))"], "Equality(Mul(Add(Symbol('C', commutative=True), Derivative(log(log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(4))), Mul(Add(Symbol('C', commutative=True), Derivative(log(log(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(3)), log(Symbol('C', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Add(Symbol('C', commutative=True), Derivative(log(Function('v_1')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(4))), Mul(Add(Symbol('C', commutative=True), Derivative(log(Function('v_1')(Symbol('C', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(Function('v_1')(Symbol('C', commutative=True)), Integer(3)), log(Symbol('C', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(\\mathbf{M},a)} = \\mathbf{M} - a, then obtain - 2 \\mathbf{M} + 2 a + 2 \\sigma_{p}{(\\mathbf{M},a)} = 0", "derivation": "\\sigma_{p}{(\\mathbf{M},a)} = \\mathbf{M} - a and - \\mathbf{M} + a + \\sigma_{p}{(\\mathbf{M},a)} = 0 and - \\mathbf{M} + a + 2 \\sigma_{p}{(\\mathbf{M},a)} = \\sigma_{p}{(\\mathbf{M},a)} and - 2 \\mathbf{M} + 2 a + 2 \\sigma_{p}{(\\mathbf{M},a)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True)), Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\mathbf{M}', commutative=True), Mul(Integer(-1), Symbol('a', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('a', commutative=True), Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True))), Integer(0))"], [["add", 2, "Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{M}', commutative=True)), Symbol('a', commutative=True), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True)))), Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Mul(Integer(2), Symbol('a', commutative=True)), Mul(Integer(2), Function('\\\\sigma_p')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('a', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} = Q t + f_{E}, then derive t^{2} + (Q t + f_{E}) \\int \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} dt = t^{2} + (Q t + f_{E}) (\\frac{Q t^{2}}{2} + \\varphi^* + f_{E} t), then obtain t^{2} + (Q t + f_{E}) \\int (Q t + f_{E}) dt = t^{2} + (Q t + f_{E}) (\\frac{Q t^{2}}{2} + \\varphi^* + f_{E} t)", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} = Q t + f_{E} and \\int \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} dt = \\int (Q t + f_{E}) dt and (Q t + f_{E}) \\int \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} dt = (Q t + f_{E}) \\int (Q t + f_{E}) dt and t^{2} + (Q t + f_{E}) \\int \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} dt = t^{2} + (Q t + f_{E}) \\int (Q t + f_{E}) dt and t^{2} + (Q t + f_{E}) \\int \\operatorname{f_{\\mathbf{p}}}{(t,f_{E},Q)} dt = t^{2} + (Q t + f_{E}) (\\frac{Q t^{2}}{2} + \\varphi^* + f_{E} t) and t^{2} + (Q t + f_{E}) \\int (Q t + f_{E}) dt = t^{2} + (Q t + f_{E}) (\\frac{Q t^{2}}{2} + \\varphi^* + f_{E} t)", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('t', commutative=True), Symbol('f_E', commutative=True), Symbol('Q', commutative=True)), Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('f_{\\\\mathbf{p}}')(Symbol('t', commutative=True), Symbol('f_E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["times", 2, "Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True))"], "Equality(Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('t', commutative=True), Symbol('f_E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('t', commutative=True)))), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('t', commutative=True)))))"], [["add", 3, "Pow(Symbol('t', commutative=True), Integer(2))"], "Equality(Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('t', commutative=True), Symbol('f_E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('t', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Function('f_{\\\\mathbf{p}}')(Symbol('t', commutative=True), Symbol('f_E', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Add(Mul(Rational(1, 2), Symbol('Q', commutative=True), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True), Mul(Symbol('f_E', commutative=True), Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Integral(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('t', commutative=True))))), Add(Pow(Symbol('t', commutative=True), Integer(2)), Mul(Add(Mul(Symbol('Q', commutative=True), Symbol('t', commutative=True)), Symbol('f_E', commutative=True)), Add(Mul(Rational(1, 2), Symbol('Q', commutative=True), Pow(Symbol('t', commutative=True), Integer(2))), Symbol('\\\\varphi^*', commutative=True), Mul(Symbol('f_E', commutative=True), Symbol('t', commutative=True))))))"]]}, {"prompt": "Given G{(C_{1},A_{1})} = A_{1} + C_{1}, then obtain G{(C_{1},A_{1})} - G^{A_{1}}{(C_{1},A_{1})} = A_{1} + C_{1} - G^{A_{1}}{(C_{1},A_{1})}", "derivation": "G{(C_{1},A_{1})} = A_{1} + C_{1} and G^{A_{1}}{(C_{1},A_{1})} = (A_{1} + C_{1})^{A_{1}} and - (A_{1} + C_{1})^{A_{1}} + G{(C_{1},A_{1})} = A_{1} + C_{1} - (A_{1} + C_{1})^{A_{1}} and G{(C_{1},A_{1})} - G^{A_{1}}{(C_{1},A_{1})} = A_{1} + C_{1} - G^{A_{1}}{(C_{1},A_{1})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True)))"], [["power", 1, "Symbol('A_1', commutative=True)"], "Equality(Pow(Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["minus", 1, "Pow(Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('A_1', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('A_1', commutative=True))), Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Mul(Integer(-1), Pow(Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True)), Symbol('A_1', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Mul(Integer(-1), Pow(Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))), Add(Symbol('A_1', commutative=True), Symbol('C_1', commutative=True), Mul(Integer(-1), Pow(Function('G')(Symbol('C_1', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(L)} = e^{\\sin{(L)}}, then derive \\frac{d}{d L} \\Omega{(L)} + 1 = e^{\\sin{(L)}} \\cos{(L)} + 1, then obtain \\frac{\\frac{d}{d L} \\Omega{(L)} + 1}{L - \\Omega{(L)} + e^{\\sin{(L)}} + 1} = \\frac{e^{\\sin{(L)}} \\cos{(L)} + 1}{L - \\Omega{(L)} + e^{\\sin{(L)}} + 1}", "derivation": "\\Omega{(L)} = e^{\\sin{(L)}} and L + \\Omega{(L)} = L + e^{\\sin{(L)}} and \\frac{d}{d L} (L + \\Omega{(L)}) = \\frac{d}{d L} (L + e^{\\sin{(L)}}) and \\frac{d}{d L} \\Omega{(L)} + 1 = e^{\\sin{(L)}} \\cos{(L)} + 1 and \\frac{\\frac{d}{d L} \\Omega{(L)} + 1}{L - \\Omega{(L)} + e^{\\sin{(L)}} + 1} = \\frac{e^{\\sin{(L)}} \\cos{(L)} + 1}{L - \\Omega{(L)} + e^{\\sin{(L)}} + 1}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('L', commutative=True)), exp(sin(Symbol('L', commutative=True))))"], [["add", 1, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Function('\\\\Omega')(Symbol('L', commutative=True))), Add(Symbol('L', commutative=True), exp(sin(Symbol('L', commutative=True)))))"], [["differentiate", 2, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Symbol('L', commutative=True), Function('\\\\Omega')(Symbol('L', commutative=True))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Symbol('L', commutative=True), exp(sin(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\Omega')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1)), Add(Mul(exp(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))), Integer(1)))"], [["divide", 4, "Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('L', commutative=True))), exp(sin(Symbol('L', commutative=True))), Integer(1))"], "Equality(Mul(Add(Derivative(Function('\\\\Omega')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Integer(1)), Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('L', commutative=True))), exp(sin(Symbol('L', commutative=True))), Integer(1)), Integer(-1))), Mul(Add(Mul(exp(sin(Symbol('L', commutative=True))), cos(Symbol('L', commutative=True))), Integer(1)), Pow(Add(Symbol('L', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('L', commutative=True))), exp(sin(Symbol('L', commutative=True))), Integer(1)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(y,G)} = \\sin{(G - y)}, then derive \\frac{\\partial}{\\partial y} \\operatorname{F_{g}}{(y,G)} = - \\cos{(G - y)}, then obtain \\cos^{2}{(\\sin^{2}{(G - y)})} \\int (- \\cos{(G - y)} - 1) dy = \\cos^{2}{(\\sin^{2}{(G - y)})} \\int (\\frac{\\partial}{\\partial y} \\sin{(G - y)} - 1) dy", "derivation": "\\operatorname{F_{g}}{(y,G)} = \\sin{(G - y)} and \\frac{\\partial}{\\partial y} \\operatorname{F_{g}}{(y,G)} = \\frac{\\partial}{\\partial y} \\sin{(G - y)} and \\frac{\\partial}{\\partial y} \\operatorname{F_{g}}{(y,G)} = - \\cos{(G - y)} and - \\cos{(G - y)} = \\frac{\\partial}{\\partial y} \\sin{(G - y)} and - \\cos{(G - y)} - 1 = \\frac{\\partial}{\\partial y} \\sin{(G - y)} - 1 and \\int (- \\cos{(G - y)} - 1) dy = \\int (\\frac{\\partial}{\\partial y} \\sin{(G - y)} - 1) dy and \\cos^{2}{(\\sin^{2}{(G - y)})} \\int (- \\cos{(G - y)} - 1) dy = \\cos^{2}{(\\sin^{2}{(G - y)})} \\int (\\frac{\\partial}{\\partial y} \\sin{(G - y)} - 1) dy", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('y', commutative=True), Symbol('G', commutative=True)), sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))))"], [["differentiate", 1, "Symbol('y', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('y', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Derivative(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('y', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('y', commutative=True), Integer(1))), Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Derivative(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Integer(-1)), Add(Derivative(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)))"], [["integrate", 5, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Integer(-1)), Tuple(Symbol('y', commutative=True))), Integral(Add(Derivative(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('y', commutative=True))))"], [["times", 6, "Pow(cos(Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Integer(2))), Integer(2))"], "Equality(Mul(Pow(cos(Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Integer(2))), Integer(2)), Integral(Add(Mul(Integer(-1), cos(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True))))), Integer(-1)), Tuple(Symbol('y', commutative=True)))), Mul(Pow(cos(Pow(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Integer(2))), Integer(2)), Integral(Add(Derivative(sin(Add(Symbol('G', commutative=True), Mul(Integer(-1), Symbol('y', commutative=True)))), Tuple(Symbol('y', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('y', commutative=True)))))"]]}, {"prompt": "Given \\dot{y}{(\\varphi,n)} = - n + \\cos{(\\varphi)} and b{(A_{2})} = \\cos{(A_{2})}, then obtain b{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)}} = \\cos{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)}}", "derivation": "\\dot{y}{(\\varphi,n)} = - n + \\cos{(\\varphi)} and \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)} = (- n + \\cos{(\\varphi)}) \\cos{(\\varphi)} and \\varphi + \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)} = \\varphi + (- n + \\cos{(\\varphi)}) \\cos{(\\varphi)} and b{(A_{2})} = \\cos{(A_{2})} and b{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + (- n + \\cos{(\\varphi)}) \\cos{(\\varphi)}} = \\cos{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + (- n + \\cos{(\\varphi)}) \\cos{(\\varphi)}} and b{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)}} = \\cos{(A_{2})} + \\frac{b{(A_{2})}}{\\varphi + \\dot{y}{(\\varphi,n)} \\cos{(\\varphi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))))"], [["times", 1, "cos(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Function('\\\\dot{y}')(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True))))"], [["add", 2, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Mul(Function('\\\\dot{y}')(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))), Add(Symbol('\\\\varphi', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True)))))"], ["get_premise", "Equality(Function('b')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["add", 4, "Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True)))), Integer(-1)), Function('b')(Symbol('A_2', commutative=True)))"], "Equality(Add(Function('b')(Symbol('A_2', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True)))), Integer(-1)), Function('b')(Symbol('A_2', commutative=True)))), Add(cos(Symbol('A_2', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True))), cos(Symbol('\\\\varphi', commutative=True)))), Integer(-1)), Function('b')(Symbol('A_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Add(Function('b')(Symbol('A_2', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Function('\\\\dot{y}')(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))), Integer(-1)), Function('b')(Symbol('A_2', commutative=True)))), Add(cos(Symbol('A_2', commutative=True)), Mul(Pow(Add(Symbol('\\\\varphi', commutative=True), Mul(Function('\\\\dot{y}')(Symbol('\\\\varphi', commutative=True), Symbol('n', commutative=True)), cos(Symbol('\\\\varphi', commutative=True)))), Integer(-1)), Function('b')(Symbol('A_2', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(z)} = \\sin{(\\sin{(z)})}, then obtain - \\frac{d}{d z} (\\operatorname{E_{\\lambda}}{(z)} - \\sin{(z)}) = - \\frac{d}{d z} (- \\sin{(z)} + \\sin{(\\sin{(z)})})", "derivation": "\\operatorname{E_{\\lambda}}{(z)} = \\sin{(\\sin{(z)})} and \\operatorname{E_{\\lambda}}{(z)} - \\sin{(z)} = - \\sin{(z)} + \\sin{(\\sin{(z)})} and \\frac{d}{d z} (\\operatorname{E_{\\lambda}}{(z)} - \\sin{(z)}) = \\frac{d}{d z} (- \\sin{(z)} + \\sin{(\\sin{(z)})}) and - \\frac{d}{d z} (\\operatorname{E_{\\lambda}}{(z)} - \\sin{(z)}) = - \\frac{d}{d z} (- \\sin{(z)} + \\sin{(\\sin{(z)})})", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('z', commutative=True)), sin(sin(Symbol('z', commutative=True))))"], [["minus", 1, "sin(Symbol('z', commutative=True))"], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), sin(sin(Symbol('z', commutative=True)))))"], [["differentiate", 2, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Function('E_{\\\\lambda}')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), sin(sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Derivative(Add(Function('E_{\\\\lambda}')(Symbol('z', commutative=True)), Mul(Integer(-1), sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), sin(Symbol('z', commutative=True))), sin(sin(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))))"]]}, {"prompt": "Given Q{(g,\\mu,t_{2})} = \\frac{\\mu + g}{t_{2}}, then obtain \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g) Q{(g,\\mu,t_{2})}}{t_{2}} - 1) = \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g)^{2}}{t_{2}^{2}} - 1)", "derivation": "Q{(g,\\mu,t_{2})} = \\frac{\\mu + g}{t_{2}} and \\frac{(\\mu + g) Q{(g,\\mu,t_{2})}}{t_{2}} = \\frac{(\\mu + g)^{2}}{t_{2}^{2}} and \\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g) Q{(g,\\mu,t_{2})}}{t_{2}} = \\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g)^{2}}{t_{2}^{2}} and \\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g) Q{(g,\\mu,t_{2})}}{t_{2}} - 1 = \\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g)^{2}}{t_{2}^{2}} - 1 and \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g) Q{(g,\\mu,t_{2})}}{t_{2}} - 1) = \\frac{\\partial}{\\partial t_{2}} (\\frac{\\partial}{\\partial t_{2}} \\frac{(\\mu + g)^{2}}{t_{2}^{2}} - 1)", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True)), Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True))))"], [["times", 1, "Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)))"], "Equality(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Function('Q')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True))), Mul(Pow(Symbol('t_2', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Integer(2))))"], [["differentiate", 2, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Function('Q')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('t_2', commutative=True), Integer(1))))"], [["minus", 3, 1], "Equality(Add(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Function('Q')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)))"], [["differentiate", 4, "Symbol('t_2', commutative=True)"], "Equality(Derivative(Add(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-1)), Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Function('Q')(Symbol('g', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('t_2', commutative=True))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1))), Derivative(Add(Derivative(Mul(Pow(Symbol('t_2', commutative=True), Integer(-2)), Pow(Add(Symbol('\\\\mu', commutative=True), Symbol('g', commutative=True)), Integer(2))), Tuple(Symbol('t_2', commutative=True), Integer(1))), Integer(-1)), Tuple(Symbol('t_2', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\tilde{g},p)} = \\tilde{g} + p, then derive \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{g}}{(\\tilde{g},p)} = 1, then obtain \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{g}}{(\\tilde{g},p)})^{\\tilde{g}} = \\frac{d}{d p} 1", "derivation": "\\operatorname{F_{g}}{(\\tilde{g},p)} = \\tilde{g} + p and \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{g}}{(\\tilde{g},p)} = \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + p) and \\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{g}}{(\\tilde{g},p)} = 1 and \\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + p) = 1 and (\\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + p))^{\\tilde{g}} = 1 and \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial \\tilde{g}} (\\tilde{g} + p))^{\\tilde{g}} = \\frac{d}{d p} 1 and \\frac{\\partial}{\\partial p} (\\frac{\\partial}{\\partial \\tilde{g}} \\operatorname{F_{g}}{(\\tilde{g},p)})^{\\tilde{g}} = \\frac{d}{d p} 1", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Derivative(Function('F_g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Integer(1))"], [["power", 4, "Symbol('\\\\tilde{g}', commutative=True)"], "Equality(Pow(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Integer(1))"], [["differentiate", 5, "Symbol('p', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('p', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 2], "Equality(Derivative(Pow(Derivative(Function('F_g')(Symbol('\\\\tilde{g}', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('\\\\tilde{g}', commutative=True), Integer(1))), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('p', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('p', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(b,\\Omega)} = \\Omega b, then derive \\Omega \\frac{\\partial}{\\partial \\Omega} S{(b,\\Omega)} + S{(b,\\Omega)} = 2 \\Omega b, then obtain \\Omega b + \\Omega \\frac{\\partial}{\\partial \\Omega} \\Omega b = 2 \\Omega b", "derivation": "S{(b,\\Omega)} = \\Omega b and \\Omega S{(b,\\Omega)} = \\Omega^{2} b and \\frac{\\partial}{\\partial \\Omega} \\Omega S{(b,\\Omega)} = \\frac{\\partial}{\\partial \\Omega} \\Omega^{2} b and \\Omega \\frac{\\partial}{\\partial \\Omega} S{(b,\\Omega)} + S{(b,\\Omega)} = 2 \\Omega b and \\Omega b + \\Omega \\frac{\\partial}{\\partial \\Omega} \\Omega b = 2 \\Omega b", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('b', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)))"], [["times", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Mul(Symbol('\\\\Omega', commutative=True), Function('S')(Symbol('b', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('b', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\Omega', commutative=True), Function('S')(Symbol('b', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\Omega', commutative=True), Integer(2)), Symbol('b', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Derivative(Function('S')(Symbol('b', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1)))), Function('S')(Symbol('b', commutative=True), Symbol('\\\\Omega', commutative=True))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Mul(Symbol('\\\\Omega', commutative=True), Derivative(Mul(Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True), Integer(1))))), Mul(Integer(2), Symbol('\\\\Omega', commutative=True), Symbol('b', commutative=True)))"]]}, {"prompt": "Given E{(\\hat{H},y^{\\prime})} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - y^{\\prime}), then derive - y^{\\prime} E{(\\hat{H},y^{\\prime})} = - y^{\\prime}, then obtain - y^{\\prime} \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - y^{\\prime}) = - y^{\\prime}", "derivation": "E{(\\hat{H},y^{\\prime})} = \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - y^{\\prime}) and - y^{\\prime} E{(\\hat{H},y^{\\prime})} = - y^{\\prime} \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - y^{\\prime}) and - y^{\\prime} E{(\\hat{H},y^{\\prime})} = - y^{\\prime} and - y^{\\prime} \\frac{\\partial}{\\partial \\hat{H}} (\\hat{H} - y^{\\prime}) = - y^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\hat{H}', commutative=True), Symbol('y^{\\\\prime}', commutative=True)), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["times", 1, "Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True), Function('E')(Symbol('\\\\hat{H}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True), Function('E')(Symbol('\\\\hat{H}', commutative=True), Symbol('y^{\\\\prime}', commutative=True))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True), Derivative(Add(Symbol('\\\\hat{H}', commutative=True), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1)))), Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)))"]]}, {"prompt": "Given s{(V_{\\mathbf{E}})} = \\int \\sin{(V_{\\mathbf{E}})} dV_{\\mathbf{E}}, then derive s{(V_{\\mathbf{E}})} = \\theta - \\cos{(V_{\\mathbf{E}})}, then derive \\hbar - \\cos{(V_{\\mathbf{E}})} = \\theta - \\cos{(V_{\\mathbf{E}})}, then obtain (\\hbar - \\cos{(V_{\\mathbf{E}})}) (t - \\cos{(V_{\\mathbf{E}})}) = (\\theta - \\cos{(V_{\\mathbf{E}})}) (t - \\cos{(V_{\\mathbf{E}})})", "derivation": "s{(V_{\\mathbf{E}})} = \\int \\sin{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and s{(V_{\\mathbf{E}})} = \\theta - \\cos{(V_{\\mathbf{E}})} and \\int \\sin{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = \\theta - \\cos{(V_{\\mathbf{E}})} and \\hbar - \\cos{(V_{\\mathbf{E}})} = \\theta - \\cos{(V_{\\mathbf{E}})} and (\\hbar - \\cos{(V_{\\mathbf{E}})}) \\int \\sin{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} = (\\theta - \\cos{(V_{\\mathbf{E}})}) \\int \\sin{(V_{\\mathbf{E}})} dV_{\\mathbf{E}} and (\\hbar - \\cos{(V_{\\mathbf{E}})}) (t - \\cos{(V_{\\mathbf{E}})}) = (\\theta - \\cos{(V_{\\mathbf{E}})}) (t - \\cos{(V_{\\mathbf{E}})})", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integral(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('s')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Integral(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["times", 4, "Integral(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Integral(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Mul(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Integral(sin(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{E}}', commutative=True)))))"], [["evaluate_integrals", 5], "Equality(Mul(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))), Mul(Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True)))), Add(Symbol('t', commutative=True), Mul(Integer(-1), cos(Symbol('V_{\\\\mathbf{E}}', commutative=True))))))"]]}, {"prompt": "Given f{(\\Psi^{\\dagger},H)} = - H + \\Psi^{\\dagger}, then obtain (\\frac{\\partial}{\\partial H} \\int f{(\\Psi^{\\dagger},H)} dH)^{H} = (\\frac{\\partial}{\\partial H} \\int (- H + \\Psi^{\\dagger}) dH)^{H}", "derivation": "f{(\\Psi^{\\dagger},H)} = - H + \\Psi^{\\dagger} and \\int f{(\\Psi^{\\dagger},H)} dH = \\int (- H + \\Psi^{\\dagger}) dH and \\frac{\\partial}{\\partial H} \\int f{(\\Psi^{\\dagger},H)} dH = \\frac{\\partial}{\\partial H} \\int (- H + \\Psi^{\\dagger}) dH and (\\frac{\\partial}{\\partial H} \\int f{(\\Psi^{\\dagger},H)} dH)^{H} = (\\frac{\\partial}{\\partial H} \\int (- H + \\Psi^{\\dagger}) dH)^{H}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["differentiate", 2, "Symbol('H', commutative=True)"], "Equality(Derivative(Integral(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["power", 3, "Symbol('H', commutative=True)"], "Equality(Pow(Derivative(Integral(Function('f')(Symbol('\\\\Psi^{\\\\dagger}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)), Pow(Derivative(Integral(Add(Mul(Integer(-1), Symbol('H', commutative=True)), Symbol('\\\\Psi^{\\\\dagger}', commutative=True)), Tuple(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))), Symbol('H', commutative=True)))"]]}, {"prompt": "Given \\mathbf{E}{(L)} = \\cos{(L)} and \\mathbf{S}{(L)} = \\cos{(L)}, then obtain \\sin{(L)} = - \\frac{d}{d L} \\mathbf{E}{(L)}", "derivation": "\\mathbf{E}{(L)} = \\cos{(L)} and \\mathbf{S}{(L)} = \\cos{(L)} and \\mathbf{S}{(L)} - \\cos{(L)} = 0 and \\mathbf{S}{(L)} - \\cos{(L)} + 1 = 1 and - \\mathbf{E}{(L)} + \\mathbf{S}{(L)} - \\cos{(L)} + 1 = 1 - \\mathbf{E}{(L)} and \\mathbf{S}{(L)} = \\mathbf{E}{(L)} and \\frac{d}{d L} (- \\mathbf{E}{(L)} + \\mathbf{S}{(L)} - \\cos{(L)} + 1) = \\frac{d}{d L} (1 - \\mathbf{E}{(L)}) and \\frac{d}{d L} (1 - \\cos{(L)}) = \\frac{d}{d L} (1 - \\mathbf{E}{(L)}) and \\sin{(L)} = - \\frac{d}{d L} \\mathbf{E}{(L)}", "srepr_derivation": [["renaming_premise", "Equality(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), cos(Symbol('L', commutative=True)))"], [["minus", 2, "cos(Symbol('L', commutative=True))"], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('L', commutative=True)))), Integer(0))"], [["add", 3, 1], "Equality(Add(Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('L', commutative=True))), Integer(1)), Integer(1))"], [["minus", 4, "Function('\\\\mathbf{E}')(Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('L', commutative=True))), Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('L', commutative=True))), Integer(1)), Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('L', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), Function('\\\\mathbf{E}')(Symbol('L', commutative=True)))"], [["differentiate", 5, "Symbol('L', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('L', commutative=True))), Function('\\\\mathbf{S}')(Symbol('L', commutative=True)), Mul(Integer(-1), cos(Symbol('L', commutative=True))), Integer(1)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Derivative(Add(Integer(1), Mul(Integer(-1), cos(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{E}')(Symbol('L', commutative=True)))), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["evaluate_derivatives", 8], "Equality(sin(Symbol('L', commutative=True)), Mul(Integer(-1), Derivative(Function('\\\\mathbf{E}')(Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given U{(\\ddot{x})} = \\cos{(\\log{(\\ddot{x})})}, then derive \\frac{d}{d \\ddot{x}} U{(\\ddot{x})} = - \\frac{\\sin{(\\log{(\\ddot{x})})}}{\\ddot{x}}, then obtain - \\frac{\\sin{(\\log{(\\ddot{x})})}}{\\ddot{x}} = \\frac{d}{d \\ddot{x}} \\cos{(\\log{(\\ddot{x})})}", "derivation": "U{(\\ddot{x})} = \\cos{(\\log{(\\ddot{x})})} and \\frac{d}{d \\ddot{x}} U{(\\ddot{x})} = \\frac{d}{d \\ddot{x}} \\cos{(\\log{(\\ddot{x})})} and \\frac{d}{d \\ddot{x}} U{(\\ddot{x})} = - \\frac{\\sin{(\\log{(\\ddot{x})})}}{\\ddot{x}} and - \\frac{\\sin{(\\log{(\\ddot{x})})}}{\\ddot{x}} = \\frac{d}{d \\ddot{x}} \\cos{(\\log{(\\ddot{x})})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\ddot{x}', commutative=True)), cos(log(Symbol('\\\\ddot{x}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\ddot{x}', commutative=True)"], "Equality(Derivative(Function('U')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Derivative(cos(log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('U')(Symbol('\\\\ddot{x}', commutative=True)), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\ddot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\ddot{x}', commutative=True), Integer(-1)), sin(log(Symbol('\\\\ddot{x}', commutative=True)))), Derivative(cos(log(Symbol('\\\\ddot{x}', commutative=True))), Tuple(Symbol('\\\\ddot{x}', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(n,G)} = \\log{(G + n)}, then derive \\frac{\\partial}{\\partial G} S{(n,G)} = \\frac{1}{G + n}, then obtain \\frac{\\partial^{2}}{\\partial n\\partial G} S{(n,G)} = \\frac{\\partial^{2}}{\\partial n\\partial G} \\log{(G + n)}", "derivation": "S{(n,G)} = \\log{(G + n)} and \\frac{\\partial}{\\partial G} S{(n,G)} = \\frac{\\partial}{\\partial G} \\log{(G + n)} and \\frac{\\partial}{\\partial G} S{(n,G)} = \\frac{1}{G + n} and \\frac{\\partial}{\\partial G} \\log{(G + n)} = \\frac{1}{G + n} and \\frac{\\partial^{2}}{\\partial n\\partial G} S{(n,G)} = \\frac{\\partial}{\\partial n} \\frac{1}{G + n} and \\frac{\\partial^{2}}{\\partial n\\partial G} S{(n,G)} = \\frac{\\partial^{2}}{\\partial n\\partial G} \\log{(G + n)}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('n', commutative=True), Symbol('G', commutative=True)), log(Add(Symbol('G', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 1, "Symbol('G', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Derivative(log(Add(Symbol('G', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('S')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))), Pow(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Add(Symbol('G', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1))), Pow(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Integer(-1)))"], [["differentiate", 3, "Symbol('n', commutative=True)"], "Equality(Derivative(Function('S')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(Pow(Add(Symbol('G', commutative=True), Symbol('n', commutative=True)), Integer(-1)), Tuple(Symbol('n', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('S')(Symbol('n', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))), Derivative(log(Add(Symbol('G', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('n', commutative=True), Integer(1))))"]]}, {"prompt": "Given V{(F_{g})} = \\cos{(\\cos{(F_{g})})}, then derive \\frac{d}{d F_{g}} V{(F_{g})} = \\sin{(F_{g})} \\sin{(\\cos{(F_{g})})}, then obtain \\cos{(\\frac{\\frac{d}{d F_{g}} \\cos{(\\cos{(F_{g})})}}{F_{g}})} = \\cos{(\\frac{\\sin{(F_{g})} \\sin{(\\cos{(F_{g})})}}{F_{g}})}", "derivation": "V{(F_{g})} = \\cos{(\\cos{(F_{g})})} and \\frac{d}{d F_{g}} V{(F_{g})} = \\frac{d}{d F_{g}} \\cos{(\\cos{(F_{g})})} and \\frac{d}{d F_{g}} V{(F_{g})} = \\sin{(F_{g})} \\sin{(\\cos{(F_{g})})} and \\frac{\\frac{d}{d F_{g}} V{(F_{g})}}{F_{g}} = \\frac{\\sin{(F_{g})} \\sin{(\\cos{(F_{g})})}}{F_{g}} and \\frac{\\frac{d}{d F_{g}} \\cos{(\\cos{(F_{g})})}}{F_{g}} = \\frac{\\sin{(F_{g})} \\sin{(\\cos{(F_{g})})}}{F_{g}} and \\cos{(\\frac{\\frac{d}{d F_{g}} \\cos{(\\cos{(F_{g})})}}{F_{g}})} = \\cos{(\\frac{\\sin{(F_{g})} \\sin{(\\cos{(F_{g})})}}{F_{g}})}", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('F_g', commutative=True)), cos(cos(Symbol('F_g', commutative=True))))"], [["differentiate", 1, "Symbol('F_g', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Derivative(cos(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1))), Mul(sin(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True)))))"], [["divide", 3, "Symbol('F_g', commutative=True)"], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(Function('V')(Symbol('F_g', commutative=True)), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), sin(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(cos(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1)))), Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), sin(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True)))))"], [["cos", 5], "Equality(cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), Derivative(cos(cos(Symbol('F_g', commutative=True))), Tuple(Symbol('F_g', commutative=True), Integer(1))))), cos(Mul(Pow(Symbol('F_g', commutative=True), Integer(-1)), sin(Symbol('F_g', commutative=True)), sin(cos(Symbol('F_g', commutative=True))))))"]]}, {"prompt": "Given \\eta^{\\prime}{(\\phi_2,\\rho,\\Psi_{nl})} = \\frac{\\Psi_{nl} + \\phi_2}{\\rho}, then derive \\frac{\\partial}{\\partial \\rho} \\eta^{\\prime}{(\\phi_2,\\rho,\\Psi_{nl})} = - \\frac{\\Psi_{nl} + \\phi_2}{\\rho^{2}}, then obtain \\frac{\\partial}{\\partial \\rho} \\frac{\\Psi_{nl} + \\phi_2}{\\rho} = - \\frac{\\Psi_{nl} + \\phi_2}{\\rho^{2}}", "derivation": "\\eta^{\\prime}{(\\phi_2,\\rho,\\Psi_{nl})} = \\frac{\\Psi_{nl} + \\phi_2}{\\rho} and \\frac{\\partial}{\\partial \\rho} \\eta^{\\prime}{(\\phi_2,\\rho,\\Psi_{nl})} = \\frac{\\partial}{\\partial \\rho} \\frac{\\Psi_{nl} + \\phi_2}{\\rho} and \\frac{\\partial}{\\partial \\rho} \\eta^{\\prime}{(\\phi_2,\\rho,\\Psi_{nl})} = - \\frac{\\Psi_{nl} + \\phi_2}{\\rho^{2}} and \\frac{\\partial}{\\partial \\rho} \\frac{\\Psi_{nl} + \\phi_2}{\\rho} = - \\frac{\\Psi_{nl} + \\phi_2}{\\rho^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta^{\\\\prime}')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\rho', commutative=True), Symbol('\\\\Psi_{nl}', commutative=True)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi_2', commutative=True))), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\rho', commutative=True), Integer(-2)), Add(Symbol('\\\\Psi_{nl}', commutative=True), Symbol('\\\\phi_2', commutative=True))))"]]}, {"prompt": "Given \\hat{p}_0{(F_{x})} = e^{F_{x}}, then obtain -1 = - (\\frac{- F_{x} + e^{F_{x}}}{- F_{x} + \\hat{p}_0{(F_{x})}})^{F_{x}}", "derivation": "\\hat{p}_0{(F_{x})} = e^{F_{x}} and - F_{x} + \\hat{p}_0{(F_{x})} = - F_{x} + e^{F_{x}} and 1 = \\frac{- F_{x} + e^{F_{x}}}{- F_{x} + \\hat{p}_0{(F_{x})}} and 1 = (\\frac{- F_{x} + e^{F_{x}}}{- F_{x} + \\hat{p}_0{(F_{x})}})^{F_{x}} and -1 = - (\\frac{- F_{x} + e^{F_{x}}}{- F_{x} + \\hat{p}_0{(F_{x})}})^{F_{x}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))"], [["minus", 1, "Symbol('F_x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True))), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))))"], [["power", 3, "Symbol('F_x', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True)))"], [["times", 4, "Integer(-1)"], "Equality(Integer(-1), Mul(Integer(-1), Pow(Mul(Pow(Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), Function('\\\\hat{p}_0')(Symbol('F_x', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_x', commutative=True)), exp(Symbol('F_x', commutative=True)))), Symbol('F_x', commutative=True))))"]]}, {"prompt": "Given \\rho_{b}{(\\mathbf{v})} = \\mathbf{v}, then derive A_{x} + \\frac{\\rho_{b}^{2}{(\\mathbf{v})}}{2} = \\int \\mathbf{v} d\\rho_{b}{(\\mathbf{v})}, then obtain (A_{x} + \\frac{\\rho_{b}^{2}{(\\mathbf{v})}}{2})^{\\mathbf{v}} = (\\int \\mathbf{v} d\\rho_{b}{(\\mathbf{v})})^{\\mathbf{v}}", "derivation": "\\rho_{b}{(\\mathbf{v})} = \\mathbf{v} and \\int \\rho_{b}{(\\mathbf{v})} d\\mathbf{v} = \\int \\mathbf{v} d\\mathbf{v} and \\int \\rho_{b}{(\\mathbf{v})} d\\rho_{b}{(\\mathbf{v})} = \\int \\mathbf{v} d\\rho_{b}{(\\mathbf{v})} and A_{x} + \\frac{\\rho_{b}^{2}{(\\mathbf{v})}}{2} = \\int \\mathbf{v} d\\rho_{b}{(\\mathbf{v})} and (A_{x} + \\frac{\\rho_{b}^{2}{(\\mathbf{v})}}{2})^{\\mathbf{v}} = (\\int \\mathbf{v} d\\rho_{b}{(\\mathbf{v})})^{\\mathbf{v}}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)), Symbol('\\\\mathbf{v}', commutative=True))"], [["integrate", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Symbol('\\\\mathbf{v}', commutative=True))), Integral(Symbol('\\\\mathbf{v}', commutative=True), Tuple(Symbol('\\\\mathbf{v}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)), Tuple(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)))), Integral(Symbol('\\\\mathbf{v}', commutative=True), Tuple(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))), Integral(Symbol('\\\\mathbf{v}', commutative=True), Tuple(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)))))"], [["power", 4, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Pow(Add(Symbol('A_x', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)), Integer(2)))), Symbol('\\\\mathbf{v}', commutative=True)), Pow(Integral(Symbol('\\\\mathbf{v}', commutative=True), Tuple(Function('\\\\rho_b')(Symbol('\\\\mathbf{v}', commutative=True)))), Symbol('\\\\mathbf{v}', commutative=True)))"]]}, {"prompt": "Given t{(\\Psi,\\theta_2)} = \\frac{\\theta_2}{\\Psi} and k{(\\theta_2)} = \\theta_2^{2}, then obtain \\theta_2 t{(\\Psi,\\theta_2)} = \\frac{k{(\\theta_2)}}{\\Psi}", "derivation": "t{(\\Psi,\\theta_2)} = \\frac{\\theta_2}{\\Psi} and \\theta_2 t{(\\Psi,\\theta_2)} = \\frac{\\theta_2^{2}}{\\Psi} and k{(\\theta_2)} = \\theta_2^{2} and \\theta_2 t{(\\Psi,\\theta_2)} = \\frac{k{(\\theta_2)}}{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Symbol('\\\\theta_2', commutative=True)))"], [["times", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Function('t')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2))))"], ["renaming_premise", "Equality(Function('k')(Symbol('\\\\theta_2', commutative=True)), Pow(Symbol('\\\\theta_2', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Symbol('\\\\theta_2', commutative=True), Function('t')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\theta_2', commutative=True))), Mul(Pow(Symbol('\\\\Psi', commutative=True), Integer(-1)), Function('k')(Symbol('\\\\theta_2', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(n,\\omega)} = e^{\\omega^{n}} and \\operatorname{E_{x}}{(n,\\omega)} = e^{- \\omega^{n}}, then obtain \\omega \\operatorname{E_{x}}{(n,\\omega)} \\bar{\\h}{(n,\\omega)} = \\omega", "derivation": "\\bar{\\h}{(n,\\omega)} = e^{\\omega^{n}} and \\omega \\bar{\\h}{(n,\\omega)} = \\omega e^{\\omega^{n}} and \\omega \\bar{\\h}{(n,\\omega)} e^{- \\omega^{n}} = \\omega and \\operatorname{E_{x}}{(n,\\omega)} = e^{- \\omega^{n}} and \\omega \\operatorname{E_{x}}{(n,\\omega)} \\bar{\\h}{(n,\\omega)} = \\omega", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))))"], [["times", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), exp(Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True)))))"], [["divide", 2, "exp(Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True))))), Symbol('\\\\omega', commutative=True))"], ["renaming_premise", "Equality(Function('E_x')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True)), exp(Mul(Integer(-1), Pow(Symbol('\\\\omega', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('\\\\omega', commutative=True), Function('E_x')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True)), Function('\\\\hbar')(Symbol('n', commutative=True), Symbol('\\\\omega', commutative=True))), Symbol('\\\\omega', commutative=True))"]]}, {"prompt": "Given \\rho{(\\Omega)} = e^{\\Omega}, then obtain ((\\iint e^{\\Omega} d\\Omega d\\Omega)^{\\Omega}) \\iiint \\rho{(\\Omega)} d\\Omega d\\Omega d\\Omega = ((\\iint e^{\\Omega} d\\Omega d\\Omega)^{\\Omega}) \\iiint e^{\\Omega} d\\Omega d\\Omega d\\Omega", "derivation": "\\rho{(\\Omega)} = e^{\\Omega} and \\int \\rho{(\\Omega)} d\\Omega = \\int e^{\\Omega} d\\Omega and \\iint \\rho{(\\Omega)} d\\Omega d\\Omega = \\iint e^{\\Omega} d\\Omega d\\Omega and (\\iint \\rho{(\\Omega)} d\\Omega d\\Omega)^{\\Omega} = (\\iint e^{\\Omega} d\\Omega d\\Omega)^{\\Omega} and \\iiint \\rho{(\\Omega)} d\\Omega d\\Omega d\\Omega = \\iiint e^{\\Omega} d\\Omega d\\Omega d\\Omega and ((\\iint \\rho{(\\Omega)} d\\Omega d\\Omega)^{\\Omega}) \\iiint \\rho{(\\Omega)} d\\Omega d\\Omega d\\Omega = ((\\iint \\rho{(\\Omega)} d\\Omega d\\Omega)^{\\Omega}) \\iiint e^{\\Omega} d\\Omega d\\Omega d\\Omega and ((\\iint e^{\\Omega} d\\Omega d\\Omega)^{\\Omega}) \\iiint \\rho{(\\Omega)} d\\Omega d\\Omega d\\Omega = ((\\iint e^{\\Omega} d\\Omega d\\Omega)^{\\Omega}) \\iiint e^{\\Omega} d\\Omega d\\Omega d\\Omega", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["integrate", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["integrate", 2, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["power", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Pow(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Pow(Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 3, "Symbol('\\\\Omega', commutative=True)"], "Equality(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))))"], [["times", 5, "Pow(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Pow(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Mul(Pow(Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Integral(Function('\\\\rho')(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))), Mul(Pow(Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True))), Symbol('\\\\Omega', commutative=True)), Integral(exp(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(\\mathbf{P})} = e^{\\mathbf{P}} and A{(\\mathbf{P})} = \\int (\\int \\Psi_{\\lambda}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P}, then obtain A{(\\mathbf{P})} = \\int (\\int e^{\\mathbf{P}} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P}", "derivation": "\\Psi_{\\lambda}{(\\mathbf{P})} = e^{\\mathbf{P}} and \\int \\Psi_{\\lambda}{(\\mathbf{P})} d\\mathbf{P} = \\int e^{\\mathbf{P}} d\\mathbf{P} and (\\int \\Psi_{\\lambda}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} = (\\int e^{\\mathbf{P}} d\\mathbf{P})^{\\mathbf{P}} and \\int (\\int \\Psi_{\\lambda}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P} = \\int (\\int e^{\\mathbf{P}} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P} and A{(\\mathbf{P})} = \\int (\\int \\Psi_{\\lambda}{(\\mathbf{P})} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P} and A{(\\mathbf{P})} = \\int (\\int e^{\\mathbf{P}} d\\mathbf{P})^{\\mathbf{P}} d\\mathbf{P}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), exp(Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["integrate", 3, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Integral(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Integral(Pow(Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], ["renaming_premise", "Equality(Function('A')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('A')(Symbol('\\\\mathbf{P}', commutative=True)), Integral(Pow(Integral(exp(Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True))))"]]}, {"prompt": "Given \\Psi_{\\lambda}{(m_{s},F_{H})} = \\frac{m_{s}}{F_{H}}, then obtain \\frac{\\partial}{\\partial m_{s}} ((\\frac{\\Psi_{\\lambda}{(m_{s},F_{H})}}{m_{s}})^{m_{s}})^{m_{s}} = \\frac{\\partial}{\\partial m_{s}} ((\\frac{1}{F_{H}})^{m_{s}})^{m_{s}}", "derivation": "\\Psi_{\\lambda}{(m_{s},F_{H})} = \\frac{m_{s}}{F_{H}} and \\frac{\\Psi_{\\lambda}{(m_{s},F_{H})}}{m_{s}} = \\frac{1}{F_{H}} and (\\frac{\\Psi_{\\lambda}{(m_{s},F_{H})}}{m_{s}})^{m_{s}} = (\\frac{1}{F_{H}})^{m_{s}} and ((\\frac{\\Psi_{\\lambda}{(m_{s},F_{H})}}{m_{s}})^{m_{s}})^{m_{s}} = ((\\frac{1}{F_{H}})^{m_{s}})^{m_{s}} and \\frac{\\partial}{\\partial m_{s}} ((\\frac{\\Psi_{\\lambda}{(m_{s},F_{H})}}{m_{s}})^{m_{s}})^{m_{s}} = \\frac{\\partial}{\\partial m_{s}} ((\\frac{1}{F_{H}})^{m_{s}})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True), Symbol('F_H', commutative=True)), Mul(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m_s', commutative=True)))"], [["divide", 1, "Symbol('m_s', commutative=True)"], "Equality(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True), Symbol('F_H', commutative=True))), Pow(Symbol('F_H', commutative=True), Integer(-1)))"], [["power", 2, "Symbol('m_s', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True), Symbol('F_H', commutative=True))), Symbol('m_s', commutative=True)), Pow(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m_s', commutative=True)))"], [["power", 3, "Symbol('m_s', commutative=True)"], "Equality(Pow(Pow(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True), Symbol('F_H', commutative=True))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(Pow(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["differentiate", 4, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Pow(Pow(Mul(Pow(Symbol('m_s', commutative=True), Integer(-1)), Function('\\\\Psi_{\\\\lambda}')(Symbol('m_s', commutative=True), Symbol('F_H', commutative=True))), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Pow(Pow(Pow(Symbol('F_H', commutative=True), Integer(-1)), Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\varphi{(\\dot{x})} = \\log{(\\dot{x})}, then obtain (\\mathbf{J}_M{(\\dot{x})} - \\int 1 d\\dot{x}) (\\int \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} d\\dot{x} - 1) = (\\mathbf{J}_M{(\\dot{x})} - \\int 1 d\\dot{x}) (\\int 1 d\\dot{x} - 1)", "derivation": "\\varphi{(\\dot{x})} = \\log{(\\dot{x})} and \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} = 1 and \\int \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} d\\dot{x} = \\int 1 d\\dot{x} and - \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} + \\int \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} d\\dot{x} = - \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} + \\int 1 d\\dot{x} and \\int \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} d\\dot{x} - 1 = \\int 1 d\\dot{x} - 1 and (\\mathbf{J}_M{(\\dot{x})} - \\int 1 d\\dot{x}) (\\int \\frac{\\varphi{(\\dot{x})}}{\\log{(\\dot{x})}} d\\dot{x} - 1) = (\\mathbf{J}_M{(\\dot{x})} - \\int 1 d\\dot{x}) (\\int 1 d\\dot{x} - 1)", "srepr_derivation": [["premise", "Equality(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), log(Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 1, "log(Symbol('\\\\dot{x}', commutative=True))"], "Equality(Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["minus", 3, "Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integral(Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Add(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)), Add(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1)))"], [["times", 5, "Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True)))))"], "Equality(Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))))), Add(Integral(Mul(Function('\\\\varphi')(Symbol('\\\\dot{x}', commutative=True)), Pow(log(Symbol('\\\\dot{x}', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))), Mul(Add(Function('\\\\mathbf{J}_M')(Symbol('\\\\dot{x}', commutative=True)), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))))), Add(Integral(Integer(1), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(J,v_{z})} = \\frac{v_{z}}{J}, then derive \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = - \\frac{v_{z}}{J^{2}}, then obtain \\frac{\\partial}{\\partial J} \\frac{v_{z}}{J} + \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = \\frac{\\partial}{\\partial J} \\frac{v_{z}}{J} - \\frac{\\operatorname{F_{N}}{(J,v_{z})}}{J}", "derivation": "\\operatorname{F_{N}}{(J,v_{z})} = \\frac{v_{z}}{J} and \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = \\frac{\\partial}{\\partial J} \\frac{v_{z}}{J} and \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = - \\frac{v_{z}}{J^{2}} and \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = - \\frac{\\operatorname{F_{N}}{(J,v_{z})}}{J} and \\frac{\\partial}{\\partial J} \\frac{v_{z}}{J} + \\frac{\\partial}{\\partial J} \\operatorname{F_{N}}{(J,v_{z})} = \\frac{\\partial}{\\partial J} \\frac{v_{z}}{J} - \\frac{\\operatorname{F_{N}}{(J,v_{z})}}{J}", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2)), Symbol('v_z', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True))))"], [["add", 4, "Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))), Add(Derivative(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('v_z', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-1)), Function('F_N')(Symbol('J', commutative=True), Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given V{(h)} = \\cos{(h)}, then derive E_{n} - \\int 1 dh - \\int - \\frac{V{(h)}}{\\cos{(h)}} dh - \\int V{(h)} dh = \\int - V{(h)} dh, then obtain E_{n} - \\int (-1) dh - \\int 1 dh - \\int \\cos{(h)} dh = \\int - \\cos{(h)} dh", "derivation": "V{(h)} = \\cos{(h)} and \\frac{V{(h)}}{\\cos{(h)}} = 1 and \\frac{V{(h)}}{\\cos{(h)}} - 1 = 0 and - V{(h)} + \\frac{V{(h)}}{\\cos{(h)}} - 1 = - V{(h)} and \\int (- V{(h)} + \\frac{V{(h)}}{\\cos{(h)}} - 1) dh = \\int - V{(h)} dh and E_{n} - \\int 1 dh - \\int - \\frac{V{(h)}}{\\cos{(h)}} dh - \\int V{(h)} dh = \\int - V{(h)} dh and E_{n} - \\int (-1) dh - \\int 1 dh - \\int \\cos{(h)} dh = \\int - \\cos{(h)} dh", "srepr_derivation": [["premise", "Equality(Function('V')(Symbol('h', commutative=True)), cos(Symbol('h', commutative=True)))"], [["divide", 1, "cos(Symbol('h', commutative=True))"], "Equality(Mul(Function('V')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Mul(Function('V')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Integer(-1)), Integer(0))"], [["minus", 3, "Function('V')(Symbol('h', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('V')(Symbol('h', commutative=True))), Mul(Function('V')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Integer(-1)), Mul(Integer(-1), Function('V')(Symbol('h', commutative=True))))"], [["integrate", 4, "Symbol('h', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Function('V')(Symbol('h', commutative=True))), Mul(Function('V')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Integer(-1)), Tuple(Symbol('h', commutative=True))), Integral(Mul(Integer(-1), Function('V')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('h', commutative=True)))), Mul(Integer(-1), Integral(Mul(Integer(-1), Function('V')(Symbol('h', commutative=True)), Pow(cos(Symbol('h', commutative=True)), Integer(-1))), Tuple(Symbol('h', commutative=True)))), Mul(Integer(-1), Integral(Function('V')(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))), Integral(Mul(Integer(-1), Function('V')(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Symbol('E_n', commutative=True), Mul(Integer(-1), Integral(Integer(-1), Tuple(Symbol('h', commutative=True)))), Mul(Integer(-1), Integral(Integer(1), Tuple(Symbol('h', commutative=True)))), Mul(Integer(-1), Integral(cos(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True))))), Integral(Mul(Integer(-1), cos(Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\pi,t_{2})} = - \\pi + t_{2}, then obtain \\int t_{2} \\operatorname{A_{y}}^{\\pi}{(\\pi,t_{2})} dt_{2} = \\int t_{2} (- \\pi + t_{2})^{\\pi} dt_{2}", "derivation": "\\operatorname{A_{y}}{(\\pi,t_{2})} = - \\pi + t_{2} and \\operatorname{A_{y}}^{\\pi}{(\\pi,t_{2})} = (- \\pi + t_{2})^{\\pi} and t_{2} \\operatorname{A_{y}}^{\\pi}{(\\pi,t_{2})} = t_{2} (- \\pi + t_{2})^{\\pi} and \\int t_{2} \\operatorname{A_{y}}^{\\pi}{(\\pi,t_{2})} dt_{2} = \\int t_{2} (- \\pi + t_{2})^{\\pi} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True)))"], [["times", 2, "Symbol('t_2', commutative=True)"], "Equality(Mul(Symbol('t_2', commutative=True), Pow(Function('A_y')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True))), Mul(Symbol('t_2', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True))))"], [["integrate", 3, "Symbol('t_2', commutative=True)"], "Equality(Integral(Mul(Symbol('t_2', commutative=True), Pow(Function('A_y')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Symbol('t_2', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('\\\\pi', commutative=True)), Symbol('t_2', commutative=True)), Symbol('\\\\pi', commutative=True))), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{v}}}{(b,V)} = \\log{(\\frac{V}{b})} and \\operatorname{E_{\\lambda}}{(b,V)} = - b + \\operatorname{f_{\\mathbf{v}}}{(b,V)}, then obtain \\frac{V \\int \\operatorname{E_{\\lambda}}{(b,V)} dV}{b \\int (- b + \\log{(\\frac{V}{b})}) dV} = \\frac{V}{b}", "derivation": "\\operatorname{f_{\\mathbf{v}}}{(b,V)} = \\log{(\\frac{V}{b})} and - b + \\operatorname{f_{\\mathbf{v}}}{(b,V)} = - b + \\log{(\\frac{V}{b})} and \\operatorname{E_{\\lambda}}{(b,V)} = - b + \\operatorname{f_{\\mathbf{v}}}{(b,V)} and \\operatorname{E_{\\lambda}}{(b,V)} = - b + \\log{(\\frac{V}{b})} and \\int \\operatorname{E_{\\lambda}}{(b,V)} dV = \\int (- b + \\log{(\\frac{V}{b})}) dV and \\frac{\\int \\operatorname{E_{\\lambda}}{(b,V)} dV}{\\int (- b + \\log{(\\frac{V}{b})}) dV} = 1 and \\frac{V \\int \\operatorname{E_{\\lambda}}{(b,V)} dV}{b \\int (- b + \\log{(\\frac{V}{b})}) dV} = \\frac{V}{b}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)))))"], [["minus", 1, "Symbol('b', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('V', commutative=True))), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), Function('f_{\\\\mathbf{v}}')(Symbol('b', commutative=True), Symbol('V', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))))"], [["integrate", 4, "Symbol('V', commutative=True)"], "Equality(Integral(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True))))"], [["divide", 5, "Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True))), Integer(-1)), Integral(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Integer(1))"], [["times", 6, "Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)))"], "Equality(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1)), Pow(Integral(Add(Mul(Integer(-1), Symbol('b', commutative=True)), log(Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))), Tuple(Symbol('V', commutative=True))), Integer(-1)), Integral(Function('E_{\\\\lambda}')(Symbol('b', commutative=True), Symbol('V', commutative=True)), Tuple(Symbol('V', commutative=True)))), Mul(Symbol('V', commutative=True), Pow(Symbol('b', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\hat{X}{(J)} = \\cos{(J)} and \\operatorname{v_{y}}{(J)} = J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}}, then obtain \\operatorname{v_{y}}^{J}{(J)} = (J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}})^{J}", "derivation": "\\hat{X}{(J)} = \\cos{(J)} and 1 = \\frac{\\cos{(J)}}{\\hat{X}{(J)}} and J + 1 = J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}} and (J + 1)^{J} = (J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}})^{J} and \\operatorname{v_{y}}{(J)} = J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}} and (J + 1)^{J} = \\operatorname{v_{y}}^{J}{(J)} and \\operatorname{v_{y}}^{J}{(J)} = (J + \\frac{\\cos{(J)}}{\\hat{X}{(J)}})^{J}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{X}')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["divide", 1, "Function('\\\\hat{X}')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True))))"], [["add", 2, "Symbol('J', commutative=True)"], "Equality(Add(Symbol('J', commutative=True), Integer(1)), Add(Symbol('J', commutative=True), Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True)))))"], [["power", 3, "Symbol('J', commutative=True)"], "Equality(Pow(Add(Symbol('J', commutative=True), Integer(1)), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"], ["renaming_premise", "Equality(Function('v_y')(Symbol('J', commutative=True)), Add(Symbol('J', commutative=True), Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Pow(Add(Symbol('J', commutative=True), Integer(1)), Symbol('J', commutative=True)), Pow(Function('v_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 6], "Equality(Pow(Function('v_y')(Symbol('J', commutative=True)), Symbol('J', commutative=True)), Pow(Add(Symbol('J', commutative=True), Mul(Pow(Function('\\\\hat{X}')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True)))), Symbol('J', commutative=True)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} = C \\Psi_{\\lambda}, then derive \\frac{\\partial^{2}}{\\partial C^{2}} \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} - 1 = -1, then obtain \\frac{\\partial^{2}}{\\partial C^{2}} C \\Psi_{\\lambda} - 1 = -1", "derivation": "\\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} = C \\Psi_{\\lambda} and \\frac{\\partial}{\\partial C} \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} = \\frac{\\partial}{\\partial C} C \\Psi_{\\lambda} and \\frac{\\partial^{2}}{\\partial C^{2}} \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} = \\frac{\\partial^{2}}{\\partial C^{2}} C \\Psi_{\\lambda} and \\frac{\\partial^{2}}{\\partial C^{2}} \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} - 1 = \\frac{\\partial^{2}}{\\partial C^{2}} C \\Psi_{\\lambda} - 1 and \\frac{\\partial^{2}}{\\partial C^{2}} \\operatorname{F_{H}}{(C,\\Psi_{\\lambda})} - 1 = -1 and \\frac{\\partial^{2}}{\\partial C^{2}} C \\Psi_{\\lambda} - 1 = -1", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Mul(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)))"], [["differentiate", 1, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))))"], [["minus", 3, 1], "Equality(Add(Derivative(Function('F_H')(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Integer(-1)), Add(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Integer(-1)))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('F_H')(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Integer(-1)), Integer(-1))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Derivative(Mul(Symbol('C', commutative=True), Symbol('\\\\Psi_{\\\\lambda}', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(2))), Integer(-1)), Integer(-1))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})}, then obtain ((\\frac{\\operatorname{F_{N}}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}})^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} = 1", "derivation": "\\operatorname{F_{N}}{(V_{\\mathbf{E}})} = \\log{(V_{\\mathbf{E}})} and \\frac{\\operatorname{F_{N}}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}} = 1 and (\\frac{\\operatorname{F_{N}}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}})^{V_{\\mathbf{E}}} = 1 and ((\\frac{\\operatorname{F_{N}}{(V_{\\mathbf{E}})}}{\\log{(V_{\\mathbf{E}})}})^{V_{\\mathbf{E}}})^{V_{\\mathbf{E}}} = 1", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), log(Symbol('V_{\\\\mathbf{E}}', commutative=True)))"], [["divide", 1, "log(Symbol('V_{\\\\mathbf{E}}', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Mul(Function('F_N')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1))"], [["power", 3, "Symbol('V_{\\\\mathbf{E}}', commutative=True)"], "Equality(Pow(Pow(Mul(Function('F_N')(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Pow(log(Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(-1))), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Symbol('V_{\\\\mathbf{E}}', commutative=True)), Integer(1))"]]}, {"prompt": "Given k{(J_{\\varepsilon})} = J_{\\varepsilon}, then obtain \\frac{\\partial}{\\partial \\mathbf{H}} (- J_{\\varepsilon} \\mathbf{H} + \\mathbf{H} + k{(J_{\\varepsilon})} - 1) = \\frac{\\partial}{\\partial \\mathbf{H}} (- J_{\\varepsilon} \\mathbf{H} + J_{\\varepsilon} + \\mathbf{H} - 1)", "derivation": "k{(J_{\\varepsilon})} = J_{\\varepsilon} and - J_{\\varepsilon} \\mathbf{H} + k{(J_{\\varepsilon})} = - J_{\\varepsilon} \\mathbf{H} + J_{\\varepsilon} and - J_{\\varepsilon} \\mathbf{H} + k{(J_{\\varepsilon})} - 1 = - J_{\\varepsilon} \\mathbf{H} + J_{\\varepsilon} - 1 and - J_{\\varepsilon} \\mathbf{H} + \\mathbf{H} + k{(J_{\\varepsilon})} - 1 = - J_{\\varepsilon} \\mathbf{H} + J_{\\varepsilon} + \\mathbf{H} - 1 and \\frac{\\partial}{\\partial \\mathbf{H}} (- J_{\\varepsilon} \\mathbf{H} + \\mathbf{H} + k{(J_{\\varepsilon})} - 1) = \\frac{\\partial}{\\partial \\mathbf{H}} (- J_{\\varepsilon} \\mathbf{H} + J_{\\varepsilon} + \\mathbf{H} - 1)", "srepr_derivation": [["renaming_premise", "Equality(Function('k')(Symbol('J_{\\\\varepsilon}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True))"], [["minus", 1, "Mul(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('k')(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True)))"], [["minus", 2, 1], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('k')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)))"], [["add", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Function('k')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)))"], [["differentiate", 4, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True), Function('k')(Symbol('J_{\\\\varepsilon}', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{H}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(p)} = \\int \\cos{(p)} dp, then derive \\frac{\\operatorname{E_{n}}{(p)}}{p} = \\frac{l + \\sin{(p)}}{p}, then derive \\frac{f^{*} + \\sin{(p)}}{p} = \\frac{l + \\sin{(p)}}{p}, then obtain \\frac{f^{*} + \\sin{(p)}}{p \\cos{(p)}} = \\frac{l + \\sin{(p)}}{p \\cos{(p)}}", "derivation": "\\operatorname{E_{n}}{(p)} = \\int \\cos{(p)} dp and \\frac{\\operatorname{E_{n}}{(p)}}{p} = \\frac{\\int \\cos{(p)} dp}{p} and \\frac{\\operatorname{E_{n}}{(p)}}{p} = \\frac{l + \\sin{(p)}}{p} and \\frac{\\int \\cos{(p)} dp}{p} = \\frac{l + \\sin{(p)}}{p} and \\frac{f^{*} + \\sin{(p)}}{p} = \\frac{l + \\sin{(p)}}{p} and \\frac{f^{*} + \\sin{(p)}}{p \\cos{(p)}} = \\frac{l + \\sin{(p)}}{p \\cos{(p)}}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('p', commutative=True)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))))"], [["divide", 1, "Symbol('p', commutative=True)"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('E_n')(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 2], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Function('E_n')(Symbol('p', commutative=True))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('l', commutative=True), sin(Symbol('p', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Integral(cos(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('l', commutative=True), sin(Symbol('p', commutative=True)))))"], [["evaluate_integrals", 4], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), sin(Symbol('p', commutative=True)))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('l', commutative=True), sin(Symbol('p', commutative=True)))))"], [["times", 5, "Pow(cos(Symbol('p', commutative=True)), Integer(-1))"], "Equality(Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('f^*', commutative=True), sin(Symbol('p', commutative=True))), Pow(cos(Symbol('p', commutative=True)), Integer(-1))), Mul(Pow(Symbol('p', commutative=True), Integer(-1)), Add(Symbol('l', commutative=True), sin(Symbol('p', commutative=True))), Pow(cos(Symbol('p', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given I{(z,s)} = s + z, then obtain (2 I^{2}{(z,s)})^{s} + \\int (2 I^{2}{(z,s)})^{s} dz = (2 I^{2}{(z,s)})^{s} + \\int (2 (s + z) I{(z,s)})^{s} dz", "derivation": "I{(z,s)} = s + z and 2 I{(z,s)} = s + z + I{(z,s)} and (s + z + I{(z,s)}) I{(z,s)} = (s + z) (s + z + I{(z,s)}) and 2 I^{2}{(z,s)} = 2 (s + z) I{(z,s)} and (2 I^{2}{(z,s)})^{s} = (2 (s + z) I{(z,s)})^{s} and \\int (2 I^{2}{(z,s)})^{s} dz = \\int (2 (s + z) I{(z,s)})^{s} dz and (2 I^{2}{(z,s)})^{s} + \\int (2 I^{2}{(z,s)})^{s} dz = (2 I^{2}{(z,s)})^{s} + \\int (2 (s + z) I{(z,s)})^{s} dz", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Add(Symbol('s', commutative=True), Symbol('z', commutative=True)))"], [["add", 1, "Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Integer(2), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Add(Symbol('s', commutative=True), Symbol('z', commutative=True), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))))"], [["times", 1, "Add(Symbol('s', commutative=True), Symbol('z', commutative=True), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)))"], "Equality(Mul(Add(Symbol('s', commutative=True), Symbol('z', commutative=True), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Mul(Add(Symbol('s', commutative=True), Symbol('z', commutative=True)), Add(Symbol('s', commutative=True), Symbol('z', commutative=True), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Mul(Integer(2), Add(Symbol('s', commutative=True), Symbol('z', commutative=True)), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))))"], [["power", 4, "Symbol('s', commutative=True)"], "Equality(Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True)), Pow(Mul(Integer(2), Add(Symbol('s', commutative=True), Symbol('z', commutative=True)), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)))"], [["integrate", 5, "Symbol('z', commutative=True)"], "Equality(Integral(Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True)), Tuple(Symbol('z', commutative=True))), Integral(Pow(Mul(Integer(2), Add(Symbol('s', commutative=True), Symbol('z', commutative=True)), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('z', commutative=True))))"], [["add", 6, "Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True))"], "Equality(Add(Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True)), Integral(Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True)), Tuple(Symbol('z', commutative=True)))), Add(Pow(Mul(Integer(2), Pow(Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True)), Integer(2))), Symbol('s', commutative=True)), Integral(Pow(Mul(Integer(2), Add(Symbol('s', commutative=True), Symbol('z', commutative=True)), Function('I')(Symbol('z', commutative=True), Symbol('s', commutative=True))), Symbol('s', commutative=True)), Tuple(Symbol('z', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(Z)} = \\sin{(Z)} and \\operatorname{a^{\\dagger}}{(Z)} = \\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}}, then obtain (\\operatorname{a^{\\dagger}}^{Z}{(Z)})^{Z} = ((\\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}})^{Z})^{Z}", "derivation": "\\theta_{1}{(Z)} = \\sin{(Z)} and \\theta_{1}^{Z}{(Z)} = \\sin^{Z}{(Z)} and \\frac{\\theta_{1}{(Z)} \\theta_{1}^{Z}{(Z)}}{\\sin{(Z)}} = \\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}} and \\operatorname{a^{\\dagger}}{(Z)} = \\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}} and \\operatorname{a^{\\dagger}}{(Z)} = \\frac{\\theta_{1}{(Z)} \\theta_{1}^{Z}{(Z)}}{\\sin{(Z)}} and \\operatorname{a^{\\dagger}}^{Z}{(Z)} = (\\frac{\\theta_{1}{(Z)} \\theta_{1}^{Z}{(Z)}}{\\sin{(Z)}})^{Z} and \\operatorname{a^{\\dagger}}^{Z}{(Z)} = (\\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}})^{Z} and (\\operatorname{a^{\\dagger}}^{Z}{(Z)})^{Z} = ((\\frac{\\theta_{1}{(Z)} \\sin^{Z}{(Z)}}{\\sin{(Z)}})^{Z})^{Z}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('Z', commutative=True)), sin(Symbol('Z', commutative=True)))"], [["power", 1, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"], [["times", 2, "Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)))"], "Equality(Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], ["renaming_premise", "Equality(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True)), Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('Z', commutative=True)"], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1))), Symbol('Z', commutative=True)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Pow(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)))"], [["power", 7, "Symbol('Z', commutative=True)"], "Equality(Pow(Pow(Function('a^{\\\\dagger}')(Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)), Pow(Pow(Mul(Function('\\\\theta_1')(Symbol('Z', commutative=True)), Pow(sin(Symbol('Z', commutative=True)), Integer(-1)), Pow(sin(Symbol('Z', commutative=True)), Symbol('Z', commutative=True))), Symbol('Z', commutative=True)), Symbol('Z', commutative=True)))"]]}, {"prompt": "Given p{(L_{\\varepsilon},C_{2})} = \\frac{C_{2}}{L_{\\varepsilon}}, then derive \\frac{\\partial}{\\partial C_{2}} p{(L_{\\varepsilon},C_{2})} = \\frac{1}{L_{\\varepsilon}}, then obtain \\frac{\\partial}{\\partial C_{2}} p{(L_{\\varepsilon},C_{2})} + 1 = 1 + \\frac{1}{L_{\\varepsilon}}", "derivation": "p{(L_{\\varepsilon},C_{2})} = \\frac{C_{2}}{L_{\\varepsilon}} and \\frac{\\partial}{\\partial C_{2}} p{(L_{\\varepsilon},C_{2})} = \\frac{\\partial}{\\partial C_{2}} \\frac{C_{2}}{L_{\\varepsilon}} and \\frac{\\partial}{\\partial C_{2}} p{(L_{\\varepsilon},C_{2})} = \\frac{1}{L_{\\varepsilon}} and \\frac{\\partial}{\\partial C_{2}} p{(L_{\\varepsilon},C_{2})} + 1 = 1 + \\frac{1}{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('C_2', commutative=True)), Mul(Symbol('C_2', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('C_2', commutative=True)"], "Equality(Derivative(Function('p')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Derivative(Mul(Symbol('C_2', commutative=True), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))), Tuple(Symbol('C_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('p')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1)))"], [["minus", 3, "Integer(-1)"], "Equality(Add(Derivative(Function('p')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('C_2', commutative=True)), Tuple(Symbol('C_2', commutative=True), Integer(1))), Integer(1)), Add(Integer(1), Pow(Symbol('L_{\\\\varepsilon}', commutative=True), Integer(-1))))"]]}, {"prompt": "Given \\Psi{(\\varepsilon,\\mathbb{I})} = - \\mathbb{I} + \\varepsilon, then derive \\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I} = J - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\varepsilon, then obtain \\cos{(\\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I})} = \\cos{(J - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\varepsilon)}", "derivation": "\\Psi{(\\varepsilon,\\mathbb{I})} = - \\mathbb{I} + \\varepsilon and \\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I} = \\int (- \\mathbb{I} + \\varepsilon) d\\mathbb{I} and \\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I} = J - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\varepsilon and \\cos{(\\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I})} = \\cos{(\\int (- \\mathbb{I} + \\varepsilon) d\\mathbb{I})} and \\cos{(J - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\varepsilon)} = \\cos{(\\int (- \\mathbb{I} + \\varepsilon) d\\mathbb{I})} and \\cos{(\\int \\Psi{(\\varepsilon,\\mathbb{I})} d\\mathbb{I})} = \\cos{(J - \\frac{\\mathbb{I}^{2}}{2} + \\mathbb{I} \\varepsilon)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\Psi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True))), Add(Symbol('J', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varepsilon', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\Psi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), cos(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))), cos(Integral(Add(Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)), Symbol('\\\\varepsilon', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(cos(Integral(Function('\\\\Psi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\mathbb{I}', commutative=True)), Tuple(Symbol('\\\\mathbb{I}', commutative=True)))), cos(Add(Symbol('J', commutative=True), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('\\\\mathbb{I}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('\\\\varepsilon', commutative=True)))))"]]}, {"prompt": "Given \\mathbb{I}{(A_{2},\\hbar)} = - A_{2} + \\hbar, then derive 2 - \\frac{\\partial}{\\partial \\hbar} \\mathbb{I}{(A_{2},\\hbar)} = 1, then obtain \\hbar - \\frac{\\partial}{\\partial \\hbar} (- A_{2} + \\hbar) + 2 = \\hbar + 1", "derivation": "\\mathbb{I}{(A_{2},\\hbar)} = - A_{2} + \\hbar and - \\mathbb{I}{(A_{2},\\hbar)} = A_{2} - \\hbar and 0 = A_{2} - \\hbar + \\mathbb{I}{(A_{2},\\hbar)} and - A_{2} + 2 \\hbar - \\mathbb{I}{(A_{2},\\hbar)} = \\hbar and \\frac{\\partial}{\\partial \\hbar} (- A_{2} + 2 \\hbar - \\mathbb{I}{(A_{2},\\hbar)}) = \\frac{d}{d \\hbar} \\hbar and 2 - \\frac{\\partial}{\\partial \\hbar} \\mathbb{I}{(A_{2},\\hbar)} = 1 and 2 - \\frac{\\partial}{\\partial \\hbar} (- A_{2} + \\hbar) = 1 and \\hbar - \\frac{\\partial}{\\partial \\hbar} (- A_{2} + \\hbar) + 2 = \\hbar + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["divide", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Integer(0), Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True))))"], [["minus", 3, "Add(Symbol('A_2', commutative=True), Mul(Integer(-1), Integer(2), Symbol('\\\\hbar', commutative=True)), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))), Symbol('\\\\hbar', commutative=True))"], [["differentiate", 4, "Symbol('\\\\hbar', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)))), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))), Derivative(Symbol('\\\\hbar', commutative=True), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Add(Integer(2), Mul(Integer(-1), Derivative(Function('\\\\mathbb{I}')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), Integer(1))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Add(Integer(2), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1))))), Integer(1))"], [["minus", 7, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Symbol('\\\\hbar', commutative=True), Mul(Integer(-1), Derivative(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\hbar', commutative=True), Integer(1)))), Integer(2)), Add(Symbol('\\\\hbar', commutative=True), Integer(1)))"]]}, {"prompt": "Given J{(\\psi^*)} = \\log{(\\psi^*)}, then derive \\int J{(\\psi^*)} d\\psi^* = E_{x} + \\psi^* \\log{(\\psi^*)} - \\psi^*, then obtain (\\psi^* \\log{(\\psi^*)} + \\int \\log{(\\psi^*)} d\\psi^*)^{\\psi^*} = (E_{x} + \\psi^* J{(\\psi^*)} + \\psi^* \\log{(\\psi^*)} - \\psi^*)^{\\psi^*}", "derivation": "J{(\\psi^*)} = \\log{(\\psi^*)} and \\int J{(\\psi^*)} d\\psi^* = \\int \\log{(\\psi^*)} d\\psi^* and \\int J{(\\psi^*)} d\\psi^* = E_{x} + \\psi^* \\log{(\\psi^*)} - \\psi^* and \\int J{(\\psi^*)} d\\psi^* = E_{x} + \\psi^* J{(\\psi^*)} - \\psi^* and \\int \\log{(\\psi^*)} d\\psi^* = E_{x} + \\psi^* J{(\\psi^*)} - \\psi^* and \\psi^* \\log{(\\psi^*)} + \\int \\log{(\\psi^*)} d\\psi^* = E_{x} + \\psi^* J{(\\psi^*)} + \\psi^* \\log{(\\psi^*)} - \\psi^* and (\\psi^* \\log{(\\psi^*)} + \\int \\log{(\\psi^*)} d\\psi^*)^{\\psi^*} = (E_{x} + \\psi^* J{(\\psi^*)} + \\psi^* \\log{(\\psi^*)} - \\psi^*)^{\\psi^*}", "srepr_derivation": [["get_premise", "Equality(Function('J')(Symbol('\\\\psi^*', commutative=True)), log(Symbol('\\\\psi^*', commutative=True)))"], [["integrate", 1, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Integral(Function('J')(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), Function('J')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), Function('J')(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["add", 5, "Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), Function('J')(Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))))"], [["power", 6, "Symbol('\\\\psi^*', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Integral(log(Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('\\\\psi^*', commutative=True)))), Symbol('\\\\psi^*', commutative=True)), Pow(Add(Symbol('E_x', commutative=True), Mul(Symbol('\\\\psi^*', commutative=True), Function('J')(Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('\\\\psi^*', commutative=True), log(Symbol('\\\\psi^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\psi^*', commutative=True))), Symbol('\\\\psi^*', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(c_{0},\\delta)} = \\delta \\log{(c_{0})} and m{(c_{0})} = \\log{(c_{0})}^{2}, then obtain \\frac{\\partial}{\\partial c_{0}} \\delta \\operatorname{v_{1}}{(c_{0},\\delta)} \\log{(c_{0})} = \\frac{\\partial}{\\partial c_{0}} \\delta^{2} m{(c_{0})}", "derivation": "\\operatorname{v_{1}}{(c_{0},\\delta)} = \\delta \\log{(c_{0})} and \\delta \\operatorname{v_{1}}{(c_{0},\\delta)} \\log{(c_{0})} = \\delta^{2} \\log{(c_{0})}^{2} and m{(c_{0})} = \\log{(c_{0})}^{2} and \\frac{\\partial}{\\partial c_{0}} \\delta \\operatorname{v_{1}}{(c_{0},\\delta)} \\log{(c_{0})} = \\frac{\\partial}{\\partial c_{0}} \\delta^{2} \\log{(c_{0})}^{2} and \\frac{\\partial}{\\partial c_{0}} \\delta \\operatorname{v_{1}}{(c_{0},\\delta)} \\log{(c_{0})} = \\frac{\\partial}{\\partial c_{0}} \\delta^{2} m{(c_{0})}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('c_0', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), log(Symbol('c_0', commutative=True))))"], [["times", 1, "Mul(Symbol('\\\\delta', commutative=True), log(Symbol('c_0', commutative=True)))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Function('v_1')(Symbol('c_0', commutative=True), Symbol('\\\\delta', commutative=True)), log(Symbol('c_0', commutative=True))), Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(log(Symbol('c_0', commutative=True)), Integer(2))))"], ["renaming_premise", "Equality(Function('m')(Symbol('c_0', commutative=True)), Pow(log(Symbol('c_0', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('c_0', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\delta', commutative=True), Function('v_1')(Symbol('c_0', commutative=True), Symbol('\\\\delta', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(log(Symbol('c_0', commutative=True)), Integer(2))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Derivative(Mul(Symbol('\\\\delta', commutative=True), Function('v_1')(Symbol('c_0', commutative=True), Symbol('\\\\delta', commutative=True)), log(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Function('m')(Symbol('c_0', commutative=True))), Tuple(Symbol('c_0', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(\\tilde{g}^*,\\theta)} = \\cos{(\\theta + \\tilde{g}^*)}, then obtain \\theta \\tilde{g}^* = \\tilde{g}^* (\\theta - f{(\\tilde{g}^*,\\theta)} + \\cos{(\\theta + \\tilde{g}^*)})", "derivation": "f{(\\tilde{g}^*,\\theta)} = \\cos{(\\theta + \\tilde{g}^*)} and \\theta + f{(\\tilde{g}^*,\\theta)} = \\theta + \\cos{(\\theta + \\tilde{g}^*)} and \\theta = \\theta - f{(\\tilde{g}^*,\\theta)} + \\cos{(\\theta + \\tilde{g}^*)} and \\theta \\tilde{g}^* = \\tilde{g}^* (\\theta - f{(\\tilde{g}^*,\\theta)} + \\cos{(\\theta + \\tilde{g}^*)})", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta', commutative=True)), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["add", 1, "Symbol('\\\\theta', commutative=True)"], "Equality(Add(Symbol('\\\\theta', commutative=True), Function('f')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta', commutative=True))), Add(Symbol('\\\\theta', commutative=True), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["minus", 2, "Function('f')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta', commutative=True))"], "Equality(Symbol('\\\\theta', commutative=True), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta', commutative=True))), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["times", 3, "Symbol('\\\\tilde{g}^*', commutative=True)"], "Equality(Mul(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Mul(Symbol('\\\\tilde{g}^*', commutative=True), Add(Symbol('\\\\theta', commutative=True), Mul(Integer(-1), Function('f')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\theta', commutative=True))), cos(Add(Symbol('\\\\theta', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))))"]]}, {"prompt": "Given \\hat{H}_{\\lambda}{(J,L)} = J + L, then obtain \\frac{- J - L + \\hat{H}_{\\lambda}{(J,L)}}{L (J + L)} = 0", "derivation": "\\hat{H}_{\\lambda}{(J,L)} = J + L and L \\hat{H}_{\\lambda}{(J,L)} = L (J + L) and - J - L + \\hat{H}_{\\lambda}{(J,L)} = 0 and \\frac{- J - L + \\hat{H}_{\\lambda}{(J,L)}}{L \\hat{H}_{\\lambda}{(J,L)}} = 0 and \\frac{- J - L + \\hat{H}_{\\lambda}{(J,L)}}{L (J + L)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True)), Add(Symbol('J', commutative=True), Symbol('L', commutative=True)))"], [["times", 1, "Symbol('L', commutative=True)"], "Equality(Mul(Symbol('L', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True))), Mul(Symbol('L', commutative=True), Add(Symbol('J', commutative=True), Symbol('L', commutative=True))))"], [["minus", 1, "Add(Symbol('J', commutative=True), Symbol('L', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True))), Integer(0))"], [["divide", 3, "Mul(Symbol('L', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True))), Pow(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True)), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('L', commutative=True), Integer(-1)), Pow(Add(Symbol('J', commutative=True), Symbol('L', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('J', commutative=True)), Mul(Integer(-1), Symbol('L', commutative=True)), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('J', commutative=True), Symbol('L', commutative=True)))), Integer(0))"]]}, {"prompt": "Given \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + z^{*}) and \\operatorname{n_{1}}{(z^{*},\\mathbf{s})} = 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})}, then derive 2 = 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})}, then obtain \\mathbf{s} + z^{*} + 2 = \\mathbf{s} + z^{*} + 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})}", "derivation": "\\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})} = \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + z^{*}) and \\operatorname{n_{1}}{(z^{*},\\mathbf{s})} = 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})} and \\operatorname{n_{1}}{(z^{*},\\mathbf{s})} = 2 \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + z^{*}) and 2 \\frac{\\partial}{\\partial \\mathbf{s}} (\\mathbf{s} + z^{*}) = 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})} and 2 = 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})} and \\mathbf{s} + z^{*} + 2 = \\mathbf{s} + z^{*} + 2 \\hat{\\mathbf{r}}{(z^{*},\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('n_1')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(2), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(2), Derivative(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["evaluate_derivatives", 4], "Equality(Integer(2), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 5, "Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('z^*', commutative=True), Mul(Integer(2), Function('\\\\hat{\\\\mathbf{r}}')(Symbol('z^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given l{(\\varepsilon,\\delta)} = \\delta \\varepsilon, then obtain \\delta \\varepsilon \\int \\frac{\\partial}{\\partial \\delta} \\varepsilon l{(\\varepsilon,\\delta)} d\\delta = \\delta \\varepsilon \\int \\frac{\\partial}{\\partial \\delta} \\delta \\varepsilon^{2} d\\delta", "derivation": "l{(\\varepsilon,\\delta)} = \\delta \\varepsilon and \\varepsilon l{(\\varepsilon,\\delta)} = \\delta \\varepsilon^{2} and \\frac{\\partial}{\\partial \\delta} \\varepsilon l{(\\varepsilon,\\delta)} = \\frac{\\partial}{\\partial \\delta} \\delta \\varepsilon^{2} and \\int \\frac{\\partial}{\\partial \\delta} \\varepsilon l{(\\varepsilon,\\delta)} d\\delta = \\int \\frac{\\partial}{\\partial \\delta} \\delta \\varepsilon^{2} d\\delta and \\delta \\varepsilon \\int \\frac{\\partial}{\\partial \\delta} \\varepsilon l{(\\varepsilon,\\delta)} d\\delta = \\delta \\varepsilon \\int \\frac{\\partial}{\\partial \\delta} \\delta \\varepsilon^{2} d\\delta", "srepr_derivation": [["premise", "Equality(Function('l')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], [["times", 1, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["times", 4, "Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Derivative(Mul(Symbol('\\\\varepsilon', commutative=True), Function('l')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True), Integral(Derivative(Mul(Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given I{(\\mathbf{S})} = e^{\\mathbf{S}}, then obtain \\frac{2 \\mathbf{S} + \\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}}{\\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}} = \\frac{2 \\mathbf{S} + \\int e^{2 \\mathbf{S}} d\\mathbf{S}}{\\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}}", "derivation": "I{(\\mathbf{S})} = e^{\\mathbf{S}} and I{(\\mathbf{S})} e^{\\mathbf{S}} = e^{2 \\mathbf{S}} and \\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S} = \\int e^{2 \\mathbf{S}} d\\mathbf{S} and 2 \\mathbf{S} + \\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S} = 2 \\mathbf{S} + \\int e^{2 \\mathbf{S}} d\\mathbf{S} and \\frac{2 \\mathbf{S} + \\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}}{\\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}} = \\frac{2 \\mathbf{S} + \\int e^{2 \\mathbf{S}} d\\mathbf{S}}{\\int I{(\\mathbf{S})} e^{\\mathbf{S}} d\\mathbf{S}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True)))"], [["times", 1, "exp(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 3, "Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["divide", 4, "Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Pow(Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))), Mul(Add(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True)), Integral(exp(Mul(Integer(2), Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Pow(Integral(Mul(Function('I')(Symbol('\\\\mathbf{S}', commutative=True)), exp(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\mathbf{s}{(\\pi,B)} = \\frac{B}{\\pi}, then obtain \\cos{(\\frac{\\partial}{\\partial \\pi} \\mathbf{s}{(\\pi,B)})} = \\cos{(\\frac{B}{\\pi^{2}})}", "derivation": "\\mathbf{s}{(\\pi,B)} = \\frac{B}{\\pi} and \\frac{\\partial}{\\partial \\pi} \\mathbf{s}{(\\pi,B)} = \\frac{\\partial}{\\partial \\pi} \\frac{B}{\\pi} and \\cos{(\\frac{\\partial}{\\partial \\pi} \\mathbf{s}{(\\pi,B)})} = \\cos{(\\frac{\\partial}{\\partial \\pi} \\frac{B}{\\pi})} and \\cos{(\\frac{\\partial}{\\partial \\pi} \\mathbf{s}{(\\pi,B)})} = \\cos{(\\frac{B}{\\pi^{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('B', commutative=True)), Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["cos", 2], "Equality(cos(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), cos(Derivative(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-1))), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(cos(Derivative(Function('\\\\mathbf{s}')(Symbol('\\\\pi', commutative=True), Symbol('B', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), cos(Mul(Symbol('B', commutative=True), Pow(Symbol('\\\\pi', commutative=True), Integer(-2)))))"]]}, {"prompt": "Given c{(I)} = \\log{(I)}, then obtain I c{(I)} c^{I}{(I)} \\log{(I)}^{- I} = I c^{I}{(I)} \\log{(I)} \\log{(I)}^{- I}", "derivation": "c{(I)} = \\log{(I)} and I c{(I)} = I \\log{(I)} and I c{(I)} \\log{(I)}^{- I} = I \\log{(I)} \\log{(I)}^{- I} and I c{(I)} c^{I}{(I)} \\log{(I)}^{- I} = I c^{I}{(I)} \\log{(I)} \\log{(I)}^{- I}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('I', commutative=True)), log(Symbol('I', commutative=True)))"], [["times", 1, "Symbol('I', commutative=True)"], "Equality(Mul(Symbol('I', commutative=True), Function('c')(Symbol('I', commutative=True))), Mul(Symbol('I', commutative=True), log(Symbol('I', commutative=True))))"], [["divide", 2, "Pow(log(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Mul(Symbol('I', commutative=True), Function('c')(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), log(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["times", 3, "Pow(Function('c')(Symbol('I', commutative=True)), Symbol('I', commutative=True))"], "Equality(Mul(Symbol('I', commutative=True), Function('c')(Symbol('I', commutative=True)), Pow(Function('c')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Symbol('I', commutative=True), Pow(Function('c')(Symbol('I', commutative=True)), Symbol('I', commutative=True)), log(Symbol('I', commutative=True)), Pow(log(Symbol('I', commutative=True)), Mul(Integer(-1), Symbol('I', commutative=True)))))"]]}, {"prompt": "Given \\hat{p}_0{(m)} = \\cos{(m)}, then obtain (- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{- m} (0^{m})^{m} = (- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{- m}", "derivation": "\\hat{p}_0{(m)} = \\cos{(m)} and \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} = 1 and 0 = - \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1 and 0^{m} = (- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{m} and (0^{m})^{m} = ((- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{m})^{m} and ((- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{m})^{m} = 1 and (0^{m})^{m} = 1 and (- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{- m} (0^{m})^{m} = (- \\frac{\\hat{p}_0{(m)}}{\\cos{(m)}} + 1)^{- m}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), cos(Symbol('m', commutative=True)))"], [["divide", 1, "cos(Symbol('m', commutative=True))"], "Equality(Mul(Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, "Mul(Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)))"], [["power", 3, "Symbol('m', commutative=True)"], "Equality(Pow(Integer(0), Symbol('m', commutative=True)), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Symbol('m', commutative=True)))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Symbol('m', commutative=True)), Symbol('m', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Pow(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True)), Integer(1))"], [["divide", 7, "Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Symbol('m', commutative=True))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Mul(Integer(-1), Symbol('m', commutative=True))), Pow(Pow(Integer(0), Symbol('m', commutative=True)), Symbol('m', commutative=True))), Pow(Add(Mul(Integer(-1), Function('\\\\hat{p}_0')(Symbol('m', commutative=True)), Pow(cos(Symbol('m', commutative=True)), Integer(-1))), Integer(1)), Mul(Integer(-1), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\mathbf{p}{(C,z,\\hat{p}_0)} = - C + \\hat{p}_0 + z, then derive \\int \\mathbf{p}{(C,z,\\hat{p}_0)} dC = - \\frac{C^{2}}{2} + C (\\hat{p}_0 + z) + F_{N}, then obtain - \\frac{C^{2}}{2} + C (\\hat{p}_0 + z) + F_{N} - \\sin{(- C + \\hat{p}_0 + z)} = - \\sin{(- C + \\hat{p}_0 + z)} + \\int (- C + \\hat{p}_0 + z) dC", "derivation": "\\mathbf{p}{(C,z,\\hat{p}_0)} = - C + \\hat{p}_0 + z and \\int \\mathbf{p}{(C,z,\\hat{p}_0)} dC = \\int (- C + \\hat{p}_0 + z) dC and \\int \\mathbf{p}{(C,z,\\hat{p}_0)} dC = - \\frac{C^{2}}{2} + C (\\hat{p}_0 + z) + F_{N} and - \\sin{(- C + \\hat{p}_0 + z)} + \\int \\mathbf{p}{(C,z,\\hat{p}_0)} dC = - \\sin{(- C + \\hat{p}_0 + z)} + \\int (- C + \\hat{p}_0 + z) dC and - \\frac{C^{2}}{2} + C (\\hat{p}_0 + z) + F_{N} - \\sin{(- C + \\hat{p}_0 + z)} = - \\sin{(- C + \\hat{p}_0 + z)} + \\int (- C + \\hat{p}_0 + z) dC", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('C', commutative=True), Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)))"], [["integrate", 1, "Symbol('C', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('C', commutative=True), Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('C', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{p}')(Symbol('C', commutative=True), Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Symbol('C', commutative=True), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True))), Symbol('F_N', commutative=True)))"], [["minus", 2, "sin(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)))"], "Equality(Add(Mul(Integer(-1), sin(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)))), Integral(Function('\\\\mathbf{p}')(Symbol('C', commutative=True), Symbol('z', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('C', commutative=True)))), Add(Mul(Integer(-1), sin(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('C', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('C', commutative=True), Integer(2))), Mul(Symbol('C', commutative=True), Add(Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True))), Symbol('F_N', commutative=True), Mul(Integer(-1), sin(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True))))), Add(Mul(Integer(-1), sin(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)))), Integral(Add(Mul(Integer(-1), Symbol('C', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('C', commutative=True)))))"]]}, {"prompt": "Given \\ddot{x}{(\\lambda)} = e^{\\lambda}, then derive \\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} = e^{\\lambda}, then obtain (e^{\\lambda} - \\frac{d}{d \\lambda} e^{\\lambda}) \\ddot{x}{(\\lambda)} e^{\\lambda} = 0", "derivation": "\\ddot{x}{(\\lambda)} = e^{\\lambda} and \\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} = \\frac{d}{d \\lambda} e^{\\lambda} and \\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} - \\frac{d}{d \\lambda} e^{\\lambda} = 0 and (\\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} - \\frac{d}{d \\lambda} e^{\\lambda}) \\ddot{x}{(\\lambda)} e^{\\lambda} = 0 and \\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} = e^{\\lambda} and (\\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} - \\frac{d^{2}}{d \\lambda^{2}} \\ddot{x}{(\\lambda)}) \\ddot{x}{(\\lambda)} \\frac{d}{d \\lambda} \\ddot{x}{(\\lambda)} = 0 and (e^{\\lambda} - \\frac{d}{d \\lambda} e^{\\lambda}) \\ddot{x}{(\\lambda)} e^{\\lambda} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\lambda', commutative=True)"], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))"], [["minus", 2, "Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Integer(0))"], [["times", 3, "Mul(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True)))"], "Equality(Mul(Add(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Integer(0))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), exp(Symbol('\\\\lambda', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Mul(Add(Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(2))))), Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Derivative(Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1)))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Mul(Add(exp(Symbol('\\\\lambda', commutative=True)), Mul(Integer(-1), Derivative(exp(Symbol('\\\\lambda', commutative=True)), Tuple(Symbol('\\\\lambda', commutative=True), Integer(1))))), Function('\\\\ddot{x}')(Symbol('\\\\lambda', commutative=True)), exp(Symbol('\\\\lambda', commutative=True))), Integer(0))"]]}, {"prompt": "Given G{(s,f^{*})} = s \\cos{(f^{*})} and \\mathbf{H}{(v_{t})} = \\cos{(v_{t})}, then obtain G{(s,f^{*})} \\mathbf{H}{(v_{t})} = G{(s,f^{*})} \\cos{(v_{t})}", "derivation": "G{(s,f^{*})} = s \\cos{(f^{*})} and \\mathbf{H}{(v_{t})} = \\cos{(v_{t})} and s \\mathbf{H}{(v_{t})} \\cos{(f^{*})} = s \\cos{(f^{*})} \\cos{(v_{t})} and G{(s,f^{*})} \\mathbf{H}{(v_{t})} = G{(s,f^{*})} \\cos{(v_{t})}", "srepr_derivation": [["premise", "Equality(Function('G')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Mul(Symbol('s', commutative=True), cos(Symbol('f^*', commutative=True))))"], ["get_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('v_t', commutative=True)), cos(Symbol('v_t', commutative=True)))"], [["times", 2, "Mul(Symbol('s', commutative=True), cos(Symbol('f^*', commutative=True)))"], "Equality(Mul(Symbol('s', commutative=True), Function('\\\\mathbf{H}')(Symbol('v_t', commutative=True)), cos(Symbol('f^*', commutative=True))), Mul(Symbol('s', commutative=True), cos(Symbol('f^*', commutative=True)), cos(Symbol('v_t', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Mul(Function('G')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), Function('\\\\mathbf{H}')(Symbol('v_t', commutative=True))), Mul(Function('G')(Symbol('s', commutative=True), Symbol('f^*', commutative=True)), cos(Symbol('v_t', commutative=True))))"]]}, {"prompt": "Given \\sigma_{p}{(A_{2})} = \\cos{(A_{2})}, then obtain (\\sigma_{p}^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\sigma_{p}{(A_{2})} = (\\sigma_{p}^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\cos{(A_{2})}", "derivation": "\\sigma_{p}{(A_{2})} = \\cos{(A_{2})} and \\sigma_{p}^{A_{2}}{(A_{2})} = \\cos^{A_{2}}{(A_{2})} and (\\sigma_{p}^{A_{2}}{(A_{2})})^{A_{2}} = (\\cos^{A_{2}}{(A_{2})})^{A_{2}} and \\frac{d}{d A_{2}} \\sigma_{p}{(A_{2})} = \\frac{d}{d A_{2}} \\cos{(A_{2})} and (\\cos^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\sigma_{p}{(A_{2})} = (\\cos^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\cos{(A_{2})} and (\\sigma_{p}^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\sigma_{p}{(A_{2})} = (\\sigma_{p}^{A_{2}}{(A_{2})})^{A_{2}} \\frac{d}{d A_{2}} \\cos{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), cos(Symbol('A_2', commutative=True)))"], [["power", 1, "Symbol('A_2', commutative=True)"], "Equality(Pow(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["power", 2, "Symbol('A_2', commutative=True)"], "Equality(Pow(Pow(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Pow(Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["times", 4, "Pow(Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True))"], "Equality(Mul(Pow(Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Pow(Pow(cos(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Pow(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))), Mul(Pow(Pow(Function('\\\\sigma_p')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Derivative(cos(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\dot{x},\\hat{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z} and \\operatorname{A_{1}}{(\\hat{x},\\dot{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z} - \\operatorname{f_{E}}{(\\dot{x},\\hat{x},v_{z})}, then obtain \\dot{x} - \\hat{x} + v_{z} + \\operatorname{A_{1}}{(\\hat{x},\\dot{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z}", "derivation": "\\operatorname{f_{E}}{(\\dot{x},\\hat{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z} and \\operatorname{A_{1}}{(\\hat{x},\\dot{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z} - \\operatorname{f_{E}}{(\\dot{x},\\hat{x},v_{z})} and \\operatorname{A_{1}}{(\\hat{x},\\dot{x},v_{z})} = 0 and \\dot{x} - \\hat{x} + v_{z} + \\operatorname{A_{1}}{(\\hat{x},\\dot{x},v_{z})} = \\dot{x} - \\hat{x} + v_{z}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True), Mul(Integer(-1), Function('f_E')(Symbol('\\\\dot{x}', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('v_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True)), Integer(0))"], [["add", 3, "Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True))"], "Equality(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True), Function('A_1')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\dot{x}', commutative=True), Symbol('v_z', commutative=True))), Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{x}', commutative=True)), Symbol('v_z', commutative=True)))"]]}, {"prompt": "Given \\mathbf{M}{(\\nabla,\\mathbf{s})} = \\mathbf{s} + \\nabla, then obtain \\mathbf{s} + \\nabla + \\iint \\mathbf{M}{(\\nabla,\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\mathbf{s} + \\nabla + \\iint (\\mathbf{s} + \\nabla) d\\mathbf{s} d\\mathbf{s}", "derivation": "\\mathbf{M}{(\\nabla,\\mathbf{s})} = \\mathbf{s} + \\nabla and \\int \\mathbf{M}{(\\nabla,\\mathbf{s})} d\\mathbf{s} = \\int (\\mathbf{s} + \\nabla) d\\mathbf{s} and \\iint \\mathbf{M}{(\\nabla,\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\iint (\\mathbf{s} + \\nabla) d\\mathbf{s} d\\mathbf{s} and \\mathbf{s} + \\nabla + \\iint \\mathbf{M}{(\\nabla,\\mathbf{s})} d\\mathbf{s} d\\mathbf{s} = \\mathbf{s} + \\nabla + \\iint (\\mathbf{s} + \\nabla) d\\mathbf{s} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["add", 3, "Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\nabla', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))), Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True), Integral(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\nabla', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} = \\frac{V_{\\mathbf{B}}}{\\mathbf{A}}, then obtain \\int (\\frac{V_{\\mathbf{B}}}{\\mathbf{A}} + \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} + 2)^{V_{\\mathbf{B}}} d\\mathbf{A} = \\int (\\frac{2 V_{\\mathbf{B}}}{\\mathbf{A}} + 2)^{V_{\\mathbf{B}}} d\\mathbf{A}", "derivation": "\\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} = \\frac{V_{\\mathbf{B}}}{\\mathbf{A}} and \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} + 1 = \\frac{V_{\\mathbf{B}}}{\\mathbf{A}} + 1 and \\frac{V_{\\mathbf{B}}}{\\mathbf{A}} + \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} + 2 = \\frac{2 V_{\\mathbf{B}}}{\\mathbf{A}} + 2 and (\\frac{V_{\\mathbf{B}}}{\\mathbf{A}} + \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} + 2)^{V_{\\mathbf{B}}} = (\\frac{2 V_{\\mathbf{B}}}{\\mathbf{A}} + 2)^{V_{\\mathbf{B}}} and \\int (\\frac{V_{\\mathbf{B}}}{\\mathbf{A}} + \\Psi_{nl}{(V_{\\mathbf{B}},\\mathbf{A})} + 2)^{V_{\\mathbf{B}}} d\\mathbf{A} = \\int (\\frac{2 V_{\\mathbf{B}}}{\\mathbf{A}} + 2)^{V_{\\mathbf{B}}} d\\mathbf{A}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(1)), Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Integer(1)))"], [["add", 2, "Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Integer(1))"], "Equality(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Integer(2)))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Pow(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Integer(2)), Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["integrate", 4, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Integral(Pow(Add(Mul(Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Function('\\\\Psi_{nl}')(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('\\\\mathbf{A}', commutative=True)), Integer(2)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))), Integral(Pow(Add(Mul(Integer(2), Symbol('V_{\\\\mathbf{B}}', commutative=True), Pow(Symbol('\\\\mathbf{A}', commutative=True), Integer(-1))), Integer(2)), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{g}{(z,A)} = z^{A}, then derive - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} \\mathbf{g}{(z,A)} = \\frac{A z^{A}}{z} - \\mathbf{g}{(z,A)}, then obtain \\frac{A z^{A}}{z} + 1 = \\frac{\\partial}{\\partial z} z^{A} + 1", "derivation": "\\mathbf{g}{(z,A)} = z^{A} and \\frac{\\partial}{\\partial z} \\mathbf{g}{(z,A)} = \\frac{\\partial}{\\partial z} z^{A} and - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} \\mathbf{g}{(z,A)} = - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} z^{A} and - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} \\mathbf{g}{(z,A)} = \\frac{A z^{A}}{z} - \\mathbf{g}{(z,A)} and \\frac{A z^{A}}{z} - \\mathbf{g}{(z,A)} = - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} z^{A} and \\frac{A z^{A}}{z} - \\mathbf{g}{(z,A)} + 1 = - \\mathbf{g}{(z,A)} + \\frac{\\partial}{\\partial z} z^{A} + 1 and \\frac{A z^{A}}{z} + 1 = \\frac{\\partial}{\\partial z} z^{A} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)), Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)))"], [["differentiate", 1, "Symbol('z', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["minus", 2, "Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Derivative(Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Derivative(Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Add(Mul(Symbol('A', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Derivative(Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["minus", 5, "Integer(-1)"], "Equality(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('A', commutative=True))), Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True))), Derivative(Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)))"], [["minus", 6, "Mul(Integer(-1), Function('\\\\mathbf{g}')(Symbol('z', commutative=True), Symbol('A', commutative=True)))"], "Equality(Add(Mul(Symbol('A', commutative=True), Pow(Symbol('z', commutative=True), Integer(-1)), Pow(Symbol('z', commutative=True), Symbol('A', commutative=True))), Integer(1)), Add(Derivative(Pow(Symbol('z', commutative=True), Symbol('A', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1))), Integer(1)))"]]}, {"prompt": "Given \\mathbf{M}{(r,f_{E})} = f_{E} - r, then obtain 0 = \\int (f_{E} - r)^{r} dr - \\int \\mathbf{M}^{r}{(r,f_{E})} dr", "derivation": "\\mathbf{M}{(r,f_{E})} = f_{E} - r and \\mathbf{M}^{r}{(r,f_{E})} = (f_{E} - r)^{r} and \\int \\mathbf{M}^{r}{(r,f_{E})} dr = \\int (f_{E} - r)^{r} dr and \\log{(\\frac{\\partial}{\\partial f_{E}} \\mathbf{M}^{r}{(r,f_{E})})} + \\int \\mathbf{M}^{r}{(r,f_{E})} dr = \\log{(\\frac{\\partial}{\\partial f_{E}} \\mathbf{M}^{r}{(r,f_{E})})} + \\int (f_{E} - r)^{r} dr and 0 = \\int (f_{E} - r)^{r} dr - \\int \\mathbf{M}^{r}{(r,f_{E})} dr", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))))"], [["power", 1, "Symbol('r', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Symbol('r', commutative=True)))"], [["integrate", 2, "Symbol('r', commutative=True)"], "Equality(Integral(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Integral(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], [["add", 3, "log(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1))))"], "Equality(Add(log(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integral(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))), Add(log(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integral(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True)))))"], [["minus", 4, "Add(log(Derivative(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('f_E', commutative=True), Integer(1)))), Integral(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))"], "Equality(Integer(0), Add(Integral(Pow(Add(Symbol('f_E', commutative=True), Mul(Integer(-1), Symbol('r', commutative=True))), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))), Mul(Integer(-1), Integral(Pow(Function('\\\\mathbf{M}')(Symbol('r', commutative=True), Symbol('f_E', commutative=True)), Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True))))))"]]}, {"prompt": "Given L{(\\pi,t_{1})} = e^{\\pi t_{1}}, then obtain - \\int L{(\\pi,t_{1})} d\\pi + \\iint L{(\\pi,t_{1})} d\\pi dt_{1} = - \\int L{(\\pi,t_{1})} d\\pi + \\iint e^{\\pi t_{1}} d\\pi dt_{1}", "derivation": "L{(\\pi,t_{1})} = e^{\\pi t_{1}} and \\int L{(\\pi,t_{1})} d\\pi = \\int e^{\\pi t_{1}} d\\pi and \\iint L{(\\pi,t_{1})} d\\pi dt_{1} = \\iint e^{\\pi t_{1}} d\\pi dt_{1} and - \\int L{(\\pi,t_{1})} d\\pi + \\iint L{(\\pi,t_{1})} d\\pi dt_{1} = - \\int L{(\\pi,t_{1})} d\\pi + \\iint e^{\\pi t_{1}} d\\pi dt_{1}", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True))))"], [["integrate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True))), Integral(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True))))"], [["integrate", 2, "Symbol('t_1', commutative=True)"], "Equality(Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('t_1', commutative=True))), Integral(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('t_1', commutative=True))))"], [["minus", 3, "Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('t_1', commutative=True)))), Add(Mul(Integer(-1), Integral(Function('L')(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True)))), Integral(exp(Mul(Symbol('\\\\pi', commutative=True), Symbol('t_1', commutative=True))), Tuple(Symbol('\\\\pi', commutative=True)), Tuple(Symbol('t_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(t)} = \\int \\log{(t)} dt, then derive \\operatorname{f^{\\prime}}{(t)} = \\mathbf{J} + t \\log{(t)} - t, then obtain \\frac{\\operatorname{f^{\\prime}}^{t}{(t)}}{\\mathbf{J} (- \\sigma_p + \\dot{y}{(\\sigma_p)})} = \\frac{(\\int \\log{(t)} dt)^{t}}{\\mathbf{J} (- \\sigma_p + \\dot{y}{(\\sigma_p)})}", "derivation": "\\operatorname{f^{\\prime}}{(t)} = \\int \\log{(t)} dt and \\operatorname{f^{\\prime}}{(t)} = \\mathbf{J} + t \\log{(t)} - t and \\operatorname{f^{\\prime}}^{t}{(t)} = (\\mathbf{J} + t \\log{(t)} - t)^{t} and (\\int \\log{(t)} dt)^{t} = (\\mathbf{J} + t \\log{(t)} - t)^{t} and \\operatorname{f^{\\prime}}^{t}{(t)} = (\\int \\log{(t)} dt)^{t} and \\frac{\\operatorname{f^{\\prime}}^{t}{(t)}}{\\mathbf{J}} = \\frac{(\\int \\log{(t)} dt)^{t}}{\\mathbf{J}} and \\frac{\\operatorname{f^{\\prime}}^{t}{(t)}}{\\mathbf{J} (- \\sigma_p + \\dot{y}{(\\sigma_p)})} = \\frac{(\\int \\log{(t)} dt)^{t}}{\\mathbf{J} (- \\sigma_p + \\dot{y}{(\\sigma_p)})}", "srepr_derivation": [["get_premise", "Equality(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["power", 2, "Symbol('t', commutative=True)"], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Pow(Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Mul(Symbol('t', commutative=True), log(Symbol('t', commutative=True))), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Pow(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["divide", 5, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], [["divide", 6, "Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('\\\\sigma_p', commutative=True)), Function('\\\\dot{y}')(Symbol('\\\\sigma_p', commutative=True))), Integer(-1)), Pow(Integral(log(Symbol('t', commutative=True)), Tuple(Symbol('t', commutative=True))), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})} = \\log{(- \\hat{\\mathbf{x}} + \\mathbf{S})}, then obtain \\hat{\\mathbf{x}} + \\log{(e^{\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})}})} = \\hat{\\mathbf{x}} + \\log{(- \\hat{\\mathbf{x}} + \\mathbf{S})}", "derivation": "\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})} = \\log{(- \\hat{\\mathbf{x}} + \\mathbf{S})} and e^{\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})}} = - \\hat{\\mathbf{x}} + \\mathbf{S} and \\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})} = \\log{(e^{\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})}})} and \\log{(e^{\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})}})} = \\log{(- \\hat{\\mathbf{x}} + \\mathbf{S})} and \\hat{\\mathbf{x}} + \\log{(e^{\\operatorname{f_{\\mathbf{p}}}{(\\hat{\\mathbf{x}},\\mathbf{S})}})} = \\hat{\\mathbf{x}} + \\log{(- \\hat{\\mathbf{x}} + \\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["exp", 1], "Equality(exp(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)), log(exp(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(log(exp(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), log(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True))))"], [["add", 4, "Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)"], "Equality(Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(exp(Function('f_{\\\\mathbf{p}}')(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))), Add(Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\hat{\\\\mathbf{x}}', commutative=True)), Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(E,H)} = \\frac{\\sin{(H)}}{E}, then obtain (\\frac{\\operatorname{V_{\\mathbf{E}}}{(E,H)}}{\\sin{(\\frac{\\partial}{\\partial H} \\frac{\\sin{(H)}}{E})}})^{E} = (\\frac{\\sin{(H)}}{E \\sin{(\\frac{\\partial}{\\partial H} \\frac{\\sin{(H)}}{E})}})^{E}", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(E,H)} = \\frac{\\sin{(H)}}{E} and \\frac{\\partial}{\\partial H} \\operatorname{V_{\\mathbf{E}}}{(E,H)} = \\frac{\\partial}{\\partial H} \\frac{\\sin{(H)}}{E} and \\frac{\\operatorname{V_{\\mathbf{E}}}{(E,H)}}{\\sin{(\\frac{\\partial}{\\partial H} \\operatorname{V_{\\mathbf{E}}}{(E,H)})}} = \\frac{\\sin{(H)}}{E \\sin{(\\frac{\\partial}{\\partial H} \\operatorname{V_{\\mathbf{E}}}{(E,H)})}} and (\\frac{\\operatorname{V_{\\mathbf{E}}}{(E,H)}}{\\sin{(\\frac{\\partial}{\\partial H} \\operatorname{V_{\\mathbf{E}}}{(E,H)})}})^{E} = (\\frac{\\sin{(H)}}{E \\sin{(\\frac{\\partial}{\\partial H} \\operatorname{V_{\\mathbf{E}}}{(E,H)})}})^{E} and (\\frac{\\operatorname{V_{\\mathbf{E}}}{(E,H)}}{\\sin{(\\frac{\\partial}{\\partial H} \\frac{\\sin{(H)}}{E})}})^{E} = (\\frac{\\sin{(H)}}{E \\sin{(\\frac{\\partial}{\\partial H} \\frac{\\sin{(H)}}{E})}})^{E}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["divide", 1, "sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], "Equality(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Pow(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))), Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True)), Pow(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))))"], [["power", 3, "Symbol('E', commutative=True)"], "Equality(Pow(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Pow(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))), Symbol('E', commutative=True)), Pow(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True)), Pow(sin(Derivative(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))), Symbol('E', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Mul(Function('V_{\\\\mathbf{E}}')(Symbol('E', commutative=True), Symbol('H', commutative=True)), Pow(sin(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))), Symbol('E', commutative=True)), Pow(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True)), Pow(sin(Derivative(Mul(Pow(Symbol('E', commutative=True), Integer(-1)), sin(Symbol('H', commutative=True))), Tuple(Symbol('H', commutative=True), Integer(1)))), Integer(-1))), Symbol('E', commutative=True)))"]]}, {"prompt": "Given \\theta_{2}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}}, then obtain - A_{y}^{J_{\\varepsilon}} + e^{\\theta_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})}} = - A_{y}^{J_{\\varepsilon}} + e^{(e^{\\sin{(g_{\\varepsilon})}})^{g_{\\varepsilon}}}", "derivation": "\\theta_{2}{(g_{\\varepsilon})} = e^{\\sin{(g_{\\varepsilon})}} and \\theta_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})} = (e^{\\sin{(g_{\\varepsilon})}})^{g_{\\varepsilon}} and e^{\\theta_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})}} = e^{(e^{\\sin{(g_{\\varepsilon})}})^{g_{\\varepsilon}}} and - A_{y}^{J_{\\varepsilon}} + e^{\\theta_{2}^{g_{\\varepsilon}}{(g_{\\varepsilon})}} = - A_{y}^{J_{\\varepsilon}} + e^{(e^{\\sin{(g_{\\varepsilon})}})^{g_{\\varepsilon}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["power", 1, "Symbol('g_{\\\\varepsilon}', commutative=True)"], "Equality(Pow(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)), Pow(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))"], [["exp", 2], "Equality(exp(Pow(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True))), exp(Pow(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True))))"], [["minus", 3, "Pow(Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), exp(Pow(Function('\\\\theta_2')(Symbol('g_{\\\\varepsilon}', commutative=True)), Symbol('g_{\\\\varepsilon}', commutative=True)))), Add(Mul(Integer(-1), Pow(Symbol('A_y', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True))), exp(Pow(exp(sin(Symbol('g_{\\\\varepsilon}', commutative=True))), Symbol('g_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given J{(i,\\varphi^*)} = \\varphi^* i, then obtain (\\varphi^*)^{2} i^{2} J^{2}{(i,\\varphi^*)} - (\\varphi^*)^{2} i^{2} = (\\varphi^*)^{4} i^{4} - (\\varphi^*)^{2} i^{2}", "derivation": "J{(i,\\varphi^*)} = \\varphi^* i and \\varphi^* i J{(i,\\varphi^*)} = (\\varphi^*)^{2} i^{2} and (\\varphi^*)^{2} i^{2} J^{2}{(i,\\varphi^*)} = (\\varphi^*)^{4} i^{4} and \\varphi^* i J^{3}{(i,\\varphi^*)} = (\\varphi^*)^{2} i^{2} J^{2}{(i,\\varphi^*)} and \\varphi^* i J^{3}{(i,\\varphi^*)} = (\\varphi^*)^{4} i^{4} and - (\\varphi^*)^{2} i^{2} + \\varphi^* i J^{3}{(i,\\varphi^*)} = (\\varphi^*)^{4} i^{4} - (\\varphi^*)^{2} i^{2} and (\\varphi^*)^{2} i^{2} J^{2}{(i,\\varphi^*)} - (\\varphi^*)^{2} i^{2} = (\\varphi^*)^{4} i^{4} - (\\varphi^*)^{2} i^{2}", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True)))"], [["times", 1, "Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True))"], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True), Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))))"], [["power", 2, 2], "Equality(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(3))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(3))), Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4))))"], [["minus", 5, "Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)))"], "Equality(Add(Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2))), Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('i', commutative=True), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(3)))), Add(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)), Pow(Function('J')(Symbol('i', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Integer(2))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)))), Add(Mul(Pow(Symbol('\\\\varphi^*', commutative=True), Integer(4)), Pow(Symbol('i', commutative=True), Integer(4))), Mul(Integer(-1), Pow(Symbol('\\\\varphi^*', commutative=True), Integer(2)), Pow(Symbol('i', commutative=True), Integer(2)))))"]]}, {"prompt": "Given \\mu_{0}{(c_{0})} = \\sin{(c_{0})}, then derive 0 = \\rho_b - \\cos{(c_{0})} - \\int \\mu_{0}{(c_{0})} dc_{0}, then obtain \\rho_b = 2 \\rho_b - \\cos{(c_{0})} - \\int \\sin{(c_{0})} dc_{0}", "derivation": "\\mu_{0}{(c_{0})} = \\sin{(c_{0})} and \\int \\mu_{0}{(c_{0})} dc_{0} = \\int \\sin{(c_{0})} dc_{0} and 0 = - \\int \\mu_{0}{(c_{0})} dc_{0} + \\int \\sin{(c_{0})} dc_{0} and 0 = \\rho_b - \\cos{(c_{0})} - \\int \\mu_{0}{(c_{0})} dc_{0} and 0 = \\rho_b - \\cos{(c_{0})} - \\int \\sin{(c_{0})} dc_{0} and \\rho_b = 2 \\rho_b - \\cos{(c_{0})} - \\int \\sin{(c_{0})} dc_{0}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('c_0', commutative=True)), sin(Symbol('c_0', commutative=True)))"], [["integrate", 1, "Symbol('c_0', commutative=True)"], "Equality(Integral(Function('\\\\mu_0')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\mu_0')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))"], "Equality(Integer(0), Add(Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Integer(0), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\mu_0')(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(0), Add(Symbol('\\\\rho_b', commutative=True), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))))"], [["add", 5, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Symbol('\\\\rho_b', commutative=True), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True)), Mul(Integer(-1), cos(Symbol('c_0', commutative=True))), Mul(Integer(-1), Integral(sin(Symbol('c_0', commutative=True)), Tuple(Symbol('c_0', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\hat{x},\\rho)} = \\rho \\sin{(\\hat{x})} and L{(\\hat{x})} = \\sin{(\\hat{x})}, then obtain \\frac{L{(\\hat{x})} \\sin{(\\hat{x})}}{2 \\rho} = \\frac{\\sin^{2}{(\\hat{x})}}{2 \\rho}", "derivation": "\\operatorname{F_{H}}{(\\hat{x},\\rho)} = \\rho \\sin{(\\hat{x})} and \\rho \\sin{(\\hat{x})} + \\operatorname{F_{H}}{(\\hat{x},\\rho)} = 2 \\rho \\sin{(\\hat{x})} and L{(\\hat{x})} = \\sin{(\\hat{x})} and \\frac{L{(\\hat{x})}}{\\rho \\sin{(\\hat{x})} + \\operatorname{F_{H}}{(\\hat{x},\\rho)}} = \\frac{\\sin{(\\hat{x})}}{\\rho \\sin{(\\hat{x})} + \\operatorname{F_{H}}{(\\hat{x},\\rho)}} and \\frac{L{(\\hat{x})}}{2 \\rho \\sin{(\\hat{x})}} = \\frac{1}{2 \\rho} and \\frac{L{(\\hat{x})} \\sin{(\\hat{x})}}{2 \\rho} = \\frac{\\sin^{2}{(\\hat{x})}}{2 \\rho}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["add", 1, "Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True)))"], "Equality(Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Function('F_H')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho', commutative=True))), Mul(Integer(2), Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('L')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True)))"], [["divide", 3, "Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Function('F_H')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Function('F_H')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1)), Function('L')(Symbol('\\\\hat{x}', commutative=True))), Mul(Pow(Add(Mul(Symbol('\\\\rho', commutative=True), sin(Symbol('\\\\hat{x}', commutative=True))), Function('F_H')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho', commutative=True))), Integer(-1)), sin(Symbol('\\\\hat{x}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\hat{x}', commutative=True)), Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Integer(-1))), Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(-1))))"], [["times", 5, "Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Integer(2))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Function('L')(Symbol('\\\\hat{x}', commutative=True)), sin(Symbol('\\\\hat{x}', commutative=True))), Mul(Rational(1, 2), Pow(Symbol('\\\\rho', commutative=True), Integer(-1)), Pow(sin(Symbol('\\\\hat{x}', commutative=True)), Integer(2))))"]]}, {"prompt": "Given y{(C_{1},M)} = \\int (C_{1} + M)^{C_{1}} dM and \\operatorname{F_{g}}{(C_{1},M)} = \\int (C_{1} + M)^{C_{1}} dM, then obtain - C_{1} - M + \\operatorname{F_{g}}{(C_{1},M)} + \\int (C_{1} + M) dC_{1} = - C_{1} - M + \\int (C_{1} + M) dC_{1} + \\int (C_{1} + M)^{C_{1}} dM", "derivation": "y{(C_{1},M)} = \\int (C_{1} + M)^{C_{1}} dM and \\operatorname{F_{g}}{(C_{1},M)} = \\int (C_{1} + M)^{C_{1}} dM and y{(C_{1},M)} + \\int (C_{1} + M) dC_{1} = \\int (C_{1} + M) dC_{1} + \\int (C_{1} + M)^{C_{1}} dM and \\operatorname{F_{g}}{(C_{1},M)} = y{(C_{1},M)} and \\operatorname{F_{g}}{(C_{1},M)} + \\int (C_{1} + M) dC_{1} = \\int (C_{1} + M) dC_{1} + \\int (C_{1} + M)^{C_{1}} dM and - C_{1} - M + \\operatorname{F_{g}}{(C_{1},M)} + \\int (C_{1} + M) dC_{1} = - C_{1} - M + \\int (C_{1} + M) dC_{1} + \\int (C_{1} + M)^{C_{1}} dM", "srepr_derivation": [["renaming_premise", "Equality(Function('y')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Integral(Pow(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('M', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Integral(Pow(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('M', commutative=True))))"], [["add", 1, "Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True)))"], "Equality(Add(Function('y')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('F_g')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Function('y')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Function('F_g')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('M', commutative=True)))))"], [["minus", 5, "Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)), Function('F_g')(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True)))), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Mul(Integer(-1), Symbol('M', commutative=True)), Integral(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Tuple(Symbol('C_1', commutative=True))), Integral(Pow(Add(Symbol('C_1', commutative=True), Symbol('M', commutative=True)), Symbol('C_1', commutative=True)), Tuple(Symbol('M', commutative=True)))))"]]}, {"prompt": "Given \\pi{(F_{N})} = \\sin{(F_{N})}, then derive \\frac{d}{d F_{N}} \\pi{(F_{N})} = \\cos{(F_{N})}, then derive \\varphi^* + \\sin{(F_{N})} = \\Psi + \\sin{(F_{N})}, then obtain \\varphi^* + \\pi{(F_{N})} - \\cos{(F_{N})} = \\Psi + \\pi{(F_{N})} - \\cos{(F_{N})}", "derivation": "\\pi{(F_{N})} = \\sin{(F_{N})} and \\frac{d}{d F_{N}} \\pi{(F_{N})} = \\frac{d}{d F_{N}} \\sin{(F_{N})} and \\frac{d}{d F_{N}} \\pi{(F_{N})} = \\cos{(F_{N})} and \\frac{d}{d F_{N}} \\sin{(F_{N})} = \\cos{(F_{N})} and \\int \\frac{d}{d F_{N}} \\sin{(F_{N})} dF_{N} = \\int \\cos{(F_{N})} dF_{N} and \\varphi^* + \\sin{(F_{N})} = \\Psi + \\sin{(F_{N})} and \\varphi^* + \\pi{(F_{N})} = \\Psi + \\pi{(F_{N})} and \\varphi^* + \\pi{(F_{N})} - \\cos{(F_{N})} = \\Psi + \\pi{(F_{N})} - \\cos{(F_{N})}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('F_N', commutative=True)), sin(Symbol('F_N', commutative=True)))"], [["differentiate", 1, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Function('\\\\pi')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\pi')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), cos(Symbol('F_N', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), cos(Symbol('F_N', commutative=True)))"], [["integrate", 4, "Symbol('F_N', commutative=True)"], "Equality(Integral(Derivative(sin(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Tuple(Symbol('F_N', commutative=True))), Integral(cos(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), sin(Symbol('F_N', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), sin(Symbol('F_N', commutative=True))))"], [["substitute_RHS_for_LHS", 6, 1], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\pi')(Symbol('F_N', commutative=True))), Add(Symbol('\\\\Psi', commutative=True), Function('\\\\pi')(Symbol('F_N', commutative=True))))"], [["minus", 7, "cos(Symbol('F_N', commutative=True))"], "Equality(Add(Symbol('\\\\varphi^*', commutative=True), Function('\\\\pi')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))), Add(Symbol('\\\\Psi', commutative=True), Function('\\\\pi')(Symbol('F_N', commutative=True)), Mul(Integer(-1), cos(Symbol('F_N', commutative=True)))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{s})} = \\cos{(\\mathbf{s})}, then derive \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} = - \\sin{(\\mathbf{s})}, then obtain \\sin{(\\mathbf{s})} + \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} = 0", "derivation": "\\theta_{1}{(\\mathbf{s})} = \\cos{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} = \\frac{d}{d \\mathbf{s}} \\cos{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} = - \\sin{(\\mathbf{s})} and \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\cos{(\\mathbf{s})} = - \\sin{(\\mathbf{s})} - \\frac{d}{d \\mathbf{s}} \\cos{(\\mathbf{s})} and \\sin{(\\mathbf{s})} + \\frac{d}{d \\mathbf{s}} \\theta_{1}{(\\mathbf{s})} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True)), cos(Symbol('\\\\mathbf{s}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["minus", 3, "Derivative(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Integer(-1), Derivative(cos(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(sin(Symbol('\\\\mathbf{s}', commutative=True)), Derivative(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\mu{(\\dot{x},n)} = \\frac{\\dot{x}}{n}, then obtain \\mu^{\\dot{x}}{(\\dot{x},n)} \\sin{(\\frac{\\dot{x}}{n})} - \\mu^{\\dot{x}}{(\\dot{x},n)} = (\\frac{\\dot{x}}{n})^{\\dot{x}} \\sin{(\\frac{\\dot{x}}{n})} - \\mu^{\\dot{x}}{(\\dot{x},n)}", "derivation": "\\mu{(\\dot{x},n)} = \\frac{\\dot{x}}{n} and \\sin{(\\mu{(\\dot{x},n)})} = \\sin{(\\frac{\\dot{x}}{n})} and \\mu^{\\dot{x}}{(\\dot{x},n)} = (\\frac{\\dot{x}}{n})^{\\dot{x}} and \\mu^{\\dot{x}}{(\\dot{x},n)} \\sin{(\\mu{(\\dot{x},n)})} = (\\frac{\\dot{x}}{n})^{\\dot{x}} \\sin{(\\mu{(\\dot{x},n)})} and \\mu^{\\dot{x}}{(\\dot{x},n)} \\sin{(\\frac{\\dot{x}}{n})} = (\\frac{\\dot{x}}{n})^{\\dot{x}} \\sin{(\\frac{\\dot{x}}{n})} and \\mu^{\\dot{x}}{(\\dot{x},n)} \\sin{(\\frac{\\dot{x}}{n})} - \\mu^{\\dot{x}}{(\\dot{x},n)} = (\\frac{\\dot{x}}{n})^{\\dot{x}} \\sin{(\\frac{\\dot{x}}{n})} - \\mu^{\\dot{x}}{(\\dot{x},n)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))"], [["sin", 1], "Equality(sin(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True))), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1)))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)))"], [["times", 3, "sin(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), sin(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)))), Mul(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), sin(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))), Mul(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))))"], [["minus", 5, "Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Add(Mul(Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))), Add(Mul(Pow(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))), Symbol('\\\\dot{x}', commutative=True)), sin(Mul(Symbol('\\\\dot{x}', commutative=True), Pow(Symbol('n', commutative=True), Integer(-1))))), Mul(Integer(-1), Pow(Function('\\\\mu')(Symbol('\\\\dot{x}', commutative=True), Symbol('n', commutative=True)), Symbol('\\\\dot{x}', commutative=True)))))"]]}, {"prompt": "Given r{(C_{1})} = \\cos{(C_{1})} and \\rho_{f}{(C_{1})} = (\\frac{d}{d C_{1}} (r{(C_{1})} - 1))^{C_{1}}, then obtain \\rho_{f}{(C_{1})} = (- \\sin{(C_{1})})^{C_{1}}", "derivation": "r{(C_{1})} = \\cos{(C_{1})} and r{(C_{1})} - 1 = \\cos{(C_{1})} - 1 and \\frac{d}{d C_{1}} (r{(C_{1})} - 1) = \\frac{d}{d C_{1}} (\\cos{(C_{1})} - 1) and (\\frac{d}{d C_{1}} (r{(C_{1})} - 1))^{C_{1}} = (\\frac{d}{d C_{1}} (\\cos{(C_{1})} - 1))^{C_{1}} and \\rho_{f}{(C_{1})} = (\\frac{d}{d C_{1}} (r{(C_{1})} - 1))^{C_{1}} and \\rho_{f}{(C_{1})} = (\\frac{d}{d C_{1}} (\\cos{(C_{1})} - 1))^{C_{1}} and \\rho_{f}{(C_{1})} = (- \\sin{(C_{1})})^{C_{1}}", "srepr_derivation": [["premise", "Equality(Function('r')(Symbol('C_1', commutative=True)), cos(Symbol('C_1', commutative=True)))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('r')(Symbol('C_1', commutative=True)), Integer(-1)), Add(cos(Symbol('C_1', commutative=True)), Integer(-1)))"], [["differentiate", 2, "Symbol('C_1', commutative=True)"], "Equality(Derivative(Add(Function('r')(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Derivative(Add(cos(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))))"], [["power", 3, "Symbol('C_1', commutative=True)"], "Equality(Pow(Derivative(Add(Function('r')(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)), Pow(Derivative(Add(cos(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Pow(Derivative(Add(Function('r')(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Pow(Derivative(Add(cos(Symbol('C_1', commutative=True)), Integer(-1)), Tuple(Symbol('C_1', commutative=True), Integer(1))), Symbol('C_1', commutative=True)))"], [["evaluate_derivatives", 6], "Equality(Function('\\\\rho_f')(Symbol('C_1', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('C_1', commutative=True))), Symbol('C_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(\\phi_2,M)} = \\log{(M \\phi_2)}, then obtain (- M \\operatorname{v_{x}}{(\\phi_2,M)} + M \\log{(M \\phi_2)} + 2 \\operatorname{v_{x}}{(\\phi_2,M)})^{\\phi_2} = (\\operatorname{v_{x}}{(\\phi_2,M)} + \\log{(M \\phi_2)})^{\\phi_2}", "derivation": "\\operatorname{v_{x}}{(\\phi_2,M)} = \\log{(M \\phi_2)} and M \\operatorname{v_{x}}{(\\phi_2,M)} = M \\log{(M \\phi_2)} and \\operatorname{v_{x}}{(\\phi_2,M)} = - M \\operatorname{v_{x}}{(\\phi_2,M)} + M \\log{(M \\phi_2)} + \\operatorname{v_{x}}{(\\phi_2,M)} and - M \\operatorname{v_{x}}{(\\phi_2,M)} + M \\log{(M \\phi_2)} + \\operatorname{v_{x}}{(\\phi_2,M)} = \\log{(M \\phi_2)} and - M \\operatorname{v_{x}}{(\\phi_2,M)} + M \\log{(M \\phi_2)} + 2 \\operatorname{v_{x}}{(\\phi_2,M)} = \\operatorname{v_{x}}{(\\phi_2,M)} + \\log{(M \\phi_2)} and (- M \\operatorname{v_{x}}{(\\phi_2,M)} + M \\log{(M \\phi_2)} + 2 \\operatorname{v_{x}}{(\\phi_2,M)})^{\\phi_2} = (\\operatorname{v_{x}}{(\\phi_2,M)} + \\log{(M \\phi_2)})^{\\phi_2}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["times", 1, "Symbol('M', commutative=True)"], "Equality(Mul(Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["add", 2, "Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)))"], "Equality(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True))))"], [["add", 4, "Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(2), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)))), Add(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))))"], [["power", 5, "Symbol('\\\\phi_2', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('M', commutative=True), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True))), Mul(Symbol('M', commutative=True), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Mul(Integer(2), Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)))), Symbol('\\\\phi_2', commutative=True)), Pow(Add(Function('v_x')(Symbol('\\\\phi_2', commutative=True), Symbol('M', commutative=True)), log(Mul(Symbol('M', commutative=True), Symbol('\\\\phi_2', commutative=True)))), Symbol('\\\\phi_2', commutative=True)))"]]}, {"prompt": "Given b{(\\dot{x})} = e^{\\sin{(\\dot{x})}}, then obtain 1 = ((\\int b^{\\dot{x}}{(\\dot{x})} d\\dot{x})^{- \\dot{x}}) (\\int (e^{\\sin{(\\dot{x})}})^{\\dot{x}} d\\dot{x})^{\\dot{x}}", "derivation": "b{(\\dot{x})} = e^{\\sin{(\\dot{x})}} and b^{\\dot{x}}{(\\dot{x})} = (e^{\\sin{(\\dot{x})}})^{\\dot{x}} and \\int b^{\\dot{x}}{(\\dot{x})} d\\dot{x} = \\int (e^{\\sin{(\\dot{x})}})^{\\dot{x}} d\\dot{x} and (\\int b^{\\dot{x}}{(\\dot{x})} d\\dot{x})^{\\dot{x}} = (\\int (e^{\\sin{(\\dot{x})}})^{\\dot{x}} d\\dot{x})^{\\dot{x}} and 1 = ((\\int b^{\\dot{x}}{(\\dot{x})} d\\dot{x})^{- \\dot{x}}) (\\int (e^{\\sin{(\\dot{x})}})^{\\dot{x}} d\\dot{x})^{\\dot{x}}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\dot{x}', commutative=True)), exp(sin(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 1, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Function('b')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Pow(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Integral(Pow(Function('b')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Integral(Pow(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))))"], [["power", 3, "Symbol('\\\\dot{x}', commutative=True)"], "Equality(Pow(Integral(Pow(Function('b')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Pow(Integral(Pow(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)))"], [["divide", 4, "Pow(Integral(Pow(Function('b')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integral(Pow(Function('b')(Symbol('\\\\dot{x}', commutative=True)), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Mul(Integer(-1), Symbol('\\\\dot{x}', commutative=True))), Pow(Integral(Pow(exp(sin(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True)), Tuple(Symbol('\\\\dot{x}', commutative=True))), Symbol('\\\\dot{x}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{M_{E}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})}, then obtain \\operatorname{M_{E}}{(\\hat{H}_{\\lambda})} - \\log{(\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})})} = \\log{(\\hat{H}_{\\lambda})} - \\log{(\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})})}", "derivation": "\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda})} and \\log{(\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})})} = \\log{(\\log{(\\hat{H}_{\\lambda})})} and \\operatorname{M_{E}}{(\\hat{H}_{\\lambda})} - \\log{(\\log{(\\hat{H}_{\\lambda})})} = \\log{(\\hat{H}_{\\lambda})} - \\log{(\\log{(\\hat{H}_{\\lambda})})} and \\operatorname{M_{E}}{(\\hat{H}_{\\lambda})} - \\log{(\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})})} = \\log{(\\hat{H}_{\\lambda})} - \\log{(\\operatorname{M_{E}}{(\\hat{H}_{\\lambda})})}", "srepr_derivation": [["premise", "Equality(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], [["log", 1], "Equality(log(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), log(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))"], [["minus", 1, "log(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))"], "Equality(Add(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Add(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), log(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), log(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))), Add(log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Mul(Integer(-1), log(Function('M_E')(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))))))"]]}, {"prompt": "Given \\psi{(\\mathbf{J}_P)} = \\sin{(\\sin{(\\mathbf{J}_P)})}, then obtain - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\psi{(\\mathbf{J}_P)} - 1 = - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\sin{(\\sin{(\\mathbf{J}_P)})} - 1", "derivation": "\\psi{(\\mathbf{J}_P)} = \\sin{(\\sin{(\\mathbf{J}_P)})} and \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} = \\mathbf{J}_P \\sin{(\\sin{(\\mathbf{J}_P)})} and - \\mathbf{J}_P \\sin{(\\sin{(\\mathbf{J}_P)})} + \\psi{(\\mathbf{J}_P)} = - \\mathbf{J}_P \\sin{(\\sin{(\\mathbf{J}_P)})} + \\sin{(\\sin{(\\mathbf{J}_P)})} and - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\psi{(\\mathbf{J}_P)} = - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\sin{(\\sin{(\\mathbf{J}_P)})} and - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\psi{(\\mathbf{J}_P)} - 1 = - \\mathbf{J}_P \\psi{(\\mathbf{J}_P)} + \\sin{(\\sin{(\\mathbf{J}_P)})} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), Mul(Symbol('\\\\mathbf{J}_P', commutative=True), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["minus", 1, "Mul(Symbol('\\\\mathbf{J}_P', commutative=True), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True))))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True)))))"], [["add", 4, "Integer(-1)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}_P', commutative=True), Function('\\\\psi')(Symbol('\\\\mathbf{J}_P', commutative=True))), sin(sin(Symbol('\\\\mathbf{J}_P', commutative=True))), Integer(-1)))"]]}, {"prompt": "Given \\psi^{*}{(f_{\\mathbf{p}})} = \\log{(e^{f_{\\mathbf{p}}})}, then obtain f_{\\mathbf{p}} + \\psi^{*}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} - \\frac{1}{f_{\\mathbf{p}}} = f_{\\mathbf{p}} + \\log{(e^{f_{\\mathbf{p}}})}^{f_{\\mathbf{p}}} - \\frac{1}{f_{\\mathbf{p}}}", "derivation": "\\psi^{*}{(f_{\\mathbf{p}})} = \\log{(e^{f_{\\mathbf{p}}})} and \\psi^{*}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} = \\log{(e^{f_{\\mathbf{p}}})}^{f_{\\mathbf{p}}} and f_{\\mathbf{p}} + \\psi^{*}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} = f_{\\mathbf{p}} + \\log{(e^{f_{\\mathbf{p}}})}^{f_{\\mathbf{p}}} and f_{\\mathbf{p}} + \\psi^{*}^{f_{\\mathbf{p}}}{(f_{\\mathbf{p}})} - \\frac{1}{f_{\\mathbf{p}}} = f_{\\mathbf{p}} + \\log{(e^{f_{\\mathbf{p}}})}^{f_{\\mathbf{p}}} - \\frac{1}{f_{\\mathbf{p}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), log(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["power", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Pow(log(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["add", 2, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(log(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True))))"], [["minus", 3, "Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1))"], "Equality(Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(Function('\\\\psi^*')(Symbol('f_{\\\\mathbf{p}}', commutative=True)), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))), Add(Symbol('f_{\\\\mathbf{p}}', commutative=True), Pow(log(exp(Symbol('f_{\\\\mathbf{p}}', commutative=True))), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Mul(Integer(-1), Pow(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{F}{(i,\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda}^{i})}, then derive \\int \\mathbf{F}{(i,\\hat{H}_{\\lambda})} di = \\hbar + \\frac{i^{2} \\log{(\\hat{H}_{\\lambda})}}{2}, then obtain 2 \\hbar + i^{2} \\log{(\\hat{H}_{\\lambda})} = \\hbar + \\frac{i^{2} \\log{(\\hat{H}_{\\lambda})}}{2} + \\int \\log{(\\hat{H}_{\\lambda}^{i})} di", "derivation": "\\mathbf{F}{(i,\\hat{H}_{\\lambda})} = \\log{(\\hat{H}_{\\lambda}^{i})} and \\int \\mathbf{F}{(i,\\hat{H}_{\\lambda})} di = \\int \\log{(\\hat{H}_{\\lambda}^{i})} di and \\int \\mathbf{F}{(i,\\hat{H}_{\\lambda})} di = \\hbar + \\frac{i^{2} \\log{(\\hat{H}_{\\lambda})}}{2} and 2 \\int \\mathbf{F}{(i,\\hat{H}_{\\lambda})} di = \\int \\mathbf{F}{(i,\\hat{H}_{\\lambda})} di + \\int \\log{(\\hat{H}_{\\lambda}^{i})} di and 2 \\hbar + i^{2} \\log{(\\hat{H}_{\\lambda})} = \\hbar + \\frac{i^{2} \\log{(\\hat{H}_{\\lambda})}}{2} + \\int \\log{(\\hat{H}_{\\lambda}^{i})} di", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), log(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('i', commutative=True))))"], [["integrate", 1, "Symbol('i', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(log(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('i', commutative=True))), Add(Symbol('\\\\hbar', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))))"], [["add", 2, "Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('i', commutative=True)))"], "Equality(Mul(Integer(2), Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('i', commutative=True)))), Add(Integral(Function('\\\\mathbf{F}')(Symbol('i', commutative=True), Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)), Tuple(Symbol('i', commutative=True))), Integral(log(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Integer(2), Symbol('\\\\hbar', commutative=True)), Mul(Pow(Symbol('i', commutative=True), Integer(2)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True)))), Add(Symbol('\\\\hbar', commutative=True), Mul(Rational(1, 2), Pow(Symbol('i', commutative=True), Integer(2)), log(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True))), Integral(log(Pow(Symbol('\\\\hat{H}_{\\\\lambda}', commutative=True), Symbol('i', commutative=True))), Tuple(Symbol('i', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{1}}{(\\rho)} = \\cos{(\\rho)}, then obtain \\frac{d}{d \\rho} 2 = \\frac{d}{d \\rho} ((\\int 0 d\\rho)^{\\rho} + 1)", "derivation": "\\operatorname{A_{1}}{(\\rho)} = \\cos{(\\rho)} and \\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)} = 0 and \\int (\\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)}) d\\rho = \\int 0 d\\rho and (\\int (\\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)}) d\\rho)^{\\rho} = (\\int 0 d\\rho)^{\\rho} and 2 (\\int (\\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)}) d\\rho)^{\\rho} = (\\int 0 d\\rho)^{\\rho} + (\\int (\\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)}) d\\rho)^{\\rho} and 2 = (\\int (\\operatorname{A_{1}}{(\\rho)} - \\cos{(\\rho)}) d\\rho)^{\\rho} + 1 and 2 = (\\int 0 d\\rho)^{\\rho} + 1 and \\frac{d}{d \\rho} 2 = \\frac{d}{d \\rho} ((\\int 0 d\\rho)^{\\rho} + 1)", "srepr_derivation": [["premise", "Equality(Function('A_1')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\rho', commutative=True))"], "Equality(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\rho', commutative=True)"], "Equality(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))))"], [["power", 3, "Symbol('\\\\rho', commutative=True)"], "Equality(Pow(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)))"], [["add", 4, "Pow(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(2), Pow(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))), Add(Pow(Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Pow(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(2), Add(Pow(Integral(Add(Function('A_1')(Symbol('\\\\rho', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\rho', commutative=True)))), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integer(2), Add(Pow(Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Integer(1)))"], [["differentiate", 7, "Symbol('\\\\rho', commutative=True)"], "Equality(Derivative(Integer(2), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))), Derivative(Add(Pow(Integral(Integer(0), Tuple(Symbol('\\\\rho', commutative=True))), Symbol('\\\\rho', commutative=True)), Integer(1)), Tuple(Symbol('\\\\rho', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\phi{(\\Psi,\\hbar)} = \\Psi - \\hbar, then derive (\\int \\phi{(\\Psi,\\hbar)} d\\Psi)^{\\hbar} = (\\frac{\\Psi^{2}}{2} - \\Psi \\hbar + n_{2})^{\\hbar}, then obtain - \\hbar + (\\frac{\\Psi^{2}}{2} - \\Psi \\hbar + n_{2})^{\\hbar} = - \\hbar + (\\int (\\Psi - \\hbar) d\\Psi)^{\\hbar}", "derivation": "\\phi{(\\Psi,\\hbar)} = \\Psi - \\hbar and \\int \\phi{(\\Psi,\\hbar)} d\\Psi = \\int (\\Psi - \\hbar) d\\Psi and (\\int \\phi{(\\Psi,\\hbar)} d\\Psi)^{\\hbar} = (\\int (\\Psi - \\hbar) d\\Psi)^{\\hbar} and - \\hbar + (\\int \\phi{(\\Psi,\\hbar)} d\\Psi)^{\\hbar} = - \\hbar + (\\int (\\Psi - \\hbar) d\\Psi)^{\\hbar} and (\\int \\phi{(\\Psi,\\hbar)} d\\Psi)^{\\hbar} = (\\frac{\\Psi^{2}}{2} - \\Psi \\hbar + n_{2})^{\\hbar} and - \\hbar + (\\frac{\\Psi^{2}}{2} - \\Psi \\hbar + n_{2})^{\\hbar} = - \\hbar + (\\int (\\Psi - \\hbar) d\\Psi)^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))"], [["integrate", 1, "Symbol('\\\\Psi', commutative=True)"], "Equality(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Integral(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))))"], [["power", 2, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["add", 3, "Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('\\\\phi')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('n_2', commutative=True)), Symbol('\\\\hbar', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\Psi', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True), Symbol('\\\\hbar', commutative=True)), Symbol('n_2', commutative=True)), Symbol('\\\\hbar', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)), Pow(Integral(Add(Symbol('\\\\Psi', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True))), Symbol('\\\\hbar', commutative=True))))"]]}, {"prompt": "Given \\phi{(\\varepsilon,\\delta)} = \\delta \\varepsilon and \\tilde{g}{(\\varepsilon,\\delta)} = 2 \\phi{(\\varepsilon,\\delta)}, then obtain - \\varepsilon (- \\delta \\varepsilon + 3 \\phi{(\\varepsilon,\\delta)}) \\phi{(\\varepsilon,\\delta)} \\tilde{g}{(\\varepsilon,\\delta)} = - 2 \\varepsilon (- \\delta \\varepsilon + 3 \\phi{(\\varepsilon,\\delta)}) \\phi^{2}{(\\varepsilon,\\delta)}", "derivation": "\\phi{(\\varepsilon,\\delta)} = \\delta \\varepsilon and \\tilde{g}{(\\varepsilon,\\delta)} = 2 \\phi{(\\varepsilon,\\delta)} and \\varepsilon \\tilde{g}{(\\varepsilon,\\delta)} = 2 \\varepsilon \\phi{(\\varepsilon,\\delta)} and - \\delta \\varepsilon^{2} \\tilde{g}{(\\varepsilon,\\delta)} = - 2 \\delta \\varepsilon^{2} \\phi{(\\varepsilon,\\delta)} and - \\delta \\varepsilon^{2} \\tilde{g}{(\\varepsilon,\\delta)} = - 2 \\delta^{2} \\varepsilon^{3} and - \\varepsilon \\phi{(\\varepsilon,\\delta)} \\tilde{g}{(\\varepsilon,\\delta)} = - 2 \\varepsilon \\phi^{2}{(\\varepsilon,\\delta)} and - \\varepsilon (- \\delta \\varepsilon + 3 \\phi{(\\varepsilon,\\delta)}) \\phi{(\\varepsilon,\\delta)} \\tilde{g}{(\\varepsilon,\\delta)} = - 2 \\varepsilon (- \\delta \\varepsilon + 3 \\phi{(\\varepsilon,\\delta)}) \\phi^{2}{(\\varepsilon,\\delta)}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Mul(Integer(2), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["times", 2, "Symbol('\\\\varepsilon', commutative=True)"], "Equality(Mul(Symbol('\\\\varepsilon', commutative=True), Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(2), Symbol('\\\\varepsilon', commutative=True), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["times", 3, "Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(2)), Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\delta', commutative=True), Integer(2)), Pow(Symbol('\\\\varepsilon', commutative=True), Integer(3))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\varepsilon', commutative=True), Pow(Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))))"], [["times", 6, "Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(3), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))))"], "Equality(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(3), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)))), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Function('\\\\tilde{g}')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True))), Mul(Integer(-1), Integer(2), Symbol('\\\\varepsilon', commutative=True), Add(Mul(Integer(-1), Symbol('\\\\delta', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Mul(Integer(3), Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)))), Pow(Function('\\\\phi')(Symbol('\\\\varepsilon', commutative=True), Symbol('\\\\delta', commutative=True)), Integer(2))))"]]}, {"prompt": "Given W{(\\hat{H},n)} = - \\hat{H} + n, then derive e^{n + \\int W{(\\hat{H},n)} dn} = e^{\\dot{x} - \\hat{H} n + \\frac{n^{2}}{2} + n}, then obtain \\log{(e^{\\dot{x} - \\hat{H} n + \\frac{n^{2}}{2} + n})} = \\log{(e^{n + \\int (- \\hat{H} + n) dn})}", "derivation": "W{(\\hat{H},n)} = - \\hat{H} + n and \\int W{(\\hat{H},n)} dn = \\int (- \\hat{H} + n) dn and n + \\int W{(\\hat{H},n)} dn = n + \\int (- \\hat{H} + n) dn and e^{n + \\int W{(\\hat{H},n)} dn} = e^{n + \\int (- \\hat{H} + n) dn} and e^{n + \\int W{(\\hat{H},n)} dn} = e^{\\dot{x} - \\hat{H} n + \\frac{n^{2}}{2} + n} and \\log{(e^{n + \\int W{(\\hat{H},n)} dn})} = \\log{(e^{n + \\int (- \\hat{H} + n) dn})} and \\log{(e^{\\dot{x} - \\hat{H} n + \\frac{n^{2}}{2} + n})} = \\log{(e^{n + \\int (- \\hat{H} + n) dn})}", "srepr_derivation": [["get_premise", "Equality(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)))"], [["integrate", 1, "Symbol('n', commutative=True)"], "Equality(Integral(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))"], [["add", 2, "Symbol('n', commutative=True)"], "Equality(Add(Symbol('n', commutative=True), Integral(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))), Add(Symbol('n', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))"], [["exp", 3], "Equality(exp(Add(Symbol('n', commutative=True), Integral(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), exp(Add(Symbol('n', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(exp(Add(Symbol('n', commutative=True), Integral(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True))))), exp(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Symbol('n', commutative=True))))"], [["log", 4], "Equality(log(exp(Add(Symbol('n', commutative=True), Integral(Function('W')(Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))), log(exp(Add(Symbol('n', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(log(exp(Add(Symbol('\\\\dot{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True), Symbol('n', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('n', commutative=True), Integer(2))), Symbol('n', commutative=True)))), log(exp(Add(Symbol('n', commutative=True), Integral(Add(Mul(Integer(-1), Symbol('\\\\hat{H}', commutative=True)), Symbol('n', commutative=True)), Tuple(Symbol('n', commutative=True)))))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\sigma_x)} = \\cos{(\\sigma_x)}, then obtain \\frac{d^{2}}{d \\sigma_x^{2}} \\frac{1}{\\cos{(\\sigma_x)}} = \\frac{d^{2}}{d \\sigma_x^{2}} \\frac{(\\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}})^{\\sigma_x}}{\\cos{(\\sigma_x)}}", "derivation": "\\operatorname{f_{E}}{(\\sigma_x)} = \\cos{(\\sigma_x)} and 1 = \\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}} and 1 = (\\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}})^{\\sigma_x} and \\frac{1}{\\cos{(\\sigma_x)}} = \\frac{(\\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}})^{\\sigma_x}}{\\cos{(\\sigma_x)}} and \\frac{d}{d \\sigma_x} \\frac{1}{\\cos{(\\sigma_x)}} = \\frac{d}{d \\sigma_x} \\frac{(\\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}})^{\\sigma_x}}{\\cos{(\\sigma_x)}} and \\frac{d^{2}}{d \\sigma_x^{2}} \\frac{1}{\\cos{(\\sigma_x)}} = \\frac{d^{2}}{d \\sigma_x^{2}} \\frac{(\\frac{\\cos{(\\sigma_x)}}{\\operatorname{f_{E}}{(\\sigma_x)}})^{\\sigma_x}}{\\cos{(\\sigma_x)}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), cos(Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 1, "Function('f_E')(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))))"], [["power", 2, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)))"], [["divide", 3, "cos(Symbol('\\\\sigma_x', commutative=True))"], "Equality(Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Mul(Pow(Mul(Pow(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))))"], [["differentiate", 4, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))), Derivative(Mul(Pow(Mul(Pow(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Derivative(Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))), Derivative(Mul(Pow(Mul(Pow(Function('f_E')(Symbol('\\\\sigma_x', commutative=True)), Integer(-1)), cos(Symbol('\\\\sigma_x', commutative=True))), Symbol('\\\\sigma_x', commutative=True)), Pow(cos(Symbol('\\\\sigma_x', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\sigma_x', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mu)} = \\log{(\\mu)}, then obtain \\frac{d}{d \\mu} \\operatorname{A_{y}}^{2}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)}^{2}", "derivation": "\\operatorname{A_{y}}{(\\mu)} = \\log{(\\mu)} and \\operatorname{A_{y}}{(\\mu)} \\log{(\\mu)} = \\log{(\\mu)}^{2} and \\frac{d}{d \\mu} \\operatorname{A_{y}}{(\\mu)} \\log{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)}^{2} and \\operatorname{A_{y}}^{2}{(\\mu)} = \\operatorname{A_{y}}{(\\mu)} \\log{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{A_{y}}^{2}{(\\mu)} = \\frac{d}{d \\mu} \\operatorname{A_{y}}{(\\mu)} \\log{(\\mu)} and \\frac{d}{d \\mu} \\operatorname{A_{y}}^{2}{(\\mu)} = \\frac{d}{d \\mu} \\log{(\\mu)}^{2}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True)))"], [["times", 1, "log(Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Function('A_y')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Mul(Function('A_y')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["times", 1, "Function('A_y')(Symbol('\\\\mu', commutative=True))"], "Equality(Pow(Function('A_y')(Symbol('\\\\mu', commutative=True)), Integer(2)), Mul(Function('A_y')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))))"], [["differentiate", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('A_y')(Symbol('\\\\mu', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Mul(Function('A_y')(Symbol('\\\\mu', commutative=True)), log(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Pow(Function('A_y')(Symbol('\\\\mu', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\mu', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{S})} = \\sin{(\\cos{(\\mathbf{S})})}, then obtain \\bar{\\h}{(\\mathbf{S})} \\iint \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} d\\mathbf{S} = \\sin{(\\cos{(\\mathbf{S})})} \\iint \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} d\\mathbf{S}", "derivation": "\\bar{\\h}{(\\mathbf{S})} = \\sin{(\\cos{(\\mathbf{S})})} and \\int \\bar{\\h}{(\\mathbf{S})} d\\mathbf{S} = \\int \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} and \\iint \\bar{\\h}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\iint \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} d\\mathbf{S} and \\bar{\\h}{(\\mathbf{S})} \\iint \\bar{\\h}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} = \\sin{(\\cos{(\\mathbf{S})})} \\iint \\bar{\\h}{(\\mathbf{S})} d\\mathbf{S} d\\mathbf{S} and \\bar{\\h}{(\\mathbf{S})} \\iint \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} d\\mathbf{S} = \\sin{(\\cos{(\\mathbf{S})})} \\iint \\sin{(\\cos{(\\mathbf{S})})} d\\mathbf{S} d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), sin(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["integrate", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 1, "Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Function('\\\\hbar')(Symbol('\\\\mathbf{S}', commutative=True)), Integral(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Mul(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Integral(sin(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\mathbf{P},v_{1})} = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{P} + v_{1}), then derive \\omega{(\\mathbf{P},v_{1})} = -1, then obtain \\frac{\\partial}{\\partial v_{1}} (\\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (- \\mathbf{P} + v_{1}))^{\\mathbf{P}} = \\frac{d}{d v_{1}} (\\frac{d}{d \\mathbf{P}} (-1))^{\\mathbf{P}}", "derivation": "\\omega{(\\mathbf{P},v_{1})} = \\frac{\\partial}{\\partial \\mathbf{P}} (- \\mathbf{P} + v_{1}) and \\omega{(\\mathbf{P},v_{1})} = -1 and \\frac{\\partial}{\\partial \\mathbf{P}} \\omega{(\\mathbf{P},v_{1})} = \\frac{d}{d \\mathbf{P}} (-1) and \\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (- \\mathbf{P} + v_{1}) = \\frac{d}{d \\mathbf{P}} (-1) and (\\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (- \\mathbf{P} + v_{1}))^{\\mathbf{P}} = (\\frac{d}{d \\mathbf{P}} (-1))^{\\mathbf{P}} and \\frac{\\partial}{\\partial v_{1}} (\\frac{\\partial^{2}}{\\partial \\mathbf{P}^{2}} (- \\mathbf{P} + v_{1}))^{\\mathbf{P}} = \\frac{d}{d v_{1}} (\\frac{d}{d \\mathbf{P}} (-1))^{\\mathbf{P}}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Integer(-1))"], [["differentiate", 2, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\omega')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["power", 4, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Symbol('\\\\mathbf{P}', commutative=True)), Pow(Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)))"], [["differentiate", 5, "Symbol('v_1', commutative=True)"], "Equality(Derivative(Pow(Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{P}', commutative=True)), Symbol('v_1', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(2))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))), Derivative(Pow(Derivative(Integer(-1), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Symbol('\\\\mathbf{P}', commutative=True)), Tuple(Symbol('v_1', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(F_{c})} = \\log{(F_{c})}, then obtain F_{c} \\hat{x}_0^{2}{(F_{c})} = F_{c} \\log{(F_{c})}^{2}", "derivation": "\\hat{x}_0{(F_{c})} = \\log{(F_{c})} and F_{c} \\hat{x}_0{(F_{c})} = F_{c} \\log{(F_{c})} and F_{c} \\hat{x}_0{(F_{c})} \\log{(F_{c})} = F_{c} \\log{(F_{c})}^{2} and F_{c} \\hat{x}_0^{2}{(F_{c})} = F_{c} \\hat{x}_0{(F_{c})} \\log{(F_{c})} and F_{c} \\hat{x}_0^{2}{(F_{c})} = F_{c} \\log{(F_{c})}^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True)))"], [["times", 1, "Symbol('F_c', commutative=True)"], "Equality(Mul(Symbol('F_c', commutative=True), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True))))"], [["times", 1, "Mul(Symbol('F_c', commutative=True), log(Symbol('F_c', commutative=True)))"], "Equality(Mul(Symbol('F_c', commutative=True), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True))), Mul(Symbol('F_c', commutative=True), Pow(log(Symbol('F_c', commutative=True)), Integer(2))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Symbol('F_c', commutative=True), Pow(Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), Integer(2))), Mul(Symbol('F_c', commutative=True), Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), log(Symbol('F_c', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Symbol('F_c', commutative=True), Pow(Function('\\\\hat{x}_0')(Symbol('F_c', commutative=True)), Integer(2))), Mul(Symbol('F_c', commutative=True), Pow(log(Symbol('F_c', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\theta_{2}{(S)} = \\sin{(S)}, then derive 2 \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)} = \\theta_{2}{(S)} \\cos{(S)} + \\theta_{2}{(S)} + \\sin{(S)} \\frac{d}{d S} \\theta_{2}{(S)}, then obtain 2 \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)} = \\theta_{2}{(S)} \\cos{(S)} + \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)}", "derivation": "\\theta_{2}{(S)} = \\sin{(S)} and \\theta_{2}^{2}{(S)} = \\theta_{2}{(S)} \\sin{(S)} and \\frac{d}{d S} \\theta_{2}^{2}{(S)} = \\frac{d}{d S} \\theta_{2}{(S)} \\sin{(S)} and \\theta_{2}{(S)} + \\frac{d}{d S} \\theta_{2}^{2}{(S)} = \\theta_{2}{(S)} + \\frac{d}{d S} \\theta_{2}{(S)} \\sin{(S)} and 2 \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)} = \\theta_{2}{(S)} \\cos{(S)} + \\theta_{2}{(S)} + \\sin{(S)} \\frac{d}{d S} \\theta_{2}{(S)} and 2 \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)} = \\theta_{2}{(S)} \\cos{(S)} + \\theta_{2}{(S)} \\frac{d}{d S} \\theta_{2}{(S)} + \\theta_{2}{(S)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True)))"], [["times", 1, "Function('\\\\theta_2')(Symbol('S', commutative=True))"], "Equality(Pow(Function('\\\\theta_2')(Symbol('S', commutative=True)), Integer(2)), Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Pow(Function('\\\\theta_2')(Symbol('S', commutative=True)), Integer(2)), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["add", 3, "Function('\\\\theta_2')(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\theta_2')(Symbol('S', commutative=True)), Derivative(Pow(Function('\\\\theta_2')(Symbol('S', commutative=True)), Integer(2)), Tuple(Symbol('S', commutative=True), Integer(1)))), Add(Function('\\\\theta_2')(Symbol('S', commutative=True)), Derivative(Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), sin(Symbol('S', commutative=True))), Tuple(Symbol('S', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 4], "Equality(Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('S', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Function('\\\\theta_2')(Symbol('S', commutative=True))), Add(Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), Function('\\\\theta_2')(Symbol('S', commutative=True)), Mul(sin(Symbol('S', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1))))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Add(Mul(Integer(2), Function('\\\\theta_2')(Symbol('S', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Function('\\\\theta_2')(Symbol('S', commutative=True))), Add(Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), cos(Symbol('S', commutative=True))), Mul(Function('\\\\theta_2')(Symbol('S', commutative=True)), Derivative(Function('\\\\theta_2')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Function('\\\\theta_2')(Symbol('S', commutative=True))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(C)} = \\frac{d}{d C} \\sin{(C)}, then derive \\Psi^{\\dagger}{(C)} = \\cos{(C)}, then obtain \\frac{\\sin{(\\frac{d}{d C} \\Psi^{\\dagger}{(C)})}}{\\cos{(C)}} = - \\frac{\\sin{(\\sin{(C)})}}{\\cos{(C)}}", "derivation": "\\Psi^{\\dagger}{(C)} = \\frac{d}{d C} \\sin{(C)} and \\Psi^{\\dagger}{(C)} = \\cos{(C)} and \\frac{d}{d C} \\Psi^{\\dagger}{(C)} = \\frac{d}{d C} \\cos{(C)} and \\sin{(\\frac{d}{d C} \\Psi^{\\dagger}{(C)})} = \\sin{(\\frac{d}{d C} \\cos{(C)})} and \\frac{\\sin{(\\frac{d}{d C} \\Psi^{\\dagger}{(C)})}}{\\cos{(C)}} = \\frac{\\sin{(\\frac{d}{d C} \\cos{(C)})}}{\\cos{(C)}} and \\frac{\\sin{(\\frac{d}{d C} \\Psi^{\\dagger}{(C)})}}{\\cos{(C)}} = - \\frac{\\sin{(\\sin{(C)})}}{\\cos{(C)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Derivative(sin(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), cos(Symbol('C', commutative=True)))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["sin", 3], "Equality(sin(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), sin(Derivative(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))))"], [["divide", 4, "cos(Symbol('C', commutative=True))"], "Equality(Mul(sin(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(cos(Symbol('C', commutative=True)), Integer(-1))), Mul(sin(Derivative(cos(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(cos(Symbol('C', commutative=True)), Integer(-1))))"], [["evaluate_derivatives", 5], "Equality(Mul(sin(Derivative(Function('\\\\Psi^{\\\\dagger}')(Symbol('C', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Pow(cos(Symbol('C', commutative=True)), Integer(-1))), Mul(Integer(-1), sin(sin(Symbol('C', commutative=True))), Pow(cos(Symbol('C', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{A_{2}}{(\\varepsilon_0,V)} = \\frac{\\log{(V)}}{\\varepsilon_0} and \\chi{(s,\\varphi^*)} = \\cos{(\\varphi^* s)}, then obtain \\operatorname{A_{2}}^{\\varepsilon_0}{(\\varepsilon_0,V)} (\\int \\chi{(s,\\varphi^*)} d\\varphi^*)^{s} = (\\frac{\\log{(V)}}{\\varepsilon_0})^{\\varepsilon_0} (\\int \\chi{(s,\\varphi^*)} d\\varphi^*)^{s}", "derivation": "\\operatorname{A_{2}}{(\\varepsilon_0,V)} = \\frac{\\log{(V)}}{\\varepsilon_0} and \\chi{(s,\\varphi^*)} = \\cos{(\\varphi^* s)} and \\int \\chi{(s,\\varphi^*)} d\\varphi^* = \\int \\cos{(\\varphi^* s)} d\\varphi^* and \\operatorname{A_{2}}^{\\varepsilon_0}{(\\varepsilon_0,V)} = (\\frac{\\log{(V)}}{\\varepsilon_0})^{\\varepsilon_0} and \\operatorname{A_{2}}^{\\varepsilon_0}{(\\varepsilon_0,V)} (\\int \\cos{(\\varphi^* s)} d\\varphi^*)^{s} = (\\frac{\\log{(V)}}{\\varepsilon_0})^{\\varepsilon_0} (\\int \\cos{(\\varphi^* s)} d\\varphi^*)^{s} and \\operatorname{A_{2}}^{\\varepsilon_0}{(\\varepsilon_0,V)} (\\int \\chi{(s,\\varphi^*)} d\\varphi^*)^{s} = (\\frac{\\log{(V)}}{\\varepsilon_0})^{\\varepsilon_0} (\\int \\chi{(s,\\varphi^*)} d\\varphi^*)^{s}", "srepr_derivation": [["premise", "Equality(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)), Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))))"], ["get_premise", "Equality(Function('\\\\chi')(Symbol('s', commutative=True), Symbol('\\\\varphi^*', commutative=True)), cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))))"], [["integrate", 2, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\chi')(Symbol('s', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["power", 1, "Symbol('\\\\varepsilon_0', commutative=True)"], "Equality(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)))"], [["times", 4, "Pow(Integral(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('s', commutative=True))"], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('s', commutative=True))), Mul(Pow(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(cos(Mul(Symbol('\\\\varphi^*', commutative=True), Symbol('s', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Function('A_2')(Symbol('\\\\varepsilon_0', commutative=True), Symbol('V', commutative=True)), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(Function('\\\\chi')(Symbol('s', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('s', commutative=True))), Mul(Pow(Mul(Pow(Symbol('\\\\varepsilon_0', commutative=True), Integer(-1)), log(Symbol('V', commutative=True))), Symbol('\\\\varepsilon_0', commutative=True)), Pow(Integral(Function('\\\\chi')(Symbol('s', commutative=True), Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Symbol('s', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + f_{\\mathbf{p}}, then derive \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = 1, then obtain \\int (\\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} = \\int 1 dV_{\\mathbf{B}}", "derivation": "\\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = V_{\\mathbf{B}} + f_{\\mathbf{p}} and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = \\frac{\\partial}{\\partial f_{\\mathbf{p}}} (V_{\\mathbf{B}} + f_{\\mathbf{p}}) and \\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})} = 1 and (\\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} = 1 and \\int (\\frac{\\partial}{\\partial f_{\\mathbf{p}}} \\operatorname{v_{x}}{(f_{\\mathbf{p}},V_{\\mathbf{B}})})^{V_{\\mathbf{B}}} dV_{\\mathbf{B}} = \\int 1 dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)))"], [["differentiate", 1, "Symbol('f_{\\\\mathbf{p}}', commutative=True)"], "Equality(Derivative(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Derivative(Add(Symbol('V_{\\\\mathbf{B}}', commutative=True), Symbol('f_{\\\\mathbf{p}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Pow(Derivative(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Derivative(Function('v_x')(Symbol('f_{\\\\mathbf{p}}', commutative=True), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('f_{\\\\mathbf{p}}', commutative=True), Integer(1))), Symbol('V_{\\\\mathbf{B}}', commutative=True)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Integer(1), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(u,p)} = p u and \\chi{(u,p)} = u + \\operatorname{P_{g}}{(u,p)}, then obtain \\frac{\\partial}{\\partial u} \\chi{(u,p)} = \\frac{\\partial}{\\partial u} (p u + u)", "derivation": "\\operatorname{P_{g}}{(u,p)} = p u and \\chi{(u,p)} = u + \\operatorname{P_{g}}{(u,p)} and \\chi{(u,p)} = p u + u and \\frac{\\partial}{\\partial u} \\chi{(u,p)} = \\frac{\\partial}{\\partial u} (p u + u)", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('u', commutative=True), Symbol('p', commutative=True)), Mul(Symbol('p', commutative=True), Symbol('u', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('u', commutative=True), Symbol('p', commutative=True)), Add(Symbol('u', commutative=True), Function('P_g')(Symbol('u', commutative=True), Symbol('p', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('u', commutative=True), Symbol('p', commutative=True)), Add(Mul(Symbol('p', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)))"], [["differentiate", 3, "Symbol('u', commutative=True)"], "Equality(Derivative(Function('\\\\chi')(Symbol('u', commutative=True), Symbol('p', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('p', commutative=True), Symbol('u', commutative=True)), Symbol('u', commutative=True)), Tuple(Symbol('u', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{H}{(B)} = \\sin{(B)} and \\chi{(B)} = \\sin{(B)}, then obtain - B + \\chi{(B)} + \\sin{(B)} = - B + \\mathbf{H}{(B)} + \\sin{(B)}", "derivation": "\\mathbf{H}{(B)} = \\sin{(B)} and \\chi{(B)} = \\sin{(B)} and \\chi{(B)} = \\mathbf{H}{(B)} and \\chi{(B)} + \\sin{(B)} = \\mathbf{H}{(B)} + \\sin{(B)} and - B + \\chi{(B)} + \\sin{(B)} = - B + \\mathbf{H}{(B)} + \\sin{(B)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\chi')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\chi')(Symbol('B', commutative=True)), Function('\\\\mathbf{H}')(Symbol('B', commutative=True)))"], [["add", 3, "sin(Symbol('B', commutative=True))"], "Equality(Add(Function('\\\\chi')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True))), Add(Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True))))"], [["minus", 4, "Symbol('B', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\chi')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True))), Add(Mul(Integer(-1), Symbol('B', commutative=True)), Function('\\\\mathbf{H}')(Symbol('B', commutative=True)), sin(Symbol('B', commutative=True))))"]]}, {"prompt": "Given \\mathbf{v}{(A_{z},H)} = A_{z}^{H}, then derive \\sin{(\\frac{\\partial}{\\partial H} \\mathbf{v}{(A_{z},H)})} = \\sin{(A_{z}^{H} \\log{(A_{z})})}, then obtain \\int (- r_{0} + \\sin{(\\frac{\\partial}{\\partial H} A_{z}^{H})}) dH = \\int (- r_{0} + \\sin{(A_{z}^{H} \\log{(A_{z})})}) dH", "derivation": "\\mathbf{v}{(A_{z},H)} = A_{z}^{H} and \\frac{\\partial}{\\partial H} \\mathbf{v}{(A_{z},H)} = \\frac{\\partial}{\\partial H} A_{z}^{H} and \\sin{(\\frac{\\partial}{\\partial H} \\mathbf{v}{(A_{z},H)})} = \\sin{(\\frac{\\partial}{\\partial H} A_{z}^{H})} and \\sin{(\\frac{\\partial}{\\partial H} \\mathbf{v}{(A_{z},H)})} = \\sin{(A_{z}^{H} \\log{(A_{z})})} and \\sin{(\\frac{\\partial}{\\partial H} A_{z}^{H})} = \\sin{(A_{z}^{H} \\log{(A_{z})})} and - r_{0} + \\sin{(\\frac{\\partial}{\\partial H} A_{z}^{H})} = - r_{0} + \\sin{(A_{z}^{H} \\log{(A_{z})})} and \\int (- r_{0} + \\sin{(\\frac{\\partial}{\\partial H} A_{z}^{H})}) dH = \\int (- r_{0} + \\sin{(A_{z}^{H} \\log{(A_{z})})}) dH", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)))"], [["differentiate", 1, "Symbol('H', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{v}')(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('\\\\mathbf{v}')(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), sin(Derivative(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(sin(Derivative(Function('\\\\mathbf{v}')(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), sin(Mul(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), log(Symbol('A_z', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(sin(Derivative(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1)))), sin(Mul(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), log(Symbol('A_z', commutative=True)))))"], [["minus", 5, "Symbol('r_0', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), sin(Derivative(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), sin(Mul(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), log(Symbol('A_z', commutative=True))))))"], [["integrate", 6, "Symbol('H', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), sin(Derivative(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True), Integer(1))))), Tuple(Symbol('H', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), sin(Mul(Pow(Symbol('A_z', commutative=True), Symbol('H', commutative=True)), log(Symbol('A_z', commutative=True))))), Tuple(Symbol('H', commutative=True))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(J)} = \\cos{(J)} and \\operatorname{E_{\\lambda}}{(J)} = \\frac{\\cos{(J)}}{\\operatorname{t_{1}}{(J)}}, then obtain \\theta_1 = \\frac{\\theta_1 \\frac{d}{d J} \\operatorname{E_{\\lambda}}{(J)}}{\\frac{d}{d J} 1}", "derivation": "\\operatorname{t_{1}}{(J)} = \\cos{(J)} and 1 = \\frac{\\cos{(J)}}{\\operatorname{t_{1}}{(J)}} and \\frac{d}{d J} 1 = \\frac{d}{d J} \\frac{\\cos{(J)}}{\\operatorname{t_{1}}{(J)}} and \\operatorname{E_{\\lambda}}{(J)} = \\frac{\\cos{(J)}}{\\operatorname{t_{1}}{(J)}} and \\frac{d}{d J} 1 = \\frac{d}{d J} \\operatorname{E_{\\lambda}}{(J)} and \\theta_1 \\frac{d}{d J} 1 = \\theta_1 \\frac{d}{d J} \\operatorname{E_{\\lambda}}{(J)} and \\theta_1 = \\frac{\\theta_1 \\frac{d}{d J} \\operatorname{E_{\\lambda}}{(J)}}{\\frac{d}{d J} 1}", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["divide", 1, "Function('t_1')(Symbol('J', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('t_1')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True))))"], [["differentiate", 2, "Symbol('J', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('t_1')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('J', commutative=True)), Mul(Pow(Function('t_1')(Symbol('J', commutative=True)), Integer(-1)), cos(Symbol('J', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(Function('E_{\\\\lambda}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["times", 5, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Mul(Symbol('\\\\theta_1', commutative=True), Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1)))), Mul(Symbol('\\\\theta_1', commutative=True), Derivative(Function('E_{\\\\lambda}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"], [["divide", 6, "Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1)))"], "Equality(Symbol('\\\\theta_1', commutative=True), Mul(Symbol('\\\\theta_1', commutative=True), Pow(Derivative(Integer(1), Tuple(Symbol('J', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('E_{\\\\lambda}')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1)))))"]]}, {"prompt": "Given B{(\\phi)} = \\log{(\\phi)}, then obtain (B^{2}{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)})^{\\phi} = (B{(\\phi)} \\log{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)})^{\\phi}", "derivation": "B{(\\phi)} = \\log{(\\phi)} and B^{2}{(\\phi)} = B{(\\phi)} \\log{(\\phi)} and B^{2}{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} = B{(\\phi)} \\log{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} and B^{2}{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)} = B{(\\phi)} \\log{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)} and (B^{2}{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)})^{\\phi} = (B{(\\phi)} \\log{(\\phi)} \\frac{d}{d \\phi} B{(\\phi)} - \\log{(\\phi)})^{\\phi}", "srepr_derivation": [["premise", "Equality(Function('B')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)))"], [["times", 1, "Function('B')(Symbol('\\\\phi', commutative=True))"], "Equality(Pow(Function('B')(Symbol('\\\\phi', commutative=True)), Integer(2)), Mul(Function('B')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True))))"], [["times", 2, "Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Function('B')(Symbol('\\\\phi', commutative=True)), Integer(2)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Function('B')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))))"], [["minus", 3, "log(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Pow(Function('B')(Symbol('\\\\phi', commutative=True)), Integer(2)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True)))), Add(Mul(Function('B')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True)))))"], [["power", 4, "Symbol('\\\\phi', commutative=True)"], "Equality(Pow(Add(Mul(Pow(Function('B')(Symbol('\\\\phi', commutative=True)), Integer(2)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)), Pow(Add(Mul(Function('B')(Symbol('\\\\phi', commutative=True)), log(Symbol('\\\\phi', commutative=True)), Derivative(Function('B')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1)))), Mul(Integer(-1), log(Symbol('\\\\phi', commutative=True)))), Symbol('\\\\phi', commutative=True)))"]]}, {"prompt": "Given \\operatorname{a^{\\dagger}}{(A_{2})} = \\sin{(A_{2})}, then derive \\frac{d}{d A_{2}} \\operatorname{a^{\\dagger}}{(A_{2})} = \\cos{(A_{2})}, then obtain \\frac{d}{d A_{2}} \\sin{(A_{2})} = \\cos{(A_{2})}", "derivation": "\\operatorname{a^{\\dagger}}{(A_{2})} = \\sin{(A_{2})} and \\frac{d}{d A_{2}} \\operatorname{a^{\\dagger}}{(A_{2})} = \\frac{d}{d A_{2}} \\sin{(A_{2})} and \\frac{d}{d A_{2}} \\operatorname{a^{\\dagger}}{(A_{2})} = \\cos{(A_{2})} and \\frac{d}{d A_{2}} \\sin{(A_{2})} = \\cos{(A_{2})}", "srepr_derivation": [["premise", "Equality(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), sin(Symbol('A_2', commutative=True)))"], [["differentiate", 1, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('a^{\\\\dagger}')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), cos(Symbol('A_2', commutative=True)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(sin(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), cos(Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\mathbf{M})} = \\log{(e^{\\mathbf{M}})} and \\operatorname{x^{{\\}'}}{(\\mathbf{M})} = \\int \\frac{d}{d \\mathbf{M}} \\operatorname{A_{y}}{(\\mathbf{M})} d\\mathbf{M}, then obtain \\operatorname{x^{{\\}'}}{(\\mathbf{M})} = \\int \\frac{d}{d \\mathbf{M}} \\log{(e^{\\mathbf{M}})} d\\mathbf{M}", "derivation": "\\operatorname{A_{y}}{(\\mathbf{M})} = \\log{(e^{\\mathbf{M}})} and \\frac{d}{d \\mathbf{M}} \\operatorname{A_{y}}{(\\mathbf{M})} = \\frac{d}{d \\mathbf{M}} \\log{(e^{\\mathbf{M}})} and \\int \\frac{d}{d \\mathbf{M}} \\operatorname{A_{y}}{(\\mathbf{M})} d\\mathbf{M} = \\int \\frac{d}{d \\mathbf{M}} \\log{(e^{\\mathbf{M}})} d\\mathbf{M} and \\operatorname{x^{{\\}'}}{(\\mathbf{M})} = \\int \\frac{d}{d \\mathbf{M}} \\operatorname{A_{y}}{(\\mathbf{M})} d\\mathbf{M} and \\operatorname{x^{{\\}'}}{(\\mathbf{M})} = \\int \\frac{d}{d \\mathbf{M}} \\log{(e^{\\mathbf{M}})} d\\mathbf{M}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\mathbf{M}', commutative=True)), log(exp(Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(log(exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Integral(Derivative(Function('A_y')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))), Integral(Derivative(log(exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], ["renaming_premise", "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True)), Integral(Derivative(Function('A_y')(Symbol('\\\\mathbf{M}', commutative=True)), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('x^\\\\prime')(Symbol('\\\\mathbf{M}', commutative=True)), Integral(Derivative(log(exp(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{M}', commutative=True))))"]]}, {"prompt": "Given \\sigma_{x}{(z^{*})} = z^{*}, then obtain \\int (- z^{*} \\sigma_{x}{(z^{*})} + \\sigma_{x}^{2}{(z^{*})}) d\\sigma_{x}{(z^{*})} = \\int 0 d\\sigma_{x}{(z^{*})}", "derivation": "\\sigma_{x}{(z^{*})} = z^{*} and \\sigma_{x}^{2}{(z^{*})} = z^{*} \\sigma_{x}{(z^{*})} and - z^{*} \\sigma_{x}{(z^{*})} + \\sigma_{x}^{2}{(z^{*})} = 0 and \\int (- z^{*} \\sigma_{x}{(z^{*})} + \\sigma_{x}^{2}{(z^{*})}) dz^{*} = \\int 0 dz^{*} and \\int (- z^{*} \\sigma_{x}{(z^{*})} + \\sigma_{x}^{2}{(z^{*})}) d\\sigma_{x}{(z^{*})} = \\int 0 d\\sigma_{x}{(z^{*})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)), Symbol('z^*', commutative=True))"], [["times", 1, "Function('\\\\sigma_x')(Symbol('z^*', commutative=True))"], "Equality(Pow(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)), Integer(2)), Mul(Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True))))"], [["minus", 2, "Mul(Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)), Integer(2))), Integer(0))"], [["integrate", 3, "Symbol('z^*', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)), Integer(2))), Tuple(Symbol('z^*', commutative=True))), Integral(Integer(0), Tuple(Symbol('z^*', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('z^*', commutative=True), Function('\\\\sigma_x')(Symbol('z^*', commutative=True))), Pow(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)), Integer(2))), Tuple(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)))), Integral(Integer(0), Tuple(Function('\\\\sigma_x')(Symbol('z^*', commutative=True)))))"]]}, {"prompt": "Given k{(I)} = \\cos{(I)}, then obtain (1 - \\frac{\\cos{(I)}}{k{(I)}}) ((1 + \\frac{\\cos{(I)}}{k{(I)}})^{1 + \\frac{\\cos{(I)}}{k{(I)}}} - \\int 3 dI) = 0", "derivation": "k{(I)} = \\cos{(I)} and 1 = \\frac{\\cos{(I)}}{k{(I)}} and 2 = 1 + \\frac{\\cos{(I)}}{k{(I)}} and 1 - \\frac{\\cos{(I)}}{k{(I)}} = 0 and 4 = (1 + \\frac{\\cos{(I)}}{k{(I)}})^{2} and 3 = (1 + \\frac{\\cos{(I)}}{k{(I)}})^{2} - 1 and \\int 3 dI = \\int ((1 + \\frac{\\cos{(I)}}{k{(I)}})^{2} - 1) dI and (1 - \\frac{\\cos{(I)}}{k{(I)}}) ((1 + \\frac{\\cos{(I)}}{k{(I)}})^{1 + \\frac{\\cos{(I)}}{k{(I)}}} - \\int ((1 + \\frac{\\cos{(I)}}{k{(I)}})^{2} - 1) dI) = 0 and (1 - \\frac{\\cos{(I)}}{k{(I)}}) ((1 + \\frac{\\cos{(I)}}{k{(I)}})^{1 + \\frac{\\cos{(I)}}{k{(I)}}} - \\int 3 dI) = 0", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["divide", 1, "Function('k')(Symbol('I', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True))))"], [["add", 2, 1], "Equality(Integer(2), Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))))"], [["minus", 3, "Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True))))"], "Equality(Add(Integer(1), Mul(Integer(-1), Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(0))"], [["power", 3, 2], "Equality(Integer(4), Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(2)))"], [["minus", 5, 1], "Equality(Integer(3), Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(2)), Integer(-1)))"], [["integrate", 6, "Symbol('I', commutative=True)"], "Equality(Integral(Integer(3), Tuple(Symbol('I', commutative=True))), Integral(Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(2)), Integer(-1)), Tuple(Symbol('I', commutative=True))))"], [["times", 4, "Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True))))), Mul(Integer(-1), Integral(Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(2)), Integer(-1)), Tuple(Symbol('I', commutative=True)))))"], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True))))), Mul(Integer(-1), Integral(Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Integer(2)), Integer(-1)), Tuple(Symbol('I', commutative=True)))))), Integer(0))"], [["substitute_RHS_for_LHS", 8, 7], "Equality(Mul(Add(Integer(1), Mul(Integer(-1), Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Add(Pow(Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True)))), Add(Integer(1), Mul(Pow(Function('k')(Symbol('I', commutative=True)), Integer(-1)), cos(Symbol('I', commutative=True))))), Mul(Integer(-1), Integral(Integer(3), Tuple(Symbol('I', commutative=True)))))), Integer(0))"]]}, {"prompt": "Given \\omega{(\\delta)} = \\cos{(\\delta)}, then obtain g + 2 \\int \\omega{(\\delta)} d\\delta = \\int \\frac{d}{d \\delta} \\int (\\omega{(\\delta)} + \\cos{(\\delta)}) d\\delta d\\delta", "derivation": "\\omega{(\\delta)} = \\cos{(\\delta)} and 2 \\omega{(\\delta)} = \\omega{(\\delta)} + \\cos{(\\delta)} and \\int 2 \\omega{(\\delta)} d\\delta = \\int (\\omega{(\\delta)} + \\cos{(\\delta)}) d\\delta and \\frac{d}{d \\delta} \\int 2 \\omega{(\\delta)} d\\delta = \\frac{d}{d \\delta} \\int (\\omega{(\\delta)} + \\cos{(\\delta)}) d\\delta and \\int \\frac{d}{d \\delta} \\int 2 \\omega{(\\delta)} d\\delta d\\delta = \\int \\frac{d}{d \\delta} \\int (\\omega{(\\delta)} + \\cos{(\\delta)}) d\\delta d\\delta and g + 2 \\int \\omega{(\\delta)} d\\delta = \\int \\frac{d}{d \\delta} \\int (\\omega{(\\delta)} + \\cos{(\\delta)}) d\\delta d\\delta", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True)))"], [["add", 1, "Function('\\\\omega')(Symbol('\\\\delta', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Add(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Add(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Integral(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Integral(Add(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Derivative(Integral(Mul(Integer(2), Function('\\\\omega')(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))), Integral(Derivative(Integral(Add(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"], [["evaluate_integrals", 5], "Equality(Add(Symbol('g', commutative=True), Mul(Integer(2), Integral(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))), Integral(Derivative(Integral(Add(Function('\\\\omega')(Symbol('\\\\delta', commutative=True)), cos(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Tuple(Symbol('\\\\delta', commutative=True))))"]]}, {"prompt": "Given Z{(B)} = \\cos{(B)} and \\operatorname{n_{1}}{(B)} = \\cos{(B)}, then obtain 1 = \\frac{\\cos{(B)}}{Z{(B)}}", "derivation": "Z{(B)} = \\cos{(B)} and \\operatorname{n_{1}}{(B)} = \\cos{(B)} and Z{(B)} = \\operatorname{n_{1}}{(B)} and 1 = \\frac{\\cos{(B)}}{\\operatorname{n_{1}}{(B)}} and 1 = \\frac{\\cos{(B)}}{Z{(B)}}", "srepr_derivation": [["premise", "Equality(Function('Z')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], ["renaming_premise", "Equality(Function('n_1')(Symbol('B', commutative=True)), cos(Symbol('B', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('Z')(Symbol('B', commutative=True)), Function('n_1')(Symbol('B', commutative=True)))"], [["divide", 2, "Function('n_1')(Symbol('B', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('n_1')(Symbol('B', commutative=True)), Integer(-1)), cos(Symbol('B', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Integer(1), Mul(Pow(Function('Z')(Symbol('B', commutative=True)), Integer(-1)), cos(Symbol('B', commutative=True))))"]]}, {"prompt": "Given h{(n_{2},\\mathbf{J})} = \\sin{(n_{2}^{\\mathbf{J}})}, then obtain \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{h{(n_{2},\\mathbf{J})} + \\sin{(n_{2}^{\\mathbf{J}})}}{\\sin{(n_{2}^{\\mathbf{J}})}} - 1 = \\frac{d}{d \\mathbf{J}} 2 - 1", "derivation": "h{(n_{2},\\mathbf{J})} = \\sin{(n_{2}^{\\mathbf{J}})} and h{(n_{2},\\mathbf{J})} + \\sin{(n_{2}^{\\mathbf{J}})} = 2 \\sin{(n_{2}^{\\mathbf{J}})} and \\frac{h{(n_{2},\\mathbf{J})} + \\sin{(n_{2}^{\\mathbf{J}})}}{\\sin{(n_{2}^{\\mathbf{J}})}} = 2 and \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{h{(n_{2},\\mathbf{J})} + \\sin{(n_{2}^{\\mathbf{J}})}}{\\sin{(n_{2}^{\\mathbf{J}})}} = \\frac{d}{d \\mathbf{J}} 2 and \\frac{\\partial}{\\partial \\mathbf{J}} \\frac{h{(n_{2},\\mathbf{J})} + \\sin{(n_{2}^{\\mathbf{J}})}}{\\sin{(n_{2}^{\\mathbf{J}})}} - 1 = \\frac{d}{d \\mathbf{J}} 2 - 1", "srepr_derivation": [["premise", "Equality(Function('h')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))))"], [["add", 1, "sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Add(Function('h')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Mul(Integer(2), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"], [["divide", 2, "sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))"], "Equality(Mul(Add(Function('h')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Pow(sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Integer(2))"], [["differentiate", 3, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Derivative(Mul(Add(Function('h')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Pow(sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Derivative(Integer(2), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))))"], [["minus", 4, 1], "Equality(Add(Derivative(Mul(Add(Function('h')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))), Pow(sin(Pow(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(Integer(2), Tuple(Symbol('\\\\mathbf{J}', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{F}{(a^{\\dagger},h)} = - h + \\cos{(a^{\\dagger})}, then derive \\int - \\mathbf{F}{(a^{\\dagger},h)} dh = \\mathbf{B} + \\frac{h^{2}}{2} - h \\cos{(a^{\\dagger})}, then obtain 2 \\int (h - \\cos{(a^{\\dagger})}) dh = 2 \\mathbf{B} + h^{2} - 2 h \\cos{(a^{\\dagger})}", "derivation": "\\mathbf{F}{(a^{\\dagger},h)} = - h + \\cos{(a^{\\dagger})} and - \\mathbf{F}{(a^{\\dagger},h)} = h - \\cos{(a^{\\dagger})} and \\int - \\mathbf{F}{(a^{\\dagger},h)} dh = \\int (h - \\cos{(a^{\\dagger})}) dh and \\int - \\mathbf{F}{(a^{\\dagger},h)} dh = \\mathbf{B} + \\frac{h^{2}}{2} - h \\cos{(a^{\\dagger})} and \\int (h - \\cos{(a^{\\dagger})}) dh = \\mathbf{B} + \\frac{h^{2}}{2} - h \\cos{(a^{\\dagger})} and 2 \\int (h - \\cos{(a^{\\dagger})}) dh = 2 \\mathbf{B} + h^{2} - 2 h \\cos{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{F}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True)), Add(Mul(Integer(-1), Symbol('h', commutative=True)), cos(Symbol('a^{\\\\dagger}', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Add(Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["integrate", 2, "Symbol('h', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Integral(Add(Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('h', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Integer(-1), Function('\\\\mathbf{F}')(Symbol('a^{\\\\dagger}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('h', commutative=True), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Add(Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('h', commutative=True))), Add(Symbol('\\\\mathbf{B}', commutative=True), Mul(Rational(1, 2), Pow(Symbol('h', commutative=True), Integer(2))), Mul(Integer(-1), Symbol('h', commutative=True), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"], [["divide", 5, "Rational(1, 2)"], "Equality(Mul(Integer(2), Integral(Add(Symbol('h', commutative=True), Mul(Integer(-1), cos(Symbol('a^{\\\\dagger}', commutative=True)))), Tuple(Symbol('h', commutative=True)))), Add(Mul(Integer(2), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('h', commutative=True), Integer(2)), Mul(Integer(-1), Integer(2), Symbol('h', commutative=True), cos(Symbol('a^{\\\\dagger}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})} = (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu}, then obtain ((- \\mathbf{v} (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} + \\mathbf{v} \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})})^{\\mathbf{g}})^{\\mu} = (0^{\\mathbf{g}})^{\\mu}", "derivation": "\\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})} = (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} and \\mathbf{v} \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})} = \\mathbf{v} (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} and - \\mathbf{v} (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} + \\mathbf{v} \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})} = 0 and (- \\mathbf{v} (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} + \\mathbf{v} \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})})^{\\mathbf{g}} = 0^{\\mathbf{g}} and ((- \\mathbf{v} (\\frac{\\mathbf{g}}{\\mathbf{v}})^{\\mu} + \\mathbf{v} \\mathbf{J}_f{(\\mathbf{v},\\mu,\\mathbf{g})})^{\\mathbf{g}})^{\\mu} = (0^{\\mathbf{g}})^{\\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{v}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{g}', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True))))"], [["minus", 2, "Mul(Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Integer(0))"], [["power", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbf{g}', commutative=True)))"], [["power", 4, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Pow(Add(Mul(Integer(-1), Symbol('\\\\mathbf{v}', commutative=True), Pow(Mul(Symbol('\\\\mathbf{g}', commutative=True), Pow(Symbol('\\\\mathbf{v}', commutative=True), Integer(-1))), Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mathbf{v}', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\mathbf{v}', commutative=True), Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)))), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mu', commutative=True)), Pow(Pow(Integer(0), Symbol('\\\\mathbf{g}', commutative=True)), Symbol('\\\\mu', commutative=True)))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(\\mu)} = \\log{(\\cos{(\\mu)})} and \\operatorname{f^{*}}{(\\mu)} = (\\mu \\log{(\\cos{(\\mu)})})^{\\mu}, then obtain (\\mu \\operatorname{f_{E}}{(\\mu)})^{\\mu} \\operatorname{f^{*}}{(\\mu)} = (\\mu \\operatorname{f_{E}}{(\\mu)})^{2 \\mu}", "derivation": "\\operatorname{f_{E}}{(\\mu)} = \\log{(\\cos{(\\mu)})} and \\mu \\operatorname{f_{E}}{(\\mu)} = \\mu \\log{(\\cos{(\\mu)})} and (\\mu \\operatorname{f_{E}}{(\\mu)})^{\\mu} = (\\mu \\log{(\\cos{(\\mu)})})^{\\mu} and \\operatorname{f^{*}}{(\\mu)} = (\\mu \\log{(\\cos{(\\mu)})})^{\\mu} and (\\mu \\log{(\\cos{(\\mu)})})^{\\mu} \\operatorname{f^{*}}{(\\mu)} = (\\mu \\log{(\\cos{(\\mu)})})^{2 \\mu} and (\\mu \\operatorname{f_{E}}{(\\mu)})^{\\mu} \\operatorname{f^{*}}{(\\mu)} = (\\mu \\operatorname{f_{E}}{(\\mu)})^{2 \\mu}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('\\\\mu', commutative=True)), log(cos(Symbol('\\\\mu', commutative=True))))"], [["times", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Mul(Symbol('\\\\mu', commutative=True), Function('f_E')(Symbol('\\\\mu', commutative=True))), Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))))"], [["power", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mu', commutative=True), Function('f_E')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], ["renaming_premise", "Equality(Function('f^*')(Symbol('\\\\mu', commutative=True)), Pow(Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)))"], [["times", 4, "Pow(Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))), Symbol('\\\\mu', commutative=True)), Function('f^*')(Symbol('\\\\mu', commutative=True))), Pow(Mul(Symbol('\\\\mu', commutative=True), log(cos(Symbol('\\\\mu', commutative=True)))), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Mul(Pow(Mul(Symbol('\\\\mu', commutative=True), Function('f_E')(Symbol('\\\\mu', commutative=True))), Symbol('\\\\mu', commutative=True)), Function('f^*')(Symbol('\\\\mu', commutative=True))), Pow(Mul(Symbol('\\\\mu', commutative=True), Function('f_E')(Symbol('\\\\mu', commutative=True))), Mul(Integer(2), Symbol('\\\\mu', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(y)} = \\log{(y)}, then derive \\int (y + \\operatorname{E_{n}}{(y)}) dy = x + \\frac{y^{2}}{2} + y \\log{(y)} - y, then obtain x \\int (y + \\operatorname{E_{n}}{(y)}) dy = x (x + \\frac{y^{2}}{2} + y \\operatorname{E_{n}}{(y)} - y)", "derivation": "\\operatorname{E_{n}}{(y)} = \\log{(y)} and y + \\operatorname{E_{n}}{(y)} = y + \\log{(y)} and \\int (y + \\operatorname{E_{n}}{(y)}) dy = \\int (y + \\log{(y)}) dy and \\int (y + \\operatorname{E_{n}}{(y)}) dy = x + \\frac{y^{2}}{2} + y \\log{(y)} - y and \\int (y + \\operatorname{E_{n}}{(y)}) dy = x + \\frac{y^{2}}{2} + y \\operatorname{E_{n}}{(y)} - y and x \\int (y + \\operatorname{E_{n}}{(y)}) dy = x (x + \\frac{y^{2}}{2} + y \\operatorname{E_{n}}{(y)} - y)", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('y', commutative=True)), log(Symbol('y', commutative=True)))"], [["add", 1, "Symbol('y', commutative=True)"], "Equality(Add(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Add(Symbol('y', commutative=True), log(Symbol('y', commutative=True))))"], [["integrate", 2, "Symbol('y', commutative=True)"], "Equality(Integral(Add(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Integral(Add(Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), log(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integral(Add(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True))), Add(Symbol('x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True))))"], [["times", 5, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Integral(Add(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Tuple(Symbol('y', commutative=True)))), Mul(Symbol('x', commutative=True), Add(Symbol('x', commutative=True), Mul(Rational(1, 2), Pow(Symbol('y', commutative=True), Integer(2))), Mul(Symbol('y', commutative=True), Function('E_n')(Symbol('y', commutative=True))), Mul(Integer(-1), Symbol('y', commutative=True)))))"]]}, {"prompt": "Given A{(\\varphi)} = \\sin{(\\varphi)}, then obtain \\int (A^{\\varphi}{(\\varphi)} \\sin{(\\varphi)} + \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)}) d\\varphi = \\int 2 \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)} d\\varphi", "derivation": "A{(\\varphi)} = \\sin{(\\varphi)} and A^{\\varphi}{(\\varphi)} = \\sin^{\\varphi}{(\\varphi)} and A^{\\varphi}{(\\varphi)} \\sin{(\\varphi)} = \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)} and A^{\\varphi}{(\\varphi)} \\sin{(\\varphi)} + \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)} = 2 \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)} and \\int (A^{\\varphi}{(\\varphi)} \\sin{(\\varphi)} + \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)}) d\\varphi = \\int 2 \\sin{(\\varphi)} \\sin^{\\varphi}{(\\varphi)} d\\varphi", "srepr_derivation": [["premise", "Equality(Function('A')(Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True)))"], [["power", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Pow(Function('A')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], [["times", 2, "sin(Symbol('\\\\varphi', commutative=True))"], "Equality(Mul(Pow(Function('A')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Mul(sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["add", 3, "Mul(sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))"], "Equality(Add(Mul(Pow(Function('A')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Mul(sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))), Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\varphi', commutative=True)"], "Equality(Integral(Add(Mul(Pow(Function('A')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)), sin(Symbol('\\\\varphi', commutative=True))), Mul(sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True)))), Tuple(Symbol('\\\\varphi', commutative=True))), Integral(Mul(Integer(2), sin(Symbol('\\\\varphi', commutative=True)), Pow(sin(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\varphi', commutative=True))), Tuple(Symbol('\\\\varphi', commutative=True))))"]]}, {"prompt": "Given \\phi{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\operatorname{J_{\\varepsilon}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}}, then obtain - e^{L_{\\varepsilon}} = \\operatorname{J_{\\varepsilon}}{(L_{\\varepsilon})} - \\phi{(L_{\\varepsilon})} - e^{L_{\\varepsilon}}", "derivation": "\\phi{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\operatorname{J_{\\varepsilon}}{(L_{\\varepsilon})} = e^{L_{\\varepsilon}} and \\phi{(L_{\\varepsilon})} = \\operatorname{J_{\\varepsilon}}{(L_{\\varepsilon})} and - e^{L_{\\varepsilon}} = \\operatorname{J_{\\varepsilon}}{(L_{\\varepsilon})} - \\phi{(L_{\\varepsilon})} - e^{L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\phi')(Symbol('L_{\\\\varepsilon}', commutative=True)), Function('J_{\\\\varepsilon}')(Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["minus", 3, "Add(Function('\\\\phi')(Symbol('L_{\\\\varepsilon}', commutative=True)), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))"], "Equality(Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True))), Add(Function('J_{\\\\varepsilon}')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), Function('\\\\phi')(Symbol('L_{\\\\varepsilon}', commutative=True))), Mul(Integer(-1), exp(Symbol('L_{\\\\varepsilon}', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{p}{(F_{x})} = \\cos{(F_{x})}, then derive \\frac{d}{d F_{x}} \\mathbf{p}{(F_{x})} = - \\sin{(F_{x})}, then obtain - \\mathbf{p}^{F_{x}}{(F_{x})} + \\frac{d}{d F_{x}} \\cos{(F_{x})} = - \\mathbf{p}^{F_{x}}{(F_{x})} - \\sin{(F_{x})}", "derivation": "\\mathbf{p}{(F_{x})} = \\cos{(F_{x})} and \\frac{d}{d F_{x}} \\mathbf{p}{(F_{x})} = \\frac{d}{d F_{x}} \\cos{(F_{x})} and \\frac{d}{d F_{x}} \\mathbf{p}{(F_{x})} = - \\sin{(F_{x})} and \\frac{d}{d F_{x}} \\cos{(F_{x})} = - \\sin{(F_{x})} and - \\mathbf{p}^{F_{x}}{(F_{x})} + \\frac{d}{d F_{x}} \\cos{(F_{x})} = - \\mathbf{p}^{F_{x}}{(F_{x})} - \\sin{(F_{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["differentiate", 1, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True))))"], [["minus", 4, "Pow(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Derivative(cos(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Pow(Function('\\\\mathbf{p}')(Symbol('F_x', commutative=True)), Symbol('F_x', commutative=True))), Mul(Integer(-1), sin(Symbol('F_x', commutative=True)))))"]]}, {"prompt": "Given \\Omega{(\\mu,\\mathbf{H})} = \\mathbf{H} + \\mu, then derive \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} = 1, then obtain \\frac{\\int \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H} + \\mu} = \\frac{\\int 1 d\\mathbf{H}}{\\mathbf{H} + \\mu}", "derivation": "\\Omega{(\\mu,\\mathbf{H})} = \\mathbf{H} + \\mu and \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} = \\frac{\\partial}{\\partial \\mu} (\\mathbf{H} + \\mu) and \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} = 1 and \\int \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} d\\mathbf{H} = \\int 1 d\\mathbf{H} and \\frac{\\int \\frac{\\partial}{\\partial \\mu} \\Omega{(\\mu,\\mathbf{H})} d\\mathbf{H}}{\\mathbf{H} + \\mu} = \\frac{\\int 1 d\\mathbf{H}}{\\mathbf{H} + \\mu}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\Omega')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Integer(1))"], [["integrate", 3, "Symbol('\\\\mathbf{H}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\Omega')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{H}', commutative=True))))"], [["divide", 4, "Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True))"], "Equality(Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Derivative(Function('\\\\Omega')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))), Mul(Pow(Add(Symbol('\\\\mathbf{H}', commutative=True), Symbol('\\\\mu', commutative=True)), Integer(-1)), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{H}', commutative=True)))))"]]}, {"prompt": "Given \\mu_{0}{(v_{t},Z)} = \\frac{\\log{(v_{t})}}{Z}, then obtain \\frac{\\partial}{\\partial v_{t}} \\mu_{0}{(v_{t},Z)} - \\frac{1}{Z v_{t}} = 0", "derivation": "\\mu_{0}{(v_{t},Z)} = \\frac{\\log{(v_{t})}}{Z} and - Z + \\mu_{0}{(v_{t},Z)} = - Z + \\frac{\\log{(v_{t})}}{Z} and - Z + \\mu_{0}{(v_{t},Z)} - \\frac{\\log{(v_{t})}}{Z} = - Z and \\frac{\\partial}{\\partial v_{t}} (- Z + \\mu_{0}{(v_{t},Z)} - \\frac{\\log{(v_{t})}}{Z}) = \\frac{d}{d v_{t}} - Z and \\frac{\\partial}{\\partial v_{t}} \\mu_{0}{(v_{t},Z)} - \\frac{1}{Z v_{t}} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True))))"], [["minus", 1, "Symbol('Z', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mu_0')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True))), Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True)))))"], [["minus", 2, "Mul(Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mu_0')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True)))), Mul(Integer(-1), Symbol('Z', commutative=True)))"], [["differentiate", 3, "Symbol('v_t', commutative=True)"], "Equality(Derivative(Add(Mul(Integer(-1), Symbol('Z', commutative=True)), Function('\\\\mu_0')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)), log(Symbol('v_t', commutative=True)))), Tuple(Symbol('v_t', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('Z', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(Derivative(Function('\\\\mu_0')(Symbol('v_t', commutative=True), Symbol('Z', commutative=True)), Tuple(Symbol('v_t', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('Z', commutative=True), Integer(-1)), Pow(Symbol('v_t', commutative=True), Integer(-1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{t_{2}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})}, then derive (\\int \\operatorname{t_{2}}{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (C_{1} + \\sin{(\\mathbf{S})})^{\\mathbf{S}}, then obtain \\cos{((\\int \\cos{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}})} = \\cos{((C_{1} + \\sin{(\\mathbf{S})})^{\\mathbf{S}})}", "derivation": "\\operatorname{t_{2}}{(\\mathbf{S})} = \\cos{(\\mathbf{S})} and \\int \\operatorname{t_{2}}{(\\mathbf{S})} d\\mathbf{S} = \\int \\cos{(\\mathbf{S})} d\\mathbf{S} and (\\int \\operatorname{t_{2}}{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (\\int \\cos{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} and (\\int \\operatorname{t_{2}}{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (C_{1} + \\sin{(\\mathbf{S})})^{\\mathbf{S}} and (\\int \\cos{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}} = (C_{1} + \\sin{(\\mathbf{S})})^{\\mathbf{S}} and \\cos{((\\int \\cos{(\\mathbf{S})} d\\mathbf{S})^{\\mathbf{S}})} = \\cos{((C_{1} + \\sin{(\\mathbf{S})})^{\\mathbf{S}})}", "srepr_derivation": [["premise", "Equality(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True)))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["power", 2, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('t_2')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Pow(Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)), Pow(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True)))"], [["cos", 5], "Equality(cos(Pow(Integral(cos(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True))), cos(Pow(Add(Symbol('C_1', commutative=True), sin(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given q{(v_{1},s)} = s v_{1} and \\operatorname{C_{2}}{(v_{1},s)} = s v_{1}, then obtain \\operatorname{C_{2}}{(v_{1},s)} q{(v_{1},s)} = \\operatorname{C_{2}}^{2}{(v_{1},s)}", "derivation": "q{(v_{1},s)} = s v_{1} and \\operatorname{C_{2}}{(v_{1},s)} = s v_{1} and \\operatorname{C_{2}}{(v_{1},s)} q{(v_{1},s)} = s v_{1} \\operatorname{C_{2}}{(v_{1},s)} and \\operatorname{C_{2}}{(v_{1},s)} q{(v_{1},s)} = \\operatorname{C_{2}}^{2}{(v_{1},s)}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v_1', commutative=True)))"], ["renaming_premise", "Equality(Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Mul(Symbol('s', commutative=True), Symbol('v_1', commutative=True)))"], [["times", 1, "Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True))"], "Equality(Mul(Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Function('q')(Symbol('v_1', commutative=True), Symbol('s', commutative=True))), Mul(Symbol('s', commutative=True), Symbol('v_1', commutative=True), Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Function('q')(Symbol('v_1', commutative=True), Symbol('s', commutative=True))), Pow(Function('C_2')(Symbol('v_1', commutative=True), Symbol('s', commutative=True)), Integer(2)))"]]}, {"prompt": "Given L{(\\chi)} = \\cos{(\\chi)}, then obtain - \\tilde{g}^*{(\\chi)} + \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi) = - \\tilde{g}^*{(\\chi)} + \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int 0 d\\chi)", "derivation": "L{(\\chi)} = \\cos{(\\chi)} and L{(\\chi)} - \\cos{(\\chi)} = 0 and \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi = \\int 0 d\\chi and \\chi + \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi = \\chi + \\int 0 d\\chi and \\frac{d}{d \\chi} (\\chi + \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi) = \\frac{d}{d \\chi} (\\chi + \\int 0 d\\chi) and \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi) = \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int 0 d\\chi) and - \\tilde{g}^*{(\\chi)} + \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int (L{(\\chi)} - \\cos{(\\chi)}) d\\chi) = - \\tilde{g}^*{(\\chi)} + \\frac{d^{2}}{d \\chi^{2}} (\\chi + \\int 0 d\\chi)", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\chi', commutative=True)), cos(Symbol('\\\\chi', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Integer(0))"], [["integrate", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True))))"], [["add", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Add(Symbol('\\\\chi', commutative=True), Integral(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Add(Symbol('\\\\chi', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))))"], [["differentiate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(1))))"], [["differentiate", 5, "Symbol('\\\\chi', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))), Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2))))"], [["minus", 6, "Function('\\\\tilde{g}^*')(Symbol('\\\\chi', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\chi', commutative=True))), Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Add(Function('L')(Symbol('\\\\chi', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))), Add(Mul(Integer(-1), Function('\\\\tilde{g}^*')(Symbol('\\\\chi', commutative=True))), Derivative(Add(Symbol('\\\\chi', commutative=True), Integral(Integer(0), Tuple(Symbol('\\\\chi', commutative=True)))), Tuple(Symbol('\\\\chi', commutative=True), Integer(2)))))"]]}, {"prompt": "Given q{(k,\\tilde{g}^*)} = - \\tilde{g}^* + \\sin{(k)} and \\hat{H}{(k,\\tilde{g}^*)} = - \\tilde{g}^* + \\sin{(k)} + \\frac{1}{\\tilde{g}^*}, then obtain 1 = (\\frac{\\hat{H}{(k,\\tilde{g}^*)}}{q{(k,\\tilde{g}^*)} + \\frac{1}{\\tilde{g}^*}})^{k}", "derivation": "q{(k,\\tilde{g}^*)} = - \\tilde{g}^* + \\sin{(k)} and q{(k,\\tilde{g}^*)} + \\frac{1}{\\tilde{g}^*} = - \\tilde{g}^* + \\sin{(k)} + \\frac{1}{\\tilde{g}^*} and 1 = \\frac{- \\tilde{g}^* + \\sin{(k)} + \\frac{1}{\\tilde{g}^*}}{q{(k,\\tilde{g}^*)} + \\frac{1}{\\tilde{g}^*}} and \\hat{H}{(k,\\tilde{g}^*)} = - \\tilde{g}^* + \\sin{(k)} + \\frac{1}{\\tilde{g}^*} and 1 = (\\frac{- \\tilde{g}^* + \\sin{(k)} + \\frac{1}{\\tilde{g}^*}}{q{(k,\\tilde{g}^*)} + \\frac{1}{\\tilde{g}^*}})^{k} and 1 = (\\frac{\\hat{H}{(k,\\tilde{g}^*)}}{q{(k,\\tilde{g}^*)} + \\frac{1}{\\tilde{g}^*}})^{k}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('k', commutative=True))))"], [["add", 1, "Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))"], "Equality(Add(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('k', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))))"], [["divide", 2, "Add(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)))"], "Equality(Integer(1), Mul(Pow(Add(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('k', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('k', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))))"], [["power", 3, "Symbol('k', commutative=True)"], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)), sin(Symbol('k', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1)))), Symbol('k', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Integer(1), Pow(Mul(Pow(Add(Function('q')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True)), Pow(Symbol('\\\\tilde{g}^*', commutative=True), Integer(-1))), Integer(-1)), Function('\\\\hat{H}')(Symbol('k', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('k', commutative=True)))"]]}, {"prompt": "Given f{(E,n)} = \\cos{(E + n)} and \\mathbf{H}{(E,n)} = E + n, then obtain (2 f{(E,n)} + 1)^{n} = (f{(E,n)} + \\cos{(E + n)} + 1)^{n}", "derivation": "f{(E,n)} = \\cos{(E + n)} and f{(E,n)} + 1 = \\cos{(E + n)} + 1 and \\mathbf{H}{(E,n)} = E + n and f{(E,n)} = \\cos{(\\mathbf{H}{(E,n)})} and f{(E,n)} + \\cos{(E + n)} + 1 = \\cos{(E + n)} + \\cos{(\\mathbf{H}{(E,n)})} + 1 and 2 f{(E,n)} + 1 = f{(E,n)} + \\cos{(\\mathbf{H}{(E,n)})} + 1 and 2 f{(E,n)} + 1 = f{(E,n)} + \\cos{(E + n)} + 1 and (2 f{(E,n)} + 1)^{n} = (f{(E,n)} + \\cos{(E + n)} + 1)^{n}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), Integer(1)), Add(cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('n', commutative=True)), Add(Symbol('E', commutative=True), Symbol('n', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('n', commutative=True))))"], [["add", 4, "Add(cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1))"], "Equality(Add(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)), Add(cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), cos(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Integer(2), Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)), Add(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Function('\\\\mathbf{H}')(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Mul(Integer(2), Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)), Add(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)))"], [["power", 7, "Symbol('n', commutative=True)"], "Equality(Pow(Add(Mul(Integer(2), Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)), Symbol('n', commutative=True)), Pow(Add(Function('f')(Symbol('E', commutative=True), Symbol('n', commutative=True)), cos(Add(Symbol('E', commutative=True), Symbol('n', commutative=True))), Integer(1)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given \\hat{H}_l{(B,\\theta_2)} = B e^{\\theta_2}, then obtain (\\int \\hat{H}_l^{\\theta_2}{(B,\\theta_2)} dB)^{\\theta_2} = (\\int (B e^{\\theta_2})^{\\theta_2} dB)^{\\theta_2}", "derivation": "\\hat{H}_l{(B,\\theta_2)} = B e^{\\theta_2} and \\hat{H}_l^{\\theta_2}{(B,\\theta_2)} = (B e^{\\theta_2})^{\\theta_2} and \\int \\hat{H}_l^{\\theta_2}{(B,\\theta_2)} dB = \\int (B e^{\\theta_2})^{\\theta_2} dB and (\\int \\hat{H}_l^{\\theta_2}{(B,\\theta_2)} dB)^{\\theta_2} = (\\int (B e^{\\theta_2})^{\\theta_2} dB)^{\\theta_2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('B', commutative=True), Symbol('\\\\theta_2', commutative=True)), Mul(Symbol('B', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))))"], [["power", 1, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Pow(Mul(Symbol('B', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"], [["integrate", 2, "Symbol('B', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('B', commutative=True))), Integral(Pow(Mul(Symbol('B', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('B', commutative=True))))"], [["power", 3, "Symbol('\\\\theta_2', commutative=True)"], "Equality(Pow(Integral(Pow(Function('\\\\hat{H}_l')(Symbol('B', commutative=True), Symbol('\\\\theta_2', commutative=True)), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Pow(Integral(Pow(Mul(Symbol('B', commutative=True), exp(Symbol('\\\\theta_2', commutative=True))), Symbol('\\\\theta_2', commutative=True)), Tuple(Symbol('B', commutative=True))), Symbol('\\\\theta_2', commutative=True)))"]]}, {"prompt": "Given m{(Q)} = \\cos{(Q)}, then derive \\frac{d}{d Q} m{(Q)} = - \\sin{(Q)}, then obtain (\\frac{d}{d Q} - \\sin{(Q)})^{Q} \\int \\operatorname{J_{\\varepsilon}}{(T)} dT = (\\frac{d^{2}}{d Q^{2}} m{(Q)})^{Q} \\int \\operatorname{J_{\\varepsilon}}{(T)} dT", "derivation": "m{(Q)} = \\cos{(Q)} and \\frac{d}{d Q} m{(Q)} = \\frac{d}{d Q} \\cos{(Q)} and \\frac{d}{d Q} m{(Q)} = - \\sin{(Q)} and \\frac{d}{d Q} \\cos{(Q)} = - \\sin{(Q)} and \\frac{d^{2}}{d Q^{2}} \\cos{(Q)} = \\frac{d}{d Q} - \\sin{(Q)} and (\\frac{d^{2}}{d Q^{2}} \\cos{(Q)})^{Q} = (\\frac{d}{d Q} - \\sin{(Q)})^{Q} and (\\frac{d^{2}}{d Q^{2}} \\cos{(Q)})^{Q} = (\\frac{d^{2}}{d Q^{2}} m{(Q)})^{Q} and (\\frac{d}{d Q} - \\sin{(Q)})^{Q} = (\\frac{d^{2}}{d Q^{2}} m{(Q)})^{Q} and (\\frac{d}{d Q} - \\sin{(Q)})^{Q} \\int \\operatorname{J_{\\varepsilon}}{(T)} dT = (\\frac{d^{2}}{d Q^{2}} m{(Q)})^{Q} \\int \\operatorname{J_{\\varepsilon}}{(T)} dT", "srepr_derivation": [["get_premise", "Equality(Function('m')(Symbol('Q', commutative=True)), cos(Symbol('Q', commutative=True)))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('m')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('m')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), sin(Symbol('Q', commutative=True))))"], [["differentiate", 4, "Symbol('Q', commutative=True)"], "Equality(Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Derivative(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["power", 5, "Symbol('Q', commutative=True)"], "Equality(Pow(Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)), Pow(Derivative(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Derivative(cos(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)), Pow(Derivative(Function('m')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)))"], [["substitute_LHS_for_RHS", 7, 6], "Equality(Pow(Derivative(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Pow(Derivative(Function('m')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)))"], [["times", 8, "Integral(Function('J_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))"], "Equality(Mul(Pow(Derivative(Mul(Integer(-1), sin(Symbol('Q', commutative=True))), Tuple(Symbol('Q', commutative=True), Integer(1))), Symbol('Q', commutative=True)), Integral(Function('J_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))), Mul(Pow(Derivative(Function('m')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(2))), Symbol('Q', commutative=True)), Integral(Function('J_{\\\\varepsilon}')(Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True)))))"]]}, {"prompt": "Given \\rho{(\\hat{p}_0)} = e^{\\hat{p}_0}, then derive - e^{\\hat{p}_0} + \\frac{d}{d \\hat{p}_0} \\rho{(\\hat{p}_0)} = 0, then obtain \\int (- e^{\\hat{p}_0} + \\frac{d}{d \\hat{p}_0} e^{\\hat{p}_0}) d\\hat{p}_0 = \\int 0 d\\hat{p}_0", "derivation": "\\rho{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\rho{(\\hat{p}_0)} - e^{\\hat{p}_0} = 0 and \\frac{d}{d \\hat{p}_0} (\\rho{(\\hat{p}_0)} - e^{\\hat{p}_0}) = \\frac{d}{d \\hat{p}_0} 0 and - e^{\\hat{p}_0} + \\frac{d}{d \\hat{p}_0} \\rho{(\\hat{p}_0)} = 0 and \\int (- e^{\\hat{p}_0} + \\frac{d}{d \\hat{p}_0} \\rho{(\\hat{p}_0)}) d\\hat{p}_0 = \\int 0 d\\hat{p}_0 and \\int (- e^{\\hat{p}_0} + \\frac{d}{d \\hat{p}_0} e^{\\hat{p}_0}) d\\hat{p}_0 = \\int 0 d\\hat{p}_0", "srepr_derivation": [["get_premise", "Equality(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["minus", 1, "exp(Symbol('\\\\hat{p}_0', commutative=True))"], "Equality(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Derivative(Add(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), Derivative(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Integer(0))"], [["integrate", 4, "Symbol('\\\\hat{p}_0', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), Derivative(Function('\\\\rho')(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 1], "Equality(Integral(Add(Mul(Integer(-1), exp(Symbol('\\\\hat{p}_0', commutative=True))), Derivative(exp(Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\hat{p}_0', commutative=True), Integer(1)))), Tuple(Symbol('\\\\hat{p}_0', commutative=True))), Integral(Integer(0), Tuple(Symbol('\\\\hat{p}_0', commutative=True))))"]]}, {"prompt": "Given E{(\\rho_f,r_{0})} = \\rho_f - r_{0}, then obtain - \\rho_f + E{(\\rho_f,r_{0})} - 1 = - r_{0} - 1", "derivation": "E{(\\rho_f,r_{0})} = \\rho_f - r_{0} and \\frac{\\partial}{\\partial r_{0}} E{(\\rho_f,r_{0})} = \\frac{\\partial}{\\partial r_{0}} (\\rho_f - r_{0}) and E{(\\rho_f,r_{0})} + \\frac{\\partial}{\\partial r_{0}} E{(\\rho_f,r_{0})} = \\rho_f - r_{0} + \\frac{\\partial}{\\partial r_{0}} E{(\\rho_f,r_{0})} and E{(\\rho_f,r_{0})} + \\frac{\\partial}{\\partial r_{0}} (\\rho_f - r_{0}) = \\rho_f - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho_f - r_{0}) and - \\rho_f + E{(\\rho_f,r_{0})} + \\frac{\\partial}{\\partial r_{0}} (\\rho_f - r_{0}) = - r_{0} + \\frac{\\partial}{\\partial r_{0}} (\\rho_f - r_{0}) and - \\rho_f + E{(\\rho_f,r_{0})} - 1 = - r_{0} - 1", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))))"], [["differentiate", 1, "Symbol('r_0', commutative=True)"], "Equality(Derivative(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1))))"], [["add", 1, "Derivative(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))"], "Equality(Add(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Derivative(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["minus", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Derivative(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('r_0', commutative=True))), Tuple(Symbol('r_0', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Add(Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)), Function('E')(Symbol('\\\\rho_f', commutative=True), Symbol('r_0', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('r_0', commutative=True)), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_f{(f_{E})} = \\cos{(f_{E})}, then obtain \\mathbf{J}_f{(f_{E})} - \\iint \\mathbf{J}_f{(f_{E})} df_{E} df_{E} = \\cos{(f_{E})} - \\iint \\mathbf{J}_f{(f_{E})} df_{E} df_{E}", "derivation": "\\mathbf{J}_f{(f_{E})} = \\cos{(f_{E})} and \\int \\mathbf{J}_f{(f_{E})} df_{E} = \\int \\cos{(f_{E})} df_{E} and \\iint \\mathbf{J}_f{(f_{E})} df_{E} df_{E} = \\iint \\cos{(f_{E})} df_{E} df_{E} and \\mathbf{J}_f{(f_{E})} - \\iint \\cos{(f_{E})} df_{E} df_{E} = \\cos{(f_{E})} - \\iint \\cos{(f_{E})} df_{E} df_{E} and \\mathbf{J}_f{(f_{E})} - \\iint \\mathbf{J}_f{(f_{E})} df_{E} df_{E} = \\cos{(f_{E})} - \\iint \\mathbf{J}_f{(f_{E})} df_{E} df_{E}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), cos(Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["integrate", 2, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["minus", 1, "Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))), Add(cos(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(cos(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))), Add(cos(Symbol('f_E', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\mathbf{J}_f')(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))))"]]}, {"prompt": "Given f{(r)} = \\frac{d}{d r} e^{r}, then derive f{(r)} = e^{r}, then obtain r f{(r)} \\frac{d^{2}}{d r^{2}} e^{r} = r f{(r)} \\frac{d^{3}}{d r^{3}} e^{r}", "derivation": "f{(r)} = \\frac{d}{d r} e^{r} and f{(r)} = e^{r} and f{(r)} = \\frac{d}{d r} f{(r)} and \\frac{d}{d r} f{(r)} = \\frac{d^{2}}{d r^{2}} f{(r)} and \\frac{d^{2}}{d r^{2}} e^{r} = \\frac{d^{3}}{d r^{3}} e^{r} and r f{(r)} \\frac{d^{2}}{d r^{2}} e^{r} = r f{(r)} \\frac{d^{3}}{d r^{3}} e^{r}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('r', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f')(Symbol('r', commutative=True)), exp(Symbol('r', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('f')(Symbol('r', commutative=True)), Derivative(Function('f')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('r', commutative=True)"], "Equality(Derivative(Function('f')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(1))), Derivative(Function('f')(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2))), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(3))))"], [["times", 5, "Mul(Symbol('r', commutative=True), Function('f')(Symbol('r', commutative=True)))"], "Equality(Mul(Symbol('r', commutative=True), Function('f')(Symbol('r', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(2)))), Mul(Symbol('r', commutative=True), Function('f')(Symbol('r', commutative=True)), Derivative(exp(Symbol('r', commutative=True)), Tuple(Symbol('r', commutative=True), Integer(3)))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(A_{2})} = e^{A_{2}} and \\operatorname{A_{1}}{(A_{2})} = (e^{A_{2}})^{A_{2}}, then obtain (\\frac{d}{d A_{2}} \\operatorname{A_{1}}{(A_{2})})^{A_{2}} = (\\frac{d}{d A_{2}} \\operatorname{v_{1}}^{A_{2}}{(A_{2})})^{A_{2}}", "derivation": "\\operatorname{v_{1}}{(A_{2})} = e^{A_{2}} and \\operatorname{A_{1}}{(A_{2})} = (e^{A_{2}})^{A_{2}} and \\operatorname{A_{1}}{(A_{2})} = \\operatorname{v_{1}}^{A_{2}}{(A_{2})} and \\frac{d}{d A_{2}} \\operatorname{A_{1}}{(A_{2})} = \\frac{d}{d A_{2}} \\operatorname{v_{1}}^{A_{2}}{(A_{2})} and (\\frac{d}{d A_{2}} \\operatorname{A_{1}}{(A_{2})})^{A_{2}} = (\\frac{d}{d A_{2}} \\operatorname{v_{1}}^{A_{2}}{(A_{2})})^{A_{2}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('A_2', commutative=True)), exp(Symbol('A_2', commutative=True)))"], ["renaming_premise", "Equality(Function('A_1')(Symbol('A_2', commutative=True)), Pow(exp(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('A_1')(Symbol('A_2', commutative=True)), Pow(Function('v_1')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)))"], [["differentiate", 3, "Symbol('A_2', commutative=True)"], "Equality(Derivative(Function('A_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Derivative(Pow(Function('v_1')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))))"], [["power", 4, "Symbol('A_2', commutative=True)"], "Equality(Pow(Derivative(Function('A_1')(Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('A_2', commutative=True)), Pow(Derivative(Pow(Function('v_1')(Symbol('A_2', commutative=True)), Symbol('A_2', commutative=True)), Tuple(Symbol('A_2', commutative=True), Integer(1))), Symbol('A_2', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(z)} = \\log{(z)}, then derive \\frac{z (\\frac{\\frac{d}{d z} \\operatorname{v_{t}}{(z)}}{z} - \\frac{\\operatorname{v_{t}}{(z)}}{z^{2}} + \\frac{\\log{(z)}}{z^{2}} - \\frac{1}{z^{2}})}{\\log{(z)}} = 0, then obtain \\frac{z (\\frac{\\frac{d}{d z} \\log{(z)}}{z} - \\frac{1}{z^{2}})}{\\log{(z)}} = 0", "derivation": "\\operatorname{v_{t}}{(z)} = \\log{(z)} and \\frac{\\operatorname{v_{t}}{(z)}}{z} = \\frac{\\log{(z)}}{z} and \\frac{\\operatorname{v_{t}}{(z)}}{z} - \\frac{\\log{(z)}}{z} = 0 and \\frac{d}{d z} (\\frac{\\operatorname{v_{t}}{(z)}}{z} - \\frac{\\log{(z)}}{z}) = \\frac{d}{d z} 0 and \\frac{z \\frac{d}{d z} (\\frac{\\operatorname{v_{t}}{(z)}}{z} - \\frac{\\log{(z)}}{z})}{\\log{(z)}} = \\frac{z \\frac{d}{d z} 0}{\\log{(z)}} and \\frac{z (\\frac{\\frac{d}{d z} \\operatorname{v_{t}}{(z)}}{z} - \\frac{\\operatorname{v_{t}}{(z)}}{z^{2}} + \\frac{\\log{(z)}}{z^{2}} - \\frac{1}{z^{2}})}{\\log{(z)}} = 0 and \\frac{z (\\frac{\\frac{d}{d z} \\log{(z)}}{z} - \\frac{1}{z^{2}})}{\\log{(z)}} = 0", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('z', commutative=True)), log(Symbol('z', commutative=True)))"], [["divide", 1, "Symbol('z', commutative=True)"], "Equality(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('v_t')(Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('v_t')(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))), Integer(0))"], [["differentiate", 3, "Symbol('z', commutative=True)"], "Equality(Derivative(Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('v_t')(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1))))"], [["divide", 4, "Mul(Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))"], "Equality(Mul(Symbol('z', commutative=True), Pow(log(Symbol('z', commutative=True)), Integer(-1)), Derivative(Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Function('v_t')(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-1)), log(Symbol('z', commutative=True)))), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Symbol('z', commutative=True), Pow(log(Symbol('z', commutative=True)), Integer(-1)), Derivative(Integer(0), Tuple(Symbol('z', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 5], "Equality(Mul(Symbol('z', commutative=True), Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(Function('v_t')(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-2)), Function('v_t')(Symbol('z', commutative=True))), Mul(Pow(Symbol('z', commutative=True), Integer(-2)), log(Symbol('z', commutative=True))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-2)))), Pow(log(Symbol('z', commutative=True)), Integer(-1))), Integer(0))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Mul(Symbol('z', commutative=True), Add(Mul(Pow(Symbol('z', commutative=True), Integer(-1)), Derivative(log(Symbol('z', commutative=True)), Tuple(Symbol('z', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('z', commutative=True), Integer(-2)))), Pow(log(Symbol('z', commutative=True)), Integer(-1))), Integer(0))"]]}, {"prompt": "Given \\rho{(F_{H},t_{2},\\delta)} = (- F_{H} + \\delta)^{t_{2}} and \\operatorname{v_{x}}{(\\delta,F_{H},t_{2})} = \\frac{t_{2} (- F_{H} + \\delta)^{t_{2}}}{- F_{H} + \\delta}, then derive \\frac{\\partial}{\\partial \\delta} \\rho{(F_{H},t_{2},\\delta)} + 1 = \\frac{t_{2} (- F_{H} + \\delta)^{t_{2}}}{- F_{H} + \\delta} + 1, then obtain \\frac{\\partial}{\\partial \\delta} (- F_{H} + \\delta)^{t_{2}} + 1 = \\operatorname{v_{x}}{(\\delta,F_{H},t_{2})} + 1", "derivation": "\\rho{(F_{H},t_{2},\\delta)} = (- F_{H} + \\delta)^{t_{2}} and \\delta + \\rho{(F_{H},t_{2},\\delta)} = \\delta + (- F_{H} + \\delta)^{t_{2}} and \\frac{\\partial}{\\partial \\delta} (\\delta + \\rho{(F_{H},t_{2},\\delta)}) = \\frac{\\partial}{\\partial \\delta} (\\delta + (- F_{H} + \\delta)^{t_{2}}) and \\frac{\\partial}{\\partial \\delta} \\rho{(F_{H},t_{2},\\delta)} + 1 = \\frac{t_{2} (- F_{H} + \\delta)^{t_{2}}}{- F_{H} + \\delta} + 1 and \\frac{\\partial}{\\partial \\delta} (- F_{H} + \\delta)^{t_{2}} + 1 = \\frac{t_{2} (- F_{H} + \\delta)^{t_{2}}}{- F_{H} + \\delta} + 1 and \\operatorname{v_{x}}{(\\delta,F_{H},t_{2})} = \\frac{t_{2} (- F_{H} + \\delta)^{t_{2}}}{- F_{H} + \\delta} and \\frac{\\partial}{\\partial \\delta} (- F_{H} + \\delta)^{t_{2}} + 1 = \\operatorname{v_{x}}{(\\delta,F_{H},t_{2})} + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\rho')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\delta', commutative=True), Function('\\\\rho')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\delta', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Derivative(Function('\\\\rho')(Symbol('F_H', commutative=True), Symbol('t_2', commutative=True), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(Mul(Symbol('t_2', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True))), Integer(1)))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Derivative(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(Mul(Symbol('t_2', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True))), Integer(1)))"], ["renaming_premise", "Equality(Function('v_x')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Mul(Symbol('t_2', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Integer(-1)), Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(Derivative(Pow(Add(Mul(Integer(-1), Symbol('F_H', commutative=True)), Symbol('\\\\delta', commutative=True)), Symbol('t_2', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integer(1)), Add(Function('v_x')(Symbol('\\\\delta', commutative=True), Symbol('F_H', commutative=True), Symbol('t_2', commutative=True)), Integer(1)))"]]}, {"prompt": "Given \\dot{z}{(\\varphi,G,\\omega)} = \\frac{\\omega \\varphi}{G} and \\mathbf{A}{(\\varphi,G,\\omega)} = \\frac{\\omega \\varphi}{G}, then obtain - \\mathbf{A}{(\\varphi,G,\\omega)} = \\omega \\varphi (- \\dot{z}{(\\varphi,G,\\omega)} + \\frac{\\omega \\varphi}{G}) - \\mathbf{A}{(\\varphi,G,\\omega)}", "derivation": "\\dot{z}{(\\varphi,G,\\omega)} = \\frac{\\omega \\varphi}{G} and \\mathbf{A}{(\\varphi,G,\\omega)} = \\frac{\\omega \\varphi}{G} and \\mathbf{A}{(\\varphi,G,\\omega)} = \\dot{z}{(\\varphi,G,\\omega)} and 0 = - \\mathbf{A}{(\\varphi,G,\\omega)} + \\frac{\\omega \\varphi}{G} and 0 = - \\dot{z}{(\\varphi,G,\\omega)} + \\frac{\\omega \\varphi}{G} and 0 = \\omega \\varphi (- \\dot{z}{(\\varphi,G,\\omega)} + \\frac{\\omega \\varphi}{G}) and - \\mathbf{A}{(\\varphi,G,\\omega)} = \\omega \\varphi (- \\dot{z}{(\\varphi,G,\\omega)} + \\frac{\\omega \\varphi}{G}) - \\mathbf{A}{(\\varphi,G,\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True)), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True)), Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True)))"], [["minus", 2, "Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))))"], [["times", 5, "Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True))"], "Equality(Integer(0), Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)))))"], [["minus", 6, "Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))"], "Equality(Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))), Add(Mul(Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True), Add(Mul(Integer(-1), Function('\\\\dot{z}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True))), Mul(Pow(Symbol('G', commutative=True), Integer(-1)), Symbol('\\\\omega', commutative=True), Symbol('\\\\varphi', commutative=True)))), Mul(Integer(-1), Function('\\\\mathbf{A}')(Symbol('\\\\varphi', commutative=True), Symbol('G', commutative=True), Symbol('\\\\omega', commutative=True)))))"]]}, {"prompt": "Given \\tilde{g}{(\\mathbf{J}_f)} = \\log{(\\log{(\\mathbf{J}_f)})}, then derive \\frac{d}{d \\mathbf{J}_f} \\tilde{g}{(\\mathbf{J}_f)} = \\frac{1}{\\mathbf{J}_f \\log{(\\mathbf{J}_f)}}, then obtain \\frac{d}{d \\mathbf{J}_f} \\log{(\\log{(\\mathbf{J}_f)})} = \\frac{1}{\\mathbf{J}_f \\log{(\\mathbf{J}_f)}}", "derivation": "\\tilde{g}{(\\mathbf{J}_f)} = \\log{(\\log{(\\mathbf{J}_f)})} and \\frac{d}{d \\mathbf{J}_f} \\tilde{g}{(\\mathbf{J}_f)} = \\frac{d}{d \\mathbf{J}_f} \\log{(\\log{(\\mathbf{J}_f)})} and \\frac{d}{d \\mathbf{J}_f} \\tilde{g}{(\\mathbf{J}_f)} = \\frac{1}{\\mathbf{J}_f \\log{(\\mathbf{J}_f)}} and \\frac{d}{d \\mathbf{J}_f} \\log{(\\log{(\\mathbf{J}_f)})} = \\frac{1}{\\mathbf{J}_f \\log{(\\mathbf{J}_f)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), log(log(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(log(log(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\tilde{g}')(Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(log(Symbol('\\\\mathbf{J}_f', commutative=True))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(-1)), Pow(log(Symbol('\\\\mathbf{J}_f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(t_{2},\\eta)} = \\eta t_{2}, then obtain 2 (\\eta t_{2})^{\\eta} \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)} = (\\eta t_{2})^{2 \\eta} + (\\eta t_{2})^{\\eta} \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)}", "derivation": "\\operatorname{E_{\\lambda}}{(t_{2},\\eta)} = \\eta t_{2} and \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)} = (\\eta t_{2})^{\\eta} and (\\eta t_{2})^{\\eta} \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)} = (\\eta t_{2})^{2 \\eta} and 2 (\\eta t_{2})^{\\eta} \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)} = (\\eta t_{2})^{2 \\eta} + (\\eta t_{2})^{\\eta} \\operatorname{E_{\\lambda}}^{\\eta}{(t_{2},\\eta)}", "srepr_derivation": [["get_premise", "Equality(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)))"], [["power", 1, "Symbol('\\\\eta', commutative=True)"], "Equality(Pow(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True)))"], [["times", 2, "Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True))"], "Equality(Mul(Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))))"], [["add", 3, "Mul(Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))"], "Equality(Mul(Integer(2), Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True))), Add(Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Mul(Integer(2), Symbol('\\\\eta', commutative=True))), Mul(Pow(Mul(Symbol('\\\\eta', commutative=True), Symbol('t_2', commutative=True)), Symbol('\\\\eta', commutative=True)), Pow(Function('E_{\\\\lambda}')(Symbol('t_2', commutative=True), Symbol('\\\\eta', commutative=True)), Symbol('\\\\eta', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(m_{s})} = e^{m_{s}}, then obtain \\operatorname{c_{0}}^{m_{s}}{(m_{s})} + \\frac{d}{d m_{s}} \\operatorname{c_{0}}^{m_{s}}{(m_{s})} = \\operatorname{c_{0}}^{m_{s}}{(m_{s})} + \\frac{d}{d m_{s}} (e^{m_{s}})^{m_{s}}", "derivation": "\\operatorname{c_{0}}{(m_{s})} = e^{m_{s}} and \\operatorname{c_{0}}^{m_{s}}{(m_{s})} = (e^{m_{s}})^{m_{s}} and \\frac{d}{d m_{s}} \\operatorname{c_{0}}^{m_{s}}{(m_{s})} = \\frac{d}{d m_{s}} (e^{m_{s}})^{m_{s}} and \\operatorname{c_{0}}^{m_{s}}{(m_{s})} + \\frac{d}{d m_{s}} \\operatorname{c_{0}}^{m_{s}}{(m_{s})} = \\operatorname{c_{0}}^{m_{s}}{(m_{s})} + \\frac{d}{d m_{s}} (e^{m_{s}})^{m_{s}}", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('m_s', commutative=True)), exp(Symbol('m_s', commutative=True)))"], [["power", 1, "Symbol('m_s', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Pow(exp(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)))"], [["differentiate", 2, "Symbol('m_s', commutative=True)"], "Equality(Derivative(Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))), Derivative(Pow(exp(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1))))"], [["add", 3, "Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True))"], "Equality(Add(Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Derivative(Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))), Add(Pow(Function('c_0')(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Derivative(Pow(exp(Symbol('m_s', commutative=True)), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\omega{(\\hat{H}_l,I)} = \\hat{H}_l^{I}, then obtain - \\hat{H}_l^{- I} = - \\frac{1}{\\omega{(\\hat{H}_l,I)}}", "derivation": "\\omega{(\\hat{H}_l,I)} = \\hat{H}_l^{I} and 1 = \\frac{\\hat{H}_l^{I}}{\\omega{(\\hat{H}_l,I)}} and \\hat{H}_l^{- I} = \\frac{1}{\\omega{(\\hat{H}_l,I)}} and - \\hat{H}_l^{- I} = - \\frac{1}{\\omega{(\\hat{H}_l,I)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)), Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)))"], [["divide", 1, "Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True))"], "Equality(Integer(1), Mul(Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)), Pow(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)), Integer(-1))))"], [["divide", 2, "Pow(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True))"], "Equality(Pow(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True))), Pow(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)), Integer(-1)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}_l', commutative=True), Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Integer(-1), Pow(Function('\\\\omega')(Symbol('\\\\hat{H}_l', commutative=True), Symbol('I', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\mathbb{I}{(\\mathbf{D},L)} = L^{\\mathbf{D}} and T{(\\mathbf{D},L)} = L^{\\mathbf{D}}, then obtain \\frac{T^{L}{(\\mathbf{D},L)}}{L + \\int L^{\\mathbf{D}} dL} = \\frac{(L^{\\mathbf{D}})^{L}}{L + \\int L^{\\mathbf{D}} dL}", "derivation": "\\mathbb{I}{(\\mathbf{D},L)} = L^{\\mathbf{D}} and \\int \\mathbb{I}{(\\mathbf{D},L)} dL = \\int L^{\\mathbf{D}} dL and L + \\int \\mathbb{I}{(\\mathbf{D},L)} dL = L + \\int L^{\\mathbf{D}} dL and T{(\\mathbf{D},L)} = L^{\\mathbf{D}} and T^{L}{(\\mathbf{D},L)} = (L^{\\mathbf{D}})^{L} and \\frac{T^{L}{(\\mathbf{D},L)}}{L + \\int \\mathbb{I}{(\\mathbf{D},L)} dL} = \\frac{(L^{\\mathbf{D}})^{L}}{L + \\int \\mathbb{I}{(\\mathbf{D},L)} dL} and \\frac{T^{L}{(\\mathbf{D},L)}}{L + \\int L^{\\mathbf{D}} dL} = \\frac{(L^{\\mathbf{D}})^{L}}{L + \\int L^{\\mathbf{D}} dL}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["integrate", 1, "Symbol('L', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('L', commutative=True))))"], [["add", 2, "Symbol('L', commutative=True)"], "Equality(Add(Symbol('L', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Add(Symbol('L', commutative=True), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('L', commutative=True)))))"], ["renaming_premise", "Equality(Function('T')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)))"], [["power", 4, "Symbol('L', commutative=True)"], "Equality(Pow(Function('T')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True)), Pow(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('L', commutative=True)))"], [["divide", 5, "Add(Symbol('L', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True))))"], "Equality(Mul(Pow(Add(Symbol('L', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integer(-1)), Pow(Function('T')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Pow(Add(Symbol('L', commutative=True), Integral(Function('\\\\mathbb{I}')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integer(-1)), Pow(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('L', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Mul(Pow(Add(Symbol('L', commutative=True), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integer(-1)), Pow(Function('T')(Symbol('\\\\mathbf{D}', commutative=True), Symbol('L', commutative=True)), Symbol('L', commutative=True))), Mul(Pow(Add(Symbol('L', commutative=True), Integral(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('L', commutative=True)))), Integer(-1)), Pow(Pow(Symbol('L', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Symbol('L', commutative=True))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(f)} = \\log{(f)}, then derive \\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)} = \\frac{1}{f}, then obtain ((\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{2 f})^{f} = ((\\frac{1}{f})^{f} (\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{f})^{f}", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(f)} = \\log{(f)} and \\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)} = \\frac{d}{d f} \\log{(f)} and \\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)} = \\frac{1}{f} and \\frac{d}{d f} \\log{(f)} = \\frac{1}{f} and (\\frac{d}{d f} \\log{(f)})^{f} = (\\frac{1}{f})^{f} and (\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{f} = (\\frac{1}{f})^{f} and (\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{2 f} = (\\frac{1}{f})^{f} (\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{f} and ((\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{2 f})^{f} = ((\\frac{1}{f})^{f} (\\frac{d}{d f} \\operatorname{V_{\\mathbf{B}}}{(f)})^{f})^{f}", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), log(Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Pow(Symbol('f', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Pow(Symbol('f', commutative=True), Integer(-1)))"], [["power", 4, "Symbol('f', commutative=True)"], "Equality(Pow(Derivative(log(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('f', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True)), Pow(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('f', commutative=True)))"], [["times", 6, "Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True))"], "Equality(Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Integer(2), Symbol('f', commutative=True))), Mul(Pow(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('f', commutative=True)), Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True))))"], [["power", 7, "Symbol('f', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Integer(2), Symbol('f', commutative=True))), Symbol('f', commutative=True)), Pow(Mul(Pow(Pow(Symbol('f', commutative=True), Integer(-1)), Symbol('f', commutative=True)), Pow(Derivative(Function('V_{\\\\mathbf{B}}')(Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Symbol('f', commutative=True))), Symbol('f', commutative=True)))"]]}, {"prompt": "Given U{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})} and \\varepsilon{(\\rho_f)} = \\rho_f^{2}, then obtain \\frac{d}{d \\rho_f} U^{2}{(\\rho_f)} \\varepsilon{(\\rho_f)} = \\frac{d}{d \\rho_f} U{(\\rho_f)} \\varepsilon{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})}", "derivation": "U{(\\rho_f)} = \\sin{(\\sin{(\\rho_f)})} and \\rho_f U{(\\rho_f)} = \\rho_f \\sin{(\\sin{(\\rho_f)})} and \\rho_f^{2} U^{2}{(\\rho_f)} = \\rho_f^{2} U{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} and \\frac{d}{d \\rho_f} \\rho_f^{2} U^{2}{(\\rho_f)} = \\frac{d}{d \\rho_f} \\rho_f^{2} U{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})} and \\varepsilon{(\\rho_f)} = \\rho_f^{2} and \\frac{d}{d \\rho_f} U^{2}{(\\rho_f)} \\varepsilon{(\\rho_f)} = \\frac{d}{d \\rho_f} U{(\\rho_f)} \\varepsilon{(\\rho_f)} \\sin{(\\sin{(\\rho_f)})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True))))"], [["times", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Mul(Symbol('\\\\rho_f', commutative=True), Function('U')(Symbol('\\\\rho_f', commutative=True))), Mul(Symbol('\\\\rho_f', commutative=True), sin(sin(Symbol('\\\\rho_f', commutative=True)))))"], [["times", 2, "Mul(Symbol('\\\\rho_f', commutative=True), Function('U')(Symbol('\\\\rho_f', commutative=True)))"], "Equality(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Pow(Function('U')(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Function('U')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))))"], [["differentiate", 3, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Pow(Function('U')(Symbol('\\\\rho_f', commutative=True)), Integer(2))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Function('U')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\varepsilon')(Symbol('\\\\rho_f', commutative=True)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Derivative(Mul(Pow(Function('U')(Symbol('\\\\rho_f', commutative=True)), Integer(2)), Function('\\\\varepsilon')(Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Mul(Function('U')(Symbol('\\\\rho_f', commutative=True)), Function('\\\\varepsilon')(Symbol('\\\\rho_f', commutative=True)), sin(sin(Symbol('\\\\rho_f', commutative=True)))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbf{B}{(C,A_{z})} = e^{C^{A_{z}}}, then derive \\frac{\\frac{\\partial}{\\partial C} \\mathbf{B}{(C,A_{z})}}{\\mathbf{B}{(C,A_{z})}} = \\frac{A_{z} C^{A_{z}}}{C}, then obtain e^{- C^{A_{z}}} \\frac{\\partial}{\\partial C} e^{C^{A_{z}}} = \\frac{A_{z} C^{A_{z}}}{C}", "derivation": "\\mathbf{B}{(C,A_{z})} = e^{C^{A_{z}}} and \\log{(\\mathbf{B}{(C,A_{z})})} = \\log{(e^{C^{A_{z}}})} and \\frac{\\partial}{\\partial C} \\log{(\\mathbf{B}{(C,A_{z})})} = \\frac{\\partial}{\\partial C} \\log{(e^{C^{A_{z}}})} and \\frac{\\frac{\\partial}{\\partial C} \\mathbf{B}{(C,A_{z})}}{\\mathbf{B}{(C,A_{z})}} = \\frac{A_{z} C^{A_{z}}}{C} and e^{- C^{A_{z}}} \\frac{\\partial}{\\partial C} e^{C^{A_{z}}} = \\frac{A_{z} C^{A_{z}}}{C}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('A_z', commutative=True)), exp(Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True))))"], [["log", 1], "Equality(log(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('A_z', commutative=True))), log(exp(Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True)))))"], [["differentiate", 2, "Symbol('C', commutative=True)"], "Equality(Derivative(log(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1))), Derivative(log(exp(Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True)))), Tuple(Symbol('C', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('A_z', commutative=True)), Integer(-1)), Derivative(Function('\\\\mathbf{B}')(Symbol('C', commutative=True), Symbol('A_z', commutative=True)), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(exp(Mul(Integer(-1), Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True)))), Derivative(exp(Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True))), Tuple(Symbol('C', commutative=True), Integer(1)))), Mul(Symbol('A_z', commutative=True), Pow(Symbol('C', commutative=True), Integer(-1)), Pow(Symbol('C', commutative=True), Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\hat{p},\\mathbf{S},v_{z})} = \\hat{p} + v_{z}^{\\mathbf{S}}, then obtain \\frac{\\partial}{\\partial v_{z}} - \\frac{\\hat{p} + v_{z}^{\\mathbf{S}}}{\\hat{p}} = \\frac{\\partial}{\\partial v_{z}} \\frac{- \\hat{p} - v_{z}^{\\mathbf{S}}}{\\hat{p}}", "derivation": "\\theta{(\\hat{p},\\mathbf{S},v_{z})} = \\hat{p} + v_{z}^{\\mathbf{S}} and - \\frac{\\theta{(\\hat{p},\\mathbf{S},v_{z})}}{\\hat{p}} = - \\frac{\\hat{p} + v_{z}^{\\mathbf{S}}}{\\hat{p}} and \\frac{\\partial}{\\partial v_{z}} - \\frac{\\theta{(\\hat{p},\\mathbf{S},v_{z})}}{\\hat{p}} = \\frac{\\partial}{\\partial v_{z}} - \\frac{\\hat{p} + v_{z}^{\\mathbf{S}}}{\\hat{p}} and \\frac{\\partial}{\\partial v_{z}} - \\frac{\\hat{p} + v_{z}^{\\mathbf{S}}}{\\hat{p}} = \\frac{\\partial}{\\partial v_{z}} \\frac{- \\hat{p} - v_{z}^{\\mathbf{S}}}{\\hat{p}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_z', commutative=True)), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))"], [["divide", 1, "Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_z', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))))"], [["differentiate", 2, "Symbol('v_z', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\mathbf{S}', commutative=True), Symbol('v_z', commutative=True))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Symbol('\\\\hat{p}', commutative=True), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True)))), Tuple(Symbol('v_z', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('\\\\hat{p}', commutative=True)), Mul(Integer(-1), Pow(Symbol('v_z', commutative=True), Symbol('\\\\mathbf{S}', commutative=True))))), Tuple(Symbol('v_z', commutative=True), Integer(1))))"]]}, {"prompt": "Given S{(x,m_{s})} = \\frac{m_{s}}{x}, then obtain - x + 2 \\int S{(x,m_{s})} dm_{s} = - x + \\int \\frac{m_{s}}{x} dm_{s} + \\int S{(x,m_{s})} dm_{s}", "derivation": "S{(x,m_{s})} = \\frac{m_{s}}{x} and \\int S{(x,m_{s})} dm_{s} = \\int \\frac{m_{s}}{x} dm_{s} and - x + \\int S{(x,m_{s})} dm_{s} = - x + \\int \\frac{m_{s}}{x} dm_{s} and - x + 2 \\int S{(x,m_{s})} dm_{s} = - x + \\int \\frac{m_{s}}{x} dm_{s} + \\int S{(x,m_{s})} dm_{s}", "srepr_derivation": [["premise", "Equality(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Mul(Symbol('m_s', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["integrate", 1, "Symbol('m_s', commutative=True)"], "Equality(Integral(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))), Integral(Mul(Symbol('m_s', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True))))"], [["minus", 2, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Mul(Symbol('m_s', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True)))))"], [["add", 3, "Integral(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Integer(2), Integral(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True))))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Integral(Mul(Symbol('m_s', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))), Tuple(Symbol('m_s', commutative=True))), Integral(Function('S')(Symbol('x', commutative=True), Symbol('m_s', commutative=True)), Tuple(Symbol('m_s', commutative=True)))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbb{I})} = \\cos{(\\mathbb{I})}, then obtain 1 = 0^{\\mathbb{I}} (\\hat{H}_l{(\\mathbb{I})} - \\cos{(\\mathbb{I})})^{- \\mathbb{I}}", "derivation": "\\hat{H}_l{(\\mathbb{I})} = \\cos{(\\mathbb{I})} and \\hat{H}_l{(\\mathbb{I})} - \\cos{(\\mathbb{I})} = 0 and (\\hat{H}_l{(\\mathbb{I})} - \\cos{(\\mathbb{I})})^{\\mathbb{I}} = 0^{\\mathbb{I}} and 1 = 0^{\\mathbb{I}} (\\hat{H}_l{(\\mathbb{I})} - \\cos{(\\mathbb{I})})^{- \\mathbb{I}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbb{I}', commutative=True)), cos(Symbol('\\\\mathbb{I}', commutative=True)))"], [["minus", 1, "cos(Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Integer(0))"], [["power", 2, "Symbol('\\\\mathbb{I}', commutative=True)"], "Equality(Pow(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Integer(0), Symbol('\\\\mathbb{I}', commutative=True)))"], [["divide", 3, "Pow(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Symbol('\\\\mathbb{I}', commutative=True))"], "Equality(Integer(1), Mul(Pow(Integer(0), Symbol('\\\\mathbb{I}', commutative=True)), Pow(Add(Function('\\\\hat{H}_l')(Symbol('\\\\mathbb{I}', commutative=True)), Mul(Integer(-1), cos(Symbol('\\\\mathbb{I}', commutative=True)))), Mul(Integer(-1), Symbol('\\\\mathbb{I}', commutative=True)))))"]]}, {"prompt": "Given \\Psi_{nl}{(f^{*})} = \\log{(f^{*})}, then obtain - \\Psi_{nl}^{f^{*}}{(f^{*})} + \\int \\Psi_{nl}^{f^{*}}{(f^{*})} df^{*} = - \\Psi_{nl}^{f^{*}}{(f^{*})} + \\int \\log{(f^{*})}^{f^{*}} df^{*}", "derivation": "\\Psi_{nl}{(f^{*})} = \\log{(f^{*})} and \\Psi_{nl}^{f^{*}}{(f^{*})} = \\log{(f^{*})}^{f^{*}} and \\int \\Psi_{nl}^{f^{*}}{(f^{*})} df^{*} = \\int \\log{(f^{*})}^{f^{*}} df^{*} and - \\log{(f^{*})}^{f^{*}} + \\int \\Psi_{nl}^{f^{*}}{(f^{*})} df^{*} = - \\log{(f^{*})}^{f^{*}} + \\int \\log{(f^{*})}^{f^{*}} df^{*} and - \\Psi_{nl}^{f^{*}}{(f^{*})} + \\int \\Psi_{nl}^{f^{*}}{(f^{*})} df^{*} = - \\Psi_{nl}^{f^{*}}{(f^{*})} + \\int \\log{(f^{*})}^{f^{*}} df^{*}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), log(Symbol('f^*', commutative=True)))"], [["power", 1, "Symbol('f^*', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)))"], [["integrate", 2, "Symbol('f^*', commutative=True)"], "Equality(Integral(Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True))))"], [["minus", 3, "Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integral(Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Mul(Integer(-1), Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integral(Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('\\\\Psi_{nl}')(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True))), Integral(Pow(log(Symbol('f^*', commutative=True)), Symbol('f^*', commutative=True)), Tuple(Symbol('f^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})}, then derive \\operatorname{f^{\\prime}}{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}}, then obtain (-1)^{\\mathbf{B}} + \\int \\operatorname{f^{\\prime}}{(\\mathbf{B})} d\\mathbf{B} = (- \\frac{1}{\\mathbf{B} \\operatorname{f^{\\prime}}{(\\mathbf{B})}})^{\\mathbf{B}} + \\int \\operatorname{f^{\\prime}}{(\\mathbf{B})} d\\mathbf{B}", "derivation": "\\operatorname{f^{\\prime}}{(\\mathbf{B})} = \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} and \\operatorname{f^{\\prime}}{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}} and \\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})} = \\frac{1}{\\mathbf{B}} and - \\frac{\\frac{d}{d \\mathbf{B}} \\log{(\\mathbf{B})}}{\\operatorname{f^{\\prime}}{(\\mathbf{B})}} = - \\frac{1}{\\mathbf{B} \\operatorname{f^{\\prime}}{(\\mathbf{B})}} and -1 = - \\frac{1}{\\mathbf{B} \\operatorname{f^{\\prime}}{(\\mathbf{B})}} and (-1)^{\\mathbf{B}} = (- \\frac{1}{\\mathbf{B} \\operatorname{f^{\\prime}}{(\\mathbf{B})}})^{\\mathbf{B}} and (-1)^{\\mathbf{B}} + \\int \\operatorname{f^{\\prime}}{(\\mathbf{B})} d\\mathbf{B} = (- \\frac{1}{\\mathbf{B} \\operatorname{f^{\\prime}}{(\\mathbf{B})}})^{\\mathbf{B}} + \\int \\operatorname{f^{\\prime}}{(\\mathbf{B})} d\\mathbf{B}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 2, 1], "Equality(Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1))), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)))"], [["divide", 3, "Mul(Integer(-1), Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)))"], "Equality(Mul(Integer(-1), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1)), Derivative(log(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True), Integer(1)))), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Integer(-1), Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))))"], [["power", 5, "Symbol('\\\\mathbf{B}', commutative=True)"], "Equality(Pow(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)))"], [["minus", 6, "Mul(Integer(-1), Integral(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True))))"], "Equality(Add(Pow(Integer(-1), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))), Add(Pow(Mul(Integer(-1), Pow(Symbol('\\\\mathbf{B}', commutative=True), Integer(-1)), Pow(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{B}', commutative=True)), Integral(Function('f^{\\\\prime}')(Symbol('\\\\mathbf{B}', commutative=True)), Tuple(Symbol('\\\\mathbf{B}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(\\sigma_x,\\mu_0)} = \\sigma_x^{\\mu_0} and y{(\\mu_0)} = \\mu_0, then obtain \\mu_0 + \\frac{- (\\sigma_x^{\\mu_0})^{\\mu_0} + y{(\\mu_0)}}{\\mu_0} = \\mu_0 + \\frac{\\mu_0 - (\\sigma_x^{\\mu_0})^{\\mu_0}}{\\mu_0}", "derivation": "\\operatorname{A_{y}}{(\\sigma_x,\\mu_0)} = \\sigma_x^{\\mu_0} and \\operatorname{A_{y}}^{\\mu_0}{(\\sigma_x,\\mu_0)} = (\\sigma_x^{\\mu_0})^{\\mu_0} and y{(\\mu_0)} = \\mu_0 and - \\operatorname{A_{y}}^{\\mu_0}{(\\sigma_x,\\mu_0)} + y{(\\mu_0)} = \\mu_0 - \\operatorname{A_{y}}^{\\mu_0}{(\\sigma_x,\\mu_0)} and \\frac{- \\operatorname{A_{y}}^{\\mu_0}{(\\sigma_x,\\mu_0)} + y{(\\mu_0)}}{\\mu_0} = \\frac{\\mu_0 - \\operatorname{A_{y}}^{\\mu_0}{(\\sigma_x,\\mu_0)}}{\\mu_0} and \\frac{- (\\sigma_x^{\\mu_0})^{\\mu_0} + y{(\\mu_0)}}{\\mu_0} = \\frac{\\mu_0 - (\\sigma_x^{\\mu_0})^{\\mu_0}}{\\mu_0} and \\mu_0 + \\frac{- (\\sigma_x^{\\mu_0})^{\\mu_0} + y{(\\mu_0)}}{\\mu_0} = \\mu_0 + \\frac{\\mu_0 - (\\sigma_x^{\\mu_0})^{\\mu_0}}{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["power", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))"], ["renaming_premise", "Equality(Function('y')(Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], [["minus", 3, "Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))"], "Equality(Add(Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Function('y')(Symbol('\\\\mu_0', commutative=True))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))))"], [["divide", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Function('y')(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Pow(Function('A_y')(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Function('y')(Symbol('\\\\mu_0', commutative=True)))), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))))))"], [["add", 6, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Add(Symbol('\\\\mu_0', commutative=True), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True))), Function('y')(Symbol('\\\\mu_0', commutative=True))))), Add(Symbol('\\\\mu_0', commutative=True), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Integer(-1)), Add(Symbol('\\\\mu_0', commutative=True), Mul(Integer(-1), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\mu_0', commutative=True)), Symbol('\\\\mu_0', commutative=True)))))))"]]}, {"prompt": "Given \\varepsilon{(M_{E})} = e^{M_{E}}, then derive \\frac{d}{d M_{E}} \\varepsilon{(M_{E})} = e^{M_{E}}, then obtain \\frac{d^{2}}{d M_{E}^{2}} \\varepsilon{(M_{E})} = e^{M_{E}}", "derivation": "\\varepsilon{(M_{E})} = e^{M_{E}} and \\frac{d}{d M_{E}} \\varepsilon{(M_{E})} = \\frac{d}{d M_{E}} e^{M_{E}} and \\frac{d}{d M_{E}} \\varepsilon{(M_{E})} = e^{M_{E}} and \\frac{d}{d M_{E}} \\varepsilon{(M_{E})} = \\varepsilon{(M_{E})} and \\frac{d^{2}}{d M_{E}^{2}} \\varepsilon{(M_{E})} = \\frac{d}{d M_{E}} e^{M_{E}} and \\frac{d^{2}}{d M_{E}^{2}} \\varepsilon{(M_{E})} = e^{M_{E}}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), exp(Symbol('M_E', commutative=True)))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), exp(Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Function('\\\\varepsilon')(Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(2))), Derivative(exp(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 5], "Equality(Derivative(Function('\\\\varepsilon')(Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(2))), exp(Symbol('M_E', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(E,J)} = \\sin{(J^{E})}, then obtain \\sigma_{x}^{3}{(E,J)} \\sin{(J^{E})} = \\sin^{4}{(J^{E})}", "derivation": "\\sigma_{x}{(E,J)} = \\sin{(J^{E})} and \\sigma_{x}{(E,J)} \\sin{(J^{E})} = \\sin^{2}{(J^{E})} and \\sigma_{x}^{2}{(E,J)} \\sin^{2}{(J^{E})} = \\sigma_{x}{(E,J)} \\sin^{3}{(J^{E})} and \\sigma_{x}^{3}{(E,J)} \\sin{(J^{E})} = \\sigma_{x}{(E,J)} \\sin^{3}{(J^{E})} and \\sigma_{x}^{2}{(E,J)} \\sin^{2}{(J^{E})} = \\sin^{4}{(J^{E})} and \\sigma_{x}^{3}{(E,J)} \\sin{(J^{E})} = \\sigma_{x}^{2}{(E,J)} \\sin^{2}{(J^{E})} and \\sigma_{x}^{3}{(E,J)} \\sin{(J^{E})} = \\sin^{4}{(J^{E})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))))"], [["times", 1, "sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True)))"], "Equality(Mul(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True)))), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(2)))"], [["times", 2, "Mul(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))))"], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(2)), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(2))), Mul(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(3))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(3)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True)))), Mul(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(3))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(2)), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(2))), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(4)))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(3)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True)))), Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(2)), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Pow(Function('\\\\sigma_x')(Symbol('E', commutative=True), Symbol('J', commutative=True)), Integer(3)), sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True)))), Pow(sin(Pow(Symbol('J', commutative=True), Symbol('E', commutative=True))), Integer(4)))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(v_{2},M_{E})} = \\frac{\\sin{(M_{E})}}{v_{2}}, then obtain M_{E} \\frac{\\partial}{\\partial M_{E}} \\operatorname{F_{H}}{(v_{2},M_{E})} = \\frac{M_{E} \\cos{(M_{E})}}{v_{2}}", "derivation": "\\operatorname{F_{H}}{(v_{2},M_{E})} = \\frac{\\sin{(M_{E})}}{v_{2}} and \\frac{\\partial}{\\partial M_{E}} \\operatorname{F_{H}}{(v_{2},M_{E})} = \\frac{\\partial}{\\partial M_{E}} \\frac{\\sin{(M_{E})}}{v_{2}} and M_{E} \\frac{\\partial}{\\partial M_{E}} \\operatorname{F_{H}}{(v_{2},M_{E})} = M_{E} \\frac{\\partial}{\\partial M_{E}} \\frac{\\sin{(M_{E})}}{v_{2}} and M_{E} \\frac{\\partial}{\\partial M_{E}} \\operatorname{F_{H}}{(v_{2},M_{E})} = \\frac{M_{E} \\cos{(M_{E})}}{v_{2}}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('v_2', commutative=True), Symbol('M_E', commutative=True)), Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), sin(Symbol('M_E', commutative=True))))"], [["differentiate", 1, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('v_2', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"], [["times", 2, "Symbol('M_E', commutative=True)"], "Equality(Mul(Symbol('M_E', commutative=True), Derivative(Function('F_H')(Symbol('v_2', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Symbol('M_E', commutative=True), Derivative(Mul(Pow(Symbol('v_2', commutative=True), Integer(-1)), sin(Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('M_E', commutative=True), Derivative(Function('F_H')(Symbol('v_2', commutative=True), Symbol('M_E', commutative=True)), Tuple(Symbol('M_E', commutative=True), Integer(1)))), Mul(Symbol('M_E', commutative=True), Pow(Symbol('v_2', commutative=True), Integer(-1)), cos(Symbol('M_E', commutative=True))))"]]}, {"prompt": "Given y{(c_{0},q)} = \\sin{(c_{0} + q)}, then obtain (c_{0} + q) y^{q}{(c_{0},q)} + y{(c_{0},q)} = (c_{0} + q) \\sin^{q}{(c_{0} + q)} + y{(c_{0},q)}", "derivation": "y{(c_{0},q)} = \\sin{(c_{0} + q)} and y^{q}{(c_{0},q)} = \\sin^{q}{(c_{0} + q)} and (c_{0} + q) y^{q}{(c_{0},q)} = (c_{0} + q) \\sin^{q}{(c_{0} + q)} and (c_{0} + q) y^{q}{(c_{0},q)} + y{(c_{0},q)} = (c_{0} + q) \\sin^{q}{(c_{0} + q)} + y{(c_{0},q)}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True)))"], [["times", 2, "Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Pow(Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Pow(sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))))"], [["add", 3, "Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True))"], "Equality(Add(Mul(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Pow(Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Add(Mul(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True)), Pow(sin(Add(Symbol('c_0', commutative=True), Symbol('q', commutative=True))), Symbol('q', commutative=True))), Function('y')(Symbol('c_0', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given U{(f^{*},\\eta)} = \\log{(\\eta - f^{*})} and A{(f^{*},\\eta)} = - \\frac{U{(f^{*},\\eta)}}{f^{*}}, then obtain A{(f^{*},\\eta)} U{(f^{*},\\eta)} = - \\frac{U^{2}{(f^{*},\\eta)}}{f^{*}}", "derivation": "U{(f^{*},\\eta)} = \\log{(\\eta - f^{*})} and - \\frac{U{(f^{*},\\eta)}}{f^{*}} = - \\frac{\\log{(\\eta - f^{*})}}{f^{*}} and A{(f^{*},\\eta)} = - \\frac{U{(f^{*},\\eta)}}{f^{*}} and A{(f^{*},\\eta)} = - \\frac{\\log{(\\eta - f^{*})}}{f^{*}} and A{(f^{*},\\eta)} \\log{(\\eta - f^{*})} = - \\frac{\\log{(\\eta - f^{*})}^{2}}{f^{*}} and A{(f^{*},\\eta)} U{(f^{*},\\eta)} = - \\frac{U^{2}{(f^{*},\\eta)}}{f^{*}}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))))"], [["divide", 1, "Mul(Integer(-1), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))))"], ["renaming_premise", "Equality(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))))"], [["times", 4, "log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], "Equality(Mul(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(log(Add(Symbol('\\\\eta', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Integer(2))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(Function('A')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True))), Mul(Integer(-1), Pow(Symbol('f^*', commutative=True), Integer(-1)), Pow(Function('U')(Symbol('f^*', commutative=True), Symbol('\\\\eta', commutative=True)), Integer(2))))"]]}, {"prompt": "Given Q{(A_{y})} = e^{\\sin{(A_{y})}} and c{(A_{y})} = - \\sin{(A_{y})}, then obtain \\frac{d}{d A_{y}} Q{(A_{y})} - 1 = \\frac{d}{d A_{y}} e^{- c{(A_{y})}} - 1", "derivation": "Q{(A_{y})} = e^{\\sin{(A_{y})}} and \\frac{d}{d A_{y}} Q{(A_{y})} = \\frac{d}{d A_{y}} e^{\\sin{(A_{y})}} and c{(A_{y})} = - \\sin{(A_{y})} and \\frac{d}{d A_{y}} Q{(A_{y})} - 1 = \\frac{d}{d A_{y}} e^{\\sin{(A_{y})}} - 1 and - c{(A_{y})} = \\sin{(A_{y})} and \\frac{d}{d A_{y}} Q{(A_{y})} - 1 = \\frac{d}{d A_{y}} e^{- c{(A_{y})}} - 1", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('A_y', commutative=True)), exp(sin(Symbol('A_y', commutative=True))))"], [["differentiate", 1, "Symbol('A_y', commutative=True)"], "Equality(Derivative(Function('Q')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Derivative(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('c')(Symbol('A_y', commutative=True)), Mul(Integer(-1), sin(Symbol('A_y', commutative=True))))"], [["add", 2, "Integer(-1)"], "Equality(Add(Derivative(Function('Q')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(exp(sin(Symbol('A_y', commutative=True))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1)))"], [["times", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('c')(Symbol('A_y', commutative=True))), sin(Symbol('A_y', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Derivative(Function('Q')(Symbol('A_y', commutative=True)), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(exp(Mul(Integer(-1), Function('c')(Symbol('A_y', commutative=True)))), Tuple(Symbol('A_y', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\dot{z}{(m,n_{2})} = \\sin{(m^{n_{2}})}, then obtain m^{n_{2}} + 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\dot{z}{(m,n_{2})} = m^{n_{2}} + 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\sin{(m^{n_{2}})}", "derivation": "\\dot{z}{(m,n_{2})} = \\sin{(m^{n_{2}})} and m^{- n_{2}} \\dot{z}{(m,n_{2})} = m^{- n_{2}} \\sin{(m^{n_{2}})} and \\sin{(m^{n_{2}})} + m^{- n_{2}} \\dot{z}{(m,n_{2})} = \\sin{(m^{n_{2}})} + m^{- n_{2}} \\sin{(m^{n_{2}})} and 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\dot{z}{(m,n_{2})} = 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\sin{(m^{n_{2}})} and m^{n_{2}} + 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\dot{z}{(m,n_{2})} = m^{n_{2}} + 2 \\sin{(m^{n_{2}})} + m^{- n_{2}} \\sin{(m^{n_{2}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('n_2', commutative=True)), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))))"], [["divide", 1, "Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))))"], [["add", 2, "sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Add(sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Add(sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))))))"], [["add", 3, "sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Add(Mul(Integer(2), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Add(Mul(Integer(2), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))))))"], [["add", 4, "Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))"], "Equality(Add(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(2), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), Function('\\\\dot{z}')(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Add(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)), Mul(Integer(2), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True)))), Mul(Pow(Symbol('m', commutative=True), Mul(Integer(-1), Symbol('n_2', commutative=True))), sin(Pow(Symbol('m', commutative=True), Symbol('n_2', commutative=True))))))"]]}, {"prompt": "Given i{(\\phi)} = \\sin{(\\phi)}, then derive \\frac{d}{d \\phi} i{(\\phi)} = \\cos{(\\phi)}, then obtain - \\sin{(\\phi)} + \\cos{(\\phi)} - 1 = - \\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)} - 1", "derivation": "i{(\\phi)} = \\sin{(\\phi)} and \\frac{d}{d \\phi} i{(\\phi)} = \\frac{d}{d \\phi} \\sin{(\\phi)} and \\frac{d}{d \\phi} i{(\\phi)} - 1 = \\frac{d}{d \\phi} \\sin{(\\phi)} - 1 and \\frac{d}{d \\phi} i{(\\phi)} = \\cos{(\\phi)} and - \\sin{(\\phi)} + \\frac{d}{d \\phi} i{(\\phi)} - 1 = - \\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)} - 1 and - \\sin{(\\phi)} + \\cos{(\\phi)} - 1 = - \\sin{(\\phi)} + \\frac{d}{d \\phi} \\sin{(\\phi)} - 1", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('\\\\phi', commutative=True)), sin(Symbol('\\\\phi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"], [["minus", 2, 1], "Equality(Add(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), cos(Symbol('\\\\phi', commutative=True)))"], [["minus", 3, "sin(Symbol('\\\\phi', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Derivative(Function('i')(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), cos(Symbol('\\\\phi', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), sin(Symbol('\\\\phi', commutative=True))), Derivative(sin(Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Integer(-1)))"]]}, {"prompt": "Given \\operatorname{P_{g}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})}, then obtain - \\operatorname{P_{g}}^{3}{(V_{\\mathbf{B}})} + \\int \\operatorname{P_{g}}^{2}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = - \\operatorname{P_{g}}^{3}{(V_{\\mathbf{B}})} + \\int \\operatorname{P_{g}}{(V_{\\mathbf{B}})} \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}", "derivation": "\\operatorname{P_{g}}{(V_{\\mathbf{B}})} = \\sin{(V_{\\mathbf{B}})} and \\operatorname{P_{g}}^{2}{(V_{\\mathbf{B}})} = \\operatorname{P_{g}}{(V_{\\mathbf{B}})} \\sin{(V_{\\mathbf{B}})} and \\int \\operatorname{P_{g}}^{2}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = \\int \\operatorname{P_{g}}{(V_{\\mathbf{B}})} \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} and - \\operatorname{P_{g}}^{3}{(V_{\\mathbf{B}})} + \\int \\operatorname{P_{g}}^{2}{(V_{\\mathbf{B}})} dV_{\\mathbf{B}} = - \\operatorname{P_{g}}^{3}{(V_{\\mathbf{B}})} + \\int \\operatorname{P_{g}}{(V_{\\mathbf{B}})} \\sin{(V_{\\mathbf{B}})} dV_{\\mathbf{B}}", "srepr_derivation": [["premise", "Equality(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True)))"], [["times", 1, "Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True))"], "Equality(Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Mul(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["integrate", 2, "Symbol('V_{\\\\mathbf{B}}', commutative=True)"], "Equality(Integral(Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Integral(Mul(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True))))"], [["minus", 3, "Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(3))"], "Equality(Add(Mul(Integer(-1), Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(3))), Integral(Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(2)), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))), Add(Mul(Integer(-1), Pow(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), Integer(3))), Integral(Mul(Function('P_g')(Symbol('V_{\\\\mathbf{B}}', commutative=True)), sin(Symbol('V_{\\\\mathbf{B}}', commutative=True))), Tuple(Symbol('V_{\\\\mathbf{B}}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(M_{E})} = \\log{(e^{M_{E}})}, then obtain 0 = \\frac{4 \\operatorname{C_{2}}{(M_{E})} - 4 \\log{(e^{M_{E}})}}{\\operatorname{C_{2}}{(M_{E})} \\log{(e^{M_{E}})}}", "derivation": "\\operatorname{C_{2}}{(M_{E})} = \\log{(e^{M_{E}})} and 0 = - \\operatorname{C_{2}}{(M_{E})} + \\log{(e^{M_{E}})} and 0 = \\operatorname{C_{2}}{(M_{E})} - \\log{(e^{M_{E}})} and 0 = \\frac{\\operatorname{C_{2}}{(M_{E})} - \\log{(e^{M_{E}})}}{\\log{(e^{M_{E}})}} and 2 \\operatorname{C_{2}}{(M_{E})} - \\log{(e^{M_{E}})} = \\operatorname{C_{2}}{(M_{E})} and 0 = \\frac{2 \\operatorname{C_{2}}{(M_{E})} - 2 \\log{(e^{M_{E}})}}{\\log{(e^{M_{E}})}} and 0 = \\frac{4 \\operatorname{C_{2}}{(M_{E})} - 4 \\log{(e^{M_{E}})}}{\\log{(e^{M_{E}})}} and 0 = \\frac{4 \\operatorname{C_{2}}{(M_{E})} - 4 \\log{(e^{M_{E}})}}{\\operatorname{C_{2}}{(M_{E})} \\log{(e^{M_{E}})}}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('M_E', commutative=True)), log(exp(Symbol('M_E', commutative=True))))"], [["minus", 1, "Function('C_2')(Symbol('M_E', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('C_2')(Symbol('M_E', commutative=True))), log(exp(Symbol('M_E', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Function('C_2')(Symbol('M_E', commutative=True)), Mul(Integer(-1), log(exp(Symbol('M_E', commutative=True))))))"], [["divide", 3, "log(exp(Symbol('M_E', commutative=True)))"], "Equality(Integer(0), Mul(Add(Function('C_2')(Symbol('M_E', commutative=True)), Mul(Integer(-1), log(exp(Symbol('M_E', commutative=True))))), Pow(log(exp(Symbol('M_E', commutative=True))), Integer(-1))))"], [["minus", 1, "Add(Mul(Integer(-1), Function('C_2')(Symbol('M_E', commutative=True))), log(exp(Symbol('M_E', commutative=True))))"], "Equality(Add(Mul(Integer(2), Function('C_2')(Symbol('M_E', commutative=True))), Mul(Integer(-1), log(exp(Symbol('M_E', commutative=True))))), Function('C_2')(Symbol('M_E', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Integer(0), Mul(Add(Mul(Integer(2), Function('C_2')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integer(2), log(exp(Symbol('M_E', commutative=True))))), Pow(log(exp(Symbol('M_E', commutative=True))), Integer(-1))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Integer(0), Mul(Add(Mul(Integer(4), Function('C_2')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integer(4), log(exp(Symbol('M_E', commutative=True))))), Pow(log(exp(Symbol('M_E', commutative=True))), Integer(-1))))"], [["divide", 7, "Function('C_2')(Symbol('M_E', commutative=True))"], "Equality(Integer(0), Mul(Add(Mul(Integer(4), Function('C_2')(Symbol('M_E', commutative=True))), Mul(Integer(-1), Integer(4), log(exp(Symbol('M_E', commutative=True))))), Pow(Function('C_2')(Symbol('M_E', commutative=True)), Integer(-1)), Pow(log(exp(Symbol('M_E', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\dot{y}{(\\tilde{g}^*,\\mathbf{s})} = \\cos{(\\mathbf{s} + \\tilde{g}^*)} and V{(\\eta,y)} = \\log{(\\eta + y)}, then obtain (- \\eta + V{(\\eta,y)}) \\dot{y}^{- \\tilde{g}^*}{(\\tilde{g}^*,\\mathbf{s})} = (- \\eta + \\log{(\\eta + y)}) \\dot{y}^{- \\tilde{g}^*}{(\\tilde{g}^*,\\mathbf{s})}", "derivation": "\\dot{y}{(\\tilde{g}^*,\\mathbf{s})} = \\cos{(\\mathbf{s} + \\tilde{g}^*)} and V{(\\eta,y)} = \\log{(\\eta + y)} and - \\eta + V{(\\eta,y)} = - \\eta + \\log{(\\eta + y)} and (- \\eta + V{(\\eta,y)}) \\cos^{- \\tilde{g}^*}{(\\mathbf{s} + \\tilde{g}^*)} = (- \\eta + \\log{(\\eta + y)}) \\cos^{- \\tilde{g}^*}{(\\mathbf{s} + \\tilde{g}^*)} and (- \\eta + V{(\\eta,y)}) \\dot{y}^{- \\tilde{g}^*}{(\\tilde{g}^*,\\mathbf{s})} = (- \\eta + \\log{(\\eta + y)}) \\dot{y}^{- \\tilde{g}^*}{(\\tilde{g}^*,\\mathbf{s})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))))"], ["get_premise", "Equality(Function('V')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))))"], [["minus", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('V')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)))))"], [["divide", 3, "Pow(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Symbol('\\\\tilde{g}^*', commutative=True))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('V')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Pow(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)))), Pow(cos(Add(Symbol('\\\\mathbf{s}', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True))), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), Function('V')(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True))), Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))), Mul(Add(Mul(Integer(-1), Symbol('\\\\eta', commutative=True)), log(Add(Symbol('\\\\eta', commutative=True), Symbol('y', commutative=True)))), Pow(Function('\\\\dot{y}')(Symbol('\\\\tilde{g}^*', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Integer(-1), Symbol('\\\\tilde{g}^*', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{A_{z}}{(V)} = V and p{(V)} = V - 1, then obtain \\int (e^{\\operatorname{A_{z}}{(V)} - 1} + 1) dV = \\int (e^{p{(V)}} + 1) dV", "derivation": "\\operatorname{A_{z}}{(V)} = V and \\operatorname{A_{z}}{(V)} - 1 = V - 1 and p{(V)} = V - 1 and \\operatorname{A_{z}}{(V)} - 1 = p{(V)} and e^{\\operatorname{A_{z}}{(V)} - 1} = e^{p{(V)}} and e^{\\operatorname{A_{z}}{(V)} - 1} + 1 = e^{p{(V)}} + 1 and \\int (e^{\\operatorname{A_{z}}{(V)} - 1} + 1) dV = \\int (e^{p{(V)}} + 1) dV", "srepr_derivation": [["premise", "Equality(Function('A_z')(Symbol('V', commutative=True)), Symbol('V', commutative=True))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('A_z')(Symbol('V', commutative=True)), Integer(-1)), Add(Symbol('V', commutative=True), Integer(-1)))"], ["renaming_premise", "Equality(Function('p')(Symbol('V', commutative=True)), Add(Symbol('V', commutative=True), Integer(-1)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Function('A_z')(Symbol('V', commutative=True)), Integer(-1)), Function('p')(Symbol('V', commutative=True)))"], [["exp", 4], "Equality(exp(Add(Function('A_z')(Symbol('V', commutative=True)), Integer(-1))), exp(Function('p')(Symbol('V', commutative=True))))"], [["add", 5, 1], "Equality(Add(exp(Add(Function('A_z')(Symbol('V', commutative=True)), Integer(-1))), Integer(1)), Add(exp(Function('p')(Symbol('V', commutative=True))), Integer(1)))"], [["integrate", 6, "Symbol('V', commutative=True)"], "Equality(Integral(Add(exp(Add(Function('A_z')(Symbol('V', commutative=True)), Integer(-1))), Integer(1)), Tuple(Symbol('V', commutative=True))), Integral(Add(exp(Function('p')(Symbol('V', commutative=True))), Integer(1)), Tuple(Symbol('V', commutative=True))))"]]}, {"prompt": "Given I{(\\hat{X})} = \\cos{(\\hat{X})}, then obtain (- \\hat{X} + \\int I^{\\hat{X}}{(\\hat{X})} d\\hat{X})^{\\hat{X}} = (- \\hat{X} + \\int \\cos^{\\hat{X}}{(\\hat{X})} d\\hat{X})^{\\hat{X}}", "derivation": "I{(\\hat{X})} = \\cos{(\\hat{X})} and I^{\\hat{X}}{(\\hat{X})} = \\cos^{\\hat{X}}{(\\hat{X})} and \\int I^{\\hat{X}}{(\\hat{X})} d\\hat{X} = \\int \\cos^{\\hat{X}}{(\\hat{X})} d\\hat{X} and - \\hat{X} + \\int I^{\\hat{X}}{(\\hat{X})} d\\hat{X} = - \\hat{X} + \\int \\cos^{\\hat{X}}{(\\hat{X})} d\\hat{X} and (- \\hat{X} + \\int I^{\\hat{X}}{(\\hat{X})} d\\hat{X})^{\\hat{X}} = (- \\hat{X} + \\int \\cos^{\\hat{X}}{(\\hat{X})} d\\hat{X})^{\\hat{X}}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('\\\\hat{X}', commutative=True)), cos(Symbol('\\\\hat{X}', commutative=True)))"], [["power", 1, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Function('I')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)))"], [["integrate", 2, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Pow(Function('I')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True))))"], [["minus", 3, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integral(Pow(Function('I')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integral(Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))))"], [["power", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integral(Pow(Function('I')(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('\\\\hat{X}', commutative=True)), Integral(Pow(cos(Symbol('\\\\hat{X}', commutative=True)), Symbol('\\\\hat{X}', commutative=True)), Tuple(Symbol('\\\\hat{X}', commutative=True)))), Symbol('\\\\hat{X}', commutative=True)))"]]}, {"prompt": "Given \\bar{\\h}{(F_{x})} = \\cos{(F_{x})}, then derive 0 = - \\sin{(F_{x})} - \\frac{d}{d F_{x}} \\bar{\\h}{(F_{x})}, then obtain \\frac{d}{d F_{x}} (- \\bar{\\h}{(F_{x})} + \\cos{(F_{x})}) = \\frac{d}{d F_{x}} (- \\sin{(F_{x})} - \\frac{d}{d F_{x}} \\bar{\\h}{(F_{x})})", "derivation": "\\bar{\\h}{(F_{x})} = \\cos{(F_{x})} and 0 = - \\bar{\\h}{(F_{x})} + \\cos{(F_{x})} and \\frac{d}{d F_{x}} 0 = \\frac{d}{d F_{x}} (- \\bar{\\h}{(F_{x})} + \\cos{(F_{x})}) and 0 = - \\sin{(F_{x})} - \\frac{d}{d F_{x}} \\bar{\\h}{(F_{x})} and \\frac{d}{d F_{x}} 0 = \\frac{d}{d F_{x}} (- \\sin{(F_{x})} - \\frac{d}{d F_{x}} \\bar{\\h}{(F_{x})}) and \\frac{d}{d F_{x}} (- \\bar{\\h}{(F_{x})} + \\cos{(F_{x})}) = \\frac{d}{d F_{x}} (- \\sin{(F_{x})} - \\frac{d}{d F_{x}} \\bar{\\h}{(F_{x})})", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('F_x', commutative=True)), cos(Symbol('F_x', commutative=True)))"], [["minus", 1, "Function('\\\\hbar')(Symbol('F_x', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))))"], [["differentiate", 2, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Integer(0), Add(Mul(Integer(-1), sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))))"], [["differentiate", 4, "Symbol('F_x', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Derivative(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('F_x', commutative=True))), cos(Symbol('F_x', commutative=True))), Tuple(Symbol('F_x', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), sin(Symbol('F_x', commutative=True))), Mul(Integer(-1), Derivative(Function('\\\\hbar')(Symbol('F_x', commutative=True)), Tuple(Symbol('F_x', commutative=True), Integer(1))))), Tuple(Symbol('F_x', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\mathbb{I}{(z,c)} = c + z, then obtain 2 \\mathbb{I}{(z,c)} + 1 = 2 c + 2 z + 1", "derivation": "\\mathbb{I}{(z,c)} = c + z and \\mathbb{I}{(z,c)} + 1 = c + z + 1 and 2 \\mathbb{I}{(z,c)} + 1 = c + z + \\mathbb{I}{(z,c)} + 1 and 2 \\mathbb{I}{(z,c)} + 1 = 2 c + 2 z + 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True)), Add(Symbol('c', commutative=True), Symbol('z', commutative=True)))"], [["minus", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True)), Integer(1)), Add(Symbol('c', commutative=True), Symbol('z', commutative=True), Integer(1)))"], [["add", 1, "Add(Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True)), Integer(1))"], "Equality(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True))), Integer(1)), Add(Symbol('c', commutative=True), Symbol('z', commutative=True), Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True)), Integer(1)))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Mul(Integer(2), Function('\\\\mathbb{I}')(Symbol('z', commutative=True), Symbol('c', commutative=True))), Integer(1)), Add(Mul(Integer(2), Symbol('c', commutative=True)), Mul(Integer(2), Symbol('z', commutative=True)), Integer(1)))"]]}, {"prompt": "Given p{(\\rho_b,m)} = \\frac{\\rho_b}{m}, then obtain \\frac{\\rho_b}{m} + 2 p^{2}{(\\rho_b,m)} = \\frac{2 \\rho_b p{(\\rho_b,m)}}{m} + \\frac{\\rho_b}{m}", "derivation": "p{(\\rho_b,m)} = \\frac{\\rho_b}{m} and p^{2}{(\\rho_b,m)} = \\frac{\\rho_b p{(\\rho_b,m)}}{m} and \\frac{\\rho_b}{m} + p^{2}{(\\rho_b,m)} = \\frac{\\rho_b p{(\\rho_b,m)}}{m} + \\frac{\\rho_b}{m} and \\frac{\\rho_b}{m} + 2 p^{2}{(\\rho_b,m)} = \\frac{\\rho_b p{(\\rho_b,m)}}{m} + \\frac{\\rho_b}{m} + p^{2}{(\\rho_b,m)} and \\frac{\\rho_b}{m} + 2 p^{2}{(\\rho_b,m)} = \\frac{2 \\rho_b p{(\\rho_b,m)}}{m} + \\frac{\\rho_b}{m}", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))))"], [["times", 1, "Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True))"], "Equality(Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2)), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True))))"], [["add", 2, "Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"], [["add", 2, "Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2)))"], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2)))), Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1))), Mul(Integer(2), Pow(Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True)), Integer(2)))), Add(Mul(Integer(2), Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)), Function('p')(Symbol('\\\\rho_b', commutative=True), Symbol('m', commutative=True))), Mul(Symbol('\\\\rho_b', commutative=True), Pow(Symbol('m', commutative=True), Integer(-1)))))"]]}, {"prompt": "Given \\mathbf{J}{(m,M)} = \\frac{\\partial}{\\partial m} M^{m}, then derive \\frac{\\mathbf{J}^{m}{(m,M)}}{\\mathbf{J}{(m,M)}} = \\frac{(M^{m} \\log{(M)})^{m}}{\\mathbf{J}{(m,M)}}, then obtain (\\frac{\\mathbf{J}^{m}{(m,M)}}{\\mathbf{J}{(m,M)}})^{m} = (\\frac{(M^{m} \\log{(M)})^{m}}{\\mathbf{J}{(m,M)}})^{m}", "derivation": "\\mathbf{J}{(m,M)} = \\frac{\\partial}{\\partial m} M^{m} and \\mathbf{J}^{m}{(m,M)} = (\\frac{\\partial}{\\partial m} M^{m})^{m} and \\frac{\\mathbf{J}^{m}{(m,M)}}{\\mathbf{J}{(m,M)}} = \\frac{(\\frac{\\partial}{\\partial m} M^{m})^{m}}{\\mathbf{J}{(m,M)}} and \\frac{\\mathbf{J}^{m}{(m,M)}}{\\mathbf{J}{(m,M)}} = \\frac{(M^{m} \\log{(M)})^{m}}{\\mathbf{J}{(m,M)}} and (\\frac{\\mathbf{J}^{m}{(m,M)}}{\\mathbf{J}{(m,M)}})^{m} = (\\frac{(M^{m} \\log{(M)})^{m}}{\\mathbf{J}{(m,M)}})^{m}", "srepr_derivation": [["get_premise", "Equality(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Derivative(Pow(Symbol('M', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Symbol('m', commutative=True)), Pow(Derivative(Pow(Symbol('M', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True)))"], [["divide", 2, "Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True))"], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Symbol('m', commutative=True))), Mul(Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Derivative(Pow(Symbol('M', commutative=True), Symbol('m', commutative=True)), Tuple(Symbol('m', commutative=True), Integer(1))), Symbol('m', commutative=True))))"], [["evaluate_derivatives", 3], "Equality(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Symbol('m', commutative=True))), Mul(Pow(Mul(Pow(Symbol('M', commutative=True), Symbol('m', commutative=True)), log(Symbol('M', commutative=True))), Symbol('m', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1))))"], [["power", 4, "Symbol('m', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1)), Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Symbol('m', commutative=True))), Symbol('m', commutative=True)), Pow(Mul(Pow(Mul(Pow(Symbol('M', commutative=True), Symbol('m', commutative=True)), log(Symbol('M', commutative=True))), Symbol('m', commutative=True)), Pow(Function('\\\\mathbf{J}')(Symbol('m', commutative=True), Symbol('M', commutative=True)), Integer(-1))), Symbol('m', commutative=True)))"]]}, {"prompt": "Given \\mathbf{A}{(\\omega)} = \\sin{(\\omega)}, then obtain \\omega \\mathbf{A}^{\\omega}{(\\omega)} \\log{(- \\omega + \\sin^{\\omega}{(\\omega)})} = \\omega \\log{(- \\omega + \\sin^{\\omega}{(\\omega)})} \\sin^{\\omega}{(\\omega)}", "derivation": "\\mathbf{A}{(\\omega)} = \\sin{(\\omega)} and \\mathbf{A}^{\\omega}{(\\omega)} = \\sin^{\\omega}{(\\omega)} and \\omega \\mathbf{A}^{\\omega}{(\\omega)} = \\omega \\sin^{\\omega}{(\\omega)} and - \\omega + \\mathbf{A}^{\\omega}{(\\omega)} = - \\omega + \\sin^{\\omega}{(\\omega)} and \\omega \\mathbf{A}^{\\omega}{(\\omega)} \\log{(- \\omega + \\mathbf{A}^{\\omega}{(\\omega)})} = \\omega \\log{(- \\omega + \\mathbf{A}^{\\omega}{(\\omega)})} \\sin^{\\omega}{(\\omega)} and \\omega \\mathbf{A}^{\\omega}{(\\omega)} \\log{(- \\omega + \\sin^{\\omega}{(\\omega)})} = \\omega \\log{(- \\omega + \\sin^{\\omega}{(\\omega)})} \\sin^{\\omega}{(\\omega)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), sin(Symbol('\\\\omega', commutative=True)))"], [["power", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))"], [["times", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Mul(Symbol('\\\\omega', commutative=True), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["minus", 2, "Symbol('\\\\omega', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["times", 3, "log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))), Mul(Symbol('\\\\omega', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('\\\\omega', commutative=True), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)), log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))), Mul(Symbol('\\\\omega', commutative=True), log(Add(Mul(Integer(-1), Symbol('\\\\omega', commutative=True)), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True)))), Pow(sin(Symbol('\\\\omega', commutative=True)), Symbol('\\\\omega', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and \\varphi^{*}{(\\varepsilon_0)} = 2 \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} and \\Psi^{\\dagger}{(\\varepsilon_0)} = \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} + \\log{(\\varepsilon_0)}, then obtain \\Psi^{\\dagger}{(\\varepsilon_0)} = 2 \\log{(\\varepsilon_0)}", "derivation": "\\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\log{(\\varepsilon_0)} and 2 \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} = \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} + \\log{(\\varepsilon_0)} and \\varphi^{*}{(\\varepsilon_0)} = 2 \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} and \\varphi^{*}{(\\varepsilon_0)} = \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} + \\log{(\\varepsilon_0)} and \\Psi^{\\dagger}{(\\varepsilon_0)} = \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} + \\log{(\\varepsilon_0)} and \\varphi^{*}{(\\varepsilon_0)} = 2 \\log{(\\varepsilon_0)} and \\operatorname{E_{\\lambda}}{(\\varepsilon_0)} + \\log{(\\varepsilon_0)} = 2 \\log{(\\varepsilon_0)} and \\Psi^{\\dagger}{(\\varepsilon_0)} = 2 \\log{(\\varepsilon_0)}", "srepr_derivation": [["premise", "Equality(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True)))"], [["add", 1, "Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True))), Add(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Add(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True)), Add(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\varphi^*')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), log(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Add(Function('E_{\\\\lambda}')(Symbol('\\\\varepsilon_0', commutative=True)), log(Symbol('\\\\varepsilon_0', commutative=True))), Mul(Integer(2), log(Symbol('\\\\varepsilon_0', commutative=True))))"], [["substitute_RHS_for_LHS", 7, 5], "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('\\\\varepsilon_0', commutative=True)), Mul(Integer(2), log(Symbol('\\\\varepsilon_0', commutative=True))))"]]}, {"prompt": "Given \\theta_{1}{(\\mathbf{s},H)} = H \\mathbf{s}, then derive 0 = \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH), then obtain \\int 0 d\\mathbf{s} = \\int \\hat{p} \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH) d\\mathbf{s}", "derivation": "\\theta_{1}{(\\mathbf{s},H)} = H \\mathbf{s} and \\int \\theta_{1}{(\\mathbf{s},H)} dH = \\int H \\mathbf{s} dH and 0 = \\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH and \\frac{d}{d H} 0 = \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH) and 0 = \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH) and 0 = \\hat{p} \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH) and \\int 0 d\\mathbf{s} = \\int \\hat{p} \\frac{\\partial}{\\partial H} (\\int H \\mathbf{s} dH - \\int \\theta_{1}{(\\mathbf{s},H)} dH) d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)))"], [["integrate", 1, "Symbol('H', commutative=True)"], "Equality(Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))), Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))))"], [["minus", 2, "Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True)))"], "Equality(Integer(0), Add(Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))))"], [["differentiate", 3, "Symbol('H', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('H', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Integer(0), Derivative(Add(Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1))))"], [["times", 5, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\hat{p}', commutative=True), Derivative(Add(Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1)))))"], [["integrate", 6, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Integer(0), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Mul(Symbol('\\\\hat{p}', commutative=True), Derivative(Add(Integral(Mul(Symbol('H', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('H', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\theta_1')(Symbol('\\\\mathbf{s}', commutative=True), Symbol('H', commutative=True)), Tuple(Symbol('H', commutative=True))))), Tuple(Symbol('H', commutative=True), Integer(1)))), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{g_{\\varepsilon}}{(v)} = e^{v}, then obtain 1 = \\frac{\\frac{d}{d v} \\frac{e^{v}}{\\operatorname{g_{\\varepsilon}}{(v)}}}{\\frac{d}{d v} 1}", "derivation": "\\operatorname{g_{\\varepsilon}}{(v)} = e^{v} and 1 = \\frac{e^{v}}{\\operatorname{g_{\\varepsilon}}{(v)}} and \\frac{d}{d v} 1 = \\frac{d}{d v} \\frac{e^{v}}{\\operatorname{g_{\\varepsilon}}{(v)}} and 1 = \\frac{\\frac{d}{d v} \\frac{e^{v}}{\\operatorname{g_{\\varepsilon}}{(v)}}}{\\frac{d}{d v} 1}", "srepr_derivation": [["premise", "Equality(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), exp(Symbol('v', commutative=True)))"], [["divide", 1, "Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), Integer(-1)), exp(Symbol('v', commutative=True))))"], [["differentiate", 2, "Symbol('v', commutative=True)"], "Equality(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), Integer(-1)), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["divide", 3, "Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1)))"], "Equality(Integer(1), Mul(Pow(Derivative(Integer(1), Tuple(Symbol('v', commutative=True), Integer(1))), Integer(-1)), Derivative(Mul(Pow(Function('g_{\\\\varepsilon}')(Symbol('v', commutative=True)), Integer(-1)), exp(Symbol('v', commutative=True))), Tuple(Symbol('v', commutative=True), Integer(1)))))"]]}, {"prompt": "Given n{(C_{d},C_{1})} = - C_{1} + C_{d}, then obtain \\frac{v - \\varphi{(v)} + \\log{(v)}^{v} + 1}{C_{d}} = \\frac{v - \\varphi{(v)} + \\log{(v)}^{v} + \\frac{\\int (- C_{1} + C_{d}) dC_{d}}{\\int n{(C_{d},C_{1})} dC_{d}}}{C_{d}}", "derivation": "n{(C_{d},C_{1})} = - C_{1} + C_{d} and \\int n{(C_{d},C_{1})} dC_{d} = \\int (- C_{1} + C_{d}) dC_{d} and 1 = \\frac{\\int (- C_{1} + C_{d}) dC_{d}}{\\int n{(C_{d},C_{1})} dC_{d}} and v + 1 = v + \\frac{\\int (- C_{1} + C_{d}) dC_{d}}{\\int n{(C_{d},C_{1})} dC_{d}} and v - \\varphi{(v)} + \\log{(v)}^{v} + 1 = v - \\varphi{(v)} + \\log{(v)}^{v} + \\frac{\\int (- C_{1} + C_{d}) dC_{d}}{\\int n{(C_{d},C_{1})} dC_{d}} and \\frac{v - \\varphi{(v)} + \\log{(v)}^{v} + 1}{C_{d}} = \\frac{v - \\varphi{(v)} + \\log{(v)}^{v} + \\frac{\\int (- C_{1} + C_{d}) dC_{d}}{\\int n{(C_{d},C_{1})} dC_{d}}}{C_{d}}", "srepr_derivation": [["get_premise", "Equality(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)))"], [["integrate", 1, "Symbol('C_d', commutative=True)"], "Equality(Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))))"], [["divide", 2, "Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True)))"], "Equality(Integer(1), Mul(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Pow(Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integer(-1))))"], [["add", 3, "Symbol('v', commutative=True)"], "Equality(Add(Symbol('v', commutative=True), Integer(1)), Add(Symbol('v', commutative=True), Mul(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Pow(Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integer(-1)))))"], [["minus", 4, "Add(Function('\\\\varphi')(Symbol('v', commutative=True)), Mul(Integer(-1), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True))))"], "Equality(Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v', commutative=True))), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Integer(1)), Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v', commutative=True))), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Mul(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Pow(Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integer(-1)))))"], [["divide", 5, "Symbol('C_d', commutative=True)"], "Equality(Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v', commutative=True))), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Integer(1))), Mul(Pow(Symbol('C_d', commutative=True), Integer(-1)), Add(Symbol('v', commutative=True), Mul(Integer(-1), Function('\\\\varphi')(Symbol('v', commutative=True))), Pow(log(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Mul(Integral(Add(Mul(Integer(-1), Symbol('C_1', commutative=True)), Symbol('C_d', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Pow(Integral(Function('n')(Symbol('C_d', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('C_d', commutative=True))), Integer(-1))))))"]]}, {"prompt": "Given E{(t_{1},v_{y})} = t_{1} v_{y} and \\Psi_{nl}{(t_{1},v_{y})} = t_{1} v_{y} + E{(t_{1},v_{y})}, then obtain \\frac{\\partial}{\\partial v_{y}} 2 E{(t_{1},v_{y})} = \\frac{\\partial}{\\partial v_{y}} 2 t_{1} v_{y}", "derivation": "E{(t_{1},v_{y})} = t_{1} v_{y} and 2 E{(t_{1},v_{y})} = t_{1} v_{y} + E{(t_{1},v_{y})} and \\Psi_{nl}{(t_{1},v_{y})} = t_{1} v_{y} + E{(t_{1},v_{y})} and \\Psi_{nl}{(t_{1},v_{y})} = 2 t_{1} v_{y} and t_{1} v_{y} + E{(t_{1},v_{y})} = 2 t_{1} v_{y} and 2 E{(t_{1},v_{y})} = 2 t_{1} v_{y} and \\frac{\\partial}{\\partial v_{y}} 2 E{(t_{1},v_{y})} = \\frac{\\partial}{\\partial v_{y}} 2 t_{1} v_{y}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Mul(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)))"], [["add", 1, "Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))"], "Equality(Mul(Integer(2), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))), Add(Mul(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Add(Mul(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Function('\\\\Psi_{nl}')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Mul(Integer(2), Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Mul(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(2), Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 5], "Equality(Mul(Integer(2), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))), Mul(Integer(2), Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)))"], [["differentiate", 6, "Symbol('v_y', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('E')(Symbol('t_1', commutative=True), Symbol('v_y', commutative=True))), Tuple(Symbol('v_y', commutative=True), Integer(1))), Derivative(Mul(Integer(2), Symbol('t_1', commutative=True), Symbol('v_y', commutative=True)), Tuple(Symbol('v_y', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\Omega{(t_{1})} = e^{e^{t_{1}}} and Q{(v_{2},z)} = v_{2} + z, then obtain Q{(v_{2},z)} - e^{e^{t_{1}}} = v_{2} + z - e^{e^{t_{1}}}", "derivation": "\\Omega{(t_{1})} = e^{e^{t_{1}}} and Q{(v_{2},z)} = v_{2} + z and Q{(v_{2},z)} - \\Omega{(t_{1})} = v_{2} + z - \\Omega{(t_{1})} and Q{(v_{2},z)} - e^{e^{t_{1}}} = v_{2} + z - e^{e^{t_{1}}}", "srepr_derivation": [["premise", "Equality(Function('\\\\Omega')(Symbol('t_1', commutative=True)), exp(exp(Symbol('t_1', commutative=True))))"], ["get_premise", "Equality(Function('Q')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True)))"], [["minus", 2, "Function('\\\\Omega')(Symbol('t_1', commutative=True))"], "Equality(Add(Function('Q')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), Function('\\\\Omega')(Symbol('t_1', commutative=True)))), Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), Function('\\\\Omega')(Symbol('t_1', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Function('Q')(Symbol('v_2', commutative=True), Symbol('z', commutative=True)), Mul(Integer(-1), exp(exp(Symbol('t_1', commutative=True))))), Add(Symbol('v_2', commutative=True), Symbol('z', commutative=True), Mul(Integer(-1), exp(exp(Symbol('t_1', commutative=True))))))"]]}, {"prompt": "Given \\dot{z}{(G,\\mathbf{D})} = \\frac{\\partial}{\\partial G} \\mathbf{D}^{G}, then obtain \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} + \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\dot{z}{(G,\\mathbf{D})} dG = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} + \\int \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} dG", "derivation": "\\dot{z}{(G,\\mathbf{D})} = \\frac{\\partial}{\\partial G} \\mathbf{D}^{G} and \\frac{\\partial}{\\partial \\mathbf{D}} \\dot{z}{(G,\\mathbf{D})} = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} and \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\dot{z}{(G,\\mathbf{D})} dG = \\int \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} dG and \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} + \\int \\frac{\\partial}{\\partial \\mathbf{D}} \\dot{z}{(G,\\mathbf{D})} dG = \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} + \\int \\frac{\\partial^{2}}{\\partial \\mathbf{D}\\partial G} \\mathbf{D}^{G} dG", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{D}', commutative=True)"], "Equality(Derivative(Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('G', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))), Integral(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True))))"], [["add", 3, "Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1)))"], "Equality(Add(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integral(Derivative(Function('\\\\dot{z}')(Symbol('G', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))), Add(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Integral(Derivative(Pow(Symbol('\\\\mathbf{D}', commutative=True), Symbol('G', commutative=True)), Tuple(Symbol('G', commutative=True), Integer(1)), Tuple(Symbol('\\\\mathbf{D}', commutative=True), Integer(1))), Tuple(Symbol('G', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{E}{(f^{\\prime},a)} = \\int (- a + f^{\\prime}) df^{\\prime}, then obtain \\sin{(\\int \\mathbf{E}{(f^{\\prime},a)} df^{\\prime} + 1)} = \\sin{(\\iint (- a + f^{\\prime}) df^{\\prime} df^{\\prime} + 1)}", "derivation": "\\mathbf{E}{(f^{\\prime},a)} = \\int (- a + f^{\\prime}) df^{\\prime} and \\int \\mathbf{E}{(f^{\\prime},a)} df^{\\prime} = \\iint (- a + f^{\\prime}) df^{\\prime} df^{\\prime} and \\int \\mathbf{E}{(f^{\\prime},a)} df^{\\prime} + 1 = \\iint (- a + f^{\\prime}) df^{\\prime} df^{\\prime} + 1 and \\sin{(\\int \\mathbf{E}{(f^{\\prime},a)} df^{\\prime} + 1)} = \\sin{(\\iint (- a + f^{\\prime}) df^{\\prime} df^{\\prime} + 1)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('a', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["integrate", 1, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["add", 2, 1], "Equality(Add(Integral(Function('\\\\mathbf{E}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)), Add(Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1)))"], [["sin", 3], "Equality(sin(Add(Integral(Function('\\\\mathbf{E}')(Symbol('f^{\\\\prime}', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1))), sin(Add(Integral(Add(Mul(Integer(-1), Symbol('a', commutative=True)), Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integer(1))))"]]}, {"prompt": "Given t{(I)} = \\int e^{I} dI, then obtain ((t{(I)} - e^{I}) e^{- I})^{I} + e^{I} - \\int e^{I} dI = ((- e^{I} + \\int e^{I} dI) e^{- I})^{I} + e^{I} - \\int e^{I} dI", "derivation": "t{(I)} = \\int e^{I} dI and t{(I)} - e^{I} = - e^{I} + \\int e^{I} dI and (t{(I)} - e^{I}) e^{- I} = (- e^{I} + \\int e^{I} dI) e^{- I} and ((t{(I)} - e^{I}) e^{- I})^{I} = ((- e^{I} + \\int e^{I} dI) e^{- I})^{I} and ((t{(I)} - e^{I}) e^{- I})^{I} + e^{I} - \\int e^{I} dI = ((- e^{I} + \\int e^{I} dI) e^{- I})^{I} + e^{I} - \\int e^{I} dI", "srepr_derivation": [["premise", "Equality(Function('t')(Symbol('I', commutative=True)), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))"], [["minus", 1, "exp(Symbol('I', commutative=True))"], "Equality(Add(Function('t')(Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('I', commutative=True)))), Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], [["divide", 2, "exp(Symbol('I', commutative=True))"], "Equality(Mul(Add(Function('t')(Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Mul(Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))))"], [["power", 3, "Symbol('I', commutative=True)"], "Equality(Pow(Mul(Add(Function('t')(Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Symbol('I', commutative=True)), Pow(Mul(Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Symbol('I', commutative=True)))"], [["add", 4, "Add(exp(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))))"], "Equality(Add(Pow(Mul(Add(Function('t')(Symbol('I', commutative=True)), Mul(Integer(-1), exp(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))), Add(Pow(Mul(Add(Mul(Integer(-1), exp(Symbol('I', commutative=True))), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True)))), exp(Mul(Integer(-1), Symbol('I', commutative=True)))), Symbol('I', commutative=True)), exp(Symbol('I', commutative=True)), Mul(Integer(-1), Integral(exp(Symbol('I', commutative=True)), Tuple(Symbol('I', commutative=True))))))"]]}, {"prompt": "Given U{(G)} = \\cos{(e^{G})} and \\operatorname{F_{g}}{(G)} = e^{G}, then obtain \\cos{(\\operatorname{F_{g}}{(G)})} = \\cos{(e^{G})}", "derivation": "U{(G)} = \\cos{(e^{G})} and \\operatorname{F_{g}}{(G)} = e^{G} and U{(G)} = \\cos{(\\operatorname{F_{g}}{(G)})} and \\cos{(\\operatorname{F_{g}}{(G)})} = \\cos{(e^{G})}", "srepr_derivation": [["premise", "Equality(Function('U')(Symbol('G', commutative=True)), cos(exp(Symbol('G', commutative=True))))"], ["renaming_premise", "Equality(Function('F_g')(Symbol('G', commutative=True)), exp(Symbol('G', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('U')(Symbol('G', commutative=True)), cos(Function('F_g')(Symbol('G', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(cos(Function('F_g')(Symbol('G', commutative=True))), cos(exp(Symbol('G', commutative=True))))"]]}, {"prompt": "Given \\hat{x}{(q)} = \\log{(\\cos{(q)})}, then obtain \\int e^{\\int (q \\hat{x}{(q)} + \\log{(\\cos{(q)})}) dq} dq = \\int e^{\\int (q \\log{(\\cos{(q)})} + \\log{(\\cos{(q)})}) dq} dq", "derivation": "\\hat{x}{(q)} = \\log{(\\cos{(q)})} and q \\hat{x}{(q)} = q \\log{(\\cos{(q)})} and q \\hat{x}{(q)} + \\log{(\\cos{(q)})} = q \\log{(\\cos{(q)})} + \\log{(\\cos{(q)})} and \\int (q \\hat{x}{(q)} + \\log{(\\cos{(q)})}) dq = \\int (q \\log{(\\cos{(q)})} + \\log{(\\cos{(q)})}) dq and e^{\\int (q \\hat{x}{(q)} + \\log{(\\cos{(q)})}) dq} = e^{\\int (q \\log{(\\cos{(q)})} + \\log{(\\cos{(q)})}) dq} and \\int e^{\\int (q \\hat{x}{(q)} + \\log{(\\cos{(q)})}) dq} dq = \\int e^{\\int (q \\log{(\\cos{(q)})} + \\log{(\\cos{(q)})}) dq} dq", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('q', commutative=True)), log(cos(Symbol('q', commutative=True))))"], [["times", 1, "Symbol('q', commutative=True)"], "Equality(Mul(Symbol('q', commutative=True), Function('\\\\hat{x}')(Symbol('q', commutative=True))), Mul(Symbol('q', commutative=True), log(cos(Symbol('q', commutative=True)))))"], [["add", 2, "log(cos(Symbol('q', commutative=True)))"], "Equality(Add(Mul(Symbol('q', commutative=True), Function('\\\\hat{x}')(Symbol('q', commutative=True))), log(cos(Symbol('q', commutative=True)))), Add(Mul(Symbol('q', commutative=True), log(cos(Symbol('q', commutative=True)))), log(cos(Symbol('q', commutative=True)))))"], [["integrate", 3, "Symbol('q', commutative=True)"], "Equality(Integral(Add(Mul(Symbol('q', commutative=True), Function('\\\\hat{x}')(Symbol('q', commutative=True))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(Add(Mul(Symbol('q', commutative=True), log(cos(Symbol('q', commutative=True)))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))))"], [["exp", 4], "Equality(exp(Integral(Add(Mul(Symbol('q', commutative=True), Function('\\\\hat{x}')(Symbol('q', commutative=True))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))), exp(Integral(Add(Mul(Symbol('q', commutative=True), log(cos(Symbol('q', commutative=True)))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))))"], [["integrate", 5, "Symbol('q', commutative=True)"], "Equality(Integral(exp(Integral(Add(Mul(Symbol('q', commutative=True), Function('\\\\hat{x}')(Symbol('q', commutative=True))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))), Integral(exp(Integral(Add(Mul(Symbol('q', commutative=True), log(cos(Symbol('q', commutative=True)))), log(cos(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True)))), Tuple(Symbol('q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(J)} = \\cos{(J)}, then derive (\\frac{d}{d J} \\operatorname{F_{N}}{(J)})^{J} = (- \\sin{(J)})^{J}, then obtain \\int (\\frac{d}{d J} \\cos{(J)})^{J} dJ = \\int (- \\sin{(J)})^{J} dJ", "derivation": "\\operatorname{F_{N}}{(J)} = \\cos{(J)} and \\frac{d}{d J} \\operatorname{F_{N}}{(J)} = \\frac{d}{d J} \\cos{(J)} and (\\frac{d}{d J} \\operatorname{F_{N}}{(J)})^{J} = (\\frac{d}{d J} \\cos{(J)})^{J} and (\\frac{d}{d J} \\operatorname{F_{N}}{(J)})^{J} = (- \\sin{(J)})^{J} and (\\frac{d}{d J} \\cos{(J)})^{J} = (- \\sin{(J)})^{J} and \\int (\\frac{d}{d J} \\cos{(J)})^{J} dJ = \\int (- \\sin{(J)})^{J} dJ", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('J', commutative=True)), cos(Symbol('J', commutative=True)))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('F_N')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["power", 2, "Symbol('J', commutative=True)"], "Equality(Pow(Derivative(Function('F_N')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)))"], [["evaluate_derivatives", 3], "Equality(Pow(Derivative(Function('F_N')(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)))"], [["integrate", 5, "Symbol('J', commutative=True)"], "Equality(Integral(Pow(Derivative(cos(Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))), Integral(Pow(Mul(Integer(-1), sin(Symbol('J', commutative=True))), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True))))"]]}, {"prompt": "Given I{(v_{2},v_{t})} = v_{2} - v_{t}, then obtain 0 = \\frac{- 2 v_{2} + 2 v_{t} + 2 I{(v_{2},v_{t})}}{(- 2 v_{2} + 2 v_{t} + I{(v_{2},v_{t})}) (- v_{2} - v_{t} + I{(v_{2},v_{t})})}", "derivation": "I{(v_{2},v_{t})} = v_{2} - v_{t} and 0 = v_{2} - v_{t} - I{(v_{2},v_{t})} and 0 = - v_{2} + v_{t} + I{(v_{2},v_{t})} and - v_{2} + v_{t} = - 2 v_{2} + 2 v_{t} + I{(v_{2},v_{t})} and 0 = \\frac{- v_{2} + v_{t} + I{(v_{2},v_{t})}}{- v_{2} + v_{t}} and 0 = \\frac{- 2 v_{2} + 2 v_{t} + 2 I{(v_{2},v_{t})}}{- 2 v_{2} + 2 v_{t} + I{(v_{2},v_{t})}} and 0 = \\frac{- 2 v_{2} + 2 v_{t} + 2 I{(v_{2},v_{t})}}{(- 2 v_{2} + 2 v_{t} + I{(v_{2},v_{t})}) (- v_{2} - v_{t} + I{(v_{2},v_{t})})}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True))))"], [["minus", 1, "Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))"], "Equality(Integer(0), Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)), Mul(Integer(-1), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))))"], [["times", 2, "Integer(-1)"], "Equality(Integer(0), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('v_t', commutative=True), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))"], [["minus", 3, "Add(Symbol('v_2', commutative=True), Mul(Integer(-1), Symbol('v_t', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('v_t', commutative=True)), Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))"], [["divide", 3, "Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('v_t', commutative=True))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('v_t', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Symbol('v_t', commutative=True), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))))))"], [["divide", 6, "Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))"], "Equality(Integer(0), Mul(Pow(Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Integer(-1)), Add(Mul(Integer(-1), Integer(2), Symbol('v_2', commutative=True)), Mul(Integer(2), Symbol('v_t', commutative=True)), Mul(Integer(2), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True)))), Pow(Add(Mul(Integer(-1), Symbol('v_2', commutative=True)), Mul(Integer(-1), Symbol('v_t', commutative=True)), Function('I')(Symbol('v_2', commutative=True), Symbol('v_t', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given \\rho_{b}{(\\ddot{x},\\hat{X})} = \\sin{(\\hat{X}^{\\ddot{x}})} and \\mathbf{P}{(\\ddot{x},\\hat{X})} = \\sin{(\\hat{X}^{\\ddot{x}})}, then obtain \\int (\\rho_{b}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}}) d\\hat{X} = \\int (\\mathbf{P}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}}) d\\hat{X}", "derivation": "\\rho_{b}{(\\ddot{x},\\hat{X})} = \\sin{(\\hat{X}^{\\ddot{x}})} and \\mathbf{P}{(\\ddot{x},\\hat{X})} = \\sin{(\\hat{X}^{\\ddot{x}})} and \\rho_{b}{(\\ddot{x},\\hat{X})} = \\mathbf{P}{(\\ddot{x},\\hat{X})} and \\rho_{b}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}} = \\mathbf{P}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}} and \\int (\\rho_{b}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}}) d\\hat{X} = \\int (\\mathbf{P}{(\\ddot{x},\\hat{X})} - e^{\\mathbf{P}{(\\ddot{x},\\hat{X})}}) d\\hat{X}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), sin(Pow(Symbol('\\\\hat{X}', commutative=True), Symbol('\\\\ddot{x}', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], [["minus", 3, "exp(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)))"], "Equality(Add(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Add(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))))"], [["integrate", 4, "Symbol('\\\\hat{X}', commutative=True)"], "Equality(Integral(Add(Function('\\\\rho_b')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True))), Integral(Add(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True)), Mul(Integer(-1), exp(Function('\\\\mathbf{P}')(Symbol('\\\\ddot{x}', commutative=True), Symbol('\\\\hat{X}', commutative=True))))), Tuple(Symbol('\\\\hat{X}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(P_{e})} = \\log{(P_{e})}, then obtain 2 (\\frac{\\psi^{*}{(P_{e})}}{P_{e}})^{P_{e}} \\log{(P_{e})} = 2 (\\frac{\\log{(P_{e})}}{P_{e}})^{P_{e}} \\log{(P_{e})}", "derivation": "\\psi^{*}{(P_{e})} = \\log{(P_{e})} and \\frac{\\psi^{*}{(P_{e})}}{P_{e}} = \\frac{\\log{(P_{e})}}{P_{e}} and (\\frac{\\psi^{*}{(P_{e})}}{P_{e}})^{P_{e}} = (\\frac{\\log{(P_{e})}}{P_{e}})^{P_{e}} and \\psi^{*}{(P_{e})} + \\log{(P_{e})} = 2 \\log{(P_{e})} and (\\frac{\\psi^{*}{(P_{e})}}{P_{e}})^{P_{e}} (\\psi^{*}{(P_{e})} + \\log{(P_{e})}) = (\\frac{\\log{(P_{e})}}{P_{e}})^{P_{e}} (\\psi^{*}{(P_{e})} + \\log{(P_{e})}) and 2 (\\frac{\\psi^{*}{(P_{e})}}{P_{e}})^{P_{e}} \\log{(P_{e})} = 2 (\\frac{\\log{(P_{e})}}{P_{e}})^{P_{e}} \\log{(P_{e})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], [["divide", 1, "Symbol('P_e', commutative=True)"], "Equality(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('P_e', commutative=True))), Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), log(Symbol('P_e', commutative=True))))"], [["power", 2, "Symbol('P_e', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)))"], [["add", 1, "log(Symbol('P_e', commutative=True))"], "Equality(Add(Function('\\\\psi^*')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Mul(Integer(2), log(Symbol('P_e', commutative=True))))"], [["times", 3, "Add(Function('\\\\psi^*')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))"], "Equality(Mul(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Add(Function('\\\\psi^*')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))), Mul(Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), Add(Function('\\\\psi^*')(Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True)))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Integer(2), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), Function('\\\\psi^*')(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))), Mul(Integer(2), Pow(Mul(Pow(Symbol('P_e', commutative=True), Integer(-1)), log(Symbol('P_e', commutative=True))), Symbol('P_e', commutative=True)), log(Symbol('P_e', commutative=True))))"]]}, {"prompt": "Given E{(\\phi_2,\\omega)} = \\cos^{\\phi_2}{(\\omega)}, then obtain ((\\int E{(\\phi_2,\\omega)} d\\omega)^{3}) \\int \\cos^{\\phi_2}{(\\omega)} d\\omega = ((\\int E{(\\phi_2,\\omega)} d\\omega)^{2}) (\\int \\cos^{\\phi_2}{(\\omega)} d\\omega)^{2}", "derivation": "E{(\\phi_2,\\omega)} = \\cos^{\\phi_2}{(\\omega)} and \\int E{(\\phi_2,\\omega)} d\\omega = \\int \\cos^{\\phi_2}{(\\omega)} d\\omega and (\\int E{(\\phi_2,\\omega)} d\\omega)^{2} = (\\int E{(\\phi_2,\\omega)} d\\omega) \\int \\cos^{\\phi_2}{(\\omega)} d\\omega and ((\\int E{(\\phi_2,\\omega)} d\\omega)^{2}) \\int \\cos^{\\phi_2}{(\\omega)} d\\omega = (\\int E{(\\phi_2,\\omega)} d\\omega) (\\int \\cos^{\\phi_2}{(\\omega)} d\\omega)^{2} and (\\int E{(\\phi_2,\\omega)} d\\omega)^{3} = ((\\int E{(\\phi_2,\\omega)} d\\omega)^{2}) \\int \\cos^{\\phi_2}{(\\omega)} d\\omega and ((\\int E{(\\phi_2,\\omega)} d\\omega)^{3}) \\int \\cos^{\\phi_2}{(\\omega)} d\\omega = ((\\int E{(\\phi_2,\\omega)} d\\omega)^{2}) (\\int \\cos^{\\phi_2}{(\\omega)} d\\omega)^{2}", "srepr_derivation": [["premise", "Equality(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)))"], [["integrate", 1, "Symbol('\\\\omega', commutative=True)"], "Equality(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))))"], [["times", 2, "Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)), Mul(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["times", 3, "Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Pow(Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(3)), Mul(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))))"], [["times", 5, "Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))"], "Equality(Mul(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(3)), Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True)))), Mul(Pow(Integral(Function('E')(Symbol('\\\\phi_2', commutative=True), Symbol('\\\\omega', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2)), Pow(Integral(Pow(cos(Symbol('\\\\omega', commutative=True)), Symbol('\\\\phi_2', commutative=True)), Tuple(Symbol('\\\\omega', commutative=True))), Integer(2))))"]]}, {"prompt": "Given x{(\\rho_b,\\mathbf{g})} = - \\mathbf{g} + \\cos{(\\rho_b)} and s{(\\mathbf{g})} = - \\mathbf{g}, then obtain \\frac{\\partial}{\\partial \\mathbf{g}} x{(\\rho_b,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g} + \\cos{(\\rho_b)})", "derivation": "x{(\\rho_b,\\mathbf{g})} = - \\mathbf{g} + \\cos{(\\rho_b)} and s{(\\mathbf{g})} = - \\mathbf{g} and x{(\\rho_b,\\mathbf{g})} = s{(\\mathbf{g})} + \\cos{(\\rho_b)} and - \\mathbf{g} + \\cos{(\\rho_b)} = s{(\\mathbf{g})} + \\cos{(\\rho_b)} and \\frac{\\partial}{\\partial \\mathbf{g}} x{(\\rho_b,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} (s{(\\mathbf{g})} + \\cos{(\\rho_b)}) and \\frac{\\partial}{\\partial \\mathbf{g}} x{(\\rho_b,\\mathbf{g})} = \\frac{\\partial}{\\partial \\mathbf{g}} (- \\mathbf{g} + \\cos{(\\rho_b)})", "srepr_derivation": [["premise", "Equality(Function('x')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))))"], ["renaming_premise", "Equality(Function('s')(Symbol('\\\\mathbf{g}', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('x')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Add(Function('s')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Add(Function('s')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{g}', commutative=True)"], "Equality(Derivative(Function('x')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Function('s')(Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Derivative(Function('x')(Symbol('\\\\rho_b', commutative=True), Symbol('\\\\mathbf{g}', commutative=True)), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('\\\\mathbf{g}', commutative=True)), cos(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\mathbf{g}', commutative=True), Integer(1))))"]]}, {"prompt": "Given y{(a)} = \\cos{(\\log{(a)})}, then obtain - \\frac{y{(a)}}{\\cos{(\\log{(a)})}} - \\log{(a)} + 1 = - \\frac{y{(a)}}{\\cos{(\\log{(a)})}} - \\frac{2 y{(a)}}{\\cos{(\\frac{y{(a)}}{\\cos{(\\log{(a)})}} + \\log{(a)} - 1)}} - \\log{(a)} + 3", "derivation": "y{(a)} = \\cos{(\\log{(a)})} and \\frac{y{(a)}}{\\cos{(\\log{(a)})}} = 1 and \\frac{y{(a)}}{\\cos{(\\log{(a)})}} + \\log{(a)} = \\log{(a)} + 1 and - \\log{(a)} = - \\frac{y{(a)}}{\\cos{(\\log{(a)})}} - \\log{(a)} + 1 and - \\log{(a)} = - \\frac{2 y{(a)}}{\\cos{(\\log{(a)})}} - \\log{(a)} + 2 and - \\frac{y{(a)}}{\\cos{(\\log{(a)})}} - \\log{(a)} + 1 = - \\frac{y{(a)}}{\\cos{(\\log{(a)})}} - \\frac{2 y{(a)}}{\\cos{(\\frac{y{(a)}}{\\cos{(\\log{(a)})}} + \\log{(a)} - 1)}} - \\log{(a)} + 3", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('a', commutative=True)), cos(log(Symbol('a', commutative=True))))"], [["divide", 1, "cos(log(Symbol('a', commutative=True)))"], "Equality(Mul(Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), Integer(1))"], [["add", 2, "log(Symbol('a', commutative=True))"], "Equality(Add(Mul(Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), log(Symbol('a', commutative=True))), Add(log(Symbol('a', commutative=True)), Integer(1)))"], [["minus", 2, "Add(Mul(Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), log(Symbol('a', commutative=True)))"], "Equality(Mul(Integer(-1), log(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), Mul(Integer(-1), log(Symbol('a', commutative=True))), Integer(1)))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Integer(-1), log(Symbol('a', commutative=True))), Add(Mul(Integer(-1), Integer(2), Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), Mul(Integer(-1), log(Symbol('a', commutative=True))), Integer(2)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Add(Mul(Integer(-1), Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), Mul(Integer(-1), log(Symbol('a', commutative=True))), Integer(1)), Add(Mul(Integer(-1), Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), Mul(Integer(-1), Integer(2), Function('y')(Symbol('a', commutative=True)), Pow(cos(Add(Mul(Function('y')(Symbol('a', commutative=True)), Pow(cos(log(Symbol('a', commutative=True))), Integer(-1))), log(Symbol('a', commutative=True)), Integer(-1))), Integer(-1))), Mul(Integer(-1), log(Symbol('a', commutative=True))), Integer(3)))"]]}, {"prompt": "Given c{(F_{c})} = e^{F_{c}}, then derive \\frac{d}{d F_{c}} c{(F_{c})} = e^{F_{c}}, then obtain \\log{(\\frac{d^{2}}{d F_{c}^{2}} c{(F_{c})})} = \\log{(\\frac{d}{d F_{c}} e^{F_{c}})}", "derivation": "c{(F_{c})} = e^{F_{c}} and \\frac{d}{d F_{c}} c{(F_{c})} = \\frac{d}{d F_{c}} e^{F_{c}} and \\frac{d}{d F_{c}} c{(F_{c})} = e^{F_{c}} and \\frac{d}{d F_{c}} c{(F_{c})} = c{(F_{c})} and \\frac{d^{2}}{d F_{c}^{2}} c{(F_{c})} = \\frac{d}{d F_{c}} e^{F_{c}} and \\log{(\\frac{d^{2}}{d F_{c}^{2}} c{(F_{c})})} = \\log{(\\frac{d}{d F_{c}} e^{F_{c}})}", "srepr_derivation": [["premise", "Equality(Function('c')(Symbol('F_c', commutative=True)), exp(Symbol('F_c', commutative=True)))"], [["differentiate", 1, "Symbol('F_c', commutative=True)"], "Equality(Derivative(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Derivative(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), exp(Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))), Function('c')(Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Derivative(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(2))), Derivative(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1))))"], [["log", 5], "Equality(log(Derivative(Function('c')(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(2)))), log(Derivative(exp(Symbol('F_c', commutative=True)), Tuple(Symbol('F_c', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}{(T)} = \\log{(\\cos{(T)})}, then obtain T + 2 \\mathbf{J}{(T)} - 2 \\log{(\\cos{(T)})} + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} = T + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}}", "derivation": "\\mathbf{J}{(T)} = \\log{(\\cos{(T)})} and \\mathbf{J}{(T)} - \\log{(\\cos{(T)})} = 0 and T + \\mathbf{J}{(T)} - \\log{(\\cos{(T)})} = T and T + \\mathbf{J}{(T)} - \\log{(\\cos{(T)})} + \\cos{(T)} = T + \\cos{(T)} and T + \\mathbf{J}{(T)} - \\log{(\\cos{(T)})} + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} = T + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} and T + 2 \\mathbf{J}{(T)} - 2 \\log{(\\cos{(T)})} + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} = T + \\mathbf{J}{(T)} - \\log{(\\cos{(T)})} + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} and T + 2 \\mathbf{J}{(T)} - 2 \\log{(\\cos{(T)})} + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}} = T + \\cos{(T)} - \\frac{1}{\\log{(\\cos{(T)})}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), log(cos(Symbol('T', commutative=True))))"], [["minus", 1, "log(cos(Symbol('T', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(cos(Symbol('T', commutative=True))))), Integer(0))"], [["add", 2, "Symbol('T', commutative=True)"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(cos(Symbol('T', commutative=True))))), Symbol('T', commutative=True))"], [["add", 3, "cos(Symbol('T', commutative=True))"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True))), Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True))))"], [["minus", 4, "Pow(log(cos(Symbol('T', commutative=True))), Integer(-1))"], "Equality(Add(Symbol('T', commutative=True), Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))), Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('T', commutative=True))), Mul(Integer(-1), Integer(2), log(cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))), Add(Symbol('T', commutative=True), Function('\\\\mathbf{J}')(Symbol('T', commutative=True)), Mul(Integer(-1), log(cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Add(Symbol('T', commutative=True), Mul(Integer(2), Function('\\\\mathbf{J}')(Symbol('T', commutative=True))), Mul(Integer(-1), Integer(2), log(cos(Symbol('T', commutative=True)))), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))), Add(Symbol('T', commutative=True), cos(Symbol('T', commutative=True)), Mul(Integer(-1), Pow(log(cos(Symbol('T', commutative=True))), Integer(-1)))))"]]}, {"prompt": "Given \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})} = \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} \\mathbf{H} and \\sigma_{p}{(x,f_{E})} = \\frac{f_{E}}{x}, then derive \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})} = \\mathbf{H}, then obtain \\mathbf{H}^{2} \\sigma_{p}{(x,f_{E})} = \\frac{\\mathbf{H}^{2} f_{E}}{x}", "derivation": "\\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})} = \\frac{\\partial}{\\partial \\hat{H}} \\hat{H} \\mathbf{H} and \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})} = \\mathbf{H} and \\sigma_{p}{(x,f_{E})} = \\frac{f_{E}}{x} and \\mathbf{H} \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})} \\sigma_{p}{(x,f_{E})} = \\frac{\\mathbf{H} f_{E} \\dot{\\mathbf{r}}{(\\hat{H},\\mathbf{H})}}{x} and \\mathbf{H}^{2} \\sigma_{p}{(x,f_{E})} = \\frac{\\mathbf{H}^{2} f_{E}}{x}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Derivative(Mul(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Symbol('\\\\mathbf{H}', commutative=True))"], ["renaming_premise", "Equality(Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('f_E', commutative=True)), Mul(Symbol('f_E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"], [["times", 3, "Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)))"], "Equality(Mul(Symbol('\\\\mathbf{H}', commutative=True), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True)), Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('f_E', commutative=True))), Mul(Symbol('\\\\mathbf{H}', commutative=True), Symbol('f_E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1)), Function('\\\\dot{\\\\mathbf{r}}')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\mathbf{H}', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Function('\\\\sigma_p')(Symbol('x', commutative=True), Symbol('f_E', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{H}', commutative=True), Integer(2)), Symbol('f_E', commutative=True), Pow(Symbol('x', commutative=True), Integer(-1))))"]]}, {"prompt": "Given z{(\\mathbf{M},E_{\\lambda})} = \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}, then obtain E_{\\lambda} + \\frac{- E_{\\lambda} + z{(\\mathbf{M},E_{\\lambda})}}{- E_{\\lambda} + \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}} - \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}} = E_{\\lambda} + 1 - \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}", "derivation": "z{(\\mathbf{M},E_{\\lambda})} = \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}} and - E_{\\lambda} + z{(\\mathbf{M},E_{\\lambda})} = - E_{\\lambda} + \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}} and \\frac{- E_{\\lambda} + z{(\\mathbf{M},E_{\\lambda})}}{- E_{\\lambda} + \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}} = 1 and E_{\\lambda} + \\frac{- E_{\\lambda} + z{(\\mathbf{M},E_{\\lambda})}}{- E_{\\lambda} + \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}} - \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}} = E_{\\lambda} + 1 - \\frac{\\cos{(\\mathbf{M})}}{E_{\\lambda}}", "srepr_derivation": [["premise", "Equality(Function('z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], [["minus", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True))), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))))"], [["divide", 2, "Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Integer(1))"], [["minus", 3, "Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True))))"], "Equality(Add(Symbol('E_{\\\\lambda}', commutative=True), Mul(Pow(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Mul(Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Integer(-1)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), Function('z')(Symbol('\\\\mathbf{M}', commutative=True), Symbol('E_{\\\\lambda}', commutative=True)))), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))), Add(Symbol('E_{\\\\lambda}', commutative=True), Integer(1), Mul(Integer(-1), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(-1)), cos(Symbol('\\\\mathbf{M}', commutative=True)))))"]]}, {"prompt": "Given \\psi{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})}, then obtain - \\mathbf{A} + 2 \\psi{(\\mathbf{A})} - \\log{(\\sin{(\\mathbf{A})})} = - \\mathbf{A} + \\psi{(\\mathbf{A})}", "derivation": "\\psi{(\\mathbf{A})} = \\log{(\\sin{(\\mathbf{A})})} and - \\mathbf{A} + \\psi{(\\mathbf{A})} = - \\mathbf{A} + \\log{(\\sin{(\\mathbf{A})})} and - \\mathbf{A} + \\psi{(\\mathbf{A})} - \\log{(\\sin{(\\mathbf{A})})} = - \\mathbf{A} and - \\mathbf{A} + 2 \\psi{(\\mathbf{A})} - \\log{(\\sin{(\\mathbf{A})})} = - \\mathbf{A} + \\psi{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), log(sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), log(sin(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["minus", 2, "log(sin(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), log(sin(Symbol('\\\\mathbf{A}', commutative=True))))), Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(-1), log(sin(Symbol('\\\\mathbf{A}', commutative=True))))), Add(Mul(Integer(-1), Symbol('\\\\mathbf{A}', commutative=True)), Function('\\\\psi')(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A_{2},\\hbar,n)} = \\frac{A_{2} + n}{\\hbar} and M{(A_{2},n,\\hbar)} = - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} - \\frac{A_{2} + n}{\\hbar}, then obtain M{(A_{2},n,\\hbar)} - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} + \\frac{- A_{2} - n}{\\hbar} = - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} + \\frac{- A_{2} - n}{\\hbar} - \\frac{2 (A_{2} + n)}{\\hbar}", "derivation": "\\operatorname{v_{y}}{(A_{2},\\hbar,n)} = \\frac{A_{2} + n}{\\hbar} and - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} = - \\frac{A_{2} + n}{\\hbar} and - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} - \\frac{A_{2} + n}{\\hbar} = - \\frac{2 (A_{2} + n)}{\\hbar} and M{(A_{2},n,\\hbar)} = - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} - \\frac{A_{2} + n}{\\hbar} and M{(A_{2},n,\\hbar)} = - \\frac{2 (A_{2} + n)}{\\hbar} and M{(A_{2},n,\\hbar)} - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} + \\frac{- A_{2} - n}{\\hbar} = - \\operatorname{v_{y}}{(A_{2},\\hbar,n)} + \\frac{- A_{2} - n}{\\hbar} - \\frac{2 (A_{2} + n)}{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True)), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True))))"], [["minus", 2, "Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True))))"], ["renaming_premise", "Equality(Function('M')(Symbol('A_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Function('M')(Symbol('A_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True))))"], [["add", 5, "Add(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True)))))"], "Equality(Add(Function('M')(Symbol('A_2', commutative=True), Symbol('n', commutative=True), Symbol('\\\\hbar', commutative=True)), Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))))), Add(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('\\\\hbar', commutative=True), Symbol('n', commutative=True))), Mul(Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True)))), Mul(Integer(-1), Integer(2), Pow(Symbol('\\\\hbar', commutative=True), Integer(-1)), Add(Symbol('A_2', commutative=True), Symbol('n', commutative=True)))))"]]}, {"prompt": "Given \\Psi^{\\dagger}{(Q)} = Q, then derive \\rho + \\frac{\\Psi^{\\dagger}^{2}{(Q)}}{2} = \\int Q d\\Psi^{\\dagger}{(Q)}, then obtain \\frac{Q^{2}}{2} + \\rho = \\int \\Psi^{\\dagger}{(Q)} dQ", "derivation": "\\Psi^{\\dagger}{(Q)} = Q and \\int \\Psi^{\\dagger}{(Q)} dQ = \\int Q dQ and \\int \\Psi^{\\dagger}{(Q)} d\\Psi^{\\dagger}{(Q)} = \\int Q d\\Psi^{\\dagger}{(Q)} and \\rho + \\frac{\\Psi^{\\dagger}^{2}{(Q)}}{2} = \\int Q d\\Psi^{\\dagger}{(Q)} and \\frac{Q^{2}}{2} + \\rho = \\int Q dQ and \\frac{Q^{2}}{2} + \\rho = \\int \\Psi^{\\dagger}{(Q)} dQ", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Symbol('Q', commutative=True))"], [["integrate", 1, "Symbol('Q', commutative=True)"], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)))), Integral(Symbol('Q', commutative=True), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('\\\\rho', commutative=True), Mul(Rational(1, 2), Pow(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Integer(2)))), Integral(Symbol('Q', commutative=True), Tuple(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True)), Integral(Symbol('Q', commutative=True), Tuple(Symbol('Q', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Add(Mul(Rational(1, 2), Pow(Symbol('Q', commutative=True), Integer(2))), Symbol('\\\\rho', commutative=True)), Integral(Function('\\\\Psi^{\\\\dagger}')(Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{H}}{(\\omega,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{\\omega}, then derive \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{F_{H}}{(\\omega,\\mathbf{J}_f)} = \\frac{1}{\\omega}, then obtain (\\int \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\mathbf{J}_f}{\\omega} d\\mathbf{J}_f)^{\\omega} = (\\int \\frac{1}{\\omega} d\\mathbf{J}_f)^{\\omega}", "derivation": "\\operatorname{F_{H}}{(\\omega,\\mathbf{J}_f)} = \\frac{\\mathbf{J}_f}{\\omega} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{F_{H}}{(\\omega,\\mathbf{J}_f)} = \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\mathbf{J}_f}{\\omega} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\operatorname{F_{H}}{(\\omega,\\mathbf{J}_f)} = \\frac{1}{\\omega} and \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\mathbf{J}_f}{\\omega} = \\frac{1}{\\omega} and \\int \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\mathbf{J}_f}{\\omega} d\\mathbf{J}_f = \\int \\frac{1}{\\omega} d\\mathbf{J}_f and (\\int \\frac{\\partial}{\\partial \\mathbf{J}_f} \\frac{\\mathbf{J}_f}{\\omega} d\\mathbf{J}_f)^{\\omega} = (\\int \\frac{1}{\\omega} d\\mathbf{J}_f)^{\\omega}", "srepr_derivation": [["premise", "Equality(Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Derivative(Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('F_H')(Symbol('\\\\omega', commutative=True), Symbol('\\\\mathbf{J}_f', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Pow(Symbol('\\\\omega', commutative=True), Integer(-1)))"], [["integrate", 4, "Symbol('\\\\mathbf{J}_f', commutative=True)"], "Equality(Integral(Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Integral(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))))"], [["power", 5, "Symbol('\\\\omega', commutative=True)"], "Equality(Pow(Integral(Derivative(Mul(Symbol('\\\\mathbf{J}_f', commutative=True), Pow(Symbol('\\\\omega', commutative=True), Integer(-1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\omega', commutative=True)), Pow(Integral(Pow(Symbol('\\\\omega', commutative=True), Integer(-1)), Tuple(Symbol('\\\\mathbf{J}_f', commutative=True))), Symbol('\\\\omega', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(x,\\psi^*)} = \\psi^* + x, then obtain - x \\hat{H}{(x,\\psi^*)} + \\hat{H}{(x,\\psi^*)} + 1 + \\frac{\\psi^* + 2 x}{x \\frac{\\partial}{\\partial x} \\hat{H}{(x,\\psi^*)}} = \\psi^* - x \\hat{H}{(x,\\psi^*)} + x + 1 + \\frac{\\psi^* + 2 x}{x \\frac{\\partial}{\\partial x} \\hat{H}{(x,\\psi^*)}}", "derivation": "\\hat{H}{(x,\\psi^*)} = \\psi^* + x and x \\hat{H}{(x,\\psi^*)} = x (\\psi^* + x) and - x (\\psi^* + x) + \\hat{H}{(x,\\psi^*)} = \\psi^* - x (\\psi^* + x) + x and - x \\hat{H}{(x,\\psi^*)} + \\hat{H}{(x,\\psi^*)} = \\psi^* - x \\hat{H}{(x,\\psi^*)} + x and - x \\hat{H}{(x,\\psi^*)} + \\hat{H}{(x,\\psi^*)} + 1 = \\psi^* - x \\hat{H}{(x,\\psi^*)} + x + 1 and - x \\hat{H}{(x,\\psi^*)} + \\hat{H}{(x,\\psi^*)} + 1 + \\frac{\\psi^* + 2 x}{x \\frac{\\partial}{\\partial x} \\hat{H}{(x,\\psi^*)}} = \\psi^* - x \\hat{H}{(x,\\psi^*)} + x + 1 + \\frac{\\psi^* + 2 x}{x \\frac{\\partial}{\\partial x} \\hat{H}{(x,\\psi^*)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Add(Symbol('\\\\psi^*', commutative=True), Symbol('x', commutative=True)))"], [["times", 1, "Symbol('x', commutative=True)"], "Equality(Mul(Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Mul(Symbol('x', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Symbol('x', commutative=True))))"], [["minus", 1, "Mul(Symbol('x', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Symbol('x', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Symbol('x', commutative=True))), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True), Add(Symbol('\\\\psi^*', commutative=True), Symbol('x', commutative=True))), Symbol('x', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('x', commutative=True)))"], [["add", 4, 1], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('x', commutative=True), Integer(1)))"], [["add", 5, "Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Integer(1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(-1), Symbol('x', commutative=True), Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True))), Symbol('x', commutative=True), Integer(1), Mul(Pow(Symbol('x', commutative=True), Integer(-1)), Add(Symbol('\\\\psi^*', commutative=True), Mul(Integer(2), Symbol('x', commutative=True))), Pow(Derivative(Function('\\\\hat{H}')(Symbol('x', commutative=True), Symbol('\\\\psi^*', commutative=True)), Tuple(Symbol('x', commutative=True), Integer(1))), Integer(-1)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(n_{2},J)} = e^{\\frac{n_{2}}{J}}, then derive \\frac{\\partial}{\\partial J} \\operatorname{v_{y}}{(n_{2},J)} = - \\frac{n_{2} e^{\\frac{n_{2}}{J}}}{J^{2}}, then obtain \\frac{\\partial}{\\partial J} \\operatorname{v_{y}}{(n_{2},J)} = - \\frac{n_{2} \\operatorname{v_{y}}{(n_{2},J)}}{J^{2}}", "derivation": "\\operatorname{v_{y}}{(n_{2},J)} = e^{\\frac{n_{2}}{J}} and \\frac{\\partial}{\\partial J} \\operatorname{v_{y}}{(n_{2},J)} = \\frac{\\partial}{\\partial J} e^{\\frac{n_{2}}{J}} and \\frac{\\partial}{\\partial J} \\operatorname{v_{y}}{(n_{2},J)} = - \\frac{n_{2} e^{\\frac{n_{2}}{J}}}{J^{2}} and \\frac{\\partial}{\\partial J} \\operatorname{v_{y}}{(n_{2},J)} = - \\frac{n_{2} \\operatorname{v_{y}}{(n_{2},J)}}{J^{2}}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('n_2', commutative=True), Symbol('J', commutative=True)), exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))))"], [["differentiate", 1, "Symbol('J', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('n_2', commutative=True))), Tuple(Symbol('J', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2)), Symbol('n_2', commutative=True), exp(Mul(Pow(Symbol('J', commutative=True), Integer(-1)), Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('v_y')(Symbol('n_2', commutative=True), Symbol('J', commutative=True)), Tuple(Symbol('J', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('J', commutative=True), Integer(-2)), Symbol('n_2', commutative=True), Function('v_y')(Symbol('n_2', commutative=True), Symbol('J', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(\\mathbf{S},U)} = e^{\\frac{U}{\\mathbf{S}}}, then derive \\frac{\\partial}{\\partial U} \\operatorname{v_{y}}{(\\mathbf{S},U)} = \\frac{e^{\\frac{U}{\\mathbf{S}}}}{\\mathbf{S}}, then obtain \\frac{\\partial}{\\partial U} e^{\\frac{U}{\\mathbf{S}}} = \\frac{e^{\\frac{U}{\\mathbf{S}}}}{\\mathbf{S}}", "derivation": "\\operatorname{v_{y}}{(\\mathbf{S},U)} = e^{\\frac{U}{\\mathbf{S}}} and \\frac{\\partial}{\\partial U} \\operatorname{v_{y}}{(\\mathbf{S},U)} = \\frac{\\partial}{\\partial U} e^{\\frac{U}{\\mathbf{S}}} and \\frac{\\partial}{\\partial U} \\operatorname{v_{y}}{(\\mathbf{S},U)} = \\frac{e^{\\frac{U}{\\mathbf{S}}}}{\\mathbf{S}} and \\frac{\\partial}{\\partial U} e^{\\frac{U}{\\mathbf{S}}} = \\frac{e^{\\frac{U}{\\mathbf{S}}}}{\\mathbf{S}}", "srepr_derivation": [["get_premise", "Equality(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('U', commutative=True)), exp(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))))"], [["differentiate", 1, "Symbol('U', commutative=True)"], "Equality(Derivative(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Derivative(exp(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('U', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('v_y')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('U', commutative=True)), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), exp(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)))), Tuple(Symbol('U', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1)), exp(Mul(Symbol('U', commutative=True), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\mathbf{v}{(s)} = \\sin{(s)}, then derive s \\int \\mathbf{v}{(s)} ds = s (\\mathbf{r} - \\cos{(s)}), then obtain s^{2} (\\int \\mathbf{v}{(s)} ds)^{2} = s^{2} (\\mathbf{r} - \\cos{(s)}) \\int \\mathbf{v}{(s)} ds", "derivation": "\\mathbf{v}{(s)} = \\sin{(s)} and \\int \\mathbf{v}{(s)} ds = \\int \\sin{(s)} ds and s \\int \\mathbf{v}{(s)} ds = s \\int \\sin{(s)} ds and s \\int \\mathbf{v}{(s)} ds = s (\\mathbf{r} - \\cos{(s)}) and s^{2} (\\int \\mathbf{v}{(s)} ds) \\int \\sin{(s)} ds = s^{2} (\\mathbf{r} - \\cos{(s)}) \\int \\sin{(s)} ds and s^{2} (\\int \\mathbf{v}{(s)} ds)^{2} = s^{2} (\\mathbf{r} - \\cos{(s)}) \\int \\mathbf{v}{(s)} ds", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), sin(Symbol('s', commutative=True)))"], [["integrate", 1, "Symbol('s', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], [["times", 2, "Symbol('s', commutative=True)"], "Equality(Mul(Symbol('s', commutative=True), Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Symbol('s', commutative=True), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["evaluate_integrals", 3], "Equality(Mul(Symbol('s', commutative=True), Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Symbol('s', commutative=True), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True))))))"], [["times", 4, "Mul(Symbol('s', commutative=True), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))))"], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(2)), Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))), Mul(Pow(Symbol('s', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Integral(sin(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"], [["substitute_RHS_for_LHS", 5, 2], "Equality(Mul(Pow(Symbol('s', commutative=True), Integer(2)), Pow(Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True))), Integer(2))), Mul(Pow(Symbol('s', commutative=True), Integer(2)), Add(Symbol('\\\\mathbf{r}', commutative=True), Mul(Integer(-1), cos(Symbol('s', commutative=True)))), Integral(Function('\\\\mathbf{v}')(Symbol('s', commutative=True)), Tuple(Symbol('s', commutative=True)))))"]]}, {"prompt": "Given \\phi_{1}{(\\theta_1,v)} = \\theta_1 v, then derive \\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)} = \\theta_1, then obtain \\theta_1^{\\theta_1} v = v (\\frac{\\partial}{\\partial v} \\theta_1 v)^{\\theta_1}", "derivation": "\\phi_{1}{(\\theta_1,v)} = \\theta_1 v and \\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)} = \\frac{\\partial}{\\partial v} \\theta_1 v and (\\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)})^{\\theta_1} = (\\frac{\\partial}{\\partial v} \\theta_1 v)^{\\theta_1} and \\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)} = \\theta_1 and v (\\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)})^{\\theta_1} = v (\\frac{\\partial}{\\partial v} \\theta_1 v)^{\\theta_1} and \\theta_1^{\\theta_1} = (\\frac{\\partial}{\\partial v} \\theta_1 v)^{\\theta_1} and v (\\frac{\\partial}{\\partial v} \\phi_{1}{(\\theta_1,v)})^{\\theta_1} = \\theta_1^{\\theta_1} v and \\theta_1^{\\theta_1} v = v (\\frac{\\partial}{\\partial v} \\theta_1 v)^{\\theta_1}", "srepr_derivation": [["premise", "Equality(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)))"], [["differentiate", 1, "Symbol('v', commutative=True)"], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))))"], [["power", 2, "Symbol('\\\\theta_1', commutative=True)"], "Equality(Pow(Derivative(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))"], [["times", 3, "Symbol('v', commutative=True)"], "Equality(Mul(Symbol('v', commutative=True), Pow(Derivative(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))), Mul(Symbol('v', commutative=True), Pow(Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Pow(Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True)))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Mul(Symbol('v', commutative=True), Pow(Derivative(Function('\\\\phi_1')(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))), Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('v', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Mul(Pow(Symbol('\\\\theta_1', commutative=True), Symbol('\\\\theta_1', commutative=True)), Symbol('v', commutative=True)), Mul(Symbol('v', commutative=True), Pow(Derivative(Mul(Symbol('\\\\theta_1', commutative=True), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True), Integer(1))), Symbol('\\\\theta_1', commutative=True))))"]]}, {"prompt": "Given \\bar{\\h}{(g,t)} = g - t and Q{(g,t)} = \\frac{(g - t)^{t}}{g}, then obtain \\log{(\\frac{\\bar{\\h}^{t}{(g,t)}}{g})} = \\log{(Q{(g,t)})}", "derivation": "\\bar{\\h}{(g,t)} = g - t and \\bar{\\h}^{t}{(g,t)} = (g - t)^{t} and \\frac{\\bar{\\h}^{t}{(g,t)}}{g} = \\frac{(g - t)^{t}}{g} and \\log{(\\frac{\\bar{\\h}^{t}{(g,t)}}{g})} = \\log{(\\frac{(g - t)^{t}}{g})} and Q{(g,t)} = \\frac{(g - t)^{t}}{g} and \\log{(\\frac{\\bar{\\h}^{t}{(g,t)}}{g})} = \\log{(Q{(g,t)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["power", 1, "Symbol('t', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)), Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True)))"], [["divide", 2, "Symbol('g', commutative=True)"], "Equality(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True))), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], [["log", 3], "Equality(log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))), log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True)))))"], ["renaming_premise", "Equality(Function('Q')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Add(Symbol('g', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Symbol('t', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Mul(Pow(Symbol('g', commutative=True), Integer(-1)), Pow(Function('\\\\hbar')(Symbol('g', commutative=True), Symbol('t', commutative=True)), Symbol('t', commutative=True)))), log(Function('Q')(Symbol('g', commutative=True), Symbol('t', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{g}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\theta{(\\mathbf{A})} = \\operatorname{F_{g}}{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})}, then derive \\operatorname{F_{g}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})}, then obtain \\theta{(\\mathbf{A})} = 2 \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})}", "derivation": "\\operatorname{F_{g}}{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\operatorname{F_{g}}{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} and - \\sin{(\\mathbf{A})} = \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\theta{(\\mathbf{A})} = \\operatorname{F_{g}}{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\theta{(\\mathbf{A})} = - \\sin{(\\mathbf{A})} + \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})} and \\theta{(\\mathbf{A})} = 2 \\frac{d}{d \\mathbf{A}} \\cos{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('F_g')(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 1], "Equality(Function('F_g')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 2], "Equality(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Function('F_g')(Symbol('\\\\mathbf{A}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{A}', commutative=True)), Add(Mul(Integer(-1), sin(Symbol('\\\\mathbf{A}', commutative=True))), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 5, 3], "Equality(Function('\\\\theta')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), Derivative(cos(Symbol('\\\\mathbf{A}', commutative=True)), Tuple(Symbol('\\\\mathbf{A}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\hat{x}{(\\nabla)} = \\nabla, then obtain 16 \\hat{x}^{4}{(\\nabla)} = (\\nabla + \\hat{x}{(\\nabla)})^{4}", "derivation": "\\hat{x}{(\\nabla)} = \\nabla and 2 \\hat{x}{(\\nabla)} = \\nabla + \\hat{x}{(\\nabla)} and 4 \\hat{x}^{2}{(\\nabla)} = (\\nabla + \\hat{x}{(\\nabla)})^{2} and 16 \\hat{x}^{4}{(\\nabla)} = (\\nabla + \\hat{x}{(\\nabla)})^{4}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Symbol('\\\\nabla', commutative=True))"], [["add", 1, "Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True))), Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True))))"], [["power", 2, 2], "Equality(Mul(Integer(4), Pow(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Integer(2))), Pow(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True))), Integer(2)))"], [["power", 3, 2], "Equality(Mul(Integer(16), Pow(Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True)), Integer(4))), Pow(Add(Symbol('\\\\nabla', commutative=True), Function('\\\\hat{x}')(Symbol('\\\\nabla', commutative=True))), Integer(4)))"]]}, {"prompt": "Given \\Psi_{nl}{(\\mathbf{P},h)} = \\sin{(\\mathbf{P} h)} and \\rho_{f}{(\\mathbf{P},h)} = 2 \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)}, then obtain \\rho_{f}{(\\mathbf{P},h)} = \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} + \\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} h)}", "derivation": "\\Psi_{nl}{(\\mathbf{P},h)} = \\sin{(\\mathbf{P} h)} and \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} = \\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} h)} and 2 \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} = \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} + \\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} h)} and \\rho_{f}{(\\mathbf{P},h)} = 2 \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} and \\rho_{f}{(\\mathbf{P},h)} = \\frac{\\partial}{\\partial \\mathbf{P}} \\Psi_{nl}{(\\mathbf{P},h)} + \\frac{\\partial}{\\partial \\mathbf{P}} \\sin{(\\mathbf{P} h)}", "srepr_derivation": [["premise", "Equality(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{P}', commutative=True)"], "Equality(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))))"], [["add", 2, "Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))"], "Equality(Mul(Integer(2), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))), Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], ["renaming_premise", "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Mul(Integer(2), Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\rho_f')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Add(Derivative(Function('\\\\Psi_{nl}')(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True)), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1))), Derivative(sin(Mul(Symbol('\\\\mathbf{P}', commutative=True), Symbol('h', commutative=True))), Tuple(Symbol('\\\\mathbf{P}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\eta{(\\mathbf{p},T)} = \\cos^{T}{(\\mathbf{p})}, then derive \\frac{\\partial}{\\partial T} \\eta{(\\mathbf{p},T)} = \\log{(\\cos{(\\mathbf{p})})} \\cos^{T}{(\\mathbf{p})}, then obtain \\int \\frac{\\partial}{\\partial T} \\eta{(\\mathbf{p},T)} d\\mathbf{p} = \\int \\log{(\\cos{(\\mathbf{p})})} \\cos^{T}{(\\mathbf{p})} d\\mathbf{p}", "derivation": "\\eta{(\\mathbf{p},T)} = \\cos^{T}{(\\mathbf{p})} and \\frac{\\partial}{\\partial T} \\eta{(\\mathbf{p},T)} = \\frac{\\partial}{\\partial T} \\cos^{T}{(\\mathbf{p})} and \\frac{\\partial}{\\partial T} \\eta{(\\mathbf{p},T)} = \\log{(\\cos{(\\mathbf{p})})} \\cos^{T}{(\\mathbf{p})} and \\int \\frac{\\partial}{\\partial T} \\eta{(\\mathbf{p},T)} d\\mathbf{p} = \\int \\log{(\\cos{(\\mathbf{p})})} \\cos^{T}{(\\mathbf{p})} d\\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('\\\\eta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('T', commutative=True)), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('T', commutative=True)))"], [["differentiate", 1, "Symbol('T', commutative=True)"], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Derivative(Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Mul(log(cos(Symbol('\\\\mathbf{p}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('T', commutative=True))))"], [["integrate", 3, "Symbol('\\\\mathbf{p}', commutative=True)"], "Equality(Integral(Derivative(Function('\\\\eta')(Symbol('\\\\mathbf{p}', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True), Integer(1))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))), Integral(Mul(log(cos(Symbol('\\\\mathbf{p}', commutative=True))), Pow(cos(Symbol('\\\\mathbf{p}', commutative=True)), Symbol('T', commutative=True))), Tuple(Symbol('\\\\mathbf{p}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{F_{N}}{(\\mu_0)} = \\log{(\\mu_0)}, then obtain \\int \\operatorname{F_{N}}{(\\mu_0)} \\log{(\\mu_0)}^{2} d\\mu_0 = \\int \\log{(\\mu_0)}^{3} d\\mu_0", "derivation": "\\operatorname{F_{N}}{(\\mu_0)} = \\log{(\\mu_0)} and \\operatorname{F_{N}}{(\\mu_0)} \\log{(\\mu_0)} = \\log{(\\mu_0)}^{2} and \\operatorname{F_{N}}^{2}{(\\mu_0)} \\log{(\\mu_0)} = \\operatorname{F_{N}}{(\\mu_0)} \\log{(\\mu_0)}^{2} and \\operatorname{F_{N}}{(\\mu_0)} \\log{(\\mu_0)}^{2} = \\log{(\\mu_0)}^{3} and \\int \\operatorname{F_{N}}{(\\mu_0)} \\log{(\\mu_0)}^{2} d\\mu_0 = \\int \\log{(\\mu_0)}^{3} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True)))"], [["times", 1, "log(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), log(Symbol('\\\\mu_0', commutative=True))), Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(2)))"], [["times", 2, "Function('F_N')(Symbol('\\\\mu_0', commutative=True))"], "Equality(Mul(Pow(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Integer(2)), log(Symbol('\\\\mu_0', commutative=True))), Mul(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(2))), Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(3)))"], [["integrate", 4, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Mul(Function('F_N')(Symbol('\\\\mu_0', commutative=True)), Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(2))), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Pow(log(Symbol('\\\\mu_0', commutative=True)), Integer(3)), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given \\pi{(F_{g},\\Omega)} = \\log{(F_{g} + \\Omega)}, then derive \\int (\\Omega + \\pi{(F_{g},\\Omega)}) dF_{g} = F_{g} (\\Omega - 1) + F_{g} \\log{(F_{g} + \\Omega)} + V + \\Omega \\log{(F_{g} + \\Omega)}, then obtain F_{g} (\\Omega - 1) + F_{g} \\log{(F_{g} + \\Omega)} + V + \\Omega \\log{(F_{g} + \\Omega)} = \\int (\\Omega + \\log{(F_{g} + \\Omega)}) dF_{g}", "derivation": "\\pi{(F_{g},\\Omega)} = \\log{(F_{g} + \\Omega)} and \\Omega + \\pi{(F_{g},\\Omega)} = \\Omega + \\log{(F_{g} + \\Omega)} and \\int (\\Omega + \\pi{(F_{g},\\Omega)}) dF_{g} = \\int (\\Omega + \\log{(F_{g} + \\Omega)}) dF_{g} and \\int (\\Omega + \\pi{(F_{g},\\Omega)}) dF_{g} = F_{g} (\\Omega - 1) + F_{g} \\log{(F_{g} + \\Omega)} + V + \\Omega \\log{(F_{g} + \\Omega)} and F_{g} (\\Omega - 1) + F_{g} \\log{(F_{g} + \\Omega)} + V + \\Omega \\log{(F_{g} + \\Omega)} = \\int (\\Omega + \\log{(F_{g} + \\Omega)}) dF_{g}", "srepr_derivation": [["premise", "Equality(Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))))"], [["integrate", 2, "Symbol('F_g', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Add(Symbol('\\\\Omega', commutative=True), Function('\\\\pi')(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))), Tuple(Symbol('F_g', commutative=True))), Add(Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Integer(-1))), Mul(Symbol('F_g', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))), Symbol('V', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Add(Mul(Symbol('F_g', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Integer(-1))), Mul(Symbol('F_g', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))), Symbol('V', commutative=True), Mul(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True))))), Integral(Add(Symbol('\\\\Omega', commutative=True), log(Add(Symbol('F_g', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('F_g', commutative=True))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(F_{x},C_{1})} = \\cos{(C_{1} + F_{x})} and I{(C_{1})} = C_{1}, then obtain \\log{(\\int F_{x} \\operatorname{A_{y}}{(F_{x},C_{1})} dI{(C_{1})})} = \\log{(\\int F_{x} \\cos{(C_{1} + F_{x})} dI{(C_{1})})}", "derivation": "\\operatorname{A_{y}}{(F_{x},C_{1})} = \\cos{(C_{1} + F_{x})} and F_{x} \\operatorname{A_{y}}{(F_{x},C_{1})} = F_{x} \\cos{(C_{1} + F_{x})} and \\int F_{x} \\operatorname{A_{y}}{(F_{x},C_{1})} dC_{1} = \\int F_{x} \\cos{(C_{1} + F_{x})} dC_{1} and \\log{(\\int F_{x} \\operatorname{A_{y}}{(F_{x},C_{1})} dC_{1})} = \\log{(\\int F_{x} \\cos{(C_{1} + F_{x})} dC_{1})} and I{(C_{1})} = C_{1} and \\log{(\\int F_{x} \\operatorname{A_{y}}{(F_{x},C_{1})} dI{(C_{1})})} = \\log{(\\int F_{x} \\cos{(C_{1} + F_{x})} dI{(C_{1})})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('F_x', commutative=True), Symbol('C_1', commutative=True)), cos(Add(Symbol('C_1', commutative=True), Symbol('F_x', commutative=True))))"], [["times", 1, "Symbol('F_x', commutative=True)"], "Equality(Mul(Symbol('F_x', commutative=True), Function('A_y')(Symbol('F_x', commutative=True), Symbol('C_1', commutative=True))), Mul(Symbol('F_x', commutative=True), cos(Add(Symbol('C_1', commutative=True), Symbol('F_x', commutative=True)))))"], [["integrate", 2, "Symbol('C_1', commutative=True)"], "Equality(Integral(Mul(Symbol('F_x', commutative=True), Function('A_y')(Symbol('F_x', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True))), Integral(Mul(Symbol('F_x', commutative=True), cos(Add(Symbol('C_1', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('C_1', commutative=True))))"], [["log", 3], "Equality(log(Integral(Mul(Symbol('F_x', commutative=True), Function('A_y')(Symbol('F_x', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('C_1', commutative=True)))), log(Integral(Mul(Symbol('F_x', commutative=True), cos(Add(Symbol('C_1', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Symbol('C_1', commutative=True)))))"], ["renaming_premise", "Equality(Function('I')(Symbol('C_1', commutative=True)), Symbol('C_1', commutative=True))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(log(Integral(Mul(Symbol('F_x', commutative=True), Function('A_y')(Symbol('F_x', commutative=True), Symbol('C_1', commutative=True))), Tuple(Function('I')(Symbol('C_1', commutative=True))))), log(Integral(Mul(Symbol('F_x', commutative=True), cos(Add(Symbol('C_1', commutative=True), Symbol('F_x', commutative=True)))), Tuple(Function('I')(Symbol('C_1', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\mathbf{M})} = \\mathbf{M}, then obtain \\frac{d}{d \\mathbf{M}} (\\mathbf{M} + \\mathbf{M}^{\\mathbf{M}} + \\operatorname{c_{0}}{(\\mathbf{M})}) = \\frac{d}{d \\mathbf{M}} (2 \\mathbf{M} + \\mathbf{M}^{\\mathbf{M}})", "derivation": "\\operatorname{c_{0}}{(\\mathbf{M})} = \\mathbf{M} and \\operatorname{c_{0}}^{\\mathbf{M}}{(\\mathbf{M})} = \\mathbf{M}^{\\mathbf{M}} and \\mathbf{M} + \\operatorname{c_{0}}{(\\mathbf{M})} + \\operatorname{c_{0}}^{\\mathbf{M}}{(\\mathbf{M})} = 2 \\mathbf{M} + \\operatorname{c_{0}}^{\\mathbf{M}}{(\\mathbf{M})} and \\frac{d}{d \\mathbf{M}} (\\mathbf{M} + \\operatorname{c_{0}}{(\\mathbf{M})} + \\operatorname{c_{0}}^{\\mathbf{M}}{(\\mathbf{M})}) = \\frac{d}{d \\mathbf{M}} (2 \\mathbf{M} + \\operatorname{c_{0}}^{\\mathbf{M}}{(\\mathbf{M})}) and \\frac{d}{d \\mathbf{M}} (\\mathbf{M} + \\mathbf{M}^{\\mathbf{M}} + \\operatorname{c_{0}}{(\\mathbf{M})}) = \\frac{d}{d \\mathbf{M}} (2 \\mathbf{M} + \\mathbf{M}^{\\mathbf{M}})", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))"], [["power", 1, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)))"], [["add", 1, "Add(Symbol('\\\\mathbf{M}', commutative=True), Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True)))"], "Equality(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))))"], [["differentiate", 3, "Symbol('\\\\mathbf{M}', commutative=True)"], "Equality(Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True)), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Derivative(Add(Symbol('\\\\mathbf{M}', commutative=True), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True)), Function('c_0')(Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(2), Symbol('\\\\mathbf{M}', commutative=True)), Pow(Symbol('\\\\mathbf{M}', commutative=True), Symbol('\\\\mathbf{M}', commutative=True))), Tuple(Symbol('\\\\mathbf{M}', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{A_{y}}{(c,a)} = e^{c^{a}}, then obtain - e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} \\operatorname{A_{y}}{(c,a)})} = - e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} e^{c^{a}})}", "derivation": "\\operatorname{A_{y}}{(c,a)} = e^{c^{a}} and \\frac{\\partial}{\\partial a} \\operatorname{A_{y}}{(c,a)} = \\frac{\\partial}{\\partial a} e^{c^{a}} and \\sin{(\\frac{\\partial}{\\partial a} \\operatorname{A_{y}}{(c,a)})} = \\sin{(\\frac{\\partial}{\\partial a} e^{c^{a}})} and e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} \\operatorname{A_{y}}{(c,a)})} = e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} e^{c^{a}})} and - e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} \\operatorname{A_{y}}{(c,a)})} = - e^{- c^{a}} \\sin{(\\frac{\\partial}{\\partial a} e^{c^{a}})}", "srepr_derivation": [["premise", "Equality(Function('A_y')(Symbol('c', commutative=True), Symbol('a', commutative=True)), exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True))))"], [["differentiate", 1, "Symbol('a', commutative=True)"], "Equality(Derivative(Function('A_y')(Symbol('c', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))), Derivative(exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))"], [["sin", 2], "Equality(sin(Derivative(Function('A_y')(Symbol('c', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1)))), sin(Derivative(exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1)))))"], [["divide", 3, "exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True)))"], "Equality(Mul(exp(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('a', commutative=True)))), sin(Derivative(Function('A_y')(Symbol('c', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Mul(exp(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('a', commutative=True)))), sin(Derivative(exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))))"], [["times", 4, "Integer(-1)"], "Equality(Mul(Integer(-1), exp(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('a', commutative=True)))), sin(Derivative(Function('A_y')(Symbol('c', commutative=True), Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True), Integer(1))))), Mul(Integer(-1), exp(Mul(Integer(-1), Pow(Symbol('c', commutative=True), Symbol('a', commutative=True)))), sin(Derivative(exp(Pow(Symbol('c', commutative=True), Symbol('a', commutative=True))), Tuple(Symbol('a', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\operatorname{t_{1}}{(U,\\hat{p}_0)} = U e^{\\hat{p}_0} and \\hat{H}_{\\lambda}{(\\hat{p}_0)} = e^{\\hat{p}_0}, then obtain \\int \\operatorname{t_{1}}{(U,\\hat{p}_0)} dU = \\int U \\hat{H}_{\\lambda}{(\\hat{p}_0)} dU", "derivation": "\\operatorname{t_{1}}{(U,\\hat{p}_0)} = U e^{\\hat{p}_0} and \\int \\operatorname{t_{1}}{(U,\\hat{p}_0)} dU = \\int U e^{\\hat{p}_0} dU and \\hat{H}_{\\lambda}{(\\hat{p}_0)} = e^{\\hat{p}_0} and \\operatorname{t_{1}}{(U,\\hat{p}_0)} = U \\hat{H}_{\\lambda}{(\\hat{p}_0)} and \\int U \\hat{H}_{\\lambda}{(\\hat{p}_0)} dU = \\int U e^{\\hat{p}_0} dU and \\int \\operatorname{t_{1}}{(U,\\hat{p}_0)} dU = \\int U \\hat{H}_{\\lambda}{(\\hat{p}_0)} dU", "srepr_derivation": [["premise", "Equality(Function('t_1')(Symbol('U', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('U', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))))"], [["integrate", 1, "Symbol('U', commutative=True)"], "Equality(Integral(Function('t_1')(Symbol('U', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('U', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('U', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True)), exp(Symbol('\\\\hat{p}_0', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('t_1')(Symbol('U', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Mul(Symbol('U', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Integral(Mul(Symbol('U', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('U', commutative=True), exp(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('U', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 5], "Equality(Integral(Function('t_1')(Symbol('U', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('U', commutative=True))), Integral(Mul(Symbol('U', commutative=True), Function('\\\\hat{H}_{\\\\lambda}')(Symbol('\\\\hat{p}_0', commutative=True))), Tuple(Symbol('U', commutative=True))))"]]}, {"prompt": "Given \\operatorname{E_{x}}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and \\mathbf{H}{(v_{z})} = \\sin{(\\sin{(v_{z})})}, then obtain \\mathbf{H}{(v_{z})} \\int \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\mathbf{H}{(v_{z})} \\int \\mathbf{H}{(v_{z})} dv_{z}", "derivation": "\\operatorname{E_{x}}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and \\mathbf{H}{(v_{z})} = \\sin{(\\sin{(v_{z})})} and \\operatorname{E_{x}}{(v_{z})} = \\mathbf{H}{(v_{z})} and \\int \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\int \\mathbf{H}{(v_{z})} dv_{z} and \\mathbf{H}{(v_{z})} \\int \\operatorname{E_{x}}{(v_{z})} dv_{z} = \\mathbf{H}{(v_{z})} \\int \\mathbf{H}{(v_{z})} dv_{z}", "srepr_derivation": [["premise", "Equality(Function('E_x')(Symbol('v_z', commutative=True)), sin(sin(Symbol('v_z', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)), sin(sin(Symbol('v_z', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E_x')(Symbol('v_z', commutative=True)), Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)))"], [["integrate", 3, "Symbol('v_z', commutative=True)"], "Equality(Integral(Function('E_x')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))), Integral(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True))))"], [["times", 4, "Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True))"], "Equality(Mul(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)), Integral(Function('E_x')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))), Mul(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)), Integral(Function('\\\\mathbf{H}')(Symbol('v_z', commutative=True)), Tuple(Symbol('v_z', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} = r^{M_{E}}, then obtain \\frac{\\partial}{\\partial M_{E}} (\\operatorname{V_{\\mathbf{E}}}^{2}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})}) = \\frac{\\partial}{\\partial M_{E}} (r^{M_{E}} \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})})", "derivation": "\\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} = r^{M_{E}} and \\operatorname{V_{\\mathbf{E}}}^{2}{(r,M_{E})} = r^{M_{E}} \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} and \\operatorname{V_{\\mathbf{E}}}^{2}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} = r^{M_{E}} \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} and \\frac{\\partial}{\\partial M_{E}} (\\operatorname{V_{\\mathbf{E}}}^{2}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})}) = \\frac{\\partial}{\\partial M_{E}} (r^{M_{E}} \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})} + \\operatorname{V_{\\mathbf{E}}}{(r,M_{E})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Pow(Symbol('r', commutative=True), Symbol('M_E', commutative=True)))"], [["times", 1, "Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Integer(2)), Mul(Pow(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))))"], [["add", 2, "Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))"], "Equality(Add(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Integer(2)), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))), Add(Mul(Pow(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))))"], [["differentiate", 3, "Symbol('M_E', commutative=True)"], "Equality(Derivative(Add(Pow(Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Integer(2)), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))), Derivative(Add(Mul(Pow(Symbol('r', commutative=True), Symbol('M_E', commutative=True)), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))), Function('V_{\\\\mathbf{E}}')(Symbol('r', commutative=True), Symbol('M_E', commutative=True))), Tuple(Symbol('M_E', commutative=True), Integer(1))))"]]}, {"prompt": "Given M{(\\rho_f,\\mathbf{D})} = \\mathbf{D} \\rho_f, then obtain M^{4}{(\\rho_f,\\mathbf{D})} = \\mathbf{D}^{3} \\rho_f^{3} M{(\\rho_f,\\mathbf{D})}", "derivation": "M{(\\rho_f,\\mathbf{D})} = \\mathbf{D} \\rho_f and M^{2}{(\\rho_f,\\mathbf{D})} = \\mathbf{D} \\rho_f M{(\\rho_f,\\mathbf{D})} and - M^{2}{(\\rho_f,\\mathbf{D})} = - \\mathbf{D} \\rho_f M{(\\rho_f,\\mathbf{D})} and M^{4}{(\\rho_f,\\mathbf{D})} = \\mathbf{D}^{2} \\rho_f^{2} M^{2}{(\\rho_f,\\mathbf{D})} and \\mathbf{D}^{2} \\rho_f^{2} M^{2}{(\\rho_f,\\mathbf{D})} = \\mathbf{D}^{3} \\rho_f^{3} M{(\\rho_f,\\mathbf{D})} and M^{4}{(\\rho_f,\\mathbf{D})} = \\mathbf{D}^{3} \\rho_f^{3} M{(\\rho_f,\\mathbf{D})}", "srepr_derivation": [["premise", "Equality(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))"], "Equality(Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2)), Mul(Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True), Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["times", 2, "Integer(-1)"], "Equality(Mul(Integer(-1), Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2))), Mul(Integer(-1), Symbol('\\\\mathbf{D}', commutative=True), Symbol('\\\\rho_f', commutative=True), Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["power", 3, 2], "Equality(Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(4)), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(2)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(2)), Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(2))), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(3)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(3)), Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 4], "Equality(Pow(Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True)), Integer(4)), Mul(Pow(Symbol('\\\\mathbf{D}', commutative=True), Integer(3)), Pow(Symbol('\\\\rho_f', commutative=True), Integer(3)), Function('M')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\mathbf{D}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(\\chi)} = \\cos{(e^{\\chi})}, then obtain \\iint (\\int \\operatorname{C_{2}}{(\\chi)} d\\chi)^{\\chi} d\\chi d\\chi = \\iint (\\int \\cos{(e^{\\chi})} d\\chi)^{\\chi} d\\chi d\\chi", "derivation": "\\operatorname{C_{2}}{(\\chi)} = \\cos{(e^{\\chi})} and \\int \\operatorname{C_{2}}{(\\chi)} d\\chi = \\int \\cos{(e^{\\chi})} d\\chi and (\\int \\operatorname{C_{2}}{(\\chi)} d\\chi)^{\\chi} = (\\int \\cos{(e^{\\chi})} d\\chi)^{\\chi} and \\int (\\int \\operatorname{C_{2}}{(\\chi)} d\\chi)^{\\chi} d\\chi = \\int (\\int \\cos{(e^{\\chi})} d\\chi)^{\\chi} d\\chi and \\iint (\\int \\operatorname{C_{2}}{(\\chi)} d\\chi)^{\\chi} d\\chi d\\chi = \\iint (\\int \\cos{(e^{\\chi})} d\\chi)^{\\chi} d\\chi d\\chi", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('\\\\chi', commutative=True)), cos(exp(Symbol('\\\\chi', commutative=True))))"], [["integrate", 1, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Function('C_2')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(cos(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))))"], [["power", 2, "Symbol('\\\\chi', commutative=True)"], "Equality(Pow(Integral(Function('C_2')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Pow(Integral(cos(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)))"], [["integrate", 3, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Integral(Function('C_2')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Integral(cos(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"], [["integrate", 4, "Symbol('\\\\chi', commutative=True)"], "Equality(Integral(Pow(Integral(Function('C_2')(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))), Integral(Pow(Integral(cos(exp(Symbol('\\\\chi', commutative=True))), Tuple(Symbol('\\\\chi', commutative=True))), Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True)), Tuple(Symbol('\\\\chi', commutative=True))))"]]}, {"prompt": "Given L{(\\Psi)} = e^{\\Psi}, then derive e^{\\Psi} - \\frac{d}{d \\Psi} L{(\\Psi)} = 2 e^{\\Psi} - 2 \\frac{d}{d \\Psi} L{(\\Psi)}, then obtain \\tilde{\\infty} (e^{\\Psi} - \\frac{d}{d \\Psi} L{(\\Psi)}) = \\tilde{\\infty} (2 e^{\\Psi} - 2 \\frac{d}{d \\Psi} L{(\\Psi)})", "derivation": "L{(\\Psi)} = e^{\\Psi} and 0 = - L{(\\Psi)} + e^{\\Psi} and \\frac{d}{d \\Psi} 0 = \\frac{d}{d \\Psi} (- L{(\\Psi)} + e^{\\Psi}) and e^{\\Psi} + \\frac{d}{d \\Psi} 0 - \\frac{d}{d \\Psi} L{(\\Psi)} = e^{\\Psi} + \\frac{d}{d \\Psi} (- L{(\\Psi)} + e^{\\Psi}) - \\frac{d}{d \\Psi} L{(\\Psi)} and e^{\\Psi} - \\frac{d}{d \\Psi} L{(\\Psi)} = 2 e^{\\Psi} - 2 \\frac{d}{d \\Psi} L{(\\Psi)} and \\tilde{\\infty} (e^{\\Psi} - \\frac{d}{d \\Psi} L{(\\Psi)}) = \\tilde{\\infty} (2 e^{\\Psi} - 2 \\frac{d}{d \\Psi} L{(\\Psi)})", "srepr_derivation": [["premise", "Equality(Function('L')(Symbol('\\\\Psi', commutative=True)), exp(Symbol('\\\\Psi', commutative=True)))"], [["minus", 1, "Function('L')(Symbol('\\\\Psi', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\Psi', commutative=True)"], "Equality(Derivative(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))"], [["add", 3, "Add(exp(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))"], "Equality(Add(exp(Symbol('\\\\Psi', commutative=True)), Derivative(Integer(0), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Add(exp(Symbol('\\\\Psi', commutative=True)), Derivative(Add(Mul(Integer(-1), Function('L')(Symbol('\\\\Psi', commutative=True))), exp(Symbol('\\\\Psi', commutative=True))), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))), Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["evaluate_derivatives", 4], "Equality(Add(exp(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))), Add(Mul(Integer(2), exp(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Integer(2), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1))))))"], [["divide", 5, 0], "Equality(Mul(zoo, Add(exp(Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))), Mul(zoo, Add(Mul(Integer(2), exp(Symbol('\\\\Psi', commutative=True))), Mul(Integer(-1), Integer(2), Derivative(Function('L')(Symbol('\\\\Psi', commutative=True)), Tuple(Symbol('\\\\Psi', commutative=True), Integer(1)))))))"]]}, {"prompt": "Given n{(\\mathbf{F})} = \\cos{(\\mathbf{F})}, then derive \\cos{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})} = - \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})}, then obtain \\frac{n{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})}}{\\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})}} = - \\frac{n{(\\mathbf{F})} \\sin{(\\mathbf{F})}}{\\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})}}", "derivation": "n{(\\mathbf{F})} = \\cos{(\\mathbf{F})} and \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})} = \\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})} and \\cos{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})} = \\cos{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})} and \\cos{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})} = - \\sin{(\\mathbf{F})} \\cos{(\\mathbf{F})} and n{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})} = - n{(\\mathbf{F})} \\sin{(\\mathbf{F})} and \\frac{n{(\\mathbf{F})} \\frac{d}{d \\mathbf{F}} n{(\\mathbf{F})}}{\\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})}} = - \\frac{n{(\\mathbf{F})} \\sin{(\\mathbf{F})}}{\\frac{d}{d \\mathbf{F}} \\cos{(\\mathbf{F})}}", "srepr_derivation": [["premise", "Equality(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\mathbf{F}', commutative=True)"], "Equality(Derivative(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Derivative(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))))"], [["times", 2, "cos(Symbol('\\\\mathbf{F}', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(cos(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 3], "Equality(Mul(cos(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Integer(-1), sin(Symbol('\\\\mathbf{F}', commutative=True)), cos(Symbol('\\\\mathbf{F}', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))), Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True))))"], [["divide", 5, "Derivative(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1)))"], "Equality(Mul(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Derivative(Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Pow(Derivative(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Integer(-1))), Mul(Integer(-1), Function('n')(Symbol('\\\\mathbf{F}', commutative=True)), sin(Symbol('\\\\mathbf{F}', commutative=True)), Pow(Derivative(cos(Symbol('\\\\mathbf{F}', commutative=True)), Tuple(Symbol('\\\\mathbf{F}', commutative=True), Integer(1))), Integer(-1))))"]]}, {"prompt": "Given q{(\\mathbf{S})} = e^{\\cos{(\\mathbf{S})}}, then derive \\frac{d}{d \\mathbf{S}} q{(\\mathbf{S})} = - e^{\\cos{(\\mathbf{S})}} \\sin{(\\mathbf{S})}, then obtain \\cos{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} q{(\\mathbf{S})} = - q{(\\mathbf{S})} \\sin{(\\mathbf{S})} \\cos{(\\mathbf{S})}", "derivation": "q{(\\mathbf{S})} = e^{\\cos{(\\mathbf{S})}} and \\frac{d}{d \\mathbf{S}} q{(\\mathbf{S})} = \\frac{d}{d \\mathbf{S}} e^{\\cos{(\\mathbf{S})}} and \\frac{d}{d \\mathbf{S}} q{(\\mathbf{S})} = - e^{\\cos{(\\mathbf{S})}} \\sin{(\\mathbf{S})} and \\frac{d}{d \\mathbf{S}} e^{\\cos{(\\mathbf{S})}} = - e^{\\cos{(\\mathbf{S})}} \\sin{(\\mathbf{S})} and \\cos{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} e^{\\cos{(\\mathbf{S})}} = - e^{\\cos{(\\mathbf{S})}} \\sin{(\\mathbf{S})} \\cos{(\\mathbf{S})} and \\cos{(\\mathbf{S})} \\frac{d}{d \\mathbf{S}} q{(\\mathbf{S})} = - q{(\\mathbf{S})} \\sin{(\\mathbf{S})} \\cos{(\\mathbf{S})}", "srepr_derivation": [["premise", "Equality(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), exp(cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Derivative(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Derivative(exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Derivative(exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1))), Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), sin(Symbol('\\\\mathbf{S}', commutative=True))))"], [["times", 4, "cos(Symbol('\\\\mathbf{S}', commutative=True))"], "Equality(Mul(cos(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Integer(-1), exp(cos(Symbol('\\\\mathbf{S}', commutative=True))), sin(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Mul(cos(Symbol('\\\\mathbf{S}', commutative=True)), Derivative(Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True), Integer(1)))), Mul(Integer(-1), Function('q')(Symbol('\\\\mathbf{S}', commutative=True)), sin(Symbol('\\\\mathbf{S}', commutative=True)), cos(Symbol('\\\\mathbf{S}', commutative=True))))"]]}, {"prompt": "Given \\tilde{g}^*{(\\Omega,A_{1})} = A_{1} \\log{(\\Omega)}, then obtain (- A_{1} + 2 \\tilde{g}^*{(\\Omega,A_{1})})^{A_{1}} = (A_{1} \\log{(\\Omega)} - A_{1} + \\tilde{g}^*{(\\Omega,A_{1})})^{A_{1}}", "derivation": "\\tilde{g}^*{(\\Omega,A_{1})} = A_{1} \\log{(\\Omega)} and 2 \\tilde{g}^*{(\\Omega,A_{1})} = A_{1} \\log{(\\Omega)} + \\tilde{g}^*{(\\Omega,A_{1})} and - A_{1} + 2 \\tilde{g}^*{(\\Omega,A_{1})} = A_{1} \\log{(\\Omega)} - A_{1} + \\tilde{g}^*{(\\Omega,A_{1})} and (- A_{1} + 2 \\tilde{g}^*{(\\Omega,A_{1})})^{A_{1}} = (A_{1} \\log{(\\Omega)} - A_{1} + \\tilde{g}^*{(\\Omega,A_{1})})^{A_{1}}", "srepr_derivation": [["premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)), Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))))"], [["add", 1, "Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))), Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))))"], [["minus", 2, "Symbol('A_1', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)))), Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))))"], [["power", 3, "Symbol('A_1', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('A_1', commutative=True)), Mul(Integer(2), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True)))), Symbol('A_1', commutative=True)), Pow(Add(Mul(Symbol('A_1', commutative=True), log(Symbol('\\\\Omega', commutative=True))), Mul(Integer(-1), Symbol('A_1', commutative=True)), Function('\\\\tilde{g}^*')(Symbol('\\\\Omega', commutative=True), Symbol('A_1', commutative=True))), Symbol('A_1', commutative=True)))"]]}, {"prompt": "Given \\operatorname{E_{n}}{(A,W)} = W \\cos{(A)} and G{(A)} = \\cos{(A)}, then obtain \\frac{\\partial}{\\partial W} \\operatorname{E_{n}}^{A}{(A,W)} = \\frac{\\partial}{\\partial W} (W \\cos{(A)})^{A}", "derivation": "\\operatorname{E_{n}}{(A,W)} = W \\cos{(A)} and G{(A)} = \\cos{(A)} and \\operatorname{E_{n}}{(A,W)} = W G{(A)} and W G{(A)} = W \\cos{(A)} and \\operatorname{E_{n}}^{A}{(A,W)} = (W G{(A)})^{A} and \\frac{\\partial}{\\partial W} \\operatorname{E_{n}}^{A}{(A,W)} = \\frac{\\partial}{\\partial W} (W G{(A)})^{A} and \\frac{\\partial}{\\partial W} \\operatorname{E_{n}}^{A}{(A,W)} = \\frac{\\partial}{\\partial W} (W \\cos{(A)})^{A}", "srepr_derivation": [["premise", "Equality(Function('E_n')(Symbol('A', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), cos(Symbol('A', commutative=True))))"], ["renaming_premise", "Equality(Function('G')(Symbol('A', commutative=True)), cos(Symbol('A', commutative=True)))"], [["substitute_RHS_for_LHS", 1, 2], "Equality(Function('E_n')(Symbol('A', commutative=True), Symbol('W', commutative=True)), Mul(Symbol('W', commutative=True), Function('G')(Symbol('A', commutative=True))))"], [["substitute_LHS_for_RHS", 1, 3], "Equality(Mul(Symbol('W', commutative=True), Function('G')(Symbol('A', commutative=True))), Mul(Symbol('W', commutative=True), cos(Symbol('A', commutative=True))))"], [["power", 3, "Symbol('A', commutative=True)"], "Equality(Pow(Function('E_n')(Symbol('A', commutative=True), Symbol('W', commutative=True)), Symbol('A', commutative=True)), Pow(Mul(Symbol('W', commutative=True), Function('G')(Symbol('A', commutative=True))), Symbol('A', commutative=True)))"], [["differentiate", 5, "Symbol('W', commutative=True)"], "Equality(Derivative(Pow(Function('E_n')(Symbol('A', commutative=True), Symbol('W', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('W', commutative=True), Function('G')(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 6, 4], "Equality(Derivative(Pow(Function('E_n')(Symbol('A', commutative=True), Symbol('W', commutative=True)), Symbol('A', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('W', commutative=True), cos(Symbol('A', commutative=True))), Symbol('A', commutative=True)), Tuple(Symbol('W', commutative=True), Integer(1))))"]]}, {"prompt": "Given T{(\\hbar,L)} = \\cos{(L - \\hbar)}, then obtain \\dot{z}{(\\varepsilon_0)} \\frac{\\partial}{\\partial L} (- 4 T{(\\hbar,L)} + 2 \\cos{(L - \\hbar)})^{\\hbar} = \\dot{z}{(\\varepsilon_0)} \\frac{\\partial}{\\partial L} (- 2 T{(\\hbar,L)})^{\\hbar}", "derivation": "T{(\\hbar,L)} = \\cos{(L - \\hbar)} and - T{(\\hbar,L)} = - \\cos{(L - \\hbar)} and - 2 T{(\\hbar,L)} = - T{(\\hbar,L)} - \\cos{(L - \\hbar)} and - 2 T{(\\hbar,L)} + \\cos{(L - \\hbar)} = - T{(\\hbar,L)} and - 4 T{(\\hbar,L)} + 2 \\cos{(L - \\hbar)} = - 2 T{(\\hbar,L)} and (- 4 T{(\\hbar,L)} + 2 \\cos{(L - \\hbar)})^{\\hbar} = (- 2 T{(\\hbar,L)})^{\\hbar} and \\frac{\\partial}{\\partial L} (- 4 T{(\\hbar,L)} + 2 \\cos{(L - \\hbar)})^{\\hbar} = \\frac{\\partial}{\\partial L} (- 2 T{(\\hbar,L)})^{\\hbar} and \\dot{z}{(\\varepsilon_0)} \\frac{\\partial}{\\partial L} (- 4 T{(\\hbar,L)} + 2 \\cos{(L - \\hbar)})^{\\hbar} = \\dot{z}{(\\varepsilon_0)} \\frac{\\partial}{\\partial L} (- 2 T{(\\hbar,L)})^{\\hbar}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], [["times", 1, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))))"], [["add", 2, "Mul(Integer(-1), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True)))"], "Equality(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Add(Mul(Integer(-1), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(-1), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))))"], [["minus", 3, "Mul(Integer(-1), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))"], "Equality(Add(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True))))), Mul(Integer(-1), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Add(Mul(Integer(-1), Integer(4), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(2), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))), Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))))"], [["power", 5, "Symbol('\\\\hbar', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Integer(4), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(2), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))), Symbol('\\\\hbar', commutative=True)), Pow(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Symbol('\\\\hbar', commutative=True)))"], [["differentiate", 6, "Symbol('L', commutative=True)"], "Equality(Derivative(Pow(Add(Mul(Integer(-1), Integer(4), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(2), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))), Derivative(Pow(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1))))"], [["times", 7, "Function('\\\\dot{z}')(Symbol('\\\\varepsilon_0', commutative=True))"], "Equality(Mul(Function('\\\\dot{z}')(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Pow(Add(Mul(Integer(-1), Integer(4), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Mul(Integer(2), cos(Add(Symbol('L', commutative=True), Mul(Integer(-1), Symbol('\\\\hbar', commutative=True)))))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))), Mul(Function('\\\\dot{z}')(Symbol('\\\\varepsilon_0', commutative=True)), Derivative(Pow(Mul(Integer(-1), Integer(2), Function('T')(Symbol('\\\\hbar', commutative=True), Symbol('L', commutative=True))), Symbol('\\\\hbar', commutative=True)), Tuple(Symbol('L', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\ddot{x}{(F_{H})} = \\cos{(\\sin{(F_{H})})}, then obtain - \\sin{(F_{H})} + \\int 0 dF_{H} + \\int (\\ddot{x}{(F_{H})} - \\cos{(\\sin{(F_{H})})}) dF_{H} = - \\sin{(F_{H})} + 2 \\int 0 dF_{H}", "derivation": "\\ddot{x}{(F_{H})} = \\cos{(\\sin{(F_{H})})} and \\ddot{x}{(F_{H})} - \\cos{(\\sin{(F_{H})})} = 0 and \\int (\\ddot{x}{(F_{H})} - \\cos{(\\sin{(F_{H})})}) dF_{H} = \\int 0 dF_{H} and \\int 0 dF_{H} + \\int (\\ddot{x}{(F_{H})} - \\cos{(\\sin{(F_{H})})}) dF_{H} = 2 \\int 0 dF_{H} and - \\sin{(F_{H})} + \\int 0 dF_{H} + \\int (\\ddot{x}{(F_{H})} - \\cos{(\\sin{(F_{H})})}) dF_{H} = - \\sin{(F_{H})} + 2 \\int 0 dF_{H}", "srepr_derivation": [["premise", "Equality(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), cos(sin(Symbol('F_H', commutative=True))))"], [["minus", 1, "cos(sin(Symbol('F_H', commutative=True)))"], "Equality(Add(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('F_H', commutative=True))))), Integer(0))"], [["integrate", 2, "Symbol('F_H', commutative=True)"], "Equality(Integral(Add(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('F_H', commutative=True))))), Tuple(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))))"], [["add", 3, "Integral(Integer(0), Tuple(Symbol('F_H', commutative=True)))"], "Equality(Add(Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('F_H', commutative=True))))), Tuple(Symbol('F_H', commutative=True)))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True)))))"], [["minus", 4, "sin(Symbol('F_H', commutative=True))"], "Equality(Add(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))), Integral(Add(Function('\\\\ddot{x}')(Symbol('F_H', commutative=True)), Mul(Integer(-1), cos(sin(Symbol('F_H', commutative=True))))), Tuple(Symbol('F_H', commutative=True)))), Add(Mul(Integer(-1), sin(Symbol('F_H', commutative=True))), Mul(Integer(2), Integral(Integer(0), Tuple(Symbol('F_H', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(E_{\\lambda},n)} = - E_{\\lambda} + \\cos{(n)}, then obtain (\\int \\operatorname{v_{z}}{(E_{\\lambda},n)} dE_{\\lambda})^{n} = (- \\frac{E_{\\lambda}^{2}}{2} + E_{\\lambda} \\cos{(n)} + v_{z})^{n}", "derivation": "\\operatorname{v_{z}}{(E_{\\lambda},n)} = - E_{\\lambda} + \\cos{(n)} and \\int \\operatorname{v_{z}}{(E_{\\lambda},n)} dE_{\\lambda} = \\int (- E_{\\lambda} + \\cos{(n)}) dE_{\\lambda} and (\\int \\operatorname{v_{z}}{(E_{\\lambda},n)} dE_{\\lambda})^{n} = (\\int (- E_{\\lambda} + \\cos{(n)}) dE_{\\lambda})^{n} and (\\int \\operatorname{v_{z}}{(E_{\\lambda},n)} dE_{\\lambda})^{n} = (- \\frac{E_{\\lambda}^{2}}{2} + E_{\\lambda} \\cos{(n)} + v_{z})^{n}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('n', commutative=True))))"], [["integrate", 1, "Symbol('E_{\\\\lambda}', commutative=True)"], "Equality(Integral(Function('v_z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('n', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))))"], [["power", 2, "Symbol('n', commutative=True)"], "Equality(Pow(Integral(Function('v_z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n', commutative=True)), Pow(Integral(Add(Mul(Integer(-1), Symbol('E_{\\\\lambda}', commutative=True)), cos(Symbol('n', commutative=True))), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n', commutative=True)))"], [["evaluate_integrals", 3], "Equality(Pow(Integral(Function('v_z')(Symbol('E_{\\\\lambda}', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('E_{\\\\lambda}', commutative=True))), Symbol('n', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('E_{\\\\lambda}', commutative=True), Integer(2))), Mul(Symbol('E_{\\\\lambda}', commutative=True), cos(Symbol('n', commutative=True))), Symbol('v_z', commutative=True)), Symbol('n', commutative=True)))"]]}, {"prompt": "Given b{(\\pi,t_{2})} = \\sin{(\\pi + t_{2})}, then obtain c_{0} + t_{2} = \\int \\frac{\\sin{(\\pi + t_{2})}}{b{(\\pi,t_{2})}} dt_{2}", "derivation": "b{(\\pi,t_{2})} = \\sin{(\\pi + t_{2})} and 1 = \\frac{\\sin{(\\pi + t_{2})}}{b{(\\pi,t_{2})}} and \\int 1 dt_{2} = \\int \\frac{\\sin{(\\pi + t_{2})}}{b{(\\pi,t_{2})}} dt_{2} and c_{0} + t_{2} = \\int \\frac{\\sin{(\\pi + t_{2})}}{b{(\\pi,t_{2})}} dt_{2}", "srepr_derivation": [["premise", "Equality(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), sin(Add(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True))))"], [["divide", 1, "Function('b')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)))))"], [["integrate", 2, "Symbol('t_2', commutative=True)"], "Equality(Integral(Integer(1), Tuple(Symbol('t_2', commutative=True))), Integral(Mul(Pow(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Add(Symbol('c_0', commutative=True), Symbol('t_2', commutative=True)), Integral(Mul(Pow(Function('b')(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)), Integer(-1)), sin(Add(Symbol('\\\\pi', commutative=True), Symbol('t_2', commutative=True)))), Tuple(Symbol('t_2', commutative=True))))"]]}, {"prompt": "Given \\rho{(m)} = \\log{(e^{m})}, then obtain 2 \\rho^{m}{(m)} - e^{m} = \\rho^{m}{(m)} - e^{m} + \\log{(e^{m})}^{m}", "derivation": "\\rho{(m)} = \\log{(e^{m})} and \\rho^{m}{(m)} = \\log{(e^{m})}^{m} and 2 \\rho^{m}{(m)} = \\rho^{m}{(m)} + \\log{(e^{m})}^{m} and 2 \\rho^{m}{(m)} - e^{m} = \\rho^{m}{(m)} - e^{m} + \\log{(e^{m})}^{m}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho')(Symbol('m', commutative=True)), log(exp(Symbol('m', commutative=True))))"], [["power", 1, "Symbol('m', commutative=True)"], "Equality(Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True)))"], [["add", 2, "Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True))"], "Equality(Mul(Integer(2), Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Add(Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"], [["minus", 3, "exp(Symbol('m', commutative=True))"], "Equality(Add(Mul(Integer(2), Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True))), Mul(Integer(-1), exp(Symbol('m', commutative=True)))), Add(Pow(Function('\\\\rho')(Symbol('m', commutative=True)), Symbol('m', commutative=True)), Mul(Integer(-1), exp(Symbol('m', commutative=True))), Pow(log(exp(Symbol('m', commutative=True))), Symbol('m', commutative=True))))"]]}, {"prompt": "Given u{(\\sigma_x)} = \\sigma_x and \\Psi_{\\lambda}{(\\sigma_x)} = \\sigma_x^{\\sigma_x}, then obtain (u^{\\sigma_x}{(\\sigma_x)})^{\\sigma_x} = (\\sigma_x^{\\sigma_x})^{\\sigma_x}", "derivation": "u{(\\sigma_x)} = \\sigma_x and u^{\\sigma_x}{(\\sigma_x)} = \\sigma_x^{\\sigma_x} and \\Psi_{\\lambda}{(\\sigma_x)} = \\sigma_x^{\\sigma_x} and \\Psi_{\\lambda}^{\\sigma_x}{(\\sigma_x)} = (\\sigma_x^{\\sigma_x})^{\\sigma_x} and \\Psi_{\\lambda}^{\\sigma_x}{(\\sigma_x)} = (u^{\\sigma_x}{(\\sigma_x)})^{\\sigma_x} and (u^{\\sigma_x}{(\\sigma_x)})^{\\sigma_x} = (\\sigma_x^{\\sigma_x})^{\\sigma_x}", "srepr_derivation": [["premise", "Equality(Function('u')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True))"], [["power", 1, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('u')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)))"], [["power", 3, "Symbol('\\\\sigma_x', commutative=True)"], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Function('\\\\Psi_{\\\\lambda}')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Function('u')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Pow(Pow(Function('u')(Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)), Pow(Pow(Symbol('\\\\sigma_x', commutative=True), Symbol('\\\\sigma_x', commutative=True)), Symbol('\\\\sigma_x', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{z}}{(\\rho_f,f^{*})} = \\rho_f - f^{*}, then obtain - \\frac{2 f^{*} (\\rho_f - f^{*})^{2}}{\\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})}} = - \\frac{f^{*} (\\rho_f - f^{*}) (2 \\rho_f - 2 f^{*})}{\\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})}}", "derivation": "\\operatorname{v_{z}}{(\\rho_f,f^{*})} = \\rho_f - f^{*} and 2 \\operatorname{v_{z}}{(\\rho_f,f^{*})} = \\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})} and - 2 f^{*} \\operatorname{v_{z}}{(\\rho_f,f^{*})} = - f^{*} (\\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})}) and - 2 f^{*} (\\rho_f - f^{*}) = - f^{*} (2 \\rho_f - 2 f^{*}) and - \\frac{2 f^{*} (\\rho_f - f^{*})^{2}}{\\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})}} = - \\frac{f^{*} (\\rho_f - f^{*}) (2 \\rho_f - 2 f^{*})}{\\rho_f - f^{*} + \\operatorname{v_{z}}{(\\rho_f,f^{*})}}", "srepr_derivation": [["premise", "Equality(Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True)), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))))"], [["add", 1, "Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(2), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))))"], [["times", 2, "Mul(Integer(-1), Symbol('f^*', commutative=True))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))), Mul(Integer(-1), Symbol('f^*', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)))), Mul(Integer(-1), Symbol('f^*', commutative=True), Add(Mul(Integer(2), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True)))))"], [["times", 4, "Mul(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))), Integer(-1)))"], "Equality(Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Integer(2)), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))), Integer(-1))), Mul(Integer(-1), Symbol('f^*', commutative=True), Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\rho_f', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('f^*', commutative=True))), Pow(Add(Symbol('\\\\rho_f', commutative=True), Mul(Integer(-1), Symbol('f^*', commutative=True)), Function('v_z')(Symbol('\\\\rho_f', commutative=True), Symbol('f^*', commutative=True))), Integer(-1))))"]]}, {"prompt": "Given i{(y^{\\prime})} = \\cos{(\\cos{(y^{\\prime})})}, then obtain i{(y^{\\prime})} + \\cos{(y^{\\prime})} = \\cos{(y^{\\prime})} + \\cos{(\\cos{(y^{\\prime})})}", "derivation": "i{(y^{\\prime})} = \\cos{(\\cos{(y^{\\prime})})} and - y^{\\prime} + i{(y^{\\prime})} = - y^{\\prime} + \\cos{(\\cos{(y^{\\prime})})} and - y^{\\prime} + i{(y^{\\prime})} + \\cos{(y^{\\prime})} = - y^{\\prime} + \\cos{(y^{\\prime})} + \\cos{(\\cos{(y^{\\prime})})} and i{(y^{\\prime})} + \\cos{(y^{\\prime})} = \\cos{(y^{\\prime})} + \\cos{(\\cos{(y^{\\prime})})}", "srepr_derivation": [["premise", "Equality(Function('i')(Symbol('y^{\\\\prime}', commutative=True)), cos(cos(Symbol('y^{\\\\prime}', commutative=True))))"], [["minus", 1, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('i')(Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), cos(cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 2, "cos(Symbol('y^{\\\\prime}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), Function('i')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True)), cos(cos(Symbol('y^{\\\\prime}', commutative=True)))))"], [["add", 3, "Symbol('y^{\\\\prime}', commutative=True)"], "Equality(Add(Function('i')(Symbol('y^{\\\\prime}', commutative=True)), cos(Symbol('y^{\\\\prime}', commutative=True))), Add(cos(Symbol('y^{\\\\prime}', commutative=True)), cos(cos(Symbol('y^{\\\\prime}', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(C_{2})} = \\sin{(C_{2})}, then obtain \\sin{(C_{2})} = \\frac{\\sin^{2}{(C_{2})}}{C_{2} (\\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} - 1) + \\operatorname{f_{E}}{(C_{2})}}", "derivation": "\\operatorname{f_{E}}{(C_{2})} = \\sin{(C_{2})} and \\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} = 1 and \\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} - 1 = 0 and \\sin{(C_{2})} = \\frac{\\sin^{2}{(C_{2})}}{\\operatorname{f_{E}}{(C_{2})}} and C_{2} (\\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} - 1) = 0 and C_{2} (\\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} - 1) + \\operatorname{f_{E}}{(C_{2})} = \\operatorname{f_{E}}{(C_{2})} and \\sin{(C_{2})} = \\frac{\\sin^{2}{(C_{2})}}{C_{2} (\\frac{\\operatorname{f_{E}}{(C_{2})}}{\\sin{(C_{2})}} - 1) + \\operatorname{f_{E}}{(C_{2})}}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('C_2', commutative=True)), sin(Symbol('C_2', commutative=True)))"], [["divide", 1, "sin(Symbol('C_2', commutative=True))"], "Equality(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1))), Integer(1))"], [["minus", 2, 1], "Equality(Add(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1))), Integer(-1)), Integer(0))"], [["divide", 1, "Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1)))"], "Equality(sin(Symbol('C_2', commutative=True)), Mul(Pow(Function('f_E')(Symbol('C_2', commutative=True)), Integer(-1)), Pow(sin(Symbol('C_2', commutative=True)), Integer(2))))"], [["times", 3, "Symbol('C_2', commutative=True)"], "Equality(Mul(Symbol('C_2', commutative=True), Add(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1))), Integer(-1))), Integer(0))"], [["add", 5, "Function('f_E')(Symbol('C_2', commutative=True))"], "Equality(Add(Mul(Symbol('C_2', commutative=True), Add(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1))), Integer(-1))), Function('f_E')(Symbol('C_2', commutative=True))), Function('f_E')(Symbol('C_2', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 6], "Equality(sin(Symbol('C_2', commutative=True)), Mul(Pow(Add(Mul(Symbol('C_2', commutative=True), Add(Mul(Function('f_E')(Symbol('C_2', commutative=True)), Pow(sin(Symbol('C_2', commutative=True)), Integer(-1))), Integer(-1))), Function('f_E')(Symbol('C_2', commutative=True))), Integer(-1)), Pow(sin(Symbol('C_2', commutative=True)), Integer(2))))"]]}, {"prompt": "Given \\dot{x}{(v_{t})} = e^{e^{v_{t}}}, then obtain - J_{\\varepsilon} + \\dot{x}{(v_{t})} - \\operatorname{Ei}{(e^{v_{t}})} = - J_{\\varepsilon} + e^{e^{v_{t}}} - \\operatorname{Ei}{(e^{v_{t}})}", "derivation": "\\dot{x}{(v_{t})} = e^{e^{v_{t}}} and \\int \\dot{x}{(v_{t})} dv_{t} = \\int e^{e^{v_{t}}} dv_{t} and \\dot{x}{(v_{t})} - \\int \\dot{x}{(v_{t})} dv_{t} = e^{e^{v_{t}}} - \\int \\dot{x}{(v_{t})} dv_{t} and \\dot{x}{(v_{t})} - \\int e^{e^{v_{t}}} dv_{t} = e^{e^{v_{t}}} - \\int e^{e^{v_{t}}} dv_{t} and - J_{\\varepsilon} + \\dot{x}{(v_{t})} - \\operatorname{Ei}{(e^{v_{t}})} = - J_{\\varepsilon} + e^{e^{v_{t}}} - \\operatorname{Ei}{(e^{v_{t}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), exp(exp(Symbol('v_t', commutative=True))))"], [["integrate", 1, "Symbol('v_t', commutative=True)"], "Equality(Integral(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))), Integral(exp(exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))"], [["minus", 1, "Integral(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True)))"], "Equality(Add(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))), Add(exp(exp(Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Tuple(Symbol('v_t', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Integral(exp(exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))), Add(exp(exp(Symbol('v_t', commutative=True))), Mul(Integer(-1), Integral(exp(exp(Symbol('v_t', commutative=True))), Tuple(Symbol('v_t', commutative=True))))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{x}')(Symbol('v_t', commutative=True)), Mul(Integer(-1), Ei(exp(Symbol('v_t', commutative=True))))), Add(Mul(Integer(-1), Symbol('J_{\\\\varepsilon}', commutative=True)), exp(exp(Symbol('v_t', commutative=True))), Mul(Integer(-1), Ei(exp(Symbol('v_t', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(\\hat{p})} = \\hat{p} and \\operatorname{J_{\\varepsilon}}{(\\hat{p})} = \\hat{p}^{\\hat{p}}, then obtain \\hat{p}^{\\hat{p}} + n{(\\hat{p},y^{\\prime},L)} \\operatorname{z^{*}}^{\\hat{p}}{(\\hat{p})} = \\hat{p}^{\\hat{p}} + \\operatorname{J_{\\varepsilon}}{(\\hat{p})} n{(\\hat{p},y^{\\prime},L)}", "derivation": "\\operatorname{z^{*}}{(\\hat{p})} = \\hat{p} and \\operatorname{z^{*}}^{\\hat{p}}{(\\hat{p})} = \\hat{p}^{\\hat{p}} and \\operatorname{J_{\\varepsilon}}{(\\hat{p})} = \\hat{p}^{\\hat{p}} and n{(\\hat{p},y^{\\prime},L)} \\operatorname{z^{*}}^{\\hat{p}}{(\\hat{p})} = \\hat{p}^{\\hat{p}} n{(\\hat{p},y^{\\prime},L)} and n{(\\hat{p},y^{\\prime},L)} \\operatorname{z^{*}}^{\\hat{p}}{(\\hat{p})} = \\operatorname{J_{\\varepsilon}}{(\\hat{p})} n{(\\hat{p},y^{\\prime},L)} and \\hat{p}^{\\hat{p}} + n{(\\hat{p},y^{\\prime},L)} \\operatorname{z^{*}}^{\\hat{p}}{(\\hat{p})} = \\hat{p}^{\\hat{p}} + \\operatorname{J_{\\varepsilon}}{(\\hat{p})} n{(\\hat{p},y^{\\prime},L)}", "srepr_derivation": [["renaming_premise", "Equality(Function('z^*')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))"], [["power", 1, "Symbol('\\\\hat{p}', commutative=True)"], "Equality(Pow(Function('z^*')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)), Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], ["renaming_premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True)), Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True)))"], [["times", 2, "Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))"], "Equality(Mul(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Pow(Function('z^*')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Mul(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 3], "Equality(Mul(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Pow(Function('z^*')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True))), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True)), Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True))))"], [["add", 5, "Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True))"], "Equality(Add(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)), Pow(Function('z^*')(Symbol('\\\\hat{p}', commutative=True)), Symbol('\\\\hat{p}', commutative=True)))), Add(Pow(Symbol('\\\\hat{p}', commutative=True), Symbol('\\\\hat{p}', commutative=True)), Mul(Function('J_{\\\\varepsilon}')(Symbol('\\\\hat{p}', commutative=True)), Function('n')(Symbol('\\\\hat{p}', commutative=True), Symbol('y^{\\\\prime}', commutative=True), Symbol('L', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{f}{(t,r_{0})} = r_{0} - t, then obtain r_{0} \\cos{(\\int \\mathbf{f}{(t,r_{0})} dt)} = r_{0} \\cos{(\\int (r_{0} - t) dt)}", "derivation": "\\mathbf{f}{(t,r_{0})} = r_{0} - t and \\int \\mathbf{f}{(t,r_{0})} dt = \\int (r_{0} - t) dt and \\cos{(\\int \\mathbf{f}{(t,r_{0})} dt)} = \\cos{(\\int (r_{0} - t) dt)} and r_{0} \\cos{(\\int \\mathbf{f}{(t,r_{0})} dt)} = r_{0} \\cos{(\\int (r_{0} - t) dt)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('r_0', commutative=True)), Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))))"], [["integrate", 1, "Symbol('t', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('t', commutative=True))), Integral(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))"], [["cos", 2], "Equality(cos(Integral(Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('t', commutative=True)))), cos(Integral(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True)))))"], [["times", 3, "Symbol('r_0', commutative=True)"], "Equality(Mul(Symbol('r_0', commutative=True), cos(Integral(Function('\\\\mathbf{f}')(Symbol('t', commutative=True), Symbol('r_0', commutative=True)), Tuple(Symbol('t', commutative=True))))), Mul(Symbol('r_0', commutative=True), cos(Integral(Add(Symbol('r_0', commutative=True), Mul(Integer(-1), Symbol('t', commutative=True))), Tuple(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\mathbf{A}{(\\pi)} = \\sin{(\\pi)}, then obtain \\frac{\\pi (1 + \\frac{\\mathbf{A}{(\\pi)}}{\\pi})}{\\mathbf{A}{(\\pi)}} = \\frac{\\pi (1 + \\frac{\\sin{(\\pi)}}{\\pi})}{\\mathbf{A}{(\\pi)}}", "derivation": "\\mathbf{A}{(\\pi)} = \\sin{(\\pi)} and \\frac{\\mathbf{A}{(\\pi)}}{\\pi} = \\frac{\\sin{(\\pi)}}{\\pi} and 1 + \\frac{\\mathbf{A}{(\\pi)}}{\\pi} = 1 + \\frac{\\sin{(\\pi)}}{\\pi} and \\frac{\\pi (1 + \\frac{\\mathbf{A}{(\\pi)}}{\\pi})}{\\mathbf{A}{(\\pi)}} = \\frac{\\pi (1 + \\frac{\\sin{(\\pi)}}{\\pi})}{\\mathbf{A}{(\\pi)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), sin(Symbol('\\\\pi', commutative=True)))"], [["divide", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True))), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integer(1), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)))), Add(Integer(1), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True)))))"], [["divide", 3, "Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)))"], "Equality(Mul(Symbol('\\\\pi', commutative=True), Add(Integer(1), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)))), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1))), Mul(Symbol('\\\\pi', commutative=True), Add(Integer(1), Mul(Pow(Symbol('\\\\pi', commutative=True), Integer(-1)), sin(Symbol('\\\\pi', commutative=True)))), Pow(Function('\\\\mathbf{A}')(Symbol('\\\\pi', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{z^{*}}{(g,Q)} = \\frac{Q}{g}, then obtain e^{\\frac{\\partial}{\\partial Q} \\operatorname{z^{*}}{(g,Q)} - \\frac{1}{g}} = e^{\\frac{\\partial}{\\partial Q} \\frac{Q}{g} - \\frac{1}{g}}", "derivation": "\\operatorname{z^{*}}{(g,Q)} = \\frac{Q}{g} and \\frac{\\partial}{\\partial Q} \\operatorname{z^{*}}{(g,Q)} = \\frac{\\partial}{\\partial Q} \\frac{Q}{g} and \\frac{\\partial}{\\partial Q} \\operatorname{z^{*}}{(g,Q)} - \\frac{1}{g} = \\frac{\\partial}{\\partial Q} \\frac{Q}{g} - \\frac{1}{g} and e^{\\frac{\\partial}{\\partial Q} \\operatorname{z^{*}}{(g,Q)} - \\frac{1}{g}} = e^{\\frac{\\partial}{\\partial Q} \\frac{Q}{g} - \\frac{1}{g}}", "srepr_derivation": [["premise", "Equality(Function('z^*')(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Mul(Symbol('Q', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))))"], [["differentiate", 1, "Symbol('Q', commutative=True)"], "Equality(Derivative(Function('z^*')(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))))"], [["minus", 2, "Pow(Symbol('g', commutative=True), Integer(-1))"], "Equality(Add(Derivative(Function('z^*')(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)))), Add(Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1)))))"], [["exp", 3], "Equality(exp(Add(Derivative(Function('z^*')(Symbol('g', commutative=True), Symbol('Q', commutative=True)), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1))))), exp(Add(Derivative(Mul(Symbol('Q', commutative=True), Pow(Symbol('g', commutative=True), Integer(-1))), Tuple(Symbol('Q', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('g', commutative=True), Integer(-1))))))"]]}, {"prompt": "Given \\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)} = \\log{(\\varphi)}^{\\hat{p}_0} and \\operatorname{M_{E}}{(\\varphi)} = \\log{(\\varphi)}, then obtain \\frac{\\frac{\\partial}{\\partial \\varphi} \\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)}}{\\frac{\\partial}{\\partial \\varphi} \\operatorname{M_{E}}^{\\hat{p}_0}{(\\varphi)}} = 1", "derivation": "\\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)} = \\log{(\\varphi)}^{\\hat{p}_0} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\varphi} \\log{(\\varphi)}^{\\hat{p}_0} and \\operatorname{M_{E}}{(\\varphi)} = \\log{(\\varphi)} and \\frac{\\partial}{\\partial \\varphi} \\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)} = \\frac{\\partial}{\\partial \\varphi} \\operatorname{M_{E}}^{\\hat{p}_0}{(\\varphi)} and \\frac{\\frac{\\partial}{\\partial \\varphi} \\operatorname{c_{0}}{(\\varphi,\\hat{p}_0)}}{\\frac{\\partial}{\\partial \\varphi} \\operatorname{M_{E}}^{\\hat{p}_0}{(\\varphi)}} = 1", "srepr_derivation": [["premise", "Equality(Function('c_0')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Function('c_0')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(log(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('M_E')(Symbol('\\\\varphi', commutative=True)), log(Symbol('\\\\varphi', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Derivative(Function('c_0')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"], [["divide", 4, "Derivative(Pow(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))"], "Equality(Mul(Pow(Derivative(Pow(Function('M_E')(Symbol('\\\\varphi', commutative=True)), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Integer(-1)), Derivative(Function('c_0')(Symbol('\\\\varphi', commutative=True), Symbol('\\\\hat{p}_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1)))), Integer(1))"]]}, {"prompt": "Given \\hat{x}{(h)} = \\sin{(h)}, then obtain (\\frac{d}{d h} \\frac{\\sin^{4}{(h)}}{\\hat{x}^{3}{(h)}})^{2} = (\\frac{d}{d h} \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}})^{2}", "derivation": "\\hat{x}{(h)} = \\sin{(h)} and \\hat{x}{(h)} \\sin{(h)} = \\sin^{2}{(h)} and \\sin{(h)} = \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}} and \\frac{d}{d h} \\sin{(h)} = \\frac{d}{d h} \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}} and (\\frac{d}{d h} \\sin{(h)})^{2} = (\\frac{d}{d h} \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}})^{2} and \\frac{d}{d h} \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}} = \\frac{d}{d h} \\frac{\\sin^{4}{(h)}}{\\hat{x}^{3}{(h)}} and \\frac{d}{d h} \\sin{(h)} = \\frac{d}{d h} \\frac{\\sin^{4}{(h)}}{\\hat{x}^{3}{(h)}} and (\\frac{d}{d h} \\frac{\\sin^{4}{(h)}}{\\hat{x}^{3}{(h)}})^{2} = (\\frac{d}{d h} \\frac{\\sin^{2}{(h)}}{\\hat{x}{(h)}})^{2}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True)))"], [["times", 1, "sin(Symbol('h', commutative=True))"], "Equality(Mul(Function('\\\\hat{x}')(Symbol('h', commutative=True)), sin(Symbol('h', commutative=True))), Pow(sin(Symbol('h', commutative=True)), Integer(2)))"], [["divide", 2, "Function('\\\\hat{x}')(Symbol('h', commutative=True))"], "Equality(sin(Symbol('h', commutative=True)), Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(2))))"], [["differentiate", 3, "Symbol('h', commutative=True)"], "Equality(Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["power", 4, 2], "Equality(Pow(Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-3)), Pow(sin(Symbol('h', commutative=True)), Integer(4))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 6, 4], "Equality(Derivative(sin(Symbol('h', commutative=True)), Tuple(Symbol('h', commutative=True), Integer(1))), Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-3)), Pow(sin(Symbol('h', commutative=True)), Integer(4))), Tuple(Symbol('h', commutative=True), Integer(1))))"], [["substitute_LHS_for_RHS", 5, 7], "Equality(Pow(Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-3)), Pow(sin(Symbol('h', commutative=True)), Integer(4))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)), Pow(Derivative(Mul(Pow(Function('\\\\hat{x}')(Symbol('h', commutative=True)), Integer(-1)), Pow(sin(Symbol('h', commutative=True)), Integer(2))), Tuple(Symbol('h', commutative=True), Integer(1))), Integer(2)))"]]}, {"prompt": "Given \\bar{\\h}{(t_{2},\\mathbf{s})} = \\frac{\\log{(t_{2})}}{\\mathbf{s}}, then obtain (\\mathbf{s} \\bar{\\h}{(t_{2},\\mathbf{s})} + \\mathbf{s})^{\\mathbf{s}} = (\\mathbf{s} + \\log{(t_{2})})^{\\mathbf{s}}", "derivation": "\\bar{\\h}{(t_{2},\\mathbf{s})} = \\frac{\\log{(t_{2})}}{\\mathbf{s}} and \\mathbf{s} \\bar{\\h}{(t_{2},\\mathbf{s})} = \\log{(t_{2})} and \\mathbf{s} \\bar{\\h}{(t_{2},\\mathbf{s})} + \\mathbf{s} = \\mathbf{s} + \\log{(t_{2})} and (\\mathbf{s} \\bar{\\h}{(t_{2},\\mathbf{s})} + \\mathbf{s})^{\\mathbf{s}} = (\\mathbf{s} + \\log{(t_{2})})^{\\mathbf{s}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True)), Mul(Pow(Symbol('\\\\mathbf{s}', commutative=True), Integer(-1)), log(Symbol('t_2', commutative=True))))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), log(Symbol('t_2', commutative=True)))"], [["add", 2, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Add(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('t_2', commutative=True))))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Add(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hbar')(Symbol('t_2', commutative=True), Symbol('\\\\mathbf{s}', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)), Symbol('\\\\mathbf{s}', commutative=True)), Pow(Add(Symbol('\\\\mathbf{s}', commutative=True), log(Symbol('t_2', commutative=True))), Symbol('\\\\mathbf{s}', commutative=True)))"]]}, {"prompt": "Given \\hat{H}{(v)} = \\cos{(v)}, then obtain \\int \\hat{H}^{v}{(v)} dv + \\frac{\\int \\hat{H}^{v}{(v)} dv}{\\cos{(v)}} = \\int \\hat{H}^{v}{(v)} dv + \\frac{\\int \\cos^{v}{(v)} dv}{\\cos{(v)}}", "derivation": "\\hat{H}{(v)} = \\cos{(v)} and \\hat{H}^{v}{(v)} = \\cos^{v}{(v)} and \\int \\hat{H}^{v}{(v)} dv = \\int \\cos^{v}{(v)} dv and \\frac{\\int \\hat{H}^{v}{(v)} dv}{\\cos{(v)}} = \\frac{\\int \\cos^{v}{(v)} dv}{\\cos{(v)}} and \\int \\hat{H}^{v}{(v)} dv + \\frac{\\int \\hat{H}^{v}{(v)} dv}{\\cos{(v)}} = \\int \\hat{H}^{v}{(v)} dv + \\frac{\\int \\cos^{v}{(v)} dv}{\\cos{(v)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('v', commutative=True)), cos(Symbol('v', commutative=True)))"], [["power", 1, "Symbol('v', commutative=True)"], "Equality(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Pow(cos(Symbol('v', commutative=True)), Symbol('v', commutative=True)))"], [["integrate", 2, "Symbol('v', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))"], [["divide", 3, "cos(Symbol('v', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('v', commutative=True)), Integer(-1)), Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))), Mul(Pow(cos(Symbol('v', commutative=True)), Integer(-1)), Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))))"], [["add", 4, "Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True)))"], "Equality(Add(Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Mul(Pow(cos(Symbol('v', commutative=True)), Integer(-1)), Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))), Add(Integral(Pow(Function('\\\\hat{H}')(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))), Mul(Pow(cos(Symbol('v', commutative=True)), Integer(-1)), Integral(Pow(cos(Symbol('v', commutative=True)), Symbol('v', commutative=True)), Tuple(Symbol('v', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{v_{2}}{(c,T)} = T + c, then derive \\int \\operatorname{v_{2}}{(c,T)} dT = A_{2} + \\frac{T^{2}}{2} + T c, then obtain \\frac{\\partial}{\\partial c} \\int \\operatorname{v_{2}}{(c,T)} dT = \\frac{\\partial}{\\partial c} (A_{2} + \\frac{T^{2}}{2} + T c)", "derivation": "\\operatorname{v_{2}}{(c,T)} = T + c and \\int \\operatorname{v_{2}}{(c,T)} dT = \\int (T + c) dT and \\int \\operatorname{v_{2}}{(c,T)} dT = A_{2} + \\frac{T^{2}}{2} + T c and \\frac{\\partial}{\\partial c} \\int \\operatorname{v_{2}}{(c,T)} dT = \\frac{\\partial}{\\partial c} (A_{2} + \\frac{T^{2}}{2} + T c)", "srepr_derivation": [["premise", "Equality(Function('v_2')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Add(Symbol('T', commutative=True), Symbol('c', commutative=True)))"], [["integrate", 1, "Symbol('T', commutative=True)"], "Equality(Integral(Function('v_2')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Integral(Add(Symbol('T', commutative=True), Symbol('c', commutative=True)), Tuple(Symbol('T', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('v_2')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Add(Symbol('A_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))))"], [["differentiate", 3, "Symbol('c', commutative=True)"], "Equality(Derivative(Integral(Function('v_2')(Symbol('c', commutative=True), Symbol('T', commutative=True)), Tuple(Symbol('T', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))), Derivative(Add(Symbol('A_2', commutative=True), Mul(Rational(1, 2), Pow(Symbol('T', commutative=True), Integer(2))), Mul(Symbol('T', commutative=True), Symbol('c', commutative=True))), Tuple(Symbol('c', commutative=True), Integer(1))))"]]}, {"prompt": "Given f{(\\dot{\\mathbf{r}},\\varepsilon)} = \\dot{\\mathbf{r}} - \\varepsilon, then obtain \\int (\\dot{\\mathbf{r}} - \\varepsilon f{(\\dot{\\mathbf{r}},\\varepsilon)} - \\varepsilon) d\\dot{\\mathbf{r}} = \\int (\\dot{\\mathbf{r}} - \\varepsilon (\\dot{\\mathbf{r}} - \\varepsilon) - \\varepsilon) d\\dot{\\mathbf{r}}", "derivation": "f{(\\dot{\\mathbf{r}},\\varepsilon)} = \\dot{\\mathbf{r}} - \\varepsilon and - \\varepsilon f{(\\dot{\\mathbf{r}},\\varepsilon)} = - \\varepsilon (\\dot{\\mathbf{r}} - \\varepsilon) and \\dot{\\mathbf{r}} - \\varepsilon f{(\\dot{\\mathbf{r}},\\varepsilon)} - \\varepsilon = \\dot{\\mathbf{r}} - \\varepsilon (\\dot{\\mathbf{r}} - \\varepsilon) - \\varepsilon and \\int (\\dot{\\mathbf{r}} - \\varepsilon f{(\\dot{\\mathbf{r}},\\varepsilon)} - \\varepsilon) d\\dot{\\mathbf{r}} = \\int (\\dot{\\mathbf{r}} - \\varepsilon (\\dot{\\mathbf{r}} - \\varepsilon) - \\varepsilon) d\\dot{\\mathbf{r}}", "srepr_derivation": [["premise", "Equality(Function('f')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True)), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["times", 1, "Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('f')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))))"], [["add", 2, "Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))"], "Equality(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('f')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))))"], [["integrate", 3, "Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)"], "Equality(Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Function('f')(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Symbol('\\\\varepsilon', commutative=True))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))), Integral(Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True), Add(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True)))), Mul(Integer(-1), Symbol('\\\\varepsilon', commutative=True))), Tuple(Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)} = \\phi_1 + n, then derive \\frac{\\partial}{\\partial \\phi_1} \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)} = 1, then obtain ((\\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + n))^{n})^{\\phi_1} \\int 1 dn = \\int 1 dn", "derivation": "\\operatorname{J_{\\varepsilon}}{(n,\\phi_1)} = \\phi_1 + n and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)} = \\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + n) and \\frac{\\partial}{\\partial \\phi_1} \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)} = 1 and (\\frac{\\partial}{\\partial \\phi_1} \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)})^{n} = 1 and ((\\frac{\\partial}{\\partial \\phi_1} \\operatorname{J_{\\varepsilon}}{(n,\\phi_1)})^{n})^{\\phi_1} = 1 and ((\\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + n))^{n})^{\\phi_1} = 1 and ((\\frac{\\partial}{\\partial \\phi_1} (\\phi_1 + n))^{n})^{\\phi_1} \\int 1 dn = \\int 1 dn", "srepr_derivation": [["premise", "Equality(Function('J_{\\\\varepsilon}')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('J_{\\\\varepsilon}')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Integer(1))"], [["power", 3, "Symbol('n', commutative=True)"], "Equality(Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('n', commutative=True)), Integer(1))"], [["power", 4, "Symbol('\\\\phi_1', commutative=True)"], "Equality(Pow(Pow(Derivative(Function('J_{\\\\varepsilon}')(Symbol('n', commutative=True), Symbol('\\\\phi_1', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('n', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 5, 2], "Equality(Pow(Pow(Derivative(Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('n', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Integer(1))"], [["times", 6, "Integral(Integer(1), Tuple(Symbol('n', commutative=True)))"], "Equality(Mul(Pow(Pow(Derivative(Add(Symbol('\\\\phi_1', commutative=True), Symbol('n', commutative=True)), Tuple(Symbol('\\\\phi_1', commutative=True), Integer(1))), Symbol('n', commutative=True)), Symbol('\\\\phi_1', commutative=True)), Integral(Integer(1), Tuple(Symbol('n', commutative=True)))), Integral(Integer(1), Tuple(Symbol('n', commutative=True))))"]]}, {"prompt": "Given J{(\\hat{H})} = \\cos{(\\hat{H})}, then obtain 0 = \\hat{H} (1 - (\\frac{J{(\\hat{H})}}{\\cos{(\\hat{H})}})^{\\hat{H}})", "derivation": "J{(\\hat{H})} = \\cos{(\\hat{H})} and \\frac{J{(\\hat{H})}}{\\cos{(\\hat{H})}} = 1 and (\\frac{J{(\\hat{H})}}{\\cos{(\\hat{H})}})^{\\hat{H}} = 1 and 0 = 1 - (\\frac{J{(\\hat{H})}}{\\cos{(\\hat{H})}})^{\\hat{H}} and 0 = \\hat{H} (1 - (\\frac{J{(\\hat{H})}}{\\cos{(\\hat{H})}})^{\\hat{H}})", "srepr_derivation": [["premise", "Equality(Function('J')(Symbol('\\\\hat{H}', commutative=True)), cos(Symbol('\\\\hat{H}', commutative=True)))"], [["divide", 1, "cos(Symbol('\\\\hat{H}', commutative=True))"], "Equality(Mul(Function('J')(Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Mul(Function('J')(Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Symbol('\\\\hat{H}', commutative=True)), Integer(1))"], [["minus", 3, "Pow(Mul(Function('J')(Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Symbol('\\\\hat{H}', commutative=True))"], "Equality(Integer(0), Add(Integer(1), Mul(Integer(-1), Pow(Mul(Function('J')(Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Symbol('\\\\hat{H}', commutative=True)))))"], [["times", 4, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Integer(0), Mul(Symbol('\\\\hat{H}', commutative=True), Add(Integer(1), Mul(Integer(-1), Pow(Mul(Function('J')(Symbol('\\\\hat{H}', commutative=True)), Pow(cos(Symbol('\\\\hat{H}', commutative=True)), Integer(-1))), Symbol('\\\\hat{H}', commutative=True))))))"]]}, {"prompt": "Given v{(\\sigma_p)} = e^{\\sigma_p} and u{(\\sigma_p)} = (\\sigma_p + v{(\\sigma_p)} + e^{\\sigma_p}) v{(\\sigma_p)}, then obtain u{(\\sigma_p)} = (\\sigma_p + 2 e^{\\sigma_p}) v{(\\sigma_p)}", "derivation": "v{(\\sigma_p)} = e^{\\sigma_p} and v{(\\sigma_p)} + e^{\\sigma_p} = 2 e^{\\sigma_p} and \\sigma_p + v{(\\sigma_p)} + e^{\\sigma_p} = \\sigma_p + 2 e^{\\sigma_p} and (\\sigma_p + v{(\\sigma_p)} + e^{\\sigma_p}) v{(\\sigma_p)} = (\\sigma_p + 2 e^{\\sigma_p}) v{(\\sigma_p)} and u{(\\sigma_p)} = (\\sigma_p + v{(\\sigma_p)} + e^{\\sigma_p}) v{(\\sigma_p)} and u{(\\sigma_p)} = (\\sigma_p + 2 e^{\\sigma_p}) v{(\\sigma_p)}", "srepr_derivation": [["premise", "Equality(Function('v')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True)))"], [["add", 1, "exp(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Add(Function('v')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True))))"], [["add", 2, "Symbol('\\\\sigma_p', commutative=True)"], "Equality(Add(Symbol('\\\\sigma_p', commutative=True), Function('v')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))))"], [["times", 3, "Function('v')(Symbol('\\\\sigma_p', commutative=True))"], "Equality(Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('v')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Function('v')(Symbol('\\\\sigma_p', commutative=True))), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))), Function('v')(Symbol('\\\\sigma_p', commutative=True))))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Function('v')(Symbol('\\\\sigma_p', commutative=True)), exp(Symbol('\\\\sigma_p', commutative=True))), Function('v')(Symbol('\\\\sigma_p', commutative=True))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Function('u')(Symbol('\\\\sigma_p', commutative=True)), Mul(Add(Symbol('\\\\sigma_p', commutative=True), Mul(Integer(2), exp(Symbol('\\\\sigma_p', commutative=True)))), Function('v')(Symbol('\\\\sigma_p', commutative=True))))"]]}, {"prompt": "Given \\mathbf{H}{(\\Psi,\\mathbf{J},H)} = H - \\Psi - \\mathbf{J}, then obtain (\\mathbf{J} + \\mathbf{H}{(\\Psi,\\mathbf{J},H)})^{\\Psi} = (H - \\Psi)^{\\Psi}", "derivation": "\\mathbf{H}{(\\Psi,\\mathbf{J},H)} = H - \\Psi - \\mathbf{J} and - \\mathbf{J} + \\mathbf{H}{(\\Psi,\\mathbf{J},H)} = H - \\Psi - 2 \\mathbf{J} and \\mathbf{J} + \\mathbf{H}{(\\Psi,\\mathbf{J},H)} = H - \\Psi and (\\mathbf{J} + \\mathbf{H}{(\\Psi,\\mathbf{J},H)})^{\\Psi} = (H - \\Psi)^{\\Psi}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{H}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('H', commutative=True)), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{J}', commutative=True)), Function('\\\\mathbf{H}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True))))"], [["minus", 2, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{J}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('H', commutative=True))), Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))))"], [["power", 3, "Symbol('\\\\Psi', commutative=True)"], "Equality(Pow(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('\\\\mathbf{H}')(Symbol('\\\\Psi', commutative=True), Symbol('\\\\mathbf{J}', commutative=True), Symbol('H', commutative=True))), Symbol('\\\\Psi', commutative=True)), Pow(Add(Symbol('H', commutative=True), Mul(Integer(-1), Symbol('\\\\Psi', commutative=True))), Symbol('\\\\Psi', commutative=True)))"]]}, {"prompt": "Given W{(x^\\prime,\\Omega)} = \\Omega + x^\\prime and \\lambda{(x^\\prime,\\Omega)} = x^\\prime (2 \\Omega + x^\\prime), then obtain \\frac{\\int x^\\prime (2 \\Omega + x^\\prime) dx^\\prime}{2 \\Omega} = \\frac{\\int x^\\prime (\\Omega + W{(x^\\prime,\\Omega)}) dx^\\prime}{2 \\Omega}", "derivation": "W{(x^\\prime,\\Omega)} = \\Omega + x^\\prime and \\Omega + W{(x^\\prime,\\Omega)} = 2 \\Omega + x^\\prime and x^\\prime (\\Omega + W{(x^\\prime,\\Omega)}) = x^\\prime (2 \\Omega + x^\\prime) and \\lambda{(x^\\prime,\\Omega)} = x^\\prime (2 \\Omega + x^\\prime) and x^\\prime (\\Omega + W{(x^\\prime,\\Omega)}) = \\lambda{(x^\\prime,\\Omega)} and \\int x^\\prime (\\Omega + W{(x^\\prime,\\Omega)}) dx^\\prime = \\int \\lambda{(x^\\prime,\\Omega)} dx^\\prime and \\int x^\\prime (2 \\Omega + x^\\prime) dx^\\prime = \\int \\lambda{(x^\\prime,\\Omega)} dx^\\prime and \\frac{\\int x^\\prime (2 \\Omega + x^\\prime) dx^\\prime}{2 \\Omega} = \\frac{\\int \\lambda{(x^\\prime,\\Omega)} dx^\\prime}{2 \\Omega} and \\frac{\\int x^\\prime (2 \\Omega + x^\\prime) dx^\\prime}{2 \\Omega} = \\frac{\\int x^\\prime (\\Omega + W{(x^\\prime,\\Omega)}) dx^\\prime}{2 \\Omega}", "srepr_derivation": [["premise", "Equality(Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)), Add(Symbol('\\\\Omega', commutative=True), Symbol('x^\\\\prime', commutative=True)))"], [["add", 1, "Symbol('\\\\Omega', commutative=True)"], "Equality(Add(Symbol('\\\\Omega', commutative=True), Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True))), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["times", 2, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)))), Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)), Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)))), Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)))"], [["integrate", 5, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 1], "Equality(Integral(Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True))), Integral(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True))))"], [["divide", 7, "Mul(Integer(2), Symbol('\\\\Omega', commutative=True))"], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Function('\\\\lambda')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)), Tuple(Symbol('x^\\\\prime', commutative=True)))))"], [["substitute_RHS_for_LHS", 8, 6], "Equality(Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), Add(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Tuple(Symbol('x^\\\\prime', commutative=True)))), Mul(Rational(1, 2), Pow(Symbol('\\\\Omega', commutative=True), Integer(-1)), Integral(Mul(Symbol('x^\\\\prime', commutative=True), Add(Symbol('\\\\Omega', commutative=True), Function('W')(Symbol('x^\\\\prime', commutative=True), Symbol('\\\\Omega', commutative=True)))), Tuple(Symbol('x^\\\\prime', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{M}{(\\varphi^*)} = \\sin{(e^{\\varphi^*})}, then obtain \\cos{(\\mathbf{M}{(\\varphi^*)} \\int \\sin{(e^{\\varphi^*})} d\\varphi^*)} = \\cos{(\\sin{(e^{\\varphi^*})} \\int \\sin{(e^{\\varphi^*})} d\\varphi^*)}", "derivation": "\\mathbf{M}{(\\varphi^*)} = \\sin{(e^{\\varphi^*})} and \\int \\mathbf{M}{(\\varphi^*)} d\\varphi^* = \\int \\sin{(e^{\\varphi^*})} d\\varphi^* and \\mathbf{M}{(\\varphi^*)} \\int \\mathbf{M}{(\\varphi^*)} d\\varphi^* = \\sin{(e^{\\varphi^*})} \\int \\mathbf{M}{(\\varphi^*)} d\\varphi^* and \\mathbf{M}{(\\varphi^*)} \\int \\sin{(e^{\\varphi^*})} d\\varphi^* = \\sin{(e^{\\varphi^*})} \\int \\sin{(e^{\\varphi^*})} d\\varphi^* and \\cos{(\\mathbf{M}{(\\varphi^*)} \\int \\sin{(e^{\\varphi^*})} d\\varphi^*)} = \\cos{(\\sin{(e^{\\varphi^*})} \\int \\sin{(e^{\\varphi^*})} d\\varphi^*)}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), sin(exp(Symbol('\\\\varphi^*', commutative=True))))"], [["integrate", 1, "Symbol('\\\\varphi^*', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))"], [["times", 1, "Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))"], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Integral(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integral(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))), Mul(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True)))))"], [["cos", 4], "Equality(cos(Mul(Function('\\\\mathbf{M}')(Symbol('\\\\varphi^*', commutative=True)), Integral(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))), cos(Mul(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Integral(sin(exp(Symbol('\\\\varphi^*', commutative=True))), Tuple(Symbol('\\\\varphi^*', commutative=True))))))"]]}, {"prompt": "Given \\dot{y}{(f_{E},S)} = - S + f_{E}, then derive \\int \\dot{y}{(f_{E},S)} df_{E} = - S f_{E} + \\frac{f_{E}^{2}}{2} + f_{\\mathbf{v}}, then obtain \\frac{\\int \\dot{y}{(f_{E},S)} df_{E}}{- S f_{E} + \\frac{f_{E}^{2}}{2} + f_{\\mathbf{v}}} = 1", "derivation": "\\dot{y}{(f_{E},S)} = - S + f_{E} and \\int \\dot{y}{(f_{E},S)} df_{E} = \\int (- S + f_{E}) df_{E} and \\int \\dot{y}{(f_{E},S)} df_{E} = - S f_{E} + \\frac{f_{E}^{2}}{2} + f_{\\mathbf{v}} and \\frac{\\int \\dot{y}{(f_{E},S)} df_{E}}{\\int (- S + f_{E}) df_{E}} = 1 and \\int (- S + f_{E}) df_{E} = - S f_{E} + \\frac{f_{E}^{2}}{2} + f_{\\mathbf{v}} and \\frac{\\int \\dot{y}{(f_{E},S)} df_{E}}{- S f_{E} + \\frac{f_{E}^{2}}{2} + f_{\\mathbf{v}}} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('f_E', commutative=True), Symbol('S', commutative=True)), Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('f_E', commutative=True)))"], [["integrate", 1, "Symbol('f_E', commutative=True)"], "Equality(Integral(Function('\\\\dot{y}')(Symbol('f_E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\dot{y}')(Symbol('f_E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["divide", 2, "Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True)))"], "Equality(Mul(Pow(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Integer(-1)), Integral(Function('\\\\dot{y}')(Symbol('f_E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Integer(1))"], [["substitute_LHS_for_RHS", 3, 1], "Equality(Integral(Add(Mul(Integer(-1), Symbol('S', commutative=True)), Symbol('f_E', commutative=True)), Tuple(Symbol('f_E', commutative=True))), Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{v}}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(Pow(Add(Mul(Integer(-1), Symbol('S', commutative=True), Symbol('f_E', commutative=True)), Mul(Rational(1, 2), Pow(Symbol('f_E', commutative=True), Integer(2))), Symbol('f_{\\\\mathbf{v}}', commutative=True)), Integer(-1)), Integral(Function('\\\\dot{y}')(Symbol('f_E', commutative=True), Symbol('S', commutative=True)), Tuple(Symbol('f_E', commutative=True)))), Integer(1))"]]}, {"prompt": "Given g{(n_{2},n)} = n + n_{2}, then obtain \\frac{\\partial^{2}}{\\partial n_{2}^{2}} g^{2}{(n_{2},n)} = \\frac{\\partial^{2}}{\\partial n_{2}^{2}} (n + n_{2}) g{(n_{2},n)}", "derivation": "g{(n_{2},n)} = n + n_{2} and g^{2}{(n_{2},n)} = (n + n_{2}) g{(n_{2},n)} and \\frac{\\partial}{\\partial n_{2}} g^{2}{(n_{2},n)} = \\frac{\\partial}{\\partial n_{2}} (n + n_{2}) g{(n_{2},n)} and \\frac{\\partial^{2}}{\\partial n_{2}^{2}} g^{2}{(n_{2},n)} = \\frac{\\partial^{2}}{\\partial n_{2}^{2}} (n + n_{2}) g{(n_{2},n)}", "srepr_derivation": [["premise", "Equality(Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True)), Add(Symbol('n', commutative=True), Symbol('n_2', commutative=True)))"], [["times", 1, "Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True))"], "Equality(Pow(Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True)), Integer(2)), Mul(Add(Symbol('n', commutative=True), Symbol('n_2', commutative=True)), Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True))))"], [["differentiate", 2, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Pow(Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('n_2', commutative=True), Integer(1))), Derivative(Mul(Add(Symbol('n', commutative=True), Symbol('n_2', commutative=True)), Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('n_2', commutative=True)"], "Equality(Derivative(Pow(Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True)), Integer(2)), Tuple(Symbol('n_2', commutative=True), Integer(2))), Derivative(Mul(Add(Symbol('n', commutative=True), Symbol('n_2', commutative=True)), Function('g')(Symbol('n_2', commutative=True), Symbol('n', commutative=True))), Tuple(Symbol('n_2', commutative=True), Integer(2))))"]]}, {"prompt": "Given \\theta_{1}{(\\mu_0,Q)} = \\log{(\\mu_0^{Q})}, then obtain \\int \\log{(\\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}})}^{2} d\\mu_0 = \\int \\log{(\\mu_0^{Q})} \\log{(\\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}})} d\\mu_0", "derivation": "\\theta_{1}{(\\mu_0,Q)} = \\log{(\\mu_0^{Q})} and \\theta_{1}^{2}{(\\mu_0,Q)} = \\theta_{1}{(\\mu_0,Q)} \\log{(\\mu_0^{Q})} and \\int \\theta_{1}^{2}{(\\mu_0,Q)} d\\mu_0 = \\int \\theta_{1}{(\\mu_0,Q)} \\log{(\\mu_0^{Q})} d\\mu_0 and 1 = \\frac{\\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}} and \\mu_0^{Q} = \\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}} and \\theta_{1}{(\\mu_0,Q)} = \\log{(\\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}})} and \\int \\log{(\\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}})}^{2} d\\mu_0 = \\int \\log{(\\mu_0^{Q})} \\log{(\\frac{\\mu_0^{Q} \\log{(\\mu_0^{Q})}}{\\theta_{1}{(\\mu_0,Q)}})} d\\mu_0", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))))"], [["times", 1, "Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(2)), Mul(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)))))"], [["integrate", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Integral(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)))), Tuple(Symbol('\\\\mu_0', commutative=True))))"], [["divide", 1, "Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)))))"], [["times", 4, "Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))"], "Equality(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Mul(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)))))"], [["substitute_LHS_for_RHS", 1, 5], "Equality(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))))))"], [["substitute_LHS_for_RHS", 3, 6], "Equality(Integral(Pow(log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))))), Integer(2)), Tuple(Symbol('\\\\mu_0', commutative=True))), Integral(Mul(log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True))), log(Mul(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Pow(Function('\\\\theta_1')(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)), Integer(-1)), log(Pow(Symbol('\\\\mu_0', commutative=True), Symbol('Q', commutative=True)))))), Tuple(Symbol('\\\\mu_0', commutative=True))))"]]}, {"prompt": "Given k{(P_{g},x)} = - P_{g} + x, then derive \\int k{(P_{g},x)} dP_{g} = - \\frac{P_{g}^{2}}{2} + P_{g} x + i, then derive - \\frac{P_{g}^{2}}{2} + P_{g} x + i = - \\frac{P_{g}^{2}}{2} + P_{g} x + \\dot{\\mathbf{r}}, then obtain (\\int (- P_{g} + x) dP_{g})^{P_{g}} = (- \\frac{P_{g}^{2}}{2} + P_{g} x + \\dot{\\mathbf{r}})^{P_{g}}", "derivation": "k{(P_{g},x)} = - P_{g} + x and \\int k{(P_{g},x)} dP_{g} = \\int (- P_{g} + x) dP_{g} and \\int k{(P_{g},x)} dP_{g} = - \\frac{P_{g}^{2}}{2} + P_{g} x + i and - \\frac{P_{g}^{2}}{2} + P_{g} x + i = \\int (- P_{g} + x) dP_{g} and - \\frac{P_{g}^{2}}{2} + P_{g} x + i = - \\frac{P_{g}^{2}}{2} + P_{g} x + \\dot{\\mathbf{r}} and \\int (- P_{g} + x) dP_{g} = - \\frac{P_{g}^{2}}{2} + P_{g} x + \\dot{\\mathbf{r}} and (\\int (- P_{g} + x) dP_{g})^{P_{g}} = (- \\frac{P_{g}^{2}}{2} + P_{g} x + \\dot{\\mathbf{r}})^{P_{g}}", "srepr_derivation": [["premise", "Equality(Function('k')(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('x', commutative=True)))"], [["integrate", 1, "Symbol('P_g', commutative=True)"], "Equality(Integral(Function('k')(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('k')(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('i', commutative=True)))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('i', commutative=True)), Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('i', commutative=True)), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)))"], [["power", 6, "Symbol('P_g', commutative=True)"], "Equality(Pow(Integral(Add(Mul(Integer(-1), Symbol('P_g', commutative=True)), Symbol('x', commutative=True)), Tuple(Symbol('P_g', commutative=True))), Symbol('P_g', commutative=True)), Pow(Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('P_g', commutative=True), Integer(2))), Mul(Symbol('P_g', commutative=True), Symbol('x', commutative=True)), Symbol('\\\\dot{\\\\mathbf{r}}', commutative=True)), Symbol('P_g', commutative=True)))"]]}, {"prompt": "Given y{(A_{z})} = \\cos{(A_{z})} and \\operatorname{y^{\\prime}}{(A_{z})} = \\int \\frac{d}{d A_{z}} y{(A_{z})} dA_{z}, then derive \\operatorname{y^{\\prime}}{(A_{z})} = S + y{(A_{z})}, then obtain S + y{(A_{z})} = \\int \\frac{d}{d A_{z}} \\cos{(A_{z})} dA_{z}", "derivation": "y{(A_{z})} = \\cos{(A_{z})} and \\frac{d}{d A_{z}} y{(A_{z})} = \\frac{d}{d A_{z}} \\cos{(A_{z})} and \\int \\frac{d}{d A_{z}} y{(A_{z})} dA_{z} = \\int \\frac{d}{d A_{z}} \\cos{(A_{z})} dA_{z} and \\operatorname{y^{\\prime}}{(A_{z})} = \\int \\frac{d}{d A_{z}} y{(A_{z})} dA_{z} and \\operatorname{y^{\\prime}}{(A_{z})} = S + y{(A_{z})} and S + y{(A_{z})} = \\int \\frac{d}{d A_{z}} y{(A_{z})} dA_{z} and S + y{(A_{z})} = \\int \\frac{d}{d A_{z}} \\cos{(A_{z})} dA_{z}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('A_z', commutative=True)), cos(Symbol('A_z', commutative=True)))"], [["differentiate", 1, "Symbol('A_z', commutative=True)"], "Equality(Derivative(Function('y')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))))"], [["integrate", 2, "Symbol('A_z', commutative=True)"], "Equality(Integral(Derivative(Function('y')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))), Integral(Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"], ["renaming_premise", "Equality(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Integral(Derivative(Function('y')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"], [["evaluate_integrals", 4], "Equality(Function('y^{\\\\prime}')(Symbol('A_z', commutative=True)), Add(Symbol('S', commutative=True), Function('y')(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Add(Symbol('S', commutative=True), Function('y')(Symbol('A_z', commutative=True))), Integral(Derivative(Function('y')(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 3], "Equality(Add(Symbol('S', commutative=True), Function('y')(Symbol('A_z', commutative=True))), Integral(Derivative(cos(Symbol('A_z', commutative=True)), Tuple(Symbol('A_z', commutative=True), Integer(1))), Tuple(Symbol('A_z', commutative=True))))"]]}, {"prompt": "Given T{(c_{0},\\mathbf{p},f_{E})} = \\mathbf{p} + c_{0} - f_{E}, then obtain - 2 c_{0} + 2 f_{E} + 2 T{(c_{0},\\mathbf{p},f_{E})} = 2 \\mathbf{p}", "derivation": "T{(c_{0},\\mathbf{p},f_{E})} = \\mathbf{p} + c_{0} - f_{E} and - \\mathbf{p} - c_{0} + f_{E} + T{(c_{0},\\mathbf{p},f_{E})} = 0 and - 2 \\mathbf{p} - c_{0} + f_{E} + T{(c_{0},\\mathbf{p},f_{E})} = - \\mathbf{p} and - 2 \\mathbf{p} - 2 c_{0} + 2 f_{E} + 2 T{(c_{0},\\mathbf{p},f_{E})} = 0 and - 2 c_{0} + 2 f_{E} + 2 T{(c_{0},\\mathbf{p},f_{E})} = 2 \\mathbf{p}", "srepr_derivation": [["premise", "Equality(Function('T')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_E', commutative=True)), Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('c_0', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True))))"], [["minus", 1, "Add(Symbol('\\\\mathbf{p}', commutative=True), Symbol('c_0', commutative=True), Mul(Integer(-1), Symbol('f_E', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f_E', commutative=True), Function('T')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_E', commutative=True))), Integer(0))"], [["add", 2, "Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Symbol('c_0', commutative=True)), Symbol('f_E', commutative=True), Function('T')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_E', commutative=True))), Mul(Integer(-1), Symbol('\\\\mathbf{p}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True)), Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), Function('T')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_E', commutative=True)))), Integer(0))"], [["minus", 4, "Mul(Integer(-1), Integer(2), Symbol('\\\\mathbf{p}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Integer(2), Symbol('c_0', commutative=True)), Mul(Integer(2), Symbol('f_E', commutative=True)), Mul(Integer(2), Function('T')(Symbol('c_0', commutative=True), Symbol('\\\\mathbf{p}', commutative=True), Symbol('f_E', commutative=True)))), Mul(Integer(2), Symbol('\\\\mathbf{p}', commutative=True)))"]]}, {"prompt": "Given y{(\\mu_0,\\varphi)} = \\varphi^{\\mu_0}, then obtain \\frac{\\partial}{\\partial \\varphi} (\\mu_0 y{(\\mu_0,\\varphi)})^{\\mu_0} = \\frac{\\partial}{\\partial \\varphi} (\\mu_0 \\varphi^{\\mu_0})^{\\mu_0}", "derivation": "y{(\\mu_0,\\varphi)} = \\varphi^{\\mu_0} and \\mu_0 y{(\\mu_0,\\varphi)} = \\mu_0 \\varphi^{\\mu_0} and (\\mu_0 y{(\\mu_0,\\varphi)})^{\\mu_0} = (\\mu_0 \\varphi^{\\mu_0})^{\\mu_0} and \\frac{\\partial}{\\partial \\varphi} (\\mu_0 y{(\\mu_0,\\varphi)})^{\\mu_0} = \\frac{\\partial}{\\partial \\varphi} (\\mu_0 \\varphi^{\\mu_0})^{\\mu_0}", "srepr_derivation": [["premise", "Equality(Function('y')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True)), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu_0', commutative=True)))"], [["times", 1, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Mul(Symbol('\\\\mu_0', commutative=True), Function('y')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu_0', commutative=True))))"], [["power", 2, "Symbol('\\\\mu_0', commutative=True)"], "Equality(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Function('y')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Pow(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)))"], [["differentiate", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Derivative(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Function('y')(Symbol('\\\\mu_0', commutative=True), Symbol('\\\\varphi', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\mu_0', commutative=True), Pow(Symbol('\\\\varphi', commutative=True), Symbol('\\\\mu_0', commutative=True))), Symbol('\\\\mu_0', commutative=True)), Tuple(Symbol('\\\\varphi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta}, then derive \\operatorname{v_{1}}{(\\eta)} + e^{\\eta} = 2 e^{\\eta}, then derive e^{\\eta} + \\frac{d}{d \\eta} \\operatorname{v_{1}}{(\\eta)} = 2 e^{\\eta}, then obtain e^{\\eta} + \\frac{d}{d \\eta} \\operatorname{v_{1}}{(\\eta)} = e^{\\eta} + \\frac{d}{d \\eta} e^{\\eta}", "derivation": "\\operatorname{v_{1}}{(\\eta)} = \\frac{d}{d \\eta} e^{\\eta} and \\operatorname{v_{1}}{(\\eta)} + e^{\\eta} = e^{\\eta} + \\frac{d}{d \\eta} e^{\\eta} and \\operatorname{v_{1}}{(\\eta)} + e^{\\eta} = 2 e^{\\eta} and \\frac{d}{d \\eta} (\\operatorname{v_{1}}{(\\eta)} + e^{\\eta}) = \\frac{d}{d \\eta} (e^{\\eta} + \\frac{d}{d \\eta} e^{\\eta}) and e^{\\eta} + \\frac{d}{d \\eta} \\operatorname{v_{1}}{(\\eta)} = 2 e^{\\eta} and e^{\\eta} + \\frac{d}{d \\eta} e^{\\eta} = 2 e^{\\eta} and e^{\\eta} + \\frac{d}{d \\eta} \\operatorname{v_{1}}{(\\eta)} = e^{\\eta} + \\frac{d}{d \\eta} e^{\\eta}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["add", 1, "exp(Symbol('\\\\eta', commutative=True))"], "Equality(Add(Function('v_1')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))), Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"], [["evaluate_derivatives", 2], "Equality(Add(Function('v_1')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))), Mul(Integer(2), exp(Symbol('\\\\eta', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\eta', commutative=True)"], "Equality(Derivative(Add(Function('v_1')(Symbol('\\\\eta', commutative=True)), exp(Symbol('\\\\eta', commutative=True))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))), Derivative(Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Tuple(Symbol('\\\\eta', commutative=True), Integer(1))))"], [["evaluate_derivatives", 4], "Equality(Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(Function('v_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\eta', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Mul(Integer(2), exp(Symbol('\\\\eta', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(Function('v_1')(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))), Add(exp(Symbol('\\\\eta', commutative=True)), Derivative(exp(Symbol('\\\\eta', commutative=True)), Tuple(Symbol('\\\\eta', commutative=True), Integer(1)))))"]]}, {"prompt": "Given a{(a^{\\dagger})} = e^{a^{\\dagger}}, then obtain a^{\\dagger} a{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} a{(a^{\\dagger})} = a^{\\dagger} e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} a{(a^{\\dagger})}", "derivation": "a{(a^{\\dagger})} = e^{a^{\\dagger}} and \\frac{d}{d a^{\\dagger}} a{(a^{\\dagger})} = \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and a^{\\dagger} a{(a^{\\dagger})} = a^{\\dagger} e^{a^{\\dagger}} and a^{\\dagger} a{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} = a^{\\dagger} e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} e^{a^{\\dagger}} and a^{\\dagger} a{(a^{\\dagger})} + \\frac{d}{d a^{\\dagger}} a{(a^{\\dagger})} = a^{\\dagger} e^{a^{\\dagger}} + \\frac{d}{d a^{\\dagger}} a{(a^{\\dagger})}", "srepr_derivation": [["premise", "Equality(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), exp(Symbol('a^{\\\\dagger}', commutative=True)))"], [["differentiate", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Derivative(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1))))"], [["times", 1, "Symbol('a^{\\\\dagger}', commutative=True)"], "Equality(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('a')(Symbol('a^{\\\\dagger}', commutative=True))), Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))))"], [["add", 3, "Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))"], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('a')(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(exp(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), Function('a')(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))), Add(Mul(Symbol('a^{\\\\dagger}', commutative=True), exp(Symbol('a^{\\\\dagger}', commutative=True))), Derivative(Function('a')(Symbol('a^{\\\\dagger}', commutative=True)), Tuple(Symbol('a^{\\\\dagger}', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\operatorname{C_{2}}{(f^{\\prime})} = \\log{(f^{\\prime})} and \\tilde{g}{(f^{\\prime})} = - f^{\\prime} + \\operatorname{C_{2}}{(f^{\\prime})} + \\log{(f^{\\prime})}, then obtain \\int (- f^{\\prime} + 2 \\operatorname{C_{2}}{(f^{\\prime})}) df^{\\prime} = \\int (- f^{\\prime} + 2 \\log{(f^{\\prime})}) df^{\\prime}", "derivation": "\\operatorname{C_{2}}{(f^{\\prime})} = \\log{(f^{\\prime})} and - f^{\\prime} + \\operatorname{C_{2}}{(f^{\\prime})} + \\log{(f^{\\prime})} = - f^{\\prime} + 2 \\log{(f^{\\prime})} and \\tilde{g}{(f^{\\prime})} = - f^{\\prime} + \\operatorname{C_{2}}{(f^{\\prime})} + \\log{(f^{\\prime})} and \\tilde{g}{(f^{\\prime})} = - f^{\\prime} + 2 \\log{(f^{\\prime})} and \\tilde{g}{(f^{\\prime})} = - f^{\\prime} + 2 \\operatorname{C_{2}}{(f^{\\prime})} and \\int \\tilde{g}{(f^{\\prime})} df^{\\prime} = \\int (- f^{\\prime} + 2 \\log{(f^{\\prime})}) df^{\\prime} and \\int (- f^{\\prime} + 2 \\operatorname{C_{2}}{(f^{\\prime})}) df^{\\prime} = \\int (- f^{\\prime} + 2 \\log{(f^{\\prime})}) df^{\\prime}", "srepr_derivation": [["premise", "Equality(Function('C_2')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], [["add", 1, "Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('C_2')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), log(Symbol('f^{\\\\prime}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Function('C_2')(Symbol('f^{\\\\prime}', commutative=True)), log(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\tilde{g}')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), log(Symbol('f^{\\\\prime}', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Function('\\\\tilde{g}')(Symbol('f^{\\\\prime}', commutative=True)), Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('C_2')(Symbol('f^{\\\\prime}', commutative=True)))))"], [["integrate", 4, "Symbol('f^{\\\\prime}', commutative=True)"], "Equality(Integral(Function('\\\\tilde{g}')(Symbol('f^{\\\\prime}', commutative=True)), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), log(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"], [["substitute_LHS_for_RHS", 6, 5], "Equality(Integral(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), Function('C_2')(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('f^{\\\\prime}', commutative=True)), Mul(Integer(2), log(Symbol('f^{\\\\prime}', commutative=True)))), Tuple(Symbol('f^{\\\\prime}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{n_{1}}{(\\mu,\\mathbf{J})} = \\mathbf{J} \\mu, then obtain \\frac{\\mathbf{J} \\operatorname{n_{1}}^{\\mathbf{J}}{(\\mu,\\mathbf{J})} \\frac{\\partial}{\\partial \\mu} \\operatorname{n_{1}}{(\\mu,\\mathbf{J})}}{\\operatorname{n_{1}}{(\\mu,\\mathbf{J})}} = \\frac{\\mathbf{J} (\\mathbf{J} \\mu)^{\\mathbf{J}}}{\\mu}", "derivation": "\\operatorname{n_{1}}{(\\mu,\\mathbf{J})} = \\mathbf{J} \\mu and \\operatorname{n_{1}}^{\\mathbf{J}}{(\\mu,\\mathbf{J})} = (\\mathbf{J} \\mu)^{\\mathbf{J}} and \\frac{\\partial}{\\partial \\mu} \\operatorname{n_{1}}^{\\mathbf{J}}{(\\mu,\\mathbf{J})} = \\frac{\\partial}{\\partial \\mu} (\\mathbf{J} \\mu)^{\\mathbf{J}} and \\frac{\\mathbf{J} \\operatorname{n_{1}}^{\\mathbf{J}}{(\\mu,\\mathbf{J})} \\frac{\\partial}{\\partial \\mu} \\operatorname{n_{1}}{(\\mu,\\mathbf{J})}}{\\operatorname{n_{1}}{(\\mu,\\mathbf{J})}} = \\frac{\\mathbf{J} (\\mathbf{J} \\mu)^{\\mathbf{J}}}{\\mu}", "srepr_derivation": [["premise", "Equality(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mu', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Pow(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)))"], [["differentiate", 2, "Symbol('\\\\mu', commutative=True)"], "Equality(Derivative(Pow(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))), Derivative(Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Integer(-1)), Pow(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True)), Derivative(Function('n_1')(Symbol('\\\\mu', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Tuple(Symbol('\\\\mu', commutative=True), Integer(1)))), Mul(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\mu', commutative=True), Integer(-1)), Pow(Mul(Symbol('\\\\mathbf{J}', commutative=True), Symbol('\\\\mu', commutative=True)), Symbol('\\\\mathbf{J}', commutative=True))))"]]}, {"prompt": "Given \\operatorname{y^{\\prime}}{(f,A_{1})} = A_{1}^{f}, then derive \\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})} = A_{1}^{f} \\log{(A_{1})}, then obtain \\frac{\\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})}}{\\cos{(\\rho_f)}} = \\frac{\\operatorname{y^{\\prime}}{(f,A_{1})} \\log{(A_{1})}}{\\cos{(\\rho_f)}}", "derivation": "\\operatorname{y^{\\prime}}{(f,A_{1})} = A_{1}^{f} and \\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})} = \\frac{\\partial}{\\partial f} A_{1}^{f} and \\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})} = A_{1}^{f} \\log{(A_{1})} and \\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})} = \\operatorname{y^{\\prime}}{(f,A_{1})} \\log{(A_{1})} and \\frac{\\frac{\\partial}{\\partial f} \\operatorname{y^{\\prime}}{(f,A_{1})}}{\\cos{(\\rho_f)}} = \\frac{\\operatorname{y^{\\prime}}{(f,A_{1})} \\log{(A_{1})}}{\\cos{(\\rho_f)}}", "srepr_derivation": [["get_premise", "Equality(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Pow(Symbol('A_1', commutative=True), Symbol('f', commutative=True)))"], [["differentiate", 1, "Symbol('f', commutative=True)"], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Derivative(Pow(Symbol('A_1', commutative=True), Symbol('f', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Pow(Symbol('A_1', commutative=True), Symbol('f', commutative=True)), log(Symbol('A_1', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1))), Mul(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True))))"], [["divide", 4, "cos(Symbol('\\\\rho_f', commutative=True))"], "Equality(Mul(Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Derivative(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), Tuple(Symbol('f', commutative=True), Integer(1)))), Mul(Function('y^{\\\\prime}')(Symbol('f', commutative=True), Symbol('A_1', commutative=True)), log(Symbol('A_1', commutative=True)), Pow(cos(Symbol('\\\\rho_f', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\bar{\\h}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)}, then obtain - \\bar{\\h}{(\\mathbf{J}_P)} + \\int \\bar{\\h}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P = - \\bar{\\h}{(\\mathbf{J}_P)} + \\int \\sin^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P", "derivation": "\\bar{\\h}{(\\mathbf{J}_P)} = \\sin{(\\mathbf{J}_P)} and \\bar{\\h}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} = \\sin^{\\mathbf{J}_P}{(\\mathbf{J}_P)} and \\int \\bar{\\h}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P = \\int \\sin^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P and - \\bar{\\h}{(\\mathbf{J}_P)} + \\int \\bar{\\h}^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P = - \\bar{\\h}{(\\mathbf{J}_P)} + \\int \\sin^{\\mathbf{J}_P}{(\\mathbf{J}_P)} d\\mathbf{J}_P", "srepr_derivation": [["premise", "Equality(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True)), sin(Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["power", 1, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)))"], [["integrate", 2, "Symbol('\\\\mathbf{J}_P', commutative=True)"], "Equality(Integral(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True))))"], [["minus", 3, "Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True))"], "Equality(Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\hbar')(Symbol('\\\\mathbf{J}_P', commutative=True))), Integral(Pow(sin(Symbol('\\\\mathbf{J}_P', commutative=True)), Symbol('\\\\mathbf{J}_P', commutative=True)), Tuple(Symbol('\\\\mathbf{J}_P', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{f_{E}}{(a)} = \\cos{(a)} and \\Psi_{\\lambda}{(a)} = \\cos{(a)}, then obtain (\\int \\Psi_{\\lambda}{(a)} da)^{a} = (\\int \\cos{(a)} da)^{a}", "derivation": "\\operatorname{f_{E}}{(a)} = \\cos{(a)} and \\Psi_{\\lambda}{(a)} = \\cos{(a)} and \\Psi_{\\lambda}{(a)} = \\operatorname{f_{E}}{(a)} and \\int \\Psi_{\\lambda}{(a)} da = \\int \\cos{(a)} da and \\int \\operatorname{f_{E}}{(a)} da = \\int \\cos{(a)} da and (\\int \\operatorname{f_{E}}{(a)} da)^{a} = (\\int \\cos{(a)} da)^{a} and (\\int \\Psi_{\\lambda}{(a)} da)^{a} = (\\int \\cos{(a)} da)^{a}", "srepr_derivation": [["premise", "Equality(Function('f_E')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), cos(Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), Function('f_E')(Symbol('a', commutative=True)))"], [["integrate", 2, "Symbol('a', commutative=True)"], "Equality(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Function('f_E')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))))"], [["power", 5, "Symbol('a', commutative=True)"], "Equality(Pow(Integral(Function('f_E')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"], [["substitute_RHS_for_LHS", 6, 3], "Equality(Pow(Integral(Function('\\\\Psi_{\\\\lambda}')(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)), Pow(Integral(cos(Symbol('a', commutative=True)), Tuple(Symbol('a', commutative=True))), Symbol('a', commutative=True)))"]]}, {"prompt": "Given V{(\\hat{x},\\rho_f)} = \\hat{x} - \\rho_f, then derive \\frac{\\partial}{\\partial \\rho_f} V{(\\hat{x},\\rho_f)} = -1, then derive (\\int (-1) d\\rho_f)^{\\rho_f} = (P_{e} - \\rho_f)^{\\rho_f}, then obtain ((\\int (-1) d\\rho_f)^{\\rho_f})^{\\rho_f} = ((P_{e} - \\rho_f)^{\\rho_f})^{\\rho_f}", "derivation": "V{(\\hat{x},\\rho_f)} = \\hat{x} - \\rho_f and \\frac{\\partial}{\\partial \\rho_f} V{(\\hat{x},\\rho_f)} = \\frac{\\partial}{\\partial \\rho_f} (\\hat{x} - \\rho_f) and \\frac{\\partial}{\\partial \\rho_f} V{(\\hat{x},\\rho_f)} = -1 and -1 = \\frac{\\partial}{\\partial \\rho_f} (\\hat{x} - \\rho_f) and \\int (-1) d\\rho_f = \\int \\frac{\\partial}{\\partial \\rho_f} (\\hat{x} - \\rho_f) d\\rho_f and (\\int (-1) d\\rho_f)^{\\rho_f} = (\\int \\frac{\\partial}{\\partial \\rho_f} (\\hat{x} - \\rho_f) d\\rho_f)^{\\rho_f} and (\\int (-1) d\\rho_f)^{\\rho_f} = (P_{e} - \\rho_f)^{\\rho_f} and ((\\int (-1) d\\rho_f)^{\\rho_f})^{\\rho_f} = ((P_{e} - \\rho_f)^{\\rho_f})^{\\rho_f}", "srepr_derivation": [["get_premise", "Equality(Function('V')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Derivative(Function('V')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('V')(Symbol('\\\\hat{x}', commutative=True), Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Integer(-1))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Integer(-1), Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Integral(Integer(-1), Tuple(Symbol('\\\\rho_f', commutative=True))), Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["power", 5, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Integral(Integer(-1), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Integral(Derivative(Add(Symbol('\\\\hat{x}', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Tuple(Symbol('\\\\rho_f', commutative=True), Integer(1))), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["evaluate_integrals", 6], "Equality(Pow(Integral(Integer(-1), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Pow(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)))"], [["power", 7, "Symbol('\\\\rho_f', commutative=True)"], "Equality(Pow(Pow(Integral(Integer(-1), Tuple(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)), Pow(Pow(Add(Symbol('P_e', commutative=True), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\rho_f', commutative=True)))"]]}, {"prompt": "Given \\varepsilon_{0}{(x^\\prime)} = e^{x^\\prime} and \\bar{\\h}{(x^\\prime)} = \\varepsilon_{0}^{x^\\prime}{(x^\\prime)}, then obtain e^{x^\\prime} (e^{x^\\prime})^{x^\\prime} = \\varepsilon_{0}^{x^\\prime}{(x^\\prime)} e^{x^\\prime}", "derivation": "\\varepsilon_{0}{(x^\\prime)} = e^{x^\\prime} and \\varepsilon_{0}^{x^\\prime}{(x^\\prime)} = (e^{x^\\prime})^{x^\\prime} and \\bar{\\h}{(x^\\prime)} = \\varepsilon_{0}^{x^\\prime}{(x^\\prime)} and \\bar{\\h}{(x^\\prime)} e^{x^\\prime} = \\varepsilon_{0}^{x^\\prime}{(x^\\prime)} e^{x^\\prime} and \\bar{\\h}{(x^\\prime)} = (e^{x^\\prime})^{x^\\prime} and e^{x^\\prime} (e^{x^\\prime})^{x^\\prime} = \\varepsilon_{0}^{x^\\prime}{(x^\\prime)} e^{x^\\prime}", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon_0')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True)))"], [["power", 1, "Symbol('x^\\\\prime', commutative=True)"], "Equality(Pow(Function('\\\\varepsilon_0')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), Pow(exp(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\hbar')(Symbol('x^\\\\prime', commutative=True)), Pow(Function('\\\\varepsilon_0')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["times", 3, "exp(Symbol('x^\\\\prime', commutative=True))"], "Equality(Mul(Function('\\\\hbar')(Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Function('\\\\hbar')(Symbol('x^\\\\prime', commutative=True)), Pow(exp(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 5], "Equality(Mul(exp(Symbol('x^\\\\prime', commutative=True)), Pow(exp(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True))), Mul(Pow(Function('\\\\varepsilon_0')(Symbol('x^\\\\prime', commutative=True)), Symbol('x^\\\\prime', commutative=True)), exp(Symbol('x^\\\\prime', commutative=True))))"]]}, {"prompt": "Given \\theta{(\\hat{H},\\phi)} = \\frac{\\log{(\\phi)}}{\\hat{H}}, then derive \\frac{\\partial}{\\partial \\hat{H}} \\theta{(\\hat{H},\\phi)} = - \\frac{\\log{(\\phi)}}{\\hat{H}^{2}}, then obtain \\frac{\\partial}{\\partial \\phi} - \\frac{\\theta{(\\hat{H},\\phi)}}{\\hat{H} \\log{(\\phi)}} = \\frac{d}{d \\phi} - \\frac{1}{\\hat{H}^{2}}", "derivation": "\\theta{(\\hat{H},\\phi)} = \\frac{\\log{(\\phi)}}{\\hat{H}} and \\frac{\\partial}{\\partial \\hat{H}} \\theta{(\\hat{H},\\phi)} = \\frac{\\partial}{\\partial \\hat{H}} \\frac{\\log{(\\phi)}}{\\hat{H}} and \\frac{\\partial}{\\partial \\hat{H}} \\theta{(\\hat{H},\\phi)} = - \\frac{\\log{(\\phi)}}{\\hat{H}^{2}} and \\frac{\\partial}{\\partial \\hat{H}} \\theta{(\\hat{H},\\phi)} = - \\frac{\\theta{(\\hat{H},\\phi)}}{\\hat{H}} and - \\frac{\\theta{(\\hat{H},\\phi)}}{\\hat{H}} = - \\frac{\\log{(\\phi)}}{\\hat{H}^{2}} and - \\frac{\\theta{(\\hat{H},\\phi)}}{\\hat{H} \\log{(\\phi)}} = - \\frac{1}{\\hat{H}^{2}} and \\frac{\\partial}{\\partial \\phi} - \\frac{\\theta{(\\hat{H},\\phi)}}{\\hat{H} \\log{(\\phi)}} = \\frac{d}{d \\phi} - \\frac{1}{\\hat{H}^{2}}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), log(Symbol('\\\\phi', commutative=True))))"], [["differentiate", 1, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Derivative(Mul(Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), log(Symbol('\\\\phi', commutative=True))), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), log(Symbol('\\\\phi', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Derivative(Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Tuple(Symbol('\\\\hat{H}', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2)), log(Symbol('\\\\phi', commutative=True))))"], [["divide", 5, "log(Symbol('\\\\phi', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1))), Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2))))"], [["differentiate", 6, "Symbol('\\\\phi', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-1)), Function('\\\\theta')(Symbol('\\\\hat{H}', commutative=True), Symbol('\\\\phi', commutative=True)), Pow(log(Symbol('\\\\phi', commutative=True)), Integer(-1))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Pow(Symbol('\\\\hat{H}', commutative=True), Integer(-2))), Tuple(Symbol('\\\\phi', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{p}_0{(\\rho)} = \\cos{(\\rho)}, then obtain \\sin{(\\frac{\\rho}{2})} = \\sin{(\\frac{\\rho \\cos{(\\rho)}}{\\hat{p}_0{(\\rho)} + \\cos{(\\rho)}})}", "derivation": "\\hat{p}_0{(\\rho)} = \\cos{(\\rho)} and 2 \\hat{p}_0{(\\rho)} = \\hat{p}_0{(\\rho)} + \\cos{(\\rho)} and \\rho \\hat{p}_0{(\\rho)} = \\rho \\cos{(\\rho)} and \\frac{\\rho}{2} = \\frac{\\rho \\cos{(\\rho)}}{2 \\hat{p}_0{(\\rho)}} and \\frac{\\rho}{2} = \\frac{\\rho \\cos{(\\rho)}}{\\hat{p}_0{(\\rho)} + \\cos{(\\rho)}} and \\sin{(\\frac{\\rho}{2})} = \\sin{(\\frac{\\rho \\cos{(\\rho)}}{\\hat{p}_0{(\\rho)} + \\cos{(\\rho)}})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True)))"], [["add", 1, "Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))), Add(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True))))"], [["times", 1, "Symbol('\\\\rho', commutative=True)"], "Equality(Mul(Symbol('\\\\rho', commutative=True), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True))), Mul(Symbol('\\\\rho', commutative=True), cos(Symbol('\\\\rho', commutative=True))))"], [["divide", 3, "Mul(Integer(2), Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)))"], "Equality(Mul(Rational(1, 2), Symbol('\\\\rho', commutative=True)), Mul(Rational(1, 2), Symbol('\\\\rho', commutative=True), Pow(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), Integer(-1)), cos(Symbol('\\\\rho', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Mul(Rational(1, 2), Symbol('\\\\rho', commutative=True)), Mul(Symbol('\\\\rho', commutative=True), Pow(Add(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True))), Integer(-1)), cos(Symbol('\\\\rho', commutative=True))))"], [["sin", 5], "Equality(sin(Mul(Rational(1, 2), Symbol('\\\\rho', commutative=True))), sin(Mul(Symbol('\\\\rho', commutative=True), Pow(Add(Function('\\\\hat{p}_0')(Symbol('\\\\rho', commutative=True)), cos(Symbol('\\\\rho', commutative=True))), Integer(-1)), cos(Symbol('\\\\rho', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{B}{(\\mathbb{I},n)} = \\log{(\\mathbb{I} n)}, then obtain (- n + \\mathbf{B}{(\\mathbb{I},n)})^{2} = (- n + \\mathbf{B}{(\\mathbb{I},n)})^{2} \\mathbf{B}^{- n}{(\\mathbb{I},n)} \\log{(\\mathbb{I} n)}^{n}", "derivation": "\\mathbf{B}{(\\mathbb{I},n)} = \\log{(\\mathbb{I} n)} and \\mathbf{B}^{n}{(\\mathbb{I},n)} = \\log{(\\mathbb{I} n)}^{n} and - n + \\mathbf{B}{(\\mathbb{I},n)} = - n + \\log{(\\mathbb{I} n)} and 1 = \\mathbf{B}^{- n}{(\\mathbb{I},n)} \\log{(\\mathbb{I} n)}^{n} and (- n + \\mathbf{B}{(\\mathbb{I},n)}) (- n + \\log{(\\mathbb{I} n)}) = (- n + \\mathbf{B}{(\\mathbb{I},n)}) (- n + \\log{(\\mathbb{I} n)}) \\mathbf{B}^{- n}{(\\mathbb{I},n)} \\log{(\\mathbb{I} n)}^{n} and (- n + \\mathbf{B}{(\\mathbb{I},n)})^{2} = (- n + \\mathbf{B}{(\\mathbb{I},n)})^{2} \\mathbf{B}^{- n}{(\\mathbb{I},n)} \\log{(\\mathbb{I} n)}^{n}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))))"], [["power", 1, "Symbol('n', commutative=True)"], "Equality(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True)), Pow(log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True)))"], [["add", 1, "Mul(Integer(-1), Symbol('n', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)))))"], [["divide", 2, "Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), Symbol('n', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))), Pow(log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True))))"], [["times", 4, "Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)))))"], "Equality(Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))))), Mul(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Add(Mul(Integer(-1), Symbol('n', commutative=True)), log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)))), Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))), Pow(log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 3], "Equality(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Integer(2)), Mul(Pow(Add(Mul(Integer(-1), Symbol('n', commutative=True)), Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Integer(2)), Pow(Function('\\\\mathbf{B}')(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True)), Mul(Integer(-1), Symbol('n', commutative=True))), Pow(log(Mul(Symbol('\\\\mathbb{I}', commutative=True), Symbol('n', commutative=True))), Symbol('n', commutative=True))))"]]}, {"prompt": "Given \\mu_{0}{(S)} = e^{S}, then obtain - e^{S} + \\frac{d}{d S} \\mu_{0}{(S)} = 0", "derivation": "\\mu_{0}{(S)} = e^{S} and \\mu_{0}{(S)} - e^{S} = 0 and \\frac{d}{d S} (\\mu_{0}{(S)} - e^{S}) = \\frac{d}{d S} 0 and - e^{S} + \\frac{d}{d S} \\mu_{0}{(S)} = 0", "srepr_derivation": [["premise", "Equality(Function('\\\\mu_0')(Symbol('S', commutative=True)), exp(Symbol('S', commutative=True)))"], [["minus", 1, "exp(Symbol('S', commutative=True))"], "Equality(Add(Function('\\\\mu_0')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Integer(0))"], [["differentiate", 2, "Symbol('S', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mu_0')(Symbol('S', commutative=True)), Mul(Integer(-1), exp(Symbol('S', commutative=True)))), Tuple(Symbol('S', commutative=True), Integer(1))), Derivative(Integer(0), Tuple(Symbol('S', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), exp(Symbol('S', commutative=True))), Derivative(Function('\\\\mu_0')(Symbol('S', commutative=True)), Tuple(Symbol('S', commutative=True), Integer(1)))), Integer(0))"]]}, {"prompt": "Given \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J}^{n_{2}}, then obtain (\\mathbf{J} + \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})}) \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J}^{n_{2}} (\\mathbf{J} + \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})})", "derivation": "\\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J}^{n_{2}} and \\mathbf{J} + \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J} + \\mathbf{J}^{n_{2}} and (\\mathbf{J} + \\mathbf{J}^{n_{2}}) \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J}^{n_{2}} (\\mathbf{J} + \\mathbf{J}^{n_{2}}) and (\\mathbf{J} + \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})}) \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})} = \\mathbf{J}^{n_{2}} (\\mathbf{J} + \\operatorname{V_{\\mathbf{B}}}{(n_{2},\\mathbf{J})})", "srepr_derivation": [["premise", "Equality(Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{J}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True))))"], [["times", 1, "Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)))"], "Equality(Mul(Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 2], "Equality(Mul(Add(Symbol('\\\\mathbf{J}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True))), Mul(Pow(Symbol('\\\\mathbf{J}', commutative=True), Symbol('n_2', commutative=True)), Add(Symbol('\\\\mathbf{J}', commutative=True), Function('V_{\\\\mathbf{B}}')(Symbol('n_2', commutative=True), Symbol('\\\\mathbf{J}', commutative=True)))))"]]}, {"prompt": "Given \\omega{(\\dot{x},J_{\\varepsilon})} = \\cos{(J_{\\varepsilon} + \\dot{x})}, then derive \\int \\omega{(\\dot{x},J_{\\varepsilon})} dJ_{\\varepsilon} = \\Psi_{\\lambda} + \\sin{(J_{\\varepsilon} + \\dot{x})}, then obtain (\\Psi_{\\lambda} + \\sin{(J_{\\varepsilon} + \\dot{x})}) \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dU{(J_{\\varepsilon})} = (\\int \\cos{(J_{\\varepsilon} + \\dot{x})} dJ_{\\varepsilon}) \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dU{(J_{\\varepsilon})}", "derivation": "\\omega{(\\dot{x},J_{\\varepsilon})} = \\cos{(J_{\\varepsilon} + \\dot{x})} and \\int \\omega{(\\dot{x},J_{\\varepsilon})} dJ_{\\varepsilon} = \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dJ_{\\varepsilon} and \\int \\omega{(\\dot{x},J_{\\varepsilon})} dJ_{\\varepsilon} = \\Psi_{\\lambda} + \\sin{(J_{\\varepsilon} + \\dot{x})} and \\Psi_{\\lambda} + \\sin{(J_{\\varepsilon} + \\dot{x})} = \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dJ_{\\varepsilon} and (\\Psi_{\\lambda} + \\sin{(J_{\\varepsilon} + \\dot{x})}) \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dU{(J_{\\varepsilon})} = (\\int \\cos{(J_{\\varepsilon} + \\dot{x})} dJ_{\\varepsilon}) \\int \\cos{(J_{\\varepsilon} + \\dot{x})} dU{(J_{\\varepsilon})}", "srepr_derivation": [["premise", "Equality(Function('\\\\omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))))"], [["integrate", 1, "Symbol('J_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\omega')(Symbol('\\\\dot{x}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))))"], [["times", 4, "Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Function('U')(Symbol('J_{\\\\varepsilon}', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\Psi_{\\\\lambda}', commutative=True), sin(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True)))), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Function('U')(Symbol('J_{\\\\varepsilon}', commutative=True))))), Mul(Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Symbol('J_{\\\\varepsilon}', commutative=True))), Integral(cos(Add(Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('\\\\dot{x}', commutative=True))), Tuple(Function('U')(Symbol('J_{\\\\varepsilon}', commutative=True))))))"]]}, {"prompt": "Given \\delta{(b,E_{n})} = e^{\\frac{b}{E_{n}}}, then derive \\frac{\\partial}{\\partial E_{n}} \\delta{(b,E_{n})} = - \\frac{b e^{\\frac{b}{E_{n}}}}{E_{n}^{2}}, then obtain \\int - \\frac{b e^{\\frac{b}{E_{n}}}}{E_{n}^{2}} db = \\int \\frac{\\partial}{\\partial E_{n}} e^{\\frac{b}{E_{n}}} db", "derivation": "\\delta{(b,E_{n})} = e^{\\frac{b}{E_{n}}} and \\frac{\\partial}{\\partial E_{n}} \\delta{(b,E_{n})} = \\frac{\\partial}{\\partial E_{n}} e^{\\frac{b}{E_{n}}} and \\frac{\\partial}{\\partial E_{n}} \\delta{(b,E_{n})} = - \\frac{b e^{\\frac{b}{E_{n}}}}{E_{n}^{2}} and - \\frac{b e^{\\frac{b}{E_{n}}}}{E_{n}^{2}} = \\frac{\\partial}{\\partial E_{n}} e^{\\frac{b}{E_{n}}} and \\int - \\frac{b e^{\\frac{b}{E_{n}}}}{E_{n}^{2}} db = \\int \\frac{\\partial}{\\partial E_{n}} e^{\\frac{b}{E_{n}}} db", "srepr_derivation": [["premise", "Equality(Function('\\\\delta')(Symbol('b', commutative=True), Symbol('E_n', commutative=True)), exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True))))"], [["differentiate", 1, "Symbol('E_n', commutative=True)"], "Equality(Derivative(Function('\\\\delta')(Symbol('b', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Derivative(exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('\\\\delta')(Symbol('b', commutative=True), Symbol('E_n', commutative=True)), Tuple(Symbol('E_n', commutative=True), Integer(1))), Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('b', commutative=True), exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True)))))"], [["substitute_LHS_for_RHS", 2, 3], "Equality(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('b', commutative=True), exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True)))), Derivative(exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))))"], [["integrate", 4, "Symbol('b', commutative=True)"], "Equality(Integral(Mul(Integer(-1), Pow(Symbol('E_n', commutative=True), Integer(-2)), Symbol('b', commutative=True), exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True)))), Tuple(Symbol('b', commutative=True))), Integral(Derivative(exp(Mul(Pow(Symbol('E_n', commutative=True), Integer(-1)), Symbol('b', commutative=True))), Tuple(Symbol('E_n', commutative=True), Integer(1))), Tuple(Symbol('b', commutative=True))))"]]}, {"prompt": "Given \\hat{p}{(Z)} = \\log{(Z)}, then obtain \\frac{\\hat{p}{(Z)} + \\log{(Z)}}{\\hat{p}{(Z)} \\log{(Z)}} = \\frac{2}{\\hat{p}{(Z)}}", "derivation": "\\hat{p}{(Z)} = \\log{(Z)} and \\hat{p}{(Z)} + \\log{(Z)} = 2 \\log{(Z)} and \\hat{p}^{2}{(Z)} = \\hat{p}{(Z)} \\log{(Z)} and \\frac{\\hat{p}{(Z)} + \\log{(Z)}}{\\hat{p}^{2}{(Z)}} = \\frac{2 \\log{(Z)}}{\\hat{p}^{2}{(Z)}} and \\frac{\\hat{p}{(Z)} + \\log{(Z)}}{\\hat{p}{(Z)} \\log{(Z)}} = \\frac{2}{\\hat{p}{(Z)}}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True)))"], [["add", 1, "log(Symbol('Z', commutative=True))"], "Equality(Add(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Mul(Integer(2), log(Symbol('Z', commutative=True))))"], [["times", 1, "Function('\\\\hat{p}')(Symbol('Z', commutative=True))"], "Equality(Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(2)), Mul(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))))"], [["divide", 2, "Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(2))"], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(-2))), Mul(Integer(2), Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(-2)), log(Symbol('Z', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Mul(Add(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), log(Symbol('Z', commutative=True))), Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(-1)), Pow(log(Symbol('Z', commutative=True)), Integer(-1))), Mul(Integer(2), Pow(Function('\\\\hat{p}')(Symbol('Z', commutative=True)), Integer(-1))))"]]}, {"prompt": "Given \\operatorname{f^{\\prime}}{(L_{\\varepsilon},L)} = L^{L_{\\varepsilon}} and \\omega{(L_{\\varepsilon})} = 2 L_{\\varepsilon}, then obtain L^{\\omega{(L_{\\varepsilon})}} = L^{2 L_{\\varepsilon}}", "derivation": "\\operatorname{f^{\\prime}}{(L_{\\varepsilon},L)} = L^{L_{\\varepsilon}} and L^{L_{\\varepsilon}} \\operatorname{f^{\\prime}}{(L_{\\varepsilon},L)} = L^{2 L_{\\varepsilon}} and \\omega{(L_{\\varepsilon})} = 2 L_{\\varepsilon} and L^{L_{\\varepsilon}} \\operatorname{f^{\\prime}}{(L_{\\varepsilon},L)} = L^{\\omega{(L_{\\varepsilon})}} and L^{\\omega{(L_{\\varepsilon})}} = L^{2 L_{\\varepsilon}}", "srepr_derivation": [["premise", "Equality(Function('f^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True)), Pow(Symbol('L', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["times", 1, "Pow(Symbol('L', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Mul(Pow(Symbol('L', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('f^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\omega')(Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 3], "Equality(Mul(Pow(Symbol('L', commutative=True), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('f^{\\\\prime}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('L', commutative=True))), Pow(Symbol('L', commutative=True), Function('\\\\omega')(Symbol('L_{\\\\varepsilon}', commutative=True))))"], [["substitute_LHS_for_RHS", 2, 4], "Equality(Pow(Symbol('L', commutative=True), Function('\\\\omega')(Symbol('L_{\\\\varepsilon}', commutative=True))), Pow(Symbol('L', commutative=True), Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given \\psi^{*}{(H)} = \\log{(\\log{(H)})}, then obtain (\\log{(e^{\\psi^{*}{(H)}})}^{H} \\log{(\\log{(H)})}^{- H})^{H} = 1", "derivation": "\\psi^{*}{(H)} = \\log{(\\log{(H)})} and \\psi^{*}^{H}{(H)} = \\log{(\\log{(H)})}^{H} and e^{\\psi^{*}{(H)}} = \\log{(H)} and H \\psi^{*}^{H}{(H)} = H \\log{(\\log{(H)})}^{H} and \\psi^{*}{(H)} = \\log{(e^{\\psi^{*}{(H)}})} and \\psi^{*}^{H}{(H)} \\log{(\\log{(H)})}^{- H} = 1 and (\\psi^{*}^{H}{(H)} \\log{(\\log{(H)})}^{- H})^{H} = 1 and (\\log{(e^{\\psi^{*}{(H)}})}^{H} \\log{(\\log{(H)})}^{- H})^{H} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\psi^*')(Symbol('H', commutative=True)), log(log(Symbol('H', commutative=True))))"], [["power", 1, "Symbol('H', commutative=True)"], "Equality(Pow(Function('\\\\psi^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(log(log(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], [["exp", 1], "Equality(exp(Function('\\\\psi^*')(Symbol('H', commutative=True))), log(Symbol('H', commutative=True)))"], [["times", 2, "Symbol('H', commutative=True)"], "Equality(Mul(Symbol('H', commutative=True), Pow(Function('\\\\psi^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True))), Mul(Symbol('H', commutative=True), Pow(log(log(Symbol('H', commutative=True))), Symbol('H', commutative=True))))"], [["substitute_RHS_for_LHS", 1, 3], "Equality(Function('\\\\psi^*')(Symbol('H', commutative=True)), log(exp(Function('\\\\psi^*')(Symbol('H', commutative=True)))))"], [["divide", 4, "Mul(Symbol('H', commutative=True), Pow(log(log(Symbol('H', commutative=True))), Symbol('H', commutative=True)))"], "Equality(Mul(Pow(Function('\\\\psi^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(log(log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)))), Integer(1))"], [["power", 6, "Symbol('H', commutative=True)"], "Equality(Pow(Mul(Pow(Function('\\\\psi^*')(Symbol('H', commutative=True)), Symbol('H', commutative=True)), Pow(log(log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Integer(1))"], [["substitute_LHS_for_RHS", 7, 5], "Equality(Pow(Mul(Pow(log(exp(Function('\\\\psi^*')(Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Pow(log(log(Symbol('H', commutative=True))), Mul(Integer(-1), Symbol('H', commutative=True)))), Symbol('H', commutative=True)), Integer(1))"]]}, {"prompt": "Given \\varepsilon{(n_{1},\\delta)} = \\cos^{n_{1}}{(\\delta)}, then obtain \\frac{\\partial}{\\partial \\delta} \\frac{\\delta + \\varepsilon{(n_{1},\\delta)}}{\\delta + \\cos^{n_{1}}{(\\delta)}} = \\frac{d}{d \\delta} 1", "derivation": "\\varepsilon{(n_{1},\\delta)} = \\cos^{n_{1}}{(\\delta)} and \\delta + \\varepsilon{(n_{1},\\delta)} = \\delta + \\cos^{n_{1}}{(\\delta)} and 1 = \\frac{\\delta + \\cos^{n_{1}}{(\\delta)}}{\\delta + \\varepsilon{(n_{1},\\delta)}} and \\frac{\\delta + \\varepsilon{(n_{1},\\delta)}}{\\delta + \\cos^{n_{1}}{(\\delta)}} = 1 and \\frac{\\partial}{\\partial \\delta} \\frac{\\delta + \\varepsilon{(n_{1},\\delta)}}{\\delta + \\cos^{n_{1}}{(\\delta)}} = \\frac{d}{d \\delta} 1", "srepr_derivation": [["premise", "Equality(Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True)), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True)))"], [["add", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True))), Add(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True))))"], [["divide", 2, "Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True)))"], "Equality(Integer(1), Mul(Pow(Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True)))))"], [["divide", 3, "Mul(Pow(Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True))), Integer(-1)), Add(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True))))"], "Equality(Mul(Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True))), Pow(Add(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True))), Integer(-1))), Integer(1))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Mul(Add(Symbol('\\\\delta', commutative=True), Function('\\\\varepsilon')(Symbol('n_1', commutative=True), Symbol('\\\\delta', commutative=True))), Pow(Add(Symbol('\\\\delta', commutative=True), Pow(cos(Symbol('\\\\delta', commutative=True)), Symbol('n_1', commutative=True))), Integer(-1))), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Integer(1), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{x}_0{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\tilde{g}^*{(\\mathbf{A})} = 2 \\cos{(\\mathbf{A})}, then obtain \\mathbf{A} = \\mathbf{A} - 2 \\hat{x}_0{(\\mathbf{A})} + \\tilde{g}^*{(\\mathbf{A})}", "derivation": "\\hat{x}_0{(\\mathbf{A})} = \\cos{(\\mathbf{A})} and \\mathbf{A} + \\hat{x}_0{(\\mathbf{A})} = \\mathbf{A} + \\cos{(\\mathbf{A})} and \\mathbf{A} + 2 \\hat{x}_0{(\\mathbf{A})} = \\mathbf{A} + \\hat{x}_0{(\\mathbf{A})} + \\cos{(\\mathbf{A})} and \\mathbf{A} + 2 \\hat{x}_0{(\\mathbf{A})} = \\mathbf{A} + 2 \\cos{(\\mathbf{A})} and \\tilde{g}^*{(\\mathbf{A})} = 2 \\cos{(\\mathbf{A})} and \\mathbf{A} = \\mathbf{A} - 2 \\hat{x}_0{(\\mathbf{A})} + 2 \\cos{(\\mathbf{A})} and \\mathbf{A} = \\mathbf{A} - 2 \\hat{x}_0{(\\mathbf{A})} + \\tilde{g}^*{(\\mathbf{A})}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True)))"], [["add", 1, "Symbol('\\\\mathbf{A}', commutative=True)"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True))), Add(Symbol('\\\\mathbf{A}', commutative=True), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["add", 2, "Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True))"], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True)), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["substitute_LHS_for_RHS", 3, 2], "Equality(Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True)))), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(2), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], ["renaming_premise", "Equality(Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{A}', commutative=True)), Mul(Integer(2), cos(Symbol('\\\\mathbf{A}', commutative=True))))"], [["minus", 4, "Mul(Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True)))"], "Equality(Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True))), Mul(Integer(2), cos(Symbol('\\\\mathbf{A}', commutative=True)))))"], [["substitute_RHS_for_LHS", 6, 5], "Equality(Symbol('\\\\mathbf{A}', commutative=True), Add(Symbol('\\\\mathbf{A}', commutative=True), Mul(Integer(-1), Integer(2), Function('\\\\hat{x}_0')(Symbol('\\\\mathbf{A}', commutative=True))), Function('\\\\tilde{g}^*')(Symbol('\\\\mathbf{A}', commutative=True))))"]]}, {"prompt": "Given \\mathbb{I}{(F_{N})} = \\cos{(e^{F_{N}})}, then obtain \\frac{d}{d F_{N}} (\\int \\mathbb{I}{(F_{N})} dF_{N} - \\int \\cos{(e^{F_{N}})} dF_{N})^{F_{N}} = \\frac{d}{d F_{N}} 0^{F_{N}}", "derivation": "\\mathbb{I}{(F_{N})} = \\cos{(e^{F_{N}})} and \\int \\mathbb{I}{(F_{N})} dF_{N} = \\int \\cos{(e^{F_{N}})} dF_{N} and \\int \\mathbb{I}{(F_{N})} dF_{N} - \\int \\cos{(e^{F_{N}})} dF_{N} = 0 and (\\int \\mathbb{I}{(F_{N})} dF_{N} - \\int \\cos{(e^{F_{N}})} dF_{N})^{F_{N}} = 0^{F_{N}} and \\frac{d}{d F_{N}} (\\int \\mathbb{I}{(F_{N})} dF_{N} - \\int \\cos{(e^{F_{N}})} dF_{N})^{F_{N}} = \\frac{d}{d F_{N}} 0^{F_{N}}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbb{I}')(Symbol('F_N', commutative=True)), cos(exp(Symbol('F_N', commutative=True))))"], [["integrate", 1, "Symbol('F_N', commutative=True)"], "Equality(Integral(Function('\\\\mathbb{I}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))"], [["minus", 2, "Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True)))"], "Equality(Add(Integral(Function('\\\\mathbb{I}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))), Integer(0))"], [["power", 3, "Symbol('F_N', commutative=True)"], "Equality(Pow(Add(Integral(Function('\\\\mathbb{I}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)), Pow(Integer(0), Symbol('F_N', commutative=True)))"], [["differentiate", 4, "Symbol('F_N', commutative=True)"], "Equality(Derivative(Pow(Add(Integral(Function('\\\\mathbb{I}')(Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True))), Mul(Integer(-1), Integral(cos(exp(Symbol('F_N', commutative=True))), Tuple(Symbol('F_N', commutative=True))))), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))), Derivative(Pow(Integer(0), Symbol('F_N', commutative=True)), Tuple(Symbol('F_N', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\hat{H}{(p)} = \\log{(\\log{(p)})}, then obtain \\int \\frac{p}{\\iint \\log{(\\log{(p)})} dp dp} dp = \\int \\frac{p - \\hat{H}{(p)} + \\log{(\\log{(p)})}}{\\iint \\log{(\\log{(p)})} dp dp} dp", "derivation": "\\hat{H}{(p)} = \\log{(\\log{(p)})} and 0 = - \\hat{H}{(p)} + \\log{(\\log{(p)})} and p = p - \\hat{H}{(p)} + \\log{(\\log{(p)})} and \\frac{p}{\\iint \\log{(\\log{(p)})} dp dp} = \\frac{p - \\hat{H}{(p)} + \\log{(\\log{(p)})}}{\\iint \\log{(\\log{(p)})} dp dp} and \\int \\frac{p}{\\iint \\log{(\\log{(p)})} dp dp} dp = \\int \\frac{p - \\hat{H}{(p)} + \\log{(\\log{(p)})}}{\\iint \\log{(\\log{(p)})} dp dp} dp", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}')(Symbol('p', commutative=True)), log(log(Symbol('p', commutative=True))))"], [["minus", 1, "Function('\\\\hat{H}')(Symbol('p', commutative=True))"], "Equality(Integer(0), Add(Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), log(log(Symbol('p', commutative=True)))))"], [["add", 2, "Symbol('p', commutative=True)"], "Equality(Symbol('p', commutative=True), Add(Symbol('p', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), log(log(Symbol('p', commutative=True)))))"], [["divide", 3, "Integral(log(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True)))"], "Equality(Mul(Symbol('p', commutative=True), Pow(Integral(log(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))), Mul(Add(Symbol('p', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), log(log(Symbol('p', commutative=True)))), Pow(Integral(log(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))))"], [["integrate", 4, "Symbol('p', commutative=True)"], "Equality(Integral(Mul(Symbol('p', commutative=True), Pow(Integral(log(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))), Tuple(Symbol('p', commutative=True))), Integral(Mul(Add(Symbol('p', commutative=True), Mul(Integer(-1), Function('\\\\hat{H}')(Symbol('p', commutative=True))), log(log(Symbol('p', commutative=True)))), Pow(Integral(log(log(Symbol('p', commutative=True))), Tuple(Symbol('p', commutative=True)), Tuple(Symbol('p', commutative=True))), Integer(-1))), Tuple(Symbol('p', commutative=True))))"]]}, {"prompt": "Given s{(\\rho_b,z)} = z e^{\\rho_b}, then derive 2 \\frac{\\partial}{\\partial \\rho_b} s{(\\rho_b,z)} = z e^{\\rho_b} + \\frac{\\partial}{\\partial \\rho_b} s{(\\rho_b,z)}, then obtain \\int 2 \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b} dz = A + z^{2} e^{\\rho_b}", "derivation": "s{(\\rho_b,z)} = z e^{\\rho_b} and 2 s{(\\rho_b,z)} = z e^{\\rho_b} + s{(\\rho_b,z)} and \\frac{\\partial}{\\partial \\rho_b} 2 s{(\\rho_b,z)} = \\frac{\\partial}{\\partial \\rho_b} (z e^{\\rho_b} + s{(\\rho_b,z)}) and 2 \\frac{\\partial}{\\partial \\rho_b} s{(\\rho_b,z)} = z e^{\\rho_b} + \\frac{\\partial}{\\partial \\rho_b} s{(\\rho_b,z)} and 2 \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b} = z e^{\\rho_b} + \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b} and \\int 2 \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b} dz = \\int (z e^{\\rho_b} + \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b}) dz and \\int 2 \\frac{\\partial}{\\partial \\rho_b} z e^{\\rho_b} dz = A + z^{2} e^{\\rho_b}", "srepr_derivation": [["premise", "Equality(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))))"], [["add", 1, "Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True))"], "Equality(Mul(Integer(2), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True))), Add(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True))))"], [["differentiate", 2, "Symbol('\\\\rho_b', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))), Derivative(Add(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Derivative(Function('s')(Symbol('\\\\rho_b', commutative=True), Symbol('z', commutative=True)), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["substitute_LHS_for_RHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Add(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Derivative(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))))"], [["integrate", 5, "Symbol('z', commutative=True)"], "Equality(Integral(Mul(Integer(2), Derivative(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))), Integral(Add(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Derivative(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))))"], [["evaluate_integrals", 6], "Equality(Integral(Mul(Integer(2), Derivative(Mul(Symbol('z', commutative=True), exp(Symbol('\\\\rho_b', commutative=True))), Tuple(Symbol('\\\\rho_b', commutative=True), Integer(1)))), Tuple(Symbol('z', commutative=True))), Add(Symbol('A', commutative=True), Mul(Pow(Symbol('z', commutative=True), Integer(2)), exp(Symbol('\\\\rho_b', commutative=True)))))"]]}, {"prompt": "Given \\mathbf{v}{(l)} = \\log{(\\log{(l)})}, then obtain \\int \\frac{d^{2}}{d l^{2}} (\\mathbf{v}{(l)} + \\log{(\\log{(l)})}) dl = \\int \\frac{d^{2}}{d l^{2}} 2 \\log{(\\log{(l)})} dl", "derivation": "\\mathbf{v}{(l)} = \\log{(\\log{(l)})} and \\mathbf{v}{(l)} + \\log{(\\log{(l)})} = 2 \\log{(\\log{(l)})} and \\frac{d}{d l} (\\mathbf{v}{(l)} + \\log{(\\log{(l)})}) = \\frac{d}{d l} 2 \\log{(\\log{(l)})} and \\frac{d^{2}}{d l^{2}} (\\mathbf{v}{(l)} + \\log{(\\log{(l)})}) = \\frac{d^{2}}{d l^{2}} 2 \\log{(\\log{(l)})} and \\int \\frac{d^{2}}{d l^{2}} (\\mathbf{v}{(l)} + \\log{(\\log{(l)})}) dl = \\int \\frac{d^{2}}{d l^{2}} 2 \\log{(\\log{(l)})} dl", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{v}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True))))"], [["add", 1, "log(log(Symbol('l', commutative=True)))"], "Equality(Add(Function('\\\\mathbf{v}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))), Mul(Integer(2), log(log(Symbol('l', commutative=True)))))"], [["differentiate", 2, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{v}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Mul(Integer(2), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(1))))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Function('\\\\mathbf{v}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(2))), Derivative(Mul(Integer(2), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(2))))"], [["integrate", 4, "Symbol('l', commutative=True)"], "Equality(Integral(Derivative(Add(Function('\\\\mathbf{v}')(Symbol('l', commutative=True)), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(2))), Tuple(Symbol('l', commutative=True))), Integral(Derivative(Mul(Integer(2), log(log(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True), Integer(2))), Tuple(Symbol('l', commutative=True))))"]]}, {"prompt": "Given \\operatorname{v_{t}}{(E,k,\\mu)} = - E + \\mu - k and \\rho{(E,k,\\mu)} = (\\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)})^{E}, then obtain \\frac{\\partial}{\\partial E} e^{\\rho{(E,k,\\mu)}} = \\frac{\\partial}{\\partial E} e^{(\\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)})^{E}}", "derivation": "\\operatorname{v_{t}}{(E,k,\\mu)} = - E + \\mu - k and \\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)} = \\frac{\\partial}{\\partial E} (- E + \\mu - k) and (\\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)})^{E} = (\\frac{\\partial}{\\partial E} (- E + \\mu - k))^{E} and \\rho{(E,k,\\mu)} = (\\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)})^{E} and \\rho{(E,k,\\mu)} = (\\frac{\\partial}{\\partial E} (- E + \\mu - k))^{E} and e^{\\rho{(E,k,\\mu)}} = e^{(\\frac{\\partial}{\\partial E} (- E + \\mu - k))^{E}} and \\frac{\\partial}{\\partial E} e^{\\rho{(E,k,\\mu)}} = \\frac{\\partial}{\\partial E} e^{(\\frac{\\partial}{\\partial E} (- E + \\mu - k))^{E}} and \\frac{\\partial}{\\partial E} e^{\\rho{(E,k,\\mu)}} = \\frac{\\partial}{\\partial E} e^{(\\frac{\\partial}{\\partial E} \\operatorname{v_{t}}{(E,k,\\mu)})^{E}}", "srepr_derivation": [["premise", "Equality(Function('v_t')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))))"], [["differentiate", 1, "Symbol('E', commutative=True)"], "Equality(Derivative(Function('v_t')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["power", 2, "Symbol('E', commutative=True)"], "Equality(Pow(Derivative(Function('v_t')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Function('v_t')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True)))"], [["exp", 5], "Equality(exp(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True))), exp(Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))))"], [["differentiate", 6, "Symbol('E', commutative=True)"], "Equality(Derivative(exp(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(exp(Pow(Derivative(Add(Mul(Integer(-1), Symbol('E', commutative=True)), Symbol('\\\\mu', commutative=True), Mul(Integer(-1), Symbol('k', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 7, 3], "Equality(Derivative(exp(Function('\\\\rho')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(exp(Pow(Derivative(Function('v_t')(Symbol('E', commutative=True), Symbol('k', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1))), Symbol('E', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\dot{y}{(L_{\\varepsilon},W)} = - L_{\\varepsilon} + \\cos{(W)}, then obtain 1 = \\frac{2 L_{\\varepsilon} - \\cos{(W)} + \\int (- 2 L_{\\varepsilon} + \\cos{(W)}) dW}{2 L_{\\varepsilon} - \\cos{(W)} + \\int (- L_{\\varepsilon} + \\dot{y}{(L_{\\varepsilon},W)}) dW}", "derivation": "\\dot{y}{(L_{\\varepsilon},W)} = - L_{\\varepsilon} + \\cos{(W)} and - L_{\\varepsilon} + \\dot{y}{(L_{\\varepsilon},W)} = - 2 L_{\\varepsilon} + \\cos{(W)} and \\int (- L_{\\varepsilon} + \\dot{y}{(L_{\\varepsilon},W)}) dW = \\int (- 2 L_{\\varepsilon} + \\cos{(W)}) dW and 2 L_{\\varepsilon} - \\cos{(W)} + \\int (- L_{\\varepsilon} + \\dot{y}{(L_{\\varepsilon},W)}) dW = 2 L_{\\varepsilon} - \\cos{(W)} + \\int (- 2 L_{\\varepsilon} + \\cos{(W)}) dW and 1 = \\frac{2 L_{\\varepsilon} - \\cos{(W)} + \\int (- 2 L_{\\varepsilon} + \\cos{(W)}) dW}{2 L_{\\varepsilon} - \\cos{(W)} + \\int (- L_{\\varepsilon} + \\dot{y}{(L_{\\varepsilon},W)}) dW}", "srepr_derivation": [["premise", "Equality(Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True)), Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True))))"], [["add", 1, "Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True))"], "Equality(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Add(Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True))))"], [["integrate", 2, "Symbol('W', commutative=True)"], "Equality(Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], [["minus", 3, "Add(Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True)))"], "Equality(Add(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Add(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))))"], [["divide", 4, "Add(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True))))"], "Equality(Integer(1), Mul(Add(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), cos(Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Pow(Add(Mul(Integer(2), Symbol('L_{\\\\varepsilon}', commutative=True)), Mul(Integer(-1), cos(Symbol('W', commutative=True))), Integral(Add(Mul(Integer(-1), Symbol('L_{\\\\varepsilon}', commutative=True)), Function('\\\\dot{y}')(Symbol('L_{\\\\varepsilon}', commutative=True), Symbol('W', commutative=True))), Tuple(Symbol('W', commutative=True)))), Integer(-1))))"]]}, {"prompt": "Given \\theta_{2}{(I)} = \\cos{(I)}, then derive \\int \\frac{\\theta_{2}{(I)}}{\\cos{(I)}} dI = I + \\tilde{g}, then obtain (I + \\tilde{g}) \\frac{d}{d I} \\int 1 dI = (I + \\tilde{g}) \\frac{\\partial}{\\partial I} (I + \\tilde{g})", "derivation": "\\theta_{2}{(I)} = \\cos{(I)} and \\frac{\\theta_{2}{(I)}}{\\cos{(I)}} = 1 and \\int \\frac{\\theta_{2}{(I)}}{\\cos{(I)}} dI = \\int 1 dI and \\int \\frac{\\theta_{2}{(I)}}{\\cos{(I)}} dI = I + \\tilde{g} and \\int 1 dI = I + \\tilde{g} and \\frac{d}{d I} \\int 1 dI = \\frac{\\partial}{\\partial I} (I + \\tilde{g}) and (I + \\tilde{g}) \\frac{d}{d I} \\int 1 dI = (I + \\tilde{g}) \\frac{\\partial}{\\partial I} (I + \\tilde{g})", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_2')(Symbol('I', commutative=True)), cos(Symbol('I', commutative=True)))"], [["divide", 1, "cos(Symbol('I', commutative=True))"], "Equality(Mul(Function('\\\\theta_2')(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1))), Integer(1))"], [["integrate", 2, "Symbol('I', commutative=True)"], "Equality(Integral(Mul(Function('\\\\theta_2')(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1))), Tuple(Symbol('I', commutative=True))), Integral(Integer(1), Tuple(Symbol('I', commutative=True))))"], [["evaluate_integrals", 3], "Equality(Integral(Mul(Function('\\\\theta_2')(Symbol('I', commutative=True)), Pow(cos(Symbol('I', commutative=True)), Integer(-1))), Tuple(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)))"], [["differentiate", 5, "Symbol('I', commutative=True)"], "Equality(Derivative(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1))), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1))))"], [["times", 6, "Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True))"], "Equality(Mul(Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Integral(Integer(1), Tuple(Symbol('I', commutative=True))), Tuple(Symbol('I', commutative=True), Integer(1)))), Mul(Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Derivative(Add(Symbol('I', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('I', commutative=True), Integer(1)))))"]]}, {"prompt": "Given \\mathbf{J}_f{(\\rho_f)} = \\int \\log{(\\rho_f)} d\\rho_f, then derive \\mathbf{J}_f{(\\rho_f)} = Q + \\rho_f \\log{(\\rho_f)} - \\rho_f, then obtain -1 = - Q + \\mathbf{g} - 1", "derivation": "\\mathbf{J}_f{(\\rho_f)} = \\int \\log{(\\rho_f)} d\\rho_f and \\mathbf{J}_f{(\\rho_f)} = Q + \\rho_f \\log{(\\rho_f)} - \\rho_f and \\mathbf{J}_f{(\\rho_f)} - 1 = \\int \\log{(\\rho_f)} d\\rho_f - 1 and - Q - \\rho_f \\log{(\\rho_f)} + \\rho_f + \\mathbf{J}_f{(\\rho_f)} - 1 = - Q - \\rho_f \\log{(\\rho_f)} + \\rho_f + \\int \\log{(\\rho_f)} d\\rho_f - 1 and -1 = - Q - \\rho_f \\log{(\\rho_f)} + \\rho_f + \\int \\log{(\\rho_f)} d\\rho_f - 1 and -1 = - Q + \\mathbf{g} - 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_f', commutative=True)), Integral(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))))"], [["evaluate_integrals", 1], "Equality(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_f', commutative=True)), Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True))))"], [["add", 1, "Integer(-1)"], "Equality(Add(Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Add(Integral(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(-1)))"], [["minus", 3, "Add(Symbol('Q', commutative=True), Mul(Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True), Function('\\\\mathbf{J}_f')(Symbol('\\\\rho_f', commutative=True)), Integer(-1)), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True), Integral(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(-1)))"], [["substitute_LHS_for_RHS", 4, 2], "Equality(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Mul(Integer(-1), Symbol('\\\\rho_f', commutative=True), log(Symbol('\\\\rho_f', commutative=True))), Symbol('\\\\rho_f', commutative=True), Integral(log(Symbol('\\\\rho_f', commutative=True)), Tuple(Symbol('\\\\rho_f', commutative=True))), Integer(-1)))"], [["evaluate_integrals", 5], "Equality(Integer(-1), Add(Mul(Integer(-1), Symbol('Q', commutative=True)), Symbol('\\\\mathbf{g}', commutative=True), Integer(-1)))"]]}, {"prompt": "Given \\mathbf{J}_P{(\\Omega)} = e^{\\Omega}, then obtain \\mathbf{J}_P^{324}{(\\Omega)} = \\mathbf{J}_P^{180}{(\\Omega)} e^{144 \\Omega}", "derivation": "\\mathbf{J}_P{(\\Omega)} = e^{\\Omega} and \\mathbf{J}_P^{2}{(\\Omega)} = \\mathbf{J}_P{(\\Omega)} e^{\\Omega} and \\mathbf{J}_P^{4}{(\\Omega)} = \\mathbf{J}_P^{2}{(\\Omega)} e^{2 \\Omega} and \\mathbf{J}_P^{16}{(\\Omega)} = \\mathbf{J}_P^{8}{(\\Omega)} e^{8 \\Omega} and \\mathbf{J}_P^{18}{(\\Omega)} = \\mathbf{J}_P^{10}{(\\Omega)} e^{8 \\Omega} and \\mathbf{J}_P^{324}{(\\Omega)} = \\mathbf{J}_P^{180}{(\\Omega)} e^{144 \\Omega}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True)))"], [["times", 1, "Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True))"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(2)), Mul(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), exp(Symbol('\\\\Omega', commutative=True))))"], [["power", 2, 2], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(4)), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(2)), exp(Mul(Integer(2), Symbol('\\\\Omega', commutative=True)))))"], [["power", 3, 4], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(16)), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(8)), exp(Mul(Integer(8), Symbol('\\\\Omega', commutative=True)))))"], [["times", 4, "Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(2))"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(18)), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(10)), exp(Mul(Integer(8), Symbol('\\\\Omega', commutative=True)))))"], [["power", 5, "Integer(18)"], "Equality(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(324)), Mul(Pow(Function('\\\\mathbf{J}_P')(Symbol('\\\\Omega', commutative=True)), Integer(180)), exp(Mul(Integer(144), Symbol('\\\\Omega', commutative=True)))))"]]}, {"prompt": "Given I{(C_{d},\\hat{x},r_{0})} = r_{0} (C_{d} + \\hat{x}), then obtain (C_{d} + \\hat{x} + I^{2}{(C_{d},\\hat{x},r_{0})})^{2} = (C_{d} + \\hat{x} + r_{0} (C_{d} + \\hat{x}) I{(C_{d},\\hat{x},r_{0})})^{2}", "derivation": "I{(C_{d},\\hat{x},r_{0})} = r_{0} (C_{d} + \\hat{x}) and I^{2}{(C_{d},\\hat{x},r_{0})} = r_{0} (C_{d} + \\hat{x}) I{(C_{d},\\hat{x},r_{0})} and C_{d} + \\hat{x} + I^{2}{(C_{d},\\hat{x},r_{0})} = C_{d} + \\hat{x} + r_{0} (C_{d} + \\hat{x}) I{(C_{d},\\hat{x},r_{0})} and (C_{d} + \\hat{x} + I^{2}{(C_{d},\\hat{x},r_{0})})^{2} = (C_{d} + \\hat{x} + r_{0} (C_{d} + \\hat{x}) I{(C_{d},\\hat{x},r_{0})})^{2}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Mul(Symbol('r_0', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True))))"], [["times", 1, "Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True))"], "Equality(Pow(Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Integer(2)), Mul(Symbol('r_0', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True))))"], [["add", 2, "Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True))"], "Equality(Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Pow(Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('r_0', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)))))"], [["power", 3, 2], "Equality(Pow(Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Pow(Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)), Integer(2))), Integer(2)), Pow(Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Mul(Symbol('r_0', commutative=True), Add(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True)), Function('I')(Symbol('C_d', commutative=True), Symbol('\\\\hat{x}', commutative=True), Symbol('r_0', commutative=True)))), Integer(2)))"]]}, {"prompt": "Given \\sigma_{x}{(\\varphi,A_{1})} = A_{1} + \\varphi and u{(\\varphi,A_{1})} = (A_{1} + \\varphi)^{A_{1}}, then obtain \\varphi + u{(\\varphi,A_{1})} = \\varphi + \\sigma_{x}^{A_{1}}{(\\varphi,A_{1})}", "derivation": "\\sigma_{x}{(\\varphi,A_{1})} = A_{1} + \\varphi and u{(\\varphi,A_{1})} = (A_{1} + \\varphi)^{A_{1}} and u{(\\varphi,A_{1})} = \\sigma_{x}^{A_{1}}{(\\varphi,A_{1})} and \\varphi + u{(\\varphi,A_{1})} = \\varphi + \\sigma_{x}^{A_{1}}{(\\varphi,A_{1})}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True)), Add(Symbol('A_1', commutative=True), Symbol('\\\\varphi', commutative=True)))"], ["renaming_premise", "Equality(Function('u')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True)), Pow(Add(Symbol('A_1', commutative=True), Symbol('\\\\varphi', commutative=True)), Symbol('A_1', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 1], "Equality(Function('u')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True)), Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True)))"], [["add", 3, "Symbol('\\\\varphi', commutative=True)"], "Equality(Add(Symbol('\\\\varphi', commutative=True), Function('u')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True))), Add(Symbol('\\\\varphi', commutative=True), Pow(Function('\\\\sigma_x')(Symbol('\\\\varphi', commutative=True), Symbol('A_1', commutative=True)), Symbol('A_1', commutative=True))))"]]}, {"prompt": "Given \\mathbf{E}{(\\mathbf{S},\\mu)} = \\mathbf{S} + \\sin{(\\mu)}, then derive \\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S} = \\frac{\\mathbf{S}^{2}}{2} + \\mathbf{S} \\sin{(\\mu)} + m_{s}, then obtain - \\int (\\mathbf{S} + \\sin{(\\mu)}) d\\mathbf{S} + (\\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S})^{\\mu} = (\\frac{\\mathbf{S}^{2}}{2} + \\mathbf{S} \\sin{(\\mu)} + m_{s})^{\\mu} - \\int (\\mathbf{S} + \\sin{(\\mu)}) d\\mathbf{S}", "derivation": "\\mathbf{E}{(\\mathbf{S},\\mu)} = \\mathbf{S} + \\sin{(\\mu)} and \\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S} = \\int (\\mathbf{S} + \\sin{(\\mu)}) d\\mathbf{S} and \\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S} = \\frac{\\mathbf{S}^{2}}{2} + \\mathbf{S} \\sin{(\\mu)} + m_{s} and (\\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S})^{\\mu} = (\\frac{\\mathbf{S}^{2}}{2} + \\mathbf{S} \\sin{(\\mu)} + m_{s})^{\\mu} and - \\int (\\mathbf{S} + \\sin{(\\mu)}) d\\mathbf{S} + (\\int \\mathbf{E}{(\\mathbf{S},\\mu)} d\\mathbf{S})^{\\mu} = (\\frac{\\mathbf{S}^{2}}{2} + \\mathbf{S} \\sin{(\\mu)} + m_{s})^{\\mu} - \\int (\\mathbf{S} + \\sin{(\\mu)}) d\\mathbf{S}", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mu', commutative=True)), Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))))"], [["integrate", 1, "Symbol('\\\\mathbf{S}', commutative=True)"], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))"], [["evaluate_integrals", 2], "Equality(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Symbol('m_s', commutative=True)))"], [["power", 3, "Symbol('\\\\mu', commutative=True)"], "Equality(Pow(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mu', commutative=True)), Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Symbol('m_s', commutative=True)), Symbol('\\\\mu', commutative=True)))"], [["minus", 4, "Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))"], "Equality(Add(Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True)))), Pow(Integral(Function('\\\\mathbf{E}')(Symbol('\\\\mathbf{S}', commutative=True), Symbol('\\\\mu', commutative=True)), Tuple(Symbol('\\\\mathbf{S}', commutative=True))), Symbol('\\\\mu', commutative=True))), Add(Pow(Add(Mul(Rational(1, 2), Pow(Symbol('\\\\mathbf{S}', commutative=True), Integer(2))), Mul(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Symbol('m_s', commutative=True)), Symbol('\\\\mu', commutative=True)), Mul(Integer(-1), Integral(Add(Symbol('\\\\mathbf{S}', commutative=True), sin(Symbol('\\\\mu', commutative=True))), Tuple(Symbol('\\\\mathbf{S}', commutative=True))))))"]]}, {"prompt": "Given \\operatorname{n_{2}}{(E,C_{1})} = e^{C_{1} - E} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})} = e^{C_{1} - E}, then derive - E \\frac{\\partial}{\\partial E} \\operatorname{n_{2}}{(E,C_{1})} - \\operatorname{n_{2}}{(E,C_{1})} = E e^{C_{1} - E} - e^{C_{1} - E}, then obtain - E \\frac{\\partial}{\\partial E} \\operatorname{n_{2}}{(E,C_{1})} - \\operatorname{n_{2}}{(E,C_{1})} = E \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})} - \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})}", "derivation": "\\operatorname{n_{2}}{(E,C_{1})} = e^{C_{1} - E} and - E \\operatorname{n_{2}}{(E,C_{1})} = - E e^{C_{1} - E} and \\frac{\\partial}{\\partial E} - E \\operatorname{n_{2}}{(E,C_{1})} = \\frac{\\partial}{\\partial E} - E e^{C_{1} - E} and - E \\frac{\\partial}{\\partial E} \\operatorname{n_{2}}{(E,C_{1})} - \\operatorname{n_{2}}{(E,C_{1})} = E e^{C_{1} - E} - e^{C_{1} - E} and \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})} = e^{C_{1} - E} and - E \\frac{\\partial}{\\partial E} \\operatorname{n_{2}}{(E,C_{1})} - \\operatorname{n_{2}}{(E,C_{1})} = E \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})} - \\operatorname{g^{\\prime}_{\\varepsilon}}{(E,C_{1})}", "srepr_derivation": [["premise", "Equality(Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True)))))"], [["times", 1, "Mul(Integer(-1), Symbol('E', commutative=True))"], "Equality(Mul(Integer(-1), Symbol('E', commutative=True), Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True))), Mul(Integer(-1), Symbol('E', commutative=True), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))))))"], [["differentiate", 2, "Symbol('E', commutative=True)"], "Equality(Derivative(Mul(Integer(-1), Symbol('E', commutative=True), Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True))), Tuple(Symbol('E', commutative=True), Integer(1))), Derivative(Mul(Integer(-1), Symbol('E', commutative=True), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))))), Tuple(Symbol('E', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Derivative(Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)))), Add(Mul(Symbol('E', commutative=True), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True))))), Mul(Integer(-1), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True)))))))"], ["renaming_premise", "Equality(Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)), exp(Add(Symbol('C_1', commutative=True), Mul(Integer(-1), Symbol('E', commutative=True)))))"], [["substitute_RHS_for_LHS", 4, 5], "Equality(Add(Mul(Integer(-1), Symbol('E', commutative=True), Derivative(Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)), Tuple(Symbol('E', commutative=True), Integer(1)))), Mul(Integer(-1), Function('n_2')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)))), Add(Mul(Symbol('E', commutative=True), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('C_1', commutative=True))), Mul(Integer(-1), Function('g^{\\\\prime}_{\\\\varepsilon}')(Symbol('E', commutative=True), Symbol('C_1', commutative=True)))))"]]}, {"prompt": "Given \\operatorname{v_{y}}{(A_{2},q)} = - A_{2} + q, then obtain - (- A_{2} + q)^{q} \\operatorname{v_{y}}{(A_{2},q)} \\operatorname{v_{y}}^{q}{(A_{2},q)} = - (- A_{2} + q)^{2 q} \\operatorname{v_{y}}{(A_{2},q)}", "derivation": "\\operatorname{v_{y}}{(A_{2},q)} = - A_{2} + q and \\operatorname{v_{y}}^{q}{(A_{2},q)} = (- A_{2} + q)^{q} and \\operatorname{v_{y}}{(A_{2},q)} \\operatorname{v_{y}}^{q}{(A_{2},q)} = (- A_{2} + q)^{q} \\operatorname{v_{y}}{(A_{2},q)} and - \\operatorname{v_{y}}{(A_{2},q)} \\operatorname{v_{y}}^{q}{(A_{2},q)} = - (- A_{2} + q)^{q} \\operatorname{v_{y}}{(A_{2},q)} and - (- A_{2} + q)^{q} \\operatorname{v_{y}}{(A_{2},q)} \\operatorname{v_{y}}^{q}{(A_{2},q)} = - (- A_{2} + q)^{2 q} \\operatorname{v_{y}}{(A_{2},q)}", "srepr_derivation": [["premise", "Equality(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)))"], [["power", 1, "Symbol('q', commutative=True)"], "Equality(Pow(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)))"], [["times", 2, "Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True))"], "Equality(Mul(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Pow(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True))))"], [["divide", 3, "Integer(-1)"], "Equality(Mul(Integer(-1), Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Pow(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True))))"], [["times", 4, "Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True))"], "Equality(Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Symbol('q', commutative=True)), Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Pow(Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True)), Symbol('q', commutative=True))), Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Symbol('A_2', commutative=True)), Symbol('q', commutative=True)), Mul(Integer(2), Symbol('q', commutative=True))), Function('v_y')(Symbol('A_2', commutative=True), Symbol('q', commutative=True))))"]]}, {"prompt": "Given p{(\\delta)} = \\sin{(\\delta)} and \\pi{(\\delta)} = \\frac{d}{d \\delta} (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta}, then obtain (\\frac{d}{d \\delta} (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta}) \\int p{(\\delta)} d\\delta = \\pi{(\\delta)} \\int p{(\\delta)} d\\delta", "derivation": "p{(\\delta)} = \\sin{(\\delta)} and \\int p{(\\delta)} d\\delta = \\int \\sin{(\\delta)} d\\delta and \\iint p{(\\delta)} d\\delta d\\delta = \\iint \\sin{(\\delta)} d\\delta d\\delta and (\\iint p{(\\delta)} d\\delta d\\delta)^{\\delta} = (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta} and \\frac{d}{d \\delta} (\\iint p{(\\delta)} d\\delta d\\delta)^{\\delta} = \\frac{d}{d \\delta} (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta} and \\pi{(\\delta)} = \\frac{d}{d \\delta} (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta} and \\frac{d}{d \\delta} (\\iint p{(\\delta)} d\\delta d\\delta)^{\\delta} = \\pi{(\\delta)} and (\\frac{d}{d \\delta} (\\iint p{(\\delta)} d\\delta d\\delta)^{\\delta}) \\int p{(\\delta)} d\\delta = \\pi{(\\delta)} \\int p{(\\delta)} d\\delta and (\\frac{d}{d \\delta} (\\iint \\sin{(\\delta)} d\\delta d\\delta)^{\\delta}) \\int p{(\\delta)} d\\delta = \\pi{(\\delta)} \\int p{(\\delta)} d\\delta", "srepr_derivation": [["premise", "Equality(Function('p')(Symbol('\\\\delta', commutative=True)), sin(Symbol('\\\\delta', commutative=True)))"], [["integrate", 1, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["integrate", 2, "Symbol('\\\\delta', commutative=True)"], "Equality(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))))"], [["power", 3, "Symbol('\\\\delta', commutative=True)"], "Equality(Pow(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Pow(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)))"], [["differentiate", 4, "Symbol('\\\\delta', commutative=True)"], "Equality(Derivative(Pow(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Derivative(Pow(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], ["renaming_premise", "Equality(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Derivative(Pow(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))))"], [["substitute_RHS_for_LHS", 5, 6], "Equality(Derivative(Pow(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Function('\\\\pi')(Symbol('\\\\delta', commutative=True)))"], [["times", 7, "Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))"], "Equality(Mul(Derivative(Pow(Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"], [["substitute_LHS_for_RHS", 8, 3], "Equality(Mul(Derivative(Pow(Integral(sin(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True))), Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True), Integer(1))), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))), Mul(Function('\\\\pi')(Symbol('\\\\delta', commutative=True)), Integral(Function('p')(Symbol('\\\\delta', commutative=True)), Tuple(Symbol('\\\\delta', commutative=True)))))"]]}, {"prompt": "Given y{(\\dot{y},l)} = \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}}, then obtain \\frac{\\partial}{\\partial l} (\\int y{(\\dot{y},l)} dl + 1) = \\frac{\\partial}{\\partial l} (\\int \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}} dl + 1)", "derivation": "y{(\\dot{y},l)} = \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}} and \\int y{(\\dot{y},l)} dl = \\int \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}} dl and \\int y{(\\dot{y},l)} dl + 1 = \\int \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}} dl + 1 and \\frac{\\partial}{\\partial l} (\\int y{(\\dot{y},l)} dl + 1) = \\frac{\\partial}{\\partial l} (\\int \\frac{\\int l^{\\dot{y}} dl}{\\dot{y}} dl + 1)", "srepr_derivation": [["renaming_premise", "Equality(Function('y')(Symbol('\\\\dot{y}', commutative=True), Symbol('l', commutative=True)), Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Pow(Symbol('l', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('l', commutative=True)))))"], [["integrate", 1, "Symbol('l', commutative=True)"], "Equality(Integral(Function('y')(Symbol('\\\\dot{y}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Pow(Symbol('l', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))))"], [["minus", 2, "Integer(-1)"], "Equality(Add(Integral(Function('y')(Symbol('\\\\dot{y}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Add(Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Pow(Symbol('l', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Integer(1)))"], [["differentiate", 3, "Symbol('l', commutative=True)"], "Equality(Derivative(Add(Integral(Function('y')(Symbol('\\\\dot{y}', commutative=True), Symbol('l', commutative=True)), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1))), Derivative(Add(Integral(Mul(Pow(Symbol('\\\\dot{y}', commutative=True), Integer(-1)), Integral(Pow(Symbol('l', commutative=True), Symbol('\\\\dot{y}', commutative=True)), Tuple(Symbol('l', commutative=True)))), Tuple(Symbol('l', commutative=True))), Integer(1)), Tuple(Symbol('l', commutative=True), Integer(1))))"]]}, {"prompt": "Given \\theta{(n_{1},m)} = m + n_{1} and \\hat{\\mathbf{x}}{(n_{1},m)} = n_{1} (m + n_{1}) + n_{1} \\theta{(n_{1},m)}, then obtain \\hat{\\mathbf{x}}{(n_{1},m)} = 2 n_{1} \\theta{(n_{1},m)}", "derivation": "\\theta{(n_{1},m)} = m + n_{1} and n_{1} \\theta{(n_{1},m)} = n_{1} (m + n_{1}) and n_{1} (m + n_{1}) + n_{1} \\theta{(n_{1},m)} = 2 n_{1} (m + n_{1}) and \\hat{\\mathbf{x}}{(n_{1},m)} = n_{1} (m + n_{1}) + n_{1} \\theta{(n_{1},m)} and \\hat{\\mathbf{x}}{(n_{1},m)} = 2 n_{1} (m + n_{1}) and \\hat{\\mathbf{x}}{(n_{1},m)} = 2 n_{1} \\theta{(n_{1},m)}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True)))"], [["times", 1, "Symbol('n_1', commutative=True)"], "Equality(Mul(Symbol('n_1', commutative=True), Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('m', commutative=True))), Mul(Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True))))"], [["add", 2, "Mul(Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True)))"], "Equality(Add(Mul(Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)))), Mul(Integer(2), Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True))))"], ["renaming_premise", "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)), Add(Mul(Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True))), Mul(Symbol('n_1', commutative=True), Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)))))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True), Add(Symbol('m', commutative=True), Symbol('n_1', commutative=True))))"], [["substitute_RHS_for_LHS", 5, 1], "Equality(Function('\\\\hat{\\\\mathbf{x}}')(Symbol('n_1', commutative=True), Symbol('m', commutative=True)), Mul(Integer(2), Symbol('n_1', commutative=True), Function('\\\\theta')(Symbol('n_1', commutative=True), Symbol('m', commutative=True))))"]]}, {"prompt": "Given \\rho_{f}{(A)} = e^{A}, then derive 2 \\frac{d}{d A} \\rho_{f}{(A)} = e^{A} + \\frac{d}{d A} \\rho_{f}{(A)}, then obtain \\sin{(2 \\frac{d}{d A} \\rho_{f}{(A)})} = \\sin{(\\rho_{f}{(A)} + \\frac{d}{d A} \\rho_{f}{(A)})}", "derivation": "\\rho_{f}{(A)} = e^{A} and 2 \\rho_{f}{(A)} = \\rho_{f}{(A)} + e^{A} and \\frac{d}{d A} 2 \\rho_{f}{(A)} = \\frac{d}{d A} (\\rho_{f}{(A)} + e^{A}) and 2 \\frac{d}{d A} \\rho_{f}{(A)} = e^{A} + \\frac{d}{d A} \\rho_{f}{(A)} and 2 \\frac{d}{d A} \\rho_{f}{(A)} = \\rho_{f}{(A)} + \\frac{d}{d A} \\rho_{f}{(A)} and \\sin{(2 \\frac{d}{d A} \\rho_{f}{(A)})} = \\sin{(\\rho_{f}{(A)} + \\frac{d}{d A} \\rho_{f}{(A)})}", "srepr_derivation": [["premise", "Equality(Function('\\\\rho_f')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True)))"], [["add", 1, "Function('\\\\rho_f')(Symbol('A', commutative=True))"], "Equality(Mul(Integer(2), Function('\\\\rho_f')(Symbol('A', commutative=True))), Add(Function('\\\\rho_f')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))))"], [["differentiate", 2, "Symbol('A', commutative=True)"], "Equality(Derivative(Mul(Integer(2), Function('\\\\rho_f')(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))), Derivative(Add(Function('\\\\rho_f')(Symbol('A', commutative=True)), exp(Symbol('A', commutative=True))), Tuple(Symbol('A', commutative=True), Integer(1))))"], [["evaluate_derivatives", 3], "Equality(Mul(Integer(2), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(exp(Symbol('A', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["substitute_RHS_for_LHS", 4, 1], "Equality(Mul(Integer(2), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))), Add(Function('\\\\rho_f')(Symbol('A', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1)))))"], [["sin", 5], "Equality(sin(Mul(Integer(2), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))), sin(Add(Function('\\\\rho_f')(Symbol('A', commutative=True)), Derivative(Function('\\\\rho_f')(Symbol('A', commutative=True)), Tuple(Symbol('A', commutative=True), Integer(1))))))"]]}, {"prompt": "Given \\mathbf{J}_M{(\\sigma_p,\\tilde{g}^*,F_{c})} = \\frac{- F_{c} + \\tilde{g}^*}{\\sigma_p} and Q{(\\sigma_p,\\tilde{g}^*,F_{c})} = \\frac{- F_{c} + \\tilde{g}^*}{\\sigma_p}, then obtain \\frac{\\sigma_p Q{(\\sigma_p,\\tilde{g}^*,F_{c})}}{- F_{c} + \\tilde{g}^*} = 1", "derivation": "\\mathbf{J}_M{(\\sigma_p,\\tilde{g}^*,F_{c})} = \\frac{- F_{c} + \\tilde{g}^*}{\\sigma_p} and \\frac{\\sigma_p \\mathbf{J}_M{(\\sigma_p,\\tilde{g}^*,F_{c})}}{- F_{c} + \\tilde{g}^*} = 1 and Q{(\\sigma_p,\\tilde{g}^*,F_{c})} = \\frac{- F_{c} + \\tilde{g}^*}{\\sigma_p} and Q{(\\sigma_p,\\tilde{g}^*,F_{c})} = \\mathbf{J}_M{(\\sigma_p,\\tilde{g}^*,F_{c})} and \\frac{\\sigma_p Q{(\\sigma_p,\\tilde{g}^*,F_{c})}}{- F_{c} + \\tilde{g}^*} = 1", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["divide", 1, "Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)))"], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True))), Integer(1))"], ["renaming_premise", "Equality(Function('Q')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True)), Mul(Pow(Symbol('\\\\sigma_p', commutative=True), Integer(-1)), Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True))))"], [["substitute_RHS_for_LHS", 3, 1], "Equality(Function('Q')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True)), Function('\\\\mathbf{J}_M')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True)))"], [["substitute_RHS_for_LHS", 2, 4], "Equality(Mul(Symbol('\\\\sigma_p', commutative=True), Pow(Add(Mul(Integer(-1), Symbol('F_c', commutative=True)), Symbol('\\\\tilde{g}^*', commutative=True)), Integer(-1)), Function('Q')(Symbol('\\\\sigma_p', commutative=True), Symbol('\\\\tilde{g}^*', commutative=True), Symbol('F_c', commutative=True))), Integer(1))"]]}, {"prompt": "Given C{(\\pi,\\tilde{g})} = \\tilde{g}^{\\pi}, then derive \\frac{\\partial}{\\partial \\pi} C{(\\pi,\\tilde{g})} = \\tilde{g}^{\\pi} \\log{(\\tilde{g})}, then obtain \\tilde{g}^{\\pi} \\log{(\\tilde{g})} + \\frac{\\partial}{\\partial \\pi} C{(\\pi,\\tilde{g})} = 2 \\tilde{g}^{\\pi} \\log{(\\tilde{g})}", "derivation": "C{(\\pi,\\tilde{g})} = \\tilde{g}^{\\pi} and \\frac{\\partial}{\\partial \\pi} C{(\\pi,\\tilde{g})} = \\frac{\\partial}{\\partial \\pi} \\tilde{g}^{\\pi} and \\frac{\\partial}{\\partial \\pi} C{(\\pi,\\tilde{g})} = \\tilde{g}^{\\pi} \\log{(\\tilde{g})} and \\tilde{g}^{\\pi} \\log{(\\tilde{g})} + \\frac{\\partial}{\\partial \\pi} C{(\\pi,\\tilde{g})} = 2 \\tilde{g}^{\\pi} \\log{(\\tilde{g})}", "srepr_derivation": [["premise", "Equality(Function('C')(Symbol('\\\\pi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)))"], [["differentiate", 1, "Symbol('\\\\pi', commutative=True)"], "Equality(Derivative(Function('C')(Symbol('\\\\pi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Derivative(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))))"], [["evaluate_derivatives", 2], "Equality(Derivative(Function('C')(Symbol('\\\\pi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1))), Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))))"], [["add", 3, "Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True)))"], "Equality(Add(Mul(Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))), Derivative(Function('C')(Symbol('\\\\pi', commutative=True), Symbol('\\\\tilde{g}', commutative=True)), Tuple(Symbol('\\\\pi', commutative=True), Integer(1)))), Mul(Integer(2), Pow(Symbol('\\\\tilde{g}', commutative=True), Symbol('\\\\pi', commutative=True)), log(Symbol('\\\\tilde{g}', commutative=True))))"]]}, {"prompt": "Given \\mathbf{r}{(t)} = \\cos{(t)}, then obtain t (t - \\mathbf{r}{(t)} + 1) = t (t - \\mathbf{r}{(t)} - 1 + \\frac{2 \\cos{(t)}}{\\mathbf{r}{(t)}})", "derivation": "\\mathbf{r}{(t)} = \\cos{(t)} and 1 = \\frac{\\cos{(t)}}{\\mathbf{r}{(t)}} and 1 - \\mathbf{r}{(t)} = - \\mathbf{r}{(t)} + \\frac{\\cos{(t)}}{\\mathbf{r}{(t)}} and t - \\mathbf{r}{(t)} + 1 = t - \\mathbf{r}{(t)} + \\frac{\\cos{(t)}}{\\mathbf{r}{(t)}} and t (t - \\mathbf{r}{(t)} + 1) = t (t - \\mathbf{r}{(t)} + \\frac{\\cos{(t)}}{\\mathbf{r}{(t)}}) and t (t - \\mathbf{r}{(t)} + \\frac{\\cos{(t)}}{\\mathbf{r}{(t)}}) = t (t - \\mathbf{r}{(t)} - 1 + \\frac{2 \\cos{(t)}}{\\mathbf{r}{(t)}}) and t (t - \\mathbf{r}{(t)} + 1) = t (t - \\mathbf{r}{(t)} - 1 + \\frac{2 \\cos{(t)}}{\\mathbf{r}{(t)}})", "srepr_derivation": [["premise", "Equality(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["divide", 1, "Function('\\\\mathbf{r}')(Symbol('t', commutative=True))"], "Equality(Integer(1), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True))))"], [["minus", 2, "Function('\\\\mathbf{r}')(Symbol('t', commutative=True))"], "Equality(Add(Integer(1), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True)))), Add(Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True)))))"], [["add", 3, "Symbol('t', commutative=True)"], "Equality(Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Integer(1)), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True)))))"], [["times", 4, "Symbol('t', commutative=True)"], "Equality(Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Integer(1))), Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 4], "Equality(Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Mul(Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True))))), Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True))))))"], [["substitute_LHS_for_RHS", 5, 6], "Equality(Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Integer(1))), Mul(Symbol('t', commutative=True), Add(Symbol('t', commutative=True), Mul(Integer(-1), Function('\\\\mathbf{r}')(Symbol('t', commutative=True))), Integer(-1), Mul(Integer(2), Pow(Function('\\\\mathbf{r}')(Symbol('t', commutative=True)), Integer(-1)), cos(Symbol('t', commutative=True))))))"]]}, {"prompt": "Given \\theta_{1}{(\\hat{x})} = \\int \\sin{(\\hat{x})} d\\hat{x}, then derive \\log{(\\theta_{1}^{2}{(\\hat{x})})} = \\log{((m_{s} - \\cos{(\\hat{x})}) \\theta_{1}{(\\hat{x})})}, then obtain \\log{((m_{s} - \\cos{(\\hat{x})}) \\theta_{1}{(\\hat{x})})} = \\log{(\\theta_{1}{(\\hat{x})} \\int \\sin{(\\hat{x})} d\\hat{x})}", "derivation": "\\theta_{1}{(\\hat{x})} = \\int \\sin{(\\hat{x})} d\\hat{x} and \\theta_{1}^{2}{(\\hat{x})} = \\theta_{1}{(\\hat{x})} \\int \\sin{(\\hat{x})} d\\hat{x} and \\log{(\\theta_{1}^{2}{(\\hat{x})})} = \\log{(\\theta_{1}{(\\hat{x})} \\int \\sin{(\\hat{x})} d\\hat{x})} and \\log{(\\theta_{1}^{2}{(\\hat{x})})} = \\log{((m_{s} - \\cos{(\\hat{x})}) \\theta_{1}{(\\hat{x})})} and \\log{((m_{s} - \\cos{(\\hat{x})}) \\theta_{1}{(\\hat{x})})} = \\log{(\\theta_{1}{(\\hat{x})} \\int \\sin{(\\hat{x})} d\\hat{x})}", "srepr_derivation": [["premise", "Equality(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))"], [["times", 1, "Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True))"], "Equality(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integer(2)), Mul(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True)))))"], [["log", 2], "Equality(log(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integer(2))), log(Mul(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))))"], [["evaluate_integrals", 3], "Equality(log(Pow(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integer(2))), log(Mul(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)))))"], [["substitute_LHS_for_RHS", 3, 4], "Equality(log(Mul(Add(Symbol('m_s', commutative=True), Mul(Integer(-1), cos(Symbol('\\\\hat{x}', commutative=True)))), Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)))), log(Mul(Function('\\\\theta_1')(Symbol('\\\\hat{x}', commutative=True)), Integral(sin(Symbol('\\\\hat{x}', commutative=True)), Tuple(Symbol('\\\\hat{x}', commutative=True))))))"]]}, {"prompt": "Given I{(t)} = \\cos{(t)}, then obtain (\\frac{I{(t)}}{I{(t)} + \\cos{(t)}})^{t} = (\\frac{1}{2})^{t}", "derivation": "I{(t)} = \\cos{(t)} and I{(t)} + \\cos{(t)} = 2 \\cos{(t)} and \\frac{I{(t)}}{2 \\cos{(t)}} = \\frac{1}{2} and (\\frac{I{(t)}}{2 \\cos{(t)}})^{t} = (\\frac{1}{2})^{t} and (\\frac{I{(t)}}{I{(t)} + \\cos{(t)}})^{t} = (\\frac{1}{2})^{t}", "srepr_derivation": [["premise", "Equality(Function('I')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True)))"], [["add", 1, "cos(Symbol('t', commutative=True))"], "Equality(Add(Function('I')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Mul(Integer(2), cos(Symbol('t', commutative=True))))"], [["divide", 1, "Mul(Integer(2), cos(Symbol('t', commutative=True)))"], "Equality(Mul(Rational(1, 2), Function('I')(Symbol('t', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-1))), Rational(1, 2))"], [["power", 3, "Symbol('t', commutative=True)"], "Equality(Pow(Mul(Rational(1, 2), Function('I')(Symbol('t', commutative=True)), Pow(cos(Symbol('t', commutative=True)), Integer(-1))), Symbol('t', commutative=True)), Pow(Rational(1, 2), Symbol('t', commutative=True)))"], [["substitute_RHS_for_LHS", 4, 2], "Equality(Pow(Mul(Pow(Add(Function('I')(Symbol('t', commutative=True)), cos(Symbol('t', commutative=True))), Integer(-1)), Function('I')(Symbol('t', commutative=True))), Symbol('t', commutative=True)), Pow(Rational(1, 2), Symbol('t', commutative=True)))"]]}, {"prompt": "Given \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})}, then obtain \\iint g^{\\prime}_{\\varepsilon} \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon} = \\iint g^{\\prime}_{\\varepsilon} \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon}", "derivation": "\\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})} and g^{\\prime}_{\\varepsilon} \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} = g^{\\prime}_{\\varepsilon} \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})} and \\int g^{\\prime}_{\\varepsilon} \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} = \\int g^{\\prime}_{\\varepsilon} \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon} and \\iint g^{\\prime}_{\\varepsilon} \\sigma_{x}{(g^{\\prime}_{\\varepsilon})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon} = \\iint g^{\\prime}_{\\varepsilon} \\cos{(\\log{(g^{\\prime}_{\\varepsilon})})} dg^{\\prime}_{\\varepsilon} dg^{\\prime}_{\\varepsilon}", "srepr_derivation": [["premise", "Equality(Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), cos(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["times", 1, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), cos(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))))"], [["integrate", 2, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), cos(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"], [["integrate", 3, "Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)"], "Equality(Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), Function('\\\\sigma_x')(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))), Integral(Mul(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True), cos(log(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)))), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True)), Tuple(Symbol('g^{\\\\prime}_{\\\\varepsilon}', commutative=True))))"]]}, {"prompt": "Given Q{(\\hat{H},J_{\\varepsilon},x)} = \\frac{\\hat{H} x}{J_{\\varepsilon}} and \\mathbf{M}{(\\hat{H},J_{\\varepsilon},x)} = (- x + Q{(\\hat{H},J_{\\varepsilon},x)})^{\\hat{H}}, then obtain \\mathbf{M}{(\\hat{H},J_{\\varepsilon},x)} = (- x + \\frac{\\hat{H} x}{J_{\\varepsilon}})^{\\hat{H}}", "derivation": "Q{(\\hat{H},J_{\\varepsilon},x)} = \\frac{\\hat{H} x}{J_{\\varepsilon}} and - x + Q{(\\hat{H},J_{\\varepsilon},x)} = - x + \\frac{\\hat{H} x}{J_{\\varepsilon}} and (- x + Q{(\\hat{H},J_{\\varepsilon},x)})^{\\hat{H}} = (- x + \\frac{\\hat{H} x}{J_{\\varepsilon}})^{\\hat{H}} and \\mathbf{M}{(\\hat{H},J_{\\varepsilon},x)} = (- x + Q{(\\hat{H},J_{\\varepsilon},x)})^{\\hat{H}} and \\mathbf{M}{(\\hat{H},J_{\\varepsilon},x)} = (- x + \\frac{\\hat{H} x}{J_{\\varepsilon}})^{\\hat{H}}", "srepr_derivation": [["premise", "Equality(Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Symbol('x', commutative=True)))"], [["minus", 1, "Symbol('x', commutative=True)"], "Equality(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Symbol('x', commutative=True))))"], [["power", 2, "Symbol('\\\\hat{H}', commutative=True)"], "Equality(Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('\\\\hat{H}', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Symbol('x', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], ["renaming_premise", "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Function('Q')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"], [["substitute_RHS_for_LHS", 3, 4], "Equality(Function('\\\\mathbf{M}')(Symbol('\\\\hat{H}', commutative=True), Symbol('J_{\\\\varepsilon}', commutative=True), Symbol('x', commutative=True)), Pow(Add(Mul(Integer(-1), Symbol('x', commutative=True)), Mul(Pow(Symbol('J_{\\\\varepsilon}', commutative=True), Integer(-1)), Symbol('\\\\hat{H}', commutative=True), Symbol('x', commutative=True))), Symbol('\\\\hat{H}', commutative=True)))"]]}, {"prompt": "Given \\operatorname{v_{1}}{(\\rho_f,\\lambda)} = \\frac{\\rho_f}{\\lambda}, then obtain (\\frac{\\rho_f}{\\lambda^{2}})^{\\lambda} + \\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda} = (\\frac{\\rho_f}{\\lambda^{2}})^{\\lambda} + \\frac{\\rho_f}{\\lambda^{2}}", "derivation": "\\operatorname{v_{1}}{(\\rho_f,\\lambda)} = \\frac{\\rho_f}{\\lambda} and \\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda} = \\frac{\\rho_f}{\\lambda^{2}} and (\\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda})^{\\lambda} = (\\frac{\\rho_f}{\\lambda^{2}})^{\\lambda} and (\\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda})^{\\lambda} + \\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda} = (\\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda})^{\\lambda} + \\frac{\\rho_f}{\\lambda^{2}} and (\\frac{\\rho_f}{\\lambda^{2}})^{\\lambda} + \\frac{\\operatorname{v_{1}}{(\\rho_f,\\lambda)}}{\\lambda} = (\\frac{\\rho_f}{\\lambda^{2}})^{\\lambda} + \\frac{\\rho_f}{\\lambda^{2}}", "srepr_derivation": [["premise", "Equality(Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Symbol('\\\\rho_f', commutative=True)))"], [["times", 1, "Pow(Symbol('\\\\lambda', commutative=True), Integer(-1))"], "Equality(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True))), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True)))"], [["power", 2, "Symbol('\\\\lambda', commutative=True)"], "Equality(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\lambda', commutative=True)))"], [["add", 2, "Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True))"], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True)))), Add(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True))), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True))))"], [["substitute_LHS_for_RHS", 4, 3], "Equality(Add(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-1)), Function('v_1')(Symbol('\\\\rho_f', commutative=True), Symbol('\\\\lambda', commutative=True)))), Add(Pow(Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True)), Symbol('\\\\lambda', commutative=True)), Mul(Pow(Symbol('\\\\lambda', commutative=True), Integer(-2)), Symbol('\\\\rho_f', commutative=True))))"]]}, {"prompt": "Given \\hat{H}_l{(\\mathbf{s})} = \\sin{(\\mathbf{s})}, then obtain \\iint (\\frac{\\hat{H}_l{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}})^{\\mathbf{s}} d\\mathbf{s} d\\mathbf{s} = \\iint 1 d\\mathbf{s} d\\mathbf{s}", "derivation": "\\hat{H}_l{(\\mathbf{s})} = \\sin{(\\mathbf{s})} and \\mathbf{s} \\hat{H}_l{(\\mathbf{s})} = \\mathbf{s} \\sin{(\\mathbf{s})} and \\frac{\\hat{H}_l{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}} = 1 and (\\frac{\\hat{H}_l{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}})^{\\mathbf{s}} = 1 and \\int (\\frac{\\hat{H}_l{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}})^{\\mathbf{s}} d\\mathbf{s} = \\int 1 d\\mathbf{s} and \\iint (\\frac{\\hat{H}_l{(\\mathbf{s})}}{\\sin{(\\mathbf{s})}})^{\\mathbf{s}} d\\mathbf{s} d\\mathbf{s} = \\iint 1 d\\mathbf{s} d\\mathbf{s}", "srepr_derivation": [["premise", "Equality(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True)), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], [["times", 1, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Mul(Symbol('\\\\mathbf{s}', commutative=True), Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True))), Mul(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True))))"], [["divide", 2, "Mul(Symbol('\\\\mathbf{s}', commutative=True), sin(Symbol('\\\\mathbf{s}', commutative=True)))"], "Equality(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Integer(1))"], [["power", 3, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Pow(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Integer(1))"], [["integrate", 4, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"], [["integrate", 5, "Symbol('\\\\mathbf{s}', commutative=True)"], "Equality(Integral(Pow(Mul(Function('\\\\hat{H}_l')(Symbol('\\\\mathbf{s}', commutative=True)), Pow(sin(Symbol('\\\\mathbf{s}', commutative=True)), Integer(-1))), Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))), Integral(Integer(1), Tuple(Symbol('\\\\mathbf{s}', commutative=True)), Tuple(Symbol('\\\\mathbf{s}', commutative=True))))"]]}]